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+Metadata-Version: 2.4 +Name: sympy +Version: 1.14.0 +Summary: Computer algebra system (CAS) in Python +Home-page: https://sympy.org +Author: SymPy development team +Author-email: sympy@googlegroups.com +License: BSD +Project-URL: Source, https://github.com/sympy/sympy +Keywords: Math CAS +Classifier: License :: OSI Approved :: BSD License +Classifier: Operating System :: OS Independent +Classifier: Programming Language :: Python +Classifier: Topic :: Scientific/Engineering +Classifier: Topic :: Scientific/Engineering :: Mathematics +Classifier: Topic :: Scientific/Engineering :: Physics +Classifier: Programming Language :: Python :: 3 +Classifier: Programming Language :: Python :: 3.9 +Classifier: Programming Language :: Python :: 3.10 +Classifier: Programming Language :: Python :: 3.11 +Classifier: Programming Language :: Python :: 3.12 +Classifier: Programming Language :: Python :: 3.13 +Classifier: Programming Language :: Python :: 3 :: Only +Classifier: Programming Language :: Python :: Implementation :: CPython +Classifier: Programming Language :: Python :: Implementation :: PyPy +Requires-Python: >=3.9 +Description-Content-Type: text/markdown +License-File: LICENSE +License-File: AUTHORS +Requires-Dist: mpmath<1.4,>=1.1.0 +Provides-Extra: dev +Requires-Dist: pytest>=7.1.0; extra == "dev" +Requires-Dist: hypothesis>=6.70.0; extra == "dev" +Dynamic: author +Dynamic: author-email +Dynamic: classifier +Dynamic: description +Dynamic: description-content-type +Dynamic: home-page +Dynamic: keywords +Dynamic: license +Dynamic: license-file +Dynamic: project-url +Dynamic: provides-extra +Dynamic: requires-dist +Dynamic: requires-python +Dynamic: summary + +# SymPy + +[![pypi version](https://img.shields.io/pypi/v/sympy.svg)](https://pypi.python.org/pypi/sympy) +[![Join the chat at https://gitter.im/sympy/sympy](https://badges.gitter.im/Join%20Chat.svg)](https://gitter.im/sympy/sympy?utm_source=badge&utm_medium=badge&utm_campaign=pr-badge&utm_content=badge) +[![Zenodo Badge](https://zenodo.org/badge/18918/sympy/sympy.svg)](https://zenodo.org/badge/latestdoi/18918/sympy/sympy) +[![Downloads](https://pepy.tech/badge/sympy/month)](https://pepy.tech/project/sympy) +[![GitHub Issues](https://img.shields.io/badge/issue_tracking-github-blue.svg)](https://github.com/sympy/sympy/issues) +[![Git Tutorial](https://img.shields.io/badge/PR-Welcome-%23FF8300.svg?)](https://git-scm.com/book/en/v2/GitHub-Contributing-to-a-Project) +[![Powered by NumFocus](https://img.shields.io/badge/powered%20by-NumFOCUS-orange.svg?style=flat&colorA=E1523D&colorB=007D8A)](https://numfocus.org) +[![Commits since last release](https://img.shields.io/github/commits-since/sympy/sympy/latest.svg?longCache=true&style=flat-square&logo=git&logoColor=fff)](https://github.com/sympy/sympy/releases) + +[![SymPy Banner](https://github.com/sympy/sympy/raw/master/banner.svg)](https://sympy.org/) + + +See the [AUTHORS](AUTHORS) file for the list of authors. + +And many more people helped on the SymPy mailing list, reported bugs, +helped organize SymPy's participation in the Google Summer of Code, the +Google Highly Open Participation Contest, Google Code-In, wrote and +blogged about SymPy... + +License: New BSD License (see the [LICENSE](LICENSE) file for details) covers all +files in the sympy repository unless stated otherwise. + +Our mailing list is at +. + +We have a community chat at [Gitter](https://gitter.im/sympy/sympy). Feel +free to ask us anything there. We have a very welcoming and helpful +community. + +## Download + +The recommended installation method is through Anaconda, + + +You can also get the latest version of SymPy from + + +To get the git version do + + $ git clone https://github.com/sympy/sympy.git + +For other options (tarballs, debs, etc.), see +. + +## Documentation and Usage + +For in-depth instructions on installation and building the +documentation, see the [SymPy Documentation Style Guide](https://docs.sympy.org/dev/documentation-style-guide.html). + +Everything is at: + + + +You can generate everything at the above site in your local copy of +SymPy by: + + $ cd doc + $ make html + +Then the docs will be in \_build/html. If +you don't want to read that, here is a short usage: + +From this directory, start Python and: + +``` python +>>> from sympy import Symbol, cos +>>> x = Symbol('x') +>>> e = 1/cos(x) +>>> print(e.series(x, 0, 10)) +1 + x**2/2 + 5*x**4/24 + 61*x**6/720 + 277*x**8/8064 + O(x**10) +``` + +SymPy also comes with a console that is a simple wrapper around the +classic python console (or IPython when available) that loads the SymPy +namespace and executes some common commands for you. + +To start it, issue: + + $ bin/isympy + +from this directory, if SymPy is not installed or simply: + + $ isympy + +if SymPy is installed. + +## Installation + +To install SymPy using PyPI, run the following command: + + $ pip install sympy + +To install SymPy using Anaconda, run the following command: + + $ conda install -c anaconda sympy + +To install SymPy from GitHub source, first clone SymPy using `git`: + + $ git clone https://github.com/sympy/sympy.git + +Then, in the `sympy` repository that you cloned, simply run: + + $ pip install . + +See for more information. + +## Contributing + +We welcome contributions from anyone, even if you are new to open +source. Please read our [Introduction to Contributing](https://docs.sympy.org/dev/contributing/introduction-to-contributing.html) +page and the [SymPy Documentation Style Guide](https://docs.sympy.org/dev/documentation-style-guide.html). If you +are new and looking for some way to contribute, a good place to start is +to look at the issues tagged [Easy to Fix](https://github.com/sympy/sympy/issues?q=is%3Aopen+is%3Aissue+label%3A%22Easy+to+Fix%22). + +Please note that all participants in this project are expected to follow +our Code of Conduct. By participating in this project you agree to abide +by its terms. See [CODE\_OF\_CONDUCT.md](CODE_OF_CONDUCT.md). + +## Tests + +To execute all tests, run: + + $./setup.py test + +in the current directory. + +For the more fine-grained running of tests or doctests, use `bin/test` +or respectively `bin/doctest`. The master branch is automatically tested +by GitHub Actions. + +To test pull requests, use +[sympy-bot](https://github.com/sympy/sympy-bot). + +## Regenerate Experimental LaTeX Parser/Lexer + +The parser and lexer were generated with the [ANTLR4](http://antlr4.org) +toolchain in `sympy/parsing/latex/_antlr` and checked into the repo. +Presently, most users should not need to regenerate these files, but +if you plan to work on this feature, you will need the `antlr4` +command-line tool (and you must ensure that it is in your `PATH`). +One way to get it is: + + $ conda install -c conda-forge antlr=4.11.1 + +Alternatively, follow the instructions on the ANTLR website and download +the `antlr-4.11.1-complete.jar`. Then export the `CLASSPATH` as instructed +and instead of creating `antlr4` as an alias, make it an executable file +with the following contents: +``` bash +#!/bin/bash +java -jar /usr/local/lib/antlr-4.11.1-complete.jar "$@" +``` + +After making changes to `sympy/parsing/latex/LaTeX.g4`, run: + + $ ./setup.py antlr + +## Clean + +To clean everything (thus getting the same tree as in the repository): + + $ git clean -Xdf + +which will clear everything ignored by `.gitignore`, and: + + $ git clean -df + +to clear all untracked files. You can revert the most recent changes in +git with: + + $ git reset --hard + +WARNING: The above commands will all clear changes you may have made, +and you will lose them forever. Be sure to check things with `git +status`, `git diff`, `git clean -Xn`, and `git clean -n` before doing any +of those. + +## Bugs + +Our issue tracker is at . Please +report any bugs that you find. Or, even better, fork the repository on +GitHub and create a pull request. We welcome all changes, big or small, +and we will help you make the pull request if you are new to git (just +ask on our mailing list or Gitter Channel). If you further have any queries, you can find answers +on Stack Overflow using the [sympy](https://stackoverflow.com/questions/tagged/sympy) tag. + +## Brief History + +SymPy was started by Ondřej Čertík in 2005, he wrote some code during +the summer, then he wrote some more code during summer 2006. In February +2007, Fabian Pedregosa joined the project and helped fix many things, +contributed documentation, and made it alive again. 5 students (Mateusz +Paprocki, Brian Jorgensen, Jason Gedge, Robert Schwarz, and Chris Wu) +improved SymPy incredibly during summer 2007 as part of the Google +Summer of Code. Pearu Peterson joined the development during the summer +2007 and he has made SymPy much more competitive by rewriting the core +from scratch, which has made it from 10x to 100x faster. Jurjen N.E. Bos +has contributed pretty-printing and other patches. Fredrik Johansson has +written mpmath and contributed a lot of patches. + +SymPy has participated in every Google Summer of Code since 2007. You +can see for +full details. Each year has improved SymPy by bounds. Most of SymPy's +development has come from Google Summer of Code students. + +In 2011, Ondřej Čertík stepped down as lead developer, with Aaron +Meurer, who also started as a Google Summer of Code student, taking his +place. Ondřej Čertík is still active in the community but is too busy +with work and family to play a lead development role. + +Since then, a lot more people have joined the development and some +people have also left. You can see the full list in doc/src/aboutus.rst, +or online at: + + + +The git history goes back to 2007 when development moved from svn to hg. +To see the history before that point, look at +. + +You can use git to see the biggest developers. The command: + + $ git shortlog -ns + +will show each developer, sorted by commits to the project. The command: + + $ git shortlog -ns --since="1 year" + +will show the top developers from the last year. + +## Citation + +To cite SymPy in publications use + +> Meurer A, Smith CP, Paprocki M, Čertík O, Kirpichev SB, Rocklin M, +> Kumar A, Ivanov S, Moore JK, Singh S, Rathnayake T, Vig S, Granger BE, +> Muller RP, Bonazzi F, Gupta H, Vats S, Johansson F, Pedregosa F, Curry +> MJ, Terrel AR, Roučka Š, Saboo A, Fernando I, Kulal S, Cimrman R, +> Scopatz A. (2017) SymPy: symbolic computing in Python. *PeerJ Computer +> Science* 3:e103 + +A BibTeX entry for LaTeX users is + +``` bibtex +@article{10.7717/peerj-cs.103, + title = {SymPy: symbolic computing in Python}, + author = {Meurer, Aaron and Smith, Christopher P. and Paprocki, Mateusz and \v{C}ert\'{i}k, Ond\v{r}ej and Kirpichev, Sergey B. and Rocklin, Matthew and Kumar, Amit and Ivanov, Sergiu and Moore, Jason K. and Singh, Sartaj and Rathnayake, Thilina and Vig, Sean and Granger, Brian E. and Muller, Richard P. and Bonazzi, Francesco and Gupta, Harsh and Vats, Shivam and Johansson, Fredrik and Pedregosa, Fabian and Curry, Matthew J. and Terrel, Andy R. and Rou\v{c}ka, \v{S}t\v{e}p\'{a}n and Saboo, Ashutosh and Fernando, Isuru and Kulal, Sumith and Cimrman, Robert and Scopatz, Anthony}, + year = 2017, + month = Jan, + keywords = {Python, Computer algebra system, Symbolics}, + abstract = { + SymPy is an open-source computer algebra system written in pure Python. It is built with a focus on extensibility and ease of use, through both interactive and programmatic applications. These characteristics have led SymPy to become a popular symbolic library for the scientific Python ecosystem. This paper presents the architecture of SymPy, a description of its features, and a discussion of select submodules. The supplementary material provides additional examples and further outlines details of the architecture and features of SymPy. + }, + volume = 3, + pages = {e103}, + journal = {PeerJ Computer Science}, + issn = {2376-5992}, + url = {https://doi.org/10.7717/peerj-cs.103}, + doi = {10.7717/peerj-cs.103} +} +``` + +SymPy is BSD licensed, so you are free to use it whatever you like, be +it academic, commercial, creating forks or derivatives, as long as you +copy the BSD statement if you redistribute it (see the LICENSE file for +details). That said, although not required by the SymPy license, if it +is convenient for you, please cite SymPy when using it in your work and +also consider contributing all your changes back, so that we can +incorporate it and all of us will benefit in the end. diff --git a/.venv/lib/python3.12/site-packages/sympy-1.14.0.dist-info/RECORD b/.venv/lib/python3.12/site-packages/sympy-1.14.0.dist-info/RECORD new file mode 100644 index 0000000000000000000000000000000000000000..a31b1622a83ebdc80916d72478b306930e36bd5e --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy-1.14.0.dist-info/RECORD @@ -0,0 +1,3105 @@ +../../../bin/isympy,sha256=M5AMTJqv7K9QGHtadgn4UEjCuxX_dVdwJ1MXAq8NAuA,237 +../../../share/man/man1/isympy.1,sha256=9DZdSOIQLikrATHlbkdDZ04LBQigZDUE0_oCXBDvdBs,6659 +__pycache__/isympy.cpython-312.pyc,, +isympy.py,sha256=haEArczb_Ny-LyjOCwx2FNGal_S0Mo4d0deiWL2g_Gc,11220 +sympy-1.14.0.dist-info/INSTALLER,sha256=zuuue4knoyJ-UwPPXg8fezS7VCrXJQrAP7zeNuwvFQg,4 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/dev/null +++ b/.venv/lib/python3.12/site-packages/sympy-1.14.0.dist-info/entry_points.txt @@ -0,0 +1,2 @@ +[console_scripts] +isympy = isympy:main diff --git a/.venv/lib/python3.12/site-packages/sympy-1.14.0.dist-info/licenses/AUTHORS b/.venv/lib/python3.12/site-packages/sympy-1.14.0.dist-info/licenses/AUTHORS new file mode 100644 index 0000000000000000000000000000000000000000..1062f3bf743c0a37bcde05b1afc1ba5e98a95822 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy-1.14.0.dist-info/licenses/AUTHORS @@ -0,0 +1,1379 @@ +All people who contributed to SymPy by sending at least a patch or +more (in the order of the date of their first contribution), except +those who explicitly didn't want to be mentioned. People with a * next +to their names are not found in the metadata of the git history. This +file is generated automatically by running `./bin/authors_update.py`. + +There are a total of 1371 authors. + +Ondřej Čertík +Fabian Pedregosa +Jurjen N.E. Bos +Mateusz Paprocki +*Marc-Etienne M.Leveille +Brian Jorgensen +Jason Gedge +Robert Schwarz +Pearu Peterson +Fredrik Johansson +Chris Wu +*Ulrich Hecht +Goutham Lakshminarayan +David Lawrence +Jaroslaw Tworek +David Marek +Bernhard R. Link +Andrej Tokarčík +Or Dvory +Saroj Adhikari +Pauli Virtanen +Robert Kern +James Aspnes +Nimish Telang +Abderrahim Kitouni +Pan Peng +Friedrich Hagedorn +Elrond der Elbenfuerst +Rizgar Mella +Felix Kaiser +Roberto Nobrega +David Roberts +Sebastian Krämer +Vinzent Steinberg +Riccardo Gori +Case Van Horsen +Stepan Roucka +Ali Raza Syed +Stefano Maggiolo +Robert Cimrman +Bastian Weber +Sebastian Krause +Sebastian Kreft +*Dan +Alan Bromborsky +Boris Timokhin +Robert +Andy R. Terrel +Hubert Tsang +Konrad Meyer +Henrik Johansson +Priit Laes +Freddie Witherden +Brian E. Granger +Andrew Straw +Kaifeng Zhu +Ted Horst +Andrew Docherty +Akshay Srinivasan +Aaron Meurer +Barry Wardell +Tomasz Buchert +Vinay Kumar +Johann Cohen-Tanugi +Jochen Voss +Luke Peterson +Chris Smith +Thomas Sidoti +Florian Mickler +Nicolas Pourcelot +Ben Goodrich +Toon Verstraelen +Ronan Lamy +James Abbatiello +Ryan Krauss +Bill Flynn +Kevin Goodsell +Jorn Baayen +Eh Tan +Renato Coutinho +Oscar Benjamin +Øyvind Jensen +Julio Idichekop Filho +Łukasz Pankowski +*Chu-Ching Huang +Fernando Perez +Raffaele De Feo +Christian Muise +Matt Curry +Kazuo Thow +Christian Schubert +Jezreel Ng +James Pearson +Matthew Brett +Addison Cugini +Nicholas J.S. Kinar +Harold Erbin +Thomas Dixon +Cristóvão Sousa +Andre de Fortier Smit +Mark Dewing +Alexey U. Gudchenko +Gary Kerr +Sherjil Ozair +Oleksandr Gituliar +Sean Vig +Prafullkumar P. Tale +Vladimir Perić +Tom Bachmann +Yuri Karadzhov +Vladimir Lagunov +Matthew Rocklin +Saptarshi Mandal +Gilbert Gede +Anatolii Koval +Tomo Lazovich +Pavel Fedotov +Jack McCaffery +Jeremias Yehdegho +Kibeom Kim +Gregory Ksionda +Tomáš Bambas +Raymond Wong +Luca Weihs +Shai 'Deshe' Wyborski +Thomas Wiecki +Óscar Nájera +Mario Pernici +Benjamin McDonald +Sam Magura +Stefan Krastanov +Bradley Froehle +Min Ragan-Kelley +Emma Hogan +Nikhil Sarda +Julien Rioux +Roberto Colistete, Jr. +Raoul Bourquin +Gert-Ludwig Ingold +Srinivas Vasudevan +Jason Moore +Miha Marolt +Tim Lahey +Luis Garcia +Matt Rajca +David Li +Alexandr Gudulin +Bilal Akhtar +Grzegorz Świrski +Matt Habel +David Ju +Nichita Utiu +Nikolay Lazarov +Steve Anton +Imran Ahmed Manzoor +Ljubiša Moćić <3rdslasher@gmail.com> +Piotr Korgul +Jim Zhang +Sam Sleight +tborisova +Chancellor Arkantos +Stepan Simsa +Tobias Lenz +Siddhanathan Shanmugam +Tiffany Zhu +Tristan Hume +Alexey Subach +Joan Creus +Geoffry Song +Puneeth Chaganti +Marcin Kostrzewa <> +Natalia Nawara +vishal +Shruti Mangipudi +Davy Mao +Swapnil Agarwal +Dhia Kennouche +jerryma1121 +Joachim Durchholz +Martin Povišer +Siddhant Jain +Kevin Hunter +Michael Mayorov +Nathan Alison +Christian Bühler +Carsten Knoll +Bharath M R +Matthias Toews +Sergiu Ivanov +Jorge E. 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IN NO EVENT SHALL THE +AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER +LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, +OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE +SOFTWARE. diff --git a/.venv/lib/python3.12/site-packages/sympy-1.14.0.dist-info/top_level.txt b/.venv/lib/python3.12/site-packages/sympy-1.14.0.dist-info/top_level.txt new file mode 100644 index 0000000000000000000000000000000000000000..0aa85c253508b68bb075f556be3c3f76dc4467ad --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy-1.14.0.dist-info/top_level.txt @@ -0,0 +1,2 @@ +isympy +sympy diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/orderings.py b/.venv/lib/python3.12/site-packages/sympy/polys/orderings.py new file mode 100644 index 0000000000000000000000000000000000000000..b6ed575d5103440e1e8ebda4c53c4149d3badf11 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/orderings.py @@ -0,0 +1,286 @@ +"""Definitions of monomial orderings. """ + +from __future__ import annotations + +__all__ = ["lex", "grlex", "grevlex", "ilex", "igrlex", "igrevlex"] + +from sympy.core import Symbol +from sympy.utilities.iterables import iterable + +class MonomialOrder: + """Base class for monomial orderings. """ + + alias: str | None = None + is_global: bool | None = None + is_default = False + + def __repr__(self): + return self.__class__.__name__ + "()" + + def __str__(self): + return self.alias + + def __call__(self, monomial): + raise NotImplementedError + + def __eq__(self, other): + return self.__class__ == other.__class__ + + def __hash__(self): + return hash(self.__class__) + + def __ne__(self, other): + return not (self == other) + +class LexOrder(MonomialOrder): + """Lexicographic order of monomials. """ + + alias = 'lex' + is_global = True + is_default = True + + def __call__(self, monomial): + return monomial + +class GradedLexOrder(MonomialOrder): + """Graded lexicographic order of monomials. """ + + alias = 'grlex' + is_global = True + + def __call__(self, monomial): + return (sum(monomial), monomial) + +class ReversedGradedLexOrder(MonomialOrder): + """Reversed graded lexicographic order of monomials. """ + + alias = 'grevlex' + is_global = True + + def __call__(self, monomial): + return (sum(monomial), tuple(reversed([-m for m in monomial]))) + +class ProductOrder(MonomialOrder): + """ + A product order built from other monomial orders. + + Given (not necessarily total) orders O1, O2, ..., On, their product order + P is defined as M1 > M2 iff there exists i such that O1(M1) = O2(M2), + ..., Oi(M1) = Oi(M2), O{i+1}(M1) > O{i+1}(M2). + + Product orders are typically built from monomial orders on different sets + of variables. + + ProductOrder is constructed by passing a list of pairs + [(O1, L1), (O2, L2), ...] where Oi are MonomialOrders and Li are callables. + Upon comparison, the Li are passed the total monomial, and should filter + out the part of the monomial to pass to Oi. + + Examples + ======== + + We can use a lexicographic order on x_1, x_2 and also on + y_1, y_2, y_3, and their product on {x_i, y_i} as follows: + + >>> from sympy.polys.orderings import lex, grlex, ProductOrder + >>> P = ProductOrder( + ... (lex, lambda m: m[:2]), # lex order on x_1 and x_2 of monomial + ... (grlex, lambda m: m[2:]) # grlex on y_1, y_2, y_3 + ... ) + >>> P((2, 1, 1, 0, 0)) > P((1, 10, 0, 2, 0)) + True + + Here the exponent `2` of `x_1` in the first monomial + (`x_1^2 x_2 y_1`) is bigger than the exponent `1` of `x_1` in the + second monomial (`x_1 x_2^10 y_2^2`), so the first monomial is greater + in the product ordering. + + >>> P((2, 1, 1, 0, 0)) < P((2, 1, 0, 2, 0)) + True + + Here the exponents of `x_1` and `x_2` agree, so the grlex order on + `y_1, y_2, y_3` is used to decide the ordering. In this case the monomial + `y_2^2` is ordered larger than `y_1`, since for the grlex order the degree + of the monomial is most important. + """ + + def __init__(self, *args): + self.args = args + + def __call__(self, monomial): + return tuple(O(lamda(monomial)) for (O, lamda) in self.args) + + def __repr__(self): + contents = [repr(x[0]) for x in self.args] + return self.__class__.__name__ + '(' + ", ".join(contents) + ')' + + def __str__(self): + contents = [str(x[0]) for x in self.args] + return self.__class__.__name__ + '(' + ", ".join(contents) + ')' + + def __eq__(self, other): + if not isinstance(other, ProductOrder): + return False + return self.args == other.args + + def __hash__(self): + return hash((self.__class__, self.args)) + + @property + def is_global(self): + if all(o.is_global is True for o, _ in self.args): + return True + if all(o.is_global is False for o, _ in self.args): + return False + return None + +class InverseOrder(MonomialOrder): + """ + The "inverse" of another monomial order. + + If O is any monomial order, we can construct another monomial order iO + such that `A >_{iO} B` if and only if `B >_O A`. This is useful for + constructing local orders. + + Note that many algorithms only work with *global* orders. + + For example, in the inverse lexicographic order on a single variable `x`, + high powers of `x` count as small: + + >>> from sympy.polys.orderings import lex, InverseOrder + >>> ilex = InverseOrder(lex) + >>> ilex((5,)) < ilex((0,)) + True + """ + + def __init__(self, O): + self.O = O + + def __str__(self): + return "i" + str(self.O) + + def __call__(self, monomial): + def inv(l): + if iterable(l): + return tuple(inv(x) for x in l) + return -l + return inv(self.O(monomial)) + + @property + def is_global(self): + if self.O.is_global is True: + return False + if self.O.is_global is False: + return True + return None + + def __eq__(self, other): + return isinstance(other, InverseOrder) and other.O == self.O + + def __hash__(self): + return hash((self.__class__, self.O)) + +lex = LexOrder() +grlex = GradedLexOrder() +grevlex = ReversedGradedLexOrder() +ilex = InverseOrder(lex) +igrlex = InverseOrder(grlex) +igrevlex = InverseOrder(grevlex) + +_monomial_key = { + 'lex': lex, + 'grlex': grlex, + 'grevlex': grevlex, + 'ilex': ilex, + 'igrlex': igrlex, + 'igrevlex': igrevlex +} + +def monomial_key(order=None, gens=None): + """ + Return a function defining admissible order on monomials. + + The result of a call to :func:`monomial_key` is a function which should + be used as a key to :func:`sorted` built-in function, to provide order + in a set of monomials of the same length. + + Currently supported monomial orderings are: + + 1. lex - lexicographic order (default) + 2. grlex - graded lexicographic order + 3. grevlex - reversed graded lexicographic order + 4. ilex, igrlex, igrevlex - the corresponding inverse orders + + If the ``order`` input argument is not a string but has ``__call__`` + attribute, then it will pass through with an assumption that the + callable object defines an admissible order on monomials. + + If the ``gens`` input argument contains a list of generators, the + resulting key function can be used to sort SymPy ``Expr`` objects. + + """ + if order is None: + order = lex + + if isinstance(order, Symbol): + order = str(order) + + if isinstance(order, str): + try: + order = _monomial_key[order] + except KeyError: + raise ValueError("supported monomial orderings are 'lex', 'grlex' and 'grevlex', got %r" % order) + if hasattr(order, '__call__'): + if gens is not None: + def _order(expr): + return order(expr.as_poly(*gens).degree_list()) + return _order + return order + else: + raise ValueError("monomial ordering specification must be a string or a callable, got %s" % order) + +class _ItemGetter: + """Helper class to return a subsequence of values.""" + + def __init__(self, seq): + self.seq = tuple(seq) + + def __call__(self, m): + return tuple(m[idx] for idx in self.seq) + + def __eq__(self, other): + if not isinstance(other, _ItemGetter): + return False + return self.seq == other.seq + +def build_product_order(arg, gens): + """ + Build a monomial order on ``gens``. + + ``arg`` should be a tuple of iterables. The first element of each iterable + should be a string or monomial order (will be passed to monomial_key), + the others should be subsets of the generators. This function will build + the corresponding product order. + + For example, build a product of two grlex orders: + + >>> from sympy.polys.orderings import build_product_order + >>> from sympy.abc import x, y, z, t + + >>> O = build_product_order((("grlex", x, y), ("grlex", z, t)), [x, y, z, t]) + >>> O((1, 2, 3, 4)) + ((3, (1, 2)), (7, (3, 4))) + + """ + gens2idx = {} + for i, g in enumerate(gens): + gens2idx[g] = i + order = [] + for expr in arg: + name = expr[0] + var = expr[1:] + + def makelambda(var): + return _ItemGetter(gens2idx[g] for g in var) + order.append((monomial_key(name), makelambda(var))) + return ProductOrder(*order) diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/orthopolys.py b/.venv/lib/python3.12/site-packages/sympy/polys/orthopolys.py new file mode 100644 index 0000000000000000000000000000000000000000..ee82457703a2be172951ee38e3cd67f221a438a0 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/orthopolys.py @@ -0,0 +1,343 @@ +"""Efficient functions for generating orthogonal polynomials.""" +from sympy.core.symbol import Dummy +from sympy.polys.densearith import (dup_mul, dup_mul_ground, + dup_lshift, dup_sub, dup_add, dup_sub_term, dup_sub_ground, dup_sqr) +from sympy.polys.domains import ZZ, QQ +from sympy.polys.polytools import named_poly +from sympy.utilities import public + +def dup_jacobi(n, a, b, K): + """Low-level implementation of Jacobi polynomials.""" + if n < 1: + return [K.one] + m2, m1 = [K.one], [(a+b)/K(2) + K.one, (a-b)/K(2)] + for i in range(2, n+1): + den = K(i)*(a + b + i)*(a + b + K(2)*i - K(2)) + f0 = (a + b + K(2)*i - K.one) * (a*a - b*b) / (K(2)*den) + f1 = (a + b + K(2)*i - K.one) * (a + b + K(2)*i - K(2)) * (a + b + K(2)*i) / (K(2)*den) + f2 = (a + i - K.one)*(b + i - K.one)*(a + b + K(2)*i) / den + p0 = dup_mul_ground(m1, f0, K) + p1 = dup_mul_ground(dup_lshift(m1, 1, K), f1, K) + p2 = dup_mul_ground(m2, f2, K) + m2, m1 = m1, dup_sub(dup_add(p0, p1, K), p2, K) + return m1 + +@public +def jacobi_poly(n, a, b, x=None, polys=False): + r"""Generates the Jacobi polynomial `P_n^{(a,b)}(x)`. + + Parameters + ========== + + n : int + Degree of the polynomial. + a + Lower limit of minimal domain for the list of coefficients. + b + Upper limit of minimal domain for the list of coefficients. + x : optional + polys : bool, optional + If True, return a Poly, otherwise (default) return an expression. + """ + return named_poly(n, dup_jacobi, None, "Jacobi polynomial", (x, a, b), polys) + +def dup_gegenbauer(n, a, K): + """Low-level implementation of Gegenbauer polynomials.""" + if n < 1: + return [K.one] + m2, m1 = [K.one], [K(2)*a, K.zero] + for i in range(2, n+1): + p1 = dup_mul_ground(dup_lshift(m1, 1, K), K(2)*(a-K.one)/K(i) + K(2), K) + p2 = dup_mul_ground(m2, K(2)*(a-K.one)/K(i) + K.one, K) + m2, m1 = m1, dup_sub(p1, p2, K) + return m1 + +def gegenbauer_poly(n, a, x=None, polys=False): + r"""Generates the Gegenbauer polynomial `C_n^{(a)}(x)`. + + Parameters + ========== + + n : int + Degree of the polynomial. + x : optional + a + Decides minimal domain for the list of coefficients. + polys : bool, optional + If True, return a Poly, otherwise (default) return an expression. + """ + return named_poly(n, dup_gegenbauer, None, "Gegenbauer polynomial", (x, a), polys) + +def dup_chebyshevt(n, K): + """Low-level implementation of Chebyshev polynomials of the first kind.""" + if n < 1: + return [K.one] + # When n is small, it is faster to directly calculate the recurrence relation. + if n < 64: # The threshold serves as a heuristic + return _dup_chebyshevt_rec(n, K) + return _dup_chebyshevt_prod(n, K) + +def _dup_chebyshevt_rec(n, K): + r""" Chebyshev polynomials of the first kind using recurrence. + + Explanation + =========== + + Chebyshev polynomials of the first kind are defined by the recurrence + relation: + + .. math:: + T_0(x) &= 1\\ + T_1(x) &= x\\ + T_n(x) &= 2xT_{n-1}(x) - T_{n-2}(x) + + This function calculates the Chebyshev polynomial of the first kind using + the above recurrence relation. + + Parameters + ========== + + n : int + n is a nonnegative integer. + K : domain + + """ + m2, m1 = [K.one], [K.one, K.zero] + for _ in range(n - 1): + m2, m1 = m1, dup_sub(dup_mul_ground(dup_lshift(m1, 1, K), K(2), K), m2, K) + return m1 + +def _dup_chebyshevt_prod(n, K): + r""" Chebyshev polynomials of the first kind using recursive products. + + Explanation + =========== + + Computes Chebyshev polynomials of the first kind using + + .. math:: + T_{2n}(x) &= 2T_n^2(x) - 1\\ + T_{2n+1}(x) &= 2T_{n+1}(x)T_n(x) - x + + This is faster than ``_dup_chebyshevt_rec`` for large ``n``. + + Parameters + ========== + + n : int + n is a nonnegative integer. + K : domain + + """ + m2, m1 = [K.one, K.zero], [K(2), K.zero, -K.one] + for i in bin(n)[3:]: + c = dup_sub_term(dup_mul_ground(dup_mul(m1, m2, K), K(2), K), K.one, 1, K) + if i == '1': + m2, m1 = c, dup_sub_ground(dup_mul_ground(dup_sqr(m1, K), K(2), K), K.one, K) + else: + m2, m1 = dup_sub_ground(dup_mul_ground(dup_sqr(m2, K), K(2), K), K.one, K), c + return m2 + +def dup_chebyshevu(n, K): + """Low-level implementation of Chebyshev polynomials of the second kind.""" + if n < 1: + return [K.one] + m2, m1 = [K.one], [K(2), K.zero] + for i in range(2, n+1): + m2, m1 = m1, dup_sub(dup_mul_ground(dup_lshift(m1, 1, K), K(2), K), m2, K) + return m1 + +@public +def chebyshevt_poly(n, x=None, polys=False): + r"""Generates the Chebyshev polynomial of the first kind `T_n(x)`. + + Parameters + ========== + + n : int + Degree of the polynomial. + x : optional + polys : bool, optional + If True, return a Poly, otherwise (default) return an expression. + """ + return named_poly(n, dup_chebyshevt, ZZ, + "Chebyshev polynomial of the first kind", (x,), polys) + +@public +def chebyshevu_poly(n, x=None, polys=False): + r"""Generates the Chebyshev polynomial of the second kind `U_n(x)`. + + Parameters + ========== + + n : int + Degree of the polynomial. + x : optional + polys : bool, optional + If True, return a Poly, otherwise (default) return an expression. + """ + return named_poly(n, dup_chebyshevu, ZZ, + "Chebyshev polynomial of the second kind", (x,), polys) + +def dup_hermite(n, K): + """Low-level implementation of Hermite polynomials.""" + if n < 1: + return [K.one] + m2, m1 = [K.one], [K(2), K.zero] + for i in range(2, n+1): + a = dup_lshift(m1, 1, K) + b = dup_mul_ground(m2, K(i-1), K) + m2, m1 = m1, dup_mul_ground(dup_sub(a, b, K), K(2), K) + return m1 + +def dup_hermite_prob(n, K): + """Low-level implementation of probabilist's Hermite polynomials.""" + if n < 1: + return [K.one] + m2, m1 = [K.one], [K.one, K.zero] + for i in range(2, n+1): + a = dup_lshift(m1, 1, K) + b = dup_mul_ground(m2, K(i-1), K) + m2, m1 = m1, dup_sub(a, b, K) + return m1 + +@public +def hermite_poly(n, x=None, polys=False): + r"""Generates the Hermite polynomial `H_n(x)`. + + Parameters + ========== + + n : int + Degree of the polynomial. + x : optional + polys : bool, optional + If True, return a Poly, otherwise (default) return an expression. + """ + return named_poly(n, dup_hermite, ZZ, "Hermite polynomial", (x,), polys) + +@public +def hermite_prob_poly(n, x=None, polys=False): + r"""Generates the probabilist's Hermite polynomial `He_n(x)`. + + Parameters + ========== + + n : int + Degree of the polynomial. + x : optional + polys : bool, optional + If True, return a Poly, otherwise (default) return an expression. + """ + return named_poly(n, dup_hermite_prob, ZZ, + "probabilist's Hermite polynomial", (x,), polys) + +def dup_legendre(n, K): + """Low-level implementation of Legendre polynomials.""" + if n < 1: + return [K.one] + m2, m1 = [K.one], [K.one, K.zero] + for i in range(2, n+1): + a = dup_mul_ground(dup_lshift(m1, 1, K), K(2*i-1, i), K) + b = dup_mul_ground(m2, K(i-1, i), K) + m2, m1 = m1, dup_sub(a, b, K) + return m1 + +@public +def legendre_poly(n, x=None, polys=False): + r"""Generates the Legendre polynomial `P_n(x)`. + + Parameters + ========== + + n : int + Degree of the polynomial. + x : optional + polys : bool, optional + If True, return a Poly, otherwise (default) return an expression. + """ + return named_poly(n, dup_legendre, QQ, "Legendre polynomial", (x,), polys) + +def dup_laguerre(n, alpha, K): + """Low-level implementation of Laguerre polynomials.""" + m2, m1 = [K.zero], [K.one] + for i in range(1, n+1): + a = dup_mul(m1, [-K.one/K(i), (alpha-K.one)/K(i) + K(2)], K) + b = dup_mul_ground(m2, (alpha-K.one)/K(i) + K.one, K) + m2, m1 = m1, dup_sub(a, b, K) + return m1 + +@public +def laguerre_poly(n, x=None, alpha=0, polys=False): + r"""Generates the Laguerre polynomial `L_n^{(\alpha)}(x)`. + + Parameters + ========== + + n : int + Degree of the polynomial. + x : optional + alpha : optional + Decides minimal domain for the list of coefficients. + polys : bool, optional + If True, return a Poly, otherwise (default) return an expression. + """ + return named_poly(n, dup_laguerre, None, "Laguerre polynomial", (x, alpha), polys) + +def dup_spherical_bessel_fn(n, K): + """Low-level implementation of fn(n, x).""" + if n < 1: + return [K.one, K.zero] + m2, m1 = [K.one], [K.one, K.zero] + for i in range(2, n+1): + m2, m1 = m1, dup_sub(dup_mul_ground(dup_lshift(m1, 1, K), K(2*i-1), K), m2, K) + return dup_lshift(m1, 1, K) + +def dup_spherical_bessel_fn_minus(n, K): + """Low-level implementation of fn(-n, x).""" + m2, m1 = [K.one, K.zero], [K.zero] + for i in range(2, n+1): + m2, m1 = m1, dup_sub(dup_mul_ground(dup_lshift(m1, 1, K), K(3-2*i), K), m2, K) + return m1 + +def spherical_bessel_fn(n, x=None, polys=False): + """ + Coefficients for the spherical Bessel functions. + + These are only needed in the jn() function. + + The coefficients are calculated from: + + fn(0, z) = 1/z + fn(1, z) = 1/z**2 + fn(n-1, z) + fn(n+1, z) == (2*n+1)/z * fn(n, z) + + Parameters + ========== + + n : int + Degree of the polynomial. + x : optional + polys : bool, optional + If True, return a Poly, otherwise (default) return an expression. + + Examples + ======== + + >>> from sympy.polys.orthopolys import spherical_bessel_fn as fn + >>> from sympy import Symbol + >>> z = Symbol("z") + >>> fn(1, z) + z**(-2) + >>> fn(2, z) + -1/z + 3/z**3 + >>> fn(3, z) + -6/z**2 + 15/z**4 + >>> fn(4, z) + 1/z - 45/z**3 + 105/z**5 + + """ + if x is None: + x = Dummy("x") + f = dup_spherical_bessel_fn_minus if n < 0 else dup_spherical_bessel_fn + return named_poly(abs(n), f, ZZ, "", (QQ(1)/x,), polys) diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/partfrac.py b/.venv/lib/python3.12/site-packages/sympy/polys/partfrac.py new file mode 100644 index 0000000000000000000000000000000000000000..dedc1bf0fba42128e869303ed9b12c598640a36c --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/partfrac.py @@ -0,0 +1,496 @@ +"""Algorithms for partial fraction decomposition of rational functions. """ + + +from sympy.core import S, Add, sympify, Function, Lambda, Dummy +from sympy.core.traversal import preorder_traversal +from sympy.polys import Poly, RootSum, cancel, factor +from sympy.polys.polyerrors import PolynomialError +from sympy.polys.polyoptions import allowed_flags, set_defaults +from sympy.polys.polytools import parallel_poly_from_expr +from sympy.utilities import numbered_symbols, take, xthreaded, public + + +@xthreaded +@public +def apart(f, x=None, full=False, **options): + """ + Compute partial fraction decomposition of a rational function. + + Given a rational function ``f``, computes the partial fraction + decomposition of ``f``. Two algorithms are available: One is based on the + undetermined coefficients method, the other is Bronstein's full partial + fraction decomposition algorithm. + + The undetermined coefficients method (selected by ``full=False``) uses + polynomial factorization (and therefore accepts the same options as + factor) for the denominator. Per default it works over the rational + numbers, therefore decomposition of denominators with non-rational roots + (e.g. irrational, complex roots) is not supported by default (see options + of factor). + + Bronstein's algorithm can be selected by using ``full=True`` and allows a + decomposition of denominators with non-rational roots. A human-readable + result can be obtained via ``doit()`` (see examples below). + + Examples + ======== + + >>> from sympy.polys.partfrac import apart + >>> from sympy.abc import x, y + + By default, using the undetermined coefficients method: + + >>> apart(y/(x + 2)/(x + 1), x) + -y/(x + 2) + y/(x + 1) + + The undetermined coefficients method does not provide a result when the + denominators roots are not rational: + + >>> apart(y/(x**2 + x + 1), x) + y/(x**2 + x + 1) + + You can choose Bronstein's algorithm by setting ``full=True``: + + >>> apart(y/(x**2 + x + 1), x, full=True) + RootSum(_w**2 + _w + 1, Lambda(_a, (-2*_a*y/3 - y/3)/(-_a + x))) + + Calling ``doit()`` yields a human-readable result: + + >>> apart(y/(x**2 + x + 1), x, full=True).doit() + (-y/3 - 2*y*(-1/2 - sqrt(3)*I/2)/3)/(x + 1/2 + sqrt(3)*I/2) + (-y/3 - + 2*y*(-1/2 + sqrt(3)*I/2)/3)/(x + 1/2 - sqrt(3)*I/2) + + + See Also + ======== + + apart_list, assemble_partfrac_list + """ + allowed_flags(options, []) + + f = sympify(f) + + if f.is_Atom: + return f + else: + P, Q = f.as_numer_denom() + + _options = options.copy() + options = set_defaults(options, extension=True) + try: + (P, Q), opt = parallel_poly_from_expr((P, Q), x, **options) + except PolynomialError as msg: + if f.is_commutative: + raise PolynomialError(msg) + # non-commutative + if f.is_Mul: + c, nc = f.args_cnc(split_1=False) + nc = f.func(*nc) + if c: + c = apart(f.func._from_args(c), x=x, full=full, **_options) + return c*nc + else: + return nc + elif f.is_Add: + c = [] + nc = [] + for i in f.args: + if i.is_commutative: + c.append(i) + else: + try: + nc.append(apart(i, x=x, full=full, **_options)) + except NotImplementedError: + nc.append(i) + return apart(f.func(*c), x=x, full=full, **_options) + f.func(*nc) + else: + reps = [] + pot = preorder_traversal(f) + next(pot) + for e in pot: + try: + reps.append((e, apart(e, x=x, full=full, **_options))) + pot.skip() # this was handled successfully + except NotImplementedError: + pass + return f.xreplace(dict(reps)) + + if P.is_multivariate: + fc = f.cancel() + if fc != f: + return apart(fc, x=x, full=full, **_options) + + raise NotImplementedError( + "multivariate partial fraction decomposition") + + common, P, Q = P.cancel(Q) + + poly, P = P.div(Q, auto=True) + P, Q = P.rat_clear_denoms(Q) + + if Q.degree() <= 1: + partial = P/Q + else: + if not full: + partial = apart_undetermined_coeffs(P, Q) + else: + partial = apart_full_decomposition(P, Q) + + terms = S.Zero + + for term in Add.make_args(partial): + if term.has(RootSum): + terms += term + else: + terms += factor(term) + + return common*(poly.as_expr() + terms) + + +def apart_undetermined_coeffs(P, Q): + """Partial fractions via method of undetermined coefficients. """ + X = numbered_symbols(cls=Dummy) + partial, symbols = [], [] + + _, factors = Q.factor_list() + + for f, k in factors: + n, q = f.degree(), Q + + for i in range(1, k + 1): + coeffs, q = take(X, n), q.quo(f) + partial.append((coeffs, q, f, i)) + symbols.extend(coeffs) + + dom = Q.get_domain().inject(*symbols) + F = Poly(0, Q.gen, domain=dom) + + for i, (coeffs, q, f, k) in enumerate(partial): + h = Poly(coeffs, Q.gen, domain=dom) + partial[i] = (h, f, k) + q = q.set_domain(dom) + F += h*q + + system, result = [], S.Zero + + for (k,), coeff in F.terms(): + system.append(coeff - P.nth(k)) + + from sympy.solvers import solve + solution = solve(system, symbols) + + for h, f, k in partial: + h = h.as_expr().subs(solution) + result += h/f.as_expr()**k + + return result + + +def apart_full_decomposition(P, Q): + """ + Bronstein's full partial fraction decomposition algorithm. + + Given a univariate rational function ``f``, performing only GCD + operations over the algebraic closure of the initial ground domain + of definition, compute full partial fraction decomposition with + fractions having linear denominators. + + Note that no factorization of the initial denominator of ``f`` is + performed. The final decomposition is formed in terms of a sum of + :class:`RootSum` instances. + + References + ========== + + .. [1] [Bronstein93]_ + + """ + return assemble_partfrac_list(apart_list(P/Q, P.gens[0])) + + +@public +def apart_list(f, x=None, dummies=None, **options): + """ + Compute partial fraction decomposition of a rational function + and return the result in structured form. + + Given a rational function ``f`` compute the partial fraction decomposition + of ``f``. Only Bronstein's full partial fraction decomposition algorithm + is supported by this method. The return value is highly structured and + perfectly suited for further algorithmic treatment rather than being + human-readable. The function returns a tuple holding three elements: + + * The first item is the common coefficient, free of the variable `x` used + for decomposition. (It is an element of the base field `K`.) + + * The second item is the polynomial part of the decomposition. This can be + the zero polynomial. (It is an element of `K[x]`.) + + * The third part itself is a list of quadruples. Each quadruple + has the following elements in this order: + + - The (not necessarily irreducible) polynomial `D` whose roots `w_i` appear + in the linear denominator of a bunch of related fraction terms. (This item + can also be a list of explicit roots. However, at the moment ``apart_list`` + never returns a result this way, but the related ``assemble_partfrac_list`` + function accepts this format as input.) + + - The numerator of the fraction, written as a function of the root `w` + + - The linear denominator of the fraction *excluding its power exponent*, + written as a function of the root `w`. + + - The power to which the denominator has to be raised. + + On can always rebuild a plain expression by using the function ``assemble_partfrac_list``. + + Examples + ======== + + A first example: + + >>> from sympy.polys.partfrac import apart_list, assemble_partfrac_list + >>> from sympy.abc import x, t + + >>> f = (2*x**3 - 2*x) / (x**2 - 2*x + 1) + >>> pfd = apart_list(f) + >>> pfd + (1, + Poly(2*x + 4, x, domain='ZZ'), + [(Poly(_w - 1, _w, domain='ZZ'), Lambda(_a, 4), Lambda(_a, -_a + x), 1)]) + + >>> assemble_partfrac_list(pfd) + 2*x + 4 + 4/(x - 1) + + Second example: + + >>> f = (-2*x - 2*x**2) / (3*x**2 - 6*x) + >>> pfd = apart_list(f) + >>> pfd + (-1, + Poly(2/3, x, domain='QQ'), + [(Poly(_w - 2, _w, domain='ZZ'), Lambda(_a, 2), Lambda(_a, -_a + x), 1)]) + + >>> assemble_partfrac_list(pfd) + -2/3 - 2/(x - 2) + + Another example, showing symbolic parameters: + + >>> pfd = apart_list(t/(x**2 + x + t), x) + >>> pfd + (1, + Poly(0, x, domain='ZZ[t]'), + [(Poly(_w**2 + _w + t, _w, domain='ZZ[t]'), + Lambda(_a, -2*_a*t/(4*t - 1) - t/(4*t - 1)), + Lambda(_a, -_a + x), + 1)]) + + >>> assemble_partfrac_list(pfd) + RootSum(_w**2 + _w + t, Lambda(_a, (-2*_a*t/(4*t - 1) - t/(4*t - 1))/(-_a + x))) + + This example is taken from Bronstein's original paper: + + >>> f = 36 / (x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2) + >>> pfd = apart_list(f) + >>> pfd + (1, + Poly(0, x, domain='ZZ'), + [(Poly(_w - 2, _w, domain='ZZ'), Lambda(_a, 4), Lambda(_a, -_a + x), 1), + (Poly(_w**2 - 1, _w, domain='ZZ'), Lambda(_a, -3*_a - 6), Lambda(_a, -_a + x), 2), + (Poly(_w + 1, _w, domain='ZZ'), Lambda(_a, -4), Lambda(_a, -_a + x), 1)]) + + >>> assemble_partfrac_list(pfd) + -4/(x + 1) - 3/(x + 1)**2 - 9/(x - 1)**2 + 4/(x - 2) + + See also + ======== + + apart, assemble_partfrac_list + + References + ========== + + .. [1] [Bronstein93]_ + + """ + allowed_flags(options, []) + + f = sympify(f) + + if f.is_Atom: + return f + else: + P, Q = f.as_numer_denom() + + options = set_defaults(options, extension=True) + (P, Q), opt = parallel_poly_from_expr((P, Q), x, **options) + + if P.is_multivariate: + raise NotImplementedError( + "multivariate partial fraction decomposition") + + common, P, Q = P.cancel(Q) + + poly, P = P.div(Q, auto=True) + P, Q = P.rat_clear_denoms(Q) + + polypart = poly + + if dummies is None: + def dummies(name): + d = Dummy(name) + while True: + yield d + + dummies = dummies("w") + + rationalpart = apart_list_full_decomposition(P, Q, dummies) + + return (common, polypart, rationalpart) + + +def apart_list_full_decomposition(P, Q, dummygen): + """ + Bronstein's full partial fraction decomposition algorithm. + + Given a univariate rational function ``f``, performing only GCD + operations over the algebraic closure of the initial ground domain + of definition, compute full partial fraction decomposition with + fractions having linear denominators. + + Note that no factorization of the initial denominator of ``f`` is + performed. The final decomposition is formed in terms of a sum of + :class:`RootSum` instances. + + References + ========== + + .. [1] [Bronstein93]_ + + """ + P_orig, Q_orig, x, U = P, Q, P.gen, [] + + u = Function('u')(x) + a = Dummy('a') + + partial = [] + + for d, n in Q.sqf_list_include(all=True): + b = d.as_expr() + U += [ u.diff(x, n - 1) ] + + h = cancel(P_orig/Q_orig.quo(d**n)) / u**n + + H, subs = [h], [] + + for j in range(1, n): + H += [ H[-1].diff(x) / j ] + + for j in range(1, n + 1): + subs += [ (U[j - 1], b.diff(x, j) / j) ] + + for j in range(0, n): + P, Q = cancel(H[j]).as_numer_denom() + + for i in range(0, j + 1): + P = P.subs(*subs[j - i]) + + Q = Q.subs(*subs[0]) + + P = Poly(P, x) + Q = Poly(Q, x) + + G = P.gcd(d) + D = d.quo(G) + + B, g = Q.half_gcdex(D) + b = (P * B.quo(g)).rem(D) + + Dw = D.subs(x, next(dummygen)) + numer = Lambda(a, b.as_expr().subs(x, a)) + denom = Lambda(a, (x - a)) + exponent = n-j + + partial.append((Dw, numer, denom, exponent)) + + return partial + + +@public +def assemble_partfrac_list(partial_list): + r"""Reassemble a full partial fraction decomposition + from a structured result obtained by the function ``apart_list``. + + Examples + ======== + + This example is taken from Bronstein's original paper: + + >>> from sympy.polys.partfrac import apart_list, assemble_partfrac_list + >>> from sympy.abc import x + + >>> f = 36 / (x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2) + >>> pfd = apart_list(f) + >>> pfd + (1, + Poly(0, x, domain='ZZ'), + [(Poly(_w - 2, _w, domain='ZZ'), Lambda(_a, 4), Lambda(_a, -_a + x), 1), + (Poly(_w**2 - 1, _w, domain='ZZ'), Lambda(_a, -3*_a - 6), Lambda(_a, -_a + x), 2), + (Poly(_w + 1, _w, domain='ZZ'), Lambda(_a, -4), Lambda(_a, -_a + x), 1)]) + + >>> assemble_partfrac_list(pfd) + -4/(x + 1) - 3/(x + 1)**2 - 9/(x - 1)**2 + 4/(x - 2) + + If we happen to know some roots we can provide them easily inside the structure: + + >>> pfd = apart_list(2/(x**2-2)) + >>> pfd + (1, + Poly(0, x, domain='ZZ'), + [(Poly(_w**2 - 2, _w, domain='ZZ'), + Lambda(_a, _a/2), + Lambda(_a, -_a + x), + 1)]) + + >>> pfda = assemble_partfrac_list(pfd) + >>> pfda + RootSum(_w**2 - 2, Lambda(_a, _a/(-_a + x)))/2 + + >>> pfda.doit() + -sqrt(2)/(2*(x + sqrt(2))) + sqrt(2)/(2*(x - sqrt(2))) + + >>> from sympy import Dummy, Poly, Lambda, sqrt + >>> a = Dummy("a") + >>> pfd = (1, Poly(0, x, domain='ZZ'), [([sqrt(2),-sqrt(2)], Lambda(a, a/2), Lambda(a, -a + x), 1)]) + + >>> assemble_partfrac_list(pfd) + -sqrt(2)/(2*(x + sqrt(2))) + sqrt(2)/(2*(x - sqrt(2))) + + See Also + ======== + + apart, apart_list + """ + # Common factor + common = partial_list[0] + + # Polynomial part + polypart = partial_list[1] + pfd = polypart.as_expr() + + # Rational parts + for r, nf, df, ex in partial_list[2]: + if isinstance(r, Poly): + # Assemble in case the roots are given implicitly by a polynomials + an, nu = nf.variables, nf.expr + ad, de = df.variables, df.expr + # Hack to make dummies equal because Lambda created new Dummies + de = de.subs(ad[0], an[0]) + func = Lambda(tuple(an), nu/de**ex) + pfd += RootSum(r, func, auto=False, quadratic=False) + else: + # Assemble in case the roots are given explicitly by a list of algebraic numbers + for root in r: + pfd += nf(root)/df(root)**ex + + return common*pfd diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/polyclasses.py b/.venv/lib/python3.12/site-packages/sympy/polys/polyclasses.py new file mode 100644 index 0000000000000000000000000000000000000000..1cc0e0f368ab07d64837e057b841ea991d9de223 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/polyclasses.py @@ -0,0 +1,3186 @@ +"""OO layer for several polynomial representations. """ + +from __future__ import annotations + +from sympy.external.gmpy import GROUND_TYPES + +from sympy.utilities.exceptions import sympy_deprecation_warning + +from sympy.core.numbers import oo +from sympy.core.sympify import CantSympify +from sympy.polys.polyutils import PicklableWithSlots, _sort_factors +from sympy.polys.domains import Domain, ZZ, QQ + +from sympy.polys.polyerrors import ( + CoercionFailed, + ExactQuotientFailed, + DomainError, + NotInvertible, +) + +from sympy.polys.densebasic import ( + ninf, + dmp_validate, + dup_normal, dmp_normal, + dup_convert, dmp_convert, + dmp_from_sympy, + dup_strip, + dmp_degree_in, + dmp_degree_list, + dmp_negative_p, + dmp_ground_LC, + dmp_ground_TC, + dmp_ground_nth, + dmp_one, dmp_ground, + dmp_zero, dmp_zero_p, dmp_one_p, dmp_ground_p, + dup_from_dict, dmp_from_dict, + dmp_to_dict, + dmp_deflate, + dmp_inject, dmp_eject, + dmp_terms_gcd, + dmp_list_terms, dmp_exclude, + dup_slice, dmp_slice_in, dmp_permute, + dmp_to_tuple,) + +from sympy.polys.densearith import ( + dmp_add_ground, + dmp_sub_ground, + dmp_mul_ground, + dmp_quo_ground, + dmp_exquo_ground, + dmp_abs, + dmp_neg, + dmp_add, + dmp_sub, + dmp_mul, + dmp_sqr, + dmp_pow, + dmp_pdiv, + dmp_prem, + dmp_pquo, + dmp_pexquo, + dmp_div, + dmp_rem, + dmp_quo, + dmp_exquo, + dmp_add_mul, dmp_sub_mul, + dmp_max_norm, + dmp_l1_norm, + dmp_l2_norm_squared) + +from sympy.polys.densetools import ( + dmp_clear_denoms, + dmp_integrate_in, + dmp_diff_in, + dmp_eval_in, + dup_revert, + dmp_ground_trunc, + dmp_ground_content, + dmp_ground_primitive, + dmp_ground_monic, + dmp_compose, + dup_decompose, + dup_shift, + dmp_shift, + dup_transform, + dmp_lift) + +from sympy.polys.euclidtools import ( + dup_half_gcdex, dup_gcdex, dup_invert, + dmp_subresultants, + dmp_resultant, + dmp_discriminant, + dmp_inner_gcd, + dmp_gcd, + dmp_lcm, + dmp_cancel) + +from sympy.polys.sqfreetools import ( + dup_gff_list, + dmp_norm, + dmp_sqf_p, + dmp_sqf_norm, + dmp_sqf_part, + dmp_sqf_list, dmp_sqf_list_include) + +from sympy.polys.factortools import ( + dup_cyclotomic_p, dmp_irreducible_p, + dmp_factor_list, dmp_factor_list_include) + +from sympy.polys.rootisolation import ( + dup_isolate_real_roots_sqf, + dup_isolate_real_roots, + dup_isolate_all_roots_sqf, + dup_isolate_all_roots, + dup_refine_real_root, + dup_count_real_roots, + dup_count_complex_roots, + dup_sturm, + dup_cauchy_upper_bound, + dup_cauchy_lower_bound, + dup_mignotte_sep_bound_squared) + +from sympy.polys.polyerrors import ( + UnificationFailed, + PolynomialError) + + +if GROUND_TYPES == 'flint': + import flint + def _supported_flint_domain(D): + return D.is_ZZ or D.is_QQ or D.is_FF and D._is_flint +else: + flint = None + def _supported_flint_domain(D): + return False + + +class DMP(CantSympify): + """Dense Multivariate Polynomials over `K`. """ + + __slots__ = () + + lev: int + dom: Domain + + def __new__(cls, rep, dom, lev=None): + + if lev is None: + rep, lev = dmp_validate(rep) + elif not isinstance(rep, list): + raise CoercionFailed("expected list, got %s" % type(rep)) + + return cls.new(rep, dom, lev) + + @classmethod + def new(cls, rep, dom, lev): + # It would be too slow to call _validate_args always at runtime. + # Ideally this checking would be handled by a static type checker. + # + #cls._validate_args(rep, dom, lev) + if flint is not None: + if lev == 0 and _supported_flint_domain(dom): + return DUP_Flint._new(rep, dom, lev) + + return DMP_Python._new(rep, dom, lev) + + @property + def rep(f): + """Get the representation of ``f``. """ + + sympy_deprecation_warning(""" + Accessing the ``DMP.rep`` attribute is deprecated. The internal + representation of ``DMP`` instances can now be ``DUP_Flint`` when the + ground types are ``flint``. In this case the ``DMP`` instance does not + have a ``rep`` attribute. Use ``DMP.to_list()`` instead. Using + ``DMP.to_list()`` also works in previous versions of SymPy. + """, + deprecated_since_version="1.13", + active_deprecations_target="dmp-rep", + ) + + return f.to_list() + + def to_best(f): + """Convert to DUP_Flint if possible. + + This method should be used when the domain or level is changed and it + potentially becomes possible to convert from DMP_Python to DUP_Flint. + """ + if flint is not None: + if isinstance(f, DMP_Python) and f.lev == 0 and _supported_flint_domain(f.dom): + return DUP_Flint.new(f._rep, f.dom, f.lev) + + return f + + @classmethod + def _validate_args(cls, rep, dom, lev): + assert isinstance(dom, Domain) + assert isinstance(lev, int) and lev >= 0 + + def validate_rep(rep, lev): + assert isinstance(rep, list) + if lev == 0: + assert all(dom.of_type(c) for c in rep) + else: + for r in rep: + validate_rep(r, lev - 1) + + validate_rep(rep, lev) + + @classmethod + def from_dict(cls, rep, lev, dom): + rep = dmp_from_dict(rep, lev, dom) + return cls.new(rep, dom, lev) + + @classmethod + def from_list(cls, rep, lev, dom): + """Create an instance of ``cls`` given a list of native coefficients. """ + return cls.new(dmp_convert(rep, lev, None, dom), dom, lev) + + @classmethod + def from_sympy_list(cls, rep, lev, dom): + """Create an instance of ``cls`` given a list of SymPy coefficients. """ + return cls.new(dmp_from_sympy(rep, lev, dom), dom, lev) + + @classmethod + def from_monoms_coeffs(cls, monoms, coeffs, lev, dom): + return cls(dict(list(zip(monoms, coeffs))), dom, lev) + + def convert(f, dom): + """Convert ``f`` to a ``DMP`` over the new domain. """ + if f.dom == dom: + return f + elif f.lev or flint is None: + return f._convert(dom) + elif isinstance(f, DUP_Flint): + if _supported_flint_domain(dom): + return f._convert(dom) + else: + return f.to_DMP_Python()._convert(dom) + elif isinstance(f, DMP_Python): + if _supported_flint_domain(dom): + return f._convert(dom).to_DUP_Flint() + else: + return f._convert(dom) + else: + raise RuntimeError("unreachable code") + + def _convert(f, dom): + raise NotImplementedError + + @classmethod + def zero(cls, lev, dom): + return DMP(dmp_zero(lev), dom, lev) + + @classmethod + def one(cls, lev, dom): + return DMP(dmp_one(lev, dom), dom, lev) + + def _one(f): + raise NotImplementedError + + def __repr__(f): + return "%s(%s, %s)" % (f.__class__.__name__, f.to_list(), f.dom) + + def __hash__(f): + return hash((f.__class__.__name__, f.to_tuple(), f.lev, f.dom)) + + def __getnewargs__(self): + return self.to_list(), self.dom, self.lev + + def ground_new(f, coeff): + """Construct a new ground instance of ``f``. """ + raise NotImplementedError + + def unify_DMP(f, g): + """Unify and return ``DMP`` instances of ``f`` and ``g``. """ + if not isinstance(g, DMP) or f.lev != g.lev: + raise UnificationFailed("Cannot unify %s with %s" % (f, g)) + + if f.dom != g.dom: + dom = f.dom.unify(g.dom) + f = f.convert(dom) + g = g.convert(dom) + + return f, g + + def to_dict(f, zero=False): + """Convert ``f`` to a dict representation with native coefficients. """ + return dmp_to_dict(f.to_list(), f.lev, f.dom, zero=zero) + + def to_sympy_dict(f, zero=False): + """Convert ``f`` to a dict representation with SymPy coefficients. """ + rep = f.to_dict(zero=zero) + + for k, v in rep.items(): + rep[k] = f.dom.to_sympy(v) + + return rep + + def to_sympy_list(f): + """Convert ``f`` to a list representation with SymPy coefficients. """ + def sympify_nested_list(rep): + out = [] + for val in rep: + if isinstance(val, list): + out.append(sympify_nested_list(val)) + else: + out.append(f.dom.to_sympy(val)) + return out + + return sympify_nested_list(f.to_list()) + + def to_list(f): + """Convert ``f`` to a list representation with native coefficients. """ + raise NotImplementedError + + def to_tuple(f): + """ + Convert ``f`` to a tuple representation with native coefficients. + + This is needed for hashing. + """ + raise NotImplementedError + + def to_ring(f): + """Make the ground domain a ring. """ + return f.convert(f.dom.get_ring()) + + def to_field(f): + """Make the ground domain a field. """ + return f.convert(f.dom.get_field()) + + def to_exact(f): + """Make the ground domain exact. """ + return f.convert(f.dom.get_exact()) + + def slice(f, m, n, j=0): + """Take a continuous subsequence of terms of ``f``. """ + if not f.lev and not j: + return f._slice(m, n) + else: + return f._slice_lev(m, n, j) + + def _slice(f, m, n): + raise NotImplementedError + + def _slice_lev(f, m, n, j): + raise NotImplementedError + + def coeffs(f, order=None): + """Returns all non-zero coefficients from ``f`` in lex order. """ + return [ c for _, c in f.terms(order=order) ] + + def monoms(f, order=None): + """Returns all non-zero monomials from ``f`` in lex order. """ + return [ m for m, _ in f.terms(order=order) ] + + def terms(f, order=None): + """Returns all non-zero terms from ``f`` in lex order. """ + if f.is_zero: + zero_monom = (0,)*(f.lev + 1) + return [(zero_monom, f.dom.zero)] + else: + return f._terms(order=order) + + def _terms(f, order=None): + raise NotImplementedError + + def all_coeffs(f): + """Returns all coefficients from ``f``. """ + if f.lev: + raise PolynomialError('multivariate polynomials not supported') + + if not f: + return [f.dom.zero] + else: + return list(f.to_list()) + + def all_monoms(f): + """Returns all monomials from ``f``. """ + if f.lev: + raise PolynomialError('multivariate polynomials not supported') + + n = f.degree() + + if n < 0: + return [(0,)] + else: + return [ (n - i,) for i, c in enumerate(f.to_list()) ] + + def all_terms(f): + """Returns all terms from a ``f``. """ + if f.lev: + raise PolynomialError('multivariate polynomials not supported') + + n = f.degree() + + if n < 0: + return [((0,), f.dom.zero)] + else: + return [ ((n - i,), c) for i, c in enumerate(f.to_list()) ] + + def lift(f): + """Convert algebraic coefficients to rationals. """ + return f._lift().to_best() + + def _lift(f): + raise NotImplementedError + + def deflate(f): + """Reduce degree of `f` by mapping `x_i^m` to `y_i`. """ + raise NotImplementedError + + def inject(f, front=False): + """Inject ground domain generators into ``f``. """ + raise NotImplementedError + + def eject(f, dom, front=False): + """Eject selected generators into the ground domain. """ + raise NotImplementedError + + def exclude(f): + r""" + Remove useless generators from ``f``. + + Returns the removed generators and the new excluded ``f``. + + Examples + ======== + + >>> from sympy.polys.polyclasses import DMP + >>> from sympy.polys.domains import ZZ + + >>> DMP([[[ZZ(1)]], [[ZZ(1)], [ZZ(2)]]], ZZ).exclude() + ([2], DMP_Python([[1], [1, 2]], ZZ)) + + """ + J, F = f._exclude() + return J, F.to_best() + + def _exclude(f): + raise NotImplementedError + + def permute(f, P): + r""" + Returns a polynomial in `K[x_{P(1)}, ..., x_{P(n)}]`. + + Examples + ======== + + >>> from sympy.polys.polyclasses import DMP + >>> from sympy.polys.domains import ZZ + + >>> DMP([[[ZZ(2)], [ZZ(1), ZZ(0)]], [[]]], ZZ).permute([1, 0, 2]) + DMP_Python([[[2], []], [[1, 0], []]], ZZ) + + >>> DMP([[[ZZ(2)], [ZZ(1), ZZ(0)]], [[]]], ZZ).permute([1, 2, 0]) + DMP_Python([[[1], []], [[2, 0], []]], ZZ) + + """ + return f._permute(P) + + def _permute(f, P): + raise NotImplementedError + + def terms_gcd(f): + """Remove GCD of terms from the polynomial ``f``. """ + raise NotImplementedError + + def abs(f): + """Make all coefficients in ``f`` positive. """ + raise NotImplementedError + + def neg(f): + """Negate all coefficients in ``f``. """ + raise NotImplementedError + + def add_ground(f, c): + """Add an element of the ground domain to ``f``. """ + return f._add_ground(f.dom.convert(c)) + + def sub_ground(f, c): + """Subtract an element of the ground domain from ``f``. """ + return f._sub_ground(f.dom.convert(c)) + + def mul_ground(f, c): + """Multiply ``f`` by a an element of the ground domain. """ + return f._mul_ground(f.dom.convert(c)) + + def quo_ground(f, c): + """Quotient of ``f`` by a an element of the ground domain. """ + return f._quo_ground(f.dom.convert(c)) + + def exquo_ground(f, c): + """Exact quotient of ``f`` by a an element of the ground domain. """ + return f._exquo_ground(f.dom.convert(c)) + + def add(f, g): + """Add two multivariate polynomials ``f`` and ``g``. """ + F, G = f.unify_DMP(g) + return F._add(G) + + def sub(f, g): + """Subtract two multivariate polynomials ``f`` and ``g``. """ + F, G = f.unify_DMP(g) + return F._sub(G) + + def mul(f, g): + """Multiply two multivariate polynomials ``f`` and ``g``. """ + F, G = f.unify_DMP(g) + return F._mul(G) + + def sqr(f): + """Square a multivariate polynomial ``f``. """ + return f._sqr() + + def pow(f, n): + """Raise ``f`` to a non-negative power ``n``. """ + if not isinstance(n, int): + raise TypeError("``int`` expected, got %s" % type(n)) + return f._pow(n) + + def pdiv(f, g): + """Polynomial pseudo-division of ``f`` and ``g``. """ + F, G = f.unify_DMP(g) + return F._pdiv(G) + + def prem(f, g): + """Polynomial pseudo-remainder of ``f`` and ``g``. """ + F, G = f.unify_DMP(g) + return F._prem(G) + + def pquo(f, g): + """Polynomial pseudo-quotient of ``f`` and ``g``. """ + F, G = f.unify_DMP(g) + return F._pquo(G) + + def pexquo(f, g): + """Polynomial exact pseudo-quotient of ``f`` and ``g``. """ + F, G = f.unify_DMP(g) + return F._pexquo(G) + + def div(f, g): + """Polynomial division with remainder of ``f`` and ``g``. """ + F, G = f.unify_DMP(g) + return F._div(G) + + def rem(f, g): + """Computes polynomial remainder of ``f`` and ``g``. """ + F, G = f.unify_DMP(g) + return F._rem(G) + + def quo(f, g): + """Computes polynomial quotient of ``f`` and ``g``. """ + F, G = f.unify_DMP(g) + return F._quo(G) + + def exquo(f, g): + """Computes polynomial exact quotient of ``f`` and ``g``. """ + F, G = f.unify_DMP(g) + return F._exquo(G) + + def _add_ground(f, c): + raise NotImplementedError + + def _sub_ground(f, c): + raise NotImplementedError + + def _mul_ground(f, c): + raise NotImplementedError + + def _quo_ground(f, c): + raise NotImplementedError + + def _exquo_ground(f, c): + raise NotImplementedError + + def _add(f, g): + raise NotImplementedError + + def _sub(f, g): + raise NotImplementedError + + def _mul(f, g): + raise NotImplementedError + + def _sqr(f): + raise NotImplementedError + + def _pow(f, n): + raise NotImplementedError + + def _pdiv(f, g): + raise NotImplementedError + + def _prem(f, g): + raise NotImplementedError + + def _pquo(f, g): + raise NotImplementedError + + def _pexquo(f, g): + raise NotImplementedError + + def _div(f, g): + raise NotImplementedError + + def _rem(f, g): + raise NotImplementedError + + def _quo(f, g): + raise NotImplementedError + + def _exquo(f, g): + raise NotImplementedError + + def degree(f, j=0): + """Returns the leading degree of ``f`` in ``x_j``. """ + if not isinstance(j, int): + raise TypeError("``int`` expected, got %s" % type(j)) + + return f._degree(j) + + def _degree(f, j): + raise NotImplementedError + + def degree_list(f): + """Returns a list of degrees of ``f``. """ + raise NotImplementedError + + def total_degree(f): + """Returns the total degree of ``f``. """ + raise NotImplementedError + + def homogenize(f, s): + """Return homogeneous polynomial of ``f``""" + td = f.total_degree() + result = {} + new_symbol = (s == len(f.terms()[0][0])) + for term in f.terms(): + d = sum(term[0]) + if d < td: + i = td - d + else: + i = 0 + if new_symbol: + result[term[0] + (i,)] = term[1] + else: + l = list(term[0]) + l[s] += i + result[tuple(l)] = term[1] + return DMP.from_dict(result, f.lev + int(new_symbol), f.dom) + + def homogeneous_order(f): + """Returns the homogeneous order of ``f``. """ + if f.is_zero: + return -oo + + monoms = f.monoms() + tdeg = sum(monoms[0]) + + for monom in monoms: + _tdeg = sum(monom) + + if _tdeg != tdeg: + return None + + return tdeg + + def LC(f): + """Returns the leading coefficient of ``f``. """ + raise NotImplementedError + + def TC(f): + """Returns the trailing coefficient of ``f``. """ + raise NotImplementedError + + def nth(f, *N): + """Returns the ``n``-th coefficient of ``f``. """ + if all(isinstance(n, int) for n in N): + return f._nth(N) + else: + raise TypeError("a sequence of integers expected") + + def _nth(f, N): + raise NotImplementedError + + def max_norm(f): + """Returns maximum norm of ``f``. """ + raise NotImplementedError + + def l1_norm(f): + """Returns l1 norm of ``f``. """ + raise NotImplementedError + + def l2_norm_squared(f): + """Return squared l2 norm of ``f``. """ + raise NotImplementedError + + def clear_denoms(f): + """Clear denominators, but keep the ground domain. """ + raise NotImplementedError + + def integrate(f, m=1, j=0): + """Computes the ``m``-th order indefinite integral of ``f`` in ``x_j``. """ + if not isinstance(m, int): + raise TypeError("``int`` expected, got %s" % type(m)) + + if not isinstance(j, int): + raise TypeError("``int`` expected, got %s" % type(j)) + + return f._integrate(m, j) + + def _integrate(f, m, j): + raise NotImplementedError + + def diff(f, m=1, j=0): + """Computes the ``m``-th order derivative of ``f`` in ``x_j``. """ + if not isinstance(m, int): + raise TypeError("``int`` expected, got %s" % type(m)) + + if not isinstance(j, int): + raise TypeError("``int`` expected, got %s" % type(j)) + + return f._diff(m, j) + + def _diff(f, m, j): + raise NotImplementedError + + def eval(f, a, j=0): + """Evaluates ``f`` at the given point ``a`` in ``x_j``. """ + if not isinstance(j, int): + raise TypeError("``int`` expected, got %s" % type(j)) + elif not (0 <= j <= f.lev): + raise ValueError("invalid variable index %s" % j) + + if f.lev: + return f._eval_lev(a, j) + else: + return f._eval(a) + + def _eval(f, a): + raise NotImplementedError + + def _eval_lev(f, a, j): + raise NotImplementedError + + def half_gcdex(f, g): + """Half extended Euclidean algorithm, if univariate. """ + F, G = f.unify_DMP(g) + + if F.lev: + raise ValueError('univariate polynomial expected') + + return F._half_gcdex(G) + + def _half_gcdex(f, g): + raise NotImplementedError + + def gcdex(f, g): + """Extended Euclidean algorithm, if univariate. """ + F, G = f.unify_DMP(g) + + if F.lev: + raise ValueError('univariate polynomial expected') + + if not F.dom.is_Field: + raise DomainError('ground domain must be a field') + + return F._gcdex(G) + + def _gcdex(f, g): + raise NotImplementedError + + def invert(f, g): + """Invert ``f`` modulo ``g``, if possible. """ + F, G = f.unify_DMP(g) + + if F.lev: + raise ValueError('univariate polynomial expected') + + return F._invert(G) + + def _invert(f, g): + raise NotImplementedError + + def revert(f, n): + """Compute ``f**(-1)`` mod ``x**n``. """ + if f.lev: + raise ValueError('univariate polynomial expected') + + return f._revert(n) + + def _revert(f, n): + raise NotImplementedError + + def subresultants(f, g): + """Computes subresultant PRS sequence of ``f`` and ``g``. """ + F, G = f.unify_DMP(g) + return F._subresultants(G) + + def _subresultants(f, g): + raise NotImplementedError + + def resultant(f, g, includePRS=False): + """Computes resultant of ``f`` and ``g`` via PRS. """ + F, G = f.unify_DMP(g) + if includePRS: + return F._resultant_includePRS(G) + else: + return F._resultant(G) + + def _resultant(f, g, includePRS=False): + raise NotImplementedError + + def discriminant(f): + """Computes discriminant of ``f``. """ + raise NotImplementedError + + def cofactors(f, g): + """Returns GCD of ``f`` and ``g`` and their cofactors. """ + F, G = f.unify_DMP(g) + return F._cofactors(G) + + def _cofactors(f, g): + raise NotImplementedError + + def gcd(f, g): + """Returns polynomial GCD of ``f`` and ``g``. """ + F, G = f.unify_DMP(g) + return F._gcd(G) + + def _gcd(f, g): + raise NotImplementedError + + def lcm(f, g): + """Returns polynomial LCM of ``f`` and ``g``. """ + F, G = f.unify_DMP(g) + return F._lcm(G) + + def _lcm(f, g): + raise NotImplementedError + + def cancel(f, g, include=True): + """Cancel common factors in a rational function ``f/g``. """ + F, G = f.unify_DMP(g) + + if include: + return F._cancel_include(G) + else: + return F._cancel(G) + + def _cancel(f, g): + raise NotImplementedError + + def _cancel_include(f, g): + raise NotImplementedError + + def trunc(f, p): + """Reduce ``f`` modulo a constant ``p``. """ + return f._trunc(f.dom.convert(p)) + + def _trunc(f, p): + raise NotImplementedError + + def monic(f): + """Divides all coefficients by ``LC(f)``. """ + raise NotImplementedError + + def content(f): + """Returns GCD of polynomial coefficients. """ + raise NotImplementedError + + def primitive(f): + """Returns content and a primitive form of ``f``. """ + raise NotImplementedError + + def compose(f, g): + """Computes functional composition of ``f`` and ``g``. """ + F, G = f.unify_DMP(g) + return F._compose(G) + + def _compose(f, g): + raise NotImplementedError + + def decompose(f): + """Computes functional decomposition of ``f``. """ + if f.lev: + raise ValueError('univariate polynomial expected') + + return f._decompose() + + def _decompose(f): + raise NotImplementedError + + def shift(f, a): + """Efficiently compute Taylor shift ``f(x + a)``. """ + if f.lev: + raise ValueError('univariate polynomial expected') + + return f._shift(f.dom.convert(a)) + + def shift_list(f, a): + """Efficiently compute Taylor shift ``f(X + A)``. """ + a = [f.dom.convert(ai) for ai in a] + return f._shift_list(a) + + def _shift(f, a): + raise NotImplementedError + + def transform(f, p, q): + """Evaluate functional transformation ``q**n * f(p/q)``.""" + if f.lev: + raise ValueError('univariate polynomial expected') + + P, Q = p.unify_DMP(q) + F, P = f.unify_DMP(P) + F, Q = F.unify_DMP(Q) + + return F._transform(P, Q) + + def _transform(f, p, q): + raise NotImplementedError + + def sturm(f): + """Computes the Sturm sequence of ``f``. """ + if f.lev: + raise ValueError('univariate polynomial expected') + + return f._sturm() + + def _sturm(f): + raise NotImplementedError + + def cauchy_upper_bound(f): + """Computes the Cauchy upper bound on the roots of ``f``. """ + if f.lev: + raise ValueError('univariate polynomial expected') + + return f._cauchy_upper_bound() + + def _cauchy_upper_bound(f): + raise NotImplementedError + + def cauchy_lower_bound(f): + """Computes the Cauchy lower bound on the nonzero roots of ``f``. """ + if f.lev: + raise ValueError('univariate polynomial expected') + + return f._cauchy_lower_bound() + + def _cauchy_lower_bound(f): + raise NotImplementedError + + def mignotte_sep_bound_squared(f): + """Computes the squared Mignotte bound on root separations of ``f``. """ + if f.lev: + raise ValueError('univariate polynomial expected') + + return f._mignotte_sep_bound_squared() + + def _mignotte_sep_bound_squared(f): + raise NotImplementedError + + def gff_list(f): + """Computes greatest factorial factorization of ``f``. """ + if f.lev: + raise ValueError('univariate polynomial expected') + + return f._gff_list() + + def _gff_list(f): + raise NotImplementedError + + def norm(f): + """Computes ``Norm(f)``.""" + raise NotImplementedError + + def sqf_norm(f): + """Computes square-free norm of ``f``. """ + raise NotImplementedError + + def sqf_part(f): + """Computes square-free part of ``f``. """ + raise NotImplementedError + + def sqf_list(f, all=False): + """Returns a list of square-free factors of ``f``. """ + raise NotImplementedError + + def sqf_list_include(f, all=False): + """Returns a list of square-free factors of ``f``. """ + raise NotImplementedError + + def factor_list(f): + """Returns a list of irreducible factors of ``f``. """ + raise NotImplementedError + + def factor_list_include(f): + """Returns a list of irreducible factors of ``f``. """ + raise NotImplementedError + + def intervals(f, all=False, eps=None, inf=None, sup=None, fast=False, sqf=False): + """Compute isolating intervals for roots of ``f``. """ + if f.lev: + raise PolynomialError("Cannot isolate roots of a multivariate polynomial") + + if all and sqf: + return f._isolate_all_roots_sqf(eps=eps, inf=inf, sup=sup, fast=fast) + elif all and not sqf: + return f._isolate_all_roots(eps=eps, inf=inf, sup=sup, fast=fast) + elif not all and sqf: + return f._isolate_real_roots_sqf(eps=eps, inf=inf, sup=sup, fast=fast) + else: + return f._isolate_real_roots(eps=eps, inf=inf, sup=sup, fast=fast) + + def _isolate_all_roots(f, eps, inf, sup, fast): + raise NotImplementedError + + def _isolate_all_roots_sqf(f, eps, inf, sup, fast): + raise NotImplementedError + + def _isolate_real_roots(f, eps, inf, sup, fast): + raise NotImplementedError + + def _isolate_real_roots_sqf(f, eps, inf, sup, fast): + raise NotImplementedError + + def refine_root(f, s, t, eps=None, steps=None, fast=False): + """ + Refine an isolating interval to the given precision. + + ``eps`` should be a rational number. + + """ + if f.lev: + raise PolynomialError( + "Cannot refine a root of a multivariate polynomial") + + return f._refine_real_root(s, t, eps=eps, steps=steps, fast=fast) + + def _refine_real_root(f, s, t, eps, steps, fast): + raise NotImplementedError + + def count_real_roots(f, inf=None, sup=None): + """Return the number of real roots of ``f`` in ``[inf, sup]``. """ + raise NotImplementedError + + def count_complex_roots(f, inf=None, sup=None): + """Return the number of complex roots of ``f`` in ``[inf, sup]``. """ + raise NotImplementedError + + @property + def is_zero(f): + """Returns ``True`` if ``f`` is a zero polynomial. """ + raise NotImplementedError + + @property + def is_one(f): + """Returns ``True`` if ``f`` is a unit polynomial. """ + raise NotImplementedError + + @property + def is_ground(f): + """Returns ``True`` if ``f`` is an element of the ground domain. """ + raise NotImplementedError + + @property + def is_sqf(f): + """Returns ``True`` if ``f`` is a square-free polynomial. """ + raise NotImplementedError + + @property + def is_monic(f): + """Returns ``True`` if the leading coefficient of ``f`` is one. """ + raise NotImplementedError + + @property + def is_primitive(f): + """Returns ``True`` if the GCD of the coefficients of ``f`` is one. """ + raise NotImplementedError + + @property + def is_linear(f): + """Returns ``True`` if ``f`` is linear in all its variables. """ + raise NotImplementedError + + @property + def is_quadratic(f): + """Returns ``True`` if ``f`` is quadratic in all its variables. """ + raise NotImplementedError + + @property + def is_monomial(f): + """Returns ``True`` if ``f`` is zero or has only one term. """ + raise NotImplementedError + + @property + def is_homogeneous(f): + """Returns ``True`` if ``f`` is a homogeneous polynomial. """ + raise NotImplementedError + + @property + def is_irreducible(f): + """Returns ``True`` if ``f`` has no factors over its domain. """ + raise NotImplementedError + + @property + def is_cyclotomic(f): + """Returns ``True`` if ``f`` is a cyclotomic polynomial. """ + raise NotImplementedError + + def __abs__(f): + return f.abs() + + def __neg__(f): + return f.neg() + + def __add__(f, g): + if isinstance(g, DMP): + return f.add(g) + else: + try: + return f.add_ground(g) + except CoercionFailed: + return NotImplemented + + def __radd__(f, g): + return f.__add__(g) + + def __sub__(f, g): + if isinstance(g, DMP): + return f.sub(g) + else: + try: + return f.sub_ground(g) + except CoercionFailed: + return NotImplemented + + def __rsub__(f, g): + return (-f).__add__(g) + + def __mul__(f, g): + if isinstance(g, DMP): + return f.mul(g) + else: + try: + return f.mul_ground(g) + except CoercionFailed: + return NotImplemented + + def __rmul__(f, g): + return f.__mul__(g) + + def __truediv__(f, g): + if isinstance(g, DMP): + return f.exquo(g) + else: + try: + return f.mul_ground(g) + except CoercionFailed: + return NotImplemented + + def __rtruediv__(f, g): + if isinstance(g, DMP): + return g.exquo(f) + else: + try: + return f._one().mul_ground(g).exquo(f) + except CoercionFailed: + return NotImplemented + + def __pow__(f, n): + return f.pow(n) + + def __divmod__(f, g): + return f.div(g) + + def __mod__(f, g): + return f.rem(g) + + def __floordiv__(f, g): + if isinstance(g, DMP): + return f.quo(g) + else: + try: + return f.quo_ground(g) + except TypeError: + return NotImplemented + + def __eq__(f, g): + if f is g: + return True + if not isinstance(g, DMP): + return NotImplemented + try: + F, G = f.unify_DMP(g) + except UnificationFailed: + return False + else: + return F._strict_eq(G) + + def _strict_eq(f, g): + raise NotImplementedError + + def eq(f, g, strict=False): + if not strict: + return f == g + else: + return f._strict_eq(g) + + def ne(f, g, strict=False): + return not f.eq(g, strict=strict) + + def __lt__(f, g): + F, G = f.unify_DMP(g) + return F.to_list() < G.to_list() + + def __le__(f, g): + F, G = f.unify_DMP(g) + return F.to_list() <= G.to_list() + + def __gt__(f, g): + F, G = f.unify_DMP(g) + return F.to_list() > G.to_list() + + def __ge__(f, g): + F, G = f.unify_DMP(g) + return F.to_list() >= G.to_list() + + def __bool__(f): + return not f.is_zero + + +class DMP_Python(DMP): + """Dense Multivariate Polynomials over `K`. """ + + __slots__ = ('_rep', 'dom', 'lev') + + @classmethod + def _new(cls, rep, dom, lev): + obj = object.__new__(cls) + obj._rep = rep + obj.lev = lev + obj.dom = dom + return obj + + def _strict_eq(f, g): + if type(f) != type(g): + return False + return f.lev == g.lev and f.dom == g.dom and f._rep == g._rep + + def per(f, rep): + """Create a DMP out of the given representation. """ + return f._new(rep, f.dom, f.lev) + + def ground_new(f, coeff): + """Construct a new ground instance of ``f``. """ + return f._new(dmp_ground(coeff, f.lev), f.dom, f.lev) + + def _one(f): + return f.one(f.lev, f.dom) + + def unify(f, g): + """Unify representations of two multivariate polynomials. """ + # XXX: This function is not really used any more since there is + # unify_DMP now. + if not isinstance(g, DMP) or f.lev != g.lev: + raise UnificationFailed("Cannot unify %s with %s" % (f, g)) + + if f.dom == g.dom: + return f.lev, f.dom, f.per, f._rep, g._rep + else: + lev, dom = f.lev, f.dom.unify(g.dom) + + F = dmp_convert(f._rep, lev, f.dom, dom) + G = dmp_convert(g._rep, lev, g.dom, dom) + + def per(rep): + return f._new(rep, dom, lev) + + return lev, dom, per, F, G + + def to_DUP_Flint(f): + """Convert ``f`` to a Flint representation. """ + return DUP_Flint._new(f._rep, f.dom, f.lev) + + def to_list(f): + """Convert ``f`` to a list representation with native coefficients. """ + return list(f._rep) + + def to_tuple(f): + """Convert ``f`` to a tuple representation with native coefficients. """ + return dmp_to_tuple(f._rep, f.lev) + + def _convert(f, dom): + """Convert the ground domain of ``f``. """ + return f._new(dmp_convert(f._rep, f.lev, f.dom, dom), dom, f.lev) + + def _slice(f, m, n): + """Take a continuous subsequence of terms of ``f``. """ + rep = dup_slice(f._rep, m, n, f.dom) + return f._new(rep, f.dom, f.lev) + + def _slice_lev(f, m, n, j): + """Take a continuous subsequence of terms of ``f``. """ + rep = dmp_slice_in(f._rep, m, n, j, f.lev, f.dom) + return f._new(rep, f.dom, f.lev) + + def _terms(f, order=None): + """Returns all non-zero terms from ``f`` in lex order. """ + return dmp_list_terms(f._rep, f.lev, f.dom, order=order) + + def _lift(f): + """Convert algebraic coefficients to rationals. """ + r = dmp_lift(f._rep, f.lev, f.dom) + return f._new(r, f.dom.dom, f.lev) + + def deflate(f): + """Reduce degree of `f` by mapping `x_i^m` to `y_i`. """ + J, F = dmp_deflate(f._rep, f.lev, f.dom) + return J, f.per(F) + + def inject(f, front=False): + """Inject ground domain generators into ``f``. """ + F, lev = dmp_inject(f._rep, f.lev, f.dom, front=front) + # XXX: domain and level changed here + return f._new(F, f.dom.dom, lev) + + def eject(f, dom, front=False): + """Eject selected generators into the ground domain. """ + F = dmp_eject(f._rep, f.lev, dom, front=front) + # XXX: domain and level changed here + return f._new(F, dom, f.lev - len(dom.symbols)) + + def _exclude(f): + """Remove useless generators from ``f``. """ + J, F, u = dmp_exclude(f._rep, f.lev, f.dom) + # XXX: level changed here + return J, f._new(F, f.dom, u) + + def _permute(f, P): + """Returns a polynomial in `K[x_{P(1)}, ..., x_{P(n)}]`. """ + return f.per(dmp_permute(f._rep, P, f.lev, f.dom)) + + def terms_gcd(f): + """Remove GCD of terms from the polynomial ``f``. """ + J, F = dmp_terms_gcd(f._rep, f.lev, f.dom) + return J, f.per(F) + + def _add_ground(f, c): + """Add an element of the ground domain to ``f``. """ + return f.per(dmp_add_ground(f._rep, c, f.lev, f.dom)) + + def _sub_ground(f, c): + """Subtract an element of the ground domain from ``f``. """ + return f.per(dmp_sub_ground(f._rep, c, f.lev, f.dom)) + + def _mul_ground(f, c): + """Multiply ``f`` by a an element of the ground domain. """ + return f.per(dmp_mul_ground(f._rep, c, f.lev, f.dom)) + + def _quo_ground(f, c): + """Quotient of ``f`` by a an element of the ground domain. """ + return f.per(dmp_quo_ground(f._rep, c, f.lev, f.dom)) + + def _exquo_ground(f, c): + """Exact quotient of ``f`` by a an element of the ground domain. """ + return f.per(dmp_exquo_ground(f._rep, c, f.lev, f.dom)) + + def abs(f): + """Make all coefficients in ``f`` positive. """ + return f.per(dmp_abs(f._rep, f.lev, f.dom)) + + def neg(f): + """Negate all coefficients in ``f``. """ + return f.per(dmp_neg(f._rep, f.lev, f.dom)) + + def _add(f, g): + """Add two multivariate polynomials ``f`` and ``g``. """ + return f.per(dmp_add(f._rep, g._rep, f.lev, f.dom)) + + def _sub(f, g): + """Subtract two multivariate polynomials ``f`` and ``g``. """ + return f.per(dmp_sub(f._rep, g._rep, f.lev, f.dom)) + + def _mul(f, g): + """Multiply two multivariate polynomials ``f`` and ``g``. """ + return f.per(dmp_mul(f._rep, g._rep, f.lev, f.dom)) + + def sqr(f): + """Square a multivariate polynomial ``f``. """ + return f.per(dmp_sqr(f._rep, f.lev, f.dom)) + + def _pow(f, n): + """Raise ``f`` to a non-negative power ``n``. """ + return f.per(dmp_pow(f._rep, n, f.lev, f.dom)) + + def _pdiv(f, g): + """Polynomial pseudo-division of ``f`` and ``g``. """ + q, r = dmp_pdiv(f._rep, g._rep, f.lev, f.dom) + return f.per(q), f.per(r) + + def _prem(f, g): + """Polynomial pseudo-remainder of ``f`` and ``g``. """ + return f.per(dmp_prem(f._rep, g._rep, f.lev, f.dom)) + + def _pquo(f, g): + """Polynomial pseudo-quotient of ``f`` and ``g``. """ + return f.per(dmp_pquo(f._rep, g._rep, f.lev, f.dom)) + + def _pexquo(f, g): + """Polynomial exact pseudo-quotient of ``f`` and ``g``. """ + return f.per(dmp_pexquo(f._rep, g._rep, f.lev, f.dom)) + + def _div(f, g): + """Polynomial division with remainder of ``f`` and ``g``. """ + q, r = dmp_div(f._rep, g._rep, f.lev, f.dom) + return f.per(q), f.per(r) + + def _rem(f, g): + """Computes polynomial remainder of ``f`` and ``g``. """ + return f.per(dmp_rem(f._rep, g._rep, f.lev, f.dom)) + + def _quo(f, g): + """Computes polynomial quotient of ``f`` and ``g``. """ + return f.per(dmp_quo(f._rep, g._rep, f.lev, f.dom)) + + def _exquo(f, g): + """Computes polynomial exact quotient of ``f`` and ``g``. """ + return f.per(dmp_exquo(f._rep, g._rep, f.lev, f.dom)) + + def _degree(f, j=0): + """Returns the leading degree of ``f`` in ``x_j``. """ + return dmp_degree_in(f._rep, j, f.lev) + + def degree_list(f): + """Returns a list of degrees of ``f``. """ + return dmp_degree_list(f._rep, f.lev) + + def total_degree(f): + """Returns the total degree of ``f``. """ + return max(sum(m) for m in f.monoms()) + + def LC(f): + """Returns the leading coefficient of ``f``. """ + return dmp_ground_LC(f._rep, f.lev, f.dom) + + def TC(f): + """Returns the trailing coefficient of ``f``. """ + return dmp_ground_TC(f._rep, f.lev, f.dom) + + def _nth(f, N): + """Returns the ``n``-th coefficient of ``f``. """ + return dmp_ground_nth(f._rep, N, f.lev, f.dom) + + def max_norm(f): + """Returns maximum norm of ``f``. """ + return dmp_max_norm(f._rep, f.lev, f.dom) + + def l1_norm(f): + """Returns l1 norm of ``f``. """ + return dmp_l1_norm(f._rep, f.lev, f.dom) + + def l2_norm_squared(f): + """Return squared l2 norm of ``f``. """ + return dmp_l2_norm_squared(f._rep, f.lev, f.dom) + + def clear_denoms(f): + """Clear denominators, but keep the ground domain. """ + coeff, F = dmp_clear_denoms(f._rep, f.lev, f.dom) + return coeff, f.per(F) + + def _integrate(f, m=1, j=0): + """Computes the ``m``-th order indefinite integral of ``f`` in ``x_j``. """ + return f.per(dmp_integrate_in(f._rep, m, j, f.lev, f.dom)) + + def _diff(f, m=1, j=0): + """Computes the ``m``-th order derivative of ``f`` in ``x_j``. """ + return f.per(dmp_diff_in(f._rep, m, j, f.lev, f.dom)) + + def _eval(f, a): + return dmp_eval_in(f._rep, f.dom.convert(a), 0, f.lev, f.dom) + + def _eval_lev(f, a, j): + rep = dmp_eval_in(f._rep, f.dom.convert(a), j, f.lev, f.dom) + return f.new(rep, f.dom, f.lev - 1) + + def _half_gcdex(f, g): + """Half extended Euclidean algorithm, if univariate. """ + s, h = dup_half_gcdex(f._rep, g._rep, f.dom) + return f.per(s), f.per(h) + + def _gcdex(f, g): + """Extended Euclidean algorithm, if univariate. """ + s, t, h = dup_gcdex(f._rep, g._rep, f.dom) + return f.per(s), f.per(t), f.per(h) + + def _invert(f, g): + """Invert ``f`` modulo ``g``, if possible. """ + s = dup_invert(f._rep, g._rep, f.dom) + return f.per(s) + + def _revert(f, n): + """Compute ``f**(-1)`` mod ``x**n``. """ + return f.per(dup_revert(f._rep, n, f.dom)) + + def _subresultants(f, g): + """Computes subresultant PRS sequence of ``f`` and ``g``. """ + R = dmp_subresultants(f._rep, g._rep, f.lev, f.dom) + return list(map(f.per, R)) + + def _resultant_includePRS(f, g): + """Computes resultant of ``f`` and ``g`` via PRS. """ + res, R = dmp_resultant(f._rep, g._rep, f.lev, f.dom, includePRS=True) + if f.lev: + res = f.new(res, f.dom, f.lev - 1) + return res, list(map(f.per, R)) + + def _resultant(f, g): + res = dmp_resultant(f._rep, g._rep, f.lev, f.dom) + if f.lev: + res = f.new(res, f.dom, f.lev - 1) + return res + + def discriminant(f): + """Computes discriminant of ``f``. """ + res = dmp_discriminant(f._rep, f.lev, f.dom) + if f.lev: + res = f.new(res, f.dom, f.lev - 1) + return res + + def _cofactors(f, g): + """Returns GCD of ``f`` and ``g`` and their cofactors. """ + h, cff, cfg = dmp_inner_gcd(f._rep, g._rep, f.lev, f.dom) + return f.per(h), f.per(cff), f.per(cfg) + + def _gcd(f, g): + """Returns polynomial GCD of ``f`` and ``g``. """ + return f.per(dmp_gcd(f._rep, g._rep, f.lev, f.dom)) + + def _lcm(f, g): + """Returns polynomial LCM of ``f`` and ``g``. """ + return f.per(dmp_lcm(f._rep, g._rep, f.lev, f.dom)) + + def _cancel(f, g): + """Cancel common factors in a rational function ``f/g``. """ + cF, cG, F, G = dmp_cancel(f._rep, g._rep, f.lev, f.dom, include=False) + return cF, cG, f.per(F), f.per(G) + + def _cancel_include(f, g): + """Cancel common factors in a rational function ``f/g``. """ + F, G = dmp_cancel(f._rep, g._rep, f.lev, f.dom, include=True) + return f.per(F), f.per(G) + + def _trunc(f, p): + """Reduce ``f`` modulo a constant ``p``. """ + return f.per(dmp_ground_trunc(f._rep, p, f.lev, f.dom)) + + def monic(f): + """Divides all coefficients by ``LC(f)``. """ + return f.per(dmp_ground_monic(f._rep, f.lev, f.dom)) + + def content(f): + """Returns GCD of polynomial coefficients. """ + return dmp_ground_content(f._rep, f.lev, f.dom) + + def primitive(f): + """Returns content and a primitive form of ``f``. """ + cont, F = dmp_ground_primitive(f._rep, f.lev, f.dom) + return cont, f.per(F) + + def _compose(f, g): + """Computes functional composition of ``f`` and ``g``. """ + return f.per(dmp_compose(f._rep, g._rep, f.lev, f.dom)) + + def _decompose(f): + """Computes functional decomposition of ``f``. """ + return list(map(f.per, dup_decompose(f._rep, f.dom))) + + def _shift(f, a): + """Efficiently compute Taylor shift ``f(x + a)``. """ + return f.per(dup_shift(f._rep, a, f.dom)) + + def _shift_list(f, a): + """Efficiently compute Taylor shift ``f(X + A)``. """ + return f.per(dmp_shift(f._rep, a, f.lev, f.dom)) + + def _transform(f, p, q): + """Evaluate functional transformation ``q**n * f(p/q)``.""" + return f.per(dup_transform(f._rep, p._rep, q._rep, f.dom)) + + def _sturm(f): + """Computes the Sturm sequence of ``f``. """ + return list(map(f.per, dup_sturm(f._rep, f.dom))) + + def _cauchy_upper_bound(f): + """Computes the Cauchy upper bound on the roots of ``f``. """ + return dup_cauchy_upper_bound(f._rep, f.dom) + + def _cauchy_lower_bound(f): + """Computes the Cauchy lower bound on the nonzero roots of ``f``. """ + return dup_cauchy_lower_bound(f._rep, f.dom) + + def _mignotte_sep_bound_squared(f): + """Computes the squared Mignotte bound on root separations of ``f``. """ + return dup_mignotte_sep_bound_squared(f._rep, f.dom) + + def _gff_list(f): + """Computes greatest factorial factorization of ``f``. """ + return [ (f.per(g), k) for g, k in dup_gff_list(f._rep, f.dom) ] + + def norm(f): + """Computes ``Norm(f)``.""" + r = dmp_norm(f._rep, f.lev, f.dom) + return f.new(r, f.dom.dom, f.lev) + + def sqf_norm(f): + """Computes square-free norm of ``f``. """ + s, g, r = dmp_sqf_norm(f._rep, f.lev, f.dom) + return s, f.per(g), f.new(r, f.dom.dom, f.lev) + + def sqf_part(f): + """Computes square-free part of ``f``. """ + return f.per(dmp_sqf_part(f._rep, f.lev, f.dom)) + + def sqf_list(f, all=False): + """Returns a list of square-free factors of ``f``. """ + coeff, factors = dmp_sqf_list(f._rep, f.lev, f.dom, all) + return coeff, [ (f.per(g), k) for g, k in factors ] + + def sqf_list_include(f, all=False): + """Returns a list of square-free factors of ``f``. """ + factors = dmp_sqf_list_include(f._rep, f.lev, f.dom, all) + return [ (f.per(g), k) for g, k in factors ] + + def factor_list(f): + """Returns a list of irreducible factors of ``f``. """ + coeff, factors = dmp_factor_list(f._rep, f.lev, f.dom) + return coeff, [ (f.per(g), k) for g, k in factors ] + + def factor_list_include(f): + """Returns a list of irreducible factors of ``f``. """ + factors = dmp_factor_list_include(f._rep, f.lev, f.dom) + return [ (f.per(g), k) for g, k in factors ] + + def _isolate_real_roots(f, eps, inf, sup, fast): + return dup_isolate_real_roots(f._rep, f.dom, eps=eps, inf=inf, sup=sup, fast=fast) + + def _isolate_real_roots_sqf(f, eps, inf, sup, fast): + return dup_isolate_real_roots_sqf(f._rep, f.dom, eps=eps, inf=inf, sup=sup, fast=fast) + + def _isolate_all_roots(f, eps, inf, sup, fast): + return dup_isolate_all_roots(f._rep, f.dom, eps=eps, inf=inf, sup=sup, fast=fast) + + def _isolate_all_roots_sqf(f, eps, inf, sup, fast): + return dup_isolate_all_roots_sqf(f._rep, f.dom, eps=eps, inf=inf, sup=sup, fast=fast) + + def _refine_real_root(f, s, t, eps, steps, fast): + return dup_refine_real_root(f._rep, s, t, f.dom, eps=eps, steps=steps, fast=fast) + + def count_real_roots(f, inf=None, sup=None): + """Return the number of real roots of ``f`` in ``[inf, sup]``. """ + return dup_count_real_roots(f._rep, f.dom, inf=inf, sup=sup) + + def count_complex_roots(f, inf=None, sup=None): + """Return the number of complex roots of ``f`` in ``[inf, sup]``. """ + return dup_count_complex_roots(f._rep, f.dom, inf=inf, sup=sup) + + @property + def is_zero(f): + """Returns ``True`` if ``f`` is a zero polynomial. """ + return dmp_zero_p(f._rep, f.lev) + + @property + def is_one(f): + """Returns ``True`` if ``f`` is a unit polynomial. """ + return dmp_one_p(f._rep, f.lev, f.dom) + + @property + def is_ground(f): + """Returns ``True`` if ``f`` is an element of the ground domain. """ + return dmp_ground_p(f._rep, None, f.lev) + + @property + def is_sqf(f): + """Returns ``True`` if ``f`` is a square-free polynomial. """ + return dmp_sqf_p(f._rep, f.lev, f.dom) + + @property + def is_monic(f): + """Returns ``True`` if the leading coefficient of ``f`` is one. """ + return f.dom.is_one(dmp_ground_LC(f._rep, f.lev, f.dom)) + + @property + def is_primitive(f): + """Returns ``True`` if the GCD of the coefficients of ``f`` is one. """ + return f.dom.is_one(dmp_ground_content(f._rep, f.lev, f.dom)) + + @property + def is_linear(f): + """Returns ``True`` if ``f`` is linear in all its variables. """ + return all(sum(monom) <= 1 for monom in dmp_to_dict(f._rep, f.lev, f.dom).keys()) + + @property + def is_quadratic(f): + """Returns ``True`` if ``f`` is quadratic in all its variables. """ + return all(sum(monom) <= 2 for monom in dmp_to_dict(f._rep, f.lev, f.dom).keys()) + + @property + def is_monomial(f): + """Returns ``True`` if ``f`` is zero or has only one term. """ + return len(f.to_dict()) <= 1 + + @property + def is_homogeneous(f): + """Returns ``True`` if ``f`` is a homogeneous polynomial. """ + return f.homogeneous_order() is not None + + @property + def is_irreducible(f): + """Returns ``True`` if ``f`` has no factors over its domain. """ + return dmp_irreducible_p(f._rep, f.lev, f.dom) + + @property + def is_cyclotomic(f): + """Returns ``True`` if ``f`` is a cyclotomic polynomial. """ + if not f.lev: + return dup_cyclotomic_p(f._rep, f.dom) + else: + return False + + +class DUP_Flint(DMP): + """Dense Multivariate Polynomials over `K`. """ + + lev = 0 + + __slots__ = ('_rep', 'dom', '_cls') + + def __reduce__(self): + return self.__class__, (self.to_list(), self.dom, self.lev) + + @classmethod + def _new(cls, rep, dom, lev): + rep = cls._flint_poly(rep[::-1], dom, lev) + return cls.from_rep(rep, dom) + + def to_list(f): + """Convert ``f`` to a list representation with native coefficients. """ + return f._rep.coeffs()[::-1] + + @classmethod + def _flint_poly(cls, rep, dom, lev): + assert _supported_flint_domain(dom) + assert lev == 0 + flint_cls = cls._get_flint_poly_cls(dom) + return flint_cls(rep) + + @classmethod + def _get_flint_poly_cls(cls, dom): + if dom.is_ZZ: + return flint.fmpz_poly + elif dom.is_QQ: + return flint.fmpq_poly + elif dom.is_FF: + return dom._poly_ctx + else: + raise RuntimeError("Domain %s is not supported with flint" % dom) + + @classmethod + def from_rep(cls, rep, dom): + """Create a DMP from the given representation. """ + + if dom.is_ZZ: + assert isinstance(rep, flint.fmpz_poly) + _cls = flint.fmpz_poly + elif dom.is_QQ: + assert isinstance(rep, flint.fmpq_poly) + _cls = flint.fmpq_poly + elif dom.is_FF: + assert isinstance(rep, (flint.nmod_poly, flint.fmpz_mod_poly)) + c = dom.characteristic() + __cls = type(rep) + _cls = lambda e: __cls(e, c) + else: + raise RuntimeError("Domain %s is not supported with flint" % dom) + + obj = object.__new__(cls) + obj.dom = dom + obj._rep = rep + obj._cls = _cls + + return obj + + def _strict_eq(f, g): + if type(f) != type(g): + return False + return f.dom == g.dom and f._rep == g._rep + + def ground_new(f, coeff): + """Construct a new ground instance of ``f``. """ + return f.from_rep(f._cls([coeff]), f.dom) + + def _one(f): + return f.ground_new(f.dom.one) + + def unify(f, g): + """Unify representations of two polynomials. """ + raise RuntimeError + + def to_DMP_Python(f): + """Convert ``f`` to a Python native representation. """ + return DMP_Python._new(f.to_list(), f.dom, f.lev) + + def to_tuple(f): + """Convert ``f`` to a tuple representation with native coefficients. """ + return tuple(f.to_list()) + + def _convert(f, dom): + """Convert the ground domain of ``f``. """ + if dom == QQ and f.dom == ZZ: + return f.from_rep(flint.fmpq_poly(f._rep), dom) + elif _supported_flint_domain(dom) and _supported_flint_domain(f.dom): + # XXX: python-flint should provide a faster way to do this. + return f.to_DMP_Python()._convert(dom).to_DUP_Flint() + else: + raise RuntimeError(f"DUP_Flint: Cannot convert {f.dom} to {dom}") + + def _slice(f, m, n): + """Take a continuous subsequence of terms of ``f``. """ + coeffs = f._rep.coeffs()[m:n] + return f.from_rep(f._cls(coeffs), f.dom) + + def _slice_lev(f, m, n, j): + """Take a continuous subsequence of terms of ``f``. """ + # Only makes sense for multivariate polynomials + raise NotImplementedError + + def _terms(f, order=None): + """Returns all non-zero terms from ``f`` in lex order. """ + if order is None or order.alias == 'lex': + terms = [ ((n,), c) for n, c in enumerate(f._rep.coeffs()) if c ] + return terms[::-1] + else: + # XXX: InverseOrder (ilex) comes here. We could handle that case + # efficiently by reversing the coefficients but it is not clear + # how to test if the order is InverseOrder. + # + # Otherwise why would the order ever be different for univariate + # polynomials? + return f.to_DMP_Python()._terms(order=order) + + def _lift(f): + """Convert algebraic coefficients to rationals. """ + # This is for algebraic number fields which DUP_Flint does not support + raise NotImplementedError + + def deflate(f): + """Reduce degree of `f` by mapping `x_i^m` to `y_i`. """ + # XXX: Check because otherwise this segfaults with python-flint: + # + # >>> flint.fmpz_poly([]).deflation() + # Exception (fmpz_poly_deflate). Division by zero. + # Aborted (core dumped + # + if f.is_zero: + return (1,), f + g, n = f._rep.deflation() + return (n,), f.from_rep(g, f.dom) + + def inject(f, front=False): + """Inject ground domain generators into ``f``. """ + # Ground domain would need to be a poly ring + raise NotImplementedError + + def eject(f, dom, front=False): + """Eject selected generators into the ground domain. """ + # Only makes sense for multivariate polynomials + raise NotImplementedError + + def _exclude(f): + """Remove useless generators from ``f``. """ + # Only makes sense for multivariate polynomials + raise NotImplementedError + + def _permute(f, P): + """Returns a polynomial in `K[x_{P(1)}, ..., x_{P(n)}]`. """ + # Only makes sense for multivariate polynomials + raise NotImplementedError + + def terms_gcd(f): + """Remove GCD of terms from the polynomial ``f``. """ + # XXX: python-flint should have primitive, content, etc methods. + J, F = f.to_DMP_Python().terms_gcd() + return J, F.to_DUP_Flint() + + def _add_ground(f, c): + """Add an element of the ground domain to ``f``. """ + return f.from_rep(f._rep + c, f.dom) + + def _sub_ground(f, c): + """Subtract an element of the ground domain from ``f``. """ + return f.from_rep(f._rep - c, f.dom) + + def _mul_ground(f, c): + """Multiply ``f`` by a an element of the ground domain. """ + return f.from_rep(f._rep * c, f.dom) + + def _quo_ground(f, c): + """Quotient of ``f`` by a an element of the ground domain. """ + return f.from_rep(f._rep // c, f.dom) + + def _exquo_ground(f, c): + """Exact quotient of ``f`` by an element of the ground domain. """ + q, r = divmod(f._rep, c) + if r: + raise ExactQuotientFailed(f, c) + return f.from_rep(q, f.dom) + + def abs(f): + """Make all coefficients in ``f`` positive. """ + return f.to_DMP_Python().abs().to_DUP_Flint() + + def neg(f): + """Negate all coefficients in ``f``. """ + return f.from_rep(-f._rep, f.dom) + + def _add(f, g): + """Add two multivariate polynomials ``f`` and ``g``. """ + return f.from_rep(f._rep + g._rep, f.dom) + + def _sub(f, g): + """Subtract two multivariate polynomials ``f`` and ``g``. """ + return f.from_rep(f._rep - g._rep, f.dom) + + def _mul(f, g): + """Multiply two multivariate polynomials ``f`` and ``g``. """ + return f.from_rep(f._rep * g._rep, f.dom) + + def sqr(f): + """Square a multivariate polynomial ``f``. """ + return f.from_rep(f._rep ** 2, f.dom) + + def _pow(f, n): + """Raise ``f`` to a non-negative power ``n``. """ + return f.from_rep(f._rep ** n, f.dom) + + def _pdiv(f, g): + """Polynomial pseudo-division of ``f`` and ``g``. """ + d = f.degree() - g.degree() + 1 + q, r = divmod(g.LC()**d * f._rep, g._rep) + return f.from_rep(q, f.dom), f.from_rep(r, f.dom) + + def _prem(f, g): + """Polynomial pseudo-remainder of ``f`` and ``g``. """ + d = f.degree() - g.degree() + 1 + q = (g.LC()**d * f._rep) % g._rep + return f.from_rep(q, f.dom) + + def _pquo(f, g): + """Polynomial pseudo-quotient of ``f`` and ``g``. """ + d = f.degree() - g.degree() + 1 + r = (g.LC()**d * f._rep) // g._rep + return f.from_rep(r, f.dom) + + def _pexquo(f, g): + """Polynomial exact pseudo-quotient of ``f`` and ``g``. """ + d = f.degree() - g.degree() + 1 + q, r = divmod(g.LC()**d * f._rep, g._rep) + if r: + raise ExactQuotientFailed(f, g) + return f.from_rep(q, f.dom) + + def _div(f, g): + """Polynomial division with remainder of ``f`` and ``g``. """ + if f.dom.is_Field: + q, r = divmod(f._rep, g._rep) + return f.from_rep(q, f.dom), f.from_rep(r, f.dom) + else: + # XXX: python-flint defines division in ZZ[x] differently + q, r = f.to_DMP_Python()._div(g.to_DMP_Python()) + return q.to_DUP_Flint(), r.to_DUP_Flint() + + def _rem(f, g): + """Computes polynomial remainder of ``f`` and ``g``. """ + return f.from_rep(f._rep % g._rep, f.dom) + + def _quo(f, g): + """Computes polynomial quotient of ``f`` and ``g``. """ + return f.from_rep(f._rep // g._rep, f.dom) + + def _exquo(f, g): + """Computes polynomial exact quotient of ``f`` and ``g``. """ + q, r = f._div(g) + if r: + raise ExactQuotientFailed(f, g) + return q + + def _degree(f, j=0): + """Returns the leading degree of ``f`` in ``x_j``. """ + d = f._rep.degree() + if d == -1: + d = ninf + return d + + def degree_list(f): + """Returns a list of degrees of ``f``. """ + return ( f._degree() ,) + + def total_degree(f): + """Returns the total degree of ``f``. """ + return f._degree() + + def LC(f): + """Returns the leading coefficient of ``f``. """ + return f._rep[f._rep.degree()] + + def TC(f): + """Returns the trailing coefficient of ``f``. """ + return f._rep[0] + + def _nth(f, N): + """Returns the ``n``-th coefficient of ``f``. """ + [n] = N + return f._rep[n] + + def max_norm(f): + """Returns maximum norm of ``f``. """ + return f.to_DMP_Python().max_norm() + + def l1_norm(f): + """Returns l1 norm of ``f``. """ + return f.to_DMP_Python().l1_norm() + + def l2_norm_squared(f): + """Return squared l2 norm of ``f``. """ + return f.to_DMP_Python().l2_norm_squared() + + def clear_denoms(f): + """Clear denominators, but keep the ground domain. """ + R = f.dom + if R.is_QQ: + denom = f._rep.denom() + numer = f.from_rep(f._cls(f._rep.numer()), f.dom) + return denom, numer + elif R.is_ZZ or R.is_FiniteField: + return R.one, f + else: + raise NotImplementedError + + def _integrate(f, m=1, j=0): + """Computes the ``m``-th order indefinite integral of ``f`` in ``x_j``. """ + assert j == 0 + if f.dom.is_Field: + rep = f._rep + for i in range(m): + rep = rep.integral() + return f.from_rep(rep, f.dom) + else: + return f.to_DMP_Python()._integrate(m=m, j=j).to_DUP_Flint() + + def _diff(f, m=1, j=0): + """Computes the ``m``-th order derivative of ``f``. """ + assert j == 0 + rep = f._rep + for i in range(m): + rep = rep.derivative() + return f.from_rep(rep, f.dom) + + def _eval(f, a): + # XXX: This method is called with many different input types. Ideally + # we could use e.g. fmpz_poly.__call__ here but more thought needs to + # go into which types this is supposed to be called with and what types + # it should return. + return f.to_DMP_Python()._eval(a) + + def _eval_lev(f, a, j): + # Only makes sense for multivariate polynomials + raise NotImplementedError + + def _half_gcdex(f, g): + """Half extended Euclidean algorithm. """ + s, h = f.to_DMP_Python()._half_gcdex(g.to_DMP_Python()) + return s.to_DUP_Flint(), h.to_DUP_Flint() + + def _gcdex(f, g): + """Extended Euclidean algorithm. """ + h, s, t = f._rep.xgcd(g._rep) + return f.from_rep(s, f.dom), f.from_rep(t, f.dom), f.from_rep(h, f.dom) + + def _invert(f, g): + """Invert ``f`` modulo ``g``, if possible. """ + R = f.dom + if R.is_Field: + gcd, F_inv, _ = f._rep.xgcd(g._rep) + # XXX: Should be gcd != 1 but nmod_poly does not compare equal to + # other types. + if gcd != 0*gcd + 1: + raise NotInvertible("zero divisor") + return f.from_rep(F_inv, R) + else: + # fmpz_poly does not have xgcd or invert and this is not well + # defined in general. + return f.to_DMP_Python()._invert(g.to_DMP_Python()).to_DUP_Flint() + + def _revert(f, n): + """Compute ``f**(-1)`` mod ``x**n``. """ + # XXX: Use fmpz_series etc for reversion? + # Maybe python-flint should provide revert for fmpz_poly... + return f.to_DMP_Python()._revert(n).to_DUP_Flint() + + def _subresultants(f, g): + """Computes subresultant PRS sequence of ``f`` and ``g``. """ + # XXX: Maybe _fmpz_poly_pseudo_rem_cohen could be used... + R = f.to_DMP_Python()._subresultants(g.to_DMP_Python()) + return [ g.to_DUP_Flint() for g in R ] + + def _resultant_includePRS(f, g): + """Computes resultant of ``f`` and ``g`` via PRS. """ + # XXX: Maybe _fmpz_poly_pseudo_rem_cohen could be used... + res, R = f.to_DMP_Python()._resultant_includePRS(g.to_DMP_Python()) + return res, [ g.to_DUP_Flint() for g in R ] + + def _resultant(f, g): + """Computes resultant of ``f`` and ``g``. """ + # XXX: Use fmpz_mpoly etc when possible... + return f.to_DMP_Python()._resultant(g.to_DMP_Python()) + + def discriminant(f): + """Computes discriminant of ``f``. """ + # XXX: Use fmpz_mpoly etc when possible... + return f.to_DMP_Python().discriminant() + + def _cofactors(f, g): + """Returns GCD of ``f`` and ``g`` and their cofactors. """ + h = f.gcd(g) + return h, f.exquo(h), g.exquo(h) + + def _gcd(f, g): + """Returns polynomial GCD of ``f`` and ``g``. """ + return f.from_rep(f._rep.gcd(g._rep), f.dom) + + def _lcm(f, g): + """Returns polynomial LCM of ``f`` and ``g``. """ + # XXX: python-flint should have a lcm method + if not (f and g): + return f.ground_new(f.dom.zero) + + l = f._mul(g)._exquo(f._gcd(g)) + + if l.dom.is_Field: + l = l.monic() + elif l.LC() < 0: + l = l.neg() + + return l + + def _cancel(f, g): + """Cancel common factors in a rational function ``f/g``. """ + assert f.dom == g.dom + R = f.dom + + # Think carefully about how to handle denominators and coefficient + # canonicalisation if more domains are permitted... + assert R.is_ZZ or R.is_QQ or R.is_FiniteField + + if R.is_FiniteField: + h = f._gcd(g) + F, G = f.exquo(h), g.exquo(h) + return R.one, R.one, F, G + + if R.is_QQ: + cG, F = f.clear_denoms() + cF, G = g.clear_denoms() + else: + cG, F = R.one, f + cF, G = R.one, g + + cH = cF.gcd(cG) + cF, cG = cF // cH, cG // cH + + H = F._gcd(G) + F, G = F.exquo(H), G.exquo(H) + + f_neg = F.LC() < 0 + g_neg = G.LC() < 0 + + if f_neg and g_neg: + F, G = F.neg(), G.neg() + elif f_neg: + cF, F = -cF, F.neg() + elif g_neg: + cF, G = -cF, G.neg() + + return cF, cG, F, G + + def _cancel_include(f, g): + """Cancel common factors in a rational function ``f/g``. """ + cF, cG, F, G = f._cancel(g) + return F._mul_ground(cF), G._mul_ground(cG) + + def _trunc(f, p): + """Reduce ``f`` modulo a constant ``p``. """ + return f.to_DMP_Python()._trunc(p).to_DUP_Flint() + + def monic(f): + """Divides all coefficients by ``LC(f)``. """ + # XXX: python-flint should add monic + return f._exquo_ground(f.LC()) + + def content(f): + """Returns GCD of polynomial coefficients. """ + # XXX: python-flint should have a content method + return f.to_DMP_Python().content() + + def primitive(f): + """Returns content and a primitive form of ``f``. """ + cont = f.content() + if f.is_zero: + return f.dom.zero, f + prim = f._exquo_ground(cont) + return cont, prim + + def _compose(f, g): + """Computes functional composition of ``f`` and ``g``. """ + return f.from_rep(f._rep(g._rep), f.dom) + + def _decompose(f): + """Computes functional decomposition of ``f``. """ + return [ g.to_DUP_Flint() for g in f.to_DMP_Python()._decompose() ] + + def _shift(f, a): + """Efficiently compute Taylor shift ``f(x + a)``. """ + x_plus_a = f._cls([a, f.dom.one]) + return f.from_rep(f._rep(x_plus_a), f.dom) + + def _transform(f, p, q): + """Evaluate functional transformation ``q**n * f(p/q)``.""" + F, P, Q = f.to_DMP_Python(), p.to_DMP_Python(), q.to_DMP_Python() + return F.transform(P, Q).to_DUP_Flint() + + def _sturm(f): + """Computes the Sturm sequence of ``f``. """ + return [ g.to_DUP_Flint() for g in f.to_DMP_Python()._sturm() ] + + def _cauchy_upper_bound(f): + """Computes the Cauchy upper bound on the roots of ``f``. """ + return f.to_DMP_Python()._cauchy_upper_bound() + + def _cauchy_lower_bound(f): + """Computes the Cauchy lower bound on the nonzero roots of ``f``. """ + return f.to_DMP_Python()._cauchy_lower_bound() + + def _mignotte_sep_bound_squared(f): + """Computes the squared Mignotte bound on root separations of ``f``. """ + return f.to_DMP_Python()._mignotte_sep_bound_squared() + + def _gff_list(f): + """Computes greatest factorial factorization of ``f``. """ + F = f.to_DMP_Python() + return [ (g.to_DUP_Flint(), k) for g, k in F.gff_list() ] + + def norm(f): + """Computes ``Norm(f)``.""" + # This is for algebraic number fields which DUP_Flint does not support + raise NotImplementedError + + def sqf_norm(f): + """Computes square-free norm of ``f``. """ + # This is for algebraic number fields which DUP_Flint does not support + raise NotImplementedError + + def sqf_part(f): + """Computes square-free part of ``f``. """ + return f._exquo(f._gcd(f._diff())) + + def sqf_list(f, all=False): + """Returns a list of square-free factors of ``f``. """ + # XXX: python-flint should provide square free factorisation. + coeff, factors = f.to_DMP_Python().sqf_list(all=all) + return coeff, [ (g.to_DUP_Flint(), k) for g, k in factors ] + + def sqf_list_include(f, all=False): + """Returns a list of square-free factors of ``f``. """ + factors = f.to_DMP_Python().sqf_list_include(all=all) + return [ (g.to_DUP_Flint(), k) for g, k in factors ] + + def factor_list(f): + """Returns a list of irreducible factors of ``f``. """ + + if f.dom.is_ZZ or f.dom.is_FF: + # python-flint matches polys here + coeff, factors = f._rep.factor() + factors = [ (f.from_rep(g, f.dom), k) for g, k in factors ] + + elif f.dom.is_QQ: + # python-flint returns monic factors over QQ whereas polys returns + # denominator free factors. + coeff, factors = f._rep.factor() + factors_monic = [ (f.from_rep(g, f.dom), k) for g, k in factors ] + + # Absorb the denominators into coeff + factors = [] + for g, k in factors_monic: + d, g = g.clear_denoms() + coeff /= d**k + factors.append((g, k)) + + else: + # Check carefully when adding more domains here... + raise RuntimeError("Domain %s is not supported with flint" % f.dom) + + # We need to match the way that polys orders the factors + factors = f._sort_factors(factors) + + return coeff, factors + + def factor_list_include(f): + """Returns a list of irreducible factors of ``f``. """ + # XXX: factor_list_include seems to be broken in general: + # + # >>> Poly(2*(x - 1)**3, x).factor_list_include() + # [(Poly(2*x - 2, x, domain='ZZ'), 3)] + # + # Let's not try to implement it here. + factors = f.to_DMP_Python().factor_list_include() + return [ (g.to_DUP_Flint(), k) for g, k in factors ] + + def _sort_factors(f, factors): + """Sort a list of factors to canonical order. """ + # Convert the factors to lists and use _sort_factors from polys + factors = [ (g.to_list(), k) for g, k in factors ] + factors = _sort_factors(factors, multiple=True) + to_dup_flint = lambda g: f.from_rep(f._cls(g[::-1]), f.dom) + return [ (to_dup_flint(g), k) for g, k in factors ] + + def _isolate_real_roots(f, eps, inf, sup, fast): + return f.to_DMP_Python()._isolate_real_roots(eps, inf, sup, fast) + + def _isolate_real_roots_sqf(f, eps, inf, sup, fast): + return f.to_DMP_Python()._isolate_real_roots_sqf(eps, inf, sup, fast) + + def _isolate_all_roots(f, eps, inf, sup, fast): + # fmpz_poly and fmpq_poly have a complex_roots method that could be + # used here. It probably makes more sense to add analogous methods in + # python-flint though. + return f.to_DMP_Python()._isolate_all_roots(eps, inf, sup, fast) + + def _isolate_all_roots_sqf(f, eps, inf, sup, fast): + return f.to_DMP_Python()._isolate_all_roots_sqf(eps, inf, sup, fast) + + def _refine_real_root(f, s, t, eps, steps, fast): + return f.to_DMP_Python()._refine_real_root(s, t, eps, steps, fast) + + def count_real_roots(f, inf=None, sup=None): + """Return the number of real roots of ``f`` in ``[inf, sup]``. """ + return f.to_DMP_Python().count_real_roots(inf=inf, sup=sup) + + def count_complex_roots(f, inf=None, sup=None): + """Return the number of complex roots of ``f`` in ``[inf, sup]``. """ + return f.to_DMP_Python().count_complex_roots(inf=inf, sup=sup) + + @property + def is_zero(f): + """Returns ``True`` if ``f`` is a zero polynomial. """ + return not f._rep + + @property + def is_one(f): + """Returns ``True`` if ``f`` is a unit polynomial. """ + return f._rep == f.dom.one + + @property + def is_ground(f): + """Returns ``True`` if ``f`` is an element of the ground domain. """ + return f._rep.degree() <= 0 + + @property + def is_linear(f): + """Returns ``True`` if ``f`` is linear in all its variables. """ + return f._rep.degree() <= 1 + + @property + def is_quadratic(f): + """Returns ``True`` if ``f`` is quadratic in all its variables. """ + return f._rep.degree() <= 2 + + @property + def is_monomial(f): + """Returns ``True`` if ``f`` is zero or has only one term. """ + fr = f._rep + return fr.degree() < 0 or not any(fr[n] for n in range(fr.degree())) + + @property + def is_monic(f): + """Returns ``True`` if the leading coefficient of ``f`` is one. """ + return f.LC() == f.dom.one + + @property + def is_primitive(f): + """Returns ``True`` if the GCD of the coefficients of ``f`` is one. """ + return f.to_DMP_Python().is_primitive + + @property + def is_homogeneous(f): + """Returns ``True`` if ``f`` is a homogeneous polynomial. """ + return f.to_DMP_Python().is_homogeneous + + @property + def is_sqf(f): + """Returns ``True`` if ``f`` is a square-free polynomial. """ + g = f._rep.gcd(f._rep.derivative()) + return g.degree() <= 0 + + @property + def is_irreducible(f): + """Returns ``True`` if ``f`` has no factors over its domain. """ + _, factors = f._rep.factor() + if len(factors) == 0: + return True + elif len(factors) == 1: + return factors[0][1] == 1 + else: + return False + + @property + def is_cyclotomic(f): + """Returns ``True`` if ``f`` is a cyclotomic polynomial. """ + if f.dom.is_QQ: + try: + f = f.convert(ZZ) + except CoercionFailed: + return False + if f.dom.is_ZZ: + return bool(f._rep.is_cyclotomic()) + else: + # This is what dup_cyclotomic_p does... + return False + + +def init_normal_DMF(num, den, lev, dom): + return DMF(dmp_normal(num, lev, dom), + dmp_normal(den, lev, dom), dom, lev) + + +class DMF(PicklableWithSlots, CantSympify): + """Dense Multivariate Fractions over `K`. """ + + __slots__ = ('num', 'den', 'lev', 'dom') + + def __init__(self, rep, dom, lev=None): + num, den, lev = self._parse(rep, dom, lev) + num, den = dmp_cancel(num, den, lev, dom) + + self.num = num + self.den = den + self.lev = lev + self.dom = dom + + @classmethod + def new(cls, rep, dom, lev=None): + num, den, lev = cls._parse(rep, dom, lev) + + obj = object.__new__(cls) + + obj.num = num + obj.den = den + obj.lev = lev + obj.dom = dom + + return obj + + def ground_new(self, rep): + return self.new(rep, self.dom, self.lev) + + @classmethod + def _parse(cls, rep, dom, lev=None): + if isinstance(rep, tuple): + num, den = rep + + if lev is not None: + if isinstance(num, dict): + num = dmp_from_dict(num, lev, dom) + + if isinstance(den, dict): + den = dmp_from_dict(den, lev, dom) + else: + num, num_lev = dmp_validate(num) + den, den_lev = dmp_validate(den) + + if num_lev == den_lev: + lev = num_lev + else: + raise ValueError('inconsistent number of levels') + + if dmp_zero_p(den, lev): + raise ZeroDivisionError('fraction denominator') + + if dmp_zero_p(num, lev): + den = dmp_one(lev, dom) + else: + if dmp_negative_p(den, lev, dom): + num = dmp_neg(num, lev, dom) + den = dmp_neg(den, lev, dom) + else: + num = rep + + if lev is not None: + if isinstance(num, dict): + num = dmp_from_dict(num, lev, dom) + elif not isinstance(num, list): + num = dmp_ground(dom.convert(num), lev) + else: + num, lev = dmp_validate(num) + + den = dmp_one(lev, dom) + + return num, den, lev + + def __repr__(f): + return "%s((%s, %s), %s)" % (f.__class__.__name__, f.num, f.den, f.dom) + + def __hash__(f): + return hash((f.__class__.__name__, dmp_to_tuple(f.num, f.lev), + dmp_to_tuple(f.den, f.lev), f.lev, f.dom)) + + def poly_unify(f, g): + """Unify a multivariate fraction and a polynomial. """ + if not isinstance(g, DMP) or f.lev != g.lev: + raise UnificationFailed("Cannot unify %s with %s" % (f, g)) + + if f.dom == g.dom: + return (f.lev, f.dom, f.per, (f.num, f.den), g._rep) + else: + lev, dom = f.lev, f.dom.unify(g.dom) + + F = (dmp_convert(f.num, lev, f.dom, dom), + dmp_convert(f.den, lev, f.dom, dom)) + + G = dmp_convert(g._rep, lev, g.dom, dom) + + def per(num, den, cancel=True, kill=False, lev=lev): + if kill: + if not lev: + return num/den + else: + lev = lev - 1 + + if cancel: + num, den = dmp_cancel(num, den, lev, dom) + + return f.__class__.new((num, den), dom, lev) + + return lev, dom, per, F, G + + def frac_unify(f, g): + """Unify representations of two multivariate fractions. """ + if not isinstance(g, DMF) or f.lev != g.lev: + raise UnificationFailed("Cannot unify %s with %s" % (f, g)) + + if f.dom == g.dom: + return (f.lev, f.dom, f.per, (f.num, f.den), + (g.num, g.den)) + else: + lev, dom = f.lev, f.dom.unify(g.dom) + + F = (dmp_convert(f.num, lev, f.dom, dom), + dmp_convert(f.den, lev, f.dom, dom)) + + G = (dmp_convert(g.num, lev, g.dom, dom), + dmp_convert(g.den, lev, g.dom, dom)) + + def per(num, den, cancel=True, kill=False, lev=lev): + if kill: + if not lev: + return num/den + else: + lev = lev - 1 + + if cancel: + num, den = dmp_cancel(num, den, lev, dom) + + return f.__class__.new((num, den), dom, lev) + + return lev, dom, per, F, G + + def per(f, num, den, cancel=True, kill=False): + """Create a DMF out of the given representation. """ + lev, dom = f.lev, f.dom + + if kill: + if not lev: + return num/den + else: + lev -= 1 + + if cancel: + num, den = dmp_cancel(num, den, lev, dom) + + return f.__class__.new((num, den), dom, lev) + + def half_per(f, rep, kill=False): + """Create a DMP out of the given representation. """ + lev = f.lev + + if kill: + if not lev: + return rep + else: + lev -= 1 + + return DMP(rep, f.dom, lev) + + @classmethod + def zero(cls, lev, dom): + return cls.new(0, dom, lev) + + @classmethod + def one(cls, lev, dom): + return cls.new(1, dom, lev) + + def numer(f): + """Returns the numerator of ``f``. """ + return f.half_per(f.num) + + def denom(f): + """Returns the denominator of ``f``. """ + return f.half_per(f.den) + + def cancel(f): + """Remove common factors from ``f.num`` and ``f.den``. """ + return f.per(f.num, f.den) + + def neg(f): + """Negate all coefficients in ``f``. """ + return f.per(dmp_neg(f.num, f.lev, f.dom), f.den, cancel=False) + + def add_ground(f, c): + """Add an element of the ground domain to ``f``. """ + return f + f.ground_new(c) + + def add(f, g): + """Add two multivariate fractions ``f`` and ``g``. """ + if isinstance(g, DMP): + lev, dom, per, (F_num, F_den), G = f.poly_unify(g) + num, den = dmp_add_mul(F_num, F_den, G, lev, dom), F_den + else: + lev, dom, per, F, G = f.frac_unify(g) + (F_num, F_den), (G_num, G_den) = F, G + + num = dmp_add(dmp_mul(F_num, G_den, lev, dom), + dmp_mul(F_den, G_num, lev, dom), lev, dom) + den = dmp_mul(F_den, G_den, lev, dom) + + return per(num, den) + + def sub(f, g): + """Subtract two multivariate fractions ``f`` and ``g``. """ + if isinstance(g, DMP): + lev, dom, per, (F_num, F_den), G = f.poly_unify(g) + num, den = dmp_sub_mul(F_num, F_den, G, lev, dom), F_den + else: + lev, dom, per, F, G = f.frac_unify(g) + (F_num, F_den), (G_num, G_den) = F, G + + num = dmp_sub(dmp_mul(F_num, G_den, lev, dom), + dmp_mul(F_den, G_num, lev, dom), lev, dom) + den = dmp_mul(F_den, G_den, lev, dom) + + return per(num, den) + + def mul(f, g): + """Multiply two multivariate fractions ``f`` and ``g``. """ + if isinstance(g, DMP): + lev, dom, per, (F_num, F_den), G = f.poly_unify(g) + num, den = dmp_mul(F_num, G, lev, dom), F_den + else: + lev, dom, per, F, G = f.frac_unify(g) + (F_num, F_den), (G_num, G_den) = F, G + + num = dmp_mul(F_num, G_num, lev, dom) + den = dmp_mul(F_den, G_den, lev, dom) + + return per(num, den) + + def pow(f, n): + """Raise ``f`` to a non-negative power ``n``. """ + if isinstance(n, int): + num, den = f.num, f.den + if n < 0: + num, den, n = den, num, -n + return f.per(dmp_pow(num, n, f.lev, f.dom), + dmp_pow(den, n, f.lev, f.dom), cancel=False) + else: + raise TypeError("``int`` expected, got %s" % type(n)) + + def quo(f, g): + """Computes quotient of fractions ``f`` and ``g``. """ + if isinstance(g, DMP): + lev, dom, per, (F_num, F_den), G = f.poly_unify(g) + num, den = F_num, dmp_mul(F_den, G, lev, dom) + else: + lev, dom, per, F, G = f.frac_unify(g) + (F_num, F_den), (G_num, G_den) = F, G + + num = dmp_mul(F_num, G_den, lev, dom) + den = dmp_mul(F_den, G_num, lev, dom) + + return per(num, den) + + exquo = quo + + def invert(f, check=True): + """Computes inverse of a fraction ``f``. """ + return f.per(f.den, f.num, cancel=False) + + @property + def is_zero(f): + """Returns ``True`` if ``f`` is a zero fraction. """ + return dmp_zero_p(f.num, f.lev) + + @property + def is_one(f): + """Returns ``True`` if ``f`` is a unit fraction. """ + return dmp_one_p(f.num, f.lev, f.dom) and \ + dmp_one_p(f.den, f.lev, f.dom) + + def __neg__(f): + return f.neg() + + def __add__(f, g): + if isinstance(g, (DMP, DMF)): + return f.add(g) + elif g in f.dom: + return f.add_ground(f.dom.convert(g)) + + try: + return f.add(f.half_per(g)) + except (TypeError, CoercionFailed, NotImplementedError): + return NotImplemented + + def __radd__(f, g): + return f.__add__(g) + + def __sub__(f, g): + if isinstance(g, (DMP, DMF)): + return f.sub(g) + + try: + return f.sub(f.half_per(g)) + except (TypeError, CoercionFailed, NotImplementedError): + return NotImplemented + + def __rsub__(f, g): + return (-f).__add__(g) + + def __mul__(f, g): + if isinstance(g, (DMP, DMF)): + return f.mul(g) + + try: + return f.mul(f.half_per(g)) + except (TypeError, CoercionFailed, NotImplementedError): + return NotImplemented + + def __rmul__(f, g): + return f.__mul__(g) + + def __pow__(f, n): + return f.pow(n) + + def __truediv__(f, g): + if isinstance(g, (DMP, DMF)): + return f.quo(g) + + try: + return f.quo(f.half_per(g)) + except (TypeError, CoercionFailed, NotImplementedError): + return NotImplemented + + def __rtruediv__(self, g): + return self.invert(check=False)*g + + def __eq__(f, g): + try: + if isinstance(g, DMP): + _, _, _, (F_num, F_den), G = f.poly_unify(g) + + if f.lev == g.lev: + return dmp_one_p(F_den, f.lev, f.dom) and F_num == G + else: + _, _, _, F, G = f.frac_unify(g) + + if f.lev == g.lev: + return F == G + except UnificationFailed: + pass + + return False + + def __ne__(f, g): + try: + if isinstance(g, DMP): + _, _, _, (F_num, F_den), G = f.poly_unify(g) + + if f.lev == g.lev: + return not (dmp_one_p(F_den, f.lev, f.dom) and F_num == G) + else: + _, _, _, F, G = f.frac_unify(g) + + if f.lev == g.lev: + return F != G + except UnificationFailed: + pass + + return True + + def __lt__(f, g): + _, _, _, F, G = f.frac_unify(g) + return F < G + + def __le__(f, g): + _, _, _, F, G = f.frac_unify(g) + return F <= G + + def __gt__(f, g): + _, _, _, F, G = f.frac_unify(g) + return F > G + + def __ge__(f, g): + _, _, _, F, G = f.frac_unify(g) + return F >= G + + def __bool__(f): + return not dmp_zero_p(f.num, f.lev) + + +def init_normal_ANP(rep, mod, dom): + return ANP(dup_normal(rep, dom), + dup_normal(mod, dom), dom) + + +class ANP(CantSympify): + """Dense Algebraic Number Polynomials over a field. """ + + __slots__ = ('_rep', '_mod', 'dom') + + def __new__(cls, rep, mod, dom): + if isinstance(rep, DMP): + pass + elif type(rep) is dict: # don't use isinstance + rep = DMP(dup_from_dict(rep, dom), dom, 0) + else: + if isinstance(rep, list): + rep = [dom.convert(a) for a in rep] + else: + rep = [dom.convert(rep)] + rep = DMP(dup_strip(rep), dom, 0) + + if isinstance(mod, DMP): + pass + elif isinstance(mod, dict): + mod = DMP(dup_from_dict(mod, dom), dom, 0) + else: + mod = DMP(dup_strip(mod), dom, 0) + + return cls.new(rep, mod, dom) + + @classmethod + def new(cls, rep, mod, dom): + if not (rep.dom == mod.dom == dom): + raise RuntimeError("Inconsistent domain") + obj = super().__new__(cls) + obj._rep = rep + obj._mod = mod + obj.dom = dom + return obj + + # XXX: It should be possible to use __getnewargs__ rather than __reduce__ + # but it doesn't work for some reason. Probably this would be easier if + # python-flint supported pickling for polynomial types. + def __reduce__(self): + return ANP, (self.rep, self.mod, self.dom) + + @property + def rep(self): + return self._rep.to_list() + + @property + def mod(self): + return self.mod_to_list() + + def to_DMP(self): + return self._rep + + def mod_to_DMP(self): + return self._mod + + def per(f, rep): + return f.new(rep, f._mod, f.dom) + + def __repr__(f): + return "%s(%s, %s, %s)" % (f.__class__.__name__, f._rep.to_list(), f._mod.to_list(), f.dom) + + def __hash__(f): + return hash((f.__class__.__name__, f.to_tuple(), f._mod.to_tuple(), f.dom)) + + def convert(f, dom): + """Convert ``f`` to a ``ANP`` over a new domain. """ + if f.dom == dom: + return f + else: + return f.new(f._rep.convert(dom), f._mod.convert(dom), dom) + + def unify(f, g): + """Unify representations of two algebraic numbers. """ + + # XXX: This unify method is not used any more because unify_ANP is used + # instead. + + if not isinstance(g, ANP) or f.mod != g.mod: + raise UnificationFailed("Cannot unify %s with %s" % (f, g)) + + if f.dom == g.dom: + return f.dom, f.per, f.rep, g.rep, f.mod + else: + dom = f.dom.unify(g.dom) + + F = dup_convert(f.rep, f.dom, dom) + G = dup_convert(g.rep, g.dom, dom) + + if dom != f.dom and dom != g.dom: + mod = dup_convert(f.mod, f.dom, dom) + else: + if dom == f.dom: + mod = f.mod + else: + mod = g.mod + + per = lambda rep: ANP(rep, mod, dom) + + return dom, per, F, G, mod + + def unify_ANP(f, g): + """Unify and return ``DMP`` instances of ``f`` and ``g``. """ + if not isinstance(g, ANP) or f._mod != g._mod: + raise UnificationFailed("Cannot unify %s with %s" % (f, g)) + + # The domain is almost always QQ but there are some tests involving ZZ + if f.dom != g.dom: + dom = f.dom.unify(g.dom) + f = f.convert(dom) + g = g.convert(dom) + + return f._rep, g._rep, f._mod, f.dom + + @classmethod + def zero(cls, mod, dom): + return ANP(0, mod, dom) + + @classmethod + def one(cls, mod, dom): + return ANP(1, mod, dom) + + def to_dict(f): + """Convert ``f`` to a dict representation with native coefficients. """ + return f._rep.to_dict() + + def to_sympy_dict(f): + """Convert ``f`` to a dict representation with SymPy coefficients. """ + rep = dmp_to_dict(f.rep, 0, f.dom) + + for k, v in rep.items(): + rep[k] = f.dom.to_sympy(v) + + return rep + + def to_list(f): + """Convert ``f`` to a list representation with native coefficients. """ + return f._rep.to_list() + + def mod_to_list(f): + """Return ``f.mod`` as a list with native coefficients. """ + return f._mod.to_list() + + def to_sympy_list(f): + """Convert ``f`` to a list representation with SymPy coefficients. """ + return [ f.dom.to_sympy(c) for c in f.to_list() ] + + def to_tuple(f): + """ + Convert ``f`` to a tuple representation with native coefficients. + + This is needed for hashing. + """ + return f._rep.to_tuple() + + @classmethod + def from_list(cls, rep, mod, dom): + return ANP(dup_strip(list(map(dom.convert, rep))), mod, dom) + + def add_ground(f, c): + """Add an element of the ground domain to ``f``. """ + return f.per(f._rep.add_ground(c)) + + def sub_ground(f, c): + """Subtract an element of the ground domain from ``f``. """ + return f.per(f._rep.sub_ground(c)) + + def mul_ground(f, c): + """Multiply ``f`` by an element of the ground domain. """ + return f.per(f._rep.mul_ground(c)) + + def quo_ground(f, c): + """Quotient of ``f`` by an element of the ground domain. """ + return f.per(f._rep.quo_ground(c)) + + def neg(f): + return f.per(f._rep.neg()) + + def add(f, g): + F, G, mod, dom = f.unify_ANP(g) + return f.new(F.add(G), mod, dom) + + def sub(f, g): + F, G, mod, dom = f.unify_ANP(g) + return f.new(F.sub(G), mod, dom) + + def mul(f, g): + F, G, mod, dom = f.unify_ANP(g) + return f.new(F.mul(G).rem(mod), mod, dom) + + def pow(f, n): + """Raise ``f`` to a non-negative power ``n``. """ + if not isinstance(n, int): + raise TypeError("``int`` expected, got %s" % type(n)) + + mod = f._mod + F = f._rep + + if n < 0: + F, n = F.invert(mod), -n + + # XXX: Need a pow_mod method for DMP + return f.new(F.pow(n).rem(f._mod), mod, f.dom) + + def exquo(f, g): + F, G, mod, dom = f.unify_ANP(g) + return f.new(F.mul(G.invert(mod)).rem(mod), mod, dom) + + def div(f, g): + return f.exquo(g), f.zero(f._mod, f.dom) + + def quo(f, g): + return f.exquo(g) + + def rem(f, g): + F, G, mod, dom = f.unify_ANP(g) + s, h = F.half_gcdex(G) + + if h.is_one: + return f.zero(mod, dom) + else: + raise NotInvertible("zero divisor") + + def LC(f): + """Returns the leading coefficient of ``f``. """ + return f._rep.LC() + + def TC(f): + """Returns the trailing coefficient of ``f``. """ + return f._rep.TC() + + @property + def is_zero(f): + """Returns ``True`` if ``f`` is a zero algebraic number. """ + return f._rep.is_zero + + @property + def is_one(f): + """Returns ``True`` if ``f`` is a unit algebraic number. """ + return f._rep.is_one + + @property + def is_ground(f): + """Returns ``True`` if ``f`` is an element of the ground domain. """ + return f._rep.is_ground + + def __pos__(f): + return f + + def __neg__(f): + return f.neg() + + def __add__(f, g): + if isinstance(g, ANP): + return f.add(g) + try: + g = f.dom.convert(g) + except CoercionFailed: + return NotImplemented + else: + return f.add_ground(g) + + def __radd__(f, g): + return f.__add__(g) + + def __sub__(f, g): + if isinstance(g, ANP): + return f.sub(g) + try: + g = f.dom.convert(g) + except CoercionFailed: + return NotImplemented + else: + return f.sub_ground(g) + + def __rsub__(f, g): + return (-f).__add__(g) + + def __mul__(f, g): + if isinstance(g, ANP): + return f.mul(g) + try: + g = f.dom.convert(g) + except CoercionFailed: + return NotImplemented + else: + return f.mul_ground(g) + + def __rmul__(f, g): + return f.__mul__(g) + + def __pow__(f, n): + return f.pow(n) + + def __divmod__(f, g): + return f.div(g) + + def __mod__(f, g): + return f.rem(g) + + def __truediv__(f, g): + if isinstance(g, ANP): + return f.quo(g) + try: + g = f.dom.convert(g) + except CoercionFailed: + return NotImplemented + else: + return f.quo_ground(g) + + def __eq__(f, g): + try: + F, G, _, _ = f.unify_ANP(g) + except UnificationFailed: + return NotImplemented + return F == G + + def __ne__(f, g): + try: + F, G, _, _ = f.unify_ANP(g) + except UnificationFailed: + return NotImplemented + return F != G + + def __lt__(f, g): + F, G, _, _ = f.unify_ANP(g) + return F < G + + def __le__(f, g): + F, G, _, _ = f.unify_ANP(g) + return F <= G + + def __gt__(f, g): + F, G, _, _ = f.unify_ANP(g) + return F > G + + def __ge__(f, g): + F, G, _, _ = f.unify_ANP(g) + return F >= G + + def __bool__(f): + return bool(f._rep) diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/polyconfig.py b/.venv/lib/python3.12/site-packages/sympy/polys/polyconfig.py new file mode 100644 index 0000000000000000000000000000000000000000..75731f7ac4e4f8784ff8f999cc3537bfa3c6659a --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/polyconfig.py @@ -0,0 +1,67 @@ +"""Configuration utilities for polynomial manipulation algorithms. """ + + +from contextlib import contextmanager + +_default_config = { + 'USE_COLLINS_RESULTANT': False, + 'USE_SIMPLIFY_GCD': True, + 'USE_HEU_GCD': True, + + 'USE_IRREDUCIBLE_IN_FACTOR': False, + 'USE_CYCLOTOMIC_FACTOR': True, + + 'EEZ_RESTART_IF_NEEDED': True, + 'EEZ_NUMBER_OF_CONFIGS': 3, + 'EEZ_NUMBER_OF_TRIES': 5, + 'EEZ_MODULUS_STEP': 2, + + 'GF_IRRED_METHOD': 'rabin', + 'GF_FACTOR_METHOD': 'zassenhaus', + + 'GROEBNER': 'buchberger', +} + +_current_config = {} + +@contextmanager +def using(**kwargs): + for k, v in kwargs.items(): + setup(k, v) + + yield + + for k in kwargs.keys(): + setup(k) + +def setup(key, value=None): + """Assign a value to (or reset) a configuration item. """ + key = key.upper() + + if value is not None: + _current_config[key] = value + else: + _current_config[key] = _default_config[key] + + +def query(key): + """Ask for a value of the given configuration item. """ + return _current_config.get(key.upper(), None) + + +def configure(): + """Initialized configuration of polys module. """ + from os import getenv + + for key, default in _default_config.items(): + value = getenv('SYMPY_' + key) + + if value is not None: + try: + _current_config[key] = eval(value) + except NameError: + _current_config[key] = value + else: + _current_config[key] = default + +configure() diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/polyerrors.py b/.venv/lib/python3.12/site-packages/sympy/polys/polyerrors.py new file mode 100644 index 0000000000000000000000000000000000000000..79385ffaf6746386f8f108c3e02992dcaf4a4f55 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/polyerrors.py @@ -0,0 +1,183 @@ +"""Definitions of common exceptions for `polys` module. """ + + +from sympy.utilities import public + +@public +class BasePolynomialError(Exception): + """Base class for polynomial related exceptions. """ + + def new(self, *args): + raise NotImplementedError("abstract base class") + +@public +class ExactQuotientFailed(BasePolynomialError): + + def __init__(self, f, g, dom=None): + self.f, self.g, self.dom = f, g, dom + + def __str__(self): # pragma: no cover + from sympy.printing.str import sstr + + if self.dom is None: + return "%s does not divide %s" % (sstr(self.g), sstr(self.f)) + else: + return "%s does not divide %s in %s" % (sstr(self.g), sstr(self.f), sstr(self.dom)) + + def new(self, f, g): + return self.__class__(f, g, self.dom) + +@public +class PolynomialDivisionFailed(BasePolynomialError): + + def __init__(self, f, g, domain): + self.f = f + self.g = g + self.domain = domain + + def __str__(self): + if self.domain.is_EX: + msg = "You may want to use a different simplification algorithm. Note " \ + "that in general it's not possible to guarantee to detect zero " \ + "in this domain." + elif not self.domain.is_Exact: + msg = "Your working precision or tolerance of computations may be set " \ + "improperly. Adjust those parameters of the coefficient domain " \ + "and try again." + else: + msg = "Zero detection is guaranteed in this coefficient domain. This " \ + "may indicate a bug in SymPy or the domain is user defined and " \ + "doesn't implement zero detection properly." + + return "couldn't reduce degree in a polynomial division algorithm when " \ + "dividing %s by %s. This can happen when it's not possible to " \ + "detect zero in the coefficient domain. The domain of computation " \ + "is %s. %s" % (self.f, self.g, self.domain, msg) + +@public +class OperationNotSupported(BasePolynomialError): + + def __init__(self, poly, func): + self.poly = poly + self.func = func + + def __str__(self): # pragma: no cover + return "`%s` operation not supported by %s representation" % (self.func, self.poly.rep.__class__.__name__) + +@public +class HeuristicGCDFailed(BasePolynomialError): + pass + +class ModularGCDFailed(BasePolynomialError): + pass + +@public +class HomomorphismFailed(BasePolynomialError): + pass + +@public +class IsomorphismFailed(BasePolynomialError): + pass + +@public +class ExtraneousFactors(BasePolynomialError): + pass + +@public +class EvaluationFailed(BasePolynomialError): + pass + +@public +class RefinementFailed(BasePolynomialError): + pass + +@public +class CoercionFailed(BasePolynomialError): + pass + +@public +class NotInvertible(BasePolynomialError): + pass + +@public +class NotReversible(BasePolynomialError): + pass + +@public +class NotAlgebraic(BasePolynomialError): + pass + +@public +class DomainError(BasePolynomialError): + pass + +@public +class PolynomialError(BasePolynomialError): + pass + +@public +class UnificationFailed(BasePolynomialError): + pass + +@public +class UnsolvableFactorError(BasePolynomialError): + """Raised if ``roots`` is called with strict=True and a polynomial + having a factor whose solutions are not expressible in radicals + is encountered.""" + +@public +class GeneratorsError(BasePolynomialError): + pass + +@public +class GeneratorsNeeded(GeneratorsError): + pass + +@public +class ComputationFailed(BasePolynomialError): + + def __init__(self, func, nargs, exc): + self.func = func + self.nargs = nargs + self.exc = exc + + def __str__(self): + return "%s(%s) failed without generators" % (self.func, ', '.join(map(str, self.exc.exprs[:self.nargs]))) + +@public +class UnivariatePolynomialError(PolynomialError): + pass + +@public +class MultivariatePolynomialError(PolynomialError): + pass + +@public +class PolificationFailed(PolynomialError): + + def __init__(self, opt, origs, exprs, seq=False): + if not seq: + self.orig = origs + self.expr = exprs + self.origs = [origs] + self.exprs = [exprs] + else: + self.origs = origs + self.exprs = exprs + + self.opt = opt + self.seq = seq + + def __str__(self): # pragma: no cover + if not self.seq: + return "Cannot construct a polynomial from %s" % str(self.orig) + else: + return "Cannot construct polynomials from %s" % ', '.join(map(str, self.origs)) + +@public +class OptionError(BasePolynomialError): + pass + +@public +class FlagError(OptionError): + pass diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/polyfuncs.py b/.venv/lib/python3.12/site-packages/sympy/polys/polyfuncs.py new file mode 100644 index 0000000000000000000000000000000000000000..b412123f7383c68177a88df8817e921d96f6d5af --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/polyfuncs.py @@ -0,0 +1,321 @@ +"""High-level polynomials manipulation functions. """ + + +from sympy.core import S, Basic, symbols, Dummy +from sympy.polys.polyerrors import ( + PolificationFailed, ComputationFailed, + MultivariatePolynomialError, OptionError) +from sympy.polys.polyoptions import allowed_flags, build_options +from sympy.polys.polytools import poly_from_expr, Poly +from sympy.polys.specialpolys import ( + symmetric_poly, interpolating_poly) +from sympy.polys.rings import sring +from sympy.utilities import numbered_symbols, take, public + +@public +def symmetrize(F, *gens, **args): + r""" + Rewrite a polynomial in terms of elementary symmetric polynomials. + + A symmetric polynomial is a multivariate polynomial that remains invariant + under any variable permutation, i.e., if `f = f(x_1, x_2, \dots, x_n)`, + then `f = f(x_{i_1}, x_{i_2}, \dots, x_{i_n})`, where + `(i_1, i_2, \dots, i_n)` is a permutation of `(1, 2, \dots, n)` (an + element of the group `S_n`). + + Returns a tuple of symmetric polynomials ``(f1, f2, ..., fn)`` such that + ``f = f1 + f2 + ... + fn``. + + Examples + ======== + + >>> from sympy.polys.polyfuncs import symmetrize + >>> from sympy.abc import x, y + + >>> symmetrize(x**2 + y**2) + (-2*x*y + (x + y)**2, 0) + + >>> symmetrize(x**2 + y**2, formal=True) + (s1**2 - 2*s2, 0, [(s1, x + y), (s2, x*y)]) + + >>> symmetrize(x**2 - y**2) + (-2*x*y + (x + y)**2, -2*y**2) + + >>> symmetrize(x**2 - y**2, formal=True) + (s1**2 - 2*s2, -2*y**2, [(s1, x + y), (s2, x*y)]) + + """ + allowed_flags(args, ['formal', 'symbols']) + + iterable = True + + if not hasattr(F, '__iter__'): + iterable = False + F = [F] + + R, F = sring(F, *gens, **args) + gens = R.symbols + + opt = build_options(gens, args) + symbols = opt.symbols + symbols = [next(symbols) for i in range(len(gens))] + + result = [] + + for f in F: + p, r, m = f.symmetrize() + result.append((p.as_expr(*symbols), r.as_expr(*gens))) + + polys = [(s, g.as_expr()) for s, (_, g) in zip(symbols, m)] + + if not opt.formal: + for i, (sym, non_sym) in enumerate(result): + result[i] = (sym.subs(polys), non_sym) + + if not iterable: + result, = result + + if not opt.formal: + return result + else: + if iterable: + return result, polys + else: + return result + (polys,) + + +@public +def horner(f, *gens, **args): + """ + Rewrite a polynomial in Horner form. + + Among other applications, evaluation of a polynomial at a point is optimal + when it is applied using the Horner scheme ([1]). + + Examples + ======== + + >>> from sympy.polys.polyfuncs import horner + >>> from sympy.abc import x, y, a, b, c, d, e + + >>> horner(9*x**4 + 8*x**3 + 7*x**2 + 6*x + 5) + x*(x*(x*(9*x + 8) + 7) + 6) + 5 + + >>> horner(a*x**4 + b*x**3 + c*x**2 + d*x + e) + e + x*(d + x*(c + x*(a*x + b))) + + >>> f = 4*x**2*y**2 + 2*x**2*y + 2*x*y**2 + x*y + + >>> horner(f, wrt=x) + x*(x*y*(4*y + 2) + y*(2*y + 1)) + + >>> horner(f, wrt=y) + y*(x*y*(4*x + 2) + x*(2*x + 1)) + + References + ========== + [1] - https://en.wikipedia.org/wiki/Horner_scheme + + """ + allowed_flags(args, []) + + try: + F, opt = poly_from_expr(f, *gens, **args) + except PolificationFailed as exc: + return exc.expr + + form, gen = S.Zero, F.gen + + if F.is_univariate: + for coeff in F.all_coeffs(): + form = form*gen + coeff + else: + F, gens = Poly(F, gen), gens[1:] + + for coeff in F.all_coeffs(): + form = form*gen + horner(coeff, *gens, **args) + + return form + + +@public +def interpolate(data, x): + """ + Construct an interpolating polynomial for the data points + evaluated at point x (which can be symbolic or numeric). + + Examples + ======== + + >>> from sympy.polys.polyfuncs import interpolate + >>> from sympy.abc import a, b, x + + A list is interpreted as though it were paired with a range starting + from 1: + + >>> interpolate([1, 4, 9, 16], x) + x**2 + + This can be made explicit by giving a list of coordinates: + + >>> interpolate([(1, 1), (2, 4), (3, 9)], x) + x**2 + + The (x, y) coordinates can also be given as keys and values of a + dictionary (and the points need not be equispaced): + + >>> interpolate([(-1, 2), (1, 2), (2, 5)], x) + x**2 + 1 + >>> interpolate({-1: 2, 1: 2, 2: 5}, x) + x**2 + 1 + + If the interpolation is going to be used only once then the + value of interest can be passed instead of passing a symbol: + + >>> interpolate([1, 4, 9], 5) + 25 + + Symbolic coordinates are also supported: + + >>> [(i,interpolate((a, b), i)) for i in range(1, 4)] + [(1, a), (2, b), (3, -a + 2*b)] + """ + n = len(data) + + if isinstance(data, dict): + if x in data: + return S(data[x]) + X, Y = list(zip(*data.items())) + else: + if isinstance(data[0], tuple): + X, Y = list(zip(*data)) + if x in X: + return S(Y[X.index(x)]) + else: + if x in range(1, n + 1): + return S(data[x - 1]) + Y = list(data) + X = list(range(1, n + 1)) + + try: + return interpolating_poly(n, x, X, Y).expand() + except ValueError: + d = Dummy() + return interpolating_poly(n, d, X, Y).expand().subs(d, x) + + +@public +def rational_interpolate(data, degnum, X=symbols('x')): + """ + Returns a rational interpolation, where the data points are element of + any integral domain. + + The first argument contains the data (as a list of coordinates). The + ``degnum`` argument is the degree in the numerator of the rational + function. Setting it too high will decrease the maximal degree in the + denominator for the same amount of data. + + Examples + ======== + + >>> from sympy.polys.polyfuncs import rational_interpolate + + >>> data = [(1, -210), (2, -35), (3, 105), (4, 231), (5, 350), (6, 465)] + >>> rational_interpolate(data, 2) + (105*x**2 - 525)/(x + 1) + + Values do not need to be integers: + + >>> from sympy import sympify + >>> x = [1, 2, 3, 4, 5, 6] + >>> y = sympify("[-1, 0, 2, 22/5, 7, 68/7]") + >>> rational_interpolate(zip(x, y), 2) + (3*x**2 - 7*x + 2)/(x + 1) + + The symbol for the variable can be changed if needed: + >>> from sympy import symbols + >>> z = symbols('z') + >>> rational_interpolate(data, 2, X=z) + (105*z**2 - 525)/(z + 1) + + References + ========== + + .. [1] Algorithm is adapted from: + http://axiom-wiki.newsynthesis.org/RationalInterpolation + + """ + from sympy.matrices.dense import ones + + xdata, ydata = list(zip(*data)) + + k = len(xdata) - degnum - 1 + if k < 0: + raise OptionError("Too few values for the required degree.") + c = ones(degnum + k + 1, degnum + k + 2) + for j in range(max(degnum, k)): + for i in range(degnum + k + 1): + c[i, j + 1] = c[i, j]*xdata[i] + for j in range(k + 1): + for i in range(degnum + k + 1): + c[i, degnum + k + 1 - j] = -c[i, k - j]*ydata[i] + r = c.nullspace()[0] + return (sum(r[i] * X**i for i in range(degnum + 1)) + / sum(r[i + degnum + 1] * X**i for i in range(k + 1))) + + +@public +def viete(f, roots=None, *gens, **args): + """ + Generate Viete's formulas for ``f``. + + Examples + ======== + + >>> from sympy.polys.polyfuncs import viete + >>> from sympy import symbols + + >>> x, a, b, c, r1, r2 = symbols('x,a:c,r1:3') + + >>> viete(a*x**2 + b*x + c, [r1, r2], x) + [(r1 + r2, -b/a), (r1*r2, c/a)] + + """ + allowed_flags(args, []) + + if isinstance(roots, Basic): + gens, roots = (roots,) + gens, None + + try: + f, opt = poly_from_expr(f, *gens, **args) + except PolificationFailed as exc: + raise ComputationFailed('viete', 1, exc) + + if f.is_multivariate: + raise MultivariatePolynomialError( + "multivariate polynomials are not allowed") + + n = f.degree() + + if n < 1: + raise ValueError( + "Cannot derive Viete's formulas for a constant polynomial") + + if roots is None: + roots = numbered_symbols('r', start=1) + + roots = take(roots, n) + + if n != len(roots): + raise ValueError("required %s roots, got %s" % (n, len(roots))) + + lc, coeffs = f.LC(), f.all_coeffs() + result, sign = [], -1 + + for i, coeff in enumerate(coeffs[1:]): + poly = symmetric_poly(i + 1, roots) + coeff = sign*(coeff/lc) + result.append((poly, coeff)) + sign = -sign + + return result diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/polymatrix.py b/.venv/lib/python3.12/site-packages/sympy/polys/polymatrix.py new file mode 100644 index 0000000000000000000000000000000000000000..fb2a58efc3ebfd85507ac2b0cfd31230e55ded66 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/polymatrix.py @@ -0,0 +1,292 @@ +from sympy.core.expr import Expr +from sympy.core.symbol import Dummy +from sympy.core.sympify import _sympify + +from sympy.polys.polyerrors import CoercionFailed +from sympy.polys.polytools import Poly, parallel_poly_from_expr +from sympy.polys.domains import QQ + +from sympy.polys.matrices import DomainMatrix +from sympy.polys.matrices.domainscalar import DomainScalar + + +class MutablePolyDenseMatrix: + """ + A mutable matrix of objects from poly module or to operate with them. + + Examples + ======== + + >>> from sympy.polys.polymatrix import PolyMatrix + >>> from sympy import Symbol, Poly + >>> x = Symbol('x') + >>> pm1 = PolyMatrix([[Poly(x**2, x), Poly(-x, x)], [Poly(x**3, x), Poly(-1 + x, x)]]) + >>> v1 = PolyMatrix([[1, 0], [-1, 0]], x) + >>> pm1*v1 + PolyMatrix([ + [ x**2 + x, 0], + [x**3 - x + 1, 0]], ring=QQ[x]) + + >>> pm1.ring + ZZ[x] + + >>> v1*pm1 + PolyMatrix([ + [ x**2, -x], + [-x**2, x]], ring=QQ[x]) + + >>> pm2 = PolyMatrix([[Poly(x**2, x, domain='QQ'), Poly(0, x, domain='QQ'), Poly(1, x, domain='QQ'), \ + Poly(x**3, x, domain='QQ'), Poly(0, x, domain='QQ'), Poly(-x**3, x, domain='QQ')]]) + >>> v2 = PolyMatrix([1, 0, 0, 0, 0, 0], x) + >>> v2.ring + QQ[x] + >>> pm2*v2 + PolyMatrix([[x**2]], ring=QQ[x]) + + """ + + def __new__(cls, *args, ring=None): + + if not args: + # PolyMatrix(ring=QQ[x]) + if ring is None: + raise TypeError("The ring needs to be specified for an empty PolyMatrix") + rows, cols, items, gens = 0, 0, [], () + elif isinstance(args[0], list): + elements, gens = args[0], args[1:] + if not elements: + # PolyMatrix([]) + rows, cols, items = 0, 0, [] + elif isinstance(elements[0], (list, tuple)): + # PolyMatrix([[1, 2]], x) + rows, cols = len(elements), len(elements[0]) + items = [e for row in elements for e in row] + else: + # PolyMatrix([1, 2], x) + rows, cols = len(elements), 1 + items = elements + elif [type(a) for a in args[:3]] == [int, int, list]: + # PolyMatrix(2, 2, [1, 2, 3, 4], x) + rows, cols, items, gens = args[0], args[1], args[2], args[3:] + elif [type(a) for a in args[:3]] == [int, int, type(lambda: 0)]: + # PolyMatrix(2, 2, lambda i, j: i+j, x) + rows, cols, func, gens = args[0], args[1], args[2], args[3:] + items = [func(i, j) for i in range(rows) for j in range(cols)] + else: + raise TypeError("Invalid arguments") + + # PolyMatrix([[1]], x, y) vs PolyMatrix([[1]], (x, y)) + if len(gens) == 1 and isinstance(gens[0], tuple): + gens = gens[0] + # gens is now a tuple (x, y) + + return cls.from_list(rows, cols, items, gens, ring) + + @classmethod + def from_list(cls, rows, cols, items, gens, ring): + + # items can be Expr, Poly, or a mix of Expr and Poly + items = [_sympify(item) for item in items] + if items and all(isinstance(item, Poly) for item in items): + polys = True + else: + polys = False + + # Identify the ring for the polys + if ring is not None: + # Parse a domain string like 'QQ[x]' + if isinstance(ring, str): + ring = Poly(0, Dummy(), domain=ring).domain + elif polys: + p = items[0] + for p2 in items[1:]: + p, _ = p.unify(p2) + ring = p.domain[p.gens] + else: + items, info = parallel_poly_from_expr(items, gens, field=True) + ring = info['domain'][info['gens']] + polys = True + + # Efficiently convert when all elements are Poly + if polys: + p_ring = Poly(0, ring.symbols, domain=ring.domain) + to_ring = ring.ring.from_list + convert_poly = lambda p: to_ring(p.unify(p_ring)[0].rep.to_list()) + elements = [convert_poly(p) for p in items] + else: + convert_expr = ring.from_sympy + elements = [convert_expr(e.as_expr()) for e in items] + + # Convert to domain elements and construct DomainMatrix + elements_lol = [[elements[i*cols + j] for j in range(cols)] for i in range(rows)] + dm = DomainMatrix(elements_lol, (rows, cols), ring) + return cls.from_dm(dm) + + @classmethod + def from_dm(cls, dm): + obj = super().__new__(cls) + dm = dm.to_sparse() + R = dm.domain + obj._dm = dm + obj.ring = R + obj.domain = R.domain + obj.gens = R.symbols + return obj + + def to_Matrix(self): + return self._dm.to_Matrix() + + @classmethod + def from_Matrix(cls, other, *gens, ring=None): + return cls(*other.shape, other.flat(), *gens, ring=ring) + + def set_gens(self, gens): + return self.from_Matrix(self.to_Matrix(), gens) + + def __repr__(self): + if self.rows * self.cols: + return 'Poly' + repr(self.to_Matrix())[:-1] + f', ring={self.ring})' + else: + return f'PolyMatrix({self.rows}, {self.cols}, [], ring={self.ring})' + + @property + def shape(self): + return self._dm.shape + + @property + def rows(self): + return self.shape[0] + + @property + def cols(self): + return self.shape[1] + + def __len__(self): + return self.rows * self.cols + + def __getitem__(self, key): + + def to_poly(v): + ground = self._dm.domain.domain + gens = self._dm.domain.symbols + return Poly(v.to_dict(), gens, domain=ground) + + dm = self._dm + + if isinstance(key, slice): + items = dm.flat()[key] + return [to_poly(item) for item in items] + elif isinstance(key, int): + i, j = divmod(key, self.cols) + e = dm[i,j] + return to_poly(e.element) + + i, j = key + if isinstance(i, int) and isinstance(j, int): + return to_poly(dm[i, j].element) + else: + return self.from_dm(dm[i, j]) + + def __eq__(self, other): + if not isinstance(self, type(other)): + return NotImplemented + return self._dm == other._dm + + def __add__(self, other): + if isinstance(other, type(self)): + return self.from_dm(self._dm + other._dm) + return NotImplemented + + def __sub__(self, other): + if isinstance(other, type(self)): + return self.from_dm(self._dm - other._dm) + return NotImplemented + + def __mul__(self, other): + if isinstance(other, type(self)): + return self.from_dm(self._dm * other._dm) + elif isinstance(other, int): + other = _sympify(other) + if isinstance(other, Expr): + Kx = self.ring + try: + other_ds = DomainScalar(Kx.from_sympy(other), Kx) + except (CoercionFailed, ValueError): + other_ds = DomainScalar.from_sympy(other) + return self.from_dm(self._dm * other_ds) + return NotImplemented + + def __rmul__(self, other): + if isinstance(other, int): + other = _sympify(other) + if isinstance(other, Expr): + other_ds = DomainScalar.from_sympy(other) + return self.from_dm(other_ds * self._dm) + return NotImplemented + + def __truediv__(self, other): + + if isinstance(other, Poly): + other = other.as_expr() + elif isinstance(other, int): + other = _sympify(other) + if not isinstance(other, Expr): + return NotImplemented + + other = self.domain.from_sympy(other) + inverse = self.ring.convert_from(1/other, self.domain) + inverse = DomainScalar(inverse, self.ring) + dm = self._dm * inverse + return self.from_dm(dm) + + def __neg__(self): + return self.from_dm(-self._dm) + + def transpose(self): + return self.from_dm(self._dm.transpose()) + + def row_join(self, other): + dm = DomainMatrix.hstack(self._dm, other._dm) + return self.from_dm(dm) + + def col_join(self, other): + dm = DomainMatrix.vstack(self._dm, other._dm) + return self.from_dm(dm) + + def applyfunc(self, func): + M = self.to_Matrix().applyfunc(func) + return self.from_Matrix(M, self.gens) + + @classmethod + def eye(cls, n, gens): + return cls.from_dm(DomainMatrix.eye(n, QQ[gens])) + + @classmethod + def zeros(cls, m, n, gens): + return cls.from_dm(DomainMatrix.zeros((m, n), QQ[gens])) + + def rref(self, simplify='ignore', normalize_last='ignore'): + # If this is K[x] then computes RREF in ground field K. + if not (self.domain.is_Field and all(p.is_ground for p in self)): + raise ValueError("PolyMatrix rref is only for ground field elements") + dm = self._dm + dm_ground = dm.convert_to(dm.domain.domain) + dm_rref, pivots = dm_ground.rref() + dm_rref = dm_rref.convert_to(dm.domain) + return self.from_dm(dm_rref), pivots + + def nullspace(self): + # If this is K[x] then computes nullspace in ground field K. + if not (self.domain.is_Field and all(p.is_ground for p in self)): + raise ValueError("PolyMatrix nullspace is only for ground field elements") + dm = self._dm + K, Kx = self.domain, self.ring + dm_null_rows = dm.convert_to(K).nullspace(divide_last=True).convert_to(Kx) + dm_null = dm_null_rows.transpose() + dm_basis = [dm_null[:,i] for i in range(dm_null.shape[1])] + return [self.from_dm(dmvec) for dmvec in dm_basis] + + def rank(self): + return self.cols - len(self.nullspace()) + +MutablePolyMatrix = PolyMatrix = MutablePolyDenseMatrix diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/polyoptions.py b/.venv/lib/python3.12/site-packages/sympy/polys/polyoptions.py new file mode 100644 index 0000000000000000000000000000000000000000..7b9bd989c4d5676aab32e65c62996137b3e0b73e --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/polyoptions.py @@ -0,0 +1,791 @@ +"""Options manager for :class:`~.Poly` and public API functions. """ + +from __future__ import annotations + +__all__ = ["Options"] + +from sympy.core.basic import Basic +from sympy.core.expr import Expr +from sympy.core.sympify import sympify +from sympy.polys.polyerrors import GeneratorsError, OptionError, FlagError +from sympy.utilities import numbered_symbols, topological_sort, public +from sympy.utilities.iterables import has_dups, is_sequence + +import sympy.polys + +import re + +class Option: + """Base class for all kinds of options. """ + + option: str | None = None + + is_Flag = False + + requires: list[str] = [] + excludes: list[str] = [] + + after: list[str] = [] + before: list[str] = [] + + @classmethod + def default(cls): + return None + + @classmethod + def preprocess(cls, option): + return None + + @classmethod + def postprocess(cls, options): + pass + + +class Flag(Option): + """Base class for all kinds of flags. """ + + is_Flag = True + + +class BooleanOption(Option): + """An option that must have a boolean value or equivalent assigned. """ + + @classmethod + def preprocess(cls, value): + if value in [True, False]: + return bool(value) + else: + raise OptionError("'%s' must have a boolean value assigned, got %s" % (cls.option, value)) + + +class OptionType(type): + """Base type for all options that does registers options. """ + + def __init__(cls, *args, **kwargs): + @property + def getter(self): + try: + return self[cls.option] + except KeyError: + return cls.default() + + setattr(Options, cls.option, getter) + Options.__options__[cls.option] = cls + + +@public +class Options(dict): + """ + Options manager for polynomial manipulation module. + + Examples + ======== + + >>> from sympy.polys.polyoptions import Options + >>> from sympy.polys.polyoptions import build_options + + >>> from sympy.abc import x, y, z + + >>> Options((x, y, z), {'domain': 'ZZ'}) + {'auto': False, 'domain': ZZ, 'gens': (x, y, z)} + + >>> build_options((x, y, z), {'domain': 'ZZ'}) + {'auto': False, 'domain': ZZ, 'gens': (x, y, z)} + + **Options** + + * Expand --- boolean option + * Gens --- option + * Wrt --- option + * Sort --- option + * Order --- option + * Field --- boolean option + * Greedy --- boolean option + * Domain --- option + * Split --- boolean option + * Gaussian --- boolean option + * Extension --- option + * Modulus --- option + * Symmetric --- boolean option + * Strict --- boolean option + + **Flags** + + * Auto --- boolean flag + * Frac --- boolean flag + * Formal --- boolean flag + * Polys --- boolean flag + * Include --- boolean flag + * All --- boolean flag + * Gen --- flag + * Series --- boolean flag + + """ + + __order__ = None + __options__: dict[str, type[Option]] = {} + + gens: tuple[Expr, ...] + domain: sympy.polys.domains.Domain + + def __init__(self, gens, args, flags=None, strict=False): + dict.__init__(self) + + if gens and args.get('gens', ()): + raise OptionError( + "both '*gens' and keyword argument 'gens' supplied") + elif gens: + args = dict(args) + args['gens'] = gens + + defaults = args.pop('defaults', {}) + + def preprocess_options(args): + for option, value in args.items(): + try: + cls = self.__options__[option] + except KeyError: + raise OptionError("'%s' is not a valid option" % option) + + if issubclass(cls, Flag): + if flags is None or option not in flags: + if strict: + raise OptionError("'%s' flag is not allowed in this context" % option) + + if value is not None: + self[option] = cls.preprocess(value) + + preprocess_options(args) + + for key in dict(defaults): + if key in self: + del defaults[key] + else: + for option in self.keys(): + cls = self.__options__[option] + + if key in cls.excludes: + del defaults[key] + break + + preprocess_options(defaults) + + for option in self.keys(): + cls = self.__options__[option] + + for require_option in cls.requires: + if self.get(require_option) is None: + raise OptionError("'%s' option is only allowed together with '%s'" % (option, require_option)) + + for exclude_option in cls.excludes: + if self.get(exclude_option) is not None: + raise OptionError("'%s' option is not allowed together with '%s'" % (option, exclude_option)) + + for option in self.__order__: + self.__options__[option].postprocess(self) + + @classmethod + def _init_dependencies_order(cls): + """Resolve the order of options' processing. """ + if cls.__order__ is None: + vertices, edges = [], set() + + for name, option in cls.__options__.items(): + vertices.append(name) + + edges.update((_name, name) for _name in option.after) + + edges.update((name, _name) for _name in option.before) + + try: + cls.__order__ = topological_sort((vertices, list(edges))) + except ValueError: + raise RuntimeError( + "cycle detected in sympy.polys options framework") + + def clone(self, updates={}): + """Clone ``self`` and update specified options. """ + obj = dict.__new__(self.__class__) + + for option, value in self.items(): + obj[option] = value + + for option, value in updates.items(): + obj[option] = value + + return obj + + def __setattr__(self, attr, value): + if attr in self.__options__: + self[attr] = value + else: + super().__setattr__(attr, value) + + @property + def args(self): + args = {} + + for option, value in self.items(): + if value is not None and option != 'gens': + cls = self.__options__[option] + + if not issubclass(cls, Flag): + args[option] = value + + return args + + @property + def options(self): + options = {} + + for option, cls in self.__options__.items(): + if not issubclass(cls, Flag): + options[option] = getattr(self, option) + + return options + + @property + def flags(self): + flags = {} + + for option, cls in self.__options__.items(): + if issubclass(cls, Flag): + flags[option] = getattr(self, option) + + return flags + + +class Expand(BooleanOption, metaclass=OptionType): + """``expand`` option to polynomial manipulation functions. """ + + option = 'expand' + + requires: list[str] = [] + excludes: list[str] = [] + + @classmethod + def default(cls): + return True + + +class Gens(Option, metaclass=OptionType): + """``gens`` option to polynomial manipulation functions. """ + + option = 'gens' + + requires: list[str] = [] + excludes: list[str] = [] + + @classmethod + def default(cls): + return () + + @classmethod + def preprocess(cls, gens): + if isinstance(gens, Basic): + gens = (gens,) + elif len(gens) == 1 and is_sequence(gens[0]): + gens = gens[0] + + if gens == (None,): + gens = () + elif has_dups(gens): + raise GeneratorsError("duplicated generators: %s" % str(gens)) + elif any(gen.is_commutative is False for gen in gens): + raise GeneratorsError("non-commutative generators: %s" % str(gens)) + + return tuple(gens) + + +class Wrt(Option, metaclass=OptionType): + """``wrt`` option to polynomial manipulation functions. """ + + option = 'wrt' + + requires: list[str] = [] + excludes: list[str] = [] + + _re_split = re.compile(r"\s*,\s*|\s+") + + @classmethod + def preprocess(cls, wrt): + if isinstance(wrt, Basic): + return [str(wrt)] + elif isinstance(wrt, str): + wrt = wrt.strip() + if wrt.endswith(','): + raise OptionError('Bad input: missing parameter.') + if not wrt: + return [] + return list(cls._re_split.split(wrt)) + elif hasattr(wrt, '__getitem__'): + return list(map(str, wrt)) + else: + raise OptionError("invalid argument for 'wrt' option") + + +class Sort(Option, metaclass=OptionType): + """``sort`` option to polynomial manipulation functions. """ + + option = 'sort' + + requires: list[str] = [] + excludes: list[str] = [] + + @classmethod + def default(cls): + return [] + + @classmethod + def preprocess(cls, sort): + if isinstance(sort, str): + return [ gen.strip() for gen in sort.split('>') ] + elif hasattr(sort, '__getitem__'): + return list(map(str, sort)) + else: + raise OptionError("invalid argument for 'sort' option") + + +class Order(Option, metaclass=OptionType): + """``order`` option to polynomial manipulation functions. """ + + option = 'order' + + requires: list[str] = [] + excludes: list[str] = [] + + @classmethod + def default(cls): + return sympy.polys.orderings.lex + + @classmethod + def preprocess(cls, order): + return sympy.polys.orderings.monomial_key(order) + + +class Field(BooleanOption, metaclass=OptionType): + """``field`` option to polynomial manipulation functions. """ + + option = 'field' + + requires: list[str] = [] + excludes = ['domain', 'split', 'gaussian'] + + +class Greedy(BooleanOption, metaclass=OptionType): + """``greedy`` option to polynomial manipulation functions. """ + + option = 'greedy' + + requires: list[str] = [] + excludes = ['domain', 'split', 'gaussian', 'extension', 'modulus', 'symmetric'] + + +class Composite(BooleanOption, metaclass=OptionType): + """``composite`` option to polynomial manipulation functions. """ + + option = 'composite' + + @classmethod + def default(cls): + return None + + requires: list[str] = [] + excludes = ['domain', 'split', 'gaussian', 'extension', 'modulus', 'symmetric'] + + +class Domain(Option, metaclass=OptionType): + """``domain`` option to polynomial manipulation functions. """ + + option = 'domain' + + requires: list[str] = [] + excludes = ['field', 'greedy', 'split', 'gaussian', 'extension'] + + after = ['gens'] + + _re_realfield = re.compile(r"^(R|RR)(_(\d+))?$") + _re_complexfield = re.compile(r"^(C|CC)(_(\d+))?$") + _re_finitefield = re.compile(r"^(FF|GF)\((\d+)\)$") + _re_polynomial = re.compile(r"^(Z|ZZ|Q|QQ|ZZ_I|QQ_I|R|RR|C|CC)\[(.+)\]$") + _re_fraction = re.compile(r"^(Z|ZZ|Q|QQ)\((.+)\)$") + _re_algebraic = re.compile(r"^(Q|QQ)\<(.+)\>$") + + @classmethod + def preprocess(cls, domain): + if isinstance(domain, sympy.polys.domains.Domain): + return domain + elif hasattr(domain, 'to_domain'): + return domain.to_domain() + elif isinstance(domain, str): + if domain in ['Z', 'ZZ']: + return sympy.polys.domains.ZZ + + if domain in ['Q', 'QQ']: + return sympy.polys.domains.QQ + + if domain == 'ZZ_I': + return sympy.polys.domains.ZZ_I + + if domain == 'QQ_I': + return sympy.polys.domains.QQ_I + + if domain == 'EX': + return sympy.polys.domains.EX + + r = cls._re_realfield.match(domain) + + if r is not None: + _, _, prec = r.groups() + + if prec is None: + return sympy.polys.domains.RR + else: + return sympy.polys.domains.RealField(int(prec)) + + r = cls._re_complexfield.match(domain) + + if r is not None: + _, _, prec = r.groups() + + if prec is None: + return sympy.polys.domains.CC + else: + return sympy.polys.domains.ComplexField(int(prec)) + + r = cls._re_finitefield.match(domain) + + if r is not None: + return sympy.polys.domains.FF(int(r.groups()[1])) + + r = cls._re_polynomial.match(domain) + + if r is not None: + ground, gens = r.groups() + + gens = list(map(sympify, gens.split(','))) + + if ground in ['Z', 'ZZ']: + return sympy.polys.domains.ZZ.poly_ring(*gens) + elif ground in ['Q', 'QQ']: + return sympy.polys.domains.QQ.poly_ring(*gens) + elif ground in ['R', 'RR']: + return sympy.polys.domains.RR.poly_ring(*gens) + elif ground == 'ZZ_I': + return sympy.polys.domains.ZZ_I.poly_ring(*gens) + elif ground == 'QQ_I': + return sympy.polys.domains.QQ_I.poly_ring(*gens) + else: + return sympy.polys.domains.CC.poly_ring(*gens) + + r = cls._re_fraction.match(domain) + + if r is not None: + ground, gens = r.groups() + + gens = list(map(sympify, gens.split(','))) + + if ground in ['Z', 'ZZ']: + return sympy.polys.domains.ZZ.frac_field(*gens) + else: + return sympy.polys.domains.QQ.frac_field(*gens) + + r = cls._re_algebraic.match(domain) + + if r is not None: + gens = list(map(sympify, r.groups()[1].split(','))) + return sympy.polys.domains.QQ.algebraic_field(*gens) + + raise OptionError('expected a valid domain specification, got %s' % domain) + + @classmethod + def postprocess(cls, options): + if 'gens' in options and 'domain' in options and options['domain'].is_Composite and \ + (set(options['domain'].symbols) & set(options['gens'])): + raise GeneratorsError( + "ground domain and generators interfere together") + elif ('gens' not in options or not options['gens']) and \ + 'domain' in options and options['domain'] == sympy.polys.domains.EX: + raise GeneratorsError("you have to provide generators because EX domain was requested") + + +class Split(BooleanOption, metaclass=OptionType): + """``split`` option to polynomial manipulation functions. """ + + option = 'split' + + requires: list[str] = [] + excludes = ['field', 'greedy', 'domain', 'gaussian', 'extension', + 'modulus', 'symmetric'] + + @classmethod + def postprocess(cls, options): + if 'split' in options: + raise NotImplementedError("'split' option is not implemented yet") + + +class Gaussian(BooleanOption, metaclass=OptionType): + """``gaussian`` option to polynomial manipulation functions. """ + + option = 'gaussian' + + requires: list[str] = [] + excludes = ['field', 'greedy', 'domain', 'split', 'extension', + 'modulus', 'symmetric'] + + @classmethod + def postprocess(cls, options): + if 'gaussian' in options and options['gaussian'] is True: + options['domain'] = sympy.polys.domains.QQ_I + Extension.postprocess(options) + + +class Extension(Option, metaclass=OptionType): + """``extension`` option to polynomial manipulation functions. """ + + option = 'extension' + + requires: list[str] = [] + excludes = ['greedy', 'domain', 'split', 'gaussian', 'modulus', + 'symmetric'] + + @classmethod + def preprocess(cls, extension): + if extension == 1: + return bool(extension) + elif extension == 0: + raise OptionError("'False' is an invalid argument for 'extension'") + else: + if not hasattr(extension, '__iter__'): + extension = {extension} + else: + if not extension: + extension = None + else: + extension = set(extension) + + return extension + + @classmethod + def postprocess(cls, options): + if 'extension' in options and options['extension'] is not True: + options['domain'] = sympy.polys.domains.QQ.algebraic_field( + *options['extension']) + + +class Modulus(Option, metaclass=OptionType): + """``modulus`` option to polynomial manipulation functions. """ + + option = 'modulus' + + requires: list[str] = [] + excludes = ['greedy', 'split', 'domain', 'gaussian', 'extension'] + + @classmethod + def preprocess(cls, modulus): + modulus = sympify(modulus) + + if modulus.is_Integer and modulus > 0: + return int(modulus) + else: + raise OptionError( + "'modulus' must a positive integer, got %s" % modulus) + + @classmethod + def postprocess(cls, options): + if 'modulus' in options: + modulus = options['modulus'] + symmetric = options.get('symmetric', True) + options['domain'] = sympy.polys.domains.FF(modulus, symmetric) + + +class Symmetric(BooleanOption, metaclass=OptionType): + """``symmetric`` option to polynomial manipulation functions. """ + + option = 'symmetric' + + requires = ['modulus'] + excludes = ['greedy', 'domain', 'split', 'gaussian', 'extension'] + + +class Strict(BooleanOption, metaclass=OptionType): + """``strict`` option to polynomial manipulation functions. """ + + option = 'strict' + + @classmethod + def default(cls): + return True + + +class Auto(BooleanOption, Flag, metaclass=OptionType): + """``auto`` flag to polynomial manipulation functions. """ + + option = 'auto' + + after = ['field', 'domain', 'extension', 'gaussian'] + + @classmethod + def default(cls): + return True + + @classmethod + def postprocess(cls, options): + if ('domain' in options or 'field' in options) and 'auto' not in options: + options['auto'] = False + + +class Frac(BooleanOption, Flag, metaclass=OptionType): + """``auto`` option to polynomial manipulation functions. """ + + option = 'frac' + + @classmethod + def default(cls): + return False + + +class Formal(BooleanOption, Flag, metaclass=OptionType): + """``formal`` flag to polynomial manipulation functions. """ + + option = 'formal' + + @classmethod + def default(cls): + return False + + +class Polys(BooleanOption, Flag, metaclass=OptionType): + """``polys`` flag to polynomial manipulation functions. """ + + option = 'polys' + + +class Include(BooleanOption, Flag, metaclass=OptionType): + """``include`` flag to polynomial manipulation functions. """ + + option = 'include' + + @classmethod + def default(cls): + return False + + +class All(BooleanOption, Flag, metaclass=OptionType): + """``all`` flag to polynomial manipulation functions. """ + + option = 'all' + + @classmethod + def default(cls): + return False + + +class Gen(Flag, metaclass=OptionType): + """``gen`` flag to polynomial manipulation functions. """ + + option = 'gen' + + @classmethod + def default(cls): + return 0 + + @classmethod + def preprocess(cls, gen): + if isinstance(gen, (Basic, int)): + return gen + else: + raise OptionError("invalid argument for 'gen' option") + + +class Series(BooleanOption, Flag, metaclass=OptionType): + """``series`` flag to polynomial manipulation functions. """ + + option = 'series' + + @classmethod + def default(cls): + return False + + +class Symbols(Flag, metaclass=OptionType): + """``symbols`` flag to polynomial manipulation functions. """ + + option = 'symbols' + + @classmethod + def default(cls): + return numbered_symbols('s', start=1) + + @classmethod + def preprocess(cls, symbols): + if hasattr(symbols, '__iter__'): + return iter(symbols) + else: + raise OptionError("expected an iterator or iterable container, got %s" % symbols) + + +class Method(Flag, metaclass=OptionType): + """``method`` flag to polynomial manipulation functions. """ + + option = 'method' + + @classmethod + def preprocess(cls, method): + if isinstance(method, str): + return method.lower() + else: + raise OptionError("expected a string, got %s" % method) + + +def build_options(gens, args=None): + """Construct options from keyword arguments or ... options. """ + if args is None: + gens, args = (), gens + + if len(args) != 1 or 'opt' not in args or gens: + return Options(gens, args) + else: + return args['opt'] + + +def allowed_flags(args, flags): + """ + Allow specified flags to be used in the given context. + + Examples + ======== + + >>> from sympy.polys.polyoptions import allowed_flags + >>> from sympy.polys.domains import ZZ + + >>> allowed_flags({'domain': ZZ}, []) + + >>> allowed_flags({'domain': ZZ, 'frac': True}, []) + Traceback (most recent call last): + ... + FlagError: 'frac' flag is not allowed in this context + + >>> allowed_flags({'domain': ZZ, 'frac': True}, ['frac']) + + """ + flags = set(flags) + + for arg in args.keys(): + try: + if Options.__options__[arg].is_Flag and arg not in flags: + raise FlagError( + "'%s' flag is not allowed in this context" % arg) + except KeyError: + raise OptionError("'%s' is not a valid option" % arg) + + +def set_defaults(options, **defaults): + """Update options with default values. """ + if 'defaults' not in options: + options = dict(options) + options['defaults'] = defaults + + return options + +Options._init_dependencies_order() diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/polyquinticconst.py b/.venv/lib/python3.12/site-packages/sympy/polys/polyquinticconst.py new file mode 100644 index 0000000000000000000000000000000000000000..3b17096fd2cf3b205c3b819eb11ffc2012ea125b --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/polyquinticconst.py @@ -0,0 +1,187 @@ +""" +Solving solvable quintics - An implementation of DS Dummit's paper + +Paper : +https://www.ams.org/journals/mcom/1991-57-195/S0025-5718-1991-1079014-X/S0025-5718-1991-1079014-X.pdf + +Mathematica notebook: +http://www.emba.uvm.edu/~ddummit/quintics/quintics.nb + +""" + + +from sympy.core import Symbol +from sympy.core.evalf import N +from sympy.core.numbers import I, Rational +from sympy.functions import sqrt +from sympy.polys.polytools import Poly +from sympy.utilities import public + +x = Symbol('x') + +@public +class PolyQuintic: + """Special functions for solvable quintics""" + def __init__(self, poly): + _, _, self.p, self.q, self.r, self.s = poly.all_coeffs() + self.zeta1 = Rational(-1, 4) + (sqrt(5)/4) + I*sqrt((sqrt(5)/8) + Rational(5, 8)) + self.zeta2 = (-sqrt(5)/4) - Rational(1, 4) + I*sqrt((-sqrt(5)/8) + Rational(5, 8)) + self.zeta3 = (-sqrt(5)/4) - Rational(1, 4) - I*sqrt((-sqrt(5)/8) + Rational(5, 8)) + self.zeta4 = Rational(-1, 4) + (sqrt(5)/4) - I*sqrt((sqrt(5)/8) + Rational(5, 8)) + + @property + def f20(self): + p, q, r, s = self.p, self.q, self.r, self.s + f20 = q**8 - 13*p*q**6*r + p**5*q**2*r**2 + 65*p**2*q**4*r**2 - 4*p**6*r**3 - 128*p**3*q**2*r**3 + 17*q**4*r**3 + 48*p**4*r**4 - 16*p*q**2*r**4 - 192*p**2*r**5 + 256*r**6 - 4*p**5*q**3*s - 12*p**2*q**5*s + 18*p**6*q*r*s + 12*p**3*q**3*r*s - 124*q**5*r*s + 196*p**4*q*r**2*s + 590*p*q**3*r**2*s - 160*p**2*q*r**3*s - 1600*q*r**4*s - 27*p**7*s**2 - 150*p**4*q**2*s**2 - 125*p*q**4*s**2 - 99*p**5*r*s**2 - 725*p**2*q**2*r*s**2 + 1200*p**3*r**2*s**2 + 3250*q**2*r**2*s**2 - 2000*p*r**3*s**2 - 1250*p*q*r*s**3 + 3125*p**2*s**4 - 9375*r*s**4-(2*p*q**6 - 19*p**2*q**4*r + 51*p**3*q**2*r**2 - 3*q**4*r**2 - 32*p**4*r**3 - 76*p*q**2*r**3 + 256*p**2*r**4 - 512*r**5 + 31*p**3*q**3*s + 58*q**5*s - 117*p**4*q*r*s - 105*p*q**3*r*s - 260*p**2*q*r**2*s + 2400*q*r**3*s + 108*p**5*s**2 + 325*p**2*q**2*s**2 - 525*p**3*r*s**2 - 2750*q**2*r*s**2 + 500*p*r**2*s**2 - 625*p*q*s**3 + 3125*s**4)*x+(p**2*q**4 - 6*p**3*q**2*r - 8*q**4*r + 9*p**4*r**2 + 76*p*q**2*r**2 - 136*p**2*r**3 + 400*r**4 - 50*p*q**3*s + 90*p**2*q*r*s - 1400*q*r**2*s + 625*q**2*s**2 + 500*p*r*s**2)*x**2-(2*q**4 - 21*p*q**2*r + 40*p**2*r**2 - 160*r**3 + 15*p**2*q*s + 400*q*r*s - 125*p*s**2)*x**3+(2*p*q**2 - 6*p**2*r + 40*r**2 - 50*q*s)*x**4 + 8*r*x**5 + x**6 + return Poly(f20, x) + + @property + def b(self): + p, q, r, s = self.p, self.q, self.r, self.s + b = ( [], [0,0,0,0,0,0], [0,0,0,0,0,0], [0,0,0,0,0,0], [0,0,0,0,0,0],) + + b[1][5] = 100*p**7*q**7 + 2175*p**4*q**9 + 10500*p*q**11 - 1100*p**8*q**5*r - 27975*p**5*q**7*r - 152950*p**2*q**9*r + 4125*p**9*q**3*r**2 + 128875*p**6*q**5*r**2 + 830525*p**3*q**7*r**2 - 59450*q**9*r**2 - 5400*p**10*q*r**3 - 243800*p**7*q**3*r**3 - 2082650*p**4*q**5*r**3 + 333925*p*q**7*r**3 + 139200*p**8*q*r**4 + 2406000*p**5*q**3*r**4 + 122600*p**2*q**5*r**4 - 1254400*p**6*q*r**5 - 3776000*p**3*q**3*r**5 - 1832000*q**5*r**5 + 4736000*p**4*q*r**6 + 6720000*p*q**3*r**6 - 6400000*p**2*q*r**7 + 900*p**9*q**4*s + 37400*p**6*q**6*s + 281625*p**3*q**8*s + 435000*q**10*s - 6750*p**10*q**2*r*s - 322300*p**7*q**4*r*s - 2718575*p**4*q**6*r*s - 4214250*p*q**8*r*s + 16200*p**11*r**2*s + 859275*p**8*q**2*r**2*s + 8925475*p**5*q**4*r**2*s + 14427875*p**2*q**6*r**2*s - 453600*p**9*r**3*s - 10038400*p**6*q**2*r**3*s - 17397500*p**3*q**4*r**3*s + 11333125*q**6*r**3*s + 4451200*p**7*r**4*s + 15850000*p**4*q**2*r**4*s - 34000000*p*q**4*r**4*s - 17984000*p**5*r**5*s + 10000000*p**2*q**2*r**5*s + 25600000*p**3*r**6*s + 8000000*q**2*r**6*s - 6075*p**11*q*s**2 + 83250*p**8*q**3*s**2 + 1282500*p**5*q**5*s**2 + 2862500*p**2*q**7*s**2 - 724275*p**9*q*r*s**2 - 9807250*p**6*q**3*r*s**2 - 28374375*p**3*q**5*r*s**2 - 22212500*q**7*r*s**2 + 8982000*p**7*q*r**2*s**2 + 39600000*p**4*q**3*r**2*s**2 + 61746875*p*q**5*r**2*s**2 + 1010000*p**5*q*r**3*s**2 + 1000000*p**2*q**3*r**3*s**2 - 78000000*p**3*q*r**4*s**2 - 30000000*q**3*r**4*s**2 - 80000000*p*q*r**5*s**2 + 759375*p**10*s**3 + 9787500*p**7*q**2*s**3 + 39062500*p**4*q**4*s**3 + 52343750*p*q**6*s**3 - 12301875*p**8*r*s**3 - 98175000*p**5*q**2*r*s**3 - 225078125*p**2*q**4*r*s**3 + 54900000*p**6*r**2*s**3 + 310000000*p**3*q**2*r**2*s**3 + 7890625*q**4*r**2*s**3 - 51250000*p**4*r**3*s**3 + 420000000*p*q**2*r**3*s**3 - 110000000*p**2*r**4*s**3 + 200000000*r**5*s**3 - 2109375*p**6*q*s**4 + 21093750*p**3*q**3*s**4 + 89843750*q**5*s**4 - 182343750*p**4*q*r*s**4 - 733203125*p*q**3*r*s**4 + 196875000*p**2*q*r**2*s**4 - 1125000000*q*r**3*s**4 + 158203125*p**5*s**5 + 566406250*p**2*q**2*s**5 - 101562500*p**3*r*s**5 + 1669921875*q**2*r*s**5 - 1250000000*p*r**2*s**5 + 1220703125*p*q*s**6 - 6103515625*s**7 + + b[1][4] = -1000*p**5*q**7 - 7250*p**2*q**9 + 10800*p**6*q**5*r + 96900*p**3*q**7*r + 52500*q**9*r - 37400*p**7*q**3*r**2 - 470850*p**4*q**5*r**2 - 640600*p*q**7*r**2 + 39600*p**8*q*r**3 + 983600*p**5*q**3*r**3 + 2848100*p**2*q**5*r**3 - 814400*p**6*q*r**4 - 6076000*p**3*q**3*r**4 - 2308000*q**5*r**4 + 5024000*p**4*q*r**5 + 9680000*p*q**3*r**5 - 9600000*p**2*q*r**6 - 13800*p**7*q**4*s - 94650*p**4*q**6*s + 26500*p*q**8*s + 86400*p**8*q**2*r*s + 816500*p**5*q**4*r*s + 257500*p**2*q**6*r*s - 91800*p**9*r**2*s - 1853700*p**6*q**2*r**2*s - 630000*p**3*q**4*r**2*s + 8971250*q**6*r**2*s + 2071200*p**7*r**3*s + 7240000*p**4*q**2*r**3*s - 29375000*p*q**4*r**3*s - 14416000*p**5*r**4*s + 5200000*p**2*q**2*r**4*s + 30400000*p**3*r**5*s + 12000000*q**2*r**5*s - 64800*p**9*q*s**2 - 567000*p**6*q**3*s**2 - 1655000*p**3*q**5*s**2 - 6987500*q**7*s**2 - 337500*p**7*q*r*s**2 - 8462500*p**4*q**3*r*s**2 + 5812500*p*q**5*r*s**2 + 24930000*p**5*q*r**2*s**2 + 69125000*p**2*q**3*r**2*s**2 - 103500000*p**3*q*r**3*s**2 - 30000000*q**3*r**3*s**2 - 90000000*p*q*r**4*s**2 + 708750*p**8*s**3 + 5400000*p**5*q**2*s**3 - 8906250*p**2*q**4*s**3 - 18562500*p**6*r*s**3 + 625000*p**3*q**2*r*s**3 - 29687500*q**4*r*s**3 + 75000000*p**4*r**2*s**3 + 416250000*p*q**2*r**2*s**3 - 60000000*p**2*r**3*s**3 + 300000000*r**4*s**3 - 71718750*p**4*q*s**4 - 189062500*p*q**3*s**4 - 210937500*p**2*q*r*s**4 - 1187500000*q*r**2*s**4 + 187500000*p**3*s**5 + 800781250*q**2*s**5 + 390625000*p*r*s**5 + + b[1][3] = 500*p**6*q**5 + 6350*p**3*q**7 + 19800*q**9 - 3750*p**7*q**3*r - 65100*p**4*q**5*r - 264950*p*q**7*r + 6750*p**8*q*r**2 + 209050*p**5*q**3*r**2 + 1217250*p**2*q**5*r**2 - 219000*p**6*q*r**3 - 2510000*p**3*q**3*r**3 - 1098500*q**5*r**3 + 2068000*p**4*q*r**4 + 5060000*p*q**3*r**4 - 5200000*p**2*q*r**5 + 6750*p**8*q**2*s + 96350*p**5*q**4*s + 346000*p**2*q**6*s - 20250*p**9*r*s - 459900*p**6*q**2*r*s - 1828750*p**3*q**4*r*s + 2930000*q**6*r*s + 594000*p**7*r**2*s + 4301250*p**4*q**2*r**2*s - 10906250*p*q**4*r**2*s - 5252000*p**5*r**3*s + 1450000*p**2*q**2*r**3*s + 12800000*p**3*r**4*s + 6500000*q**2*r**4*s - 74250*p**7*q*s**2 - 1418750*p**4*q**3*s**2 - 5956250*p*q**5*s**2 + 4297500*p**5*q*r*s**2 + 29906250*p**2*q**3*r*s**2 - 31500000*p**3*q*r**2*s**2 - 12500000*q**3*r**2*s**2 - 35000000*p*q*r**3*s**2 - 1350000*p**6*s**3 - 6093750*p**3*q**2*s**3 - 17500000*q**4*s**3 + 7031250*p**4*r*s**3 + 127812500*p*q**2*r*s**3 - 18750000*p**2*r**2*s**3 + 162500000*r**3*s**3 - 107812500*p**2*q*s**4 - 460937500*q*r*s**4 + 214843750*p*s**5 + + b[1][2] = -1950*p**4*q**5 - 14100*p*q**7 + 14350*p**5*q**3*r + 125600*p**2*q**5*r - 27900*p**6*q*r**2 - 402250*p**3*q**3*r**2 - 288250*q**5*r**2 + 436000*p**4*q*r**3 + 1345000*p*q**3*r**3 - 1400000*p**2*q*r**4 - 9450*p**6*q**2*s + 1250*p**3*q**4*s + 465000*q**6*s + 49950*p**7*r*s + 302500*p**4*q**2*r*s - 1718750*p*q**4*r*s - 834000*p**5*r**2*s - 437500*p**2*q**2*r**2*s + 3100000*p**3*r**3*s + 1750000*q**2*r**3*s + 292500*p**5*q*s**2 + 1937500*p**2*q**3*s**2 - 3343750*p**3*q*r*s**2 - 1875000*q**3*r*s**2 - 8125000*p*q*r**2*s**2 + 1406250*p**4*s**3 + 12343750*p*q**2*s**3 - 5312500*p**2*r*s**3 + 43750000*r**2*s**3 - 74218750*q*s**4 + + b[1][1] = 300*p**5*q**3 + 2150*p**2*q**5 - 1350*p**6*q*r - 21500*p**3*q**3*r - 61500*q**5*r + 42000*p**4*q*r**2 + 290000*p*q**3*r**2 - 300000*p**2*q*r**3 + 4050*p**7*s + 45000*p**4*q**2*s + 125000*p*q**4*s - 108000*p**5*r*s - 643750*p**2*q**2*r*s + 700000*p**3*r**2*s + 375000*q**2*r**2*s + 93750*p**3*q*s**2 + 312500*q**3*s**2 - 1875000*p*q*r*s**2 + 1406250*p**2*s**3 + 9375000*r*s**3 + + b[1][0] = -1250*p**3*q**3 - 9000*q**5 + 4500*p**4*q*r + 46250*p*q**3*r - 50000*p**2*q*r**2 - 6750*p**5*s - 43750*p**2*q**2*s + 75000*p**3*r*s + 62500*q**2*r*s - 156250*p*q*s**2 + 1562500*s**3 + + b[2][5] = 200*p**6*q**11 - 250*p**3*q**13 - 10800*q**15 - 3900*p**7*q**9*r - 3325*p**4*q**11*r + 181800*p*q**13*r + 26950*p**8*q**7*r**2 + 69625*p**5*q**9*r**2 - 1214450*p**2*q**11*r**2 - 78725*p**9*q**5*r**3 - 368675*p**6*q**7*r**3 + 4166325*p**3*q**9*r**3 + 1131100*q**11*r**3 + 73400*p**10*q**3*r**4 + 661950*p**7*q**5*r**4 - 9151950*p**4*q**7*r**4 - 16633075*p*q**9*r**4 + 36000*p**11*q*r**5 + 135600*p**8*q**3*r**5 + 17321400*p**5*q**5*r**5 + 85338300*p**2*q**7*r**5 - 832000*p**9*q*r**6 - 21379200*p**6*q**3*r**6 - 176044000*p**3*q**5*r**6 - 1410000*q**7*r**6 + 6528000*p**7*q*r**7 + 129664000*p**4*q**3*r**7 + 47344000*p*q**5*r**7 - 21504000*p**5*q*r**8 - 115200000*p**2*q**3*r**8 + 25600000*p**3*q*r**9 + 64000000*q**3*r**9 + 15700*p**8*q**8*s + 120525*p**5*q**10*s + 113250*p**2*q**12*s - 196900*p**9*q**6*r*s - 1776925*p**6*q**8*r*s - 3062475*p**3*q**10*r*s - 4153500*q**12*r*s + 857925*p**10*q**4*r**2*s + 10562775*p**7*q**6*r**2*s + 34866250*p**4*q**8*r**2*s + 73486750*p*q**10*r**2*s - 1333800*p**11*q**2*r**3*s - 29212625*p**8*q**4*r**3*s - 168729675*p**5*q**6*r**3*s - 427230750*p**2*q**8*r**3*s + 108000*p**12*r**4*s + 30384200*p**9*q**2*r**4*s + 324535100*p**6*q**4*r**4*s + 952666750*p**3*q**6*r**4*s - 38076875*q**8*r**4*s - 4296000*p**10*r**5*s - 213606400*p**7*q**2*r**5*s - 842060000*p**4*q**4*r**5*s - 95285000*p*q**6*r**5*s + 61184000*p**8*r**6*s + 567520000*p**5*q**2*r**6*s + 547000000*p**2*q**4*r**6*s - 390912000*p**6*r**7*s - 812800000*p**3*q**2*r**7*s - 924000000*q**4*r**7*s + 1152000000*p**4*r**8*s + 800000000*p*q**2*r**8*s - 1280000000*p**2*r**9*s + 141750*p**10*q**5*s**2 - 31500*p**7*q**7*s**2 - 11325000*p**4*q**9*s**2 - 31687500*p*q**11*s**2 - 1293975*p**11*q**3*r*s**2 - 4803800*p**8*q**5*r*s**2 + 71398250*p**5*q**7*r*s**2 + 227625000*p**2*q**9*r*s**2 + 3256200*p**12*q*r**2*s**2 + 43870125*p**9*q**3*r**2*s**2 + 64581500*p**6*q**5*r**2*s**2 + 56090625*p**3*q**7*r**2*s**2 + 260218750*q**9*r**2*s**2 - 74610000*p**10*q*r**3*s**2 - 662186500*p**7*q**3*r**3*s**2 - 1987747500*p**4*q**5*r**3*s**2 - 811928125*p*q**7*r**3*s**2 + 471286000*p**8*q*r**4*s**2 + 2106040000*p**5*q**3*r**4*s**2 + 792687500*p**2*q**5*r**4*s**2 - 135120000*p**6*q*r**5*s**2 + 2479000000*p**3*q**3*r**5*s**2 + 5242250000*q**5*r**5*s**2 - 6400000000*p**4*q*r**6*s**2 - 8620000000*p*q**3*r**6*s**2 + 13280000000*p**2*q*r**7*s**2 + 1600000000*q*r**8*s**2 + 273375*p**12*q**2*s**3 - 13612500*p**9*q**4*s**3 - 177250000*p**6*q**6*s**3 - 511015625*p**3*q**8*s**3 - 320937500*q**10*s**3 - 2770200*p**13*r*s**3 + 12595500*p**10*q**2*r*s**3 + 543950000*p**7*q**4*r*s**3 + 1612281250*p**4*q**6*r*s**3 + 968125000*p*q**8*r*s**3 + 77031000*p**11*r**2*s**3 + 373218750*p**8*q**2*r**2*s**3 + 1839765625*p**5*q**4*r**2*s**3 + 1818515625*p**2*q**6*r**2*s**3 - 776745000*p**9*r**3*s**3 - 6861075000*p**6*q**2*r**3*s**3 - 20014531250*p**3*q**4*r**3*s**3 - 13747812500*q**6*r**3*s**3 + 3768000000*p**7*r**4*s**3 + 35365000000*p**4*q**2*r**4*s**3 + 34441875000*p*q**4*r**4*s**3 - 9628000000*p**5*r**5*s**3 - 63230000000*p**2*q**2*r**5*s**3 + 13600000000*p**3*r**6*s**3 - 15000000000*q**2*r**6*s**3 - 10400000000*p*r**7*s**3 - 45562500*p**11*q*s**4 - 525937500*p**8*q**3*s**4 - 1364218750*p**5*q**5*s**4 - 1382812500*p**2*q**7*s**4 + 572062500*p**9*q*r*s**4 + 2473515625*p**6*q**3*r*s**4 + 13192187500*p**3*q**5*r*s**4 + 12703125000*q**7*r*s**4 - 451406250*p**7*q*r**2*s**4 - 18153906250*p**4*q**3*r**2*s**4 - 36908203125*p*q**5*r**2*s**4 - 9069375000*p**5*q*r**3*s**4 + 79957812500*p**2*q**3*r**3*s**4 + 5512500000*p**3*q*r**4*s**4 + 50656250000*q**3*r**4*s**4 + 74750000000*p*q*r**5*s**4 + 56953125*p**10*s**5 + 1381640625*p**7*q**2*s**5 - 781250000*p**4*q**4*s**5 + 878906250*p*q**6*s**5 - 2655703125*p**8*r*s**5 - 3223046875*p**5*q**2*r*s**5 - 35117187500*p**2*q**4*r*s**5 + 26573437500*p**6*r**2*s**5 + 14785156250*p**3*q**2*r**2*s**5 - 52050781250*q**4*r**2*s**5 - 103062500000*p**4*r**3*s**5 - 281796875000*p*q**2*r**3*s**5 + 146875000000*p**2*r**4*s**5 - 37500000000*r**5*s**5 - 8789062500*p**6*q*s**6 - 3906250000*p**3*q**3*s**6 + 1464843750*q**5*s**6 + 102929687500*p**4*q*r*s**6 + 297119140625*p*q**3*r*s**6 - 217773437500*p**2*q*r**2*s**6 + 167968750000*q*r**3*s**6 + 10986328125*p**5*s**7 + 98876953125*p**2*q**2*s**7 - 188964843750*p**3*r*s**7 - 278320312500*q**2*r*s**7 + 517578125000*p*r**2*s**7 - 610351562500*p*q*s**8 + 762939453125*s**9 + + b[2][4] = -200*p**7*q**9 + 1850*p**4*q**11 + 21600*p*q**13 + 3200*p**8*q**7*r - 19200*p**5*q**9*r - 316350*p**2*q**11*r - 19050*p**9*q**5*r**2 + 37400*p**6*q**7*r**2 + 1759250*p**3*q**9*r**2 + 440100*q**11*r**2 + 48750*p**10*q**3*r**3 + 190200*p**7*q**5*r**3 - 4604200*p**4*q**7*r**3 - 6072800*p*q**9*r**3 - 43200*p**11*q*r**4 - 834500*p**8*q**3*r**4 + 4916000*p**5*q**5*r**4 + 27926850*p**2*q**7*r**4 + 969600*p**9*q*r**5 + 2467200*p**6*q**3*r**5 - 45393200*p**3*q**5*r**5 - 5399500*q**7*r**5 - 7283200*p**7*q*r**6 + 10536000*p**4*q**3*r**6 + 41656000*p*q**5*r**6 + 22784000*p**5*q*r**7 - 35200000*p**2*q**3*r**7 - 25600000*p**3*q*r**8 + 96000000*q**3*r**8 - 3000*p**9*q**6*s + 40400*p**6*q**8*s + 136550*p**3*q**10*s - 1647000*q**12*s + 40500*p**10*q**4*r*s - 173600*p**7*q**6*r*s - 126500*p**4*q**8*r*s + 23969250*p*q**10*r*s - 153900*p**11*q**2*r**2*s - 486150*p**8*q**4*r**2*s - 4115800*p**5*q**6*r**2*s - 112653250*p**2*q**8*r**2*s + 129600*p**12*r**3*s + 2683350*p**9*q**2*r**3*s + 10906650*p**6*q**4*r**3*s + 187289500*p**3*q**6*r**3*s + 44098750*q**8*r**3*s - 4384800*p**10*r**4*s - 35660800*p**7*q**2*r**4*s - 175420000*p**4*q**4*r**4*s - 426538750*p*q**6*r**4*s + 60857600*p**8*r**5*s + 349436000*p**5*q**2*r**5*s + 900600000*p**2*q**4*r**5*s - 429568000*p**6*r**6*s - 1511200000*p**3*q**2*r**6*s - 1286000000*q**4*r**6*s + 1472000000*p**4*r**7*s + 1440000000*p*q**2*r**7*s - 1920000000*p**2*r**8*s - 36450*p**11*q**3*s**2 - 188100*p**8*q**5*s**2 - 5504750*p**5*q**7*s**2 - 37968750*p**2*q**9*s**2 + 255150*p**12*q*r*s**2 + 2754000*p**9*q**3*r*s**2 + 49196500*p**6*q**5*r*s**2 + 323587500*p**3*q**7*r*s**2 - 83250000*q**9*r*s**2 - 465750*p**10*q*r**2*s**2 - 31881500*p**7*q**3*r**2*s**2 - 415585000*p**4*q**5*r**2*s**2 + 1054775000*p*q**7*r**2*s**2 - 96823500*p**8*q*r**3*s**2 - 701490000*p**5*q**3*r**3*s**2 - 2953531250*p**2*q**5*r**3*s**2 + 1454560000*p**6*q*r**4*s**2 + 7670500000*p**3*q**3*r**4*s**2 + 5661062500*q**5*r**4*s**2 - 7785000000*p**4*q*r**5*s**2 - 9450000000*p*q**3*r**5*s**2 + 14000000000*p**2*q*r**6*s**2 + 2400000000*q*r**7*s**2 - 437400*p**13*s**3 - 10145250*p**10*q**2*s**3 - 121912500*p**7*q**4*s**3 - 576531250*p**4*q**6*s**3 - 528593750*p*q**8*s**3 + 12939750*p**11*r*s**3 + 313368750*p**8*q**2*r*s**3 + 2171812500*p**5*q**4*r*s**3 + 2381718750*p**2*q**6*r*s**3 - 124638750*p**9*r**2*s**3 - 3001575000*p**6*q**2*r**2*s**3 - 12259375000*p**3*q**4*r**2*s**3 - 9985312500*q**6*r**2*s**3 + 384000000*p**7*r**3*s**3 + 13997500000*p**4*q**2*r**3*s**3 + 20749531250*p*q**4*r**3*s**3 - 553500000*p**5*r**4*s**3 - 41835000000*p**2*q**2*r**4*s**3 + 5420000000*p**3*r**5*s**3 - 16300000000*q**2*r**5*s**3 - 17600000000*p*r**6*s**3 - 7593750*p**9*q*s**4 + 289218750*p**6*q**3*s**4 + 3591406250*p**3*q**5*s**4 + 5992187500*q**7*s**4 + 658125000*p**7*q*r*s**4 - 269531250*p**4*q**3*r*s**4 - 15882812500*p*q**5*r*s**4 - 4785000000*p**5*q*r**2*s**4 + 54375781250*p**2*q**3*r**2*s**4 - 5668750000*p**3*q*r**3*s**4 + 35867187500*q**3*r**3*s**4 + 113875000000*p*q*r**4*s**4 - 544218750*p**8*s**5 - 5407031250*p**5*q**2*s**5 - 14277343750*p**2*q**4*s**5 + 5421093750*p**6*r*s**5 - 24941406250*p**3*q**2*r*s**5 - 25488281250*q**4*r*s**5 - 11500000000*p**4*r**2*s**5 - 231894531250*p*q**2*r**2*s**5 - 6250000000*p**2*r**3*s**5 - 43750000000*r**4*s**5 + 35449218750*p**4*q*s**6 + 137695312500*p*q**3*s**6 + 34667968750*p**2*q*r*s**6 + 202148437500*q*r**2*s**6 - 33691406250*p**3*s**7 - 214843750000*q**2*s**7 - 31738281250*p*r*s**7 + + b[2][3] = -800*p**5*q**9 - 5400*p**2*q**11 + 5800*p**6*q**7*r + 48750*p**3*q**9*r + 16200*q**11*r - 3000*p**7*q**5*r**2 - 108350*p**4*q**7*r**2 - 263250*p*q**9*r**2 - 60700*p**8*q**3*r**3 - 386250*p**5*q**5*r**3 + 253100*p**2*q**7*r**3 + 127800*p**9*q*r**4 + 2326700*p**6*q**3*r**4 + 6565550*p**3*q**5*r**4 - 705750*q**7*r**4 - 2903200*p**7*q*r**5 - 21218000*p**4*q**3*r**5 + 1057000*p*q**5*r**5 + 20368000*p**5*q*r**6 + 33000000*p**2*q**3*r**6 - 43200000*p**3*q*r**7 + 52000000*q**3*r**7 + 6200*p**7*q**6*s + 188250*p**4*q**8*s + 931500*p*q**10*s - 73800*p**8*q**4*r*s - 1466850*p**5*q**6*r*s - 6894000*p**2*q**8*r*s + 315900*p**9*q**2*r**2*s + 4547000*p**6*q**4*r**2*s + 20362500*p**3*q**6*r**2*s + 15018750*q**8*r**2*s - 653400*p**10*r**3*s - 13897550*p**7*q**2*r**3*s - 76757500*p**4*q**4*r**3*s - 124207500*p*q**6*r**3*s + 18567600*p**8*r**4*s + 175911000*p**5*q**2*r**4*s + 253787500*p**2*q**4*r**4*s - 183816000*p**6*r**5*s - 706900000*p**3*q**2*r**5*s - 665750000*q**4*r**5*s + 740000000*p**4*r**6*s + 890000000*p*q**2*r**6*s - 1040000000*p**2*r**7*s - 763000*p**6*q**5*s**2 - 12375000*p**3*q**7*s**2 - 40500000*q**9*s**2 + 364500*p**10*q*r*s**2 + 15537000*p**7*q**3*r*s**2 + 154392500*p**4*q**5*r*s**2 + 372206250*p*q**7*r*s**2 - 25481250*p**8*q*r**2*s**2 - 386300000*p**5*q**3*r**2*s**2 - 996343750*p**2*q**5*r**2*s**2 + 459872500*p**6*q*r**3*s**2 + 2943937500*p**3*q**3*r**3*s**2 + 2437781250*q**5*r**3*s**2 - 2883750000*p**4*q*r**4*s**2 - 4343750000*p*q**3*r**4*s**2 + 5495000000*p**2*q*r**5*s**2 + 1300000000*q*r**6*s**2 - 364500*p**11*s**3 - 13668750*p**8*q**2*s**3 - 113406250*p**5*q**4*s**3 - 159062500*p**2*q**6*s**3 + 13972500*p**9*r*s**3 + 61537500*p**6*q**2*r*s**3 - 1622656250*p**3*q**4*r*s**3 - 2720625000*q**6*r*s**3 - 201656250*p**7*r**2*s**3 + 1949687500*p**4*q**2*r**2*s**3 + 4979687500*p*q**4*r**2*s**3 + 497125000*p**5*r**3*s**3 - 11150625000*p**2*q**2*r**3*s**3 + 2982500000*p**3*r**4*s**3 - 6612500000*q**2*r**4*s**3 - 10450000000*p*r**5*s**3 + 126562500*p**7*q*s**4 + 1443750000*p**4*q**3*s**4 + 281250000*p*q**5*s**4 - 1648125000*p**5*q*r*s**4 + 11271093750*p**2*q**3*r*s**4 - 4785156250*p**3*q*r**2*s**4 + 8808593750*q**3*r**2*s**4 + 52390625000*p*q*r**3*s**4 - 611718750*p**6*s**5 - 13027343750*p**3*q**2*s**5 - 1464843750*q**4*s**5 + 6492187500*p**4*r*s**5 - 65351562500*p*q**2*r*s**5 - 13476562500*p**2*r**2*s**5 - 24218750000*r**3*s**5 + 41992187500*p**2*q*s**6 + 69824218750*q*r*s**6 - 34179687500*p*s**7 + + b[2][2] = -1000*p**6*q**7 - 5150*p**3*q**9 + 10800*q**11 + 11000*p**7*q**5*r + 66450*p**4*q**7*r - 127800*p*q**9*r - 41250*p**8*q**3*r**2 - 368400*p**5*q**5*r**2 + 204200*p**2*q**7*r**2 + 54000*p**9*q*r**3 + 1040950*p**6*q**3*r**3 + 2096500*p**3*q**5*r**3 + 200000*q**7*r**3 - 1140000*p**7*q*r**4 - 7691000*p**4*q**3*r**4 - 2281000*p*q**5*r**4 + 7296000*p**5*q*r**5 + 13300000*p**2*q**3*r**5 - 14400000*p**3*q*r**6 + 14000000*q**3*r**6 - 9000*p**8*q**4*s + 52100*p**5*q**6*s + 710250*p**2*q**8*s + 67500*p**9*q**2*r*s - 256100*p**6*q**4*r*s - 5753000*p**3*q**6*r*s + 292500*q**8*r*s - 162000*p**10*r**2*s - 1432350*p**7*q**2*r**2*s + 5410000*p**4*q**4*r**2*s - 7408750*p*q**6*r**2*s + 4401000*p**8*r**3*s + 24185000*p**5*q**2*r**3*s + 20781250*p**2*q**4*r**3*s - 43012000*p**6*r**4*s - 146300000*p**3*q**2*r**4*s - 165875000*q**4*r**4*s + 182000000*p**4*r**5*s + 250000000*p*q**2*r**5*s - 280000000*p**2*r**6*s + 60750*p**10*q*s**2 + 2414250*p**7*q**3*s**2 + 15770000*p**4*q**5*s**2 + 15825000*p*q**7*s**2 - 6021000*p**8*q*r*s**2 - 62252500*p**5*q**3*r*s**2 - 74718750*p**2*q**5*r*s**2 + 90888750*p**6*q*r**2*s**2 + 471312500*p**3*q**3*r**2*s**2 + 525875000*q**5*r**2*s**2 - 539375000*p**4*q*r**3*s**2 - 1030000000*p*q**3*r**3*s**2 + 1142500000*p**2*q*r**4*s**2 + 350000000*q*r**5*s**2 - 303750*p**9*s**3 - 35943750*p**6*q**2*s**3 - 331875000*p**3*q**4*s**3 - 505937500*q**6*s**3 + 8437500*p**7*r*s**3 + 530781250*p**4*q**2*r*s**3 + 1150312500*p*q**4*r*s**3 - 154500000*p**5*r**2*s**3 - 2059062500*p**2*q**2*r**2*s**3 + 1150000000*p**3*r**3*s**3 - 1343750000*q**2*r**3*s**3 - 2900000000*p*r**4*s**3 + 30937500*p**5*q*s**4 + 1166406250*p**2*q**3*s**4 - 1496875000*p**3*q*r*s**4 + 1296875000*q**3*r*s**4 + 10640625000*p*q*r**2*s**4 - 281250000*p**4*s**5 - 9746093750*p*q**2*s**5 + 1269531250*p**2*r*s**5 - 7421875000*r**2*s**5 + 15625000000*q*s**6 + + b[2][1] = -1600*p**4*q**7 - 10800*p*q**9 + 9800*p**5*q**5*r + 80550*p**2*q**7*r - 4600*p**6*q**3*r**2 - 112700*p**3*q**5*r**2 + 40500*q**7*r**2 - 34200*p**7*q*r**3 - 279500*p**4*q**3*r**3 - 665750*p*q**5*r**3 + 632000*p**5*q*r**4 + 3200000*p**2*q**3*r**4 - 2800000*p**3*q*r**5 + 3000000*q**3*r**5 - 18600*p**6*q**4*s - 51750*p**3*q**6*s + 405000*q**8*s + 21600*p**7*q**2*r*s - 122500*p**4*q**4*r*s - 2891250*p*q**6*r*s + 156600*p**8*r**2*s + 1569750*p**5*q**2*r**2*s + 6943750*p**2*q**4*r**2*s - 3774000*p**6*r**3*s - 27100000*p**3*q**2*r**3*s - 30187500*q**4*r**3*s + 28000000*p**4*r**4*s + 52500000*p*q**2*r**4*s - 60000000*p**2*r**5*s - 81000*p**8*q*s**2 - 240000*p**5*q**3*s**2 + 937500*p**2*q**5*s**2 + 3273750*p**6*q*r*s**2 + 30406250*p**3*q**3*r*s**2 + 55687500*q**5*r*s**2 - 42187500*p**4*q*r**2*s**2 - 112812500*p*q**3*r**2*s**2 + 152500000*p**2*q*r**3*s**2 + 75000000*q*r**4*s**2 - 4218750*p**4*q**2*s**3 + 15156250*p*q**4*s**3 + 5906250*p**5*r*s**3 - 206562500*p**2*q**2*r*s**3 + 107500000*p**3*r**2*s**3 - 159375000*q**2*r**2*s**3 - 612500000*p*r**3*s**3 + 135937500*p**3*q*s**4 + 46875000*q**3*s**4 + 1175781250*p*q*r*s**4 - 292968750*p**2*s**5 - 1367187500*r*s**5 + + b[2][0] = -800*p**5*q**5 - 5400*p**2*q**7 + 6000*p**6*q**3*r + 51700*p**3*q**5*r + 27000*q**7*r - 10800*p**7*q*r**2 - 163250*p**4*q**3*r**2 - 285750*p*q**5*r**2 + 192000*p**5*q*r**3 + 1000000*p**2*q**3*r**3 - 800000*p**3*q*r**4 + 500000*q**3*r**4 - 10800*p**7*q**2*s - 57500*p**4*q**4*s + 67500*p*q**6*s + 32400*p**8*r*s + 279000*p**5*q**2*r*s - 131250*p**2*q**4*r*s - 729000*p**6*r**2*s - 4100000*p**3*q**2*r**2*s - 5343750*q**4*r**2*s + 5000000*p**4*r**3*s + 10000000*p*q**2*r**3*s - 10000000*p**2*r**4*s + 641250*p**6*q*s**2 + 5812500*p**3*q**3*s**2 + 10125000*q**5*s**2 - 7031250*p**4*q*r*s**2 - 20625000*p*q**3*r*s**2 + 17500000*p**2*q*r**2*s**2 + 12500000*q*r**3*s**2 - 843750*p**5*s**3 - 19375000*p**2*q**2*s**3 + 30000000*p**3*r*s**3 - 20312500*q**2*r*s**3 - 112500000*p*r**2*s**3 + 183593750*p*q*s**4 - 292968750*s**5 + + b[3][5] = 500*p**11*q**6 + 9875*p**8*q**8 + 42625*p**5*q**10 - 35000*p**2*q**12 - 4500*p**12*q**4*r - 108375*p**9*q**6*r - 516750*p**6*q**8*r + 1110500*p**3*q**10*r + 2730000*q**12*r + 10125*p**13*q**2*r**2 + 358250*p**10*q**4*r**2 + 1908625*p**7*q**6*r**2 - 11744250*p**4*q**8*r**2 - 43383250*p*q**10*r**2 - 313875*p**11*q**2*r**3 - 2074875*p**8*q**4*r**3 + 52094750*p**5*q**6*r**3 + 264567500*p**2*q**8*r**3 + 796125*p**9*q**2*r**4 - 92486250*p**6*q**4*r**4 - 757957500*p**3*q**6*r**4 - 29354375*q**8*r**4 + 60970000*p**7*q**2*r**5 + 1112462500*p**4*q**4*r**5 + 571094375*p*q**6*r**5 - 685290000*p**5*q**2*r**6 - 2037800000*p**2*q**4*r**6 + 2279600000*p**3*q**2*r**7 + 849000000*q**4*r**7 - 1480000000*p*q**2*r**8 + 13500*p**13*q**3*s + 363000*p**10*q**5*s + 2861250*p**7*q**7*s + 8493750*p**4*q**9*s + 17031250*p*q**11*s - 60750*p**14*q*r*s - 2319750*p**11*q**3*r*s - 22674250*p**8*q**5*r*s - 74368750*p**5*q**7*r*s - 170578125*p**2*q**9*r*s + 2760750*p**12*q*r**2*s + 46719000*p**9*q**3*r**2*s + 163356375*p**6*q**5*r**2*s + 360295625*p**3*q**7*r**2*s - 195990625*q**9*r**2*s - 37341750*p**10*q*r**3*s - 194739375*p**7*q**3*r**3*s - 105463125*p**4*q**5*r**3*s - 415825000*p*q**7*r**3*s + 90180000*p**8*q*r**4*s - 990552500*p**5*q**3*r**4*s + 3519212500*p**2*q**5*r**4*s + 1112220000*p**6*q*r**5*s - 4508750000*p**3*q**3*r**5*s - 8159500000*q**5*r**5*s - 4356000000*p**4*q*r**6*s + 14615000000*p*q**3*r**6*s - 2160000000*p**2*q*r**7*s + 91125*p**15*s**2 + 3290625*p**12*q**2*s**2 + 35100000*p**9*q**4*s**2 + 175406250*p**6*q**6*s**2 + 629062500*p**3*q**8*s**2 + 910937500*q**10*s**2 - 5710500*p**13*r*s**2 - 100423125*p**10*q**2*r*s**2 - 604743750*p**7*q**4*r*s**2 - 2954843750*p**4*q**6*r*s**2 - 4587578125*p*q**8*r*s**2 + 116194500*p**11*r**2*s**2 + 1280716250*p**8*q**2*r**2*s**2 + 7401190625*p**5*q**4*r**2*s**2 + 11619937500*p**2*q**6*r**2*s**2 - 952173125*p**9*r**3*s**2 - 6519712500*p**6*q**2*r**3*s**2 - 10238593750*p**3*q**4*r**3*s**2 + 29984609375*q**6*r**3*s**2 + 2558300000*p**7*r**4*s**2 + 16225000000*p**4*q**2*r**4*s**2 - 64994140625*p*q**4*r**4*s**2 + 4202250000*p**5*r**5*s**2 + 46925000000*p**2*q**2*r**5*s**2 - 28950000000*p**3*r**6*s**2 - 1000000000*q**2*r**6*s**2 + 37000000000*p*r**7*s**2 - 48093750*p**11*q*s**3 - 673359375*p**8*q**3*s**3 - 2170312500*p**5*q**5*s**3 - 2466796875*p**2*q**7*s**3 + 647578125*p**9*q*r*s**3 + 597031250*p**6*q**3*r*s**3 - 7542578125*p**3*q**5*r*s**3 - 41125000000*q**7*r*s**3 - 2175828125*p**7*q*r**2*s**3 - 7101562500*p**4*q**3*r**2*s**3 + 100596875000*p*q**5*r**2*s**3 - 8984687500*p**5*q*r**3*s**3 - 120070312500*p**2*q**3*r**3*s**3 + 57343750000*p**3*q*r**4*s**3 + 9500000000*q**3*r**4*s**3 - 342875000000*p*q*r**5*s**3 + 400781250*p**10*s**4 + 8531250000*p**7*q**2*s**4 + 34033203125*p**4*q**4*s**4 + 42724609375*p*q**6*s**4 - 6289453125*p**8*r*s**4 - 24037109375*p**5*q**2*r*s**4 - 62626953125*p**2*q**4*r*s**4 + 17299218750*p**6*r**2*s**4 + 108357421875*p**3*q**2*r**2*s**4 - 55380859375*q**4*r**2*s**4 + 105648437500*p**4*r**3*s**4 + 1204228515625*p*q**2*r**3*s**4 - 365000000000*p**2*r**4*s**4 + 184375000000*r**5*s**4 - 32080078125*p**6*q*s**5 - 98144531250*p**3*q**3*s**5 + 93994140625*q**5*s**5 - 178955078125*p**4*q*r*s**5 - 1299804687500*p*q**3*r*s**5 + 332421875000*p**2*q*r**2*s**5 - 1195312500000*q*r**3*s**5 + 72021484375*p**5*s**6 + 323486328125*p**2*q**2*s**6 + 682373046875*p**3*r*s**6 + 2447509765625*q**2*r*s**6 - 3011474609375*p*r**2*s**6 + 3051757812500*p*q*s**7 - 7629394531250*s**8 + + b[3][4] = 1500*p**9*q**6 + 69625*p**6*q**8 + 590375*p**3*q**10 + 1035000*q**12 - 13500*p**10*q**4*r - 760625*p**7*q**6*r - 7904500*p**4*q**8*r - 18169250*p*q**10*r + 30375*p**11*q**2*r**2 + 2628625*p**8*q**4*r**2 + 37879000*p**5*q**6*r**2 + 121367500*p**2*q**8*r**2 - 2699250*p**9*q**2*r**3 - 76776875*p**6*q**4*r**3 - 403583125*p**3*q**6*r**3 - 78865625*q**8*r**3 + 60907500*p**7*q**2*r**4 + 735291250*p**4*q**4*r**4 + 781142500*p*q**6*r**4 - 558270000*p**5*q**2*r**5 - 2150725000*p**2*q**4*r**5 + 2015400000*p**3*q**2*r**6 + 1181000000*q**4*r**6 - 2220000000*p*q**2*r**7 + 40500*p**11*q**3*s + 1376500*p**8*q**5*s + 9953125*p**5*q**7*s + 9765625*p**2*q**9*s - 182250*p**12*q*r*s - 8859000*p**9*q**3*r*s - 82854500*p**6*q**5*r*s - 71511250*p**3*q**7*r*s + 273631250*q**9*r*s + 10233000*p**10*q*r**2*s + 179627500*p**7*q**3*r**2*s + 25164375*p**4*q**5*r**2*s - 2927290625*p*q**7*r**2*s - 171305000*p**8*q*r**3*s - 544768750*p**5*q**3*r**3*s + 7583437500*p**2*q**5*r**3*s + 1139860000*p**6*q*r**4*s - 6489375000*p**3*q**3*r**4*s - 9625375000*q**5*r**4*s - 1838000000*p**4*q*r**5*s + 19835000000*p*q**3*r**5*s - 3240000000*p**2*q*r**6*s + 273375*p**13*s**2 + 9753750*p**10*q**2*s**2 + 82575000*p**7*q**4*s**2 + 202265625*p**4*q**6*s**2 + 556093750*p*q**8*s**2 - 11552625*p**11*r*s**2 - 115813125*p**8*q**2*r*s**2 + 630590625*p**5*q**4*r*s**2 + 1347015625*p**2*q**6*r*s**2 + 157578750*p**9*r**2*s**2 - 689206250*p**6*q**2*r**2*s**2 - 4299609375*p**3*q**4*r**2*s**2 + 23896171875*q**6*r**2*s**2 - 1022437500*p**7*r**3*s**2 + 6648125000*p**4*q**2*r**3*s**2 - 52895312500*p*q**4*r**3*s**2 + 4401750000*p**5*r**4*s**2 + 26500000000*p**2*q**2*r**4*s**2 - 22125000000*p**3*r**5*s**2 - 1500000000*q**2*r**5*s**2 + 55500000000*p*r**6*s**2 - 137109375*p**9*q*s**3 - 1955937500*p**6*q**3*s**3 - 6790234375*p**3*q**5*s**3 - 16996093750*q**7*s**3 + 2146218750*p**7*q*r*s**3 + 6570312500*p**4*q**3*r*s**3 + 39918750000*p*q**5*r*s**3 - 7673281250*p**5*q*r**2*s**3 - 52000000000*p**2*q**3*r**2*s**3 + 50796875000*p**3*q*r**3*s**3 + 18750000000*q**3*r**3*s**3 - 399875000000*p*q*r**4*s**3 + 780468750*p**8*s**4 + 14455078125*p**5*q**2*s**4 + 10048828125*p**2*q**4*s**4 - 15113671875*p**6*r*s**4 + 39298828125*p**3*q**2*r*s**4 - 52138671875*q**4*r*s**4 + 45964843750*p**4*r**2*s**4 + 914414062500*p*q**2*r**2*s**4 + 1953125000*p**2*r**3*s**4 + 334375000000*r**4*s**4 - 149169921875*p**4*q*s**5 - 459716796875*p*q**3*s**5 - 325585937500*p**2*q*r*s**5 - 1462890625000*q*r**2*s**5 + 296630859375*p**3*s**6 + 1324462890625*q**2*s**6 + 307617187500*p*r*s**6 + + b[3][3] = -20750*p**7*q**6 - 290125*p**4*q**8 - 993000*p*q**10 + 146125*p**8*q**4*r + 2721500*p**5*q**6*r + 11833750*p**2*q**8*r - 237375*p**9*q**2*r**2 - 8167500*p**6*q**4*r**2 - 54605625*p**3*q**6*r**2 - 23802500*q**8*r**2 + 8927500*p**7*q**2*r**3 + 131184375*p**4*q**4*r**3 + 254695000*p*q**6*r**3 - 121561250*p**5*q**2*r**4 - 728003125*p**2*q**4*r**4 + 702550000*p**3*q**2*r**5 + 597312500*q**4*r**5 - 1202500000*p*q**2*r**6 - 194625*p**9*q**3*s - 1568875*p**6*q**5*s + 9685625*p**3*q**7*s + 74662500*q**9*s + 327375*p**10*q*r*s + 1280000*p**7*q**3*r*s - 123703750*p**4*q**5*r*s - 850121875*p*q**7*r*s - 7436250*p**8*q*r**2*s + 164820000*p**5*q**3*r**2*s + 2336659375*p**2*q**5*r**2*s + 32202500*p**6*q*r**3*s - 2429765625*p**3*q**3*r**3*s - 4318609375*q**5*r**3*s + 148000000*p**4*q*r**4*s + 9902812500*p*q**3*r**4*s - 1755000000*p**2*q*r**5*s + 1154250*p**11*s**2 + 36821250*p**8*q**2*s**2 + 372825000*p**5*q**4*s**2 + 1170921875*p**2*q**6*s**2 - 38913750*p**9*r*s**2 - 797071875*p**6*q**2*r*s**2 - 2848984375*p**3*q**4*r*s**2 + 7651406250*q**6*r*s**2 + 415068750*p**7*r**2*s**2 + 3151328125*p**4*q**2*r**2*s**2 - 17696875000*p*q**4*r**2*s**2 - 725968750*p**5*r**3*s**2 + 5295312500*p**2*q**2*r**3*s**2 - 8581250000*p**3*r**4*s**2 - 812500000*q**2*r**4*s**2 + 30062500000*p*r**5*s**2 - 110109375*p**7*q*s**3 - 1976562500*p**4*q**3*s**3 - 6329296875*p*q**5*s**3 + 2256328125*p**5*q*r*s**3 + 8554687500*p**2*q**3*r*s**3 + 12947265625*p**3*q*r**2*s**3 + 7984375000*q**3*r**2*s**3 - 167039062500*p*q*r**3*s**3 + 1181250000*p**6*s**4 + 17873046875*p**3*q**2*s**4 - 20449218750*q**4*s**4 - 16265625000*p**4*r*s**4 + 260869140625*p*q**2*r*s**4 + 21025390625*p**2*r**2*s**4 + 207617187500*r**3*s**4 - 207177734375*p**2*q*s**5 - 615478515625*q*r*s**5 + 301513671875*p*s**6 + + b[3][2] = 53125*p**5*q**6 + 425000*p**2*q**8 - 394375*p**6*q**4*r - 4301875*p**3*q**6*r - 3225000*q**8*r + 851250*p**7*q**2*r**2 + 16910625*p**4*q**4*r**2 + 44210000*p*q**6*r**2 - 20474375*p**5*q**2*r**3 - 147190625*p**2*q**4*r**3 + 163975000*p**3*q**2*r**4 + 156812500*q**4*r**4 - 323750000*p*q**2*r**5 - 99375*p**7*q**3*s - 6395000*p**4*q**5*s - 49243750*p*q**7*s - 1164375*p**8*q*r*s + 4465625*p**5*q**3*r*s + 205546875*p**2*q**5*r*s + 12163750*p**6*q*r**2*s - 315546875*p**3*q**3*r**2*s - 946453125*q**5*r**2*s - 23500000*p**4*q*r**3*s + 2313437500*p*q**3*r**3*s - 472500000*p**2*q*r**4*s + 1316250*p**9*s**2 + 22715625*p**6*q**2*s**2 + 206953125*p**3*q**4*s**2 + 1220000000*q**6*s**2 - 20953125*p**7*r*s**2 - 277656250*p**4*q**2*r*s**2 - 3317187500*p*q**4*r*s**2 + 293734375*p**5*r**2*s**2 + 1351562500*p**2*q**2*r**2*s**2 - 2278125000*p**3*r**3*s**2 - 218750000*q**2*r**3*s**2 + 8093750000*p*r**4*s**2 - 9609375*p**5*q*s**3 + 240234375*p**2*q**3*s**3 + 2310546875*p**3*q*r*s**3 + 1171875000*q**3*r*s**3 - 33460937500*p*q*r**2*s**3 + 2185546875*p**4*s**4 + 32578125000*p*q**2*s**4 - 8544921875*p**2*r*s**4 + 58398437500*r**2*s**4 - 114013671875*q*s**5 + + b[3][1] = -16250*p**6*q**4 - 191875*p**3*q**6 - 495000*q**8 + 73125*p**7*q**2*r + 1437500*p**4*q**4*r + 5866250*p*q**6*r - 2043125*p**5*q**2*r**2 - 17218750*p**2*q**4*r**2 + 19106250*p**3*q**2*r**3 + 34015625*q**4*r**3 - 69375000*p*q**2*r**4 - 219375*p**8*q*s - 2846250*p**5*q**3*s - 8021875*p**2*q**5*s + 3420000*p**6*q*r*s - 1640625*p**3*q**3*r*s - 152468750*q**5*r*s + 3062500*p**4*q*r**2*s + 381171875*p*q**3*r**2*s - 101250000*p**2*q*r**3*s + 2784375*p**7*s**2 + 43515625*p**4*q**2*s**2 + 115625000*p*q**4*s**2 - 48140625*p**5*r*s**2 - 307421875*p**2*q**2*r*s**2 - 25781250*p**3*r**2*s**2 - 46875000*q**2*r**2*s**2 + 1734375000*p*r**3*s**2 - 128906250*p**3*q*s**3 + 339843750*q**3*s**3 - 4583984375*p*q*r*s**3 + 2236328125*p**2*s**4 + 12255859375*r*s**4 + + b[3][0] = 31875*p**4*q**4 + 255000*p*q**6 - 82500*p**5*q**2*r - 1106250*p**2*q**4*r + 1653125*p**3*q**2*r**2 + 5187500*q**4*r**2 - 11562500*p*q**2*r**3 - 118125*p**6*q*s - 3593750*p**3*q**3*s - 23812500*q**5*s + 4656250*p**4*q*r*s + 67109375*p*q**3*r*s - 16875000*p**2*q*r**2*s - 984375*p**5*s**2 - 19531250*p**2*q**2*s**2 - 37890625*p**3*r*s**2 - 7812500*q**2*r*s**2 + 289062500*p*r**2*s**2 - 529296875*p*q*s**3 + 2343750000*s**4 + + b[4][5] = 600*p**10*q**10 + 13850*p**7*q**12 + 106150*p**4*q**14 + 270000*p*q**16 - 9300*p**11*q**8*r - 234075*p**8*q**10*r - 1942825*p**5*q**12*r - 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23582031250*p**5*q*r**2*s**5 + 202441406250*p**2*q**3*r**2*s**5 - 383203125000*p**3*q*r**3*s**5 + 2232910156250*q**3*r**3*s**5 + 1500000000000*p*q*r**4*s**5 - 13710937500*p**8*s**6 - 202832031250*p**5*q**2*s**6 - 531738281250*p**2*q**4*s**6 + 73330078125*p**6*r*s**6 - 3906250000*p**3*q**2*r*s**6 - 1275878906250*q**4*r*s**6 - 121093750000*p**4*r**2*s**6 - 3308593750000*p*q**2*r**2*s**6 + 18066406250*p**2*r**3*s**6 - 244140625000*r**4*s**6 + 327148437500*p**4*q*s**7 + 1672363281250*p*q**3*s**7 + 446777343750*p**2*q*r*s**7 + 1232910156250*q*r**2*s**7 - 274658203125*p**3*s**8 - 1068115234375*q**2*s**8 - 61035156250*p*r*s**8 + + b[4][3] = 200*p**9*q**8 + 7550*p**6*q**10 + 78650*p**3*q**12 + 248400*q**14 - 4800*p**10*q**6*r - 164300*p**7*q**8*r - 1709575*p**4*q**10*r - 5566500*p*q**12*r + 31050*p**11*q**4*r**2 + 1116175*p**8*q**6*r**2 + 12674650*p**5*q**8*r**2 + 45333850*p**2*q**10*r**2 - 60750*p**12*q**2*r**3 - 2872725*p**9*q**4*r**3 - 40403050*p**6*q**6*r**3 - 173564375*p**3*q**8*r**3 - 11242250*q**10*r**3 + 2174100*p**10*q**2*r**4 + 54010000*p**7*q**4*r**4 + 331074875*p**4*q**6*r**4 + 114173750*p*q**8*r**4 - 24858500*p**8*q**2*r**5 - 300875000*p**5*q**4*r**5 - 319430625*p**2*q**6*r**5 + 69810000*p**6*q**2*r**6 - 23900000*p**3*q**4*r**6 - 294662500*q**6*r**6 + 524200000*p**4*q**2*r**7 + 1432000000*p*q**4*r**7 - 2340000000*p**2*q**2*r**8 + 5400*p**11*q**5*s + 310400*p**8*q**7*s + 3591725*p**5*q**9*s + 11556750*p**2*q**11*s - 105300*p**12*q**3*r*s - 4234650*p**9*q**5*r*s - 49928875*p**6*q**7*r*s - 174078125*p**3*q**9*r*s + 18000000*q**11*r*s + 364500*p**13*q*r**2*s + 15763050*p**10*q**3*r**2*s + 220187400*p**7*q**5*r**2*s + 929609375*p**4*q**7*r**2*s - 43653125*p*q**9*r**2*s - 13427100*p**11*q*r**3*s - 346066250*p**8*q**3*r**3*s - 2287673375*p**5*q**5*r**3*s - 1403903125*p**2*q**7*r**3*s + 184586000*p**9*q*r**4*s + 2983460000*p**6*q**3*r**4*s + 8725818750*p**3*q**5*r**4*s + 2527734375*q**7*r**4*s - 1284480000*p**7*q*r**5*s - 13138250000*p**4*q**3*r**5*s - 14001625000*p*q**5*r**5*s + 4224800000*p**5*q*r**6*s + 27460000000*p**2*q**3*r**6*s - 3760000000*p**3*q*r**7*s + 3900000000*q**3*r**7*s + 36450*p**13*q**2*s**2 + 2765475*p**10*q**4*s**2 + 34027625*p**7*q**6*s**2 + 97375000*p**4*q**8*s**2 - 88275000*p*q**10*s**2 - 546750*p**14*r*s**2 - 21961125*p**11*q**2*r*s**2 - 273059375*p**8*q**4*r*s**2 - 761562500*p**5*q**6*r*s**2 + 1869656250*p**2*q**8*r*s**2 + 20545650*p**12*r**2*s**2 + 473934375*p**9*q**2*r**2*s**2 + 1758053125*p**6*q**4*r**2*s**2 - 8743359375*p**3*q**6*r**2*s**2 - 4154375000*q**8*r**2*s**2 - 296559000*p**10*r**3*s**2 - 4065056250*p**7*q**2*r**3*s**2 - 186328125*p**4*q**4*r**3*s**2 + 19419453125*p*q**6*r**3*s**2 + 2326262500*p**8*r**4*s**2 + 21189375000*p**5*q**2*r**4*s**2 - 26301953125*p**2*q**4*r**4*s**2 - 10513250000*p**6*r**5*s**2 - 69937500000*p**3*q**2*r**5*s**2 - 42257812500*q**4*r**5*s**2 + 23375000000*p**4*r**6*s**2 + 40750000000*p*q**2*r**6*s**2 - 19500000000*p**2*r**7*s**2 + 4009500*p**12*q*s**3 + 36140625*p**9*q**3*s**3 - 335459375*p**6*q**5*s**3 - 2695312500*p**3*q**7*s**3 - 1486250000*q**9*s**3 + 102515625*p**10*q*r*s**3 + 4006812500*p**7*q**3*r*s**3 + 27589609375*p**4*q**5*r*s**3 + 20195312500*p*q**7*r*s**3 - 2792812500*p**8*q*r**2*s**3 - 44115156250*p**5*q**3*r**2*s**3 - 72609453125*p**2*q**5*r**2*s**3 + 18752500000*p**6*q*r**3*s**3 + 218140625000*p**3*q**3*r**3*s**3 + 109940234375*q**5*r**3*s**3 - 21893750000*p**4*q*r**4*s**3 - 65187500000*p*q**3*r**4*s**3 - 31000000000*p**2*q*r**5*s**3 + 97500000000*q*r**6*s**3 - 86568750*p**11*s**4 - 1955390625*p**8*q**2*s**4 - 8960781250*p**5*q**4*s**4 - 1357812500*p**2*q**6*s**4 + 1657968750*p**9*r*s**4 + 10467187500*p**6*q**2*r*s**4 - 55292968750*p**3*q**4*r*s**4 - 60683593750*q**6*r*s**4 - 11473593750*p**7*r**2*s**4 - 123281250000*p**4*q**2*r**2*s**4 - 164912109375*p*q**4*r**2*s**4 + 13150000000*p**5*r**3*s**4 + 190751953125*p**2*q**2*r**3*s**4 + 61875000000*p**3*r**4*s**4 - 467773437500*q**2*r**4*s**4 - 118750000000*p*r**5*s**4 + 7583203125*p**7*q*s**5 + 54638671875*p**4*q**3*s**5 + 39423828125*p*q**5*s**5 + 32392578125*p**5*q*r*s**5 + 278515625000*p**2*q**3*r*s**5 - 298339843750*p**3*q*r**2*s**5 + 560791015625*q**3*r**2*s**5 + 720703125000*p*q*r**3*s**5 - 19687500000*p**6*s**6 - 159667968750*p**3*q**2*s**6 - 72265625000*q**4*s**6 + 116699218750*p**4*r*s**6 - 924072265625*p*q**2*r*s**6 - 156005859375*p**2*r**2*s**6 - 112304687500*r**3*s**6 + 349121093750*p**2*q*s**7 + 396728515625*q*r*s**7 - 213623046875*p*s**8 + + b[4][2] = -600*p**10*q**6 - 18450*p**7*q**8 - 174000*p**4*q**10 - 518400*p*q**12 + 5400*p**11*q**4*r + 197550*p**8*q**6*r + 2147775*p**5*q**8*r + 7219800*p**2*q**10*r - 12150*p**12*q**2*r**2 - 662200*p**9*q**4*r**2 - 9274775*p**6*q**6*r**2 - 38330625*p**3*q**8*r**2 - 5508000*q**10*r**2 + 656550*p**10*q**2*r**3 + 16233750*p**7*q**4*r**3 + 97335875*p**4*q**6*r**3 + 58271250*p*q**8*r**3 - 9845500*p**8*q**2*r**4 - 119464375*p**5*q**4*r**4 - 194431875*p**2*q**6*r**4 + 49465000*p**6*q**2*r**5 + 166000000*p**3*q**4*r**5 - 80793750*q**6*r**5 + 54400000*p**4*q**2*r**6 + 377750000*p*q**4*r**6 - 630000000*p**2*q**2*r**7 - 16200*p**12*q**3*s - 459300*p**9*q**5*s - 4207225*p**6*q**7*s - 10827500*p**3*q**9*s + 13635000*q**11*s + 72900*p**13*q*r*s + 2877300*p**10*q**3*r*s + 33239700*p**7*q**5*r*s + 107080625*p**4*q**7*r*s - 114975000*p*q**9*r*s - 3601800*p**11*q*r**2*s - 75214375*p**8*q**3*r**2*s - 387073250*p**5*q**5*r**2*s + 55540625*p**2*q**7*r**2*s + 53793000*p**9*q*r**3*s + 687176875*p**6*q**3*r**3*s + 1670018750*p**3*q**5*r**3*s + 665234375*q**7*r**3*s - 391570000*p**7*q*r**4*s - 3420125000*p**4*q**3*r**4*s - 3609625000*p*q**5*r**4*s + 1365600000*p**5*q*r**5*s + 7236250000*p**2*q**3*r**5*s - 1220000000*p**3*q*r**6*s + 1050000000*q**3*r**6*s - 109350*p**14*s**2 - 3065850*p**11*q**2*s**2 - 26908125*p**8*q**4*s**2 - 44606875*p**5*q**6*s**2 + 269812500*p**2*q**8*s**2 + 5200200*p**12*r*s**2 + 81826875*p**9*q**2*r*s**2 + 155378125*p**6*q**4*r*s**2 - 1936203125*p**3*q**6*r*s**2 - 998437500*q**8*r*s**2 - 77145750*p**10*r**2*s**2 - 745528125*p**7*q**2*r**2*s**2 + 683437500*p**4*q**4*r**2*s**2 + 4083359375*p*q**6*r**2*s**2 + 593287500*p**8*r**3*s**2 + 4799375000*p**5*q**2*r**3*s**2 - 4167578125*p**2*q**4*r**3*s**2 - 2731125000*p**6*r**4*s**2 - 18668750000*p**3*q**2*r**4*s**2 - 10480468750*q**4*r**4*s**2 + 6200000000*p**4*r**5*s**2 + 11750000000*p*q**2*r**5*s**2 - 5250000000*p**2*r**6*s**2 + 26527500*p**10*q*s**3 + 526031250*p**7*q**3*s**3 + 3160703125*p**4*q**5*s**3 + 2650312500*p*q**7*s**3 - 448031250*p**8*q*r*s**3 - 6682968750*p**5*q**3*r*s**3 - 11642812500*p**2*q**5*r*s**3 + 2553203125*p**6*q*r**2*s**3 + 37234375000*p**3*q**3*r**2*s**3 + 21871484375*q**5*r**2*s**3 + 2803125000*p**4*q*r**3*s**3 - 10796875000*p*q**3*r**3*s**3 - 16656250000*p**2*q*r**4*s**3 + 26250000000*q*r**5*s**3 - 75937500*p**9*s**4 - 704062500*p**6*q**2*s**4 - 8363281250*p**3*q**4*s**4 - 10398437500*q**6*s**4 + 197578125*p**7*r*s**4 - 16441406250*p**4*q**2*r*s**4 - 24277343750*p*q**4*r*s**4 - 5716015625*p**5*r**2*s**4 + 31728515625*p**2*q**2*r**2*s**4 + 27031250000*p**3*r**3*s**4 - 92285156250*q**2*r**3*s**4 - 33593750000*p*r**4*s**4 + 10394531250*p**5*q*s**5 + 38037109375*p**2*q**3*s**5 - 48144531250*p**3*q*r*s**5 + 74462890625*q**3*r*s**5 + 121093750000*p*q*r**2*s**5 - 2197265625*p**4*s**6 - 92529296875*p*q**2*s**6 + 15380859375*p**2*r*s**6 - 31738281250*r**2*s**6 + 54931640625*q*s**7 + + b[4][1] = 200*p**8*q**6 + 2950*p**5*q**8 + 10800*p**2*q**10 - 1800*p**9*q**4*r - 49650*p**6*q**6*r - 403375*p**3*q**8*r - 999000*q**10*r + 4050*p**10*q**2*r**2 + 236625*p**7*q**4*r**2 + 3109500*p**4*q**6*r**2 + 11463750*p*q**8*r**2 - 331500*p**8*q**2*r**3 - 7818125*p**5*q**4*r**3 - 41411250*p**2*q**6*r**3 + 4782500*p**6*q**2*r**4 + 47475000*p**3*q**4*r**4 - 16728125*q**6*r**4 - 8700000*p**4*q**2*r**5 + 81750000*p*q**4*r**5 - 135000000*p**2*q**2*r**6 + 5400*p**10*q**3*s + 144200*p**7*q**5*s + 939375*p**4*q**7*s + 1012500*p*q**9*s - 24300*p**11*q*r*s - 1169250*p**8*q**3*r*s - 14027250*p**5*q**5*r*s - 44446875*p**2*q**7*r*s + 2011500*p**9*q*r**2*s + 49330625*p**6*q**3*r**2*s + 272009375*p**3*q**5*r**2*s + 104062500*q**7*r**2*s - 34660000*p**7*q*r**3*s - 455062500*p**4*q**3*r**3*s - 625906250*p*q**5*r**3*s + 210200000*p**5*q*r**4*s + 1298750000*p**2*q**3*r**4*s - 240000000*p**3*q*r**5*s + 225000000*q**3*r**5*s + 36450*p**12*s**2 + 1231875*p**9*q**2*s**2 + 10712500*p**6*q**4*s**2 + 21718750*p**3*q**6*s**2 + 16875000*q**8*s**2 - 2814750*p**10*r*s**2 - 67612500*p**7*q**2*r*s**2 - 345156250*p**4*q**4*r*s**2 - 283125000*p*q**6*r*s**2 + 51300000*p**8*r**2*s**2 + 734531250*p**5*q**2*r**2*s**2 + 1267187500*p**2*q**4*r**2*s**2 - 384312500*p**6*r**3*s**2 - 3912500000*p**3*q**2*r**3*s**2 - 1822265625*q**4*r**3*s**2 + 1112500000*p**4*r**4*s**2 + 2437500000*p*q**2*r**4*s**2 - 1125000000*p**2*r**5*s**2 - 72578125*p**5*q**3*s**3 - 189296875*p**2*q**5*s**3 + 127265625*p**6*q*r*s**3 + 1415625000*p**3*q**3*r*s**3 + 1229687500*q**5*r*s**3 + 1448437500*p**4*q*r**2*s**3 + 2218750000*p*q**3*r**2*s**3 - 4031250000*p**2*q*r**3*s**3 + 5625000000*q*r**4*s**3 - 132890625*p**7*s**4 - 529296875*p**4*q**2*s**4 - 175781250*p*q**4*s**4 - 401953125*p**5*r*s**4 - 4482421875*p**2*q**2*r*s**4 + 4140625000*p**3*r**2*s**4 - 10498046875*q**2*r**2*s**4 - 7031250000*p*r**3*s**4 + 1220703125*p**3*q*s**5 + 1953125000*q**3*s**5 + 14160156250*p*q*r*s**5 - 1708984375*p**2*s**6 - 3662109375*r*s**6 + + b[4][0] = -4600*p**6*q**6 - 67850*p**3*q**8 - 248400*q**10 + 38900*p**7*q**4*r + 679575*p**4*q**6*r + 2866500*p*q**8*r - 81900*p**8*q**2*r**2 - 2009750*p**5*q**4*r**2 - 10783750*p**2*q**6*r**2 + 1478750*p**6*q**2*r**3 + 14165625*p**3*q**4*r**3 - 2743750*q**6*r**3 - 5450000*p**4*q**2*r**4 + 12687500*p*q**4*r**4 - 22500000*p**2*q**2*r**5 - 101700*p**8*q**3*s - 1700975*p**5*q**5*s - 7061250*p**2*q**7*s + 423900*p**9*q*r*s + 9292375*p**6*q**3*r*s + 50438750*p**3*q**5*r*s + 20475000*q**7*r*s - 7852500*p**7*q*r**2*s - 87765625*p**4*q**3*r**2*s - 121609375*p*q**5*r**2*s + 47700000*p**5*q*r**3*s + 264687500*p**2*q**3*r**3*s - 65000000*p**3*q*r**4*s + 37500000*q**3*r**4*s - 534600*p**10*s**2 - 10344375*p**7*q**2*s**2 - 54859375*p**4*q**4*s**2 - 40312500*p*q**6*s**2 + 10158750*p**8*r*s**2 + 117778125*p**5*q**2*r*s**2 + 192421875*p**2*q**4*r*s**2 - 70593750*p**6*r**2*s**2 - 685312500*p**3*q**2*r**2*s**2 - 334375000*q**4*r**2*s**2 + 193750000*p**4*r**3*s**2 + 500000000*p*q**2*r**3*s**2 - 187500000*p**2*r**4*s**2 + 8437500*p**6*q*s**3 + 159218750*p**3*q**3*s**3 + 220625000*q**5*s**3 + 353828125*p**4*q*r*s**3 + 412500000*p*q**3*r*s**3 - 1023437500*p**2*q*r**2*s**3 + 937500000*q*r**3*s**3 - 206015625*p**5*s**4 - 701171875*p**2*q**2*s**4 + 998046875*p**3*r*s**4 - 1308593750*q**2*r*s**4 - 1367187500*p*r**2*s**4 + 1708984375*p*q*s**5 - 976562500*s**6 + + return b + + @property + def o(self): + p, q, r, s = self.p, self.q, self.r, self.s + o = [0]*6 + + o[5] = -1600*p**10*q**10 - 23600*p**7*q**12 - 86400*p**4*q**14 + 24800*p**11*q**8*r + 419200*p**8*q**10*r + 1850450*p**5*q**12*r + 896400*p**2*q**14*r - 138800*p**12*q**6*r**2 - 2921900*p**9*q**8*r**2 - 17295200*p**6*q**10*r**2 - 27127750*p**3*q**12*r**2 - 26076600*q**14*r**2 + 325800*p**13*q**4*r**3 + 9993850*p**10*q**6*r**3 + 88010500*p**7*q**8*r**3 + 274047650*p**4*q**10*r**3 + 410171400*p*q**12*r**3 - 259200*p**14*q**2*r**4 - 17147100*p**11*q**4*r**4 - 254289150*p**8*q**6*r**4 - 1318548225*p**5*q**8*r**4 - 2633598475*p**2*q**10*r**4 + 12636000*p**12*q**2*r**5 + 388911000*p**9*q**4*r**5 + 3269704725*p**6*q**6*r**5 + 8791192300*p**3*q**8*r**5 + 93560575*q**10*r**5 - 228361600*p**10*q**2*r**6 - 3951199200*p**7*q**4*r**6 - 16276981100*p**4*q**6*r**6 - 1597227000*p*q**8*r**6 + 1947899200*p**8*q**2*r**7 + 17037648000*p**5*q**4*r**7 + 8919740000*p**2*q**6*r**7 - 7672160000*p**6*q**2*r**8 - 15496000000*p**3*q**4*r**8 + 4224000000*q**6*r**8 + 9968000000*p**4*q**2*r**9 - 8640000000*p*q**4*r**9 + 4800000000*p**2*q**2*r**10 - 55200*p**12*q**7*s - 685600*p**9*q**9*s + 1028250*p**6*q**11*s + 37650000*p**3*q**13*s + 111375000*q**15*s + 583200*p**13*q**5*r*s + 9075600*p**10*q**7*r*s - 883150*p**7*q**9*r*s - 506830750*p**4*q**11*r*s - 1793137500*p*q**13*r*s - 1852200*p**14*q**3*r**2*s - 41435250*p**11*q**5*r**2*s - 80566700*p**8*q**7*r**2*s + 2485673600*p**5*q**9*r**2*s + 11442286125*p**2*q**11*r**2*s + 1555200*p**15*q*r**3*s + 80846100*p**12*q**3*r**3*s + 564906800*p**9*q**5*r**3*s - 4493012400*p**6*q**7*r**3*s - 35492391250*p**3*q**9*r**3*s - 789931875*q**11*r**3*s - 71766000*p**13*q*r**4*s - 1551149200*p**10*q**3*r**4*s - 1773437900*p**7*q**5*r**4*s + 51957593125*p**4*q**7*r**4*s + 14964765625*p*q**9*r**4*s + 1231569600*p**11*q*r**5*s + 12042977600*p**8*q**3*r**5*s - 27151011200*p**5*q**5*r**5*s - 88080610000*p**2*q**7*r**5*s - 9912995200*p**9*q*r**6*s - 29448104000*p**6*q**3*r**6*s + 144954840000*p**3*q**5*r**6*s - 44601300000*q**7*r**6*s + 35453760000*p**7*q*r**7*s - 63264000000*p**4*q**3*r**7*s + 60544000000*p*q**5*r**7*s - 30048000000*p**5*q*r**8*s + 37040000000*p**2*q**3*r**8*s - 60800000000*p**3*q*r**9*s - 48000000000*q**3*r**9*s - 615600*p**14*q**4*s**2 - 10524500*p**11*q**6*s**2 - 33831250*p**8*q**8*s**2 + 222806250*p**5*q**10*s**2 + 1099687500*p**2*q**12*s**2 + 3353400*p**15*q**2*r*s**2 + 74269350*p**12*q**4*r*s**2 + 276445750*p**9*q**6*r*s**2 - 2618600000*p**6*q**8*r*s**2 - 14473243750*p**3*q**10*r*s**2 + 1383750000*q**12*r*s**2 - 2332800*p**16*r**2*s**2 - 132750900*p**13*q**2*r**2*s**2 - 900775150*p**10*q**4*r**2*s**2 + 8249244500*p**7*q**6*r**2*s**2 + 59525796875*p**4*q**8*r**2*s**2 - 40292868750*p*q**10*r**2*s**2 + 128304000*p**14*r**3*s**2 + 3160232100*p**11*q**2*r**3*s**2 + 8329580000*p**8*q**4*r**3*s**2 - 45558458750*p**5*q**6*r**3*s**2 + 297252890625*p**2*q**8*r**3*s**2 - 2769854400*p**12*r**4*s**2 - 37065970000*p**9*q**2*r**4*s**2 - 90812546875*p**6*q**4*r**4*s**2 - 627902000000*p**3*q**6*r**4*s**2 + 181347421875*q**8*r**4*s**2 + 30946932800*p**10*r**5*s**2 + 249954680000*p**7*q**2*r**5*s**2 + 802954812500*p**4*q**4*r**5*s**2 - 80900000000*p*q**6*r**5*s**2 - 192137320000*p**8*r**6*s**2 - 932641600000*p**5*q**2*r**6*s**2 - 943242500000*p**2*q**4*r**6*s**2 + 658412000000*p**6*r**7*s**2 + 1930720000000*p**3*q**2*r**7*s**2 + 593800000000*q**4*r**7*s**2 - 1162800000000*p**4*r**8*s**2 - 280000000000*p*q**2*r**8*s**2 + 840000000000*p**2*r**9*s**2 - 2187000*p**16*q*s**3 - 47418750*p**13*q**3*s**3 - 180618750*p**10*q**5*s**3 + 2231250000*p**7*q**7*s**3 + 17857734375*p**4*q**9*s**3 + 29882812500*p*q**11*s**3 + 24664500*p**14*q*r*s**3 - 853368750*p**11*q**3*r*s**3 - 25939693750*p**8*q**5*r*s**3 - 177541562500*p**5*q**7*r*s**3 - 297978828125*p**2*q**9*r*s**3 - 153468000*p**12*q*r**2*s**3 + 30188125000*p**9*q**3*r**2*s**3 + 344049821875*p**6*q**5*r**2*s**3 + 534026875000*p**3*q**7*r**2*s**3 - 340726484375*q**9*r**2*s**3 - 9056190000*p**10*q*r**3*s**3 - 322314687500*p**7*q**3*r**3*s**3 - 769632109375*p**4*q**5*r**3*s**3 - 83276875000*p*q**7*r**3*s**3 + 164061000000*p**8*q*r**4*s**3 + 1381358750000*p**5*q**3*r**4*s**3 + 3088020000000*p**2*q**5*r**4*s**3 - 1267655000000*p**6*q*r**5*s**3 - 7642630000000*p**3*q**3*r**5*s**3 - 2759877500000*q**5*r**5*s**3 + 4597760000000*p**4*q*r**6*s**3 + 1846200000000*p*q**3*r**6*s**3 - 7006000000000*p**2*q*r**7*s**3 - 1200000000000*q*r**8*s**3 + 18225000*p**15*s**4 + 1328906250*p**12*q**2*s**4 + 24729140625*p**9*q**4*s**4 + 169467187500*p**6*q**6*s**4 + 413281250000*p**3*q**8*s**4 + 223828125000*q**10*s**4 + 710775000*p**13*r*s**4 - 18611015625*p**10*q**2*r*s**4 - 314344375000*p**7*q**4*r*s**4 - 828439843750*p**4*q**6*r*s**4 + 460937500000*p*q**8*r*s**4 - 25674975000*p**11*r**2*s**4 - 52223515625*p**8*q**2*r**2*s**4 - 387160000000*p**5*q**4*r**2*s**4 - 4733680078125*p**2*q**6*r**2*s**4 + 343911875000*p**9*r**3*s**4 + 3328658359375*p**6*q**2*r**3*s**4 + 16532406250000*p**3*q**4*r**3*s**4 + 5980613281250*q**6*r**3*s**4 - 2295497500000*p**7*r**4*s**4 - 14809820312500*p**4*q**2*r**4*s**4 - 6491406250000*p*q**4*r**4*s**4 + 7768470000000*p**5*r**5*s**4 + 34192562500000*p**2*q**2*r**5*s**4 - 11859000000000*p**3*r**6*s**4 + 10530000000000*q**2*r**6*s**4 + 6000000000000*p*r**7*s**4 + 11453906250*p**11*q*s**5 + 149765625000*p**8*q**3*s**5 + 545537109375*p**5*q**5*s**5 + 527343750000*p**2*q**7*s**5 - 371313281250*p**9*q*r*s**5 - 3461455078125*p**6*q**3*r*s**5 - 7920878906250*p**3*q**5*r*s**5 - 4747314453125*q**7*r*s**5 + 2417815625000*p**7*q*r**2*s**5 + 5465576171875*p**4*q**3*r**2*s**5 + 5937128906250*p*q**5*r**2*s**5 - 10661156250000*p**5*q*r**3*s**5 - 63574218750000*p**2*q**3*r**3*s**5 + 24059375000000*p**3*q*r**4*s**5 - 33023437500000*q**3*r**4*s**5 - 43125000000000*p*q*r**5*s**5 + 94394531250*p**10*s**6 + 1097167968750*p**7*q**2*s**6 + 2829833984375*p**4*q**4*s**6 - 1525878906250*p*q**6*s**6 + 2724609375*p**8*r*s**6 + 13998535156250*p**5*q**2*r*s**6 + 57094482421875*p**2*q**4*r*s**6 - 8512509765625*p**6*r**2*s**6 - 37941406250000*p**3*q**2*r**2*s**6 + 33191894531250*q**4*r**2*s**6 + 50534179687500*p**4*r**3*s**6 + 156656250000000*p*q**2*r**3*s**6 - 85023437500000*p**2*r**4*s**6 + 10125000000000*r**5*s**6 - 2717285156250*p**6*q*s**7 - 11352539062500*p**3*q**3*s**7 - 2593994140625*q**5*s**7 - 47154541015625*p**4*q*r*s**7 - 160644531250000*p*q**3*r*s**7 + 142500000000000*p**2*q*r**2*s**7 - 26757812500000*q*r**3*s**7 - 4364013671875*p**5*s**8 - 94604492187500*p**2*q**2*s**8 + 114379882812500*p**3*r*s**8 + 51116943359375*q**2*r*s**8 - 346435546875000*p*r**2*s**8 + 476837158203125*p*q*s**9 - 476837158203125*s**10 + + o[4] = 1600*p**11*q**8 + 20800*p**8*q**10 + 45100*p**5*q**12 - 151200*p**2*q**14 - 19200*p**12*q**6*r - 293200*p**9*q**8*r - 794600*p**6*q**10*r + 2634675*p**3*q**12*r + 2640600*q**14*r + 75600*p**13*q**4*r**2 + 1529100*p**10*q**6*r**2 + 6233350*p**7*q**8*r**2 - 12013350*p**4*q**10*r**2 - 29069550*p*q**12*r**2 - 97200*p**14*q**2*r**3 - 3562500*p**11*q**4*r**3 - 26984900*p**8*q**6*r**3 - 15900325*p**5*q**8*r**3 + 76267100*p**2*q**10*r**3 + 3272400*p**12*q**2*r**4 + 59486850*p**9*q**4*r**4 + 221270075*p**6*q**6*r**4 + 74065250*p**3*q**8*r**4 - 300564375*q**10*r**4 - 45569400*p**10*q**2*r**5 - 438666000*p**7*q**4*r**5 - 444821250*p**4*q**6*r**5 + 2448256250*p*q**8*r**5 + 290640000*p**8*q**2*r**6 + 855850000*p**5*q**4*r**6 - 5741875000*p**2*q**6*r**6 - 644000000*p**6*q**2*r**7 + 5574000000*p**3*q**4*r**7 + 4643000000*q**6*r**7 - 1696000000*p**4*q**2*r**8 - 12660000000*p*q**4*r**8 + 7200000000*p**2*q**2*r**9 + 43200*p**13*q**5*s + 572000*p**10*q**7*s - 59800*p**7*q**9*s - 24174625*p**4*q**11*s - 74587500*p*q**13*s - 324000*p**14*q**3*r*s - 5531400*p**11*q**5*r*s - 3712100*p**8*q**7*r*s + 293009275*p**5*q**9*r*s + 1115548875*p**2*q**11*r*s + 583200*p**15*q*r**2*s + 18343800*p**12*q**3*r**2*s + 77911100*p**9*q**5*r**2*s - 957488825*p**6*q**7*r**2*s - 5449661250*p**3*q**9*r**2*s + 960120000*q**11*r**2*s - 23684400*p**13*q*r**3*s - 373761900*p**10*q**3*r**3*s - 27944975*p**7*q**5*r**3*s + 10375740625*p**4*q**7*r**3*s - 4649093750*p*q**9*r**3*s + 395816400*p**11*q*r**4*s + 2910968000*p**8*q**3*r**4*s - 9126162500*p**5*q**5*r**4*s - 11696118750*p**2*q**7*r**4*s - 3028640000*p**9*q*r**5*s - 3251550000*p**6*q**3*r**5*s + 47914250000*p**3*q**5*r**5*s - 30255625000*q**7*r**5*s + 9304000000*p**7*q*r**6*s - 42970000000*p**4*q**3*r**6*s + 31475000000*p*q**5*r**6*s + 2176000000*p**5*q*r**7*s + 62100000000*p**2*q**3*r**7*s - 43200000000*p**3*q*r**8*s - 72000000000*q**3*r**8*s + 291600*p**15*q**2*s**2 + 2702700*p**12*q**4*s**2 - 38692250*p**9*q**6*s**2 - 538903125*p**6*q**8*s**2 - 1613112500*p**3*q**10*s**2 + 320625000*q**12*s**2 - 874800*p**16*r*s**2 - 14166900*p**13*q**2*r*s**2 + 193284900*p**10*q**4*r*s**2 + 3688520500*p**7*q**6*r*s**2 + 11613390625*p**4*q**8*r*s**2 - 15609881250*p*q**10*r*s**2 + 44031600*p**14*r**2*s**2 + 482345550*p**11*q**2*r**2*s**2 - 2020881875*p**8*q**4*r**2*s**2 - 7407026250*p**5*q**6*r**2*s**2 + 136175750000*p**2*q**8*r**2*s**2 - 1000884600*p**12*r**3*s**2 - 8888950000*p**9*q**2*r**3*s**2 - 30101703125*p**6*q**4*r**3*s**2 - 319761000000*p**3*q**6*r**3*s**2 + 51519218750*q**8*r**3*s**2 + 12622395000*p**10*r**4*s**2 + 97032450000*p**7*q**2*r**4*s**2 + 469929218750*p**4*q**4*r**4*s**2 + 291342187500*p*q**6*r**4*s**2 - 96382000000*p**8*r**5*s**2 - 598070000000*p**5*q**2*r**5*s**2 - 1165021875000*p**2*q**4*r**5*s**2 + 446500000000*p**6*r**6*s**2 + 1651500000000*p**3*q**2*r**6*s**2 + 789375000000*q**4*r**6*s**2 - 1152000000000*p**4*r**7*s**2 - 600000000000*p*q**2*r**7*s**2 + 1260000000000*p**2*r**8*s**2 - 24786000*p**14*q*s**3 - 660487500*p**11*q**3*s**3 - 5886356250*p**8*q**5*s**3 - 18137187500*p**5*q**7*s**3 - 5120546875*p**2*q**9*s**3 + 827658000*p**12*q*r*s**3 + 13343062500*p**9*q**3*r*s**3 + 39782068750*p**6*q**5*r*s**3 - 111288437500*p**3*q**7*r*s**3 - 15438750000*q**9*r*s**3 - 14540782500*p**10*q*r**2*s**3 - 135889750000*p**7*q**3*r**2*s**3 - 176892578125*p**4*q**5*r**2*s**3 - 934462656250*p*q**7*r**2*s**3 + 171669250000*p**8*q*r**3*s**3 + 1164538125000*p**5*q**3*r**3*s**3 + 3192346406250*p**2*q**5*r**3*s**3 - 1295476250000*p**6*q*r**4*s**3 - 6540712500000*p**3*q**3*r**4*s**3 - 2957828125000*q**5*r**4*s**3 + 5366750000000*p**4*q*r**5*s**3 + 3165000000000*p*q**3*r**5*s**3 - 8862500000000*p**2*q*r**6*s**3 - 1800000000000*q*r**7*s**3 + 236925000*p**13*s**4 + 8895234375*p**10*q**2*s**4 + 106180781250*p**7*q**4*s**4 + 474221875000*p**4*q**6*s**4 + 616210937500*p*q**8*s**4 - 6995868750*p**11*r*s**4 - 184190625000*p**8*q**2*r*s**4 - 1299254453125*p**5*q**4*r*s**4 - 2475458593750*p**2*q**6*r*s**4 + 63049218750*p**9*r**2*s**4 + 1646791484375*p**6*q**2*r**2*s**4 + 9086886718750*p**3*q**4*r**2*s**4 + 4673421875000*q**6*r**2*s**4 - 215665000000*p**7*r**3*s**4 - 7864589843750*p**4*q**2*r**3*s**4 - 5987890625000*p*q**4*r**3*s**4 + 594843750000*p**5*r**4*s**4 + 27791171875000*p**2*q**2*r**4*s**4 - 3881250000000*p**3*r**5*s**4 + 12203125000000*q**2*r**5*s**4 + 10312500000000*p*r**6*s**4 - 34720312500*p**9*q*s**5 - 545126953125*p**6*q**3*s**5 - 2176425781250*p**3*q**5*s**5 - 2792968750000*q**7*s**5 - 1395703125*p**7*q*r*s**5 - 1957568359375*p**4*q**3*r*s**5 + 5122636718750*p*q**5*r*s**5 + 858210937500*p**5*q*r**2*s**5 - 42050097656250*p**2*q**3*r**2*s**5 + 7088281250000*p**3*q*r**3*s**5 - 25974609375000*q**3*r**3*s**5 - 69296875000000*p*q*r**4*s**5 + 384697265625*p**8*s**6 + 6403320312500*p**5*q**2*s**6 + 16742675781250*p**2*q**4*s**6 - 3467080078125*p**6*r*s**6 + 11009765625000*p**3*q**2*r*s**6 + 16451660156250*q**4*r*s**6 + 6979003906250*p**4*r**2*s**6 + 145403320312500*p*q**2*r**2*s**6 + 4076171875000*p**2*r**3*s**6 + 22265625000000*r**4*s**6 - 21915283203125*p**4*q*s**7 - 86608886718750*p*q**3*s**7 - 22785644531250*p**2*q*r*s**7 - 103466796875000*q*r**2*s**7 + 18798828125000*p**3*s**8 + 106048583984375*q**2*s**8 + 17761230468750*p*r*s**8 + + o[3] = 2800*p**9*q**8 + 55700*p**6*q**10 + 363600*p**3*q**12 + 777600*q**14 - 27200*p**10*q**6*r - 700200*p**7*q**8*r - 5726550*p**4*q**10*r - 15066000*p*q**12*r + 74700*p**11*q**4*r**2 + 2859575*p**8*q**6*r**2 + 31175725*p**5*q**8*r**2 + 103147650*p**2*q**10*r**2 - 40500*p**12*q**2*r**3 - 4274400*p**9*q**4*r**3 - 76065825*p**6*q**6*r**3 - 365623750*p**3*q**8*r**3 - 132264000*q**10*r**3 + 2192400*p**10*q**2*r**4 + 92562500*p**7*q**4*r**4 + 799193875*p**4*q**6*r**4 + 1188193125*p*q**8*r**4 - 41231500*p**8*q**2*r**5 - 914210000*p**5*q**4*r**5 - 3318853125*p**2*q**6*r**5 + 398850000*p**6*q**2*r**6 + 3944000000*p**3*q**4*r**6 + 2211312500*q**6*r**6 - 1817000000*p**4*q**2*r**7 - 6720000000*p*q**4*r**7 + 3900000000*p**2*q**2*r**8 + 75600*p**11*q**5*s + 1823100*p**8*q**7*s + 14534150*p**5*q**9*s + 38265750*p**2*q**11*s - 394200*p**12*q**3*r*s - 11453850*p**9*q**5*r*s - 101213000*p**6*q**7*r*s - 223565625*p**3*q**9*r*s + 415125000*q**11*r*s + 243000*p**13*q*r**2*s + 13654575*p**10*q**3*r**2*s + 163811725*p**7*q**5*r**2*s + 173461250*p**4*q**7*r**2*s - 3008671875*p*q**9*r**2*s - 2016900*p**11*q*r**3*s - 86576250*p**8*q**3*r**3*s - 324146625*p**5*q**5*r**3*s + 3378506250*p**2*q**7*r**3*s - 89211000*p**9*q*r**4*s - 55207500*p**6*q**3*r**4*s + 1493950000*p**3*q**5*r**4*s - 12573609375*q**7*r**4*s + 1140100000*p**7*q*r**5*s + 42500000*p**4*q**3*r**5*s + 21511250000*p*q**5*r**5*s - 4058000000*p**5*q*r**6*s + 6725000000*p**2*q**3*r**6*s - 1400000000*p**3*q*r**7*s - 39000000000*q**3*r**7*s + 510300*p**13*q**2*s**2 + 4814775*p**10*q**4*s**2 - 70265125*p**7*q**6*s**2 - 1016484375*p**4*q**8*s**2 - 3221100000*p*q**10*s**2 - 364500*p**14*r*s**2 + 30314250*p**11*q**2*r*s**2 + 1106765625*p**8*q**4*r*s**2 + 10984203125*p**5*q**6*r*s**2 + 33905812500*p**2*q**8*r*s**2 - 37980900*p**12*r**2*s**2 - 2142905625*p**9*q**2*r**2*s**2 - 26896125000*p**6*q**4*r**2*s**2 - 95551328125*p**3*q**6*r**2*s**2 + 11320312500*q**8*r**2*s**2 + 1743781500*p**10*r**3*s**2 + 35432262500*p**7*q**2*r**3*s**2 + 177855859375*p**4*q**4*r**3*s**2 + 121260546875*p*q**6*r**3*s**2 - 25943162500*p**8*r**4*s**2 - 249165500000*p**5*q**2*r**4*s**2 - 461739453125*p**2*q**4*r**4*s**2 + 177823750000*p**6*r**5*s**2 + 726225000000*p**3*q**2*r**5*s**2 + 404195312500*q**4*r**5*s**2 - 565875000000*p**4*r**6*s**2 - 407500000000*p*q**2*r**6*s**2 + 682500000000*p**2*r**7*s**2 - 59140125*p**12*q*s**3 - 1290515625*p**9*q**3*s**3 - 8785071875*p**6*q**5*s**3 - 15588281250*p**3*q**7*s**3 + 17505000000*q**9*s**3 + 896062500*p**10*q*r*s**3 + 2589750000*p**7*q**3*r*s**3 - 82700156250*p**4*q**5*r*s**3 - 347683593750*p*q**7*r*s**3 + 17022656250*p**8*q*r**2*s**3 + 320923593750*p**5*q**3*r**2*s**3 + 1042116875000*p**2*q**5*r**2*s**3 - 353262812500*p**6*q*r**3*s**3 - 2212664062500*p**3*q**3*r**3*s**3 - 1252408984375*q**5*r**3*s**3 + 1967362500000*p**4*q*r**4*s**3 + 1583343750000*p*q**3*r**4*s**3 - 3560625000000*p**2*q*r**5*s**3 - 975000000000*q*r**6*s**3 + 462459375*p**11*s**4 + 14210859375*p**8*q**2*s**4 + 99521718750*p**5*q**4*s**4 + 114955468750*p**2*q**6*s**4 - 17720859375*p**9*r*s**4 - 100320703125*p**6*q**2*r*s**4 + 1021943359375*p**3*q**4*r*s**4 + 1193203125000*q**6*r*s**4 + 171371250000*p**7*r**2*s**4 - 1113390625000*p**4*q**2*r**2*s**4 - 1211474609375*p*q**4*r**2*s**4 - 274056250000*p**5*r**3*s**4 + 8285166015625*p**2*q**2*r**3*s**4 - 2079375000000*p**3*r**4*s**4 + 5137304687500*q**2*r**4*s**4 + 6187500000000*p*r**5*s**4 - 135675000000*p**7*q*s**5 - 1275244140625*p**4*q**3*s**5 - 28388671875*p*q**5*s**5 + 1015166015625*p**5*q*r*s**5 - 10584423828125*p**2*q**3*r*s**5 + 3559570312500*p**3*q*r**2*s**5 - 6929931640625*q**3*r**2*s**5 - 32304687500000*p*q*r**3*s**5 + 430576171875*p**6*s**6 + 9397949218750*p**3*q**2*s**6 + 575195312500*q**4*s**6 - 4086425781250*p**4*r*s**6 + 42183837890625*p*q**2*r*s**6 + 8156494140625*p**2*r**2*s**6 + 12612304687500*r**3*s**6 - 25513916015625*p**2*q*s**7 - 37017822265625*q*r*s**7 + 18981933593750*p*s**8 + + o[2] = 1600*p**10*q**6 + 9200*p**7*q**8 - 126000*p**4*q**10 - 777600*p*q**12 - 14400*p**11*q**4*r - 119300*p**8*q**6*r + 1203225*p**5*q**8*r + 9412200*p**2*q**10*r + 32400*p**12*q**2*r**2 + 417950*p**9*q**4*r**2 - 4543725*p**6*q**6*r**2 - 49008125*p**3*q**8*r**2 - 24192000*q**10*r**2 - 292050*p**10*q**2*r**3 + 8760000*p**7*q**4*r**3 + 137506625*p**4*q**6*r**3 + 225438750*p*q**8*r**3 - 4213250*p**8*q**2*r**4 - 173595625*p**5*q**4*r**4 - 653003125*p**2*q**6*r**4 + 82575000*p**6*q**2*r**5 + 838125000*p**3*q**4*r**5 + 578562500*q**6*r**5 - 421500000*p**4*q**2*r**6 - 1796250000*p*q**4*r**6 + 1050000000*p**2*q**2*r**7 + 43200*p**12*q**3*s + 807300*p**9*q**5*s + 5328225*p**6*q**7*s + 16946250*p**3*q**9*s + 29565000*q**11*s - 194400*p**13*q*r*s - 5505300*p**10*q**3*r*s - 49886700*p**7*q**5*r*s - 178821875*p**4*q**7*r*s - 222750000*p*q**9*r*s + 6814800*p**11*q*r**2*s + 120525625*p**8*q**3*r**2*s + 526694500*p**5*q**5*r**2*s + 84065625*p**2*q**7*r**2*s - 123670500*p**9*q*r**3*s - 1106731875*p**6*q**3*r**3*s - 669556250*p**3*q**5*r**3*s - 2869265625*q**7*r**3*s + 1004350000*p**7*q*r**4*s + 3384375000*p**4*q**3*r**4*s + 5665625000*p*q**5*r**4*s - 3411000000*p**5*q*r**5*s - 418750000*p**2*q**3*r**5*s + 1700000000*p**3*q*r**6*s - 10500000000*q**3*r**6*s + 291600*p**14*s**2 + 9829350*p**11*q**2*s**2 + 114151875*p**8*q**4*s**2 + 522169375*p**5*q**6*s**2 + 716906250*p**2*q**8*s**2 - 18625950*p**12*r*s**2 - 387703125*p**9*q**2*r*s**2 - 2056109375*p**6*q**4*r*s**2 - 760203125*p**3*q**6*r*s**2 + 3071250000*q**8*r*s**2 + 512419500*p**10*r**2*s**2 + 5859053125*p**7*q**2*r**2*s**2 + 12154062500*p**4*q**4*r**2*s**2 + 15931640625*p*q**6*r**2*s**2 - 6598393750*p**8*r**3*s**2 - 43549625000*p**5*q**2*r**3*s**2 - 82011328125*p**2*q**4*r**3*s**2 + 43538125000*p**6*r**4*s**2 + 160831250000*p**3*q**2*r**4*s**2 + 99070312500*q**4*r**4*s**2 - 141812500000*p**4*r**5*s**2 - 117500000000*p*q**2*r**5*s**2 + 183750000000*p**2*r**6*s**2 - 154608750*p**10*q*s**3 - 3309468750*p**7*q**3*s**3 - 20834140625*p**4*q**5*s**3 - 34731562500*p*q**7*s**3 + 5970375000*p**8*q*r*s**3 + 68533281250*p**5*q**3*r*s**3 + 142698281250*p**2*q**5*r*s**3 - 74509140625*p**6*q*r**2*s**3 - 389148437500*p**3*q**3*r**2*s**3 - 270937890625*q**5*r**2*s**3 + 366696875000*p**4*q*r**3*s**3 + 400031250000*p*q**3*r**3*s**3 - 735156250000*p**2*q*r**4*s**3 - 262500000000*q*r**5*s**3 + 371250000*p**9*s**4 + 21315000000*p**6*q**2*s**4 + 179515625000*p**3*q**4*s**4 + 238406250000*q**6*s**4 - 9071015625*p**7*r*s**4 - 268945312500*p**4*q**2*r*s**4 - 379785156250*p*q**4*r*s**4 + 140262890625*p**5*r**2*s**4 + 1486259765625*p**2*q**2*r**2*s**4 - 806484375000*p**3*r**3*s**4 + 1066210937500*q**2*r**3*s**4 + 1722656250000*p*r**4*s**4 - 125648437500*p**5*q*s**5 - 1236279296875*p**2*q**3*s**5 + 1267871093750*p**3*q*r*s**5 - 1044677734375*q**3*r*s**5 - 6630859375000*p*q*r**2*s**5 + 160888671875*p**4*s**6 + 6352294921875*p*q**2*s**6 - 708740234375*p**2*r*s**6 + 3901367187500*r**2*s**6 - 8050537109375*q*s**7 + + o[1] = 2800*p**8*q**6 + 41300*p**5*q**8 + 151200*p**2*q**10 - 25200*p**9*q**4*r - 542600*p**6*q**6*r - 3397875*p**3*q**8*r - 5751000*q**10*r + 56700*p**10*q**2*r**2 + 1972125*p**7*q**4*r**2 + 18624250*p**4*q**6*r**2 + 50253750*p*q**8*r**2 - 1701000*p**8*q**2*r**3 - 32630625*p**5*q**4*r**3 - 139868750*p**2*q**6*r**3 + 18162500*p**6*q**2*r**4 + 177125000*p**3*q**4*r**4 + 121734375*q**6*r**4 - 100500000*p**4*q**2*r**5 - 386250000*p*q**4*r**5 + 225000000*p**2*q**2*r**6 + 75600*p**10*q**3*s + 1708800*p**7*q**5*s + 12836875*p**4*q**7*s + 32062500*p*q**9*s - 340200*p**11*q*r*s - 10185750*p**8*q**3*r*s - 97502750*p**5*q**5*r*s - 301640625*p**2*q**7*r*s + 7168500*p**9*q*r**2*s + 135960625*p**6*q**3*r**2*s + 587471875*p**3*q**5*r**2*s - 384750000*q**7*r**2*s - 29325000*p**7*q*r**3*s - 320625000*p**4*q**3*r**3*s + 523437500*p*q**5*r**3*s - 42000000*p**5*q*r**4*s + 343750000*p**2*q**3*r**4*s + 150000000*p**3*q*r**5*s - 2250000000*q**3*r**5*s + 510300*p**12*s**2 + 12808125*p**9*q**2*s**2 + 107062500*p**6*q**4*s**2 + 270312500*p**3*q**6*s**2 - 168750000*q**8*s**2 - 2551500*p**10*r*s**2 - 5062500*p**7*q**2*r*s**2 + 712343750*p**4*q**4*r*s**2 + 4788281250*p*q**6*r*s**2 - 256837500*p**8*r**2*s**2 - 3574812500*p**5*q**2*r**2*s**2 - 14967968750*p**2*q**4*r**2*s**2 + 4040937500*p**6*r**3*s**2 + 26400000000*p**3*q**2*r**3*s**2 + 17083984375*q**4*r**3*s**2 - 21812500000*p**4*r**4*s**2 - 24375000000*p*q**2*r**4*s**2 + 39375000000*p**2*r**5*s**2 - 127265625*p**5*q**3*s**3 - 680234375*p**2*q**5*s**3 - 2048203125*p**6*q*r*s**3 - 18794531250*p**3*q**3*r*s**3 - 25050000000*q**5*r*s**3 + 26621875000*p**4*q*r**2*s**3 + 37007812500*p*q**3*r**2*s**3 - 105468750000*p**2*q*r**3*s**3 - 56250000000*q*r**4*s**3 + 1124296875*p**7*s**4 + 9251953125*p**4*q**2*s**4 - 8007812500*p*q**4*s**4 - 4004296875*p**5*r*s**4 + 179931640625*p**2*q**2*r*s**4 - 75703125000*p**3*r**2*s**4 + 133447265625*q**2*r**2*s**4 + 363281250000*p*r**3*s**4 - 91552734375*p**3*q*s**5 - 19531250000*q**3*s**5 - 751953125000*p*q*r*s**5 + 157958984375*p**2*s**6 + 748291015625*r*s**6 + + o[0] = -14400*p**6*q**6 - 212400*p**3*q**8 - 777600*q**10 + 92100*p**7*q**4*r + 1689675*p**4*q**6*r + 7371000*p*q**8*r - 122850*p**8*q**2*r**2 - 3735250*p**5*q**4*r**2 - 22432500*p**2*q**6*r**2 + 2298750*p**6*q**2*r**3 + 29390625*p**3*q**4*r**3 + 18000000*q**6*r**3 - 17750000*p**4*q**2*r**4 - 62812500*p*q**4*r**4 + 37500000*p**2*q**2*r**5 - 51300*p**8*q**3*s - 768025*p**5*q**5*s - 2801250*p**2*q**7*s - 275400*p**9*q*r*s - 5479875*p**6*q**3*r*s - 35538750*p**3*q**5*r*s - 68850000*q**7*r*s + 12757500*p**7*q*r**2*s + 133640625*p**4*q**3*r**2*s + 222609375*p*q**5*r**2*s - 108500000*p**5*q*r**3*s - 290312500*p**2*q**3*r**3*s + 275000000*p**3*q*r**4*s - 375000000*q**3*r**4*s + 1931850*p**10*s**2 + 40213125*p**7*q**2*s**2 + 253921875*p**4*q**4*s**2 + 464062500*p*q**6*s**2 - 71077500*p**8*r*s**2 - 818746875*p**5*q**2*r*s**2 - 1882265625*p**2*q**4*r*s**2 + 826031250*p**6*r**2*s**2 + 4369687500*p**3*q**2*r**2*s**2 + 3107812500*q**4*r**2*s**2 - 3943750000*p**4*r**3*s**2 - 5000000000*p*q**2*r**3*s**2 + 6562500000*p**2*r**4*s**2 - 295312500*p**6*q*s**3 - 2938906250*p**3*q**3*s**3 - 4848750000*q**5*s**3 + 3791484375*p**4*q*r*s**3 + 7556250000*p*q**3*r*s**3 - 11960937500*p**2*q*r**2*s**3 - 9375000000*q*r**3*s**3 + 1668515625*p**5*s**4 + 20447265625*p**2*q**2*s**4 - 21955078125*p**3*r*s**4 + 18984375000*q**2*r*s**4 + 67382812500*p*r**2*s**4 - 120849609375*p*q*s**5 + 157226562500*s**6 + + return o + + @property + def a(self): + p, q, r, s = self.p, self.q, self.r, self.s + a = [0]*6 + + a[5] = -100*p**7*q**7 - 2175*p**4*q**9 - 10500*p*q**11 + 1100*p**8*q**5*r + 27975*p**5*q**7*r + 152950*p**2*q**9*r - 4125*p**9*q**3*r**2 - 128875*p**6*q**5*r**2 - 830525*p**3*q**7*r**2 + 59450*q**9*r**2 + 5400*p**10*q*r**3 + 243800*p**7*q**3*r**3 + 2082650*p**4*q**5*r**3 - 333925*p*q**7*r**3 - 139200*p**8*q*r**4 - 2406000*p**5*q**3*r**4 - 122600*p**2*q**5*r**4 + 1254400*p**6*q*r**5 + 3776000*p**3*q**3*r**5 + 1832000*q**5*r**5 - 4736000*p**4*q*r**6 - 6720000*p*q**3*r**6 + 6400000*p**2*q*r**7 - 900*p**9*q**4*s - 37400*p**6*q**6*s - 281625*p**3*q**8*s - 435000*q**10*s + 6750*p**10*q**2*r*s + 322300*p**7*q**4*r*s + 2718575*p**4*q**6*r*s + 4214250*p*q**8*r*s - 16200*p**11*r**2*s - 859275*p**8*q**2*r**2*s - 8925475*p**5*q**4*r**2*s - 14427875*p**2*q**6*r**2*s + 453600*p**9*r**3*s + 10038400*p**6*q**2*r**3*s + 17397500*p**3*q**4*r**3*s - 11333125*q**6*r**3*s - 4451200*p**7*r**4*s - 15850000*p**4*q**2*r**4*s + 34000000*p*q**4*r**4*s + 17984000*p**5*r**5*s - 10000000*p**2*q**2*r**5*s - 25600000*p**3*r**6*s - 8000000*q**2*r**6*s + 6075*p**11*q*s**2 - 83250*p**8*q**3*s**2 - 1282500*p**5*q**5*s**2 - 2862500*p**2*q**7*s**2 + 724275*p**9*q*r*s**2 + 9807250*p**6*q**3*r*s**2 + 28374375*p**3*q**5*r*s**2 + 22212500*q**7*r*s**2 - 8982000*p**7*q*r**2*s**2 - 39600000*p**4*q**3*r**2*s**2 - 61746875*p*q**5*r**2*s**2 - 1010000*p**5*q*r**3*s**2 - 1000000*p**2*q**3*r**3*s**2 + 78000000*p**3*q*r**4*s**2 + 30000000*q**3*r**4*s**2 + 80000000*p*q*r**5*s**2 - 759375*p**10*s**3 - 9787500*p**7*q**2*s**3 - 39062500*p**4*q**4*s**3 - 52343750*p*q**6*s**3 + 12301875*p**8*r*s**3 + 98175000*p**5*q**2*r*s**3 + 225078125*p**2*q**4*r*s**3 - 54900000*p**6*r**2*s**3 - 310000000*p**3*q**2*r**2*s**3 - 7890625*q**4*r**2*s**3 + 51250000*p**4*r**3*s**3 - 420000000*p*q**2*r**3*s**3 + 110000000*p**2*r**4*s**3 - 200000000*r**5*s**3 + 2109375*p**6*q*s**4 - 21093750*p**3*q**3*s**4 - 89843750*q**5*s**4 + 182343750*p**4*q*r*s**4 + 733203125*p*q**3*r*s**4 - 196875000*p**2*q*r**2*s**4 + 1125000000*q*r**3*s**4 - 158203125*p**5*s**5 - 566406250*p**2*q**2*s**5 + 101562500*p**3*r*s**5 - 1669921875*q**2*r*s**5 + 1250000000*p*r**2*s**5 - 1220703125*p*q*s**6 + 6103515625*s**7 + + a[4] = 1000*p**5*q**7 + 7250*p**2*q**9 - 10800*p**6*q**5*r - 96900*p**3*q**7*r - 52500*q**9*r + 37400*p**7*q**3*r**2 + 470850*p**4*q**5*r**2 + 640600*p*q**7*r**2 - 39600*p**8*q*r**3 - 983600*p**5*q**3*r**3 - 2848100*p**2*q**5*r**3 + 814400*p**6*q*r**4 + 6076000*p**3*q**3*r**4 + 2308000*q**5*r**4 - 5024000*p**4*q*r**5 - 9680000*p*q**3*r**5 + 9600000*p**2*q*r**6 + 13800*p**7*q**4*s + 94650*p**4*q**6*s - 26500*p*q**8*s - 86400*p**8*q**2*r*s - 816500*p**5*q**4*r*s - 257500*p**2*q**6*r*s + 91800*p**9*r**2*s + 1853700*p**6*q**2*r**2*s + 630000*p**3*q**4*r**2*s - 8971250*q**6*r**2*s - 2071200*p**7*r**3*s - 7240000*p**4*q**2*r**3*s + 29375000*p*q**4*r**3*s + 14416000*p**5*r**4*s - 5200000*p**2*q**2*r**4*s - 30400000*p**3*r**5*s - 12000000*q**2*r**5*s + 64800*p**9*q*s**2 + 567000*p**6*q**3*s**2 + 1655000*p**3*q**5*s**2 + 6987500*q**7*s**2 + 337500*p**7*q*r*s**2 + 8462500*p**4*q**3*r*s**2 - 5812500*p*q**5*r*s**2 - 24930000*p**5*q*r**2*s**2 - 69125000*p**2*q**3*r**2*s**2 + 103500000*p**3*q*r**3*s**2 + 30000000*q**3*r**3*s**2 + 90000000*p*q*r**4*s**2 - 708750*p**8*s**3 - 5400000*p**5*q**2*s**3 + 8906250*p**2*q**4*s**3 + 18562500*p**6*r*s**3 - 625000*p**3*q**2*r*s**3 + 29687500*q**4*r*s**3 - 75000000*p**4*r**2*s**3 - 416250000*p*q**2*r**2*s**3 + 60000000*p**2*r**3*s**3 - 300000000*r**4*s**3 + 71718750*p**4*q*s**4 + 189062500*p*q**3*s**4 + 210937500*p**2*q*r*s**4 + 1187500000*q*r**2*s**4 - 187500000*p**3*s**5 - 800781250*q**2*s**5 - 390625000*p*r*s**5 + + a[3] = -500*p**6*q**5 - 6350*p**3*q**7 - 19800*q**9 + 3750*p**7*q**3*r + 65100*p**4*q**5*r + 264950*p*q**7*r - 6750*p**8*q*r**2 - 209050*p**5*q**3*r**2 - 1217250*p**2*q**5*r**2 + 219000*p**6*q*r**3 + 2510000*p**3*q**3*r**3 + 1098500*q**5*r**3 - 2068000*p**4*q*r**4 - 5060000*p*q**3*r**4 + 5200000*p**2*q*r**5 - 6750*p**8*q**2*s - 96350*p**5*q**4*s - 346000*p**2*q**6*s + 20250*p**9*r*s + 459900*p**6*q**2*r*s + 1828750*p**3*q**4*r*s - 2930000*q**6*r*s - 594000*p**7*r**2*s - 4301250*p**4*q**2*r**2*s + 10906250*p*q**4*r**2*s + 5252000*p**5*r**3*s - 1450000*p**2*q**2*r**3*s - 12800000*p**3*r**4*s - 6500000*q**2*r**4*s + 74250*p**7*q*s**2 + 1418750*p**4*q**3*s**2 + 5956250*p*q**5*s**2 - 4297500*p**5*q*r*s**2 - 29906250*p**2*q**3*r*s**2 + 31500000*p**3*q*r**2*s**2 + 12500000*q**3*r**2*s**2 + 35000000*p*q*r**3*s**2 + 1350000*p**6*s**3 + 6093750*p**3*q**2*s**3 + 17500000*q**4*s**3 - 7031250*p**4*r*s**3 - 127812500*p*q**2*r*s**3 + 18750000*p**2*r**2*s**3 - 162500000*r**3*s**3 + 107812500*p**2*q*s**4 + 460937500*q*r*s**4 - 214843750*p*s**5 + + a[2] = 1950*p**4*q**5 + 14100*p*q**7 - 14350*p**5*q**3*r - 125600*p**2*q**5*r + 27900*p**6*q*r**2 + 402250*p**3*q**3*r**2 + 288250*q**5*r**2 - 436000*p**4*q*r**3 - 1345000*p*q**3*r**3 + 1400000*p**2*q*r**4 + 9450*p**6*q**2*s - 1250*p**3*q**4*s - 465000*q**6*s - 49950*p**7*r*s - 302500*p**4*q**2*r*s + 1718750*p*q**4*r*s + 834000*p**5*r**2*s + 437500*p**2*q**2*r**2*s - 3100000*p**3*r**3*s - 1750000*q**2*r**3*s - 292500*p**5*q*s**2 - 1937500*p**2*q**3*s**2 + 3343750*p**3*q*r*s**2 + 1875000*q**3*r*s**2 + 8125000*p*q*r**2*s**2 - 1406250*p**4*s**3 - 12343750*p*q**2*s**3 + 5312500*p**2*r*s**3 - 43750000*r**2*s**3 + 74218750*q*s**4 + + a[1] = -300*p**5*q**3 - 2150*p**2*q**5 + 1350*p**6*q*r + 21500*p**3*q**3*r + 61500*q**5*r - 42000*p**4*q*r**2 - 290000*p*q**3*r**2 + 300000*p**2*q*r**3 - 4050*p**7*s - 45000*p**4*q**2*s - 125000*p*q**4*s + 108000*p**5*r*s + 643750*p**2*q**2*r*s - 700000*p**3*r**2*s - 375000*q**2*r**2*s - 93750*p**3*q*s**2 - 312500*q**3*s**2 + 1875000*p*q*r*s**2 - 1406250*p**2*s**3 - 9375000*r*s**3 + + a[0] = 1250*p**3*q**3 + 9000*q**5 - 4500*p**4*q*r - 46250*p*q**3*r + 50000*p**2*q*r**2 + 6750*p**5*s + 43750*p**2*q**2*s - 75000*p**3*r*s - 62500*q**2*r*s + 156250*p*q*s**2 - 1562500*s**3 + + return a + + @property + def c(self): + p, q, r, s = self.p, self.q, self.r, self.s + c = [0]*6 + + c[5] = -40*p**5*q**11 - 270*p**2*q**13 + 700*p**6*q**9*r + 5165*p**3*q**11*r + 540*q**13*r - 4230*p**7*q**7*r**2 - 31845*p**4*q**9*r**2 + 20880*p*q**11*r**2 + 9645*p**8*q**5*r**3 + 57615*p**5*q**7*r**3 - 358255*p**2*q**9*r**3 - 1880*p**9*q**3*r**4 + 114020*p**6*q**5*r**4 + 2012190*p**3*q**7*r**4 - 26855*q**9*r**4 - 14400*p**10*q*r**5 - 470400*p**7*q**3*r**5 - 5088640*p**4*q**5*r**5 + 920*p*q**7*r**5 + 332800*p**8*q*r**6 + 5797120*p**5*q**3*r**6 + 1608000*p**2*q**5*r**6 - 2611200*p**6*q*r**7 - 7424000*p**3*q**3*r**7 - 2323200*q**5*r**7 + 8601600*p**4*q*r**8 + 9472000*p*q**3*r**8 - 10240000*p**2*q*r**9 - 3060*p**7*q**8*s - 39085*p**4*q**10*s - 132300*p*q**12*s + 36580*p**8*q**6*r*s + 520185*p**5*q**8*r*s + 1969860*p**2*q**10*r*s - 144045*p**9*q**4*r**2*s - 2438425*p**6*q**6*r**2*s - 10809475*p**3*q**8*r**2*s + 518850*q**10*r**2*s + 182520*p**10*q**2*r**3*s + 4533930*p**7*q**4*r**3*s + 26196770*p**4*q**6*r**3*s - 4542325*p*q**8*r**3*s + 21600*p**11*r**4*s - 2208080*p**8*q**2*r**4*s - 24787960*p**5*q**4*r**4*s + 10813900*p**2*q**6*r**4*s - 499200*p**9*r**5*s + 3827840*p**6*q**2*r**5*s + 9596000*p**3*q**4*r**5*s + 22662000*q**6*r**5*s + 3916800*p**7*r**6*s - 29952000*p**4*q**2*r**6*s - 90800000*p*q**4*r**6*s - 12902400*p**5*r**7*s + 87040000*p**2*q**2*r**7*s + 15360000*p**3*r**8*s + 12800000*q**2*r**8*s - 38070*p**9*q**5*s**2 - 566700*p**6*q**7*s**2 - 2574375*p**3*q**9*s**2 - 1822500*q**11*s**2 + 292815*p**10*q**3*r*s**2 + 5170280*p**7*q**5*r*s**2 + 27918125*p**4*q**7*r*s**2 + 21997500*p*q**9*r*s**2 - 573480*p**11*q*r**2*s**2 - 14566350*p**8*q**3*r**2*s**2 - 104851575*p**5*q**5*r**2*s**2 - 96448750*p**2*q**7*r**2*s**2 + 11001240*p**9*q*r**3*s**2 + 147798600*p**6*q**3*r**3*s**2 + 158632750*p**3*q**5*r**3*s**2 - 78222500*q**7*r**3*s**2 - 62819200*p**7*q*r**4*s**2 - 136160000*p**4*q**3*r**4*s**2 + 317555000*p*q**5*r**4*s**2 + 160224000*p**5*q*r**5*s**2 - 267600000*p**2*q**3*r**5*s**2 - 153600000*p**3*q*r**6*s**2 - 120000000*q**3*r**6*s**2 - 32000000*p*q*r**7*s**2 - 127575*p**11*q**2*s**3 - 2148750*p**8*q**4*s**3 - 13652500*p**5*q**6*s**3 - 19531250*p**2*q**8*s**3 + 495720*p**12*r*s**3 + 11856375*p**9*q**2*r*s**3 + 107807500*p**6*q**4*r*s**3 + 222334375*p**3*q**6*r*s**3 + 105062500*q**8*r*s**3 - 11566800*p**10*r**2*s**3 - 216787500*p**7*q**2*r**2*s**3 - 633437500*p**4*q**4*r**2*s**3 - 504484375*p*q**6*r**2*s**3 + 90918000*p**8*r**3*s**3 + 567080000*p**5*q**2*r**3*s**3 + 692937500*p**2*q**4*r**3*s**3 - 326640000*p**6*r**4*s**3 - 339000000*p**3*q**2*r**4*s**3 + 369250000*q**4*r**4*s**3 + 560000000*p**4*r**5*s**3 + 508000000*p*q**2*r**5*s**3 - 480000000*p**2*r**6*s**3 + 320000000*r**7*s**3 - 455625*p**10*q*s**4 - 27562500*p**7*q**3*s**4 - 120593750*p**4*q**5*s**4 - 60312500*p*q**7*s**4 + 110615625*p**8*q*r*s**4 + 662984375*p**5*q**3*r*s**4 + 528515625*p**2*q**5*r*s**4 - 541687500*p**6*q*r**2*s**4 - 1262343750*p**3*q**3*r**2*s**4 - 466406250*q**5*r**2*s**4 + 633000000*p**4*q*r**3*s**4 - 1264375000*p*q**3*r**3*s**4 + 1085000000*p**2*q*r**4*s**4 - 2700000000*q*r**5*s**4 - 68343750*p**9*s**5 - 478828125*p**6*q**2*s**5 - 355468750*p**3*q**4*s**5 - 11718750*q**6*s**5 + 718031250*p**7*r*s**5 + 1658593750*p**4*q**2*r*s**5 + 2212890625*p*q**4*r*s**5 - 2855625000*p**5*r**2*s**5 - 4273437500*p**2*q**2*r**2*s**5 + 4537500000*p**3*r**3*s**5 + 8031250000*q**2*r**3*s**5 - 1750000000*p*r**4*s**5 + 1353515625*p**5*q*s**6 + 1562500000*p**2*q**3*s**6 - 3964843750*p**3*q*r*s**6 - 7226562500*q**3*r*s**6 + 1953125000*p*q*r**2*s**6 - 1757812500*p**4*s**7 - 3173828125*p*q**2*s**7 + 6445312500*p**2*r*s**7 - 3906250000*r**2*s**7 + 6103515625*q*s**8 + + c[4] = 40*p**6*q**9 + 110*p**3*q**11 - 1080*q**13 - 560*p**7*q**7*r - 1780*p**4*q**9*r + 17370*p*q**11*r + 2850*p**8*q**5*r**2 + 10520*p**5*q**7*r**2 - 115910*p**2*q**9*r**2 - 6090*p**9*q**3*r**3 - 25330*p**6*q**5*r**3 + 448740*p**3*q**7*r**3 + 128230*q**9*r**3 + 4320*p**10*q*r**4 + 16960*p**7*q**3*r**4 - 1143600*p**4*q**5*r**4 - 1410310*p*q**7*r**4 + 3840*p**8*q*r**5 + 1744480*p**5*q**3*r**5 + 5619520*p**2*q**5*r**5 - 1198080*p**6*q*r**6 - 10579200*p**3*q**3*r**6 - 2940800*q**5*r**6 + 8294400*p**4*q*r**7 + 13568000*p*q**3*r**7 - 15360000*p**2*q*r**8 + 840*p**8*q**6*s + 7580*p**5*q**8*s + 24420*p**2*q**10*s - 8100*p**9*q**4*r*s - 94100*p**6*q**6*r*s - 473000*p**3*q**8*r*s - 473400*q**10*r*s + 22680*p**10*q**2*r**2*s + 374370*p**7*q**4*r**2*s + 2888020*p**4*q**6*r**2*s + 5561050*p*q**8*r**2*s - 12960*p**11*r**3*s - 485820*p**8*q**2*r**3*s - 6723440*p**5*q**4*r**3*s - 23561400*p**2*q**6*r**3*s + 190080*p**9*r**4*s + 5894880*p**6*q**2*r**4*s + 50882000*p**3*q**4*r**4*s + 22411500*q**6*r**4*s - 258560*p**7*r**5*s - 46248000*p**4*q**2*r**5*s - 103800000*p*q**4*r**5*s - 3737600*p**5*r**6*s + 119680000*p**2*q**2*r**6*s + 10240000*p**3*r**7*s + 19200000*q**2*r**7*s + 7290*p**10*q**3*s**2 + 117360*p**7*q**5*s**2 + 691250*p**4*q**7*s**2 - 198750*p*q**9*s**2 - 36450*p**11*q*r*s**2 - 854550*p**8*q**3*r*s**2 - 7340700*p**5*q**5*r*s**2 - 2028750*p**2*q**7*r*s**2 + 995490*p**9*q*r**2*s**2 + 18896600*p**6*q**3*r**2*s**2 + 5026500*p**3*q**5*r**2*s**2 - 52272500*q**7*r**2*s**2 - 16636800*p**7*q*r**3*s**2 - 43200000*p**4*q**3*r**3*s**2 + 223426250*p*q**5*r**3*s**2 + 112068000*p**5*q*r**4*s**2 - 177000000*p**2*q**3*r**4*s**2 - 244000000*p**3*q*r**5*s**2 - 156000000*q**3*r**5*s**2 + 43740*p**12*s**3 + 1032750*p**9*q**2*s**3 + 8602500*p**6*q**4*s**3 + 15606250*p**3*q**6*s**3 + 39625000*q**8*s**3 - 1603800*p**10*r*s**3 - 26932500*p**7*q**2*r*s**3 - 19562500*p**4*q**4*r*s**3 - 152000000*p*q**6*r*s**3 + 25555500*p**8*r**2*s**3 + 16230000*p**5*q**2*r**2*s**3 + 42187500*p**2*q**4*r**2*s**3 - 165660000*p**6*r**3*s**3 + 373500000*p**3*q**2*r**3*s**3 + 332937500*q**4*r**3*s**3 + 465000000*p**4*r**4*s**3 + 586000000*p*q**2*r**4*s**3 - 592000000*p**2*r**5*s**3 + 480000000*r**6*s**3 - 1518750*p**8*q*s**4 - 62531250*p**5*q**3*s**4 + 7656250*p**2*q**5*s**4 + 184781250*p**6*q*r*s**4 - 15781250*p**3*q**3*r*s**4 - 135156250*q**5*r*s**4 - 1148250000*p**4*q*r**2*s**4 - 2121406250*p*q**3*r**2*s**4 + 1990000000*p**2*q*r**3*s**4 - 3150000000*q*r**4*s**4 - 2531250*p**7*s**5 + 660937500*p**4*q**2*s**5 + 1339843750*p*q**4*s**5 - 33750000*p**5*r*s**5 - 679687500*p**2*q**2*r*s**5 + 6250000*p**3*r**2*s**5 + 6195312500*q**2*r**2*s**5 + 1125000000*p*r**3*s**5 - 996093750*p**3*q*s**6 - 3125000000*q**3*s**6 - 3222656250*p*q*r*s**6 + 1171875000*p**2*s**7 + 976562500*r*s**7 + + c[3] = 80*p**4*q**9 + 540*p*q**11 - 600*p**5*q**7*r - 4770*p**2*q**9*r + 1230*p**6*q**5*r**2 + 20900*p**3*q**7*r**2 + 47250*q**9*r**2 - 710*p**7*q**3*r**3 - 84950*p**4*q**5*r**3 - 526310*p*q**7*r**3 + 720*p**8*q*r**4 + 216280*p**5*q**3*r**4 + 2068020*p**2*q**5*r**4 - 198080*p**6*q*r**5 - 3703200*p**3*q**3*r**5 - 1423600*q**5*r**5 + 2860800*p**4*q*r**6 + 7056000*p*q**3*r**6 - 8320000*p**2*q*r**7 - 2720*p**6*q**6*s - 46350*p**3*q**8*s - 178200*q**10*s + 25740*p**7*q**4*r*s + 489490*p**4*q**6*r*s + 2152350*p*q**8*r*s - 61560*p**8*q**2*r**2*s - 1568150*p**5*q**4*r**2*s - 9060500*p**2*q**6*r**2*s + 24840*p**9*r**3*s + 1692380*p**6*q**2*r**3*s + 18098250*p**3*q**4*r**3*s + 9387750*q**6*r**3*s - 382560*p**7*r**4*s - 16818000*p**4*q**2*r**4*s - 49325000*p*q**4*r**4*s + 1212800*p**5*r**5*s + 64840000*p**2*q**2*r**5*s - 320000*p**3*r**6*s + 10400000*q**2*r**6*s - 36450*p**8*q**3*s**2 - 588350*p**5*q**5*s**2 - 2156250*p**2*q**7*s**2 + 123930*p**9*q*r*s**2 + 2879700*p**6*q**3*r*s**2 + 12548000*p**3*q**5*r*s**2 - 14445000*q**7*r*s**2 - 3233250*p**7*q*r**2*s**2 - 28485000*p**4*q**3*r**2*s**2 + 72231250*p*q**5*r**2*s**2 + 32093000*p**5*q*r**3*s**2 - 61275000*p**2*q**3*r**3*s**2 - 107500000*p**3*q*r**4*s**2 - 78500000*q**3*r**4*s**2 + 22000000*p*q*r**5*s**2 - 72900*p**10*s**3 - 1215000*p**7*q**2*s**3 - 2937500*p**4*q**4*s**3 + 9156250*p*q**6*s**3 + 2612250*p**8*r*s**3 + 16560000*p**5*q**2*r*s**3 - 75468750*p**2*q**4*r*s**3 - 32737500*p**6*r**2*s**3 + 169062500*p**3*q**2*r**2*s**3 + 121718750*q**4*r**2*s**3 + 160250000*p**4*r**3*s**3 + 219750000*p*q**2*r**3*s**3 - 317000000*p**2*r**4*s**3 + 260000000*r**5*s**3 + 2531250*p**6*q*s**4 + 22500000*p**3*q**3*s**4 + 39843750*q**5*s**4 - 266343750*p**4*q*r*s**4 - 776406250*p*q**3*r*s**4 + 789062500*p**2*q*r**2*s**4 - 1368750000*q*r**3*s**4 + 67500000*p**5*s**5 + 441406250*p**2*q**2*s**5 - 311718750*p**3*r*s**5 + 1785156250*q**2*r*s**5 + 546875000*p*r**2*s**5 - 1269531250*p*q*s**6 + 488281250*s**7 + + c[2] = 120*p**5*q**7 + 810*p**2*q**9 - 1280*p**6*q**5*r - 9160*p**3*q**7*r + 3780*q**9*r + 4530*p**7*q**3*r**2 + 36640*p**4*q**5*r**2 - 45270*p*q**7*r**2 - 5400*p**8*q*r**3 - 60920*p**5*q**3*r**3 + 200050*p**2*q**5*r**3 + 31200*p**6*q*r**4 - 476000*p**3*q**3*r**4 - 378200*q**5*r**4 + 521600*p**4*q*r**5 + 1872000*p*q**3*r**5 - 2240000*p**2*q*r**6 + 1440*p**7*q**4*s + 15310*p**4*q**6*s + 59400*p*q**8*s - 9180*p**8*q**2*r*s - 115240*p**5*q**4*r*s - 589650*p**2*q**6*r*s + 16200*p**9*r**2*s + 316710*p**6*q**2*r**2*s + 2547750*p**3*q**4*r**2*s + 2178000*q**6*r**2*s - 259200*p**7*r**3*s - 4123000*p**4*q**2*r**3*s - 11700000*p*q**4*r**3*s + 937600*p**5*r**4*s + 16340000*p**2*q**2*r**4*s - 640000*p**3*r**5*s + 2800000*q**2*r**5*s - 2430*p**9*q*s**2 - 54450*p**6*q**3*s**2 - 285500*p**3*q**5*s**2 - 2767500*q**7*s**2 + 43200*p**7*q*r*s**2 - 916250*p**4*q**3*r*s**2 + 14482500*p*q**5*r*s**2 + 4806000*p**5*q*r**2*s**2 - 13212500*p**2*q**3*r**2*s**2 - 25400000*p**3*q*r**3*s**2 - 18750000*q**3*r**3*s**2 + 8000000*p*q*r**4*s**2 + 121500*p**8*s**3 + 2058750*p**5*q**2*s**3 - 6656250*p**2*q**4*s**3 - 6716250*p**6*r*s**3 + 24125000*p**3*q**2*r*s**3 + 23875000*q**4*r*s**3 + 43125000*p**4*r**2*s**3 + 45750000*p*q**2*r**2*s**3 - 87500000*p**2*r**3*s**3 + 70000000*r**4*s**3 - 44437500*p**4*q*s**4 - 107968750*p*q**3*s**4 + 159531250*p**2*q*r*s**4 - 284375000*q*r**2*s**4 + 7031250*p**3*s**5 + 265625000*q**2*s**5 + 31250000*p*r*s**5 + + c[1] = 160*p**3*q**7 + 1080*q**9 - 1080*p**4*q**5*r - 8730*p*q**7*r + 1510*p**5*q**3*r**2 + 20420*p**2*q**5*r**2 + 720*p**6*q*r**3 - 23200*p**3*q**3*r**3 - 79900*q**5*r**3 + 35200*p**4*q*r**4 + 404000*p*q**3*r**4 - 480000*p**2*q*r**5 + 960*p**5*q**4*s + 2850*p**2*q**6*s + 540*p**6*q**2*r*s + 63500*p**3*q**4*r*s + 319500*q**6*r*s - 7560*p**7*r**2*s - 253500*p**4*q**2*r**2*s - 1806250*p*q**4*r**2*s + 91200*p**5*r**3*s + 2600000*p**2*q**2*r**3*s - 80000*p**3*r**4*s + 600000*q**2*r**4*s - 4050*p**7*q*s**2 - 120000*p**4*q**3*s**2 - 273750*p*q**5*s**2 + 425250*p**5*q*r*s**2 + 2325000*p**2*q**3*r*s**2 - 5400000*p**3*q*r**2*s**2 - 2875000*q**3*r**2*s**2 + 1500000*p*q*r**3*s**2 - 303750*p**6*s**3 - 843750*p**3*q**2*s**3 - 812500*q**4*s**3 + 5062500*p**4*r*s**3 + 13312500*p*q**2*r*s**3 - 14500000*p**2*r**2*s**3 + 15000000*r**3*s**3 - 3750000*p**2*q*s**4 - 35937500*q*r*s**4 + 11718750*p*s**5 + + c[0] = 80*p**4*q**5 + 540*p*q**7 - 600*p**5*q**3*r - 4770*p**2*q**5*r + 1080*p**6*q*r**2 + 11200*p**3*q**3*r**2 - 12150*q**5*r**2 - 4800*p**4*q*r**3 + 64000*p*q**3*r**3 - 80000*p**2*q*r**4 + 1080*p**6*q**2*s + 13250*p**3*q**4*s + 54000*q**6*s - 3240*p**7*r*s - 56250*p**4*q**2*r*s - 337500*p*q**4*r*s + 43200*p**5*r**2*s + 560000*p**2*q**2*r**2*s - 80000*p**3*r**3*s + 100000*q**2*r**3*s + 6750*p**5*q*s**2 + 225000*p**2*q**3*s**2 - 900000*p**3*q*r*s**2 - 562500*q**3*r*s**2 + 500000*p*q*r**2*s**2 + 843750*p**4*s**3 + 1937500*p*q**2*s**3 - 3000000*p**2*r*s**3 + 2500000*r**2*s**3 - 5468750*q*s**4 + + return c + + @property + def F(self): + p, q, r, s = self.p, self.q, self.r, self.s + F = 4*p**6*q**6 + 59*p**3*q**8 + 216*q**10 - 36*p**7*q**4*r - 623*p**4*q**6*r - 2610*p*q**8*r + 81*p**8*q**2*r**2 + 2015*p**5*q**4*r**2 + 10825*p**2*q**6*r**2 - 1800*p**6*q**2*r**3 - 17500*p**3*q**4*r**3 + 625*q**6*r**3 + 10000*p**4*q**2*r**4 + 108*p**8*q**3*s + 1584*p**5*q**5*s + 5700*p**2*q**7*s - 486*p**9*q*r*s - 9720*p**6*q**3*r*s - 45050*p**3*q**5*r*s - 9000*q**7*r*s + 10800*p**7*q*r**2*s + 92500*p**4*q**3*r**2*s + 32500*p*q**5*r**2*s - 60000*p**5*q*r**3*s - 50000*p**2*q**3*r**3*s + 729*p**10*s**2 + 12150*p**7*q**2*s**2 + 60000*p**4*q**4*s**2 + 93750*p*q**6*s**2 - 18225*p**8*r*s**2 - 175500*p**5*q**2*r*s**2 - 478125*p**2*q**4*r*s**2 + 135000*p**6*r**2*s**2 + 850000*p**3*q**2*r**2*s**2 + 15625*q**4*r**2*s**2 - 250000*p**4*r**3*s**2 + 225000*p**3*q**3*s**3 + 175000*q**5*s**3 - 1012500*p**4*q*r*s**3 - 1187500*p*q**3*r*s**3 + 1250000*p**2*q*r**2*s**3 + 928125*p**5*s**4 + 1875000*p**2*q**2*s**4 - 2812500*p**3*r*s**4 - 390625*q**2*r*s**4 - 9765625*s**6 + return F + + def l0(self, theta): + F = self.F + a = self.a + l0 = Poly(a, x).eval(theta)/F + return l0 + + def T(self, theta, d): + F = self.F + T = [0]*5 + b = self.b + # Note that the order of sublists of the b's has been reversed compared to the paper + T[1] = -Poly(b[1], x).eval(theta)/(2*F) + T[2] = Poly(b[2], x).eval(theta)/(2*d*F) + T[3] = Poly(b[3], x).eval(theta)/(2*F) + T[4] = Poly(b[4], x).eval(theta)/(2*d*F) + return T + + def order(self, theta, d): + F = self.F + o = self.o + order = Poly(o, x).eval(theta)/(d*F) + return N(order) + + def uv(self, theta, d): + c = self.c + u = self.q*Rational(-25, 2) + v = Poly(c, x).eval(theta)/(2*d*self.F) + return N(u), N(v) + + @property + def zeta(self): + return [self.zeta1, self.zeta2, self.zeta3, self.zeta4] diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/polyroots.py b/.venv/lib/python3.12/site-packages/sympy/polys/polyroots.py new file mode 100644 index 0000000000000000000000000000000000000000..4def1312eb5b94a13e511d2d4f9b15f1d51fd63f --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/polyroots.py @@ -0,0 +1,1227 @@ +"""Algorithms for computing symbolic roots of polynomials. """ + + +import math +from functools import reduce + +from sympy.core import S, I, pi +from sympy.core.exprtools import factor_terms +from sympy.core.function import _mexpand +from sympy.core.logic import fuzzy_not +from sympy.core.mul import expand_2arg, Mul +from sympy.core.intfunc import igcd +from sympy.core.numbers import Rational, comp +from sympy.core.power import Pow +from sympy.core.relational import Eq +from sympy.core.sorting import ordered +from sympy.core.symbol import Dummy, Symbol, symbols +from sympy.core.sympify import sympify +from sympy.functions import exp, im, cos, acos, Piecewise +from sympy.functions.elementary.miscellaneous import root, sqrt +from sympy.ntheory import divisors, isprime, nextprime +from sympy.polys.domains import EX +from sympy.polys.polyerrors import (PolynomialError, GeneratorsNeeded, + DomainError, UnsolvableFactorError) +from sympy.polys.polyquinticconst import PolyQuintic +from sympy.polys.polytools import Poly, cancel, factor, gcd_list, discriminant +from sympy.polys.rationaltools import together +from sympy.polys.specialpolys import cyclotomic_poly +from sympy.utilities import public +from sympy.utilities.misc import filldedent + + + +z = Symbol('z') # importing from abc cause O to be lost as clashing symbol + + +def roots_linear(f): + """Returns a list of roots of a linear polynomial.""" + r = -f.nth(0)/f.nth(1) + dom = f.get_domain() + + if not dom.is_Numerical: + if dom.is_Composite: + r = factor(r) + else: + from sympy.simplify.simplify import simplify + r = simplify(r) + + return [r] + + +def roots_quadratic(f): + """Returns a list of roots of a quadratic polynomial. If the domain is ZZ + then the roots will be sorted with negatives coming before positives. + The ordering will be the same for any numerical coefficients as long as + the assumptions tested are correct, otherwise the ordering will not be + sorted (but will be canonical). + """ + + a, b, c = f.all_coeffs() + dom = f.get_domain() + + def _sqrt(d): + # remove squares from square root since both will be represented + # in the results; a similar thing is happening in roots() but + # must be duplicated here because not all quadratics are binomials + co = [] + other = [] + for di in Mul.make_args(d): + if di.is_Pow and di.exp.is_Integer and di.exp % 2 == 0: + co.append(Pow(di.base, di.exp//2)) + else: + other.append(di) + if co: + d = Mul(*other) + co = Mul(*co) + return co*sqrt(d) + return sqrt(d) + + def _simplify(expr): + if dom.is_Composite: + return factor(expr) + else: + from sympy.simplify.simplify import simplify + return simplify(expr) + + if c is S.Zero: + r0, r1 = S.Zero, -b/a + + if not dom.is_Numerical: + r1 = _simplify(r1) + elif r1.is_negative: + r0, r1 = r1, r0 + elif b is S.Zero: + r = -c/a + if not dom.is_Numerical: + r = _simplify(r) + + R = _sqrt(r) + r0 = -R + r1 = R + else: + d = b**2 - 4*a*c + A = 2*a + B = -b/A + + if not dom.is_Numerical: + d = _simplify(d) + B = _simplify(B) + + D = factor_terms(_sqrt(d)/A) + r0 = B - D + r1 = B + D + if a.is_negative: + r0, r1 = r1, r0 + elif not dom.is_Numerical: + r0, r1 = [expand_2arg(i) for i in (r0, r1)] + + return [r0, r1] + + +def roots_cubic(f, trig=False): + """Returns a list of roots of a cubic polynomial. + + References + ========== + [1] https://en.wikipedia.org/wiki/Cubic_function, General formula for roots, + (accessed November 17, 2014). + """ + if trig: + a, b, c, d = f.all_coeffs() + p = (3*a*c - b**2)/(3*a**2) + q = (2*b**3 - 9*a*b*c + 27*a**2*d)/(27*a**3) + D = 18*a*b*c*d - 4*b**3*d + b**2*c**2 - 4*a*c**3 - 27*a**2*d**2 + if (D > 0) == True: + rv = [] + for k in range(3): + rv.append(2*sqrt(-p/3)*cos(acos(q/p*sqrt(-3/p)*Rational(3, 2))/3 - k*pi*Rational(2, 3))) + return [i - b/3/a for i in rv] + + # a*x**3 + b*x**2 + c*x + d -> x**3 + a*x**2 + b*x + c + _, a, b, c = f.monic().all_coeffs() + + if c is S.Zero: + x1, x2 = roots([1, a, b], multiple=True) + return [x1, S.Zero, x2] + + # x**3 + a*x**2 + b*x + c -> u**3 + p*u + q + p = b - a**2/3 + q = c - a*b/3 + 2*a**3/27 + + pon3 = p/3 + aon3 = a/3 + + u1 = None + if p is S.Zero: + if q is S.Zero: + return [-aon3]*3 + u1 = -root(q, 3) if q.is_positive else root(-q, 3) + elif q is S.Zero: + y1, y2 = roots([1, 0, p], multiple=True) + return [tmp - aon3 for tmp in [y1, S.Zero, y2]] + elif q.is_real and q.is_negative: + u1 = -root(-q/2 + sqrt(q**2/4 + pon3**3), 3) + + coeff = I*sqrt(3)/2 + if u1 is None: + u1 = S.One + u2 = Rational(-1, 2) + coeff + u3 = Rational(-1, 2) - coeff + b, c, d = a, b, c # a, b, c, d = S.One, a, b, c + D0 = b**2 - 3*c # b**2 - 3*a*c + D1 = 2*b**3 - 9*b*c + 27*d # 2*b**3 - 9*a*b*c + 27*a**2*d + C = root((D1 + sqrt(D1**2 - 4*D0**3))/2, 3) + return [-(b + uk*C + D0/C/uk)/3 for uk in [u1, u2, u3]] # -(b + uk*C + D0/C/uk)/3/a + + u2 = u1*(Rational(-1, 2) + coeff) + u3 = u1*(Rational(-1, 2) - coeff) + + if p is S.Zero: + return [u1 - aon3, u2 - aon3, u3 - aon3] + + soln = [ + -u1 + pon3/u1 - aon3, + -u2 + pon3/u2 - aon3, + -u3 + pon3/u3 - aon3 + ] + + return soln + +def _roots_quartic_euler(p, q, r, a): + """ + Descartes-Euler solution of the quartic equation + + Parameters + ========== + + p, q, r: coefficients of ``x**4 + p*x**2 + q*x + r`` + a: shift of the roots + + Notes + ===== + + This is a helper function for ``roots_quartic``. + + Look for solutions of the form :: + + ``x1 = sqrt(R) - sqrt(A + B*sqrt(R))`` + ``x2 = -sqrt(R) - sqrt(A - B*sqrt(R))`` + ``x3 = -sqrt(R) + sqrt(A - B*sqrt(R))`` + ``x4 = sqrt(R) + sqrt(A + B*sqrt(R))`` + + To satisfy the quartic equation one must have + ``p = -2*(R + A); q = -4*B*R; r = (R - A)**2 - B**2*R`` + so that ``R`` must satisfy the Descartes-Euler resolvent equation + ``64*R**3 + 32*p*R**2 + (4*p**2 - 16*r)*R - q**2 = 0`` + + If the resolvent does not have a rational solution, return None; + in that case it is likely that the Ferrari method gives a simpler + solution. + + Examples + ======== + + >>> from sympy import S + >>> from sympy.polys.polyroots import _roots_quartic_euler + >>> p, q, r = -S(64)/5, -S(512)/125, -S(1024)/3125 + >>> _roots_quartic_euler(p, q, r, S(0))[0] + -sqrt(32*sqrt(5)/125 + 16/5) + 4*sqrt(5)/5 + """ + # solve the resolvent equation + x = Dummy('x') + eq = 64*x**3 + 32*p*x**2 + (4*p**2 - 16*r)*x - q**2 + xsols = list(roots(Poly(eq, x), cubics=False).keys()) + xsols = [sol for sol in xsols if sol.is_rational and sol.is_nonzero] + if not xsols: + return None + R = max(xsols) + c1 = sqrt(R) + B = -q*c1/(4*R) + A = -R - p/2 + c2 = sqrt(A + B) + c3 = sqrt(A - B) + return [c1 - c2 - a, -c1 - c3 - a, -c1 + c3 - a, c1 + c2 - a] + + +def roots_quartic(f): + r""" + Returns a list of roots of a quartic polynomial. + + There are many references for solving quartic expressions available [1-5]. + This reviewer has found that many of them require one to select from among + 2 or more possible sets of solutions and that some solutions work when one + is searching for real roots but do not work when searching for complex roots + (though this is not always stated clearly). The following routine has been + tested and found to be correct for 0, 2 or 4 complex roots. + + The quasisymmetric case solution [6] looks for quartics that have the form + `x**4 + A*x**3 + B*x**2 + C*x + D = 0` where `(C/A)**2 = D`. + + Although no general solution that is always applicable for all + coefficients is known to this reviewer, certain conditions are tested + to determine the simplest 4 expressions that can be returned: + + 1) `f = c + a*(a**2/8 - b/2) == 0` + 2) `g = d - a*(a*(3*a**2/256 - b/16) + c/4) = 0` + 3) if `f != 0` and `g != 0` and `p = -d + a*c/4 - b**2/12` then + a) `p == 0` + b) `p != 0` + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.polys.polyroots import roots_quartic + + >>> r = roots_quartic(Poly('x**4-6*x**3+17*x**2-26*x+20')) + + >>> # 4 complex roots: 1+-I*sqrt(3), 2+-I + >>> sorted(str(tmp.evalf(n=2)) for tmp in r) + ['1.0 + 1.7*I', '1.0 - 1.7*I', '2.0 + 1.0*I', '2.0 - 1.0*I'] + + References + ========== + + 1. http://mathforum.org/dr.math/faq/faq.cubic.equations.html + 2. https://en.wikipedia.org/wiki/Quartic_function#Summary_of_Ferrari.27s_method + 3. https://planetmath.org/encyclopedia/GaloisTheoreticDerivationOfTheQuarticFormula.html + 4. https://people.bath.ac.uk/masjhd/JHD-CA.pdf + 5. http://www.albmath.org/files/Math_5713.pdf + 6. https://web.archive.org/web/20171002081448/http://www.statemaster.com/encyclopedia/Quartic-equation + 7. https://eqworld.ipmnet.ru/en/solutions/ae/ae0108.pdf + """ + _, a, b, c, d = f.monic().all_coeffs() + + if not d: + return [S.Zero] + roots([1, a, b, c], multiple=True) + elif (c/a)**2 == d: + x, m = f.gen, c/a + + g = Poly(x**2 + a*x + b - 2*m, x) + + z1, z2 = roots_quadratic(g) + + h1 = Poly(x**2 - z1*x + m, x) + h2 = Poly(x**2 - z2*x + m, x) + + r1 = roots_quadratic(h1) + r2 = roots_quadratic(h2) + + return r1 + r2 + else: + a2 = a**2 + e = b - 3*a2/8 + f = _mexpand(c + a*(a2/8 - b/2)) + aon4 = a/4 + g = _mexpand(d - aon4*(a*(3*a2/64 - b/4) + c)) + + if f.is_zero: + y1, y2 = [sqrt(tmp) for tmp in + roots([1, e, g], multiple=True)] + return [tmp - aon4 for tmp in [-y1, -y2, y1, y2]] + if g.is_zero: + y = [S.Zero] + roots([1, 0, e, f], multiple=True) + return [tmp - aon4 for tmp in y] + else: + # Descartes-Euler method, see [7] + sols = _roots_quartic_euler(e, f, g, aon4) + if sols: + return sols + # Ferrari method, see [1, 2] + p = -e**2/12 - g + q = -e**3/108 + e*g/3 - f**2/8 + TH = Rational(1, 3) + + def _ans(y): + w = sqrt(e + 2*y) + arg1 = 3*e + 2*y + arg2 = 2*f/w + ans = [] + for s in [-1, 1]: + root = sqrt(-(arg1 + s*arg2)) + for t in [-1, 1]: + ans.append((s*w - t*root)/2 - aon4) + return ans + + # whether a Piecewise is returned or not + # depends on knowing p, so try to put + # in a simple form + p = _mexpand(p) + + + # p == 0 case + y1 = e*Rational(-5, 6) - q**TH + if p.is_zero: + return _ans(y1) + + # if p != 0 then u below is not 0 + root = sqrt(q**2/4 + p**3/27) + r = -q/2 + root # or -q/2 - root + u = r**TH # primary root of solve(x**3 - r, x) + y2 = e*Rational(-5, 6) + u - p/u/3 + if fuzzy_not(p.is_zero): + return _ans(y2) + + # sort it out once they know the values of the coefficients + return [Piecewise((a1, Eq(p, 0)), (a2, True)) + for a1, a2 in zip(_ans(y1), _ans(y2))] + + +def roots_binomial(f): + """Returns a list of roots of a binomial polynomial. If the domain is ZZ + then the roots will be sorted with negatives coming before positives. + The ordering will be the same for any numerical coefficients as long as + the assumptions tested are correct, otherwise the ordering will not be + sorted (but will be canonical). + """ + n = f.degree() + + a, b = f.nth(n), f.nth(0) + base = -cancel(b/a) + alpha = root(base, n) + + if alpha.is_number: + alpha = alpha.expand(complex=True) + + # define some parameters that will allow us to order the roots. + # If the domain is ZZ this is guaranteed to return roots sorted + # with reals before non-real roots and non-real sorted according + # to real part and imaginary part, e.g. -1, 1, -1 + I, 2 - I + neg = base.is_negative + even = n % 2 == 0 + if neg: + if even == True and (base + 1).is_positive: + big = True + else: + big = False + + # get the indices in the right order so the computed + # roots will be sorted when the domain is ZZ + ks = [] + imax = n//2 + if even: + ks.append(imax) + imax -= 1 + if not neg: + ks.append(0) + for i in range(imax, 0, -1): + if neg: + ks.extend([i, -i]) + else: + ks.extend([-i, i]) + if neg: + ks.append(0) + if big: + for i in range(0, len(ks), 2): + pair = ks[i: i + 2] + pair = list(reversed(pair)) + + # compute the roots + roots, d = [], 2*I*pi/n + for k in ks: + zeta = exp(k*d).expand(complex=True) + roots.append((alpha*zeta).expand(power_base=False)) + + return roots + + +def _inv_totient_estimate(m): + """ + Find ``(L, U)`` such that ``L <= phi^-1(m) <= U``. + + Examples + ======== + + >>> from sympy.polys.polyroots import _inv_totient_estimate + + >>> _inv_totient_estimate(192) + (192, 840) + >>> _inv_totient_estimate(400) + (400, 1750) + + """ + primes = [ d + 1 for d in divisors(m) if isprime(d + 1) ] + + a, b = 1, 1 + + for p in primes: + a *= p + b *= p - 1 + + L = m + U = int(math.ceil(m*(float(a)/b))) + + P = p = 2 + primes = [] + + while P <= U: + p = nextprime(p) + primes.append(p) + P *= p + + P //= p + b = 1 + + for p in primes[:-1]: + b *= p - 1 + + U = int(math.ceil(m*(float(P)/b))) + + return L, U + + +def roots_cyclotomic(f, factor=False): + """Compute roots of cyclotomic polynomials. """ + L, U = _inv_totient_estimate(f.degree()) + + for n in range(L, U + 1): + g = cyclotomic_poly(n, f.gen, polys=True) + + if f.expr == g.expr: + break + else: # pragma: no cover + raise RuntimeError("failed to find index of a cyclotomic polynomial") + + roots = [] + + if not factor: + # get the indices in the right order so the computed + # roots will be sorted + h = n//2 + ks = [i for i in range(1, n + 1) if igcd(i, n) == 1] + ks.sort(key=lambda x: (x, -1) if x <= h else (abs(x - n), 1)) + d = 2*I*pi/n + for k in reversed(ks): + roots.append(exp(k*d).expand(complex=True)) + else: + g = Poly(f, extension=root(-1, n)) + + for h, _ in ordered(g.factor_list()[1]): + roots.append(-h.TC()) + + return roots + + +def roots_quintic(f): + """ + Calculate exact roots of a solvable irreducible quintic with rational coefficients. + Return an empty list if the quintic is reducible or not solvable. + """ + result = [] + + coeff_5, coeff_4, p_, q_, r_, s_ = f.all_coeffs() + + if not all(coeff.is_Rational for coeff in (coeff_5, coeff_4, p_, q_, r_, s_)): + return result + + if coeff_5 != 1: + f = Poly(f / coeff_5) + _, coeff_4, p_, q_, r_, s_ = f.all_coeffs() + + # Cancel coeff_4 to form x^5 + px^3 + qx^2 + rx + s + if coeff_4: + p = p_ - 2*coeff_4*coeff_4/5 + q = q_ - 3*coeff_4*p_/5 + 4*coeff_4**3/25 + r = r_ - 2*coeff_4*q_/5 + 3*coeff_4**2*p_/25 - 3*coeff_4**4/125 + s = s_ - coeff_4*r_/5 + coeff_4**2*q_/25 - coeff_4**3*p_/125 + 4*coeff_4**5/3125 + x = f.gen + f = Poly(x**5 + p*x**3 + q*x**2 + r*x + s) + else: + p, q, r, s = p_, q_, r_, s_ + + quintic = PolyQuintic(f) + + # Eqn standardized. Algo for solving starts here + if not f.is_irreducible: + return result + f20 = quintic.f20 + # Check if f20 has linear factors over domain Z + if f20.is_irreducible: + return result + # Now, we know that f is solvable + for _factor in f20.factor_list()[1]: + if _factor[0].is_linear: + theta = _factor[0].root(0) + break + d = discriminant(f) + delta = sqrt(d) + # zeta = a fifth root of unity + zeta1, zeta2, zeta3, zeta4 = quintic.zeta + T = quintic.T(theta, d) + tol = S(1e-10) + alpha = T[1] + T[2]*delta + alpha_bar = T[1] - T[2]*delta + beta = T[3] + T[4]*delta + beta_bar = T[3] - T[4]*delta + + disc = alpha**2 - 4*beta + disc_bar = alpha_bar**2 - 4*beta_bar + + l0 = quintic.l0(theta) + Stwo = S(2) + l1 = _quintic_simplify((-alpha + sqrt(disc)) / Stwo) + l4 = _quintic_simplify((-alpha - sqrt(disc)) / Stwo) + + l2 = _quintic_simplify((-alpha_bar + sqrt(disc_bar)) / Stwo) + l3 = _quintic_simplify((-alpha_bar - sqrt(disc_bar)) / Stwo) + + order = quintic.order(theta, d) + test = (order*delta.n()) - ( (l1.n() - l4.n())*(l2.n() - l3.n()) ) + # Comparing floats + if not comp(test, 0, tol): + l2, l3 = l3, l2 + + # Now we have correct order of l's + R1 = l0 + l1*zeta1 + l2*zeta2 + l3*zeta3 + l4*zeta4 + R2 = l0 + l3*zeta1 + l1*zeta2 + l4*zeta3 + l2*zeta4 + R3 = l0 + l2*zeta1 + l4*zeta2 + l1*zeta3 + l3*zeta4 + R4 = l0 + l4*zeta1 + l3*zeta2 + l2*zeta3 + l1*zeta4 + + Res = [None, [None]*5, [None]*5, [None]*5, [None]*5] + Res_n = [None, [None]*5, [None]*5, [None]*5, [None]*5] + + # Simplifying improves performance a lot for exact expressions + R1 = _quintic_simplify(R1) + R2 = _quintic_simplify(R2) + R3 = _quintic_simplify(R3) + R4 = _quintic_simplify(R4) + + # hard-coded results for [factor(i) for i in _vsolve(x**5 - a - I*b, x)] + x0 = z**(S(1)/5) + x1 = sqrt(2) + x2 = sqrt(5) + x3 = sqrt(5 - x2) + x4 = I*x2 + x5 = x4 + I + x6 = I*x0/4 + x7 = x1*sqrt(x2 + 5) + sol = [x0, -x6*(x1*x3 - x5), x6*(x1*x3 + x5), -x6*(x4 + x7 - I), x6*(-x4 + x7 + I)] + + R1 = R1.as_real_imag() + R2 = R2.as_real_imag() + R3 = R3.as_real_imag() + R4 = R4.as_real_imag() + + for i, s in enumerate(sol): + Res[1][i] = _quintic_simplify(s.xreplace({z: R1[0] + I*R1[1]})) + Res[2][i] = _quintic_simplify(s.xreplace({z: R2[0] + I*R2[1]})) + Res[3][i] = _quintic_simplify(s.xreplace({z: R3[0] + I*R3[1]})) + Res[4][i] = _quintic_simplify(s.xreplace({z: R4[0] + I*R4[1]})) + + for i in range(1, 5): + for j in range(5): + Res_n[i][j] = Res[i][j].n() + Res[i][j] = _quintic_simplify(Res[i][j]) + r1 = Res[1][0] + r1_n = Res_n[1][0] + + for i in range(5): + if comp(im(r1_n*Res_n[4][i]), 0, tol): + r4 = Res[4][i] + break + + # Now we have various Res values. Each will be a list of five + # values. We have to pick one r value from those five for each Res + u, v = quintic.uv(theta, d) + testplus = (u + v*delta*sqrt(5)).n() + testminus = (u - v*delta*sqrt(5)).n() + + # Evaluated numbers suffixed with _n + # We will use evaluated numbers for calculation. Much faster. + r4_n = r4.n() + r2 = r3 = None + + for i in range(5): + r2temp_n = Res_n[2][i] + for j in range(5): + # Again storing away the exact number and using + # evaluated numbers in computations + r3temp_n = Res_n[3][j] + if (comp((r1_n*r2temp_n**2 + r4_n*r3temp_n**2 - testplus).n(), 0, tol) and + comp((r3temp_n*r1_n**2 + r2temp_n*r4_n**2 - testminus).n(), 0, tol)): + r2 = Res[2][i] + r3 = Res[3][j] + break + if r2 is not None: + break + else: + return [] # fall back to normal solve + + # Now, we have r's so we can get roots + x1 = (r1 + r2 + r3 + r4)/5 + x2 = (r1*zeta4 + r2*zeta3 + r3*zeta2 + r4*zeta1)/5 + x3 = (r1*zeta3 + r2*zeta1 + r3*zeta4 + r4*zeta2)/5 + x4 = (r1*zeta2 + r2*zeta4 + r3*zeta1 + r4*zeta3)/5 + x5 = (r1*zeta1 + r2*zeta2 + r3*zeta3 + r4*zeta4)/5 + result = [x1, x2, x3, x4, x5] + + # Now check if solutions are distinct + + saw = set() + for r in result: + r = r.n(2) + if r in saw: + # Roots were identical. Abort, return [] + # and fall back to usual solve + return [] + saw.add(r) + + # Restore to original equation where coeff_4 is nonzero + if coeff_4: + result = [x - coeff_4 / 5 for x in result] + return result + + +def _quintic_simplify(expr): + from sympy.simplify.simplify import powsimp + expr = powsimp(expr) + expr = cancel(expr) + return together(expr) + + +def _integer_basis(poly): + """Compute coefficient basis for a polynomial over integers. + + Returns the integer ``div`` such that substituting ``x = div*y`` + ``p(x) = m*q(y)`` where the coefficients of ``q`` are smaller + than those of ``p``. + + For example ``x**5 + 512*x + 1024 = 0`` + with ``div = 4`` becomes ``y**5 + 2*y + 1 = 0`` + + Returns the integer ``div`` or ``None`` if there is no possible scaling. + + Examples + ======== + + >>> from sympy.polys import Poly + >>> from sympy.abc import x + >>> from sympy.polys.polyroots import _integer_basis + >>> p = Poly(x**5 + 512*x + 1024, x, domain='ZZ') + >>> _integer_basis(p) + 4 + """ + monoms, coeffs = list(zip(*poly.terms())) + + monoms, = list(zip(*monoms)) + coeffs = list(map(abs, coeffs)) + + if coeffs[0] < coeffs[-1]: + coeffs = list(reversed(coeffs)) + n = monoms[0] + monoms = [n - i for i in reversed(monoms)] + else: + return None + + monoms = monoms[:-1] + coeffs = coeffs[:-1] + + # Special case for two-term polynominals + if len(monoms) == 1: + r = Pow(coeffs[0], S.One/monoms[0]) + if r.is_Integer: + return int(r) + else: + return None + + divs = reversed(divisors(gcd_list(coeffs))[1:]) + + try: + div = next(divs) + except StopIteration: + return None + + while True: + for monom, coeff in zip(monoms, coeffs): + if coeff % div**monom != 0: + try: + div = next(divs) + except StopIteration: + return None + else: + break + else: + return div + + +def preprocess_roots(poly): + """Try to get rid of symbolic coefficients from ``poly``. """ + coeff = S.One + + poly_func = poly.func + try: + _, poly = poly.clear_denoms(convert=True) + except DomainError: + return coeff, poly + + poly = poly.primitive()[1] + poly = poly.retract() + + # TODO: This is fragile. Figure out how to make this independent of construct_domain(). + if poly.get_domain().is_Poly and all(c.is_term for c in poly.rep.coeffs()): + poly = poly.inject() + + strips = list(zip(*poly.monoms())) + gens = list(poly.gens[1:]) + + base, strips = strips[0], strips[1:] + + for gen, strip in zip(list(gens), strips): + reverse = False + + if strip[0] < strip[-1]: + strip = reversed(strip) + reverse = True + + ratio = None + + for a, b in zip(base, strip): + if not a and not b: + continue + elif not a or not b: + break + elif b % a != 0: + break + else: + _ratio = b // a + + if ratio is None: + ratio = _ratio + elif ratio != _ratio: + break + else: + if reverse: + ratio = -ratio + + poly = poly.eval(gen, 1) + coeff *= gen**(-ratio) + gens.remove(gen) + + if gens: + poly = poly.eject(*gens) + + if poly.is_univariate and poly.get_domain().is_ZZ: + basis = _integer_basis(poly) + + if basis is not None: + n = poly.degree() + + def func(k, coeff): + return coeff//basis**(n - k[0]) + + poly = poly.termwise(func) + coeff *= basis + + if not isinstance(poly, poly_func): + poly = poly_func(poly) + return coeff, poly + + +@public +def roots(f, *gens, + auto=True, + cubics=True, + trig=False, + quartics=True, + quintics=False, + multiple=False, + filter=None, + predicate=None, + strict=False, + **flags): + """ + Computes symbolic roots of a univariate polynomial. + + Given a univariate polynomial f with symbolic coefficients (or + a list of the polynomial's coefficients), returns a dictionary + with its roots and their multiplicities. + + Only roots expressible via radicals will be returned. To get + a complete set of roots use RootOf class or numerical methods + instead. By default cubic and quartic formulas are used in + the algorithm. To disable them because of unreadable output + set ``cubics=False`` or ``quartics=False`` respectively. If cubic + roots are real but are expressed in terms of complex numbers + (casus irreducibilis [1]) the ``trig`` flag can be set to True to + have the solutions returned in terms of cosine and inverse cosine + functions. + + To get roots from a specific domain set the ``filter`` flag with + one of the following specifiers: Z, Q, R, I, C. By default all + roots are returned (this is equivalent to setting ``filter='C'``). + + By default a dictionary is returned giving a compact result in + case of multiple roots. However to get a list containing all + those roots set the ``multiple`` flag to True; the list will + have identical roots appearing next to each other in the result. + (For a given Poly, the all_roots method will give the roots in + sorted numerical order.) + + If the ``strict`` flag is True, ``UnsolvableFactorError`` will be + raised if the roots found are known to be incomplete (because + some roots are not expressible in radicals). + + Examples + ======== + + >>> from sympy import Poly, roots, degree + >>> from sympy.abc import x, y + + >>> roots(x**2 - 1, x) + {-1: 1, 1: 1} + + >>> p = Poly(x**2-1, x) + >>> roots(p) + {-1: 1, 1: 1} + + >>> p = Poly(x**2-y, x, y) + + >>> roots(Poly(p, x)) + {-sqrt(y): 1, sqrt(y): 1} + + >>> roots(x**2 - y, x) + {-sqrt(y): 1, sqrt(y): 1} + + >>> roots([1, 0, -1]) + {-1: 1, 1: 1} + + ``roots`` will only return roots expressible in radicals. If + the given polynomial has some or all of its roots inexpressible in + radicals, the result of ``roots`` will be incomplete or empty + respectively. + + Example where result is incomplete: + + >>> roots((x-1)*(x**5-x+1), x) + {1: 1} + + In this case, the polynomial has an unsolvable quintic factor + whose roots cannot be expressed by radicals. The polynomial has a + rational root (due to the factor `(x-1)`), which is returned since + ``roots`` always finds all rational roots. + + Example where result is empty: + + >>> roots(x**7-3*x**2+1, x) + {} + + Here, the polynomial has no roots expressible in radicals, so + ``roots`` returns an empty dictionary. + + The result produced by ``roots`` is complete if and only if the + sum of the multiplicity of each root is equal to the degree of + the polynomial. If strict=True, UnsolvableFactorError will be + raised if the result is incomplete. + + The result can be be checked for completeness as follows: + + >>> f = x**3-2*x**2+1 + >>> sum(roots(f, x).values()) == degree(f, x) + True + >>> f = (x-1)*(x**5-x+1) + >>> sum(roots(f, x).values()) == degree(f, x) + False + + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Cubic_equation#Trigonometric_and_hyperbolic_solutions + + """ + from sympy.polys.polytools import to_rational_coeffs + flags = dict(flags) + + if isinstance(f, list): + if gens: + raise ValueError('redundant generators given') + + x = Dummy('x') + + poly, i = {}, len(f) - 1 + + for coeff in f: + poly[i], i = sympify(coeff), i - 1 + + f = Poly(poly, x, field=True) + else: + try: + F = Poly(f, *gens, **flags) + if not isinstance(f, Poly) and not F.gen.is_Symbol: + raise PolynomialError("generator must be a Symbol") + f = F + except GeneratorsNeeded: + if multiple: + return [] + else: + return {} + else: + n = f.degree() + if f.length() == 2 and n > 2: + # check for foo**n in constant if dep is c*gen**m + con, dep = f.as_expr().as_independent(*f.gens) + fcon = -(-con).factor() + if fcon != con: + con = fcon + bases = [] + for i in Mul.make_args(con): + if i.is_Pow: + b, e = i.as_base_exp() + if e.is_Integer and b.is_Add: + bases.append((b, Dummy(positive=True))) + if bases: + rv = roots(Poly((dep + con).xreplace(dict(bases)), + *f.gens), *F.gens, + auto=auto, + cubics=cubics, + trig=trig, + quartics=quartics, + quintics=quintics, + multiple=multiple, + filter=filter, + predicate=predicate, + **flags) + return {factor_terms(k.xreplace( + {v: k for k, v in bases}) + ): v for k, v in rv.items()} + + if f.is_multivariate: + raise PolynomialError('multivariate polynomials are not supported') + + def _update_dict(result, zeros, currentroot, k): + if currentroot == S.Zero: + if S.Zero in zeros: + zeros[S.Zero] += k + else: + zeros[S.Zero] = k + if currentroot in result: + result[currentroot] += k + else: + result[currentroot] = k + + def _try_decompose(f): + """Find roots using functional decomposition. """ + factors, roots = f.decompose(), [] + + for currentroot in _try_heuristics(factors[0]): + roots.append(currentroot) + + for currentfactor in factors[1:]: + previous, roots = list(roots), [] + + for currentroot in previous: + g = currentfactor - Poly(currentroot, f.gen) + + for currentroot in _try_heuristics(g): + roots.append(currentroot) + + return roots + + def _try_heuristics(f): + """Find roots using formulas and some tricks. """ + if f.is_ground: + return [] + if f.is_monomial: + return [S.Zero]*f.degree() + + if f.length() == 2: + if f.degree() == 1: + return list(map(cancel, roots_linear(f))) + else: + return roots_binomial(f) + + result = [] + + for i in [-1, 1]: + if not f.eval(i): + f = f.quo(Poly(f.gen - i, f.gen)) + result.append(i) + break + + n = f.degree() + + if n == 1: + result += list(map(cancel, roots_linear(f))) + elif n == 2: + result += list(map(cancel, roots_quadratic(f))) + elif f.is_cyclotomic: + result += roots_cyclotomic(f) + elif n == 3 and cubics: + result += roots_cubic(f, trig=trig) + elif n == 4 and quartics: + result += roots_quartic(f) + elif n == 5 and quintics: + result += roots_quintic(f) + + return result + + # Convert the generators to symbols + dumgens = symbols('x:%d' % len(f.gens), cls=Dummy) + f = f.per(f.rep, dumgens) + + (k,), f = f.terms_gcd() + + if not k: + zeros = {} + else: + zeros = {S.Zero: k} + + coeff, f = preprocess_roots(f) + + if auto and f.get_domain().is_Ring: + f = f.to_field() + + # Use EX instead of ZZ_I or QQ_I + if f.get_domain().is_QQ_I: + f = f.per(f.rep.convert(EX)) + + rescale_x = None + translate_x = None + + result = {} + + if not f.is_ground: + dom = f.get_domain() + if not dom.is_Exact and dom.is_Numerical: + for r in f.nroots(): + _update_dict(result, zeros, r, 1) + elif f.degree() == 1: + _update_dict(result, zeros, roots_linear(f)[0], 1) + elif f.length() == 2: + roots_fun = roots_quadratic if f.degree() == 2 else roots_binomial + for r in roots_fun(f): + _update_dict(result, zeros, r, 1) + else: + _, factors = Poly(f.as_expr()).factor_list() + if len(factors) == 1 and f.degree() == 2: + for r in roots_quadratic(f): + _update_dict(result, zeros, r, 1) + else: + if len(factors) == 1 and factors[0][1] == 1: + if f.get_domain().is_EX: + res = to_rational_coeffs(f) + if res: + if res[0] is None: + translate_x, f = res[2:] + else: + rescale_x, f = res[1], res[-1] + result = roots(f) + if not result: + for currentroot in _try_decompose(f): + _update_dict(result, zeros, currentroot, 1) + else: + for r in _try_heuristics(f): + _update_dict(result, zeros, r, 1) + else: + for currentroot in _try_decompose(f): + _update_dict(result, zeros, currentroot, 1) + else: + for currentfactor, k in factors: + for r in _try_heuristics(Poly(currentfactor, f.gen, field=True)): + _update_dict(result, zeros, r, k) + + if coeff is not S.One: + _result, result, = result, {} + + for currentroot, k in _result.items(): + result[coeff*currentroot] = k + + if filter not in [None, 'C']: + handlers = { + 'Z': lambda r: r.is_Integer, + 'Q': lambda r: r.is_Rational, + 'R': lambda r: all(a.is_real for a in r.as_numer_denom()), + 'I': lambda r: r.is_imaginary, + } + + try: + query = handlers[filter] + except KeyError: + raise ValueError("Invalid filter: %s" % filter) + + for zero in dict(result).keys(): + if not query(zero): + del result[zero] + + if predicate is not None: + for zero in dict(result).keys(): + if not predicate(zero): + del result[zero] + if rescale_x: + result1 = {} + for k, v in result.items(): + result1[k*rescale_x] = v + result = result1 + if translate_x: + result1 = {} + for k, v in result.items(): + result1[k + translate_x] = v + result = result1 + + # adding zero roots after non-trivial roots have been translated + result.update(zeros) + + if strict and sum(result.values()) < f.degree(): + raise UnsolvableFactorError(filldedent(''' + Strict mode: some factors cannot be solved in radicals, so + a complete list of solutions cannot be returned. Call + roots with strict=False to get solutions expressible in + radicals (if there are any). + ''')) + + if not multiple: + return result + else: + zeros = [] + + for zero in ordered(result): + zeros.extend([zero]*result[zero]) + + return zeros + + +def root_factors(f, *gens, filter=None, **args): + """ + Returns all factors of a univariate polynomial. + + Examples + ======== + + >>> from sympy.abc import x, y + >>> from sympy.polys.polyroots import root_factors + + >>> root_factors(x**2 - y, x) + [x - sqrt(y), x + sqrt(y)] + + """ + args = dict(args) + + F = Poly(f, *gens, **args) + + if not F.is_Poly: + return [f] + + if F.is_multivariate: + raise ValueError('multivariate polynomials are not supported') + + x = F.gens[0] + + zeros = roots(F, filter=filter) + + if not zeros: + factors = [F] + else: + factors, N = [], 0 + + for r, n in ordered(zeros.items()): + factors, N = factors + [Poly(x - r, x)]*n, N + n + + if N < F.degree(): + G = reduce(lambda p, q: p*q, factors) + factors.append(F.quo(G)) + + if not isinstance(f, Poly): + factors = [ f.as_expr() for f in factors ] + + return factors diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/polytools.py b/.venv/lib/python3.12/site-packages/sympy/polys/polytools.py new file mode 100644 index 0000000000000000000000000000000000000000..11b9dd3435f8dc68ea3b0578df9fccfb07dd0f4c --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/polytools.py @@ -0,0 +1,7960 @@ +"""User-friendly public interface to polynomial functions. """ + +from __future__ import annotations + +from functools import wraps, reduce +from operator import mul +from typing import Optional +from collections import Counter, defaultdict + +from sympy.core import ( + S, Expr, Add, Tuple +) +from sympy.core.basic import Basic +from sympy.core.decorators import _sympifyit +from sympy.core.exprtools import Factors, factor_nc, factor_terms +from sympy.core.evalf import ( + pure_complex, evalf, fastlog, _evalf_with_bounded_error, quad_to_mpmath) +from sympy.core.function import Derivative +from sympy.core.mul import Mul, _keep_coeff +from sympy.core.intfunc import ilcm +from sympy.core.numbers import I, Integer, equal_valued +from sympy.core.relational import Relational, Equality +from sympy.core.sorting import ordered +from sympy.core.symbol import Dummy, Symbol +from sympy.core.sympify import sympify, _sympify +from sympy.core.traversal import preorder_traversal, bottom_up +from sympy.logic.boolalg import BooleanAtom +from sympy.polys import polyoptions as options +from sympy.polys.constructor import construct_domain +from sympy.polys.domains import FF, QQ, ZZ +from sympy.polys.domains.domainelement import DomainElement +from sympy.polys.fglmtools import matrix_fglm +from sympy.polys.groebnertools import groebner as _groebner +from sympy.polys.monomials import Monomial +from sympy.polys.orderings import monomial_key +from sympy.polys.polyclasses import DMP, DMF, ANP +from sympy.polys.polyerrors import ( + OperationNotSupported, DomainError, + CoercionFailed, UnificationFailed, + GeneratorsNeeded, PolynomialError, + MultivariatePolynomialError, + ExactQuotientFailed, + PolificationFailed, + ComputationFailed, + GeneratorsError, +) +from sympy.polys.polyutils import ( + basic_from_dict, + _sort_gens, + _unify_gens, + _dict_reorder, + _dict_from_expr, + _parallel_dict_from_expr, +) +from sympy.polys.rationaltools import together +from sympy.polys.rootisolation import dup_isolate_real_roots_list +from sympy.utilities import group, public, filldedent +from sympy.utilities.exceptions import sympy_deprecation_warning +from sympy.utilities.iterables import iterable, sift + +# Required to avoid errors +import sympy.polys + +import mpmath +from mpmath.libmp.libhyper import NoConvergence + + + +def _polifyit(func): + @wraps(func) + def wrapper(f, g): + g = _sympify(g) + if isinstance(g, Poly): + return func(f, g) + elif isinstance(g, Integer): + g = f.from_expr(g, *f.gens, domain=f.domain) + return func(f, g) + elif isinstance(g, Expr): + try: + g = f.from_expr(g, *f.gens) + except PolynomialError: + if g.is_Matrix: + return NotImplemented + expr_method = getattr(f.as_expr(), func.__name__) + result = expr_method(g) + if result is not NotImplemented: + sympy_deprecation_warning( + """ + Mixing Poly with non-polynomial expressions in binary + operations is deprecated. Either explicitly convert + the non-Poly operand to a Poly with as_poly() or + convert the Poly to an Expr with as_expr(). + """, + deprecated_since_version="1.6", + active_deprecations_target="deprecated-poly-nonpoly-binary-operations", + ) + return result + else: + return func(f, g) + else: + return NotImplemented + return wrapper + + + +@public +class Poly(Basic): + """ + Generic class for representing and operating on polynomial expressions. + + See :ref:`polys-docs` for general documentation. + + Poly is a subclass of Basic rather than Expr but instances can be + converted to Expr with the :py:meth:`~.Poly.as_expr` method. + + .. deprecated:: 1.6 + + Combining Poly with non-Poly objects in binary operations is + deprecated. Explicitly convert both objects to either Poly or Expr + first. See :ref:`deprecated-poly-nonpoly-binary-operations`. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + Create a univariate polynomial: + + >>> Poly(x*(x**2 + x - 1)**2) + Poly(x**5 + 2*x**4 - x**3 - 2*x**2 + x, x, domain='ZZ') + + Create a univariate polynomial with specific domain: + + >>> from sympy import sqrt + >>> Poly(x**2 + 2*x + sqrt(3), domain='R') + Poly(1.0*x**2 + 2.0*x + 1.73205080756888, x, domain='RR') + + Create a multivariate polynomial: + + >>> Poly(y*x**2 + x*y + 1) + Poly(x**2*y + x*y + 1, x, y, domain='ZZ') + + Create a univariate polynomial, where y is a constant: + + >>> Poly(y*x**2 + x*y + 1,x) + Poly(y*x**2 + y*x + 1, x, domain='ZZ[y]') + + You can evaluate the above polynomial as a function of y: + + >>> Poly(y*x**2 + x*y + 1,x).eval(2) + 6*y + 1 + + See Also + ======== + + sympy.core.expr.Expr + + """ + + __slots__ = ('rep', 'gens') + + is_commutative = True + is_Poly = True + _op_priority = 10.001 + + rep: DMP + gens: tuple[Expr, ...] + + def __new__(cls, rep, *gens, **args) -> Poly: + """Create a new polynomial instance out of something useful. """ + opt = options.build_options(gens, args) + + if 'order' in opt: + raise NotImplementedError("'order' keyword is not implemented yet") + + if isinstance(rep, (DMP, DMF, ANP, DomainElement)): + return cls._from_domain_element(rep, opt) + elif iterable(rep, exclude=str): + if isinstance(rep, dict): + return cls._from_dict(rep, opt) + else: + return cls._from_list(list(rep), opt) + else: + rep = sympify(rep, evaluate=type(rep) is not str) # type: ignore + + if rep.is_Poly: + return cls._from_poly(rep, opt) + else: + return cls._from_expr(rep, opt) + + # Poly does not pass its args to Basic.__new__ to be stored in _args so we + # have to emulate them here with an args property that derives from rep + # and gens which are instance attributes. This also means we need to + # define _hashable_content. The _hashable_content is rep and gens but args + # uses expr instead of rep (expr is the Basic version of rep). Passing + # expr in args means that Basic methods like subs should work. Using rep + # otherwise means that Poly can remain more efficient than Basic by + # avoiding creating a Basic instance just to be hashable. + + @classmethod + def new(cls, rep, *gens): + """Construct :class:`Poly` instance from raw representation. """ + if not isinstance(rep, DMP): + raise PolynomialError( + "invalid polynomial representation: %s" % rep) + elif rep.lev != len(gens) - 1: + raise PolynomialError("invalid arguments: %s, %s" % (rep, gens)) + + obj = Basic.__new__(cls) + obj.rep = rep + obj.gens = gens + + return obj + + @property + def expr(self): + return basic_from_dict(self.rep.to_sympy_dict(), *self.gens) + + @property + def args(self): + return (self.expr,) + self.gens + + def _hashable_content(self): + return (self.rep,) + self.gens + + @classmethod + def from_dict(cls, rep, *gens, **args): + """Construct a polynomial from a ``dict``. """ + opt = options.build_options(gens, args) + return cls._from_dict(rep, opt) + + @classmethod + def from_list(cls, rep, *gens, **args): + """Construct a polynomial from a ``list``. """ + opt = options.build_options(gens, args) + return cls._from_list(rep, opt) + + @classmethod + def from_poly(cls, rep, *gens, **args): + """Construct a polynomial from a polynomial. """ + opt = options.build_options(gens, args) + return cls._from_poly(rep, opt) + + @classmethod + def from_expr(cls, rep, *gens, **args): + """Construct a polynomial from an expression. """ + opt = options.build_options(gens, args) + return cls._from_expr(rep, opt) + + @classmethod + def _from_dict(cls, rep, opt): + """Construct a polynomial from a ``dict``. """ + gens = opt.gens + + if not gens: + raise GeneratorsNeeded( + "Cannot initialize from 'dict' without generators") + + level = len(gens) - 1 + domain = opt.domain + + if domain is None: + domain, rep = construct_domain(rep, opt=opt) + else: + for monom, coeff in rep.items(): + rep[monom] = domain.convert(coeff) + + return cls.new(DMP.from_dict(rep, level, domain), *gens) + + @classmethod + def _from_list(cls, rep, opt): + """Construct a polynomial from a ``list``. """ + gens = opt.gens + + if not gens: + raise GeneratorsNeeded( + "Cannot initialize from 'list' without generators") + elif len(gens) != 1: + raise MultivariatePolynomialError( + "'list' representation not supported") + + level = len(gens) - 1 + domain = opt.domain + + if domain is None: + domain, rep = construct_domain(rep, opt=opt) + else: + rep = list(map(domain.convert, rep)) + + return cls.new(DMP.from_list(rep, level, domain), *gens) + + @classmethod + def _from_poly(cls, rep, opt): + """Construct a polynomial from a polynomial. """ + if cls != rep.__class__: + rep = cls.new(rep.rep, *rep.gens) + + gens = opt.gens + field = opt.field + domain = opt.domain + + if gens and rep.gens != gens: + if set(rep.gens) != set(gens): + return cls._from_expr(rep.as_expr(), opt) + else: + rep = rep.reorder(*gens) + + if 'domain' in opt and domain: + rep = rep.set_domain(domain) + elif field is True: + rep = rep.to_field() + + return rep + + @classmethod + def _from_expr(cls, rep, opt): + """Construct a polynomial from an expression. """ + rep, opt = _dict_from_expr(rep, opt) + return cls._from_dict(rep, opt) + + @classmethod + def _from_domain_element(cls, rep, opt): + gens = opt.gens + domain = opt.domain + + level = len(gens) - 1 + rep = [domain.convert(rep)] + + return cls.new(DMP.from_list(rep, level, domain), *gens) + + def __hash__(self): + return super().__hash__() + + @property + def free_symbols(self): + """ + Free symbols of a polynomial expression. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y, z + + >>> Poly(x**2 + 1).free_symbols + {x} + >>> Poly(x**2 + y).free_symbols + {x, y} + >>> Poly(x**2 + y, x).free_symbols + {x, y} + >>> Poly(x**2 + y, x, z).free_symbols + {x, y} + + """ + symbols = set() + gens = self.gens + for i in range(len(gens)): + for monom in self.monoms(): + if monom[i]: + symbols |= gens[i].free_symbols + break + + return symbols | self.free_symbols_in_domain + + @property + def free_symbols_in_domain(self): + """ + Free symbols of the domain of ``self``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + >>> Poly(x**2 + 1).free_symbols_in_domain + set() + >>> Poly(x**2 + y).free_symbols_in_domain + set() + >>> Poly(x**2 + y, x).free_symbols_in_domain + {y} + + """ + domain, symbols = self.rep.dom, set() + + if domain.is_Composite: + for gen in domain.symbols: + symbols |= gen.free_symbols + elif domain.is_EX: + for coeff in self.coeffs(): + symbols |= coeff.free_symbols + + return symbols + + @property + def gen(self): + """ + Return the principal generator. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**2 + 1, x).gen + x + + """ + return self.gens[0] + + @property + def domain(self): + """Get the ground domain of a :py:class:`~.Poly` + + Returns + ======= + + :py:class:`~.Domain`: + Ground domain of the :py:class:`~.Poly`. + + Examples + ======== + + >>> from sympy import Poly, Symbol + >>> x = Symbol('x') + >>> p = Poly(x**2 + x) + >>> p + Poly(x**2 + x, x, domain='ZZ') + >>> p.domain + ZZ + """ + return self.get_domain() + + @property + def zero(self): + """Return zero polynomial with ``self``'s properties. """ + return self.new(self.rep.zero(self.rep.lev, self.rep.dom), *self.gens) + + @property + def one(self): + """Return one polynomial with ``self``'s properties. """ + return self.new(self.rep.one(self.rep.lev, self.rep.dom), *self.gens) + + def unify(f, g): + """ + Make ``f`` and ``g`` belong to the same domain. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> f, g = Poly(x/2 + 1), Poly(2*x + 1) + + >>> f + Poly(1/2*x + 1, x, domain='QQ') + >>> g + Poly(2*x + 1, x, domain='ZZ') + + >>> F, G = f.unify(g) + + >>> F + Poly(1/2*x + 1, x, domain='QQ') + >>> G + Poly(2*x + 1, x, domain='QQ') + + """ + _, per, F, G = f._unify(g) + return per(F), per(G) + + def _unify(f, g): + g = sympify(g) + + if not g.is_Poly: + try: + g_coeff = f.rep.dom.from_sympy(g) + except CoercionFailed: + raise UnificationFailed("Cannot unify %s with %s" % (f, g)) + else: + return f.rep.dom, f.per, f.rep, f.rep.ground_new(g_coeff) + + if isinstance(f.rep, DMP) and isinstance(g.rep, DMP): + gens = _unify_gens(f.gens, g.gens) + + dom, lev = f.rep.dom.unify(g.rep.dom, gens), len(gens) - 1 + + if f.gens != gens: + f_monoms, f_coeffs = _dict_reorder( + f.rep.to_dict(), f.gens, gens) + + if f.rep.dom != dom: + f_coeffs = [dom.convert(c, f.rep.dom) for c in f_coeffs] + + F = DMP.from_dict(dict(list(zip(f_monoms, f_coeffs))), lev, dom) + else: + F = f.rep.convert(dom) + + if g.gens != gens: + g_monoms, g_coeffs = _dict_reorder( + g.rep.to_dict(), g.gens, gens) + + if g.rep.dom != dom: + g_coeffs = [dom.convert(c, g.rep.dom) for c in g_coeffs] + + G = DMP.from_dict(dict(list(zip(g_monoms, g_coeffs))), lev, dom) + else: + G = g.rep.convert(dom) + else: + raise UnificationFailed("Cannot unify %s with %s" % (f, g)) + + cls = f.__class__ + + def per(rep, dom=dom, gens=gens, remove=None): + if remove is not None: + gens = gens[:remove] + gens[remove + 1:] + + if not gens: + return dom.to_sympy(rep) + + return cls.new(rep, *gens) + + return dom, per, F, G + + def per(f, rep, gens=None, remove=None): + """ + Create a Poly out of the given representation. + + Examples + ======== + + >>> from sympy import Poly, ZZ + >>> from sympy.abc import x, y + + >>> from sympy.polys.polyclasses import DMP + + >>> a = Poly(x**2 + 1) + + >>> a.per(DMP([ZZ(1), ZZ(1)], ZZ), gens=[y]) + Poly(y + 1, y, domain='ZZ') + + """ + if gens is None: + gens = f.gens + + if remove is not None: + gens = gens[:remove] + gens[remove + 1:] + + if not gens: + return f.rep.dom.to_sympy(rep) + + return f.__class__.new(rep, *gens) + + def set_domain(f, domain): + """Set the ground domain of ``f``. """ + opt = options.build_options(f.gens, {'domain': domain}) + return f.per(f.rep.convert(opt.domain)) + + def get_domain(f): + """Get the ground domain of ``f``. """ + return f.rep.dom + + def set_modulus(f, modulus): + """ + Set the modulus of ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(5*x**2 + 2*x - 1, x).set_modulus(2) + Poly(x**2 + 1, x, modulus=2) + + """ + modulus = options.Modulus.preprocess(modulus) + return f.set_domain(FF(modulus)) + + def get_modulus(f): + """ + Get the modulus of ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**2 + 1, modulus=2).get_modulus() + 2 + + """ + domain = f.get_domain() + + if domain.is_FiniteField: + return Integer(domain.characteristic()) + else: + raise PolynomialError("not a polynomial over a Galois field") + + def _eval_subs(f, old, new): + """Internal implementation of :func:`subs`. """ + if old in f.gens: + if new.is_number: + return f.eval(old, new) + else: + try: + return f.replace(old, new) + except PolynomialError: + pass + + return f.as_expr().subs(old, new) + + def exclude(f): + """ + Remove unnecessary generators from ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import a, b, c, d, x + + >>> Poly(a + x, a, b, c, d, x).exclude() + Poly(a + x, a, x, domain='ZZ') + + """ + J, new = f.rep.exclude() + gens = [gen for j, gen in enumerate(f.gens) if j not in J] + + return f.per(new, gens=gens) + + def replace(f, x, y=None, **_ignore): + # XXX this does not match Basic's signature + """ + Replace ``x`` with ``y`` in generators list. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + >>> Poly(x**2 + 1, x).replace(x, y) + Poly(y**2 + 1, y, domain='ZZ') + + """ + if y is None: + if f.is_univariate: + x, y = f.gen, x + else: + raise PolynomialError( + "syntax supported only in univariate case") + + if x == y or x not in f.gens: + return f + + if x in f.gens and y not in f.gens: + dom = f.get_domain() + + if not dom.is_Composite or y not in dom.symbols: + gens = list(f.gens) + gens[gens.index(x)] = y + return f.per(f.rep, gens=gens) + + raise PolynomialError("Cannot replace %s with %s in %s" % (x, y, f)) + + def match(f, *args, **kwargs): + """Match expression from Poly. See Basic.match()""" + return f.as_expr().match(*args, **kwargs) + + def reorder(f, *gens, **args): + """ + Efficiently apply new order of generators. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + >>> Poly(x**2 + x*y**2, x, y).reorder(y, x) + Poly(y**2*x + x**2, y, x, domain='ZZ') + + """ + opt = options.Options((), args) + + if not gens: + gens = _sort_gens(f.gens, opt=opt) + elif set(f.gens) != set(gens): + raise PolynomialError( + "generators list can differ only up to order of elements") + + rep = dict(list(zip(*_dict_reorder(f.rep.to_dict(), f.gens, gens)))) + + return f.per(DMP.from_dict(rep, len(gens) - 1, f.rep.dom), gens=gens) + + def ltrim(f, gen): + """ + Remove dummy generators from ``f`` that are to the left of + specified ``gen`` in the generators as ordered. When ``gen`` + is an integer, it refers to the generator located at that + position within the tuple of generators of ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y, z + + >>> Poly(y**2 + y*z**2, x, y, z).ltrim(y) + Poly(y**2 + y*z**2, y, z, domain='ZZ') + >>> Poly(z, x, y, z).ltrim(-1) + Poly(z, z, domain='ZZ') + + """ + rep = f.as_dict(native=True) + j = f._gen_to_level(gen) + + terms = {} + + for monom, coeff in rep.items(): + + if any(monom[:j]): + # some generator is used in the portion to be trimmed + raise PolynomialError("Cannot left trim %s" % f) + + terms[monom[j:]] = coeff + + gens = f.gens[j:] + + return f.new(DMP.from_dict(terms, len(gens) - 1, f.rep.dom), *gens) + + def has_only_gens(f, *gens): + """ + Return ``True`` if ``Poly(f, *gens)`` retains ground domain. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y, z + + >>> Poly(x*y + 1, x, y, z).has_only_gens(x, y) + True + >>> Poly(x*y + z, x, y, z).has_only_gens(x, y) + False + + """ + indices = set() + + for gen in gens: + try: + index = f.gens.index(gen) + except ValueError: + raise GeneratorsError( + "%s doesn't have %s as generator" % (f, gen)) + else: + indices.add(index) + + for monom in f.monoms(): + for i, elt in enumerate(monom): + if i not in indices and elt: + return False + + return True + + def to_ring(f): + """ + Make the ground domain a ring. + + Examples + ======== + + >>> from sympy import Poly, QQ + >>> from sympy.abc import x + + >>> Poly(x**2 + 1, domain=QQ).to_ring() + Poly(x**2 + 1, x, domain='ZZ') + + """ + if hasattr(f.rep, 'to_ring'): + result = f.rep.to_ring() + else: # pragma: no cover + raise OperationNotSupported(f, 'to_ring') + + return f.per(result) + + def to_field(f): + """ + Make the ground domain a field. + + Examples + ======== + + >>> from sympy import Poly, ZZ + >>> from sympy.abc import x + + >>> Poly(x**2 + 1, x, domain=ZZ).to_field() + Poly(x**2 + 1, x, domain='QQ') + + """ + if hasattr(f.rep, 'to_field'): + result = f.rep.to_field() + else: # pragma: no cover + raise OperationNotSupported(f, 'to_field') + + return f.per(result) + + def to_exact(f): + """ + Make the ground domain exact. + + Examples + ======== + + >>> from sympy import Poly, RR + >>> from sympy.abc import x + + >>> Poly(x**2 + 1.0, x, domain=RR).to_exact() + Poly(x**2 + 1, x, domain='QQ') + + """ + if hasattr(f.rep, 'to_exact'): + result = f.rep.to_exact() + else: # pragma: no cover + raise OperationNotSupported(f, 'to_exact') + + return f.per(result) + + def retract(f, field=None): + """ + Recalculate the ground domain of a polynomial. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> f = Poly(x**2 + 1, x, domain='QQ[y]') + >>> f + Poly(x**2 + 1, x, domain='QQ[y]') + + >>> f.retract() + Poly(x**2 + 1, x, domain='ZZ') + >>> f.retract(field=True) + Poly(x**2 + 1, x, domain='QQ') + + """ + dom, rep = construct_domain(f.as_dict(zero=True), + field=field, composite=f.domain.is_Composite or None) + return f.from_dict(rep, f.gens, domain=dom) + + def slice(f, x, m, n=None): + """Take a continuous subsequence of terms of ``f``. """ + if n is None: + j, m, n = 0, x, m + else: + j = f._gen_to_level(x) + + m, n = int(m), int(n) + + if hasattr(f.rep, 'slice'): + result = f.rep.slice(m, n, j) + else: # pragma: no cover + raise OperationNotSupported(f, 'slice') + + return f.per(result) + + def coeffs(f, order=None): + """ + Returns all non-zero coefficients from ``f`` in lex order. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**3 + 2*x + 3, x).coeffs() + [1, 2, 3] + + See Also + ======== + all_coeffs + coeff_monomial + nth + + """ + return [f.rep.dom.to_sympy(c) for c in f.rep.coeffs(order=order)] + + def monoms(f, order=None): + """ + Returns all non-zero monomials from ``f`` in lex order. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + >>> Poly(x**2 + 2*x*y**2 + x*y + 3*y, x, y).monoms() + [(2, 0), (1, 2), (1, 1), (0, 1)] + + See Also + ======== + all_monoms + + """ + return f.rep.monoms(order=order) + + def terms(f, order=None): + """ + Returns all non-zero terms from ``f`` in lex order. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + >>> Poly(x**2 + 2*x*y**2 + x*y + 3*y, x, y).terms() + [((2, 0), 1), ((1, 2), 2), ((1, 1), 1), ((0, 1), 3)] + + See Also + ======== + all_terms + + """ + return [(m, f.rep.dom.to_sympy(c)) for m, c in f.rep.terms(order=order)] + + def all_coeffs(f): + """ + Returns all coefficients from a univariate polynomial ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**3 + 2*x - 1, x).all_coeffs() + [1, 0, 2, -1] + + """ + return [f.rep.dom.to_sympy(c) for c in f.rep.all_coeffs()] + + def all_monoms(f): + """ + Returns all monomials from a univariate polynomial ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**3 + 2*x - 1, x).all_monoms() + [(3,), (2,), (1,), (0,)] + + See Also + ======== + all_terms + + """ + return f.rep.all_monoms() + + def all_terms(f): + """ + Returns all terms from a univariate polynomial ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**3 + 2*x - 1, x).all_terms() + [((3,), 1), ((2,), 0), ((1,), 2), ((0,), -1)] + + """ + return [(m, f.rep.dom.to_sympy(c)) for m, c in f.rep.all_terms()] + + def termwise(f, func, *gens, **args): + """ + Apply a function to all terms of ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> def func(k, coeff): + ... k = k[0] + ... return coeff//10**(2-k) + + >>> Poly(x**2 + 20*x + 400).termwise(func) + Poly(x**2 + 2*x + 4, x, domain='ZZ') + + """ + terms = {} + + for monom, coeff in f.terms(): + result = func(monom, coeff) + + if isinstance(result, tuple): + monom, coeff = result + else: + coeff = result + + if coeff: + if monom not in terms: + terms[monom] = coeff + else: + raise PolynomialError( + "%s monomial was generated twice" % monom) + + return f.from_dict(terms, *(gens or f.gens), **args) + + def length(f): + """ + Returns the number of non-zero terms in ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**2 + 2*x - 1).length() + 3 + + """ + return len(f.as_dict()) + + def as_dict(f, native=False, zero=False): + """ + Switch to a ``dict`` representation. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + >>> Poly(x**2 + 2*x*y**2 - y, x, y).as_dict() + {(0, 1): -1, (1, 2): 2, (2, 0): 1} + + """ + if native: + return f.rep.to_dict(zero=zero) + else: + return f.rep.to_sympy_dict(zero=zero) + + def as_list(f, native=False): + """Switch to a ``list`` representation. """ + if native: + return f.rep.to_list() + else: + return f.rep.to_sympy_list() + + def as_expr(f, *gens): + """ + Convert a Poly instance to an Expr instance. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + >>> f = Poly(x**2 + 2*x*y**2 - y, x, y) + + >>> f.as_expr() + x**2 + 2*x*y**2 - y + >>> f.as_expr({x: 5}) + 10*y**2 - y + 25 + >>> f.as_expr(5, 6) + 379 + + """ + if not gens: + return f.expr + + if len(gens) == 1 and isinstance(gens[0], dict): + mapping = gens[0] + gens = list(f.gens) + + for gen, value in mapping.items(): + try: + index = gens.index(gen) + except ValueError: + raise GeneratorsError( + "%s doesn't have %s as generator" % (f, gen)) + else: + gens[index] = value + + return basic_from_dict(f.rep.to_sympy_dict(), *gens) + + def as_poly(self, *gens, **args): + """Converts ``self`` to a polynomial or returns ``None``. + + >>> from sympy import sin + >>> from sympy.abc import x, y + + >>> print((x**2 + x*y).as_poly()) + Poly(x**2 + x*y, x, y, domain='ZZ') + + >>> print((x**2 + x*y).as_poly(x, y)) + Poly(x**2 + x*y, x, y, domain='ZZ') + + >>> print((x**2 + sin(y)).as_poly(x, y)) + None + + """ + try: + poly = Poly(self, *gens, **args) + + if not poly.is_Poly: + return None + else: + return poly + except PolynomialError: + return None + + def lift(f): + """ + Convert algebraic coefficients to rationals. + + Examples + ======== + + >>> from sympy import Poly, I + >>> from sympy.abc import x + + >>> Poly(x**2 + I*x + 1, x, extension=I).lift() + Poly(x**4 + 3*x**2 + 1, x, domain='QQ') + + """ + if hasattr(f.rep, 'lift'): + result = f.rep.lift() + else: # pragma: no cover + raise OperationNotSupported(f, 'lift') + + return f.per(result) + + def deflate(f): + """ + Reduce degree of ``f`` by mapping ``x_i**m`` to ``y_i``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + >>> Poly(x**6*y**2 + x**3 + 1, x, y).deflate() + ((3, 2), Poly(x**2*y + x + 1, x, y, domain='ZZ')) + + """ + if hasattr(f.rep, 'deflate'): + J, result = f.rep.deflate() + else: # pragma: no cover + raise OperationNotSupported(f, 'deflate') + + return J, f.per(result) + + def inject(f, front=False): + """ + Inject ground domain generators into ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + >>> f = Poly(x**2*y + x*y**3 + x*y + 1, x) + + >>> f.inject() + Poly(x**2*y + x*y**3 + x*y + 1, x, y, domain='ZZ') + >>> f.inject(front=True) + Poly(y**3*x + y*x**2 + y*x + 1, y, x, domain='ZZ') + + """ + dom = f.rep.dom + + if dom.is_Numerical: + return f + elif not dom.is_Poly: + raise DomainError("Cannot inject generators over %s" % dom) + + if hasattr(f.rep, 'inject'): + result = f.rep.inject(front=front) + else: # pragma: no cover + raise OperationNotSupported(f, 'inject') + + if front: + gens = dom.symbols + f.gens + else: + gens = f.gens + dom.symbols + + return f.new(result, *gens) + + def eject(f, *gens): + """ + Eject selected generators into the ground domain. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + >>> f = Poly(x**2*y + x*y**3 + x*y + 1, x, y) + + >>> f.eject(x) + Poly(x*y**3 + (x**2 + x)*y + 1, y, domain='ZZ[x]') + >>> f.eject(y) + Poly(y*x**2 + (y**3 + y)*x + 1, x, domain='ZZ[y]') + + """ + dom = f.rep.dom + + if not dom.is_Numerical: + raise DomainError("Cannot eject generators over %s" % dom) + + k = len(gens) + + if f.gens[:k] == gens: + _gens, front = f.gens[k:], True + elif f.gens[-k:] == gens: + _gens, front = f.gens[:-k], False + else: + raise NotImplementedError( + "can only eject front or back generators") + + dom = dom.inject(*gens) + + if hasattr(f.rep, 'eject'): + result = f.rep.eject(dom, front=front) + else: # pragma: no cover + raise OperationNotSupported(f, 'eject') + + return f.new(result, *_gens) + + def terms_gcd(f): + """ + Remove GCD of terms from the polynomial ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + >>> Poly(x**6*y**2 + x**3*y, x, y).terms_gcd() + ((3, 1), Poly(x**3*y + 1, x, y, domain='ZZ')) + + """ + if hasattr(f.rep, 'terms_gcd'): + J, result = f.rep.terms_gcd() + else: # pragma: no cover + raise OperationNotSupported(f, 'terms_gcd') + + return J, f.per(result) + + def add_ground(f, coeff): + """ + Add an element of the ground domain to ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x + 1).add_ground(2) + Poly(x + 3, x, domain='ZZ') + + """ + if hasattr(f.rep, 'add_ground'): + result = f.rep.add_ground(coeff) + else: # pragma: no cover + raise OperationNotSupported(f, 'add_ground') + + return f.per(result) + + def sub_ground(f, coeff): + """ + Subtract an element of the ground domain from ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x + 1).sub_ground(2) + Poly(x - 1, x, domain='ZZ') + + """ + if hasattr(f.rep, 'sub_ground'): + result = f.rep.sub_ground(coeff) + else: # pragma: no cover + raise OperationNotSupported(f, 'sub_ground') + + return f.per(result) + + def mul_ground(f, coeff): + """ + Multiply ``f`` by a an element of the ground domain. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x + 1).mul_ground(2) + Poly(2*x + 2, x, domain='ZZ') + + """ + if hasattr(f.rep, 'mul_ground'): + result = f.rep.mul_ground(coeff) + else: # pragma: no cover + raise OperationNotSupported(f, 'mul_ground') + + return f.per(result) + + def quo_ground(f, coeff): + """ + Quotient of ``f`` by a an element of the ground domain. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(2*x + 4).quo_ground(2) + Poly(x + 2, x, domain='ZZ') + + >>> Poly(2*x + 3).quo_ground(2) + Poly(x + 1, x, domain='ZZ') + + """ + if hasattr(f.rep, 'quo_ground'): + result = f.rep.quo_ground(coeff) + else: # pragma: no cover + raise OperationNotSupported(f, 'quo_ground') + + return f.per(result) + + def exquo_ground(f, coeff): + """ + Exact quotient of ``f`` by a an element of the ground domain. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(2*x + 4).exquo_ground(2) + Poly(x + 2, x, domain='ZZ') + + >>> Poly(2*x + 3).exquo_ground(2) + Traceback (most recent call last): + ... + ExactQuotientFailed: 2 does not divide 3 in ZZ + + """ + if hasattr(f.rep, 'exquo_ground'): + result = f.rep.exquo_ground(coeff) + else: # pragma: no cover + raise OperationNotSupported(f, 'exquo_ground') + + return f.per(result) + + def abs(f): + """ + Make all coefficients in ``f`` positive. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**2 - 1, x).abs() + Poly(x**2 + 1, x, domain='ZZ') + + """ + if hasattr(f.rep, 'abs'): + result = f.rep.abs() + else: # pragma: no cover + raise OperationNotSupported(f, 'abs') + + return f.per(result) + + def neg(f): + """ + Negate all coefficients in ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**2 - 1, x).neg() + Poly(-x**2 + 1, x, domain='ZZ') + + >>> -Poly(x**2 - 1, x) + Poly(-x**2 + 1, x, domain='ZZ') + + """ + if hasattr(f.rep, 'neg'): + result = f.rep.neg() + else: # pragma: no cover + raise OperationNotSupported(f, 'neg') + + return f.per(result) + + def add(f, g): + """ + Add two polynomials ``f`` and ``g``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**2 + 1, x).add(Poly(x - 2, x)) + Poly(x**2 + x - 1, x, domain='ZZ') + + >>> Poly(x**2 + 1, x) + Poly(x - 2, x) + Poly(x**2 + x - 1, x, domain='ZZ') + + """ + g = sympify(g) + + if not g.is_Poly: + return f.add_ground(g) + + _, per, F, G = f._unify(g) + + if hasattr(f.rep, 'add'): + result = F.add(G) + else: # pragma: no cover + raise OperationNotSupported(f, 'add') + + return per(result) + + def sub(f, g): + """ + Subtract two polynomials ``f`` and ``g``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**2 + 1, x).sub(Poly(x - 2, x)) + Poly(x**2 - x + 3, x, domain='ZZ') + + >>> Poly(x**2 + 1, x) - Poly(x - 2, x) + Poly(x**2 - x + 3, x, domain='ZZ') + + """ + g = sympify(g) + + if not g.is_Poly: + return f.sub_ground(g) + + _, per, F, G = f._unify(g) + + if hasattr(f.rep, 'sub'): + result = F.sub(G) + else: # pragma: no cover + raise OperationNotSupported(f, 'sub') + + return per(result) + + def mul(f, g): + """ + Multiply two polynomials ``f`` and ``g``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**2 + 1, x).mul(Poly(x - 2, x)) + Poly(x**3 - 2*x**2 + x - 2, x, domain='ZZ') + + >>> Poly(x**2 + 1, x)*Poly(x - 2, x) + Poly(x**3 - 2*x**2 + x - 2, x, domain='ZZ') + + """ + g = sympify(g) + + if not g.is_Poly: + return f.mul_ground(g) + + _, per, F, G = f._unify(g) + + if hasattr(f.rep, 'mul'): + result = F.mul(G) + else: # pragma: no cover + raise OperationNotSupported(f, 'mul') + + return per(result) + + def sqr(f): + """ + Square a polynomial ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x - 2, x).sqr() + Poly(x**2 - 4*x + 4, x, domain='ZZ') + + >>> Poly(x - 2, x)**2 + Poly(x**2 - 4*x + 4, x, domain='ZZ') + + """ + if hasattr(f.rep, 'sqr'): + result = f.rep.sqr() + else: # pragma: no cover + raise OperationNotSupported(f, 'sqr') + + return f.per(result) + + def pow(f, n): + """ + Raise ``f`` to a non-negative power ``n``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x - 2, x).pow(3) + Poly(x**3 - 6*x**2 + 12*x - 8, x, domain='ZZ') + + >>> Poly(x - 2, x)**3 + Poly(x**3 - 6*x**2 + 12*x - 8, x, domain='ZZ') + + """ + n = int(n) + + if hasattr(f.rep, 'pow'): + result = f.rep.pow(n) + else: # pragma: no cover + raise OperationNotSupported(f, 'pow') + + return f.per(result) + + def pdiv(f, g): + """ + Polynomial pseudo-division of ``f`` by ``g``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**2 + 1, x).pdiv(Poly(2*x - 4, x)) + (Poly(2*x + 4, x, domain='ZZ'), Poly(20, x, domain='ZZ')) + + """ + _, per, F, G = f._unify(g) + + if hasattr(f.rep, 'pdiv'): + q, r = F.pdiv(G) + else: # pragma: no cover + raise OperationNotSupported(f, 'pdiv') + + return per(q), per(r) + + def prem(f, g): + """ + Polynomial pseudo-remainder of ``f`` by ``g``. + + Caveat: The function prem(f, g, x) can be safely used to compute + in Z[x] _only_ subresultant polynomial remainder sequences (prs's). + + To safely compute Euclidean and Sturmian prs's in Z[x] + employ anyone of the corresponding functions found in + the module sympy.polys.subresultants_qq_zz. The functions + in the module with suffix _pg compute prs's in Z[x] employing + rem(f, g, x), whereas the functions with suffix _amv + compute prs's in Z[x] employing rem_z(f, g, x). + + The function rem_z(f, g, x) differs from prem(f, g, x) in that + to compute the remainder polynomials in Z[x] it premultiplies + the divident times the absolute value of the leading coefficient + of the divisor raised to the power degree(f, x) - degree(g, x) + 1. + + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**2 + 1, x).prem(Poly(2*x - 4, x)) + Poly(20, x, domain='ZZ') + + """ + _, per, F, G = f._unify(g) + + if hasattr(f.rep, 'prem'): + result = F.prem(G) + else: # pragma: no cover + raise OperationNotSupported(f, 'prem') + + return per(result) + + def pquo(f, g): + """ + Polynomial pseudo-quotient of ``f`` by ``g``. + + See the Caveat note in the function prem(f, g). + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**2 + 1, x).pquo(Poly(2*x - 4, x)) + Poly(2*x + 4, x, domain='ZZ') + + >>> Poly(x**2 - 1, x).pquo(Poly(2*x - 2, x)) + Poly(2*x + 2, x, domain='ZZ') + + """ + _, per, F, G = f._unify(g) + + if hasattr(f.rep, 'pquo'): + result = F.pquo(G) + else: # pragma: no cover + raise OperationNotSupported(f, 'pquo') + + return per(result) + + def pexquo(f, g): + """ + Polynomial exact pseudo-quotient of ``f`` by ``g``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**2 - 1, x).pexquo(Poly(2*x - 2, x)) + Poly(2*x + 2, x, domain='ZZ') + + >>> Poly(x**2 + 1, x).pexquo(Poly(2*x - 4, x)) + Traceback (most recent call last): + ... + ExactQuotientFailed: 2*x - 4 does not divide x**2 + 1 + + """ + _, per, F, G = f._unify(g) + + if hasattr(f.rep, 'pexquo'): + try: + result = F.pexquo(G) + except ExactQuotientFailed as exc: + raise exc.new(f.as_expr(), g.as_expr()) + else: # pragma: no cover + raise OperationNotSupported(f, 'pexquo') + + return per(result) + + def div(f, g, auto=True): + """ + Polynomial division with remainder of ``f`` by ``g``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**2 + 1, x).div(Poly(2*x - 4, x)) + (Poly(1/2*x + 1, x, domain='QQ'), Poly(5, x, domain='QQ')) + + >>> Poly(x**2 + 1, x).div(Poly(2*x - 4, x), auto=False) + (Poly(0, x, domain='ZZ'), Poly(x**2 + 1, x, domain='ZZ')) + + """ + dom, per, F, G = f._unify(g) + retract = False + + if auto and dom.is_Ring and not dom.is_Field: + F, G = F.to_field(), G.to_field() + retract = True + + if hasattr(f.rep, 'div'): + q, r = F.div(G) + else: # pragma: no cover + raise OperationNotSupported(f, 'div') + + if retract: + try: + Q, R = q.to_ring(), r.to_ring() + except CoercionFailed: + pass + else: + q, r = Q, R + + return per(q), per(r) + + def rem(f, g, auto=True): + """ + Computes the polynomial remainder of ``f`` by ``g``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**2 + 1, x).rem(Poly(2*x - 4, x)) + Poly(5, x, domain='ZZ') + + >>> Poly(x**2 + 1, x).rem(Poly(2*x - 4, x), auto=False) + Poly(x**2 + 1, x, domain='ZZ') + + """ + dom, per, F, G = f._unify(g) + retract = False + + if auto and dom.is_Ring and not dom.is_Field: + F, G = F.to_field(), G.to_field() + retract = True + + if hasattr(f.rep, 'rem'): + r = F.rem(G) + else: # pragma: no cover + raise OperationNotSupported(f, 'rem') + + if retract: + try: + r = r.to_ring() + except CoercionFailed: + pass + + return per(r) + + def quo(f, g, auto=True): + """ + Computes polynomial quotient of ``f`` by ``g``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**2 + 1, x).quo(Poly(2*x - 4, x)) + Poly(1/2*x + 1, x, domain='QQ') + + >>> Poly(x**2 - 1, x).quo(Poly(x - 1, x)) + Poly(x + 1, x, domain='ZZ') + + """ + dom, per, F, G = f._unify(g) + retract = False + + if auto and dom.is_Ring and not dom.is_Field: + F, G = F.to_field(), G.to_field() + retract = True + + if hasattr(f.rep, 'quo'): + q = F.quo(G) + else: # pragma: no cover + raise OperationNotSupported(f, 'quo') + + if retract: + try: + q = q.to_ring() + except CoercionFailed: + pass + + return per(q) + + def exquo(f, g, auto=True): + """ + Computes polynomial exact quotient of ``f`` by ``g``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**2 - 1, x).exquo(Poly(x - 1, x)) + Poly(x + 1, x, domain='ZZ') + + >>> Poly(x**2 + 1, x).exquo(Poly(2*x - 4, x)) + Traceback (most recent call last): + ... + ExactQuotientFailed: 2*x - 4 does not divide x**2 + 1 + + """ + dom, per, F, G = f._unify(g) + retract = False + + if auto and dom.is_Ring and not dom.is_Field: + F, G = F.to_field(), G.to_field() + retract = True + + if hasattr(f.rep, 'exquo'): + try: + q = F.exquo(G) + except ExactQuotientFailed as exc: + raise exc.new(f.as_expr(), g.as_expr()) + else: # pragma: no cover + raise OperationNotSupported(f, 'exquo') + + if retract: + try: + q = q.to_ring() + except CoercionFailed: + pass + + return per(q) + + def _gen_to_level(f, gen): + """Returns level associated with the given generator. """ + if isinstance(gen, int): + length = len(f.gens) + + if -length <= gen < length: + if gen < 0: + return length + gen + else: + return gen + else: + raise PolynomialError("-%s <= gen < %s expected, got %s" % + (length, length, gen)) + else: + try: + return f.gens.index(sympify(gen)) + except ValueError: + raise PolynomialError( + "a valid generator expected, got %s" % gen) + + def degree(f, gen=0): + """ + Returns degree of ``f`` in ``x_j``. + + The degree of 0 is negative infinity. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + >>> Poly(x**2 + y*x + 1, x, y).degree() + 2 + >>> Poly(x**2 + y*x + y, x, y).degree(y) + 1 + >>> Poly(0, x).degree() + -oo + + """ + j = f._gen_to_level(gen) + + if hasattr(f.rep, 'degree'): + d = f.rep.degree(j) + if d < 0: + d = S.NegativeInfinity + return d + else: # pragma: no cover + raise OperationNotSupported(f, 'degree') + + def degree_list(f): + """ + Returns a list of degrees of ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + >>> Poly(x**2 + y*x + 1, x, y).degree_list() + (2, 1) + + """ + if hasattr(f.rep, 'degree_list'): + return f.rep.degree_list() + else: # pragma: no cover + raise OperationNotSupported(f, 'degree_list') + + def total_degree(f): + """ + Returns the total degree of ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + >>> Poly(x**2 + y*x + 1, x, y).total_degree() + 2 + >>> Poly(x + y**5, x, y).total_degree() + 5 + + """ + if hasattr(f.rep, 'total_degree'): + return f.rep.total_degree() + else: # pragma: no cover + raise OperationNotSupported(f, 'total_degree') + + def homogenize(f, s): + """ + Returns the homogeneous polynomial of ``f``. + + A homogeneous polynomial is a polynomial whose all monomials with + non-zero coefficients have the same total degree. If you only + want to check if a polynomial is homogeneous, then use + :func:`Poly.is_homogeneous`. If you want not only to check if a + polynomial is homogeneous but also compute its homogeneous order, + then use :func:`Poly.homogeneous_order`. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y, z + + >>> f = Poly(x**5 + 2*x**2*y**2 + 9*x*y**3) + >>> f.homogenize(z) + Poly(x**5 + 2*x**2*y**2*z + 9*x*y**3*z, x, y, z, domain='ZZ') + + """ + if not isinstance(s, Symbol): + raise TypeError("``Symbol`` expected, got %s" % type(s)) + if s in f.gens: + i = f.gens.index(s) + gens = f.gens + else: + i = len(f.gens) + gens = f.gens + (s,) + if hasattr(f.rep, 'homogenize'): + return f.per(f.rep.homogenize(i), gens=gens) + raise OperationNotSupported(f, 'homogeneous_order') + + def homogeneous_order(f): + """ + Returns the homogeneous order of ``f``. + + A homogeneous polynomial is a polynomial whose all monomials with + non-zero coefficients have the same total degree. This degree is + the homogeneous order of ``f``. If you only want to check if a + polynomial is homogeneous, then use :func:`Poly.is_homogeneous`. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + >>> f = Poly(x**5 + 2*x**3*y**2 + 9*x*y**4) + >>> f.homogeneous_order() + 5 + + """ + if hasattr(f.rep, 'homogeneous_order'): + return f.rep.homogeneous_order() + else: # pragma: no cover + raise OperationNotSupported(f, 'homogeneous_order') + + def LC(f, order=None): + """ + Returns the leading coefficient of ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(4*x**3 + 2*x**2 + 3*x, x).LC() + 4 + + """ + if order is not None: + return f.coeffs(order)[0] + + if hasattr(f.rep, 'LC'): + result = f.rep.LC() + else: # pragma: no cover + raise OperationNotSupported(f, 'LC') + + return f.rep.dom.to_sympy(result) + + def TC(f): + """ + Returns the trailing coefficient of ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**3 + 2*x**2 + 3*x, x).TC() + 0 + + """ + if hasattr(f.rep, 'TC'): + result = f.rep.TC() + else: # pragma: no cover + raise OperationNotSupported(f, 'TC') + + return f.rep.dom.to_sympy(result) + + def EC(f, order=None): + """ + Returns the last non-zero coefficient of ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**3 + 2*x**2 + 3*x, x).EC() + 3 + + """ + if hasattr(f.rep, 'coeffs'): + return f.coeffs(order)[-1] + else: # pragma: no cover + raise OperationNotSupported(f, 'EC') + + def coeff_monomial(f, monom): + """ + Returns the coefficient of ``monom`` in ``f`` if there, else None. + + Examples + ======== + + >>> from sympy import Poly, exp + >>> from sympy.abc import x, y + + >>> p = Poly(24*x*y*exp(8) + 23*x, x, y) + + >>> p.coeff_monomial(x) + 23 + >>> p.coeff_monomial(y) + 0 + >>> p.coeff_monomial(x*y) + 24*exp(8) + + Note that ``Expr.coeff()`` behaves differently, collecting terms + if possible; the Poly must be converted to an Expr to use that + method, however: + + >>> p.as_expr().coeff(x) + 24*y*exp(8) + 23 + >>> p.as_expr().coeff(y) + 24*x*exp(8) + >>> p.as_expr().coeff(x*y) + 24*exp(8) + + See Also + ======== + nth: more efficient query using exponents of the monomial's generators + + """ + return f.nth(*Monomial(monom, f.gens).exponents) + + def nth(f, *N): + """ + Returns the ``n``-th coefficient of ``f`` where ``N`` are the + exponents of the generators in the term of interest. + + Examples + ======== + + >>> from sympy import Poly, sqrt + >>> from sympy.abc import x, y + + >>> Poly(x**3 + 2*x**2 + 3*x, x).nth(2) + 2 + >>> Poly(x**3 + 2*x*y**2 + y**2, x, y).nth(1, 2) + 2 + >>> Poly(4*sqrt(x)*y) + Poly(4*y*(sqrt(x)), y, sqrt(x), domain='ZZ') + >>> _.nth(1, 1) + 4 + + See Also + ======== + coeff_monomial + + """ + if hasattr(f.rep, 'nth'): + if len(N) != len(f.gens): + raise ValueError('exponent of each generator must be specified') + result = f.rep.nth(*list(map(int, N))) + else: # pragma: no cover + raise OperationNotSupported(f, 'nth') + + return f.rep.dom.to_sympy(result) + + def coeff(f, x, n=1, right=False): + # the semantics of coeff_monomial and Expr.coeff are different; + # if someone is working with a Poly, they should be aware of the + # differences and chose the method best suited for the query. + # Alternatively, a pure-polys method could be written here but + # at this time the ``right`` keyword would be ignored because Poly + # doesn't work with non-commutatives. + raise NotImplementedError( + 'Either convert to Expr with `as_expr` method ' + 'to use Expr\'s coeff method or else use the ' + '`coeff_monomial` method of Polys.') + + def LM(f, order=None): + """ + Returns the leading monomial of ``f``. + + The Leading monomial signifies the monomial having + the highest power of the principal generator in the + expression f. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + >>> Poly(4*x**2 + 2*x*y**2 + x*y + 3*y, x, y).LM() + x**2*y**0 + + """ + return Monomial(f.monoms(order)[0], f.gens) + + def EM(f, order=None): + """ + Returns the last non-zero monomial of ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + >>> Poly(4*x**2 + 2*x*y**2 + x*y + 3*y, x, y).EM() + x**0*y**1 + + """ + return Monomial(f.monoms(order)[-1], f.gens) + + def LT(f, order=None): + """ + Returns the leading term of ``f``. + + The Leading term signifies the term having + the highest power of the principal generator in the + expression f along with its coefficient. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + >>> Poly(4*x**2 + 2*x*y**2 + x*y + 3*y, x, y).LT() + (x**2*y**0, 4) + + """ + monom, coeff = f.terms(order)[0] + return Monomial(monom, f.gens), coeff + + def ET(f, order=None): + """ + Returns the last non-zero term of ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + >>> Poly(4*x**2 + 2*x*y**2 + x*y + 3*y, x, y).ET() + (x**0*y**1, 3) + + """ + monom, coeff = f.terms(order)[-1] + return Monomial(monom, f.gens), coeff + + def max_norm(f): + """ + Returns maximum norm of ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(-x**2 + 2*x - 3, x).max_norm() + 3 + + """ + if hasattr(f.rep, 'max_norm'): + result = f.rep.max_norm() + else: # pragma: no cover + raise OperationNotSupported(f, 'max_norm') + + return f.rep.dom.to_sympy(result) + + def l1_norm(f): + """ + Returns l1 norm of ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(-x**2 + 2*x - 3, x).l1_norm() + 6 + + """ + if hasattr(f.rep, 'l1_norm'): + result = f.rep.l1_norm() + else: # pragma: no cover + raise OperationNotSupported(f, 'l1_norm') + + return f.rep.dom.to_sympy(result) + + def clear_denoms(self, convert=False): + """ + Clear denominators, but keep the ground domain. + + Examples + ======== + + >>> from sympy import Poly, S, QQ + >>> from sympy.abc import x + + >>> f = Poly(x/2 + S(1)/3, x, domain=QQ) + + >>> f.clear_denoms() + (6, Poly(3*x + 2, x, domain='QQ')) + >>> f.clear_denoms(convert=True) + (6, Poly(3*x + 2, x, domain='ZZ')) + + """ + f = self + + if not f.rep.dom.is_Field: + return S.One, f + + dom = f.get_domain() + if dom.has_assoc_Ring: + dom = f.rep.dom.get_ring() + + if hasattr(f.rep, 'clear_denoms'): + coeff, result = f.rep.clear_denoms() + else: # pragma: no cover + raise OperationNotSupported(f, 'clear_denoms') + + coeff, f = dom.to_sympy(coeff), f.per(result) + + if not convert or not dom.has_assoc_Ring: + return coeff, f + else: + return coeff, f.to_ring() + + def rat_clear_denoms(self, g): + """ + Clear denominators in a rational function ``f/g``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + >>> f = Poly(x**2/y + 1, x) + >>> g = Poly(x**3 + y, x) + + >>> p, q = f.rat_clear_denoms(g) + + >>> p + Poly(x**2 + y, x, domain='ZZ[y]') + >>> q + Poly(y*x**3 + y**2, x, domain='ZZ[y]') + + """ + f = self + + dom, per, f, g = f._unify(g) + + f = per(f) + g = per(g) + + if not (dom.is_Field and dom.has_assoc_Ring): + return f, g + + a, f = f.clear_denoms(convert=True) + b, g = g.clear_denoms(convert=True) + + f = f.mul_ground(b) + g = g.mul_ground(a) + + return f, g + + def integrate(self, *specs, **args): + """ + Computes indefinite integral of ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + >>> Poly(x**2 + 2*x + 1, x).integrate() + Poly(1/3*x**3 + x**2 + x, x, domain='QQ') + + >>> Poly(x*y**2 + x, x, y).integrate((0, 1), (1, 0)) + Poly(1/2*x**2*y**2 + 1/2*x**2, x, y, domain='QQ') + + """ + f = self + + if args.get('auto', True) and f.rep.dom.is_Ring: + f = f.to_field() + + if hasattr(f.rep, 'integrate'): + if not specs: + return f.per(f.rep.integrate(m=1)) + + rep = f.rep + + for spec in specs: + if isinstance(spec, tuple): + gen, m = spec + else: + gen, m = spec, 1 + + rep = rep.integrate(int(m), f._gen_to_level(gen)) + + return f.per(rep) + else: # pragma: no cover + raise OperationNotSupported(f, 'integrate') + + def diff(f, *specs, **kwargs): + """ + Computes partial derivative of ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + >>> Poly(x**2 + 2*x + 1, x).diff() + Poly(2*x + 2, x, domain='ZZ') + + >>> Poly(x*y**2 + x, x, y).diff((0, 0), (1, 1)) + Poly(2*x*y, x, y, domain='ZZ') + + """ + if not kwargs.get('evaluate', True): + return Derivative(f, *specs, **kwargs) + + if hasattr(f.rep, 'diff'): + if not specs: + return f.per(f.rep.diff(m=1)) + + rep = f.rep + + for spec in specs: + if isinstance(spec, tuple): + gen, m = spec + else: + gen, m = spec, 1 + + rep = rep.diff(int(m), f._gen_to_level(gen)) + + return f.per(rep) + else: # pragma: no cover + raise OperationNotSupported(f, 'diff') + + _eval_derivative = diff + + def eval(self, x, a=None, auto=True): + """ + Evaluate ``f`` at ``a`` in the given variable. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y, z + + >>> Poly(x**2 + 2*x + 3, x).eval(2) + 11 + + >>> Poly(2*x*y + 3*x + y + 2, x, y).eval(x, 2) + Poly(5*y + 8, y, domain='ZZ') + + >>> f = Poly(2*x*y + 3*x + y + 2*z, x, y, z) + + >>> f.eval({x: 2}) + Poly(5*y + 2*z + 6, y, z, domain='ZZ') + >>> f.eval({x: 2, y: 5}) + Poly(2*z + 31, z, domain='ZZ') + >>> f.eval({x: 2, y: 5, z: 7}) + 45 + + >>> f.eval((2, 5)) + Poly(2*z + 31, z, domain='ZZ') + >>> f(2, 5) + Poly(2*z + 31, z, domain='ZZ') + + """ + f = self + + if a is None: + if isinstance(x, dict): + mapping = x + + for gen, value in mapping.items(): + f = f.eval(gen, value) + + return f + elif isinstance(x, (tuple, list)): + values = x + + if len(values) > len(f.gens): + raise ValueError("too many values provided") + + for gen, value in zip(f.gens, values): + f = f.eval(gen, value) + + return f + else: + j, a = 0, x + else: + j = f._gen_to_level(x) + + if not hasattr(f.rep, 'eval'): # pragma: no cover + raise OperationNotSupported(f, 'eval') + + try: + result = f.rep.eval(a, j) + except CoercionFailed: + if not auto: + raise DomainError("Cannot evaluate at %s in %s" % (a, f.rep.dom)) + else: + a_domain, [a] = construct_domain([a]) + new_domain = f.get_domain().unify_with_symbols(a_domain, f.gens) + + f = f.set_domain(new_domain) + a = new_domain.convert(a, a_domain) + + result = f.rep.eval(a, j) + + return f.per(result, remove=j) + + def __call__(f, *values): + """ + Evaluate ``f`` at the give values. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y, z + + >>> f = Poly(2*x*y + 3*x + y + 2*z, x, y, z) + + >>> f(2) + Poly(5*y + 2*z + 6, y, z, domain='ZZ') + >>> f(2, 5) + Poly(2*z + 31, z, domain='ZZ') + >>> f(2, 5, 7) + 45 + + """ + return f.eval(values) + + def half_gcdex(f, g, auto=True): + """ + Half extended Euclidean algorithm of ``f`` and ``g``. + + Returns ``(s, h)`` such that ``h = gcd(f, g)`` and ``s*f = h (mod g)``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15 + >>> g = x**3 + x**2 - 4*x - 4 + + >>> Poly(f).half_gcdex(Poly(g)) + (Poly(-1/5*x + 3/5, x, domain='QQ'), Poly(x + 1, x, domain='QQ')) + + """ + dom, per, F, G = f._unify(g) + + if auto and dom.is_Ring: + F, G = F.to_field(), G.to_field() + + if hasattr(f.rep, 'half_gcdex'): + s, h = F.half_gcdex(G) + else: # pragma: no cover + raise OperationNotSupported(f, 'half_gcdex') + + return per(s), per(h) + + def gcdex(f, g, auto=True): + """ + Extended Euclidean algorithm of ``f`` and ``g``. + + Returns ``(s, t, h)`` such that ``h = gcd(f, g)`` and ``s*f + t*g = h``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15 + >>> g = x**3 + x**2 - 4*x - 4 + + >>> Poly(f).gcdex(Poly(g)) + (Poly(-1/5*x + 3/5, x, domain='QQ'), + Poly(1/5*x**2 - 6/5*x + 2, x, domain='QQ'), + Poly(x + 1, x, domain='QQ')) + + """ + dom, per, F, G = f._unify(g) + + if auto and dom.is_Ring: + F, G = F.to_field(), G.to_field() + + if hasattr(f.rep, 'gcdex'): + s, t, h = F.gcdex(G) + else: # pragma: no cover + raise OperationNotSupported(f, 'gcdex') + + return per(s), per(t), per(h) + + def invert(f, g, auto=True): + """ + Invert ``f`` modulo ``g`` when possible. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**2 - 1, x).invert(Poly(2*x - 1, x)) + Poly(-4/3, x, domain='QQ') + + >>> Poly(x**2 - 1, x).invert(Poly(x - 1, x)) + Traceback (most recent call last): + ... + NotInvertible: zero divisor + + """ + dom, per, F, G = f._unify(g) + + if auto and dom.is_Ring: + F, G = F.to_field(), G.to_field() + + if hasattr(f.rep, 'invert'): + result = F.invert(G) + else: # pragma: no cover + raise OperationNotSupported(f, 'invert') + + return per(result) + + def revert(f, n): + """ + Compute ``f**(-1)`` mod ``x**n``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(1, x).revert(2) + Poly(1, x, domain='ZZ') + + >>> Poly(1 + x, x).revert(1) + Poly(1, x, domain='ZZ') + + >>> Poly(x**2 - 2, x).revert(2) + Traceback (most recent call last): + ... + NotReversible: only units are reversible in a ring + + >>> Poly(1/x, x).revert(1) + Traceback (most recent call last): + ... + PolynomialError: 1/x contains an element of the generators set + + """ + if hasattr(f.rep, 'revert'): + result = f.rep.revert(int(n)) + else: # pragma: no cover + raise OperationNotSupported(f, 'revert') + + return f.per(result) + + def subresultants(f, g): + """ + Computes the subresultant PRS of ``f`` and ``g``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**2 + 1, x).subresultants(Poly(x**2 - 1, x)) + [Poly(x**2 + 1, x, domain='ZZ'), + Poly(x**2 - 1, x, domain='ZZ'), + Poly(-2, x, domain='ZZ')] + + """ + _, per, F, G = f._unify(g) + + if hasattr(f.rep, 'subresultants'): + result = F.subresultants(G) + else: # pragma: no cover + raise OperationNotSupported(f, 'subresultants') + + return list(map(per, result)) + + def resultant(f, g, includePRS=False): + """ + Computes the resultant of ``f`` and ``g`` via PRS. + + If includePRS=True, it includes the subresultant PRS in the result. + Because the PRS is used to calculate the resultant, this is more + efficient than calling :func:`subresultants` separately. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> f = Poly(x**2 + 1, x) + + >>> f.resultant(Poly(x**2 - 1, x)) + 4 + >>> f.resultant(Poly(x**2 - 1, x), includePRS=True) + (4, [Poly(x**2 + 1, x, domain='ZZ'), Poly(x**2 - 1, x, domain='ZZ'), + Poly(-2, x, domain='ZZ')]) + + """ + _, per, F, G = f._unify(g) + + if hasattr(f.rep, 'resultant'): + if includePRS: + result, R = F.resultant(G, includePRS=includePRS) + else: + result = F.resultant(G) + else: # pragma: no cover + raise OperationNotSupported(f, 'resultant') + + if includePRS: + return (per(result, remove=0), list(map(per, R))) + return per(result, remove=0) + + def discriminant(f): + """ + Computes the discriminant of ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**2 + 2*x + 3, x).discriminant() + -8 + + """ + if hasattr(f.rep, 'discriminant'): + result = f.rep.discriminant() + else: # pragma: no cover + raise OperationNotSupported(f, 'discriminant') + + return f.per(result, remove=0) + + def dispersionset(f, g=None): + r"""Compute the *dispersion set* of two polynomials. + + For two polynomials `f(x)` and `g(x)` with `\deg f > 0` + and `\deg g > 0` the dispersion set `\operatorname{J}(f, g)` is defined as: + + .. math:: + \operatorname{J}(f, g) + & := \{a \in \mathbb{N}_0 | \gcd(f(x), g(x+a)) \neq 1\} \\ + & = \{a \in \mathbb{N}_0 | \deg \gcd(f(x), g(x+a)) \geq 1\} + + For a single polynomial one defines `\operatorname{J}(f) := \operatorname{J}(f, f)`. + + Examples + ======== + + >>> from sympy import poly + >>> from sympy.polys.dispersion import dispersion, dispersionset + >>> from sympy.abc import x + + Dispersion set and dispersion of a simple polynomial: + + >>> fp = poly((x - 3)*(x + 3), x) + >>> sorted(dispersionset(fp)) + [0, 6] + >>> dispersion(fp) + 6 + + Note that the definition of the dispersion is not symmetric: + + >>> fp = poly(x**4 - 3*x**2 + 1, x) + >>> gp = fp.shift(-3) + >>> sorted(dispersionset(fp, gp)) + [2, 3, 4] + >>> dispersion(fp, gp) + 4 + >>> sorted(dispersionset(gp, fp)) + [] + >>> dispersion(gp, fp) + -oo + + Computing the dispersion also works over field extensions: + + >>> from sympy import sqrt + >>> fp = poly(x**2 + sqrt(5)*x - 1, x, domain='QQ') + >>> gp = poly(x**2 + (2 + sqrt(5))*x + sqrt(5), x, domain='QQ') + >>> sorted(dispersionset(fp, gp)) + [2] + >>> sorted(dispersionset(gp, fp)) + [1, 4] + + We can even perform the computations for polynomials + having symbolic coefficients: + + >>> from sympy.abc import a + >>> fp = poly(4*x**4 + (4*a + 8)*x**3 + (a**2 + 6*a + 4)*x**2 + (a**2 + 2*a)*x, x) + >>> sorted(dispersionset(fp)) + [0, 1] + + See Also + ======== + + dispersion + + References + ========== + + 1. [ManWright94]_ + 2. [Koepf98]_ + 3. [Abramov71]_ + 4. [Man93]_ + """ + from sympy.polys.dispersion import dispersionset + return dispersionset(f, g) + + def dispersion(f, g=None): + r"""Compute the *dispersion* of polynomials. + + For two polynomials `f(x)` and `g(x)` with `\deg f > 0` + and `\deg g > 0` the dispersion `\operatorname{dis}(f, g)` is defined as: + + .. math:: + \operatorname{dis}(f, g) + & := \max\{ J(f,g) \cup \{0\} \} \\ + & = \max\{ \{a \in \mathbb{N} | \gcd(f(x), g(x+a)) \neq 1\} \cup \{0\} \} + + and for a single polynomial `\operatorname{dis}(f) := \operatorname{dis}(f, f)`. + + Examples + ======== + + >>> from sympy import poly + >>> from sympy.polys.dispersion import dispersion, dispersionset + >>> from sympy.abc import x + + Dispersion set and dispersion of a simple polynomial: + + >>> fp = poly((x - 3)*(x + 3), x) + >>> sorted(dispersionset(fp)) + [0, 6] + >>> dispersion(fp) + 6 + + Note that the definition of the dispersion is not symmetric: + + >>> fp = poly(x**4 - 3*x**2 + 1, x) + >>> gp = fp.shift(-3) + >>> sorted(dispersionset(fp, gp)) + [2, 3, 4] + >>> dispersion(fp, gp) + 4 + >>> sorted(dispersionset(gp, fp)) + [] + >>> dispersion(gp, fp) + -oo + + Computing the dispersion also works over field extensions: + + >>> from sympy import sqrt + >>> fp = poly(x**2 + sqrt(5)*x - 1, x, domain='QQ') + >>> gp = poly(x**2 + (2 + sqrt(5))*x + sqrt(5), x, domain='QQ') + >>> sorted(dispersionset(fp, gp)) + [2] + >>> sorted(dispersionset(gp, fp)) + [1, 4] + + We can even perform the computations for polynomials + having symbolic coefficients: + + >>> from sympy.abc import a + >>> fp = poly(4*x**4 + (4*a + 8)*x**3 + (a**2 + 6*a + 4)*x**2 + (a**2 + 2*a)*x, x) + >>> sorted(dispersionset(fp)) + [0, 1] + + See Also + ======== + + dispersionset + + References + ========== + + 1. [ManWright94]_ + 2. [Koepf98]_ + 3. [Abramov71]_ + 4. [Man93]_ + """ + from sympy.polys.dispersion import dispersion + return dispersion(f, g) + + def cofactors(f, g): + """ + Returns the GCD of ``f`` and ``g`` and their cofactors. + + Returns polynomials ``(h, cff, cfg)`` such that ``h = gcd(f, g)``, and + ``cff = quo(f, h)`` and ``cfg = quo(g, h)`` are, so called, cofactors + of ``f`` and ``g``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**2 - 1, x).cofactors(Poly(x**2 - 3*x + 2, x)) + (Poly(x - 1, x, domain='ZZ'), + Poly(x + 1, x, domain='ZZ'), + Poly(x - 2, x, domain='ZZ')) + + """ + _, per, F, G = f._unify(g) + + if hasattr(f.rep, 'cofactors'): + h, cff, cfg = F.cofactors(G) + else: # pragma: no cover + raise OperationNotSupported(f, 'cofactors') + + return per(h), per(cff), per(cfg) + + def gcd(f, g): + """ + Returns the polynomial GCD of ``f`` and ``g``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**2 - 1, x).gcd(Poly(x**2 - 3*x + 2, x)) + Poly(x - 1, x, domain='ZZ') + + """ + _, per, F, G = f._unify(g) + + if hasattr(f.rep, 'gcd'): + result = F.gcd(G) + else: # pragma: no cover + raise OperationNotSupported(f, 'gcd') + + return per(result) + + def lcm(f, g): + """ + Returns polynomial LCM of ``f`` and ``g``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**2 - 1, x).lcm(Poly(x**2 - 3*x + 2, x)) + Poly(x**3 - 2*x**2 - x + 2, x, domain='ZZ') + + """ + _, per, F, G = f._unify(g) + + if hasattr(f.rep, 'lcm'): + result = F.lcm(G) + else: # pragma: no cover + raise OperationNotSupported(f, 'lcm') + + return per(result) + + def trunc(f, p): + """ + Reduce ``f`` modulo a constant ``p``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(2*x**3 + 3*x**2 + 5*x + 7, x).trunc(3) + Poly(-x**3 - x + 1, x, domain='ZZ') + + """ + p = f.rep.dom.convert(p) + + if hasattr(f.rep, 'trunc'): + result = f.rep.trunc(p) + else: # pragma: no cover + raise OperationNotSupported(f, 'trunc') + + return f.per(result) + + def monic(self, auto=True): + """ + Divides all coefficients by ``LC(f)``. + + Examples + ======== + + >>> from sympy import Poly, ZZ + >>> from sympy.abc import x + + >>> Poly(3*x**2 + 6*x + 9, x, domain=ZZ).monic() + Poly(x**2 + 2*x + 3, x, domain='QQ') + + >>> Poly(3*x**2 + 4*x + 2, x, domain=ZZ).monic() + Poly(x**2 + 4/3*x + 2/3, x, domain='QQ') + + """ + f = self + + if auto and f.rep.dom.is_Ring: + f = f.to_field() + + if hasattr(f.rep, 'monic'): + result = f.rep.monic() + else: # pragma: no cover + raise OperationNotSupported(f, 'monic') + + return f.per(result) + + def content(f): + """ + Returns the GCD of polynomial coefficients. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(6*x**2 + 8*x + 12, x).content() + 2 + + """ + if hasattr(f.rep, 'content'): + result = f.rep.content() + else: # pragma: no cover + raise OperationNotSupported(f, 'content') + + return f.rep.dom.to_sympy(result) + + def primitive(f): + """ + Returns the content and a primitive form of ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(2*x**2 + 8*x + 12, x).primitive() + (2, Poly(x**2 + 4*x + 6, x, domain='ZZ')) + + """ + if hasattr(f.rep, 'primitive'): + cont, result = f.rep.primitive() + else: # pragma: no cover + raise OperationNotSupported(f, 'primitive') + + return f.rep.dom.to_sympy(cont), f.per(result) + + def compose(f, g): + """ + Computes the functional composition of ``f`` and ``g``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**2 + x, x).compose(Poly(x - 1, x)) + Poly(x**2 - x, x, domain='ZZ') + + """ + _, per, F, G = f._unify(g) + + if hasattr(f.rep, 'compose'): + result = F.compose(G) + else: # pragma: no cover + raise OperationNotSupported(f, 'compose') + + return per(result) + + def decompose(f): + """ + Computes a functional decomposition of ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**4 + 2*x**3 - x - 1, x, domain='ZZ').decompose() + [Poly(x**2 - x - 1, x, domain='ZZ'), Poly(x**2 + x, x, domain='ZZ')] + + """ + if hasattr(f.rep, 'decompose'): + result = f.rep.decompose() + else: # pragma: no cover + raise OperationNotSupported(f, 'decompose') + + return list(map(f.per, result)) + + def shift(f, a): + """ + Efficiently compute Taylor shift ``f(x + a)``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**2 - 2*x + 1, x).shift(2) + Poly(x**2 + 2*x + 1, x, domain='ZZ') + + See Also + ======== + + shift_list: Analogous method for multivariate polynomials. + """ + return f.per(f.rep.shift(a)) + + def shift_list(f, a): + """ + Efficiently compute Taylor shift ``f(X + A)``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + >>> Poly(x*y, [x,y]).shift_list([1, 2]) == Poly((x+1)*(y+2), [x,y]) + True + + See Also + ======== + + shift: Analogous method for univariate polynomials. + """ + return f.per(f.rep.shift_list(a)) + + def transform(f, p, q): + """ + Efficiently evaluate the functional transformation ``q**n * f(p/q)``. + + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**2 - 2*x + 1, x).transform(Poly(x + 1, x), Poly(x - 1, x)) + Poly(4, x, domain='ZZ') + + """ + P, Q = p.unify(q) + F, P = f.unify(P) + F, Q = F.unify(Q) + + if hasattr(F.rep, 'transform'): + result = F.rep.transform(P.rep, Q.rep) + else: # pragma: no cover + raise OperationNotSupported(F, 'transform') + + return F.per(result) + + def sturm(self, auto=True): + """ + Computes the Sturm sequence of ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**3 - 2*x**2 + x - 3, x).sturm() + [Poly(x**3 - 2*x**2 + x - 3, x, domain='QQ'), + Poly(3*x**2 - 4*x + 1, x, domain='QQ'), + Poly(2/9*x + 25/9, x, domain='QQ'), + Poly(-2079/4, x, domain='QQ')] + + """ + f = self + + if auto and f.rep.dom.is_Ring: + f = f.to_field() + + if hasattr(f.rep, 'sturm'): + result = f.rep.sturm() + else: # pragma: no cover + raise OperationNotSupported(f, 'sturm') + + return list(map(f.per, result)) + + def gff_list(f): + """ + Computes greatest factorial factorization of ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> f = x**5 + 2*x**4 - x**3 - 2*x**2 + + >>> Poly(f).gff_list() + [(Poly(x, x, domain='ZZ'), 1), (Poly(x + 2, x, domain='ZZ'), 4)] + + """ + if hasattr(f.rep, 'gff_list'): + result = f.rep.gff_list() + else: # pragma: no cover + raise OperationNotSupported(f, 'gff_list') + + return [(f.per(g), k) for g, k in result] + + def norm(f): + """ + Computes the product, ``Norm(f)``, of the conjugates of + a polynomial ``f`` defined over a number field ``K``. + + Examples + ======== + + >>> from sympy import Poly, sqrt + >>> from sympy.abc import x + + >>> a, b = sqrt(2), sqrt(3) + + A polynomial over a quadratic extension. + Two conjugates x - a and x + a. + + >>> f = Poly(x - a, x, extension=a) + >>> f.norm() + Poly(x**2 - 2, x, domain='QQ') + + A polynomial over a quartic extension. + Four conjugates x - a, x - a, x + a and x + a. + + >>> f = Poly(x - a, x, extension=(a, b)) + >>> f.norm() + Poly(x**4 - 4*x**2 + 4, x, domain='QQ') + + """ + if hasattr(f.rep, 'norm'): + r = f.rep.norm() + else: # pragma: no cover + raise OperationNotSupported(f, 'norm') + + return f.per(r) + + def sqf_norm(f): + """ + Computes square-free norm of ``f``. + + Returns ``s``, ``f``, ``r``, such that ``g(x) = f(x-sa)`` and + ``r(x) = Norm(g(x))`` is a square-free polynomial over ``K``, + where ``a`` is the algebraic extension of the ground domain. + + Examples + ======== + + >>> from sympy import Poly, sqrt + >>> from sympy.abc import x + + >>> s, f, r = Poly(x**2 + 1, x, extension=[sqrt(3)]).sqf_norm() + + >>> s + [1] + >>> f + Poly(x**2 - 2*sqrt(3)*x + 4, x, domain='QQ') + >>> r + Poly(x**4 - 4*x**2 + 16, x, domain='QQ') + + """ + if hasattr(f.rep, 'sqf_norm'): + s, g, r = f.rep.sqf_norm() + else: # pragma: no cover + raise OperationNotSupported(f, 'sqf_norm') + + return s, f.per(g), f.per(r) + + def sqf_part(f): + """ + Computes square-free part of ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**3 - 3*x - 2, x).sqf_part() + Poly(x**2 - x - 2, x, domain='ZZ') + + """ + if hasattr(f.rep, 'sqf_part'): + result = f.rep.sqf_part() + else: # pragma: no cover + raise OperationNotSupported(f, 'sqf_part') + + return f.per(result) + + def sqf_list(f, all=False): + """ + Returns a list of square-free factors of ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> f = 2*x**5 + 16*x**4 + 50*x**3 + 76*x**2 + 56*x + 16 + + >>> Poly(f).sqf_list() + (2, [(Poly(x + 1, x, domain='ZZ'), 2), + (Poly(x + 2, x, domain='ZZ'), 3)]) + + >>> Poly(f).sqf_list(all=True) + (2, [(Poly(1, x, domain='ZZ'), 1), + (Poly(x + 1, x, domain='ZZ'), 2), + (Poly(x + 2, x, domain='ZZ'), 3)]) + + """ + if hasattr(f.rep, 'sqf_list'): + coeff, factors = f.rep.sqf_list(all) + else: # pragma: no cover + raise OperationNotSupported(f, 'sqf_list') + + return f.rep.dom.to_sympy(coeff), [(f.per(g), k) for g, k in factors] + + def sqf_list_include(f, all=False): + """ + Returns a list of square-free factors of ``f``. + + Examples + ======== + + >>> from sympy import Poly, expand + >>> from sympy.abc import x + + >>> f = expand(2*(x + 1)**3*x**4) + >>> f + 2*x**7 + 6*x**6 + 6*x**5 + 2*x**4 + + >>> Poly(f).sqf_list_include() + [(Poly(2, x, domain='ZZ'), 1), + (Poly(x + 1, x, domain='ZZ'), 3), + (Poly(x, x, domain='ZZ'), 4)] + + >>> Poly(f).sqf_list_include(all=True) + [(Poly(2, x, domain='ZZ'), 1), + (Poly(1, x, domain='ZZ'), 2), + (Poly(x + 1, x, domain='ZZ'), 3), + (Poly(x, x, domain='ZZ'), 4)] + + """ + if hasattr(f.rep, 'sqf_list_include'): + factors = f.rep.sqf_list_include(all) + else: # pragma: no cover + raise OperationNotSupported(f, 'sqf_list_include') + + return [(f.per(g), k) for g, k in factors] + + def factor_list(f) -> tuple[Expr, list[tuple[Poly, int]]]: + """ + Returns a list of irreducible factors of ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + >>> f = 2*x**5 + 2*x**4*y + 4*x**3 + 4*x**2*y + 2*x + 2*y + + >>> Poly(f).factor_list() + (2, [(Poly(x + y, x, y, domain='ZZ'), 1), + (Poly(x**2 + 1, x, y, domain='ZZ'), 2)]) + + """ + if hasattr(f.rep, 'factor_list'): + try: + coeff, factors = f.rep.factor_list() + except DomainError: + if f.degree() == 0: + return f.as_expr(), [] + else: + return S.One, [(f, 1)] + else: # pragma: no cover + raise OperationNotSupported(f, 'factor_list') + + return f.rep.dom.to_sympy(coeff), [(f.per(g), k) for g, k in factors] + + def factor_list_include(f): + """ + Returns a list of irreducible factors of ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + >>> f = 2*x**5 + 2*x**4*y + 4*x**3 + 4*x**2*y + 2*x + 2*y + + >>> Poly(f).factor_list_include() + [(Poly(2*x + 2*y, x, y, domain='ZZ'), 1), + (Poly(x**2 + 1, x, y, domain='ZZ'), 2)] + + """ + if hasattr(f.rep, 'factor_list_include'): + try: + factors = f.rep.factor_list_include() + except DomainError: + return [(f, 1)] + else: # pragma: no cover + raise OperationNotSupported(f, 'factor_list_include') + + return [(f.per(g), k) for g, k in factors] + + def intervals(f, all=False, eps=None, inf=None, sup=None, fast=False, sqf=False): + """ + Compute isolating intervals for roots of ``f``. + + For real roots the Vincent-Akritas-Strzebonski (VAS) continued fractions method is used. + + References + ========== + .. [#] Alkiviadis G. Akritas and Adam W. Strzebonski: A Comparative Study of Two Real Root + Isolation Methods . Nonlinear Analysis: Modelling and Control, Vol. 10, No. 4, 297-304, 2005. + .. [#] Alkiviadis G. Akritas, Adam W. Strzebonski and Panagiotis S. Vigklas: Improving the + Performance of the Continued Fractions Method Using new Bounds of Positive Roots. Nonlinear + Analysis: Modelling and Control, Vol. 13, No. 3, 265-279, 2008. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**2 - 3, x).intervals() + [((-2, -1), 1), ((1, 2), 1)] + >>> Poly(x**2 - 3, x).intervals(eps=1e-2) + [((-26/15, -19/11), 1), ((19/11, 26/15), 1)] + + """ + if eps is not None: + eps = QQ.convert(eps) + + if eps <= 0: + raise ValueError("'eps' must be a positive rational") + + if inf is not None: + inf = QQ.convert(inf) + if sup is not None: + sup = QQ.convert(sup) + + if hasattr(f.rep, 'intervals'): + result = f.rep.intervals( + all=all, eps=eps, inf=inf, sup=sup, fast=fast, sqf=sqf) + else: # pragma: no cover + raise OperationNotSupported(f, 'intervals') + + if sqf: + def _real(interval): + s, t = interval + return (QQ.to_sympy(s), QQ.to_sympy(t)) + + if not all: + return list(map(_real, result)) + + def _complex(rectangle): + (u, v), (s, t) = rectangle + return (QQ.to_sympy(u) + I*QQ.to_sympy(v), + QQ.to_sympy(s) + I*QQ.to_sympy(t)) + + real_part, complex_part = result + + return list(map(_real, real_part)), list(map(_complex, complex_part)) + else: + def _real(interval): + (s, t), k = interval + return ((QQ.to_sympy(s), QQ.to_sympy(t)), k) + + if not all: + return list(map(_real, result)) + + def _complex(rectangle): + ((u, v), (s, t)), k = rectangle + return ((QQ.to_sympy(u) + I*QQ.to_sympy(v), + QQ.to_sympy(s) + I*QQ.to_sympy(t)), k) + + real_part, complex_part = result + + return list(map(_real, real_part)), list(map(_complex, complex_part)) + + def refine_root(f, s, t, eps=None, steps=None, fast=False, check_sqf=False): + """ + Refine an isolating interval of a root to the given precision. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**2 - 3, x).refine_root(1, 2, eps=1e-2) + (19/11, 26/15) + + """ + if check_sqf and not f.is_sqf: + raise PolynomialError("only square-free polynomials supported") + + s, t = QQ.convert(s), QQ.convert(t) + + if eps is not None: + eps = QQ.convert(eps) + + if eps <= 0: + raise ValueError("'eps' must be a positive rational") + + if steps is not None: + steps = int(steps) + elif eps is None: + steps = 1 + + if hasattr(f.rep, 'refine_root'): + S, T = f.rep.refine_root(s, t, eps=eps, steps=steps, fast=fast) + else: # pragma: no cover + raise OperationNotSupported(f, 'refine_root') + + return QQ.to_sympy(S), QQ.to_sympy(T) + + def count_roots(f, inf=None, sup=None): + """ + Return the number of roots of ``f`` in ``[inf, sup]`` interval. + + Examples + ======== + + >>> from sympy import Poly, I + >>> from sympy.abc import x + + >>> Poly(x**4 - 4, x).count_roots(-3, 3) + 2 + >>> Poly(x**4 - 4, x).count_roots(0, 1 + 3*I) + 1 + + """ + inf_real, sup_real = True, True + + if inf is not None: + inf = sympify(inf) + + if inf is S.NegativeInfinity: + inf = None + else: + re, im = inf.as_real_imag() + + if not im: + inf = QQ.convert(inf) + else: + inf, inf_real = list(map(QQ.convert, (re, im))), False + + if sup is not None: + sup = sympify(sup) + + if sup is S.Infinity: + sup = None + else: + re, im = sup.as_real_imag() + + if not im: + sup = QQ.convert(sup) + else: + sup, sup_real = list(map(QQ.convert, (re, im))), False + + if inf_real and sup_real: + if hasattr(f.rep, 'count_real_roots'): + count = f.rep.count_real_roots(inf=inf, sup=sup) + else: # pragma: no cover + raise OperationNotSupported(f, 'count_real_roots') + else: + if inf_real and inf is not None: + inf = (inf, QQ.zero) + + if sup_real and sup is not None: + sup = (sup, QQ.zero) + + if hasattr(f.rep, 'count_complex_roots'): + count = f.rep.count_complex_roots(inf=inf, sup=sup) + else: # pragma: no cover + raise OperationNotSupported(f, 'count_complex_roots') + + return Integer(count) + + def root(f, index, radicals=True): + """ + Get an indexed root of a polynomial. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> f = Poly(2*x**3 - 7*x**2 + 4*x + 4) + + >>> f.root(0) + -1/2 + >>> f.root(1) + 2 + >>> f.root(2) + 2 + >>> f.root(3) + Traceback (most recent call last): + ... + IndexError: root index out of [-3, 2] range, got 3 + + >>> Poly(x**5 + x + 1).root(0) + CRootOf(x**3 - x**2 + 1, 0) + + """ + return sympy.polys.rootoftools.rootof(f, index, radicals=radicals) + + def real_roots(f, multiple=True, radicals=True): + """ + Return a list of real roots with multiplicities. + + See :func:`real_roots` for more explanation. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(2*x**3 - 7*x**2 + 4*x + 4).real_roots() + [-1/2, 2, 2] + >>> Poly(x**3 + x + 1).real_roots() + [CRootOf(x**3 + x + 1, 0)] + """ + reals = sympy.polys.rootoftools.CRootOf.real_roots(f, radicals=radicals) + + if multiple: + return reals + else: + return group(reals, multiple=False) + + def all_roots(f, multiple=True, radicals=True): + """ + Return a list of real and complex roots with multiplicities. + + See :func:`all_roots` for more explanation. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(2*x**3 - 7*x**2 + 4*x + 4).all_roots() + [-1/2, 2, 2] + >>> Poly(x**3 + x + 1).all_roots() + [CRootOf(x**3 + x + 1, 0), + CRootOf(x**3 + x + 1, 1), + CRootOf(x**3 + x + 1, 2)] + + """ + roots = sympy.polys.rootoftools.CRootOf.all_roots(f, radicals=radicals) + + if multiple: + return roots + else: + return group(roots, multiple=False) + + def nroots(f, n=15, maxsteps=50, cleanup=True): + """ + Compute numerical approximations of roots of ``f``. + + Parameters + ========== + + n ... the number of digits to calculate + maxsteps ... the maximum number of iterations to do + + If the accuracy `n` cannot be reached in `maxsteps`, it will raise an + exception. You need to rerun with higher maxsteps. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**2 - 3).nroots(n=15) + [-1.73205080756888, 1.73205080756888] + >>> Poly(x**2 - 3).nroots(n=30) + [-1.73205080756887729352744634151, 1.73205080756887729352744634151] + + """ + if f.is_multivariate: + raise MultivariatePolynomialError( + "Cannot compute numerical roots of %s" % f) + + if f.degree() <= 0: + return [] + + # For integer and rational coefficients, convert them to integers only + # (for accuracy). Otherwise just try to convert the coefficients to + # mpmath.mpc and raise an exception if the conversion fails. + if f.rep.dom is ZZ: + coeffs = [int(coeff) for coeff in f.all_coeffs()] + elif f.rep.dom is QQ: + denoms = [coeff.q for coeff in f.all_coeffs()] + fac = ilcm(*denoms) + coeffs = [int(coeff*fac) for coeff in f.all_coeffs()] + else: + coeffs = [coeff.evalf(n=n).as_real_imag() + for coeff in f.all_coeffs()] + with mpmath.workdps(n): + try: + coeffs = [mpmath.mpc(*coeff) for coeff in coeffs] + except TypeError: + raise DomainError("Numerical domain expected, got %s" % \ + f.rep.dom) + + dps = mpmath.mp.dps + mpmath.mp.dps = n + + from sympy.functions.elementary.complexes import sign + try: + # We need to add extra precision to guard against losing accuracy. + # 10 times the degree of the polynomial seems to work well. + roots = mpmath.polyroots(coeffs, maxsteps=maxsteps, + cleanup=cleanup, error=False, extraprec=f.degree()*10) + + # Mpmath puts real roots first, then complex ones (as does all_roots) + # so we make sure this convention holds here, too. + roots = list(map(sympify, + sorted(roots, key=lambda r: (1 if r.imag else 0, r.real, abs(r.imag), sign(r.imag))))) + except NoConvergence: + try: + # If roots did not converge try again with more extra precision. + roots = mpmath.polyroots(coeffs, maxsteps=maxsteps, + cleanup=cleanup, error=False, extraprec=f.degree()*15) + roots = list(map(sympify, + sorted(roots, key=lambda r: (1 if r.imag else 0, r.real, abs(r.imag), sign(r.imag))))) + except NoConvergence: + raise NoConvergence( + 'convergence to root failed; try n < %s or maxsteps > %s' % ( + n, maxsteps)) + finally: + mpmath.mp.dps = dps + + return roots + + def ground_roots(f): + """ + Compute roots of ``f`` by factorization in the ground domain. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**6 - 4*x**4 + 4*x**3 - x**2).ground_roots() + {0: 2, 1: 2} + + """ + if f.is_multivariate: + raise MultivariatePolynomialError( + "Cannot compute ground roots of %s" % f) + + roots = {} + + for factor, k in f.factor_list()[1]: + if factor.is_linear: + a, b = factor.all_coeffs() + roots[-b/a] = k + + return roots + + def nth_power_roots_poly(f, n): + """ + Construct a polynomial with n-th powers of roots of ``f``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> f = Poly(x**4 - x**2 + 1) + + >>> f.nth_power_roots_poly(2) + Poly(x**4 - 2*x**3 + 3*x**2 - 2*x + 1, x, domain='ZZ') + >>> f.nth_power_roots_poly(3) + Poly(x**4 + 2*x**2 + 1, x, domain='ZZ') + >>> f.nth_power_roots_poly(4) + Poly(x**4 + 2*x**3 + 3*x**2 + 2*x + 1, x, domain='ZZ') + >>> f.nth_power_roots_poly(12) + Poly(x**4 - 4*x**3 + 6*x**2 - 4*x + 1, x, domain='ZZ') + + """ + if f.is_multivariate: + raise MultivariatePolynomialError( + "must be a univariate polynomial") + + N = sympify(n) + + if N.is_Integer and N >= 1: + n = int(N) + else: + raise ValueError("'n' must an integer and n >= 1, got %s" % n) + + x = f.gen + t = Dummy('t') + + r = f.resultant(f.__class__.from_expr(x**n - t, x, t)) + + return r.replace(t, x) + + def which_real_roots(f, candidates): + """ + Find roots of a square-free polynomial ``f`` from ``candidates``. + + Explanation + =========== + + If ``f`` is a square-free polynomial and ``candidates`` is a superset + of the roots of ``f``, then ``f.which_real_roots(candidates)`` returns a + list containing exactly the set of roots of ``f``. The domain must be + :ref:`ZZ`, :ref:`QQ`, or :ref:`QQ(a)` and``f`` must be univariate and + square-free. + + The list ``candidates`` must be a superset of the real roots of ``f`` + and ``f.which_real_roots(candidates)`` returns the set of real roots + of ``f``. The output preserves the order of the order of ``candidates``. + + Examples + ======== + + >>> from sympy import Poly, sqrt + >>> from sympy.abc import x + + >>> f = Poly(x**4 - 1) + >>> f.which_real_roots([-1, 1, 0, -2, 2]) + [-1, 1] + >>> f.which_real_roots([-1, 1, 1, 1, 1]) + [-1, 1] + + This method is useful as lifting to rational coefficients + produced extraneous roots, which we can filter out with + this method. + + >>> f = Poly(sqrt(2)*x**3 + x**2 - 1, x, extension=True) + >>> f.lift() + Poly(-2*x**6 + x**4 - 2*x**2 + 1, x, domain='QQ') + >>> f.lift().real_roots() + [-sqrt(2)/2, sqrt(2)/2] + >>> f.which_real_roots(f.lift().real_roots()) + [sqrt(2)/2] + + This procedure is already done internally when calling + `.real_roots()` on a polynomial with algebraic coefficients. + + >>> f.real_roots() + [sqrt(2)/2] + + See Also + ======== + + same_root + which_all_roots + """ + if f.is_multivariate: + raise MultivariatePolynomialError( + "Must be a univariate polynomial") + + dom = f.get_domain() + + if not (dom.is_ZZ or dom.is_QQ or dom.is_AlgebraicField): + raise NotImplementedError( + "root counting not supported over %s" % dom) + + return f._which_roots(candidates, f.count_roots()) + + def which_all_roots(f, candidates): + """ + Find roots of a square-free polynomial ``f`` from ``candidates``. + + Explanation + =========== + + If ``f`` is a square-free polynomial and ``candidates`` is a superset + of the roots of ``f``, then ``f.which_all_roots(candidates)`` returns a + list containing exactly the set of roots of ``f``. The polynomial``f`` + must be univariate and square-free. + + The list ``candidates`` must be a superset of the complex roots of + ``f`` and ``f.which_all_roots(candidates)`` returns exactly the + set of all complex roots of ``f``. The output preserves the order of + the order of ``candidates``. + + Examples + ======== + + >>> from sympy import Poly, I + >>> from sympy.abc import x + + >>> f = Poly(x**4 - 1) + >>> f.which_all_roots([-1, 1, -I, I, 0]) + [-1, 1, -I, I] + >>> f.which_all_roots([-1, 1, -I, I, I, I]) + [-1, 1, -I, I] + + This method is useful as lifting to rational coefficients + produced extraneous roots, which we can filter out with + this method. + + >>> f = Poly(x**2 + I*x - 1, x, extension=True) + >>> f.lift() + Poly(x**4 - x**2 + 1, x, domain='ZZ') + >>> f.lift().all_roots() + [CRootOf(x**4 - x**2 + 1, 0), + CRootOf(x**4 - x**2 + 1, 1), + CRootOf(x**4 - x**2 + 1, 2), + CRootOf(x**4 - x**2 + 1, 3)] + >>> f.which_all_roots(f.lift().all_roots()) + [CRootOf(x**4 - x**2 + 1, 0), CRootOf(x**4 - x**2 + 1, 2)] + + This procedure is already done internally when calling + `.all_roots()` on a polynomial with algebraic coefficients, + or polynomials with Gaussian domains. + + >>> f.all_roots() + [CRootOf(x**4 - x**2 + 1, 0), CRootOf(x**4 - x**2 + 1, 2)] + + See Also + ======== + + same_root + which_real_roots + """ + if f.is_multivariate: + raise MultivariatePolynomialError( + "Must be a univariate polynomial") + + return f._which_roots(candidates, f.degree()) + + def _which_roots(f, candidates, num_roots): + prec = 10 + # using Counter bc its like an ordered set + root_counts = Counter(candidates) + + while len(root_counts) > num_roots: + for r in list(root_counts.keys()): + # If f(r) != 0 then f(r).evalf() gives a float/complex with precision. + f_r = f(r).evalf(prec, maxn=2*prec) + if abs(f_r)._prec >= 2: + root_counts.pop(r) + + prec *= 2 + + return list(root_counts.keys()) + + def same_root(f, a, b): + """ + Decide whether two roots of this polynomial are equal. + + Examples + ======== + + >>> from sympy import Poly, cyclotomic_poly, exp, I, pi + >>> f = Poly(cyclotomic_poly(5)) + >>> r0 = exp(2*I*pi/5) + >>> indices = [i for i, r in enumerate(f.all_roots()) if f.same_root(r, r0)] + >>> print(indices) + [3] + + Raises + ====== + + DomainError + If the domain of the polynomial is not :ref:`ZZ`, :ref:`QQ`, + :ref:`RR`, or :ref:`CC`. + MultivariatePolynomialError + If the polynomial is not univariate. + PolynomialError + If the polynomial is of degree < 2. + + See Also + ======== + + which_real_roots + which_all_roots + """ + if f.is_multivariate: + raise MultivariatePolynomialError( + "Must be a univariate polynomial") + + dom_delta_sq = f.rep.mignotte_sep_bound_squared() + delta_sq = f.domain.get_field().to_sympy(dom_delta_sq) + # We have delta_sq = delta**2, where delta is a lower bound on the + # minimum separation between any two roots of this polynomial. + # Let eps = delta/3, and define eps_sq = eps**2 = delta**2/9. + eps_sq = delta_sq / 9 + + r, _, _, _ = evalf(1/eps_sq, 1, {}) + n = fastlog(r) + # Then 2^n > 1/eps**2. + m = (n // 2) + (n % 2) + # Then 2^(-m) < eps. + ev = lambda x: quad_to_mpmath(_evalf_with_bounded_error(x, m=m)) + + # Then for any complex numbers a, b we will have + # |a - ev(a)| < eps and |b - ev(b)| < eps. + # So if |ev(a) - ev(b)|**2 < eps**2, then + # |ev(a) - ev(b)| < eps, hence |a - b| < 3*eps = delta. + A, B = ev(a), ev(b) + return (A.real - B.real)**2 + (A.imag - B.imag)**2 < eps_sq + + def cancel(f, g, include=False): + """ + Cancel common factors in a rational function ``f/g``. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(2*x**2 - 2, x).cancel(Poly(x**2 - 2*x + 1, x)) + (1, Poly(2*x + 2, x, domain='ZZ'), Poly(x - 1, x, domain='ZZ')) + + >>> Poly(2*x**2 - 2, x).cancel(Poly(x**2 - 2*x + 1, x), include=True) + (Poly(2*x + 2, x, domain='ZZ'), Poly(x - 1, x, domain='ZZ')) + + """ + dom, per, F, G = f._unify(g) + + if hasattr(F, 'cancel'): + result = F.cancel(G, include=include) + else: # pragma: no cover + raise OperationNotSupported(f, 'cancel') + + if not include: + if dom.has_assoc_Ring: + dom = dom.get_ring() + + cp, cq, p, q = result + + cp = dom.to_sympy(cp) + cq = dom.to_sympy(cq) + + return cp/cq, per(p), per(q) + else: + return tuple(map(per, result)) + + def make_monic_over_integers_by_scaling_roots(f): + """ + Turn any univariate polynomial over :ref:`QQ` or :ref:`ZZ` into a monic + polynomial over :ref:`ZZ`, by scaling the roots as necessary. + + Explanation + =========== + + This operation can be performed whether or not *f* is irreducible; when + it is, this can be understood as determining an algebraic integer + generating the same field as a root of *f*. + + Examples + ======== + + >>> from sympy import Poly, S + >>> from sympy.abc import x + >>> f = Poly(x**2/2 + S(1)/4 * x + S(1)/8, x, domain='QQ') + >>> f.make_monic_over_integers_by_scaling_roots() + (Poly(x**2 + 2*x + 4, x, domain='ZZ'), 4) + + Returns + ======= + + Pair ``(g, c)`` + g is the polynomial + + c is the integer by which the roots had to be scaled + + """ + if not f.is_univariate or f.domain not in [ZZ, QQ]: + raise ValueError('Polynomial must be univariate over ZZ or QQ.') + if f.is_monic and f.domain == ZZ: + return f, ZZ.one + else: + fm = f.monic() + c, _ = fm.clear_denoms() + return fm.transform(Poly(fm.gen), c).to_ring(), c + + def galois_group(f, by_name=False, max_tries=30, randomize=False): + """ + Compute the Galois group of this polynomial. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + >>> f = Poly(x**4 - 2) + >>> G, _ = f.galois_group(by_name=True) + >>> print(G) + S4TransitiveSubgroups.D4 + + See Also + ======== + + sympy.polys.numberfields.galoisgroups.galois_group + + """ + from sympy.polys.numberfields.galoisgroups import ( + _galois_group_degree_3, _galois_group_degree_4_lookup, + _galois_group_degree_5_lookup_ext_factor, + _galois_group_degree_6_lookup, + ) + if (not f.is_univariate + or not f.is_irreducible + or f.domain not in [ZZ, QQ] + ): + raise ValueError('Polynomial must be irreducible and univariate over ZZ or QQ.') + gg = { + 3: _galois_group_degree_3, + 4: _galois_group_degree_4_lookup, + 5: _galois_group_degree_5_lookup_ext_factor, + 6: _galois_group_degree_6_lookup, + } + max_supported = max(gg.keys()) + n = f.degree() + if n > max_supported: + raise ValueError(f"Only polynomials up to degree {max_supported} are supported.") + elif n < 1: + raise ValueError("Constant polynomial has no Galois group.") + elif n == 1: + from sympy.combinatorics.galois import S1TransitiveSubgroups + name, alt = S1TransitiveSubgroups.S1, True + elif n == 2: + from sympy.combinatorics.galois import S2TransitiveSubgroups + name, alt = S2TransitiveSubgroups.S2, False + else: + g, _ = f.make_monic_over_integers_by_scaling_roots() + name, alt = gg[n](g, max_tries=max_tries, randomize=randomize) + G = name if by_name else name.get_perm_group() + return G, alt + + @property + def is_zero(f): + """ + Returns ``True`` if ``f`` is a zero polynomial. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(0, x).is_zero + True + >>> Poly(1, x).is_zero + False + + """ + return f.rep.is_zero + + @property + def is_one(f): + """ + Returns ``True`` if ``f`` is a unit polynomial. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(0, x).is_one + False + >>> Poly(1, x).is_one + True + + """ + return f.rep.is_one + + @property + def is_sqf(f): + """ + Returns ``True`` if ``f`` is a square-free polynomial. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**2 - 2*x + 1, x).is_sqf + False + >>> Poly(x**2 - 1, x).is_sqf + True + + """ + return f.rep.is_sqf + + @property + def is_monic(f): + """ + Returns ``True`` if the leading coefficient of ``f`` is one. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x + 2, x).is_monic + True + >>> Poly(2*x + 2, x).is_monic + False + + """ + return f.rep.is_monic + + @property + def is_primitive(f): + """ + Returns ``True`` if GCD of the coefficients of ``f`` is one. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(2*x**2 + 6*x + 12, x).is_primitive + False + >>> Poly(x**2 + 3*x + 6, x).is_primitive + True + + """ + return f.rep.is_primitive + + @property + def is_ground(f): + """ + Returns ``True`` if ``f`` is an element of the ground domain. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + >>> Poly(x, x).is_ground + False + >>> Poly(2, x).is_ground + True + >>> Poly(y, x).is_ground + True + + """ + return f.rep.is_ground + + @property + def is_linear(f): + """ + Returns ``True`` if ``f`` is linear in all its variables. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + >>> Poly(x + y + 2, x, y).is_linear + True + >>> Poly(x*y + 2, x, y).is_linear + False + + """ + return f.rep.is_linear + + @property + def is_quadratic(f): + """ + Returns ``True`` if ``f`` is quadratic in all its variables. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + >>> Poly(x*y + 2, x, y).is_quadratic + True + >>> Poly(x*y**2 + 2, x, y).is_quadratic + False + + """ + return f.rep.is_quadratic + + @property + def is_monomial(f): + """ + Returns ``True`` if ``f`` is zero or has only one term. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(3*x**2, x).is_monomial + True + >>> Poly(3*x**2 + 1, x).is_monomial + False + + """ + return f.rep.is_monomial + + @property + def is_homogeneous(f): + """ + Returns ``True`` if ``f`` is a homogeneous polynomial. + + A homogeneous polynomial is a polynomial whose all monomials with + non-zero coefficients have the same total degree. If you want not + only to check if a polynomial is homogeneous but also compute its + homogeneous order, then use :func:`Poly.homogeneous_order`. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + >>> Poly(x**2 + x*y, x, y).is_homogeneous + True + >>> Poly(x**3 + x*y, x, y).is_homogeneous + False + + """ + return f.rep.is_homogeneous + + @property + def is_irreducible(f): + """ + Returns ``True`` if ``f`` has no factors over its domain. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> Poly(x**2 + x + 1, x, modulus=2).is_irreducible + True + >>> Poly(x**2 + 1, x, modulus=2).is_irreducible + False + + """ + return f.rep.is_irreducible + + @property + def is_univariate(f): + """ + Returns ``True`` if ``f`` is a univariate polynomial. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + >>> Poly(x**2 + x + 1, x).is_univariate + True + >>> Poly(x*y**2 + x*y + 1, x, y).is_univariate + False + >>> Poly(x*y**2 + x*y + 1, x).is_univariate + True + >>> Poly(x**2 + x + 1, x, y).is_univariate + False + + """ + return len(f.gens) == 1 + + @property + def is_multivariate(f): + """ + Returns ``True`` if ``f`` is a multivariate polynomial. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x, y + + >>> Poly(x**2 + x + 1, x).is_multivariate + False + >>> Poly(x*y**2 + x*y + 1, x, y).is_multivariate + True + >>> Poly(x*y**2 + x*y + 1, x).is_multivariate + False + >>> Poly(x**2 + x + 1, x, y).is_multivariate + True + + """ + return len(f.gens) != 1 + + @property + def is_cyclotomic(f): + """ + Returns ``True`` if ``f`` is a cyclotomic polnomial. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.abc import x + + >>> f = x**16 + x**14 - x**10 + x**8 - x**6 + x**2 + 1 + + >>> Poly(f).is_cyclotomic + False + + >>> g = x**16 + x**14 - x**10 - x**8 - x**6 + x**2 + 1 + + >>> Poly(g).is_cyclotomic + True + + """ + return f.rep.is_cyclotomic + + def __abs__(f): + return f.abs() + + def __neg__(f): + return f.neg() + + @_polifyit + def __add__(f, g): + return f.add(g) + + @_polifyit + def __radd__(f, g): + return g.add(f) + + @_polifyit + def __sub__(f, g): + return f.sub(g) + + @_polifyit + def __rsub__(f, g): + return g.sub(f) + + @_polifyit + def __mul__(f, g): + return f.mul(g) + + @_polifyit + def __rmul__(f, g): + return g.mul(f) + + @_sympifyit('n', NotImplemented) + def __pow__(f, n): + if n.is_Integer and n >= 0: + return f.pow(n) + else: + return NotImplemented + + @_polifyit + def __divmod__(f, g): + return f.div(g) + + @_polifyit + def __rdivmod__(f, g): + return g.div(f) + + @_polifyit + def __mod__(f, g): + return f.rem(g) + + @_polifyit + def __rmod__(f, g): + return g.rem(f) + + @_polifyit + def __floordiv__(f, g): + return f.quo(g) + + @_polifyit + def __rfloordiv__(f, g): + return g.quo(f) + + @_sympifyit('g', NotImplemented) + def __truediv__(f, g): + return f.as_expr()/g.as_expr() + + @_sympifyit('g', NotImplemented) + def __rtruediv__(f, g): + return g.as_expr()/f.as_expr() + + @_sympifyit('other', NotImplemented) + def __eq__(self, other): + f, g = self, other + + if not g.is_Poly: + try: + g = f.__class__(g, f.gens, domain=f.get_domain()) + except (PolynomialError, DomainError, CoercionFailed): + return False + + if f.gens != g.gens: + return False + + if f.rep.dom != g.rep.dom: + return False + + return f.rep == g.rep + + @_sympifyit('g', NotImplemented) + def __ne__(f, g): + return not f == g + + def __bool__(f): + return not f.is_zero + + def eq(f, g, strict=False): + if not strict: + return f == g + else: + return f._strict_eq(sympify(g)) + + def ne(f, g, strict=False): + return not f.eq(g, strict=strict) + + def _strict_eq(f, g): + return isinstance(g, f.__class__) and f.gens == g.gens and f.rep.eq(g.rep, strict=True) + + +@public +class PurePoly(Poly): + """Class for representing pure polynomials. """ + + def _hashable_content(self): + """Allow SymPy to hash Poly instances. """ + return (self.rep,) + + def __hash__(self): + return super().__hash__() + + @property + def free_symbols(self): + """ + Free symbols of a polynomial. + + Examples + ======== + + >>> from sympy import PurePoly + >>> from sympy.abc import x, y + + >>> PurePoly(x**2 + 1).free_symbols + set() + >>> PurePoly(x**2 + y).free_symbols + set() + >>> PurePoly(x**2 + y, x).free_symbols + {y} + + """ + return self.free_symbols_in_domain + + @_sympifyit('other', NotImplemented) + def __eq__(self, other): + f, g = self, other + + if not g.is_Poly: + try: + g = f.__class__(g, f.gens, domain=f.get_domain()) + except (PolynomialError, DomainError, CoercionFailed): + return False + + if len(f.gens) != len(g.gens): + return False + + if f.rep.dom != g.rep.dom: + try: + dom = f.rep.dom.unify(g.rep.dom, f.gens) + except UnificationFailed: + return False + + f = f.set_domain(dom) + g = g.set_domain(dom) + + return f.rep == g.rep + + def _strict_eq(f, g): + return isinstance(g, f.__class__) and f.rep.eq(g.rep, strict=True) + + def _unify(f, g): + g = sympify(g) + + if not g.is_Poly: + try: + return f.rep.dom, f.per, f.rep, f.rep.per(f.rep.dom.from_sympy(g)) + except CoercionFailed: + raise UnificationFailed("Cannot unify %s with %s" % (f, g)) + + if len(f.gens) != len(g.gens): + raise UnificationFailed("Cannot unify %s with %s" % (f, g)) + + if not (isinstance(f.rep, DMP) and isinstance(g.rep, DMP)): + raise UnificationFailed("Cannot unify %s with %s" % (f, g)) + + cls = f.__class__ + gens = f.gens + + dom = f.rep.dom.unify(g.rep.dom, gens) + + F = f.rep.convert(dom) + G = g.rep.convert(dom) + + def per(rep, dom=dom, gens=gens, remove=None): + if remove is not None: + gens = gens[:remove] + gens[remove + 1:] + + if not gens: + return dom.to_sympy(rep) + + return cls.new(rep, *gens) + + return dom, per, F, G + + +@public +def poly_from_expr(expr, *gens, **args): + """Construct a polynomial from an expression. """ + opt = options.build_options(gens, args) + return _poly_from_expr(expr, opt) + + +def _poly_from_expr(expr, opt): + """Construct a polynomial from an expression. """ + orig, expr = expr, sympify(expr) + + if not isinstance(expr, Basic): + raise PolificationFailed(opt, orig, expr) + elif expr.is_Poly: + poly = expr.__class__._from_poly(expr, opt) + + opt.gens = poly.gens + opt.domain = poly.domain + + if opt.polys is None: + opt.polys = True + + return poly, opt + elif opt.expand: + expr = expr.expand() + + rep, opt = _dict_from_expr(expr, opt) + if not opt.gens: + raise PolificationFailed(opt, orig, expr) + + monoms, coeffs = list(zip(*list(rep.items()))) + domain = opt.domain + + if domain is None: + opt.domain, coeffs = construct_domain(coeffs, opt=opt) + else: + coeffs = list(map(domain.from_sympy, coeffs)) + + rep = dict(list(zip(monoms, coeffs))) + poly = Poly._from_dict(rep, opt) + + if opt.polys is None: + opt.polys = False + + return poly, opt + + +@public +def parallel_poly_from_expr(exprs, *gens, **args): + """Construct polynomials from expressions. """ + opt = options.build_options(gens, args) + return _parallel_poly_from_expr(exprs, opt) + + +def _parallel_poly_from_expr(exprs, opt): + """Construct polynomials from expressions. """ + if len(exprs) == 2: + f, g = exprs + + if isinstance(f, Poly) and isinstance(g, Poly): + f = f.__class__._from_poly(f, opt) + g = g.__class__._from_poly(g, opt) + + f, g = f.unify(g) + + opt.gens = f.gens + opt.domain = f.domain + + if opt.polys is None: + opt.polys = True + + return [f, g], opt + + origs, exprs = list(exprs), [] + _exprs, _polys = [], [] + + failed = False + + for i, expr in enumerate(origs): + expr = sympify(expr) + + if isinstance(expr, Basic): + if expr.is_Poly: + _polys.append(i) + else: + _exprs.append(i) + + if opt.expand: + expr = expr.expand() + else: + failed = True + + exprs.append(expr) + + if failed: + raise PolificationFailed(opt, origs, exprs, True) + + if _polys: + # XXX: this is a temporary solution + for i in _polys: + exprs[i] = exprs[i].as_expr() + + reps, opt = _parallel_dict_from_expr(exprs, opt) + if not opt.gens: + raise PolificationFailed(opt, origs, exprs, True) + + from sympy.functions.elementary.piecewise import Piecewise + for k in opt.gens: + if isinstance(k, Piecewise): + raise PolynomialError("Piecewise generators do not make sense") + + coeffs_list, lengths = [], [] + + all_monoms = [] + all_coeffs = [] + + for rep in reps: + monoms, coeffs = list(zip(*list(rep.items()))) + + coeffs_list.extend(coeffs) + all_monoms.append(monoms) + + lengths.append(len(coeffs)) + + domain = opt.domain + + if domain is None: + opt.domain, coeffs_list = construct_domain(coeffs_list, opt=opt) + else: + coeffs_list = list(map(domain.from_sympy, coeffs_list)) + + for k in lengths: + all_coeffs.append(coeffs_list[:k]) + coeffs_list = coeffs_list[k:] + + polys = [] + + for monoms, coeffs in zip(all_monoms, all_coeffs): + rep = dict(list(zip(monoms, coeffs))) + poly = Poly._from_dict(rep, opt) + polys.append(poly) + + if opt.polys is None: + opt.polys = bool(_polys) + + return polys, opt + + +def _update_args(args, key, value): + """Add a new ``(key, value)`` pair to arguments ``dict``. """ + args = dict(args) + + if key not in args: + args[key] = value + + return args + + +@public +def degree(f, gen=0): + """ + Return the degree of ``f`` in the given variable. + + The degree of 0 is negative infinity. + + Examples + ======== + + >>> from sympy import degree + >>> from sympy.abc import x, y + + >>> degree(x**2 + y*x + 1, gen=x) + 2 + >>> degree(x**2 + y*x + 1, gen=y) + 1 + >>> degree(0, x) + -oo + + See also + ======== + + sympy.polys.polytools.Poly.total_degree + degree_list + """ + + f = sympify(f, strict=True) + gen_is_Num = sympify(gen, strict=True).is_Number + if f.is_Poly: + p = f + isNum = p.as_expr().is_Number + else: + isNum = f.is_Number + if not isNum: + if gen_is_Num: + p, _ = poly_from_expr(f) + else: + p, _ = poly_from_expr(f, gen) + + if isNum: + return S.Zero if f else S.NegativeInfinity + + if not gen_is_Num: + if f.is_Poly and gen not in p.gens: + # try recast without explicit gens + p, _ = poly_from_expr(f.as_expr()) + if gen not in p.gens: + return S.Zero + elif not f.is_Poly and len(f.free_symbols) > 1: + raise TypeError(filldedent(''' + A symbolic generator of interest is required for a multivariate + expression like func = %s, e.g. degree(func, gen = %s) instead of + degree(func, gen = %s). + ''' % (f, next(ordered(f.free_symbols)), gen))) + result = p.degree(gen) + return Integer(result) if isinstance(result, int) else S.NegativeInfinity + + +@public +def total_degree(f, *gens): + """ + Return the total_degree of ``f`` in the given variables. + + Examples + ======== + >>> from sympy import total_degree, Poly + >>> from sympy.abc import x, y + + >>> total_degree(1) + 0 + >>> total_degree(x + x*y) + 2 + >>> total_degree(x + x*y, x) + 1 + + If the expression is a Poly and no variables are given + then the generators of the Poly will be used: + + >>> p = Poly(x + x*y, y) + >>> total_degree(p) + 1 + + To deal with the underlying expression of the Poly, convert + it to an Expr: + + >>> total_degree(p.as_expr()) + 2 + + This is done automatically if any variables are given: + + >>> total_degree(p, x) + 1 + + See also + ======== + degree + """ + + p = sympify(f) + if p.is_Poly: + p = p.as_expr() + if p.is_Number: + rv = 0 + else: + if f.is_Poly: + gens = gens or f.gens + rv = Poly(p, gens).total_degree() + + return Integer(rv) + + +@public +def degree_list(f, *gens, **args): + """ + Return a list of degrees of ``f`` in all variables. + + Examples + ======== + + >>> from sympy import degree_list + >>> from sympy.abc import x, y + + >>> degree_list(x**2 + y*x + 1) + (2, 1) + + """ + options.allowed_flags(args, ['polys']) + + try: + F, opt = poly_from_expr(f, *gens, **args) + except PolificationFailed as exc: + raise ComputationFailed('degree_list', 1, exc) + + degrees = F.degree_list() + + return tuple(map(Integer, degrees)) + + +@public +def LC(f, *gens, **args): + """ + Return the leading coefficient of ``f``. + + Examples + ======== + + >>> from sympy import LC + >>> from sympy.abc import x, y + + >>> LC(4*x**2 + 2*x*y**2 + x*y + 3*y) + 4 + + """ + options.allowed_flags(args, ['polys']) + + try: + F, opt = poly_from_expr(f, *gens, **args) + except PolificationFailed as exc: + raise ComputationFailed('LC', 1, exc) + + return F.LC(order=opt.order) + + +@public +def LM(f, *gens, **args): + """ + Return the leading monomial of ``f``. + + Examples + ======== + + >>> from sympy import LM + >>> from sympy.abc import x, y + + >>> LM(4*x**2 + 2*x*y**2 + x*y + 3*y) + x**2 + + """ + options.allowed_flags(args, ['polys']) + + try: + F, opt = poly_from_expr(f, *gens, **args) + except PolificationFailed as exc: + raise ComputationFailed('LM', 1, exc) + + monom = F.LM(order=opt.order) + return monom.as_expr() + + +@public +def LT(f, *gens, **args): + """ + Return the leading term of ``f``. + + Examples + ======== + + >>> from sympy import LT + >>> from sympy.abc import x, y + + >>> LT(4*x**2 + 2*x*y**2 + x*y + 3*y) + 4*x**2 + + """ + options.allowed_flags(args, ['polys']) + + try: + F, opt = poly_from_expr(f, *gens, **args) + except PolificationFailed as exc: + raise ComputationFailed('LT', 1, exc) + + monom, coeff = F.LT(order=opt.order) + return coeff*monom.as_expr() + + +@public +def pdiv(f, g, *gens, **args): + """ + Compute polynomial pseudo-division of ``f`` and ``g``. + + Examples + ======== + + >>> from sympy import pdiv + >>> from sympy.abc import x + + >>> pdiv(x**2 + 1, 2*x - 4) + (2*x + 4, 20) + + """ + options.allowed_flags(args, ['polys']) + + try: + (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) + except PolificationFailed as exc: + raise ComputationFailed('pdiv', 2, exc) + + q, r = F.pdiv(G) + + if not opt.polys: + return q.as_expr(), r.as_expr() + else: + return q, r + + +@public +def prem(f, g, *gens, **args): + """ + Compute polynomial pseudo-remainder of ``f`` and ``g``. + + Examples + ======== + + >>> from sympy import prem + >>> from sympy.abc import x + + >>> prem(x**2 + 1, 2*x - 4) + 20 + + """ + options.allowed_flags(args, ['polys']) + + try: + (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) + except PolificationFailed as exc: + raise ComputationFailed('prem', 2, exc) + + r = F.prem(G) + + if not opt.polys: + return r.as_expr() + else: + return r + + +@public +def pquo(f, g, *gens, **args): + """ + Compute polynomial pseudo-quotient of ``f`` and ``g``. + + Examples + ======== + + >>> from sympy import pquo + >>> from sympy.abc import x + + >>> pquo(x**2 + 1, 2*x - 4) + 2*x + 4 + >>> pquo(x**2 - 1, 2*x - 1) + 2*x + 1 + + """ + options.allowed_flags(args, ['polys']) + + try: + (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) + except PolificationFailed as exc: + raise ComputationFailed('pquo', 2, exc) + + try: + q = F.pquo(G) + except ExactQuotientFailed: + raise ExactQuotientFailed(f, g) + + if not opt.polys: + return q.as_expr() + else: + return q + + +@public +def pexquo(f, g, *gens, **args): + """ + Compute polynomial exact pseudo-quotient of ``f`` and ``g``. + + Examples + ======== + + >>> from sympy import pexquo + >>> from sympy.abc import x + + >>> pexquo(x**2 - 1, 2*x - 2) + 2*x + 2 + + >>> pexquo(x**2 + 1, 2*x - 4) + Traceback (most recent call last): + ... + ExactQuotientFailed: 2*x - 4 does not divide x**2 + 1 + + """ + options.allowed_flags(args, ['polys']) + + try: + (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) + except PolificationFailed as exc: + raise ComputationFailed('pexquo', 2, exc) + + q = F.pexquo(G) + + if not opt.polys: + return q.as_expr() + else: + return q + + +@public +def div(f, g, *gens, **args): + """ + Compute polynomial division of ``f`` and ``g``. + + Examples + ======== + + >>> from sympy import div, ZZ, QQ + >>> from sympy.abc import x + + >>> div(x**2 + 1, 2*x - 4, domain=ZZ) + (0, x**2 + 1) + >>> div(x**2 + 1, 2*x - 4, domain=QQ) + (x/2 + 1, 5) + + """ + options.allowed_flags(args, ['auto', 'polys']) + + try: + (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) + except PolificationFailed as exc: + raise ComputationFailed('div', 2, exc) + + q, r = F.div(G, auto=opt.auto) + + if not opt.polys: + return q.as_expr(), r.as_expr() + else: + return q, r + + +@public +def rem(f, g, *gens, **args): + """ + Compute polynomial remainder of ``f`` and ``g``. + + Examples + ======== + + >>> from sympy import rem, ZZ, QQ + >>> from sympy.abc import x + + >>> rem(x**2 + 1, 2*x - 4, domain=ZZ) + x**2 + 1 + >>> rem(x**2 + 1, 2*x - 4, domain=QQ) + 5 + + """ + options.allowed_flags(args, ['auto', 'polys']) + + try: + (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) + except PolificationFailed as exc: + raise ComputationFailed('rem', 2, exc) + + r = F.rem(G, auto=opt.auto) + + if not opt.polys: + return r.as_expr() + else: + return r + + +@public +def quo(f, g, *gens, **args): + """ + Compute polynomial quotient of ``f`` and ``g``. + + Examples + ======== + + >>> from sympy import quo + >>> from sympy.abc import x + + >>> quo(x**2 + 1, 2*x - 4) + x/2 + 1 + >>> quo(x**2 - 1, x - 1) + x + 1 + + """ + options.allowed_flags(args, ['auto', 'polys']) + + try: + (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) + except PolificationFailed as exc: + raise ComputationFailed('quo', 2, exc) + + q = F.quo(G, auto=opt.auto) + + if not opt.polys: + return q.as_expr() + else: + return q + + +@public +def exquo(f, g, *gens, **args): + """ + Compute polynomial exact quotient of ``f`` and ``g``. + + Examples + ======== + + >>> from sympy import exquo + >>> from sympy.abc import x + + >>> exquo(x**2 - 1, x - 1) + x + 1 + + >>> exquo(x**2 + 1, 2*x - 4) + Traceback (most recent call last): + ... + ExactQuotientFailed: 2*x - 4 does not divide x**2 + 1 + + """ + options.allowed_flags(args, ['auto', 'polys']) + + try: + (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) + except PolificationFailed as exc: + raise ComputationFailed('exquo', 2, exc) + + q = F.exquo(G, auto=opt.auto) + + if not opt.polys: + return q.as_expr() + else: + return q + + +@public +def half_gcdex(f, g, *gens, **args): + """ + Half extended Euclidean algorithm of ``f`` and ``g``. + + Returns ``(s, h)`` such that ``h = gcd(f, g)`` and ``s*f = h (mod g)``. + + Examples + ======== + + >>> from sympy import half_gcdex + >>> from sympy.abc import x + + >>> half_gcdex(x**4 - 2*x**3 - 6*x**2 + 12*x + 15, x**3 + x**2 - 4*x - 4) + (3/5 - x/5, x + 1) + + """ + options.allowed_flags(args, ['auto', 'polys']) + + try: + (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) + except PolificationFailed as exc: + domain, (a, b) = construct_domain(exc.exprs) + + try: + s, h = domain.half_gcdex(a, b) + except NotImplementedError: + raise ComputationFailed('half_gcdex', 2, exc) + else: + return domain.to_sympy(s), domain.to_sympy(h) + + s, h = F.half_gcdex(G, auto=opt.auto) + + if not opt.polys: + return s.as_expr(), h.as_expr() + else: + return s, h + + +@public +def gcdex(f, g, *gens, **args): + """ + Extended Euclidean algorithm of ``f`` and ``g``. + + Returns ``(s, t, h)`` such that ``h = gcd(f, g)`` and ``s*f + t*g = h``. + + Examples + ======== + + >>> from sympy import gcdex + >>> from sympy.abc import x + + >>> gcdex(x**4 - 2*x**3 - 6*x**2 + 12*x + 15, x**3 + x**2 - 4*x - 4) + (3/5 - x/5, x**2/5 - 6*x/5 + 2, x + 1) + + """ + options.allowed_flags(args, ['auto', 'polys']) + + try: + (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) + except PolificationFailed as exc: + domain, (a, b) = construct_domain(exc.exprs) + + try: + s, t, h = domain.gcdex(a, b) + except NotImplementedError: + raise ComputationFailed('gcdex', 2, exc) + else: + return domain.to_sympy(s), domain.to_sympy(t), domain.to_sympy(h) + + s, t, h = F.gcdex(G, auto=opt.auto) + + if not opt.polys: + return s.as_expr(), t.as_expr(), h.as_expr() + else: + return s, t, h + + +@public +def invert(f, g, *gens, **args): + """ + Invert ``f`` modulo ``g`` when possible. + + Examples + ======== + + >>> from sympy import invert, S, mod_inverse + >>> from sympy.abc import x + + >>> invert(x**2 - 1, 2*x - 1) + -4/3 + + >>> invert(x**2 - 1, x - 1) + Traceback (most recent call last): + ... + NotInvertible: zero divisor + + For more efficient inversion of Rationals, + use the :obj:`sympy.core.intfunc.mod_inverse` function: + + >>> mod_inverse(3, 5) + 2 + >>> (S(2)/5).invert(S(7)/3) + 5/2 + + See Also + ======== + sympy.core.intfunc.mod_inverse + + """ + options.allowed_flags(args, ['auto', 'polys']) + + try: + (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) + except PolificationFailed as exc: + domain, (a, b) = construct_domain(exc.exprs) + + try: + return domain.to_sympy(domain.invert(a, b)) + except NotImplementedError: + raise ComputationFailed('invert', 2, exc) + + h = F.invert(G, auto=opt.auto) + + if not opt.polys: + return h.as_expr() + else: + return h + + +@public +def subresultants(f, g, *gens, **args): + """ + Compute subresultant PRS of ``f`` and ``g``. + + Examples + ======== + + >>> from sympy import subresultants + >>> from sympy.abc import x + + >>> subresultants(x**2 + 1, x**2 - 1) + [x**2 + 1, x**2 - 1, -2] + + """ + options.allowed_flags(args, ['polys']) + + try: + (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) + except PolificationFailed as exc: + raise ComputationFailed('subresultants', 2, exc) + + result = F.subresultants(G) + + if not opt.polys: + return [r.as_expr() for r in result] + else: + return result + + +@public +def resultant(f, g, *gens, includePRS=False, **args): + """ + Compute resultant of ``f`` and ``g``. + + Examples + ======== + + >>> from sympy import resultant + >>> from sympy.abc import x + + >>> resultant(x**2 + 1, x**2 - 1) + 4 + + """ + options.allowed_flags(args, ['polys']) + + try: + (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) + except PolificationFailed as exc: + raise ComputationFailed('resultant', 2, exc) + + if includePRS: + result, R = F.resultant(G, includePRS=includePRS) + else: + result = F.resultant(G) + + if not opt.polys: + if includePRS: + return result.as_expr(), [r.as_expr() for r in R] + return result.as_expr() + else: + if includePRS: + return result, R + return result + + +@public +def discriminant(f, *gens, **args): + """ + Compute discriminant of ``f``. + + Examples + ======== + + >>> from sympy import discriminant + >>> from sympy.abc import x + + >>> discriminant(x**2 + 2*x + 3) + -8 + + """ + options.allowed_flags(args, ['polys']) + + try: + F, opt = poly_from_expr(f, *gens, **args) + except PolificationFailed as exc: + raise ComputationFailed('discriminant', 1, exc) + + result = F.discriminant() + + if not opt.polys: + return result.as_expr() + else: + return result + + +@public +def cofactors(f, g, *gens, **args): + """ + Compute GCD and cofactors of ``f`` and ``g``. + + Returns polynomials ``(h, cff, cfg)`` such that ``h = gcd(f, g)``, and + ``cff = quo(f, h)`` and ``cfg = quo(g, h)`` are, so called, cofactors + of ``f`` and ``g``. + + Examples + ======== + + >>> from sympy import cofactors + >>> from sympy.abc import x + + >>> cofactors(x**2 - 1, x**2 - 3*x + 2) + (x - 1, x + 1, x - 2) + + """ + options.allowed_flags(args, ['polys']) + + try: + (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) + except PolificationFailed as exc: + domain, (a, b) = construct_domain(exc.exprs) + + try: + h, cff, cfg = domain.cofactors(a, b) + except NotImplementedError: + raise ComputationFailed('cofactors', 2, exc) + else: + return domain.to_sympy(h), domain.to_sympy(cff), domain.to_sympy(cfg) + + h, cff, cfg = F.cofactors(G) + + if not opt.polys: + return h.as_expr(), cff.as_expr(), cfg.as_expr() + else: + return h, cff, cfg + + +@public +def gcd_list(seq, *gens, **args): + """ + Compute GCD of a list of polynomials. + + Examples + ======== + + >>> from sympy import gcd_list + >>> from sympy.abc import x + + >>> gcd_list([x**3 - 1, x**2 - 1, x**2 - 3*x + 2]) + x - 1 + + """ + seq = sympify(seq) + + def try_non_polynomial_gcd(seq): + if not gens and not args: + domain, numbers = construct_domain(seq) + + if not numbers: + return domain.zero + elif domain.is_Numerical: + result, numbers = numbers[0], numbers[1:] + + for number in numbers: + result = domain.gcd(result, number) + + if domain.is_one(result): + break + + return domain.to_sympy(result) + + return None + + result = try_non_polynomial_gcd(seq) + + if result is not None: + return result + + options.allowed_flags(args, ['polys']) + + try: + polys, opt = parallel_poly_from_expr(seq, *gens, **args) + + # gcd for domain Q[irrational] (purely algebraic irrational) + if len(seq) > 1 and all(elt.is_algebraic and elt.is_irrational for elt in seq): + a = seq[-1] + lst = [ (a/elt).ratsimp() for elt in seq[:-1] ] + if all(frc.is_rational for frc in lst): + lc = 1 + for frc in lst: + lc = lcm(lc, frc.as_numer_denom()[0]) + # abs ensures that the gcd is always non-negative + return abs(a/lc) + + except PolificationFailed as exc: + result = try_non_polynomial_gcd(exc.exprs) + + if result is not None: + return result + else: + raise ComputationFailed('gcd_list', len(seq), exc) + + if not polys: + if not opt.polys: + return S.Zero + else: + return Poly(0, opt=opt) + + result, polys = polys[0], polys[1:] + + for poly in polys: + result = result.gcd(poly) + + if result.is_one: + break + + if not opt.polys: + return result.as_expr() + else: + return result + + +@public +def gcd(f, g=None, *gens, **args): + """ + Compute GCD of ``f`` and ``g``. + + Examples + ======== + + >>> from sympy import gcd + >>> from sympy.abc import x + + >>> gcd(x**2 - 1, x**2 - 3*x + 2) + x - 1 + + """ + if hasattr(f, '__iter__'): + if g is not None: + gens = (g,) + gens + + return gcd_list(f, *gens, **args) + elif g is None: + raise TypeError("gcd() takes 2 arguments or a sequence of arguments") + + options.allowed_flags(args, ['polys']) + + try: + (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) + + # gcd for domain Q[irrational] (purely algebraic irrational) + a, b = map(sympify, (f, g)) + if a.is_algebraic and a.is_irrational and b.is_algebraic and b.is_irrational: + frc = (a/b).ratsimp() + if frc.is_rational: + # abs ensures that the returned gcd is always non-negative + return abs(a/frc.as_numer_denom()[0]) + + except PolificationFailed as exc: + domain, (a, b) = construct_domain(exc.exprs) + + try: + return domain.to_sympy(domain.gcd(a, b)) + except NotImplementedError: + raise ComputationFailed('gcd', 2, exc) + + result = F.gcd(G) + + if not opt.polys: + return result.as_expr() + else: + return result + + +@public +def lcm_list(seq, *gens, **args): + """ + Compute LCM of a list of polynomials. + + Examples + ======== + + >>> from sympy import lcm_list + >>> from sympy.abc import x + + >>> lcm_list([x**3 - 1, x**2 - 1, x**2 - 3*x + 2]) + x**5 - x**4 - 2*x**3 - x**2 + x + 2 + + """ + seq = sympify(seq) + + def try_non_polynomial_lcm(seq) -> Optional[Expr]: + if not gens and not args: + domain, numbers = construct_domain(seq) + + if not numbers: + return domain.to_sympy(domain.one) + elif domain.is_Numerical: + result, numbers = numbers[0], numbers[1:] + + for number in numbers: + result = domain.lcm(result, number) + + return domain.to_sympy(result) + + return None + + result = try_non_polynomial_lcm(seq) + + if result is not None: + return result + + options.allowed_flags(args, ['polys']) + + try: + polys, opt = parallel_poly_from_expr(seq, *gens, **args) + + # lcm for domain Q[irrational] (purely algebraic irrational) + if len(seq) > 1 and all(elt.is_algebraic and elt.is_irrational for elt in seq): + a = seq[-1] + lst = [ (a/elt).ratsimp() for elt in seq[:-1] ] + if all(frc.is_rational for frc in lst): + lc = 1 + for frc in lst: + lc = lcm(lc, frc.as_numer_denom()[1]) + return a*lc + + except PolificationFailed as exc: + result = try_non_polynomial_lcm(exc.exprs) + + if result is not None: + return result + else: + raise ComputationFailed('lcm_list', len(seq), exc) + + if not polys: + if not opt.polys: + return S.One + else: + return Poly(1, opt=opt) + + result, polys = polys[0], polys[1:] + + for poly in polys: + result = result.lcm(poly) + + if not opt.polys: + return result.as_expr() + else: + return result + + +@public +def lcm(f, g=None, *gens, **args): + """ + Compute LCM of ``f`` and ``g``. + + Examples + ======== + + >>> from sympy import lcm + >>> from sympy.abc import x + + >>> lcm(x**2 - 1, x**2 - 3*x + 2) + x**3 - 2*x**2 - x + 2 + + """ + if hasattr(f, '__iter__'): + if g is not None: + gens = (g,) + gens + + return lcm_list(f, *gens, **args) + elif g is None: + raise TypeError("lcm() takes 2 arguments or a sequence of arguments") + + options.allowed_flags(args, ['polys']) + + try: + (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) + + # lcm for domain Q[irrational] (purely algebraic irrational) + a, b = map(sympify, (f, g)) + if a.is_algebraic and a.is_irrational and b.is_algebraic and b.is_irrational: + frc = (a/b).ratsimp() + if frc.is_rational: + return a*frc.as_numer_denom()[1] + + except PolificationFailed as exc: + domain, (a, b) = construct_domain(exc.exprs) + + try: + return domain.to_sympy(domain.lcm(a, b)) + except NotImplementedError: + raise ComputationFailed('lcm', 2, exc) + + result = F.lcm(G) + + if not opt.polys: + return result.as_expr() + else: + return result + + +@public +def terms_gcd(f, *gens, **args): + """ + Remove GCD of terms from ``f``. + + If the ``deep`` flag is True, then the arguments of ``f`` will have + terms_gcd applied to them. + + If a fraction is factored out of ``f`` and ``f`` is an Add, then + an unevaluated Mul will be returned so that automatic simplification + does not redistribute it. The hint ``clear``, when set to False, can be + used to prevent such factoring when all coefficients are not fractions. + + Examples + ======== + + >>> from sympy import terms_gcd, cos + >>> from sympy.abc import x, y + >>> terms_gcd(x**6*y**2 + x**3*y, x, y) + x**3*y*(x**3*y + 1) + + The default action of polys routines is to expand the expression + given to them. terms_gcd follows this behavior: + + >>> terms_gcd((3+3*x)*(x+x*y)) + 3*x*(x*y + x + y + 1) + + If this is not desired then the hint ``expand`` can be set to False. + In this case the expression will be treated as though it were comprised + of one or more terms: + + >>> terms_gcd((3+3*x)*(x+x*y), expand=False) + (3*x + 3)*(x*y + x) + + In order to traverse factors of a Mul or the arguments of other + functions, the ``deep`` hint can be used: + + >>> terms_gcd((3 + 3*x)*(x + x*y), expand=False, deep=True) + 3*x*(x + 1)*(y + 1) + >>> terms_gcd(cos(x + x*y), deep=True) + cos(x*(y + 1)) + + Rationals are factored out by default: + + >>> terms_gcd(x + y/2) + (2*x + y)/2 + + Only the y-term had a coefficient that was a fraction; if one + does not want to factor out the 1/2 in cases like this, the + flag ``clear`` can be set to False: + + >>> terms_gcd(x + y/2, clear=False) + x + y/2 + >>> terms_gcd(x*y/2 + y**2, clear=False) + y*(x/2 + y) + + The ``clear`` flag is ignored if all coefficients are fractions: + + >>> terms_gcd(x/3 + y/2, clear=False) + (2*x + 3*y)/6 + + See Also + ======== + sympy.core.exprtools.gcd_terms, sympy.core.exprtools.factor_terms + + """ + + orig = sympify(f) + + if isinstance(f, Equality): + return Equality(*(terms_gcd(s, *gens, **args) for s in [f.lhs, f.rhs])) + elif isinstance(f, Relational): + raise TypeError("Inequalities cannot be used with terms_gcd. Found: %s" %(f,)) + + if not isinstance(f, Expr) or f.is_Atom: + return orig + + if args.get('deep', False): + new = f.func(*[terms_gcd(a, *gens, **args) for a in f.args]) + args.pop('deep') + args['expand'] = False + return terms_gcd(new, *gens, **args) + + clear = args.pop('clear', True) + options.allowed_flags(args, ['polys']) + + try: + F, opt = poly_from_expr(f, *gens, **args) + except PolificationFailed as exc: + return exc.expr + + J, f = F.terms_gcd() + + if opt.domain.is_Ring: + if opt.domain.is_Field: + denom, f = f.clear_denoms(convert=True) + + coeff, f = f.primitive() + + if opt.domain.is_Field: + coeff /= denom + else: + coeff = S.One + + term = Mul(*[x**j for x, j in zip(f.gens, J)]) + if equal_valued(coeff, 1): + coeff = S.One + if term == 1: + return orig + + if clear: + return _keep_coeff(coeff, term*f.as_expr()) + # base the clearing on the form of the original expression, not + # the (perhaps) Mul that we have now + coeff, f = _keep_coeff(coeff, f.as_expr(), clear=False).as_coeff_Mul() + return _keep_coeff(coeff, term*f, clear=False) + + +@public +def trunc(f, p, *gens, **args): + """ + Reduce ``f`` modulo a constant ``p``. + + Examples + ======== + + >>> from sympy import trunc + >>> from sympy.abc import x + + >>> trunc(2*x**3 + 3*x**2 + 5*x + 7, 3) + -x**3 - x + 1 + + """ + options.allowed_flags(args, ['auto', 'polys']) + + try: + F, opt = poly_from_expr(f, *gens, **args) + except PolificationFailed as exc: + raise ComputationFailed('trunc', 1, exc) + + result = F.trunc(sympify(p)) + + if not opt.polys: + return result.as_expr() + else: + return result + + +@public +def monic(f, *gens, **args): + """ + Divide all coefficients of ``f`` by ``LC(f)``. + + Examples + ======== + + >>> from sympy import monic + >>> from sympy.abc import x + + >>> monic(3*x**2 + 4*x + 2) + x**2 + 4*x/3 + 2/3 + + """ + options.allowed_flags(args, ['auto', 'polys']) + + try: + F, opt = poly_from_expr(f, *gens, **args) + except PolificationFailed as exc: + raise ComputationFailed('monic', 1, exc) + + result = F.monic(auto=opt.auto) + + if not opt.polys: + return result.as_expr() + else: + return result + + +@public +def content(f, *gens, **args): + """ + Compute GCD of coefficients of ``f``. + + Examples + ======== + + >>> from sympy import content + >>> from sympy.abc import x + + >>> content(6*x**2 + 8*x + 12) + 2 + + """ + options.allowed_flags(args, ['polys']) + + try: + F, opt = poly_from_expr(f, *gens, **args) + except PolificationFailed as exc: + raise ComputationFailed('content', 1, exc) + + return F.content() + + +@public +def primitive(f, *gens, **args): + """ + Compute content and the primitive form of ``f``. + + Examples + ======== + + >>> from sympy.polys.polytools import primitive + >>> from sympy.abc import x + + >>> primitive(6*x**2 + 8*x + 12) + (2, 3*x**2 + 4*x + 6) + + >>> eq = (2 + 2*x)*x + 2 + + Expansion is performed by default: + + >>> primitive(eq) + (2, x**2 + x + 1) + + Set ``expand`` to False to shut this off. Note that the + extraction will not be recursive; use the as_content_primitive method + for recursive, non-destructive Rational extraction. + + >>> primitive(eq, expand=False) + (1, x*(2*x + 2) + 2) + + >>> eq.as_content_primitive() + (2, x*(x + 1) + 1) + + """ + options.allowed_flags(args, ['polys']) + + try: + F, opt = poly_from_expr(f, *gens, **args) + except PolificationFailed as exc: + raise ComputationFailed('primitive', 1, exc) + + cont, result = F.primitive() + if not opt.polys: + return cont, result.as_expr() + else: + return cont, result + + +@public +def compose(f, g, *gens, **args): + """ + Compute functional composition ``f(g)``. + + Examples + ======== + + >>> from sympy import compose + >>> from sympy.abc import x + + >>> compose(x**2 + x, x - 1) + x**2 - x + + """ + options.allowed_flags(args, ['polys']) + + try: + (F, G), opt = parallel_poly_from_expr((f, g), *gens, **args) + except PolificationFailed as exc: + raise ComputationFailed('compose', 2, exc) + + result = F.compose(G) + + if not opt.polys: + return result.as_expr() + else: + return result + + +@public +def decompose(f, *gens, **args): + """ + Compute functional decomposition of ``f``. + + Examples + ======== + + >>> from sympy import decompose + >>> from sympy.abc import x + + >>> decompose(x**4 + 2*x**3 - x - 1) + [x**2 - x - 1, x**2 + x] + + """ + options.allowed_flags(args, ['polys']) + + try: + F, opt = poly_from_expr(f, *gens, **args) + except PolificationFailed as exc: + raise ComputationFailed('decompose', 1, exc) + + result = F.decompose() + + if not opt.polys: + return [r.as_expr() for r in result] + else: + return result + + +@public +def sturm(f, *gens, **args): + """ + Compute Sturm sequence of ``f``. + + Examples + ======== + + >>> from sympy import sturm + >>> from sympy.abc import x + + >>> sturm(x**3 - 2*x**2 + x - 3) + [x**3 - 2*x**2 + x - 3, 3*x**2 - 4*x + 1, 2*x/9 + 25/9, -2079/4] + + """ + options.allowed_flags(args, ['auto', 'polys']) + + try: + F, opt = poly_from_expr(f, *gens, **args) + except PolificationFailed as exc: + raise ComputationFailed('sturm', 1, exc) + + result = F.sturm(auto=opt.auto) + + if not opt.polys: + return [r.as_expr() for r in result] + else: + return result + + +@public +def gff_list(f, *gens, **args): + """ + Compute a list of greatest factorial factors of ``f``. + + Note that the input to ff() and rf() should be Poly instances to use the + definitions here. + + Examples + ======== + + >>> from sympy import gff_list, ff, Poly + >>> from sympy.abc import x + + >>> f = Poly(x**5 + 2*x**4 - x**3 - 2*x**2, x) + + >>> gff_list(f) + [(Poly(x, x, domain='ZZ'), 1), (Poly(x + 2, x, domain='ZZ'), 4)] + + >>> (ff(Poly(x), 1)*ff(Poly(x + 2), 4)) == f + True + + >>> f = Poly(x**12 + 6*x**11 - 11*x**10 - 56*x**9 + 220*x**8 + 208*x**7 - \ + 1401*x**6 + 1090*x**5 + 2715*x**4 - 6720*x**3 - 1092*x**2 + 5040*x, x) + + >>> gff_list(f) + [(Poly(x**3 + 7, x, domain='ZZ'), 2), (Poly(x**2 + 5*x, x, domain='ZZ'), 3)] + + >>> ff(Poly(x**3 + 7, x), 2)*ff(Poly(x**2 + 5*x, x), 3) == f + True + + """ + options.allowed_flags(args, ['polys']) + + try: + F, opt = poly_from_expr(f, *gens, **args) + except PolificationFailed as exc: + raise ComputationFailed('gff_list', 1, exc) + + factors = F.gff_list() + + if not opt.polys: + return [(g.as_expr(), k) for g, k in factors] + else: + return factors + + +@public +def gff(f, *gens, **args): + """Compute greatest factorial factorization of ``f``. """ + raise NotImplementedError('symbolic falling factorial') + + +@public +def sqf_norm(f, *gens, **args): + """ + Compute square-free norm of ``f``. + + Returns ``s``, ``f``, ``r``, such that ``g(x) = f(x-sa)`` and + ``r(x) = Norm(g(x))`` is a square-free polynomial over ``K``, + where ``a`` is the algebraic extension of the ground domain. + + Examples + ======== + + >>> from sympy import sqf_norm, sqrt + >>> from sympy.abc import x + + >>> sqf_norm(x**2 + 1, extension=[sqrt(3)]) + ([1], x**2 - 2*sqrt(3)*x + 4, x**4 - 4*x**2 + 16) + + """ + options.allowed_flags(args, ['polys']) + + try: + F, opt = poly_from_expr(f, *gens, **args) + except PolificationFailed as exc: + raise ComputationFailed('sqf_norm', 1, exc) + + s, g, r = F.sqf_norm() + + s_expr = [Integer(si) for si in s] + + if not opt.polys: + return s_expr, g.as_expr(), r.as_expr() + else: + return s_expr, g, r + + +@public +def sqf_part(f, *gens, **args): + """ + Compute square-free part of ``f``. + + Examples + ======== + + >>> from sympy import sqf_part + >>> from sympy.abc import x + + >>> sqf_part(x**3 - 3*x - 2) + x**2 - x - 2 + + """ + options.allowed_flags(args, ['polys']) + + try: + F, opt = poly_from_expr(f, *gens, **args) + except PolificationFailed as exc: + raise ComputationFailed('sqf_part', 1, exc) + + result = F.sqf_part() + + if not opt.polys: + return result.as_expr() + else: + return result + + +def _poly_sort_key(poly): + """Sort a list of polys.""" + rep = poly.rep.to_list() + return (len(rep), len(poly.gens), str(poly.domain), rep) + + +def _sorted_factors(factors, method): + """Sort a list of ``(expr, exp)`` pairs. """ + if method == 'sqf': + def key(obj): + poly, exp = obj + rep = poly.rep.to_list() + return (exp, len(rep), len(poly.gens), str(poly.domain), rep) + else: + def key(obj): + poly, exp = obj + rep = poly.rep.to_list() + return (len(rep), len(poly.gens), exp, str(poly.domain), rep) + + return sorted(factors, key=key) + + +def _factors_product(factors): + """Multiply a list of ``(expr, exp)`` pairs. """ + return Mul(*[f.as_expr()**k for f, k in factors]) + + +def _symbolic_factor_list(expr, opt, method): + """Helper function for :func:`_symbolic_factor`. """ + coeff, factors = S.One, [] + + args = [i._eval_factor() if hasattr(i, '_eval_factor') else i + for i in Mul.make_args(expr)] + for arg in args: + if arg.is_Number or (isinstance(arg, Expr) and pure_complex(arg)): + coeff *= arg + continue + elif arg.is_Pow and arg.base != S.Exp1: + base, exp = arg.args + if base.is_Number and exp.is_Number: + coeff *= arg + continue + if base.is_Number: + factors.append((base, exp)) + continue + else: + base, exp = arg, S.One + + try: + poly, _ = _poly_from_expr(base, opt) + except PolificationFailed as exc: + factors.append((exc.expr, exp)) + else: + func = getattr(poly, method + '_list') + + _coeff, _factors = func() + if _coeff is not S.One: + if exp.is_Integer: + coeff *= _coeff**exp + elif _coeff.is_positive: + factors.append((_coeff, exp)) + else: + _factors.append((_coeff, S.One)) + + if exp is S.One: + factors.extend(_factors) + elif exp.is_integer: + factors.extend([(f, k*exp) for f, k in _factors]) + else: + other = [] + + for f, k in _factors: + if f.as_expr().is_positive: + factors.append((f, k*exp)) + else: + other.append((f, k)) + + factors.append((_factors_product(other), exp)) + if method == 'sqf': + factors = [(reduce(mul, (f for f, _ in factors if _ == k)), k) + for k in {i for _, i in factors}] + #collect duplicates + rv = defaultdict(int) + for k, v in factors: + rv[k] += v + return coeff, list(rv.items()) + + +def _symbolic_factor(expr, opt, method): + """Helper function for :func:`_factor`. """ + if isinstance(expr, Expr): + if hasattr(expr,'_eval_factor'): + return expr._eval_factor() + coeff, factors = _symbolic_factor_list(together(expr, fraction=opt['fraction']), opt, method) + return _keep_coeff(coeff, _factors_product(factors)) + elif hasattr(expr, 'args'): + return expr.func(*[_symbolic_factor(arg, opt, method) for arg in expr.args]) + elif hasattr(expr, '__iter__'): + return expr.__class__([_symbolic_factor(arg, opt, method) for arg in expr]) + else: + return expr + + +def _generic_factor_list(expr, gens, args, method): + """Helper function for :func:`sqf_list` and :func:`factor_list`. """ + options.allowed_flags(args, ['frac', 'polys']) + opt = options.build_options(gens, args) + + expr = sympify(expr) + + if isinstance(expr, (Expr, Poly)): + if isinstance(expr, Poly): + numer, denom = expr, 1 + else: + numer, denom = together(expr).as_numer_denom() + + cp, fp = _symbolic_factor_list(numer, opt, method) + cq, fq = _symbolic_factor_list(denom, opt, method) + + if fq and not opt.frac: + raise PolynomialError("a polynomial expected, got %s" % expr) + + _opt = opt.clone({"expand": True}) + + for factors in (fp, fq): + for i, (f, k) in enumerate(factors): + if not f.is_Poly: + f, _ = _poly_from_expr(f, _opt) + factors[i] = (f, k) + + fp = _sorted_factors(fp, method) + fq = _sorted_factors(fq, method) + + if not opt.polys: + fp = [(f.as_expr(), k) for f, k in fp] + fq = [(f.as_expr(), k) for f, k in fq] + + coeff = cp/cq + + if not opt.frac: + return coeff, fp + else: + return coeff, fp, fq + else: + raise PolynomialError("a polynomial expected, got %s" % expr) + + +def _generic_factor(expr, gens, args, method): + """Helper function for :func:`sqf` and :func:`factor`. """ + fraction = args.pop('fraction', True) + options.allowed_flags(args, []) + opt = options.build_options(gens, args) + opt['fraction'] = fraction + return _symbolic_factor(sympify(expr), opt, method) + + +def to_rational_coeffs(f): + """ + try to transform a polynomial to have rational coefficients + + try to find a transformation ``x = alpha*y`` + + ``f(x) = lc*alpha**n * g(y)`` where ``g`` is a polynomial with + rational coefficients, ``lc`` the leading coefficient. + + If this fails, try ``x = y + beta`` + ``f(x) = g(y)`` + + Returns ``None`` if ``g`` not found; + ``(lc, alpha, None, g)`` in case of rescaling + ``(None, None, beta, g)`` in case of translation + + Notes + ===== + + Currently it transforms only polynomials without roots larger than 2. + + Examples + ======== + + >>> from sympy import sqrt, Poly, simplify + >>> from sympy.polys.polytools import to_rational_coeffs + >>> from sympy.abc import x + >>> p = Poly(((x**2-1)*(x-2)).subs({x:x*(1 + sqrt(2))}), x, domain='EX') + >>> lc, r, _, g = to_rational_coeffs(p) + >>> lc, r + (7 + 5*sqrt(2), 2 - 2*sqrt(2)) + >>> g + Poly(x**3 + x**2 - 1/4*x - 1/4, x, domain='QQ') + >>> r1 = simplify(1/r) + >>> Poly(lc*r**3*(g.as_expr()).subs({x:x*r1}), x, domain='EX') == p + True + + """ + from sympy.simplify.simplify import simplify + + def _try_rescale(f, f1=None): + """ + try rescaling ``x -> alpha*x`` to convert f to a polynomial + with rational coefficients. + Returns ``alpha, f``; if the rescaling is successful, + ``alpha`` is the rescaling factor, and ``f`` is the rescaled + polynomial; else ``alpha`` is ``None``. + """ + if not len(f.gens) == 1 or not (f.gens[0]).is_Atom: + return None, f + n = f.degree() + lc = f.LC() + f1 = f1 or f1.monic() + coeffs = f1.all_coeffs()[1:] + coeffs = [simplify(coeffx) for coeffx in coeffs] + if len(coeffs) > 1 and coeffs[-2]: + rescale1_x = simplify(coeffs[-2]/coeffs[-1]) + coeffs1 = [] + for i in range(len(coeffs)): + coeffx = simplify(coeffs[i]*rescale1_x**(i + 1)) + if not coeffx.is_rational: + break + coeffs1.append(coeffx) + else: + rescale_x = simplify(1/rescale1_x) + x = f.gens[0] + v = [x**n] + for i in range(1, n + 1): + v.append(coeffs1[i - 1]*x**(n - i)) + f = Add(*v) + f = Poly(f) + return lc, rescale_x, f + return None + + def _try_translate(f, f1=None): + """ + try translating ``x -> x + alpha`` to convert f to a polynomial + with rational coefficients. + Returns ``alpha, f``; if the translating is successful, + ``alpha`` is the translating factor, and ``f`` is the shifted + polynomial; else ``alpha`` is ``None``. + """ + if not len(f.gens) == 1 or not (f.gens[0]).is_Atom: + return None, f + n = f.degree() + f1 = f1 or f1.monic() + coeffs = f1.all_coeffs()[1:] + c = simplify(coeffs[0]) + if c.is_Add and not c.is_rational: + rat, nonrat = sift(c.args, + lambda z: z.is_rational is True, binary=True) + alpha = -c.func(*nonrat)/n + f2 = f1.shift(alpha) + return alpha, f2 + return None + + def _has_square_roots(p): + """ + Return True if ``f`` is a sum with square roots but no other root + """ + coeffs = p.coeffs() + has_sq = False + for y in coeffs: + for x in Add.make_args(y): + f = Factors(x).factors + r = [wx.q for b, wx in f.items() if + b.is_number and wx.is_Rational and wx.q >= 2] + if not r: + continue + if min(r) == 2: + has_sq = True + if max(r) > 2: + return False + return has_sq + + if f.get_domain().is_EX and _has_square_roots(f): + f1 = f.monic() + r = _try_rescale(f, f1) + if r: + return r[0], r[1], None, r[2] + else: + r = _try_translate(f, f1) + if r: + return None, None, r[0], r[1] + return None + + +def _torational_factor_list(p, x): + """ + helper function to factor polynomial using to_rational_coeffs + + Examples + ======== + + >>> from sympy.polys.polytools import _torational_factor_list + >>> from sympy.abc import x + >>> from sympy import sqrt, expand, Mul + >>> p = expand(((x**2-1)*(x-2)).subs({x:x*(1 + sqrt(2))})) + >>> factors = _torational_factor_list(p, x); factors + (-2, [(-x*(1 + sqrt(2))/2 + 1, 1), (-x*(1 + sqrt(2)) - 1, 1), (-x*(1 + sqrt(2)) + 1, 1)]) + >>> expand(factors[0]*Mul(*[z[0] for z in factors[1]])) == p + True + >>> p = expand(((x**2-1)*(x-2)).subs({x:x + sqrt(2)})) + >>> factors = _torational_factor_list(p, x); factors + (1, [(x - 2 + sqrt(2), 1), (x - 1 + sqrt(2), 1), (x + 1 + sqrt(2), 1)]) + >>> expand(factors[0]*Mul(*[z[0] for z in factors[1]])) == p + True + + """ + from sympy.simplify.simplify import simplify + p1 = Poly(p, x, domain='EX') + n = p1.degree() + res = to_rational_coeffs(p1) + if not res: + return None + lc, r, t, g = res + factors = factor_list(g.as_expr()) + if lc: + c = simplify(factors[0]*lc*r**n) + r1 = simplify(1/r) + a = [] + for z in factors[1:][0]: + a.append((simplify(z[0].subs({x: x*r1})), z[1])) + else: + c = factors[0] + a = [] + for z in factors[1:][0]: + a.append((z[0].subs({x: x - t}), z[1])) + return (c, a) + + +@public +def sqf_list(f, *gens, **args): + """ + Compute a list of square-free factors of ``f``. + + Examples + ======== + + >>> from sympy import sqf_list + >>> from sympy.abc import x + + >>> sqf_list(2*x**5 + 16*x**4 + 50*x**3 + 76*x**2 + 56*x + 16) + (2, [(x + 1, 2), (x + 2, 3)]) + + """ + return _generic_factor_list(f, gens, args, method='sqf') + + +@public +def sqf(f, *gens, **args): + """ + Compute square-free factorization of ``f``. + + Examples + ======== + + >>> from sympy import sqf + >>> from sympy.abc import x + + >>> sqf(2*x**5 + 16*x**4 + 50*x**3 + 76*x**2 + 56*x + 16) + 2*(x + 1)**2*(x + 2)**3 + + """ + return _generic_factor(f, gens, args, method='sqf') + + +@public +def factor_list(f, *gens, **args): + """ + Compute a list of irreducible factors of ``f``. + + Examples + ======== + + >>> from sympy import factor_list + >>> from sympy.abc import x, y + + >>> factor_list(2*x**5 + 2*x**4*y + 4*x**3 + 4*x**2*y + 2*x + 2*y) + (2, [(x + y, 1), (x**2 + 1, 2)]) + + """ + return _generic_factor_list(f, gens, args, method='factor') + + +@public +def factor(f, *gens, deep=False, **args): + """ + Compute the factorization of expression, ``f``, into irreducibles. (To + factor an integer into primes, use ``factorint``.) + + There two modes implemented: symbolic and formal. If ``f`` is not an + instance of :class:`Poly` and generators are not specified, then the + former mode is used. Otherwise, the formal mode is used. + + In symbolic mode, :func:`factor` will traverse the expression tree and + factor its components without any prior expansion, unless an instance + of :class:`~.Add` is encountered (in this case formal factorization is + used). This way :func:`factor` can handle large or symbolic exponents. + + By default, the factorization is computed over the rationals. To factor + over other domain, e.g. an algebraic or finite field, use appropriate + options: ``extension``, ``modulus`` or ``domain``. + + Examples + ======== + + >>> from sympy import factor, sqrt, exp + >>> from sympy.abc import x, y + + >>> factor(2*x**5 + 2*x**4*y + 4*x**3 + 4*x**2*y + 2*x + 2*y) + 2*(x + y)*(x**2 + 1)**2 + + >>> factor(x**2 + 1) + x**2 + 1 + >>> factor(x**2 + 1, modulus=2) + (x + 1)**2 + >>> factor(x**2 + 1, gaussian=True) + (x - I)*(x + I) + + >>> factor(x**2 - 2, extension=sqrt(2)) + (x - sqrt(2))*(x + sqrt(2)) + + >>> factor((x**2 - 1)/(x**2 + 4*x + 4)) + (x - 1)*(x + 1)/(x + 2)**2 + >>> factor((x**2 + 4*x + 4)**10000000*(x**2 + 1)) + (x + 2)**20000000*(x**2 + 1) + + By default, factor deals with an expression as a whole: + + >>> eq = 2**(x**2 + 2*x + 1) + >>> factor(eq) + 2**(x**2 + 2*x + 1) + + If the ``deep`` flag is True then subexpressions will + be factored: + + >>> factor(eq, deep=True) + 2**((x + 1)**2) + + If the ``fraction`` flag is False then rational expressions + will not be combined. By default it is True. + + >>> factor(5*x + 3*exp(2 - 7*x), deep=True) + (5*x*exp(7*x) + 3*exp(2))*exp(-7*x) + >>> factor(5*x + 3*exp(2 - 7*x), deep=True, fraction=False) + 5*x + 3*exp(2)*exp(-7*x) + + See Also + ======== + sympy.ntheory.factor_.factorint + + """ + f = sympify(f) + if deep: + def _try_factor(expr): + """ + Factor, but avoid changing the expression when unable to. + """ + fac = factor(expr, *gens, **args) + if fac.is_Mul or fac.is_Pow: + return fac + return expr + + f = bottom_up(f, _try_factor) + # clean up any subexpressions that may have been expanded + # while factoring out a larger expression + partials = {} + muladd = f.atoms(Mul, Add) + for p in muladd: + fac = factor(p, *gens, **args) + if (fac.is_Mul or fac.is_Pow) and fac != p: + partials[p] = fac + return f.xreplace(partials) + + try: + return _generic_factor(f, gens, args, method='factor') + except PolynomialError: + if not f.is_commutative: + return factor_nc(f) + else: + raise + + +@public +def intervals(F, all=False, eps=None, inf=None, sup=None, strict=False, fast=False, sqf=False): + """ + Compute isolating intervals for roots of ``f``. + + Examples + ======== + + >>> from sympy import intervals + >>> from sympy.abc import x + + >>> intervals(x**2 - 3) + [((-2, -1), 1), ((1, 2), 1)] + >>> intervals(x**2 - 3, eps=1e-2) + [((-26/15, -19/11), 1), ((19/11, 26/15), 1)] + + """ + if not hasattr(F, '__iter__'): + try: + F = Poly(F) + except GeneratorsNeeded: + return [] + + return F.intervals(all=all, eps=eps, inf=inf, sup=sup, fast=fast, sqf=sqf) + else: + polys, opt = parallel_poly_from_expr(F, domain='QQ') + + if len(opt.gens) > 1: + raise MultivariatePolynomialError + + for i, poly in enumerate(polys): + polys[i] = poly.rep.to_list() + + if eps is not None: + eps = opt.domain.convert(eps) + + if eps <= 0: + raise ValueError("'eps' must be a positive rational") + + if inf is not None: + inf = opt.domain.convert(inf) + if sup is not None: + sup = opt.domain.convert(sup) + + intervals = dup_isolate_real_roots_list(polys, opt.domain, + eps=eps, inf=inf, sup=sup, strict=strict, fast=fast) + + result = [] + + for (s, t), indices in intervals: + s, t = opt.domain.to_sympy(s), opt.domain.to_sympy(t) + result.append(((s, t), indices)) + + return result + + +@public +def refine_root(f, s, t, eps=None, steps=None, fast=False, check_sqf=False): + """ + Refine an isolating interval of a root to the given precision. + + Examples + ======== + + >>> from sympy import refine_root + >>> from sympy.abc import x + + >>> refine_root(x**2 - 3, 1, 2, eps=1e-2) + (19/11, 26/15) + + """ + try: + F = Poly(f) + if not isinstance(f, Poly) and not F.gen.is_Symbol: + # root of sin(x) + 1 is -1 but when someone + # passes an Expr instead of Poly they may not expect + # that the generator will be sin(x), not x + raise PolynomialError("generator must be a Symbol") + except GeneratorsNeeded: + raise PolynomialError( + "Cannot refine a root of %s, not a polynomial" % f) + + return F.refine_root(s, t, eps=eps, steps=steps, fast=fast, check_sqf=check_sqf) + + +@public +def count_roots(f, inf=None, sup=None): + """ + Return the number of roots of ``f`` in ``[inf, sup]`` interval. + + If one of ``inf`` or ``sup`` is complex, it will return the number of roots + in the complex rectangle with corners at ``inf`` and ``sup``. + + Examples + ======== + + >>> from sympy import count_roots, I + >>> from sympy.abc import x + + >>> count_roots(x**4 - 4, -3, 3) + 2 + >>> count_roots(x**4 - 4, 0, 1 + 3*I) + 1 + + """ + try: + F = Poly(f, greedy=False) + if not isinstance(f, Poly) and not F.gen.is_Symbol: + # root of sin(x) + 1 is -1 but when someone + # passes an Expr instead of Poly they may not expect + # that the generator will be sin(x), not x + raise PolynomialError("generator must be a Symbol") + except GeneratorsNeeded: + raise PolynomialError("Cannot count roots of %s, not a polynomial" % f) + + return F.count_roots(inf=inf, sup=sup) + + +@public +def all_roots(f, multiple=True, radicals=True, extension=False): + """ + Returns the real and complex roots of ``f`` with multiplicities. + + Explanation + =========== + + Finds all real and complex roots of a univariate polynomial with rational + coefficients of any degree exactly. The roots are represented in the form + given by :func:`~.rootof`. This is equivalent to using :func:`~.rootof` to + find each of the indexed roots. + + Examples + ======== + + >>> from sympy import all_roots + >>> from sympy.abc import x, y + + >>> print(all_roots(x**3 + 1)) + [-1, 1/2 - sqrt(3)*I/2, 1/2 + sqrt(3)*I/2] + + Simple radical formulae are used in some cases but the cubic and quartic + formulae are avoided. Instead most non-rational roots will be represented + as :class:`~.ComplexRootOf`: + + >>> print(all_roots(x**3 + x + 1)) + [CRootOf(x**3 + x + 1, 0), CRootOf(x**3 + x + 1, 1), CRootOf(x**3 + x + 1, 2)] + + All roots of any polynomial with rational coefficients of any degree can be + represented using :py:class:`~.ComplexRootOf`. The use of + :py:class:`~.ComplexRootOf` bypasses limitations on the availability of + radical formulae for quintic and higher degree polynomials _[1]: + + >>> p = x**5 - x - 1 + >>> for r in all_roots(p): print(r) + CRootOf(x**5 - x - 1, 0) + CRootOf(x**5 - x - 1, 1) + CRootOf(x**5 - x - 1, 2) + CRootOf(x**5 - x - 1, 3) + CRootOf(x**5 - x - 1, 4) + >>> [r.evalf(3) for r in all_roots(p)] + [1.17, -0.765 - 0.352*I, -0.765 + 0.352*I, 0.181 - 1.08*I, 0.181 + 1.08*I] + + Irrational algebraic coefficients are handled by :func:`all_roots` + if `extension=True` is set. + + >>> from sympy import sqrt, expand + >>> p = expand((x - sqrt(2))*(x - sqrt(3))) + >>> print(p) + x**2 - sqrt(3)*x - sqrt(2)*x + sqrt(6) + >>> all_roots(p) + Traceback (most recent call last): + ... + NotImplementedError: sorted roots not supported over EX + >>> all_roots(p, extension=True) + [sqrt(2), sqrt(3)] + + Algebraic coefficients can be complex as well. + + >>> from sympy import I + >>> all_roots(x**2 - I, extension=True) + [-sqrt(2)/2 - sqrt(2)*I/2, sqrt(2)/2 + sqrt(2)*I/2] + >>> all_roots(x**2 - sqrt(2)*I, extension=True) + [-2**(3/4)/2 - 2**(3/4)*I/2, 2**(3/4)/2 + 2**(3/4)*I/2] + + Transcendental coefficients cannot currently be handled by + :func:`all_roots`. In the case of algebraic or transcendental coefficients + :func:`~.ground_roots` might be able to find some roots by factorisation: + + >>> from sympy import ground_roots + >>> ground_roots(p, x, extension=True) + {sqrt(2): 1, sqrt(3): 1} + + If the coefficients are numeric then :func:`~.nroots` can be used to find + all roots approximately: + + >>> from sympy import nroots + >>> nroots(p, 5) + [1.4142, 1.732] + + If the coefficients are symbolic then :func:`sympy.polys.polyroots.roots` + or :func:`~.ground_roots` should be used instead: + + >>> from sympy import roots, ground_roots + >>> p = x**2 - 3*x*y + 2*y**2 + >>> roots(p, x) + {y: 1, 2*y: 1} + >>> ground_roots(p, x) + {y: 1, 2*y: 1} + + Parameters + ========== + + f : :class:`~.Expr` or :class:`~.Poly` + A univariate polynomial with rational (or ``Float``) coefficients. + multiple : ``bool`` (default ``True``). + Whether to return a ``list`` of roots or a list of root/multiplicity + pairs. + radicals : ``bool`` (default ``True``) + Use simple radical formulae rather than :py:class:`~.ComplexRootOf` for + some irrational roots. + extension: ``bool`` (default ``False``) + Whether to construct an algebraic extension domain before computing + the roots. Setting to ``True`` is necessary for finding roots of a + polynomial with (irrational) algebraic coefficients but can be slow. + + Returns + ======= + + A list of :class:`~.Expr` (usually :class:`~.ComplexRootOf`) representing + the roots is returned with each root repeated according to its multiplicity + as a root of ``f``. The roots are always uniquely ordered with real roots + coming before complex roots. The real roots are in increasing order. + Complex roots are ordered by increasing real part and then increasing + imaginary part. + + If ``multiple=False`` is passed then a list of root/multiplicity pairs is + returned instead. + + If ``radicals=False`` is passed then all roots will be represented as + either rational numbers or :class:`~.ComplexRootOf`. + + See also + ======== + + Poly.all_roots: + The underlying :class:`Poly` method used by :func:`~.all_roots`. + rootof: + Compute a single numbered root of a univariate polynomial. + real_roots: + Compute all the real roots using :func:`~.rootof`. + ground_roots: + Compute some roots in the ground domain by factorisation. + nroots: + Compute all roots using approximate numerical techniques. + sympy.polys.polyroots.roots: + Compute symbolic expressions for roots using radical formulae. + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem + """ + try: + if isinstance(f, Poly): + if extension and not f.domain.is_AlgebraicField: + F = Poly(f.expr, extension=True) + else: + F = f + else: + if extension: + F = Poly(f, extension=True) + else: + F = Poly(f, greedy=False) + + if not isinstance(f, Poly) and not F.gen.is_Symbol: + # root of sin(x) + 1 is -1 but when someone + # passes an Expr instead of Poly they may not expect + # that the generator will be sin(x), not x + raise PolynomialError("generator must be a Symbol") + except GeneratorsNeeded: + raise PolynomialError( + "Cannot compute real roots of %s, not a polynomial" % f) + + return F.all_roots(multiple=multiple, radicals=radicals) + + +@public +def real_roots(f, multiple=True, radicals=True, extension=False): + """ + Returns the real roots of ``f`` with multiplicities. + + Explanation + =========== + + Finds all real roots of a univariate polynomial with rational coefficients + of any degree exactly. The roots are represented in the form given by + :func:`~.rootof`. This is equivalent to using :func:`~.rootof` or + :func:`~.all_roots` and filtering out only the real roots. However if only + the real roots are needed then :func:`real_roots` is more efficient than + :func:`~.all_roots` because it computes only the real roots and avoids + costly complex root isolation routines. + + Examples + ======== + + >>> from sympy import real_roots + >>> from sympy.abc import x, y + + >>> real_roots(2*x**3 - 7*x**2 + 4*x + 4) + [-1/2, 2, 2] + >>> real_roots(2*x**3 - 7*x**2 + 4*x + 4, multiple=False) + [(-1/2, 1), (2, 2)] + + Real roots of any polynomial with rational coefficients of any degree can + be represented using :py:class:`~.ComplexRootOf`: + + >>> p = x**9 + 2*x + 2 + >>> print(real_roots(p)) + [CRootOf(x**9 + 2*x + 2, 0)] + >>> [r.evalf(3) for r in real_roots(p)] + [-0.865] + + All rational roots will be returned as rational numbers. Roots of some + simple factors will be expressed using radical or other formulae (unless + ``radicals=False`` is passed). All other roots will be expressed as + :class:`~.ComplexRootOf`. + + >>> p = (x + 7)*(x**2 - 2)*(x**3 + x + 1) + >>> print(real_roots(p)) + [-7, -sqrt(2), CRootOf(x**3 + x + 1, 0), sqrt(2)] + >>> print(real_roots(p, radicals=False)) + [-7, CRootOf(x**2 - 2, 0), CRootOf(x**3 + x + 1, 0), CRootOf(x**2 - 2, 1)] + + All returned root expressions will numerically evaluate to real numbers + with no imaginary part. This is in contrast to the expressions generated by + the cubic or quartic formulae as used by :func:`~.roots` which suffer from + casus irreducibilis [1]_: + + >>> from sympy import roots + >>> p = 2*x**3 - 9*x**2 - 6*x + 3 + >>> [r.evalf(5) for r in roots(p, multiple=True)] + [5.0365 - 0.e-11*I, 0.33984 + 0.e-13*I, -0.87636 + 0.e-10*I] + >>> [r.evalf(5) for r in real_roots(p, x)] + [-0.87636, 0.33984, 5.0365] + >>> [r.is_real for r in roots(p, multiple=True)] + [None, None, None] + >>> [r.is_real for r in real_roots(p)] + [True, True, True] + + Using :func:`real_roots` is equivalent to using :func:`~.all_roots` (or + :func:`~.rootof`) and filtering out only the real roots: + + >>> from sympy import all_roots + >>> r = [r for r in all_roots(p) if r.is_real] + >>> real_roots(p) == r + True + + If only the real roots are wanted then using :func:`real_roots` is faster + than using :func:`~.all_roots`. Using :func:`real_roots` avoids complex root + isolation which can be a lot slower than real root isolation especially for + polynomials of high degree which typically have many more complex roots + than real roots. + + Irrational algebraic coefficients are handled by :func:`real_roots` + if `extension=True` is set. + + >>> from sympy import sqrt, expand + >>> p = expand((x - sqrt(2))*(x - sqrt(3))) + >>> print(p) + x**2 - sqrt(3)*x - sqrt(2)*x + sqrt(6) + >>> real_roots(p) + Traceback (most recent call last): + ... + NotImplementedError: sorted roots not supported over EX + >>> real_roots(p, extension=True) + [sqrt(2), sqrt(3)] + + Transcendental coefficients cannot currently be handled by + :func:`real_roots`. In the case of algebraic or transcendental coefficients + :func:`~.ground_roots` might be able to find some roots by factorisation: + + >>> from sympy import ground_roots + >>> ground_roots(p, x, extension=True) + {sqrt(2): 1, sqrt(3): 1} + + If the coefficients are numeric then :func:`~.nroots` can be used to find + all roots approximately: + + >>> from sympy import nroots + >>> nroots(p, 5) + [1.4142, 1.732] + + If the coefficients are symbolic then :func:`sympy.polys.polyroots.roots` + or :func:`~.ground_roots` should be used instead. + + >>> from sympy import roots, ground_roots + >>> p = x**2 - 3*x*y + 2*y**2 + >>> roots(p, x) + {y: 1, 2*y: 1} + >>> ground_roots(p, x) + {y: 1, 2*y: 1} + + Parameters + ========== + + f : :class:`~.Expr` or :class:`~.Poly` + A univariate polynomial with rational (or ``Float``) coefficients. + multiple : ``bool`` (default ``True``). + Whether to return a ``list`` of roots or a list of root/multiplicity + pairs. + radicals : ``bool`` (default ``True``) + Use simple radical formulae rather than :py:class:`~.ComplexRootOf` for + some irrational roots. + extension: ``bool`` (default ``False``) + Whether to construct an algebraic extension domain before computing + the roots. Setting to ``True`` is necessary for finding roots of a + polynomial with (irrational) algebraic coefficients but can be slow. + + Returns + ======= + + A list of :class:`~.Expr` (usually :class:`~.ComplexRootOf`) representing + the real roots is returned. The roots are arranged in increasing order and + are repeated according to their multiplicities as roots of ``f``. + + If ``multiple=False`` is passed then a list of root/multiplicity pairs is + returned instead. + + If ``radicals=False`` is passed then all roots will be represented as + either rational numbers or :class:`~.ComplexRootOf`. + + See also + ======== + + Poly.real_roots: + The underlying :class:`Poly` method used by :func:`real_roots`. + rootof: + Compute a single numbered root of a univariate polynomial. + all_roots: + Compute all real and non-real roots using :func:`~.rootof`. + ground_roots: + Compute some roots in the ground domain by factorisation. + nroots: + Compute all roots using approximate numerical techniques. + sympy.polys.polyroots.roots: + Compute symbolic expressions for roots using radical formulae. + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Casus_irreducibilis + """ + try: + if isinstance(f, Poly): + if extension and not f.domain.is_AlgebraicField: + F = Poly(f.expr, extension=True) + else: + F = f + else: + if extension: + F = Poly(f, extension=True) + else: + F = Poly(f, greedy=False) + + if not isinstance(f, Poly) and not F.gen.is_Symbol: + # root of sin(x) + 1 is -1 but when someone + # passes an Expr instead of Poly they may not expect + # that the generator will be sin(x), not x + raise PolynomialError("generator must be a Symbol") + except GeneratorsNeeded: + raise PolynomialError( + "Cannot compute real roots of %s, not a polynomial" % f) + + return F.real_roots(multiple=multiple, radicals=radicals) + + +@public +def nroots(f, n=15, maxsteps=50, cleanup=True): + """ + Compute numerical approximations of roots of ``f``. + + Examples + ======== + + >>> from sympy import nroots + >>> from sympy.abc import x + + >>> nroots(x**2 - 3, n=15) + [-1.73205080756888, 1.73205080756888] + >>> nroots(x**2 - 3, n=30) + [-1.73205080756887729352744634151, 1.73205080756887729352744634151] + + """ + try: + F = Poly(f, greedy=False) + if not isinstance(f, Poly) and not F.gen.is_Symbol: + # root of sin(x) + 1 is -1 but when someone + # passes an Expr instead of Poly they may not expect + # that the generator will be sin(x), not x + raise PolynomialError("generator must be a Symbol") + except GeneratorsNeeded: + raise PolynomialError( + "Cannot compute numerical roots of %s, not a polynomial" % f) + + return F.nroots(n=n, maxsteps=maxsteps, cleanup=cleanup) + + +@public +def ground_roots(f, *gens, **args): + """ + Compute roots of ``f`` by factorization in the ground domain. + + Examples + ======== + + >>> from sympy import ground_roots + >>> from sympy.abc import x + + >>> ground_roots(x**6 - 4*x**4 + 4*x**3 - x**2) + {0: 2, 1: 2} + + """ + options.allowed_flags(args, []) + + try: + F, opt = poly_from_expr(f, *gens, **args) + if not isinstance(f, Poly) and not F.gen.is_Symbol: + # root of sin(x) + 1 is -1 but when someone + # passes an Expr instead of Poly they may not expect + # that the generator will be sin(x), not x + raise PolynomialError("generator must be a Symbol") + except PolificationFailed as exc: + raise ComputationFailed('ground_roots', 1, exc) + + return F.ground_roots() + + +@public +def nth_power_roots_poly(f, n, *gens, **args): + """ + Construct a polynomial with n-th powers of roots of ``f``. + + Examples + ======== + + >>> from sympy import nth_power_roots_poly, factor, roots + >>> from sympy.abc import x + + >>> f = x**4 - x**2 + 1 + >>> g = factor(nth_power_roots_poly(f, 2)) + + >>> g + (x**2 - x + 1)**2 + + >>> R_f = [ (r**2).expand() for r in roots(f) ] + >>> R_g = roots(g).keys() + + >>> set(R_f) == set(R_g) + True + + """ + options.allowed_flags(args, []) + + try: + F, opt = poly_from_expr(f, *gens, **args) + if not isinstance(f, Poly) and not F.gen.is_Symbol: + # root of sin(x) + 1 is -1 but when someone + # passes an Expr instead of Poly they may not expect + # that the generator will be sin(x), not x + raise PolynomialError("generator must be a Symbol") + except PolificationFailed as exc: + raise ComputationFailed('nth_power_roots_poly', 1, exc) + + result = F.nth_power_roots_poly(n) + + if not opt.polys: + return result.as_expr() + else: + return result + + +@public +def cancel(f, *gens, _signsimp=True, **args): + """ + Cancel common factors in a rational function ``f``. + + Examples + ======== + + >>> from sympy import cancel, sqrt, Symbol, together + >>> from sympy.abc import x + >>> A = Symbol('A', commutative=False) + + >>> cancel((2*x**2 - 2)/(x**2 - 2*x + 1)) + (2*x + 2)/(x - 1) + >>> cancel((sqrt(3) + sqrt(15)*A)/(sqrt(2) + sqrt(10)*A)) + sqrt(6)/2 + + Note: due to automatic distribution of Rationals, a sum divided by an integer + will appear as a sum. To recover a rational form use `together` on the result: + + >>> cancel(x/2 + 1) + x/2 + 1 + >>> together(_) + (x + 2)/2 + """ + from sympy.simplify.simplify import signsimp + from sympy.polys.rings import sring + options.allowed_flags(args, ['polys']) + + f = sympify(f) + if _signsimp: + f = signsimp(f) + opt = {} + if 'polys' in args: + opt['polys'] = args['polys'] + + if not isinstance(f, Tuple): + if f.is_Number or isinstance(f, Relational) or not isinstance(f, Expr): + return f + f = factor_terms(f, radical=True) + p, q = f.as_numer_denom() + + elif len(f) == 2: + p, q = f + if isinstance(p, Poly) and isinstance(q, Poly): + opt['gens'] = p.gens + opt['domain'] = p.domain + opt['polys'] = opt.get('polys', True) + p, q = p.as_expr(), q.as_expr() + else: + raise ValueError('unexpected argument: %s' % f) + + from sympy.functions.elementary.piecewise import Piecewise + try: + if f.has(Piecewise): + raise PolynomialError() + R, (F, G) = sring((p, q), *gens, **args) + if not R.ngens: + if not isinstance(f, Tuple): + return f.expand() + else: + return S.One, p, q + except PolynomialError as msg: + if f.is_commutative and not f.has(Piecewise): + raise PolynomialError(msg) + # Handling of noncommutative and/or piecewise expressions + if f.is_Add or f.is_Mul: + c, nc = sift(f.args, lambda x: + x.is_commutative is True and not x.has(Piecewise), + binary=True) + nc = [cancel(i) for i in nc] + return f.func(cancel(f.func(*c)), *nc) + else: + reps = [] + pot = preorder_traversal(f) + next(pot) + for e in pot: + if isinstance(e, BooleanAtom) or not isinstance(e, Expr): + continue + try: + reps.append((e, cancel(e))) + pot.skip() # this was handled successfully + except NotImplementedError: + pass + return f.xreplace(dict(reps)) + + c, (P, Q) = 1, F.cancel(G) + if opt.get('polys', False) and 'gens' not in opt: + opt['gens'] = R.symbols + + if not isinstance(f, Tuple): + return c*(P.as_expr()/Q.as_expr()) + else: + P, Q = P.as_expr(), Q.as_expr() + if not opt.get('polys', False): + return c, P, Q + else: + return c, Poly(P, *gens, **opt), Poly(Q, *gens, **opt) + + +@public +def reduced(f, G, *gens, **args): + """ + Reduces a polynomial ``f`` modulo a set of polynomials ``G``. + + Given a polynomial ``f`` and a set of polynomials ``G = (g_1, ..., g_n)``, + computes a set of quotients ``q = (q_1, ..., q_n)`` and the remainder ``r`` + such that ``f = q_1*g_1 + ... + q_n*g_n + r``, where ``r`` vanishes or ``r`` + is a completely reduced polynomial with respect to ``G``. + + Examples + ======== + + >>> from sympy import reduced + >>> from sympy.abc import x, y + + >>> reduced(2*x**4 + y**2 - x**2 + y**3, [x**3 - x, y**3 - y]) + ([2*x, 1], x**2 + y**2 + y) + + """ + options.allowed_flags(args, ['polys', 'auto']) + + try: + polys, opt = parallel_poly_from_expr([f] + list(G), *gens, **args) + except PolificationFailed as exc: + raise ComputationFailed('reduced', 0, exc) + + domain = opt.domain + retract = False + + if opt.auto and domain.is_Ring and not domain.is_Field: + opt = opt.clone({"domain": domain.get_field()}) + retract = True + + from sympy.polys.rings import xring + _ring, _ = xring(opt.gens, opt.domain, opt.order) + + for i, poly in enumerate(polys): + poly = poly.set_domain(opt.domain).rep.to_dict() + polys[i] = _ring.from_dict(poly) + + Q, r = polys[0].div(polys[1:]) + + Q = [Poly._from_dict(dict(q), opt) for q in Q] + r = Poly._from_dict(dict(r), opt) + + if retract: + try: + _Q, _r = [q.to_ring() for q in Q], r.to_ring() + except CoercionFailed: + pass + else: + Q, r = _Q, _r + + if not opt.polys: + return [q.as_expr() for q in Q], r.as_expr() + else: + return Q, r + + +@public +def groebner(F, *gens, **args): + """ + Computes the reduced Groebner basis for a set of polynomials. + + Use the ``order`` argument to set the monomial ordering that will be + used to compute the basis. Allowed orders are ``lex``, ``grlex`` and + ``grevlex``. If no order is specified, it defaults to ``lex``. + + For more information on Groebner bases, see the references and the docstring + of :func:`~.solve_poly_system`. + + Examples + ======== + + Example taken from [1]. + + >>> from sympy import groebner + >>> from sympy.abc import x, y + + >>> F = [x*y - 2*y, 2*y**2 - x**2] + + >>> groebner(F, x, y, order='lex') + GroebnerBasis([x**2 - 2*y**2, x*y - 2*y, y**3 - 2*y], x, y, + domain='ZZ', order='lex') + >>> groebner(F, x, y, order='grlex') + GroebnerBasis([y**3 - 2*y, x**2 - 2*y**2, x*y - 2*y], x, y, + domain='ZZ', order='grlex') + >>> groebner(F, x, y, order='grevlex') + GroebnerBasis([y**3 - 2*y, x**2 - 2*y**2, x*y - 2*y], x, y, + domain='ZZ', order='grevlex') + + By default, an improved implementation of the Buchberger algorithm is + used. Optionally, an implementation of the F5B algorithm can be used. The + algorithm can be set using the ``method`` flag or with the + :func:`sympy.polys.polyconfig.setup` function. + + >>> F = [x**2 - x - 1, (2*x - 1) * y - (x**10 - (1 - x)**10)] + + >>> groebner(F, x, y, method='buchberger') + GroebnerBasis([x**2 - x - 1, y - 55], x, y, domain='ZZ', order='lex') + >>> groebner(F, x, y, method='f5b') + GroebnerBasis([x**2 - x - 1, y - 55], x, y, domain='ZZ', order='lex') + + References + ========== + + 1. [Buchberger01]_ + 2. [Cox97]_ + + """ + return GroebnerBasis(F, *gens, **args) + + +@public +def is_zero_dimensional(F, *gens, **args): + """ + Checks if the ideal generated by a Groebner basis is zero-dimensional. + + The algorithm checks if the set of monomials not divisible by the + leading monomial of any element of ``F`` is bounded. + + References + ========== + + David A. Cox, John B. Little, Donal O'Shea. Ideals, Varieties and + Algorithms, 3rd edition, p. 230 + + """ + return GroebnerBasis(F, *gens, **args).is_zero_dimensional + + +@public +class GroebnerBasis(Basic): + """Represents a reduced Groebner basis. """ + + def __new__(cls, F, *gens, **args): + """Compute a reduced Groebner basis for a system of polynomials. """ + options.allowed_flags(args, ['polys', 'method']) + + try: + polys, opt = parallel_poly_from_expr(F, *gens, **args) + except PolificationFailed as exc: + raise ComputationFailed('groebner', len(F), exc) + + from sympy.polys.rings import PolyRing + ring = PolyRing(opt.gens, opt.domain, opt.order) + + polys = [ring.from_dict(poly.rep.to_dict()) for poly in polys if poly] + + G = _groebner(polys, ring, method=opt.method) + G = [Poly._from_dict(g, opt) for g in G] + + return cls._new(G, opt) + + @classmethod + def _new(cls, basis, options): + obj = Basic.__new__(cls) + + obj._basis = tuple(basis) + obj._options = options + + return obj + + @property + def args(self): + basis = (p.as_expr() for p in self._basis) + return (Tuple(*basis), Tuple(*self._options.gens)) + + @property + def exprs(self): + return [poly.as_expr() for poly in self._basis] + + @property + def polys(self): + return list(self._basis) + + @property + def gens(self): + return self._options.gens + + @property + def domain(self): + return self._options.domain + + @property + def order(self): + return self._options.order + + def __len__(self): + return len(self._basis) + + def __iter__(self): + if self._options.polys: + return iter(self.polys) + else: + return iter(self.exprs) + + def __getitem__(self, item): + if self._options.polys: + basis = self.polys + else: + basis = self.exprs + + return basis[item] + + def __hash__(self): + return hash((self._basis, tuple(self._options.items()))) + + def __eq__(self, other): + if isinstance(other, self.__class__): + return self._basis == other._basis and self._options == other._options + elif iterable(other): + return self.polys == list(other) or self.exprs == list(other) + else: + return False + + def __ne__(self, other): + return not self == other + + @property + def is_zero_dimensional(self): + """ + Checks if the ideal generated by a Groebner basis is zero-dimensional. + + The algorithm checks if the set of monomials not divisible by the + leading monomial of any element of ``F`` is bounded. + + References + ========== + + David A. Cox, John B. Little, Donal O'Shea. Ideals, Varieties and + Algorithms, 3rd edition, p. 230 + + """ + def single_var(monomial): + return sum(map(bool, monomial)) == 1 + + exponents = Monomial([0]*len(self.gens)) + order = self._options.order + + for poly in self.polys: + monomial = poly.LM(order=order) + + if single_var(monomial): + exponents *= monomial + + # If any element of the exponents vector is zero, then there's + # a variable for which there's no degree bound and the ideal + # generated by this Groebner basis isn't zero-dimensional. + return all(exponents) + + def fglm(self, order): + """ + Convert a Groebner basis from one ordering to another. + + The FGLM algorithm converts reduced Groebner bases of zero-dimensional + ideals from one ordering to another. This method is often used when it + is infeasible to compute a Groebner basis with respect to a particular + ordering directly. + + Examples + ======== + + >>> from sympy.abc import x, y + >>> from sympy import groebner + + >>> F = [x**2 - 3*y - x + 1, y**2 - 2*x + y - 1] + >>> G = groebner(F, x, y, order='grlex') + + >>> list(G.fglm('lex')) + [2*x - y**2 - y + 1, y**4 + 2*y**3 - 3*y**2 - 16*y + 7] + >>> list(groebner(F, x, y, order='lex')) + [2*x - y**2 - y + 1, y**4 + 2*y**3 - 3*y**2 - 16*y + 7] + + References + ========== + + .. [1] J.C. Faugere, P. Gianni, D. Lazard, T. Mora (1994). Efficient + Computation of Zero-dimensional Groebner Bases by Change of + Ordering + + """ + opt = self._options + + src_order = opt.order + dst_order = monomial_key(order) + + if src_order == dst_order: + return self + + if not self.is_zero_dimensional: + raise NotImplementedError("Cannot convert Groebner bases of ideals with positive dimension") + + polys = list(self._basis) + domain = opt.domain + + opt = opt.clone({ + "domain": domain.get_field(), + "order": dst_order, + }) + + from sympy.polys.rings import xring + _ring, _ = xring(opt.gens, opt.domain, src_order) + + for i, poly in enumerate(polys): + poly = poly.set_domain(opt.domain).rep.to_dict() + polys[i] = _ring.from_dict(poly) + + G = matrix_fglm(polys, _ring, dst_order) + G = [Poly._from_dict(dict(g), opt) for g in G] + + if not domain.is_Field: + G = [g.clear_denoms(convert=True)[1] for g in G] + opt.domain = domain + + return self._new(G, opt) + + def reduce(self, expr, auto=True): + """ + Reduces a polynomial modulo a Groebner basis. + + Given a polynomial ``f`` and a set of polynomials ``G = (g_1, ..., g_n)``, + computes a set of quotients ``q = (q_1, ..., q_n)`` and the remainder ``r`` + such that ``f = q_1*f_1 + ... + q_n*f_n + r``, where ``r`` vanishes or ``r`` + is a completely reduced polynomial with respect to ``G``. + + Examples + ======== + + >>> from sympy import groebner, expand, Poly + >>> from sympy.abc import x, y + + >>> f = 2*x**4 - x**2 + y**3 + y**2 + >>> G = groebner([x**3 - x, y**3 - y]) + + >>> G.reduce(f) + ([2*x, 1], x**2 + y**2 + y) + >>> Q, r = _ + + >>> expand(sum(q*g for q, g in zip(Q, G)) + r) + 2*x**4 - x**2 + y**3 + y**2 + >>> _ == f + True + + # Using Poly input + >>> f_poly = Poly(f, x, y) + >>> G = groebner([Poly(x**3 - x), Poly(y**3 - y)]) + + >>> G.reduce(f_poly) + ([Poly(2*x, x, y, domain='ZZ'), Poly(1, x, y, domain='ZZ')], Poly(x**2 + y**2 + y, x, y, domain='ZZ')) + + """ + if isinstance(expr, Poly): + + if expr.gens != self._options.gens: + raise ValueError("Polynomial generators don't match Groebner basis generators") + poly = expr.set_domain(self._options.domain) + else: + + poly = Poly._from_expr(expr, self._options) + + polys = [poly] + list(self._basis) + + opt = self._options + domain = opt.domain + + retract = False + + if auto and domain.is_Ring and not domain.is_Field: + opt = opt.clone({"domain": domain.get_field()}) + retract = True + + from sympy.polys.rings import xring + _ring, _ = xring(opt.gens, opt.domain, opt.order) + + for i, poly in enumerate(polys): + poly = poly.set_domain(opt.domain).rep.to_dict() + polys[i] = _ring.from_dict(poly) + + Q, r = polys[0].div(polys[1:]) + + Q = [Poly._from_dict(dict(q), opt) for q in Q] + r = Poly._from_dict(dict(r), opt) + + if retract: + try: + _Q, _r = [q.to_ring() for q in Q], r.to_ring() + except CoercionFailed: + pass + else: + Q, r = _Q, _r + + if not opt.polys: + return [q.as_expr() for q in Q], r.as_expr() + else: + return Q, r + + def contains(self, poly): + """ + Check if ``poly`` belongs the ideal generated by ``self``. + + Examples + ======== + + >>> from sympy import groebner + >>> from sympy.abc import x, y + + >>> f = 2*x**3 + y**3 + 3*y + >>> G = groebner([x**2 + y**2 - 1, x*y - 2]) + + >>> G.contains(f) + True + >>> G.contains(f + 1) + False + + """ + return self.reduce(poly)[1] == 0 + + +@public +def poly(expr, *gens, **args): + """ + Efficiently transform an expression into a polynomial. + + Examples + ======== + + >>> from sympy import poly + >>> from sympy.abc import x + + >>> poly(x*(x**2 + x - 1)**2) + Poly(x**5 + 2*x**4 - x**3 - 2*x**2 + x, x, domain='ZZ') + + """ + options.allowed_flags(args, []) + + def _poly(expr, opt): + terms, poly_terms = [], [] + + for term in Add.make_args(expr): + factors, poly_factors = [], [] + + for factor in Mul.make_args(term): + if factor.is_Add: + poly_factors.append(_poly(factor, opt)) + elif factor.is_Pow and factor.base.is_Add and \ + factor.exp.is_Integer and factor.exp >= 0: + poly_factors.append( + _poly(factor.base, opt).pow(factor.exp)) + else: + factors.append(factor) + + if not poly_factors: + terms.append(term) + else: + product = poly_factors[0] + + for factor in poly_factors[1:]: + product = product.mul(factor) + + if factors: + factor = Mul(*factors) + + if factor.is_Number: + product *= factor + else: + product = product.mul(Poly._from_expr(factor, opt)) + + poly_terms.append(product) + + if not poly_terms: + result = Poly._from_expr(expr, opt) + else: + result = poly_terms[0] + + for term in poly_terms[1:]: + result = result.add(term) + + if terms: + term = Add(*terms) + + if term.is_Number: + result += term + else: + result = result.add(Poly._from_expr(term, opt)) + + return result.reorder(*opt.get('gens', ()), **args) + + expr = sympify(expr) + + if expr.is_Poly: + return Poly(expr, *gens, **args) + + if 'expand' not in args: + args['expand'] = False + + opt = options.build_options(gens, args) + + return _poly(expr, opt) + + +def named_poly(n, f, K, name, x, polys): + r"""Common interface to the low-level polynomial generating functions + in orthopolys and appellseqs. + + Parameters + ========== + + n : int + Index of the polynomial, which may or may not equal its degree. + f : callable + Low-level generating function to use. + K : Domain or None + Domain in which to perform the computations. If None, use the smallest + field containing the rationals and the extra parameters of x (see below). + name : str + Name of an arbitrary individual polynomial in the sequence generated + by f, only used in the error message for invalid n. + x : seq + The first element of this argument is the main variable of all + polynomials in this sequence. Any further elements are extra + parameters required by f. + polys : bool, optional + If True, return a Poly, otherwise (default) return an expression. + """ + if n < 0: + raise ValueError("Cannot generate %s of index %s" % (name, n)) + head, tail = x[0], x[1:] + if K is None: + K, tail = construct_domain(tail, field=True) + poly = DMP(f(int(n), *tail, K), K) + if head is None: + poly = PurePoly.new(poly, Dummy('x')) + else: + poly = Poly.new(poly, head) + return poly if polys else poly.as_expr() diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/polyutils.py b/.venv/lib/python3.12/site-packages/sympy/polys/polyutils.py new file mode 100644 index 0000000000000000000000000000000000000000..6a2019d3b195891d84ce8e0b368f6bdc5f45d4b3 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/polyutils.py @@ -0,0 +1,584 @@ +"""Useful utilities for higher level polynomial classes. """ + +from __future__ import annotations + +from sympy.external.gmpy import GROUND_TYPES + +from sympy.core import (S, Add, Mul, Pow, Eq, Expr, + expand_mul, expand_multinomial) +from sympy.core.exprtools import decompose_power, decompose_power_rat +from sympy.core.numbers import _illegal +from sympy.polys.polyerrors import PolynomialError, GeneratorsError +from sympy.polys.polyoptions import build_options + +import re + + +_gens_order = { + 'a': 301, 'b': 302, 'c': 303, 'd': 304, + 'e': 305, 'f': 306, 'g': 307, 'h': 308, + 'i': 309, 'j': 310, 'k': 311, 'l': 312, + 'm': 313, 'n': 314, 'o': 315, 'p': 216, + 'q': 217, 'r': 218, 's': 219, 't': 220, + 'u': 221, 'v': 222, 'w': 223, 'x': 124, + 'y': 125, 'z': 126, +} + +_max_order = 1000 +_re_gen = re.compile(r"^(.*?)(\d*)$", re.MULTILINE) + + +def _nsort(roots, separated=False): + """Sort the numerical roots putting the real roots first, then sorting + according to real and imaginary parts. If ``separated`` is True, then + the real and imaginary roots will be returned in two lists, respectively. + + This routine tries to avoid issue 6137 by separating the roots into real + and imaginary parts before evaluation. In addition, the sorting will raise + an error if any computation cannot be done with precision. + """ + if not all(r.is_number for r in roots): + raise NotImplementedError + if not len(roots): + return [] if not separated else ([], []) + # see issue 6137: + # get the real part of the evaluated real and imaginary parts of each root + key = [[i.n(2).as_real_imag()[0] for i in r.as_real_imag()] for r in roots] + # make sure the parts were computed with precision + if len(roots) > 1 and any(i._prec == 1 for k in key for i in k): + raise NotImplementedError("could not compute root with precision") + # insert a key to indicate if the root has an imaginary part + key = [(1 if i else 0, r, i) for r, i in key] + key = sorted(zip(key, roots)) + # return the real and imaginary roots separately if desired + if separated: + r = [] + i = [] + for (im, _, _), v in key: + if im: + i.append(v) + else: + r.append(v) + return r, i + _, roots = zip(*key) + return list(roots) + + +def _sort_gens(gens, **args): + """Sort generators in a reasonably intelligent way. """ + opt = build_options(args) + + gens_order, wrt = {}, None + + if opt is not None: + gens_order, wrt = {}, opt.wrt + + for i, gen in enumerate(opt.sort): + gens_order[gen] = i + 1 + + def order_key(gen): + gen = str(gen) + + if wrt is not None: + try: + return (-len(wrt) + wrt.index(gen), gen, 0) + except ValueError: + pass + + name, index = _re_gen.match(gen).groups() + + if index: + index = int(index) + else: + index = 0 + + try: + return ( gens_order[name], name, index) + except KeyError: + pass + + try: + return (_gens_order[name], name, index) + except KeyError: + pass + + return (_max_order, name, index) + + try: + gens = sorted(gens, key=order_key) + except TypeError: # pragma: no cover + pass + + return tuple(gens) + + +def _unify_gens(f_gens, g_gens): + """Unify generators in a reasonably intelligent way. """ + f_gens = list(f_gens) + g_gens = list(g_gens) + + if f_gens == g_gens: + return tuple(f_gens) + + gens, common, k = [], [], 0 + + for gen in f_gens: + if gen in g_gens: + common.append(gen) + + for i, gen in enumerate(g_gens): + if gen in common: + g_gens[i], k = common[k], k + 1 + + for gen in common: + i = f_gens.index(gen) + + gens.extend(f_gens[:i]) + f_gens = f_gens[i + 1:] + + i = g_gens.index(gen) + + gens.extend(g_gens[:i]) + g_gens = g_gens[i + 1:] + + gens.append(gen) + + gens.extend(f_gens) + gens.extend(g_gens) + + return tuple(gens) + + +def _analyze_gens(gens): + """Support for passing generators as `*gens` and `[gens]`. """ + if len(gens) == 1 and hasattr(gens[0], '__iter__'): + return tuple(gens[0]) + else: + return tuple(gens) + + +def _sort_factors(factors, **args): + """Sort low-level factors in increasing 'complexity' order. """ + + # XXX: GF(p) does not support comparisons so we need a key function to sort + # the factors if python-flint is being used. A better solution might be to + # add a sort key method to each domain. + def order_key(factor): + if isinstance(factor, _GF_types): + return int(factor) + elif isinstance(factor, list): + return [order_key(f) for f in factor] + else: + return factor + + def order_if_multiple_key(factor): + (f, n) = factor + return (len(f), n, order_key(f)) + + def order_no_multiple_key(f): + return (len(f), order_key(f)) + + if args.get('multiple', True): + return sorted(factors, key=order_if_multiple_key) + else: + return sorted(factors, key=order_no_multiple_key) + + +illegal_types = [type(obj) for obj in _illegal] +finf = [float(i) for i in _illegal[1:3]] + + +def _not_a_coeff(expr): + """Do not treat NaN and infinities as valid polynomial coefficients. """ + if type(expr) in illegal_types or expr in finf: + return True + if isinstance(expr, float) and float(expr) != expr: + return True # nan + return # could be + + +def _parallel_dict_from_expr_if_gens(exprs, opt): + """Transform expressions into a multinomial form given generators. """ + k, indices = len(opt.gens), {} + + for i, g in enumerate(opt.gens): + indices[g] = i + + polys = [] + + for expr in exprs: + poly = {} + + if expr.is_Equality: + expr = expr.lhs - expr.rhs + + for term in Add.make_args(expr): + coeff, monom = [], [0]*k + + for factor in Mul.make_args(term): + if not _not_a_coeff(factor) and factor.is_Number: + coeff.append(factor) + else: + try: + if opt.series is False: + base, exp = decompose_power(factor) + + if exp < 0: + exp, base = -exp, Pow(base, -S.One) + else: + base, exp = decompose_power_rat(factor) + + monom[indices[base]] = exp + except KeyError: + if not factor.has_free(*opt.gens): + coeff.append(factor) + else: + raise PolynomialError("%s contains an element of " + "the set of generators." % factor) + + monom = tuple(monom) + + if monom in poly: + poly[monom] += Mul(*coeff) + else: + poly[monom] = Mul(*coeff) + + polys.append(poly) + + return polys, opt.gens + + +def _parallel_dict_from_expr_no_gens(exprs, opt): + """Transform expressions into a multinomial form and figure out generators. """ + if opt.domain is not None: + def _is_coeff(factor): + return factor in opt.domain + elif opt.extension is True: + def _is_coeff(factor): + return factor.is_algebraic + elif opt.greedy is not False: + def _is_coeff(factor): + return factor is S.ImaginaryUnit + else: + def _is_coeff(factor): + return factor.is_number + + gens, reprs = set(), [] + + for expr in exprs: + terms = [] + + if expr.is_Equality: + expr = expr.lhs - expr.rhs + + for term in Add.make_args(expr): + coeff, elements = [], {} + + for factor in Mul.make_args(term): + if not _not_a_coeff(factor) and (factor.is_Number or _is_coeff(factor)): + coeff.append(factor) + else: + if opt.series is False: + base, exp = decompose_power(factor) + + if exp < 0: + exp, base = -exp, Pow(base, -S.One) + else: + base, exp = decompose_power_rat(factor) + + elements[base] = elements.setdefault(base, 0) + exp + gens.add(base) + + terms.append((coeff, elements)) + + reprs.append(terms) + + gens = _sort_gens(gens, opt=opt) + k, indices = len(gens), {} + + for i, g in enumerate(gens): + indices[g] = i + + polys = [] + + for terms in reprs: + poly = {} + + for coeff, term in terms: + monom = [0]*k + + for base, exp in term.items(): + monom[indices[base]] = exp + + monom = tuple(monom) + + if monom in poly: + poly[monom] += Mul(*coeff) + else: + poly[monom] = Mul(*coeff) + + polys.append(poly) + + return polys, tuple(gens) + + +def _dict_from_expr_if_gens(expr, opt): + """Transform an expression into a multinomial form given generators. """ + (poly,), gens = _parallel_dict_from_expr_if_gens((expr,), opt) + return poly, gens + + +def _dict_from_expr_no_gens(expr, opt): + """Transform an expression into a multinomial form and figure out generators. """ + (poly,), gens = _parallel_dict_from_expr_no_gens((expr,), opt) + return poly, gens + + +def parallel_dict_from_expr(exprs, **args): + """Transform expressions into a multinomial form. """ + reps, opt = _parallel_dict_from_expr(exprs, build_options(args)) + return reps, opt.gens + + +def _parallel_dict_from_expr(exprs, opt): + """Transform expressions into a multinomial form. """ + if opt.expand is not False: + exprs = [ expr.expand() for expr in exprs ] + + if any(expr.is_commutative is False for expr in exprs): + raise PolynomialError('non-commutative expressions are not supported') + + if opt.gens: + reps, gens = _parallel_dict_from_expr_if_gens(exprs, opt) + else: + reps, gens = _parallel_dict_from_expr_no_gens(exprs, opt) + + return reps, opt.clone({'gens': gens}) + + +def dict_from_expr(expr, **args): + """Transform an expression into a multinomial form. """ + rep, opt = _dict_from_expr(expr, build_options(args)) + return rep, opt.gens + + +def _dict_from_expr(expr, opt): + """Transform an expression into a multinomial form. """ + if expr.is_commutative is False: + raise PolynomialError('non-commutative expressions are not supported') + + def _is_expandable_pow(expr): + return (expr.is_Pow and expr.exp.is_positive and expr.exp.is_Integer + and expr.base.is_Add) + + if opt.expand is not False: + if not isinstance(expr, (Expr, Eq)): + raise PolynomialError('expression must be of type Expr') + expr = expr.expand() + # TODO: Integrate this into expand() itself + while any(_is_expandable_pow(i) or i.is_Mul and + any(_is_expandable_pow(j) for j in i.args) for i in + Add.make_args(expr)): + + expr = expand_multinomial(expr) + while any(i.is_Mul and any(j.is_Add for j in i.args) for i in Add.make_args(expr)): + expr = expand_mul(expr) + + if opt.gens: + rep, gens = _dict_from_expr_if_gens(expr, opt) + else: + rep, gens = _dict_from_expr_no_gens(expr, opt) + + return rep, opt.clone({'gens': gens}) + + +def expr_from_dict(rep, *gens): + """Convert a multinomial form into an expression. """ + result = [] + + for monom, coeff in rep.items(): + term = [coeff] + for g, m in zip(gens, monom): + if m: + term.append(Pow(g, m)) + + result.append(Mul(*term)) + + return Add(*result) + +parallel_dict_from_basic = parallel_dict_from_expr +dict_from_basic = dict_from_expr +basic_from_dict = expr_from_dict + + +def _dict_reorder(rep, gens, new_gens): + """Reorder levels using dict representation. """ + gens = list(gens) + + monoms = rep.keys() + coeffs = rep.values() + + new_monoms = [ [] for _ in range(len(rep)) ] + used_indices = set() + + for gen in new_gens: + try: + j = gens.index(gen) + used_indices.add(j) + + for M, new_M in zip(monoms, new_monoms): + new_M.append(M[j]) + except ValueError: + for new_M in new_monoms: + new_M.append(0) + + for i, _ in enumerate(gens): + if i not in used_indices: + for monom in monoms: + if monom[i]: + raise GeneratorsError("unable to drop generators") + + return map(tuple, new_monoms), coeffs + + +class PicklableWithSlots: + """ + Mixin class that allows to pickle objects with ``__slots__``. + + Examples + ======== + + First define a class that mixes :class:`PicklableWithSlots` in:: + + >>> from sympy.polys.polyutils import PicklableWithSlots + >>> class Some(PicklableWithSlots): + ... __slots__ = ('foo', 'bar') + ... + ... def __init__(self, foo, bar): + ... self.foo = foo + ... self.bar = bar + + To make :mod:`pickle` happy in doctest we have to use these hacks:: + + >>> import builtins + >>> builtins.Some = Some + >>> from sympy.polys import polyutils + >>> polyutils.Some = Some + + Next lets see if we can create an instance, pickle it and unpickle:: + + >>> some = Some('abc', 10) + >>> some.foo, some.bar + ('abc', 10) + + >>> from pickle import dumps, loads + >>> some2 = loads(dumps(some)) + + >>> some2.foo, some2.bar + ('abc', 10) + + """ + + __slots__ = () + + def __getstate__(self, cls=None): + if cls is None: + # This is the case for the instance that gets pickled + cls = self.__class__ + + d = {} + + # Get all data that should be stored from super classes + for c in cls.__bases__: + # XXX: Python 3.11 defines object.__getstate__ and it does not + # accept any arguments so we need to make sure not to call it with + # an argument here. To be compatible with Python < 3.11 we need to + # be careful not to assume that c or object has a __getstate__ + # method though. + getstate = getattr(c, "__getstate__", None) + objstate = getattr(object, "__getstate__", None) + if getstate is not None and getstate is not objstate: + d.update(getstate(self, c)) + + # Get all information that should be stored from cls and return the dict + for name in cls.__slots__: + if hasattr(self, name): + d[name] = getattr(self, name) + + return d + + def __setstate__(self, d): + # All values that were pickled are now assigned to a fresh instance + for name, value in d.items(): + setattr(self, name, value) + + +class IntegerPowerable: + r""" + Mixin class for classes that define a `__mul__` method, and want to be + raised to integer powers in the natural way that follows. Implements + powering via binary expansion, for efficiency. + + By default, only integer powers $\geq 2$ are supported. To support the + first, zeroth, or negative powers, override the corresponding methods, + `_first_power`, `_zeroth_power`, `_negative_power`, below. + """ + + def __pow__(self, e, modulo=None): + if e < 2: + try: + if e == 1: + return self._first_power() + elif e == 0: + return self._zeroth_power() + else: + return self._negative_power(e, modulo=modulo) + except NotImplementedError: + return NotImplemented + else: + bits = [int(d) for d in reversed(bin(e)[2:])] + n = len(bits) + p = self + first = True + for i in range(n): + if bits[i]: + if first: + r = p + first = False + else: + r *= p + if modulo is not None: + r %= modulo + if i < n - 1: + p *= p + if modulo is not None: + p %= modulo + return r + + def _negative_power(self, e, modulo=None): + """ + Compute inverse of self, then raise that to the abs(e) power. + For example, if the class has an `inv()` method, + return self.inv() ** abs(e) % modulo + """ + raise NotImplementedError + + def _zeroth_power(self): + """Return unity element of algebraic struct to which self belongs.""" + raise NotImplementedError + + def _first_power(self): + """Return a copy of self.""" + raise NotImplementedError + + +_GF_types: tuple[type, ...] + + +if GROUND_TYPES == 'flint': + import flint + _GF_types = (flint.nmod, flint.fmpz_mod) +else: + from sympy.polys.domains.modularinteger import ModularInteger + flint = None + _GF_types = (ModularInteger,) diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/puiseux.py b/.venv/lib/python3.12/site-packages/sympy/polys/puiseux.py new file mode 100644 index 0000000000000000000000000000000000000000..446dc9c1a5e0d873cdf23da37d2c2430ba0bac6e --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/puiseux.py @@ -0,0 +1,795 @@ +""" +Puiseux rings. These are used by the ring_series module to represented +truncated Puiseux series. Elements of a Puiseux ring are like polynomials +except that the exponents can be negative or rational rather than just +non-negative integers. +""" + +# Previously the ring_series module used PolyElement to represent Puiseux +# series. This is problematic because it means that PolyElement has to support +# negative and non-integer exponents which most polynomial representations do +# not support. This module provides an implementation of a ring for Puiseux +# series that can be used by ring_series without breaking the basic invariants +# of polynomial rings. +# +# Ideally there would be more of a proper series type that can keep track of +# not just the leading terms of a truncated series but also the precision +# of the series. For now the rings here are just introduced to keep the +# interface that ring_series was using before. + +from __future__ import annotations + +from sympy.polys.domains import QQ +from sympy.polys.rings import PolyRing, PolyElement +from sympy.core.add import Add +from sympy.core.mul import Mul +from sympy.external.gmpy import gcd, lcm + + +from typing import TYPE_CHECKING + + +if TYPE_CHECKING: + from typing import Any, Unpack + from sympy.core.expr import Expr + from sympy.polys.domains import Domain + from collections.abc import Iterable, Iterator + + +def puiseux_ring( + symbols: str | list[Expr], domain: Domain +) -> tuple[PuiseuxRing, Unpack[tuple[PuiseuxPoly, ...]]]: + """Construct a Puiseux ring. + + This function constructs a Puiseux ring with the given symbols and domain. + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.puiseux import puiseux_ring + >>> R, x, y = puiseux_ring('x y', QQ) + >>> R + PuiseuxRing((x, y), QQ) + >>> p = 5*x**QQ(1,2) + 7/y + >>> p + 7*y**(-1) + 5*x**(1/2) + """ + ring = PuiseuxRing(symbols, domain) + return (ring,) + ring.gens # type: ignore + + +class PuiseuxRing: + """Ring of Puiseux polynomials. + + A Puiseux polynomial is a truncated Puiseux series. The exponents of the + monomials can be negative or rational numbers. This ring is used by the + ring_series module: + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.puiseux import puiseux_ring + >>> from sympy.polys.ring_series import rs_exp, rs_nth_root + >>> ring, x, y = puiseux_ring('x y', QQ) + >>> f = x**2 + y**3 + >>> f + y**3 + x**2 + >>> f.diff(x) + 2*x + >>> rs_exp(x, x, 5) + 1 + x + 1/2*x**2 + 1/6*x**3 + 1/24*x**4 + + Importantly the Puiseux ring can represent truncated series with negative + and fractional exponents: + + >>> f = 1/x + 1/y**2 + >>> f + x**(-1) + y**(-2) + >>> f.diff(x) + -1*x**(-2) + + >>> rs_nth_root(8*x + x**2 + x**3, 3, x, 5) + 2*x**(1/3) + 1/12*x**(4/3) + 23/288*x**(7/3) + -139/20736*x**(10/3) + + See Also + ======== + + sympy.polys.ring_series.rs_series + PuiseuxPoly + """ + def __init__(self, symbols: str | list[Expr], domain: Domain): + + poly_ring = PolyRing(symbols, domain) + + domain = poly_ring.domain + ngens = poly_ring.ngens + + self.poly_ring = poly_ring + self.domain = domain + + self.symbols = poly_ring.symbols + self.gens = tuple([self.from_poly(g) for g in poly_ring.gens]) + self.ngens = ngens + + self.zero = self.from_poly(poly_ring.zero) + self.one = self.from_poly(poly_ring.one) + + self.zero_monom = poly_ring.zero_monom # type: ignore + self.monomial_mul = poly_ring.monomial_mul # type: ignore + + def __repr__(self) -> str: + return f"PuiseuxRing({self.symbols}, {self.domain})" + + def __eq__(self, other: Any) -> bool: + if not isinstance(other, PuiseuxRing): + return NotImplemented + return self.symbols == other.symbols and self.domain == other.domain + + def from_poly(self, poly: PolyElement) -> PuiseuxPoly: + """Create a Puiseux polynomial from a polynomial. + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.puiseux import puiseux_ring + >>> R1, x1 = ring('x', QQ) + >>> R2, x2 = puiseux_ring('x', QQ) + >>> R2.from_poly(x1**2) + x**2 + """ + return PuiseuxPoly(poly, self) + + def from_dict(self, terms: dict[tuple[int, ...], Any]) -> PuiseuxPoly: + """Create a Puiseux polynomial from a dictionary of terms. + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.puiseux import puiseux_ring + >>> R, x = puiseux_ring('x', QQ) + >>> R.from_dict({(QQ(1,2),): QQ(3)}) + 3*x**(1/2) + """ + return PuiseuxPoly.from_dict(terms, self) + + def from_int(self, n: int) -> PuiseuxPoly: + """Create a Puiseux polynomial from an integer. + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.puiseux import puiseux_ring + >>> R, x = puiseux_ring('x', QQ) + >>> R.from_int(3) + 3 + """ + return self.from_poly(self.poly_ring(n)) + + def domain_new(self, arg: Any) -> Any: + """Create a new element of the domain. + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.puiseux import puiseux_ring + >>> R, x = puiseux_ring('x', QQ) + >>> R.domain_new(3) + 3 + >>> QQ.of_type(_) + True + """ + return self.poly_ring.domain_new(arg) + + def ground_new(self, arg: Any) -> PuiseuxPoly: + """Create a new element from a ground element. + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.puiseux import puiseux_ring, PuiseuxPoly + >>> R, x = puiseux_ring('x', QQ) + >>> R.ground_new(3) + 3 + >>> isinstance(_, PuiseuxPoly) + True + """ + return self.from_poly(self.poly_ring.ground_new(arg)) + + def __call__(self, arg: Any) -> PuiseuxPoly: + """Coerce an element into the ring. + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.puiseux import puiseux_ring + >>> R, x = puiseux_ring('x', QQ) + >>> R(3) + 3 + >>> R({(QQ(1,2),): QQ(3)}) + 3*x**(1/2) + """ + if isinstance(arg, dict): + return self.from_dict(arg) + else: + return self.from_poly(self.poly_ring(arg)) + + def index(self, x: PuiseuxPoly) -> int: + """Return the index of a generator. + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.puiseux import puiseux_ring + >>> R, x, y = puiseux_ring('x y', QQ) + >>> R.index(x) + 0 + >>> R.index(y) + 1 + """ + return self.gens.index(x) + + +def _div_poly_monom(poly: PolyElement, monom: Iterable[int]) -> PolyElement: + ring = poly.ring + div = ring.monomial_div + return ring.from_dict({div(m, monom): c for m, c in poly.terms()}) + + +def _mul_poly_monom(poly: PolyElement, monom: Iterable[int]) -> PolyElement: + ring = poly.ring + mul = ring.monomial_mul + return ring.from_dict({mul(m, monom): c for m, c in poly.terms()}) + + +def _div_monom(monom: Iterable[int], div: Iterable[int]) -> tuple[int, ...]: + return tuple(mi - di for mi, di in zip(monom, div)) + + +class PuiseuxPoly: + """Puiseux polynomial. Represents a truncated Puiseux series. + + See the :class:`PuiseuxRing` class for more information. + + >>> from sympy import QQ + >>> from sympy.polys.puiseux import puiseux_ring + >>> R, x, y = puiseux_ring('x, y', QQ) + >>> p = 5*x**2 + 7*y**3 + >>> p + 7*y**3 + 5*x**2 + + The internal representation of a Puiseux polynomial wraps a normal + polynomial. To support negative powers the polynomial is considered to be + divided by a monomial. + + >>> p2 = 1/x + 1/y**2 + >>> p2.monom # x*y**2 + (1, 2) + >>> p2.poly + x + y**2 + >>> (y**2 + x) / (x*y**2) == p2 + True + + To support fractional powers the polynomial is considered to be a function + of ``x**(1/nx), y**(1/ny), ...``. The representation keeps track of a + monomial and a list of exponent denominators so that the polynomial can be + used to represent both negative and fractional powers. + + >>> p3 = x**QQ(1,2) + y**QQ(2,3) + >>> p3.ns + (2, 3) + >>> p3.poly + x + y**2 + + See Also + ======== + + sympy.polys.puiseux.PuiseuxRing + sympy.polys.rings.PolyElement + """ + + ring: PuiseuxRing + poly: PolyElement + monom: tuple[int, ...] | None + ns: tuple[int, ...] | None + + def __new__(cls, poly: PolyElement, ring: PuiseuxRing) -> PuiseuxPoly: + return cls._new(ring, poly, None, None) + + @classmethod + def _new( + cls, + ring: PuiseuxRing, + poly: PolyElement, + monom: tuple[int, ...] | None, + ns: tuple[int, ...] | None, + ) -> PuiseuxPoly: + poly, monom, ns = cls._normalize(poly, monom, ns) + return cls._new_raw(ring, poly, monom, ns) + + @classmethod + def _new_raw( + cls, + ring: PuiseuxRing, + poly: PolyElement, + monom: tuple[int, ...] | None, + ns: tuple[int, ...] | None, + ) -> PuiseuxPoly: + obj = object.__new__(cls) + obj.ring = ring + obj.poly = poly + obj.monom = monom + obj.ns = ns + return obj + + def __eq__(self, other: Any) -> bool: + if isinstance(other, PuiseuxPoly): + return ( + self.poly == other.poly + and self.monom == other.monom + and self.ns == other.ns + ) + elif self.monom is None and self.ns is None: + return self.poly.__eq__(other) + else: + return NotImplemented + + @classmethod + def _normalize( + cls, + poly: PolyElement, + monom: tuple[int, ...] | None, + ns: tuple[int, ...] | None, + ) -> tuple[PolyElement, tuple[int, ...] | None, tuple[int, ...] | None]: + if monom is None and ns is None: + return poly, None, None + + if monom is not None: + degs = [max(d, 0) for d in poly.tail_degrees()] + if all(di >= mi for di, mi in zip(degs, monom)): + poly = _div_poly_monom(poly, monom) + monom = None + elif any(degs): + poly = _div_poly_monom(poly, degs) + monom = _div_monom(monom, degs) + + if ns is not None: + factors_d, [poly_d] = poly.deflate() + degrees = poly.degrees() + monom_d = monom if monom is not None else [0] * len(degrees) + ns_new = [] + monom_new = [] + inflations = [] + for fi, ni, di, mi in zip(factors_d, ns, degrees, monom_d): + if di == 0: + g = gcd(ni, mi) + else: + g = gcd(fi, ni, mi) + ns_new.append(ni // g) + monom_new.append(mi // g) + inflations.append(fi // g) + + if any(infl > 1 for infl in inflations): + poly_d = poly_d.inflate(inflations) + + poly = poly_d + + if monom is not None: + monom = tuple(monom_new) + + if all(n == 1 for n in ns_new): + ns = None + else: + ns = tuple(ns_new) + + return poly, monom, ns + + @classmethod + def _monom_fromint( + cls, + monom: tuple[int, ...], + dmonom: tuple[int, ...] | None, + ns: tuple[int, ...] | None, + ) -> tuple[Any, ...]: + if dmonom is not None and ns is not None: + return tuple(QQ(mi - di, ni) for mi, di, ni in zip(monom, dmonom, ns)) + elif dmonom is not None: + return tuple(QQ(mi - di) for mi, di in zip(monom, dmonom)) + elif ns is not None: + return tuple(QQ(mi, ni) for mi, ni in zip(monom, ns)) + else: + return tuple(QQ(mi) for mi in monom) + + @classmethod + def _monom_toint( + cls, + monom: tuple[Any, ...], + dmonom: tuple[int, ...] | None, + ns: tuple[int, ...] | None, + ) -> tuple[int, ...]: + if dmonom is not None and ns is not None: + return tuple( + int((mi * ni).numerator + di) for mi, di, ni in zip(monom, dmonom, ns) + ) + elif dmonom is not None: + return tuple(int(mi.numerator + di) for mi, di in zip(monom, dmonom)) + elif ns is not None: + return tuple(int((mi * ni).numerator) for mi, ni in zip(monom, ns)) + else: + return tuple(int(mi.numerator) for mi in monom) + + def itermonoms(self) -> Iterator[tuple[Any, ...]]: + """Iterate over the monomials of a Puiseux polynomial. + + >>> from sympy import QQ + >>> from sympy.polys.puiseux import puiseux_ring + >>> R, x, y = puiseux_ring('x, y', QQ) + >>> p = 5*x**2 + 7*y**3 + >>> list(p.itermonoms()) + [(2, 0), (0, 3)] + >>> p[(2, 0)] + 5 + """ + monom, ns = self.monom, self.ns + for m in self.poly.itermonoms(): + yield self._monom_fromint(m, monom, ns) + + def monoms(self) -> list[tuple[Any, ...]]: + """Return a list of the monomials of a Puiseux polynomial.""" + return list(self.itermonoms()) + + def __iter__(self) -> Iterator[tuple[tuple[Any, ...], Any]]: + return self.itermonoms() + + def __getitem__(self, monom: tuple[int, ...]) -> Any: + monom = self._monom_toint(monom, self.monom, self.ns) + return self.poly[monom] + + def __len__(self) -> int: + return len(self.poly) + + def iterterms(self) -> Iterator[tuple[tuple[Any, ...], Any]]: + """Iterate over the terms of a Puiseux polynomial. + + >>> from sympy import QQ + >>> from sympy.polys.puiseux import puiseux_ring + >>> R, x, y = puiseux_ring('x, y', QQ) + >>> p = 5*x**2 + 7*y**3 + >>> list(p.iterterms()) + [((2, 0), 5), ((0, 3), 7)] + """ + monom, ns = self.monom, self.ns + for m, coeff in self.poly.iterterms(): + mq = self._monom_fromint(m, monom, ns) + yield mq, coeff + + def terms(self) -> list[tuple[tuple[Any, ...], Any]]: + """Return a list of the terms of a Puiseux polynomial.""" + return list(self.iterterms()) + + @property + def is_term(self) -> bool: + """Return True if the Puiseux polynomial is a single term.""" + return self.poly.is_term + + def to_dict(self) -> dict[tuple[int, ...], Any]: + """Return a dictionary representation of a Puiseux polynomial.""" + return dict(self.iterterms()) + + @classmethod + def from_dict( + cls, terms: dict[tuple[Any, ...], Any], ring: PuiseuxRing + ) -> PuiseuxPoly: + """Create a Puiseux polynomial from a dictionary of terms. + + >>> from sympy import QQ + >>> from sympy.polys.puiseux import puiseux_ring, PuiseuxPoly + >>> R, x = puiseux_ring('x', QQ) + >>> PuiseuxPoly.from_dict({(QQ(1,2),): QQ(3)}, R) + 3*x**(1/2) + >>> R.from_dict({(QQ(1,2),): QQ(3)}) + 3*x**(1/2) + """ + ns = [1] * ring.ngens + mon = [0] * ring.ngens + for mo in terms: + ns = [lcm(n, m.denominator) for n, m in zip(ns, mo)] + mon = [min(m, n) for m, n in zip(mo, mon)] + + if not any(mon): + monom = None + else: + monom = tuple(-int((m * n).numerator) for m, n in zip(mon, ns)) + + if all(n == 1 for n in ns): + ns_final = None + else: + ns_final = tuple(ns) + + terms_p = {cls._monom_toint(m, monom, ns_final): coeff for m, coeff in terms.items()} + + poly = ring.poly_ring.from_dict(terms_p) + + return cls._new(ring, poly, monom, ns_final) + + def as_expr(self) -> Expr: + """Convert a Puiseux polynomial to :class:`~sympy.core.expr.Expr`. + + >>> from sympy import QQ, Expr + >>> from sympy.polys.puiseux import puiseux_ring + >>> R, x = puiseux_ring('x', QQ) + >>> p = 5*x**2 + 7*x**3 + >>> p.as_expr() + 7*x**3 + 5*x**2 + >>> isinstance(_, Expr) + True + """ + ring = self.ring + dom = ring.domain + symbols = ring.symbols + terms = [] + for monom, coeff in self.iterterms(): + coeff_expr = dom.to_sympy(coeff) + monoms_expr = [] + for i, m in enumerate(monom): + monoms_expr.append(symbols[i] ** m) + terms.append(Mul(coeff_expr, *monoms_expr)) + return Add(*terms) + + def __repr__(self) -> str: + + def format_power(base: str, exp: int) -> str: + if exp == 1: + return base + elif exp >= 0 and int(exp) == exp: + return f"{base}**{exp}" + else: + return f"{base}**({exp})" + + ring = self.ring + dom = ring.domain + + syms = [str(s) for s in ring.symbols] + terms_str = [] + for monom, coeff in sorted(self.terms()): + monom_str = "*".join(format_power(s, e) for s, e in zip(syms, monom) if e) + if coeff == dom.one: + if monom_str: + terms_str.append(monom_str) + else: + terms_str.append("1") + elif not monom_str: + terms_str.append(str(coeff)) + else: + terms_str.append(f"{coeff}*{monom_str}") + + return " + ".join(terms_str) + + def _unify( + self, other: PuiseuxPoly + ) -> tuple[ + PolyElement, PolyElement, tuple[int, ...] | None, tuple[int, ...] | None + ]: + """Bring two Puiseux polynomials to a common monom and ns.""" + poly1, monom1, ns1 = self.poly, self.monom, self.ns + poly2, monom2, ns2 = other.poly, other.monom, other.ns + + if monom1 == monom2 and ns1 == ns2: + return poly1, poly2, monom1, ns1 + + if ns1 == ns2: + ns = ns1 + elif ns1 is not None and ns2 is not None: + ns = tuple(lcm(n1, n2) for n1, n2 in zip(ns1, ns2)) + f1 = [n // n1 for n, n1 in zip(ns, ns1)] + f2 = [n // n2 for n, n2 in zip(ns, ns2)] + poly1 = poly1.inflate(f1) + poly2 = poly2.inflate(f2) + if monom1 is not None: + monom1 = tuple(m * f for m, f in zip(monom1, f1)) + if monom2 is not None: + monom2 = tuple(m * f for m, f in zip(monom2, f2)) + elif ns2 is not None: + ns = ns2 + poly1 = poly1.inflate(ns) + if monom1 is not None: + monom1 = tuple(m * n for m, n in zip(monom1, ns)) + elif ns1 is not None: + ns = ns1 + poly2 = poly2.inflate(ns) + if monom2 is not None: + monom2 = tuple(m * n for m, n in zip(monom2, ns)) + else: + assert False + + if monom1 == monom2: + monom = monom1 + elif monom1 is not None and monom2 is not None: + monom = tuple(max(m1, m2) for m1, m2 in zip(monom1, monom2)) + poly1 = _mul_poly_monom(poly1, _div_monom(monom, monom1)) + poly2 = _mul_poly_monom(poly2, _div_monom(monom, monom2)) + elif monom2 is not None: + monom = monom2 + poly1 = _mul_poly_monom(poly1, monom2) + elif monom1 is not None: + monom = monom1 + poly2 = _mul_poly_monom(poly2, monom1) + else: + assert False + + return poly1, poly2, monom, ns + + def __pos__(self) -> PuiseuxPoly: + return self + + def __neg__(self) -> PuiseuxPoly: + return self._new_raw(self.ring, -self.poly, self.monom, self.ns) + + def __add__(self, other: Any) -> PuiseuxPoly: + if isinstance(other, PuiseuxPoly): + if self.ring != other.ring: + raise ValueError("Cannot add Puiseux polynomials from different rings") + return self._add(other) + domain = self.ring.domain + if isinstance(other, int): + return self._add_ground(domain.convert_from(QQ(other), QQ)) + elif domain.of_type(other): + return self._add_ground(other) + else: + return NotImplemented + + def __radd__(self, other: Any) -> PuiseuxPoly: + domain = self.ring.domain + if isinstance(other, int): + return self._add_ground(domain.convert_from(QQ(other), QQ)) + elif domain.of_type(other): + return self._add_ground(other) + else: + return NotImplemented + + def __sub__(self, other: Any) -> PuiseuxPoly: + if isinstance(other, PuiseuxPoly): + if self.ring != other.ring: + raise ValueError( + "Cannot subtract Puiseux polynomials from different rings" + ) + return self._sub(other) + domain = self.ring.domain + if isinstance(other, int): + return self._sub_ground(domain.convert_from(QQ(other), QQ)) + elif domain.of_type(other): + return self._sub_ground(other) + else: + return NotImplemented + + def __rsub__(self, other: Any) -> PuiseuxPoly: + domain = self.ring.domain + if isinstance(other, int): + return self._rsub_ground(domain.convert_from(QQ(other), QQ)) + elif domain.of_type(other): + return self._rsub_ground(other) + else: + return NotImplemented + + def __mul__(self, other: Any) -> PuiseuxPoly: + if isinstance(other, PuiseuxPoly): + if self.ring != other.ring: + raise ValueError( + "Cannot multiply Puiseux polynomials from different rings" + ) + return self._mul(other) + domain = self.ring.domain + if isinstance(other, int): + return self._mul_ground(domain.convert_from(QQ(other), QQ)) + elif domain.of_type(other): + return self._mul_ground(other) + else: + return NotImplemented + + def __rmul__(self, other: Any) -> PuiseuxPoly: + domain = self.ring.domain + if isinstance(other, int): + return self._mul_ground(domain.convert_from(QQ(other), QQ)) + elif domain.of_type(other): + return self._mul_ground(other) + else: + return NotImplemented + + def __pow__(self, other: Any) -> PuiseuxPoly: + if isinstance(other, int): + if other >= 0: + return self._pow_pint(other) + else: + return self._pow_nint(-other) + elif QQ.of_type(other): + return self._pow_rational(other) + else: + return NotImplemented + + def __truediv__(self, other: Any) -> PuiseuxPoly: + if isinstance(other, PuiseuxPoly): + if self.ring != other.ring: + raise ValueError( + "Cannot divide Puiseux polynomials from different rings" + ) + return self._mul(other._inv()) + domain = self.ring.domain + if isinstance(other, int): + return self._mul_ground(domain.convert_from(QQ(1, other), QQ)) + elif domain.of_type(other): + return self._div_ground(other) + else: + return NotImplemented + + def __rtruediv__(self, other: Any) -> PuiseuxPoly: + if isinstance(other, int): + return self._inv()._mul_ground(self.ring.domain.convert_from(QQ(other), QQ)) + elif self.ring.domain.of_type(other): + return self._inv()._mul_ground(other) + else: + return NotImplemented + + def _add(self, other: PuiseuxPoly) -> PuiseuxPoly: + poly1, poly2, monom, ns = self._unify(other) + return self._new(self.ring, poly1 + poly2, monom, ns) + + def _add_ground(self, ground: Any) -> PuiseuxPoly: + return self._add(self.ring.ground_new(ground)) + + def _sub(self, other: PuiseuxPoly) -> PuiseuxPoly: + poly1, poly2, monom, ns = self._unify(other) + return self._new(self.ring, poly1 - poly2, monom, ns) + + def _sub_ground(self, ground: Any) -> PuiseuxPoly: + return self._sub(self.ring.ground_new(ground)) + + def _rsub_ground(self, ground: Any) -> PuiseuxPoly: + return self.ring.ground_new(ground)._sub(self) + + def _mul(self, other: PuiseuxPoly) -> PuiseuxPoly: + poly1, poly2, monom, ns = self._unify(other) + if monom is not None: + monom = tuple(2 * e for e in monom) + return self._new(self.ring, poly1 * poly2, monom, ns) + + def _mul_ground(self, ground: Any) -> PuiseuxPoly: + return self._new_raw(self.ring, self.poly * ground, self.monom, self.ns) + + def _div_ground(self, ground: Any) -> PuiseuxPoly: + return self._new_raw(self.ring, self.poly / ground, self.monom, self.ns) + + def _pow_pint(self, n: int) -> PuiseuxPoly: + assert n >= 0 + monom = self.monom + if monom is not None: + monom = tuple(m * n for m in monom) + return self._new(self.ring, self.poly**n, monom, self.ns) + + def _pow_nint(self, n: int) -> PuiseuxPoly: + return self._inv()._pow_pint(n) + + def _pow_rational(self, n: Any) -> PuiseuxPoly: + if not self.is_term: + raise ValueError("Only monomials can be raised to a rational power") + [(monom, coeff)] = self.terms() + domain = self.ring.domain + if not domain.is_one(coeff): + raise ValueError("Only monomials can be raised to a rational power") + monom = tuple(m * n for m in monom) + return self.ring.from_dict({monom: domain.one}) + + def _inv(self) -> PuiseuxPoly: + if not self.is_term: + raise ValueError("Only terms can be inverted") + [(monom, coeff)] = self.terms() + domain = self.ring.domain + if not domain.is_Field and not domain.is_one(coeff): + raise ValueError("Cannot invert non-unit coefficient") + monom = tuple(-m for m in monom) + coeff = 1 / coeff + return self.ring.from_dict({monom: coeff}) + + def diff(self, x: PuiseuxPoly) -> PuiseuxPoly: + """Differentiate a Puiseux polynomial with respect to a variable. + + >>> from sympy import QQ + >>> from sympy.polys.puiseux import puiseux_ring + >>> R, x, y = puiseux_ring('x, y', QQ) + >>> p = 5*x**2 + 7*y**3 + >>> p.diff(x) + 10*x + >>> p.diff(y) + 21*y**2 + """ + ring = self.ring + i = ring.index(x) + g = {} + for expv, coeff in self.iterterms(): + n = expv[i] + if n: + e = list(expv) + e[i] -= 1 + g[tuple(e)] = coeff * n + return ring(g) diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/rationaltools.py b/.venv/lib/python3.12/site-packages/sympy/polys/rationaltools.py new file mode 100644 index 0000000000000000000000000000000000000000..0ca513ff2d4af96baaaf1c82caf501750b1524da --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/rationaltools.py @@ -0,0 +1,85 @@ +"""Tools for manipulation of rational expressions. """ + + +from sympy.core import Basic, Add, sympify +from sympy.core.exprtools import gcd_terms +from sympy.utilities import public +from sympy.utilities.iterables import iterable + + +@public +def together(expr, deep=False, fraction=True): + """ + Denest and combine rational expressions using symbolic methods. + + This function takes an expression or a container of expressions + and puts it (them) together by denesting and combining rational + subexpressions. No heroic measures are taken to minimize degree + of the resulting numerator and denominator. To obtain completely + reduced expression use :func:`~.cancel`. However, :func:`~.together` + can preserve as much as possible of the structure of the input + expression in the output (no expansion is performed). + + A wide variety of objects can be put together including lists, + tuples, sets, relational objects, integrals and others. It is + also possible to transform interior of function applications, + by setting ``deep`` flag to ``True``. + + By definition, :func:`~.together` is a complement to :func:`~.apart`, + so ``apart(together(expr))`` should return expr unchanged. Note + however, that :func:`~.together` uses only symbolic methods, so + it might be necessary to use :func:`~.cancel` to perform algebraic + simplification and minimize degree of the numerator and denominator. + + Examples + ======== + + >>> from sympy import together, exp + >>> from sympy.abc import x, y, z + + >>> together(1/x + 1/y) + (x + y)/(x*y) + >>> together(1/x + 1/y + 1/z) + (x*y + x*z + y*z)/(x*y*z) + + >>> together(1/(x*y) + 1/y**2) + (x + y)/(x*y**2) + + >>> together(1/(1 + 1/x) + 1/(1 + 1/y)) + (x*(y + 1) + y*(x + 1))/((x + 1)*(y + 1)) + + >>> together(exp(1/x + 1/y)) + exp(1/y + 1/x) + >>> together(exp(1/x + 1/y), deep=True) + exp((x + y)/(x*y)) + + >>> together(1/exp(x) + 1/(x*exp(x))) + (x + 1)*exp(-x)/x + + >>> together(1/exp(2*x) + 1/(x*exp(3*x))) + (x*exp(x) + 1)*exp(-3*x)/x + + """ + def _together(expr): + if isinstance(expr, Basic): + if expr.is_Atom or (expr.is_Function and not deep): + return expr + elif expr.is_Add: + return gcd_terms(list(map(_together, Add.make_args(expr))), fraction=fraction) + elif expr.is_Pow: + base = _together(expr.base) + + if deep: + exp = _together(expr.exp) + else: + exp = expr.exp + + return expr.func(base, exp) + else: + return expr.func(*[ _together(arg) for arg in expr.args ]) + elif iterable(expr): + return expr.__class__([ _together(ex) for ex in expr ]) + + return expr + + return _together(sympify(expr)) diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/ring_series.py b/.venv/lib/python3.12/site-packages/sympy/polys/ring_series.py new file mode 100644 index 0000000000000000000000000000000000000000..b4333f0add9365991794d920a2699722900e8a5e --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/ring_series.py @@ -0,0 +1,2127 @@ +"""Power series evaluation and manipulation using sparse Polynomials + +Implementing a new function +--------------------------- + +There are a few things to be kept in mind when adding a new function here:: + + - The implementation should work on all possible input domains/rings. + Special cases include the ``EX`` ring and a constant term in the series + to be expanded. There can be two types of constant terms in the series: + + + A constant value or symbol. + + A term of a multivariate series not involving the generator, with + respect to which the series is to expanded. + + Strictly speaking, a generator of a ring should not be considered a + constant. However, for series expansion both the cases need similar + treatment (as the user does not care about inner details), i.e, use an + addition formula to separate the constant part and the variable part (see + rs_sin for reference). + + - All the algorithms used here are primarily designed to work for Taylor + series (number of iterations in the algo equals the required order). + Hence, it becomes tricky to get the series of the right order if a + Puiseux series is input. Use rs_puiseux? in your function if your + algorithm is not designed to handle fractional powers. + +Extending rs_series +------------------- + +To make a function work with rs_series you need to do two things:: + + - Many sure it works with a constant term (as explained above). + - If the series contains constant terms, you might need to extend its ring. + You do so by adding the new terms to the rings as generators. + ``PolyRing.compose`` and ``PolyRing.add_gens`` are two functions that do + so and need to be called every time you expand a series containing a + constant term. + +Look at rs_sin and rs_series for further reference. + +""" + +from sympy.polys.domains import QQ, EX +from sympy.polys.rings import PolyElement, ring, sring +from sympy.polys.puiseux import PuiseuxPoly +from sympy.polys.polyerrors import DomainError +from sympy.polys.monomials import (monomial_min, monomial_mul, monomial_div, + monomial_ldiv) +from mpmath.libmp.libintmath import ifac +from sympy.core import PoleError, Function, Expr +from sympy.core.numbers import Rational +from sympy.core.intfunc import igcd +from sympy.functions import (sin, cos, tan, atan, exp, atanh, asinh, tanh, log, + ceiling, sinh, cosh) +from sympy.utilities.misc import as_int +from mpmath.libmp.libintmath import giant_steps +import math + + +def _invert_monoms(p1): + """ + Compute ``x**n * p1(1/x)`` for a univariate polynomial ``p1`` in ``x``. + + Examples + ======== + + >>> from sympy.polys.domains import ZZ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import _invert_monoms + >>> R, x = ring('x', ZZ) + >>> p = x**2 + 2*x + 3 + >>> _invert_monoms(p) + 3*x**2 + 2*x + 1 + + See Also + ======== + + sympy.polys.densebasic.dup_reverse + """ + terms = list(p1.items()) + terms.sort() + deg = p1.degree() + R = p1.ring + p = R.zero + cv = p1.listcoeffs() + mv = p1.listmonoms() + for mvi, cvi in zip(mv, cv): + p[(deg - mvi[0],)] = cvi + return p + +def _giant_steps(target): + """Return a list of precision steps for the Newton's method""" + # We use ceil here because giant_steps cannot handle flint.fmpq + res = giant_steps(2, math.ceil(target)) + if res[0] != 2: + res = [2] + res + return res + +def rs_trunc(p1, x, prec): + """ + Truncate the series in the ``x`` variable with precision ``prec``, + that is, modulo ``O(x**prec)`` + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import rs_trunc + >>> R, x = ring('x', QQ) + >>> p = x**10 + x**5 + x + 1 + >>> rs_trunc(p, x, 12) + x**10 + x**5 + x + 1 + >>> rs_trunc(p, x, 10) + x**5 + x + 1 + """ + R = p1.ring + p = {} + i = R.gens.index(x) + for exp1 in p1: + if exp1[i] >= prec: + continue + p[exp1] = p1[exp1] + return R(p) + +def rs_is_puiseux(p, x): + """ + Test if ``p`` is Puiseux series in ``x``. + + Raise an exception if it has a negative power in ``x``. + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.puiseux import puiseux_ring + >>> from sympy.polys.ring_series import rs_is_puiseux + >>> R, x = puiseux_ring('x', QQ) + >>> p = x**QQ(2,5) + x**QQ(2,3) + x + >>> rs_is_puiseux(p, x) + True + """ + index = p.ring.gens.index(x) + for k in p.itermonoms(): + if k[index] != int(k[index]): + return True + if k[index] < 0: + raise ValueError('The series is not regular in %s' % x) + return False + +def rs_puiseux(f, p, x, prec): + """ + Return the puiseux series for `f(p, x, prec)`. + + To be used when function ``f`` is implemented only for regular series. + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.puiseux import puiseux_ring + >>> from sympy.polys.ring_series import rs_puiseux, rs_exp + >>> R, x = puiseux_ring('x', QQ) + >>> p = x**QQ(2,5) + x**QQ(2,3) + x + >>> rs_puiseux(rs_exp,p, x, 1) + 1 + x**(2/5) + x**(2/3) + 1/2*x**(4/5) + """ + index = p.ring.gens.index(x) + n = 1 + for k in p: + power = k[index] + if isinstance(power, Rational): + num, den = power.as_numer_denom() + n = int(n*den // igcd(n, den)) + elif power != int(power): + den = power.denominator + n = int(n*den // igcd(n, den)) + if n != 1: + p1 = pow_xin(p, index, n) + r = f(p1, x, prec*n) + n1 = QQ(1, n) + if isinstance(r, tuple): + r = tuple([pow_xin(rx, index, n1) for rx in r]) + else: + r = pow_xin(r, index, n1) + else: + r = f(p, x, prec) + return r + +def rs_puiseux2(f, p, q, x, prec): + """ + Return the puiseux series for `f(p, q, x, prec)`. + + To be used when function ``f`` is implemented only for regular series. + """ + index = p.ring.gens.index(x) + n = 1 + for k in p: + power = k[index] + if isinstance(power, Rational): + num, den = power.as_numer_denom() + n = n*den // igcd(n, den) + elif power != int(power): + den = power.denominator + n = n*den // igcd(n, den) + if n != 1: + p1 = pow_xin(p, index, n) + r = f(p1, q, x, prec*n) + n1 = QQ(1, n) + r = pow_xin(r, index, n1) + else: + r = f(p, q, x, prec) + return r + +def rs_mul(p1, p2, x, prec): + """ + Return the product of the given two series, modulo ``O(x**prec)``. + + ``x`` is the series variable or its position in the generators. + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import rs_mul + >>> R, x = ring('x', QQ) + >>> p1 = x**2 + 2*x + 1 + >>> p2 = x + 1 + >>> rs_mul(p1, p2, x, 3) + 3*x**2 + 3*x + 1 + """ + R = p1.ring + p = {} + if R.__class__ != p2.ring.__class__ or R != p2.ring: + raise ValueError('p1 and p2 must have the same ring') + iv = R.gens.index(x) + if not isinstance(p2, (PolyElement, PuiseuxPoly)): + raise ValueError('p2 must be a polynomial') + if R == p2.ring: + get = p.get + items2 = p2.terms() + items2.sort(key=lambda e: e[0][iv]) + if R.ngens == 1: + for exp1, v1 in p1.iterterms(): + for exp2, v2 in items2: + exp = exp1[0] + exp2[0] + if exp < prec: + exp = (exp, ) + p[exp] = get(exp, 0) + v1*v2 + else: + break + else: + monomial_mul = R.monomial_mul + for exp1, v1 in p1.iterterms(): + for exp2, v2 in items2: + if exp1[iv] + exp2[iv] < prec: + exp = monomial_mul(exp1, exp2) + p[exp] = get(exp, 0) + v1*v2 + else: + break + + return R(p) + +def rs_square(p1, x, prec): + """ + Square the series modulo ``O(x**prec)`` + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import rs_square + >>> R, x = ring('x', QQ) + >>> p = x**2 + 2*x + 1 + >>> rs_square(p, x, 3) + 6*x**2 + 4*x + 1 + """ + R = p1.ring + p = {} + iv = R.gens.index(x) + get = p.get + items = p1.terms() + items.sort(key=lambda e: e[0][iv]) + monomial_mul = R.monomial_mul + for i in range(len(items)): + exp1, v1 = items[i] + for j in range(i): + exp2, v2 = items[j] + if exp1[iv] + exp2[iv] < prec: + exp = monomial_mul(exp1, exp2) + p[exp] = get(exp, 0) + v1*v2 + else: + break + p = {m: 2*v for m, v in p.items()} + get = p.get + for expv, v in p1.iterterms(): + if 2*expv[iv] < prec: + e2 = monomial_mul(expv, expv) + p[e2] = get(e2, 0) + v**2 + return R(p) + +def rs_pow(p1, n, x, prec): + """ + Return ``p1**n`` modulo ``O(x**prec)`` + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import rs_pow + >>> R, x = ring('x', QQ) + >>> p = x + 1 + >>> rs_pow(p, 4, x, 3) + 6*x**2 + 4*x + 1 + """ + R = p1.ring + if isinstance(n, Rational): + np = int(n.p) + nq = int(n.q) + if nq != 1: + res = rs_nth_root(p1, nq, x, prec) + if np != 1: + res = rs_pow(res, np, x, prec) + else: + res = rs_pow(p1, np, x, prec) + return res + + n = as_int(n) + if n == 0: + if p1: + return R(1) + else: + raise ValueError('0**0 is undefined') + if n < 0: + p1 = rs_pow(p1, -n, x, prec) + return rs_series_inversion(p1, x, prec) + if n == 1: + return rs_trunc(p1, x, prec) + if n == 2: + return rs_square(p1, x, prec) + if n == 3: + p2 = rs_square(p1, x, prec) + return rs_mul(p1, p2, x, prec) + p = R(1) + while 1: + if n & 1: + p = rs_mul(p1, p, x, prec) + n -= 1 + if not n: + break + p1 = rs_square(p1, x, prec) + n = n // 2 + return p + +def rs_subs(p, rules, x, prec): + """ + Substitution with truncation according to the mapping in ``rules``. + + Return a series with precision ``prec`` in the generator ``x`` + + Note that substitutions are not done one after the other + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import rs_subs + >>> R, x, y = ring('x, y', QQ) + >>> p = x**2 + y**2 + >>> rs_subs(p, {x: x+ y, y: x+ 2*y}, x, 3) + 2*x**2 + 6*x*y + 5*y**2 + >>> (x + y)**2 + (x + 2*y)**2 + 2*x**2 + 6*x*y + 5*y**2 + + which differs from + + >>> rs_subs(rs_subs(p, {x: x+ y}, x, 3), {y: x+ 2*y}, x, 3) + 5*x**2 + 12*x*y + 8*y**2 + + Parameters + ---------- + p : :class:`~.PolyElement` Input series. + rules : ``dict`` with substitution mappings. + x : :class:`~.PolyElement` in which the series truncation is to be done. + prec : :class:`~.Integer` order of the series after truncation. + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import rs_subs + >>> R, x, y = ring('x, y', QQ) + >>> rs_subs(x**2+y**2, {y: (x+y)**2}, x, 3) + 6*x**2*y**2 + x**2 + 4*x*y**3 + y**4 + """ + R = p.ring + ngens = R.ngens + d = R(0) + for i in range(ngens): + d[(i, 1)] = R.gens[i] + for var in rules: + d[(R.index(var), 1)] = rules[var] + p1 = R(0) + p_keys = sorted(p.keys()) + for expv in p_keys: + p2 = R(1) + for i in range(ngens): + power = expv[i] + if power == 0: + continue + if (i, power) not in d: + q, r = divmod(power, 2) + if r == 0 and (i, q) in d: + d[(i, power)] = rs_square(d[(i, q)], x, prec) + elif (i, power - 1) in d: + d[(i, power)] = rs_mul(d[(i, power - 1)], d[(i, 1)], + x, prec) + else: + d[(i, power)] = rs_pow(d[(i, 1)], power, x, prec) + p2 = rs_mul(p2, d[(i, power)], x, prec) + p1 += p2*p[expv] + return p1 + +def _has_constant_term(p, x): + """ + Check if ``p`` has a constant term in ``x`` + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import _has_constant_term + >>> R, x = ring('x', QQ) + >>> p = x**2 + x + 1 + >>> _has_constant_term(p, x) + True + """ + R = p.ring + iv = R.gens.index(x) + zm = R.zero_monom + a = [0]*R.ngens + a[iv] = 1 + miv = tuple(a) + return any(monomial_min(expv, miv) == zm for expv in p) + +def _get_constant_term(p, x): + """Return constant term in p with respect to x + + Note that it is not simply `p[R.zero_monom]` as there might be multiple + generators in the ring R. We want the `x`-free term which can contain other + generators. + """ + R = p.ring + i = R.gens.index(x) + zm = R.zero_monom + a = [0]*R.ngens + a[i] = 1 + miv = tuple(a) + c = 0 + for expv in p: + if monomial_min(expv, miv) == zm: + c += R({expv: p[expv]}) + return c + +def _check_series_var(p, x, name): + index = p.ring.gens.index(x) + m = min(p, key=lambda k: k[index])[index] + if m < 0: + raise PoleError("Asymptotic expansion of %s around [oo] not " + "implemented." % name) + return index, m + +def _series_inversion1(p, x, prec): + """ + Univariate series inversion ``1/p`` modulo ``O(x**prec)``. + + The Newton method is used. + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import _series_inversion1 + >>> R, x = ring('x', QQ) + >>> p = x + 1 + >>> _series_inversion1(p, x, 4) + -x**3 + x**2 - x + 1 + """ + if rs_is_puiseux(p, x): + return rs_puiseux(_series_inversion1, p, x, prec) + R = p.ring + zm = R.zero_monom + c = p[zm] + + # giant_steps does not seem to work with PythonRational numbers with 1 as + # denominator. This makes sure such a number is converted to integer. + if prec == int(prec): + prec = int(prec) + + if zm not in p: + raise ValueError("No constant term in series") + if _has_constant_term(p - c, x): + raise ValueError("p cannot contain a constant term depending on " + "parameters") + if not R.domain.is_unit(c): + raise ValueError(f"Constant term {c} must be a unit in {R.domain}") + + one = R(1) + if R.domain is EX: + one = 1 + if c != one: + p1 = R(1)/c + else: + p1 = R(1) + for precx in _giant_steps(prec): + t = 1 - rs_mul(p1, p, x, precx) + p1 = p1 + rs_mul(p1, t, x, precx) + return p1 + +def rs_series_inversion(p, x, prec): + """ + Multivariate series inversion ``1/p`` modulo ``O(x**prec)``. + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import rs_series_inversion + >>> R, x, y = ring('x, y', QQ) + >>> rs_series_inversion(1 + x*y**2, x, 4) + -x**3*y**6 + x**2*y**4 - x*y**2 + 1 + >>> rs_series_inversion(1 + x*y**2, y, 4) + -x*y**2 + 1 + >>> rs_series_inversion(x + x**2, x, 4) + x**3 - x**2 + x - 1 + x**(-1) + """ + R = p.ring + if p == R.zero: + raise ZeroDivisionError + zm = R.zero_monom + index = R.gens.index(x) + m = min(p, key=lambda k: k[index])[index] + if m: + p = mul_xin(p, index, -m) + prec = prec + m + if zm not in p: + raise NotImplementedError("No constant term in series") + + if _has_constant_term(p - p[zm], x): + raise NotImplementedError("p - p[0] must not have a constant term in " + "the series variables") + r = _series_inversion1(p, x, prec) + if m != 0: + r = mul_xin(r, index, -m) + return r + +def _coefficient_t(p, t): + r"""Coefficient of `x_i**j` in p, where ``t`` = (i, j)""" + i, j = t + R = p.ring + expv1 = [0]*R.ngens + expv1[i] = j + expv1 = tuple(expv1) + p1 = R(0) + for expv in p: + if expv[i] == j: + p1[monomial_div(expv, expv1)] = p[expv] + return p1 + +def rs_series_reversion(p, x, n, y): + r""" + Reversion of a series. + + ``p`` is a series with ``O(x**n)`` of the form $p = ax + f(x)$ + where $a$ is a number different from 0. + + $f(x) = \sum_{k=2}^{n-1} a_kx_k$ + + Parameters + ========== + + a_k : Can depend polynomially on other variables, not indicated. + x : Variable with name x. + y : Variable with name y. + + Returns + ======= + + Solve $p = y$, that is, given $ax + f(x) - y = 0$, + find the solution $x = r(y)$ up to $O(y^n)$. + + Algorithm + ========= + + If $r_i$ is the solution at order $i$, then: + $ar_i + f(r_i) - y = O\left(y^{i + 1}\right)$ + + and if $r_{i + 1}$ is the solution at order $i + 1$, then: + $ar_{i + 1} + f(r_{i + 1}) - y = O\left(y^{i + 2}\right)$ + + We have, $r_{i + 1} = r_i + e$, such that, + $ae + f(r_i) = O\left(y^{i + 2}\right)$ + or $e = -f(r_i)/a$ + + So we use the recursion relation: + $r_{i + 1} = r_i - f(r_i)/a$ + with the boundary condition: $r_1 = y$ + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import rs_series_reversion, rs_trunc + >>> R, x, y, a, b = ring('x, y, a, b', QQ) + >>> p = x - x**2 - 2*b*x**2 + 2*a*b*x**2 + >>> p1 = rs_series_reversion(p, x, 3, y); p1 + -2*y**2*a*b + 2*y**2*b + y**2 + y + >>> rs_trunc(p.compose(x, p1), y, 3) + y + """ + if rs_is_puiseux(p, x): + raise NotImplementedError + R = p.ring + nx = R.gens.index(x) + y = R(y) + ny = R.gens.index(y) + if _has_constant_term(p, x): + raise ValueError("p must not contain a constant term in the series " + "variable") + a = _coefficient_t(p, (nx, 1)) + zm = R.zero_monom + assert zm in a and len(a) == 1 + a = a[zm] + r = y/a + for i in range(2, n): + sp = rs_subs(p, {x: r}, y, i + 1) + sp = _coefficient_t(sp, (ny, i))*y**i + r -= sp/a + return r + +def rs_series_from_list(p, c, x, prec, concur=1): + """ + Return a series `sum c[n]*p**n` modulo `O(x**prec)`. + + It reduces the number of multiplications by summing concurrently. + + `ax = [1, p, p**2, .., p**(J - 1)]` + `s = sum(c[i]*ax[i]` for i in `range(r, (r + 1)*J))*p**((K - 1)*J)` + with `K >= (n + 1)/J` + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import rs_series_from_list, rs_trunc + >>> R, x = ring('x', QQ) + >>> p = x**2 + x + 1 + >>> c = [1, 2, 3] + >>> rs_series_from_list(p, c, x, 4) + 6*x**3 + 11*x**2 + 8*x + 6 + >>> rs_trunc(1 + 2*p + 3*p**2, x, 4) + 6*x**3 + 11*x**2 + 8*x + 6 + >>> pc = R.from_list(list(reversed(c))) + >>> rs_trunc(pc.compose(x, p), x, 4) + 6*x**3 + 11*x**2 + 8*x + 6 + + See Also + ======== + + sympy.polys.rings.PolyRing.compose + + """ + R = p.ring + n = len(c) + if not concur: + q = R(1) + s = c[0]*q + for i in range(1, n): + q = rs_mul(q, p, x, prec) + s += c[i]*q + return s + J = int(math.sqrt(n) + 1) + K, r = divmod(n, J) + if r: + K += 1 + ax = [R(1)] + q = R(1) + if len(p) < 20: + for i in range(1, J): + q = rs_mul(q, p, x, prec) + ax.append(q) + else: + for i in range(1, J): + if i % 2 == 0: + q = rs_square(ax[i//2], x, prec) + else: + q = rs_mul(q, p, x, prec) + ax.append(q) + # optimize using rs_square + pj = rs_mul(ax[-1], p, x, prec) + b = R(1) + s = R(0) + for k in range(K - 1): + r = J*k + s1 = c[r] + for j in range(1, J): + s1 += c[r + j]*ax[j] + s1 = rs_mul(s1, b, x, prec) + s += s1 + b = rs_mul(b, pj, x, prec) + if not b: + break + k = K - 1 + r = J*k + if r < n: + s1 = c[r]*R(1) + for j in range(1, J): + if r + j >= n: + break + s1 += c[r + j]*ax[j] + s1 = rs_mul(s1, b, x, prec) + s += s1 + return s + +def rs_diff(p, x): + """ + Return partial derivative of ``p`` with respect to ``x``. + + Parameters + ========== + + x : :class:`~.PolyElement` with respect to which ``p`` is differentiated. + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import rs_diff + >>> R, x, y = ring('x, y', QQ) + >>> p = x + x**2*y**3 + >>> rs_diff(p, x) + 2*x*y**3 + 1 + """ + R = p.ring + n = R.gens.index(x) + p1 = {} + mn = [0]*R.ngens + mn[n] = 1 + mn = tuple(mn) + for expv in p: + if expv[n]: + e = monomial_ldiv(expv, mn) + p1[e] = R.domain_new(p[expv]*expv[n]) + return R(p1) + +def rs_integrate(p, x): + """ + Integrate ``p`` with respect to ``x``. + + Parameters + ========== + + x : :class:`~.PolyElement` with respect to which ``p`` is integrated. + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import rs_integrate + >>> R, x, y = ring('x, y', QQ) + >>> p = x + x**2*y**3 + >>> rs_integrate(p, x) + 1/3*x**3*y**3 + 1/2*x**2 + """ + R = p.ring + p1 = {} + n = R.gens.index(x) + mn = [0]*R.ngens + mn[n] = 1 + mn = tuple(mn) + + for expv in p: + e = monomial_mul(expv, mn) + p1[e] = R.domain_new(p[expv]/(expv[n] + 1)) + return R(p1) + +def rs_fun(p, f, *args): + r""" + Function of a multivariate series computed by substitution. + + The case with f method name is used to compute `rs\_tan` and `rs\_nth\_root` + of a multivariate series: + + `rs\_fun(p, tan, iv, prec)` + + tan series is first computed for a dummy variable _x, + i.e, `rs\_tan(\_x, iv, prec)`. Then we substitute _x with p to get the + desired series + + Parameters + ========== + + p : :class:`~.PolyElement` The multivariate series to be expanded. + f : `ring\_series` function to be applied on `p`. + args[-2] : :class:`~.PolyElement` with respect to which, the series is to be expanded. + args[-1] : Required order of the expanded series. + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import rs_fun, _tan1 + >>> R, x, y = ring('x, y', QQ) + >>> p = x + x*y + x**2*y + x**3*y**2 + >>> rs_fun(p, _tan1, x, 4) + 1/3*x**3*y**3 + 2*x**3*y**2 + x**3*y + 1/3*x**3 + x**2*y + x*y + x + """ + _R = p.ring + R1, _x = ring('_x', _R.domain) + h = int(args[-1]) + args1 = args[:-2] + (_x, h) + zm = _R.zero_monom + # separate the constant term of the series + # compute the univariate series f(_x, .., 'x', sum(nv)) + if zm in p: + x1 = _x + p[zm] + p1 = p - p[zm] + else: + x1 = _x + p1 = p + if isinstance(f, str): + q = getattr(x1, f)(*args1) + else: + q = f(x1, *args1) + a = sorted(q.items()) + c = [0]*h + for x in a: + c[x[0][0]] = x[1] + p1 = rs_series_from_list(p1, c, args[-2], args[-1]) + return p1 + +def mul_xin(p, i, n): + r""" + Return `p*x_i**n`. + + `x\_i` is the ith variable in ``p``. + """ + R = p.ring + q = {} + for k, v in p.terms(): + k1 = list(k) + k1[i] += n + q[tuple(k1)] = v + return R(q) + +def pow_xin(p, i, n): + """ + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.puiseux import puiseux_ring + >>> from sympy.polys.ring_series import pow_xin + >>> R, x, y = puiseux_ring('x, y', QQ) + >>> p = x**QQ(2,5) + x + x**QQ(2,3) + >>> index = p.ring.gens.index(x) + >>> pow_xin(p, index, 15) + x**6 + x**10 + x**15 + """ + R = p.ring + q = {} + for k, v in p.terms(): + k1 = list(k) + k1[i] *= n + q[tuple(k1)] = v + return R(q) + +def _nth_root1(p, n, x, prec): + """ + Univariate series expansion of the nth root of ``p``. + + The Newton method is used. + """ + if rs_is_puiseux(p, x): + return rs_puiseux2(_nth_root1, p, n, x, prec) + R = p.ring + zm = R.zero_monom + if zm not in p: + raise NotImplementedError('No constant term in series') + n = as_int(n) + assert p[zm] == 1 + p1 = R(1) + if p == 1: + return p + if n == 0: + return R(1) + if n == 1: + return p + if n < 0: + n = -n + sign = 1 + else: + sign = 0 + for precx in _giant_steps(prec): + tmp = rs_pow(p1, n + 1, x, precx) + tmp = rs_mul(tmp, p, x, precx) + p1 += p1/n - tmp/n + if sign: + return p1 + else: + return _series_inversion1(p1, x, prec) + +def rs_nth_root(p, n, x, prec): + """ + Multivariate series expansion of the nth root of ``p``. + + Parameters + ========== + + p : Expr + The polynomial to computer the root of. + n : integer + The order of the root to be computed. + x : :class:`~.PolyElement` + prec : integer + Order of the expanded series. + + Notes + ===== + + The result of this function is dependent on the ring over which the + polynomial has been defined. If the answer involves a root of a constant, + make sure that the polynomial is over a real field. It cannot yet handle + roots of symbols. + + Examples + ======== + + >>> from sympy.polys.domains import QQ, RR + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import rs_nth_root + >>> R, x, y = ring('x, y', QQ) + >>> rs_nth_root(1 + x + x*y, -3, x, 3) + 2/9*x**2*y**2 + 4/9*x**2*y + 2/9*x**2 - 1/3*x*y - 1/3*x + 1 + >>> R, x, y = ring('x, y', RR) + >>> rs_nth_root(3 + x + x*y, 3, x, 2) + 0.160249952256379*x*y + 0.160249952256379*x + 1.44224957030741 + """ + if n == 0: + if p == 0: + raise ValueError('0**0 expression') + else: + return p.ring(1) + if n == 1: + return rs_trunc(p, x, prec) + R = p.ring + index = R.gens.index(x) + m = min(p, key=lambda k: k[index])[index] + p = mul_xin(p, index, -m) + prec -= m + + if _has_constant_term(p - 1, x): + zm = R.zero_monom + c = p[zm] + if isinstance(c, PolyElement): + try: + c_expr = c.as_expr() + const = R(c_expr**(QQ(1, n))) + except ValueError: + raise DomainError("The given series cannot be expanded in " + "this domain.") + else: + try: # RealElement doesn't support + const = R(c**Rational(1, n)) # exponentiation with mpq object + except ValueError: # as exponent + raise DomainError("The given series cannot be expanded in " + "this domain.") + res = rs_nth_root(p/c, n, x, prec)*const + else: + res = _nth_root1(p, n, x, prec) + if m: + m = QQ(m) / n + res = mul_xin(res, index, m) + return res + +def rs_log(p, x, prec): + """ + The Logarithm of ``p`` modulo ``O(x**prec)``. + + Notes + ===== + + Truncation of ``integral dx p**-1*d p/dx`` is used. + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.puiseux import puiseux_ring + >>> from sympy.polys.ring_series import rs_log + >>> R, x = puiseux_ring('x', QQ) + >>> rs_log(1 + x, x, 8) + x + -1/2*x**2 + 1/3*x**3 + -1/4*x**4 + 1/5*x**5 + -1/6*x**6 + 1/7*x**7 + >>> rs_log(x**QQ(3, 2) + 1, x, 5) + x**(3/2) + -1/2*x**3 + 1/3*x**(9/2) + """ + if rs_is_puiseux(p, x): + return rs_puiseux(rs_log, p, x, prec) + R = p.ring + if p == 1: + return R.zero + c = _get_constant_term(p, x) + if c: + const = 0 + if c == 1: + pass + try: + c_expr = c.as_expr() + const = R(log(c_expr)) + except ValueError: + R = R.add_gens([log(c_expr)]) + p = p.set_ring(R) + x = x.set_ring(R) + c = c.set_ring(R) + const = R(log(c_expr)) + + dlog = p.diff(x) + dlog = rs_mul(dlog, _series_inversion1(p, x, prec), x, prec - 1) + return rs_integrate(dlog, x) + const + else: + raise NotImplementedError + +def rs_LambertW(p, x, prec): + """ + Calculate the series expansion of the principal branch of the Lambert W + function. + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import rs_LambertW + >>> R, x, y = ring('x, y', QQ) + >>> rs_LambertW(x + x*y, x, 3) + -x**2*y**2 - 2*x**2*y - x**2 + x*y + x + + See Also + ======== + + LambertW + """ + if rs_is_puiseux(p, x): + return rs_puiseux(rs_LambertW, p, x, prec) + R = p.ring + p1 = R(0) + if _has_constant_term(p, x): + raise NotImplementedError("Polynomial must not have constant term in " + "the series variables") + if x in R.gens: + for precx in _giant_steps(prec): + e = rs_exp(p1, x, precx) + p2 = rs_mul(e, p1, x, precx) - p + p3 = rs_mul(e, p1 + 1, x, precx) + p3 = rs_series_inversion(p3, x, precx) + tmp = rs_mul(p2, p3, x, precx) + p1 -= tmp + return p1 + else: + raise NotImplementedError + +def _exp1(p, x, prec): + r"""Helper function for `rs\_exp`. """ + R = p.ring + p1 = R(1) + for precx in _giant_steps(prec): + pt = p - rs_log(p1, x, precx) + tmp = rs_mul(pt, p1, x, precx) + p1 += tmp + return p1 + +def rs_exp(p, x, prec): + """ + Exponentiation of a series modulo ``O(x**prec)`` + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import rs_exp + >>> R, x = ring('x', QQ) + >>> rs_exp(x**2, x, 7) + 1/6*x**6 + 1/2*x**4 + x**2 + 1 + """ + if rs_is_puiseux(p, x): + return rs_puiseux(rs_exp, p, x, prec) + R = p.ring + c = _get_constant_term(p, x) + if c: + try: + c_expr = c.as_expr() + const = R(exp(c_expr)) + except ValueError: + R = R.add_gens([exp(c_expr)]) + p = p.set_ring(R) + x = x.set_ring(R) + c = c.set_ring(R) + const = R(exp(c_expr)) + + p1 = p - c + + # Makes use of SymPy functions to evaluate the values of the cos/sin + # of the constant term. + return const*rs_exp(p1, x, prec) + + if len(p) > 20: + return _exp1(p, x, prec) + one = R(1) + n = 1 + c = [] + for k in range(prec): + c.append(one/n) + k += 1 + n *= k + + r = rs_series_from_list(p, c, x, prec) + return r + +def _atan(p, iv, prec): + """ + Expansion using formula. + + Faster on very small and univariate series. + """ + R = p.ring + mo = R(-1) + c = [-mo] + p2 = rs_square(p, iv, prec) + for k in range(1, prec): + c.append(mo**k/(2*k + 1)) + s = rs_series_from_list(p2, c, iv, prec) + s = rs_mul(s, p, iv, prec) + return s + +def rs_atan(p, x, prec): + """ + The arctangent of a series + + Return the series expansion of the atan of ``p``, about 0. + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import rs_atan + >>> R, x, y = ring('x, y', QQ) + >>> rs_atan(x + x*y, x, 4) + -1/3*x**3*y**3 - x**3*y**2 - x**3*y - 1/3*x**3 + x*y + x + + See Also + ======== + + atan + """ + if rs_is_puiseux(p, x): + return rs_puiseux(rs_atan, p, x, prec) + R = p.ring + const = 0 + c = _get_constant_term(p, x) + if c: + try: + c_expr = c.as_expr() + const = R(atan(c_expr)) + except ValueError: + R = R.add_gens([atan(c_expr)]) + p = p.set_ring(R) + x = x.set_ring(R) + c = c.set_ring(R) + const = R(atan(c_expr)) + + # Instead of using a closed form formula, we differentiate atan(p) to get + # `1/(1+p**2) * dp`, whose series expansion is much easier to calculate. + # Finally we integrate to get back atan + dp = p.diff(x) + p1 = rs_square(p, x, prec) + R(1) + p1 = rs_series_inversion(p1, x, prec - 1) + p1 = rs_mul(dp, p1, x, prec - 1) + return rs_integrate(p1, x) + const + +def rs_asin(p, x, prec): + """ + Arcsine of a series + + Return the series expansion of the asin of ``p``, about 0. + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import rs_asin + >>> R, x, y = ring('x, y', QQ) + >>> rs_asin(x, x, 8) + 5/112*x**7 + 3/40*x**5 + 1/6*x**3 + x + + See Also + ======== + + asin + """ + if rs_is_puiseux(p, x): + return rs_puiseux(rs_asin, p, x, prec) + if _has_constant_term(p, x): + raise NotImplementedError("Polynomial must not have constant term in " + "series variables") + R = p.ring + if x in R.gens: + # get a good value + if len(p) > 20: + dp = rs_diff(p, x) + p1 = 1 - rs_square(p, x, prec - 1) + p1 = rs_nth_root(p1, -2, x, prec - 1) + p1 = rs_mul(dp, p1, x, prec - 1) + return rs_integrate(p1, x) + one = R(1) + c = [0, one, 0] + for k in range(3, prec, 2): + c.append((k - 2)**2*c[-2]/(k*(k - 1))) + c.append(0) + return rs_series_from_list(p, c, x, prec) + + else: + raise NotImplementedError + +def _tan1(p, x, prec): + r""" + Helper function of :func:`rs_tan`. + + Return the series expansion of tan of a univariate series using Newton's + method. It takes advantage of the fact that series expansion of atan is + easier than that of tan. + + Consider `f(x) = y - \arctan(x)` + Let r be a root of f(x) found using Newton's method. + Then `f(r) = 0` + Or `y = \arctan(x)` where `x = \tan(y)` as required. + """ + R = p.ring + p1 = R(0) + for precx in _giant_steps(prec): + tmp = p - rs_atan(p1, x, precx) + tmp = rs_mul(tmp, 1 + rs_square(p1, x, precx), x, precx) + p1 += tmp + return p1 + +def rs_tan(p, x, prec): + """ + Tangent of a series. + + Return the series expansion of the tan of ``p``, about 0. + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import rs_tan + >>> R, x, y = ring('x, y', QQ) + >>> rs_tan(x + x*y, x, 4) + 1/3*x**3*y**3 + x**3*y**2 + x**3*y + 1/3*x**3 + x*y + x + + See Also + ======== + + _tan1, tan + """ + if rs_is_puiseux(p, x): + r = rs_puiseux(rs_tan, p, x, prec) + return r + R = p.ring + const = 0 + c = _get_constant_term(p, x) + if c: + try: + c_expr = c.as_expr() + const = R(tan(c_expr)) + except ValueError: + R = R.add_gens([tan(c_expr, )]) + p = p.set_ring(R) + x = x.set_ring(R) + c = c.set_ring(R) + const = R(tan(c_expr)) + + p1 = p - c + + # Makes use of SymPy functions to evaluate the values of the cos/sin + # of the constant term. + t2 = rs_tan(p1, x, prec) + t = rs_series_inversion(1 - const*t2, x, prec) + return rs_mul(const + t2, t, x, prec) + + if R.ngens == 1: + return _tan1(p, x, prec) + else: + return rs_fun(p, rs_tan, x, prec) + +def rs_cot(p, x, prec): + """ + Cotangent of a series + + Return the series expansion of the cot of ``p``, about 0. + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import rs_cot + >>> R, x, y = ring('x, y', QQ) + >>> rs_cot(x, x, 6) + -2/945*x**5 - 1/45*x**3 - 1/3*x + x**(-1) + + See Also + ======== + + cot + """ + # It can not handle series like `p = x + x*y` where the coefficient of the + # linear term in the series variable is symbolic. + if rs_is_puiseux(p, x): + r = rs_puiseux(rs_cot, p, x, prec) + return r + i, m = _check_series_var(p, x, 'cot') + prec1 = int(prec + 2*m) + c, s = rs_cos_sin(p, x, prec1) + s = mul_xin(s, i, -m) + s = rs_series_inversion(s, x, prec1) + res = rs_mul(c, s, x, prec1) + res = mul_xin(res, i, -m) + res = rs_trunc(res, x, prec) + return res + +def rs_sin(p, x, prec): + """ + Sine of a series + + Return the series expansion of the sin of ``p``, about 0. + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.puiseux import puiseux_ring + >>> from sympy.polys.ring_series import rs_sin + >>> R, x, y = puiseux_ring('x, y', QQ) + >>> rs_sin(x + x*y, x, 4) + x + x*y + -1/6*x**3 + -1/2*x**3*y + -1/2*x**3*y**2 + -1/6*x**3*y**3 + >>> rs_sin(x**QQ(3, 2) + x*y**QQ(7, 5), x, 4) + x*y**(7/5) + x**(3/2) + -1/6*x**3*y**(21/5) + -1/2*x**(7/2)*y**(14/5) + + See Also + ======== + + sin + """ + if rs_is_puiseux(p, x): + return rs_puiseux(rs_sin, p, x, prec) + R = x.ring + if not p: + return R(0) + c = _get_constant_term(p, x) + if c: + try: + c_expr = c.as_expr() + t1, t2 = R(sin(c_expr)), R(cos(c_expr)) + except ValueError: + R = R.add_gens([sin(c_expr), cos(c_expr)]) + p = p.set_ring(R) + x = x.set_ring(R) + c = c.set_ring(R) + t1, t2 = R(sin(c_expr)), R(cos(c_expr)) + + p1 = p - c + + # Makes use of SymPy cos, sin functions to evaluate the values of the + # cos/sin of the constant term. + p_cos, p_sin = rs_cos_sin(p1, x, prec) + return p_sin*t2 + p_cos*t1 + + # Series is calculated in terms of tan as its evaluation is fast. + if len(p) > 20 and R.ngens == 1: + t = rs_tan(p/2, x, prec) + t2 = rs_square(t, x, prec) + p1 = rs_series_inversion(1 + t2, x, prec) + return rs_mul(p1, 2*t, x, prec) + one = R(1) + n = 1 + c = [0] + for k in range(2, prec + 2, 2): + c.append(one/n) + c.append(0) + n *= -k*(k + 1) + return rs_series_from_list(p, c, x, prec) + +def rs_cos(p, x, prec): + """ + Cosine of a series + + Return the series expansion of the cos of ``p``, about 0. + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.puiseux import puiseux_ring + >>> from sympy.polys.ring_series import rs_cos + >>> R, x, y = puiseux_ring('x, y', QQ) + >>> rs_cos(x + x*y, x, 4) + 1 + -1/2*x**2 + -1*x**2*y + -1/2*x**2*y**2 + >>> rs_cos(x + x*y, x, 4)/x**QQ(7, 5) + x**(-7/5) + -1/2*x**(3/5) + -1*x**(3/5)*y + -1/2*x**(3/5)*y**2 + + See Also + ======== + + cos + """ + if rs_is_puiseux(p, x): + return rs_puiseux(rs_cos, p, x, prec) + R = p.ring + c = _get_constant_term(p, x) + if c: + try: + c_expr = c.as_expr() + t1, t2 = R(sin(c_expr)), R(cos(c_expr)) + except ValueError: + R = R.add_gens([sin(c_expr), cos(c_expr)]) + p = p.set_ring(R) + x = x.set_ring(R) + c = c.set_ring(R) + t1, t2 = R(sin(c_expr)), R(cos(c_expr)) + + p1 = p - c + # Makes use of SymPy cos, sin functions to evaluate the values of the + # cos/sin of the constant term. + p_cos, p_sin = rs_cos_sin(p1, x, prec) + return p_cos*t2 - p_sin*t1 + + # Series is calculated in terms of tan as its evaluation is fast. + if len(p) > 20 and R.ngens == 1: + t = rs_tan(p/2, x, prec) + t2 = rs_square(t, x, prec) + p1 = rs_series_inversion(1+t2, x, prec) + return rs_mul(p1, 1 - t2, x, prec) + one = R(1) + n = 1 + c = [] + for k in range(2, prec + 2, 2): + c.append(one/n) + c.append(0) + n *= -k*(k - 1) + return rs_series_from_list(p, c, x, prec) + +def rs_cos_sin(p, x, prec): + """ + Cosine and sine of a series + + Return the series expansion of the cosine and sine of ``p``, about 0. + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import rs_cos_sin + >>> R, x, y = ring('x, y', QQ) + >>> c, s = rs_cos_sin(x + x*y, x, 4) + >>> c + -1/2*x**2*y**2 - x**2*y - 1/2*x**2 + 1 + >>> s + -1/6*x**3*y**3 - 1/2*x**3*y**2 - 1/2*x**3*y - 1/6*x**3 + x*y + x + + See Also + ======== + + rs_cos, rs_sin + """ + if rs_is_puiseux(p, x): + return rs_puiseux(rs_cos_sin, p, x, prec) + R = p.ring + if not p: + return R(0), R(0) + c = _get_constant_term(p, x) + if c: + try: + c_expr = c.as_expr() + t1, t2 = R(sin(c_expr)), R(cos(c_expr)) + except ValueError: + R = R.add_gens([sin(c_expr), cos(c_expr)]) + p = p.set_ring(R) + x = x.set_ring(R) + c = c.set_ring(R) + t1, t2 = R(sin(c_expr)), R(cos(c_expr)) + + p1 = p - c + p_cos, p_sin = rs_cos_sin(p1, x, prec) + return p_cos*t2 - p_sin*t1, p_cos*t1 + p_sin*t2 + + if len(p) > 20 and R.ngens == 1: + t = rs_tan(p/2, x, prec) + t2 = rs_square(t, x, prec) + p1 = rs_series_inversion(1 + t2, x, prec) + return (rs_mul(p1, 1 - t2, x, prec), rs_mul(p1, 2*t, x, prec)) + + one = R(1) + coeffs = [] + cn, sn = 1, 1 + for k in range(2, prec+2, 2): + coeffs.extend([(one/cn, 0), (0, one/sn)]) + cn, sn = -cn*k*(k - 1), -sn*k*(k + 1) + + c, s = zip(*coeffs) + return (rs_series_from_list(p, c, x, prec), rs_series_from_list(p, s, x, prec)) + +def _atanh(p, x, prec): + """ + Expansion using formula + + Faster for very small and univariate series + """ + R = p.ring + one = R(1) + c = [one] + p2 = rs_square(p, x, prec) + for k in range(1, prec): + c.append(one/(2*k + 1)) + s = rs_series_from_list(p2, c, x, prec) + s = rs_mul(s, p, x, prec) + return s + +def rs_atanh(p, x, prec): + """ + Hyperbolic arctangent of a series + + Return the series expansion of the atanh of ``p``, about 0. + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import rs_atanh + >>> R, x, y = ring('x, y', QQ) + >>> rs_atanh(x + x*y, x, 4) + 1/3*x**3*y**3 + x**3*y**2 + x**3*y + 1/3*x**3 + x*y + x + + See Also + ======== + + atanh + """ + if rs_is_puiseux(p, x): + return rs_puiseux(rs_atanh, p, x, prec) + R = p.ring + const = 0 + c = _get_constant_term(p, x) + if c: + try: + c_expr = c.as_expr() + const = R(atanh(c_expr)) + except ValueError: + raise DomainError("The given series cannot be expanded in " + "this domain.") + + # Instead of using a closed form formula, we differentiate atanh(p) to get + # `1/(1-p**2) * dp`, whose series expansion is much easier to calculate. + # Finally we integrate to get back atanh + dp = rs_diff(p, x) + p1 = - rs_square(p, x, prec) + 1 + p1 = rs_series_inversion(p1, x, prec - 1) + p1 = rs_mul(dp, p1, x, prec - 1) + return rs_integrate(p1, x) + const + +def rs_asinh(p, x, prec): + """ + Hyperbolic arcsine of a series + + Return the series expansion of the arcsinh of ``p``, about 0. + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import rs_asinh + >>> R, x = ring('x', QQ) + >>> rs_asinh(x, x, 9) + -5/112*x**7 + 3/40*x**5 - 1/6*x**3 + x + + See Also + ======== + + asinh + """ + if rs_is_puiseux(p, x): + return rs_puiseux(rs_asinh, p, x, prec) + R = p.ring + const = 0 + c = _get_constant_term(p, x) + if c: + try: + c_expr = c.as_expr() + const = R(asinh(c_expr)) + except ValueError: + raise DomainError("The given series cannot be expanded in " + "this domain.") + + # Instead of using a closed form formula, we differentiate asinh(p) to get + # `1/sqrt(1+p**2) * dp`, whose series expansion is much easier to calculate. + # Finally we integrate to get back asinh + dp = rs_diff(p, x) + p_squared = rs_square(p, x, prec) + denom = p_squared + R(1) + p1 = rs_nth_root(denom, -2, x, prec - 1) + p1 = rs_mul(dp, p1, x, prec - 1) + return rs_integrate(p1, x) + const + +def rs_sinh(p, x, prec): + """ + Hyperbolic sine of a series + + Return the series expansion of the sinh of ``p``, about 0. + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import rs_sinh + >>> R, x, y = ring('x, y', QQ) + >>> rs_sinh(x + x*y, x, 4) + 1/6*x**3*y**3 + 1/2*x**3*y**2 + 1/2*x**3*y + 1/6*x**3 + x*y + x + + See Also + ======== + + sinh + """ + if rs_is_puiseux(p, x): + return rs_puiseux(rs_sinh, p, x, prec) + R = p.ring + if not p: + return R(0) + c = _get_constant_term(p, x) + if c: + try: + c_expr = c.as_expr() + t1, t2 = R(sinh(c_expr)), R(cosh(c_expr)) + except ValueError: + R = R.add_gens([sinh(c_expr), cosh(c_expr)]) + p = p.set_ring(R) + x = x.set_ring(R) + c = c.set_ring(R) + t1, t2 = R(sinh(c_expr)), R(cosh(c_expr)) + + p1 = p - c + p_cosh, p_sinh = rs_cosh_sinh(p1, x, prec) + return p_sinh * t2 + p_cosh * t1 + + t = rs_exp(p, x, prec) + t1 = rs_series_inversion(t, x, prec) + return (t - t1)/2 + +def rs_cosh(p, x, prec): + """ + Hyperbolic cosine of a series + + Return the series expansion of the cosh of ``p``, about 0. + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import rs_cosh + >>> R, x, y = ring('x, y', QQ) + >>> rs_cosh(x + x*y, x, 4) + 1/2*x**2*y**2 + x**2*y + 1/2*x**2 + 1 + + See Also + ======== + + cosh + """ + if rs_is_puiseux(p, x): + return rs_puiseux(rs_cosh, p, x, prec) + R = p.ring + if not p: + return R(0) + c = _get_constant_term(p, x) + if c: + try: + c_expr = c.as_expr() + t1, t2 = R(sinh(c_expr)), R(cosh(c_expr)) + except ValueError: + R = R.add_gens([sinh(c_expr), cosh(c_expr)]) + p = p.set_ring(R) + x = x.set_ring(R) + c = c.set_ring(R) + t1, t2 = R(sinh(c_expr)), R(cosh(c_expr)) + + p1 = p - c + p_cosh, p_sinh = rs_cosh_sinh(p1, x, prec) + return p_cosh * t2 + p_sinh * t1 + + t = rs_exp(p, x, prec) + t1 = rs_series_inversion(t, x, prec) + return (t + t1)/2 + +def rs_cosh_sinh(p, x, prec): + """ + Hyperbolic cosine and sine of a series + + Return the series expansion of the hyperbolic cosine and sine of ``p``, about 0. + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import rs_cosh_sinh + >>> R, x, y = ring('x, y', QQ) + >>> c, s = rs_cosh_sinh(x + x*y, x, 4) + >>> c + 1/2*x**2*y**2 + x**2*y + 1/2*x**2 + 1 + >>> s + 1/6*x**3*y**3 + 1/2*x**3*y**2 + 1/2*x**3*y + 1/6*x**3 + x*y + x + + See Also + ======== + + rs_cosh, rs_sinh + """ + if rs_is_puiseux(p, x): + return rs_puiseux(rs_cosh_sinh, p, x, prec) + R = p.ring + if not p: + return R(0), R(0) + c = _get_constant_term(p, x) + if c: + try: + c_expr = c.as_expr() + t1, t2 = R(sinh(c_expr)), R(cosh(c_expr)) + except ValueError: + R = R.add_gens([sinh(c_expr), cosh(c_expr)]) + p = p.set_ring(R) + x = x.set_ring(R) + c = c.set_ring(R) + t1, t2 = R(sinh(c_expr)), R(cosh(c_expr)) + + p1 = p - c + p_cosh, p_sinh = rs_cosh_sinh(p1, x, prec) + return p_cosh * t2 + p_sinh * t1, p_sinh * t2 + p_cosh * t1 + + t = rs_exp(p, x, prec) + t1 = rs_series_inversion(t, x, prec) + return (t + t1)/2, (t - t1)/2 + + +def _tanh(p, x, prec): + r""" + Helper function of :func:`rs_tanh` + + Return the series expansion of tanh of a univariate series using Newton's + method. It takes advantage of the fact that series expansion of atanh is + easier than that of tanh. + + See Also + ======== + + _tanh + """ + R = p.ring + p1 = R(0) + for precx in _giant_steps(prec): + tmp = p - rs_atanh(p1, x, precx) + tmp = rs_mul(tmp, 1 - rs_square(p1, x, prec), x, precx) + p1 += tmp + return p1 + +def rs_tanh(p, x, prec): + """ + Hyperbolic tangent of a series + + Return the series expansion of the tanh of ``p``, about 0. + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import rs_tanh + >>> R, x, y = ring('x, y', QQ) + >>> rs_tanh(x + x*y, x, 4) + -1/3*x**3*y**3 - x**3*y**2 - x**3*y - 1/3*x**3 + x*y + x + + See Also + ======== + + tanh + """ + if rs_is_puiseux(p, x): + return rs_puiseux(rs_tanh, p, x, prec) + R = p.ring + const = 0 + c = _get_constant_term(p, x) + if c: + try: + c_expr = c.as_expr() + const = R(tanh(c_expr)) + except ValueError: + R = R.add_gens([tanh(c_expr)]) + p = p.set_ring(R) + x = x.set_ring(R) + c = c.set_ring(R) + const = R(tanh(c_expr)) + + p1 = p - c + t1 = rs_tanh(p1, x, prec) + t = rs_series_inversion(1 + const*t1, x, prec) + return rs_mul(const + t1, t, x, prec) + + if R.ngens == 1: + return _tanh(p, x, prec) + else: + return rs_fun(p, _tanh, x, prec) + +def rs_newton(p, x, prec): + """ + Compute the truncated Newton sum of the polynomial ``p`` + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import rs_newton + >>> R, x = ring('x', QQ) + >>> p = x**2 - 2 + >>> rs_newton(p, x, 5) + 8*x**4 + 4*x**2 + 2 + """ + deg = p.degree() + p1 = _invert_monoms(p) + p2 = rs_series_inversion(p1, x, prec) + p3 = rs_mul(p1.diff(x), p2, x, prec) + res = deg - p3*x + return res + +def rs_hadamard_exp(p1, inverse=False): + """ + Return ``sum f_i/i!*x**i`` from ``sum f_i*x**i``, + where ``x`` is the first variable. + + If ``inverse=True`` return ``sum f_i*i!*x**i`` + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import rs_hadamard_exp + >>> R, x = ring('x', QQ) + >>> p = 1 + x + x**2 + x**3 + >>> rs_hadamard_exp(p) + 1/6*x**3 + 1/2*x**2 + x + 1 + """ + R = p1.ring + if R.domain != QQ: + raise NotImplementedError + p = R.zero + if not inverse: + for exp1, v1 in p1.items(): + p[exp1] = v1/int(ifac(exp1[0])) + else: + for exp1, v1 in p1.items(): + p[exp1] = v1*int(ifac(exp1[0])) + return p + +def rs_compose_add(p1, p2): + """ + compute the composed sum ``prod(p2(x - beta) for beta root of p1)`` + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + >>> from sympy.polys.ring_series import rs_compose_add + >>> R, x = ring('x', QQ) + >>> f = x**2 - 2 + >>> g = x**2 - 3 + >>> rs_compose_add(f, g) + x**4 - 10*x**2 + 1 + + References + ========== + + .. [1] A. Bostan, P. Flajolet, B. Salvy and E. Schost + "Fast Computation with Two Algebraic Numbers", + (2002) Research Report 4579, Institut + National de Recherche en Informatique et en Automatique + """ + R = p1.ring + x = R.gens[0] + prec = p1.degree()*p2.degree() + 1 + np1 = rs_newton(p1, x, prec) + np1e = rs_hadamard_exp(np1) + np2 = rs_newton(p2, x, prec) + np2e = rs_hadamard_exp(np2) + np3e = rs_mul(np1e, np2e, x, prec) + np3 = rs_hadamard_exp(np3e, True) + np3a = (np3[(0,)] - np3) / x + q = rs_integrate(np3a, x) + q = rs_exp(q, x, prec) + q = _invert_monoms(q) + q = q.primitive()[1] + dp = p1.degree()*p2.degree() - q.degree() + # `dp` is the multiplicity of the zeroes of the resultant; + # these zeroes are missed in this computation so they are put here. + # if p1 and p2 are monic irreducible polynomials, + # there are zeroes in the resultant + # if and only if p1 = p2 ; in fact in that case p1 and p2 have a + # root in common, so gcd(p1, p2) != 1; being p1 and p2 irreducible + # this means p1 = p2 + if dp: + q = q*x**dp + return q + + +_convert_func = { + 'sin': 'rs_sin', + 'cos': 'rs_cos', + 'exp': 'rs_exp', + 'tan': 'rs_tan', + 'log': 'rs_log', + 'atan': 'rs_atan', + 'sinh': 'rs_sinh', + 'cosh': 'rs_cosh', + 'tanh': 'rs_tanh' + } + +def rs_min_pow(expr, series_rs, a): + """Find the minimum power of `a` in the series expansion of expr""" + series = 0 + n = 2 + while series == 0: + series = _rs_series(expr, series_rs, a, n) + n *= 2 + R = series.ring + a = R(a) + i = R.gens.index(a) + return min(series, key=lambda t: t[i])[i] + + +def _rs_series(expr, series_rs, a, prec): + # TODO Use _parallel_dict_from_expr instead of sring as sring is + # inefficient. For details, read the todo in sring. + args = expr.args + R = series_rs.ring + + # expr does not contain any function to be expanded + if not any(arg.has(Function) for arg in args) and not expr.is_Function: + return series_rs + + if not expr.has(a): + return series_rs + + elif expr.is_Function: + arg = args[0] + if len(args) > 1: + raise NotImplementedError + R1, series = sring(arg, domain=QQ, expand=False, series=True) + series_inner = _rs_series(arg, series, a, prec) + + # Why do we need to compose these three rings? + # + # We want to use a simple domain (like ``QQ`` or ``RR``) but they don't + # support symbolic coefficients. We need a ring that for example lets + # us have `sin(1)` and `cos(1)` as coefficients if we are expanding + # `sin(x + 1)`. The ``EX`` domain allows all symbolic coefficients, but + # that makes it very complex and hence slow. + # + # To solve this problem, we add only those symbolic elements as + # generators to our ring, that we need. Here, series_inner might + # involve terms like `sin(4)`, `exp(a)`, etc, which are not there in + # R1 or R. Hence, we compose these three rings to create one that has + # the generators of all three. + R = R.compose(R1).compose(series_inner.ring) + series_inner = series_inner.set_ring(R) + series = eval(_convert_func[str(expr.func)])(series_inner, + R(a), prec) + return series + + elif expr.is_Mul: + n = len(args) + for arg in args: # XXX Looks redundant + if not arg.is_Number: + R1, _ = sring(arg, expand=False, series=True) + R = R.compose(R1) + min_pows = list(map(rs_min_pow, args, [R(arg) for arg in args], + [a]*len(args))) + sum_pows = sum(min_pows) + series = R(1) + + for i in range(n): + _series = _rs_series(args[i], R(args[i]), a, ceiling(prec + - sum_pows + min_pows[i])) + R = R.compose(_series.ring) + _series = _series.set_ring(R) + series = series.set_ring(R) + series *= _series + series = rs_trunc(series, R(a), prec) + return series + + elif expr.is_Add: + n = len(args) + series = R(0) + for i in range(n): + _series = _rs_series(args[i], R(args[i]), a, prec) + R = R.compose(_series.ring) + _series = _series.set_ring(R) + series = series.set_ring(R) + series += _series + return series + + elif expr.is_Pow: + R1, _ = sring(expr.base, domain=QQ, expand=False, series=True) + R = R.compose(R1) + series_inner = _rs_series(expr.base, R(expr.base), a, prec) + return rs_pow(series_inner, expr.exp, series_inner.ring(a), prec) + + # The `is_constant` method is buggy hence we check it at the end. + # See issue #9786 for details. + elif isinstance(expr, Expr) and expr.is_constant(): + return sring(expr, domain=QQ, expand=False, series=True)[1] + + else: + raise NotImplementedError + +def rs_series(expr, a, prec): + """Return the series expansion of an expression about 0. + + Parameters + ========== + + expr : :class:`~.Expr` + a : :class:`~.Symbol` with respect to which expr is to be expanded + prec : order of the series expansion + + Currently supports multivariate Taylor series expansion. This is much + faster that SymPy's series method as it uses sparse polynomial operations. + + It automatically creates the simplest ring required to represent the series + expansion through repeated calls to sring. + + Examples + ======== + + >>> from sympy.polys.ring_series import rs_series + >>> from sympy import sin, cos, exp, tan, symbols, QQ + >>> a, b, c = symbols('a, b, c') + >>> rs_series(sin(a) + exp(a), a, 5) + 1/24*a**4 + 1/2*a**2 + 2*a + 1 + >>> series = rs_series(tan(a + b)*cos(a + c), a, 2) + >>> series.as_expr() + -a*sin(c)*tan(b) + a*cos(c)*tan(b)**2 + a*cos(c) + cos(c)*tan(b) + >>> series = rs_series(exp(a**QQ(1,3) + a**QQ(2, 5)), a, 1) + >>> series.as_expr() + a**(11/15) + a**(4/5)/2 + a**(2/5) + a**(2/3)/2 + a**(1/3) + 1 + + """ + R, series = sring(expr, domain=QQ, expand=False, series=True) + if a not in R.symbols: + R = R.add_gens([a, ]) + series = series.set_ring(R) + series = _rs_series(expr, series, a, prec) + R = series.ring + gen = R(a) + prec_got = series.degree(gen) + 1 + + if prec_got >= prec: + return rs_trunc(series, gen, prec) + else: + # increase the requested number of terms to get the desired + # number keep increasing (up to 9) until the received order + # is different than the original order and then predict how + # many additional terms are needed + for more in range(1, 9): + p1 = _rs_series(expr, series, a, prec=prec + more) + gen = gen.set_ring(p1.ring) + new_prec = p1.degree(gen) + 1 + if new_prec != prec_got: + prec_do = ceiling(prec + (prec - prec_got)*more/(new_prec - + prec_got)) + p1 = _rs_series(expr, series, a, prec=prec_do) + while p1.degree(gen) + 1 < prec: + p1 = _rs_series(expr, series, a, prec=prec_do) + gen = gen.set_ring(p1.ring) + prec_do *= 2 + break + else: + break + else: + raise ValueError('Could not calculate %s terms for %s' + % (str(prec), expr)) + return rs_trunc(p1, gen, prec) diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/rings.py b/.venv/lib/python3.12/site-packages/sympy/polys/rings.py new file mode 100644 index 0000000000000000000000000000000000000000..9df84dcf1691ac9bcd4aa01a85ca34b7ffc53e5d --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/rings.py @@ -0,0 +1,3096 @@ +"""Sparse polynomial rings. """ + +from __future__ import annotations + +from operator import add, mul, lt, le, gt, ge +from functools import reduce +from types import GeneratorType + +from sympy.core.cache import cacheit +from sympy.core.expr import Expr +from sympy.core.intfunc import igcd +from sympy.core.symbol import Symbol, symbols as _symbols +from sympy.core.sympify import CantSympify, sympify +from sympy.ntheory.multinomial import multinomial_coefficients +from sympy.polys.compatibility import IPolys +from sympy.polys.constructor import construct_domain +from sympy.polys.densebasic import ninf, dmp_to_dict, dmp_from_dict +from sympy.polys.domains.domain import Domain +from sympy.polys.domains.domainelement import DomainElement +from sympy.polys.domains.polynomialring import PolynomialRing +from sympy.polys.heuristicgcd import heugcd +from sympy.polys.monomials import MonomialOps +from sympy.polys.orderings import lex, MonomialOrder +from sympy.polys.polyerrors import ( + CoercionFailed, GeneratorsError, + ExactQuotientFailed, MultivariatePolynomialError) +from sympy.polys.polyoptions import (Domain as DomainOpt, + Order as OrderOpt, build_options) +from sympy.polys.polyutils import (expr_from_dict, _dict_reorder, + _parallel_dict_from_expr) +from sympy.printing.defaults import DefaultPrinting +from sympy.utilities import public, subsets +from sympy.utilities.iterables import is_sequence +from sympy.utilities.magic import pollute + +@public +def ring(symbols, domain, order: MonomialOrder|str = lex): + """Construct a polynomial ring returning ``(ring, x_1, ..., x_n)``. + + Parameters + ========== + + symbols : str + Symbol/Expr or sequence of str, Symbol/Expr (non-empty) + domain : :class:`~.Domain` or coercible + order : :class:`~.MonomialOrder` or coercible, optional, defaults to ``lex`` + + Examples + ======== + + >>> from sympy.polys.rings import ring + >>> from sympy.polys.domains import ZZ + >>> from sympy.polys.orderings import lex + + >>> R, x, y, z = ring("x,y,z", ZZ, lex) + >>> R + Polynomial ring in x, y, z over ZZ with lex order + >>> x + y + z + x + y + z + >>> type(_) + + + """ + _ring = PolyRing(symbols, domain, order) + return (_ring,) + _ring.gens + +@public +def xring(symbols, domain, order=lex): + """Construct a polynomial ring returning ``(ring, (x_1, ..., x_n))``. + + Parameters + ========== + + symbols : str + Symbol/Expr or sequence of str, Symbol/Expr (non-empty) + domain : :class:`~.Domain` or coercible + order : :class:`~.MonomialOrder` or coercible, optional, defaults to ``lex`` + + Examples + ======== + + >>> from sympy.polys.rings import xring + >>> from sympy.polys.domains import ZZ + >>> from sympy.polys.orderings import lex + + >>> R, (x, y, z) = xring("x,y,z", ZZ, lex) + >>> R + Polynomial ring in x, y, z over ZZ with lex order + >>> x + y + z + x + y + z + >>> type(_) + + + """ + _ring = PolyRing(symbols, domain, order) + return (_ring, _ring.gens) + +@public +def vring(symbols, domain, order=lex): + """Construct a polynomial ring and inject ``x_1, ..., x_n`` into the global namespace. + + Parameters + ========== + + symbols : str + Symbol/Expr or sequence of str, Symbol/Expr (non-empty) + domain : :class:`~.Domain` or coercible + order : :class:`~.MonomialOrder` or coercible, optional, defaults to ``lex`` + + Examples + ======== + + >>> from sympy.polys.rings import vring + >>> from sympy.polys.domains import ZZ + >>> from sympy.polys.orderings import lex + + >>> vring("x,y,z", ZZ, lex) + Polynomial ring in x, y, z over ZZ with lex order + >>> x + y + z # noqa: + x + y + z + >>> type(_) + + + """ + _ring = PolyRing(symbols, domain, order) + pollute([ sym.name for sym in _ring.symbols ], _ring.gens) + return _ring + +@public +def sring(exprs, *symbols, **options): + """Construct a ring deriving generators and domain from options and input expressions. + + Parameters + ========== + + exprs : :class:`~.Expr` or sequence of :class:`~.Expr` (sympifiable) + symbols : sequence of :class:`~.Symbol`/:class:`~.Expr` + options : keyword arguments understood by :class:`~.Options` + + Examples + ======== + + >>> from sympy import sring, symbols + + >>> x, y, z = symbols("x,y,z") + >>> R, f = sring(x + 2*y + 3*z) + >>> R + Polynomial ring in x, y, z over ZZ with lex order + >>> f + x + 2*y + 3*z + >>> type(_) + + + """ + single = False + + if not is_sequence(exprs): + exprs, single = [exprs], True + + exprs = list(map(sympify, exprs)) + opt = build_options(symbols, options) + + # TODO: rewrite this so that it doesn't use expand() (see poly()). + reps, opt = _parallel_dict_from_expr(exprs, opt) + + if opt.domain is None: + coeffs = sum([ list(rep.values()) for rep in reps ], []) + + opt.domain, coeffs_dom = construct_domain(coeffs, opt=opt) + + coeff_map = dict(zip(coeffs, coeffs_dom)) + reps = [{m: coeff_map[c] for m, c in rep.items()} for rep in reps] + + _ring = PolyRing(opt.gens, opt.domain, opt.order) + polys = list(map(_ring.from_dict, reps)) + + if single: + return (_ring, polys[0]) + else: + return (_ring, polys) + +def _parse_symbols(symbols): + if isinstance(symbols, str): + return _symbols(symbols, seq=True) if symbols else () + elif isinstance(symbols, Expr): + return (symbols,) + elif is_sequence(symbols): + if all(isinstance(s, str) for s in symbols): + return _symbols(symbols) + elif all(isinstance(s, Expr) for s in symbols): + return symbols + + raise GeneratorsError("expected a string, Symbol or expression or a non-empty sequence of strings, Symbols or expressions") + + +class PolyRing(DefaultPrinting, IPolys): + """Multivariate distributed polynomial ring. """ + + gens: tuple[PolyElement, ...] + symbols: tuple[Expr, ...] + ngens: int + domain: Domain + order: MonomialOrder + + def __new__(cls, symbols, domain, order=lex): + symbols = tuple(_parse_symbols(symbols)) + ngens = len(symbols) + domain = DomainOpt.preprocess(domain) + order = OrderOpt.preprocess(order) + + _hash_tuple = (cls.__name__, symbols, ngens, domain, order) + + if domain.is_Composite and set(symbols) & set(domain.symbols): + raise GeneratorsError("polynomial ring and it's ground domain share generators") + + obj = object.__new__(cls) + obj._hash_tuple = _hash_tuple + obj._hash = hash(_hash_tuple) + obj.symbols = symbols + obj.ngens = ngens + obj.domain = domain + obj.order = order + + obj.dtype = PolyElement(obj, ()).new + + obj.zero_monom = (0,)*ngens + obj.gens = obj._gens() + obj._gens_set = set(obj.gens) + + obj._one = [(obj.zero_monom, domain.one)] + + if ngens: + # These expect monomials in at least one variable + codegen = MonomialOps(ngens) + obj.monomial_mul = codegen.mul() + obj.monomial_pow = codegen.pow() + obj.monomial_mulpow = codegen.mulpow() + obj.monomial_ldiv = codegen.ldiv() + obj.monomial_div = codegen.div() + obj.monomial_lcm = codegen.lcm() + obj.monomial_gcd = codegen.gcd() + else: + monunit = lambda a, b: () + obj.monomial_mul = monunit + obj.monomial_pow = monunit + obj.monomial_mulpow = lambda a, b, c: () + obj.monomial_ldiv = monunit + obj.monomial_div = monunit + obj.monomial_lcm = monunit + obj.monomial_gcd = monunit + + + if order is lex: + obj.leading_expv = max + else: + obj.leading_expv = lambda f: max(f, key=order) + + for symbol, generator in zip(obj.symbols, obj.gens): + if isinstance(symbol, Symbol): + name = symbol.name + + if not hasattr(obj, name): + setattr(obj, name, generator) + + return obj + + def _gens(self): + """Return a list of polynomial generators. """ + one = self.domain.one + _gens = [] + for i in range(self.ngens): + expv = self.monomial_basis(i) + poly = self.zero + poly[expv] = one + _gens.append(poly) + return tuple(_gens) + + def __getnewargs__(self): + return (self.symbols, self.domain, self.order) + + def __getstate__(self): + state = self.__dict__.copy() + del state["leading_expv"] + + for key in state: + if key.startswith("monomial_"): + del state[key] + + return state + + def __hash__(self): + return self._hash + + def __eq__(self, other): + return isinstance(other, PolyRing) and \ + (self.symbols, self.domain, self.ngens, self.order) == \ + (other.symbols, other.domain, other.ngens, other.order) + + def __ne__(self, other): + return not self == other + + def clone(self, symbols=None, domain=None, order=None): + # Need a hashable tuple for cacheit to work + if symbols is not None and isinstance(symbols, list): + symbols = tuple(symbols) + return self._clone(symbols, domain, order) + + @cacheit + def _clone(self, symbols, domain, order): + return self.__class__(symbols or self.symbols, domain or self.domain, order or self.order) + + def monomial_basis(self, i): + """Return the ith-basis element. """ + basis = [0]*self.ngens + basis[i] = 1 + return tuple(basis) + + @property + def zero(self): + return self.dtype([]) + + @property + def one(self): + return self.dtype(self._one) + + def is_element(self, element): + """True if ``element`` is an element of this ring. False otherwise. """ + return isinstance(element, PolyElement) and element.ring == self + + def domain_new(self, element, orig_domain=None): + return self.domain.convert(element, orig_domain) + + def ground_new(self, coeff): + return self.term_new(self.zero_monom, coeff) + + def term_new(self, monom, coeff): + coeff = self.domain_new(coeff) + poly = self.zero + if coeff: + poly[monom] = coeff + return poly + + def ring_new(self, element): + if isinstance(element, PolyElement): + if self == element.ring: + return element + elif isinstance(self.domain, PolynomialRing) and self.domain.ring == element.ring: + return self.ground_new(element) + else: + raise NotImplementedError("conversion") + elif isinstance(element, str): + raise NotImplementedError("parsing") + elif isinstance(element, dict): + return self.from_dict(element) + elif isinstance(element, list): + try: + return self.from_terms(element) + except ValueError: + return self.from_list(element) + elif isinstance(element, Expr): + return self.from_expr(element) + else: + return self.ground_new(element) + + __call__ = ring_new + + def from_dict(self, element, orig_domain=None): + domain_new = self.domain_new + poly = self.zero + + for monom, coeff in element.items(): + coeff = domain_new(coeff, orig_domain) + if coeff: + poly[monom] = coeff + + return poly + + def from_terms(self, element, orig_domain=None): + return self.from_dict(dict(element), orig_domain) + + def from_list(self, element): + return self.from_dict(dmp_to_dict(element, self.ngens-1, self.domain)) + + def _rebuild_expr(self, expr, mapping): + domain = self.domain + + def _rebuild(expr): + generator = mapping.get(expr) + + if generator is not None: + return generator + elif expr.is_Add: + return reduce(add, list(map(_rebuild, expr.args))) + elif expr.is_Mul: + return reduce(mul, list(map(_rebuild, expr.args))) + else: + # XXX: Use as_base_exp() to handle Pow(x, n) and also exp(n) + # XXX: E can be a generator e.g. sring([exp(2)]) -> ZZ[E] + base, exp = expr.as_base_exp() + if exp.is_Integer and exp > 1: + return _rebuild(base)**int(exp) + else: + return self.ground_new(domain.convert(expr)) + + return _rebuild(sympify(expr)) + + def from_expr(self, expr): + mapping = dict(list(zip(self.symbols, self.gens))) + + try: + poly = self._rebuild_expr(expr, mapping) + except CoercionFailed: + raise ValueError("expected an expression convertible to a polynomial in %s, got %s" % (self, expr)) + else: + return self.ring_new(poly) + + def index(self, gen): + """Compute index of ``gen`` in ``self.gens``. """ + if gen is None: + if self.ngens: + i = 0 + else: + i = -1 # indicate impossible choice + elif isinstance(gen, int): + i = gen + + if 0 <= i and i < self.ngens: + pass + elif -self.ngens <= i and i <= -1: + i = -i - 1 + else: + raise ValueError("invalid generator index: %s" % gen) + elif self.is_element(gen): + try: + i = self.gens.index(gen) + except ValueError: + raise ValueError("invalid generator: %s" % gen) + elif isinstance(gen, str): + try: + i = self.symbols.index(gen) + except ValueError: + raise ValueError("invalid generator: %s" % gen) + else: + raise ValueError("expected a polynomial generator, an integer, a string or None, got %s" % gen) + + return i + + def drop(self, *gens): + """Remove specified generators from this ring. """ + indices = set(map(self.index, gens)) + symbols = [ s for i, s in enumerate(self.symbols) if i not in indices ] + + if not symbols: + return self.domain + else: + return self.clone(symbols=symbols) + + def __getitem__(self, key): + symbols = self.symbols[key] + + if not symbols: + return self.domain + else: + return self.clone(symbols=symbols) + + def to_ground(self): + # TODO: should AlgebraicField be a Composite domain? + if self.domain.is_Composite or hasattr(self.domain, 'domain'): + return self.clone(domain=self.domain.domain) + else: + raise ValueError("%s is not a composite domain" % self.domain) + + def to_domain(self): + return PolynomialRing(self) + + def to_field(self): + from sympy.polys.fields import FracField + return FracField(self.symbols, self.domain, self.order) + + @property + def is_univariate(self): + return len(self.gens) == 1 + + @property + def is_multivariate(self): + return len(self.gens) > 1 + + def add(self, *objs): + """ + Add a sequence of polynomials or containers of polynomials. + + Examples + ======== + + >>> from sympy.polys.rings import ring + >>> from sympy.polys.domains import ZZ + + >>> R, x = ring("x", ZZ) + >>> R.add([ x**2 + 2*i + 3 for i in range(4) ]) + 4*x**2 + 24 + >>> _.factor_list() + (4, [(x**2 + 6, 1)]) + + """ + p = self.zero + + for obj in objs: + if is_sequence(obj, include=GeneratorType): + p += self.add(*obj) + else: + p += obj + + return p + + def mul(self, *objs): + """ + Multiply a sequence of polynomials or containers of polynomials. + + Examples + ======== + + >>> from sympy.polys.rings import ring + >>> from sympy.polys.domains import ZZ + + >>> R, x = ring("x", ZZ) + >>> R.mul([ x**2 + 2*i + 3 for i in range(4) ]) + x**8 + 24*x**6 + 206*x**4 + 744*x**2 + 945 + >>> _.factor_list() + (1, [(x**2 + 3, 1), (x**2 + 5, 1), (x**2 + 7, 1), (x**2 + 9, 1)]) + + """ + p = self.one + + for obj in objs: + if is_sequence(obj, include=GeneratorType): + p *= self.mul(*obj) + else: + p *= obj + + return p + + def drop_to_ground(self, *gens): + r""" + Remove specified generators from the ring and inject them into + its domain. + """ + indices = set(map(self.index, gens)) + symbols = [s for i, s in enumerate(self.symbols) if i not in indices] + gens = [gen for i, gen in enumerate(self.gens) if i not in indices] + + if not symbols: + return self + else: + return self.clone(symbols=symbols, domain=self.drop(*gens)) + + def compose(self, other): + """Add the generators of ``other`` to ``self``""" + if self != other: + syms = set(self.symbols).union(set(other.symbols)) + return self.clone(symbols=list(syms)) + else: + return self + + def add_gens(self, symbols): + """Add the elements of ``symbols`` as generators to ``self``""" + syms = set(self.symbols).union(set(symbols)) + return self.clone(symbols=list(syms)) + + def symmetric_poly(self, n): + """ + Return the elementary symmetric polynomial of degree *n* over + this ring's generators. + """ + if n < 0 or n > self.ngens: + raise ValueError("Cannot generate symmetric polynomial of order %s for %s" % (n, self.gens)) + elif not n: + return self.one + else: + poly = self.zero + for s in subsets(range(self.ngens), int(n)): + monom = tuple(int(i in s) for i in range(self.ngens)) + poly += self.term_new(monom, self.domain.one) + return poly + + +class PolyElement(DomainElement, DefaultPrinting, CantSympify, dict): + """Element of multivariate distributed polynomial ring. """ + + def __init__(self, ring, init): + super().__init__(init) + self.ring = ring + # This check would be too slow to run every time: + # self._check() + + def _check(self): + assert isinstance(self, PolyElement) + assert isinstance(self.ring, PolyRing) + dom = self.ring.domain + assert isinstance(dom, Domain) + for monom, coeff in self.items(): + assert dom.of_type(coeff) + assert len(monom) == self.ring.ngens + assert all(isinstance(exp, int) and exp >= 0 for exp in monom) + + def new(self, init): + return self.__class__(self.ring, init) + + def parent(self): + return self.ring.to_domain() + + def __getnewargs__(self): + return (self.ring, list(self.iterterms())) + + _hash = None + + def __hash__(self): + # XXX: This computes a hash of a dictionary, but currently we don't + # protect dictionary from being changed so any use site modifications + # will make hashing go wrong. Use this feature with caution until we + # figure out how to make a safe API without compromising speed of this + # low-level class. + _hash = self._hash + if _hash is None: + self._hash = _hash = hash((self.ring, frozenset(self.items()))) + return _hash + + def copy(self): + """Return a copy of polynomial self. + + Polynomials are mutable; if one is interested in preserving + a polynomial, and one plans to use inplace operations, one + can copy the polynomial. This method makes a shallow copy. + + Examples + ======== + + >>> from sympy.polys.domains import ZZ + >>> from sympy.polys.rings import ring + + >>> R, x, y = ring('x, y', ZZ) + >>> p = (x + y)**2 + >>> p1 = p.copy() + >>> p2 = p + >>> p[R.zero_monom] = 3 + >>> p + x**2 + 2*x*y + y**2 + 3 + >>> p1 + x**2 + 2*x*y + y**2 + >>> p2 + x**2 + 2*x*y + y**2 + 3 + + """ + return self.new(self) + + def set_ring(self, new_ring): + if self.ring == new_ring: + return self + elif self.ring.symbols != new_ring.symbols: + terms = list(zip(*_dict_reorder(self, self.ring.symbols, new_ring.symbols))) + return new_ring.from_terms(terms, self.ring.domain) + else: + return new_ring.from_dict(self, self.ring.domain) + + def as_expr(self, *symbols): + if not symbols: + symbols = self.ring.symbols + elif len(symbols) != self.ring.ngens: + raise ValueError( + "Wrong number of symbols, expected %s got %s" % + (self.ring.ngens, len(symbols)) + ) + + return expr_from_dict(self.as_expr_dict(), *symbols) + + def as_expr_dict(self): + to_sympy = self.ring.domain.to_sympy + return {monom: to_sympy(coeff) for monom, coeff in self.iterterms()} + + def clear_denoms(self): + domain = self.ring.domain + + if not domain.is_Field or not domain.has_assoc_Ring: + return domain.one, self + + ground_ring = domain.get_ring() + common = ground_ring.one + lcm = ground_ring.lcm + denom = domain.denom + + for coeff in self.values(): + common = lcm(common, denom(coeff)) + + poly = self.new([ (k, v*common) for k, v in self.items() ]) + return common, poly + + def strip_zero(self): + """Eliminate monomials with zero coefficient. """ + for k, v in list(self.items()): + if not v: + del self[k] + + def __eq__(p1, p2): + """Equality test for polynomials. + + Examples + ======== + + >>> from sympy.polys.domains import ZZ + >>> from sympy.polys.rings import ring + + >>> _, x, y = ring('x, y', ZZ) + >>> p1 = (x + y)**2 + (x - y)**2 + >>> p1 == 4*x*y + False + >>> p1 == 2*(x**2 + y**2) + True + + """ + if not p2: + return not p1 + elif p1.ring.is_element(p2): + return dict.__eq__(p1, p2) + elif len(p1) > 1: + return False + else: + return p1.get(p1.ring.zero_monom) == p2 + + def __ne__(p1, p2): + return not p1 == p2 + + def almosteq(p1, p2, tolerance=None): + """Approximate equality test for polynomials. """ + ring = p1.ring + + if ring.is_element(p2): + if set(p1.keys()) != set(p2.keys()): + return False + + almosteq = ring.domain.almosteq + + for k in p1.keys(): + if not almosteq(p1[k], p2[k], tolerance): + return False + return True + elif len(p1) > 1: + return False + else: + try: + p2 = ring.domain.convert(p2) + except CoercionFailed: + return False + else: + return ring.domain.almosteq(p1.const(), p2, tolerance) + + def sort_key(self): + return (len(self), self.terms()) + + def _cmp(p1, p2, op): + if p1.ring.is_element(p2): + return op(p1.sort_key(), p2.sort_key()) + else: + return NotImplemented + + def __lt__(p1, p2): + return p1._cmp(p2, lt) + def __le__(p1, p2): + return p1._cmp(p2, le) + def __gt__(p1, p2): + return p1._cmp(p2, gt) + def __ge__(p1, p2): + return p1._cmp(p2, ge) + + def _drop(self, gen): + ring = self.ring + i = ring.index(gen) + + if ring.ngens == 1: + return i, ring.domain + else: + symbols = list(ring.symbols) + del symbols[i] + return i, ring.clone(symbols=symbols) + + def drop(self, gen): + i, ring = self._drop(gen) + + if self.ring.ngens == 1: + if self.is_ground: + return self.coeff(1) + else: + raise ValueError("Cannot drop %s" % gen) + else: + poly = ring.zero + + for k, v in self.items(): + if k[i] == 0: + K = list(k) + del K[i] + poly[tuple(K)] = v + else: + raise ValueError("Cannot drop %s" % gen) + + return poly + + def _drop_to_ground(self, gen): + ring = self.ring + i = ring.index(gen) + + symbols = list(ring.symbols) + del symbols[i] + return i, ring.clone(symbols=symbols, domain=ring[i]) + + def drop_to_ground(self, gen): + if self.ring.ngens == 1: + raise ValueError("Cannot drop only generator to ground") + + i, ring = self._drop_to_ground(gen) + poly = ring.zero + gen = ring.domain.gens[0] + + for monom, coeff in self.iterterms(): + mon = monom[:i] + monom[i+1:] + if mon not in poly: + poly[mon] = (gen**monom[i]).mul_ground(coeff) + else: + poly[mon] += (gen**monom[i]).mul_ground(coeff) + + return poly + + def to_dense(self): + return dmp_from_dict(self, self.ring.ngens-1, self.ring.domain) + + def to_dict(self): + return dict(self) + + def str(self, printer, precedence, exp_pattern, mul_symbol): + if not self: + return printer._print(self.ring.domain.zero) + prec_mul = precedence["Mul"] + prec_atom = precedence["Atom"] + ring = self.ring + symbols = ring.symbols + ngens = ring.ngens + zm = ring.zero_monom + sexpvs = [] + for expv, coeff in self.terms(): + negative = ring.domain.is_negative(coeff) + sign = " - " if negative else " + " + sexpvs.append(sign) + if expv == zm: + scoeff = printer._print(coeff) + if negative and scoeff.startswith("-"): + scoeff = scoeff[1:] + else: + if negative: + coeff = -coeff + if coeff != self.ring.domain.one: + scoeff = printer.parenthesize(coeff, prec_mul, strict=True) + else: + scoeff = '' + sexpv = [] + for i in range(ngens): + exp = expv[i] + if not exp: + continue + symbol = printer.parenthesize(symbols[i], prec_atom, strict=True) + if exp != 1: + if exp != int(exp) or exp < 0: + sexp = printer.parenthesize(exp, prec_atom, strict=False) + else: + sexp = exp + sexpv.append(exp_pattern % (symbol, sexp)) + else: + sexpv.append('%s' % symbol) + if scoeff: + sexpv = [scoeff] + sexpv + sexpvs.append(mul_symbol.join(sexpv)) + if sexpvs[0] in [" + ", " - "]: + head = sexpvs.pop(0) + if head == " - ": + sexpvs.insert(0, "-") + return "".join(sexpvs) + + @property + def is_generator(self): + return self in self.ring._gens_set + + @property + def is_ground(self): + return not self or (len(self) == 1 and self.ring.zero_monom in self) + + @property + def is_monomial(self): + return not self or (len(self) == 1 and self.LC == 1) + + @property + def is_term(self): + return len(self) <= 1 + + @property + def is_negative(self): + return self.ring.domain.is_negative(self.LC) + + @property + def is_positive(self): + return self.ring.domain.is_positive(self.LC) + + @property + def is_nonnegative(self): + return self.ring.domain.is_nonnegative(self.LC) + + @property + def is_nonpositive(self): + return self.ring.domain.is_nonpositive(self.LC) + + @property + def is_zero(f): + return not f + + @property + def is_one(f): + return f == f.ring.one + + @property + def is_monic(f): + return f.ring.domain.is_one(f.LC) + + @property + def is_primitive(f): + return f.ring.domain.is_one(f.content()) + + @property + def is_linear(f): + return all(sum(monom) <= 1 for monom in f.itermonoms()) + + @property + def is_quadratic(f): + return all(sum(monom) <= 2 for monom in f.itermonoms()) + + @property + def is_squarefree(f): + if not f.ring.ngens: + return True + return f.ring.dmp_sqf_p(f) + + @property + def is_irreducible(f): + if not f.ring.ngens: + return True + return f.ring.dmp_irreducible_p(f) + + @property + def is_cyclotomic(f): + if f.ring.is_univariate: + return f.ring.dup_cyclotomic_p(f) + else: + raise MultivariatePolynomialError("cyclotomic polynomial") + + def __neg__(self): + return self.new([ (monom, -coeff) for monom, coeff in self.iterterms() ]) + + def __pos__(self): + return self + + def __add__(p1, p2): + """Add two polynomials. + + Examples + ======== + + >>> from sympy.polys.domains import ZZ + >>> from sympy.polys.rings import ring + + >>> _, x, y = ring('x, y', ZZ) + >>> (x + y)**2 + (x - y)**2 + 2*x**2 + 2*y**2 + + """ + if not p2: + return p1.copy() + ring = p1.ring + if ring.is_element(p2): + p = p1.copy() + get = p.get + zero = ring.domain.zero + for k, v in p2.items(): + v = get(k, zero) + v + if v: + p[k] = v + else: + del p[k] + return p + elif isinstance(p2, PolyElement): + if isinstance(ring.domain, PolynomialRing) and ring.domain.ring == p2.ring: + pass + elif isinstance(p2.ring.domain, PolynomialRing) and p2.ring.domain.ring == ring: + return p2.__radd__(p1) + else: + return NotImplemented + + try: + cp2 = ring.domain_new(p2) + except CoercionFailed: + return NotImplemented + else: + p = p1.copy() + if not cp2: + return p + zm = ring.zero_monom + if zm not in p1.keys(): + p[zm] = cp2 + else: + if p2 == -p[zm]: + del p[zm] + else: + p[zm] += cp2 + return p + + def __radd__(p1, n): + p = p1.copy() + if not n: + return p + ring = p1.ring + try: + n = ring.domain_new(n) + except CoercionFailed: + return NotImplemented + else: + zm = ring.zero_monom + if zm not in p1.keys(): + p[zm] = n + else: + if n == -p[zm]: + del p[zm] + else: + p[zm] += n + return p + + def __sub__(p1, p2): + """Subtract polynomial p2 from p1. + + Examples + ======== + + >>> from sympy.polys.domains import ZZ + >>> from sympy.polys.rings import ring + + >>> _, x, y = ring('x, y', ZZ) + >>> p1 = x + y**2 + >>> p2 = x*y + y**2 + >>> p1 - p2 + -x*y + x + + """ + if not p2: + return p1.copy() + ring = p1.ring + if ring.is_element(p2): + p = p1.copy() + get = p.get + zero = ring.domain.zero + for k, v in p2.items(): + v = get(k, zero) - v + if v: + p[k] = v + else: + del p[k] + return p + elif isinstance(p2, PolyElement): + if isinstance(ring.domain, PolynomialRing) and ring.domain.ring == p2.ring: + pass + elif isinstance(p2.ring.domain, PolynomialRing) and p2.ring.domain.ring == ring: + return p2.__rsub__(p1) + else: + return NotImplemented + + try: + p2 = ring.domain_new(p2) + except CoercionFailed: + return NotImplemented + else: + p = p1.copy() + zm = ring.zero_monom + if zm not in p1.keys(): + p[zm] = -p2 + else: + if p2 == p[zm]: + del p[zm] + else: + p[zm] -= p2 + return p + + def __rsub__(p1, n): + """n - p1 with n convertible to the coefficient domain. + + Examples + ======== + + >>> from sympy.polys.domains import ZZ + >>> from sympy.polys.rings import ring + + >>> _, x, y = ring('x, y', ZZ) + >>> p = x + y + >>> 4 - p + -x - y + 4 + + """ + ring = p1.ring + try: + n = ring.domain_new(n) + except CoercionFailed: + return NotImplemented + else: + p = ring.zero + for expv in p1: + p[expv] = -p1[expv] + p += n + # p._check() + return p + + def __mul__(p1, p2): + """Multiply two polynomials. + + Examples + ======== + + >>> from sympy.polys.domains import QQ + >>> from sympy.polys.rings import ring + + >>> _, x, y = ring('x, y', QQ) + >>> p1 = x + y + >>> p2 = x - y + >>> p1*p2 + x**2 - y**2 + + """ + ring = p1.ring + p = ring.zero + if not p1 or not p2: + return p + elif ring.is_element(p2): + get = p.get + zero = ring.domain.zero + monomial_mul = ring.monomial_mul + p2it = list(p2.items()) + for exp1, v1 in p1.items(): + for exp2, v2 in p2it: + exp = monomial_mul(exp1, exp2) + p[exp] = get(exp, zero) + v1*v2 + p.strip_zero() + # p._check() + return p + elif isinstance(p2, PolyElement): + if isinstance(ring.domain, PolynomialRing) and ring.domain.ring == p2.ring: + pass + elif isinstance(p2.ring.domain, PolynomialRing) and p2.ring.domain.ring == ring: + return p2.__rmul__(p1) + else: + return NotImplemented + + try: + p2 = ring.domain_new(p2) + except CoercionFailed: + return NotImplemented + else: + for exp1, v1 in p1.items(): + v = v1*p2 + if v: + p[exp1] = v + # p._check() + return p + + def __rmul__(p1, p2): + """p2 * p1 with p2 in the coefficient domain of p1. + + Examples + ======== + + >>> from sympy.polys.domains import ZZ + >>> from sympy.polys.rings import ring + + >>> _, x, y = ring('x, y', ZZ) + >>> p = x + y + >>> 4 * p + 4*x + 4*y + + """ + p = p1.ring.zero + if not p2: + return p + try: + p2 = p.ring.domain_new(p2) + except CoercionFailed: + return NotImplemented + else: + for exp1, v1 in p1.items(): + v = p2*v1 + if v: + p[exp1] = v + return p + + def __pow__(self, n): + """raise polynomial to power `n` + + Examples + ======== + + >>> from sympy.polys.domains import ZZ + >>> from sympy.polys.rings import ring + + >>> _, x, y = ring('x, y', ZZ) + >>> p = x + y**2 + >>> p**3 + x**3 + 3*x**2*y**2 + 3*x*y**4 + y**6 + + """ + if not isinstance(n, int): + raise TypeError("exponent must be an integer, got %s" % n) + elif n < 0: + raise ValueError("exponent must be a non-negative integer, got %s" % n) + + ring = self.ring + + if not n: + if self: + return ring.one + else: + raise ValueError("0**0") + elif len(self) == 1: + monom, coeff = list(self.items())[0] + p = ring.zero + if coeff == ring.domain.one: + p[ring.monomial_pow(monom, n)] = coeff + else: + p[ring.monomial_pow(monom, n)] = coeff**n + # p._check() + return p + + # For ring series, we need negative and rational exponent support only + # with monomials. + n = int(n) + if n < 0: + raise ValueError("Negative exponent") + + elif n == 1: + return self.copy() + elif n == 2: + return self.square() + elif n == 3: + return self*self.square() + elif len(self) <= 5: # TODO: use an actual density measure + return self._pow_multinomial(n) + else: + return self._pow_generic(n) + + def _pow_generic(self, n): + p = self.ring.one + c = self + + while True: + if n & 1: + p = p*c + n -= 1 + if not n: + break + + c = c.square() + n = n // 2 + + return p + + def _pow_multinomial(self, n): + multinomials = multinomial_coefficients(len(self), n).items() + monomial_mulpow = self.ring.monomial_mulpow + zero_monom = self.ring.zero_monom + terms = self.items() + zero = self.ring.domain.zero + poly = self.ring.zero + + for multinomial, multinomial_coeff in multinomials: + product_monom = zero_monom + product_coeff = multinomial_coeff + + for exp, (monom, coeff) in zip(multinomial, terms): + if exp: + product_monom = monomial_mulpow(product_monom, monom, exp) + product_coeff *= coeff**exp + + monom = tuple(product_monom) + coeff = product_coeff + + coeff = poly.get(monom, zero) + coeff + + if coeff: + poly[monom] = coeff + elif monom in poly: + del poly[monom] + + return poly + + def square(self): + """square of a polynomial + + Examples + ======== + + >>> from sympy.polys.rings import ring + >>> from sympy.polys.domains import ZZ + + >>> _, x, y = ring('x, y', ZZ) + >>> p = x + y**2 + >>> p.square() + x**2 + 2*x*y**2 + y**4 + + """ + ring = self.ring + p = ring.zero + get = p.get + keys = list(self.keys()) + zero = ring.domain.zero + monomial_mul = ring.monomial_mul + for i in range(len(keys)): + k1 = keys[i] + pk = self[k1] + for j in range(i): + k2 = keys[j] + exp = monomial_mul(k1, k2) + p[exp] = get(exp, zero) + pk*self[k2] + p = p.imul_num(2) + get = p.get + for k, v in self.items(): + k2 = monomial_mul(k, k) + p[k2] = get(k2, zero) + v**2 + p.strip_zero() + # p._check() + return p + + def __divmod__(p1, p2): + ring = p1.ring + + if not p2: + raise ZeroDivisionError("polynomial division") + elif ring.is_element(p2): + return p1.div(p2) + elif isinstance(p2, PolyElement): + if isinstance(ring.domain, PolynomialRing) and ring.domain.ring == p2.ring: + pass + elif isinstance(p2.ring.domain, PolynomialRing) and p2.ring.domain.ring == ring: + return p2.__rdivmod__(p1) + else: + return NotImplemented + + try: + p2 = ring.domain_new(p2) + except CoercionFailed: + return NotImplemented + else: + return (p1.quo_ground(p2), p1.rem_ground(p2)) + + def __rdivmod__(p1, p2): + ring = p1.ring + try: + p2 = ring.ground_new(p2) + except CoercionFailed: + return NotImplemented + else: + return p2.div(p1) + + def __mod__(p1, p2): + ring = p1.ring + + if not p2: + raise ZeroDivisionError("polynomial division") + elif ring.is_element(p2): + return p1.rem(p2) + elif isinstance(p2, PolyElement): + if isinstance(ring.domain, PolynomialRing) and ring.domain.ring == p2.ring: + pass + elif isinstance(p2.ring.domain, PolynomialRing) and p2.ring.domain.ring == ring: + return p2.__rmod__(p1) + else: + return NotImplemented + + try: + p2 = ring.domain_new(p2) + except CoercionFailed: + return NotImplemented + else: + return p1.rem_ground(p2) + + def __rmod__(p1, p2): + ring = p1.ring + try: + p2 = ring.ground_new(p2) + except CoercionFailed: + return NotImplemented + else: + return p2.rem(p1) + + def __floordiv__(p1, p2): + ring = p1.ring + + if not p2: + raise ZeroDivisionError("polynomial division") + elif ring.is_element(p2): + return p1.quo(p2) + elif isinstance(p2, PolyElement): + if isinstance(ring.domain, PolynomialRing) and ring.domain.ring == p2.ring: + pass + elif isinstance(p2.ring.domain, PolynomialRing) and p2.ring.domain.ring == ring: + return p2.__rtruediv__(p1) + else: + return NotImplemented + + try: + p2 = ring.domain_new(p2) + except CoercionFailed: + return NotImplemented + else: + return p1.quo_ground(p2) + + def __rfloordiv__(p1, p2): + ring = p1.ring + try: + p2 = ring.ground_new(p2) + except CoercionFailed: + return NotImplemented + else: + return p2.quo(p1) + + def __truediv__(p1, p2): + ring = p1.ring + + if not p2: + raise ZeroDivisionError("polynomial division") + elif ring.is_element(p2): + return p1.exquo(p2) + elif isinstance(p2, PolyElement): + if isinstance(ring.domain, PolynomialRing) and ring.domain.ring == p2.ring: + pass + elif isinstance(p2.ring.domain, PolynomialRing) and p2.ring.domain.ring == ring: + return p2.__rtruediv__(p1) + else: + return NotImplemented + + try: + p2 = ring.domain_new(p2) + except CoercionFailed: + return NotImplemented + else: + return p1.quo_ground(p2) + + def __rtruediv__(p1, p2): + ring = p1.ring + try: + p2 = ring.ground_new(p2) + except CoercionFailed: + return NotImplemented + else: + return p2.exquo(p1) + + def _term_div(self): + zm = self.ring.zero_monom + domain = self.ring.domain + domain_quo = domain.quo + monomial_div = self.ring.monomial_div + + if domain.is_Field: + def term_div(a_lm_a_lc, b_lm_b_lc): + a_lm, a_lc = a_lm_a_lc + b_lm, b_lc = b_lm_b_lc + if b_lm == zm: # apparently this is a very common case + monom = a_lm + else: + monom = monomial_div(a_lm, b_lm) + if monom is not None: + return monom, domain_quo(a_lc, b_lc) + else: + return None + else: + def term_div(a_lm_a_lc, b_lm_b_lc): + a_lm, a_lc = a_lm_a_lc + b_lm, b_lc = b_lm_b_lc + if b_lm == zm: # apparently this is a very common case + monom = a_lm + else: + monom = monomial_div(a_lm, b_lm) + if not (monom is None or a_lc % b_lc): + return monom, domain_quo(a_lc, b_lc) + else: + return None + + return term_div + + def div(self, fv): + """Division algorithm, see [CLO] p64. + + fv array of polynomials + return qv, r such that + self = sum(fv[i]*qv[i]) + r + + All polynomials are required not to be Laurent polynomials. + + Examples + ======== + + >>> from sympy.polys.rings import ring + >>> from sympy.polys.domains import ZZ + + >>> _, x, y = ring('x, y', ZZ) + >>> f = x**3 + >>> f0 = x - y**2 + >>> f1 = x - y + >>> qv, r = f.div((f0, f1)) + >>> qv[0] + x**2 + x*y**2 + y**4 + >>> qv[1] + 0 + >>> r + y**6 + + """ + ring = self.ring + ret_single = False + if isinstance(fv, PolyElement): + ret_single = True + fv = [fv] + if not all(fv): + raise ZeroDivisionError("polynomial division") + if not self: + if ret_single: + return ring.zero, ring.zero + else: + return [], ring.zero + for f in fv: + if f.ring != ring: + raise ValueError('self and f must have the same ring') + s = len(fv) + qv = [ring.zero for i in range(s)] + p = self.copy() + r = ring.zero + term_div = self._term_div() + expvs = [fx.leading_expv() for fx in fv] + while p: + i = 0 + divoccurred = 0 + while i < s and divoccurred == 0: + expv = p.leading_expv() + term = term_div((expv, p[expv]), (expvs[i], fv[i][expvs[i]])) + if term is not None: + expv1, c = term + qv[i] = qv[i]._iadd_monom((expv1, c)) + p = p._iadd_poly_monom(fv[i], (expv1, -c)) + divoccurred = 1 + else: + i += 1 + if not divoccurred: + expv = p.leading_expv() + r = r._iadd_monom((expv, p[expv])) + del p[expv] + if expv == ring.zero_monom: + r += p + if ret_single: + if not qv: + return ring.zero, r + else: + return qv[0], r + else: + return qv, r + + def rem(self, G): + f = self + if isinstance(G, PolyElement): + G = [G] + if not all(G): + raise ZeroDivisionError("polynomial division") + ring = f.ring + domain = ring.domain + zero = domain.zero + monomial_mul = ring.monomial_mul + r = ring.zero + term_div = f._term_div() + ltf = f.LT + f = f.copy() + get = f.get + while f: + for g in G: + tq = term_div(ltf, g.LT) + if tq is not None: + m, c = tq + for mg, cg in g.iterterms(): + m1 = monomial_mul(mg, m) + c1 = get(m1, zero) - c*cg + if not c1: + del f[m1] + else: + f[m1] = c1 + ltm = f.leading_expv() + if ltm is not None: + ltf = ltm, f[ltm] + + break + else: + ltm, ltc = ltf + if ltm in r: + r[ltm] += ltc + else: + r[ltm] = ltc + del f[ltm] + ltm = f.leading_expv() + if ltm is not None: + ltf = ltm, f[ltm] + + return r + + def quo(f, G): + return f.div(G)[0] + + def exquo(f, G): + q, r = f.div(G) + + if not r: + return q + else: + raise ExactQuotientFailed(f, G) + + def _iadd_monom(self, mc): + """add to self the monomial coeff*x0**i0*x1**i1*... + unless self is a generator -- then just return the sum of the two. + + mc is a tuple, (monom, coeff), where monomial is (i0, i1, ...) + + Examples + ======== + + >>> from sympy.polys.rings import ring + >>> from sympy.polys.domains import ZZ + + >>> _, x, y = ring('x, y', ZZ) + >>> p = x**4 + 2*y + >>> m = (1, 2) + >>> p1 = p._iadd_monom((m, 5)) + >>> p1 + x**4 + 5*x*y**2 + 2*y + >>> p1 is p + True + >>> p = x + >>> p1 = p._iadd_monom((m, 5)) + >>> p1 + 5*x*y**2 + x + >>> p1 is p + False + + """ + if self in self.ring._gens_set: + cpself = self.copy() + else: + cpself = self + expv, coeff = mc + c = cpself.get(expv) + if c is None: + cpself[expv] = coeff + else: + c += coeff + if c: + cpself[expv] = c + else: + del cpself[expv] + return cpself + + def _iadd_poly_monom(self, p2, mc): + """add to self the product of (p)*(coeff*x0**i0*x1**i1*...) + unless self is a generator -- then just return the sum of the two. + + mc is a tuple, (monom, coeff), where monomial is (i0, i1, ...) + + Examples + ======== + + >>> from sympy.polys.rings import ring + >>> from sympy.polys.domains import ZZ + + >>> _, x, y, z = ring('x, y, z', ZZ) + >>> p1 = x**4 + 2*y + >>> p2 = y + z + >>> m = (1, 2, 3) + >>> p1 = p1._iadd_poly_monom(p2, (m, 3)) + >>> p1 + x**4 + 3*x*y**3*z**3 + 3*x*y**2*z**4 + 2*y + + """ + p1 = self + if p1 in p1.ring._gens_set: + p1 = p1.copy() + (m, c) = mc + get = p1.get + zero = p1.ring.domain.zero + monomial_mul = p1.ring.monomial_mul + for k, v in p2.items(): + ka = monomial_mul(k, m) + coeff = get(ka, zero) + v*c + if coeff: + p1[ka] = coeff + else: + del p1[ka] + return p1 + + def degree(f, x=None): + """ + The leading degree in ``x`` or the main variable. + + Note that the degree of 0 is negative infinity (``float('-inf')``) + + """ + i = f.ring.index(x) + + if not f: + return ninf + elif i < 0: + return 0 + else: + return max(monom[i] for monom in f.itermonoms()) + + def degrees(f): + """ + A tuple containing leading degrees in all variables. + + Note that the degree of 0 is negative infinity (``float('-inf')``) + + """ + if not f: + return (ninf,)*f.ring.ngens + else: + return tuple(map(max, list(zip(*f.itermonoms())))) + + def tail_degree(f, x=None): + """ + The tail degree in ``x`` or the main variable. + + Note that the degree of 0 is negative infinity (``float('-inf')``) + + """ + i = f.ring.index(x) + + if not f: + return ninf + elif i < 0: + return 0 + else: + return min(monom[i] for monom in f.itermonoms()) + + def tail_degrees(f): + """ + A tuple containing tail degrees in all variables. + + Note that the degree of 0 is negative infinity (``float('-inf')``) + + """ + if not f: + return (ninf,)*f.ring.ngens + else: + return tuple(map(min, list(zip(*f.itermonoms())))) + + def leading_expv(self): + """Leading monomial tuple according to the monomial ordering. + + Examples + ======== + + >>> from sympy.polys.rings import ring + >>> from sympy.polys.domains import ZZ + + >>> _, x, y, z = ring('x, y, z', ZZ) + >>> p = x**4 + x**3*y + x**2*z**2 + z**7 + >>> p.leading_expv() + (4, 0, 0) + + """ + if self: + return self.ring.leading_expv(self) + else: + return None + + def _get_coeff(self, expv): + return self.get(expv, self.ring.domain.zero) + + def coeff(self, element): + """ + Returns the coefficient that stands next to the given monomial. + + Parameters + ========== + + element : PolyElement (with ``is_monomial = True``) or 1 + + Examples + ======== + + >>> from sympy.polys.rings import ring + >>> from sympy.polys.domains import ZZ + + >>> _, x, y, z = ring("x,y,z", ZZ) + >>> f = 3*x**2*y - x*y*z + 7*z**3 + 23 + + >>> f.coeff(x**2*y) + 3 + >>> f.coeff(x*y) + 0 + >>> f.coeff(1) + 23 + + """ + if element == 1: + return self._get_coeff(self.ring.zero_monom) + elif self.ring.is_element(element): + terms = list(element.iterterms()) + if len(terms) == 1: + monom, coeff = terms[0] + if coeff == self.ring.domain.one: + return self._get_coeff(monom) + + raise ValueError("expected a monomial, got %s" % element) + + def const(self): + """Returns the constant coefficient. """ + return self._get_coeff(self.ring.zero_monom) + + @property + def LC(self): + return self._get_coeff(self.leading_expv()) + + @property + def LM(self): + expv = self.leading_expv() + if expv is None: + return self.ring.zero_monom + else: + return expv + + def leading_monom(self): + """ + Leading monomial as a polynomial element. + + Examples + ======== + + >>> from sympy.polys.rings import ring + >>> from sympy.polys.domains import ZZ + + >>> _, x, y = ring('x, y', ZZ) + >>> (3*x*y + y**2).leading_monom() + x*y + + """ + p = self.ring.zero + expv = self.leading_expv() + if expv: + p[expv] = self.ring.domain.one + return p + + @property + def LT(self): + expv = self.leading_expv() + if expv is None: + return (self.ring.zero_monom, self.ring.domain.zero) + else: + return (expv, self._get_coeff(expv)) + + def leading_term(self): + """Leading term as a polynomial element. + + Examples + ======== + + >>> from sympy.polys.rings import ring + >>> from sympy.polys.domains import ZZ + + >>> _, x, y = ring('x, y', ZZ) + >>> (3*x*y + y**2).leading_term() + 3*x*y + + """ + p = self.ring.zero + expv = self.leading_expv() + if expv is not None: + p[expv] = self[expv] + return p + + def _sorted(self, seq, order): + if order is None: + order = self.ring.order + else: + order = OrderOpt.preprocess(order) + + if order is lex: + return sorted(seq, key=lambda monom: monom[0], reverse=True) + else: + return sorted(seq, key=lambda monom: order(monom[0]), reverse=True) + + def coeffs(self, order=None): + """Ordered list of polynomial coefficients. + + Parameters + ========== + + order : :class:`~.MonomialOrder` or coercible, optional + + Examples + ======== + + >>> from sympy.polys.rings import ring + >>> from sympy.polys.domains import ZZ + >>> from sympy.polys.orderings import lex, grlex + + >>> _, x, y = ring("x, y", ZZ, lex) + >>> f = x*y**7 + 2*x**2*y**3 + + >>> f.coeffs() + [2, 1] + >>> f.coeffs(grlex) + [1, 2] + + """ + return [ coeff for _, coeff in self.terms(order) ] + + def monoms(self, order=None): + """Ordered list of polynomial monomials. + + Parameters + ========== + + order : :class:`~.MonomialOrder` or coercible, optional + + Examples + ======== + + >>> from sympy.polys.rings import ring + >>> from sympy.polys.domains import ZZ + >>> from sympy.polys.orderings import lex, grlex + + >>> _, x, y = ring("x, y", ZZ, lex) + >>> f = x*y**7 + 2*x**2*y**3 + + >>> f.monoms() + [(2, 3), (1, 7)] + >>> f.monoms(grlex) + [(1, 7), (2, 3)] + + """ + return [ monom for monom, _ in self.terms(order) ] + + def terms(self, order=None): + """Ordered list of polynomial terms. + + Parameters + ========== + + order : :class:`~.MonomialOrder` or coercible, optional + + Examples + ======== + + >>> from sympy.polys.rings import ring + >>> from sympy.polys.domains import ZZ + >>> from sympy.polys.orderings import lex, grlex + + >>> _, x, y = ring("x, y", ZZ, lex) + >>> f = x*y**7 + 2*x**2*y**3 + + >>> f.terms() + [((2, 3), 2), ((1, 7), 1)] + >>> f.terms(grlex) + [((1, 7), 1), ((2, 3), 2)] + + """ + return self._sorted(list(self.items()), order) + + def itercoeffs(self): + """Iterator over coefficients of a polynomial. """ + return iter(self.values()) + + def itermonoms(self): + """Iterator over monomials of a polynomial. """ + return iter(self.keys()) + + def iterterms(self): + """Iterator over terms of a polynomial. """ + return iter(self.items()) + + def listcoeffs(self): + """Unordered list of polynomial coefficients. """ + return list(self.values()) + + def listmonoms(self): + """Unordered list of polynomial monomials. """ + return list(self.keys()) + + def listterms(self): + """Unordered list of polynomial terms. """ + return list(self.items()) + + def imul_num(p, c): + """multiply inplace the polynomial p by an element in the + coefficient ring, provided p is not one of the generators; + else multiply not inplace + + Examples + ======== + + >>> from sympy.polys.rings import ring + >>> from sympy.polys.domains import ZZ + + >>> _, x, y = ring('x, y', ZZ) + >>> p = x + y**2 + >>> p1 = p.imul_num(3) + >>> p1 + 3*x + 3*y**2 + >>> p1 is p + True + >>> p = x + >>> p1 = p.imul_num(3) + >>> p1 + 3*x + >>> p1 is p + False + + """ + if p in p.ring._gens_set: + return p*c + if not c: + p.clear() + return + for exp in p: + p[exp] *= c + return p + + def content(f): + """Returns GCD of polynomial's coefficients. """ + domain = f.ring.domain + cont = domain.zero + gcd = domain.gcd + + for coeff in f.itercoeffs(): + cont = gcd(cont, coeff) + + return cont + + def primitive(f): + """Returns content and a primitive polynomial. """ + cont = f.content() + if cont == f.ring.domain.zero: + return (cont, f) + return cont, f.quo_ground(cont) + + def monic(f): + """Divides all coefficients by the leading coefficient. """ + if not f: + return f + else: + return f.quo_ground(f.LC) + + def mul_ground(f, x): + if not x: + return f.ring.zero + + terms = [ (monom, coeff*x) for monom, coeff in f.iterterms() ] + return f.new(terms) + + def mul_monom(f, monom): + monomial_mul = f.ring.monomial_mul + terms = [ (monomial_mul(f_monom, monom), f_coeff) for f_monom, f_coeff in f.items() ] + return f.new(terms) + + def mul_term(f, term): + monom, coeff = term + + if not f or not coeff: + return f.ring.zero + elif monom == f.ring.zero_monom: + return f.mul_ground(coeff) + + monomial_mul = f.ring.monomial_mul + terms = [ (monomial_mul(f_monom, monom), f_coeff*coeff) for f_monom, f_coeff in f.items() ] + return f.new(terms) + + def quo_ground(f, x): + domain = f.ring.domain + + if not x: + raise ZeroDivisionError('polynomial division') + if not f or x == domain.one: + return f + + if domain.is_Field: + quo = domain.quo + terms = [ (monom, quo(coeff, x)) for monom, coeff in f.iterterms() ] + else: + terms = [ (monom, coeff // x) for monom, coeff in f.iterterms() if not (coeff % x) ] + + return f.new(terms) + + def quo_term(f, term): + monom, coeff = term + + if not coeff: + raise ZeroDivisionError("polynomial division") + elif not f: + return f.ring.zero + elif monom == f.ring.zero_monom: + return f.quo_ground(coeff) + + term_div = f._term_div() + + terms = [ term_div(t, term) for t in f.iterterms() ] + return f.new([ t for t in terms if t is not None ]) + + def trunc_ground(f, p): + if f.ring.domain.is_ZZ: + terms = [] + + for monom, coeff in f.iterterms(): + coeff = coeff % p + + if coeff > p // 2: + coeff = coeff - p + + terms.append((monom, coeff)) + else: + terms = [ (monom, coeff % p) for monom, coeff in f.iterterms() ] + + poly = f.new(terms) + poly.strip_zero() + return poly + + rem_ground = trunc_ground + + def extract_ground(self, g): + f = self + fc = f.content() + gc = g.content() + + gcd = f.ring.domain.gcd(fc, gc) + + f = f.quo_ground(gcd) + g = g.quo_ground(gcd) + + return gcd, f, g + + def _norm(f, norm_func): + if not f: + return f.ring.domain.zero + else: + ground_abs = f.ring.domain.abs + return norm_func([ ground_abs(coeff) for coeff in f.itercoeffs() ]) + + def max_norm(f): + return f._norm(max) + + def l1_norm(f): + return f._norm(sum) + + def deflate(f, *G): + ring = f.ring + polys = [f] + list(G) + + J = [0]*ring.ngens + + for p in polys: + for monom in p.itermonoms(): + for i, m in enumerate(monom): + J[i] = igcd(J[i], m) + + for i, b in enumerate(J): + if not b: + J[i] = 1 + + J = tuple(J) + + if all(b == 1 for b in J): + return J, polys + + H = [] + + for p in polys: + h = ring.zero + + for I, coeff in p.iterterms(): + N = [ i // j for i, j in zip(I, J) ] + h[tuple(N)] = coeff + + H.append(h) + + return J, H + + def inflate(f, J): + poly = f.ring.zero + + for I, coeff in f.iterterms(): + N = [ i*j for i, j in zip(I, J) ] + poly[tuple(N)] = coeff + + return poly + + def lcm(self, g): + f = self + domain = f.ring.domain + + if not domain.is_Field: + fc, f = f.primitive() + gc, g = g.primitive() + c = domain.lcm(fc, gc) + + h = (f*g).quo(f.gcd(g)) + + if not domain.is_Field: + return h.mul_ground(c) + else: + return h.monic() + + def gcd(f, g): + return f.cofactors(g)[0] + + def cofactors(f, g): + if not f and not g: + zero = f.ring.zero + return zero, zero, zero + elif not f: + h, cff, cfg = f._gcd_zero(g) + return h, cff, cfg + elif not g: + h, cfg, cff = g._gcd_zero(f) + return h, cff, cfg + elif len(f) == 1: + h, cff, cfg = f._gcd_monom(g) + return h, cff, cfg + elif len(g) == 1: + h, cfg, cff = g._gcd_monom(f) + return h, cff, cfg + + J, (f, g) = f.deflate(g) + h, cff, cfg = f._gcd(g) + + return (h.inflate(J), cff.inflate(J), cfg.inflate(J)) + + def _gcd_zero(f, g): + one, zero = f.ring.one, f.ring.zero + if g.is_nonnegative: + return g, zero, one + else: + return -g, zero, -one + + def _gcd_monom(f, g): + ring = f.ring + ground_gcd = ring.domain.gcd + ground_quo = ring.domain.quo + monomial_gcd = ring.monomial_gcd + monomial_ldiv = ring.monomial_ldiv + mf, cf = list(f.iterterms())[0] + _mgcd, _cgcd = mf, cf + for mg, cg in g.iterterms(): + _mgcd = monomial_gcd(_mgcd, mg) + _cgcd = ground_gcd(_cgcd, cg) + h = f.new([(_mgcd, _cgcd)]) + cff = f.new([(monomial_ldiv(mf, _mgcd), ground_quo(cf, _cgcd))]) + cfg = f.new([(monomial_ldiv(mg, _mgcd), ground_quo(cg, _cgcd)) for mg, cg in g.iterterms()]) + return h, cff, cfg + + def _gcd(f, g): + ring = f.ring + + if ring.domain.is_QQ: + return f._gcd_QQ(g) + elif ring.domain.is_ZZ: + return f._gcd_ZZ(g) + else: # TODO: don't use dense representation (port PRS algorithms) + return ring.dmp_inner_gcd(f, g) + + def _gcd_ZZ(f, g): + return heugcd(f, g) + + def _gcd_QQ(self, g): + f = self + ring = f.ring + new_ring = ring.clone(domain=ring.domain.get_ring()) + + cf, f = f.clear_denoms() + cg, g = g.clear_denoms() + + f = f.set_ring(new_ring) + g = g.set_ring(new_ring) + + h, cff, cfg = f._gcd_ZZ(g) + + h = h.set_ring(ring) + c, h = h.LC, h.monic() + + cff = cff.set_ring(ring).mul_ground(ring.domain.quo(c, cf)) + cfg = cfg.set_ring(ring).mul_ground(ring.domain.quo(c, cg)) + + return h, cff, cfg + + def cancel(self, g): + """ + Cancel common factors in a rational function ``f/g``. + + Examples + ======== + + >>> from sympy.polys import ring, ZZ + >>> R, x,y = ring("x,y", ZZ) + + >>> (2*x**2 - 2).cancel(x**2 - 2*x + 1) + (2*x + 2, x - 1) + + """ + f = self + ring = f.ring + + if not f: + return f, ring.one + + domain = ring.domain + + if not (domain.is_Field and domain.has_assoc_Ring): + _, p, q = f.cofactors(g) + else: + new_ring = ring.clone(domain=domain.get_ring()) + + cq, f = f.clear_denoms() + cp, g = g.clear_denoms() + + f = f.set_ring(new_ring) + g = g.set_ring(new_ring) + + _, p, q = f.cofactors(g) + _, cp, cq = new_ring.domain.cofactors(cp, cq) + + p = p.set_ring(ring) + q = q.set_ring(ring) + + p = p.mul_ground(cp) + q = q.mul_ground(cq) + + # Make canonical with respect to sign or quadrant in the case of ZZ_I + # or QQ_I. This ensures that the LC of the denominator is canonical by + # multiplying top and bottom by a unit of the ring. + u = q.canonical_unit() + if u == domain.one: + pass + elif u == -domain.one: + p, q = -p, -q + else: + p = p.mul_ground(u) + q = q.mul_ground(u) + + return p, q + + def canonical_unit(f): + domain = f.ring.domain + return domain.canonical_unit(f.LC) + + def diff(f, x): + """Computes partial derivative in ``x``. + + Examples + ======== + + >>> from sympy.polys.rings import ring + >>> from sympy.polys.domains import ZZ + + >>> _, x, y = ring("x,y", ZZ) + >>> p = x + x**2*y**3 + >>> p.diff(x) + 2*x*y**3 + 1 + + """ + ring = f.ring + i = ring.index(x) + m = ring.monomial_basis(i) + g = ring.zero + for expv, coeff in f.iterterms(): + if expv[i]: + e = ring.monomial_ldiv(expv, m) + g[e] = ring.domain_new(coeff*expv[i]) + return g + + def __call__(f, *values): + if 0 < len(values) <= f.ring.ngens: + return f.evaluate(list(zip(f.ring.gens, values))) + else: + raise ValueError("expected at least 1 and at most %s values, got %s" % (f.ring.ngens, len(values))) + + def evaluate(self, x, a=None): + f = self + + if isinstance(x, list) and a is None: + (X, a), x = x[0], x[1:] + f = f.evaluate(X, a) + + if not x: + return f + else: + x = [ (Y.drop(X), a) for (Y, a) in x ] + return f.evaluate(x) + + ring = f.ring + i = ring.index(x) + a = ring.domain.convert(a) + + if ring.ngens == 1: + result = ring.domain.zero + + for (n,), coeff in f.iterterms(): + result += coeff*a**n + + return result + else: + poly = ring.drop(x).zero + + for monom, coeff in f.iterterms(): + n, monom = monom[i], monom[:i] + monom[i+1:] + coeff = coeff*a**n + + if monom in poly: + coeff = coeff + poly[monom] + + if coeff: + poly[monom] = coeff + else: + del poly[monom] + else: + if coeff: + poly[monom] = coeff + + return poly + + def subs(self, x, a=None): + f = self + + if isinstance(x, list) and a is None: + for X, a in x: + f = f.subs(X, a) + return f + + ring = f.ring + i = ring.index(x) + a = ring.domain.convert(a) + + if ring.ngens == 1: + result = ring.domain.zero + + for (n,), coeff in f.iterterms(): + result += coeff*a**n + + return ring.ground_new(result) + else: + poly = ring.zero + + for monom, coeff in f.iterterms(): + n, monom = monom[i], monom[:i] + (0,) + monom[i+1:] + coeff = coeff*a**n + + if monom in poly: + coeff = coeff + poly[monom] + + if coeff: + poly[monom] = coeff + else: + del poly[monom] + else: + if coeff: + poly[monom] = coeff + + return poly + + def symmetrize(self): + r""" + Rewrite *self* in terms of elementary symmetric polynomials. + + Explanation + =========== + + If this :py:class:`~.PolyElement` belongs to a ring of $n$ variables, + we can try to write it as a function of the elementary symmetric + polynomials on $n$ variables. We compute a symmetric part, and a + remainder for any part we were not able to symmetrize. + + Examples + ======== + + >>> from sympy.polys.rings import ring + >>> from sympy.polys.domains import ZZ + >>> R, x, y = ring("x,y", ZZ) + + >>> f = x**2 + y**2 + >>> f.symmetrize() + (x**2 - 2*y, 0, [(x, x + y), (y, x*y)]) + + >>> f = x**2 - y**2 + >>> f.symmetrize() + (x**2 - 2*y, -2*y**2, [(x, x + y), (y, x*y)]) + + Returns + ======= + + Triple ``(p, r, m)`` + ``p`` is a :py:class:`~.PolyElement` that represents our attempt + to express *self* as a function of elementary symmetric + polynomials. Each variable in ``p`` stands for one of the + elementary symmetric polynomials. The correspondence is given + by ``m``. + + ``r`` is the remainder. + + ``m`` is a list of pairs, giving the mapping from variables in + ``p`` to elementary symmetric polynomials. + + The triple satisfies the equation ``p.compose(m) + r == self``. + If the remainder ``r`` is zero, *self* is symmetric. If it is + nonzero, we were not able to represent *self* as symmetric. + + See Also + ======== + + sympy.polys.polyfuncs.symmetrize + + References + ========== + + .. [1] Lauer, E. Algorithms for symmetrical polynomials, Proc. 1976 + ACM Symp. on Symbolic and Algebraic Computing, NY 242-247. + https://dl.acm.org/doi/pdf/10.1145/800205.806342 + + """ + f = self.copy() + ring = f.ring + n = ring.ngens + + if not n: + return f, ring.zero, [] + + polys = [ring.symmetric_poly(i+1) for i in range(n)] + + poly_powers = {} + def get_poly_power(i, n): + if (i, n) not in poly_powers: + poly_powers[(i, n)] = polys[i]**n + return poly_powers[(i, n)] + + indices = list(range(n - 1)) + weights = list(range(n, 0, -1)) + + symmetric = ring.zero + + while f: + _height, _monom, _coeff = -1, None, None + + for i, (monom, coeff) in enumerate(f.terms()): + if all(monom[i] >= monom[i + 1] for i in indices): + height = max(n*m for n, m in zip(weights, monom)) + + if height > _height: + _height, _monom, _coeff = height, monom, coeff + + if _height != -1: + monom, coeff = _monom, _coeff + else: + break + + exponents = [] + for m1, m2 in zip(monom, monom[1:] + (0,)): + exponents.append(m1 - m2) + + symmetric += ring.term_new(tuple(exponents), coeff) + + product = coeff + for i, n in enumerate(exponents): + product *= get_poly_power(i, n) + f -= product + + mapping = list(zip(ring.gens, polys)) + + return symmetric, f, mapping + + def compose(f, x, a=None): + ring = f.ring + poly = ring.zero + gens_map = dict(zip(ring.gens, range(ring.ngens))) + + if a is not None: + replacements = [(x, a)] + else: + if isinstance(x, list): + replacements = list(x) + elif isinstance(x, dict): + replacements = sorted(x.items(), key=lambda k: gens_map[k[0]]) + else: + raise ValueError("expected a generator, value pair a sequence of such pairs") + + for k, (x, g) in enumerate(replacements): + replacements[k] = (gens_map[x], ring.ring_new(g)) + + for monom, coeff in f.iterterms(): + monom = list(monom) + subpoly = ring.one + + for i, g in replacements: + n, monom[i] = monom[i], 0 + if n: + subpoly *= g**n + + subpoly = subpoly.mul_term((tuple(monom), coeff)) + poly += subpoly + + return poly + + def coeff_wrt(self, x, deg): + """ + Coefficient of ``self`` with respect to ``x**deg``. + + Treating ``self`` as a univariate polynomial in ``x`` this finds the + coefficient of ``x**deg`` as a polynomial in the other generators. + + Parameters + ========== + + x : generator or generator index + The generator or generator index to compute the expression for. + deg : int + The degree of the monomial to compute the expression for. + + Returns + ======= + + :py:class:`~.PolyElement` + The coefficient of ``x**deg`` as a polynomial in the same ring. + + Examples + ======== + + >>> from sympy.polys import ring, ZZ + >>> R, x, y, z = ring("x, y, z", ZZ) + + >>> p = 2*x**4 + 3*y**4 + 10*z**2 + 10*x*z**2 + >>> deg = 2 + >>> p.coeff_wrt(2, deg) # Using the generator index + 10*x + 10 + >>> p.coeff_wrt(z, deg) # Using the generator + 10*x + 10 + >>> p.coeff(z**2) # shows the difference between coeff and coeff_wrt + 10 + + See Also + ======== + + coeff, coeffs + + """ + p = self + i = p.ring.index(x) + terms = [(m, c) for m, c in p.iterterms() if m[i] == deg] + + if not terms: + return p.ring.zero + + monoms, coeffs = zip(*terms) + monoms = [m[:i] + (0,) + m[i + 1:] for m in monoms] + return p.ring.from_dict(dict(zip(monoms, coeffs))) + + def prem(self, g, x=None): + """ + Pseudo-remainder of the polynomial ``self`` with respect to ``g``. + + The pseudo-quotient ``q`` and pseudo-remainder ``r`` with respect to + ``z`` when dividing ``f`` by ``g`` satisfy ``m*f = g*q + r``, + where ``deg(r,z) < deg(g,z)`` and + ``m = LC(g,z)**(deg(f,z) - deg(g,z)+1)``. + + See :meth:`pdiv` for explanation of pseudo-division. + + + Parameters + ========== + + g : :py:class:`~.PolyElement` + The polynomial to divide ``self`` by. + x : generator or generator index, optional + The main variable of the polynomials and default is first generator. + + Returns + ======= + + :py:class:`~.PolyElement` + The pseudo-remainder polynomial. + + Raises + ====== + + ZeroDivisionError : If ``g`` is the zero polynomial. + + Examples + ======== + + >>> from sympy.polys import ring, ZZ + >>> R, x, y = ring("x, y", ZZ) + + >>> f = x**2 + x*y + >>> g = 2*x + 2 + >>> f.prem(g) # first generator is chosen by default if it is not given + -4*y + 4 + >>> f.rem(g) # shows the difference between prem and rem + x**2 + x*y + >>> f.prem(g, y) # generator is given + 0 + >>> f.prem(g, 1) # generator index is given + 0 + + See Also + ======== + + pdiv, pquo, pexquo, sympy.polys.domains.ring.Ring.rem + + """ + f = self + x = f.ring.index(x) + df = f.degree(x) + dg = g.degree(x) + + if dg < 0: + raise ZeroDivisionError('polynomial division') + + r, dr = f, df + + if df < dg: + return r + + N = df - dg + 1 + + lc_g = g.coeff_wrt(x, dg) + + xp = f.ring.gens[x] + + while True: + + lc_r = r.coeff_wrt(x, dr) + j, N = dr - dg, N - 1 + + R = r * lc_g + G = g * lc_r * xp**j + r = R - G + + dr = r.degree(x) + + if dr < dg: + break + + c = lc_g ** N + + return r * c + + def pdiv(self, g, x=None): + """ + Computes the pseudo-division of the polynomial ``self`` with respect to ``g``. + + The pseudo-division algorithm is used to find the pseudo-quotient ``q`` + and pseudo-remainder ``r`` such that ``m*f = g*q + r``, where ``m`` + represents the multiplier and ``f`` is the dividend polynomial. + + The pseudo-quotient ``q`` and pseudo-remainder ``r`` are polynomials in + the variable ``x``, with the degree of ``r`` with respect to ``x`` + being strictly less than the degree of ``g`` with respect to ``x``. + + The multiplier ``m`` is defined as + ``LC(g, x) ^ (deg(f, x) - deg(g, x) + 1)``, + where ``LC(g, x)`` represents the leading coefficient of ``g``. + + It is important to note that in the context of the ``prem`` method, + multivariate polynomials in a ring, such as ``R[x,y,z]``, are treated + as univariate polynomials with coefficients that are polynomials, + such as ``R[x,y][z]``. When dividing ``f`` by ``g`` with respect to the + variable ``z``, the pseudo-quotient ``q`` and pseudo-remainder ``r`` + satisfy ``m*f = g*q + r``, where ``deg(r, z) < deg(g, z)`` + and ``m = LC(g, z)^(deg(f, z) - deg(g, z) + 1)``. + + In this function, the pseudo-remainder ``r`` can be obtained using the + ``prem`` method, the pseudo-quotient ``q`` can + be obtained using the ``pquo`` method, and + the function ``pdiv`` itself returns a tuple ``(q, r)``. + + + Parameters + ========== + + g : :py:class:`~.PolyElement` + The polynomial to divide ``self`` by. + x : generator or generator index, optional + The main variable of the polynomials and default is first generator. + + Returns + ======= + + :py:class:`~.PolyElement` + The pseudo-division polynomial (tuple of ``q`` and ``r``). + + Raises + ====== + + ZeroDivisionError : If ``g`` is the zero polynomial. + + Examples + ======== + + >>> from sympy.polys import ring, ZZ + >>> R, x, y = ring("x, y", ZZ) + + >>> f = x**2 + x*y + >>> g = 2*x + 2 + >>> f.pdiv(g) # first generator is chosen by default if it is not given + (2*x + 2*y - 2, -4*y + 4) + >>> f.div(g) # shows the difference between pdiv and div + (0, x**2 + x*y) + >>> f.pdiv(g, y) # generator is given + (2*x**3 + 2*x**2*y + 6*x**2 + 2*x*y + 8*x + 4, 0) + >>> f.pdiv(g, 1) # generator index is given + (2*x**3 + 2*x**2*y + 6*x**2 + 2*x*y + 8*x + 4, 0) + + See Also + ======== + + prem + Computes only the pseudo-remainder more efficiently than + `f.pdiv(g)[1]`. + pquo + Returns only the pseudo-quotient. + pexquo + Returns only an exact pseudo-quotient having no remainder. + div + Returns quotient and remainder of f and g polynomials. + + """ + f = self + x = f.ring.index(x) + + df = f.degree(x) + dg = g.degree(x) + + if dg < 0: + raise ZeroDivisionError("polynomial division") + + q, r, dr = x, f, df + + if df < dg: + return q, r + + N = df - dg + 1 + lc_g = g.coeff_wrt(x, dg) + + xp = f.ring.gens[x] + + while True: + + lc_r = r.coeff_wrt(x, dr) + j, N = dr - dg, N - 1 + + Q = q * lc_g + + q = Q + (lc_r)*xp**j + + R = r * lc_g + + G = g * lc_r * xp**j + + r = R - G + + dr = r.degree(x) + + if dr < dg: + break + + c = lc_g**N + + q = q * c + r = r * c + + return q, r + + def pquo(self, g, x=None): + """ + Polynomial pseudo-quotient in multivariate polynomial ring. + + Examples + ======== + >>> from sympy.polys import ring, ZZ + >>> R, x,y = ring("x,y", ZZ) + + >>> f = x**2 + x*y + >>> g = 2*x + 2*y + >>> h = 2*x + 2 + >>> f.pquo(g) + 2*x + >>> f.quo(g) # shows the difference between pquo and quo + 0 + >>> f.pquo(h) + 2*x + 2*y - 2 + >>> f.quo(h) # shows the difference between pquo and quo + 0 + + See Also + ======== + + prem, pdiv, pexquo, sympy.polys.domains.ring.Ring.quo + + """ + f = self + return f.pdiv(g, x)[0] + + def pexquo(self, g, x=None): + """ + Polynomial exact pseudo-quotient in multivariate polynomial ring. + + Examples + ======== + >>> from sympy.polys import ring, ZZ + >>> R, x,y = ring("x,y", ZZ) + + >>> f = x**2 + x*y + >>> g = 2*x + 2*y + >>> h = 2*x + 2 + >>> f.pexquo(g) + 2*x + >>> f.exquo(g) # shows the difference between pexquo and exquo + Traceback (most recent call last): + ... + ExactQuotientFailed: 2*x + 2*y does not divide x**2 + x*y + >>> f.pexquo(h) + Traceback (most recent call last): + ... + ExactQuotientFailed: 2*x + 2 does not divide x**2 + x*y + + See Also + ======== + + prem, pdiv, pquo, sympy.polys.domains.ring.Ring.exquo + + """ + f = self + q, r = f.pdiv(g, x) + + if r.is_zero: + return q + else: + raise ExactQuotientFailed(f, g) + + def subresultants(self, g, x=None): + """ + Computes the subresultant PRS of two polynomials ``self`` and ``g``. + + Parameters + ========== + + g : :py:class:`~.PolyElement` + The second polynomial. + x : generator or generator index + The variable with respect to which the subresultant sequence is computed. + + Returns + ======= + + R : list + Returns a list polynomials representing the subresultant PRS. + + Examples + ======== + + >>> from sympy.polys import ring, ZZ + >>> R, x, y = ring("x, y", ZZ) + + >>> f = x**2*y + x*y + >>> g = x + y + >>> f.subresultants(g) # first generator is chosen by default if not given + [x**2*y + x*y, x + y, y**3 - y**2] + >>> f.subresultants(g, 0) # generator index is given + [x**2*y + x*y, x + y, y**3 - y**2] + >>> f.subresultants(g, y) # generator is given + [x**2*y + x*y, x + y, x**3 + x**2] + + """ + f = self + x = f.ring.index(x) + n = f.degree(x) + m = g.degree(x) + + if n < m: + f, g = g, f + n, m = m, n + + if f == 0: + return [0, 0] + + if g == 0: + return [f, 1] + + R = [f, g] + + d = n - m + b = (-1) ** (d + 1) + + # Compute the pseudo-remainder for f and g + h = f.prem(g, x) + h = h * b + + # Compute the coefficient of g with respect to x**m + lc = g.coeff_wrt(x, m) + + c = lc ** d + + S = [1, c] + + c = -c + + while h: + k = h.degree(x) + + R.append(h) + f, g, m, d = g, h, k, m - k + + b = -lc * c ** d + h = f.prem(g, x) + h = h.exquo(b) + + lc = g.coeff_wrt(x, k) + + if d > 1: + p = (-lc) ** d + q = c ** (d - 1) + c = p.exquo(q) + else: + c = -lc + + S.append(-c) + + return R + + # TODO: following methods should point to polynomial + # representation independent algorithm implementations. + + def half_gcdex(f, g): + return f.ring.dmp_half_gcdex(f, g) + + def gcdex(f, g): + return f.ring.dmp_gcdex(f, g) + + def resultant(f, g): + return f.ring.dmp_resultant(f, g) + + def discriminant(f): + return f.ring.dmp_discriminant(f) + + def decompose(f): + if f.ring.is_univariate: + return f.ring.dup_decompose(f) + else: + raise MultivariatePolynomialError("polynomial decomposition") + + def shift(f, a): + if f.ring.is_univariate: + return f.ring.dup_shift(f, a) + else: + raise MultivariatePolynomialError("shift: use shift_list instead") + + def shift_list(f, a): + return f.ring.dmp_shift(f, a) + + def sturm(f): + if f.ring.is_univariate: + return f.ring.dup_sturm(f) + else: + raise MultivariatePolynomialError("sturm sequence") + + def gff_list(f): + return f.ring.dmp_gff_list(f) + + def norm(f): + return f.ring.dmp_norm(f) + + def sqf_norm(f): + return f.ring.dmp_sqf_norm(f) + + def sqf_part(f): + return f.ring.dmp_sqf_part(f) + + def sqf_list(f, all=False): + return f.ring.dmp_sqf_list(f, all=all) + + def factor_list(f): + return f.ring.dmp_factor_list(f) diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/rootisolation.py b/.venv/lib/python3.12/site-packages/sympy/polys/rootisolation.py new file mode 100644 index 0000000000000000000000000000000000000000..b2f8fd115e49ce8dcf4db8659a60c3361818b7bb --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/rootisolation.py @@ -0,0 +1,2190 @@ +"""Real and complex root isolation and refinement algorithms. """ + + +from sympy.polys.densearith import ( + dup_neg, dup_rshift, dup_rem, + dup_l2_norm_squared) +from sympy.polys.densebasic import ( + dup_LC, dup_TC, dup_degree, + dup_strip, dup_reverse, + dup_convert, + dup_terms_gcd) +from sympy.polys.densetools import ( + dup_clear_denoms, + dup_mirror, dup_scale, dup_shift, + dup_transform, + dup_diff, + dup_eval, dmp_eval_in, + dup_sign_variations, + dup_real_imag) +from sympy.polys.euclidtools import ( + dup_discriminant) +from sympy.polys.factortools import ( + dup_factor_list) +from sympy.polys.polyerrors import ( + RefinementFailed, + DomainError, + PolynomialError) +from sympy.polys.sqfreetools import ( + dup_sqf_part, dup_sqf_list) + + +def dup_sturm(f, K): + """ + Computes the Sturm sequence of ``f`` in ``F[x]``. + + Given a univariate, square-free polynomial ``f(x)`` returns the + associated Sturm sequence ``f_0(x), ..., f_n(x)`` defined by:: + + f_0(x), f_1(x) = f(x), f'(x) + f_n = -rem(f_{n-2}(x), f_{n-1}(x)) + + Examples + ======== + + >>> from sympy.polys import ring, QQ + >>> R, x = ring("x", QQ) + + >>> R.dup_sturm(x**3 - 2*x**2 + x - 3) + [x**3 - 2*x**2 + x - 3, 3*x**2 - 4*x + 1, 2/9*x + 25/9, -2079/4] + + References + ========== + + .. [1] [Davenport88]_ + + """ + if not K.is_Field: + raise DomainError("Cannot compute Sturm sequence over %s" % K) + + f = dup_sqf_part(f, K) + + sturm = [f, dup_diff(f, 1, K)] + + while sturm[-1]: + s = dup_rem(sturm[-2], sturm[-1], K) + sturm.append(dup_neg(s, K)) + + return sturm[:-1] + +def dup_root_upper_bound(f, K): + """Compute the LMQ upper bound for the positive roots of `f`; + LMQ (Local Max Quadratic) was developed by Akritas-Strzebonski-Vigklas. + + References + ========== + .. [1] Alkiviadis G. Akritas: "Linear and Quadratic Complexity Bounds on the + Values of the Positive Roots of Polynomials" + Journal of Universal Computer Science, Vol. 15, No. 3, 523-537, 2009. + """ + n, P = len(f), [] + t = n * [K.one] + if dup_LC(f, K) < 0: + f = dup_neg(f, K) + f = list(reversed(f)) + + for i in range(0, n): + if f[i] >= 0: + continue + + a, QL = K.log(-f[i], 2), [] + + for j in range(i + 1, n): + + if f[j] <= 0: + continue + + q = t[j] + a - K.log(f[j], 2) + QL.append([q // (j - i), j]) + + if not QL: + continue + + q = min(QL) + + t[q[1]] = t[q[1]] + 1 + + P.append(q[0]) + + if not P: + return None + else: + return K.get_field()(2)**(max(P) + 1) + +def dup_root_lower_bound(f, K): + """Compute the LMQ lower bound for the positive roots of `f`; + LMQ (Local Max Quadratic) was developed by Akritas-Strzebonski-Vigklas. + + References + ========== + .. [1] Alkiviadis G. Akritas: "Linear and Quadratic Complexity Bounds on the + Values of the Positive Roots of Polynomials" + Journal of Universal Computer Science, Vol. 15, No. 3, 523-537, 2009. + """ + bound = dup_root_upper_bound(dup_reverse(f), K) + + if bound is not None: + return 1/bound + else: + return None + +def dup_cauchy_upper_bound(f, K): + """ + Compute the Cauchy upper bound on the absolute value of all roots of f, + real or complex. + + References + ========== + .. [1] https://en.wikipedia.org/wiki/Geometrical_properties_of_polynomial_roots#Lagrange's_and_Cauchy's_bounds + """ + n = dup_degree(f) + if n < 1: + raise PolynomialError('Polynomial has no roots.') + + if K.is_ZZ: + L = K.get_field() + f, K = dup_convert(f, K, L), L + elif not K.is_QQ or K.is_RR or K.is_CC: + # We need to compute absolute value, and we are not supporting cases + # where this would take us outside the domain (or its quotient field). + raise DomainError('Cauchy bound not supported over %s' % K) + else: + f = f[:] + + while K.is_zero(f[-1]): + f.pop() + if len(f) == 1: + # Monomial. All roots are zero. + return K.zero + + lc = f[0] + return K.one + max(abs(n / lc) for n in f[1:]) + +def dup_cauchy_lower_bound(f, K): + """Compute the Cauchy lower bound on the absolute value of all non-zero + roots of f, real or complex.""" + g = dup_reverse(f) + if len(g) < 2: + raise PolynomialError('Polynomial has no non-zero roots.') + if K.is_ZZ: + K = K.get_field() + b = dup_cauchy_upper_bound(g, K) + return K.one / b + +def dup_mignotte_sep_bound_squared(f, K): + """ + Return the square of the Mignotte lower bound on separation between + distinct roots of f. The square is returned so that the bound lies in + K or its quotient field. + + References + ========== + + .. [1] Mignotte, Maurice. "Some useful bounds." Computer algebra. + Springer, Vienna, 1982. 259-263. + https://people.dm.unipi.it/gianni/AC-EAG/Mignotte.pdf + """ + n = dup_degree(f) + if n < 2: + raise PolynomialError('Polynomials of degree < 2 have no distinct roots.') + + if K.is_ZZ: + L = K.get_field() + f, K = dup_convert(f, K, L), L + elif not K.is_QQ or K.is_RR or K.is_CC: + # We need to compute absolute value, and we are not supporting cases + # where this would take us outside the domain (or its quotient field). + raise DomainError('Mignotte bound not supported over %s' % K) + + D = dup_discriminant(f, K) + l2sq = dup_l2_norm_squared(f, K) + return K(3)*K.abs(D) / ( K(n)**(n+1) * l2sq**(n-1) ) + +def _mobius_from_interval(I, field): + """Convert an open interval to a Mobius transform. """ + s, t = I + + a, c = field.numer(s), field.denom(s) + b, d = field.numer(t), field.denom(t) + + return a, b, c, d + +def _mobius_to_interval(M, field): + """Convert a Mobius transform to an open interval. """ + a, b, c, d = M + + s, t = field(a, c), field(b, d) + + if s <= t: + return (s, t) + else: + return (t, s) + +def dup_step_refine_real_root(f, M, K, fast=False): + """One step of positive real root refinement algorithm. """ + a, b, c, d = M + + if a == b and c == d: + return f, (a, b, c, d) + + A = dup_root_lower_bound(f, K) + + if A is not None: + A = K(int(A)) + else: + A = K.zero + + if fast and A > 16: + f = dup_scale(f, A, K) + a, c, A = A*a, A*c, K.one + + if A >= K.one: + f = dup_shift(f, A, K) + b, d = A*a + b, A*c + d + + if not dup_eval(f, K.zero, K): + return f, (b, b, d, d) + + f, g = dup_shift(f, K.one, K), f + + a1, b1, c1, d1 = a, a + b, c, c + d + + if not dup_eval(f, K.zero, K): + return f, (b1, b1, d1, d1) + + k = dup_sign_variations(f, K) + + if k == 1: + a, b, c, d = a1, b1, c1, d1 + else: + f = dup_shift(dup_reverse(g), K.one, K) + + if not dup_eval(f, K.zero, K): + f = dup_rshift(f, 1, K) + + a, b, c, d = b, a + b, d, c + d + + return f, (a, b, c, d) + +def dup_inner_refine_real_root(f, M, K, eps=None, steps=None, disjoint=None, fast=False, mobius=False): + """Refine a positive root of `f` given a Mobius transform or an interval. """ + F = K.get_field() + + if len(M) == 2: + a, b, c, d = _mobius_from_interval(M, F) + else: + a, b, c, d = M + + while not c: + f, (a, b, c, d) = dup_step_refine_real_root(f, (a, b, c, + d), K, fast=fast) + + if eps is not None and steps is not None: + for i in range(0, steps): + if abs(F(a, c) - F(b, d)) >= eps: + f, (a, b, c, d) = dup_step_refine_real_root(f, (a, b, c, d), K, fast=fast) + else: + break + else: + if eps is not None: + while abs(F(a, c) - F(b, d)) >= eps: + f, (a, b, c, d) = dup_step_refine_real_root(f, (a, b, c, d), K, fast=fast) + + if steps is not None: + for i in range(0, steps): + f, (a, b, c, d) = dup_step_refine_real_root(f, (a, b, c, d), K, fast=fast) + + if disjoint is not None: + while True: + u, v = _mobius_to_interval((a, b, c, d), F) + + if v <= disjoint or disjoint <= u: + break + else: + f, (a, b, c, d) = dup_step_refine_real_root(f, (a, b, c, d), K, fast=fast) + + if not mobius: + return _mobius_to_interval((a, b, c, d), F) + else: + return f, (a, b, c, d) + +def dup_outer_refine_real_root(f, s, t, K, eps=None, steps=None, disjoint=None, fast=False): + """Refine a positive root of `f` given an interval `(s, t)`. """ + a, b, c, d = _mobius_from_interval((s, t), K.get_field()) + + f = dup_transform(f, dup_strip([a, b]), + dup_strip([c, d]), K) + + if dup_sign_variations(f, K) != 1: + raise RefinementFailed("there should be exactly one root in (%s, %s) interval" % (s, t)) + + return dup_inner_refine_real_root(f, (a, b, c, d), K, eps=eps, steps=steps, disjoint=disjoint, fast=fast) + +def dup_refine_real_root(f, s, t, K, eps=None, steps=None, disjoint=None, fast=False): + """Refine real root's approximating interval to the given precision. """ + if K.is_QQ: + (_, f), K = dup_clear_denoms(f, K, convert=True), K.get_ring() + elif not K.is_ZZ: + raise DomainError("real root refinement not supported over %s" % K) + + if s == t: + return (s, t) + + if s > t: + s, t = t, s + + negative = False + + if s < 0: + if t <= 0: + f, s, t, negative = dup_mirror(f, K), -t, -s, True + else: + raise ValueError("Cannot refine a real root in (%s, %s)" % (s, t)) + + if negative and disjoint is not None: + if disjoint < 0: + disjoint = -disjoint + else: + disjoint = None + + s, t = dup_outer_refine_real_root( + f, s, t, K, eps=eps, steps=steps, disjoint=disjoint, fast=fast) + + if negative: + return (-t, -s) + else: + return ( s, t) + +def dup_inner_isolate_real_roots(f, K, eps=None, fast=False): + """Internal function for isolation positive roots up to given precision. + + References + ========== + 1. Alkiviadis G. Akritas and Adam W. Strzebonski: A Comparative Study of Two Real Root + Isolation Methods . Nonlinear Analysis: Modelling and Control, Vol. 10, No. 4, 297-304, 2005. + 2. Alkiviadis G. Akritas, Adam W. Strzebonski and Panagiotis S. Vigklas: Improving the + Performance of the Continued Fractions Method Using new Bounds of Positive Roots. Nonlinear + Analysis: Modelling and Control, Vol. 13, No. 3, 265-279, 2008. + """ + a, b, c, d = K.one, K.zero, K.zero, K.one + + k = dup_sign_variations(f, K) + + if k == 0: + return [] + if k == 1: + roots = [dup_inner_refine_real_root( + f, (a, b, c, d), K, eps=eps, fast=fast, mobius=True)] + else: + roots, stack = [], [(a, b, c, d, f, k)] + + while stack: + a, b, c, d, f, k = stack.pop() + + A = dup_root_lower_bound(f, K) + + if A is not None: + A = K(int(A)) + else: + A = K.zero + + if fast and A > 16: + f = dup_scale(f, A, K) + a, c, A = A*a, A*c, K.one + + if A >= K.one: + f = dup_shift(f, A, K) + b, d = A*a + b, A*c + d + + if not dup_TC(f, K): + roots.append((f, (b, b, d, d))) + f = dup_rshift(f, 1, K) + + k = dup_sign_variations(f, K) + + if k == 0: + continue + if k == 1: + roots.append(dup_inner_refine_real_root( + f, (a, b, c, d), K, eps=eps, fast=fast, mobius=True)) + continue + + f1 = dup_shift(f, K.one, K) + + a1, b1, c1, d1, r = a, a + b, c, c + d, 0 + + if not dup_TC(f1, K): + roots.append((f1, (b1, b1, d1, d1))) + f1, r = dup_rshift(f1, 1, K), 1 + + k1 = dup_sign_variations(f1, K) + k2 = k - k1 - r + + a2, b2, c2, d2 = b, a + b, d, c + d + + if k2 > 1: + f2 = dup_shift(dup_reverse(f), K.one, K) + + if not dup_TC(f2, K): + f2 = dup_rshift(f2, 1, K) + + k2 = dup_sign_variations(f2, K) + else: + f2 = None + + if k1 < k2: + a1, a2, b1, b2 = a2, a1, b2, b1 + c1, c2, d1, d2 = c2, c1, d2, d1 + f1, f2, k1, k2 = f2, f1, k2, k1 + + if not k1: + continue + + if f1 is None: + f1 = dup_shift(dup_reverse(f), K.one, K) + + if not dup_TC(f1, K): + f1 = dup_rshift(f1, 1, K) + + if k1 == 1: + roots.append(dup_inner_refine_real_root( + f1, (a1, b1, c1, d1), K, eps=eps, fast=fast, mobius=True)) + else: + stack.append((a1, b1, c1, d1, f1, k1)) + + if not k2: + continue + + if f2 is None: + f2 = dup_shift(dup_reverse(f), K.one, K) + + if not dup_TC(f2, K): + f2 = dup_rshift(f2, 1, K) + + if k2 == 1: + roots.append(dup_inner_refine_real_root( + f2, (a2, b2, c2, d2), K, eps=eps, fast=fast, mobius=True)) + else: + stack.append((a2, b2, c2, d2, f2, k2)) + + return roots + +def _discard_if_outside_interval(f, M, inf, sup, K, negative, fast, mobius): + """Discard an isolating interval if outside ``(inf, sup)``. """ + F = K.get_field() + + while True: + u, v = _mobius_to_interval(M, F) + + if negative: + u, v = -v, -u + + if (inf is None or u >= inf) and (sup is None or v <= sup): + if not mobius: + return u, v + else: + return f, M + elif (sup is not None and u > sup) or (inf is not None and v < inf): + return None + else: + f, M = dup_step_refine_real_root(f, M, K, fast=fast) + +def dup_inner_isolate_positive_roots(f, K, eps=None, inf=None, sup=None, fast=False, mobius=False): + """Iteratively compute disjoint positive root isolation intervals. """ + if sup is not None and sup < 0: + return [] + + roots = dup_inner_isolate_real_roots(f, K, eps=eps, fast=fast) + + F, results = K.get_field(), [] + + if inf is not None or sup is not None: + for f, M in roots: + result = _discard_if_outside_interval(f, M, inf, sup, K, False, fast, mobius) + + if result is not None: + results.append(result) + elif not mobius: + results.extend(_mobius_to_interval(M, F) for _, M in roots) + else: + results = roots + + return results + +def dup_inner_isolate_negative_roots(f, K, inf=None, sup=None, eps=None, fast=False, mobius=False): + """Iteratively compute disjoint negative root isolation intervals. """ + if inf is not None and inf >= 0: + return [] + + roots = dup_inner_isolate_real_roots(dup_mirror(f, K), K, eps=eps, fast=fast) + + F, results = K.get_field(), [] + + if inf is not None or sup is not None: + for f, M in roots: + result = _discard_if_outside_interval(f, M, inf, sup, K, True, fast, mobius) + + if result is not None: + results.append(result) + elif not mobius: + for f, M in roots: + u, v = _mobius_to_interval(M, F) + results.append((-v, -u)) + else: + results = roots + + return results + +def _isolate_zero(f, K, inf, sup, basis=False, sqf=False): + """Handle special case of CF algorithm when ``f`` is homogeneous. """ + j, f = dup_terms_gcd(f, K) + + if j > 0: + F = K.get_field() + + if (inf is None or inf <= 0) and (sup is None or 0 <= sup): + if not sqf: + if not basis: + return [((F.zero, F.zero), j)], f + else: + return [((F.zero, F.zero), j, [K.one, K.zero])], f + else: + return [(F.zero, F.zero)], f + + return [], f + +def dup_isolate_real_roots_sqf(f, K, eps=None, inf=None, sup=None, fast=False, blackbox=False): + """Isolate real roots of a square-free polynomial using the Vincent-Akritas-Strzebonski (VAS) CF approach. + + References + ========== + .. [1] Alkiviadis G. Akritas and Adam W. Strzebonski: A Comparative + Study of Two Real Root Isolation Methods. Nonlinear Analysis: + Modelling and Control, Vol. 10, No. 4, 297-304, 2005. + .. [2] Alkiviadis G. Akritas, Adam W. Strzebonski and Panagiotis S. + Vigklas: Improving the Performance of the Continued Fractions + Method Using New Bounds of Positive Roots. Nonlinear Analysis: + Modelling and Control, Vol. 13, No. 3, 265-279, 2008. + + """ + if K.is_QQ: + (_, f), K = dup_clear_denoms(f, K, convert=True), K.get_ring() + elif not K.is_ZZ: + raise DomainError("isolation of real roots not supported over %s" % K) + + if dup_degree(f) <= 0: + return [] + + I_zero, f = _isolate_zero(f, K, inf, sup, basis=False, sqf=True) + + I_neg = dup_inner_isolate_negative_roots(f, K, eps=eps, inf=inf, sup=sup, fast=fast) + I_pos = dup_inner_isolate_positive_roots(f, K, eps=eps, inf=inf, sup=sup, fast=fast) + + roots = sorted(I_neg + I_zero + I_pos) + + if not blackbox: + return roots + else: + return [ RealInterval((a, b), f, K) for (a, b) in roots ] + +def dup_isolate_real_roots(f, K, eps=None, inf=None, sup=None, basis=False, fast=False): + """Isolate real roots using Vincent-Akritas-Strzebonski (VAS) continued fractions approach. + + References + ========== + + .. [1] Alkiviadis G. Akritas and Adam W. Strzebonski: A Comparative + Study of Two Real Root Isolation Methods. Nonlinear Analysis: + Modelling and Control, Vol. 10, No. 4, 297-304, 2005. + .. [2] Alkiviadis G. Akritas, Adam W. Strzebonski and Panagiotis S. + Vigklas: Improving the Performance of the Continued Fractions + Method Using New Bounds of Positive Roots. + Nonlinear Analysis: Modelling and Control, Vol. 13, No. 3, 265-279, 2008. + + """ + if K.is_QQ: + (_, f), K = dup_clear_denoms(f, K, convert=True), K.get_ring() + elif not K.is_ZZ: + raise DomainError("isolation of real roots not supported over %s" % K) + + if dup_degree(f) <= 0: + return [] + + I_zero, f = _isolate_zero(f, K, inf, sup, basis=basis, sqf=False) + + _, factors = dup_sqf_list(f, K) + + if len(factors) == 1: + ((f, k),) = factors + + I_neg = dup_inner_isolate_negative_roots(f, K, eps=eps, inf=inf, sup=sup, fast=fast) + I_pos = dup_inner_isolate_positive_roots(f, K, eps=eps, inf=inf, sup=sup, fast=fast) + + I_neg = [ ((u, v), k) for u, v in I_neg ] + I_pos = [ ((u, v), k) for u, v in I_pos ] + else: + I_neg, I_pos = _real_isolate_and_disjoin(factors, K, + eps=eps, inf=inf, sup=sup, basis=basis, fast=fast) + + return sorted(I_neg + I_zero + I_pos) + +def dup_isolate_real_roots_list(polys, K, eps=None, inf=None, sup=None, strict=False, basis=False, fast=False): + """Isolate real roots of a list of polynomial using Vincent-Akritas-Strzebonski (VAS) CF approach. + + References + ========== + + .. [1] Alkiviadis G. Akritas and Adam W. Strzebonski: A Comparative + Study of Two Real Root Isolation Methods. Nonlinear Analysis: + Modelling and Control, Vol. 10, No. 4, 297-304, 2005. + .. [2] Alkiviadis G. Akritas, Adam W. Strzebonski and Panagiotis S. + Vigklas: Improving the Performance of the Continued Fractions + Method Using New Bounds of Positive Roots. + Nonlinear Analysis: Modelling and Control, Vol. 13, No. 3, 265-279, 2008. + + """ + if K.is_QQ: + K, F, polys = K.get_ring(), K, polys[:] + + for i, p in enumerate(polys): + polys[i] = dup_clear_denoms(p, F, K, convert=True)[1] + elif not K.is_ZZ: + raise DomainError("isolation of real roots not supported over %s" % K) + + zeros, factors_dict = False, {} + + if (inf is None or inf <= 0) and (sup is None or 0 <= sup): + zeros, zero_indices = True, {} + + for i, p in enumerate(polys): + j, p = dup_terms_gcd(p, K) + + if zeros and j > 0: + zero_indices[i] = j + + for f, k in dup_factor_list(p, K)[1]: + f = tuple(f) + + if f not in factors_dict: + factors_dict[f] = {i: k} + else: + factors_dict[f][i] = k + + factors_list = [(list(f), indices) for f, indices in factors_dict.items()] + I_neg, I_pos = _real_isolate_and_disjoin(factors_list, K, eps=eps, + inf=inf, sup=sup, strict=strict, basis=basis, fast=fast) + + F = K.get_field() + + if not zeros or not zero_indices: + I_zero = [] + else: + if not basis: + I_zero = [((F.zero, F.zero), zero_indices)] + else: + I_zero = [((F.zero, F.zero), zero_indices, [K.one, K.zero])] + + return sorted(I_neg + I_zero + I_pos) + +def _disjoint_p(M, N, strict=False): + """Check if Mobius transforms define disjoint intervals. """ + a1, b1, c1, d1 = M + a2, b2, c2, d2 = N + + a1d1, b1c1 = a1*d1, b1*c1 + a2d2, b2c2 = a2*d2, b2*c2 + + if a1d1 == b1c1 and a2d2 == b2c2: + return True + + if a1d1 > b1c1: + a1, c1, b1, d1 = b1, d1, a1, c1 + + if a2d2 > b2c2: + a2, c2, b2, d2 = b2, d2, a2, c2 + + if not strict: + return a2*d1 >= c2*b1 or b2*c1 <= d2*a1 + else: + return a2*d1 > c2*b1 or b2*c1 < d2*a1 + +def _real_isolate_and_disjoin(factors, K, eps=None, inf=None, sup=None, strict=False, basis=False, fast=False): + """Isolate real roots of a list of polynomials and disjoin intervals. """ + I_pos, I_neg = [], [] + + for i, (f, k) in enumerate(factors): + for F, M in dup_inner_isolate_positive_roots(f, K, eps=eps, inf=inf, sup=sup, fast=fast, mobius=True): + I_pos.append((F, M, k, f)) + + for G, N in dup_inner_isolate_negative_roots(f, K, eps=eps, inf=inf, sup=sup, fast=fast, mobius=True): + I_neg.append((G, N, k, f)) + + for i, (f, M, k, F) in enumerate(I_pos): + for j, (g, N, m, G) in enumerate(I_pos[i + 1:]): + while not _disjoint_p(M, N, strict=strict): + f, M = dup_inner_refine_real_root(f, M, K, steps=1, fast=fast, mobius=True) + g, N = dup_inner_refine_real_root(g, N, K, steps=1, fast=fast, mobius=True) + + I_pos[i + j + 1] = (g, N, m, G) + + I_pos[i] = (f, M, k, F) + + for i, (f, M, k, F) in enumerate(I_neg): + for j, (g, N, m, G) in enumerate(I_neg[i + 1:]): + while not _disjoint_p(M, N, strict=strict): + f, M = dup_inner_refine_real_root(f, M, K, steps=1, fast=fast, mobius=True) + g, N = dup_inner_refine_real_root(g, N, K, steps=1, fast=fast, mobius=True) + + I_neg[i + j + 1] = (g, N, m, G) + + I_neg[i] = (f, M, k, F) + + if strict: + for i, (f, M, k, F) in enumerate(I_neg): + if not M[0]: + while not M[0]: + f, M = dup_inner_refine_real_root(f, M, K, steps=1, fast=fast, mobius=True) + + I_neg[i] = (f, M, k, F) + break + + for j, (g, N, m, G) in enumerate(I_pos): + if not N[0]: + while not N[0]: + g, N = dup_inner_refine_real_root(g, N, K, steps=1, fast=fast, mobius=True) + + I_pos[j] = (g, N, m, G) + break + + field = K.get_field() + + I_neg = [ (_mobius_to_interval(M, field), k, f) for (_, M, k, f) in I_neg ] + I_pos = [ (_mobius_to_interval(M, field), k, f) for (_, M, k, f) in I_pos ] + + I_neg = [((-v, -u), k, f) for ((u, v), k, f) in I_neg] + + if not basis: + I_neg = [((u, v), k) for ((u, v), k, _) in I_neg] + I_pos = [((u, v), k) for ((u, v), k, _) in I_pos] + + return I_neg, I_pos + +def dup_count_real_roots(f, K, inf=None, sup=None): + """Returns the number of distinct real roots of ``f`` in ``[inf, sup]``. """ + if dup_degree(f) <= 0: + return 0 + + if not K.is_Field: + R, K = K, K.get_field() + f = dup_convert(f, R, K) + + sturm = dup_sturm(f, K) + + if inf is None: + signs_inf = dup_sign_variations([ dup_LC(s, K)*(-1)**dup_degree(s) for s in sturm ], K) + else: + signs_inf = dup_sign_variations([ dup_eval(s, inf, K) for s in sturm ], K) + + if sup is None: + signs_sup = dup_sign_variations([ dup_LC(s, K) for s in sturm ], K) + else: + signs_sup = dup_sign_variations([ dup_eval(s, sup, K) for s in sturm ], K) + + count = abs(signs_inf - signs_sup) + + if inf is not None and not dup_eval(f, inf, K): + count += 1 + + return count + +OO = 'OO' # Origin of (re, im) coordinate system + +Q1 = 'Q1' # Quadrant #1 (++): re > 0 and im > 0 +Q2 = 'Q2' # Quadrant #2 (-+): re < 0 and im > 0 +Q3 = 'Q3' # Quadrant #3 (--): re < 0 and im < 0 +Q4 = 'Q4' # Quadrant #4 (+-): re > 0 and im < 0 + +A1 = 'A1' # Axis #1 (+0): re > 0 and im = 0 +A2 = 'A2' # Axis #2 (0+): re = 0 and im > 0 +A3 = 'A3' # Axis #3 (-0): re < 0 and im = 0 +A4 = 'A4' # Axis #4 (0-): re = 0 and im < 0 + +_rules_simple = { + # Q --> Q (same) => no change + (Q1, Q1): 0, + (Q2, Q2): 0, + (Q3, Q3): 0, + (Q4, Q4): 0, + + # A -- CCW --> Q => +1/4 (CCW) + (A1, Q1): 1, + (A2, Q2): 1, + (A3, Q3): 1, + (A4, Q4): 1, + + # A -- CW --> Q => -1/4 (CCW) + (A1, Q4): 2, + (A2, Q1): 2, + (A3, Q2): 2, + (A4, Q3): 2, + + # Q -- CCW --> A => +1/4 (CCW) + (Q1, A2): 3, + (Q2, A3): 3, + (Q3, A4): 3, + (Q4, A1): 3, + + # Q -- CW --> A => -1/4 (CCW) + (Q1, A1): 4, + (Q2, A2): 4, + (Q3, A3): 4, + (Q4, A4): 4, + + # Q -- CCW --> Q => +1/2 (CCW) + (Q1, Q2): +5, + (Q2, Q3): +5, + (Q3, Q4): +5, + (Q4, Q1): +5, + + # Q -- CW --> Q => -1/2 (CW) + (Q1, Q4): -5, + (Q2, Q1): -5, + (Q3, Q2): -5, + (Q4, Q3): -5, +} + +_rules_ambiguous = { + # A -- CCW --> Q => { +1/4 (CCW), -9/4 (CW) } + (A1, OO, Q1): -1, + (A2, OO, Q2): -1, + (A3, OO, Q3): -1, + (A4, OO, Q4): -1, + + # A -- CW --> Q => { -1/4 (CCW), +7/4 (CW) } + (A1, OO, Q4): -2, + (A2, OO, Q1): -2, + (A3, OO, Q2): -2, + (A4, OO, Q3): -2, + + # Q -- CCW --> A => { +1/4 (CCW), -9/4 (CW) } + (Q1, OO, A2): -3, + (Q2, OO, A3): -3, + (Q3, OO, A4): -3, + (Q4, OO, A1): -3, + + # Q -- CW --> A => { -1/4 (CCW), +7/4 (CW) } + (Q1, OO, A1): -4, + (Q2, OO, A2): -4, + (Q3, OO, A3): -4, + (Q4, OO, A4): -4, + + # A -- OO --> A => { +1 (CCW), -1 (CW) } + (A1, A3): 7, + (A2, A4): 7, + (A3, A1): 7, + (A4, A2): 7, + + (A1, OO, A3): 7, + (A2, OO, A4): 7, + (A3, OO, A1): 7, + (A4, OO, A2): 7, + + # Q -- DIA --> Q => { +1 (CCW), -1 (CW) } + (Q1, Q3): 8, + (Q2, Q4): 8, + (Q3, Q1): 8, + (Q4, Q2): 8, + + (Q1, OO, Q3): 8, + (Q2, OO, Q4): 8, + (Q3, OO, Q1): 8, + (Q4, OO, Q2): 8, + + # A --- R ---> A => { +1/2 (CCW), -3/2 (CW) } + (A1, A2): 9, + (A2, A3): 9, + (A3, A4): 9, + (A4, A1): 9, + + (A1, OO, A2): 9, + (A2, OO, A3): 9, + (A3, OO, A4): 9, + (A4, OO, A1): 9, + + # A --- L ---> A => { +3/2 (CCW), -1/2 (CW) } + (A1, A4): 10, + (A2, A1): 10, + (A3, A2): 10, + (A4, A3): 10, + + (A1, OO, A4): 10, + (A2, OO, A1): 10, + (A3, OO, A2): 10, + (A4, OO, A3): 10, + + # Q --- 1 ---> A => { +3/4 (CCW), -5/4 (CW) } + (Q1, A3): 11, + (Q2, A4): 11, + (Q3, A1): 11, + (Q4, A2): 11, + + (Q1, OO, A3): 11, + (Q2, OO, A4): 11, + (Q3, OO, A1): 11, + (Q4, OO, A2): 11, + + # Q --- 2 ---> A => { +5/4 (CCW), -3/4 (CW) } + (Q1, A4): 12, + (Q2, A1): 12, + (Q3, A2): 12, + (Q4, A3): 12, + + (Q1, OO, A4): 12, + (Q2, OO, A1): 12, + (Q3, OO, A2): 12, + (Q4, OO, A3): 12, + + # A --- 1 ---> Q => { +5/4 (CCW), -3/4 (CW) } + (A1, Q3): 13, + (A2, Q4): 13, + (A3, Q1): 13, + (A4, Q2): 13, + + (A1, OO, Q3): 13, + (A2, OO, Q4): 13, + (A3, OO, Q1): 13, + (A4, OO, Q2): 13, + + # A --- 2 ---> Q => { +3/4 (CCW), -5/4 (CW) } + (A1, Q2): 14, + (A2, Q3): 14, + (A3, Q4): 14, + (A4, Q1): 14, + + (A1, OO, Q2): 14, + (A2, OO, Q3): 14, + (A3, OO, Q4): 14, + (A4, OO, Q1): 14, + + # Q --> OO --> Q => { +1/2 (CCW), -3/2 (CW) } + (Q1, OO, Q2): 15, + (Q2, OO, Q3): 15, + (Q3, OO, Q4): 15, + (Q4, OO, Q1): 15, + + # Q --> OO --> Q => { +3/2 (CCW), -1/2 (CW) } + (Q1, OO, Q4): 16, + (Q2, OO, Q1): 16, + (Q3, OO, Q2): 16, + (Q4, OO, Q3): 16, + + # A --> OO --> A => { +2 (CCW), 0 (CW) } + (A1, OO, A1): 17, + (A2, OO, A2): 17, + (A3, OO, A3): 17, + (A4, OO, A4): 17, + + # Q --> OO --> Q => { +2 (CCW), 0 (CW) } + (Q1, OO, Q1): 18, + (Q2, OO, Q2): 18, + (Q3, OO, Q3): 18, + (Q4, OO, Q4): 18, +} + +_values = { + 0: [( 0, 1)], + 1: [(+1, 4)], + 2: [(-1, 4)], + 3: [(+1, 4)], + 4: [(-1, 4)], + -1: [(+9, 4), (+1, 4)], + -2: [(+7, 4), (-1, 4)], + -3: [(+9, 4), (+1, 4)], + -4: [(+7, 4), (-1, 4)], + +5: [(+1, 2)], + -5: [(-1, 2)], + 7: [(+1, 1), (-1, 1)], + 8: [(+1, 1), (-1, 1)], + 9: [(+1, 2), (-3, 2)], + 10: [(+3, 2), (-1, 2)], + 11: [(+3, 4), (-5, 4)], + 12: [(+5, 4), (-3, 4)], + 13: [(+5, 4), (-3, 4)], + 14: [(+3, 4), (-5, 4)], + 15: [(+1, 2), (-3, 2)], + 16: [(+3, 2), (-1, 2)], + 17: [(+2, 1), ( 0, 1)], + 18: [(+2, 1), ( 0, 1)], +} + +def _classify_point(re, im): + """Return the half-axis (or origin) on which (re, im) point is located. """ + if not re and not im: + return OO + + if not re: + if im > 0: + return A2 + else: + return A4 + elif not im: + if re > 0: + return A1 + else: + return A3 + +def _intervals_to_quadrants(intervals, f1, f2, s, t, F): + """Generate a sequence of extended quadrants from a list of critical points. """ + if not intervals: + return [] + + Q = [] + + if not f1: + (a, b), _, _ = intervals[0] + + if a == b == s: + if len(intervals) == 1: + if dup_eval(f2, t, F) > 0: + return [OO, A2] + else: + return [OO, A4] + else: + (a, _), _, _ = intervals[1] + + if dup_eval(f2, (s + a)/2, F) > 0: + Q.extend([OO, A2]) + f2_sgn = +1 + else: + Q.extend([OO, A4]) + f2_sgn = -1 + + intervals = intervals[1:] + else: + if dup_eval(f2, s, F) > 0: + Q.append(A2) + f2_sgn = +1 + else: + Q.append(A4) + f2_sgn = -1 + + for (a, _), indices, _ in intervals: + Q.append(OO) + + if indices[1] % 2 == 1: + f2_sgn = -f2_sgn + + if a != t: + if f2_sgn > 0: + Q.append(A2) + else: + Q.append(A4) + + return Q + + if not f2: + (a, b), _, _ = intervals[0] + + if a == b == s: + if len(intervals) == 1: + if dup_eval(f1, t, F) > 0: + return [OO, A1] + else: + return [OO, A3] + else: + (a, _), _, _ = intervals[1] + + if dup_eval(f1, (s + a)/2, F) > 0: + Q.extend([OO, A1]) + f1_sgn = +1 + else: + Q.extend([OO, A3]) + f1_sgn = -1 + + intervals = intervals[1:] + else: + if dup_eval(f1, s, F) > 0: + Q.append(A1) + f1_sgn = +1 + else: + Q.append(A3) + f1_sgn = -1 + + for (a, _), indices, _ in intervals: + Q.append(OO) + + if indices[0] % 2 == 1: + f1_sgn = -f1_sgn + + if a != t: + if f1_sgn > 0: + Q.append(A1) + else: + Q.append(A3) + + return Q + + re = dup_eval(f1, s, F) + im = dup_eval(f2, s, F) + + if not re or not im: + Q.append(_classify_point(re, im)) + + if len(intervals) == 1: + re = dup_eval(f1, t, F) + im = dup_eval(f2, t, F) + else: + (a, _), _, _ = intervals[1] + + re = dup_eval(f1, (s + a)/2, F) + im = dup_eval(f2, (s + a)/2, F) + + intervals = intervals[1:] + + if re > 0: + f1_sgn = +1 + else: + f1_sgn = -1 + + if im > 0: + f2_sgn = +1 + else: + f2_sgn = -1 + + sgn = { + (+1, +1): Q1, + (-1, +1): Q2, + (-1, -1): Q3, + (+1, -1): Q4, + } + + Q.append(sgn[(f1_sgn, f2_sgn)]) + + for (a, b), indices, _ in intervals: + if a == b: + re = dup_eval(f1, a, F) + im = dup_eval(f2, a, F) + + cls = _classify_point(re, im) + + if cls is not None: + Q.append(cls) + + if 0 in indices: + if indices[0] % 2 == 1: + f1_sgn = -f1_sgn + + if 1 in indices: + if indices[1] % 2 == 1: + f2_sgn = -f2_sgn + + if not (a == b and b == t): + Q.append(sgn[(f1_sgn, f2_sgn)]) + + return Q + +def _traverse_quadrants(Q_L1, Q_L2, Q_L3, Q_L4, exclude=None): + """Transform sequences of quadrants to a sequence of rules. """ + if exclude is True: + edges = [1, 1, 0, 0] + + corners = { + (0, 1): 1, + (1, 2): 1, + (2, 3): 0, + (3, 0): 1, + } + else: + edges = [0, 0, 0, 0] + + corners = { + (0, 1): 0, + (1, 2): 0, + (2, 3): 0, + (3, 0): 0, + } + + if exclude is not None and exclude is not True: + exclude = set(exclude) + + for i, edge in enumerate(['S', 'E', 'N', 'W']): + if edge in exclude: + edges[i] = 1 + + for i, corner in enumerate(['SW', 'SE', 'NE', 'NW']): + if corner in exclude: + corners[((i - 1) % 4, i)] = 1 + + QQ, rules = [Q_L1, Q_L2, Q_L3, Q_L4], [] + + for i, Q in enumerate(QQ): + if not Q: + continue + + if Q[-1] == OO: + Q = Q[:-1] + + if Q[0] == OO: + j, Q = (i - 1) % 4, Q[1:] + qq = (QQ[j][-2], OO, Q[0]) + + if qq in _rules_ambiguous: + rules.append((_rules_ambiguous[qq], corners[(j, i)])) + else: + raise NotImplementedError("3 element rule (corner): " + str(qq)) + + q1, k = Q[0], 1 + + while k < len(Q): + q2, k = Q[k], k + 1 + + if q2 != OO: + qq = (q1, q2) + + if qq in _rules_simple: + rules.append((_rules_simple[qq], 0)) + elif qq in _rules_ambiguous: + rules.append((_rules_ambiguous[qq], edges[i])) + else: + raise NotImplementedError("2 element rule (inside): " + str(qq)) + else: + qq, k = (q1, q2, Q[k]), k + 1 + + if qq in _rules_ambiguous: + rules.append((_rules_ambiguous[qq], edges[i])) + else: + raise NotImplementedError("3 element rule (edge): " + str(qq)) + + q1 = qq[-1] + + return rules + +def _reverse_intervals(intervals): + """Reverse intervals for traversal from right to left and from top to bottom. """ + return [ ((b, a), indices, f) for (a, b), indices, f in reversed(intervals) ] + +def _winding_number(T, field): + """Compute the winding number of the input polynomial, i.e. the number of roots. """ + return int(sum(field(*_values[t][i]) for t, i in T) / field(2)) + +def dup_count_complex_roots(f, K, inf=None, sup=None, exclude=None): + """Count all roots in [u + v*I, s + t*I] rectangle using Collins-Krandick algorithm. """ + if not K.is_ZZ and not K.is_QQ: + raise DomainError("complex root counting is not supported over %s" % K) + + if K.is_ZZ: + R, F = K, K.get_field() + else: + R, F = K.get_ring(), K + + f = dup_convert(f, K, F) + + if inf is None or sup is None: + _, lc = dup_degree(f), abs(dup_LC(f, F)) + B = 2*max(F.quo(abs(c), lc) for c in f) + + if inf is None: + (u, v) = (-B, -B) + else: + (u, v) = inf + + if sup is None: + (s, t) = (+B, +B) + else: + (s, t) = sup + + f1, f2 = dup_real_imag(f, F) + + f1L1F = dmp_eval_in(f1, v, 1, 1, F) + f2L1F = dmp_eval_in(f2, v, 1, 1, F) + + _, f1L1R = dup_clear_denoms(f1L1F, F, R, convert=True) + _, f2L1R = dup_clear_denoms(f2L1F, F, R, convert=True) + + f1L2F = dmp_eval_in(f1, s, 0, 1, F) + f2L2F = dmp_eval_in(f2, s, 0, 1, F) + + _, f1L2R = dup_clear_denoms(f1L2F, F, R, convert=True) + _, f2L2R = dup_clear_denoms(f2L2F, F, R, convert=True) + + f1L3F = dmp_eval_in(f1, t, 1, 1, F) + f2L3F = dmp_eval_in(f2, t, 1, 1, F) + + _, f1L3R = dup_clear_denoms(f1L3F, F, R, convert=True) + _, f2L3R = dup_clear_denoms(f2L3F, F, R, convert=True) + + f1L4F = dmp_eval_in(f1, u, 0, 1, F) + f2L4F = dmp_eval_in(f2, u, 0, 1, F) + + _, f1L4R = dup_clear_denoms(f1L4F, F, R, convert=True) + _, f2L4R = dup_clear_denoms(f2L4F, F, R, convert=True) + + S_L1 = [f1L1R, f2L1R] + S_L2 = [f1L2R, f2L2R] + S_L3 = [f1L3R, f2L3R] + S_L4 = [f1L4R, f2L4R] + + I_L1 = dup_isolate_real_roots_list(S_L1, R, inf=u, sup=s, fast=True, basis=True, strict=True) + I_L2 = dup_isolate_real_roots_list(S_L2, R, inf=v, sup=t, fast=True, basis=True, strict=True) + I_L3 = dup_isolate_real_roots_list(S_L3, R, inf=u, sup=s, fast=True, basis=True, strict=True) + I_L4 = dup_isolate_real_roots_list(S_L4, R, inf=v, sup=t, fast=True, basis=True, strict=True) + + I_L3 = _reverse_intervals(I_L3) + I_L4 = _reverse_intervals(I_L4) + + Q_L1 = _intervals_to_quadrants(I_L1, f1L1F, f2L1F, u, s, F) + Q_L2 = _intervals_to_quadrants(I_L2, f1L2F, f2L2F, v, t, F) + Q_L3 = _intervals_to_quadrants(I_L3, f1L3F, f2L3F, s, u, F) + Q_L4 = _intervals_to_quadrants(I_L4, f1L4F, f2L4F, t, v, F) + + T = _traverse_quadrants(Q_L1, Q_L2, Q_L3, Q_L4, exclude=exclude) + + return _winding_number(T, F) + +def _vertical_bisection(N, a, b, I, Q, F1, F2, f1, f2, F): + """Vertical bisection step in Collins-Krandick root isolation algorithm. """ + (u, v), (s, t) = a, b + + I_L1, I_L2, I_L3, I_L4 = I + Q_L1, Q_L2, Q_L3, Q_L4 = Q + + f1L1F, f1L2F, f1L3F, f1L4F = F1 + f2L1F, f2L2F, f2L3F, f2L4F = F2 + + x = (u + s) / 2 + + f1V = dmp_eval_in(f1, x, 0, 1, F) + f2V = dmp_eval_in(f2, x, 0, 1, F) + + I_V = dup_isolate_real_roots_list([f1V, f2V], F, inf=v, sup=t, fast=True, strict=True, basis=True) + + I_L1_L, I_L1_R = [], [] + I_L2_L, I_L2_R = I_V, I_L2 + I_L3_L, I_L3_R = [], [] + I_L4_L, I_L4_R = I_L4, _reverse_intervals(I_V) + + for I in I_L1: + (a, b), indices, h = I + + if a == b: + if a == x: + I_L1_L.append(I) + I_L1_R.append(I) + elif a < x: + I_L1_L.append(I) + else: + I_L1_R.append(I) + else: + if b <= x: + I_L1_L.append(I) + elif a >= x: + I_L1_R.append(I) + else: + a, b = dup_refine_real_root(h, a, b, F.get_ring(), disjoint=x, fast=True) + + if b <= x: + I_L1_L.append(((a, b), indices, h)) + if a >= x: + I_L1_R.append(((a, b), indices, h)) + + for I in I_L3: + (b, a), indices, h = I + + if a == b: + if a == x: + I_L3_L.append(I) + I_L3_R.append(I) + elif a < x: + I_L3_L.append(I) + else: + I_L3_R.append(I) + else: + if b <= x: + I_L3_L.append(I) + elif a >= x: + I_L3_R.append(I) + else: + a, b = dup_refine_real_root(h, a, b, F.get_ring(), disjoint=x, fast=True) + + if b <= x: + I_L3_L.append(((b, a), indices, h)) + if a >= x: + I_L3_R.append(((b, a), indices, h)) + + Q_L1_L = _intervals_to_quadrants(I_L1_L, f1L1F, f2L1F, u, x, F) + Q_L2_L = _intervals_to_quadrants(I_L2_L, f1V, f2V, v, t, F) + Q_L3_L = _intervals_to_quadrants(I_L3_L, f1L3F, f2L3F, x, u, F) + Q_L4_L = Q_L4 + + Q_L1_R = _intervals_to_quadrants(I_L1_R, f1L1F, f2L1F, x, s, F) + Q_L2_R = Q_L2 + Q_L3_R = _intervals_to_quadrants(I_L3_R, f1L3F, f2L3F, s, x, F) + Q_L4_R = _intervals_to_quadrants(I_L4_R, f1V, f2V, t, v, F) + + T_L = _traverse_quadrants(Q_L1_L, Q_L2_L, Q_L3_L, Q_L4_L, exclude=True) + T_R = _traverse_quadrants(Q_L1_R, Q_L2_R, Q_L3_R, Q_L4_R, exclude=True) + + N_L = _winding_number(T_L, F) + N_R = _winding_number(T_R, F) + + I_L = (I_L1_L, I_L2_L, I_L3_L, I_L4_L) + Q_L = (Q_L1_L, Q_L2_L, Q_L3_L, Q_L4_L) + + I_R = (I_L1_R, I_L2_R, I_L3_R, I_L4_R) + Q_R = (Q_L1_R, Q_L2_R, Q_L3_R, Q_L4_R) + + F1_L = (f1L1F, f1V, f1L3F, f1L4F) + F2_L = (f2L1F, f2V, f2L3F, f2L4F) + + F1_R = (f1L1F, f1L2F, f1L3F, f1V) + F2_R = (f2L1F, f2L2F, f2L3F, f2V) + + a, b = (u, v), (x, t) + c, d = (x, v), (s, t) + + D_L = (N_L, a, b, I_L, Q_L, F1_L, F2_L) + D_R = (N_R, c, d, I_R, Q_R, F1_R, F2_R) + + return D_L, D_R + +def _horizontal_bisection(N, a, b, I, Q, F1, F2, f1, f2, F): + """Horizontal bisection step in Collins-Krandick root isolation algorithm. """ + (u, v), (s, t) = a, b + + I_L1, I_L2, I_L3, I_L4 = I + Q_L1, Q_L2, Q_L3, Q_L4 = Q + + f1L1F, f1L2F, f1L3F, f1L4F = F1 + f2L1F, f2L2F, f2L3F, f2L4F = F2 + + y = (v + t) / 2 + + f1H = dmp_eval_in(f1, y, 1, 1, F) + f2H = dmp_eval_in(f2, y, 1, 1, F) + + I_H = dup_isolate_real_roots_list([f1H, f2H], F, inf=u, sup=s, fast=True, strict=True, basis=True) + + I_L1_B, I_L1_U = I_L1, I_H + I_L2_B, I_L2_U = [], [] + I_L3_B, I_L3_U = _reverse_intervals(I_H), I_L3 + I_L4_B, I_L4_U = [], [] + + for I in I_L2: + (a, b), indices, h = I + + if a == b: + if a == y: + I_L2_B.append(I) + I_L2_U.append(I) + elif a < y: + I_L2_B.append(I) + else: + I_L2_U.append(I) + else: + if b <= y: + I_L2_B.append(I) + elif a >= y: + I_L2_U.append(I) + else: + a, b = dup_refine_real_root(h, a, b, F.get_ring(), disjoint=y, fast=True) + + if b <= y: + I_L2_B.append(((a, b), indices, h)) + if a >= y: + I_L2_U.append(((a, b), indices, h)) + + for I in I_L4: + (b, a), indices, h = I + + if a == b: + if a == y: + I_L4_B.append(I) + I_L4_U.append(I) + elif a < y: + I_L4_B.append(I) + else: + I_L4_U.append(I) + else: + if b <= y: + I_L4_B.append(I) + elif a >= y: + I_L4_U.append(I) + else: + a, b = dup_refine_real_root(h, a, b, F.get_ring(), disjoint=y, fast=True) + + if b <= y: + I_L4_B.append(((b, a), indices, h)) + if a >= y: + I_L4_U.append(((b, a), indices, h)) + + Q_L1_B = Q_L1 + Q_L2_B = _intervals_to_quadrants(I_L2_B, f1L2F, f2L2F, v, y, F) + Q_L3_B = _intervals_to_quadrants(I_L3_B, f1H, f2H, s, u, F) + Q_L4_B = _intervals_to_quadrants(I_L4_B, f1L4F, f2L4F, y, v, F) + + Q_L1_U = _intervals_to_quadrants(I_L1_U, f1H, f2H, u, s, F) + Q_L2_U = _intervals_to_quadrants(I_L2_U, f1L2F, f2L2F, y, t, F) + Q_L3_U = Q_L3 + Q_L4_U = _intervals_to_quadrants(I_L4_U, f1L4F, f2L4F, t, y, F) + + T_B = _traverse_quadrants(Q_L1_B, Q_L2_B, Q_L3_B, Q_L4_B, exclude=True) + T_U = _traverse_quadrants(Q_L1_U, Q_L2_U, Q_L3_U, Q_L4_U, exclude=True) + + N_B = _winding_number(T_B, F) + N_U = _winding_number(T_U, F) + + I_B = (I_L1_B, I_L2_B, I_L3_B, I_L4_B) + Q_B = (Q_L1_B, Q_L2_B, Q_L3_B, Q_L4_B) + + I_U = (I_L1_U, I_L2_U, I_L3_U, I_L4_U) + Q_U = (Q_L1_U, Q_L2_U, Q_L3_U, Q_L4_U) + + F1_B = (f1L1F, f1L2F, f1H, f1L4F) + F2_B = (f2L1F, f2L2F, f2H, f2L4F) + + F1_U = (f1H, f1L2F, f1L3F, f1L4F) + F2_U = (f2H, f2L2F, f2L3F, f2L4F) + + a, b = (u, v), (s, y) + c, d = (u, y), (s, t) + + D_B = (N_B, a, b, I_B, Q_B, F1_B, F2_B) + D_U = (N_U, c, d, I_U, Q_U, F1_U, F2_U) + + return D_B, D_U + +def _depth_first_select(rectangles): + """Find a rectangle of minimum area for bisection. """ + min_area, j = None, None + + for i, (_, (u, v), (s, t), _, _, _, _) in enumerate(rectangles): + area = (s - u)*(t - v) + + if min_area is None or area < min_area: + min_area, j = area, i + + return rectangles.pop(j) + +def _rectangle_small_p(a, b, eps): + """Return ``True`` if the given rectangle is small enough. """ + (u, v), (s, t) = a, b + + if eps is not None: + return s - u < eps and t - v < eps + else: + return True + +def dup_isolate_complex_roots_sqf(f, K, eps=None, inf=None, sup=None, blackbox=False): + """Isolate complex roots of a square-free polynomial using Collins-Krandick algorithm. """ + if not K.is_ZZ and not K.is_QQ: + raise DomainError("isolation of complex roots is not supported over %s" % K) + + if dup_degree(f) <= 0: + return [] + + if K.is_ZZ: + F = K.get_field() + else: + F = K + + f = dup_convert(f, K, F) + + lc = abs(dup_LC(f, F)) + B = 2*max(F.quo(abs(c), lc) for c in f) + + (u, v), (s, t) = (-B, F.zero), (B, B) + + if inf is not None: + u = inf + + if sup is not None: + s = sup + + if v < 0 or t <= v or s <= u: + raise ValueError("not a valid complex isolation rectangle") + + f1, f2 = dup_real_imag(f, F) + + f1L1 = dmp_eval_in(f1, v, 1, 1, F) + f2L1 = dmp_eval_in(f2, v, 1, 1, F) + + f1L2 = dmp_eval_in(f1, s, 0, 1, F) + f2L2 = dmp_eval_in(f2, s, 0, 1, F) + + f1L3 = dmp_eval_in(f1, t, 1, 1, F) + f2L3 = dmp_eval_in(f2, t, 1, 1, F) + + f1L4 = dmp_eval_in(f1, u, 0, 1, F) + f2L4 = dmp_eval_in(f2, u, 0, 1, F) + + S_L1 = [f1L1, f2L1] + S_L2 = [f1L2, f2L2] + S_L3 = [f1L3, f2L3] + S_L4 = [f1L4, f2L4] + + I_L1 = dup_isolate_real_roots_list(S_L1, F, inf=u, sup=s, fast=True, strict=True, basis=True) + I_L2 = dup_isolate_real_roots_list(S_L2, F, inf=v, sup=t, fast=True, strict=True, basis=True) + I_L3 = dup_isolate_real_roots_list(S_L3, F, inf=u, sup=s, fast=True, strict=True, basis=True) + I_L4 = dup_isolate_real_roots_list(S_L4, F, inf=v, sup=t, fast=True, strict=True, basis=True) + + I_L3 = _reverse_intervals(I_L3) + I_L4 = _reverse_intervals(I_L4) + + Q_L1 = _intervals_to_quadrants(I_L1, f1L1, f2L1, u, s, F) + Q_L2 = _intervals_to_quadrants(I_L2, f1L2, f2L2, v, t, F) + Q_L3 = _intervals_to_quadrants(I_L3, f1L3, f2L3, s, u, F) + Q_L4 = _intervals_to_quadrants(I_L4, f1L4, f2L4, t, v, F) + + T = _traverse_quadrants(Q_L1, Q_L2, Q_L3, Q_L4) + N = _winding_number(T, F) + + if not N: + return [] + + I = (I_L1, I_L2, I_L3, I_L4) + Q = (Q_L1, Q_L2, Q_L3, Q_L4) + + F1 = (f1L1, f1L2, f1L3, f1L4) + F2 = (f2L1, f2L2, f2L3, f2L4) + + rectangles, roots = [(N, (u, v), (s, t), I, Q, F1, F2)], [] + + while rectangles: + N, (u, v), (s, t), I, Q, F1, F2 = _depth_first_select(rectangles) + + if s - u > t - v: + D_L, D_R = _vertical_bisection(N, (u, v), (s, t), I, Q, F1, F2, f1, f2, F) + + N_L, a, b, I_L, Q_L, F1_L, F2_L = D_L + N_R, c, d, I_R, Q_R, F1_R, F2_R = D_R + + if N_L >= 1: + if N_L == 1 and _rectangle_small_p(a, b, eps): + roots.append(ComplexInterval(a, b, I_L, Q_L, F1_L, F2_L, f1, f2, F)) + else: + rectangles.append(D_L) + + if N_R >= 1: + if N_R == 1 and _rectangle_small_p(c, d, eps): + roots.append(ComplexInterval(c, d, I_R, Q_R, F1_R, F2_R, f1, f2, F)) + else: + rectangles.append(D_R) + else: + D_B, D_U = _horizontal_bisection(N, (u, v), (s, t), I, Q, F1, F2, f1, f2, F) + + N_B, a, b, I_B, Q_B, F1_B, F2_B = D_B + N_U, c, d, I_U, Q_U, F1_U, F2_U = D_U + + if N_B >= 1: + if N_B == 1 and _rectangle_small_p(a, b, eps): + roots.append(ComplexInterval( + a, b, I_B, Q_B, F1_B, F2_B, f1, f2, F)) + else: + rectangles.append(D_B) + + if N_U >= 1: + if N_U == 1 and _rectangle_small_p(c, d, eps): + roots.append(ComplexInterval( + c, d, I_U, Q_U, F1_U, F2_U, f1, f2, F)) + else: + rectangles.append(D_U) + + _roots, roots = sorted(roots, key=lambda r: (r.ax, r.ay)), [] + + for root in _roots: + roots.extend([root.conjugate(), root]) + + if blackbox: + return roots + else: + return [ r.as_tuple() for r in roots ] + +def dup_isolate_all_roots_sqf(f, K, eps=None, inf=None, sup=None, fast=False, blackbox=False): + """Isolate real and complex roots of a square-free polynomial ``f``. """ + return ( + dup_isolate_real_roots_sqf( f, K, eps=eps, inf=inf, sup=sup, fast=fast, blackbox=blackbox), + dup_isolate_complex_roots_sqf(f, K, eps=eps, inf=inf, sup=sup, blackbox=blackbox)) + +def dup_isolate_all_roots(f, K, eps=None, inf=None, sup=None, fast=False): + """Isolate real and complex roots of a non-square-free polynomial ``f``. """ + if not K.is_ZZ and not K.is_QQ: + raise DomainError("isolation of real and complex roots is not supported over %s" % K) + + _, factors = dup_sqf_list(f, K) + + if len(factors) == 1: + ((f, k),) = factors + + real_part, complex_part = dup_isolate_all_roots_sqf( + f, K, eps=eps, inf=inf, sup=sup, fast=fast) + + real_part = [ ((a, b), k) for (a, b) in real_part ] + complex_part = [ ((a, b), k) for (a, b) in complex_part ] + + return real_part, complex_part + else: + raise NotImplementedError( "only trivial square-free polynomials are supported") + +class RealInterval: + """A fully qualified representation of a real isolation interval. """ + + def __init__(self, data, f, dom): + """Initialize new real interval with complete information. """ + if len(data) == 2: + s, t = data + + self.neg = False + + if s < 0: + if t <= 0: + f, s, t, self.neg = dup_mirror(f, dom), -t, -s, True + else: + raise ValueError("Cannot refine a real root in (%s, %s)" % (s, t)) + + a, b, c, d = _mobius_from_interval((s, t), dom.get_field()) + + f = dup_transform(f, dup_strip([a, b]), + dup_strip([c, d]), dom) + + self.mobius = a, b, c, d + else: + self.mobius = data[:-1] + self.neg = data[-1] + + self.f, self.dom = f, dom + + @property + def func(self): + return RealInterval + + @property + def args(self): + i = self + return (i.mobius + (i.neg,), i.f, i.dom) + + def __eq__(self, other): + if type(other) is not type(self): + return False + return self.args == other.args + + @property + def a(self): + """Return the position of the left end. """ + field = self.dom.get_field() + a, b, c, d = self.mobius + + if not self.neg: + if a*d < b*c: + return field(a, c) + return field(b, d) + else: + if a*d > b*c: + return -field(a, c) + return -field(b, d) + + @property + def b(self): + """Return the position of the right end. """ + was = self.neg + self.neg = not was + rv = -self.a + self.neg = was + return rv + + @property + def dx(self): + """Return width of the real isolating interval. """ + return self.b - self.a + + @property + def center(self): + """Return the center of the real isolating interval. """ + return (self.a + self.b)/2 + + @property + def max_denom(self): + """Return the largest denominator occurring in either endpoint. """ + return max(self.a.denominator, self.b.denominator) + + def as_tuple(self): + """Return tuple representation of real isolating interval. """ + return (self.a, self.b) + + def __repr__(self): + return "(%s, %s)" % (self.a, self.b) + + def __contains__(self, item): + """ + Say whether a complex number belongs to this real interval. + + Parameters + ========== + + item : pair (re, im) or number re + Either a pair giving the real and imaginary parts of the number, + or else a real number. + + """ + if isinstance(item, tuple): + re, im = item + else: + re, im = item, 0 + return im == 0 and self.a <= re <= self.b + + def is_disjoint(self, other): + """Return ``True`` if two isolation intervals are disjoint. """ + if isinstance(other, RealInterval): + return (self.b < other.a or other.b < self.a) + assert isinstance(other, ComplexInterval) + return (self.b < other.ax or other.bx < self.a + or other.ay*other.by > 0) + + def _inner_refine(self): + """Internal one step real root refinement procedure. """ + if self.mobius is None: + return self + + f, mobius = dup_inner_refine_real_root( + self.f, self.mobius, self.dom, steps=1, mobius=True) + + return RealInterval(mobius + (self.neg,), f, self.dom) + + def refine_disjoint(self, other): + """Refine an isolating interval until it is disjoint with another one. """ + expr = self + while not expr.is_disjoint(other): + expr, other = expr._inner_refine(), other._inner_refine() + + return expr, other + + def refine_size(self, dx): + """Refine an isolating interval until it is of sufficiently small size. """ + expr = self + while not (expr.dx < dx): + expr = expr._inner_refine() + + return expr + + def refine_step(self, steps=1): + """Perform several steps of real root refinement algorithm. """ + expr = self + for _ in range(steps): + expr = expr._inner_refine() + + return expr + + def refine(self): + """Perform one step of real root refinement algorithm. """ + return self._inner_refine() + + +class ComplexInterval: + """A fully qualified representation of a complex isolation interval. + The printed form is shown as (ax, bx) x (ay, by) where (ax, ay) + and (bx, by) are the coordinates of the southwest and northeast + corners of the interval's rectangle, respectively. + + Examples + ======== + + >>> from sympy import CRootOf, S + >>> from sympy.abc import x + >>> CRootOf.clear_cache() # for doctest reproducibility + >>> root = CRootOf(x**10 - 2*x + 3, 9) + >>> i = root._get_interval(); i + (3/64, 3/32) x (9/8, 75/64) + + The real part of the root lies within the range [0, 3/4] while + the imaginary part lies within the range [9/8, 3/2]: + + >>> root.n(3) + 0.0766 + 1.14*I + + The width of the ranges in the x and y directions on the complex + plane are: + + >>> i.dx, i.dy + (3/64, 3/64) + + The center of the range is + + >>> i.center + (9/128, 147/128) + + The northeast coordinate of the rectangle bounding the root in the + complex plane is given by attribute b and the x and y components + are accessed by bx and by: + + >>> i.b, i.bx, i.by + ((3/32, 75/64), 3/32, 75/64) + + The southwest coordinate is similarly given by i.a + + >>> i.a, i.ax, i.ay + ((3/64, 9/8), 3/64, 9/8) + + Although the interval prints to show only the real and imaginary + range of the root, all the information of the underlying root + is contained as properties of the interval. + + For example, an interval with a nonpositive imaginary range is + considered to be the conjugate. Since the y values of y are in the + range [0, 1/4] it is not the conjugate: + + >>> i.conj + False + + The conjugate's interval is + + >>> ic = i.conjugate(); ic + (3/64, 3/32) x (-75/64, -9/8) + + NOTE: the values printed still represent the x and y range + in which the root -- conjugate, in this case -- is located, + but the underlying a and b values of a root and its conjugate + are the same: + + >>> assert i.a == ic.a and i.b == ic.b + + What changes are the reported coordinates of the bounding rectangle: + + >>> (i.ax, i.ay), (i.bx, i.by) + ((3/64, 9/8), (3/32, 75/64)) + >>> (ic.ax, ic.ay), (ic.bx, ic.by) + ((3/64, -75/64), (3/32, -9/8)) + + The interval can be refined once: + + >>> i # for reference, this is the current interval + (3/64, 3/32) x (9/8, 75/64) + + >>> i.refine() + (3/64, 3/32) x (9/8, 147/128) + + Several refinement steps can be taken: + + >>> i.refine_step(2) # 2 steps + (9/128, 3/32) x (9/8, 147/128) + + It is also possible to refine to a given tolerance: + + >>> tol = min(i.dx, i.dy)/2 + >>> i.refine_size(tol) + (9/128, 21/256) x (9/8, 291/256) + + A disjoint interval is one whose bounding rectangle does not + overlap with another. An interval, necessarily, is not disjoint with + itself, but any interval is disjoint with a conjugate since the + conjugate rectangle will always be in the lower half of the complex + plane and the non-conjugate in the upper half: + + >>> i.is_disjoint(i), i.is_disjoint(i.conjugate()) + (False, True) + + The following interval j is not disjoint from i: + + >>> close = CRootOf(x**10 - 2*x + 300/S(101), 9) + >>> j = close._get_interval(); j + (75/1616, 75/808) x (225/202, 1875/1616) + >>> i.is_disjoint(j) + False + + The two can be made disjoint, however: + + >>> newi, newj = i.refine_disjoint(j) + >>> newi + (39/512, 159/2048) x (2325/2048, 4653/4096) + >>> newj + (3975/51712, 2025/25856) x (29325/25856, 117375/103424) + + Even though the real ranges overlap, the imaginary do not, so + the roots have been resolved as distinct. Intervals are disjoint + when either the real or imaginary component of the intervals is + distinct. In the case above, the real components have not been + resolved (so we do not know, yet, which root has the smaller real + part) but the imaginary part of ``close`` is larger than ``root``: + + >>> close.n(3) + 0.0771 + 1.13*I + >>> root.n(3) + 0.0766 + 1.14*I + """ + + def __init__(self, a, b, I, Q, F1, F2, f1, f2, dom, conj=False): + """Initialize new complex interval with complete information. """ + # a and b are the SW and NE corner of the bounding interval, + # (ax, ay) and (bx, by), respectively, for the NON-CONJUGATE + # root (the one with the positive imaginary part); when working + # with the conjugate, the a and b value are still non-negative + # but the ay, by are reversed and have oppositite sign + self.a, self.b = a, b + self.I, self.Q = I, Q + + self.f1, self.F1 = f1, F1 + self.f2, self.F2 = f2, F2 + + self.dom = dom + self.conj = conj + + @property + def func(self): + return ComplexInterval + + @property + def args(self): + i = self + return (i.a, i.b, i.I, i.Q, i.F1, i.F2, i.f1, i.f2, i.dom, i.conj) + + def __eq__(self, other): + if type(other) is not type(self): + return False + return self.args == other.args + + @property + def ax(self): + """Return ``x`` coordinate of south-western corner. """ + return self.a[0] + + @property + def ay(self): + """Return ``y`` coordinate of south-western corner. """ + if not self.conj: + return self.a[1] + else: + return -self.b[1] + + @property + def bx(self): + """Return ``x`` coordinate of north-eastern corner. """ + return self.b[0] + + @property + def by(self): + """Return ``y`` coordinate of north-eastern corner. """ + if not self.conj: + return self.b[1] + else: + return -self.a[1] + + @property + def dx(self): + """Return width of the complex isolating interval. """ + return self.b[0] - self.a[0] + + @property + def dy(self): + """Return height of the complex isolating interval. """ + return self.b[1] - self.a[1] + + @property + def center(self): + """Return the center of the complex isolating interval. """ + return ((self.ax + self.bx)/2, (self.ay + self.by)/2) + + @property + def max_denom(self): + """Return the largest denominator occurring in either endpoint. """ + return max(self.ax.denominator, self.bx.denominator, + self.ay.denominator, self.by.denominator) + + def as_tuple(self): + """Return tuple representation of the complex isolating + interval's SW and NE corners, respectively. """ + return ((self.ax, self.ay), (self.bx, self.by)) + + def __repr__(self): + return "(%s, %s) x (%s, %s)" % (self.ax, self.bx, self.ay, self.by) + + def conjugate(self): + """This complex interval really is located in lower half-plane. """ + return ComplexInterval(self.a, self.b, self.I, self.Q, + self.F1, self.F2, self.f1, self.f2, self.dom, conj=True) + + def __contains__(self, item): + """ + Say whether a complex number belongs to this complex rectangular + region. + + Parameters + ========== + + item : pair (re, im) or number re + Either a pair giving the real and imaginary parts of the number, + or else a real number. + + """ + if isinstance(item, tuple): + re, im = item + else: + re, im = item, 0 + return self.ax <= re <= self.bx and self.ay <= im <= self.by + + def is_disjoint(self, other): + """Return ``True`` if two isolation intervals are disjoint. """ + if isinstance(other, RealInterval): + return other.is_disjoint(self) + if self.conj != other.conj: # above and below real axis + return True + re_distinct = (self.bx < other.ax or other.bx < self.ax) + if re_distinct: + return True + im_distinct = (self.by < other.ay or other.by < self.ay) + return im_distinct + + def _inner_refine(self): + """Internal one step complex root refinement procedure. """ + (u, v), (s, t) = self.a, self.b + + I, Q = self.I, self.Q + + f1, F1 = self.f1, self.F1 + f2, F2 = self.f2, self.F2 + + dom = self.dom + + if s - u > t - v: + D_L, D_R = _vertical_bisection(1, (u, v), (s, t), I, Q, F1, F2, f1, f2, dom) + + if D_L[0] == 1: + _, a, b, I, Q, F1, F2 = D_L + else: + _, a, b, I, Q, F1, F2 = D_R + else: + D_B, D_U = _horizontal_bisection(1, (u, v), (s, t), I, Q, F1, F2, f1, f2, dom) + + if D_B[0] == 1: + _, a, b, I, Q, F1, F2 = D_B + else: + _, a, b, I, Q, F1, F2 = D_U + + return ComplexInterval(a, b, I, Q, F1, F2, f1, f2, dom, self.conj) + + def refine_disjoint(self, other): + """Refine an isolating interval until it is disjoint with another one. """ + expr = self + while not expr.is_disjoint(other): + expr, other = expr._inner_refine(), other._inner_refine() + + return expr, other + + def refine_size(self, dx, dy=None): + """Refine an isolating interval until it is of sufficiently small size. """ + if dy is None: + dy = dx + expr = self + while not (expr.dx < dx and expr.dy < dy): + expr = expr._inner_refine() + + return expr + + def refine_step(self, steps=1): + """Perform several steps of complex root refinement algorithm. """ + expr = self + for _ in range(steps): + expr = expr._inner_refine() + + return expr + + def refine(self): + """Perform one step of complex root refinement algorithm. """ + return self._inner_refine() diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/rootoftools.py b/.venv/lib/python3.12/site-packages/sympy/polys/rootoftools.py new file mode 100644 index 0000000000000000000000000000000000000000..d68d8b008281c7e9b5aac618c6c76f74fa236d9e --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/rootoftools.py @@ -0,0 +1,1298 @@ +"""Implementation of RootOf class and related tools. """ + + + +from sympy.core.basic import Basic +from sympy.core import (S, Expr, Integer, Float, I, oo, Add, Lambda, + symbols, sympify, Rational, Dummy) +from sympy.core.cache import cacheit +from sympy.core.relational import is_le +from sympy.core.sorting import ordered +from sympy.polys.domains import QQ +from sympy.polys.polyerrors import ( + MultivariatePolynomialError, + GeneratorsNeeded, + PolynomialError, + DomainError) +from sympy.polys.polyfuncs import symmetrize, viete +from sympy.polys.polyroots import ( + roots_linear, roots_quadratic, roots_binomial, + preprocess_roots, roots) +from sympy.polys.polytools import Poly, PurePoly, factor +from sympy.polys.rationaltools import together +from sympy.polys.rootisolation import ( + dup_isolate_complex_roots_sqf, + dup_isolate_real_roots_sqf) +from sympy.utilities import lambdify, public, sift, numbered_symbols + +from mpmath import mpf, mpc, findroot, workprec +from mpmath.libmp.libmpf import dps_to_prec, prec_to_dps +from sympy.multipledispatch import dispatch +from itertools import chain + + +__all__ = ['CRootOf'] + + + +class _pure_key_dict: + """A minimal dictionary that makes sure that the key is a + univariate PurePoly instance. + + Examples + ======== + + Only the following actions are guaranteed: + + >>> from sympy.polys.rootoftools import _pure_key_dict + >>> from sympy import PurePoly + >>> from sympy.abc import x, y + + 1) creation + + >>> P = _pure_key_dict() + + 2) assignment for a PurePoly or univariate polynomial + + >>> P[x] = 1 + >>> P[PurePoly(x - y, x)] = 2 + + 3) retrieval based on PurePoly key comparison (use this + instead of the get method) + + >>> P[y] + 1 + + 4) KeyError when trying to retrieve a nonexisting key + + >>> P[y + 1] + Traceback (most recent call last): + ... + KeyError: PurePoly(y + 1, y, domain='ZZ') + + 5) ability to query with ``in`` + + >>> x + 1 in P + False + + NOTE: this is a *not* a dictionary. It is a very basic object + for internal use that makes sure to always address its cache + via PurePoly instances. It does not, for example, implement + ``get`` or ``setdefault``. + """ + def __init__(self): + self._dict = {} + + def __getitem__(self, k): + if not isinstance(k, PurePoly): + if not (isinstance(k, Expr) and len(k.free_symbols) == 1): + raise KeyError + k = PurePoly(k, expand=False) + return self._dict[k] + + def __setitem__(self, k, v): + if not isinstance(k, PurePoly): + if not (isinstance(k, Expr) and len(k.free_symbols) == 1): + raise ValueError('expecting univariate expression') + k = PurePoly(k, expand=False) + self._dict[k] = v + + def __contains__(self, k): + try: + self[k] + return True + except KeyError: + return False + +_reals_cache = _pure_key_dict() +_complexes_cache = _pure_key_dict() + + +def _pure_factors(poly): + _, factors = poly.factor_list() + return [(PurePoly(f, expand=False), m) for f, m in factors] + + +def _imag_count_of_factor(f): + """Return the number of imaginary roots for irreducible + univariate polynomial ``f``. + """ + terms = [(i, j) for (i,), j in f.terms()] + if any(i % 2 for i, j in terms): + return 0 + # update signs + even = [(i, I**i*j) for i, j in terms] + even = Poly.from_dict(dict(even), Dummy('x')) + return int(even.count_roots(-oo, oo)) + + +@public +def rootof(f, x, index=None, radicals=True, expand=True): + """An indexed root of a univariate polynomial. + + Returns either a :obj:`ComplexRootOf` object or an explicit + expression involving radicals. + + Parameters + ========== + + f : Expr + Univariate polynomial. + x : Symbol, optional + Generator for ``f``. + index : int or Integer + radicals : bool + Return a radical expression if possible. + expand : bool + Expand ``f``. + """ + return CRootOf(f, x, index=index, radicals=radicals, expand=expand) + + +@public +class RootOf(Expr): + """Represents a root of a univariate polynomial. + + Base class for roots of different kinds of polynomials. + Only complex roots are currently supported. + """ + + __slots__ = ('poly',) + + def __new__(cls, f, x, index=None, radicals=True, expand=True): + """Construct a new ``CRootOf`` object for ``k``-th root of ``f``.""" + return rootof(f, x, index=index, radicals=radicals, expand=expand) + +@public +class ComplexRootOf(RootOf): + """Represents an indexed complex root of a polynomial. + + Roots of a univariate polynomial separated into disjoint + real or complex intervals and indexed in a fixed order: + + * real roots come first and are sorted in increasing order; + * complex roots come next and are sorted primarily by increasing + real part, secondarily by increasing imaginary part. + + Currently only rational coefficients are allowed. + Can be imported as ``CRootOf``. To avoid confusion, the + generator must be a Symbol. + + + Examples + ======== + + >>> from sympy import CRootOf, rootof + >>> from sympy.abc import x + + CRootOf is a way to reference a particular root of a + polynomial. If there is a rational root, it will be returned: + + >>> CRootOf.clear_cache() # for doctest reproducibility + >>> CRootOf(x**2 - 4, 0) + -2 + + Whether roots involving radicals are returned or not + depends on whether the ``radicals`` flag is true (which is + set to True with rootof): + + >>> CRootOf(x**2 - 3, 0) + CRootOf(x**2 - 3, 0) + >>> CRootOf(x**2 - 3, 0, radicals=True) + -sqrt(3) + >>> rootof(x**2 - 3, 0) + -sqrt(3) + + The following cannot be expressed in terms of radicals: + + >>> r = rootof(4*x**5 + 16*x**3 + 12*x**2 + 7, 0); r + CRootOf(4*x**5 + 16*x**3 + 12*x**2 + 7, 0) + + The root bounds can be seen, however, and they are used by the + evaluation methods to get numerical approximations for the root. + + >>> interval = r._get_interval(); interval + (-1, 0) + >>> r.evalf(2) + -0.98 + + The evalf method refines the width of the root bounds until it + guarantees that any decimal approximation within those bounds + will satisfy the desired precision. It then stores the refined + interval so subsequent requests at or below the requested + precision will not have to recompute the root bounds and will + return very quickly. + + Before evaluation above, the interval was + + >>> interval + (-1, 0) + + After evaluation it is now + + >>> r._get_interval() # doctest: +SKIP + (-165/169, -206/211) + + To reset all intervals for a given polynomial, the :meth:`_reset` method + can be called from any CRootOf instance of the polynomial: + + >>> r._reset() + >>> r._get_interval() + (-1, 0) + + The :meth:`eval_approx` method will also find the root to a given + precision but the interval is not modified unless the search + for the root fails to converge within the root bounds. And + the secant method is used to find the root. (The ``evalf`` + method uses bisection and will always update the interval.) + + >>> r.eval_approx(2) + -0.98 + + The interval needed to be slightly updated to find that root: + + >>> r._get_interval() + (-1, -1/2) + + The ``evalf_rational`` will compute a rational approximation + of the root to the desired accuracy or precision. + + >>> r.eval_rational(n=2) + -69629/71318 + + >>> t = CRootOf(x**3 + 10*x + 1, 1) + >>> t.eval_rational(1e-1) + 15/256 - 805*I/256 + >>> t.eval_rational(1e-1, 1e-4) + 3275/65536 - 414645*I/131072 + >>> t.eval_rational(1e-4, 1e-4) + 6545/131072 - 414645*I/131072 + >>> t.eval_rational(n=2) + 104755/2097152 - 6634255*I/2097152 + + Notes + ===== + + Although a PurePoly can be constructed from a non-symbol generator + RootOf instances of non-symbols are disallowed to avoid confusion + over what root is being represented. + + >>> from sympy import exp, PurePoly + >>> PurePoly(x) == PurePoly(exp(x)) + True + >>> CRootOf(x - 1, 0) + 1 + >>> CRootOf(exp(x) - 1, 0) # would correspond to x == 0 + Traceback (most recent call last): + ... + sympy.polys.polyerrors.PolynomialError: generator must be a Symbol + + See Also + ======== + + eval_approx + eval_rational + + """ + + __slots__ = ('index',) + is_complex = True + is_number = True + is_finite = True + is_algebraic = True + + def __new__(cls, f, x, index=None, radicals=False, expand=True): + """ Construct an indexed complex root of a polynomial. + + See ``rootof`` for the parameters. + + The default value of ``radicals`` is ``False`` to satisfy + ``eval(srepr(expr) == expr``. + """ + x = sympify(x) + + if index is None and x.is_Integer: + x, index = None, x + else: + index = sympify(index) + + if index is not None and index.is_Integer: + index = int(index) + else: + raise ValueError("expected an integer root index, got %s" % index) + + poly = PurePoly(f, x, greedy=False, expand=expand) + + if not poly.is_univariate: + raise PolynomialError("only univariate polynomials are allowed") + + if not poly.gen.is_Symbol: + # PurePoly(sin(x) + 1) == PurePoly(x + 1) but the roots of + # x for each are not the same: issue 8617 + raise PolynomialError("generator must be a Symbol") + + degree = poly.degree() + + if degree <= 0: + raise PolynomialError("Cannot construct CRootOf object for %s" % f) + + if index < -degree or index >= degree: + raise IndexError("root index out of [%d, %d] range, got %d" % + (-degree, degree - 1, index)) + elif index < 0: + index += degree + + dom = poly.get_domain() + + if not dom.is_Exact: + poly = poly.to_exact() + + roots = cls._roots_trivial(poly, radicals) + + if roots is not None: + return roots[index] + + coeff, poly = preprocess_roots(poly) + dom = poly.get_domain() + + if not dom.is_ZZ: + raise NotImplementedError("CRootOf is not supported over %s" % dom) + + root = cls._indexed_root(poly, index, lazy=True) + return coeff * cls._postprocess_root(root, radicals) + + @classmethod + def _new(cls, poly, index): + """Construct new ``CRootOf`` object from raw data. """ + obj = Expr.__new__(cls) + + obj.poly = PurePoly(poly) + obj.index = index + + try: + _reals_cache[obj.poly] = _reals_cache[poly] + _complexes_cache[obj.poly] = _complexes_cache[poly] + except KeyError: + pass + + return obj + + def _hashable_content(self): + return (self.poly, self.index) + + @property + def expr(self): + return self.poly.as_expr() + + @property + def args(self): + return (self.expr, Integer(self.index)) + + @property + def free_symbols(self): + # CRootOf currently only works with univariate expressions + # whose poly attribute should be a PurePoly with no free + # symbols + return set() + + def _eval_is_real(self): + """Return ``True`` if the root is real. """ + self._ensure_reals_init() + return self.index < len(_reals_cache[self.poly]) + + def _eval_is_imaginary(self): + """Return ``True`` if the root is imaginary. """ + self._ensure_reals_init() + if self.index >= len(_reals_cache[self.poly]): + ivl = self._get_interval() + return ivl.ax*ivl.bx <= 0 # all others are on one side or the other + return False # XXX is this necessary? + + @classmethod + def real_roots(cls, poly, radicals=True): + """Get real roots of a polynomial. """ + return cls._get_roots("_real_roots", poly, radicals) + + @classmethod + def all_roots(cls, poly, radicals=True): + """Get real and complex roots of a polynomial. """ + return cls._get_roots("_all_roots", poly, radicals) + + @classmethod + def _get_reals_sqf(cls, currentfactor, use_cache=True): + """Get real root isolating intervals for a square-free factor.""" + if use_cache and currentfactor in _reals_cache: + real_part = _reals_cache[currentfactor] + else: + _reals_cache[currentfactor] = real_part = \ + dup_isolate_real_roots_sqf( + currentfactor.rep.to_list(), currentfactor.rep.dom, blackbox=True) + + return real_part + + @classmethod + def _get_complexes_sqf(cls, currentfactor, use_cache=True): + """Get complex root isolating intervals for a square-free factor.""" + if use_cache and currentfactor in _complexes_cache: + complex_part = _complexes_cache[currentfactor] + else: + _complexes_cache[currentfactor] = complex_part = \ + dup_isolate_complex_roots_sqf( + currentfactor.rep.to_list(), currentfactor.rep.dom, blackbox=True) + return complex_part + + @classmethod + def _get_reals(cls, factors, use_cache=True): + """Compute real root isolating intervals for a list of factors. """ + reals = [] + + for currentfactor, k in factors: + try: + if not use_cache: + raise KeyError + r = _reals_cache[currentfactor] + reals.extend([(i, currentfactor, k) for i in r]) + except KeyError: + real_part = cls._get_reals_sqf(currentfactor, use_cache) + new = [(root, currentfactor, k) for root in real_part] + reals.extend(new) + + reals = cls._reals_sorted(reals) + return reals + + @classmethod + def _get_complexes(cls, factors, use_cache=True): + """Compute complex root isolating intervals for a list of factors. """ + complexes = [] + + for currentfactor, k in ordered(factors): + try: + if not use_cache: + raise KeyError + c = _complexes_cache[currentfactor] + complexes.extend([(i, currentfactor, k) for i in c]) + except KeyError: + complex_part = cls._get_complexes_sqf(currentfactor, use_cache) + new = [(root, currentfactor, k) for root in complex_part] + complexes.extend(new) + + complexes = cls._complexes_sorted(complexes) + return complexes + + @classmethod + def _reals_sorted(cls, reals): + """Make real isolating intervals disjoint and sort roots. """ + cache = {} + + for i, (u, f, k) in enumerate(reals): + for j, (v, g, m) in enumerate(reals[i + 1:]): + u, v = u.refine_disjoint(v) + reals[i + j + 1] = (v, g, m) + + reals[i] = (u, f, k) + + reals = sorted(reals, key=lambda r: r[0].a) + + for root, currentfactor, _ in reals: + if currentfactor in cache: + cache[currentfactor].append(root) + else: + cache[currentfactor] = [root] + + for currentfactor, root in cache.items(): + _reals_cache[currentfactor] = root + + return reals + + @classmethod + def _refine_imaginary(cls, complexes): + sifted = sift(complexes, lambda c: c[1]) + complexes = [] + for f in ordered(sifted): + nimag = _imag_count_of_factor(f) + if nimag == 0: + # refine until xbounds are neg or pos + for u, f, k in sifted[f]: + while u.ax*u.bx <= 0: + u = u._inner_refine() + complexes.append((u, f, k)) + else: + # refine until all but nimag xbounds are neg or pos + potential_imag = list(range(len(sifted[f]))) + while True: + assert len(potential_imag) > 1 + for i in list(potential_imag): + u, f, k = sifted[f][i] + if u.ax*u.bx > 0: + potential_imag.remove(i) + elif u.ax != u.bx: + u = u._inner_refine() + sifted[f][i] = u, f, k + if len(potential_imag) == nimag: + break + complexes.extend(sifted[f]) + return complexes + + @classmethod + def _refine_complexes(cls, complexes): + """return complexes such that no bounding rectangles of non-conjugate + roots would intersect. In addition, assure that neither ay nor by is + 0 to guarantee that non-real roots are distinct from real roots in + terms of the y-bounds. + """ + # get the intervals pairwise-disjoint. + # If rectangles were drawn around the coordinates of the bounding + # rectangles, no rectangles would intersect after this procedure. + for i, (u, f, k) in enumerate(complexes): + for j, (v, g, m) in enumerate(complexes[i + 1:]): + u, v = u.refine_disjoint(v) + complexes[i + j + 1] = (v, g, m) + + complexes[i] = (u, f, k) + + # refine until the x-bounds are unambiguously positive or negative + # for non-imaginary roots + complexes = cls._refine_imaginary(complexes) + + # make sure that all y bounds are off the real axis + # and on the same side of the axis + for i, (u, f, k) in enumerate(complexes): + while u.ay*u.by <= 0: + u = u.refine() + complexes[i] = u, f, k + return complexes + + @classmethod + def _complexes_sorted(cls, complexes): + """Make complex isolating intervals disjoint and sort roots. """ + complexes = cls._refine_complexes(complexes) + # XXX don't sort until you are sure that it is compatible + # with the indexing method but assert that the desired state + # is not broken + C, F = 0, 1 # location of ComplexInterval and factor + fs = {i[F] for i in complexes} + for i in range(1, len(complexes)): + if complexes[i][F] != complexes[i - 1][F]: + # if this fails the factors of a root were not + # contiguous because a discontinuity should only + # happen once + fs.remove(complexes[i - 1][F]) + for i, cmplx in enumerate(complexes): + # negative im part (conj=True) comes before + # positive im part (conj=False) + assert cmplx[C].conj is (i % 2 == 0) + + # update cache + cache = {} + # -- collate + for root, currentfactor, _ in complexes: + cache.setdefault(currentfactor, []).append(root) + # -- store + for currentfactor, root in cache.items(): + _complexes_cache[currentfactor] = root + + return complexes + + @classmethod + def _reals_index(cls, reals, index): + """ + Map initial real root index to an index in a factor where + the root belongs. + """ + i = 0 + + for j, (_, currentfactor, k) in enumerate(reals): + if index < i + k: + poly, index = currentfactor, 0 + + for _, currentfactor, _ in reals[:j]: + if currentfactor == poly: + index += 1 + + return poly, index + else: + i += k + + @classmethod + def _complexes_index(cls, complexes, index): + """ + Map initial complex root index to an index in a factor where + the root belongs. + """ + i = 0 + for j, (_, currentfactor, k) in enumerate(complexes): + if index < i + k: + poly, index = currentfactor, 0 + + for _, currentfactor, _ in complexes[:j]: + if currentfactor == poly: + index += 1 + + index += len(_reals_cache[poly]) + + return poly, index + else: + i += k + + @classmethod + def _count_roots(cls, roots): + """Count the number of real or complex roots with multiplicities.""" + return sum(k for _, _, k in roots) + + @classmethod + def _indexed_root(cls, poly, index, lazy=False): + """Get a root of a composite polynomial by index. """ + factors = _pure_factors(poly) + + # If the given poly is already irreducible, then the index does not + # need to be adjusted, and we can postpone the heavy lifting of + # computing and refining isolating intervals until that is needed. + # Note, however, that `_pure_factors()` extracts a negative leading + # coeff if present, so `factors[0][0]` may differ from `poly`, and + # is the "normalized" version of `poly` that we must return. + if lazy and len(factors) == 1 and factors[0][1] == 1: + return factors[0][0], index + + reals = cls._get_reals(factors) + reals_count = cls._count_roots(reals) + + if index < reals_count: + return cls._reals_index(reals, index) + else: + complexes = cls._get_complexes(factors) + return cls._complexes_index(complexes, index - reals_count) + + def _ensure_reals_init(self): + """Ensure that our poly has entries in the reals cache. """ + if self.poly not in _reals_cache: + self._indexed_root(self.poly, self.index) + + def _ensure_complexes_init(self): + """Ensure that our poly has entries in the complexes cache. """ + if self.poly not in _complexes_cache: + self._indexed_root(self.poly, self.index) + + @classmethod + def _real_roots(cls, poly): + """Get real roots of a composite polynomial. """ + factors = _pure_factors(poly) + + reals = cls._get_reals(factors) + reals_count = cls._count_roots(reals) + + roots = [] + + for index in range(0, reals_count): + roots.append(cls._reals_index(reals, index)) + + return roots + + def _reset(self): + """ + Reset all intervals + """ + self._all_roots(self.poly, use_cache=False) + + @classmethod + def _all_roots(cls, poly, use_cache=True): + """Get real and complex roots of a composite polynomial. """ + factors = _pure_factors(poly) + + reals = cls._get_reals(factors, use_cache=use_cache) + reals_count = cls._count_roots(reals) + + roots = [] + + for index in range(0, reals_count): + roots.append(cls._reals_index(reals, index)) + + complexes = cls._get_complexes(factors, use_cache=use_cache) + complexes_count = cls._count_roots(complexes) + + for index in range(0, complexes_count): + roots.append(cls._complexes_index(complexes, index)) + + return roots + + @classmethod + @cacheit + def _roots_trivial(cls, poly, radicals): + """Compute roots in linear, quadratic and binomial cases. """ + if poly.degree() == 1: + return roots_linear(poly) + + if not radicals: + return None + + if poly.degree() == 2: + return roots_quadratic(poly) + elif poly.length() == 2 and poly.TC(): + return roots_binomial(poly) + else: + return None + + @classmethod + def _preprocess_roots(cls, poly): + """Take heroic measures to make ``poly`` compatible with ``CRootOf``.""" + dom = poly.get_domain() + + if not dom.is_Exact: + poly = poly.to_exact() + + coeff, poly = preprocess_roots(poly) + dom = poly.get_domain() + + if not dom.is_ZZ: + raise NotImplementedError( + "sorted roots not supported over %s" % dom) + + return coeff, poly + + @classmethod + def _postprocess_root(cls, root, radicals): + """Return the root if it is trivial or a ``CRootOf`` object. """ + poly, index = root + roots = cls._roots_trivial(poly, radicals) + + if roots is not None: + return roots[index] + else: + return cls._new(poly, index) + + @classmethod + def _get_roots(cls, method, poly, radicals): + """Return postprocessed roots of specified kind. """ + if not poly.is_univariate: + raise PolynomialError("only univariate polynomials are allowed") + + dom = poly.get_domain() + + # get rid of gen and it's free symbol + d = Dummy() + poly = poly.subs(poly.gen, d) + x = symbols('x') + # see what others are left and select x or a numbered x + # that doesn't clash + free_names = {str(i) for i in poly.free_symbols} + for x in chain((symbols('x'),), numbered_symbols('x')): + if x.name not in free_names: + poly = poly.replace(d, x) + break + + if dom.is_QQ or dom.is_ZZ: + return cls._get_roots_qq(method, poly, radicals) + elif dom.is_AlgebraicField or dom.is_ZZ_I or dom.is_QQ_I: + return cls._get_roots_alg(method, poly, radicals) + else: + # XXX: not sure how to handle ZZ[x] which appears in some tests? + # this makes the tests pass alright but has to be a better way? + return cls._get_roots_qq(method, poly, radicals) + + + @classmethod + def _get_roots_qq(cls, method, poly, radicals): + """Return postprocessed roots of specified kind + for polynomials with rational coefficients. """ + coeff, poly = cls._preprocess_roots(poly) + roots = [] + + for root in getattr(cls, method)(poly): + roots.append(coeff*cls._postprocess_root(root, radicals)) + + return roots + + @classmethod + def _get_roots_alg(cls, method, poly, radicals): + """Return postprocessed roots of specified kind + for polynomials with algebraic coefficients. It assumes + the domain is already an algebraic field. First it + finds the roots using _get_roots_qq, then uses the + square-free factors to filter roots and get the correct + multiplicity. + """ + + # Existing QQ code can find and sort the roots + roots = cls._get_roots_qq(method, poly.lift(), radicals) + + subroots = {} + for f, m in poly.sqf_list()[1]: + if method == "_real_roots": + roots_filt = f.which_real_roots(roots) + elif method == "_all_roots": + roots_filt = f.which_all_roots(roots) + for r in roots_filt: + subroots[r] = m + + roots_seen = set() + roots_flat = [] + for r in roots: + if r in subroots and r not in roots_seen: + m = subroots[r] + roots_flat.extend([r] * m) + roots_seen.add(r) + + return roots_flat + + @classmethod + def clear_cache(cls): + """Reset cache for reals and complexes. + + The intervals used to approximate a root instance are updated + as needed. When a request is made to see the intervals, the + most current values are shown. `clear_cache` will reset all + CRootOf instances back to their original state. + + See Also + ======== + + _reset + """ + global _reals_cache, _complexes_cache + _reals_cache = _pure_key_dict() + _complexes_cache = _pure_key_dict() + + def _get_interval(self): + """Internal function for retrieving isolation interval from cache. """ + self._ensure_reals_init() + if self.is_real: + return _reals_cache[self.poly][self.index] + else: + reals_count = len(_reals_cache[self.poly]) + self._ensure_complexes_init() + return _complexes_cache[self.poly][self.index - reals_count] + + def _set_interval(self, interval): + """Internal function for updating isolation interval in cache. """ + self._ensure_reals_init() + if self.is_real: + _reals_cache[self.poly][self.index] = interval + else: + reals_count = len(_reals_cache[self.poly]) + self._ensure_complexes_init() + _complexes_cache[self.poly][self.index - reals_count] = interval + + def _eval_subs(self, old, new): + # don't allow subs to change anything + return self + + def _eval_conjugate(self): + if self.is_real: + return self + expr, i = self.args + return self.func(expr, i + (1 if self._get_interval().conj else -1)) + + def eval_approx(self, n, return_mpmath=False): + """Evaluate this complex root to the given precision. + + This uses secant method and root bounds are used to both + generate an initial guess and to check that the root + returned is valid. If ever the method converges outside the + root bounds, the bounds will be made smaller and updated. + """ + prec = dps_to_prec(n) + with workprec(prec): + g = self.poly.gen + if not g.is_Symbol: + d = Dummy('x') + if self.is_imaginary: + d *= I + func = lambdify(d, self.expr.subs(g, d)) + else: + expr = self.expr + if self.is_imaginary: + expr = self.expr.subs(g, I*g) + func = lambdify(g, expr) + + interval = self._get_interval() + while True: + if self.is_real: + a = mpf(str(interval.a)) + b = mpf(str(interval.b)) + if a == b: + root = a + break + x0 = mpf(str(interval.center)) + x1 = x0 + mpf(str(interval.dx))/4 + elif self.is_imaginary: + a = mpf(str(interval.ay)) + b = mpf(str(interval.by)) + if a == b: + root = mpc(mpf('0'), a) + break + x0 = mpf(str(interval.center[1])) + x1 = x0 + mpf(str(interval.dy))/4 + else: + ax = mpf(str(interval.ax)) + bx = mpf(str(interval.bx)) + ay = mpf(str(interval.ay)) + by = mpf(str(interval.by)) + if ax == bx and ay == by: + root = mpc(ax, ay) + break + x0 = mpc(*map(str, interval.center)) + x1 = x0 + mpc(*map(str, (interval.dx, interval.dy)))/4 + try: + # without a tolerance, this will return when (to within + # the given precision) x_i == x_{i-1} + root = findroot(func, (x0, x1)) + # If the (real or complex) root is not in the 'interval', + # then keep refining the interval. This happens if findroot + # accidentally finds a different root outside of this + # interval because our initial estimate 'x0' was not close + # enough. It is also possible that the secant method will + # get trapped by a max/min in the interval; the root + # verification by findroot will raise a ValueError in this + # case and the interval will then be tightened -- and + # eventually the root will be found. + # + # It is also possible that findroot will not have any + # successful iterations to process (in which case it + # will fail to initialize a variable that is tested + # after the iterations and raise an UnboundLocalError). + if self.is_real or self.is_imaginary: + if not bool(root.imag) == self.is_real and ( + a <= root <= b): + if self.is_imaginary: + root = mpc(mpf('0'), root.real) + break + elif (ax <= root.real <= bx and ay <= root.imag <= by): + break + except (UnboundLocalError, ValueError): + pass + interval = interval.refine() + + # update the interval so we at least (for this precision or + # less) don't have much work to do to recompute the root + self._set_interval(interval) + if return_mpmath: + return root + return (Float._new(root.real._mpf_, prec) + + I*Float._new(root.imag._mpf_, prec)) + + def _eval_evalf(self, prec, **kwargs): + """Evaluate this complex root to the given precision.""" + # all kwargs are ignored + return self.eval_rational(n=prec_to_dps(prec))._evalf(prec) + + def eval_rational(self, dx=None, dy=None, n=15): + """ + Return a Rational approximation of ``self`` that has real + and imaginary component approximations that are within ``dx`` + and ``dy`` of the true values, respectively. Alternatively, + ``n`` digits of precision can be specified. + + The interval is refined with bisection and is sure to + converge. The root bounds are updated when the refinement + is complete so recalculation at the same or lesser precision + will not have to repeat the refinement and should be much + faster. + + The following example first obtains Rational approximation to + 1e-8 accuracy for all roots of the 4-th order Legendre + polynomial. Since the roots are all less than 1, this will + ensure the decimal representation of the approximation will be + correct (including rounding) to 6 digits: + + >>> from sympy import legendre_poly, Symbol + >>> x = Symbol("x") + >>> p = legendre_poly(4, x, polys=True) + >>> r = p.real_roots()[-1] + >>> r.eval_rational(10**-8).n(6) + 0.861136 + + It is not necessary to a two-step calculation, however: the + decimal representation can be computed directly: + + >>> r.evalf(17) + 0.86113631159405258 + + """ + dy = dy or dx + if dx: + rtol = None + dx = dx if isinstance(dx, Rational) else Rational(str(dx)) + dy = dy if isinstance(dy, Rational) else Rational(str(dy)) + else: + # 5 binary (or 2 decimal) digits are needed to ensure that + # a given digit is correctly rounded + # prec_to_dps(dps_to_prec(n) + 5) - n <= 2 (tested for + # n in range(1000000) + rtol = S(10)**-(n + 2) # +2 for guard digits + interval = self._get_interval() + while True: + if self.is_real: + if rtol: + dx = abs(interval.center*rtol) + interval = interval.refine_size(dx=dx) + c = interval.center + real = Rational(c) + imag = S.Zero + if not rtol or interval.dx < abs(c*rtol): + break + elif self.is_imaginary: + if rtol: + dy = abs(interval.center[1]*rtol) + dx = 1 + interval = interval.refine_size(dx=dx, dy=dy) + c = interval.center[1] + imag = Rational(c) + real = S.Zero + if not rtol or interval.dy < abs(c*rtol): + break + else: + if rtol: + dx = abs(interval.center[0]*rtol) + dy = abs(interval.center[1]*rtol) + interval = interval.refine_size(dx, dy) + c = interval.center + real, imag = map(Rational, c) + if not rtol or ( + interval.dx < abs(c[0]*rtol) and + interval.dy < abs(c[1]*rtol)): + break + + # update the interval so we at least (for this precision or + # less) don't have much work to do to recompute the root + self._set_interval(interval) + return real + I*imag + + +CRootOf = ComplexRootOf + + +@dispatch(ComplexRootOf, ComplexRootOf) +def _eval_is_eq(lhs, rhs): # noqa:F811 + # if we use is_eq to check here, we get infinite recursion + return lhs == rhs + + +@dispatch(ComplexRootOf, Basic) # type:ignore +def _eval_is_eq(lhs, rhs): # noqa:F811 + # CRootOf represents a Root, so if rhs is that root, it should set + # the expression to zero *and* it should be in the interval of the + # CRootOf instance. It must also be a number that agrees with the + # is_real value of the CRootOf instance. + if not rhs.is_number: + return None + if not rhs.is_finite: + return False + z = lhs.expr.subs(lhs.expr.free_symbols.pop(), rhs).is_zero + if z is False: # all roots will make z True but we don't know + # whether this is the right root if z is True + return False + o = rhs.is_real, rhs.is_imaginary + s = lhs.is_real, lhs.is_imaginary + assert None not in s # this is part of initial refinement + if o != s and None not in o: + return False + re, im = rhs.as_real_imag() + if lhs.is_real: + if im: + return False + i = lhs._get_interval() + a, b = [Rational(str(_)) for _ in (i.a, i.b)] + return sympify(a <= rhs and rhs <= b) + i = lhs._get_interval() + r1, r2, i1, i2 = [Rational(str(j)) for j in ( + i.ax, i.bx, i.ay, i.by)] + return is_le(r1, re) and is_le(re,r2) and is_le(i1,im) and is_le(im,i2) + + +@public +class RootSum(Expr): + """Represents a sum of all roots of a univariate polynomial. """ + + __slots__ = ('poly', 'fun', 'auto') + + def __new__(cls, expr, func=None, x=None, auto=True, quadratic=False): + """Construct a new ``RootSum`` instance of roots of a polynomial.""" + coeff, poly = cls._transform(expr, x) + + if not poly.is_univariate: + raise MultivariatePolynomialError( + "only univariate polynomials are allowed") + + if func is None: + func = Lambda(poly.gen, poly.gen) + else: + is_func = getattr(func, 'is_Function', False) + + if is_func and 1 in func.nargs: + if not isinstance(func, Lambda): + func = Lambda(poly.gen, func(poly.gen)) + else: + raise ValueError( + "expected a univariate function, got %s" % func) + + var, expr = func.variables[0], func.expr + + if coeff is not S.One: + expr = expr.subs(var, coeff*var) + + deg = poly.degree() + + if not expr.has(var): + return deg*expr + + if expr.is_Add: + add_const, expr = expr.as_independent(var) + else: + add_const = S.Zero + + if expr.is_Mul: + mul_const, expr = expr.as_independent(var) + else: + mul_const = S.One + + func = Lambda(var, expr) + + rational = cls._is_func_rational(poly, func) + factors, terms = _pure_factors(poly), [] + + for poly, k in factors: + if poly.is_linear: + term = func(roots_linear(poly)[0]) + elif quadratic and poly.is_quadratic: + term = sum(map(func, roots_quadratic(poly))) + else: + if not rational or not auto: + term = cls._new(poly, func, auto) + else: + term = cls._rational_case(poly, func) + + terms.append(k*term) + + return mul_const*Add(*terms) + deg*add_const + + @classmethod + def _new(cls, poly, func, auto=True): + """Construct new raw ``RootSum`` instance. """ + obj = Expr.__new__(cls) + + obj.poly = poly + obj.fun = func + obj.auto = auto + + return obj + + @classmethod + def new(cls, poly, func, auto=True): + """Construct new ``RootSum`` instance. """ + if not func.expr.has(*func.variables): + return func.expr + + rational = cls._is_func_rational(poly, func) + + if not rational or not auto: + return cls._new(poly, func, auto) + else: + return cls._rational_case(poly, func) + + @classmethod + def _transform(cls, expr, x): + """Transform an expression to a polynomial. """ + poly = PurePoly(expr, x, greedy=False) + return preprocess_roots(poly) + + @classmethod + def _is_func_rational(cls, poly, func): + """Check if a lambda is a rational function. """ + var, expr = func.variables[0], func.expr + return expr.is_rational_function(var) + + @classmethod + def _rational_case(cls, poly, func): + """Handle the rational function case. """ + roots = symbols('r:%d' % poly.degree()) + var, expr = func.variables[0], func.expr + + f = sum(expr.subs(var, r) for r in roots) + p, q = together(f).as_numer_denom() + + domain = QQ[roots] + + p = p.expand() + q = q.expand() + + try: + p = Poly(p, domain=domain, expand=False) + except GeneratorsNeeded: + p, p_coeff = None, (p,) + else: + p_monom, p_coeff = zip(*p.terms()) + + try: + q = Poly(q, domain=domain, expand=False) + except GeneratorsNeeded: + q, q_coeff = None, (q,) + else: + q_monom, q_coeff = zip(*q.terms()) + + coeffs, mapping = symmetrize(p_coeff + q_coeff, formal=True) + formulas, values = viete(poly, roots), [] + + for (sym, _), (_, val) in zip(mapping, formulas): + values.append((sym, val)) + + for i, (coeff, _) in enumerate(coeffs): + coeffs[i] = coeff.subs(values) + + n = len(p_coeff) + + p_coeff = coeffs[:n] + q_coeff = coeffs[n:] + + if p is not None: + p = Poly(dict(zip(p_monom, p_coeff)), *p.gens).as_expr() + else: + (p,) = p_coeff + + if q is not None: + q = Poly(dict(zip(q_monom, q_coeff)), *q.gens).as_expr() + else: + (q,) = q_coeff + + return factor(p/q) + + def _hashable_content(self): + return (self.poly, self.fun) + + @property + def expr(self): + return self.poly.as_expr() + + @property + def args(self): + return (self.expr, self.fun, self.poly.gen) + + @property + def free_symbols(self): + return self.poly.free_symbols | self.fun.free_symbols + + @property + def is_commutative(self): + return True + + def doit(self, **hints): + if not hints.get('roots', True): + return self + + _roots = roots(self.poly, multiple=True) + + if len(_roots) < self.poly.degree(): + return self + else: + return Add(*[self.fun(r) for r in _roots]) + + def _eval_evalf(self, prec): + try: + _roots = self.poly.nroots(n=prec_to_dps(prec)) + except (DomainError, PolynomialError): + return self + else: + return Add(*[self.fun(r) for r in _roots]) + + def _eval_derivative(self, x): + var, expr = self.fun.args + func = Lambda(var, expr.diff(x)) + return self.new(self.poly, func, self.auto) diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/solvers.py b/.venv/lib/python3.12/site-packages/sympy/polys/solvers.py new file mode 100644 index 0000000000000000000000000000000000000000..b333e81d975a8cd71e7eb683c2b943d8538f6ac5 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/solvers.py @@ -0,0 +1,435 @@ +"""Low-level linear systems solver. """ + + +from sympy.utilities.exceptions import sympy_deprecation_warning +from sympy.utilities.iterables import connected_components + +from sympy.core.sympify import sympify +from sympy.core.numbers import Integer, Rational +from sympy.matrices.dense import MutableDenseMatrix +from sympy.polys.domains import ZZ, QQ + +from sympy.polys.domains import EX +from sympy.polys.rings import sring +from sympy.polys.polyerrors import NotInvertible +from sympy.polys.domainmatrix import DomainMatrix + + +class PolyNonlinearError(Exception): + """Raised by solve_lin_sys for nonlinear equations""" + pass + + +class RawMatrix(MutableDenseMatrix): + """ + .. deprecated:: 1.9 + + This class fundamentally is broken by design. Use ``DomainMatrix`` if + you want a matrix over the polys domains or ``Matrix`` for a matrix + with ``Expr`` elements. The ``RawMatrix`` class will be removed/broken + in future in order to reestablish the invariant that the elements of a + Matrix should be of type ``Expr``. + + """ + _sympify = staticmethod(lambda x, *args, **kwargs: x) + + def __init__(self, *args, **kwargs): + sympy_deprecation_warning( + """ + The RawMatrix class is deprecated. Use either DomainMatrix or + Matrix instead. + """, + deprecated_since_version="1.9", + active_deprecations_target="deprecated-rawmatrix", + ) + + domain = ZZ + for i in range(self.rows): + for j in range(self.cols): + val = self[i,j] + if getattr(val, 'is_Poly', False): + K = val.domain[val.gens] + val_sympy = val.as_expr() + elif hasattr(val, 'parent'): + K = val.parent() + val_sympy = K.to_sympy(val) + elif isinstance(val, (int, Integer)): + K = ZZ + val_sympy = sympify(val) + elif isinstance(val, Rational): + K = QQ + val_sympy = val + else: + for K in ZZ, QQ: + if K.of_type(val): + val_sympy = K.to_sympy(val) + break + else: + raise TypeError + domain = domain.unify(K) + self[i,j] = val_sympy + self.ring = domain + + +def eqs_to_matrix(eqs_coeffs, eqs_rhs, gens, domain): + """Get matrix from linear equations in dict format. + + Explanation + =========== + + Get the matrix representation of a system of linear equations represented + as dicts with low-level DomainElement coefficients. This is an + *internal* function that is used by solve_lin_sys. + + Parameters + ========== + + eqs_coeffs: list[dict[Symbol, DomainElement]] + The left hand sides of the equations as dicts mapping from symbols to + coefficients where the coefficients are instances of + DomainElement. + eqs_rhs: list[DomainElements] + The right hand sides of the equations as instances of + DomainElement. + gens: list[Symbol] + The unknowns in the system of equations. + domain: Domain + The domain for coefficients of both lhs and rhs. + + Returns + ======= + + The augmented matrix representation of the system as a DomainMatrix. + + Examples + ======== + + >>> from sympy import symbols, ZZ + >>> from sympy.polys.solvers import eqs_to_matrix + >>> x, y = symbols('x, y') + >>> eqs_coeff = [{x:ZZ(1), y:ZZ(1)}, {x:ZZ(1), y:ZZ(-1)}] + >>> eqs_rhs = [ZZ(0), ZZ(-1)] + >>> eqs_to_matrix(eqs_coeff, eqs_rhs, [x, y], ZZ) + DomainMatrix([[1, 1, 0], [1, -1, 1]], (2, 3), ZZ) + + See also + ======== + + solve_lin_sys: Uses :func:`~eqs_to_matrix` internally + """ + sym2index = {x: n for n, x in enumerate(gens)} + nrows = len(eqs_coeffs) + ncols = len(gens) + 1 + rows = [[domain.zero] * ncols for _ in range(nrows)] + for row, eq_coeff, eq_rhs in zip(rows, eqs_coeffs, eqs_rhs): + for sym, coeff in eq_coeff.items(): + row[sym2index[sym]] = domain.convert(coeff) + row[-1] = -domain.convert(eq_rhs) + + return DomainMatrix(rows, (nrows, ncols), domain) + + +def sympy_eqs_to_ring(eqs, symbols): + """Convert a system of equations from Expr to a PolyRing + + Explanation + =========== + + High-level functions like ``solve`` expect Expr as inputs but can use + ``solve_lin_sys`` internally. This function converts equations from + ``Expr`` to the low-level poly types used by the ``solve_lin_sys`` + function. + + Parameters + ========== + + eqs: List of Expr + A list of equations as Expr instances + symbols: List of Symbol + A list of the symbols that are the unknowns in the system of + equations. + + Returns + ======= + + Tuple[List[PolyElement], Ring]: The equations as PolyElement instances + and the ring of polynomials within which each equation is represented. + + Examples + ======== + + >>> from sympy import symbols + >>> from sympy.polys.solvers import sympy_eqs_to_ring + >>> a, x, y = symbols('a, x, y') + >>> eqs = [x-y, x+a*y] + >>> eqs_ring, ring = sympy_eqs_to_ring(eqs, [x, y]) + >>> eqs_ring + [x - y, x + a*y] + >>> type(eqs_ring[0]) + + >>> ring + ZZ(a)[x,y] + + With the equations in this form they can be passed to ``solve_lin_sys``: + + >>> from sympy.polys.solvers import solve_lin_sys + >>> solve_lin_sys(eqs_ring, ring) + {y: 0, x: 0} + """ + try: + K, eqs_K = sring(eqs, symbols, field=True, extension=True) + except NotInvertible: + # https://github.com/sympy/sympy/issues/18874 + K, eqs_K = sring(eqs, symbols, domain=EX) + return eqs_K, K.to_domain() + + +def solve_lin_sys(eqs, ring, _raw=True): + """Solve a system of linear equations from a PolynomialRing + + Explanation + =========== + + Solves a system of linear equations given as PolyElement instances of a + PolynomialRing. The basic arithmetic is carried out using instance of + DomainElement which is more efficient than :class:`~sympy.core.expr.Expr` + for the most common inputs. + + While this is a public function it is intended primarily for internal use + so its interface is not necessarily convenient. Users are suggested to use + the :func:`sympy.solvers.solveset.linsolve` function (which uses this + function internally) instead. + + Parameters + ========== + + eqs: list[PolyElement] + The linear equations to be solved as elements of a + PolynomialRing (assumed equal to zero). + ring: PolynomialRing + The polynomial ring from which eqs are drawn. The generators of this + ring are the unknowns to be solved for and the domain of the ring is + the domain of the coefficients of the system of equations. + _raw: bool + If *_raw* is False, the keys and values in the returned dictionary + will be of type Expr (and the unit of the field will be removed from + the keys) otherwise the low-level polys types will be returned, e.g. + PolyElement: PythonRational. + + Returns + ======= + + ``None`` if the system has no solution. + + dict[Symbol, Expr] if _raw=False + + dict[Symbol, DomainElement] if _raw=True. + + Examples + ======== + + >>> from sympy import symbols + >>> from sympy.polys.solvers import solve_lin_sys, sympy_eqs_to_ring + >>> x, y = symbols('x, y') + >>> eqs = [x - y, x + y - 2] + >>> eqs_ring, ring = sympy_eqs_to_ring(eqs, [x, y]) + >>> solve_lin_sys(eqs_ring, ring) + {y: 1, x: 1} + + Passing ``_raw=False`` returns the same result except that the keys are + ``Expr`` rather than low-level poly types. + + >>> solve_lin_sys(eqs_ring, ring, _raw=False) + {x: 1, y: 1} + + See also + ======== + + sympy_eqs_to_ring: prepares the inputs to ``solve_lin_sys``. + linsolve: ``linsolve`` uses ``solve_lin_sys`` internally. + sympy.solvers.solvers.solve: ``solve`` uses ``solve_lin_sys`` internally. + """ + as_expr = not _raw + + assert ring.domain.is_Field + + eqs_dict = [dict(eq) for eq in eqs] + + one_monom = ring.one.monoms()[0] + zero = ring.domain.zero + + eqs_rhs = [] + eqs_coeffs = [] + for eq_dict in eqs_dict: + eq_rhs = eq_dict.pop(one_monom, zero) + eq_coeffs = {} + for monom, coeff in eq_dict.items(): + if sum(monom) != 1: + msg = "Nonlinear term encountered in solve_lin_sys" + raise PolyNonlinearError(msg) + eq_coeffs[ring.gens[monom.index(1)]] = coeff + if not eq_coeffs: + if not eq_rhs: + continue + else: + return None + eqs_rhs.append(eq_rhs) + eqs_coeffs.append(eq_coeffs) + + result = _solve_lin_sys(eqs_coeffs, eqs_rhs, ring) + + if result is not None and as_expr: + + def to_sympy(x): + as_expr = getattr(x, 'as_expr', None) + if as_expr: + return as_expr() + else: + return ring.domain.to_sympy(x) + + tresult = {to_sympy(sym): to_sympy(val) for sym, val in result.items()} + + # Remove 1.0x + result = {} + for k, v in tresult.items(): + if k.is_Mul: + c, s = k.as_coeff_Mul() + result[s] = v/c + else: + result[k] = v + + return result + + +def _solve_lin_sys(eqs_coeffs, eqs_rhs, ring): + """Solve a linear system from dict of PolynomialRing coefficients + + Explanation + =========== + + This is an **internal** function used by :func:`solve_lin_sys` after the + equations have been preprocessed. The role of this function is to split + the system into connected components and pass those to + :func:`_solve_lin_sys_component`. + + Examples + ======== + + Setup a system for $x-y=0$ and $x+y=2$ and solve: + + >>> from sympy import symbols, sring + >>> from sympy.polys.solvers import _solve_lin_sys + >>> x, y = symbols('x, y') + >>> R, (xr, yr) = sring([x, y], [x, y]) + >>> eqs = [{xr:R.one, yr:-R.one}, {xr:R.one, yr:R.one}] + >>> eqs_rhs = [R.zero, -2*R.one] + >>> _solve_lin_sys(eqs, eqs_rhs, R) + {y: 1, x: 1} + + See also + ======== + + solve_lin_sys: This function is used internally by :func:`solve_lin_sys`. + """ + V = ring.gens + E = [] + for eq_coeffs in eqs_coeffs: + syms = list(eq_coeffs) + E.extend(zip(syms[:-1], syms[1:])) + G = V, E + + components = connected_components(G) + + sym2comp = {} + for n, component in enumerate(components): + for sym in component: + sym2comp[sym] = n + + subsystems = [([], []) for _ in range(len(components))] + for eq_coeff, eq_rhs in zip(eqs_coeffs, eqs_rhs): + sym = next(iter(eq_coeff), None) + sub_coeff, sub_rhs = subsystems[sym2comp[sym]] + sub_coeff.append(eq_coeff) + sub_rhs.append(eq_rhs) + + sol = {} + for subsystem in subsystems: + subsol = _solve_lin_sys_component(subsystem[0], subsystem[1], ring) + if subsol is None: + return None + sol.update(subsol) + + return sol + + +def _solve_lin_sys_component(eqs_coeffs, eqs_rhs, ring): + """Solve a linear system from dict of PolynomialRing coefficients + + Explanation + =========== + + This is an **internal** function used by :func:`solve_lin_sys` after the + equations have been preprocessed. After :func:`_solve_lin_sys` splits the + system into connected components this function is called for each + component. The system of equations is solved using Gauss-Jordan + elimination with division followed by back-substitution. + + Examples + ======== + + Setup a system for $x-y=0$ and $x+y=2$ and solve: + + >>> from sympy import symbols, sring + >>> from sympy.polys.solvers import _solve_lin_sys_component + >>> x, y = symbols('x, y') + >>> R, (xr, yr) = sring([x, y], [x, y]) + >>> eqs = [{xr:R.one, yr:-R.one}, {xr:R.one, yr:R.one}] + >>> eqs_rhs = [R.zero, -2*R.one] + >>> _solve_lin_sys_component(eqs, eqs_rhs, R) + {y: 1, x: 1} + + See also + ======== + + solve_lin_sys: This function is used internally by :func:`solve_lin_sys`. + """ + + # transform from equations to matrix form + matrix = eqs_to_matrix(eqs_coeffs, eqs_rhs, ring.gens, ring.domain) + + # convert to a field for rref + if not matrix.domain.is_Field: + matrix = matrix.to_field() + + # solve by row-reduction + echelon, pivots = matrix.rref() + + # construct the returnable form of the solutions + keys = ring.gens + + if pivots and pivots[-1] == len(keys): + return None + + if len(pivots) == len(keys): + sol = [] + for s in [row[-1] for row in echelon.rep.to_ddm()]: + a = s + sol.append(a) + sols = dict(zip(keys, sol)) + else: + sols = {} + g = ring.gens + # Extract ground domain coefficients and convert to the ring: + if hasattr(ring, 'ring'): + convert = ring.ring.ground_new + else: + convert = ring.ground_new + echelon = echelon.rep.to_ddm() + vals_set = {v for row in echelon for v in row} + vals_map = {v: convert(v) for v in vals_set} + echelon = [[vals_map[eij] for eij in ei] for ei in echelon] + for i, p in enumerate(pivots): + v = echelon[i][-1] - sum(echelon[i][j]*g[j] for j in range(p+1, len(g)) if echelon[i][j]) + sols[keys[p]] = v + + return sols diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/specialpolys.py b/.venv/lib/python3.12/site-packages/sympy/polys/specialpolys.py new file mode 100644 index 0000000000000000000000000000000000000000..3e85de8679cda3084f1c263a045f4d8f817bed98 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/specialpolys.py @@ -0,0 +1,340 @@ +"""Functions for generating interesting polynomials, e.g. for benchmarking. """ + + +from sympy.core import Add, Mul, Symbol, sympify, Dummy, symbols +from sympy.core.containers import Tuple +from sympy.core.singleton import S +from sympy.ntheory import nextprime +from sympy.polys.densearith import ( + dmp_add_term, dmp_neg, dmp_mul, dmp_sqr +) +from sympy.polys.densebasic import ( + dmp_zero, dmp_one, dmp_ground, + dup_from_raw_dict, dmp_raise, dup_random +) +from sympy.polys.domains import ZZ +from sympy.polys.factortools import dup_zz_cyclotomic_poly +from sympy.polys.polyclasses import DMP +from sympy.polys.polytools import Poly, PurePoly +from sympy.polys.polyutils import _analyze_gens +from sympy.utilities import subsets, public, filldedent + + +@public +def swinnerton_dyer_poly(n, x=None, polys=False): + """Generates n-th Swinnerton-Dyer polynomial in `x`. + + Parameters + ---------- + n : int + `n` decides the order of polynomial + x : optional + polys : bool, optional + ``polys=True`` returns an expression, otherwise + (default) returns an expression. + """ + if n <= 0: + raise ValueError( + "Cannot generate Swinnerton-Dyer polynomial of order %s" % n) + + if x is not None: + sympify(x) + else: + x = Dummy('x') + + if n > 3: + from sympy.functions.elementary.miscellaneous import sqrt + from .numberfields import minimal_polynomial + p = 2 + a = [sqrt(2)] + for i in range(2, n + 1): + p = nextprime(p) + a.append(sqrt(p)) + return minimal_polynomial(Add(*a), x, polys=polys) + + if n == 1: + ex = x**2 - 2 + elif n == 2: + ex = x**4 - 10*x**2 + 1 + elif n == 3: + ex = x**8 - 40*x**6 + 352*x**4 - 960*x**2 + 576 + + return PurePoly(ex, x) if polys else ex + + +@public +def cyclotomic_poly(n, x=None, polys=False): + """Generates cyclotomic polynomial of order `n` in `x`. + + Parameters + ---------- + n : int + `n` decides the order of polynomial + x : optional + polys : bool, optional + ``polys=True`` returns an expression, otherwise + (default) returns an expression. + """ + if n <= 0: + raise ValueError( + "Cannot generate cyclotomic polynomial of order %s" % n) + + poly = DMP(dup_zz_cyclotomic_poly(int(n), ZZ), ZZ) + + if x is not None: + poly = Poly.new(poly, x) + else: + poly = PurePoly.new(poly, Dummy('x')) + + return poly if polys else poly.as_expr() + + +@public +def symmetric_poly(n, *gens, polys=False): + """ + Generates symmetric polynomial of order `n`. + + Parameters + ========== + + polys: bool, optional (default: False) + Returns a Poly object when ``polys=True``, otherwise + (default) returns an expression. + """ + gens = _analyze_gens(gens) + + if n < 0 or n > len(gens) or not gens: + raise ValueError("Cannot generate symmetric polynomial of order %s for %s" % (n, gens)) + elif not n: + poly = S.One + else: + poly = Add(*[Mul(*s) for s in subsets(gens, int(n))]) + + return Poly(poly, *gens) if polys else poly + + +@public +def random_poly(x, n, inf, sup, domain=ZZ, polys=False): + """Generates a polynomial of degree ``n`` with coefficients in + ``[inf, sup]``. + + Parameters + ---------- + x + `x` is the independent term of polynomial + n : int + `n` decides the order of polynomial + inf + Lower limit of range in which coefficients lie + sup + Upper limit of range in which coefficients lie + domain : optional + Decides what ring the coefficients are supposed + to belong. Default is set to Integers. + polys : bool, optional + ``polys=True`` returns an expression, otherwise + (default) returns an expression. + """ + poly = Poly(dup_random(n, inf, sup, domain), x, domain=domain) + + return poly if polys else poly.as_expr() + + +@public +def interpolating_poly(n, x, X='x', Y='y'): + """Construct Lagrange interpolating polynomial for ``n`` + data points. If a sequence of values are given for ``X`` and ``Y`` + then the first ``n`` values will be used. + """ + ok = getattr(x, 'free_symbols', None) + + if isinstance(X, str): + X = symbols("%s:%s" % (X, n)) + elif ok and ok & Tuple(*X).free_symbols: + ok = False + + if isinstance(Y, str): + Y = symbols("%s:%s" % (Y, n)) + elif ok and ok & Tuple(*Y).free_symbols: + ok = False + + if not ok: + raise ValueError(filldedent(''' + Expecting symbol for x that does not appear in X or Y. + Use `interpolate(list(zip(X, Y)), x)` instead.''')) + + coeffs = [] + numert = Mul(*[x - X[i] for i in range(n)]) + + for i in range(n): + numer = numert/(x - X[i]) + denom = Mul(*[(X[i] - X[j]) for j in range(n) if i != j]) + coeffs.append(numer/denom) + + return Add(*[coeff*y for coeff, y in zip(coeffs, Y)]) + + +def fateman_poly_F_1(n): + """Fateman's GCD benchmark: trivial GCD """ + Y = [Symbol('y_' + str(i)) for i in range(n + 1)] + + y_0, y_1 = Y[0], Y[1] + + u = y_0 + Add(*Y[1:]) + v = y_0**2 + Add(*[y**2 for y in Y[1:]]) + + F = ((u + 1)*(u + 2)).as_poly(*Y) + G = ((v + 1)*(-3*y_1*y_0**2 + y_1**2 - 1)).as_poly(*Y) + + H = Poly(1, *Y) + + return F, G, H + + +def dmp_fateman_poly_F_1(n, K): + """Fateman's GCD benchmark: trivial GCD """ + u = [K(1), K(0)] + + for i in range(n): + u = [dmp_one(i, K), u] + + v = [K(1), K(0), K(0)] + + for i in range(0, n): + v = [dmp_one(i, K), dmp_zero(i), v] + + m = n - 1 + + U = dmp_add_term(u, dmp_ground(K(1), m), 0, n, K) + V = dmp_add_term(u, dmp_ground(K(2), m), 0, n, K) + + f = [[-K(3), K(0)], [], [K(1), K(0), -K(1)]] + + W = dmp_add_term(v, dmp_ground(K(1), m), 0, n, K) + Y = dmp_raise(f, m, 1, K) + + F = dmp_mul(U, V, n, K) + G = dmp_mul(W, Y, n, K) + + H = dmp_one(n, K) + + return F, G, H + + +def fateman_poly_F_2(n): + """Fateman's GCD benchmark: linearly dense quartic inputs """ + Y = [Symbol('y_' + str(i)) for i in range(n + 1)] + + y_0 = Y[0] + + u = Add(*Y[1:]) + + H = Poly((y_0 + u + 1)**2, *Y) + + F = Poly((y_0 - u - 2)**2, *Y) + G = Poly((y_0 + u + 2)**2, *Y) + + return H*F, H*G, H + + +def dmp_fateman_poly_F_2(n, K): + """Fateman's GCD benchmark: linearly dense quartic inputs """ + u = [K(1), K(0)] + + for i in range(n - 1): + u = [dmp_one(i, K), u] + + m = n - 1 + + v = dmp_add_term(u, dmp_ground(K(2), m - 1), 0, n, K) + + f = dmp_sqr([dmp_one(m, K), dmp_neg(v, m, K)], n, K) + g = dmp_sqr([dmp_one(m, K), v], n, K) + + v = dmp_add_term(u, dmp_one(m - 1, K), 0, n, K) + + h = dmp_sqr([dmp_one(m, K), v], n, K) + + return dmp_mul(f, h, n, K), dmp_mul(g, h, n, K), h + + +def fateman_poly_F_3(n): + """Fateman's GCD benchmark: sparse inputs (deg f ~ vars f) """ + Y = [Symbol('y_' + str(i)) for i in range(n + 1)] + + y_0 = Y[0] + + u = Add(*[y**(n + 1) for y in Y[1:]]) + + H = Poly((y_0**(n + 1) + u + 1)**2, *Y) + + F = Poly((y_0**(n + 1) - u - 2)**2, *Y) + G = Poly((y_0**(n + 1) + u + 2)**2, *Y) + + return H*F, H*G, H + + +def dmp_fateman_poly_F_3(n, K): + """Fateman's GCD benchmark: sparse inputs (deg f ~ vars f) """ + u = dup_from_raw_dict({n + 1: K.one}, K) + + for i in range(0, n - 1): + u = dmp_add_term([u], dmp_one(i, K), n + 1, i + 1, K) + + v = dmp_add_term(u, dmp_ground(K(2), n - 2), 0, n, K) + + f = dmp_sqr( + dmp_add_term([dmp_neg(v, n - 1, K)], dmp_one(n - 1, K), n + 1, n, K), n, K) + g = dmp_sqr(dmp_add_term([v], dmp_one(n - 1, K), n + 1, n, K), n, K) + + v = dmp_add_term(u, dmp_one(n - 2, K), 0, n - 1, K) + + h = dmp_sqr(dmp_add_term([v], dmp_one(n - 1, K), n + 1, n, K), n, K) + + return dmp_mul(f, h, n, K), dmp_mul(g, h, n, K), h + +# A few useful polynomials from Wang's paper ('78). + +from sympy.polys.rings import ring + +def _f_0(): + R, x, y, z = ring("x,y,z", ZZ) + return x**2*y*z**2 + 2*x**2*y*z + 3*x**2*y + 2*x**2 + 3*x + 4*y**2*z**2 + 5*y**2*z + 6*y**2 + y*z**2 + 2*y*z + y + 1 + +def _f_1(): + R, x, y, z = ring("x,y,z", ZZ) + return x**3*y*z + x**2*y**2*z**2 + x**2*y**2 + 20*x**2*y*z + 30*x**2*y + x**2*z**2 + 10*x**2*z + x*y**3*z + 30*x*y**2*z + 20*x*y**2 + x*y*z**3 + 10*x*y*z**2 + x*y*z + 610*x*y + 20*x*z**2 + 230*x*z + 300*x + y**2*z**2 + 10*y**2*z + 30*y*z**2 + 320*y*z + 200*y + 600*z + 6000 + +def _f_2(): + R, x, y, z = ring("x,y,z", ZZ) + return x**5*y**3 + x**5*y**2*z + x**5*y*z**2 + x**5*z**3 + x**3*y**2 + x**3*y*z + 90*x**3*y + 90*x**3*z + x**2*y**2*z - 11*x**2*y**2 + x**2*z**3 - 11*x**2*z**2 + y*z - 11*y + 90*z - 990 + +def _f_3(): + R, x, y, z = ring("x,y,z", ZZ) + return x**5*y**2 + x**4*z**4 + x**4 + x**3*y**3*z + x**3*z + x**2*y**4 + x**2*y**3*z**3 + x**2*y*z**5 + x**2*y*z + x*y**2*z**4 + x*y**2 + x*y*z**7 + x*y*z**3 + x*y*z**2 + y**2*z + y*z**4 + +def _f_4(): + R, x, y, z = ring("x,y,z", ZZ) + return -x**9*y**8*z - x**8*y**5*z**3 - x**7*y**12*z**2 - 5*x**7*y**8 - x**6*y**9*z**4 + x**6*y**7*z**3 + 3*x**6*y**7*z - 5*x**6*y**5*z**2 - x**6*y**4*z**3 + x**5*y**4*z**5 + 3*x**5*y**4*z**3 - x**5*y*z**5 + x**4*y**11*z**4 + 3*x**4*y**11*z**2 - x**4*y**8*z**4 + 5*x**4*y**7*z**2 + 15*x**4*y**7 - 5*x**4*y**4*z**2 + x**3*y**8*z**6 + 3*x**3*y**8*z**4 - x**3*y**5*z**6 + 5*x**3*y**4*z**4 + 15*x**3*y**4*z**2 + x**3*y**3*z**5 + 3*x**3*y**3*z**3 - 5*x**3*y*z**4 + x**2*z**7 + 3*x**2*z**5 + x*y**7*z**6 + 3*x*y**7*z**4 + 5*x*y**3*z**4 + 15*x*y**3*z**2 + y**4*z**8 + 3*y**4*z**6 + 5*z**6 + 15*z**4 + +def _f_5(): + R, x, y, z = ring("x,y,z", ZZ) + return -x**3 - 3*x**2*y + 3*x**2*z - 3*x*y**2 + 6*x*y*z - 3*x*z**2 - y**3 + 3*y**2*z - 3*y*z**2 + z**3 + +def _f_6(): + R, x, y, z, t = ring("x,y,z,t", ZZ) + return 2115*x**4*y + 45*x**3*z**3*t**2 - 45*x**3*t**2 - 423*x*y**4 - 47*x*y**3 + 141*x*y*z**3 + 94*x*y*z*t - 9*y**3*z**3*t**2 + 9*y**3*t**2 - y**2*z**3*t**2 + y**2*t**2 + 3*z**6*t**2 + 2*z**4*t**3 - 3*z**3*t**2 - 2*z*t**3 + +def _w_1(): + R, x, y, z = ring("x,y,z", ZZ) + return 4*x**6*y**4*z**2 + 4*x**6*y**3*z**3 - 4*x**6*y**2*z**4 - 4*x**6*y*z**5 + x**5*y**4*z**3 + 12*x**5*y**3*z - x**5*y**2*z**5 + 12*x**5*y**2*z**2 - 12*x**5*y*z**3 - 12*x**5*z**4 + 8*x**4*y**4 + 6*x**4*y**3*z**2 + 8*x**4*y**3*z - 4*x**4*y**2*z**4 + 4*x**4*y**2*z**3 - 8*x**4*y**2*z**2 - 4*x**4*y*z**5 - 2*x**4*y*z**4 - 8*x**4*y*z**3 + 2*x**3*y**4*z + x**3*y**3*z**3 - x**3*y**2*z**5 - 2*x**3*y**2*z**3 + 9*x**3*y**2*z - 12*x**3*y*z**3 + 12*x**3*y*z**2 - 12*x**3*z**4 + 3*x**3*z**3 + 6*x**2*y**3 - 6*x**2*y**2*z**2 + 8*x**2*y**2*z - 2*x**2*y*z**4 - 8*x**2*y*z**3 + 2*x**2*y*z**2 + 2*x*y**3*z - 2*x*y**2*z**3 - 3*x*y*z + 3*x*z**3 - 2*y**2 + 2*y*z**2 + +def _w_2(): + R, x, y = ring("x,y", ZZ) + return 24*x**8*y**3 + 48*x**8*y**2 + 24*x**7*y**5 - 72*x**7*y**2 + 25*x**6*y**4 + 2*x**6*y**3 + 4*x**6*y + 8*x**6 + x**5*y**6 + x**5*y**3 - 12*x**5 + x**4*y**5 - x**4*y**4 - 2*x**4*y**3 + 292*x**4*y**2 - x**3*y**6 + 3*x**3*y**3 - x**2*y**5 + 12*x**2*y**3 + 48*x**2 - 12*y**3 + +def f_polys(): + return _f_0(), _f_1(), _f_2(), _f_3(), _f_4(), _f_5(), _f_6() + +def w_polys(): + return _w_1(), _w_2() diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/sqfreetools.py b/.venv/lib/python3.12/site-packages/sympy/polys/sqfreetools.py new file mode 100644 index 0000000000000000000000000000000000000000..b2bf434cab542a42c0f7d67058e1a3c01857335d --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/sqfreetools.py @@ -0,0 +1,795 @@ +"""Square-free decomposition algorithms and related tools. """ + + +from sympy.polys.densearith import ( + dup_neg, dmp_neg, + dup_sub, dmp_sub, + dup_mul, dmp_mul, + dup_quo, dmp_quo, + dup_mul_ground, dmp_mul_ground) +from sympy.polys.densebasic import ( + dup_strip, + dup_LC, dmp_ground_LC, + dmp_zero_p, + dmp_ground, + dup_degree, dmp_degree, dmp_degree_in, dmp_degree_list, + dmp_raise, dmp_inject, + dup_convert) +from sympy.polys.densetools import ( + dup_diff, dmp_diff, dmp_diff_in, + dup_shift, dmp_shift, + dup_monic, dmp_ground_monic, + dup_primitive, dmp_ground_primitive) +from sympy.polys.euclidtools import ( + dup_inner_gcd, dmp_inner_gcd, + dup_gcd, dmp_gcd, + dmp_resultant, dmp_primitive) +from sympy.polys.galoistools import ( + gf_sqf_list, gf_sqf_part) +from sympy.polys.polyerrors import ( + MultivariatePolynomialError, + DomainError) + + +def _dup_check_degrees(f, result): + """Sanity check the degrees of a computed factorization in K[x].""" + deg = sum(k * dup_degree(fac) for (fac, k) in result) + assert deg == dup_degree(f) + + +def _dmp_check_degrees(f, u, result): + """Sanity check the degrees of a computed factorization in K[X].""" + degs = [0] * (u + 1) + for fac, k in result: + degs_fac = dmp_degree_list(fac, u) + degs = [d1 + k * d2 for d1, d2 in zip(degs, degs_fac)] + assert tuple(degs) == dmp_degree_list(f, u) + + +def dup_sqf_p(f, K): + """ + Return ``True`` if ``f`` is a square-free polynomial in ``K[x]``. + + Examples + ======== + + >>> from sympy.polys import ring, ZZ + >>> R, x = ring("x", ZZ) + + >>> R.dup_sqf_p(x**2 - 2*x + 1) + False + >>> R.dup_sqf_p(x**2 - 1) + True + + """ + if not f: + return True + else: + return not dup_degree(dup_gcd(f, dup_diff(f, 1, K), K)) + + +def dmp_sqf_p(f, u, K): + """ + Return ``True`` if ``f`` is a square-free polynomial in ``K[X]``. + + Examples + ======== + + >>> from sympy.polys import ring, ZZ + >>> R, x,y = ring("x,y", ZZ) + + >>> R.dmp_sqf_p(x**2 + 2*x*y + y**2) + False + >>> R.dmp_sqf_p(x**2 + y**2) + True + + """ + if dmp_zero_p(f, u): + return True + + for i in range(u+1): + + fp = dmp_diff_in(f, 1, i, u, K) + + if dmp_zero_p(fp, u): + continue + + gcd = dmp_gcd(f, fp, u, K) + + if dmp_degree_in(gcd, i, u) != 0: + return False + + return True + + +def dup_sqf_norm(f, K): + r""" + Find a shift of `f` in `K[x]` that has square-free norm. + + The domain `K` must be an algebraic number field `k(a)` (see :ref:`QQ(a)`). + + Returns `(s,g,r)`, such that `g(x)=f(x-sa)`, `r(x)=\text{Norm}(g(x))` and + `r` is a square-free polynomial over `k`. + + Examples + ======== + + We first create the algebraic number field `K=k(a)=\mathbb{Q}(\sqrt{3})` + and rings `K[x]` and `k[x]`: + + >>> from sympy.polys import ring, QQ + >>> from sympy import sqrt + + >>> K = QQ.algebraic_field(sqrt(3)) + >>> R, x = ring("x", K) + >>> _, X = ring("x", QQ) + + We can now find a square free norm for a shift of `f`: + + >>> f = x**2 - 1 + >>> s, g, r = R.dup_sqf_norm(f) + + The choice of shift `s` is arbitrary and the particular values returned for + `g` and `r` are determined by `s`. + + >>> s == 1 + True + >>> g == x**2 - 2*sqrt(3)*x + 2 + True + >>> r == X**4 - 8*X**2 + 4 + True + + The invariants are: + + >>> g == f.shift(-s*K.unit) + True + >>> g.norm() == r + True + >>> r.is_squarefree + True + + Explanation + =========== + + This is part of Trager's algorithm for factorizing polynomials over + algebraic number fields. In particular this function is algorithm + ``sqfr_norm`` from [Trager76]_. + + See Also + ======== + + dmp_sqf_norm: + Analogous function for multivariate polynomials over ``k(a)``. + dmp_norm: + Computes the norm of `f` directly without any shift. + dup_ext_factor: + Function implementing Trager's algorithm that uses this. + sympy.polys.polytools.sqf_norm: + High-level interface for using this function. + """ + if not K.is_Algebraic: + raise DomainError("ground domain must be algebraic") + + s, g = 0, dmp_raise(K.mod.to_list(), 1, 0, K.dom) + + while True: + h, _ = dmp_inject(f, 0, K, front=True) + r = dmp_resultant(g, h, 1, K.dom) + + if dup_sqf_p(r, K.dom): + break + else: + f, s = dup_shift(f, -K.unit, K), s + 1 + + return s, f, r + + +def _dmp_sqf_norm_shifts(f, u, K): + """Generate a sequence of candidate shifts for dmp_sqf_norm.""" + # + # We want to find a minimal shift if possible because shifting high degree + # variables can be expensive e.g. x**10 -> (x + 1)**10. We try a few easy + # cases first before the final infinite loop that is guaranteed to give + # only finitely many bad shifts (see Trager76 for proof of this in the + # univariate case). + # + + # First the trivial shift [0, 0, ...] + n = u + 1 + s0 = [0] * n + yield s0, f + + # Shift in multiples of the generator of the extension field K + a = K.unit + + # Variables of degree > 0 ordered by increasing degree + d = dmp_degree_list(f, u) + var_indices = [i for di, i in sorted(zip(d, range(u+1))) if di > 0] + + # Now try [1, 0, 0, ...], [0, 1, 0, ...] + for i in var_indices: + s1 = s0.copy() + s1[i] = 1 + a1 = [-a*s1i for s1i in s1] + f1 = dmp_shift(f, a1, u, K) + yield s1, f1 + + # Now try [1, 1, 1, ...], [2, 2, 2, ...] + j = 0 + while True: + j += 1 + sj = [j] * n + aj = [-a*j] * n + fj = dmp_shift(f, aj, u, K) + yield sj, fj + + +def dmp_sqf_norm(f, u, K): + r""" + Find a shift of ``f`` in ``K[X]`` that has square-free norm. + + The domain `K` must be an algebraic number field `k(a)` (see :ref:`QQ(a)`). + + Returns `(s,g,r)`, such that `g(x_1,x_2,\cdots)=f(x_1-s_1 a, x_2 - s_2 a, + \cdots)`, `r(x)=\text{Norm}(g(x))` and `r` is a square-free polynomial over + `k`. + + Examples + ======== + + We first create the algebraic number field `K=k(a)=\mathbb{Q}(i)` and rings + `K[x,y]` and `k[x,y]`: + + >>> from sympy.polys import ring, QQ + >>> from sympy import I + + >>> K = QQ.algebraic_field(I) + >>> R, x, y = ring("x,y", K) + >>> _, X, Y = ring("x,y", QQ) + + We can now find a square free norm for a shift of `f`: + + >>> f = x*y + y**2 + >>> s, g, r = R.dmp_sqf_norm(f) + + The choice of shifts ``s`` is arbitrary and the particular values returned + for ``g`` and ``r`` are determined by ``s``. + + >>> s + [0, 1] + >>> g == x*y - I*x + y**2 - 2*I*y - 1 + True + >>> r == X**2*Y**2 + X**2 + 2*X*Y**3 + 2*X*Y + Y**4 + 2*Y**2 + 1 + True + + The required invariants are: + + >>> g == f.shift_list([-si*K.unit for si in s]) + True + >>> g.norm() == r + True + >>> r.is_squarefree + True + + Explanation + =========== + + This is part of Trager's algorithm for factorizing polynomials over + algebraic number fields. In particular this function is a multivariate + generalization of algorithm ``sqfr_norm`` from [Trager76]_. + + See Also + ======== + + dup_sqf_norm: + Analogous function for univariate polynomials over ``k(a)``. + dmp_norm: + Computes the norm of `f` directly without any shift. + dmp_ext_factor: + Function implementing Trager's algorithm that uses this. + sympy.polys.polytools.sqf_norm: + High-level interface for using this function. + """ + if not u: + s, g, r = dup_sqf_norm(f, K) + return [s], g, r + + if not K.is_Algebraic: + raise DomainError("ground domain must be algebraic") + + g = dmp_raise(K.mod.to_list(), u + 1, 0, K.dom) + + for s, f in _dmp_sqf_norm_shifts(f, u, K): + + h, _ = dmp_inject(f, u, K, front=True) + r = dmp_resultant(g, h, u + 1, K.dom) + + if dmp_sqf_p(r, u, K.dom): + break + + return s, f, r + + +def dmp_norm(f, u, K): + r""" + Norm of ``f`` in ``K[X]``, often not square-free. + + The domain `K` must be an algebraic number field `k(a)` (see :ref:`QQ(a)`). + + Examples + ======== + + We first define the algebraic number field `K = k(a) = \mathbb{Q}(\sqrt{2})`: + + >>> from sympy import QQ, sqrt + >>> from sympy.polys.sqfreetools import dmp_norm + >>> k = QQ + >>> K = k.algebraic_field(sqrt(2)) + + We can now compute the norm of a polynomial `p` in `K[x,y]`: + + >>> p = [[K(1)], [K(1),K.unit]] # x + y + sqrt(2) + >>> N = [[k(1)], [k(2),k(0)], [k(1),k(0),k(-2)]] # x**2 + 2*x*y + y**2 - 2 + >>> dmp_norm(p, 1, K) == N + True + + In higher level functions that is: + + >>> from sympy import expand, roots, minpoly + >>> from sympy.abc import x, y + >>> from math import prod + >>> a = sqrt(2) + >>> e = (x + y + a) + >>> e.as_poly([x, y], extension=a).norm() + Poly(x**2 + 2*x*y + y**2 - 2, x, y, domain='QQ') + + This is equal to the product of the expressions `x + y + a_i` where the + `a_i` are the conjugates of `a`: + + >>> pa = minpoly(a) + >>> pa + _x**2 - 2 + >>> rs = roots(pa, multiple=True) + >>> rs + [sqrt(2), -sqrt(2)] + >>> n = prod(e.subs(a, r) for r in rs) + >>> n + (x + y - sqrt(2))*(x + y + sqrt(2)) + >>> expand(n) + x**2 + 2*x*y + y**2 - 2 + + Explanation + =========== + + Given an algebraic number field `K = k(a)` any element `b` of `K` can be + represented as polynomial function `b=g(a)` where `g` is in `k[x]`. If the + minimal polynomial of `a` over `k` is `p_a` then the roots `a_1`, `a_2`, + `\cdots` of `p_a(x)` are the conjugates of `a`. The norm of `b` is the + product `g(a1) \times g(a2) \times \cdots` and is an element of `k`. + + As in [Trager76]_ we extend this norm to multivariate polynomials over `K`. + If `b(x)` is a polynomial in `k(a)[X]` then we can think of `b` as being + alternately a function `g_X(a)` where `g_X` is an element of `k[X][y]` i.e. + a polynomial function with coefficients that are elements of `k[X]`. Then + the norm of `b` is the product `g_X(a1) \times g_X(a2) \times \cdots` and + will be an element of `k[X]`. + + See Also + ======== + + dmp_sqf_norm: + Compute a shift of `f` so that the `\text{Norm}(f)` is square-free. + sympy.polys.polytools.Poly.norm: + Higher-level function that calls this. + """ + if not K.is_Algebraic: + raise DomainError("ground domain must be algebraic") + + g = dmp_raise(K.mod.to_list(), u + 1, 0, K.dom) + h, _ = dmp_inject(f, u, K, front=True) + + return dmp_resultant(g, h, u + 1, K.dom) + + +def dup_gf_sqf_part(f, K): + """Compute square-free part of ``f`` in ``GF(p)[x]``. """ + f = dup_convert(f, K, K.dom) + g = gf_sqf_part(f, K.mod, K.dom) + return dup_convert(g, K.dom, K) + + +def dmp_gf_sqf_part(f, u, K): + """Compute square-free part of ``f`` in ``GF(p)[X]``. """ + raise NotImplementedError('multivariate polynomials over finite fields') + + +def dup_sqf_part(f, K): + """ + Returns square-free part of a polynomial in ``K[x]``. + + Examples + ======== + + >>> from sympy.polys import ring, ZZ + >>> R, x = ring("x", ZZ) + + >>> R.dup_sqf_part(x**3 - 3*x - 2) + x**2 - x - 2 + + See Also + ======== + + sympy.polys.polytools.Poly.sqf_part + """ + if K.is_FiniteField: + return dup_gf_sqf_part(f, K) + + if not f: + return f + + if K.is_negative(dup_LC(f, K)): + f = dup_neg(f, K) + + gcd = dup_gcd(f, dup_diff(f, 1, K), K) + sqf = dup_quo(f, gcd, K) + + if K.is_Field: + return dup_monic(sqf, K) + else: + return dup_primitive(sqf, K)[1] + + +def dmp_sqf_part(f, u, K): + """ + Returns square-free part of a polynomial in ``K[X]``. + + Examples + ======== + + >>> from sympy.polys import ring, ZZ + >>> R, x,y = ring("x,y", ZZ) + + >>> R.dmp_sqf_part(x**3 + 2*x**2*y + x*y**2) + x**2 + x*y + + """ + if not u: + return dup_sqf_part(f, K) + + if K.is_FiniteField: + return dmp_gf_sqf_part(f, u, K) + + if dmp_zero_p(f, u): + return f + + if K.is_negative(dmp_ground_LC(f, u, K)): + f = dmp_neg(f, u, K) + + gcd = f + for i in range(u+1): + gcd = dmp_gcd(gcd, dmp_diff_in(f, 1, i, u, K), u, K) + sqf = dmp_quo(f, gcd, u, K) + + if K.is_Field: + return dmp_ground_monic(sqf, u, K) + else: + return dmp_ground_primitive(sqf, u, K)[1] + + +def dup_gf_sqf_list(f, K, all=False): + """Compute square-free decomposition of ``f`` in ``GF(p)[x]``. """ + f_orig = f + + f = dup_convert(f, K, K.dom) + + coeff, factors = gf_sqf_list(f, K.mod, K.dom, all=all) + + for i, (f, k) in enumerate(factors): + factors[i] = (dup_convert(f, K.dom, K), k) + + _dup_check_degrees(f_orig, factors) + + return K.convert(coeff, K.dom), factors + + +def dmp_gf_sqf_list(f, u, K, all=False): + """Compute square-free decomposition of ``f`` in ``GF(p)[X]``. """ + raise NotImplementedError('multivariate polynomials over finite fields') + + +def dup_sqf_list(f, K, all=False): + """ + Return square-free decomposition of a polynomial in ``K[x]``. + + Uses Yun's algorithm from [Yun76]_. + + Examples + ======== + + >>> from sympy.polys import ring, ZZ + >>> R, x = ring("x", ZZ) + + >>> f = 2*x**5 + 16*x**4 + 50*x**3 + 76*x**2 + 56*x + 16 + + >>> R.dup_sqf_list(f) + (2, [(x + 1, 2), (x + 2, 3)]) + >>> R.dup_sqf_list(f, all=True) + (2, [(1, 1), (x + 1, 2), (x + 2, 3)]) + + See Also + ======== + + dmp_sqf_list: + Corresponding function for multivariate polynomials. + sympy.polys.polytools.sqf_list: + High-level function for square-free factorization of expressions. + sympy.polys.polytools.Poly.sqf_list: + Analogous method on :class:`~.Poly`. + + References + ========== + + [Yun76]_ + """ + if K.is_FiniteField: + return dup_gf_sqf_list(f, K, all=all) + + f_orig = f + + if K.is_Field: + coeff = dup_LC(f, K) + f = dup_monic(f, K) + else: + coeff, f = dup_primitive(f, K) + + if K.is_negative(dup_LC(f, K)): + f = dup_neg(f, K) + coeff = -coeff + + if dup_degree(f) <= 0: + return coeff, [] + + result, i = [], 1 + + h = dup_diff(f, 1, K) + g, p, q = dup_inner_gcd(f, h, K) + + while True: + d = dup_diff(p, 1, K) + h = dup_sub(q, d, K) + + if not h: + result.append((p, i)) + break + + g, p, q = dup_inner_gcd(p, h, K) + + if all or dup_degree(g) > 0: + result.append((g, i)) + + i += 1 + + _dup_check_degrees(f_orig, result) + + return coeff, result + + +def dup_sqf_list_include(f, K, all=False): + """ + Return square-free decomposition of a polynomial in ``K[x]``. + + Examples + ======== + + >>> from sympy.polys import ring, ZZ + >>> R, x = ring("x", ZZ) + + >>> f = 2*x**5 + 16*x**4 + 50*x**3 + 76*x**2 + 56*x + 16 + + >>> R.dup_sqf_list_include(f) + [(2, 1), (x + 1, 2), (x + 2, 3)] + >>> R.dup_sqf_list_include(f, all=True) + [(2, 1), (x + 1, 2), (x + 2, 3)] + + """ + coeff, factors = dup_sqf_list(f, K, all=all) + + if factors and factors[0][1] == 1: + g = dup_mul_ground(factors[0][0], coeff, K) + return [(g, 1)] + factors[1:] + else: + g = dup_strip([coeff]) + return [(g, 1)] + factors + + +def dmp_sqf_list(f, u, K, all=False): + """ + Return square-free decomposition of a polynomial in `K[X]`. + + Examples + ======== + + >>> from sympy.polys import ring, ZZ + >>> R, x,y = ring("x,y", ZZ) + + >>> f = x**5 + 2*x**4*y + x**3*y**2 + + >>> R.dmp_sqf_list(f) + (1, [(x + y, 2), (x, 3)]) + >>> R.dmp_sqf_list(f, all=True) + (1, [(1, 1), (x + y, 2), (x, 3)]) + + Explanation + =========== + + Uses Yun's algorithm for univariate polynomials from [Yun76]_ recursively. + The multivariate polynomial is treated as a univariate polynomial in its + leading variable. Then Yun's algorithm computes the square-free + factorization of the primitive and the content is factored recursively. + + It would be better to use a dedicated algorithm for multivariate + polynomials instead. + + See Also + ======== + + dup_sqf_list: + Corresponding function for univariate polynomials. + sympy.polys.polytools.sqf_list: + High-level function for square-free factorization of expressions. + sympy.polys.polytools.Poly.sqf_list: + Analogous method on :class:`~.Poly`. + """ + if not u: + return dup_sqf_list(f, K, all=all) + + if K.is_FiniteField: + return dmp_gf_sqf_list(f, u, K, all=all) + + f_orig = f + + if K.is_Field: + coeff = dmp_ground_LC(f, u, K) + f = dmp_ground_monic(f, u, K) + else: + coeff, f = dmp_ground_primitive(f, u, K) + + if K.is_negative(dmp_ground_LC(f, u, K)): + f = dmp_neg(f, u, K) + coeff = -coeff + + deg = dmp_degree(f, u) + if deg < 0: + return coeff, [] + + # Yun's algorithm requires the polynomial to be primitive as a univariate + # polynomial in its main variable. + content, f = dmp_primitive(f, u, K) + + result = {} + + if deg != 0: + + h = dmp_diff(f, 1, u, K) + g, p, q = dmp_inner_gcd(f, h, u, K) + + i = 1 + + while True: + d = dmp_diff(p, 1, u, K) + h = dmp_sub(q, d, u, K) + + if dmp_zero_p(h, u): + result[i] = p + break + + g, p, q = dmp_inner_gcd(p, h, u, K) + + if all or dmp_degree(g, u) > 0: + result[i] = g + + i += 1 + + coeff_content, result_content = dmp_sqf_list(content, u-1, K, all=all) + + coeff *= coeff_content + + # Combine factors of the content and primitive part that have the same + # multiplicity to produce a list in ascending order of multiplicity. + for fac, i in result_content: + fac = [fac] + if i in result: + result[i] = dmp_mul(result[i], fac, u, K) + else: + result[i] = fac + + result = [(result[i], i) for i in sorted(result)] + + _dmp_check_degrees(f_orig, u, result) + + return coeff, result + + +def dmp_sqf_list_include(f, u, K, all=False): + """ + Return square-free decomposition of a polynomial in ``K[x]``. + + Examples + ======== + + >>> from sympy.polys import ring, ZZ + >>> R, x,y = ring("x,y", ZZ) + + >>> f = x**5 + 2*x**4*y + x**3*y**2 + + >>> R.dmp_sqf_list_include(f) + [(1, 1), (x + y, 2), (x, 3)] + >>> R.dmp_sqf_list_include(f, all=True) + [(1, 1), (x + y, 2), (x, 3)] + + """ + if not u: + return dup_sqf_list_include(f, K, all=all) + + coeff, factors = dmp_sqf_list(f, u, K, all=all) + + if factors and factors[0][1] == 1: + g = dmp_mul_ground(factors[0][0], coeff, u, K) + return [(g, 1)] + factors[1:] + else: + g = dmp_ground(coeff, u) + return [(g, 1)] + factors + + +def dup_gff_list(f, K): + """ + Compute greatest factorial factorization of ``f`` in ``K[x]``. + + Examples + ======== + + >>> from sympy.polys import ring, ZZ + >>> R, x = ring("x", ZZ) + + >>> R.dup_gff_list(x**5 + 2*x**4 - x**3 - 2*x**2) + [(x, 1), (x + 2, 4)] + + """ + if not f: + raise ValueError("greatest factorial factorization doesn't exist for a zero polynomial") + + f = dup_monic(f, K) + + if not dup_degree(f): + return [] + else: + g = dup_gcd(f, dup_shift(f, K.one, K), K) + H = dup_gff_list(g, K) + + for i, (h, k) in enumerate(H): + g = dup_mul(g, dup_shift(h, -K(k), K), K) + H[i] = (h, k + 1) + + f = dup_quo(f, g, K) + + if not dup_degree(f): + return H + else: + return [(f, 1)] + H + + +def dmp_gff_list(f, u, K): + """ + Compute greatest factorial factorization of ``f`` in ``K[X]``. + + Examples + ======== + + >>> from sympy.polys import ring, ZZ + >>> R, x,y = ring("x,y", ZZ) + + """ + if not u: + return dup_gff_list(f, K) + else: + raise MultivariatePolynomialError(f) diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/subresultants_qq_zz.py b/.venv/lib/python3.12/site-packages/sympy/polys/subresultants_qq_zz.py new file mode 100644 index 0000000000000000000000000000000000000000..9ce8d5c88d44022621d13d8e82956e676a7e75ae --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/subresultants_qq_zz.py @@ -0,0 +1,2558 @@ +""" +This module contains functions for the computation +of Euclidean, (generalized) Sturmian, (modified) subresultant +polynomial remainder sequences (prs's) of two polynomials; +included are also three functions for the computation of the +resultant of two polynomials. + +Except for the function res_z(), which computes the resultant +of two polynomials, the pseudo-remainder function prem() +of sympy is _not_ used by any of the functions in the module. + +Instead of prem() we use the function + +rem_z(). + +Included is also the function quo_z(). + +An explanation of why we avoid prem() can be found in the +references stated in the docstring of rem_z(). + +1. Theoretical background: +========================== +Consider the polynomials f, g in Z[x] of degrees deg(f) = n and +deg(g) = m with n >= m. + +Definition 1: +============= +The sign sequence of a polynomial remainder sequence (prs) is the +sequence of signs of the leading coefficients of its polynomials. + +Sign sequences can be computed with the function: + +sign_seq(poly_seq, x) + +Definition 2: +============= +A polynomial remainder sequence (prs) is called complete if the +degree difference between any two consecutive polynomials is 1; +otherwise, it called incomplete. + +It is understood that f, g belong to the sequences mentioned in +the two definitions above. + +1A. Euclidean and subresultant prs's: +===================================== +The subresultant prs of f, g is a sequence of polynomials in Z[x] +analogous to the Euclidean prs, the sequence obtained by applying +on f, g Euclid's algorithm for polynomial greatest common divisors +(gcd) in Q[x]. + +The subresultant prs differs from the Euclidean prs in that the +coefficients of each polynomial in the former sequence are determinants +--- also referred to as subresultants --- of appropriately selected +sub-matrices of sylvester1(f, g, x), Sylvester's matrix of 1840 of +dimensions (n + m) * (n + m). + +Recall that the determinant of sylvester1(f, g, x) itself is +called the resultant of f, g and serves as a criterion of whether +the two polynomials have common roots or not. + +In SymPy the resultant is computed with the function +resultant(f, g, x). This function does _not_ evaluate the +determinant of sylvester(f, g, x, 1); instead, it returns +the last member of the subresultant prs of f, g, multiplied +(if needed) by an appropriate power of -1; see the caveat below. + +In this module we use three functions to compute the +resultant of f, g: +a) res(f, g, x) computes the resultant by evaluating +the determinant of sylvester(f, g, x, 1); +b) res_q(f, g, x) computes the resultant recursively, by +performing polynomial divisions in Q[x] with the function rem(); +c) res_z(f, g, x) computes the resultant recursively, by +performing polynomial divisions in Z[x] with the function prem(). + +Caveat: If Df = degree(f, x) and Dg = degree(g, x), then: + +resultant(f, g, x) = (-1)**(Df*Dg) * resultant(g, f, x). + +For complete prs's the sign sequence of the Euclidean prs of f, g +is identical to the sign sequence of the subresultant prs of f, g +and the coefficients of one sequence are easily computed from the +coefficients of the other. + +For incomplete prs's the polynomials in the subresultant prs, generally +differ in sign from those of the Euclidean prs, and --- unlike the +case of complete prs's --- it is not at all obvious how to compute +the coefficients of one sequence from the coefficients of the other. + +1B. Sturmian and modified subresultant prs's: +============================================= +For the same polynomials f, g in Z[x] mentioned above, their ``modified'' +subresultant prs is a sequence of polynomials similar to the Sturmian +prs, the sequence obtained by applying in Q[x] Sturm's algorithm on f, g. + +The two sequences differ in that the coefficients of each polynomial +in the modified subresultant prs are the determinants --- also referred +to as modified subresultants --- of appropriately selected sub-matrices +of sylvester2(f, g, x), Sylvester's matrix of 1853 of dimensions 2n x 2n. + +The determinant of sylvester2 itself is called the modified resultant +of f, g and it also can serve as a criterion of whether the two +polynomials have common roots or not. + +For complete prs's the sign sequence of the Sturmian prs of f, g is +identical to the sign sequence of the modified subresultant prs of +f, g and the coefficients of one sequence are easily computed from +the coefficients of the other. + +For incomplete prs's the polynomials in the modified subresultant prs, +generally differ in sign from those of the Sturmian prs, and --- unlike +the case of complete prs's --- it is not at all obvious how to compute +the coefficients of one sequence from the coefficients of the other. + +As Sylvester pointed out, the coefficients of the polynomial remainders +obtained as (modified) subresultants are the smallest possible without +introducing rationals and without computing (integer) greatest common +divisors. + +1C. On terminology: +=================== +Whence the terminology? Well generalized Sturmian prs's are +``modifications'' of Euclidean prs's; the hint came from the title +of the Pell-Gordon paper of 1917. + +In the literature one also encounters the name ``non signed'' and +``signed'' prs for Euclidean and Sturmian prs respectively. + +Likewise ``non signed'' and ``signed'' subresultant prs for +subresultant and modified subresultant prs respectively. + +2. Functions in the module: +=========================== +No function utilizes SymPy's function prem(). + +2A. Matrices: +============= +The functions sylvester(f, g, x, method=1) and +sylvester(f, g, x, method=2) compute either Sylvester matrix. +They can be used to compute (modified) subresultant prs's by +direct determinant evaluation. + +The function bezout(f, g, x, method='prs') provides a matrix of +smaller dimensions than either Sylvester matrix. It is the function +of choice for computing (modified) subresultant prs's by direct +determinant evaluation. + +sylvester(f, g, x, method=1) +sylvester(f, g, x, method=2) +bezout(f, g, x, method='prs') + +The following identity holds: + +bezout(f, g, x, method='prs') = +backward_eye(deg(f))*bezout(f, g, x, method='bz')*backward_eye(deg(f)) + +2B. Subresultant and modified subresultant prs's by +=================================================== +determinant evaluations: +======================= +We use the Sylvester matrices of 1840 and 1853 to +compute, respectively, subresultant and modified +subresultant polynomial remainder sequences. However, +for large matrices this approach takes a lot of time. + +Instead of utilizing the Sylvester matrices, we can +employ the Bezout matrix which is of smaller dimensions. + +subresultants_sylv(f, g, x) +modified_subresultants_sylv(f, g, x) +subresultants_bezout(f, g, x) +modified_subresultants_bezout(f, g, x) + +2C. Subresultant prs's by ONE determinant evaluation: +===================================================== +All three functions in this section evaluate one determinant +per remainder polynomial; this is the determinant of an +appropriately selected sub-matrix of sylvester1(f, g, x), +Sylvester's matrix of 1840. + +To compute the remainder polynomials the function +subresultants_rem(f, g, x) employs rem(f, g, x). +By contrast, the other two functions implement Van Vleck's ideas +of 1900 and compute the remainder polynomials by trinagularizing +sylvester2(f, g, x), Sylvester's matrix of 1853. + + +subresultants_rem(f, g, x) +subresultants_vv(f, g, x) +subresultants_vv_2(f, g, x). + +2E. Euclidean, Sturmian prs's in Q[x]: +====================================== +euclid_q(f, g, x) +sturm_q(f, g, x) + +2F. Euclidean, Sturmian and (modified) subresultant prs's P-G: +============================================================== +All functions in this section are based on the Pell-Gordon (P-G) +theorem of 1917. +Computations are done in Q[x], employing the function rem(f, g, x) +for the computation of the remainder polynomials. + +euclid_pg(f, g, x) +sturm pg(f, g, x) +subresultants_pg(f, g, x) +modified_subresultants_pg(f, g, x) + +2G. Euclidean, Sturmian and (modified) subresultant prs's A-M-V: +================================================================ +All functions in this section are based on the Akritas-Malaschonok- +Vigklas (A-M-V) theorem of 2015. +Computations are done in Z[x], employing the function rem_z(f, g, x) +for the computation of the remainder polynomials. + +euclid_amv(f, g, x) +sturm_amv(f, g, x) +subresultants_amv(f, g, x) +modified_subresultants_amv(f, g, x) + +2Ga. Exception: +=============== +subresultants_amv_q(f, g, x) + +This function employs rem(f, g, x) for the computation of +the remainder polynomials, despite the fact that it implements +the A-M-V Theorem. + +It is included in our module in order to show that theorems P-G +and A-M-V can be implemented utilizing either the function +rem(f, g, x) or the function rem_z(f, g, x). + +For clearly historical reasons --- since the Collins-Brown-Traub +coefficients-reduction factor beta_i was not available in 1917 --- +we have implemented the Pell-Gordon theorem with the function +rem(f, g, x) and the A-M-V Theorem with the function rem_z(f, g, x). + +2H. Resultants: +=============== +res(f, g, x) +res_q(f, g, x) +res_z(f, g, x) +""" + + +from sympy.concrete.summations import summation +from sympy.core.function import expand +from sympy.core.numbers import nan +from sympy.core.singleton import S +from sympy.core.symbol import Dummy as var +from sympy.functions.elementary.complexes import Abs, sign +from sympy.functions.elementary.integers import floor +from sympy.matrices.dense import eye, Matrix, zeros +from sympy.printing.pretty.pretty import pretty_print as pprint +from sympy.simplify.simplify import simplify +from sympy.polys.domains import QQ +from sympy.polys.polytools import degree, LC, Poly, pquo, quo, prem, rem +from sympy.polys.polyerrors import PolynomialError + + +def sylvester(f, g, x, method = 1): + ''' + The input polynomials f, g are in Z[x] or in Q[x]. Let m = degree(f, x), + n = degree(g, x) and mx = max(m, n). + + a. If method = 1 (default), computes sylvester1, Sylvester's matrix of 1840 + of dimension (m + n) x (m + n). The determinants of properly chosen + submatrices of this matrix (a.k.a. subresultants) can be + used to compute the coefficients of the Euclidean PRS of f, g. + + b. If method = 2, computes sylvester2, Sylvester's matrix of 1853 + of dimension (2*mx) x (2*mx). The determinants of properly chosen + submatrices of this matrix (a.k.a. ``modified'' subresultants) can be + used to compute the coefficients of the Sturmian PRS of f, g. + + Applications of these Matrices can be found in the references below. + Especially, for applications of sylvester2, see the first reference!! + + References + ========== + 1. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``On a Theorem + by Van Vleck Regarding Sturm Sequences. Serdica Journal of Computing, + Vol. 7, No 4, 101-134, 2013. + + 2. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``Sturm Sequences + and Modified Subresultant Polynomial Remainder Sequences.'' + Serdica Journal of Computing, Vol. 8, No 1, 29-46, 2014. + + ''' + # obtain degrees of polys + m, n = degree( Poly(f, x), x), degree( Poly(g, x), x) + + # Special cases: + # A:: case m = n < 0 (i.e. both polys are 0) + if m == n and n < 0: + return Matrix([]) + + # B:: case m = n = 0 (i.e. both polys are constants) + if m == n and n == 0: + return Matrix([]) + + # C:: m == 0 and n < 0 or m < 0 and n == 0 + # (i.e. one poly is constant and the other is 0) + if m == 0 and n < 0: + return Matrix([]) + elif m < 0 and n == 0: + return Matrix([]) + + # D:: m >= 1 and n < 0 or m < 0 and n >=1 + # (i.e. one poly is of degree >=1 and the other is 0) + if m >= 1 and n < 0: + return Matrix([0]) + elif m < 0 and n >= 1: + return Matrix([0]) + + fp = Poly(f, x).all_coeffs() + gp = Poly(g, x).all_coeffs() + + # Sylvester's matrix of 1840 (default; a.k.a. sylvester1) + if method <= 1: + M = zeros(m + n) + k = 0 + for i in range(n): + j = k + for coeff in fp: + M[i, j] = coeff + j = j + 1 + k = k + 1 + k = 0 + for i in range(n, m + n): + j = k + for coeff in gp: + M[i, j] = coeff + j = j + 1 + k = k + 1 + return M + + # Sylvester's matrix of 1853 (a.k.a sylvester2) + else: + if len(fp) < len(gp): + h = [] + for i in range(len(gp) - len(fp)): + h.append(0) + fp[ : 0] = h + else: + h = [] + for i in range(len(fp) - len(gp)): + h.append(0) + gp[ : 0] = h + mx = max(m, n) + dim = 2*mx + M = zeros( dim ) + k = 0 + for i in range( mx ): + j = k + for coeff in fp: + M[2*i, j] = coeff + j = j + 1 + j = k + for coeff in gp: + M[2*i + 1, j] = coeff + j = j + 1 + k = k + 1 + return M + +def process_matrix_output(poly_seq, x): + """ + poly_seq is a polynomial remainder sequence computed either by + (modified_)subresultants_bezout or by (modified_)subresultants_sylv. + + This function removes from poly_seq all zero polynomials as well + as all those whose degree is equal to the degree of a preceding + polynomial in poly_seq, as we scan it from left to right. + + """ + L = poly_seq[:] # get a copy of the input sequence + d = degree(L[1], x) + i = 2 + while i < len(L): + d_i = degree(L[i], x) + if d_i < 0: # zero poly + L.remove(L[i]) + i = i - 1 + if d == d_i: # poly degree equals degree of previous poly + L.remove(L[i]) + i = i - 1 + if d_i >= 0: + d = d_i + i = i + 1 + + return L + +def subresultants_sylv(f, g, x): + """ + The input polynomials f, g are in Z[x] or in Q[x]. It is assumed + that deg(f) >= deg(g). + + Computes the subresultant polynomial remainder sequence (prs) + of f, g by evaluating determinants of appropriately selected + submatrices of sylvester(f, g, x, 1). The dimensions of the + latter are (deg(f) + deg(g)) x (deg(f) + deg(g)). + + Each coefficient is computed by evaluating the determinant of the + corresponding submatrix of sylvester(f, g, x, 1). + + If the subresultant prs is complete, then the output coincides + with the Euclidean sequence of the polynomials f, g. + + References: + =========== + 1. G.M.Diaz-Toca,L.Gonzalez-Vega: Various New Expressions for Subresultants + and Their Applications. Appl. Algebra in Engin., Communic. and Comp., + Vol. 15, 233-266, 2004. + + """ + + # make sure neither f nor g is 0 + if f == 0 or g == 0: + return [f, g] + + n = degF = degree(f, x) + m = degG = degree(g, x) + + # make sure proper degrees + if n == 0 and m == 0: + return [f, g] + if n < m: + n, m, degF, degG, f, g = m, n, degG, degF, g, f + if n > 0 and m == 0: + return [f, g] + + SR_L = [f, g] # subresultant list + + # form matrix sylvester(f, g, x, 1) + S = sylvester(f, g, x, 1) + + # pick appropriate submatrices of S + # and form subresultant polys + j = m - 1 + + while j > 0: + Sp = S[:, :] # copy of S + # delete last j rows of coeffs of g + for ind in range(m + n - j, m + n): + Sp.row_del(m + n - j) + # delete last j rows of coeffs of f + for ind in range(m - j, m): + Sp.row_del(m - j) + + # evaluate determinants and form coefficients list + coeff_L, k, l = [], Sp.rows, 0 + while l <= j: + coeff_L.append(Sp[:, 0:k].det()) + Sp.col_swap(k - 1, k + l) + l += 1 + + # form poly and append to SP_L + SR_L.append(Poly(coeff_L, x).as_expr()) + j -= 1 + + # j = 0 + SR_L.append(S.det()) + + return process_matrix_output(SR_L, x) + +def modified_subresultants_sylv(f, g, x): + """ + The input polynomials f, g are in Z[x] or in Q[x]. It is assumed + that deg(f) >= deg(g). + + Computes the modified subresultant polynomial remainder sequence (prs) + of f, g by evaluating determinants of appropriately selected + submatrices of sylvester(f, g, x, 2). The dimensions of the + latter are (2*deg(f)) x (2*deg(f)). + + Each coefficient is computed by evaluating the determinant of the + corresponding submatrix of sylvester(f, g, x, 2). + + If the modified subresultant prs is complete, then the output coincides + with the Sturmian sequence of the polynomials f, g. + + References: + =========== + 1. A. G. Akritas,G.I. Malaschonok and P.S. Vigklas: + Sturm Sequences and Modified Subresultant Polynomial Remainder + Sequences. Serdica Journal of Computing, Vol. 8, No 1, 29--46, 2014. + + """ + + # make sure neither f nor g is 0 + if f == 0 or g == 0: + return [f, g] + + n = degF = degree(f, x) + m = degG = degree(g, x) + + # make sure proper degrees + if n == 0 and m == 0: + return [f, g] + if n < m: + n, m, degF, degG, f, g = m, n, degG, degF, g, f + if n > 0 and m == 0: + return [f, g] + + SR_L = [f, g] # modified subresultant list + + # form matrix sylvester(f, g, x, 2) + S = sylvester(f, g, x, 2) + + # pick appropriate submatrices of S + # and form modified subresultant polys + j = m - 1 + + while j > 0: + # delete last 2*j rows of pairs of coeffs of f, g + Sp = S[0:2*n - 2*j, :] # copy of first 2*n - 2*j rows of S + + # evaluate determinants and form coefficients list + coeff_L, k, l = [], Sp.rows, 0 + while l <= j: + coeff_L.append(Sp[:, 0:k].det()) + Sp.col_swap(k - 1, k + l) + l += 1 + + # form poly and append to SP_L + SR_L.append(Poly(coeff_L, x).as_expr()) + j -= 1 + + # j = 0 + SR_L.append(S.det()) + + return process_matrix_output(SR_L, x) + +def res(f, g, x): + """ + The input polynomials f, g are in Z[x] or in Q[x]. + + The output is the resultant of f, g computed by evaluating + the determinant of the matrix sylvester(f, g, x, 1). + + References: + =========== + 1. J. S. Cohen: Computer Algebra and Symbolic Computation + - Mathematical Methods. A. K. Peters, 2003. + + """ + if f == 0 or g == 0: + raise PolynomialError("The resultant of %s and %s is not defined" % (f, g)) + else: + return sylvester(f, g, x, 1).det() + +def res_q(f, g, x): + """ + The input polynomials f, g are in Z[x] or in Q[x]. + + The output is the resultant of f, g computed recursively + by polynomial divisions in Q[x], using the function rem. + See Cohen's book p. 281. + + References: + =========== + 1. J. S. Cohen: Computer Algebra and Symbolic Computation + - Mathematical Methods. A. K. Peters, 2003. + """ + m = degree(f, x) + n = degree(g, x) + if m < n: + return (-1)**(m*n) * res_q(g, f, x) + elif n == 0: # g is a constant + return g**m + else: + r = rem(f, g, x) + if r == 0: + return 0 + else: + s = degree(r, x) + l = LC(g, x) + return (-1)**(m*n) * l**(m-s)*res_q(g, r, x) + +def res_z(f, g, x): + """ + The input polynomials f, g are in Z[x] or in Q[x]. + + The output is the resultant of f, g computed recursively + by polynomial divisions in Z[x], using the function prem(). + See Cohen's book p. 283. + + References: + =========== + 1. J. S. Cohen: Computer Algebra and Symbolic Computation + - Mathematical Methods. A. K. Peters, 2003. + """ + m = degree(f, x) + n = degree(g, x) + if m < n: + return (-1)**(m*n) * res_z(g, f, x) + elif n == 0: # g is a constant + return g**m + else: + r = prem(f, g, x) + if r == 0: + return 0 + else: + delta = m - n + 1 + w = (-1)**(m*n) * res_z(g, r, x) + s = degree(r, x) + l = LC(g, x) + k = delta * n - m + s + return quo(w, l**k, x) + +def sign_seq(poly_seq, x): + """ + Given a sequence of polynomials poly_seq, it returns + the sequence of signs of the leading coefficients of + the polynomials in poly_seq. + + """ + return [sign(LC(poly_seq[i], x)) for i in range(len(poly_seq))] + +def bezout(p, q, x, method='bz'): + """ + The input polynomials p, q are in Z[x] or in Q[x]. Let + mx = max(degree(p, x), degree(q, x)). + + The default option bezout(p, q, x, method='bz') returns Bezout's + symmetric matrix of p and q, of dimensions (mx) x (mx). The + determinant of this matrix is equal to the determinant of sylvester2, + Sylvester's matrix of 1853, whose dimensions are (2*mx) x (2*mx); + however the subresultants of these two matrices may differ. + + The other option, bezout(p, q, x, 'prs'), is of interest to us + in this module because it returns a matrix equivalent to sylvester2. + In this case all subresultants of the two matrices are identical. + + Both the subresultant polynomial remainder sequence (prs) and + the modified subresultant prs of p and q can be computed by + evaluating determinants of appropriately selected submatrices of + bezout(p, q, x, 'prs') --- one determinant per coefficient of the + remainder polynomials. + + The matrices bezout(p, q, x, 'bz') and bezout(p, q, x, 'prs') + are related by the formula + + bezout(p, q, x, 'prs') = + backward_eye(deg(p)) * bezout(p, q, x, 'bz') * backward_eye(deg(p)), + + where backward_eye() is the backward identity function. + + References + ========== + 1. G.M.Diaz-Toca,L.Gonzalez-Vega: Various New Expressions for Subresultants + and Their Applications. Appl. Algebra in Engin., Communic. and Comp., + Vol. 15, 233-266, 2004. + + """ + # obtain degrees of polys + m, n = degree( Poly(p, x), x), degree( Poly(q, x), x) + + # Special cases: + # A:: case m = n < 0 (i.e. both polys are 0) + if m == n and n < 0: + return Matrix([]) + + # B:: case m = n = 0 (i.e. both polys are constants) + if m == n and n == 0: + return Matrix([]) + + # C:: m == 0 and n < 0 or m < 0 and n == 0 + # (i.e. one poly is constant and the other is 0) + if m == 0 and n < 0: + return Matrix([]) + elif m < 0 and n == 0: + return Matrix([]) + + # D:: m >= 1 and n < 0 or m < 0 and n >=1 + # (i.e. one poly is of degree >=1 and the other is 0) + if m >= 1 and n < 0: + return Matrix([0]) + elif m < 0 and n >= 1: + return Matrix([0]) + + y = var('y') + + # expr is 0 when x = y + expr = p * q.subs({x:y}) - p.subs({x:y}) * q + + # hence expr is exactly divisible by x - y + poly = Poly( quo(expr, x-y), x, y) + + # form Bezout matrix and store them in B as indicated to get + # the LC coefficient of each poly either in the first position + # of each row (method='prs') or in the last (method='bz'). + mx = max(m, n) + B = zeros(mx) + for i in range(mx): + for j in range(mx): + if method == 'prs': + B[mx - 1 - i, mx - 1 - j] = poly.nth(i, j) + else: + B[i, j] = poly.nth(i, j) + return B + +def backward_eye(n): + ''' + Returns the backward identity matrix of dimensions n x n. + + Needed to "turn" the Bezout matrices + so that the leading coefficients are first. + See docstring of the function bezout(p, q, x, method='bz'). + ''' + M = eye(n) # identity matrix of order n + + for i in range(int(M.rows / 2)): + M.row_swap(0 + i, M.rows - 1 - i) + + return M + +def subresultants_bezout(p, q, x): + """ + The input polynomials p, q are in Z[x] or in Q[x]. It is assumed + that degree(p, x) >= degree(q, x). + + Computes the subresultant polynomial remainder sequence + of p, q by evaluating determinants of appropriately selected + submatrices of bezout(p, q, x, 'prs'). The dimensions of the + latter are deg(p) x deg(p). + + Each coefficient is computed by evaluating the determinant of the + corresponding submatrix of bezout(p, q, x, 'prs'). + + bezout(p, q, x, 'prs) is used instead of sylvester(p, q, x, 1), + Sylvester's matrix of 1840, because the dimensions of the latter + are (deg(p) + deg(q)) x (deg(p) + deg(q)). + + If the subresultant prs is complete, then the output coincides + with the Euclidean sequence of the polynomials p, q. + + References + ========== + 1. G.M.Diaz-Toca,L.Gonzalez-Vega: Various New Expressions for Subresultants + and Their Applications. Appl. Algebra in Engin., Communic. and Comp., + Vol. 15, 233-266, 2004. + + """ + # make sure neither p nor q is 0 + if p == 0 or q == 0: + return [p, q] + + f, g = p, q + n = degF = degree(f, x) + m = degG = degree(g, x) + + # make sure proper degrees + if n == 0 and m == 0: + return [f, g] + if n < m: + n, m, degF, degG, f, g = m, n, degG, degF, g, f + if n > 0 and m == 0: + return [f, g] + + SR_L = [f, g] # subresultant list + F = LC(f, x)**(degF - degG) + + # form the bezout matrix + B = bezout(f, g, x, 'prs') + + # pick appropriate submatrices of B + # and form subresultant polys + if degF > degG: + j = 2 + if degF == degG: + j = 1 + while j <= degF: + M = B[0:j, :] + k, coeff_L = j - 1, [] + while k <= degF - 1: + coeff_L.append(M[:, 0:j].det()) + if k < degF - 1: + M.col_swap(j - 1, k + 1) + k = k + 1 + + # apply Theorem 2.1 in the paper by Toca & Vega 2004 + # to get correct signs + SR_L.append(int((-1)**(j*(j-1)/2)) * (Poly(coeff_L, x) / F).as_expr()) + j = j + 1 + + return process_matrix_output(SR_L, x) + +def modified_subresultants_bezout(p, q, x): + """ + The input polynomials p, q are in Z[x] or in Q[x]. It is assumed + that degree(p, x) >= degree(q, x). + + Computes the modified subresultant polynomial remainder sequence + of p, q by evaluating determinants of appropriately selected + submatrices of bezout(p, q, x, 'prs'). The dimensions of the + latter are deg(p) x deg(p). + + Each coefficient is computed by evaluating the determinant of the + corresponding submatrix of bezout(p, q, x, 'prs'). + + bezout(p, q, x, 'prs') is used instead of sylvester(p, q, x, 2), + Sylvester's matrix of 1853, because the dimensions of the latter + are 2*deg(p) x 2*deg(p). + + If the modified subresultant prs is complete, and LC( p ) > 0, the output + coincides with the (generalized) Sturm's sequence of the polynomials p, q. + + References + ========== + 1. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``Sturm Sequences + and Modified Subresultant Polynomial Remainder Sequences.'' + Serdica Journal of Computing, Vol. 8, No 1, 29-46, 2014. + + 2. G.M.Diaz-Toca,L.Gonzalez-Vega: Various New Expressions for Subresultants + and Their Applications. Appl. Algebra in Engin., Communic. and Comp., + Vol. 15, 233-266, 2004. + + + """ + # make sure neither p nor q is 0 + if p == 0 or q == 0: + return [p, q] + + f, g = p, q + n = degF = degree(f, x) + m = degG = degree(g, x) + + # make sure proper degrees + if n == 0 and m == 0: + return [f, g] + if n < m: + n, m, degF, degG, f, g = m, n, degG, degF, g, f + if n > 0 and m == 0: + return [f, g] + + SR_L = [f, g] # subresultant list + + # form the bezout matrix + B = bezout(f, g, x, 'prs') + + # pick appropriate submatrices of B + # and form subresultant polys + if degF > degG: + j = 2 + if degF == degG: + j = 1 + while j <= degF: + M = B[0:j, :] + k, coeff_L = j - 1, [] + while k <= degF - 1: + coeff_L.append(M[:, 0:j].det()) + if k < degF - 1: + M.col_swap(j - 1, k + 1) + k = k + 1 + + ## Theorem 2.1 in the paper by Toca & Vega 2004 is _not needed_ + ## in this case since + ## the bezout matrix is equivalent to sylvester2 + SR_L.append(( Poly(coeff_L, x)).as_expr()) + j = j + 1 + + return process_matrix_output(SR_L, x) + +def sturm_pg(p, q, x, method=0): + """ + p, q are polynomials in Z[x] or Q[x]. It is assumed + that degree(p, x) >= degree(q, x). + + Computes the (generalized) Sturm sequence of p and q in Z[x] or Q[x]. + If q = diff(p, x, 1) it is the usual Sturm sequence. + + A. If method == 0, default, the remainder coefficients of the sequence + are (in absolute value) ``modified'' subresultants, which for non-monic + polynomials are greater than the coefficients of the corresponding + subresultants by the factor Abs(LC(p)**( deg(p)- deg(q))). + + B. If method == 1, the remainder coefficients of the sequence are (in + absolute value) subresultants, which for non-monic polynomials are + smaller than the coefficients of the corresponding ``modified'' + subresultants by the factor Abs(LC(p)**( deg(p)- deg(q))). + + If the Sturm sequence is complete, method=0 and LC( p ) > 0, the coefficients + of the polynomials in the sequence are ``modified'' subresultants. + That is, they are determinants of appropriately selected submatrices of + sylvester2, Sylvester's matrix of 1853. In this case the Sturm sequence + coincides with the ``modified'' subresultant prs, of the polynomials + p, q. + + If the Sturm sequence is incomplete and method=0 then the signs of the + coefficients of the polynomials in the sequence may differ from the signs + of the coefficients of the corresponding polynomials in the ``modified'' + subresultant prs; however, the absolute values are the same. + + To compute the coefficients, no determinant evaluation takes place. Instead, + polynomial divisions in Q[x] are performed, using the function rem(p, q, x); + the coefficients of the remainders computed this way become (``modified'') + subresultants with the help of the Pell-Gordon Theorem of 1917. + See also the function euclid_pg(p, q, x). + + References + ========== + 1. Pell A. J., R. L. Gordon. The Modified Remainders Obtained in Finding + the Highest Common Factor of Two Polynomials. Annals of MatheMatics, + Second Series, 18 (1917), No. 4, 188-193. + + 2. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``Sturm Sequences + and Modified Subresultant Polynomial Remainder Sequences.'' + Serdica Journal of Computing, Vol. 8, No 1, 29-46, 2014. + + """ + # make sure neither p nor q is 0 + if p == 0 or q == 0: + return [p, q] + + # make sure proper degrees + d0 = degree(p, x) + d1 = degree(q, x) + if d0 == 0 and d1 == 0: + return [p, q] + if d1 > d0: + d0, d1 = d1, d0 + p, q = q, p + if d0 > 0 and d1 == 0: + return [p,q] + + # make sure LC(p) > 0 + flag = 0 + if LC(p,x) < 0: + flag = 1 + p = -p + q = -q + + # initialize + lcf = LC(p, x)**(d0 - d1) # lcf * subr = modified subr + a0, a1 = p, q # the input polys + sturm_seq = [a0, a1] # the output list + del0 = d0 - d1 # degree difference + rho1 = LC(a1, x) # leading coeff of a1 + exp_deg = d1 - 1 # expected degree of a2 + a2 = - rem(a0, a1, domain=QQ) # first remainder + rho2 = LC(a2,x) # leading coeff of a2 + d2 = degree(a2, x) # actual degree of a2 + deg_diff_new = exp_deg - d2 # expected - actual degree + del1 = d1 - d2 # degree difference + + # mul_fac is the factor by which a2 is multiplied to + # get integer coefficients + mul_fac_old = rho1**(del0 + del1 - deg_diff_new) + + # append accordingly + if method == 0: + sturm_seq.append( simplify(lcf * a2 * Abs(mul_fac_old))) + else: + sturm_seq.append( simplify( a2 * Abs(mul_fac_old))) + + # main loop + deg_diff_old = deg_diff_new + while d2 > 0: + a0, a1, d0, d1 = a1, a2, d1, d2 # update polys and degrees + del0 = del1 # update degree difference + exp_deg = d1 - 1 # new expected degree + a2 = - rem(a0, a1, domain=QQ) # new remainder + rho3 = LC(a2, x) # leading coeff of a2 + d2 = degree(a2, x) # actual degree of a2 + deg_diff_new = exp_deg - d2 # expected - actual degree + del1 = d1 - d2 # degree difference + + # take into consideration the power + # rho1**deg_diff_old that was "left out" + expo_old = deg_diff_old # rho1 raised to this power + expo_new = del0 + del1 - deg_diff_new # rho2 raised to this power + + # update variables and append + mul_fac_new = rho2**(expo_new) * rho1**(expo_old) * mul_fac_old + deg_diff_old, mul_fac_old = deg_diff_new, mul_fac_new + rho1, rho2 = rho2, rho3 + if method == 0: + sturm_seq.append( simplify(lcf * a2 * Abs(mul_fac_old))) + else: + sturm_seq.append( simplify( a2 * Abs(mul_fac_old))) + + if flag: # change the sign of the sequence + sturm_seq = [-i for i in sturm_seq] + + # gcd is of degree > 0 ? + m = len(sturm_seq) + if sturm_seq[m - 1] == nan or sturm_seq[m - 1] == 0: + sturm_seq.pop(m - 1) + + return sturm_seq + +def sturm_q(p, q, x): + """ + p, q are polynomials in Z[x] or Q[x]. It is assumed + that degree(p, x) >= degree(q, x). + + Computes the (generalized) Sturm sequence of p and q in Q[x]. + Polynomial divisions in Q[x] are performed, using the function rem(p, q, x). + + The coefficients of the polynomials in the Sturm sequence can be uniquely + determined from the corresponding coefficients of the polynomials found + either in: + + (a) the ``modified'' subresultant prs, (references 1, 2) + + or in + + (b) the subresultant prs (reference 3). + + References + ========== + 1. Pell A. J., R. L. Gordon. The Modified Remainders Obtained in Finding + the Highest Common Factor of Two Polynomials. Annals of MatheMatics, + Second Series, 18 (1917), No. 4, 188-193. + + 2 Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``Sturm Sequences + and Modified Subresultant Polynomial Remainder Sequences.'' + Serdica Journal of Computing, Vol. 8, No 1, 29-46, 2014. + + 3. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``A Basic Result + on the Theory of Subresultants.'' Serdica Journal of Computing 10 (2016), No.1, 31-48. + + """ + # make sure neither p nor q is 0 + if p == 0 or q == 0: + return [p, q] + + # make sure proper degrees + d0 = degree(p, x) + d1 = degree(q, x) + if d0 == 0 and d1 == 0: + return [p, q] + if d1 > d0: + d0, d1 = d1, d0 + p, q = q, p + if d0 > 0 and d1 == 0: + return [p,q] + + # make sure LC(p) > 0 + flag = 0 + if LC(p,x) < 0: + flag = 1 + p = -p + q = -q + + # initialize + a0, a1 = p, q # the input polys + sturm_seq = [a0, a1] # the output list + a2 = -rem(a0, a1, domain=QQ) # first remainder + d2 = degree(a2, x) # degree of a2 + sturm_seq.append( a2 ) + + # main loop + while d2 > 0: + a0, a1, d0, d1 = a1, a2, d1, d2 # update polys and degrees + a2 = -rem(a0, a1, domain=QQ) # new remainder + d2 = degree(a2, x) # actual degree of a2 + sturm_seq.append( a2 ) + + if flag: # change the sign of the sequence + sturm_seq = [-i for i in sturm_seq] + + # gcd is of degree > 0 ? + m = len(sturm_seq) + if sturm_seq[m - 1] == nan or sturm_seq[m - 1] == 0: + sturm_seq.pop(m - 1) + + return sturm_seq + +def sturm_amv(p, q, x, method=0): + """ + p, q are polynomials in Z[x] or Q[x]. It is assumed + that degree(p, x) >= degree(q, x). + + Computes the (generalized) Sturm sequence of p and q in Z[x] or Q[x]. + If q = diff(p, x, 1) it is the usual Sturm sequence. + + A. If method == 0, default, the remainder coefficients of the + sequence are (in absolute value) ``modified'' subresultants, which + for non-monic polynomials are greater than the coefficients of the + corresponding subresultants by the factor Abs(LC(p)**( deg(p)- deg(q))). + + B. If method == 1, the remainder coefficients of the sequence are (in + absolute value) subresultants, which for non-monic polynomials are + smaller than the coefficients of the corresponding ``modified'' + subresultants by the factor Abs( LC(p)**( deg(p)- deg(q)) ). + + If the Sturm sequence is complete, method=0 and LC( p ) > 0, then the + coefficients of the polynomials in the sequence are ``modified'' subresultants. + That is, they are determinants of appropriately selected submatrices of + sylvester2, Sylvester's matrix of 1853. In this case the Sturm sequence + coincides with the ``modified'' subresultant prs, of the polynomials + p, q. + + If the Sturm sequence is incomplete and method=0 then the signs of the + coefficients of the polynomials in the sequence may differ from the signs + of the coefficients of the corresponding polynomials in the ``modified'' + subresultant prs; however, the absolute values are the same. + + To compute the coefficients, no determinant evaluation takes place. + Instead, we first compute the euclidean sequence of p and q using + euclid_amv(p, q, x) and then: (a) change the signs of the remainders in the + Euclidean sequence according to the pattern "-, -, +, +, -, -, +, +,..." + (see Lemma 1 in the 1st reference or Theorem 3 in the 2nd reference) + and (b) if method=0, assuming deg(p) > deg(q), we multiply the remainder + coefficients of the Euclidean sequence times the factor + Abs( LC(p)**( deg(p)- deg(q)) ) to make them modified subresultants. + See also the function sturm_pg(p, q, x). + + References + ========== + 1. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``A Basic Result + on the Theory of Subresultants.'' Serdica Journal of Computing 10 (2016), No.1, 31-48. + + 2. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``On the Remainders + Obtained in Finding the Greatest Common Divisor of Two Polynomials.'' Serdica + Journal of Computing 9(2) (2015), 123-138. + + 3. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``Subresultant Polynomial + Remainder Sequences Obtained by Polynomial Divisions in Q[x] or in Z[x].'' + Serdica Journal of Computing 10 (2016), No.3-4, 197-217. + + """ + # compute the euclidean sequence + prs = euclid_amv(p, q, x) + + # defensive + if prs == [] or len(prs) == 2: + return prs + + # the coefficients in prs are subresultants and hence are smaller + # than the corresponding subresultants by the factor + # Abs( LC(prs[0])**( deg(prs[0]) - deg(prs[1])) ); Theorem 2, 2nd reference. + lcf = Abs( LC(prs[0])**( degree(prs[0], x) - degree(prs[1], x) ) ) + + # the signs of the first two polys in the sequence stay the same + sturm_seq = [prs[0], prs[1]] + + # change the signs according to "-, -, +, +, -, -, +, +,..." + # and multiply times lcf if needed + flag = 0 + m = len(prs) + i = 2 + while i <= m-1: + if flag == 0: + sturm_seq.append( - prs[i] ) + i = i + 1 + if i == m: + break + sturm_seq.append( - prs[i] ) + i = i + 1 + flag = 1 + elif flag == 1: + sturm_seq.append( prs[i] ) + i = i + 1 + if i == m: + break + sturm_seq.append( prs[i] ) + i = i + 1 + flag = 0 + + # subresultants or modified subresultants? + if method == 0 and lcf > 1: + aux_seq = [sturm_seq[0], sturm_seq[1]] + for i in range(2, m): + aux_seq.append(simplify(sturm_seq[i] * lcf )) + sturm_seq = aux_seq + + return sturm_seq + +def euclid_pg(p, q, x): + """ + p, q are polynomials in Z[x] or Q[x]. It is assumed + that degree(p, x) >= degree(q, x). + + Computes the Euclidean sequence of p and q in Z[x] or Q[x]. + + If the Euclidean sequence is complete the coefficients of the polynomials + in the sequence are subresultants. That is, they are determinants of + appropriately selected submatrices of sylvester1, Sylvester's matrix of 1840. + In this case the Euclidean sequence coincides with the subresultant prs + of the polynomials p, q. + + If the Euclidean sequence is incomplete the signs of the coefficients of the + polynomials in the sequence may differ from the signs of the coefficients of + the corresponding polynomials in the subresultant prs; however, the absolute + values are the same. + + To compute the Euclidean sequence, no determinant evaluation takes place. + We first compute the (generalized) Sturm sequence of p and q using + sturm_pg(p, q, x, 1), in which case the coefficients are (in absolute value) + equal to subresultants. Then we change the signs of the remainders in the + Sturm sequence according to the pattern "-, -, +, +, -, -, +, +,..." ; + see Lemma 1 in the 1st reference or Theorem 3 in the 2nd reference as well as + the function sturm_pg(p, q, x). + + References + ========== + 1. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``A Basic Result + on the Theory of Subresultants.'' Serdica Journal of Computing 10 (2016), No.1, 31-48. + + 2. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``On the Remainders + Obtained in Finding the Greatest Common Divisor of Two Polynomials.'' Serdica + Journal of Computing 9(2) (2015), 123-138. + + 3. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``Subresultant Polynomial + Remainder Sequences Obtained by Polynomial Divisions in Q[x] or in Z[x].'' + Serdica Journal of Computing 10 (2016), No.3-4, 197-217. + """ + # compute the sturmian sequence using the Pell-Gordon (or AMV) theorem + # with the coefficients in the prs being (in absolute value) subresultants + prs = sturm_pg(p, q, x, 1) ## any other method would do + + # defensive + if prs == [] or len(prs) == 2: + return prs + + # the signs of the first two polys in the sequence stay the same + euclid_seq = [prs[0], prs[1]] + + # change the signs according to "-, -, +, +, -, -, +, +,..." + flag = 0 + m = len(prs) + i = 2 + while i <= m-1: + if flag == 0: + euclid_seq.append(- prs[i] ) + i = i + 1 + if i == m: + break + euclid_seq.append(- prs[i] ) + i = i + 1 + flag = 1 + elif flag == 1: + euclid_seq.append(prs[i] ) + i = i + 1 + if i == m: + break + euclid_seq.append(prs[i] ) + i = i + 1 + flag = 0 + + return euclid_seq + +def euclid_q(p, q, x): + """ + p, q are polynomials in Z[x] or Q[x]. It is assumed + that degree(p, x) >= degree(q, x). + + Computes the Euclidean sequence of p and q in Q[x]. + Polynomial divisions in Q[x] are performed, using the function rem(p, q, x). + + The coefficients of the polynomials in the Euclidean sequence can be uniquely + determined from the corresponding coefficients of the polynomials found + either in: + + (a) the ``modified'' subresultant polynomial remainder sequence, + (references 1, 2) + + or in + + (b) the subresultant polynomial remainder sequence (references 3). + + References + ========== + 1. Pell A. J., R. L. Gordon. The Modified Remainders Obtained in Finding + the Highest Common Factor of Two Polynomials. Annals of MatheMatics, + Second Series, 18 (1917), No. 4, 188-193. + + 2. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``Sturm Sequences + and Modified Subresultant Polynomial Remainder Sequences.'' + Serdica Journal of Computing, Vol. 8, No 1, 29-46, 2014. + + 3. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``A Basic Result + on the Theory of Subresultants.'' Serdica Journal of Computing 10 (2016), No.1, 31-48. + + """ + # make sure neither p nor q is 0 + if p == 0 or q == 0: + return [p, q] + + # make sure proper degrees + d0 = degree(p, x) + d1 = degree(q, x) + if d0 == 0 and d1 == 0: + return [p, q] + if d1 > d0: + d0, d1 = d1, d0 + p, q = q, p + if d0 > 0 and d1 == 0: + return [p,q] + + # make sure LC(p) > 0 + flag = 0 + if LC(p,x) < 0: + flag = 1 + p = -p + q = -q + + # initialize + a0, a1 = p, q # the input polys + euclid_seq = [a0, a1] # the output list + a2 = rem(a0, a1, domain=QQ) # first remainder + d2 = degree(a2, x) # degree of a2 + euclid_seq.append( a2 ) + + # main loop + while d2 > 0: + a0, a1, d0, d1 = a1, a2, d1, d2 # update polys and degrees + a2 = rem(a0, a1, domain=QQ) # new remainder + d2 = degree(a2, x) # actual degree of a2 + euclid_seq.append( a2 ) + + if flag: # change the sign of the sequence + euclid_seq = [-i for i in euclid_seq] + + # gcd is of degree > 0 ? + m = len(euclid_seq) + if euclid_seq[m - 1] == nan or euclid_seq[m - 1] == 0: + euclid_seq.pop(m - 1) + + return euclid_seq + +def euclid_amv(f, g, x): + """ + f, g are polynomials in Z[x] or Q[x]. It is assumed + that degree(f, x) >= degree(g, x). + + Computes the Euclidean sequence of p and q in Z[x] or Q[x]. + + If the Euclidean sequence is complete the coefficients of the polynomials + in the sequence are subresultants. That is, they are determinants of + appropriately selected submatrices of sylvester1, Sylvester's matrix of 1840. + In this case the Euclidean sequence coincides with the subresultant prs, + of the polynomials p, q. + + If the Euclidean sequence is incomplete the signs of the coefficients of the + polynomials in the sequence may differ from the signs of the coefficients of + the corresponding polynomials in the subresultant prs; however, the absolute + values are the same. + + To compute the coefficients, no determinant evaluation takes place. + Instead, polynomial divisions in Z[x] or Q[x] are performed, using + the function rem_z(f, g, x); the coefficients of the remainders + computed this way become subresultants with the help of the + Collins-Brown-Traub formula for coefficient reduction. + + References + ========== + 1. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``A Basic Result + on the Theory of Subresultants.'' Serdica Journal of Computing 10 (2016), No.1, 31-48. + + 2. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``Subresultant Polynomial + remainder Sequences Obtained by Polynomial Divisions in Q[x] or in Z[x].'' + Serdica Journal of Computing 10 (2016), No.3-4, 197-217. + + """ + # make sure neither f nor g is 0 + if f == 0 or g == 0: + return [f, g] + + # make sure proper degrees + d0 = degree(f, x) + d1 = degree(g, x) + if d0 == 0 and d1 == 0: + return [f, g] + if d1 > d0: + d0, d1 = d1, d0 + f, g = g, f + if d0 > 0 and d1 == 0: + return [f, g] + + # initialize + a0 = f + a1 = g + euclid_seq = [a0, a1] + deg_dif_p1, c = degree(a0, x) - degree(a1, x) + 1, -1 + + # compute the first polynomial of the prs + i = 1 + a2 = rem_z(a0, a1, x) / Abs( (-1)**deg_dif_p1 ) # first remainder + euclid_seq.append( a2 ) + d2 = degree(a2, x) # actual degree of a2 + + # main loop + while d2 >= 1: + a0, a1, d0, d1 = a1, a2, d1, d2 # update polys and degrees + i += 1 + sigma0 = -LC(a0) + c = (sigma0**(deg_dif_p1 - 1)) / (c**(deg_dif_p1 - 2)) + deg_dif_p1 = degree(a0, x) - d2 + 1 + a2 = rem_z(a0, a1, x) / Abs( (c**(deg_dif_p1 - 1)) * sigma0 ) + euclid_seq.append( a2 ) + d2 = degree(a2, x) # actual degree of a2 + + # gcd is of degree > 0 ? + m = len(euclid_seq) + if euclid_seq[m - 1] == nan or euclid_seq[m - 1] == 0: + euclid_seq.pop(m - 1) + + return euclid_seq + +def modified_subresultants_pg(p, q, x): + """ + p, q are polynomials in Z[x] or Q[x]. It is assumed + that degree(p, x) >= degree(q, x). + + Computes the ``modified'' subresultant prs of p and q in Z[x] or Q[x]; + the coefficients of the polynomials in the sequence are + ``modified'' subresultants. That is, they are determinants of appropriately + selected submatrices of sylvester2, Sylvester's matrix of 1853. + + To compute the coefficients, no determinant evaluation takes place. Instead, + polynomial divisions in Q[x] are performed, using the function rem(p, q, x); + the coefficients of the remainders computed this way become ``modified'' + subresultants with the help of the Pell-Gordon Theorem of 1917. + + If the ``modified'' subresultant prs is complete, and LC( p ) > 0, it coincides + with the (generalized) Sturm sequence of the polynomials p, q. + + References + ========== + 1. Pell A. J., R. L. Gordon. The Modified Remainders Obtained in Finding + the Highest Common Factor of Two Polynomials. Annals of MatheMatics, + Second Series, 18 (1917), No. 4, 188-193. + + 2. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``Sturm Sequences + and Modified Subresultant Polynomial Remainder Sequences.'' + Serdica Journal of Computing, Vol. 8, No 1, 29-46, 2014. + + """ + # make sure neither p nor q is 0 + if p == 0 or q == 0: + return [p, q] + + # make sure proper degrees + d0 = degree(p,x) + d1 = degree(q,x) + if d0 == 0 and d1 == 0: + return [p, q] + if d1 > d0: + d0, d1 = d1, d0 + p, q = q, p + if d0 > 0 and d1 == 0: + return [p,q] + + # initialize + k = var('k') # index in summation formula + u_list = [] # of elements (-1)**u_i + subres_l = [p, q] # mod. subr. prs output list + a0, a1 = p, q # the input polys + del0 = d0 - d1 # degree difference + degdif = del0 # save it + rho_1 = LC(a0) # lead. coeff (a0) + + # Initialize Pell-Gordon variables + rho_list_minus_1 = sign( LC(a0, x)) # sign of LC(a0) + rho1 = LC(a1, x) # leading coeff of a1 + rho_list = [ sign(rho1)] # of signs + p_list = [del0] # of degree differences + u = summation(k, (k, 1, p_list[0])) # value of u + u_list.append(u) # of u values + v = sum(p_list) # v value + + # first remainder + exp_deg = d1 - 1 # expected degree of a2 + a2 = - rem(a0, a1, domain=QQ) # first remainder + rho2 = LC(a2, x) # leading coeff of a2 + d2 = degree(a2, x) # actual degree of a2 + deg_diff_new = exp_deg - d2 # expected - actual degree + del1 = d1 - d2 # degree difference + + # mul_fac is the factor by which a2 is multiplied to + # get integer coefficients + mul_fac_old = rho1**(del0 + del1 - deg_diff_new) + + # update Pell-Gordon variables + p_list.append(1 + deg_diff_new) # deg_diff_new is 0 for complete seq + + # apply Pell-Gordon formula (7) in second reference + num = 1 # numerator of fraction + for u in u_list: + num *= (-1)**u + num = num * (-1)**v + + # denominator depends on complete / incomplete seq + if deg_diff_new == 0: # complete seq + den = 1 + for k in range(len(rho_list)): + den *= rho_list[k]**(p_list[k] + p_list[k + 1]) + den = den * rho_list_minus_1 + else: # incomplete seq + den = 1 + for k in range(len(rho_list)-1): + den *= rho_list[k]**(p_list[k] + p_list[k + 1]) + den = den * rho_list_minus_1 + expo = (p_list[len(rho_list) - 1] + p_list[len(rho_list)] - deg_diff_new) + den = den * rho_list[len(rho_list) - 1]**expo + + # the sign of the determinant depends on sg(num / den) + if sign(num / den) > 0: + subres_l.append( simplify(rho_1**degdif*a2* Abs(mul_fac_old) ) ) + else: + subres_l.append(- simplify(rho_1**degdif*a2* Abs(mul_fac_old) ) ) + + # update Pell-Gordon variables + k = var('k') + rho_list.append( sign(rho2)) + u = summation(k, (k, 1, p_list[len(p_list) - 1])) + u_list.append(u) + v = sum(p_list) + deg_diff_old=deg_diff_new + + # main loop + while d2 > 0: + a0, a1, d0, d1 = a1, a2, d1, d2 # update polys and degrees + del0 = del1 # update degree difference + exp_deg = d1 - 1 # new expected degree + a2 = - rem(a0, a1, domain=QQ) # new remainder + rho3 = LC(a2, x) # leading coeff of a2 + d2 = degree(a2, x) # actual degree of a2 + deg_diff_new = exp_deg - d2 # expected - actual degree + del1 = d1 - d2 # degree difference + + # take into consideration the power + # rho1**deg_diff_old that was "left out" + expo_old = deg_diff_old # rho1 raised to this power + expo_new = del0 + del1 - deg_diff_new # rho2 raised to this power + + mul_fac_new = rho2**(expo_new) * rho1**(expo_old) * mul_fac_old + + # update variables + deg_diff_old, mul_fac_old = deg_diff_new, mul_fac_new + rho1, rho2 = rho2, rho3 + + # update Pell-Gordon variables + p_list.append(1 + deg_diff_new) # deg_diff_new is 0 for complete seq + + # apply Pell-Gordon formula (7) in second reference + num = 1 # numerator + for u in u_list: + num *= (-1)**u + num = num * (-1)**v + + # denominator depends on complete / incomplete seq + if deg_diff_new == 0: # complete seq + den = 1 + for k in range(len(rho_list)): + den *= rho_list[k]**(p_list[k] + p_list[k + 1]) + den = den * rho_list_minus_1 + else: # incomplete seq + den = 1 + for k in range(len(rho_list)-1): + den *= rho_list[k]**(p_list[k] + p_list[k + 1]) + den = den * rho_list_minus_1 + expo = (p_list[len(rho_list) - 1] + p_list[len(rho_list)] - deg_diff_new) + den = den * rho_list[len(rho_list) - 1]**expo + + # the sign of the determinant depends on sg(num / den) + if sign(num / den) > 0: + subres_l.append( simplify(rho_1**degdif*a2* Abs(mul_fac_old) ) ) + else: + subres_l.append(- simplify(rho_1**degdif*a2* Abs(mul_fac_old) ) ) + + # update Pell-Gordon variables + k = var('k') + rho_list.append( sign(rho2)) + u = summation(k, (k, 1, p_list[len(p_list) - 1])) + u_list.append(u) + v = sum(p_list) + + # gcd is of degree > 0 ? + m = len(subres_l) + if subres_l[m - 1] == nan or subres_l[m - 1] == 0: + subres_l.pop(m - 1) + + # LC( p ) < 0 + m = len(subres_l) # list may be shorter now due to deg(gcd ) > 0 + if LC( p ) < 0: + aux_seq = [subres_l[0], subres_l[1]] + for i in range(2, m): + aux_seq.append(simplify(subres_l[i] * (-1) )) + subres_l = aux_seq + + return subres_l + +def subresultants_pg(p, q, x): + """ + p, q are polynomials in Z[x] or Q[x]. It is assumed + that degree(p, x) >= degree(q, x). + + Computes the subresultant prs of p and q in Z[x] or Q[x], from + the modified subresultant prs of p and q. + + The coefficients of the polynomials in these two sequences differ only + in sign and the factor LC(p)**( deg(p)- deg(q)) as stated in + Theorem 2 of the reference. + + The coefficients of the polynomials in the output sequence are + subresultants. That is, they are determinants of appropriately + selected submatrices of sylvester1, Sylvester's matrix of 1840. + + If the subresultant prs is complete, then it coincides with the + Euclidean sequence of the polynomials p, q. + + References + ========== + 1. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: "On the Remainders + Obtained in Finding the Greatest Common Divisor of Two Polynomials." + Serdica Journal of Computing 9(2) (2015), 123-138. + + """ + # compute the modified subresultant prs + lst = modified_subresultants_pg(p,q,x) ## any other method would do + + # defensive + if lst == [] or len(lst) == 2: + return lst + + # the coefficients in lst are modified subresultants and, hence, are + # greater than those of the corresponding subresultants by the factor + # LC(lst[0])**( deg(lst[0]) - deg(lst[1])); see Theorem 2 in reference. + lcf = LC(lst[0])**( degree(lst[0], x) - degree(lst[1], x) ) + + # Initialize the subresultant prs list + subr_seq = [lst[0], lst[1]] + + # compute the degree sequences m_i and j_i of Theorem 2 in reference. + deg_seq = [degree(Poly(poly, x), x) for poly in lst] + deg = deg_seq[0] + deg_seq_s = deg_seq[1:-1] + m_seq = [m-1 for m in deg_seq_s] + j_seq = [deg - m for m in m_seq] + + # compute the AMV factors of Theorem 2 in reference. + fact = [(-1)**( j*(j-1)/S(2) ) for j in j_seq] + + # shortened list without the first two polys + lst_s = lst[2:] + + # poly lst_s[k] is multiplied times fact[k], divided by lcf + # and appended to the subresultant prs list + m = len(fact) + for k in range(m): + if sign(fact[k]) == -1: + subr_seq.append(-lst_s[k] / lcf) + else: + subr_seq.append(lst_s[k] / lcf) + + return subr_seq + +def subresultants_amv_q(p, q, x): + """ + p, q are polynomials in Z[x] or Q[x]. It is assumed + that degree(p, x) >= degree(q, x). + + Computes the subresultant prs of p and q in Q[x]; + the coefficients of the polynomials in the sequence are + subresultants. That is, they are determinants of appropriately + selected submatrices of sylvester1, Sylvester's matrix of 1840. + + To compute the coefficients, no determinant evaluation takes place. + Instead, polynomial divisions in Q[x] are performed, using the + function rem(p, q, x); the coefficients of the remainders + computed this way become subresultants with the help of the + Akritas-Malaschonok-Vigklas Theorem of 2015. + + If the subresultant prs is complete, then it coincides with the + Euclidean sequence of the polynomials p, q. + + References + ========== + 1. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``A Basic Result + on the Theory of Subresultants.'' Serdica Journal of Computing 10 (2016), No.1, 31-48. + + 2. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``Subresultant Polynomial + remainder Sequences Obtained by Polynomial Divisions in Q[x] or in Z[x].'' + Serdica Journal of Computing 10 (2016), No.3-4, 197-217. + + """ + # make sure neither p nor q is 0 + if p == 0 or q == 0: + return [p, q] + + # make sure proper degrees + d0 = degree(p, x) + d1 = degree(q, x) + if d0 == 0 and d1 == 0: + return [p, q] + if d1 > d0: + d0, d1 = d1, d0 + p, q = q, p + if d0 > 0 and d1 == 0: + return [p, q] + + # initialize + i, s = 0, 0 # counters for remainders & odd elements + p_odd_index_sum = 0 # contains the sum of p_1, p_3, etc + subres_l = [p, q] # subresultant prs output list + a0, a1 = p, q # the input polys + sigma1 = LC(a1, x) # leading coeff of a1 + p0 = d0 - d1 # degree difference + if p0 % 2 == 1: + s += 1 + phi = floor( (s + 1) / 2 ) + mul_fac = 1 + d2 = d1 + + # main loop + while d2 > 0: + i += 1 + a2 = rem(a0, a1, domain= QQ) # new remainder + if i == 1: + sigma2 = LC(a2, x) + else: + sigma3 = LC(a2, x) + sigma1, sigma2 = sigma2, sigma3 + d2 = degree(a2, x) + p1 = d1 - d2 + psi = i + phi + p_odd_index_sum + + # new mul_fac + mul_fac = sigma1**(p0 + 1) * mul_fac + + ## compute the sign of the first fraction in formula (9) of the paper + # numerator + num = (-1)**psi + # denominator + den = sign(mul_fac) + + # the sign of the determinant depends on sign( num / den ) != 0 + if sign(num / den) > 0: + subres_l.append( simplify(expand(a2* Abs(mul_fac)))) + else: + subres_l.append(- simplify(expand(a2* Abs(mul_fac)))) + + ## bring into mul_fac the missing power of sigma if there was a degree gap + if p1 - 1 > 0: + mul_fac = mul_fac * sigma1**(p1 - 1) + + # update AMV variables + a0, a1, d0, d1 = a1, a2, d1, d2 + p0 = p1 + if p0 % 2 ==1: + s += 1 + phi = floor( (s + 1) / 2 ) + if i%2 == 1: + p_odd_index_sum += p0 # p_i has odd index + + # gcd is of degree > 0 ? + m = len(subres_l) + if subres_l[m - 1] == nan or subres_l[m - 1] == 0: + subres_l.pop(m - 1) + + return subres_l + +def compute_sign(base, expo): + ''' + base != 0 and expo >= 0 are integers; + + returns the sign of base**expo without + evaluating the power itself! + ''' + sb = sign(base) + if sb == 1: + return 1 + pe = expo % 2 + if pe == 0: + return -sb + else: + return sb + +def rem_z(p, q, x): + ''' + Intended mainly for p, q polynomials in Z[x] so that, + on dividing p by q, the remainder will also be in Z[x]. (However, + it also works fine for polynomials in Q[x].) It is assumed + that degree(p, x) >= degree(q, x). + + It premultiplies p by the _absolute_ value of the leading coefficient + of q, raised to the power deg(p) - deg(q) + 1 and then performs + polynomial division in Q[x], using the function rem(p, q, x). + + By contrast the function prem(p, q, x) does _not_ use the absolute + value of the leading coefficient of q. + This results not only in ``messing up the signs'' of the Euclidean and + Sturmian prs's as mentioned in the second reference, + but also in violation of the main results of the first and third + references --- Theorem 4 and Theorem 1 respectively. Theorems 4 and 1 + establish a one-to-one correspondence between the Euclidean and the + Sturmian prs of p, q, on one hand, and the subresultant prs of p, q, + on the other. + + References + ========== + 1. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``On the Remainders + Obtained in Finding the Greatest Common Divisor of Two Polynomials.'' + Serdica Journal of Computing, 9(2) (2015), 123-138. + + 2. https://planetMath.org/sturmstheorem + + 3. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``A Basic Result on + the Theory of Subresultants.'' Serdica Journal of Computing 10 (2016), No.1, 31-48. + + ''' + if (p.as_poly().is_univariate and q.as_poly().is_univariate and + p.as_poly().gens == q.as_poly().gens): + delta = (degree(p, x) - degree(q, x) + 1) + return rem(Abs(LC(q, x))**delta * p, q, x) + else: + return prem(p, q, x) + +def quo_z(p, q, x): + """ + Intended mainly for p, q polynomials in Z[x] so that, + on dividing p by q, the quotient will also be in Z[x]. (However, + it also works fine for polynomials in Q[x].) It is assumed + that degree(p, x) >= degree(q, x). + + It premultiplies p by the _absolute_ value of the leading coefficient + of q, raised to the power deg(p) - deg(q) + 1 and then performs + polynomial division in Q[x], using the function quo(p, q, x). + + By contrast the function pquo(p, q, x) does _not_ use the absolute + value of the leading coefficient of q. + + See also function rem_z(p, q, x) for additional comments and references. + + """ + if (p.as_poly().is_univariate and q.as_poly().is_univariate and + p.as_poly().gens == q.as_poly().gens): + delta = (degree(p, x) - degree(q, x) + 1) + return quo(Abs(LC(q, x))**delta * p, q, x) + else: + return pquo(p, q, x) + +def subresultants_amv(f, g, x): + """ + p, q are polynomials in Z[x] or Q[x]. It is assumed + that degree(f, x) >= degree(g, x). + + Computes the subresultant prs of p and q in Z[x] or Q[x]; + the coefficients of the polynomials in the sequence are + subresultants. That is, they are determinants of appropriately + selected submatrices of sylvester1, Sylvester's matrix of 1840. + + To compute the coefficients, no determinant evaluation takes place. + Instead, polynomial divisions in Z[x] or Q[x] are performed, using + the function rem_z(p, q, x); the coefficients of the remainders + computed this way become subresultants with the help of the + Akritas-Malaschonok-Vigklas Theorem of 2015 and the Collins-Brown- + Traub formula for coefficient reduction. + + If the subresultant prs is complete, then it coincides with the + Euclidean sequence of the polynomials p, q. + + References + ========== + 1. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``A Basic Result + on the Theory of Subresultants.'' Serdica Journal of Computing 10 (2016), No.1, 31-48. + + 2. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``Subresultant Polynomial + remainder Sequences Obtained by Polynomial Divisions in Q[x] or in Z[x].'' + Serdica Journal of Computing 10 (2016), No.3-4, 197-217. + + """ + # make sure neither f nor g is 0 + if f == 0 or g == 0: + return [f, g] + + # make sure proper degrees + d0 = degree(f, x) + d1 = degree(g, x) + if d0 == 0 and d1 == 0: + return [f, g] + if d1 > d0: + d0, d1 = d1, d0 + f, g = g, f + if d0 > 0 and d1 == 0: + return [f, g] + + # initialize + a0 = f + a1 = g + subres_l = [a0, a1] + deg_dif_p1, c = degree(a0, x) - degree(a1, x) + 1, -1 + + # initialize AMV variables + sigma1 = LC(a1, x) # leading coeff of a1 + i, s = 0, 0 # counters for remainders & odd elements + p_odd_index_sum = 0 # contains the sum of p_1, p_3, etc + p0 = deg_dif_p1 - 1 + if p0 % 2 == 1: + s += 1 + phi = floor( (s + 1) / 2 ) + + # compute the first polynomial of the prs + i += 1 + a2 = rem_z(a0, a1, x) / Abs( (-1)**deg_dif_p1 ) # first remainder + sigma2 = LC(a2, x) # leading coeff of a2 + d2 = degree(a2, x) # actual degree of a2 + p1 = d1 - d2 # degree difference + + # sgn_den is the factor, the denominator 1st fraction of (9), + # by which a2 is multiplied to get integer coefficients + sgn_den = compute_sign( sigma1, p0 + 1 ) + + ## compute sign of the 1st fraction in formula (9) of the paper + # numerator + psi = i + phi + p_odd_index_sum + num = (-1)**psi + # denominator + den = sgn_den + + # the sign of the determinant depends on sign(num / den) != 0 + if sign(num / den) > 0: + subres_l.append( a2 ) + else: + subres_l.append( -a2 ) + + # update AMV variable + if p1 % 2 == 1: + s += 1 + + # bring in the missing power of sigma if there was gap + if p1 - 1 > 0: + sgn_den = sgn_den * compute_sign( sigma1, p1 - 1 ) + + # main loop + while d2 >= 1: + phi = floor( (s + 1) / 2 ) + if i%2 == 1: + p_odd_index_sum += p1 # p_i has odd index + a0, a1, d0, d1 = a1, a2, d1, d2 # update polys and degrees + p0 = p1 # update degree difference + i += 1 + sigma0 = -LC(a0) + c = (sigma0**(deg_dif_p1 - 1)) / (c**(deg_dif_p1 - 2)) + deg_dif_p1 = degree(a0, x) - d2 + 1 + a2 = rem_z(a0, a1, x) / Abs( (c**(deg_dif_p1 - 1)) * sigma0 ) + sigma3 = LC(a2, x) # leading coeff of a2 + d2 = degree(a2, x) # actual degree of a2 + p1 = d1 - d2 # degree difference + psi = i + phi + p_odd_index_sum + + # update variables + sigma1, sigma2 = sigma2, sigma3 + + # new sgn_den + sgn_den = compute_sign( sigma1, p0 + 1 ) * sgn_den + + # compute the sign of the first fraction in formula (9) of the paper + # numerator + num = (-1)**psi + # denominator + den = sgn_den + + # the sign of the determinant depends on sign( num / den ) != 0 + if sign(num / den) > 0: + subres_l.append( a2 ) + else: + subres_l.append( -a2 ) + + # update AMV variable + if p1 % 2 ==1: + s += 1 + + # bring in the missing power of sigma if there was gap + if p1 - 1 > 0: + sgn_den = sgn_den * compute_sign( sigma1, p1 - 1 ) + + # gcd is of degree > 0 ? + m = len(subres_l) + if subres_l[m - 1] == nan or subres_l[m - 1] == 0: + subres_l.pop(m - 1) + + return subres_l + +def modified_subresultants_amv(p, q, x): + """ + p, q are polynomials in Z[x] or Q[x]. It is assumed + that degree(p, x) >= degree(q, x). + + Computes the modified subresultant prs of p and q in Z[x] or Q[x], + from the subresultant prs of p and q. + The coefficients of the polynomials in the two sequences differ only + in sign and the factor LC(p)**( deg(p)- deg(q)) as stated in + Theorem 2 of the reference. + + The coefficients of the polynomials in the output sequence are + modified subresultants. That is, they are determinants of appropriately + selected submatrices of sylvester2, Sylvester's matrix of 1853. + + If the modified subresultant prs is complete, and LC( p ) > 0, it coincides + with the (generalized) Sturm's sequence of the polynomials p, q. + + References + ========== + 1. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: "On the Remainders + Obtained in Finding the Greatest Common Divisor of Two Polynomials." + Serdica Journal of Computing, Serdica Journal of Computing, 9(2) (2015), 123-138. + + """ + # compute the subresultant prs + lst = subresultants_amv(p,q,x) ## any other method would do + + # defensive + if lst == [] or len(lst) == 2: + return lst + + # the coefficients in lst are subresultants and, hence, smaller than those + # of the corresponding modified subresultants by the factor + # LC(lst[0])**( deg(lst[0]) - deg(lst[1])); see Theorem 2. + lcf = LC(lst[0])**( degree(lst[0], x) - degree(lst[1], x) ) + + # Initialize the modified subresultant prs list + subr_seq = [lst[0], lst[1]] + + # compute the degree sequences m_i and j_i of Theorem 2 + deg_seq = [degree(Poly(poly, x), x) for poly in lst] + deg = deg_seq[0] + deg_seq_s = deg_seq[1:-1] + m_seq = [m-1 for m in deg_seq_s] + j_seq = [deg - m for m in m_seq] + + # compute the AMV factors of Theorem 2 + fact = [(-1)**( j*(j-1)/S(2) ) for j in j_seq] + + # shortened list without the first two polys + lst_s = lst[2:] + + # poly lst_s[k] is multiplied times fact[k] and times lcf + # and appended to the subresultant prs list + m = len(fact) + for k in range(m): + if sign(fact[k]) == -1: + subr_seq.append( simplify(-lst_s[k] * lcf) ) + else: + subr_seq.append( simplify(lst_s[k] * lcf) ) + + return subr_seq + +def correct_sign(deg_f, deg_g, s1, rdel, cdel): + """ + Used in various subresultant prs algorithms. + + Evaluates the determinant, (a.k.a. subresultant) of a properly selected + submatrix of s1, Sylvester's matrix of 1840, to get the correct sign + and value of the leading coefficient of a given polynomial remainder. + + deg_f, deg_g are the degrees of the original polynomials p, q for which the + matrix s1 = sylvester(p, q, x, 1) was constructed. + + rdel denotes the expected degree of the remainder; it is the number of + rows to be deleted from each group of rows in s1 as described in the + reference below. + + cdel denotes the expected degree minus the actual degree of the remainder; + it is the number of columns to be deleted --- starting with the last column + forming the square matrix --- from the matrix resulting after the row deletions. + + References + ========== + Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``Sturm Sequences + and Modified Subresultant Polynomial Remainder Sequences.'' + Serdica Journal of Computing, Vol. 8, No 1, 29-46, 2014. + + """ + M = s1[:, :] # copy of matrix s1 + + # eliminate rdel rows from the first deg_g rows + for i in range(M.rows - deg_f - 1, M.rows - deg_f - rdel - 1, -1): + M.row_del(i) + + # eliminate rdel rows from the last deg_f rows + for i in range(M.rows - 1, M.rows - rdel - 1, -1): + M.row_del(i) + + # eliminate cdel columns + for i in range(cdel): + M.col_del(M.rows - 1) + + # define submatrix + Md = M[:, 0: M.rows] + + return Md.det() + +def subresultants_rem(p, q, x): + """ + p, q are polynomials in Z[x] or Q[x]. It is assumed + that degree(p, x) >= degree(q, x). + + Computes the subresultant prs of p and q in Z[x] or Q[x]; + the coefficients of the polynomials in the sequence are + subresultants. That is, they are determinants of appropriately + selected submatrices of sylvester1, Sylvester's matrix of 1840. + + To compute the coefficients polynomial divisions in Q[x] are + performed, using the function rem(p, q, x). The coefficients + of the remainders computed this way become subresultants by evaluating + one subresultant per remainder --- that of the leading coefficient. + This way we obtain the correct sign and value of the leading coefficient + of the remainder and we easily ``force'' the rest of the coefficients + to become subresultants. + + If the subresultant prs is complete, then it coincides with the + Euclidean sequence of the polynomials p, q. + + References + ========== + 1. Akritas, A. G.:``Three New Methods for Computing Subresultant + Polynomial Remainder Sequences (PRS's).'' Serdica Journal of Computing 9(1) (2015), 1-26. + + """ + # make sure neither p nor q is 0 + if p == 0 or q == 0: + return [p, q] + + # make sure proper degrees + f, g = p, q + n = deg_f = degree(f, x) + m = deg_g = degree(g, x) + if n == 0 and m == 0: + return [f, g] + if n < m: + n, m, deg_f, deg_g, f, g = m, n, deg_g, deg_f, g, f + if n > 0 and m == 0: + return [f, g] + + # initialize + s1 = sylvester(f, g, x, 1) + sr_list = [f, g] # subresultant list + + # main loop + while deg_g > 0: + r = rem(p, q, x) + d = degree(r, x) + if d < 0: + return sr_list + + # make coefficients subresultants evaluating ONE determinant + exp_deg = deg_g - 1 # expected degree + sign_value = correct_sign(n, m, s1, exp_deg, exp_deg - d) + r = simplify((r / LC(r, x)) * sign_value) + + # append poly with subresultant coeffs + sr_list.append(r) + + # update degrees and polys + deg_f, deg_g = deg_g, d + p, q = q, r + + # gcd is of degree > 0 ? + m = len(sr_list) + if sr_list[m - 1] == nan or sr_list[m - 1] == 0: + sr_list.pop(m - 1) + + return sr_list + +def pivot(M, i, j): + ''' + M is a matrix, and M[i, j] specifies the pivot element. + + All elements below M[i, j], in the j-th column, will + be zeroed, if they are not already 0, according to + Dodgson-Bareiss' integer preserving transformations. + + References + ========== + 1. Akritas, A. G.: ``A new method for computing polynomial greatest + common divisors and polynomial remainder sequences.'' + Numerische MatheMatik 52, 119-127, 1988. + + 2. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``On a Theorem + by Van Vleck Regarding Sturm Sequences.'' + Serdica Journal of Computing, 7, No 4, 101-134, 2013. + + ''' + ma = M[:, :] # copy of matrix M + rs = ma.rows # No. of rows + cs = ma.cols # No. of cols + for r in range(i+1, rs): + if ma[r, j] != 0: + for c in range(j + 1, cs): + ma[r, c] = ma[i, j] * ma[r, c] - ma[i, c] * ma[r, j] + ma[r, j] = 0 + return ma + +def rotate_r(L, k): + ''' + Rotates right by k. L is a row of a matrix or a list. + + ''' + ll = list(L) + if ll == []: + return [] + for i in range(k): + el = ll.pop(len(ll) - 1) + ll.insert(0, el) + return ll if isinstance(L, list) else Matrix([ll]) + +def rotate_l(L, k): + ''' + Rotates left by k. L is a row of a matrix or a list. + + ''' + ll = list(L) + if ll == []: + return [] + for i in range(k): + el = ll.pop(0) + ll.insert(len(ll) - 1, el) + return ll if isinstance(L, list) else Matrix([ll]) + +def row2poly(row, deg, x): + ''' + Converts the row of a matrix to a poly of degree deg and variable x. + Some entries at the beginning and/or at the end of the row may be zero. + + ''' + k = 0 + poly = [] + leng = len(row) + + # find the beginning of the poly ; i.e. the first + # non-zero element of the row + while row[k] == 0: + k = k + 1 + + # append the next deg + 1 elements to poly + for j in range( deg + 1): + if k + j <= leng: + poly.append(row[k + j]) + + return Poly(poly, x) + +def create_ma(deg_f, deg_g, row1, row2, col_num): + ''' + Creates a ``small'' matrix M to be triangularized. + + deg_f, deg_g are the degrees of the divident and of the + divisor polynomials respectively, deg_g > deg_f. + + The coefficients of the divident poly are the elements + in row2 and those of the divisor poly are the elements + in row1. + + col_num defines the number of columns of the matrix M. + + ''' + if deg_g - deg_f >= 1: + print('Reverse degrees') + return + + m = zeros(deg_f - deg_g + 2, col_num) + + for i in range(deg_f - deg_g + 1): + m[i, :] = rotate_r(row1, i) + m[deg_f - deg_g + 1, :] = row2 + + return m + +def find_degree(M, deg_f): + ''' + Finds the degree of the poly corresponding (after triangularization) + to the _last_ row of the ``small'' matrix M, created by create_ma(). + + deg_f is the degree of the divident poly. + If _last_ row is all 0's returns None. + + ''' + j = deg_f + for i in range(0, M.cols): + if M[M.rows - 1, i] == 0: + j = j - 1 + else: + return max(j, 0) + +def final_touches(s2, r, deg_g): + """ + s2 is sylvester2, r is the row pointer in s2, + deg_g is the degree of the poly last inserted in s2. + + After a gcd of degree > 0 has been found with Van Vleck's + method, and was inserted into s2, if its last term is not + in the last column of s2, then it is inserted as many + times as needed, rotated right by one each time, until + the condition is met. + + """ + R = s2.row(r-1) + + # find the first non zero term + for i in range(s2.cols): + if R[0,i] == 0: + continue + else: + break + + # missing rows until last term is in last column + mr = s2.cols - (i + deg_g + 1) + + # insert them by replacing the existing entries in the row + i = 0 + while mr != 0 and r + i < s2.rows : + s2[r + i, : ] = rotate_r(R, i + 1) + i += 1 + mr -= 1 + + return s2 + +def subresultants_vv(p, q, x, method = 0): + """ + p, q are polynomials in Z[x] (intended) or Q[x]. It is assumed + that degree(p, x) >= degree(q, x). + + Computes the subresultant prs of p, q by triangularizing, + in Z[x] or in Q[x], all the smaller matrices encountered in the + process of triangularizing sylvester2, Sylvester's matrix of 1853; + see references 1 and 2 for Van Vleck's method. With each remainder, + sylvester2 gets updated and is prepared to be printed if requested. + + If sylvester2 has small dimensions and you want to see the final, + triangularized matrix use this version with method=1; otherwise, + use either this version with method=0 (default) or the faster version, + subresultants_vv_2(p, q, x), where sylvester2 is used implicitly. + + Sylvester's matrix sylvester1 is also used to compute one + subresultant per remainder; namely, that of the leading + coefficient, in order to obtain the correct sign and to + force the remainder coefficients to become subresultants. + + If the subresultant prs is complete, then it coincides with the + Euclidean sequence of the polynomials p, q. + + If the final, triangularized matrix s2 is printed, then: + (a) if deg(p) - deg(q) > 1 or deg( gcd(p, q) ) > 0, several + of the last rows in s2 will remain unprocessed; + (b) if deg(p) - deg(q) == 0, p will not appear in the final matrix. + + References + ========== + 1. Akritas, A. G.: ``A new method for computing polynomial greatest + common divisors and polynomial remainder sequences.'' + Numerische MatheMatik 52, 119-127, 1988. + + 2. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``On a Theorem + by Van Vleck Regarding Sturm Sequences.'' + Serdica Journal of Computing, 7, No 4, 101-134, 2013. + + 3. Akritas, A. G.:``Three New Methods for Computing Subresultant + Polynomial Remainder Sequences (PRS's).'' Serdica Journal of Computing 9(1) (2015), 1-26. + + """ + # make sure neither p nor q is 0 + if p == 0 or q == 0: + return [p, q] + + # make sure proper degrees + f, g = p, q + n = deg_f = degree(f, x) + m = deg_g = degree(g, x) + if n == 0 and m == 0: + return [f, g] + if n < m: + n, m, deg_f, deg_g, f, g = m, n, deg_g, deg_f, g, f + if n > 0 and m == 0: + return [f, g] + + # initialize + s1 = sylvester(f, g, x, 1) + s2 = sylvester(f, g, x, 2) + sr_list = [f, g] + col_num = 2 * n # columns in s2 + + # make two rows (row0, row1) of poly coefficients + row0 = Poly(f, x, domain = QQ).all_coeffs() + leng0 = len(row0) + for i in range(col_num - leng0): + row0.append(0) + row0 = Matrix([row0]) + row1 = Poly(g,x, domain = QQ).all_coeffs() + leng1 = len(row1) + for i in range(col_num - leng1): + row1.append(0) + row1 = Matrix([row1]) + + # row pointer for deg_f - deg_g == 1; may be reset below + r = 2 + + # modify first rows of s2 matrix depending on poly degrees + if deg_f - deg_g > 1: + r = 1 + # replacing the existing entries in the rows of s2, + # insert row0 (deg_f - deg_g - 1) times, rotated each time + for i in range(deg_f - deg_g - 1): + s2[r + i, : ] = rotate_r(row0, i + 1) + r = r + deg_f - deg_g - 1 + # insert row1 (deg_f - deg_g) times, rotated each time + for i in range(deg_f - deg_g): + s2[r + i, : ] = rotate_r(row1, r + i) + r = r + deg_f - deg_g + + if deg_f - deg_g == 0: + r = 0 + + # main loop + while deg_g > 0: + # create a small matrix M, and triangularize it; + M = create_ma(deg_f, deg_g, row1, row0, col_num) + # will need only the first and last rows of M + for i in range(deg_f - deg_g + 1): + M1 = pivot(M, i, i) + M = M1[:, :] + + # treat last row of M as poly; find its degree + d = find_degree(M, deg_f) + if d is None: + break + exp_deg = deg_g - 1 + + # evaluate one determinant & make coefficients subresultants + sign_value = correct_sign(n, m, s1, exp_deg, exp_deg - d) + poly = row2poly(M[M.rows - 1, :], d, x) + temp2 = LC(poly, x) + poly = simplify((poly / temp2) * sign_value) + + # update s2 by inserting first row of M as needed + row0 = M[0, :] + for i in range(deg_g - d): + s2[r + i, :] = rotate_r(row0, r + i) + r = r + deg_g - d + + # update s2 by inserting last row of M as needed + row1 = rotate_l(M[M.rows - 1, :], deg_f - d) + row1 = (row1 / temp2) * sign_value + for i in range(deg_g - d): + s2[r + i, :] = rotate_r(row1, r + i) + r = r + deg_g - d + + # update degrees + deg_f, deg_g = deg_g, d + + # append poly with subresultant coeffs + sr_list.append(poly) + + # final touches to print the s2 matrix + if method != 0 and s2.rows > 2: + s2 = final_touches(s2, r, deg_g) + pprint(s2) + elif method != 0 and s2.rows == 2: + s2[1, :] = rotate_r(s2.row(1), 1) + pprint(s2) + + return sr_list + +def subresultants_vv_2(p, q, x): + """ + p, q are polynomials in Z[x] (intended) or Q[x]. It is assumed + that degree(p, x) >= degree(q, x). + + Computes the subresultant prs of p, q by triangularizing, + in Z[x] or in Q[x], all the smaller matrices encountered in the + process of triangularizing sylvester2, Sylvester's matrix of 1853; + see references 1 and 2 for Van Vleck's method. + + If the sylvester2 matrix has big dimensions use this version, + where sylvester2 is used implicitly. If you want to see the final, + triangularized matrix sylvester2, then use the first version, + subresultants_vv(p, q, x, 1). + + sylvester1, Sylvester's matrix of 1840, is also used to compute + one subresultant per remainder; namely, that of the leading + coefficient, in order to obtain the correct sign and to + ``force'' the remainder coefficients to become subresultants. + + If the subresultant prs is complete, then it coincides with the + Euclidean sequence of the polynomials p, q. + + References + ========== + 1. Akritas, A. G.: ``A new method for computing polynomial greatest + common divisors and polynomial remainder sequences.'' + Numerische MatheMatik 52, 119-127, 1988. + + 2. Akritas, A. G., G.I. Malaschonok and P.S. Vigklas: ``On a Theorem + by Van Vleck Regarding Sturm Sequences.'' + Serdica Journal of Computing, 7, No 4, 101-134, 2013. + + 3. Akritas, A. G.:``Three New Methods for Computing Subresultant + Polynomial Remainder Sequences (PRS's).'' Serdica Journal of Computing 9(1) (2015), 1-26. + + """ + # make sure neither p nor q is 0 + if p == 0 or q == 0: + return [p, q] + + # make sure proper degrees + f, g = p, q + n = deg_f = degree(f, x) + m = deg_g = degree(g, x) + if n == 0 and m == 0: + return [f, g] + if n < m: + n, m, deg_f, deg_g, f, g = m, n, deg_g, deg_f, g, f + if n > 0 and m == 0: + return [f, g] + + # initialize + s1 = sylvester(f, g, x, 1) + sr_list = [f, g] # subresultant list + col_num = 2 * n # columns in sylvester2 + + # make two rows (row0, row1) of poly coefficients + row0 = Poly(f, x, domain = QQ).all_coeffs() + leng0 = len(row0) + for i in range(col_num - leng0): + row0.append(0) + row0 = Matrix([row0]) + row1 = Poly(g,x, domain = QQ).all_coeffs() + leng1 = len(row1) + for i in range(col_num - leng1): + row1.append(0) + row1 = Matrix([row1]) + + # main loop + while deg_g > 0: + # create a small matrix M, and triangularize it + M = create_ma(deg_f, deg_g, row1, row0, col_num) + for i in range(deg_f - 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for generating Appell sequences.""" +from sympy.core.numbers import Rational as Q +from sympy.polys.polytools import Poly +from sympy.testing.pytest import raises +from sympy.polys.appellseqs import (bernoulli_poly, bernoulli_c_poly, + euler_poly, genocchi_poly, andre_poly) +from sympy.abc import x + +def test_bernoulli_poly(): + raises(ValueError, lambda: bernoulli_poly(-1, x)) + assert bernoulli_poly(1, x, polys=True) == Poly(x - Q(1,2)) + + assert bernoulli_poly(0, x) == 1 + assert bernoulli_poly(1, x) == x - Q(1,2) + assert bernoulli_poly(2, x) == x**2 - x + Q(1,6) + assert bernoulli_poly(3, x) == x**3 - Q(3,2)*x**2 + Q(1,2)*x + assert bernoulli_poly(4, x) == x**4 - 2*x**3 + x**2 - Q(1,30) + assert bernoulli_poly(5, x) == x**5 - Q(5,2)*x**4 + Q(5,3)*x**3 - Q(1,6)*x + assert bernoulli_poly(6, x) == x**6 - 3*x**5 + Q(5,2)*x**4 - Q(1,2)*x**2 + Q(1,42) + + assert bernoulli_poly(1).dummy_eq(x - Q(1,2)) + assert bernoulli_poly(1, polys=True) == Poly(x - Q(1,2)) + +def test_bernoulli_c_poly(): + raises(ValueError, lambda: bernoulli_c_poly(-1, x)) + assert bernoulli_c_poly(1, x, polys=True) == Poly(x, domain='QQ') + + assert bernoulli_c_poly(0, x) == 1 + assert bernoulli_c_poly(1, x) == x + assert bernoulli_c_poly(2, x) == x**2 - Q(1,3) + assert bernoulli_c_poly(3, x) == x**3 - x + assert bernoulli_c_poly(4, x) == x**4 - 2*x**2 + Q(7,15) + assert bernoulli_c_poly(5, x) == x**5 - Q(10,3)*x**3 + Q(7,3)*x + assert bernoulli_c_poly(6, x) == x**6 - 5*x**4 + 7*x**2 - Q(31,21) + + assert bernoulli_c_poly(1).dummy_eq(x) + assert bernoulli_c_poly(1, polys=True) == Poly(x, domain='QQ') + + assert 2**8 * bernoulli_poly(8, (x+1)/2).expand() == bernoulli_c_poly(8, x) + assert 2**9 * bernoulli_poly(9, (x+1)/2).expand() == bernoulli_c_poly(9, x) + +def test_genocchi_poly(): + raises(ValueError, lambda: genocchi_poly(-1, x)) + assert genocchi_poly(2, x, polys=True) == Poly(-2*x + 1) + + assert genocchi_poly(0, x) == 0 + assert genocchi_poly(1, x) == -1 + assert genocchi_poly(2, x) == 1 - 2*x + assert genocchi_poly(3, x) == 3*x - 3*x**2 + assert genocchi_poly(4, x) == -1 + 6*x**2 - 4*x**3 + assert genocchi_poly(5, x) == -5*x + 10*x**3 - 5*x**4 + assert genocchi_poly(6, x) == 3 - 15*x**2 + 15*x**4 - 6*x**5 + + assert genocchi_poly(2).dummy_eq(-2*x + 1) + assert genocchi_poly(2, polys=True) == Poly(-2*x + 1) + + assert 2 * (bernoulli_poly(8, x) - bernoulli_c_poly(8, x)) == genocchi_poly(8, x) + assert 2 * (bernoulli_poly(9, x) - bernoulli_c_poly(9, x)) == genocchi_poly(9, x) + +def test_euler_poly(): + raises(ValueError, lambda: euler_poly(-1, x)) + assert euler_poly(1, x, polys=True) == Poly(x - Q(1,2)) + + assert euler_poly(0, x) == 1 + assert euler_poly(1, x) == x - Q(1,2) + assert euler_poly(2, x) == x**2 - x + assert euler_poly(3, x) == x**3 - Q(3,2)*x**2 + Q(1,4) + assert euler_poly(4, x) == x**4 - 2*x**3 + x + assert euler_poly(5, x) == x**5 - Q(5,2)*x**4 + Q(5,2)*x**2 - Q(1,2) + assert euler_poly(6, x) == x**6 - 3*x**5 + 5*x**3 - 3*x + + assert euler_poly(1).dummy_eq(x - Q(1,2)) + assert euler_poly(1, polys=True) == Poly(x - Q(1,2)) + + assert genocchi_poly(9, x) == euler_poly(8, x) * -9 + assert genocchi_poly(10, x) == euler_poly(9, x) * -10 + +def test_andre_poly(): + raises(ValueError, lambda: andre_poly(-1, x)) + assert andre_poly(1, x, polys=True) == Poly(x) + + assert andre_poly(0, x) == 1 + assert andre_poly(1, x) == x + assert andre_poly(2, x) == x**2 - 1 + assert andre_poly(3, x) == x**3 - 3*x + assert andre_poly(4, x) == x**4 - 6*x**2 + 5 + assert andre_poly(5, x) == x**5 - 10*x**3 + 25*x + assert andre_poly(6, x) == x**6 - 15*x**4 + 75*x**2 - 61 + + assert andre_poly(1).dummy_eq(x) + assert andre_poly(1, polys=True) == Poly(x) diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_constructor.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_constructor.py new file mode 100644 index 0000000000000000000000000000000000000000..b02d8a4b360dd09b993bbed80cdec307d09908fc --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_constructor.py @@ -0,0 +1,236 @@ +"""Tests for tools for constructing domains for expressions. """ + +from sympy.testing.pytest import tooslow + +from sympy.polys.constructor import construct_domain +from sympy.polys.domains import ZZ, QQ, ZZ_I, QQ_I, RR, CC, EX +from sympy.polys.domains.realfield import RealField +from sympy.polys.domains.complexfield import ComplexField + +from sympy.core import (Catalan, GoldenRatio) +from sympy.core.numbers import (E, Float, I, Rational, pi) +from sympy.core.singleton import S +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import sin +from sympy import rootof + +from sympy.abc import x, y + + +def test_construct_domain(): + + assert construct_domain([1, 2, 3]) == (ZZ, [ZZ(1), ZZ(2), ZZ(3)]) + assert construct_domain([1, 2, 3], field=True) == (QQ, [QQ(1), QQ(2), QQ(3)]) + + assert construct_domain([S.One, S(2), S(3)]) == (ZZ, [ZZ(1), ZZ(2), ZZ(3)]) + assert construct_domain([S.One, S(2), S(3)], field=True) == (QQ, [QQ(1), QQ(2), QQ(3)]) + + assert construct_domain([S.Half, S(2)]) == (QQ, [QQ(1, 2), QQ(2)]) + result = construct_domain([3.14, 1, S.Half]) + assert isinstance(result[0], RealField) + assert result[1] == [RR(3.14), RR(1.0), RR(0.5)] + + result = construct_domain([3.14, I, S.Half]) + assert isinstance(result[0], ComplexField) + assert result[1] == [CC(3.14), CC(1.0j), CC(0.5)] + + assert construct_domain([1.0+I]) == (CC, [CC(1.0, 1.0)]) + assert construct_domain([2.0+3.0*I]) == (CC, [CC(2.0, 3.0)]) + + assert construct_domain([1, I]) == (ZZ_I, [ZZ_I(1, 0), ZZ_I(0, 1)]) + assert construct_domain([1, I/2]) == (QQ_I, [QQ_I(1, 0), QQ_I(0, S.Half)]) + + assert construct_domain([3.14, sqrt(2)], extension=None) == (EX, [EX(3.14), EX(sqrt(2))]) + assert construct_domain([3.14, sqrt(2)], extension=True) == (EX, [EX(3.14), EX(sqrt(2))]) + + assert construct_domain([1, sqrt(2)], extension=None) == (EX, [EX(1), EX(sqrt(2))]) + + assert construct_domain([x, sqrt(x)]) == (EX, [EX(x), EX(sqrt(x))]) + assert construct_domain([x, sqrt(x), sqrt(y)]) == (EX, [EX(x), EX(sqrt(x)), EX(sqrt(y))]) + + alg = QQ.algebraic_field(sqrt(2)) + + assert construct_domain([7, S.Half, sqrt(2)], extension=True) == \ + (alg, [alg.convert(7), alg.convert(S.Half), alg.convert(sqrt(2))]) + + alg = QQ.algebraic_field(sqrt(2) + sqrt(3)) + + assert construct_domain([7, sqrt(2), sqrt(3)], extension=True) == \ + (alg, [alg.convert(7), alg.convert(sqrt(2)), alg.convert(sqrt(3))]) + + dom = ZZ[x] + + assert construct_domain([2*x, 3]) == \ + (dom, [dom.convert(2*x), dom.convert(3)]) + + dom = ZZ[x, y] + + assert construct_domain([2*x, 3*y]) == \ + (dom, [dom.convert(2*x), dom.convert(3*y)]) + + dom = QQ[x] + + assert construct_domain([x/2, 3]) == \ + (dom, [dom.convert(x/2), dom.convert(3)]) + + dom = QQ[x, y] + + assert construct_domain([x/2, 3*y]) == \ + (dom, [dom.convert(x/2), dom.convert(3*y)]) + + dom = ZZ_I[x] + + assert construct_domain([2*x, I]) == \ + (dom, [dom.convert(2*x), dom.convert(I)]) + + dom = ZZ_I[x, y] + + assert construct_domain([2*x, I*y]) == \ + (dom, [dom.convert(2*x), dom.convert(I*y)]) + + dom = QQ_I[x] + + assert construct_domain([x/2, I]) == \ + (dom, [dom.convert(x/2), dom.convert(I)]) + + dom = QQ_I[x, y] + + assert construct_domain([x/2, I*y]) == \ + (dom, [dom.convert(x/2), dom.convert(I*y)]) + + dom = RR[x] + + assert construct_domain([x/2, 3.5]) == \ + (dom, [dom.convert(x/2), dom.convert(3.5)]) + + dom = RR[x, y] + + assert construct_domain([x/2, 3.5*y]) == \ + (dom, [dom.convert(x/2), dom.convert(3.5*y)]) + + dom = CC[x] + + assert construct_domain([I*x/2, 3.5]) == \ + (dom, [dom.convert(I*x/2), dom.convert(3.5)]) + + dom = CC[x, y] + + assert construct_domain([I*x/2, 3.5*y]) == \ + (dom, [dom.convert(I*x/2), dom.convert(3.5*y)]) + + dom = CC[x] + + assert construct_domain([x/2, I*3.5]) == \ + (dom, [dom.convert(x/2), dom.convert(I*3.5)]) + + dom = CC[x, y] + + assert construct_domain([x/2, I*3.5*y]) == \ + (dom, [dom.convert(x/2), dom.convert(I*3.5*y)]) + + dom = ZZ.frac_field(x) + + assert construct_domain([2/x, 3]) == \ + (dom, [dom.convert(2/x), dom.convert(3)]) + + dom = ZZ.frac_field(x, y) + + assert construct_domain([2/x, 3*y]) == \ + (dom, [dom.convert(2/x), dom.convert(3*y)]) + + dom = RR.frac_field(x) + + assert construct_domain([2/x, 3.5]) == \ + (dom, [dom.convert(2/x), dom.convert(3.5)]) + + dom = RR.frac_field(x, y) + + assert construct_domain([2/x, 3.5*y]) == \ + (dom, [dom.convert(2/x), dom.convert(3.5*y)]) + + dom = RealField(prec=336)[x] + + assert construct_domain([pi.evalf(100)*x]) == \ + (dom, [dom.convert(pi.evalf(100)*x)]) + + assert construct_domain(2) == (ZZ, ZZ(2)) + assert construct_domain(S(2)/3) == (QQ, QQ(2, 3)) + assert construct_domain(Rational(2, 3)) == (QQ, QQ(2, 3)) + + assert construct_domain({}) == (ZZ, {}) + + +def test_complex_exponential(): + w = exp(-I*2*pi/3, evaluate=False) + alg = QQ.algebraic_field(w) + assert construct_domain([w**2, w, 1], extension=True) == ( + alg, + [alg.convert(w**2), + alg.convert(w), + alg.convert(1)] + ) + + +def test_rootof(): + r1 = rootof(x**3 + x + 1, 0) + r2 = rootof(x**3 + x + 1, 1) + K1 = QQ.algebraic_field(r1) + K2 = QQ.algebraic_field(r2) + assert construct_domain([r1]) == (EX, [EX(r1)]) + assert construct_domain([r2]) == (EX, [EX(r2)]) + assert construct_domain([r1, r2]) == (EX, [EX(r1), EX(r2)]) + + assert construct_domain([r1], extension=True) == ( + K1, [K1.from_sympy(r1)]) + assert construct_domain([r2], extension=True) == ( + K2, [K2.from_sympy(r2)]) + + +@tooslow +def test_rootof_primitive_element(): + r1 = rootof(x**3 + x + 1, 0) + r2 = rootof(x**3 + x + 1, 1) + K12 = QQ.algebraic_field(r1 + r2) + assert construct_domain([r1, r2], extension=True) == ( + K12, [K12.from_sympy(r1), K12.from_sympy(r2)]) + + +def test_composite_option(): + assert construct_domain({(1,): sin(y)}, composite=False) == \ + (EX, {(1,): EX(sin(y))}) + + assert construct_domain({(1,): y}, composite=False) == \ + (EX, {(1,): EX(y)}) + + assert construct_domain({(1, 1): 1}, composite=False) == \ + (ZZ, {(1, 1): 1}) + + assert construct_domain({(1, 0): y}, composite=False) == \ + (EX, {(1, 0): EX(y)}) + + +def test_precision(): + f1 = Float("1.01") + f2 = Float("1.0000000000000000000001") + for u in [1, 1e-2, 1e-6, 1e-13, 1e-14, 1e-16, 1e-20, 1e-100, 1e-300, + f1, f2]: + result = construct_domain([u]) + v = float(result[1][0]) + assert abs(u - v) / u < 1e-14 # Test relative accuracy + + result = construct_domain([f1]) + y = result[1][0] + assert y-1 > 1e-50 + + result = construct_domain([f2]) + y = result[1][0] + assert y-1 > 1e-50 + + +def test_issue_11538(): + for n in [E, pi, Catalan]: + assert construct_domain(n)[0] == ZZ[n] + assert construct_domain(x + n)[0] == ZZ[x, n] + assert construct_domain(GoldenRatio)[0] == EX + assert construct_domain(x + GoldenRatio)[0] == EX diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_densearith.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_densearith.py new file mode 100644 index 0000000000000000000000000000000000000000..ebb29d50867ad578274ed11c766e0515d8e4da35 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_densearith.py @@ -0,0 +1,1007 @@ +"""Tests for dense recursive polynomials' arithmetics. """ + +from sympy.external.gmpy import GROUND_TYPES + +from sympy.polys.densebasic import ( + dup_normal, dmp_normal, +) + +from sympy.polys.densearith import ( + dup_add_term, dmp_add_term, + dup_sub_term, dmp_sub_term, + dup_mul_term, dmp_mul_term, + dup_add_ground, dmp_add_ground, + dup_sub_ground, dmp_sub_ground, + dup_mul_ground, dmp_mul_ground, + dup_quo_ground, dmp_quo_ground, + dup_exquo_ground, dmp_exquo_ground, + dup_lshift, dup_rshift, + dup_abs, dmp_abs, + dup_neg, dmp_neg, + dup_add, dmp_add, + dup_sub, dmp_sub, + dup_mul, dmp_mul, + dup_sqr, dmp_sqr, + dup_pow, dmp_pow, + dup_add_mul, dmp_add_mul, + dup_sub_mul, dmp_sub_mul, + dup_pdiv, dup_prem, dup_pquo, dup_pexquo, + dmp_pdiv, dmp_prem, dmp_pquo, dmp_pexquo, + dup_rr_div, dmp_rr_div, + dup_ff_div, dmp_ff_div, + dup_div, dup_rem, dup_quo, dup_exquo, + dmp_div, dmp_rem, dmp_quo, dmp_exquo, + dup_max_norm, dmp_max_norm, + dup_l1_norm, dmp_l1_norm, + dup_l2_norm_squared, dmp_l2_norm_squared, + dup_expand, dmp_expand, +) + +from sympy.polys.polyerrors import ( + ExactQuotientFailed, +) + +from sympy.polys.specialpolys import f_polys, Symbol, Poly +from sympy.polys.domains import FF, ZZ, QQ, CC + +from sympy.testing.pytest import raises + +x = Symbol('x') + +f_0, f_1, f_2, f_3, f_4, f_5, f_6 = [ f.to_dense() for f in f_polys() ] +F_0 = dmp_mul_ground(dmp_normal(f_0, 2, QQ), QQ(1, 7), 2, QQ) + +def test_dup_add_term(): + f = dup_normal([], ZZ) + + assert dup_add_term(f, ZZ(0), 0, ZZ) == dup_normal([], ZZ) + + assert dup_add_term(f, ZZ(1), 0, ZZ) == dup_normal([1], ZZ) + assert dup_add_term(f, ZZ(1), 1, ZZ) == dup_normal([1, 0], ZZ) + assert dup_add_term(f, ZZ(1), 2, ZZ) == dup_normal([1, 0, 0], ZZ) + + f = dup_normal([1, 1, 1], ZZ) + + assert dup_add_term(f, ZZ(1), 0, ZZ) == dup_normal([1, 1, 2], ZZ) + assert dup_add_term(f, ZZ(1), 1, ZZ) == dup_normal([1, 2, 1], ZZ) + assert dup_add_term(f, ZZ(1), 2, ZZ) == dup_normal([2, 1, 1], ZZ) + + assert dup_add_term(f, ZZ(1), 3, ZZ) == dup_normal([1, 1, 1, 1], ZZ) + assert dup_add_term(f, ZZ(1), 4, ZZ) == dup_normal([1, 0, 1, 1, 1], ZZ) + assert dup_add_term(f, ZZ(1), 5, ZZ) == dup_normal([1, 0, 0, 1, 1, 1], ZZ) + assert dup_add_term( + f, ZZ(1), 6, ZZ) == dup_normal([1, 0, 0, 0, 1, 1, 1], ZZ) + + assert dup_add_term(f, ZZ(-1), 2, ZZ) == dup_normal([1, 1], ZZ) + + +def test_dmp_add_term(): + assert dmp_add_term([ZZ(1), ZZ(1), ZZ(1)], ZZ(1), 2, 0, ZZ) == \ + dup_add_term([ZZ(1), ZZ(1), ZZ(1)], ZZ(1), 2, ZZ) + assert dmp_add_term(f_0, [[]], 3, 2, ZZ) == f_0 + assert dmp_add_term(F_0, [[]], 3, 2, QQ) == F_0 + + +def test_dup_sub_term(): + f = dup_normal([], ZZ) + + assert dup_sub_term(f, ZZ(0), 0, ZZ) == dup_normal([], ZZ) + + assert dup_sub_term(f, ZZ(1), 0, ZZ) == dup_normal([-1], ZZ) + assert dup_sub_term(f, ZZ(1), 1, ZZ) == dup_normal([-1, 0], ZZ) + assert dup_sub_term(f, ZZ(1), 2, ZZ) == dup_normal([-1, 0, 0], ZZ) + + f = dup_normal([1, 1, 1], ZZ) + + assert dup_sub_term(f, ZZ(2), 0, ZZ) == dup_normal([ 1, 1, -1], ZZ) + assert dup_sub_term(f, ZZ(2), 1, ZZ) == dup_normal([ 1, -1, 1], ZZ) + assert dup_sub_term(f, ZZ(2), 2, ZZ) == dup_normal([-1, 1, 1], ZZ) + + assert dup_sub_term(f, ZZ(1), 3, ZZ) == dup_normal([-1, 1, 1, 1], ZZ) + assert dup_sub_term(f, ZZ(1), 4, ZZ) == dup_normal([-1, 0, 1, 1, 1], ZZ) + assert dup_sub_term(f, ZZ(1), 5, ZZ) == dup_normal([-1, 0, 0, 1, 1, 1], ZZ) + assert dup_sub_term( + f, ZZ(1), 6, ZZ) == dup_normal([-1, 0, 0, 0, 1, 1, 1], ZZ) + + assert dup_sub_term(f, ZZ(1), 2, ZZ) == dup_normal([1, 1], ZZ) + + +def test_dmp_sub_term(): + assert dmp_sub_term([ZZ(1), ZZ(1), ZZ(1)], ZZ(1), 2, 0, ZZ) == \ + dup_sub_term([ZZ(1), ZZ(1), ZZ(1)], ZZ(1), 2, ZZ) + assert dmp_sub_term(f_0, [[]], 3, 2, ZZ) == f_0 + assert dmp_sub_term(F_0, [[]], 3, 2, QQ) == F_0 + + +def test_dup_mul_term(): + f = dup_normal([], ZZ) + + assert dup_mul_term(f, ZZ(2), 3, ZZ) == dup_normal([], ZZ) + + f = dup_normal([1, 1], ZZ) + + assert dup_mul_term(f, ZZ(0), 3, ZZ) == dup_normal([], ZZ) + + f = dup_normal([1, 2, 3], ZZ) + + assert dup_mul_term(f, ZZ(2), 0, ZZ) == dup_normal([2, 4, 6], ZZ) + assert dup_mul_term(f, ZZ(2), 1, ZZ) == dup_normal([2, 4, 6, 0], ZZ) + assert dup_mul_term(f, ZZ(2), 2, ZZ) == dup_normal([2, 4, 6, 0, 0], ZZ) + assert dup_mul_term(f, ZZ(2), 3, ZZ) == dup_normal([2, 4, 6, 0, 0, 0], ZZ) + + +def test_dmp_mul_term(): + assert dmp_mul_term([ZZ(1), ZZ(2), ZZ(3)], ZZ(2), 1, 0, ZZ) == \ + dup_mul_term([ZZ(1), ZZ(2), ZZ(3)], ZZ(2), 1, ZZ) + + assert dmp_mul_term([[]], [ZZ(2)], 3, 1, ZZ) == [[]] + assert dmp_mul_term([[ZZ(1)]], [], 3, 1, ZZ) == [[]] + + assert dmp_mul_term([[ZZ(1), ZZ(2)], [ZZ(3)]], [ZZ(2)], 2, 1, ZZ) == \ + [[ZZ(2), ZZ(4)], [ZZ(6)], [], []] + + assert dmp_mul_term([[]], [QQ(2, 3)], 3, 1, QQ) == [[]] + assert dmp_mul_term([[QQ(1, 2)]], [], 3, 1, QQ) == [[]] + + assert dmp_mul_term([[QQ(1, 5), QQ(2, 5)], [QQ(3, 5)]], [QQ(2, 3)], 2, 1, QQ) == \ + [[QQ(2, 15), QQ(4, 15)], [QQ(6, 15)], [], []] + + +def test_dup_add_ground(): + f = ZZ.map([1, 2, 3, 4]) + g = ZZ.map([1, 2, 3, 8]) + + assert dup_add_ground(f, ZZ(4), ZZ) == g + + +def test_dmp_add_ground(): + f = ZZ.map([[1], [2], [3], [4]]) + g = ZZ.map([[1], [2], [3], [8]]) + + assert dmp_add_ground(f, ZZ(4), 1, ZZ) == g + + +def test_dup_sub_ground(): + f = ZZ.map([1, 2, 3, 4]) + g = ZZ.map([1, 2, 3, 0]) + + assert dup_sub_ground(f, ZZ(4), ZZ) == g + + +def test_dmp_sub_ground(): + f = ZZ.map([[1], [2], [3], [4]]) + g = ZZ.map([[1], [2], [3], []]) + + assert dmp_sub_ground(f, ZZ(4), 1, ZZ) == g + + +def test_dup_mul_ground(): + f = dup_normal([], ZZ) + + assert dup_mul_ground(f, ZZ(2), ZZ) == dup_normal([], ZZ) + + f = dup_normal([1, 2, 3], ZZ) + + assert dup_mul_ground(f, ZZ(0), ZZ) == dup_normal([], ZZ) + assert dup_mul_ground(f, ZZ(2), ZZ) == dup_normal([2, 4, 6], ZZ) + + +def test_dmp_mul_ground(): + assert dmp_mul_ground(f_0, ZZ(2), 2, ZZ) == [ + [[ZZ(2), ZZ(4), ZZ(6)], [ZZ(4)]], + [[ZZ(6)]], + [[ZZ(8), ZZ(10), ZZ(12)], [ZZ(2), ZZ(4), ZZ(2)], [ZZ(2)]] + ] + + assert dmp_mul_ground(F_0, QQ(1, 2), 2, QQ) == [ + [[QQ(1, 14), QQ(2, 14), QQ(3, 14)], [QQ(2, 14)]], + [[QQ(3, 14)]], + [[QQ(4, 14), QQ(5, 14), QQ(6, 14)], [QQ(1, 14), QQ(2, 14), + QQ(1, 14)], [QQ(1, 14)]] + ] + + +def test_dup_quo_ground(): + raises(ZeroDivisionError, lambda: dup_quo_ground(dup_normal([1, 2, + 3], ZZ), ZZ(0), ZZ)) + + f = dup_normal([], ZZ) + + assert dup_quo_ground(f, ZZ(3), ZZ) == dup_normal([], ZZ) + + f = dup_normal([6, 2, 8], ZZ) + + assert dup_quo_ground(f, ZZ(1), ZZ) == f + assert dup_quo_ground(f, ZZ(2), ZZ) == dup_normal([3, 1, 4], ZZ) + + assert dup_quo_ground(f, ZZ(3), ZZ) == dup_normal([2, 0, 2], ZZ) + + f = dup_normal([6, 2, 8], QQ) + + assert dup_quo_ground(f, QQ(1), QQ) == f + assert dup_quo_ground(f, QQ(2), QQ) == [QQ(3), QQ(1), QQ(4)] + assert dup_quo_ground(f, QQ(7), QQ) == [QQ(6, 7), QQ(2, 7), QQ(8, 7)] + + +def test_dup_exquo_ground(): + raises(ZeroDivisionError, lambda: dup_exquo_ground(dup_normal([1, + 2, 3], ZZ), ZZ(0), ZZ)) + raises(ExactQuotientFailed, lambda: dup_exquo_ground(dup_normal([1, + 2, 3], ZZ), ZZ(3), ZZ)) + + f = dup_normal([], ZZ) + + assert dup_exquo_ground(f, ZZ(3), ZZ) == dup_normal([], ZZ) + + f = dup_normal([6, 2, 8], ZZ) + + assert dup_exquo_ground(f, ZZ(1), ZZ) == f + assert dup_exquo_ground(f, ZZ(2), ZZ) == dup_normal([3, 1, 4], ZZ) + + f = dup_normal([6, 2, 8], QQ) + + assert dup_exquo_ground(f, QQ(1), QQ) == f + assert dup_exquo_ground(f, QQ(2), QQ) == [QQ(3), QQ(1), QQ(4)] + assert dup_exquo_ground(f, QQ(7), QQ) == [QQ(6, 7), QQ(2, 7), QQ(8, 7)] + + +def test_dmp_quo_ground(): + f = dmp_normal([[6], [2], [8]], 1, ZZ) + + assert dmp_quo_ground(f, ZZ(1), 1, ZZ) == f + assert dmp_quo_ground( + f, ZZ(2), 1, ZZ) == dmp_normal([[3], [1], [4]], 1, ZZ) + + assert dmp_normal(dmp_quo_ground( + f, ZZ(3), 1, ZZ), 1, ZZ) == dmp_normal([[2], [], [2]], 1, ZZ) + + +def test_dmp_exquo_ground(): + f = dmp_normal([[6], [2], [8]], 1, ZZ) + + assert dmp_exquo_ground(f, ZZ(1), 1, ZZ) == f + assert dmp_exquo_ground( + f, ZZ(2), 1, ZZ) == dmp_normal([[3], [1], [4]], 1, ZZ) + + +def test_dup_lshift(): + assert dup_lshift([], 3, ZZ) == [] + assert dup_lshift([1], 3, ZZ) == [1, 0, 0, 0] + + +def test_dup_rshift(): + assert dup_rshift([], 3, ZZ) == [] + assert dup_rshift([1, 0, 0, 0], 3, ZZ) == [1] + + +def test_dup_abs(): + assert dup_abs([], ZZ) == [] + assert dup_abs([ZZ( 1)], ZZ) == [ZZ(1)] + assert dup_abs([ZZ(-7)], ZZ) == [ZZ(7)] + assert dup_abs([ZZ(-1), ZZ(2), ZZ(3)], ZZ) == [ZZ(1), ZZ(2), ZZ(3)] + + assert dup_abs([], QQ) == [] + assert dup_abs([QQ( 1, 2)], QQ) == [QQ(1, 2)] + assert dup_abs([QQ(-7, 3)], QQ) == [QQ(7, 3)] + assert dup_abs( + [QQ(-1, 7), QQ(2, 7), QQ(3, 7)], QQ) == [QQ(1, 7), QQ(2, 7), QQ(3, 7)] + + +def test_dmp_abs(): + assert dmp_abs([ZZ(-1)], 0, ZZ) == [ZZ(1)] + assert dmp_abs([QQ(-1, 2)], 0, QQ) == [QQ(1, 2)] + + assert dmp_abs([[[]]], 2, ZZ) == [[[]]] + assert dmp_abs([[[ZZ(1)]]], 2, ZZ) == [[[ZZ(1)]]] + assert dmp_abs([[[ZZ(-7)]]], 2, ZZ) == [[[ZZ(7)]]] + + assert dmp_abs([[[]]], 2, QQ) == [[[]]] + assert dmp_abs([[[QQ(1, 2)]]], 2, QQ) == [[[QQ(1, 2)]]] + assert dmp_abs([[[QQ(-7, 9)]]], 2, QQ) == [[[QQ(7, 9)]]] + + +def test_dup_neg(): + assert dup_neg([], ZZ) == [] + assert dup_neg([ZZ(1)], ZZ) == [ZZ(-1)] + assert dup_neg([ZZ(-7)], ZZ) == [ZZ(7)] + assert dup_neg([ZZ(-1), ZZ(2), ZZ(3)], ZZ) == [ZZ(1), ZZ(-2), ZZ(-3)] + + assert dup_neg([], QQ) == [] + assert dup_neg([QQ(1, 2)], QQ) == [QQ(-1, 2)] + assert dup_neg([QQ(-7, 9)], QQ) == [QQ(7, 9)] + assert dup_neg([QQ( + -1, 7), QQ(2, 7), QQ(3, 7)], QQ) == [QQ(1, 7), QQ(-2, 7), QQ(-3, 7)] + + +def test_dmp_neg(): + assert dmp_neg([ZZ(-1)], 0, ZZ) == [ZZ(1)] + assert dmp_neg([QQ(-1, 2)], 0, QQ) == [QQ(1, 2)] + + assert dmp_neg([[[]]], 2, ZZ) == [[[]]] + assert dmp_neg([[[ZZ(1)]]], 2, ZZ) == [[[ZZ(-1)]]] + assert dmp_neg([[[ZZ(-7)]]], 2, ZZ) == [[[ZZ(7)]]] + + assert dmp_neg([[[]]], 2, QQ) == [[[]]] + assert dmp_neg([[[QQ(1, 9)]]], 2, QQ) == [[[QQ(-1, 9)]]] + assert dmp_neg([[[QQ(-7, 9)]]], 2, QQ) == [[[QQ(7, 9)]]] + + +def test_dup_add(): + assert dup_add([], [], ZZ) == [] + assert dup_add([ZZ(1)], [], ZZ) == [ZZ(1)] + assert dup_add([], [ZZ(1)], ZZ) == [ZZ(1)] + assert dup_add([ZZ(1)], [ZZ(1)], ZZ) == [ZZ(2)] + assert dup_add([ZZ(1)], [ZZ(2)], ZZ) == [ZZ(3)] + + assert dup_add([ZZ(1), ZZ(2)], [ZZ(1)], ZZ) == [ZZ(1), ZZ(3)] + assert dup_add([ZZ(1)], [ZZ(1), ZZ(2)], ZZ) == [ZZ(1), ZZ(3)] + + assert dup_add([ZZ(1), ZZ( + 2), ZZ(3)], [ZZ(8), ZZ(9), ZZ(10)], ZZ) == [ZZ(9), ZZ(11), ZZ(13)] + + assert dup_add([], [], QQ) == [] + assert dup_add([QQ(1, 2)], [], QQ) == [QQ(1, 2)] + assert dup_add([], [QQ(1, 2)], QQ) == [QQ(1, 2)] + assert dup_add([QQ(1, 4)], [QQ(1, 4)], QQ) == [QQ(1, 2)] + assert dup_add([QQ(1, 4)], [QQ(1, 2)], QQ) == [QQ(3, 4)] + + assert dup_add([QQ(1, 2), QQ(2, 3)], [QQ(1)], QQ) == [QQ(1, 2), QQ(5, 3)] + assert dup_add([QQ(1)], [QQ(1, 2), QQ(2, 3)], QQ) == [QQ(1, 2), QQ(5, 3)] + + assert dup_add([QQ(1, 7), QQ(2, 7), QQ(3, 7)], [QQ( + 8, 7), QQ(9, 7), QQ(10, 7)], QQ) == [QQ(9, 7), QQ(11, 7), QQ(13, 7)] + + +def test_dmp_add(): + assert dmp_add([ZZ(1), ZZ(2)], [ZZ(1)], 0, ZZ) == \ + dup_add([ZZ(1), ZZ(2)], [ZZ(1)], ZZ) + assert dmp_add([QQ(1, 2), QQ(2, 3)], [QQ(1)], 0, QQ) == \ + dup_add([QQ(1, 2), QQ(2, 3)], [QQ(1)], QQ) + + assert dmp_add([[[]]], [[[]]], 2, ZZ) == [[[]]] + assert dmp_add([[[ZZ(1)]]], [[[]]], 2, ZZ) == [[[ZZ(1)]]] + assert dmp_add([[[]]], [[[ZZ(1)]]], 2, ZZ) == [[[ZZ(1)]]] + assert dmp_add([[[ZZ(2)]]], [[[ZZ(1)]]], 2, ZZ) == [[[ZZ(3)]]] + assert dmp_add([[[ZZ(1)]]], [[[ZZ(2)]]], 2, ZZ) == [[[ZZ(3)]]] + + assert dmp_add([[[]]], [[[]]], 2, QQ) == [[[]]] + assert dmp_add([[[QQ(1, 2)]]], [[[]]], 2, QQ) == [[[QQ(1, 2)]]] + assert dmp_add([[[]]], [[[QQ(1, 2)]]], 2, QQ) == [[[QQ(1, 2)]]] + assert dmp_add([[[QQ(2, 7)]]], [[[QQ(1, 7)]]], 2, QQ) == [[[QQ(3, 7)]]] + assert dmp_add([[[QQ(1, 7)]]], [[[QQ(2, 7)]]], 2, QQ) == [[[QQ(3, 7)]]] + + +def test_dup_sub(): + assert dup_sub([], [], ZZ) == [] + assert dup_sub([ZZ(1)], [], ZZ) == [ZZ(1)] + assert dup_sub([], [ZZ(1)], ZZ) == [ZZ(-1)] + assert dup_sub([ZZ(1)], [ZZ(1)], ZZ) == [] + assert dup_sub([ZZ(1)], [ZZ(2)], ZZ) == [ZZ(-1)] + + assert dup_sub([ZZ(1), ZZ(2)], [ZZ(1)], ZZ) == [ZZ(1), ZZ(1)] + assert dup_sub([ZZ(1)], [ZZ(1), ZZ(2)], ZZ) == [ZZ(-1), ZZ(-1)] + + assert dup_sub([ZZ(3), ZZ( + 2), ZZ(1)], [ZZ(8), ZZ(9), ZZ(10)], ZZ) == [ZZ(-5), ZZ(-7), ZZ(-9)] + + assert dup_sub([], [], QQ) == [] + assert dup_sub([QQ(1, 2)], [], QQ) == [QQ(1, 2)] + assert dup_sub([], [QQ(1, 2)], QQ) == [QQ(-1, 2)] + assert dup_sub([QQ(1, 3)], [QQ(1, 3)], QQ) == [] + assert dup_sub([QQ(1, 3)], [QQ(2, 3)], QQ) == [QQ(-1, 3)] + + assert dup_sub([QQ(1, 7), QQ(2, 7)], [QQ(1)], QQ) == [QQ(1, 7), QQ(-5, 7)] + assert dup_sub([QQ(1)], [QQ(1, 7), QQ(2, 7)], QQ) == [QQ(-1, 7), QQ(5, 7)] + + assert dup_sub([QQ(3, 7), QQ(2, 7), QQ(1, 7)], [QQ( + 8, 7), QQ(9, 7), QQ(10, 7)], QQ) == [QQ(-5, 7), QQ(-7, 7), QQ(-9, 7)] + + +def test_dmp_sub(): + assert dmp_sub([ZZ(1), ZZ(2)], [ZZ(1)], 0, ZZ) == \ + dup_sub([ZZ(1), ZZ(2)], [ZZ(1)], ZZ) + assert dmp_sub([QQ(1, 2), QQ(2, 3)], [QQ(1)], 0, QQ) == \ + dup_sub([QQ(1, 2), QQ(2, 3)], [QQ(1)], QQ) + + assert dmp_sub([[[]]], [[[]]], 2, ZZ) == [[[]]] + assert dmp_sub([[[ZZ(1)]]], [[[]]], 2, ZZ) == [[[ZZ(1)]]] + assert dmp_sub([[[]]], [[[ZZ(1)]]], 2, ZZ) == [[[ZZ(-1)]]] + assert dmp_sub([[[ZZ(2)]]], [[[ZZ(1)]]], 2, ZZ) == [[[ZZ(1)]]] + assert dmp_sub([[[ZZ(1)]]], [[[ZZ(2)]]], 2, ZZ) == [[[ZZ(-1)]]] + + assert dmp_sub([[[]]], [[[]]], 2, QQ) == [[[]]] + assert dmp_sub([[[QQ(1, 2)]]], [[[]]], 2, QQ) == [[[QQ(1, 2)]]] + assert dmp_sub([[[]]], [[[QQ(1, 2)]]], 2, QQ) == [[[QQ(-1, 2)]]] + assert dmp_sub([[[QQ(2, 7)]]], [[[QQ(1, 7)]]], 2, QQ) == [[[QQ(1, 7)]]] + assert dmp_sub([[[QQ(1, 7)]]], [[[QQ(2, 7)]]], 2, QQ) == [[[QQ(-1, 7)]]] + + +def test_dup_add_mul(): + assert dup_add_mul([ZZ(1), ZZ(2), ZZ(3)], [ZZ(3), ZZ(2), ZZ(1)], + [ZZ(1), ZZ(2)], ZZ) == [ZZ(3), ZZ(9), ZZ(7), ZZ(5)] + assert dmp_add_mul([[ZZ(1), ZZ(2)], [ZZ(3)]], [[ZZ(3)], [ZZ(2), ZZ(1)]], + [[ZZ(1)], [ZZ(2)]], 1, ZZ) == [[ZZ(3)], [ZZ(3), ZZ(9)], [ZZ(4), ZZ(5)]] + + +def test_dup_sub_mul(): + assert dup_sub_mul([ZZ(1), ZZ(2), ZZ(3)], [ZZ(3), ZZ(2), ZZ(1)], + [ZZ(1), ZZ(2)], ZZ) == [ZZ(-3), ZZ(-7), ZZ(-3), ZZ(1)] + assert dmp_sub_mul([[ZZ(1), ZZ(2)], [ZZ(3)]], [[ZZ(3)], [ZZ(2), ZZ(1)]], + [[ZZ(1)], [ZZ(2)]], 1, ZZ) == [[ZZ(-3)], [ZZ(-1), ZZ(-5)], [ZZ(-4), ZZ(1)]] + + +def test_dup_mul(): + assert dup_mul([], [], ZZ) == [] + assert dup_mul([], [ZZ(1)], ZZ) == [] + assert dup_mul([ZZ(1)], [], ZZ) == [] + assert dup_mul([ZZ(1)], [ZZ(1)], ZZ) == [ZZ(1)] + assert dup_mul([ZZ(5)], [ZZ(7)], ZZ) == [ZZ(35)] + + assert dup_mul([], [], QQ) == [] + assert dup_mul([], [QQ(1, 2)], QQ) == [] + assert dup_mul([QQ(1, 2)], [], QQ) == [] + assert dup_mul([QQ(1, 2)], [QQ(4, 7)], QQ) == [QQ(2, 7)] + assert dup_mul([QQ(5, 7)], [QQ(3, 7)], QQ) == [QQ(15, 49)] + + f = dup_normal([3, 0, 0, 6, 1, 2], ZZ) + g = dup_normal([4, 0, 1, 0], ZZ) + h = dup_normal([12, 0, 3, 24, 4, 14, 1, 2, 0], ZZ) + + assert dup_mul(f, g, ZZ) == h + assert dup_mul(g, f, ZZ) == h + + f = dup_normal([2, 0, 0, 1, 7], ZZ) + h = dup_normal([4, 0, 0, 4, 28, 0, 1, 14, 49], ZZ) + + assert dup_mul(f, f, ZZ) == h + + K = FF(6) + + assert dup_mul([K(2), K(1)], [K(3), K(4)], K) == [K(5), K(4)] + + p1 = dup_normal([79, -1, 78, -94, -10, 11, 32, -19, 78, 2, -89, 30, 73, 42, + 85, 77, 83, -30, -34, -2, 95, -81, 37, -49, -46, -58, -16, 37, 35, -11, + -57, -15, -31, 67, -20, 27, 76, 2, 70, 67, -65, 65, -26, -93, -44, -12, + -92, 57, -90, -57, -11, -67, -98, -69, 97, -41, 89, 33, 89, -50, 81, + -31, 60, -27, 43, 29, -77, 44, 21, -91, 32, -57, 33, 3, 53, -51, -38, + -99, -84, 23, -50, 66, -100, 1, -75, -25, 27, -60, 98, -51, -87, 6, 8, + 78, -28, -95, -88, 12, -35, 26, -9, 16, -92, 55, -7, -86, 68, -39, -46, + 84, 94, 45, 60, 92, 68, -75, -74, -19, 8, 75, 78, 91, 57, 34, 14, -3, + -49, 65, 78, -18, 6, -29, -80, -98, 17, 13, 58, 21, 20, 9, 37, 7, -30, + -53, -20, 34, 67, -42, 89, -22, 73, 43, -6, 5, 51, -8, -15, -52, -22, + -58, -72, -3, 43, -92, 82, 83, -2, -13, -23, -60, 16, -94, -8, -28, + -95, -72, 63, -90, 76, 6, -43, -100, -59, 76, 3, 3, 46, -85, 75, 62, + -71, -76, 88, 97, -72, -1, 30, -64, 72, -48, 14, -78, 58, 63, -91, 24, + -87, -27, -80, -100, -44, 98, 70, 100, -29, -38, 11, 77, 100, 52, 86, + 65, -5, -42, -81, -38, -42, 43, -2, -70, -63, -52], ZZ) + p2 = dup_normal([65, -19, -47, 1, 90, 81, -15, -34, 25, -75, 9, -83, 50, -5, + -44, 31, 1, 70, -7, 78, 74, 80, 85, 65, 21, 41, 66, 19, -40, 63, -21, + -27, 32, 69, 83, 34, -35, 14, 81, 57, -75, 32, -67, -89, -100, -61, 46, + 84, -78, -29, -50, -94, -24, -32, -68, -16, 100, -7, -72, -89, 35, 82, + 58, 81, -92, 62, 5, -47, -39, -58, -72, -13, 84, 44, 55, -25, 48, -54, + -31, -56, -11, -50, -84, 10, 67, 17, 13, -14, 61, 76, -64, -44, -40, + -96, 11, -11, -94, 2, 6, 27, -6, 68, -54, 66, -74, -14, -1, -24, -73, + 96, 89, -11, -89, 56, -53, 72, -43, 96, 25, 63, -31, 29, 68, 83, 91, + -93, -19, -38, -40, 40, -12, -19, -79, 44, 100, -66, -29, -77, 62, 39, + -8, 11, -97, 14, 87, 64, 21, -18, 13, 15, -59, -75, -99, -88, 57, 54, + 56, -67, 6, -63, -59, -14, 28, 87, -20, -39, 84, -91, -2, 49, -75, 11, + -24, -95, 36, 66, 5, 25, -72, -40, 86, 90, 37, -33, 57, -35, 29, -18, + 4, -79, 64, -17, -27, 21, 29, -5, -44, -87, -24, 52, 78, 11, -23, -53, + 36, 42, 21, -68, 94, -91, -51, -21, 51, -76, 72, 31, 24, -48, -80, -9, + 37, -47, -6, -8, -63, -91, 79, -79, -100, 38, -20, 38, 100, 83, -90, + 87, 63, -36, 82, -19, 18, -98, -38, 26, 98, -70, 79, 92, 12, 12, 70, + 74, 36, 48, -13, 31, 31, -47, -71, -12, -64, 36, -42, 32, -86, 60, 83, + 70, 55, 0, 1, 29, -35, 8, -82, 8, -73, -46, -50, 43, 48, -5, -86, -72, + 44, -90, 19, 19, 5, -20, 97, -13, -66, -5, 5, -69, 64, -30, 41, 51, 36, + 13, -99, -61, 94, -12, 74, 98, 68, 24, 46, -97, -87, -6, -27, 82, 62, + -11, -77, 86, 66, -47, -49, -50, 13, 18, 89, -89, 46, -80, 13, 98, -35, + -36, -25, 12, 20, 26, -52, 79, 27, 79, 100, 8, 62, -58, -28, 37], ZZ) + res = dup_normal([5135, -1566, 1376, -7466, 4579, 11710, 8001, -7183, + -3737, -7439, 345, -10084, 24522, -1201, 1070, -10245, 9582, 9264, + 1903, 23312, 18953, 10037, -15268, -5450, 6442, -6243, -3777, 5110, + 10936, -16649, -6022, 16255, 31300, 24818, 31922, 32760, 7854, 27080, + 15766, 29596, 7139, 31945, -19810, 465, -38026, -3971, 9641, 465, + -19375, 5524, -30112, -11960, -12813, 13535, 30670, 5925, -43725, + -14089, 11503, -22782, 6371, 43881, 37465, -33529, -33590, -39798, + -37854, -18466, -7908, -35825, -26020, -36923, -11332, -5699, 25166, + -3147, 19885, 12962, -20659, -1642, 27723, -56331, -24580, -11010, + -20206, 20087, -23772, -16038, 38580, 20901, -50731, 32037, -4299, + 26508, 18038, -28357, 31846, -7405, -20172, -15894, 2096, 25110, + -45786, 45918, -55333, -31928, -49428, -29824, -58796, -24609, -15408, + 69, -35415, -18439, 10123, -20360, -65949, 33356, -20333, 26476, + -32073, 33621, 930, 28803, -42791, 44716, 38164, 12302, -1739, 11421, + 73385, -7613, 14297, 38155, -414, 77587, 24338, -21415, 29367, 42639, + 13901, -288, 51027, -11827, 91260, 43407, 88521, -15186, 70572, -12049, + 5090, -12208, -56374, 15520, -623, -7742, 50825, 11199, -14894, 40892, + 59591, -31356, -28696, -57842, -87751, -33744, -28436, -28945, -40287, + 37957, -35638, 33401, -61534, 14870, 40292, 70366, -10803, 102290, + -71719, -85251, 7902, -22409, 75009, 99927, 35298, -1175, -762, -34744, + -10587, -47574, -62629, -19581, -43659, -54369, -32250, -39545, 15225, + -24454, 11241, -67308, -30148, 39929, 37639, 14383, -73475, -77636, + -81048, -35992, 41601, -90143, 76937, -8112, 56588, 9124, -40094, + -32340, 13253, 10898, -51639, 36390, 12086, -1885, 100714, -28561, + -23784, -18735, 18916, 16286, 10742, -87360, -13697, 10689, -19477, + -29770, 5060, 20189, -8297, 112407, 47071, 47743, 45519, -4109, 17468, + -68831, 78325, -6481, -21641, -19459, 30919, 96115, 8607, 53341, 32105, + -16211, 23538, 57259, -76272, -40583, 62093, 38511, -34255, -40665, + -40604, -37606, -15274, 33156, -13885, 103636, 118678, -14101, -92682, + -100791, 2634, 63791, 98266, 19286, -34590, -21067, -71130, 25380, + -40839, -27614, -26060, 52358, -15537, 27138, -6749, 36269, -33306, + 13207, -91084, -5540, -57116, 69548, 44169, -57742, -41234, -103327, + -62904, -8566, 41149, -12866, 71188, 23980, 1838, 58230, 73950, 5594, + 43113, -8159, -15925, 6911, 85598, -75016, -16214, -62726, -39016, + 8618, -63882, -4299, 23182, 49959, 49342, -3238, -24913, -37138, 78361, + 32451, 6337, -11438, -36241, -37737, 8169, -3077, -24829, 57953, 53016, + -31511, -91168, 12599, -41849, 41576, 55275, -62539, 47814, -62319, + 12300, -32076, -55137, -84881, -27546, 4312, -3433, -54382, 113288, + -30157, 74469, 18219, 79880, -2124, 98911, 17655, -33499, -32861, + 47242, -37393, 99765, 14831, -44483, 10800, -31617, -52710, 37406, + 22105, 29704, -20050, 13778, 43683, 36628, 8494, 60964, -22644, 31550, + -17693, 33805, -124879, -12302, 19343, 20400, -30937, -21574, -34037, + -33380, 56539, -24993, -75513, -1527, 53563, 65407, -101, 53577, 37991, + 18717, -23795, -8090, -47987, -94717, 41967, 5170, -14815, -94311, + 17896, -17734, -57718, -774, -38410, 24830, 29682, 76480, 58802, + -46416, -20348, -61353, -68225, -68306, 23822, -31598, 42972, 36327, + 28968, -65638, -21638, 24354, -8356, 26777, 52982, -11783, -44051, + -26467, -44721, -28435, -53265, -25574, -2669, 44155, 22946, -18454, + -30718, -11252, 58420, 8711, 67447, 4425, 41749, 67543, 43162, 11793, + -41907, 20477, -13080, 6559, -6104, -13244, 42853, 42935, 29793, 36730, + -28087, 28657, 17946, 7503, 7204, 21491, -27450, -24241, -98156, + -18082, -42613, -24928, 10775, -14842, -44127, 55910, 14777, 31151, -2194, + 39206, -2100, -4211, 11827, -8918, -19471, 72567, 36447, -65590, -34861, + -17147, -45303, 9025, -7333, -35473, 11101, 11638, 3441, 6626, -41800, + 9416, 13679, 33508, 40502, -60542, 16358, 8392, -43242, -35864, -34127, + -48721, 35878, 30598, 28630, 20279, -19983, -14638, -24455, -1851, -11344, + 45150, 42051, 26034, -28889, -32382, -3527, -14532, 22564, -22346, 477, + 11706, 28338, -25972, -9185, -22867, -12522, 32120, -4424, 11339, -33913, + -7184, 5101, -23552, -17115, -31401, -6104, 21906, 25708, 8406, 6317, + -7525, 5014, 20750, 20179, 22724, 11692, 13297, 2493, -253, -16841, -17339, + -6753, -4808, 2976, -10881, -10228, -13816, -12686, 1385, 2316, 2190, -875, + -1924], ZZ) + + assert dup_mul(p1, p2, ZZ) == res + + p1 = dup_normal([83, -61, -86, -24, 12, 43, -88, -9, 42, 55, -66, 74, 95, + -25, -12, 68, -99, 4, 45, 6, -15, -19, 78, 65, -55, 47, -13, 17, 86, + 81, -58, -27, 50, -40, -24, 39, -41, -92, 75, 90, -1, 40, -15, -27, + -35, 68, 70, -64, -40, 78, -88, -58, -39, 69, 46, 12, 28, -94, -37, + -50, -80, -96, -61, 25, 1, 71, 4, 12, 48, 4, 34, -47, -75, 5, 48, 82, + 88, 23, 98, 35, 17, -10, 48, -61, -95, 47, 65, -19, -66, -57, -6, -51, + -42, -89, 66, -13, 18, 37, 90, -23, 72, 96, -53, 0, 40, -73, -52, -68, + 32, -25, -53, 79, -52, 18, 44, 73, -81, 31, -90, 70, 3, 36, 48, 76, + -24, -44, 23, 98, -4, 73, 69, 88, -70, 14, -68, 94, -78, -15, -64, -97, + -70, -35, 65, 88, 49, -53, -7, 12, -45, -7, 59, -94, 99, -2, 67, -60, + -71, 29, -62, -77, 1, 51, 17, 80, -20, -47, -19, 24, -9, 39, -23, 21, + -84, 10, 84, 56, -17, -21, -66, 85, 70, 46, -51, -22, -95, 78, -60, + -96, -97, -45, 72, 35, 30, -61, -92, -93, -60, -61, 4, -4, -81, -73, + 46, 53, -11, 26, 94, 45, 14, -78, 55, 84, -68, 98, 60, 23, 100, -63, + 68, 96, -16, 3, 56, 21, -58, 62, -67, 66, 85, 41, -79, -22, 97, -67, + 82, 82, -96, -20, -7, 48, -67, 48, -9, -39, 78], ZZ) + p2 = dup_normal([52, 88, 76, 66, 9, -64, 46, -20, -28, 69, 60, 96, -36, + -92, -30, -11, -35, 35, 55, 63, -92, -7, 25, -58, 74, 55, -6, 4, 47, + -92, -65, 67, -45, 74, -76, 59, -6, 69, 39, 24, -71, -7, 39, -45, 60, + -68, 98, 97, -79, 17, 4, 94, -64, 68, -100, -96, -2, 3, 22, 96, 54, + -77, -86, 67, 6, 57, 37, 40, 89, -78, 64, -94, -45, -92, 57, 87, -26, + 36, 19, 97, 25, 77, -87, 24, 43, -5, 35, 57, 83, 71, 35, 63, 61, 96, + -22, 8, -1, 96, 43, 45, 94, -93, 36, 71, -41, -99, 85, -48, 59, 52, + -17, 5, 87, -16, -68, -54, 76, -18, 100, 91, -42, -70, -66, -88, -12, + 1, 95, -82, 52, 43, -29, 3, 12, 72, -99, -43, -32, -93, -51, 16, -20, + -12, -11, 5, 33, -38, 93, -5, -74, 25, 74, -58, 93, 59, -63, -86, 63, + -20, -4, -74, -73, -95, 29, -28, 93, -91, -2, -38, -62, 77, -58, -85, + -28, 95, 38, 19, -69, 86, 94, 25, -2, -4, 47, 34, -59, 35, -48, 29, + -63, -53, 34, 29, 66, 73, 6, 92, -84, 89, 15, 81, 93, 97, 51, -72, -78, + 25, 60, 90, -45, 39, 67, -84, -62, 57, 26, -32, -56, -14, -83, 76, 5, + -2, 99, -100, 28, 46, 94, -7, 53, -25, 16, -23, -36, 89, -78, -63, 31, + 1, 84, -99, -52, 76, 48, 90, -76, 44, -19, 54, -36, -9, -73, -100, -69, + 31, 42, 25, -39, 76, -26, -8, -14, 51, 3, 37, 45, 2, -54, 13, -34, -92, + 17, -25, -65, 53, -63, 30, 4, -70, -67, 90, 52, 51, 18, -3, 31, -45, + -9, 59, 63, -87, 22, -32, 29, -38, 21, 36, -82, 27, -11], ZZ) + res = dup_normal([4316, 4132, -3532, -7974, -11303, -10069, 5484, -3330, + -5874, 7734, 4673, 11327, -9884, -8031, 17343, 21035, -10570, -9285, + 15893, 3780, -14083, 8819, 17592, 10159, 7174, -11587, 8598, -16479, + 3602, 25596, 9781, 12163, 150, 18749, -21782, -12307, 27578, -2757, + -12573, 12565, 6345, -18956, 19503, -15617, 1443, -16778, 36851, 23588, + -28474, 5749, 40695, -7521, -53669, -2497, -18530, 6770, 57038, 3926, + -6927, -15399, 1848, -64649, -27728, 3644, 49608, 15187, -8902, -9480, + -7398, -40425, 4824, 23767, -7594, -6905, 33089, 18786, 12192, 24670, + 31114, 35334, -4501, -14676, 7107, -59018, -21352, 20777, 19661, 20653, + 33754, -885, -43758, 6269, 51897, -28719, -97488, -9527, 13746, 11644, + 17644, -21720, 23782, -10481, 47867, 20752, 33810, -1875, 39918, -7710, + -40840, 19808, -47075, 23066, 46616, 25201, 9287, 35436, -1602, 9645, + -11978, 13273, 15544, 33465, 20063, 44539, 11687, 27314, -6538, -37467, + 14031, 32970, -27086, 41323, 29551, 65910, -39027, -37800, -22232, + 8212, 46316, -28981, -55282, 50417, -44929, -44062, 73879, 37573, + -2596, -10877, -21893, -133218, -33707, -25753, -9531, 17530, 61126, + 2748, -56235, 43874, -10872, -90459, -30387, 115267, -7264, -44452, + 122626, 14839, -599, 10337, 57166, -67467, -54957, 63669, 1202, 18488, + 52594, 7205, -97822, 612, 78069, -5403, -63562, 47236, 36873, -154827, + -26188, 82427, -39521, 5628, 7416, 5276, -53095, 47050, 26121, -42207, + 79021, -13035, 2499, -66943, 29040, -72355, -23480, 23416, -12885, + -44225, -42688, -4224, 19858, 55299, 15735, 11465, 101876, -39169, + 51786, 14723, 43280, -68697, 16410, 92295, 56767, 7183, 111850, 4550, + 115451, -38443, -19642, -35058, 10230, 93829, 8925, 63047, 3146, 29250, + 8530, 5255, -98117, -115517, -76817, -8724, 41044, 1312, -35974, 79333, + -28567, 7547, -10580, -24559, -16238, 10794, -3867, 24848, 57770, + -51536, -35040, 71033, 29853, 62029, -7125, -125585, -32169, -47907, + 156811, -65176, -58006, -15757, -57861, 11963, 30225, -41901, -41681, + 31310, 27982, 18613, 61760, 60746, -59096, 33499, 30097, -17997, 24032, + 56442, -83042, 23747, -20931, -21978, -158752, -9883, -73598, -7987, + -7333, -125403, -116329, 30585, 53281, 51018, -29193, 88575, 8264, + -40147, -16289, 113088, 12810, -6508, 101552, -13037, 34440, -41840, + 101643, 24263, 80532, 61748, 65574, 6423, -20672, 6591, -10834, -71716, + 86919, -92626, 39161, 28490, 81319, 46676, 106720, 43530, 26998, 57456, + -8862, 60989, 13982, 3119, -2224, 14743, 55415, -49093, -29303, 28999, + 1789, 55953, -84043, -7780, -65013, 57129, -47251, 61484, 61994, + -78361, -82778, 22487, -26894, 9756, -74637, -15519, -4360, 30115, + 42433, 35475, 15286, 69768, 21509, -20214, 78675, -21163, 13596, 11443, + -10698, -53621, -53867, -24155, 64500, -42784, -33077, -16500, 873, + -52788, 14546, -38011, 36974, -39849, -34029, -94311, 83068, -50437, + -26169, -46746, 59185, 42259, -101379, -12943, 30089, -59086, 36271, + 22723, -30253, -52472, -70826, -23289, 3331, -31687, 14183, -857, + -28627, 35246, -51284, 5636, -6933, 66539, 36654, 50927, 24783, 3457, + 33276, 45281, 45650, -4938, -9968, -22590, 47995, 69229, 5214, -58365, + -17907, -14651, 18668, 18009, 12649, -11851, -13387, 20339, 52472, + -1087, -21458, -68647, 52295, 15849, 40608, 15323, 25164, -29368, + 10352, -7055, 7159, 21695, -5373, -54849, 101103, -24963, -10511, + 33227, 7659, 41042, -69588, 26718, -20515, 6441, 38135, -63, 24088, + -35364, -12785, -18709, 47843, 48533, -48575, 17251, -19394, 32878, + -9010, -9050, 504, -12407, 28076, -3429, 25324, -4210, -26119, 752, + -29203, 28251, -11324, -32140, -3366, -25135, 18702, -31588, -7047, + -24267, 49987, -14975, -33169, 37744, -7720, -9035, 16964, -2807, -421, + 14114, -17097, -13662, 40628, -12139, -9427, 5369, 17551, -13232, -16211, + 9804, -7422, 2677, 28635, -8280, -4906, 2908, -22558, 5604, 12459, 8756, + -3980, -4745, -18525, 7913, 5970, -16457, 20230, -6247, -13812, 2505, + 11899, 1409, -15094, 22540, -18863, 137, 11123, -4516, 2290, -8594, 12150, + -10380, 3005, 5235, -7350, 2535, -858], ZZ) + + assert dup_mul(p1, p2, ZZ) == res + + +def test_dmp_mul(): + assert dmp_mul([ZZ(5)], [ZZ(7)], 0, ZZ) == \ + dup_mul([ZZ(5)], [ZZ(7)], ZZ) + assert dmp_mul([QQ(5, 7)], [QQ(3, 7)], 0, QQ) == \ + dup_mul([QQ(5, 7)], [QQ(3, 7)], QQ) + + assert dmp_mul([[[]]], [[[]]], 2, ZZ) == [[[]]] + assert dmp_mul([[[ZZ(1)]]], [[[]]], 2, ZZ) == [[[]]] + assert dmp_mul([[[]]], [[[ZZ(1)]]], 2, ZZ) == [[[]]] + assert dmp_mul([[[ZZ(2)]]], [[[ZZ(1)]]], 2, ZZ) == [[[ZZ(2)]]] + assert dmp_mul([[[ZZ(1)]]], [[[ZZ(2)]]], 2, ZZ) == [[[ZZ(2)]]] + + assert dmp_mul([[[]]], [[[]]], 2, QQ) == [[[]]] + assert dmp_mul([[[QQ(1, 2)]]], [[[]]], 2, QQ) == [[[]]] + assert dmp_mul([[[]]], [[[QQ(1, 2)]]], 2, QQ) == [[[]]] + assert dmp_mul([[[QQ(2, 7)]]], [[[QQ(1, 3)]]], 2, QQ) == [[[QQ(2, 21)]]] + assert dmp_mul([[[QQ(1, 7)]]], [[[QQ(2, 3)]]], 2, QQ) == [[[QQ(2, 21)]]] + + K = FF(6) + + assert dmp_mul( + [[K(2)], [K(1)]], [[K(3)], [K(4)]], 1, K) == [[K(5)], [K(4)]] + + +def test_dup_sqr(): + assert dup_sqr([], ZZ) == [] + assert dup_sqr([ZZ(2)], ZZ) == [ZZ(4)] + assert dup_sqr([ZZ(1), ZZ(2)], ZZ) == [ZZ(1), ZZ(4), ZZ(4)] + + assert dup_sqr([], QQ) == [] + assert dup_sqr([QQ(2, 3)], QQ) == [QQ(4, 9)] + assert dup_sqr([QQ(1, 3), QQ(2, 3)], QQ) == [QQ(1, 9), QQ(4, 9), QQ(4, 9)] + + f = dup_normal([2, 0, 0, 1, 7], ZZ) + + assert dup_sqr(f, ZZ) == dup_normal([4, 0, 0, 4, 28, 0, 1, 14, 49], ZZ) + + K = FF(9) + + assert dup_sqr([K(3), K(4)], K) == [K(6), K(7)] + + +def test_dmp_sqr(): + assert dmp_sqr([ZZ(1), ZZ(2)], 0, ZZ) == \ + dup_sqr([ZZ(1), ZZ(2)], ZZ) + + assert dmp_sqr([[[]]], 2, ZZ) == [[[]]] + assert dmp_sqr([[[ZZ(2)]]], 2, ZZ) == [[[ZZ(4)]]] + + assert dmp_sqr([[[]]], 2, QQ) == [[[]]] + assert dmp_sqr([[[QQ(2, 3)]]], 2, QQ) == [[[QQ(4, 9)]]] + + K = FF(9) + + assert dmp_sqr([[K(3)], [K(4)]], 1, K) == [[K(6)], [K(7)]] + + +def test_dup_pow(): + assert dup_pow([], 0, ZZ) == [ZZ(1)] + assert dup_pow([], 0, QQ) == [QQ(1)] + + assert dup_pow([], 1, ZZ) == [] + assert dup_pow([], 7, ZZ) == [] + + assert dup_pow([ZZ(1)], 0, ZZ) == [ZZ(1)] + assert dup_pow([ZZ(1)], 1, ZZ) == [ZZ(1)] + assert dup_pow([ZZ(1)], 7, ZZ) == [ZZ(1)] + + assert dup_pow([ZZ(3)], 0, ZZ) == [ZZ(1)] + assert dup_pow([ZZ(3)], 1, ZZ) == [ZZ(3)] + assert dup_pow([ZZ(3)], 7, ZZ) == [ZZ(2187)] + + assert dup_pow([QQ(1, 1)], 0, QQ) == [QQ(1, 1)] + assert dup_pow([QQ(1, 1)], 1, QQ) == [QQ(1, 1)] + assert dup_pow([QQ(1, 1)], 7, QQ) == [QQ(1, 1)] + + assert dup_pow([QQ(3, 7)], 0, QQ) == [QQ(1, 1)] + assert dup_pow([QQ(3, 7)], 1, QQ) == [QQ(3, 7)] + assert dup_pow([QQ(3, 7)], 7, QQ) == [QQ(2187, 823543)] + + f = dup_normal([2, 0, 0, 1, 7], ZZ) + + assert dup_pow(f, 0, ZZ) == dup_normal([1], ZZ) + assert dup_pow(f, 1, ZZ) == dup_normal([2, 0, 0, 1, 7], ZZ) + assert dup_pow(f, 2, ZZ) == dup_normal([4, 0, 0, 4, 28, 0, 1, 14, 49], ZZ) + assert dup_pow(f, 3, ZZ) == dup_normal( + [8, 0, 0, 12, 84, 0, 6, 84, 294, 1, 21, 147, 343], ZZ) + + +def test_dmp_pow(): + assert dmp_pow([[]], 0, 1, ZZ) == [[ZZ(1)]] + assert dmp_pow([[]], 0, 1, QQ) == [[QQ(1)]] + + assert dmp_pow([[]], 1, 1, ZZ) == [[]] + assert dmp_pow([[]], 7, 1, ZZ) == [[]] + + assert dmp_pow([[ZZ(1)]], 0, 1, ZZ) == [[ZZ(1)]] + assert dmp_pow([[ZZ(1)]], 1, 1, ZZ) == [[ZZ(1)]] + assert dmp_pow([[ZZ(1)]], 7, 1, ZZ) == [[ZZ(1)]] + + assert dmp_pow([[QQ(3, 7)]], 0, 1, QQ) == [[QQ(1, 1)]] + assert dmp_pow([[QQ(3, 7)]], 1, 1, QQ) == [[QQ(3, 7)]] + assert dmp_pow([[QQ(3, 7)]], 7, 1, QQ) == [[QQ(2187, 823543)]] + + f = dup_normal([2, 0, 0, 1, 7], ZZ) + + assert dmp_pow(f, 2, 0, ZZ) == dup_pow(f, 2, ZZ) + + +def test_dup_pdiv(): + f = dup_normal([3, 1, 1, 5], ZZ) + g = dup_normal([5, -3, 1], ZZ) + + q = dup_normal([15, 14], ZZ) + r = dup_normal([52, 111], ZZ) + + assert dup_pdiv(f, g, ZZ) == (q, r) + assert dup_pquo(f, g, ZZ) == q + assert dup_prem(f, g, ZZ) == r + + raises(ExactQuotientFailed, lambda: dup_pexquo(f, g, ZZ)) + + f = dup_normal([3, 1, 1, 5], QQ) + g = dup_normal([5, -3, 1], QQ) + + q = dup_normal([15, 14], QQ) + r = dup_normal([52, 111], QQ) + + assert dup_pdiv(f, g, QQ) == (q, r) + assert dup_pquo(f, g, QQ) == q + assert dup_prem(f, g, QQ) == r + + raises(ExactQuotientFailed, lambda: dup_pexquo(f, g, QQ)) + + +def test_dmp_pdiv(): + f = dmp_normal([[1], [], [1, 0, 0]], 1, ZZ) + g = dmp_normal([[1], [-1, 0]], 1, ZZ) + + q = dmp_normal([[1], [1, 0]], 1, ZZ) + r = dmp_normal([[2, 0, 0]], 1, ZZ) + + assert dmp_pdiv(f, g, 1, ZZ) == (q, r) + assert dmp_pquo(f, g, 1, ZZ) == q + assert dmp_prem(f, g, 1, ZZ) == r + + raises(ExactQuotientFailed, lambda: dmp_pexquo(f, g, 1, ZZ)) + + f = dmp_normal([[1], [], [1, 0, 0]], 1, ZZ) + g = dmp_normal([[2], [-2, 0]], 1, ZZ) + + q = dmp_normal([[2], [2, 0]], 1, ZZ) + r = dmp_normal([[8, 0, 0]], 1, ZZ) + + assert dmp_pdiv(f, g, 1, ZZ) == (q, r) + assert dmp_pquo(f, g, 1, ZZ) == q + assert dmp_prem(f, g, 1, ZZ) == r + + raises(ExactQuotientFailed, lambda: dmp_pexquo(f, g, 1, ZZ)) + + +def test_dup_rr_div(): + raises(ZeroDivisionError, lambda: dup_rr_div([1, 2, 3], [], ZZ)) + + f = dup_normal([3, 1, 1, 5], ZZ) + g = dup_normal([5, -3, 1], ZZ) + + q, r = [], f + + assert dup_rr_div(f, g, ZZ) == (q, r) + + +def test_dmp_rr_div(): + raises(ZeroDivisionError, lambda: dmp_rr_div([[1, 2], [3]], [[]], 1, ZZ)) + + f = dmp_normal([[1], [], [1, 0, 0]], 1, ZZ) + g = dmp_normal([[1], [-1, 0]], 1, ZZ) + + q = dmp_normal([[1], [1, 0]], 1, ZZ) + r = dmp_normal([[2, 0, 0]], 1, ZZ) + + assert dmp_rr_div(f, g, 1, ZZ) == (q, r) + + f = dmp_normal([[1], [], [1, 0, 0]], 1, ZZ) + g = dmp_normal([[-1], [1, 0]], 1, ZZ) + + q = dmp_normal([[-1], [-1, 0]], 1, ZZ) + r = dmp_normal([[2, 0, 0]], 1, ZZ) + + assert dmp_rr_div(f, g, 1, ZZ) == (q, r) + + f = dmp_normal([[1], [], [1, 0, 0]], 1, ZZ) + g = dmp_normal([[2], [-2, 0]], 1, ZZ) + + q, r = [[]], f + + assert dmp_rr_div(f, g, 1, ZZ) == (q, r) + + +def test_dup_ff_div(): + raises(ZeroDivisionError, lambda: dup_ff_div([1, 2, 3], [], QQ)) + + f = dup_normal([3, 1, 1, 5], QQ) + g = dup_normal([5, -3, 1], QQ) + + q = [QQ(3, 5), QQ(14, 25)] + r = [QQ(52, 25), QQ(111, 25)] + + assert dup_ff_div(f, g, QQ) == (q, r) + +def test_dup_ff_div_gmpy2(): + if GROUND_TYPES != 'gmpy2': + return + + from gmpy2 import mpq + from sympy.polys.domains import GMPYRationalField + K = GMPYRationalField() + + f = [mpq(1,3), mpq(3,2)] + g = [mpq(2,1)] + assert dmp_ff_div(f, g, 0, K) == ([mpq(1,6), mpq(3,4)], []) + + f = [mpq(1,2), mpq(1,3), mpq(1,4), mpq(1,5)] + g = [mpq(-1,1), mpq(1,1), mpq(-1,1)] + assert dmp_ff_div(f, g, 0, K) == ([mpq(-1,2), mpq(-5,6)], [mpq(7,12), mpq(-19,30)]) + +def test_dmp_ff_div(): + raises(ZeroDivisionError, lambda: dmp_ff_div([[1, 2], [3]], [[]], 1, QQ)) + + f = dmp_normal([[1], [], [1, 0, 0]], 1, QQ) + g = dmp_normal([[1], [-1, 0]], 1, QQ) + + q = [[QQ(1, 1)], [QQ(1, 1), QQ(0, 1)]] + r = [[QQ(2, 1), QQ(0, 1), QQ(0, 1)]] + + assert dmp_ff_div(f, g, 1, QQ) == (q, r) + + f = dmp_normal([[1], [], [1, 0, 0]], 1, QQ) + g = dmp_normal([[-1], [1, 0]], 1, QQ) + + q = [[QQ(-1, 1)], [QQ(-1, 1), QQ(0, 1)]] + r = [[QQ(2, 1), QQ(0, 1), QQ(0, 1)]] + + assert dmp_ff_div(f, g, 1, QQ) == (q, r) + + f = dmp_normal([[1], [], [1, 0, 0]], 1, QQ) + g = dmp_normal([[2], [-2, 0]], 1, QQ) + + q = [[QQ(1, 2)], [QQ(1, 2), QQ(0, 1)]] + r = [[QQ(2, 1), QQ(0, 1), QQ(0, 1)]] + + assert dmp_ff_div(f, g, 1, QQ) == (q, r) + + +def test_dup_div(): + f, g, q, r = [5, 4, 3, 2, 1], [1, 2, 3], [5, -6, 0], [20, 1] + + assert dup_div(f, g, ZZ) == (q, r) + assert dup_quo(f, g, ZZ) == q + assert dup_rem(f, g, ZZ) == r + + raises(ExactQuotientFailed, lambda: dup_exquo(f, g, ZZ)) + + f, g, q, r = [5, 4, 3, 2, 1, 0], [1, 2, 0, 0, 9], [5, -6], [15, 2, -44, 54] + + assert dup_div(f, g, ZZ) == (q, r) + assert dup_quo(f, g, ZZ) == q + assert dup_rem(f, g, ZZ) == r + + raises(ExactQuotientFailed, lambda: dup_exquo(f, g, ZZ)) + + +def test_dmp_div(): + f, g, q, r = [5, 4, 3, 2, 1], [1, 2, 3], [5, -6, 0], [20, 1] + + assert dmp_div(f, g, 0, ZZ) == (q, r) + assert dmp_quo(f, g, 0, ZZ) == q + assert dmp_rem(f, g, 0, ZZ) == r + + raises(ExactQuotientFailed, lambda: dmp_exquo(f, g, 0, ZZ)) + + f, g, q, r = [[[1]]], [[[2]], [1]], [[[]]], [[[1]]] + + assert dmp_div(f, g, 2, ZZ) == (q, r) + assert dmp_quo(f, g, 2, ZZ) == q + assert dmp_rem(f, g, 2, ZZ) == r + + raises(ExactQuotientFailed, lambda: dmp_exquo(f, g, 2, ZZ)) + + +def test_dup_max_norm(): + assert dup_max_norm([], ZZ) == 0 + assert dup_max_norm([1], ZZ) == 1 + + assert dup_max_norm([1, 4, 2, 3], ZZ) == 4 + + +def test_dmp_max_norm(): + assert dmp_max_norm([[[]]], 2, ZZ) == 0 + assert dmp_max_norm([[[1]]], 2, ZZ) == 1 + + assert dmp_max_norm(f_0, 2, ZZ) == 6 + + +def test_dup_l1_norm(): + assert dup_l1_norm([], ZZ) == 0 + assert dup_l1_norm([1], ZZ) == 1 + assert dup_l1_norm([1, 4, 2, 3], ZZ) == 10 + + +def test_dmp_l1_norm(): + assert dmp_l1_norm([[[]]], 2, ZZ) == 0 + assert dmp_l1_norm([[[1]]], 2, ZZ) == 1 + + assert dmp_l1_norm(f_0, 2, ZZ) == 31 + + +def test_dup_l2_norm_squared(): + assert dup_l2_norm_squared([], ZZ) == 0 + assert dup_l2_norm_squared([1], ZZ) == 1 + assert dup_l2_norm_squared([1, 4, 2, 3], ZZ) == 30 + + +def test_dmp_l2_norm_squared(): + assert dmp_l2_norm_squared([[[]]], 2, ZZ) == 0 + assert dmp_l2_norm_squared([[[1]]], 2, ZZ) == 1 + assert dmp_l2_norm_squared(f_0, 2, ZZ) == 111 + + +def test_dup_expand(): + assert dup_expand((), ZZ) == [1] + assert dup_expand(([1, 2, 3], [1, 2], [7, 5, 4, 3]), ZZ) == \ + dup_mul([1, 2, 3], dup_mul([1, 2], [7, 5, 4, 3], ZZ), ZZ) + + +def test_dmp_expand(): + assert dmp_expand((), 1, ZZ) == [[1]] + assert dmp_expand(([[1], [2], [3]], [[1], [2]], [[7], [5], [4], [3]]), 1, ZZ) == \ + dmp_mul([[1], [2], [3]], dmp_mul([[1], [2]], [[7], [5], [ + 4], [3]], 1, ZZ), 1, ZZ) + +def test_dup_mul_poly(): + p = Poly(18786186952704.0*x**165 + 9.31746684052255e+31*x**82, x, domain='RR') + px = Poly(18786186952704.0*x**166 + 9.31746684052255e+31*x**83, x, domain='RR') + + assert p * x == px + assert p.set_domain(QQ) * x == px.set_domain(QQ) + assert p.set_domain(CC) * x == px.set_domain(CC) diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_densebasic.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_densebasic.py new file mode 100644 index 0000000000000000000000000000000000000000..43386d86d0e6ec7b20d3962d8063aa6402165f9a --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_densebasic.py @@ -0,0 +1,730 @@ +"""Tests for dense recursive polynomials' basic tools. """ + +from sympy.polys.densebasic import ( + ninf, + dup_LC, dmp_LC, + dup_TC, dmp_TC, + dmp_ground_LC, dmp_ground_TC, + dmp_true_LT, + dup_degree, dmp_degree, + dmp_degree_in, dmp_degree_list, + dup_strip, dmp_strip, + dmp_validate, + dup_reverse, + dup_copy, dmp_copy, + dup_normal, dmp_normal, + dup_convert, dmp_convert, + dup_from_sympy, dmp_from_sympy, + dup_nth, dmp_nth, dmp_ground_nth, + dmp_zero_p, dmp_zero, + dmp_one_p, dmp_one, + dmp_ground_p, dmp_ground, + dmp_negative_p, dmp_positive_p, + dmp_zeros, dmp_grounds, + dup_from_dict, dup_from_raw_dict, + dup_to_dict, dup_to_raw_dict, + dmp_from_dict, dmp_to_dict, + dmp_swap, dmp_permute, + dmp_nest, dmp_raise, + dup_deflate, dmp_deflate, + dup_multi_deflate, dmp_multi_deflate, + dup_inflate, dmp_inflate, + dmp_exclude, dmp_include, + dmp_inject, dmp_eject, + dup_terms_gcd, dmp_terms_gcd, + dmp_list_terms, dmp_apply_pairs, + dup_slice, + dup_random, +) + +from sympy.polys.specialpolys import f_polys +from sympy.polys.domains import ZZ, QQ +from sympy.polys.rings import ring + +from sympy.core.singleton import S +from sympy.testing.pytest import raises + +from sympy.core.numbers import oo + +f_0, f_1, f_2, f_3, f_4, f_5, f_6 = [ f.to_dense() for f in f_polys() ] + +def test_dup_LC(): + assert dup_LC([], ZZ) == 0 + assert dup_LC([2, 3, 4, 5], ZZ) == 2 + + +def test_dup_TC(): + assert dup_TC([], ZZ) == 0 + assert dup_TC([2, 3, 4, 5], ZZ) == 5 + + +def test_dmp_LC(): + assert dmp_LC([[]], ZZ) == [] + assert dmp_LC([[2, 3, 4], [5]], ZZ) == [2, 3, 4] + assert dmp_LC([[[]]], ZZ) == [[]] + assert dmp_LC([[[2], [3, 4]], [[5]]], ZZ) == [[2], [3, 4]] + + +def test_dmp_TC(): + assert dmp_TC([[]], ZZ) == [] + assert dmp_TC([[2, 3, 4], [5]], ZZ) == [5] + assert dmp_TC([[[]]], ZZ) == [[]] + assert dmp_TC([[[2], [3, 4]], [[5]]], ZZ) == [[5]] + + +def test_dmp_ground_LC(): + assert dmp_ground_LC([[]], 1, ZZ) == 0 + assert dmp_ground_LC([[2, 3, 4], [5]], 1, ZZ) == 2 + assert dmp_ground_LC([[[]]], 2, ZZ) == 0 + assert dmp_ground_LC([[[2], [3, 4]], [[5]]], 2, ZZ) == 2 + + +def test_dmp_ground_TC(): + assert dmp_ground_TC([[]], 1, ZZ) == 0 + assert dmp_ground_TC([[2, 3, 4], [5]], 1, ZZ) == 5 + assert dmp_ground_TC([[[]]], 2, ZZ) == 0 + assert dmp_ground_TC([[[2], [3, 4]], [[5]]], 2, ZZ) == 5 + + +def test_dmp_true_LT(): + assert dmp_true_LT([[]], 1, ZZ) == ((0, 0), 0) + assert dmp_true_LT([[7]], 1, ZZ) == ((0, 0), 7) + + assert dmp_true_LT([[1, 0]], 1, ZZ) == ((0, 1), 1) + assert dmp_true_LT([[1], []], 1, ZZ) == ((1, 0), 1) + assert dmp_true_LT([[1, 0], []], 1, ZZ) == ((1, 1), 1) + + +def test_dup_degree(): + assert ninf == float('-inf') + assert dup_degree([]) is ninf + assert dup_degree([1]) == 0 + assert dup_degree([1, 0]) == 1 + assert dup_degree([1, 0, 0, 0, 1]) == 4 + + +def test_dmp_degree(): + assert dmp_degree([[]], 1) is ninf + assert dmp_degree([[[]]], 2) is ninf + + assert dmp_degree([[1]], 1) == 0 + assert dmp_degree([[2], [1]], 1) == 1 + + +def test_dmp_degree_in(): + assert dmp_degree_in([[[]]], 0, 2) is ninf + assert dmp_degree_in([[[]]], 1, 2) is ninf + assert dmp_degree_in([[[]]], 2, 2) is ninf + + assert dmp_degree_in([[[1]]], 0, 2) == 0 + assert dmp_degree_in([[[1]]], 1, 2) == 0 + assert dmp_degree_in([[[1]]], 2, 2) == 0 + + assert dmp_degree_in(f_4, 0, 2) == 9 + assert dmp_degree_in(f_4, 1, 2) == 12 + assert dmp_degree_in(f_4, 2, 2) == 8 + + assert dmp_degree_in(f_6, 0, 2) == 4 + assert dmp_degree_in(f_6, 1, 2) == 4 + assert dmp_degree_in(f_6, 2, 2) == 6 + assert dmp_degree_in(f_6, 3, 3) == 3 + + raises(IndexError, lambda: dmp_degree_in([[1]], -5, 1)) + + +def test_dmp_degree_list(): + assert dmp_degree_list([[[[ ]]]], 3) == (-oo, -oo, -oo, -oo) + assert dmp_degree_list([[[[1]]]], 3) == ( 0, 0, 0, 0) + + assert dmp_degree_list(f_0, 2) == (2, 2, 2) + assert dmp_degree_list(f_1, 2) == (3, 3, 3) + assert dmp_degree_list(f_2, 2) == (5, 3, 3) + assert dmp_degree_list(f_3, 2) == (5, 4, 7) + assert dmp_degree_list(f_4, 2) == (9, 12, 8) + assert dmp_degree_list(f_5, 2) == (3, 3, 3) + assert dmp_degree_list(f_6, 3) == (4, 4, 6, 3) + + +def test_dup_strip(): + assert dup_strip([]) == [] + assert dup_strip([0]) == [] + assert dup_strip([0, 0, 0]) == [] + + assert dup_strip([1]) == [1] + assert dup_strip([0, 1]) == [1] + assert dup_strip([0, 0, 0, 1]) == [1] + + assert dup_strip([1, 2, 0]) == [1, 2, 0] + assert dup_strip([0, 1, 2, 0]) == [1, 2, 0] + assert dup_strip([0, 0, 0, 1, 2, 0]) == [1, 2, 0] + + +def test_dmp_strip(): + assert dmp_strip([0, 1, 0], 0) == [1, 0] + + assert dmp_strip([[]], 1) == [[]] + assert dmp_strip([[], []], 1) == [[]] + assert dmp_strip([[], [], []], 1) == [[]] + + assert dmp_strip([[[]]], 2) == [[[]]] + assert dmp_strip([[[]], [[]]], 2) == [[[]]] + assert dmp_strip([[[]], [[]], [[]]], 2) == [[[]]] + + assert dmp_strip([[[1]]], 2) == [[[1]]] + assert dmp_strip([[[]], [[1]]], 2) == [[[1]]] + assert dmp_strip([[[]], [[1]], [[]]], 2) == [[[1]], [[]]] + + +def test_dmp_validate(): + assert dmp_validate([]) == ([], 0) + assert dmp_validate([0, 0, 0, 1, 0]) == ([1, 0], 0) + + assert dmp_validate([[[]]]) == ([[[]]], 2) + assert dmp_validate([[0], [], [0], [1], [0]]) == ([[1], []], 1) + + raises(ValueError, lambda: dmp_validate([[0], 0, [0], [1], [0]])) + + +def test_dup_reverse(): + assert dup_reverse([1, 2, 0, 3]) == [3, 0, 2, 1] + assert dup_reverse([1, 2, 3, 0]) == [3, 2, 1] + + +def test_dup_copy(): + f = [ZZ(1), ZZ(0), ZZ(2)] + g = dup_copy(f) + + g[0], g[2] = ZZ(7), ZZ(0) + + assert f != g + + +def test_dmp_copy(): + f = [[ZZ(1)], [ZZ(2), ZZ(0)]] + g = dmp_copy(f, 1) + + g[0][0], g[1][1] = ZZ(7), ZZ(1) + + assert f != g + + +def test_dup_normal(): + assert dup_normal([0, 0, 2, 1, 0, 11, 0], ZZ) == \ + [ZZ(2), ZZ(1), ZZ(0), ZZ(11), ZZ(0)] + + +def test_dmp_normal(): + assert dmp_normal([[0], [], [0, 2, 1], [0], [11], []], 1, ZZ) == \ + [[ZZ(2), ZZ(1)], [], [ZZ(11)], []] + + +def test_dup_convert(): + K0, K1 = ZZ['x'], ZZ + + f = [K0(1), K0(2), K0(0), K0(3)] + + assert dup_convert(f, K0, K1) == \ + [ZZ(1), ZZ(2), ZZ(0), ZZ(3)] + + +def test_dmp_convert(): + K0, K1 = ZZ['x'], ZZ + + f = [[K0(1)], [K0(2)], [], [K0(3)]] + + assert dmp_convert(f, 1, K0, K1) == \ + [[ZZ(1)], [ZZ(2)], [], [ZZ(3)]] + + +def test_dup_from_sympy(): + assert dup_from_sympy([S.One, S(2)], ZZ) == \ + [ZZ(1), ZZ(2)] + assert dup_from_sympy([S.Half, S(3)], QQ) == \ + [QQ(1, 2), QQ(3, 1)] + + +def test_dmp_from_sympy(): + assert dmp_from_sympy([[S.One, S(2)], [S.Zero]], 1, ZZ) == \ + [[ZZ(1), ZZ(2)], []] + assert dmp_from_sympy([[S.Half, S(2)]], 1, QQ) == \ + [[QQ(1, 2), QQ(2, 1)]] + + +def test_dup_nth(): + assert dup_nth([1, 2, 3], 0, ZZ) == 3 + assert dup_nth([1, 2, 3], 1, ZZ) == 2 + assert dup_nth([1, 2, 3], 2, ZZ) == 1 + + assert dup_nth([1, 2, 3], 9, ZZ) == 0 + + raises(IndexError, lambda: dup_nth([3, 4, 5], -1, ZZ)) + + +def test_dmp_nth(): + assert dmp_nth([[1], [2], [3]], 0, 1, ZZ) == [3] + assert dmp_nth([[1], [2], [3]], 1, 1, ZZ) == [2] + assert dmp_nth([[1], [2], [3]], 2, 1, ZZ) == [1] + + assert dmp_nth([[1], [2], [3]], 9, 1, ZZ) == [] + + raises(IndexError, lambda: dmp_nth([[3], [4], [5]], -1, 1, ZZ)) + + +def test_dmp_ground_nth(): + assert dmp_ground_nth([[]], (0, 0), 1, ZZ) == 0 + assert dmp_ground_nth([[1], [2], [3]], (0, 0), 1, ZZ) == 3 + assert dmp_ground_nth([[1], [2], [3]], (1, 0), 1, ZZ) == 2 + assert dmp_ground_nth([[1], [2], [3]], (2, 0), 1, ZZ) == 1 + + assert dmp_ground_nth([[1], [2], [3]], (2, 1), 1, ZZ) == 0 + assert dmp_ground_nth([[1], [2], [3]], (3, 0), 1, ZZ) == 0 + + raises(IndexError, lambda: dmp_ground_nth([[3], [4], [5]], (2, -1), 1, ZZ)) + + +def test_dmp_zero_p(): + assert dmp_zero_p([], 0) is True + assert dmp_zero_p([[]], 1) is True + + assert dmp_zero_p([[[]]], 2) is True + assert dmp_zero_p([[[1]]], 2) is False + + +def test_dmp_zero(): + assert dmp_zero(0) == [] + assert dmp_zero(2) == [[[]]] + + +def test_dmp_one_p(): + assert dmp_one_p([1], 0, ZZ) is True + assert dmp_one_p([[1]], 1, ZZ) is True + assert dmp_one_p([[[1]]], 2, ZZ) is True + assert dmp_one_p([[[12]]], 2, ZZ) is False + + +def test_dmp_one(): + assert dmp_one(0, ZZ) == [ZZ(1)] + assert dmp_one(2, ZZ) == [[[ZZ(1)]]] + + +def test_dmp_ground_p(): + assert dmp_ground_p([], 0, 0) is True + assert dmp_ground_p([[]], 0, 1) is True + assert dmp_ground_p([[]], 1, 1) is False + + assert dmp_ground_p([[ZZ(1)]], 1, 1) is True + assert dmp_ground_p([[[ZZ(2)]]], 2, 2) is True + + assert dmp_ground_p([[[ZZ(2)]]], 3, 2) is False + assert dmp_ground_p([[[ZZ(3)], []]], 3, 2) is False + + assert dmp_ground_p([], None, 0) is True + assert dmp_ground_p([[]], None, 1) is True + + assert dmp_ground_p([ZZ(1)], None, 0) is True + assert dmp_ground_p([[[ZZ(1)]]], None, 2) is True + + assert dmp_ground_p([[[ZZ(3)], []]], None, 2) is False + + +def test_dmp_ground(): + assert dmp_ground(ZZ(0), 2) == [[[]]] + + assert dmp_ground(ZZ(7), -1) == ZZ(7) + assert dmp_ground(ZZ(7), 0) == [ZZ(7)] + assert dmp_ground(ZZ(7), 2) == [[[ZZ(7)]]] + + +def test_dmp_zeros(): + assert dmp_zeros(4, 0, ZZ) == [[], [], [], []] + + assert dmp_zeros(0, 2, ZZ) == [] + assert dmp_zeros(1, 2, ZZ) == [[[[]]]] + assert dmp_zeros(2, 2, ZZ) == [[[[]]], [[[]]]] + assert dmp_zeros(3, 2, ZZ) == [[[[]]], [[[]]], [[[]]]] + + assert dmp_zeros(3, -1, ZZ) == [0, 0, 0] + + +def test_dmp_grounds(): + assert dmp_grounds(ZZ(7), 0, 2) == [] + + assert dmp_grounds(ZZ(7), 1, 2) == [[[[7]]]] + assert dmp_grounds(ZZ(7), 2, 2) == [[[[7]]], [[[7]]]] + assert dmp_grounds(ZZ(7), 3, 2) == [[[[7]]], [[[7]]], [[[7]]]] + + assert dmp_grounds(ZZ(7), 3, -1) == [7, 7, 7] + + +def test_dmp_negative_p(): + assert dmp_negative_p([[[]]], 2, ZZ) is False + assert dmp_negative_p([[[1], [2]]], 2, ZZ) is False + assert dmp_negative_p([[[-1], [2]]], 2, ZZ) is True + + +def test_dmp_positive_p(): + assert dmp_positive_p([[[]]], 2, ZZ) is False + assert dmp_positive_p([[[1], [2]]], 2, ZZ) is True + assert dmp_positive_p([[[-1], [2]]], 2, ZZ) is False + + +def test_dup_from_to_dict(): + assert dup_from_raw_dict({}, ZZ) == [] + assert dup_from_dict({}, ZZ) == [] + + assert dup_to_raw_dict([]) == {} + assert dup_to_dict([]) == {} + + assert dup_to_raw_dict([], ZZ, zero=True) == {0: ZZ(0)} + assert dup_to_dict([], ZZ, zero=True) == {(0,): ZZ(0)} + + f = [3, 0, 0, 2, 0, 0, 0, 0, 8] + g = {8: 3, 5: 2, 0: 8} + h = {(8,): 3, (5,): 2, (0,): 8} + + assert dup_from_raw_dict(g, ZZ) == f + assert dup_from_dict(h, ZZ) == f + + assert dup_to_raw_dict(f) == g + assert dup_to_dict(f) == h + + R, x,y = ring("x,y", ZZ) + K = R.to_domain() + + f = [R(3), R(0), R(2), R(0), R(0), R(8)] + g = {5: R(3), 3: R(2), 0: R(8)} + h = {(5,): R(3), (3,): R(2), (0,): R(8)} + + assert dup_from_raw_dict(g, K) == f + assert dup_from_dict(h, K) == f + + assert dup_to_raw_dict(f) == g + assert dup_to_dict(f) == h + + +def test_dmp_from_to_dict(): + assert dmp_from_dict({}, 1, ZZ) == [[]] + assert dmp_to_dict([[]], 1) == {} + + assert dmp_to_dict([], 0, ZZ, zero=True) == {(0,): ZZ(0)} + assert dmp_to_dict([[]], 1, ZZ, zero=True) == {(0, 0): ZZ(0)} + + f = [[3], [], [], [2], [], [], [], [], [8]] + g = {(8, 0): 3, (5, 0): 2, (0, 0): 8} + + assert dmp_from_dict(g, 1, ZZ) == f + assert dmp_to_dict(f, 1) == g + + +def test_dmp_swap(): + f = dmp_normal([[1, 0, 0], [], [1, 0], [], [1]], 1, ZZ) + g = dmp_normal([[1, 0, 0, 0, 0], [1, 0, 0], [1]], 1, ZZ) + + assert dmp_swap(f, 1, 1, 1, ZZ) == f + + assert dmp_swap(f, 0, 1, 1, ZZ) == g + assert dmp_swap(g, 0, 1, 1, ZZ) == f + + raises(IndexError, lambda: dmp_swap(f, -1, -7, 1, ZZ)) + + +def test_dmp_permute(): + f = dmp_normal([[1, 0, 0], [], [1, 0], [], [1]], 1, ZZ) + g = dmp_normal([[1, 0, 0, 0, 0], [1, 0, 0], [1]], 1, ZZ) + + assert dmp_permute(f, [0, 1], 1, ZZ) == f + assert dmp_permute(g, [0, 1], 1, ZZ) == g + + assert dmp_permute(f, [1, 0], 1, ZZ) == g + assert dmp_permute(g, [1, 0], 1, ZZ) == f + + +def test_dmp_nest(): + assert dmp_nest(ZZ(1), 2, ZZ) == [[[1]]] + + assert dmp_nest([[1]], 0, ZZ) == [[1]] + assert dmp_nest([[1]], 1, ZZ) == [[[1]]] + assert dmp_nest([[1]], 2, ZZ) == [[[[1]]]] + + +def test_dmp_raise(): + assert dmp_raise([], 2, 0, ZZ) == [[[]]] + assert dmp_raise([[1]], 0, 1, ZZ) == [[1]] + + assert dmp_raise([[1, 2, 3], [], [2, 3]], 2, 1, ZZ) == \ + [[[[1]], [[2]], [[3]]], [[[]]], [[[2]], [[3]]]] + + +def test_dup_deflate(): + assert dup_deflate([], ZZ) == (1, []) + assert dup_deflate([2], ZZ) == (1, [2]) + assert dup_deflate([1, 2, 3], ZZ) == (1, [1, 2, 3]) + assert dup_deflate([1, 0, 2, 0, 3], ZZ) == (2, [1, 2, 3]) + + assert dup_deflate(dup_from_raw_dict({7: 1, 1: 1}, ZZ), ZZ) == \ + (1, [1, 0, 0, 0, 0, 0, 1, 0]) + assert dup_deflate(dup_from_raw_dict({7: 1, 0: 1}, ZZ), ZZ) == \ + (7, [1, 1]) + assert dup_deflate(dup_from_raw_dict({7: 1, 3: 1}, ZZ), ZZ) == \ + (1, [1, 0, 0, 0, 1, 0, 0, 0]) + + assert dup_deflate(dup_from_raw_dict({7: 1, 4: 1}, ZZ), ZZ) == \ + (1, [1, 0, 0, 1, 0, 0, 0, 0]) + assert dup_deflate(dup_from_raw_dict({8: 1, 4: 1}, ZZ), ZZ) == \ + (4, [1, 1, 0]) + + assert dup_deflate(dup_from_raw_dict({8: 1}, ZZ), ZZ) == \ + (8, [1, 0]) + assert dup_deflate(dup_from_raw_dict({7: 1}, ZZ), ZZ) == \ + (7, [1, 0]) + assert dup_deflate(dup_from_raw_dict({1: 1}, ZZ), ZZ) == \ + (1, [1, 0]) + + +def test_dmp_deflate(): + assert dmp_deflate([[]], 1, ZZ) == ((1, 1), [[]]) + assert dmp_deflate([[2]], 1, ZZ) == ((1, 1), [[2]]) + + f = [[1, 0, 0], [], [1, 0], [], [1]] + + assert dmp_deflate(f, 1, ZZ) == ((2, 1), [[1, 0, 0], [1, 0], [1]]) + + +def test_dup_multi_deflate(): + assert dup_multi_deflate(([2],), ZZ) == (1, ([2],)) + assert dup_multi_deflate(([], []), ZZ) == (1, ([], [])) + + assert dup_multi_deflate(([1, 2, 3],), ZZ) == (1, ([1, 2, 3],)) + assert dup_multi_deflate(([1, 0, 2, 0, 3],), ZZ) == (2, ([1, 2, 3],)) + + assert dup_multi_deflate(([1, 0, 2, 0, 3], [2, 0, 0]), ZZ) == \ + (2, ([1, 2, 3], [2, 0])) + assert dup_multi_deflate(([1, 0, 2, 0, 3], [2, 1, 0]), ZZ) == \ + (1, ([1, 0, 2, 0, 3], [2, 1, 0])) + + +def test_dmp_multi_deflate(): + assert dmp_multi_deflate(([[]],), 1, ZZ) == \ + ((1, 1), ([[]],)) + assert dmp_multi_deflate(([[]], [[]]), 1, ZZ) == \ + ((1, 1), ([[]], [[]])) + + assert dmp_multi_deflate(([[1]], [[]]), 1, ZZ) == \ + ((1, 1), ([[1]], [[]])) + assert dmp_multi_deflate(([[1]], [[2]]), 1, ZZ) == \ + ((1, 1), ([[1]], [[2]])) + assert dmp_multi_deflate(([[1]], [[2, 0]]), 1, ZZ) == \ + ((1, 1), ([[1]], [[2, 0]])) + + assert dmp_multi_deflate(([[2, 0]], [[2, 0]]), 1, ZZ) == \ + ((1, 1), ([[2, 0]], [[2, 0]])) + + assert dmp_multi_deflate( + ([[2]], [[2, 0, 0]]), 1, ZZ) == ((1, 2), ([[2]], [[2, 0]])) + assert dmp_multi_deflate( + ([[2, 0, 0]], [[2, 0, 0]]), 1, ZZ) == ((1, 2), ([[2, 0]], [[2, 0]])) + + assert dmp_multi_deflate(([2, 0, 0], [1, 0, 4, 0, 1]), 0, ZZ) == \ + ((2,), ([2, 0], [1, 4, 1])) + + f = [[1, 0, 0], [], [1, 0], [], [1]] + g = [[1, 0, 1, 0], [], [1]] + + assert dmp_multi_deflate((f,), 1, ZZ) == \ + ((2, 1), ([[1, 0, 0], [1, 0], [1]],)) + + assert dmp_multi_deflate((f, g), 1, ZZ) == \ + ((2, 1), ([[1, 0, 0], [1, 0], [1]], + [[1, 0, 1, 0], [1]])) + + +def test_dup_inflate(): + assert dup_inflate([], 17, ZZ) == [] + + assert dup_inflate([1, 2, 3], 1, ZZ) == [1, 2, 3] + assert dup_inflate([1, 2, 3], 2, ZZ) == [1, 0, 2, 0, 3] + assert dup_inflate([1, 2, 3], 3, ZZ) == [1, 0, 0, 2, 0, 0, 3] + assert dup_inflate([1, 2, 3], 4, ZZ) == [1, 0, 0, 0, 2, 0, 0, 0, 3] + + raises(IndexError, lambda: dup_inflate([1, 2, 3], 0, ZZ)) + + +def test_dmp_inflate(): + assert dmp_inflate([1], (3,), 0, ZZ) == [1] + + assert dmp_inflate([[]], (3, 7), 1, ZZ) == [[]] + assert dmp_inflate([[2]], (1, 2), 1, ZZ) == [[2]] + + assert dmp_inflate([[2, 0]], (1, 1), 1, ZZ) == [[2, 0]] + assert dmp_inflate([[2, 0]], (1, 2), 1, ZZ) == [[2, 0, 0]] + assert dmp_inflate([[2, 0]], (1, 3), 1, ZZ) == [[2, 0, 0, 0]] + + assert dmp_inflate([[1, 0, 0], [1], [1, 0]], (2, 1), 1, ZZ) == \ + [[1, 0, 0], [], [1], [], [1, 0]] + + raises(IndexError, lambda: dmp_inflate([[]], (-3, 7), 1, ZZ)) + + +def test_dmp_exclude(): + assert dmp_exclude([[[]]], 2, ZZ) == ([], [[[]]], 2) + assert dmp_exclude([[[7]]], 2, ZZ) == ([], [[[7]]], 2) + + assert dmp_exclude([1, 2, 3], 0, ZZ) == ([], [1, 2, 3], 0) + assert dmp_exclude([[1], [2, 3]], 1, ZZ) == ([], [[1], [2, 3]], 1) + + assert dmp_exclude([[1, 2, 3]], 1, ZZ) == ([0], [1, 2, 3], 0) + assert dmp_exclude([[1], [2], [3]], 1, ZZ) == ([1], [1, 2, 3], 0) + + assert dmp_exclude([[[1, 2, 3]]], 2, ZZ) == ([0, 1], [1, 2, 3], 0) + assert dmp_exclude([[[1]], [[2]], [[3]]], 2, ZZ) == ([1, 2], [1, 2, 3], 0) + + +def test_dmp_include(): + assert dmp_include([1, 2, 3], [], 0, ZZ) == [1, 2, 3] + + assert dmp_include([1, 2, 3], [0], 0, ZZ) == [[1, 2, 3]] + assert dmp_include([1, 2, 3], [1], 0, ZZ) == [[1], [2], [3]] + + assert dmp_include([1, 2, 3], [0, 1], 0, ZZ) == [[[1, 2, 3]]] + assert dmp_include([1, 2, 3], [1, 2], 0, ZZ) == [[[1]], [[2]], [[3]]] + + +def test_dmp_inject(): + R, x,y = ring("x,y", ZZ) + K = R.to_domain() + + assert dmp_inject([], 0, K) == ([[[]]], 2) + assert dmp_inject([[]], 1, K) == ([[[[]]]], 3) + + assert dmp_inject([R(1)], 0, K) == ([[[1]]], 2) + assert dmp_inject([[R(1)]], 1, K) == ([[[[1]]]], 3) + + assert dmp_inject([R(1), 2*x + 3*y + 4], 0, K) == ([[[1]], [[2], [3, 4]]], 2) + + f = [3*x**2 + 7*x*y + 5*y**2, 2*x, R(0), x*y**2 + 11] + g = [[[3], [7, 0], [5, 0, 0]], [[2], []], [[]], [[1, 0, 0], [11]]] + + assert dmp_inject(f, 0, K) == (g, 2) + + +def test_dmp_eject(): + R, x,y = ring("x,y", ZZ) + K = R.to_domain() + + assert dmp_eject([[[]]], 2, K) == [] + assert dmp_eject([[[[]]]], 3, K) == [[]] + + assert dmp_eject([[[1]]], 2, K) == [R(1)] + assert dmp_eject([[[[1]]]], 3, K) == [[R(1)]] + + assert dmp_eject([[[1]], [[2], [3, 4]]], 2, K) == [R(1), 2*x + 3*y + 4] + + f = [3*x**2 + 7*x*y + 5*y**2, 2*x, R(0), x*y**2 + 11] + g = [[[3], [7, 0], [5, 0, 0]], [[2], []], [[]], [[1, 0, 0], [11]]] + + assert dmp_eject(g, 2, K) == f + + +def test_dup_terms_gcd(): + assert dup_terms_gcd([], ZZ) == (0, []) + assert dup_terms_gcd([1, 0, 1], ZZ) == (0, [1, 0, 1]) + assert dup_terms_gcd([1, 0, 1, 0], ZZ) == (1, [1, 0, 1]) + + +def test_dmp_terms_gcd(): + assert dmp_terms_gcd([[]], 1, ZZ) == ((0, 0), [[]]) + + assert dmp_terms_gcd([1, 0, 1, 0], 0, ZZ) == ((1,), [1, 0, 1]) + assert dmp_terms_gcd([[1], [], [1], []], 1, ZZ) == ((1, 0), [[1], [], [1]]) + + assert dmp_terms_gcd( + [[1, 0], [], [1]], 1, ZZ) == ((0, 0), [[1, 0], [], [1]]) + assert dmp_terms_gcd( + [[1, 0], [1, 0, 0], [], []], 1, ZZ) == ((2, 1), [[1], [1, 0]]) + + +def test_dmp_list_terms(): + assert dmp_list_terms([[[]]], 2, ZZ) == [((0, 0, 0), 0)] + assert dmp_list_terms([[[1]]], 2, ZZ) == [((0, 0, 0), 1)] + + assert dmp_list_terms([1, 2, 4, 3, 5], 0, ZZ) == \ + [((4,), 1), ((3,), 2), ((2,), 4), ((1,), 3), ((0,), 5)] + + assert dmp_list_terms([[1], [2, 4], [3, 5, 0]], 1, ZZ) == \ + [((2, 0), 1), ((1, 1), 2), ((1, 0), 4), ((0, 2), 3), ((0, 1), 5)] + + f = [[2, 0, 0, 0], [1, 0, 0], []] + + assert dmp_list_terms(f, 1, ZZ, order='lex') == [((2, 3), 2), ((1, 2), 1)] + assert dmp_list_terms( + f, 1, ZZ, order='grlex') == [((2, 3), 2), ((1, 2), 1)] + + f = [[2, 0, 0, 0], [1, 0, 0, 0, 0, 0], []] + + assert dmp_list_terms(f, 1, ZZ, order='lex') == [((2, 3), 2), ((1, 5), 1)] + assert dmp_list_terms( + f, 1, ZZ, order='grlex') == [((1, 5), 1), ((2, 3), 2)] + + +def test_dmp_apply_pairs(): + h = lambda a, b: a*b + + assert dmp_apply_pairs([1, 2, 3], [4, 5, 6], h, [], 0, ZZ) == [4, 10, 18] + + assert dmp_apply_pairs([2, 3], [4, 5, 6], h, [], 0, ZZ) == [10, 18] + assert dmp_apply_pairs([1, 2, 3], [5, 6], h, [], 0, ZZ) == [10, 18] + + assert dmp_apply_pairs( + [[1, 2], [3]], [[4, 5], [6]], h, [], 1, ZZ) == [[4, 10], [18]] + + assert dmp_apply_pairs( + [[1, 2], [3]], [[4], [5, 6]], h, [], 1, ZZ) == [[8], [18]] + assert dmp_apply_pairs( + [[1], [2, 3]], [[4, 5], [6]], h, [], 1, ZZ) == [[5], [18]] + + +def test_dup_slice(): + f = [1, 2, 3, 4] + + assert dup_slice(f, 0, 0, ZZ) == [] + assert dup_slice(f, 0, 1, ZZ) == [4] + assert dup_slice(f, 0, 2, ZZ) == [3, 4] + assert dup_slice(f, 0, 3, ZZ) == [2, 3, 4] + assert dup_slice(f, 0, 4, ZZ) == [1, 2, 3, 4] + + assert dup_slice(f, 0, 4, ZZ) == f + assert dup_slice(f, 0, 9, ZZ) == f + + assert dup_slice(f, 1, 0, ZZ) == [] + assert dup_slice(f, 1, 1, ZZ) == [] + assert dup_slice(f, 1, 2, ZZ) == [3, 0] + assert dup_slice(f, 1, 3, ZZ) == [2, 3, 0] + assert dup_slice(f, 1, 4, ZZ) == [1, 2, 3, 0] + + assert dup_slice([1, 2], 0, 3, ZZ) == [1, 2] + + g = [1, 0, 0, 2] + + assert dup_slice(g, 0, 3, ZZ) == [2] + + +def test_dup_random(): + f = dup_random(0, -10, 10, ZZ) + + assert dup_degree(f) == 0 + assert all(-10 <= c <= 10 for c in f) + + f = dup_random(1, -20, 20, ZZ) + + assert dup_degree(f) == 1 + assert all(-20 <= c <= 20 for c in f) + + f = dup_random(2, -30, 30, ZZ) + + assert dup_degree(f) == 2 + assert all(-30 <= c <= 30 for c in f) + + f = dup_random(3, -40, 40, ZZ) + + assert dup_degree(f) == 3 + assert all(-40 <= c <= 40 for c in f) diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_densetools.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_densetools.py new file mode 100644 index 0000000000000000000000000000000000000000..b4bebd2a6f061a13a7d34b7689c696456310f62e --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_densetools.py @@ -0,0 +1,714 @@ +"""Tests for dense recursive polynomials' tools. """ + +from sympy.polys.densebasic import ( + dup_normal, dmp_normal, + dup_from_raw_dict, + dmp_convert, dmp_swap, +) + +from sympy.polys.densearith import dmp_mul_ground + +from sympy.polys.densetools import ( + dup_clear_denoms, dmp_clear_denoms, + dup_integrate, dmp_integrate, dmp_integrate_in, + dup_diff, dmp_diff, dmp_diff_in, + dup_eval, dmp_eval, dmp_eval_in, + dmp_eval_tail, dmp_diff_eval_in, + dup_trunc, dmp_trunc, dmp_ground_trunc, + dup_monic, dmp_ground_monic, + dup_content, dmp_ground_content, + dup_primitive, dmp_ground_primitive, + dup_extract, dmp_ground_extract, + dup_real_imag, + dup_mirror, dup_scale, dup_shift, dmp_shift, + dup_transform, + dup_compose, dmp_compose, + dup_decompose, + dmp_lift, + dup_sign_variations, + dup_revert, dmp_revert, +) +from sympy.polys.polyclasses import ANP + +from sympy.polys.polyerrors import ( + MultivariatePolynomialError, + ExactQuotientFailed, + NotReversible, + DomainError, +) + +from sympy.polys.specialpolys import f_polys + +from sympy.polys.domains import FF, ZZ, QQ, ZZ_I, QQ_I, EX, RR +from sympy.polys.rings import ring + +from sympy.core.numbers import I +from sympy.core.singleton import S +from sympy.functions.elementary.trigonometric import sin + +from sympy.abc import x +from sympy.testing.pytest import raises + +f_0, f_1, f_2, f_3, f_4, f_5, f_6 = [ f.to_dense() for f in f_polys() ] + +def test_dup_integrate(): + assert dup_integrate([], 1, QQ) == [] + assert dup_integrate([], 2, QQ) == [] + + assert dup_integrate([QQ(1)], 1, QQ) == [QQ(1), QQ(0)] + assert dup_integrate([QQ(1)], 2, QQ) == [QQ(1, 2), QQ(0), QQ(0)] + + assert dup_integrate([QQ(1), QQ(2), QQ(3)], 0, QQ) == \ + [QQ(1), QQ(2), QQ(3)] + assert dup_integrate([QQ(1), QQ(2), QQ(3)], 1, QQ) == \ + [QQ(1, 3), QQ(1), QQ(3), QQ(0)] + assert dup_integrate([QQ(1), QQ(2), QQ(3)], 2, QQ) == \ + [QQ(1, 12), QQ(1, 3), QQ(3, 2), QQ(0), QQ(0)] + assert dup_integrate([QQ(1), QQ(2), QQ(3)], 3, QQ) == \ + [QQ(1, 60), QQ(1, 12), QQ(1, 2), QQ(0), QQ(0), QQ(0)] + + assert dup_integrate(dup_from_raw_dict({29: QQ(17)}, QQ), 3, QQ) == \ + dup_from_raw_dict({32: QQ(17, 29760)}, QQ) + + assert dup_integrate(dup_from_raw_dict({29: QQ(17), 5: QQ(1, 2)}, QQ), 3, QQ) == \ + dup_from_raw_dict({32: QQ(17, 29760), 8: QQ(1, 672)}, QQ) + + +def test_dmp_integrate(): + assert dmp_integrate([QQ(1)], 2, 0, QQ) == [QQ(1, 2), QQ(0), QQ(0)] + + assert dmp_integrate([[[]]], 1, 2, QQ) == [[[]]] + assert dmp_integrate([[[]]], 2, 2, QQ) == [[[]]] + + assert dmp_integrate([[[QQ(1)]]], 1, 2, QQ) == [[[QQ(1)]], [[]]] + assert dmp_integrate([[[QQ(1)]]], 2, 2, QQ) == [[[QQ(1, 2)]], [[]], [[]]] + + assert dmp_integrate([[QQ(1)], [QQ(2)], [QQ(3)]], 0, 1, QQ) == \ + [[QQ(1)], [QQ(2)], [QQ(3)]] + assert dmp_integrate([[QQ(1)], [QQ(2)], [QQ(3)]], 1, 1, QQ) == \ + [[QQ(1, 3)], [QQ(1)], [QQ(3)], []] + assert dmp_integrate([[QQ(1)], [QQ(2)], [QQ(3)]], 2, 1, QQ) == \ + [[QQ(1, 12)], [QQ(1, 3)], [QQ(3, 2)], [], []] + assert dmp_integrate([[QQ(1)], [QQ(2)], [QQ(3)]], 3, 1, QQ) == \ + [[QQ(1, 60)], [QQ(1, 12)], [QQ(1, 2)], [], [], []] + + +def test_dmp_integrate_in(): + f = dmp_convert(f_6, 3, ZZ, QQ) + + assert dmp_integrate_in(f, 2, 1, 3, QQ) == \ + dmp_swap( + dmp_integrate(dmp_swap(f, 0, 1, 3, QQ), 2, 3, QQ), 0, 1, 3, QQ) + assert dmp_integrate_in(f, 3, 1, 3, QQ) == \ + dmp_swap( + dmp_integrate(dmp_swap(f, 0, 1, 3, QQ), 3, 3, QQ), 0, 1, 3, QQ) + assert dmp_integrate_in(f, 2, 2, 3, QQ) == \ + dmp_swap( + dmp_integrate(dmp_swap(f, 0, 2, 3, QQ), 2, 3, QQ), 0, 2, 3, QQ) + assert dmp_integrate_in(f, 3, 2, 3, QQ) == \ + dmp_swap( + dmp_integrate(dmp_swap(f, 0, 2, 3, QQ), 3, 3, QQ), 0, 2, 3, QQ) + + raises(IndexError, lambda: dmp_integrate_in(f, 1, -1, 3, QQ)) + raises(IndexError, lambda: dmp_integrate_in(f, 1, 4, 3, QQ)) + + +def test_dup_diff(): + assert dup_diff([], 1, ZZ) == [] + assert dup_diff([7], 1, ZZ) == [] + assert dup_diff([2, 7], 1, ZZ) == [2] + assert dup_diff([1, 2, 1], 1, ZZ) == [2, 2] + assert dup_diff([1, 2, 3, 4], 1, ZZ) == [3, 4, 3] + assert dup_diff([1, -1, 0, 0, 2], 1, ZZ) == [4, -3, 0, 0] + + f = dup_normal([17, 34, 56, -345, 23, 76, 0, 0, 12, 3, 7], ZZ) + + assert dup_diff(f, 0, ZZ) == f + assert dup_diff(f, 1, ZZ) == [170, 306, 448, -2415, 138, 380, 0, 0, 24, 3] + assert dup_diff(f, 2, ZZ) == dup_diff(dup_diff(f, 1, ZZ), 1, ZZ) + assert dup_diff( + f, 3, ZZ) == dup_diff(dup_diff(dup_diff(f, 1, ZZ), 1, ZZ), 1, ZZ) + + K = FF(3) + f = dup_normal([17, 34, 56, -345, 23, 76, 0, 0, 12, 3, 7], K) + + assert dup_diff(f, 1, K) == dup_normal([2, 0, 1, 0, 0, 2, 0, 0, 0, 0], K) + assert dup_diff(f, 2, K) == dup_normal([1, 0, 0, 2, 0, 0, 0], K) + assert dup_diff(f, 3, K) == dup_normal([], K) + + assert dup_diff(f, 0, K) == f + assert dup_diff(f, 2, K) == dup_diff(dup_diff(f, 1, K), 1, K) + assert dup_diff( + f, 3, K) == dup_diff(dup_diff(dup_diff(f, 1, K), 1, K), 1, K) + + +def test_dmp_diff(): + assert dmp_diff([], 1, 0, ZZ) == [] + assert dmp_diff([[]], 1, 1, ZZ) == [[]] + assert dmp_diff([[[]]], 1, 2, ZZ) == [[[]]] + + assert dmp_diff([[[1], [2]]], 1, 2, ZZ) == [[[]]] + + assert dmp_diff([[[1]], [[]]], 1, 2, ZZ) == [[[1]]] + assert dmp_diff([[[3]], [[1]], [[]]], 1, 2, ZZ) == [[[6]], [[1]]] + + assert dmp_diff([1, -1, 0, 0, 2], 1, 0, ZZ) == \ + dup_diff([1, -1, 0, 0, 2], 1, ZZ) + + assert dmp_diff(f_6, 0, 3, ZZ) == f_6 + assert dmp_diff(f_6, 1, 3, ZZ) == [[[[8460]], [[]]], + [[[135, 0, 0], [], [], [-135, 0, 0]]], + [[[]]], + [[[-423]], [[-47]], [[]], [[141], [], [94, 0], []], [[]]]] + assert dmp_diff( + f_6, 2, 3, ZZ) == dmp_diff(dmp_diff(f_6, 1, 3, ZZ), 1, 3, ZZ) + assert dmp_diff(f_6, 3, 3, ZZ) == dmp_diff( + dmp_diff(dmp_diff(f_6, 1, 3, ZZ), 1, 3, ZZ), 1, 3, ZZ) + + K = FF(23) + F_6 = dmp_normal(f_6, 3, K) + + assert dmp_diff(F_6, 0, 3, K) == F_6 + assert dmp_diff(F_6, 1, 3, K) == dmp_diff(F_6, 1, 3, K) + assert dmp_diff(F_6, 2, 3, K) == dmp_diff(dmp_diff(F_6, 1, 3, K), 1, 3, K) + assert dmp_diff(F_6, 3, 3, K) == dmp_diff( + dmp_diff(dmp_diff(F_6, 1, 3, K), 1, 3, K), 1, 3, K) + + +def test_dmp_diff_in(): + assert dmp_diff_in(f_6, 2, 1, 3, ZZ) == \ + dmp_swap(dmp_diff(dmp_swap(f_6, 0, 1, 3, ZZ), 2, 3, ZZ), 0, 1, 3, ZZ) + assert dmp_diff_in(f_6, 3, 1, 3, ZZ) == \ + dmp_swap(dmp_diff(dmp_swap(f_6, 0, 1, 3, ZZ), 3, 3, ZZ), 0, 1, 3, ZZ) + assert dmp_diff_in(f_6, 2, 2, 3, ZZ) == \ + dmp_swap(dmp_diff(dmp_swap(f_6, 0, 2, 3, ZZ), 2, 3, ZZ), 0, 2, 3, ZZ) + assert dmp_diff_in(f_6, 3, 2, 3, ZZ) == \ + dmp_swap(dmp_diff(dmp_swap(f_6, 0, 2, 3, ZZ), 3, 3, ZZ), 0, 2, 3, ZZ) + + raises(IndexError, lambda: dmp_diff_in(f_6, 1, -1, 3, ZZ)) + raises(IndexError, lambda: dmp_diff_in(f_6, 1, 4, 3, ZZ)) + +def test_dup_eval(): + assert dup_eval([], 7, ZZ) == 0 + assert dup_eval([1, 2], 0, ZZ) == 2 + assert dup_eval([1, 2, 3], 7, ZZ) == 66 + + +def test_dmp_eval(): + assert dmp_eval([], 3, 0, ZZ) == 0 + + assert dmp_eval([[]], 3, 1, ZZ) == [] + assert dmp_eval([[[]]], 3, 2, ZZ) == [[]] + + assert dmp_eval([[1, 2]], 0, 1, ZZ) == [1, 2] + + assert dmp_eval([[[1]]], 3, 2, ZZ) == [[1]] + assert dmp_eval([[[1, 2]]], 3, 2, ZZ) == [[1, 2]] + + assert dmp_eval([[3, 2], [1, 2]], 3, 1, ZZ) == [10, 8] + assert dmp_eval([[[3, 2]], [[1, 2]]], 3, 2, ZZ) == [[10, 8]] + + +def test_dmp_eval_in(): + assert dmp_eval_in( + f_6, -2, 1, 3, ZZ) == dmp_eval(dmp_swap(f_6, 0, 1, 3, ZZ), -2, 3, ZZ) + assert dmp_eval_in( + f_6, 7, 1, 3, ZZ) == dmp_eval(dmp_swap(f_6, 0, 1, 3, ZZ), 7, 3, ZZ) + assert dmp_eval_in(f_6, -2, 2, 3, ZZ) == dmp_swap( + dmp_eval(dmp_swap(f_6, 0, 2, 3, ZZ), -2, 3, ZZ), 0, 1, 2, ZZ) + assert dmp_eval_in(f_6, 7, 2, 3, ZZ) == dmp_swap( + dmp_eval(dmp_swap(f_6, 0, 2, 3, ZZ), 7, 3, ZZ), 0, 1, 2, ZZ) + + f = [[[int(45)]], [[]], [[]], [[int(-9)], [-1], [], [int(3), int(0), int(10), int(0)]]] + + assert dmp_eval_in(f, -2, 2, 2, ZZ) == \ + [[45], [], [], [-9, -1, 0, -44]] + + raises(IndexError, lambda: dmp_eval_in(f_6, ZZ(1), -1, 3, ZZ)) + raises(IndexError, lambda: dmp_eval_in(f_6, ZZ(1), 4, 3, ZZ)) + + +def test_dmp_eval_tail(): + assert dmp_eval_tail([[]], [1], 1, ZZ) == [] + assert dmp_eval_tail([[[]]], [1], 2, ZZ) == [[]] + assert dmp_eval_tail([[[]]], [1, 2], 2, ZZ) == [] + + assert dmp_eval_tail(f_0, [], 2, ZZ) == f_0 + + assert dmp_eval_tail(f_0, [1, -17, 8], 2, ZZ) == 84496 + assert dmp_eval_tail(f_0, [-17, 8], 2, ZZ) == [-1409, 3, 85902] + assert dmp_eval_tail(f_0, [8], 2, ZZ) == [[83, 2], [3], [302, 81, 1]] + + assert dmp_eval_tail(f_1, [-17, 8], 2, ZZ) == [-136, 15699, 9166, -27144] + + assert dmp_eval_tail( + f_2, [-12, 3], 2, ZZ) == [-1377, 0, -702, -1224, 0, -624] + assert dmp_eval_tail( + f_3, [-12, 3], 2, ZZ) == [144, 82, -5181, -28872, -14868, -540] + + assert dmp_eval_tail( + f_4, [25, -1], 2, ZZ) == [152587890625, 9765625, -59605407714843750, + -3839159765625, -1562475, 9536712644531250, 610349546750, -4, 24414375000, 1562520] + assert dmp_eval_tail(f_5, [25, -1], 2, ZZ) == [-1, -78, -2028, -17576] + + assert dmp_eval_tail(f_6, [0, 2, 4], 3, ZZ) == [5040, 0, 0, 4480] + + +def test_dmp_diff_eval_in(): + assert dmp_diff_eval_in(f_6, 2, 7, 1, 3, ZZ) == \ + dmp_eval(dmp_diff(dmp_swap(f_6, 0, 1, 3, ZZ), 2, 3, ZZ), 7, 3, ZZ) + + assert dmp_diff_eval_in(f_6, 2, 7, 0, 3, ZZ) == \ + dmp_eval(dmp_diff(f_6, 2, 3, ZZ), 7, 3, ZZ) + + raises(IndexError, lambda: dmp_diff_eval_in(f_6, 1, ZZ(1), 4, 3, ZZ)) + + +def test_dup_revert(): + f = [-QQ(1, 720), QQ(0), QQ(1, 24), QQ(0), -QQ(1, 2), QQ(0), QQ(1)] + g = [QQ(61, 720), QQ(0), QQ(5, 24), QQ(0), QQ(1, 2), QQ(0), QQ(1)] + + assert dup_revert(f, 8, QQ) == g + + raises(NotReversible, lambda: dup_revert([QQ(1), QQ(0)], 3, QQ)) + + +def test_dmp_revert(): + f = [-QQ(1, 720), QQ(0), QQ(1, 24), QQ(0), -QQ(1, 2), QQ(0), QQ(1)] + g = [QQ(61, 720), QQ(0), QQ(5, 24), QQ(0), QQ(1, 2), QQ(0), QQ(1)] + + assert dmp_revert(f, 8, 0, QQ) == g + + raises(MultivariatePolynomialError, lambda: dmp_revert([[1]], 2, 1, QQ)) + + +def test_dup_trunc(): + assert dup_trunc([1, 2, 3, 4, 5, 6], ZZ(3), ZZ) == [1, -1, 0, 1, -1, 0] + assert dup_trunc([6, 5, 4, 3, 2, 1], ZZ(3), ZZ) == [-1, 1, 0, -1, 1] + + R = ZZ_I + assert dup_trunc([R(3), R(4), R(5)], R(3), R) == [R(1), R(-1)] + + K = FF(5) + assert dup_trunc([K(3), K(4), K(5)], K(3), K) == [K(1), K(0)] + + +def test_dmp_trunc(): + assert dmp_trunc([[]], [1, 2], 2, ZZ) == [[]] + assert dmp_trunc([[1, 2], [1, 4, 1], [1]], [1, 2], 1, ZZ) == [[-3], [1]] + + +def test_dmp_ground_trunc(): + assert dmp_ground_trunc(f_0, ZZ(3), 2, ZZ) == \ + dmp_normal( + [[[1, -1, 0], [-1]], [[]], [[1, -1, 0], [1, -1, 1], [1]]], 2, ZZ) + + +def test_dup_monic(): + assert dup_monic([3, 6, 9], ZZ) == [1, 2, 3] + + raises(ExactQuotientFailed, lambda: dup_monic([3, 4, 5], ZZ)) + + assert dup_monic([], QQ) == [] + assert dup_monic([QQ(1)], QQ) == [QQ(1)] + assert dup_monic([QQ(7), QQ(1), QQ(21)], QQ) == [QQ(1), QQ(1, 7), QQ(3)] + + +def test_dmp_ground_monic(): + assert dmp_ground_monic([3, 6, 9], 0, ZZ) == [1, 2, 3] + + assert dmp_ground_monic([[3], [6], [9]], 1, ZZ) == [[1], [2], [3]] + + raises( + ExactQuotientFailed, lambda: dmp_ground_monic([[3], [4], [5]], 1, ZZ)) + + assert dmp_ground_monic([[]], 1, QQ) == [[]] + assert dmp_ground_monic([[QQ(1)]], 1, QQ) == [[QQ(1)]] + assert dmp_ground_monic( + [[QQ(7)], [QQ(1)], [QQ(21)]], 1, QQ) == [[QQ(1)], [QQ(1, 7)], [QQ(3)]] + + +def test_dup_content(): + assert dup_content([], ZZ) == ZZ(0) + assert dup_content([1], ZZ) == ZZ(1) + assert dup_content([-1], ZZ) == ZZ(1) + assert dup_content([1, 1], ZZ) == ZZ(1) + assert dup_content([2, 2], ZZ) == ZZ(2) + assert dup_content([1, 2, 1], ZZ) == ZZ(1) + assert dup_content([2, 4, 2], ZZ) == ZZ(2) + + assert dup_content([QQ(2, 3), QQ(4, 9)], QQ) == QQ(2, 9) + assert dup_content([QQ(2, 3), QQ(4, 5)], QQ) == QQ(2, 15) + + +def test_dmp_ground_content(): + assert dmp_ground_content([[]], 1, ZZ) == ZZ(0) + assert dmp_ground_content([[]], 1, QQ) == QQ(0) + assert dmp_ground_content([[1]], 1, ZZ) == ZZ(1) + assert dmp_ground_content([[-1]], 1, ZZ) == ZZ(1) + assert dmp_ground_content([[1], [1]], 1, ZZ) == ZZ(1) + assert dmp_ground_content([[2], [2]], 1, ZZ) == ZZ(2) + assert dmp_ground_content([[1], [2], [1]], 1, ZZ) == ZZ(1) + assert dmp_ground_content([[2], [4], [2]], 1, ZZ) == ZZ(2) + + assert dmp_ground_content([[QQ(2, 3)], [QQ(4, 9)]], 1, QQ) == QQ(2, 9) + assert dmp_ground_content([[QQ(2, 3)], [QQ(4, 5)]], 1, QQ) == QQ(2, 15) + + assert dmp_ground_content(f_0, 2, ZZ) == ZZ(1) + assert dmp_ground_content( + dmp_mul_ground(f_0, ZZ(2), 2, ZZ), 2, ZZ) == ZZ(2) + + assert dmp_ground_content(f_1, 2, ZZ) == ZZ(1) + assert dmp_ground_content( + dmp_mul_ground(f_1, ZZ(3), 2, ZZ), 2, ZZ) == ZZ(3) + + assert dmp_ground_content(f_2, 2, ZZ) == ZZ(1) + assert dmp_ground_content( + dmp_mul_ground(f_2, ZZ(4), 2, ZZ), 2, ZZ) == ZZ(4) + + assert dmp_ground_content(f_3, 2, ZZ) == ZZ(1) + assert dmp_ground_content( + dmp_mul_ground(f_3, ZZ(5), 2, ZZ), 2, ZZ) == ZZ(5) + + assert dmp_ground_content(f_4, 2, ZZ) == ZZ(1) + assert dmp_ground_content( + dmp_mul_ground(f_4, ZZ(6), 2, ZZ), 2, ZZ) == ZZ(6) + + assert dmp_ground_content(f_5, 2, ZZ) == ZZ(1) + assert dmp_ground_content( + dmp_mul_ground(f_5, ZZ(7), 2, ZZ), 2, ZZ) == ZZ(7) + + assert dmp_ground_content(f_6, 3, ZZ) == ZZ(1) + assert dmp_ground_content( + dmp_mul_ground(f_6, ZZ(8), 3, ZZ), 3, ZZ) == ZZ(8) + + +def test_dup_primitive(): + assert dup_primitive([], ZZ) == (ZZ(0), []) + assert dup_primitive([ZZ(1)], ZZ) == (ZZ(1), [ZZ(1)]) + assert dup_primitive([ZZ(1), ZZ(1)], ZZ) == (ZZ(1), [ZZ(1), ZZ(1)]) + assert dup_primitive([ZZ(2), ZZ(2)], ZZ) == (ZZ(2), [ZZ(1), ZZ(1)]) + assert dup_primitive( + [ZZ(1), ZZ(2), ZZ(1)], ZZ) == (ZZ(1), [ZZ(1), ZZ(2), ZZ(1)]) + assert dup_primitive( + [ZZ(2), ZZ(4), ZZ(2)], ZZ) == (ZZ(2), [ZZ(1), ZZ(2), ZZ(1)]) + + assert dup_primitive([], QQ) == (QQ(0), []) + assert dup_primitive([QQ(1)], QQ) == (QQ(1), [QQ(1)]) + assert dup_primitive([QQ(1), QQ(1)], QQ) == (QQ(1), [QQ(1), QQ(1)]) + assert dup_primitive([QQ(2), QQ(2)], QQ) == (QQ(2), [QQ(1), QQ(1)]) + assert dup_primitive( + [QQ(1), QQ(2), QQ(1)], QQ) == (QQ(1), [QQ(1), QQ(2), QQ(1)]) + assert dup_primitive( + [QQ(2), QQ(4), QQ(2)], QQ) == (QQ(2), [QQ(1), QQ(2), QQ(1)]) + + assert dup_primitive( + [QQ(2, 3), QQ(4, 9)], QQ) == (QQ(2, 9), [QQ(3), QQ(2)]) + assert dup_primitive( + [QQ(2, 3), QQ(4, 5)], QQ) == (QQ(2, 15), [QQ(5), QQ(6)]) + + +def test_dmp_ground_primitive(): + assert dmp_ground_primitive([ZZ(1)], 0, ZZ) == (ZZ(1), [ZZ(1)]) + + assert dmp_ground_primitive([[]], 1, ZZ) == (ZZ(0), [[]]) + + assert dmp_ground_primitive(f_0, 2, ZZ) == (ZZ(1), f_0) + assert dmp_ground_primitive( + dmp_mul_ground(f_0, ZZ(2), 2, ZZ), 2, ZZ) == (ZZ(2), f_0) + + assert dmp_ground_primitive(f_1, 2, ZZ) == (ZZ(1), f_1) + assert dmp_ground_primitive( + dmp_mul_ground(f_1, ZZ(3), 2, ZZ), 2, ZZ) == (ZZ(3), f_1) + + assert dmp_ground_primitive(f_2, 2, ZZ) == (ZZ(1), f_2) + assert dmp_ground_primitive( + dmp_mul_ground(f_2, ZZ(4), 2, ZZ), 2, ZZ) == (ZZ(4), f_2) + + assert dmp_ground_primitive(f_3, 2, ZZ) == (ZZ(1), f_3) + assert dmp_ground_primitive( + dmp_mul_ground(f_3, ZZ(5), 2, ZZ), 2, ZZ) == (ZZ(5), f_3) + + assert dmp_ground_primitive(f_4, 2, ZZ) == (ZZ(1), f_4) + assert dmp_ground_primitive( + dmp_mul_ground(f_4, ZZ(6), 2, ZZ), 2, ZZ) == (ZZ(6), f_4) + + assert dmp_ground_primitive(f_5, 2, ZZ) == (ZZ(1), f_5) + assert dmp_ground_primitive( + dmp_mul_ground(f_5, ZZ(7), 2, ZZ), 2, ZZ) == (ZZ(7), f_5) + + assert dmp_ground_primitive(f_6, 3, ZZ) == (ZZ(1), f_6) + assert dmp_ground_primitive( + dmp_mul_ground(f_6, ZZ(8), 3, ZZ), 3, ZZ) == (ZZ(8), f_6) + + assert dmp_ground_primitive([[ZZ(2)]], 1, ZZ) == (ZZ(2), [[ZZ(1)]]) + assert dmp_ground_primitive([[QQ(2)]], 1, QQ) == (QQ(2), [[QQ(1)]]) + + assert dmp_ground_primitive( + [[QQ(2, 3)], [QQ(4, 9)]], 1, QQ) == (QQ(2, 9), [[QQ(3)], [QQ(2)]]) + assert dmp_ground_primitive( + [[QQ(2, 3)], [QQ(4, 5)]], 1, QQ) == (QQ(2, 15), [[QQ(5)], [QQ(6)]]) + + +def test_dup_extract(): + f = dup_normal([2930944, 0, 2198208, 0, 549552, 0, 45796], ZZ) + g = dup_normal([17585664, 0, 8792832, 0, 1099104, 0], ZZ) + + F = dup_normal([64, 0, 48, 0, 12, 0, 1], ZZ) + G = dup_normal([384, 0, 192, 0, 24, 0], ZZ) + + assert dup_extract(f, g, ZZ) == (45796, F, G) + + +def test_dmp_ground_extract(): + f = dmp_normal( + [[2930944], [], [2198208], [], [549552], [], [45796]], 1, ZZ) + g = dmp_normal([[17585664], [], [8792832], [], [1099104], []], 1, ZZ) + + F = dmp_normal([[64], [], [48], [], [12], [], [1]], 1, ZZ) + G = dmp_normal([[384], [], [192], [], [24], []], 1, ZZ) + + assert dmp_ground_extract(f, g, 1, ZZ) == (45796, F, G) + + +def test_dup_real_imag(): + assert dup_real_imag([], ZZ) == ([[]], [[]]) + assert dup_real_imag([1], ZZ) == ([[1]], [[]]) + + assert dup_real_imag([1, 1], ZZ) == ([[1], [1]], [[1, 0]]) + assert dup_real_imag([1, 2], ZZ) == ([[1], [2]], [[1, 0]]) + + assert dup_real_imag( + [1, 2, 3], ZZ) == ([[1], [2], [-1, 0, 3]], [[2, 0], [2, 0]]) + + assert dup_real_imag([ZZ(1), ZZ(0), ZZ(1), ZZ(3)], ZZ) == ( + [[ZZ(1)], [], [ZZ(-3), ZZ(0), ZZ(1)], [ZZ(3)]], + [[ZZ(3), ZZ(0)], [], [ZZ(-1), ZZ(0), ZZ(1), ZZ(0)]] + ) + + raises(DomainError, lambda: dup_real_imag([EX(1), EX(2)], EX)) + + + +def test_dup_mirror(): + assert dup_mirror([], ZZ) == [] + assert dup_mirror([1], ZZ) == [1] + + assert dup_mirror([1, 2, 3, 4, 5], ZZ) == [1, -2, 3, -4, 5] + assert dup_mirror([1, 2, 3, 4, 5, 6], ZZ) == [-1, 2, -3, 4, -5, 6] + + +def test_dup_scale(): + assert dup_scale([], -1, ZZ) == [] + assert dup_scale([1], -1, ZZ) == [1] + + assert dup_scale([1, 2, 3, 4, 5], -1, ZZ) == [1, -2, 3, -4, 5] + assert dup_scale([1, 2, 3, 4, 5], -7, ZZ) == [2401, -686, 147, -28, 5] + + +def test_dup_shift(): + assert dup_shift([], 1, ZZ) == [] + assert dup_shift([1], 1, ZZ) == [1] + + assert dup_shift([1, 2, 3, 4, 5], 1, ZZ) == [1, 6, 15, 20, 15] + assert dup_shift([1, 2, 3, 4, 5], 7, ZZ) == [1, 30, 339, 1712, 3267] + + +def test_dmp_shift(): + assert dmp_shift([ZZ(1), ZZ(2)], [ZZ(1)], 0, ZZ) == [ZZ(1), ZZ(3)] + + assert dmp_shift([[]], [ZZ(1), ZZ(2)], 1, ZZ) == [[]] + + xy = [[ZZ(1), ZZ(0)], []] # x*y + x1y2 = [[ZZ(1), ZZ(2)], [ZZ(1), ZZ(2)]] # (x+1)*(y+2) + assert dmp_shift(xy, [ZZ(1), ZZ(2)], 1, ZZ) == x1y2 + + +def test_dup_transform(): + assert dup_transform([], [], [1, 1], ZZ) == [] + assert dup_transform([], [1], [1, 1], ZZ) == [] + assert dup_transform([], [1, 2], [1, 1], ZZ) == [] + + assert dup_transform([6, -5, 4, -3, 17], [1, -3, 4], [2, -3], ZZ) == \ + [6, -82, 541, -2205, 6277, -12723, 17191, -13603, 4773] + + +def test_dup_compose(): + assert dup_compose([], [], ZZ) == [] + assert dup_compose([], [1], ZZ) == [] + assert dup_compose([], [1, 2], ZZ) == [] + + assert dup_compose([1], [], ZZ) == [1] + + assert dup_compose([1, 2, 0], [], ZZ) == [] + assert dup_compose([1, 2, 1], [], ZZ) == [1] + + assert dup_compose([1, 2, 1], [1], ZZ) == [4] + assert dup_compose([1, 2, 1], [7], ZZ) == [64] + + assert dup_compose([1, 2, 1], [1, -1], ZZ) == [1, 0, 0] + assert dup_compose([1, 2, 1], [1, 1], ZZ) == [1, 4, 4] + assert dup_compose([1, 2, 1], [1, 2, 1], ZZ) == [1, 4, 8, 8, 4] + + +def test_dmp_compose(): + assert dmp_compose([1, 2, 1], [1, 2, 1], 0, ZZ) == [1, 4, 8, 8, 4] + + assert dmp_compose([[[]]], [[[]]], 2, ZZ) == [[[]]] + assert dmp_compose([[[]]], [[[1]]], 2, ZZ) == [[[]]] + assert dmp_compose([[[]]], [[[1]], [[2]]], 2, ZZ) == [[[]]] + + assert dmp_compose([[[1]]], [], 2, ZZ) == [[[1]]] + + assert dmp_compose([[1], [2], [ ]], [[]], 1, ZZ) == [[]] + assert dmp_compose([[1], [2], [1]], [[]], 1, ZZ) == [[1]] + + assert dmp_compose([[1], [2], [1]], [[1]], 1, ZZ) == [[4]] + assert dmp_compose([[1], [2], [1]], [[7]], 1, ZZ) == [[64]] + + assert dmp_compose([[1], [2], [1]], [[1], [-1]], 1, ZZ) == [[1], [ ], [ ]] + assert dmp_compose([[1], [2], [1]], [[1], [ 1]], 1, ZZ) == [[1], [4], [4]] + + assert dmp_compose( + [[1], [2], [1]], [[1], [2], [1]], 1, ZZ) == [[1], [4], [8], [8], [4]] + + +def test_dup_decompose(): + assert dup_decompose([1], ZZ) == [[1]] + + assert dup_decompose([1, 0], ZZ) == [[1, 0]] + assert dup_decompose([1, 0, 0, 0], ZZ) == [[1, 0, 0, 0]] + + assert dup_decompose([1, 0, 0, 0, 0], ZZ) == [[1, 0, 0], [1, 0, 0]] + assert dup_decompose( + [1, 0, 0, 0, 0, 0, 0], ZZ) == [[1, 0, 0, 0], [1, 0, 0]] + + assert dup_decompose([7, 0, 0, 0, 1], ZZ) == [[7, 0, 1], [1, 0, 0]] + assert dup_decompose([4, 0, 3, 0, 2], ZZ) == [[4, 3, 2], [1, 0, 0]] + + f = [1, 0, 20, 0, 150, 0, 500, 0, 625, -2, 0, -10, 9] + + assert dup_decompose(f, ZZ) == [[1, 0, 0, -2, 9], [1, 0, 5, 0]] + + f = [2, 0, 40, 0, 300, 0, 1000, 0, 1250, -4, 0, -20, 18] + + assert dup_decompose(f, ZZ) == [[2, 0, 0, -4, 18], [1, 0, 5, 0]] + + f = [1, 0, 20, -8, 150, -120, 524, -600, 865, -1034, 600, -170, 29] + + assert dup_decompose(f, ZZ) == [[1, -8, 24, -34, 29], [1, 0, 5, 0]] + + R, t = ring("t", ZZ) + f = [6*t**2 - 42, + 48*t**2 + 96, + 144*t**2 + 648*t + 288, + 624*t**2 + 864*t + 384, + 108*t**3 + 312*t**2 + 432*t + 192] + + assert dup_decompose(f, R.to_domain()) == [f] + + +def test_dmp_lift(): + q = [QQ(1, 1), QQ(0, 1), QQ(1, 1)] + + f_a = [ANP([QQ(1, 1)], q, QQ), ANP([], q, QQ), ANP([], q, QQ), + ANP([QQ(1, 1), QQ(0, 1)], q, QQ), ANP([QQ(17, 1), QQ(0, 1)], q, QQ)] + + f_lift = QQ.map([1, 0, 0, 0, 0, 0, 1, 34, 289]) + + assert dmp_lift(f_a, 0, QQ.algebraic_field(I)) == f_lift + + f_g = [QQ_I(1), QQ_I(0), QQ_I(0), QQ_I(0, 1), QQ_I(0, 17)] + + assert dmp_lift(f_g, 0, QQ_I) == f_lift + + raises(DomainError, lambda: dmp_lift([EX(1), EX(2)], 0, EX)) + + +def test_dup_sign_variations(): + assert dup_sign_variations([], ZZ) == 0 + assert dup_sign_variations([1, 0], ZZ) == 0 + assert dup_sign_variations([1, 0, 2], ZZ) == 0 + assert dup_sign_variations([1, 0, 3, 0], ZZ) == 0 + assert dup_sign_variations([1, 0, 4, 0, 5], ZZ) == 0 + + assert dup_sign_variations([-1, 0, 2], ZZ) == 1 + assert dup_sign_variations([-1, 0, 3, 0], ZZ) == 1 + assert dup_sign_variations([-1, 0, 4, 0, 5], ZZ) == 1 + + assert dup_sign_variations([-1, -4, -5], ZZ) == 0 + assert dup_sign_variations([ 1, -4, -5], ZZ) == 1 + assert dup_sign_variations([ 1, 4, -5], ZZ) == 1 + assert dup_sign_variations([ 1, -4, 5], ZZ) == 2 + assert dup_sign_variations([-1, 4, -5], ZZ) == 2 + assert dup_sign_variations([-1, 4, 5], ZZ) == 1 + assert dup_sign_variations([-1, -4, 5], ZZ) == 1 + assert dup_sign_variations([ 1, 4, 5], ZZ) == 0 + + assert dup_sign_variations([-1, 0, -4, 0, -5], ZZ) == 0 + assert dup_sign_variations([ 1, 0, -4, 0, -5], ZZ) == 1 + assert dup_sign_variations([ 1, 0, 4, 0, -5], ZZ) == 1 + assert dup_sign_variations([ 1, 0, -4, 0, 5], ZZ) == 2 + assert dup_sign_variations([-1, 0, 4, 0, -5], ZZ) == 2 + assert dup_sign_variations([-1, 0, 4, 0, 5], ZZ) == 1 + assert dup_sign_variations([-1, 0, -4, 0, 5], ZZ) == 1 + assert dup_sign_variations([ 1, 0, 4, 0, 5], ZZ) == 0 + + +def test_dup_clear_denoms(): + assert dup_clear_denoms([], QQ, ZZ) == (ZZ(1), []) + + assert dup_clear_denoms([QQ(1)], QQ, ZZ) == (ZZ(1), [QQ(1)]) + assert dup_clear_denoms([QQ(7)], QQ, ZZ) == (ZZ(1), [QQ(7)]) + + assert dup_clear_denoms([QQ(7, 3)], QQ) == (ZZ(3), [QQ(7)]) + assert dup_clear_denoms([QQ(7, 3)], QQ, ZZ) == (ZZ(3), [QQ(7)]) + + assert dup_clear_denoms( + [QQ(3), QQ(1), QQ(0)], QQ, ZZ) == (ZZ(1), [QQ(3), QQ(1), QQ(0)]) + assert dup_clear_denoms( + [QQ(1), QQ(1, 2), QQ(0)], QQ, ZZ) == (ZZ(2), [QQ(2), QQ(1), QQ(0)]) + + assert dup_clear_denoms([QQ(3), QQ( + 1), QQ(0)], QQ, ZZ, convert=True) == (ZZ(1), [ZZ(3), ZZ(1), ZZ(0)]) + assert dup_clear_denoms([QQ(1), QQ( + 1, 2), QQ(0)], QQ, ZZ, convert=True) == (ZZ(2), [ZZ(2), ZZ(1), ZZ(0)]) + + assert dup_clear_denoms( + [EX(S(3)/2), EX(S(9)/4)], EX) == (EX(4), [EX(6), EX(9)]) + + assert dup_clear_denoms([EX(7)], EX) == (EX(1), [EX(7)]) + assert dup_clear_denoms([EX(sin(x)/x), EX(0)], EX) == (EX(x), [EX(sin(x)), EX(0)]) + + F = RR.frac_field(x) + result = dup_clear_denoms([F(8.48717/(8.0089*x + 2.83)), F(0.0)], F) + assert str(result) == "(x + 0.353356890459364, [1.05971731448763, 0.0])" + +def test_dmp_clear_denoms(): + assert dmp_clear_denoms([[]], 1, QQ, ZZ) == (ZZ(1), [[]]) + + assert dmp_clear_denoms([[QQ(1)]], 1, QQ, ZZ) == (ZZ(1), [[QQ(1)]]) + assert dmp_clear_denoms([[QQ(7)]], 1, QQ, ZZ) == (ZZ(1), [[QQ(7)]]) + + assert dmp_clear_denoms([[QQ(7, 3)]], 1, QQ) == (ZZ(3), [[QQ(7)]]) + assert dmp_clear_denoms([[QQ(7, 3)]], 1, QQ, ZZ) == (ZZ(3), [[QQ(7)]]) + + assert dmp_clear_denoms( + [[QQ(3)], [QQ(1)], []], 1, QQ, ZZ) == (ZZ(1), [[QQ(3)], [QQ(1)], []]) + assert dmp_clear_denoms([[QQ( + 1)], [QQ(1, 2)], []], 1, QQ, ZZ) == (ZZ(2), [[QQ(2)], [QQ(1)], []]) + + assert dmp_clear_denoms([QQ(3), QQ( + 1), QQ(0)], 0, QQ, ZZ, convert=True) == (ZZ(1), [ZZ(3), ZZ(1), ZZ(0)]) + assert dmp_clear_denoms([QQ(1), QQ(1, 2), QQ( + 0)], 0, QQ, ZZ, convert=True) == (ZZ(2), [ZZ(2), ZZ(1), ZZ(0)]) + + assert dmp_clear_denoms([[QQ(3)], [QQ( + 1)], []], 1, QQ, ZZ, convert=True) == (ZZ(1), [[QQ(3)], [QQ(1)], []]) + assert dmp_clear_denoms([[QQ(1)], [QQ(1, 2)], []], 1, QQ, ZZ, + convert=True) == (ZZ(2), [[QQ(2)], [QQ(1)], []]) + + assert dmp_clear_denoms( + [[EX(S(3)/2)], [EX(S(9)/4)]], 1, EX) == (EX(4), [[EX(6)], [EX(9)]]) + assert dmp_clear_denoms([[EX(7)]], 1, EX) == (EX(1), [[EX(7)]]) + assert dmp_clear_denoms([[EX(sin(x)/x), EX(0)]], 1, EX) == (EX(x), [[EX(sin(x)), EX(0)]]) diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_dispersion.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_dispersion.py new file mode 100644 index 0000000000000000000000000000000000000000..ad56b7bebd73c38e037085d36625a41729c0369a --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_dispersion.py @@ -0,0 +1,95 @@ +from sympy.core import Symbol, S, oo +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.polys import poly +from sympy.polys.dispersion import dispersion, dispersionset + + +def test_dispersion(): + x = Symbol("x") + a = Symbol("a") + + fp = poly(S.Zero, x) + assert sorted(dispersionset(fp)) == [0] + + fp = poly(S(2), x) + assert sorted(dispersionset(fp)) == [0] + + fp = poly(x + 1, x) + assert sorted(dispersionset(fp)) == [0] + assert dispersion(fp) == 0 + + fp = poly((x + 1)*(x + 2), x) + assert sorted(dispersionset(fp)) == [0, 1] + assert dispersion(fp) == 1 + + fp = poly(x*(x + 3), x) + assert sorted(dispersionset(fp)) == [0, 3] + assert dispersion(fp) == 3 + + fp = poly((x - 3)*(x + 3), x) + assert sorted(dispersionset(fp)) == [0, 6] + assert dispersion(fp) == 6 + + fp = poly(x**4 - 3*x**2 + 1, x) + gp = fp.shift(-3) + assert sorted(dispersionset(fp, gp)) == [2, 3, 4] + assert dispersion(fp, gp) == 4 + assert sorted(dispersionset(gp, fp)) == [] + assert dispersion(gp, fp) is -oo + + fp = poly(x*(3*x**2+a)*(x-2536)*(x**3+a), x) + gp = fp.as_expr().subs(x, x-345).as_poly(x) + assert sorted(dispersionset(fp, gp)) == [345, 2881] + assert sorted(dispersionset(gp, fp)) == [2191] + + gp = poly((x-2)**2*(x-3)**3*(x-5)**3, x) + assert sorted(dispersionset(gp)) == [0, 1, 2, 3] + assert sorted(dispersionset(gp, (gp+4)**2)) == [1, 2] + + fp = poly(x*(x+2)*(x-1), x) + assert sorted(dispersionset(fp)) == [0, 1, 2, 3] + + fp = poly(x**2 + sqrt(5)*x - 1, x, domain='QQ') + gp = poly(x**2 + (2 + sqrt(5))*x + sqrt(5), x, domain='QQ') + assert sorted(dispersionset(fp, gp)) == [2] + assert sorted(dispersionset(gp, fp)) == [1, 4] + + # There are some difficulties if we compute over Z[a] + # and alpha happens to lie in Z[a] instead of simply Z. + # Hence we can not decide if alpha is indeed integral + # in general. + + fp = poly(4*x**4 + (4*a + 8)*x**3 + (a**2 + 6*a + 4)*x**2 + (a**2 + 2*a)*x, x) + assert sorted(dispersionset(fp)) == [0, 1] + + # For any specific value of a, the dispersion is 3*a + # but the algorithm can not find this in general. + # This is the point where the resultant based Ansatz + # is superior to the current one. + fp = poly(a**2*x**3 + (a**3 + a**2 + a + 1)*x, x) + gp = fp.as_expr().subs(x, x - 3*a).as_poly(x) + assert sorted(dispersionset(fp, gp)) == [] + + fpa = fp.as_expr().subs(a, 2).as_poly(x) + gpa = gp.as_expr().subs(a, 2).as_poly(x) + assert sorted(dispersionset(fpa, gpa)) == [6] + + # Work with Expr instead of Poly + f = (x + 1)*(x + 2) + assert sorted(dispersionset(f)) == [0, 1] + assert dispersion(f) == 1 + + f = x**4 - 3*x**2 + 1 + g = x**4 - 12*x**3 + 51*x**2 - 90*x + 55 + assert sorted(dispersionset(f, g)) == [2, 3, 4] + assert dispersion(f, g) == 4 + + # Work with Expr and specify a generator + f = (x + 1)*(x + 2) + assert sorted(dispersionset(f, None, x)) == [0, 1] + assert dispersion(f, None, x) == 1 + + f = x**4 - 3*x**2 + 1 + g = x**4 - 12*x**3 + 51*x**2 - 90*x + 55 + assert sorted(dispersionset(f, g, x)) == [2, 3, 4] + assert dispersion(f, g, x) == 4 diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_distributedmodules.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_distributedmodules.py new file mode 100644 index 0000000000000000000000000000000000000000..c95672f99f878f3def660aadec901afbde9adf8b --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_distributedmodules.py @@ -0,0 +1,208 @@ +"""Tests for sparse distributed modules. """ + +from sympy.polys.distributedmodules import ( + sdm_monomial_mul, sdm_monomial_deg, sdm_monomial_divides, + sdm_add, sdm_LM, sdm_LT, sdm_mul_term, sdm_zero, sdm_deg, + sdm_LC, sdm_from_dict, + sdm_spoly, sdm_ecart, sdm_nf_mora, sdm_groebner, + sdm_from_vector, sdm_to_vector, sdm_monomial_lcm +) + +from sympy.polys.orderings import lex, grlex, InverseOrder +from sympy.polys.domains import QQ + +from sympy.abc import x, y, z + + +def test_sdm_monomial_mul(): + assert sdm_monomial_mul((1, 1, 0), (1, 3)) == (1, 2, 3) + + +def test_sdm_monomial_deg(): + assert sdm_monomial_deg((5, 2, 1)) == 3 + + +def test_sdm_monomial_lcm(): + assert sdm_monomial_lcm((1, 2, 3), (1, 5, 0)) == (1, 5, 3) + + +def test_sdm_monomial_divides(): + assert sdm_monomial_divides((1, 0, 0), (1, 0, 0)) is True + assert sdm_monomial_divides((1, 0, 0), (1, 2, 1)) is True + assert sdm_monomial_divides((5, 1, 1), (5, 2, 1)) is True + + assert sdm_monomial_divides((1, 0, 0), (2, 0, 0)) is False + assert sdm_monomial_divides((1, 1, 0), (1, 0, 0)) is False + assert sdm_monomial_divides((5, 1, 2), (5, 0, 1)) is False + + +def test_sdm_LC(): + assert sdm_LC([((1, 2, 3), QQ(5))], QQ) == QQ(5) + + +def test_sdm_from_dict(): + dic = {(1, 2, 1, 1): QQ(1), (1, 1, 2, 1): QQ(1), (1, 0, 2, 1): QQ(1), + (1, 0, 0, 3): QQ(1), (1, 1, 1, 0): QQ(1)} + assert sdm_from_dict(dic, grlex) == \ + [((1, 2, 1, 1), QQ(1)), ((1, 1, 2, 1), QQ(1)), + ((1, 0, 2, 1), QQ(1)), ((1, 0, 0, 3), QQ(1)), ((1, 1, 1, 0), QQ(1))] + +# TODO test to_dict? + + +def test_sdm_add(): + assert sdm_add([((1, 1, 1), QQ(1))], [((2, 0, 0), QQ(1))], lex, QQ) == \ + [((2, 0, 0), QQ(1)), ((1, 1, 1), QQ(1))] + assert sdm_add([((1, 1, 1), QQ(1))], [((1, 1, 1), QQ(-1))], lex, QQ) == [] + assert sdm_add([((1, 0, 0), QQ(1))], [((1, 0, 0), QQ(2))], lex, QQ) == \ + [((1, 0, 0), QQ(3))] + assert sdm_add([((1, 0, 1), QQ(1))], [((1, 1, 0), QQ(1))], lex, QQ) == \ + [((1, 1, 0), QQ(1)), ((1, 0, 1), QQ(1))] + + +def test_sdm_LM(): + dic = {(1, 2, 3): QQ(1), (4, 0, 0): QQ(1), (4, 0, 1): QQ(1)} + assert sdm_LM(sdm_from_dict(dic, lex)) == (4, 0, 1) + + +def test_sdm_LT(): + dic = {(1, 2, 3): QQ(1), (4, 0, 0): QQ(2), (4, 0, 1): QQ(3)} + assert sdm_LT(sdm_from_dict(dic, lex)) == ((4, 0, 1), QQ(3)) + + +def test_sdm_mul_term(): + assert sdm_mul_term([((1, 0, 0), QQ(1))], ((0, 0), QQ(0)), lex, QQ) == [] + assert sdm_mul_term([], ((1, 0), QQ(1)), lex, QQ) == [] + assert sdm_mul_term([((1, 0, 0), QQ(1))], ((1, 0), QQ(1)), lex, QQ) == \ + [((1, 1, 0), QQ(1))] + f = [((2, 0, 1), QQ(4)), ((1, 1, 0), QQ(3))] + assert sdm_mul_term(f, ((1, 1), QQ(2)), lex, QQ) == \ + [((2, 1, 2), QQ(8)), ((1, 2, 1), QQ(6))] + + +def test_sdm_zero(): + assert sdm_zero() == [] + + +def test_sdm_deg(): + assert sdm_deg([((1, 2, 3), 1), ((10, 0, 1), 1), ((2, 3, 4), 4)]) == 7 + + +def test_sdm_spoly(): + f = [((2, 1, 1), QQ(1)), ((1, 0, 1), QQ(1))] + g = [((2, 3, 0), QQ(1))] + h = [((1, 2, 3), QQ(1))] + assert sdm_spoly(f, h, lex, QQ) == [] + assert sdm_spoly(f, g, lex, QQ) == [((1, 2, 1), QQ(1))] + + +def test_sdm_ecart(): + assert sdm_ecart([((1, 2, 3), 1), ((1, 0, 1), 1)]) == 0 + assert sdm_ecart([((2, 2, 1), 1), ((1, 5, 1), 1)]) == 3 + + +def test_sdm_nf_mora(): + f = sdm_from_dict({(1, 2, 1, 1): QQ(1), (1, 1, 2, 1): QQ(1), + (1, 0, 2, 1): QQ(1), (1, 0, 0, 3): QQ(1), (1, 1, 1, 0): QQ(1)}, + grlex) + f1 = sdm_from_dict({(1, 1, 1, 0): QQ(1), (1, 0, 2, 0): QQ(1), + (1, 0, 0, 0): QQ(-1)}, grlex) + f2 = sdm_from_dict({(1, 1, 1, 0): QQ(1)}, grlex) + (id0, id1, id2) = [sdm_from_dict({(i, 0, 0, 0): QQ(1)}, grlex) + for i in range(3)] + + assert sdm_nf_mora(f, [f1, f2], grlex, QQ, phantom=(id0, [id1, id2])) == \ + ([((1, 0, 2, 1), QQ(1)), ((1, 0, 0, 3), QQ(1)), ((1, 1, 1, 0), QQ(1)), + ((1, 1, 0, 1), QQ(1))], + [((1, 1, 0, 1), QQ(-1)), ((0, 0, 0, 0), QQ(1))]) + assert sdm_nf_mora(f, [f2, f1], grlex, QQ, phantom=(id0, [id2, id1])) == \ + ([((1, 0, 2, 1), QQ(1)), ((1, 0, 0, 3), QQ(1)), ((1, 1, 1, 0), QQ(1))], + [((2, 1, 0, 1), QQ(-1)), ((2, 0, 1, 1), QQ(-1)), ((0, 0, 0, 0), QQ(1))]) + + f = sdm_from_vector([x*z, y**2 + y*z - z, y], lex, QQ, gens=[x, y, z]) + f1 = sdm_from_vector([x, y, 1], lex, QQ, gens=[x, y, z]) + f2 = sdm_from_vector([x*y, z, z**2], lex, QQ, gens=[x, y, z]) + assert sdm_nf_mora(f, [f1, f2], lex, QQ) == \ + sdm_nf_mora(f, [f2, f1], lex, QQ) == \ + [((1, 0, 1, 1), QQ(1)), ((1, 0, 0, 1), QQ(-1)), ((0, 1, 1, 0), QQ(-1)), + ((0, 1, 0, 1), QQ(1))] + + +def test_conversion(): + f = [x**2 + y**2, 2*z] + g = [((1, 0, 0, 1), QQ(2)), ((0, 2, 0, 0), QQ(1)), ((0, 0, 2, 0), QQ(1))] + assert sdm_to_vector(g, [x, y, z], QQ) == f + assert sdm_from_vector(f, lex, QQ) == g + assert sdm_from_vector( + [x, 1], lex, QQ) == [((1, 0), QQ(1)), ((0, 1), QQ(1))] + assert sdm_to_vector([((1, 1, 0, 0), 1)], [x, y, z], QQ, n=3) == [0, x, 0] + assert sdm_from_vector([0, 0], lex, QQ, gens=[x, y]) == sdm_zero() + + +def test_nontrivial(): + gens = [x, y, z] + + def contains(I, f): + S = [sdm_from_vector([g], lex, QQ, gens=gens) for g in I] + G = sdm_groebner(S, sdm_nf_mora, lex, QQ) + return sdm_nf_mora(sdm_from_vector([f], lex, QQ, gens=gens), + G, lex, QQ) == sdm_zero() + + assert contains([x, y], x) + assert contains([x, y], x + y) + assert not contains([x, y], 1) + assert not contains([x, y], z) + assert contains([x**2 + y, x**2 + x], x - y) + assert not contains([x + y + z, x*y + x*z + y*z, x*y*z], x**2) + assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x**3) + assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x**4) + assert not contains([x + y + z, x*y + x*z + y*z, x*y*z], x*y**2) + assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x**4 + y**3 + 2*z*y*x) + assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x*y*z) + assert contains([x, 1 + x + y, 5 - 7*y], 1) + assert contains( + [x**3 + y**3, y**3 + z**3, z**3 + x**3, x**2*y + x**2*z + y**2*z], + x**3) + assert not contains( + [x**3 + y**3, y**3 + z**3, z**3 + x**3, x**2*y + x**2*z + y**2*z], + x**2 + y**2) + + # compare local order + assert not contains([x*(1 + x + y), y*(1 + z)], x) + assert not contains([x*(1 + x + y), y*(1 + z)], x + y) + + +def test_local(): + igrlex = InverseOrder(grlex) + gens = [x, y, z] + + def contains(I, f): + S = [sdm_from_vector([g], igrlex, QQ, gens=gens) for g in I] + G = sdm_groebner(S, sdm_nf_mora, igrlex, QQ) + return sdm_nf_mora(sdm_from_vector([f], lex, QQ, gens=gens), + G, lex, QQ) == sdm_zero() + assert contains([x, y], x) + assert contains([x, y], x + y) + assert not contains([x, y], 1) + assert not contains([x, y], z) + assert contains([x**2 + y, x**2 + x], x - y) + assert not contains([x + y + z, x*y + x*z + y*z, x*y*z], x**2) + assert contains([x*(1 + x + y), y*(1 + z)], x) + assert contains([x*(1 + x + y), y*(1 + z)], x + y) + + +def test_uncovered_line(): + gens = [x, y] + f1 = sdm_zero() + f2 = sdm_from_vector([x, 0], lex, QQ, gens=gens) + f3 = sdm_from_vector([0, y], lex, QQ, gens=gens) + + assert sdm_spoly(f1, f2, lex, QQ) == sdm_zero() + assert sdm_spoly(f3, f2, lex, QQ) == sdm_zero() + + +def test_chain_criterion(): + gens = [x] + f1 = sdm_from_vector([1, x], grlex, QQ, gens=gens) + f2 = sdm_from_vector([0, x - 2], grlex, QQ, gens=gens) + assert len(sdm_groebner([f1, f2], sdm_nf_mora, grlex, QQ)) == 2 diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_euclidtools.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_euclidtools.py new file mode 100644 index 0000000000000000000000000000000000000000..3061be73f987163951a5836ff50125d29abc60c7 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_euclidtools.py @@ -0,0 +1,712 @@ +"""Tests for Euclidean algorithms, GCDs, LCMs and polynomial remainder sequences. """ + +from sympy.polys.rings import ring +from sympy.polys.domains import ZZ, QQ, RR + +from sympy.polys.specialpolys import ( + f_polys, + dmp_fateman_poly_F_1, + dmp_fateman_poly_F_2, + dmp_fateman_poly_F_3) + +f_0, f_1, f_2, f_3, f_4, f_5, f_6 = f_polys() + +def test_dup_gcdex(): + R, x = ring("x", QQ) + + f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15 + g = x**3 + x**2 - 4*x - 4 + + s = -QQ(1,5)*x + QQ(3,5) + t = QQ(1,5)*x**2 - QQ(6,5)*x + 2 + h = x + 1 + + assert R.dup_half_gcdex(f, g) == (s, h) + assert R.dup_gcdex(f, g) == (s, t, h) + + f = x**4 + 4*x**3 - x + 1 + g = x**3 - x + 1 + + s, t, h = R.dup_gcdex(f, g) + S, T, H = R.dup_gcdex(g, f) + + assert R.dup_add(R.dup_mul(s, f), + R.dup_mul(t, g)) == h + assert R.dup_add(R.dup_mul(S, g), + R.dup_mul(T, f)) == H + + f = 2*x + g = x**2 - 16 + + s = QQ(1,32)*x + t = -QQ(1,16) + h = 1 + + assert R.dup_half_gcdex(f, g) == (s, h) + assert R.dup_gcdex(f, g) == (s, t, h) + + +def test_dup_invert(): + R, x = ring("x", QQ) + assert R.dup_invert(2*x, x**2 - 16) == QQ(1,32)*x + + +def test_dup_euclidean_prs(): + R, x = ring("x", QQ) + + f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 + g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 + + assert R.dup_euclidean_prs(f, g) == [ + f, + g, + -QQ(5,9)*x**4 + QQ(1,9)*x**2 - QQ(1,3), + -QQ(117,25)*x**2 - 9*x + QQ(441,25), + QQ(233150,19773)*x - QQ(102500,6591), + -QQ(1288744821,543589225)] + + +def test_dup_primitive_prs(): + R, x = ring("x", ZZ) + + f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 + g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 + + assert R.dup_primitive_prs(f, g) == [ + f, + g, + -5*x**4 + x**2 - 3, + 13*x**2 + 25*x - 49, + 4663*x - 6150, + 1] + + +def test_dup_subresultants(): + R, x = ring("x", ZZ) + + assert R.dup_resultant(0, 0) == 0 + + assert R.dup_resultant(1, 0) == 0 + assert R.dup_resultant(0, 1) == 0 + + f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 + g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 + + a = 15*x**4 - 3*x**2 + 9 + b = 65*x**2 + 125*x - 245 + c = 9326*x - 12300 + d = 260708 + + assert R.dup_subresultants(f, g) == [f, g, a, b, c, d] + assert R.dup_resultant(f, g) == R.dup_LC(d) + + f = x**2 - 2*x + 1 + g = x**2 - 1 + + a = 2*x - 2 + + assert R.dup_subresultants(f, g) == [f, g, a] + assert R.dup_resultant(f, g) == 0 + + f = x**2 + 1 + g = x**2 - 1 + + a = -2 + + assert R.dup_subresultants(f, g) == [f, g, a] + assert R.dup_resultant(f, g) == 4 + + f = x**2 - 1 + g = x**3 - x**2 + 2 + + assert R.dup_resultant(f, g) == 0 + + f = 3*x**3 - x + g = 5*x**2 + 1 + + assert R.dup_resultant(f, g) == 64 + + f = x**2 - 2*x + 7 + g = x**3 - x + 5 + + assert R.dup_resultant(f, g) == 265 + + f = x**3 - 6*x**2 + 11*x - 6 + g = x**3 - 15*x**2 + 74*x - 120 + + assert R.dup_resultant(f, g) == -8640 + + f = x**3 - 6*x**2 + 11*x - 6 + g = x**3 - 10*x**2 + 29*x - 20 + + assert R.dup_resultant(f, g) == 0 + + f = x**3 - 1 + g = x**3 + 2*x**2 + 2*x - 1 + + assert R.dup_resultant(f, g) == 16 + + f = x**8 - 2 + g = x - 1 + + assert R.dup_resultant(f, g) == -1 + + +def test_dmp_subresultants(): + R, x, y = ring("x,y", ZZ) + + assert R.dmp_resultant(0, 0) == 0 + assert R.dmp_prs_resultant(0, 0)[0] == 0 + assert R.dmp_zz_collins_resultant(0, 0) == 0 + assert R.dmp_qq_collins_resultant(0, 0) == 0 + + assert R.dmp_resultant(1, 0) == 0 + assert R.dmp_resultant(1, 0) == 0 + assert R.dmp_resultant(1, 0) == 0 + + assert R.dmp_resultant(0, 1) == 0 + assert R.dmp_prs_resultant(0, 1)[0] == 0 + assert R.dmp_zz_collins_resultant(0, 1) == 0 + assert R.dmp_qq_collins_resultant(0, 1) == 0 + + f = 3*x**2*y - y**3 - 4 + g = x**2 + x*y**3 - 9 + + a = 3*x*y**4 + y**3 - 27*y + 4 + b = -3*y**10 - 12*y**7 + y**6 - 54*y**4 + 8*y**3 + 729*y**2 - 216*y + 16 + + r = R.dmp_LC(b) + + assert R.dmp_subresultants(f, g) == [f, g, a, b] + + assert R.dmp_resultant(f, g) == r + assert R.dmp_prs_resultant(f, g)[0] == r + assert R.dmp_zz_collins_resultant(f, g) == r + assert R.dmp_qq_collins_resultant(f, g) == r + + f = -x**3 + 5 + g = 3*x**2*y + x**2 + + a = 45*y**2 + 30*y + 5 + b = 675*y**3 + 675*y**2 + 225*y + 25 + + r = R.dmp_LC(b) + + assert R.dmp_subresultants(f, g) == [f, g, a] + assert R.dmp_resultant(f, g) == r + assert R.dmp_prs_resultant(f, g)[0] == r + assert R.dmp_zz_collins_resultant(f, g) == r + assert R.dmp_qq_collins_resultant(f, g) == r + + R, x, y, z, u, v = ring("x,y,z,u,v", ZZ) + + f = 6*x**2 - 3*x*y - 2*x*z + y*z + g = x**2 - x*u - x*v + u*v + + r = y**2*z**2 - 3*y**2*z*u - 3*y**2*z*v + 9*y**2*u*v - 2*y*z**2*u \ + - 2*y*z**2*v + 6*y*z*u**2 + 12*y*z*u*v + 6*y*z*v**2 - 18*y*u**2*v \ + - 18*y*u*v**2 + 4*z**2*u*v - 12*z*u**2*v - 12*z*u*v**2 + 36*u**2*v**2 + + assert R.dmp_zz_collins_resultant(f, g) == r.drop(x) + + R, x, y, z, u, v = ring("x,y,z,u,v", QQ) + + f = x**2 - QQ(1,2)*x*y - QQ(1,3)*x*z + QQ(1,6)*y*z + g = x**2 - x*u - x*v + u*v + + r = QQ(1,36)*y**2*z**2 - QQ(1,12)*y**2*z*u - QQ(1,12)*y**2*z*v + QQ(1,4)*y**2*u*v \ + - QQ(1,18)*y*z**2*u - QQ(1,18)*y*z**2*v + QQ(1,6)*y*z*u**2 + QQ(1,3)*y*z*u*v \ + + QQ(1,6)*y*z*v**2 - QQ(1,2)*y*u**2*v - QQ(1,2)*y*u*v**2 + QQ(1,9)*z**2*u*v \ + - QQ(1,3)*z*u**2*v - QQ(1,3)*z*u*v**2 + u**2*v**2 + + assert R.dmp_qq_collins_resultant(f, g) == r.drop(x) + + Rt, t = ring("t", ZZ) + Rx, x = ring("x", Rt) + + f = x**6 - 5*x**4 + 5*x**2 + 4 + g = -6*t*x**5 + x**4 + 20*t*x**3 - 3*x**2 - 10*t*x + 6 + + assert Rx.dup_resultant(f, g) == 2930944*t**6 + 2198208*t**4 + 549552*t**2 + 45796 + + +def test_dup_discriminant(): + R, x = ring("x", ZZ) + + assert R.dup_discriminant(0) == 0 + assert R.dup_discriminant(x) == 1 + + assert R.dup_discriminant(x**3 + 3*x**2 + 9*x - 13) == -11664 + assert R.dup_discriminant(5*x**5 + x**3 + 2) == 31252160 + assert R.dup_discriminant(x**4 + 2*x**3 + 6*x**2 - 22*x + 13) == 0 + assert R.dup_discriminant(12*x**7 + 15*x**4 + 30*x**3 + x**2 + 1) == -220289699947514112 + + +def test_dmp_discriminant(): + R, x = ring("x", ZZ) + + assert R.dmp_discriminant(0) == 0 + + R, x, y = ring("x,y", ZZ) + + assert R.dmp_discriminant(0) == 0 + assert R.dmp_discriminant(y) == 0 + + assert R.dmp_discriminant(x**3 + 3*x**2 + 9*x - 13) == -11664 + assert R.dmp_discriminant(5*x**5 + x**3 + 2) == 31252160 + assert R.dmp_discriminant(x**4 + 2*x**3 + 6*x**2 - 22*x + 13) == 0 + assert R.dmp_discriminant(12*x**7 + 15*x**4 + 30*x**3 + x**2 + 1) == -220289699947514112 + + assert R.dmp_discriminant(x**2*y + 2*y) == (-8*y**2).drop(x) + assert R.dmp_discriminant(x*y**2 + 2*x) == 1 + + R, x, y, z = ring("x,y,z", ZZ) + assert R.dmp_discriminant(x*y + z) == 1 + + R, x, y, z, u = ring("x,y,z,u", ZZ) + assert R.dmp_discriminant(x**2*y + x*z + u) == (-4*y*u + z**2).drop(x) + + R, x, y, z, u, v = ring("x,y,z,u,v", ZZ) + assert R.dmp_discriminant(x**3*y + x**2*z + x*u + v) == \ + (-27*y**2*v**2 + 18*y*z*u*v - 4*y*u**3 - 4*z**3*v + z**2*u**2).drop(x) + + +def test_dup_gcd(): + R, x = ring("x", ZZ) + + f, g = 0, 0 + assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (0, 0, 0) + + f, g = 2, 0 + assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, 1, 0) + + f, g = -2, 0 + assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, -1, 0) + + f, g = 0, -2 + assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, 0, -1) + + f, g = 0, 2*x + 4 + assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2*x + 4, 0, 1) + + f, g = 2*x + 4, 0 + assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2*x + 4, 1, 0) + + f, g = 2, 2 + assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, 1, 1) + + f, g = -2, 2 + assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, -1, 1) + + f, g = 2, -2 + assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, 1, -1) + + f, g = -2, -2 + assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, -1, -1) + + f, g = x**2 + 2*x + 1, 1 + assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (1, x**2 + 2*x + 1, 1) + + f, g = x**2 + 2*x + 1, 2 + assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (1, x**2 + 2*x + 1, 2) + + f, g = 2*x**2 + 4*x + 2, 2 + assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, x**2 + 2*x + 1, 1) + + f, g = 2, 2*x**2 + 4*x + 2 + assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, 1, x**2 + 2*x + 1) + + f, g = 2*x**2 + 4*x + 2, x + 1 + assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (x + 1, 2*x + 2, 1) + + f, g = x + 1, 2*x**2 + 4*x + 2 + assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (x + 1, 1, 2*x + 2) + + f, g = x - 31, x + assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (1, f, g) + + f = x**4 + 8*x**3 + 21*x**2 + 22*x + 8 + g = x**3 + 6*x**2 + 11*x + 6 + + h = x**2 + 3*x + 2 + + cff = x**2 + 5*x + 4 + cfg = x + 3 + + assert R.dup_zz_heu_gcd(f, g) == (h, cff, cfg) + assert R.dup_rr_prs_gcd(f, g) == (h, cff, cfg) + + f = x**4 - 4 + g = x**4 + 4*x**2 + 4 + + h = x**2 + 2 + + cff = x**2 - 2 + cfg = x**2 + 2 + + assert R.dup_zz_heu_gcd(f, g) == (h, cff, cfg) + assert R.dup_rr_prs_gcd(f, g) == (h, cff, cfg) + + f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 + g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 + + h = 1 + + cff = f + cfg = g + + assert R.dup_zz_heu_gcd(f, g) == (h, cff, cfg) + assert R.dup_rr_prs_gcd(f, g) == (h, cff, cfg) + + R, x = ring("x", QQ) + + f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 + g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 + + h = 1 + + cff = f + cfg = g + + assert R.dup_qq_heu_gcd(f, g) == (h, cff, cfg) + assert R.dup_ff_prs_gcd(f, g) == (h, cff, cfg) + + R, x = ring("x", ZZ) + + f = - 352518131239247345597970242177235495263669787845475025293906825864749649589178600387510272*x**49 \ + + 46818041807522713962450042363465092040687472354933295397472942006618953623327997952*x**42 \ + + 378182690892293941192071663536490788434899030680411695933646320291525827756032*x**35 \ + + 112806468807371824947796775491032386836656074179286744191026149539708928*x**28 \ + - 12278371209708240950316872681744825481125965781519138077173235712*x**21 \ + + 289127344604779611146960547954288113529690984687482920704*x**14 \ + + 19007977035740498977629742919480623972236450681*x**7 \ + + 311973482284542371301330321821976049 + + g = 365431878023781158602430064717380211405897160759702125019136*x**21 \ + + 197599133478719444145775798221171663643171734081650688*x**14 \ + - 9504116979659010018253915765478924103928886144*x**7 \ + - 311973482284542371301330321821976049 + + assert R.dup_zz_heu_gcd(f, R.dup_diff(f, 1))[0] == g + assert R.dup_rr_prs_gcd(f, R.dup_diff(f, 1))[0] == g + + R, x = ring("x", QQ) + + f = QQ(1,2)*x**2 + x + QQ(1,2) + g = QQ(1,2)*x + QQ(1,2) + + h = x + 1 + + assert R.dup_qq_heu_gcd(f, g) == (h, g, QQ(1,2)) + assert R.dup_ff_prs_gcd(f, g) == (h, g, QQ(1,2)) + + R, x = ring("x", ZZ) + + f = 1317378933230047068160*x + 2945748836994210856960 + g = 120352542776360960*x + 269116466014453760 + + h = 120352542776360960*x + 269116466014453760 + cff = 10946 + cfg = 1 + + assert R.dup_zz_heu_gcd(f, g) == (h, cff, cfg) + + +def test_dmp_gcd(): + R, x, y = ring("x,y", ZZ) + + f, g = 0, 0 + assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (0, 0, 0) + + f, g = 2, 0 + assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 1, 0) + + f, g = -2, 0 + assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, -1, 0) + + f, g = 0, -2 + assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 0, -1) + + f, g = 0, 2*x + 4 + assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2*x + 4, 0, 1) + + f, g = 2*x + 4, 0 + assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2*x + 4, 1, 0) + + f, g = 2, 2 + assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 1, 1) + + f, g = -2, 2 + assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, -1, 1) + + f, g = 2, -2 + assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 1, -1) + + f, g = -2, -2 + assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, -1, -1) + + f, g = x**2 + 2*x + 1, 1 + assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (1, x**2 + 2*x + 1, 1) + + f, g = x**2 + 2*x + 1, 2 + assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (1, x**2 + 2*x + 1, 2) + + f, g = 2*x**2 + 4*x + 2, 2 + assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, x**2 + 2*x + 1, 1) + + f, g = 2, 2*x**2 + 4*x + 2 + assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 1, x**2 + 2*x + 1) + + f, g = 2*x**2 + 4*x + 2, x + 1 + assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (x + 1, 2*x + 2, 1) + + f, g = x + 1, 2*x**2 + 4*x + 2 + assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (x + 1, 1, 2*x + 2) + + R, x, y, z, u = ring("x,y,z,u", ZZ) + + f, g = u**2 + 2*u + 1, 2*u + 2 + assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (u + 1, u + 1, 2) + + f, g = z**2*u**2 + 2*z**2*u + z**2 + z*u + z, u**2 + 2*u + 1 + h, cff, cfg = u + 1, z**2*u + z**2 + z, u + 1 + + assert R.dmp_zz_heu_gcd(f, g) == (h, cff, cfg) + assert R.dmp_rr_prs_gcd(f, g) == (h, cff, cfg) + + assert R.dmp_zz_heu_gcd(g, f) == (h, cfg, cff) + assert R.dmp_rr_prs_gcd(g, f) == (h, cfg, cff) + + R, x, y, z = ring("x,y,z", ZZ) + + f, g, h = map(R.from_dense, dmp_fateman_poly_F_1(2, ZZ)) + H, cff, cfg = R.dmp_zz_heu_gcd(f, g) + + assert H == h and R.dmp_mul(H, cff) == f \ + and R.dmp_mul(H, cfg) == g + + H, cff, cfg = R.dmp_rr_prs_gcd(f, g) + + assert H == h and R.dmp_mul(H, cff) == f \ + and R.dmp_mul(H, cfg) == g + + R, x, y, z, u, v = ring("x,y,z,u,v", ZZ) + + f, g, h = map(R.from_dense, dmp_fateman_poly_F_1(4, ZZ)) + H, cff, cfg = R.dmp_zz_heu_gcd(f, g) + + assert H == h and R.dmp_mul(H, cff) == f \ + and R.dmp_mul(H, cfg) == g + + R, x, y, z, u, v, a, b = ring("x,y,z,u,v,a,b", ZZ) + + f, g, h = map(R.from_dense, dmp_fateman_poly_F_1(6, ZZ)) + H, cff, cfg = R.dmp_zz_heu_gcd(f, g) + + assert H == h and R.dmp_mul(H, cff) == f \ + and R.dmp_mul(H, cfg) == g + + R, x, y, z, u, v, a, b, c, d = ring("x,y,z,u,v,a,b,c,d", ZZ) + + f, g, h = map(R.from_dense, dmp_fateman_poly_F_1(8, ZZ)) + H, cff, cfg = R.dmp_zz_heu_gcd(f, g) + + assert H == h and R.dmp_mul(H, cff) == f \ + and R.dmp_mul(H, cfg) == g + + R, x, y, z = ring("x,y,z", ZZ) + + f, g, h = map(R.from_dense, dmp_fateman_poly_F_2(2, ZZ)) + H, cff, cfg = R.dmp_zz_heu_gcd(f, g) + + assert H == h and R.dmp_mul(H, cff) == f \ + and R.dmp_mul(H, cfg) == g + + H, cff, cfg = R.dmp_rr_prs_gcd(f, g) + + assert H == h and R.dmp_mul(H, cff) == f \ + and R.dmp_mul(H, cfg) == g + + f, g, h = map(R.from_dense, dmp_fateman_poly_F_3(2, ZZ)) + H, cff, cfg = R.dmp_zz_heu_gcd(f, g) + + assert H == h and R.dmp_mul(H, cff) == f \ + and R.dmp_mul(H, cfg) == g + + H, cff, cfg = R.dmp_rr_prs_gcd(f, g) + + assert H == h and R.dmp_mul(H, cff) == f \ + and R.dmp_mul(H, cfg) == g + + R, x, y, z, u, v = ring("x,y,z,u,v", ZZ) + + f, g, h = map(R.from_dense, dmp_fateman_poly_F_3(4, ZZ)) + H, cff, cfg = R.dmp_inner_gcd(f, g) + + assert H == h and R.dmp_mul(H, cff) == f \ + and R.dmp_mul(H, cfg) == g + + R, x, y = ring("x,y", QQ) + + f = QQ(1,2)*x**2 + x + QQ(1,2) + g = QQ(1,2)*x + QQ(1,2) + + h = x + 1 + + assert R.dmp_qq_heu_gcd(f, g) == (h, g, QQ(1,2)) + assert R.dmp_ff_prs_gcd(f, g) == (h, g, QQ(1,2)) + + R, x, y = ring("x,y", RR) + + f = 2.1*x*y**2 - 2.2*x*y + 2.1*x + g = 1.0*x**3 + + assert R.dmp_ff_prs_gcd(f, g) == \ + (1.0*x, 2.1*y**2 - 2.2*y + 2.1, 1.0*x**2) + + +def test_dup_lcm(): + R, x = ring("x", ZZ) + + assert R.dup_lcm(2, 6) == 6 + + assert R.dup_lcm(2*x**3, 6*x) == 6*x**3 + assert R.dup_lcm(2*x**3, 3*x) == 6*x**3 + + assert R.dup_lcm(x**2 + x, x) == x**2 + x + assert R.dup_lcm(x**2 + x, 2*x) == 2*x**2 + 2*x + assert R.dup_lcm(x**2 + 2*x, x) == x**2 + 2*x + assert R.dup_lcm(2*x**2 + x, x) == 2*x**2 + x + assert R.dup_lcm(2*x**2 + x, 2*x) == 4*x**2 + 2*x + + +def test_dmp_lcm(): + R, x, y = ring("x,y", ZZ) + + assert R.dmp_lcm(2, 6) == 6 + assert R.dmp_lcm(x, y) == x*y + + assert R.dmp_lcm(2*x**3, 6*x*y**2) == 6*x**3*y**2 + assert R.dmp_lcm(2*x**3, 3*x*y**2) == 6*x**3*y**2 + + assert R.dmp_lcm(x**2*y, x*y**2) == x**2*y**2 + + f = 2*x*y**5 - 3*x*y**4 - 2*x*y**3 + 3*x*y**2 + g = y**5 - 2*y**3 + y + h = 2*x*y**7 - 3*x*y**6 - 4*x*y**5 + 6*x*y**4 + 2*x*y**3 - 3*x*y**2 + + assert R.dmp_lcm(f, g) == h + + f = x**3 - 3*x**2*y - 9*x*y**2 - 5*y**3 + g = x**4 + 6*x**3*y + 12*x**2*y**2 + 10*x*y**3 + 3*y**4 + h = x**5 + x**4*y - 18*x**3*y**2 - 50*x**2*y**3 - 47*x*y**4 - 15*y**5 + + assert R.dmp_lcm(f, g) == h + + +def test_dmp_content(): + R, x,y = ring("x,y", ZZ) + + assert R.dmp_content(-2) == 2 + + f, g, F = 3*y**2 + 2*y + 1, 1, 0 + + for i in range(0, 5): + g *= f + F += x**i*g + + assert R.dmp_content(F) == f.drop(x) + + R, x,y,z = ring("x,y,z", ZZ) + + assert R.dmp_content(f_4) == 1 + assert R.dmp_content(f_5) == 1 + + R, x,y,z,t = ring("x,y,z,t", ZZ) + assert R.dmp_content(f_6) == 1 + + +def test_dmp_primitive(): + R, x,y = ring("x,y", ZZ) + + assert R.dmp_primitive(0) == (0, 0) + assert R.dmp_primitive(1) == (1, 1) + + f, g, F = 3*y**2 + 2*y + 1, 1, 0 + + for i in range(0, 5): + g *= f + F += x**i*g + + assert R.dmp_primitive(F) == (f.drop(x), F / f) + + R, x,y,z = ring("x,y,z", ZZ) + + cont, f = R.dmp_primitive(f_4) + assert cont == 1 and f == f_4 + cont, f = R.dmp_primitive(f_5) + assert cont == 1 and f == f_5 + + R, x,y,z,t = ring("x,y,z,t", ZZ) + + cont, f = R.dmp_primitive(f_6) + assert cont == 1 and f == f_6 + + +def test_dup_cancel(): + R, x = ring("x", ZZ) + + f = 2*x**2 - 2 + g = x**2 - 2*x + 1 + + p = 2*x + 2 + q = x - 1 + + assert R.dup_cancel(f, g) == (p, q) + assert R.dup_cancel(f, g, include=False) == (1, 1, p, q) + + f = -x - 2 + g = 3*x - 4 + + F = x + 2 + G = -3*x + 4 + + assert R.dup_cancel(f, g) == (f, g) + assert R.dup_cancel(F, G) == (f, g) + + assert R.dup_cancel(0, 0) == (0, 0) + assert R.dup_cancel(0, 0, include=False) == (1, 1, 0, 0) + + assert R.dup_cancel(x, 0) == (1, 0) + assert R.dup_cancel(x, 0, include=False) == (1, 1, 1, 0) + + assert R.dup_cancel(0, x) == (0, 1) + assert R.dup_cancel(0, x, include=False) == (1, 1, 0, 1) + + f = 0 + g = x + one = 1 + + assert R.dup_cancel(f, g, include=True) == (f, one) + + +def test_dmp_cancel(): + R, x, y = ring("x,y", ZZ) + + f = 2*x**2 - 2 + g = x**2 - 2*x + 1 + + p = 2*x + 2 + q = x - 1 + + assert R.dmp_cancel(f, g) == (p, q) + assert R.dmp_cancel(f, g, include=False) == (1, 1, p, q) + + assert R.dmp_cancel(0, 0) == (0, 0) + assert R.dmp_cancel(0, 0, include=False) == (1, 1, 0, 0) + + assert R.dmp_cancel(y, 0) == (1, 0) + assert R.dmp_cancel(y, 0, include=False) == (1, 1, 1, 0) + + assert R.dmp_cancel(0, y) == (0, 1) + assert R.dmp_cancel(0, y, include=False) == (1, 1, 0, 1) diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_factortools.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_factortools.py new file mode 100644 index 0000000000000000000000000000000000000000..7f99097c71e9cde761a800b01b149ec5c9896266 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_factortools.py @@ -0,0 +1,784 @@ +"""Tools for polynomial factorization routines in characteristic zero. """ + +from sympy.polys.rings import ring, xring +from sympy.polys.domains import FF, ZZ, QQ, ZZ_I, QQ_I, RR, EX + +from sympy.polys import polyconfig as config +from sympy.polys.polyerrors import DomainError +from sympy.polys.polyclasses import ANP +from sympy.polys.specialpolys import f_polys, w_polys + +from sympy.core.numbers import I +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import sin +from sympy.ntheory.generate import nextprime +from sympy.testing.pytest import raises, XFAIL + + +f_0, f_1, f_2, f_3, f_4, f_5, f_6 = f_polys() +w_1, w_2 = w_polys() + +def test_dup_trial_division(): + R, x = ring("x", ZZ) + assert R.dup_trial_division(x**5 + 8*x**4 + 25*x**3 + 38*x**2 + 28*x + 8, (x + 1, x + 2)) == [(x + 1, 2), (x + 2, 3)] + + +def test_dmp_trial_division(): + R, x, y = ring("x,y", ZZ) + assert R.dmp_trial_division(x**5 + 8*x**4 + 25*x**3 + 38*x**2 + 28*x + 8, (x + 1, x + 2)) == [(x + 1, 2), (x + 2, 3)] + + +def test_dup_zz_mignotte_bound(): + R, x = ring("x", ZZ) + assert R.dup_zz_mignotte_bound(2*x**2 + 3*x + 4) == 6 + assert R.dup_zz_mignotte_bound(x**3 + 14*x**2 + 56*x + 64) == 152 + + +def test_dmp_zz_mignotte_bound(): + R, x, y = ring("x,y", ZZ) + assert R.dmp_zz_mignotte_bound(2*x**2 + 3*x + 4) == 32 + + +def test_dup_zz_hensel_step(): + R, x = ring("x", ZZ) + + f = x**4 - 1 + g = x**3 + 2*x**2 - x - 2 + h = x - 2 + s = -2 + t = 2*x**2 - 2*x - 1 + + G, H, S, T = R.dup_zz_hensel_step(5, f, g, h, s, t) + + assert G == x**3 + 7*x**2 - x - 7 + assert H == x - 7 + assert S == 8 + assert T == -8*x**2 - 12*x - 1 + + +def test_dup_zz_hensel_lift(): + R, x = ring("x", ZZ) + + f = x**4 - 1 + F = [x - 1, x - 2, x + 2, x + 1] + + assert R.dup_zz_hensel_lift(ZZ(5), f, F, 4) == \ + [x - 1, x - 182, x + 182, x + 1] + + +def test_dup_zz_irreducible_p(): + R, x = ring("x", ZZ) + + assert R.dup_zz_irreducible_p(3*x**4 + 2*x**3 + 6*x**2 + 8*x + 7) is None + assert R.dup_zz_irreducible_p(3*x**4 + 2*x**3 + 6*x**2 + 8*x + 4) is None + + assert R.dup_zz_irreducible_p(3*x**4 + 2*x**3 + 6*x**2 + 8*x + 10) is True + assert R.dup_zz_irreducible_p(3*x**4 + 2*x**3 + 6*x**2 + 8*x + 14) is True + + +def test_dup_cyclotomic_p(): + R, x = ring("x", ZZ) + + assert R.dup_cyclotomic_p(x - 1) is True + assert R.dup_cyclotomic_p(x + 1) is True + assert R.dup_cyclotomic_p(x**2 + x + 1) is True + assert R.dup_cyclotomic_p(x**2 + 1) is True + assert R.dup_cyclotomic_p(x**4 + x**3 + x**2 + x + 1) is True + assert R.dup_cyclotomic_p(x**2 - x + 1) is True + assert R.dup_cyclotomic_p(x**6 + x**5 + x**4 + x**3 + x**2 + x + 1) is True + assert R.dup_cyclotomic_p(x**4 + 1) is True + assert R.dup_cyclotomic_p(x**6 + x**3 + 1) is True + + assert R.dup_cyclotomic_p(0) is False + assert R.dup_cyclotomic_p(1) is False + assert R.dup_cyclotomic_p(x) is False + assert R.dup_cyclotomic_p(x + 2) is False + assert R.dup_cyclotomic_p(3*x + 1) is False + assert R.dup_cyclotomic_p(x**2 - 1) is False + + f = x**16 + x**14 - x**10 + x**8 - x**6 + x**2 + 1 + assert R.dup_cyclotomic_p(f) is False + + g = x**16 + x**14 - x**10 - x**8 - x**6 + x**2 + 1 + assert R.dup_cyclotomic_p(g) is True + + R, x = ring("x", QQ) + assert R.dup_cyclotomic_p(x**2 + x + 1) is True + assert R.dup_cyclotomic_p(QQ(1,2)*x**2 + x + 1) is False + + R, x = ring("x", ZZ["y"]) + assert R.dup_cyclotomic_p(x**2 + x + 1) is False + + +def test_dup_zz_cyclotomic_poly(): + R, x = ring("x", ZZ) + + assert R.dup_zz_cyclotomic_poly(1) == x - 1 + assert R.dup_zz_cyclotomic_poly(2) == x + 1 + assert R.dup_zz_cyclotomic_poly(3) == x**2 + x + 1 + assert R.dup_zz_cyclotomic_poly(4) == x**2 + 1 + assert R.dup_zz_cyclotomic_poly(5) == x**4 + x**3 + x**2 + x + 1 + assert R.dup_zz_cyclotomic_poly(6) == x**2 - x + 1 + assert R.dup_zz_cyclotomic_poly(7) == x**6 + x**5 + x**4 + x**3 + x**2 + x + 1 + assert R.dup_zz_cyclotomic_poly(8) == x**4 + 1 + assert R.dup_zz_cyclotomic_poly(9) == x**6 + x**3 + 1 + + +def test_dup_zz_cyclotomic_factor(): + R, x = ring("x", ZZ) + + assert R.dup_zz_cyclotomic_factor(0) is None + assert R.dup_zz_cyclotomic_factor(1) is None + + assert R.dup_zz_cyclotomic_factor(2*x**10 - 1) is None + assert R.dup_zz_cyclotomic_factor(x**10 - 3) is None + assert R.dup_zz_cyclotomic_factor(x**10 + x**5 - 1) is None + + assert R.dup_zz_cyclotomic_factor(x + 1) == [x + 1] + assert R.dup_zz_cyclotomic_factor(x - 1) == [x - 1] + + assert R.dup_zz_cyclotomic_factor(x**2 + 1) == [x**2 + 1] + assert R.dup_zz_cyclotomic_factor(x**2 - 1) == [x - 1, x + 1] + + assert R.dup_zz_cyclotomic_factor(x**27 + 1) == \ + [x + 1, x**2 - x + 1, x**6 - x**3 + 1, x**18 - x**9 + 1] + assert R.dup_zz_cyclotomic_factor(x**27 - 1) == \ + [x - 1, x**2 + x + 1, x**6 + x**3 + 1, x**18 + x**9 + 1] + + +def test_dup_zz_factor(): + R, x = ring("x", ZZ) + + assert R.dup_zz_factor(0) == (0, []) + assert R.dup_zz_factor(7) == (7, []) + assert R.dup_zz_factor(-7) == (-7, []) + + assert R.dup_zz_factor_sqf(0) == (0, []) + assert R.dup_zz_factor_sqf(7) == (7, []) + assert R.dup_zz_factor_sqf(-7) == (-7, []) + + assert R.dup_zz_factor(2*x + 4) == (2, [(x + 2, 1)]) + assert R.dup_zz_factor_sqf(2*x + 4) == (2, [x + 2]) + + f = x**4 + x + 1 + + for i in range(0, 20): + assert R.dup_zz_factor(f) == (1, [(f, 1)]) + + assert R.dup_zz_factor(x**2 + 2*x + 2) == \ + (1, [(x**2 + 2*x + 2, 1)]) + + assert R.dup_zz_factor(18*x**2 + 12*x + 2) == \ + (2, [(3*x + 1, 2)]) + + assert R.dup_zz_factor(-9*x**2 + 1) == \ + (-1, [(3*x - 1, 1), + (3*x + 1, 1)]) + + assert R.dup_zz_factor_sqf(-9*x**2 + 1) == \ + (-1, [3*x - 1, + 3*x + 1]) + + # The order of the factors will be different when the ground types are + # flint. At the higher level dup_factor_list will sort the factors. + c, factors = R.dup_zz_factor(x**3 - 6*x**2 + 11*x - 6) + assert c == 1 + assert set(factors) == {(x - 3, 1), (x - 2, 1), (x - 1, 1)} + + assert R.dup_zz_factor_sqf(x**3 - 6*x**2 + 11*x - 6) == \ + (1, [x - 3, + x - 2, + x - 1]) + + assert R.dup_zz_factor(3*x**3 + 10*x**2 + 13*x + 10) == \ + (1, [(x + 2, 1), + (3*x**2 + 4*x + 5, 1)]) + + assert R.dup_zz_factor_sqf(3*x**3 + 10*x**2 + 13*x + 10) == \ + (1, [x + 2, + 3*x**2 + 4*x + 5]) + + c, factors = R.dup_zz_factor(-x**6 + x**2) + assert c == -1 + assert set(factors) == {(x, 2), (x - 1, 1), (x + 1, 1), (x**2 + 1, 1)} + + f = 1080*x**8 + 5184*x**7 + 2099*x**6 + 744*x**5 + 2736*x**4 - 648*x**3 + 129*x**2 - 324 + + assert R.dup_zz_factor(f) == \ + (1, [(5*x**4 + 24*x**3 + 9*x**2 + 12, 1), + (216*x**4 + 31*x**2 - 27, 1)]) + + f = -29802322387695312500000000000000000000*x**25 \ + + 2980232238769531250000000000000000*x**20 \ + + 1743435859680175781250000000000*x**15 \ + + 114142894744873046875000000*x**10 \ + - 210106372833251953125*x**5 \ + + 95367431640625 + + c, factors = R.dup_zz_factor(f) + assert c == -95367431640625 + assert set(factors) == { + (5*x - 1, 1), + (100*x**2 + 10*x - 1, 2), + (625*x**4 + 125*x**3 + 25*x**2 + 5*x + 1, 1), + (10000*x**4 - 3000*x**3 + 400*x**2 - 20*x + 1, 2), + (10000*x**4 + 2000*x**3 + 400*x**2 + 30*x + 1, 2), + } + + f = x**10 - 1 + + config.setup('USE_CYCLOTOMIC_FACTOR', True) + c0, F_0 = R.dup_zz_factor(f) + + config.setup('USE_CYCLOTOMIC_FACTOR', False) + c1, F_1 = R.dup_zz_factor(f) + + assert c0 == c1 == 1 + assert set(F_0) == set(F_1) == { + (x - 1, 1), + (x + 1, 1), + (x**4 - x**3 + x**2 - x + 1, 1), + (x**4 + x**3 + x**2 + x + 1, 1), + } + + config.setup('USE_CYCLOTOMIC_FACTOR') + + f = x**10 + 1 + + config.setup('USE_CYCLOTOMIC_FACTOR', True) + F_0 = R.dup_zz_factor(f) + + config.setup('USE_CYCLOTOMIC_FACTOR', False) + F_1 = R.dup_zz_factor(f) + + assert F_0 == F_1 == \ + (1, [(x**2 + 1, 1), + (x**8 - x**6 + x**4 - x**2 + 1, 1)]) + + config.setup('USE_CYCLOTOMIC_FACTOR') + +def test_dmp_zz_wang(): + R, x,y,z = ring("x,y,z", ZZ) + UV, _x = ring("x", ZZ) + + p = ZZ(nextprime(R.dmp_zz_mignotte_bound(w_1))) + assert p == 6291469 + + t_1, k_1, e_1 = y, 1, ZZ(-14) + t_2, k_2, e_2 = z, 2, ZZ(3) + t_3, k_3, e_3 = y + z, 2, ZZ(-11) + t_4, k_4, e_4 = y - z, 1, ZZ(-17) + + T = [t_1, t_2, t_3, t_4] + K = [k_1, k_2, k_3, k_4] + E = [e_1, e_2, e_3, e_4] + + T = zip([ t.drop(x) for t in T ], K) + + A = [ZZ(-14), ZZ(3)] + + S = R.dmp_eval_tail(w_1, A) + cs, s = UV.dup_primitive(S) + + assert cs == 1 and s == S == \ + 1036728*_x**6 + 915552*_x**5 + 55748*_x**4 + 105621*_x**3 - 17304*_x**2 - 26841*_x - 644 + + assert R.dmp_zz_wang_non_divisors(E, cs, ZZ(4)) == [7, 3, 11, 17] + assert UV.dup_sqf_p(s) and UV.dup_degree(s) == R.dmp_degree(w_1) + + _, H = UV.dup_zz_factor_sqf(s) + + h_1 = 44*_x**2 + 42*_x + 1 + h_2 = 126*_x**2 - 9*_x + 28 + h_3 = 187*_x**2 - 23 + + assert H == [h_1, h_2, h_3] + + LC = [ lc.drop(x) for lc in [-4*y - 4*z, -y*z**2, y**2 - z**2] ] + + assert R.dmp_zz_wang_lead_coeffs(w_1, T, cs, E, H, A) == (w_1, H, LC) + + factors = R.dmp_zz_wang_hensel_lifting(w_1, H, LC, A, p) + assert R.dmp_expand(factors) == w_1 + + +@XFAIL +def test_dmp_zz_wang_fail(): + R, x,y,z = ring("x,y,z", ZZ) + UV, _x = ring("x", ZZ) + + p = ZZ(nextprime(R.dmp_zz_mignotte_bound(w_1))) + assert p == 6291469 + + H_1 = [44*x**2 + 42*x + 1, 126*x**2 - 9*x + 28, 187*x**2 - 23] + H_2 = [-4*x**2*y - 12*x**2 - 3*x*y + 1, -9*x**2*y - 9*x - 2*y, x**2*y**2 - 9*x**2 + y - 9] + H_3 = [-4*x**2*y - 12*x**2 - 3*x*y + 1, -9*x**2*y - 9*x - 2*y, x**2*y**2 - 9*x**2 + y - 9] + + c_1 = -70686*x**5 - 5863*x**4 - 17826*x**3 + 2009*x**2 + 5031*x + 74 + c_2 = 9*x**5*y**4 + 12*x**5*y**3 - 45*x**5*y**2 - 108*x**5*y - 324*x**5 + 18*x**4*y**3 - 216*x**4*y**2 - 810*x**4*y + 2*x**3*y**4 + 9*x**3*y**3 - 252*x**3*y**2 - 288*x**3*y - 945*x**3 - 30*x**2*y**2 - 414*x**2*y + 2*x*y**3 - 54*x*y**2 - 3*x*y + 81*x + 12*y + c_3 = -36*x**4*y**2 - 108*x**4*y - 27*x**3*y**2 - 36*x**3*y - 108*x**3 - 8*x**2*y**2 - 42*x**2*y - 6*x*y**2 + 9*x + 2*y + + assert R.dmp_zz_diophantine(H_1, c_1, [], 5, p) == [-3*x, -2, 1] + assert R.dmp_zz_diophantine(H_2, c_2, [ZZ(-14)], 5, p) == [-x*y, -3*x, -6] + assert R.dmp_zz_diophantine(H_3, c_3, [ZZ(-14)], 5, p) == [0, 0, -1] + + +def test_issue_6355(): + # This tests a bug in the Wang algorithm that occurred only with a very + # specific set of random numbers. + random_sequence = [-1, -1, 0, 0, 0, 0, -1, -1, 0, -1, 3, -1, 3, 3, 3, 3, -1, 3] + + R, x, y, z = ring("x,y,z", ZZ) + f = 2*x**2 + y*z - y - z**2 + z + + assert R.dmp_zz_wang(f, seed=random_sequence) == [f] + + +def test_dmp_zz_factor(): + R, x = ring("x", ZZ) + assert R.dmp_zz_factor(0) == (0, []) + assert R.dmp_zz_factor(7) == (7, []) + assert R.dmp_zz_factor(-7) == (-7, []) + + assert R.dmp_zz_factor(x**2 - 9) == (1, [(x - 3, 1), (x + 3, 1)]) + + R, x, y = ring("x,y", ZZ) + assert R.dmp_zz_factor(0) == (0, []) + assert R.dmp_zz_factor(7) == (7, []) + assert R.dmp_zz_factor(-7) == (-7, []) + + assert R.dmp_zz_factor(x) == (1, [(x, 1)]) + assert R.dmp_zz_factor(4*x) == (4, [(x, 1)]) + assert R.dmp_zz_factor(4*x + 2) == (2, [(2*x + 1, 1)]) + assert R.dmp_zz_factor(x*y + 1) == (1, [(x*y + 1, 1)]) + assert R.dmp_zz_factor(y**2 + 1) == (1, [(y**2 + 1, 1)]) + assert R.dmp_zz_factor(y**2 - 1) == (1, [(y - 1, 1), (y + 1, 1)]) + + assert R.dmp_zz_factor(x**2*y**2 + 6*x**2*y + 9*x**2 - 1) == (1, [(x*y + 3*x - 1, 1), (x*y + 3*x + 1, 1)]) + assert R.dmp_zz_factor(x**2*y**2 - 9) == (1, [(x*y - 3, 1), (x*y + 3, 1)]) + + R, x, y, z = ring("x,y,z", ZZ) + assert R.dmp_zz_factor(x**2*y**2*z**2 - 9) == \ + (1, [(x*y*z - 3, 1), + (x*y*z + 3, 1)]) + + R, x, y, z, u = ring("x,y,z,u", ZZ) + assert R.dmp_zz_factor(x**2*y**2*z**2*u**2 - 9) == \ + (1, [(x*y*z*u - 3, 1), + (x*y*z*u + 3, 1)]) + + R, x, y, z = ring("x,y,z", ZZ) + assert R.dmp_zz_factor(f_1) == \ + (1, [(x + y*z + 20, 1), + (x*y + z + 10, 1), + (x*z + y + 30, 1)]) + + assert R.dmp_zz_factor(f_2) == \ + (1, [(x**2*y**2 + x**2*z**2 + y + 90, 1), + (x**3*y + x**3*z + z - 11, 1)]) + + assert R.dmp_zz_factor(f_3) == \ + (1, [(x**2*y**2 + x*z**4 + x + z, 1), + (x**3 + x*y*z + y**2 + y*z**3, 1)]) + + assert R.dmp_zz_factor(f_4) == \ + (-1, [(x*y**3 + z**2, 1), + (x**2*z + y**4*z**2 + 5, 1), + (x**3*y - z**2 - 3, 1), + (x**3*y**4 + z**2, 1)]) + + assert R.dmp_zz_factor(f_5) == \ + (-1, [(x + y - z, 3)]) + + R, x, y, z, t = ring("x,y,z,t", ZZ) + assert R.dmp_zz_factor(f_6) == \ + (1, [(47*x*y + z**3*t**2 - t**2, 1), + (45*x**3 - 9*y**3 - y**2 + 3*z**3 + 2*z*t, 1)]) + + R, x, y, z = ring("x,y,z", ZZ) + assert R.dmp_zz_factor(w_1) == \ + (1, [(x**2*y**2 - x**2*z**2 + y - z**2, 1), + (x**2*y*z**2 + 3*x*z + 2*y, 1), + (4*x**2*y + 4*x**2*z + x*y*z - 1, 1)]) + + R, x, y = ring("x,y", ZZ) + f = -12*x**16*y + 240*x**12*y**3 - 768*x**10*y**4 + 1080*x**8*y**5 - 768*x**6*y**6 + 240*x**4*y**7 - 12*y**9 + + assert R.dmp_zz_factor(f) == \ + (-12, [(y, 1), + (x**2 - y, 6), + (x**4 + 6*x**2*y + y**2, 1)]) + + +def test_dup_qq_i_factor(): + R, x = ring("x", QQ_I) + i = QQ_I(0, 1) + + assert R.dup_qq_i_factor(x**2 - 2) == (QQ_I(1, 0), [(x**2 - 2, 1)]) + + assert R.dup_qq_i_factor(x**2 - 1) == (QQ_I(1, 0), [(x - 1, 1), (x + 1, 1)]) + + assert R.dup_qq_i_factor(x**2 + 1) == (QQ_I(1, 0), [(x - i, 1), (x + i, 1)]) + + assert R.dup_qq_i_factor(x**2/4 + 1) == \ + (QQ_I(QQ(1, 4), 0), [(x - 2*i, 1), (x + 2*i, 1)]) + + assert R.dup_qq_i_factor(x**2 + 4) == \ + (QQ_I(1, 0), [(x - 2*i, 1), (x + 2*i, 1)]) + + assert R.dup_qq_i_factor(x**2 + 2*x + 1) == \ + (QQ_I(1, 0), [(x + 1, 2)]) + + assert R.dup_qq_i_factor(x**2 + 2*i*x - 1) == \ + (QQ_I(1, 0), [(x + i, 2)]) + + f = 8192*x**2 + x*(22656 + 175232*i) - 921416 + 242313*i + + assert R.dup_qq_i_factor(f) == \ + (QQ_I(8192, 0), [(x + QQ_I(QQ(177, 128), QQ(1369, 128)), 2)]) + + +def test_dmp_qq_i_factor(): + R, x, y = ring("x, y", QQ_I) + i = QQ_I(0, 1) + + assert R.dmp_qq_i_factor(x**2 + 2*y**2) == \ + (QQ_I(1, 0), [(x**2 + 2*y**2, 1)]) + + assert R.dmp_qq_i_factor(x**2 + y**2) == \ + (QQ_I(1, 0), [(x - i*y, 1), (x + i*y, 1)]) + + assert R.dmp_qq_i_factor(x**2 + y**2/4) == \ + (QQ_I(1, 0), [(x - i*y/2, 1), (x + i*y/2, 1)]) + + assert R.dmp_qq_i_factor(4*x**2 + y**2) == \ + (QQ_I(4, 0), [(x - i*y/2, 1), (x + i*y/2, 1)]) + + +def test_dup_zz_i_factor(): + R, x = ring("x", ZZ_I) + i = ZZ_I(0, 1) + + assert R.dup_zz_i_factor(x**2 - 2) == (ZZ_I(1, 0), [(x**2 - 2, 1)]) + + assert R.dup_zz_i_factor(x**2 - 1) == (ZZ_I(1, 0), [(x - 1, 1), (x + 1, 1)]) + + assert R.dup_zz_i_factor(x**2 + 1) == (ZZ_I(1, 0), [(x - i, 1), (x + i, 1)]) + + assert R.dup_zz_i_factor(x**2 + 4) == \ + (ZZ_I(1, 0), [(x - 2*i, 1), (x + 2*i, 1)]) + + assert R.dup_zz_i_factor(x**2 + 2*x + 1) == \ + (ZZ_I(1, 0), [(x + 1, 2)]) + + assert R.dup_zz_i_factor(x**2 + 2*i*x - 1) == \ + (ZZ_I(1, 0), [(x + i, 2)]) + + f = 8192*x**2 + x*(22656 + 175232*i) - 921416 + 242313*i + + assert R.dup_zz_i_factor(f) == \ + (ZZ_I(0, 1), [((64 - 64*i)*x + (773 + 596*i), 2)]) + + +def test_dmp_zz_i_factor(): + R, x, y = ring("x, y", ZZ_I) + i = ZZ_I(0, 1) + + assert R.dmp_zz_i_factor(x**2 + 2*y**2) == \ + (ZZ_I(1, 0), [(x**2 + 2*y**2, 1)]) + + assert R.dmp_zz_i_factor(x**2 + y**2) == \ + (ZZ_I(1, 0), [(x - i*y, 1), (x + i*y, 1)]) + + assert R.dmp_zz_i_factor(4*x**2 + y**2) == \ + (ZZ_I(1, 0), [(2*x - i*y, 1), (2*x + i*y, 1)]) + + +def test_dup_ext_factor(): + R, x = ring("x", QQ.algebraic_field(I)) + def anp(element): + return ANP(element, [QQ(1), QQ(0), QQ(1)], QQ) + + assert R.dup_ext_factor(0) == (anp([]), []) + + f = anp([QQ(1)])*x + anp([QQ(1)]) + + assert R.dup_ext_factor(f) == (anp([QQ(1)]), [(f, 1)]) + + g = anp([QQ(2)])*x + anp([QQ(2)]) + + assert R.dup_ext_factor(g) == (anp([QQ(2)]), [(f, 1)]) + + f = anp([QQ(7)])*x**4 + anp([QQ(1, 1)]) + g = anp([QQ(1)])*x**4 + anp([QQ(1, 7)]) + + assert R.dup_ext_factor(f) == (anp([QQ(7)]), [(g, 1)]) + + f = anp([QQ(1)])*x**4 + anp([QQ(1)]) + + assert R.dup_ext_factor(f) == \ + (anp([QQ(1, 1)]), [(anp([QQ(1)])*x**2 + anp([QQ(-1), QQ(0)]), 1), + (anp([QQ(1)])*x**2 + anp([QQ( 1), QQ(0)]), 1)]) + + f = anp([QQ(4, 1)])*x**2 + anp([QQ(9, 1)]) + + assert R.dup_ext_factor(f) == \ + (anp([QQ(4, 1)]), [(anp([QQ(1, 1)])*x + anp([-QQ(3, 2), QQ(0, 1)]), 1), + (anp([QQ(1, 1)])*x + anp([ QQ(3, 2), QQ(0, 1)]), 1)]) + + f = anp([QQ(4, 1)])*x**4 + anp([QQ(8, 1)])*x**3 + anp([QQ(77, 1)])*x**2 + anp([QQ(18, 1)])*x + anp([QQ(153, 1)]) + + assert R.dup_ext_factor(f) == \ + (anp([QQ(4, 1)]), [(anp([QQ(1, 1)])*x + anp([-QQ(4, 1), QQ(1, 1)]), 1), + (anp([QQ(1, 1)])*x + anp([-QQ(3, 2), QQ(0, 1)]), 1), + (anp([QQ(1, 1)])*x + anp([ QQ(3, 2), QQ(0, 1)]), 1), + (anp([QQ(1, 1)])*x + anp([ QQ(4, 1), QQ(1, 1)]), 1)]) + + R, x = ring("x", QQ.algebraic_field(sqrt(2))) + def anp(element): + return ANP(element, [QQ(1), QQ(0), QQ(-2)], QQ) + + f = anp([QQ(1)])*x**4 + anp([QQ(1, 1)]) + + assert R.dup_ext_factor(f) == \ + (anp([QQ(1)]), [(anp([QQ(1)])*x**2 + anp([QQ(-1), QQ(0)])*x + anp([QQ(1)]), 1), + (anp([QQ(1)])*x**2 + anp([QQ( 1), QQ(0)])*x + anp([QQ(1)]), 1)]) + + f = anp([QQ(1, 1)])*x**2 + anp([QQ(2), QQ(0)])*x + anp([QQ(2, 1)]) + + assert R.dup_ext_factor(f) == \ + (anp([QQ(1, 1)]), [(anp([1])*x + anp([1, 0]), 2)]) + + assert R.dup_ext_factor(f**3) == \ + (anp([QQ(1, 1)]), [(anp([1])*x + anp([1, 0]), 6)]) + + f *= anp([QQ(2, 1)]) + + assert R.dup_ext_factor(f) == \ + (anp([QQ(2, 1)]), [(anp([1])*x + anp([1, 0]), 2)]) + + assert R.dup_ext_factor(f**3) == \ + (anp([QQ(8, 1)]), [(anp([1])*x + anp([1, 0]), 6)]) + + +def test_dmp_ext_factor(): + K = QQ.algebraic_field(sqrt(2)) + R, x,y = ring("x,y", K) + sqrt2 = K.unit + + def anp(x): + return ANP(x, [QQ(1), QQ(0), QQ(-2)], QQ) + + assert R.dmp_ext_factor(0) == (anp([]), []) + + f = anp([QQ(1)])*x + anp([QQ(1)]) + + assert R.dmp_ext_factor(f) == (anp([QQ(1)]), [(f, 1)]) + + g = anp([QQ(2)])*x + anp([QQ(2)]) + + assert R.dmp_ext_factor(g) == (anp([QQ(2)]), [(f, 1)]) + + f = anp([QQ(1)])*x**2 + anp([QQ(-2)])*y**2 + + assert R.dmp_ext_factor(f) == \ + (anp([QQ(1)]), [(anp([QQ(1)])*x + anp([QQ(-1), QQ(0)])*y, 1), + (anp([QQ(1)])*x + anp([QQ( 1), QQ(0)])*y, 1)]) + + f = anp([QQ(2)])*x**2 + anp([QQ(-4)])*y**2 + + assert R.dmp_ext_factor(f) == \ + (anp([QQ(2)]), [(anp([QQ(1)])*x + anp([QQ(-1), QQ(0)])*y, 1), + (anp([QQ(1)])*x + anp([QQ( 1), QQ(0)])*y, 1)]) + + f1 = y + 1 + f2 = y + sqrt2 + f3 = x**2 + x + 2 + 3*sqrt2 + f = f1**2 * f2**2 * f3**2 + assert R.dmp_ext_factor(f) == (K.one, [(f1, 2), (f2, 2), (f3, 2)]) + + +def test_dup_factor_list(): + R, x = ring("x", ZZ) + assert R.dup_factor_list(0) == (0, []) + assert R.dup_factor_list(7) == (7, []) + + R, x = ring("x", QQ) + assert R.dup_factor_list(0) == (0, []) + assert R.dup_factor_list(QQ(1, 7)) == (QQ(1, 7), []) + + R, x = ring("x", ZZ['t']) + assert R.dup_factor_list(0) == (0, []) + assert R.dup_factor_list(7) == (7, []) + + R, x = ring("x", QQ['t']) + assert R.dup_factor_list(0) == (0, []) + assert R.dup_factor_list(QQ(1, 7)) == (QQ(1, 7), []) + + R, x = ring("x", ZZ) + assert R.dup_factor_list_include(0) == [(0, 1)] + assert R.dup_factor_list_include(7) == [(7, 1)] + + assert R.dup_factor_list(x**2 + 2*x + 1) == (1, [(x + 1, 2)]) + assert R.dup_factor_list_include(x**2 + 2*x + 1) == [(x + 1, 2)] + # issue 8037 + assert R.dup_factor_list(6*x**2 - 5*x - 6) == (1, [(2*x - 3, 1), (3*x + 2, 1)]) + + R, x = ring("x", QQ) + assert R.dup_factor_list(QQ(1,2)*x**2 + x + QQ(1,2)) == (QQ(1, 2), [(x + 1, 2)]) + + R, x = ring("x", FF(2)) + assert R.dup_factor_list(x**2 + 1) == (1, [(x + 1, 2)]) + + R, x = ring("x", RR) + assert R.dup_factor_list(1.0*x**2 + 2.0*x + 1.0) == (1.0, [(1.0*x + 1.0, 2)]) + assert R.dup_factor_list(2.0*x**2 + 4.0*x + 2.0) == (2.0, [(1.0*x + 1.0, 2)]) + + f = 6.7225336055071*x**2 - 10.6463972754741*x - 0.33469524022264 + coeff, factors = R.dup_factor_list(f) + assert coeff == RR(10.6463972754741) + assert len(factors) == 1 + assert factors[0][0].max_norm() == RR(1.0) + assert factors[0][1] == 1 + + Rt, t = ring("t", ZZ) + R, x = ring("x", Rt) + + f = 4*t*x**2 + 4*t**2*x + + assert R.dup_factor_list(f) == \ + (4*t, [(x, 1), + (x + t, 1)]) + + Rt, t = ring("t", QQ) + R, x = ring("x", Rt) + + f = QQ(1, 2)*t*x**2 + QQ(1, 2)*t**2*x + + assert R.dup_factor_list(f) == \ + (QQ(1, 2)*t, [(x, 1), + (x + t, 1)]) + + R, x = ring("x", QQ.algebraic_field(I)) + def anp(element): + return ANP(element, [QQ(1), QQ(0), QQ(1)], QQ) + + f = anp([QQ(1, 1)])*x**4 + anp([QQ(2, 1)])*x**2 + + assert R.dup_factor_list(f) == \ + (anp([QQ(1, 1)]), [(anp([QQ(1, 1)])*x, 2), + (anp([QQ(1, 1)])*x**2 + anp([])*x + anp([QQ(2, 1)]), 1)]) + + R, x = ring("x", EX) + raises(DomainError, lambda: R.dup_factor_list(EX(sin(1)))) + + +def test_dmp_factor_list(): + R, x, y = ring("x,y", ZZ) + assert R.dmp_factor_list(0) == (ZZ(0), []) + assert R.dmp_factor_list(7) == (7, []) + + R, x, y = ring("x,y", QQ) + assert R.dmp_factor_list(0) == (QQ(0), []) + assert R.dmp_factor_list(QQ(1, 7)) == (QQ(1, 7), []) + + Rt, t = ring("t", ZZ) + R, x, y = ring("x,y", Rt) + assert R.dmp_factor_list(0) == (0, []) + assert R.dmp_factor_list(7) == (ZZ(7), []) + + Rt, t = ring("t", QQ) + R, x, y = ring("x,y", Rt) + assert R.dmp_factor_list(0) == (0, []) + assert R.dmp_factor_list(QQ(1, 7)) == (QQ(1, 7), []) + + R, x, y = ring("x,y", ZZ) + assert R.dmp_factor_list_include(0) == [(0, 1)] + assert R.dmp_factor_list_include(7) == [(7, 1)] + + R, X = xring("x:200", ZZ) + + f, g = X[0]**2 + 2*X[0] + 1, X[0] + 1 + assert R.dmp_factor_list(f) == (1, [(g, 2)]) + + f, g = X[-1]**2 + 2*X[-1] + 1, X[-1] + 1 + assert R.dmp_factor_list(f) == (1, [(g, 2)]) + + R, x = ring("x", ZZ) + assert R.dmp_factor_list(x**2 + 2*x + 1) == (1, [(x + 1, 2)]) + R, x = ring("x", QQ) + assert R.dmp_factor_list(QQ(1,2)*x**2 + x + QQ(1,2)) == (QQ(1,2), [(x + 1, 2)]) + + R, x, y = ring("x,y", ZZ) + assert R.dmp_factor_list(x**2 + 2*x + 1) == (1, [(x + 1, 2)]) + R, x, y = ring("x,y", QQ) + assert R.dmp_factor_list(QQ(1,2)*x**2 + x + QQ(1,2)) == (QQ(1,2), [(x + 1, 2)]) + + R, x, y = ring("x,y", ZZ) + f = 4*x**2*y + 4*x*y**2 + + assert R.dmp_factor_list(f) == \ + (4, [(y, 1), + (x, 1), + (x + y, 1)]) + + assert R.dmp_factor_list_include(f) == \ + [(4*y, 1), + (x, 1), + (x + y, 1)] + + R, x, y = ring("x,y", QQ) + f = QQ(1,2)*x**2*y + QQ(1,2)*x*y**2 + + assert R.dmp_factor_list(f) == \ + (QQ(1,2), [(y, 1), + (x, 1), + (x + y, 1)]) + + R, x, y = ring("x,y", RR) + f = 2.0*x**2 - 8.0*y**2 + + assert R.dmp_factor_list(f) == \ + (RR(8.0), [(0.5*x - y, 1), + (0.5*x + y, 1)]) + + f = 6.7225336055071*x**2*y**2 - 10.6463972754741*x*y - 0.33469524022264 + coeff, factors = R.dmp_factor_list(f) + assert coeff == RR(10.6463972754741) + assert len(factors) == 1 + assert factors[0][0].max_norm() == RR(1.0) + assert factors[0][1] == 1 + + Rt, t = ring("t", ZZ) + R, x, y = ring("x,y", Rt) + f = 4*t*x**2 + 4*t**2*x + + assert R.dmp_factor_list(f) == \ + (4*t, [(x, 1), + (x + t, 1)]) + + Rt, t = ring("t", QQ) + R, x, y = ring("x,y", Rt) + f = QQ(1, 2)*t*x**2 + QQ(1, 2)*t**2*x + + assert R.dmp_factor_list(f) == \ + (QQ(1, 2)*t, [(x, 1), + (x + t, 1)]) + + R, x, y = ring("x,y", FF(2)) + raises(NotImplementedError, lambda: R.dmp_factor_list(x**2 + y**2)) + + R, x, y = ring("x,y", EX) + raises(DomainError, lambda: R.dmp_factor_list(EX(sin(1)))) + + +def test_dup_irreducible_p(): + R, x = ring("x", ZZ) + assert R.dup_irreducible_p(x**2 + x + 1) is True + assert R.dup_irreducible_p(x**2 + 2*x + 1) is False + + +def test_dmp_irreducible_p(): + R, x, y = ring("x,y", ZZ) + assert R.dmp_irreducible_p(x**2 + x + 1) is True + assert R.dmp_irreducible_p(x**2 + 2*x + 1) is False diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_fields.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_fields.py new file mode 100644 index 0000000000000000000000000000000000000000..4f85a00d75dc02ab794ff94c83ba18ddc2023313 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_fields.py @@ -0,0 +1,353 @@ +"""Test sparse rational functions. """ + +from sympy.polys.fields import field, sfield, FracField, FracElement +from sympy.polys.rings import ring +from sympy.polys.domains import ZZ, QQ +from sympy.polys.orderings import lex + +from sympy.testing.pytest import raises, XFAIL +from sympy.core import symbols, E +from sympy.core.numbers import Rational +from sympy.functions.elementary.exponential import (exp, log) +from sympy.functions.elementary.miscellaneous import sqrt + +def test_FracField___init__(): + F1 = FracField("x,y", ZZ, lex) + F2 = FracField("x,y", ZZ, lex) + F3 = FracField("x,y,z", ZZ, lex) + + assert F1.x == F1.gens[0] + assert F1.y == F1.gens[1] + assert F1.x == F2.x + assert F1.y == F2.y + assert F1.x != F3.x + assert F1.y != F3.y + +def test_FracField___hash__(): + F, x, y, z = field("x,y,z", QQ) + assert hash(F) + +def test_FracField___eq__(): + assert field("x,y,z", QQ)[0] == field("x,y,z", QQ)[0] + assert field("x,y,z", QQ)[0] != field("x,y,z", ZZ)[0] + assert field("x,y,z", ZZ)[0] != field("x,y,z", QQ)[0] + assert field("x,y,z", QQ)[0] != field("x,y", QQ)[0] + assert field("x,y", QQ)[0] != field("x,y,z", QQ)[0] + +def test_sfield(): + x = symbols("x") + + F = FracField((E, exp(exp(x)), exp(x)), ZZ, lex) + e, exex, ex = F.gens + assert sfield(exp(x)*exp(exp(x) + 1 + log(exp(x) + 3)/2)**2/(exp(x) + 3)) \ + == (F, e**2*exex**2*ex) + + F = FracField((x, exp(1/x), log(x), x**QQ(1, 3)), ZZ, lex) + _, ex, lg, x3 = F.gens + assert sfield(((x-3)*log(x)+4*x**2)*exp(1/x+log(x)/3)/x**2) == \ + (F, (4*F.x**2*ex + F.x*ex*lg - 3*ex*lg)/x3**5) + + F = FracField((x, log(x), sqrt(x + log(x))), ZZ, lex) + _, lg, srt = F.gens + assert sfield((x + 1) / (x * (x + log(x))**QQ(3, 2)) - 1/(x * log(x)**2)) \ + == (F, (F.x*lg**2 - F.x*srt + lg**2 - lg*srt)/ + (F.x**2*lg**2*srt + F.x*lg**3*srt)) + +def test_FracElement___hash__(): + F, x, y, z = field("x,y,z", QQ) + assert hash(x*y/z) + +def test_FracElement_copy(): + F, x, y, z = field("x,y,z", ZZ) + + f = x*y/3*z + g = f.copy() + + assert f == g + g.numer[(1, 1, 1)] = 7 + assert f != g + +def test_FracElement_as_expr(): + F, x, y, z = field("x,y,z", ZZ) + f = (3*x**2*y - x*y*z)/(7*z**3 + 1) + + X, Y, Z = F.symbols + g = (3*X**2*Y - X*Y*Z)/(7*Z**3 + 1) + + assert f != g + assert f.as_expr() == g + + X, Y, Z = symbols("x,y,z") + g = (3*X**2*Y - X*Y*Z)/(7*Z**3 + 1) + + assert f != g + assert f.as_expr(X, Y, Z) == g + + raises(ValueError, lambda: f.as_expr(X)) + +def test_FracElement_from_expr(): + x, y, z = symbols("x,y,z") + F, X, Y, Z = field((x, y, z), ZZ) + + f = F.from_expr(1) + assert f == 1 and F.is_element(f) + + f = F.from_expr(Rational(3, 7)) + assert f == F(3)/7 and F.is_element(f) + + f = F.from_expr(x) + assert f == X and F.is_element(f) + + f = F.from_expr(Rational(3,7)*x) + assert f == X*Rational(3, 7) and F.is_element(f) + + f = F.from_expr(1/x) + assert f == 1/X and F.is_element(f) + + f = F.from_expr(x*y*z) + assert f == X*Y*Z and F.is_element(f) + + f = F.from_expr(x*y/z) + assert f == X*Y/Z and F.is_element(f) + + f = F.from_expr(x*y*z + x*y + x) + assert f == X*Y*Z + X*Y + X and F.is_element(f) + + f = F.from_expr((x*y*z + x*y + x)/(x*y + 7)) + assert f == (X*Y*Z + X*Y + X)/(X*Y + 7) and F.is_element(f) + + f = F.from_expr(x**3*y*z + x**2*y**7 + 1) + assert f == X**3*Y*Z + X**2*Y**7 + 1 and F.is_element(f) + + raises(ValueError, lambda: F.from_expr(2**x)) + raises(ValueError, lambda: F.from_expr(7*x + sqrt(2))) + + assert isinstance(ZZ[2**x].get_field().convert(2**(-x)), + FracElement) + assert isinstance(ZZ[x**2].get_field().convert(x**(-6)), + FracElement) + assert isinstance(ZZ[exp(Rational(1, 3))].get_field().convert(E), + FracElement) + + +def test_FracField_nested(): + a, b, x = symbols('a b x') + F1 = ZZ.frac_field(a, b) + F2 = F1.frac_field(x) + frac = F2(a + b) + assert frac.numer == F1.poly_ring(x)(a + b) + assert frac.numer.coeffs() == [F1(a + b)] + assert frac.denom == F1.poly_ring(x)(1) + + F3 = ZZ.poly_ring(a, b) + F4 = F3.frac_field(x) + frac = F4(a + b) + assert frac.numer == F3.poly_ring(x)(a + b) + assert frac.numer.coeffs() == [F3(a + b)] + assert frac.denom == F3.poly_ring(x)(1) + + frac = F2(F3(a + b)) + assert frac.numer == F1.poly_ring(x)(a + b) + assert frac.numer.coeffs() == [F1(a + b)] + assert frac.denom == F1.poly_ring(x)(1) + + frac = F4(F1(a + b)) + assert frac.numer == F3.poly_ring(x)(a + b) + assert frac.numer.coeffs() == [F3(a + b)] + assert frac.denom == F3.poly_ring(x)(1) + + +def test_FracElement__lt_le_gt_ge__(): + F, x, y = field("x,y", ZZ) + + assert F(1) < 1/x < 1/x**2 < 1/x**3 + assert F(1) <= 1/x <= 1/x**2 <= 1/x**3 + + assert -7/x < 1/x < 3/x < y/x < 1/x**2 + assert -7/x <= 1/x <= 3/x <= y/x <= 1/x**2 + + assert 1/x**3 > 1/x**2 > 1/x > F(1) + assert 1/x**3 >= 1/x**2 >= 1/x >= F(1) + + assert 1/x**2 > y/x > 3/x > 1/x > -7/x + assert 1/x**2 >= y/x >= 3/x >= 1/x >= -7/x + +def test_FracElement___neg__(): + F, x,y = field("x,y", QQ) + + f = (7*x - 9)/y + g = (-7*x + 9)/y + + assert -f == g + assert -g == f + +def test_FracElement___add__(): + F, x,y = field("x,y", QQ) + + f, g = 1/x, 1/y + assert f + g == g + f == (x + y)/(x*y) + + assert x + F.ring.gens[0] == F.ring.gens[0] + x == 2*x + + F, x,y = field("x,y", ZZ) + assert x + 3 == 3 + x + assert x + QQ(3,7) == QQ(3,7) + x == (7*x + 3)/7 + + Fuv, u,v = field("u,v", ZZ) + Fxyzt, x,y,z,t = field("x,y,z,t", Fuv) + + f = (u*v + x)/(y + u*v) + assert dict(f.numer) == {(1, 0, 0, 0): 1, (0, 0, 0, 0): u*v} + assert dict(f.denom) == {(0, 1, 0, 0): 1, (0, 0, 0, 0): u*v} + + Ruv, u,v = ring("u,v", ZZ) + Fxyzt, x,y,z,t = field("x,y,z,t", Ruv) + + f = (u*v + x)/(y + u*v) + assert dict(f.numer) == {(1, 0, 0, 0): 1, (0, 0, 0, 0): u*v} + assert dict(f.denom) == {(0, 1, 0, 0): 1, (0, 0, 0, 0): u*v} + +def test_FracElement___sub__(): + F, x,y = field("x,y", QQ) + + f, g = 1/x, 1/y + assert f - g == (-x + y)/(x*y) + + assert x - F.ring.gens[0] == F.ring.gens[0] - x == 0 + + F, x,y = field("x,y", ZZ) + assert x - 3 == -(3 - x) + assert x - QQ(3,7) == -(QQ(3,7) - x) == (7*x - 3)/7 + + Fuv, u,v = field("u,v", ZZ) + Fxyzt, x,y,z,t = field("x,y,z,t", Fuv) + + f = (u*v - x)/(y - u*v) + assert dict(f.numer) == {(1, 0, 0, 0):-1, (0, 0, 0, 0): u*v} + assert dict(f.denom) == {(0, 1, 0, 0): 1, (0, 0, 0, 0):-u*v} + + Ruv, u,v = ring("u,v", ZZ) + Fxyzt, x,y,z,t = field("x,y,z,t", Ruv) + + f = (u*v - x)/(y - u*v) + assert dict(f.numer) == {(1, 0, 0, 0):-1, (0, 0, 0, 0): u*v} + assert dict(f.denom) == {(0, 1, 0, 0): 1, (0, 0, 0, 0):-u*v} + +def test_FracElement___mul__(): + F, x,y = field("x,y", QQ) + + f, g = 1/x, 1/y + assert f*g == g*f == 1/(x*y) + + assert x*F.ring.gens[0] == F.ring.gens[0]*x == x**2 + + F, x,y = field("x,y", ZZ) + assert x*3 == 3*x + assert x*QQ(3,7) == QQ(3,7)*x == x*Rational(3, 7) + + Fuv, u,v = field("u,v", ZZ) + Fxyzt, x,y,z,t = field("x,y,z,t", Fuv) + + f = ((u + 1)*x*y + 1)/((v - 1)*z - t*u*v - 1) + assert dict(f.numer) == {(1, 1, 0, 0): u + 1, (0, 0, 0, 0): 1} + assert dict(f.denom) == {(0, 0, 1, 0): v - 1, (0, 0, 0, 1): -u*v, (0, 0, 0, 0): -1} + + Ruv, u,v = ring("u,v", ZZ) + Fxyzt, x,y,z,t = field("x,y,z,t", Ruv) + + f = ((u + 1)*x*y + 1)/((v - 1)*z - t*u*v - 1) + assert dict(f.numer) == {(1, 1, 0, 0): u + 1, (0, 0, 0, 0): 1} + assert dict(f.denom) == {(0, 0, 1, 0): v - 1, (0, 0, 0, 1): -u*v, (0, 0, 0, 0): -1} + +def test_FracElement___truediv__(): + F, x,y = field("x,y", QQ) + + f, g = 1/x, 1/y + assert f/g == y/x + + assert x/F.ring.gens[0] == F.ring.gens[0]/x == 1 + + F, x,y = field("x,y", ZZ) + assert x*3 == 3*x + assert x/QQ(3,7) == (QQ(3,7)/x)**-1 == x*Rational(7, 3) + + raises(ZeroDivisionError, lambda: x/0) + raises(ZeroDivisionError, lambda: 1/(x - x)) + raises(ZeroDivisionError, lambda: x/(x - x)) + + Fuv, u,v = field("u,v", ZZ) + Fxyzt, x,y,z,t = field("x,y,z,t", Fuv) + + f = (u*v)/(x*y) + assert dict(f.numer) == {(0, 0, 0, 0): u*v} + assert dict(f.denom) == {(1, 1, 0, 0): 1} + + g = (x*y)/(u*v) + assert dict(g.numer) == {(1, 1, 0, 0): 1} + assert dict(g.denom) == {(0, 0, 0, 0): u*v} + + Ruv, u,v = ring("u,v", ZZ) + Fxyzt, x,y,z,t = field("x,y,z,t", Ruv) + + f = (u*v)/(x*y) + assert dict(f.numer) == {(0, 0, 0, 0): u*v} + assert dict(f.denom) == {(1, 1, 0, 0): 1} + + g = (x*y)/(u*v) + assert dict(g.numer) == {(1, 1, 0, 0): 1} + assert dict(g.denom) == {(0, 0, 0, 0): u*v} + +def test_FracElement___pow__(): + F, x,y = field("x,y", QQ) + + f, g = 1/x, 1/y + + assert f**3 == 1/x**3 + assert g**3 == 1/y**3 + + assert (f*g)**3 == 1/(x**3*y**3) + assert (f*g)**-3 == (x*y)**3 + + raises(ZeroDivisionError, lambda: (x - x)**-3) + +def test_FracElement_diff(): + F, x,y,z = field("x,y,z", ZZ) + + assert ((x**2 + y)/(z + 1)).diff(x) == 2*x/(z + 1) + +@XFAIL +def test_FracElement___call__(): + F, x,y,z = field("x,y,z", ZZ) + f = (x**2 + 3*y)/z + + r = f(1, 1, 1) + assert r == 4 and not isinstance(r, FracElement) + raises(ZeroDivisionError, lambda: f(1, 1, 0)) + +def test_FracElement_evaluate(): + F, x,y,z = field("x,y,z", ZZ) + Fyz = field("y,z", ZZ)[0] + f = (x**2 + 3*y)/z + + assert f.evaluate(x, 0) == 3*Fyz.y/Fyz.z + raises(ZeroDivisionError, lambda: f.evaluate(z, 0)) + +def test_FracElement_subs(): + F, x,y,z = field("x,y,z", ZZ) + f = (x**2 + 3*y)/z + + assert f.subs(x, 0) == 3*y/z + raises(ZeroDivisionError, lambda: f.subs(z, 0)) + +def test_FracElement_compose(): + pass + +def test_FracField_index(): + a = symbols("a") + F, x, y, z = field('x y z', QQ) + assert F.index(x) == 0 + assert F.index(y) == 1 + + raises(ValueError, lambda: F.index(1)) + raises(ValueError, lambda: F.index(a)) + pass diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_galoistools.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_galoistools.py new file mode 100644 index 0000000000000000000000000000000000000000..e512bdd865c300bb138cb40b4ff78f393b323c22 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_galoistools.py @@ -0,0 +1,875 @@ +from sympy.polys.galoistools import ( + gf_crt, gf_crt1, gf_crt2, gf_int, + gf_degree, gf_strip, gf_trunc, gf_normal, + gf_from_dict, gf_to_dict, + gf_from_int_poly, gf_to_int_poly, + gf_neg, gf_add_ground, gf_sub_ground, gf_mul_ground, + gf_add, gf_sub, gf_add_mul, gf_sub_mul, gf_mul, gf_sqr, + gf_div, gf_rem, gf_quo, gf_exquo, + gf_lshift, gf_rshift, gf_expand, + gf_pow, gf_pow_mod, + gf_gcdex, gf_gcd, gf_lcm, gf_cofactors, + gf_LC, gf_TC, gf_monic, + gf_eval, gf_multi_eval, + gf_compose, gf_compose_mod, + gf_trace_map, + gf_diff, + gf_irreducible, gf_irreducible_p, + gf_irred_p_ben_or, gf_irred_p_rabin, + gf_sqf_list, gf_sqf_part, gf_sqf_p, + gf_Qmatrix, gf_Qbasis, + gf_ddf_zassenhaus, gf_ddf_shoup, + gf_edf_zassenhaus, gf_edf_shoup, + gf_berlekamp, + gf_factor_sqf, gf_factor, + gf_value, linear_congruence, _csolve_prime_las_vegas, + csolve_prime, gf_csolve, gf_frobenius_map, gf_frobenius_monomial_base +) + +from sympy.polys.polyerrors import ( + ExactQuotientFailed, +) + +from sympy.polys import polyconfig as config + +from sympy.polys.domains import ZZ +from sympy.core.numbers import pi +from sympy.ntheory.generate import nextprime +from sympy.testing.pytest import raises + + +def test_gf_crt(): + U = [49, 76, 65] + M = [99, 97, 95] + + p = 912285 + u = 639985 + + assert gf_crt(U, M, ZZ) == u + + E = [9215, 9405, 9603] + S = [62, 24, 12] + + assert gf_crt1(M, ZZ) == (p, E, S) + assert gf_crt2(U, M, p, E, S, ZZ) == u + + +def test_gf_int(): + assert gf_int(0, 5) == 0 + assert gf_int(1, 5) == 1 + assert gf_int(2, 5) == 2 + assert gf_int(3, 5) == -2 + assert gf_int(4, 5) == -1 + assert gf_int(5, 5) == 0 + + +def test_gf_degree(): + assert gf_degree([]) == -1 + assert gf_degree([1]) == 0 + assert gf_degree([1, 0]) == 1 + assert gf_degree([1, 0, 0, 0, 1]) == 4 + + +def test_gf_strip(): + assert gf_strip([]) == [] + assert gf_strip([0]) == [] + assert gf_strip([0, 0, 0]) == [] + + assert gf_strip([1]) == [1] + assert gf_strip([0, 1]) == [1] + assert gf_strip([0, 0, 0, 1]) == [1] + + assert gf_strip([1, 2, 0]) == [1, 2, 0] + assert gf_strip([0, 1, 2, 0]) == [1, 2, 0] + assert gf_strip([0, 0, 0, 1, 2, 0]) == [1, 2, 0] + + +def test_gf_trunc(): + assert gf_trunc([], 11) == [] + assert gf_trunc([1], 11) == [1] + assert gf_trunc([22], 11) == [] + assert gf_trunc([12], 11) == [1] + + assert gf_trunc([11, 22, 17, 1, 0], 11) == [6, 1, 0] + assert gf_trunc([12, 23, 17, 1, 0], 11) == [1, 1, 6, 1, 0] + + +def test_gf_normal(): + assert gf_normal([11, 22, 17, 1, 0], 11, ZZ) == [6, 1, 0] + + +def test_gf_from_to_dict(): + f = {11: 12, 6: 2, 0: 25} + F = {11: 1, 6: 2, 0: 3} + g = [1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3] + + assert gf_from_dict(f, 11, ZZ) == g + assert gf_to_dict(g, 11) == F + + f = {11: -5, 4: 0, 3: 1, 0: 12} + F = {11: -5, 3: 1, 0: 1} + g = [6, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1] + + assert gf_from_dict(f, 11, ZZ) == g + assert gf_to_dict(g, 11) == F + + assert gf_to_dict([10], 11, symmetric=True) == {0: -1} + assert gf_to_dict([10], 11, symmetric=False) == {0: 10} + + +def test_gf_from_to_int_poly(): + assert gf_from_int_poly([1, 0, 7, 2, 20], 5) == [1, 0, 2, 2, 0] + assert gf_to_int_poly([1, 0, 4, 2, 3], 5) == [1, 0, -1, 2, -2] + + assert gf_to_int_poly([10], 11, symmetric=True) == [-1] + assert gf_to_int_poly([10], 11, symmetric=False) == [10] + + +def test_gf_LC(): + assert gf_LC([], ZZ) == 0 + assert gf_LC([1], ZZ) == 1 + assert gf_LC([1, 2], ZZ) == 1 + + +def test_gf_TC(): + assert gf_TC([], ZZ) == 0 + assert gf_TC([1], ZZ) == 1 + assert gf_TC([1, 2], ZZ) == 2 + + +def test_gf_monic(): + assert gf_monic(ZZ.map([]), 11, ZZ) == (0, []) + + assert gf_monic(ZZ.map([1]), 11, ZZ) == (1, [1]) + assert gf_monic(ZZ.map([2]), 11, ZZ) == (2, [1]) + + assert gf_monic(ZZ.map([1, 2, 3, 4]), 11, ZZ) == (1, [1, 2, 3, 4]) + assert gf_monic(ZZ.map([2, 3, 4, 5]), 11, ZZ) == (2, [1, 7, 2, 8]) + + +def test_gf_arith(): + assert gf_neg([], 11, ZZ) == [] + assert gf_neg([1], 11, ZZ) == [10] + assert gf_neg([1, 2, 3], 11, ZZ) == [10, 9, 8] + + assert gf_add_ground([], 0, 11, ZZ) == [] + assert gf_sub_ground([], 0, 11, ZZ) == [] + + assert gf_add_ground([], 3, 11, ZZ) == [3] + assert gf_sub_ground([], 3, 11, ZZ) == [8] + + assert gf_add_ground([1], 3, 11, ZZ) == [4] + assert gf_sub_ground([1], 3, 11, ZZ) == [9] + + assert gf_add_ground([8], 3, 11, ZZ) == [] + assert gf_sub_ground([3], 3, 11, ZZ) == [] + + assert gf_add_ground([1, 2, 3], 3, 11, ZZ) == [1, 2, 6] + assert gf_sub_ground([1, 2, 3], 3, 11, ZZ) == [1, 2, 0] + + assert gf_mul_ground([], 0, 11, ZZ) == [] + assert gf_mul_ground([], 1, 11, ZZ) == [] + + assert gf_mul_ground([1], 0, 11, ZZ) == [] + assert gf_mul_ground([1], 1, 11, ZZ) == [1] + + assert gf_mul_ground([1, 2, 3], 0, 11, ZZ) == [] + assert gf_mul_ground([1, 2, 3], 1, 11, ZZ) == [1, 2, 3] + assert gf_mul_ground([1, 2, 3], 7, 11, ZZ) == [7, 3, 10] + + assert gf_add([], [], 11, ZZ) == [] + assert gf_add([1], [], 11, ZZ) == [1] + assert gf_add([], [1], 11, ZZ) == [1] + assert gf_add([1], [1], 11, ZZ) == [2] + assert gf_add([1], [2], 11, ZZ) == [3] + + assert gf_add([1, 2], [1], 11, ZZ) == [1, 3] + assert gf_add([1], [1, 2], 11, ZZ) == [1, 3] + + assert gf_add([1, 2, 3], [8, 9, 10], 11, ZZ) == [9, 0, 2] + + assert gf_sub([], [], 11, ZZ) == [] + assert gf_sub([1], [], 11, ZZ) == [1] + assert gf_sub([], [1], 11, ZZ) == [10] + assert gf_sub([1], [1], 11, ZZ) == [] + assert gf_sub([1], [2], 11, ZZ) == [10] + + assert gf_sub([1, 2], [1], 11, ZZ) == [1, 1] + assert gf_sub([1], [1, 2], 11, ZZ) == [10, 10] + + assert gf_sub([3, 2, 1], [8, 9, 10], 11, ZZ) == [6, 4, 2] + + assert gf_add_mul( + [1, 5, 6], [7, 3], [8, 0, 6, 1], 11, ZZ) == [1, 2, 10, 8, 9] + assert gf_sub_mul( + [1, 5, 6], [7, 3], [8, 0, 6, 1], 11, ZZ) == [10, 9, 3, 2, 3] + + assert gf_mul([], [], 11, ZZ) == [] + assert gf_mul([], [1], 11, ZZ) == [] + assert gf_mul([1], [], 11, ZZ) == [] + assert gf_mul([1], [1], 11, ZZ) == [1] + assert gf_mul([5], [7], 11, ZZ) == [2] + + assert gf_mul([3, 0, 0, 6, 1, 2], [4, 0, 1, 0], 11, ZZ) == [1, 0, + 3, 2, 4, 3, 1, 2, 0] + assert gf_mul([4, 0, 1, 0], [3, 0, 0, 6, 1, 2], 11, ZZ) == [1, 0, + 3, 2, 4, 3, 1, 2, 0] + + assert gf_mul([2, 0, 0, 1, 7], [2, 0, 0, 1, 7], 11, ZZ) == [4, 0, + 0, 4, 6, 0, 1, 3, 5] + + assert gf_sqr([], 11, ZZ) == [] + assert gf_sqr([2], 11, ZZ) == [4] + assert gf_sqr([1, 2], 11, ZZ) == [1, 4, 4] + + assert gf_sqr([2, 0, 0, 1, 7], 11, ZZ) == [4, 0, 0, 4, 6, 0, 1, 3, 5] + + +def test_gf_division(): + raises(ZeroDivisionError, lambda: gf_div([1, 2, 3], [], 11, ZZ)) + raises(ZeroDivisionError, lambda: gf_rem([1, 2, 3], [], 11, ZZ)) + raises(ZeroDivisionError, lambda: gf_quo([1, 2, 3], [], 11, ZZ)) + raises(ZeroDivisionError, lambda: gf_quo([1, 2, 3], [], 11, ZZ)) + + assert gf_div([1], [1, 2, 3], 7, ZZ) == ([], [1]) + assert gf_rem([1], [1, 2, 3], 7, ZZ) == [1] + assert gf_quo([1], [1, 2, 3], 7, ZZ) == [] + + f = ZZ.map([5, 4, 3, 2, 1, 0]) + g = ZZ.map([1, 2, 3]) + q = [5, 1, 0, 6] + r = [3, 3] + + assert gf_div(f, g, 7, ZZ) == (q, r) + assert gf_rem(f, g, 7, ZZ) == r + assert gf_quo(f, g, 7, ZZ) == q + + raises(ExactQuotientFailed, lambda: gf_exquo(f, g, 7, ZZ)) + + f = ZZ.map([5, 4, 3, 2, 1, 0]) + g = ZZ.map([1, 2, 3, 0]) + q = [5, 1, 0] + r = [6, 1, 0] + + assert gf_div(f, g, 7, ZZ) == (q, r) + assert gf_rem(f, g, 7, ZZ) == r + assert gf_quo(f, g, 7, ZZ) == q + + raises(ExactQuotientFailed, lambda: gf_exquo(f, g, 7, ZZ)) + + assert gf_quo(ZZ.map([1, 2, 1]), ZZ.map([1, 1]), 11, ZZ) == [1, 1] + + +def test_gf_shift(): + f = [1, 2, 3, 4, 5] + + assert gf_lshift([], 5, ZZ) == [] + assert gf_rshift([], 5, ZZ) == ([], []) + + assert gf_lshift(f, 1, ZZ) == [1, 2, 3, 4, 5, 0] + assert gf_lshift(f, 2, ZZ) == [1, 2, 3, 4, 5, 0, 0] + + assert gf_rshift(f, 0, ZZ) == (f, []) + assert gf_rshift(f, 1, ZZ) == ([1, 2, 3, 4], [5]) + assert gf_rshift(f, 3, ZZ) == ([1, 2], [3, 4, 5]) + assert gf_rshift(f, 5, ZZ) == ([], f) + + +def test_gf_expand(): + F = [([1, 1], 2), ([1, 2], 3)] + + assert gf_expand(F, 11, ZZ) == [1, 8, 3, 5, 6, 8] + assert gf_expand((4, F), 11, ZZ) == [4, 10, 1, 9, 2, 10] + + +def test_gf_powering(): + assert gf_pow([1, 0, 0, 1, 8], 0, 11, ZZ) == [1] + assert gf_pow([1, 0, 0, 1, 8], 1, 11, ZZ) == [1, 0, 0, 1, 8] + assert gf_pow([1, 0, 0, 1, 8], 2, 11, ZZ) == [1, 0, 0, 2, 5, 0, 1, 5, 9] + + assert gf_pow([1, 0, 0, 1, 8], 5, 11, ZZ) == \ + [1, 0, 0, 5, 7, 0, 10, 6, 2, 10, 9, 6, 10, 6, 6, 0, 5, 2, 5, 9, 10] + + assert gf_pow([1, 0, 0, 1, 8], 8, 11, ZZ) == \ + [1, 0, 0, 8, 9, 0, 6, 8, 10, 1, 2, 5, 10, 7, 7, 9, 1, 2, 0, 0, 6, 2, + 5, 2, 5, 7, 7, 9, 10, 10, 7, 5, 5] + + assert gf_pow([1, 0, 0, 1, 8], 45, 11, ZZ) == \ + [ 1, 0, 0, 1, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, + 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 4, 10, 0, 0, 0, 0, 0, 0, + 10, 0, 0, 10, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, + 6, 0, 0, 6, 4, 0, 0, 0, 0, 0, 0, 8, 0, 0, 8, 9, 0, 0, 0, 0, 0, 0, + 10, 0, 0, 10, 3, 0, 0, 0, 0, 0, 0, 4, 0, 0, 4, 10, 0, 0, 0, 0, 0, 0, + 8, 0, 0, 8, 9, 0, 0, 0, 0, 0, 0, 9, 0, 0, 9, 6, 0, 0, 0, 0, 0, 0, + 3, 0, 0, 3, 2, 0, 0, 0, 0, 0, 0, 10, 0, 0, 10, 3, 0, 0, 0, 0, 0, 0, + 10, 0, 0, 10, 3, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 5, 0, 0, 0, 0, 0, 0, + 4, 0, 0, 4, 10] + + assert gf_pow_mod(ZZ.map([1, 0, 0, 1, 8]), 0, ZZ.map([2, 0, 7]), 11, ZZ) == [1] + assert gf_pow_mod(ZZ.map([1, 0, 0, 1, 8]), 1, ZZ.map([2, 0, 7]), 11, ZZ) == [1, 1] + assert gf_pow_mod(ZZ.map([1, 0, 0, 1, 8]), 2, ZZ.map([2, 0, 7]), 11, ZZ) == [2, 3] + assert gf_pow_mod(ZZ.map([1, 0, 0, 1, 8]), 5, ZZ.map([2, 0, 7]), 11, ZZ) == [7, 8] + assert gf_pow_mod(ZZ.map([1, 0, 0, 1, 8]), 8, ZZ.map([2, 0, 7]), 11, ZZ) == [1, 5] + assert gf_pow_mod(ZZ.map([1, 0, 0, 1, 8]), 45, ZZ.map([2, 0, 7]), 11, ZZ) == [5, 4] + + +def test_gf_gcdex(): + assert gf_gcdex(ZZ.map([]), ZZ.map([]), 11, ZZ) == ([1], [], []) + assert gf_gcdex(ZZ.map([2]), ZZ.map([]), 11, ZZ) == ([6], [], [1]) + assert gf_gcdex(ZZ.map([]), ZZ.map([2]), 11, ZZ) == ([], [6], [1]) + assert gf_gcdex(ZZ.map([2]), ZZ.map([2]), 11, ZZ) == ([], [6], [1]) + + assert gf_gcdex(ZZ.map([]), ZZ.map([3, 0]), 11, ZZ) == ([], [4], [1, 0]) + assert gf_gcdex(ZZ.map([3, 0]), ZZ.map([]), 11, ZZ) == ([4], [], [1, 0]) + + assert gf_gcdex(ZZ.map([3, 0]), ZZ.map([3, 0]), 11, ZZ) == ([], [4], [1, 0]) + + assert gf_gcdex(ZZ.map([1, 8, 7]), ZZ.map([1, 7, 1, 7]), 11, ZZ) == ([5, 6], [6], [1, 7]) + + +def test_gf_gcd(): + assert gf_gcd(ZZ.map([]), ZZ.map([]), 11, ZZ) == [] + assert gf_gcd(ZZ.map([2]), ZZ.map([]), 11, ZZ) == [1] + assert gf_gcd(ZZ.map([]), ZZ.map([2]), 11, ZZ) == [1] + assert gf_gcd(ZZ.map([2]), ZZ.map([2]), 11, ZZ) == [1] + + assert gf_gcd(ZZ.map([]), ZZ.map([1, 0]), 11, ZZ) == [1, 0] + assert gf_gcd(ZZ.map([1, 0]), ZZ.map([]), 11, ZZ) == [1, 0] + + assert gf_gcd(ZZ.map([3, 0]), ZZ.map([3, 0]), 11, ZZ) == [1, 0] + assert gf_gcd(ZZ.map([1, 8, 7]), ZZ.map([1, 7, 1, 7]), 11, ZZ) == [1, 7] + + +def test_gf_lcm(): + assert gf_lcm(ZZ.map([]), ZZ.map([]), 11, ZZ) == [] + assert gf_lcm(ZZ.map([2]), ZZ.map([]), 11, ZZ) == [] + assert gf_lcm(ZZ.map([]), ZZ.map([2]), 11, ZZ) == [] + assert gf_lcm(ZZ.map([2]), ZZ.map([2]), 11, ZZ) == [1] + + assert gf_lcm(ZZ.map([]), ZZ.map([1, 0]), 11, ZZ) == [] + assert gf_lcm(ZZ.map([1, 0]), ZZ.map([]), 11, ZZ) == [] + + assert gf_lcm(ZZ.map([3, 0]), ZZ.map([3, 0]), 11, ZZ) == [1, 0] + assert gf_lcm(ZZ.map([1, 8, 7]), ZZ.map([1, 7, 1, 7]), 11, ZZ) == [1, 8, 8, 8, 7] + + +def test_gf_cofactors(): + assert gf_cofactors(ZZ.map([]), ZZ.map([]), 11, ZZ) == ([], [], []) + assert gf_cofactors(ZZ.map([2]), ZZ.map([]), 11, ZZ) == ([1], [2], []) + assert gf_cofactors(ZZ.map([]), ZZ.map([2]), 11, ZZ) == ([1], [], [2]) + assert gf_cofactors(ZZ.map([2]), ZZ.map([2]), 11, ZZ) == ([1], [2], [2]) + + assert gf_cofactors(ZZ.map([]), ZZ.map([1, 0]), 11, ZZ) == ([1, 0], [], [1]) + assert gf_cofactors(ZZ.map([1, 0]), ZZ.map([]), 11, ZZ) == ([1, 0], [1], []) + + assert gf_cofactors(ZZ.map([3, 0]), ZZ.map([3, 0]), 11, ZZ) == ( + [1, 0], [3], [3]) + assert gf_cofactors(ZZ.map([1, 8, 7]), ZZ.map([1, 7, 1, 7]), 11, ZZ) == ( + ([1, 7], [1, 1], [1, 0, 1])) + + +def test_gf_diff(): + assert gf_diff([], 11, ZZ) == [] + assert gf_diff([7], 11, ZZ) == [] + + assert gf_diff([7, 3], 11, ZZ) == [7] + assert gf_diff([7, 3, 1], 11, ZZ) == [3, 3] + + assert gf_diff([1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], 11, ZZ) == [] + + +def test_gf_eval(): + assert gf_eval([], 4, 11, ZZ) == 0 + assert gf_eval([], 27, 11, ZZ) == 0 + assert gf_eval([7], 4, 11, ZZ) == 7 + assert gf_eval([7], 27, 11, ZZ) == 7 + + assert gf_eval([1, 0, 3, 2, 4, 3, 1, 2, 0], 0, 11, ZZ) == 0 + assert gf_eval([1, 0, 3, 2, 4, 3, 1, 2, 0], 4, 11, ZZ) == 9 + assert gf_eval([1, 0, 3, 2, 4, 3, 1, 2, 0], 27, 11, ZZ) == 5 + + assert gf_eval([4, 0, 0, 4, 6, 0, 1, 3, 5], 0, 11, ZZ) == 5 + assert gf_eval([4, 0, 0, 4, 6, 0, 1, 3, 5], 4, 11, ZZ) == 3 + assert gf_eval([4, 0, 0, 4, 6, 0, 1, 3, 5], 27, 11, ZZ) == 9 + + assert gf_multi_eval([3, 2, 1], [0, 1, 2, 3], 11, ZZ) == [1, 6, 6, 1] + + +def test_gf_compose(): + assert gf_compose([], [1, 0], 11, ZZ) == [] + assert gf_compose_mod([], [1, 0], [1, 0], 11, ZZ) == [] + + assert gf_compose([1], [], 11, ZZ) == [1] + assert gf_compose([1, 0], [], 11, ZZ) == [] + assert gf_compose([1, 0], [1, 0], 11, ZZ) == [1, 0] + + f = ZZ.map([1, 1, 4, 9, 1]) + g = ZZ.map([1, 1, 1]) + h = ZZ.map([1, 0, 0, 2]) + + assert gf_compose(g, h, 11, ZZ) == [1, 0, 0, 5, 0, 0, 7] + assert gf_compose_mod(g, h, f, 11, ZZ) == [3, 9, 6, 10] + + +def test_gf_trace_map(): + f = ZZ.map([1, 1, 4, 9, 1]) + a = [1, 1, 1] + c = ZZ.map([1, 0]) + b = gf_pow_mod(c, 11, f, 11, ZZ) + + assert gf_trace_map(a, b, c, 0, f, 11, ZZ) == \ + ([1, 1, 1], [1, 1, 1]) + assert gf_trace_map(a, b, c, 1, f, 11, ZZ) == \ + ([5, 2, 10, 3], [5, 3, 0, 4]) + assert gf_trace_map(a, b, c, 2, f, 11, ZZ) == \ + ([5, 9, 5, 3], [10, 1, 5, 7]) + assert gf_trace_map(a, b, c, 3, f, 11, ZZ) == \ + ([1, 10, 6, 0], [7]) + assert gf_trace_map(a, b, c, 4, f, 11, ZZ) == \ + ([1, 1, 1], [1, 1, 8]) + assert gf_trace_map(a, b, c, 5, f, 11, ZZ) == \ + ([5, 2, 10, 3], [5, 3, 0, 0]) + assert gf_trace_map(a, b, c, 11, f, 11, ZZ) == \ + ([1, 10, 6, 0], [10]) + + +def test_gf_irreducible(): + assert gf_irreducible_p(gf_irreducible(1, 11, ZZ), 11, ZZ) is True + assert gf_irreducible_p(gf_irreducible(2, 11, ZZ), 11, ZZ) is True + assert gf_irreducible_p(gf_irreducible(3, 11, ZZ), 11, ZZ) is True + assert gf_irreducible_p(gf_irreducible(4, 11, ZZ), 11, ZZ) is True + assert gf_irreducible_p(gf_irreducible(5, 11, ZZ), 11, ZZ) is True + assert gf_irreducible_p(gf_irreducible(6, 11, ZZ), 11, ZZ) is True + assert gf_irreducible_p(gf_irreducible(7, 11, ZZ), 11, ZZ) is True + + +def test_gf_irreducible_p(): + assert gf_irred_p_ben_or(ZZ.map([7]), 11, ZZ) is True + assert gf_irred_p_ben_or(ZZ.map([7, 3]), 11, ZZ) is True + assert gf_irred_p_ben_or(ZZ.map([7, 3, 1]), 11, ZZ) is False + + assert gf_irred_p_rabin(ZZ.map([7]), 11, ZZ) is True + assert gf_irred_p_rabin(ZZ.map([7, 3]), 11, ZZ) is True + assert gf_irred_p_rabin(ZZ.map([7, 3, 1]), 11, ZZ) is False + + config.setup('GF_IRRED_METHOD', 'ben-or') + + assert gf_irreducible_p(ZZ.map([7]), 11, ZZ) is True + assert gf_irreducible_p(ZZ.map([7, 3]), 11, ZZ) is True + assert gf_irreducible_p(ZZ.map([7, 3, 1]), 11, ZZ) is False + + config.setup('GF_IRRED_METHOD', 'rabin') + + assert gf_irreducible_p(ZZ.map([7]), 11, ZZ) is True + assert gf_irreducible_p(ZZ.map([7, 3]), 11, ZZ) is True + assert gf_irreducible_p(ZZ.map([7, 3, 1]), 11, ZZ) is False + + config.setup('GF_IRRED_METHOD', 'other') + raises(KeyError, lambda: gf_irreducible_p([7], 11, ZZ)) + config.setup('GF_IRRED_METHOD') + + f = ZZ.map([1, 9, 9, 13, 16, 15, 6, 7, 7, 7, 10]) + g = ZZ.map([1, 7, 16, 7, 15, 13, 13, 11, 16, 10, 9]) + + h = gf_mul(f, g, 17, ZZ) + + assert gf_irred_p_ben_or(f, 17, ZZ) is True + assert gf_irred_p_ben_or(g, 17, ZZ) is True + + assert gf_irred_p_ben_or(h, 17, ZZ) is False + + assert gf_irred_p_rabin(f, 17, ZZ) is True + assert gf_irred_p_rabin(g, 17, ZZ) is True + + assert gf_irred_p_rabin(h, 17, ZZ) is False + + +def test_gf_squarefree(): + assert gf_sqf_list([], 11, ZZ) == (0, []) + assert gf_sqf_list([1], 11, ZZ) == (1, []) + assert gf_sqf_list([1, 1], 11, ZZ) == (1, [([1, 1], 1)]) + + assert gf_sqf_p([], 11, ZZ) is True + assert gf_sqf_p([1], 11, ZZ) is True + assert gf_sqf_p([1, 1], 11, ZZ) is True + + f = gf_from_dict({11: 1, 0: 1}, 11, ZZ) + + assert gf_sqf_p(f, 11, ZZ) is False + + assert gf_sqf_list(f, 11, ZZ) == \ + (1, [([1, 1], 11)]) + + f = [1, 5, 8, 4] + + assert gf_sqf_p(f, 11, ZZ) is False + + assert gf_sqf_list(f, 11, ZZ) == \ + (1, [([1, 1], 1), + ([1, 2], 2)]) + + assert gf_sqf_part(f, 11, ZZ) == [1, 3, 2] + + f = [1, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0] + + assert gf_sqf_list(f, 3, ZZ) == \ + (1, [([1, 0], 1), + ([1, 1], 3), + ([1, 2], 6)]) + +def test_gf_frobenius_map(): + f = ZZ.map([2, 0, 1, 0, 2, 2, 0, 2, 2, 2]) + g = ZZ.map([1,1,0,2,0,1,0,2,0,1]) + p = 3 + b = gf_frobenius_monomial_base(g, p, ZZ) + h = gf_frobenius_map(f, g, b, p, ZZ) + h1 = gf_pow_mod(f, p, g, p, ZZ) + assert h == h1 + + +def test_gf_berlekamp(): + f = gf_from_int_poly([1, -3, 1, -3, -1, -3, 1], 11) + + Q = [[1, 0, 0, 0, 0, 0], + [3, 5, 8, 8, 6, 5], + [3, 6, 6, 1, 10, 0], + [9, 4, 10, 3, 7, 9], + [7, 8, 10, 0, 0, 8], + [8, 10, 7, 8, 10, 8]] + + V = [[1, 0, 0, 0, 0, 0], + [0, 1, 1, 1, 1, 0], + [0, 0, 7, 9, 0, 1]] + + assert gf_Qmatrix(f, 11, ZZ) == Q + assert gf_Qbasis(Q, 11, ZZ) == V + + assert gf_berlekamp(f, 11, ZZ) == \ + [[1, 1], [1, 5, 3], [1, 2, 3, 4]] + + f = ZZ.map([1, 0, 1, 0, 10, 10, 8, 2, 8]) + + Q = ZZ.map([[1, 0, 0, 0, 0, 0, 0, 0], + [2, 1, 7, 11, 10, 12, 5, 11], + [3, 6, 4, 3, 0, 4, 7, 2], + [4, 3, 6, 5, 1, 6, 2, 3], + [2, 11, 8, 8, 3, 1, 3, 11], + [6, 11, 8, 6, 2, 7, 10, 9], + [5, 11, 7, 10, 0, 11, 7, 12], + [3, 3, 12, 5, 0, 11, 9, 12]]) + + V = [[1, 0, 0, 0, 0, 0, 0, 0], + [0, 5, 5, 0, 9, 5, 1, 0], + [0, 9, 11, 9, 10, 12, 0, 1]] + + assert gf_Qmatrix(f, 13, ZZ) == Q + assert gf_Qbasis(Q, 13, ZZ) == V + + assert gf_berlekamp(f, 13, ZZ) == \ + [[1, 3], [1, 8, 4, 12], [1, 2, 3, 4, 6]] + + +def test_gf_ddf(): + f = gf_from_dict({15: ZZ(1), 0: ZZ(-1)}, 11, ZZ) + g = [([1, 0, 0, 0, 0, 10], 1), + ([1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1], 2)] + + assert gf_ddf_zassenhaus(f, 11, ZZ) == g + assert gf_ddf_shoup(f, 11, ZZ) == g + + f = gf_from_dict({63: ZZ(1), 0: ZZ(1)}, 2, ZZ) + g = [([1, 1], 1), + ([1, 1, 1], 2), + ([1, 1, 1, 1, 1, 1, 1], 3), + ([1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, + 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, + 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1], 6)] + + assert gf_ddf_zassenhaus(f, 2, ZZ) == g + assert gf_ddf_shoup(f, 2, ZZ) == g + + f = gf_from_dict({6: ZZ(1), 5: ZZ(-1), 4: ZZ(1), 3: ZZ(1), 1: ZZ(-1)}, 3, ZZ) + g = [([1, 1, 0], 1), + ([1, 1, 0, 1, 2], 2)] + + assert gf_ddf_zassenhaus(f, 3, ZZ) == g + assert gf_ddf_shoup(f, 3, ZZ) == g + + f = ZZ.map([1, 2, 5, 26, 677, 436, 791, 325, 456, 24, 577]) + g = [([1, 701], 1), + ([1, 110, 559, 532, 694, 151, 110, 70, 735, 122], 9)] + + assert gf_ddf_zassenhaus(f, 809, ZZ) == g + assert gf_ddf_shoup(f, 809, ZZ) == g + + p = ZZ(nextprime(int((2**15 * pi).evalf()))) + f = gf_from_dict({15: 1, 1: 1, 0: 1}, p, ZZ) + g = [([1, 22730, 68144], 2), + ([1, 64876, 83977, 10787, 12561, 68608, 52650, 88001, 84356], 4), + ([1, 15347, 95022, 84569, 94508, 92335], 5)] + + assert gf_ddf_zassenhaus(f, p, ZZ) == g + assert gf_ddf_shoup(f, p, ZZ) == g + + +def test_gf_edf(): + f = ZZ.map([1, 1, 0, 1, 2]) + g = ZZ.map([[1, 0, 1], [1, 1, 2]]) + + assert gf_edf_zassenhaus(f, 2, 3, ZZ) == g + assert gf_edf_shoup(f, 2, 3, ZZ) == g + + +def test_issue_23174(): + f = ZZ.map([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]) + g = ZZ.map([[1, 0, 0, 1, 1, 1, 0, 0, 1], [1, 1, 1, 0, 1, 0, 1, 1, 1]]) + + assert gf_edf_zassenhaus(f, 8, 2, ZZ) == g + + +def test_gf_factor(): + assert gf_factor([], 11, ZZ) == (0, []) + assert gf_factor([1], 11, ZZ) == (1, []) + assert gf_factor([1, 1], 11, ZZ) == (1, [([1, 1], 1)]) + + assert gf_factor_sqf([], 11, ZZ) == (0, []) + assert gf_factor_sqf([1], 11, ZZ) == (1, []) + assert gf_factor_sqf([1, 1], 11, ZZ) == (1, [[1, 1]]) + + config.setup('GF_FACTOR_METHOD', 'berlekamp') + + assert gf_factor_sqf([], 11, ZZ) == (0, []) + assert gf_factor_sqf([1], 11, ZZ) == (1, []) + assert gf_factor_sqf([1, 1], 11, ZZ) == (1, [[1, 1]]) + + config.setup('GF_FACTOR_METHOD', 'zassenhaus') + + assert gf_factor_sqf([], 11, ZZ) == (0, []) + assert gf_factor_sqf([1], 11, ZZ) == (1, []) + assert gf_factor_sqf([1, 1], 11, ZZ) == (1, [[1, 1]]) + + config.setup('GF_FACTOR_METHOD', 'shoup') + + assert gf_factor_sqf(ZZ.map([]), 11, ZZ) == (0, []) + assert gf_factor_sqf(ZZ.map([1]), 11, ZZ) == (1, []) + assert gf_factor_sqf(ZZ.map([1, 1]), 11, ZZ) == (1, [[1, 1]]) + + f, p = ZZ.map([1, 0, 0, 1, 0]), 2 + + g = (1, [([1, 0], 1), + ([1, 1], 1), + ([1, 1, 1], 1)]) + + config.setup('GF_FACTOR_METHOD', 'berlekamp') + assert gf_factor(f, p, ZZ) == g + + config.setup('GF_FACTOR_METHOD', 'zassenhaus') + assert gf_factor(f, p, ZZ) == g + + config.setup('GF_FACTOR_METHOD', 'shoup') + assert gf_factor(f, p, ZZ) == g + + g = (1, [[1, 0], + [1, 1], + [1, 1, 1]]) + + config.setup('GF_FACTOR_METHOD', 'berlekamp') + assert gf_factor_sqf(f, p, ZZ) == g + + config.setup('GF_FACTOR_METHOD', 'zassenhaus') + assert gf_factor_sqf(f, p, ZZ) == g + + config.setup('GF_FACTOR_METHOD', 'shoup') + assert gf_factor_sqf(f, p, ZZ) == g + + f, p = gf_from_int_poly([1, -3, 1, -3, -1, -3, 1], 11), 11 + + g = (1, [([1, 1], 1), + ([1, 5, 3], 1), + ([1, 2, 3, 4], 1)]) + + config.setup('GF_FACTOR_METHOD', 'berlekamp') + assert gf_factor(f, p, ZZ) == g + + config.setup('GF_FACTOR_METHOD', 'zassenhaus') + assert gf_factor(f, p, ZZ) == g + + config.setup('GF_FACTOR_METHOD', 'shoup') + assert gf_factor(f, p, ZZ) == g + + f, p = [1, 5, 8, 4], 11 + + g = (1, [([1, 1], 1), ([1, 2], 2)]) + + config.setup('GF_FACTOR_METHOD', 'berlekamp') + assert gf_factor(f, p, ZZ) == g + + config.setup('GF_FACTOR_METHOD', 'zassenhaus') + assert gf_factor(f, p, ZZ) == g + + config.setup('GF_FACTOR_METHOD', 'shoup') + assert gf_factor(f, p, ZZ) == g + + f, p = [1, 1, 10, 1, 0, 10, 10, 10, 0, 0], 11 + + g = (1, [([1, 0], 2), ([1, 9, 5], 1), ([1, 3, 0, 8, 5, 2], 1)]) + + config.setup('GF_FACTOR_METHOD', 'berlekamp') + assert gf_factor(f, p, ZZ) == g + + config.setup('GF_FACTOR_METHOD', 'zassenhaus') + assert gf_factor(f, p, ZZ) == g + + config.setup('GF_FACTOR_METHOD', 'shoup') + assert gf_factor(f, p, ZZ) == g + + f, p = gf_from_dict({32: 1, 0: 1}, 11, ZZ), 11 + + g = (1, [([1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 10], 1), + ([1, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 10], 1)]) + + config.setup('GF_FACTOR_METHOD', 'berlekamp') + assert gf_factor(f, p, ZZ) == g + + config.setup('GF_FACTOR_METHOD', 'zassenhaus') + assert gf_factor(f, p, ZZ) == g + + config.setup('GF_FACTOR_METHOD', 'shoup') + assert gf_factor(f, p, ZZ) == g + + f, p = gf_from_dict({32: ZZ(8), 0: ZZ(5)}, 11, ZZ), 11 + + g = (8, [([1, 3], 1), + ([1, 8], 1), + ([1, 0, 9], 1), + ([1, 2, 2], 1), + ([1, 9, 2], 1), + ([1, 0, 5, 0, 7], 1), + ([1, 0, 6, 0, 7], 1), + ([1, 0, 0, 0, 1, 0, 0, 0, 6], 1), + ([1, 0, 0, 0, 10, 0, 0, 0, 6], 1)]) + + config.setup('GF_FACTOR_METHOD', 'berlekamp') + assert gf_factor(f, p, ZZ) == g + + config.setup('GF_FACTOR_METHOD', 'zassenhaus') + assert gf_factor(f, p, ZZ) == g + + config.setup('GF_FACTOR_METHOD', 'shoup') + assert gf_factor(f, p, ZZ) == g + + f, p = gf_from_dict({63: ZZ(8), 0: ZZ(5)}, 11, ZZ), 11 + + g = (8, [([1, 7], 1), + ([1, 4, 5], 1), + ([1, 6, 8, 2], 1), + ([1, 9, 9, 2], 1), + ([1, 0, 0, 9, 0, 0, 4], 1), + ([1, 2, 0, 8, 4, 6, 4], 1), + ([1, 2, 3, 8, 0, 6, 4], 1), + ([1, 2, 6, 0, 8, 4, 4], 1), + ([1, 3, 3, 1, 6, 8, 4], 1), + ([1, 5, 6, 0, 8, 6, 4], 1), + ([1, 6, 2, 7, 9, 8, 4], 1), + ([1, 10, 4, 7, 10, 7, 4], 1), + ([1, 10, 10, 1, 4, 9, 4], 1)]) + + config.setup('GF_FACTOR_METHOD', 'berlekamp') + assert gf_factor(f, p, ZZ) == g + + config.setup('GF_FACTOR_METHOD', 'zassenhaus') + assert gf_factor(f, p, ZZ) == g + + config.setup('GF_FACTOR_METHOD', 'shoup') + assert gf_factor(f, p, ZZ) == g + + # Gathen polynomials: x**n + x + 1 (mod p > 2**n * pi) + + p = ZZ(nextprime(int((2**15 * pi).evalf()))) + f = gf_from_dict({15: 1, 1: 1, 0: 1}, p, ZZ) + + assert gf_sqf_p(f, p, ZZ) is True + + g = (1, [([1, 22730, 68144], 1), + ([1, 81553, 77449, 86810, 4724], 1), + ([1, 86276, 56779, 14859, 31575], 1), + ([1, 15347, 95022, 84569, 94508, 92335], 1)]) + + config.setup('GF_FACTOR_METHOD', 'zassenhaus') + assert gf_factor(f, p, ZZ) == g + + config.setup('GF_FACTOR_METHOD', 'shoup') + assert gf_factor(f, p, ZZ) == g + + g = (1, [[1, 22730, 68144], + [1, 81553, 77449, 86810, 4724], + [1, 86276, 56779, 14859, 31575], + [1, 15347, 95022, 84569, 94508, 92335]]) + + config.setup('GF_FACTOR_METHOD', 'zassenhaus') + assert gf_factor_sqf(f, p, ZZ) == g + + config.setup('GF_FACTOR_METHOD', 'shoup') + assert gf_factor_sqf(f, p, ZZ) == g + + # Shoup polynomials: f = a_0 x**n + a_1 x**(n-1) + ... + a_n + # (mod p > 2**(n-2) * pi), where a_n = a_{n-1}**2 + 1, a_0 = 1 + + p = ZZ(nextprime(int((2**4 * pi).evalf()))) + f = ZZ.map([1, 2, 5, 26, 41, 39, 38]) + + assert gf_sqf_p(f, p, ZZ) is True + + g = (1, [([1, 44, 26], 1), + ([1, 11, 25, 18, 30], 1)]) + + config.setup('GF_FACTOR_METHOD', 'zassenhaus') + assert gf_factor(f, p, ZZ) == g + + config.setup('GF_FACTOR_METHOD', 'shoup') + assert gf_factor(f, p, ZZ) == g + + g = (1, [[1, 44, 26], + [1, 11, 25, 18, 30]]) + + config.setup('GF_FACTOR_METHOD', 'zassenhaus') + assert gf_factor_sqf(f, p, ZZ) == g + + config.setup('GF_FACTOR_METHOD', 'shoup') + assert gf_factor_sqf(f, p, ZZ) == g + + config.setup('GF_FACTOR_METHOD', 'other') + raises(KeyError, lambda: gf_factor([1, 1], 11, ZZ)) + config.setup('GF_FACTOR_METHOD') + + +def test_gf_csolve(): + assert gf_value([1, 7, 2, 4], 11) == 2204 + + assert linear_congruence(4, 3, 5) == [2] + assert linear_congruence(0, 3, 5) == [] + assert linear_congruence(6, 1, 4) == [] + assert linear_congruence(0, 5, 5) == [0, 1, 2, 3, 4] + assert linear_congruence(3, 12, 15) == [4, 9, 14] + assert linear_congruence(6, 0, 18) == [0, 3, 6, 9, 12, 15] + # _csolve_prime_las_vegas + assert _csolve_prime_las_vegas([2, 3, 1], 5) == [2, 4] + assert _csolve_prime_las_vegas([2, 0, 1], 5) == [] + from sympy.ntheory import primerange + for p in primerange(2, 100): + # f = x**(p-1) - 1 + f = gf_sub_ground(gf_pow([1, 0], p - 1, p, ZZ), 1, p, ZZ) + assert _csolve_prime_las_vegas(f, p) == list(range(1, p)) + # with power = 1 + assert csolve_prime([1, 3, 2, 17], 7) == [3] + assert csolve_prime([1, 3, 1, 5], 5) == [0, 1] + assert csolve_prime([3, 6, 9, 3], 3) == [0, 1, 2] + # with power > 1 + assert csolve_prime( + [1, 1, 223], 3, 4) == [4, 13, 22, 31, 40, 49, 58, 67, 76] + assert csolve_prime([3, 5, 2, 25], 5, 3) == [16, 50, 99] + assert csolve_prime([3, 2, 2, 49], 7, 3) == [147, 190, 234] + + assert gf_csolve([1, 1, 7], 189) == [13, 49, 76, 112, 139, 175] + assert gf_csolve([1, 3, 4, 1, 30], 60) == [10, 30] + assert gf_csolve([1, 1, 7], 15) == [] diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_groebnertools.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_groebnertools.py new file mode 100644 index 0000000000000000000000000000000000000000..b7d0fc112047ac26f67d096db02eb8a1c91cab89 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_groebnertools.py @@ -0,0 +1,533 @@ +"""Tests for Groebner bases. """ + +from sympy.polys.groebnertools import ( + groebner, sig, sig_key, + lbp, lbp_key, critical_pair, + cp_key, is_rewritable_or_comparable, + Sign, Polyn, Num, s_poly, f5_reduce, + groebner_lcm, groebner_gcd, is_groebner, + is_reduced +) + +from sympy.polys.fglmtools import _representing_matrices +from sympy.polys.orderings import lex, grlex + +from sympy.polys.rings import ring, xring +from sympy.polys.domains import ZZ, QQ + +from sympy.testing.pytest import slow +from sympy.polys import polyconfig as config + +def _do_test_groebner(): + R, x,y = ring("x,y", QQ, lex) + f = x**2 + 2*x*y**2 + g = x*y + 2*y**3 - 1 + + assert groebner([f, g], R) == [x, y**3 - QQ(1,2)] + + R, y,x = ring("y,x", QQ, lex) + f = 2*x**2*y + y**2 + g = 2*x**3 + x*y - 1 + + assert groebner([f, g], R) == [y, x**3 - QQ(1,2)] + + R, x,y,z = ring("x,y,z", QQ, lex) + f = x - z**2 + g = y - z**3 + + assert groebner([f, g], R) == [f, g] + + R, x,y = ring("x,y", QQ, grlex) + f = x**3 - 2*x*y + g = x**2*y + x - 2*y**2 + + assert groebner([f, g], R) == [x**2, x*y, -QQ(1,2)*x + y**2] + + R, x,y,z = ring("x,y,z", QQ, lex) + f = -x**2 + y + g = -x**3 + z + + assert groebner([f, g], R) == [x**2 - y, x*y - z, x*z - y**2, y**3 - z**2] + + R, x,y,z = ring("x,y,z", QQ, grlex) + f = -x**2 + y + g = -x**3 + z + + assert groebner([f, g], R) == [y**3 - z**2, x**2 - y, x*y - z, x*z - y**2] + + R, x,y,z = ring("x,y,z", QQ, lex) + f = -x**2 + z + g = -x**3 + y + + assert groebner([f, g], R) == [x**2 - z, x*y - z**2, x*z - y, y**2 - z**3] + + R, x,y,z = ring("x,y,z", QQ, grlex) + f = -x**2 + z + g = -x**3 + y + + assert groebner([f, g], R) == [-y**2 + z**3, x**2 - z, x*y - z**2, x*z - y] + + R, x,y,z = ring("x,y,z", QQ, lex) + f = x - y**2 + g = -y**3 + z + + assert groebner([f, g], R) == [x - y**2, y**3 - z] + + R, x,y,z = ring("x,y,z", QQ, grlex) + f = x - y**2 + g = -y**3 + z + + assert groebner([f, g], R) == [x**2 - y*z, x*y - z, -x + y**2] + + R, x,y,z = ring("x,y,z", QQ, lex) + f = x - z**2 + g = y - z**3 + + assert groebner([f, g], R) == [x - z**2, y - z**3] + + R, x,y,z = ring("x,y,z", QQ, grlex) + f = x - z**2 + g = y - z**3 + + assert groebner([f, g], R) == [x**2 - y*z, x*z - y, -x + z**2] + + R, x,y,z = ring("x,y,z", QQ, lex) + f = -y**2 + z + g = x - y**3 + + assert groebner([f, g], R) == [x - y*z, y**2 - z] + + R, x,y,z = ring("x,y,z", QQ, grlex) + f = -y**2 + z + g = x - y**3 + + assert groebner([f, g], R) == [-x**2 + z**3, x*y - z**2, y**2 - z, -x + y*z] + + R, x,y,z = ring("x,y,z", QQ, lex) + f = y - z**2 + g = x - z**3 + + assert groebner([f, g], R) == [x - z**3, y - z**2] + + R, x,y,z = ring("x,y,z", QQ, grlex) + f = y - z**2 + g = x - z**3 + + assert groebner([f, g], R) == [-x**2 + y**3, x*z - y**2, -x + y*z, -y + z**2] + + R, x,y,z = ring("x,y,z", QQ, lex) + f = 4*x**2*y**2 + 4*x*y + 1 + g = x**2 + y**2 - 1 + + assert groebner([f, g], R) == [ + x - 4*y**7 + 8*y**5 - 7*y**3 + 3*y, + y**8 - 2*y**6 + QQ(3,2)*y**4 - QQ(1,2)*y**2 + QQ(1,16), + ] + +def test_groebner_buchberger(): + with config.using(groebner='buchberger'): + _do_test_groebner() + +def test_groebner_f5b(): + with config.using(groebner='f5b'): + _do_test_groebner() + +def _do_test_benchmark_minpoly(): + R, x,y,z = ring("x,y,z", QQ, lex) + + F = [x**3 + x + 1, y**2 + y + 1, (x + y) * z - (x**2 + y)] + G = [x + QQ(155,2067)*z**5 - QQ(355,689)*z**4 + QQ(6062,2067)*z**3 - QQ(3687,689)*z**2 + QQ(6878,2067)*z - QQ(25,53), + y + QQ(4,53)*z**5 - QQ(91,159)*z**4 + QQ(523,159)*z**3 - QQ(387,53)*z**2 + QQ(1043,159)*z - QQ(308,159), + z**6 - 7*z**5 + 41*z**4 - 82*z**3 + 89*z**2 - 46*z + 13] + + assert groebner(F, R) == G + +def test_benchmark_minpoly_buchberger(): + with config.using(groebner='buchberger'): + _do_test_benchmark_minpoly() + +def test_benchmark_minpoly_f5b(): + with config.using(groebner='f5b'): + _do_test_benchmark_minpoly() + + +def test_benchmark_coloring(): + V = range(1, 12 + 1) + E = [(1, 2), (2, 3), (1, 4), (1, 6), (1, 12), (2, 5), (2, 7), (3, 8), (3, 10), + (4, 11), (4, 9), (5, 6), (6, 7), (7, 8), (8, 9), (9, 10), (10, 11), + (11, 12), (5, 12), (5, 9), (6, 10), (7, 11), (8, 12), (3, 4)] + + R, V = xring([ "x%d" % v for v in V ], QQ, lex) + E = [(V[i - 1], V[j - 1]) for i, j in E] + + x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12 = V + + I3 = [x**3 - 1 for x in V] + Ig = [x**2 + x*y + y**2 for x, y in E] + + I = I3 + Ig + + assert groebner(I[:-1], R) == [ + x1 + x11 + x12, + x2 - x11, + x3 - x12, + x4 - x12, + x5 + x11 + x12, + x6 - x11, + x7 - x12, + x8 + x11 + x12, + x9 - x11, + x10 + x11 + x12, + x11**2 + x11*x12 + x12**2, + x12**3 - 1, + ] + + assert groebner(I, R) == [1] + + +def _do_test_benchmark_katsura_3(): + R, x0,x1,x2 = ring("x:3", ZZ, lex) + I = [x0 + 2*x1 + 2*x2 - 1, + x0**2 + 2*x1**2 + 2*x2**2 - x0, + 2*x0*x1 + 2*x1*x2 - x1] + + assert groebner(I, R) == [ + -7 + 7*x0 + 8*x2 + 158*x2**2 - 420*x2**3, + 7*x1 + 3*x2 - 79*x2**2 + 210*x2**3, + x2 + x2**2 - 40*x2**3 + 84*x2**4, + ] + + R, x0,x1,x2 = ring("x:3", ZZ, grlex) + I = [ i.set_ring(R) for i in I ] + + assert groebner(I, R) == [ + 7*x1 + 3*x2 - 79*x2**2 + 210*x2**3, + -x1 + x2 - 3*x2**2 + 5*x1**2, + -x1 - 4*x2 + 10*x1*x2 + 12*x2**2, + -1 + x0 + 2*x1 + 2*x2, + ] + +def test_benchmark_katsura3_buchberger(): + with config.using(groebner='buchberger'): + _do_test_benchmark_katsura_3() + +def test_benchmark_katsura3_f5b(): + with config.using(groebner='f5b'): + _do_test_benchmark_katsura_3() + +def _do_test_benchmark_katsura_4(): + R, x0,x1,x2,x3 = ring("x:4", ZZ, lex) + I = [x0 + 2*x1 + 2*x2 + 2*x3 - 1, + x0**2 + 2*x1**2 + 2*x2**2 + 2*x3**2 - x0, + 2*x0*x1 + 2*x1*x2 + 2*x2*x3 - x1, + x1**2 + 2*x0*x2 + 2*x1*x3 - x2] + + assert groebner(I, R) == [ + 5913075*x0 - 159690237696*x3**7 + 31246269696*x3**6 + 27439610544*x3**5 - 6475723368*x3**4 - 838935856*x3**3 + 275119624*x3**2 + 4884038*x3 - 5913075, + 1971025*x1 - 97197721632*x3**7 + 73975630752*x3**6 - 12121915032*x3**5 - 2760941496*x3**4 + 814792828*x3**3 - 1678512*x3**2 - 9158924*x3, + 5913075*x2 + 371438283744*x3**7 - 237550027104*x3**6 + 22645939824*x3**5 + 11520686172*x3**4 - 2024910556*x3**3 - 132524276*x3**2 + 30947828*x3, + 128304*x3**8 - 93312*x3**7 + 15552*x3**6 + 3144*x3**5 - + 1120*x3**4 + 36*x3**3 + 15*x3**2 - x3, + ] + + R, x0,x1,x2,x3 = ring("x:4", ZZ, grlex) + I = [ i.set_ring(R) for i in I ] + + assert groebner(I, R) == [ + 393*x1 - 4662*x2**2 + 4462*x2*x3 - 59*x2 + 224532*x3**4 - 91224*x3**3 - 678*x3**2 + 2046*x3, + -x1 + 196*x2**3 - 21*x2**2 + 60*x2*x3 - 18*x2 - 168*x3**3 + 83*x3**2 - 9*x3, + -6*x1 + 1134*x2**2*x3 - 189*x2**2 - 466*x2*x3 + 32*x2 - 630*x3**3 + 57*x3**2 + 51*x3, + 33*x1 + 63*x2**2 + 2268*x2*x3**2 - 188*x2*x3 + 34*x2 + 2520*x3**3 - 849*x3**2 + 3*x3, + 7*x1**2 - x1 - 7*x2**2 - 24*x2*x3 + 3*x2 - 15*x3**2 + 5*x3, + 14*x1*x2 - x1 + 14*x2**2 + 18*x2*x3 - 4*x2 + 6*x3**2 - 2*x3, + 14*x1*x3 - x1 + 7*x2**2 + 32*x2*x3 - 4*x2 + 27*x3**2 - 9*x3, + x0 + 2*x1 + 2*x2 + 2*x3 - 1, + ] + +def test_benchmark_kastura_4_buchberger(): + with config.using(groebner='buchberger'): + _do_test_benchmark_katsura_4() + +def test_benchmark_kastura_4_f5b(): + with config.using(groebner='f5b'): + _do_test_benchmark_katsura_4() + +def _do_test_benchmark_czichowski(): + R, x,t = ring("x,t", ZZ, lex) + I = [9*x**8 + 36*x**7 - 32*x**6 - 252*x**5 - 78*x**4 + 468*x**3 + 288*x**2 - 108*x + 9, + (-72 - 72*t)*x**7 + (-256 - 252*t)*x**6 + (192 + 192*t)*x**5 + (1280 + 1260*t)*x**4 + (312 + 312*t)*x**3 + (-404*t)*x**2 + (-576 - 576*t)*x + 96 + 108*t] + + assert groebner(I, R) == [ + 3725588592068034903797967297424801242396746870413359539263038139343329273586196480000*x - + 160420835591776763325581422211936558925462474417709511019228211783493866564923546661604487873*t**7 - + 1406108495478033395547109582678806497509499966197028487131115097902188374051595011248311352864*t**6 - + 5241326875850889518164640374668786338033653548841427557880599579174438246266263602956254030352*t**5 - + 10758917262823299139373269714910672770004760114329943852726887632013485035262879510837043892416*t**4 - + 13119383576444715672578819534846747735372132018341964647712009275306635391456880068261130581248*t**3 - + 9491412317016197146080450036267011389660653495578680036574753839055748080962214787557853941760*t**2 - + 3767520915562795326943800040277726397326609797172964377014046018280260848046603967211258368000*t - + 632314652371226552085897259159210286886724229880266931574701654721512325555116066073245696000, + 610733380717522355121*t**8 + + 6243748742141230639968*t**7 + + 27761407182086143225024*t**6 + + 70066148869420956398592*t**5 + + 109701225644313784229376*t**4 + + 109009005495588442152960*t**3 + + 67072101084384786432000*t**2 + + 23339979742629593088000*t + + 3513592776846090240000, + ] + + R, x,t = ring("x,t", ZZ, grlex) + I = [ i.set_ring(R) for i in I ] + + assert groebner(I, R) == [ + 16996618586000601590732959134095643086442*t**3*x - + 32936701459297092865176560282688198064839*t**3 + + 78592411049800639484139414821529525782364*t**2*x - + 120753953358671750165454009478961405619916*t**2 + + 120988399875140799712152158915653654637280*t*x - + 144576390266626470824138354942076045758736*t + + 60017634054270480831259316163620768960*x**2 + + 61976058033571109604821862786675242894400*x - + 56266268491293858791834120380427754600960, + 576689018321912327136790519059646508441672750656050290242749*t**4 + + 2326673103677477425562248201573604572527893938459296513327336*t**3 + + 110743790416688497407826310048520299245819959064297990236000*t**2*x + + 3308669114229100853338245486174247752683277925010505284338016*t**2 + + 323150205645687941261103426627818874426097912639158572428800*t*x + + 1914335199925152083917206349978534224695445819017286960055680*t + + 861662882561803377986838989464278045397192862768588480000*x**2 + + 235296483281783440197069672204341465480107019878814196672000*x + + 361850798943225141738895123621685122544503614946436727532800, + -117584925286448670474763406733005510014188341867*t**3 + + 68566565876066068463853874568722190223721653044*t**2*x - + 435970731348366266878180788833437896139920683940*t**2 + + 196297602447033751918195568051376792491869233408*t*x - + 525011527660010557871349062870980202067479780112*t + + 517905853447200553360289634770487684447317120*x**3 + + 569119014870778921949288951688799397569321920*x**2 + + 138877356748142786670127389526667463202210102080*x - + 205109210539096046121625447192779783475018619520, + -3725142681462373002731339445216700112264527*t**3 + + 583711207282060457652784180668273817487940*t**2*x - + 12381382393074485225164741437227437062814908*t**2 + + 151081054097783125250959636747516827435040*t*x**2 + + 1814103857455163948531448580501928933873280*t*x - + 13353115629395094645843682074271212731433648*t + + 236415091385250007660606958022544983766080*x**2 + + 1390443278862804663728298060085399578417600*x - + 4716885828494075789338754454248931750698880, + ] + +# NOTE: This is very slow (> 2 minutes on 3.4 GHz) without GMPY +@slow +def test_benchmark_czichowski_buchberger(): + with config.using(groebner='buchberger'): + _do_test_benchmark_czichowski() + +def test_benchmark_czichowski_f5b(): + with config.using(groebner='f5b'): + _do_test_benchmark_czichowski() + +def _do_test_benchmark_cyclic_4(): + R, a,b,c,d = ring("a,b,c,d", ZZ, lex) + + I = [a + b + c + d, + a*b + a*d + b*c + b*d, + a*b*c + a*b*d + a*c*d + b*c*d, + a*b*c*d - 1] + + assert groebner(I, R) == [ + 4*a + 3*d**9 - 4*d**5 - 3*d, + 4*b + 4*c - 3*d**9 + 4*d**5 + 7*d, + 4*c**2 + 3*d**10 - 4*d**6 - 3*d**2, + 4*c*d**4 + 4*c - d**9 + 4*d**5 + 5*d, d**12 - d**8 - d**4 + 1 + ] + + R, a,b,c,d = ring("a,b,c,d", ZZ, grlex) + I = [ i.set_ring(R) for i in I ] + + assert groebner(I, R) == [ + 3*b*c - c**2 + d**6 - 3*d**2, + -b + 3*c**2*d**3 - c - d**5 - 4*d, + -b + 3*c*d**4 + 2*c + 2*d**5 + 2*d, + c**4 + 2*c**2*d**2 - d**4 - 2, + c**3*d + c*d**3 + d**4 + 1, + b*c**2 - c**3 - c**2*d - 2*c*d**2 - d**3, + b**2 - c**2, b*d + c**2 + c*d + d**2, + a + b + c + d + ] + +def test_benchmark_cyclic_4_buchberger(): + with config.using(groebner='buchberger'): + _do_test_benchmark_cyclic_4() + +def test_benchmark_cyclic_4_f5b(): + with config.using(groebner='f5b'): + _do_test_benchmark_cyclic_4() + +def test_sig_key(): + s1 = sig((0,) * 3, 2) + s2 = sig((1,) * 3, 4) + s3 = sig((2,) * 3, 2) + + assert sig_key(s1, lex) > sig_key(s2, lex) + assert sig_key(s2, lex) < sig_key(s3, lex) + + +def test_lbp_key(): + R, x,y,z,t = ring("x,y,z,t", ZZ, lex) + + p1 = lbp(sig((0,) * 4, 3), R.zero, 12) + p2 = lbp(sig((0,) * 4, 4), R.zero, 13) + p3 = lbp(sig((0,) * 4, 4), R.zero, 12) + + assert lbp_key(p1) > lbp_key(p2) + assert lbp_key(p2) < lbp_key(p3) + + +def test_critical_pair(): + # from cyclic4 with grlex + R, x,y,z,t = ring("x,y,z,t", QQ, grlex) + + p1 = (((0, 0, 0, 0), 4), y*z*t**2 + z**2*t**2 - t**4 - 1, 4) + q1 = (((0, 0, 0, 0), 2), -y**2 - y*t - z*t - t**2, 2) + + p2 = (((0, 0, 0, 2), 3), z**3*t**2 + z**2*t**3 - z - t, 5) + q2 = (((0, 0, 2, 2), 2), y*z + z*t**5 + z*t + t**6, 13) + + assert critical_pair(p1, q1, R) == ( + ((0, 0, 1, 2), 2), ((0, 0, 1, 2), QQ(-1, 1)), (((0, 0, 0, 0), 2), -y**2 - y*t - z*t - t**2, 2), + ((0, 1, 0, 0), 4), ((0, 1, 0, 0), QQ(1, 1)), (((0, 0, 0, 0), 4), y*z*t**2 + z**2*t**2 - t**4 - 1, 4) + ) + assert critical_pair(p2, q2, R) == ( + ((0, 0, 4, 2), 2), ((0, 0, 2, 0), QQ(1, 1)), (((0, 0, 2, 2), 2), y*z + z*t**5 + z*t + t**6, 13), + ((0, 0, 0, 5), 3), ((0, 0, 0, 3), QQ(1, 1)), (((0, 0, 0, 2), 3), z**3*t**2 + z**2*t**3 - z - t, 5) + ) + +def test_cp_key(): + # from cyclic4 with grlex + R, x,y,z,t = ring("x,y,z,t", QQ, grlex) + + p1 = (((0, 0, 0, 0), 4), y*z*t**2 + z**2*t**2 - t**4 - 1, 4) + q1 = (((0, 0, 0, 0), 2), -y**2 - y*t - z*t - t**2, 2) + + p2 = (((0, 0, 0, 2), 3), z**3*t**2 + z**2*t**3 - z - t, 5) + q2 = (((0, 0, 2, 2), 2), y*z + z*t**5 + z*t + t**6, 13) + + cp1 = critical_pair(p1, q1, R) + cp2 = critical_pair(p2, q2, R) + + assert cp_key(cp1, R) < cp_key(cp2, R) + + cp1 = critical_pair(p1, p2, R) + cp2 = critical_pair(q1, q2, R) + + assert cp_key(cp1, R) < cp_key(cp2, R) + + +def test_is_rewritable_or_comparable(): + # from katsura4 with grlex + R, x,y,z,t = ring("x,y,z,t", QQ, grlex) + + p = lbp(sig((0, 0, 2, 1), 2), R.zero, 2) + B = [lbp(sig((0, 0, 0, 1), 2), QQ(2,45)*y**2 + QQ(1,5)*y*z + QQ(5,63)*y*t + z**2*t + QQ(4,45)*z**2 + QQ(76,35)*z*t**2 - QQ(32,105)*z*t + QQ(13,7)*t**3 - QQ(13,21)*t**2, 6)] + + # rewritable: + assert is_rewritable_or_comparable(Sign(p), Num(p), B) is True + + p = lbp(sig((0, 1, 1, 0), 2), R.zero, 7) + B = [lbp(sig((0, 0, 0, 0), 3), QQ(10,3)*y*z + QQ(4,3)*y*t - QQ(1,3)*y + 4*z**2 + QQ(22,3)*z*t - QQ(4,3)*z + 4*t**2 - QQ(4,3)*t, 3)] + + # comparable: + assert is_rewritable_or_comparable(Sign(p), Num(p), B) is True + + +def test_f5_reduce(): + # katsura3 with lex + R, x,y,z = ring("x,y,z", QQ, lex) + + F = [(((0, 0, 0), 1), x + 2*y + 2*z - 1, 1), + (((0, 0, 0), 2), 6*y**2 + 8*y*z - 2*y + 6*z**2 - 2*z, 2), + (((0, 0, 0), 3), QQ(10,3)*y*z - QQ(1,3)*y + 4*z**2 - QQ(4,3)*z, 3), + (((0, 0, 1), 2), y + 30*z**3 - QQ(79,7)*z**2 + QQ(3,7)*z, 4), + (((0, 0, 2), 2), z**4 - QQ(10,21)*z**3 + QQ(1,84)*z**2 + QQ(1,84)*z, 5)] + + cp = critical_pair(F[0], F[1], R) + s = s_poly(cp) + + assert f5_reduce(s, F) == (((0, 2, 0), 1), R.zero, 1) + + s = lbp(sig(Sign(s)[0], 100), Polyn(s), Num(s)) + assert f5_reduce(s, F) == s + + +def test_representing_matrices(): + R, x,y = ring("x,y", QQ, grlex) + + basis = [(0, 0), (0, 1), (1, 0), (1, 1)] + F = [x**2 - x - 3*y + 1, -2*x + y**2 + y - 1] + + assert _representing_matrices(basis, F, R) == [ + [[QQ(0, 1), QQ(0, 1),-QQ(1, 1), QQ(3, 1)], + [QQ(0, 1), QQ(0, 1), QQ(3, 1),-QQ(4, 1)], + [QQ(1, 1), QQ(0, 1), QQ(1, 1), QQ(6, 1)], + [QQ(0, 1), QQ(1, 1), QQ(0, 1), QQ(1, 1)]], + [[QQ(0, 1), QQ(1, 1), QQ(0, 1),-QQ(2, 1)], + [QQ(1, 1),-QQ(1, 1), QQ(0, 1), QQ(6, 1)], + [QQ(0, 1), QQ(2, 1), QQ(0, 1), QQ(3, 1)], + [QQ(0, 1), QQ(0, 1), QQ(1, 1),-QQ(1, 1)]]] + +def test_groebner_lcm(): + R, x,y,z = ring("x,y,z", ZZ) + + assert groebner_lcm(x**2 - y**2, x - y) == x**2 - y**2 + assert groebner_lcm(2*x**2 - 2*y**2, 2*x - 2*y) == 2*x**2 - 2*y**2 + + R, x,y,z = ring("x,y,z", QQ) + + assert groebner_lcm(x**2 - y**2, x - y) == x**2 - y**2 + assert groebner_lcm(2*x**2 - 2*y**2, 2*x - 2*y) == 2*x**2 - 2*y**2 + + R, x,y = ring("x,y", ZZ) + + assert groebner_lcm(x**2*y, x*y**2) == x**2*y**2 + + f = 2*x*y**5 - 3*x*y**4 - 2*x*y**3 + 3*x*y**2 + g = y**5 - 2*y**3 + y + h = 2*x*y**7 - 3*x*y**6 - 4*x*y**5 + 6*x*y**4 + 2*x*y**3 - 3*x*y**2 + + assert groebner_lcm(f, g) == h + + f = x**3 - 3*x**2*y - 9*x*y**2 - 5*y**3 + g = x**4 + 6*x**3*y + 12*x**2*y**2 + 10*x*y**3 + 3*y**4 + h = x**5 + x**4*y - 18*x**3*y**2 - 50*x**2*y**3 - 47*x*y**4 - 15*y**5 + + assert groebner_lcm(f, g) == h + +def test_groebner_gcd(): + R, x,y,z = ring("x,y,z", ZZ) + + assert groebner_gcd(x**2 - y**2, x - y) == x - y + assert groebner_gcd(2*x**2 - 2*y**2, 2*x - 2*y) == 2*x - 2*y + + R, x,y,z = ring("x,y,z", QQ) + + assert groebner_gcd(x**2 - y**2, x - y) == x - y + assert groebner_gcd(2*x**2 - 2*y**2, 2*x - 2*y) == x - y + +def test_is_groebner(): + R, x,y = ring("x,y", QQ, grlex) + valid_groebner = [x**2, x*y, -QQ(1,2)*x + y**2] + invalid_groebner = [x**3, x*y, -QQ(1,2)*x + y**2] + assert is_groebner(valid_groebner, R) is True + assert is_groebner(invalid_groebner, R) is False + +def test_is_reduced(): + R, x, y = ring("x,y", QQ, lex) + f = x**2 + 2*x*y**2 + g = x*y + 2*y**3 - 1 + assert is_reduced([f, g], R) == False + G = groebner([f, g], R) + assert is_reduced(G, R) == True diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_heuristicgcd.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_heuristicgcd.py new file mode 100644 index 0000000000000000000000000000000000000000..7ff6bd6ea4effbd49c5e942ea8925cfcca4ba162 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_heuristicgcd.py @@ -0,0 +1,152 @@ +from sympy.polys.rings import ring +from sympy.polys.domains import ZZ +from sympy.polys.heuristicgcd import heugcd + + +def test_heugcd_univariate_integers(): + R, x = ring("x", ZZ) + + f = x**4 + 8*x**3 + 21*x**2 + 22*x + 8 + g = x**3 + 6*x**2 + 11*x + 6 + + h = x**2 + 3*x + 2 + + cff = x**2 + 5*x + 4 + cfg = x + 3 + + assert heugcd(f, g) == (h, cff, cfg) + + f = x**4 - 4 + g = x**4 + 4*x**2 + 4 + + h = x**2 + 2 + + cff = x**2 - 2 + cfg = x**2 + 2 + + assert heugcd(f, g) == (h, cff, cfg) + + f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 + g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 + + h = 1 + + cff = f + cfg = g + + assert heugcd(f, g) == (h, cff, cfg) + + f = - 352518131239247345597970242177235495263669787845475025293906825864749649589178600387510272*x**49 \ + + 46818041807522713962450042363465092040687472354933295397472942006618953623327997952*x**42 \ + + 378182690892293941192071663536490788434899030680411695933646320291525827756032*x**35 \ + + 112806468807371824947796775491032386836656074179286744191026149539708928*x**28 \ + - 12278371209708240950316872681744825481125965781519138077173235712*x**21 \ + + 289127344604779611146960547954288113529690984687482920704*x**14 \ + + 19007977035740498977629742919480623972236450681*x**7 \ + + 311973482284542371301330321821976049 + + g = 365431878023781158602430064717380211405897160759702125019136*x**21 \ + + 197599133478719444145775798221171663643171734081650688*x**14 \ + - 9504116979659010018253915765478924103928886144*x**7 \ + - 311973482284542371301330321821976049 + + # TODO: assert heugcd(f, f.diff(x))[0] == g + + f = 1317378933230047068160*x + 2945748836994210856960 + g = 120352542776360960*x + 269116466014453760 + + h = 120352542776360960*x + 269116466014453760 + cff = 10946 + cfg = 1 + + assert heugcd(f, g) == (h, cff, cfg) + +def test_heugcd_multivariate_integers(): + R, x, y = ring("x,y", ZZ) + + f, g = 2*x**2 + 4*x + 2, x + 1 + assert heugcd(f, g) == (x + 1, 2*x + 2, 1) + + f, g = x + 1, 2*x**2 + 4*x + 2 + assert heugcd(f, g) == (x + 1, 1, 2*x + 2) + + R, x, y, z, u = ring("x,y,z,u", ZZ) + + f, g = u**2 + 2*u + 1, 2*u + 2 + assert heugcd(f, g) == (u + 1, u + 1, 2) + + f, g = z**2*u**2 + 2*z**2*u + z**2 + z*u + z, u**2 + 2*u + 1 + h, cff, cfg = u + 1, z**2*u + z**2 + z, u + 1 + + assert heugcd(f, g) == (h, cff, cfg) + assert heugcd(g, f) == (h, cfg, cff) + + R, x, y, z = ring("x,y,z", ZZ) + + f, g, h = R.fateman_poly_F_1() + H, cff, cfg = heugcd(f, g) + + assert H == h and H*cff == f and H*cfg == g + + R, x, y, z, u, v = ring("x,y,z,u,v", ZZ) + + f, g, h = R.fateman_poly_F_1() + H, cff, cfg = heugcd(f, g) + + assert H == h and H*cff == f and H*cfg == g + + R, x, y, z, u, v, a, b = ring("x,y,z,u,v,a,b", ZZ) + + f, g, h = R.fateman_poly_F_1() + H, cff, cfg = heugcd(f, g) + + assert H == h and H*cff == f and H*cfg == g + + R, x, y, z, u, v, a, b, c, d = ring("x,y,z,u,v,a,b,c,d", ZZ) + + f, g, h = R.fateman_poly_F_1() + H, cff, cfg = heugcd(f, g) + + assert H == h and H*cff == f and H*cfg == g + + R, x, y, z = ring("x,y,z", ZZ) + + f, g, h = R.fateman_poly_F_2() + H, cff, cfg = heugcd(f, g) + + assert H == h and H*cff == f and H*cfg == g + + f, g, h = R.fateman_poly_F_3() + H, cff, cfg = heugcd(f, g) + + assert H == h and H*cff == f and H*cfg == g + + R, x, y, z, t = ring("x,y,z,t", ZZ) + + f, g, h = R.fateman_poly_F_3() + H, cff, cfg = heugcd(f, g) + + assert H == h and H*cff == f and H*cfg == g + + +def test_issue_10996(): + R, x, y, z = ring("x,y,z", ZZ) + + f = 12*x**6*y**7*z**3 - 3*x**4*y**9*z**3 + 12*x**3*y**5*z**4 + g = -48*x**7*y**8*z**3 + 12*x**5*y**10*z**3 - 48*x**5*y**7*z**2 + \ + 36*x**4*y**7*z - 48*x**4*y**6*z**4 + 12*x**3*y**9*z**2 - 48*x**3*y**4 \ + - 9*x**2*y**9*z - 48*x**2*y**5*z**3 + 12*x*y**6 + 36*x*y**5*z**2 - 48*y**2*z + + H, cff, cfg = heugcd(f, g) + + assert H == 12*x**3*y**4 - 3*x*y**6 + 12*y**2*z + assert H*cff == f and H*cfg == g + + +def test_issue_25793(): + R, x = ring("x", ZZ) + f = x - 4851 # failure starts for values more than 4850 + g = f*(2*x + 1) + H, cff, cfg = R.dup_zz_heu_gcd(f, g) + assert H == f + # needs a test for dmp, too, that fails in master before this change diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_hypothesis.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_hypothesis.py new file mode 100644 index 0000000000000000000000000000000000000000..78c2369179c3f0ea4d34b8a7868417506177e3c5 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_hypothesis.py @@ -0,0 +1,36 @@ +from hypothesis import given +from hypothesis import strategies as st +from sympy.abc import x +from sympy.polys.polytools import Poly + + +def polys(*, nonzero=False, domain="ZZ"): + # This is a simple strategy, but sufficient the tests below + elems = {"ZZ": st.integers(), "QQ": st.fractions()} + coeff_st = st.lists(elems[domain]) + if nonzero: + coeff_st = coeff_st.filter(any) + return st.builds(Poly, coeff_st, st.just(x), domain=st.just(domain)) + + +@given(f=polys(), g=polys(), r=polys()) +def test_gcd_hypothesis(f, g, r): + gcd_1 = f.gcd(g) + gcd_2 = g.gcd(f) + assert gcd_1 == gcd_2 + + # multiply by r + gcd_3 = g.gcd(f + r * g) + assert gcd_1 == gcd_3 + + +@given(f_z=polys(), g_z=polys(nonzero=True)) +def test_poly_hypothesis_integers(f_z, g_z): + remainder_z = f_z.rem(g_z) + assert g_z.degree() >= remainder_z.degree() or remainder_z.degree() == 0 + + +@given(f_q=polys(domain="QQ"), g_q=polys(nonzero=True, domain="QQ")) +def test_poly_hypothesis_rationals(f_q, g_q): + remainder_q = f_q.rem(g_q) + assert g_q.degree() >= remainder_q.degree() or remainder_q.degree() == 0 diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_injections.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_injections.py new file mode 100644 index 0000000000000000000000000000000000000000..63a5537c94f00e52a3899c97f0d78bfadab78a67 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_injections.py @@ -0,0 +1,39 @@ +"""Tests for functions that inject symbols into the global namespace. """ + +from sympy.polys.rings import vring +from sympy.polys.fields import vfield +from sympy.polys.domains import QQ + +def test_vring(): + ns = {'vring':vring, 'QQ':QQ} + exec('R = vring("r", QQ)', ns) + exec('assert r == R.gens[0]', ns) + + exec('R = vring("rb rbb rcc rzz _rx", QQ)', ns) + exec('assert rb == R.gens[0]', ns) + exec('assert rbb == R.gens[1]', ns) + exec('assert rcc == R.gens[2]', ns) + exec('assert rzz == R.gens[3]', ns) + exec('assert _rx == R.gens[4]', ns) + + exec('R = vring(["rd", "re", "rfg"], QQ)', ns) + exec('assert rd == R.gens[0]', ns) + exec('assert re == R.gens[1]', ns) + exec('assert rfg == R.gens[2]', ns) + +def test_vfield(): + ns = {'vfield':vfield, 'QQ':QQ} + exec('F = vfield("f", QQ)', ns) + exec('assert f == F.gens[0]', ns) + + exec('F = vfield("fb fbb fcc fzz _fx", QQ)', ns) + exec('assert fb == F.gens[0]', ns) + exec('assert fbb == F.gens[1]', ns) + exec('assert fcc == F.gens[2]', ns) + exec('assert fzz == F.gens[3]', ns) + exec('assert _fx == F.gens[4]', ns) + + exec('F = vfield(["fd", "fe", "ffg"], QQ)', ns) + exec('assert fd == F.gens[0]', ns) + exec('assert fe == F.gens[1]', ns) + exec('assert ffg == F.gens[2]', ns) diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_modulargcd.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_modulargcd.py new file mode 100644 index 0000000000000000000000000000000000000000..20510f59186524ed4008ade943fab526a9ae7194 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_modulargcd.py @@ -0,0 +1,325 @@ +from sympy.polys.rings import ring +from sympy.polys.domains import ZZ, QQ, AlgebraicField +from sympy.polys.modulargcd import ( + modgcd_univariate, + modgcd_bivariate, + _chinese_remainder_reconstruction_multivariate, + modgcd_multivariate, + _to_ZZ_poly, + _to_ANP_poly, + func_field_modgcd, + _func_field_modgcd_m) +from sympy.functions.elementary.miscellaneous import sqrt + + +def test_modgcd_univariate_integers(): + R, x = ring("x", ZZ) + + f, g = R.zero, R.zero + assert modgcd_univariate(f, g) == (0, 0, 0) + + f, g = R.zero, x + assert modgcd_univariate(f, g) == (x, 0, 1) + assert modgcd_univariate(g, f) == (x, 1, 0) + + f, g = R.zero, -x + assert modgcd_univariate(f, g) == (x, 0, -1) + assert modgcd_univariate(g, f) == (x, -1, 0) + + f, g = 2*x, R(2) + assert modgcd_univariate(f, g) == (2, x, 1) + + f, g = 2*x + 2, 6*x**2 - 6 + assert modgcd_univariate(f, g) == (2*x + 2, 1, 3*x - 3) + + f = x**4 + 8*x**3 + 21*x**2 + 22*x + 8 + g = x**3 + 6*x**2 + 11*x + 6 + + h = x**2 + 3*x + 2 + + cff = x**2 + 5*x + 4 + cfg = x + 3 + + assert modgcd_univariate(f, g) == (h, cff, cfg) + + f = x**4 - 4 + g = x**4 + 4*x**2 + 4 + + h = x**2 + 2 + + cff = x**2 - 2 + cfg = x**2 + 2 + + assert modgcd_univariate(f, g) == (h, cff, cfg) + + f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 + g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 + + h = 1 + + cff = f + cfg = g + + assert modgcd_univariate(f, g) == (h, cff, cfg) + + f = - 352518131239247345597970242177235495263669787845475025293906825864749649589178600387510272*x**49 \ + + 46818041807522713962450042363465092040687472354933295397472942006618953623327997952*x**42 \ + + 378182690892293941192071663536490788434899030680411695933646320291525827756032*x**35 \ + + 112806468807371824947796775491032386836656074179286744191026149539708928*x**28 \ + - 12278371209708240950316872681744825481125965781519138077173235712*x**21 \ + + 289127344604779611146960547954288113529690984687482920704*x**14 \ + + 19007977035740498977629742919480623972236450681*x**7 \ + + 311973482284542371301330321821976049 + + g = 365431878023781158602430064717380211405897160759702125019136*x**21 \ + + 197599133478719444145775798221171663643171734081650688*x**14 \ + - 9504116979659010018253915765478924103928886144*x**7 \ + - 311973482284542371301330321821976049 + + assert modgcd_univariate(f, f.diff(x))[0] == g + + f = 1317378933230047068160*x + 2945748836994210856960 + g = 120352542776360960*x + 269116466014453760 + + h = 120352542776360960*x + 269116466014453760 + cff = 10946 + cfg = 1 + + assert modgcd_univariate(f, g) == (h, cff, cfg) + + +def test_modgcd_bivariate_integers(): + R, x, y = ring("x,y", ZZ) + + f, g = R.zero, R.zero + assert modgcd_bivariate(f, g) == (0, 0, 0) + + f, g = 2*x, R(2) + assert modgcd_bivariate(f, g) == (2, x, 1) + + f, g = x + 2*y, x + y + assert modgcd_bivariate(f, g) == (1, f, g) + + f, g = x**2 + 2*x*y + y**2, x**3 + y**3 + assert modgcd_bivariate(f, g) == (x + y, x + y, x**2 - x*y + y**2) + + f, g = x*y**2 + 2*x*y + x, x*y**3 + x + assert modgcd_bivariate(f, g) == (x*y + x, y + 1, y**2 - y + 1) + + f, g = x**2*y**2 + x**2*y + 1, x*y**2 + x*y + 1 + assert modgcd_bivariate(f, g) == (1, f, g) + + f = 2*x*y**2 + 4*x*y + 2*x + y**2 + 2*y + 1 + g = 2*x*y**3 + 2*x + y**3 + 1 + assert modgcd_bivariate(f, g) == (2*x*y + 2*x + y + 1, y + 1, y**2 - y + 1) + + f, g = 2*x**2 + 4*x + 2, x + 1 + assert modgcd_bivariate(f, g) == (x + 1, 2*x + 2, 1) + + f, g = x + 1, 2*x**2 + 4*x + 2 + assert modgcd_bivariate(f, g) == (x + 1, 1, 2*x + 2) + + f = 2*x**2 + 4*x*y - 2*x - 4*y + g = x**2 + x - 2 + assert modgcd_bivariate(f, g) == (x - 1, 2*x + 4*y, x + 2) + + f = 2*x**2 + 2*x*y - 3*x - 3*y + g = 4*x*y - 2*x + 4*y**2 - 2*y + assert modgcd_bivariate(f, g) == (x + y, 2*x - 3, 4*y - 2) + + +def test_chinese_remainder(): + R, x, y = ring("x, y", ZZ) + p, q = 3, 5 + + hp = x**3*y - x**2 - 1 + hq = -x**3*y - 2*x*y**2 + 2 + + hpq = _chinese_remainder_reconstruction_multivariate(hp, hq, p, q) + + assert hpq.trunc_ground(p) == hp + assert hpq.trunc_ground(q) == hq + + T, z = ring("z", R) + p, q = 3, 7 + + hp = (x*y + 1)*z**2 + x + hq = (x**2 - 3*y)*z + 2 + + hpq = _chinese_remainder_reconstruction_multivariate(hp, hq, p, q) + + assert hpq.trunc_ground(p) == hp + assert hpq.trunc_ground(q) == hq + + +def test_modgcd_multivariate_integers(): + R, x, y = ring("x,y", ZZ) + + f, g = R.zero, R.zero + assert modgcd_multivariate(f, g) == (0, 0, 0) + + f, g = 2*x**2 + 4*x + 2, x + 1 + assert modgcd_multivariate(f, g) == (x + 1, 2*x + 2, 1) + + f, g = x + 1, 2*x**2 + 4*x + 2 + assert modgcd_multivariate(f, g) == (x + 1, 1, 2*x + 2) + + f = 2*x**2 + 2*x*y - 3*x - 3*y + g = 4*x*y - 2*x + 4*y**2 - 2*y + assert modgcd_multivariate(f, g) == (x + y, 2*x - 3, 4*y - 2) + + f, g = x*y**2 + 2*x*y + x, x*y**3 + x + assert modgcd_multivariate(f, g) == (x*y + x, y + 1, y**2 - y + 1) + + f, g = x**2*y**2 + x**2*y + 1, x*y**2 + x*y + 1 + assert modgcd_multivariate(f, g) == (1, f, g) + + f = x**4 + 8*x**3 + 21*x**2 + 22*x + 8 + g = x**3 + 6*x**2 + 11*x + 6 + + h = x**2 + 3*x + 2 + + cff = x**2 + 5*x + 4 + cfg = x + 3 + + assert modgcd_multivariate(f, g) == (h, cff, cfg) + + R, x, y, z, u = ring("x,y,z,u", ZZ) + + f, g = x + y + z, -x - y - z - u + assert modgcd_multivariate(f, g) == (1, f, g) + + f, g = u**2 + 2*u + 1, 2*u + 2 + assert modgcd_multivariate(f, g) == (u + 1, u + 1, 2) + + f, g = z**2*u**2 + 2*z**2*u + z**2 + z*u + z, u**2 + 2*u + 1 + h, cff, cfg = u + 1, z**2*u + z**2 + z, u + 1 + + assert modgcd_multivariate(f, g) == (h, cff, cfg) + assert modgcd_multivariate(g, f) == (h, cfg, cff) + + R, x, y, z = ring("x,y,z", ZZ) + + f, g = x - y*z, x - y*z + assert modgcd_multivariate(f, g) == (x - y*z, 1, 1) + + f, g, h = R.fateman_poly_F_1() + H, cff, cfg = modgcd_multivariate(f, g) + + assert H == h and H*cff == f and H*cfg == g + + R, x, y, z, u, v = ring("x,y,z,u,v", ZZ) + + f, g, h = R.fateman_poly_F_1() + H, cff, cfg = modgcd_multivariate(f, g) + + assert H == h and H*cff == f and H*cfg == g + + R, x, y, z, u, v, a, b = ring("x,y,z,u,v,a,b", ZZ) + + f, g, h = R.fateman_poly_F_1() + H, cff, cfg = modgcd_multivariate(f, g) + + assert H == h and H*cff == f and H*cfg == g + + R, x, y, z, u, v, a, b, c, d = ring("x,y,z,u,v,a,b,c,d", ZZ) + + f, g, h = R.fateman_poly_F_1() + H, cff, cfg = modgcd_multivariate(f, g) + + assert H == h and H*cff == f and H*cfg == g + + R, x, y, z = ring("x,y,z", ZZ) + + f, g, h = R.fateman_poly_F_2() + H, cff, cfg = modgcd_multivariate(f, g) + + assert H == h and H*cff == f and H*cfg == g + + f, g, h = R.fateman_poly_F_3() + H, cff, cfg = modgcd_multivariate(f, g) + + assert H == h and H*cff == f and H*cfg == g + + R, x, y, z, t = ring("x,y,z,t", ZZ) + + f, g, h = R.fateman_poly_F_3() + H, cff, cfg = modgcd_multivariate(f, g) + + assert H == h and H*cff == f and H*cfg == g + + +def test_to_ZZ_ANP_poly(): + A = AlgebraicField(QQ, sqrt(2)) + R, x = ring("x", A) + f = x*(sqrt(2) + 1) + + T, x_, z_ = ring("x_, z_", ZZ) + f_ = x_*z_ + x_ + + assert _to_ZZ_poly(f, T) == f_ + assert _to_ANP_poly(f_, R) == f + + R, x, t, s = ring("x, t, s", A) + f = x*t**2 + x*s + sqrt(2) + + D, t_, s_ = ring("t_, s_", ZZ) + T, x_, z_ = ring("x_, z_", D) + f_ = (t_**2 + s_)*x_ + z_ + + assert _to_ZZ_poly(f, T) == f_ + assert _to_ANP_poly(f_, R) == f + + +def test_modgcd_algebraic_field(): + A = AlgebraicField(QQ, sqrt(2)) + R, x = ring("x", A) + one = A.one + + f, g = 2*x, R(2) + assert func_field_modgcd(f, g) == (one, f, g) + + f, g = 2*x, R(sqrt(2)) + assert func_field_modgcd(f, g) == (one, f, g) + + f, g = 2*x + 2, 6*x**2 - 6 + assert func_field_modgcd(f, g) == (x + 1, R(2), 6*x - 6) + + R, x, y = ring("x, y", A) + + f, g = x + sqrt(2)*y, x + y + assert func_field_modgcd(f, g) == (one, f, g) + + f, g = x*y + sqrt(2)*y**2, R(sqrt(2))*y + assert func_field_modgcd(f, g) == (y, x + sqrt(2)*y, R(sqrt(2))) + + f, g = x**2 + 2*sqrt(2)*x*y + 2*y**2, x + sqrt(2)*y + assert func_field_modgcd(f, g) == (g, g, one) + + A = AlgebraicField(QQ, sqrt(2), sqrt(3)) + R, x, y, z = ring("x, y, z", A) + + h = x**2*y**7 + sqrt(6)/21*z + f, g = h*(27*y**3 + 1), h*(y + x) + assert func_field_modgcd(f, g) == (h, 27*y**3+1, y+x) + + h = x**13*y**3 + 1/2*x**10 + 1/sqrt(2) + f, g = h*(x + 1), h*sqrt(2)/sqrt(3) + assert func_field_modgcd(f, g) == (h, x + 1, R(sqrt(2)/sqrt(3))) + + A = AlgebraicField(QQ, sqrt(2)**(-1)*sqrt(3)) + R, x = ring("x", A) + + f, g = x + 1, x - 1 + assert func_field_modgcd(f, g) == (A.one, f, g) + + +# when func_field_modgcd supports function fields, this test can be changed +def test_modgcd_func_field(): + D, t = ring("t", ZZ) + R, x, z = ring("x, z", D) + + minpoly = (z**2*t**2 + z**2*t - 1).drop(0) + f, g = x + 1, x - 1 + + assert _func_field_modgcd_m(f, g, minpoly) == R.one diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_monomials.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_monomials.py new file mode 100644 index 0000000000000000000000000000000000000000..c5ed28ba0e8e3f8e9f85c543a4fffcaef855fff8 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_monomials.py @@ -0,0 +1,269 @@ +"""Tests for tools and arithmetics for monomials of distributed polynomials. """ + +from sympy.polys.monomials import ( + itermonomials, monomial_count, + monomial_mul, monomial_div, + monomial_gcd, monomial_lcm, + monomial_max, monomial_min, + monomial_divides, monomial_pow, + Monomial, +) + +from sympy.polys.polyerrors import ExactQuotientFailed + +from sympy.abc import a, b, c, x, y, z +from sympy.core import S, symbols +from sympy.testing.pytest import raises + +def test_monomials(): + + # total_degree tests + assert set(itermonomials([], 0)) == {S.One} + assert set(itermonomials([], 1)) == {S.One} + assert set(itermonomials([], 2)) == {S.One} + + assert set(itermonomials([], 0, 0)) == {S.One} + assert set(itermonomials([], 1, 0)) == {S.One} + assert set(itermonomials([], 2, 0)) == {S.One} + + raises(StopIteration, lambda: next(itermonomials([], 0, 1))) + raises(StopIteration, lambda: next(itermonomials([], 0, 2))) + raises(StopIteration, lambda: next(itermonomials([], 0, 3))) + + assert set(itermonomials([], 0, 1)) == set() + assert set(itermonomials([], 0, 2)) == set() + assert set(itermonomials([], 0, 3)) == set() + + raises(ValueError, lambda: set(itermonomials([], -1))) + raises(ValueError, lambda: set(itermonomials([x], -1))) + raises(ValueError, lambda: set(itermonomials([x, y], -1))) + + assert set(itermonomials([x], 0)) == {S.One} + assert set(itermonomials([x], 1)) == {S.One, x} + assert set(itermonomials([x], 2)) == {S.One, x, x**2} + assert set(itermonomials([x], 3)) == {S.One, x, x**2, x**3} + + assert set(itermonomials([x, y], 0)) == {S.One} + assert set(itermonomials([x, y], 1)) == {S.One, x, y} + assert set(itermonomials([x, y], 2)) == {S.One, x, y, x**2, y**2, x*y} + assert set(itermonomials([x, y], 3)) == \ + {S.One, x, y, x**2, x**3, y**2, y**3, x*y, x*y**2, y*x**2} + + i, j, k = symbols('i j k', commutative=False) + assert set(itermonomials([i, j, k], 0)) == {S.One} + assert set(itermonomials([i, j, k], 1)) == {S.One, i, j, k} + assert set(itermonomials([i, j, k], 2)) == \ + {S.One, i, j, k, i**2, j**2, k**2, i*j, i*k, j*i, j*k, k*i, k*j} + + assert set(itermonomials([i, j, k], 3)) == \ + {S.One, i, j, k, i**2, j**2, k**2, i*j, i*k, j*i, j*k, k*i, k*j, + i**3, j**3, k**3, + i**2 * j, i**2 * k, j * i**2, k * i**2, + j**2 * i, j**2 * k, i * j**2, k * j**2, + k**2 * i, k**2 * j, i * k**2, j * k**2, + i*j*i, i*k*i, j*i*j, j*k*j, k*i*k, k*j*k, + i*j*k, i*k*j, j*i*k, j*k*i, k*i*j, k*j*i, + } + + assert set(itermonomials([x, i, j], 0)) == {S.One} + assert set(itermonomials([x, i, j], 1)) == {S.One, x, i, j} + assert set(itermonomials([x, i, j], 2)) == {S.One, x, i, j, x*i, x*j, i*j, j*i, x**2, i**2, j**2} + assert set(itermonomials([x, i, j], 3)) == \ + {S.One, x, i, j, x*i, x*j, i*j, j*i, x**2, i**2, j**2, + x**3, i**3, j**3, + x**2 * i, x**2 * j, + x * i**2, j * i**2, i**2 * j, i*j*i, + x * j**2, i * j**2, j**2 * i, j*i*j, + x * i * j, x * j * i + } + + # degree_list tests + assert set(itermonomials([], [])) == {S.One} + + raises(ValueError, lambda: set(itermonomials([], [0]))) + raises(ValueError, lambda: set(itermonomials([], [1]))) + raises(ValueError, lambda: set(itermonomials([], [2]))) + + raises(ValueError, lambda: set(itermonomials([x], [1], []))) + raises(ValueError, lambda: set(itermonomials([x], [1, 2], []))) + raises(ValueError, lambda: set(itermonomials([x], [1, 2, 3], []))) + + raises(ValueError, lambda: set(itermonomials([x], [], [1]))) + raises(ValueError, lambda: set(itermonomials([x], [], [1, 2]))) + raises(ValueError, lambda: set(itermonomials([x], [], [1, 2, 3]))) + + raises(ValueError, lambda: set(itermonomials([x, y], [1, 2], [1, 2, 3]))) + raises(ValueError, lambda: set(itermonomials([x, y, z], [1, 2, 3], [0, 1]))) + + raises(ValueError, lambda: set(itermonomials([x], [1], [-1]))) + raises(ValueError, lambda: set(itermonomials([x, y], [1, 2], [1, -1]))) + + raises(ValueError, lambda: set(itermonomials([], [], 1))) + raises(ValueError, lambda: set(itermonomials([], [], 2))) + raises(ValueError, lambda: set(itermonomials([], [], 3))) + + raises(ValueError, lambda: set(itermonomials([x, y], [0, 1], [1, 2]))) + raises(ValueError, lambda: set(itermonomials([x, y, z], [0, 0, 3], [0, 1, 2]))) + + assert set(itermonomials([x], [0])) == {S.One} + assert set(itermonomials([x], [1])) == {S.One, x} + assert set(itermonomials([x], [2])) == {S.One, x, x**2} + assert set(itermonomials([x], [3])) == {S.One, x, x**2, x**3} + + assert set(itermonomials([x], [3], [1])) == {x, x**3, x**2} + assert set(itermonomials([x], [3], [2])) == {x**3, x**2} + + assert set(itermonomials([x, y], 3, 3)) == {x**3, x**2*y, x*y**2, y**3} + assert set(itermonomials([x, y], 3, 2)) == {x**2, x*y, y**2, x**3, x**2*y, x*y**2, y**3} + + assert set(itermonomials([x, y], [0, 0])) == {S.One} + assert set(itermonomials([x, y], [0, 1])) == {S.One, y} + assert set(itermonomials([x, y], [0, 2])) == {S.One, y, y**2} + assert set(itermonomials([x, y], [0, 2], [0, 1])) == {y, y**2} + assert set(itermonomials([x, y], [0, 2], [0, 2])) == {y**2} + + assert set(itermonomials([x, y], [1, 0])) == {S.One, x} + assert set(itermonomials([x, y], [1, 1])) == {S.One, x, y, x*y} + assert set(itermonomials([x, y], [1, 2])) == {S.One, x, y, x*y, y**2, x*y**2} + assert set(itermonomials([x, y], [1, 2], [1, 1])) == {x*y, x*y**2} + assert set(itermonomials([x, y], [1, 2], [1, 2])) == {x*y**2} + + assert set(itermonomials([x, y], [2, 0])) == {S.One, x, x**2} + assert set(itermonomials([x, y], [2, 1])) == {S.One, x, y, x*y, x**2, x**2*y} + assert set(itermonomials([x, y], [2, 2])) == \ + {S.One, y**2, x*y**2, x, x*y, x**2, x**2*y**2, y, x**2*y} + + i, j, k = symbols('i j k', commutative=False) + assert set(itermonomials([i, j, k], 2, 2)) == \ + {k*i, i**2, i*j, j*k, j*i, k**2, j**2, k*j, i*k} + assert set(itermonomials([i, j, k], 3, 2)) == \ + {j*k**2, i*k**2, k*i*j, k*i**2, k**2, j*k*j, k*j**2, i*k*i, i*j, + j**2*k, i**2*j, j*i*k, j**3, i**3, k*j*i, j*k*i, j*i, + k**2*j, j*i**2, k*j, k*j*k, i*j*i, j*i*j, i*j**2, j**2, + k*i*k, i**2, j*k, i*k, i*k*j, k**3, i**2*k, j**2*i, k**2*i, + i*j*k, k*i + } + assert set(itermonomials([i, j, k], [0, 0, 0])) == {S.One} + assert set(itermonomials([i, j, k], [0, 0, 1])) == {1, k} + assert set(itermonomials([i, j, k], [0, 1, 0])) == {1, j} + assert set(itermonomials([i, j, k], [1, 0, 0])) == {i, 1} + assert set(itermonomials([i, j, k], [0, 0, 2])) == {k**2, 1, k} + assert set(itermonomials([i, j, k], [0, 2, 0])) == {1, j, j**2} + assert set(itermonomials([i, j, k], [2, 0, 0])) == {i, 1, i**2} + assert set(itermonomials([i, j, k], [1, 1, 1])) == {1, k, j, j*k, i*k, i, i*j, i*j*k} + assert set(itermonomials([i, j, k], [2, 2, 2])) == \ + {1, k, i**2*k**2, j*k, j**2, i, i*k, j*k**2, i*j**2*k**2, + i**2*j, i**2*j**2, k**2, j**2*k, i*j**2*k, + j**2*k**2, i*j, i**2*k, i**2*j**2*k, j, i**2*j*k, + i*j**2, i*k**2, i*j*k, i**2*j**2*k**2, i*j*k**2, i**2, i**2*j*k**2 + } + + assert set(itermonomials([x, j, k], [0, 0, 0])) == {S.One} + assert set(itermonomials([x, j, k], [0, 0, 1])) == {1, k} + assert set(itermonomials([x, j, k], [0, 1, 0])) == {1, j} + assert set(itermonomials([x, j, k], [1, 0, 0])) == {x, 1} + assert set(itermonomials([x, j, k], [0, 0, 2])) == {k**2, 1, k} + assert set(itermonomials([x, j, k], [0, 2, 0])) == {1, j, j**2} + assert set(itermonomials([x, j, k], [2, 0, 0])) == {x, 1, x**2} + assert set(itermonomials([x, j, k], [1, 1, 1])) == {1, k, j, j*k, x*k, x, x*j, x*j*k} + assert set(itermonomials([x, j, k], [2, 2, 2])) == \ + {1, k, x**2*k**2, j*k, j**2, x, x*k, j*k**2, x*j**2*k**2, + x**2*j, x**2*j**2, k**2, j**2*k, x*j**2*k, + j**2*k**2, x*j, x**2*k, x**2*j**2*k, j, x**2*j*k, + x*j**2, x*k**2, x*j*k, x**2*j**2*k**2, x*j*k**2, x**2, x**2*j*k**2 + } + +def test_monomial_count(): + assert monomial_count(2, 2) == 6 + assert monomial_count(2, 3) == 10 + +def test_monomial_mul(): + assert monomial_mul((3, 4, 1), (1, 2, 0)) == (4, 6, 1) + +def test_monomial_div(): + assert monomial_div((3, 4, 1), (1, 2, 0)) == (2, 2, 1) + +def test_monomial_gcd(): + assert monomial_gcd((3, 4, 1), (1, 2, 0)) == (1, 2, 0) + +def test_monomial_lcm(): + assert monomial_lcm((3, 4, 1), (1, 2, 0)) == (3, 4, 1) + +def test_monomial_max(): + assert monomial_max((3, 4, 5), (0, 5, 1), (6, 3, 9)) == (6, 5, 9) + +def test_monomial_pow(): + assert monomial_pow((1, 2, 3), 3) == (3, 6, 9) + +def test_monomial_min(): + assert monomial_min((3, 4, 5), (0, 5, 1), (6, 3, 9)) == (0, 3, 1) + +def test_monomial_divides(): + assert monomial_divides((1, 2, 3), (4, 5, 6)) is True + assert monomial_divides((1, 2, 3), (0, 5, 6)) is False + +def test_Monomial(): + m = Monomial((3, 4, 1), (x, y, z)) + n = Monomial((1, 2, 0), (x, y, z)) + + assert m.as_expr() == x**3*y**4*z + assert n.as_expr() == x**1*y**2 + + assert m.as_expr(a, b, c) == a**3*b**4*c + assert n.as_expr(a, b, c) == a**1*b**2 + + assert m.exponents == (3, 4, 1) + assert m.gens == (x, y, z) + + assert n.exponents == (1, 2, 0) + assert n.gens == (x, y, z) + + assert m == (3, 4, 1) + assert n != (3, 4, 1) + assert m != (1, 2, 0) + assert n == (1, 2, 0) + assert (m == 1) is False + + assert m[0] == m[-3] == 3 + assert m[1] == m[-2] == 4 + assert m[2] == m[-1] == 1 + + assert n[0] == n[-3] == 1 + assert n[1] == n[-2] == 2 + assert n[2] == n[-1] == 0 + + assert m[:2] == (3, 4) + assert n[:2] == (1, 2) + + assert m*n == Monomial((4, 6, 1)) + assert m/n == Monomial((2, 2, 1)) + + assert m*(1, 2, 0) == Monomial((4, 6, 1)) + assert m/(1, 2, 0) == Monomial((2, 2, 1)) + + assert m.gcd(n) == Monomial((1, 2, 0)) + assert m.lcm(n) == Monomial((3, 4, 1)) + + assert m.gcd((1, 2, 0)) == Monomial((1, 2, 0)) + assert m.lcm((1, 2, 0)) == Monomial((3, 4, 1)) + + assert m**0 == Monomial((0, 0, 0)) + assert m**1 == m + assert m**2 == Monomial((6, 8, 2)) + assert m**3 == Monomial((9, 12, 3)) + _a = Monomial((0, 0, 0)) + for n in range(10): + assert _a == m**n + _a *= m + + raises(ExactQuotientFailed, lambda: m/Monomial((5, 2, 0))) + + mm = Monomial((1, 2, 3)) + raises(ValueError, lambda: mm.as_expr()) + assert str(mm) == 'Monomial((1, 2, 3))' + assert str(m) == 'x**3*y**4*z**1' + raises(NotImplementedError, lambda: m*1) + raises(NotImplementedError, lambda: m/1) + raises(ValueError, lambda: m**-1) + raises(TypeError, lambda: m.gcd(3)) + raises(TypeError, lambda: m.lcm(3)) diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_multivariate_resultants.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_multivariate_resultants.py new file mode 100644 index 0000000000000000000000000000000000000000..0799feb41fc875cf038723916a3efd62ff31b1b4 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_multivariate_resultants.py @@ -0,0 +1,294 @@ +"""Tests for Dixon's and Macaulay's classes. """ + +from sympy.matrices.dense import Matrix +from sympy.polys.polytools import factor +from sympy.core import symbols +from sympy.tensor.indexed import IndexedBase + +from sympy.polys.multivariate_resultants import (DixonResultant, + MacaulayResultant) + +c, d = symbols("a, b") +x, y = symbols("x, y") + +p = c * x + y +q = x + d * y + +dixon = DixonResultant(polynomials=[p, q], variables=[x, y]) +macaulay = MacaulayResultant(polynomials=[p, q], variables=[x, y]) + +def test_dixon_resultant_init(): + """Test init method of DixonResultant.""" + a = IndexedBase("alpha") + + assert dixon.polynomials == [p, q] + assert dixon.variables == [x, y] + assert dixon.n == 2 + assert dixon.m == 2 + assert dixon.dummy_variables == [a[0], a[1]] + +def test_get_dixon_polynomial_numerical(): + """Test Dixon's polynomial for a numerical example.""" + a = IndexedBase("alpha") + + p = x + y + q = x ** 2 + y **3 + h = x ** 2 + y + + dixon = DixonResultant([p, q, h], [x, y]) + polynomial = -x * y ** 2 * a[0] - x * y ** 2 * a[1] - x * y * a[0] \ + * a[1] - x * y * a[1] ** 2 - x * a[0] * a[1] ** 2 + x * a[0] - \ + y ** 2 * a[0] * a[1] + y ** 2 * a[1] - y * a[0] * a[1] ** 2 + y * \ + a[1] ** 2 + + assert dixon.get_dixon_polynomial().as_expr().expand() == polynomial + +def test_get_max_degrees(): + """Tests max degrees function.""" + + p = x + y + q = x ** 2 + y **3 + h = x ** 2 + y + + dixon = DixonResultant(polynomials=[p, q, h], variables=[x, y]) + dixon_polynomial = dixon.get_dixon_polynomial() + + assert dixon.get_max_degrees(dixon_polynomial) == [1, 2] + +def test_get_dixon_matrix(): + """Test Dixon's resultant for a numerical example.""" + + x, y = symbols('x, y') + + p = x + y + q = x ** 2 + y ** 3 + h = x ** 2 + y + + dixon = DixonResultant([p, q, h], [x, y]) + polynomial = dixon.get_dixon_polynomial() + + assert dixon.get_dixon_matrix(polynomial).det() == 0 + +def test_get_dixon_matrix_example_two(): + """Test Dixon's matrix for example from [Palancz08]_.""" + x, y, z = symbols('x, y, z') + + f = x ** 2 + y ** 2 - 1 + z * 0 + g = x ** 2 + z ** 2 - 1 + y * 0 + h = y ** 2 + z ** 2 - 1 + + example_two = DixonResultant([f, g, h], [y, z]) + poly = example_two.get_dixon_polynomial() + matrix = example_two.get_dixon_matrix(poly) + + expr = 1 - 8 * x ** 2 + 24 * x ** 4 - 32 * x ** 6 + 16 * x ** 8 + assert (matrix.det() - expr).expand() == 0 + +def test_KSY_precondition(): + """Tests precondition for KSY Resultant.""" + A, B, C = symbols('A, B, C') + + m1 = Matrix([[1, 2, 3], + [4, 5, 12], + [6, 7, 18]]) + + m2 = Matrix([[0, C**2], + [-2 * C, -C ** 2]]) + + m3 = Matrix([[1, 0], + [0, 1]]) + + m4 = Matrix([[A**2, 0, 1], + [A, 1, 1 / A]]) + + m5 = Matrix([[5, 1], + [2, B], + [0, 1], + [0, 0]]) + + assert dixon.KSY_precondition(m1) == False + assert dixon.KSY_precondition(m2) == True + assert dixon.KSY_precondition(m3) == True + assert dixon.KSY_precondition(m4) == False + assert dixon.KSY_precondition(m5) == True + +def test_delete_zero_rows_and_columns(): + """Tests method for deleting rows and columns containing only zeros.""" + A, B, C = symbols('A, B, C') + + m1 = Matrix([[0, 0], + [0, 0], + [1, 2]]) + + m2 = Matrix([[0, 1, 2], + [0, 3, 4], + [0, 5, 6]]) + + m3 = Matrix([[0, 0, 0, 0], + [0, 1, 2, 0], + [0, 3, 4, 0], + [0, 0, 0, 0]]) + + m4 = Matrix([[1, 0, 2], + [0, 0, 0], + [3, 0, 4]]) + + m5 = Matrix([[0, 0, 0, 1], + [0, 0, 0, 2], + [0, 0, 0, 3], + [0, 0, 0, 4]]) + + m6 = Matrix([[0, 0, A], + [B, 0, 0], + [0, 0, C]]) + + assert dixon.delete_zero_rows_and_columns(m1) == Matrix([[1, 2]]) + + assert dixon.delete_zero_rows_and_columns(m2) == Matrix([[1, 2], + [3, 4], + [5, 6]]) + + assert dixon.delete_zero_rows_and_columns(m3) == Matrix([[1, 2], + [3, 4]]) + + assert dixon.delete_zero_rows_and_columns(m4) == Matrix([[1, 2], + [3, 4]]) + + assert dixon.delete_zero_rows_and_columns(m5) == Matrix([[1], + [2], + [3], + [4]]) + + assert dixon.delete_zero_rows_and_columns(m6) == Matrix([[0, A], + [B, 0], + [0, C]]) + +def test_product_leading_entries(): + """Tests product of leading entries method.""" + A, B = symbols('A, B') + + m1 = Matrix([[1, 2, 3], + [0, 4, 5], + [0, 0, 6]]) + + m2 = Matrix([[0, 0, 1], + [2, 0, 3]]) + + m3 = Matrix([[0, 0, 0], + [1, 2, 3], + [0, 0, 0]]) + + m4 = Matrix([[0, 0, A], + [1, 2, 3], + [B, 0, 0]]) + + assert dixon.product_leading_entries(m1) == 24 + assert dixon.product_leading_entries(m2) == 2 + assert dixon.product_leading_entries(m3) == 1 + assert dixon.product_leading_entries(m4) == A * B + +def test_get_KSY_Dixon_resultant_example_one(): + """Tests the KSY Dixon resultant for example one""" + x, y, z = symbols('x, y, z') + + p = x * y * z + q = x**2 - z**2 + h = x + y + z + dixon = DixonResultant([p, q, h], [x, y]) + dixon_poly = dixon.get_dixon_polynomial() + dixon_matrix = dixon.get_dixon_matrix(dixon_poly) + D = dixon.get_KSY_Dixon_resultant(dixon_matrix) + + assert D == -z**3 + +def test_get_KSY_Dixon_resultant_example_two(): + """Tests the KSY Dixon resultant for example two""" + x, y, A = symbols('x, y, A') + + p = x * y + x * A + x - A**2 - A + y**2 + y + q = x**2 + x * A - x + x * y + y * A - y + h = x**2 + x * y + 2 * x - x * A - y * A - 2 * A + + dixon = DixonResultant([p, q, h], [x, y]) + dixon_poly = dixon.get_dixon_polynomial() + dixon_matrix = dixon.get_dixon_matrix(dixon_poly) + D = factor(dixon.get_KSY_Dixon_resultant(dixon_matrix)) + + assert D == -8*A*(A - 1)*(A + 2)*(2*A - 1)**2 + +def test_macaulay_resultant_init(): + """Test init method of MacaulayResultant.""" + + assert macaulay.polynomials == [p, q] + assert macaulay.variables == [x, y] + assert macaulay.n == 2 + assert macaulay.degrees == [1, 1] + assert macaulay.degree_m == 1 + assert macaulay.monomials_size == 2 + +def test_get_degree_m(): + assert macaulay._get_degree_m() == 1 + +def test_get_size(): + assert macaulay.get_size() == 2 + +def test_macaulay_example_one(): + """Tests the Macaulay for example from [Bruce97]_""" + + x, y, z = symbols('x, y, z') + a_1_1, a_1_2, a_1_3 = symbols('a_1_1, a_1_2, a_1_3') + a_2_2, a_2_3, a_3_3 = symbols('a_2_2, a_2_3, a_3_3') + b_1_1, b_1_2, b_1_3 = symbols('b_1_1, b_1_2, b_1_3') + b_2_2, b_2_3, b_3_3 = symbols('b_2_2, b_2_3, b_3_3') + c_1, c_2, c_3 = symbols('c_1, c_2, c_3') + + f_1 = a_1_1 * x ** 2 + a_1_2 * x * y + a_1_3 * x * z + \ + a_2_2 * y ** 2 + a_2_3 * y * z + a_3_3 * z ** 2 + f_2 = b_1_1 * x ** 2 + b_1_2 * x * y + b_1_3 * x * z + \ + b_2_2 * y ** 2 + b_2_3 * y * z + b_3_3 * z ** 2 + f_3 = c_1 * x + c_2 * y + c_3 * z + + mac = MacaulayResultant([f_1, f_2, f_3], [x, y, z]) + + assert mac.degrees == [2, 2, 1] + assert mac.degree_m == 3 + + assert mac.monomial_set == [x ** 3, x ** 2 * y, x ** 2 * z, + x * y ** 2, + x * y * z, x * z ** 2, y ** 3, + y ** 2 *z, y * z ** 2, z ** 3] + assert mac.monomials_size == 10 + assert mac.get_row_coefficients() == [[x, y, z], [x, y, z], + [x * y, x * z, y * z, z ** 2]] + + matrix = mac.get_matrix() + assert matrix.shape == (mac.monomials_size, mac.monomials_size) + assert mac.get_submatrix(matrix) == Matrix([[a_1_1, a_2_2], + [b_1_1, b_2_2]]) + +def test_macaulay_example_two(): + """Tests the Macaulay formulation for example from [Stiller96]_.""" + + x, y, z = symbols('x, y, z') + a_0, a_1, a_2 = symbols('a_0, a_1, a_2') + b_0, b_1, b_2 = symbols('b_0, b_1, b_2') + c_0, c_1, c_2, c_3, c_4 = symbols('c_0, c_1, c_2, c_3, c_4') + + f = a_0 * y - a_1 * x + a_2 * z + g = b_1 * x ** 2 + b_0 * y ** 2 - b_2 * z ** 2 + h = c_0 * y - c_1 * x ** 3 + c_2 * x ** 2 * z - c_3 * x * z ** 2 + \ + c_4 * z ** 3 + + mac = MacaulayResultant([f, g, h], [x, y, z]) + + assert mac.degrees == [1, 2, 3] + assert mac.degree_m == 4 + assert mac.monomials_size == 15 + assert len(mac.get_row_coefficients()) == mac.n + + matrix = mac.get_matrix() + assert matrix.shape == (mac.monomials_size, mac.monomials_size) + assert mac.get_submatrix(matrix) == Matrix([[-a_1, a_0, a_2, 0], + [0, -a_1, 0, 0], + [0, 0, -a_1, 0], + [0, 0, 0, -a_1]]) diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_orderings.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_orderings.py new file mode 100644 index 0000000000000000000000000000000000000000..d61d4887754c9d9f49905c2e131d253a45cf2ffd --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_orderings.py @@ -0,0 +1,124 @@ +"""Tests of monomial orderings. """ + +from sympy.polys.orderings import ( + monomial_key, lex, grlex, grevlex, ilex, igrlex, + LexOrder, InverseOrder, ProductOrder, build_product_order, +) + +from sympy.abc import x, y, z, t +from sympy.core import S +from sympy.testing.pytest import raises + +def test_lex_order(): + assert lex((1, 2, 3)) == (1, 2, 3) + assert str(lex) == 'lex' + + assert lex((1, 2, 3)) == lex((1, 2, 3)) + + assert lex((2, 2, 3)) > lex((1, 2, 3)) + assert lex((1, 3, 3)) > lex((1, 2, 3)) + assert lex((1, 2, 4)) > lex((1, 2, 3)) + + assert lex((0, 2, 3)) < lex((1, 2, 3)) + assert lex((1, 1, 3)) < lex((1, 2, 3)) + assert lex((1, 2, 2)) < lex((1, 2, 3)) + + assert lex.is_global is True + assert lex == LexOrder() + assert lex != grlex + +def test_grlex_order(): + assert grlex((1, 2, 3)) == (6, (1, 2, 3)) + assert str(grlex) == 'grlex' + + assert grlex((1, 2, 3)) == grlex((1, 2, 3)) + + assert grlex((2, 2, 3)) > grlex((1, 2, 3)) + assert grlex((1, 3, 3)) > grlex((1, 2, 3)) + assert grlex((1, 2, 4)) > grlex((1, 2, 3)) + + assert grlex((0, 2, 3)) < grlex((1, 2, 3)) + assert grlex((1, 1, 3)) < grlex((1, 2, 3)) + assert grlex((1, 2, 2)) < grlex((1, 2, 3)) + + assert grlex((2, 2, 3)) > grlex((1, 2, 4)) + assert grlex((1, 3, 3)) > grlex((1, 2, 4)) + + assert grlex((0, 2, 3)) < grlex((1, 2, 2)) + assert grlex((1, 1, 3)) < grlex((1, 2, 2)) + + assert grlex((0, 1, 1)) > grlex((0, 0, 2)) + assert grlex((0, 3, 1)) < grlex((2, 2, 1)) + + assert grlex.is_global is True + +def test_grevlex_order(): + assert grevlex((1, 2, 3)) == (6, (-3, -2, -1)) + assert str(grevlex) == 'grevlex' + + assert grevlex((1, 2, 3)) == grevlex((1, 2, 3)) + + assert grevlex((2, 2, 3)) > grevlex((1, 2, 3)) + assert grevlex((1, 3, 3)) > grevlex((1, 2, 3)) + assert grevlex((1, 2, 4)) > grevlex((1, 2, 3)) + + assert grevlex((0, 2, 3)) < grevlex((1, 2, 3)) + assert grevlex((1, 1, 3)) < grevlex((1, 2, 3)) + assert grevlex((1, 2, 2)) < grevlex((1, 2, 3)) + + assert grevlex((2, 2, 3)) > grevlex((1, 2, 4)) + assert grevlex((1, 3, 3)) > grevlex((1, 2, 4)) + + assert grevlex((0, 2, 3)) < grevlex((1, 2, 2)) + assert grevlex((1, 1, 3)) < grevlex((1, 2, 2)) + + assert grevlex((0, 1, 1)) > grevlex((0, 0, 2)) + assert grevlex((0, 3, 1)) < grevlex((2, 2, 1)) + + assert grevlex.is_global is True + +def test_InverseOrder(): + ilex = InverseOrder(lex) + igrlex = InverseOrder(grlex) + + assert ilex((1, 2, 3)) > ilex((2, 0, 3)) + assert igrlex((1, 2, 3)) < igrlex((0, 2, 3)) + assert str(ilex) == "ilex" + assert str(igrlex) == "igrlex" + assert ilex.is_global is False + assert igrlex.is_global is False + assert ilex != igrlex + assert ilex == InverseOrder(LexOrder()) + +def test_ProductOrder(): + P = ProductOrder((grlex, lambda m: m[:2]), (grlex, lambda m: m[2:])) + assert P((1, 3, 3, 4, 5)) > P((2, 1, 5, 5, 5)) + assert str(P) == "ProductOrder(grlex, grlex)" + assert P.is_global is True + assert ProductOrder((grlex, None), (ilex, None)).is_global is None + assert ProductOrder((igrlex, None), (ilex, None)).is_global is False + +def test_monomial_key(): + assert monomial_key() == lex + + assert monomial_key('lex') == lex + assert monomial_key('grlex') == grlex + assert monomial_key('grevlex') == grevlex + + raises(ValueError, lambda: monomial_key('foo')) + raises(ValueError, lambda: monomial_key(1)) + + M = [x, x**2*z**2, x*y, x**2, S.One, y**2, x**3, y, z, x*y**2*z, x**2*y**2] + assert sorted(M, key=monomial_key('lex', [z, y, x])) == \ + [S.One, x, x**2, x**3, y, x*y, y**2, x**2*y**2, z, x*y**2*z, x**2*z**2] + assert sorted(M, key=monomial_key('grlex', [z, y, x])) == \ + [S.One, x, y, z, x**2, x*y, y**2, x**3, x**2*y**2, x*y**2*z, x**2*z**2] + assert sorted(M, key=monomial_key('grevlex', [z, y, x])) == \ + [S.One, x, y, z, x**2, x*y, y**2, x**3, x**2*y**2, x**2*z**2, x*y**2*z] + +def test_build_product_order(): + assert build_product_order((("grlex", x, y), ("grlex", z, t)), [x, y, z, t])((4, 5, 6, 7)) == \ + ((9, (4, 5)), (13, (6, 7))) + + assert build_product_order((("grlex", x, y), ("grlex", z, t)), [x, y, z, t]) == \ + build_product_order((("grlex", x, y), ("grlex", z, t)), [x, y, z, t]) diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_orthopolys.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_orthopolys.py new file mode 100644 index 0000000000000000000000000000000000000000..e81fbe75aa6285d229ba817026f44b23b76abd6a --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_orthopolys.py @@ -0,0 +1,175 @@ +"""Tests for efficient functions for generating orthogonal polynomials. """ + +from sympy.core.numbers import Rational as Q +from sympy.core.singleton import S +from sympy.core.symbol import symbols +from sympy.polys.polytools import Poly +from sympy.testing.pytest import raises + +from sympy.polys.orthopolys import ( + jacobi_poly, + gegenbauer_poly, + chebyshevt_poly, + chebyshevu_poly, + hermite_poly, + hermite_prob_poly, + legendre_poly, + laguerre_poly, + spherical_bessel_fn, +) + +from sympy.abc import x, a, b + + +def test_jacobi_poly(): + raises(ValueError, lambda: jacobi_poly(-1, a, b, x)) + + assert jacobi_poly(1, a, b, x, polys=True) == Poly( + (a/2 + b/2 + 1)*x + a/2 - b/2, x, domain='ZZ(a,b)') + + assert jacobi_poly(0, a, b, x) == 1 + assert jacobi_poly(1, a, b, x) == a/2 - b/2 + x*(a/2 + b/2 + 1) + assert jacobi_poly(2, a, b, x) == (a**2/8 - a*b/4 - a/8 + b**2/8 - b/8 + + x**2*(a**2/8 + a*b/4 + a*Q(7, 8) + b**2/8 + + b*Q(7, 8) + Q(3, 2)) + x*(a**2/4 + + a*Q(3, 4) - b**2/4 - b*Q(3, 4)) - S.Half) + + assert jacobi_poly(1, a, b, polys=True) == Poly( + (a/2 + b/2 + 1)*x + a/2 - b/2, x, domain='ZZ(a,b)') + + +def test_gegenbauer_poly(): + raises(ValueError, lambda: gegenbauer_poly(-1, a, x)) + + assert gegenbauer_poly( + 1, a, x, polys=True) == Poly(2*a*x, x, domain='ZZ(a)') + + assert gegenbauer_poly(0, a, x) == 1 + assert gegenbauer_poly(1, a, x) == 2*a*x + assert gegenbauer_poly(2, a, x) == -a + x**2*(2*a**2 + 2*a) + assert gegenbauer_poly( + 3, a, x) == x**3*(4*a**3/3 + 4*a**2 + a*Q(8, 3)) + x*(-2*a**2 - 2*a) + + assert gegenbauer_poly(1, S.Half).dummy_eq(x) + assert gegenbauer_poly(1, a, polys=True) == Poly(2*a*x, x, domain='ZZ(a)') + + +def test_chebyshevt_poly(): + raises(ValueError, lambda: chebyshevt_poly(-1, x)) + + assert chebyshevt_poly(1, x, polys=True) == Poly(x) + + assert chebyshevt_poly(0, x) == 1 + assert chebyshevt_poly(1, x) == x + assert chebyshevt_poly(2, x) == 2*x**2 - 1 + assert chebyshevt_poly(3, x) == 4*x**3 - 3*x + assert chebyshevt_poly(4, x) == 8*x**4 - 8*x**2 + 1 + assert chebyshevt_poly(5, x) == 16*x**5 - 20*x**3 + 5*x + assert chebyshevt_poly(6, x) == 32*x**6 - 48*x**4 + 18*x**2 - 1 + assert chebyshevt_poly(75, x) == (2*chebyshevt_poly(37, x)*chebyshevt_poly(38, x) - x).expand() + assert chebyshevt_poly(100, x) == (2*chebyshevt_poly(50, x)**2 - 1).expand() + + assert chebyshevt_poly(1).dummy_eq(x) + assert chebyshevt_poly(1, polys=True) == Poly(x) + + +def test_chebyshevu_poly(): + raises(ValueError, lambda: chebyshevu_poly(-1, x)) + + assert chebyshevu_poly(1, x, polys=True) == Poly(2*x) + + assert chebyshevu_poly(0, x) == 1 + assert chebyshevu_poly(1, x) == 2*x + assert chebyshevu_poly(2, x) == 4*x**2 - 1 + assert chebyshevu_poly(3, x) == 8*x**3 - 4*x + assert chebyshevu_poly(4, x) == 16*x**4 - 12*x**2 + 1 + assert chebyshevu_poly(5, x) == 32*x**5 - 32*x**3 + 6*x + assert chebyshevu_poly(6, x) == 64*x**6 - 80*x**4 + 24*x**2 - 1 + + assert chebyshevu_poly(1).dummy_eq(2*x) + assert chebyshevu_poly(1, polys=True) == Poly(2*x) + + +def test_hermite_poly(): + raises(ValueError, lambda: hermite_poly(-1, x)) + + assert hermite_poly(1, x, polys=True) == Poly(2*x) + + assert hermite_poly(0, x) == 1 + assert hermite_poly(1, x) == 2*x + assert hermite_poly(2, x) == 4*x**2 - 2 + assert hermite_poly(3, x) == 8*x**3 - 12*x + assert hermite_poly(4, x) == 16*x**4 - 48*x**2 + 12 + assert hermite_poly(5, x) == 32*x**5 - 160*x**3 + 120*x + assert hermite_poly(6, x) == 64*x**6 - 480*x**4 + 720*x**2 - 120 + + assert hermite_poly(1).dummy_eq(2*x) + assert hermite_poly(1, polys=True) == Poly(2*x) + + +def test_hermite_prob_poly(): + raises(ValueError, lambda: hermite_prob_poly(-1, x)) + + assert hermite_prob_poly(1, x, polys=True) == Poly(x) + + assert hermite_prob_poly(0, x) == 1 + assert hermite_prob_poly(1, x) == x + assert hermite_prob_poly(2, x) == x**2 - 1 + assert hermite_prob_poly(3, x) == x**3 - 3*x + assert hermite_prob_poly(4, x) == x**4 - 6*x**2 + 3 + assert hermite_prob_poly(5, x) == x**5 - 10*x**3 + 15*x + assert hermite_prob_poly(6, x) == x**6 - 15*x**4 + 45*x**2 - 15 + + assert hermite_prob_poly(1).dummy_eq(x) + assert hermite_prob_poly(1, polys=True) == Poly(x) + + +def test_legendre_poly(): + raises(ValueError, lambda: legendre_poly(-1, x)) + + assert legendre_poly(1, x, polys=True) == Poly(x, domain='QQ') + + assert legendre_poly(0, x) == 1 + assert legendre_poly(1, x) == x + assert legendre_poly(2, x) == Q(3, 2)*x**2 - Q(1, 2) + assert legendre_poly(3, x) == Q(5, 2)*x**3 - Q(3, 2)*x + assert legendre_poly(4, x) == Q(35, 8)*x**4 - Q(30, 8)*x**2 + Q(3, 8) + assert legendre_poly(5, x) == Q(63, 8)*x**5 - Q(70, 8)*x**3 + Q(15, 8)*x + assert legendre_poly(6, x) == Q( + 231, 16)*x**6 - Q(315, 16)*x**4 + Q(105, 16)*x**2 - Q(5, 16) + + assert legendre_poly(1).dummy_eq(x) + assert legendre_poly(1, polys=True) == Poly(x) + + +def test_laguerre_poly(): + raises(ValueError, lambda: laguerre_poly(-1, x)) + + assert laguerre_poly(1, x, polys=True) == Poly(-x + 1, domain='QQ') + + assert laguerre_poly(0, x) == 1 + assert laguerre_poly(1, x) == -x + 1 + assert laguerre_poly(2, x) == Q(1, 2)*x**2 - Q(4, 2)*x + 1 + assert laguerre_poly(3, x) == -Q(1, 6)*x**3 + Q(9, 6)*x**2 - Q(18, 6)*x + 1 + assert laguerre_poly(4, x) == Q( + 1, 24)*x**4 - Q(16, 24)*x**3 + Q(72, 24)*x**2 - Q(96, 24)*x + 1 + assert laguerre_poly(5, x) == -Q(1, 120)*x**5 + Q(25, 120)*x**4 - Q( + 200, 120)*x**3 + Q(600, 120)*x**2 - Q(600, 120)*x + 1 + assert laguerre_poly(6, x) == Q(1, 720)*x**6 - Q(36, 720)*x**5 + Q(450, 720)*x**4 - Q(2400, 720)*x**3 + Q(5400, 720)*x**2 - Q(4320, 720)*x + 1 + + assert laguerre_poly(0, x, a) == 1 + assert laguerre_poly(1, x, a) == -x + a + 1 + assert laguerre_poly(2, x, a) == x**2/2 + (-a - 2)*x + a**2/2 + a*Q(3, 2) + 1 + assert laguerre_poly(3, x, a) == -x**3/6 + (a/2 + Q( + 3)/2)*x**2 + (-a**2/2 - a*Q(5, 2) - 3)*x + a**3/6 + a**2 + a*Q(11, 6) + 1 + + assert laguerre_poly(1).dummy_eq(-x + 1) + assert laguerre_poly(1, polys=True) == Poly(-x + 1) + + +def test_spherical_bessel_fn(): + x, z = symbols("x z") + assert spherical_bessel_fn(1, z) == 1/z**2 + assert spherical_bessel_fn(2, z) == -1/z + 3/z**3 + assert spherical_bessel_fn(3, z) == -6/z**2 + 15/z**4 + assert spherical_bessel_fn(4, z) == 1/z - 45/z**3 + 105/z**5 diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_partfrac.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_partfrac.py new file mode 100644 index 0000000000000000000000000000000000000000..83c5d48383d20e67dbb53c081093ad35e654c9a0 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_partfrac.py @@ -0,0 +1,249 @@ +"""Tests for algorithms for partial fraction decomposition of rational +functions. """ + +from sympy.polys.partfrac import ( + apart_undetermined_coeffs, + apart, + apart_list, assemble_partfrac_list +) + +from sympy.core.expr import Expr +from sympy.core.function import Lambda +from sympy.core.numbers import (E, I, Rational, pi, all_close) +from sympy.core.relational import Eq +from sympy.core.singleton import S +from sympy.core.symbol import (Dummy, Symbol) +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.matrices.dense import Matrix +from sympy.polys.polytools import (Poly, factor) +from sympy.polys.rationaltools import together +from sympy.polys.rootoftools import RootSum +from sympy.testing.pytest import raises, XFAIL +from sympy.abc import x, y, a, b, c + + +def test_apart(): + assert apart(1) == 1 + assert apart(1, x) == 1 + + f, g = (x**2 + 1)/(x + 1), 2/(x + 1) + x - 1 + + assert apart(f, full=False) == g + assert apart(f, full=True) == g + + f, g = 1/(x + 2)/(x + 1), 1/(1 + x) - 1/(2 + x) + + assert apart(f, full=False) == g + assert apart(f, full=True) == g + + f, g = 1/(x + 1)/(x + 5), -1/(5 + x)/4 + 1/(1 + x)/4 + + assert apart(f, full=False) == g + assert apart(f, full=True) == g + + assert apart((E*x + 2)/(x - pi)*(x - 1), x) == \ + 2 - E + E*pi + E*x + (E*pi + 2)*(pi - 1)/(x - pi) + + assert apart(Eq((x**2 + 1)/(x + 1), x), x) == Eq(x - 1 + 2/(x + 1), x) + + assert apart(x/2, y) == x/2 + + f, g = (x+y)/(2*x - y), Rational(3, 2)*y/(2*x - y) + S.Half + + assert apart(f, x, full=False) == g + assert apart(f, x, full=True) == g + + f, g = (x+y)/(2*x - y), 3*x/(2*x - y) - 1 + + assert apart(f, y, full=False) == g + assert apart(f, y, full=True) == g + + raises(NotImplementedError, lambda: apart(1/(x + 1)/(y + 2))) + + +def test_apart_matrix(): + M = Matrix(2, 2, lambda i, j: 1/(x + i + 1)/(x + j)) + + assert apart(M) == Matrix([ + [1/x - 1/(x + 1), (x + 1)**(-2)], + [1/(2*x) - (S.Half)/(x + 2), 1/(x + 1) - 1/(x + 2)], + ]) + + +def test_apart_symbolic(): + f = a*x**4 + (2*b + 2*a*c)*x**3 + (4*b*c - a**2 + a*c**2)*x**2 + \ + (-2*a*b + 2*b*c**2)*x - b**2 + g = a**2*x**4 + (2*a*b + 2*c*a**2)*x**3 + (4*a*b*c + b**2 + + a**2*c**2)*x**2 + (2*c*b**2 + 2*a*b*c**2)*x + b**2*c**2 + + assert apart(f/g, x) == 1/a - 1/(x + c)**2 - b**2/(a*(a*x + b)**2) + + assert apart(1/((x + a)*(x + b)*(x + c)), x) == \ + 1/((a - c)*(b - c)*(c + x)) - 1/((a - b)*(b - c)*(b + x)) + \ + 1/((a - b)*(a - c)*(a + x)) + + +def _make_extension_example(): + # https://github.com/sympy/sympy/issues/18531 + from sympy.core import Mul + def mul2(expr): + # 2-arg mul hack... + return Mul(2, expr, evaluate=False) + + f = ((x**2 + 1)**3/((x - 1)**2*(x + 1)**2*(-x**2 + 2*x + 1)*(x**2 + 2*x - 1))) + g = (1/mul2(x - sqrt(2) + 1) + - 1/mul2(x - sqrt(2) - 1) + + 1/mul2(x + 1 + sqrt(2)) + - 1/mul2(x - 1 + sqrt(2)) + + 1/mul2((x + 1)**2) + + 1/mul2((x - 1)**2)) + return f, g + + +def test_apart_extension(): + f = 2/(x**2 + 1) + g = I/(x + I) - I/(x - I) + + assert apart(f, extension=I) == g + assert apart(f, gaussian=True) == g + + f = x/((x - 2)*(x + I)) + + assert factor(together(apart(f)).expand()) == f + + f, g = _make_extension_example() + + # XXX: Only works with dotprodsimp. See test_apart_extension_xfail below + from sympy.matrices import dotprodsimp + with dotprodsimp(True): + assert apart(f, x, extension={sqrt(2)}) == g + + +def test_apart_extension_xfail(): + f, g = _make_extension_example() + assert apart(f, x, extension={sqrt(2)}) == g + + +def test_apart_full(): + f = 1/(x**2 + 1) + + assert apart(f, full=False) == f + assert apart(f, full=True).dummy_eq( + -RootSum(x**2 + 1, Lambda(a, a/(x - a)), auto=False)/2) + + f = 1/(x**3 + x + 1) + + assert apart(f, full=False) == f + assert apart(f, full=True).dummy_eq( + RootSum(x**3 + x + 1, + Lambda(a, (a**2*Rational(6, 31) - a*Rational(9, 31) + Rational(4, 31))/(x - a)), auto=False)) + + f = 1/(x**5 + 1) + + assert apart(f, full=False) == \ + (Rational(-1, 5))*((x**3 - 2*x**2 + 3*x - 4)/(x**4 - x**3 + x**2 - + x + 1)) + (Rational(1, 5))/(x + 1) + assert apart(f, full=True).dummy_eq( + -RootSum(x**4 - x**3 + x**2 - x + 1, + Lambda(a, a/(x - a)), auto=False)/5 + (Rational(1, 5))/(x + 1)) + + +def test_apart_full_floats(): + # https://github.com/sympy/sympy/issues/26648 + f = ( + 6.43369157032015e-9*x**3 + 1.35203404799555e-5*x**2 + + 0.00357538393743079*x + 0.085 + )/( + 4.74334912634438e-11*x**4 + 4.09576274286244e-6*x**3 + + 0.00334241812250921*x**2 + 0.15406018058983*x + 1.0 + ) + + expected = ( + 133.599202650992/(x + 85524.0054884464) + + 1.07757928431867/(x + 774.88576677949) + + 0.395006955518971/(x + 40.7977016133126) + + 0.564264854137341/(x + 7.79746609204661) + ) + + f_apart = apart(f, full=True).evalf() + + # There is a significant floating point error in this operation. + assert all_close(f_apart, expected, rtol=1e-3, atol=1e-5) + + +def test_apart_undetermined_coeffs(): + p = Poly(2*x - 3) + q = Poly(x**9 - x**8 - x**6 + x**5 - 2*x**2 + 3*x - 1) + r = (-x**7 - x**6 - x**5 + 4)/(x**8 - x**5 - 2*x + 1) + 1/(x - 1) + + assert apart_undetermined_coeffs(p, q) == r + + p = Poly(1, x, domain='ZZ[a,b]') + q = Poly((x + a)*(x + b), x, domain='ZZ[a,b]') + r = 1/((a - b)*(b + x)) - 1/((a - b)*(a + x)) + + assert apart_undetermined_coeffs(p, q) == r + + +def test_apart_list(): + from sympy.utilities.iterables import numbered_symbols + def dummy_eq(i, j): + if type(i) in (list, tuple): + return all(dummy_eq(i, j) for i, j in zip(i, j)) + return i == j or i.dummy_eq(j) + + w0, w1, w2 = Symbol("w0"), Symbol("w1"), Symbol("w2") + _a = Dummy("a") + + f = (-2*x - 2*x**2) / (3*x**2 - 6*x) + got = apart_list(f, x, dummies=numbered_symbols("w")) + ans = (-1, Poly(Rational(2, 3), x, domain='QQ'), + [(Poly(w0 - 2, w0, domain='ZZ'), Lambda(_a, 2), Lambda(_a, -_a + x), 1)]) + assert dummy_eq(got, ans) + + got = apart_list(2/(x**2-2), x, dummies=numbered_symbols("w")) + ans = (1, Poly(0, x, domain='ZZ'), [(Poly(w0**2 - 2, w0, domain='ZZ'), + Lambda(_a, _a/2), + Lambda(_a, -_a + x), 1)]) + assert dummy_eq(got, ans) + + f = 36 / (x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2) + got = apart_list(f, x, dummies=numbered_symbols("w")) + ans = (1, Poly(0, x, domain='ZZ'), + [(Poly(w0 - 2, w0, domain='ZZ'), Lambda(_a, 4), Lambda(_a, -_a + x), 1), + (Poly(w1**2 - 1, w1, domain='ZZ'), Lambda(_a, -3*_a - 6), Lambda(_a, -_a + x), 2), + (Poly(w2 + 1, w2, domain='ZZ'), Lambda(_a, -4), Lambda(_a, -_a + x), 1)]) + assert dummy_eq(got, ans) + + +def test_assemble_partfrac_list(): + f = 36 / (x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2) + pfd = apart_list(f) + assert assemble_partfrac_list(pfd) == -4/(x + 1) - 3/(x + 1)**2 - 9/(x - 1)**2 + 4/(x - 2) + + a = Dummy("a") + pfd = (1, Poly(0, x, domain='ZZ'), [([sqrt(2),-sqrt(2)], Lambda(a, a/2), Lambda(a, -a + x), 1)]) + assert assemble_partfrac_list(pfd) == -1/(sqrt(2)*(x + sqrt(2))) + 1/(sqrt(2)*(x - sqrt(2))) + + +@XFAIL +def test_noncommutative_pseudomultivariate(): + # apart doesn't go inside noncommutative expressions + class foo(Expr): + is_commutative=False + e = x/(x + x*y) + c = 1/(1 + y) + assert apart(e + foo(e)) == c + foo(c) + assert apart(e*foo(e)) == c*foo(c) + +def test_noncommutative(): + class foo(Expr): + is_commutative=False + e = x/(x + x*y) + c = 1/(1 + y) + assert apart(e + foo()) == c + foo() + +def test_issue_5798(): + assert apart( + 2*x/(x**2 + 1) - (x - 1)/(2*(x**2 + 1)) + 1/(2*(x + 1)) - 2/x) == \ + (3*x + 1)/(x**2 + 1)/2 + 1/(x + 1)/2 - 2/x diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_polyclasses.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_polyclasses.py new file mode 100644 index 0000000000000000000000000000000000000000..5e2c8f2c3ca94c42fc524c3ec1c0300d881cf3a5 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_polyclasses.py @@ -0,0 +1,601 @@ +"""Tests for OO layer of several polynomial representations. """ + +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.polys.domains import ZZ, QQ +from sympy.polys.polyclasses import DMP, DMF, ANP +from sympy.polys.polyerrors import (CoercionFailed, ExactQuotientFailed, + NotInvertible) +from sympy.polys.specialpolys import f_polys +from sympy.testing.pytest import raises, warns_deprecated_sympy + +f_0, f_1, f_2, f_3, f_4, f_5, f_6 = [ f.to_dense() for f in f_polys() ] + +def test_DMP___init__(): + f = DMP([[ZZ(0)], [], [ZZ(0), ZZ(1), ZZ(2)], [ZZ(3)]], ZZ) + + assert f._rep == [[1, 2], [3]] + assert f.dom == ZZ + assert f.lev == 1 + + f = DMP([[ZZ(1), ZZ(2)], [ZZ(3)]], ZZ, 1) + + assert f._rep == [[1, 2], [3]] + assert f.dom == ZZ + assert f.lev == 1 + + f = DMP.from_dict({(1, 1): ZZ(1), (0, 0): ZZ(2)}, 1, ZZ) + + assert f._rep == [[1, 0], [2]] + assert f.dom == ZZ + assert f.lev == 1 + + +def test_DMP_rep_deprecation(): + f = DMP([1, 2, 3], ZZ) + + with warns_deprecated_sympy(): + assert f.rep == [1, 2, 3] + + +def test_DMP___eq__(): + assert DMP([[ZZ(1), ZZ(2)], [ZZ(3)]], ZZ) == \ + DMP([[ZZ(1), ZZ(2)], [ZZ(3)]], ZZ) + + assert DMP([[ZZ(1), ZZ(2)], [ZZ(3)]], ZZ) == \ + DMP([[QQ(1), QQ(2)], [QQ(3)]], QQ) + assert DMP([[QQ(1), QQ(2)], [QQ(3)]], QQ) == \ + DMP([[ZZ(1), ZZ(2)], [ZZ(3)]], ZZ) + + assert DMP([[[ZZ(1)]]], ZZ) != DMP([[ZZ(1)]], ZZ) + assert DMP([[ZZ(1)]], ZZ) != DMP([[[ZZ(1)]]], ZZ) + + +def test_DMP___bool__(): + assert bool(DMP([[]], ZZ)) is False + assert bool(DMP([[ZZ(1)]], ZZ)) is True + + +def test_DMP_to_dict(): + f = DMP([[ZZ(3)], [], [ZZ(2)], [], [ZZ(8)]], ZZ) + + assert f.to_dict() == \ + {(4, 0): 3, (2, 0): 2, (0, 0): 8} + assert f.to_sympy_dict() == \ + {(4, 0): ZZ.to_sympy(3), (2, 0): ZZ.to_sympy(2), (0, 0): + ZZ.to_sympy(8)} + + +def test_DMP_properties(): + assert DMP([[]], ZZ).is_zero is True + assert DMP([[ZZ(1)]], ZZ).is_zero is False + + assert DMP([[ZZ(1)]], ZZ).is_one is True + assert DMP([[ZZ(2)]], ZZ).is_one is False + + assert DMP([[ZZ(1)]], ZZ).is_ground is True + assert DMP([[ZZ(1)], [ZZ(2)], [ZZ(1)]], ZZ).is_ground is False + + assert DMP([[ZZ(1)], [ZZ(2), ZZ(0)], [ZZ(1), ZZ(0)]], ZZ).is_sqf is True + assert DMP([[ZZ(1)], [ZZ(2), ZZ(0)], [ZZ(1), ZZ(0), ZZ(0)]], ZZ).is_sqf is False + + assert DMP([[ZZ(1), ZZ(2)], [ZZ(3)]], ZZ).is_monic is True + assert DMP([[ZZ(2), ZZ(2)], [ZZ(3)]], ZZ).is_monic is False + + assert DMP([[ZZ(1), ZZ(2)], [ZZ(3)]], ZZ).is_primitive is True + assert DMP([[ZZ(2), ZZ(4)], [ZZ(6)]], ZZ).is_primitive is False + + +def test_DMP_arithmetics(): + f = DMP([[ZZ(2)], [ZZ(2), ZZ(0)]], ZZ) + + assert f.mul_ground(2) == DMP([[ZZ(4)], [ZZ(4), ZZ(0)]], ZZ) + assert f.quo_ground(2) == DMP([[ZZ(1)], [ZZ(1), ZZ(0)]], ZZ) + + raises(ExactQuotientFailed, lambda: f.exquo_ground(3)) + + f = DMP([[ZZ(-5)]], ZZ) + g = DMP([[ZZ(5)]], ZZ) + + assert f.abs() == g + assert abs(f) == g + + assert g.neg() == f + assert -g == f + + h = DMP([[]], ZZ) + + assert f.add(g) == h + assert f + g == h + assert g + f == h + assert f + 5 == h + assert 5 + f == h + + h = DMP([[ZZ(-10)]], ZZ) + + assert f.sub(g) == h + assert f - g == h + assert g - f == -h + assert f - 5 == h + assert 5 - f == -h + + h = DMP([[ZZ(-25)]], ZZ) + + assert f.mul(g) == h + assert f * g == h + assert g * f == h + assert f * 5 == h + assert 5 * f == h + + h = DMP([[ZZ(25)]], ZZ) + + assert f.sqr() == h + assert f.pow(2) == h + assert f**2 == h + + raises(TypeError, lambda: f.pow('x')) + + f = DMP([[ZZ(1)], [], [ZZ(1), ZZ(0), ZZ(0)]], ZZ) + g = DMP([[ZZ(2)], [ZZ(-2), ZZ(0)]], ZZ) + + q = DMP([[ZZ(2)], [ZZ(2), ZZ(0)]], ZZ) + r = DMP([[ZZ(8), ZZ(0), ZZ(0)]], ZZ) + + assert f.pdiv(g) == (q, r) + assert f.pquo(g) == q + assert f.prem(g) == r + + raises(ExactQuotientFailed, lambda: f.pexquo(g)) + + f = DMP([[ZZ(1)], [], [ZZ(1), ZZ(0), ZZ(0)]], ZZ) + g = DMP([[ZZ(1)], [ZZ(-1), ZZ(0)]], ZZ) + + q = DMP([[ZZ(1)], [ZZ(1), ZZ(0)]], ZZ) + r = DMP([[ZZ(2), ZZ(0), ZZ(0)]], ZZ) + + assert f.div(g) == (q, r) + assert f.quo(g) == q + assert f.rem(g) == r + + assert divmod(f, g) == (q, r) + assert f // g == q + assert f % g == r + + raises(ExactQuotientFailed, lambda: f.exquo(g)) + + f = DMP([ZZ(1), ZZ(0), ZZ(-1)], ZZ) + g = DMP([ZZ(2), ZZ(-2)], ZZ) + + q = DMP([], ZZ) + r = f + + pq = DMP([ZZ(2), ZZ(2)], ZZ) + pr = DMP([], ZZ) + + assert f.div(g) == (q, r) + assert f.quo(g) == q + assert f.rem(g) == r + + assert divmod(f, g) == (q, r) + assert f // g == q + assert f % g == r + + raises(ExactQuotientFailed, lambda: f.exquo(g)) + + assert f.pdiv(g) == (pq, pr) + assert f.pquo(g) == pq + assert f.prem(g) == pr + assert f.pexquo(g) == pq + + +def test_DMP_functionality(): + f = DMP([[ZZ(1)], [ZZ(2), ZZ(0)], [ZZ(1), ZZ(0), ZZ(0)]], ZZ) + g = DMP([[ZZ(1)], [ZZ(1), ZZ(0)]], ZZ) + h = DMP([[ZZ(1)]], ZZ) + + assert f.degree() == 2 + assert f.degree_list() == (2, 2) + assert f.total_degree() == 2 + + assert f.LC() == ZZ(1) + assert f.TC() == ZZ(0) + assert f.nth(1, 1) == ZZ(2) + + raises(TypeError, lambda: f.nth(0, 'x')) + + assert f.max_norm() == 2 + assert f.l1_norm() == 4 + + u = DMP([[ZZ(2)], [ZZ(2), ZZ(0)]], ZZ) + + assert f.diff(m=1, j=0) == u + assert f.diff(m=1, j=1) == u + + raises(TypeError, lambda: f.diff(m='x', j=0)) + + u = DMP([ZZ(1), ZZ(2), ZZ(1)], ZZ) + v = DMP([ZZ(1), ZZ(2), ZZ(1)], ZZ) + + assert f.eval(a=1, j=0) == u + assert f.eval(a=1, j=1) == v + + assert f.eval(1).eval(1) == ZZ(4) + + assert f.cofactors(g) == (g, g, h) + assert f.gcd(g) == g + assert f.lcm(g) == f + + u = DMP([[QQ(45), QQ(30), QQ(5)]], QQ) + v = DMP([[QQ(1), QQ(2, 3), QQ(1, 9)]], QQ) + + assert u.monic() == v + + assert (4*f).content() == ZZ(4) + assert (4*f).primitive() == (ZZ(4), f) + + f = DMP([QQ(1,3), QQ(1)], QQ) + g = DMP([QQ(1,7), QQ(1)], QQ) + + assert f.cancel(g) == f.cancel(g, include=True) == ( + DMP([QQ(7), QQ(21)], QQ), + DMP([QQ(3), QQ(21)], QQ) + ) + assert f.cancel(g, include=False) == ( + QQ(7), + QQ(3), + DMP([QQ(1), QQ(3)], QQ), + DMP([QQ(1), QQ(7)], QQ) + ) + + f = DMP([[ZZ(1)], [ZZ(2)], [ZZ(3)], [ZZ(4)], [ZZ(5)], [ZZ(6)]], ZZ) + + assert f.trunc(3) == DMP([[ZZ(1)], [ZZ(-1)], [], [ZZ(1)], [ZZ(-1)], []], ZZ) + + f = DMP(f_4, ZZ) + + assert f.sqf_part() == -f + assert f.sqf_list() == (ZZ(-1), [(-f, 1)]) + + f = DMP([[ZZ(-1)], [], [], [ZZ(5)]], ZZ) + g = DMP([[ZZ(3), ZZ(1)], [], []], ZZ) + h = DMP([[ZZ(45), ZZ(30), ZZ(5)]], ZZ) + + r = DMP([ZZ(675), ZZ(675), ZZ(225), ZZ(25)], ZZ) + + assert f.subresultants(g) == [f, g, h] + assert f.resultant(g) == r + + f = DMP([ZZ(1), ZZ(3), ZZ(9), ZZ(-13)], ZZ) + + assert f.discriminant() == -11664 + + f = DMP([QQ(2), QQ(0)], QQ) + g = DMP([QQ(1), QQ(0), QQ(-16)], QQ) + + s = DMP([QQ(1, 32), QQ(0)], QQ) + t = DMP([QQ(-1, 16)], QQ) + h = DMP([QQ(1)], QQ) + + assert f.half_gcdex(g) == (s, h) + assert f.gcdex(g) == (s, t, h) + + assert f.invert(g) == s + + f = DMP([[QQ(1)], [QQ(2)], [QQ(3)]], QQ) + + raises(ValueError, lambda: f.half_gcdex(f)) + raises(ValueError, lambda: f.gcdex(f)) + + raises(ValueError, lambda: f.invert(f)) + + f = DMP(ZZ.map([1, 0, 20, 0, 150, 0, 500, 0, 625, -2, 0, -10, 9]), ZZ) + g = DMP([ZZ(1), ZZ(0), ZZ(0), ZZ(-2), ZZ(9)], ZZ) + h = DMP([ZZ(1), ZZ(0), ZZ(5), ZZ(0)], ZZ) + + assert g.compose(h) == f + assert f.decompose() == [g, h] + + f = DMP([[QQ(1)], [QQ(2)], [QQ(3)]], QQ) + + raises(ValueError, lambda: f.decompose()) + raises(ValueError, lambda: f.sturm()) + + +def test_DMP_exclude(): + f = [[[[[[[[[[[[[[[[[[[[[[[[[[ZZ(1)]], [[]]]]]]]]]]]]]]]]]]]]]]]]]] + J = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, + 18, 19, 20, 21, 22, 24, 25] + + assert DMP(f, ZZ).exclude() == (J, DMP([ZZ(1), ZZ(0)], ZZ)) + assert DMP([[ZZ(1)], [ZZ(1), ZZ(0)]], ZZ).exclude() ==\ + ([], DMP([[ZZ(1)], [ZZ(1), ZZ(0)]], ZZ)) + + +def test_DMF__init__(): + f = DMF(([[0], [], [0, 1, 2], [3]], [[1, 2, 3]]), ZZ) + + assert f.num == [[1, 2], [3]] + assert f.den == [[1, 2, 3]] + assert f.lev == 1 + assert f.dom == ZZ + + f = DMF(([[1, 2], [3]], [[1, 2, 3]]), ZZ, 1) + + assert f.num == [[1, 2], [3]] + assert f.den == [[1, 2, 3]] + assert f.lev == 1 + assert f.dom == ZZ + + f = DMF(([[-1], [-2]], [[3], [-4]]), ZZ) + + assert f.num == [[-1], [-2]] + assert f.den == [[3], [-4]] + assert f.lev == 1 + assert f.dom == ZZ + + f = DMF(([[1], [2]], [[-3], [4]]), ZZ) + + assert f.num == [[-1], [-2]] + assert f.den == [[3], [-4]] + assert f.lev == 1 + assert f.dom == ZZ + + f = DMF(([[1], [2]], [[-3], [4]]), ZZ) + + assert f.num == [[-1], [-2]] + assert f.den == [[3], [-4]] + assert f.lev == 1 + assert f.dom == ZZ + + f = DMF(([[]], [[-3], [4]]), ZZ) + + assert f.num == [[]] + assert f.den == [[1]] + assert f.lev == 1 + assert f.dom == ZZ + + f = DMF(17, ZZ, 1) + + assert f.num == [[17]] + assert f.den == [[1]] + assert f.lev == 1 + assert f.dom == ZZ + + f = DMF(([[1], [2]]), ZZ) + + assert f.num == [[1], [2]] + assert f.den == [[1]] + assert f.lev == 1 + assert f.dom == ZZ + + f = DMF([[0], [], [0, 1, 2], [3]], ZZ) + + assert f.num == [[1, 2], [3]] + assert f.den == [[1]] + assert f.lev == 1 + assert f.dom == ZZ + + f = DMF({(1, 1): 1, (0, 0): 2}, ZZ, 1) + + assert f.num == [[1, 0], [2]] + assert f.den == [[1]] + assert f.lev == 1 + assert f.dom == ZZ + + f = DMF(([[QQ(1)], [QQ(2)]], [[-QQ(3)], [QQ(4)]]), QQ) + + assert f.num == [[-QQ(1)], [-QQ(2)]] + assert f.den == [[QQ(3)], [-QQ(4)]] + assert f.lev == 1 + assert f.dom == QQ + + f = DMF(([[QQ(1, 5)], [QQ(2, 5)]], [[-QQ(3, 7)], [QQ(4, 7)]]), QQ) + + assert f.num == [[-QQ(7)], [-QQ(14)]] + assert f.den == [[QQ(15)], [-QQ(20)]] + assert f.lev == 1 + assert f.dom == QQ + + raises(ValueError, lambda: DMF(([1], [[1]]), ZZ)) + raises(ZeroDivisionError, lambda: DMF(([1], []), ZZ)) + + +def test_DMF__bool__(): + assert bool(DMF([[]], ZZ)) is False + assert bool(DMF([[1]], ZZ)) is True + + +def test_DMF_properties(): + assert DMF([[]], ZZ).is_zero is True + assert DMF([[]], ZZ).is_one is False + + assert DMF([[1]], ZZ).is_zero is False + assert DMF([[1]], ZZ).is_one is True + + assert DMF(([[1]], [[2]]), ZZ).is_one is False + + +def test_DMF_arithmetics(): + f = DMF([[7], [-9]], ZZ) + g = DMF([[-7], [9]], ZZ) + + assert f.neg() == -f == g + + f = DMF(([[1]], [[1], []]), ZZ) + g = DMF(([[1]], [[1, 0]]), ZZ) + + h = DMF(([[1], [1, 0]], [[1, 0], []]), ZZ) + + assert f.add(g) == f + g == h + assert g.add(f) == g + f == h + + h = DMF(([[-1], [1, 0]], [[1, 0], []]), ZZ) + + assert f.sub(g) == f - g == h + + h = DMF(([[1]], [[1, 0], []]), ZZ) + + assert f.mul(g) == f*g == h + assert g.mul(f) == g*f == h + + h = DMF(([[1, 0]], [[1], []]), ZZ) + + assert f.quo(g) == f/g == h + + h = DMF(([[1]], [[1], [], [], []]), ZZ) + + assert f.pow(3) == f**3 == h + + h = DMF(([[1]], [[1, 0, 0, 0]]), ZZ) + + assert g.pow(3) == g**3 == h + + h = DMF(([[1, 0]], [[1]]), ZZ) + + assert g.pow(-1) == g**-1 == h + + +def test_ANP___init__(): + rep = [QQ(1), QQ(1)] + mod = [QQ(1), QQ(0), QQ(1)] + + f = ANP(rep, mod, QQ) + + assert f.to_list() == [QQ(1), QQ(1)] + assert f.mod_to_list() == [QQ(1), QQ(0), QQ(1)] + assert f.dom == QQ + + rep = {1: QQ(1), 0: QQ(1)} + mod = {2: QQ(1), 0: QQ(1)} + + f = ANP(rep, mod, QQ) + + assert f.to_list() == [QQ(1), QQ(1)] + assert f.mod_to_list() == [QQ(1), QQ(0), QQ(1)] + assert f.dom == QQ + + f = ANP(1, mod, QQ) + + assert f.to_list() == [QQ(1)] + assert f.mod_to_list() == [QQ(1), QQ(0), QQ(1)] + assert f.dom == QQ + + f = ANP([1, 0.5], mod, QQ) + + assert all(QQ.of_type(a) for a in f.to_list()) + + raises(CoercionFailed, lambda: ANP([sqrt(2)], mod, QQ)) + + +def test_ANP___eq__(): + a = ANP([QQ(1), QQ(1)], [QQ(1), QQ(0), QQ(1)], QQ) + b = ANP([QQ(1), QQ(1)], [QQ(1), QQ(0), QQ(2)], QQ) + + assert (a == a) is True + assert (a != a) is False + + assert (a == b) is False + assert (a != b) is True + + b = ANP([QQ(1), QQ(2)], [QQ(1), QQ(0), QQ(1)], QQ) + + assert (a == b) is False + assert (a != b) is True + + +def test_ANP___bool__(): + assert bool(ANP([], [QQ(1), QQ(0), QQ(1)], QQ)) is False + assert bool(ANP([QQ(1)], [QQ(1), QQ(0), QQ(1)], QQ)) is True + + +def test_ANP_properties(): + mod = [QQ(1), QQ(0), QQ(1)] + + assert ANP([QQ(0)], mod, QQ).is_zero is True + assert ANP([QQ(1)], mod, QQ).is_zero is False + + assert ANP([QQ(1)], mod, QQ).is_one is True + assert ANP([QQ(2)], mod, QQ).is_one is False + + +def test_ANP_arithmetics(): + mod = [QQ(1), QQ(0), QQ(0), QQ(-2)] + + a = ANP([QQ(2), QQ(-1), QQ(1)], mod, QQ) + b = ANP([QQ(1), QQ(2)], mod, QQ) + + c = ANP([QQ(-2), QQ(1), QQ(-1)], mod, QQ) + + assert a.neg() == -a == c + + c = ANP([QQ(2), QQ(0), QQ(3)], mod, QQ) + + assert a.add(b) == a + b == c + assert b.add(a) == b + a == c + + c = ANP([QQ(2), QQ(-2), QQ(-1)], mod, QQ) + + assert a.sub(b) == a - b == c + + c = ANP([QQ(-2), QQ(2), QQ(1)], mod, QQ) + + assert b.sub(a) == b - a == c + + c = ANP([QQ(3), QQ(-1), QQ(6)], mod, QQ) + + assert a.mul(b) == a*b == c + assert b.mul(a) == b*a == c + + c = ANP([QQ(-1, 43), QQ(9, 43), QQ(5, 43)], mod, QQ) + + assert a.pow(0) == a**(0) == ANP(1, mod, QQ) + assert a.pow(1) == a**(1) == a + + assert a.pow(-1) == a**(-1) == c + + assert a.quo(a) == a.mul(a.pow(-1)) == a*a**(-1) == ANP(1, mod, QQ) + + c = ANP([], [1, 0, 0, -2], QQ) + r1 = a.rem(b) + + (q, r2) = a.div(b) + + assert r1 == r2 == c == a % b + + raises(NotInvertible, lambda: a.div(c)) + raises(NotInvertible, lambda: a.rem(c)) + + # Comparison with "hard-coded" value fails despite looking identical + # from sympy import Rational + # c = ANP([Rational(11, 10), Rational(-1, 5), Rational(-3, 5)], [1, 0, 0, -2], QQ) + + assert q == a/b # == c + +def test_ANP_unify(): + mod_z = [ZZ(1), ZZ(0), ZZ(-2)] + mod_q = [QQ(1), QQ(0), QQ(-2)] + + a = ANP([QQ(1)], mod_q, QQ) + b = ANP([ZZ(1)], mod_z, ZZ) + + assert a.unify(b)[0] == QQ + assert b.unify(a)[0] == QQ + assert a.unify(a)[0] == QQ + assert b.unify(b)[0] == ZZ + + assert a.unify_ANP(b)[-1] == QQ + assert b.unify_ANP(a)[-1] == QQ + assert a.unify_ANP(a)[-1] == QQ + assert b.unify_ANP(b)[-1] == ZZ + + +def test_zero_poly(): + from sympy import Symbol + x = Symbol('x') + + R_old = ZZ.old_poly_ring(x) + zero_poly_old = R_old(0) + cont_old, prim_old = zero_poly_old.primitive() + + assert cont_old == 0 + assert prim_old == zero_poly_old + assert prim_old.is_primitive is False diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_polyfuncs.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_polyfuncs.py new file mode 100644 index 0000000000000000000000000000000000000000..496f63bf14e4dd9f68cf653004eb35a3ed7615ca --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_polyfuncs.py @@ -0,0 +1,126 @@ +"""Tests for high-level polynomials manipulation functions. """ + +from sympy.polys.polyfuncs import ( + symmetrize, horner, interpolate, rational_interpolate, viete, +) + +from sympy.polys.polyerrors import ( + MultivariatePolynomialError, +) + +from sympy.core.singleton import S +from sympy.core.symbol import symbols +from sympy.testing.pytest import raises + +from sympy.abc import a, b, c, d, e, x, y, z + + +def test_symmetrize(): + assert symmetrize(0, x, y, z) == (0, 0) + assert symmetrize(1, x, y, z) == (1, 0) + + s1 = x + y + z + s2 = x*y + x*z + y*z + + assert symmetrize(1) == (1, 0) + assert symmetrize(1, formal=True) == (1, 0, []) + + assert symmetrize(x) == (x, 0) + assert symmetrize(x + 1) == (x + 1, 0) + + assert symmetrize(x, x, y) == (x + y, -y) + assert symmetrize(x + 1, x, y) == (x + y + 1, -y) + + assert symmetrize(x, x, y, z) == (s1, -y - z) + assert symmetrize(x + 1, x, y, z) == (s1 + 1, -y - z) + + assert symmetrize(x**2, x, y, z) == (s1**2 - 2*s2, -y**2 - z**2) + + assert symmetrize(x**2 + y**2) == (-2*x*y + (x + y)**2, 0) + assert symmetrize(x**2 - y**2) == (-2*x*y + (x + y)**2, -2*y**2) + + assert symmetrize(x**3 + y**2 + a*x**2 + b*y**3, x, y) == \ + (-3*x*y*(x + y) - 2*a*x*y + a*(x + y)**2 + (x + y)**3, + y**2*(1 - a) + y**3*(b - 1)) + + U = [u0, u1, u2] = symbols('u:3') + + assert symmetrize(x + 1, x, y, z, formal=True, symbols=U) == \ + (u0 + 1, -y - z, [(u0, x + y + z), (u1, x*y + x*z + y*z), (u2, x*y*z)]) + + assert symmetrize([1, 2, 3]) == [(1, 0), (2, 0), (3, 0)] + assert symmetrize([1, 2, 3], formal=True) == ([(1, 0), (2, 0), (3, 0)], []) + + assert symmetrize([x + y, x - y]) == [(x + y, 0), (x + y, -2*y)] + + +def test_horner(): + assert horner(0) == 0 + assert horner(1) == 1 + assert horner(x) == x + + assert horner(x + 1) == x + 1 + assert horner(x**2 + 1) == x**2 + 1 + assert horner(x**2 + x) == (x + 1)*x + assert horner(x**2 + x + 1) == (x + 1)*x + 1 + + assert horner( + 9*x**4 + 8*x**3 + 7*x**2 + 6*x + 5) == (((9*x + 8)*x + 7)*x + 6)*x + 5 + assert horner( + a*x**4 + b*x**3 + c*x**2 + d*x + e) == (((a*x + b)*x + c)*x + d)*x + e + + assert horner(4*x**2*y**2 + 2*x**2*y + 2*x*y**2 + x*y, wrt=x) == (( + 4*y + 2)*x*y + (2*y + 1)*y)*x + assert horner(4*x**2*y**2 + 2*x**2*y + 2*x*y**2 + x*y, wrt=y) == (( + 4*x + 2)*y*x + (2*x + 1)*x)*y + + +def test_interpolate(): + assert interpolate([1, 4, 9, 16], x) == x**2 + assert interpolate([1, 4, 9, 25], x) == S(3)*x**3/2 - S(8)*x**2 + S(33)*x/2 - 9 + assert interpolate([(1, 1), (2, 4), (3, 9)], x) == x**2 + assert interpolate([(1, 2), (2, 5), (3, 10)], x) == 1 + x**2 + assert interpolate({1: 2, 2: 5, 3: 10}, x) == 1 + x**2 + assert interpolate({5: 2, 7: 5, 8: 10, 9: 13}, x) == \ + -S(13)*x**3/24 + S(12)*x**2 - S(2003)*x/24 + 187 + assert interpolate([(1, 3), (0, 6), (2, 5), (5, 7), (-2, 4)], x) == \ + S(-61)*x**4/280 + S(247)*x**3/210 + S(139)*x**2/280 - S(1871)*x/420 + 6 + assert interpolate((9, 4, 9), 3) == 9 + assert interpolate((1, 9, 16), 1) is S.One + assert interpolate(((x, 1), (2, 3)), x) is S.One + assert interpolate({x: 1, 2: 3}, x) is S.One + assert interpolate(((2, x), (1, 3)), x) == x**2 - 4*x + 6 + + +def test_rational_interpolate(): + x, y = symbols('x,y') + xdata = [1, 2, 3, 4, 5, 6] + ydata1 = [120, 150, 200, 255, 312, 370] + ydata2 = [-210, -35, 105, 231, 350, 465] + assert rational_interpolate(list(zip(xdata, ydata1)), 2) == ( + (60*x**2 + 60)/x ) + assert rational_interpolate(list(zip(xdata, ydata1)), 3) == ( + (60*x**2 + 60)/x ) + assert rational_interpolate(list(zip(xdata, ydata2)), 2, X=y) == ( + (105*y**2 - 525)/(y + 1) ) + xdata = list(range(1,11)) + ydata = [-1923885361858460, -5212158811973685, -9838050145867125, + -15662936261217245, -22469424125057910, -30073793365223685, + -38332297297028735, -47132954289530109, -56387719094026320, + -66026548943876885] + assert rational_interpolate(list(zip(xdata, ydata)), 5) == ( + (-12986226192544605*x**4 + + 8657484128363070*x**3 - 30301194449270745*x**2 + 4328742064181535*x + - 4328742064181535)/(x**3 + 9*x**2 - 3*x + 11)) + + +def test_viete(): + r1, r2 = symbols('r1, r2') + + assert viete( + a*x**2 + b*x + c, [r1, r2], x) == [(r1 + r2, -b/a), (r1*r2, c/a)] + + raises(ValueError, lambda: viete(1, [], x)) + raises(ValueError, lambda: viete(x**2 + 1, [r1])) + + raises(MultivariatePolynomialError, lambda: viete(x + y, [r1])) diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_polymatrix.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_polymatrix.py new file mode 100644 index 0000000000000000000000000000000000000000..287f23d537392510acda094e764a8c3dbbd1ef73 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_polymatrix.py @@ -0,0 +1,185 @@ +from sympy.testing.pytest import raises + +from sympy.polys.polymatrix import PolyMatrix +from sympy.polys import Poly + +from sympy.core.singleton import S +from sympy.matrices.dense import Matrix +from sympy.polys.domains.integerring import ZZ +from sympy.polys.domains.rationalfield import QQ + +from sympy.abc import x, y + + +def _test_polymatrix(): + pm1 = PolyMatrix([[Poly(x**2, x), Poly(-x, x)], [Poly(x**3, x), Poly(-1 + x, x)]]) + v1 = PolyMatrix([[1, 0], [-1, 0]], ring='ZZ[x]') + m1 = PolyMatrix([[1, 0], [-1, 0]], ring='ZZ[x]') + A = PolyMatrix([[Poly(x**2 + x, x), Poly(0, x)], \ + [Poly(x**3 - x + 1, x), Poly(0, x)]]) + B = PolyMatrix([[Poly(x**2, x), Poly(-x, x)], [Poly(-x**2, x), Poly(x, x)]]) + assert A.ring == ZZ[x] + assert isinstance(pm1*v1, PolyMatrix) + assert pm1*v1 == A + assert pm1*m1 == A + assert v1*pm1 == B + + pm2 = PolyMatrix([[Poly(x**2, x, domain='QQ'), Poly(0, x, domain='QQ'), Poly(-x**2, x, domain='QQ'), \ + Poly(x**3, x, domain='QQ'), Poly(0, x, domain='QQ'), Poly(-x**3, x, domain='QQ')]]) + assert pm2.ring == QQ[x] + v2 = PolyMatrix([1, 0, 0, 0, 0, 0], ring='ZZ[x]') + m2 = PolyMatrix([1, 0, 0, 0, 0, 0], ring='ZZ[x]') + C = PolyMatrix([[Poly(x**2, x, domain='QQ')]]) + assert pm2*v2 == C + assert pm2*m2 == C + + pm3 = PolyMatrix([[Poly(x**2, x), S.One]], ring='ZZ[x]') + v3 = S.Half*pm3 + assert v3 == PolyMatrix([[Poly(S.Half*x**2, x, domain='QQ'), S.Half]], ring='QQ[x]') + assert pm3*S.Half == v3 + assert v3.ring == QQ[x] + + pm4 = PolyMatrix([[Poly(x**2, x, domain='ZZ'), Poly(-x**2, x, domain='ZZ')]]) + v4 = PolyMatrix([1, -1], ring='ZZ[x]') + assert pm4*v4 == PolyMatrix([[Poly(2*x**2, x, domain='ZZ')]]) + + assert len(PolyMatrix(ring=ZZ[x])) == 0 + assert PolyMatrix([1, 0, 0, 1], x)/(-1) == PolyMatrix([-1, 0, 0, -1], x) + + +def test_polymatrix_constructor(): + M1 = PolyMatrix([[x, y]], ring=QQ[x,y]) + assert M1.ring == QQ[x,y] + assert M1.domain == QQ + assert M1.gens == (x, y) + assert M1.shape == (1, 2) + assert M1.rows == 1 + assert M1.cols == 2 + assert len(M1) == 2 + assert list(M1) == [Poly(x, (x, y), domain=QQ), Poly(y, (x, y), domain=QQ)] + + M2 = PolyMatrix([[x, y]], ring=QQ[x][y]) + assert M2.ring == QQ[x][y] + assert M2.domain == QQ[x] + assert M2.gens == (y,) + assert M2.shape == (1, 2) + assert M2.rows == 1 + assert M2.cols == 2 + assert len(M2) == 2 + assert list(M2) == [Poly(x, (y,), domain=QQ[x]), Poly(y, (y,), domain=QQ[x])] + + assert PolyMatrix([[x, y]], y) == PolyMatrix([[x, y]], ring=ZZ.frac_field(x)[y]) + assert PolyMatrix([[x, y]], ring='ZZ[x,y]') == PolyMatrix([[x, y]], ring=ZZ[x,y]) + + assert PolyMatrix([[x, y]], (x, y)) == PolyMatrix([[x, y]], ring=QQ[x,y]) + assert PolyMatrix([[x, y]], x, y) == PolyMatrix([[x, y]], ring=QQ[x,y]) + assert PolyMatrix([x, y]) == PolyMatrix([[x], [y]], ring=QQ[x,y]) + assert PolyMatrix(1, 2, [x, y]) == PolyMatrix([[x, y]], ring=QQ[x,y]) + assert PolyMatrix(1, 2, lambda i,j: [x,y][j]) == PolyMatrix([[x, y]], ring=QQ[x,y]) + assert PolyMatrix(0, 2, [], x, y).shape == (0, 2) + assert PolyMatrix(2, 0, [], x, y).shape == (2, 0) + assert PolyMatrix([[], []], x, y).shape == (2, 0) + assert PolyMatrix(ring=QQ[x,y]) == PolyMatrix(0, 0, [], ring=QQ[x,y]) == PolyMatrix([], ring=QQ[x,y]) + raises(TypeError, lambda: PolyMatrix()) + raises(TypeError, lambda: PolyMatrix(1)) + + assert PolyMatrix([Poly(x), Poly(y)]) == PolyMatrix([[x], [y]], ring=ZZ[x,y]) + + # XXX: Maybe a bug in parallel_poly_from_expr (x lost from gens and domain): + assert PolyMatrix([Poly(y, x), 1]) == PolyMatrix([[y], [1]], ring=QQ[y]) + + +def test_polymatrix_eq(): + assert (PolyMatrix([x]) == PolyMatrix([x])) is True + assert (PolyMatrix([y]) == PolyMatrix([x])) is False + assert (PolyMatrix([x]) != PolyMatrix([x])) is False + assert (PolyMatrix([y]) != PolyMatrix([x])) is True + + assert PolyMatrix([[x, y]]) != PolyMatrix([x, y]) == PolyMatrix([[x], [y]]) + + assert PolyMatrix([x], ring=QQ[x]) != PolyMatrix([x], ring=ZZ[x]) + + assert PolyMatrix([x]) != Matrix([x]) + assert PolyMatrix([x]).to_Matrix() == Matrix([x]) + + assert PolyMatrix([1], x) == PolyMatrix([1], x) + assert PolyMatrix([1], x) != PolyMatrix([1], y) + + +def test_polymatrix_from_Matrix(): + assert PolyMatrix.from_Matrix(Matrix([1, 2]), x) == PolyMatrix([1, 2], x, ring=QQ[x]) + assert PolyMatrix.from_Matrix(Matrix([1]), ring=QQ[x]) == PolyMatrix([1], x) + pmx = PolyMatrix([1, 2], x) + pmy = PolyMatrix([1, 2], y) + assert pmx != pmy + assert pmx.set_gens(y) == pmy + + +def test_polymatrix_repr(): + assert repr(PolyMatrix([[1, 2]], x)) == 'PolyMatrix([[1, 2]], ring=QQ[x])' + assert repr(PolyMatrix(0, 2, [], x)) == 'PolyMatrix(0, 2, [], ring=QQ[x])' + + +def test_polymatrix_getitem(): + M = PolyMatrix([[1, 2], [3, 4]], x) + assert M[:, :] == M + assert M[0, :] == PolyMatrix([[1, 2]], x) + assert M[:, 0] == PolyMatrix([1, 3], x) + assert M[0, 0] == Poly(1, x, domain=QQ) + assert M[0] == Poly(1, x, domain=QQ) + assert M[:2] == [Poly(1, x, domain=QQ), Poly(2, x, domain=QQ)] + + +def test_polymatrix_arithmetic(): + M = PolyMatrix([[1, 2], [3, 4]], x) + assert M + M == PolyMatrix([[2, 4], [6, 8]], x) + assert M - M == PolyMatrix([[0, 0], [0, 0]], x) + assert -M == PolyMatrix([[-1, -2], [-3, -4]], x) + raises(TypeError, lambda: M + 1) + raises(TypeError, lambda: M - 1) + raises(TypeError, lambda: 1 + M) + raises(TypeError, lambda: 1 - M) + + assert M * M == PolyMatrix([[7, 10], [15, 22]], x) + assert 2 * M == PolyMatrix([[2, 4], [6, 8]], x) + assert M * 2 == PolyMatrix([[2, 4], [6, 8]], x) + assert S(2) * M == PolyMatrix([[2, 4], [6, 8]], x) + assert M * S(2) == PolyMatrix([[2, 4], [6, 8]], x) + raises(TypeError, lambda: [] * M) + raises(TypeError, lambda: M * []) + M2 = PolyMatrix([[1, 2]], ring=ZZ[x]) + assert S.Half * M2 == PolyMatrix([[S.Half, 1]], ring=QQ[x]) + assert M2 * S.Half == PolyMatrix([[S.Half, 1]], ring=QQ[x]) + + assert M / 2 == PolyMatrix([[S(1)/2, 1], [S(3)/2, 2]], x) + assert M / Poly(2, x) == PolyMatrix([[S(1)/2, 1], [S(3)/2, 2]], x) + raises(TypeError, lambda: M / []) + + +def test_polymatrix_manipulations(): + M1 = PolyMatrix([[1, 2], [3, 4]], x) + assert M1.transpose() == PolyMatrix([[1, 3], [2, 4]], x) + M2 = PolyMatrix([[5, 6], [7, 8]], x) + assert M1.row_join(M2) == PolyMatrix([[1, 2, 5, 6], [3, 4, 7, 8]], x) + assert M1.col_join(M2) == PolyMatrix([[1, 2], [3, 4], [5, 6], [7, 8]], x) + assert M1.applyfunc(lambda e: 2*e) == PolyMatrix([[2, 4], [6, 8]], x) + + +def test_polymatrix_ones_zeros(): + assert PolyMatrix.zeros(1, 2, x) == PolyMatrix([[0, 0]], x) + assert PolyMatrix.eye(2, x) == PolyMatrix([[1, 0], [0, 1]], x) + + +def test_polymatrix_rref(): + M = PolyMatrix([[1, 2], [3, 4]], x) + assert M.rref() == (PolyMatrix.eye(2, x), (0, 1)) + raises(ValueError, lambda: PolyMatrix([1, 2], ring=ZZ[x]).rref()) + raises(ValueError, lambda: PolyMatrix([1, x], ring=QQ[x]).rref()) + + +def test_polymatrix_nullspace(): + M = PolyMatrix([[1, 2], [3, 6]], x) + assert M.nullspace() == [PolyMatrix([-2, 1], x)] + raises(ValueError, lambda: PolyMatrix([1, 2], ring=ZZ[x]).nullspace()) + raises(ValueError, lambda: PolyMatrix([1, x], ring=QQ[x]).nullspace()) + assert M.rank() == 1 diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_polyoptions.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_polyoptions.py new file mode 100644 index 0000000000000000000000000000000000000000..fa2e6054bad43aef5470949180ea5c2ffdc11f30 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_polyoptions.py @@ -0,0 +1,485 @@ +"""Tests for options manager for :class:`Poly` and public API functions. """ + +from sympy.polys.polyoptions import ( + Options, Expand, Gens, Wrt, Sort, Order, Field, Greedy, Domain, + Split, Gaussian, Extension, Modulus, Symmetric, Strict, Auto, + Frac, Formal, Polys, Include, All, Gen, Symbols, Method) + +from sympy.polys.orderings import lex +from sympy.polys.domains import FF, GF, ZZ, QQ, QQ_I, RR, CC, EX + +from sympy.polys.polyerrors import OptionError, GeneratorsError + +from sympy.core.numbers import (I, Integer) +from sympy.core.symbol import Symbol +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.testing.pytest import raises +from sympy.abc import x, y, z + + +def test_Options_clone(): + opt = Options((x, y, z), {'domain': 'ZZ'}) + + assert opt.gens == (x, y, z) + assert opt.domain == ZZ + assert ('order' in opt) is False + + new_opt = opt.clone({'gens': (x, y), 'order': 'lex'}) + + assert opt.gens == (x, y, z) + assert opt.domain == ZZ + assert ('order' in opt) is False + + assert new_opt.gens == (x, y) + assert new_opt.domain == ZZ + assert ('order' in new_opt) is True + + +def test_Expand_preprocess(): + assert Expand.preprocess(False) is False + assert Expand.preprocess(True) is True + + assert Expand.preprocess(0) is False + assert Expand.preprocess(1) is True + + raises(OptionError, lambda: Expand.preprocess(x)) + + +def test_Expand_postprocess(): + opt = {'expand': True} + Expand.postprocess(opt) + + assert opt == {'expand': True} + + +def test_Gens_preprocess(): + assert Gens.preprocess((None,)) == () + assert Gens.preprocess((x, y, z)) == (x, y, z) + assert Gens.preprocess(((x, y, z),)) == (x, y, z) + + a = Symbol('a', commutative=False) + + raises(GeneratorsError, lambda: Gens.preprocess((x, x, y))) + raises(GeneratorsError, lambda: Gens.preprocess((x, y, a))) + + +def test_Gens_postprocess(): + opt = {'gens': (x, y)} + Gens.postprocess(opt) + + assert opt == {'gens': (x, y)} + + +def test_Wrt_preprocess(): + assert Wrt.preprocess(x) == ['x'] + assert Wrt.preprocess('') == [] + assert Wrt.preprocess(' ') == [] + assert Wrt.preprocess('x,y') == ['x', 'y'] + assert Wrt.preprocess('x y') == ['x', 'y'] + assert Wrt.preprocess('x, y') == ['x', 'y'] + assert Wrt.preprocess('x , y') == ['x', 'y'] + assert Wrt.preprocess(' x, y') == ['x', 'y'] + assert Wrt.preprocess(' x, y') == ['x', 'y'] + assert Wrt.preprocess([x, y]) == ['x', 'y'] + + raises(OptionError, lambda: Wrt.preprocess(',')) + raises(OptionError, lambda: Wrt.preprocess(0)) + + +def test_Wrt_postprocess(): + opt = {'wrt': ['x']} + Wrt.postprocess(opt) + + assert opt == {'wrt': ['x']} + + +def test_Sort_preprocess(): + assert Sort.preprocess([x, y, z]) == ['x', 'y', 'z'] + assert Sort.preprocess((x, y, z)) == ['x', 'y', 'z'] + + assert Sort.preprocess('x > y > z') == ['x', 'y', 'z'] + assert Sort.preprocess('x>y>z') == ['x', 'y', 'z'] + + raises(OptionError, lambda: Sort.preprocess(0)) + raises(OptionError, lambda: Sort.preprocess({x, y, z})) + + +def test_Sort_postprocess(): + opt = {'sort': 'x > y'} + Sort.postprocess(opt) + + assert opt == {'sort': 'x > y'} + + +def test_Order_preprocess(): + assert Order.preprocess('lex') == lex + + +def test_Order_postprocess(): + opt = {'order': True} + Order.postprocess(opt) + + assert opt == {'order': True} + + +def test_Field_preprocess(): + assert Field.preprocess(False) is False + assert Field.preprocess(True) is True + + assert Field.preprocess(0) is False + assert Field.preprocess(1) is True + + raises(OptionError, lambda: Field.preprocess(x)) + + +def test_Field_postprocess(): + opt = {'field': True} + Field.postprocess(opt) + + assert opt == {'field': True} + + +def test_Greedy_preprocess(): + assert Greedy.preprocess(False) is False + assert Greedy.preprocess(True) is True + + assert Greedy.preprocess(0) is False + assert Greedy.preprocess(1) is True + + raises(OptionError, lambda: Greedy.preprocess(x)) + + +def test_Greedy_postprocess(): + opt = {'greedy': True} + Greedy.postprocess(opt) + + assert opt == {'greedy': True} + + +def test_Domain_preprocess(): + assert Domain.preprocess(ZZ) == ZZ + assert Domain.preprocess(QQ) == QQ + assert Domain.preprocess(EX) == EX + assert Domain.preprocess(FF(2)) == FF(2) + assert Domain.preprocess(ZZ[x, y]) == ZZ[x, y] + + assert Domain.preprocess('Z') == ZZ + assert Domain.preprocess('Q') == QQ + + assert Domain.preprocess('ZZ') == ZZ + assert Domain.preprocess('QQ') == QQ + + assert Domain.preprocess('EX') == EX + + assert Domain.preprocess('FF(23)') == FF(23) + assert Domain.preprocess('GF(23)') == GF(23) + + raises(OptionError, lambda: Domain.preprocess('Z[]')) + + assert Domain.preprocess('Z[x]') == ZZ[x] + assert Domain.preprocess('Q[x]') == QQ[x] + assert Domain.preprocess('R[x]') == RR[x] + assert Domain.preprocess('C[x]') == CC[x] + + assert Domain.preprocess('ZZ[x]') == ZZ[x] + assert Domain.preprocess('QQ[x]') == QQ[x] + assert Domain.preprocess('RR[x]') == RR[x] + assert Domain.preprocess('CC[x]') == CC[x] + + assert Domain.preprocess('Z[x,y]') == ZZ[x, y] + assert Domain.preprocess('Q[x,y]') == QQ[x, y] + assert Domain.preprocess('R[x,y]') == RR[x, y] + assert Domain.preprocess('C[x,y]') == CC[x, y] + + assert Domain.preprocess('ZZ[x,y]') == ZZ[x, y] + assert Domain.preprocess('QQ[x,y]') == QQ[x, y] + assert Domain.preprocess('RR[x,y]') == RR[x, y] + assert Domain.preprocess('CC[x,y]') == CC[x, y] + + raises(OptionError, lambda: Domain.preprocess('Z()')) + + assert Domain.preprocess('Z(x)') == ZZ.frac_field(x) + assert Domain.preprocess('Q(x)') == QQ.frac_field(x) + + assert Domain.preprocess('ZZ(x)') == ZZ.frac_field(x) + assert Domain.preprocess('QQ(x)') == QQ.frac_field(x) + + assert Domain.preprocess('Z(x,y)') == ZZ.frac_field(x, y) + assert Domain.preprocess('Q(x,y)') == QQ.frac_field(x, y) + + assert Domain.preprocess('ZZ(x,y)') == ZZ.frac_field(x, y) + assert Domain.preprocess('QQ(x,y)') == QQ.frac_field(x, y) + + assert Domain.preprocess('Q') == QQ.algebraic_field(I) + assert Domain.preprocess('QQ') == QQ.algebraic_field(I) + + assert Domain.preprocess('Q') == QQ.algebraic_field(sqrt(2), I) + assert Domain.preprocess( + 'QQ') == QQ.algebraic_field(sqrt(2), I) + + raises(OptionError, lambda: Domain.preprocess('abc')) + + +def test_Domain_postprocess(): + raises(GeneratorsError, lambda: Domain.postprocess({'gens': (x, y), + 'domain': ZZ[y, z]})) + + raises(GeneratorsError, lambda: Domain.postprocess({'gens': (), + 'domain': EX})) + raises(GeneratorsError, lambda: Domain.postprocess({'domain': EX})) + + +def test_Split_preprocess(): + assert Split.preprocess(False) is False + assert Split.preprocess(True) is True + + assert Split.preprocess(0) is False + assert Split.preprocess(1) is True + + raises(OptionError, lambda: Split.preprocess(x)) + + +def test_Split_postprocess(): + raises(NotImplementedError, lambda: Split.postprocess({'split': True})) + + +def test_Gaussian_preprocess(): + assert Gaussian.preprocess(False) is False + assert Gaussian.preprocess(True) is True + + assert Gaussian.preprocess(0) is False + assert Gaussian.preprocess(1) is True + + raises(OptionError, lambda: Gaussian.preprocess(x)) + + +def test_Gaussian_postprocess(): + opt = {'gaussian': True} + Gaussian.postprocess(opt) + + assert opt == { + 'gaussian': True, + 'domain': QQ_I, + } + + +def test_Extension_preprocess(): + assert Extension.preprocess(True) is True + assert Extension.preprocess(1) is True + + assert Extension.preprocess([]) is None + + assert Extension.preprocess(sqrt(2)) == {sqrt(2)} + assert Extension.preprocess([sqrt(2)]) == {sqrt(2)} + + assert Extension.preprocess([sqrt(2), I]) == {sqrt(2), I} + + raises(OptionError, lambda: Extension.preprocess(False)) + raises(OptionError, lambda: Extension.preprocess(0)) + + +def test_Extension_postprocess(): + opt = {'extension': {sqrt(2)}} + Extension.postprocess(opt) + + assert opt == { + 'extension': {sqrt(2)}, + 'domain': QQ.algebraic_field(sqrt(2)), + } + + opt = {'extension': True} + Extension.postprocess(opt) + + assert opt == {'extension': True} + + +def test_Modulus_preprocess(): + assert Modulus.preprocess(23) == 23 + assert Modulus.preprocess(Integer(23)) == 23 + + raises(OptionError, lambda: Modulus.preprocess(0)) + raises(OptionError, lambda: Modulus.preprocess(x)) + + +def test_Modulus_postprocess(): + opt = {'modulus': 5} + Modulus.postprocess(opt) + + assert opt == { + 'modulus': 5, + 'domain': FF(5), + } + + opt = {'modulus': 5, 'symmetric': False} + Modulus.postprocess(opt) + + assert opt == { + 'modulus': 5, + 'domain': FF(5, False), + 'symmetric': False, + } + + +def test_Symmetric_preprocess(): + assert Symmetric.preprocess(False) is False + assert Symmetric.preprocess(True) is True + + assert Symmetric.preprocess(0) is False + assert Symmetric.preprocess(1) is True + + raises(OptionError, lambda: Symmetric.preprocess(x)) + + +def test_Symmetric_postprocess(): + opt = {'symmetric': True} + Symmetric.postprocess(opt) + + assert opt == {'symmetric': True} + + +def test_Strict_preprocess(): + assert Strict.preprocess(False) is False + assert Strict.preprocess(True) is True + + assert Strict.preprocess(0) is False + assert Strict.preprocess(1) is True + + raises(OptionError, lambda: Strict.preprocess(x)) + + +def test_Strict_postprocess(): + opt = {'strict': True} + Strict.postprocess(opt) + + assert opt == {'strict': True} + + +def test_Auto_preprocess(): + assert Auto.preprocess(False) is False + assert Auto.preprocess(True) is True + + assert Auto.preprocess(0) is False + assert Auto.preprocess(1) is True + + raises(OptionError, lambda: Auto.preprocess(x)) + + +def test_Auto_postprocess(): + opt = {'auto': True} + Auto.postprocess(opt) + + assert opt == {'auto': True} + + +def test_Frac_preprocess(): + assert Frac.preprocess(False) is False + assert Frac.preprocess(True) is True + + assert Frac.preprocess(0) is False + assert Frac.preprocess(1) is True + + raises(OptionError, lambda: Frac.preprocess(x)) + + +def test_Frac_postprocess(): + opt = {'frac': True} + Frac.postprocess(opt) + + assert opt == {'frac': True} + + +def test_Formal_preprocess(): + assert Formal.preprocess(False) is False + assert Formal.preprocess(True) is True + + assert Formal.preprocess(0) is False + assert Formal.preprocess(1) is True + + raises(OptionError, lambda: Formal.preprocess(x)) + + +def test_Formal_postprocess(): + opt = {'formal': True} + Formal.postprocess(opt) + + assert opt == {'formal': True} + + +def test_Polys_preprocess(): + assert Polys.preprocess(False) is False + assert Polys.preprocess(True) is True + + assert Polys.preprocess(0) is False + assert Polys.preprocess(1) is True + + raises(OptionError, lambda: Polys.preprocess(x)) + + +def test_Polys_postprocess(): + opt = {'polys': True} + Polys.postprocess(opt) + + assert opt == {'polys': True} + + +def test_Include_preprocess(): + assert Include.preprocess(False) is False + assert Include.preprocess(True) is True + + assert Include.preprocess(0) is False + assert Include.preprocess(1) is True + + raises(OptionError, lambda: Include.preprocess(x)) + + +def test_Include_postprocess(): + opt = {'include': True} + Include.postprocess(opt) + + assert opt == {'include': True} + + +def test_All_preprocess(): + assert All.preprocess(False) is False + assert All.preprocess(True) is True + + assert All.preprocess(0) is False + assert All.preprocess(1) is True + + raises(OptionError, lambda: All.preprocess(x)) + + +def test_All_postprocess(): + opt = {'all': True} + All.postprocess(opt) + + assert opt == {'all': True} + + +def test_Gen_postprocess(): + opt = {'gen': x} + Gen.postprocess(opt) + + assert opt == {'gen': x} + + +def test_Symbols_preprocess(): + raises(OptionError, lambda: Symbols.preprocess(x)) + + +def test_Symbols_postprocess(): + opt = {'symbols': [x, y, z]} + Symbols.postprocess(opt) + + assert opt == {'symbols': [x, y, z]} + + +def test_Method_preprocess(): + raises(OptionError, lambda: Method.preprocess(10)) + + +def test_Method_postprocess(): + opt = {'method': 'f5b'} + Method.postprocess(opt) + + assert opt == {'method': 'f5b'} diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_polyroots.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_polyroots.py new file mode 100644 index 0000000000000000000000000000000000000000..7f96b1930f6789ce3150ae2c920ba7d9faa68791 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_polyroots.py @@ -0,0 +1,758 @@ +"""Tests for algorithms for computing symbolic roots of polynomials. """ + +from sympy.core.numbers import (I, Rational, pi) +from sympy.core.singleton import S +from sympy.core.symbol import (Symbol, Wild, symbols) +from sympy.functions.elementary.complexes import (conjugate, im, re) +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.miscellaneous import (root, sqrt) +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.elementary.trigonometric import (acos, cos, sin) +from sympy.polys.domains.integerring import ZZ +from sympy.sets.sets import Interval +from sympy.simplify.powsimp import powsimp + +from sympy.polys import Poly, cyclotomic_poly, intervals, nroots, rootof + +from sympy.polys.polyroots import (root_factors, roots_linear, + roots_quadratic, roots_cubic, roots_quartic, roots_quintic, + roots_cyclotomic, roots_binomial, preprocess_roots, roots) + +from sympy.polys.orthopolys import legendre_poly +from sympy.polys.polyerrors import PolynomialError, \ + UnsolvableFactorError +from sympy.polys.polyutils import _nsort + +from sympy.testing.pytest import raises, slow +from sympy.core.random import verify_numerically +import mpmath +from itertools import product + + + +a, b, c, d, e, q, t, x, y, z = symbols('a,b,c,d,e,q,t,x,y,z') + + +def _check(roots): + # this is the desired invariant for roots returned + # by all_roots. It is trivially true for linear + # polynomials. + nreal = sum(1 if i.is_real else 0 for i in roots) + assert sorted(roots[:nreal]) == list(roots[:nreal]) + for ix in range(nreal, len(roots), 2): + if not ( + roots[ix + 1] == roots[ix] or + roots[ix + 1] == conjugate(roots[ix])): + return False + return True + + +def test_roots_linear(): + assert roots_linear(Poly(2*x + 1, x)) == [Rational(-1, 2)] + + +def test_roots_quadratic(): + assert roots_quadratic(Poly(2*x**2, x)) == [0, 0] + assert roots_quadratic(Poly(2*x**2 + 3*x, x)) == [Rational(-3, 2), 0] + assert roots_quadratic(Poly(2*x**2 + 3, x)) == [-I*sqrt(6)/2, I*sqrt(6)/2] + assert roots_quadratic(Poly(2*x**2 + 4*x + 3, x)) == [-1 - I*sqrt(2)/2, -1 + I*sqrt(2)/2] + _check(Poly(2*x**2 + 4*x + 3, x).all_roots()) + + f = x**2 + (2*a*e + 2*c*e)/(a - c)*x + (d - b + a*e**2 - c*e**2)/(a - c) + assert roots_quadratic(Poly(f, x)) == \ + [-e*(a + c)/(a - c) - sqrt(a*b + c*d - a*d - b*c + 4*a*c*e**2)/(a - c), + -e*(a + c)/(a - c) + sqrt(a*b + c*d - a*d - b*c + 4*a*c*e**2)/(a - c)] + + # check for simplification + f = Poly(y*x**2 - 2*x - 2*y, x) + assert roots_quadratic(f) == \ + [-sqrt(2*y**2 + 1)/y + 1/y, sqrt(2*y**2 + 1)/y + 1/y] + f = Poly(x**2 + (-y**2 - 2)*x + y**2 + 1, x) + assert roots_quadratic(f) == \ + [1,y**2 + 1] + + f = Poly(sqrt(2)*x**2 - 1, x) + r = roots_quadratic(f) + assert r == _nsort(r) + + # issue 8255 + f = Poly(-24*x**2 - 180*x + 264) + assert [w.n(2) for w in f.all_roots(radicals=True)] == \ + [w.n(2) for w in f.all_roots(radicals=False)] + for _a, _b, _c in product((-2, 2), (-2, 2), (0, -1)): + f = Poly(_a*x**2 + _b*x + _c) + roots = roots_quadratic(f) + assert roots == _nsort(roots) + + +def test_issue_7724(): + eq = Poly(x**4*I + x**2 + I, x) + assert roots(eq) == { + sqrt(I/2 + sqrt(5)*I/2): 1, + sqrt(-sqrt(5)*I/2 + I/2): 1, + -sqrt(I/2 + sqrt(5)*I/2): 1, + -sqrt(-sqrt(5)*I/2 + I/2): 1} + + +def test_issue_8438(): + p = Poly([1, y, -2, -3], x).as_expr() + roots = roots_cubic(Poly(p, x), x) + z = Rational(-3, 2) - I*7/2 # this will fail in code given in commit msg + post = [r.subs(y, z) for r in roots] + assert set(post) == \ + set(roots_cubic(Poly(p.subs(y, z), x))) + # /!\ if p is not made an expression, this is *very* slow + assert all(p.subs({y: z, x: i}).n(2, chop=True) == 0 for i in post) + + +def test_issue_8285(): + roots = (Poly(4*x**8 - 1, x)*Poly(x**2 + 1)).all_roots() + assert _check(roots) + f = Poly(x**4 + 5*x**2 + 6, x) + ro = [rootof(f, i) for i in range(4)] + roots = Poly(x**4 + 5*x**2 + 6, x).all_roots() + assert roots == ro + assert _check(roots) + # more than 2 complex roots from which to identify the + # imaginary ones + roots = Poly(2*x**8 - 1).all_roots() + assert _check(roots) + assert len(Poly(2*x**10 - 1).all_roots()) == 10 # doesn't fail + + +def test_issue_8289(): + roots = (Poly(x**2 + 2)*Poly(x**4 + 2)).all_roots() + assert _check(roots) + roots = Poly(x**6 + 3*x**3 + 2, x).all_roots() + assert _check(roots) + roots = Poly(x**6 - x + 1).all_roots() + assert _check(roots) + # all imaginary roots with multiplicity of 2 + roots = Poly(x**4 + 4*x**2 + 4, x).all_roots() + assert _check(roots) + + +def test_issue_14291(): + assert Poly(((x - 1)**2 + 1)*((x - 1)**2 + 2)*(x - 1) + ).all_roots() == [1, 1 - I, 1 + I, 1 - sqrt(2)*I, 1 + sqrt(2)*I] + p = x**4 + 10*x**2 + 1 + ans = [rootof(p, i) for i in range(4)] + assert Poly(p).all_roots() == ans + _check(ans) + + +def test_issue_13340(): + eq = Poly(y**3 + exp(x)*y + x, y, domain='EX') + roots_d = roots(eq) + assert len(roots_d) == 3 + + +def test_issue_14522(): + eq = Poly(x**4 + x**3*(16 + 32*I) + x**2*(-285 + 386*I) + x*(-2824 - 448*I) - 2058 - 6053*I, x) + roots_eq = roots(eq) + assert all(eq(r) == 0 for r in roots_eq) + + +def test_issue_15076(): + sol = roots_quartic(Poly(t**4 - 6*t**2 + t/x - 3, t)) + assert sol[0].has(x) + + +def test_issue_16589(): + eq = Poly(x**4 - 8*sqrt(2)*x**3 + 4*x**3 - 64*sqrt(2)*x**2 + 1024*x, x) + roots_eq = roots(eq) + assert 0 in roots_eq + + +def test_roots_cubic(): + assert roots_cubic(Poly(2*x**3, x)) == [0, 0, 0] + assert roots_cubic(Poly(x**3 - 3*x**2 + 3*x - 1, x)) == [1, 1, 1] + + # valid for arbitrary y (issue 21263) + r = root(y, 3) + assert roots_cubic(Poly(x**3 - y, x)) == [r, + r*(-S.Half + sqrt(3)*I/2), + r*(-S.Half - sqrt(3)*I/2)] + # simpler form when y is negative + assert roots_cubic(Poly(x**3 - -1, x)) == \ + [-1, S.Half - I*sqrt(3)/2, S.Half + I*sqrt(3)/2] + assert roots_cubic(Poly(2*x**3 - 3*x**2 - 3*x - 1, x))[0] == \ + S.Half + 3**Rational(1, 3)/2 + 3**Rational(2, 3)/2 + eq = -x**3 + 2*x**2 + 3*x - 2 + assert roots(eq, trig=True, multiple=True) == \ + roots_cubic(Poly(eq, x), trig=True) == [ + Rational(2, 3) + 2*sqrt(13)*cos(acos(8*sqrt(13)/169)/3)/3, + -2*sqrt(13)*sin(-acos(8*sqrt(13)/169)/3 + pi/6)/3 + Rational(2, 3), + -2*sqrt(13)*cos(-acos(8*sqrt(13)/169)/3 + pi/3)/3 + Rational(2, 3), + ] + + +def test_roots_quartic(): + assert roots_quartic(Poly(x**4, x)) == [0, 0, 0, 0] + assert roots_quartic(Poly(x**4 + x**3, x)) in [ + [-1, 0, 0, 0], + [0, -1, 0, 0], + [0, 0, -1, 0], + [0, 0, 0, -1] + ] + assert roots_quartic(Poly(x**4 - x**3, x)) in [ + [1, 0, 0, 0], + [0, 1, 0, 0], + [0, 0, 1, 0], + [0, 0, 0, 1] + ] + + lhs = roots_quartic(Poly(x**4 + x, x)) + rhs = [S.Half + I*sqrt(3)/2, S.Half - I*sqrt(3)/2, S.Zero, -S.One] + + assert sorted(lhs, key=hash) == sorted(rhs, key=hash) + + # test of all branches of roots quartic + for i, (a, b, c, d) in enumerate([(1, 2, 3, 0), + (3, -7, -9, 9), + (1, 2, 3, 4), + (1, 2, 3, 4), + (-7, -3, 3, -6), + (-3, 5, -6, -4), + (6, -5, -10, -3)]): + if i == 2: + c = -a*(a**2/S(8) - b/S(2)) + elif i == 3: + d = a*(a*(a**2*Rational(3, 256) - b/S(16)) + c/S(4)) + eq = x**4 + a*x**3 + b*x**2 + c*x + d + ans = roots_quartic(Poly(eq, x)) + assert all(eq.subs(x, ai).n(chop=True) == 0 for ai in ans) + + # not all symbolic quartics are unresolvable + eq = Poly(q*x + q/4 + x**4 + x**3 + 2*x**2 - Rational(1, 3), x) + sol = roots_quartic(eq) + assert all(verify_numerically(eq.subs(x, i), 0) for i in sol) + z = symbols('z', negative=True) + eq = x**4 + 2*x**3 + 3*x**2 + x*(z + 11) + 5 + zans = roots_quartic(Poly(eq, x)) + assert all(verify_numerically(eq.subs(((x, i), (z, -1))), 0) for i in zans) + # but some are (see also issue 4989) + # it's ok if the solution is not Piecewise, but the tests below should pass + eq = Poly(y*x**4 + x**3 - x + z, x) + ans = roots_quartic(eq) + assert all(type(i) == Piecewise for i in ans) + reps = ( + {"y": Rational(-1, 3), "z": Rational(-1, 4)}, # 4 real + {"y": Rational(-1, 3), "z": Rational(-1, 2)}, # 2 real + {"y": Rational(-1, 3), "z": -2}) # 0 real + for rep in reps: + sol = roots_quartic(Poly(eq.subs(rep), x)) + assert all(verify_numerically(w.subs(rep) - s, 0) for w, s in zip(ans, sol)) + + +def test_issue_21287(): + assert not any(isinstance(i, Piecewise) for i in roots_quartic( + Poly(x**4 - x**2*(3 + 5*I) + 2*x*(-1 + I) - 1 + 3*I, x))) + + +def test_roots_quintic(): + eqs = (x**5 - 2, + (x/2 + 1)**5 - 5*(x/2 + 1) + 12, + x**5 - 110*x**3 - 55*x**2 + 2310*x + 979) + for eq in eqs: + roots = roots_quintic(Poly(eq)) + assert len(roots) == 5 + assert all(eq.subs(x, r.n(10)).n(chop = 1e-5) == 0 for r in roots) + + +def test_roots_cyclotomic(): + assert roots_cyclotomic(cyclotomic_poly(1, x, polys=True)) == [1] + assert roots_cyclotomic(cyclotomic_poly(2, x, polys=True)) == [-1] + assert roots_cyclotomic(cyclotomic_poly( + 3, x, polys=True)) == [Rational(-1, 2) - I*sqrt(3)/2, Rational(-1, 2) + I*sqrt(3)/2] + assert roots_cyclotomic(cyclotomic_poly(4, x, polys=True)) == [-I, I] + assert roots_cyclotomic(cyclotomic_poly( + 6, x, polys=True)) == [S.Half - I*sqrt(3)/2, S.Half + I*sqrt(3)/2] + + assert roots_cyclotomic(cyclotomic_poly(7, x, polys=True)) == [ + -cos(pi/7) - I*sin(pi/7), + -cos(pi/7) + I*sin(pi/7), + -cos(pi*Rational(3, 7)) - I*sin(pi*Rational(3, 7)), + -cos(pi*Rational(3, 7)) + I*sin(pi*Rational(3, 7)), + cos(pi*Rational(2, 7)) - I*sin(pi*Rational(2, 7)), + cos(pi*Rational(2, 7)) + I*sin(pi*Rational(2, 7)), + ] + + assert roots_cyclotomic(cyclotomic_poly(8, x, polys=True)) == [ + -sqrt(2)/2 - I*sqrt(2)/2, + -sqrt(2)/2 + I*sqrt(2)/2, + sqrt(2)/2 - I*sqrt(2)/2, + sqrt(2)/2 + I*sqrt(2)/2, + ] + + assert roots_cyclotomic(cyclotomic_poly(12, x, polys=True)) == [ + -sqrt(3)/2 - I/2, + -sqrt(3)/2 + I/2, + sqrt(3)/2 - I/2, + sqrt(3)/2 + I/2, + ] + + assert roots_cyclotomic( + cyclotomic_poly(1, x, polys=True), factor=True) == [1] + assert roots_cyclotomic( + cyclotomic_poly(2, x, polys=True), factor=True) == [-1] + + assert roots_cyclotomic(cyclotomic_poly(3, x, polys=True), factor=True) == \ + [-root(-1, 3), -1 + root(-1, 3)] + assert roots_cyclotomic(cyclotomic_poly(4, x, polys=True), factor=True) == \ + [-I, I] + assert roots_cyclotomic(cyclotomic_poly(5, x, polys=True), factor=True) == \ + [-root(-1, 5), -root(-1, 5)**3, root(-1, 5)**2, -1 - root(-1, 5)**2 + root(-1, 5) + root(-1, 5)**3] + + assert roots_cyclotomic(cyclotomic_poly(6, x, polys=True), factor=True) == \ + [1 - root(-1, 3), root(-1, 3)] + + +def test_roots_binomial(): + assert roots_binomial(Poly(5*x, x)) == [0] + assert roots_binomial(Poly(5*x**4, x)) == [0, 0, 0, 0] + assert roots_binomial(Poly(5*x + 2, x)) == [Rational(-2, 5)] + + A = 10**Rational(3, 4)/10 + + assert roots_binomial(Poly(5*x**4 + 2, x)) == \ + [-A - A*I, -A + A*I, A - A*I, A + A*I] + _check(roots_binomial(Poly(x**8 - 2))) + + a1 = Symbol('a1', nonnegative=True) + b1 = Symbol('b1', nonnegative=True) + + r0 = roots_quadratic(Poly(a1*x**2 + b1, x)) + r1 = roots_binomial(Poly(a1*x**2 + b1, x)) + + assert powsimp(r0[0]) == powsimp(r1[0]) + assert powsimp(r0[1]) == powsimp(r1[1]) + for a, b, s, n in product((1, 2), (1, 2), (-1, 1), (2, 3, 4, 5)): + if a == b and a != 1: # a == b == 1 is sufficient + continue + p = Poly(a*x**n + s*b) + ans = roots_binomial(p) + assert ans == _nsort(ans) + + # issue 8813 + assert roots(Poly(2*x**3 - 16*y**3, x)) == { + 2*y*(Rational(-1, 2) - sqrt(3)*I/2): 1, + 2*y: 1, + 2*y*(Rational(-1, 2) + sqrt(3)*I/2): 1} + + +def test_roots_preprocessing(): + f = a*y*x**2 + y - b + + coeff, poly = preprocess_roots(Poly(f, x)) + + assert coeff == 1 + assert poly == Poly(a*y*x**2 + y - b, x) + + f = c**3*x**3 + c**2*x**2 + c*x + a + + coeff, poly = preprocess_roots(Poly(f, x)) + + assert coeff == 1/c + assert poly == Poly(x**3 + x**2 + x + a, x) + + f = c**3*x**3 + c**2*x**2 + a + + coeff, poly = preprocess_roots(Poly(f, x)) + + assert coeff == 1/c + assert poly == Poly(x**3 + x**2 + a, x) + + f = c**3*x**3 + c*x + a + + coeff, poly = preprocess_roots(Poly(f, x)) + + assert coeff == 1/c + assert poly == Poly(x**3 + x + a, x) + + f = c**3*x**3 + a + + coeff, poly = preprocess_roots(Poly(f, x)) + + assert coeff == 1/c + assert poly == Poly(x**3 + a, x) + + E, F, J, L = symbols("E,F,J,L") + + f = -21601054687500000000*E**8*J**8/L**16 + \ + 508232812500000000*F*x*E**7*J**7/L**14 - \ + 4269543750000000*E**6*F**2*J**6*x**2/L**12 + \ + 16194716250000*E**5*F**3*J**5*x**3/L**10 - \ + 27633173750*E**4*F**4*J**4*x**4/L**8 + \ + 14840215*E**3*F**5*J**3*x**5/L**6 + \ + 54794*E**2*F**6*J**2*x**6/(5*L**4) - \ + 1153*E*J*F**7*x**7/(80*L**2) + \ + 633*F**8*x**8/160000 + + coeff, poly = preprocess_roots(Poly(f, x)) + + assert coeff == 20*E*J/(F*L**2) + assert poly == 633*x**8 - 115300*x**7 + 4383520*x**6 + 296804300*x**5 - 27633173750*x**4 + \ + 809735812500*x**3 - 10673859375000*x**2 + 63529101562500*x - 135006591796875 + + f = Poly(-y**2 + x**2*exp(x), y, domain=ZZ[x, exp(x)]) + g = Poly(-y**2 + exp(x), y, domain=ZZ[exp(x)]) + + assert preprocess_roots(f) == (x, g) + + +def test_roots0(): + assert roots(1, x) == {} + assert roots(x, x) == {S.Zero: 1} + assert roots(x**9, x) == {S.Zero: 9} + assert roots(((x - 2)*(x + 3)*(x - 4)).expand(), x) == {-S(3): 1, S(2): 1, S(4): 1} + + assert roots(2*x + 1, x) == {Rational(-1, 2): 1} + assert roots((2*x + 1)**2, x) == {Rational(-1, 2): 2} + assert roots((2*x + 1)**5, x) == {Rational(-1, 2): 5} + assert roots((2*x + 1)**10, x) == {Rational(-1, 2): 10} + + assert roots(x**4 - 1, x) == {I: 1, S.One: 1, -S.One: 1, -I: 1} + assert roots((x**4 - 1)**2, x) == {I: 2, S.One: 2, -S.One: 2, -I: 2} + + assert roots(((2*x - 3)**2).expand(), x) == {Rational( 3, 2): 2} + assert roots(((2*x + 3)**2).expand(), x) == {Rational(-3, 2): 2} + + assert roots(((2*x - 3)**3).expand(), x) == {Rational( 3, 2): 3} + assert roots(((2*x + 3)**3).expand(), x) == {Rational(-3, 2): 3} + + assert roots(((2*x - 3)**5).expand(), x) == {Rational( 3, 2): 5} + assert roots(((2*x + 3)**5).expand(), x) == {Rational(-3, 2): 5} + + assert roots(((a*x - b)**5).expand(), x) == { b/a: 5} + assert roots(((a*x + b)**5).expand(), x) == {-b/a: 5} + + assert roots(x**2 + (-a - 1)*x + a, x) == {a: 1, S.One: 1} + + assert roots(x**4 - 2*x**2 + 1, x) == {S.One: 2, S.NegativeOne: 2} + + assert roots(x**6 - 4*x**4 + 4*x**3 - x**2, x) == \ + {S.One: 2, -1 - sqrt(2): 1, S.Zero: 2, -1 + sqrt(2): 1} + + assert roots(x**8 - 1, x) == { + sqrt(2)/2 + I*sqrt(2)/2: 1, + sqrt(2)/2 - I*sqrt(2)/2: 1, + -sqrt(2)/2 + I*sqrt(2)/2: 1, + -sqrt(2)/2 - I*sqrt(2)/2: 1, + S.One: 1, -S.One: 1, I: 1, -I: 1 + } + + f = -2016*x**2 - 5616*x**3 - 2056*x**4 + 3324*x**5 + 2176*x**6 - \ + 224*x**7 - 384*x**8 - 64*x**9 + + assert roots(f) == {S.Zero: 2, -S(2): 2, S(2): 1, Rational(-7, 2): 1, + Rational(-3, 2): 1, Rational(-1, 2): 1, Rational(3, 2): 1} + + assert roots((a + b + c)*x - (a + b + c + d), x) == {(a + b + c + d)/(a + b + c): 1} + + assert roots(x**3 + x**2 - x + 1, x, cubics=False) == {} + assert roots(((x - 2)*( + x + 3)*(x - 4)).expand(), x, cubics=False) == {-S(3): 1, S(2): 1, S(4): 1} + assert roots(((x - 2)*(x + 3)*(x - 4)*(x - 5)).expand(), x, cubics=False) == \ + {-S(3): 1, S(2): 1, S(4): 1, S(5): 1} + assert roots(x**3 + 2*x**2 + 4*x + 8, x) == {-S(2): 1, -2*I: 1, 2*I: 1} + assert roots(x**3 + 2*x**2 + 4*x + 8, x, cubics=True) == \ + {-2*I: 1, 2*I: 1, -S(2): 1} + assert roots((x**2 - x)*(x**3 + 2*x**2 + 4*x + 8), x ) == \ + {S.One: 1, S.Zero: 1, -S(2): 1, -2*I: 1, 2*I: 1} + + r1_2, r1_3 = S.Half, Rational(1, 3) + + x0 = (3*sqrt(33) + 19)**r1_3 + x1 = 4/x0/3 + x2 = x0/3 + x3 = sqrt(3)*I/2 + x4 = x3 - r1_2 + x5 = -x3 - r1_2 + assert roots(x**3 + x**2 - x + 1, x, cubics=True) == { + -x1 - x2 - r1_3: 1, + -x1/x4 - x2*x4 - r1_3: 1, + -x1/x5 - x2*x5 - r1_3: 1, + } + + f = (x**2 + 2*x + 3).subs(x, 2*x**2 + 3*x).subs(x, 5*x - 4) + + r13_20, r1_20 = [ Rational(*r) + for r in ((13, 20), (1, 20)) ] + + s2 = sqrt(2) + assert roots(f, x) == { + r13_20 + r1_20*sqrt(1 - 8*I*s2): 1, + r13_20 - r1_20*sqrt(1 - 8*I*s2): 1, + r13_20 + r1_20*sqrt(1 + 8*I*s2): 1, + r13_20 - r1_20*sqrt(1 + 8*I*s2): 1, + } + + f = x**4 + x**3 + x**2 + x + 1 + + r1_4, r1_8, r5_8 = [ Rational(*r) for r in ((1, 4), (1, 8), (5, 8)) ] + + assert roots(f, x) == { + -r1_4 + r1_4*5**r1_2 + I*(r5_8 + r1_8*5**r1_2)**r1_2: 1, + -r1_4 + r1_4*5**r1_2 - I*(r5_8 + r1_8*5**r1_2)**r1_2: 1, + -r1_4 - r1_4*5**r1_2 + I*(r5_8 - r1_8*5**r1_2)**r1_2: 1, + -r1_4 - r1_4*5**r1_2 - I*(r5_8 - r1_8*5**r1_2)**r1_2: 1, + } + + f = z**3 + (-2 - y)*z**2 + (1 + 2*y - 2*x**2)*z - y + 2*x**2 + + assert roots(f, z) == { + S.One: 1, + S.Half + S.Half*y + S.Half*sqrt(1 - 2*y + y**2 + 8*x**2): 1, + S.Half + S.Half*y - S.Half*sqrt(1 - 2*y + y**2 + 8*x**2): 1, + } + + assert roots(a*b*c*x**3 + 2*x**2 + 4*x + 8, x, cubics=False) == {} + assert roots(a*b*c*x**3 + 2*x**2 + 4*x + 8, x, cubics=True) != {} + + assert roots(x**4 - 1, x, filter='Z') == {S.One: 1, -S.One: 1} + assert roots(x**4 - 1, x, filter='I') == {I: 1, -I: 1} + + assert roots((x - 1)*(x + 1), x) == {S.One: 1, -S.One: 1} + assert roots( + (x - 1)*(x + 1), x, predicate=lambda r: r.is_positive) == {S.One: 1} + + assert roots(x**4 - 1, x, filter='Z', multiple=True) == [-S.One, S.One] + assert roots(x**4 - 1, x, filter='I', multiple=True) == [I, -I] + + ar, br = symbols('a, b', real=True) + p = x**2*(ar-br)**2 + 2*x*(br-ar) + 1 + assert roots(p, x, filter='R') == {1/(ar - br): 2} + + assert roots(x**3, x, multiple=True) == [S.Zero, S.Zero, S.Zero] + assert roots(1234, x, multiple=True) == [] + + f = x**6 - x**5 + x**4 - x**3 + x**2 - x + 1 + + assert roots(f) == { + -I*sin(pi/7) + cos(pi/7): 1, + -I*sin(pi*Rational(2, 7)) - cos(pi*Rational(2, 7)): 1, + -I*sin(pi*Rational(3, 7)) + cos(pi*Rational(3, 7)): 1, + I*sin(pi/7) + cos(pi/7): 1, + I*sin(pi*Rational(2, 7)) - cos(pi*Rational(2, 7)): 1, + I*sin(pi*Rational(3, 7)) + cos(pi*Rational(3, 7)): 1, + } + + g = ((x**2 + 1)*f**2).expand() + + assert roots(g) == { + -I*sin(pi/7) + cos(pi/7): 2, + -I*sin(pi*Rational(2, 7)) - cos(pi*Rational(2, 7)): 2, + -I*sin(pi*Rational(3, 7)) + cos(pi*Rational(3, 7)): 2, + I*sin(pi/7) + cos(pi/7): 2, + I*sin(pi*Rational(2, 7)) - cos(pi*Rational(2, 7)): 2, + I*sin(pi*Rational(3, 7)) + cos(pi*Rational(3, 7)): 2, + -I: 1, I: 1, + } + + r = roots(x**3 + 40*x + 64) + real_root = [rx for rx in r if rx.is_real][0] + cr = 108 + 6*sqrt(1074) + assert real_root == -2*root(cr, 3)/3 + 20/root(cr, 3) + + eq = Poly((7 + 5*sqrt(2))*x**3 + (-6 - 4*sqrt(2))*x**2 + (-sqrt(2) - 1)*x + 2, x, domain='EX') + assert roots(eq) == {-1 + sqrt(2): 1, -2 + 2*sqrt(2): 1, -sqrt(2) + 1: 1} + + eq = Poly(41*x**5 + 29*sqrt(2)*x**5 - 153*x**4 - 108*sqrt(2)*x**4 + + 175*x**3 + 125*sqrt(2)*x**3 - 45*x**2 - 30*sqrt(2)*x**2 - 26*sqrt(2)*x - + 26*x + 24, x, domain='EX') + assert roots(eq) == {-sqrt(2) + 1: 1, -2 + 2*sqrt(2): 1, -1 + sqrt(2): 1, + -4 + 4*sqrt(2): 1, -3 + 3*sqrt(2): 1} + + eq = Poly(x**3 - 2*x**2 + 6*sqrt(2)*x**2 - 8*sqrt(2)*x + 23*x - 14 + + 14*sqrt(2), x, domain='EX') + assert roots(eq) == {-2*sqrt(2) + 2: 1, -2*sqrt(2) + 1: 1, -2*sqrt(2) - 1: 1} + + assert roots(Poly((x + sqrt(2))**3 - 7, x, domain='EX')) == \ + {-sqrt(2) + root(7, 3)*(-S.Half - sqrt(3)*I/2): 1, + -sqrt(2) + root(7, 3)*(-S.Half + sqrt(3)*I/2): 1, + -sqrt(2) + root(7, 3): 1} + +def test_roots_slow(): + """Just test that calculating these roots does not hang. """ + a, b, c, d, x = symbols("a,b,c,d,x") + + f1 = x**2*c + (a/b) + x*c*d - a + f2 = x**2*(a + b*(c - d)*a) + x*a*b*c/(b*d - d) + (a*d - c/d) + + assert list(roots(f1, x).values()) == [1, 1] + assert list(roots(f2, x).values()) == [1, 1] + + (zz, yy, xx, zy, zx, yx, k) = symbols("zz,yy,xx,zy,zx,yx,k") + + e1 = (zz - k)*(yy - k)*(xx - k) + zy*yx*zx + zx - zy - yx + e2 = (zz - k)*yx*yx + zx*(yy - k)*zx + zy*zy*(xx - k) + + assert list(roots(e1 - e2, k).values()) == [1, 1, 1] + + f = x**3 + 2*x**2 + 8 + R = list(roots(f).keys()) + + assert not any(i for i in [f.subs(x, ri).n(chop=True) for ri in R]) + + +def test_roots_inexact(): + R1 = roots(x**2 + x + 1, x, multiple=True) + R2 = roots(x**2 + x + 1.0, x, multiple=True) + + for r1, r2 in zip(R1, R2): + assert abs(r1 - r2) < 1e-12 + + f = x**4 + 3.0*sqrt(2.0)*x**3 - (78.0 + 24.0*sqrt(3.0))*x**2 \ + + 144.0*(2*sqrt(3.0) + 9.0) + + R1 = roots(f, multiple=True) + R2 = (-12.7530479110482, -3.85012393732929, + 4.89897948556636, 7.46155167569183) + + for r1, r2 in zip(R1, R2): + assert abs(r1 - r2) < 1e-10 + + +def test_roots_preprocessed(): + E, F, J, L = symbols("E,F,J,L") + + f = -21601054687500000000*E**8*J**8/L**16 + \ + 508232812500000000*F*x*E**7*J**7/L**14 - \ + 4269543750000000*E**6*F**2*J**6*x**2/L**12 + \ + 16194716250000*E**5*F**3*J**5*x**3/L**10 - \ + 27633173750*E**4*F**4*J**4*x**4/L**8 + \ + 14840215*E**3*F**5*J**3*x**5/L**6 + \ + 54794*E**2*F**6*J**2*x**6/(5*L**4) - \ + 1153*E*J*F**7*x**7/(80*L**2) + \ + 633*F**8*x**8/160000 + + assert roots(f, x) == {} + + R1 = roots(f.evalf(), x, multiple=True) + R2 = [-1304.88375606366, 97.1168816800648, 186.946430171876, 245.526792947065, + 503.441004174773, 791.549343830097, 1273.16678129348, 1850.10650616851] + + w = Wild('w') + p = w*E*J/(F*L**2) + + assert len(R1) == len(R2) + + for r1, r2 in zip(R1, R2): + match = r1.match(p) + assert match is not None and abs(match[w] - r2) < 1e-10 + + +def test_roots_strict(): + assert roots(x**2 - 2*x + 1, strict=False) == {1: 2} + assert roots(x**2 - 2*x + 1, strict=True) == {1: 2} + + assert roots(x**6 - 2*x**5 - x**2 + 3*x - 2, strict=False) == {2: 1} + raises(UnsolvableFactorError, lambda: roots(x**6 - 2*x**5 - x**2 + 3*x - 2, strict=True)) + + +def test_roots_mixed(): + f = -1936 - 5056*x - 7592*x**2 + 2704*x**3 - 49*x**4 + + _re, _im = intervals(f, all=True) + _nroots = nroots(f) + _sroots = roots(f, multiple=True) + + _re = [ Interval(a, b) for (a, b), _ in _re ] + _im = [ Interval(re(a), re(b))*Interval(im(a), im(b)) for (a, b), + _ in _im ] + + _intervals = _re + _im + _sroots = [ r.evalf() for r in _sroots ] + + _nroots = sorted(_nroots, key=lambda x: x.sort_key()) + _sroots = sorted(_sroots, key=lambda x: x.sort_key()) + + for _roots in (_nroots, _sroots): + for i, r in zip(_intervals, _roots): + if r.is_real: + assert r in i + else: + assert (re(r), im(r)) in i + + +def test_root_factors(): + assert root_factors(Poly(1, x)) == [Poly(1, x)] + assert root_factors(Poly(x, x)) == [Poly(x, x)] + + assert root_factors(x**2 - 1, x) == [x + 1, x - 1] + assert root_factors(x**2 - y, x) == [x - sqrt(y), x + sqrt(y)] + + assert root_factors((x**4 - 1)**2) == \ + [x + 1, x + 1, x - 1, x - 1, x - I, x - I, x + I, x + I] + + assert root_factors(Poly(x**4 - 1, x), filter='Z') == \ + [Poly(x + 1, x), Poly(x - 1, x), Poly(x**2 + 1, x)] + assert root_factors(8*x**2 + 12*x**4 + 6*x**6 + x**8, x, filter='Q') == \ + [x, x, x**6 + 6*x**4 + 12*x**2 + 8] + + +@slow +def test_nroots1(): + n = 64 + p = legendre_poly(n, x, polys=True) + + raises(mpmath.mp.NoConvergence, lambda: p.nroots(n=3, maxsteps=5)) + + roots = p.nroots(n=3) + # The order of roots matters. They are ordered from smallest to the + # largest. + assert [str(r) for r in roots] == \ + ['-0.999', '-0.996', '-0.991', '-0.983', '-0.973', '-0.961', + '-0.946', '-0.930', '-0.911', '-0.889', '-0.866', '-0.841', + '-0.813', '-0.784', '-0.753', '-0.720', '-0.685', '-0.649', + '-0.611', '-0.572', '-0.531', '-0.489', '-0.446', '-0.402', + '-0.357', '-0.311', '-0.265', '-0.217', '-0.170', '-0.121', + '-0.0730', '-0.0243', '0.0243', '0.0730', '0.121', '0.170', + '0.217', '0.265', '0.311', '0.357', '0.402', '0.446', '0.489', + '0.531', '0.572', '0.611', '0.649', '0.685', '0.720', '0.753', + '0.784', '0.813', '0.841', '0.866', '0.889', '0.911', '0.930', + '0.946', '0.961', '0.973', '0.983', '0.991', '0.996', '0.999'] + +def test_nroots2(): + p = Poly(x**5 + 3*x + 1, x) + + roots = p.nroots(n=3) + # The order of roots matters. The roots are ordered by their real + # components (if they agree, then by their imaginary components), + # with real roots appearing first. + assert [str(r) for r in roots] == \ + ['-0.332', '-0.839 - 0.944*I', '-0.839 + 0.944*I', + '1.01 - 0.937*I', '1.01 + 0.937*I'] + + roots = p.nroots(n=5) + assert [str(r) for r in roots] == \ + ['-0.33199', '-0.83907 - 0.94385*I', '-0.83907 + 0.94385*I', + '1.0051 - 0.93726*I', '1.0051 + 0.93726*I'] + + +def test_roots_composite(): + assert len(roots(Poly(y**3 + y**2*sqrt(x) + y + x, y, composite=True))) == 3 + + +def test_issue_19113(): + eq = cos(x)**3 - cos(x) + 1 + raises(PolynomialError, lambda: roots(eq)) + + +def test_issue_17454(): + assert roots([1, -3*(-4 - 4*I)**2/8 + 12*I, 0], multiple=True) == [0, 0] + + +def test_issue_20913(): + assert Poly(x + 9671406556917067856609794, x).real_roots() == [-9671406556917067856609794] + assert Poly(x**3 + 4, x).real_roots() == [-2**(S(2)/3)] + + +def test_issue_22768(): + e = Rational(1, 3) + r = (-1/a)**e*(a + 1)**(5*e) + assert roots(Poly(a*x**3 + (a + 1)**5, x)) == { + r: 1, + -r*(1 + sqrt(3)*I)/2: 1, + r*(-1 + sqrt(3)*I)/2: 1} diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_polytools.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_polytools.py new file mode 100644 index 0000000000000000000000000000000000000000..a4096447cecea9db6e7559c305af6312b2a72725 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_polytools.py @@ -0,0 +1,3976 @@ +"""Tests for user-friendly public interface to polynomial functions. """ + +import pickle + +from sympy.polys.polytools import ( + Poly, PurePoly, poly, + parallel_poly_from_expr, + degree, degree_list, + total_degree, + LC, LM, LT, + pdiv, prem, pquo, pexquo, + div, rem, quo, exquo, + half_gcdex, gcdex, invert, + subresultants, + resultant, discriminant, + terms_gcd, cofactors, + gcd, gcd_list, + lcm, lcm_list, + trunc, + monic, content, primitive, + compose, decompose, + sturm, + gff_list, gff, + sqf_norm, sqf_part, sqf_list, sqf, + factor_list, factor, + intervals, refine_root, count_roots, + all_roots, real_roots, nroots, ground_roots, + nth_power_roots_poly, + cancel, reduced, groebner, + GroebnerBasis, is_zero_dimensional, + _torational_factor_list, + to_rational_coeffs) + +from sympy.polys.polyerrors import ( + MultivariatePolynomialError, + ExactQuotientFailed, + PolificationFailed, + ComputationFailed, + UnificationFailed, + RefinementFailed, + GeneratorsNeeded, + GeneratorsError, + PolynomialError, + CoercionFailed, + DomainError, + OptionError, + FlagError) + +from sympy.polys.polyclasses import DMP + +from sympy.polys.fields import field +from sympy.polys.domains import FF, ZZ, QQ, ZZ_I, QQ_I, RR, EX +from sympy.polys.domains.realfield import RealField +from sympy.polys.domains.complexfield import ComplexField +from sympy.polys.orderings import lex, grlex, grevlex + +from sympy.combinatorics.galois import S4TransitiveSubgroups +from sympy.core.add import Add +from sympy.core.basic import _aresame +from sympy.core.containers import Tuple +from sympy.core.expr import Expr +from sympy.core.function import (Derivative, diff, expand) +from sympy.core.mul import _keep_coeff, Mul +from sympy.core.numbers import (Float, I, Integer, Rational, oo, pi) +from sympy.core.power import Pow +from sympy.core.relational import Eq +from sympy.core.singleton import S +from sympy.core.symbol import Symbol +from sympy.functions.elementary.complexes import (im, re) +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.hyperbolic import tanh +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.elementary.trigonometric import sin +from sympy.matrices.dense import Matrix +from sympy.matrices.expressions.matexpr import MatrixSymbol +from sympy.polys.rootoftools import rootof +from sympy.simplify.simplify import signsimp +from sympy.utilities.iterables import iterable +from sympy.utilities.exceptions import SymPyDeprecationWarning + +from sympy.testing.pytest import ( + raises, warns_deprecated_sympy, warns, tooslow, XFAIL +) + +from sympy.abc import a, b, c, d, p, q, t, w, x, y, z + + +def _epsilon_eq(a, b): + for u, v in zip(a, b): + if abs(u - v) > 1e-10: + return False + return True + + +def _strict_eq(a, b): + if type(a) == type(b): + if iterable(a): + if len(a) == len(b): + return all(_strict_eq(c, d) for c, d in zip(a, b)) + else: + return False + else: + return isinstance(a, Poly) and a.eq(b, strict=True) + else: + return False + + +def test_Poly_mixed_operations(): + p = Poly(x, x) + with warns_deprecated_sympy(): + p * exp(x) + with warns_deprecated_sympy(): + p + exp(x) + with warns_deprecated_sympy(): + p - exp(x) + + +def test_Poly_from_dict(): + K = FF(3) + + assert Poly.from_dict( + {0: 1, 1: 2}, gens=x, domain=K).rep == DMP([K(2), K(1)], K) + assert Poly.from_dict( + {0: 1, 1: 5}, gens=x, domain=K).rep == DMP([K(2), K(1)], K) + + assert Poly.from_dict( + {(0,): 1, (1,): 2}, gens=x, domain=K).rep == DMP([K(2), K(1)], K) + assert Poly.from_dict( + {(0,): 1, (1,): 5}, gens=x, domain=K).rep == DMP([K(2), K(1)], K) + + assert Poly.from_dict({(0, 0): 1, (1, 1): 2}, gens=( + x, y), domain=K).rep == DMP([[K(2), K(0)], [K(1)]], K) + + assert Poly.from_dict({0: 1, 1: 2}, gens=x).rep == DMP([ZZ(2), ZZ(1)], ZZ) + assert Poly.from_dict( + {0: 1, 1: 2}, gens=x, field=True).rep == DMP([QQ(2), QQ(1)], QQ) + + assert Poly.from_dict( + {0: 1, 1: 2}, gens=x, domain=ZZ).rep == DMP([ZZ(2), ZZ(1)], ZZ) + assert Poly.from_dict( + {0: 1, 1: 2}, gens=x, domain=QQ).rep == DMP([QQ(2), QQ(1)], QQ) + + assert Poly.from_dict( + {(0,): 1, (1,): 2}, gens=x).rep == DMP([ZZ(2), ZZ(1)], ZZ) + assert Poly.from_dict( + {(0,): 1, (1,): 2}, gens=x, field=True).rep == DMP([QQ(2), QQ(1)], QQ) + + assert Poly.from_dict( + {(0,): 1, (1,): 2}, gens=x, domain=ZZ).rep == DMP([ZZ(2), ZZ(1)], ZZ) + assert Poly.from_dict( + {(0,): 1, (1,): 2}, gens=x, domain=QQ).rep == DMP([QQ(2), QQ(1)], QQ) + + assert Poly.from_dict({(1,): sin(y)}, gens=x, composite=False) == \ + Poly(sin(y)*x, x, domain='EX') + assert Poly.from_dict({(1,): y}, gens=x, composite=False) == \ + Poly(y*x, x, domain='EX') + assert Poly.from_dict({(1, 1): 1}, gens=(x, y), composite=False) == \ + Poly(x*y, x, y, domain='ZZ') + assert Poly.from_dict({(1, 0): y}, gens=(x, z), composite=False) == \ + Poly(y*x, x, z, domain='EX') + + +def test_Poly_from_list(): + K = FF(3) + + assert Poly.from_list([2, 1], gens=x, domain=K).rep == DMP([K(2), K(1)], K) + assert Poly.from_list([5, 1], gens=x, domain=K).rep == DMP([K(2), K(1)], K) + + assert Poly.from_list([2, 1], gens=x).rep == DMP([ZZ(2), ZZ(1)], ZZ) + assert Poly.from_list([2, 1], gens=x, field=True).rep == DMP([QQ(2), QQ(1)], QQ) + + assert Poly.from_list([2, 1], gens=x, domain=ZZ).rep == DMP([ZZ(2), ZZ(1)], ZZ) + assert Poly.from_list([2, 1], gens=x, domain=QQ).rep == DMP([QQ(2), QQ(1)], QQ) + + assert Poly.from_list([0, 1.0], gens=x).rep == DMP([RR(1.0)], RR) + assert Poly.from_list([1.0, 0], gens=x).rep == DMP([RR(1.0), RR(0.0)], RR) + + raises(MultivariatePolynomialError, lambda: Poly.from_list([[]], gens=(x, y))) + + +def test_Poly_from_poly(): + f = Poly(x + 7, x, domain=ZZ) + g = Poly(x + 2, x, modulus=3) + h = Poly(x + y, x, y, domain=ZZ) + + K = FF(3) + + assert Poly.from_poly(f) == f + assert Poly.from_poly(f, domain=K).rep == DMP([K(1), K(1)], K) + assert Poly.from_poly(f, domain=ZZ).rep == DMP([ZZ(1), ZZ(7)], ZZ) + assert Poly.from_poly(f, domain=QQ).rep == DMP([QQ(1), QQ(7)], QQ) + + assert Poly.from_poly(f, gens=x) == f + assert Poly.from_poly(f, gens=x, domain=K).rep == DMP([K(1), K(1)], K) + assert Poly.from_poly(f, gens=x, domain=ZZ).rep == DMP([ZZ(1), ZZ(7)], ZZ) + assert Poly.from_poly(f, gens=x, domain=QQ).rep == DMP([QQ(1), QQ(7)], QQ) + + assert Poly.from_poly(f, gens=y) == Poly(x + 7, y, domain='ZZ[x]') + raises(CoercionFailed, lambda: Poly.from_poly(f, gens=y, domain=K)) + raises(CoercionFailed, lambda: Poly.from_poly(f, gens=y, domain=ZZ)) + raises(CoercionFailed, lambda: Poly.from_poly(f, gens=y, domain=QQ)) + + assert Poly.from_poly(f, gens=(x, y)) == Poly(x + 7, x, y, domain='ZZ') + assert Poly.from_poly( + f, gens=(x, y), domain=ZZ) == Poly(x + 7, x, y, domain='ZZ') + assert Poly.from_poly( + f, gens=(x, y), domain=QQ) == Poly(x + 7, x, y, domain='QQ') + assert Poly.from_poly( + f, gens=(x, y), modulus=3) == Poly(x + 7, x, y, domain='FF(3)') + + K = FF(2) + + assert Poly.from_poly(g) == g + assert Poly.from_poly(g, domain=ZZ).rep == DMP([ZZ(1), ZZ(-1)], ZZ) + raises(CoercionFailed, lambda: Poly.from_poly(g, domain=QQ)) + assert Poly.from_poly(g, domain=K).rep == DMP([K(1), K(0)], K) + + assert Poly.from_poly(g, gens=x) == g + assert Poly.from_poly(g, gens=x, domain=ZZ).rep == DMP([ZZ(1), ZZ(-1)], ZZ) + raises(CoercionFailed, lambda: Poly.from_poly(g, gens=x, domain=QQ)) + assert Poly.from_poly(g, gens=x, domain=K).rep == DMP([K(1), K(0)], K) + + K = FF(3) + + assert Poly.from_poly(h) == h + assert Poly.from_poly( + h, domain=ZZ).rep == DMP([[ZZ(1)], [ZZ(1), ZZ(0)]], ZZ) + assert Poly.from_poly( + h, domain=QQ).rep == DMP([[QQ(1)], [QQ(1), QQ(0)]], QQ) + assert Poly.from_poly(h, domain=K).rep == DMP([[K(1)], [K(1), K(0)]], K) + + assert Poly.from_poly(h, gens=x) == Poly(x + y, x, domain=ZZ[y]) + raises(CoercionFailed, lambda: Poly.from_poly(h, gens=x, domain=ZZ)) + assert Poly.from_poly( + h, gens=x, domain=ZZ[y]) == Poly(x + y, x, domain=ZZ[y]) + raises(CoercionFailed, lambda: Poly.from_poly(h, gens=x, domain=QQ)) + assert Poly.from_poly( + h, gens=x, domain=QQ[y]) == Poly(x + y, x, domain=QQ[y]) + raises(CoercionFailed, lambda: Poly.from_poly(h, gens=x, modulus=3)) + + assert Poly.from_poly(h, gens=y) == Poly(x + y, y, domain=ZZ[x]) + raises(CoercionFailed, lambda: Poly.from_poly(h, gens=y, domain=ZZ)) + assert Poly.from_poly( + h, gens=y, domain=ZZ[x]) == Poly(x + y, y, domain=ZZ[x]) + raises(CoercionFailed, lambda: Poly.from_poly(h, gens=y, domain=QQ)) + assert Poly.from_poly( + h, gens=y, domain=QQ[x]) == Poly(x + y, y, domain=QQ[x]) + raises(CoercionFailed, lambda: Poly.from_poly(h, gens=y, modulus=3)) + + assert Poly.from_poly(h, gens=(x, y)) == h + assert Poly.from_poly( + h, gens=(x, y), domain=ZZ).rep == DMP([[ZZ(1)], [ZZ(1), ZZ(0)]], ZZ) + assert Poly.from_poly( + h, gens=(x, y), domain=QQ).rep == DMP([[QQ(1)], [QQ(1), QQ(0)]], QQ) + assert Poly.from_poly( + h, gens=(x, y), domain=K).rep == DMP([[K(1)], [K(1), K(0)]], K) + + assert Poly.from_poly( + h, gens=(y, x)).rep == DMP([[ZZ(1)], [ZZ(1), ZZ(0)]], ZZ) + assert Poly.from_poly( + h, gens=(y, x), domain=ZZ).rep == DMP([[ZZ(1)], [ZZ(1), ZZ(0)]], ZZ) + assert Poly.from_poly( + h, gens=(y, x), domain=QQ).rep == DMP([[QQ(1)], [QQ(1), QQ(0)]], QQ) + assert Poly.from_poly( + h, gens=(y, x), domain=K).rep == DMP([[K(1)], [K(1), K(0)]], K) + + assert Poly.from_poly( + h, gens=(x, y), field=True).rep == DMP([[QQ(1)], [QQ(1), QQ(0)]], QQ) + assert Poly.from_poly( + h, gens=(x, y), field=True).rep == DMP([[QQ(1)], [QQ(1), QQ(0)]], QQ) + + +def test_Poly_from_expr(): + raises(GeneratorsNeeded, lambda: Poly.from_expr(S.Zero)) + raises(GeneratorsNeeded, lambda: Poly.from_expr(S(7))) + + F3 = FF(3) + + assert Poly.from_expr(x + 5, domain=F3).rep == DMP([F3(1), F3(2)], F3) + assert Poly.from_expr(y + 5, domain=F3).rep == DMP([F3(1), F3(2)], F3) + + assert Poly.from_expr(x + 5, x, domain=F3).rep == DMP([F3(1), F3(2)], F3) + assert Poly.from_expr(y + 5, y, domain=F3).rep == DMP([F3(1), F3(2)], F3) + + assert Poly.from_expr(x + y, domain=F3).rep == DMP([[F3(1)], [F3(1), F3(0)]], F3) + assert Poly.from_expr(x + y, x, y, domain=F3).rep == DMP([[F3(1)], [F3(1), F3(0)]], F3) + + assert Poly.from_expr(x + 5).rep == DMP([ZZ(1), ZZ(5)], ZZ) + assert Poly.from_expr(y + 5).rep == DMP([ZZ(1), ZZ(5)], ZZ) + + assert Poly.from_expr(x + 5, x).rep == DMP([ZZ(1), ZZ(5)], ZZ) + assert Poly.from_expr(y + 5, y).rep == DMP([ZZ(1), ZZ(5)], ZZ) + + assert Poly.from_expr(x + 5, domain=ZZ).rep == DMP([ZZ(1), ZZ(5)], ZZ) + assert Poly.from_expr(y + 5, domain=ZZ).rep == DMP([ZZ(1), ZZ(5)], ZZ) + + assert Poly.from_expr(x + 5, x, domain=ZZ).rep == DMP([ZZ(1), ZZ(5)], ZZ) + assert Poly.from_expr(y + 5, y, domain=ZZ).rep == DMP([ZZ(1), ZZ(5)], ZZ) + + assert Poly.from_expr(x + 5, x, y, domain=ZZ).rep == DMP([[ZZ(1)], [ZZ(5)]], ZZ) + assert Poly.from_expr(y + 5, x, y, domain=ZZ).rep == DMP([[ZZ(1), ZZ(5)]], ZZ) + + +def test_Poly_rootof_extension(): + r1 = rootof(x**3 + x + 3, 0) + r2 = rootof(x**3 + x + 3, 1) + K1 = QQ.algebraic_field(r1) + K2 = QQ.algebraic_field(r2) + assert Poly(r1, y) == Poly(r1, y, domain=EX) + assert Poly(r2, y) == Poly(r2, y, domain=EX) + assert Poly(r1, y, extension=True) == Poly(r1, y, domain=K1) + assert Poly(r2, y, extension=True) == Poly(r2, y, domain=K2) + + +@tooslow +def test_Poly_rootof_extension_primitive_element(): + r1 = rootof(x**3 + x + 3, 0) + r2 = rootof(x**3 + x + 3, 1) + K12 = QQ.algebraic_field(r1 + r2) + assert Poly(r1*y + r2, y, extension=True) == Poly(r1*y + r2, y, domain=K12) + + +@XFAIL +def test_Poly_rootof_same_symbol_issue_26808(): + # XXX: This fails because r1 contains x. + r1 = rootof(x**3 + x + 3, 0) + K1 = QQ.algebraic_field(r1) + assert Poly(r1, x) == Poly(r1, x, domain=EX) + assert Poly(r1, x, extension=True) == Poly(r1, x, domain=K1) + + +def test_Poly_rootof_extension_to_sympy(): + # Verify that when primitive elements and RootOf are used, the expression + # is not exploded on the way back to sympy. + r1 = rootof(y**3 + y**2 - 1, 0) + r2 = rootof(z**5 + z**2 - 1, 0) + p = -x**5 + x**2 + x*r1 - r2 + 3*r1**2 + assert p.as_poly(x, extension=True).as_expr() == p + + +def test_poly_from_domain_element(): + dom = ZZ[x] + assert Poly(dom(x+1), y, domain=dom).rep == DMP([dom(x+1)], dom) + dom = dom.get_field() + assert Poly(dom(x+1), y, domain=dom).rep == DMP([dom(x+1)], dom) + + dom = QQ[x] + assert Poly(dom(x+1), y, domain=dom).rep == DMP([dom(x+1)], dom) + dom = dom.get_field() + assert Poly(dom(x+1), y, domain=dom).rep == DMP([dom(x+1)], dom) + + dom = ZZ.old_poly_ring(x) + assert Poly(dom([ZZ(1), ZZ(1)]), y, domain=dom).rep == DMP([dom([ZZ(1), ZZ(1)])], dom) + dom = dom.get_field() + assert Poly(dom([ZZ(1), ZZ(1)]), y, domain=dom).rep == DMP([dom([ZZ(1), ZZ(1)])], dom) + + dom = QQ.old_poly_ring(x) + assert Poly(dom([QQ(1), QQ(1)]), y, domain=dom).rep == DMP([dom([QQ(1), QQ(1)])], dom) + dom = dom.get_field() + assert Poly(dom([QQ(1), QQ(1)]), y, domain=dom).rep == DMP([dom([QQ(1), QQ(1)])], dom) + + dom = QQ.algebraic_field(I) + assert Poly(dom([1, 1]), x, domain=dom).rep == DMP([dom([1, 1])], dom) + + +def test_Poly__new__(): + raises(GeneratorsError, lambda: Poly(x + 1, x, x)) + + raises(GeneratorsError, lambda: Poly(x + y, x, y, domain=ZZ[x])) + raises(GeneratorsError, lambda: Poly(x + y, x, y, domain=ZZ[y])) + + raises(OptionError, lambda: Poly(x, x, symmetric=True)) + raises(OptionError, lambda: Poly(x + 2, x, modulus=3, domain=QQ)) + + raises(OptionError, lambda: Poly(x + 2, x, domain=ZZ, gaussian=True)) + raises(OptionError, lambda: Poly(x + 2, x, modulus=3, gaussian=True)) + + raises(OptionError, lambda: Poly(x + 2, x, domain=ZZ, extension=[sqrt(3)])) + raises(OptionError, lambda: Poly(x + 2, x, modulus=3, extension=[sqrt(3)])) + + raises(OptionError, lambda: Poly(x + 2, x, domain=ZZ, extension=True)) + raises(OptionError, lambda: Poly(x + 2, x, modulus=3, extension=True)) + + raises(OptionError, lambda: Poly(x + 2, x, domain=ZZ, greedy=True)) + raises(OptionError, lambda: Poly(x + 2, x, domain=QQ, field=True)) + + raises(OptionError, lambda: Poly(x + 2, x, domain=ZZ, greedy=False)) + raises(OptionError, lambda: Poly(x + 2, x, domain=QQ, field=False)) + + raises(NotImplementedError, lambda: Poly(x + 1, x, modulus=3, order='grlex')) + raises(NotImplementedError, lambda: Poly(x + 1, x, order='grlex')) + + raises(GeneratorsNeeded, lambda: Poly({1: 2, 0: 1})) + raises(GeneratorsNeeded, lambda: Poly([2, 1])) + raises(GeneratorsNeeded, lambda: Poly((2, 1))) + + raises(GeneratorsNeeded, lambda: Poly(1)) + + assert Poly('x-x') == Poly(0, x) + + f = a*x**2 + b*x + c + + assert Poly({2: a, 1: b, 0: c}, x) == f + assert Poly(iter([a, b, c]), x) == f + assert Poly([a, b, c], x) == f + assert Poly((a, b, c), x) == f + + f = Poly({}, x, y, z) + + assert f.gens == (x, y, z) and f.as_expr() == 0 + + assert Poly(Poly(a*x + b*y, x, y), x) == Poly(a*x + b*y, x) + + assert Poly(3*x**2 + 2*x + 1, domain='ZZ').all_coeffs() == [3, 2, 1] + assert Poly(3*x**2 + 2*x + 1, domain='QQ').all_coeffs() == [3, 2, 1] + assert Poly(3*x**2 + 2*x + 1, domain='RR').all_coeffs() == [3.0, 2.0, 1.0] + + raises(CoercionFailed, lambda: Poly(3*x**2/5 + x*Rational(2, 5) + 1, domain='ZZ')) + assert Poly( + 3*x**2/5 + x*Rational(2, 5) + 1, domain='QQ').all_coeffs() == [Rational(3, 5), Rational(2, 5), 1] + assert _epsilon_eq( + Poly(3*x**2/5 + x*Rational(2, 5) + 1, domain='RR').all_coeffs(), [0.6, 0.4, 1.0]) + + assert Poly(3.0*x**2 + 2.0*x + 1, domain='ZZ').all_coeffs() == [3, 2, 1] + assert Poly(3.0*x**2 + 2.0*x + 1, domain='QQ').all_coeffs() == [3, 2, 1] + assert Poly( + 3.0*x**2 + 2.0*x + 1, domain='RR').all_coeffs() == [3.0, 2.0, 1.0] + + raises(CoercionFailed, lambda: Poly(3.1*x**2 + 2.1*x + 1, domain='ZZ')) + assert Poly(3.1*x**2 + 2.1*x + 1, domain='QQ').all_coeffs() == [Rational(31, 10), Rational(21, 10), 1] + assert Poly(3.1*x**2 + 2.1*x + 1, domain='RR').all_coeffs() == [3.1, 2.1, 1.0] + + assert Poly({(2, 1): 1, (1, 2): 2, (1, 1): 3}, x, y) == \ + Poly(x**2*y + 2*x*y**2 + 3*x*y, x, y) + + assert Poly(x**2 + 1, extension=I).get_domain() == QQ.algebraic_field(I) + + f = 3*x**5 - x**4 + x**3 - x** 2 + 65538 + + assert Poly(f, x, modulus=65537, symmetric=True) == \ + Poly(3*x**5 - x**4 + x**3 - x** 2 + 1, x, modulus=65537, + symmetric=True) + assert Poly(f, x, modulus=65537, symmetric=False) == \ + Poly(3*x**5 + 65536*x**4 + x**3 + 65536*x** 2 + 1, x, + modulus=65537, symmetric=False) + + N = 10**100 + assert Poly(-1, x, modulus=N, symmetric=False).as_expr() == N - 1 + + assert isinstance(Poly(x**2 + x + 1.0).get_domain(), RealField) + assert isinstance(Poly(x**2 + x + I + 1.0).get_domain(), ComplexField) + + +def test_Poly__args(): + assert Poly(x**2 + 1).args == (x**2 + 1, x) + + +def test_Poly__gens(): + assert Poly((x - p)*(x - q), x).gens == (x,) + assert Poly((x - p)*(x - q), p).gens == (p,) + assert Poly((x - p)*(x - q), q).gens == (q,) + + assert Poly((x - p)*(x - q), x, p).gens == (x, p) + assert Poly((x - p)*(x - q), x, q).gens == (x, q) + + assert Poly((x - p)*(x - q), x, p, q).gens == (x, p, q) + assert Poly((x - p)*(x - q), p, x, q).gens == (p, x, q) + assert Poly((x - p)*(x - q), p, q, x).gens == (p, q, x) + + assert Poly((x - p)*(x - q)).gens == (x, p, q) + + assert Poly((x - p)*(x - q), sort='x > p > q').gens == (x, p, q) + assert Poly((x - p)*(x - q), sort='p > x > q').gens == (p, x, q) + assert Poly((x - p)*(x - q), sort='p > q > x').gens == (p, q, x) + + assert Poly((x - p)*(x - q), x, p, q, sort='p > q > x').gens == (x, p, q) + + assert Poly((x - p)*(x - q), wrt='x').gens == (x, p, q) + assert Poly((x - p)*(x - q), wrt='p').gens == (p, x, q) + assert Poly((x - p)*(x - q), wrt='q').gens == (q, x, p) + + assert Poly((x - p)*(x - q), wrt=x).gens == (x, p, q) + assert Poly((x - p)*(x - q), wrt=p).gens == (p, x, q) + assert Poly((x - p)*(x - q), wrt=q).gens == (q, x, p) + + assert Poly((x - p)*(x - q), x, p, q, wrt='p').gens == (x, p, q) + + assert Poly((x - p)*(x - q), wrt='p', sort='q > x').gens == (p, q, x) + assert Poly((x - p)*(x - q), wrt='q', sort='p > x').gens == (q, p, x) + + +def test_Poly_zero(): + assert Poly(x).zero == Poly(0, x, domain=ZZ) + assert Poly(x/2).zero == Poly(0, x, domain=QQ) + + +def test_Poly_one(): + assert Poly(x).one == Poly(1, x, domain=ZZ) + assert Poly(x/2).one == Poly(1, x, domain=QQ) + + +def test_Poly__unify(): + raises(UnificationFailed, lambda: Poly(x)._unify(y)) + + F3 = FF(3) + + assert Poly(x, x, modulus=3)._unify(Poly(y, y, modulus=3))[2:] == ( + DMP([[F3(1)], []], F3), DMP([[F3(1), F3(0)]], F3)) + raises(UnificationFailed, lambda: Poly(x, x, modulus=3)._unify(Poly(y, y, modulus=5))) + + raises(UnificationFailed, lambda: Poly(y, x, y)._unify(Poly(x, x, modulus=3))) + raises(UnificationFailed, lambda: Poly(x, x, modulus=3)._unify(Poly(y, x, y))) + + assert Poly(x + 1, x)._unify(Poly(x + 2, x))[2:] ==\ + (DMP([ZZ(1), ZZ(1)], ZZ), DMP([ZZ(1), ZZ(2)], ZZ)) + assert Poly(x + 1, x, domain='QQ')._unify(Poly(x + 2, x))[2:] ==\ + (DMP([QQ(1), QQ(1)], QQ), DMP([QQ(1), QQ(2)], QQ)) + assert Poly(x + 1, x)._unify(Poly(x + 2, x, domain='QQ'))[2:] ==\ + (DMP([QQ(1), QQ(1)], QQ), DMP([QQ(1), QQ(2)], QQ)) + + assert Poly(x + 1, x)._unify(Poly(x + 2, x, y))[2:] ==\ + (DMP([[ZZ(1)], [ZZ(1)]], ZZ), DMP([[ZZ(1)], [ZZ(2)]], ZZ)) + assert Poly(x + 1, x, domain='QQ')._unify(Poly(x + 2, x, y))[2:] ==\ + (DMP([[QQ(1)], [QQ(1)]], QQ), DMP([[QQ(1)], [QQ(2)]], QQ)) + assert Poly(x + 1, x)._unify(Poly(x + 2, x, y, domain='QQ'))[2:] ==\ + (DMP([[QQ(1)], [QQ(1)]], QQ), DMP([[QQ(1)], [QQ(2)]], QQ)) + + assert Poly(x + 1, x, y)._unify(Poly(x + 2, x))[2:] ==\ + (DMP([[ZZ(1)], [ZZ(1)]], ZZ), DMP([[ZZ(1)], [ZZ(2)]], ZZ)) + assert Poly(x + 1, x, y, domain='QQ')._unify(Poly(x + 2, x))[2:] ==\ + (DMP([[QQ(1)], [QQ(1)]], QQ), DMP([[QQ(1)], [QQ(2)]], QQ)) + assert Poly(x + 1, x, y)._unify(Poly(x + 2, x, domain='QQ'))[2:] ==\ + (DMP([[QQ(1)], [QQ(1)]], QQ), DMP([[QQ(1)], [QQ(2)]], QQ)) + + assert Poly(x + 1, x, y)._unify(Poly(x + 2, x, y))[2:] ==\ + (DMP([[ZZ(1)], [ZZ(1)]], ZZ), DMP([[ZZ(1)], [ZZ(2)]], ZZ)) + assert Poly(x + 1, x, y, domain='QQ')._unify(Poly(x + 2, x, y))[2:] ==\ + (DMP([[QQ(1)], [QQ(1)]], QQ), DMP([[QQ(1)], [QQ(2)]], QQ)) + assert Poly(x + 1, x, y)._unify(Poly(x + 2, x, y, domain='QQ'))[2:] ==\ + (DMP([[QQ(1)], [QQ(1)]], QQ), DMP([[QQ(1)], [QQ(2)]], QQ)) + + assert Poly(x + 1, x)._unify(Poly(x + 2, y, x))[2:] ==\ + (DMP([[ZZ(1), ZZ(1)]], ZZ), DMP([[ZZ(1), ZZ(2)]], ZZ)) + assert Poly(x + 1, x, domain='QQ')._unify(Poly(x + 2, y, x))[2:] ==\ + (DMP([[QQ(1), QQ(1)]], QQ), DMP([[QQ(1), QQ(2)]], QQ)) + assert Poly(x + 1, x)._unify(Poly(x + 2, y, x, domain='QQ'))[2:] ==\ + (DMP([[QQ(1), QQ(1)]], QQ), DMP([[QQ(1), QQ(2)]], QQ)) + + assert Poly(x + 1, y, x)._unify(Poly(x + 2, x))[2:] ==\ + (DMP([[ZZ(1), ZZ(1)]], ZZ), DMP([[ZZ(1), ZZ(2)]], ZZ)) + assert Poly(x + 1, y, x, domain='QQ')._unify(Poly(x + 2, x))[2:] ==\ + (DMP([[QQ(1), QQ(1)]], QQ), DMP([[QQ(1), QQ(2)]], QQ)) + assert Poly(x + 1, y, x)._unify(Poly(x + 2, x, domain='QQ'))[2:] ==\ + (DMP([[QQ(1), QQ(1)]], QQ), DMP([[QQ(1), QQ(2)]], QQ)) + + assert Poly(x + 1, x, y)._unify(Poly(x + 2, y, x))[2:] ==\ + (DMP([[ZZ(1)], [ZZ(1)]], ZZ), DMP([[ZZ(1)], [ZZ(2)]], ZZ)) + assert Poly(x + 1, x, y, domain='QQ')._unify(Poly(x + 2, y, x))[2:] ==\ + (DMP([[QQ(1)], [QQ(1)]], QQ), DMP([[QQ(1)], [QQ(2)]], QQ)) + assert Poly(x + 1, x, y)._unify(Poly(x + 2, y, x, domain='QQ'))[2:] ==\ + (DMP([[QQ(1)], [QQ(1)]], QQ), DMP([[QQ(1)], [QQ(2)]], QQ)) + + assert Poly(x + 1, y, x)._unify(Poly(x + 2, x, y))[2:] ==\ + (DMP([[ZZ(1), ZZ(1)]], ZZ), DMP([[ZZ(1), ZZ(2)]], ZZ)) + assert Poly(x + 1, y, x, domain='QQ')._unify(Poly(x + 2, x, y))[2:] ==\ + (DMP([[QQ(1), QQ(1)]], QQ), DMP([[QQ(1), QQ(2)]], QQ)) + assert Poly(x + 1, y, x)._unify(Poly(x + 2, x, y, domain='QQ'))[2:] ==\ + (DMP([[QQ(1), QQ(1)]], QQ), DMP([[QQ(1), QQ(2)]], QQ)) + + assert Poly(x**2 + I, x, domain=ZZ_I).unify(Poly(x**2 + sqrt(2), x, extension=True)) == \ + (Poly(x**2 + I, x, domain='QQ'), Poly(x**2 + sqrt(2), x, domain='QQ')) + + F, A, B = field("a,b", ZZ) + + assert Poly(a*x, x, domain='ZZ[a]')._unify(Poly(a*b*x, x, domain='ZZ(a,b)'))[2:] == \ + (DMP([A, F(0)], F.to_domain()), DMP([A*B, F(0)], F.to_domain())) + + assert Poly(a*x, x, domain='ZZ(a)')._unify(Poly(a*b*x, x, domain='ZZ(a,b)'))[2:] == \ + (DMP([A, F(0)], F.to_domain()), DMP([A*B, F(0)], F.to_domain())) + + raises(CoercionFailed, lambda: Poly(Poly(x**2 + x**2*z, y, field=True), domain='ZZ(x)')) + + f = Poly(t**2 + t/3 + x, t, domain='QQ(x)') + g = Poly(t**2 + t/3 + x, t, domain='QQ[x]') + + assert f._unify(g)[2:] == (f.rep, f.rep) + + +def test_Poly_free_symbols(): + assert Poly(x**2 + 1).free_symbols == {x} + assert Poly(x**2 + y*z).free_symbols == {x, y, z} + assert Poly(x**2 + y*z, x).free_symbols == {x, y, z} + assert Poly(x**2 + sin(y*z)).free_symbols == {x, y, z} + assert Poly(x**2 + sin(y*z), x).free_symbols == {x, y, z} + assert Poly(x**2 + sin(y*z), x, domain=EX).free_symbols == {x, y, z} + assert Poly(1 + x + x**2, x, y, z).free_symbols == {x} + assert Poly(x + sin(y), z).free_symbols == {x, y} + + +def test_PurePoly_free_symbols(): + assert PurePoly(x**2 + 1).free_symbols == set() + assert PurePoly(x**2 + y*z).free_symbols == set() + assert PurePoly(x**2 + y*z, x).free_symbols == {y, z} + assert PurePoly(x**2 + sin(y*z)).free_symbols == set() + assert PurePoly(x**2 + sin(y*z), x).free_symbols == {y, z} + assert PurePoly(x**2 + sin(y*z), x, domain=EX).free_symbols == {y, z} + + +def test_Poly__eq__(): + assert (Poly(x, x) == Poly(x, x)) is True + assert (Poly(x, x, domain=QQ) == Poly(x, x)) is False + assert (Poly(x, x) == Poly(x, x, domain=QQ)) is False + + assert (Poly(x, x, domain=ZZ[a]) == Poly(x, x)) is False + assert (Poly(x, x) == Poly(x, x, domain=ZZ[a])) is False + + assert (Poly(x*y, x, y) == Poly(x, x)) is False + + assert (Poly(x, x, y) == Poly(x, x)) is False + assert (Poly(x, x) == Poly(x, x, y)) is False + + assert (Poly(x**2 + 1, x) == Poly(y**2 + 1, y)) is False + assert (Poly(y**2 + 1, y) == Poly(x**2 + 1, x)) is False + + f = Poly(x, x, domain=ZZ) + g = Poly(x, x, domain=QQ) + + assert f.eq(g) is False + assert f.ne(g) is True + + assert f.eq(g, strict=True) is False + assert f.ne(g, strict=True) is True + + t0 = Symbol('t0') + + f = Poly((t0/2 + x**2)*t**2 - x**2*t, t, domain='QQ[x,t0]') + g = Poly((t0/2 + x**2)*t**2 - x**2*t, t, domain='ZZ(x,t0)') + + assert (f == g) is False + + +def test_PurePoly__eq__(): + assert (PurePoly(x, x) == PurePoly(x, x)) is True + assert (PurePoly(x, x, domain=QQ) == PurePoly(x, x)) is True + assert (PurePoly(x, x) == PurePoly(x, x, domain=QQ)) is True + + assert (PurePoly(x, x, domain=ZZ[a]) == PurePoly(x, x)) is True + assert (PurePoly(x, x) == PurePoly(x, x, domain=ZZ[a])) is True + + assert (PurePoly(x*y, x, y) == PurePoly(x, x)) is False + + assert (PurePoly(x, x, y) == PurePoly(x, x)) is False + assert (PurePoly(x, x) == PurePoly(x, x, y)) is False + + assert (PurePoly(x**2 + 1, x) == PurePoly(y**2 + 1, y)) is True + assert (PurePoly(y**2 + 1, y) == PurePoly(x**2 + 1, x)) is True + + f = PurePoly(x, x, domain=ZZ) + g = PurePoly(x, x, domain=QQ) + + assert f.eq(g) is True + assert f.ne(g) is False + + assert f.eq(g, strict=True) is False + assert f.ne(g, strict=True) is True + + f = PurePoly(x, x, domain=ZZ) + g = PurePoly(y, y, domain=QQ) + + assert f.eq(g) is True + assert f.ne(g) is False + + assert f.eq(g, strict=True) is False + assert f.ne(g, strict=True) is True + + +def test_PurePoly_Poly(): + assert isinstance(PurePoly(Poly(x**2 + 1)), PurePoly) is True + assert isinstance(Poly(PurePoly(x**2 + 1)), Poly) is True + + +def test_Poly_get_domain(): + assert Poly(2*x).get_domain() == ZZ + + assert Poly(2*x, domain='ZZ').get_domain() == ZZ + assert Poly(2*x, domain='QQ').get_domain() == QQ + + assert Poly(x/2).get_domain() == QQ + + raises(CoercionFailed, lambda: Poly(x/2, domain='ZZ')) + assert Poly(x/2, domain='QQ').get_domain() == QQ + + assert isinstance(Poly(0.2*x).get_domain(), RealField) + + +def test_Poly_set_domain(): + assert Poly(2*x + 1).set_domain(ZZ) == Poly(2*x + 1) + assert Poly(2*x + 1).set_domain('ZZ') == Poly(2*x + 1) + + assert Poly(2*x + 1).set_domain(QQ) == Poly(2*x + 1, domain='QQ') + assert Poly(2*x + 1).set_domain('QQ') == Poly(2*x + 1, domain='QQ') + + assert Poly(Rational(2, 10)*x + Rational(1, 10)).set_domain('RR') == Poly(0.2*x + 0.1) + assert Poly(0.2*x + 0.1).set_domain('QQ') == Poly(Rational(2, 10)*x + Rational(1, 10)) + + raises(CoercionFailed, lambda: Poly(x/2 + 1).set_domain(ZZ)) + raises(CoercionFailed, lambda: Poly(x + 1, modulus=2).set_domain(QQ)) + + raises(GeneratorsError, lambda: Poly(x*y, x, y).set_domain(ZZ[y])) + + +def test_Poly_get_modulus(): + assert Poly(x**2 + 1, modulus=2).get_modulus() == 2 + raises(PolynomialError, lambda: Poly(x**2 + 1).get_modulus()) + + +def test_Poly_set_modulus(): + assert Poly( + x**2 + 1, modulus=2).set_modulus(7) == Poly(x**2 + 1, modulus=7) + assert Poly( + x**2 + 5, modulus=7).set_modulus(2) == Poly(x**2 + 1, modulus=2) + + assert Poly(x**2 + 1).set_modulus(2) == Poly(x**2 + 1, modulus=2) + + raises(CoercionFailed, lambda: Poly(x/2 + 1).set_modulus(2)) + + +def test_Poly_add_ground(): + assert Poly(x + 1).add_ground(2) == Poly(x + 3) + + +def test_Poly_sub_ground(): + assert Poly(x + 1).sub_ground(2) == Poly(x - 1) + + +def test_Poly_mul_ground(): + assert Poly(x + 1).mul_ground(2) == Poly(2*x + 2) + + +def test_Poly_quo_ground(): + assert Poly(2*x + 4).quo_ground(2) == Poly(x + 2) + assert Poly(2*x + 3).quo_ground(2) == Poly(x + 1) + + +def test_Poly_exquo_ground(): + assert Poly(2*x + 4).exquo_ground(2) == Poly(x + 2) + raises(ExactQuotientFailed, lambda: Poly(2*x + 3).exquo_ground(2)) + + +def test_Poly_abs(): + assert Poly(-x + 1, x).abs() == abs(Poly(-x + 1, x)) == Poly(x + 1, x) + + +def test_Poly_neg(): + assert Poly(-x + 1, x).neg() == -Poly(-x + 1, x) == Poly(x - 1, x) + + +def test_Poly_add(): + assert Poly(0, x).add(Poly(0, x)) == Poly(0, x) + assert Poly(0, x) + Poly(0, x) == Poly(0, x) + + assert Poly(1, x).add(Poly(0, x)) == Poly(1, x) + assert Poly(1, x, y) + Poly(0, x) == Poly(1, x, y) + assert Poly(0, x).add(Poly(1, x, y)) == Poly(1, x, y) + assert Poly(0, x, y) + Poly(1, x, y) == Poly(1, x, y) + + assert Poly(1, x) + x == Poly(x + 1, x) + with warns_deprecated_sympy(): + Poly(1, x) + sin(x) + + assert Poly(x, x) + 1 == Poly(x + 1, x) + assert 1 + Poly(x, x) == Poly(x + 1, x) + + +def test_Poly_sub(): + assert Poly(0, x).sub(Poly(0, x)) == Poly(0, x) + assert Poly(0, x) - Poly(0, x) == Poly(0, x) + + assert Poly(1, x).sub(Poly(0, x)) == Poly(1, x) + assert Poly(1, x, y) - Poly(0, x) == Poly(1, x, y) + assert Poly(0, x).sub(Poly(1, x, y)) == Poly(-1, x, y) + assert Poly(0, x, y) - Poly(1, x, y) == Poly(-1, x, y) + + assert Poly(1, x) - x == Poly(1 - x, x) + with warns_deprecated_sympy(): + Poly(1, x) - sin(x) + + assert Poly(x, x) - 1 == Poly(x - 1, x) + assert 1 - Poly(x, x) == Poly(1 - x, x) + + +def test_Poly_mul(): + assert Poly(0, x).mul(Poly(0, x)) == Poly(0, x) + assert Poly(0, x) * Poly(0, x) == Poly(0, x) + + assert Poly(2, x).mul(Poly(4, x)) == Poly(8, x) + assert Poly(2, x, y) * Poly(4, x) == Poly(8, x, y) + assert Poly(4, x).mul(Poly(2, x, y)) == Poly(8, x, y) + assert Poly(4, x, y) * Poly(2, x, y) == Poly(8, x, y) + + assert Poly(1, x) * x == Poly(x, x) + with warns_deprecated_sympy(): + Poly(1, x) * sin(x) + + assert Poly(x, x) * 2 == Poly(2*x, x) + assert 2 * Poly(x, x) == Poly(2*x, x) + +def test_issue_13079(): + assert Poly(x)*x == Poly(x**2, x, domain='ZZ') + assert x*Poly(x) == Poly(x**2, x, domain='ZZ') + assert -2*Poly(x) == Poly(-2*x, x, domain='ZZ') + assert S(-2)*Poly(x) == Poly(-2*x, x, domain='ZZ') + assert Poly(x)*S(-2) == Poly(-2*x, x, domain='ZZ') + +def test_Poly_sqr(): + assert Poly(x*y, x, y).sqr() == Poly(x**2*y**2, x, y) + + +def test_Poly_pow(): + assert Poly(x, x).pow(10) == Poly(x**10, x) + assert Poly(x, x).pow(Integer(10)) == Poly(x**10, x) + + assert Poly(2*y, x, y).pow(4) == Poly(16*y**4, x, y) + assert Poly(2*y, x, y).pow(Integer(4)) == Poly(16*y**4, x, y) + + assert Poly(7*x*y, x, y)**3 == Poly(343*x**3*y**3, x, y) + + raises(TypeError, lambda: Poly(x*y + 1, x, y)**(-1)) + raises(TypeError, lambda: Poly(x*y + 1, x, y)**x) + + +def test_Poly_divmod(): + f, g = Poly(x**2), Poly(x) + q, r = g, Poly(0, x) + + assert divmod(f, g) == (q, r) + assert f // g == q + assert f % g == r + + assert divmod(f, x) == (q, r) + assert f // x == q + assert f % x == r + + q, r = Poly(0, x), Poly(2, x) + + assert divmod(2, g) == (q, r) + assert 2 // g == q + assert 2 % g == r + + assert Poly(x)/Poly(x) == 1 + assert Poly(x**2)/Poly(x) == x + assert Poly(x)/Poly(x**2) == 1/x + + +def test_Poly_eq_ne(): + assert (Poly(x + y, x, y) == Poly(x + y, x, y)) is True + assert (Poly(x + y, x) == Poly(x + y, x, y)) is False + assert (Poly(x + y, x, y) == Poly(x + y, x)) is False + assert (Poly(x + y, x) == Poly(x + y, x)) is True + assert (Poly(x + y, y) == Poly(x + y, y)) is True + + assert (Poly(x + y, x, y) == x + y) is True + assert (Poly(x + y, x) == x + y) is True + assert (Poly(x + y, x, y) == x + y) is True + assert (Poly(x + y, x) == x + y) is True + assert (Poly(x + y, y) == x + y) is True + + assert (Poly(x + y, x, y) != Poly(x + y, x, y)) is False + assert (Poly(x + y, x) != Poly(x + y, x, y)) is True + assert (Poly(x + y, x, y) != Poly(x + y, x)) is True + assert (Poly(x + y, x) != Poly(x + y, x)) is False + assert (Poly(x + y, y) != Poly(x + y, y)) is False + + assert (Poly(x + y, x, y) != x + y) is False + assert (Poly(x + y, x) != x + y) is False + assert (Poly(x + y, x, y) != x + y) is False + assert (Poly(x + y, x) != x + y) is False + assert (Poly(x + y, y) != x + y) is False + + assert (Poly(x, x) == sin(x)) is False + assert (Poly(x, x) != sin(x)) is True + + +def test_Poly_nonzero(): + assert not bool(Poly(0, x)) is True + assert not bool(Poly(1, x)) is False + + +def test_Poly_properties(): + assert Poly(0, x).is_zero is True + assert Poly(1, x).is_zero is False + + assert Poly(1, x).is_one is True + assert Poly(2, x).is_one is False + + assert Poly(x - 1, x).is_sqf is True + assert Poly((x - 1)**2, x).is_sqf is False + + assert Poly(x - 1, x).is_monic is True + assert Poly(2*x - 1, x).is_monic is False + + assert Poly(3*x + 2, x).is_primitive is True + assert Poly(4*x + 2, x).is_primitive is False + + assert Poly(1, x).is_ground is True + assert Poly(x, x).is_ground is False + + assert Poly(x + y + z + 1).is_linear is True + assert Poly(x*y*z + 1).is_linear is False + + assert Poly(x*y + z + 1).is_quadratic is True + assert Poly(x*y*z + 1).is_quadratic is False + + assert Poly(x*y).is_monomial is True + assert Poly(x*y + 1).is_monomial is False + + assert Poly(x**2 + x*y).is_homogeneous is True + assert Poly(x**3 + x*y).is_homogeneous is False + + assert Poly(x).is_univariate is True + assert Poly(x*y).is_univariate is False + + assert Poly(x*y).is_multivariate is True + assert Poly(x).is_multivariate is False + + assert Poly( + x**16 + x**14 - x**10 + x**8 - x**6 + x**2 + 1).is_cyclotomic is False + assert Poly( + x**16 + x**14 - x**10 - x**8 - x**6 + x**2 + 1).is_cyclotomic is True + + +def test_Poly_is_irreducible(): + assert Poly(x**2 + x + 1).is_irreducible is True + assert Poly(x**2 + 2*x + 1).is_irreducible is False + + assert Poly(7*x + 3, modulus=11).is_irreducible is True + assert Poly(7*x**2 + 3*x + 1, modulus=11).is_irreducible is False + + +def test_Poly_subs(): + assert Poly(x + 1).subs(x, 0) == 1 + + assert Poly(x + 1).subs(x, x) == Poly(x + 1) + assert Poly(x + 1).subs(x, y) == Poly(y + 1) + + assert Poly(x*y, x).subs(y, x) == x**2 + assert Poly(x*y, x).subs(x, y) == y**2 + + +def test_Poly_replace(): + assert Poly(x + 1).replace(x) == Poly(x + 1) + assert Poly(x + 1).replace(y) == Poly(y + 1) + + raises(PolynomialError, lambda: Poly(x + y).replace(z)) + + assert Poly(x + 1).replace(x, x) == Poly(x + 1) + assert Poly(x + 1).replace(x, y) == Poly(y + 1) + + assert Poly(x + y).replace(x, x) == Poly(x + y) + assert Poly(x + y).replace(x, z) == Poly(z + y, z, y) + + assert Poly(x + y).replace(y, y) == Poly(x + y) + assert Poly(x + y).replace(y, z) == Poly(x + z, x, z) + assert Poly(x + y).replace(z, t) == Poly(x + y) + + raises(PolynomialError, lambda: Poly(x + y).replace(x, y)) + + assert Poly(x + y, x).replace(x, z) == Poly(z + y, z) + assert Poly(x + y, y).replace(y, z) == Poly(x + z, z) + + raises(PolynomialError, lambda: Poly(x + y, x).replace(x, y)) + raises(PolynomialError, lambda: Poly(x + y, y).replace(y, x)) + + +def test_Poly_reorder(): + raises(PolynomialError, lambda: Poly(x + y).reorder(x, z)) + + assert Poly(x + y, x, y).reorder(x, y) == Poly(x + y, x, y) + assert Poly(x + y, x, y).reorder(y, x) == Poly(x + y, y, x) + + assert Poly(x + y, y, x).reorder(x, y) == Poly(x + y, x, y) + assert Poly(x + y, y, x).reorder(y, x) == Poly(x + y, y, x) + + assert Poly(x + y, x, y).reorder(wrt=x) == Poly(x + y, x, y) + assert Poly(x + y, x, y).reorder(wrt=y) == Poly(x + y, y, x) + + +def test_Poly_ltrim(): + f = Poly(y**2 + y*z**2, x, y, z).ltrim(y) + assert f.as_expr() == y**2 + y*z**2 and f.gens == (y, z) + assert Poly(x*y - x, z, x, y).ltrim(1) == Poly(x*y - x, x, y) + + raises(PolynomialError, lambda: Poly(x*y**2 + y**2, x, y).ltrim(y)) + raises(PolynomialError, lambda: Poly(x*y - x, x, y).ltrim(-1)) + +def test_Poly_has_only_gens(): + assert Poly(x*y + 1, x, y, z).has_only_gens(x, y) is True + assert Poly(x*y + z, x, y, z).has_only_gens(x, y) is False + + raises(GeneratorsError, lambda: Poly(x*y**2 + y**2, x, y).has_only_gens(t)) + + +def test_Poly_to_ring(): + assert Poly(2*x + 1, domain='ZZ').to_ring() == Poly(2*x + 1, domain='ZZ') + assert Poly(2*x + 1, domain='QQ').to_ring() == Poly(2*x + 1, domain='ZZ') + + raises(CoercionFailed, lambda: Poly(x/2 + 1).to_ring()) + raises(DomainError, lambda: Poly(2*x + 1, modulus=3).to_ring()) + + +def test_Poly_to_field(): + assert Poly(2*x + 1, domain='ZZ').to_field() == Poly(2*x + 1, domain='QQ') + assert Poly(2*x + 1, domain='QQ').to_field() == Poly(2*x + 1, domain='QQ') + + assert Poly(x/2 + 1, domain='QQ').to_field() == Poly(x/2 + 1, domain='QQ') + assert Poly(2*x + 1, modulus=3).to_field() == Poly(2*x + 1, modulus=3) + + assert Poly(2.0*x + 1.0).to_field() == Poly(2.0*x + 1.0) + + +def test_Poly_to_exact(): + assert Poly(2*x).to_exact() == Poly(2*x) + assert Poly(x/2).to_exact() == Poly(x/2) + + assert Poly(0.1*x).to_exact() == Poly(x/10) + + +def test_Poly_retract(): + f = Poly(x**2 + 1, x, domain=QQ[y]) + + assert f.retract() == Poly(x**2 + 1, x, domain='ZZ') + assert f.retract(field=True) == Poly(x**2 + 1, x, domain='QQ') + + assert Poly(0, x, y).retract() == Poly(0, x, y) + + +def test_Poly_slice(): + f = Poly(x**3 + 2*x**2 + 3*x + 4) + + assert f.slice(0, 0) == Poly(0, x) + assert f.slice(0, 1) == Poly(4, x) + assert f.slice(0, 2) == Poly(3*x + 4, x) + assert f.slice(0, 3) == Poly(2*x**2 + 3*x + 4, x) + assert f.slice(0, 4) == Poly(x**3 + 2*x**2 + 3*x + 4, x) + + assert f.slice(x, 0, 0) == Poly(0, x) + assert f.slice(x, 0, 1) == Poly(4, x) + assert f.slice(x, 0, 2) == Poly(3*x + 4, x) + assert f.slice(x, 0, 3) == Poly(2*x**2 + 3*x + 4, x) + assert f.slice(x, 0, 4) == Poly(x**3 + 2*x**2 + 3*x + 4, x) + + g = Poly(x**3 + 1) + + assert g.slice(0, 3) == Poly(1, x) + + +def test_Poly_coeffs(): + assert Poly(0, x).coeffs() == [0] + assert Poly(1, x).coeffs() == [1] + + assert Poly(2*x + 1, x).coeffs() == [2, 1] + + assert Poly(7*x**2 + 2*x + 1, x).coeffs() == [7, 2, 1] + assert Poly(7*x**4 + 2*x + 1, x).coeffs() == [7, 2, 1] + + assert Poly(x*y**7 + 2*x**2*y**3).coeffs('lex') == [2, 1] + assert Poly(x*y**7 + 2*x**2*y**3).coeffs('grlex') == [1, 2] + + +def test_Poly_monoms(): + assert Poly(0, x).monoms() == [(0,)] + assert Poly(1, x).monoms() == [(0,)] + + assert Poly(2*x + 1, x).monoms() == [(1,), (0,)] + + assert Poly(7*x**2 + 2*x + 1, x).monoms() == [(2,), (1,), (0,)] + assert Poly(7*x**4 + 2*x + 1, x).monoms() == [(4,), (1,), (0,)] + + assert Poly(x*y**7 + 2*x**2*y**3).monoms('lex') == [(2, 3), (1, 7)] + assert Poly(x*y**7 + 2*x**2*y**3).monoms('grlex') == [(1, 7), (2, 3)] + + +def test_Poly_terms(): + assert Poly(0, x).terms() == [((0,), 0)] + assert Poly(1, x).terms() == [((0,), 1)] + + assert Poly(2*x + 1, x).terms() == [((1,), 2), ((0,), 1)] + + assert Poly(7*x**2 + 2*x + 1, x).terms() == [((2,), 7), ((1,), 2), ((0,), 1)] + assert Poly(7*x**4 + 2*x + 1, x).terms() == [((4,), 7), ((1,), 2), ((0,), 1)] + + assert Poly( + x*y**7 + 2*x**2*y**3).terms('lex') == [((2, 3), 2), ((1, 7), 1)] + assert Poly( + x*y**7 + 2*x**2*y**3).terms('grlex') == [((1, 7), 1), ((2, 3), 2)] + + +def test_Poly_all_coeffs(): + assert Poly(0, x).all_coeffs() == [0] + assert Poly(1, x).all_coeffs() == [1] + + assert Poly(2*x + 1, x).all_coeffs() == [2, 1] + + assert Poly(7*x**2 + 2*x + 1, x).all_coeffs() == [7, 2, 1] + assert Poly(7*x**4 + 2*x + 1, x).all_coeffs() == [7, 0, 0, 2, 1] + + +def test_Poly_all_monoms(): + assert Poly(0, x).all_monoms() == [(0,)] + assert Poly(1, x).all_monoms() == [(0,)] + + assert Poly(2*x + 1, x).all_monoms() == [(1,), (0,)] + + assert Poly(7*x**2 + 2*x + 1, x).all_monoms() == [(2,), (1,), (0,)] + assert Poly(7*x**4 + 2*x + 1, x).all_monoms() == [(4,), (3,), (2,), (1,), (0,)] + + +def test_Poly_all_terms(): + assert Poly(0, x).all_terms() == [((0,), 0)] + assert Poly(1, x).all_terms() == [((0,), 1)] + + assert Poly(2*x + 1, x).all_terms() == [((1,), 2), ((0,), 1)] + + assert Poly(7*x**2 + 2*x + 1, x).all_terms() == \ + [((2,), 7), ((1,), 2), ((0,), 1)] + assert Poly(7*x**4 + 2*x + 1, x).all_terms() == \ + [((4,), 7), ((3,), 0), ((2,), 0), ((1,), 2), ((0,), 1)] + + +def test_Poly_termwise(): + f = Poly(x**2 + 20*x + 400) + g = Poly(x**2 + 2*x + 4) + + def func(monom, coeff): + (k,) = monom + return coeff//10**(2 - k) + + assert f.termwise(func) == g + + def func(monom, coeff): + (k,) = monom + return (k,), coeff//10**(2 - k) + + assert f.termwise(func) == g + + +def test_Poly_length(): + assert Poly(0, x).length() == 0 + assert Poly(1, x).length() == 1 + assert Poly(x, x).length() == 1 + + assert Poly(x + 1, x).length() == 2 + assert Poly(x**2 + 1, x).length() == 2 + assert Poly(x**2 + x + 1, x).length() == 3 + + +def test_Poly_as_dict(): + assert Poly(0, x).as_dict() == {} + assert Poly(0, x, y, z).as_dict() == {} + + assert Poly(1, x).as_dict() == {(0,): 1} + assert Poly(1, x, y, z).as_dict() == {(0, 0, 0): 1} + + assert Poly(x**2 + 3, x).as_dict() == {(2,): 1, (0,): 3} + assert Poly(x**2 + 3, x, y, z).as_dict() == {(2, 0, 0): 1, (0, 0, 0): 3} + + assert Poly(3*x**2*y*z**3 + 4*x*y + 5*x*z).as_dict() == {(2, 1, 3): 3, + (1, 1, 0): 4, (1, 0, 1): 5} + + +def test_Poly_as_expr(): + assert Poly(0, x).as_expr() == 0 + assert Poly(0, x, y, z).as_expr() == 0 + + assert Poly(1, x).as_expr() == 1 + assert Poly(1, x, y, z).as_expr() == 1 + + assert Poly(x**2 + 3, x).as_expr() == x**2 + 3 + assert Poly(x**2 + 3, x, y, z).as_expr() == x**2 + 3 + + assert Poly( + 3*x**2*y*z**3 + 4*x*y + 5*x*z).as_expr() == 3*x**2*y*z**3 + 4*x*y + 5*x*z + + f = Poly(x**2 + 2*x*y**2 - y, x, y) + + assert f.as_expr() == -y + x**2 + 2*x*y**2 + + assert f.as_expr({x: 5}) == 25 - y + 10*y**2 + assert f.as_expr({y: 6}) == -6 + 72*x + x**2 + + assert f.as_expr({x: 5, y: 6}) == 379 + assert f.as_expr(5, 6) == 379 + + raises(GeneratorsError, lambda: f.as_expr({z: 7})) + + +def test_Poly_lift(): + p = Poly(x**4 - I*x + 17*I, x, gaussian=True) + assert p.lift() == Poly(x**8 + x**2 - 34*x + 289, x, domain='QQ') + + +def test_Poly_lift_multiple(): + + r1 = rootof(y**3 + y**2 - 1, 0) + r2 = rootof(z**5 + z**2 - 1, 0) + p = Poly(r1*x + 3*r1**2 - r2 + x**2 - x**5, x, extension=True) + + assert p.lift() == Poly( + -x**75 + 15*x**72 - 5*x**71 + 15*x**70 - 105*x**69 + 70*x**68 - + 220*x**67 + 560*x**66 - 635*x**65 + 1495*x**64 - 2735*x**63 + + 4415*x**62 - 7410*x**61 + 12741*x**60 - 22090*x**59 + 32125*x**58 - + 56281*x**57 + 88157*x**56 - 126842*x**55 + 214223*x**54 - 311802*x**53 + + 462667*x**52 - 700883*x**51 + 1006278*x**50 - 1480950*x**49 + + 2078055*x**48 - 3004675*x**47 + 4140410*x**46 - 5664222*x**45 + + 8029445*x**44 - 10528785*x**43 + 14309614*x**42 - 19032988*x**41 + + 24570573*x**40 - 32530459*x**39 + 41239581*x**38 - 52968051*x**37 + + 65891606*x**36 - 81997276*x**35 + 102530732*x**34 - 122009994*x**33 + + 150227996*x**32 - 176452478*x**31 + 206393768*x**30 - 245291426*x**29 + + 276598718*x**28 - 320005297*x**27 + 353649032*x**26 + - 393246309*x**25 + 434566186*x**24 - 460608964*x**23 + 508052079*x**22 + - 513976618*x**21 + 539374498*x**20 - 557851717*x**19 + 540788016*x**18 + - 564949060*x**17 + 520866566*x**16 + - 507861375*x**15 + 474999819*x**14 - 423619160*x**13 + 414540540*x**12 + - 322522367*x**11 + 311586511*x**10 - 238812299*x**9 + 184482053*x**8 + - 189265274*x**7 + 93619528*x**6 - 106852385*x**5 + 57294385*x**4 - + 26486666*x**3 + 42614683*x**2 - 1511583*x + 15975845, x, domain='QQ' + ) + + +def test_Poly_deflate(): + assert Poly(0, x).deflate() == ((1,), Poly(0, x)) + assert Poly(1, x).deflate() == ((1,), Poly(1, x)) + assert Poly(x, x).deflate() == ((1,), Poly(x, x)) + + assert Poly(x**2, x).deflate() == ((2,), Poly(x, x)) + assert Poly(x**17, x).deflate() == ((17,), Poly(x, x)) + + assert Poly( + x**2*y*z**11 + x**4*z**11).deflate() == ((2, 1, 11), Poly(x*y*z + x**2*z)) + + +def test_Poly_inject(): + f = Poly(x**2*y + x*y**3 + x*y + 1, x) + + assert f.inject() == Poly(x**2*y + x*y**3 + x*y + 1, x, y) + assert f.inject(front=True) == Poly(y**3*x + y*x**2 + y*x + 1, y, x) + + +def test_Poly_eject(): + f = Poly(x**2*y + x*y**3 + x*y + 1, x, y) + + assert f.eject(x) == Poly(x*y**3 + (x**2 + x)*y + 1, y, domain='ZZ[x]') + assert f.eject(y) == Poly(y*x**2 + (y**3 + y)*x + 1, x, domain='ZZ[y]') + + ex = x + y + z + t + w + g = Poly(ex, x, y, z, t, w) + + assert g.eject(x) == Poly(ex, y, z, t, w, domain='ZZ[x]') + assert g.eject(x, y) == Poly(ex, z, t, w, domain='ZZ[x, y]') + assert g.eject(x, y, z) == Poly(ex, t, w, domain='ZZ[x, y, z]') + assert g.eject(w) == Poly(ex, x, y, z, t, domain='ZZ[w]') + assert g.eject(t, w) == Poly(ex, x, y, z, domain='ZZ[t, w]') + assert g.eject(z, t, w) == Poly(ex, x, y, domain='ZZ[z, t, w]') + + raises(DomainError, lambda: Poly(x*y, x, y, domain=ZZ[z]).eject(y)) + raises(NotImplementedError, lambda: Poly(x*y, x, y, z).eject(y)) + + +def test_Poly_exclude(): + assert Poly(x, x, y).exclude() == Poly(x, x) + assert Poly(x*y, x, y).exclude() == Poly(x*y, x, y) + assert Poly(1, x, y).exclude() == Poly(1, x, y) + + +def test_Poly__gen_to_level(): + assert Poly(1, x, y)._gen_to_level(-2) == 0 + assert Poly(1, x, y)._gen_to_level(-1) == 1 + assert Poly(1, x, y)._gen_to_level( 0) == 0 + assert Poly(1, x, y)._gen_to_level( 1) == 1 + + raises(PolynomialError, lambda: Poly(1, x, y)._gen_to_level(-3)) + raises(PolynomialError, lambda: Poly(1, x, y)._gen_to_level( 2)) + + assert Poly(1, x, y)._gen_to_level(x) == 0 + assert Poly(1, x, y)._gen_to_level(y) == 1 + + assert Poly(1, x, y)._gen_to_level('x') == 0 + assert Poly(1, x, y)._gen_to_level('y') == 1 + + raises(PolynomialError, lambda: Poly(1, x, y)._gen_to_level(z)) + raises(PolynomialError, lambda: Poly(1, x, y)._gen_to_level('z')) + + +def test_Poly_degree(): + assert Poly(0, x).degree() is -oo + assert Poly(1, x).degree() == 0 + assert Poly(x, x).degree() == 1 + + assert Poly(0, x).degree(gen=0) is -oo + assert Poly(1, x).degree(gen=0) == 0 + assert Poly(x, x).degree(gen=0) == 1 + + assert Poly(0, x).degree(gen=x) is -oo + assert Poly(1, x).degree(gen=x) == 0 + assert Poly(x, x).degree(gen=x) == 1 + + assert Poly(0, x).degree(gen='x') is -oo + assert Poly(1, x).degree(gen='x') == 0 + assert Poly(x, x).degree(gen='x') == 1 + + raises(PolynomialError, lambda: Poly(1, x).degree(gen=1)) + raises(PolynomialError, lambda: Poly(1, x).degree(gen=y)) + raises(PolynomialError, lambda: Poly(1, x).degree(gen='y')) + + assert Poly(1, x, y).degree() == 0 + assert Poly(2*y, x, y).degree() == 0 + assert Poly(x*y, x, y).degree() == 1 + + assert Poly(1, x, y).degree(gen=x) == 0 + assert Poly(2*y, x, y).degree(gen=x) == 0 + assert Poly(x*y, x, y).degree(gen=x) == 1 + + assert Poly(1, x, y).degree(gen=y) == 0 + assert Poly(2*y, x, y).degree(gen=y) == 1 + assert Poly(x*y, x, y).degree(gen=y) == 1 + + assert degree(0, x) is -oo + assert degree(1, x) == 0 + assert degree(x, x) == 1 + + assert degree(x*y**2, x) == 1 + assert degree(x*y**2, y) == 2 + assert degree(x*y**2, z) == 0 + + assert degree(pi) == 1 + + raises(TypeError, lambda: degree(y**2 + x**3)) + raises(TypeError, lambda: degree(y**2 + x**3, 1)) + raises(PolynomialError, lambda: degree(x, 1.1)) + raises(PolynomialError, lambda: degree(x**2/(x**3 + 1), x)) + + assert degree(Poly(0,x),z) is -oo + assert degree(Poly(1,x),z) == 0 + assert degree(Poly(x**2+y**3,y)) == 3 + assert degree(Poly(y**2 + x**3, y, x), 1) == 3 + assert degree(Poly(y**2 + x**3, x), z) == 0 + assert degree(Poly(y**2 + x**3 + z**4, x), z) == 4 + +def test_Poly_degree_list(): + assert Poly(0, x).degree_list() == (-oo,) + assert Poly(0, x, y).degree_list() == (-oo, -oo) + assert Poly(0, x, y, z).degree_list() == (-oo, -oo, -oo) + + assert Poly(1, x).degree_list() == (0,) + assert Poly(1, x, y).degree_list() == (0, 0) + assert Poly(1, x, y, z).degree_list() == (0, 0, 0) + + assert Poly(x**2*y + x**3*z**2 + 1).degree_list() == (3, 1, 2) + + assert degree_list(1, x) == (0,) + assert degree_list(x, x) == (1,) + + assert degree_list(x*y**2) == (1, 2) + + raises(ComputationFailed, lambda: degree_list(1)) + + +def test_Poly_total_degree(): + assert Poly(x**2*y + x**3*z**2 + 1).total_degree() == 5 + assert Poly(x**2 + z**3).total_degree() == 3 + assert Poly(x*y*z + z**4).total_degree() == 4 + assert Poly(x**3 + x + 1).total_degree() == 3 + + assert total_degree(x*y + z**3) == 3 + assert total_degree(x*y + z**3, x, y) == 2 + assert total_degree(1) == 0 + assert total_degree(Poly(y**2 + x**3 + z**4)) == 4 + assert total_degree(Poly(y**2 + x**3 + z**4, x)) == 3 + assert total_degree(Poly(y**2 + x**3 + z**4, x), z) == 4 + assert total_degree(Poly(x**9 + x*z*y + x**3*z**2 + z**7,x), z) == 7 + +def test_Poly_homogenize(): + assert Poly(x**2+y).homogenize(z) == Poly(x**2+y*z) + assert Poly(x+y).homogenize(z) == Poly(x+y, x, y, z) + assert Poly(x+y**2).homogenize(y) == Poly(x*y+y**2) + + +def test_Poly_homogeneous_order(): + assert Poly(0, x, y).homogeneous_order() is -oo + assert Poly(1, x, y).homogeneous_order() == 0 + assert Poly(x, x, y).homogeneous_order() == 1 + assert Poly(x*y, x, y).homogeneous_order() == 2 + + assert Poly(x + 1, x, y).homogeneous_order() is None + assert Poly(x*y + x, x, y).homogeneous_order() is None + + assert Poly(x**5 + 2*x**3*y**2 + 9*x*y**4).homogeneous_order() == 5 + assert Poly(x**5 + 2*x**3*y**3 + 9*x*y**4).homogeneous_order() is None + + +def test_Poly_LC(): + assert Poly(0, x).LC() == 0 + assert Poly(1, x).LC() == 1 + assert Poly(2*x**2 + x, x).LC() == 2 + + assert Poly(x*y**7 + 2*x**2*y**3).LC('lex') == 2 + assert Poly(x*y**7 + 2*x**2*y**3).LC('grlex') == 1 + + assert LC(x*y**7 + 2*x**2*y**3, order='lex') == 2 + assert LC(x*y**7 + 2*x**2*y**3, order='grlex') == 1 + + +def test_Poly_TC(): + assert Poly(0, x).TC() == 0 + assert Poly(1, x).TC() == 1 + assert Poly(2*x**2 + x, x).TC() == 0 + + +def test_Poly_EC(): + assert Poly(0, x).EC() == 0 + assert Poly(1, x).EC() == 1 + assert Poly(2*x**2 + x, x).EC() == 1 + + assert Poly(x*y**7 + 2*x**2*y**3).EC('lex') == 1 + assert Poly(x*y**7 + 2*x**2*y**3).EC('grlex') == 2 + + +def test_Poly_coeff(): + assert Poly(0, x).coeff_monomial(1) == 0 + assert Poly(0, x).coeff_monomial(x) == 0 + + assert Poly(1, x).coeff_monomial(1) == 1 + assert Poly(1, x).coeff_monomial(x) == 0 + + assert Poly(x**8, x).coeff_monomial(1) == 0 + assert Poly(x**8, x).coeff_monomial(x**7) == 0 + assert Poly(x**8, x).coeff_monomial(x**8) == 1 + assert Poly(x**8, x).coeff_monomial(x**9) == 0 + + assert Poly(3*x*y**2 + 1, x, y).coeff_monomial(1) == 1 + assert Poly(3*x*y**2 + 1, x, y).coeff_monomial(x*y**2) == 3 + + p = Poly(24*x*y*exp(8) + 23*x, x, y) + + assert p.coeff_monomial(x) == 23 + assert p.coeff_monomial(y) == 0 + assert p.coeff_monomial(x*y) == 24*exp(8) + + assert p.as_expr().coeff(x) == 24*y*exp(8) + 23 + raises(NotImplementedError, lambda: p.coeff(x)) + + raises(ValueError, lambda: Poly(x + 1).coeff_monomial(0)) + raises(ValueError, lambda: Poly(x + 1).coeff_monomial(3*x)) + raises(ValueError, lambda: Poly(x + 1).coeff_monomial(3*x*y)) + + +def test_Poly_nth(): + assert Poly(0, x).nth(0) == 0 + assert Poly(0, x).nth(1) == 0 + + assert Poly(1, x).nth(0) == 1 + assert Poly(1, x).nth(1) == 0 + + assert Poly(x**8, x).nth(0) == 0 + assert Poly(x**8, x).nth(7) == 0 + assert Poly(x**8, x).nth(8) == 1 + assert Poly(x**8, x).nth(9) == 0 + + assert Poly(3*x*y**2 + 1, x, y).nth(0, 0) == 1 + assert Poly(3*x*y**2 + 1, x, y).nth(1, 2) == 3 + + raises(ValueError, lambda: Poly(x*y + 1, x, y).nth(1)) + + +def test_Poly_LM(): + assert Poly(0, x).LM() == (0,) + assert Poly(1, x).LM() == (0,) + assert Poly(2*x**2 + x, x).LM() == (2,) + + assert Poly(x*y**7 + 2*x**2*y**3).LM('lex') == (2, 3) + assert Poly(x*y**7 + 2*x**2*y**3).LM('grlex') == (1, 7) + + assert LM(x*y**7 + 2*x**2*y**3, order='lex') == x**2*y**3 + assert LM(x*y**7 + 2*x**2*y**3, order='grlex') == x*y**7 + + +def test_Poly_LM_custom_order(): + f = Poly(x**2*y**3*z + x**2*y*z**3 + x*y*z + 1) + rev_lex = lambda monom: tuple(reversed(monom)) + + assert f.LM(order='lex') == (2, 3, 1) + assert f.LM(order=rev_lex) == (2, 1, 3) + + +def test_Poly_EM(): + assert Poly(0, x).EM() == (0,) + assert Poly(1, x).EM() == (0,) + assert Poly(2*x**2 + x, x).EM() == (1,) + + assert Poly(x*y**7 + 2*x**2*y**3).EM('lex') == (1, 7) + assert Poly(x*y**7 + 2*x**2*y**3).EM('grlex') == (2, 3) + + +def test_Poly_LT(): + assert Poly(0, x).LT() == ((0,), 0) + assert Poly(1, x).LT() == ((0,), 1) + assert Poly(2*x**2 + x, x).LT() == ((2,), 2) + + assert Poly(x*y**7 + 2*x**2*y**3).LT('lex') == ((2, 3), 2) + assert Poly(x*y**7 + 2*x**2*y**3).LT('grlex') == ((1, 7), 1) + + assert LT(x*y**7 + 2*x**2*y**3, order='lex') == 2*x**2*y**3 + assert LT(x*y**7 + 2*x**2*y**3, order='grlex') == x*y**7 + + +def test_Poly_ET(): + assert Poly(0, x).ET() == ((0,), 0) + assert Poly(1, x).ET() == ((0,), 1) + assert Poly(2*x**2 + x, x).ET() == ((1,), 1) + + assert Poly(x*y**7 + 2*x**2*y**3).ET('lex') == ((1, 7), 1) + assert Poly(x*y**7 + 2*x**2*y**3).ET('grlex') == ((2, 3), 2) + + +def test_Poly_max_norm(): + assert Poly(-1, x).max_norm() == 1 + assert Poly( 0, x).max_norm() == 0 + assert Poly( 1, x).max_norm() == 1 + + +def test_Poly_l1_norm(): + assert Poly(-1, x).l1_norm() == 1 + assert Poly( 0, x).l1_norm() == 0 + assert Poly( 1, x).l1_norm() == 1 + + +def test_Poly_clear_denoms(): + coeff, poly = Poly(x + 2, x).clear_denoms() + assert coeff == 1 and poly == Poly( + x + 2, x, domain='ZZ') and poly.get_domain() == ZZ + + coeff, poly = Poly(x/2 + 1, x).clear_denoms() + assert coeff == 2 and poly == Poly( + x + 2, x, domain='QQ') and poly.get_domain() == QQ + + coeff, poly = Poly(2*x**2 + 3, modulus=5).clear_denoms() + assert coeff == 1 and poly == Poly( + 2*x**2 + 3, x, modulus=5) and poly.get_domain() == FF(5) + + coeff, poly = Poly(x/2 + 1, x).clear_denoms(convert=True) + assert coeff == 2 and poly == Poly( + x + 2, x, domain='ZZ') and poly.get_domain() == ZZ + + coeff, poly = Poly(x/y + 1, x).clear_denoms(convert=True) + assert coeff == y and poly == Poly( + x + y, x, domain='ZZ[y]') and poly.get_domain() == ZZ[y] + + coeff, poly = Poly(x/3 + sqrt(2), x, domain='EX').clear_denoms() + assert coeff == 3 and poly == Poly( + x + 3*sqrt(2), x, domain='EX') and poly.get_domain() == EX + + coeff, poly = Poly( + x/3 + sqrt(2), x, domain='EX').clear_denoms(convert=True) + assert coeff == 3 and poly == Poly( + x + 3*sqrt(2), x, domain='EX') and poly.get_domain() == EX + + +def test_Poly_rat_clear_denoms(): + f = Poly(x**2/y + 1, x) + g = Poly(x**3 + y, x) + + assert f.rat_clear_denoms(g) == \ + (Poly(x**2 + y, x), Poly(y*x**3 + y**2, x)) + + f = f.set_domain(EX) + g = g.set_domain(EX) + + assert f.rat_clear_denoms(g) == (f, g) + + +def test_issue_20427(): + f = Poly(-117968192370600*18**(S(1)/3)/(217603955769048*(24201 + + 253*sqrt(9165))**(S(1)/3) + 2273005839412*sqrt(9165)*(24201 + + 253*sqrt(9165))**(S(1)/3)) - 15720318185*2**(S(2)/3)*3**(S(1)/3)*(24201 + + 253*sqrt(9165))**(S(2)/3)/(217603955769048*(24201 + 253*sqrt(9165))** + (S(1)/3) + 2273005839412*sqrt(9165)*(24201 + 253*sqrt(9165))**(S(1)/3)) + + 15720318185*12**(S(1)/3)*(24201 + 253*sqrt(9165))**(S(2)/3)/( + 217603955769048*(24201 + 253*sqrt(9165))**(S(1)/3) + 2273005839412* + sqrt(9165)*(24201 + 253*sqrt(9165))**(S(1)/3)) + 117968192370600*2**( + S(1)/3)*3**(S(2)/3)/(217603955769048*(24201 + 253*sqrt(9165))**(S(1)/3) + + 2273005839412*sqrt(9165)*(24201 + 253*sqrt(9165))**(S(1)/3)), x) + assert f == Poly(0, x, domain='EX') + + +def test_Poly_integrate(): + assert Poly(x + 1).integrate() == Poly(x**2/2 + x) + assert Poly(x + 1).integrate(x) == Poly(x**2/2 + x) + assert Poly(x + 1).integrate((x, 1)) == Poly(x**2/2 + x) + + assert Poly(x*y + 1).integrate(x) == Poly(x**2*y/2 + x) + assert Poly(x*y + 1).integrate(y) == Poly(x*y**2/2 + y) + + assert Poly(x*y + 1).integrate(x, x) == Poly(x**3*y/6 + x**2/2) + assert Poly(x*y + 1).integrate(y, y) == Poly(x*y**3/6 + y**2/2) + + assert Poly(x*y + 1).integrate((x, 2)) == Poly(x**3*y/6 + x**2/2) + assert Poly(x*y + 1).integrate((y, 2)) == Poly(x*y**3/6 + y**2/2) + + assert Poly(x*y + 1).integrate(x, y) == Poly(x**2*y**2/4 + x*y) + assert Poly(x*y + 1).integrate(y, x) == Poly(x**2*y**2/4 + x*y) + + +def test_Poly_diff(): + assert Poly(x**2 + x).diff() == Poly(2*x + 1) + assert Poly(x**2 + x).diff(x) == Poly(2*x + 1) + assert Poly(x**2 + x).diff((x, 1)) == Poly(2*x + 1) + + assert Poly(x**2*y**2 + x*y).diff(x) == Poly(2*x*y**2 + y) + assert Poly(x**2*y**2 + x*y).diff(y) == Poly(2*x**2*y + x) + + assert Poly(x**2*y**2 + x*y).diff(x, x) == Poly(2*y**2, x, y) + assert Poly(x**2*y**2 + x*y).diff(y, y) == Poly(2*x**2, x, y) + + assert Poly(x**2*y**2 + x*y).diff((x, 2)) == Poly(2*y**2, x, y) + assert Poly(x**2*y**2 + x*y).diff((y, 2)) == Poly(2*x**2, x, y) + + assert Poly(x**2*y**2 + x*y).diff(x, y) == Poly(4*x*y + 1) + assert Poly(x**2*y**2 + x*y).diff(y, x) == Poly(4*x*y + 1) + + +def test_issue_9585(): + assert diff(Poly(x**2 + x)) == Poly(2*x + 1) + assert diff(Poly(x**2 + x), x, evaluate=False) == \ + Derivative(Poly(x**2 + x), x) + assert Derivative(Poly(x**2 + x), x).doit() == Poly(2*x + 1) + + +def test_Poly_eval(): + assert Poly(0, x).eval(7) == 0 + assert Poly(1, x).eval(7) == 1 + assert Poly(x, x).eval(7) == 7 + + assert Poly(0, x).eval(0, 7) == 0 + assert Poly(1, x).eval(0, 7) == 1 + assert Poly(x, x).eval(0, 7) == 7 + + assert Poly(0, x).eval(x, 7) == 0 + assert Poly(1, x).eval(x, 7) == 1 + assert Poly(x, x).eval(x, 7) == 7 + + assert Poly(0, x).eval('x', 7) == 0 + assert Poly(1, x).eval('x', 7) == 1 + assert Poly(x, x).eval('x', 7) == 7 + + raises(PolynomialError, lambda: Poly(1, x).eval(1, 7)) + raises(PolynomialError, lambda: Poly(1, x).eval(y, 7)) + raises(PolynomialError, lambda: Poly(1, x).eval('y', 7)) + + assert Poly(123, x, y).eval(7) == Poly(123, y) + assert Poly(2*y, x, y).eval(7) == Poly(2*y, y) + assert Poly(x*y, x, y).eval(7) == Poly(7*y, y) + + assert Poly(123, x, y).eval(x, 7) == Poly(123, y) + assert Poly(2*y, x, y).eval(x, 7) == Poly(2*y, y) + assert Poly(x*y, x, y).eval(x, 7) == Poly(7*y, y) + + assert Poly(123, x, y).eval(y, 7) == Poly(123, x) + assert Poly(2*y, x, y).eval(y, 7) == Poly(14, x) + assert Poly(x*y, x, y).eval(y, 7) == Poly(7*x, x) + + assert Poly(x*y + y, x, y).eval({x: 7}) == Poly(8*y, y) + assert Poly(x*y + y, x, y).eval({y: 7}) == Poly(7*x + 7, x) + + assert Poly(x*y + y, x, y).eval({x: 6, y: 7}) == 49 + assert Poly(x*y + y, x, y).eval({x: 7, y: 6}) == 48 + + assert Poly(x*y + y, x, y).eval((6, 7)) == 49 + assert Poly(x*y + y, x, y).eval([6, 7]) == 49 + + assert Poly(x + 1, domain='ZZ').eval(S.Half) == Rational(3, 2) + assert Poly(x + 1, domain='ZZ').eval(sqrt(2)) == sqrt(2) + 1 + + raises(ValueError, lambda: Poly(x*y + y, x, y).eval((6, 7, 8))) + raises(DomainError, lambda: Poly(x + 1, domain='ZZ').eval(S.Half, auto=False)) + + # issue 6344 + alpha = Symbol('alpha') + result = (2*alpha*z - 2*alpha + z**2 + 3)/(z**2 - 2*z + 1) + + f = Poly(x**2 + (alpha - 1)*x - alpha + 1, x, domain='ZZ[alpha]') + assert f.eval((z + 1)/(z - 1)) == result + + g = Poly(x**2 + (alpha - 1)*x - alpha + 1, x, y, domain='ZZ[alpha]') + assert g.eval((z + 1)/(z - 1)) == Poly(result, y, domain='ZZ(alpha,z)') + +def test_Poly___call__(): + f = Poly(2*x*y + 3*x + y + 2*z) + + assert f(2) == Poly(5*y + 2*z + 6) + assert f(2, 5) == Poly(2*z + 31) + assert f(2, 5, 7) == 45 + + +def test_parallel_poly_from_expr(): + assert parallel_poly_from_expr( + [x - 1, x**2 - 1], x)[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)] + assert parallel_poly_from_expr( + [Poly(x - 1, x), x**2 - 1], x)[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)] + assert parallel_poly_from_expr( + [x - 1, Poly(x**2 - 1, x)], x)[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)] + assert parallel_poly_from_expr([Poly( + x - 1, x), Poly(x**2 - 1, x)], x)[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)] + + assert parallel_poly_from_expr( + [x - 1, x**2 - 1], x, y)[0] == [Poly(x - 1, x, y), Poly(x**2 - 1, x, y)] + assert parallel_poly_from_expr([Poly( + x - 1, x), x**2 - 1], x, y)[0] == [Poly(x - 1, x, y), Poly(x**2 - 1, x, y)] + assert parallel_poly_from_expr([x - 1, Poly( + x**2 - 1, x)], x, y)[0] == [Poly(x - 1, x, y), Poly(x**2 - 1, x, y)] + assert parallel_poly_from_expr([Poly(x - 1, x), Poly( + x**2 - 1, x)], x, y)[0] == [Poly(x - 1, x, y), Poly(x**2 - 1, x, y)] + + assert parallel_poly_from_expr( + [x - 1, x**2 - 1])[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)] + assert parallel_poly_from_expr( + [Poly(x - 1, x), x**2 - 1])[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)] + assert parallel_poly_from_expr( + [x - 1, Poly(x**2 - 1, x)])[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)] + assert parallel_poly_from_expr( + [Poly(x - 1, x), Poly(x**2 - 1, x)])[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)] + + assert parallel_poly_from_expr( + [1, x**2 - 1])[0] == [Poly(1, x), Poly(x**2 - 1, x)] + assert parallel_poly_from_expr( + [1, x**2 - 1])[0] == [Poly(1, x), Poly(x**2 - 1, x)] + assert parallel_poly_from_expr( + [1, Poly(x**2 - 1, x)])[0] == [Poly(1, x), Poly(x**2 - 1, x)] + assert parallel_poly_from_expr( + [1, Poly(x**2 - 1, x)])[0] == [Poly(1, x), Poly(x**2 - 1, x)] + + assert parallel_poly_from_expr( + [x**2 - 1, 1])[0] == [Poly(x**2 - 1, x), Poly(1, x)] + assert parallel_poly_from_expr( + [x**2 - 1, 1])[0] == [Poly(x**2 - 1, x), Poly(1, x)] + assert parallel_poly_from_expr( + [Poly(x**2 - 1, x), 1])[0] == [Poly(x**2 - 1, x), Poly(1, x)] + assert parallel_poly_from_expr( + [Poly(x**2 - 1, x), 1])[0] == [Poly(x**2 - 1, x), Poly(1, x)] + + assert parallel_poly_from_expr([Poly(x, x, y), Poly(y, x, y)], x, y, order='lex')[0] == \ + [Poly(x, x, y, domain='ZZ'), Poly(y, x, y, domain='ZZ')] + + raises(PolificationFailed, lambda: parallel_poly_from_expr([0, 1])) + + +def test_pdiv(): + f, g = x**2 - y**2, x - y + q, r = x + y, 0 + + F, G, Q, R = [ Poly(h, x, y) for h in (f, g, q, r) ] + + assert F.pdiv(G) == (Q, R) + assert F.prem(G) == R + assert F.pquo(G) == Q + assert F.pexquo(G) == Q + + assert pdiv(f, g) == (q, r) + assert prem(f, g) == r + assert pquo(f, g) == q + assert pexquo(f, g) == q + + assert pdiv(f, g, x, y) == (q, r) + assert prem(f, g, x, y) == r + assert pquo(f, g, x, y) == q + assert pexquo(f, g, x, y) == q + + assert pdiv(f, g, (x, y)) == (q, r) + assert prem(f, g, (x, y)) == r + assert pquo(f, g, (x, y)) == q + assert pexquo(f, g, (x, y)) == q + + assert pdiv(F, G) == (Q, R) + assert prem(F, G) == R + assert pquo(F, G) == Q + assert pexquo(F, G) == Q + + assert pdiv(f, g, polys=True) == (Q, R) + assert prem(f, g, polys=True) == R + assert pquo(f, g, polys=True) == Q + assert pexquo(f, g, polys=True) == Q + + assert pdiv(F, G, polys=False) == (q, r) + assert prem(F, G, polys=False) == r + assert pquo(F, G, polys=False) == q + assert pexquo(F, G, polys=False) == q + + raises(ComputationFailed, lambda: pdiv(4, 2)) + raises(ComputationFailed, lambda: prem(4, 2)) + raises(ComputationFailed, lambda: pquo(4, 2)) + raises(ComputationFailed, lambda: pexquo(4, 2)) + + +def test_div(): + f, g = x**2 - y**2, x - y + q, r = x + y, 0 + + F, G, Q, R = [ Poly(h, x, y) for h in (f, g, q, r) ] + + assert F.div(G) == (Q, R) + assert F.rem(G) == R + assert F.quo(G) == Q + assert F.exquo(G) == Q + + assert div(f, g) == (q, r) + assert rem(f, g) == r + assert quo(f, g) == q + assert exquo(f, g) == q + + assert div(f, g, x, y) == (q, r) + assert rem(f, g, x, y) == r + assert quo(f, g, x, y) == q + assert exquo(f, g, x, y) == q + + assert div(f, g, (x, y)) == (q, r) + assert rem(f, g, (x, y)) == r + assert quo(f, g, (x, y)) == q + assert exquo(f, g, (x, y)) == q + + assert div(F, G) == (Q, R) + assert rem(F, G) == R + assert quo(F, G) == Q + assert exquo(F, G) == Q + + assert div(f, g, polys=True) == (Q, R) + assert rem(f, g, polys=True) == R + assert quo(f, g, polys=True) == Q + assert exquo(f, g, polys=True) == Q + + assert div(F, G, polys=False) == (q, r) + assert rem(F, G, polys=False) == r + assert quo(F, G, polys=False) == q + assert exquo(F, G, polys=False) == q + + raises(ComputationFailed, lambda: div(4, 2)) + raises(ComputationFailed, lambda: rem(4, 2)) + raises(ComputationFailed, lambda: quo(4, 2)) + raises(ComputationFailed, lambda: exquo(4, 2)) + + f, g = x**2 + 1, 2*x - 4 + + qz, rz = 0, x**2 + 1 + qq, rq = x/2 + 1, 5 + + assert div(f, g) == (qq, rq) + assert div(f, g, auto=True) == (qq, rq) + assert div(f, g, auto=False) == (qz, rz) + assert div(f, g, domain=ZZ) == (qz, rz) + assert div(f, g, domain=QQ) == (qq, rq) + assert div(f, g, domain=ZZ, auto=True) == (qq, rq) + assert div(f, g, domain=ZZ, auto=False) == (qz, rz) + assert div(f, g, domain=QQ, auto=True) == (qq, rq) + assert div(f, g, domain=QQ, auto=False) == (qq, rq) + + assert rem(f, g) == rq + assert rem(f, g, auto=True) == rq + assert rem(f, g, auto=False) == rz + assert rem(f, g, domain=ZZ) == rz + assert rem(f, g, domain=QQ) == rq + assert rem(f, g, domain=ZZ, auto=True) == rq + assert rem(f, g, domain=ZZ, auto=False) == rz + assert rem(f, g, domain=QQ, auto=True) == rq + assert rem(f, g, domain=QQ, auto=False) == rq + + assert quo(f, g) == qq + assert quo(f, g, auto=True) == qq + assert quo(f, g, auto=False) == qz + assert quo(f, g, domain=ZZ) == qz + assert quo(f, g, domain=QQ) == qq + assert quo(f, g, domain=ZZ, auto=True) == qq + assert quo(f, g, domain=ZZ, auto=False) == qz + assert quo(f, g, domain=QQ, auto=True) == qq + assert quo(f, g, domain=QQ, auto=False) == qq + + f, g, q = x**2, 2*x, x/2 + + assert exquo(f, g) == q + assert exquo(f, g, auto=True) == q + raises(ExactQuotientFailed, lambda: exquo(f, g, auto=False)) + raises(ExactQuotientFailed, lambda: exquo(f, g, domain=ZZ)) + assert exquo(f, g, domain=QQ) == q + assert exquo(f, g, domain=ZZ, auto=True) == q + raises(ExactQuotientFailed, lambda: exquo(f, g, domain=ZZ, auto=False)) + assert exquo(f, g, domain=QQ, auto=True) == q + assert exquo(f, g, domain=QQ, auto=False) == q + + f, g = Poly(x**2), Poly(x) + + q, r = f.div(g) + assert q.get_domain().is_ZZ and r.get_domain().is_ZZ + r = f.rem(g) + assert r.get_domain().is_ZZ + q = f.quo(g) + assert q.get_domain().is_ZZ + q = f.exquo(g) + assert q.get_domain().is_ZZ + + f, g = Poly(x+y, x), Poly(2*x+y, x) + q, r = f.div(g) + assert q.get_domain().is_Frac and r.get_domain().is_Frac + + # https://github.com/sympy/sympy/issues/19579 + p = Poly(2+3*I, x, domain=ZZ_I) + q = Poly(1-I, x, domain=ZZ_I) + assert p.div(q, auto=False) == \ + (Poly(0, x, domain='ZZ_I'), Poly(2 + 3*I, x, domain='ZZ_I')) + assert p.div(q, auto=True) == \ + (Poly(-S(1)/2 + 5*I/2, x, domain='QQ_I'), Poly(0, x, domain='QQ_I')) + + f = 5*x**2 + 10*x + 3 + g = 2*x + 2 + assert div(f, g, domain=ZZ) == (0, f) + + +def test_issue_7864(): + q, r = div(a, .408248290463863*a) + assert abs(q - 2.44948974278318) < 1e-14 + assert r == 0 + + +def test_gcdex(): + f, g = 2*x, x**2 - 16 + s, t, h = x/32, Rational(-1, 16), 1 + + F, G, S, T, H = [ Poly(u, x, domain='QQ') for u in (f, g, s, t, h) ] + + assert F.half_gcdex(G) == (S, H) + assert F.gcdex(G) == (S, T, H) + assert F.invert(G) == S + + assert half_gcdex(f, g) == (s, h) + assert gcdex(f, g) == (s, t, h) + assert invert(f, g) == s + + assert half_gcdex(f, g, x) == (s, h) + assert gcdex(f, g, x) == (s, t, h) + assert invert(f, g, x) == s + + assert half_gcdex(f, g, (x,)) == (s, h) + assert gcdex(f, g, (x,)) == (s, t, h) + assert invert(f, g, (x,)) == s + + assert half_gcdex(F, G) == (S, H) + assert gcdex(F, G) == (S, T, H) + assert invert(F, G) == S + + assert half_gcdex(f, g, polys=True) == (S, H) + assert gcdex(f, g, polys=True) == (S, T, H) + assert invert(f, g, polys=True) == S + + assert half_gcdex(F, G, polys=False) == (s, h) + assert gcdex(F, G, polys=False) == (s, t, h) + assert invert(F, G, polys=False) == s + + assert half_gcdex(100, 2004) == (-20, 4) + assert gcdex(100, 2004) == (-20, 1, 4) + assert invert(3, 7) == 5 + + raises(DomainError, lambda: half_gcdex(x + 1, 2*x + 1, auto=False)) + raises(DomainError, lambda: gcdex(x + 1, 2*x + 1, auto=False)) + raises(DomainError, lambda: invert(x + 1, 2*x + 1, auto=False)) + + +def test_revert(): + f = Poly(1 - x**2/2 + x**4/24 - x**6/720) + g = Poly(61*x**6/720 + 5*x**4/24 + x**2/2 + 1) + + assert f.revert(8) == g + + +def test_subresultants(): + f, g, h = x**2 - 2*x + 1, x**2 - 1, 2*x - 2 + F, G, H = Poly(f), Poly(g), Poly(h) + + assert F.subresultants(G) == [F, G, H] + assert subresultants(f, g) == [f, g, h] + assert subresultants(f, g, x) == [f, g, h] + assert subresultants(f, g, (x,)) == [f, g, h] + assert subresultants(F, G) == [F, G, H] + assert subresultants(f, g, polys=True) == [F, G, H] + assert subresultants(F, G, polys=False) == [f, g, h] + + raises(ComputationFailed, lambda: subresultants(4, 2)) + + +def test_resultant(): + f, g, h = x**2 - 2*x + 1, x**2 - 1, 0 + F, G = Poly(f), Poly(g) + + assert F.resultant(G) == h + assert resultant(f, g) == h + assert resultant(f, g, x) == h + assert resultant(f, g, (x,)) == h + assert resultant(F, G) == h + assert resultant(f, g, polys=True) == h + assert resultant(F, G, polys=False) == h + assert resultant(f, g, includePRS=True) == (h, [f, g, 2*x - 2]) + + f, g, h = x - a, x - b, a - b + F, G, H = Poly(f), Poly(g), Poly(h) + + assert F.resultant(G) == H + assert resultant(f, g) == h + assert resultant(f, g, x) == h + assert resultant(f, g, (x,)) == h + assert resultant(F, G) == H + assert resultant(f, g, polys=True) == H + assert resultant(F, G, polys=False) == h + + raises(ComputationFailed, lambda: resultant(4, 2)) + + +def test_discriminant(): + f, g = x**3 + 3*x**2 + 9*x - 13, -11664 + F = Poly(f) + + assert F.discriminant() == g + assert discriminant(f) == g + assert discriminant(f, x) == g + assert discriminant(f, (x,)) == g + assert discriminant(F) == g + assert discriminant(f, polys=True) == g + assert discriminant(F, polys=False) == g + + f, g = a*x**2 + b*x + c, b**2 - 4*a*c + F, G = Poly(f), Poly(g) + + assert F.discriminant() == G + assert discriminant(f) == g + assert discriminant(f, x, a, b, c) == g + assert discriminant(f, (x, a, b, c)) == g + assert discriminant(F) == G + assert discriminant(f, polys=True) == G + assert discriminant(F, polys=False) == g + + raises(ComputationFailed, lambda: discriminant(4)) + + +def test_dispersion(): + # We test only the API here. For more mathematical + # tests see the dedicated test file. + fp = poly((x + 1)*(x + 2), x) + assert sorted(fp.dispersionset()) == [0, 1] + assert fp.dispersion() == 1 + + fp = poly(x**4 - 3*x**2 + 1, x) + gp = fp.shift(-3) + assert sorted(fp.dispersionset(gp)) == [2, 3, 4] + assert fp.dispersion(gp) == 4 + + +def test_gcd_list(): + F = [x**3 - 1, x**2 - 1, x**2 - 3*x + 2] + + assert gcd_list(F) == x - 1 + assert gcd_list(F, polys=True) == Poly(x - 1) + + assert gcd_list([]) == 0 + assert gcd_list([1, 2]) == 1 + assert gcd_list([4, 6, 8]) == 2 + + assert gcd_list([x*(y + 42) - x*y - x*42]) == 0 + + gcd = gcd_list([], x) + assert gcd.is_Number and gcd is S.Zero + + gcd = gcd_list([], x, polys=True) + assert gcd.is_Poly and gcd.is_zero + + a = sqrt(2) + assert gcd_list([a, -a]) == gcd_list([-a, a]) == a + + raises(ComputationFailed, lambda: gcd_list([], polys=True)) + + +def test_lcm_list(): + F = [x**3 - 1, x**2 - 1, x**2 - 3*x + 2] + + assert lcm_list(F) == x**5 - x**4 - 2*x**3 - x**2 + x + 2 + assert lcm_list(F, polys=True) == Poly(x**5 - x**4 - 2*x**3 - x**2 + x + 2) + + assert lcm_list([]) == 1 + assert lcm_list([1, 2]) == 2 + assert lcm_list([4, 6, 8]) == 24 + + assert lcm_list([x*(y + 42) - x*y - x*42]) == 0 + + lcm = lcm_list([], x) + assert lcm.is_Number and lcm is S.One + + lcm = lcm_list([], x, polys=True) + assert lcm.is_Poly and lcm.is_one + + raises(ComputationFailed, lambda: lcm_list([], polys=True)) + + +def test_gcd(): + f, g = x**3 - 1, x**2 - 1 + s, t = x**2 + x + 1, x + 1 + h, r = x - 1, x**4 + x**3 - x - 1 + + F, G, S, T, H, R = [ Poly(u) for u in (f, g, s, t, h, r) ] + + assert F.cofactors(G) == (H, S, T) + assert F.gcd(G) == H + assert F.lcm(G) == R + + assert cofactors(f, g) == (h, s, t) + assert gcd(f, g) == h + assert lcm(f, g) == r + + assert cofactors(f, g, x) == (h, s, t) + assert gcd(f, g, x) == h + assert lcm(f, g, x) == r + + assert cofactors(f, g, (x,)) == (h, s, t) + assert gcd(f, g, (x,)) == h + assert lcm(f, g, (x,)) == r + + assert cofactors(F, G) == (H, S, T) + assert gcd(F, G) == H + assert lcm(F, G) == R + + assert cofactors(f, g, polys=True) == (H, S, T) + assert gcd(f, g, polys=True) == H + assert lcm(f, g, polys=True) == R + + assert cofactors(F, G, polys=False) == (h, s, t) + assert gcd(F, G, polys=False) == h + assert lcm(F, G, polys=False) == r + + f, g = 1.0*x**2 - 1.0, 1.0*x - 1.0 + h, s, t = g, 1.0*x + 1.0, 1.0 + + assert cofactors(f, g) == (h, s, t) + assert gcd(f, g) == h + assert lcm(f, g) == f + + f, g = 1.0*x**2 - 1.0, 1.0*x - 1.0 + h, s, t = g, 1.0*x + 1.0, 1.0 + + assert cofactors(f, g) == (h, s, t) + assert gcd(f, g) == h + assert lcm(f, g) == f + + assert cofactors(8, 6) == (2, 4, 3) + assert gcd(8, 6) == 2 + assert lcm(8, 6) == 24 + + f, g = x**2 - 3*x - 4, x**3 - 4*x**2 + x - 4 + l = x**4 - 3*x**3 - 3*x**2 - 3*x - 4 + h, s, t = x - 4, x + 1, x**2 + 1 + + assert cofactors(f, g, modulus=11) == (h, s, t) + assert gcd(f, g, modulus=11) == h + assert lcm(f, g, modulus=11) == l + + f, g = x**2 + 8*x + 7, x**3 + 7*x**2 + x + 7 + l = x**4 + 8*x**3 + 8*x**2 + 8*x + 7 + h, s, t = x + 7, x + 1, x**2 + 1 + + assert cofactors(f, g, modulus=11, symmetric=False) == (h, s, t) + assert gcd(f, g, modulus=11, symmetric=False) == h + assert lcm(f, g, modulus=11, symmetric=False) == l + + a, b = sqrt(2), -sqrt(2) + assert gcd(a, b) == gcd(b, a) == sqrt(2) + + a, b = sqrt(-2), -sqrt(-2) + assert gcd(a, b) == gcd(b, a) == sqrt(2) + + assert gcd(Poly(x - 2, x), Poly(I*x, x)) == Poly(1, x, domain=ZZ_I) + + raises(TypeError, lambda: gcd(x)) + raises(TypeError, lambda: lcm(x)) + + f = Poly(-1, x) + g = Poly(1, x) + assert lcm(f, g) == Poly(1, x) + + f = Poly(0, x) + g = Poly([1, 1], x) + for i in (f, g): + assert lcm(i, 0) == 0 + assert lcm(0, i) == 0 + assert lcm(i, f) == 0 + assert lcm(f, i) == 0 + + f = 4*x**2 + x + 2 + pfz = Poly(f, domain=ZZ) + pfq = Poly(f, domain=QQ) + + assert pfz.gcd(pfz) == pfz + assert pfz.lcm(pfz) == pfz + assert pfq.gcd(pfq) == pfq.monic() + assert pfq.lcm(pfq) == pfq.monic() + assert gcd(f, f) == f + assert lcm(f, f) == f + assert gcd(f, f, domain=QQ) == monic(f) + assert lcm(f, f, domain=QQ) == monic(f) + + +def test_gcd_numbers_vs_polys(): + assert isinstance(gcd(3, 9), Integer) + assert isinstance(gcd(3*x, 9), Integer) + + assert gcd(3, 9) == 3 + assert gcd(3*x, 9) == 3 + + assert isinstance(gcd(Rational(3, 2), Rational(9, 4)), Rational) + assert isinstance(gcd(Rational(3, 2)*x, Rational(9, 4)), Rational) + + assert gcd(Rational(3, 2), Rational(9, 4)) == Rational(3, 4) + assert gcd(Rational(3, 2)*x, Rational(9, 4)) == 1 + + assert isinstance(gcd(3.0, 9.0), Float) + assert isinstance(gcd(3.0*x, 9.0), Float) + + assert gcd(3.0, 9.0) == 1.0 + assert gcd(3.0*x, 9.0) == 1.0 + + # partial fix of 20597 + assert gcd(Mul(2, 3, evaluate=False), 2) == 2 + + +def test_terms_gcd(): + assert terms_gcd(1) == 1 + assert terms_gcd(1, x) == 1 + + assert terms_gcd(x - 1) == x - 1 + assert terms_gcd(-x - 1) == -x - 1 + + assert terms_gcd(2*x + 3) == 2*x + 3 + assert terms_gcd(6*x + 4) == Mul(2, 3*x + 2, evaluate=False) + + assert terms_gcd(x**3*y + x*y**3) == x*y*(x**2 + y**2) + assert terms_gcd(2*x**3*y + 2*x*y**3) == 2*x*y*(x**2 + y**2) + assert terms_gcd(x**3*y/2 + x*y**3/2) == x*y/2*(x**2 + y**2) + + assert terms_gcd(x**3*y + 2*x*y**3) == x*y*(x**2 + 2*y**2) + assert terms_gcd(2*x**3*y + 4*x*y**3) == 2*x*y*(x**2 + 2*y**2) + assert terms_gcd(2*x**3*y/3 + 4*x*y**3/5) == x*y*Rational(2, 15)*(5*x**2 + 6*y**2) + + assert terms_gcd(2.0*x**3*y + 4.1*x*y**3) == x*y*(2.0*x**2 + 4.1*y**2) + assert _aresame(terms_gcd(2.0*x + 3), 2.0*x + 3) + + assert terms_gcd((3 + 3*x)*(x + x*y), expand=False) == \ + (3*x + 3)*(x*y + x) + assert terms_gcd((3 + 3*x)*(x + x*sin(3 + 3*y)), expand=False, deep=True) == \ + 3*x*(x + 1)*(sin(Mul(3, y + 1, evaluate=False)) + 1) + assert terms_gcd(sin(x + x*y), deep=True) == \ + sin(x*(y + 1)) + + eq = Eq(2*x, 2*y + 2*z*y) + assert terms_gcd(eq) == Eq(2*x, 2*y*(z + 1)) + assert terms_gcd(eq, deep=True) == Eq(2*x, 2*y*(z + 1)) + + raises(TypeError, lambda: terms_gcd(x < 2)) + + +def test_trunc(): + f, g = x**5 + 2*x**4 + 3*x**3 + 4*x**2 + 5*x + 6, x**5 - x**4 + x**2 - x + F, G = Poly(f), Poly(g) + + assert F.trunc(3) == G + assert trunc(f, 3) == g + assert trunc(f, 3, x) == g + assert trunc(f, 3, (x,)) == g + assert trunc(F, 3) == G + assert trunc(f, 3, polys=True) == G + assert trunc(F, 3, polys=False) == g + + f, g = 6*x**5 + 5*x**4 + 4*x**3 + 3*x**2 + 2*x + 1, -x**4 + x**3 - x + 1 + F, G = Poly(f), Poly(g) + + assert F.trunc(3) == G + assert trunc(f, 3) == g + assert trunc(f, 3, x) == g + assert trunc(f, 3, (x,)) == g + assert trunc(F, 3) == G + assert trunc(f, 3, polys=True) == G + assert trunc(F, 3, polys=False) == g + + f = Poly(x**2 + 2*x + 3, modulus=5) + + assert f.trunc(2) == Poly(x**2 + 1, modulus=5) + + +def test_monic(): + f, g = 2*x - 1, x - S.Half + F, G = Poly(f, domain='QQ'), Poly(g) + + assert F.monic() == G + assert monic(f) == g + assert monic(f, x) == g + assert monic(f, (x,)) == g + assert monic(F) == G + assert monic(f, polys=True) == G + assert monic(F, polys=False) == g + + raises(ComputationFailed, lambda: monic(4)) + + assert monic(2*x**2 + 6*x + 4, auto=False) == x**2 + 3*x + 2 + raises(ExactQuotientFailed, lambda: monic(2*x + 6*x + 1, auto=False)) + + assert monic(2.0*x**2 + 6.0*x + 4.0) == 1.0*x**2 + 3.0*x + 2.0 + assert monic(2*x**2 + 3*x + 4, modulus=5) == x**2 - x + 2 + + +def test_content(): + f, F = 4*x + 2, Poly(4*x + 2) + + assert F.content() == 2 + assert content(f) == 2 + + raises(ComputationFailed, lambda: content(4)) + + f = Poly(2*x, modulus=3) + + assert f.content() == 1 + + +def test_primitive(): + f, g = 4*x + 2, 2*x + 1 + F, G = Poly(f), Poly(g) + + assert F.primitive() == (2, G) + assert primitive(f) == (2, g) + assert primitive(f, x) == (2, g) + assert primitive(f, (x,)) == (2, g) + assert primitive(F) == (2, G) + assert primitive(f, polys=True) == (2, G) + assert primitive(F, polys=False) == (2, g) + + raises(ComputationFailed, lambda: primitive(4)) + + f = Poly(2*x, modulus=3) + g = Poly(2.0*x, domain=RR) + + assert f.primitive() == (1, f) + assert g.primitive() == (1.0, g) + + assert primitive(S('-3*x/4 + y + 11/8')) == \ + S('(1/8, -6*x + 8*y + 11)') + + +def test_compose(): + f = x**12 + 20*x**10 + 150*x**8 + 500*x**6 + 625*x**4 - 2*x**3 - 10*x + 9 + g = x**4 - 2*x + 9 + h = x**3 + 5*x + + F, G, H = map(Poly, (f, g, h)) + + assert G.compose(H) == F + assert compose(g, h) == f + assert compose(g, h, x) == f + assert compose(g, h, (x,)) == f + assert compose(G, H) == F + assert compose(g, h, polys=True) == F + assert compose(G, H, polys=False) == f + + assert F.decompose() == [G, H] + assert decompose(f) == [g, h] + assert decompose(f, x) == [g, h] + assert decompose(f, (x,)) == [g, h] + assert decompose(F) == [G, H] + assert decompose(f, polys=True) == [G, H] + assert decompose(F, polys=False) == [g, h] + + raises(ComputationFailed, lambda: compose(4, 2)) + raises(ComputationFailed, lambda: decompose(4)) + + assert compose(x**2 - y**2, x - y, x, y) == x**2 - 2*x*y + assert compose(x**2 - y**2, x - y, y, x) == -y**2 + 2*x*y + + +def test_shift(): + assert Poly(x**2 - 2*x + 1, x).shift(2) == Poly(x**2 + 2*x + 1, x) + + +def test_shift_list(): + assert Poly(x*y, [x,y]).shift_list([1,2]) == Poly((x+1)*(y+2), [x,y]) + + +def test_transform(): + # Also test that 3-way unification is done correctly + assert Poly(x**2 - 2*x + 1, x).transform(Poly(x + 1), Poly(x - 1)) == \ + Poly(4, x) == \ + cancel((x - 1)**2*(x**2 - 2*x + 1).subs(x, (x + 1)/(x - 1))) + + assert Poly(x**2 - x/2 + 1, x).transform(Poly(x + 1), Poly(x - 1)) == \ + Poly(3*x**2/2 + Rational(5, 2), x) == \ + cancel((x - 1)**2*(x**2 - x/2 + 1).subs(x, (x + 1)/(x - 1))) + + assert Poly(x**2 - 2*x + 1, x).transform(Poly(x + S.Half), Poly(x - 1)) == \ + Poly(Rational(9, 4), x) == \ + cancel((x - 1)**2*(x**2 - 2*x + 1).subs(x, (x + S.Half)/(x - 1))) + + assert Poly(x**2 - 2*x + 1, x).transform(Poly(x + 1), Poly(x - S.Half)) == \ + Poly(Rational(9, 4), x) == \ + cancel((x - S.Half)**2*(x**2 - 2*x + 1).subs(x, (x + 1)/(x - S.Half))) + + # Unify ZZ, QQ, and RR + assert Poly(x**2 - 2*x + 1, x).transform(Poly(x + 1.0), Poly(x - S.Half)) == \ + Poly(Rational(9, 4), x, domain='RR') == \ + cancel((x - S.Half)**2*(x**2 - 2*x + 1).subs(x, (x + 1.0)/(x - S.Half))) + + raises(ValueError, lambda: Poly(x*y).transform(Poly(x + 1), Poly(x - 1))) + raises(ValueError, lambda: Poly(x).transform(Poly(y + 1), Poly(x - 1))) + raises(ValueError, lambda: Poly(x).transform(Poly(x + 1), Poly(y - 1))) + raises(ValueError, lambda: Poly(x).transform(Poly(x*y + 1), Poly(x - 1))) + raises(ValueError, lambda: Poly(x).transform(Poly(x + 1), Poly(x*y - 1))) + + +def test_sturm(): + f, F = x, Poly(x, domain='QQ') + g, G = 1, Poly(1, x, domain='QQ') + + assert F.sturm() == [F, G] + assert sturm(f) == [f, g] + assert sturm(f, x) == [f, g] + assert sturm(f, (x,)) == [f, g] + assert sturm(F) == [F, G] + assert sturm(f, polys=True) == [F, G] + assert sturm(F, polys=False) == [f, g] + + raises(ComputationFailed, lambda: sturm(4)) + raises(DomainError, lambda: sturm(f, auto=False)) + + f = Poly(S(1024)/(15625*pi**8)*x**5 + - S(4096)/(625*pi**8)*x**4 + + S(32)/(15625*pi**4)*x**3 + - S(128)/(625*pi**4)*x**2 + + Rational(1, 62500)*x + - Rational(1, 625), x, domain='ZZ(pi)') + + assert sturm(f) == \ + [Poly(x**3 - 100*x**2 + pi**4/64*x - 25*pi**4/16, x, domain='ZZ(pi)'), + Poly(3*x**2 - 200*x + pi**4/64, x, domain='ZZ(pi)'), + Poly((Rational(20000, 9) - pi**4/96)*x + 25*pi**4/18, x, domain='ZZ(pi)'), + Poly((-3686400000000*pi**4 - 11520000*pi**8 - 9*pi**12)/(26214400000000 - 245760000*pi**4 + 576*pi**8), x, domain='ZZ(pi)')] + + +def test_gff(): + f = x**5 + 2*x**4 - x**3 - 2*x**2 + + assert Poly(f).gff_list() == [(Poly(x), 1), (Poly(x + 2), 4)] + assert gff_list(f) == [(x, 1), (x + 2, 4)] + + raises(NotImplementedError, lambda: gff(f)) + + f = x*(x - 1)**3*(x - 2)**2*(x - 4)**2*(x - 5) + + assert Poly(f).gff_list() == [( + Poly(x**2 - 5*x + 4), 1), (Poly(x**2 - 5*x + 4), 2), (Poly(x), 3)] + assert gff_list(f) == [(x**2 - 5*x + 4, 1), (x**2 - 5*x + 4, 2), (x, 3)] + + raises(NotImplementedError, lambda: gff(f)) + + +def test_norm(): + a, b = sqrt(2), sqrt(3) + f = Poly(a*x + b*y, x, y, extension=(a, b)) + assert f.norm() == Poly(4*x**4 - 12*x**2*y**2 + 9*y**4, x, y, domain='QQ') + + +def test_sqf_norm(): + assert sqf_norm(x**2 - 2, extension=sqrt(3)) == \ + ([1], x**2 - 2*sqrt(3)*x + 1, x**4 - 10*x**2 + 1) + assert sqf_norm(x**2 - 3, extension=sqrt(2)) == \ + ([1], x**2 - 2*sqrt(2)*x - 1, x**4 - 10*x**2 + 1) + + assert Poly(x**2 - 2, extension=sqrt(3)).sqf_norm() == \ + ([1], Poly(x**2 - 2*sqrt(3)*x + 1, x, extension=sqrt(3)), + Poly(x**4 - 10*x**2 + 1, x, domain='QQ')) + + assert Poly(x**2 - 3, extension=sqrt(2)).sqf_norm() == \ + ([1], Poly(x**2 - 2*sqrt(2)*x - 1, x, extension=sqrt(2)), + Poly(x**4 - 10*x**2 + 1, x, domain='QQ')) + + +def test_sqf(): + f = x**5 - x**3 - x**2 + 1 + g = x**3 + 2*x**2 + 2*x + 1 + h = x - 1 + + p = x**4 + x**3 - x - 1 + + F, G, H, P = map(Poly, (f, g, h, p)) + + assert F.sqf_part() == P + assert sqf_part(f) == p + assert sqf_part(f, x) == p + assert sqf_part(f, (x,)) == p + assert sqf_part(F) == P + assert sqf_part(f, polys=True) == P + assert sqf_part(F, polys=False) == p + + assert F.sqf_list() == (1, [(G, 1), (H, 2)]) + assert sqf_list(f) == (1, [(g, 1), (h, 2)]) + assert sqf_list(f, x) == (1, [(g, 1), (h, 2)]) + assert sqf_list(f, (x,)) == (1, [(g, 1), (h, 2)]) + assert sqf_list(F) == (1, [(G, 1), (H, 2)]) + assert sqf_list(f, polys=True) == (1, [(G, 1), (H, 2)]) + assert sqf_list(F, polys=False) == (1, [(g, 1), (h, 2)]) + + assert F.sqf_list_include() == [(G, 1), (H, 2)] + + raises(ComputationFailed, lambda: sqf_part(4)) + + assert sqf(1) == 1 + assert sqf_list(1) == (1, []) + + assert sqf((2*x**2 + 2)**7) == 128*(x**2 + 1)**7 + + assert sqf(f) == g*h**2 + assert sqf(f, x) == g*h**2 + assert sqf(f, (x,)) == g*h**2 + + d = x**2 + y**2 + + assert sqf(f/d) == (g*h**2)/d + assert sqf(f/d, x) == (g*h**2)/d + assert sqf(f/d, (x,)) == (g*h**2)/d + + assert sqf(x - 1) == x - 1 + assert sqf(-x - 1) == -x - 1 + + assert sqf(x - 1) == x - 1 + assert sqf(6*x - 10) == Mul(2, 3*x - 5, evaluate=False) + + assert sqf((6*x - 10)/(3*x - 6)) == Rational(2, 3)*((3*x - 5)/(x - 2)) + assert sqf(Poly(x**2 - 2*x + 1)) == (x - 1)**2 + + f = 3 + x - x*(1 + x) + x**2 + + assert sqf(f) == 3 + + f = (x**2 + 2*x + 1)**20000000000 + + assert sqf(f) == (x + 1)**40000000000 + assert sqf_list(f) == (1, [(x + 1, 40000000000)]) + + # https://github.com/sympy/sympy/issues/26497 + assert sqf(expand(((y - 2)**2 * (y + 2) * (x + 1)))) == \ + (y - 2)**2 * expand((y + 2) * (x + 1)) + assert sqf(expand(((y - 2)**2 * (y + 2) * (z + 1)))) == \ + (y - 2)**2 * expand((y + 2) * (z + 1)) + assert sqf(expand(((y - I)**2 * (y + I) * (x + 1)))) == \ + (y - I)**2 * expand((y + I) * (x + 1)) + assert sqf(expand(((y - I)**2 * (y + I) * (z + 1)))) == \ + (y - I)**2 * expand((y + I) * (z + 1)) + + # Check that factors are combined and sorted. + p = (x - 2)**2*(x - 1)*(x + y)**2*(y - 2)**2*(y - 1) + assert Poly(p).sqf_list() == (1, [ + (Poly(x*y - x - y + 1), 1), + (Poly(x**2*y - 2*x**2 + x*y**2 - 4*x*y + 4*x - 2*y**2 + 4*y), 2) + ]) + + +def test_factor(): + f = x**5 - x**3 - x**2 + 1 + + u = x + 1 + v = x - 1 + w = x**2 + x + 1 + + F, U, V, W = map(Poly, (f, u, v, w)) + + assert F.factor_list() == (1, [(U, 1), (V, 2), (W, 1)]) + assert factor_list(f) == (1, [(u, 1), (v, 2), (w, 1)]) + assert factor_list(f, x) == (1, [(u, 1), (v, 2), (w, 1)]) + assert factor_list(f, (x,)) == (1, [(u, 1), (v, 2), (w, 1)]) + assert factor_list(F) == (1, [(U, 1), (V, 2), (W, 1)]) + assert factor_list(f, polys=True) == (1, [(U, 1), (V, 2), (W, 1)]) + assert factor_list(F, polys=False) == (1, [(u, 1), (v, 2), (w, 1)]) + + assert F.factor_list_include() == [(U, 1), (V, 2), (W, 1)] + + assert factor_list(1) == (1, []) + assert factor_list(6) == (6, []) + assert factor_list(sqrt(3), x) == (sqrt(3), []) + assert factor_list((-1)**x, x) == (1, [(-1, x)]) + assert factor_list((2*x)**y, x) == (1, [(2, y), (x, y)]) + assert factor_list(sqrt(x*y), x) == (1, [(x*y, S.Half)]) + + assert factor(6) == 6 and factor(6).is_Integer + + assert factor_list(3*x) == (3, [(x, 1)]) + assert factor_list(3*x**2) == (3, [(x, 2)]) + + assert factor(3*x) == 3*x + assert factor(3*x**2) == 3*x**2 + + assert factor((2*x**2 + 2)**7) == 128*(x**2 + 1)**7 + + assert factor(f) == u*v**2*w + assert factor(f, x) == u*v**2*w + assert factor(f, (x,)) == u*v**2*w + + g, p, q, r = x**2 - y**2, x - y, x + y, x**2 + 1 + + assert factor(f/g) == (u*v**2*w)/(p*q) + assert factor(f/g, x) == (u*v**2*w)/(p*q) + assert factor(f/g, (x,)) == (u*v**2*w)/(p*q) + + p = Symbol('p', positive=True) + i = Symbol('i', integer=True) + r = Symbol('r', real=True) + + assert factor(sqrt(x*y)).is_Pow is True + + assert factor(sqrt(3*x**2 - 3)) == sqrt(3)*sqrt((x - 1)*(x + 1)) + assert factor(sqrt(3*x**2 + 3)) == sqrt(3)*sqrt(x**2 + 1) + + assert factor((y*x**2 - y)**i) == y**i*(x - 1)**i*(x + 1)**i + assert factor((y*x**2 + y)**i) == y**i*(x**2 + 1)**i + + assert factor((y*x**2 - y)**t) == (y*(x - 1)*(x + 1))**t + assert factor((y*x**2 + y)**t) == (y*(x**2 + 1))**t + + f = sqrt(expand((r**2 + 1)*(p + 1)*(p - 1)*(p - 2)**3)) + g = sqrt((p - 2)**3*(p - 1))*sqrt(p + 1)*sqrt(r**2 + 1) + + assert factor(f) == g + assert factor(g) == g + + g = (x - 1)**5*(r**2 + 1) + f = sqrt(expand(g)) + + assert factor(f) == sqrt(g) + + f = Poly(sin(1)*x + 1, x, domain=EX) + + assert f.factor_list() == (1, [(f, 1)]) + + f = x**4 + 1 + + assert factor(f) == f + assert factor(f, extension=I) == (x**2 - I)*(x**2 + I) + assert factor(f, gaussian=True) == (x**2 - I)*(x**2 + I) + assert factor( + f, extension=sqrt(2)) == (x**2 + sqrt(2)*x + 1)*(x**2 - sqrt(2)*x + 1) + + assert factor(x**2 + 4*I*x - 4) == (x + 2*I)**2 + + f = x**2 + 2*I*x - 4 + + assert factor(f) == f + + f = 8192*x**2 + x*(22656 + 175232*I) - 921416 + 242313*I + f_zzi = I*(x*(64 - 64*I) + 773 + 596*I)**2 + f_qqi = 8192*(x + S(177)/128 + 1369*I/128)**2 + + assert factor(f) == f_zzi + assert factor(f, domain=ZZ_I) == f_zzi + assert factor(f, domain=QQ_I) == f_qqi + + f = x**2 + 2*sqrt(2)*x + 2 + + assert factor(f, extension=sqrt(2)) == (x + sqrt(2))**2 + assert factor(f**3, extension=sqrt(2)) == (x + sqrt(2))**6 + + assert factor(x**2 - 2*y**2, extension=sqrt(2)) == \ + (x + sqrt(2)*y)*(x - sqrt(2)*y) + assert factor(2*x**2 - 4*y**2, extension=sqrt(2)) == \ + 2*((x + sqrt(2)*y)*(x - sqrt(2)*y)) + + assert factor(x - 1) == x - 1 + assert factor(-x - 1) == -x - 1 + + assert factor(x - 1) == x - 1 + + assert factor(6*x - 10) == Mul(2, 3*x - 5, evaluate=False) + + assert factor(x**11 + x + 1, modulus=65537, symmetric=True) == \ + (x**2 + x + 1)*(x**9 - x**8 + x**6 - x**5 + x**3 - x** 2 + 1) + assert factor(x**11 + x + 1, modulus=65537, symmetric=False) == \ + (x**2 + x + 1)*(x**9 + 65536*x**8 + x**6 + 65536*x**5 + + x**3 + 65536*x** 2 + 1) + + f = x/pi + x*sin(x)/pi + g = y/(pi**2 + 2*pi + 1) + y*sin(x)/(pi**2 + 2*pi + 1) + + assert factor(f) == x*(sin(x) + 1)/pi + assert factor(g) == y*(sin(x) + 1)/(pi + 1)**2 + + assert factor(Eq( + x**2 + 2*x + 1, x**3 + 1)) == Eq((x + 1)**2, (x + 1)*(x**2 - x + 1)) + + f = (x**2 - 1)/(x**2 + 4*x + 4) + + assert factor(f) == (x + 1)*(x - 1)/(x + 2)**2 + assert factor(f, x) == (x + 1)*(x - 1)/(x + 2)**2 + + f = 3 + x - x*(1 + x) + x**2 + + assert factor(f) == 3 + assert factor(f, x) == 3 + + assert factor(1/(x**2 + 2*x + 1/x) - 1) == -((1 - x + 2*x**2 + + x**3)/(1 + 2*x**2 + x**3)) + + assert factor(f, expand=False) == f + raises(PolynomialError, lambda: factor(f, x, expand=False)) + + raises(FlagError, lambda: factor(x**2 - 1, polys=True)) + + assert factor([x, Eq(x**2 - y**2, Tuple(x**2 - z**2, 1/x + 1/y))]) == \ + [x, Eq((x - y)*(x + y), Tuple((x - z)*(x + z), (x + y)/x/y))] + + assert not isinstance( + Poly(x**3 + x + 1).factor_list()[1][0][0], PurePoly) is True + assert isinstance( + PurePoly(x**3 + x + 1).factor_list()[1][0][0], PurePoly) is True + + assert factor(sqrt(-x)) == sqrt(-x) + + # issue 5917 + e = (-2*x*(-x + 1)*(x - 1)*(-x*(-x + 1)*(x - 1) - x*(x - 1)**2)*(x**2*(x - + 1) - x*(x - 1) - x) - (-2*x**2*(x - 1)**2 - x*(-x + 1)*(-x*(-x + 1) + + x*(x - 1)))*(x**2*(x - 1)**4 - x*(-x*(-x + 1)*(x - 1) - x*(x - 1)**2))) + assert factor(e) == 0 + + # deep option + assert factor(sin(x**2 + x) + x, deep=True) == sin(x*(x + 1)) + x + assert factor(sin(x**2 + x)*x, deep=True) == sin(x*(x + 1))*x + + assert factor(sqrt(x**2)) == sqrt(x**2) + + # issue 13149 + assert factor(expand((0.5*x+1)*(0.5*y+1))) == Mul(1.0, 0.5*x + 1.0, + 0.5*y + 1.0, evaluate = False) + assert factor(expand((0.5*x+0.5)**2)) == 0.25*(1.0*x + 1.0)**2 + + eq = x**2*y**2 + 11*x**2*y + 30*x**2 + 7*x*y**2 + 77*x*y + 210*x + 12*y**2 + 132*y + 360 + assert factor(eq, x) == (x + 3)*(x + 4)*(y**2 + 11*y + 30) + assert factor(eq, x, deep=True) == (x + 3)*(x + 4)*(y**2 + 11*y + 30) + assert factor(eq, y, deep=True) == (y + 5)*(y + 6)*(x**2 + 7*x + 12) + + # fraction option + f = 5*x + 3*exp(2 - 7*x) + assert factor(f, deep=True) == factor(f, deep=True, fraction=True) + assert factor(f, deep=True, fraction=False) == 5*x + 3*exp(2)*exp(-7*x) + + assert factor_list(x**3 - x*y**2, t, w, x) == ( + 1, [(x, 1), (x - y, 1), (x + y, 1)]) + assert factor_list((x+1)*(x**6-1)) == ( + 1, [(x - 1, 1), (x + 1, 2), (x**2 - x + 1, 1), (x**2 + x + 1, 1)]) + + # https://github.com/sympy/sympy/issues/24952 + s2, s2p, s2n = sqrt(2), 1 + sqrt(2), 1 - sqrt(2) + pip, pin = 1 + pi, 1 - pi + assert factor_list(s2p*s2n) == (-1, [(-s2n, 1), (s2p, 1)]) + assert factor_list(pip*pin) == (-1, [(-pin, 1), (pip, 1)]) + # Not sure about this one. Maybe coeff should be 1 or -1? + assert factor_list(s2*s2n) == (-s2, [(-s2n, 1)]) + assert factor_list(pi*pin) == (-1, [(-pin, 1), (pi, 1)]) + assert factor_list(s2p*s2n, x) == (s2p*s2n, []) + assert factor_list(pip*pin, x) == (pip*pin, []) + assert factor_list(s2*s2n, x) == (s2*s2n, []) + assert factor_list(pi*pin, x) == (pi*pin, []) + assert factor_list((x - sqrt(2)*pi)*(x + sqrt(2)*pi), x) == ( + 1, [(x - sqrt(2)*pi, 1), (x + sqrt(2)*pi, 1)]) + + # https://github.com/sympy/sympy/issues/26497 + p = ((y - I)**2 * (y + I) * (x + 1)) + assert factor(expand(p)) == p + + p = ((x - I)**2 * (x + I) * (y + 1)) + assert factor(expand(p)) == p + + p = (y + 1)**2*(y + sqrt(2))**2*(x**2 + x + 2 + 3*sqrt(2))**2 + assert factor(expand(p), extension=True) == p + + e = ( + -x**2*y**4/(y**2 + 1) + 2*I*x**2*y**3/(y**2 + 1) + 2*I*x**2*y/(y**2 + 1) + + x**2/(y**2 + 1) - 2*x*y**4/(y**2 + 1) + 4*I*x*y**3/(y**2 + 1) + + 4*I*x*y/(y**2 + 1) + 2*x/(y**2 + 1) - y**4 - y**4/(y**2 + 1) + 2*I*y**3 + + 2*I*y**3/(y**2 + 1) + 2*I*y + 2*I*y/(y**2 + 1) + 1 + 1/(y**2 + 1) + ) + assert factor(e) == -(y - I)**3*(y + I)*(x**2 + 2*x + y**2 + 2)/(y**2 + 1) + + # issue 27506 + e = (I*t*x*y - 3*I*t - I*x*y*z - 6*x*y + 3*I*z + 18) + assert factor(e) == -I*(x*y - 3)*(-t + z - 6*I) + + e = (8*x**2*z**2 - 32*x**2*z*t + 24*x**2*t**2 - 4*I*x*y*z**2 + 16*I*x*y*z*t - + 12*I*x*y*t**2 + z**4 - 8*z**3*t + 22*z**2*t**2 - 24*z*t**3 + 9*t**4) + assert factor(e) == (-3*t + z)*(-t + z)*(3*t**2 - 4*t*z + 8*x**2 - 4*I*x*y + z**2) + + +def test_factor_large(): + f = (x**2 + 4*x + 4)**10000000*(x**2 + 1)*(x**2 + 2*x + 1)**1234567 + g = ((x**2 + 2*x + 1)**3000*y**2 + (x**2 + 2*x + 1)**3000*2*y + ( + x**2 + 2*x + 1)**3000) + + assert factor(f) == (x + 2)**20000000*(x**2 + 1)*(x + 1)**2469134 + assert factor(g) == (x + 1)**6000*(y + 1)**2 + + assert factor_list( + f) == (1, [(x + 1, 2469134), (x + 2, 20000000), (x**2 + 1, 1)]) + assert factor_list(g) == (1, [(y + 1, 2), (x + 1, 6000)]) + + f = (x**2 - y**2)**200000*(x**7 + 1) + g = (x**2 + y**2)**200000*(x**7 + 1) + + assert factor(f) == \ + (x + 1)*(x - y)**200000*(x + y)**200000*(x**6 - x**5 + + x**4 - x**3 + x**2 - x + 1) + assert factor(g, gaussian=True) == \ + (x + 1)*(x - I*y)**200000*(x + I*y)**200000*(x**6 - x**5 + + x**4 - x**3 + x**2 - x + 1) + + assert factor_list(f) == \ + (1, [(x + 1, 1), (x - y, 200000), (x + y, 200000), (x**6 - + x**5 + x**4 - x**3 + x**2 - x + 1, 1)]) + assert factor_list(g, gaussian=True) == \ + (1, [(x + 1, 1), (x - I*y, 200000), (x + I*y, 200000), ( + x**6 - x**5 + x**4 - x**3 + x**2 - x + 1, 1)]) + + +def test_factor_noeval(): + assert factor(6*x - 10) == Mul(2, 3*x - 5, evaluate=False) + assert factor((6*x - 10)/(3*x - 6)) == Mul(Rational(2, 3), 3*x - 5, 1/(x - 2)) + + +def test_intervals(): + assert intervals(0) == [] + assert intervals(1) == [] + + assert intervals(x, sqf=True) == [(0, 0)] + assert intervals(x) == [((0, 0), 1)] + + assert intervals(x**128) == [((0, 0), 128)] + assert intervals([x**2, x**4]) == [((0, 0), {0: 2, 1: 4})] + + f = Poly((x*Rational(2, 5) - Rational(17, 3))*(4*x + Rational(1, 257))) + + assert f.intervals(sqf=True) == [(-1, 0), (14, 15)] + assert f.intervals() == [((-1, 0), 1), ((14, 15), 1)] + + assert f.intervals(fast=True, sqf=True) == [(-1, 0), (14, 15)] + assert f.intervals(fast=True) == [((-1, 0), 1), ((14, 15), 1)] + + assert f.intervals(eps=Rational(1, 10)) == f.intervals(eps=0.1) == \ + [((Rational(-1, 258), 0), 1), ((Rational(85, 6), Rational(85, 6)), 1)] + assert f.intervals(eps=Rational(1, 100)) == f.intervals(eps=0.01) == \ + [((Rational(-1, 258), 0), 1), ((Rational(85, 6), Rational(85, 6)), 1)] + assert f.intervals(eps=Rational(1, 1000)) == f.intervals(eps=0.001) == \ + [((Rational(-1, 1002), 0), 1), ((Rational(85, 6), Rational(85, 6)), 1)] + assert f.intervals(eps=Rational(1, 10000)) == f.intervals(eps=0.0001) == \ + [((Rational(-1, 1028), Rational(-1, 1028)), 1), ((Rational(85, 6), Rational(85, 6)), 1)] + + f = (x*Rational(2, 5) - Rational(17, 3))*(4*x + Rational(1, 257)) + + assert intervals(f, sqf=True) == [(-1, 0), (14, 15)] + assert intervals(f) == [((-1, 0), 1), ((14, 15), 1)] + + assert intervals(f, eps=Rational(1, 10)) == intervals(f, eps=0.1) == \ + [((Rational(-1, 258), 0), 1), ((Rational(85, 6), Rational(85, 6)), 1)] + assert intervals(f, eps=Rational(1, 100)) == intervals(f, eps=0.01) == \ + [((Rational(-1, 258), 0), 1), ((Rational(85, 6), Rational(85, 6)), 1)] + assert intervals(f, eps=Rational(1, 1000)) == intervals(f, eps=0.001) == \ + [((Rational(-1, 1002), 0), 1), ((Rational(85, 6), Rational(85, 6)), 1)] + assert intervals(f, eps=Rational(1, 10000)) == intervals(f, eps=0.0001) == \ + [((Rational(-1, 1028), Rational(-1, 1028)), 1), ((Rational(85, 6), Rational(85, 6)), 1)] + + f = Poly((x**2 - 2)*(x**2 - 3)**7*(x + 1)*(7*x + 3)**3) + + assert f.intervals() == \ + [((-2, Rational(-3, 2)), 7), ((Rational(-3, 2), -1), 1), + ((-1, -1), 1), ((-1, 0), 3), + ((1, Rational(3, 2)), 1), ((Rational(3, 2), 2), 7)] + + assert intervals([x**5 - 200, x**5 - 201]) == \ + [((Rational(75, 26), Rational(101, 35)), {0: 1}), ((Rational(309, 107), Rational(26, 9)), {1: 1})] + + assert intervals([x**5 - 200, x**5 - 201], fast=True) == \ + [((Rational(75, 26), Rational(101, 35)), {0: 1}), ((Rational(309, 107), Rational(26, 9)), {1: 1})] + + assert intervals([x**2 - 200, x**2 - 201]) == \ + [((Rational(-71, 5), Rational(-85, 6)), {1: 1}), ((Rational(-85, 6), -14), {0: 1}), + ((14, Rational(85, 6)), {0: 1}), ((Rational(85, 6), Rational(71, 5)), {1: 1})] + + assert intervals([x + 1, x + 2, x - 1, x + 1, 1, x - 1, x - 1, (x - 2)**2]) == \ + [((-2, -2), {1: 1}), ((-1, -1), {0: 1, 3: 1}), ((1, 1), {2: + 1, 5: 1, 6: 1}), ((2, 2), {7: 2})] + + f, g, h = x**2 - 2, x**4 - 4*x**2 + 4, x - 1 + + assert intervals(f, inf=Rational(7, 4), sqf=True) == [] + assert intervals(f, inf=Rational(7, 5), sqf=True) == [(Rational(7, 5), Rational(3, 2))] + assert intervals(f, sup=Rational(7, 4), sqf=True) == [(-2, -1), (1, Rational(3, 2))] + assert intervals(f, sup=Rational(7, 5), sqf=True) == [(-2, -1)] + + assert intervals(g, inf=Rational(7, 4)) == [] + assert intervals(g, inf=Rational(7, 5)) == [((Rational(7, 5), Rational(3, 2)), 2)] + assert intervals(g, sup=Rational(7, 4)) == [((-2, -1), 2), ((1, Rational(3, 2)), 2)] + assert intervals(g, sup=Rational(7, 5)) == [((-2, -1), 2)] + + assert intervals([g, h], inf=Rational(7, 4)) == [] + assert intervals([g, h], inf=Rational(7, 5)) == [((Rational(7, 5), Rational(3, 2)), {0: 2})] + assert intervals([g, h], sup=S( + 7)/4) == [((-2, -1), {0: 2}), ((1, 1), {1: 1}), ((1, Rational(3, 2)), {0: 2})] + assert intervals( + [g, h], sup=Rational(7, 5)) == [((-2, -1), {0: 2}), ((1, 1), {1: 1})] + + assert intervals([x + 2, x**2 - 2]) == \ + [((-2, -2), {0: 1}), ((-2, -1), {1: 1}), ((1, 2), {1: 1})] + assert intervals([x + 2, x**2 - 2], strict=True) == \ + [((-2, -2), {0: 1}), ((Rational(-3, 2), -1), {1: 1}), ((1, 2), {1: 1})] + + f = 7*z**4 - 19*z**3 + 20*z**2 + 17*z + 20 + + assert intervals(f) == [] + + real_part, complex_part = intervals(f, all=True, sqf=True) + + assert real_part == [] + assert all(re(a) < re(r) < re(b) and im( + a) < im(r) < im(b) for (a, b), r in zip(complex_part, nroots(f))) + + assert complex_part == [(Rational(-40, 7) - I*40/7, 0), + (Rational(-40, 7), I*40/7), + (I*Rational(-40, 7), Rational(40, 7)), + (0, Rational(40, 7) + I*40/7)] + + real_part, complex_part = intervals(f, all=True, sqf=True, eps=Rational(1, 10)) + + assert real_part == [] + assert all(re(a) < re(r) < re(b) and im( + a) < im(r) < im(b) for (a, b), r in zip(complex_part, nroots(f))) + + raises(ValueError, lambda: intervals(x**2 - 2, eps=10**-100000)) + raises(ValueError, lambda: Poly(x**2 - 2).intervals(eps=10**-100000)) + raises( + ValueError, lambda: intervals([x**2 - 2, x**2 - 3], eps=10**-100000)) + + +def test_refine_root(): + f = Poly(x**2 - 2) + + assert f.refine_root(1, 2, steps=0) == (1, 2) + assert f.refine_root(-2, -1, steps=0) == (-2, -1) + + assert f.refine_root(1, 2, steps=None) == (1, Rational(3, 2)) + assert f.refine_root(-2, -1, steps=None) == (Rational(-3, 2), -1) + + assert f.refine_root(1, 2, steps=1) == (1, Rational(3, 2)) + assert f.refine_root(-2, -1, steps=1) == (Rational(-3, 2), -1) + + assert f.refine_root(1, 2, steps=1, fast=True) == (1, Rational(3, 2)) + assert f.refine_root(-2, -1, steps=1, fast=True) == (Rational(-3, 2), -1) + + assert f.refine_root(1, 2, eps=Rational(1, 100)) == (Rational(24, 17), Rational(17, 12)) + assert f.refine_root(1, 2, eps=1e-2) == (Rational(24, 17), Rational(17, 12)) + + raises(PolynomialError, lambda: (f**2).refine_root(1, 2, check_sqf=True)) + + raises(RefinementFailed, lambda: (f**2).refine_root(1, 2)) + raises(RefinementFailed, lambda: (f**2).refine_root(2, 3)) + + f = x**2 - 2 + + assert refine_root(f, 1, 2, steps=1) == (1, Rational(3, 2)) + assert refine_root(f, -2, -1, steps=1) == (Rational(-3, 2), -1) + + assert refine_root(f, 1, 2, steps=1, fast=True) == (1, Rational(3, 2)) + assert refine_root(f, -2, -1, steps=1, fast=True) == (Rational(-3, 2), -1) + + assert refine_root(f, 1, 2, eps=Rational(1, 100)) == (Rational(24, 17), Rational(17, 12)) + assert refine_root(f, 1, 2, eps=1e-2) == (Rational(24, 17), Rational(17, 12)) + + raises(PolynomialError, lambda: refine_root(1, 7, 8, eps=Rational(1, 100))) + + raises(ValueError, lambda: Poly(f).refine_root(1, 2, eps=10**-100000)) + raises(ValueError, lambda: refine_root(f, 1, 2, eps=10**-100000)) + + +def test_count_roots(): + assert count_roots(x**2 - 2) == 2 + + assert count_roots(x**2 - 2, inf=-oo) == 2 + assert count_roots(x**2 - 2, sup=+oo) == 2 + assert count_roots(x**2 - 2, inf=-oo, sup=+oo) == 2 + + assert count_roots(x**2 - 2, inf=-2) == 2 + assert count_roots(x**2 - 2, inf=-1) == 1 + + assert count_roots(x**2 - 2, sup=1) == 1 + assert count_roots(x**2 - 2, sup=2) == 2 + + assert count_roots(x**2 - 2, inf=-1, sup=1) == 0 + assert count_roots(x**2 - 2, inf=-2, sup=2) == 2 + + assert count_roots(x**2 - 2, inf=-1, sup=1) == 0 + assert count_roots(x**2 - 2, inf=-2, sup=2) == 2 + + assert count_roots(x**2 + 2) == 0 + assert count_roots(x**2 + 2, inf=-2*I) == 2 + assert count_roots(x**2 + 2, sup=+2*I) == 2 + assert count_roots(x**2 + 2, inf=-2*I, sup=+2*I) == 2 + + assert count_roots(x**2 + 2, inf=0) == 0 + assert count_roots(x**2 + 2, sup=0) == 0 + + assert count_roots(x**2 + 2, inf=-I) == 1 + assert count_roots(x**2 + 2, sup=+I) == 1 + + assert count_roots(x**2 + 2, inf=+I/2, sup=+I) == 0 + assert count_roots(x**2 + 2, inf=-I, sup=-I/2) == 0 + + raises(PolynomialError, lambda: count_roots(1)) + + +def test_count_roots_extension(): + + p1 = Poly(sqrt(2)*x**2 - 2, x, extension=True) + assert p1.count_roots() == 2 + assert p1.count_roots(inf=0) == 1 + assert p1.count_roots(sup=0) == 1 + + p2 = Poly(x**2 + sqrt(2), x, extension=True) + assert p2.count_roots() == 0 + + p3 = Poly(x**2 + 2*sqrt(2)*x + 1, x, extension=True) + assert p3.count_roots() == 2 + assert p3.count_roots(inf=-10, sup=10) == 2 + assert p3.count_roots(inf=-10, sup=0) == 2 + assert p3.count_roots(inf=-10, sup=-3) == 0 + assert p3.count_roots(inf=-3, sup=-2) == 1 + assert p3.count_roots(inf=-1, sup=0) == 1 + + +def test_Poly_root(): + f = Poly(2*x**3 - 7*x**2 + 4*x + 4) + + assert f.root(0) == Rational(-1, 2) + assert f.root(1) == 2 + assert f.root(2) == 2 + raises(IndexError, lambda: f.root(3)) + + assert Poly(x**5 + x + 1).root(0) == rootof(x**3 - x**2 + 1, 0) + + +def test_real_roots(): + + assert real_roots(x) == [0] + assert real_roots(x, multiple=False) == [(0, 1)] + + assert real_roots(x**3) == [0, 0, 0] + assert real_roots(x**3, multiple=False) == [(0, 3)] + + assert real_roots(x*(x**3 + x + 3)) == [rootof(x**3 + x + 3, 0), 0] + assert real_roots(x*(x**3 + x + 3), multiple=False) == [(rootof( + x**3 + x + 3, 0), 1), (0, 1)] + + assert real_roots( + x**3*(x**3 + x + 3)) == [rootof(x**3 + x + 3, 0), 0, 0, 0] + assert real_roots(x**3*(x**3 + x + 3), multiple=False) == [(rootof( + x**3 + x + 3, 0), 1), (0, 3)] + + assert real_roots(x**2 - 2, radicals=False) == [ + rootof(x**2 - 2, 0, radicals=False), + rootof(x**2 - 2, 1, radicals=False), + ] + + f = 2*x**3 - 7*x**2 + 4*x + 4 + g = x**3 + x + 1 + + assert Poly(f).real_roots() == [Rational(-1, 2), 2, 2] + assert Poly(g).real_roots() == [rootof(g, 0)] + + # testing extension + f = x**2 - sqrt(2) + roots = [-2**(S(1)/4), 2**(S(1)/4)] + raises(NotImplementedError, lambda: real_roots(f)) + raises(NotImplementedError, lambda: real_roots(Poly(f, x))) + assert real_roots(f, extension=True) == roots + assert real_roots(Poly(f, extension=True)) == roots + assert real_roots(Poly(f), extension=True) == roots + + +def test_all_roots(): + + f = 2*x**3 - 7*x**2 + 4*x + 4 + froots = [Rational(-1, 2), 2, 2] + assert all_roots(f) == Poly(f).all_roots() == froots + + g = x**3 + x + 1 + groots = [rootof(g, 0), rootof(g, 1), rootof(g, 2)] + assert all_roots(g) == Poly(g).all_roots() == groots + + assert all_roots(x**2 - 2) == [-sqrt(2), sqrt(2)] + assert all_roots(x**2 - 2, multiple=False) == [(-sqrt(2), 1), (sqrt(2), 1)] + assert all_roots(x**2 - 2, radicals=False) == [ + rootof(x**2 - 2, 0, radicals=False), + rootof(x**2 - 2, 1, radicals=False), + ] + + p = x**5 - x - 1 + assert all_roots(p) == [ + rootof(p, 0), rootof(p, 1), rootof(p, 2), rootof(p, 3), rootof(p, 4) + ] + + # testing extension + f = x**2 + sqrt(2) + roots = [-2**(S(1)/4)*I, 2**(S(1)/4)*I] + raises(NotImplementedError, lambda: all_roots(f)) + raises(NotImplementedError, lambda : all_roots(Poly(f, x))) + assert all_roots(f, extension=True) == roots + assert all_roots(Poly(f, extension=True)) == roots + assert all_roots(Poly(f), extension=True) == roots + + +def test_nroots(): + assert Poly(0, x).nroots() == [] + assert Poly(1, x).nroots() == [] + + assert Poly(x**2 - 1, x).nroots() == [-1.0, 1.0] + assert Poly(x**2 + 1, x).nroots() == [-1.0*I, 1.0*I] + + roots = Poly(x**2 - 1, x).nroots() + assert roots == [-1.0, 1.0] + + roots = Poly(x**2 + 1, x).nroots() + assert roots == [-1.0*I, 1.0*I] + + roots = Poly(x**2/3 - Rational(1, 3), x).nroots() + assert roots == [-1.0, 1.0] + + roots = Poly(x**2/3 + Rational(1, 3), x).nroots() + assert roots == [-1.0*I, 1.0*I] + + assert Poly(x**2 + 2*I, x).nroots() == [-1.0 + 1.0*I, 1.0 - 1.0*I] + assert Poly( + x**2 + 2*I, x, extension=I).nroots() == [-1.0 + 1.0*I, 1.0 - 1.0*I] + + assert Poly(0.2*x + 0.1).nroots() == [-0.5] + + roots = nroots(x**5 + x + 1, n=5) + eps = Float("1e-5") + + assert re(roots[0]).epsilon_eq(-0.75487, eps) is S.true + assert im(roots[0]) == 0 + assert re(roots[1]) == Float(-0.5, 5) + assert im(roots[1]).epsilon_eq(-0.86602, eps) is S.true + assert re(roots[2]) == Float(-0.5, 5) + assert im(roots[2]).epsilon_eq(+0.86602, eps) is S.true + assert re(roots[3]).epsilon_eq(+0.87743, eps) is S.true + assert im(roots[3]).epsilon_eq(-0.74486, eps) is S.true + assert re(roots[4]).epsilon_eq(+0.87743, eps) is S.true + assert im(roots[4]).epsilon_eq(+0.74486, eps) is S.true + + eps = Float("1e-6") + + assert re(roots[0]).epsilon_eq(-0.75487, eps) is S.false + assert im(roots[0]) == 0 + assert re(roots[1]) == Float(-0.5, 5) + assert im(roots[1]).epsilon_eq(-0.86602, eps) is S.false + assert re(roots[2]) == Float(-0.5, 5) + assert im(roots[2]).epsilon_eq(+0.86602, eps) is S.false + assert re(roots[3]).epsilon_eq(+0.87743, eps) is S.false + assert im(roots[3]).epsilon_eq(-0.74486, eps) is S.false + assert re(roots[4]).epsilon_eq(+0.87743, eps) is S.false + assert im(roots[4]).epsilon_eq(+0.74486, eps) is S.false + + raises(DomainError, lambda: Poly(x + y, x).nroots()) + raises(MultivariatePolynomialError, lambda: Poly(x + y).nroots()) + + assert nroots(x**2 - 1) == [-1.0, 1.0] + + roots = nroots(x**2 - 1) + assert roots == [-1.0, 1.0] + + assert nroots(x + I) == [-1.0*I] + assert nroots(x + 2*I) == [-2.0*I] + + raises(PolynomialError, lambda: nroots(0)) + + # issue 8296 + f = Poly(x**4 - 1) + assert f.nroots(2) == [w.n(2) for w in f.all_roots()] + + assert str(Poly(x**16 + 32*x**14 + 508*x**12 + 5440*x**10 + + 39510*x**8 + 204320*x**6 + 755548*x**4 + 1434496*x**2 + + 877969).nroots(2)) == ('[-1.7 - 1.9*I, -1.7 + 1.9*I, -1.7 ' + '- 2.5*I, -1.7 + 2.5*I, -1.0*I, 1.0*I, -1.7*I, 1.7*I, -2.8*I, ' + '2.8*I, -3.4*I, 3.4*I, 1.7 - 1.9*I, 1.7 + 1.9*I, 1.7 - 2.5*I, ' + '1.7 + 2.5*I]') + assert str(Poly(1e-15*x**2 -1).nroots()) == ('[-31622776.6016838, 31622776.6016838]') + + # https://github.com/sympy/sympy/issues/23861 + + i = Float('3.000000000000000000000000000000000000000000000000001') + [r] = nroots(x + I*i, n=300) + assert abs(r + I*i) < 1e-300 + + +def test_ground_roots(): + f = x**6 - 4*x**4 + 4*x**3 - x**2 + + assert Poly(f).ground_roots() == {S.One: 2, S.Zero: 2} + assert ground_roots(f) == {S.One: 2, S.Zero: 2} + + +def test_nth_power_roots_poly(): + f = x**4 - x**2 + 1 + + f_2 = (x**2 - x + 1)**2 + f_3 = (x**2 + 1)**2 + f_4 = (x**2 + x + 1)**2 + f_12 = (x - 1)**4 + + assert nth_power_roots_poly(f, 1) == f + + raises(ValueError, lambda: nth_power_roots_poly(f, 0)) + raises(ValueError, lambda: nth_power_roots_poly(f, x)) + + assert factor(nth_power_roots_poly(f, 2)) == f_2 + assert factor(nth_power_roots_poly(f, 3)) == f_3 + assert factor(nth_power_roots_poly(f, 4)) == f_4 + assert factor(nth_power_roots_poly(f, 12)) == f_12 + + raises(MultivariatePolynomialError, lambda: nth_power_roots_poly( + x + y, 2, x, y)) + +def test_which_real_roots(): + f = Poly(x**4 - 1) + + assert f.which_real_roots([1, -1]) == [1, -1] + assert f.which_real_roots([1, -1, 2, 4]) == [1, -1] + assert f.which_real_roots([1, -1, -1, 1, 2, 5]) == [1, -1] + assert f.which_real_roots([10, 8, 7, -1, 1]) == [-1, 1] + + # no real roots + # (technically its still a superset) + f = Poly(x**2 + 1) + assert f.which_real_roots([5, 10]) == [] + + # not square free + f = Poly((x-1)**2) + assert f.which_real_roots([1, 1, -1, 2]) == [1] + + # candidates not superset + f = Poly(x**2 - 1) + assert f.which_real_roots([0, 2]) == [0, 2] + +def test_which_all_roots(): + f = Poly(x**4 - 1) + + assert f.which_all_roots([1, -1, I, -I]) == [1, -1, I, -I] + assert f.which_all_roots([I, I, -I, 1, -1]) == [I, -I, 1, -1] + + f = Poly(x**2 + 1) + assert f.which_all_roots([I, -I, I/2]) == [I, -I] + + # not square free + f = Poly((x-I)**2) + assert f.which_all_roots([I, I, 1, -1, 0]) == [I] + + # candidates not superset + f = Poly(x**2 + 1) + assert f.which_all_roots([I/2, -I/2]) == [I/2, -I/2] + +def test_same_root(): + f = Poly(x**4 + x**3 + x**2 + x + 1) + eq = f.same_root + r0 = exp(2 * I * pi / 5) + assert [i for i, r in enumerate(f.all_roots()) if eq(r, r0)] == [3] + + raises(PolynomialError, + lambda: Poly(x + 1, domain=QQ).same_root(0, 0)) + raises(DomainError, + lambda: Poly(x**2 + 1, domain=FF(7)).same_root(0, 0)) + raises(DomainError, + lambda: Poly(x ** 2 + 1, domain=ZZ_I).same_root(0, 0)) + raises(DomainError, + lambda: Poly(y * x**2 + 1, domain=ZZ[y]).same_root(0, 0)) + raises(MultivariatePolynomialError, + lambda: Poly(x * y + 1, domain=ZZ).same_root(0, 0)) + + +def test_torational_factor_list(): + p = expand(((x**2-1)*(x-2)).subs({x:x*(1 + sqrt(2))})) + assert _torational_factor_list(p, x) == (-2, [ + (-x*(1 + sqrt(2))/2 + 1, 1), + (-x*(1 + sqrt(2)) - 1, 1), + (-x*(1 + sqrt(2)) + 1, 1)]) + + + p = expand(((x**2-1)*(x-2)).subs({x:x*(1 + 2**Rational(1, 4))})) + assert _torational_factor_list(p, x) is None + + +def test_cancel(): + assert cancel(0) == 0 + assert cancel(7) == 7 + assert cancel(x) == x + + assert cancel(oo) is oo + + raises(ValueError, lambda: cancel((1, 2, 3))) + + # test first tuple returnr + assert (t:=cancel((2, 3))) == (1, 2, 3) + assert isinstance(t, tuple) + + # tests 2nd tuple return + assert (t:=cancel((1, 0), x)) == (1, 1, 0) + assert isinstance(t, tuple) + assert cancel((0, 1), x) == (1, 0, 1) + + f, g, p, q = 4*x**2 - 4, 2*x - 2, 2*x + 2, 1 + F, G, P, Q = [ Poly(u, x) for u in (f, g, p, q) ] + + assert F.cancel(G) == (1, P, Q) + assert cancel((f, g)) == (1, p, q) + assert cancel((f, g), x) == (1, p, q) + assert cancel((f, g), (x,)) == (1, p, q) + # tests 3rd tuple return + assert (t:=cancel((F, G))) == (1, P, Q) + assert isinstance(t, tuple) + assert cancel((f, g), polys=True) == (1, P, Q) + assert cancel((F, G), polys=False) == (1, p, q) + + f = (x**2 - 2)/(x + sqrt(2)) + + assert cancel(f) == f + assert cancel(f, greedy=False) == x - sqrt(2) + + f = (x**2 - 2)/(x - sqrt(2)) + + assert cancel(f) == f + assert cancel(f, greedy=False) == x + sqrt(2) + + assert cancel((x**2/4 - 1, x/2 - 1)) == (1, x + 2, 2) + # assert cancel((x**2/4 - 1, x/2 - 1)) == (S.Half, x + 2, 1) + + assert cancel((x**2 - y)/(x - y)) == 1/(x - y)*(x**2 - y) + + assert cancel((x**2 - y**2)/(x - y), x) == x + y + assert cancel((x**2 - y**2)/(x - y), y) == x + y + assert cancel((x**2 - y**2)/(x - y)) == x + y + + assert cancel((x**3 - 1)/(x**2 - 1)) == (x**2 + x + 1)/(x + 1) + assert cancel((x**3/2 - S.Half)/(x**2 - 1)) == (x**2 + x + 1)/(2*x + 2) + + assert cancel((exp(2*x) + 2*exp(x) + 1)/(exp(x) + 1)) == exp(x) + 1 + + f = Poly(x**2 - a**2, x) + g = Poly(x - a, x) + + F = Poly(x + a, x, domain='ZZ[a]') + G = Poly(1, x, domain='ZZ[a]') + + assert cancel((f, g)) == (1, F, G) + + f = x**3 + (sqrt(2) - 2)*x**2 - (2*sqrt(2) + 3)*x - 3*sqrt(2) + g = x**2 - 2 + + assert cancel((f, g), extension=True) == (1, x**2 - 2*x - 3, x - sqrt(2)) + + f = Poly(-2*x + 3, x) + g = Poly(-x**9 + x**8 + x**6 - x**5 + 2*x**2 - 3*x + 1, x) + + assert cancel((f, g)) == (1, -f, -g) + + f = Poly(x/3 + 1, x) + g = Poly(x/7 + 1, x) + + assert f.cancel(g) == (S(7)/3, + Poly(x + 3, x, domain=QQ), + Poly(x + 7, x, domain=QQ)) + assert f.cancel(g, include=True) == ( + Poly(7*x + 21, x, domain=QQ), + Poly(3*x + 21, x, domain=QQ)) + + pairs = [ + (1 + x, 1 + x, 1, 1, 1), + (1 + x, 1 - x, -1, -1-x, -1+x), + (1 - x, 1 + x, -1, 1-x, 1+x), + (1 - x, 1 - x, 1, 1, 1), + ] + for f, g, coeff, p, q in pairs: + assert cancel((f, g)) == (1, p, q) + pf = Poly(f, x) + pg = Poly(g, x) + pp = Poly(p, x) + pq = Poly(q, x) + assert pf.cancel(pg) == (coeff, coeff*pp, pq) + assert pf.rep.cancel(pg.rep) == (pp.rep, pq.rep) + assert pf.rep.cancel(pg.rep, include=True) == (pp.rep, pq.rep) + + f = Poly(y, y, domain='ZZ(x)') + g = Poly(1, y, domain='ZZ[x]') + + assert f.cancel( + g) == (1, Poly(y, y, domain='ZZ(x)'), Poly(1, y, domain='ZZ(x)')) + assert f.cancel(g, include=True) == ( + Poly(y, y, domain='ZZ(x)'), Poly(1, y, domain='ZZ(x)')) + + f = Poly(5*x*y + x, y, domain='ZZ(x)') + g = Poly(2*x**2*y, y, domain='ZZ(x)') + + assert f.cancel(g, include=True) == ( + Poly(5*y + 1, y, domain='ZZ(x)'), Poly(2*x*y, y, domain='ZZ(x)')) + + f = -(-2*x - 4*y + 0.005*(z - y)**2)/((z - y)*(-z + y + 2)) + assert cancel(f).is_Mul == True + + P = tanh(x - 3.0) + Q = tanh(x + 3.0) + f = ((-2*P**2 + 2)*(-P**2 + 1)*Q**2/2 + (-2*P**2 + 2)*(-2*Q**2 + 2)*P*Q - (-2*P**2 + 2)*P**2*Q**2 + (-2*Q**2 + 2)*(-Q**2 + 1)*P**2/2 - (-2*Q**2 + 2)*P**2*Q**2)/(2*sqrt(P**2*Q**2 + 0.0001)) \ + + (-(-2*P**2 + 2)*P*Q**2/2 - (-2*Q**2 + 2)*P**2*Q/2)*((-2*P**2 + 2)*P*Q**2/2 + (-2*Q**2 + 2)*P**2*Q/2)/(2*(P**2*Q**2 + 0.0001)**Rational(3, 2)) + assert cancel(f).is_Mul == True + + # issue 7022 + A = Symbol('A', commutative=False) + p1 = Piecewise((A*(x**2 - 1)/(x + 1), x > 1), ((x + 2)/(x**2 + 2*x), True)) + p2 = Piecewise((A*(x - 1), x > 1), (1/x, True)) + assert cancel(p1) == p2 + assert cancel(2*p1) == 2*p2 + assert cancel(1 + p1) == 1 + p2 + assert cancel((x**2 - 1)/(x + 1)*p1) == (x - 1)*p2 + assert cancel((x**2 - 1)/(x + 1) + p1) == (x - 1) + p2 + p3 = Piecewise(((x**2 - 1)/(x + 1), x > 1), ((x + 2)/(x**2 + 2*x), True)) + p4 = Piecewise(((x - 1), x > 1), (1/x, True)) + assert cancel(p3) == p4 + assert cancel(2*p3) == 2*p4 + assert cancel(1 + p3) == 1 + p4 + assert cancel((x**2 - 1)/(x + 1)*p3) == (x - 1)*p4 + assert cancel((x**2 - 1)/(x + 1) + p3) == (x - 1) + p4 + + # issue 4077 + q = S('''(2*1*(x - 1/x)/(x*(2*x - (-x + 1/x)/(x**2*(x - 1/x)**2) - 1/(x**2*(x - + 1/x)) - 2/x)) - 2*1*((x - 1/x)/((x*(x - 1/x)**2)) - 1/(x*(x - + 1/x)))*((-x + 1/x)*((x - 1/x)/((x*(x - 1/x)**2)) - 1/(x*(x - + 1/x)))/(2*x - (-x + 1/x)/(x**2*(x - 1/x)**2) - 1/(x**2*(x - 1/x)) - + 2/x) + 1)*((x - 1/x)/((x - 1/x)**2) - ((x - 1/x)/((x*(x - 1/x)**2)) - + 1/(x*(x - 1/x)))**2/(2*x - (-x + 1/x)/(x**2*(x - 1/x)**2) - 1/(x**2*(x + - 1/x)) - 2/x) - 1/(x - 1/x))*(2*x - (-x + 1/x)/(x**2*(x - 1/x)**2) - + 1/(x**2*(x - 1/x)) - 2/x)/x - 1/x)*(((-x + 1/x)/((x*(x - 1/x)**2)) + + 1/(x*(x - 1/x)))*((-(x - 1/x)/(x*(x - 1/x)) - 1/x)*((x - 1/x)/((x*(x - + 1/x)**2)) - 1/(x*(x - 1/x)))/(2*x - (-x + 1/x)/(x**2*(x - 1/x)**2) - + 1/(x**2*(x - 1/x)) - 2/x) - 1 + (x - 1/x)/(x - 1/x))/((x*((x - + 1/x)/((x - 1/x)**2) - ((x - 1/x)/((x*(x - 1/x)**2)) - 1/(x*(x - + 1/x)))**2/(2*x - (-x + 1/x)/(x**2*(x - 1/x)**2) - 1/(x**2*(x - 1/x)) - + 2/x) - 1/(x - 1/x))*(2*x - (-x + 1/x)/(x**2*(x - 1/x)**2) - 1/(x**2*(x + - 1/x)) - 2/x))) + ((x - 1/x)/((x*(x - 1/x))) + 1/x)/((x*(2*x - (-x + + 1/x)/(x**2*(x - 1/x)**2) - 1/(x**2*(x - 1/x)) - 2/x))) + 1/x)/(2*x + + 2*((x - 1/x)/((x*(x - 1/x)**2)) - 1/(x*(x - 1/x)))*((-(x - 1/x)/(x*(x + - 1/x)) - 1/x)*((x - 1/x)/((x*(x - 1/x)**2)) - 1/(x*(x - 1/x)))/(2*x - + (-x + 1/x)/(x**2*(x - 1/x)**2) - 1/(x**2*(x - 1/x)) - 2/x) - 1 + (x - + 1/x)/(x - 1/x))/((x*((x - 1/x)/((x - 1/x)**2) - ((x - 1/x)/((x*(x - + 1/x)**2)) - 1/(x*(x - 1/x)))**2/(2*x - (-x + 1/x)/(x**2*(x - 1/x)**2) + - 1/(x**2*(x - 1/x)) - 2/x) - 1/(x - 1/x))*(2*x - (-x + 1/x)/(x**2*(x + - 1/x)**2) - 1/(x**2*(x - 1/x)) - 2/x))) - 2*((x - 1/x)/((x*(x - + 1/x))) + 1/x)/(x*(2*x - (-x + 1/x)/(x**2*(x - 1/x)**2) - 1/(x**2*(x - + 1/x)) - 2/x)) - 2/x) - ((x - 1/x)/((x*(x - 1/x)**2)) - 1/(x*(x - + 1/x)))*((-x + 1/x)*((x - 1/x)/((x*(x - 1/x)**2)) - 1/(x*(x - + 1/x)))/(2*x - (-x + 1/x)/(x**2*(x - 1/x)**2) - 1/(x**2*(x - 1/x)) - + 2/x) + 1)/(x*((x - 1/x)/((x - 1/x)**2) - ((x - 1/x)/((x*(x - 1/x)**2)) + - 1/(x*(x - 1/x)))**2/(2*x - (-x + 1/x)/(x**2*(x - 1/x)**2) - + 1/(x**2*(x - 1/x)) - 2/x) - 1/(x - 1/x))*(2*x - (-x + 1/x)/(x**2*(x - + 1/x)**2) - 1/(x**2*(x - 1/x)) - 2/x)) + (x - 1/x)/((x*(2*x - (-x + + 1/x)/(x**2*(x - 1/x)**2) - 1/(x**2*(x - 1/x)) - 2/x))) - 1/x''', + evaluate=False) + assert cancel(q, _signsimp=False) is S.NaN + assert q.subs(x, 2) is S.NaN + assert signsimp(q) is S.NaN + + # issue 9363 + M = MatrixSymbol('M', 5, 5) + assert cancel(M[0,0] + 7) == M[0,0] + 7 + expr = sin(M[1, 4] + M[2, 1] * 5 * M[4, 0]) - 5 * M[1, 2] / z + assert cancel(expr) == (z*sin(M[1, 4] + M[2, 1] * 5 * M[4, 0]) - 5 * M[1, 2]) / z + + assert cancel((x**2 + 1)/(x - I)) == x + I + + +def test_cancel_modulus(): + assert cancel((x**2 - 1)/(x + 1), modulus=2) == x + 1 + assert Poly(x**2 - 1, modulus=2).cancel(Poly(x + 1, modulus=2)) ==\ + (1, Poly(x + 1, modulus=2), Poly(1, x, modulus=2)) + + +def test_make_monic_over_integers_by_scaling_roots(): + f = Poly(x**2 + 3*x + 4, x, domain='ZZ') + g, c = f.make_monic_over_integers_by_scaling_roots() + assert g == f + assert c == ZZ.one + + f = Poly(x**2 + 3*x + 4, x, domain='QQ') + g, c = f.make_monic_over_integers_by_scaling_roots() + assert g == f.to_ring() + assert c == ZZ.one + + f = Poly(x**2/2 + S(1)/4 * x + S(1)/8, x, domain='QQ') + g, c = f.make_monic_over_integers_by_scaling_roots() + assert g == Poly(x**2 + 2*x + 4, x, domain='ZZ') + assert c == 4 + + f = Poly(x**3/2 + S(1)/4 * x + S(1)/8, x, domain='QQ') + g, c = f.make_monic_over_integers_by_scaling_roots() + assert g == Poly(x**3 + 8*x + 16, x, domain='ZZ') + assert c == 4 + + f = Poly(x*y, x, y) + raises(ValueError, lambda: f.make_monic_over_integers_by_scaling_roots()) + + f = Poly(x, domain='RR') + raises(ValueError, lambda: f.make_monic_over_integers_by_scaling_roots()) + + +def test_galois_group(): + f = Poly(x ** 4 - 2) + G, alt = f.galois_group(by_name=True) + assert G == S4TransitiveSubgroups.D4 + assert alt is False + + +def test_reduced(): + f = 2*x**4 + y**2 - x**2 + y**3 + G = [x**3 - x, y**3 - y] + + Q = [2*x, 1] + r = x**2 + y**2 + y + + assert reduced(f, G) == (Q, r) + assert reduced(f, G, x, y) == (Q, r) + + H = groebner(G) + + assert H.reduce(f) == (Q, r) + + Q = [Poly(2*x, x, y), Poly(1, x, y)] + r = Poly(x**2 + y**2 + y, x, y) + + assert _strict_eq(reduced(f, G, polys=True), (Q, r)) + assert _strict_eq(reduced(f, G, x, y, polys=True), (Q, r)) + + H = groebner(G, polys=True) + + assert _strict_eq(H.reduce(f), (Q, r)) + + f = 2*x**3 + y**3 + 3*y + G = groebner([x**2 + y**2 - 1, x*y - 2]) + + Q = [x**2 - x*y**3/2 + x*y/2 + y**6/4 - y**4/2 + y**2/4, -y**5/4 + y**3/2 + y*Rational(3, 4)] + r = 0 + + assert reduced(f, G) == (Q, r) + assert G.reduce(f) == (Q, r) + + assert reduced(f, G, auto=False)[1] != 0 + assert G.reduce(f, auto=False)[1] != 0 + + assert G.contains(f) is True + assert G.contains(f + 1) is False + + assert reduced(1, [1], x) == ([1], 0) + raises(ComputationFailed, lambda: reduced(1, [1])) + + f_poly = Poly(2*x**3 + y**3 + 3*y) + G_poly = groebner([Poly(x**2 + y**2 - 1), Poly(x*y - 2)]) + + Q_poly = [Poly(x**2 - 1/2*x*y**3 + 1/2*x*y + 1/4*y**6 - 1/2*y**4 + 1/4*y**2, x, y, domain='QQ'), + Poly(-1/4*y**5 + 1/2*y**3 + 3/4*y, x, y, domain='QQ')] + r_poly = Poly(0, x, y, domain='QQ') + + assert G_poly.reduce(f_poly) == (Q_poly, r_poly) + + Q, r = G_poly.reduce(f) + assert all(isinstance(q, Poly) for q in Q) + assert isinstance(r, Poly) + + f_wrong_gens = Poly(2*x**3 + y**3 + 3*y, x, y, z) + raises(ValueError, lambda: G_poly.reduce(f_wrong_gens)) + + zero_poly = Poly(0, x, y) + Q, r = G_poly.reduce(zero_poly) + assert all(q.is_zero for q in Q) + assert r.is_zero + + const_poly = Poly(1, x, y) + Q, r = G_poly.reduce(const_poly) + assert isinstance(r, Poly) + assert r.as_expr() == 1 + assert all(q.is_zero for q in Q) + + +def test_groebner(): + assert groebner([], x, y, z) == [] + + assert groebner([x**2 + 1, y**4*x + x**3], x, y, order='lex') == [1 + x**2, -1 + y**4] + assert groebner([x**2 + 1, y**4*x + x**3, x*y*z**3], x, y, z, order='grevlex') == [-1 + y**4, z**3, 1 + x**2] + + assert groebner([x**2 + 1, y**4*x + x**3], x, y, order='lex', polys=True) == \ + [Poly(1 + x**2, x, y), Poly(-1 + y**4, x, y)] + assert groebner([x**2 + 1, y**4*x + x**3, x*y*z**3], x, y, z, order='grevlex', polys=True) == \ + [Poly(-1 + y**4, x, y, z), Poly(z**3, x, y, z), Poly(1 + x**2, x, y, z)] + + assert groebner([x**3 - 1, x**2 - 1]) == [x - 1] + assert groebner([Eq(x**3, 1), Eq(x**2, 1)]) == [x - 1] + + F = [3*x**2 + y*z - 5*x - 1, 2*x + 3*x*y + y**2, x - 3*y + x*z - 2*z**2] + f = z**9 - x**2*y**3 - 3*x*y**2*z + 11*y*z**2 + x**2*z**2 - 5 + + G = groebner(F, x, y, z, modulus=7, symmetric=False) + + assert G == [1 + x + y + 3*z + 2*z**2 + 2*z**3 + 6*z**4 + z**5, + 1 + 3*y + y**2 + 6*z**2 + 3*z**3 + 3*z**4 + 3*z**5 + 4*z**6, + 1 + 4*y + 4*z + y*z + 4*z**3 + z**4 + z**6, + 6 + 6*z + z**2 + 4*z**3 + 3*z**4 + 6*z**5 + 3*z**6 + z**7] + + Q, r = reduced(f, G, x, y, z, modulus=7, symmetric=False, polys=True) + + assert sum([ q*g for q, g in zip(Q, G.polys)], r) == Poly(f, modulus=7) + + F = [x*y - 2*y, 2*y**2 - x**2] + + assert groebner(F, x, y, order='grevlex') == \ + [y**3 - 2*y, x**2 - 2*y**2, x*y - 2*y] + assert groebner(F, y, x, order='grevlex') == \ + [x**3 - 2*x**2, -x**2 + 2*y**2, x*y - 2*y] + assert groebner(F, order='grevlex', field=True) == \ + [y**3 - 2*y, x**2 - 2*y**2, x*y - 2*y] + + assert groebner([1], x) == [1] + + assert groebner([x**2 + 2.0*y], x, y) == [1.0*x**2 + 2.0*y] + raises(ComputationFailed, lambda: groebner([1])) + + assert groebner([x**2 - 1, x**3 + 1], method='buchberger') == [x + 1] + assert groebner([x**2 - 1, x**3 + 1], method='f5b') == [x + 1] + + raises(ValueError, lambda: groebner([x, y], method='unknown')) + + +def test_fglm(): + F = [a + b + c + d, a*b + a*d + b*c + b*d, a*b*c + a*b*d + a*c*d + b*c*d, a*b*c*d - 1] + G = groebner(F, a, b, c, d, order=grlex) + + B = [ + 4*a + 3*d**9 - 4*d**5 - 3*d, + 4*b + 4*c - 3*d**9 + 4*d**5 + 7*d, + 4*c**2 + 3*d**10 - 4*d**6 - 3*d**2, + 4*c*d**4 + 4*c - d**9 + 4*d**5 + 5*d, + d**12 - d**8 - d**4 + 1, + ] + + assert groebner(F, a, b, c, d, order=lex) == B + assert G.fglm(lex) == B + + F = [9*x**8 + 36*x**7 - 32*x**6 - 252*x**5 - 78*x**4 + 468*x**3 + 288*x**2 - 108*x + 9, + -72*t*x**7 - 252*t*x**6 + 192*t*x**5 + 1260*t*x**4 + 312*t*x**3 - 404*t*x**2 - 576*t*x + \ + 108*t - 72*x**7 - 256*x**6 + 192*x**5 + 1280*x**4 + 312*x**3 - 576*x + 96] + G = groebner(F, t, x, order=grlex) + + B = [ + 203577793572507451707*t + 627982239411707112*x**7 - 666924143779443762*x**6 - \ + 10874593056632447619*x**5 + 5119998792707079562*x**4 + 72917161949456066376*x**3 + \ + 20362663855832380362*x**2 - 142079311455258371571*x + 183756699868981873194, + 9*x**8 + 36*x**7 - 32*x**6 - 252*x**5 - 78*x**4 + 468*x**3 + 288*x**2 - 108*x + 9, + ] + + assert groebner(F, t, x, order=lex) == B + assert G.fglm(lex) == B + + F = [x**2 - x - 3*y + 1, -2*x + y**2 + y - 1] + G = groebner(F, x, y, order=lex) + + B = [ + x**2 - x - 3*y + 1, + y**2 - 2*x + y - 1, + ] + + assert groebner(F, x, y, order=grlex) == B + assert G.fglm(grlex) == B + + +def test_is_zero_dimensional(): + assert is_zero_dimensional([x, y], x, y) is True + assert is_zero_dimensional([x**3 + y**2], x, y) is False + + assert is_zero_dimensional([x, y, z], x, y, z) is True + assert is_zero_dimensional([x, y, z], x, y, z, t) is False + + F = [x*y - z, y*z - x, x*y - y] + assert is_zero_dimensional(F, x, y, z) is True + + F = [x**2 - 2*x*z + 5, x*y**2 + y*z**3, 3*y**2 - 8*z**2] + assert is_zero_dimensional(F, x, y, z) is True + + +def test_GroebnerBasis(): + F = [x*y - 2*y, 2*y**2 - x**2] + + G = groebner(F, x, y, order='grevlex') + H = [y**3 - 2*y, x**2 - 2*y**2, x*y - 2*y] + P = [ Poly(h, x, y) for h in H ] + + assert groebner(F + [0], x, y, order='grevlex') == G + assert isinstance(G, GroebnerBasis) is True + + assert len(G) == 3 + + assert G[0] == H[0] and not G[0].is_Poly + assert G[1] == H[1] and not G[1].is_Poly + assert G[2] == H[2] and not G[2].is_Poly + + assert G[1:] == H[1:] and not any(g.is_Poly for g in G[1:]) + assert G[:2] == H[:2] and not any(g.is_Poly for g in G[1:]) + + assert G.exprs == H + assert G.polys == P + assert G.gens == (x, y) + assert G.domain == ZZ + assert G.order == grevlex + + assert G == H + assert G == tuple(H) + assert G == P + assert G == tuple(P) + + assert G != [] + + G = groebner(F, x, y, order='grevlex', polys=True) + + assert G[0] == P[0] and G[0].is_Poly + assert G[1] == P[1] and G[1].is_Poly + assert G[2] == P[2] and G[2].is_Poly + + assert G[1:] == P[1:] and all(g.is_Poly for g in G[1:]) + assert G[:2] == P[:2] and all(g.is_Poly for g in G[1:]) + + +def test_poly(): + assert poly(x) == Poly(x, x) + assert poly(y) == Poly(y, y) + + assert poly(x + y) == Poly(x + y, x, y) + assert poly(x + sin(x)) == Poly(x + sin(x), x, sin(x)) + + assert poly(x + y, wrt=y) == Poly(x + y, y, x) + assert poly(x + sin(x), wrt=sin(x)) == Poly(x + sin(x), sin(x), x) + + assert poly(x*y + 2*x*z**2 + 17) == Poly(x*y + 2*x*z**2 + 17, x, y, z) + + assert poly(2*(y + z)**2 - 1) == Poly(2*y**2 + 4*y*z + 2*z**2 - 1, y, z) + assert poly( + x*(y + z)**2 - 1) == Poly(x*y**2 + 2*x*y*z + x*z**2 - 1, x, y, z) + assert poly(2*x*( + y + z)**2 - 1) == Poly(2*x*y**2 + 4*x*y*z + 2*x*z**2 - 1, x, y, z) + + assert poly(2*( + y + z)**2 - x - 1) == Poly(2*y**2 + 4*y*z + 2*z**2 - x - 1, x, y, z) + assert poly(x*( + y + z)**2 - x - 1) == Poly(x*y**2 + 2*x*y*z + x*z**2 - x - 1, x, y, z) + assert poly(2*x*(y + z)**2 - x - 1) == Poly(2*x*y**2 + 4*x*y*z + 2* + x*z**2 - x - 1, x, y, z) + + assert poly(x*y + (x + y)**2 + (x + z)**2) == \ + Poly(2*x*z + 3*x*y + y**2 + z**2 + 2*x**2, x, y, z) + assert poly(x*y*(x + y)*(x + z)**2) == \ + Poly(x**3*y**2 + x*y**2*z**2 + y*x**2*z**2 + 2*z*x**2* + y**2 + 2*y*z*x**3 + y*x**4, x, y, z) + + assert poly(Poly(x + y + z, y, x, z)) == Poly(x + y + z, y, x, z) + + assert poly((x + y)**2, x) == Poly(x**2 + 2*x*y + y**2, x, domain=ZZ[y]) + assert poly((x + y)**2, y) == Poly(x**2 + 2*x*y + y**2, y, domain=ZZ[x]) + + assert poly(1, x) == Poly(1, x) + raises(GeneratorsNeeded, lambda: poly(1)) + + # issue 6184 + assert poly(x + y, x, y) == Poly(x + y, x, y) + assert poly(x + y, y, x) == Poly(x + y, y, x) + + # https://github.com/sympy/sympy/issues/19755 + expr1 = x + (2*x + 3)**2/5 + S(6)/5 + assert poly(expr1).as_expr() == expr1.expand() + expr2 = y*(y+1) + S(1)/3 + assert poly(expr2).as_expr() == expr2.expand() + + +def test_keep_coeff(): + u = Mul(2, x + 1, evaluate=False) + assert _keep_coeff(S.One, x) == x + assert _keep_coeff(S.NegativeOne, x) == -x + assert _keep_coeff(S(1.0), x) == 1.0*x + assert _keep_coeff(S(-1.0), x) == -1.0*x + assert _keep_coeff(S.One, 2*x) == 2*x + assert _keep_coeff(S(2), x/2) == x + assert _keep_coeff(S(2), sin(x)) == 2*sin(x) + assert _keep_coeff(S(2), x + 1) == u + assert _keep_coeff(x, 1/x) == 1 + assert _keep_coeff(x + 1, S(2)) == u + assert _keep_coeff(S.Half, S.One) == S.Half + p = Pow(2, 3, evaluate=False) + assert _keep_coeff(S(-1), p) == Mul(-1, p, evaluate=False) + a = Add(2, p, evaluate=False) + assert _keep_coeff(S.Half, a, clear=True + ) == Mul(S.Half, a, evaluate=False) + assert _keep_coeff(S.Half, a, clear=False + ) == Add(1, Mul(S.Half, p, evaluate=False), evaluate=False) + + +def test_poly_matching_consistency(): + # Test for this issue: + # https://github.com/sympy/sympy/issues/5514 + assert I * Poly(x, x) == Poly(I*x, x) + assert Poly(x, x) * I == Poly(I*x, x) + + +def test_issue_5786(): + assert expand(factor(expand( + (x - I*y)*(z - I*t)), extension=[I])) == -I*t*x - t*y + x*z - I*y*z + + +def test_noncommutative(): + class foo(Expr): + is_commutative=False + e = x/(x + x*y) + c = 1/( 1 + y) + assert cancel(foo(e)) == foo(c) + assert cancel(e + foo(e)) == c + foo(c) + assert cancel(e*foo(c)) == c*foo(c) + + +def test_to_rational_coeffs(): + assert to_rational_coeffs( + Poly(x**3 + y*x**2 + sqrt(y), x, domain='EX')) is None + # issue 21268 + assert to_rational_coeffs( + Poly(y**3 + sqrt(2)*y**2*sin(x) + 1, y)) is None + + assert to_rational_coeffs(Poly(x, y)) is None + assert to_rational_coeffs(Poly(sqrt(2)*y)) is None + + +def test_factor_terms(): + # issue 7067 + assert factor_list(x*(x + y)) == (1, [(x, 1), (x + y, 1)]) + assert sqf_list(x*(x + y)) == (1, [(x**2 + x*y, 1)]) + + +def test_as_list(): + # issue 14496 + assert Poly(x**3 + 2, x, domain='ZZ').as_list() == [1, 0, 0, 2] + assert Poly(x**2 + y + 1, x, y, domain='ZZ').as_list() == [[1], [], [1, 1]] + assert Poly(x**2 + y + 1, x, y, z, domain='ZZ').as_list() == \ + [[[1]], [[]], [[1], [1]]] + + +def test_issue_11198(): + assert factor_list(sqrt(2)*x) == (sqrt(2), [(x, 1)]) + assert factor_list(sqrt(2)*sin(x), sin(x)) == (sqrt(2), [(sin(x), 1)]) + + +def test_Poly_precision(): + # Make sure Poly doesn't lose precision + p = Poly(pi.evalf(100)*x) + assert p.as_expr() == pi.evalf(100)*x + + +def test_issue_12400(): + # Correction of check for negative exponents + assert poly(1/(1+sqrt(2)), x) == \ + Poly(1/(1+sqrt(2)), x, domain='EX') + +def test_issue_14364(): + assert gcd(S(6)*(1 + sqrt(3))/5, S(3)*(1 + sqrt(3))/10) == Rational(3, 10) * (1 + sqrt(3)) + assert gcd(sqrt(5)*Rational(4, 7), sqrt(5)*Rational(2, 3)) == sqrt(5)*Rational(2, 21) + + assert lcm(Rational(2, 3)*sqrt(3), Rational(5, 6)*sqrt(3)) == S(10)*sqrt(3)/3 + assert lcm(3*sqrt(3), 4/sqrt(3)) == 12*sqrt(3) + assert lcm(S(5)*(1 + 2**Rational(1, 3))/6, S(3)*(1 + 2**Rational(1, 3))/8) == Rational(15, 2) * (1 + 2**Rational(1, 3)) + + assert gcd(Rational(2, 3)*sqrt(3), Rational(5, 6)/sqrt(3)) == sqrt(3)/18 + assert gcd(S(4)*sqrt(13)/7, S(3)*sqrt(13)/14) == sqrt(13)/14 + + # gcd_list and lcm_list + assert gcd([S(2)*sqrt(47)/7, S(6)*sqrt(47)/5, S(8)*sqrt(47)/5]) == sqrt(47)*Rational(2, 35) + assert gcd([S(6)*(1 + sqrt(7))/5, S(2)*(1 + sqrt(7))/7, S(4)*(1 + sqrt(7))/13]) == (1 + sqrt(7))*Rational(2, 455) + assert lcm((Rational(7, 2)/sqrt(15), Rational(5, 6)/sqrt(15), Rational(5, 8)/sqrt(15))) == Rational(35, 2)/sqrt(15) + assert lcm([S(5)*(2 + 2**Rational(5, 7))/6, S(7)*(2 + 2**Rational(5, 7))/2, S(13)*(2 + 2**Rational(5, 7))/4]) == Rational(455, 2) * (2 + 2**Rational(5, 7)) + + +def test_issue_15669(): + x = Symbol("x", positive=True) + expr = (16*x**3/(-x**2 + sqrt(8*x**2 + (x**2 - 2)**2) + 2)**2 - + 2*2**Rational(4, 5)*x*(-x**2 + sqrt(8*x**2 + (x**2 - 2)**2) + 2)**Rational(3, 5) + 10*x) + assert factor(expr, deep=True) == x*(x**2 + 2) + + +def test_issue_17988(): + x = Symbol('x') + p = poly(x - 1) + with warns_deprecated_sympy(): + M = Matrix([[poly(x + 1), poly(x + 1)]]) + with warns(SymPyDeprecationWarning, test_stacklevel=False): + assert p * M == M * p == Matrix([[poly(x**2 - 1), poly(x**2 - 1)]]) + + +def test_issue_18205(): + assert cancel((2 + I)*(3 - I)) == 7 + I + assert cancel((2 + I)*(2 - I)) == 5 + + +def test_issue_8695(): + p = (x**2 + 1) * (x - 1)**2 * (x - 2)**3 * (x - 3)**3 + result = (1, [(x**2 + 1, 1), (x - 1, 2), (x**2 - 5*x + 6, 3)]) + assert sqf_list(p) == result + + +def test_issue_19113(): + eq = sin(x)**3 - sin(x) + 1 + raises(PolynomialError, lambda: refine_root(eq, 1, 2, 1e-2)) + raises(PolynomialError, lambda: count_roots(eq, -1, 1)) + raises(PolynomialError, lambda: real_roots(eq)) + raises(PolynomialError, lambda: nroots(eq)) + raises(PolynomialError, lambda: ground_roots(eq)) + raises(PolynomialError, lambda: nth_power_roots_poly(eq, 2)) + + +def test_issue_19360(): + f = 2*x**2 - 2*sqrt(2)*x*y + y**2 + assert factor(f, extension=sqrt(2)) == 2*(x - (sqrt(2)*y/2))**2 + + f = -I*t*x - t*y + x*z - I*y*z + assert factor(f, extension=I) == (x - I*y)*(-I*t + z) + + +def test_poly_copy_equals_original(): + poly = Poly(x + y, x, y, z) + copy = poly.copy() + assert poly == copy, ( + "Copied polynomial not equal to original.") + assert poly.gens == copy.gens, ( + "Copied polynomial has different generators than original.") + + +def test_deserialized_poly_equals_original(): + poly = Poly(x + y, x, y, z) + deserialized = pickle.loads(pickle.dumps(poly)) + assert poly == deserialized, ( + "Deserialized polynomial not equal to original.") + assert poly.gens == deserialized.gens, ( + "Deserialized polynomial has different generators than original.") + + +def test_issue_20389(): + result = degree(x * (x + 1) - x ** 2 - x, x) + assert result == -oo + + +def test_issue_20985(): + from sympy.core.symbol import symbols + w, R = symbols('w R') + poly = Poly(1.0 + I*w/R, w, 1/R) + assert poly.degree() == S(1) diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_polyutils.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_polyutils.py new file mode 100644 index 0000000000000000000000000000000000000000..f39561a1c5035fed52add5e49476d0eea91bdae0 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_polyutils.py @@ -0,0 +1,300 @@ +"""Tests for useful utilities for higher level polynomial classes. """ + +from sympy.core.mul import Mul +from sympy.core.numbers import (Integer, pi) +from sympy.core.relational import Eq +from sympy.core.singleton import S +from sympy.core.symbol import (Symbol, symbols) +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.integrals.integrals import Integral +from sympy.testing.pytest import raises + +from sympy.polys.polyutils import ( + _nsort, + _sort_gens, + _unify_gens, + _analyze_gens, + _sort_factors, + parallel_dict_from_expr, + dict_from_expr, +) + +from sympy.polys.polyerrors import PolynomialError + +from sympy.polys.domains import ZZ + +x, y, z, p, q, r, s, t, u, v, w = symbols('x,y,z,p,q,r,s,t,u,v,w') +A, B = symbols('A,B', commutative=False) + + +def test__nsort(): + # issue 6137 + r = S('''[3/2 + sqrt(-14/3 - 2*(-415/216 + 13*I/12)**(1/3) - 4/sqrt(-7/3 + + 61/(18*(-415/216 + 13*I/12)**(1/3)) + 2*(-415/216 + 13*I/12)**(1/3)) - + 61/(18*(-415/216 + 13*I/12)**(1/3)))/2 - sqrt(-7/3 + 61/(18*(-415/216 + + 13*I/12)**(1/3)) + 2*(-415/216 + 13*I/12)**(1/3))/2, 3/2 - sqrt(-7/3 + + 61/(18*(-415/216 + 13*I/12)**(1/3)) + 2*(-415/216 + + 13*I/12)**(1/3))/2 - sqrt(-14/3 - 2*(-415/216 + 13*I/12)**(1/3) - + 4/sqrt(-7/3 + 61/(18*(-415/216 + 13*I/12)**(1/3)) + 2*(-415/216 + + 13*I/12)**(1/3)) - 61/(18*(-415/216 + 13*I/12)**(1/3)))/2, 3/2 + + sqrt(-14/3 - 2*(-415/216 + 13*I/12)**(1/3) + 4/sqrt(-7/3 + + 61/(18*(-415/216 + 13*I/12)**(1/3)) + 2*(-415/216 + 13*I/12)**(1/3)) - + 61/(18*(-415/216 + 13*I/12)**(1/3)))/2 + sqrt(-7/3 + 61/(18*(-415/216 + + 13*I/12)**(1/3)) + 2*(-415/216 + 13*I/12)**(1/3))/2, 3/2 + sqrt(-7/3 + + 61/(18*(-415/216 + 13*I/12)**(1/3)) + 2*(-415/216 + + 13*I/12)**(1/3))/2 - sqrt(-14/3 - 2*(-415/216 + 13*I/12)**(1/3) + + 4/sqrt(-7/3 + 61/(18*(-415/216 + 13*I/12)**(1/3)) + 2*(-415/216 + + 13*I/12)**(1/3)) - 61/(18*(-415/216 + 13*I/12)**(1/3)))/2]''') + ans = [r[1], r[0], r[-1], r[-2]] + assert _nsort(r) == ans + assert len(_nsort(r, separated=True)[0]) == 0 + b, c, a = exp(-1000), exp(-999), exp(-1001) + assert _nsort((b, c, a)) == [a, b, c] + # issue 12560 + a = cos(1)**2 + sin(1)**2 - 1 + assert _nsort([a]) == [a] + + +def test__sort_gens(): + assert _sort_gens([]) == () + + assert _sort_gens([x]) == (x,) + assert _sort_gens([p]) == (p,) + assert _sort_gens([q]) == (q,) + + assert _sort_gens([x, p]) == (x, p) + assert _sort_gens([p, x]) == (x, p) + assert _sort_gens([q, p]) == (p, q) + + assert _sort_gens([q, p, x]) == (x, p, q) + + assert _sort_gens([x, p, q], wrt=x) == (x, p, q) + assert _sort_gens([x, p, q], wrt=p) == (p, x, q) + assert _sort_gens([x, p, q], wrt=q) == (q, x, p) + + assert _sort_gens([x, p, q], wrt='x') == (x, p, q) + assert _sort_gens([x, p, q], wrt='p') == (p, x, q) + assert _sort_gens([x, p, q], wrt='q') == (q, x, p) + + assert _sort_gens([x, p, q], wrt='x,q') == (x, q, p) + assert _sort_gens([x, p, q], wrt='q,x') == (q, x, p) + assert _sort_gens([x, p, q], wrt='p,q') == (p, q, x) + assert _sort_gens([x, p, q], wrt='q,p') == (q, p, x) + + assert _sort_gens([x, p, q], wrt='x, q') == (x, q, p) + assert _sort_gens([x, p, q], wrt='q, x') == (q, x, p) + assert _sort_gens([x, p, q], wrt='p, q') == (p, q, x) + assert _sort_gens([x, p, q], wrt='q, p') == (q, p, x) + + assert _sort_gens([x, p, q], wrt=[x, 'q']) == (x, q, p) + assert _sort_gens([x, p, q], wrt=[q, 'x']) == (q, x, p) + assert _sort_gens([x, p, q], wrt=[p, 'q']) == (p, q, x) + assert _sort_gens([x, p, q], wrt=[q, 'p']) == (q, p, x) + + assert _sort_gens([x, p, q], wrt=['x', 'q']) == (x, q, p) + assert _sort_gens([x, p, q], wrt=['q', 'x']) == (q, x, p) + assert _sort_gens([x, p, q], wrt=['p', 'q']) == (p, q, x) + assert _sort_gens([x, p, q], wrt=['q', 'p']) == (q, p, x) + + assert _sort_gens([x, p, q], sort='x > p > q') == (x, p, q) + assert _sort_gens([x, p, q], sort='p > x > q') == (p, x, q) + assert _sort_gens([x, p, q], sort='p > q > x') == (p, q, x) + + assert _sort_gens([x, p, q], wrt='x', sort='q > p') == (x, q, p) + assert _sort_gens([x, p, q], wrt='p', sort='q > x') == (p, q, x) + assert _sort_gens([x, p, q], wrt='q', sort='p > x') == (q, p, x) + + # https://github.com/sympy/sympy/issues/19353 + n1 = Symbol('\n1') + assert _sort_gens([n1]) == (n1,) + assert _sort_gens([x, n1]) == (x, n1) + + X = symbols('x0,x1,x2,x10,x11,x12,x20,x21,x22') + + assert _sort_gens(X) == X + + +def test__unify_gens(): + assert _unify_gens([], []) == () + + assert _unify_gens([x], [x]) == (x,) + assert _unify_gens([y], [y]) == (y,) + + assert _unify_gens([x, y], [x]) == (x, y) + assert _unify_gens([x], [x, y]) == (x, y) + + assert _unify_gens([x, y], [x, y]) == (x, y) + assert _unify_gens([y, x], [y, x]) == (y, x) + + assert _unify_gens([x], [y]) == (x, y) + assert _unify_gens([y], [x]) == (y, x) + + assert _unify_gens([x], [y, x]) == (y, x) + assert _unify_gens([y, x], [x]) == (y, x) + + assert _unify_gens([x, y, z], [x, y, z]) == (x, y, z) + assert _unify_gens([z, y, x], [x, y, z]) == (z, y, x) + assert _unify_gens([x, y, z], [z, y, x]) == (x, y, z) + assert _unify_gens([z, y, x], [z, y, x]) == (z, y, x) + + assert _unify_gens([x, y, z], [t, x, p, q, z]) == (t, x, y, p, q, z) + + +def test__analyze_gens(): + assert _analyze_gens((x, y, z)) == (x, y, z) + assert _analyze_gens([x, y, z]) == (x, y, z) + + assert _analyze_gens(([x, y, z],)) == (x, y, z) + assert _analyze_gens(((x, y, z),)) == (x, y, z) + + +def test__sort_factors(): + assert _sort_factors([], multiple=True) == [] + assert _sort_factors([], multiple=False) == [] + + F = [[1, 2, 3], [1, 2], [1]] + G = [[1], [1, 2], [1, 2, 3]] + + assert _sort_factors(F, multiple=False) == G + + F = [[1, 2], [1, 2, 3], [1, 2], [1]] + G = [[1], [1, 2], [1, 2], [1, 2, 3]] + + assert _sort_factors(F, multiple=False) == G + + F = [[2, 2], [1, 2, 3], [1, 2], [1]] + G = [[1], [1, 2], [2, 2], [1, 2, 3]] + + assert _sort_factors(F, multiple=False) == G + + F = [([1, 2, 3], 1), ([1, 2], 1), ([1], 1)] + G = [([1], 1), ([1, 2], 1), ([1, 2, 3], 1)] + + assert _sort_factors(F, multiple=True) == G + + F = [([1, 2], 1), ([1, 2, 3], 1), ([1, 2], 1), ([1], 1)] + G = [([1], 1), ([1, 2], 1), ([1, 2], 1), ([1, 2, 3], 1)] + + assert _sort_factors(F, multiple=True) == G + + F = [([2, 2], 1), ([1, 2, 3], 1), ([1, 2], 1), ([1], 1)] + G = [([1], 1), ([1, 2], 1), ([2, 2], 1), ([1, 2, 3], 1)] + + assert _sort_factors(F, multiple=True) == G + + F = [([2, 2], 1), ([1, 2, 3], 1), ([1, 2], 2), ([1], 1)] + G = [([1], 1), ([2, 2], 1), ([1, 2], 2), ([1, 2, 3], 1)] + + assert _sort_factors(F, multiple=True) == G + + +def test__dict_from_expr_if_gens(): + assert dict_from_expr( + Integer(17), gens=(x,)) == ({(0,): Integer(17)}, (x,)) + assert dict_from_expr( + Integer(17), gens=(x, y)) == ({(0, 0): Integer(17)}, (x, y)) + assert dict_from_expr( + Integer(17), gens=(x, y, z)) == ({(0, 0, 0): Integer(17)}, (x, y, z)) + + assert dict_from_expr( + Integer(-17), gens=(x,)) == ({(0,): Integer(-17)}, (x,)) + assert dict_from_expr( + Integer(-17), gens=(x, y)) == ({(0, 0): Integer(-17)}, (x, y)) + assert dict_from_expr(Integer( + -17), gens=(x, y, z)) == ({(0, 0, 0): Integer(-17)}, (x, y, z)) + + assert dict_from_expr( + Integer(17)*x, gens=(x,)) == ({(1,): Integer(17)}, (x,)) + assert dict_from_expr( + Integer(17)*x, gens=(x, y)) == ({(1, 0): Integer(17)}, (x, y)) + assert dict_from_expr(Integer( + 17)*x, gens=(x, y, z)) == ({(1, 0, 0): Integer(17)}, (x, y, z)) + + assert dict_from_expr( + Integer(17)*x**7, gens=(x,)) == ({(7,): Integer(17)}, (x,)) + assert dict_from_expr( + Integer(17)*x**7*y, gens=(x, y)) == ({(7, 1): Integer(17)}, (x, y)) + assert dict_from_expr(Integer(17)*x**7*y*z**12, gens=( + x, y, z)) == ({(7, 1, 12): Integer(17)}, (x, y, z)) + + assert dict_from_expr(x + 2*y + 3*z, gens=(x,)) == \ + ({(1,): Integer(1), (0,): 2*y + 3*z}, (x,)) + assert dict_from_expr(x + 2*y + 3*z, gens=(x, y)) == \ + ({(1, 0): Integer(1), (0, 1): Integer(2), (0, 0): 3*z}, (x, y)) + assert dict_from_expr(x + 2*y + 3*z, gens=(x, y, z)) == \ + ({(1, 0, 0): Integer( + 1), (0, 1, 0): Integer(2), (0, 0, 1): Integer(3)}, (x, y, z)) + + assert dict_from_expr(x*y + 2*x*z + 3*y*z, gens=(x,)) == \ + ({(1,): y + 2*z, (0,): 3*y*z}, (x,)) + assert dict_from_expr(x*y + 2*x*z + 3*y*z, gens=(x, y)) == \ + ({(1, 1): Integer(1), (1, 0): 2*z, (0, 1): 3*z}, (x, y)) + assert dict_from_expr(x*y + 2*x*z + 3*y*z, gens=(x, y, z)) == \ + ({(1, 1, 0): Integer( + 1), (1, 0, 1): Integer(2), (0, 1, 1): Integer(3)}, (x, y, z)) + + assert dict_from_expr(2**y*x, gens=(x,)) == ({(1,): 2**y}, (x,)) + assert dict_from_expr(Integral(x, (x, 1, 2)) + x) == ( + {(0, 1): 1, (1, 0): 1}, (x, Integral(x, (x, 1, 2)))) + raises(PolynomialError, lambda: dict_from_expr(2**y*x, gens=(x, y))) + + +def test__dict_from_expr_no_gens(): + assert dict_from_expr(Integer(17)) == ({(): Integer(17)}, ()) + + assert dict_from_expr(x) == ({(1,): Integer(1)}, (x,)) + assert dict_from_expr(y) == ({(1,): Integer(1)}, (y,)) + + assert dict_from_expr(x*y) == ({(1, 1): Integer(1)}, (x, y)) + assert dict_from_expr( + x + y) == ({(1, 0): Integer(1), (0, 1): Integer(1)}, (x, y)) + + assert dict_from_expr(sqrt(2)) == ({(1,): Integer(1)}, (sqrt(2),)) + assert dict_from_expr(sqrt(2), greedy=False) == ({(): sqrt(2)}, ()) + + assert dict_from_expr(x*y, domain=ZZ[x]) == ({(1,): x}, (y,)) + assert dict_from_expr(x*y, domain=ZZ[y]) == ({(1,): y}, (x,)) + + assert dict_from_expr(3*sqrt( + 2)*pi*x*y, extension=None) == ({(1, 1, 1, 1): 3}, (x, y, pi, sqrt(2))) + assert dict_from_expr(3*sqrt( + 2)*pi*x*y, extension=True) == ({(1, 1, 1): 3*sqrt(2)}, (x, y, pi)) + + assert dict_from_expr(3*sqrt( + 2)*pi*x*y, extension=True) == ({(1, 1, 1): 3*sqrt(2)}, (x, y, pi)) + + f = cos(x)*sin(x) + cos(x)*sin(y) + cos(y)*sin(x) + cos(y)*sin(y) + + assert dict_from_expr(f) == ({(0, 1, 0, 1): 1, (0, 1, 1, 0): 1, + (1, 0, 0, 1): 1, (1, 0, 1, 0): 1}, (cos(x), cos(y), sin(x), sin(y))) + + +def test__parallel_dict_from_expr_if_gens(): + assert parallel_dict_from_expr([x + 2*y + 3*z, Integer(7)], gens=(x,)) == \ + ([{(1,): Integer(1), (0,): 2*y + 3*z}, {(0,): Integer(7)}], (x,)) + + +def test__parallel_dict_from_expr_no_gens(): + assert parallel_dict_from_expr([x*y, Integer(3)]) == \ + ([{(1, 1): Integer(1)}, {(0, 0): Integer(3)}], (x, y)) + assert parallel_dict_from_expr([x*y, 2*z, Integer(3)]) == \ + ([{(1, 1, 0): Integer( + 1)}, {(0, 0, 1): Integer(2)}, {(0, 0, 0): Integer(3)}], (x, y, z)) + assert parallel_dict_from_expr((Mul(x, x**2, evaluate=False),)) == \ + ([{(3,): 1}], (x,)) + + +def test_parallel_dict_from_expr(): + assert parallel_dict_from_expr([Eq(x, 1), Eq( + x**2, 2)]) == ([{(0,): -Integer(1), (1,): Integer(1)}, + {(0,): -Integer(2), (2,): Integer(1)}], (x,)) + raises(PolynomialError, lambda: parallel_dict_from_expr([A*B - B*A])) + + +def test_dict_from_expr(): + assert dict_from_expr(Eq(x, 1)) == \ + ({(0,): -Integer(1), (1,): Integer(1)}, (x,)) + raises(PolynomialError, lambda: dict_from_expr(A*B - B*A)) + raises(PolynomialError, lambda: dict_from_expr(S.true)) diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_puiseux.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_puiseux.py new file mode 100644 index 0000000000000000000000000000000000000000..031881e9d12c53053d8ec7136374bd8b3a385df0 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_puiseux.py @@ -0,0 +1,204 @@ +# +# Tests for PuiseuxRing and PuiseuxPoly +# + +from sympy.testing.pytest import raises + +from sympy import ZZ, QQ, ring +from sympy.polys.puiseux import PuiseuxRing, PuiseuxPoly, puiseux_ring + +from sympy.abc import x, y + + +def test_puiseux_ring(): + R, px = puiseux_ring('x', QQ) + R2, px2 = puiseux_ring([x], QQ) + assert isinstance(R, PuiseuxRing) + assert isinstance(px, PuiseuxPoly) + assert R == R2 + assert px == px2 + assert R == PuiseuxRing('x', QQ) + assert R == PuiseuxRing([x], QQ) + assert R != PuiseuxRing('y', QQ) + assert R != PuiseuxRing('x', ZZ) + assert R != PuiseuxRing('x, y', QQ) + assert R != QQ + assert str(R) == 'PuiseuxRing((x,), QQ)' + + +def test_puiseux_ring_attributes(): + R1, px1, py1 = ring('x, y', QQ) + R2, px2, py2 = puiseux_ring('x, y', QQ) + assert R2.domain == QQ + assert R2.symbols == (x, y) + assert R2.gens == (px2, py2) + assert R2.ngens == 2 + assert R2.poly_ring == R1 + assert R2.zero == PuiseuxPoly(R1.zero, R2) + assert R2.one == PuiseuxPoly(R1.one, R2) + assert R2.zero_monom == R1.zero_monom == (0, 0) # type: ignore + assert R2.monomial_mul((1, 2), (3, 4)) == (4, 6) + + +def test_puiseux_ring_methods(): + R1, px1, py1 = ring('x, y', QQ) + R2, px2, py2 = puiseux_ring('x, y', QQ) + assert R2({(1, 2): 3}) == 3*px2*py2**2 + assert R2(px1) == px2 + assert R2(1) == R2.one + assert R2(QQ(1,2)) == QQ(1,2)*R2.one + assert R2.from_poly(px1) == px2 + assert R2.from_poly(px1) != py2 + assert R2.from_dict({(1, 2): QQ(3)}) == 3*px2*py2**2 + assert R2.from_dict({(QQ(1,2), 2): QQ(3)}) == 3*px2**QQ(1,2)*py2**2 + assert R2.from_int(3) == 3*R2.one + assert R2.domain_new(3) == QQ(3) + assert QQ.of_type(R2.domain_new(3)) + assert R2.ground_new(3) == 3*R2.one + assert isinstance(R2.ground_new(3), PuiseuxPoly) + assert R2.index(px2) == 0 + assert R2.index(py2) == 1 + + +def test_puiseux_poly(): + R1, px1 = ring('x', QQ) + R2, px2 = puiseux_ring('x', QQ) + assert PuiseuxPoly(px1, R2) == px2 + assert px2.ring == R2 + assert px2.as_expr() == px1.as_expr() == x + assert px1 != px2 + assert R2.one == px2**0 == 1 + assert px2 == px1 + assert px2 != 2.0 + assert px2**QQ(1,2) != px1 + + +def test_puiseux_poly_normalization(): + R, x = puiseux_ring('x', QQ) + assert (x**2 + 1) / x == x + 1/x == R({(1,): 1, (-1,): 1}) + assert (x**QQ(1,6))**2 == x**QQ(1,3) == R({(QQ(1,3),): 1}) + assert (x**QQ(1,6))**(-2) == x**(-QQ(1,3)) == R({(-QQ(1,3),): 1}) + assert (x**QQ(1,6))**QQ(1,2) == x**QQ(1,12) == R({(QQ(1,12),): 1}) + assert (x**QQ(1,6))**6 == x == R({(1,): 1}) + assert x**QQ(1,6) * x**QQ(1,3) == x**QQ(1,2) == R({(QQ(1,2),): 1}) + assert 1/x * x**2 == x == R({(1,): 1}) + assert 1/x**QQ(1,3) * x**QQ(1,3) == 1 == R({(0,): 1}) + + +def test_puiseux_poly_monoms(): + R, x = puiseux_ring('x', QQ) + assert x.monoms() == [(1,)] + assert list(x) == [(1,)] + assert (x**2 + 1).monoms() == [(2,), (0,)] + assert R({(1,): 1, (-1,): 1}).monoms() == [(1,), (-1,)] + assert R({(QQ(1,3),): 1}).monoms() == [(QQ(1,3),)] + assert R({(-QQ(1,3),): 1}).monoms() == [(-QQ(1,3),)] + p = x**QQ(1,6) + assert p[(QQ(1,6),)] == 1 + raises(KeyError, lambda: p[(1,)]) + assert p.to_dict() == {(QQ(1,6),): 1} + assert R(p.to_dict()) == p + assert PuiseuxPoly.from_dict({(QQ(1,6),): 1}, R) == p + + +def test_puiseux_poly_repr(): + R, x = puiseux_ring('x', QQ) + assert repr(x) == 'x' + assert repr(x**QQ(1,2)) == 'x**(1/2)' + assert repr(1/x) == 'x**(-1)' + assert repr(2*x**2 + 1) == '1 + 2*x**2' + assert repr(R.one) == '1' + assert repr(2*R.one) == '2' + + +def test_puiseux_poly_unify(): + R, x = puiseux_ring('x', QQ) + assert 1/x + x == x + 1/x == R({(1,): 1, (-1,): 1}) + assert repr(1/x + x) == 'x**(-1) + x' + assert 1/x + 1/x == 2/x == R({(-1,): 2}) + assert repr(1/x + 1/x) == '2*x**(-1)' + assert x**QQ(1,2) + x**QQ(1,2) == 2*x**QQ(1,2) == R({(QQ(1,2),): 2}) + assert repr(x**QQ(1,2) + x**QQ(1,2)) == '2*x**(1/2)' + assert x**QQ(1,2) + x**QQ(1,3) == R({(QQ(1,2),): 1, (QQ(1,3),): 1}) + assert repr(x**QQ(1,2) + x**QQ(1,3)) == 'x**(1/3) + x**(1/2)' + assert x + x**QQ(1,2) == R({(1,): 1, (QQ(1,2),): 1}) + assert repr(x + x**QQ(1,2)) == 'x**(1/2) + x' + assert 1/x**QQ(1,2) + 1/x**QQ(1,3) == R({(-QQ(1,2),): 1, (-QQ(1,3),): 1}) + assert repr(1/x**QQ(1,2) + 1/x**QQ(1,3)) == 'x**(-1/2) + x**(-1/3)' + assert 1/x + x**QQ(1,2) == x**QQ(1,2) + 1/x == R({(-1,): 1, (QQ(1,2),): 1}) + assert repr(1/x + x**QQ(1,2)) == 'x**(-1) + x**(1/2)' + + +def test_puiseux_poly_arit(): + R, x = puiseux_ring('x', QQ) + R2, y = puiseux_ring('y', QQ) + p = x**2 + 1 + assert +p == p + assert -p == -1 - x**2 + assert p + p == 2*p == 2*x**2 + 2 + assert p + 1 == 1 + p == x**2 + 2 + assert p + QQ(1,2) == QQ(1,2) + p == x**2 + QQ(3,2) + assert p - p == 0 + assert p - 1 == -1 + p == x**2 + assert p - QQ(1,2) == -QQ(1,2) + p == x**2 + QQ(1,2) + assert 1 - p == -p + 1 == -x**2 + assert QQ(1,2) - p == -p + QQ(1,2) == -x**2 - QQ(1,2) + assert p * p == x**4 + 2*x**2 + 1 + assert p * 1 == 1 * p == p + assert 2 * p == p * 2 == 2*x**2 + 2 + assert p * QQ(1,2) == QQ(1,2) * p == QQ(1,2)*x**2 + QQ(1,2) + assert x**QQ(1,2) * x**QQ(1,2) == x + raises(ValueError, lambda: x + y) + raises(ValueError, lambda: x - y) + raises(ValueError, lambda: x * y) + raises(TypeError, lambda: x + None) + raises(TypeError, lambda: x - None) + raises(TypeError, lambda: x * None) + raises(TypeError, lambda: None + x) + raises(TypeError, lambda: None - x) + raises(TypeError, lambda: None * x) + + +def test_puiseux_poly_div(): + R, x = puiseux_ring('x', QQ) + R2, y = puiseux_ring('y', QQ) + p = x**2 - 1 + assert p / 1 == p + assert p / QQ(1,2) == 2*p == 2*x**2 - 2 + assert p / x == x - 1/x == R({(1,): 1, (-1,): -1}) + assert 2 / x == 2*x**-1 == R({(-1,): 2}) + assert QQ(1,2) / x == QQ(1,2)*x**-1 == 1/(2*x) == 1/x/2 == R({(-1,): QQ(1,2)}) + raises(ZeroDivisionError, lambda: p / 0) + raises(ValueError, lambda: (x + 1) / (x + 2)) + raises(ValueError, lambda: (x + 1) / (x + 1)) + raises(ValueError, lambda: x / y) + raises(TypeError, lambda: x / None) + raises(TypeError, lambda: None / x) + + +def test_puiseux_poly_pow(): + R, x = puiseux_ring('x', QQ) + Rz, xz = puiseux_ring('x', ZZ) + assert x**0 == 1 == R({(0,): 1}) + assert x**1 == x == R({(1,): 1}) + assert x**2 == x*x == R({(2,): 1}) + assert x**QQ(1,2) == R({(QQ(1,2),): 1}) + assert x**-1 == 1/x == R({(-1,): 1}) + assert x**-QQ(1,2) == 1/x**QQ(1,2) == R({(-QQ(1,2),): 1}) + assert (2*x)**-1 == 1/(2*x) == QQ(1,2)/x == QQ(1,2)*x**-1 == R({(-1,): QQ(1,2)}) + assert 2/x**2 == 2*x**-2 == R({(-2,): 2}) + assert 2/xz**2 == 2*xz**-2 == Rz({(-2,): 2}) + raises(TypeError, lambda: x**None) + raises(ValueError, lambda: (x + 1)**-1) + raises(ValueError, lambda: (x + 1)**QQ(1,2)) + raises(ValueError, lambda: (2*x)**QQ(1,2)) + raises(ValueError, lambda: (2*xz)**-1) + + +def test_puiseux_poly_diff(): + R, x, y = puiseux_ring('x, y', QQ) + assert (x**2 + 1).diff(x) == 2*x + assert (x**2 + 1).diff(y) == 0 + assert (x**2 + y**2).diff(x) == 2*x + assert (x**QQ(1,2) + y**QQ(1,2)).diff(x) == QQ(1,2)*x**-QQ(1,2) + assert ((x*y)**QQ(1,2)).diff(x) == QQ(1,2)*y**QQ(1,2)*x**-QQ(1,2) diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_pythonrational.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_pythonrational.py new file mode 100644 index 0000000000000000000000000000000000000000..547a5679626fd3a6165b151364bb506a574bb1db --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_pythonrational.py @@ -0,0 +1,139 @@ +"""Tests for PythonRational type. """ + +from sympy.polys.domains import PythonRational as QQ +from sympy.testing.pytest import raises + +def test_PythonRational__init__(): + assert QQ(0).numerator == 0 + assert QQ(0).denominator == 1 + assert QQ(0, 1).numerator == 0 + assert QQ(0, 1).denominator == 1 + assert QQ(0, -1).numerator == 0 + assert QQ(0, -1).denominator == 1 + + assert QQ(1).numerator == 1 + assert QQ(1).denominator == 1 + assert QQ(1, 1).numerator == 1 + assert QQ(1, 1).denominator == 1 + assert QQ(-1, -1).numerator == 1 + assert QQ(-1, -1).denominator == 1 + + assert QQ(-1).numerator == -1 + assert QQ(-1).denominator == 1 + assert QQ(-1, 1).numerator == -1 + assert QQ(-1, 1).denominator == 1 + assert QQ( 1, -1).numerator == -1 + assert QQ( 1, -1).denominator == 1 + + assert QQ(1, 2).numerator == 1 + assert QQ(1, 2).denominator == 2 + assert QQ(3, 4).numerator == 3 + assert QQ(3, 4).denominator == 4 + + assert QQ(2, 2).numerator == 1 + assert QQ(2, 2).denominator == 1 + assert QQ(2, 4).numerator == 1 + assert QQ(2, 4).denominator == 2 + +def test_PythonRational__hash__(): + assert hash(QQ(0)) == hash(0) + assert hash(QQ(1)) == hash(1) + assert hash(QQ(117)) == hash(117) + +def test_PythonRational__int__(): + assert int(QQ(-1, 4)) == 0 + assert int(QQ( 1, 4)) == 0 + assert int(QQ(-5, 4)) == -1 + assert int(QQ( 5, 4)) == 1 + +def test_PythonRational__float__(): + assert float(QQ(-1, 2)) == -0.5 + assert float(QQ( 1, 2)) == 0.5 + +def test_PythonRational__abs__(): + assert abs(QQ(-1, 2)) == QQ(1, 2) + assert abs(QQ( 1, 2)) == QQ(1, 2) + +def test_PythonRational__pos__(): + assert +QQ(-1, 2) == QQ(-1, 2) + assert +QQ( 1, 2) == QQ( 1, 2) + +def test_PythonRational__neg__(): + assert -QQ(-1, 2) == QQ( 1, 2) + assert -QQ( 1, 2) == QQ(-1, 2) + +def test_PythonRational__add__(): + assert QQ(-1, 2) + QQ( 1, 2) == QQ(0) + assert QQ( 1, 2) + QQ(-1, 2) == QQ(0) + + assert QQ(1, 2) + QQ(1, 2) == QQ(1) + assert QQ(1, 2) + QQ(3, 2) == QQ(2) + assert QQ(3, 2) + QQ(1, 2) == QQ(2) + assert QQ(3, 2) + QQ(3, 2) == QQ(3) + + assert 1 + QQ(1, 2) == QQ(3, 2) + assert QQ(1, 2) + 1 == QQ(3, 2) + +def test_PythonRational__sub__(): + assert QQ(-1, 2) - QQ( 1, 2) == QQ(-1) + assert QQ( 1, 2) - QQ(-1, 2) == QQ( 1) + + assert QQ(1, 2) - QQ(1, 2) == QQ( 0) + assert QQ(1, 2) - QQ(3, 2) == QQ(-1) + assert QQ(3, 2) - QQ(1, 2) == QQ( 1) + assert QQ(3, 2) - QQ(3, 2) == QQ( 0) + + assert 1 - QQ(1, 2) == QQ( 1, 2) + assert QQ(1, 2) - 1 == QQ(-1, 2) + +def test_PythonRational__mul__(): + assert QQ(-1, 2) * QQ( 1, 2) == QQ(-1, 4) + assert QQ( 1, 2) * QQ(-1, 2) == QQ(-1, 4) + + assert QQ(1, 2) * QQ(1, 2) == QQ(1, 4) + assert QQ(1, 2) * QQ(3, 2) == QQ(3, 4) + assert QQ(3, 2) * QQ(1, 2) == QQ(3, 4) + assert QQ(3, 2) * QQ(3, 2) == QQ(9, 4) + + assert 2 * QQ(1, 2) == QQ(1) + assert QQ(1, 2) * 2 == QQ(1) + +def test_PythonRational__truediv__(): + assert QQ(-1, 2) / QQ( 1, 2) == QQ(-1) + assert QQ( 1, 2) / QQ(-1, 2) == QQ(-1) + + assert QQ(1, 2) / QQ(1, 2) == QQ(1) + assert QQ(1, 2) / QQ(3, 2) == QQ(1, 3) + assert QQ(3, 2) / QQ(1, 2) == QQ(3) + assert QQ(3, 2) / QQ(3, 2) == QQ(1) + + assert 2 / QQ(1, 2) == QQ(4) + assert QQ(1, 2) / 2 == QQ(1, 4) + + raises(ZeroDivisionError, lambda: QQ(1, 2) / QQ(0)) + raises(ZeroDivisionError, lambda: QQ(1, 2) / 0) + +def test_PythonRational__pow__(): + assert QQ(1)**10 == QQ(1) + assert QQ(2)**10 == QQ(1024) + + assert QQ(1)**(-10) == QQ(1) + assert QQ(2)**(-10) == QQ(1, 1024) + +def test_PythonRational__eq__(): + assert (QQ(1, 2) == QQ(1, 2)) is True + assert (QQ(1, 2) != QQ(1, 2)) is False + + assert (QQ(1, 2) == QQ(1, 3)) is False + assert (QQ(1, 2) != QQ(1, 3)) is True + +def test_PythonRational__lt_le_gt_ge__(): + assert (QQ(1, 2) < QQ(1, 4)) is False + assert (QQ(1, 2) <= QQ(1, 4)) is False + assert (QQ(1, 2) > QQ(1, 4)) is True + assert (QQ(1, 2) >= QQ(1, 4)) is True + + assert (QQ(1, 4) < QQ(1, 2)) is True + assert (QQ(1, 4) <= QQ(1, 2)) is True + assert (QQ(1, 4) > QQ(1, 2)) is False + assert (QQ(1, 4) >= QQ(1, 2)) is False diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_rationaltools.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_rationaltools.py new file mode 100644 index 0000000000000000000000000000000000000000..3ee0192a3fbc8997347df081663015afd91dd8ad --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_rationaltools.py @@ -0,0 +1,63 @@ +"""Tests for tools for manipulation of rational expressions. """ + +from sympy.polys.rationaltools import together + +from sympy.core.mul import Mul +from sympy.core.numbers import Rational +from sympy.core.relational import Eq +from sympy.core.singleton import S +from sympy.core.symbol import symbols +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.trigonometric import sin +from sympy.integrals.integrals import Integral +from sympy.abc import x, y, z + +A, B = symbols('A,B', commutative=False) + + +def test_together(): + assert together(0) == 0 + assert together(1) == 1 + + assert together(x*y*z) == x*y*z + assert together(x + y) == x + y + + assert together(1/x) == 1/x + + assert together(1/x + 1) == (x + 1)/x + assert together(1/x + 3) == (3*x + 1)/x + assert together(1/x + x) == (x**2 + 1)/x + + assert together(1/x + S.Half) == (x + 2)/(2*x) + assert together(S.Half + x/2) == Mul(S.Half, x + 1, evaluate=False) + + assert together(1/x + 2/y) == (2*x + y)/(y*x) + assert together(1/(1 + 1/x)) == x/(1 + x) + assert together(x/(1 + 1/x)) == x**2/(1 + x) + + assert together(1/x + 1/y + 1/z) == (x*y + x*z + y*z)/(x*y*z) + assert together(1/(1 + x + 1/y + 1/z)) == y*z/(y + z + y*z + x*y*z) + + assert together(1/(x*y) + 1/(x*y)**2) == y**(-2)*x**(-2)*(1 + x*y) + assert together(1/(x*y) + 1/(x*y)**4) == y**(-4)*x**(-4)*(1 + x**3*y**3) + assert together(1/(x**7*y) + 1/(x*y)**4) == y**(-4)*x**(-7)*(x**3 + y**3) + + assert together(5/(2 + 6/(3 + 7/(4 + 8/(5 + 9/x))))) == \ + Rational(5, 2)*((171 + 119*x)/(279 + 203*x)) + + assert together(1 + 1/(x + 1)**2) == (1 + (x + 1)**2)/(x + 1)**2 + assert together(1 + 1/(x*(1 + x))) == (1 + x*(1 + x))/(x*(1 + x)) + assert together( + 1/(x*(x + 1)) + 1/(x*(x + 2))) == (3 + 2*x)/(x*(1 + x)*(2 + x)) + assert together(1 + 1/(2*x + 2)**2) == (4*(x + 1)**2 + 1)/(4*(x + 1)**2) + + assert together(sin(1/x + 1/y)) == sin(1/x + 1/y) + assert together(sin(1/x + 1/y), deep=True) == sin((x + y)/(x*y)) + + assert together(1/exp(x) + 1/(x*exp(x))) == (1 + x)/(x*exp(x)) + assert together(1/exp(2*x) + 1/(x*exp(3*x))) == (1 + exp(x)*x)/(x*exp(3*x)) + + assert together(Integral(1/x + 1/y, x)) == Integral((x + y)/(x*y), x) + assert together(Eq(1/x + 1/y, 1 + 1/z)) == Eq((x + y)/(x*y), (z + 1)/z) + + assert together((A*B)**-1 + (B*A)**-1) == (A*B)**-1 + (B*A)**-1 diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_ring_series.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_ring_series.py new file mode 100644 index 0000000000000000000000000000000000000000..d983fc99f8ffcf9361d8d069f1d381928ac0aada --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_ring_series.py @@ -0,0 +1,831 @@ +from sympy.polys.domains import ZZ, QQ, EX, RR +from sympy.polys.rings import ring +from sympy.polys.puiseux import puiseux_ring +from sympy.polys.ring_series import (_invert_monoms, rs_integrate, + rs_trunc, rs_mul, rs_square, rs_pow, _has_constant_term, rs_hadamard_exp, + rs_series_from_list, rs_exp, rs_log, rs_newton, rs_series_inversion, + rs_compose_add, rs_asin, _atan, rs_atan, _atanh, rs_atanh, rs_asinh, rs_tan, + rs_cot, rs_sin, rs_cos, rs_cos_sin, rs_sinh, rs_cosh, rs_cosh_sinh, rs_tanh, + _tan1, rs_fun, rs_nth_root, rs_LambertW, rs_series_reversion, rs_is_puiseux, + rs_series) +from sympy.testing.pytest import raises, slow +from sympy.core.symbol import symbols +from sympy.functions import (sin, cos, exp, tan, cot, sinh, cosh, atan, atanh, + asinh, tanh, log, sqrt) +from sympy.core.numbers import Rational, pi +from sympy.core import expand, S + +def is_close(a, b): + tol = 10**(-10) + assert abs(a - b) < tol + + +def test_ring_series1(): + R, x = ring('x', QQ) + p = x**4 + 2*x**3 + 3*x + 4 + assert _invert_monoms(p) == 4*x**4 + 3*x**3 + 2*x + 1 + assert rs_hadamard_exp(p) == x**4/24 + x**3/3 + 3*x + 4 + R, x = ring('x', QQ) + p = x**4 + 2*x**3 + 3*x + 4 + assert rs_integrate(p, x) == x**5/5 + x**4/2 + 3*x**2/2 + 4*x + R, x, y = ring('x, y', QQ) + p = x**2*y**2 + x + 1 + assert rs_integrate(p, x) == x**3*y**2/3 + x**2/2 + x + assert rs_integrate(p, y) == x**2*y**3/3 + x*y + y + + +def test_trunc(): + R, x, y, t = ring('x, y, t', QQ) + p = (y + t*x)**4 + p1 = rs_trunc(p, x, 3) + assert p1 == y**4 + 4*y**3*t*x + 6*y**2*t**2*x**2 + + +def test_mul_trunc(): + R, x, y, t = ring('x, y, t', QQ) + p = 1 + t*x + t*y + for i in range(2): + p = rs_mul(p, p, t, 3) + + assert p == 6*x**2*t**2 + 12*x*y*t**2 + 6*y**2*t**2 + 4*x*t + 4*y*t + 1 + p = 1 + t*x + t*y + t**2*x*y + p1 = rs_mul(p, p, t, 2) + assert p1 == 1 + 2*t*x + 2*t*y + R1, z = ring('z', QQ) + raises(ValueError, lambda: rs_mul(p, z, x, 2)) + + p1 = 2 + 2*x + 3*x**2 + p2 = 3 + x**2 + assert rs_mul(p1, p2, x, 4) == 2*x**3 + 11*x**2 + 6*x + 6 + + +def test_square_trunc(): + R, x, y, t = ring('x, y, t', QQ) + p = (1 + t*x + t*y)*2 + p1 = rs_mul(p, p, x, 3) + p2 = rs_square(p, x, 3) + assert p1 == p2 + p = 1 + x + x**2 + x**3 + assert rs_square(p, x, 4) == 4*x**3 + 3*x**2 + 2*x + 1 + + +def test_pow_trunc(): + R, x, y, z = ring('x, y, z', QQ) + p0 = y + x*z + p = p0**16 + for xx in (x, y, z): + p1 = rs_trunc(p, xx, 8) + p2 = rs_pow(p0, 16, xx, 8) + assert p1 == p2 + + p = 1 + x + p1 = rs_pow(p, 3, x, 2) + assert p1 == 1 + 3*x + assert rs_pow(p, 0, x, 2) == 1 + assert rs_pow(p, -2, x, 2) == 1 - 2*x + p = x + y + assert rs_pow(p, 3, y, 3) == x**3 + 3*x**2*y + 3*x*y**2 + assert rs_pow(1 + x, Rational(2, 3), x, 4) == 4*x**3/81 - x**2/9 + x*Rational(2, 3) + 1 + + +def test_has_constant_term(): + R, x, y, z = ring('x, y, z', QQ) + p = y + x*z + assert _has_constant_term(p, x) + p = x + x**4 + assert not _has_constant_term(p, x) + p = 1 + x + x**4 + assert _has_constant_term(p, x) + p = x + y + x*z + + +def test_inversion(): + R, x = ring('x', QQ) + p = 2 + x + 2*x**2 + n = 5 + p1 = rs_series_inversion(p, x, n) + assert rs_trunc(p*p1, x, n) == 1 + R, x, y = ring('x, y', QQ) + p = 2 + x + 2*x**2 + y*x + x**2*y + p1 = rs_series_inversion(p, x, n) + assert rs_trunc(p*p1, x, n) == 1 + + R, x, y = ring('x, y', QQ) + p = 1 + x + y + raises(NotImplementedError, lambda: rs_series_inversion(p, x, 4)) + p = R.zero + raises(ZeroDivisionError, lambda: rs_series_inversion(p, x, 3)) + + R, x = ring('x', ZZ) + p = 2 + x + raises(ValueError, lambda: rs_series_inversion(p, x, 3)) + + +def test_series_reversion(): + R, x, y = ring('x, y', QQ) + + p = rs_tan(x, x, 10) + assert rs_series_reversion(p, x, 8, y) == rs_atan(y, y, 8) + + p = rs_sin(x, x, 10) + assert rs_series_reversion(p, x, 8, y) == 5*y**7/112 + 3*y**5/40 + \ + y**3/6 + y + + +def test_series_from_list(): + R, x = ring('x', QQ) + p = 1 + 2*x + x**2 + 3*x**3 + c = [1, 2, 0, 4, 4] + r = rs_series_from_list(p, c, x, 5) + pc = R.from_list(list(reversed(c))) + r1 = rs_trunc(pc.compose(x, p), x, 5) + assert r == r1 + R, x, y = ring('x, y', QQ) + c = [1, 3, 5, 7] + p1 = rs_series_from_list(x + y, c, x, 3, concur=0) + p2 = rs_trunc((1 + 3*(x+y) + 5*(x+y)**2 + 7*(x+y)**3), x, 3) + assert p1 == p2 + + R, x = ring('x', QQ) + h = 25 + p = rs_exp(x, x, h) - 1 + p1 = rs_series_from_list(p, c, x, h) + p2 = 0 + for i, cx in enumerate(c): + p2 += cx*rs_pow(p, i, x, h) + assert p1 == p2 + + +def test_log(): + R, x = ring('x', QQ) + p = 1 + x + assert rs_log(p, x, 4) == x - x**2/2 + x**3/3 + p = 1 + x +2*x**2/3 + p1 = rs_log(p, x, 9) + assert p1 == -17*x**8/648 + 13*x**7/189 - 11*x**6/162 - x**5/45 + \ + 7*x**4/36 - x**3/3 + x**2/6 + x + p2 = rs_series_inversion(p, x, 9) + p3 = rs_log(p2, x, 9) + assert p3 == -p1 + + R, x, y = ring('x, y', QQ) + p = 1 + x + 2*y*x**2 + p1 = rs_log(p, x, 6) + assert p1 == (4*x**5*y**2 - 2*x**5*y - 2*x**4*y**2 + x**5/5 + 2*x**4*y - + x**4/4 - 2*x**3*y + x**3/3 + 2*x**2*y - x**2/2 + x) + + # Constant term in series + a = symbols('a') + R, x, y = ring('x, y', EX) + assert rs_log(x + a, x, 5) == -EX(1/(4*a**4))*x**4 + EX(1/(3*a**3))*x**3 \ + - EX(1/(2*a**2))*x**2 + EX(1/a)*x + EX(log(a)) + assert rs_log(x + x**2*y + a, x, 4) == -EX(a**(-2))*x**3*y + \ + EX(1/(3*a**3))*x**3 + EX(1/a)*x**2*y - EX(1/(2*a**2))*x**2 + \ + EX(1/a)*x + EX(log(a)) + + p = x + x**2 + 3 + assert rs_log(p, x, 10).compose(x, 5) == EX(log(3) + Rational(19281291595, 9920232)) + + +def test_exp(): + R, x = ring('x', QQ) + p = x + x**4 + for h in [10, 30]: + q = rs_series_inversion(1 + p, x, h) - 1 + p1 = rs_exp(q, x, h) + q1 = rs_log(p1, x, h) + assert q1 == q + p1 = rs_exp(p, x, 30) + assert p1.coeff(x**29) == QQ(74274246775059676726972369, 353670479749588078181744640000) + prec = 21 + p = rs_log(1 + x, x, prec) + p1 = rs_exp(p, x, prec) + assert p1 == x + 1 + + # Constant term in series + a = symbols('a') + R, x, y = ring('x, y', QQ[exp(a), a]) + assert rs_exp(x + a, x, 5) == exp(a)*x**4/24 + exp(a)*x**3/6 + \ + exp(a)*x**2/2 + exp(a)*x + exp(a) + assert rs_exp(x + x**2*y + a, x, 5) == exp(a)*x**4*y**2/2 + \ + exp(a)*x**4*y/2 + exp(a)*x**4/24 + exp(a)*x**3*y + \ + exp(a)*x**3/6 + exp(a)*x**2*y + exp(a)*x**2/2 + exp(a)*x + exp(a) + + R, x, y = ring('x, y', EX) + assert rs_exp(x + a, x, 5) == EX(exp(a)/24)*x**4 + EX(exp(a)/6)*x**3 + \ + EX(exp(a)/2)*x**2 + EX(exp(a))*x + EX(exp(a)) + assert rs_exp(x + x**2*y + a, x, 5) == EX(exp(a)/2)*x**4*y**2 + \ + EX(exp(a)/2)*x**4*y + EX(exp(a)/24)*x**4 + EX(exp(a))*x**3*y + \ + EX(exp(a)/6)*x**3 + EX(exp(a))*x**2*y + EX(exp(a)/2)*x**2 + \ + EX(exp(a))*x + EX(exp(a)) + + +def test_newton(): + R, x = ring('x', QQ) + p = x**2 - 2 + r = rs_newton(p, x, 4) + assert r == 8*x**4 + 4*x**2 + 2 + + +def test_compose_add(): + R, x = ring('x', QQ) + p1 = x**3 - 1 + p2 = x**2 - 2 + assert rs_compose_add(p1, p2) == x**6 - 6*x**4 - 2*x**3 + 12*x**2 - 12*x - 7 + + +def test_fun(): + R, x, y = ring('x, y', QQ) + p = x*y + x**2*y**3 + x**5*y + assert rs_fun(p, rs_tan, x, 10) == rs_tan(p, x, 10) + assert rs_fun(p, _tan1, x, 10) == _tan1(p, x, 10) + + +def test_nth_root(): + R, x, y = puiseux_ring('x, y', QQ) + assert rs_nth_root(1 + x**2*y, 4, x, 10) == -77*x**8*y**4/2048 + \ + 7*x**6*y**3/128 - 3*x**4*y**2/32 + x**2*y/4 + 1 + assert rs_nth_root(1 + x*y + x**2*y**3, 3, x, 5) == -x**4*y**6/9 + \ + 5*x**4*y**5/27 - 10*x**4*y**4/243 - 2*x**3*y**4/9 + 5*x**3*y**3/81 + \ + x**2*y**3/3 - x**2*y**2/9 + x*y/3 + 1 + assert rs_nth_root(8*x, 3, x, 3) == 2*x**QQ(1, 3) + assert rs_nth_root(8*x + x**2 + x**3, 3, x, 3) == x**QQ(4,3)/12 + 2*x**QQ(1,3) + r = rs_nth_root(8*x + x**2*y + x**3, 3, x, 4) + assert r == -x**QQ(7,3)*y**2/288 + x**QQ(7,3)/12 + x**QQ(4,3)*y/12 + 2*x**QQ(1,3) + + # Constant term in series + a = symbols('a') + R, x, y = puiseux_ring('x, y', EX) + assert rs_nth_root(x + EX(a), 3, x, 4) == EX(5/(81*a**QQ(8, 3)))*x**3 - \ + EX(1/(9*a**QQ(5, 3)))*x**2 + EX(1/(3*a**QQ(2, 3)))*x + EX(a**QQ(1, 3)) + assert rs_nth_root(x**QQ(2, 3) + x**2*y + 5, 2, x, 3) == -EX(sqrt(5)/100)*\ + x**QQ(8, 3)*y - EX(sqrt(5)/16000)*x**QQ(8, 3) + EX(sqrt(5)/10)*x**2*y + \ + EX(sqrt(5)/2000)*x**2 - EX(sqrt(5)/200)*x**QQ(4, 3) + \ + EX(sqrt(5)/10)*x**QQ(2, 3) + EX(sqrt(5)) + + +def test_atan(): + R, x, y = ring('x, y', QQ) + assert rs_atan(x, x, 9) == -x**7/7 + x**5/5 - x**3/3 + x + assert rs_atan(x*y + x**2*y**3, x, 9) == 2*x**8*y**11 - x**8*y**9 + \ + 2*x**7*y**9 - x**7*y**7/7 - x**6*y**9/3 + x**6*y**7 - x**5*y**7 + \ + x**5*y**5/5 - x**4*y**5 - x**3*y**3/3 + x**2*y**3 + x*y + + # Constant term in series + a = symbols('a') + R, x, y = ring('x, y', EX) + assert rs_atan(x + a, x, 5) == -EX((a**3 - a)/(a**8 + 4*a**6 + 6*a**4 + \ + 4*a**2 + 1))*x**4 + EX((3*a**2 - 1)/(3*a**6 + 9*a**4 + \ + 9*a**2 + 3))*x**3 - EX(a/(a**4 + 2*a**2 + 1))*x**2 + \ + EX(1/(a**2 + 1))*x + EX(atan(a)) + assert rs_atan(x + x**2*y + a, x, 4) == -EX(2*a/(a**4 + 2*a**2 + 1)) \ + *x**3*y + EX((3*a**2 - 1)/(3*a**6 + 9*a**4 + 9*a**2 + 3))*x**3 + \ + EX(1/(a**2 + 1))*x**2*y - EX(a/(a**4 + 2*a**2 + 1))*x**2 + EX(1/(a**2 \ + + 1))*x + EX(atan(a)) + + # Test for _atan faster for small and univariate series + R, x = ring('x', QQ) + p = x**2 + 2*x + assert _atan(p, x, 5) == rs_atan(p, x, 5) + + R, x = ring('x', EX) + p = x**2 + 2*x + assert _atan(p, x, 9) == rs_atan(p, x, 9) + + +def test_asin(): + R, x, y = ring('x, y', QQ) + assert rs_asin(x + x*y, x, 5) == x**3*y**3/6 + x**3*y**2/2 + x**3*y/2 + \ + x**3/6 + x*y + x + assert rs_asin(x*y + x**2*y**3, x, 6) == x**5*y**7/2 + 3*x**5*y**5/40 + \ + x**4*y**5/2 + x**3*y**3/6 + x**2*y**3 + x*y + + +def test_tan(): + R, x, y = ring('x, y', QQ) + assert rs_tan(x, x, 9) == x + x**3/3 + QQ(2,15)*x**5 + QQ(17,315)*x**7 + assert rs_tan(x*y + x**2*y**3, x, 9) == 4*x**8*y**11/3 + 17*x**8*y**9/45 + \ + 4*x**7*y**9/3 + 17*x**7*y**7/315 + x**6*y**9/3 + 2*x**6*y**7/3 + \ + x**5*y**7 + 2*x**5*y**5/15 + x**4*y**5 + x**3*y**3/3 + x**2*y**3 + x*y + + # Constant term in series + a = symbols('a') + R, x, y = ring('x, y', QQ[tan(a), a]) + assert rs_tan(x + a, x, 5) == (tan(a)**5 + 5*tan(a)**3/3 + + 2*tan(a)/3)*x**4 + (tan(a)**4 + 4*tan(a)**2/3 + Rational(1, 3))*x**3 + \ + (tan(a)**3 + tan(a))*x**2 + (tan(a)**2 + 1)*x + tan(a) + assert rs_tan(x + x**2*y + a, x, 4) == (2*tan(a)**3 + 2*tan(a))*x**3*y + \ + (tan(a)**4 + Rational(4, 3)*tan(a)**2 + Rational(1, 3))*x**3 + (tan(a)**2 + 1)*x**2*y + \ + (tan(a)**3 + tan(a))*x**2 + (tan(a)**2 + 1)*x + tan(a) + + R, x, y = ring('x, y', EX) + assert rs_tan(x + a, x, 5) == EX(tan(a)**5 + 5*tan(a)**3/3 + + 2*tan(a)/3)*x**4 + EX(tan(a)**4 + 4*tan(a)**2/3 + EX(1)/3)*x**3 + \ + EX(tan(a)**3 + tan(a))*x**2 + EX(tan(a)**2 + 1)*x + EX(tan(a)) + assert rs_tan(x + x**2*y + a, x, 4) == EX(2*tan(a)**3 + + 2*tan(a))*x**3*y + EX(tan(a)**4 + 4*tan(a)**2/3 + EX(1)/3)*x**3 + \ + EX(tan(a)**2 + 1)*x**2*y + EX(tan(a)**3 + tan(a))*x**2 + \ + EX(tan(a)**2 + 1)*x + EX(tan(a)) + + p = x + x**2 + 5 + assert rs_atan(p, x, 10).compose(x, 10) == EX(atan(5) + S(67701870330562640) / \ + 668083460499) + + +def test_cot(): + R, x, y = puiseux_ring('x, y', QQ) + assert rs_cot(x**6 + x**7, x, 8) == x**(-6) - x**(-5) + x**(-4) - \ + x**(-3) + x**(-2) - x**(-1) + 1 - x + x**2 - x**3 + x**4 - x**5 + \ + 2*x**6/3 - 4*x**7/3 + assert rs_cot(x + x**2*y, x, 5) == -x**4*y**5 - x**4*y/15 + x**3*y**4 - \ + x**3/45 - x**2*y**3 - x**2*y/3 + x*y**2 - x/3 - y + x**(-1) + + +def test_sin(): + R, x, y = ring('x, y', QQ) + assert rs_sin(x, x, 9) == x - x**3/6 + x**5/120 - x**7/5040 + assert rs_sin(x*y + x**2*y**3, x, 9) == x**8*y**11/12 - \ + x**8*y**9/720 + x**7*y**9/12 - x**7*y**7/5040 - x**6*y**9/6 + \ + x**6*y**7/24 - x**5*y**7/2 + x**5*y**5/120 - x**4*y**5/2 - \ + x**3*y**3/6 + x**2*y**3 + x*y + + # Constant term in series + a = symbols('a') + R, x, y = ring('x, y', QQ[sin(a), cos(a), a]) + assert rs_sin(x + a, x, 5) == sin(a)*x**4/24 - cos(a)*x**3/6 - \ + sin(a)*x**2/2 + cos(a)*x + sin(a) + assert rs_sin(x + x**2*y + a, x, 5) == -sin(a)*x**4*y**2/2 - \ + cos(a)*x**4*y/2 + sin(a)*x**4/24 - sin(a)*x**3*y - cos(a)*x**3/6 + \ + cos(a)*x**2*y - sin(a)*x**2/2 + cos(a)*x + sin(a) + + R, x, y = ring('x, y', EX) + assert rs_sin(x + a, x, 5) == EX(sin(a)/24)*x**4 - EX(cos(a)/6)*x**3 - \ + EX(sin(a)/2)*x**2 + EX(cos(a))*x + EX(sin(a)) + assert rs_sin(x + x**2*y + a, x, 5) == -EX(sin(a)/2)*x**4*y**2 - \ + EX(cos(a)/2)*x**4*y + EX(sin(a)/24)*x**4 - EX(sin(a))*x**3*y - \ + EX(cos(a)/6)*x**3 + EX(cos(a))*x**2*y - EX(sin(a)/2)*x**2 + \ + EX(cos(a))*x + EX(sin(a)) + + +def test_cos(): + R, x, y = ring('x, y', QQ) + assert rs_cos(x, x, 9) == 1 - x**2/2 + x**4/24 - x**6/720 + x**8/40320 + assert rs_cos(x*y + x**2*y**3, x, 9) == x**8*y**12/24 - \ + x**8*y**10/48 + x**8*y**8/40320 + x**7*y**10/6 - \ + x**7*y**8/120 + x**6*y**8/4 - x**6*y**6/720 + x**5*y**6/6 - \ + x**4*y**6/2 + x**4*y**4/24 - x**3*y**4 - x**2*y**2/2 + 1 + + # Constant term in series + a = symbols('a') + R, x, y = ring('x, y', QQ[sin(a), cos(a), a]) + assert rs_cos(x + a, x, 5) == cos(a)*x**4/24 + sin(a)*x**3/6 - \ + cos(a)*x**2/2 - sin(a)*x + cos(a) + assert rs_cos(x + x**2*y + a, x, 5) == -cos(a)*x**4*y**2/2 + \ + sin(a)*x**4*y/2 + cos(a)*x**4/24 - cos(a)*x**3*y + sin(a)*x**3/6 - \ + sin(a)*x**2*y - cos(a)*x**2/2 - sin(a)*x + cos(a) + + R, x, y = ring('x, y', EX) + assert rs_cos(x + a, x, 5) == EX(cos(a)/24)*x**4 + EX(sin(a)/6)*x**3 - \ + EX(cos(a)/2)*x**2 - EX(sin(a))*x + EX(cos(a)) + assert rs_cos(x + x**2*y + a, x, 5) == -EX(cos(a)/2)*x**4*y**2 + \ + EX(sin(a)/2)*x**4*y + EX(cos(a)/24)*x**4 - EX(cos(a))*x**3*y + \ + EX(sin(a)/6)*x**3 - EX(sin(a))*x**2*y - EX(cos(a)/2)*x**2 - \ + EX(sin(a))*x + EX(cos(a)) + + +def test_cos_sin(): + R, x, y = ring('x, y', QQ) + c, s = rs_cos_sin(x, x, 9) + assert c == rs_cos(x, x, 9) + assert s == rs_sin(x, x, 9) + c, s = rs_cos_sin(x + x*y, x, 5) + assert c == rs_cos(x + x*y, x, 5) + assert s == rs_sin(x + x*y, x, 5) + + # constant term in series + c, s = rs_cos_sin(1 + x + x**2, x, 5) + assert c == rs_cos(1 + x + x**2, x, 5) + assert s == rs_sin(1 + x + x**2, x, 5) + + a = symbols('a') + R, x, y = ring('x, y', QQ[sin(a), cos(a), a]) + c, s = rs_cos_sin(x + a, x, 5) + assert c == rs_cos(x + a, x, 5) + assert s == rs_sin(x + a, x, 5) + + R, x, y = ring('x, y', EX) + c, s = rs_cos_sin(x + a, x, 5) + assert c == rs_cos(x + a, x, 5) + assert s == rs_sin(x + a, x, 5) + + +def test_atanh(): + R, x, y = ring('x, y', QQ) + assert rs_atanh(x, x, 9) == x + x**3/3 + x**5/5 + x**7/7 + assert rs_atanh(x*y + x**2*y**3, x, 9) == 2*x**8*y**11 + x**8*y**9 + \ + 2*x**7*y**9 + x**7*y**7/7 + x**6*y**9/3 + x**6*y**7 + x**5*y**7 + \ + x**5*y**5/5 + x**4*y**5 + x**3*y**3/3 + x**2*y**3 + x*y + + # Constant term in series + a = symbols('a') + R, x, y = ring('x, y', EX) + assert rs_atanh(x + a, x, 5) == EX((a**3 + a)/(a**8 - 4*a**6 + 6*a**4 - \ + 4*a**2 + 1))*x**4 - EX((3*a**2 + 1)/(3*a**6 - 9*a**4 + \ + 9*a**2 - 3))*x**3 + EX(a/(a**4 - 2*a**2 + 1))*x**2 - EX(1/(a**2 - \ + 1))*x + EX(atanh(a)) + assert rs_atanh(x + x**2*y + a, x, 4) == EX(2*a/(a**4 - 2*a**2 + \ + 1))*x**3*y - EX((3*a**2 + 1)/(3*a**6 - 9*a**4 + 9*a**2 - 3))*x**3 - \ + EX(1/(a**2 - 1))*x**2*y + EX(a/(a**4 - 2*a**2 + 1))*x**2 - \ + EX(1/(a**2 - 1))*x + EX(atanh(a)) + + p = x + x**2 + 5 + assert rs_atanh(p, x, 10).compose(x, 10) == EX(Rational(-733442653682135, 5079158784) \ + + atanh(5)) + + # Test for _atanh faster for small and univariate series + R,x = ring('x', QQ) + p = x**2 + 2*x + assert _atanh(p, x, 5) == rs_atanh(p, x, 5) + + R,x = ring('x', EX) + p = x**2 + 2*x + assert _atanh(p, x, 9) == rs_atanh(p, x, 9) + + +def test_asinh(): + R, x, y = ring('x, y', QQ) + assert rs_asinh(x, x, 9) == -5/112*x**7 + 3/40*x**5 - 1/6*x**3 + x + assert rs_asinh(x*y + x**2*y**3, x, 9) == 3/4*x**8*y**11 - 5/16*x**8*y**9 + \ + 3/4*x**7*y**9 - 5/112*x**7*y**7 - 1/6*x**6*y**9 + 3/8*x**6*y**7 - 1/2*x \ + **5*y**7 + 3/40*x**5*y**5 - 1/2*x**4*y**5 - 1/6*x**3*y**3 + x**2*y**3 + x*y + + # Constant term in series + a = symbols('a') + R, x, y = ring('x, y', EX) + assert rs_asinh(x + a, x, 3) == -EX(a/(2*a**2*sqrt(a**2 + 1) + 2*sqrt(a**2 + 1))) \ + *x**2 + EX(1/sqrt(a**2 + 1))*x + EX(asinh(a)) + assert rs_asinh(x + x**2*y + a, x, 3) == EX(1/sqrt(a**2 + 1))*x**2*y - EX(a/(2*a**2 \ + *sqrt(a**2 + 1) + 2*sqrt(a**2 + 1)))*x**2 + EX(1/sqrt(a**2 + 1))*x + EX(asinh(a)) + + p = x + x ** 2 + 5 + assert rs_asinh(p, x, 10).compose(x, 10) == EX(asinh(5) + 4643789843094995*sqrt(26)/\ + 205564141692) + + +def test_sinh(): + R, x, y = ring('x, y', QQ) + assert rs_sinh(x, x, 9) == x + x**3/6 + x**5/120 + x**7/5040 + assert rs_sinh(x*y + x**2*y**3, x, 9) == x**8*y**11/12 + \ + x**8*y**9/720 + x**7*y**9/12 + x**7*y**7/5040 + x**6*y**9/6 + \ + x**6*y**7/24 + x**5*y**7/2 + x**5*y**5/120 + x**4*y**5/2 + \ + x**3*y**3/6 + x**2*y**3 + x*y + + # constant term in series + a = symbols('a') + R, x, y = ring('x, y', QQ[sinh(a), cosh(a), a]) + assert rs_sinh(x + a, x, 5) == 1/24*x**4*(sinh(a)) + 1/6*x**3*(cosh(a)) + 1/\ + 2*x**2*(sinh(a)) + x*(cosh(a)) + (sinh(a)) + assert rs_sinh(x + x**2*y + a, x, 5) == 1/2*(sinh(a))*x**4*y**2 + 1/2*(cosh(a))\ + *x**4*y + 1/24*(sinh(a))*x**4 + (sinh(a))*x**3*y + 1/6*(cosh(a))*x**3 + \ + (cosh(a))*x**2*y + 1/2*(sinh(a))*x**2 + (cosh(a))*x + (sinh(a)) + + R, x, y = ring('x, y', EX) + assert rs_sinh(x + a, x, 5) == EX(sinh(a)/24)*x**4 + EX(cosh(a)/6)*x**3 + \ + EX(sinh(a)/2)*x**2 + EX(cosh(a))*x + EX(sinh(a)) + assert rs_sinh(x + x**2*y + a, x, 5) == EX(sinh(a)/2)*x**4*y**2 + EX(cosh(a)/\ + 2)*x**4*y + EX(sinh(a)/24)*x**4 + EX(sinh(a))*x**3*y + EX(cosh(a)/6)*x**3 \ + + EX(cosh(a))*x**2*y + EX(sinh(a)/2)*x**2 + EX(cosh(a))*x + EX(sinh(a)) + + +def test_cosh(): + R, x, y = ring('x, y', QQ) + assert rs_cosh(x, x, 9) == 1 + x**2/2 + x**4/24 + x**6/720 + x**8/40320 + assert rs_cosh(x*y + x**2*y**3, x, 9) == x**8*y**12/24 + \ + x**8*y**10/48 + x**8*y**8/40320 + x**7*y**10/6 + \ + x**7*y**8/120 + x**6*y**8/4 + x**6*y**6/720 + x**5*y**6/6 + \ + x**4*y**6/2 + x**4*y**4/24 + x**3*y**4 + x**2*y**2/2 + 1 + + # constant term in series + a = symbols('a') + R, x, y = ring('x, y', QQ[sinh(a), cosh(a), a]) + assert rs_cosh(x + a, x, 5) == 1/24*(cosh(a))*x**4 + 1/6*(sinh(a))*x**3 + \ + 1/2*(cosh(a))*x**2 + (sinh(a))*x + (cosh(a)) + assert rs_cosh(x + x**2*y + a, x, 5) == 1/2*(cosh(a))*x**4*y**2 + 1/2*(sinh(a))\ + *x**4*y + 1/24*(cosh(a))*x**4 + (cosh(a))*x**3*y + 1/6*(sinh(a))*x**3 + \ + (sinh(a))*x**2*y + 1/2*(cosh(a))*x**2 + (sinh(a))*x + (cosh(a)) + R, x, y = ring('x, y', EX) + assert rs_cosh(x + a, x, 5) == EX(cosh(a)/24)*x**4 + EX(sinh(a)/6)*x**3 + \ + EX(cosh(a)/2)*x**2 + EX(sinh(a))*x + EX(cosh(a)) + assert rs_cosh(x + x**2*y + a, x, 5) == EX(cosh(a)/2)*x**4*y**2 + EX(sinh(a)/\ + 2)*x**4*y + EX(cosh(a)/24)*x**4 + EX(cosh(a))*x**3*y + EX(sinh(a)/6)*x**3 \ + + EX(sinh(a))*x**2*y + EX(cosh(a)/2)*x**2 + EX(sinh(a))*x + EX(cosh(a)) + + +def test_cosh_sinh(): + R, x, y = ring('x, y', QQ) + ch, sh = rs_cosh_sinh(x, x, 9) + assert ch == rs_cosh(x, x, 9) + assert sh == rs_sinh(x, x, 9) + ch, sh = rs_cosh_sinh(x + x*y, x, 5) + assert ch == rs_cosh(x + x*y, x, 5) + assert sh == rs_sinh(x + x*y, x, 5) + + # constant term in series + c, s = rs_cosh_sinh(1 + x + x**2, x, 5) + assert c == rs_cosh(1 + x + x**2, x, 5) + assert s == rs_sinh(1 + x + x**2, x, 5) + + a = symbols('a') + R, x, y = ring('x, y', QQ[sinh(a), cosh(a), a]) + ch, sh = rs_cosh_sinh(x + a, x, 5) + assert ch == rs_cosh(x + a, x, 5) + assert sh == rs_sinh(x + a, x, 5) + R, x, y = ring('x, y', EX) + ch, sh = rs_cosh_sinh(x + a, x, 5) + assert ch == rs_cosh(x + a, x, 5) + assert sh == rs_sinh(x + a, x, 5) + + +def test_tanh(): + R, x, y = ring('x, y', QQ) + assert rs_tanh(x, x, 9) == x - QQ(1,3)*x**3 + QQ(2,15)*x**5 - QQ(17,315)*x**7 + assert rs_tanh(x*y + x**2*y**3, x, 9) == 4*x**8*y**11/3 - \ + 17*x**8*y**9/45 + 4*x**7*y**9/3 - 17*x**7*y**7/315 - x**6*y**9/3 + \ + 2*x**6*y**7/3 - x**5*y**7 + 2*x**5*y**5/15 - x**4*y**5 - \ + x**3*y**3/3 + x**2*y**3 + x*y + + # Constant term in series + a = symbols('a') + R, x, y = ring('x, y', EX) + assert rs_tanh(x + a, x, 5) == EX(tanh(a)**5 - 5*tanh(a)**3/3 + + 2*tanh(a)/3)*x**4 + EX(-tanh(a)**4 + 4*tanh(a)**2/3 - QQ(1, 3))*x**3 + \ + EX(tanh(a)**3 - tanh(a))*x**2 + EX(-tanh(a)**2 + 1)*x + EX(tanh(a)) + + p = rs_tanh(x + x**2*y + a, x, 4) + assert (p.compose(x, 10)).compose(y, 5) == EX(-1000*tanh(a)**4 + \ + 10100*tanh(a)**3 + 2470*tanh(a)**2/3 - 10099*tanh(a) + QQ(530, 3)) + + +def test_RR(): + rs_funcs = [rs_sin, rs_cos, rs_tan, rs_cot, rs_atan, rs_tanh] + sympy_funcs = [sin, cos, tan, cot, atan, tanh] + R, x, y = ring('x, y', RR) + a = symbols('a') + for rs_func, sympy_func in zip(rs_funcs, sympy_funcs): + p = rs_func(2 + x, x, 5).compose(x, 5) + q = sympy_func(2 + a).series(a, 0, 5).removeO() + is_close(p.as_expr(), q.subs(a, 5).n()) + + p = rs_nth_root(2 + x, 5, x, 5).compose(x, 5) + q = ((2 + a)**QQ(1, 5)).series(a, 0, 5).removeO() + is_close(p.as_expr(), q.subs(a, 5).n()) + + +def test_is_regular(): + R, x, y = puiseux_ring('x, y', QQ) + p = 1 + 2*x + x**2 + 3*x**3 + assert not rs_is_puiseux(p, x) + + p = x + x**QQ(1,5)*y + assert rs_is_puiseux(p, x) + assert not rs_is_puiseux(p, y) + + p = x + x**2*y**QQ(1,5)*y + assert not rs_is_puiseux(p, x) + + +def test_puiseux(): + R, x, y = puiseux_ring('x, y', QQ) + p = x**QQ(2,5) + x**QQ(2,3) + x + + r = rs_series_inversion(p, x, 1) + r1 = -x**QQ(14,15) + x**QQ(4,5) - 3*x**QQ(11,15) + x**QQ(2,3) + \ + 2*x**QQ(7,15) - x**QQ(2,5) - x**QQ(1,5) + x**QQ(2,15) - x**QQ(-2,15) \ + + x**QQ(-2,5) + assert r == r1 + + r = rs_nth_root(1 + p, 3, x, 1) + assert r == -x**QQ(4,5)/9 + x**QQ(2,3)/3 + x**QQ(2,5)/3 + 1 + + r = rs_log(1 + p, x, 1) + assert r == -x**QQ(4,5)/2 + x**QQ(2,3) + x**QQ(2,5) + + r = rs_LambertW(p, x, 1) + assert r == -x**QQ(4,5) + x**QQ(2,3) + x**QQ(2,5) + + p1 = x + x**QQ(1,5)*y + r = rs_exp(p1, x, 1) + assert r == x**QQ(4,5)*y**4/24 + x**QQ(3,5)*y**3/6 + x**QQ(2,5)*y**2/2 + \ + x**QQ(1,5)*y + 1 + + r = rs_atan(p, x, 2) + assert r == -x**QQ(9,5) - x**QQ(26,15) - x**QQ(22,15) - x**QQ(6,5)/3 + \ + x + x**QQ(2,3) + x**QQ(2,5) + + r = rs_atan(p1, x, 2) + assert r == x**QQ(9,5)*y**9/9 + x**QQ(9,5)*y**4 - x**QQ(7,5)*y**7/7 - \ + x**QQ(7,5)*y**2 + x*y**5/5 + x - x**QQ(3,5)*y**3/3 + x**QQ(1,5)*y + + r = rs_tan(p, x, 2) + assert r == x**QQ(2,5) + x**QQ(2,3) + x + QQ(1,3)*x**QQ(6,5) + x**QQ(22,15)\ + + x**QQ(26,15) + x**QQ(9,5) + + r = rs_sin(p, x, 2) + assert r == x**QQ(2,5) + x**QQ(2,3) + x - QQ(1,6)*x**QQ(6,5) - QQ(1,2)*x**\ + QQ(22,15) - QQ(1,2)*x**QQ(26,15) - QQ(1,2)*x**QQ(9,5) + + r = rs_cos(p, x, 2) + assert r == 1 - QQ(1,2)*x**QQ(4,5) - x**QQ(16,15) - QQ(1,2)*x**QQ(4,3) - \ + x**QQ(7,5) + QQ(1,24)*x**QQ(8,5) - x**QQ(5,3) + QQ(1,6)*x**QQ(28,15) + + r = rs_asin(p, x, 2) + assert r == x**QQ(9,5)/2 + x**QQ(26,15)/2 + x**QQ(22,15)/2 + \ + x**QQ(6,5)/6 + x + x**QQ(2,3) + x**QQ(2,5) + + r = rs_cot(p, x, 1) + assert r == -x**QQ(14,15) + x**QQ(4,5) - 3*x**QQ(11,15) + \ + 2*x**QQ(2,3)/3 + 2*x**QQ(7,15) - 4*x**QQ(2,5)/3 - x**QQ(1,5) + \ + x**QQ(2,15) - x**QQ(-2,15) + x**QQ(-2,5) + + r = rs_cos_sin(p, x, 2) + assert r[0] == x**QQ(28,15)/6 - x**QQ(5,3) + x**QQ(8,5)/24 - x**QQ(7,5) - \ + x**QQ(4,3)/2 - x**QQ(16,15) - x**QQ(4,5)/2 + 1 + assert r[1] == -x**QQ(9,5)/2 - x**QQ(26,15)/2 - x**QQ(22,15)/2 - \ + x**QQ(6,5)/6 + x + x**QQ(2,3) + x**QQ(2,5) + + r = rs_atanh(p, x, 2) + assert r == x**QQ(9,5) + x**QQ(26,15) + x**QQ(22,15) + x**QQ(6,5)/3 + x + \ + x**QQ(2,3) + x**QQ(2,5) + + r = rs_asinh(p, x, 2) + assert r == x**QQ(2,5) + x**QQ(2,3) + x - QQ(1,6)*x**QQ(6,5) - QQ(1,2)*x**\ + QQ(22,15) - QQ(1,2)*x**QQ(26,15) - QQ(1,2)*x**QQ(9,5) + + r = rs_cosh(p, x, 2) + assert r == x**QQ(28,15)/6 + x**QQ(5,3) + x**QQ(8,5)/24 + x**QQ(7,5) + \ + x**QQ(4,3)/2 + x**QQ(16,15) + x**QQ(4,5)/2 + 1 + + r = rs_sinh(p, x, 2) + assert r == x**QQ(9,5)/2 + x**QQ(26,15)/2 + x**QQ(22,15)/2 + \ + x**QQ(6,5)/6 + x + x**QQ(2,3) + x**QQ(2,5) + + r = rs_cosh_sinh(p, x, 2) + assert r[0] == x**QQ(28,15)/6 + x**QQ(5,3) + x**QQ(8,5)/24 + x**QQ(7,5) + \ + x**QQ(4,3)/2 + x**QQ(16,15) + x**QQ(4,5)/2 + 1 + assert r[1] == x**QQ(9,5)/2 + x**QQ(26,15)/2 + x**QQ(22,15)/2 + \ + x**QQ(6,5)/6 + x + x**QQ(2,3) + x**QQ(2,5) + + r = rs_tanh(p, x, 2) + assert r == -x**QQ(9,5) - x**QQ(26,15) - x**QQ(22,15) - x**QQ(6,5)/3 + \ + x + x**QQ(2,3) + x**QQ(2,5) + + +def test_puiseux_algebraic(): # https://github.com/sympy/sympy/issues/24395 + + K = QQ.algebraic_field(sqrt(2)) + sqrt2 = K.from_sympy(sqrt(2)) + x, y = symbols('x, y') + R, xr, yr = puiseux_ring([x, y], K) + p = (1+sqrt2)*xr**QQ(1,2) + (1-sqrt2)*yr**QQ(2,3) + + assert p.to_dict() == {(QQ(1,2),QQ(0)):1+sqrt2, (QQ(0),QQ(2,3)):1-sqrt2} + assert p.as_expr() == (1 + sqrt(2))*x**(S(1)/2) + (1 - sqrt(2))*y**(S(2)/3) + + +def test1(): + R, x = puiseux_ring('x', QQ) + r = rs_sin(x, x, 15)*x**(-5) + assert r == x**8/6227020800 - x**6/39916800 + x**4/362880 - x**2/5040 + \ + QQ(1,120) - x**-2/6 + x**-4 + + p = rs_sin(x, x, 10) + r = rs_nth_root(p, 2, x, 10) + assert r == -67*x**QQ(17,2)/29030400 - x**QQ(13,2)/24192 + \ + x**QQ(9,2)/1440 - x**QQ(5,2)/12 + x**QQ(1,2) + + p = rs_sin(x, x, 10) + r = rs_nth_root(p, 7, x, 10) + r = rs_pow(r, 5, x, 10) + assert r == -97*x**QQ(61,7)/124467840 - x**QQ(47,7)/16464 + \ + 11*x**QQ(33,7)/3528 - 5*x**QQ(19,7)/42 + x**QQ(5,7) + + r = rs_exp(x**QQ(1,2), x, 10) + assert r == x**QQ(19,2)/121645100408832000 + x**9/6402373705728000 + \ + x**QQ(17,2)/355687428096000 + x**8/20922789888000 + \ + x**QQ(15,2)/1307674368000 + x**7/87178291200 + \ + x**QQ(13,2)/6227020800 + x**6/479001600 + x**QQ(11,2)/39916800 + \ + x**5/3628800 + x**QQ(9,2)/362880 + x**4/40320 + x**QQ(7,2)/5040 + \ + x**3/720 + x**QQ(5,2)/120 + x**2/24 + x**QQ(3,2)/6 + x/2 + \ + x**QQ(1,2) + 1 + + +def test_puiseux2(): + R, y = ring('y', QQ) + S, x = puiseux_ring('x', R.to_domain()) + + p = x + x**QQ(1,5)*y + r = rs_atan(p, x, 3) + assert r == (y**13/13 + y**8 + 2*y**3)*x**QQ(13,5) - (y**11/11 + y**6 + + y)*x**QQ(11,5) + (y**9/9 + y**4)*x**QQ(9,5) - (y**7/7 + + y**2)*x**QQ(7,5) + (y**5/5 + 1)*x - y**3*x**QQ(3,5)/3 + y*x**QQ(1,5) + + +@slow +def test_rs_series(): + x, a, b, c = symbols('x, a, b, c') + + assert rs_series(a, a, 5).as_expr() == a + assert rs_series(sin(a), a, 5).as_expr() == (sin(a).series(a, 0, + 5)).removeO() + assert rs_series(sin(a) + cos(a), a, 5).as_expr() == ((sin(a) + + cos(a)).series(a, 0, 5)).removeO() + assert rs_series(sin(a)*cos(a), a, 5).as_expr() == ((sin(a)* + cos(a)).series(a, 0, 5)).removeO() + + p = (sin(a) - a)*(cos(a**2) + a**4/2) + assert expand(rs_series(p, a, 10).as_expr()) == expand(p.series(a, 0, + 10).removeO()) + + p = sin(a**2/2 + a/3) + cos(a/5)*sin(a/2)**3 + assert expand(rs_series(p, a, 5).as_expr()) == expand(p.series(a, 0, + 5).removeO()) + + p = sin(x**2 + a)*(cos(x**3 - 1) - a - a**2) + assert expand(rs_series(p, a, 5).as_expr()) == expand(p.series(a, 0, + 5).removeO()) + + p = sin(a**2 - a/3 + 2)**5*exp(a**3 - a/2) + assert expand(rs_series(p, a, 10).as_expr()) == expand(p.series(a, 0, + 10).removeO()) + + p = sin(a + b + c) + assert expand(rs_series(p, a, 5).as_expr()) == expand(p.series(a, 0, + 5).removeO()) + + p = tan(sin(a**2 + 4) + b + c) + assert expand(rs_series(p, a, 6).as_expr()) == expand(p.series(a, 0, + 6).removeO()) + + p = a**QQ(2,5) + a**QQ(2,3) + a + + r = rs_series(tan(p), a, 2) + assert r.as_expr() == a**QQ(9,5) + a**QQ(26,15) + a**QQ(22,15) + a**QQ(6,5)/3 + \ + a + a**QQ(2,3) + a**QQ(2,5) + + r = rs_series(exp(p), a, 1) + assert r.as_expr() == a**QQ(4,5)/2 + a**QQ(2,3) + a**QQ(2,5) + 1 + + r = rs_series(sin(p), a, 2) + assert r.as_expr() == -a**QQ(9,5)/2 - a**QQ(26,15)/2 - a**QQ(22,15)/2 - \ + a**QQ(6,5)/6 + a + a**QQ(2,3) + a**QQ(2,5) + + r = rs_series(cos(p), a, 2) + assert r.as_expr() == a**QQ(28,15)/6 - a**QQ(5,3) + a**QQ(8,5)/24 - a**QQ(7,5) - \ + a**QQ(4,3)/2 - a**QQ(16,15) - a**QQ(4,5)/2 + 1 + + assert rs_series(sin(a)/7, a, 5).as_expr() == (sin(a)/7).series(a, 0, + 5).removeO() + + +def test_rs_series_ConstantInExpr(): + x, a = symbols('x a') + assert rs_series(log(1 + x), x, 5).as_expr() == -x**4/4 + x**3/3 - \ + x**2/2 + x + assert rs_series(log(1 + 4*x), x, 5).as_expr() == -64*x**4 + 64*x**3/3 - \ + 8*x**2 + 4*x + assert rs_series(log(1 + x + x**2), x, 10).as_expr() == -2*x**9/9 + \ + x**8/8 + x**7/7 - x**6/3 + x**5/5 + x**4/4 - 2*x**3/3 + x**2/2 + x + assert rs_series(log(1 + x*a**2), x, 7).as_expr() == -x**6*a**12/6 + \ + x**5*a**10/5 - x**4*a**8/4 + x**3*a**6/3 - x**2*a**4/2 + x*a**2 + + assert rs_series(atan(1 + x), x, 9).as_expr() == -x**7/112 + x**6/48 - x**5/40 \ + + x**3/12 - x**2/4 + x/2 + pi/4 + assert rs_series(atan(1 + x + x**2),x, 9).as_expr() == -15*x**7/112 - x**6/48 + \ + 9*x**5/40 - 5*x**3/12 + x**2/4 + x/2 + pi/4 + assert rs_series(atan(1 + x * a), x, 9).as_expr() == -a**7*x**7/112 + a**6*x**6/48 \ + - a**5*x**5/40 + a**3*x**3/12 - a**2*x**2/4 + a*x/2 + pi/4 + + assert rs_series(tanh(1 + x), x, 5).as_expr() == -5*x**4*tanh(1)**3/3 + x**4* \ + tanh(1)**5 + 2*x**4*tanh(1)/3 - x**3*tanh(1)**4 - x**3/3 + 4*x**3*tanh(1) \ + **2/3 - x**2*tanh(1) + x**2*tanh(1)**3 - x*tanh(1)**2 + x + tanh(1) + assert rs_series(tanh(1 + x * a), x, 3).as_expr() == -a**2*x**2*tanh(1) + a**2*x** \ + 2*tanh(1)**3 - a*x*tanh(1)**2 + a*x + tanh(1) + + assert rs_series(sinh(1 + x), x, 5).as_expr() == x**4*sinh(1)/24 + x**3*cosh(1)/6 + \ + x**2*sinh(1)/2 + x*cosh(1) + sinh(1) + assert rs_series(sinh(1 + x * a), x, 5).as_expr() == a**4*x**4*sinh(1)/24 + \ + a**3*x**3*cosh(1)/6 + a**2*x**2*sinh(1)/2 + a*x*cosh(1) + sinh(1) + + assert rs_series(cosh(1 + x), x, 5).as_expr() == x**4*cosh(1)/24 + x**3*sinh(1)/6 + \ + x**2*cosh(1)/2 + x*sinh(1) + cosh(1) + assert rs_series(cosh(1 + x * a), x, 5).as_expr() == a**4*x**4*cosh(1)/24 + \ + a**3*x**3*sinh(1)/6 + a**2*x**2*cosh(1)/2 + a*x*sinh(1) + cosh(1) + + +def test_issue(): + # https://github.com/sympy/sympy/issues/10191 + # https://github.com/sympy/sympy/issues/19543 + + a, b = symbols('a b') + assert rs_series(sin(a**QQ(3,7))*exp(a + b**QQ(6,7)), a,2).as_expr() == \ + a**QQ(10,7)*exp(b**QQ(6,7)) - a**QQ(9,7)*exp(b**QQ(6,7))/6 + a**QQ(3,7)*exp(b**QQ(6,7)) diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_rings.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_rings.py new file mode 100644 index 0000000000000000000000000000000000000000..455cc319908d0173737531b339e22def8e4a26fc --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_rings.py @@ -0,0 +1,1591 @@ +"""Test sparse polynomials. """ + +from functools import reduce +from operator import add, mul + +from sympy.polys.rings import ring, xring, sring, PolyRing, PolyElement +from sympy.polys.fields import field, FracField +from sympy.polys.densebasic import ninf +from sympy.polys.domains import ZZ, QQ, RR, FF, EX +from sympy.polys.orderings import lex, grlex +from sympy.polys.polyerrors import GeneratorsError, \ + ExactQuotientFailed, MultivariatePolynomialError, CoercionFailed + +from sympy.testing.pytest import raises +from sympy.core import Symbol, symbols +from sympy.core.singleton import S +from sympy.core.numbers import pi +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.miscellaneous import sqrt + +def test_PolyRing___init__(): + x, y, z, t = map(Symbol, "xyzt") + + assert len(PolyRing("x,y,z", ZZ, lex).gens) == 3 + assert len(PolyRing(x, ZZ, lex).gens) == 1 + assert len(PolyRing(("x", "y", "z"), ZZ, lex).gens) == 3 + assert len(PolyRing((x, y, z), ZZ, lex).gens) == 3 + assert len(PolyRing("", ZZ, lex).gens) == 0 + assert len(PolyRing([], ZZ, lex).gens) == 0 + + raises(GeneratorsError, lambda: PolyRing(0, ZZ, lex)) + + assert PolyRing("x", ZZ[t], lex).domain == ZZ[t] + assert PolyRing("x", 'ZZ[t]', lex).domain == ZZ[t] + assert PolyRing("x", PolyRing("t", ZZ, lex), lex).domain == ZZ[t] + + raises(GeneratorsError, lambda: PolyRing("x", PolyRing("x", ZZ, lex), lex)) + + _lex = Symbol("lex") + assert PolyRing("x", ZZ, lex).order == lex + assert PolyRing("x", ZZ, _lex).order == lex + assert PolyRing("x", ZZ, 'lex').order == lex + + R1 = PolyRing("x,y", ZZ, lex) + R2 = PolyRing("x,y", ZZ, lex) + R3 = PolyRing("x,y,z", ZZ, lex) + + assert R1.x == R1.gens[0] + assert R1.y == R1.gens[1] + assert R1.x == R2.x + assert R1.y == R2.y + assert R1.x != R3.x + assert R1.y != R3.y + +def test_PolyRing___hash__(): + R, x, y, z = ring("x,y,z", QQ) + assert hash(R) + +def test_PolyRing___eq__(): + assert ring("x,y,z", QQ)[0] == ring("x,y,z", QQ)[0] + assert ring("x,y,z", QQ)[0] != ring("x,y,z", ZZ)[0] + assert ring("x,y,z", ZZ)[0] != ring("x,y,z", QQ)[0] + assert ring("x,y,z", QQ)[0] != ring("x,y", QQ)[0] + assert ring("x,y", QQ)[0] != ring("x,y,z", QQ)[0] + +def test_PolyRing_ring_new(): + R, x, y, z = ring("x,y,z", QQ) + + assert R.ring_new(7) == R(7) + assert R.ring_new(7*x*y*z) == 7*x*y*z + + f = x**2 + 2*x*y + 3*x + 4*z**2 + 5*z + 6 + + assert R.ring_new([[[1]], [[2], [3]], [[4, 5, 6]]]) == f + assert R.ring_new({(2, 0, 0): 1, (1, 1, 0): 2, (1, 0, 0): 3, (0, 0, 2): 4, (0, 0, 1): 5, (0, 0, 0): 6}) == f + assert R.ring_new([((2, 0, 0), 1), ((1, 1, 0), 2), ((1, 0, 0), 3), ((0, 0, 2), 4), ((0, 0, 1), 5), ((0, 0, 0), 6)]) == f + + R, = ring("", QQ) + assert R.ring_new([((), 7)]) == R(7) + +def test_PolyRing_drop(): + R, x,y,z = ring("x,y,z", ZZ) + + assert R.drop(x) == PolyRing("y,z", ZZ, lex) + assert R.drop(y) == PolyRing("x,z", ZZ, lex) + assert R.drop(z) == PolyRing("x,y", ZZ, lex) + + assert R.drop(0) == PolyRing("y,z", ZZ, lex) + assert R.drop(0).drop(0) == PolyRing("z", ZZ, lex) + assert R.drop(0).drop(0).drop(0) == ZZ + + assert R.drop(1) == PolyRing("x,z", ZZ, lex) + + assert R.drop(2) == PolyRing("x,y", ZZ, lex) + assert R.drop(2).drop(1) == PolyRing("x", ZZ, lex) + assert R.drop(2).drop(1).drop(0) == ZZ + + raises(ValueError, lambda: R.drop(3)) + raises(ValueError, lambda: R.drop(x).drop(y)) + +def test_PolyRing___getitem__(): + R, x,y,z = ring("x,y,z", ZZ) + + assert R[0:] == PolyRing("x,y,z", ZZ, lex) + assert R[1:] == PolyRing("y,z", ZZ, lex) + assert R[2:] == PolyRing("z", ZZ, lex) + assert R[3:] == ZZ + +def test_PolyRing_is_(): + R = PolyRing("x", QQ, lex) + + assert R.is_univariate is True + assert R.is_multivariate is False + + R = PolyRing("x,y,z", QQ, lex) + + assert R.is_univariate is False + assert R.is_multivariate is True + + R = PolyRing("", QQ, lex) + + assert R.is_univariate is False + assert R.is_multivariate is False + +def test_PolyRing_add(): + R, x = ring("x", ZZ) + F = [ x**2 + 2*i + 3 for i in range(4) ] + + assert R.add(F) == reduce(add, F) == 4*x**2 + 24 + + R, = ring("", ZZ) + + assert R.add([2, 5, 7]) == 14 + +def test_PolyRing_mul(): + R, x = ring("x", ZZ) + F = [ x**2 + 2*i + 3 for i in range(4) ] + + assert R.mul(F) == reduce(mul, F) == x**8 + 24*x**6 + 206*x**4 + 744*x**2 + 945 + + R, = ring("", ZZ) + + assert R.mul([2, 3, 5]) == 30 + +def test_PolyRing_symmetric_poly(): + R, x, y, z, t = ring("x,y,z,t", ZZ) + + raises(ValueError, lambda: R.symmetric_poly(-1)) + raises(ValueError, lambda: R.symmetric_poly(5)) + + assert R.symmetric_poly(0) == R.one + assert R.symmetric_poly(1) == x + y + z + t + assert R.symmetric_poly(2) == x*y + x*z + x*t + y*z + y*t + z*t + assert R.symmetric_poly(3) == x*y*z + x*y*t + x*z*t + y*z*t + assert R.symmetric_poly(4) == x*y*z*t + +def test_sring(): + x, y, z, t = symbols("x,y,z,t") + + R = PolyRing("x,y,z", ZZ, lex) + assert sring(x + 2*y + 3*z) == (R, R.x + 2*R.y + 3*R.z) + + R = PolyRing("x,y,z", QQ, lex) + assert sring(x + 2*y + z/3) == (R, R.x + 2*R.y + R.z/3) + assert sring([x, 2*y, z/3]) == (R, [R.x, 2*R.y, R.z/3]) + + Rt = PolyRing("t", ZZ, lex) + R = PolyRing("x,y,z", Rt, lex) + assert sring(x + 2*t*y + 3*t**2*z, x, y, z) == (R, R.x + 2*Rt.t*R.y + 3*Rt.t**2*R.z) + + Rt = PolyRing("t", QQ, lex) + R = PolyRing("x,y,z", Rt, lex) + assert sring(x + t*y/2 + t**2*z/3, x, y, z) == (R, R.x + Rt.t*R.y/2 + Rt.t**2*R.z/3) + + Rt = FracField("t", ZZ, lex) + R = PolyRing("x,y,z", Rt, lex) + assert sring(x + 2*y/t + t**2*z/3, x, y, z) == (R, R.x + 2*R.y/Rt.t + Rt.t**2*R.z/3) + + r = sqrt(2) - sqrt(3) + R, a = sring(r, extension=True) + assert R.domain == QQ.algebraic_field(sqrt(2) + sqrt(3)) + assert R.gens == () + assert a == R.domain.from_sympy(r) + +def test_PolyElement___hash__(): + R, x, y, z = ring("x,y,z", QQ) + assert hash(x*y*z) + +def test_PolyElement___eq__(): + R, x, y = ring("x,y", ZZ, lex) + + assert ((x*y + 5*x*y) == 6) == False + assert ((x*y + 5*x*y) == 6*x*y) == True + assert (6 == (x*y + 5*x*y)) == False + assert (6*x*y == (x*y + 5*x*y)) == True + + assert ((x*y - x*y) == 0) == True + assert (0 == (x*y - x*y)) == True + + assert ((x*y - x*y) == 1) == False + assert (1 == (x*y - x*y)) == False + + assert ((x*y - x*y) == 1) == False + assert (1 == (x*y - x*y)) == False + + assert ((x*y + 5*x*y) != 6) == True + assert ((x*y + 5*x*y) != 6*x*y) == False + assert (6 != (x*y + 5*x*y)) == True + assert (6*x*y != (x*y + 5*x*y)) == False + + assert ((x*y - x*y) != 0) == False + assert (0 != (x*y - x*y)) == False + + assert ((x*y - x*y) != 1) == True + assert (1 != (x*y - x*y)) == True + + assert R.one == QQ(1, 1) == R.one + assert R.one == 1 == R.one + + Rt, t = ring("t", ZZ) + R, x, y = ring("x,y", Rt) + + assert (t**3*x/x == t**3) == True + assert (t**3*x/x == t**4) == False + +def test_PolyElement__lt_le_gt_ge__(): + R, x, y = ring("x,y", ZZ) + + assert R(1) < x < x**2 < x**3 + assert R(1) <= x <= x**2 <= x**3 + + assert x**3 > x**2 > x > R(1) + assert x**3 >= x**2 >= x >= R(1) + +def test_PolyElement__str__(): + x, y = symbols('x, y') + + for dom in [ZZ, QQ, ZZ[x], ZZ[x,y], ZZ[x][y]]: + R, t = ring('t', dom) + assert str(2*t**2 + 1) == '2*t**2 + 1' + + for dom in [EX, EX[x]]: + R, t = ring('t', dom) + assert str(2*t**2 + 1) == 'EX(2)*t**2 + EX(1)' + +def test_PolyElement_copy(): + R, x, y, z = ring("x,y,z", ZZ) + + f = x*y + 3*z + g = f.copy() + + assert f == g + g[(1, 1, 1)] = 7 + assert f != g + +def test_PolyElement_as_expr(): + R, x, y, z = ring("x,y,z", ZZ) + f = 3*x**2*y - x*y*z + 7*z**3 + 1 + + X, Y, Z = R.symbols + g = 3*X**2*Y - X*Y*Z + 7*Z**3 + 1 + + assert f != g + assert f.as_expr() == g + + U, V, W = symbols("u,v,w") + g = 3*U**2*V - U*V*W + 7*W**3 + 1 + + assert f != g + assert f.as_expr(U, V, W) == g + + raises(ValueError, lambda: f.as_expr(X)) + + R, = ring("", ZZ) + assert R(3).as_expr() == 3 + +def test_PolyElement_from_expr(): + x, y, z = symbols("x,y,z") + R, X, Y, Z = ring((x, y, z), ZZ) + + f = R.from_expr(1) + assert f == 1 and R.is_element(f) + + f = R.from_expr(x) + assert f == X and R.is_element(f) + + f = R.from_expr(x*y*z) + assert f == X*Y*Z and R.is_element(f) + + f = R.from_expr(x*y*z + x*y + x) + assert f == X*Y*Z + X*Y + X and R.is_element(f) + + f = R.from_expr(x**3*y*z + x**2*y**7 + 1) + assert f == X**3*Y*Z + X**2*Y**7 + 1 and R.is_element(f) + + r, F = sring([exp(2)]) + f = r.from_expr(exp(2)) + assert f == F[0] and r.is_element(f) + + raises(ValueError, lambda: R.from_expr(1/x)) + raises(ValueError, lambda: R.from_expr(2**x)) + raises(ValueError, lambda: R.from_expr(7*x + sqrt(2))) + + R, = ring("", ZZ) + f = R.from_expr(1) + assert f == 1 and R.is_element(f) + +def test_PolyElement_degree(): + R, x,y,z = ring("x,y,z", ZZ) + + assert ninf == float('-inf') + + assert R(0).degree() is ninf + assert R(1).degree() == 0 + assert (x + 1).degree() == 1 + assert (2*y**3 + z).degree() == 0 + assert (x*y**3 + z).degree() == 1 + assert (x**5*y**3 + z).degree() == 5 + + assert R(0).degree(x) is ninf + assert R(1).degree(x) == 0 + assert (x + 1).degree(x) == 1 + assert (2*y**3 + z).degree(x) == 0 + assert (x*y**3 + z).degree(x) == 1 + assert (7*x**5*y**3 + z).degree(x) == 5 + + assert R(0).degree(y) is ninf + assert R(1).degree(y) == 0 + assert (x + 1).degree(y) == 0 + assert (2*y**3 + z).degree(y) == 3 + assert (x*y**3 + z).degree(y) == 3 + assert (7*x**5*y**3 + z).degree(y) == 3 + + assert R(0).degree(z) is ninf + assert R(1).degree(z) == 0 + assert (x + 1).degree(z) == 0 + assert (2*y**3 + z).degree(z) == 1 + assert (x*y**3 + z).degree(z) == 1 + assert (7*x**5*y**3 + z).degree(z) == 1 + + R, = ring("", ZZ) + assert R(0).degree() is ninf + assert R(1).degree() == 0 + +def test_PolyElement_tail_degree(): + R, x,y,z = ring("x,y,z", ZZ) + + assert R(0).tail_degree() is ninf + assert R(1).tail_degree() == 0 + assert (x + 1).tail_degree() == 0 + assert (2*y**3 + x**3*z).tail_degree() == 0 + assert (x*y**3 + x**3*z).tail_degree() == 1 + assert (x**5*y**3 + x**3*z).tail_degree() == 3 + + assert R(0).tail_degree(x) is ninf + assert R(1).tail_degree(x) == 0 + assert (x + 1).tail_degree(x) == 0 + assert (2*y**3 + x**3*z).tail_degree(x) == 0 + assert (x*y**3 + x**3*z).tail_degree(x) == 1 + assert (7*x**5*y**3 + x**3*z).tail_degree(x) == 3 + + assert R(0).tail_degree(y) is ninf + assert R(1).tail_degree(y) == 0 + assert (x + 1).tail_degree(y) == 0 + assert (2*y**3 + x**3*z).tail_degree(y) == 0 + assert (x*y**3 + x**3*z).tail_degree(y) == 0 + assert (7*x**5*y**3 + x**3*z).tail_degree(y) == 0 + + assert R(0).tail_degree(z) is ninf + assert R(1).tail_degree(z) == 0 + assert (x + 1).tail_degree(z) == 0 + assert (2*y**3 + x**3*z).tail_degree(z) == 0 + assert (x*y**3 + x**3*z).tail_degree(z) == 0 + assert (7*x**5*y**3 + x**3*z).tail_degree(z) == 0 + + R, = ring("", ZZ) + assert R(0).tail_degree() is ninf + assert R(1).tail_degree() == 0 + +def test_PolyElement_degrees(): + R, x,y,z = ring("x,y,z", ZZ) + + assert R(0).degrees() == (ninf, ninf, ninf) + assert R(1).degrees() == (0, 0, 0) + assert (x**2*y + x**3*z**2).degrees() == (3, 1, 2) + +def test_PolyElement_tail_degrees(): + R, x,y,z = ring("x,y,z", ZZ) + + assert R(0).tail_degrees() == (ninf, ninf, ninf) + assert R(1).tail_degrees() == (0, 0, 0) + assert (x**2*y + x**3*z**2).tail_degrees() == (2, 0, 0) + +def test_PolyElement_coeff(): + R, x, y, z = ring("x,y,z", ZZ, lex) + f = 3*x**2*y - x*y*z + 7*z**3 + 23 + + assert f.coeff(1) == 23 + raises(ValueError, lambda: f.coeff(3)) + + assert f.coeff(x) == 0 + assert f.coeff(y) == 0 + assert f.coeff(z) == 0 + + assert f.coeff(x**2*y) == 3 + assert f.coeff(x*y*z) == -1 + assert f.coeff(z**3) == 7 + + raises(ValueError, lambda: f.coeff(3*x**2*y)) + raises(ValueError, lambda: f.coeff(-x*y*z)) + raises(ValueError, lambda: f.coeff(7*z**3)) + + R, = ring("", ZZ) + assert R(3).coeff(1) == 3 + +def test_PolyElement_LC(): + R, x, y = ring("x,y", QQ, lex) + assert R(0).LC == QQ(0) + assert (QQ(1,2)*x).LC == QQ(1, 2) + assert (QQ(1,4)*x*y + QQ(1,2)*x).LC == QQ(1, 4) + +def test_PolyElement_LM(): + R, x, y = ring("x,y", QQ, lex) + assert R(0).LM == (0, 0) + assert (QQ(1,2)*x).LM == (1, 0) + assert (QQ(1,4)*x*y + QQ(1,2)*x).LM == (1, 1) + +def test_PolyElement_LT(): + R, x, y = ring("x,y", QQ, lex) + assert R(0).LT == ((0, 0), QQ(0)) + assert (QQ(1,2)*x).LT == ((1, 0), QQ(1, 2)) + assert (QQ(1,4)*x*y + QQ(1,2)*x).LT == ((1, 1), QQ(1, 4)) + + R, = ring("", ZZ) + assert R(0).LT == ((), 0) + assert R(1).LT == ((), 1) + +def test_PolyElement_leading_monom(): + R, x, y = ring("x,y", QQ, lex) + assert R(0).leading_monom() == 0 + assert (QQ(1,2)*x).leading_monom() == x + assert (QQ(1,4)*x*y + QQ(1,2)*x).leading_monom() == x*y + +def test_PolyElement_leading_term(): + R, x, y = ring("x,y", QQ, lex) + assert R(0).leading_term() == 0 + assert (QQ(1,2)*x).leading_term() == QQ(1,2)*x + assert (QQ(1,4)*x*y + QQ(1,2)*x).leading_term() == QQ(1,4)*x*y + +def test_PolyElement_terms(): + R, x,y,z = ring("x,y,z", QQ) + terms = (x**2/3 + y**3/4 + z**4/5).terms() + assert terms == [((2,0,0), QQ(1,3)), ((0,3,0), QQ(1,4)), ((0,0,4), QQ(1,5))] + + R, x,y = ring("x,y", ZZ, lex) + f = x*y**7 + 2*x**2*y**3 + + assert f.terms() == f.terms(lex) == f.terms('lex') == [((2, 3), 2), ((1, 7), 1)] + assert f.terms(grlex) == f.terms('grlex') == [((1, 7), 1), ((2, 3), 2)] + + R, x,y = ring("x,y", ZZ, grlex) + f = x*y**7 + 2*x**2*y**3 + + assert f.terms() == f.terms(grlex) == f.terms('grlex') == [((1, 7), 1), ((2, 3), 2)] + assert f.terms(lex) == f.terms('lex') == [((2, 3), 2), ((1, 7), 1)] + + R, = ring("", ZZ) + assert R(3).terms() == [((), 3)] + +def test_PolyElement_monoms(): + R, x,y,z = ring("x,y,z", QQ) + monoms = (x**2/3 + y**3/4 + z**4/5).monoms() + assert monoms == [(2,0,0), (0,3,0), (0,0,4)] + + R, x,y = ring("x,y", ZZ, lex) + f = x*y**7 + 2*x**2*y**3 + + assert f.monoms() == f.monoms(lex) == f.monoms('lex') == [(2, 3), (1, 7)] + assert f.monoms(grlex) == f.monoms('grlex') == [(1, 7), (2, 3)] + + R, x,y = ring("x,y", ZZ, grlex) + f = x*y**7 + 2*x**2*y**3 + + assert f.monoms() == f.monoms(grlex) == f.monoms('grlex') == [(1, 7), (2, 3)] + assert f.monoms(lex) == f.monoms('lex') == [(2, 3), (1, 7)] + +def test_PolyElement_coeffs(): + R, x,y,z = ring("x,y,z", QQ) + coeffs = (x**2/3 + y**3/4 + z**4/5).coeffs() + assert coeffs == [QQ(1,3), QQ(1,4), QQ(1,5)] + + R, x,y = ring("x,y", ZZ, lex) + f = x*y**7 + 2*x**2*y**3 + + assert f.coeffs() == f.coeffs(lex) == f.coeffs('lex') == [2, 1] + assert f.coeffs(grlex) == f.coeffs('grlex') == [1, 2] + + R, x,y = ring("x,y", ZZ, grlex) + f = x*y**7 + 2*x**2*y**3 + + assert f.coeffs() == f.coeffs(grlex) == f.coeffs('grlex') == [1, 2] + assert f.coeffs(lex) == f.coeffs('lex') == [2, 1] + +def test_PolyElement___add__(): + Rt, t = ring("t", ZZ) + Ruv, u,v = ring("u,v", ZZ) + Rxyz, x,y,z = ring("x,y,z", Ruv) + + assert dict(x + 3*y) == {(1, 0, 0): 1, (0, 1, 0): 3} + + assert dict(u + x) == dict(x + u) == {(1, 0, 0): 1, (0, 0, 0): u} + assert dict(u + x*y) == dict(x*y + u) == {(1, 1, 0): 1, (0, 0, 0): u} + assert dict(u + x*y + z) == dict(x*y + z + u) == {(1, 1, 0): 1, (0, 0, 1): 1, (0, 0, 0): u} + + assert dict(u*x + x) == dict(x + u*x) == {(1, 0, 0): u + 1} + assert dict(u*x + x*y) == dict(x*y + u*x) == {(1, 1, 0): 1, (1, 0, 0): u} + assert dict(u*x + x*y + z) == dict(x*y + z + u*x) == {(1, 1, 0): 1, (0, 0, 1): 1, (1, 0, 0): u} + + raises(TypeError, lambda: t + x) + raises(TypeError, lambda: x + t) + raises(TypeError, lambda: t + u) + raises(TypeError, lambda: u + t) + + Fuv, u,v = field("u,v", ZZ) + Rxyz, x,y,z = ring("x,y,z", Fuv) + + assert dict(u + x) == dict(x + u) == {(1, 0, 0): 1, (0, 0, 0): u} + + Rxyz, x,y,z = ring("x,y,z", EX) + + assert dict(EX(pi) + x*y*z) == dict(x*y*z + EX(pi)) == {(1, 1, 1): EX(1), (0, 0, 0): EX(pi)} + +def test_PolyElement___sub__(): + Rt, t = ring("t", ZZ) + Ruv, u,v = ring("u,v", ZZ) + Rxyz, x,y,z = ring("x,y,z", Ruv) + + assert dict(x - 3*y) == {(1, 0, 0): 1, (0, 1, 0): -3} + + assert dict(-u + x) == dict(x - u) == {(1, 0, 0): 1, (0, 0, 0): -u} + assert dict(-u + x*y) == dict(x*y - u) == {(1, 1, 0): 1, (0, 0, 0): -u} + assert dict(-u + x*y + z) == dict(x*y + z - u) == {(1, 1, 0): 1, (0, 0, 1): 1, (0, 0, 0): -u} + + assert dict(-u*x + x) == dict(x - u*x) == {(1, 0, 0): -u + 1} + assert dict(-u*x + x*y) == dict(x*y - u*x) == {(1, 1, 0): 1, (1, 0, 0): -u} + assert dict(-u*x + x*y + z) == dict(x*y + z - u*x) == {(1, 1, 0): 1, (0, 0, 1): 1, (1, 0, 0): -u} + + raises(TypeError, lambda: t - x) + raises(TypeError, lambda: x - t) + raises(TypeError, lambda: t - u) + raises(TypeError, lambda: u - t) + + Fuv, u,v = field("u,v", ZZ) + Rxyz, x,y,z = ring("x,y,z", Fuv) + + assert dict(-u + x) == dict(x - u) == {(1, 0, 0): 1, (0, 0, 0): -u} + + Rxyz, x,y,z = ring("x,y,z", EX) + + assert dict(-EX(pi) + x*y*z) == dict(x*y*z - EX(pi)) == {(1, 1, 1): EX(1), (0, 0, 0): -EX(pi)} + +def test_PolyElement___mul__(): + Rt, t = ring("t", ZZ) + Ruv, u,v = ring("u,v", ZZ) + Rxyz, x,y,z = ring("x,y,z", Ruv) + + assert dict(u*x) == dict(x*u) == {(1, 0, 0): u} + + assert dict(2*u*x + z) == dict(x*2*u + z) == {(1, 0, 0): 2*u, (0, 0, 1): 1} + assert dict(u*2*x + z) == dict(2*x*u + z) == {(1, 0, 0): 2*u, (0, 0, 1): 1} + assert dict(2*u*x + z) == dict(x*2*u + z) == {(1, 0, 0): 2*u, (0, 0, 1): 1} + assert dict(u*x*2 + z) == dict(x*u*2 + z) == {(1, 0, 0): 2*u, (0, 0, 1): 1} + + assert dict(2*u*x*y + z) == dict(x*y*2*u + z) == {(1, 1, 0): 2*u, (0, 0, 1): 1} + assert dict(u*2*x*y + z) == dict(2*x*y*u + z) == {(1, 1, 0): 2*u, (0, 0, 1): 1} + assert dict(2*u*x*y + z) == dict(x*y*2*u + z) == {(1, 1, 0): 2*u, (0, 0, 1): 1} + assert dict(u*x*y*2 + z) == dict(x*y*u*2 + z) == {(1, 1, 0): 2*u, (0, 0, 1): 1} + + assert dict(2*u*y*x + z) == dict(y*x*2*u + z) == {(1, 1, 0): 2*u, (0, 0, 1): 1} + assert dict(u*2*y*x + z) == dict(2*y*x*u + z) == {(1, 1, 0): 2*u, (0, 0, 1): 1} + assert dict(2*u*y*x + z) == dict(y*x*2*u + z) == {(1, 1, 0): 2*u, (0, 0, 1): 1} + assert dict(u*y*x*2 + z) == dict(y*x*u*2 + z) == {(1, 1, 0): 2*u, (0, 0, 1): 1} + + assert dict(3*u*(x + y) + z) == dict((x + y)*3*u + z) == {(1, 0, 0): 3*u, (0, 1, 0): 3*u, (0, 0, 1): 1} + + raises(TypeError, lambda: t*x + z) + raises(TypeError, lambda: x*t + z) + raises(TypeError, lambda: t*u + z) + raises(TypeError, lambda: u*t + z) + + Fuv, u,v = field("u,v", ZZ) + Rxyz, x,y,z = ring("x,y,z", Fuv) + + assert dict(u*x) == dict(x*u) == {(1, 0, 0): u} + + Rxyz, x,y,z = ring("x,y,z", EX) + + assert dict(EX(pi)*x*y*z) == dict(x*y*z*EX(pi)) == {(1, 1, 1): EX(pi)} + +def test_PolyElement___truediv__(): + R, x,y,z = ring("x,y,z", ZZ) + + assert (2*x**2 - 4)/2 == x**2 - 2 + assert (2*x**2 - 3)/2 == x**2 + + assert (x**2 - 1).quo(x) == x + assert (x**2 - x).quo(x) == x - 1 + + raises(ExactQuotientFailed, lambda: (x**2 - 1)/x) + assert (x**2 - x)/x == x - 1 + raises(ExactQuotientFailed, lambda: (x**2 - 1)/(2*x)) + + assert (x**2 - 1).quo(2*x) == 0 + assert (x**2 - x)/(x - 1) == (x**2 - x).quo(x - 1) == x + + + R, x,y,z = ring("x,y,z", ZZ) + assert len((x**2/3 + y**3/4 + z**4/5).terms()) == 0 + + R, x,y,z = ring("x,y,z", QQ) + assert len((x**2/3 + y**3/4 + z**4/5).terms()) == 3 + + Rt, t = ring("t", ZZ) + Ruv, u,v = ring("u,v", ZZ) + Rxyz, x,y,z = ring("x,y,z", Ruv) + + assert dict((u**2*x + u)/u) == {(1, 0, 0): u, (0, 0, 0): 1} + raises(ExactQuotientFailed, lambda: u/(u**2*x + u)) + + raises(TypeError, lambda: t/x) + raises(TypeError, lambda: x/t) + raises(TypeError, lambda: t/u) + raises(TypeError, lambda: u/t) + + R, x = ring("x", ZZ) + f, g = x**2 + 2*x + 3, R(0) + + raises(ZeroDivisionError, lambda: f.div(g)) + raises(ZeroDivisionError, lambda: divmod(f, g)) + raises(ZeroDivisionError, lambda: f.rem(g)) + raises(ZeroDivisionError, lambda: f % g) + raises(ZeroDivisionError, lambda: f.quo(g)) + raises(ZeroDivisionError, lambda: f / g) + raises(ZeroDivisionError, lambda: f.exquo(g)) + + R, x, y = ring("x,y", ZZ) + f, g = x*y + 2*x + 3, R(0) + + raises(ZeroDivisionError, lambda: f.div(g)) + raises(ZeroDivisionError, lambda: divmod(f, g)) + raises(ZeroDivisionError, lambda: f.rem(g)) + raises(ZeroDivisionError, lambda: f % g) + raises(ZeroDivisionError, lambda: f.quo(g)) + raises(ZeroDivisionError, lambda: f / g) + raises(ZeroDivisionError, lambda: f.exquo(g)) + + R, x = ring("x", ZZ) + + f, g = x**2 + 1, 2*x - 4 + q, r = R(0), x**2 + 1 + + assert f.div(g) == divmod(f, g) == (q, r) + assert f.rem(g) == f % g == r + assert f.quo(g) == q + raises(ExactQuotientFailed, lambda: f / g) + raises(ExactQuotientFailed, lambda: f.exquo(g)) + + f, g = 3*x**3 + x**2 + x + 5, 5*x**2 - 3*x + 1 + q, r = R(0), f + + assert f.div(g) == divmod(f, g) == (q, r) + assert f.rem(g) == f % g == r + assert f.quo(g) == q + raises(ExactQuotientFailed, lambda: f / g) + raises(ExactQuotientFailed, lambda: f.exquo(g)) + + f, g = 5*x**4 + 4*x**3 + 3*x**2 + 2*x + 1, x**2 + 2*x + 3 + q, r = 5*x**2 - 6*x, 20*x + 1 + + assert f.div(g) == divmod(f, g) == (q, r) + assert f.rem(g) == f % g == r + assert f.quo(g) == q + raises(ExactQuotientFailed, lambda: f / g) + raises(ExactQuotientFailed, lambda: f.exquo(g)) + + f, g = 5*x**5 + 4*x**4 + 3*x**3 + 2*x**2 + x, x**4 + 2*x**3 + 9 + q, r = 5*x - 6, 15*x**3 + 2*x**2 - 44*x + 54 + + assert f.div(g) == divmod(f, g) == (q, r) + assert f.rem(g) == f % g == r + assert f.quo(g) == q + raises(ExactQuotientFailed, lambda: f / g) + raises(ExactQuotientFailed, lambda: f.exquo(g)) + + R, x = ring("x", QQ) + + f, g = x**2 + 1, 2*x - 4 + q, r = x/2 + 1, R(5) + + assert f.div(g) == divmod(f, g) == (q, r) + assert f.rem(g) == f % g == r + assert f.quo(g) == q + raises(ExactQuotientFailed, lambda: f / g) + raises(ExactQuotientFailed, lambda: f.exquo(g)) + + f, g = 3*x**3 + x**2 + x + 5, 5*x**2 - 3*x + 1 + q, r = QQ(3, 5)*x + QQ(14, 25), QQ(52, 25)*x + QQ(111, 25) + + assert f.div(g) == divmod(f, g) == (q, r) + assert f.rem(g) == f % g == r + assert f.quo(g) == q + raises(ExactQuotientFailed, lambda: f / g) + raises(ExactQuotientFailed, lambda: f.exquo(g)) + + R, x,y = ring("x,y", ZZ) + + f, g = x**2 - y**2, x - y + q, r = x + y, R(0) + + assert f.div(g) == divmod(f, g) == (q, r) + assert f.rem(g) == f % g == r + assert f.quo(g) == q + assert f.exquo(g) == f / g == q + + f, g = x**2 + y**2, x - y + q, r = x + y, 2*y**2 + + assert f.div(g) == divmod(f, g) == (q, r) + assert f.rem(g) == f % g == r + assert f.quo(g) == q + raises(ExactQuotientFailed, lambda: f / g) + raises(ExactQuotientFailed, lambda: f.exquo(g)) + + f, g = x**2 + y**2, -x + y + q, r = -x - y, 2*y**2 + + assert f.div(g) == divmod(f, g) == (q, r) + assert f.rem(g) == f % g == r + assert f.quo(g) == q + raises(ExactQuotientFailed, lambda: f / g) + raises(ExactQuotientFailed, lambda: f.exquo(g)) + + f, g = x**2 + y**2, 2*x - 2*y + q, r = R(0), f + + assert f.div(g) == divmod(f, g) == (q, r) + assert f.rem(g) == f % g == r + assert f.quo(g) == q + raises(ExactQuotientFailed, lambda: f / g) + raises(ExactQuotientFailed, lambda: f.exquo(g)) + + R, x,y = ring("x,y", QQ) + + f, g = x**2 - y**2, x - y + q, r = x + y, R(0) + + assert f.div(g) == divmod(f, g) == (q, r) + assert f.rem(g) == f % g == r + assert f.quo(g) == q + assert f.exquo(g) == f / g == q + + f, g = x**2 + y**2, x - y + q, r = x + y, 2*y**2 + + assert f.div(g) == divmod(f, g) == (q, r) + assert f.rem(g) == f % g == r + assert f.quo(g) == q + raises(ExactQuotientFailed, lambda: f / g) + raises(ExactQuotientFailed, lambda: f.exquo(g)) + + f, g = x**2 + y**2, -x + y + q, r = -x - y, 2*y**2 + + assert f.div(g) == divmod(f, g) == (q, r) + assert f.rem(g) == f % g == r + assert f.quo(g) == q + raises(ExactQuotientFailed, lambda: f / g) + raises(ExactQuotientFailed, lambda: f.exquo(g)) + + f, g = x**2 + y**2, 2*x - 2*y + q, r = x/2 + y/2, 2*y**2 + + assert f.div(g) == divmod(f, g) == (q, r) + assert f.rem(g) == f % g == r + assert f.quo(g) == q + raises(ExactQuotientFailed, lambda: f / g) + raises(ExactQuotientFailed, lambda: f.exquo(g)) + +def test_PolyElement___pow__(): + R, x = ring("x", ZZ, grlex) + f = 2*x + 3 + + assert f**0 == 1 + assert f**1 == f + raises(ValueError, lambda: f**(-1)) + + assert f**2 == f._pow_generic(2) == f._pow_multinomial(2) == 4*x**2 + 12*x + 9 + assert f**3 == f._pow_generic(3) == f._pow_multinomial(3) == 8*x**3 + 36*x**2 + 54*x + 27 + assert f**4 == f._pow_generic(4) == f._pow_multinomial(4) == 16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81 + assert f**5 == f._pow_generic(5) == f._pow_multinomial(5) == 32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243 + + R, x,y,z = ring("x,y,z", ZZ, grlex) + f = x**3*y - 2*x*y**2 - 3*z + 1 + g = x**6*y**2 - 4*x**4*y**3 - 6*x**3*y*z + 2*x**3*y + 4*x**2*y**4 + 12*x*y**2*z - 4*x*y**2 + 9*z**2 - 6*z + 1 + + assert f**2 == f._pow_generic(2) == f._pow_multinomial(2) == g + + R, t = ring("t", ZZ) + f = -11200*t**4 - 2604*t**2 + 49 + g = 15735193600000000*t**16 + 14633730048000000*t**14 + 4828147466240000*t**12 \ + + 598976863027200*t**10 + 3130812416256*t**8 - 2620523775744*t**6 \ + + 92413760096*t**4 - 1225431984*t**2 + 5764801 + + assert f**4 == f._pow_generic(4) == f._pow_multinomial(4) == g + +def test_PolyElement_div(): + R, x = ring("x", ZZ, grlex) + + f = x**3 - 12*x**2 - 42 + g = x - 3 + + q = x**2 - 9*x - 27 + r = -123 + + assert f.div([g]) == ([q], r) + + R, x = ring("x", ZZ, grlex) + f = x**2 + 2*x + 2 + assert f.div([R(1)]) == ([f], 0) + + R, x = ring("x", QQ, grlex) + f = x**2 + 2*x + 2 + assert f.div([R(2)]) == ([QQ(1,2)*x**2 + x + 1], 0) + + R, x,y = ring("x,y", ZZ, grlex) + f = 4*x**2*y - 2*x*y + 4*x - 2*y + 8 + + assert f.div([R(2)]) == ([2*x**2*y - x*y + 2*x - y + 4], 0) + assert f.div([2*y]) == ([2*x**2 - x - 1], 4*x + 8) + + f = x - 1 + g = y - 1 + + assert f.div([g]) == ([0], f) + + f = x*y**2 + 1 + G = [x*y + 1, y + 1] + + Q = [y, -1] + r = 2 + + assert f.div(G) == (Q, r) + + f = x**2*y + x*y**2 + y**2 + G = [x*y - 1, y**2 - 1] + + Q = [x + y, 1] + r = x + y + 1 + + assert f.div(G) == (Q, r) + + G = [y**2 - 1, x*y - 1] + + Q = [x + 1, x] + r = 2*x + 1 + + assert f.div(G) == (Q, r) + + R, = ring("", ZZ) + assert R(3).div(R(2)) == (0, 3) + + R, = ring("", QQ) + assert R(3).div(R(2)) == (QQ(3, 2), 0) + +def test_PolyElement_rem(): + R, x = ring("x", ZZ, grlex) + + f = x**3 - 12*x**2 - 42 + g = x - 3 + r = -123 + + assert f.rem([g]) == f.div([g])[1] == r + + R, x,y = ring("x,y", ZZ, grlex) + + f = 4*x**2*y - 2*x*y + 4*x - 2*y + 8 + + assert f.rem([R(2)]) == f.div([R(2)])[1] == 0 + assert f.rem([2*y]) == f.div([2*y])[1] == 4*x + 8 + + f = x - 1 + g = y - 1 + + assert f.rem([g]) == f.div([g])[1] == f + + f = x*y**2 + 1 + G = [x*y + 1, y + 1] + r = 2 + + assert f.rem(G) == f.div(G)[1] == r + + f = x**2*y + x*y**2 + y**2 + G = [x*y - 1, y**2 - 1] + r = x + y + 1 + + assert f.rem(G) == f.div(G)[1] == r + + G = [y**2 - 1, x*y - 1] + r = 2*x + 1 + + assert f.rem(G) == f.div(G)[1] == r + +def test_PolyElement_deflate(): + R, x = ring("x", ZZ) + + assert (2*x**2).deflate(x**4 + 4*x**2 + 1) == ((2,), [2*x, x**2 + 4*x + 1]) + + R, x,y = ring("x,y", ZZ) + + assert R(0).deflate(R(0)) == ((1, 1), [0, 0]) + assert R(1).deflate(R(0)) == ((1, 1), [1, 0]) + assert R(1).deflate(R(2)) == ((1, 1), [1, 2]) + assert R(1).deflate(2*y) == ((1, 1), [1, 2*y]) + assert (2*y).deflate(2*y) == ((1, 1), [2*y, 2*y]) + assert R(2).deflate(2*y**2) == ((1, 2), [2, 2*y]) + assert (2*y**2).deflate(2*y**2) == ((1, 2), [2*y, 2*y]) + + f = x**4*y**2 + x**2*y + 1 + g = x**2*y**3 + x**2*y + 1 + + assert f.deflate(g) == ((2, 1), [x**2*y**2 + x*y + 1, x*y**3 + x*y + 1]) + +def test_PolyElement_clear_denoms(): + R, x,y = ring("x,y", QQ) + + assert R(1).clear_denoms() == (ZZ(1), 1) + assert R(7).clear_denoms() == (ZZ(1), 7) + + assert R(QQ(7,3)).clear_denoms() == (3, 7) + assert R(QQ(7,3)).clear_denoms() == (3, 7) + + assert (3*x**2 + x).clear_denoms() == (1, 3*x**2 + x) + assert (x**2 + QQ(1,2)*x).clear_denoms() == (2, 2*x**2 + x) + + rQQ, x,t = ring("x,t", QQ, lex) + rZZ, X,T = ring("x,t", ZZ, lex) + + F = [x - QQ(17824537287975195925064602467992950991718052713078834557692023531499318507213727406844943097,413954288007559433755329699713866804710749652268151059918115348815925474842910720000)*t**7 + - QQ(4882321164854282623427463828745855894130208215961904469205260756604820743234704900167747753,12936071500236232304854053116058337647210926633379720622441104650497671088840960000)*t**6 + - QQ(36398103304520066098365558157422127347455927422509913596393052633155821154626830576085097433,25872143000472464609708106232116675294421853266759441244882209300995342177681920000)*t**5 + - QQ(168108082231614049052707339295479262031324376786405372698857619250210703675982492356828810819,58212321751063045371843239022262519412449169850208742800984970927239519899784320000)*t**4 + - QQ(5694176899498574510667890423110567593477487855183144378347226247962949388653159751849449037,1617008937529529038106756639507292205901365829172465077805138081312208886105120000)*t**3 + - QQ(154482622347268833757819824809033388503591365487934245386958884099214649755244381307907779,60637835157357338929003373981523457721301218593967440417692678049207833228942000)*t**2 + - QQ(2452813096069528207645703151222478123259511586701148682951852876484544822947007791153163,2425513406294293557160134959260938308852048743758697616707707121968313329157680)*t + - QQ(34305265428126440542854669008203683099323146152358231964773310260498715579162112959703,202126117191191129763344579938411525737670728646558134725642260164026110763140), + t**8 + QQ(693749860237914515552,67859264524169150569)*t**7 + + QQ(27761407182086143225024,610733380717522355121)*t**6 + + QQ(7785127652157884044288,67859264524169150569)*t**5 + + QQ(36567075214771261409792,203577793572507451707)*t**4 + + QQ(36336335165196147384320,203577793572507451707)*t**3 + + QQ(7452455676042754048000,67859264524169150569)*t**2 + + QQ(2593331082514399232000,67859264524169150569)*t + + QQ(390399197427343360000,67859264524169150569)] + + G = [3725588592068034903797967297424801242396746870413359539263038139343329273586196480000*X - + 160420835591776763325581422211936558925462474417709511019228211783493866564923546661604487873*T**7 - + 1406108495478033395547109582678806497509499966197028487131115097902188374051595011248311352864*T**6 - + 5241326875850889518164640374668786338033653548841427557880599579174438246266263602956254030352*T**5 - + 10758917262823299139373269714910672770004760114329943852726887632013485035262879510837043892416*T**4 - + 13119383576444715672578819534846747735372132018341964647712009275306635391456880068261130581248*T**3 - + 9491412317016197146080450036267011389660653495578680036574753839055748080962214787557853941760*T**2 - + 3767520915562795326943800040277726397326609797172964377014046018280260848046603967211258368000*T - + 632314652371226552085897259159210286886724229880266931574701654721512325555116066073245696000, + 610733380717522355121*T**8 + + 6243748742141230639968*T**7 + + 27761407182086143225024*T**6 + + 70066148869420956398592*T**5 + + 109701225644313784229376*T**4 + + 109009005495588442152960*T**3 + + 67072101084384786432000*T**2 + + 23339979742629593088000*T + + 3513592776846090240000] + + assert [ f.clear_denoms()[1].set_ring(rZZ) for f in F ] == G + +def test_PolyElement_cofactors(): + R, x, y = ring("x,y", ZZ) + + f, g = R(0), R(0) + assert f.cofactors(g) == (0, 0, 0) + + f, g = R(2), R(0) + assert f.cofactors(g) == (2, 1, 0) + + f, g = R(-2), R(0) + assert f.cofactors(g) == (2, -1, 0) + + f, g = R(0), R(-2) + assert f.cofactors(g) == (2, 0, -1) + + f, g = R(0), 2*x + 4 + assert f.cofactors(g) == (2*x + 4, 0, 1) + + f, g = 2*x + 4, R(0) + assert f.cofactors(g) == (2*x + 4, 1, 0) + + f, g = R(2), R(2) + assert f.cofactors(g) == (2, 1, 1) + + f, g = R(-2), R(2) + assert f.cofactors(g) == (2, -1, 1) + + f, g = R(2), R(-2) + assert f.cofactors(g) == (2, 1, -1) + + f, g = R(-2), R(-2) + assert f.cofactors(g) == (2, -1, -1) + + f, g = x**2 + 2*x + 1, R(1) + assert f.cofactors(g) == (1, x**2 + 2*x + 1, 1) + + f, g = x**2 + 2*x + 1, R(2) + assert f.cofactors(g) == (1, x**2 + 2*x + 1, 2) + + f, g = 2*x**2 + 4*x + 2, R(2) + assert f.cofactors(g) == (2, x**2 + 2*x + 1, 1) + + f, g = R(2), 2*x**2 + 4*x + 2 + assert f.cofactors(g) == (2, 1, x**2 + 2*x + 1) + + f, g = 2*x**2 + 4*x + 2, x + 1 + assert f.cofactors(g) == (x + 1, 2*x + 2, 1) + + f, g = x + 1, 2*x**2 + 4*x + 2 + assert f.cofactors(g) == (x + 1, 1, 2*x + 2) + + R, x, y, z, t = ring("x,y,z,t", ZZ) + + f, g = t**2 + 2*t + 1, 2*t + 2 + assert f.cofactors(g) == (t + 1, t + 1, 2) + + f, g = z**2*t**2 + 2*z**2*t + z**2 + z*t + z, t**2 + 2*t + 1 + h, cff, cfg = t + 1, z**2*t + z**2 + z, t + 1 + + assert f.cofactors(g) == (h, cff, cfg) + assert g.cofactors(f) == (h, cfg, cff) + + R, x, y = ring("x,y", QQ) + + f = QQ(1,2)*x**2 + x + QQ(1,2) + g = QQ(1,2)*x + QQ(1,2) + + h = x + 1 + + assert f.cofactors(g) == (h, g, QQ(1,2)) + assert g.cofactors(f) == (h, QQ(1,2), g) + + R, x, y = ring("x,y", RR) + + f = 2.1*x*y**2 - 2.1*x*y + 2.1*x + g = 2.1*x**3 + h = 1.0*x + + assert f.cofactors(g) == (h, f/h, g/h) + assert g.cofactors(f) == (h, g/h, f/h) + +def test_PolyElement_gcd(): + R, x, y = ring("x,y", QQ) + + f = QQ(1,2)*x**2 + x + QQ(1,2) + g = QQ(1,2)*x + QQ(1,2) + + assert f.gcd(g) == x + 1 + +def test_PolyElement_cancel(): + R, x, y = ring("x,y", ZZ) + + f = 2*x**3 + 4*x**2 + 2*x + g = 3*x**2 + 3*x + F = 2*x + 2 + G = 3 + + assert f.cancel(g) == (F, G) + + assert (-f).cancel(g) == (-F, G) + assert f.cancel(-g) == (-F, G) + + R, x, y = ring("x,y", QQ) + + f = QQ(1,2)*x**3 + x**2 + QQ(1,2)*x + g = QQ(1,3)*x**2 + QQ(1,3)*x + F = 3*x + 3 + G = 2 + + assert f.cancel(g) == (F, G) + + assert (-f).cancel(g) == (-F, G) + assert f.cancel(-g) == (-F, G) + + Fx, x = field("x", ZZ) + Rt, t = ring("t", Fx) + + f = (-x**2 - 4)/4*t + g = t**2 + (x**2 + 2)/2 + + assert f.cancel(g) == ((-x**2 - 4)*t, 4*t**2 + 2*x**2 + 4) + +def test_PolyElement_max_norm(): + R, x, y = ring("x,y", ZZ) + + assert R(0).max_norm() == 0 + assert R(1).max_norm() == 1 + + assert (x**3 + 4*x**2 + 2*x + 3).max_norm() == 4 + +def test_PolyElement_l1_norm(): + R, x, y = ring("x,y", ZZ) + + assert R(0).l1_norm() == 0 + assert R(1).l1_norm() == 1 + + assert (x**3 + 4*x**2 + 2*x + 3).l1_norm() == 10 + +def test_PolyElement_diff(): + R, X = xring("x:11", QQ) + + f = QQ(288,5)*X[0]**8*X[1]**6*X[4]**3*X[10]**2 + 8*X[0]**2*X[2]**3*X[4]**3 +2*X[0]**2 - 2*X[1]**2 + + assert f.diff(X[0]) == QQ(2304,5)*X[0]**7*X[1]**6*X[4]**3*X[10]**2 + 16*X[0]*X[2]**3*X[4]**3 + 4*X[0] + assert f.diff(X[4]) == QQ(864,5)*X[0]**8*X[1]**6*X[4]**2*X[10]**2 + 24*X[0]**2*X[2]**3*X[4]**2 + assert f.diff(X[10]) == QQ(576,5)*X[0]**8*X[1]**6*X[4]**3*X[10] + +def test_PolyElement___call__(): + R, x = ring("x", ZZ) + f = 3*x + 1 + + assert f(0) == 1 + assert f(1) == 4 + + raises(ValueError, lambda: f()) + raises(ValueError, lambda: f(0, 1)) + + raises(CoercionFailed, lambda: f(QQ(1,7))) + + R, x,y = ring("x,y", ZZ) + f = 3*x + y**2 + 1 + + assert f(0, 0) == 1 + assert f(1, 7) == 53 + + Ry = R.drop(x) + + assert f(0) == Ry.y**2 + 1 + assert f(1) == Ry.y**2 + 4 + + raises(ValueError, lambda: f()) + raises(ValueError, lambda: f(0, 1, 2)) + + raises(CoercionFailed, lambda: f(1, QQ(1,7))) + raises(CoercionFailed, lambda: f(QQ(1,7), 1)) + raises(CoercionFailed, lambda: f(QQ(1,7), QQ(1,7))) + +def test_PolyElement_evaluate(): + R, x = ring("x", ZZ) + f = x**3 + 4*x**2 + 2*x + 3 + + r = f.evaluate(x, 0) + assert r == 3 and not isinstance(r, PolyElement) + + raises(CoercionFailed, lambda: f.evaluate(x, QQ(1,7))) + + R, x, y, z = ring("x,y,z", ZZ) + f = (x*y)**3 + 4*(x*y)**2 + 2*x*y + 3 + + r = f.evaluate(x, 0) + assert r == 3 and R.drop(x).is_element(r) + r = f.evaluate([(x, 0), (y, 0)]) + assert r == 3 and R.drop(x, y).is_element(r) + r = f.evaluate(y, 0) + assert r == 3 and R.drop(y).is_element(r) + r = f.evaluate([(y, 0), (x, 0)]) + assert r == 3 and R.drop(y, x).is_element(r) + + r = f.evaluate([(x, 0), (y, 0), (z, 0)]) + assert r == 3 and not isinstance(r, PolyElement) + + raises(CoercionFailed, lambda: f.evaluate([(x, 1), (y, QQ(1,7))])) + raises(CoercionFailed, lambda: f.evaluate([(x, QQ(1,7)), (y, 1)])) + raises(CoercionFailed, lambda: f.evaluate([(x, QQ(1,7)), (y, QQ(1,7))])) + +def test_PolyElement_subs(): + R, x = ring("x", ZZ) + f = x**3 + 4*x**2 + 2*x + 3 + + r = f.subs(x, 0) + assert r == 3 and R.is_element(r) + + raises(CoercionFailed, lambda: f.subs(x, QQ(1,7))) + + R, x, y, z = ring("x,y,z", ZZ) + f = x**3 + 4*x**2 + 2*x + 3 + + r = f.subs(x, 0) + assert r == 3 and R.is_element(r) + r = f.subs([(x, 0), (y, 0)]) + assert r == 3 and R.is_element(r) + + raises(CoercionFailed, lambda: f.subs([(x, 1), (y, QQ(1,7))])) + raises(CoercionFailed, lambda: f.subs([(x, QQ(1,7)), (y, 1)])) + raises(CoercionFailed, lambda: f.subs([(x, QQ(1,7)), (y, QQ(1,7))])) + +def test_PolyElement_symmetrize(): + R, x, y = ring("x,y", ZZ) + + # Homogeneous, symmetric + f = x**2 + y**2 + sym, rem, m = f.symmetrize() + assert rem == 0 + assert sym.compose(m) + rem == f + + # Homogeneous, asymmetric + f = x**2 - y**2 + sym, rem, m = f.symmetrize() + assert rem != 0 + assert sym.compose(m) + rem == f + + # Inhomogeneous, symmetric + f = x*y + 7 + sym, rem, m = f.symmetrize() + assert rem == 0 + assert sym.compose(m) + rem == f + + # Inhomogeneous, asymmetric + f = y + 7 + sym, rem, m = f.symmetrize() + assert rem != 0 + assert sym.compose(m) + rem == f + + # Constant + f = R.from_expr(3) + sym, rem, m = f.symmetrize() + assert rem == 0 + assert sym.compose(m) + rem == f + + # Constant constructed from sring + R, f = sring(3) + sym, rem, m = f.symmetrize() + assert rem == 0 + assert sym.compose(m) + rem == f + +def test_PolyElement_compose(): + R, x = ring("x", ZZ) + f = x**3 + 4*x**2 + 2*x + 3 + + r = f.compose(x, 0) + assert r == 3 and R.is_element(r) + + assert f.compose(x, x) == f + assert f.compose(x, x**2) == x**6 + 4*x**4 + 2*x**2 + 3 + + raises(CoercionFailed, lambda: f.compose(x, QQ(1,7))) + + R, x, y, z = ring("x,y,z", ZZ) + f = x**3 + 4*x**2 + 2*x + 3 + + r = f.compose(x, 0) + assert r == 3 and R.is_element(r) + r = f.compose([(x, 0), (y, 0)]) + assert r == 3 and R.is_element(r) + + r = (x**3 + 4*x**2 + 2*x*y*z + 3).compose(x, y*z**2 - 1) + q = (y*z**2 - 1)**3 + 4*(y*z**2 - 1)**2 + 2*(y*z**2 - 1)*y*z + 3 + assert r == q and R.is_element(r) + +def test_PolyElement_is_(): + R, x,y,z = ring("x,y,z", QQ) + + assert (x - x).is_generator == False + assert (x - x).is_ground == True + assert (x - x).is_monomial == True + assert (x - x).is_term == True + + assert (x - x + 1).is_generator == False + assert (x - x + 1).is_ground == True + assert (x - x + 1).is_monomial == True + assert (x - x + 1).is_term == True + + assert x.is_generator == True + assert x.is_ground == False + assert x.is_monomial == True + assert x.is_term == True + + assert (x*y).is_generator == False + assert (x*y).is_ground == False + assert (x*y).is_monomial == True + assert (x*y).is_term == True + + assert (3*x).is_generator == False + assert (3*x).is_ground == False + assert (3*x).is_monomial == False + assert (3*x).is_term == True + + assert (3*x + 1).is_generator == False + assert (3*x + 1).is_ground == False + assert (3*x + 1).is_monomial == False + assert (3*x + 1).is_term == False + + assert R(0).is_zero is True + assert R(1).is_zero is False + + assert R(0).is_one is False + assert R(1).is_one is True + + assert (x - 1).is_monic is True + assert (2*x - 1).is_monic is False + + assert (3*x + 2).is_primitive is True + assert (4*x + 2).is_primitive is False + + assert (x + y + z + 1).is_linear is True + assert (x*y*z + 1).is_linear is False + + assert (x*y + z + 1).is_quadratic is True + assert (x*y*z + 1).is_quadratic is False + + assert (x - 1).is_squarefree is True + assert ((x - 1)**2).is_squarefree is False + + assert (x**2 + x + 1).is_irreducible is True + assert (x**2 + 2*x + 1).is_irreducible is False + + _, t = ring("t", FF(11)) + + assert (7*t + 3).is_irreducible is True + assert (7*t**2 + 3*t + 1).is_irreducible is False + + _, u = ring("u", ZZ) + f = u**16 + u**14 - u**10 - u**8 - u**6 + u**2 + + assert f.is_cyclotomic is False + assert (f + 1).is_cyclotomic is True + + raises(MultivariatePolynomialError, lambda: x.is_cyclotomic) + + R, = ring("", ZZ) + assert R(4).is_squarefree is True + assert R(6).is_irreducible is True + +def test_PolyElement_drop(): + R, x,y,z = ring("x,y,z", ZZ) + + assert R(1).drop(0).ring == PolyRing("y,z", ZZ, lex) + assert R(1).drop(0).drop(0).ring == PolyRing("z", ZZ, lex) + assert R.is_element(R(1).drop(0).drop(0).drop(0)) is False + + raises(ValueError, lambda: z.drop(0).drop(0).drop(0)) + raises(ValueError, lambda: x.drop(0)) + +def test_PolyElement_coeff_wrt(): + R, x, y, z = ring("x, y, z", ZZ) + + p = 4*x**3 + 5*y**2 + 6*y**2*z + 7 + assert p.coeff_wrt(1, 2) == 6*z + 5 # using generator index + assert p.coeff_wrt(x, 3) == 4 # using generator + + p = 2*x**4 + 3*x*y**2*z + 10*y**2 + 10*x*z**2 + assert p.coeff_wrt(x, 1) == 3*y**2*z + 10*z**2 + assert p.coeff_wrt(y, 2) == 3*x*z + 10 + + p = 4*x**2 + 2*x*y + 5 + assert p.coeff_wrt(z, 1) == R(0) + assert p.coeff_wrt(y, 2) == R(0) + +def test_PolyElement_prem(): + R, x, y = ring("x, y", ZZ) + + f, g = x**2 + x*y, 2*x + 2 + assert f.prem(g) == -4*y + 4 # first generator is chosen by default if it is not given + + f, g = x**2 + 1, 2*x - 4 + assert f.prem(g) == f.prem(g, x) == 20 + assert f.prem(g, 1) == R(0) + + f, g = x*y + 2*x + 1, x + y + assert f.prem(g) == -y**2 - 2*y + 1 + assert f.prem(g, 1) == f.prem(g, y) == -x**2 + 2*x + 1 + + raises(ZeroDivisionError, lambda: f.prem(R(0))) + +def test_PolyElement_pdiv(): + R, x, y = ring("x,y", ZZ) + + f, g = x**4 + 5*x**3 + 7*x**2, 2*x**2 + 3 + assert f.pdiv(g) == f.pdiv(g, x) == (4*x**2 + 20*x + 22, -60*x - 66) + + f, g = x**2 - y**2, x - y + assert f.pdiv(g) == f.pdiv(g, 0) == (x + y, 0) + + f, g = x*y + 2*x + 1, x + y + assert f.pdiv(g) == (y + 2, -y**2 - 2*y + 1) + assert f.pdiv(g, y) == f.pdiv(g, 1) == (x + 1, -x**2 + 2*x + 1) + + assert R(0).pdiv(g) == (0, 0) + raises(ZeroDivisionError, lambda: f.prem(R(0))) + +def test_PolyElement_pquo(): + R, x, y = ring("x, y", ZZ) + + f, g = x**4 - 4*x**2*y + 4*y**2, x**2 - 2*y + assert f.pquo(g) == f.pquo(g, x) == x**2 - 2*y + assert f.pquo(g, y) == 4*x**2 - 8*y + 4 + + f, g = x**4 - y**4, x**2 - y**2 + assert f.pquo(g) == f.pquo(g, 0) == x**2 + y**2 + +def test_PolyElement_pexquo(): + R, x, y = ring("x, y", ZZ) + + f, g = x**2 - y**2, x - y + assert f.pexquo(g) == f.pexquo(g, x) == x + y + assert f.pexquo(g, y) == f.pexquo(g, 1) == x + y + 1 + + f, g = x**2 + 3*x + 6, x + 2 + raises(ExactQuotientFailed, lambda: f.pexquo(g)) + +def test_PolyElement_gcdex(): + _, x = ring("x", QQ) + + f, g = 2*x, x**2 - 16 + s, t, h = x/32, -QQ(1, 16), 1 + + assert f.half_gcdex(g) == (s, h) + assert f.gcdex(g) == (s, t, h) + +def test_PolyElement_subresultants(): + R, x, y = ring("x, y", ZZ) + + f, g = x**2*y + x*y, x + y # degree(f, x) > degree(g, x) + h = y**3 - y**2 + assert f.subresultants(g) == [f, g, h] # first generator is chosen default + + # generator index or generator is given + assert f.subresultants(g, 0) == f.subresultants(g, x) == [f, g, h] + + assert f.subresultants(g, y) == [x**2*y + x*y, x + y, x**3 + x**2] + + f, g = 2*x - y, x**2 + 2*y + x # degree(f, x) < degree(g, x) + assert f.subresultants(g) == [x**2 + x + 2*y, 2*x - y, y**2 + 10*y] + + f, g = R(0), y**3 - y**2 # f = 0 + assert f.subresultants(g) == [y**3 - y**2, 1] + + f, g = x**2*y + x*y, R(0) # g = 0 + assert f.subresultants(g) == [x**2*y + x*y, 1] + + f, g = R(0), R(0) # f = 0 and g = 0 + assert f.subresultants(g) == [0, 0] + + f, g = x**2 + x, x**2 + x # f and g are same polynomial + assert f.subresultants(g) == [x**2 + x, x**2 + x] + +def test_PolyElement_resultant(): + _, x = ring("x", ZZ) + f, g, h = x**2 - 2*x + 1, x**2 - 1, 0 + + assert f.resultant(g) == h + +def test_PolyElement_discriminant(): + _, x = ring("x", ZZ) + f, g = x**3 + 3*x**2 + 9*x - 13, -11664 + + assert f.discriminant() == g + + F, a, b, c = ring("a,b,c", ZZ) + _, x = ring("x", F) + + f, g = a*x**2 + b*x + c, b**2 - 4*a*c + + assert f.discriminant() == g + +def test_PolyElement_decompose(): + _, x = ring("x", ZZ) + + f = x**12 + 20*x**10 + 150*x**8 + 500*x**6 + 625*x**4 - 2*x**3 - 10*x + 9 + g = x**4 - 2*x + 9 + h = x**3 + 5*x + + assert g.compose(x, h) == f + assert f.decompose() == [g, h] + +def test_PolyElement_shift(): + _, x = ring("x", ZZ) + assert (x**2 - 2*x + 1).shift(2) == x**2 + 2*x + 1 + assert (x**2 - 2*x + 1).shift_list([2]) == x**2 + 2*x + 1 + + R, x, y = ring("x, y", ZZ) + assert (x*y).shift_list([1, 2]) == (x+1)*(y+2) + + raises(MultivariatePolynomialError, lambda: (x*y).shift(1)) + +def test_PolyElement_sturm(): + F, t = field("t", ZZ) + _, x = ring("x", F) + + f = 1024/(15625*t**8)*x**5 - 4096/(625*t**8)*x**4 + 32/(15625*t**4)*x**3 - 128/(625*t**4)*x**2 + F(1)/62500*x - F(1)/625 + + assert f.sturm() == [ + x**3 - 100*x**2 + t**4/64*x - 25*t**4/16, + 3*x**2 - 200*x + t**4/64, + (-t**4/96 + F(20000)/9)*x + 25*t**4/18, + (-9*t**12 - 11520000*t**8 - 3686400000000*t**4)/(576*t**8 - 245760000*t**4 + 26214400000000), + ] + +def test_PolyElement_gff_list(): + _, x = ring("x", ZZ) + + f = x**5 + 2*x**4 - x**3 - 2*x**2 + assert f.gff_list() == [(x, 1), (x + 2, 4)] + + f = x*(x - 1)**3*(x - 2)**2*(x - 4)**2*(x - 5) + assert f.gff_list() == [(x**2 - 5*x + 4, 1), (x**2 - 5*x + 4, 2), (x, 3)] + +def test_PolyElement_norm(): + k = QQ + K = QQ.algebraic_field(sqrt(2)) + sqrt2 = K.unit + _, X, Y = ring("x,y", k) + _, x, y = ring("x,y", K) + + assert (x*y + sqrt2).norm() == X**2*Y**2 - 2 + +def test_PolyElement_sqf_norm(): + R, x = ring("x", QQ.algebraic_field(sqrt(3))) + X = R.to_ground().x + + assert (x**2 - 2).sqf_norm() == ([1], x**2 - 2*sqrt(3)*x + 1, X**4 - 10*X**2 + 1) + + R, x = ring("x", QQ.algebraic_field(sqrt(2))) + X = R.to_ground().x + + assert (x**2 - 3).sqf_norm() == ([1], x**2 - 2*sqrt(2)*x - 1, X**4 - 10*X**2 + 1) + +def test_PolyElement_sqf_list(): + _, x = ring("x", ZZ) + + f = x**5 - x**3 - x**2 + 1 + g = x**3 + 2*x**2 + 2*x + 1 + h = x - 1 + p = x**4 + x**3 - x - 1 + + assert f.sqf_part() == p + assert f.sqf_list() == (1, [(g, 1), (h, 2)]) + +def test_issue_18894(): + items = [S(3)/16 + sqrt(3*sqrt(3) + 10)/8, S(1)/8 + 3*sqrt(3)/16, S(1)/8 + 3*sqrt(3)/16, -S(3)/16 + sqrt(3*sqrt(3) + 10)/8] + R, a = sring(items, extension=True) + assert R.domain == QQ.algebraic_field(sqrt(3)+sqrt(3*sqrt(3)+10)) + assert R.gens == () + result = [] + for item in items: + result.append(R.domain.from_sympy(item)) + assert a == result + +def test_PolyElement_factor_list(): + _, x = ring("x", ZZ) + + f = x**5 - x**3 - x**2 + 1 + + u = x + 1 + v = x - 1 + w = x**2 + x + 1 + + assert f.factor_list() == (1, [(u, 1), (v, 2), (w, 1)]) + + +def test_issue_21410(): + R, x = ring('x', FF(2)) + p = x**6 + x**5 + x**4 + x**3 + 1 + assert p._pow_multinomial(4) == x**24 + x**20 + x**16 + x**12 + 1 + + +def test_zero_polynomial_primitive(): + + x = symbols('x') + + R = ZZ[x] + zero_poly = R(0) + cont, prim = zero_poly.primitive() + assert cont == 0 + assert prim == zero_poly + assert prim.is_primitive is False diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_rootisolation.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_rootisolation.py new file mode 100644 index 0000000000000000000000000000000000000000..9661c1d6b63bfb941157c7e904ba4e048afbc538 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_rootisolation.py @@ -0,0 +1,823 @@ +"""Tests for real and complex root isolation and refinement algorithms. """ + +from sympy.polys.rings import ring +from sympy.polys.domains import ZZ, QQ, ZZ_I, EX +from sympy.polys.polyerrors import DomainError, RefinementFailed, PolynomialError +from sympy.polys.rootisolation import ( + dup_cauchy_upper_bound, dup_cauchy_lower_bound, + dup_mignotte_sep_bound_squared, +) +from sympy.testing.pytest import raises + +def test_dup_sturm(): + R, x = ring("x", QQ) + + assert R.dup_sturm(5) == [1] + assert R.dup_sturm(x) == [x, 1] + + f = x**3 - 2*x**2 + 3*x - 5 + assert R.dup_sturm(f) == [f, 3*x**2 - 4*x + 3, -QQ(10,9)*x + QQ(13,3), -QQ(3303,100)] + + +def test_dup_cauchy_upper_bound(): + raises(PolynomialError, lambda: dup_cauchy_upper_bound([], QQ)) + raises(PolynomialError, lambda: dup_cauchy_upper_bound([QQ(1)], QQ)) + raises(DomainError, lambda: dup_cauchy_upper_bound([ZZ_I(1), ZZ_I(1)], ZZ_I)) + + assert dup_cauchy_upper_bound([QQ(1), QQ(0), QQ(0)], QQ) == QQ.zero + assert dup_cauchy_upper_bound([QQ(1), QQ(0), QQ(-2)], QQ) == QQ(3) + + +def test_dup_cauchy_lower_bound(): + raises(PolynomialError, lambda: dup_cauchy_lower_bound([], QQ)) + raises(PolynomialError, lambda: dup_cauchy_lower_bound([QQ(1)], QQ)) + raises(PolynomialError, lambda: dup_cauchy_lower_bound([QQ(1), QQ(0), QQ(0)], QQ)) + raises(DomainError, lambda: dup_cauchy_lower_bound([ZZ_I(1), ZZ_I(1)], ZZ_I)) + + assert dup_cauchy_lower_bound([QQ(1), QQ(0), QQ(-2)], QQ) == QQ(2, 3) + + +def test_dup_mignotte_sep_bound_squared(): + raises(PolynomialError, lambda: dup_mignotte_sep_bound_squared([], QQ)) + raises(PolynomialError, lambda: dup_mignotte_sep_bound_squared([QQ(1)], QQ)) + + assert dup_mignotte_sep_bound_squared([QQ(1), QQ(0), QQ(-2)], QQ) == QQ(3, 5) + + +def test_dup_refine_real_root(): + R, x = ring("x", ZZ) + f = x**2 - 2 + + assert R.dup_refine_real_root(f, QQ(1), QQ(1), steps=1) == (QQ(1), QQ(1)) + assert R.dup_refine_real_root(f, QQ(1), QQ(1), steps=9) == (QQ(1), QQ(1)) + + raises(ValueError, lambda: R.dup_refine_real_root(f, QQ(-2), QQ(2))) + + s, t = QQ(1, 1), QQ(2, 1) + + assert R.dup_refine_real_root(f, s, t, steps=0) == (QQ(1, 1), QQ(2, 1)) + assert R.dup_refine_real_root(f, s, t, steps=1) == (QQ(1, 1), QQ(3, 2)) + assert R.dup_refine_real_root(f, s, t, steps=2) == (QQ(4, 3), QQ(3, 2)) + assert R.dup_refine_real_root(f, s, t, steps=3) == (QQ(7, 5), QQ(3, 2)) + assert R.dup_refine_real_root(f, s, t, steps=4) == (QQ(7, 5), QQ(10, 7)) + + s, t = QQ(1, 1), QQ(3, 2) + + assert R.dup_refine_real_root(f, s, t, steps=0) == (QQ(1, 1), QQ(3, 2)) + assert R.dup_refine_real_root(f, s, t, steps=1) == (QQ(4, 3), QQ(3, 2)) + assert R.dup_refine_real_root(f, s, t, steps=2) == (QQ(7, 5), QQ(3, 2)) + assert R.dup_refine_real_root(f, s, t, steps=3) == (QQ(7, 5), QQ(10, 7)) + assert R.dup_refine_real_root(f, s, t, steps=4) == (QQ(7, 5), QQ(17, 12)) + + s, t = QQ(1, 1), QQ(5, 3) + + assert R.dup_refine_real_root(f, s, t, steps=0) == (QQ(1, 1), QQ(5, 3)) + assert R.dup_refine_real_root(f, s, t, steps=1) == (QQ(1, 1), QQ(3, 2)) + assert R.dup_refine_real_root(f, s, t, steps=2) == (QQ(7, 5), QQ(3, 2)) + assert R.dup_refine_real_root(f, s, t, steps=3) == (QQ(7, 5), QQ(13, 9)) + assert R.dup_refine_real_root(f, s, t, steps=4) == (QQ(7, 5), QQ(27, 19)) + + s, t = QQ(-1, 1), QQ(-2, 1) + + assert R.dup_refine_real_root(f, s, t, steps=0) == (-QQ(2, 1), -QQ(1, 1)) + assert R.dup_refine_real_root(f, s, t, steps=1) == (-QQ(3, 2), -QQ(1, 1)) + assert R.dup_refine_real_root(f, s, t, steps=2) == (-QQ(3, 2), -QQ(4, 3)) + assert R.dup_refine_real_root(f, s, t, steps=3) == (-QQ(3, 2), -QQ(7, 5)) + assert R.dup_refine_real_root(f, s, t, steps=4) == (-QQ(10, 7), -QQ(7, 5)) + + raises(RefinementFailed, lambda: R.dup_refine_real_root(f, QQ(0), QQ(1))) + + s, t, u, v, w = QQ(1), QQ(2), QQ(24, 17), QQ(17, 12), QQ(7, 5) + + assert R.dup_refine_real_root(f, s, t, eps=QQ(1, 100)) == (u, v) + assert R.dup_refine_real_root(f, s, t, steps=6) == (u, v) + + assert R.dup_refine_real_root(f, s, t, eps=QQ(1, 100), steps=5) == (w, v) + assert R.dup_refine_real_root(f, s, t, eps=QQ(1, 100), steps=6) == (u, v) + assert R.dup_refine_real_root(f, s, t, eps=QQ(1, 100), steps=7) == (u, v) + + s, t, u, v = QQ(-2), QQ(-1), QQ(-3, 2), QQ(-4, 3) + + assert R.dup_refine_real_root(f, s, t, disjoint=QQ(-5)) == (s, t) + assert R.dup_refine_real_root(f, s, t, disjoint=-v) == (s, t) + assert R.dup_refine_real_root(f, s, t, disjoint=v) == (u, v) + + s, t, u, v = QQ(1), QQ(2), QQ(4, 3), QQ(3, 2) + + assert R.dup_refine_real_root(f, s, t, disjoint=QQ(5)) == (s, t) + assert R.dup_refine_real_root(f, s, t, disjoint=-u) == (s, t) + assert R.dup_refine_real_root(f, s, t, disjoint=u) == (u, v) + + +def test_dup_isolate_real_roots_sqf(): + R, x = ring("x", ZZ) + + assert R.dup_isolate_real_roots_sqf(0) == [] + assert R.dup_isolate_real_roots_sqf(5) == [] + + assert R.dup_isolate_real_roots_sqf(x**2 + x) == [(-1, -1), (0, 0)] + assert R.dup_isolate_real_roots_sqf(x**2 - x) == [( 0, 0), (1, 1)] + + assert R.dup_isolate_real_roots_sqf(x**4 + x + 1) == [] + + I = [(-2, -1), (1, 2)] + + assert R.dup_isolate_real_roots_sqf(x**2 - 2) == I + assert R.dup_isolate_real_roots_sqf(-x**2 + 2) == I + + assert R.dup_isolate_real_roots_sqf(x - 1) == \ + [(1, 1)] + assert R.dup_isolate_real_roots_sqf(x**2 - 3*x + 2) == \ + [(1, 1), (2, 2)] + assert R.dup_isolate_real_roots_sqf(x**3 - 6*x**2 + 11*x - 6) == \ + [(1, 1), (2, 2), (3, 3)] + assert R.dup_isolate_real_roots_sqf(x**4 - 10*x**3 + 35*x**2 - 50*x + 24) == \ + [(1, 1), (2, 2), (3, 3), (4, 4)] + assert R.dup_isolate_real_roots_sqf(x**5 - 15*x**4 + 85*x**3 - 225*x**2 + 274*x - 120) == \ + [(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)] + + assert R.dup_isolate_real_roots_sqf(x - 10) == \ + [(10, 10)] + assert R.dup_isolate_real_roots_sqf(x**2 - 30*x + 200) == \ + [(10, 10), (20, 20)] + assert R.dup_isolate_real_roots_sqf(x**3 - 60*x**2 + 1100*x - 6000) == \ + [(10, 10), (20, 20), (30, 30)] + assert R.dup_isolate_real_roots_sqf(x**4 - 100*x**3 + 3500*x**2 - 50000*x + 240000) == \ + [(10, 10), (20, 20), (30, 30), (40, 40)] + assert R.dup_isolate_real_roots_sqf(x**5 - 150*x**4 + 8500*x**3 - 225000*x**2 + 2740000*x - 12000000) == \ + [(10, 10), (20, 20), (30, 30), (40, 40), (50, 50)] + + assert R.dup_isolate_real_roots_sqf(x + 1) == \ + [(-1, -1)] + assert R.dup_isolate_real_roots_sqf(x**2 + 3*x + 2) == \ + [(-2, -2), (-1, -1)] + assert R.dup_isolate_real_roots_sqf(x**3 + 6*x**2 + 11*x + 6) == \ + [(-3, -3), (-2, -2), (-1, -1)] + assert R.dup_isolate_real_roots_sqf(x**4 + 10*x**3 + 35*x**2 + 50*x + 24) == \ + [(-4, -4), (-3, -3), (-2, -2), (-1, -1)] + assert R.dup_isolate_real_roots_sqf(x**5 + 15*x**4 + 85*x**3 + 225*x**2 + 274*x + 120) == \ + [(-5, -5), (-4, -4), (-3, -3), (-2, -2), (-1, -1)] + + assert R.dup_isolate_real_roots_sqf(x + 10) == \ + [(-10, -10)] + assert R.dup_isolate_real_roots_sqf(x**2 + 30*x + 200) == \ + [(-20, -20), (-10, -10)] + assert R.dup_isolate_real_roots_sqf(x**3 + 60*x**2 + 1100*x + 6000) == \ + [(-30, -30), (-20, -20), (-10, -10)] + assert R.dup_isolate_real_roots_sqf(x**4 + 100*x**3 + 3500*x**2 + 50000*x + 240000) == \ + [(-40, -40), (-30, -30), (-20, -20), (-10, -10)] + assert R.dup_isolate_real_roots_sqf(x**5 + 150*x**4 + 8500*x**3 + 225000*x**2 + 2740000*x + 12000000) == \ + [(-50, -50), (-40, -40), (-30, -30), (-20, -20), (-10, -10)] + + assert R.dup_isolate_real_roots_sqf(x**2 - 5) == [(-3, -2), (2, 3)] + assert R.dup_isolate_real_roots_sqf(x**3 - 5) == [(1, 2)] + assert R.dup_isolate_real_roots_sqf(x**4 - 5) == [(-2, -1), (1, 2)] + assert R.dup_isolate_real_roots_sqf(x**5 - 5) == [(1, 2)] + assert R.dup_isolate_real_roots_sqf(x**6 - 5) == [(-2, -1), (1, 2)] + assert R.dup_isolate_real_roots_sqf(x**7 - 5) == [(1, 2)] + assert R.dup_isolate_real_roots_sqf(x**8 - 5) == [(-2, -1), (1, 2)] + assert R.dup_isolate_real_roots_sqf(x**9 - 5) == [(1, 2)] + + assert R.dup_isolate_real_roots_sqf(x**2 - 1) == \ + [(-1, -1), (1, 1)] + assert R.dup_isolate_real_roots_sqf(x**3 + 2*x**2 - x - 2) == \ + [(-2, -2), (-1, -1), (1, 1)] + assert R.dup_isolate_real_roots_sqf(x**4 - 5*x**2 + 4) == \ + [(-2, -2), (-1, -1), (1, 1), (2, 2)] + assert R.dup_isolate_real_roots_sqf(x**5 + 3*x**4 - 5*x**3 - 15*x**2 + 4*x + 12) == \ + [(-3, -3), (-2, -2), (-1, -1), (1, 1), (2, 2)] + assert R.dup_isolate_real_roots_sqf(x**6 - 14*x**4 + 49*x**2 - 36) == \ + [(-3, -3), (-2, -2), (-1, -1), (1, 1), (2, 2), (3, 3)] + assert R.dup_isolate_real_roots_sqf(2*x**7 + x**6 - 28*x**5 - 14*x**4 + 98*x**3 + 49*x**2 - 72*x - 36) == \ + [(-3, -3), (-2, -2), (-1, -1), (-1, 0), (1, 1), (2, 2), (3, 3)] + assert R.dup_isolate_real_roots_sqf(4*x**8 - 57*x**6 + 210*x**4 - 193*x**2 + 36) == \ + [(-3, -3), (-2, -2), (-1, -1), (-1, 0), (0, 1), (1, 1), (2, 2), (3, 3)] + + f = 9*x**2 - 2 + + assert R.dup_isolate_real_roots_sqf(f) == \ + [(-1, 0), (0, 1)] + + assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 10)) == \ + [(QQ(-1, 2), QQ(-3, 7)), (QQ(3, 7), QQ(1, 2))] + assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 100)) == \ + [(QQ(-9, 19), QQ(-8, 17)), (QQ(8, 17), QQ(9, 19))] + assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 1000)) == \ + [(QQ(-33, 70), QQ(-8, 17)), (QQ(8, 17), QQ(33, 70))] + assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 10000)) == \ + [(QQ(-33, 70), QQ(-107, 227)), (QQ(107, 227), QQ(33, 70))] + assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 100000)) == \ + [(QQ(-305, 647), QQ(-272, 577)), (QQ(272, 577), QQ(305, 647))] + assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 1000000)) == \ + [(QQ(-1121, 2378), QQ(-272, 577)), (QQ(272, 577), QQ(1121, 2378))] + + f = 200100012*x**5 - 700390052*x**4 + 700490079*x**3 - 200240054*x**2 + 40017*x - 2 + + assert R.dup_isolate_real_roots_sqf(f) == \ + [(QQ(0), QQ(1, 10002)), (QQ(1, 10002), QQ(1, 10002)), + (QQ(1, 2), QQ(1, 2)), (QQ(1), QQ(1)), (QQ(2), QQ(2))] + + assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 100000)) == \ + [(QQ(1, 10003), QQ(1, 10003)), (QQ(1, 10002), QQ(1, 10002)), + (QQ(1, 2), QQ(1, 2)), (QQ(1), QQ(1)), (QQ(2), QQ(2))] + + a, b, c, d = 10000090000001, 2000100003, 10000300007, 10000005000008 + + f = 20001600074001600021*x**4 \ + + 1700135866278935491773999857*x**3 \ + - 2000179008931031182161141026995283662899200197*x**2 \ + - 800027600594323913802305066986600025*x \ + + 100000950000540000725000008 + + assert R.dup_isolate_real_roots_sqf(f) == \ + [(-a, -a), (-1, 0), (0, 1), (d, d)] + + assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 100000000000)) == \ + [(-QQ(a), -QQ(a)), (-QQ(1, b), -QQ(1, b)), (QQ(1, c), QQ(1, c)), (QQ(d), QQ(d))] + + (u, v), B, C, (s, t) = R.dup_isolate_real_roots_sqf(f, fast=True) + + assert u < -a < v and B == (-QQ(1), QQ(0)) and C == (QQ(0), QQ(1)) and s < d < t + + assert R.dup_isolate_real_roots_sqf(f, fast=True, eps=QQ(1, 100000000000000000000000000000)) == \ + [(-QQ(a), -QQ(a)), (-QQ(1, b), -QQ(1, b)), (QQ(1, c), QQ(1, c)), (QQ(d), QQ(d))] + + f = -10*x**4 + 8*x**3 + 80*x**2 - 32*x - 160 + + assert R.dup_isolate_real_roots_sqf(f) == \ + [(-2, -2), (-2, -1), (2, 2), (2, 3)] + + assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 100)) == \ + [(-QQ(2), -QQ(2)), (-QQ(23, 14), -QQ(18, 11)), (QQ(2), QQ(2)), (QQ(39, 16), QQ(22, 9))] + + f = x - 1 + + assert R.dup_isolate_real_roots_sqf(f, inf=2) == [] + assert R.dup_isolate_real_roots_sqf(f, sup=0) == [] + + assert R.dup_isolate_real_roots_sqf(f) == [(1, 1)] + assert R.dup_isolate_real_roots_sqf(f, inf=1) == [(1, 1)] + assert R.dup_isolate_real_roots_sqf(f, sup=1) == [(1, 1)] + assert R.dup_isolate_real_roots_sqf(f, inf=1, sup=1) == [(1, 1)] + + f = x**2 - 2 + + assert R.dup_isolate_real_roots_sqf(f, inf=QQ(7, 4)) == [] + assert R.dup_isolate_real_roots_sqf(f, inf=QQ(7, 5)) == [(QQ(7, 5), QQ(3, 2))] + assert R.dup_isolate_real_roots_sqf(f, sup=QQ(7, 5)) == [(-2, -1)] + assert R.dup_isolate_real_roots_sqf(f, sup=QQ(7, 4)) == [(-2, -1), (1, QQ(3, 2))] + assert R.dup_isolate_real_roots_sqf(f, sup=-QQ(7, 4)) == [] + assert R.dup_isolate_real_roots_sqf(f, sup=-QQ(7, 5)) == [(-QQ(3, 2), -QQ(7, 5))] + assert R.dup_isolate_real_roots_sqf(f, inf=-QQ(7, 5)) == [(1, 2)] + assert R.dup_isolate_real_roots_sqf(f, inf=-QQ(7, 4)) == [(-QQ(3, 2), -1), (1, 2)] + + I = [(-2, -1), (1, 2)] + + assert R.dup_isolate_real_roots_sqf(f, inf=-2) == I + assert R.dup_isolate_real_roots_sqf(f, sup=+2) == I + + assert R.dup_isolate_real_roots_sqf(f, inf=-2, sup=2) == I + + R, x = ring("x", QQ) + f = QQ(8, 5)*x**2 - QQ(87374, 3855)*x - QQ(17, 771) + + assert R.dup_isolate_real_roots_sqf(f) == [(-1, 0), (14, 15)] + + R, x = ring("x", EX) + raises(DomainError, lambda: R.dup_isolate_real_roots_sqf(x + 3)) + +def test_dup_isolate_real_roots(): + R, x = ring("x", ZZ) + + assert R.dup_isolate_real_roots(0) == [] + assert R.dup_isolate_real_roots(3) == [] + + assert R.dup_isolate_real_roots(5*x) == [((0, 0), 1)] + assert R.dup_isolate_real_roots(7*x**4) == [((0, 0), 4)] + + assert R.dup_isolate_real_roots(x**2 + x) == [((-1, -1), 1), ((0, 0), 1)] + assert R.dup_isolate_real_roots(x**2 - x) == [((0, 0), 1), ((1, 1), 1)] + + assert R.dup_isolate_real_roots(x**4 + x + 1) == [] + + I = [((-2, -1), 1), ((1, 2), 1)] + + assert R.dup_isolate_real_roots(x**2 - 2) == I + assert R.dup_isolate_real_roots(-x**2 + 2) == I + + f = 16*x**14 - 96*x**13 + 24*x**12 + 936*x**11 - 1599*x**10 - 2880*x**9 + 9196*x**8 \ + + 552*x**7 - 21831*x**6 + 13968*x**5 + 21690*x**4 - 26784*x**3 - 2916*x**2 + 15552*x - 5832 + g = R.dup_sqf_part(f) + + assert R.dup_isolate_real_roots(f) == \ + [((-QQ(2), -QQ(3, 2)), 2), ((-QQ(3, 2), -QQ(1, 1)), 3), ((QQ(1), QQ(3, 2)), 3), + ((QQ(3, 2), QQ(3, 2)), 4), ((QQ(5, 3), QQ(2)), 2)] + + assert R.dup_isolate_real_roots_sqf(g) == \ + [(-QQ(2), -QQ(3, 2)), (-QQ(3, 2), -QQ(1, 1)), (QQ(1), QQ(3, 2)), + (QQ(3, 2), QQ(3, 2)), (QQ(3, 2), QQ(2))] + assert R.dup_isolate_real_roots(g) == \ + [((-QQ(2), -QQ(3, 2)), 1), ((-QQ(3, 2), -QQ(1, 1)), 1), ((QQ(1), QQ(3, 2)), 1), + ((QQ(3, 2), QQ(3, 2)), 1), ((QQ(3, 2), QQ(2)), 1)] + + f = x - 1 + + assert R.dup_isolate_real_roots(f, inf=2) == [] + assert R.dup_isolate_real_roots(f, sup=0) == [] + + assert R.dup_isolate_real_roots(f) == [((1, 1), 1)] + assert R.dup_isolate_real_roots(f, inf=1) == [((1, 1), 1)] + assert R.dup_isolate_real_roots(f, sup=1) == [((1, 1), 1)] + assert R.dup_isolate_real_roots(f, inf=1, sup=1) == [((1, 1), 1)] + + f = x**4 - 4*x**2 + 4 + + assert R.dup_isolate_real_roots(f, inf=QQ(7, 4)) == [] + assert R.dup_isolate_real_roots(f, inf=QQ(7, 5)) == [((QQ(7, 5), QQ(3, 2)), 2)] + assert R.dup_isolate_real_roots(f, sup=QQ(7, 5)) == [((-2, -1), 2)] + assert R.dup_isolate_real_roots(f, sup=QQ(7, 4)) == [((-2, -1), 2), ((1, QQ(3, 2)), 2)] + assert R.dup_isolate_real_roots(f, sup=-QQ(7, 4)) == [] + assert R.dup_isolate_real_roots(f, sup=-QQ(7, 5)) == [((-QQ(3, 2), -QQ(7, 5)), 2)] + assert R.dup_isolate_real_roots(f, inf=-QQ(7, 5)) == [((1, 2), 2)] + assert R.dup_isolate_real_roots(f, inf=-QQ(7, 4)) == [((-QQ(3, 2), -1), 2), ((1, 2), 2)] + + I = [((-2, -1), 2), ((1, 2), 2)] + + assert R.dup_isolate_real_roots(f, inf=-2) == I + assert R.dup_isolate_real_roots(f, sup=+2) == I + + assert R.dup_isolate_real_roots(f, inf=-2, sup=2) == I + + f = x**11 - 3*x**10 - x**9 + 11*x**8 - 8*x**7 - 8*x**6 + 12*x**5 - 4*x**4 + + assert R.dup_isolate_real_roots(f, basis=False) == \ + [((-2, -1), 2), ((0, 0), 4), ((1, 1), 3), ((1, 2), 2)] + assert R.dup_isolate_real_roots(f, basis=True) == \ + [((-2, -1), 2, [1, 0, -2]), ((0, 0), 4, [1, 0]), ((1, 1), 3, [1, -1]), ((1, 2), 2, [1, 0, -2])] + + f = (x**45 - 45*x**44 + 990*x**43 - 1) + g = (x**46 - 15180*x**43 + 9366819*x**40 - 53524680*x**39 + 260932815*x**38 - 1101716330*x**37 + 4076350421*x**36 - 13340783196*x**35 + 38910617655*x**34 - 101766230790*x**33 + 239877544005*x**32 - 511738760544*x**31 + 991493848554*x**30 - 1749695026860*x**29 + 2818953098830*x**28 - 4154246671960*x**27 + 5608233007146*x**26 - 6943526580276*x**25 + 7890371113950*x**24 - 8233430727600*x**23 + 7890371113950*x**22 - 6943526580276*x**21 + 5608233007146*x**20 - 4154246671960*x**19 + 2818953098830*x**18 - 1749695026860*x**17 + 991493848554*x**16 - 511738760544*x**15 + 239877544005*x**14 - 101766230790*x**13 + 38910617655*x**12 - 13340783196*x**11 + 4076350421*x**10 - 1101716330*x**9 + 260932815*x**8 - 53524680*x**7 + 9366819*x**6 - 1370754*x**5 + 163185*x**4 - 15180*x**3 + 1035*x**2 - 47*x + 1) + + assert R.dup_isolate_real_roots(f*g) == \ + [((0, QQ(1, 2)), 1), ((QQ(2, 3), QQ(3, 4)), 1), ((QQ(3, 4), 1), 1), ((6, 7), 1), ((24, 25), 1)] + + R, x = ring("x", EX) + raises(DomainError, lambda: R.dup_isolate_real_roots(x + 3)) + + +def test_dup_isolate_real_roots_list(): + R, x = ring("x", ZZ) + + assert R.dup_isolate_real_roots_list([x**2 + x, x]) == \ + [((-1, -1), {0: 1}), ((0, 0), {0: 1, 1: 1})] + assert R.dup_isolate_real_roots_list([x**2 - x, x]) == \ + [((0, 0), {0: 1, 1: 1}), ((1, 1), {0: 1})] + + assert R.dup_isolate_real_roots_list([x + 1, x + 2, x - 1, x + 1, x - 1, x - 1]) == \ + [((-QQ(2), -QQ(2)), {1: 1}), ((-QQ(1), -QQ(1)), {0: 1, 3: 1}), ((QQ(1), QQ(1)), {2: 1, 4: 1, 5: 1})] + + assert R.dup_isolate_real_roots_list([x + 1, x + 2, x - 1, x + 1, x - 1, x + 2]) == \ + [((-QQ(2), -QQ(2)), {1: 1, 5: 1}), ((-QQ(1), -QQ(1)), {0: 1, 3: 1}), ((QQ(1), QQ(1)), {2: 1, 4: 1})] + + f, g = x**4 - 4*x**2 + 4, x - 1 + + assert R.dup_isolate_real_roots_list([f, g], inf=QQ(7, 4)) == [] + assert R.dup_isolate_real_roots_list([f, g], inf=QQ(7, 5)) == \ + [((QQ(7, 5), QQ(3, 2)), {0: 2})] + assert R.dup_isolate_real_roots_list([f, g], sup=QQ(7, 5)) == \ + [((-2, -1), {0: 2}), ((1, 1), {1: 1})] + assert R.dup_isolate_real_roots_list([f, g], sup=QQ(7, 4)) == \ + [((-2, -1), {0: 2}), ((1, 1), {1: 1}), ((1, QQ(3, 2)), {0: 2})] + assert R.dup_isolate_real_roots_list([f, g], sup=-QQ(7, 4)) == [] + assert R.dup_isolate_real_roots_list([f, g], sup=-QQ(7, 5)) == \ + [((-QQ(3, 2), -QQ(7, 5)), {0: 2})] + assert R.dup_isolate_real_roots_list([f, g], inf=-QQ(7, 5)) == \ + [((1, 1), {1: 1}), ((1, 2), {0: 2})] + assert R.dup_isolate_real_roots_list([f, g], inf=-QQ(7, 4)) == \ + [((-QQ(3, 2), -1), {0: 2}), ((1, 1), {1: 1}), ((1, 2), {0: 2})] + + f, g = 2*x**2 - 1, x**2 - 2 + + assert R.dup_isolate_real_roots_list([f, g]) == \ + [((-QQ(2), -QQ(1)), {1: 1}), ((-QQ(1), QQ(0)), {0: 1}), + ((QQ(0), QQ(1)), {0: 1}), ((QQ(1), QQ(2)), {1: 1})] + assert R.dup_isolate_real_roots_list([f, g], strict=True) == \ + [((-QQ(3, 2), -QQ(4, 3)), {1: 1}), ((-QQ(1), -QQ(2, 3)), {0: 1}), + ((QQ(2, 3), QQ(1)), {0: 1}), ((QQ(4, 3), QQ(3, 2)), {1: 1})] + + f, g = x**2 - 2, x**3 - x**2 - 2*x + 2 + + assert R.dup_isolate_real_roots_list([f, g]) == \ + [((-QQ(2), -QQ(1)), {1: 1, 0: 1}), ((QQ(1), QQ(1)), {1: 1}), ((QQ(1), QQ(2)), {1: 1, 0: 1})] + + f, g = x**3 - 2*x, x**5 - x**4 - 2*x**3 + 2*x**2 + + assert R.dup_isolate_real_roots_list([f, g]) == \ + [((-QQ(2), -QQ(1)), {1: 1, 0: 1}), ((QQ(0), QQ(0)), {0: 1, 1: 2}), + ((QQ(1), QQ(1)), {1: 1}), ((QQ(1), QQ(2)), {1: 1, 0: 1})] + + f, g = x**9 - 3*x**8 - x**7 + 11*x**6 - 8*x**5 - 8*x**4 + 12*x**3 - 4*x**2, x**5 - 2*x**4 + 3*x**3 - 4*x**2 + 2*x + + assert R.dup_isolate_real_roots_list([f, g], basis=False) == \ + [((-2, -1), {0: 2}), ((0, 0), {0: 2, 1: 1}), ((1, 1), {0: 3, 1: 2}), ((1, 2), {0: 2})] + assert R.dup_isolate_real_roots_list([f, g], basis=True) == \ + [((-2, -1), {0: 2}, [1, 0, -2]), ((0, 0), {0: 2, 1: 1}, [1, 0]), + ((1, 1), {0: 3, 1: 2}, [1, -1]), ((1, 2), {0: 2}, [1, 0, -2])] + + R, x = ring("x", EX) + raises(DomainError, lambda: R.dup_isolate_real_roots_list([x + 3])) + + +def test_dup_isolate_real_roots_list_QQ(): + R, x = ring("x", ZZ) + + f = x**5 - 200 + g = x**5 - 201 + + assert R.dup_isolate_real_roots_list([f, g]) == \ + [((QQ(75, 26), QQ(101, 35)), {0: 1}), ((QQ(309, 107), QQ(26, 9)), {1: 1})] + + R, x = ring("x", QQ) + + f = -QQ(1, 200)*x**5 + 1 + g = -QQ(1, 201)*x**5 + 1 + + assert R.dup_isolate_real_roots_list([f, g]) == \ + [((QQ(75, 26), QQ(101, 35)), {0: 1}), ((QQ(309, 107), QQ(26, 9)), {1: 1})] + + +def test_dup_count_real_roots(): + R, x = ring("x", ZZ) + + assert R.dup_count_real_roots(0) == 0 + assert R.dup_count_real_roots(7) == 0 + + f = x - 1 + assert R.dup_count_real_roots(f) == 1 + assert R.dup_count_real_roots(f, inf=1) == 1 + assert R.dup_count_real_roots(f, sup=0) == 0 + assert R.dup_count_real_roots(f, sup=1) == 1 + assert R.dup_count_real_roots(f, inf=0, sup=1) == 1 + assert R.dup_count_real_roots(f, inf=0, sup=2) == 1 + assert R.dup_count_real_roots(f, inf=1, sup=2) == 1 + + f = x**2 - 2 + assert R.dup_count_real_roots(f) == 2 + assert R.dup_count_real_roots(f, sup=0) == 1 + assert R.dup_count_real_roots(f, inf=-1, sup=1) == 0 + + +# parameters for test_dup_count_complex_roots_n(): n = 1..8 +a, b = (-QQ(1), -QQ(1)), (QQ(1), QQ(1)) +c, d = ( QQ(0), QQ(0)), (QQ(1), QQ(1)) + +def test_dup_count_complex_roots_1(): + R, x = ring("x", ZZ) + + # z-1 + f = x - 1 + assert R.dup_count_complex_roots(f, a, b) == 1 + assert R.dup_count_complex_roots(f, c, d) == 1 + + # z+1 + f = x + 1 + assert R.dup_count_complex_roots(f, a, b) == 1 + assert R.dup_count_complex_roots(f, c, d) == 0 + + +def test_dup_count_complex_roots_2(): + R, x = ring("x", ZZ) + + # (z-1)*(z) + f = x**2 - x + assert R.dup_count_complex_roots(f, a, b) == 2 + assert R.dup_count_complex_roots(f, c, d) == 2 + + # (z-1)*(-z) + f = -x**2 + x + assert R.dup_count_complex_roots(f, a, b) == 2 + assert R.dup_count_complex_roots(f, c, d) == 2 + + # (z+1)*(z) + f = x**2 + x + assert R.dup_count_complex_roots(f, a, b) == 2 + assert R.dup_count_complex_roots(f, c, d) == 1 + + # (z+1)*(-z) + f = -x**2 - x + assert R.dup_count_complex_roots(f, a, b) == 2 + assert R.dup_count_complex_roots(f, c, d) == 1 + + +def test_dup_count_complex_roots_3(): + R, x = ring("x", ZZ) + + # (z-1)*(z+1) + f = x**2 - 1 + assert R.dup_count_complex_roots(f, a, b) == 2 + assert R.dup_count_complex_roots(f, c, d) == 1 + + # (z-1)*(z+1)*(z) + f = x**3 - x + assert R.dup_count_complex_roots(f, a, b) == 3 + assert R.dup_count_complex_roots(f, c, d) == 2 + + # (z-1)*(z+1)*(-z) + f = -x**3 + x + assert R.dup_count_complex_roots(f, a, b) == 3 + assert R.dup_count_complex_roots(f, c, d) == 2 + + +def test_dup_count_complex_roots_4(): + R, x = ring("x", ZZ) + + # (z-I)*(z+I) + f = x**2 + 1 + assert R.dup_count_complex_roots(f, a, b) == 2 + assert R.dup_count_complex_roots(f, c, d) == 1 + + # (z-I)*(z+I)*(z) + f = x**3 + x + assert R.dup_count_complex_roots(f, a, b) == 3 + assert R.dup_count_complex_roots(f, c, d) == 2 + + # (z-I)*(z+I)*(-z) + f = -x**3 - x + assert R.dup_count_complex_roots(f, a, b) == 3 + assert R.dup_count_complex_roots(f, c, d) == 2 + + # (z-I)*(z+I)*(z-1) + f = x**3 - x**2 + x - 1 + assert R.dup_count_complex_roots(f, a, b) == 3 + assert R.dup_count_complex_roots(f, c, d) == 2 + + # (z-I)*(z+I)*(z-1)*(z) + f = x**4 - x**3 + x**2 - x + assert R.dup_count_complex_roots(f, a, b) == 4 + assert R.dup_count_complex_roots(f, c, d) == 3 + + # (z-I)*(z+I)*(z-1)*(-z) + f = -x**4 + x**3 - x**2 + x + assert R.dup_count_complex_roots(f, a, b) == 4 + assert R.dup_count_complex_roots(f, c, d) == 3 + + # (z-I)*(z+I)*(z-1)*(z+1) + f = x**4 - 1 + assert R.dup_count_complex_roots(f, a, b) == 4 + assert R.dup_count_complex_roots(f, c, d) == 2 + + # (z-I)*(z+I)*(z-1)*(z+1)*(z) + f = x**5 - x + assert R.dup_count_complex_roots(f, a, b) == 5 + assert R.dup_count_complex_roots(f, c, d) == 3 + + # (z-I)*(z+I)*(z-1)*(z+1)*(-z) + f = -x**5 + x + assert R.dup_count_complex_roots(f, a, b) == 5 + assert R.dup_count_complex_roots(f, c, d) == 3 + + +def test_dup_count_complex_roots_5(): + R, x = ring("x", ZZ) + + # (z-I+1)*(z+I+1) + f = x**2 + 2*x + 2 + assert R.dup_count_complex_roots(f, a, b) == 2 + assert R.dup_count_complex_roots(f, c, d) == 0 + + # (z-I+1)*(z+I+1)*(z-1) + f = x**3 + x**2 - 2 + assert R.dup_count_complex_roots(f, a, b) == 3 + assert R.dup_count_complex_roots(f, c, d) == 1 + + # (z-I+1)*(z+I+1)*(z-1)*z + f = x**4 + x**3 - 2*x + assert R.dup_count_complex_roots(f, a, b) == 4 + assert R.dup_count_complex_roots(f, c, d) == 2 + + # (z-I+1)*(z+I+1)*(z+1) + f = x**3 + 3*x**2 + 4*x + 2 + assert R.dup_count_complex_roots(f, a, b) == 3 + assert R.dup_count_complex_roots(f, c, d) == 0 + + # (z-I+1)*(z+I+1)*(z+1)*z + f = x**4 + 3*x**3 + 4*x**2 + 2*x + assert R.dup_count_complex_roots(f, a, b) == 4 + assert R.dup_count_complex_roots(f, c, d) == 1 + + # (z-I+1)*(z+I+1)*(z-1)*(z+1) + f = x**4 + 2*x**3 + x**2 - 2*x - 2 + assert R.dup_count_complex_roots(f, a, b) == 4 + assert R.dup_count_complex_roots(f, c, d) == 1 + + # (z-I+1)*(z+I+1)*(z-1)*(z+1)*z + f = x**5 + 2*x**4 + x**3 - 2*x**2 - 2*x + assert R.dup_count_complex_roots(f, a, b) == 5 + assert R.dup_count_complex_roots(f, c, d) == 2 + + +def test_dup_count_complex_roots_6(): + R, x = ring("x", ZZ) + + # (z-I-1)*(z+I-1) + f = x**2 - 2*x + 2 + assert R.dup_count_complex_roots(f, a, b) == 2 + assert R.dup_count_complex_roots(f, c, d) == 1 + + # (z-I-1)*(z+I-1)*(z-1) + f = x**3 - 3*x**2 + 4*x - 2 + assert R.dup_count_complex_roots(f, a, b) == 3 + assert R.dup_count_complex_roots(f, c, d) == 2 + + # (z-I-1)*(z+I-1)*(z-1)*z + f = x**4 - 3*x**3 + 4*x**2 - 2*x + assert R.dup_count_complex_roots(f, a, b) == 4 + assert R.dup_count_complex_roots(f, c, d) == 3 + + # (z-I-1)*(z+I-1)*(z+1) + f = x**3 - x**2 + 2 + assert R.dup_count_complex_roots(f, a, b) == 3 + assert R.dup_count_complex_roots(f, c, d) == 1 + + # (z-I-1)*(z+I-1)*(z+1)*z + f = x**4 - x**3 + 2*x + assert R.dup_count_complex_roots(f, a, b) == 4 + assert R.dup_count_complex_roots(f, c, d) == 2 + + # (z-I-1)*(z+I-1)*(z-1)*(z+1) + f = x**4 - 2*x**3 + x**2 + 2*x - 2 + assert R.dup_count_complex_roots(f, a, b) == 4 + assert R.dup_count_complex_roots(f, c, d) == 2 + + # (z-I-1)*(z+I-1)*(z-1)*(z+1)*z + f = x**5 - 2*x**4 + x**3 + 2*x**2 - 2*x + assert R.dup_count_complex_roots(f, a, b) == 5 + assert R.dup_count_complex_roots(f, c, d) == 3 + + +def test_dup_count_complex_roots_7(): + R, x = ring("x", ZZ) + + # (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1) + f = x**4 + 4 + assert R.dup_count_complex_roots(f, a, b) == 4 + assert R.dup_count_complex_roots(f, c, d) == 1 + + # (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z-2) + f = x**5 - 2*x**4 + 4*x - 8 + assert R.dup_count_complex_roots(f, a, b) == 4 + assert R.dup_count_complex_roots(f, c, d) == 1 + + # (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z**2-2) + f = x**6 - 2*x**4 + 4*x**2 - 8 + assert R.dup_count_complex_roots(f, a, b) == 4 + assert R.dup_count_complex_roots(f, c, d) == 1 + + # (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z-1) + f = x**5 - x**4 + 4*x - 4 + assert R.dup_count_complex_roots(f, a, b) == 5 + assert R.dup_count_complex_roots(f, c, d) == 2 + + # (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z-1)*z + f = x**6 - x**5 + 4*x**2 - 4*x + assert R.dup_count_complex_roots(f, a, b) == 6 + assert R.dup_count_complex_roots(f, c, d) == 3 + + # (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z+1) + f = x**5 + x**4 + 4*x + 4 + assert R.dup_count_complex_roots(f, a, b) == 5 + assert R.dup_count_complex_roots(f, c, d) == 1 + + # (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z+1)*z + f = x**6 + x**5 + 4*x**2 + 4*x + assert R.dup_count_complex_roots(f, a, b) == 6 + assert R.dup_count_complex_roots(f, c, d) == 2 + + # (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z-1)*(z+1) + f = x**6 - x**4 + 4*x**2 - 4 + assert R.dup_count_complex_roots(f, a, b) == 6 + assert R.dup_count_complex_roots(f, c, d) == 2 + + # (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z-1)*(z+1)*z + f = x**7 - x**5 + 4*x**3 - 4*x + assert R.dup_count_complex_roots(f, a, b) == 7 + assert R.dup_count_complex_roots(f, c, d) == 3 + + # (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z-1)*(z+1)*(z-I)*(z+I) + f = x**8 + 3*x**4 - 4 + assert R.dup_count_complex_roots(f, a, b) == 8 + assert R.dup_count_complex_roots(f, c, d) == 3 + + +def test_dup_count_complex_roots_8(): + R, x = ring("x", ZZ) + + # (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z-1)*(z+1)*(z-I)*(z+I)*z + f = x**9 + 3*x**5 - 4*x + assert R.dup_count_complex_roots(f, a, b) == 9 + assert R.dup_count_complex_roots(f, c, d) == 4 + + # (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z-1)*(z+1)*(z-I)*(z+I)*(z**2-2)*z + f = x**11 - 2*x**9 + 3*x**7 - 6*x**5 - 4*x**3 + 8*x + assert R.dup_count_complex_roots(f, a, b) == 9 + assert R.dup_count_complex_roots(f, c, d) == 4 + + +def test_dup_count_complex_roots_implicit(): + R, x = ring("x", ZZ) + + # z*(z-1)*(z+1)*(z-I)*(z+I) + f = x**5 - x + + assert R.dup_count_complex_roots(f) == 5 + + assert R.dup_count_complex_roots(f, sup=(0, 0)) == 3 + assert R.dup_count_complex_roots(f, inf=(0, 0)) == 3 + + +def test_dup_count_complex_roots_exclude(): + R, x = ring("x", ZZ) + + # z*(z-1)*(z+1)*(z-I)*(z+I) + f = x**5 - x + + a, b = (-QQ(1), QQ(0)), (QQ(1), QQ(1)) + + assert R.dup_count_complex_roots(f, a, b) == 4 + + assert R.dup_count_complex_roots(f, a, b, exclude=['S']) == 3 + assert R.dup_count_complex_roots(f, a, b, exclude=['N']) == 3 + + assert R.dup_count_complex_roots(f, a, b, exclude=['S', 'N']) == 2 + + assert R.dup_count_complex_roots(f, a, b, exclude=['E']) == 4 + assert R.dup_count_complex_roots(f, a, b, exclude=['W']) == 4 + + assert R.dup_count_complex_roots(f, a, b, exclude=['E', 'W']) == 4 + + assert R.dup_count_complex_roots(f, a, b, exclude=['N', 'S', 'E', 'W']) == 2 + + assert R.dup_count_complex_roots(f, a, b, exclude=['SW']) == 3 + assert R.dup_count_complex_roots(f, a, b, exclude=['SE']) == 3 + + assert R.dup_count_complex_roots(f, a, b, exclude=['SW', 'SE']) == 2 + assert R.dup_count_complex_roots(f, a, b, exclude=['SW', 'SE', 'S']) == 1 + assert R.dup_count_complex_roots(f, a, b, exclude=['SW', 'SE', 'S', 'N']) == 0 + + a, b = (QQ(0), QQ(0)), (QQ(1), QQ(1)) + + assert R.dup_count_complex_roots(f, a, b, exclude=True) == 1 + + +def test_dup_isolate_complex_roots_sqf(): + R, x = ring("x", ZZ) + f = x**2 - 2*x + 3 + + assert R.dup_isolate_complex_roots_sqf(f) == \ + [((0, -6), (6, 0)), ((0, 0), (6, 6))] + assert [ r.as_tuple() for r in R.dup_isolate_complex_roots_sqf(f, blackbox=True) ] == \ + [((0, -6), (6, 0)), ((0, 0), (6, 6))] + + assert R.dup_isolate_complex_roots_sqf(f, eps=QQ(1, 10)) == \ + [((QQ(15, 16), -QQ(3, 2)), (QQ(33, 32), -QQ(45, 32))), + ((QQ(15, 16), QQ(45, 32)), (QQ(33, 32), QQ(3, 2)))] + assert R.dup_isolate_complex_roots_sqf(f, eps=QQ(1, 100)) == \ + [((QQ(255, 256), -QQ(363, 256)), (QQ(513, 512), -QQ(723, 512))), + ((QQ(255, 256), QQ(723, 512)), (QQ(513, 512), QQ(363, 256)))] + + f = 7*x**4 - 19*x**3 + 20*x**2 + 17*x + 20 + + assert R.dup_isolate_complex_roots_sqf(f) == \ + [((-QQ(40, 7), -QQ(40, 7)), (0, 0)), ((-QQ(40, 7), 0), (0, QQ(40, 7))), + ((0, -QQ(40, 7)), (QQ(40, 7), 0)), ((0, 0), (QQ(40, 7), QQ(40, 7)))] + + +def test_dup_isolate_all_roots_sqf(): + R, x = ring("x", ZZ) + f = 4*x**4 - x**3 + 2*x**2 + 5*x + + assert R.dup_isolate_all_roots_sqf(f) == \ + ([(-1, 0), (0, 0)], + [((0, -QQ(5, 2)), (QQ(5, 2), 0)), ((0, 0), (QQ(5, 2), QQ(5, 2)))]) + + assert R.dup_isolate_all_roots_sqf(f, eps=QQ(1, 10)) == \ + ([(QQ(-7, 8), QQ(-6, 7)), (0, 0)], + [((QQ(35, 64), -QQ(35, 32)), (QQ(5, 8), -QQ(65, 64))), ((QQ(35, 64), QQ(65, 64)), (QQ(5, 8), QQ(35, 32)))]) + + +def test_dup_isolate_all_roots(): + R, x = ring("x", ZZ) + f = 4*x**4 - x**3 + 2*x**2 + 5*x + + assert R.dup_isolate_all_roots(f) == \ + ([((-1, 0), 1), ((0, 0), 1)], + [(((0, -QQ(5, 2)), (QQ(5, 2), 0)), 1), + (((0, 0), (QQ(5, 2), QQ(5, 2))), 1)]) + + assert R.dup_isolate_all_roots(f, eps=QQ(1, 10)) == \ + ([((QQ(-7, 8), QQ(-6, 7)), 1), ((0, 0), 1)], + [(((QQ(35, 64), -QQ(35, 32)), (QQ(5, 8), -QQ(65, 64))), 1), + (((QQ(35, 64), QQ(65, 64)), (QQ(5, 8), QQ(35, 32))), 1)]) + + f = x**5 + x**4 - 2*x**3 - 2*x**2 + x + 1 + raises(NotImplementedError, lambda: R.dup_isolate_all_roots(f)) diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_rootoftools.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_rootoftools.py new file mode 100644 index 0000000000000000000000000000000000000000..de9dbcabd0a7e2bed0c5adb7127041b4be058379 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_rootoftools.py @@ -0,0 +1,697 @@ +"""Tests for the implementation of RootOf class and related tools. """ + +from sympy.polys.polytools import Poly +import sympy.polys.rootoftools as rootoftools +from sympy.polys.rootoftools import (rootof, RootOf, CRootOf, RootSum, + _pure_key_dict as D) + +from sympy.polys.polyerrors import ( + MultivariatePolynomialError, + GeneratorsNeeded, + PolynomialError, +) + +from sympy.core.function import (Function, Lambda) +from sympy.core.numbers import (Float, I, Rational) +from sympy.core.relational import Eq +from sympy.core.singleton import S +from sympy.functions.elementary.exponential import (exp, log) +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import tan +from sympy.integrals.integrals import Integral +from sympy.polys.orthopolys import legendre_poly +from sympy.solvers.solvers import solve + + +from sympy.testing.pytest import raises, slow +from sympy.core.expr import unchanged + +from sympy.abc import a, b, x, y, z, r + + +def test_CRootOf___new__(): + assert rootof(x, 0) == 0 + assert rootof(x, -1) == 0 + + assert rootof(x, S.Zero) == 0 + + assert rootof(x - 1, 0) == 1 + assert rootof(x - 1, -1) == 1 + + assert rootof(x + 1, 0) == -1 + assert rootof(x + 1, -1) == -1 + + assert rootof(x**2 + 2*x + 3, 0) == -1 - I*sqrt(2) + assert rootof(x**2 + 2*x + 3, 1) == -1 + I*sqrt(2) + assert rootof(x**2 + 2*x + 3, -1) == -1 + I*sqrt(2) + assert rootof(x**2 + 2*x + 3, -2) == -1 - I*sqrt(2) + + r = rootof(x**2 + 2*x + 3, 0, radicals=False) + assert isinstance(r, RootOf) is True + + r = rootof(x**2 + 2*x + 3, 1, radicals=False) + assert isinstance(r, RootOf) is True + + r = rootof(x**2 + 2*x + 3, -1, radicals=False) + assert isinstance(r, RootOf) is True + + r = rootof(x**2 + 2*x + 3, -2, radicals=False) + assert isinstance(r, RootOf) is True + + assert rootof((x - 1)*(x + 1), 0, radicals=False) == -1 + assert rootof((x - 1)*(x + 1), 1, radicals=False) == 1 + assert rootof((x - 1)*(x + 1), -1, radicals=False) == 1 + assert rootof((x - 1)*(x + 1), -2, radicals=False) == -1 + + assert rootof((x - 1)*(x + 1), 0, radicals=True) == -1 + assert rootof((x - 1)*(x + 1), 1, radicals=True) == 1 + assert rootof((x - 1)*(x + 1), -1, radicals=True) == 1 + assert rootof((x - 1)*(x + 1), -2, radicals=True) == -1 + + assert rootof((x - 1)*(x**3 + x + 3), 0) == rootof(x**3 + x + 3, 0) + assert rootof((x - 1)*(x**3 + x + 3), 1) == 1 + assert rootof((x - 1)*(x**3 + x + 3), 2) == rootof(x**3 + x + 3, 1) + assert rootof((x - 1)*(x**3 + x + 3), 3) == rootof(x**3 + x + 3, 2) + assert rootof((x - 1)*(x**3 + x + 3), -1) == rootof(x**3 + x + 3, 2) + assert rootof((x - 1)*(x**3 + x + 3), -2) == rootof(x**3 + x + 3, 1) + assert rootof((x - 1)*(x**3 + x + 3), -3) == 1 + assert rootof((x - 1)*(x**3 + x + 3), -4) == rootof(x**3 + x + 3, 0) + + assert rootof(x**4 + 3*x**3, 0) == -3 + assert rootof(x**4 + 3*x**3, 1) == 0 + assert rootof(x**4 + 3*x**3, 2) == 0 + assert rootof(x**4 + 3*x**3, 3) == 0 + + raises(GeneratorsNeeded, lambda: rootof(0, 0)) + raises(GeneratorsNeeded, lambda: rootof(1, 0)) + + raises(PolynomialError, lambda: rootof(Poly(0, x), 0)) + raises(PolynomialError, lambda: rootof(Poly(1, x), 0)) + raises(PolynomialError, lambda: rootof(x - y, 0)) + # issue 8617 + raises(PolynomialError, lambda: rootof(exp(x), 0)) + + raises(NotImplementedError, lambda: rootof(x**3 - x + sqrt(2), 0)) + raises(NotImplementedError, lambda: rootof(x**3 - x + I, 0)) + + raises(IndexError, lambda: rootof(x**2 - 1, -4)) + raises(IndexError, lambda: rootof(x**2 - 1, -3)) + raises(IndexError, lambda: rootof(x**2 - 1, 2)) + raises(IndexError, lambda: rootof(x**2 - 1, 3)) + raises(ValueError, lambda: rootof(x**2 - 1, x)) + + assert rootof(Poly(x - y, x), 0) == y + + assert rootof(Poly(x**2 - y, x), 0) == -sqrt(y) + assert rootof(Poly(x**2 - y, x), 1) == sqrt(y) + + assert rootof(Poly(x**3 - y, x), 0) == y**Rational(1, 3) + + assert rootof(y*x**3 + y*x + 2*y, x, 0) == -1 + raises(NotImplementedError, lambda: rootof(x**3 + x + 2*y, x, 0)) + + assert rootof(x**3 + x + 1, 0).is_commutative is True + + +def test_CRootOf_attributes(): + r = rootof(x**3 + x + 3, 0) + assert r.is_number + assert r.free_symbols == set() + # if the following assertion fails then multivariate polynomials + # are apparently supported and the RootOf.free_symbols routine + # should be changed to return whatever symbols would not be + # the PurePoly dummy symbol + raises(NotImplementedError, lambda: rootof(Poly(x**3 + y*x + 1, x), 0)) + + +def test_CRootOf___eq__(): + assert (rootof(x**3 + x + 3, 0) == rootof(x**3 + x + 3, 0)) is True + assert (rootof(x**3 + x + 3, 0) == rootof(x**3 + x + 3, 1)) is False + assert (rootof(x**3 + x + 3, 1) == rootof(x**3 + x + 3, 1)) is True + assert (rootof(x**3 + x + 3, 1) == rootof(x**3 + x + 3, 2)) is False + assert (rootof(x**3 + x + 3, 2) == rootof(x**3 + x + 3, 2)) is True + + assert (rootof(x**3 + x + 3, 0) == rootof(y**3 + y + 3, 0)) is True + assert (rootof(x**3 + x + 3, 0) == rootof(y**3 + y + 3, 1)) is False + assert (rootof(x**3 + x + 3, 1) == rootof(y**3 + y + 3, 1)) is True + assert (rootof(x**3 + x + 3, 1) == rootof(y**3 + y + 3, 2)) is False + assert (rootof(x**3 + x + 3, 2) == rootof(y**3 + y + 3, 2)) is True + + +def test_CRootOf___eval_Eq__(): + f = Function('f') + eq = x**3 + x + 3 + r = rootof(eq, 2) + r1 = rootof(eq, 1) + assert Eq(r, r1) is S.false + assert Eq(r, r) is S.true + assert unchanged(Eq, r, x) + assert Eq(r, 0) is S.false + assert Eq(r, S.Infinity) is S.false + assert Eq(r, I) is S.false + assert unchanged(Eq, r, f(0)) + sol = solve(eq) + for s in sol: + if s.is_real: + assert Eq(r, s) is S.false + r = rootof(eq, 0) + for s in sol: + if s.is_real: + assert Eq(r, s) is S.true + eq = x**3 + x + 1 + sol = solve(eq) + assert [Eq(rootof(eq, i), j) for i in range(3) for j in sol + ].count(True) == 3 + assert Eq(rootof(eq, 0), 1 + S.ImaginaryUnit) == False + + +def test_CRootOf_is_real(): + assert rootof(x**3 + x + 3, 0).is_real is True + assert rootof(x**3 + x + 3, 1).is_real is False + assert rootof(x**3 + x + 3, 2).is_real is False + + +def test_CRootOf_is_complex(): + assert rootof(x**3 + x + 3, 0).is_complex is True + + +def test_CRootOf_is_algebraic(): + assert rootof(x**3 + x + 3, 0).is_algebraic is True + assert rootof(x**3 + x + 3, 1).is_algebraic is True + assert rootof(x**3 + x + 3, 2).is_algebraic is True + + +def test_CRootOf_subs(): + assert rootof(x**3 + x + 1, 0).subs(x, y) == rootof(y**3 + y + 1, 0) + + +def test_CRootOf_diff(): + assert rootof(x**3 + x + 1, 0).diff(x) == 0 + assert rootof(x**3 + x + 1, 0).diff(y) == 0 + +@slow +def test_CRootOf_evalf(): + real = rootof(x**3 + x + 3, 0).evalf(n=20) + + assert real.epsilon_eq(Float("-1.2134116627622296341")) + + re, im = rootof(x**3 + x + 3, 1).evalf(n=20).as_real_imag() + + assert re.epsilon_eq( Float("0.60670583138111481707")) + assert im.epsilon_eq(-Float("1.45061224918844152650")) + + re, im = rootof(x**3 + x + 3, 2).evalf(n=20).as_real_imag() + + assert re.epsilon_eq(Float("0.60670583138111481707")) + assert im.epsilon_eq(Float("1.45061224918844152650")) + + p = legendre_poly(4, x, polys=True) + roots = [str(r.n(17)) for r in p.real_roots()] + # magnitudes are given by + # sqrt(3/S(7) - 2*sqrt(6/S(5))/7) + # and + # sqrt(3/S(7) + 2*sqrt(6/S(5))/7) + assert roots == [ + "-0.86113631159405258", + "-0.33998104358485626", + "0.33998104358485626", + "0.86113631159405258", + ] + + re = rootof(x**5 - 5*x + 12, 0).evalf(n=20) + assert re.epsilon_eq(Float("-1.84208596619025438271")) + + re, im = rootof(x**5 - 5*x + 12, 1).evalf(n=20).as_real_imag() + assert re.epsilon_eq(Float("-0.351854240827371999559")) + assert im.epsilon_eq(Float("-1.709561043370328882010")) + + re, im = rootof(x**5 - 5*x + 12, 2).evalf(n=20).as_real_imag() + assert re.epsilon_eq(Float("-0.351854240827371999559")) + assert im.epsilon_eq(Float("+1.709561043370328882010")) + + re, im = rootof(x**5 - 5*x + 12, 3).evalf(n=20).as_real_imag() + assert re.epsilon_eq(Float("+1.272897223922499190910")) + assert im.epsilon_eq(Float("-0.719798681483861386681")) + + re, im = rootof(x**5 - 5*x + 12, 4).evalf(n=20).as_real_imag() + assert re.epsilon_eq(Float("+1.272897223922499190910")) + assert im.epsilon_eq(Float("+0.719798681483861386681")) + + # issue 6393 + assert str(rootof(x**5 + 2*x**4 + x**3 - 68719476736, 0).n(3)) == '147.' + eq = (531441*x**11 + 3857868*x**10 + 13730229*x**9 + 32597882*x**8 + + 55077472*x**7 + 60452000*x**6 + 32172064*x**5 - 4383808*x**4 - + 11942912*x**3 - 1506304*x**2 + 1453312*x + 512) + a, b = rootof(eq, 1).n(2).as_real_imag() + c, d = rootof(eq, 2).n(2).as_real_imag() + assert a == c + assert b < d + assert b == -d + # issue 6451 + r = rootof(legendre_poly(64, x), 7) + assert r.n(2) == r.n(100).n(2) + # issue 9019 + r0 = rootof(x**2 + 1, 0, radicals=False) + r1 = rootof(x**2 + 1, 1, radicals=False) + assert r0.n(4) == Float(-1.0, 4) * I + assert r1.n(4) == Float(1.0, 4) * I + + # make sure verification is used in case a max/min traps the "root" + assert str(rootof(4*x**5 + 16*x**3 + 12*x**2 + 7, 0).n(3)) == '-0.976' + + # watch out for UnboundLocalError + c = CRootOf(90720*x**6 - 4032*x**4 + 84*x**2 - 1, 0) + assert c._eval_evalf(2) # doesn't fail + + # watch out for imaginary parts that don't want to evaluate + assert str(RootOf(x**16 + 32*x**14 + 508*x**12 + 5440*x**10 + + 39510*x**8 + 204320*x**6 + 755548*x**4 + 1434496*x**2 + + 877969, 10).n(2)) == '-3.4*I' + assert abs(RootOf(x**4 + 10*x**2 + 1, 0).n(2)) < 0.4 + + # check reset and args + r = [RootOf(x**3 + x + 3, i) for i in range(3)] + r[0]._reset() + for ri in r: + i = ri._get_interval() + ri.n(2) + assert i != ri._get_interval() + ri._reset() + assert i == ri._get_interval() + assert i == i.func(*i.args) + + +def test_issue_24978(): + # Irreducible poly with negative leading coeff is normalized + # (factor of -1 is extracted), before being stored as CRootOf.poly. + f = -x**2 + 2 + r = CRootOf(f, 0) + assert r.poly.as_expr() == x**2 - 2 + # An action that prompts calculation of an interval puts r.poly in + # the cache. + r.n() + assert r.poly in rootoftools._reals_cache + + +def test_CRootOf_evalf_caching_bug(): + r = rootof(x**5 - 5*x + 12, 1) + r.n() + a = r._get_interval() + r = rootof(x**5 - 5*x + 12, 1) + r.n() + b = r._get_interval() + assert a == b + + +def test_CRootOf_real_roots(): + assert Poly(x**5 + x + 1).real_roots() == [rootof(x**3 - x**2 + 1, 0)] + assert Poly(x**5 + x + 1).real_roots(radicals=False) == [rootof( + x**3 - x**2 + 1, 0)] + + # https://github.com/sympy/sympy/issues/20902 + p = Poly(-3*x**4 - 10*x**3 - 12*x**2 - 6*x - 1, x, domain='ZZ') + assert CRootOf.real_roots(p) == [S(-1), S(-1), S(-1), S(-1)/3] + + # with real algebraic coefficients + assert Poly(x**3 + sqrt(2)*x**2 - 1, x, extension=True).real_roots() == [ + rootof(x**6 - 2*x**4 - 2*x**3 + 1, 0) + ] + assert Poly(x**5 + sqrt(2) * x**3 - 1, x, extension=True).real_roots() == [ + rootof(x**10 - 2*x**6 - 2*x**5 + 1, 0) + ] + r = rootof(y**5 + y**3 - 1, 0) + assert Poly(x**5 + r*x - 1, x, extension=True).real_roots() ==\ + [ + rootof(x**25 - 5*x**20 + x**17 + 10*x**15 - 3*x**12 - + 10*x**10 + 3*x**7 + 6*x**5 - x**2 - 1, 0) + ] + # roots with multiplicity + assert Poly((x-1) * (x-sqrt(2))**2, x, extension=True).real_roots() ==\ + [ + S(1), sqrt(2), sqrt(2) + ] + + +def test_CRootOf_all_roots(): + assert Poly(x**5 + x + 1).all_roots() == [ + rootof(x**3 - x**2 + 1, 0), + Rational(-1, 2) - sqrt(3)*I/2, + Rational(-1, 2) + sqrt(3)*I/2, + rootof(x**3 - x**2 + 1, 1), + rootof(x**3 - x**2 + 1, 2), + ] + + assert Poly(x**5 + x + 1).all_roots(radicals=False) == [ + rootof(x**3 - x**2 + 1, 0), + rootof(x**2 + x + 1, 0, radicals=False), + rootof(x**2 + x + 1, 1, radicals=False), + rootof(x**3 - x**2 + 1, 1), + rootof(x**3 - x**2 + 1, 2), + ] + + # with real algebraic coefficients + assert Poly(x**3 + sqrt(2)*x**2 - 1, x, extension=True).all_roots() ==\ + [ + rootof(x**6 - 2*x**4 - 2*x**3 + 1, 0), + rootof(x**6 - 2*x**4 - 2*x**3 + 1, 2), + rootof(x**6 - 2*x**4 - 2*x**3 + 1, 3) + ] + # roots with multiplicity + assert Poly((x-1) * (x-sqrt(2))**2 * (x-I) * (x+I), x, extension=True).all_roots() ==\ + [ + S(1), sqrt(2), sqrt(2), -I, I + ] + + # imaginary algebraic coeffs (gaussian domain) + assert Poly(x**2 - I/2, x, extension=True).all_roots() ==\ + [ + S(1)/2 + I/2, + -S(1)/2 - I/2 + ] + + +def test_CRootOf_eval_rational(): + p = legendre_poly(4, x, polys=True) + roots = [r.eval_rational(n=18) for r in p.real_roots()] + for root in roots: + assert isinstance(root, Rational) + roots = [str(root.n(17)) for root in roots] + assert roots == [ + "-0.86113631159405258", + "-0.33998104358485626", + "0.33998104358485626", + "0.86113631159405258", + ] + + +def test_CRootOf_lazy(): + # irreducible poly with both real and complex roots: + f = Poly(x**3 + 2*x + 2) + + # real root: + CRootOf.clear_cache() + r = CRootOf(f, 0) + # Not yet in cache, after construction: + assert r.poly not in rootoftools._reals_cache + assert r.poly not in rootoftools._complexes_cache + r.evalf() + # In cache after evaluation: + assert r.poly in rootoftools._reals_cache + assert r.poly not in rootoftools._complexes_cache + + # complex root: + CRootOf.clear_cache() + r = CRootOf(f, 1) + # Not yet in cache, after construction: + assert r.poly not in rootoftools._reals_cache + assert r.poly not in rootoftools._complexes_cache + r.evalf() + # In cache after evaluation: + assert r.poly in rootoftools._reals_cache + assert r.poly in rootoftools._complexes_cache + + # composite poly with both real and complex roots: + f = Poly((x**2 - 2)*(x**2 + 1)) + + # real root: + CRootOf.clear_cache() + r = CRootOf(f, 0) + # In cache immediately after construction: + assert r.poly in rootoftools._reals_cache + assert r.poly not in rootoftools._complexes_cache + + # complex root: + CRootOf.clear_cache() + r = CRootOf(f, 2) + # In cache immediately after construction: + assert r.poly in rootoftools._reals_cache + assert r.poly in rootoftools._complexes_cache + + +def test_RootSum___new__(): + f = x**3 + x + 3 + + g = Lambda(r, log(r*x)) + s = RootSum(f, g) + + assert isinstance(s, RootSum) is True + + assert RootSum(f**2, g) == 2*RootSum(f, g) + assert RootSum((x - 7)*f**3, g) == log(7*x) + 3*RootSum(f, g) + + # issue 5571 + assert hash(RootSum((x - 7)*f**3, g)) == hash(log(7*x) + 3*RootSum(f, g)) + + raises(MultivariatePolynomialError, lambda: RootSum(x**3 + x + y)) + raises(ValueError, lambda: RootSum(x**2 + 3, lambda x: x)) + + assert RootSum(f, exp) == RootSum(f, Lambda(x, exp(x))) + assert RootSum(f, log) == RootSum(f, Lambda(x, log(x))) + + assert isinstance(RootSum(f, auto=False), RootSum) is True + + assert RootSum(f) == 0 + assert RootSum(f, Lambda(x, x)) == 0 + assert RootSum(f, Lambda(x, x**2)) == -2 + + assert RootSum(f, Lambda(x, 1)) == 3 + assert RootSum(f, Lambda(x, 2)) == 6 + + assert RootSum(f, auto=False).is_commutative is True + + assert RootSum(f, Lambda(x, 1/(x + x**2))) == Rational(11, 3) + assert RootSum(f, Lambda(x, y/(x + x**2))) == Rational(11, 3)*y + + assert RootSum(x**2 - 1, Lambda(x, 3*x**2), x) == 6 + assert RootSum(x**2 - y, Lambda(x, 3*x**2), x) == 6*y + + assert RootSum(x**2 - 1, Lambda(x, z*x**2), x) == 2*z + assert RootSum(x**2 - y, Lambda(x, z*x**2), x) == 2*z*y + + assert RootSum( + x**2 - 1, Lambda(x, exp(x)), quadratic=True) == exp(-1) + exp(1) + + assert RootSum(x**3 + a*x + a**3, tan, x) == \ + RootSum(x**3 + x + 1, Lambda(x, tan(a*x))) + assert RootSum(a**3*x**3 + a*x + 1, tan, x) == \ + RootSum(x**3 + x + 1, Lambda(x, tan(x/a))) + + +def test_RootSum_free_symbols(): + assert RootSum(x**3 + x + 3, Lambda(r, exp(r))).free_symbols == set() + assert RootSum(x**3 + x + 3, Lambda(r, exp(a*r))).free_symbols == {a} + assert RootSum( + x**3 + x + y, Lambda(r, exp(a*r)), x).free_symbols == {a, y} + + +def test_RootSum___eq__(): + f = Lambda(x, exp(x)) + + assert (RootSum(x**3 + x + 1, f) == RootSum(x**3 + x + 1, f)) is True + assert (RootSum(x**3 + x + 1, f) == RootSum(y**3 + y + 1, f)) is True + + assert (RootSum(x**3 + x + 1, f) == RootSum(x**3 + x + 2, f)) is False + assert (RootSum(x**3 + x + 1, f) == RootSum(y**3 + y + 2, f)) is False + + +def test_RootSum_doit(): + rs = RootSum(x**2 + 1, exp) + + assert isinstance(rs, RootSum) is True + assert rs.doit() == exp(-I) + exp(I) + + rs = RootSum(x**2 + a, exp, x) + + assert isinstance(rs, RootSum) is True + assert rs.doit() == exp(-sqrt(-a)) + exp(sqrt(-a)) + + +def test_RootSum_evalf(): + rs = RootSum(x**2 + 1, exp) + + assert rs.evalf(n=20, chop=True).epsilon_eq(Float("1.0806046117362794348")) + assert rs.evalf(n=15, chop=True).epsilon_eq(Float("1.08060461173628")) + + rs = RootSum(x**2 + a, exp, x) + + assert rs.evalf() == rs + + +def test_RootSum_diff(): + f = x**3 + x + 3 + + g = Lambda(r, exp(r*x)) + h = Lambda(r, r*exp(r*x)) + + assert RootSum(f, g).diff(x) == RootSum(f, h) + + +def test_RootSum_subs(): + f = x**3 + x + 3 + g = Lambda(r, exp(r*x)) + + F = y**3 + y + 3 + G = Lambda(r, exp(r*y)) + + assert RootSum(f, g).subs(y, 1) == RootSum(f, g) + assert RootSum(f, g).subs(x, y) == RootSum(F, G) + + +def test_RootSum_rational(): + assert RootSum( + z**5 - z + 1, Lambda(z, z/(x - z))) == (4*x - 5)/(x**5 - x + 1) + + f = 161*z**3 + 115*z**2 + 19*z + 1 + g = Lambda(z, z*log( + -3381*z**4/4 - 3381*z**3/4 - 625*z**2/2 - z*Rational(125, 2) - 5 + exp(x))) + + assert RootSum(f, g).diff(x) == -( + (5*exp(2*x) - 6*exp(x) + 4)*exp(x)/(exp(3*x) - exp(2*x) + 1))/7 + + +def test_RootSum_independent(): + f = (x**3 - a)**2*(x**4 - b)**3 + + g = Lambda(x, 5*tan(x) + 7) + h = Lambda(x, tan(x)) + + r0 = RootSum(x**3 - a, h, x) + r1 = RootSum(x**4 - b, h, x) + + assert RootSum(f, g, x).as_ordered_terms() == [10*r0, 15*r1, 126] + + +def test_issue_7876(): + l1 = Poly(x**6 - x + 1, x).all_roots() + l2 = [rootof(x**6 - x + 1, i) for i in range(6)] + assert frozenset(l1) == frozenset(l2) + + +def test_issue_8316(): + f = Poly(7*x**8 - 9) + assert len(f.all_roots()) == 8 + f = Poly(7*x**8 - 10) + assert len(f.all_roots()) == 8 + + +def test__imag_count(): + from sympy.polys.rootoftools import _imag_count_of_factor + def imag_count(p): + return sum(_imag_count_of_factor(f)*m for f, m in + p.factor_list()[1]) + assert imag_count(Poly(x**6 + 10*x**2 + 1)) == 2 + assert imag_count(Poly(x**2)) == 0 + assert imag_count(Poly([1]*3 + [-1], x)) == 0 + assert imag_count(Poly(x**3 + 1)) == 0 + assert imag_count(Poly(x**2 + 1)) == 2 + assert imag_count(Poly(x**2 - 1)) == 0 + assert imag_count(Poly(x**4 - 1)) == 2 + assert imag_count(Poly(x**4 + 1)) == 0 + assert imag_count(Poly([1, 2, 3], x)) == 0 + assert imag_count(Poly(x**3 + x + 1)) == 0 + assert imag_count(Poly(x**4 + x + 1)) == 0 + def q(r1, r2, p): + return Poly(((x - r1)*(x - r2)).subs(x, x**p), x) + assert imag_count(q(-1, -2, 2)) == 4 + assert imag_count(q(-1, 2, 2)) == 2 + assert imag_count(q(1, 2, 2)) == 0 + assert imag_count(q(1, 2, 4)) == 4 + assert imag_count(q(-1, 2, 4)) == 2 + assert imag_count(q(-1, -2, 4)) == 0 + + +def test_RootOf_is_imaginary(): + r = RootOf(x**4 + 4*x**2 + 1, 1) + i = r._get_interval() + assert r.is_imaginary and i.ax*i.bx <= 0 + + +def test_is_disjoint(): + eq = x**3 + 5*x + 1 + ir = rootof(eq, 0)._get_interval() + ii = rootof(eq, 1)._get_interval() + assert ir.is_disjoint(ii) + assert ii.is_disjoint(ir) + + +def test_pure_key_dict(): + p = D() + assert (x in p) is False + assert (1 in p) is False + p[x] = 1 + assert x in p + assert y in p + assert p[y] == 1 + raises(KeyError, lambda: p[1]) + def dont(k): + p[k] = 2 + raises(ValueError, lambda: dont(1)) + + +@slow +def test_eval_approx_relative(): + CRootOf.clear_cache() + t = [CRootOf(x**3 + 10*x + 1, i) for i in range(3)] + assert [i.eval_rational(1e-1) for i in t] == [ + Rational(-21, 220), Rational(15, 256) - I*805/256, + Rational(15, 256) + I*805/256] + t[0]._reset() + assert [i.eval_rational(1e-1, 1e-4) for i in t] == [ + Rational(-21, 220), Rational(3275, 65536) - I*414645/131072, + Rational(3275, 65536) + I*414645/131072] + assert S(t[0]._get_interval().dx) < 1e-1 + assert S(t[1]._get_interval().dx) < 1e-1 + assert S(t[1]._get_interval().dy) < 1e-4 + assert S(t[2]._get_interval().dx) < 1e-1 + assert S(t[2]._get_interval().dy) < 1e-4 + t[0]._reset() + assert [i.eval_rational(1e-4, 1e-4) for i in t] == [ + Rational(-2001, 20020), Rational(6545, 131072) - I*414645/131072, + Rational(6545, 131072) + I*414645/131072] + assert S(t[0]._get_interval().dx) < 1e-4 + assert S(t[1]._get_interval().dx) < 1e-4 + assert S(t[1]._get_interval().dy) < 1e-4 + assert S(t[2]._get_interval().dx) < 1e-4 + assert S(t[2]._get_interval().dy) < 1e-4 + # in the following, the actual relative precision is + # less than tested, but it should never be greater + t[0]._reset() + assert [i.eval_rational(n=2) for i in t] == [ + Rational(-202201, 2024022), Rational(104755, 2097152) - I*6634255/2097152, + Rational(104755, 2097152) + I*6634255/2097152] + assert abs(S(t[0]._get_interval().dx)/t[0]) < 1e-2 + assert abs(S(t[1]._get_interval().dx)/t[1]).n() < 1e-2 + assert abs(S(t[1]._get_interval().dy)/t[1]).n() < 1e-2 + assert abs(S(t[2]._get_interval().dx)/t[2]).n() < 1e-2 + assert abs(S(t[2]._get_interval().dy)/t[2]).n() < 1e-2 + t[0]._reset() + assert [i.eval_rational(n=3) for i in t] == [ + Rational(-202201, 2024022), Rational(1676045, 33554432) - I*106148135/33554432, + Rational(1676045, 33554432) + I*106148135/33554432] + assert abs(S(t[0]._get_interval().dx)/t[0]) < 1e-3 + assert abs(S(t[1]._get_interval().dx)/t[1]).n() < 1e-3 + assert abs(S(t[1]._get_interval().dy)/t[1]).n() < 1e-3 + assert abs(S(t[2]._get_interval().dx)/t[2]).n() < 1e-3 + assert abs(S(t[2]._get_interval().dy)/t[2]).n() < 1e-3 + + t[0]._reset() + a = [i.eval_approx(2) for i in t] + assert [str(i) for i in a] == [ + '-0.10', '0.05 - 3.2*I', '0.05 + 3.2*I'] + assert all(abs(((a[i] - t[i])/t[i]).n()) < 1e-2 for i in range(len(a))) + + +def test_issue_15920(): + r = rootof(x**5 - x + 1, 0) + p = Integral(x, (x, 1, y)) + assert unchanged(Eq, r, p) + + +def test_issue_19113(): + eq = y**3 - y + 1 + # generator is a canonical x in RootOf + assert str(Poly(eq).real_roots()) == '[CRootOf(x**3 - x + 1, 0)]' + assert str(Poly(eq.subs(y, tan(y))).real_roots() + ) == '[CRootOf(x**3 - x + 1, 0)]' + assert str(Poly(eq.subs(y, tan(x))).real_roots() + ) == '[CRootOf(x**3 - x + 1, 0)]' diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_solvers.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_solvers.py new file mode 100644 index 0000000000000000000000000000000000000000..bf8708314466b6a8676ba1a4438eb84924d0030c --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_solvers.py @@ -0,0 +1,112 @@ +"""Tests for low-level linear systems solver. """ + +from sympy.matrices import Matrix +from sympy.polys.domains import ZZ, QQ +from sympy.polys.fields import field +from sympy.polys.rings import ring +from sympy.polys.solvers import solve_lin_sys, eqs_to_matrix + + +def test_solve_lin_sys_2x2_one(): + domain, x1,x2 = ring("x1,x2", QQ) + eqs = [x1 + x2 - 5, + 2*x1 - x2] + sol = {x1: QQ(5, 3), x2: QQ(10, 3)} + _sol = solve_lin_sys(eqs, domain) + assert _sol == sol and all(s.ring == domain for s in _sol) + +def test_solve_lin_sys_2x4_none(): + domain, x1,x2 = ring("x1,x2", QQ) + eqs = [x1 - 1, + x1 - x2, + x1 - 2*x2, + x2 - 1] + assert solve_lin_sys(eqs, domain) is None + + +def test_solve_lin_sys_3x4_one(): + domain, x1,x2,x3 = ring("x1,x2,x3", QQ) + eqs = [x1 + 2*x2 + 3*x3, + 2*x1 - x2 + x3, + 3*x1 + x2 + x3, + 5*x2 + 2*x3] + sol = {x1: 0, x2: 0, x3: 0} + assert solve_lin_sys(eqs, domain) == sol + +def test_solve_lin_sys_3x3_inf(): + domain, x1,x2,x3 = ring("x1,x2,x3", QQ) + eqs = [x1 - x2 + 2*x3 - 1, + 2*x1 + x2 + x3 - 8, + x1 + x2 - 5] + sol = {x1: -x3 + 3, x2: x3 + 2} + assert solve_lin_sys(eqs, domain) == sol + +def test_solve_lin_sys_3x4_none(): + domain, x1,x2,x3,x4 = ring("x1,x2,x3,x4", QQ) + eqs = [2*x1 + x2 + 7*x3 - 7*x4 - 2, + -3*x1 + 4*x2 - 5*x3 - 6*x4 - 3, + x1 + x2 + 4*x3 - 5*x4 - 2] + assert solve_lin_sys(eqs, domain) is None + + +def test_solve_lin_sys_4x7_inf(): + domain, x1,x2,x3,x4,x5,x6,x7 = ring("x1,x2,x3,x4,x5,x6,x7", QQ) + eqs = [x1 + 4*x2 - x4 + 7*x6 - 9*x7 - 3, + 2*x1 + 8*x2 - x3 + 3*x4 + 9*x5 - 13*x6 + 7*x7 - 9, + 2*x3 - 3*x4 - 4*x5 + 12*x6 - 8*x7 - 1, + -x1 - 4*x2 + 2*x3 + 4*x4 + 8*x5 - 31*x6 + 37*x7 - 4] + sol = {x1: 4 - 4*x2 - 2*x5 - x6 + 3*x7, + x3: 2 - x5 + 3*x6 - 5*x7, + x4: 1 - 2*x5 + 6*x6 - 6*x7} + assert solve_lin_sys(eqs, domain) == sol + +def test_solve_lin_sys_5x5_inf(): + domain, x1,x2,x3,x4,x5 = ring("x1,x2,x3,x4,x5", QQ) + eqs = [x1 - x2 - 2*x3 + x4 + 11*x5 - 13, + x1 - x2 + x3 + x4 + 5*x5 - 16, + 2*x1 - 2*x2 + x4 + 10*x5 - 21, + 2*x1 - 2*x2 - x3 + 3*x4 + 20*x5 - 38, + 2*x1 - 2*x2 + x3 + x4 + 8*x5 - 22] + sol = {x1: 6 + x2 - 3*x5, + x3: 1 + 2*x5, + x4: 9 - 4*x5} + assert solve_lin_sys(eqs, domain) == sol + +def test_solve_lin_sys_6x6_1(): + ground, d,r,e,g,i,j,l,o,m,p,q = field("d,r,e,g,i,j,l,o,m,p,q", ZZ) + domain, c,f,h,k,n,b = ring("c,f,h,k,n,b", ground) + + eqs = [b + q/d - c/d, c*(1/d + 1/e + 1/g) - f/g - q/d, f*(1/g + 1/i + 1/j) - c/g - h/i, h*(1/i + 1/l + 1/m) - f/i - k/m, k*(1/m + 1/o + 1/p) - h/m - n/p, n/p - k/p] + sol = { + b: (e*i*l*q + e*i*m*q + e*i*o*q + e*j*l*q + e*j*m*q + e*j*o*q + e*l*m*q + e*l*o*q + g*i*l*q + g*i*m*q + g*i*o*q + g*j*l*q + g*j*m*q + g*j*o*q + g*l*m*q + g*l*o*q + i*j*l*q + i*j*m*q + i*j*o*q + j*l*m*q + j*l*o*q)/(-d*e*i*l - d*e*i*m - d*e*i*o - d*e*j*l - d*e*j*m - d*e*j*o - d*e*l*m - d*e*l*o - d*g*i*l - d*g*i*m - d*g*i*o - d*g*j*l - d*g*j*m - d*g*j*o - d*g*l*m - d*g*l*o - d*i*j*l - d*i*j*m - d*i*j*o - d*j*l*m - d*j*l*o - e*g*i*l - e*g*i*m - e*g*i*o - e*g*j*l - e*g*j*m - e*g*j*o - e*g*l*m - e*g*l*o - e*i*j*l - e*i*j*m - e*i*j*o - e*j*l*m - e*j*l*o), + c: (-e*g*i*l*q - e*g*i*m*q - e*g*i*o*q - e*g*j*l*q - e*g*j*m*q - e*g*j*o*q - e*g*l*m*q - e*g*l*o*q - e*i*j*l*q - e*i*j*m*q - e*i*j*o*q - e*j*l*m*q - e*j*l*o*q)/(-d*e*i*l - d*e*i*m - d*e*i*o - d*e*j*l - d*e*j*m - d*e*j*o - d*e*l*m - d*e*l*o - d*g*i*l - d*g*i*m - d*g*i*o - d*g*j*l - d*g*j*m - d*g*j*o - d*g*l*m - d*g*l*o - d*i*j*l - d*i*j*m - d*i*j*o - d*j*l*m - d*j*l*o - e*g*i*l - e*g*i*m - e*g*i*o - e*g*j*l - e*g*j*m - e*g*j*o - e*g*l*m - e*g*l*o - e*i*j*l - e*i*j*m - e*i*j*o - e*j*l*m - e*j*l*o), + f: (-e*i*j*l*q - e*i*j*m*q - e*i*j*o*q - e*j*l*m*q - e*j*l*o*q)/(-d*e*i*l - d*e*i*m - d*e*i*o - d*e*j*l - d*e*j*m - d*e*j*o - d*e*l*m - d*e*l*o - d*g*i*l - d*g*i*m - d*g*i*o - d*g*j*l - d*g*j*m - d*g*j*o - d*g*l*m - d*g*l*o - d*i*j*l - d*i*j*m - d*i*j*o - d*j*l*m - d*j*l*o - e*g*i*l - e*g*i*m - e*g*i*o - e*g*j*l - e*g*j*m - e*g*j*o - e*g*l*m - e*g*l*o - e*i*j*l - e*i*j*m - e*i*j*o - e*j*l*m - e*j*l*o), + h: (-e*j*l*m*q - e*j*l*o*q)/(-d*e*i*l - d*e*i*m - d*e*i*o - d*e*j*l - d*e*j*m - d*e*j*o - d*e*l*m - d*e*l*o - d*g*i*l - d*g*i*m - d*g*i*o - d*g*j*l - d*g*j*m - d*g*j*o - d*g*l*m - d*g*l*o - d*i*j*l - d*i*j*m - d*i*j*o - d*j*l*m - d*j*l*o - e*g*i*l - e*g*i*m - e*g*i*o - e*g*j*l - e*g*j*m - e*g*j*o - e*g*l*m - e*g*l*o - e*i*j*l - e*i*j*m - e*i*j*o - e*j*l*m - e*j*l*o), + k: e*j*l*o*q/(d*e*i*l + d*e*i*m + d*e*i*o + d*e*j*l + d*e*j*m + d*e*j*o + d*e*l*m + d*e*l*o + d*g*i*l + d*g*i*m + d*g*i*o + d*g*j*l + d*g*j*m + d*g*j*o + d*g*l*m + d*g*l*o + d*i*j*l + d*i*j*m + d*i*j*o + d*j*l*m + d*j*l*o + e*g*i*l + e*g*i*m + e*g*i*o + e*g*j*l + e*g*j*m + e*g*j*o + e*g*l*m + e*g*l*o + e*i*j*l + e*i*j*m + e*i*j*o + e*j*l*m + e*j*l*o), + n: e*j*l*o*q/(d*e*i*l + d*e*i*m + d*e*i*o + d*e*j*l + d*e*j*m + d*e*j*o + d*e*l*m + d*e*l*o + d*g*i*l + d*g*i*m + d*g*i*o + d*g*j*l + d*g*j*m + d*g*j*o + d*g*l*m + d*g*l*o + d*i*j*l + d*i*j*m + d*i*j*o + d*j*l*m + d*j*l*o + e*g*i*l + e*g*i*m + e*g*i*o + e*g*j*l + e*g*j*m + e*g*j*o + e*g*l*m + e*g*l*o + e*i*j*l + e*i*j*m + e*i*j*o + e*j*l*m + e*j*l*o), + } + + assert solve_lin_sys(eqs, domain) == sol + +def test_solve_lin_sys_6x6_2(): + ground, d,r,e,g,i,j,l,o,m,p,q = field("d,r,e,g,i,j,l,o,m,p,q", ZZ) + domain, c,f,h,k,n,b = ring("c,f,h,k,n,b", ground) + + eqs = [b + r/d - c/d, c*(1/d + 1/e + 1/g) - f/g - r/d, f*(1/g + 1/i + 1/j) - c/g - h/i, h*(1/i + 1/l + 1/m) - f/i - k/m, k*(1/m + 1/o + 1/p) - h/m - n/p, n*(1/p + 1/q) - k/p] + sol = { + b: -((l*q*e*o + l*q*g*o + i*m*q*e + i*l*q*e + i*l*p*e + i*j*o*q + j*e*o*q + g*j*o*q + i*e*o*q + g*i*o*q + e*l*o*p + e*l*m*p + e*l*m*o + e*i*o*p + e*i*m*p + e*i*m*o + e*i*l*o + j*e*o*p + j*e*m*q + j*e*m*p + j*e*m*o + j*l*m*q + j*l*m*p + j*l*m*o + i*j*m*p + i*j*m*o + i*j*l*q + i*j*l*o + i*j*m*q + j*l*o*p + j*e*l*o + g*j*o*p + g*j*m*q + g*j*m*p + i*j*l*p + i*j*o*p + j*e*l*q + j*e*l*p + j*l*o*q + g*j*m*o + g*j*l*q + g*j*l*p + g*j*l*o + g*l*o*p + g*l*m*p + g*l*m*o + g*i*m*o + g*i*o*p + g*i*m*q + g*i*m*p + g*i*l*q + g*i*l*p + g*i*l*o + l*m*q*e + l*m*q*g)*r)/(l*q*d*e*o + l*q*d*g*o + l*q*e*g*o + i*j*d*o*q + i*j*e*o*q + j*d*e*o*q + g*j*d*o*q + g*j*e*o*q + g*i*e*o*q + i*d*e*o*q + g*i*d*o*q + g*i*d*o*p + g*i*d*m*q + g*i*d*m*p + g*i*d*m*o + g*i*d*l*q + g*i*d*l*p + g*i*d*l*o + g*e*l*m*p + g*e*l*o*p + g*j*e*l*q + g*e*l*m*o + g*j*e*m*p + g*j*e*m*o + d*e*l*m*p + d*e*l*m*o + i*d*e*m*p + g*j*e*l*p + g*j*e*l*o + d*e*l*o*p + i*j*d*l*o + i*j*e*o*p + i*j*e*m*q + i*j*d*m*q + i*j*d*m*p + i*j*d*m*o + i*j*d*l*q + i*j*d*l*p + i*j*e*m*p + i*j*e*m*o + i*j*e*l*q + i*j*e*l*p + i*j*e*l*o + i*d*e*m*q + i*d*e*m*o + i*d*e*l*q + i*d*e*l*p + j*d*l*o*p + j*d*e*l*o + g*j*d*o*p + g*j*d*m*q + g*j*d*m*p + g*j*d*m*o + g*j*d*l*q + g*j*d*l*p + g*j*d*l*o + g*j*e*o*p + g*j*e*m*q + g*d*l*o*p + g*d*l*m*p + g*d*l*m*o + j*d*e*m*p + i*d*e*o*p + j*e*o*q*l + j*e*o*p*l + j*e*m*q*l + j*d*e*o*p + j*d*e*m*q + i*j*d*o*p + g*i*e*o*p + j*d*e*m*o + j*d*e*l*q + j*d*e*l*p + j*e*m*p*l + j*e*m*o*l + g*i*e*m*q + g*i*e*m*p + g*i*e*m*o + g*i*e*l*q + g*i*e*l*p + g*i*e*l*o + j*d*l*o*q + j*d*l*m*q + j*d*l*m*p + j*d*l*m*o + i*d*e*l*o + l*m*q*d*e + l*m*q*d*g + l*m*q*e*g), + c: (r*e*(l*q*g*o + i*j*o*q + g*j*o*q + g*i*o*q + j*l*m*q + j*l*m*p + j*l*m*o + i*j*m*p + i*j*m*o + i*j*l*q + i*j*l*o + i*j*m*q + j*l*o*p + g*j*o*p + g*j*m*q + g*j*m*p + i*j*l*p + i*j*o*p + j*l*o*q + g*j*m*o + g*j*l*q + g*j*l*p + g*j*l*o + g*l*o*p + g*l*m*p + g*l*m*o + g*i*m*o + g*i*o*p + g*i*m*q + g*i*m*p + g*i*l*q + g*i*l*p + g*i*l*o + l*m*q*g))/(l*q*d*e*o + l*q*d*g*o + l*q*e*g*o + i*j*d*o*q + i*j*e*o*q + j*d*e*o*q + g*j*d*o*q + g*j*e*o*q + g*i*e*o*q + i*d*e*o*q + g*i*d*o*q + g*i*d*o*p + g*i*d*m*q + g*i*d*m*p + g*i*d*m*o + g*i*d*l*q + g*i*d*l*p + g*i*d*l*o + g*e*l*m*p + g*e*l*o*p + g*j*e*l*q + g*e*l*m*o + g*j*e*m*p + g*j*e*m*o + d*e*l*m*p + d*e*l*m*o + i*d*e*m*p + g*j*e*l*p + g*j*e*l*o + d*e*l*o*p + i*j*d*l*o + i*j*e*o*p + i*j*e*m*q + i*j*d*m*q + i*j*d*m*p + i*j*d*m*o + i*j*d*l*q + i*j*d*l*p + i*j*e*m*p + i*j*e*m*o + i*j*e*l*q + i*j*e*l*p + i*j*e*l*o + i*d*e*m*q + i*d*e*m*o + i*d*e*l*q + i*d*e*l*p + j*d*l*o*p + j*d*e*l*o + g*j*d*o*p + g*j*d*m*q + g*j*d*m*p + g*j*d*m*o + g*j*d*l*q + g*j*d*l*p + g*j*d*l*o + g*j*e*o*p + g*j*e*m*q + g*d*l*o*p + g*d*l*m*p + g*d*l*m*o + j*d*e*m*p + i*d*e*o*p + j*e*o*q*l + j*e*o*p*l + j*e*m*q*l + j*d*e*o*p + j*d*e*m*q + i*j*d*o*p + g*i*e*o*p + j*d*e*m*o + j*d*e*l*q + j*d*e*l*p + j*e*m*p*l + j*e*m*o*l + g*i*e*m*q + g*i*e*m*p + g*i*e*m*o + g*i*e*l*q + g*i*e*l*p + g*i*e*l*o + j*d*l*o*q + j*d*l*m*q + j*d*l*m*p + j*d*l*m*o + i*d*e*l*o + l*m*q*d*e + l*m*q*d*g + l*m*q*e*g), + f: (r*e*j*(l*q*o + l*o*p + l*m*q + l*m*p + l*m*o + i*o*q + i*o*p + i*m*q + i*m*p + i*m*o + i*l*q + i*l*p + i*l*o))/(l*q*d*e*o + l*q*d*g*o + l*q*e*g*o + i*j*d*o*q + i*j*e*o*q + j*d*e*o*q + g*j*d*o*q + g*j*e*o*q + g*i*e*o*q + i*d*e*o*q + g*i*d*o*q + g*i*d*o*p + g*i*d*m*q + g*i*d*m*p + g*i*d*m*o + g*i*d*l*q + g*i*d*l*p + g*i*d*l*o + g*e*l*m*p + g*e*l*o*p + g*j*e*l*q + g*e*l*m*o + g*j*e*m*p + g*j*e*m*o + d*e*l*m*p + d*e*l*m*o + i*d*e*m*p + g*j*e*l*p + g*j*e*l*o + d*e*l*o*p + i*j*d*l*o + i*j*e*o*p + i*j*e*m*q + i*j*d*m*q + i*j*d*m*p + i*j*d*m*o + i*j*d*l*q + i*j*d*l*p + i*j*e*m*p + i*j*e*m*o + i*j*e*l*q + i*j*e*l*p + i*j*e*l*o + i*d*e*m*q + i*d*e*m*o + i*d*e*l*q + i*d*e*l*p + j*d*l*o*p + j*d*e*l*o + g*j*d*o*p + g*j*d*m*q + g*j*d*m*p + g*j*d*m*o + g*j*d*l*q + g*j*d*l*p + g*j*d*l*o + g*j*e*o*p + g*j*e*m*q + g*d*l*o*p + g*d*l*m*p + g*d*l*m*o + j*d*e*m*p + i*d*e*o*p + j*e*o*q*l + j*e*o*p*l + j*e*m*q*l + j*d*e*o*p + j*d*e*m*q + i*j*d*o*p + g*i*e*o*p + j*d*e*m*o + j*d*e*l*q + j*d*e*l*p + j*e*m*p*l + j*e*m*o*l + g*i*e*m*q + g*i*e*m*p + g*i*e*m*o + g*i*e*l*q + g*i*e*l*p + g*i*e*l*o + j*d*l*o*q + j*d*l*m*q + j*d*l*m*p + j*d*l*m*o + i*d*e*l*o + l*m*q*d*e + l*m*q*d*g + l*m*q*e*g), + h: (j*e*r*l*(o*q + o*p + m*q + m*p + m*o))/(l*q*d*e*o + l*q*d*g*o + l*q*e*g*o + i*j*d*o*q + i*j*e*o*q + j*d*e*o*q + g*j*d*o*q + g*j*e*o*q + g*i*e*o*q + i*d*e*o*q + g*i*d*o*q + g*i*d*o*p + g*i*d*m*q + g*i*d*m*p + g*i*d*m*o + g*i*d*l*q + g*i*d*l*p + g*i*d*l*o + g*e*l*m*p + g*e*l*o*p + g*j*e*l*q + g*e*l*m*o + g*j*e*m*p + g*j*e*m*o + d*e*l*m*p + d*e*l*m*o + i*d*e*m*p + g*j*e*l*p + g*j*e*l*o + d*e*l*o*p + i*j*d*l*o + i*j*e*o*p + i*j*e*m*q + i*j*d*m*q + i*j*d*m*p + i*j*d*m*o + i*j*d*l*q + i*j*d*l*p + i*j*e*m*p + i*j*e*m*o + i*j*e*l*q + i*j*e*l*p + i*j*e*l*o + i*d*e*m*q + i*d*e*m*o + i*d*e*l*q + i*d*e*l*p + j*d*l*o*p + j*d*e*l*o + g*j*d*o*p + g*j*d*m*q + g*j*d*m*p + g*j*d*m*o + g*j*d*l*q + g*j*d*l*p + g*j*d*l*o + g*j*e*o*p + g*j*e*m*q + g*d*l*o*p + g*d*l*m*p + g*d*l*m*o + j*d*e*m*p + i*d*e*o*p + j*e*o*q*l + j*e*o*p*l + j*e*m*q*l + j*d*e*o*p + j*d*e*m*q + i*j*d*o*p + g*i*e*o*p + j*d*e*m*o + j*d*e*l*q + j*d*e*l*p + j*e*m*p*l + j*e*m*o*l + g*i*e*m*q + g*i*e*m*p + g*i*e*m*o + g*i*e*l*q + g*i*e*l*p + g*i*e*l*o + j*d*l*o*q + j*d*l*m*q + j*d*l*m*p + j*d*l*m*o + i*d*e*l*o + l*m*q*d*e + l*m*q*d*g + l*m*q*e*g), + k: (j*e*r*o*l*(q + p))/(l*q*d*e*o + l*q*d*g*o + l*q*e*g*o + i*j*d*o*q + i*j*e*o*q + j*d*e*o*q + g*j*d*o*q + g*j*e*o*q + g*i*e*o*q + i*d*e*o*q + g*i*d*o*q + g*i*d*o*p + g*i*d*m*q + g*i*d*m*p + g*i*d*m*o + g*i*d*l*q + g*i*d*l*p + g*i*d*l*o + g*e*l*m*p + g*e*l*o*p + g*j*e*l*q + g*e*l*m*o + g*j*e*m*p + g*j*e*m*o + d*e*l*m*p + d*e*l*m*o + i*d*e*m*p + g*j*e*l*p + g*j*e*l*o + d*e*l*o*p + i*j*d*l*o + i*j*e*o*p + i*j*e*m*q + i*j*d*m*q + i*j*d*m*p + i*j*d*m*o + i*j*d*l*q + i*j*d*l*p + i*j*e*m*p + i*j*e*m*o + i*j*e*l*q + i*j*e*l*p + i*j*e*l*o + i*d*e*m*q + i*d*e*m*o + i*d*e*l*q + i*d*e*l*p + j*d*l*o*p + j*d*e*l*o + g*j*d*o*p + g*j*d*m*q + g*j*d*m*p + g*j*d*m*o + g*j*d*l*q + g*j*d*l*p + g*j*d*l*o + g*j*e*o*p + g*j*e*m*q + g*d*l*o*p + g*d*l*m*p + g*d*l*m*o + j*d*e*m*p + i*d*e*o*p + j*e*o*q*l + j*e*o*p*l + j*e*m*q*l + j*d*e*o*p + j*d*e*m*q + i*j*d*o*p + g*i*e*o*p + j*d*e*m*o + j*d*e*l*q + j*d*e*l*p + j*e*m*p*l + j*e*m*o*l + g*i*e*m*q + g*i*e*m*p + g*i*e*m*o + g*i*e*l*q + g*i*e*l*p + g*i*e*l*o + j*d*l*o*q + j*d*l*m*q + j*d*l*m*p + j*d*l*m*o + i*d*e*l*o + l*m*q*d*e + l*m*q*d*g + l*m*q*e*g), + n: (j*e*r*o*q*l)/(l*q*d*e*o + l*q*d*g*o + l*q*e*g*o + i*j*d*o*q + i*j*e*o*q + j*d*e*o*q + g*j*d*o*q + g*j*e*o*q + g*i*e*o*q + i*d*e*o*q + g*i*d*o*q + g*i*d*o*p + g*i*d*m*q + g*i*d*m*p + g*i*d*m*o + g*i*d*l*q + g*i*d*l*p + g*i*d*l*o + g*e*l*m*p + g*e*l*o*p + g*j*e*l*q + g*e*l*m*o + g*j*e*m*p + g*j*e*m*o + d*e*l*m*p + d*e*l*m*o + i*d*e*m*p + g*j*e*l*p + g*j*e*l*o + d*e*l*o*p + i*j*d*l*o + i*j*e*o*p + i*j*e*m*q + i*j*d*m*q + i*j*d*m*p + i*j*d*m*o + i*j*d*l*q + i*j*d*l*p + i*j*e*m*p + i*j*e*m*o + i*j*e*l*q + i*j*e*l*p + i*j*e*l*o + i*d*e*m*q + i*d*e*m*o + i*d*e*l*q + i*d*e*l*p + j*d*l*o*p + j*d*e*l*o + g*j*d*o*p + g*j*d*m*q + g*j*d*m*p + g*j*d*m*o + g*j*d*l*q + g*j*d*l*p + g*j*d*l*o + g*j*e*o*p + g*j*e*m*q + g*d*l*o*p + g*d*l*m*p + g*d*l*m*o + j*d*e*m*p + i*d*e*o*p + j*e*o*q*l + j*e*o*p*l + j*e*m*q*l + j*d*e*o*p + j*d*e*m*q + i*j*d*o*p + g*i*e*o*p + j*d*e*m*o + j*d*e*l*q + j*d*e*l*p + j*e*m*p*l + j*e*m*o*l + g*i*e*m*q + g*i*e*m*p + g*i*e*m*o + g*i*e*l*q + g*i*e*l*p + g*i*e*l*o + j*d*l*o*q + j*d*l*m*q + j*d*l*m*p + j*d*l*m*o + i*d*e*l*o + l*m*q*d*e + l*m*q*d*g + l*m*q*e*g), + } + + assert solve_lin_sys(eqs, domain) == sol + +def test_eqs_to_matrix(): + domain, x1,x2 = ring("x1,x2", QQ) + eqs_coeff = [{x1: QQ(1), x2: QQ(1)}, {x1: QQ(2), x2: QQ(-1)}] + eqs_rhs = [QQ(-5), QQ(0)] + M = eqs_to_matrix(eqs_coeff, eqs_rhs, [x1, x2], QQ) + assert M.to_Matrix() == Matrix([[1, 1, 5], [2, -1, 0]]) diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_specialpolys.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_specialpolys.py new file mode 100644 index 0000000000000000000000000000000000000000..39f551c9e70b5c2bae748ea681b9c8a8cb349fe1 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_specialpolys.py @@ -0,0 +1,152 @@ +"""Tests for functions for generating interesting polynomials. """ + +from sympy.core.add import Add +from sympy.core.symbol import symbols +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.ntheory.generate import prime +from sympy.polys.domains.integerring import ZZ +from sympy.polys.polytools import Poly +from sympy.utilities.iterables import permute_signs +from sympy.testing.pytest import raises + +from sympy.polys.specialpolys import ( + swinnerton_dyer_poly, + cyclotomic_poly, + symmetric_poly, + random_poly, + interpolating_poly, + fateman_poly_F_1, + dmp_fateman_poly_F_1, + fateman_poly_F_2, + dmp_fateman_poly_F_2, + fateman_poly_F_3, + dmp_fateman_poly_F_3, +) + +from sympy.abc import x, y, z + + +def test_swinnerton_dyer_poly(): + raises(ValueError, lambda: swinnerton_dyer_poly(0, x)) + + assert swinnerton_dyer_poly(1, x, polys=True) == Poly(x**2 - 2) + + assert swinnerton_dyer_poly(1, x) == x**2 - 2 + assert swinnerton_dyer_poly(2, x) == x**4 - 10*x**2 + 1 + assert swinnerton_dyer_poly( + 3, x) == x**8 - 40*x**6 + 352*x**4 - 960*x**2 + 576 + # we only need to check that the polys arg works but + # we may as well test that the roots are correct + p = [sqrt(prime(i)) for i in range(1, 5)] + assert str([i.n(3) for i in + swinnerton_dyer_poly(4, polys=True).all_roots()] + ) == str(sorted([Add(*i).n(3) for i in permute_signs(p)])) + + +def test_cyclotomic_poly(): + raises(ValueError, lambda: cyclotomic_poly(0, x)) + + assert cyclotomic_poly(1, x, polys=True) == Poly(x - 1) + + assert cyclotomic_poly(1, x) == x - 1 + assert cyclotomic_poly(2, x) == x + 1 + assert cyclotomic_poly(3, x) == x**2 + x + 1 + assert cyclotomic_poly(4, x) == x**2 + 1 + assert cyclotomic_poly(5, x) == x**4 + x**3 + x**2 + x + 1 + assert cyclotomic_poly(6, x) == x**2 - x + 1 + + +def test_symmetric_poly(): + raises(ValueError, lambda: symmetric_poly(-1, x, y, z)) + raises(ValueError, lambda: symmetric_poly(5, x, y, z)) + + assert symmetric_poly(1, x, y, z, polys=True) == Poly(x + y + z) + assert symmetric_poly(1, (x, y, z), polys=True) == Poly(x + y + z) + + assert symmetric_poly(0, x, y, z) == 1 + assert symmetric_poly(1, x, y, z) == x + y + z + assert symmetric_poly(2, x, y, z) == x*y + x*z + y*z + assert symmetric_poly(3, x, y, z) == x*y*z + + +def test_random_poly(): + poly = random_poly(x, 10, -100, 100, polys=False) + + assert Poly(poly).degree() == 10 + assert all(-100 <= coeff <= 100 for coeff in Poly(poly).coeffs()) is True + + poly = random_poly(x, 10, -100, 100, polys=True) + + assert poly.degree() == 10 + assert all(-100 <= coeff <= 100 for coeff in poly.coeffs()) is True + + +def test_interpolating_poly(): + x0, x1, x2, x3, y0, y1, y2, y3 = symbols('x:4, y:4') + + assert interpolating_poly(0, x) == 0 + assert interpolating_poly(1, x) == y0 + + assert interpolating_poly(2, x) == \ + y0*(x - x1)/(x0 - x1) + y1*(x - x0)/(x1 - x0) + + assert interpolating_poly(3, x) == \ + y0*(x - x1)*(x - x2)/((x0 - x1)*(x0 - x2)) + \ + y1*(x - x0)*(x - x2)/((x1 - x0)*(x1 - x2)) + \ + y2*(x - x0)*(x - x1)/((x2 - x0)*(x2 - x1)) + + assert interpolating_poly(4, x) == \ + y0*(x - x1)*(x - x2)*(x - x3)/((x0 - x1)*(x0 - x2)*(x0 - x3)) + \ + y1*(x - x0)*(x - x2)*(x - x3)/((x1 - x0)*(x1 - x2)*(x1 - x3)) + \ + y2*(x - x0)*(x - x1)*(x - x3)/((x2 - x0)*(x2 - x1)*(x2 - x3)) + \ + y3*(x - x0)*(x - x1)*(x - x2)/((x3 - x0)*(x3 - x1)*(x3 - x2)) + + raises(ValueError, lambda: + interpolating_poly(2, x, (x, 2), (1, 3))) + raises(ValueError, lambda: + interpolating_poly(2, x, (x + y, 2), (1, 3))) + raises(ValueError, lambda: + interpolating_poly(2, x + y, (x, 2), (1, 3))) + raises(ValueError, lambda: + interpolating_poly(2, 3, (4, 5), (6, 7))) + raises(ValueError, lambda: + interpolating_poly(2, 3, (4, 5), (6, 7, 8))) + assert interpolating_poly(0, x, (1, 2), (3, 4)) == 0 + assert interpolating_poly(1, x, (1, 2), (3, 4)) == 3 + assert interpolating_poly(2, x, (1, 2), (3, 4)) == x + 2 + + +def test_fateman_poly_F_1(): + f, g, h = fateman_poly_F_1(1) + F, G, H = dmp_fateman_poly_F_1(1, ZZ) + + assert [ t.rep.to_list() for t in [f, g, h] ] == [F, G, H] + + f, g, h = fateman_poly_F_1(3) + F, G, H = dmp_fateman_poly_F_1(3, ZZ) + + assert [ t.rep.to_list() for t in [f, g, h] ] == [F, G, H] + + +def test_fateman_poly_F_2(): + f, g, h = fateman_poly_F_2(1) + F, G, H = dmp_fateman_poly_F_2(1, ZZ) + + assert [ t.rep.to_list() for t in [f, g, h] ] == [F, G, H] + + f, g, h = fateman_poly_F_2(3) + F, G, H = dmp_fateman_poly_F_2(3, ZZ) + + assert [ t.rep.to_list() for t in [f, g, h] ] == [F, G, H] + + +def test_fateman_poly_F_3(): + f, g, h = fateman_poly_F_3(1) + F, G, H = dmp_fateman_poly_F_3(1, ZZ) + + assert [ t.rep.to_list() for t in [f, g, h] ] == [F, G, H] + + f, g, h = fateman_poly_F_3(3) + F, G, H = dmp_fateman_poly_F_3(3, ZZ) + + assert [ t.rep.to_list() for t in [f, g, h] ] == [F, G, H] diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_sqfreetools.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_sqfreetools.py new file mode 100644 index 0000000000000000000000000000000000000000..b772a05a50e2eacd5a7c80352b1eadd52c69c3fa --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_sqfreetools.py @@ -0,0 +1,160 @@ +"""Tests for square-free decomposition algorithms and related tools. """ + +from sympy.polys.rings import ring +from sympy.polys.domains import FF, ZZ, QQ +from sympy.polys.specialpolys import f_polys + +from sympy.testing.pytest import raises +from sympy.external.gmpy import MPQ + +f_0, f_1, f_2, f_3, f_4, f_5, f_6 = f_polys() + +def test_dup_sqf(): + R, x = ring("x", ZZ) + + assert R.dup_sqf_part(0) == 0 + assert R.dup_sqf_p(0) is True + + assert R.dup_sqf_part(7) == 1 + assert R.dup_sqf_p(7) is True + + assert R.dup_sqf_part(2*x + 2) == x + 1 + assert R.dup_sqf_p(2*x + 2) is True + + assert R.dup_sqf_part(x**3 + x + 1) == x**3 + x + 1 + assert R.dup_sqf_p(x**3 + x + 1) is True + + assert R.dup_sqf_part(-x**3 + x + 1) == x**3 - x - 1 + assert R.dup_sqf_p(-x**3 + x + 1) is True + + assert R.dup_sqf_part(2*x**3 + 3*x**2) == 2*x**2 + 3*x + assert R.dup_sqf_p(2*x**3 + 3*x**2) is False + + assert R.dup_sqf_part(-2*x**3 + 3*x**2) == 2*x**2 - 3*x + assert R.dup_sqf_p(-2*x**3 + 3*x**2) is False + + assert R.dup_sqf_list(0) == (0, []) + assert R.dup_sqf_list(1) == (1, []) + + assert R.dup_sqf_list(x) == (1, [(x, 1)]) + assert R.dup_sqf_list(2*x**2) == (2, [(x, 2)]) + assert R.dup_sqf_list(3*x**3) == (3, [(x, 3)]) + + assert R.dup_sqf_list(-x**5 + x**4 + x - 1) == \ + (-1, [(x**3 + x**2 + x + 1, 1), (x - 1, 2)]) + assert R.dup_sqf_list(x**8 + 6*x**6 + 12*x**4 + 8*x**2) == \ + ( 1, [(x, 2), (x**2 + 2, 3)]) + + assert R.dup_sqf_list(2*x**2 + 4*x + 2) == (2, [(x + 1, 2)]) + + R, x = ring("x", QQ) + assert R.dup_sqf_list(2*x**2 + 4*x + 2) == (2, [(x + 1, 2)]) + + R, x = ring("x", FF(2)) + assert R.dup_sqf_list(x**2 + 1) == (1, [(x + 1, 2)]) + + R, x = ring("x", FF(3)) + assert R.dup_sqf_list(x**10 + 2*x**7 + 2*x**4 + x) == \ + (1, [(x, 1), + (x + 1, 3), + (x + 2, 6)]) + + R1, x = ring("x", ZZ) + R2, y = ring("y", FF(3)) + + f = x**3 + 1 + g = y**3 + 1 + + assert R1.dup_sqf_part(f) == f + assert R2.dup_sqf_part(g) == y + 1 + + assert R1.dup_sqf_p(f) is True + assert R2.dup_sqf_p(g) is False + + R, x, y = ring("x,y", ZZ) + + A = x**4 - 3*x**2 + 6 + D = x**6 - 5*x**4 + 5*x**2 + 4 + + f, g = D, R.dmp_sub(A, R.dmp_mul(R.dmp_diff(D, 1), y)) + res = R.dmp_resultant(f, g) + h = (4*y**2 + 1).drop(x) + + assert R.drop(x).dup_sqf_list(res) == (45796, [(h, 3)]) + + Rt, t = ring("t", ZZ) + R, x = ring("x", Rt) + assert R.dup_sqf_list_include(t**3*x**2) == [(t**3, 1), (x, 2)] + + +def test_dmp_sqf(): + R, x, y = ring("x,y", ZZ) + assert R.dmp_sqf_part(0) == 0 + assert R.dmp_sqf_p(0) is True + + assert R.dmp_sqf_part(7) == 1 + assert R.dmp_sqf_p(7) is True + + assert R.dmp_sqf_list(3) == (3, []) + assert R.dmp_sqf_list_include(3) == [(3, 1)] + + R, x, y, z = ring("x,y,z", ZZ) + assert R.dmp_sqf_p(f_0) is True + assert R.dmp_sqf_p(f_0**2) is False + assert R.dmp_sqf_p(f_1) is True + assert R.dmp_sqf_p(f_1**2) is False + assert R.dmp_sqf_p(f_2) is True + assert R.dmp_sqf_p(f_2**2) is False + assert R.dmp_sqf_p(f_3) is True + assert R.dmp_sqf_p(f_3**2) is False + assert R.dmp_sqf_p(f_5) is False + assert R.dmp_sqf_p(f_5**2) is False + + assert R.dmp_sqf_p(f_4) is True + assert R.dmp_sqf_part(f_4) == -f_4 + + assert R.dmp_sqf_part(f_5) == x + y - z + + R, x, y, z, t = ring("x,y,z,t", ZZ) + assert R.dmp_sqf_p(f_6) is True + assert R.dmp_sqf_part(f_6) == f_6 + + R, x = ring("x", ZZ) + f = -x**5 + x**4 + x - 1 + + assert R.dmp_sqf_list(f) == (-1, [(x**3 + x**2 + x + 1, 1), (x - 1, 2)]) + assert R.dmp_sqf_list_include(f) == [(-x**3 - x**2 - x - 1, 1), (x - 1, 2)] + + R, x, y = ring("x,y", ZZ) + f = -x**5 + x**4 + x - 1 + + assert R.dmp_sqf_list(f) == (-1, [(x**3 + x**2 + x + 1, 1), (x - 1, 2)]) + assert R.dmp_sqf_list_include(f) == [(-x**3 - x**2 - x - 1, 1), (x - 1, 2)] + + f = -x**2 + 2*x - 1 + assert R.dmp_sqf_list_include(f) == [(-1, 1), (x - 1, 2)] + + f = (y**2 + 1)**2*(x**2 + 2*x + 2) + assert R.dmp_sqf_p(f) is False + assert R.dmp_sqf_list(f) == (1, [(x**2 + 2*x + 2, 1), (y**2 + 1, 2)]) + + R, x, y = ring("x,y", FF(2)) + raises(NotImplementedError, lambda: R.dmp_sqf_list(y**2 + 1)) + + +def test_dup_gff_list(): + R, x = ring("x", ZZ) + + f = x**5 + 2*x**4 - x**3 - 2*x**2 + assert R.dup_gff_list(f) == [(x, 1), (x + 2, 4)] + + g = x**9 - 20*x**8 + 166*x**7 - 744*x**6 + 1965*x**5 - 3132*x**4 + 2948*x**3 - 1504*x**2 + 320*x + assert R.dup_gff_list(g) == [(x**2 - 5*x + 4, 1), (x**2 - 5*x + 4, 2), (x, 3)] + + raises(ValueError, lambda: R.dup_gff_list(0)) + +def test_issue_26178(): + R, x, y, z = ring(['x', 'y', 'z'], QQ) + assert (x**2 - 2*y**2 + 1).sqf_list() == (MPQ(1,1), [(x**2 - 2*y**2 + 1, 1)]) + assert (x**2 - 2*z**2 + 1).sqf_list() == (MPQ(1,1), [(x**2 - 2*z**2 + 1, 1)]) + assert (y**2 - 2*z**2 + 1).sqf_list() == (MPQ(1,1), [(y**2 - 2*z**2 + 1, 1)]) diff --git a/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_subresultants_qq_zz.py b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_subresultants_qq_zz.py new file mode 100644 index 0000000000000000000000000000000000000000..7f7560dfeaf93b20f7cf68cdc597c024cb519cca --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/polys/tests/test_subresultants_qq_zz.py @@ -0,0 +1,347 @@ +from sympy.core.symbol import Symbol +from sympy.polys.polytools import (pquo, prem, sturm, subresultants) +from sympy.matrices import Matrix +from sympy.polys.subresultants_qq_zz import (sylvester, res, res_q, res_z, bezout, + subresultants_sylv, modified_subresultants_sylv, + subresultants_bezout, modified_subresultants_bezout, + backward_eye, + sturm_pg, sturm_q, sturm_amv, euclid_pg, euclid_q, + euclid_amv, modified_subresultants_pg, subresultants_pg, + subresultants_amv_q, quo_z, rem_z, subresultants_amv, + modified_subresultants_amv, subresultants_rem, + subresultants_vv, subresultants_vv_2) + + +def test_sylvester(): + x = Symbol('x') + + assert sylvester(x**3 -7, 0, x) == sylvester(x**3 -7, 0, x, 1) == Matrix([[0]]) + assert sylvester(0, x**3 -7, x) == sylvester(0, x**3 -7, x, 1) == Matrix([[0]]) + assert sylvester(x**3 -7, 0, x, 2) == Matrix([[0]]) + assert sylvester(0, x**3 -7, x, 2) == Matrix([[0]]) + + assert sylvester(x**3 -7, 7, x).det() == sylvester(x**3 -7, 7, x, 1).det() == 343 + assert sylvester(7, x**3 -7, x).det() == sylvester(7, x**3 -7, x, 1).det() == 343 + assert sylvester(x**3 -7, 7, x, 2).det() == -343 + assert sylvester(7, x**3 -7, x, 2).det() == 343 + + assert sylvester(3, 7, x).det() == sylvester(3, 7, x, 1).det() == sylvester(3, 7, x, 2).det() == 1 + + assert sylvester(3, 0, x).det() == sylvester(3, 0, x, 1).det() == sylvester(3, 0, x, 2).det() == 1 + + assert sylvester(x - 3, x - 8, x) == sylvester(x - 3, x - 8, x, 1) == sylvester(x - 3, x - 8, x, 2) == Matrix([[1, -3], [1, -8]]) + + assert sylvester(x**3 - 7*x + 7, 3*x**2 - 7, x) == sylvester(x**3 - 7*x + 7, 3*x**2 - 7, x, 1) == Matrix([[1, 0, -7, 7, 0], [0, 1, 0, -7, 7], [3, 0, -7, 0, 0], [0, 3, 0, -7, 0], [0, 0, 3, 0, -7]]) + + assert sylvester(x**3 - 7*x + 7, 3*x**2 - 7, x, 2) == Matrix([ +[1, 0, -7, 7, 0, 0], [0, 3, 0, -7, 0, 0], [0, 1, 0, -7, 7, 0], [0, 0, 3, 0, -7, 0], [0, 0, 1, 0, -7, 7], [0, 0, 0, 3, 0, -7]]) + +def test_subresultants_sylv(): + x = Symbol('x') + + p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 + q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 + assert subresultants_sylv(p, q, x) == subresultants(p, q, x) + assert subresultants_sylv(p, q, x)[-1] == res(p, q, x) + assert subresultants_sylv(p, q, x) != euclid_amv(p, q, x) + amv_factors = [1, 1, -1, 1, -1, 1] + assert subresultants_sylv(p, q, x) == [i*j for i, j in zip(amv_factors, modified_subresultants_amv(p, q, x))] + + p = x**3 - 7*x + 7 + q = 3*x**2 - 7 + assert subresultants_sylv(p, q, x) == euclid_amv(p, q, x) + +def test_modified_subresultants_sylv(): + x = Symbol('x') + + p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 + q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 + amv_factors = [1, 1, -1, 1, -1, 1] + assert modified_subresultants_sylv(p, q, x) == [i*j for i, j in zip(amv_factors, subresultants_amv(p, q, x))] + assert modified_subresultants_sylv(p, q, x)[-1] != res_q(p + x**8, q, x) + assert modified_subresultants_sylv(p, q, x) != sturm_amv(p, q, x) + + p = x**3 - 7*x + 7 + q = 3*x**2 - 7 + assert modified_subresultants_sylv(p, q, x) == sturm_amv(p, q, x) + assert modified_subresultants_sylv(-p, q, x) != sturm_amv(-p, q, x) + +def test_res(): + x = Symbol('x') + + assert res(3, 5, x) == 1 + +def test_res_q(): + x = Symbol('x') + + assert res_q(3, 5, x) == 1 + +def test_res_z(): + x = Symbol('x') + + assert res_z(3, 5, x) == 1 + assert res(3, 5, x) == res_q(3, 5, x) == res_z(3, 5, x) + +def test_bezout(): + x = Symbol('x') + + p = -2*x**5+7*x**3+9*x**2-3*x+1 + q = -10*x**4+21*x**2+18*x-3 + assert bezout(p, q, x, 'bz').det() == sylvester(p, q, x, 2).det() + assert bezout(p, q, x, 'bz').det() != sylvester(p, q, x, 1).det() + assert bezout(p, q, x, 'prs') == backward_eye(5) * bezout(p, q, x, 'bz') * backward_eye(5) + +def test_subresultants_bezout(): + x = Symbol('x') + + p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 + q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 + assert subresultants_bezout(p, q, x) == subresultants(p, q, x) + assert subresultants_bezout(p, q, x)[-1] == sylvester(p, q, x).det() + assert subresultants_bezout(p, q, x) != euclid_amv(p, q, x) + amv_factors = [1, 1, -1, 1, -1, 1] + assert subresultants_bezout(p, q, x) == [i*j for i, j in zip(amv_factors, modified_subresultants_amv(p, q, x))] + + p = x**3 - 7*x + 7 + q = 3*x**2 - 7 + assert subresultants_bezout(p, q, x) == euclid_amv(p, q, x) + +def test_modified_subresultants_bezout(): + x = Symbol('x') + + p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 + q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 + amv_factors = [1, 1, -1, 1, -1, 1] + assert modified_subresultants_bezout(p, q, x) == [i*j for i, j in zip(amv_factors, subresultants_amv(p, q, x))] + assert modified_subresultants_bezout(p, q, x)[-1] != sylvester(p + x**8, q, x).det() + assert modified_subresultants_bezout(p, q, x) != sturm_amv(p, q, x) + + p = x**3 - 7*x + 7 + q = 3*x**2 - 7 + assert modified_subresultants_bezout(p, q, x) == sturm_amv(p, q, x) + assert modified_subresultants_bezout(-p, q, x) != sturm_amv(-p, q, x) + +def test_sturm_pg(): + x = Symbol('x') + + p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 + q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 + assert sturm_pg(p, q, x)[-1] != sylvester(p, q, x, 2).det() + sam_factors = [1, 1, -1, -1, 1, 1] + assert sturm_pg(p, q, x) == [i*j for i,j in zip(sam_factors, euclid_pg(p, q, x))] + + p = -9*x**5 - 5*x**3 - 9 + q = -45*x**4 - 15*x**2 + assert sturm_pg(p, q, x, 1)[-1] == sylvester(p, q, x, 1).det() + assert sturm_pg(p, q, x)[-1] != sylvester(p, q, x, 2).det() + assert sturm_pg(-p, q, x)[-1] == sylvester(-p, q, x, 2).det() + assert sturm_pg(-p, q, x) == modified_subresultants_pg(-p, q, x) + +def test_sturm_q(): + x = Symbol('x') + + p = x**3 - 7*x + 7 + q = 3*x**2 - 7 + assert sturm_q(p, q, x) == sturm(p) + assert sturm_q(-p, -q, x) != sturm(-p) + + +def test_sturm_amv(): + x = Symbol('x') + + p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 + q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 + assert sturm_amv(p, q, x)[-1] != sylvester(p, q, x, 2).det() + sam_factors = [1, 1, -1, -1, 1, 1] + assert sturm_amv(p, q, x) == [i*j for i,j in zip(sam_factors, euclid_amv(p, q, x))] + + p = -9*x**5 - 5*x**3 - 9 + q = -45*x**4 - 15*x**2 + assert sturm_amv(p, q, x, 1)[-1] == sylvester(p, q, x, 1).det() + assert sturm_amv(p, q, x)[-1] != sylvester(p, q, x, 2).det() + assert sturm_amv(-p, q, x)[-1] == sylvester(-p, q, x, 2).det() + assert sturm_pg(-p, q, x) == modified_subresultants_pg(-p, q, x) + + +def test_euclid_pg(): + x = Symbol('x') + + p = x**6+x**5-x**4-x**3+x**2-x+1 + q = 6*x**5+5*x**4-4*x**3-3*x**2+2*x-1 + assert euclid_pg(p, q, x)[-1] == sylvester(p, q, x).det() + assert euclid_pg(p, q, x) == subresultants_pg(p, q, x) + + p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 + q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 + assert euclid_pg(p, q, x)[-1] != sylvester(p, q, x, 2).det() + sam_factors = [1, 1, -1, -1, 1, 1] + assert euclid_pg(p, q, x) == [i*j for i,j in zip(sam_factors, sturm_pg(p, q, x))] + + +def test_euclid_q(): + x = Symbol('x') + + p = x**3 - 7*x + 7 + q = 3*x**2 - 7 + assert euclid_q(p, q, x)[-1] == -sturm(p)[-1] + + +def test_euclid_amv(): + x = Symbol('x') + + p = x**3 - 7*x + 7 + q = 3*x**2 - 7 + assert euclid_amv(p, q, x)[-1] == sylvester(p, q, x).det() + assert euclid_amv(p, q, x) == subresultants_amv(p, q, x) + + p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 + q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 + assert euclid_amv(p, q, x)[-1] != sylvester(p, q, x, 2).det() + sam_factors = [1, 1, -1, -1, 1, 1] + assert euclid_amv(p, q, x) == [i*j for i,j in zip(sam_factors, sturm_amv(p, q, x))] + + +def test_modified_subresultants_pg(): + x = Symbol('x') + + p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 + q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 + amv_factors = [1, 1, -1, 1, -1, 1] + assert modified_subresultants_pg(p, q, x) == [i*j for i, j in zip(amv_factors, subresultants_pg(p, q, x))] + assert modified_subresultants_pg(p, q, x)[-1] != sylvester(p + x**8, q, x).det() + assert modified_subresultants_pg(p, q, x) != sturm_pg(p, q, x) + + p = x**3 - 7*x + 7 + q = 3*x**2 - 7 + assert modified_subresultants_pg(p, q, x) == sturm_pg(p, q, x) + assert modified_subresultants_pg(-p, q, x) != sturm_pg(-p, q, x) + + +def test_subresultants_pg(): + x = Symbol('x') + + p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 + q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 + assert subresultants_pg(p, q, x) == subresultants(p, q, x) + assert subresultants_pg(p, q, x)[-1] == sylvester(p, q, x).det() + assert subresultants_pg(p, q, x) != euclid_pg(p, q, x) + amv_factors = [1, 1, -1, 1, -1, 1] + assert subresultants_pg(p, q, x) == [i*j for i, j in zip(amv_factors, modified_subresultants_amv(p, q, x))] + + p = x**3 - 7*x + 7 + q = 3*x**2 - 7 + assert subresultants_pg(p, q, x) == euclid_pg(p, q, x) + + +def test_subresultants_amv_q(): + x = Symbol('x') + + p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 + q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 + assert subresultants_amv_q(p, q, x) == subresultants(p, q, x) + assert subresultants_amv_q(p, q, x)[-1] == sylvester(p, q, x).det() + assert subresultants_amv_q(p, q, x) != euclid_amv(p, q, x) + amv_factors = [1, 1, -1, 1, -1, 1] + assert subresultants_amv_q(p, q, x) == [i*j for i, j in zip(amv_factors, modified_subresultants_amv(p, q, x))] + + p = x**3 - 7*x + 7 + q = 3*x**2 - 7 + assert subresultants_amv(p, q, x) == euclid_amv(p, q, x) + + +def test_rem_z(): + x = Symbol('x') + + p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 + q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 + assert rem_z(p, -q, x) != prem(p, -q, x) + +def test_quo_z(): + x = Symbol('x') + + p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 + q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 + assert quo_z(p, -q, x) != pquo(p, -q, x) + + y = Symbol('y') + q = 3*x**6 + 5*y**4 - 4*x**2 - 9*x + 21 + assert quo_z(p, -q, x) == pquo(p, -q, x) + +def test_subresultants_amv(): + x = Symbol('x') + + p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 + q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 + assert subresultants_amv(p, q, x) == subresultants(p, q, x) + assert subresultants_amv(p, q, x)[-1] == sylvester(p, q, x).det() + assert subresultants_amv(p, q, x) != euclid_amv(p, q, x) + amv_factors = [1, 1, -1, 1, -1, 1] + assert subresultants_amv(p, q, x) == [i*j for i, j in zip(amv_factors, modified_subresultants_amv(p, q, x))] + + p = x**3 - 7*x + 7 + q = 3*x**2 - 7 + assert subresultants_amv(p, q, x) == euclid_amv(p, q, x) + + +def test_modified_subresultants_amv(): + x = Symbol('x') + + p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 + q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 + amv_factors = [1, 1, -1, 1, -1, 1] + assert modified_subresultants_amv(p, q, x) == [i*j for i, j in zip(amv_factors, subresultants_amv(p, q, x))] + assert modified_subresultants_amv(p, q, x)[-1] != sylvester(p + x**8, q, x).det() + assert modified_subresultants_amv(p, q, x) != sturm_amv(p, q, x) + + p = x**3 - 7*x + 7 + q = 3*x**2 - 7 + assert modified_subresultants_amv(p, q, x) == sturm_amv(p, q, x) + assert modified_subresultants_amv(-p, q, x) != sturm_amv(-p, q, x) + + +def test_subresultants_rem(): + x = Symbol('x') + + p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 + q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 + assert subresultants_rem(p, q, x) == subresultants(p, q, x) + assert subresultants_rem(p, q, x)[-1] == sylvester(p, q, x).det() + assert subresultants_rem(p, q, x) != euclid_amv(p, q, x) + amv_factors = [1, 1, -1, 1, -1, 1] + assert subresultants_rem(p, q, x) == [i*j for i, j in zip(amv_factors, modified_subresultants_amv(p, q, x))] + + p = x**3 - 7*x + 7 + q = 3*x**2 - 7 + assert subresultants_rem(p, q, x) == euclid_amv(p, q, x) + + +def test_subresultants_vv(): + x = Symbol('x') + + p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 + q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 + assert subresultants_vv(p, q, x) == subresultants(p, q, x) + assert subresultants_vv(p, q, x)[-1] == sylvester(p, q, x).det() + assert subresultants_vv(p, q, x) != euclid_amv(p, q, x) + amv_factors = [1, 1, -1, 1, -1, 1] + assert subresultants_vv(p, q, x) == [i*j for i, j in zip(amv_factors, modified_subresultants_amv(p, q, x))] + + p = x**3 - 7*x + 7 + q = 3*x**2 - 7 + assert subresultants_vv(p, q, x) == euclid_amv(p, q, x) + + +def test_subresultants_vv_2(): + x = Symbol('x') + + p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 + q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 + assert subresultants_vv_2(p, q, x) == subresultants(p, q, x) + assert subresultants_vv_2(p, q, x)[-1] == sylvester(p, q, x).det() + assert subresultants_vv_2(p, q, x) != euclid_amv(p, q, x) + amv_factors = [1, 1, -1, 1, -1, 1] + assert subresultants_vv_2(p, q, x) == [i*j for i, j in zip(amv_factors, modified_subresultants_amv(p, q, x))] + + p = x**3 - 7*x + 7 + q = 3*x**2 - 7 + assert subresultants_vv_2(p, q, x) == euclid_amv(p, q, x) diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/__init__.py b/.venv/lib/python3.12/site-packages/sympy/printing/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..15dfaf70eb3777195b7c9a0930894bb2187bbb50 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/__init__.py @@ -0,0 +1,111 @@ +"""Printing subsystem""" + +from .pretty import pager_print, pretty, pretty_print, pprint, pprint_use_unicode, pprint_try_use_unicode + +from .latex import latex, print_latex, multiline_latex + +from .mathml import mathml, print_mathml + +from .python import python, print_python + +from .pycode import pycode + +from .codeprinter import print_ccode, print_fcode + +from .codeprinter import ccode, fcode, cxxcode, rust_code # noqa:F811 + +from .smtlib import smtlib_code + +from .glsl import glsl_code, print_glsl + +from .rcode import rcode, print_rcode + +from .jscode import jscode, print_jscode + +from .julia import julia_code + +from .mathematica import mathematica_code + +from .octave import octave_code + +from .gtk import print_gtk + +from .preview import preview + +from .repr import srepr + +from .tree import print_tree + +from .str import StrPrinter, sstr, sstrrepr + +from .tableform import TableForm + +from .dot import dotprint + +from .maple import maple_code, print_maple_code + +__all__ = [ + # sympy.printing.pretty + 'pager_print', 'pretty', 'pretty_print', 'pprint', 'pprint_use_unicode', + 'pprint_try_use_unicode', + + # sympy.printing.latex + 'latex', 'print_latex', 'multiline_latex', + + # sympy.printing.mathml + 'mathml', 'print_mathml', + + # sympy.printing.python + 'python', 'print_python', + + # sympy.printing.pycode + 'pycode', + + # sympy.printing.codeprinter + 'ccode', 'print_ccode', 'cxxcode', 'fcode', 'print_fcode', 'rust_code', + + # sympy.printing.smtlib + 'smtlib_code', + + # sympy.printing.glsl + 'glsl_code', 'print_glsl', + + # sympy.printing.rcode + 'rcode', 'print_rcode', + + # sympy.printing.jscode + 'jscode', 'print_jscode', + + # sympy.printing.julia + 'julia_code', + + # sympy.printing.mathematica + 'mathematica_code', + + # sympy.printing.octave + 'octave_code', + + # sympy.printing.gtk + 'print_gtk', + + # sympy.printing.preview + 'preview', + + # sympy.printing.repr + 'srepr', + + # sympy.printing.tree + 'print_tree', + + # sympy.printing.str + 'StrPrinter', 'sstr', 'sstrrepr', + + # sympy.printing.tableform + 'TableForm', + + # sympy.printing.dot + 'dotprint', + + # sympy.printing.maple + 'maple_code', 'print_maple_code', +] diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/__pycache__/__init__.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/printing/__pycache__/__init__.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..e924a9fab952651d3592c9a89985248b0ee0b6af Binary files /dev/null and b/.venv/lib/python3.12/site-packages/sympy/printing/__pycache__/__init__.cpython-312.pyc differ diff --git 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annotations +import math +from typing import Any + +from sympy.external import import_module +from sympy.printing.printer import Printer +from sympy.utilities.exceptions import sympy_deprecation_warning +from sympy.utilities.iterables import is_sequence +import sympy +from functools import partial + + +aesara = import_module('aesara') + +if aesara: + aes = aesara.scalar + aet = aesara.tensor + from aesara.tensor import nlinalg + from aesara.tensor.elemwise import Elemwise + from aesara.tensor.elemwise import DimShuffle + + # `true_divide` replaced `true_div` in Aesara 2.8.11 (released 2023) to + # match NumPy + # XXX: Remove this when not needed to support older versions. + true_divide = getattr(aet, 'true_divide', None) + if true_divide is None: + true_divide = aet.true_div + + mapping = { + sympy.Add: aet.add, + sympy.Mul: aet.mul, + sympy.Abs: aet.abs, + sympy.sign: aet.sgn, + sympy.ceiling: aet.ceil, + sympy.floor: aet.floor, + sympy.log: aet.log, + sympy.exp: aet.exp, + sympy.sqrt: aet.sqrt, + sympy.cos: aet.cos, + sympy.acos: aet.arccos, + sympy.sin: aet.sin, + sympy.asin: aet.arcsin, + sympy.tan: aet.tan, + sympy.atan: aet.arctan, + sympy.atan2: aet.arctan2, + sympy.cosh: aet.cosh, + sympy.acosh: aet.arccosh, + sympy.sinh: aet.sinh, + sympy.asinh: aet.arcsinh, + sympy.tanh: aet.tanh, + sympy.atanh: aet.arctanh, + sympy.re: aet.real, + sympy.im: aet.imag, + sympy.arg: aet.angle, + sympy.erf: aet.erf, + sympy.gamma: aet.gamma, + sympy.loggamma: aet.gammaln, + sympy.Pow: aet.pow, + sympy.Eq: aet.eq, + sympy.StrictGreaterThan: aet.gt, + sympy.StrictLessThan: aet.lt, + sympy.LessThan: aet.le, + sympy.GreaterThan: aet.ge, + sympy.And: aet.bitwise_and, # bitwise + sympy.Or: aet.bitwise_or, # bitwise + sympy.Not: aet.invert, # bitwise + sympy.Xor: aet.bitwise_xor, # bitwise + sympy.Max: aet.maximum, # Sympy accept >2 inputs, Aesara only 2 + sympy.Min: aet.minimum, # Sympy accept >2 inputs, Aesara only 2 + sympy.conjugate: aet.conj, + sympy.core.numbers.ImaginaryUnit: lambda:aet.complex(0,1), + # Matrices + sympy.MatAdd: Elemwise(aes.add), + sympy.HadamardProduct: Elemwise(aes.mul), + sympy.Trace: nlinalg.trace, + sympy.Determinant : nlinalg.det, + sympy.Inverse: nlinalg.matrix_inverse, + sympy.Transpose: DimShuffle((False, False), [1, 0]), + } + + +class AesaraPrinter(Printer): + """ + .. deprecated:: 1.14. + The ``Aesara Code printing`` is deprecated.See its documentation for + more information. See :ref:`deprecated-aesaraprinter` for details. + + Code printer which creates Aesara symbolic expression graphs. + + Parameters + ========== + + cache : dict + Cache dictionary to use. If None (default) will use + the global cache. To create a printer which does not depend on or alter + global state pass an empty dictionary. Note: the dictionary is not + copied on initialization of the printer and will be updated in-place, + so using the same dict object when creating multiple printers or making + multiple calls to :func:`.aesara_code` or :func:`.aesara_function` means + the cache is shared between all these applications. + + Attributes + ========== + + cache : dict + A cache of Aesara variables which have been created for SymPy + symbol-like objects (e.g. :class:`sympy.core.symbol.Symbol` or + :class:`sympy.matrices.expressions.MatrixSymbol`). This is used to + ensure that all references to a given symbol in an expression (or + multiple expressions) are printed as the same Aesara variable, which is + created only once. Symbols are differentiated only by name and type. The + format of the cache's contents should be considered opaque to the user. + """ + printmethod = "_aesara" + + def __init__(self, *args, **kwargs): + self.cache = kwargs.pop('cache', {}) + super().__init__(*args, **kwargs) + + def _get_key(self, s, name=None, dtype=None, broadcastable=None): + """ Get the cache key for a SymPy object. + + Parameters + ========== + + s : sympy.core.basic.Basic + SymPy object to get key for. + + name : str + Name of object, if it does not have a ``name`` attribute. + """ + + if name is None: + name = s.name + + return (name, type(s), s.args, dtype, broadcastable) + + def _get_or_create(self, s, name=None, dtype=None, broadcastable=None): + """ + Get the Aesara variable for a SymPy symbol from the cache, or create it + if it does not exist. + """ + + # Defaults + if name is None: + name = s.name + if dtype is None: + dtype = 'floatX' + if broadcastable is None: + broadcastable = () + + key = self._get_key(s, name, dtype=dtype, broadcastable=broadcastable) + + if key in self.cache: + return self.cache[key] + + value = aet.tensor(name=name, dtype=dtype, shape=broadcastable) + self.cache[key] = value + return value + + def _print_Symbol(self, s, **kwargs): + dtype = kwargs.get('dtypes', {}).get(s) + bc = kwargs.get('broadcastables', {}).get(s) + return self._get_or_create(s, dtype=dtype, broadcastable=bc) + + def _print_AppliedUndef(self, s, **kwargs): + name = str(type(s)) + '_' + str(s.args[0]) + dtype = kwargs.get('dtypes', {}).get(s) + bc = kwargs.get('broadcastables', {}).get(s) + return self._get_or_create(s, name=name, dtype=dtype, broadcastable=bc) + + def _print_Basic(self, expr, **kwargs): + op = mapping[type(expr)] + children = [self._print(arg, **kwargs) for arg in expr.args] + return op(*children) + + def _print_Number(self, n, **kwargs): + # Integers already taken care of below, interpret as float + return float(n.evalf()) + + def _print_MatrixSymbol(self, X, **kwargs): + dtype = kwargs.get('dtypes', {}).get(X) + return self._get_or_create(X, dtype=dtype, broadcastable=(None, None)) + + def _print_DenseMatrix(self, X, **kwargs): + if not hasattr(aet, 'stacklists'): + raise NotImplementedError( + "Matrix translation not yet supported in this version of Aesara") + + return aet.stacklists([ + [self._print(arg, **kwargs) for arg in L] + for L in X.tolist() + ]) + + _print_ImmutableMatrix = _print_ImmutableDenseMatrix = _print_DenseMatrix + + def _print_MatMul(self, expr, **kwargs): + children = [self._print(arg, **kwargs) for arg in expr.args] + result = children[0] + for child in children[1:]: + result = aet.dot(result, child) + return result + + def _print_MatPow(self, expr, **kwargs): + children = [self._print(arg, **kwargs) for arg in expr.args] + result = 1 + if isinstance(children[1], int) and children[1] > 0: + for i in range(children[1]): + result = aet.dot(result, children[0]) + else: + raise NotImplementedError('''Only non-negative integer + powers of matrices can be handled by Aesara at the moment''') + return result + + def _print_MatrixSlice(self, expr, **kwargs): + parent = self._print(expr.parent, **kwargs) + rowslice = self._print(slice(*expr.rowslice), **kwargs) + colslice = self._print(slice(*expr.colslice), **kwargs) + return parent[rowslice, colslice] + + def _print_BlockMatrix(self, expr, **kwargs): + nrows, ncols = expr.blocks.shape + blocks = [[self._print(expr.blocks[r, c], **kwargs) + for c in range(ncols)] + for r in range(nrows)] + return aet.join(0, *[aet.join(1, *row) for row in blocks]) + + + def _print_slice(self, expr, **kwargs): + return slice(*[self._print(i, **kwargs) + if isinstance(i, sympy.Basic) else i + for i in (expr.start, expr.stop, expr.step)]) + + def _print_Pi(self, expr, **kwargs): + return math.pi + + def _print_Piecewise(self, expr, **kwargs): + import numpy as np + e, cond = expr.args[0].args # First condition and corresponding value + + # Print conditional expression and value for first condition + p_cond = self._print(cond, **kwargs) + p_e = self._print(e, **kwargs) + + # One condition only + if len(expr.args) == 1: + # Return value if condition else NaN + return aet.switch(p_cond, p_e, np.nan) + + # Return value_1 if condition_1 else evaluate remaining conditions + p_remaining = self._print(sympy.Piecewise(*expr.args[1:]), **kwargs) + return aet.switch(p_cond, p_e, p_remaining) + + def _print_Rational(self, expr, **kwargs): + return true_divide(self._print(expr.p, **kwargs), + self._print(expr.q, **kwargs)) + + def _print_Integer(self, expr, **kwargs): + return expr.p + + def _print_factorial(self, expr, **kwargs): + return self._print(sympy.gamma(expr.args[0] + 1), **kwargs) + + def _print_Derivative(self, deriv, **kwargs): + from aesara.gradient import Rop + + rv = self._print(deriv.expr, **kwargs) + for var in deriv.variables: + var = self._print(var, **kwargs) + rv = Rop(rv, var, aet.ones_like(var)) + return rv + + def emptyPrinter(self, expr): + return expr + + def doprint(self, expr, dtypes=None, broadcastables=None): + """ Convert a SymPy expression to a Aesara graph variable. + + The ``dtypes`` and ``broadcastables`` arguments are used to specify the + data type, dimension, and broadcasting behavior of the Aesara variables + corresponding to the free symbols in ``expr``. Each is a mapping from + SymPy symbols to the value of the corresponding argument to + ``aesara.tensor.var.TensorVariable``. + + See the corresponding `documentation page`__ for more information on + broadcasting in Aesara. + + + .. __: https://aesara.readthedocs.io/en/latest/reference/tensor/shapes.html#broadcasting + + Parameters + ========== + + expr : sympy.core.expr.Expr + SymPy expression to print. + + dtypes : dict + Mapping from SymPy symbols to Aesara datatypes to use when creating + new Aesara variables for those symbols. Corresponds to the ``dtype`` + argument to ``aesara.tensor.var.TensorVariable``. Defaults to ``'floatX'`` + for symbols not included in the mapping. + + broadcastables : dict + Mapping from SymPy symbols to the value of the ``broadcastable`` + argument to ``aesara.tensor.var.TensorVariable`` to use when creating Aesara + variables for those symbols. Defaults to the empty tuple for symbols + not included in the mapping (resulting in a scalar). + + Returns + ======= + + aesara.graph.basic.Variable + A variable corresponding to the expression's value in a Aesara + symbolic expression graph. + + """ + if dtypes is None: + dtypes = {} + if broadcastables is None: + broadcastables = {} + + return self._print(expr, dtypes=dtypes, broadcastables=broadcastables) + + +global_cache: dict[Any, Any] = {} + + +def aesara_code(expr, cache=None, **kwargs): + """ + Convert a SymPy expression into a Aesara graph variable. + + Parameters + ========== + + expr : sympy.core.expr.Expr + SymPy expression object to convert. + + cache : dict + Cached Aesara variables (see :class:`AesaraPrinter.cache + `). Defaults to the module-level global cache. + + dtypes : dict + Passed to :meth:`.AesaraPrinter.doprint`. + + broadcastables : dict + Passed to :meth:`.AesaraPrinter.doprint`. + + Returns + ======= + + aesara.graph.basic.Variable + A variable corresponding to the expression's value in a Aesara symbolic + expression graph. + + """ + sympy_deprecation_warning( + """ + The aesara_code function is deprecated. + """, + deprecated_since_version="1.14", + active_deprecations_target='deprecated-aesaraprinter', + ) + + if not aesara: + raise ImportError("aesara is required for aesara_code") + + if cache is None: + cache = global_cache + + return AesaraPrinter(cache=cache, settings={}).doprint(expr, **kwargs) + + +def dim_handling(inputs, dim=None, dims=None, broadcastables=None): + r""" + Get value of ``broadcastables`` argument to :func:`.aesara_code` from + keyword arguments to :func:`.aesara_function`. + + Included for backwards compatibility. + + Parameters + ========== + + inputs + Sequence of input symbols. + + dim : int + Common number of dimensions for all inputs. Overrides other arguments + if given. + + dims : dict + Mapping from input symbols to number of dimensions. Overrides + ``broadcastables`` argument if given. + + broadcastables : dict + Explicit value of ``broadcastables`` argument to + :meth:`.AesaraPrinter.doprint`. If not None function will return this value unchanged. + + Returns + ======= + dict + Dictionary mapping elements of ``inputs`` to their "broadcastable" + values (tuple of ``bool``\ s). + """ + if dim is not None: + return dict.fromkeys(inputs, (False,) * dim) + + if dims is not None: + maxdim = max(dims.values()) + return { + s: (False,) * d + (True,) * (maxdim - d) + for s, d in dims.items() + } + + if broadcastables is not None: + return broadcastables + + return {} + + +def aesara_function(inputs, outputs, scalar=False, *, + dim=None, dims=None, broadcastables=None, **kwargs): + """ + Create a Aesara function from SymPy expressions. + + The inputs and outputs are converted to Aesara variables using + :func:`.aesara_code` and then passed to ``aesara.function``. + + Parameters + ========== + + inputs + Sequence of symbols which constitute the inputs of the function. + + outputs + Sequence of expressions which constitute the outputs(s) of the + function. The free symbols of each expression must be a subset of + ``inputs``. + + scalar : bool + Convert 0-dimensional arrays in output to scalars. This will return a + Python wrapper function around the Aesara function object. + + cache : dict + Cached Aesara variables (see :class:`AesaraPrinter.cache + `). Defaults to the module-level global cache. + + dtypes : dict + Passed to :meth:`.AesaraPrinter.doprint`. + + broadcastables : dict + Passed to :meth:`.AesaraPrinter.doprint`. + + dims : dict + Alternative to ``broadcastables`` argument. Mapping from elements of + ``inputs`` to integers indicating the dimension of their associated + arrays/tensors. Overrides ``broadcastables`` argument if given. + + dim : int + Another alternative to the ``broadcastables`` argument. Common number of + dimensions to use for all arrays/tensors. + ``aesara_function([x, y], [...], dim=2)`` is equivalent to using + ``broadcastables={x: (False, False), y: (False, False)}``. + + Returns + ======= + callable + A callable object which takes values of ``inputs`` as positional + arguments and returns an output array for each of the expressions + in ``outputs``. If ``outputs`` is a single expression the function will + return a Numpy array, if it is a list of multiple expressions the + function will return a list of arrays. See description of the ``squeeze`` + argument above for the behavior when a single output is passed in a list. + The returned object will either be an instance of + ``aesara.compile.function.types.Function`` or a Python wrapper + function around one. In both cases, the returned value will have a + ``aesara_function`` attribute which points to the return value of + ``aesara.function``. + + Examples + ======== + + >>> from sympy.abc import x, y, z + >>> from sympy.printing.aesaracode import aesara_function + + A simple function with one input and one output: + + >>> f1 = aesara_function([x], [x**2 - 1], scalar=True) + >>> f1(3) + 8.0 + + A function with multiple inputs and one output: + + >>> f2 = aesara_function([x, y, z], [(x**z + y**z)**(1/z)], scalar=True) + >>> f2(3, 4, 2) + 5.0 + + A function with multiple inputs and multiple outputs: + + >>> f3 = aesara_function([x, y], [x**2 + y**2, x**2 - y**2], scalar=True) + >>> f3(2, 3) + [13.0, -5.0] + + See also + ======== + + dim_handling + + """ + sympy_deprecation_warning( + """ + The aesara_function function is deprecated. + """, + deprecated_since_version="1.14", + active_deprecations_target='deprecated-aesaraprinter', + ) + + if not aesara: + raise ImportError("Aesara is required for aesara_function") + + # Pop off non-aesara keyword args + cache = kwargs.pop('cache', {}) + dtypes = kwargs.pop('dtypes', {}) + + broadcastables = dim_handling( + inputs, dim=dim, dims=dims, broadcastables=broadcastables, + ) + + # Print inputs/outputs + code = partial(aesara_code, cache=cache, dtypes=dtypes, + broadcastables=broadcastables) + tinputs = list(map(code, inputs)) + toutputs = list(map(code, outputs)) + + #fix constant expressions as variables + toutputs = [output if isinstance(output, aesara.graph.basic.Variable) else aet.as_tensor_variable(output) for output in toutputs] + + if len(toutputs) == 1: + toutputs = toutputs[0] + + # Compile aesara func + func = aesara.function(tinputs, toutputs, **kwargs) + + is_0d = [len(o.variable.broadcastable) == 0 for o in func.outputs] + + # No wrapper required + if not scalar or not any(is_0d): + func.aesara_function = func + return func + + # Create wrapper to convert 0-dimensional outputs to scalars + def wrapper(*args): + out = func(*args) + # out can be array(1.0) or [array(1.0), array(2.0)] + + if is_sequence(out): + return [o[()] if is_0d[i] else o for i, o in enumerate(out)] + else: + return out[()] + + wrapper.__wrapped__ = func + wrapper.__doc__ = func.__doc__ + wrapper.aesara_function = func + return wrapper diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/c.py b/.venv/lib/python3.12/site-packages/sympy/printing/c.py new file mode 100644 index 0000000000000000000000000000000000000000..34c4b8f021073aeee7672248838557b5fa85fbae --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/c.py @@ -0,0 +1,747 @@ +""" +C code printer + +The C89CodePrinter & C99CodePrinter converts single SymPy expressions into +single C expressions, using the functions defined in math.h where possible. + +A complete code generator, which uses ccode extensively, can be found in +sympy.utilities.codegen. The codegen module can be used to generate complete +source code files that are compilable without further modifications. + + +""" + +from __future__ import annotations +from typing import Any + +from functools import wraps +from itertools import chain + +from sympy.core import S +from sympy.core.numbers import equal_valued, Float +from sympy.codegen.ast import ( + Assignment, Pointer, Variable, Declaration, Type, + real, complex_, integer, bool_, float32, float64, float80, + complex64, complex128, intc, value_const, pointer_const, + int8, int16, int32, int64, uint8, uint16, uint32, uint64, untyped, + none +) +from sympy.printing.codeprinter import CodePrinter, requires +from sympy.printing.precedence import precedence, PRECEDENCE +from sympy.sets.fancysets import Range + +# These are defined in the other file so we can avoid importing sympy.codegen +# from the top-level 'import sympy'. Export them here as well. +from sympy.printing.codeprinter import ccode, print_ccode # noqa:F401 + +# dictionary mapping SymPy function to (argument_conditions, C_function). +# Used in C89CodePrinter._print_Function(self) +known_functions_C89 = { + "Abs": [(lambda x: not x.is_integer, "fabs"), (lambda x: x.is_integer, "abs")], + "sin": "sin", + "cos": "cos", + "tan": "tan", + "asin": "asin", + "acos": "acos", + "atan": "atan", + "atan2": "atan2", + "exp": "exp", + "log": "log", + "log10": "log10", + "sinh": "sinh", + "cosh": "cosh", + "tanh": "tanh", + "floor": "floor", + "ceiling": "ceil", + "sqrt": "sqrt", # To enable automatic rewrites +} + +known_functions_C99 = dict(known_functions_C89, **{ + 'exp2': 'exp2', + 'expm1': 'expm1', + 'log2': 'log2', + 'log1p': 'log1p', + 'Cbrt': 'cbrt', + 'hypot': 'hypot', + 'fma': 'fma', + 'loggamma': 'lgamma', + 'erfc': 'erfc', + 'Max': 'fmax', + 'Min': 'fmin', + "asinh": "asinh", + "acosh": "acosh", + "atanh": "atanh", + "erf": "erf", + "gamma": "tgamma", +}) + +# These are the core reserved words in the C language. Taken from: +# https://en.cppreference.com/w/c/keyword + +reserved_words = [ + 'auto', 'break', 'case', 'char', 'const', 'continue', 'default', 'do', + 'double', 'else', 'enum', 'extern', 'float', 'for', 'goto', 'if', 'int', + 'long', 'register', 'return', 'short', 'signed', 'sizeof', 'static', + 'struct', 'entry', # never standardized, we'll leave it here anyway + 'switch', 'typedef', 'union', 'unsigned', 'void', 'volatile', 'while' +] + +reserved_words_c99 = ['inline', 'restrict'] + +def get_math_macros(): + """ Returns a dictionary with math-related macros from math.h/cmath + + Note that these macros are not strictly required by the C/C++-standard. + For MSVC they are enabled by defining "_USE_MATH_DEFINES" (preferably + via a compilation flag). + + Returns + ======= + + Dictionary mapping SymPy expressions to strings (macro names) + + """ + from sympy.codegen.cfunctions import log2, Sqrt + from sympy.functions.elementary.exponential import log + from sympy.functions.elementary.miscellaneous import sqrt + + return { + S.Exp1: 'M_E', + log2(S.Exp1): 'M_LOG2E', + 1/log(2): 'M_LOG2E', + log(2): 'M_LN2', + log(10): 'M_LN10', + S.Pi: 'M_PI', + S.Pi/2: 'M_PI_2', + S.Pi/4: 'M_PI_4', + 1/S.Pi: 'M_1_PI', + 2/S.Pi: 'M_2_PI', + 2/sqrt(S.Pi): 'M_2_SQRTPI', + 2/Sqrt(S.Pi): 'M_2_SQRTPI', + sqrt(2): 'M_SQRT2', + Sqrt(2): 'M_SQRT2', + 1/sqrt(2): 'M_SQRT1_2', + 1/Sqrt(2): 'M_SQRT1_2' + } + + +def _as_macro_if_defined(meth): + """ Decorator for printer methods + + When a Printer's method is decorated using this decorator the expressions printed + will first be looked for in the attribute ``math_macros``, and if present it will + print the macro name in ``math_macros`` followed by a type suffix for the type + ``real``. e.g. printing ``sympy.pi`` would print ``M_PIl`` if real is mapped to float80. + + """ + @wraps(meth) + def _meth_wrapper(self, expr, **kwargs): + if expr in self.math_macros: + return '%s%s' % (self.math_macros[expr], self._get_math_macro_suffix(real)) + else: + return meth(self, expr, **kwargs) + + return _meth_wrapper + + +class C89CodePrinter(CodePrinter): + """A printer to convert Python expressions to strings of C code""" + printmethod = "_ccode" + language = "C" + standard = "C89" + reserved_words = set(reserved_words) + + _default_settings: dict[str, Any] = dict(CodePrinter._default_settings, **{ + 'precision': 17, + 'user_functions': {}, + 'contract': True, + 'dereference': set(), + 'error_on_reserved': False, + }) + + type_aliases = { + real: float64, + complex_: complex128, + integer: intc + } + + type_mappings: dict[Type, Any] = { + real: 'double', + intc: 'int', + float32: 'float', + float64: 'double', + integer: 'int', + bool_: 'bool', + int8: 'int8_t', + int16: 'int16_t', + int32: 'int32_t', + int64: 'int64_t', + uint8: 'int8_t', + uint16: 'int16_t', + uint32: 'int32_t', + uint64: 'int64_t', + } + + type_headers = { + bool_: {'stdbool.h'}, + int8: {'stdint.h'}, + int16: {'stdint.h'}, + int32: {'stdint.h'}, + int64: {'stdint.h'}, + uint8: {'stdint.h'}, + uint16: {'stdint.h'}, + uint32: {'stdint.h'}, + uint64: {'stdint.h'}, + } + + # Macros needed to be defined when using a Type + type_macros: dict[Type, tuple[str, ...]] = {} + + type_func_suffixes = { + float32: 'f', + float64: '', + float80: 'l' + } + + type_literal_suffixes = { + float32: 'F', + float64: '', + float80: 'L' + } + + type_math_macro_suffixes = { + float80: 'l' + } + + math_macros = None + + _ns = '' # namespace, C++ uses 'std::' + # known_functions-dict to copy + _kf: dict[str, Any] = known_functions_C89 + + def __init__(self, settings=None): + settings = settings or {} + if self.math_macros is None: + self.math_macros = settings.pop('math_macros', get_math_macros()) + self.type_aliases = dict(chain(self.type_aliases.items(), + settings.pop('type_aliases', {}).items())) + self.type_mappings = dict(chain(self.type_mappings.items(), + settings.pop('type_mappings', {}).items())) + self.type_headers = dict(chain(self.type_headers.items(), + settings.pop('type_headers', {}).items())) + self.type_macros = dict(chain(self.type_macros.items(), + settings.pop('type_macros', {}).items())) + self.type_func_suffixes = dict(chain(self.type_func_suffixes.items(), + settings.pop('type_func_suffixes', {}).items())) + self.type_literal_suffixes = dict(chain(self.type_literal_suffixes.items(), + settings.pop('type_literal_suffixes', {}).items())) + self.type_math_macro_suffixes = dict(chain(self.type_math_macro_suffixes.items(), + settings.pop('type_math_macro_suffixes', {}).items())) + super().__init__(settings) + self.known_functions = dict(self._kf, **settings.get('user_functions', {})) + self._dereference = set(settings.get('dereference', [])) + self.headers = set() + self.libraries = set() + self.macros = set() + + def _rate_index_position(self, p): + return p*5 + + def _get_statement(self, codestring): + """ Get code string as a statement - i.e. ending with a semicolon. """ + return codestring if codestring.endswith(';') else codestring + ';' + + def _get_comment(self, text): + return "/* {} */".format(text) + + def _declare_number_const(self, name, value): + type_ = self.type_aliases[real] + var = Variable(name, type=type_, value=value.evalf(type_.decimal_dig), attrs={value_const}) + decl = Declaration(var) + return self._get_statement(self._print(decl)) + + def _format_code(self, lines): + return self.indent_code(lines) + + def _traverse_matrix_indices(self, mat): + rows, cols = mat.shape + return ((i, j) for i in range(rows) for j in range(cols)) + + @_as_macro_if_defined + def _print_Mul(self, expr, **kwargs): + return super()._print_Mul(expr, **kwargs) + + @_as_macro_if_defined + def _print_Pow(self, expr): + if "Pow" in self.known_functions: + return self._print_Function(expr) + PREC = precedence(expr) + suffix = self._get_func_suffix(real) + if equal_valued(expr.exp, -1): + return '%s/%s' % (self._print_Float(Float(1.0)), self.parenthesize(expr.base, PREC)) + elif equal_valued(expr.exp, 0.5): + return '%ssqrt%s(%s)' % (self._ns, suffix, self._print(expr.base)) + elif expr.exp == S.One/3 and self.standard != 'C89': + return '%scbrt%s(%s)' % (self._ns, suffix, self._print(expr.base)) + else: + return '%spow%s(%s, %s)' % (self._ns, suffix, self._print(expr.base), + self._print(expr.exp)) + + def _print_Mod(self, expr): + num, den = expr.args + if num.is_integer and den.is_integer: + PREC = precedence(expr) + snum, sden = [self.parenthesize(arg, PREC) for arg in expr.args] + # % is remainder (same sign as numerator), not modulo (same sign as + # denominator), in C. Hence, % only works as modulo if both numbers + # have the same sign + if (num.is_nonnegative and den.is_nonnegative or + num.is_nonpositive and den.is_nonpositive): + return f"{snum} % {sden}" + return f"(({snum} % {sden}) + {sden}) % {sden}" + # Not guaranteed integer + return self._print_math_func(expr, known='fmod') + + def _print_Rational(self, expr): + p, q = int(expr.p), int(expr.q) + suffix = self._get_literal_suffix(real) + return '%d.0%s/%d.0%s' % (p, suffix, q, suffix) + + def _print_Indexed(self, expr): + # calculate index for 1d array + offset = getattr(expr.base, 'offset', S.Zero) + strides = getattr(expr.base, 'strides', None) + indices = expr.indices + + if strides is None or isinstance(strides, str): + dims = expr.shape + shift = S.One + temp = () + if strides == 'C' or strides is None: + traversal = reversed(range(expr.rank)) + indices = indices[::-1] + elif strides == 'F': + traversal = range(expr.rank) + + for i in traversal: + temp += (shift,) + shift *= dims[i] + strides = temp + flat_index = sum(x[0]*x[1] for x in zip(indices, strides)) + offset + return "%s[%s]" % (self._print(expr.base.label), + self._print(flat_index)) + + @_as_macro_if_defined + def _print_NumberSymbol(self, expr): + return super()._print_NumberSymbol(expr) + + def _print_Infinity(self, expr): + return 'HUGE_VAL' + + def _print_NegativeInfinity(self, expr): + return '-HUGE_VAL' + + def _print_Piecewise(self, expr): + if expr.args[-1].cond != True: + # We need the last conditional to be a True, otherwise the resulting + # function may not return a result. + raise ValueError("All Piecewise expressions must contain an " + "(expr, True) statement to be used as a default " + "condition. Without one, the generated " + "expression may not evaluate to anything under " + "some condition.") + lines = [] + if expr.has(Assignment): + for i, (e, c) in enumerate(expr.args): + if i == 0: + lines.append("if (%s) {" % self._print(c)) + elif i == len(expr.args) - 1 and c == True: + lines.append("else {") + else: + lines.append("else if (%s) {" % self._print(c)) + code0 = self._print(e) + lines.append(code0) + lines.append("}") + return "\n".join(lines) + else: + # The piecewise was used in an expression, need to do inline + # operators. This has the downside that inline operators will + # not work for statements that span multiple lines (Matrix or + # Indexed expressions). + ecpairs = ["((%s) ? (\n%s\n)\n" % (self._print(c), + self._print(e)) + for e, c in expr.args[:-1]] + last_line = ": (\n%s\n)" % self._print(expr.args[-1].expr) + return ": ".join(ecpairs) + last_line + " ".join([")"*len(ecpairs)]) + + def _print_ITE(self, expr): + from sympy.functions import Piecewise + return self._print(expr.rewrite(Piecewise, deep=False)) + + def _print_MatrixElement(self, expr): + return "{}[{}]".format(self.parenthesize(expr.parent, PRECEDENCE["Atom"], + strict=True), expr.j + expr.i*expr.parent.shape[1]) + + def _print_Symbol(self, expr): + name = super()._print_Symbol(expr) + if expr in self._settings['dereference']: + return '(*{})'.format(name) + else: + return name + + def _print_Relational(self, expr): + lhs_code = self._print(expr.lhs) + rhs_code = self._print(expr.rhs) + op = expr.rel_op + return "{} {} {}".format(lhs_code, op, rhs_code) + + def _print_For(self, expr): + target = self._print(expr.target) + if isinstance(expr.iterable, Range): + start, stop, step = expr.iterable.args + else: + raise NotImplementedError("Only iterable currently supported is Range") + body = self._print(expr.body) + return ('for ({target} = {start}; {target} < {stop}; {target} += ' + '{step}) {{\n{body}\n}}').format(target=target, start=start, + stop=stop, step=step, body=body) + + def _print_sign(self, func): + return '((({0}) > 0) - (({0}) < 0))'.format(self._print(func.args[0])) + + def _print_Max(self, expr): + if "Max" in self.known_functions: + return self._print_Function(expr) + def inner_print_max(args): # The more natural abstraction of creating + if len(args) == 1: # and printing smaller Max objects is slow + return self._print(args[0]) # when there are many arguments. + half = len(args) // 2 + return "((%(a)s > %(b)s) ? %(a)s : %(b)s)" % { + 'a': inner_print_max(args[:half]), + 'b': inner_print_max(args[half:]) + } + return inner_print_max(expr.args) + + def _print_Min(self, expr): + if "Min" in self.known_functions: + return self._print_Function(expr) + def inner_print_min(args): # The more natural abstraction of creating + if len(args) == 1: # and printing smaller Min objects is slow + return self._print(args[0]) # when there are many arguments. + half = len(args) // 2 + return "((%(a)s < %(b)s) ? %(a)s : %(b)s)" % { + 'a': inner_print_min(args[:half]), + 'b': inner_print_min(args[half:]) + } + return inner_print_min(expr.args) + + def indent_code(self, code): + """Accepts a string of code or a list of code lines""" + + if isinstance(code, str): + code_lines = self.indent_code(code.splitlines(True)) + return ''.join(code_lines) + + tab = " " + inc_token = ('{', '(', '{\n', '(\n') + dec_token = ('}', ')') + + code = [line.lstrip(' \t') for line in code] + + increase = [int(any(map(line.endswith, inc_token))) for line in code] + decrease = [int(any(map(line.startswith, dec_token))) for line in code] + + pretty = [] + level = 0 + for n, line in enumerate(code): + if line in ('', '\n'): + pretty.append(line) + continue + level -= decrease[n] + pretty.append("%s%s" % (tab*level, line)) + level += increase[n] + return pretty + + def _get_func_suffix(self, type_): + return self.type_func_suffixes[self.type_aliases.get(type_, type_)] + + def _get_literal_suffix(self, type_): + return self.type_literal_suffixes[self.type_aliases.get(type_, type_)] + + def _get_math_macro_suffix(self, type_): + alias = self.type_aliases.get(type_, type_) + dflt = self.type_math_macro_suffixes.get(alias, '') + return self.type_math_macro_suffixes.get(type_, dflt) + + def _print_Tuple(self, expr): + return '{'+', '.join(self._print(e) for e in expr)+'}' + + _print_List = _print_Tuple + + def _print_Type(self, type_): + self.headers.update(self.type_headers.get(type_, set())) + self.macros.update(self.type_macros.get(type_, set())) + return self._print(self.type_mappings.get(type_, type_.name)) + + def _print_Declaration(self, decl): + from sympy.codegen.cnodes import restrict + var = decl.variable + val = var.value + if var.type == untyped: + raise ValueError("C does not support untyped variables") + + if isinstance(var, Pointer): + result = '{vc}{t} *{pc} {r}{s}'.format( + vc='const ' if value_const in var.attrs else '', + t=self._print(var.type), + pc=' const' if pointer_const in var.attrs else '', + r='restrict ' if restrict in var.attrs else '', + s=self._print(var.symbol) + ) + elif isinstance(var, Variable): + result = '{vc}{t} {s}'.format( + vc='const ' if value_const in var.attrs else '', + t=self._print(var.type), + s=self._print(var.symbol) + ) + else: + raise NotImplementedError("Unknown type of var: %s" % type(var)) + if val != None: # Must be "!= None", cannot be "is not None" + result += ' = %s' % self._print(val) + return result + + def _print_Float(self, flt): + type_ = self.type_aliases.get(real, real) + self.macros.update(self.type_macros.get(type_, set())) + suffix = self._get_literal_suffix(type_) + num = str(flt.evalf(type_.decimal_dig)) + if 'e' not in num and '.' not in num: + num += '.0' + num_parts = num.split('e') + num_parts[0] = num_parts[0].rstrip('0') + if num_parts[0].endswith('.'): + num_parts[0] += '0' + return 'e'.join(num_parts) + suffix + + @requires(headers={'stdbool.h'}) + def _print_BooleanTrue(self, expr): + return 'true' + + @requires(headers={'stdbool.h'}) + def _print_BooleanFalse(self, expr): + return 'false' + + def _print_Element(self, elem): + if elem.strides == None: # Must be "== None", cannot be "is None" + if elem.offset != None: # Must be "!= None", cannot be "is not None" + raise ValueError("Expected strides when offset is given") + idxs = ']['.join((self._print(arg) for arg in elem.indices)) + else: + global_idx = sum(i*s for i, s in zip(elem.indices, elem.strides)) + if elem.offset != None: # Must be "!= None", cannot be "is not None" + global_idx += elem.offset + idxs = self._print(global_idx) + + return "{symb}[{idxs}]".format( + symb=self._print(elem.symbol), + idxs=idxs + ) + + def _print_CodeBlock(self, expr): + """ Elements of code blocks printed as statements. """ + return '\n'.join([self._get_statement(self._print(i)) for i in expr.args]) + + def _print_While(self, expr): + return 'while ({condition}) {{\n{body}\n}}'.format(**expr.kwargs( + apply=lambda arg: self._print(arg))) + + def _print_Scope(self, expr): + return '{\n%s\n}' % self._print_CodeBlock(expr.body) + + @requires(headers={'stdio.h'}) + def _print_Print(self, expr): + if expr.file == none: + template = 'printf({fmt}, {pargs})' + else: + template = 'fprintf(%(out)s, {fmt}, {pargs})' % { + 'out': self._print(expr.file) + } + return template.format( + fmt="%s\n" if expr.format_string == none else self._print(expr.format_string), + pargs=', '.join((self._print(arg) for arg in expr.print_args)) + ) + + def _print_Stream(self, strm): + return strm.name + + def _print_FunctionPrototype(self, expr): + pars = ', '.join((self._print(Declaration(arg)) for arg in expr.parameters)) + return "%s %s(%s)" % ( + tuple((self._print(arg) for arg in (expr.return_type, expr.name))) + (pars,) + ) + + def _print_FunctionDefinition(self, expr): + return "%s%s" % (self._print_FunctionPrototype(expr), + self._print_Scope(expr)) + + def _print_Return(self, expr): + arg, = expr.args + return 'return %s' % self._print(arg) + + def _print_CommaOperator(self, expr): + return '(%s)' % ', '.join((self._print(arg) for arg in expr.args)) + + def _print_Label(self, expr): + if expr.body == none: + return '%s:' % str(expr.name) + if len(expr.body.args) == 1: + return '%s:\n%s' % (str(expr.name), self._print_CodeBlock(expr.body)) + return '%s:\n{\n%s\n}' % (str(expr.name), self._print_CodeBlock(expr.body)) + + def _print_goto(self, expr): + return 'goto %s' % expr.label.name + + def _print_PreIncrement(self, expr): + arg, = expr.args + return '++(%s)' % self._print(arg) + + def _print_PostIncrement(self, expr): + arg, = expr.args + return '(%s)++' % self._print(arg) + + def _print_PreDecrement(self, expr): + arg, = expr.args + return '--(%s)' % self._print(arg) + + def _print_PostDecrement(self, expr): + arg, = expr.args + return '(%s)--' % self._print(arg) + + def _print_struct(self, expr): + return "%(keyword)s %(name)s {\n%(lines)s}" % { + "keyword": expr.__class__.__name__, "name": expr.name, "lines": ';\n'.join( + [self._print(decl) for decl in expr.declarations] + ['']) + } + + def _print_BreakToken(self, _): + return 'break' + + def _print_ContinueToken(self, _): + return 'continue' + + _print_union = _print_struct + +class C99CodePrinter(C89CodePrinter): + standard = 'C99' + reserved_words = set(reserved_words + reserved_words_c99) + type_mappings=dict(chain(C89CodePrinter.type_mappings.items(), { + complex64: 'float complex', + complex128: 'double complex', + }.items())) + type_headers = dict(chain(C89CodePrinter.type_headers.items(), { + complex64: {'complex.h'}, + complex128: {'complex.h'} + }.items())) + + # known_functions-dict to copy + _kf: dict[str, Any] = known_functions_C99 + + # functions with versions with 'f' and 'l' suffixes: + _prec_funcs = ('fabs fmod remainder remquo fma fmax fmin fdim nan exp exp2' + ' expm1 log log10 log2 log1p pow sqrt cbrt hypot sin cos tan' + ' asin acos atan atan2 sinh cosh tanh asinh acosh atanh erf' + ' erfc tgamma lgamma ceil floor trunc round nearbyint rint' + ' frexp ldexp modf scalbn ilogb logb nextafter copysign').split() + + def _print_Infinity(self, expr): + return 'INFINITY' + + def _print_NegativeInfinity(self, expr): + return '-INFINITY' + + def _print_NaN(self, expr): + return 'NAN' + + # tgamma was already covered by 'known_functions' dict + + @requires(headers={'math.h'}, libraries={'m'}) + @_as_macro_if_defined + def _print_math_func(self, expr, nest=False, known=None): + if known is None: + known = self.known_functions[expr.__class__.__name__] + if not isinstance(known, str): + for cb, name in known: + if cb(*expr.args): + known = name + break + else: + raise ValueError("No matching printer") + try: + return known(self, *expr.args) + except TypeError: + suffix = self._get_func_suffix(real) if self._ns + known in self._prec_funcs else '' + + if nest: + args = self._print(expr.args[0]) + if len(expr.args) > 1: + paren_pile = '' + for curr_arg in expr.args[1:-1]: + paren_pile += ')' + args += ', {ns}{name}{suffix}({next}'.format( + ns=self._ns, + name=known, + suffix=suffix, + next = self._print(curr_arg) + ) + args += ', %s%s' % ( + self._print(expr.func(expr.args[-1])), + paren_pile + ) + else: + args = ', '.join((self._print(arg) for arg in expr.args)) + return '{ns}{name}{suffix}({args})'.format( + ns=self._ns, + name=known, + suffix=suffix, + args=args + ) + + def _print_Max(self, expr): + return self._print_math_func(expr, nest=True) + + def _print_Min(self, expr): + return self._print_math_func(expr, nest=True) + + def _get_loop_opening_ending(self, indices): + open_lines = [] + close_lines = [] + loopstart = "for (int %(var)s=%(start)s; %(var)s<%(end)s; %(var)s++){" # C99 + for i in indices: + # C arrays start at 0 and end at dimension-1 + open_lines.append(loopstart % { + 'var': self._print(i.label), + 'start': self._print(i.lower), + 'end': self._print(i.upper + 1)}) + close_lines.append("}") + return open_lines, close_lines + + +for k in ('Abs Sqrt exp exp2 expm1 log log10 log2 log1p Cbrt hypot fma' + ' loggamma sin cos tan asin acos atan atan2 sinh cosh tanh asinh acosh ' + 'atanh erf erfc loggamma gamma ceiling floor').split(): + setattr(C99CodePrinter, '_print_%s' % k, C99CodePrinter._print_math_func) + + +class C11CodePrinter(C99CodePrinter): + + @requires(headers={'stdalign.h'}) + def _print_alignof(self, expr): + arg, = expr.args + return 'alignof(%s)' % self._print(arg) + + +c_code_printers = { + 'c89': C89CodePrinter, + 'c99': C99CodePrinter, + 'c11': C11CodePrinter +} diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/codeprinter.py b/.venv/lib/python3.12/site-packages/sympy/printing/codeprinter.py new file mode 100644 index 0000000000000000000000000000000000000000..1faaa0f054cbd8ff438b90e914808f720d2da90a --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/codeprinter.py @@ -0,0 +1,1039 @@ +from __future__ import annotations +from typing import Any + +from functools import wraps + +from sympy.core import Add, Mul, Pow, S, sympify, Float +from sympy.core.basic import Basic +from sympy.core.expr import Expr, UnevaluatedExpr +from sympy.core.function import Lambda +from sympy.core.mul import _keep_coeff +from sympy.core.sorting import default_sort_key +from sympy.core.symbol import Symbol +from sympy.functions.elementary.complexes import re +from sympy.printing.str import StrPrinter +from sympy.printing.precedence import precedence, PRECEDENCE + + +class requires: + """ Decorator for registering requirements on print methods. """ + def __init__(self, **kwargs): + self._req = kwargs + + def __call__(self, method): + def _method_wrapper(self_, *args, **kwargs): + for k, v in self._req.items(): + getattr(self_, k).update(v) + return method(self_, *args, **kwargs) + return wraps(method)(_method_wrapper) + + +class AssignmentError(Exception): + """ + Raised if an assignment variable for a loop is missing. + """ + pass + +class PrintMethodNotImplementedError(NotImplementedError): + """ + Raised if a _print_* method is missing in the Printer. + """ + pass + +def _convert_python_lists(arg): + if isinstance(arg, list): + from sympy.codegen.abstract_nodes import List + return List(*(_convert_python_lists(e) for e in arg)) + elif isinstance(arg, tuple): + return tuple(_convert_python_lists(e) for e in arg) + else: + return arg + + +class CodePrinter(StrPrinter): + """ + The base class for code-printing subclasses. + """ + + _operators = { + 'and': '&&', + 'or': '||', + 'not': '!', + } + + _default_settings: dict[str, Any] = { + 'order': None, + 'full_prec': 'auto', + 'error_on_reserved': False, + 'reserved_word_suffix': '_', + 'human': True, + 'inline': False, + 'allow_unknown_functions': False, + 'strict': None # True or False; None => True if human == True + } + + # Functions which are "simple" to rewrite to other functions that + # may be supported + # function_to_rewrite : (function_to_rewrite_to, iterable_with_other_functions_required) + _rewriteable_functions = { + 'cot': ('tan', []), + 'csc': ('sin', []), + 'sec': ('cos', []), + 'acot': ('atan', []), + 'acsc': ('asin', []), + 'asec': ('acos', []), + 'coth': ('exp', []), + 'csch': ('exp', []), + 'sech': ('exp', []), + 'acoth': ('log', []), + 'acsch': ('log', []), + 'asech': ('log', []), + 'catalan': ('gamma', []), + 'fibonacci': ('sqrt', []), + 'lucas': ('sqrt', []), + 'beta': ('gamma', []), + 'sinc': ('sin', ['Piecewise']), + 'Mod': ('floor', []), + 'factorial': ('gamma', []), + 'factorial2': ('gamma', ['Piecewise']), + 'subfactorial': ('uppergamma', []), + 'RisingFactorial': ('gamma', ['Piecewise']), + 'FallingFactorial': ('gamma', ['Piecewise']), + 'binomial': ('gamma', []), + 'frac': ('floor', []), + 'Max': ('Piecewise', []), + 'Min': ('Piecewise', []), + 'Heaviside': ('Piecewise', []), + 'erf2': ('erf', []), + 'erfc': ('erf', []), + 'Li': ('li', []), + 'Ei': ('li', []), + 'dirichlet_eta': ('zeta', []), + 'riemann_xi': ('zeta', ['gamma']), + 'SingularityFunction': ('Piecewise', []), + } + + def __init__(self, settings=None): + super().__init__(settings=settings) + if self._settings.get('strict', True) == None: + # for backwards compatibility, human=False need not to throw: + self._settings['strict'] = self._settings.get('human', True) == True + if not hasattr(self, 'reserved_words'): + self.reserved_words = set() + + def _handle_UnevaluatedExpr(self, expr): + return expr.replace(re, lambda arg: arg if isinstance( + arg, UnevaluatedExpr) and arg.args[0].is_real else re(arg)) + + def doprint(self, expr, assign_to=None): + """ + Print the expression as code. + + Parameters + ---------- + expr : Expression + The expression to be printed. + + assign_to : Symbol, string, MatrixSymbol, list of strings or Symbols (optional) + If provided, the printed code will set the expression to a variable or multiple variables + with the name or names given in ``assign_to``. + """ + from sympy.matrices.expressions.matexpr import MatrixSymbol + from sympy.codegen.ast import CodeBlock, Assignment + + def _handle_assign_to(expr, assign_to): + if assign_to is None: + return sympify(expr) + if isinstance(assign_to, (list, tuple)): + if len(expr) != len(assign_to): + raise ValueError('Failed to assign an expression of length {} to {} variables'.format(len(expr), len(assign_to))) + return CodeBlock(*[_handle_assign_to(lhs, rhs) for lhs, rhs in zip(expr, assign_to)]) + if isinstance(assign_to, str): + if expr.is_Matrix: + assign_to = MatrixSymbol(assign_to, *expr.shape) + else: + assign_to = Symbol(assign_to) + elif not isinstance(assign_to, Basic): + raise TypeError("{} cannot assign to object of type {}".format( + type(self).__name__, type(assign_to))) + return Assignment(assign_to, expr) + + expr = _convert_python_lists(expr) + expr = _handle_assign_to(expr, assign_to) + + # Remove re(...) nodes due to UnevaluatedExpr.is_real always is None: + expr = self._handle_UnevaluatedExpr(expr) + + # keep a set of expressions that are not strictly translatable to Code + # and number constants that must be declared and initialized + self._not_supported = set() + self._number_symbols = set() + + lines = self._print(expr).splitlines() + + # format the output + if self._settings["human"]: + frontlines = [] + if self._not_supported: + frontlines.append(self._get_comment( + "Not supported in {}:".format(self.language))) + for expr in sorted(self._not_supported, key=str): + frontlines.append(self._get_comment(type(expr).__name__)) + for name, value in sorted(self._number_symbols, key=str): + frontlines.append(self._declare_number_const(name, value)) + lines = frontlines + lines + lines = self._format_code(lines) + result = "\n".join(lines) + else: + lines = self._format_code(lines) + num_syms = {(k, self._print(v)) for k, v in self._number_symbols} + result = (num_syms, self._not_supported, "\n".join(lines)) + self._not_supported = set() + self._number_symbols = set() + return result + + def _doprint_loops(self, expr, assign_to=None): + # Here we print an expression that contains Indexed objects, they + # correspond to arrays in the generated code. The low-level implementation + # involves looping over array elements and possibly storing results in temporary + # variables or accumulate it in the assign_to object. + + if self._settings.get('contract', True): + from sympy.tensor import get_contraction_structure + # Setup loops over non-dummy indices -- all terms need these + indices = self._get_expression_indices(expr, assign_to) + # Setup loops over dummy indices -- each term needs separate treatment + dummies = get_contraction_structure(expr) + else: + indices = [] + dummies = {None: (expr,)} + openloop, closeloop = self._get_loop_opening_ending(indices) + + # terms with no summations first + if None in dummies: + text = StrPrinter.doprint(self, Add(*dummies[None])) + else: + # If all terms have summations we must initialize array to Zero + text = StrPrinter.doprint(self, 0) + + # skip redundant assignments (where lhs == rhs) + lhs_printed = self._print(assign_to) + lines = [] + if text != lhs_printed: + lines.extend(openloop) + if assign_to is not None: + text = self._get_statement("%s = %s" % (lhs_printed, text)) + lines.append(text) + lines.extend(closeloop) + + # then terms with summations + for d in dummies: + if isinstance(d, tuple): + indices = self._sort_optimized(d, expr) + openloop_d, closeloop_d = self._get_loop_opening_ending( + indices) + + for term in dummies[d]: + if term in dummies and not ([list(f.keys()) for f in dummies[term]] + == [[None] for f in dummies[term]]): + # If one factor in the term has it's own internal + # contractions, those must be computed first. + # (temporary variables?) + raise NotImplementedError( + "FIXME: no support for contractions in factor yet") + else: + + # We need the lhs expression as an accumulator for + # the loops, i.e + # + # for (int d=0; d < dim; d++){ + # lhs[] = lhs[] + term[][d] + # } ^.................. the accumulator + # + # We check if the expression already contains the + # lhs, and raise an exception if it does, as that + # syntax is currently undefined. FIXME: What would be + # a good interpretation? + if assign_to is None: + raise AssignmentError( + "need assignment variable for loops") + if term.has(assign_to): + raise ValueError("FIXME: lhs present in rhs,\ + this is undefined in CodePrinter") + + lines.extend(openloop) + lines.extend(openloop_d) + text = "%s = %s" % (lhs_printed, StrPrinter.doprint( + self, assign_to + term)) + lines.append(self._get_statement(text)) + lines.extend(closeloop_d) + lines.extend(closeloop) + + return "\n".join(lines) + + def _get_expression_indices(self, expr, assign_to): + from sympy.tensor import get_indices + rinds, junk = get_indices(expr) + linds, junk = get_indices(assign_to) + + # support broadcast of scalar + if linds and not rinds: + rinds = linds + if rinds != linds: + raise ValueError("lhs indices must match non-dummy" + " rhs indices in %s" % expr) + + return self._sort_optimized(rinds, assign_to) + + def _sort_optimized(self, indices, expr): + + from sympy.tensor.indexed import Indexed + + if not indices: + return [] + + # determine optimized loop order by giving a score to each index + # the index with the highest score are put in the innermost loop. + score_table = {} + for i in indices: + score_table[i] = 0 + + arrays = expr.atoms(Indexed) + for arr in arrays: + for p, ind in enumerate(arr.indices): + try: + score_table[ind] += self._rate_index_position(p) + except KeyError: + pass + + return sorted(indices, key=lambda x: score_table[x]) + + def _rate_index_position(self, p): + """function to calculate score based on position among indices + + This method is used to sort loops in an optimized order, see + CodePrinter._sort_optimized() + """ + raise NotImplementedError("This function must be implemented by " + "subclass of CodePrinter.") + + def _get_statement(self, codestring): + """Formats a codestring with the proper line ending.""" + raise NotImplementedError("This function must be implemented by " + "subclass of CodePrinter.") + + def _get_comment(self, text): + """Formats a text string as a comment.""" + raise NotImplementedError("This function must be implemented by " + "subclass of CodePrinter.") + + def _declare_number_const(self, name, value): + """Declare a numeric constant at the top of a function""" + raise NotImplementedError("This function must be implemented by " + "subclass of CodePrinter.") + + def _format_code(self, lines): + """Take in a list of lines of code, and format them accordingly. + + This may include indenting, wrapping long lines, etc...""" + raise NotImplementedError("This function must be implemented by " + "subclass of CodePrinter.") + + def _get_loop_opening_ending(self, indices): + """Returns a tuple (open_lines, close_lines) containing lists + of codelines""" + raise NotImplementedError("This function must be implemented by " + "subclass of CodePrinter.") + + def _print_Dummy(self, expr): + if expr.name.startswith('Dummy_'): + return '_' + expr.name + else: + return '%s_%d' % (expr.name, expr.dummy_index) + + def _print_Idx(self, expr): + return self._print(expr.label) + + def _print_CodeBlock(self, expr): + return '\n'.join([self._print(i) for i in expr.args]) + + def _print_String(self, string): + return str(string) + + def _print_QuotedString(self, arg): + return '"%s"' % arg.text + + def _print_Comment(self, string): + return self._get_comment(str(string)) + + def _print_Assignment(self, expr): + from sympy.codegen.ast import Assignment + from sympy.functions.elementary.piecewise import Piecewise + from sympy.matrices.expressions.matexpr import MatrixSymbol + from sympy.tensor.indexed import IndexedBase + lhs = expr.lhs + rhs = expr.rhs + # We special case assignments that take multiple lines + if isinstance(expr.rhs, Piecewise): + # Here we modify Piecewise so each expression is now + # an Assignment, and then continue on the print. + expressions = [] + conditions = [] + for (e, c) in rhs.args: + expressions.append(Assignment(lhs, e)) + conditions.append(c) + temp = Piecewise(*zip(expressions, conditions)) + return self._print(temp) + elif isinstance(lhs, MatrixSymbol): + # Here we form an Assignment for each element in the array, + # printing each one. + lines = [] + for (i, j) in self._traverse_matrix_indices(lhs): + temp = Assignment(lhs[i, j], rhs[i, j]) + code0 = self._print(temp) + lines.append(code0) + return "\n".join(lines) + elif self._settings.get("contract", False) and (lhs.has(IndexedBase) or + rhs.has(IndexedBase)): + # Here we check if there is looping to be done, and if so + # print the required loops. + return self._doprint_loops(rhs, lhs) + else: + lhs_code = self._print(lhs) + rhs_code = self._print(rhs) + return self._get_statement("%s = %s" % (lhs_code, rhs_code)) + + def _print_AugmentedAssignment(self, expr): + lhs_code = self._print(expr.lhs) + rhs_code = self._print(expr.rhs) + return self._get_statement("{} {} {}".format( + *(self._print(arg) for arg in [lhs_code, expr.op, rhs_code]))) + + def _print_FunctionCall(self, expr): + return '%s(%s)' % ( + expr.name, + ', '.join((self._print(arg) for arg in expr.function_args))) + + def _print_Variable(self, expr): + return self._print(expr.symbol) + + def _print_Symbol(self, expr): + name = super()._print_Symbol(expr) + + if name in self.reserved_words: + if self._settings['error_on_reserved']: + msg = ('This expression includes the symbol "{}" which is a ' + 'reserved keyword in this language.') + raise ValueError(msg.format(name)) + return name + self._settings['reserved_word_suffix'] + else: + return name + + def _can_print(self, name): + """ Check if function ``name`` is either a known function or has its own + printing method. Used to check if rewriting is possible.""" + return name in self.known_functions or getattr(self, '_print_{}'.format(name), False) + + def _print_Function(self, expr): + if expr.func.__name__ in self.known_functions: + cond_func = self.known_functions[expr.func.__name__] + if isinstance(cond_func, str): + return "%s(%s)" % (cond_func, self.stringify(expr.args, ", ")) + else: + for cond, func in cond_func: + if cond(*expr.args): + break + if func is not None: + try: + return func(*[self.parenthesize(item, 0) for item in expr.args]) + except TypeError: + return "%s(%s)" % (func, self.stringify(expr.args, ", ")) + elif hasattr(expr, '_imp_') and isinstance(expr._imp_, Lambda): + # inlined function + return self._print(expr._imp_(*expr.args)) + elif expr.func.__name__ in self._rewriteable_functions: + # Simple rewrite to supported function possible + target_f, required_fs = self._rewriteable_functions[expr.func.__name__] + if self._can_print(target_f) and all(self._can_print(f) for f in required_fs): + return '(' + self._print(expr.rewrite(target_f)) + ')' + + if expr.is_Function and self._settings.get('allow_unknown_functions', False): + return '%s(%s)' % (self._print(expr.func), ', '.join(map(self._print, expr.args))) + else: + return self._print_not_supported(expr) + + _print_Expr = _print_Function + + def _print_Derivative(self, expr): + obj, *wrt_order_pairs = expr.args + for func_arg in obj.args: + if not func_arg.is_Symbol: + raise ValueError("%s._print_Derivative(...) only supports functions with symbols as arguments." % + self.__class__.__name__) + meth_name = '_print_Derivative_%s' % obj.func.__name__ + pmeth = getattr(self, meth_name, None) + if pmeth is None: + if self._settings.get('strict', False): + raise PrintMethodNotImplementedError( + f"Unsupported by {type(self)}: {type(expr)}" + + f"\nPrinter has no method: {meth_name}" + + "\nSet the printer option 'strict' to False in order to generate partially printed code." + ) + return self._print_not_supported(expr) + orders = dict(wrt_order_pairs) + seq_orders = [orders[arg] for arg in obj.args] + return pmeth(obj.args, seq_orders) + + # Don't inherit the str-printer method for Heaviside to the code printers + _print_Heaviside = None + + def _print_NumberSymbol(self, expr): + if self._settings.get("inline", False): + return self._print(Float(expr.evalf(self._settings["precision"]))) + else: + # A Number symbol that is not implemented here or with _printmethod + # is registered and evaluated + self._number_symbols.add((expr, + Float(expr.evalf(self._settings["precision"])))) + return str(expr) + + def _print_Catalan(self, expr): + return self._print_NumberSymbol(expr) + def _print_EulerGamma(self, expr): + return self._print_NumberSymbol(expr) + def _print_GoldenRatio(self, expr): + return self._print_NumberSymbol(expr) + def _print_TribonacciConstant(self, expr): + return self._print_NumberSymbol(expr) + def _print_Exp1(self, expr): + return self._print_NumberSymbol(expr) + def _print_Pi(self, expr): + return self._print_NumberSymbol(expr) + + def _print_And(self, expr): + PREC = precedence(expr) + return (" %s " % self._operators['and']).join(self.parenthesize(a, PREC) + for a in sorted(expr.args, key=default_sort_key)) + + def _print_Or(self, expr): + PREC = precedence(expr) + return (" %s " % self._operators['or']).join(self.parenthesize(a, PREC) + for a in sorted(expr.args, key=default_sort_key)) + + def _print_Xor(self, expr): + if self._operators.get('xor') is None: + return self._print(expr.to_nnf()) + PREC = precedence(expr) + return (" %s " % self._operators['xor']).join(self.parenthesize(a, PREC) + for a in expr.args) + + def _print_Equivalent(self, expr): + if self._operators.get('equivalent') is None: + return self._print(expr.to_nnf()) + PREC = precedence(expr) + return (" %s " % self._operators['equivalent']).join(self.parenthesize(a, PREC) + for a in expr.args) + + def _print_Not(self, expr): + PREC = precedence(expr) + return self._operators['not'] + self.parenthesize(expr.args[0], PREC) + + def _print_BooleanFunction(self, expr): + return self._print(expr.to_nnf()) + + def _print_isnan(self, arg): + return 'isnan(%s)' % self._print(*arg.args) + + def _print_isinf(self, arg): + return 'isinf(%s)' % self._print(*arg.args) + + def _print_Mul(self, expr): + + prec = precedence(expr) + + c, e = expr.as_coeff_Mul() + if c < 0: + expr = _keep_coeff(-c, e) + sign = "-" + else: + sign = "" + + a = [] # items in the numerator + b = [] # items that are in the denominator (if any) + + pow_paren = [] # Will collect all pow with more than one base element and exp = -1 + + if self.order not in ('old', 'none'): + args = expr.as_ordered_factors() + else: + # use make_args in case expr was something like -x -> x + args = Mul.make_args(expr) + + # Gather args for numerator/denominator + for item in args: + if item.is_commutative and item.is_Pow and item.exp.is_Rational and item.exp.is_negative: + if item.exp != -1: + b.append(Pow(item.base, -item.exp, evaluate=False)) + else: + if len(item.args[0].args) != 1 and isinstance(item.base, Mul): # To avoid situations like #14160 + pow_paren.append(item) + b.append(Pow(item.base, -item.exp)) + else: + a.append(item) + + a = a or [S.One] + + if len(a) == 1 and sign == "-": + # Unary minus does not have a SymPy class, and hence there's no + # precedence weight associated with it, Python's unary minus has + # an operator precedence between multiplication and exponentiation, + # so we use this to compute a weight. + a_str = [self.parenthesize(a[0], 0.5*(PRECEDENCE["Pow"]+PRECEDENCE["Mul"]))] + else: + a_str = [self.parenthesize(x, prec) for x in a] + b_str = [self.parenthesize(x, prec) for x in b] + + # To parenthesize Pow with exp = -1 and having more than one Symbol + for item in pow_paren: + if item.base in b: + b_str[b.index(item.base)] = "(%s)" % b_str[b.index(item.base)] + + if not b: + return sign + '*'.join(a_str) + elif len(b) == 1: + return sign + '*'.join(a_str) + "/" + b_str[0] + else: + return sign + '*'.join(a_str) + "/(%s)" % '*'.join(b_str) + + def _print_not_supported(self, expr): + if self._settings.get('strict', False): + raise PrintMethodNotImplementedError( + f"Unsupported by {type(self)}: {type(expr)}" + + "\nSet the printer option 'strict' to False in order to generate partially printed code." + ) + try: + self._not_supported.add(expr) + except TypeError: + # not hashable + pass + return self.emptyPrinter(expr) + + # The following can not be simply translated into C or Fortran + _print_Basic = _print_not_supported + _print_ComplexInfinity = _print_not_supported + _print_ExprCondPair = _print_not_supported + _print_GeometryEntity = _print_not_supported + _print_Infinity = _print_not_supported + _print_Integral = _print_not_supported + _print_Interval = _print_not_supported + _print_AccumulationBounds = _print_not_supported + _print_Limit = _print_not_supported + _print_MatrixBase = _print_not_supported + _print_DeferredVector = _print_not_supported + _print_NaN = _print_not_supported + _print_NegativeInfinity = _print_not_supported + _print_Order = _print_not_supported + _print_RootOf = _print_not_supported + _print_RootsOf = _print_not_supported + _print_RootSum = _print_not_supported + _print_Uniform = _print_not_supported + _print_Unit = _print_not_supported + _print_Wild = _print_not_supported + _print_WildFunction = _print_not_supported + _print_Relational = _print_not_supported + + +# Code printer functions. These are included in this file so that they can be +# imported in the top-level __init__.py without importing the sympy.codegen +# module. + +def ccode(expr, assign_to=None, standard='c99', **settings): + """Converts an expr to a string of c code + + Parameters + ========== + + expr : Expr + A SymPy expression to be converted. + assign_to : optional + When given, the argument is used as the name of the variable to which + the expression is assigned. Can be a string, ``Symbol``, + ``MatrixSymbol``, or ``Indexed`` type. This is helpful in case of + line-wrapping, or for expressions that generate multi-line statements. + standard : str, optional + String specifying the standard. If your compiler supports a more modern + standard you may set this to 'c99' to allow the printer to use more math + functions. [default='c89']. + precision : integer, optional + The precision for numbers such as pi [default=17]. + user_functions : dict, optional + A dictionary where the keys are string representations of either + ``FunctionClass`` or ``UndefinedFunction`` instances and the values + are their desired C string representations. Alternatively, the + dictionary value can be a list of tuples i.e. [(argument_test, + cfunction_string)] or [(argument_test, cfunction_formater)]. See below + for examples. + dereference : iterable, optional + An iterable of symbols that should be dereferenced in the printed code + expression. These would be values passed by address to the function. + For example, if ``dereference=[a]``, the resulting code would print + ``(*a)`` instead of ``a``. + human : bool, optional + If True, the result is a single string that may contain some constant + declarations for the number symbols. If False, the same information is + returned in a tuple of (symbols_to_declare, not_supported_functions, + code_text). [default=True]. + contract: bool, optional + If True, ``Indexed`` instances are assumed to obey tensor contraction + rules and the corresponding nested loops over indices are generated. + Setting contract=False will not generate loops, instead the user is + responsible to provide values for the indices in the code. + [default=True]. + + Examples + ======== + + >>> from sympy import ccode, symbols, Rational, sin, ceiling, Abs, Function + >>> x, tau = symbols("x, tau") + >>> expr = (2*tau)**Rational(7, 2) + >>> ccode(expr) + '8*M_SQRT2*pow(tau, 7.0/2.0)' + >>> ccode(expr, math_macros={}) + '8*sqrt(2)*pow(tau, 7.0/2.0)' + >>> ccode(sin(x), assign_to="s") + 's = sin(x);' + >>> from sympy.codegen.ast import real, float80 + >>> ccode(expr, type_aliases={real: float80}) + '8*M_SQRT2l*powl(tau, 7.0L/2.0L)' + + Simple custom printing can be defined for certain types by passing a + dictionary of {"type" : "function"} to the ``user_functions`` kwarg. + Alternatively, the dictionary value can be a list of tuples i.e. + [(argument_test, cfunction_string)]. + + >>> custom_functions = { + ... "ceiling": "CEIL", + ... "Abs": [(lambda x: not x.is_integer, "fabs"), + ... (lambda x: x.is_integer, "ABS")], + ... "func": "f" + ... } + >>> func = Function('func') + >>> ccode(func(Abs(x) + ceiling(x)), standard='C89', user_functions=custom_functions) + 'f(fabs(x) + CEIL(x))' + + or if the C-function takes a subset of the original arguments: + + >>> ccode(2**x + 3**x, standard='C99', user_functions={'Pow': [ + ... (lambda b, e: b == 2, lambda b, e: 'exp2(%s)' % e), + ... (lambda b, e: b != 2, 'pow')]}) + 'exp2(x) + pow(3, x)' + + ``Piecewise`` expressions are converted into conditionals. If an + ``assign_to`` variable is provided an if statement is created, otherwise + the ternary operator is used. Note that if the ``Piecewise`` lacks a + default term, represented by ``(expr, True)`` then an error will be thrown. + This is to prevent generating an expression that may not evaluate to + anything. + + >>> from sympy import Piecewise + >>> expr = Piecewise((x + 1, x > 0), (x, True)) + >>> print(ccode(expr, tau, standard='C89')) + if (x > 0) { + tau = x + 1; + } + else { + tau = x; + } + + Support for loops is provided through ``Indexed`` types. With + ``contract=True`` these expressions will be turned into loops, whereas + ``contract=False`` will just print the assignment expression that should be + looped over: + + >>> from sympy import Eq, IndexedBase, Idx + >>> len_y = 5 + >>> y = IndexedBase('y', shape=(len_y,)) + >>> t = IndexedBase('t', shape=(len_y,)) + >>> Dy = IndexedBase('Dy', shape=(len_y-1,)) + >>> i = Idx('i', len_y-1) + >>> e=Eq(Dy[i], (y[i+1]-y[i])/(t[i+1]-t[i])) + >>> ccode(e.rhs, assign_to=e.lhs, contract=False, standard='C89') + 'Dy[i] = (y[i + 1] - y[i])/(t[i + 1] - t[i]);' + + Matrices are also supported, but a ``MatrixSymbol`` of the same dimensions + must be provided to ``assign_to``. Note that any expression that can be + generated normally can also exist inside a Matrix: + + >>> from sympy import Matrix, MatrixSymbol + >>> mat = Matrix([x**2, Piecewise((x + 1, x > 0), (x, True)), sin(x)]) + >>> A = MatrixSymbol('A', 3, 1) + >>> print(ccode(mat, A, standard='C89')) + A[0] = pow(x, 2); + if (x > 0) { + A[1] = x + 1; + } + else { + A[1] = x; + } + A[2] = sin(x); + """ + from sympy.printing.c import c_code_printers + return c_code_printers[standard.lower()](settings).doprint(expr, assign_to) + +def print_ccode(expr, **settings): + """Prints C representation of the given expression.""" + print(ccode(expr, **settings)) + +def fcode(expr, assign_to=None, **settings): + """Converts an expr to a string of fortran code + + Parameters + ========== + + expr : Expr + A SymPy expression to be converted. + assign_to : optional + When given, the argument is used as the name of the variable to which + the expression is assigned. Can be a string, ``Symbol``, + ``MatrixSymbol``, or ``Indexed`` type. This is helpful in case of + line-wrapping, or for expressions that generate multi-line statements. + precision : integer, optional + DEPRECATED. Use type_mappings instead. The precision for numbers such + as pi [default=17]. + user_functions : dict, optional + A dictionary where keys are ``FunctionClass`` instances and values are + their string representations. Alternatively, the dictionary value can + be a list of tuples i.e. [(argument_test, cfunction_string)]. See below + for examples. + human : bool, optional + If True, the result is a single string that may contain some constant + declarations for the number symbols. If False, the same information is + returned in a tuple of (symbols_to_declare, not_supported_functions, + code_text). [default=True]. + contract: bool, optional + If True, ``Indexed`` instances are assumed to obey tensor contraction + rules and the corresponding nested loops over indices are generated. + Setting contract=False will not generate loops, instead the user is + responsible to provide values for the indices in the code. + [default=True]. + source_format : optional + The source format can be either 'fixed' or 'free'. [default='fixed'] + standard : integer, optional + The Fortran standard to be followed. This is specified as an integer. + Acceptable standards are 66, 77, 90, 95, 2003, and 2008. Default is 77. + Note that currently the only distinction internally is between + standards before 95, and those 95 and after. This may change later as + more features are added. + name_mangling : bool, optional + If True, then the variables that would become identical in + case-insensitive Fortran are mangled by appending different number + of ``_`` at the end. If False, SymPy Will not interfere with naming of + variables. [default=True] + + Examples + ======== + + >>> from sympy import fcode, symbols, Rational, sin, ceiling, floor + >>> x, tau = symbols("x, tau") + >>> fcode((2*tau)**Rational(7, 2)) + ' 8*sqrt(2.0d0)*tau**(7.0d0/2.0d0)' + >>> fcode(sin(x), assign_to="s") + ' s = sin(x)' + + Custom printing can be defined for certain types by passing a dictionary of + "type" : "function" to the ``user_functions`` kwarg. Alternatively, the + dictionary value can be a list of tuples i.e. [(argument_test, + cfunction_string)]. + + >>> custom_functions = { + ... "ceiling": "CEIL", + ... "floor": [(lambda x: not x.is_integer, "FLOOR1"), + ... (lambda x: x.is_integer, "FLOOR2")] + ... } + >>> fcode(floor(x) + ceiling(x), user_functions=custom_functions) + ' CEIL(x) + FLOOR1(x)' + + ``Piecewise`` expressions are converted into conditionals. If an + ``assign_to`` variable is provided an if statement is created, otherwise + the ternary operator is used. Note that if the ``Piecewise`` lacks a + default term, represented by ``(expr, True)`` then an error will be thrown. + This is to prevent generating an expression that may not evaluate to + anything. + + >>> from sympy import Piecewise + >>> expr = Piecewise((x + 1, x > 0), (x, True)) + >>> print(fcode(expr, tau)) + if (x > 0) then + tau = x + 1 + else + tau = x + end if + + Support for loops is provided through ``Indexed`` types. With + ``contract=True`` these expressions will be turned into loops, whereas + ``contract=False`` will just print the assignment expression that should be + looped over: + + >>> from sympy import Eq, IndexedBase, Idx + >>> len_y = 5 + >>> y = IndexedBase('y', shape=(len_y,)) + >>> t = IndexedBase('t', shape=(len_y,)) + >>> Dy = IndexedBase('Dy', shape=(len_y-1,)) + >>> i = Idx('i', len_y-1) + >>> e=Eq(Dy[i], (y[i+1]-y[i])/(t[i+1]-t[i])) + >>> fcode(e.rhs, assign_to=e.lhs, contract=False) + ' Dy(i) = (y(i + 1) - y(i))/(t(i + 1) - t(i))' + + Matrices are also supported, but a ``MatrixSymbol`` of the same dimensions + must be provided to ``assign_to``. Note that any expression that can be + generated normally can also exist inside a Matrix: + + >>> from sympy import Matrix, MatrixSymbol + >>> mat = Matrix([x**2, Piecewise((x + 1, x > 0), (x, True)), sin(x)]) + >>> A = MatrixSymbol('A', 3, 1) + >>> print(fcode(mat, A)) + A(1, 1) = x**2 + if (x > 0) then + A(2, 1) = x + 1 + else + A(2, 1) = x + end if + A(3, 1) = sin(x) + """ + from sympy.printing.fortran import FCodePrinter + return FCodePrinter(settings).doprint(expr, assign_to) + + +def print_fcode(expr, **settings): + """Prints the Fortran representation of the given expression. + + See fcode for the meaning of the optional arguments. + """ + print(fcode(expr, **settings)) + +def cxxcode(expr, assign_to=None, standard='c++11', **settings): + """ C++ equivalent of :func:`~.ccode`. """ + from sympy.printing.cxx import cxx_code_printers + return cxx_code_printers[standard.lower()](settings).doprint(expr, assign_to) + + +def rust_code(expr, assign_to=None, **settings): + """Converts an expr to a string of Rust code + + Parameters + ========== + + expr : Expr + A SymPy expression to be converted. + assign_to : optional + When given, the argument is used as the name of the variable to which + the expression is assigned. Can be a string, ``Symbol``, + ``MatrixSymbol``, or ``Indexed`` type. This is helpful in case of + line-wrapping, or for expressions that generate multi-line statements. + precision : integer, optional + The precision for numbers such as pi [default=15]. + user_functions : dict, optional + A dictionary where the keys are string representations of either + ``FunctionClass`` or ``UndefinedFunction`` instances and the values + are their desired C string representations. Alternatively, the + dictionary value can be a list of tuples i.e. [(argument_test, + cfunction_string)]. See below for examples. + dereference : iterable, optional + An iterable of symbols that should be dereferenced in the printed code + expression. These would be values passed by address to the function. + For example, if ``dereference=[a]``, the resulting code would print + ``(*a)`` instead of ``a``. + human : bool, optional + If True, the result is a single string that may contain some constant + declarations for the number symbols. If False, the same information is + returned in a tuple of (symbols_to_declare, not_supported_functions, + code_text). [default=True]. + contract: bool, optional + If True, ``Indexed`` instances are assumed to obey tensor contraction + rules and the corresponding nested loops over indices are generated. + Setting contract=False will not generate loops, instead the user is + responsible to provide values for the indices in the code. + [default=True]. + + Examples + ======== + + >>> from sympy import rust_code, symbols, Rational, sin, ceiling, Abs, Function + >>> x, tau = symbols("x, tau") + >>> rust_code((2*tau)**Rational(7, 2)) + '8.0*1.4142135623731*tau.powf(7_f64/2.0)' + >>> rust_code(sin(x), assign_to="s") + 's = x.sin();' + + Simple custom printing can be defined for certain types by passing a + dictionary of {"type" : "function"} to the ``user_functions`` kwarg. + Alternatively, the dictionary value can be a list of tuples i.e. + [(argument_test, cfunction_string)]. + + >>> custom_functions = { + ... "ceiling": "CEIL", + ... "Abs": [(lambda x: not x.is_integer, "fabs", 4), + ... (lambda x: x.is_integer, "ABS", 4)], + ... "func": "f" + ... } + >>> func = Function('func') + >>> rust_code(func(Abs(x) + ceiling(x)), user_functions=custom_functions) + '(fabs(x) + x.ceil()).f()' + + ``Piecewise`` expressions are converted into conditionals. If an + ``assign_to`` variable is provided an if statement is created, otherwise + the ternary operator is used. Note that if the ``Piecewise`` lacks a + default term, represented by ``(expr, True)`` then an error will be thrown. + This is to prevent generating an expression that may not evaluate to + anything. + + >>> from sympy import Piecewise + >>> expr = Piecewise((x + 1, x > 0), (x, True)) + >>> print(rust_code(expr, tau)) + tau = if (x > 0.0) { + x + 1 + } else { + x + }; + + Support for loops is provided through ``Indexed`` types. With + ``contract=True`` these expressions will be turned into loops, whereas + ``contract=False`` will just print the assignment expression that should be + looped over: + + >>> from sympy import Eq, IndexedBase, Idx + >>> len_y = 5 + >>> y = IndexedBase('y', shape=(len_y,)) + >>> t = IndexedBase('t', shape=(len_y,)) + >>> Dy = IndexedBase('Dy', shape=(len_y-1,)) + >>> i = Idx('i', len_y-1) + >>> e=Eq(Dy[i], (y[i+1]-y[i])/(t[i+1]-t[i])) + >>> rust_code(e.rhs, assign_to=e.lhs, contract=False) + 'Dy[i] = (y[i + 1] - y[i])/(t[i + 1] - t[i]);' + + Matrices are also supported, but a ``MatrixSymbol`` of the same dimensions + must be provided to ``assign_to``. Note that any expression that can be + generated normally can also exist inside a Matrix: + + >>> from sympy import Matrix, MatrixSymbol + >>> mat = Matrix([x**2, Piecewise((x + 1, x > 0), (x, True)), sin(x)]) + >>> A = MatrixSymbol('A', 3, 1) + >>> print(rust_code(mat, A)) + A = [x.powi(2), if (x > 0.0) { + x + 1 + } else { + x + }, x.sin()]; + """ + from sympy.printing.rust import RustCodePrinter + printer = RustCodePrinter(settings) + expr = printer._rewrite_known_functions(expr) + if isinstance(expr, Expr): + for src_func, dst_func in printer.function_overrides.values(): + expr = expr.replace(src_func, dst_func) + return printer.doprint(expr, assign_to) + + +def print_rust_code(expr, **settings): + """Prints Rust representation of the given expression.""" + print(rust_code(expr, **settings)) diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/conventions.py b/.venv/lib/python3.12/site-packages/sympy/printing/conventions.py new file mode 100644 index 0000000000000000000000000000000000000000..4f5545ae38511e0bb0366da5c9fb4e59156095c8 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/conventions.py @@ -0,0 +1,88 @@ +""" +A few practical conventions common to all printers. +""" + +import re + +from collections.abc import Iterable +from sympy.core.function import Derivative + +_name_with_digits_p = re.compile(r'^([^\W\d_]+)(\d+)$', re.UNICODE) + + +def split_super_sub(text): + """Split a symbol name into a name, superscripts and subscripts + + The first part of the symbol name is considered to be its actual + 'name', followed by super- and subscripts. Each superscript is + preceded with a "^" character or by "__". Each subscript is preceded + by a "_" character. The three return values are the actual name, a + list with superscripts and a list with subscripts. + + Examples + ======== + + >>> from sympy.printing.conventions import split_super_sub + >>> split_super_sub('a_x^1') + ('a', ['1'], ['x']) + >>> split_super_sub('var_sub1__sup_sub2') + ('var', ['sup'], ['sub1', 'sub2']) + + """ + if not text: + return text, [], [] + + pos = 0 + name = None + supers = [] + subs = [] + while pos < len(text): + start = pos + 1 + if text[pos:pos + 2] == "__": + start += 1 + pos_hat = text.find("^", start) + if pos_hat < 0: + pos_hat = len(text) + pos_usc = text.find("_", start) + if pos_usc < 0: + pos_usc = len(text) + pos_next = min(pos_hat, pos_usc) + part = text[pos:pos_next] + pos = pos_next + if name is None: + name = part + elif part.startswith("^"): + supers.append(part[1:]) + elif part.startswith("__"): + supers.append(part[2:]) + elif part.startswith("_"): + subs.append(part[1:]) + else: + raise RuntimeError("This should never happen.") + + # Make a little exception when a name ends with digits, i.e. treat them + # as a subscript too. + m = _name_with_digits_p.match(name) + if m: + name, sub = m.groups() + subs.insert(0, sub) + + return name, supers, subs + + +def requires_partial(expr): + """Return whether a partial derivative symbol is required for printing + + This requires checking how many free variables there are, + filtering out the ones that are integers. Some expressions do not have + free variables. In that case, check its variable list explicitly to + get the context of the expression. + """ + + if isinstance(expr, Derivative): + return requires_partial(expr.expr) + + if not isinstance(expr.free_symbols, Iterable): + return len(set(expr.variables)) > 1 + + return sum(not s.is_integer for s in expr.free_symbols) > 1 diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/cxx.py b/.venv/lib/python3.12/site-packages/sympy/printing/cxx.py new file mode 100644 index 0000000000000000000000000000000000000000..0ed4f468b866e1b44aba6ae94a85c740dd324689 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/cxx.py @@ -0,0 +1,181 @@ +""" +C++ code printer +""" + +from itertools import chain +from sympy.codegen.ast import Type, none +from .codeprinter import requires +from .c import C89CodePrinter, C99CodePrinter + +# These are defined in the other file so we can avoid importing sympy.codegen +# from the top-level 'import sympy'. Export them here as well. +from sympy.printing.codeprinter import cxxcode # noqa:F401 + +# from https://en.cppreference.com/w/cpp/keyword +reserved = { + 'C++98': [ + 'and', 'and_eq', 'asm', 'auto', 'bitand', 'bitor', 'bool', 'break', + 'case', 'catch,', 'char', 'class', 'compl', 'const', 'const_cast', + 'continue', 'default', 'delete', 'do', 'double', 'dynamic_cast', + 'else', 'enum', 'explicit', 'export', 'extern', 'false', 'float', + 'for', 'friend', 'goto', 'if', 'inline', 'int', 'long', 'mutable', + 'namespace', 'new', 'not', 'not_eq', 'operator', 'or', 'or_eq', + 'private', 'protected', 'public', 'register', 'reinterpret_cast', + 'return', 'short', 'signed', 'sizeof', 'static', 'static_cast', + 'struct', 'switch', 'template', 'this', 'throw', 'true', 'try', + 'typedef', 'typeid', 'typename', 'union', 'unsigned', 'using', + 'virtual', 'void', 'volatile', 'wchar_t', 'while', 'xor', 'xor_eq' + ] +} + +reserved['C++11'] = reserved['C++98'][:] + [ + 'alignas', 'alignof', 'char16_t', 'char32_t', 'constexpr', 'decltype', + 'noexcept', 'nullptr', 'static_assert', 'thread_local' +] +reserved['C++17'] = reserved['C++11'][:] +reserved['C++17'].remove('register') +# TM TS: atomic_cancel, atomic_commit, atomic_noexcept, synchronized +# concepts TS: concept, requires +# module TS: import, module + + +_math_functions = { + 'C++98': { + 'Mod': 'fmod', + 'ceiling': 'ceil', + }, + 'C++11': { + 'gamma': 'tgamma', + }, + 'C++17': { + 'beta': 'beta', + 'Ei': 'expint', + 'zeta': 'riemann_zeta', + } +} + +# from https://en.cppreference.com/w/cpp/header/cmath +for k in ('Abs', 'exp', 'log', 'log10', 'sqrt', 'sin', 'cos', 'tan', # 'Pow' + 'asin', 'acos', 'atan', 'atan2', 'sinh', 'cosh', 'tanh', 'floor'): + _math_functions['C++98'][k] = k.lower() + + +for k in ('asinh', 'acosh', 'atanh', 'erf', 'erfc'): + _math_functions['C++11'][k] = k.lower() + + +def _attach_print_method(cls, sympy_name, func_name): + meth_name = '_print_%s' % sympy_name + if hasattr(cls, meth_name): + raise ValueError("Edit method (or subclass) instead of overwriting.") + def _print_method(self, expr): + return '{}{}({})'.format(self._ns, func_name, ', '.join(map(self._print, expr.args))) + _print_method.__doc__ = "Prints code for %s" % k + setattr(cls, meth_name, _print_method) + + +def _attach_print_methods(cls, cont): + for sympy_name, cxx_name in cont[cls.standard].items(): + _attach_print_method(cls, sympy_name, cxx_name) + + +class _CXXCodePrinterBase: + printmethod = "_cxxcode" + language = 'C++' + _ns = 'std::' # namespace + + def __init__(self, settings=None): + super().__init__(settings or {}) + + @requires(headers={'algorithm'}) + def _print_Max(self, expr): + from sympy.functions.elementary.miscellaneous import Max + if len(expr.args) == 1: + return self._print(expr.args[0]) + return "%smax(%s, %s)" % (self._ns, self._print(expr.args[0]), + self._print(Max(*expr.args[1:]))) + + @requires(headers={'algorithm'}) + def _print_Min(self, expr): + from sympy.functions.elementary.miscellaneous import Min + if len(expr.args) == 1: + return self._print(expr.args[0]) + return "%smin(%s, %s)" % (self._ns, self._print(expr.args[0]), + self._print(Min(*expr.args[1:]))) + + def _print_using(self, expr): + if expr.alias == none: + return 'using %s' % expr.type + else: + raise ValueError("C++98 does not support type aliases") + + def _print_Raise(self, rs): + arg, = rs.args + return 'throw %s' % self._print(arg) + + @requires(headers={'stdexcept'}) + def _print_RuntimeError_(self, re): + message, = re.args + return "%sruntime_error(%s)" % (self._ns, self._print(message)) + + +class CXX98CodePrinter(_CXXCodePrinterBase, C89CodePrinter): + standard = 'C++98' + reserved_words = set(reserved['C++98']) + + +# _attach_print_methods(CXX98CodePrinter, _math_functions) + + +class CXX11CodePrinter(_CXXCodePrinterBase, C99CodePrinter): + standard = 'C++11' + reserved_words = set(reserved['C++11']) + type_mappings = dict(chain( + CXX98CodePrinter.type_mappings.items(), + { + Type('int8'): ('int8_t', {'cstdint'}), + Type('int16'): ('int16_t', {'cstdint'}), + Type('int32'): ('int32_t', {'cstdint'}), + Type('int64'): ('int64_t', {'cstdint'}), + Type('uint8'): ('uint8_t', {'cstdint'}), + Type('uint16'): ('uint16_t', {'cstdint'}), + Type('uint32'): ('uint32_t', {'cstdint'}), + Type('uint64'): ('uint64_t', {'cstdint'}), + Type('complex64'): ('std::complex', {'complex'}), + Type('complex128'): ('std::complex', {'complex'}), + Type('bool'): ('bool', None), + }.items() + )) + + def _print_using(self, expr): + if expr.alias == none: + return super()._print_using(expr) + else: + return 'using %(alias)s = %(type)s' % expr.kwargs(apply=self._print) + +# _attach_print_methods(CXX11CodePrinter, _math_functions) + + +class CXX17CodePrinter(_CXXCodePrinterBase, C99CodePrinter): + standard = 'C++17' + reserved_words = set(reserved['C++17']) + + _kf = dict(C99CodePrinter._kf, **_math_functions['C++17']) + + def _print_beta(self, expr): + return self._print_math_func(expr) + + def _print_Ei(self, expr): + return self._print_math_func(expr) + + def _print_zeta(self, expr): + return self._print_math_func(expr) + + +# _attach_print_methods(CXX17CodePrinter, _math_functions) + +cxx_code_printers = { + 'c++98': CXX98CodePrinter, + 'c++11': CXX11CodePrinter, + 'c++17': CXX17CodePrinter +} diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/defaults.py b/.venv/lib/python3.12/site-packages/sympy/printing/defaults.py new file mode 100644 index 0000000000000000000000000000000000000000..77a88d353fed4bd70496456ddd03cc429a4ba5e7 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/defaults.py @@ -0,0 +1,5 @@ +from sympy.core._print_helpers import Printable + +# alias for compatibility +Printable.__module__ = __name__ +DefaultPrinting = Printable diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/dot.py b/.venv/lib/python3.12/site-packages/sympy/printing/dot.py new file mode 100644 index 0000000000000000000000000000000000000000..c968fee389c16108b757b8fcad531ac6fa4ddb2f --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/dot.py @@ -0,0 +1,294 @@ +from sympy.core.basic import Basic +from sympy.core.expr import Expr +from sympy.core.symbol import Symbol +from sympy.core.numbers import Integer, Rational, Float +from sympy.printing.repr import srepr + +__all__ = ['dotprint'] + +default_styles = ( + (Basic, {'color': 'blue', 'shape': 'ellipse'}), + (Expr, {'color': 'black'}) +) + +slotClasses = (Symbol, Integer, Rational, Float) +def purestr(x, with_args=False): + """A string that follows ```obj = type(obj)(*obj.args)``` exactly. + + Parameters + ========== + + with_args : boolean, optional + If ``True``, there will be a second argument for the return + value, which is a tuple containing ``purestr`` applied to each + of the subnodes. + + If ``False``, there will not be a second argument for the + return. + + Default is ``False`` + + Examples + ======== + + >>> from sympy import Float, Symbol, MatrixSymbol + >>> from sympy import Integer # noqa: F401 + >>> from sympy.core.symbol import Str # noqa: F401 + >>> from sympy.printing.dot import purestr + + Applying ``purestr`` for basic symbolic object: + >>> code = purestr(Symbol('x')) + >>> code + "Symbol('x')" + >>> eval(code) == Symbol('x') + True + + For basic numeric object: + >>> purestr(Float(2)) + "Float('2.0', precision=53)" + + For matrix symbol: + >>> code = purestr(MatrixSymbol('x', 2, 2)) + >>> code + "MatrixSymbol(Str('x'), Integer(2), Integer(2))" + >>> eval(code) == MatrixSymbol('x', 2, 2) + True + + With ``with_args=True``: + >>> purestr(Float(2), with_args=True) + ("Float('2.0', precision=53)", ()) + >>> purestr(MatrixSymbol('x', 2, 2), with_args=True) + ("MatrixSymbol(Str('x'), Integer(2), Integer(2))", + ("Str('x')", 'Integer(2)', 'Integer(2)')) + """ + sargs = () + if not isinstance(x, Basic): + rv = str(x) + elif not x.args: + rv = srepr(x) + else: + args = x.args + sargs = tuple(map(purestr, args)) + rv = "%s(%s)"%(type(x).__name__, ', '.join(sargs)) + if with_args: + rv = rv, sargs + return rv + + +def styleof(expr, styles=default_styles): + """ Merge style dictionaries in order + + Examples + ======== + + >>> from sympy import Symbol, Basic, Expr, S + >>> from sympy.printing.dot import styleof + >>> styles = [(Basic, {'color': 'blue', 'shape': 'ellipse'}), + ... (Expr, {'color': 'black'})] + + >>> styleof(Basic(S(1)), styles) + {'color': 'blue', 'shape': 'ellipse'} + + >>> x = Symbol('x') + >>> styleof(x + 1, styles) # this is an Expr + {'color': 'black', 'shape': 'ellipse'} + """ + style = {} + for typ, sty in styles: + if isinstance(expr, typ): + style.update(sty) + return style + + +def attrprint(d, delimiter=', '): + """ Print a dictionary of attributes + + Examples + ======== + + >>> from sympy.printing.dot import attrprint + >>> print(attrprint({'color': 'blue', 'shape': 'ellipse'})) + "color"="blue", "shape"="ellipse" + """ + return delimiter.join('"%s"="%s"'%item for item in sorted(d.items())) + + +def dotnode(expr, styles=default_styles, labelfunc=str, pos=(), repeat=True): + """ String defining a node + + Examples + ======== + + >>> from sympy.printing.dot import dotnode + >>> from sympy.abc import x + >>> print(dotnode(x)) + "Symbol('x')_()" ["color"="black", "label"="x", "shape"="ellipse"]; + """ + style = styleof(expr, styles) + + if isinstance(expr, Basic) and not expr.is_Atom: + label = str(expr.__class__.__name__) + else: + label = labelfunc(expr) + style['label'] = label + expr_str = purestr(expr) + if repeat: + expr_str += '_%s' % str(pos) + return '"%s" [%s];' % (expr_str, attrprint(style)) + + +def dotedges(expr, atom=lambda x: not isinstance(x, Basic), pos=(), repeat=True): + """ List of strings for all expr->expr.arg pairs + + See the docstring of dotprint for explanations of the options. + + Examples + ======== + + >>> from sympy.printing.dot import dotedges + >>> from sympy.abc import x + >>> for e in dotedges(x+2): + ... print(e) + "Add(Integer(2), Symbol('x'))_()" -> "Integer(2)_(0,)"; + "Add(Integer(2), Symbol('x'))_()" -> "Symbol('x')_(1,)"; + """ + if atom(expr): + return [] + else: + expr_str, arg_strs = purestr(expr, with_args=True) + if repeat: + expr_str += '_%s' % str(pos) + arg_strs = ['%s_%s' % (a, str(pos + (i,))) + for i, a in enumerate(arg_strs)] + return ['"%s" -> "%s";' % (expr_str, a) for a in arg_strs] + +template = \ +"""digraph{ + +# Graph style +%(graphstyle)s + +######### +# Nodes # +######### + +%(nodes)s + +######### +# Edges # +######### + +%(edges)s +}""" + +_graphstyle = {'rankdir': 'TD', 'ordering': 'out'} + +def dotprint(expr, + styles=default_styles, atom=lambda x: not isinstance(x, Basic), + maxdepth=None, repeat=True, labelfunc=str, **kwargs): + """DOT description of a SymPy expression tree + + Parameters + ========== + + styles : list of lists composed of (Class, mapping), optional + Styles for different classes. + + The default is + + .. code-block:: python + + ( + (Basic, {'color': 'blue', 'shape': 'ellipse'}), + (Expr, {'color': 'black'}) + ) + + atom : function, optional + Function used to determine if an arg is an atom. + + A good choice is ``lambda x: not x.args``. + + The default is ``lambda x: not isinstance(x, Basic)``. + + maxdepth : integer, optional + The maximum depth. + + The default is ``None``, meaning no limit. + + repeat : boolean, optional + Whether to use different nodes for common subexpressions. + + The default is ``True``. + + For example, for ``x + x*y`` with ``repeat=True``, it will have + two nodes for ``x``; with ``repeat=False``, it will have one + node. + + .. warning:: + Even if a node appears twice in the same object like ``x`` in + ``Pow(x, x)``, it will still only appear once. + Hence, with ``repeat=False``, the number of arrows out of an + object might not equal the number of args it has. + + labelfunc : function, optional + A function to create a label for a given leaf node. + + The default is ``str``. + + Another good option is ``srepr``. + + For example with ``str``, the leaf nodes of ``x + 1`` are labeled, + ``x`` and ``1``. With ``srepr``, they are labeled ``Symbol('x')`` + and ``Integer(1)``. + + **kwargs : optional + Additional keyword arguments are included as styles for the graph. + + Examples + ======== + + >>> from sympy import dotprint + >>> from sympy.abc import x + >>> print(dotprint(x+2)) # doctest: +NORMALIZE_WHITESPACE + digraph{ + + # Graph style + "ordering"="out" + "rankdir"="TD" + + ######### + # Nodes # + ######### + + "Add(Integer(2), Symbol('x'))_()" ["color"="black", "label"="Add", "shape"="ellipse"]; + "Integer(2)_(0,)" ["color"="black", "label"="2", "shape"="ellipse"]; + "Symbol('x')_(1,)" ["color"="black", "label"="x", "shape"="ellipse"]; + + ######### + # Edges # + ######### + + "Add(Integer(2), Symbol('x'))_()" -> "Integer(2)_(0,)"; + "Add(Integer(2), Symbol('x'))_()" -> "Symbol('x')_(1,)"; + } + + """ + # repeat works by adding a signature tuple to the end of each node for its + # position in the graph. For example, for expr = Add(x, Pow(x, 2)), the x in the + # Pow will have the tuple (1, 0), meaning it is expr.args[1].args[0]. + graphstyle = _graphstyle.copy() + graphstyle.update(kwargs) + + nodes = [] + edges = [] + def traverse(e, depth, pos=()): + nodes.append(dotnode(e, styles, labelfunc=labelfunc, pos=pos, repeat=repeat)) + if maxdepth and depth >= maxdepth: + return + edges.extend(dotedges(e, atom=atom, pos=pos, repeat=repeat)) + [traverse(arg, depth+1, pos + (i,)) for i, arg in enumerate(e.args) if not atom(arg)] + traverse(expr, 0) + + return template%{'graphstyle': attrprint(graphstyle, delimiter='\n'), + 'nodes': '\n'.join(nodes), + 'edges': '\n'.join(edges)} diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/fortran.py b/.venv/lib/python3.12/site-packages/sympy/printing/fortran.py new file mode 100644 index 0000000000000000000000000000000000000000..7cea812d72ddcd3ccb56c7258f74e6fe3b8d5211 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/fortran.py @@ -0,0 +1,779 @@ +""" +Fortran code printer + +The FCodePrinter converts single SymPy expressions into single Fortran +expressions, using the functions defined in the Fortran 77 standard where +possible. Some useful pointers to Fortran can be found on wikipedia: + +https://en.wikipedia.org/wiki/Fortran + +Most of the code below is based on the "Professional Programmer\'s Guide to +Fortran77" by Clive G. Page: + +https://www.star.le.ac.uk/~cgp/prof77.html + +Fortran is a case-insensitive language. This might cause trouble because +SymPy is case sensitive. So, fcode adds underscores to variable names when +it is necessary to make them different for Fortran. +""" + +from __future__ import annotations +from typing import Any + +from collections import defaultdict +from itertools import chain +import string + +from sympy.codegen.ast import ( + Assignment, Declaration, Pointer, value_const, + float32, float64, float80, complex64, complex128, int8, int16, int32, + int64, intc, real, integer, bool_, complex_, none, stderr, stdout +) +from sympy.codegen.fnodes import ( + allocatable, isign, dsign, cmplx, merge, literal_dp, elemental, pure, + intent_in, intent_out, intent_inout +) +from sympy.core import S, Add, N, Float, Symbol +from sympy.core.function import Function +from sympy.core.numbers import equal_valued +from sympy.core.relational import Eq +from sympy.sets import Range +from sympy.printing.codeprinter import CodePrinter +from sympy.printing.precedence import precedence, PRECEDENCE +from sympy.printing.printer import printer_context + +# These are defined in the other file so we can avoid importing sympy.codegen +# from the top-level 'import sympy'. Export them here as well. +from sympy.printing.codeprinter import fcode, print_fcode # noqa:F401 + +known_functions = { + "sin": "sin", + "cos": "cos", + "tan": "tan", + "asin": "asin", + "acos": "acos", + "atan": "atan", + "atan2": "atan2", + "sinh": "sinh", + "cosh": "cosh", + "tanh": "tanh", + "log": "log", + "exp": "exp", + "erf": "erf", + "Abs": "abs", + "conjugate": "conjg", + "Max": "max", + "Min": "min", +} + + +class FCodePrinter(CodePrinter): + """A printer to convert SymPy expressions to strings of Fortran code""" + printmethod = "_fcode" + language = "Fortran" + + type_aliases = { + integer: int32, + real: float64, + complex_: complex128, + } + + type_mappings = { + intc: 'integer(c_int)', + float32: 'real*4', # real(kind(0.e0)) + float64: 'real*8', # real(kind(0.d0)) + float80: 'real*10', # real(kind(????)) + complex64: 'complex*8', + complex128: 'complex*16', + int8: 'integer*1', + int16: 'integer*2', + int32: 'integer*4', + int64: 'integer*8', + bool_: 'logical' + } + + type_modules = { + intc: {'iso_c_binding': 'c_int'} + } + + _default_settings: dict[str, Any] = dict(CodePrinter._default_settings, **{ + 'precision': 17, + 'user_functions': {}, + 'source_format': 'fixed', + 'contract': True, + 'standard': 77, + 'name_mangling': True, + }) + + _operators = { + 'and': '.and.', + 'or': '.or.', + 'xor': '.neqv.', + 'equivalent': '.eqv.', + 'not': '.not. ', + } + + _relationals = { + '!=': '/=', + } + + def __init__(self, settings=None): + if not settings: + settings = {} + self.mangled_symbols = {} # Dict showing mapping of all words + self.used_name = [] + self.type_aliases = dict(chain(self.type_aliases.items(), + settings.pop('type_aliases', {}).items())) + self.type_mappings = dict(chain(self.type_mappings.items(), + settings.pop('type_mappings', {}).items())) + super().__init__(settings) + self.known_functions = dict(known_functions) + userfuncs = settings.get('user_functions', {}) + self.known_functions.update(userfuncs) + # leading columns depend on fixed or free format + standards = {66, 77, 90, 95, 2003, 2008} + if self._settings['standard'] not in standards: + raise ValueError("Unknown Fortran standard: %s" % self._settings[ + 'standard']) + self.module_uses = defaultdict(set) # e.g.: use iso_c_binding, only: c_int + + @property + def _lead(self): + if self._settings['source_format'] == 'fixed': + return {'code': " ", 'cont': " @ ", 'comment': "C "} + elif self._settings['source_format'] == 'free': + return {'code': "", 'cont': " ", 'comment': "! "} + else: + raise ValueError("Unknown source format: %s" % self._settings['source_format']) + + def _print_Symbol(self, expr): + if self._settings['name_mangling'] == True: + if expr not in self.mangled_symbols: + name = expr.name + while name.lower() in self.used_name: + name += '_' + self.used_name.append(name.lower()) + if name == expr.name: + self.mangled_symbols[expr] = expr + else: + self.mangled_symbols[expr] = Symbol(name) + + expr = expr.xreplace(self.mangled_symbols) + + name = super()._print_Symbol(expr) + return name + + def _rate_index_position(self, p): + return -p*5 + + def _get_statement(self, codestring): + return codestring + + def _get_comment(self, text): + return "! {}".format(text) + + def _declare_number_const(self, name, value): + return "parameter ({} = {})".format(name, self._print(value)) + + def _print_NumberSymbol(self, expr): + # A Number symbol that is not implemented here or with _printmethod + # is registered and evaluated + self._number_symbols.add((expr, Float(expr.evalf(self._settings['precision'])))) + return str(expr) + + def _format_code(self, lines): + return self._wrap_fortran(self.indent_code(lines)) + + def _traverse_matrix_indices(self, mat): + rows, cols = mat.shape + return ((i, j) for j in range(cols) for i in range(rows)) + + def _get_loop_opening_ending(self, indices): + open_lines = [] + close_lines = [] + for i in indices: + # fortran arrays start at 1 and end at dimension + var, start, stop = map(self._print, + [i.label, i.lower + 1, i.upper + 1]) + open_lines.append("do %s = %s, %s" % (var, start, stop)) + close_lines.append("end do") + return open_lines, close_lines + + def _print_sign(self, expr): + from sympy.functions.elementary.complexes import Abs + arg, = expr.args + if arg.is_integer: + new_expr = merge(0, isign(1, arg), Eq(arg, 0)) + elif (arg.is_complex or arg.is_infinite): + new_expr = merge(cmplx(literal_dp(0), literal_dp(0)), arg/Abs(arg), Eq(Abs(arg), literal_dp(0))) + else: + new_expr = merge(literal_dp(0), dsign(literal_dp(1), arg), Eq(arg, literal_dp(0))) + return self._print(new_expr) + + + def _print_Piecewise(self, expr): + if expr.args[-1].cond != True: + # We need the last conditional to be a True, otherwise the resulting + # function may not return a result. + raise ValueError("All Piecewise expressions must contain an " + "(expr, True) statement to be used as a default " + "condition. Without one, the generated " + "expression may not evaluate to anything under " + "some condition.") + lines = [] + if expr.has(Assignment): + for i, (e, c) in enumerate(expr.args): + if i == 0: + lines.append("if (%s) then" % self._print(c)) + elif i == len(expr.args) - 1 and c == True: + lines.append("else") + else: + lines.append("else if (%s) then" % self._print(c)) + lines.append(self._print(e)) + lines.append("end if") + return "\n".join(lines) + elif self._settings["standard"] >= 95: + # Only supported in F95 and newer: + # The piecewise was used in an expression, need to do inline + # operators. This has the downside that inline operators will + # not work for statements that span multiple lines (Matrix or + # Indexed expressions). + pattern = "merge({T}, {F}, {COND})" + code = self._print(expr.args[-1].expr) + terms = list(expr.args[:-1]) + while terms: + e, c = terms.pop() + expr = self._print(e) + cond = self._print(c) + code = pattern.format(T=expr, F=code, COND=cond) + return code + else: + # `merge` is not supported prior to F95 + raise NotImplementedError("Using Piecewise as an expression using " + "inline operators is not supported in " + "standards earlier than Fortran95.") + + def _print_MatrixElement(self, expr): + return "{}({}, {})".format(self.parenthesize(expr.parent, + PRECEDENCE["Atom"], strict=True), expr.i + 1, expr.j + 1) + + def _print_Add(self, expr): + # purpose: print complex numbers nicely in Fortran. + # collect the purely real and purely imaginary parts: + pure_real = [] + pure_imaginary = [] + mixed = [] + for arg in expr.args: + if arg.is_number and arg.is_real: + pure_real.append(arg) + elif arg.is_number and arg.is_imaginary: + pure_imaginary.append(arg) + else: + mixed.append(arg) + if pure_imaginary: + if mixed: + PREC = precedence(expr) + term = Add(*mixed) + t = self._print(term) + if t.startswith('-'): + sign = "-" + t = t[1:] + else: + sign = "+" + if precedence(term) < PREC: + t = "(%s)" % t + + return "cmplx(%s,%s) %s %s" % ( + self._print(Add(*pure_real)), + self._print(-S.ImaginaryUnit*Add(*pure_imaginary)), + sign, t, + ) + else: + return "cmplx(%s,%s)" % ( + self._print(Add(*pure_real)), + self._print(-S.ImaginaryUnit*Add(*pure_imaginary)), + ) + else: + return CodePrinter._print_Add(self, expr) + + def _print_Function(self, expr): + # All constant function args are evaluated as floats + prec = self._settings['precision'] + args = [N(a, prec) for a in expr.args] + eval_expr = expr.func(*args) + if not isinstance(eval_expr, Function): + return self._print(eval_expr) + else: + return CodePrinter._print_Function(self, expr.func(*args)) + + def _print_Mod(self, expr): + # NOTE : Fortran has the functions mod() and modulo(). modulo() behaves + # the same wrt to the sign of the arguments as Python and SymPy's + # modulus computations (% and Mod()) but is not available in Fortran 66 + # or Fortran 77, thus we raise an error. + if self._settings['standard'] in [66, 77]: + msg = ("Python % operator and SymPy's Mod() function are not " + "supported by Fortran 66 or 77 standards.") + raise NotImplementedError(msg) + else: + x, y = expr.args + return " modulo({}, {})".format(self._print(x), self._print(y)) + + def _print_ImaginaryUnit(self, expr): + # purpose: print complex numbers nicely in Fortran. + return "cmplx(0,1)" + + def _print_int(self, expr): + return str(expr) + + def _print_Mul(self, expr): + # purpose: print complex numbers nicely in Fortran. + if expr.is_number and expr.is_imaginary: + return "cmplx(0,%s)" % ( + self._print(-S.ImaginaryUnit*expr) + ) + else: + return CodePrinter._print_Mul(self, expr) + + def _print_Pow(self, expr): + PREC = precedence(expr) + if equal_valued(expr.exp, -1): + return '%s/%s' % ( + self._print(literal_dp(1)), + self.parenthesize(expr.base, PREC) + ) + elif equal_valued(expr.exp, 0.5): + if expr.base.is_integer: + # Fortran intrinsic sqrt() does not accept integer argument + if expr.base.is_Number: + return 'sqrt(%s.0d0)' % self._print(expr.base) + else: + return 'sqrt(dble(%s))' % self._print(expr.base) + else: + return 'sqrt(%s)' % self._print(expr.base) + else: + return CodePrinter._print_Pow(self, expr) + + def _print_Rational(self, expr): + p, q = int(expr.p), int(expr.q) + return "%d.0d0/%d.0d0" % (p, q) + + def _print_Float(self, expr): + printed = CodePrinter._print_Float(self, expr) + e = printed.find('e') + if e > -1: + return "%sd%s" % (printed[:e], printed[e + 1:]) + return "%sd0" % printed + + def _print_Relational(self, expr): + lhs_code = self._print(expr.lhs) + rhs_code = self._print(expr.rhs) + op = expr.rel_op + op = op if op not in self._relationals else self._relationals[op] + return "{} {} {}".format(lhs_code, op, rhs_code) + + def _print_Indexed(self, expr): + inds = [ self._print(i) for i in expr.indices ] + return "%s(%s)" % (self._print(expr.base.label), ", ".join(inds)) + + def _print_AugmentedAssignment(self, expr): + lhs_code = self._print(expr.lhs) + rhs_code = self._print(expr.rhs) + return self._get_statement("{0} = {0} {1} {2}".format( + self._print(lhs_code), self._print(expr.binop), self._print(rhs_code))) + + def _print_sum_(self, sm): + params = self._print(sm.array) + if sm.dim != None: # Must use '!= None', cannot use 'is not None' + params += ', ' + self._print(sm.dim) + if sm.mask != None: # Must use '!= None', cannot use 'is not None' + params += ', mask=' + self._print(sm.mask) + return '%s(%s)' % (sm.__class__.__name__.rstrip('_'), params) + + def _print_product_(self, prod): + return self._print_sum_(prod) + + def _print_Do(self, do): + excl = ['concurrent'] + if do.step == 1: + excl.append('step') + step = '' + else: + step = ', {step}' + + return ( + 'do {concurrent}{counter} = {first}, {last}'+step+'\n' + '{body}\n' + 'end do\n' + ).format( + concurrent='concurrent ' if do.concurrent else '', + **do.kwargs(apply=lambda arg: self._print(arg), exclude=excl) + ) + + def _print_ImpliedDoLoop(self, idl): + step = '' if idl.step == 1 else ', {step}' + return ('({expr}, {counter} = {first}, {last}'+step+')').format( + **idl.kwargs(apply=lambda arg: self._print(arg)) + ) + + def _print_For(self, expr): + target = self._print(expr.target) + if isinstance(expr.iterable, Range): + start, stop, step = expr.iterable.args + else: + raise NotImplementedError("Only iterable currently supported is Range") + body = self._print(expr.body) + return ('do {target} = {start}, {stop}, {step}\n' + '{body}\n' + 'end do').format(target=target, start=start, stop=stop - 1, + step=step, body=body) + + def _print_Type(self, type_): + type_ = self.type_aliases.get(type_, type_) + type_str = self.type_mappings.get(type_, type_.name) + module_uses = self.type_modules.get(type_) + if module_uses: + for k, v in module_uses: + self.module_uses[k].add(v) + return type_str + + def _print_Element(self, elem): + return '{symbol}({idxs})'.format( + symbol=self._print(elem.symbol), + idxs=', '.join((self._print(arg) for arg in elem.indices)) + ) + + def _print_Extent(self, ext): + return str(ext) + + def _print_Declaration(self, expr): + var = expr.variable + val = var.value + dim = var.attr_params('dimension') + intents = [intent in var.attrs for intent in (intent_in, intent_out, intent_inout)] + if intents.count(True) == 0: + intent = '' + elif intents.count(True) == 1: + intent = ', intent(%s)' % ['in', 'out', 'inout'][intents.index(True)] + else: + raise ValueError("Multiple intents specified for %s" % self) + + if isinstance(var, Pointer): + raise NotImplementedError("Pointers are not available by default in Fortran.") + if self._settings["standard"] >= 90: + result = '{t}{vc}{dim}{intent}{alloc} :: {s}'.format( + t=self._print(var.type), + vc=', parameter' if value_const in var.attrs else '', + dim=', dimension(%s)' % ', '.join((self._print(arg) for arg in dim)) if dim else '', + intent=intent, + alloc=', allocatable' if allocatable in var.attrs else '', + s=self._print(var.symbol) + ) + if val != None: # Must be "!= None", cannot be "is not None" + result += ' = %s' % self._print(val) + else: + if value_const in var.attrs or val: + raise NotImplementedError("F77 init./parameter statem. req. multiple lines.") + result = ' '.join((self._print(arg) for arg in [var.type, var.symbol])) + + return result + + + def _print_Infinity(self, expr): + return '(huge(%s) + 1)' % self._print(literal_dp(0)) + + def _print_While(self, expr): + return 'do while ({condition})\n{body}\nend do'.format(**expr.kwargs( + apply=lambda arg: self._print(arg))) + + def _print_BooleanTrue(self, expr): + return '.true.' + + def _print_BooleanFalse(self, expr): + return '.false.' + + def _pad_leading_columns(self, lines): + result = [] + for line in lines: + if line.startswith('!'): + result.append(self._lead['comment'] + line[1:].lstrip()) + else: + result.append(self._lead['code'] + line) + return result + + def _wrap_fortran(self, lines): + """Wrap long Fortran lines + + Argument: + lines -- a list of lines (without \\n character) + + A comment line is split at white space. Code lines are split with a more + complex rule to give nice results. + """ + # routine to find split point in a code line + my_alnum = set("_+-." + string.digits + string.ascii_letters) + my_white = set(" \t()") + + def split_pos_code(line, endpos): + if len(line) <= endpos: + return len(line) + pos = endpos + split = lambda pos: \ + (line[pos] in my_alnum and line[pos - 1] not in my_alnum) or \ + (line[pos] not in my_alnum and line[pos - 1] in my_alnum) or \ + (line[pos] in my_white and line[pos - 1] not in my_white) or \ + (line[pos] not in my_white and line[pos - 1] in my_white) + while not split(pos): + pos -= 1 + if pos == 0: + return endpos + return pos + # split line by line and add the split lines to result + result = [] + if self._settings['source_format'] == 'free': + trailing = ' &' + else: + trailing = '' + for line in lines: + if line.startswith(self._lead['comment']): + # comment line + if len(line) > 72: + pos = line.rfind(" ", 6, 72) + if pos == -1: + pos = 72 + hunk = line[:pos] + line = line[pos:].lstrip() + result.append(hunk) + while line: + pos = line.rfind(" ", 0, 66) + if pos == -1 or len(line) < 66: + pos = 66 + hunk = line[:pos] + line = line[pos:].lstrip() + result.append("%s%s" % (self._lead['comment'], hunk)) + else: + result.append(line) + elif line.startswith(self._lead['code']): + # code line + pos = split_pos_code(line, 72) + hunk = line[:pos].rstrip() + line = line[pos:].lstrip() + if line: + hunk += trailing + result.append(hunk) + while line: + pos = split_pos_code(line, 65) + hunk = line[:pos].rstrip() + line = line[pos:].lstrip() + if line: + hunk += trailing + result.append("%s%s" % (self._lead['cont'], hunk)) + else: + result.append(line) + return result + + def indent_code(self, code): + """Accepts a string of code or a list of code lines""" + if isinstance(code, str): + code_lines = self.indent_code(code.splitlines(True)) + return ''.join(code_lines) + + free = self._settings['source_format'] == 'free' + code = [ line.lstrip(' \t') for line in code ] + + inc_keyword = ('do ', 'if(', 'if ', 'do\n', 'else', 'program', 'interface') + dec_keyword = ('end do', 'enddo', 'end if', 'endif', 'else', 'end program', 'end interface') + + increase = [ int(any(map(line.startswith, inc_keyword))) + for line in code ] + decrease = [ int(any(map(line.startswith, dec_keyword))) + for line in code ] + continuation = [ int(any(map(line.endswith, ['&', '&\n']))) + for line in code ] + + level = 0 + cont_padding = 0 + tabwidth = 3 + new_code = [] + for i, line in enumerate(code): + if line in ('', '\n'): + new_code.append(line) + continue + level -= decrease[i] + + if free: + padding = " "*(level*tabwidth + cont_padding) + else: + padding = " "*level*tabwidth + + line = "%s%s" % (padding, line) + if not free: + line = self._pad_leading_columns([line])[0] + + new_code.append(line) + + if continuation[i]: + cont_padding = 2*tabwidth + else: + cont_padding = 0 + level += increase[i] + + if not free: + return self._wrap_fortran(new_code) + return new_code + + def _print_GoTo(self, goto): + if goto.expr: # computed goto + return "go to ({labels}), {expr}".format( + labels=', '.join((self._print(arg) for arg in goto.labels)), + expr=self._print(goto.expr) + ) + else: + lbl, = goto.labels + return "go to %s" % self._print(lbl) + + def _print_Program(self, prog): + return ( + "program {name}\n" + "{body}\n" + "end program\n" + ).format(**prog.kwargs(apply=lambda arg: self._print(arg))) + + def _print_Module(self, mod): + return ( + "module {name}\n" + "{declarations}\n" + "\ncontains\n\n" + "{definitions}\n" + "end module\n" + ).format(**mod.kwargs(apply=lambda arg: self._print(arg))) + + def _print_Stream(self, strm): + if strm.name == 'stdout' and self._settings["standard"] >= 2003: + self.module_uses['iso_c_binding'].add('stdint=>input_unit') + return 'input_unit' + elif strm.name == 'stderr' and self._settings["standard"] >= 2003: + self.module_uses['iso_c_binding'].add('stdint=>error_unit') + return 'error_unit' + else: + if strm.name == 'stdout': + return '*' + else: + return strm.name + + def _print_Print(self, ps): + if ps.format_string == none: # Must be '!= None', cannot be 'is not None' + template = "print {fmt}, {iolist}" + fmt = '*' + else: + template = 'write(%(out)s, fmt="{fmt}", advance="no"), {iolist}' % { + 'out': {stderr: '0', stdout: '6'}.get(ps.file, '*') + } + fmt = self._print(ps.format_string) + return template.format(fmt=fmt, iolist=', '.join( + (self._print(arg) for arg in ps.print_args))) + + def _print_Return(self, rs): + arg, = rs.args + return "{result_name} = {arg}".format( + result_name=self._context.get('result_name', 'sympy_result'), + arg=self._print(arg) + ) + + def _print_FortranReturn(self, frs): + arg, = frs.args + if arg: + return 'return %s' % self._print(arg) + else: + return 'return' + + def _head(self, entity, fp, **kwargs): + bind_C_params = fp.attr_params('bind_C') + if bind_C_params is None: + bind = '' + else: + bind = ' bind(C, name="%s")' % bind_C_params[0] if bind_C_params else ' bind(C)' + result_name = self._settings.get('result_name', None) + return ( + "{entity}{name}({arg_names}){result}{bind}\n" + "{arg_declarations}" + ).format( + entity=entity, + name=self._print(fp.name), + arg_names=', '.join([self._print(arg.symbol) for arg in fp.parameters]), + result=(' result(%s)' % result_name) if result_name else '', + bind=bind, + arg_declarations='\n'.join((self._print(Declaration(arg)) for arg in fp.parameters)) + ) + + def _print_FunctionPrototype(self, fp): + entity = "{} function ".format(self._print(fp.return_type)) + return ( + "interface\n" + "{function_head}\n" + "end function\n" + "end interface" + ).format(function_head=self._head(entity, fp)) + + def _print_FunctionDefinition(self, fd): + if elemental in fd.attrs: + prefix = 'elemental ' + elif pure in fd.attrs: + prefix = 'pure ' + else: + prefix = '' + + entity = "{} function ".format(self._print(fd.return_type)) + with printer_context(self, result_name=fd.name): + return ( + "{prefix}{function_head}\n" + "{body}\n" + "end function\n" + ).format( + prefix=prefix, + function_head=self._head(entity, fd), + body=self._print(fd.body) + ) + + def _print_Subroutine(self, sub): + return ( + '{subroutine_head}\n' + '{body}\n' + 'end subroutine\n' + ).format( + subroutine_head=self._head('subroutine ', sub), + body=self._print(sub.body) + ) + + def _print_SubroutineCall(self, scall): + return 'call {name}({args})'.format( + name=self._print(scall.name), + args=', '.join((self._print(arg) for arg in scall.subroutine_args)) + ) + + def _print_use_rename(self, rnm): + return "%s => %s" % tuple((self._print(arg) for arg in rnm.args)) + + def _print_use(self, use): + result = 'use %s' % self._print(use.namespace) + if use.rename != None: # Must be '!= None', cannot be 'is not None' + result += ', ' + ', '.join([self._print(rnm) for rnm in use.rename]) + if use.only != None: # Must be '!= None', cannot be 'is not None' + result += ', only: ' + ', '.join([self._print(nly) for nly in use.only]) + return result + + def _print_BreakToken(self, _): + return 'exit' + + def _print_ContinueToken(self, _): + return 'cycle' + + def _print_ArrayConstructor(self, ac): + fmtstr = "[%s]" if self._settings["standard"] >= 2003 else '(/%s/)' + return fmtstr % ', '.join((self._print(arg) for arg in ac.elements)) + + def _print_ArrayElement(self, elem): + return '{symbol}({idxs})'.format( + symbol=self._print(elem.name), + idxs=', '.join((self._print(arg) for arg in elem.indices)) + ) diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/glsl.py b/.venv/lib/python3.12/site-packages/sympy/printing/glsl.py new file mode 100644 index 0000000000000000000000000000000000000000..f98df8d46abec5f891b6bc9836a13ca69934275c --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/glsl.py @@ -0,0 +1,548 @@ +from __future__ import annotations + +from sympy.core import Basic, S +from sympy.core.function import Lambda +from sympy.core.numbers import equal_valued +from sympy.printing.codeprinter import CodePrinter +from sympy.printing.precedence import precedence +from functools import reduce + +known_functions = { + 'Abs': 'abs', + 'sin': 'sin', + 'cos': 'cos', + 'tan': 'tan', + 'acos': 'acos', + 'asin': 'asin', + 'atan': 'atan', + 'atan2': 'atan', + 'ceiling': 'ceil', + 'floor': 'floor', + 'sign': 'sign', + 'exp': 'exp', + 'log': 'log', + 'add': 'add', + 'sub': 'sub', + 'mul': 'mul', + 'pow': 'pow' +} + +class GLSLPrinter(CodePrinter): + """ + Rudimentary, generic GLSL printing tools. + + Additional settings: + 'use_operators': Boolean (should the printer use operators for +,-,*, or functions?) + """ + _not_supported: set[Basic] = set() + printmethod = "_glsl" + language = "GLSL" + + _default_settings = dict(CodePrinter._default_settings, **{ + 'use_operators': True, + 'zero': 0, + 'mat_nested': False, + 'mat_separator': ',\n', + 'mat_transpose': False, + 'array_type': 'float', + 'glsl_types': True, + + 'precision': 9, + 'user_functions': {}, + 'contract': True, + }) + + def __init__(self, settings={}): + CodePrinter.__init__(self, settings) + self.known_functions = dict(known_functions) + userfuncs = settings.get('user_functions', {}) + self.known_functions.update(userfuncs) + + def _rate_index_position(self, p): + return p*5 + + def _get_statement(self, codestring): + return "%s;" % codestring + + def _get_comment(self, text): + return "// {}".format(text) + + def _declare_number_const(self, name, value): + return "float {} = {};".format(name, value) + + def _format_code(self, lines): + return self.indent_code(lines) + + def indent_code(self, code): + """Accepts a string of code or a list of code lines""" + + if isinstance(code, str): + code_lines = self.indent_code(code.splitlines(True)) + return ''.join(code_lines) + + tab = " " + inc_token = ('{', '(', '{\n', '(\n') + dec_token = ('}', ')') + + code = [line.lstrip(' \t') for line in code] + + increase = [int(any(map(line.endswith, inc_token))) for line in code] + decrease = [int(any(map(line.startswith, dec_token))) for line in code] + + pretty = [] + level = 0 + for n, line in enumerate(code): + if line in ('', '\n'): + pretty.append(line) + continue + level -= decrease[n] + pretty.append("%s%s" % (tab*level, line)) + level += increase[n] + return pretty + + def _print_MatrixBase(self, mat): + mat_separator = self._settings['mat_separator'] + mat_transpose = self._settings['mat_transpose'] + column_vector = (mat.rows == 1) if mat_transpose else (mat.cols == 1) + A = mat.transpose() if mat_transpose != column_vector else mat + + glsl_types = self._settings['glsl_types'] + array_type = self._settings['array_type'] + array_size = A.cols*A.rows + array_constructor = "{}[{}]".format(array_type, array_size) + + if A.cols == 1: + return self._print(A[0]) + if A.rows <= 4 and A.cols <= 4 and glsl_types: + if A.rows == 1: + return "vec{}{}".format( + A.cols, A.table(self,rowstart='(',rowend=')') + ) + elif A.rows == A.cols: + return "mat{}({})".format( + A.rows, A.table(self,rowsep=', ', + rowstart='',rowend='') + ) + else: + return "mat{}x{}({})".format( + A.cols, A.rows, + A.table(self,rowsep=', ', + rowstart='',rowend='') + ) + elif S.One in A.shape: + return "{}({})".format( + array_constructor, + A.table(self,rowsep=mat_separator,rowstart='',rowend='') + ) + elif not self._settings['mat_nested']: + return "{}(\n{}\n) /* a {}x{} matrix */".format( + array_constructor, + A.table(self,rowsep=mat_separator,rowstart='',rowend=''), + A.rows, A.cols + ) + elif self._settings['mat_nested']: + return "{}[{}][{}](\n{}\n)".format( + array_type, A.rows, A.cols, + A.table(self,rowsep=mat_separator,rowstart='float[](',rowend=')') + ) + + def _print_SparseRepMatrix(self, mat): + # do not allow sparse matrices to be made dense + return self._print_not_supported(mat) + + def _traverse_matrix_indices(self, mat): + mat_transpose = self._settings['mat_transpose'] + if mat_transpose: + rows,cols = mat.shape + else: + cols,rows = mat.shape + return ((i, j) for i in range(cols) for j in range(rows)) + + def _print_MatrixElement(self, expr): + # print('begin _print_MatrixElement') + nest = self._settings['mat_nested'] + glsl_types = self._settings['glsl_types'] + mat_transpose = self._settings['mat_transpose'] + if mat_transpose: + cols,rows = expr.parent.shape + i,j = expr.j,expr.i + else: + rows,cols = expr.parent.shape + i,j = expr.i,expr.j + pnt = self._print(expr.parent) + if glsl_types and ((rows <= 4 and cols <=4) or nest): + return "{}[{}][{}]".format(pnt, i, j) + else: + return "{}[{}]".format(pnt, i + j*rows) + + def _print_list(self, expr): + l = ', '.join(self._print(item) for item in expr) + glsl_types = self._settings['glsl_types'] + array_type = self._settings['array_type'] + array_size = len(expr) + array_constructor = '{}[{}]'.format(array_type, array_size) + + if array_size <= 4 and glsl_types: + return 'vec{}({})'.format(array_size, l) + else: + return '{}({})'.format(array_constructor, l) + + _print_tuple = _print_list + _print_Tuple = _print_list + + def _get_loop_opening_ending(self, indices): + open_lines = [] + close_lines = [] + loopstart = "for (int %(varble)s=%(start)s; %(varble)s<%(end)s; %(varble)s++){" + for i in indices: + # GLSL arrays start at 0 and end at dimension-1 + open_lines.append(loopstart % { + 'varble': self._print(i.label), + 'start': self._print(i.lower), + 'end': self._print(i.upper + 1)}) + close_lines.append("}") + return open_lines, close_lines + + def _print_Function_with_args(self, func, func_args): + if func in self.known_functions: + cond_func = self.known_functions[func] + func = None + if isinstance(cond_func, str): + func = cond_func + else: + for cond, func in cond_func: + if cond(func_args): + break + if func is not None: + try: + return func(*[self.parenthesize(item, 0) for item in func_args]) + except TypeError: + return '{}({})'.format(func, self.stringify(func_args, ", ")) + elif isinstance(func, Lambda): + # inlined function + return self._print(func(*func_args)) + else: + return self._print_not_supported(func) + + def _print_Piecewise(self, expr): + from sympy.codegen.ast import Assignment + if expr.args[-1].cond != True: + # We need the last conditional to be a True, otherwise the resulting + # function may not return a result. + raise ValueError("All Piecewise expressions must contain an " + "(expr, True) statement to be used as a default " + "condition. Without one, the generated " + "expression may not evaluate to anything under " + "some condition.") + lines = [] + if expr.has(Assignment): + for i, (e, c) in enumerate(expr.args): + if i == 0: + lines.append("if (%s) {" % self._print(c)) + elif i == len(expr.args) - 1 and c == True: + lines.append("else {") + else: + lines.append("else if (%s) {" % self._print(c)) + code0 = self._print(e) + lines.append(code0) + lines.append("}") + return "\n".join(lines) + else: + # The piecewise was used in an expression, need to do inline + # operators. This has the downside that inline operators will + # not work for statements that span multiple lines (Matrix or + # Indexed expressions). + ecpairs = ["((%s) ? (\n%s\n)\n" % (self._print(c), + self._print(e)) + for e, c in expr.args[:-1]] + last_line = ": (\n%s\n)" % self._print(expr.args[-1].expr) + return ": ".join(ecpairs) + last_line + " ".join([")"*len(ecpairs)]) + + def _print_Indexed(self, expr): + # calculate index for 1d array + dims = expr.shape + elem = S.Zero + offset = S.One + for i in reversed(range(expr.rank)): + elem += expr.indices[i]*offset + offset *= dims[i] + return "{}[{}]".format( + self._print(expr.base.label), + self._print(elem) + ) + + def _print_Pow(self, expr): + PREC = precedence(expr) + if equal_valued(expr.exp, -1): + return '1.0/%s' % (self.parenthesize(expr.base, PREC)) + elif equal_valued(expr.exp, 0.5): + return 'sqrt(%s)' % self._print(expr.base) + else: + try: + e = self._print(float(expr.exp)) + except TypeError: + e = self._print(expr.exp) + return self._print_Function_with_args('pow', ( + self._print(expr.base), + e + )) + + def _print_int(self, expr): + return str(float(expr)) + + def _print_Rational(self, expr): + return "{}.0/{}.0".format(expr.p, expr.q) + + def _print_Relational(self, expr): + lhs_code = self._print(expr.lhs) + rhs_code = self._print(expr.rhs) + op = expr.rel_op + return "{} {} {}".format(lhs_code, op, rhs_code) + + def _print_Add(self, expr, order=None): + if self._settings['use_operators']: + return CodePrinter._print_Add(self, expr, order=order) + + terms = expr.as_ordered_terms() + + def partition(p,l): + return reduce(lambda x, y: (x[0]+[y], x[1]) if p(y) else (x[0], x[1]+[y]), l, ([], [])) + def add(a,b): + return self._print_Function_with_args('add', (a, b)) + # return self.known_functions['add']+'(%s, %s)' % (a,b) + neg, pos = partition(lambda arg: arg.could_extract_minus_sign(), terms) + if pos: + s = pos = reduce(lambda a,b: add(a,b), (self._print(t) for t in pos)) + else: + s = pos = self._print(self._settings['zero']) + + if neg: + # sum the absolute values of the negative terms + neg = reduce(lambda a,b: add(a,b), (self._print(-n) for n in neg)) + # then subtract them from the positive terms + s = self._print_Function_with_args('sub', (pos,neg)) + # s = self.known_functions['sub']+'(%s, %s)' % (pos,neg) + return s + + def _print_Mul(self, expr, **kwargs): + if self._settings['use_operators']: + return CodePrinter._print_Mul(self, expr, **kwargs) + terms = expr.as_ordered_factors() + def mul(a,b): + # return self.known_functions['mul']+'(%s, %s)' % (a,b) + return self._print_Function_with_args('mul', (a,b)) + + s = reduce(lambda a,b: mul(a,b), (self._print(t) for t in terms)) + return s + +def glsl_code(expr,assign_to=None,**settings): + """Converts an expr to a string of GLSL code + + Parameters + ========== + + expr : Expr + A SymPy expression to be converted. + assign_to : optional + When given, the argument is used for naming the variable or variables + to which the expression is assigned. Can be a string, ``Symbol``, + ``MatrixSymbol`` or ``Indexed`` type object. In cases where ``expr`` + would be printed as an array, a list of string or ``Symbol`` objects + can also be passed. + + This is helpful in case of line-wrapping, or for expressions that + generate multi-line statements. It can also be used to spread an array-like + expression into multiple assignments. + use_operators: bool, optional + If set to False, then *,/,+,- operators will be replaced with functions + mul, add, and sub, which must be implemented by the user, e.g. for + implementing non-standard rings or emulated quad/octal precision. + [default=True] + glsl_types: bool, optional + Set this argument to ``False`` in order to avoid using the ``vec`` and ``mat`` + types. The printer will instead use arrays (or nested arrays). + [default=True] + mat_nested: bool, optional + GLSL version 4.3 and above support nested arrays (arrays of arrays). Set this to ``True`` + to render matrices as nested arrays. + [default=False] + mat_separator: str, optional + By default, matrices are rendered with newlines using this separator, + making them easier to read, but less compact. By removing the newline + this option can be used to make them more vertically compact. + [default=',\n'] + mat_transpose: bool, optional + GLSL's matrix multiplication implementation assumes column-major indexing. + By default, this printer ignores that convention. Setting this option to + ``True`` transposes all matrix output. + [default=False] + array_type: str, optional + The GLSL array constructor type. + [default='float'] + precision : integer, optional + The precision for numbers such as pi [default=15]. + user_functions : dict, optional + A dictionary where keys are ``FunctionClass`` instances and values are + their string representations. Alternatively, the dictionary value can + be a list of tuples i.e. [(argument_test, js_function_string)]. See + below for examples. + human : bool, optional + If True, the result is a single string that may contain some constant + declarations for the number symbols. If False, the same information is + returned in a tuple of (symbols_to_declare, not_supported_functions, + code_text). [default=True]. + contract: bool, optional + If True, ``Indexed`` instances are assumed to obey tensor contraction + rules and the corresponding nested loops over indices are generated. + Setting contract=False will not generate loops, instead the user is + responsible to provide values for the indices in the code. + [default=True]. + + Examples + ======== + + >>> from sympy import glsl_code, symbols, Rational, sin, ceiling, Abs + >>> x, tau = symbols("x, tau") + >>> glsl_code((2*tau)**Rational(7, 2)) + '8*sqrt(2)*pow(tau, 3.5)' + >>> glsl_code(sin(x), assign_to="float y") + 'float y = sin(x);' + + Various GLSL types are supported: + >>> from sympy import Matrix, glsl_code + >>> glsl_code(Matrix([1,2,3])) + 'vec3(1, 2, 3)' + + >>> glsl_code(Matrix([[1, 2],[3, 4]])) + 'mat2(1, 2, 3, 4)' + + Pass ``mat_transpose = True`` to switch to column-major indexing: + >>> glsl_code(Matrix([[1, 2],[3, 4]]), mat_transpose = True) + 'mat2(1, 3, 2, 4)' + + By default, larger matrices get collapsed into float arrays: + >>> print(glsl_code( Matrix([[1,2,3,4,5],[6,7,8,9,10]]) )) + float[10]( + 1, 2, 3, 4, 5, + 6, 7, 8, 9, 10 + ) /* a 2x5 matrix */ + + The type of array constructor used to print GLSL arrays can be controlled + via the ``array_type`` parameter: + >>> glsl_code(Matrix([1,2,3,4,5]), array_type='int') + 'int[5](1, 2, 3, 4, 5)' + + Passing a list of strings or ``symbols`` to the ``assign_to`` parameter will yield + a multi-line assignment for each item in an array-like expression: + >>> x_struct_members = symbols('x.a x.b x.c x.d') + >>> print(glsl_code(Matrix([1,2,3,4]), assign_to=x_struct_members)) + x.a = 1; + x.b = 2; + x.c = 3; + x.d = 4; + + This could be useful in cases where it's desirable to modify members of a + GLSL ``Struct``. It could also be used to spread items from an array-like + expression into various miscellaneous assignments: + >>> misc_assignments = ('x[0]', 'x[1]', 'float y', 'float z') + >>> print(glsl_code(Matrix([1,2,3,4]), assign_to=misc_assignments)) + x[0] = 1; + x[1] = 2; + float y = 3; + float z = 4; + + Passing ``mat_nested = True`` instead prints out nested float arrays, which are + supported in GLSL 4.3 and above. + >>> mat = Matrix([ + ... [ 0, 1, 2], + ... [ 3, 4, 5], + ... [ 6, 7, 8], + ... [ 9, 10, 11], + ... [12, 13, 14]]) + >>> print(glsl_code( mat, mat_nested = True )) + float[5][3]( + float[]( 0, 1, 2), + float[]( 3, 4, 5), + float[]( 6, 7, 8), + float[]( 9, 10, 11), + float[](12, 13, 14) + ) + + + + Custom printing can be defined for certain types by passing a dictionary of + "type" : "function" to the ``user_functions`` kwarg. Alternatively, the + dictionary value can be a list of tuples i.e. [(argument_test, + js_function_string)]. + + >>> custom_functions = { + ... "ceiling": "CEIL", + ... "Abs": [(lambda x: not x.is_integer, "fabs"), + ... (lambda x: x.is_integer, "ABS")] + ... } + >>> glsl_code(Abs(x) + ceiling(x), user_functions=custom_functions) + 'fabs(x) + CEIL(x)' + + If further control is needed, addition, subtraction, multiplication and + division operators can be replaced with ``add``, ``sub``, and ``mul`` + functions. This is done by passing ``use_operators = False``: + + >>> x,y,z = symbols('x,y,z') + >>> glsl_code(x*(y+z), use_operators = False) + 'mul(x, add(y, z))' + >>> glsl_code(x*(y+z*(x-y)**z), use_operators = False) + 'mul(x, add(y, mul(z, pow(sub(x, y), z))))' + + ``Piecewise`` expressions are converted into conditionals. If an + ``assign_to`` variable is provided an if statement is created, otherwise + the ternary operator is used. Note that if the ``Piecewise`` lacks a + default term, represented by ``(expr, True)`` then an error will be thrown. + This is to prevent generating an expression that may not evaluate to + anything. + + >>> from sympy import Piecewise + >>> expr = Piecewise((x + 1, x > 0), (x, True)) + >>> print(glsl_code(expr, tau)) + if (x > 0) { + tau = x + 1; + } + else { + tau = x; + } + + Support for loops is provided through ``Indexed`` types. With + ``contract=True`` these expressions will be turned into loops, whereas + ``contract=False`` will just print the assignment expression that should be + looped over: + + >>> from sympy import Eq, IndexedBase, Idx + >>> len_y = 5 + >>> y = IndexedBase('y', shape=(len_y,)) + >>> t = IndexedBase('t', shape=(len_y,)) + >>> Dy = IndexedBase('Dy', shape=(len_y-1,)) + >>> i = Idx('i', len_y-1) + >>> e=Eq(Dy[i], (y[i+1]-y[i])/(t[i+1]-t[i])) + >>> glsl_code(e.rhs, assign_to=e.lhs, contract=False) + 'Dy[i] = (y[i + 1] - y[i])/(t[i + 1] - t[i]);' + + >>> from sympy import Matrix, MatrixSymbol + >>> mat = Matrix([x**2, Piecewise((x + 1, x > 0), (x, True)), sin(x)]) + >>> A = MatrixSymbol('A', 3, 1) + >>> print(glsl_code(mat, A)) + A[0][0] = pow(x, 2.0); + if (x > 0) { + A[1][0] = x + 1; + } + else { + A[1][0] = x; + } + A[2][0] = sin(x); + """ + return GLSLPrinter(settings).doprint(expr,assign_to) + +def print_glsl(expr, **settings): + """Prints the GLSL representation of the given expression. + + See GLSLPrinter init function for settings. + """ + print(glsl_code(expr, **settings)) diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/gtk.py b/.venv/lib/python3.12/site-packages/sympy/printing/gtk.py new file mode 100644 index 0000000000000000000000000000000000000000..4123d7231c730bbde28e33f441470c28b21c78d0 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/gtk.py @@ -0,0 +1,16 @@ +from sympy.printing.mathml import mathml +from sympy.utilities.mathml import c2p +import tempfile +import subprocess + + +def print_gtk(x, start_viewer=True): + """Print to Gtkmathview, a gtk widget capable of rendering MathML. + + Needs libgtkmathview-bin""" + with tempfile.NamedTemporaryFile('w') as file: + file.write(c2p(mathml(x), simple=True)) + file.flush() + + if start_viewer: + subprocess.check_call(('mathmlviewer', file.name)) diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/jscode.py b/.venv/lib/python3.12/site-packages/sympy/printing/jscode.py new file mode 100644 index 0000000000000000000000000000000000000000..753eb3291dd719ff53b06584de8ebe76c4471a3f --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/jscode.py @@ -0,0 +1,332 @@ +""" +Javascript code printer + +The JavascriptCodePrinter converts single SymPy expressions into single +Javascript expressions, using the functions defined in the Javascript +Math object where possible. + +""" + +from __future__ import annotations +from typing import Any + +from sympy.core import S +from sympy.core.numbers import equal_valued +from sympy.printing.codeprinter import CodePrinter +from sympy.printing.precedence import precedence, PRECEDENCE + + +# dictionary mapping SymPy function to (argument_conditions, Javascript_function). +# Used in JavascriptCodePrinter._print_Function(self) +known_functions = { + 'Abs': 'Math.abs', + 'acos': 'Math.acos', + 'acosh': 'Math.acosh', + 'asin': 'Math.asin', + 'asinh': 'Math.asinh', + 'atan': 'Math.atan', + 'atan2': 'Math.atan2', + 'atanh': 'Math.atanh', + 'ceiling': 'Math.ceil', + 'cos': 'Math.cos', + 'cosh': 'Math.cosh', + 'exp': 'Math.exp', + 'floor': 'Math.floor', + 'log': 'Math.log', + 'Max': 'Math.max', + 'Min': 'Math.min', + 'sign': 'Math.sign', + 'sin': 'Math.sin', + 'sinh': 'Math.sinh', + 'tan': 'Math.tan', + 'tanh': 'Math.tanh', +} + + +class JavascriptCodePrinter(CodePrinter): + """"A Printer to convert Python expressions to strings of JavaScript code + """ + printmethod = '_javascript' + language = 'JavaScript' + + _default_settings: dict[str, Any] = dict(CodePrinter._default_settings, **{ + 'precision': 17, + 'user_functions': {}, + 'contract': True, + }) + + def __init__(self, settings={}): + CodePrinter.__init__(self, settings) + self.known_functions = dict(known_functions) + userfuncs = settings.get('user_functions', {}) + self.known_functions.update(userfuncs) + + def _rate_index_position(self, p): + return p*5 + + def _get_statement(self, codestring): + return "%s;" % codestring + + def _get_comment(self, text): + return "// {}".format(text) + + def _declare_number_const(self, name, value): + return "var {} = {};".format(name, value.evalf(self._settings['precision'])) + + def _format_code(self, lines): + return self.indent_code(lines) + + def _traverse_matrix_indices(self, mat): + rows, cols = mat.shape + return ((i, j) for i in range(rows) for j in range(cols)) + + def _get_loop_opening_ending(self, indices): + open_lines = [] + close_lines = [] + loopstart = "for (var %(varble)s=%(start)s; %(varble)s<%(end)s; %(varble)s++){" + for i in indices: + # Javascript arrays start at 0 and end at dimension-1 + open_lines.append(loopstart % { + 'varble': self._print(i.label), + 'start': self._print(i.lower), + 'end': self._print(i.upper + 1)}) + close_lines.append("}") + return open_lines, close_lines + + def _print_Pow(self, expr): + PREC = precedence(expr) + if equal_valued(expr.exp, -1): + return '1/%s' % (self.parenthesize(expr.base, PREC)) + elif equal_valued(expr.exp, 0.5): + return 'Math.sqrt(%s)' % self._print(expr.base) + elif expr.exp == S.One/3: + return 'Math.cbrt(%s)' % self._print(expr.base) + else: + return 'Math.pow(%s, %s)' % (self._print(expr.base), + self._print(expr.exp)) + + def _print_Rational(self, expr): + p, q = int(expr.p), int(expr.q) + return '%d/%d' % (p, q) + + def _print_Mod(self, expr): + num, den = expr.args + PREC = precedence(expr) + snum, sden = [self.parenthesize(arg, PREC) for arg in expr.args] + # % is remainder (same sign as numerator), not modulo (same sign as + # denominator), in js. Hence, % only works as modulo if both numbers + # have the same sign + if (num.is_nonnegative and den.is_nonnegative or + num.is_nonpositive and den.is_nonpositive): + return f"{snum} % {sden}" + return f"(({snum} % {sden}) + {sden}) % {sden}" + + def _print_Relational(self, expr): + lhs_code = self._print(expr.lhs) + rhs_code = self._print(expr.rhs) + op = expr.rel_op + return "{} {} {}".format(lhs_code, op, rhs_code) + + def _print_Indexed(self, expr): + # calculate index for 1d array + dims = expr.shape + elem = S.Zero + offset = S.One + for i in reversed(range(expr.rank)): + elem += expr.indices[i]*offset + offset *= dims[i] + return "%s[%s]" % (self._print(expr.base.label), self._print(elem)) + + def _print_Exp1(self, expr): + return "Math.E" + + def _print_Pi(self, expr): + return 'Math.PI' + + def _print_Infinity(self, expr): + return 'Number.POSITIVE_INFINITY' + + def _print_NegativeInfinity(self, expr): + return 'Number.NEGATIVE_INFINITY' + + def _print_Piecewise(self, expr): + from sympy.codegen.ast import Assignment + if expr.args[-1].cond != True: + # We need the last conditional to be a True, otherwise the resulting + # function may not return a result. + raise ValueError("All Piecewise expressions must contain an " + "(expr, True) statement to be used as a default " + "condition. Without one, the generated " + "expression may not evaluate to anything under " + "some condition.") + lines = [] + if expr.has(Assignment): + for i, (e, c) in enumerate(expr.args): + if i == 0: + lines.append("if (%s) {" % self._print(c)) + elif i == len(expr.args) - 1 and c == True: + lines.append("else {") + else: + lines.append("else if (%s) {" % self._print(c)) + code0 = self._print(e) + lines.append(code0) + lines.append("}") + return "\n".join(lines) + else: + # The piecewise was used in an expression, need to do inline + # operators. This has the downside that inline operators will + # not work for statements that span multiple lines (Matrix or + # Indexed expressions). + ecpairs = ["((%s) ? (\n%s\n)\n" % (self._print(c), self._print(e)) + for e, c in expr.args[:-1]] + last_line = ": (\n%s\n)" % self._print(expr.args[-1].expr) + return ": ".join(ecpairs) + last_line + " ".join([")"*len(ecpairs)]) + + def _print_MatrixElement(self, expr): + return "{}[{}]".format(self.parenthesize(expr.parent, + PRECEDENCE["Atom"], strict=True), + expr.j + expr.i*expr.parent.shape[1]) + + def indent_code(self, code): + """Accepts a string of code or a list of code lines""" + + if isinstance(code, str): + code_lines = self.indent_code(code.splitlines(True)) + return ''.join(code_lines) + + tab = " " + inc_token = ('{', '(', '{\n', '(\n') + dec_token = ('}', ')') + + code = [ line.lstrip(' \t') for line in code ] + + increase = [ int(any(map(line.endswith, inc_token))) for line in code ] + decrease = [ int(any(map(line.startswith, dec_token))) + for line in code ] + + pretty = [] + level = 0 + for n, line in enumerate(code): + if line in ('', '\n'): + pretty.append(line) + continue + level -= decrease[n] + pretty.append("%s%s" % (tab*level, line)) + level += increase[n] + return pretty + + +def jscode(expr, assign_to=None, **settings): + """Converts an expr to a string of javascript code + + Parameters + ========== + + expr : Expr + A SymPy expression to be converted. + assign_to : optional + When given, the argument is used as the name of the variable to which + the expression is assigned. Can be a string, ``Symbol``, + ``MatrixSymbol``, or ``Indexed`` type. This is helpful in case of + line-wrapping, or for expressions that generate multi-line statements. + precision : integer, optional + The precision for numbers such as pi [default=15]. + user_functions : dict, optional + A dictionary where keys are ``FunctionClass`` instances and values are + their string representations. Alternatively, the dictionary value can + be a list of tuples i.e. [(argument_test, js_function_string)]. See + below for examples. + human : bool, optional + If True, the result is a single string that may contain some constant + declarations for the number symbols. If False, the same information is + returned in a tuple of (symbols_to_declare, not_supported_functions, + code_text). [default=True]. + contract: bool, optional + If True, ``Indexed`` instances are assumed to obey tensor contraction + rules and the corresponding nested loops over indices are generated. + Setting contract=False will not generate loops, instead the user is + responsible to provide values for the indices in the code. + [default=True]. + + Examples + ======== + + >>> from sympy import jscode, symbols, Rational, sin, ceiling, Abs + >>> x, tau = symbols("x, tau") + >>> jscode((2*tau)**Rational(7, 2)) + '8*Math.sqrt(2)*Math.pow(tau, 7/2)' + >>> jscode(sin(x), assign_to="s") + 's = Math.sin(x);' + + Custom printing can be defined for certain types by passing a dictionary of + "type" : "function" to the ``user_functions`` kwarg. Alternatively, the + dictionary value can be a list of tuples i.e. [(argument_test, + js_function_string)]. + + >>> custom_functions = { + ... "ceiling": "CEIL", + ... "Abs": [(lambda x: not x.is_integer, "fabs"), + ... (lambda x: x.is_integer, "ABS")] + ... } + >>> jscode(Abs(x) + ceiling(x), user_functions=custom_functions) + 'fabs(x) + CEIL(x)' + + ``Piecewise`` expressions are converted into conditionals. If an + ``assign_to`` variable is provided an if statement is created, otherwise + the ternary operator is used. Note that if the ``Piecewise`` lacks a + default term, represented by ``(expr, True)`` then an error will be thrown. + This is to prevent generating an expression that may not evaluate to + anything. + + >>> from sympy import Piecewise + >>> expr = Piecewise((x + 1, x > 0), (x, True)) + >>> print(jscode(expr, tau)) + if (x > 0) { + tau = x + 1; + } + else { + tau = x; + } + + Support for loops is provided through ``Indexed`` types. With + ``contract=True`` these expressions will be turned into loops, whereas + ``contract=False`` will just print the assignment expression that should be + looped over: + + >>> from sympy import Eq, IndexedBase, Idx + >>> len_y = 5 + >>> y = IndexedBase('y', shape=(len_y,)) + >>> t = IndexedBase('t', shape=(len_y,)) + >>> Dy = IndexedBase('Dy', shape=(len_y-1,)) + >>> i = Idx('i', len_y-1) + >>> e=Eq(Dy[i], (y[i+1]-y[i])/(t[i+1]-t[i])) + >>> jscode(e.rhs, assign_to=e.lhs, contract=False) + 'Dy[i] = (y[i + 1] - y[i])/(t[i + 1] - t[i]);' + + Matrices are also supported, but a ``MatrixSymbol`` of the same dimensions + must be provided to ``assign_to``. Note that any expression that can be + generated normally can also exist inside a Matrix: + + >>> from sympy import Matrix, MatrixSymbol + >>> mat = Matrix([x**2, Piecewise((x + 1, x > 0), (x, True)), sin(x)]) + >>> A = MatrixSymbol('A', 3, 1) + >>> print(jscode(mat, A)) + A[0] = Math.pow(x, 2); + if (x > 0) { + A[1] = x + 1; + } + else { + A[1] = x; + } + A[2] = Math.sin(x); + """ + + return JavascriptCodePrinter(settings).doprint(expr, assign_to) + + +def print_jscode(expr, **settings): + """Prints the Javascript representation of the given expression. + + See jscode for the meaning of the optional arguments. + """ + print(jscode(expr, **settings)) diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/julia.py b/.venv/lib/python3.12/site-packages/sympy/printing/julia.py new file mode 100644 index 0000000000000000000000000000000000000000..3ab815add4d87fe953f646409e3a7bb383b1bbc6 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/julia.py @@ -0,0 +1,652 @@ +""" +Julia code printer + +The `JuliaCodePrinter` converts SymPy expressions into Julia expressions. + +A complete code generator, which uses `julia_code` extensively, can be found +in `sympy.utilities.codegen`. The `codegen` module can be used to generate +complete source code files. + +""" + +from __future__ import annotations +from typing import Any + +from sympy.core import Mul, Pow, S, Rational +from sympy.core.mul import _keep_coeff +from sympy.core.numbers import equal_valued +from sympy.printing.codeprinter import CodePrinter +from sympy.printing.precedence import precedence, PRECEDENCE +from re import search + +# List of known functions. First, those that have the same name in +# SymPy and Julia. This is almost certainly incomplete! +known_fcns_src1 = ["sin", "cos", "tan", "cot", "sec", "csc", + "asin", "acos", "atan", "acot", "asec", "acsc", + "sinh", "cosh", "tanh", "coth", "sech", "csch", + "asinh", "acosh", "atanh", "acoth", "asech", "acsch", + "atan2", "sign", "floor", "log", "exp", + "cbrt", "sqrt", "erf", "erfc", "erfi", + "factorial", "gamma", "digamma", "trigamma", + "polygamma", "beta", + "airyai", "airyaiprime", "airybi", "airybiprime", + "besselj", "bessely", "besseli", "besselk", + "erfinv", "erfcinv"] +# These functions have different names ("SymPy": "Julia"), more +# generally a mapping to (argument_conditions, julia_function). +known_fcns_src2 = { + "Abs": "abs", + "ceiling": "ceil", + "conjugate": "conj", + "hankel1": "hankelh1", + "hankel2": "hankelh2", + "im": "imag", + "re": "real" +} + + +class JuliaCodePrinter(CodePrinter): + """ + A printer to convert expressions to strings of Julia code. + """ + printmethod = "_julia" + language = "Julia" + + _operators = { + 'and': '&&', + 'or': '||', + 'not': '!', + } + + _default_settings: dict[str, Any] = dict(CodePrinter._default_settings, **{ + 'precision': 17, + 'user_functions': {}, + 'contract': True, + 'inline': True, + }) + # Note: contract is for expressing tensors as loops (if True), or just + # assignment (if False). FIXME: this should be looked a more carefully + # for Julia. + + def __init__(self, settings={}): + super().__init__(settings) + self.known_functions = dict(zip(known_fcns_src1, known_fcns_src1)) + self.known_functions.update(dict(known_fcns_src2)) + userfuncs = settings.get('user_functions', {}) + self.known_functions.update(userfuncs) + + + def _rate_index_position(self, p): + return p*5 + + + def _get_statement(self, codestring): + return "%s" % codestring + + + def _get_comment(self, text): + return "# {}".format(text) + + + def _declare_number_const(self, name, value): + return "const {} = {}".format(name, value) + + + def _format_code(self, lines): + return self.indent_code(lines) + + + def _traverse_matrix_indices(self, mat): + # Julia uses Fortran order (column-major) + rows, cols = mat.shape + return ((i, j) for j in range(cols) for i in range(rows)) + + + def _get_loop_opening_ending(self, indices): + open_lines = [] + close_lines = [] + for i in indices: + # Julia arrays start at 1 and end at dimension + var, start, stop = map(self._print, + [i.label, i.lower + 1, i.upper + 1]) + open_lines.append("for %s = %s:%s" % (var, start, stop)) + close_lines.append("end") + return open_lines, close_lines + + + def _print_Mul(self, expr): + # print complex numbers nicely in Julia + if (expr.is_number and expr.is_imaginary and + expr.as_coeff_Mul()[0].is_integer): + return "%sim" % self._print(-S.ImaginaryUnit*expr) + + # cribbed from str.py + prec = precedence(expr) + + c, e = expr.as_coeff_Mul() + if c < 0: + expr = _keep_coeff(-c, e) + sign = "-" + else: + sign = "" + + a = [] # items in the numerator + b = [] # items that are in the denominator (if any) + + pow_paren = [] # Will collect all pow with more than one base element and exp = -1 + + if self.order not in ('old', 'none'): + args = expr.as_ordered_factors() + else: + # use make_args in case expr was something like -x -> x + args = Mul.make_args(expr) + + # Gather args for numerator/denominator + for item in args: + if (item.is_commutative and item.is_Pow and item.exp.is_Rational + and item.exp.is_negative): + if item.exp != -1: + b.append(Pow(item.base, -item.exp, evaluate=False)) + else: + if len(item.args[0].args) != 1 and isinstance(item.base, Mul): # To avoid situations like #14160 + pow_paren.append(item) + b.append(Pow(item.base, -item.exp)) + elif item.is_Rational and item is not S.Infinity and item.p == 1: + # Save the Rational type in julia Unless the numerator is 1. + # For example: + # julia_code(Rational(3, 7)*x) --> (3 // 7) * x + # julia_code(x/3) --> x / 3 but not x * (1 // 3) + b.append(Rational(item.q)) + else: + a.append(item) + + a = a or [S.One] + + a_str = [self.parenthesize(x, prec) for x in a] + b_str = [self.parenthesize(x, prec) for x in b] + + # To parenthesize Pow with exp = -1 and having more than one Symbol + for item in pow_paren: + if item.base in b: + b_str[b.index(item.base)] = "(%s)" % b_str[b.index(item.base)] + + # from here it differs from str.py to deal with "*" and ".*" + def multjoin(a, a_str): + # here we probably are assuming the constants will come first + r = a_str[0] + for i in range(1, len(a)): + mulsym = '*' if a[i-1].is_number else '.*' + r = "%s %s %s" % (r, mulsym, a_str[i]) + return r + + if not b: + return sign + multjoin(a, a_str) + elif len(b) == 1: + divsym = '/' if b[0].is_number else './' + return "%s %s %s" % (sign+multjoin(a, a_str), divsym, b_str[0]) + else: + divsym = '/' if all(bi.is_number for bi in b) else './' + return "%s %s (%s)" % (sign + multjoin(a, a_str), divsym, multjoin(b, b_str)) + + def _print_Relational(self, expr): + lhs_code = self._print(expr.lhs) + rhs_code = self._print(expr.rhs) + op = expr.rel_op + return "{} {} {}".format(lhs_code, op, rhs_code) + + def _print_Pow(self, expr): + powsymbol = '^' if all(x.is_number for x in expr.args) else '.^' + + PREC = precedence(expr) + + if equal_valued(expr.exp, 0.5): + return "sqrt(%s)" % self._print(expr.base) + + if expr.is_commutative: + if equal_valued(expr.exp, -0.5): + sym = '/' if expr.base.is_number else './' + return "1 %s sqrt(%s)" % (sym, self._print(expr.base)) + if equal_valued(expr.exp, -1): + sym = '/' if expr.base.is_number else './' + return "1 %s %s" % (sym, self.parenthesize(expr.base, PREC)) + + return '%s %s %s' % (self.parenthesize(expr.base, PREC), powsymbol, + self.parenthesize(expr.exp, PREC)) + + + def _print_MatPow(self, expr): + PREC = precedence(expr) + return '%s ^ %s' % (self.parenthesize(expr.base, PREC), + self.parenthesize(expr.exp, PREC)) + + + def _print_Pi(self, expr): + if self._settings["inline"]: + return "pi" + else: + return super()._print_NumberSymbol(expr) + + + def _print_ImaginaryUnit(self, expr): + return "im" + + + def _print_Exp1(self, expr): + if self._settings["inline"]: + return "e" + else: + return super()._print_NumberSymbol(expr) + + + def _print_EulerGamma(self, expr): + if self._settings["inline"]: + return "eulergamma" + else: + return super()._print_NumberSymbol(expr) + + + def _print_Catalan(self, expr): + if self._settings["inline"]: + return "catalan" + else: + return super()._print_NumberSymbol(expr) + + + def _print_GoldenRatio(self, expr): + if self._settings["inline"]: + return "golden" + else: + return super()._print_NumberSymbol(expr) + + + def _print_Assignment(self, expr): + from sympy.codegen.ast import Assignment + from sympy.functions.elementary.piecewise import Piecewise + from sympy.tensor.indexed import IndexedBase + # Copied from codeprinter, but remove special MatrixSymbol treatment + lhs = expr.lhs + rhs = expr.rhs + # We special case assignments that take multiple lines + if not self._settings["inline"] and isinstance(expr.rhs, Piecewise): + # Here we modify Piecewise so each expression is now + # an Assignment, and then continue on the print. + expressions = [] + conditions = [] + for (e, c) in rhs.args: + expressions.append(Assignment(lhs, e)) + conditions.append(c) + temp = Piecewise(*zip(expressions, conditions)) + return self._print(temp) + if self._settings["contract"] and (lhs.has(IndexedBase) or + rhs.has(IndexedBase)): + # Here we check if there is looping to be done, and if so + # print the required loops. + return self._doprint_loops(rhs, lhs) + else: + lhs_code = self._print(lhs) + rhs_code = self._print(rhs) + return self._get_statement("%s = %s" % (lhs_code, rhs_code)) + + + def _print_Infinity(self, expr): + return 'Inf' + + + def _print_NegativeInfinity(self, expr): + return '-Inf' + + + def _print_NaN(self, expr): + return 'NaN' + + + def _print_list(self, expr): + return 'Any[' + ', '.join(self._print(a) for a in expr) + ']' + + + def _print_tuple(self, expr): + if len(expr) == 1: + return "(%s,)" % self._print(expr[0]) + else: + return "(%s)" % self.stringify(expr, ", ") + _print_Tuple = _print_tuple + + + def _print_BooleanTrue(self, expr): + return "true" + + + def _print_BooleanFalse(self, expr): + return "false" + + + def _print_bool(self, expr): + return str(expr).lower() + + + # Could generate quadrature code for definite Integrals? + #_print_Integral = _print_not_supported + + + def _print_MatrixBase(self, A): + # Handle zero dimensions: + if S.Zero in A.shape: + return 'zeros(%s, %s)' % (A.rows, A.cols) + elif (A.rows, A.cols) == (1, 1): + return "[%s]" % A[0, 0] + elif A.rows == 1: + return "[%s]" % A.table(self, rowstart='', rowend='', colsep=' ') + elif A.cols == 1: + # note .table would unnecessarily equispace the rows + return "[%s]" % ", ".join([self._print(a) for a in A]) + return "[%s]" % A.table(self, rowstart='', rowend='', + rowsep=';\n', colsep=' ') + + + def _print_SparseRepMatrix(self, A): + from sympy.matrices import Matrix + L = A.col_list() + # make row vectors of the indices and entries + I = Matrix([k[0] + 1 for k in L]) + J = Matrix([k[1] + 1 for k in L]) + AIJ = Matrix([k[2] for k in L]) + return "sparse(%s, %s, %s, %s, %s)" % (self._print(I), self._print(J), + self._print(AIJ), A.rows, A.cols) + + + def _print_MatrixElement(self, expr): + return self.parenthesize(expr.parent, PRECEDENCE["Atom"], strict=True) \ + + '[%s,%s]' % (expr.i + 1, expr.j + 1) + + + def _print_MatrixSlice(self, expr): + def strslice(x, lim): + l = x[0] + 1 + h = x[1] + step = x[2] + lstr = self._print(l) + hstr = 'end' if h == lim else self._print(h) + if step == 1: + if l == 1 and h == lim: + return ':' + if l == h: + return lstr + else: + return lstr + ':' + hstr + else: + return ':'.join((lstr, self._print(step), hstr)) + return (self._print(expr.parent) + '[' + + strslice(expr.rowslice, expr.parent.shape[0]) + ',' + + strslice(expr.colslice, expr.parent.shape[1]) + ']') + + + def _print_Indexed(self, expr): + inds = [ self._print(i) for i in expr.indices ] + return "%s[%s]" % (self._print(expr.base.label), ",".join(inds)) + + def _print_Identity(self, expr): + return "eye(%s)" % self._print(expr.shape[0]) + + def _print_HadamardProduct(self, expr): + return ' .* '.join([self.parenthesize(arg, precedence(expr)) + for arg in expr.args]) + + def _print_HadamardPower(self, expr): + PREC = precedence(expr) + return '.**'.join([ + self.parenthesize(expr.base, PREC), + self.parenthesize(expr.exp, PREC) + ]) + + def _print_Rational(self, expr): + if expr.q == 1: + return str(expr.p) + return "%s // %s" % (expr.p, expr.q) + + # Note: as of 2022, Julia doesn't have spherical Bessel functions + def _print_jn(self, expr): + from sympy.functions import sqrt, besselj + x = expr.argument + expr2 = sqrt(S.Pi/(2*x))*besselj(expr.order + S.Half, x) + return self._print(expr2) + + + def _print_yn(self, expr): + from sympy.functions import sqrt, bessely + x = expr.argument + expr2 = sqrt(S.Pi/(2*x))*bessely(expr.order + S.Half, x) + return self._print(expr2) + + def _print_sinc(self, expr): + # Julia has the normalized sinc function + return "sinc({})".format(self._print(expr.args[0] / S.Pi)) + + def _print_Piecewise(self, expr): + if expr.args[-1].cond != True: + # We need the last conditional to be a True, otherwise the resulting + # function may not return a result. + raise ValueError("All Piecewise expressions must contain an " + "(expr, True) statement to be used as a default " + "condition. Without one, the generated " + "expression may not evaluate to anything under " + "some condition.") + lines = [] + if self._settings["inline"]: + # Express each (cond, expr) pair in a nested Horner form: + # (condition) .* (expr) + (not cond) .* () + # Expressions that result in multiple statements won't work here. + ecpairs = ["({}) ? ({}) :".format + (self._print(c), self._print(e)) + for e, c in expr.args[:-1]] + elast = " (%s)" % self._print(expr.args[-1].expr) + pw = "\n".join(ecpairs) + elast + # Note: current need these outer brackets for 2*pw. Would be + # nicer to teach parenthesize() to do this for us when needed! + return "(" + pw + ")" + else: + for i, (e, c) in enumerate(expr.args): + if i == 0: + lines.append("if (%s)" % self._print(c)) + elif i == len(expr.args) - 1 and c == True: + lines.append("else") + else: + lines.append("elseif (%s)" % self._print(c)) + code0 = self._print(e) + lines.append(code0) + if i == len(expr.args) - 1: + lines.append("end") + return "\n".join(lines) + + def _print_MatMul(self, expr): + c, m = expr.as_coeff_mmul() + + sign = "" + if c.is_number: + re, im = c.as_real_imag() + if im.is_zero and re.is_negative: + expr = _keep_coeff(-c, m) + sign = "-" + elif re.is_zero and im.is_negative: + expr = _keep_coeff(-c, m) + sign = "-" + + return sign + ' * '.join( + (self.parenthesize(arg, precedence(expr)) for arg in expr.args) + ) + + + def indent_code(self, code): + """Accepts a string of code or a list of code lines""" + + # code mostly copied from ccode + if isinstance(code, str): + code_lines = self.indent_code(code.splitlines(True)) + return ''.join(code_lines) + + tab = " " + inc_regex = ('^function ', '^if ', '^elseif ', '^else$', '^for ') + dec_regex = ('^end$', '^elseif ', '^else$') + + # pre-strip left-space from the code + code = [ line.lstrip(' \t') for line in code ] + + increase = [ int(any(search(re, line) for re in inc_regex)) + for line in code ] + decrease = [ int(any(search(re, line) for re in dec_regex)) + for line in code ] + + pretty = [] + level = 0 + for n, line in enumerate(code): + if line in ('', '\n'): + pretty.append(line) + continue + level -= decrease[n] + pretty.append("%s%s" % (tab*level, line)) + level += increase[n] + return pretty + + +def julia_code(expr, assign_to=None, **settings): + r"""Converts `expr` to a string of Julia code. + + Parameters + ========== + + expr : Expr + A SymPy expression to be converted. + assign_to : optional + When given, the argument is used as the name of the variable to which + the expression is assigned. Can be a string, ``Symbol``, + ``MatrixSymbol``, or ``Indexed`` type. This can be helpful for + expressions that generate multi-line statements. + precision : integer, optional + The precision for numbers such as pi [default=16]. + user_functions : dict, optional + A dictionary where keys are ``FunctionClass`` instances and values are + their string representations. Alternatively, the dictionary value can + be a list of tuples i.e. [(argument_test, cfunction_string)]. See + below for examples. + human : bool, optional + If True, the result is a single string that may contain some constant + declarations for the number symbols. If False, the same information is + returned in a tuple of (symbols_to_declare, not_supported_functions, + code_text). [default=True]. + contract: bool, optional + If True, ``Indexed`` instances are assumed to obey tensor contraction + rules and the corresponding nested loops over indices are generated. + Setting contract=False will not generate loops, instead the user is + responsible to provide values for the indices in the code. + [default=True]. + inline: bool, optional + If True, we try to create single-statement code instead of multiple + statements. [default=True]. + + Examples + ======== + + >>> from sympy import julia_code, symbols, sin, pi + >>> x = symbols('x') + >>> julia_code(sin(x).series(x).removeO()) + 'x .^ 5 / 120 - x .^ 3 / 6 + x' + + >>> from sympy import Rational, ceiling + >>> x, y, tau = symbols("x, y, tau") + >>> julia_code((2*tau)**Rational(7, 2)) + '8 * sqrt(2) * tau .^ (7 // 2)' + + Note that element-wise (Hadamard) operations are used by default between + symbols. This is because its possible in Julia to write "vectorized" + code. It is harmless if the values are scalars. + + >>> julia_code(sin(pi*x*y), assign_to="s") + 's = sin(pi * x .* y)' + + If you need a matrix product "*" or matrix power "^", you can specify the + symbol as a ``MatrixSymbol``. + + >>> from sympy import Symbol, MatrixSymbol + >>> n = Symbol('n', integer=True, positive=True) + >>> A = MatrixSymbol('A', n, n) + >>> julia_code(3*pi*A**3) + '(3 * pi) * A ^ 3' + + This class uses several rules to decide which symbol to use a product. + Pure numbers use "*", Symbols use ".*" and MatrixSymbols use "*". + A HadamardProduct can be used to specify componentwise multiplication ".*" + of two MatrixSymbols. There is currently there is no easy way to specify + scalar symbols, so sometimes the code might have some minor cosmetic + issues. For example, suppose x and y are scalars and A is a Matrix, then + while a human programmer might write "(x^2*y)*A^3", we generate: + + >>> julia_code(x**2*y*A**3) + '(x .^ 2 .* y) * A ^ 3' + + Matrices are supported using Julia inline notation. When using + ``assign_to`` with matrices, the name can be specified either as a string + or as a ``MatrixSymbol``. The dimensions must align in the latter case. + + >>> from sympy import Matrix, MatrixSymbol + >>> mat = Matrix([[x**2, sin(x), ceiling(x)]]) + >>> julia_code(mat, assign_to='A') + 'A = [x .^ 2 sin(x) ceil(x)]' + + ``Piecewise`` expressions are implemented with logical masking by default. + Alternatively, you can pass "inline=False" to use if-else conditionals. + Note that if the ``Piecewise`` lacks a default term, represented by + ``(expr, True)`` then an error will be thrown. This is to prevent + generating an expression that may not evaluate to anything. + + >>> from sympy import Piecewise + >>> pw = Piecewise((x + 1, x > 0), (x, True)) + >>> julia_code(pw, assign_to=tau) + 'tau = ((x > 0) ? (x + 1) : (x))' + + Note that any expression that can be generated normally can also exist + inside a Matrix: + + >>> mat = Matrix([[x**2, pw, sin(x)]]) + >>> julia_code(mat, assign_to='A') + 'A = [x .^ 2 ((x > 0) ? (x + 1) : (x)) sin(x)]' + + Custom printing can be defined for certain types by passing a dictionary of + "type" : "function" to the ``user_functions`` kwarg. Alternatively, the + dictionary value can be a list of tuples i.e., [(argument_test, + cfunction_string)]. This can be used to call a custom Julia function. + + >>> from sympy import Function + >>> f = Function('f') + >>> g = Function('g') + >>> custom_functions = { + ... "f": "existing_julia_fcn", + ... "g": [(lambda x: x.is_Matrix, "my_mat_fcn"), + ... (lambda x: not x.is_Matrix, "my_fcn")] + ... } + >>> mat = Matrix([[1, x]]) + >>> julia_code(f(x) + g(x) + g(mat), user_functions=custom_functions) + 'existing_julia_fcn(x) + my_fcn(x) + my_mat_fcn([1 x])' + + Support for loops is provided through ``Indexed`` types. With + ``contract=True`` these expressions will be turned into loops, whereas + ``contract=False`` will just print the assignment expression that should be + looped over: + + >>> from sympy import Eq, IndexedBase, Idx + >>> len_y = 5 + >>> y = IndexedBase('y', shape=(len_y,)) + >>> t = IndexedBase('t', shape=(len_y,)) + >>> Dy = IndexedBase('Dy', shape=(len_y-1,)) + >>> i = Idx('i', len_y-1) + >>> e = Eq(Dy[i], (y[i+1]-y[i])/(t[i+1]-t[i])) + >>> julia_code(e.rhs, assign_to=e.lhs, contract=False) + 'Dy[i] = (y[i + 1] - y[i]) ./ (t[i + 1] - t[i])' + """ + return JuliaCodePrinter(settings).doprint(expr, assign_to) + + +def print_julia_code(expr, **settings): + """Prints the Julia representation of the given expression. + + See `julia_code` for the meaning of the optional arguments. + """ + print(julia_code(expr, **settings)) diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/lambdarepr.py b/.venv/lib/python3.12/site-packages/sympy/printing/lambdarepr.py new file mode 100644 index 0000000000000000000000000000000000000000..87fa0988d138d54d68ab8aef1bbc0f27b243b472 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/lambdarepr.py @@ -0,0 +1,251 @@ +from .pycode import ( + PythonCodePrinter, + MpmathPrinter, +) +from .numpy import NumPyPrinter # NumPyPrinter is imported for backward compatibility +from sympy.core.sorting import default_sort_key + + +__all__ = [ + 'PythonCodePrinter', + 'MpmathPrinter', # MpmathPrinter is published for backward compatibility + 'NumPyPrinter', + 'LambdaPrinter', + 'NumPyPrinter', + 'IntervalPrinter', + 'lambdarepr', +] + + +class LambdaPrinter(PythonCodePrinter): + """ + This printer converts expressions into strings that can be used by + lambdify. + """ + printmethod = "_lambdacode" + + + def _print_And(self, expr): + result = ['('] + for arg in sorted(expr.args, key=default_sort_key): + result.extend(['(', self._print(arg), ')']) + result.append(' and ') + result = result[:-1] + result.append(')') + return ''.join(result) + + def _print_Or(self, expr): + result = ['('] + for arg in sorted(expr.args, key=default_sort_key): + result.extend(['(', self._print(arg), ')']) + result.append(' or ') + result = result[:-1] + result.append(')') + return ''.join(result) + + def _print_Not(self, expr): + result = ['(', 'not (', self._print(expr.args[0]), '))'] + return ''.join(result) + + def _print_BooleanTrue(self, expr): + return "True" + + def _print_BooleanFalse(self, expr): + return "False" + + def _print_ITE(self, expr): + result = [ + '((', self._print(expr.args[1]), + ') if (', self._print(expr.args[0]), + ') else (', self._print(expr.args[2]), '))' + ] + return ''.join(result) + + def _print_NumberSymbol(self, expr): + return str(expr) + + def _print_Pow(self, expr, **kwargs): + # XXX Temporary workaround. Should Python math printer be + # isolated from PythonCodePrinter? + return super(PythonCodePrinter, self)._print_Pow(expr, **kwargs) + + +# numexpr works by altering the string passed to numexpr.evaluate +# rather than by populating a namespace. Thus a special printer... +class NumExprPrinter(LambdaPrinter): + # key, value pairs correspond to SymPy name and numexpr name + # functions not appearing in this dict will raise a TypeError + printmethod = "_numexprcode" + + _numexpr_functions = { + 'sin' : 'sin', + 'cos' : 'cos', + 'tan' : 'tan', + 'asin': 'arcsin', + 'acos': 'arccos', + 'atan': 'arctan', + 'atan2' : 'arctan2', + 'sinh' : 'sinh', + 'cosh' : 'cosh', + 'tanh' : 'tanh', + 'asinh': 'arcsinh', + 'acosh': 'arccosh', + 'atanh': 'arctanh', + 'ln' : 'log', + 'log': 'log', + 'exp': 'exp', + 'sqrt' : 'sqrt', + 'Abs' : 'abs', + 'conjugate' : 'conj', + 'im' : 'imag', + 're' : 'real', + 'where' : 'where', + 'complex' : 'complex', + 'contains' : 'contains', + } + + module = 'numexpr' + + def _print_ImaginaryUnit(self, expr): + return '1j' + + def _print_seq(self, seq, delimiter=', '): + # simplified _print_seq taken from pretty.py + s = [self._print(item) for item in seq] + if s: + return delimiter.join(s) + else: + return "" + + def _print_Function(self, e): + func_name = e.func.__name__ + + nstr = self._numexpr_functions.get(func_name, None) + if nstr is None: + # check for implemented_function + if hasattr(e, '_imp_'): + return "(%s)" % self._print(e._imp_(*e.args)) + else: + raise TypeError("numexpr does not support function '%s'" % + func_name) + return "%s(%s)" % (nstr, self._print_seq(e.args)) + + def _print_Piecewise(self, expr): + "Piecewise function printer" + exprs = [self._print(arg.expr) for arg in expr.args] + conds = [self._print(arg.cond) for arg in expr.args] + # If [default_value, True] is a (expr, cond) sequence in a Piecewise object + # it will behave the same as passing the 'default' kwarg to select() + # *as long as* it is the last element in expr.args. + # If this is not the case, it may be triggered prematurely. + ans = [] + parenthesis_count = 0 + is_last_cond_True = False + for cond, expr in zip(conds, exprs): + if cond == 'True': + ans.append(expr) + is_last_cond_True = True + break + else: + ans.append('where(%s, %s, ' % (cond, expr)) + parenthesis_count += 1 + if not is_last_cond_True: + # See https://github.com/pydata/numexpr/issues/298 + # + # simplest way to put a nan but raises + # 'RuntimeWarning: invalid value encountered in log' + # + # There are other ways to do this such as + # + # >>> import numexpr as ne + # >>> nan = float('nan') + # >>> ne.evaluate('where(x < 0, -1, nan)', {'x': [-1, 2, 3], 'nan':nan}) + # array([-1., nan, nan]) + # + # That needs to be handled in the lambdified function though rather + # than here in the printer. + ans.append('log(-1)') + return ''.join(ans) + ')' * parenthesis_count + + def _print_ITE(self, expr): + from sympy.functions.elementary.piecewise import Piecewise + return self._print(expr.rewrite(Piecewise)) + + def blacklisted(self, expr): + raise TypeError("numexpr cannot be used with %s" % + expr.__class__.__name__) + + # blacklist all Matrix printing + _print_SparseRepMatrix = \ + _print_MutableSparseMatrix = \ + _print_ImmutableSparseMatrix = \ + _print_Matrix = \ + _print_DenseMatrix = \ + _print_MutableDenseMatrix = \ + _print_ImmutableMatrix = \ + _print_ImmutableDenseMatrix = \ + blacklisted + # blacklist some Python expressions + _print_list = \ + _print_tuple = \ + _print_Tuple = \ + _print_dict = \ + _print_Dict = \ + blacklisted + + def _print_NumExprEvaluate(self, expr): + evaluate = self._module_format(self.module +".evaluate") + return "%s('%s', truediv=True)" % (evaluate, self._print(expr.expr)) + + def doprint(self, expr): + from sympy.codegen.ast import CodegenAST + from sympy.codegen.pynodes import NumExprEvaluate + if not isinstance(expr, CodegenAST): + expr = NumExprEvaluate(expr) + return super().doprint(expr) + + def _print_Return(self, expr): + from sympy.codegen.pynodes import NumExprEvaluate + r, = expr.args + if not isinstance(r, NumExprEvaluate): + expr = expr.func(NumExprEvaluate(r)) + return super()._print_Return(expr) + + def _print_Assignment(self, expr): + from sympy.codegen.pynodes import NumExprEvaluate + lhs, rhs, *args = expr.args + if not isinstance(rhs, NumExprEvaluate): + expr = expr.func(lhs, NumExprEvaluate(rhs), *args) + return super()._print_Assignment(expr) + + def _print_CodeBlock(self, expr): + from sympy.codegen.ast import CodegenAST + from sympy.codegen.pynodes import NumExprEvaluate + args = [ arg if isinstance(arg, CodegenAST) else NumExprEvaluate(arg) for arg in expr.args ] + return super()._print_CodeBlock(self, expr.func(*args)) + + +class IntervalPrinter(MpmathPrinter, LambdaPrinter): + """Use ``lambda`` printer but print numbers as ``mpi`` intervals. """ + + def _print_Integer(self, expr): + return "mpi('%s')" % super(PythonCodePrinter, self)._print_Integer(expr) + + def _print_Rational(self, expr): + return "mpi('%s')" % super(PythonCodePrinter, self)._print_Rational(expr) + + def _print_Half(self, expr): + return "mpi('%s')" % super(PythonCodePrinter, self)._print_Rational(expr) + + def _print_Pow(self, expr): + return super(MpmathPrinter, self)._print_Pow(expr, rational=True) + + +for k in NumExprPrinter._numexpr_functions: + setattr(NumExprPrinter, '_print_%s' % k, NumExprPrinter._print_Function) + +def lambdarepr(expr, **settings): + """ + Returns a string usable for lambdifying. + """ + return LambdaPrinter(settings).doprint(expr) diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/latex.py b/.venv/lib/python3.12/site-packages/sympy/printing/latex.py new file mode 100644 index 0000000000000000000000000000000000000000..724df719d560e001deb175649a3769703bdf5ca5 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/latex.py @@ -0,0 +1,3318 @@ +""" +A Printer which converts an expression into its LaTeX equivalent. +""" +from __future__ import annotations +from typing import Any, Callable, TYPE_CHECKING + +import itertools + +from sympy.core import Add, Float, Mod, Mul, Number, S, Symbol, Expr +from sympy.core.alphabets import greeks +from sympy.core.containers import Tuple +from sympy.core.function import Function, AppliedUndef, Derivative +from sympy.core.operations import AssocOp +from sympy.core.power import Pow +from sympy.core.sorting import default_sort_key +from sympy.core.sympify import SympifyError +from sympy.logic.boolalg import true, BooleanTrue, BooleanFalse + + +# sympy.printing imports +from sympy.printing.precedence import precedence_traditional +from sympy.printing.printer import Printer, print_function +from sympy.printing.conventions import split_super_sub, requires_partial +from sympy.printing.precedence import precedence, PRECEDENCE + +from mpmath.libmp.libmpf import prec_to_dps, to_str as mlib_to_str + +from sympy.utilities.iterables import has_variety, sift + +import re + +if TYPE_CHECKING: + from sympy.tensor.array import NDimArray + from sympy.vector.basisdependent import BasisDependent + +# Hand-picked functions which can be used directly in both LaTeX and MathJax +# Complete list at +# https://docs.mathjax.org/en/latest/tex.html#supported-latex-commands +# This variable only contains those functions which SymPy uses. +accepted_latex_functions = ['arcsin', 'arccos', 'arctan', 'sin', 'cos', 'tan', + 'sinh', 'cosh', 'tanh', 'sqrt', 'ln', 'log', 'sec', + 'csc', 'cot', 'coth', 're', 'im', 'frac', 'root', + 'arg', + ] + +tex_greek_dictionary = { + 'Alpha': r'\mathrm{A}', + 'Beta': r'\mathrm{B}', + 'Gamma': r'\Gamma', + 'Delta': r'\Delta', + 'Epsilon': r'\mathrm{E}', + 'Zeta': r'\mathrm{Z}', + 'Eta': r'\mathrm{H}', + 'Theta': r'\Theta', + 'Iota': r'\mathrm{I}', + 'Kappa': r'\mathrm{K}', + 'Lambda': r'\Lambda', + 'Mu': r'\mathrm{M}', + 'Nu': r'\mathrm{N}', + 'Xi': r'\Xi', + 'omicron': 'o', + 'Omicron': r'\mathrm{O}', + 'Pi': r'\Pi', + 'Rho': r'\mathrm{P}', + 'Sigma': r'\Sigma', + 'Tau': r'\mathrm{T}', + 'Upsilon': r'\Upsilon', + 'Phi': r'\Phi', + 'Chi': r'\mathrm{X}', + 'Psi': r'\Psi', + 'Omega': r'\Omega', + 'lamda': r'\lambda', + 'Lamda': r'\Lambda', + 'khi': r'\chi', + 'Khi': r'\mathrm{X}', + 'varepsilon': r'\varepsilon', + 'varkappa': r'\varkappa', + 'varphi': r'\varphi', + 'varpi': r'\varpi', + 'varrho': r'\varrho', + 'varsigma': r'\varsigma', + 'vartheta': r'\vartheta', +} + +other_symbols = {'aleph', 'beth', 'daleth', 'gimel', 'ell', 'eth', 'hbar', + 'hslash', 'mho', 'wp'} + +# Variable name modifiers +modifier_dict: dict[str, Callable[[str], str]] = { + # Accents + 'mathring': lambda s: r'\mathring{'+s+r'}', + 'ddddot': lambda s: r'\ddddot{'+s+r'}', + 'dddot': lambda s: r'\dddot{'+s+r'}', + 'ddot': lambda s: r'\ddot{'+s+r'}', + 'dot': lambda s: r'\dot{'+s+r'}', + 'check': lambda s: r'\check{'+s+r'}', + 'breve': lambda s: r'\breve{'+s+r'}', + 'acute': lambda s: r'\acute{'+s+r'}', + 'grave': lambda s: r'\grave{'+s+r'}', + 'tilde': lambda s: r'\tilde{'+s+r'}', + 'hat': lambda s: r'\hat{'+s+r'}', + 'bar': lambda s: r'\bar{'+s+r'}', + 'vec': lambda s: r'\vec{'+s+r'}', + 'prime': lambda s: "{"+s+"}'", + 'prm': lambda s: "{"+s+"}'", + # Faces + 'bold': lambda s: r'\boldsymbol{'+s+r'}', + 'bm': lambda s: r'\boldsymbol{'+s+r'}', + 'cal': lambda s: r'\mathcal{'+s+r'}', + 'scr': lambda s: r'\mathscr{'+s+r'}', + 'frak': lambda s: r'\mathfrak{'+s+r'}', + # Brackets + 'norm': lambda s: r'\left\|{'+s+r'}\right\|', + 'avg': lambda s: r'\left\langle{'+s+r'}\right\rangle', + 'abs': lambda s: r'\left|{'+s+r'}\right|', + 'mag': lambda s: r'\left|{'+s+r'}\right|', +} + +greek_letters_set = frozenset(greeks) + +_between_two_numbers_p = ( + re.compile(r'[0-9][} ]*$'), # search + re.compile(r'(\d|\\frac{\d+}{\d+})'), # match +) + + +def latex_escape(s: str) -> str: + """ + Escape a string such that latex interprets it as plaintext. + + We cannot use verbatim easily with mathjax, so escaping is easier. + Rules from https://tex.stackexchange.com/a/34586/41112. + """ + s = s.replace('\\', r'\textbackslash') + for c in '&%$#_{}': + s = s.replace(c, '\\' + c) + s = s.replace('~', r'\textasciitilde') + s = s.replace('^', r'\textasciicircum') + return s + + +class LatexPrinter(Printer): + printmethod = "_latex" + + _default_settings: dict[str, Any] = { + "full_prec": False, + "fold_frac_powers": False, + "fold_func_brackets": False, + "fold_short_frac": None, + "inv_trig_style": "abbreviated", + "itex": False, + "ln_notation": False, + "long_frac_ratio": None, + "mat_delim": "[", + "mat_str": None, + "mode": "plain", + "mul_symbol": None, + "order": None, + "symbol_names": {}, + "root_notation": True, + "mat_symbol_style": "plain", + "imaginary_unit": "i", + "gothic_re_im": False, + "decimal_separator": "period", + "perm_cyclic": True, + "parenthesize_super": True, + "min": None, + "max": None, + "diff_operator": "d", + "adjoint_style": "dagger", + "disable_split_super_sub": False, + } + + def __init__(self, settings=None): + Printer.__init__(self, settings) + + if 'mode' in self._settings: + valid_modes = ['inline', 'plain', 'equation', + 'equation*'] + if self._settings['mode'] not in valid_modes: + raise ValueError("'mode' must be one of 'inline', 'plain', " + "'equation' or 'equation*'") + + if self._settings['fold_short_frac'] is None and \ + self._settings['mode'] == 'inline': + self._settings['fold_short_frac'] = True + + mul_symbol_table = { + None: r" ", + "ldot": r" \,.\, ", + "dot": r" \cdot ", + "times": r" \times " + } + try: + self._settings['mul_symbol_latex'] = \ + mul_symbol_table[self._settings['mul_symbol']] + except KeyError: + self._settings['mul_symbol_latex'] = \ + self._settings['mul_symbol'] + try: + self._settings['mul_symbol_latex_numbers'] = \ + mul_symbol_table[self._settings['mul_symbol'] or 'dot'] + except KeyError: + if (self._settings['mul_symbol'].strip() in + ['', ' ', '\\', '\\,', '\\:', '\\;', '\\quad']): + self._settings['mul_symbol_latex_numbers'] = \ + mul_symbol_table['dot'] + else: + self._settings['mul_symbol_latex_numbers'] = \ + self._settings['mul_symbol'] + + self._delim_dict = {'(': ')', '[': ']'} + + imaginary_unit_table = { + None: r"i", + "i": r"i", + "ri": r"\mathrm{i}", + "ti": r"\text{i}", + "j": r"j", + "rj": r"\mathrm{j}", + "tj": r"\text{j}", + } + imag_unit = self._settings['imaginary_unit'] + self._settings['imaginary_unit_latex'] = imaginary_unit_table.get(imag_unit, imag_unit) + + diff_operator_table = { + None: r"d", + "d": r"d", + "rd": r"\mathrm{d}", + "td": r"\text{d}", + } + diff_operator = self._settings['diff_operator'] + self._settings["diff_operator_latex"] = diff_operator_table.get(diff_operator, diff_operator) + + def _add_parens(self, s) -> str: + return r"\left({}\right)".format(s) + + # TODO: merge this with the above, which requires a lot of test changes + def _add_parens_lspace(self, s) -> str: + return r"\left( {}\right)".format(s) + + def parenthesize(self, item, level, is_neg=False, strict=False) -> str: + prec_val = precedence_traditional(item) + if is_neg and strict: + return self._add_parens(self._print(item)) + + if (prec_val < level) or ((not strict) and prec_val <= level): + return self._add_parens(self._print(item)) + else: + return self._print(item) + + def parenthesize_super(self, s): + """ + Protect superscripts in s + + If the parenthesize_super option is set, protect with parentheses, else + wrap in braces. + """ + if "^" in s: + if self._settings['parenthesize_super']: + return self._add_parens(s) + else: + return "{{{}}}".format(s) + return s + + def doprint(self, expr) -> str: + tex = Printer.doprint(self, expr) + + if self._settings['mode'] == 'plain': + return tex + elif self._settings['mode'] == 'inline': + return r"$%s$" % tex + elif self._settings['itex']: + return r"$$%s$$" % tex + else: + env_str = self._settings['mode'] + return r"\begin{%s}%s\end{%s}" % (env_str, tex, env_str) + + def _needs_brackets(self, expr) -> bool: + """ + Returns True if the expression needs to be wrapped in brackets when + printed, False otherwise. For example: a + b => True; a => False; + 10 => False; -10 => True. + """ + return not ((expr.is_Integer and expr.is_nonnegative) + or (expr.is_Atom and (expr is not S.NegativeOne + and expr.is_Rational is False))) + + def _needs_function_brackets(self, expr) -> bool: + """ + Returns True if the expression needs to be wrapped in brackets when + passed as an argument to a function, False otherwise. This is a more + liberal version of _needs_brackets, in that many expressions which need + to be wrapped in brackets when added/subtracted/raised to a power do + not need them when passed to a function. Such an example is a*b. + """ + if not self._needs_brackets(expr): + return False + else: + # Muls of the form a*b*c... can be folded + if expr.is_Mul and not self._mul_is_clean(expr): + return True + # Pows which don't need brackets can be folded + elif expr.is_Pow and not self._pow_is_clean(expr): + return True + # Add and Function always need brackets + elif expr.is_Add or expr.is_Function: + return True + else: + return False + + def _needs_mul_brackets(self, expr, first=False, last=False) -> bool: + """ + Returns True if the expression needs to be wrapped in brackets when + printed as part of a Mul, False otherwise. This is True for Add, + but also for some container objects that would not need brackets + when appearing last in a Mul, e.g. an Integral. ``last=True`` + specifies that this expr is the last to appear in a Mul. + ``first=True`` specifies that this expr is the first to appear in + a Mul. + """ + from sympy.concrete.products import Product + from sympy.concrete.summations import Sum + from sympy.integrals.integrals import Integral + + if expr.is_Mul: + if not first and expr.could_extract_minus_sign(): + return True + elif precedence_traditional(expr) < PRECEDENCE["Mul"]: + return True + elif expr.is_Relational: + return True + if expr.is_Piecewise: + return True + if any(expr.has(x) for x in (Mod,)): + return True + if (not last and + any(expr.has(x) for x in (Integral, Product, Sum))): + return True + + return False + + def _needs_add_brackets(self, expr) -> bool: + """ + Returns True if the expression needs to be wrapped in brackets when + printed as part of an Add, False otherwise. This is False for most + things. + """ + if expr.is_Relational: + return True + if any(expr.has(x) for x in (Mod,)): + return True + if expr.is_Add: + return True + return False + + def _mul_is_clean(self, expr) -> bool: + for arg in expr.args: + if arg.is_Function: + return False + return True + + def _pow_is_clean(self, expr) -> bool: + return not self._needs_brackets(expr.base) + + def _do_exponent(self, expr: str, exp): + if exp is not None: + return r"\left(%s\right)^{%s}" % (expr, exp) + else: + return expr + + def _print_Basic(self, expr): + name = self._deal_with_super_sub(expr.__class__.__name__) + if expr.args: + ls = [self._print(o) for o in expr.args] + s = r"\operatorname{{{}}}\left({}\right)" + return s.format(name, ", ".join(ls)) + else: + return r"\text{{{}}}".format(name) + + def _print_bool(self, e: bool | BooleanTrue | BooleanFalse): + return r"\text{%s}" % e + + _print_BooleanTrue = _print_bool + _print_BooleanFalse = _print_bool + + def _print_NoneType(self, e): + return r"\text{%s}" % e + + def _print_Add(self, expr, order=None): + terms = self._as_ordered_terms(expr, order=order) + + tex = "" + for i, term in enumerate(terms): + if i == 0: + pass + elif term.could_extract_minus_sign(): + tex += " - " + term = -term + else: + tex += " + " + term_tex = self._print(term) + if self._needs_add_brackets(term): + term_tex = r"\left(%s\right)" % term_tex + tex += term_tex + + return tex + + def _print_Cycle(self, expr): + from sympy.combinatorics.permutations import Permutation + if expr.size == 0: + return r"\left( \right)" + expr = Permutation(expr) + expr_perm = expr.cyclic_form + siz = expr.size + if expr.array_form[-1] == siz - 1: + expr_perm = expr_perm + [[siz - 1]] + term_tex = '' + for i in expr_perm: + term_tex += str(i).replace(',', r"\;") + term_tex = term_tex.replace('[', r"\left( ") + term_tex = term_tex.replace(']', r"\right)") + return term_tex + + def _print_Permutation(self, expr): + from sympy.combinatorics.permutations import Permutation + from sympy.utilities.exceptions import sympy_deprecation_warning + + perm_cyclic = Permutation.print_cyclic + if perm_cyclic is not None: + sympy_deprecation_warning( + f""" + Setting Permutation.print_cyclic is deprecated. Instead use + init_printing(perm_cyclic={perm_cyclic}). + """, + deprecated_since_version="1.6", + active_deprecations_target="deprecated-permutation-print_cyclic", + stacklevel=8, + ) + else: + perm_cyclic = self._settings.get("perm_cyclic", True) + + if perm_cyclic: + return self._print_Cycle(expr) + + if expr.size == 0: + return r"\left( \right)" + + lower = [self._print(arg) for arg in expr.array_form] + upper = [self._print(arg) for arg in range(len(lower))] + + row1 = " & ".join(upper) + row2 = " & ".join(lower) + mat = r" \\ ".join((row1, row2)) + return r"\begin{pmatrix} %s \end{pmatrix}" % mat + + + def _print_AppliedPermutation(self, expr): + perm, var = expr.args + return r"\sigma_{%s}(%s)" % (self._print(perm), self._print(var)) + + def _print_Float(self, expr): + # Based off of that in StrPrinter + dps = prec_to_dps(expr._prec) + strip = False if self._settings['full_prec'] else True + low = self._settings["min"] if "min" in self._settings else None + high = self._settings["max"] if "max" in self._settings else None + str_real = mlib_to_str(expr._mpf_, dps, strip_zeros=strip, min_fixed=low, max_fixed=high) + + # Must always have a mul symbol (as 2.5 10^{20} just looks odd) + # thus we use the number separator + separator = self._settings['mul_symbol_latex_numbers'] + + if 'e' in str_real: + (mant, exp) = str_real.split('e') + + if exp[0] == '+': + exp = exp[1:] + if self._settings['decimal_separator'] == 'comma': + mant = mant.replace('.','{,}') + + return r"%s%s10^{%s}" % (mant, separator, exp) + elif str_real == "+inf": + return r"\infty" + elif str_real == "-inf": + return r"- \infty" + else: + if self._settings['decimal_separator'] == 'comma': + str_real = str_real.replace('.','{,}') + return str_real + + def _print_Cross(self, expr): + vec1 = expr._expr1 + vec2 = expr._expr2 + return r"%s \times %s" % (self.parenthesize(vec1, PRECEDENCE['Mul']), + self.parenthesize(vec2, PRECEDENCE['Mul'])) + + def _print_Curl(self, expr): + vec = expr._expr + return r"\nabla\times %s" % self.parenthesize(vec, PRECEDENCE['Mul']) + + def _print_Divergence(self, expr): + vec = expr._expr + return r"\nabla\cdot %s" % self.parenthesize(vec, PRECEDENCE['Mul']) + + def _print_Dot(self, expr): + vec1 = expr._expr1 + vec2 = expr._expr2 + return r"%s \cdot %s" % (self.parenthesize(vec1, PRECEDENCE['Mul']), + self.parenthesize(vec2, PRECEDENCE['Mul'])) + + def _print_Gradient(self, expr): + func = expr._expr + return r"\nabla %s" % self.parenthesize(func, PRECEDENCE['Mul']) + + def _print_Laplacian(self, expr): + func = expr._expr + return r"\Delta %s" % self.parenthesize(func, PRECEDENCE['Mul']) + + def _print_Mul(self, expr: Expr): + from sympy.simplify import fraction + separator: str = self._settings['mul_symbol_latex'] + numbersep: str = self._settings['mul_symbol_latex_numbers'] + + def convert(expr) -> str: + if not expr.is_Mul: + return str(self._print(expr)) + else: + if self.order not in ('old', 'none'): + args = expr.as_ordered_factors() + else: + args = list(expr.args) + + # If there are quantities or prefixes, append them at the back. + units, nonunits = sift(args, lambda x: (hasattr(x, "_scale_factor") or hasattr(x, "is_physical_constant")) or + (isinstance(x, Pow) and + hasattr(x.base, "is_physical_constant")), binary=True) + prefixes, units = sift(units, lambda x: hasattr(x, "_scale_factor"), binary=True) + return convert_args(nonunits + prefixes + units) + + def convert_args(args) -> str: + _tex = last_term_tex = "" + + for i, term in enumerate(args): + term_tex = self._print(term) + if not (hasattr(term, "_scale_factor") or hasattr(term, "is_physical_constant")): + if self._needs_mul_brackets(term, first=(i == 0), + last=(i == len(args) - 1)): + term_tex = r"\left(%s\right)" % term_tex + + if _between_two_numbers_p[0].search(last_term_tex) and \ + _between_two_numbers_p[1].match(term_tex): + # between two numbers + _tex += numbersep + elif _tex: + _tex += separator + elif _tex: + _tex += separator + + _tex += term_tex + last_term_tex = term_tex + return _tex + + # Check for unevaluated Mul. In this case we need to make sure the + # identities are visible, multiple Rational factors are not combined + # etc so we display in a straight-forward form that fully preserves all + # args and their order. + # XXX: _print_Pow calls this routine with instances of Pow... + if isinstance(expr, Mul): + args = expr.args + if args[0] is S.One or any(isinstance(arg, Number) for arg in args[1:]): + return convert_args(args) + + include_parens = False + if expr.could_extract_minus_sign(): + expr = -expr + tex = "- " + if expr.is_Add: + tex += "(" + include_parens = True + else: + tex = "" + + numer, denom = fraction(expr, exact=True) + + if denom is S.One and Pow(1, -1, evaluate=False) not in expr.args: + # use the original expression here, since fraction() may have + # altered it when producing numer and denom + tex += convert(expr) + + else: + snumer = convert(numer) + sdenom = convert(denom) + ldenom = len(sdenom.split()) + ratio = self._settings['long_frac_ratio'] + if self._settings['fold_short_frac'] and ldenom <= 2 and \ + "^" not in sdenom: + # handle short fractions + if self._needs_mul_brackets(numer, last=False): + tex += r"\left(%s\right) / %s" % (snumer, sdenom) + else: + tex += r"%s / %s" % (snumer, sdenom) + elif ratio is not None and \ + len(snumer.split()) > ratio*ldenom: + # handle long fractions + if self._needs_mul_brackets(numer, last=True): + tex += r"\frac{1}{%s}%s\left(%s\right)" \ + % (sdenom, separator, snumer) + elif numer.is_Mul: + # split a long numerator + a = S.One + b = S.One + for x in numer.args: + if self._needs_mul_brackets(x, last=False) or \ + len(convert(a*x).split()) > ratio*ldenom or \ + (b.is_commutative is x.is_commutative is False): + b *= x + else: + a *= x + if self._needs_mul_brackets(b, last=True): + tex += r"\frac{%s}{%s}%s\left(%s\right)" \ + % (convert(a), sdenom, separator, convert(b)) + else: + tex += r"\frac{%s}{%s}%s%s" \ + % (convert(a), sdenom, separator, convert(b)) + else: + tex += r"\frac{1}{%s}%s%s" % (sdenom, separator, snumer) + else: + tex += r"\frac{%s}{%s}" % (snumer, sdenom) + + if include_parens: + tex += ")" + return tex + + def _print_AlgebraicNumber(self, expr): + if expr.is_aliased: + return self._print(expr.as_poly().as_expr()) + else: + return self._print(expr.as_expr()) + + def _print_PrimeIdeal(self, expr): + p = self._print(expr.p) + if expr.is_inert: + return rf'\left({p}\right)' + alpha = self._print(expr.alpha.as_expr()) + return rf'\left({p}, {alpha}\right)' + + def _print_Pow(self, expr: Pow): + # Treat x**Rational(1,n) as special case + if expr.exp.is_Rational: + p: int = expr.exp.p # type: ignore + q: int = expr.exp.q # type: ignore + if abs(p) == 1 and q != 1 and self._settings['root_notation']: + base = self._print(expr.base) + if q == 2: + tex = r"\sqrt{%s}" % base + elif self._settings['itex']: + tex = r"\root{%d}{%s}" % (q, base) + else: + tex = r"\sqrt[%d]{%s}" % (q, base) + if expr.exp.is_negative: + return r"\frac{1}{%s}" % tex + else: + return tex + elif self._settings['fold_frac_powers'] and q != 1: + base = self.parenthesize(expr.base, PRECEDENCE['Pow']) + # issue #12886: add parentheses for superscripts raised to powers + if expr.base.is_Symbol: + base = self.parenthesize_super(base) + if expr.base.is_Function: + return self._print(expr.base, exp="%s/%s" % (p, q)) + return r"%s^{%s/%s}" % (base, p, q) + elif expr.exp.is_negative and expr.base.is_commutative: + # special case for 1^(-x), issue 9216 + if expr.base == 1: + return r"%s^{%s}" % (expr.base, expr.exp) + # special case for (1/x)^(-y) and (-1/-x)^(-y), issue 20252 + if expr.base.is_Rational: + base_p: int = expr.base.p # type: ignore + base_q: int = expr.base.q # type: ignore + if base_p * base_q == abs(base_q): + if expr.exp == -1: + return r"\frac{1}{\frac{%s}{%s}}" % (base_p, base_q) + else: + return r"\frac{1}{(\frac{%s}{%s})^{%s}}" % (base_p, base_q, abs(expr.exp)) + # things like 1/x + return self._print_Mul(expr) + if expr.base.is_Function: + return self._print(expr.base, exp=self._print(expr.exp)) + tex = r"%s^{%s}" + return self._helper_print_standard_power(expr, tex) + + def _helper_print_standard_power(self, expr, template: str) -> str: + exp = self._print(expr.exp) + # issue #12886: add parentheses around superscripts raised + # to powers + base = self.parenthesize(expr.base, PRECEDENCE['Pow']) + if expr.base.is_Symbol: + base = self.parenthesize_super(base) + elif expr.base.is_Float: + base = r"{%s}" % base + elif (isinstance(expr.base, Derivative) + and base.startswith(r'\left(') + and re.match(r'\\left\(\\d?d?dot', base) + and base.endswith(r'\right)')): + # don't use parentheses around dotted derivative + base = base[6: -7] # remove outermost added parens + return template % (base, exp) + + def _print_UnevaluatedExpr(self, expr): + return self._print(expr.args[0]) + + def _print_Sum(self, expr): + if len(expr.limits) == 1: + tex = r"\sum_{%s=%s}^{%s} " % \ + tuple([self._print(i) for i in expr.limits[0]]) + else: + def _format_ineq(l): + return r"%s \leq %s \leq %s" % \ + tuple([self._print(s) for s in (l[1], l[0], l[2])]) + + tex = r"\sum_{\substack{%s}} " % \ + str.join('\\\\', [_format_ineq(l) for l in expr.limits]) + + if isinstance(expr.function, Add): + tex += r"\left(%s\right)" % self._print(expr.function) + else: + tex += self._print(expr.function) + + return tex + + def _print_Product(self, expr): + if len(expr.limits) == 1: + tex = r"\prod_{%s=%s}^{%s} " % \ + tuple([self._print(i) for i in expr.limits[0]]) + else: + def _format_ineq(l): + return r"%s \leq %s \leq %s" % \ + tuple([self._print(s) for s in (l[1], l[0], l[2])]) + + tex = r"\prod_{\substack{%s}} " % \ + str.join('\\\\', [_format_ineq(l) for l in expr.limits]) + + if isinstance(expr.function, Add): + tex += r"\left(%s\right)" % self._print(expr.function) + else: + tex += self._print(expr.function) + + return tex + + def _print_BasisDependent(self, expr: 'BasisDependent'): + from sympy.vector import Vector + + o1: list[str] = [] + if expr == expr.zero: + return expr.zero._latex_form + if isinstance(expr, Vector): + items = expr.separate().items() + else: + items = [(0, expr)] + + for system, vect in items: + inneritems = list(vect.components.items()) + inneritems.sort(key=lambda x: x[0].__str__()) + for k, v in inneritems: + if v == 1: + o1.append(' + ' + k._latex_form) + elif v == -1: + o1.append(' - ' + k._latex_form) + else: + arg_str = r'\left(' + self._print(v) + r'\right)' + o1.append(' + ' + arg_str + k._latex_form) + + outstr = (''.join(o1)) + if outstr[1] != '-': + outstr = outstr[3:] + else: + outstr = outstr[1:] + return outstr + + def _print_Indexed(self, expr): + tex_base = self._print(expr.base) + tex = '{'+tex_base+'}'+'_{%s}' % ','.join( + map(self._print, expr.indices)) + return tex + + def _print_IndexedBase(self, expr): + return self._print(expr.label) + + def _print_Idx(self, expr): + label = self._print(expr.label) + if expr.upper is not None: + upper = self._print(expr.upper) + if expr.lower is not None: + lower = self._print(expr.lower) + else: + lower = self._print(S.Zero) + interval = '{lower}\\mathrel{{..}}\\nobreak {upper}'.format( + lower = lower, upper = upper) + return '{{{label}}}_{{{interval}}}'.format( + label = label, interval = interval) + #if no bounds are defined this just prints the label + return label + + def _print_Derivative(self, expr): + if requires_partial(expr.expr): + diff_symbol = r'\partial' + else: + diff_symbol = self._settings["diff_operator_latex"] + + tex = "" + dim = 0 + for x, num in reversed(expr.variable_count): + dim += num + if num == 1: + tex += r"%s %s" % (diff_symbol, self._print(x)) + else: + tex += r"%s %s^{%s}" % (diff_symbol, + self.parenthesize_super(self._print(x)), + self._print(num)) + + if dim == 1: + tex = r"\frac{%s}{%s}" % (diff_symbol, tex) + else: + tex = r"\frac{%s^{%s}}{%s}" % (diff_symbol, self._print(dim), tex) + + if any(i.could_extract_minus_sign() for i in expr.args): + return r"%s %s" % (tex, self.parenthesize(expr.expr, + PRECEDENCE["Mul"], + is_neg=True, + strict=True)) + + return r"%s %s" % (tex, self.parenthesize(expr.expr, + PRECEDENCE["Mul"], + is_neg=False, + strict=True)) + + def _print_Subs(self, subs): + expr, old, new = subs.args + latex_expr = self._print(expr) + latex_old = (self._print(e) for e in old) + latex_new = (self._print(e) for e in new) + latex_subs = r'\\ '.join( + e[0] + '=' + e[1] for e in zip(latex_old, latex_new)) + return r'\left. %s \right|_{\substack{ %s }}' % (latex_expr, + latex_subs) + + def _print_Integral(self, expr): + tex, symbols = "", [] + diff_symbol = self._settings["diff_operator_latex"] + + # Only up to \iiiint exists + if len(expr.limits) <= 4 and all(len(lim) == 1 for lim in expr.limits): + # Use len(expr.limits)-1 so that syntax highlighters don't think + # \" is an escaped quote + tex = r"\i" + "i"*(len(expr.limits) - 1) + "nt" + symbols = [r"\, %s%s" % (diff_symbol, self._print(symbol[0])) + for symbol in expr.limits] + + else: + for lim in reversed(expr.limits): + symbol = lim[0] + tex += r"\int" + + if len(lim) > 1: + if self._settings['mode'] != 'inline' \ + and not self._settings['itex']: + tex += r"\limits" + + if len(lim) == 3: + tex += "_{%s}^{%s}" % (self._print(lim[1]), + self._print(lim[2])) + if len(lim) == 2: + tex += "^{%s}" % (self._print(lim[1])) + + symbols.insert(0, r"\, %s%s" % (diff_symbol, self._print(symbol))) + + return r"%s %s%s" % (tex, self.parenthesize(expr.function, + PRECEDENCE["Mul"], + is_neg=any(i.could_extract_minus_sign() for i in expr.args), + strict=True), + "".join(symbols)) + + def _print_Limit(self, expr): + e, z, z0, dir = expr.args + + tex = r"\lim_{%s \to " % self._print(z) + if str(dir) == '+-' or z0 in (S.Infinity, S.NegativeInfinity): + tex += r"%s}" % self._print(z0) + else: + tex += r"%s^%s}" % (self._print(z0), self._print(dir)) + + if isinstance(e, AssocOp): + return r"%s\left(%s\right)" % (tex, self._print(e)) + else: + return r"%s %s" % (tex, self._print(e)) + + def _hprint_Function(self, func: str) -> str: + r''' + Logic to decide how to render a function to latex + - if it is a recognized latex name, use the appropriate latex command + - if it is a single letter, excluding sub- and superscripts, just use that letter + - if it is a longer name, then put \operatorname{} around it and be + mindful of undercores in the name + ''' + func = self._deal_with_super_sub(func) + superscriptidx = func.find("^") + subscriptidx = func.find("_") + if func in accepted_latex_functions: + name = r"\%s" % func + elif len(func) == 1 or func.startswith('\\') or subscriptidx == 1 or superscriptidx == 1: + name = func + else: + if superscriptidx > 0 and subscriptidx > 0: + name = r"\operatorname{%s}%s" %( + func[:min(subscriptidx,superscriptidx)], + func[min(subscriptidx,superscriptidx):]) + elif superscriptidx > 0: + name = r"\operatorname{%s}%s" %( + func[:superscriptidx], + func[superscriptidx:]) + elif subscriptidx > 0: + name = r"\operatorname{%s}%s" %( + func[:subscriptidx], + func[subscriptidx:]) + else: + name = r"\operatorname{%s}" % func + return name + + def _print_Function(self, expr: Function, exp=None) -> str: + r''' + Render functions to LaTeX, handling functions that LaTeX knows about + e.g., sin, cos, ... by using the proper LaTeX command (\sin, \cos, ...). + For single-letter function names, render them as regular LaTeX math + symbols. For multi-letter function names that LaTeX does not know + about, (e.g., Li, sech) use \operatorname{} so that the function name + is rendered in Roman font and LaTeX handles spacing properly. + + expr is the expression involving the function + exp is an exponent + ''' + func = expr.func.__name__ + if hasattr(self, '_print_' + func) and \ + not isinstance(expr, AppliedUndef): + return getattr(self, '_print_' + func)(expr, exp) + else: + args = [str(self._print(arg)) for arg in expr.args] + # How inverse trig functions should be displayed, formats are: + # abbreviated: asin, full: arcsin, power: sin^-1 + inv_trig_style = self._settings['inv_trig_style'] + # If we are dealing with a power-style inverse trig function + inv_trig_power_case = False + # If it is applicable to fold the argument brackets + can_fold_brackets = self._settings['fold_func_brackets'] and \ + len(args) == 1 and \ + not self._needs_function_brackets(expr.args[0]) + + inv_trig_table = [ + "asin", "acos", "atan", + "acsc", "asec", "acot", + "asinh", "acosh", "atanh", + "acsch", "asech", "acoth", + ] + + # If the function is an inverse trig function, handle the style + if func in inv_trig_table: + if inv_trig_style == "abbreviated": + pass + elif inv_trig_style == "full": + func = ("ar" if func[-1] == "h" else "arc") + func[1:] + elif inv_trig_style == "power": + func = func[1:] + inv_trig_power_case = True + + # Can never fold brackets if we're raised to a power + if exp is not None: + can_fold_brackets = False + + if inv_trig_power_case: + if func in accepted_latex_functions: + name = r"\%s^{-1}" % func + else: + name = r"\operatorname{%s}^{-1}" % func + elif exp is not None: + func_tex = self._hprint_Function(func) + func_tex = self.parenthesize_super(func_tex) + name = r'%s^{%s}' % (func_tex, exp) + else: + name = self._hprint_Function(func) + + if can_fold_brackets: + if func in accepted_latex_functions: + # Wrap argument safely to avoid parse-time conflicts + # with the function name itself + name += r" {%s}" + else: + name += r"%s" + else: + name += r"{\left(%s \right)}" + + if inv_trig_power_case and exp is not None: + name += r"^{%s}" % exp + + return name % ",".join(args) + + def _print_UndefinedFunction(self, expr): + return self._hprint_Function(str(expr)) + + def _print_ElementwiseApplyFunction(self, expr): + return r"{%s}_{\circ}\left({%s}\right)" % ( + self._print(expr.function), + self._print(expr.expr), + ) + + @property + def _special_function_classes(self): + from sympy.functions.special.tensor_functions import KroneckerDelta + from sympy.functions.special.gamma_functions import gamma, lowergamma + from sympy.functions.special.beta_functions import beta + from sympy.functions.special.delta_functions import DiracDelta + from sympy.functions.special.error_functions import Chi + return {KroneckerDelta: r'\delta', + gamma: r'\Gamma', + lowergamma: r'\gamma', + beta: r'\operatorname{B}', + DiracDelta: r'\delta', + Chi: r'\operatorname{Chi}'} + + def _print_FunctionClass(self, expr): + for cls in self._special_function_classes: + if issubclass(expr, cls) and expr.__name__ == cls.__name__: + return self._special_function_classes[cls] + return self._hprint_Function(str(expr)) + + def _print_Lambda(self, expr): + symbols, expr = expr.args + + if len(symbols) == 1: + symbols = self._print(symbols[0]) + else: + symbols = self._print(tuple(symbols)) + + tex = r"\left( %s \mapsto %s \right)" % (symbols, self._print(expr)) + + return tex + + def _print_IdentityFunction(self, expr): + return r"\left( x \mapsto x \right)" + + def _hprint_variadic_function(self, expr, exp=None) -> str: + args = sorted(expr.args, key=default_sort_key) + texargs = [r"%s" % self._print(symbol) for symbol in args] + tex = r"\%s\left(%s\right)" % (str(expr.func).lower(), + ", ".join(texargs)) + if exp is not None: + return r"%s^{%s}" % (tex, exp) + else: + return tex + + _print_Min = _print_Max = _hprint_variadic_function + + def _print_floor(self, expr, exp=None): + tex = r"\left\lfloor{%s}\right\rfloor" % self._print(expr.args[0]) + + if exp is not None: + return r"%s^{%s}" % (tex, exp) + else: + return tex + + def _print_ceiling(self, expr, exp=None): + tex = r"\left\lceil{%s}\right\rceil" % self._print(expr.args[0]) + + if exp is not None: + return r"%s^{%s}" % (tex, exp) + else: + return tex + + def _print_log(self, expr, exp=None): + if not self._settings["ln_notation"]: + tex = r"\log{\left(%s \right)}" % self._print(expr.args[0]) + else: + tex = r"\ln{\left(%s \right)}" % self._print(expr.args[0]) + + if exp is not None: + return r"%s^{%s}" % (tex, exp) + else: + return tex + + def _print_Abs(self, expr, exp=None): + tex = r"\left|{%s}\right|" % self._print(expr.args[0]) + + if exp is not None: + return r"%s^{%s}" % (tex, exp) + else: + return tex + + def _print_re(self, expr, exp=None): + if self._settings['gothic_re_im']: + tex = r"\Re{%s}" % self.parenthesize(expr.args[0], PRECEDENCE['Atom']) + else: + tex = r"\operatorname{{re}}{{{}}}".format(self.parenthesize(expr.args[0], PRECEDENCE['Atom'])) + + return self._do_exponent(tex, exp) + + def _print_im(self, expr, exp=None): + if self._settings['gothic_re_im']: + tex = r"\Im{%s}" % self.parenthesize(expr.args[0], PRECEDENCE['Atom']) + else: + tex = r"\operatorname{{im}}{{{}}}".format(self.parenthesize(expr.args[0], PRECEDENCE['Atom'])) + + return self._do_exponent(tex, exp) + + def _print_Not(self, e): + from sympy.logic.boolalg import (Equivalent, Implies) + if isinstance(e.args[0], Equivalent): + return self._print_Equivalent(e.args[0], r"\not\Leftrightarrow") + if isinstance(e.args[0], Implies): + return self._print_Implies(e.args[0], r"\not\Rightarrow") + if (e.args[0].is_Boolean): + return r"\neg \left(%s\right)" % self._print(e.args[0]) + else: + return r"\neg %s" % self._print(e.args[0]) + + def _print_LogOp(self, args, char): + arg = args[0] + if arg.is_Boolean and not arg.is_Not: + tex = r"\left(%s\right)" % self._print(arg) + else: + tex = r"%s" % self._print(arg) + + for arg in args[1:]: + if arg.is_Boolean and not arg.is_Not: + tex += r" %s \left(%s\right)" % (char, self._print(arg)) + else: + tex += r" %s %s" % (char, self._print(arg)) + + return tex + + def _print_And(self, e): + args = sorted(e.args, key=default_sort_key) + return self._print_LogOp(args, r"\wedge") + + def _print_Or(self, e): + args = sorted(e.args, key=default_sort_key) + return self._print_LogOp(args, r"\vee") + + def _print_Xor(self, e): + args = sorted(e.args, key=default_sort_key) + return self._print_LogOp(args, r"\veebar") + + def _print_Implies(self, e, altchar=None): + return self._print_LogOp(e.args, altchar or r"\Rightarrow") + + def _print_Equivalent(self, e, altchar=None): + args = sorted(e.args, key=default_sort_key) + return self._print_LogOp(args, altchar or r"\Leftrightarrow") + + def _print_conjugate(self, expr, exp=None): + tex = r"\overline{%s}" % self._print(expr.args[0]) + + if exp is not None: + return r"%s^{%s}" % (tex, exp) + else: + return tex + + def _print_polar_lift(self, expr, exp=None): + func = r"\operatorname{polar\_lift}" + arg = r"{\left(%s \right)}" % self._print(expr.args[0]) + + if exp is not None: + return r"%s^{%s}%s" % (func, exp, arg) + else: + return r"%s%s" % (func, arg) + + def _print_ExpBase(self, expr, exp=None): + # TODO should exp_polar be printed differently? + # what about exp_polar(0), exp_polar(1)? + tex = r"e^{%s}" % self._print(expr.args[0]) + return self._do_exponent(tex, exp) + + def _print_Exp1(self, expr, exp=None): + return "e" + + def _print_elliptic_k(self, expr, exp=None): + tex = r"\left(%s\right)" % self._print(expr.args[0]) + if exp is not None: + return r"K^{%s}%s" % (exp, tex) + else: + return r"K%s" % tex + + def _print_elliptic_f(self, expr, exp=None): + tex = r"\left(%s\middle| %s\right)" % \ + (self._print(expr.args[0]), self._print(expr.args[1])) + if exp is not None: + return r"F^{%s}%s" % (exp, tex) + else: + return r"F%s" % tex + + def _print_elliptic_e(self, expr, exp=None): + if len(expr.args) == 2: + tex = r"\left(%s\middle| %s\right)" % \ + (self._print(expr.args[0]), self._print(expr.args[1])) + else: + tex = r"\left(%s\right)" % self._print(expr.args[0]) + if exp is not None: + return r"E^{%s}%s" % (exp, tex) + else: + return r"E%s" % tex + + def _print_elliptic_pi(self, expr, exp=None): + if len(expr.args) == 3: + tex = r"\left(%s; %s\middle| %s\right)" % \ + (self._print(expr.args[0]), self._print(expr.args[1]), + self._print(expr.args[2])) + else: + tex = r"\left(%s\middle| %s\right)" % \ + (self._print(expr.args[0]), self._print(expr.args[1])) + if exp is not None: + return r"\Pi^{%s}%s" % (exp, tex) + else: + return r"\Pi%s" % tex + + def _print_beta(self, expr, exp=None): + x = expr.args[0] + # Deal with unevaluated single argument beta + y = expr.args[0] if len(expr.args) == 1 else expr.args[1] + tex = rf"\left({x}, {y}\right)" + + if exp is not None: + return r"\operatorname{B}^{%s}%s" % (exp, tex) + else: + return r"\operatorname{B}%s" % tex + + def _print_betainc(self, expr, exp=None, operator='B'): + largs = [self._print(arg) for arg in expr.args] + tex = r"\left(%s, %s\right)" % (largs[0], largs[1]) + + if exp is not None: + return r"\operatorname{%s}_{(%s, %s)}^{%s}%s" % (operator, largs[2], largs[3], exp, tex) + else: + return r"\operatorname{%s}_{(%s, %s)}%s" % (operator, largs[2], largs[3], tex) + + def _print_betainc_regularized(self, expr, exp=None): + return self._print_betainc(expr, exp, operator='I') + + def _print_uppergamma(self, expr, exp=None): + tex = r"\left(%s, %s\right)" % (self._print(expr.args[0]), + self._print(expr.args[1])) + + if exp is not None: + return r"\Gamma^{%s}%s" % (exp, tex) + else: + return r"\Gamma%s" % tex + + def _print_lowergamma(self, expr, exp=None): + tex = r"\left(%s, %s\right)" % (self._print(expr.args[0]), + self._print(expr.args[1])) + + if exp is not None: + return r"\gamma^{%s}%s" % (exp, tex) + else: + return r"\gamma%s" % tex + + def _hprint_one_arg_func(self, expr, exp=None) -> str: + tex = r"\left(%s\right)" % self._print(expr.args[0]) + + if exp is not None: + return r"%s^{%s}%s" % (self._print(expr.func), exp, tex) + else: + return r"%s%s" % (self._print(expr.func), tex) + + _print_gamma = _hprint_one_arg_func + + def _print_Chi(self, expr, exp=None): + tex = r"\left(%s\right)" % self._print(expr.args[0]) + + if exp is not None: + return r"\operatorname{Chi}^{%s}%s" % (exp, tex) + else: + return r"\operatorname{Chi}%s" % tex + + def _print_expint(self, expr, exp=None): + tex = r"\left(%s\right)" % self._print(expr.args[1]) + nu = self._print(expr.args[0]) + + if exp is not None: + return r"\operatorname{E}_{%s}^{%s}%s" % (nu, exp, tex) + else: + return r"\operatorname{E}_{%s}%s" % (nu, tex) + + def _print_fresnels(self, expr, exp=None): + tex = r"\left(%s\right)" % self._print(expr.args[0]) + + if exp is not None: + return r"S^{%s}%s" % (exp, tex) + else: + return r"S%s" % tex + + def _print_fresnelc(self, expr, exp=None): + tex = r"\left(%s\right)" % self._print(expr.args[0]) + + if exp is not None: + return r"C^{%s}%s" % (exp, tex) + else: + return r"C%s" % tex + + def _print_subfactorial(self, expr, exp=None): + tex = r"!%s" % self.parenthesize(expr.args[0], PRECEDENCE["Func"]) + + if exp is not None: + return r"\left(%s\right)^{%s}" % (tex, exp) + else: + return tex + + def _print_factorial(self, expr, exp=None): + tex = r"%s!" % self.parenthesize(expr.args[0], PRECEDENCE["Func"]) + + if exp is not None: + return r"%s^{%s}" % (tex, exp) + else: + return tex + + def _print_factorial2(self, expr, exp=None): + tex = r"%s!!" % self.parenthesize(expr.args[0], PRECEDENCE["Func"]) + + if exp is not None: + return r"%s^{%s}" % (tex, exp) + else: + return tex + + def _print_binomial(self, expr, exp=None): + tex = r"{\binom{%s}{%s}}" % (self._print(expr.args[0]), + self._print(expr.args[1])) + + if exp is not None: + return r"%s^{%s}" % (tex, exp) + else: + return tex + + def _print_RisingFactorial(self, expr, exp=None): + n, k = expr.args + base = r"%s" % self.parenthesize(n, PRECEDENCE['Func']) + + tex = r"{%s}^{\left(%s\right)}" % (base, self._print(k)) + + return self._do_exponent(tex, exp) + + def _print_FallingFactorial(self, expr, exp=None): + n, k = expr.args + sub = r"%s" % self.parenthesize(k, PRECEDENCE['Func']) + + tex = r"{\left(%s\right)}_{%s}" % (self._print(n), sub) + + return self._do_exponent(tex, exp) + + def _hprint_BesselBase(self, expr, exp, sym: str) -> str: + tex = r"%s" % (sym) + + need_exp = False + if exp is not None: + if tex.find('^') == -1: + tex = r"%s^{%s}" % (tex, exp) + else: + need_exp = True + + tex = r"%s_{%s}\left(%s\right)" % (tex, self._print(expr.order), + self._print(expr.argument)) + + if need_exp: + tex = self._do_exponent(tex, exp) + return tex + + def _hprint_vec(self, vec) -> str: + if not vec: + return "" + s = "" + for i in vec[:-1]: + s += "%s, " % self._print(i) + s += self._print(vec[-1]) + return s + + def _print_besselj(self, expr, exp=None): + return self._hprint_BesselBase(expr, exp, 'J') + + def _print_besseli(self, expr, exp=None): + return self._hprint_BesselBase(expr, exp, 'I') + + def _print_besselk(self, expr, exp=None): + return self._hprint_BesselBase(expr, exp, 'K') + + def _print_bessely(self, expr, exp=None): + return self._hprint_BesselBase(expr, exp, 'Y') + + def _print_yn(self, expr, exp=None): + return self._hprint_BesselBase(expr, exp, 'y') + + def _print_jn(self, expr, exp=None): + return self._hprint_BesselBase(expr, exp, 'j') + + def _print_hankel1(self, expr, exp=None): + return self._hprint_BesselBase(expr, exp, 'H^{(1)}') + + def _print_hankel2(self, expr, exp=None): + return self._hprint_BesselBase(expr, exp, 'H^{(2)}') + + def _print_hn1(self, expr, exp=None): + return self._hprint_BesselBase(expr, exp, 'h^{(1)}') + + def _print_hn2(self, expr, exp=None): + return self._hprint_BesselBase(expr, exp, 'h^{(2)}') + + def _hprint_airy(self, expr, exp=None, notation="") -> str: + tex = r"\left(%s\right)" % self._print(expr.args[0]) + + if exp is not None: + return r"%s^{%s}%s" % (notation, exp, tex) + else: + return r"%s%s" % (notation, tex) + + def _hprint_airy_prime(self, expr, exp=None, notation="") -> str: + tex = r"\left(%s\right)" % self._print(expr.args[0]) + + if exp is not None: + return r"{%s^\prime}^{%s}%s" % (notation, exp, tex) + else: + return r"%s^\prime%s" % (notation, tex) + + def _print_airyai(self, expr, exp=None): + return self._hprint_airy(expr, exp, 'Ai') + + def _print_airybi(self, expr, exp=None): + return self._hprint_airy(expr, exp, 'Bi') + + def _print_airyaiprime(self, expr, exp=None): + return self._hprint_airy_prime(expr, exp, 'Ai') + + def _print_airybiprime(self, expr, exp=None): + return self._hprint_airy_prime(expr, exp, 'Bi') + + def _print_hyper(self, expr, exp=None): + tex = r"{{}_{%s}F_{%s}\left(\begin{matrix} %s \\ %s \end{matrix}" \ + r"\middle| {%s} \right)}" % \ + (self._print(len(expr.ap)), self._print(len(expr.bq)), + self._hprint_vec(expr.ap), self._hprint_vec(expr.bq), + self._print(expr.argument)) + + if exp is not None: + tex = r"{%s}^{%s}" % (tex, exp) + return tex + + def _print_meijerg(self, expr, exp=None): + tex = r"{G_{%s, %s}^{%s, %s}\left(\begin{matrix} %s & %s \\" \ + r"%s & %s \end{matrix} \middle| {%s} \right)}" % \ + (self._print(len(expr.ap)), self._print(len(expr.bq)), + self._print(len(expr.bm)), self._print(len(expr.an)), + self._hprint_vec(expr.an), self._hprint_vec(expr.aother), + self._hprint_vec(expr.bm), self._hprint_vec(expr.bother), + self._print(expr.argument)) + + if exp is not None: + tex = r"{%s}^{%s}" % (tex, exp) + return tex + + def _print_dirichlet_eta(self, expr, exp=None): + tex = r"\left(%s\right)" % self._print(expr.args[0]) + if exp is not None: + return r"\eta^{%s}%s" % (exp, tex) + return r"\eta%s" % tex + + def _print_zeta(self, expr, exp=None): + if len(expr.args) == 2: + tex = r"\left(%s, %s\right)" % tuple(map(self._print, expr.args)) + else: + tex = r"\left(%s\right)" % self._print(expr.args[0]) + if exp is not None: + return r"\zeta^{%s}%s" % (exp, tex) + return r"\zeta%s" % tex + + def _print_stieltjes(self, expr, exp=None): + if len(expr.args) == 2: + tex = r"_{%s}\left(%s\right)" % tuple(map(self._print, expr.args)) + else: + tex = r"_{%s}" % self._print(expr.args[0]) + if exp is not None: + return r"\gamma%s^{%s}" % (tex, exp) + return r"\gamma%s" % tex + + def _print_lerchphi(self, expr, exp=None): + tex = r"\left(%s, %s, %s\right)" % tuple(map(self._print, expr.args)) + if exp is None: + return r"\Phi%s" % tex + return r"\Phi^{%s}%s" % (exp, tex) + + def _print_polylog(self, expr, exp=None): + s, z = map(self._print, expr.args) + tex = r"\left(%s\right)" % z + if exp is None: + return r"\operatorname{Li}_{%s}%s" % (s, tex) + return r"\operatorname{Li}_{%s}^{%s}%s" % (s, exp, tex) + + def _print_jacobi(self, expr, exp=None): + n, a, b, x = map(self._print, expr.args) + tex = r"P_{%s}^{\left(%s,%s\right)}\left(%s\right)" % (n, a, b, x) + if exp is not None: + tex = r"\left(" + tex + r"\right)^{%s}" % (exp) + return tex + + def _print_gegenbauer(self, expr, exp=None): + n, a, x = map(self._print, expr.args) + tex = r"C_{%s}^{\left(%s\right)}\left(%s\right)" % (n, a, x) + if exp is not None: + tex = r"\left(" + tex + r"\right)^{%s}" % (exp) + return tex + + def _print_chebyshevt(self, expr, exp=None): + n, x = map(self._print, expr.args) + tex = r"T_{%s}\left(%s\right)" % (n, x) + if exp is not None: + tex = r"\left(" + tex + r"\right)^{%s}" % (exp) + return tex + + def _print_chebyshevu(self, expr, exp=None): + n, x = map(self._print, expr.args) + tex = r"U_{%s}\left(%s\right)" % (n, x) + if exp is not None: + tex = r"\left(" + tex + r"\right)^{%s}" % (exp) + return tex + + def _print_legendre(self, expr, exp=None): + n, x = map(self._print, expr.args) + tex = r"P_{%s}\left(%s\right)" % (n, x) + if exp is not None: + tex = r"\left(" + tex + r"\right)^{%s}" % (exp) + return tex + + def _print_assoc_legendre(self, expr, exp=None): + n, a, x = map(self._print, expr.args) + tex = r"P_{%s}^{\left(%s\right)}\left(%s\right)" % (n, a, x) + if exp is not None: + tex = r"\left(" + tex + r"\right)^{%s}" % (exp) + return tex + + def _print_hermite(self, expr, exp=None): + n, x = map(self._print, expr.args) + tex = r"H_{%s}\left(%s\right)" % (n, x) + if exp is not None: + tex = r"\left(" + tex + r"\right)^{%s}" % (exp) + return tex + + def _print_laguerre(self, expr, exp=None): + n, x = map(self._print, expr.args) + tex = r"L_{%s}\left(%s\right)" % (n, x) + if exp is not None: + tex = r"\left(" + tex + r"\right)^{%s}" % (exp) + return tex + + def _print_assoc_laguerre(self, expr, exp=None): + n, a, x = map(self._print, expr.args) + tex = r"L_{%s}^{\left(%s\right)}\left(%s\right)" % (n, a, x) + if exp is not None: + tex = r"\left(" + tex + r"\right)^{%s}" % (exp) + return tex + + def _print_Ynm(self, expr, exp=None): + n, m, theta, phi = map(self._print, expr.args) + tex = r"Y_{%s}^{%s}\left(%s,%s\right)" % (n, m, theta, phi) + if exp is not None: + tex = r"\left(" + tex + r"\right)^{%s}" % (exp) + return tex + + def _print_Znm(self, expr, exp=None): + n, m, theta, phi = map(self._print, expr.args) + tex = r"Z_{%s}^{%s}\left(%s,%s\right)" % (n, m, theta, phi) + if exp is not None: + tex = r"\left(" + tex + r"\right)^{%s}" % (exp) + return tex + + def __print_mathieu_functions(self, character, args, prime=False, exp=None): + a, q, z = map(self._print, args) + sup = r"^{\prime}" if prime else "" + exp = "" if not exp else "^{%s}" % exp + return r"%s%s\left(%s, %s, %s\right)%s" % (character, sup, a, q, z, exp) + + def _print_mathieuc(self, expr, exp=None): + return self.__print_mathieu_functions("C", expr.args, exp=exp) + + def _print_mathieus(self, expr, exp=None): + return self.__print_mathieu_functions("S", expr.args, exp=exp) + + def _print_mathieucprime(self, expr, exp=None): + return self.__print_mathieu_functions("C", expr.args, prime=True, exp=exp) + + def _print_mathieusprime(self, expr, exp=None): + return self.__print_mathieu_functions("S", expr.args, prime=True, exp=exp) + + def _print_Rational(self, expr): + if expr.q != 1: + sign = "" + p = expr.p + if expr.p < 0: + sign = "- " + p = -p + if self._settings['fold_short_frac']: + return r"%s%d / %d" % (sign, p, expr.q) + return r"%s\frac{%d}{%d}" % (sign, p, expr.q) + else: + return self._print(expr.p) + + def _print_Order(self, expr): + s = self._print(expr.expr) + if expr.point and any(p != S.Zero for p in expr.point) or \ + len(expr.variables) > 1: + s += '; ' + if len(expr.variables) > 1: + s += self._print(expr.variables) + elif expr.variables: + s += self._print(expr.variables[0]) + s += r'\rightarrow ' + if len(expr.point) > 1: + s += self._print(expr.point) + else: + s += self._print(expr.point[0]) + return r"O\left(%s\right)" % s + + def _print_Symbol(self, expr: Symbol, style='plain'): + name: str = self._settings['symbol_names'].get(expr) + if name is not None: + return name + + return self._deal_with_super_sub(expr.name, style=style) + + _print_RandomSymbol = _print_Symbol + + def _split_super_sub(self, name: str) -> tuple[str, list[str], list[str]]: + if name is None or '{' in name: + return (name, [], []) + elif self._settings["disable_split_super_sub"]: + name, supers, subs = (name.replace('_', '\\_').replace('^', '\\^'), [], []) + else: + name, supers, subs = split_super_sub(name) + name = translate(name) + supers = [translate(sup) for sup in supers] + subs = [translate(sub) for sub in subs] + return (name, supers, subs) + + def _deal_with_super_sub(self, string: str, style='plain') -> str: + name, supers, subs = self._split_super_sub(string) + + # apply the style only to the name + if style == 'bold': + name = "\\mathbf{{{}}}".format(name) + + # glue all items together: + if supers: + name += "^{%s}" % " ".join(supers) + if subs: + name += "_{%s}" % " ".join(subs) + + return name + + def _print_Relational(self, expr): + if self._settings['itex']: + gt = r"\gt" + lt = r"\lt" + else: + gt = ">" + lt = "<" + + charmap = { + "==": "=", + ">": gt, + "<": lt, + ">=": r"\geq", + "<=": r"\leq", + "!=": r"\neq", + } + + return "%s %s %s" % (self._print(expr.lhs), + charmap[expr.rel_op], self._print(expr.rhs)) + + def _print_Piecewise(self, expr): + ecpairs = [r"%s & \text{for}\: %s" % (self._print(e), self._print(c)) + for e, c in expr.args[:-1]] + if expr.args[-1].cond == true: + ecpairs.append(r"%s & \text{otherwise}" % + self._print(expr.args[-1].expr)) + else: + ecpairs.append(r"%s & \text{for}\: %s" % + (self._print(expr.args[-1].expr), + self._print(expr.args[-1].cond))) + tex = r"\begin{cases} %s \end{cases}" + return tex % r" \\".join(ecpairs) + + def _print_matrix_contents(self, expr): + lines = [] + + for line in range(expr.rows): # horrible, should be 'rows' + lines.append(" & ".join([self._print(i) for i in expr[line, :]])) + + mat_str = self._settings['mat_str'] + if mat_str is None: + if self._settings['mode'] == 'inline': + mat_str = 'smallmatrix' + else: + if (expr.cols <= 10) is True: + mat_str = 'matrix' + else: + mat_str = 'array' + + out_str = r'\begin{%MATSTR%}%s\end{%MATSTR%}' + out_str = out_str.replace('%MATSTR%', mat_str) + if mat_str == 'array': + out_str = out_str.replace('%s', '{' + 'c'*expr.cols + '}%s') + return out_str % r"\\".join(lines) + + def _print_MatrixBase(self, expr): + out_str = self._print_matrix_contents(expr) + if self._settings['mat_delim']: + left_delim = self._settings['mat_delim'] + right_delim = self._delim_dict[left_delim] + out_str = r'\left' + left_delim + out_str + \ + r'\right' + right_delim + return out_str + + def _print_MatrixElement(self, expr): + matrix_part = self.parenthesize(expr.parent, PRECEDENCE['Atom'], strict=True) + index_part = f"{self._print(expr.i)},{self._print(expr.j)}" + return f"{{{matrix_part}}}_{{{index_part}}}" + + def _print_MatrixSlice(self, expr): + def latexslice(x, dim): + x = list(x) + if x[2] == 1: + del x[2] + if x[0] == 0: + x[0] = None + if x[1] == dim: + x[1] = None + return ':'.join(self._print(xi) if xi is not None else '' for xi in x) + return (self.parenthesize(expr.parent, PRECEDENCE["Atom"], strict=True) + r'\left[' + + latexslice(expr.rowslice, expr.parent.rows) + ', ' + + latexslice(expr.colslice, expr.parent.cols) + r'\right]') + + def _print_BlockMatrix(self, expr): + return self._print(expr.blocks) + + def _print_Transpose(self, expr): + mat = expr.arg + from sympy.matrices import MatrixSymbol, BlockMatrix + if (not isinstance(mat, MatrixSymbol) and + not isinstance(mat, BlockMatrix) and mat.is_MatrixExpr): + return r"\left(%s\right)^{T}" % self._print(mat) + else: + s = self.parenthesize(mat, precedence_traditional(expr), True) + if '^' in s: + return r"\left(%s\right)^{T}" % s + else: + return "%s^{T}" % s + + def _print_Trace(self, expr): + mat = expr.arg + return r"\operatorname{tr}\left(%s \right)" % self._print(mat) + + def _print_Adjoint(self, expr): + style_to_latex = { + "dagger" : r"\dagger", + "star" : r"\ast", + "hermitian": r"\mathsf{H}" + } + adjoint_style = style_to_latex.get(self._settings["adjoint_style"], r"\dagger") + mat = expr.arg + from sympy.matrices import MatrixSymbol, BlockMatrix + if (not isinstance(mat, MatrixSymbol) and + not isinstance(mat, BlockMatrix) and mat.is_MatrixExpr): + return r"\left(%s\right)^{%s}" % (self._print(mat), adjoint_style) + else: + s = self.parenthesize(mat, precedence_traditional(expr), True) + if '^' in s: + return r"\left(%s\right)^{%s}" % (s, adjoint_style) + else: + return r"%s^{%s}" % (s, adjoint_style) + + def _print_MatMul(self, expr): + from sympy import MatMul + + # Parenthesize nested MatMul but not other types of Mul objects: + parens = lambda x: self._print(x) if isinstance(x, Mul) and not isinstance(x, MatMul) else \ + self.parenthesize(x, precedence_traditional(expr), False) + + args = list(expr.args) + if expr.could_extract_minus_sign(): + if args[0] == -1: + args = args[1:] + else: + args[0] = -args[0] + return '- ' + ' '.join(map(parens, args)) + else: + return ' '.join(map(parens, args)) + + def _print_DotProduct(self, expr): + level = precedence_traditional(expr) + left, right = expr.args + return rf"{self.parenthesize(left, level)} \cdot {self.parenthesize(right, level)}" + + def _print_Determinant(self, expr): + mat = expr.arg + if mat.is_MatrixExpr: + from sympy.matrices.expressions.blockmatrix import BlockMatrix + if isinstance(mat, BlockMatrix): + return r"\left|{%s}\right|" % self._print_matrix_contents(mat.blocks) + return r"\left|{%s}\right|" % self._print(mat) + return r"\left|{%s}\right|" % self._print_matrix_contents(mat) + + + def _print_Mod(self, expr, exp=None): + if exp is not None: + return r'\left(%s \bmod %s\right)^{%s}' % \ + (self.parenthesize(expr.args[0], PRECEDENCE['Mul'], + strict=True), + self.parenthesize(expr.args[1], PRECEDENCE['Mul'], + strict=True), + exp) + return r'%s \bmod %s' % (self.parenthesize(expr.args[0], + PRECEDENCE['Mul'], + strict=True), + self.parenthesize(expr.args[1], + PRECEDENCE['Mul'], + strict=True)) + + def _print_HadamardProduct(self, expr): + args = expr.args + prec = PRECEDENCE['Pow'] + parens = self.parenthesize + + return r' \circ '.join( + (parens(arg, prec, strict=True) for arg in args)) + + def _print_HadamardPower(self, expr): + if precedence_traditional(expr.exp) < PRECEDENCE["Mul"]: + template = r"%s^{\circ \left({%s}\right)}" + else: + template = r"%s^{\circ {%s}}" + return self._helper_print_standard_power(expr, template) + + def _print_KroneckerProduct(self, expr): + args = expr.args + prec = PRECEDENCE['Pow'] + parens = self.parenthesize + + return r' \otimes '.join( + (parens(arg, prec, strict=True) for arg in args)) + + def _print_MatPow(self, expr): + base, exp = expr.base, expr.exp + from sympy.matrices import MatrixSymbol + if not isinstance(base, MatrixSymbol) and base.is_MatrixExpr: + return "\\left(%s\\right)^{%s}" % (self._print(base), + self._print(exp)) + else: + base_str = self._print(base) + if '^' in base_str: + return r"\left(%s\right)^{%s}" % (base_str, self._print(exp)) + else: + return "%s^{%s}" % (base_str, self._print(exp)) + + def _print_MatrixSymbol(self, expr): + return self._print_Symbol(expr, style=self._settings[ + 'mat_symbol_style']) + + def _print_ZeroMatrix(self, Z): + return "0" if self._settings[ + 'mat_symbol_style'] == 'plain' else r"\mathbf{0}" + + def _print_OneMatrix(self, O): + return "1" if self._settings[ + 'mat_symbol_style'] == 'plain' else r"\mathbf{1}" + + def _print_Identity(self, I): + return r"\mathbb{I}" if self._settings[ + 'mat_symbol_style'] == 'plain' else r"\mathbf{I}" + + def _print_PermutationMatrix(self, P): + perm_str = self._print(P.args[0]) + return "P_{%s}" % perm_str + + def _print_NDimArray(self, expr: NDimArray): + + if expr.rank() == 0: + return self._print(expr[()]) + + mat_str = self._settings['mat_str'] + if mat_str is None: + if self._settings['mode'] == 'inline': + mat_str = 'smallmatrix' + else: + if (expr.rank() == 0) or (expr.shape[-1] <= 10): + mat_str = 'matrix' + else: + mat_str = 'array' + block_str = r'\begin{%MATSTR%}%s\end{%MATSTR%}' + block_str = block_str.replace('%MATSTR%', mat_str) + if mat_str == 'array': + block_str = block_str.replace('%s', '{' + 'c'*expr.shape[0] + '}%s') + + if self._settings['mat_delim']: + left_delim: str = self._settings['mat_delim'] + right_delim = self._delim_dict[left_delim] + block_str = r'\left' + left_delim + block_str + \ + r'\right' + right_delim + + if expr.rank() == 0: + return block_str % "" + + level_str: list[list[str]] = [[] for i in range(expr.rank() + 1)] + shape_ranges = [list(range(i)) for i in expr.shape] + for outer_i in itertools.product(*shape_ranges): + level_str[-1].append(self._print(expr[outer_i])) + even = True + for back_outer_i in range(expr.rank()-1, -1, -1): + if len(level_str[back_outer_i+1]) < expr.shape[back_outer_i]: + break + if even: + level_str[back_outer_i].append( + r" & ".join(level_str[back_outer_i+1])) + else: + level_str[back_outer_i].append( + block_str % (r"\\".join(level_str[back_outer_i+1]))) + if len(level_str[back_outer_i+1]) == 1: + level_str[back_outer_i][-1] = r"\left[" + \ + level_str[back_outer_i][-1] + r"\right]" + even = not even + level_str[back_outer_i+1] = [] + + out_str = level_str[0][0] + + if expr.rank() % 2 == 1: + out_str = block_str % out_str + + return out_str + + def _printer_tensor_indices(self, name, indices, index_map: dict): + out_str = self._print(name) + last_valence = None + prev_map = None + for index in indices: + new_valence = index.is_up + if ((index in index_map) or prev_map) and \ + last_valence == new_valence: + out_str += "," + if last_valence != new_valence: + if last_valence is not None: + out_str += "}" + if index.is_up: + out_str += "{}^{" + else: + out_str += "{}_{" + out_str += self._print(index.args[0]) + if index in index_map: + out_str += "=" + out_str += self._print(index_map[index]) + prev_map = True + else: + prev_map = False + last_valence = new_valence + if last_valence is not None: + out_str += "}" + return out_str + + def _print_Tensor(self, expr): + name = expr.args[0].args[0] + indices = expr.get_indices() + return self._printer_tensor_indices(name, indices, {}) + + def _print_TensorElement(self, expr): + name = expr.expr.args[0].args[0] + indices = expr.expr.get_indices() + index_map = expr.index_map + return self._printer_tensor_indices(name, indices, index_map) + + def _print_TensMul(self, expr): + # prints expressions like "A(a)", "3*A(a)", "(1+x)*A(a)" + sign, args = expr._get_args_for_traditional_printer() + return sign + "".join( + [self.parenthesize(arg, precedence(expr)) for arg in args] + ) + + def _print_TensAdd(self, expr): + a = [] + args = expr.args + for x in args: + a.append(self.parenthesize(x, precedence(expr))) + a.sort() + s = ' + '.join(a) + s = s.replace('+ -', '- ') + return s + + def _print_TensorIndex(self, expr): + return "{}%s{%s}" % ( + "^" if expr.is_up else "_", + self._print(expr.args[0]) + ) + + def _print_PartialDerivative(self, expr): + if len(expr.variables) == 1: + return r"\frac{\partial}{\partial {%s}}{%s}" % ( + self._print(expr.variables[0]), + self.parenthesize(expr.expr, PRECEDENCE["Mul"], False) + ) + else: + return r"\frac{\partial^{%s}}{%s}{%s}" % ( + len(expr.variables), + " ".join([r"\partial {%s}" % self._print(i) for i in expr.variables]), + self.parenthesize(expr.expr, PRECEDENCE["Mul"], False) + ) + + def _print_ArraySymbol(self, expr): + return self._print(expr.name) + + def _print_ArrayElement(self, expr): + return "{{%s}_{%s}}" % ( + self.parenthesize(expr.name, PRECEDENCE["Func"], True), + ", ".join([f"{self._print(i)}" for i in expr.indices])) + + def _print_UniversalSet(self, expr): + return r"\mathbb{U}" + + def _print_frac(self, expr, exp=None): + if exp is None: + return r"\operatorname{frac}{\left(%s\right)}" % self._print(expr.args[0]) + else: + return r"\operatorname{frac}{\left(%s\right)}^{%s}" % ( + self._print(expr.args[0]), exp) + + def _print_tuple(self, expr): + if self._settings['decimal_separator'] == 'comma': + sep = ";" + elif self._settings['decimal_separator'] == 'period': + sep = "," + else: + raise ValueError('Unknown Decimal Separator') + + if len(expr) == 1: + # 1-tuple needs a trailing separator + return self._add_parens_lspace(self._print(expr[0]) + sep) + else: + return self._add_parens_lspace( + (sep + r" \ ").join([self._print(i) for i in expr])) + + def _print_TensorProduct(self, expr): + elements = [self._print(a) for a in expr.args] + return r' \otimes '.join(elements) + + def _print_WedgeProduct(self, expr): + elements = [self._print(a) for a in expr.args] + return r' \wedge '.join(elements) + + def _print_Tuple(self, expr): + return self._print_tuple(expr) + + def _print_list(self, expr): + if self._settings['decimal_separator'] == 'comma': + return r"\left[ %s\right]" % \ + r"; \ ".join([self._print(i) for i in expr]) + elif self._settings['decimal_separator'] == 'period': + return r"\left[ %s\right]" % \ + r", \ ".join([self._print(i) for i in expr]) + else: + raise ValueError('Unknown Decimal Separator') + + + def _print_dict(self, d): + keys = sorted(d.keys(), key=default_sort_key) + items = [] + + for key in keys: + val = d[key] + items.append("%s : %s" % (self._print(key), self._print(val))) + + return r"\left\{ %s\right\}" % r", \ ".join(items) + + def _print_Dict(self, expr): + return self._print_dict(expr) + + def _print_DiracDelta(self, expr, exp=None): + if len(expr.args) == 1 or expr.args[1] == 0: + tex = r"\delta\left(%s\right)" % self._print(expr.args[0]) + else: + tex = r"\delta^{\left( %s \right)}\left( %s \right)" % ( + self._print(expr.args[1]), self._print(expr.args[0])) + if exp: + tex = r"\left(%s\right)^{%s}" % (tex, exp) + return tex + + def _print_SingularityFunction(self, expr, exp=None): + shift = self._print(expr.args[0] - expr.args[1]) + power = self._print(expr.args[2]) + tex = r"{\left\langle %s \right\rangle}^{%s}" % (shift, power) + if exp is not None: + tex = r"{\left({\langle %s \rangle}^{%s}\right)}^{%s}" % (shift, power, exp) + return tex + + def _print_Heaviside(self, expr, exp=None): + pargs = ', '.join(self._print(arg) for arg in expr.pargs) + tex = r"\theta\left(%s\right)" % pargs + if exp: + tex = r"\left(%s\right)^{%s}" % (tex, exp) + return tex + + def _print_KroneckerDelta(self, expr, exp=None): + i = self._print(expr.args[0]) + j = self._print(expr.args[1]) + if expr.args[0].is_Atom and expr.args[1].is_Atom: + tex = r'\delta_{%s %s}' % (i, j) + else: + tex = r'\delta_{%s, %s}' % (i, j) + if exp is not None: + tex = r'\left(%s\right)^{%s}' % (tex, exp) + return tex + + def _print_LeviCivita(self, expr, exp=None): + indices = map(self._print, expr.args) + if all(x.is_Atom for x in expr.args): + tex = r'\varepsilon_{%s}' % " ".join(indices) + else: + tex = r'\varepsilon_{%s}' % ", ".join(indices) + if exp: + tex = r'\left(%s\right)^{%s}' % (tex, exp) + return tex + + def _print_RandomDomain(self, d): + if hasattr(d, 'as_boolean'): + return '\\text{Domain: }' + self._print(d.as_boolean()) + elif hasattr(d, 'set'): + return ('\\text{Domain: }' + self._print(d.symbols) + ' \\in ' + + self._print(d.set)) + elif hasattr(d, 'symbols'): + return '\\text{Domain on }' + self._print(d.symbols) + else: + return self._print(None) + + def _print_FiniteSet(self, s): + items = sorted(s.args, key=default_sort_key) + return self._print_set(items) + + def _print_set(self, s): + items = sorted(s, key=default_sort_key) + if self._settings['decimal_separator'] == 'comma': + items = "; ".join(map(self._print, items)) + elif self._settings['decimal_separator'] == 'period': + items = ", ".join(map(self._print, items)) + else: + raise ValueError('Unknown Decimal Separator') + return r"\left\{%s\right\}" % items + + + _print_frozenset = _print_set + + def _print_Range(self, s): + def _print_symbolic_range(): + # Symbolic Range that cannot be resolved + if s.args[0] == 0: + if s.args[2] == 1: + cont = self._print(s.args[1]) + else: + cont = ", ".join(self._print(arg) for arg in s.args) + else: + if s.args[2] == 1: + cont = ", ".join(self._print(arg) for arg in s.args[:2]) + else: + cont = ", ".join(self._print(arg) for arg in s.args) + + return(f"\\text{{Range}}\\left({cont}\\right)") + + dots = object() + + if s.start.is_infinite and s.stop.is_infinite: + if s.step.is_positive: + printset = dots, -1, 0, 1, dots + else: + printset = dots, 1, 0, -1, dots + elif s.start.is_infinite: + printset = dots, s[-1] - s.step, s[-1] + elif s.stop.is_infinite: + it = iter(s) + printset = next(it), next(it), dots + elif s.is_empty is not None: + if (s.size < 4) == True: + printset = tuple(s) + elif s.is_iterable: + it = iter(s) + printset = next(it), next(it), dots, s[-1] + else: + return _print_symbolic_range() + else: + return _print_symbolic_range() + return (r"\left\{" + + r", ".join(self._print(el) if el is not dots else r'\ldots' for el in printset) + + r"\right\}") + + def __print_number_polynomial(self, expr, letter, exp=None): + if len(expr.args) == 2: + if exp is not None: + return r"%s_{%s}^{%s}\left(%s\right)" % (letter, + self._print(expr.args[0]), exp, + self._print(expr.args[1])) + return r"%s_{%s}\left(%s\right)" % (letter, + self._print(expr.args[0]), self._print(expr.args[1])) + + tex = r"%s_{%s}" % (letter, self._print(expr.args[0])) + if exp is not None: + tex = r"%s^{%s}" % (tex, exp) + return tex + + def _print_bernoulli(self, expr, exp=None): + return self.__print_number_polynomial(expr, "B", exp) + + def _print_genocchi(self, expr, exp=None): + return self.__print_number_polynomial(expr, "G", exp) + + def _print_bell(self, expr, exp=None): + if len(expr.args) == 3: + tex1 = r"B_{%s, %s}" % (self._print(expr.args[0]), + self._print(expr.args[1])) + tex2 = r"\left(%s\right)" % r", ".join(self._print(el) for + el in expr.args[2]) + if exp is not None: + tex = r"%s^{%s}%s" % (tex1, exp, tex2) + else: + tex = tex1 + tex2 + return tex + return self.__print_number_polynomial(expr, "B", exp) + + def _print_fibonacci(self, expr, exp=None): + return self.__print_number_polynomial(expr, "F", exp) + + def _print_lucas(self, expr, exp=None): + tex = r"L_{%s}" % self._print(expr.args[0]) + if exp is not None: + tex = r"%s^{%s}" % (tex, exp) + return tex + + def _print_tribonacci(self, expr, exp=None): + return self.__print_number_polynomial(expr, "T", exp) + + def _print_mobius(self, expr, exp=None): + if exp is None: + return r'\mu\left(%s\right)' % self._print(expr.args[0]) + return r'\mu^{%s}\left(%s\right)' % (exp, self._print(expr.args[0])) + + def _print_SeqFormula(self, s): + dots = object() + if len(s.start.free_symbols) > 0 or len(s.stop.free_symbols) > 0: + return r"\left\{%s\right\}_{%s=%s}^{%s}" % ( + self._print(s.formula), + self._print(s.variables[0]), + self._print(s.start), + self._print(s.stop) + ) + if s.start is S.NegativeInfinity: + stop = s.stop + printset = (dots, s.coeff(stop - 3), s.coeff(stop - 2), + s.coeff(stop - 1), s.coeff(stop)) + elif s.stop is S.Infinity or s.length > 4: + printset = s[:4] + printset.append(dots) + else: + printset = tuple(s) + + return (r"\left[" + + r", ".join(self._print(el) if el is not dots else r'\ldots' for el in printset) + + r"\right]") + + _print_SeqPer = _print_SeqFormula + _print_SeqAdd = _print_SeqFormula + _print_SeqMul = _print_SeqFormula + + def _print_Interval(self, i): + if i.start == i.end: + return r"\left\{%s\right\}" % self._print(i.start) + + else: + if i.left_open: + left = '(' + else: + left = '[' + + if i.right_open: + right = ')' + else: + right = ']' + + return r"\left%s%s, %s\right%s" % \ + (left, self._print(i.start), self._print(i.end), right) + + def _print_AccumulationBounds(self, i): + return r"\left\langle %s, %s\right\rangle" % \ + (self._print(i.min), self._print(i.max)) + + def _print_Union(self, u): + prec = precedence_traditional(u) + args_str = [self.parenthesize(i, prec) for i in u.args] + return r" \cup ".join(args_str) + + def _print_Complement(self, u): + prec = precedence_traditional(u) + args_str = [self.parenthesize(i, prec) for i in u.args] + return r" \setminus ".join(args_str) + + def _print_Intersection(self, u): + prec = precedence_traditional(u) + args_str = [self.parenthesize(i, prec) for i in u.args] + return r" \cap ".join(args_str) + + def _print_SymmetricDifference(self, u): + prec = precedence_traditional(u) + args_str = [self.parenthesize(i, prec) for i in u.args] + return r" \triangle ".join(args_str) + + def _print_ProductSet(self, p): + prec = precedence_traditional(p) + if len(p.sets) >= 1 and not has_variety(p.sets): + return self.parenthesize(p.sets[0], prec) + "^{%d}" % len(p.sets) + return r" \times ".join( + self.parenthesize(set, prec) for set in p.sets) + + def _print_EmptySet(self, e): + return r"\emptyset" + + def _print_Naturals(self, n): + return r"\mathbb{N}" + + def _print_Naturals0(self, n): + return r"\mathbb{N}_0" + + def _print_Integers(self, i): + return r"\mathbb{Z}" + + def _print_Rationals(self, i): + return r"\mathbb{Q}" + + def _print_Reals(self, i): + return r"\mathbb{R}" + + def _print_Complexes(self, i): + return r"\mathbb{C}" + + def _print_ImageSet(self, s): + expr = s.lamda.expr + sig = s.lamda.signature + xys = ((self._print(x), self._print(y)) for x, y in zip(sig, s.base_sets)) + xinys = r", ".join(r"%s \in %s" % xy for xy in xys) + return r"\left\{%s\; \middle|\; %s\right\}" % (self._print(expr), xinys) + + def _print_ConditionSet(self, s): + vars_print = ', '.join([self._print(var) for var in Tuple(s.sym)]) + if s.base_set is S.UniversalSet: + return r"\left\{%s\; \middle|\; %s \right\}" % \ + (vars_print, self._print(s.condition)) + + return r"\left\{%s\; \middle|\; %s \in %s \wedge %s \right\}" % ( + vars_print, + vars_print, + self._print(s.base_set), + self._print(s.condition)) + + def _print_PowerSet(self, expr): + arg_print = self._print(expr.args[0]) + return r"\mathcal{{P}}\left({}\right)".format(arg_print) + + def _print_ComplexRegion(self, s): + vars_print = ', '.join([self._print(var) for var in s.variables]) + return r"\left\{%s\; \middle|\; %s \in %s \right\}" % ( + self._print(s.expr), + vars_print, + self._print(s.sets)) + + def _print_Contains(self, e): + return r"%s \in %s" % tuple(self._print(a) for a in e.args) + + def _print_FourierSeries(self, s): + if s.an.formula is S.Zero and s.bn.formula is S.Zero: + return self._print(s.a0) + return self._print_Add(s.truncate()) + r' + \ldots' + + def _print_FormalPowerSeries(self, s): + return self._print_Add(s.infinite) + + def _print_FiniteField(self, expr): + return r"\mathbb{F}_{%s}" % expr.mod + + def _print_IntegerRing(self, expr): + return r"\mathbb{Z}" + + def _print_RationalField(self, expr): + return r"\mathbb{Q}" + + def _print_RealField(self, expr): + return r"\mathbb{R}" + + def _print_ComplexField(self, expr): + return r"\mathbb{C}" + + def _print_PolynomialRing(self, expr): + domain = self._print(expr.domain) + symbols = ", ".join(map(self._print, expr.symbols)) + return r"%s\left[%s\right]" % (domain, symbols) + + def _print_FractionField(self, expr): + domain = self._print(expr.domain) + symbols = ", ".join(map(self._print, expr.symbols)) + return r"%s\left(%s\right)" % (domain, symbols) + + def _print_PolynomialRingBase(self, expr): + domain = self._print(expr.domain) + symbols = ", ".join(map(self._print, expr.symbols)) + inv = "" + if not expr.is_Poly: + inv = r"S_<^{-1}" + return r"%s%s\left[%s\right]" % (inv, domain, symbols) + + def _print_Poly(self, poly): + cls = poly.__class__.__name__ + terms = [] + for monom, coeff in poly.terms(): + s_monom = '' + for i, exp in enumerate(monom): + if exp > 0: + if exp == 1: + s_monom += self._print(poly.gens[i]) + else: + s_monom += self._print(pow(poly.gens[i], exp)) + + if coeff.is_Add: + if s_monom: + s_coeff = r"\left(%s\right)" % self._print(coeff) + else: + s_coeff = self._print(coeff) + else: + if s_monom: + if coeff is S.One: + terms.extend(['+', s_monom]) + continue + + if coeff is S.NegativeOne: + terms.extend(['-', s_monom]) + continue + + s_coeff = self._print(coeff) + + if not s_monom: + s_term = s_coeff + else: + s_term = s_coeff + " " + s_monom + + if s_term.startswith('-'): + terms.extend(['-', s_term[1:]]) + else: + terms.extend(['+', s_term]) + + if terms[0] in ('-', '+'): + modifier = terms.pop(0) + + if modifier == '-': + terms[0] = '-' + terms[0] + + expr = ' '.join(terms) + gens = list(map(self._print, poly.gens)) + domain = "domain=%s" % self._print(poly.get_domain()) + + args = ", ".join([expr] + gens + [domain]) + if cls in accepted_latex_functions: + tex = r"\%s {\left(%s \right)}" % (cls, args) + else: + tex = r"\operatorname{%s}{\left( %s \right)}" % (cls, args) + + return tex + + def _print_ComplexRootOf(self, root): + cls = root.__class__.__name__ + if cls == "ComplexRootOf": + cls = "CRootOf" + expr = self._print(root.expr) + index = root.index + if cls in accepted_latex_functions: + return r"\%s {\left(%s, %d\right)}" % (cls, expr, index) + else: + return r"\operatorname{%s} {\left(%s, %d\right)}" % (cls, expr, + index) + + def _print_RootSum(self, expr): + cls = expr.__class__.__name__ + args = [self._print(expr.expr)] + + if expr.fun is not S.IdentityFunction: + args.append(self._print(expr.fun)) + + if cls in accepted_latex_functions: + return r"\%s {\left(%s\right)}" % (cls, ", ".join(args)) + else: + return r"\operatorname{%s} {\left(%s\right)}" % (cls, + ", ".join(args)) + + def _print_OrdinalOmega(self, expr): + return r"\omega" + + def _print_OmegaPower(self, expr): + exp, mul = expr.args + if mul != 1: + if exp != 1: + return r"{} \omega^{{{}}}".format(mul, exp) + else: + return r"{} \omega".format(mul) + else: + if exp != 1: + return r"\omega^{{{}}}".format(exp) + else: + return r"\omega" + + def _print_Ordinal(self, expr): + return " + ".join([self._print(arg) for arg in expr.args]) + + def _print_PolyElement(self, poly): + mul_symbol = self._settings['mul_symbol_latex'] + return poly.str(self, PRECEDENCE, "{%s}^{%d}", mul_symbol) + + def _print_FracElement(self, frac): + if frac.denom == 1: + return self._print(frac.numer) + else: + numer = self._print(frac.numer) + denom = self._print(frac.denom) + return r"\frac{%s}{%s}" % (numer, denom) + + def _print_euler(self, expr, exp=None): + m, x = (expr.args[0], None) if len(expr.args) == 1 else expr.args + tex = r"E_{%s}" % self._print(m) + if exp is not None: + tex = r"%s^{%s}" % (tex, exp) + if x is not None: + tex = r"%s\left(%s\right)" % (tex, self._print(x)) + return tex + + def _print_catalan(self, expr, exp=None): + tex = r"C_{%s}" % self._print(expr.args[0]) + if exp is not None: + tex = r"%s^{%s}" % (tex, exp) + return tex + + def _print_UnifiedTransform(self, expr, s, inverse=False): + return r"\mathcal{{{}}}{}_{{{}}}\left[{}\right]\left({}\right)".format(s, '^{-1}' if inverse else '', self._print(expr.args[1]), self._print(expr.args[0]), self._print(expr.args[2])) + + def _print_MellinTransform(self, expr): + return self._print_UnifiedTransform(expr, 'M') + + def _print_InverseMellinTransform(self, expr): + return self._print_UnifiedTransform(expr, 'M', True) + + def _print_LaplaceTransform(self, expr): + return self._print_UnifiedTransform(expr, 'L') + + def _print_InverseLaplaceTransform(self, expr): + return self._print_UnifiedTransform(expr, 'L', True) + + def _print_FourierTransform(self, expr): + return self._print_UnifiedTransform(expr, 'F') + + def _print_InverseFourierTransform(self, expr): + return self._print_UnifiedTransform(expr, 'F', True) + + def _print_SineTransform(self, expr): + return self._print_UnifiedTransform(expr, 'SIN') + + def _print_InverseSineTransform(self, expr): + return self._print_UnifiedTransform(expr, 'SIN', True) + + def _print_CosineTransform(self, expr): + return self._print_UnifiedTransform(expr, 'COS') + + def _print_InverseCosineTransform(self, expr): + return self._print_UnifiedTransform(expr, 'COS', True) + + def _print_DMP(self, p): + try: + if p.ring is not None: + # TODO incorporate order + return self._print(p.ring.to_sympy(p)) + except SympifyError: + pass + return self._print(repr(p)) + + def _print_DMF(self, p): + return self._print_DMP(p) + + def _print_Object(self, object): + return self._print(Symbol(object.name)) + + def _print_LambertW(self, expr, exp=None): + arg0 = self._print(expr.args[0]) + exp = r"^{%s}" % (exp,) if exp is not None else "" + if len(expr.args) == 1: + result = r"W%s\left(%s\right)" % (exp, arg0) + else: + arg1 = self._print(expr.args[1]) + result = "W{0}_{{{1}}}\\left({2}\\right)".format(exp, arg1, arg0) + return result + + def _print_Expectation(self, expr): + return r"\operatorname{{E}}\left[{}\right]".format(self._print(expr.args[0])) + + def _print_Variance(self, expr): + return r"\operatorname{{Var}}\left({}\right)".format(self._print(expr.args[0])) + + def _print_Covariance(self, expr): + return r"\operatorname{{Cov}}\left({}\right)".format(", ".join(self._print(arg) for arg in expr.args)) + + def _print_Probability(self, expr): + return r"\operatorname{{P}}\left({}\right)".format(self._print(expr.args[0])) + + def _print_Morphism(self, morphism): + domain = self._print(morphism.domain) + codomain = self._print(morphism.codomain) + return "%s\\rightarrow %s" % (domain, codomain) + + def _print_TransferFunction(self, expr): + num, den = self._print(expr.num), self._print(expr.den) + return r"\frac{%s}{%s}" % (num, den) + + def _print_Series(self, expr): + args = list(expr.args) + parens = lambda x: self.parenthesize(x, precedence_traditional(expr), + False) + return ' '.join(map(parens, args)) + + def _print_MIMOSeries(self, expr): + from sympy.physics.control.lti import MIMOParallel + args = list(expr.args)[::-1] + parens = lambda x: self.parenthesize(x, precedence_traditional(expr), + False) if isinstance(x, MIMOParallel) else self._print(x) + return r"\cdot".join(map(parens, args)) + + def _print_Parallel(self, expr): + return ' + '.join(map(self._print, expr.args)) + + def _print_MIMOParallel(self, expr): + return ' + '.join(map(self._print, expr.args)) + + def _print_Feedback(self, expr): + from sympy.physics.control import TransferFunction, Series + + num, tf = expr.sys1, TransferFunction(1, 1, expr.var) + num_arg_list = list(num.args) if isinstance(num, Series) else [num] + den_arg_list = list(expr.sys2.args) if \ + isinstance(expr.sys2, Series) else [expr.sys2] + den_term_1 = tf + + if isinstance(num, Series) and isinstance(expr.sys2, Series): + den_term_2 = Series(*num_arg_list, *den_arg_list) + elif isinstance(num, Series) and isinstance(expr.sys2, TransferFunction): + if expr.sys2 == tf: + den_term_2 = Series(*num_arg_list) + else: + den_term_2 = tf, Series(*num_arg_list, expr.sys2) + elif isinstance(num, TransferFunction) and isinstance(expr.sys2, Series): + if num == tf: + den_term_2 = Series(*den_arg_list) + else: + den_term_2 = Series(num, *den_arg_list) + else: + if num == tf: + den_term_2 = Series(*den_arg_list) + elif expr.sys2 == tf: + den_term_2 = Series(*num_arg_list) + else: + den_term_2 = Series(*num_arg_list, *den_arg_list) + + numer = self._print(num) + denom_1 = self._print(den_term_1) + denom_2 = self._print(den_term_2) + _sign = "+" if expr.sign == -1 else "-" + + return r"\frac{%s}{%s %s %s}" % (numer, denom_1, _sign, denom_2) + + def _print_MIMOFeedback(self, expr): + from sympy.physics.control import MIMOSeries + inv_mat = self._print(MIMOSeries(expr.sys2, expr.sys1)) + sys1 = self._print(expr.sys1) + _sign = "+" if expr.sign == -1 else "-" + return r"\left(I_{\tau} %s %s\right)^{-1} \cdot %s" % (_sign, inv_mat, sys1) + + def _print_TransferFunctionMatrix(self, expr): + mat = self._print(expr._expr_mat) + return r"%s_\tau" % mat + + def _print_DFT(self, expr): + return r"\text{{{}}}_{{{}}}".format(expr.__class__.__name__, expr.n) + _print_IDFT = _print_DFT + + def _print_NamedMorphism(self, morphism): + pretty_name = self._print(Symbol(morphism.name)) + pretty_morphism = self._print_Morphism(morphism) + return "%s:%s" % (pretty_name, pretty_morphism) + + def _print_IdentityMorphism(self, morphism): + from sympy.categories import NamedMorphism + return self._print_NamedMorphism(NamedMorphism( + morphism.domain, morphism.codomain, "id")) + + def _print_CompositeMorphism(self, morphism): + # All components of the morphism have names and it is thus + # possible to build the name of the composite. + component_names_list = [self._print(Symbol(component.name)) for + component in morphism.components] + component_names_list.reverse() + component_names = "\\circ ".join(component_names_list) + ":" + + pretty_morphism = self._print_Morphism(morphism) + return component_names + pretty_morphism + + def _print_Category(self, morphism): + return r"\mathbf{{{}}}".format(self._print(Symbol(morphism.name))) + + def _print_Diagram(self, diagram): + if not diagram.premises: + # This is an empty diagram. + return self._print(S.EmptySet) + + latex_result = self._print(diagram.premises) + if diagram.conclusions: + latex_result += "\\Longrightarrow %s" % \ + self._print(diagram.conclusions) + + return latex_result + + def _print_DiagramGrid(self, grid): + latex_result = "\\begin{array}{%s}\n" % ("c" * grid.width) + + for i in range(grid.height): + for j in range(grid.width): + if grid[i, j]: + latex_result += latex(grid[i, j]) + latex_result += " " + if j != grid.width - 1: + latex_result += "& " + + if i != grid.height - 1: + latex_result += "\\\\" + latex_result += "\n" + + latex_result += "\\end{array}\n" + return latex_result + + def _print_FreeModule(self, M): + return '{{{}}}^{{{}}}'.format(self._print(M.ring), self._print(M.rank)) + + def _print_FreeModuleElement(self, m): + # Print as row vector for convenience, for now. + return r"\left[ {} \right]".format(",".join( + '{' + self._print(x) + '}' for x in m)) + + def _print_SubModule(self, m): + gens = [[self._print(m.ring.to_sympy(x)) for x in g] for g in m.gens] + curly = lambda o: r"{" + o + r"}" + square = lambda o: r"\left[ " + o + r" \right]" + gens_latex = ",".join(curly(square(",".join(curly(x) for x in g))) for g in gens) + return r"\left\langle {} \right\rangle".format(gens_latex) + + def _print_SubQuotientModule(self, m): + gens_latex = ",".join(["{" + self._print(g) + "}" for g in m.gens]) + return r"\left\langle {} \right\rangle".format(gens_latex) + + def _print_ModuleImplementedIdeal(self, m): + gens = [m.ring.to_sympy(x) for [x] in m._module.gens] + gens_latex = ",".join('{' + self._print(x) + '}' for x in gens) + return r"\left\langle {} \right\rangle".format(gens_latex) + + def _print_Quaternion(self, expr): + # TODO: This expression is potentially confusing, + # shall we print it as `Quaternion( ... )`? + s = [self.parenthesize(i, PRECEDENCE["Mul"], strict=True) + for i in expr.args] + a = [s[0]] + [i+" "+j for i, j in zip(s[1:], "ijk")] + return " + ".join(a) + + def _print_QuotientRing(self, R): + # TODO nicer fractions for few generators... + return r"\frac{{{}}}{{{}}}".format(self._print(R.ring), + self._print(R.base_ideal)) + + def _print_QuotientRingElement(self, x): + x_latex = self._print(x.ring.to_sympy(x)) + return r"{{{}}} + {{{}}}".format(x_latex, + self._print(x.ring.base_ideal)) + + def _print_QuotientModuleElement(self, m): + data = [m.module.ring.to_sympy(x) for x in m.data] + data_latex = r"\left[ {} \right]".format(",".join( + '{' + self._print(x) + '}' for x in data)) + return r"{{{}}} + {{{}}}".format(data_latex, + self._print(m.module.killed_module)) + + def _print_QuotientModule(self, M): + # TODO nicer fractions for few generators... + return r"\frac{{{}}}{{{}}}".format(self._print(M.base), + self._print(M.killed_module)) + + def _print_MatrixHomomorphism(self, h): + return r"{{{}}} : {{{}}} \to {{{}}}".format(self._print(h._sympy_matrix()), + self._print(h.domain), self._print(h.codomain)) + + def _print_Manifold(self, manifold): + name, supers, subs = self._split_super_sub(manifold.name.name) + + name = r'\text{%s}' % name + if supers: + name += "^{%s}" % " ".join(supers) + if subs: + name += "_{%s}" % " ".join(subs) + + return name + + def _print_Patch(self, patch): + return r'\text{%s}_{%s}' % (self._print(patch.name), self._print(patch.manifold)) + + def _print_CoordSystem(self, coordsys): + return r'\text{%s}^{\text{%s}}_{%s}' % ( + self._print(coordsys.name), self._print(coordsys.patch.name), self._print(coordsys.manifold) + ) + + def _print_CovarDerivativeOp(self, cvd): + return r'\mathbb{\nabla}_{%s}' % self._print(cvd._wrt) + + def _print_BaseScalarField(self, field): + string = field._coord_sys.symbols[field._index].name + return r'\mathbf{{{}}}'.format(self._print(Symbol(string))) + + def _print_BaseVectorField(self, field): + string = field._coord_sys.symbols[field._index].name + return r'\partial_{{{}}}'.format(self._print(Symbol(string))) + + def _print_Differential(self, diff): + field = diff._form_field + if hasattr(field, '_coord_sys'): + string = field._coord_sys.symbols[field._index].name + return r'\operatorname{{d}}{}'.format(self._print(Symbol(string))) + else: + string = self._print(field) + return r'\operatorname{{d}}\left({}\right)'.format(string) + + def _print_Tr(self, p): + # TODO: Handle indices + contents = self._print(p.args[0]) + return r'\operatorname{{tr}}\left({}\right)'.format(contents) + + def _print_totient(self, expr, exp=None): + if exp is not None: + return r'\left(\phi\left(%s\right)\right)^{%s}' % \ + (self._print(expr.args[0]), exp) + return r'\phi\left(%s\right)' % self._print(expr.args[0]) + + def _print_reduced_totient(self, expr, exp=None): + if exp is not None: + return r'\left(\lambda\left(%s\right)\right)^{%s}' % \ + (self._print(expr.args[0]), exp) + return r'\lambda\left(%s\right)' % self._print(expr.args[0]) + + def _print_divisor_sigma(self, expr, exp=None): + if len(expr.args) == 2: + tex = r"_%s\left(%s\right)" % tuple(map(self._print, + (expr.args[1], expr.args[0]))) + else: + tex = r"\left(%s\right)" % self._print(expr.args[0]) + if exp is not None: + return r"\sigma^{%s}%s" % (exp, tex) + return r"\sigma%s" % tex + + def _print_udivisor_sigma(self, expr, exp=None): + if len(expr.args) == 2: + tex = r"_%s\left(%s\right)" % tuple(map(self._print, + (expr.args[1], expr.args[0]))) + else: + tex = r"\left(%s\right)" % self._print(expr.args[0]) + if exp is not None: + return r"\sigma^*^{%s}%s" % (exp, tex) + return r"\sigma^*%s" % tex + + def _print_primenu(self, expr, exp=None): + if exp is not None: + return r'\left(\nu\left(%s\right)\right)^{%s}' % \ + (self._print(expr.args[0]), exp) + return r'\nu\left(%s\right)' % self._print(expr.args[0]) + + def _print_primeomega(self, expr, exp=None): + if exp is not None: + return r'\left(\Omega\left(%s\right)\right)^{%s}' % \ + (self._print(expr.args[0]), exp) + return r'\Omega\left(%s\right)' % self._print(expr.args[0]) + + def _print_Str(self, s): + return str(s.name) + + def _print_float(self, expr): + return self._print(Float(expr)) + + def _print_int(self, expr): + return str(expr) + + def _print_mpz(self, expr): + return str(expr) + + def _print_mpq(self, expr): + return str(expr) + + def _print_fmpz(self, expr): + return str(expr) + + def _print_fmpq(self, expr): + return str(expr) + + def _print_Predicate(self, expr): + return r"\operatorname{{Q}}_{{\text{{{}}}}}".format(latex_escape(str(expr.name))) + + def _print_AppliedPredicate(self, expr): + pred = expr.function + args = expr.arguments + pred_latex = self._print(pred) + args_latex = ', '.join([self._print(a) for a in args]) + return '%s(%s)' % (pred_latex, args_latex) + + def emptyPrinter(self, expr): + # default to just printing as monospace, like would normally be shown + s = super().emptyPrinter(expr) + + return r"\mathtt{\text{%s}}" % latex_escape(s) + + +def translate(s: str) -> str: + r''' + Check for a modifier ending the string. If present, convert the + modifier to latex and translate the rest recursively. + + Given a description of a Greek letter or other special character, + return the appropriate latex. + + Let everything else pass as given. + + >>> from sympy.printing.latex import translate + >>> translate('alphahatdotprime') + "{\\dot{\\hat{\\alpha}}}'" + ''' + # Process the rest + tex = tex_greek_dictionary.get(s) + if tex: + return tex + elif s.lower() in greek_letters_set: + return "\\" + s.lower() + elif s in other_symbols: + return "\\" + s + else: + # Process modifiers, if any, and recurse + for key in sorted(modifier_dict.keys(), key=len, reverse=True): + if s.lower().endswith(key) and len(s) > len(key): + return modifier_dict[key](translate(s[:-len(key)])) + return s + + + +@print_function(LatexPrinter) +def latex(expr, **settings): + r"""Convert the given expression to LaTeX string representation. + + Parameters + ========== + full_prec: boolean, optional + If set to True, a floating point number is printed with full precision. + fold_frac_powers : boolean, optional + Emit ``^{p/q}`` instead of ``^{\frac{p}{q}}`` for fractional powers. + fold_func_brackets : boolean, optional + Fold function brackets where applicable. + fold_short_frac : boolean, optional + Emit ``p / q`` instead of ``\frac{p}{q}`` when the denominator is + simple enough (at most two terms and no powers). The default value is + ``True`` for inline mode, ``False`` otherwise. + inv_trig_style : string, optional + How inverse trig functions should be displayed. Can be one of + ``'abbreviated'``, ``'full'``, or ``'power'``. Defaults to + ``'abbreviated'``. + itex : boolean, optional + Specifies if itex-specific syntax is used, including emitting + ``$$...$$``. + ln_notation : boolean, optional + If set to ``True``, ``\ln`` is used instead of default ``\log``. + long_frac_ratio : float or None, optional + The allowed ratio of the width of the numerator to the width of the + denominator before the printer breaks off long fractions. If ``None`` + (the default value), long fractions are not broken up. + mat_delim : string, optional + The delimiter to wrap around matrices. Can be one of ``'['``, ``'('``, + or the empty string ``''``. Defaults to ``'['``. + mat_str : string, optional + Which matrix environment string to emit. ``'smallmatrix'``, + ``'matrix'``, ``'array'``, etc. Defaults to ``'smallmatrix'`` for + inline mode, ``'matrix'`` for matrices of no more than 10 columns, and + ``'array'`` otherwise. + mode: string, optional + Specifies how the generated code will be delimited. ``mode`` can be one + of ``'plain'``, ``'inline'``, ``'equation'`` or ``'equation*'``. If + ``mode`` is set to ``'plain'``, then the resulting code will not be + delimited at all (this is the default). If ``mode`` is set to + ``'inline'`` then inline LaTeX ``$...$`` will be used. If ``mode`` is + set to ``'equation'`` or ``'equation*'``, the resulting code will be + enclosed in the ``equation`` or ``equation*`` environment (remember to + import ``amsmath`` for ``equation*``), unless the ``itex`` option is + set. In the latter case, the ``$$...$$`` syntax is used. + mul_symbol : string or None, optional + The symbol to use for multiplication. Can be one of ``None``, + ``'ldot'``, ``'dot'``, or ``'times'``. + order: string, optional + Any of the supported monomial orderings (currently ``'lex'``, + ``'grlex'``, or ``'grevlex'``), ``'old'``, and ``'none'``. This + parameter does nothing for `~.Mul` objects. Setting order to ``'old'`` + uses the compatibility ordering for ``~.Add`` defined in Printer. For + very large expressions, set the ``order`` keyword to ``'none'`` if + speed is a concern. + symbol_names : dictionary of strings mapped to symbols, optional + Dictionary of symbols and the custom strings they should be emitted as. + root_notation : boolean, optional + If set to ``False``, exponents of the form 1/n are printed in fractonal + form. Default is ``True``, to print exponent in root form. + mat_symbol_style : string, optional + Can be either ``'plain'`` (default) or ``'bold'``. If set to + ``'bold'``, a `~.MatrixSymbol` A will be printed as ``\mathbf{A}``, + otherwise as ``A``. + imaginary_unit : string, optional + String to use for the imaginary unit. Defined options are ``'i'`` + (default) and ``'j'``. Adding ``r`` or ``t`` in front gives ``\mathrm`` + or ``\text``, so ``'ri'`` leads to ``\mathrm{i}`` which gives + `\mathrm{i}`. + gothic_re_im : boolean, optional + If set to ``True``, `\Re` and `\Im` is used for ``re`` and ``im``, respectively. + The default is ``False`` leading to `\operatorname{re}` and `\operatorname{im}`. + decimal_separator : string, optional + Specifies what separator to use to separate the whole and fractional parts of a + floating point number as in `2.5` for the default, ``period`` or `2{,}5` + when ``comma`` is specified. Lists, sets, and tuple are printed with semicolon + separating the elements when ``comma`` is chosen. For example, [1; 2; 3] when + ``comma`` is chosen and [1,2,3] for when ``period`` is chosen. + parenthesize_super : boolean, optional + If set to ``False``, superscripted expressions will not be parenthesized when + powered. Default is ``True``, which parenthesizes the expression when powered. + min: Integer or None, optional + Sets the lower bound for the exponent to print floating point numbers in + fixed-point format. + max: Integer or None, optional + Sets the upper bound for the exponent to print floating point numbers in + fixed-point format. + diff_operator: string, optional + String to use for differential operator. Default is ``'d'``, to print in italic + form. ``'rd'``, ``'td'`` are shortcuts for ``\mathrm{d}`` and ``\text{d}``. + adjoint_style: string, optional + String to use for the adjoint symbol. Defined options are ``'dagger'`` + (default),``'star'``, and ``'hermitian'``. + + Notes + ===== + + Not using a print statement for printing, results in double backslashes for + latex commands since that's the way Python escapes backslashes in strings. + + >>> from sympy import latex, Rational + >>> from sympy.abc import tau + >>> latex((2*tau)**Rational(7,2)) + '8 \\sqrt{2} \\tau^{\\frac{7}{2}}' + >>> print(latex((2*tau)**Rational(7,2))) + 8 \sqrt{2} \tau^{\frac{7}{2}} + + Examples + ======== + + >>> from sympy import latex, pi, sin, asin, Integral, Matrix, Rational, log + >>> from sympy.abc import x, y, mu, r, tau + + Basic usage: + + >>> print(latex((2*tau)**Rational(7,2))) + 8 \sqrt{2} \tau^{\frac{7}{2}} + + ``mode`` and ``itex`` options: + + >>> print(latex((2*mu)**Rational(7,2), mode='plain')) + 8 \sqrt{2} \mu^{\frac{7}{2}} + >>> print(latex((2*tau)**Rational(7,2), mode='inline')) + $8 \sqrt{2} \tau^{7 / 2}$ + >>> print(latex((2*mu)**Rational(7,2), mode='equation*')) + \begin{equation*}8 \sqrt{2} \mu^{\frac{7}{2}}\end{equation*} + >>> print(latex((2*mu)**Rational(7,2), mode='equation')) + \begin{equation}8 \sqrt{2} \mu^{\frac{7}{2}}\end{equation} + >>> print(latex((2*mu)**Rational(7,2), mode='equation', itex=True)) + $$8 \sqrt{2} \mu^{\frac{7}{2}}$$ + >>> print(latex((2*mu)**Rational(7,2), mode='plain')) + 8 \sqrt{2} \mu^{\frac{7}{2}} + >>> print(latex((2*tau)**Rational(7,2), mode='inline')) + $8 \sqrt{2} \tau^{7 / 2}$ + >>> print(latex((2*mu)**Rational(7,2), mode='equation*')) + \begin{equation*}8 \sqrt{2} \mu^{\frac{7}{2}}\end{equation*} + >>> print(latex((2*mu)**Rational(7,2), mode='equation')) + \begin{equation}8 \sqrt{2} \mu^{\frac{7}{2}}\end{equation} + >>> print(latex((2*mu)**Rational(7,2), mode='equation', itex=True)) + $$8 \sqrt{2} \mu^{\frac{7}{2}}$$ + + Fraction options: + + >>> print(latex((2*tau)**Rational(7,2), fold_frac_powers=True)) + 8 \sqrt{2} \tau^{7/2} + >>> print(latex((2*tau)**sin(Rational(7,2)))) + \left(2 \tau\right)^{\sin{\left(\frac{7}{2} \right)}} + >>> print(latex((2*tau)**sin(Rational(7,2)), fold_func_brackets=True)) + \left(2 \tau\right)^{\sin {\frac{7}{2}}} + >>> print(latex(3*x**2/y)) + \frac{3 x^{2}}{y} + >>> print(latex(3*x**2/y, fold_short_frac=True)) + 3 x^{2} / y + >>> print(latex(Integral(r, r)/2/pi, long_frac_ratio=2)) + \frac{\int r\, dr}{2 \pi} + >>> print(latex(Integral(r, r)/2/pi, long_frac_ratio=0)) + \frac{1}{2 \pi} \int r\, dr + + Multiplication options: + + >>> print(latex((2*tau)**sin(Rational(7,2)), mul_symbol="times")) + \left(2 \times \tau\right)^{\sin{\left(\frac{7}{2} \right)}} + + Trig options: + + >>> print(latex(asin(Rational(7,2)))) + \operatorname{asin}{\left(\frac{7}{2} \right)} + >>> print(latex(asin(Rational(7,2)), inv_trig_style="full")) + \arcsin{\left(\frac{7}{2} \right)} + >>> print(latex(asin(Rational(7,2)), inv_trig_style="power")) + \sin^{-1}{\left(\frac{7}{2} \right)} + + Matrix options: + + >>> print(latex(Matrix(2, 1, [x, y]))) + \left[\begin{matrix}x\\y\end{matrix}\right] + >>> print(latex(Matrix(2, 1, [x, y]), mat_str = "array")) + \left[\begin{array}{c}x\\y\end{array}\right] + >>> print(latex(Matrix(2, 1, [x, y]), mat_delim="(")) + \left(\begin{matrix}x\\y\end{matrix}\right) + + Custom printing of symbols: + + >>> print(latex(x**2, symbol_names={x: 'x_i'})) + x_i^{2} + + Logarithms: + + >>> print(latex(log(10))) + \log{\left(10 \right)} + >>> print(latex(log(10), ln_notation=True)) + \ln{\left(10 \right)} + + ``latex()`` also supports the builtin container types :class:`list`, + :class:`tuple`, and :class:`dict`: + + >>> print(latex([2/x, y], mode='inline')) + $\left[ 2 / x, \ y\right]$ + + Unsupported types are rendered as monospaced plaintext: + + >>> print(latex(int)) + \mathtt{\text{}} + >>> print(latex("plain % text")) + \mathtt{\text{plain \% text}} + + See :ref:`printer_method_example` for an example of how to override + this behavior for your own types by implementing ``_latex``. + + .. versionchanged:: 1.7.0 + Unsupported types no longer have their ``str`` representation treated as valid latex. + + """ + return LatexPrinter(settings).doprint(expr) + + +def print_latex(expr, **settings): + """Prints LaTeX representation of the given expression. Takes the same + settings as ``latex()``.""" + + print(latex(expr, **settings)) + + +def multiline_latex(lhs, rhs, terms_per_line=1, environment="align*", use_dots=False, **settings): + r""" + This function generates a LaTeX equation with a multiline right-hand side + in an ``align*``, ``eqnarray`` or ``IEEEeqnarray`` environment. + + Parameters + ========== + + lhs : Expr + Left-hand side of equation + + rhs : Expr + Right-hand side of equation + + terms_per_line : integer, optional + Number of terms per line to print. Default is 1. + + environment : "string", optional + Which LaTeX wnvironment to use for the output. Options are "align*" + (default), "eqnarray", and "IEEEeqnarray". + + use_dots : boolean, optional + If ``True``, ``\\dots`` is added to the end of each line. Default is ``False``. + + Examples + ======== + + >>> from sympy import multiline_latex, symbols, sin, cos, exp, log, I + >>> x, y, alpha = symbols('x y alpha') + >>> expr = sin(alpha*y) + exp(I*alpha) - cos(log(y)) + >>> print(multiline_latex(x, expr)) + \begin{align*} + x = & e^{i \alpha} \\ + & + \sin{\left(\alpha y \right)} \\ + & - \cos{\left(\log{\left(y \right)} \right)} + \end{align*} + + Using at most two terms per line: + >>> print(multiline_latex(x, expr, 2)) + \begin{align*} + x = & e^{i \alpha} + \sin{\left(\alpha y \right)} \\ + & - \cos{\left(\log{\left(y \right)} \right)} + \end{align*} + + Using ``eqnarray`` and dots: + >>> print(multiline_latex(x, expr, terms_per_line=2, environment="eqnarray", use_dots=True)) + \begin{eqnarray} + x & = & e^{i \alpha} + \sin{\left(\alpha y \right)} \dots\nonumber\\ + & & - \cos{\left(\log{\left(y \right)} \right)} + \end{eqnarray} + + Using ``IEEEeqnarray``: + >>> print(multiline_latex(x, expr, environment="IEEEeqnarray")) + \begin{IEEEeqnarray}{rCl} + x & = & e^{i \alpha} \nonumber\\ + & & + \sin{\left(\alpha y \right)} \nonumber\\ + & & - \cos{\left(\log{\left(y \right)} \right)} + \end{IEEEeqnarray} + + Notes + ===== + + All optional parameters from ``latex`` can also be used. + + """ + + # Based on code from https://github.com/sympy/sympy/issues/3001 + l = LatexPrinter(**settings) + if environment == "eqnarray": + result = r'\begin{eqnarray}' + '\n' + first_term = '& = &' + nonumber = r'\nonumber' + end_term = '\n\\end{eqnarray}' + doubleet = True + elif environment == "IEEEeqnarray": + result = r'\begin{IEEEeqnarray}{rCl}' + '\n' + first_term = '& = &' + nonumber = r'\nonumber' + end_term = '\n\\end{IEEEeqnarray}' + doubleet = True + elif environment == "align*": + result = r'\begin{align*}' + '\n' + first_term = '= &' + nonumber = '' + end_term = '\n\\end{align*}' + doubleet = False + else: + raise ValueError("Unknown environment: {}".format(environment)) + dots = '' + if use_dots: + dots=r'\dots' + terms = rhs.as_ordered_terms() + n_terms = len(terms) + term_count = 1 + for i in range(n_terms): + term = terms[i] + term_start = '' + term_end = '' + sign = '+' + if term_count > terms_per_line: + if doubleet: + term_start = '& & ' + else: + term_start = '& ' + term_count = 1 + if term_count == terms_per_line: + # End of line + if i < n_terms-1: + # There are terms remaining + term_end = dots + nonumber + r'\\' + '\n' + else: + term_end = '' + + if term.as_ordered_factors()[0] == -1: + term = -1*term + sign = r'-' + if i == 0: # beginning + if sign == '+': + sign = '' + result += r'{:s} {:s}{:s} {:s} {:s}'.format(l.doprint(lhs), + first_term, sign, l.doprint(term), term_end) + else: + result += r'{:s}{:s} {:s} {:s}'.format(term_start, sign, + l.doprint(term), term_end) + term_count += 1 + result += end_term + return result diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/llvmjitcode.py b/.venv/lib/python3.12/site-packages/sympy/printing/llvmjitcode.py new file mode 100644 index 0000000000000000000000000000000000000000..0e657f3b854be62fb4b6b4be96f82579b718f6ca --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/llvmjitcode.py @@ -0,0 +1,490 @@ +''' +Use llvmlite to create executable functions from SymPy expressions + +This module requires llvmlite (https://github.com/numba/llvmlite). +''' + +import ctypes + +from sympy.external import import_module +from sympy.printing.printer import Printer +from sympy.core.singleton import S +from sympy.tensor.indexed import IndexedBase +from sympy.utilities.decorator import doctest_depends_on + +llvmlite = import_module('llvmlite') +if llvmlite: + ll = import_module('llvmlite.ir').ir + llvm = import_module('llvmlite.binding').binding + llvm.initialize() + llvm.initialize_native_target() + llvm.initialize_native_asmprinter() + + +__doctest_requires__ = {('llvm_callable'): ['llvmlite']} + + +class LLVMJitPrinter(Printer): + '''Convert expressions to LLVM IR''' + def __init__(self, module, builder, fn, *args, **kwargs): + self.func_arg_map = kwargs.pop("func_arg_map", {}) + if not llvmlite: + raise ImportError("llvmlite is required for LLVMJITPrinter") + super().__init__(*args, **kwargs) + self.fp_type = ll.DoubleType() + self.module = module + self.builder = builder + self.fn = fn + self.ext_fn = {} # keep track of wrappers to external functions + self.tmp_var = {} + + def _add_tmp_var(self, name, value): + self.tmp_var[name] = value + + def _print_Number(self, n): + return ll.Constant(self.fp_type, float(n)) + + def _print_Integer(self, expr): + return ll.Constant(self.fp_type, float(expr.p)) + + def _print_Symbol(self, s): + val = self.tmp_var.get(s) + if not val: + # look up parameter with name s + val = self.func_arg_map.get(s) + if not val: + raise LookupError("Symbol not found: %s" % s) + return val + + def _print_Pow(self, expr): + base0 = self._print(expr.base) + if expr.exp == S.NegativeOne: + return self.builder.fdiv(ll.Constant(self.fp_type, 1.0), base0) + if expr.exp == S.Half: + fn = self.ext_fn.get("sqrt") + if not fn: + fn_type = ll.FunctionType(self.fp_type, [self.fp_type]) + fn = ll.Function(self.module, fn_type, "sqrt") + self.ext_fn["sqrt"] = fn + return self.builder.call(fn, [base0], "sqrt") + if expr.exp == 2: + return self.builder.fmul(base0, base0) + + exp0 = self._print(expr.exp) + fn = self.ext_fn.get("pow") + if not fn: + fn_type = ll.FunctionType(self.fp_type, [self.fp_type, self.fp_type]) + fn = ll.Function(self.module, fn_type, "pow") + self.ext_fn["pow"] = fn + return self.builder.call(fn, [base0, exp0], "pow") + + def _print_Mul(self, expr): + nodes = [self._print(a) for a in expr.args] + e = nodes[0] + for node in nodes[1:]: + e = self.builder.fmul(e, node) + return e + + def _print_Add(self, expr): + nodes = [self._print(a) for a in expr.args] + e = nodes[0] + for node in nodes[1:]: + e = self.builder.fadd(e, node) + return e + + # TODO - assumes all called functions take one double precision argument. + # Should have a list of math library functions to validate this. + def _print_Function(self, expr): + name = expr.func.__name__ + e0 = self._print(expr.args[0]) + fn = self.ext_fn.get(name) + if not fn: + fn_type = ll.FunctionType(self.fp_type, [self.fp_type]) + fn = ll.Function(self.module, fn_type, name) + self.ext_fn[name] = fn + return self.builder.call(fn, [e0], name) + + def emptyPrinter(self, expr): + raise TypeError("Unsupported type for LLVM JIT conversion: %s" + % type(expr)) + + +# Used when parameters are passed by array. Often used in callbacks to +# handle a variable number of parameters. +class LLVMJitCallbackPrinter(LLVMJitPrinter): + def __init__(self, *args, **kwargs): + super().__init__(*args, **kwargs) + + def _print_Indexed(self, expr): + array, idx = self.func_arg_map[expr.base] + offset = int(expr.indices[0].evalf()) + array_ptr = self.builder.gep(array, [ll.Constant(ll.IntType(32), offset)]) + fp_array_ptr = self.builder.bitcast(array_ptr, ll.PointerType(self.fp_type)) + value = self.builder.load(fp_array_ptr) + return value + + def _print_Symbol(self, s): + val = self.tmp_var.get(s) + if val: + return val + + array, idx = self.func_arg_map.get(s, [None, 0]) + if not array: + raise LookupError("Symbol not found: %s" % s) + array_ptr = self.builder.gep(array, [ll.Constant(ll.IntType(32), idx)]) + fp_array_ptr = self.builder.bitcast(array_ptr, + ll.PointerType(self.fp_type)) + value = self.builder.load(fp_array_ptr) + return value + + +# ensure lifetime of the execution engine persists (else call to compiled +# function will seg fault) +exe_engines = [] + +# ensure names for generated functions are unique +link_names = set() +current_link_suffix = 0 + + +class LLVMJitCode: + def __init__(self, signature): + self.signature = signature + self.fp_type = ll.DoubleType() + self.module = ll.Module('mod1') + self.fn = None + self.llvm_arg_types = [] + self.llvm_ret_type = self.fp_type + self.param_dict = {} # map symbol name to LLVM function argument + self.link_name = '' + + def _from_ctype(self, ctype): + if ctype == ctypes.c_int: + return ll.IntType(32) + if ctype == ctypes.c_double: + return self.fp_type + if ctype == ctypes.POINTER(ctypes.c_double): + return ll.PointerType(self.fp_type) + if ctype == ctypes.c_void_p: + return ll.PointerType(ll.IntType(32)) + if ctype == ctypes.py_object: + return ll.PointerType(ll.IntType(32)) + + print("Unhandled ctype = %s" % str(ctype)) + + def _create_args(self, func_args): + """Create types for function arguments""" + self.llvm_ret_type = self._from_ctype(self.signature.ret_type) + self.llvm_arg_types = \ + [self._from_ctype(a) for a in self.signature.arg_ctypes] + + def _create_function_base(self): + """Create function with name and type signature""" + global current_link_suffix + default_link_name = 'jit_func' + current_link_suffix += 1 + self.link_name = default_link_name + str(current_link_suffix) + link_names.add(self.link_name) + + fn_type = ll.FunctionType(self.llvm_ret_type, self.llvm_arg_types) + self.fn = ll.Function(self.module, fn_type, name=self.link_name) + + def _create_param_dict(self, func_args): + """Mapping of symbolic values to function arguments""" + for i, a in enumerate(func_args): + self.fn.args[i].name = str(a) + self.param_dict[a] = self.fn.args[i] + + def _create_function(self, expr): + """Create function body and return LLVM IR""" + bb_entry = self.fn.append_basic_block('entry') + builder = ll.IRBuilder(bb_entry) + + lj = LLVMJitPrinter(self.module, builder, self.fn, + func_arg_map=self.param_dict) + + ret = self._convert_expr(lj, expr) + lj.builder.ret(self._wrap_return(lj, ret)) + + strmod = str(self.module) + return strmod + + def _wrap_return(self, lj, vals): + # Return a single double if there is one return value, + # else return a tuple of doubles. + + # Don't wrap return value in this case + if self.signature.ret_type == ctypes.c_double: + return vals[0] + + # Use this instead of a real PyObject* + void_ptr = ll.PointerType(ll.IntType(32)) + + # Create a wrapped double: PyObject* PyFloat_FromDouble(double v) + wrap_type = ll.FunctionType(void_ptr, [self.fp_type]) + wrap_fn = ll.Function(lj.module, wrap_type, "PyFloat_FromDouble") + + wrapped_vals = [lj.builder.call(wrap_fn, [v]) for v in vals] + if len(vals) == 1: + final_val = wrapped_vals[0] + else: + # Create a tuple: PyObject* PyTuple_Pack(Py_ssize_t n, ...) + + # This should be Py_ssize_t + tuple_arg_types = [ll.IntType(32)] + + tuple_arg_types.extend([void_ptr]*len(vals)) + tuple_type = ll.FunctionType(void_ptr, tuple_arg_types) + tuple_fn = ll.Function(lj.module, tuple_type, "PyTuple_Pack") + + tuple_args = [ll.Constant(ll.IntType(32), len(wrapped_vals))] + tuple_args.extend(wrapped_vals) + + final_val = lj.builder.call(tuple_fn, tuple_args) + + return final_val + + def _convert_expr(self, lj, expr): + try: + # Match CSE return data structure. + if len(expr) == 2: + tmp_exprs = expr[0] + final_exprs = expr[1] + if len(final_exprs) != 1 and self.signature.ret_type == ctypes.c_double: + raise NotImplementedError("Return of multiple expressions not supported for this callback") + for name, e in tmp_exprs: + val = lj._print(e) + lj._add_tmp_var(name, val) + except TypeError: + final_exprs = [expr] + + vals = [lj._print(e) for e in final_exprs] + + return vals + + def _compile_function(self, strmod): + llmod = llvm.parse_assembly(strmod) + + pmb = llvm.create_pass_manager_builder() + pmb.opt_level = 2 + pass_manager = llvm.create_module_pass_manager() + pmb.populate(pass_manager) + + pass_manager.run(llmod) + + target_machine = \ + llvm.Target.from_default_triple().create_target_machine() + exe_eng = llvm.create_mcjit_compiler(llmod, target_machine) + exe_eng.finalize_object() + exe_engines.append(exe_eng) + + if False: + print("Assembly") + print(target_machine.emit_assembly(llmod)) + + fptr = exe_eng.get_function_address(self.link_name) + + return fptr + + +class LLVMJitCodeCallback(LLVMJitCode): + def __init__(self, signature): + super().__init__(signature) + + def _create_param_dict(self, func_args): + for i, a in enumerate(func_args): + if isinstance(a, IndexedBase): + self.param_dict[a] = (self.fn.args[i], i) + self.fn.args[i].name = str(a) + else: + self.param_dict[a] = (self.fn.args[self.signature.input_arg], + i) + + def _create_function(self, expr): + """Create function body and return LLVM IR""" + bb_entry = self.fn.append_basic_block('entry') + builder = ll.IRBuilder(bb_entry) + + lj = LLVMJitCallbackPrinter(self.module, builder, self.fn, + func_arg_map=self.param_dict) + + ret = self._convert_expr(lj, expr) + + if self.signature.ret_arg: + output_fp_ptr = builder.bitcast(self.fn.args[self.signature.ret_arg], + ll.PointerType(self.fp_type)) + for i, val in enumerate(ret): + index = ll.Constant(ll.IntType(32), i) + output_array_ptr = builder.gep(output_fp_ptr, [index]) + builder.store(val, output_array_ptr) + builder.ret(ll.Constant(ll.IntType(32), 0)) # return success + else: + lj.builder.ret(self._wrap_return(lj, ret)) + + strmod = str(self.module) + return strmod + + +class CodeSignature: + def __init__(self, ret_type): + self.ret_type = ret_type + self.arg_ctypes = [] + + # Input argument array element index + self.input_arg = 0 + + # For the case output value is referenced through a parameter rather + # than the return value + self.ret_arg = None + + +def _llvm_jit_code(args, expr, signature, callback_type): + """Create a native code function from a SymPy expression""" + if callback_type is None: + jit = LLVMJitCode(signature) + else: + jit = LLVMJitCodeCallback(signature) + + jit._create_args(args) + jit._create_function_base() + jit._create_param_dict(args) + strmod = jit._create_function(expr) + if False: + print("LLVM IR") + print(strmod) + fptr = jit._compile_function(strmod) + return fptr + + +@doctest_depends_on(modules=('llvmlite', 'scipy')) +def llvm_callable(args, expr, callback_type=None): + '''Compile function from a SymPy expression + + Expressions are evaluated using double precision arithmetic. + Some single argument math functions (exp, sin, cos, etc.) are supported + in expressions. + + Parameters + ========== + + args : List of Symbol + Arguments to the generated function. Usually the free symbols in + the expression. Currently each one is assumed to convert to + a double precision scalar. + expr : Expr, or (Replacements, Expr) as returned from 'cse' + Expression to compile. + callback_type : string + Create function with signature appropriate to use as a callback. + Currently supported: + 'scipy.integrate' + 'scipy.integrate.test' + 'cubature' + + Returns + ======= + + Compiled function that can evaluate the expression. + + Examples + ======== + + >>> import sympy.printing.llvmjitcode as jit + >>> from sympy.abc import a + >>> e = a*a + a + 1 + >>> e1 = jit.llvm_callable([a], e) + >>> e.subs(a, 1.1) # Evaluate via substitution + 3.31000000000000 + >>> e1(1.1) # Evaluate using JIT-compiled code + 3.3100000000000005 + + + Callbacks for integration functions can be JIT compiled. + + >>> import sympy.printing.llvmjitcode as jit + >>> from sympy.abc import a + >>> from sympy import integrate + >>> from scipy.integrate import quad + >>> e = a*a + >>> e1 = jit.llvm_callable([a], e, callback_type='scipy.integrate') + >>> integrate(e, (a, 0.0, 2.0)) + 2.66666666666667 + >>> quad(e1, 0.0, 2.0)[0] + 2.66666666666667 + + The 'cubature' callback is for the Python wrapper around the + cubature package ( https://github.com/saullocastro/cubature ) + and ( http://ab-initio.mit.edu/wiki/index.php/Cubature ) + + There are two signatures for the SciPy integration callbacks. + The first ('scipy.integrate') is the function to be passed to the + integration routine, and will pass the signature checks. + The second ('scipy.integrate.test') is only useful for directly calling + the function using ctypes variables. It will not pass the signature checks + for scipy.integrate. + + The return value from the cse module can also be compiled. This + can improve the performance of the compiled function. If multiple + expressions are given to cse, the compiled function returns a tuple. + The 'cubature' callback handles multiple expressions (set `fdim` + to match in the integration call.) + + >>> import sympy.printing.llvmjitcode as jit + >>> from sympy import cse + >>> from sympy.abc import x,y + >>> e1 = x*x + y*y + >>> e2 = 4*(x*x + y*y) + 8.0 + >>> after_cse = cse([e1,e2]) + >>> after_cse + ([(x0, x**2), (x1, y**2)], [x0 + x1, 4*x0 + 4*x1 + 8.0]) + >>> j1 = jit.llvm_callable([x,y], after_cse) + >>> j1(1.0, 2.0) + (5.0, 28.0) + ''' + + if not llvmlite: + raise ImportError("llvmlite is required for llvmjitcode") + + signature = CodeSignature(ctypes.py_object) + + arg_ctypes = [] + if callback_type is None: + for _ in args: + arg_ctype = ctypes.c_double + arg_ctypes.append(arg_ctype) + elif callback_type in ('scipy.integrate', 'scipy.integrate.test'): + signature.ret_type = ctypes.c_double + arg_ctypes = [ctypes.c_int, ctypes.POINTER(ctypes.c_double)] + arg_ctypes_formal = [ctypes.c_int, ctypes.c_double] + signature.input_arg = 1 + elif callback_type == 'cubature': + arg_ctypes = [ctypes.c_int, + ctypes.POINTER(ctypes.c_double), + ctypes.c_void_p, + ctypes.c_int, + ctypes.POINTER(ctypes.c_double) + ] + signature.ret_type = ctypes.c_int + signature.input_arg = 1 + signature.ret_arg = 4 + else: + raise ValueError("Unknown callback type: %s" % callback_type) + + signature.arg_ctypes = arg_ctypes + + fptr = _llvm_jit_code(args, expr, signature, callback_type) + + if callback_type and callback_type == 'scipy.integrate': + arg_ctypes = arg_ctypes_formal + + # PYFUNCTYPE holds the GIL which is needed to prevent a segfault when + # calling PyFloat_FromDouble on Python 3.10. Probably it is better to use + # ctypes.c_double when returning a float rather than using ctypes.py_object + # and returning a PyFloat from inside the jitted function (i.e. let ctypes + # handle the conversion from double to PyFloat). + if signature.ret_type == ctypes.py_object: + FUNCTYPE = ctypes.PYFUNCTYPE + else: + FUNCTYPE = ctypes.CFUNCTYPE + + cfunc = FUNCTYPE(signature.ret_type, *arg_ctypes)(fptr) + return cfunc diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/maple.py b/.venv/lib/python3.12/site-packages/sympy/printing/maple.py new file mode 100644 index 0000000000000000000000000000000000000000..2c937cd262ab7f3ee5f32b3f4b5eb5633bc6bb3c --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/maple.py @@ -0,0 +1,311 @@ +""" +Maple code printer + +The MapleCodePrinter converts single SymPy expressions into single +Maple expressions, using the functions defined in the Maple objects where possible. + + +FIXME: This module is still under actively developed. Some functions may be not completed. +""" + +from sympy.core import S +from sympy.core.numbers import Integer, IntegerConstant, equal_valued +from sympy.printing.codeprinter import CodePrinter +from sympy.printing.precedence import precedence, PRECEDENCE + +import sympy + +_known_func_same_name = ( + 'sin', 'cos', 'tan', 'sec', 'csc', 'cot', 'sinh', 'cosh', 'tanh', 'sech', + 'csch', 'coth', 'exp', 'floor', 'factorial', 'bernoulli', 'euler', + 'fibonacci', 'gcd', 'lcm', 'conjugate', 'Ci', 'Chi', 'Ei', 'Li', 'Si', 'Shi', + 'erf', 'erfc', 'harmonic', 'LambertW', + 'sqrt', # For automatic rewrites +) + +known_functions = { + # SymPy -> Maple + 'Abs': 'abs', + 'log': 'ln', + 'asin': 'arcsin', + 'acos': 'arccos', + 'atan': 'arctan', + 'asec': 'arcsec', + 'acsc': 'arccsc', + 'acot': 'arccot', + 'asinh': 'arcsinh', + 'acosh': 'arccosh', + 'atanh': 'arctanh', + 'asech': 'arcsech', + 'acsch': 'arccsch', + 'acoth': 'arccoth', + 'ceiling': 'ceil', + 'Max' : 'max', + 'Min' : 'min', + + 'factorial2': 'doublefactorial', + 'RisingFactorial': 'pochhammer', + 'besseli': 'BesselI', + 'besselj': 'BesselJ', + 'besselk': 'BesselK', + 'bessely': 'BesselY', + 'hankelh1': 'HankelH1', + 'hankelh2': 'HankelH2', + 'airyai': 'AiryAi', + 'airybi': 'AiryBi', + 'appellf1': 'AppellF1', + 'fresnelc': 'FresnelC', + 'fresnels': 'FresnelS', + 'lerchphi' : 'LerchPhi', +} + +for _func in _known_func_same_name: + known_functions[_func] = _func + +number_symbols = { + # SymPy -> Maple + S.Pi: 'Pi', + S.Exp1: 'exp(1)', + S.Catalan: 'Catalan', + S.EulerGamma: 'gamma', + S.GoldenRatio: '(1/2 + (1/2)*sqrt(5))' +} + +spec_relational_ops = { + # SymPy -> Maple + '==': '=', + '!=': '<>' +} + +not_supported_symbol = [ + S.ComplexInfinity +] + +class MapleCodePrinter(CodePrinter): + """ + Printer which converts a SymPy expression into a maple code. + """ + printmethod = "_maple" + language = "maple" + + _operators = { + 'and': 'and', + 'or': 'or', + 'not': 'not ', + } + + _default_settings = dict(CodePrinter._default_settings, **{ + 'inline': True, + 'allow_unknown_functions': True, + }) + + def __init__(self, settings=None): + if settings is None: + settings = {} + super().__init__(settings) + self.known_functions = dict(known_functions) + userfuncs = settings.get('user_functions', {}) + self.known_functions.update(userfuncs) + + def _get_statement(self, codestring): + return "%s;" % codestring + + def _get_comment(self, text): + return "# {}".format(text) + + def _declare_number_const(self, name, value): + return "{} := {};".format(name, + value.evalf(self._settings['precision'])) + + def _format_code(self, lines): + return lines + + def _print_tuple(self, expr): + return self._print(list(expr)) + + def _print_Tuple(self, expr): + return self._print(list(expr)) + + def _print_Assignment(self, expr): + lhs = self._print(expr.lhs) + rhs = self._print(expr.rhs) + return "{lhs} := {rhs}".format(lhs=lhs, rhs=rhs) + + def _print_Pow(self, expr, **kwargs): + PREC = precedence(expr) + if equal_valued(expr.exp, -1): + return '1/%s' % (self.parenthesize(expr.base, PREC)) + elif equal_valued(expr.exp, 0.5): + return 'sqrt(%s)' % self._print(expr.base) + elif equal_valued(expr.exp, -0.5): + return '1/sqrt(%s)' % self._print(expr.base) + else: + return '{base}^{exp}'.format( + base=self.parenthesize(expr.base, PREC), + exp=self.parenthesize(expr.exp, PREC)) + + def _print_Piecewise(self, expr): + if (expr.args[-1].cond is not True) and (expr.args[-1].cond != S.BooleanTrue): + # We need the last conditional to be a True, otherwise the resulting + # function may not return a result. + raise ValueError("All Piecewise expressions must contain an " + "(expr, True) statement to be used as a default " + "condition. Without one, the generated " + "expression may not evaluate to anything under " + "some condition.") + _coup_list = [ + ("{c}, {e}".format(c=self._print(c), + e=self._print(e)) if c is not True and c is not S.BooleanTrue else "{e}".format( + e=self._print(e))) + for e, c in expr.args] + _inbrace = ', '.join(_coup_list) + return 'piecewise({_inbrace})'.format(_inbrace=_inbrace) + + def _print_Rational(self, expr): + p, q = int(expr.p), int(expr.q) + return "{p}/{q}".format(p=str(p), q=str(q)) + + def _print_Relational(self, expr): + PREC=precedence(expr) + lhs_code = self.parenthesize(expr.lhs, PREC) + rhs_code = self.parenthesize(expr.rhs, PREC) + op = expr.rel_op + if op in spec_relational_ops: + op = spec_relational_ops[op] + return "{lhs} {rel_op} {rhs}".format(lhs=lhs_code, rel_op=op, rhs=rhs_code) + + def _print_NumberSymbol(self, expr): + return number_symbols[expr] + + def _print_NegativeInfinity(self, expr): + return '-infinity' + + def _print_Infinity(self, expr): + return 'infinity' + + def _print_BooleanTrue(self, expr): + return "true" + + def _print_BooleanFalse(self, expr): + return "false" + + def _print_bool(self, expr): + return 'true' if expr else 'false' + + def _print_NaN(self, expr): + return 'undefined' + + def _get_matrix(self, expr, sparse=False): + if S.Zero in expr.shape: + _strM = 'Matrix([], storage = {storage})'.format( + storage='sparse' if sparse else 'rectangular') + else: + _strM = 'Matrix({list}, storage = {storage})'.format( + list=self._print(expr.tolist()), + storage='sparse' if sparse else 'rectangular') + return _strM + + def _print_MatrixElement(self, expr): + return "{parent}[{i_maple}, {j_maple}]".format( + parent=self.parenthesize(expr.parent, PRECEDENCE["Atom"], strict=True), + i_maple=self._print(expr.i + 1), + j_maple=self._print(expr.j + 1)) + + def _print_MatrixBase(self, expr): + return self._get_matrix(expr, sparse=False) + + def _print_SparseRepMatrix(self, expr): + return self._get_matrix(expr, sparse=True) + + def _print_Identity(self, expr): + if isinstance(expr.rows, (Integer, IntegerConstant)): + return self._print(sympy.SparseMatrix(expr)) + else: + return "Matrix({var_size}, shape = identity)".format(var_size=self._print(expr.rows)) + + def _print_MatMul(self, expr): + PREC=precedence(expr) + _fact_list = list(expr.args) + _const = None + if not isinstance(_fact_list[0], (sympy.MatrixBase, sympy.MatrixExpr, + sympy.MatrixSlice, sympy.MatrixSymbol)): + _const, _fact_list = _fact_list[0], _fact_list[1:] + + if _const is None or _const == 1: + return '.'.join(self.parenthesize(_m, PREC) for _m in _fact_list) + else: + return '{c}*{m}'.format(c=_const, m='.'.join(self.parenthesize(_m, PREC) for _m in _fact_list)) + + def _print_MatPow(self, expr): + # This function requires LinearAlgebra Function in Maple + return 'MatrixPower({A}, {n})'.format(A=self._print(expr.base), n=self._print(expr.exp)) + + def _print_HadamardProduct(self, expr): + PREC = precedence(expr) + _fact_list = list(expr.args) + return '*'.join(self.parenthesize(_m, PREC) for _m in _fact_list) + + def _print_Derivative(self, expr): + _f, (_var, _order) = expr.args + + if _order != 1: + _second_arg = '{var}${order}'.format(var=self._print(_var), + order=self._print(_order)) + else: + _second_arg = '{var}'.format(var=self._print(_var)) + return 'diff({func_expr}, {sec_arg})'.format(func_expr=self._print(_f), sec_arg=_second_arg) + + +def maple_code(expr, assign_to=None, **settings): + r"""Converts ``expr`` to a string of Maple code. + + Parameters + ========== + + expr : Expr + A SymPy expression to be converted. + assign_to : optional + When given, the argument is used as the name of the variable to which + the expression is assigned. Can be a string, ``Symbol``, + ``MatrixSymbol``, or ``Indexed`` type. This can be helpful for + expressions that generate multi-line statements. + precision : integer, optional + The precision for numbers such as pi [default=16]. + user_functions : dict, optional + A dictionary where keys are ``FunctionClass`` instances and values are + their string representations. Alternatively, the dictionary value can + be a list of tuples i.e. [(argument_test, cfunction_string)]. See + below for examples. + human : bool, optional + If True, the result is a single string that may contain some constant + declarations for the number symbols. If False, the same information is + returned in a tuple of (symbols_to_declare, not_supported_functions, + code_text). [default=True]. + contract: bool, optional + If True, ``Indexed`` instances are assumed to obey tensor contraction + rules and the corresponding nested loops over indices are generated. + Setting contract=False will not generate loops, instead the user is + responsible to provide values for the indices in the code. + [default=True]. + inline: bool, optional + If True, we try to create single-statement code instead of multiple + statements. [default=True]. + + """ + return MapleCodePrinter(settings).doprint(expr, assign_to) + + +def print_maple_code(expr, **settings): + """Prints the Maple representation of the given expression. + + See :func:`maple_code` for the meaning of the optional arguments. + + Examples + ======== + + >>> from sympy import print_maple_code, symbols + >>> x, y = symbols('x y') + >>> print_maple_code(x, assign_to=y) + y := x + """ + print(maple_code(expr, **settings)) diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/mathematica.py b/.venv/lib/python3.12/site-packages/sympy/printing/mathematica.py new file mode 100644 index 0000000000000000000000000000000000000000..064925ec1a9a477d7509110c57311239bac9fcaf --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/mathematica.py @@ -0,0 +1,353 @@ +""" +Mathematica code printer +""" + +from __future__ import annotations +from typing import Any + +from sympy.core import Basic, Expr, Float +from sympy.core.sorting import default_sort_key + +from sympy.printing.codeprinter import CodePrinter +from sympy.printing.precedence import precedence + +# Used in MCodePrinter._print_Function(self) +known_functions = { + "exp": [(lambda x: True, "Exp")], + "log": [(lambda x: True, "Log")], + "sin": [(lambda x: True, "Sin")], + "cos": [(lambda x: True, "Cos")], + "tan": [(lambda x: True, "Tan")], + "cot": [(lambda x: True, "Cot")], + "sec": [(lambda x: True, "Sec")], + "csc": [(lambda x: True, "Csc")], + "asin": [(lambda x: True, "ArcSin")], + "acos": [(lambda x: True, "ArcCos")], + "atan": [(lambda x: True, "ArcTan")], + "acot": [(lambda x: True, "ArcCot")], + "asec": [(lambda x: True, "ArcSec")], + "acsc": [(lambda x: True, "ArcCsc")], + "sinh": [(lambda x: True, "Sinh")], + "cosh": [(lambda x: True, "Cosh")], + "tanh": [(lambda x: True, "Tanh")], + "coth": [(lambda x: True, "Coth")], + "sech": [(lambda x: True, "Sech")], + "csch": [(lambda x: True, "Csch")], + "asinh": [(lambda x: True, "ArcSinh")], + "acosh": [(lambda x: True, "ArcCosh")], + "atanh": [(lambda x: True, "ArcTanh")], + "acoth": [(lambda x: True, "ArcCoth")], + "asech": [(lambda x: True, "ArcSech")], + "acsch": [(lambda x: True, "ArcCsch")], + "sinc": [(lambda x: True, "Sinc")], + "conjugate": [(lambda x: True, "Conjugate")], + "Max": [(lambda *x: True, "Max")], + "Min": [(lambda *x: True, "Min")], + "erf": [(lambda x: True, "Erf")], + "erf2": [(lambda *x: True, "Erf")], + "erfc": [(lambda x: True, "Erfc")], + "erfi": [(lambda x: True, "Erfi")], + "erfinv": [(lambda x: True, "InverseErf")], + "erfcinv": [(lambda x: True, "InverseErfc")], + "erf2inv": [(lambda *x: True, "InverseErf")], + "expint": [(lambda *x: True, "ExpIntegralE")], + "Ei": [(lambda x: True, "ExpIntegralEi")], + "fresnelc": [(lambda x: True, "FresnelC")], + "fresnels": [(lambda x: True, "FresnelS")], + "gamma": [(lambda x: True, "Gamma")], + "uppergamma": [(lambda *x: True, "Gamma")], + "polygamma": [(lambda *x: True, "PolyGamma")], + "loggamma": [(lambda x: True, "LogGamma")], + "beta": [(lambda *x: True, "Beta")], + "Ci": [(lambda x: True, "CosIntegral")], + "Si": [(lambda x: True, "SinIntegral")], + "Chi": [(lambda x: True, "CoshIntegral")], + "Shi": [(lambda x: True, "SinhIntegral")], + "li": [(lambda x: True, "LogIntegral")], + "factorial": [(lambda x: True, "Factorial")], + "factorial2": [(lambda x: True, "Factorial2")], + "subfactorial": [(lambda x: True, "Subfactorial")], + "catalan": [(lambda x: True, "CatalanNumber")], + "harmonic": [(lambda *x: True, "HarmonicNumber")], + "lucas": [(lambda x: True, "LucasL")], + "RisingFactorial": [(lambda *x: True, "Pochhammer")], + "FallingFactorial": [(lambda *x: True, "FactorialPower")], + "laguerre": [(lambda *x: True, "LaguerreL")], + "assoc_laguerre": [(lambda *x: True, "LaguerreL")], + "hermite": [(lambda *x: True, "HermiteH")], + "jacobi": [(lambda *x: True, "JacobiP")], + "gegenbauer": [(lambda *x: True, "GegenbauerC")], + "chebyshevt": [(lambda *x: True, "ChebyshevT")], + "chebyshevu": [(lambda *x: True, "ChebyshevU")], + "legendre": [(lambda *x: True, "LegendreP")], + "assoc_legendre": [(lambda *x: True, "LegendreP")], + "mathieuc": [(lambda *x: True, "MathieuC")], + "mathieus": [(lambda *x: True, "MathieuS")], + "mathieucprime": [(lambda *x: True, "MathieuCPrime")], + "mathieusprime": [(lambda *x: True, "MathieuSPrime")], + "stieltjes": [(lambda x: True, "StieltjesGamma")], + "elliptic_e": [(lambda *x: True, "EllipticE")], + "elliptic_f": [(lambda *x: True, "EllipticE")], + "elliptic_k": [(lambda x: True, "EllipticK")], + "elliptic_pi": [(lambda *x: True, "EllipticPi")], + "zeta": [(lambda *x: True, "Zeta")], + "dirichlet_eta": [(lambda x: True, "DirichletEta")], + "riemann_xi": [(lambda x: True, "RiemannXi")], + "besseli": [(lambda *x: True, "BesselI")], + "besselj": [(lambda *x: True, "BesselJ")], + "besselk": [(lambda *x: True, "BesselK")], + "bessely": [(lambda *x: True, "BesselY")], + "hankel1": [(lambda *x: True, "HankelH1")], + "hankel2": [(lambda *x: True, "HankelH2")], + "airyai": [(lambda x: True, "AiryAi")], + "airybi": [(lambda x: True, "AiryBi")], + "airyaiprime": [(lambda x: True, "AiryAiPrime")], + "airybiprime": [(lambda x: True, "AiryBiPrime")], + "polylog": [(lambda *x: True, "PolyLog")], + "lerchphi": [(lambda *x: True, "LerchPhi")], + "gcd": [(lambda *x: True, "GCD")], + "lcm": [(lambda *x: True, "LCM")], + "jn": [(lambda *x: True, "SphericalBesselJ")], + "yn": [(lambda *x: True, "SphericalBesselY")], + "hyper": [(lambda *x: True, "HypergeometricPFQ")], + "meijerg": [(lambda *x: True, "MeijerG")], + "appellf1": [(lambda *x: True, "AppellF1")], + "DiracDelta": [(lambda x: True, "DiracDelta")], + "Heaviside": [(lambda x: True, "HeavisideTheta")], + "KroneckerDelta": [(lambda *x: True, "KroneckerDelta")], + "sqrt": [(lambda x: True, "Sqrt")], # For automatic rewrites +} + + +class MCodePrinter(CodePrinter): + """A printer to convert Python expressions to + strings of the Wolfram's Mathematica code + """ + printmethod = "_mcode" + language = "Wolfram Language" + + _default_settings: dict[str, Any] = dict(CodePrinter._default_settings, **{ + 'precision': 15, + 'user_functions': {}, + }) + + _number_symbols: set[tuple[Expr, Float]] = set() + _not_supported: set[Basic] = set() + + def __init__(self, settings={}): + """Register function mappings supplied by user""" + CodePrinter.__init__(self, settings) + self.known_functions = dict(known_functions) + userfuncs = settings.get('user_functions', {}).copy() + for k, v in userfuncs.items(): + if not isinstance(v, list): + userfuncs[k] = [(lambda *x: True, v)] + self.known_functions.update(userfuncs) + + def _format_code(self, lines): + return lines + + def _print_Pow(self, expr): + PREC = precedence(expr) + return '%s^%s' % (self.parenthesize(expr.base, PREC), + self.parenthesize(expr.exp, PREC)) + + def _print_Mul(self, expr): + PREC = precedence(expr) + c, nc = expr.args_cnc() + res = super()._print_Mul(expr.func(*c)) + if nc: + res += '*' + res += '**'.join(self.parenthesize(a, PREC) for a in nc) + return res + + def _print_Relational(self, expr): + lhs_code = self._print(expr.lhs) + rhs_code = self._print(expr.rhs) + op = expr.rel_op + return "{} {} {}".format(lhs_code, op, rhs_code) + + # Primitive numbers + def _print_Zero(self, expr): + return '0' + + def _print_One(self, expr): + return '1' + + def _print_NegativeOne(self, expr): + return '-1' + + def _print_Half(self, expr): + return '1/2' + + def _print_ImaginaryUnit(self, expr): + return 'I' + + + # Infinity and invalid numbers + def _print_Infinity(self, expr): + return 'Infinity' + + def _print_NegativeInfinity(self, expr): + return '-Infinity' + + def _print_ComplexInfinity(self, expr): + return 'ComplexInfinity' + + def _print_NaN(self, expr): + return 'Indeterminate' + + + # Mathematical constants + def _print_Exp1(self, expr): + return 'E' + + def _print_Pi(self, expr): + return 'Pi' + + def _print_GoldenRatio(self, expr): + return 'GoldenRatio' + + def _print_TribonacciConstant(self, expr): + expanded = expr.expand(func=True) + PREC = precedence(expr) + return self.parenthesize(expanded, PREC) + + def _print_EulerGamma(self, expr): + return 'EulerGamma' + + def _print_Catalan(self, expr): + return 'Catalan' + + + def _print_list(self, expr): + return '{' + ', '.join(self.doprint(a) for a in expr) + '}' + _print_tuple = _print_list + _print_Tuple = _print_list + + def _print_ImmutableDenseMatrix(self, expr): + return self.doprint(expr.tolist()) + + def _print_ImmutableSparseMatrix(self, expr): + + def print_rule(pos, val): + return '{} -> {}'.format( + self.doprint((pos[0]+1, pos[1]+1)), self.doprint(val)) + + def print_data(): + items = sorted(expr.todok().items(), key=default_sort_key) + return '{' + \ + ', '.join(print_rule(k, v) for k, v in items) + \ + '}' + + def print_dims(): + return self.doprint(expr.shape) + + return 'SparseArray[{}, {}]'.format(print_data(), print_dims()) + + def _print_ImmutableDenseNDimArray(self, expr): + return self.doprint(expr.tolist()) + + def _print_ImmutableSparseNDimArray(self, expr): + def print_string_list(string_list): + return '{' + ', '.join(a for a in string_list) + '}' + + def to_mathematica_index(*args): + """Helper function to change Python style indexing to + Pathematica indexing. + + Python indexing (0, 1 ... n-1) + -> Mathematica indexing (1, 2 ... n) + """ + return tuple(i + 1 for i in args) + + def print_rule(pos, val): + """Helper function to print a rule of Mathematica""" + return '{} -> {}'.format(self.doprint(pos), self.doprint(val)) + + def print_data(): + """Helper function to print data part of Mathematica + sparse array. + + It uses the fourth notation ``SparseArray[data,{d1,d2,...}]`` + from + https://reference.wolfram.com/language/ref/SparseArray.html + + ``data`` must be formatted with rule. + """ + return print_string_list( + [print_rule( + to_mathematica_index(*(expr._get_tuple_index(key))), + value) + for key, value in sorted(expr._sparse_array.items())] + ) + + def print_dims(): + """Helper function to print dimensions part of Mathematica + sparse array. + + It uses the fourth notation ``SparseArray[data,{d1,d2,...}]`` + from + https://reference.wolfram.com/language/ref/SparseArray.html + """ + return self.doprint(expr.shape) + + return 'SparseArray[{}, {}]'.format(print_data(), print_dims()) + + def _print_Function(self, expr): + if expr.func.__name__ in self.known_functions: + cond_mfunc = self.known_functions[expr.func.__name__] + for cond, mfunc in cond_mfunc: + if cond(*expr.args): + return "%s[%s]" % (mfunc, self.stringify(expr.args, ", ")) + elif expr.func.__name__ in self._rewriteable_functions: + # Simple rewrite to supported function possible + target_f, required_fs = self._rewriteable_functions[expr.func.__name__] + if self._can_print(target_f) and all(self._can_print(f) for f in required_fs): + return self._print(expr.rewrite(target_f)) + return expr.func.__name__ + "[%s]" % self.stringify(expr.args, ", ") + + _print_MinMaxBase = _print_Function + + def _print_LambertW(self, expr): + if len(expr.args) == 1: + return "ProductLog[{}]".format(self._print(expr.args[0])) + return "ProductLog[{}, {}]".format( + self._print(expr.args[1]), self._print(expr.args[0])) + + def _print_atan2(self, expr): + return "ArcTan[{}, {}]".format( + self._print(expr.args[1]), self._print(expr.args[0])) + + def _print_Integral(self, expr): + if len(expr.variables) == 1 and not expr.limits[0][1:]: + args = [expr.args[0], expr.variables[0]] + else: + args = expr.args + return "Hold[Integrate[" + ', '.join(self.doprint(a) for a in args) + "]]" + + def _print_Sum(self, expr): + return "Hold[Sum[" + ', '.join(self.doprint(a) for a in expr.args) + "]]" + + def _print_Derivative(self, expr): + dexpr = expr.expr + dvars = [i[0] if i[1] == 1 else i for i in expr.variable_count] + return "Hold[D[" + ', '.join(self.doprint(a) for a in [dexpr] + dvars) + "]]" + + + def _get_comment(self, text): + return "(* {} *)".format(text) + + +def mathematica_code(expr, **settings): + r"""Converts an expr to a string of the Wolfram Mathematica code + + Examples + ======== + + >>> from sympy import mathematica_code as mcode, symbols, sin + >>> x = symbols('x') + >>> mcode(sin(x).series(x).removeO()) + '(1/120)*x^5 - 1/6*x^3 + x' + """ + return MCodePrinter(settings).doprint(expr) diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/mathml.py b/.venv/lib/python3.12/site-packages/sympy/printing/mathml.py new file mode 100644 index 0000000000000000000000000000000000000000..4dff74cd64b17036d3ff2a766253c9af850f088d --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/mathml.py @@ -0,0 +1,2157 @@ +""" +A MathML printer. +""" + +from __future__ import annotations +from typing import Any + +from sympy.core.mul import Mul +from sympy.core.singleton import S +from sympy.core.sorting import default_sort_key +from sympy.core.sympify import sympify +from sympy.printing.conventions import split_super_sub, requires_partial +from sympy.printing.precedence import \ + precedence_traditional, PRECEDENCE, PRECEDENCE_TRADITIONAL +from sympy.printing.pretty.pretty_symbology import greek_unicode +from sympy.printing.printer import Printer, print_function + +from mpmath.libmp import prec_to_dps, repr_dps, to_str as mlib_to_str + + +class MathMLPrinterBase(Printer): + """Contains common code required for MathMLContentPrinter and + MathMLPresentationPrinter. + """ + + _default_settings: dict[str, Any] = { + "order": None, + "encoding": "utf-8", + "fold_frac_powers": False, + "fold_func_brackets": False, + "fold_short_frac": None, + "inv_trig_style": "abbreviated", + "ln_notation": False, + "long_frac_ratio": None, + "mat_delim": "[", + "mat_symbol_style": "plain", + "mul_symbol": None, + "root_notation": True, + "symbol_names": {}, + "mul_symbol_mathml_numbers": '·', + "disable_split_super_sub": False, + } + + def __init__(self, settings=None): + Printer.__init__(self, settings) + from xml.dom.minidom import Document, Text + + self.dom = Document() + + # Workaround to allow strings to remain unescaped + # Based on + # https://stackoverflow.com/questions/38015864/python-xml-dom-minidom-\ + # please-dont-escape-my-strings/38041194 + class RawText(Text): + def writexml(self, writer, indent='', addindent='', newl=''): + if self.data: + writer.write('{}{}{}'.format(indent, self.data, newl)) + + def createRawTextNode(data): + r = RawText() + r.data = data + r.ownerDocument = self.dom + return r + + self.dom.createTextNode = createRawTextNode + + def doprint(self, expr): + """ + Prints the expression as MathML. + """ + mathML = Printer._print(self, expr) + unistr = mathML.toxml() + xmlbstr = unistr.encode('ascii', 'xmlcharrefreplace') + res = xmlbstr.decode() + return res + + def _split_super_sub(self, name): + if self._settings["disable_split_super_sub"]: + return (name, [], []) + else: + return split_super_sub(name) + + +class MathMLContentPrinter(MathMLPrinterBase): + """Prints an expression to the Content MathML markup language. + + References: https://www.w3.org/TR/MathML2/chapter4.html + """ + printmethod = "_mathml_content" + + def mathml_tag(self, e): + """Returns the MathML tag for an expression.""" + translate = { + 'Add': 'plus', + 'Mul': 'times', + 'Derivative': 'diff', + 'Number': 'cn', + 'int': 'cn', + 'Pow': 'power', + 'Max': 'max', + 'Min': 'min', + 'Abs': 'abs', + 'And': 'and', + 'Or': 'or', + 'Xor': 'xor', + 'Not': 'not', + 'Implies': 'implies', + 'Symbol': 'ci', + 'MatrixSymbol': 'ci', + 'RandomSymbol': 'ci', + 'Integral': 'int', + 'Sum': 'sum', + 'sin': 'sin', + 'cos': 'cos', + 'tan': 'tan', + 'cot': 'cot', + 'csc': 'csc', + 'sec': 'sec', + 'sinh': 'sinh', + 'cosh': 'cosh', + 'tanh': 'tanh', + 'coth': 'coth', + 'csch': 'csch', + 'sech': 'sech', + 'asin': 'arcsin', + 'asinh': 'arcsinh', + 'acos': 'arccos', + 'acosh': 'arccosh', + 'atan': 'arctan', + 'atanh': 'arctanh', + 'atan2': 'arctan', + 'acot': 'arccot', + 'acoth': 'arccoth', + 'asec': 'arcsec', + 'asech': 'arcsech', + 'acsc': 'arccsc', + 'acsch': 'arccsch', + 'log': 'ln', + 'Equality': 'eq', + 'Unequality': 'neq', + 'GreaterThan': 'geq', + 'LessThan': 'leq', + 'StrictGreaterThan': 'gt', + 'StrictLessThan': 'lt', + 'Union': 'union', + 'Intersection': 'intersect', + } + + for cls in e.__class__.__mro__: + n = cls.__name__ + if n in translate: + return translate[n] + # Not found in the MRO set + n = e.__class__.__name__ + return n.lower() + + def _print_Mul(self, expr): + + if expr.could_extract_minus_sign(): + x = self.dom.createElement('apply') + x.appendChild(self.dom.createElement('minus')) + x.appendChild(self._print_Mul(-expr)) + return x + + from sympy.simplify import fraction + numer, denom = fraction(expr) + + if denom is not S.One: + x = self.dom.createElement('apply') + x.appendChild(self.dom.createElement('divide')) + x.appendChild(self._print(numer)) + x.appendChild(self._print(denom)) + return x + + coeff, terms = expr.as_coeff_mul() + if coeff is S.One and len(terms) == 1: + # XXX since the negative coefficient has been handled, I don't + # think a coeff of 1 can remain + return self._print(terms[0]) + + if self.order != 'old': + terms = Mul._from_args(terms).as_ordered_factors() + + x = self.dom.createElement('apply') + x.appendChild(self.dom.createElement('times')) + if coeff != 1: + x.appendChild(self._print(coeff)) + for term in terms: + x.appendChild(self._print(term)) + return x + + def _print_Add(self, expr, order=None): + args = self._as_ordered_terms(expr, order=order) + lastProcessed = self._print(args[0]) + plusNodes = [] + for arg in args[1:]: + if arg.could_extract_minus_sign(): + # use minus + x = self.dom.createElement('apply') + x.appendChild(self.dom.createElement('minus')) + x.appendChild(lastProcessed) + x.appendChild(self._print(-arg)) + # invert expression since this is now minused + lastProcessed = x + if arg == args[-1]: + plusNodes.append(lastProcessed) + else: + plusNodes.append(lastProcessed) + lastProcessed = self._print(arg) + if arg == args[-1]: + plusNodes.append(self._print(arg)) + if len(plusNodes) == 1: + return lastProcessed + x = self.dom.createElement('apply') + x.appendChild(self.dom.createElement('plus')) + while plusNodes: + x.appendChild(plusNodes.pop(0)) + return x + + def _print_Piecewise(self, expr): + if expr.args[-1].cond != True: + # We need the last conditional to be a True, otherwise the resulting + # function may not return a result. + raise ValueError("All Piecewise expressions must contain an " + "(expr, True) statement to be used as a default " + "condition. Without one, the generated " + "expression may not evaluate to anything under " + "some condition.") + root = self.dom.createElement('piecewise') + for i, (e, c) in enumerate(expr.args): + if i == len(expr.args) - 1 and c == True: + piece = self.dom.createElement('otherwise') + piece.appendChild(self._print(e)) + else: + piece = self.dom.createElement('piece') + piece.appendChild(self._print(e)) + piece.appendChild(self._print(c)) + root.appendChild(piece) + return root + + def _print_MatrixBase(self, m): + x = self.dom.createElement('matrix') + for i in range(m.rows): + x_r = self.dom.createElement('matrixrow') + for j in range(m.cols): + x_r.appendChild(self._print(m[i, j])) + x.appendChild(x_r) + return x + + def _print_Rational(self, e): + if e.q == 1: + # don't divide + x = self.dom.createElement('cn') + x.appendChild(self.dom.createTextNode(str(e.p))) + return x + x = self.dom.createElement('apply') + x.appendChild(self.dom.createElement('divide')) + # numerator + xnum = self.dom.createElement('cn') + xnum.appendChild(self.dom.createTextNode(str(e.p))) + # denominator + xdenom = self.dom.createElement('cn') + xdenom.appendChild(self.dom.createTextNode(str(e.q))) + x.appendChild(xnum) + x.appendChild(xdenom) + return x + + def _print_Limit(self, e): + x = self.dom.createElement('apply') + x.appendChild(self.dom.createElement(self.mathml_tag(e))) + + x_1 = self.dom.createElement('bvar') + x_2 = self.dom.createElement('lowlimit') + x_1.appendChild(self._print(e.args[1])) + x_2.appendChild(self._print(e.args[2])) + + x.appendChild(x_1) + x.appendChild(x_2) + x.appendChild(self._print(e.args[0])) + return x + + def _print_ImaginaryUnit(self, e): + return self.dom.createElement('imaginaryi') + + def _print_EulerGamma(self, e): + return self.dom.createElement('eulergamma') + + def _print_GoldenRatio(self, e): + """We use unicode #x3c6 for Greek letter phi as defined here + https://www.w3.org/2003/entities/2007doc/isogrk1.html""" + x = self.dom.createElement('cn') + x.appendChild(self.dom.createTextNode("\N{GREEK SMALL LETTER PHI}")) + return x + + def _print_Exp1(self, e): + return self.dom.createElement('exponentiale') + + def _print_Pi(self, e): + return self.dom.createElement('pi') + + def _print_Infinity(self, e): + return self.dom.createElement('infinity') + + def _print_NaN(self, e): + return self.dom.createElement('notanumber') + + def _print_EmptySet(self, e): + return self.dom.createElement('emptyset') + + def _print_BooleanTrue(self, e): + return self.dom.createElement('true') + + def _print_BooleanFalse(self, e): + return self.dom.createElement('false') + + def _print_NegativeInfinity(self, e): + x = self.dom.createElement('apply') + x.appendChild(self.dom.createElement('minus')) + x.appendChild(self.dom.createElement('infinity')) + return x + + def _print_Integral(self, e): + def lime_recur(limits): + x = self.dom.createElement('apply') + x.appendChild(self.dom.createElement(self.mathml_tag(e))) + bvar_elem = self.dom.createElement('bvar') + bvar_elem.appendChild(self._print(limits[0][0])) + x.appendChild(bvar_elem) + + if len(limits[0]) == 3: + low_elem = self.dom.createElement('lowlimit') + low_elem.appendChild(self._print(limits[0][1])) + x.appendChild(low_elem) + up_elem = self.dom.createElement('uplimit') + up_elem.appendChild(self._print(limits[0][2])) + x.appendChild(up_elem) + if len(limits[0]) == 2: + up_elem = self.dom.createElement('uplimit') + up_elem.appendChild(self._print(limits[0][1])) + x.appendChild(up_elem) + if len(limits) == 1: + x.appendChild(self._print(e.function)) + else: + x.appendChild(lime_recur(limits[1:])) + return x + + limits = list(e.limits) + limits.reverse() + return lime_recur(limits) + + def _print_Sum(self, e): + # Printer can be shared because Sum and Integral have the + # same internal representation. + return self._print_Integral(e) + + def _print_Symbol(self, sym): + ci = self.dom.createElement(self.mathml_tag(sym)) + + def join(items): + if len(items) > 1: + mrow = self.dom.createElement('mml:mrow') + for i, item in enumerate(items): + if i > 0: + mo = self.dom.createElement('mml:mo') + mo.appendChild(self.dom.createTextNode(" ")) + mrow.appendChild(mo) + mi = self.dom.createElement('mml:mi') + mi.appendChild(self.dom.createTextNode(item)) + mrow.appendChild(mi) + return mrow + else: + mi = self.dom.createElement('mml:mi') + mi.appendChild(self.dom.createTextNode(items[0])) + return mi + + # translate name, supers and subs to unicode characters + def translate(s): + if s in greek_unicode: + return greek_unicode.get(s) + else: + return s + + name, supers, subs = self._split_super_sub(sym.name) + name = translate(name) + supers = [translate(sup) for sup in supers] + subs = [translate(sub) for sub in subs] + + mname = self.dom.createElement('mml:mi') + mname.appendChild(self.dom.createTextNode(name)) + if not supers: + if not subs: + ci.appendChild(self.dom.createTextNode(name)) + else: + msub = self.dom.createElement('mml:msub') + msub.appendChild(mname) + msub.appendChild(join(subs)) + ci.appendChild(msub) + else: + if not subs: + msup = self.dom.createElement('mml:msup') + msup.appendChild(mname) + msup.appendChild(join(supers)) + ci.appendChild(msup) + else: + msubsup = self.dom.createElement('mml:msubsup') + msubsup.appendChild(mname) + msubsup.appendChild(join(subs)) + msubsup.appendChild(join(supers)) + ci.appendChild(msubsup) + return ci + + _print_MatrixSymbol = _print_Symbol + _print_RandomSymbol = _print_Symbol + + def _print_Pow(self, e): + # Here we use root instead of power if the exponent is the reciprocal + # of an integer + if (self._settings['root_notation'] and e.exp.is_Rational + and e.exp.p == 1): + x = self.dom.createElement('apply') + x.appendChild(self.dom.createElement('root')) + if e.exp.q != 2: + xmldeg = self.dom.createElement('degree') + xmlcn = self.dom.createElement('cn') + xmlcn.appendChild(self.dom.createTextNode(str(e.exp.q))) + xmldeg.appendChild(xmlcn) + x.appendChild(xmldeg) + x.appendChild(self._print(e.base)) + return x + + x = self.dom.createElement('apply') + x_1 = self.dom.createElement(self.mathml_tag(e)) + x.appendChild(x_1) + x.appendChild(self._print(e.base)) + x.appendChild(self._print(e.exp)) + return x + + def _print_Number(self, e): + x = self.dom.createElement(self.mathml_tag(e)) + x.appendChild(self.dom.createTextNode(str(e))) + return x + + def _print_Float(self, e): + x = self.dom.createElement(self.mathml_tag(e)) + repr_e = mlib_to_str(e._mpf_, repr_dps(e._prec)) + x.appendChild(self.dom.createTextNode(repr_e)) + return x + + def _print_Derivative(self, e): + x = self.dom.createElement('apply') + diff_symbol = self.mathml_tag(e) + if requires_partial(e.expr): + diff_symbol = 'partialdiff' + x.appendChild(self.dom.createElement(diff_symbol)) + x_1 = self.dom.createElement('bvar') + + for sym, times in reversed(e.variable_count): + x_1.appendChild(self._print(sym)) + if times > 1: + degree = self.dom.createElement('degree') + degree.appendChild(self._print(sympify(times))) + x_1.appendChild(degree) + + x.appendChild(x_1) + x.appendChild(self._print(e.expr)) + return x + + def _print_Function(self, e): + x = self.dom.createElement("apply") + x.appendChild(self.dom.createElement(self.mathml_tag(e))) + for arg in e.args: + x.appendChild(self._print(arg)) + return x + + def _print_Basic(self, e): + x = self.dom.createElement(self.mathml_tag(e)) + for arg in e.args: + x.appendChild(self._print(arg)) + return x + + def _print_AssocOp(self, e): + x = self.dom.createElement('apply') + x_1 = self.dom.createElement(self.mathml_tag(e)) + x.appendChild(x_1) + for arg in e.args: + x.appendChild(self._print(arg)) + return x + + def _print_Relational(self, e): + x = self.dom.createElement('apply') + x.appendChild(self.dom.createElement(self.mathml_tag(e))) + x.appendChild(self._print(e.lhs)) + x.appendChild(self._print(e.rhs)) + return x + + def _print_list(self, seq): + """MathML reference for the element: + https://www.w3.org/TR/MathML2/chapter4.html#contm.list""" + dom_element = self.dom.createElement('list') + for item in seq: + dom_element.appendChild(self._print(item)) + return dom_element + + def _print_int(self, p): + dom_element = self.dom.createElement(self.mathml_tag(p)) + dom_element.appendChild(self.dom.createTextNode(str(p))) + return dom_element + + _print_Implies = _print_AssocOp + _print_Not = _print_AssocOp + _print_Xor = _print_AssocOp + + def _print_FiniteSet(self, e): + x = self.dom.createElement('set') + for arg in e.args: + x.appendChild(self._print(arg)) + return x + + def _print_Complement(self, e): + x = self.dom.createElement('apply') + x.appendChild(self.dom.createElement('setdiff')) + for arg in e.args: + x.appendChild(self._print(arg)) + return x + + def _print_ProductSet(self, e): + x = self.dom.createElement('apply') + x.appendChild(self.dom.createElement('cartesianproduct')) + for arg in e.args: + x.appendChild(self._print(arg)) + return x + + def _print_Lambda(self, e): + # MathML reference for the lambda element: + # https://www.w3.org/TR/MathML2/chapter4.html#id.4.2.1.7 + x = self.dom.createElement(self.mathml_tag(e)) + for arg in e.signature: + x_1 = self.dom.createElement('bvar') + x_1.appendChild(self._print(arg)) + x.appendChild(x_1) + x.appendChild(self._print(e.expr)) + return x + + # XXX Symmetric difference is not supported for MathML content printers. + + +class MathMLPresentationPrinter(MathMLPrinterBase): + """Prints an expression to the Presentation MathML markup language. + + References: https://www.w3.org/TR/MathML2/chapter3.html + """ + printmethod = "_mathml_presentation" + + def mathml_tag(self, e): + """Returns the MathML tag for an expression.""" + translate = { + 'Number': 'mn', + 'Limit': '→', + 'Derivative': 'ⅆ', + 'int': 'mn', + 'Symbol': 'mi', + 'Integral': '∫', + 'Sum': '∑', + 'sin': 'sin', + 'cos': 'cos', + 'tan': 'tan', + 'cot': 'cot', + 'asin': 'arcsin', + 'asinh': 'arcsinh', + 'acos': 'arccos', + 'acosh': 'arccosh', + 'atan': 'arctan', + 'atanh': 'arctanh', + 'acot': 'arccot', + 'atan2': 'arctan', + 'Equality': '=', + 'Unequality': '≠', + 'GreaterThan': '≥', + 'LessThan': '≤', + 'StrictGreaterThan': '>', + 'StrictLessThan': '<', + 'lerchphi': 'Φ', + 'zeta': 'ζ', + 'dirichlet_eta': 'η', + 'elliptic_k': 'Κ', + 'lowergamma': 'γ', + 'uppergamma': 'Γ', + 'gamma': 'Γ', + 'totient': 'ϕ', + 'reduced_totient': 'λ', + 'primenu': 'ν', + 'primeomega': 'Ω', + 'fresnels': 'S', + 'fresnelc': 'C', + 'LambertW': 'W', + 'Heaviside': 'Θ', + 'BooleanTrue': 'True', + 'BooleanFalse': 'False', + 'NoneType': 'None', + 'mathieus': 'S', + 'mathieuc': 'C', + 'mathieusprime': 'S′', + 'mathieucprime': 'C′', + 'Lambda': 'lambda', + } + + def mul_symbol_selection(): + if (self._settings["mul_symbol"] is None or + self._settings["mul_symbol"] == 'None'): + return '⁢' + elif self._settings["mul_symbol"] == 'times': + return '×' + elif self._settings["mul_symbol"] == 'dot': + return '·' + elif self._settings["mul_symbol"] == 'ldot': + return '․' + elif not isinstance(self._settings["mul_symbol"], str): + raise TypeError + else: + return self._settings["mul_symbol"] + for cls in e.__class__.__mro__: + n = cls.__name__ + if n in translate: + return translate[n] + # Not found in the MRO set + if e.__class__.__name__ == "Mul": + return mul_symbol_selection() + n = e.__class__.__name__ + return n.lower() + + def _l_paren(self): + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('(')) + return mo + + def _r_paren(self): + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode(')')) + return mo + + def _l_brace(self): + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('{')) + return mo + + def _r_brace(self): + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('}')) + return mo + + def _comma(self): + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode(',')) + return mo + + def _bar(self): + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('|')) + return mo + + def _semicolon(self): + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode(';')) + return mo + + def _paren_comma_separated(self, *args): + mrow = self.dom.createElement('mrow') + mrow.appendChild(self._l_paren()) + for i, arg in enumerate(args): + if i: + mrow.appendChild(self._comma()) + mrow.appendChild(self._print(arg)) + mrow.appendChild(self._r_paren()) + return mrow + + def _paren_bar_separated(self, *args): + mrow = self.dom.createElement('mrow') + mrow.appendChild(self._l_paren()) + for i, arg in enumerate(args): + if i: + mrow.appendChild(self._bar()) + mrow.appendChild(self._print(arg)) + mrow.appendChild(self._r_paren()) + return mrow + + def parenthesize(self, item, level, strict=False): + prec_val = precedence_traditional(item) + if (prec_val < level) or ((not strict) and prec_val <= level): + mrow = self.dom.createElement('mrow') + mrow.appendChild(self._l_paren()) + mrow.appendChild(self._print(item)) + mrow.appendChild(self._r_paren()) + return mrow + return self._print(item) + + def _print_Mul(self, expr): + + def multiply(expr, mrow): + from sympy.simplify import fraction + numer, denom = fraction(expr) + if denom is not S.One: + frac = self.dom.createElement('mfrac') + if self._settings["fold_short_frac"] and len(str(expr)) < 7: + frac.setAttribute('bevelled', 'true') + xnum = self._print(numer) + xden = self._print(denom) + frac.appendChild(xnum) + frac.appendChild(xden) + mrow.appendChild(frac) + return mrow + + coeff, terms = expr.as_coeff_mul() + if coeff is S.One and len(terms) == 1: + mrow.appendChild(self._print(terms[0])) + return mrow + if self.order != 'old': + terms = Mul._from_args(terms).as_ordered_factors() + + if coeff != 1: + x = self._print(coeff) + y = self.dom.createElement('mo') + y.appendChild(self.dom.createTextNode(self.mathml_tag(expr))) + mrow.appendChild(x) + mrow.appendChild(y) + for term in terms: + mrow.appendChild(self.parenthesize(term, PRECEDENCE['Mul'])) + if not term == terms[-1]: + y = self.dom.createElement('mo') + y.appendChild(self.dom.createTextNode(self.mathml_tag(expr))) + mrow.appendChild(y) + return mrow + mrow = self.dom.createElement('mrow') + if expr.could_extract_minus_sign(): + x = self.dom.createElement('mo') + x.appendChild(self.dom.createTextNode('-')) + mrow.appendChild(x) + mrow = multiply(-expr, mrow) + else: + mrow = multiply(expr, mrow) + + return mrow + + def _print_Add(self, expr, order=None): + mrow = self.dom.createElement('mrow') + args = self._as_ordered_terms(expr, order=order) + mrow.appendChild(self._print(args[0])) + for arg in args[1:]: + if arg.could_extract_minus_sign(): + # use minus + x = self.dom.createElement('mo') + x.appendChild(self.dom.createTextNode('-')) + y = self._print(-arg) + # invert expression since this is now minused + else: + x = self.dom.createElement('mo') + x.appendChild(self.dom.createTextNode('+')) + y = self._print(arg) + mrow.appendChild(x) + mrow.appendChild(y) + + return mrow + + def _print_MatrixBase(self, m): + table = self.dom.createElement('mtable') + for i in range(m.rows): + x = self.dom.createElement('mtr') + for j in range(m.cols): + y = self.dom.createElement('mtd') + y.appendChild(self._print(m[i, j])) + x.appendChild(y) + table.appendChild(x) + mat_delim = self._settings["mat_delim"] + if mat_delim == '': + return table + left = self.dom.createElement('mo') + right = self.dom.createElement('mo') + if mat_delim == "[": + left.appendChild(self.dom.createTextNode("[")) + right.appendChild(self.dom.createTextNode("]")) + else: + left.appendChild(self.dom.createTextNode("(")) + right.appendChild(self.dom.createTextNode(")")) + mrow = self.dom.createElement('mrow') + mrow.appendChild(left) + mrow.appendChild(table) + mrow.appendChild(right) + return mrow + + def _get_printed_Rational(self, e, folded=None): + if e.p < 0: + p = -e.p + else: + p = e.p + x = self.dom.createElement('mfrac') + if folded or self._settings["fold_short_frac"]: + x.setAttribute('bevelled', 'true') + x.appendChild(self._print(p)) + x.appendChild(self._print(e.q)) + if e.p < 0: + mrow = self.dom.createElement('mrow') + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('-')) + mrow.appendChild(mo) + mrow.appendChild(x) + return mrow + else: + return x + + def _print_Rational(self, e): + if e.q == 1: + # don't divide + return self._print(e.p) + + return self._get_printed_Rational(e, self._settings["fold_short_frac"]) + + def _print_Limit(self, e): + mrow = self.dom.createElement('mrow') + munder = self.dom.createElement('munder') + mi = self.dom.createElement('mi') + mi.appendChild(self.dom.createTextNode('lim')) + + x = self.dom.createElement('mrow') + x_1 = self._print(e.args[1]) + arrow = self.dom.createElement('mo') + arrow.appendChild(self.dom.createTextNode(self.mathml_tag(e))) + x_2 = self._print(e.args[2]) + x.appendChild(x_1) + x.appendChild(arrow) + x.appendChild(x_2) + + munder.appendChild(mi) + munder.appendChild(x) + mrow.appendChild(munder) + mrow.appendChild(self._print(e.args[0])) + + return mrow + + def _print_ImaginaryUnit(self, e): + x = self.dom.createElement('mi') + x.appendChild(self.dom.createTextNode('ⅈ')) + return x + + def _print_GoldenRatio(self, e): + x = self.dom.createElement('mi') + x.appendChild(self.dom.createTextNode('Φ')) + return x + + def _print_Exp1(self, e): + x = self.dom.createElement('mi') + x.appendChild(self.dom.createTextNode('ⅇ')) + return x + + def _print_Pi(self, e): + x = self.dom.createElement('mi') + x.appendChild(self.dom.createTextNode('π')) + return x + + def _print_Infinity(self, e): + x = self.dom.createElement('mi') + x.appendChild(self.dom.createTextNode('∞')) + return x + + def _print_NegativeInfinity(self, e): + mrow = self.dom.createElement('mrow') + y = self.dom.createElement('mo') + y.appendChild(self.dom.createTextNode('-')) + x = self._print_Infinity(e) + mrow.appendChild(y) + mrow.appendChild(x) + return mrow + + def _print_HBar(self, e): + x = self.dom.createElement('mi') + x.appendChild(self.dom.createTextNode('ℏ')) + return x + + def _print_EulerGamma(self, e): + x = self.dom.createElement('mi') + x.appendChild(self.dom.createTextNode('γ')) + return x + + def _print_TribonacciConstant(self, e): + x = self.dom.createElement('mi') + x.appendChild(self.dom.createTextNode('TribonacciConstant')) + return x + + def _print_Dagger(self, e): + msup = self.dom.createElement('msup') + msup.appendChild(self._print(e.args[0])) + msup.appendChild(self.dom.createTextNode('†')) + return msup + + def _print_Contains(self, e): + mrow = self.dom.createElement('mrow') + mrow.appendChild(self._print(e.args[0])) + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('∈')) + mrow.appendChild(mo) + mrow.appendChild(self._print(e.args[1])) + return mrow + + def _print_HilbertSpace(self, e): + x = self.dom.createElement('mi') + x.appendChild(self.dom.createTextNode('ℋ')) + return x + + def _print_ComplexSpace(self, e): + msup = self.dom.createElement('msup') + msup.appendChild(self.dom.createTextNode('𝒞')) + msup.appendChild(self._print(e.args[0])) + return msup + + def _print_FockSpace(self, e): + x = self.dom.createElement('mi') + x.appendChild(self.dom.createTextNode('ℱ')) + return x + + + def _print_Integral(self, expr): + intsymbols = {1: "∫", 2: "∬", 3: "∭"} + + mrow = self.dom.createElement('mrow') + if len(expr.limits) <= 3 and all(len(lim) == 1 for lim in expr.limits): + # Only up to three-integral signs exists + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode(intsymbols[len(expr.limits)])) + mrow.appendChild(mo) + else: + # Either more than three or limits provided + for lim in reversed(expr.limits): + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode(intsymbols[1])) + if len(lim) == 1: + mrow.appendChild(mo) + if len(lim) == 2: + msup = self.dom.createElement('msup') + msup.appendChild(mo) + msup.appendChild(self._print(lim[1])) + mrow.appendChild(msup) + if len(lim) == 3: + msubsup = self.dom.createElement('msubsup') + msubsup.appendChild(mo) + msubsup.appendChild(self._print(lim[1])) + msubsup.appendChild(self._print(lim[2])) + mrow.appendChild(msubsup) + # print function + mrow.appendChild(self.parenthesize(expr.function, PRECEDENCE["Mul"], + strict=True)) + # print integration variables + for lim in reversed(expr.limits): + d = self.dom.createElement('mo') + d.appendChild(self.dom.createTextNode('ⅆ')) + mrow.appendChild(d) + mrow.appendChild(self._print(lim[0])) + return mrow + + def _print_Sum(self, e): + limits = list(e.limits) + subsup = self.dom.createElement('munderover') + low_elem = self._print(limits[0][1]) + up_elem = self._print(limits[0][2]) + summand = self.dom.createElement('mo') + summand.appendChild(self.dom.createTextNode(self.mathml_tag(e))) + + low = self.dom.createElement('mrow') + var = self._print(limits[0][0]) + equal = self.dom.createElement('mo') + equal.appendChild(self.dom.createTextNode('=')) + low.appendChild(var) + low.appendChild(equal) + low.appendChild(low_elem) + + subsup.appendChild(summand) + subsup.appendChild(low) + subsup.appendChild(up_elem) + + mrow = self.dom.createElement('mrow') + mrow.appendChild(subsup) + mrow.appendChild(self.parenthesize(e.function, precedence_traditional(e))) + return mrow + + def _print_Symbol(self, sym, style='plain'): + def join(items): + if len(items) > 1: + mrow = self.dom.createElement('mrow') + for i, item in enumerate(items): + if i > 0: + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode(" ")) + mrow.appendChild(mo) + mi = self.dom.createElement('mi') + mi.appendChild(self.dom.createTextNode(item)) + mrow.appendChild(mi) + return mrow + else: + mi = self.dom.createElement('mi') + mi.appendChild(self.dom.createTextNode(items[0])) + return mi + + # translate name, supers and subs to unicode characters + def translate(s): + if s in greek_unicode: + return greek_unicode.get(s) + else: + return s + + name, supers, subs = self._split_super_sub(sym.name) + name = translate(name) + supers = [translate(sup) for sup in supers] + subs = [translate(sub) for sub in subs] + + mname = self.dom.createElement('mi') + mname.appendChild(self.dom.createTextNode(name)) + if len(supers) == 0: + if len(subs) == 0: + x = mname + else: + x = self.dom.createElement('msub') + x.appendChild(mname) + x.appendChild(join(subs)) + else: + if len(subs) == 0: + x = self.dom.createElement('msup') + x.appendChild(mname) + x.appendChild(join(supers)) + else: + x = self.dom.createElement('msubsup') + x.appendChild(mname) + x.appendChild(join(subs)) + x.appendChild(join(supers)) + # Set bold font? + if style == 'bold': + x.setAttribute('mathvariant', 'bold') + return x + + def _print_MatrixSymbol(self, sym): + return self._print_Symbol(sym, + style=self._settings['mat_symbol_style']) + + _print_RandomSymbol = _print_Symbol + + def _print_conjugate(self, expr): + enc = self.dom.createElement('menclose') + enc.setAttribute('notation', 'top') + enc.appendChild(self._print(expr.args[0])) + return enc + + def _print_operator_after(self, op, expr): + row = self.dom.createElement('mrow') + row.appendChild(self.parenthesize(expr, PRECEDENCE["Func"])) + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode(op)) + row.appendChild(mo) + return row + + def _print_factorial(self, expr): + return self._print_operator_after('!', expr.args[0]) + + def _print_factorial2(self, expr): + return self._print_operator_after('!!', expr.args[0]) + + def _print_binomial(self, expr): + frac = self.dom.createElement('mfrac') + frac.setAttribute('linethickness', '0') + frac.appendChild(self._print(expr.args[0])) + frac.appendChild(self._print(expr.args[1])) + brac = self.dom.createElement('mrow') + brac.appendChild(self._l_paren()) + brac.appendChild(frac) + brac.appendChild(self._r_paren()) + return brac + + def _print_Pow(self, e): + # Here we use root instead of power if the exponent is the + # reciprocal of an integer + if (e.exp.is_Rational and abs(e.exp.p) == 1 and e.exp.q != 1 and + self._settings['root_notation']): + if e.exp.q == 2: + x = self.dom.createElement('msqrt') + x.appendChild(self._print(e.base)) + if e.exp.q != 2: + x = self.dom.createElement('mroot') + x.appendChild(self._print(e.base)) + x.appendChild(self._print(e.exp.q)) + if e.exp.p == -1: + frac = self.dom.createElement('mfrac') + frac.appendChild(self._print(1)) + frac.appendChild(x) + return frac + else: + return x + + if e.exp.is_Rational and e.exp.q != 1: + if e.exp.is_negative: + top = self.dom.createElement('mfrac') + top.appendChild(self._print(1)) + x = self.dom.createElement('msup') + x.appendChild(self.parenthesize(e.base, PRECEDENCE['Pow'])) + x.appendChild(self._get_printed_Rational(-e.exp, + self._settings['fold_frac_powers'])) + top.appendChild(x) + return top + else: + x = self.dom.createElement('msup') + x.appendChild(self.parenthesize(e.base, PRECEDENCE['Pow'])) + x.appendChild(self._get_printed_Rational(e.exp, + self._settings['fold_frac_powers'])) + return x + + if e.exp.is_negative: + top = self.dom.createElement('mfrac') + top.appendChild(self._print(1)) + if e.exp == -1: + top.appendChild(self._print(e.base)) + else: + x = self.dom.createElement('msup') + x.appendChild(self.parenthesize(e.base, PRECEDENCE['Pow'])) + x.appendChild(self._print(-e.exp)) + top.appendChild(x) + return top + + x = self.dom.createElement('msup') + x.appendChild(self.parenthesize(e.base, PRECEDENCE['Pow'])) + x.appendChild(self._print(e.exp)) + return x + + def _print_Number(self, e): + x = self.dom.createElement(self.mathml_tag(e)) + x.appendChild(self.dom.createTextNode(str(e))) + return x + + def _print_AccumulationBounds(self, i): + left = self.dom.createElement('mo') + left.appendChild(self.dom.createTextNode('\u27e8')) + right = self.dom.createElement('mo') + right.appendChild(self.dom.createTextNode('\u27e9')) + brac = self.dom.createElement('mrow') + brac.appendChild(left) + brac.appendChild(self._print(i.min)) + brac.appendChild(self._comma()) + brac.appendChild(self._print(i.max)) + brac.appendChild(right) + return brac + + def _print_Derivative(self, e): + + if requires_partial(e.expr): + d = '∂' + else: + d = self.mathml_tag(e) + + # Determine denominator + m = self.dom.createElement('mrow') + dim = 0 # Total diff dimension, for numerator + for sym, num in reversed(e.variable_count): + dim += num + if num >= 2: + x = self.dom.createElement('msup') + xx = self.dom.createElement('mo') + xx.appendChild(self.dom.createTextNode(d)) + x.appendChild(xx) + x.appendChild(self._print(num)) + else: + x = self.dom.createElement('mo') + x.appendChild(self.dom.createTextNode(d)) + m.appendChild(x) + y = self._print(sym) + m.appendChild(y) + + mnum = self.dom.createElement('mrow') + if dim >= 2: + x = self.dom.createElement('msup') + xx = self.dom.createElement('mo') + xx.appendChild(self.dom.createTextNode(d)) + x.appendChild(xx) + x.appendChild(self._print(dim)) + else: + x = self.dom.createElement('mo') + x.appendChild(self.dom.createTextNode(d)) + + mnum.appendChild(x) + mrow = self.dom.createElement('mrow') + frac = self.dom.createElement('mfrac') + frac.appendChild(mnum) + frac.appendChild(m) + mrow.appendChild(frac) + + # Print function + mrow.appendChild(self._print(e.expr)) + + return mrow + + def _print_Function(self, e): + x = self.dom.createElement('mi') + if self.mathml_tag(e) == 'log' and self._settings["ln_notation"]: + x.appendChild(self.dom.createTextNode('ln')) + else: + x.appendChild(self.dom.createTextNode(self.mathml_tag(e))) + mrow = self.dom.createElement('mrow') + mrow.appendChild(x) + mrow.appendChild(self._paren_comma_separated(*e.args)) + return mrow + + def _print_Float(self, expr): + # Based off of that in StrPrinter + dps = prec_to_dps(expr._prec) + str_real = mlib_to_str(expr._mpf_, dps, strip_zeros=True) + + # Must always have a mul symbol (as 2.5 10^{20} just looks odd) + # thus we use the number separator + separator = self._settings['mul_symbol_mathml_numbers'] + mrow = self.dom.createElement('mrow') + if 'e' in str_real: + (mant, exp) = str_real.split('e') + + if exp[0] == '+': + exp = exp[1:] + + mn = self.dom.createElement('mn') + mn.appendChild(self.dom.createTextNode(mant)) + mrow.appendChild(mn) + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode(separator)) + mrow.appendChild(mo) + msup = self.dom.createElement('msup') + mn = self.dom.createElement('mn') + mn.appendChild(self.dom.createTextNode("10")) + msup.appendChild(mn) + mn = self.dom.createElement('mn') + mn.appendChild(self.dom.createTextNode(exp)) + msup.appendChild(mn) + mrow.appendChild(msup) + return mrow + elif str_real == "+inf": + return self._print_Infinity(None) + elif str_real == "-inf": + return self._print_NegativeInfinity(None) + else: + mn = self.dom.createElement('mn') + mn.appendChild(self.dom.createTextNode(str_real)) + return mn + + def _print_polylog(self, expr): + mrow = self.dom.createElement('mrow') + m = self.dom.createElement('msub') + + mi = self.dom.createElement('mi') + mi.appendChild(self.dom.createTextNode('Li')) + m.appendChild(mi) + m.appendChild(self._print(expr.args[0])) + mrow.appendChild(m) + brac = self.dom.createElement('mrow') + brac.appendChild(self._l_paren()) + brac.appendChild(self._print(expr.args[1])) + brac.appendChild(self._r_paren()) + mrow.appendChild(brac) + return mrow + + def _print_Basic(self, e): + mrow = self.dom.createElement('mrow') + mi = self.dom.createElement('mi') + mi.appendChild(self.dom.createTextNode(self.mathml_tag(e))) + mrow.appendChild(mi) + mrow.appendChild(self._paren_comma_separated(*e.args)) + return mrow + + def _print_Tuple(self, e): + return self._paren_comma_separated(*e.args) + + def _print_Interval(self, i): + right = self.dom.createElement('mo') + if i.right_open: + right.appendChild(self.dom.createTextNode(')')) + else: + right.appendChild(self.dom.createTextNode(']')) + left = self.dom.createElement('mo') + if i.left_open: + left.appendChild(self.dom.createTextNode('(')) + else: + left.appendChild(self.dom.createTextNode('[')) + mrow = self.dom.createElement('mrow') + mrow.appendChild(left) + mrow.appendChild(self._print(i.start)) + mrow.appendChild(self._comma()) + mrow.appendChild(self._print(i.end)) + mrow.appendChild(right) + return mrow + + def _print_Abs(self, expr, exp=None): + mrow = self.dom.createElement('mrow') + mrow.appendChild(self._bar()) + mrow.appendChild(self._print(expr.args[0])) + mrow.appendChild(self._bar()) + return mrow + + _print_Determinant = _print_Abs + + def _print_re_im(self, c, expr): + brac = self.dom.createElement('mrow') + brac.appendChild(self._l_paren()) + brac.appendChild(self._print(expr)) + brac.appendChild(self._r_paren()) + mi = self.dom.createElement('mi') + mi.appendChild(self.dom.createTextNode(c)) + mrow = self.dom.createElement('mrow') + mrow.appendChild(mi) + mrow.appendChild(brac) + return mrow + + def _print_re(self, expr, exp=None): + return self._print_re_im('\u211C', expr.args[0]) + + def _print_im(self, expr, exp=None): + return self._print_re_im('\u2111', expr.args[0]) + + def _print_AssocOp(self, e): + mrow = self.dom.createElement('mrow') + mi = self.dom.createElement('mi') + mi.appendChild(self.dom.createTextNode(self.mathml_tag(e))) + mrow.appendChild(mi) + for arg in e.args: + mrow.appendChild(self._print(arg)) + return mrow + + def _print_SetOp(self, expr, symbol, prec): + mrow = self.dom.createElement('mrow') + mrow.appendChild(self.parenthesize(expr.args[0], prec)) + for arg in expr.args[1:]: + x = self.dom.createElement('mo') + x.appendChild(self.dom.createTextNode(symbol)) + y = self.parenthesize(arg, prec) + mrow.appendChild(x) + mrow.appendChild(y) + return mrow + + def _print_Union(self, expr): + prec = PRECEDENCE_TRADITIONAL['Union'] + return self._print_SetOp(expr, '∪', prec) + + def _print_Intersection(self, expr): + prec = PRECEDENCE_TRADITIONAL['Intersection'] + return self._print_SetOp(expr, '∩', prec) + + def _print_Complement(self, expr): + prec = PRECEDENCE_TRADITIONAL['Complement'] + return self._print_SetOp(expr, '∖', prec) + + def _print_SymmetricDifference(self, expr): + prec = PRECEDENCE_TRADITIONAL['SymmetricDifference'] + return self._print_SetOp(expr, '∆', prec) + + def _print_ProductSet(self, expr): + prec = PRECEDENCE_TRADITIONAL['ProductSet'] + return self._print_SetOp(expr, '×', prec) + + def _print_FiniteSet(self, s): + return self._print_set(s.args) + + def _print_set(self, s): + items = sorted(s, key=default_sort_key) + brac = self.dom.createElement('mrow') + brac.appendChild(self._l_brace()) + for i, item in enumerate(items): + if i: + brac.appendChild(self._comma()) + brac.appendChild(self._print(item)) + brac.appendChild(self._r_brace()) + return brac + + _print_frozenset = _print_set + + def _print_LogOp(self, args, symbol): + mrow = self.dom.createElement('mrow') + if args[0].is_Boolean and not args[0].is_Not: + brac = self.dom.createElement('mrow') + brac.appendChild(self._l_paren()) + brac.appendChild(self._print(args[0])) + brac.appendChild(self._r_paren()) + mrow.appendChild(brac) + else: + mrow.appendChild(self._print(args[0])) + for arg in args[1:]: + x = self.dom.createElement('mo') + x.appendChild(self.dom.createTextNode(symbol)) + if arg.is_Boolean and not arg.is_Not: + y = self.dom.createElement('mrow') + y.appendChild(self._l_paren()) + y.appendChild(self._print(arg)) + y.appendChild(self._r_paren()) + else: + y = self._print(arg) + mrow.appendChild(x) + mrow.appendChild(y) + return mrow + + def _print_BasisDependent(self, expr): + from sympy.vector import Vector + + if expr == expr.zero: + # Not clear if this is ever called + return self._print(expr.zero) + if isinstance(expr, Vector): + items = expr.separate().items() + else: + items = [(0, expr)] + + mrow = self.dom.createElement('mrow') + for system, vect in items: + inneritems = list(vect.components.items()) + inneritems.sort(key = lambda x:x[0].__str__()) + for i, (k, v) in enumerate(inneritems): + if v == 1: + if i: # No + for first item + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('+')) + mrow.appendChild(mo) + mrow.appendChild(self._print(k)) + elif v == -1: + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('-')) + mrow.appendChild(mo) + mrow.appendChild(self._print(k)) + else: + if i: # No + for first item + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('+')) + mrow.appendChild(mo) + mbrac = self.dom.createElement('mrow') + mbrac.appendChild(self._l_paren()) + mbrac.appendChild(self._print(v)) + mbrac.appendChild(self._r_paren()) + mrow.appendChild(mbrac) + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('⁢')) + mrow.appendChild(mo) + mrow.appendChild(self._print(k)) + return mrow + + + def _print_And(self, expr): + args = sorted(expr.args, key=default_sort_key) + return self._print_LogOp(args, '∧') + + def _print_Or(self, expr): + args = sorted(expr.args, key=default_sort_key) + return self._print_LogOp(args, '∨') + + def _print_Xor(self, expr): + args = sorted(expr.args, key=default_sort_key) + return self._print_LogOp(args, '⊻') + + def _print_Implies(self, expr): + return self._print_LogOp(expr.args, '⇒') + + def _print_Equivalent(self, expr): + args = sorted(expr.args, key=default_sort_key) + return self._print_LogOp(args, '⇔') + + def _print_Not(self, e): + mrow = self.dom.createElement('mrow') + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('¬')) + mrow.appendChild(mo) + if (e.args[0].is_Boolean): + x = self.dom.createElement('mrow') + x.appendChild(self._l_paren()) + x.appendChild(self._print(e.args[0])) + x.appendChild(self._r_paren()) + else: + x = self._print(e.args[0]) + mrow.appendChild(x) + return mrow + + def _print_bool(self, e): + mi = self.dom.createElement('mi') + mi.appendChild(self.dom.createTextNode(self.mathml_tag(e))) + return mi + + _print_BooleanTrue = _print_bool + _print_BooleanFalse = _print_bool + + def _print_NoneType(self, e): + mi = self.dom.createElement('mi') + mi.appendChild(self.dom.createTextNode(self.mathml_tag(e))) + return mi + + def _print_Range(self, s): + dots = "\u2026" + if s.start.is_infinite and s.stop.is_infinite: + if s.step.is_positive: + printset = dots, -1, 0, 1, dots + else: + printset = dots, 1, 0, -1, dots + elif s.start.is_infinite: + printset = dots, s[-1] - s.step, s[-1] + elif s.stop.is_infinite: + it = iter(s) + printset = next(it), next(it), dots + elif len(s) > 4: + it = iter(s) + printset = next(it), next(it), dots, s[-1] + else: + printset = tuple(s) + brac = self.dom.createElement('mrow') + brac.appendChild(self._l_brace()) + for i, el in enumerate(printset): + if i: + brac.appendChild(self._comma()) + if el == dots: + mi = self.dom.createElement('mi') + mi.appendChild(self.dom.createTextNode(dots)) + brac.appendChild(mi) + else: + brac.appendChild(self._print(el)) + brac.appendChild(self._r_brace()) + return brac + + def _hprint_variadic_function(self, expr): + args = sorted(expr.args, key=default_sort_key) + mrow = self.dom.createElement('mrow') + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode((str(expr.func)).lower())) + mrow.appendChild(mo) + mrow.appendChild(self._paren_comma_separated(*args)) + return mrow + + _print_Min = _print_Max = _hprint_variadic_function + + def _print_exp(self, expr): + msup = self.dom.createElement('msup') + msup.appendChild(self._print_Exp1(None)) + msup.appendChild(self._print(expr.args[0])) + return msup + + def _print_Relational(self, e): + mrow = self.dom.createElement('mrow') + mrow.appendChild(self._print(e.lhs)) + x = self.dom.createElement('mo') + x.appendChild(self.dom.createTextNode(self.mathml_tag(e))) + mrow.appendChild(x) + mrow.appendChild(self._print(e.rhs)) + return mrow + + def _print_int(self, p): + dom_element = self.dom.createElement(self.mathml_tag(p)) + dom_element.appendChild(self.dom.createTextNode(str(p))) + return dom_element + + def _print_BaseScalar(self, e): + msub = self.dom.createElement('msub') + index, system = e._id + mi = self.dom.createElement('mi') + mi.setAttribute('mathvariant', 'bold') + mi.appendChild(self.dom.createTextNode(system._variable_names[index])) + msub.appendChild(mi) + mi = self.dom.createElement('mi') + mi.setAttribute('mathvariant', 'bold') + mi.appendChild(self.dom.createTextNode(system._name)) + msub.appendChild(mi) + return msub + + def _print_BaseVector(self, e): + msub = self.dom.createElement('msub') + index, system = e._id + mover = self.dom.createElement('mover') + mi = self.dom.createElement('mi') + mi.setAttribute('mathvariant', 'bold') + mi.appendChild(self.dom.createTextNode(system._vector_names[index])) + mover.appendChild(mi) + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('^')) + mover.appendChild(mo) + msub.appendChild(mover) + mi = self.dom.createElement('mi') + mi.setAttribute('mathvariant', 'bold') + mi.appendChild(self.dom.createTextNode(system._name)) + msub.appendChild(mi) + return msub + + def _print_VectorZero(self, e): + mover = self.dom.createElement('mover') + mi = self.dom.createElement('mi') + mi.setAttribute('mathvariant', 'bold') + mi.appendChild(self.dom.createTextNode("0")) + mover.appendChild(mi) + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('^')) + mover.appendChild(mo) + return mover + + def _print_Cross(self, expr): + mrow = self.dom.createElement('mrow') + vec1 = expr._expr1 + vec2 = expr._expr2 + mrow.appendChild(self.parenthesize(vec1, PRECEDENCE['Mul'])) + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('×')) + mrow.appendChild(mo) + mrow.appendChild(self.parenthesize(vec2, PRECEDENCE['Mul'])) + return mrow + + def _print_Curl(self, expr): + mrow = self.dom.createElement('mrow') + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('∇')) + mrow.appendChild(mo) + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('×')) + mrow.appendChild(mo) + mrow.appendChild(self.parenthesize(expr._expr, PRECEDENCE['Mul'])) + return mrow + + def _print_Divergence(self, expr): + mrow = self.dom.createElement('mrow') + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('∇')) + mrow.appendChild(mo) + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('·')) + mrow.appendChild(mo) + mrow.appendChild(self.parenthesize(expr._expr, PRECEDENCE['Mul'])) + return mrow + + def _print_Dot(self, expr): + mrow = self.dom.createElement('mrow') + vec1 = expr._expr1 + vec2 = expr._expr2 + mrow.appendChild(self.parenthesize(vec1, PRECEDENCE['Mul'])) + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('·')) + mrow.appendChild(mo) + mrow.appendChild(self.parenthesize(vec2, PRECEDENCE['Mul'])) + return mrow + + def _print_Gradient(self, expr): + mrow = self.dom.createElement('mrow') + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('∇')) + mrow.appendChild(mo) + mrow.appendChild(self.parenthesize(expr._expr, PRECEDENCE['Mul'])) + return mrow + + def _print_Laplacian(self, expr): + mrow = self.dom.createElement('mrow') + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('∆')) + mrow.appendChild(mo) + mrow.appendChild(self.parenthesize(expr._expr, PRECEDENCE['Mul'])) + return mrow + + def _print_Integers(self, e): + x = self.dom.createElement('mi') + x.setAttribute('mathvariant', 'normal') + x.appendChild(self.dom.createTextNode('ℤ')) + return x + + def _print_Complexes(self, e): + x = self.dom.createElement('mi') + x.setAttribute('mathvariant', 'normal') + x.appendChild(self.dom.createTextNode('ℂ')) + return x + + def _print_Reals(self, e): + x = self.dom.createElement('mi') + x.setAttribute('mathvariant', 'normal') + x.appendChild(self.dom.createTextNode('ℝ')) + return x + + def _print_Naturals(self, e): + x = self.dom.createElement('mi') + x.setAttribute('mathvariant', 'normal') + x.appendChild(self.dom.createTextNode('ℕ')) + return x + + def _print_Naturals0(self, e): + sub = self.dom.createElement('msub') + x = self.dom.createElement('mi') + x.setAttribute('mathvariant', 'normal') + x.appendChild(self.dom.createTextNode('ℕ')) + sub.appendChild(x) + sub.appendChild(self._print(S.Zero)) + return sub + + def _print_SingularityFunction(self, expr): + shift = expr.args[0] - expr.args[1] + power = expr.args[2] + left = self.dom.createElement('mo') + left.appendChild(self.dom.createTextNode('\u27e8')) + right = self.dom.createElement('mo') + right.appendChild(self.dom.createTextNode('\u27e9')) + brac = self.dom.createElement('mrow') + brac.appendChild(left) + brac.appendChild(self._print(shift)) + brac.appendChild(right) + sup = self.dom.createElement('msup') + sup.appendChild(brac) + sup.appendChild(self._print(power)) + return sup + + def _print_NaN(self, e): + x = self.dom.createElement('mi') + x.appendChild(self.dom.createTextNode('NaN')) + return x + + def _print_number_function(self, e, name): + # Print name_arg[0] for one argument or name_arg[0](arg[1]) + # for more than one argument + sub = self.dom.createElement('msub') + mi = self.dom.createElement('mi') + mi.appendChild(self.dom.createTextNode(name)) + sub.appendChild(mi) + sub.appendChild(self._print(e.args[0])) + if len(e.args) == 1: + return sub + mrow = self.dom.createElement('mrow') + mrow.appendChild(sub) + mrow.appendChild(self._paren_comma_separated(*e.args[1:])) + return mrow + + def _print_bernoulli(self, e): + return self._print_number_function(e, 'B') + + _print_bell = _print_bernoulli + + def _print_catalan(self, e): + return self._print_number_function(e, 'C') + + def _print_euler(self, e): + return self._print_number_function(e, 'E') + + def _print_fibonacci(self, e): + return self._print_number_function(e, 'F') + + def _print_lucas(self, e): + return self._print_number_function(e, 'L') + + def _print_stieltjes(self, e): + return self._print_number_function(e, 'γ') + + def _print_tribonacci(self, e): + return self._print_number_function(e, 'T') + + def _print_ComplexInfinity(self, e): + x = self.dom.createElement('mover') + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('∞')) + x.appendChild(mo) + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('~')) + x.appendChild(mo) + return x + + def _print_EmptySet(self, e): + x = self.dom.createElement('mo') + x.appendChild(self.dom.createTextNode('∅')) + return x + + def _print_UniversalSet(self, e): + x = self.dom.createElement('mo') + x.appendChild(self.dom.createTextNode('𝕌')) + return x + + def _print_Adjoint(self, expr): + from sympy.matrices import MatrixSymbol + mat = expr.arg + sup = self.dom.createElement('msup') + if not isinstance(mat, MatrixSymbol): + brac = self.dom.createElement('mrow') + brac.appendChild(self._l_paren()) + brac.appendChild(self._print(mat)) + brac.appendChild(self._r_paren()) + sup.appendChild(brac) + else: + sup.appendChild(self._print(mat)) + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('†')) + sup.appendChild(mo) + return sup + + def _print_Transpose(self, expr): + from sympy.matrices import MatrixSymbol + mat = expr.arg + sup = self.dom.createElement('msup') + if not isinstance(mat, MatrixSymbol): + brac = self.dom.createElement('mrow') + brac.appendChild(self._l_paren()) + brac.appendChild(self._print(mat)) + brac.appendChild(self._r_paren()) + sup.appendChild(brac) + else: + sup.appendChild(self._print(mat)) + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('T')) + sup.appendChild(mo) + return sup + + def _print_Inverse(self, expr): + from sympy.matrices import MatrixSymbol + mat = expr.arg + sup = self.dom.createElement('msup') + if not isinstance(mat, MatrixSymbol): + brac = self.dom.createElement('mrow') + brac.appendChild(self._l_paren()) + brac.appendChild(self._print(mat)) + brac.appendChild(self._r_paren()) + sup.appendChild(brac) + else: + sup.appendChild(self._print(mat)) + sup.appendChild(self._print(-1)) + return sup + + def _print_MatMul(self, expr): + from sympy.matrices.expressions.matmul import MatMul + + x = self.dom.createElement('mrow') + args = expr.args + if isinstance(args[0], Mul): + args = args[0].as_ordered_factors() + list(args[1:]) + else: + args = list(args) + + if isinstance(expr, MatMul) and expr.could_extract_minus_sign(): + if args[0] == -1: + args = args[1:] + else: + args[0] = -args[0] + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('-')) + x.appendChild(mo) + + for arg in args[:-1]: + x.appendChild(self.parenthesize(arg, precedence_traditional(expr), + False)) + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('⁢')) + x.appendChild(mo) + x.appendChild(self.parenthesize(args[-1], precedence_traditional(expr), + False)) + return x + + def _print_MatPow(self, expr): + from sympy.matrices import MatrixSymbol + base, exp = expr.base, expr.exp + sup = self.dom.createElement('msup') + if not isinstance(base, MatrixSymbol): + brac = self.dom.createElement('mrow') + brac.appendChild(self._l_paren()) + brac.appendChild(self._print(base)) + brac.appendChild(self._r_paren()) + sup.appendChild(brac) + else: + sup.appendChild(self._print(base)) + sup.appendChild(self._print(exp)) + return sup + + def _print_HadamardProduct(self, expr): + x = self.dom.createElement('mrow') + args = expr.args + for arg in args[:-1]: + x.appendChild( + self.parenthesize(arg, precedence_traditional(expr), False)) + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('∘')) + x.appendChild(mo) + x.appendChild( + self.parenthesize(args[-1], precedence_traditional(expr), False)) + return x + + def _print_ZeroMatrix(self, Z): + x = self.dom.createElement('mn') + x.appendChild(self.dom.createTextNode('𝟘')) + return x + + def _print_OneMatrix(self, Z): + x = self.dom.createElement('mn') + x.appendChild(self.dom.createTextNode('𝟙')) + return x + + def _print_Identity(self, I): + x = self.dom.createElement('mi') + x.appendChild(self.dom.createTextNode('𝕀')) + return x + + def _print_floor(self, e): + left = self.dom.createElement('mo') + left.appendChild(self.dom.createTextNode('\u230A')) + right = self.dom.createElement('mo') + right.appendChild(self.dom.createTextNode('\u230B')) + mrow = self.dom.createElement('mrow') + mrow.appendChild(left) + mrow.appendChild(self._print(e.args[0])) + mrow.appendChild(right) + return mrow + + def _print_ceiling(self, e): + left = self.dom.createElement('mo') + left.appendChild(self.dom.createTextNode('\u2308')) + right = self.dom.createElement('mo') + right.appendChild(self.dom.createTextNode('\u2309')) + mrow = self.dom.createElement('mrow') + mrow.appendChild(left) + mrow.appendChild(self._print(e.args[0])) + mrow.appendChild(right) + return mrow + + def _print_Lambda(self, e): + mrow = self.dom.createElement('mrow') + symbols = e.args[0] + if len(symbols) == 1: + symbols = self._print(symbols[0]) + else: + symbols = self._print(symbols) + mrow.appendChild(self._l_paren()) + mrow.appendChild(symbols) + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('↦')) + mrow.appendChild(mo) + mrow.appendChild(self._print(e.args[1])) + mrow.appendChild(self._r_paren()) + return mrow + + def _print_tuple(self, e): + return self._paren_comma_separated(*e) + + def _print_IndexedBase(self, e): + return self._print(e.label) + + def _print_Indexed(self, e): + x = self.dom.createElement('msub') + x.appendChild(self._print(e.base)) + if len(e.indices) == 1: + x.appendChild(self._print(e.indices[0])) + return x + x.appendChild(self._print(e.indices)) + return x + + def _print_MatrixElement(self, e): + x = self.dom.createElement('msub') + x.appendChild(self.parenthesize(e.parent, PRECEDENCE["Atom"], strict = True)) + brac = self.dom.createElement('mrow') + for i, arg in enumerate(e.indices): + if i: + brac.appendChild(self._comma()) + brac.appendChild(self._print(arg)) + x.appendChild(brac) + return x + + def _print_elliptic_f(self, e): + x = self.dom.createElement('mrow') + mi = self.dom.createElement('mi') + mi.appendChild(self.dom.createTextNode('𝖥')) + x.appendChild(mi) + x.appendChild(self._paren_bar_separated(*e.args)) + return x + + def _print_elliptic_e(self, e): + x = self.dom.createElement('mrow') + mi = self.dom.createElement('mi') + mi.appendChild(self.dom.createTextNode('𝖤')) + x.appendChild(mi) + x.appendChild(self._paren_bar_separated(*e.args)) + return x + + def _print_elliptic_pi(self, e): + x = self.dom.createElement('mrow') + mi = self.dom.createElement('mi') + mi.appendChild(self.dom.createTextNode('𝛱')) + x.appendChild(mi) + y = self.dom.createElement('mrow') + y.appendChild(self._l_paren()) + if len(e.args) == 2: + n, m = e.args + y.appendChild(self._print(n)) + y.appendChild(self._bar()) + y.appendChild(self._print(m)) + else: + n, m, z = e.args + y.appendChild(self._print(n)) + y.appendChild(self._semicolon()) + y.appendChild(self._print(m)) + y.appendChild(self._bar()) + y.appendChild(self._print(z)) + y.appendChild(self._r_paren()) + x.appendChild(y) + return x + + def _print_Ei(self, e): + x = self.dom.createElement('mrow') + mi = self.dom.createElement('mi') + mi.appendChild(self.dom.createTextNode('Ei')) + x.appendChild(mi) + x.appendChild(self._print(e.args)) + return x + + def _print_expint(self, e): + x = self.dom.createElement('mrow') + y = self.dom.createElement('msub') + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('E')) + y.appendChild(mo) + y.appendChild(self._print(e.args[0])) + x.appendChild(y) + x.appendChild(self._print(e.args[1:])) + return x + + def _print_jacobi(self, e): + x = self.dom.createElement('mrow') + y = self.dom.createElement('msubsup') + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('P')) + y.appendChild(mo) + y.appendChild(self._print(e.args[0])) + y.appendChild(self._print(e.args[1:3])) + x.appendChild(y) + x.appendChild(self._print(e.args[3:])) + return x + + def _print_gegenbauer(self, e): + x = self.dom.createElement('mrow') + y = self.dom.createElement('msubsup') + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('C')) + y.appendChild(mo) + y.appendChild(self._print(e.args[0])) + y.appendChild(self._print(e.args[1:2])) + x.appendChild(y) + x.appendChild(self._print(e.args[2:])) + return x + + def _print_chebyshevt(self, e): + x = self.dom.createElement('mrow') + y = self.dom.createElement('msub') + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('T')) + y.appendChild(mo) + y.appendChild(self._print(e.args[0])) + x.appendChild(y) + x.appendChild(self._print(e.args[1:])) + return x + + def _print_chebyshevu(self, e): + x = self.dom.createElement('mrow') + y = self.dom.createElement('msub') + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('U')) + y.appendChild(mo) + y.appendChild(self._print(e.args[0])) + x.appendChild(y) + x.appendChild(self._print(e.args[1:])) + return x + + def _print_legendre(self, e): + x = self.dom.createElement('mrow') + y = self.dom.createElement('msub') + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('P')) + y.appendChild(mo) + y.appendChild(self._print(e.args[0])) + x.appendChild(y) + x.appendChild(self._print(e.args[1:])) + return x + + def _print_assoc_legendre(self, e): + x = self.dom.createElement('mrow') + y = self.dom.createElement('msubsup') + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('P')) + y.appendChild(mo) + y.appendChild(self._print(e.args[0])) + y.appendChild(self._print(e.args[1:2])) + x.appendChild(y) + x.appendChild(self._print(e.args[2:])) + return x + + def _print_laguerre(self, e): + x = self.dom.createElement('mrow') + y = self.dom.createElement('msub') + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('L')) + y.appendChild(mo) + y.appendChild(self._print(e.args[0])) + x.appendChild(y) + x.appendChild(self._print(e.args[1:])) + return x + + def _print_assoc_laguerre(self, e): + x = self.dom.createElement('mrow') + y = self.dom.createElement('msubsup') + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('L')) + y.appendChild(mo) + y.appendChild(self._print(e.args[0])) + y.appendChild(self._print(e.args[1:2])) + x.appendChild(y) + x.appendChild(self._print(e.args[2:])) + return x + + def _print_hermite(self, e): + x = self.dom.createElement('mrow') + y = self.dom.createElement('msub') + mo = self.dom.createElement('mo') + mo.appendChild(self.dom.createTextNode('H')) + y.appendChild(mo) + y.appendChild(self._print(e.args[0])) + x.appendChild(y) + x.appendChild(self._print(e.args[1:])) + return x + + +@print_function(MathMLPrinterBase) +def mathml(expr, printer='content', **settings): + """Returns the MathML representation of expr. If printer is presentation + then prints Presentation MathML else prints content MathML. + """ + if printer == 'presentation': + return MathMLPresentationPrinter(settings).doprint(expr) + else: + return MathMLContentPrinter(settings).doprint(expr) + + +def print_mathml(expr, printer='content', **settings): + """ + Prints a pretty representation of the MathML code for expr. If printer is + presentation then prints Presentation MathML else prints content MathML. + + Examples + ======== + + >>> ## + >>> from sympy import print_mathml + >>> from sympy.abc import x + >>> print_mathml(x+1) #doctest: +NORMALIZE_WHITESPACE + + + x + 1 + + >>> print_mathml(x+1, printer='presentation') + + x + + + 1 + + + """ + if printer == 'presentation': + s = MathMLPresentationPrinter(settings) + else: + s = MathMLContentPrinter(settings) + xml = s._print(sympify(expr)) + pretty_xml = xml.toprettyxml() + + print(pretty_xml) + + +# For backward compatibility +MathMLPrinter = MathMLContentPrinter diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/numpy.py b/.venv/lib/python3.12/site-packages/sympy/printing/numpy.py new file mode 100644 index 0000000000000000000000000000000000000000..1ff68454bb287bc0a1d2dfc1fe68fb05b3c22a74 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/numpy.py @@ -0,0 +1,541 @@ +from sympy.core import S +from sympy.core.function import Lambda +from sympy.core.power import Pow +from .pycode import PythonCodePrinter, _known_functions_math, _print_known_const, _print_known_func, _unpack_integral_limits, ArrayPrinter +from .codeprinter import CodePrinter + + +_not_in_numpy = 'erf erfc factorial gamma loggamma'.split() +_in_numpy = [(k, v) for k, v in _known_functions_math.items() if k not in _not_in_numpy] +_known_functions_numpy = dict(_in_numpy, **{ + 'acos': 'arccos', + 'acosh': 'arccosh', + 'asin': 'arcsin', + 'asinh': 'arcsinh', + 'atan': 'arctan', + 'atan2': 'arctan2', + 'atanh': 'arctanh', + 'exp2': 'exp2', + 'sign': 'sign', + 'logaddexp': 'logaddexp', + 'logaddexp2': 'logaddexp2', + 'isinf': 'isinf', + 'isnan': 'isnan', + +}) +_known_constants_numpy = { + 'Exp1': 'e', + 'Pi': 'pi', + 'EulerGamma': 'euler_gamma', + 'NaN': 'nan', + 'Infinity': 'inf', +} + +_numpy_known_functions = {k: 'numpy.' + v for k, v in _known_functions_numpy.items()} +_numpy_known_constants = {k: 'numpy.' + v for k, v in _known_constants_numpy.items()} + +class NumPyPrinter(ArrayPrinter, PythonCodePrinter): + """ + Numpy printer which handles vectorized piecewise functions, + logical operators, etc. + """ + + _module = 'numpy' + _kf = _numpy_known_functions + _kc = _numpy_known_constants + + def __init__(self, settings=None): + """ + `settings` is passed to CodePrinter.__init__() + `module` specifies the array module to use, currently 'NumPy', 'CuPy' + or 'JAX'. + """ + self.language = "Python with {}".format(self._module) + self.printmethod = "_{}code".format(self._module) + + self._kf = {**PythonCodePrinter._kf, **self._kf} + + super().__init__(settings=settings) + + + def _print_seq(self, seq): + "General sequence printer: converts to tuple" + # Print tuples here instead of lists because numba supports + # tuples in nopython mode. + delimiter=', ' + return '({},)'.format(delimiter.join(self._print(item) for item in seq)) + + def _print_NegativeInfinity(self, expr): + return '-' + self._print(S.Infinity) + + def _print_MatMul(self, expr): + "Matrix multiplication printer" + if expr.as_coeff_matrices()[0] is not S.One: + expr_list = expr.as_coeff_matrices()[1]+[(expr.as_coeff_matrices()[0])] + return '({})'.format(').dot('.join(self._print(i) for i in expr_list)) + return '({})'.format(').dot('.join(self._print(i) for i in expr.args)) + + def _print_MatPow(self, expr): + "Matrix power printer" + return '{}({}, {})'.format(self._module_format(self._module + '.linalg.matrix_power'), + self._print(expr.args[0]), self._print(expr.args[1])) + + def _print_Inverse(self, expr): + "Matrix inverse printer" + return '{}({})'.format(self._module_format(self._module + '.linalg.inv'), + self._print(expr.args[0])) + + def _print_DotProduct(self, expr): + # DotProduct allows any shape order, but numpy.dot does matrix + # multiplication, so we have to make sure it gets 1 x n by n x 1. + arg1, arg2 = expr.args + if arg1.shape[0] != 1: + arg1 = arg1.T + if arg2.shape[1] != 1: + arg2 = arg2.T + + return "%s(%s, %s)" % (self._module_format(self._module + '.dot'), + self._print(arg1), + self._print(arg2)) + + def _print_MatrixSolve(self, expr): + return "%s(%s, %s)" % (self._module_format(self._module + '.linalg.solve'), + self._print(expr.matrix), + self._print(expr.vector)) + + def _print_ZeroMatrix(self, expr): + return '{}({})'.format(self._module_format(self._module + '.zeros'), + self._print(expr.shape)) + + def _print_OneMatrix(self, expr): + return '{}({})'.format(self._module_format(self._module + '.ones'), + self._print(expr.shape)) + + def _print_FunctionMatrix(self, expr): + from sympy.abc import i, j + lamda = expr.lamda + if not isinstance(lamda, Lambda): + lamda = Lambda((i, j), lamda(i, j)) + return '{}(lambda {}: {}, {})'.format(self._module_format(self._module + '.fromfunction'), + ', '.join(self._print(arg) for arg in lamda.args[0]), + self._print(lamda.args[1]), self._print(expr.shape)) + + def _print_HadamardProduct(self, expr): + func = self._module_format(self._module + '.multiply') + return ''.join('{}({}, '.format(func, self._print(arg)) \ + for arg in expr.args[:-1]) + "{}{}".format(self._print(expr.args[-1]), + ')' * (len(expr.args) - 1)) + + def _print_KroneckerProduct(self, expr): + func = self._module_format(self._module + '.kron') + return ''.join('{}({}, '.format(func, self._print(arg)) \ + for arg in expr.args[:-1]) + "{}{}".format(self._print(expr.args[-1]), + ')' * (len(expr.args) - 1)) + + def _print_Adjoint(self, expr): + return '{}({}({}))'.format( + self._module_format(self._module + '.conjugate'), + self._module_format(self._module + '.transpose'), + self._print(expr.args[0])) + + def _print_DiagonalOf(self, expr): + vect = '{}({})'.format( + self._module_format(self._module + '.diag'), + self._print(expr.arg)) + return '{}({}, (-1, 1))'.format( + self._module_format(self._module + '.reshape'), vect) + + def _print_DiagMatrix(self, expr): + return '{}({})'.format(self._module_format(self._module + '.diagflat'), + self._print(expr.args[0])) + + def _print_DiagonalMatrix(self, expr): + return '{}({}, {}({}, {}))'.format(self._module_format(self._module + '.multiply'), + self._print(expr.arg), self._module_format(self._module + '.eye'), + self._print(expr.shape[0]), self._print(expr.shape[1])) + + def _print_Piecewise(self, expr): + "Piecewise function printer" + from sympy.logic.boolalg import ITE, simplify_logic + def print_cond(cond): + """ Problem having an ITE in the cond. """ + if cond.has(ITE): + return self._print(simplify_logic(cond)) + else: + return self._print(cond) + exprs = '[{}]'.format(','.join(self._print(arg.expr) for arg in expr.args)) + conds = '[{}]'.format(','.join(print_cond(arg.cond) for arg in expr.args)) + # If [default_value, True] is a (expr, cond) sequence in a Piecewise object + # it will behave the same as passing the 'default' kwarg to select() + # *as long as* it is the last element in expr.args. + # If this is not the case, it may be triggered prematurely. + return '{}({}, {}, default={})'.format( + self._module_format(self._module + '.select'), conds, exprs, + self._print(S.NaN)) + + def _print_Relational(self, expr): + "Relational printer for Equality and Unequality" + op = { + '==' :'equal', + '!=' :'not_equal', + '<' :'less', + '<=' :'less_equal', + '>' :'greater', + '>=' :'greater_equal', + } + if expr.rel_op in op: + lhs = self._print(expr.lhs) + rhs = self._print(expr.rhs) + return '{op}({lhs}, {rhs})'.format(op=self._module_format(self._module + '.'+op[expr.rel_op]), + lhs=lhs, rhs=rhs) + return super()._print_Relational(expr) + + def _print_And(self, expr): + "Logical And printer" + # We have to override LambdaPrinter because it uses Python 'and' keyword. + # If LambdaPrinter didn't define it, we could use StrPrinter's + # version of the function and add 'logical_and' to NUMPY_TRANSLATIONS. + return '{}.reduce(({}))'.format(self._module_format(self._module + '.logical_and'), ','.join(self._print(i) for i in expr.args)) + + def _print_Or(self, expr): + "Logical Or printer" + # We have to override LambdaPrinter because it uses Python 'or' keyword. + # If LambdaPrinter didn't define it, we could use StrPrinter's + # version of the function and add 'logical_or' to NUMPY_TRANSLATIONS. + return '{}.reduce(({}))'.format(self._module_format(self._module + '.logical_or'), ','.join(self._print(i) for i in expr.args)) + + def _print_Not(self, expr): + "Logical Not printer" + # We have to override LambdaPrinter because it uses Python 'not' keyword. + # If LambdaPrinter didn't define it, we would still have to define our + # own because StrPrinter doesn't define it. + return '{}({})'.format(self._module_format(self._module + '.logical_not'), ','.join(self._print(i) for i in expr.args)) + + def _print_Pow(self, expr, rational=False): + # XXX Workaround for negative integer power error + if expr.exp.is_integer and expr.exp.is_negative: + expr = Pow(expr.base, expr.exp.evalf(), evaluate=False) + return self._hprint_Pow(expr, rational=rational, sqrt=self._module + '.sqrt') + + def _helper_minimum_maximum(self, op: str, *args): + if len(args) == 0: + raise NotImplementedError(f"Need at least one argument for {op}") + elif len(args) == 1: + return self._print(args[0]) + _reduce = self._module_format('functools.reduce') + s_args = [self._print(arg) for arg in args] + return f"{_reduce}({op}, [{', '.join(s_args)}])" + + def _print_Min(self, expr): + return self._print_minimum(expr) + + def _print_amin(self, expr): + return '{}({}, axis={})'.format(self._module_format(self._module + '.amin'), self._print(expr.array), self._print(expr.axis)) + + def _print_minimum(self, expr): + op = self._module_format(self._module + '.minimum') + return self._helper_minimum_maximum(op, *expr.args) + + def _print_Max(self, expr): + return self._print_maximum(expr) + + def _print_amax(self, expr): + return '{}({}, axis={})'.format(self._module_format(self._module + '.amax'), self._print(expr.array), self._print(expr.axis)) + + def _print_maximum(self, expr): + op = self._module_format(self._module + '.maximum') + return self._helper_minimum_maximum(op, *expr.args) + + def _print_arg(self, expr): + return "%s(%s)" % (self._module_format(self._module + '.angle'), self._print(expr.args[0])) + + def _print_im(self, expr): + return "%s(%s)" % (self._module_format(self._module + '.imag'), self._print(expr.args[0])) + + def _print_Mod(self, expr): + return "%s(%s)" % (self._module_format(self._module + '.mod'), ', '.join( + (self._print(arg) for arg in expr.args))) + + def _print_re(self, expr): + return "%s(%s)" % (self._module_format(self._module + '.real'), self._print(expr.args[0])) + + def _print_sinc(self, expr): + return "%s(%s)" % (self._module_format(self._module + '.sinc'), self._print(expr.args[0]/S.Pi)) + + def _print_MatrixBase(self, expr): + if 0 in expr.shape: + func = self._module_format(f'{self._module}.{self._zeros}') + return f"{func}({self._print(expr.shape)})" + func = self.known_functions.get(expr.__class__.__name__, None) + if func is None: + func = self._module_format(f'{self._module}.array') + return "%s(%s)" % (func, self._print(expr.tolist())) + + def _print_Identity(self, expr): + shape = expr.shape + if all(dim.is_Integer for dim in shape): + return "%s(%s)" % (self._module_format(self._module + '.eye'), self._print(expr.shape[0])) + else: + raise NotImplementedError("Symbolic matrix dimensions are not yet supported for identity matrices") + + def _print_BlockMatrix(self, expr): + return '{}({})'.format(self._module_format(self._module + '.block'), + self._print(expr.args[0].tolist())) + + def _print_NDimArray(self, expr): + if expr.rank() == 0: + func = self._module_format(f'{self._module}.array') + return f"{func}({self._print(expr[()])})" + if 0 in expr.shape: + func = self._module_format(f'{self._module}.{self._zeros}') + return f"{func}({self._print(expr.shape)})" + func = self._module_format(f'{self._module}.array') + return f"{func}({self._print(expr.tolist())})" + + _add = "add" + _einsum = "einsum" + _transpose = "transpose" + _ones = "ones" + _zeros = "zeros" + + _print_lowergamma = CodePrinter._print_not_supported + _print_uppergamma = CodePrinter._print_not_supported + _print_fresnelc = CodePrinter._print_not_supported + _print_fresnels = CodePrinter._print_not_supported + +for func in _numpy_known_functions: + setattr(NumPyPrinter, f'_print_{func}', _print_known_func) + +for const in _numpy_known_constants: + setattr(NumPyPrinter, f'_print_{const}', _print_known_const) + + +_known_functions_scipy_special = { + 'Ei': 'expi', + 'erf': 'erf', + 'erfc': 'erfc', + 'besselj': 'jv', + 'bessely': 'yv', + 'besseli': 'iv', + 'besselk': 'kv', + 'cosm1': 'cosm1', + 'powm1': 'powm1', + 'factorial': 'factorial', + 'gamma': 'gamma', + 'loggamma': 'gammaln', + 'digamma': 'psi', + 'polygamma': 'polygamma', + 'RisingFactorial': 'poch', + 'jacobi': 'eval_jacobi', + 'gegenbauer': 'eval_gegenbauer', + 'chebyshevt': 'eval_chebyt', + 'chebyshevu': 'eval_chebyu', + 'legendre': 'eval_legendre', + 'hermite': 'eval_hermite', + 'laguerre': 'eval_laguerre', + 'assoc_laguerre': 'eval_genlaguerre', + 'beta': 'beta', + 'LambertW' : 'lambertw', +} + +_known_constants_scipy_constants = { + 'GoldenRatio': 'golden_ratio', + 'Pi': 'pi', +} +_scipy_known_functions = {k : "scipy.special." + v for k, v in _known_functions_scipy_special.items()} +_scipy_known_constants = {k : "scipy.constants." + v for k, v in _known_constants_scipy_constants.items()} + +class SciPyPrinter(NumPyPrinter): + + _kf = {**NumPyPrinter._kf, **_scipy_known_functions} + _kc = {**NumPyPrinter._kc, **_scipy_known_constants} + + def __init__(self, settings=None): + super().__init__(settings=settings) + self.language = "Python with SciPy and NumPy" + + def _print_SparseRepMatrix(self, expr): + i, j, data = [], [], [] + for (r, c), v in expr.todok().items(): + i.append(r) + j.append(c) + data.append(v) + + return "{name}(({data}, ({i}, {j})), shape={shape})".format( + name=self._module_format('scipy.sparse.coo_matrix'), + data=data, i=i, j=j, shape=expr.shape + ) + + _print_ImmutableSparseMatrix = _print_SparseRepMatrix + + # SciPy's lpmv has a different order of arguments from assoc_legendre + def _print_assoc_legendre(self, expr): + return "{0}({2}, {1}, {3})".format( + self._module_format('scipy.special.lpmv'), + self._print(expr.args[0]), + self._print(expr.args[1]), + self._print(expr.args[2])) + + def _print_lowergamma(self, expr): + return "{0}({2})*{1}({2}, {3})".format( + self._module_format('scipy.special.gamma'), + self._module_format('scipy.special.gammainc'), + self._print(expr.args[0]), + self._print(expr.args[1])) + + def _print_uppergamma(self, expr): + return "{0}({2})*{1}({2}, {3})".format( + self._module_format('scipy.special.gamma'), + self._module_format('scipy.special.gammaincc'), + self._print(expr.args[0]), + self._print(expr.args[1])) + + def _print_betainc(self, expr): + betainc = self._module_format('scipy.special.betainc') + beta = self._module_format('scipy.special.beta') + args = [self._print(arg) for arg in expr.args] + return f"({betainc}({args[0]}, {args[1]}, {args[3]}) - {betainc}({args[0]}, {args[1]}, {args[2]})) \ + * {beta}({args[0]}, {args[1]})" + + def _print_betainc_regularized(self, expr): + return "{0}({1}, {2}, {4}) - {0}({1}, {2}, {3})".format( + self._module_format('scipy.special.betainc'), + self._print(expr.args[0]), + self._print(expr.args[1]), + self._print(expr.args[2]), + self._print(expr.args[3])) + + def _print_fresnels(self, expr): + return "{}({})[0]".format( + self._module_format("scipy.special.fresnel"), + self._print(expr.args[0])) + + def _print_fresnelc(self, expr): + return "{}({})[1]".format( + self._module_format("scipy.special.fresnel"), + self._print(expr.args[0])) + + def _print_airyai(self, expr): + return "{}({})[0]".format( + self._module_format("scipy.special.airy"), + self._print(expr.args[0])) + + def _print_airyaiprime(self, expr): + return "{}({})[1]".format( + self._module_format("scipy.special.airy"), + self._print(expr.args[0])) + + def _print_airybi(self, expr): + return "{}({})[2]".format( + self._module_format("scipy.special.airy"), + self._print(expr.args[0])) + + def _print_airybiprime(self, expr): + return "{}({})[3]".format( + self._module_format("scipy.special.airy"), + self._print(expr.args[0])) + + def _print_bernoulli(self, expr): + # scipy's bernoulli is inconsistent with SymPy's so rewrite + return self._print(expr._eval_rewrite_as_zeta(*expr.args)) + + def _print_harmonic(self, expr): + return self._print(expr._eval_rewrite_as_zeta(*expr.args)) + + def _print_Integral(self, e): + integration_vars, limits = _unpack_integral_limits(e) + + if len(limits) == 1: + # nicer (but not necessary) to prefer quad over nquad for 1D case + module_str = self._module_format("scipy.integrate.quad") + limit_str = "%s, %s" % tuple(map(self._print, limits[0])) + else: + module_str = self._module_format("scipy.integrate.nquad") + limit_str = "({})".format(", ".join( + "(%s, %s)" % tuple(map(self._print, l)) for l in limits)) + + return "{}(lambda {}: {}, {})[0]".format( + module_str, + ", ".join(map(self._print, integration_vars)), + self._print(e.args[0]), + limit_str) + + def _print_Si(self, expr): + return "{}({})[0]".format( + self._module_format("scipy.special.sici"), + self._print(expr.args[0])) + + def _print_Ci(self, expr): + return "{}({})[1]".format( + self._module_format("scipy.special.sici"), + self._print(expr.args[0])) + +for func in _scipy_known_functions: + setattr(SciPyPrinter, f'_print_{func}', _print_known_func) + +for const in _scipy_known_constants: + setattr(SciPyPrinter, f'_print_{const}', _print_known_const) + + +_cupy_known_functions = {k : "cupy." + v for k, v in _known_functions_numpy.items()} +_cupy_known_constants = {k : "cupy." + v for k, v in _known_constants_numpy.items()} + +class CuPyPrinter(NumPyPrinter): + """ + CuPy printer which handles vectorized piecewise functions, + logical operators, etc. + """ + + _module = 'cupy' + _kf = _cupy_known_functions + _kc = _cupy_known_constants + + def __init__(self, settings=None): + super().__init__(settings=settings) + +for func in _cupy_known_functions: + setattr(CuPyPrinter, f'_print_{func}', _print_known_func) + +for const in _cupy_known_constants: + setattr(CuPyPrinter, f'_print_{const}', _print_known_const) + + +_jax_known_functions = {k: 'jax.numpy.' + v for k, v in _known_functions_numpy.items()} +_jax_known_constants = {k: 'jax.numpy.' + v for k, v in _known_constants_numpy.items()} + +class JaxPrinter(NumPyPrinter): + """ + JAX printer which handles vectorized piecewise functions, + logical operators, etc. + """ + _module = "jax.numpy" + + _kf = _jax_known_functions + _kc = _jax_known_constants + + def __init__(self, settings=None): + super().__init__(settings=settings) + self.printmethod = '_jaxcode' + + # These need specific override to allow for the lack of "jax.numpy.reduce" + def _print_And(self, expr): + "Logical And printer" + return "{}({}.asarray([{}]), axis=0)".format( + self._module_format(self._module + ".all"), + self._module_format(self._module), + ",".join(self._print(i) for i in expr.args), + ) + + def _print_Or(self, expr): + "Logical Or printer" + return "{}({}.asarray([{}]), axis=0)".format( + self._module_format(self._module + ".any"), + self._module_format(self._module), + ",".join(self._print(i) for i in expr.args), + ) + +for func in _jax_known_functions: + setattr(JaxPrinter, f'_print_{func}', _print_known_func) + +for const in _jax_known_constants: + setattr(JaxPrinter, f'_print_{const}', _print_known_const) diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/octave.py b/.venv/lib/python3.12/site-packages/sympy/printing/octave.py new file mode 100644 index 0000000000000000000000000000000000000000..2cf2d6a5754668d7a95ef5dc7b27b4864756a9e5 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/octave.py @@ -0,0 +1,711 @@ +""" +Octave (and Matlab) code printer + +The `OctaveCodePrinter` converts SymPy expressions into Octave expressions. +It uses a subset of the Octave language for Matlab compatibility. + +A complete code generator, which uses `octave_code` extensively, can be found +in `sympy.utilities.codegen`. The `codegen` module can be used to generate +complete source code files. + +""" + +from __future__ import annotations +from typing import Any + +from sympy.core import Mul, Pow, S, Rational +from sympy.core.mul import _keep_coeff +from sympy.core.numbers import equal_valued +from sympy.printing.codeprinter import CodePrinter +from sympy.printing.precedence import precedence, PRECEDENCE +from re import search + +# List of known functions. First, those that have the same name in +# SymPy and Octave. This is almost certainly incomplete! +known_fcns_src1 = ["sin", "cos", "tan", "cot", "sec", "csc", + "asin", "acos", "acot", "atan", "atan2", "asec", "acsc", + "sinh", "cosh", "tanh", "coth", "csch", "sech", + "asinh", "acosh", "atanh", "acoth", "asech", "acsch", + "erfc", "erfi", "erf", "erfinv", "erfcinv", + "besseli", "besselj", "besselk", "bessely", + "bernoulli", "beta", "euler", "exp", "factorial", "floor", + "fresnelc", "fresnels", "gamma", "harmonic", "log", + "polylog", "sign", "zeta", "legendre"] + +# These functions have different names ("SymPy": "Octave"), more +# generally a mapping to (argument_conditions, octave_function). +known_fcns_src2 = { + "Abs": "abs", + "arg": "angle", # arg/angle ok in Octave but only angle in Matlab + "binomial": "bincoeff", + "ceiling": "ceil", + "chebyshevu": "chebyshevU", + "chebyshevt": "chebyshevT", + "Chi": "coshint", + "Ci": "cosint", + "conjugate": "conj", + "DiracDelta": "dirac", + "Heaviside": "heaviside", + "im": "imag", + "laguerre": "laguerreL", + "LambertW": "lambertw", + "li": "logint", + "loggamma": "gammaln", + "Max": "max", + "Min": "min", + "Mod": "mod", + "polygamma": "psi", + "re": "real", + "RisingFactorial": "pochhammer", + "Shi": "sinhint", + "Si": "sinint", +} + + +class OctaveCodePrinter(CodePrinter): + """ + A printer to convert expressions to strings of Octave/Matlab code. + """ + printmethod = "_octave" + language = "Octave" + + _operators = { + 'and': '&', + 'or': '|', + 'not': '~', + } + + _default_settings: dict[str, Any] = dict(CodePrinter._default_settings, **{ + 'precision': 17, + 'user_functions': {}, + 'contract': True, + 'inline': True, + }) + # Note: contract is for expressing tensors as loops (if True), or just + # assignment (if False). FIXME: this should be looked a more carefully + # for Octave. + + + def __init__(self, settings={}): + super().__init__(settings) + self.known_functions = dict(zip(known_fcns_src1, known_fcns_src1)) + self.known_functions.update(dict(known_fcns_src2)) + userfuncs = settings.get('user_functions', {}) + self.known_functions.update(userfuncs) + + + def _rate_index_position(self, p): + return p*5 + + + def _get_statement(self, codestring): + return "%s;" % codestring + + + def _get_comment(self, text): + return "% {}".format(text) + + + def _declare_number_const(self, name, value): + return "{} = {};".format(name, value) + + + def _format_code(self, lines): + return self.indent_code(lines) + + + def _traverse_matrix_indices(self, mat): + # Octave uses Fortran order (column-major) + rows, cols = mat.shape + return ((i, j) for j in range(cols) for i in range(rows)) + + + def _get_loop_opening_ending(self, indices): + open_lines = [] + close_lines = [] + for i in indices: + # Octave arrays start at 1 and end at dimension + var, start, stop = map(self._print, + [i.label, i.lower + 1, i.upper + 1]) + open_lines.append("for %s = %s:%s" % (var, start, stop)) + close_lines.append("end") + return open_lines, close_lines + + + def _print_Mul(self, expr): + # print complex numbers nicely in Octave + if (expr.is_number and expr.is_imaginary and + (S.ImaginaryUnit*expr).is_Integer): + return "%si" % self._print(-S.ImaginaryUnit*expr) + + # cribbed from str.py + prec = precedence(expr) + + c, e = expr.as_coeff_Mul() + if c < 0: + expr = _keep_coeff(-c, e) + sign = "-" + else: + sign = "" + + a = [] # items in the numerator + b = [] # items that are in the denominator (if any) + + pow_paren = [] # Will collect all pow with more than one base element and exp = -1 + + if self.order not in ('old', 'none'): + args = expr.as_ordered_factors() + else: + # use make_args in case expr was something like -x -> x + args = Mul.make_args(expr) + + # Gather args for numerator/denominator + for item in args: + if (item.is_commutative and item.is_Pow and item.exp.is_Rational + and item.exp.is_negative): + if item.exp != -1: + b.append(Pow(item.base, -item.exp, evaluate=False)) + else: + if len(item.args[0].args) != 1 and isinstance(item.base, Mul): # To avoid situations like #14160 + pow_paren.append(item) + b.append(Pow(item.base, -item.exp)) + elif item.is_Rational and item is not S.Infinity: + if item.p != 1: + a.append(Rational(item.p)) + if item.q != 1: + b.append(Rational(item.q)) + else: + a.append(item) + + a = a or [S.One] + + a_str = [self.parenthesize(x, prec) for x in a] + b_str = [self.parenthesize(x, prec) for x in b] + + # To parenthesize Pow with exp = -1 and having more than one Symbol + for item in pow_paren: + if item.base in b: + b_str[b.index(item.base)] = "(%s)" % b_str[b.index(item.base)] + + # from here it differs from str.py to deal with "*" and ".*" + def multjoin(a, a_str): + # here we probably are assuming the constants will come first + r = a_str[0] + for i in range(1, len(a)): + mulsym = '*' if a[i-1].is_number else '.*' + r = r + mulsym + a_str[i] + return r + + if not b: + return sign + multjoin(a, a_str) + elif len(b) == 1: + divsym = '/' if b[0].is_number else './' + return sign + multjoin(a, a_str) + divsym + b_str[0] + else: + divsym = '/' if all(bi.is_number for bi in b) else './' + return (sign + multjoin(a, a_str) + + divsym + "(%s)" % multjoin(b, b_str)) + + def _print_Relational(self, expr): + lhs_code = self._print(expr.lhs) + rhs_code = self._print(expr.rhs) + op = expr.rel_op + return "{} {} {}".format(lhs_code, op, rhs_code) + + def _print_Pow(self, expr): + powsymbol = '^' if all(x.is_number for x in expr.args) else '.^' + + PREC = precedence(expr) + + if equal_valued(expr.exp, 0.5): + return "sqrt(%s)" % self._print(expr.base) + + if expr.is_commutative: + if equal_valued(expr.exp, -0.5): + sym = '/' if expr.base.is_number else './' + return "1" + sym + "sqrt(%s)" % self._print(expr.base) + if equal_valued(expr.exp, -1): + sym = '/' if expr.base.is_number else './' + return "1" + sym + "%s" % self.parenthesize(expr.base, PREC) + + return '%s%s%s' % (self.parenthesize(expr.base, PREC), powsymbol, + self.parenthesize(expr.exp, PREC)) + + + def _print_MatPow(self, expr): + PREC = precedence(expr) + return '%s^%s' % (self.parenthesize(expr.base, PREC), + self.parenthesize(expr.exp, PREC)) + + def _print_MatrixSolve(self, expr): + PREC = precedence(expr) + return "%s \\ %s" % (self.parenthesize(expr.matrix, PREC), + self.parenthesize(expr.vector, PREC)) + + def _print_Pi(self, expr): + return 'pi' + + + def _print_ImaginaryUnit(self, expr): + return "1i" + + + def _print_Exp1(self, expr): + return "exp(1)" + + + def _print_GoldenRatio(self, expr): + # FIXME: how to do better, e.g., for octave_code(2*GoldenRatio)? + #return self._print((1+sqrt(S(5)))/2) + return "(1+sqrt(5))/2" + + + def _print_Assignment(self, expr): + from sympy.codegen.ast import Assignment + from sympy.functions.elementary.piecewise import Piecewise + from sympy.tensor.indexed import IndexedBase + # Copied from codeprinter, but remove special MatrixSymbol treatment + lhs = expr.lhs + rhs = expr.rhs + # We special case assignments that take multiple lines + if not self._settings["inline"] and isinstance(expr.rhs, Piecewise): + # Here we modify Piecewise so each expression is now + # an Assignment, and then continue on the print. + expressions = [] + conditions = [] + for (e, c) in rhs.args: + expressions.append(Assignment(lhs, e)) + conditions.append(c) + temp = Piecewise(*zip(expressions, conditions)) + return self._print(temp) + if self._settings["contract"] and (lhs.has(IndexedBase) or + rhs.has(IndexedBase)): + # Here we check if there is looping to be done, and if so + # print the required loops. + return self._doprint_loops(rhs, lhs) + else: + lhs_code = self._print(lhs) + rhs_code = self._print(rhs) + return self._get_statement("%s = %s" % (lhs_code, rhs_code)) + + + def _print_Infinity(self, expr): + return 'inf' + + + def _print_NegativeInfinity(self, expr): + return '-inf' + + + def _print_NaN(self, expr): + return 'NaN' + + + def _print_list(self, expr): + return '{' + ', '.join(self._print(a) for a in expr) + '}' + _print_tuple = _print_list + _print_Tuple = _print_list + _print_List = _print_list + + + def _print_BooleanTrue(self, expr): + return "true" + + + def _print_BooleanFalse(self, expr): + return "false" + + + def _print_bool(self, expr): + return str(expr).lower() + + + # Could generate quadrature code for definite Integrals? + #_print_Integral = _print_not_supported + + + def _print_MatrixBase(self, A): + # Handle zero dimensions: + if (A.rows, A.cols) == (0, 0): + return '[]' + elif S.Zero in A.shape: + return 'zeros(%s, %s)' % (A.rows, A.cols) + elif (A.rows, A.cols) == (1, 1): + # Octave does not distinguish between scalars and 1x1 matrices + return self._print(A[0, 0]) + return "[%s]" % "; ".join(" ".join([self._print(a) for a in A[r, :]]) + for r in range(A.rows)) + + + def _print_SparseRepMatrix(self, A): + from sympy.matrices import Matrix + L = A.col_list() + # make row vectors of the indices and entries + I = Matrix([[k[0] + 1 for k in L]]) + J = Matrix([[k[1] + 1 for k in L]]) + AIJ = Matrix([[k[2] for k in L]]) + return "sparse(%s, %s, %s, %s, %s)" % (self._print(I), self._print(J), + self._print(AIJ), A.rows, A.cols) + + + def _print_MatrixElement(self, expr): + return self.parenthesize(expr.parent, PRECEDENCE["Atom"], strict=True) \ + + '(%s, %s)' % (expr.i + 1, expr.j + 1) + + + def _print_MatrixSlice(self, expr): + def strslice(x, lim): + l = x[0] + 1 + h = x[1] + step = x[2] + lstr = self._print(l) + hstr = 'end' if h == lim else self._print(h) + if step == 1: + if l == 1 and h == lim: + return ':' + if l == h: + return lstr + else: + return lstr + ':' + hstr + else: + return ':'.join((lstr, self._print(step), hstr)) + return (self._print(expr.parent) + '(' + + strslice(expr.rowslice, expr.parent.shape[0]) + ', ' + + strslice(expr.colslice, expr.parent.shape[1]) + ')') + + + def _print_Indexed(self, expr): + inds = [ self._print(i) for i in expr.indices ] + return "%s(%s)" % (self._print(expr.base.label), ", ".join(inds)) + + + def _print_KroneckerDelta(self, expr): + prec = PRECEDENCE["Pow"] + return "double(%s == %s)" % tuple(self.parenthesize(x, prec) + for x in expr.args) + + def _print_HadamardProduct(self, expr): + return '.*'.join([self.parenthesize(arg, precedence(expr)) + for arg in expr.args]) + + def _print_HadamardPower(self, expr): + PREC = precedence(expr) + return '.**'.join([ + self.parenthesize(expr.base, PREC), + self.parenthesize(expr.exp, PREC) + ]) + + def _print_Identity(self, expr): + shape = expr.shape + if len(shape) == 2 and shape[0] == shape[1]: + shape = [shape[0]] + s = ", ".join(self._print(n) for n in shape) + return "eye(" + s + ")" + + def _print_lowergamma(self, expr): + # Octave implements regularized incomplete gamma function + return "(gammainc({1}, {0}).*gamma({0}))".format( + self._print(expr.args[0]), self._print(expr.args[1])) + + + def _print_uppergamma(self, expr): + return "(gammainc({1}, {0}, 'upper').*gamma({0}))".format( + self._print(expr.args[0]), self._print(expr.args[1])) + + + def _print_sinc(self, expr): + #Note: Divide by pi because Octave implements normalized sinc function. + return "sinc(%s)" % self._print(expr.args[0]/S.Pi) + + + def _print_hankel1(self, expr): + return "besselh(%s, 1, %s)" % (self._print(expr.order), + self._print(expr.argument)) + + + def _print_hankel2(self, expr): + return "besselh(%s, 2, %s)" % (self._print(expr.order), + self._print(expr.argument)) + + + # Note: as of 2015, Octave doesn't have spherical Bessel functions + def _print_jn(self, expr): + from sympy.functions import sqrt, besselj + x = expr.argument + expr2 = sqrt(S.Pi/(2*x))*besselj(expr.order + S.Half, x) + return self._print(expr2) + + + def _print_yn(self, expr): + from sympy.functions import sqrt, bessely + x = expr.argument + expr2 = sqrt(S.Pi/(2*x))*bessely(expr.order + S.Half, x) + return self._print(expr2) + + + def _print_airyai(self, expr): + return "airy(0, %s)" % self._print(expr.args[0]) + + + def _print_airyaiprime(self, expr): + return "airy(1, %s)" % self._print(expr.args[0]) + + + def _print_airybi(self, expr): + return "airy(2, %s)" % self._print(expr.args[0]) + + + def _print_airybiprime(self, expr): + return "airy(3, %s)" % self._print(expr.args[0]) + + + def _print_expint(self, expr): + mu, x = expr.args + if mu != 1: + return self._print_not_supported(expr) + return "expint(%s)" % self._print(x) + + + def _one_or_two_reversed_args(self, expr): + assert len(expr.args) <= 2 + return '{name}({args})'.format( + name=self.known_functions[expr.__class__.__name__], + args=", ".join([self._print(x) for x in reversed(expr.args)]) + ) + + + _print_DiracDelta = _print_LambertW = _one_or_two_reversed_args + + + def _nested_binary_math_func(self, expr): + return '{name}({arg1}, {arg2})'.format( + name=self.known_functions[expr.__class__.__name__], + arg1=self._print(expr.args[0]), + arg2=self._print(expr.func(*expr.args[1:])) + ) + + _print_Max = _print_Min = _nested_binary_math_func + + + def _print_Piecewise(self, expr): + if expr.args[-1].cond != True: + # We need the last conditional to be a True, otherwise the resulting + # function may not return a result. + raise ValueError("All Piecewise expressions must contain an " + "(expr, True) statement to be used as a default " + "condition. Without one, the generated " + "expression may not evaluate to anything under " + "some condition.") + lines = [] + if self._settings["inline"]: + # Express each (cond, expr) pair in a nested Horner form: + # (condition) .* (expr) + (not cond) .* () + # Expressions that result in multiple statements won't work here. + ecpairs = ["({0}).*({1}) + (~({0})).*(".format + (self._print(c), self._print(e)) + for e, c in expr.args[:-1]] + elast = "%s" % self._print(expr.args[-1].expr) + pw = " ...\n".join(ecpairs) + elast + ")"*len(ecpairs) + # Note: current need these outer brackets for 2*pw. Would be + # nicer to teach parenthesize() to do this for us when needed! + return "(" + pw + ")" + else: + for i, (e, c) in enumerate(expr.args): + if i == 0: + lines.append("if (%s)" % self._print(c)) + elif i == len(expr.args) - 1 and c == True: + lines.append("else") + else: + lines.append("elseif (%s)" % self._print(c)) + code0 = self._print(e) + lines.append(code0) + if i == len(expr.args) - 1: + lines.append("end") + return "\n".join(lines) + + + def _print_zeta(self, expr): + if len(expr.args) == 1: + return "zeta(%s)" % self._print(expr.args[0]) + else: + # Matlab two argument zeta is not equivalent to SymPy's + return self._print_not_supported(expr) + + + def indent_code(self, code): + """Accepts a string of code or a list of code lines""" + + # code mostly copied from ccode + if isinstance(code, str): + code_lines = self.indent_code(code.splitlines(True)) + return ''.join(code_lines) + + tab = " " + inc_regex = ('^function ', '^if ', '^elseif ', '^else$', '^for ') + dec_regex = ('^end$', '^elseif ', '^else$') + + # pre-strip left-space from the code + code = [ line.lstrip(' \t') for line in code ] + + increase = [ int(any(search(re, line) for re in inc_regex)) + for line in code ] + decrease = [ int(any(search(re, line) for re in dec_regex)) + for line in code ] + + pretty = [] + level = 0 + for n, line in enumerate(code): + if line in ('', '\n'): + pretty.append(line) + continue + level -= decrease[n] + pretty.append("%s%s" % (tab*level, line)) + level += increase[n] + return pretty + + +def octave_code(expr, assign_to=None, **settings): + r"""Converts `expr` to a string of Octave (or Matlab) code. + + The string uses a subset of the Octave language for Matlab compatibility. + + Parameters + ========== + + expr : Expr + A SymPy expression to be converted. + assign_to : optional + When given, the argument is used as the name of the variable to which + the expression is assigned. Can be a string, ``Symbol``, + ``MatrixSymbol``, or ``Indexed`` type. This can be helpful for + expressions that generate multi-line statements. + precision : integer, optional + The precision for numbers such as pi [default=16]. + user_functions : dict, optional + A dictionary where keys are ``FunctionClass`` instances and values are + their string representations. Alternatively, the dictionary value can + be a list of tuples i.e. [(argument_test, cfunction_string)]. See + below for examples. + human : bool, optional + If True, the result is a single string that may contain some constant + declarations for the number symbols. If False, the same information is + returned in a tuple of (symbols_to_declare, not_supported_functions, + code_text). [default=True]. + contract: bool, optional + If True, ``Indexed`` instances are assumed to obey tensor contraction + rules and the corresponding nested loops over indices are generated. + Setting contract=False will not generate loops, instead the user is + responsible to provide values for the indices in the code. + [default=True]. + inline: bool, optional + If True, we try to create single-statement code instead of multiple + statements. [default=True]. + + Examples + ======== + + >>> from sympy import octave_code, symbols, sin, pi + >>> x = symbols('x') + >>> octave_code(sin(x).series(x).removeO()) + 'x.^5/120 - x.^3/6 + x' + + >>> from sympy import Rational, ceiling + >>> x, y, tau = symbols("x, y, tau") + >>> octave_code((2*tau)**Rational(7, 2)) + '8*sqrt(2)*tau.^(7/2)' + + Note that element-wise (Hadamard) operations are used by default between + symbols. This is because its very common in Octave to write "vectorized" + code. It is harmless if the values are scalars. + + >>> octave_code(sin(pi*x*y), assign_to="s") + 's = sin(pi*x.*y);' + + If you need a matrix product "*" or matrix power "^", you can specify the + symbol as a ``MatrixSymbol``. + + >>> from sympy import Symbol, MatrixSymbol + >>> n = Symbol('n', integer=True, positive=True) + >>> A = MatrixSymbol('A', n, n) + >>> octave_code(3*pi*A**3) + '(3*pi)*A^3' + + This class uses several rules to decide which symbol to use a product. + Pure numbers use "*", Symbols use ".*" and MatrixSymbols use "*". + A HadamardProduct can be used to specify componentwise multiplication ".*" + of two MatrixSymbols. There is currently there is no easy way to specify + scalar symbols, so sometimes the code might have some minor cosmetic + issues. For example, suppose x and y are scalars and A is a Matrix, then + while a human programmer might write "(x^2*y)*A^3", we generate: + + >>> octave_code(x**2*y*A**3) + '(x.^2.*y)*A^3' + + Matrices are supported using Octave inline notation. When using + ``assign_to`` with matrices, the name can be specified either as a string + or as a ``MatrixSymbol``. The dimensions must align in the latter case. + + >>> from sympy import Matrix, MatrixSymbol + >>> mat = Matrix([[x**2, sin(x), ceiling(x)]]) + >>> octave_code(mat, assign_to='A') + 'A = [x.^2 sin(x) ceil(x)];' + + ``Piecewise`` expressions are implemented with logical masking by default. + Alternatively, you can pass "inline=False" to use if-else conditionals. + Note that if the ``Piecewise`` lacks a default term, represented by + ``(expr, True)`` then an error will be thrown. This is to prevent + generating an expression that may not evaluate to anything. + + >>> from sympy import Piecewise + >>> pw = Piecewise((x + 1, x > 0), (x, True)) + >>> octave_code(pw, assign_to=tau) + 'tau = ((x > 0).*(x + 1) + (~(x > 0)).*(x));' + + Note that any expression that can be generated normally can also exist + inside a Matrix: + + >>> mat = Matrix([[x**2, pw, sin(x)]]) + >>> octave_code(mat, assign_to='A') + 'A = [x.^2 ((x > 0).*(x + 1) + (~(x > 0)).*(x)) sin(x)];' + + Custom printing can be defined for certain types by passing a dictionary of + "type" : "function" to the ``user_functions`` kwarg. Alternatively, the + dictionary value can be a list of tuples i.e., [(argument_test, + cfunction_string)]. This can be used to call a custom Octave function. + + >>> from sympy import Function + >>> f = Function('f') + >>> g = Function('g') + >>> custom_functions = { + ... "f": "existing_octave_fcn", + ... "g": [(lambda x: x.is_Matrix, "my_mat_fcn"), + ... (lambda x: not x.is_Matrix, "my_fcn")] + ... } + >>> mat = Matrix([[1, x]]) + >>> octave_code(f(x) + g(x) + g(mat), user_functions=custom_functions) + 'existing_octave_fcn(x) + my_fcn(x) + my_mat_fcn([1 x])' + + Support for loops is provided through ``Indexed`` types. With + ``contract=True`` these expressions will be turned into loops, whereas + ``contract=False`` will just print the assignment expression that should be + looped over: + + >>> from sympy import Eq, IndexedBase, Idx + >>> len_y = 5 + >>> y = IndexedBase('y', shape=(len_y,)) + >>> t = IndexedBase('t', shape=(len_y,)) + >>> Dy = IndexedBase('Dy', shape=(len_y-1,)) + >>> i = Idx('i', len_y-1) + >>> e = Eq(Dy[i], (y[i+1]-y[i])/(t[i+1]-t[i])) + >>> octave_code(e.rhs, assign_to=e.lhs, contract=False) + 'Dy(i) = (y(i + 1) - y(i))./(t(i + 1) - t(i));' + """ + return OctaveCodePrinter(settings).doprint(expr, assign_to) + + +def print_octave_code(expr, **settings): + """Prints the Octave (or Matlab) representation of the given expression. + + See `octave_code` for the meaning of the optional arguments. + """ + print(octave_code(expr, **settings)) diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/precedence.py b/.venv/lib/python3.12/site-packages/sympy/printing/precedence.py new file mode 100644 index 0000000000000000000000000000000000000000..d22d5746aeee51bddcf273d4575c30c3c27db71a --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/precedence.py @@ -0,0 +1,180 @@ +"""A module providing information about the necessity of brackets""" + + +# Default precedence values for some basic types +PRECEDENCE = { + "Lambda": 1, + "Xor": 10, + "Or": 20, + "And": 30, + "Relational": 35, + "Add": 40, + "Mul": 50, + "Pow": 60, + "Func": 70, + "Not": 100, + "Atom": 1000, + "BitwiseOr": 36, + "BitwiseXor": 37, + "BitwiseAnd": 38 +} + +# A dictionary assigning precedence values to certain classes. These values are +# treated like they were inherited, so not every single class has to be named +# here. +# Do not use this with printers other than StrPrinter +PRECEDENCE_VALUES = { + "Equivalent": PRECEDENCE["Xor"], + "Xor": PRECEDENCE["Xor"], + "Implies": PRECEDENCE["Xor"], + "Or": PRECEDENCE["Or"], + "And": PRECEDENCE["And"], + "Add": PRECEDENCE["Add"], + "Pow": PRECEDENCE["Pow"], + "Relational": PRECEDENCE["Relational"], + "Sub": PRECEDENCE["Add"], + "Not": PRECEDENCE["Not"], + "Function" : PRECEDENCE["Func"], + "NegativeInfinity": PRECEDENCE["Add"], + "MatAdd": PRECEDENCE["Add"], + "MatPow": PRECEDENCE["Pow"], + "MatrixSolve": PRECEDENCE["Mul"], + "Mod": PRECEDENCE["Mul"], + "TensAdd": PRECEDENCE["Add"], + # As soon as `TensMul` is a subclass of `Mul`, remove this: + "TensMul": PRECEDENCE["Mul"], + "HadamardProduct": PRECEDENCE["Mul"], + "HadamardPower": PRECEDENCE["Pow"], + "KroneckerProduct": PRECEDENCE["Mul"], + "Equality": PRECEDENCE["Mul"], + "Unequality": PRECEDENCE["Mul"], +} + +# Sometimes it's not enough to assign a fixed precedence value to a +# class. Then a function can be inserted in this dictionary that takes +# an instance of this class as argument and returns the appropriate +# precedence value. + +# Precedence functions + + +def precedence_Mul(item): + from sympy.core.function import Function + if any(hasattr(arg, 'precedence') and isinstance(arg, Function) and + arg.precedence < PRECEDENCE["Mul"] for arg in item.args): + return PRECEDENCE["Mul"] + + if item.could_extract_minus_sign(): + return PRECEDENCE["Add"] + return PRECEDENCE["Mul"] + + +def precedence_Rational(item): + if item.p < 0: + return PRECEDENCE["Add"] + return PRECEDENCE["Mul"] + + +def precedence_Integer(item): + if item.p < 0: + return PRECEDENCE["Add"] + return PRECEDENCE["Atom"] + + +def precedence_Float(item): + if item < 0: + return PRECEDENCE["Add"] + return PRECEDENCE["Atom"] + + +def precedence_PolyElement(item): + if item.is_generator: + return PRECEDENCE["Atom"] + elif item.is_ground: + return precedence(item.coeff(1)) + elif item.is_term: + return PRECEDENCE["Mul"] + else: + return PRECEDENCE["Add"] + + +def precedence_FracElement(item): + if item.denom == 1: + return precedence_PolyElement(item.numer) + else: + return PRECEDENCE["Mul"] + + +def precedence_UnevaluatedExpr(item): + return precedence(item.args[0]) - 0.5 + + +PRECEDENCE_FUNCTIONS = { + "Integer": precedence_Integer, + "Mul": precedence_Mul, + "Rational": precedence_Rational, + "Float": precedence_Float, + "PolyElement": precedence_PolyElement, + "FracElement": precedence_FracElement, + "UnevaluatedExpr": precedence_UnevaluatedExpr, +} + + +def precedence(item): + """Returns the precedence of a given object. + + This is the precedence for StrPrinter. + """ + if hasattr(item, "precedence"): + return item.precedence + if not isinstance(item, type): + for i in type(item).mro(): + n = i.__name__ + if n in PRECEDENCE_FUNCTIONS: + return PRECEDENCE_FUNCTIONS[n](item) + elif n in PRECEDENCE_VALUES: + return PRECEDENCE_VALUES[n] + return PRECEDENCE["Atom"] + + +PRECEDENCE_TRADITIONAL = PRECEDENCE.copy() +PRECEDENCE_TRADITIONAL['Integral'] = PRECEDENCE["Mul"] +PRECEDENCE_TRADITIONAL['Sum'] = PRECEDENCE["Mul"] +PRECEDENCE_TRADITIONAL['Product'] = PRECEDENCE["Mul"] +PRECEDENCE_TRADITIONAL['Limit'] = PRECEDENCE["Mul"] +PRECEDENCE_TRADITIONAL['Derivative'] = PRECEDENCE["Mul"] +PRECEDENCE_TRADITIONAL['TensorProduct'] = PRECEDENCE["Mul"] +PRECEDENCE_TRADITIONAL['Transpose'] = PRECEDENCE["Pow"] +PRECEDENCE_TRADITIONAL['Adjoint'] = PRECEDENCE["Pow"] +PRECEDENCE_TRADITIONAL['Dot'] = PRECEDENCE["Mul"] - 1 +PRECEDENCE_TRADITIONAL['Cross'] = PRECEDENCE["Mul"] - 1 +PRECEDENCE_TRADITIONAL['Gradient'] = PRECEDENCE["Mul"] - 1 +PRECEDENCE_TRADITIONAL['Divergence'] = PRECEDENCE["Mul"] - 1 +PRECEDENCE_TRADITIONAL['Curl'] = PRECEDENCE["Mul"] - 1 +PRECEDENCE_TRADITIONAL['Laplacian'] = PRECEDENCE["Mul"] - 1 +PRECEDENCE_TRADITIONAL['Union'] = PRECEDENCE['Xor'] +PRECEDENCE_TRADITIONAL['Intersection'] = PRECEDENCE['Xor'] +PRECEDENCE_TRADITIONAL['Complement'] = PRECEDENCE['Xor'] +PRECEDENCE_TRADITIONAL['SymmetricDifference'] = PRECEDENCE['Xor'] +PRECEDENCE_TRADITIONAL['ProductSet'] = PRECEDENCE['Xor'] +PRECEDENCE_TRADITIONAL['DotProduct'] = PRECEDENCE_TRADITIONAL['Dot'] + + +def precedence_traditional(item): + """Returns the precedence of a given object according to the + traditional rules of mathematics. + + This is the precedence for the LaTeX and pretty printer. + """ + # Integral, Sum, Product, Limit have the precedence of Mul in LaTeX, + # the precedence of Atom for other printers: + from sympy.core.expr import UnevaluatedExpr + + if isinstance(item, UnevaluatedExpr): + return precedence_traditional(item.args[0]) + + n = item.__class__.__name__ + if n in PRECEDENCE_TRADITIONAL: + return PRECEDENCE_TRADITIONAL[n] + + return precedence(item) diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/pretty/__init__.py b/.venv/lib/python3.12/site-packages/sympy/printing/pretty/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..cbabc649152a3c353a37225d342064634fbb5805 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/pretty/__init__.py @@ -0,0 +1,12 @@ +"""ASCII-ART 2D pretty-printer""" + +from .pretty import (pretty, pretty_print, pprint, pprint_use_unicode, + pprint_try_use_unicode, pager_print) + +# if unicode output is available -- let's use it +pprint_try_use_unicode() + +__all__ = [ + 'pretty', 'pretty_print', 'pprint', 'pprint_use_unicode', + 'pprint_try_use_unicode', 'pager_print', +] diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/pretty/__pycache__/__init__.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/printing/pretty/__pycache__/__init__.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..934119acebca57f9e6aa2feb51ccd9f3a6dbc452 Binary files /dev/null and b/.venv/lib/python3.12/site-packages/sympy/printing/pretty/__pycache__/__init__.cpython-312.pyc differ diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/pretty/__pycache__/pretty.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/printing/pretty/__pycache__/pretty.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..f2624cc2d4448bf7b46c200feac0d17a7e2872c1 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/pretty/__pycache__/pretty.cpython-312.pyc @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:56495a6b0c9c490de2ecce5f0bdd6a74d677caf38383c2563b431f62a4d5e90c +size 156851 diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/pretty/__pycache__/pretty_symbology.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/printing/pretty/__pycache__/pretty_symbology.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..131597352df3c4abb9b5b5b9cb45762998c5f283 Binary files /dev/null and b/.venv/lib/python3.12/site-packages/sympy/printing/pretty/__pycache__/pretty_symbology.cpython-312.pyc differ diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/pretty/__pycache__/stringpict.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/printing/pretty/__pycache__/stringpict.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..8f5fc23bfca075e3f3cb27bf610ea25228aa0466 Binary files /dev/null and b/.venv/lib/python3.12/site-packages/sympy/printing/pretty/__pycache__/stringpict.cpython-312.pyc differ diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/pretty/pretty.py b/.venv/lib/python3.12/site-packages/sympy/printing/pretty/pretty.py new file mode 100644 index 0000000000000000000000000000000000000000..b945f009119b24fc95e8452d91359957baba26a8 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/pretty/pretty.py @@ -0,0 +1,2937 @@ +import itertools + +from sympy.core import S +from sympy.core.add import Add +from sympy.core.containers import Tuple +from sympy.core.function import Function +from sympy.core.mul import Mul +from sympy.core.numbers import Number, Rational +from sympy.core.power import Pow +from sympy.core.sorting import default_sort_key +from sympy.core.symbol import Symbol +from sympy.core.sympify import SympifyError +from sympy.printing.conventions import requires_partial +from sympy.printing.precedence import PRECEDENCE, precedence, precedence_traditional +from sympy.printing.printer import Printer, print_function +from sympy.printing.str import sstr +from sympy.utilities.iterables import has_variety +from sympy.utilities.exceptions import sympy_deprecation_warning + +from sympy.printing.pretty.stringpict import prettyForm, stringPict +from sympy.printing.pretty.pretty_symbology import hobj, vobj, xobj, \ + xsym, pretty_symbol, pretty_atom, pretty_use_unicode, greek_unicode, U, \ + pretty_try_use_unicode, annotated, is_subscriptable_in_unicode, center_pad, root as nth_root + +# rename for usage from outside +pprint_use_unicode = pretty_use_unicode +pprint_try_use_unicode = pretty_try_use_unicode + + +class PrettyPrinter(Printer): + """Printer, which converts an expression into 2D ASCII-art figure.""" + printmethod = "_pretty" + + _default_settings = { + "order": None, + "full_prec": "auto", + "use_unicode": None, + "wrap_line": True, + "num_columns": None, + "use_unicode_sqrt_char": True, + "root_notation": True, + "mat_symbol_style": "plain", + "imaginary_unit": "i", + "perm_cyclic": True + } + + def __init__(self, settings=None): + Printer.__init__(self, settings) + + if not isinstance(self._settings['imaginary_unit'], str): + raise TypeError("'imaginary_unit' must a string, not {}".format(self._settings['imaginary_unit'])) + elif self._settings['imaginary_unit'] not in ("i", "j"): + raise ValueError("'imaginary_unit' must be either 'i' or 'j', not '{}'".format(self._settings['imaginary_unit'])) + + def emptyPrinter(self, expr): + return prettyForm(str(expr)) + + @property + def _use_unicode(self): + if self._settings['use_unicode']: + return True + else: + return pretty_use_unicode() + + def doprint(self, expr): + return self._print(expr).render(**self._settings) + + # empty op so _print(stringPict) returns the same + def _print_stringPict(self, e): + return e + + def _print_basestring(self, e): + return prettyForm(e) + + def _print_atan2(self, e): + pform = prettyForm(*self._print_seq(e.args).parens()) + pform = prettyForm(*pform.left('atan2')) + return pform + + def _print_Symbol(self, e, bold_name=False): + symb = pretty_symbol(e.name, bold_name) + return prettyForm(symb) + _print_RandomSymbol = _print_Symbol + def _print_MatrixSymbol(self, e): + return self._print_Symbol(e, self._settings['mat_symbol_style'] == "bold") + + def _print_Float(self, e): + # we will use StrPrinter's Float printer, but we need to handle the + # full_prec ourselves, according to the self._print_level + full_prec = self._settings["full_prec"] + if full_prec == "auto": + full_prec = self._print_level == 1 + return prettyForm(sstr(e, full_prec=full_prec)) + + def _print_Cross(self, e): + vec1 = e._expr1 + vec2 = e._expr2 + pform = self._print(vec2) + pform = prettyForm(*pform.left('(')) + pform = prettyForm(*pform.right(')')) + pform = prettyForm(*pform.left(self._print(U('MULTIPLICATION SIGN')))) + pform = prettyForm(*pform.left(')')) + pform = prettyForm(*pform.left(self._print(vec1))) + pform = prettyForm(*pform.left('(')) + return pform + + def _print_Curl(self, e): + vec = e._expr + pform = self._print(vec) + pform = prettyForm(*pform.left('(')) + pform = prettyForm(*pform.right(')')) + pform = prettyForm(*pform.left(self._print(U('MULTIPLICATION SIGN')))) + pform = prettyForm(*pform.left(self._print(U('NABLA')))) + return pform + + def _print_Divergence(self, e): + vec = e._expr + pform = self._print(vec) + pform = prettyForm(*pform.left('(')) + pform = prettyForm(*pform.right(')')) + pform = prettyForm(*pform.left(self._print(U('DOT OPERATOR')))) + pform = prettyForm(*pform.left(self._print(U('NABLA')))) + return pform + + def _print_Dot(self, e): + vec1 = e._expr1 + vec2 = e._expr2 + pform = self._print(vec2) + pform = prettyForm(*pform.left('(')) + pform = prettyForm(*pform.right(')')) + pform = prettyForm(*pform.left(self._print(U('DOT OPERATOR')))) + pform = prettyForm(*pform.left(')')) + pform = prettyForm(*pform.left(self._print(vec1))) + pform = prettyForm(*pform.left('(')) + return pform + + def _print_Gradient(self, e): + func = e._expr + pform = self._print(func) + pform = prettyForm(*pform.left('(')) + pform = prettyForm(*pform.right(')')) + pform = prettyForm(*pform.left(self._print(U('NABLA')))) + return pform + + def _print_Laplacian(self, e): + func = e._expr + pform = self._print(func) + pform = prettyForm(*pform.left('(')) + pform = prettyForm(*pform.right(')')) + pform = prettyForm(*pform.left(self._print(U('INCREMENT')))) + return pform + + def _print_Atom(self, e): + try: + # print atoms like Exp1 or Pi + return prettyForm(pretty_atom(e.__class__.__name__, printer=self)) + except KeyError: + return self.emptyPrinter(e) + + # Infinity inherits from Number, so we have to override _print_XXX order + _print_Infinity = _print_Atom + _print_NegativeInfinity = _print_Atom + _print_EmptySet = _print_Atom + _print_Naturals = _print_Atom + _print_Naturals0 = _print_Atom + _print_Integers = _print_Atom + _print_Rationals = _print_Atom + _print_Complexes = _print_Atom + + _print_EmptySequence = _print_Atom + + def _print_Reals(self, e): + if self._use_unicode: + return self._print_Atom(e) + else: + inf_list = ['-oo', 'oo'] + return self._print_seq(inf_list, '(', ')') + + def _print_subfactorial(self, e): + x = e.args[0] + pform = self._print(x) + # Add parentheses if needed + if not ((x.is_Integer and x.is_nonnegative) or x.is_Symbol): + pform = prettyForm(*pform.parens()) + pform = prettyForm(*pform.left('!')) + return pform + + def _print_factorial(self, e): + x = e.args[0] + pform = self._print(x) + # Add parentheses if needed + if not ((x.is_Integer and x.is_nonnegative) or x.is_Symbol): + pform = prettyForm(*pform.parens()) + pform = prettyForm(*pform.right('!')) + return pform + + def _print_factorial2(self, e): + x = e.args[0] + pform = self._print(x) + # Add parentheses if needed + if not ((x.is_Integer and x.is_nonnegative) or x.is_Symbol): + pform = prettyForm(*pform.parens()) + pform = prettyForm(*pform.right('!!')) + return pform + + def _print_binomial(self, e): + n, k = e.args + + n_pform = self._print(n) + k_pform = self._print(k) + + bar = ' '*max(n_pform.width(), k_pform.width()) + + pform = prettyForm(*k_pform.above(bar)) + pform = prettyForm(*pform.above(n_pform)) + pform = prettyForm(*pform.parens('(', ')')) + + pform.baseline = (pform.baseline + 1)//2 + + return pform + + def _print_Relational(self, e): + op = prettyForm(' ' + xsym(e.rel_op) + ' ') + + l = self._print(e.lhs) + r = self._print(e.rhs) + pform = prettyForm(*stringPict.next(l, op, r), binding=prettyForm.OPEN) + return pform + + def _print_Not(self, e): + from sympy.logic.boolalg import (Equivalent, Implies) + if self._use_unicode: + arg = e.args[0] + pform = self._print(arg) + if isinstance(arg, Equivalent): + return self._print_Equivalent(arg, altchar=pretty_atom('NotEquiv')) + if isinstance(arg, Implies): + return self._print_Implies(arg, altchar=pretty_atom('NotArrow')) + + if arg.is_Boolean and not arg.is_Not: + pform = prettyForm(*pform.parens()) + + return prettyForm(*pform.left(pretty_atom('Not'))) + else: + return self._print_Function(e) + + def __print_Boolean(self, e, char, sort=True): + args = e.args + if sort: + args = sorted(e.args, key=default_sort_key) + arg = args[0] + pform = self._print(arg) + + if arg.is_Boolean and not arg.is_Not: + pform = prettyForm(*pform.parens()) + + for arg in args[1:]: + pform_arg = self._print(arg) + + if arg.is_Boolean and not arg.is_Not: + pform_arg = prettyForm(*pform_arg.parens()) + + pform = prettyForm(*pform.right(' %s ' % char)) + pform = prettyForm(*pform.right(pform_arg)) + + return pform + + def _print_And(self, e): + if self._use_unicode: + return self.__print_Boolean(e, pretty_atom('And')) + else: + return self._print_Function(e, sort=True) + + def _print_Or(self, e): + if self._use_unicode: + return self.__print_Boolean(e, pretty_atom('Or')) + else: + return self._print_Function(e, sort=True) + + def _print_Xor(self, e): + if self._use_unicode: + return self.__print_Boolean(e, pretty_atom("Xor")) + else: + return self._print_Function(e, sort=True) + + def _print_Nand(self, e): + if self._use_unicode: + return self.__print_Boolean(e, pretty_atom('Nand')) + else: + return self._print_Function(e, sort=True) + + def _print_Nor(self, e): + if self._use_unicode: + return self.__print_Boolean(e, pretty_atom('Nor')) + else: + return self._print_Function(e, sort=True) + + def _print_Implies(self, e, altchar=None): + if self._use_unicode: + return self.__print_Boolean(e, altchar or pretty_atom('Arrow'), sort=False) + else: + return self._print_Function(e) + + def _print_Equivalent(self, e, altchar=None): + if self._use_unicode: + return self.__print_Boolean(e, altchar or pretty_atom('Equiv')) + else: + return self._print_Function(e, sort=True) + + def _print_conjugate(self, e): + pform = self._print(e.args[0]) + return prettyForm( *pform.above( hobj('_', pform.width())) ) + + def _print_Abs(self, e): + pform = self._print(e.args[0]) + pform = prettyForm(*pform.parens('|', '|')) + return pform + + def _print_floor(self, e): + if self._use_unicode: + pform = self._print(e.args[0]) + pform = prettyForm(*pform.parens('lfloor', 'rfloor')) + return pform + else: + return self._print_Function(e) + + def _print_ceiling(self, e): + if self._use_unicode: + pform = self._print(e.args[0]) + pform = prettyForm(*pform.parens('lceil', 'rceil')) + return pform + else: + return self._print_Function(e) + + def _print_Derivative(self, deriv): + if requires_partial(deriv.expr) and self._use_unicode: + deriv_symbol = U('PARTIAL DIFFERENTIAL') + else: + deriv_symbol = r'd' + x = None + count_total_deriv = 0 + + for sym, num in reversed(deriv.variable_count): + s = self._print(sym) + ds = prettyForm(*s.left(deriv_symbol)) + count_total_deriv += num + + if (not num.is_Integer) or (num > 1): + ds = ds**prettyForm(str(num)) + + if x is None: + x = ds + else: + x = prettyForm(*x.right(' ')) + x = prettyForm(*x.right(ds)) + + f = prettyForm( + binding=prettyForm.FUNC, *self._print(deriv.expr).parens()) + + pform = prettyForm(deriv_symbol) + + if (count_total_deriv > 1) != False: + pform = pform**prettyForm(str(count_total_deriv)) + + pform = prettyForm(*pform.below(stringPict.LINE, x)) + pform.baseline = pform.baseline + 1 + pform = prettyForm(*stringPict.next(pform, f)) + pform.binding = prettyForm.MUL + + return pform + + def _print_Cycle(self, dc): + from sympy.combinatorics.permutations import Permutation, Cycle + # for Empty Cycle + if dc == Cycle(): + cyc = stringPict('') + return prettyForm(*cyc.parens()) + + dc_list = Permutation(dc.list()).cyclic_form + # for Identity Cycle + if dc_list == []: + cyc = self._print(dc.size - 1) + return prettyForm(*cyc.parens()) + + cyc = stringPict('') + for i in dc_list: + l = self._print(str(tuple(i)).replace(',', '')) + cyc = prettyForm(*cyc.right(l)) + return cyc + + def _print_Permutation(self, expr): + from sympy.combinatorics.permutations import Permutation, Cycle + + perm_cyclic = Permutation.print_cyclic + if perm_cyclic is not None: + sympy_deprecation_warning( + f""" + Setting Permutation.print_cyclic is deprecated. Instead use + init_printing(perm_cyclic={perm_cyclic}). + """, + deprecated_since_version="1.6", + active_deprecations_target="deprecated-permutation-print_cyclic", + stacklevel=7, + ) + else: + perm_cyclic = self._settings.get("perm_cyclic", True) + + if perm_cyclic: + return self._print_Cycle(Cycle(expr)) + + lower = expr.array_form + upper = list(range(len(lower))) + + result = stringPict('') + first = True + for u, l in zip(upper, lower): + s1 = self._print(u) + s2 = self._print(l) + col = prettyForm(*s1.below(s2)) + if first: + first = False + else: + col = prettyForm(*col.left(" ")) + result = prettyForm(*result.right(col)) + return prettyForm(*result.parens()) + + + def _print_Integral(self, integral): + f = integral.function + + # Add parentheses if arg involves addition of terms and + # create a pretty form for the argument + prettyF = self._print(f) + # XXX generalize parens + if f.is_Add: + prettyF = prettyForm(*prettyF.parens()) + + # dx dy dz ... + arg = prettyF + for x in integral.limits: + prettyArg = self._print(x[0]) + # XXX qparens (parens if needs-parens) + if prettyArg.width() > 1: + prettyArg = prettyForm(*prettyArg.parens()) + + arg = prettyForm(*arg.right(' d', prettyArg)) + + # \int \int \int ... + firstterm = True + s = None + for lim in integral.limits: + # Create bar based on the height of the argument + h = arg.height() + H = h + 2 + + # XXX hack! + ascii_mode = not self._use_unicode + if ascii_mode: + H += 2 + + vint = vobj('int', H) + + # Construct the pretty form with the integral sign and the argument + pform = prettyForm(vint) + pform.baseline = arg.baseline + ( + H - h)//2 # covering the whole argument + + if len(lim) > 1: + # Create pretty forms for endpoints, if definite integral. + # Do not print empty endpoints. + if len(lim) == 2: + prettyA = prettyForm("") + prettyB = self._print(lim[1]) + if len(lim) == 3: + prettyA = self._print(lim[1]) + prettyB = self._print(lim[2]) + + if ascii_mode: # XXX hack + # Add spacing so that endpoint can more easily be + # identified with the correct integral sign + spc = max(1, 3 - prettyB.width()) + prettyB = prettyForm(*prettyB.left(' ' * spc)) + + spc = max(1, 4 - prettyA.width()) + prettyA = prettyForm(*prettyA.right(' ' * spc)) + + pform = prettyForm(*pform.above(prettyB)) + pform = prettyForm(*pform.below(prettyA)) + + if not ascii_mode: # XXX hack + pform = prettyForm(*pform.right(' ')) + + if firstterm: + s = pform # first term + firstterm = False + else: + s = prettyForm(*s.left(pform)) + + pform = prettyForm(*arg.left(s)) + pform.binding = prettyForm.MUL + return pform + + def _print_Product(self, expr): + func = expr.term + pretty_func = self._print(func) + + horizontal_chr = xobj('_', 1) + corner_chr = xobj('_', 1) + vertical_chr = xobj('|', 1) + + if self._use_unicode: + # use unicode corners + horizontal_chr = xobj('-', 1) + corner_chr = xobj('UpTack', 1) + + func_height = pretty_func.height() + + first = True + max_upper = 0 + sign_height = 0 + + for lim in expr.limits: + pretty_lower, pretty_upper = self.__print_SumProduct_Limits(lim) + + width = (func_height + 2) * 5 // 3 - 2 + sign_lines = [horizontal_chr + corner_chr + (horizontal_chr * (width-2)) + corner_chr + horizontal_chr] + for _ in range(func_height + 1): + sign_lines.append(' ' + vertical_chr + (' ' * (width-2)) + vertical_chr + ' ') + + pretty_sign = stringPict('') + pretty_sign = prettyForm(*pretty_sign.stack(*sign_lines)) + + + max_upper = max(max_upper, pretty_upper.height()) + + if first: + sign_height = pretty_sign.height() + + pretty_sign = prettyForm(*pretty_sign.above(pretty_upper)) + pretty_sign = prettyForm(*pretty_sign.below(pretty_lower)) + + if first: + pretty_func.baseline = 0 + first = False + + height = pretty_sign.height() + padding = stringPict('') + padding = prettyForm(*padding.stack(*[' ']*(height - 1))) + pretty_sign = prettyForm(*pretty_sign.right(padding)) + + pretty_func = prettyForm(*pretty_sign.right(pretty_func)) + + pretty_func.baseline = max_upper + sign_height//2 + pretty_func.binding = prettyForm.MUL + return pretty_func + + def __print_SumProduct_Limits(self, lim): + def print_start(lhs, rhs): + op = prettyForm(' ' + xsym("==") + ' ') + l = self._print(lhs) + r = self._print(rhs) + pform = prettyForm(*stringPict.next(l, op, r)) + return pform + + prettyUpper = self._print(lim[2]) + prettyLower = print_start(lim[0], lim[1]) + return prettyLower, prettyUpper + + def _print_Sum(self, expr): + ascii_mode = not self._use_unicode + + def asum(hrequired, lower, upper, use_ascii): + def adjust(s, wid=None, how='<^>'): + if not wid or len(s) > wid: + return s + need = wid - len(s) + if how in ('<^>', "<") or how not in list('<^>'): + return s + ' '*need + half = need//2 + lead = ' '*half + if how == ">": + return " "*need + s + return lead + s + ' '*(need - len(lead)) + + h = max(hrequired, 2) + d = h//2 + w = d + 1 + more = hrequired % 2 + + lines = [] + if use_ascii: + lines.append("_"*(w) + ' ') + lines.append(r"\%s`" % (' '*(w - 1))) + for i in range(1, d): + lines.append('%s\\%s' % (' '*i, ' '*(w - i))) + if more: + lines.append('%s)%s' % (' '*(d), ' '*(w - d))) + for i in reversed(range(1, d)): + lines.append('%s/%s' % (' '*i, ' '*(w - i))) + lines.append("/" + "_"*(w - 1) + ',') + return d, h + more, lines, more + else: + w = w + more + d = d + more + vsum = vobj('sum', 4) + lines.append("_"*(w)) + for i in range(0, d): + lines.append('%s%s%s' % (' '*i, vsum[2], ' '*(w - i - 1))) + for i in reversed(range(0, d)): + lines.append('%s%s%s' % (' '*i, vsum[4], ' '*(w - i - 1))) + lines.append(vsum[8]*(w)) + return d, h + 2*more, lines, more + + f = expr.function + + prettyF = self._print(f) + + if f.is_Add: # add parens + prettyF = prettyForm(*prettyF.parens()) + + H = prettyF.height() + 2 + + # \sum \sum \sum ... + first = True + max_upper = 0 + sign_height = 0 + + for lim in expr.limits: + prettyLower, prettyUpper = self.__print_SumProduct_Limits(lim) + + max_upper = max(max_upper, prettyUpper.height()) + + # Create sum sign based on the height of the argument + d, h, slines, adjustment = asum( + H, prettyLower.width(), prettyUpper.width(), ascii_mode) + prettySign = stringPict('') + prettySign = prettyForm(*prettySign.stack(*slines)) + + if first: + sign_height = prettySign.height() + + prettySign = prettyForm(*prettySign.above(prettyUpper)) + prettySign = prettyForm(*prettySign.below(prettyLower)) + + if first: + # change F baseline so it centers on the sign + prettyF.baseline -= d - (prettyF.height()//2 - + prettyF.baseline) + first = False + + # put padding to the right + pad = stringPict('') + pad = prettyForm(*pad.stack(*[' ']*h)) + prettySign = prettyForm(*prettySign.right(pad)) + # put the present prettyF to the right + prettyF = prettyForm(*prettySign.right(prettyF)) + + # adjust baseline of ascii mode sigma with an odd height so that it is + # exactly through the center + ascii_adjustment = ascii_mode if not adjustment else 0 + prettyF.baseline = max_upper + sign_height//2 + ascii_adjustment + + prettyF.binding = prettyForm.MUL + return prettyF + + def _print_Limit(self, l): + e, z, z0, dir = l.args + + E = self._print(e) + if precedence(e) <= PRECEDENCE["Mul"]: + E = prettyForm(*E.parens('(', ')')) + Lim = prettyForm('lim') + + LimArg = self._print(z) + if self._use_unicode: + LimArg = prettyForm(*LimArg.right(f"{xobj('-', 1)}{pretty_atom('Arrow')}")) + else: + LimArg = prettyForm(*LimArg.right('->')) + LimArg = prettyForm(*LimArg.right(self._print(z0))) + + if str(dir) == '+-' or z0 in (S.Infinity, S.NegativeInfinity): + dir = "" + else: + if self._use_unicode: + dir = pretty_atom('SuperscriptPlus') if str(dir) == "+" else pretty_atom('SuperscriptMinus') + + LimArg = prettyForm(*LimArg.right(self._print(dir))) + + Lim = prettyForm(*Lim.below(LimArg)) + Lim = prettyForm(*Lim.right(E), binding=prettyForm.MUL) + + return Lim + + def _print_matrix_contents(self, e): + """ + This method factors out what is essentially grid printing. + """ + M = e # matrix + Ms = {} # i,j -> pretty(M[i,j]) + for i in range(M.rows): + for j in range(M.cols): + Ms[i, j] = self._print(M[i, j]) + + # h- and v- spacers + hsep = 2 + vsep = 1 + + # max width for columns + maxw = [-1] * M.cols + + for j in range(M.cols): + maxw[j] = max([Ms[i, j].width() for i in range(M.rows)] or [0]) + + # drawing result + D = None + + for i in range(M.rows): + + D_row = None + for j in range(M.cols): + s = Ms[i, j] + + # reshape s to maxw + # XXX this should be generalized, and go to stringPict.reshape ? + assert s.width() <= maxw[j] + + # hcenter it, +0.5 to the right 2 + # ( it's better to align formula starts for say 0 and r ) + # XXX this is not good in all cases -- maybe introduce vbaseline? + left, right = center_pad(s.width(), maxw[j]) + + s = prettyForm(*s.right(right)) + s = prettyForm(*s.left(left)) + + # we don't need vcenter cells -- this is automatically done in + # a pretty way because when their baselines are taking into + # account in .right() + + if D_row is None: + D_row = s # first box in a row + continue + + D_row = prettyForm(*D_row.right(' '*hsep)) # h-spacer + D_row = prettyForm(*D_row.right(s)) + + if D is None: + D = D_row # first row in a picture + continue + + # v-spacer + for _ in range(vsep): + D = prettyForm(*D.below(' ')) + + D = prettyForm(*D.below(D_row)) + + if D is None: + D = prettyForm('') # Empty Matrix + + return D + + def _print_MatrixBase(self, e, lparens='[', rparens=']'): + D = self._print_matrix_contents(e) + D.baseline = D.height()//2 + D = prettyForm(*D.parens(lparens, rparens)) + return D + + def _print_Determinant(self, e): + mat = e.arg + if mat.is_MatrixExpr: + from sympy.matrices.expressions.blockmatrix import BlockMatrix + if isinstance(mat, BlockMatrix): + return self._print_MatrixBase(mat.blocks, lparens='|', rparens='|') + D = self._print(mat) + D.baseline = D.height()//2 + return prettyForm(*D.parens('|', '|')) + else: + return self._print_MatrixBase(mat, lparens='|', rparens='|') + + def _print_TensorProduct(self, expr): + # This should somehow share the code with _print_WedgeProduct: + if self._use_unicode: + circled_times = "\u2297" + else: + circled_times = ".*" + return self._print_seq(expr.args, None, None, circled_times, + parenthesize=lambda x: precedence_traditional(x) <= PRECEDENCE["Mul"]) + + def _print_WedgeProduct(self, expr): + # This should somehow share the code with _print_TensorProduct: + if self._use_unicode: + wedge_symbol = "\u2227" + else: + wedge_symbol = '/\\' + return self._print_seq(expr.args, None, None, wedge_symbol, + parenthesize=lambda x: precedence_traditional(x) <= PRECEDENCE["Mul"]) + + def _print_Trace(self, e): + D = self._print(e.arg) + D = prettyForm(*D.parens('(',')')) + D.baseline = D.height()//2 + D = prettyForm(*D.left('\n'*(0) + 'tr')) + return D + + + def _print_MatrixElement(self, expr): + from sympy.matrices import MatrixSymbol + if (isinstance(expr.parent, MatrixSymbol) + and expr.i.is_number and expr.j.is_number): + return self._print( + Symbol(expr.parent.name + '_%d%d' % (expr.i, expr.j))) + else: + prettyFunc = self._print(expr.parent) + prettyFunc = prettyForm(*prettyFunc.parens()) + prettyIndices = self._print_seq((expr.i, expr.j), delimiter=', ' + ).parens(left='[', right=']')[0] + pform = prettyForm(binding=prettyForm.FUNC, + *stringPict.next(prettyFunc, prettyIndices)) + + # store pform parts so it can be reassembled e.g. when powered + pform.prettyFunc = prettyFunc + pform.prettyArgs = prettyIndices + + return pform + + + def _print_MatrixSlice(self, m): + # XXX works only for applied functions + from sympy.matrices import MatrixSymbol + prettyFunc = self._print(m.parent) + if not isinstance(m.parent, MatrixSymbol): + prettyFunc = prettyForm(*prettyFunc.parens()) + def ppslice(x, dim): + x = list(x) + if x[2] == 1: + del x[2] + if x[0] == 0: + x[0] = '' + if x[1] == dim: + x[1] = '' + return prettyForm(*self._print_seq(x, delimiter=':')) + prettyArgs = self._print_seq((ppslice(m.rowslice, m.parent.rows), + ppslice(m.colslice, m.parent.cols)), delimiter=', ').parens(left='[', right=']')[0] + + pform = prettyForm( + binding=prettyForm.FUNC, *stringPict.next(prettyFunc, prettyArgs)) + + # store pform parts so it can be reassembled e.g. when powered + pform.prettyFunc = prettyFunc + pform.prettyArgs = prettyArgs + + return pform + + def _print_Transpose(self, expr): + mat = expr.arg + pform = self._print(mat) + from sympy.matrices import MatrixSymbol, BlockMatrix + if (not isinstance(mat, MatrixSymbol) and + not isinstance(mat, BlockMatrix) and mat.is_MatrixExpr): + pform = prettyForm(*pform.parens()) + pform = pform**(prettyForm('T')) + return pform + + def _print_Adjoint(self, expr): + mat = expr.arg + pform = self._print(mat) + if self._use_unicode: + dag = prettyForm(pretty_atom('Dagger')) + else: + dag = prettyForm('+') + from sympy.matrices import MatrixSymbol, BlockMatrix + if (not isinstance(mat, MatrixSymbol) and + not isinstance(mat, BlockMatrix) and mat.is_MatrixExpr): + pform = prettyForm(*pform.parens()) + pform = pform**dag + return pform + + def _print_BlockMatrix(self, B): + if B.blocks.shape == (1, 1): + return self._print(B.blocks[0, 0]) + return self._print(B.blocks) + + def _print_MatAdd(self, expr): + s = None + for item in expr.args: + pform = self._print(item) + if s is None: + s = pform # First element + else: + coeff = item.as_coeff_mmul()[0] + if S(coeff).could_extract_minus_sign(): + s = prettyForm(*stringPict.next(s, ' ')) + pform = self._print(item) + else: + s = prettyForm(*stringPict.next(s, ' + ')) + s = prettyForm(*stringPict.next(s, pform)) + + return s + + def _print_MatMul(self, expr): + args = list(expr.args) + from sympy.matrices.expressions.hadamard import HadamardProduct + from sympy.matrices.expressions.kronecker import KroneckerProduct + from sympy.matrices.expressions.matadd import MatAdd + for i, a in enumerate(args): + if (isinstance(a, (Add, MatAdd, HadamardProduct, KroneckerProduct)) + and len(expr.args) > 1): + args[i] = prettyForm(*self._print(a).parens()) + else: + args[i] = self._print(a) + + return prettyForm.__mul__(*args) + + def _print_Identity(self, expr): + if self._use_unicode: + return prettyForm(pretty_atom('IdentityMatrix')) + else: + return prettyForm('I') + + def _print_ZeroMatrix(self, expr): + if self._use_unicode: + return prettyForm(pretty_atom('ZeroMatrix')) + else: + return prettyForm('0') + + def _print_OneMatrix(self, expr): + if self._use_unicode: + return prettyForm(pretty_atom("OneMatrix")) + else: + return prettyForm('1') + + def _print_DotProduct(self, expr): + args = list(expr.args) + + for i, a in enumerate(args): + args[i] = self._print(a) + return prettyForm.__mul__(*args) + + def _print_MatPow(self, expr): + pform = self._print(expr.base) + from sympy.matrices import MatrixSymbol + if not isinstance(expr.base, MatrixSymbol) and expr.base.is_MatrixExpr: + pform = prettyForm(*pform.parens()) + pform = pform**(self._print(expr.exp)) + return pform + + def _print_HadamardProduct(self, expr): + from sympy.matrices.expressions.hadamard import HadamardProduct + from sympy.matrices.expressions.matadd import MatAdd + from sympy.matrices.expressions.matmul import MatMul + if self._use_unicode: + delim = pretty_atom('Ring') + else: + delim = '.*' + return self._print_seq(expr.args, None, None, delim, + parenthesize=lambda x: isinstance(x, (MatAdd, MatMul, HadamardProduct))) + + def _print_HadamardPower(self, expr): + # from sympy import MatAdd, MatMul + if self._use_unicode: + circ = pretty_atom('Ring') + else: + circ = self._print('.') + pretty_base = self._print(expr.base) + pretty_exp = self._print(expr.exp) + if precedence(expr.exp) < PRECEDENCE["Mul"]: + pretty_exp = prettyForm(*pretty_exp.parens()) + pretty_circ_exp = prettyForm( + binding=prettyForm.LINE, + *stringPict.next(circ, pretty_exp) + ) + return pretty_base**pretty_circ_exp + + def _print_KroneckerProduct(self, expr): + from sympy.matrices.expressions.matadd import MatAdd + from sympy.matrices.expressions.matmul import MatMul + if self._use_unicode: + delim = f" {pretty_atom('TensorProduct')} " + else: + delim = ' x ' + return self._print_seq(expr.args, None, None, delim, + parenthesize=lambda x: isinstance(x, (MatAdd, MatMul))) + + def _print_FunctionMatrix(self, X): + D = self._print(X.lamda.expr) + D = prettyForm(*D.parens('[', ']')) + return D + + def _print_TransferFunction(self, expr): + if not expr.num == 1: + num, den = expr.num, expr.den + res = Mul(num, Pow(den, -1, evaluate=False), evaluate=False) + return self._print_Mul(res) + else: + return self._print(1)/self._print(expr.den) + + def _print_Series(self, expr): + args = list(expr.args) + for i, a in enumerate(expr.args): + args[i] = prettyForm(*self._print(a).parens()) + return prettyForm.__mul__(*args) + + def _print_MIMOSeries(self, expr): + from sympy.physics.control.lti import MIMOParallel + args = list(expr.args) + pretty_args = [] + for a in reversed(args): + if (isinstance(a, MIMOParallel) and len(expr.args) > 1): + expression = self._print(a) + expression.baseline = expression.height()//2 + pretty_args.append(prettyForm(*expression.parens())) + else: + expression = self._print(a) + expression.baseline = expression.height()//2 + pretty_args.append(expression) + return prettyForm.__mul__(*pretty_args) + + def _print_Parallel(self, expr): + s = None + for item in expr.args: + pform = self._print(item) + if s is None: + s = pform # First element + else: + s = prettyForm(*stringPict.next(s)) + s.baseline = s.height()//2 + s = prettyForm(*stringPict.next(s, ' + ')) + s = prettyForm(*stringPict.next(s, pform)) + return s + + def _print_MIMOParallel(self, expr): + from sympy.physics.control.lti import TransferFunctionMatrix + s = None + for item in expr.args: + pform = self._print(item) + if s is None: + s = pform # First element + else: + s = prettyForm(*stringPict.next(s)) + s.baseline = s.height()//2 + s = prettyForm(*stringPict.next(s, ' + ')) + if isinstance(item, TransferFunctionMatrix): + s.baseline = s.height() - 1 + s = prettyForm(*stringPict.next(s, pform)) + # s.baseline = s.height()//2 + return s + + def _print_Feedback(self, expr): + from sympy.physics.control import TransferFunction, Series + + num, tf = expr.sys1, TransferFunction(1, 1, expr.var) + num_arg_list = list(num.args) if isinstance(num, Series) else [num] + den_arg_list = list(expr.sys2.args) if \ + isinstance(expr.sys2, Series) else [expr.sys2] + + if isinstance(num, Series) and isinstance(expr.sys2, Series): + den = Series(*num_arg_list, *den_arg_list) + elif isinstance(num, Series) and isinstance(expr.sys2, TransferFunction): + if expr.sys2 == tf: + den = Series(*num_arg_list) + else: + den = Series(*num_arg_list, expr.sys2) + elif isinstance(num, TransferFunction) and isinstance(expr.sys2, Series): + if num == tf: + den = Series(*den_arg_list) + else: + den = Series(num, *den_arg_list) + else: + if num == tf: + den = Series(*den_arg_list) + elif expr.sys2 == tf: + den = Series(*num_arg_list) + else: + den = Series(*num_arg_list, *den_arg_list) + + denom = prettyForm(*stringPict.next(self._print(tf))) + denom.baseline = denom.height()//2 + denom = prettyForm(*stringPict.next(denom, ' + ')) if expr.sign == -1 \ + else prettyForm(*stringPict.next(denom, ' - ')) + denom = prettyForm(*stringPict.next(denom, self._print(den))) + + return self._print(num)/denom + + def _print_MIMOFeedback(self, expr): + from sympy.physics.control import MIMOSeries, TransferFunctionMatrix + + inv_mat = self._print(MIMOSeries(expr.sys2, expr.sys1)) + plant = self._print(expr.sys1) + _feedback = prettyForm(*stringPict.next(inv_mat)) + _feedback = prettyForm(*stringPict.right("I + ", _feedback)) if expr.sign == -1 \ + else prettyForm(*stringPict.right("I - ", _feedback)) + _feedback = prettyForm(*stringPict.parens(_feedback)) + _feedback.baseline = 0 + _feedback = prettyForm(*stringPict.right(_feedback, '-1 ')) + _feedback.baseline = _feedback.height()//2 + _feedback = prettyForm.__mul__(_feedback, prettyForm(" ")) + if isinstance(expr.sys1, TransferFunctionMatrix): + _feedback.baseline = _feedback.height() - 1 + _feedback = prettyForm(*stringPict.next(_feedback, plant)) + return _feedback + + def _print_TransferFunctionMatrix(self, expr): + mat = self._print(expr._expr_mat) + mat.baseline = mat.height() - 1 + subscript = greek_unicode['tau'] if self._use_unicode else r'{t}' + mat = prettyForm(*mat.right(subscript)) + return mat + + def _print_StateSpace(self, expr): + from sympy.matrices.expressions.blockmatrix import BlockMatrix + A = expr._A + B = expr._B + C = expr._C + D = expr._D + mat = BlockMatrix([[A, B], [C, D]]) + return self._print(mat.blocks) + + def _print_BasisDependent(self, expr): + from sympy.vector import Vector + + if not self._use_unicode: + raise NotImplementedError("ASCII pretty printing of BasisDependent is not implemented") + + if expr == expr.zero: + return prettyForm(expr.zero._pretty_form) + o1 = [] + vectstrs = [] + if isinstance(expr, Vector): + items = expr.separate().items() + else: + items = [(0, expr)] + for system, vect in items: + inneritems = list(vect.components.items()) + inneritems.sort(key = lambda x: x[0].__str__()) + for k, v in inneritems: + #if the coef of the basis vector is 1 + #we skip the 1 + if v == 1: + o1.append("" + + k._pretty_form) + #Same for -1 + elif v == -1: + o1.append("(-1) " + + k._pretty_form) + #For a general expr + else: + #We always wrap the measure numbers in + #parentheses + arg_str = self._print( + v).parens()[0] + + o1.append(arg_str + ' ' + k._pretty_form) + vectstrs.append(k._pretty_form) + + #outstr = u("").join(o1) + if o1[0].startswith(" + "): + o1[0] = o1[0][3:] + elif o1[0].startswith(" "): + o1[0] = o1[0][1:] + #Fixing the newlines + lengths = [] + strs = [''] + flag = [] + for i, partstr in enumerate(o1): + flag.append(0) + # XXX: What is this hack? + if '\n' in partstr: + tempstr = partstr + tempstr = tempstr.replace(vectstrs[i], '') + if xobj(')_ext', 1) in tempstr: # If scalar is a fraction + for paren in range(len(tempstr)): + flag[i] = 1 + if tempstr[paren] == xobj(')_ext', 1) and tempstr[paren + 1] == '\n': + # We want to place the vector string after all the right parentheses, because + # otherwise, the vector will be in the middle of the string + tempstr = tempstr[:paren] + xobj(')_ext', 1)\ + + ' ' + vectstrs[i] + tempstr[paren + 1:] + break + elif xobj(')_lower_hook', 1) in tempstr: + # We want to place the vector string after all the right parentheses, because + # otherwise, the vector will be in the middle of the string. For this reason, + # we insert the vector string at the rightmost index. + index = tempstr.rfind(xobj(')_lower_hook', 1)) + if index != -1: # then this character was found in this string + flag[i] = 1 + tempstr = tempstr[:index] + xobj(')_lower_hook', 1)\ + + ' ' + vectstrs[i] + tempstr[index + 1:] + o1[i] = tempstr + + o1 = [x.split('\n') for x in o1] + n_newlines = max(len(x) for x in o1) # Width of part in its pretty form + + if 1 in flag: # If there was a fractional scalar + for i, parts in enumerate(o1): + if len(parts) == 1: # If part has no newline + parts.insert(0, ' ' * (len(parts[0]))) + flag[i] = 1 + + for i, parts in enumerate(o1): + lengths.append(len(parts[flag[i]])) + for j in range(n_newlines): + if j+1 <= len(parts): + if j >= len(strs): + strs.append(' ' * (sum(lengths[:-1]) + + 3*(len(lengths)-1))) + if j == flag[i]: + strs[flag[i]] += parts[flag[i]] + ' + ' + else: + strs[j] += parts[j] + ' '*(lengths[-1] - + len(parts[j])+ + 3) + else: + if j >= len(strs): + strs.append(' ' * (sum(lengths[:-1]) + + 3*(len(lengths)-1))) + strs[j] += ' '*(lengths[-1]+3) + + return prettyForm('\n'.join([s[:-3] for s in strs])) + + def _print_NDimArray(self, expr): + from sympy.matrices.immutable import ImmutableMatrix + + if expr.rank() == 0: + return self._print(expr[()]) + + level_str = [[]] + [[] for i in range(expr.rank())] + shape_ranges = [list(range(i)) for i in expr.shape] + # leave eventual matrix elements unflattened + mat = lambda x: ImmutableMatrix(x, evaluate=False) + for outer_i in itertools.product(*shape_ranges): + level_str[-1].append(expr[outer_i]) + even = True + for back_outer_i in range(expr.rank()-1, -1, -1): + if len(level_str[back_outer_i+1]) < expr.shape[back_outer_i]: + break + if even: + level_str[back_outer_i].append(level_str[back_outer_i+1]) + else: + level_str[back_outer_i].append(mat( + level_str[back_outer_i+1])) + if len(level_str[back_outer_i + 1]) == 1: + level_str[back_outer_i][-1] = mat( + [[level_str[back_outer_i][-1]]]) + even = not even + level_str[back_outer_i+1] = [] + + out_expr = level_str[0][0] + if expr.rank() % 2 == 1: + out_expr = mat([out_expr]) + + return self._print(out_expr) + + def _printer_tensor_indices(self, name, indices, index_map={}): + center = stringPict(name) + top = stringPict(" "*center.width()) + bot = stringPict(" "*center.width()) + + last_valence = None + prev_map = None + + for index in indices: + indpic = self._print(index.args[0]) + if ((index in index_map) or prev_map) and last_valence == index.is_up: + if index.is_up: + top = prettyForm(*stringPict.next(top, ",")) + else: + bot = prettyForm(*stringPict.next(bot, ",")) + if index in index_map: + indpic = prettyForm(*stringPict.next(indpic, "=")) + indpic = prettyForm(*stringPict.next(indpic, self._print(index_map[index]))) + prev_map = True + else: + prev_map = False + if index.is_up: + top = stringPict(*top.right(indpic)) + center = stringPict(*center.right(" "*indpic.width())) + bot = stringPict(*bot.right(" "*indpic.width())) + else: + bot = stringPict(*bot.right(indpic)) + center = stringPict(*center.right(" "*indpic.width())) + top = stringPict(*top.right(" "*indpic.width())) + last_valence = index.is_up + + pict = prettyForm(*center.above(top)) + pict = prettyForm(*pict.below(bot)) + return pict + + def _print_Tensor(self, expr): + name = expr.args[0].name + indices = expr.get_indices() + return self._printer_tensor_indices(name, indices) + + def _print_TensorElement(self, expr): + name = expr.expr.args[0].name + indices = expr.expr.get_indices() + index_map = expr.index_map + return self._printer_tensor_indices(name, indices, index_map) + + def _print_TensMul(self, expr): + sign, args = expr._get_args_for_traditional_printer() + args = [ + prettyForm(*self._print(i).parens()) if + precedence_traditional(i) < PRECEDENCE["Mul"] else self._print(i) + for i in args + ] + pform = prettyForm.__mul__(*args) + if sign: + return prettyForm(*pform.left(sign)) + else: + return pform + + def _print_TensAdd(self, expr): + args = [ + prettyForm(*self._print(i).parens()) if + precedence_traditional(i) < PRECEDENCE["Mul"] else self._print(i) + for i in expr.args + ] + return prettyForm.__add__(*args) + + def _print_TensorIndex(self, expr): + sym = expr.args[0] + if not expr.is_up: + sym = -sym + return self._print(sym) + + def _print_PartialDerivative(self, deriv): + if self._use_unicode: + deriv_symbol = U('PARTIAL DIFFERENTIAL') + else: + deriv_symbol = r'd' + x = None + + for variable in reversed(deriv.variables): + s = self._print(variable) + ds = prettyForm(*s.left(deriv_symbol)) + + if x is None: + x = ds + else: + x = prettyForm(*x.right(' ')) + x = prettyForm(*x.right(ds)) + + f = prettyForm( + binding=prettyForm.FUNC, *self._print(deriv.expr).parens()) + + pform = prettyForm(deriv_symbol) + + if len(deriv.variables) > 1: + pform = pform**self._print(len(deriv.variables)) + + pform = prettyForm(*pform.below(stringPict.LINE, x)) + pform.baseline = pform.baseline + 1 + pform = prettyForm(*stringPict.next(pform, f)) + pform.binding = prettyForm.MUL + + return pform + + def _print_Piecewise(self, pexpr): + + P = {} + for n, ec in enumerate(pexpr.args): + P[n, 0] = self._print(ec.expr) + if ec.cond == True: + P[n, 1] = prettyForm('otherwise') + else: + P[n, 1] = prettyForm( + *prettyForm('for ').right(self._print(ec.cond))) + hsep = 2 + vsep = 1 + len_args = len(pexpr.args) + + # max widths + maxw = [max(P[i, j].width() for i in range(len_args)) + for j in range(2)] + + # FIXME: Refactor this code and matrix into some tabular environment. + # drawing result + D = None + + for i in range(len_args): + D_row = None + for j in range(2): + p = P[i, j] + assert p.width() <= maxw[j] + + wdelta = maxw[j] - p.width() + wleft = wdelta // 2 + wright = wdelta - wleft + + p = prettyForm(*p.right(' '*wright)) + p = prettyForm(*p.left(' '*wleft)) + + if D_row is None: + D_row = p + continue + + D_row = prettyForm(*D_row.right(' '*hsep)) # h-spacer + D_row = prettyForm(*D_row.right(p)) + if D is None: + D = D_row # first row in a picture + continue + + # v-spacer + for _ in range(vsep): + D = prettyForm(*D.below(' ')) + + D = prettyForm(*D.below(D_row)) + + D = prettyForm(*D.parens('{', '')) + D.baseline = D.height()//2 + D.binding = prettyForm.OPEN + return D + + def _print_ITE(self, ite): + from sympy.functions.elementary.piecewise import Piecewise + return self._print(ite.rewrite(Piecewise)) + + def _hprint_vec(self, v): + D = None + + for a in v: + p = a + if D is None: + D = p + else: + D = prettyForm(*D.right(', ')) + D = prettyForm(*D.right(p)) + if D is None: + D = stringPict(' ') + + return D + + def _hprint_vseparator(self, p1, p2, left=None, right=None, delimiter='', ifascii_nougly=False): + if ifascii_nougly and not self._use_unicode: + return self._print_seq((p1, '|', p2), left=left, right=right, + delimiter=delimiter, ifascii_nougly=True) + tmp = self._print_seq((p1, p2,), left=left, right=right, delimiter=delimiter) + sep = stringPict(vobj('|', tmp.height()), baseline=tmp.baseline) + return self._print_seq((p1, sep, p2), left=left, right=right, + delimiter=delimiter) + + def _print_hyper(self, e): + # FIXME refactor Matrix, Piecewise, and this into a tabular environment + ap = [self._print(a) for a in e.ap] + bq = [self._print(b) for b in e.bq] + + P = self._print(e.argument) + P.baseline = P.height()//2 + + # Drawing result - first create the ap, bq vectors + D = None + for v in [ap, bq]: + D_row = self._hprint_vec(v) + if D is None: + D = D_row # first row in a picture + else: + D = prettyForm(*D.below(' ')) + D = prettyForm(*D.below(D_row)) + + # make sure that the argument `z' is centred vertically + D.baseline = D.height()//2 + + # insert horizontal separator + P = prettyForm(*P.left(' ')) + D = prettyForm(*D.right(' ')) + + # insert separating `|` + D = self._hprint_vseparator(D, P) + + # add parens + D = prettyForm(*D.parens('(', ')')) + + # create the F symbol + above = D.height()//2 - 1 + below = D.height() - above - 1 + + sz, t, b, add, img = annotated('F') + F = prettyForm('\n' * (above - t) + img + '\n' * (below - b), + baseline=above + sz) + add = (sz + 1)//2 + + F = prettyForm(*F.left(self._print(len(e.ap)))) + F = prettyForm(*F.right(self._print(len(e.bq)))) + F.baseline = above + add + + D = prettyForm(*F.right(' ', D)) + + return D + + def _print_meijerg(self, e): + # FIXME refactor Matrix, Piecewise, and this into a tabular environment + + v = {} + v[(0, 0)] = [self._print(a) for a in e.an] + v[(0, 1)] = [self._print(a) for a in e.aother] + v[(1, 0)] = [self._print(b) for b in e.bm] + v[(1, 1)] = [self._print(b) for b in e.bother] + + P = self._print(e.argument) + P.baseline = P.height()//2 + + vp = {} + for idx in v: + vp[idx] = self._hprint_vec(v[idx]) + + for i in range(2): + maxw = max(vp[(0, i)].width(), vp[(1, i)].width()) + for j in range(2): + s = vp[(j, i)] + left = (maxw - s.width()) // 2 + right = maxw - left - s.width() + s = prettyForm(*s.left(' ' * left)) + s = prettyForm(*s.right(' ' * right)) + vp[(j, i)] = s + + D1 = prettyForm(*vp[(0, 0)].right(' ', vp[(0, 1)])) + D1 = prettyForm(*D1.below(' ')) + D2 = prettyForm(*vp[(1, 0)].right(' ', vp[(1, 1)])) + D = prettyForm(*D1.below(D2)) + + # make sure that the argument `z' is centred vertically + D.baseline = D.height()//2 + + # insert horizontal separator + P = prettyForm(*P.left(' ')) + D = prettyForm(*D.right(' ')) + + # insert separating `|` + D = self._hprint_vseparator(D, P) + + # add parens + D = prettyForm(*D.parens('(', ')')) + + # create the G symbol + above = D.height()//2 - 1 + below = D.height() - above - 1 + + sz, t, b, add, img = annotated('G') + F = prettyForm('\n' * (above - t) + img + '\n' * (below - b), + baseline=above + sz) + + pp = self._print(len(e.ap)) + pq = self._print(len(e.bq)) + pm = self._print(len(e.bm)) + pn = self._print(len(e.an)) + + def adjust(p1, p2): + diff = p1.width() - p2.width() + if diff == 0: + return p1, p2 + elif diff > 0: + return p1, prettyForm(*p2.left(' '*diff)) + else: + return prettyForm(*p1.left(' '*-diff)), p2 + pp, pm = adjust(pp, pm) + pq, pn = adjust(pq, pn) + pu = prettyForm(*pm.right(', ', pn)) + pl = prettyForm(*pp.right(', ', pq)) + + ht = F.baseline - above - 2 + if ht > 0: + pu = prettyForm(*pu.below('\n'*ht)) + p = prettyForm(*pu.below(pl)) + + F.baseline = above + F = prettyForm(*F.right(p)) + + F.baseline = above + add + + D = prettyForm(*F.right(' ', D)) + + return D + + def _print_ExpBase(self, e): + # TODO should exp_polar be printed differently? + # what about exp_polar(0), exp_polar(1)? + base = prettyForm(pretty_atom('Exp1', 'e')) + return base ** self._print(e.args[0]) + + def _print_Exp1(self, e): + return prettyForm(pretty_atom('Exp1', 'e')) + + def _print_Function(self, e, sort=False, func_name=None, left='(', + right=')'): + # optional argument func_name for supplying custom names + # XXX works only for applied functions + return self._helper_print_function(e.func, e.args, sort=sort, func_name=func_name, left=left, right=right) + + def _print_mathieuc(self, e): + return self._print_Function(e, func_name='C') + + def _print_mathieus(self, e): + return self._print_Function(e, func_name='S') + + def _print_mathieucprime(self, e): + return self._print_Function(e, func_name="C'") + + def _print_mathieusprime(self, e): + return self._print_Function(e, func_name="S'") + + def _helper_print_function(self, func, args, sort=False, func_name=None, + delimiter=', ', elementwise=False, left='(', + right=')'): + if sort: + args = sorted(args, key=default_sort_key) + + if not func_name and hasattr(func, "__name__"): + func_name = func.__name__ + + if func_name: + prettyFunc = self._print(Symbol(func_name)) + else: + prettyFunc = prettyForm(*self._print(func).parens()) + + if elementwise: + if self._use_unicode: + circ = pretty_atom('Modifier Letter Low Ring') + else: + circ = '.' + circ = self._print(circ) + prettyFunc = prettyForm( + binding=prettyForm.LINE, + *stringPict.next(prettyFunc, circ) + ) + + prettyArgs = prettyForm(*self._print_seq(args, delimiter=delimiter).parens( + left=left, right=right)) + + pform = prettyForm( + binding=prettyForm.FUNC, *stringPict.next(prettyFunc, prettyArgs)) + + # store pform parts so it can be reassembled e.g. when powered + pform.prettyFunc = prettyFunc + pform.prettyArgs = prettyArgs + + return pform + + def _print_ElementwiseApplyFunction(self, e): + func = e.function + arg = e.expr + args = [arg] + return self._helper_print_function(func, args, delimiter="", elementwise=True) + + @property + def _special_function_classes(self): + from sympy.functions.special.tensor_functions import KroneckerDelta + from sympy.functions.special.gamma_functions import gamma, lowergamma + from sympy.functions.special.zeta_functions import lerchphi + from sympy.functions.special.beta_functions import beta + from sympy.functions.special.delta_functions import DiracDelta + from sympy.functions.special.error_functions import Chi + return {KroneckerDelta: [greek_unicode['delta'], 'delta'], + gamma: [greek_unicode['Gamma'], 'Gamma'], + lerchphi: [greek_unicode['Phi'], 'lerchphi'], + lowergamma: [greek_unicode['gamma'], 'gamma'], + beta: [greek_unicode['Beta'], 'B'], + DiracDelta: [greek_unicode['delta'], 'delta'], + Chi: ['Chi', 'Chi']} + + def _print_FunctionClass(self, expr): + for cls in self._special_function_classes: + if issubclass(expr, cls) and expr.__name__ == cls.__name__: + if self._use_unicode: + return prettyForm(self._special_function_classes[cls][0]) + else: + return prettyForm(self._special_function_classes[cls][1]) + func_name = expr.__name__ + return prettyForm(pretty_symbol(func_name)) + + def _print_GeometryEntity(self, expr): + # GeometryEntity is based on Tuple but should not print like a Tuple + return self.emptyPrinter(expr) + + def _print_polylog(self, e): + subscript = self._print(e.args[0]) + if self._use_unicode and is_subscriptable_in_unicode(subscript): + return self._print_Function(Function('Li_%s' % subscript)(e.args[1])) + return self._print_Function(e) + + def _print_lerchphi(self, e): + func_name = greek_unicode['Phi'] if self._use_unicode else 'lerchphi' + return self._print_Function(e, func_name=func_name) + + def _print_dirichlet_eta(self, e): + func_name = greek_unicode['eta'] if self._use_unicode else 'dirichlet_eta' + return self._print_Function(e, func_name=func_name) + + def _print_Heaviside(self, e): + func_name = greek_unicode['theta'] if self._use_unicode else 'Heaviside' + if e.args[1] is S.Half: + pform = prettyForm(*self._print(e.args[0]).parens()) + pform = prettyForm(*pform.left(func_name)) + return pform + else: + return self._print_Function(e, func_name=func_name) + + def _print_fresnels(self, e): + return self._print_Function(e, func_name="S") + + def _print_fresnelc(self, e): + return self._print_Function(e, func_name="C") + + def _print_airyai(self, e): + return self._print_Function(e, func_name="Ai") + + def _print_airybi(self, e): + return self._print_Function(e, func_name="Bi") + + def _print_airyaiprime(self, e): + return self._print_Function(e, func_name="Ai'") + + def _print_airybiprime(self, e): + return self._print_Function(e, func_name="Bi'") + + def _print_LambertW(self, e): + return self._print_Function(e, func_name="W") + + def _print_Covariance(self, e): + return self._print_Function(e, func_name="Cov") + + def _print_Variance(self, e): + return self._print_Function(e, func_name="Var") + + def _print_Probability(self, e): + return self._print_Function(e, func_name="P") + + def _print_Expectation(self, e): + return self._print_Function(e, func_name="E", left='[', right=']') + + def _print_Lambda(self, e): + expr = e.expr + sig = e.signature + if self._use_unicode: + arrow = f" {pretty_atom('ArrowFromBar')} " + else: + arrow = " -> " + if len(sig) == 1 and sig[0].is_symbol: + sig = sig[0] + var_form = self._print(sig) + + return prettyForm(*stringPict.next(var_form, arrow, self._print(expr)), binding=8) + + def _print_Order(self, expr): + pform = self._print(expr.expr) + if (expr.point and any(p != S.Zero for p in expr.point)) or \ + len(expr.variables) > 1: + pform = prettyForm(*pform.right("; ")) + if len(expr.variables) > 1: + pform = prettyForm(*pform.right(self._print(expr.variables))) + elif len(expr.variables): + pform = prettyForm(*pform.right(self._print(expr.variables[0]))) + if self._use_unicode: + pform = prettyForm(*pform.right(f" {pretty_atom('Arrow')} ")) + else: + pform = prettyForm(*pform.right(" -> ")) + if len(expr.point) > 1: + pform = prettyForm(*pform.right(self._print(expr.point))) + else: + pform = prettyForm(*pform.right(self._print(expr.point[0]))) + pform = prettyForm(*pform.parens()) + pform = prettyForm(*pform.left("O")) + return pform + + def _print_SingularityFunction(self, e): + if self._use_unicode: + shift = self._print(e.args[0]-e.args[1]) + n = self._print(e.args[2]) + base = prettyForm("<") + base = prettyForm(*base.right(shift)) + base = prettyForm(*base.right(">")) + pform = base**n + return pform + else: + n = self._print(e.args[2]) + shift = self._print(e.args[0]-e.args[1]) + base = self._print_seq(shift, "<", ">", ' ') + return base**n + + def _print_beta(self, e): + func_name = greek_unicode['Beta'] if self._use_unicode else 'B' + return self._print_Function(e, func_name=func_name) + + def _print_betainc(self, e): + func_name = "B'" + return self._print_Function(e, func_name=func_name) + + def _print_betainc_regularized(self, e): + func_name = 'I' + return self._print_Function(e, func_name=func_name) + + def _print_gamma(self, e): + func_name = greek_unicode['Gamma'] if self._use_unicode else 'Gamma' + return self._print_Function(e, func_name=func_name) + + def _print_uppergamma(self, e): + func_name = greek_unicode['Gamma'] if self._use_unicode else 'Gamma' + return self._print_Function(e, func_name=func_name) + + def _print_lowergamma(self, e): + func_name = greek_unicode['gamma'] if self._use_unicode else 'lowergamma' + return self._print_Function(e, func_name=func_name) + + def _print_DiracDelta(self, e): + if self._use_unicode: + if len(e.args) == 2: + a = prettyForm(greek_unicode['delta']) + b = self._print(e.args[1]) + b = prettyForm(*b.parens()) + c = self._print(e.args[0]) + c = prettyForm(*c.parens()) + pform = a**b + pform = prettyForm(*pform.right(' ')) + pform = prettyForm(*pform.right(c)) + return pform + pform = self._print(e.args[0]) + pform = prettyForm(*pform.parens()) + pform = prettyForm(*pform.left(greek_unicode['delta'])) + return pform + else: + return self._print_Function(e) + + def _print_expint(self, e): + subscript = self._print(e.args[0]) + if self._use_unicode and is_subscriptable_in_unicode(subscript): + return self._print_Function(Function('E_%s' % subscript)(e.args[1])) + return self._print_Function(e) + + def _print_Chi(self, e): + # This needs a special case since otherwise it comes out as greek + # letter chi... + prettyFunc = prettyForm("Chi") + prettyArgs = prettyForm(*self._print_seq(e.args).parens()) + + pform = prettyForm( + binding=prettyForm.FUNC, *stringPict.next(prettyFunc, prettyArgs)) + + # store pform parts so it can be reassembled e.g. when powered + pform.prettyFunc = prettyFunc + pform.prettyArgs = prettyArgs + + return pform + + def _print_elliptic_e(self, e): + pforma0 = self._print(e.args[0]) + if len(e.args) == 1: + pform = pforma0 + else: + pforma1 = self._print(e.args[1]) + pform = self._hprint_vseparator(pforma0, pforma1) + pform = prettyForm(*pform.parens()) + pform = prettyForm(*pform.left('E')) + return pform + + def _print_elliptic_k(self, e): + pform = self._print(e.args[0]) + pform = prettyForm(*pform.parens()) + pform = prettyForm(*pform.left('K')) + return pform + + def _print_elliptic_f(self, e): + pforma0 = self._print(e.args[0]) + pforma1 = self._print(e.args[1]) + pform = self._hprint_vseparator(pforma0, pforma1) + pform = prettyForm(*pform.parens()) + pform = prettyForm(*pform.left('F')) + return pform + + def _print_elliptic_pi(self, e): + name = greek_unicode['Pi'] if self._use_unicode else 'Pi' + pforma0 = self._print(e.args[0]) + pforma1 = self._print(e.args[1]) + if len(e.args) == 2: + pform = self._hprint_vseparator(pforma0, pforma1) + else: + pforma2 = self._print(e.args[2]) + pforma = self._hprint_vseparator(pforma1, pforma2, ifascii_nougly=False) + pforma = prettyForm(*pforma.left('; ')) + pform = prettyForm(*pforma.left(pforma0)) + pform = prettyForm(*pform.parens()) + pform = prettyForm(*pform.left(name)) + return pform + + def _print_GoldenRatio(self, expr): + if self._use_unicode: + return prettyForm(pretty_symbol('phi')) + return self._print(Symbol("GoldenRatio")) + + def _print_EulerGamma(self, expr): + if self._use_unicode: + return prettyForm(pretty_symbol('gamma')) + return self._print(Symbol("EulerGamma")) + + def _print_Catalan(self, expr): + return self._print(Symbol("G")) + + def _print_Mod(self, expr): + pform = self._print(expr.args[0]) + if pform.binding > prettyForm.MUL: + pform = prettyForm(*pform.parens()) + pform = prettyForm(*pform.right(' mod ')) + pform = prettyForm(*pform.right(self._print(expr.args[1]))) + pform.binding = prettyForm.OPEN + return pform + + def _print_Add(self, expr, order=None): + terms = self._as_ordered_terms(expr, order=order) + pforms, indices = [], [] + + def pretty_negative(pform, index): + """Prepend a minus sign to a pretty form. """ + #TODO: Move this code to prettyForm + if index == 0: + if pform.height() > 1: + pform_neg = '- ' + else: + pform_neg = '-' + else: + pform_neg = ' - ' + + if (pform.binding > prettyForm.NEG + or pform.binding == prettyForm.ADD): + p = stringPict(*pform.parens()) + else: + p = pform + p = stringPict.next(pform_neg, p) + # Lower the binding to NEG, even if it was higher. Otherwise, it + # will print as a + ( - (b)), instead of a - (b). + return prettyForm(binding=prettyForm.NEG, *p) + + for i, term in enumerate(terms): + if term.is_Mul and term.could_extract_minus_sign(): + coeff, other = term.as_coeff_mul(rational=False) + if coeff == -1: + negterm = Mul(*other, evaluate=False) + else: + negterm = Mul(-coeff, *other, evaluate=False) + pform = self._print(negterm) + pforms.append(pretty_negative(pform, i)) + elif term.is_Rational and term.q > 1: + pforms.append(None) + indices.append(i) + elif term.is_Number and term < 0: + pform = self._print(-term) + pforms.append(pretty_negative(pform, i)) + elif term.is_Relational: + pforms.append(prettyForm(*self._print(term).parens())) + else: + pforms.append(self._print(term)) + + if indices: + large = True + + for pform in pforms: + if pform is not None and pform.height() > 1: + break + else: + large = False + + for i in indices: + term, negative = terms[i], False + + if term < 0: + term, negative = -term, True + + if large: + pform = prettyForm(str(term.p))/prettyForm(str(term.q)) + else: + pform = self._print(term) + + if negative: + pform = pretty_negative(pform, i) + + pforms[i] = pform + + return prettyForm.__add__(*pforms) + + def _print_Mul(self, product): + from sympy.physics.units import Quantity + + # Check for unevaluated Mul. In this case we need to make sure the + # identities are visible, multiple Rational factors are not combined + # etc so we display in a straight-forward form that fully preserves all + # args and their order. + args = product.args + if args[0] is S.One or any(isinstance(arg, Number) for arg in args[1:]): + strargs = list(map(self._print, args)) + # XXX: This is a hack to work around the fact that + # prettyForm.__mul__ absorbs a leading -1 in the args. Probably it + # would be better to fix this in prettyForm.__mul__ instead. + negone = strargs[0] == '-1' + if negone: + strargs[0] = prettyForm('1', 0, 0) + obj = prettyForm.__mul__(*strargs) + if negone: + obj = prettyForm('-' + obj.s, obj.baseline, obj.binding) + return obj + + a = [] # items in the numerator + b = [] # items that are in the denominator (if any) + + if self.order not in ('old', 'none'): + args = product.as_ordered_factors() + else: + args = list(product.args) + + # If quantities are present append them at the back + args = sorted(args, key=lambda x: isinstance(x, Quantity) or + (isinstance(x, Pow) and isinstance(x.base, Quantity))) + + # Gather terms for numerator/denominator + for item in args: + if item.is_commutative and item.is_Pow and item.exp.is_Rational and item.exp.is_negative: + if item.exp != -1: + b.append(Pow(item.base, -item.exp, evaluate=False)) + else: + b.append(Pow(item.base, -item.exp)) + elif item.is_Rational and item is not S.Infinity: + if item.p != 1: + a.append( Rational(item.p) ) + if item.q != 1: + b.append( Rational(item.q) ) + else: + a.append(item) + + # Convert to pretty forms. Parentheses are added by `__mul__`. + a = [self._print(ai) for ai in a] + b = [self._print(bi) for bi in b] + + # Construct a pretty form + if len(b) == 0: + return prettyForm.__mul__(*a) + else: + if len(a) == 0: + a.append( self._print(S.One) ) + return prettyForm.__mul__(*a)/prettyForm.__mul__(*b) + + # A helper function for _print_Pow to print x**(1/n) + def _print_nth_root(self, base, root): + bpretty = self._print(base) + + # In very simple cases, use a single-char root sign + if (self._settings['use_unicode_sqrt_char'] and self._use_unicode + and root == 2 and bpretty.height() == 1 + and (bpretty.width() == 1 + or (base.is_Integer and base.is_nonnegative))): + return prettyForm(*bpretty.left(nth_root[2])) + + # Construct root sign, start with the \/ shape + _zZ = xobj('/', 1) + rootsign = xobj('\\', 1) + _zZ + # Constructing the number to put on root + rpretty = self._print(root) + # roots look bad if they are not a single line + if rpretty.height() != 1: + return self._print(base)**self._print(1/root) + # If power is half, no number should appear on top of root sign + exp = '' if root == 2 else str(rpretty).ljust(2) + if len(exp) > 2: + rootsign = ' '*(len(exp) - 2) + rootsign + # Stack the exponent + rootsign = stringPict(exp + '\n' + rootsign) + rootsign.baseline = 0 + # Diagonal: length is one less than height of base + linelength = bpretty.height() - 1 + diagonal = stringPict('\n'.join( + ' '*(linelength - i - 1) + _zZ + ' '*i + for i in range(linelength) + )) + # Put baseline just below lowest line: next to exp + diagonal.baseline = linelength - 1 + # Make the root symbol + rootsign = prettyForm(*rootsign.right(diagonal)) + # Det the baseline to match contents to fix the height + # but if the height of bpretty is one, the rootsign must be one higher + rootsign.baseline = max(1, bpretty.baseline) + #build result + s = prettyForm(hobj('_', 2 + bpretty.width())) + s = prettyForm(*bpretty.above(s)) + s = prettyForm(*s.left(rootsign)) + return s + + def _print_Pow(self, power): + from sympy.simplify.simplify import fraction + b, e = power.as_base_exp() + if power.is_commutative: + if e is S.NegativeOne: + return prettyForm("1")/self._print(b) + n, d = fraction(e) + if n is S.One and d.is_Atom and not e.is_Integer and (e.is_Rational or d.is_Symbol) \ + and self._settings['root_notation']: + return self._print_nth_root(b, d) + if e.is_Rational and e < 0: + return prettyForm("1")/self._print(Pow(b, -e, evaluate=False)) + + if b.is_Relational: + return prettyForm(*self._print(b).parens()).__pow__(self._print(e)) + + return self._print(b)**self._print(e) + + def _print_UnevaluatedExpr(self, expr): + return self._print(expr.args[0]) + + def __print_numer_denom(self, p, q): + if q == 1: + if p < 0: + return prettyForm(str(p), binding=prettyForm.NEG) + else: + return prettyForm(str(p)) + elif abs(p) >= 10 and abs(q) >= 10: + # If more than one digit in numer and denom, print larger fraction + if p < 0: + return prettyForm(str(p), binding=prettyForm.NEG)/prettyForm(str(q)) + # Old printing method: + #pform = prettyForm(str(-p))/prettyForm(str(q)) + #return prettyForm(binding=prettyForm.NEG, *pform.left('- ')) + else: + return prettyForm(str(p))/prettyForm(str(q)) + else: + return None + + def _print_Rational(self, expr): + result = self.__print_numer_denom(expr.p, expr.q) + + if result is not None: + return result + else: + return self.emptyPrinter(expr) + + def _print_Fraction(self, expr): + result = self.__print_numer_denom(expr.numerator, expr.denominator) + + if result is not None: + return result + else: + return self.emptyPrinter(expr) + + def _print_ProductSet(self, p): + if len(p.sets) >= 1 and not has_variety(p.sets): + return self._print(p.sets[0]) ** self._print(len(p.sets)) + else: + prod_char = pretty_atom('Multiplication') if self._use_unicode else 'x' + return self._print_seq(p.sets, None, None, ' %s ' % prod_char, + parenthesize=lambda set: set.is_Union or + set.is_Intersection or set.is_ProductSet) + + def _print_FiniteSet(self, s): + items = sorted(s.args, key=default_sort_key) + return self._print_seq(items, '{', '}', ', ' ) + + def _print_Range(self, s): + + if self._use_unicode: + dots = pretty_atom('Dots') + else: + dots = '...' + + if s.start.is_infinite and s.stop.is_infinite: + if s.step.is_positive: + printset = dots, -1, 0, 1, dots + else: + printset = dots, 1, 0, -1, dots + elif s.start.is_infinite: + printset = dots, s[-1] - s.step, s[-1] + elif s.stop.is_infinite: + it = iter(s) + printset = next(it), next(it), dots + elif len(s) > 4: + it = iter(s) + printset = next(it), next(it), dots, s[-1] + else: + printset = tuple(s) + + return self._print_seq(printset, '{', '}', ', ' ) + + def _print_Interval(self, i): + if i.start == i.end: + return self._print_seq(i.args[:1], '{', '}') + + else: + if i.left_open: + left = '(' + else: + left = '[' + + if i.right_open: + right = ')' + else: + right = ']' + + return self._print_seq(i.args[:2], left, right) + + def _print_AccumulationBounds(self, i): + left = '<' + right = '>' + + return self._print_seq(i.args[:2], left, right) + + def _print_Intersection(self, u): + + delimiter = ' %s ' % pretty_atom('Intersection', 'n') + + return self._print_seq(u.args, None, None, delimiter, + parenthesize=lambda set: set.is_ProductSet or + set.is_Union or set.is_Complement) + + def _print_Union(self, u): + + union_delimiter = ' %s ' % pretty_atom('Union', 'U') + + return self._print_seq(u.args, None, None, union_delimiter, + parenthesize=lambda set: set.is_ProductSet or + set.is_Intersection or set.is_Complement) + + def _print_SymmetricDifference(self, u): + if not self._use_unicode: + raise NotImplementedError("ASCII pretty printing of SymmetricDifference is not implemented") + + sym_delimeter = ' %s ' % pretty_atom('SymmetricDifference') + + return self._print_seq(u.args, None, None, sym_delimeter) + + def _print_Complement(self, u): + + delimiter = r' \ ' + + return self._print_seq(u.args, None, None, delimiter, + parenthesize=lambda set: set.is_ProductSet or set.is_Intersection + or set.is_Union) + + def _print_ImageSet(self, ts): + if self._use_unicode: + inn = pretty_atom("SmallElementOf") + else: + inn = 'in' + fun = ts.lamda + sets = ts.base_sets + signature = fun.signature + expr = self._print(fun.expr) + + # TODO: the stuff to the left of the | and the stuff to the right of + # the | should have independent baselines, that way something like + # ImageSet(Lambda(x, 1/x**2), S.Naturals) prints the "x in N" part + # centered on the right instead of aligned with the fraction bar on + # the left. The same also applies to ConditionSet and ComplexRegion + if len(signature) == 1: + S = self._print_seq((signature[0], inn, sets[0]), + delimiter=' ') + return self._hprint_vseparator(expr, S, + left='{', right='}', + ifascii_nougly=True, delimiter=' ') + else: + pargs = tuple(j for var, setv in zip(signature, sets) for j in + (var, ' ', inn, ' ', setv, ", ")) + S = self._print_seq(pargs[:-1], delimiter='') + return self._hprint_vseparator(expr, S, + left='{', right='}', + ifascii_nougly=True, delimiter=' ') + + def _print_ConditionSet(self, ts): + if self._use_unicode: + inn = pretty_atom('SmallElementOf') + # using _and because and is a keyword and it is bad practice to + # overwrite them + _and = pretty_atom('And') + else: + inn = 'in' + _and = 'and' + + variables = self._print_seq(Tuple(ts.sym)) + as_expr = getattr(ts.condition, 'as_expr', None) + if as_expr is not None: + cond = self._print(ts.condition.as_expr()) + else: + cond = self._print(ts.condition) + if self._use_unicode: + cond = self._print(cond) + cond = prettyForm(*cond.parens()) + + if ts.base_set is S.UniversalSet: + return self._hprint_vseparator(variables, cond, left="{", + right="}", ifascii_nougly=True, + delimiter=' ') + + base = self._print(ts.base_set) + C = self._print_seq((variables, inn, base, _and, cond), + delimiter=' ') + return self._hprint_vseparator(variables, C, left="{", right="}", + ifascii_nougly=True, delimiter=' ') + + def _print_ComplexRegion(self, ts): + if self._use_unicode: + inn = pretty_atom('SmallElementOf') + else: + inn = 'in' + variables = self._print_seq(ts.variables) + expr = self._print(ts.expr) + prodsets = self._print(ts.sets) + + C = self._print_seq((variables, inn, prodsets), + delimiter=' ') + return self._hprint_vseparator(expr, C, left="{", right="}", + ifascii_nougly=True, delimiter=' ') + + def _print_Contains(self, e): + var, set = e.args + if self._use_unicode: + el = f" {pretty_atom('ElementOf')} " + return prettyForm(*stringPict.next(self._print(var), + el, self._print(set)), binding=8) + else: + return prettyForm(sstr(e)) + + def _print_FourierSeries(self, s): + if s.an.formula is S.Zero and s.bn.formula is S.Zero: + return self._print(s.a0) + if self._use_unicode: + dots = pretty_atom('Dots') + else: + dots = '...' + return self._print_Add(s.truncate()) + self._print(dots) + + def _print_FormalPowerSeries(self, s): + return self._print_Add(s.infinite) + + def _print_SetExpr(self, se): + pretty_set = prettyForm(*self._print(se.set).parens()) + pretty_name = self._print(Symbol("SetExpr")) + return prettyForm(*pretty_name.right(pretty_set)) + + def _print_SeqFormula(self, s): + if self._use_unicode: + dots = pretty_atom('Dots') + else: + dots = '...' + + if len(s.start.free_symbols) > 0 or len(s.stop.free_symbols) > 0: + raise NotImplementedError("Pretty printing of sequences with symbolic bound not implemented") + + if s.start is S.NegativeInfinity: + stop = s.stop + printset = (dots, s.coeff(stop - 3), s.coeff(stop - 2), + s.coeff(stop - 1), s.coeff(stop)) + elif s.stop is S.Infinity or s.length > 4: + printset = s[:4] + printset.append(dots) + printset = tuple(printset) + else: + printset = tuple(s) + return self._print_list(printset) + + _print_SeqPer = _print_SeqFormula + _print_SeqAdd = _print_SeqFormula + _print_SeqMul = _print_SeqFormula + + def _print_seq(self, seq, left=None, right=None, delimiter=', ', + parenthesize=lambda x: False, ifascii_nougly=True): + + pforms = [] + for item in seq: + pform = self._print(item) + if parenthesize(item): + pform = prettyForm(*pform.parens()) + if pforms: + pforms.append(delimiter) + pforms.append(pform) + + if not pforms: + s = stringPict('') + else: + s = prettyForm(*stringPict.next(*pforms)) + + s = prettyForm(*s.parens(left, right, ifascii_nougly=ifascii_nougly)) + return s + + def join(self, delimiter, args): + pform = None + + for arg in args: + if pform is None: + pform = arg + else: + pform = prettyForm(*pform.right(delimiter)) + pform = prettyForm(*pform.right(arg)) + + if pform is None: + return prettyForm("") + else: + return pform + + def _print_list(self, l): + return self._print_seq(l, '[', ']') + + def _print_tuple(self, t): + if len(t) == 1: + ptuple = prettyForm(*stringPict.next(self._print(t[0]), ',')) + return prettyForm(*ptuple.parens('(', ')', ifascii_nougly=True)) + else: + return self._print_seq(t, '(', ')') + + def _print_Tuple(self, expr): + return self._print_tuple(expr) + + def _print_dict(self, d): + keys = sorted(d.keys(), key=default_sort_key) + items = [] + + for k in keys: + K = self._print(k) + V = self._print(d[k]) + s = prettyForm(*stringPict.next(K, ': ', V)) + + items.append(s) + + return self._print_seq(items, '{', '}') + + def _print_Dict(self, d): + return self._print_dict(d) + + def _print_set(self, s): + if not s: + return prettyForm('set()') + items = sorted(s, key=default_sort_key) + pretty = self._print_seq(items) + pretty = prettyForm(*pretty.parens('{', '}', ifascii_nougly=True)) + return pretty + + def _print_frozenset(self, s): + if not s: + return prettyForm('frozenset()') + items = sorted(s, key=default_sort_key) + pretty = self._print_seq(items) + pretty = prettyForm(*pretty.parens('{', '}', ifascii_nougly=True)) + pretty = prettyForm(*pretty.parens('(', ')', ifascii_nougly=True)) + pretty = prettyForm(*stringPict.next(type(s).__name__, pretty)) + return pretty + + def _print_UniversalSet(self, s): + if self._use_unicode: + return prettyForm(pretty_atom('Universe')) + else: + return prettyForm('UniversalSet') + + def _print_PolyRing(self, ring): + return prettyForm(sstr(ring)) + + def _print_FracField(self, field): + return prettyForm(sstr(field)) + + def _print_FreeGroupElement(self, elm): + return prettyForm(str(elm)) + + def _print_PolyElement(self, poly): + return prettyForm(sstr(poly)) + + def _print_FracElement(self, frac): + return prettyForm(sstr(frac)) + + def _print_AlgebraicNumber(self, expr): + if expr.is_aliased: + return self._print(expr.as_poly().as_expr()) + else: + return self._print(expr.as_expr()) + + def _print_ComplexRootOf(self, expr): + args = [self._print_Add(expr.expr, order='lex'), expr.index] + pform = prettyForm(*self._print_seq(args).parens()) + pform = prettyForm(*pform.left('CRootOf')) + return pform + + def _print_RootSum(self, expr): + args = [self._print_Add(expr.expr, order='lex')] + + if expr.fun is not S.IdentityFunction: + args.append(self._print(expr.fun)) + + pform = prettyForm(*self._print_seq(args).parens()) + pform = prettyForm(*pform.left('RootSum')) + + return pform + + def _print_FiniteField(self, expr): + if self._use_unicode: + form = f"{pretty_atom('Integers')}_%d" + else: + form = 'GF(%d)' + + return prettyForm(pretty_symbol(form % expr.mod)) + + def _print_IntegerRing(self, expr): + if self._use_unicode: + return prettyForm(pretty_atom('Integers')) + else: + return prettyForm('ZZ') + + def _print_RationalField(self, expr): + if self._use_unicode: + return prettyForm(pretty_atom('Rationals')) + else: + return prettyForm('QQ') + + def _print_RealField(self, domain): + if self._use_unicode: + prefix = pretty_atom("Reals") + else: + prefix = 'RR' + + if domain.has_default_precision: + return prettyForm(prefix) + else: + return self._print(pretty_symbol(prefix + "_" + str(domain.precision))) + + def _print_ComplexField(self, domain): + if self._use_unicode: + prefix = pretty_atom('Complexes') + else: + prefix = 'CC' + + if domain.has_default_precision: + return prettyForm(prefix) + else: + return self._print(pretty_symbol(prefix + "_" + str(domain.precision))) + + def _print_PolynomialRing(self, expr): + args = list(expr.symbols) + + if not expr.order.is_default: + order = prettyForm(*prettyForm("order=").right(self._print(expr.order))) + args.append(order) + + pform = self._print_seq(args, '[', ']') + pform = prettyForm(*pform.left(self._print(expr.domain))) + + return pform + + def _print_FractionField(self, expr): + args = list(expr.symbols) + + if not expr.order.is_default: + order = prettyForm(*prettyForm("order=").right(self._print(expr.order))) + args.append(order) + + pform = self._print_seq(args, '(', ')') + pform = prettyForm(*pform.left(self._print(expr.domain))) + + return pform + + def _print_PolynomialRingBase(self, expr): + g = expr.symbols + if str(expr.order) != str(expr.default_order): + g = g + ("order=" + str(expr.order),) + pform = self._print_seq(g, '[', ']') + pform = prettyForm(*pform.left(self._print(expr.domain))) + + return pform + + def _print_GroebnerBasis(self, basis): + exprs = [ self._print_Add(arg, order=basis.order) + for arg in basis.exprs ] + exprs = prettyForm(*self.join(", ", exprs).parens(left="[", right="]")) + + gens = [ self._print(gen) for gen in basis.gens ] + + domain = prettyForm( + *prettyForm("domain=").right(self._print(basis.domain))) + order = prettyForm( + *prettyForm("order=").right(self._print(basis.order))) + + pform = self.join(", ", [exprs] + gens + [domain, order]) + + pform = prettyForm(*pform.parens()) + pform = prettyForm(*pform.left(basis.__class__.__name__)) + + return pform + + def _print_Subs(self, e): + pform = self._print(e.expr) + pform = prettyForm(*pform.parens()) + + h = pform.height() if pform.height() > 1 else 2 + rvert = stringPict(vobj('|', h), baseline=pform.baseline) + pform = prettyForm(*pform.right(rvert)) + + b = pform.baseline + pform.baseline = pform.height() - 1 + pform = prettyForm(*pform.right(self._print_seq([ + self._print_seq((self._print(v[0]), xsym('=='), self._print(v[1])), + delimiter='') for v in zip(e.variables, e.point) ]))) + + pform.baseline = b + return pform + + def _print_number_function(self, e, name): + # Print name_arg[0] for one argument or name_arg[0](arg[1]) + # for more than one argument + pform = prettyForm(name) + arg = self._print(e.args[0]) + pform_arg = prettyForm(" "*arg.width()) + pform_arg = prettyForm(*pform_arg.below(arg)) + pform = prettyForm(*pform.right(pform_arg)) + if len(e.args) == 1: + return pform + m, x = e.args + # TODO: copy-pasted from _print_Function: can we do better? + prettyFunc = pform + prettyArgs = prettyForm(*self._print_seq([x]).parens()) + pform = prettyForm( + binding=prettyForm.FUNC, *stringPict.next(prettyFunc, prettyArgs)) + pform.prettyFunc = prettyFunc + pform.prettyArgs = prettyArgs + return pform + + def _print_euler(self, e): + return self._print_number_function(e, "E") + + def _print_catalan(self, e): + return self._print_number_function(e, "C") + + def _print_bernoulli(self, e): + return self._print_number_function(e, "B") + + _print_bell = _print_bernoulli + + def _print_lucas(self, e): + return self._print_number_function(e, "L") + + def _print_fibonacci(self, e): + return self._print_number_function(e, "F") + + def _print_tribonacci(self, e): + return self._print_number_function(e, "T") + + def _print_stieltjes(self, e): + if self._use_unicode: + return self._print_number_function(e, greek_unicode['gamma']) + else: + return self._print_number_function(e, "stieltjes") + + def _print_KroneckerDelta(self, e): + pform = self._print(e.args[0]) + pform = prettyForm(*pform.right(prettyForm(','))) + pform = prettyForm(*pform.right(self._print(e.args[1]))) + if self._use_unicode: + a = stringPict(pretty_symbol('delta')) + else: + a = stringPict('d') + b = pform + top = stringPict(*b.left(' '*a.width())) + bot = stringPict(*a.right(' '*b.width())) + return prettyForm(binding=prettyForm.POW, *bot.below(top)) + + def _print_RandomDomain(self, d): + if hasattr(d, 'as_boolean'): + pform = self._print('Domain: ') + pform = prettyForm(*pform.right(self._print(d.as_boolean()))) + return pform + elif hasattr(d, 'set'): + pform = self._print('Domain: ') + pform = prettyForm(*pform.right(self._print(d.symbols))) + pform = prettyForm(*pform.right(self._print(' in '))) + pform = prettyForm(*pform.right(self._print(d.set))) + return pform + elif hasattr(d, 'symbols'): + pform = self._print('Domain on ') + pform = prettyForm(*pform.right(self._print(d.symbols))) + return pform + else: + return self._print(None) + + def _print_DMP(self, p): + try: + if p.ring is not None: + # TODO incorporate order + return self._print(p.ring.to_sympy(p)) + except SympifyError: + pass + return self._print(repr(p)) + + def _print_DMF(self, p): + return self._print_DMP(p) + + def _print_Object(self, object): + return self._print(pretty_symbol(object.name)) + + def _print_Morphism(self, morphism): + arrow = xsym("-->") + + domain = self._print(morphism.domain) + codomain = self._print(morphism.codomain) + tail = domain.right(arrow, codomain)[0] + + return prettyForm(tail) + + def _print_NamedMorphism(self, morphism): + pretty_name = self._print(pretty_symbol(morphism.name)) + pretty_morphism = self._print_Morphism(morphism) + return prettyForm(pretty_name.right(":", pretty_morphism)[0]) + + def _print_IdentityMorphism(self, morphism): + from sympy.categories import NamedMorphism + return self._print_NamedMorphism( + NamedMorphism(morphism.domain, morphism.codomain, "id")) + + def _print_CompositeMorphism(self, morphism): + + circle = xsym(".") + + # All components of the morphism have names and it is thus + # possible to build the name of the composite. + component_names_list = [pretty_symbol(component.name) for + component in morphism.components] + component_names_list.reverse() + component_names = circle.join(component_names_list) + ":" + + pretty_name = self._print(component_names) + pretty_morphism = self._print_Morphism(morphism) + return prettyForm(pretty_name.right(pretty_morphism)[0]) + + def _print_Category(self, category): + return self._print(pretty_symbol(category.name)) + + def _print_Diagram(self, diagram): + if not diagram.premises: + # This is an empty diagram. + return self._print(S.EmptySet) + + pretty_result = self._print(diagram.premises) + if diagram.conclusions: + results_arrow = " %s " % xsym("==>") + + pretty_conclusions = self._print(diagram.conclusions)[0] + pretty_result = pretty_result.right( + results_arrow, pretty_conclusions) + + return prettyForm(pretty_result[0]) + + def _print_DiagramGrid(self, grid): + from sympy.matrices import Matrix + matrix = Matrix([[grid[i, j] if grid[i, j] else Symbol(" ") + for j in range(grid.width)] + for i in range(grid.height)]) + return self._print_matrix_contents(matrix) + + def _print_FreeModuleElement(self, m): + # Print as row vector for convenience, for now. + return self._print_seq(m, '[', ']') + + def _print_SubModule(self, M): + gens = [[M.ring.to_sympy(g) for g in gen] for gen in M.gens] + return self._print_seq(gens, '<', '>') + + def _print_FreeModule(self, M): + return self._print(M.ring)**self._print(M.rank) + + def _print_ModuleImplementedIdeal(self, M): + sym = M.ring.to_sympy + return self._print_seq([sym(x) for [x] in M._module.gens], '<', '>') + + def _print_QuotientRing(self, R): + return self._print(R.ring) / self._print(R.base_ideal) + + def _print_QuotientRingElement(self, R): + return self._print(R.ring.to_sympy(R)) + self._print(R.ring.base_ideal) + + def _print_QuotientModuleElement(self, m): + return self._print(m.data) + self._print(m.module.killed_module) + + def _print_QuotientModule(self, M): + return self._print(M.base) / self._print(M.killed_module) + + def _print_MatrixHomomorphism(self, h): + matrix = self._print(h._sympy_matrix()) + matrix.baseline = matrix.height() // 2 + pform = prettyForm(*matrix.right(' : ', self._print(h.domain), + ' %s> ' % hobj('-', 2), self._print(h.codomain))) + return pform + + def _print_Manifold(self, manifold): + return self._print(manifold.name) + + def _print_Patch(self, patch): + return self._print(patch.name) + + def _print_CoordSystem(self, coords): + return self._print(coords.name) + + def _print_BaseScalarField(self, field): + string = field._coord_sys.symbols[field._index].name + return self._print(pretty_symbol(string)) + + def _print_BaseVectorField(self, field): + s = U('PARTIAL DIFFERENTIAL') + '_' + field._coord_sys.symbols[field._index].name + return self._print(pretty_symbol(s)) + + def _print_Differential(self, diff): + if self._use_unicode: + d = pretty_atom('Differential') + else: + d = 'd' + field = diff._form_field + if hasattr(field, '_coord_sys'): + string = field._coord_sys.symbols[field._index].name + return self._print(d + ' ' + pretty_symbol(string)) + else: + pform = self._print(field) + pform = prettyForm(*pform.parens()) + return prettyForm(*pform.left(d)) + + def _print_Tr(self, p): + #TODO: Handle indices + pform = self._print(p.args[0]) + pform = prettyForm(*pform.left('%s(' % (p.__class__.__name__))) + pform = prettyForm(*pform.right(')')) + return pform + + def _print_primenu(self, e): + pform = self._print(e.args[0]) + pform = prettyForm(*pform.parens()) + if self._use_unicode: + pform = prettyForm(*pform.left(greek_unicode['nu'])) + else: + pform = prettyForm(*pform.left('nu')) + return pform + + def _print_primeomega(self, e): + pform = self._print(e.args[0]) + pform = prettyForm(*pform.parens()) + if self._use_unicode: + pform = prettyForm(*pform.left(greek_unicode['Omega'])) + else: + pform = prettyForm(*pform.left('Omega')) + return pform + + def _print_Quantity(self, e): + if e.name.name == 'degree': + if self._use_unicode: + pform = self._print(pretty_atom('Degree')) + else: + pform = self._print(chr(176)) + return pform + else: + return self.emptyPrinter(e) + + def _print_AssignmentBase(self, e): + + op = prettyForm(' ' + xsym(e.op) + ' ') + + l = self._print(e.lhs) + r = self._print(e.rhs) + pform = prettyForm(*stringPict.next(l, op, r)) + return pform + + def _print_Str(self, s): + return self._print(s.name) + + +@print_function(PrettyPrinter) +def pretty(expr, **settings): + """Returns a string containing the prettified form of expr. + + For information on keyword arguments see pretty_print function. + + """ + pp = PrettyPrinter(settings) + + # XXX: this is an ugly hack, but at least it works + use_unicode = pp._settings['use_unicode'] + uflag = pretty_use_unicode(use_unicode) + + try: + return pp.doprint(expr) + finally: + pretty_use_unicode(uflag) + + +def pretty_print(expr, **kwargs): + """Prints expr in pretty form. + + pprint is just a shortcut for this function. + + Parameters + ========== + + expr : expression + The expression to print. + + wrap_line : bool, optional (default=True) + Line wrapping enabled/disabled. + + num_columns : int or None, optional (default=None) + Number of columns before line breaking (default to None which reads + the terminal width), useful when using SymPy without terminal. + + use_unicode : bool or None, optional (default=None) + Use unicode characters, such as the Greek letter pi instead of + the string pi. + + full_prec : bool or string, optional (default="auto") + Use full precision. + + order : bool or string, optional (default=None) + Set to 'none' for long expressions if slow; default is None. + + use_unicode_sqrt_char : bool, optional (default=True) + Use compact single-character square root symbol (when unambiguous). + + root_notation : bool, optional (default=True) + Set to 'False' for printing exponents of the form 1/n in fractional form. + By default exponent is printed in root form. + + mat_symbol_style : string, optional (default="plain") + Set to "bold" for printing MatrixSymbols using a bold mathematical symbol face. + By default the standard face is used. + + imaginary_unit : string, optional (default="i") + Letter to use for imaginary unit when use_unicode is True. + Can be "i" (default) or "j". + """ + print(pretty(expr, **kwargs)) + +pprint = pretty_print + + +def pager_print(expr, **settings): + """Prints expr using the pager, in pretty form. + + This invokes a pager command using pydoc. Lines are not wrapped + automatically. This routine is meant to be used with a pager that allows + sideways scrolling, like ``less -S``. + + Parameters are the same as for ``pretty_print``. If you wish to wrap lines, + pass ``num_columns=None`` to auto-detect the width of the terminal. + + """ + from pydoc import pager + from locale import getpreferredencoding + if 'num_columns' not in settings: + settings['num_columns'] = 500000 # disable line wrap + pager(pretty(expr, **settings).encode(getpreferredencoding())) diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/pretty/pretty_symbology.py b/.venv/lib/python3.12/site-packages/sympy/printing/pretty/pretty_symbology.py new file mode 100644 index 0000000000000000000000000000000000000000..bdb6ec556c6ed7b15dfcddcfc3da189102d5395b --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/pretty/pretty_symbology.py @@ -0,0 +1,731 @@ +"""Symbolic primitives + unicode/ASCII abstraction for pretty.py""" + +import sys +import warnings +from string import ascii_lowercase, ascii_uppercase +import unicodedata + +unicode_warnings = '' + +def U(name): + """ + Get a unicode character by name or, None if not found. + + This exists because older versions of Python use older unicode databases. + """ + try: + return unicodedata.lookup(name) + except KeyError: + global unicode_warnings + unicode_warnings += 'No \'%s\' in unicodedata\n' % name + return None + +from sympy.printing.conventions import split_super_sub +from sympy.core.alphabets import greeks +from sympy.utilities.exceptions import sympy_deprecation_warning + +# prefix conventions when constructing tables +# L - LATIN i +# G - GREEK beta +# D - DIGIT 0 +# S - SYMBOL + + + +__all__ = ['greek_unicode', 'sub', 'sup', 'xsym', 'vobj', 'hobj', 'pretty_symbol', + 'annotated', 'center_pad', 'center'] + + +_use_unicode = False + + +def pretty_use_unicode(flag=None): + """Set whether pretty-printer should use unicode by default""" + global _use_unicode, unicode_warnings + if flag is None: + return _use_unicode + + if flag and unicode_warnings: + # print warnings (if any) on first unicode usage + warnings.warn(unicode_warnings) + unicode_warnings = '' + + use_unicode_prev = _use_unicode + _use_unicode = flag + return use_unicode_prev + + +def pretty_try_use_unicode(): + """See if unicode output is available and leverage it if possible""" + + encoding = getattr(sys.stdout, 'encoding', None) + + # this happens when e.g. stdout is redirected through a pipe, or is + # e.g. a cStringIO.StringO + if encoding is None: + return # sys.stdout has no encoding + + symbols = [] + + # see if we can represent greek alphabet + symbols += greek_unicode.values() + + # and atoms + symbols += atoms_table.values() + + for s in symbols: + if s is None: + return # common symbols not present! + + try: + s.encode(encoding) + except UnicodeEncodeError: + return + + # all the characters were present and encodable + pretty_use_unicode(True) + + +def xstr(*args): + sympy_deprecation_warning( + """ + The sympy.printing.pretty.pretty_symbology.xstr() function is + deprecated. Use str() instead. + """, + deprecated_since_version="1.7", + active_deprecations_target="deprecated-pretty-printing-functions" + ) + return str(*args) + +# GREEK +g = lambda l: U('GREEK SMALL LETTER %s' % l.upper()) +G = lambda l: U('GREEK CAPITAL LETTER %s' % l.upper()) + +greek_letters = list(greeks) # make a copy +# deal with Unicode's funny spelling of lambda +greek_letters[greek_letters.index('lambda')] = 'lamda' + +# {} greek letter -> (g,G) +greek_unicode = {L: g(L) for L in greek_letters} +greek_unicode.update((L[0].upper() + L[1:], G(L)) for L in greek_letters) + +# aliases +greek_unicode['lambda'] = greek_unicode['lamda'] +greek_unicode['Lambda'] = greek_unicode['Lamda'] +greek_unicode['varsigma'] = '\N{GREEK SMALL LETTER FINAL SIGMA}' + +# BOLD +b = lambda l: U('MATHEMATICAL BOLD SMALL %s' % l.upper()) +B = lambda l: U('MATHEMATICAL BOLD CAPITAL %s' % l.upper()) + +bold_unicode = {l: b(l) for l in ascii_lowercase} +bold_unicode.update((L, B(L)) for L in ascii_uppercase) + +# GREEK BOLD +gb = lambda l: U('MATHEMATICAL BOLD SMALL %s' % l.upper()) +GB = lambda l: U('MATHEMATICAL BOLD CAPITAL %s' % l.upper()) + +greek_bold_letters = list(greeks) # make a copy, not strictly required here +# deal with Unicode's funny spelling of lambda +greek_bold_letters[greek_bold_letters.index('lambda')] = 'lamda' + +# {} greek letter -> (g,G) +greek_bold_unicode = {L: g(L) for L in greek_bold_letters} +greek_bold_unicode.update((L[0].upper() + L[1:], G(L)) for L in greek_bold_letters) +greek_bold_unicode['lambda'] = greek_unicode['lamda'] +greek_bold_unicode['Lambda'] = greek_unicode['Lamda'] +greek_bold_unicode['varsigma'] = '\N{MATHEMATICAL BOLD SMALL FINAL SIGMA}' + +digit_2txt = { + '0': 'ZERO', + '1': 'ONE', + '2': 'TWO', + '3': 'THREE', + '4': 'FOUR', + '5': 'FIVE', + '6': 'SIX', + '7': 'SEVEN', + '8': 'EIGHT', + '9': 'NINE', +} + +symb_2txt = { + '+': 'PLUS SIGN', + '-': 'MINUS', + '=': 'EQUALS SIGN', + '(': 'LEFT PARENTHESIS', + ')': 'RIGHT PARENTHESIS', + '[': 'LEFT SQUARE BRACKET', + ']': 'RIGHT SQUARE BRACKET', + '{': 'LEFT CURLY BRACKET', + '}': 'RIGHT CURLY BRACKET', + + # non-std + '{}': 'CURLY BRACKET', + 'sum': 'SUMMATION', + 'int': 'INTEGRAL', +} + +# SUBSCRIPT & SUPERSCRIPT +LSUB = lambda letter: U('LATIN SUBSCRIPT SMALL LETTER %s' % letter.upper()) +GSUB = lambda letter: U('GREEK SUBSCRIPT SMALL LETTER %s' % letter.upper()) +DSUB = lambda digit: U('SUBSCRIPT %s' % digit_2txt[digit]) +SSUB = lambda symb: U('SUBSCRIPT %s' % symb_2txt[symb]) + +LSUP = lambda letter: U('SUPERSCRIPT LATIN SMALL LETTER %s' % letter.upper()) +DSUP = lambda digit: U('SUPERSCRIPT %s' % digit_2txt[digit]) +SSUP = lambda symb: U('SUPERSCRIPT %s' % symb_2txt[symb]) + +sub = {} # symb -> subscript symbol +sup = {} # symb -> superscript symbol + +# latin subscripts +for l in 'aeioruvxhklmnpst': + sub[l] = LSUB(l) + +for l in 'in': + sup[l] = LSUP(l) + +for gl in ['beta', 'gamma', 'rho', 'phi', 'chi']: + sub[gl] = GSUB(gl) + +for d in [str(i) for i in range(10)]: + sub[d] = DSUB(d) + sup[d] = DSUP(d) + +for s in '+-=()': + sub[s] = SSUB(s) + sup[s] = SSUP(s) + +# Variable modifiers +# TODO: Make brackets adjust to height of contents +modifier_dict = { + # Accents + 'mathring': lambda s: center_accent(s, '\N{COMBINING RING ABOVE}'), + 'ddddot': lambda s: center_accent(s, '\N{COMBINING FOUR DOTS ABOVE}'), + 'dddot': lambda s: center_accent(s, '\N{COMBINING THREE DOTS ABOVE}'), + 'ddot': lambda s: center_accent(s, '\N{COMBINING DIAERESIS}'), + 'dot': lambda s: center_accent(s, '\N{COMBINING DOT ABOVE}'), + 'check': lambda s: center_accent(s, '\N{COMBINING CARON}'), + 'breve': lambda s: center_accent(s, '\N{COMBINING BREVE}'), + 'acute': lambda s: center_accent(s, '\N{COMBINING ACUTE ACCENT}'), + 'grave': lambda s: center_accent(s, '\N{COMBINING GRAVE ACCENT}'), + 'tilde': lambda s: center_accent(s, '\N{COMBINING TILDE}'), + 'hat': lambda s: center_accent(s, '\N{COMBINING CIRCUMFLEX ACCENT}'), + 'bar': lambda s: center_accent(s, '\N{COMBINING OVERLINE}'), + 'vec': lambda s: center_accent(s, '\N{COMBINING RIGHT ARROW ABOVE}'), + 'prime': lambda s: s+'\N{PRIME}', + 'prm': lambda s: s+'\N{PRIME}', + # # Faces -- these are here for some compatibility with latex printing + # 'bold': lambda s: s, + # 'bm': lambda s: s, + # 'cal': lambda s: s, + # 'scr': lambda s: s, + # 'frak': lambda s: s, + # Brackets + 'norm': lambda s: '\N{DOUBLE VERTICAL LINE}'+s+'\N{DOUBLE VERTICAL LINE}', + 'avg': lambda s: '\N{MATHEMATICAL LEFT ANGLE BRACKET}'+s+'\N{MATHEMATICAL RIGHT ANGLE BRACKET}', + 'abs': lambda s: '\N{VERTICAL LINE}'+s+'\N{VERTICAL LINE}', + 'mag': lambda s: '\N{VERTICAL LINE}'+s+'\N{VERTICAL LINE}', +} + +# VERTICAL OBJECTS +HUP = lambda symb: U('%s UPPER HOOK' % symb_2txt[symb]) +CUP = lambda symb: U('%s UPPER CORNER' % symb_2txt[symb]) +MID = lambda symb: U('%s MIDDLE PIECE' % symb_2txt[symb]) +EXT = lambda symb: U('%s EXTENSION' % symb_2txt[symb]) +HLO = lambda symb: U('%s LOWER HOOK' % symb_2txt[symb]) +CLO = lambda symb: U('%s LOWER CORNER' % symb_2txt[symb]) +TOP = lambda symb: U('%s TOP' % symb_2txt[symb]) +BOT = lambda symb: U('%s BOTTOM' % symb_2txt[symb]) + +# {} '(' -> (extension, start, end, middle) 1-character +_xobj_unicode = { + + # vertical symbols + # (( ext, top, bot, mid ), c1) + '(': (( EXT('('), HUP('('), HLO('(') ), '('), + ')': (( EXT(')'), HUP(')'), HLO(')') ), ')'), + '[': (( EXT('['), CUP('['), CLO('[') ), '['), + ']': (( EXT(']'), CUP(']'), CLO(']') ), ']'), + '{': (( EXT('{}'), HUP('{'), HLO('{'), MID('{') ), '{'), + '}': (( EXT('{}'), HUP('}'), HLO('}'), MID('}') ), '}'), + '|': U('BOX DRAWINGS LIGHT VERTICAL'), + 'Tee': U('BOX DRAWINGS LIGHT UP AND HORIZONTAL'), + 'UpTack': U('BOX DRAWINGS LIGHT DOWN AND HORIZONTAL'), + 'corner_up_centre' + '(_ext': U('LEFT PARENTHESIS EXTENSION'), + ')_ext': U('RIGHT PARENTHESIS EXTENSION'), + '(_lower_hook': U('LEFT PARENTHESIS LOWER HOOK'), + ')_lower_hook': U('RIGHT PARENTHESIS LOWER HOOK'), + '(_upper_hook': U('LEFT PARENTHESIS UPPER HOOK'), + ')_upper_hook': U('RIGHT PARENTHESIS UPPER HOOK'), + '<': ((U('BOX DRAWINGS LIGHT VERTICAL'), + U('BOX DRAWINGS LIGHT DIAGONAL UPPER RIGHT TO LOWER LEFT'), + U('BOX DRAWINGS LIGHT DIAGONAL UPPER LEFT TO LOWER RIGHT')), '<'), + + '>': ((U('BOX DRAWINGS LIGHT VERTICAL'), + U('BOX DRAWINGS LIGHT DIAGONAL UPPER LEFT TO LOWER RIGHT'), + U('BOX DRAWINGS LIGHT DIAGONAL UPPER RIGHT TO LOWER LEFT')), '>'), + + 'lfloor': (( EXT('['), EXT('['), CLO('[') ), U('LEFT FLOOR')), + 'rfloor': (( EXT(']'), EXT(']'), CLO(']') ), U('RIGHT FLOOR')), + 'lceil': (( EXT('['), CUP('['), EXT('[') ), U('LEFT CEILING')), + 'rceil': (( EXT(']'), CUP(']'), EXT(']') ), U('RIGHT CEILING')), + + 'int': (( EXT('int'), U('TOP HALF INTEGRAL'), U('BOTTOM HALF INTEGRAL') ), U('INTEGRAL')), + 'sum': (( U('BOX DRAWINGS LIGHT DIAGONAL UPPER LEFT TO LOWER RIGHT'), '_', U('OVERLINE'), U('BOX DRAWINGS LIGHT DIAGONAL UPPER RIGHT TO LOWER LEFT')), U('N-ARY SUMMATION')), + + # horizontal objects + #'-': '-', + '-': U('BOX DRAWINGS LIGHT HORIZONTAL'), + '_': U('LOW LINE'), + # We used to use this, but LOW LINE looks better for roots, as it's a + # little lower (i.e., it lines up with the / perfectly. But perhaps this + # one would still be wanted for some cases? + # '_': U('HORIZONTAL SCAN LINE-9'), + + # diagonal objects '\' & '/' ? + '/': U('BOX DRAWINGS LIGHT DIAGONAL UPPER RIGHT TO LOWER LEFT'), + '\\': U('BOX DRAWINGS LIGHT DIAGONAL UPPER LEFT TO LOWER RIGHT'), +} + +_xobj_ascii = { + # vertical symbols + # (( ext, top, bot, mid ), c1) + '(': (( '|', '/', '\\' ), '('), + ')': (( '|', '\\', '/' ), ')'), + +# XXX this looks ugly +# '[': (( '|', '-', '-' ), '['), +# ']': (( '|', '-', '-' ), ']'), +# XXX not so ugly :( + '[': (( '[', '[', '[' ), '['), + ']': (( ']', ']', ']' ), ']'), + + '{': (( '|', '/', '\\', '<' ), '{'), + '}': (( '|', '\\', '/', '>' ), '}'), + '|': '|', + + '<': (( '|', '/', '\\' ), '<'), + '>': (( '|', '\\', '/' ), '>'), + + 'int': ( ' | ', ' /', '/ ' ), + + # horizontal objects + '-': '-', + '_': '_', + + # diagonal objects '\' & '/' ? + '/': '/', + '\\': '\\', +} + + +def xobj(symb, length): + """Construct spatial object of given length. + + return: [] of equal-length strings + """ + + if length <= 0: + raise ValueError("Length should be greater than 0") + + # TODO robustify when no unicodedat available + if _use_unicode: + _xobj = _xobj_unicode + else: + _xobj = _xobj_ascii + + vinfo = _xobj[symb] + + c1 = top = bot = mid = None + + if not isinstance(vinfo, tuple): # 1 entry + ext = vinfo + else: + if isinstance(vinfo[0], tuple): # (vlong), c1 + vlong = vinfo[0] + c1 = vinfo[1] + else: # (vlong), c1 + vlong = vinfo + + ext = vlong[0] + + try: + top = vlong[1] + bot = vlong[2] + mid = vlong[3] + except IndexError: + pass + + if c1 is None: + c1 = ext + if top is None: + top = ext + if bot is None: + bot = ext + if mid is not None: + if (length % 2) == 0: + # even height, but we have to print it somehow anyway... + # XXX is it ok? + length += 1 + + else: + mid = ext + + if length == 1: + return c1 + + res = [] + next = (length - 2)//2 + nmid = (length - 2) - next*2 + + res += [top] + res += [ext]*next + res += [mid]*nmid + res += [ext]*next + res += [bot] + + return res + + +def vobj(symb, height): + """Construct vertical object of a given height + + see: xobj + """ + return '\n'.join( xobj(symb, height) ) + + +def hobj(symb, width): + """Construct horizontal object of a given width + + see: xobj + """ + return ''.join( xobj(symb, width) ) + +# RADICAL +# n -> symbol +root = { + 2: U('SQUARE ROOT'), # U('RADICAL SYMBOL BOTTOM') + 3: U('CUBE ROOT'), + 4: U('FOURTH ROOT'), +} + + +# RATIONAL +VF = lambda txt: U('VULGAR FRACTION %s' % txt) + +# (p,q) -> symbol +frac = { + (1, 2): VF('ONE HALF'), + (1, 3): VF('ONE THIRD'), + (2, 3): VF('TWO THIRDS'), + (1, 4): VF('ONE QUARTER'), + (3, 4): VF('THREE QUARTERS'), + (1, 5): VF('ONE FIFTH'), + (2, 5): VF('TWO FIFTHS'), + (3, 5): VF('THREE FIFTHS'), + (4, 5): VF('FOUR FIFTHS'), + (1, 6): VF('ONE SIXTH'), + (5, 6): VF('FIVE SIXTHS'), + (1, 8): VF('ONE EIGHTH'), + (3, 8): VF('THREE EIGHTHS'), + (5, 8): VF('FIVE EIGHTHS'), + (7, 8): VF('SEVEN EIGHTHS'), +} + + +# atom symbols +_xsym = { + '==': ('=', '='), + '<': ('<', '<'), + '>': ('>', '>'), + '<=': ('<=', U('LESS-THAN OR EQUAL TO')), + '>=': ('>=', U('GREATER-THAN OR EQUAL TO')), + '!=': ('!=', U('NOT EQUAL TO')), + ':=': (':=', ':='), + '+=': ('+=', '+='), + '-=': ('-=', '-='), + '*=': ('*=', '*='), + '/=': ('/=', '/='), + '%=': ('%=', '%='), + '*': ('*', U('DOT OPERATOR')), + '-->': ('-->', U('EM DASH') + U('EM DASH') + + U('BLACK RIGHT-POINTING TRIANGLE') if U('EM DASH') + and U('BLACK RIGHT-POINTING TRIANGLE') else None), + '==>': ('==>', U('BOX DRAWINGS DOUBLE HORIZONTAL') + + U('BOX DRAWINGS DOUBLE HORIZONTAL') + + U('BLACK RIGHT-POINTING TRIANGLE') if + U('BOX DRAWINGS DOUBLE HORIZONTAL') and + U('BOX DRAWINGS DOUBLE HORIZONTAL') and + U('BLACK RIGHT-POINTING TRIANGLE') else None), + '.': ('*', U('RING OPERATOR')), +} + + +def xsym(sym): + """get symbology for a 'character'""" + op = _xsym[sym] + + if _use_unicode: + return op[1] + else: + return op[0] + + +# SYMBOLS + +atoms_table = { + # class how-to-display + 'Exp1': U('SCRIPT SMALL E'), + 'Pi': U('GREEK SMALL LETTER PI'), + 'Infinity': U('INFINITY'), + 'NegativeInfinity': U('INFINITY') and ('-' + U('INFINITY')), # XXX what to do here + #'ImaginaryUnit': U('GREEK SMALL LETTER IOTA'), + #'ImaginaryUnit': U('MATHEMATICAL ITALIC SMALL I'), + 'ImaginaryUnit': U('DOUBLE-STRUCK ITALIC SMALL I'), + 'EmptySet': U('EMPTY SET'), + 'Naturals': U('DOUBLE-STRUCK CAPITAL N'), + 'Naturals0': (U('DOUBLE-STRUCK CAPITAL N') and + (U('DOUBLE-STRUCK CAPITAL N') + + U('SUBSCRIPT ZERO'))), + 'Integers': U('DOUBLE-STRUCK CAPITAL Z'), + 'Rationals': U('DOUBLE-STRUCK CAPITAL Q'), + 'Reals': U('DOUBLE-STRUCK CAPITAL R'), + 'Complexes': U('DOUBLE-STRUCK CAPITAL C'), + 'Universe': U('MATHEMATICAL DOUBLE-STRUCK CAPITAL U'), + 'IdentityMatrix': U('MATHEMATICAL DOUBLE-STRUCK CAPITAL I'), + 'ZeroMatrix': U('MATHEMATICAL DOUBLE-STRUCK DIGIT ZERO'), + 'OneMatrix': U('MATHEMATICAL DOUBLE-STRUCK DIGIT ONE'), + 'Differential': U('DOUBLE-STRUCK ITALIC SMALL D'), + 'Union': U('UNION'), + 'ElementOf': U('ELEMENT OF'), + 'SmallElementOf': U('SMALL ELEMENT OF'), + 'SymmetricDifference': U('INCREMENT'), + 'Intersection': U('INTERSECTION'), + 'Ring': U('RING OPERATOR'), + 'Multiplication': U('MULTIPLICATION SIGN'), + 'TensorProduct': U('N-ARY CIRCLED TIMES OPERATOR'), + 'Dots': U('HORIZONTAL ELLIPSIS'), + 'Modifier Letter Low Ring':U('Modifier Letter Low Ring'), + 'EmptySequence': 'EmptySequence', + 'SuperscriptPlus': U('SUPERSCRIPT PLUS SIGN'), + 'SuperscriptMinus': U('SUPERSCRIPT MINUS'), + 'Dagger': U('DAGGER'), + 'Degree': U('DEGREE SIGN'), + #Logic Symbols + 'And': U('LOGICAL AND'), + 'Or': U('LOGICAL OR'), + 'Not': U('NOT SIGN'), + 'Nor': U('NOR'), + 'Nand': U('NAND'), + 'Xor': U('XOR'), + 'Equiv': U('LEFT RIGHT DOUBLE ARROW'), + 'NotEquiv': U('LEFT RIGHT DOUBLE ARROW WITH STROKE'), + 'Implies': U('LEFT RIGHT DOUBLE ARROW'), + 'NotImplies': U('LEFT RIGHT DOUBLE ARROW WITH STROKE'), + 'Arrow': U('RIGHTWARDS ARROW'), + 'ArrowFromBar': U('RIGHTWARDS ARROW FROM BAR'), + 'NotArrow': U('RIGHTWARDS ARROW WITH STROKE'), + 'Tautology': U('BOX DRAWINGS LIGHT UP AND HORIZONTAL'), + 'Contradiction': U('BOX DRAWINGS LIGHT DOWN AND HORIZONTAL') +} + + +def pretty_atom(atom_name, default=None, printer=None): + """return pretty representation of an atom""" + if _use_unicode: + if printer is not None and atom_name == 'ImaginaryUnit' and printer._settings['imaginary_unit'] == 'j': + return U('DOUBLE-STRUCK ITALIC SMALL J') + else: + return atoms_table[atom_name] + else: + if default is not None: + return default + + raise KeyError('only unicode') # send it default printer + + +def pretty_symbol(symb_name, bold_name=False): + """return pretty representation of a symbol""" + # let's split symb_name into symbol + index + # UC: beta1 + # UC: f_beta + + if not _use_unicode: + return symb_name + + name, sups, subs = split_super_sub(symb_name) + + def translate(s, bold_name) : + if bold_name: + gG = greek_bold_unicode.get(s) + else: + gG = greek_unicode.get(s) + if gG is not None: + return gG + for key in sorted(modifier_dict.keys(), key=lambda k:len(k), reverse=True) : + if s.lower().endswith(key) and len(s)>len(key): + return modifier_dict[key](translate(s[:-len(key)], bold_name)) + if bold_name: + return ''.join([bold_unicode[c] for c in s]) + return s + + name = translate(name, bold_name) + + # Let's prettify sups/subs. If it fails at one of them, pretty sups/subs are + # not used at all. + def pretty_list(l, mapping): + result = [] + for s in l: + pretty = mapping.get(s) + if pretty is None: + try: # match by separate characters + pretty = ''.join([mapping[c] for c in s]) + except (TypeError, KeyError): + return None + result.append(pretty) + return result + + pretty_sups = pretty_list(sups, sup) + if pretty_sups is not None: + pretty_subs = pretty_list(subs, sub) + else: + pretty_subs = None + + # glue the results into one string + if pretty_subs is None: # nice formatting of sups/subs did not work + if subs: + name += '_'+'_'.join([translate(s, bold_name) for s in subs]) + if sups: + name += '__'+'__'.join([translate(s, bold_name) for s in sups]) + return name + else: + sups_result = ' '.join(pretty_sups) + subs_result = ' '.join(pretty_subs) + + return ''.join([name, sups_result, subs_result]) + + +def annotated(letter): + """ + Return a stylised drawing of the letter ``letter``, together with + information on how to put annotations (super- and subscripts to the + left and to the right) on it. + + See pretty.py functions _print_meijerg, _print_hyper on how to use this + information. + """ + ucode_pics = { + 'F': (2, 0, 2, 0, '\N{BOX DRAWINGS LIGHT DOWN AND RIGHT}\N{BOX DRAWINGS LIGHT HORIZONTAL}\n' + '\N{BOX DRAWINGS LIGHT VERTICAL AND RIGHT}\N{BOX DRAWINGS LIGHT HORIZONTAL}\n' + '\N{BOX DRAWINGS LIGHT UP}'), + 'G': (3, 0, 3, 1, '\N{BOX DRAWINGS LIGHT ARC DOWN AND RIGHT}\N{BOX DRAWINGS LIGHT HORIZONTAL}\N{BOX DRAWINGS LIGHT ARC DOWN AND LEFT}\n' + '\N{BOX DRAWINGS LIGHT VERTICAL}\N{BOX DRAWINGS LIGHT RIGHT}\N{BOX DRAWINGS LIGHT DOWN AND LEFT}\n' + '\N{BOX DRAWINGS LIGHT ARC UP AND RIGHT}\N{BOX DRAWINGS LIGHT HORIZONTAL}\N{BOX DRAWINGS LIGHT ARC UP AND LEFT}') + } + ascii_pics = { + 'F': (3, 0, 3, 0, ' _\n|_\n|\n'), + 'G': (3, 0, 3, 1, ' __\n/__\n\\_|') + } + + if _use_unicode: + return ucode_pics[letter] + else: + return ascii_pics[letter] + +_remove_combining = dict.fromkeys(list(range(ord('\N{COMBINING GRAVE ACCENT}'), ord('\N{COMBINING LATIN SMALL LETTER X}'))) + + list(range(ord('\N{COMBINING LEFT HARPOON ABOVE}'), ord('\N{COMBINING ASTERISK ABOVE}')))) + +def is_combining(sym): + """Check whether symbol is a unicode modifier. """ + + return ord(sym) in _remove_combining + + +def center_accent(string, accent): + """ + Returns a string with accent inserted on the middle character. Useful to + put combining accents on symbol names, including multi-character names. + + Parameters + ========== + + string : string + The string to place the accent in. + accent : string + The combining accent to insert + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Combining_character + .. [2] https://en.wikipedia.org/wiki/Combining_Diacritical_Marks + + """ + + # Accent is placed on the previous character, although it may not always look + # like that depending on console + midpoint = len(string) // 2 + 1 + firstpart = string[:midpoint] + secondpart = string[midpoint:] + return firstpart + accent + secondpart + + +def line_width(line): + """Unicode combining symbols (modifiers) are not ever displayed as + separate symbols and thus should not be counted + """ + return len(line.translate(_remove_combining)) + + +def is_subscriptable_in_unicode(subscript): + """ + Checks whether a string is subscriptable in unicode or not. + + Parameters + ========== + + subscript: the string which needs to be checked + + Examples + ======== + + >>> from sympy.printing.pretty.pretty_symbology import is_subscriptable_in_unicode + >>> is_subscriptable_in_unicode('abc') + False + >>> is_subscriptable_in_unicode('123') + True + + """ + return all(character in sub for character in subscript) + + +def center_pad(wstring, wtarget, fillchar=' '): + """ + Return the padding strings necessary to center a string of + wstring characters wide in a wtarget wide space. + + The line_width wstring should always be less or equal to wtarget + or else a ValueError will be raised. + """ + if wstring > wtarget: + raise ValueError('not enough space for string') + wdelta = wtarget - wstring + + wleft = wdelta // 2 # favor left '1 ' + wright = wdelta - wleft + + left = fillchar * wleft + right = fillchar * wright + + return left, right + + +def center(string, width, fillchar=' '): + """Return a centered string of length determined by `line_width` + that uses `fillchar` for padding. + """ + left, right = center_pad(line_width(string), width, fillchar) + return ''.join([left, string, right]) diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/pretty/stringpict.py b/.venv/lib/python3.12/site-packages/sympy/printing/pretty/stringpict.py new file mode 100644 index 0000000000000000000000000000000000000000..b6055f09c83b2abbe0c492991aaee4dff5b34f49 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/pretty/stringpict.py @@ -0,0 +1,537 @@ +"""Prettyprinter by Jurjen Bos. +(I hate spammers: mail me at pietjepuk314 at the reverse of ku.oc.oohay). +All objects have a method that create a "stringPict", +that can be used in the str method for pretty printing. + +Updates by Jason Gedge (email at cs mun ca) + - terminal_string() method + - minor fixes and changes (mostly to prettyForm) + +TODO: + - Allow left/center/right alignment options for above/below and + top/center/bottom alignment options for left/right +""" + +import shutil + +from .pretty_symbology import hobj, vobj, xsym, xobj, pretty_use_unicode, line_width, center +from sympy.utilities.exceptions import sympy_deprecation_warning + +_GLOBAL_WRAP_LINE = None + +class stringPict: + """An ASCII picture. + The pictures are represented as a list of equal length strings. + """ + #special value for stringPict.below + LINE = 'line' + + def __init__(self, s, baseline=0): + """Initialize from string. + Multiline strings are centered. + """ + self.s = s + #picture is a string that just can be printed + self.picture = stringPict.equalLengths(s.splitlines()) + #baseline is the line number of the "base line" + self.baseline = baseline + self.binding = None + + @staticmethod + def equalLengths(lines): + # empty lines + if not lines: + return [''] + + width = max(line_width(line) for line in lines) + return [center(line, width) for line in lines] + + def height(self): + """The height of the picture in characters.""" + return len(self.picture) + + def width(self): + """The width of the picture in characters.""" + return line_width(self.picture[0]) + + @staticmethod + def next(*args): + """Put a string of stringPicts next to each other. + Returns string, baseline arguments for stringPict. + """ + #convert everything to stringPicts + objects = [] + for arg in args: + if isinstance(arg, str): + arg = stringPict(arg) + objects.append(arg) + + #make a list of pictures, with equal height and baseline + newBaseline = max(obj.baseline for obj in objects) + newHeightBelowBaseline = max( + obj.height() - obj.baseline + for obj in objects) + newHeight = newBaseline + newHeightBelowBaseline + + pictures = [] + for obj in objects: + oneEmptyLine = [' '*obj.width()] + basePadding = newBaseline - obj.baseline + totalPadding = newHeight - obj.height() + pictures.append( + oneEmptyLine * basePadding + + obj.picture + + oneEmptyLine * (totalPadding - basePadding)) + + result = [''.join(lines) for lines in zip(*pictures)] + return '\n'.join(result), newBaseline + + def right(self, *args): + r"""Put pictures next to this one. + Returns string, baseline arguments for stringPict. + (Multiline) strings are allowed, and are given a baseline of 0. + + Examples + ======== + + >>> from sympy.printing.pretty.stringpict import stringPict + >>> print(stringPict("10").right(" + ",stringPict("1\r-\r2",1))[0]) + 1 + 10 + - + 2 + + """ + return stringPict.next(self, *args) + + def left(self, *args): + """Put pictures (left to right) at left. + Returns string, baseline arguments for stringPict. + """ + return stringPict.next(*(args + (self,))) + + @staticmethod + def stack(*args): + """Put pictures on top of each other, + from top to bottom. + Returns string, baseline arguments for stringPict. + The baseline is the baseline of the second picture. + Everything is centered. + Baseline is the baseline of the second picture. + Strings are allowed. + The special value stringPict.LINE is a row of '-' extended to the width. + """ + #convert everything to stringPicts; keep LINE + objects = [] + for arg in args: + if arg is not stringPict.LINE and isinstance(arg, str): + arg = stringPict(arg) + objects.append(arg) + + #compute new width + newWidth = max( + obj.width() + for obj in objects + if obj is not stringPict.LINE) + + lineObj = stringPict(hobj('-', newWidth)) + + #replace LINE with proper lines + for i, obj in enumerate(objects): + if obj is stringPict.LINE: + objects[i] = lineObj + + #stack the pictures, and center the result + newPicture = [center(line, newWidth) for obj in objects for line in obj.picture] + newBaseline = objects[0].height() + objects[1].baseline + return '\n'.join(newPicture), newBaseline + + def below(self, *args): + """Put pictures under this picture. + Returns string, baseline arguments for stringPict. + Baseline is baseline of top picture + + Examples + ======== + + >>> from sympy.printing.pretty.stringpict import stringPict + >>> print(stringPict("x+3").below( + ... stringPict.LINE, '3')[0]) #doctest: +NORMALIZE_WHITESPACE + x+3 + --- + 3 + + """ + s, baseline = stringPict.stack(self, *args) + return s, self.baseline + + def above(self, *args): + """Put pictures above this picture. + Returns string, baseline arguments for stringPict. + Baseline is baseline of bottom picture. + """ + string, baseline = stringPict.stack(*(args + (self,))) + baseline = len(string.splitlines()) - self.height() + self.baseline + return string, baseline + + def parens(self, left='(', right=')', ifascii_nougly=False): + """Put parentheses around self. + Returns string, baseline arguments for stringPict. + + left or right can be None or empty string which means 'no paren from + that side' + """ + h = self.height() + b = self.baseline + + # XXX this is a hack -- ascii parens are ugly! + if ifascii_nougly and not pretty_use_unicode(): + h = 1 + b = 0 + + res = self + + if left: + lparen = stringPict(vobj(left, h), baseline=b) + res = stringPict(*lparen.right(self)) + if right: + rparen = stringPict(vobj(right, h), baseline=b) + res = stringPict(*res.right(rparen)) + + return ('\n'.join(res.picture), res.baseline) + + def leftslash(self): + """Precede object by a slash of the proper size. + """ + # XXX not used anywhere ? + height = max( + self.baseline, + self.height() - 1 - self.baseline)*2 + 1 + slash = '\n'.join( + ' '*(height - i - 1) + xobj('/', 1) + ' '*i + for i in range(height) + ) + return self.left(stringPict(slash, height//2)) + + def root(self, n=None): + """Produce a nice root symbol. + Produces ugly results for big n inserts. + """ + # XXX not used anywhere + # XXX duplicate of root drawing in pretty.py + #put line over expression + result = self.above('_'*self.width()) + #construct right half of root symbol + height = self.height() + slash = '\n'.join( + ' ' * (height - i - 1) + '/' + ' ' * i + for i in range(height) + ) + slash = stringPict(slash, height - 1) + #left half of root symbol + if height > 2: + downline = stringPict('\\ \n \\', 1) + else: + downline = stringPict('\\') + #put n on top, as low as possible + if n is not None and n.width() > downline.width(): + downline = downline.left(' '*(n.width() - downline.width())) + downline = downline.above(n) + #build root symbol + root = downline.right(slash) + #glue it on at the proper height + #normally, the root symbel is as high as self + #which is one less than result + #this moves the root symbol one down + #if the root became higher, the baseline has to grow too + root.baseline = result.baseline - result.height() + root.height() + return result.left(root) + + def render(self, * args, **kwargs): + """Return the string form of self. + + Unless the argument line_break is set to False, it will + break the expression in a form that can be printed + on the terminal without being broken up. + """ + if _GLOBAL_WRAP_LINE is not None: + kwargs["wrap_line"] = _GLOBAL_WRAP_LINE + + if kwargs["wrap_line"] is False: + return "\n".join(self.picture) + + if kwargs["num_columns"] is not None: + # Read the argument num_columns if it is not None + ncols = kwargs["num_columns"] + else: + # Attempt to get a terminal width + ncols = self.terminal_width() + + if ncols <= 0: + ncols = 80 + + # If smaller than the terminal width, no need to correct + if self.width() <= ncols: + return type(self.picture[0])(self) + + """ + Break long-lines in a visually pleasing format. + without overflow indicators | with overflow indicators + | 2 2 3 | | 2 2 3 ↪| + |6*x *y + 4*x*y + | |6*x *y + 4*x*y + ↪| + | | | | + | 3 4 4 | |↪ 3 4 4 | + |4*y*x + x + y | |↪ 4*y*x + x + y | + |a*c*e + a*c*f + a*d | |a*c*e + a*c*f + a*d ↪| + |*e + a*d*f + b*c*e | | | + |+ b*c*f + b*d*e + b | |↪ *e + a*d*f + b*c* ↪| + |*d*f | | | + | | |↪ e + b*c*f + b*d*e ↪| + | | | | + | | |↪ + b*d*f | + """ + + overflow_first = "" + if kwargs["use_unicode"] or pretty_use_unicode(): + overflow_start = "\N{RIGHTWARDS ARROW WITH HOOK} " + overflow_end = " \N{RIGHTWARDS ARROW WITH HOOK}" + else: + overflow_start = "> " + overflow_end = " >" + + def chunks(line): + """Yields consecutive chunks of line_width ncols""" + prefix = overflow_first + width, start = line_width(prefix + overflow_end), 0 + for i, x in enumerate(line): + wx = line_width(x) + # Only flush the screen when the current character overflows. + # This way, combining marks can be appended even when width == ncols. + if width + wx > ncols: + yield prefix + line[start:i] + overflow_end + prefix = overflow_start + width, start = line_width(prefix + overflow_end), i + width += wx + yield prefix + line[start:] + + # Concurrently assemble chunks of all lines into individual screens + pictures = zip(*map(chunks, self.picture)) + + # Join lines of each screen into sub-pictures + pictures = ["\n".join(picture) for picture in pictures] + + # Add spacers between sub-pictures + return "\n\n".join(pictures) + + def terminal_width(self): + """Return the terminal width if possible, otherwise return 0. + """ + size = shutil.get_terminal_size(fallback=(0, 0)) + return size.columns + + def __eq__(self, o): + if isinstance(o, str): + return '\n'.join(self.picture) == o + elif isinstance(o, stringPict): + return o.picture == self.picture + return False + + def __hash__(self): + return super().__hash__() + + def __str__(self): + return '\n'.join(self.picture) + + def __repr__(self): + return "stringPict(%r,%d)" % ('\n'.join(self.picture), self.baseline) + + def __getitem__(self, index): + return self.picture[index] + + def __len__(self): + return len(self.s) + + +class prettyForm(stringPict): + """ + Extension of the stringPict class that knows about basic math applications, + optimizing double minus signs. + + "Binding" is interpreted as follows:: + + ATOM this is an atom: never needs to be parenthesized + FUNC this is a function application: parenthesize if added (?) + DIV this is a division: make wider division if divided + POW this is a power: only parenthesize if exponent + MUL this is a multiplication: parenthesize if powered + ADD this is an addition: parenthesize if multiplied or powered + NEG this is a negative number: optimize if added, parenthesize if + multiplied or powered + OPEN this is an open object: parenthesize if added, multiplied, or + powered (example: Piecewise) + """ + ATOM, FUNC, DIV, POW, MUL, ADD, NEG, OPEN = range(8) + + def __init__(self, s, baseline=0, binding=0, unicode=None): + """Initialize from stringPict and binding power.""" + stringPict.__init__(self, s, baseline) + self.binding = binding + if unicode is not None: + sympy_deprecation_warning( + """ + The unicode argument to prettyForm is deprecated. Only the s + argument (the first positional argument) should be passed. + """, + deprecated_since_version="1.7", + active_deprecations_target="deprecated-pretty-printing-functions") + self._unicode = unicode or s + + @property + def unicode(self): + sympy_deprecation_warning( + """ + The prettyForm.unicode attribute is deprecated. Use the + prettyForm.s attribute instead. + """, + deprecated_since_version="1.7", + active_deprecations_target="deprecated-pretty-printing-functions") + return self._unicode + + # Note: code to handle subtraction is in _print_Add + + def __add__(self, *others): + """Make a pretty addition. + Addition of negative numbers is simplified. + """ + arg = self + if arg.binding > prettyForm.NEG: + arg = stringPict(*arg.parens()) + result = [arg] + for arg in others: + #add parentheses for weak binders + if arg.binding > prettyForm.NEG: + arg = stringPict(*arg.parens()) + #use existing minus sign if available + if arg.binding != prettyForm.NEG: + result.append(' + ') + result.append(arg) + return prettyForm(binding=prettyForm.ADD, *stringPict.next(*result)) + + def __truediv__(self, den, slashed=False): + """Make a pretty division; stacked or slashed. + """ + if slashed: + raise NotImplementedError("Can't do slashed fraction yet") + num = self + if num.binding == prettyForm.DIV: + num = stringPict(*num.parens()) + if den.binding == prettyForm.DIV: + den = stringPict(*den.parens()) + + if num.binding==prettyForm.NEG: + num = num.right(" ")[0] + + return prettyForm(binding=prettyForm.DIV, *stringPict.stack( + num, + stringPict.LINE, + den)) + + def __mul__(self, *others): + """Make a pretty multiplication. + Parentheses are needed around +, - and neg. + """ + quantity = { + 'degree': "\N{DEGREE SIGN}" + } + + if len(others) == 0: + return self # We aren't actually multiplying... So nothing to do here. + + # add parens on args that need them + arg = self + if arg.binding > prettyForm.MUL and arg.binding != prettyForm.NEG: + arg = stringPict(*arg.parens()) + result = [arg] + for arg in others: + if arg.picture[0] not in quantity.values(): + result.append(xsym('*')) + #add parentheses for weak binders + if arg.binding > prettyForm.MUL and arg.binding != prettyForm.NEG: + arg = stringPict(*arg.parens()) + result.append(arg) + + len_res = len(result) + for i in range(len_res): + if i < len_res - 1 and result[i] == '-1' and result[i + 1] == xsym('*'): + # substitute -1 by -, like in -1*x -> -x + result.pop(i) + result.pop(i) + result.insert(i, '-') + if result[0][0] == '-': + # if there is a - sign in front of all + # This test was failing to catch a prettyForm.__mul__(prettyForm("-1", 0, 6)) being negative + bin = prettyForm.NEG + if result[0] == '-': + right = result[1] + if right.picture[right.baseline][0] == '-': + result[0] = '- ' + else: + bin = prettyForm.MUL + return prettyForm(binding=bin, *stringPict.next(*result)) + + def __repr__(self): + return "prettyForm(%r,%d,%d)" % ( + '\n'.join(self.picture), + self.baseline, + self.binding) + + def __pow__(self, b): + """Make a pretty power. + """ + a = self + use_inline_func_form = False + if b.binding == prettyForm.POW: + b = stringPict(*b.parens()) + if a.binding > prettyForm.FUNC: + a = stringPict(*a.parens()) + elif a.binding == prettyForm.FUNC: + # heuristic for when to use inline power + if b.height() > 1: + a = stringPict(*a.parens()) + else: + use_inline_func_form = True + + if use_inline_func_form: + # 2 + # sin + + (x) + b.baseline = a.prettyFunc.baseline + b.height() + func = stringPict(*a.prettyFunc.right(b)) + return prettyForm(*func.right(a.prettyArgs)) + else: + # 2 <-- top + # (x+y) <-- bot + top = stringPict(*b.left(' '*a.width())) + bot = stringPict(*a.right(' '*b.width())) + + return prettyForm(binding=prettyForm.POW, *bot.above(top)) + + simpleFunctions = ["sin", "cos", "tan"] + + @staticmethod + def apply(function, *args): + """Functions of one or more variables. + """ + if function in prettyForm.simpleFunctions: + #simple function: use only space if possible + assert len( + args) == 1, "Simple function %s must have 1 argument" % function + arg = args[0].__pretty__() + if arg.binding <= prettyForm.DIV: + #optimization: no parentheses necessary + return prettyForm(binding=prettyForm.FUNC, *arg.left(function + ' ')) + argumentList = [] + for arg in args: + argumentList.append(',') + argumentList.append(arg.__pretty__()) + argumentList = stringPict(*stringPict.next(*argumentList[1:])) + argumentList = stringPict(*argumentList.parens()) + return prettyForm(binding=prettyForm.ATOM, *argumentList.left(function)) diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/pretty/tests/__init__.py b/.venv/lib/python3.12/site-packages/sympy/printing/pretty/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/pretty/tests/__pycache__/__init__.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/printing/pretty/tests/__pycache__/__init__.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..e1d4e6983373afe989ff45cf675de0ef73688c8a Binary files /dev/null and b/.venv/lib/python3.12/site-packages/sympy/printing/pretty/tests/__pycache__/__init__.cpython-312.pyc differ diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/pretty/tests/__pycache__/test_pretty.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/printing/pretty/tests/__pycache__/test_pretty.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..f13a304b26d0830b5d4c0f7f6e855a86675b3d52 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/pretty/tests/__pycache__/test_pretty.cpython-312.pyc @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e3adbec5ebb25e6d2a2e35af63818c6d2409b41eed466a44fa20a4ea7765b937 +size 243927 diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/pretty/tests/test_pretty.py b/.venv/lib/python3.12/site-packages/sympy/printing/pretty/tests/test_pretty.py new file mode 100644 index 0000000000000000000000000000000000000000..1cca79bd1dc5c3ba81483c8fe2e87c35926d1b94 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/pretty/tests/test_pretty.py @@ -0,0 +1,7972 @@ +# -*- coding: utf-8 -*- +from sympy.concrete.products import Product +from sympy.concrete.summations import Sum +from sympy.core.add import Add +from sympy.core.basic import Basic +from sympy.core.containers import (Dict, Tuple) +from sympy.core.function import (Derivative, Function, Lambda, Subs) +from sympy.core.mul import Mul +from sympy.core import (EulerGamma, GoldenRatio, Catalan) +from sympy.core.numbers import (I, Rational, oo, pi) +from sympy.core.power import Pow +from sympy.core.relational import (Eq, Ge, Gt, Le, Lt, Ne) +from sympy.core.singleton import S +from sympy.core.symbol import (Symbol, symbols) +from sympy.functions.elementary.complexes import conjugate +from sympy.functions.elementary.exponential import LambertW +from sympy.functions.special.bessel import (airyai, airyaiprime, airybi, airybiprime) +from sympy.functions.special.delta_functions import Heaviside +from sympy.functions.special.error_functions import (fresnelc, fresnels) +from sympy.functions.special.singularity_functions import SingularityFunction +from sympy.functions.special.zeta_functions import dirichlet_eta +from sympy.geometry.line import (Ray, Segment) +from sympy.integrals.integrals import Integral +from sympy.logic.boolalg import (And, Equivalent, ITE, Implies, Nand, Nor, Not, Or, Xor) +from sympy.matrices.dense import (Matrix, diag) +from sympy.matrices.expressions.slice import MatrixSlice +from sympy.matrices.expressions.trace import Trace +from sympy.polys.domains.finitefield import FF +from sympy.polys.domains.integerring import ZZ +from sympy.polys.domains.rationalfield import QQ +from sympy.polys.domains.realfield import RR +from sympy.polys.orderings import (grlex, ilex) +from sympy.polys.polytools import groebner +from sympy.polys.rootoftools import (RootSum, rootof) +from sympy.series.formal import fps +from sympy.series.fourier import fourier_series +from sympy.series.limits import Limit +from sympy.series.order import O +from sympy.series.sequences import (SeqAdd, SeqFormula, SeqMul, SeqPer) +from sympy.sets.contains import Contains +from sympy.sets.fancysets import Range +from sympy.sets.sets import (Complement, FiniteSet, Intersection, Interval, Union) +from sympy.codegen.ast import (Assignment, AddAugmentedAssignment, + SubAugmentedAssignment, MulAugmentedAssignment, DivAugmentedAssignment, ModAugmentedAssignment) +from sympy.core.expr import UnevaluatedExpr +from sympy.physics.quantum.trace import Tr + +from sympy.functions import (Abs, Chi, Ci, Ei, KroneckerDelta, + Piecewise, Shi, Si, atan2, beta, binomial, catalan, ceiling, cos, + euler, exp, expint, factorial, factorial2, floor, gamma, hyper, log, + meijerg, sin, sqrt, subfactorial, tan, uppergamma, lerchphi, polylog, + elliptic_k, elliptic_f, elliptic_e, elliptic_pi, DiracDelta, bell, + bernoulli, fibonacci, tribonacci, lucas, stieltjes, mathieuc, mathieus, + mathieusprime, mathieucprime) + +from sympy.matrices import (Adjoint, Inverse, MatrixSymbol, Transpose, + KroneckerProduct, BlockMatrix, OneMatrix, ZeroMatrix) +from sympy.matrices.expressions import hadamard_power + +from sympy.physics import mechanics +from sympy.physics.control.lti import (TransferFunction, Feedback, TransferFunctionMatrix, + Series, Parallel, MIMOSeries, MIMOParallel, MIMOFeedback, StateSpace) +from sympy.physics.units import joule, degree +from sympy.printing.pretty import pprint, pretty as xpretty +from sympy.printing.pretty.pretty_symbology import center_accent, is_combining, center +from sympy.sets.conditionset import ConditionSet + +from sympy.sets import ImageSet, ProductSet +from sympy.sets.setexpr import SetExpr +from sympy.stats.crv_types import Normal +from sympy.stats.symbolic_probability import (Covariance, Expectation, + Probability, Variance) +from sympy.tensor.array import (ImmutableDenseNDimArray, ImmutableSparseNDimArray, + MutableDenseNDimArray, MutableSparseNDimArray, tensorproduct) +from sympy.tensor.functions import TensorProduct +from sympy.tensor.tensor import (TensorIndexType, tensor_indices, TensorHead, + TensorElement, tensor_heads) + +from sympy.testing.pytest import raises, _both_exp_pow, warns_deprecated_sympy + +from sympy.vector import CoordSys3D, Gradient, Curl, Divergence, Dot, Cross, Laplacian + + + +import sympy as sym +class lowergamma(sym.lowergamma): + pass # testing notation inheritance by a subclass with same name + +a, b, c, d, x, y, z, k, n, s, p = symbols('a,b,c,d,x,y,z,k,n,s,p') +f = Function("f") +th = Symbol('theta') +ph = Symbol('phi') + +""" +Expressions whose pretty-printing is tested here: +(A '#' to the right of an expression indicates that its various acceptable +orderings are accounted for by the tests.) + + +BASIC EXPRESSIONS: + +oo +(x**2) +1/x +y*x**-2 +x**Rational(-5,2) +(-2)**x +Pow(3, 1, evaluate=False) +(x**2 + x + 1) # +1-x # +1-2*x # +x/y +-x/y +(x+2)/y # +(1+x)*y #3 +-5*x/(x+10) # correct placement of negative sign +1 - Rational(3,2)*(x+1) +-(-x + 5)*(-x - 2*sqrt(2) + 5) - (-y + 5)*(-y + 5) # issue 5524 + + +ORDERING: + +x**2 + x + 1 +1 - x +1 - 2*x +2*x**4 + y**2 - x**2 + y**3 + + +RELATIONAL: + +Eq(x, y) +Lt(x, y) +Gt(x, y) +Le(x, y) +Ge(x, y) +Ne(x/(y+1), y**2) # + + +RATIONAL NUMBERS: + +y*x**-2 +y**Rational(3,2) * x**Rational(-5,2) +sin(x)**3/tan(x)**2 + + +FUNCTIONS (ABS, CONJ, EXP, FUNCTION BRACES, FACTORIAL, FLOOR, CEILING): + +(2*x + exp(x)) # +Abs(x) +Abs(x/(x**2+1)) # +Abs(1 / (y - Abs(x))) +factorial(n) +factorial(2*n) +subfactorial(n) +subfactorial(2*n) +factorial(factorial(factorial(n))) +factorial(n+1) # +conjugate(x) +conjugate(f(x+1)) # +f(x) +f(x, y) +f(x/(y+1), y) # +f(x**x**x**x**x**x) +sin(x)**2 +conjugate(a+b*I) +conjugate(exp(a+b*I)) +conjugate( f(1 + conjugate(f(x))) ) # +f(x/(y+1), y) # denom of first arg +floor(1 / (y - floor(x))) +ceiling(1 / (y - ceiling(x))) + + +SQRT: + +sqrt(2) +2**Rational(1,3) +2**Rational(1,1000) +sqrt(x**2 + 1) +(1 + sqrt(5))**Rational(1,3) +2**(1/x) +sqrt(2+pi) +(2+(1+x**2)/(2+x))**Rational(1,4)+(1+x**Rational(1,1000))/sqrt(3+x**2) + + +DERIVATIVES: + +Derivative(log(x), x, evaluate=False) +Derivative(log(x), x, evaluate=False) + x # +Derivative(log(x) + x**2, x, y, evaluate=False) +Derivative(2*x*y, y, x, evaluate=False) + x**2 # +beta(alpha).diff(alpha) + + +INTEGRALS: + +Integral(log(x), x) +Integral(x**2, x) +Integral((sin(x))**2 / (tan(x))**2) +Integral(x**(2**x), x) +Integral(x**2, (x,1,2)) +Integral(x**2, (x,Rational(1,2),10)) +Integral(x**2*y**2, x,y) +Integral(x**2, (x, None, 1)) +Integral(x**2, (x, 1, None)) +Integral(sin(th)/cos(ph), (th,0,pi), (ph, 0, 2*pi)) + + +MATRICES: + +Matrix([[x**2+1, 1], [y, x+y]]) # +Matrix([[x/y, y, th], [0, exp(I*k*ph), 1]]) + + +PIECEWISE: + +Piecewise((x,x<1),(x**2,True)) + +ITE: + +ITE(x, y, z) + +SEQUENCES (TUPLES, LISTS, DICTIONARIES): + +() +[] +{} +(1/x,) +[x**2, 1/x, x, y, sin(th)**2/cos(ph)**2] +(x**2, 1/x, x, y, sin(th)**2/cos(ph)**2) +{x: sin(x)} +{1/x: 1/y, x: sin(x)**2} # +[x**2] +(x**2,) +{x**2: 1} + + +LIMITS: + +Limit(x, x, oo) +Limit(x**2, x, 0) +Limit(1/x, x, 0) +Limit(sin(x)/x, x, 0) + + +UNITS: + +joule => kg*m**2/s + + +SUBS: + +Subs(f(x), x, ph**2) +Subs(f(x).diff(x), x, 0) +Subs(f(x).diff(x)/y, (x, y), (0, Rational(1, 2))) + + +ORDER: + +O(1) +O(1/x) +O(x**2 + y**2) + +""" + + +def pretty(expr, order=None): + """ASCII pretty-printing""" + return xpretty(expr, order=order, use_unicode=False, wrap_line=False) + + +def upretty(expr, order=None): + """Unicode pretty-printing""" + return xpretty(expr, order=order, use_unicode=True, wrap_line=False) + + +def test_pretty_ascii_str(): + assert pretty( 'xxx' ) == 'xxx' + assert pretty( "xxx" ) == 'xxx' + assert pretty( 'xxx\'xxx' ) == 'xxx\'xxx' + assert pretty( 'xxx"xxx' ) == 'xxx\"xxx' + assert pretty( 'xxx\"xxx' ) == 'xxx\"xxx' + assert pretty( "xxx'xxx" ) == 'xxx\'xxx' + assert pretty( "xxx\'xxx" ) == 'xxx\'xxx' + assert pretty( "xxx\"xxx" ) == 'xxx\"xxx' + assert pretty( "xxx\"xxx\'xxx" ) == 'xxx"xxx\'xxx' + assert pretty( "xxx\nxxx" ) == 'xxx\nxxx' + + +def test_pretty_unicode_str(): + assert pretty( 'xxx' ) == 'xxx' + assert pretty( 'xxx' ) == 'xxx' + assert pretty( 'xxx\'xxx' ) == 'xxx\'xxx' + assert pretty( 'xxx"xxx' ) == 'xxx\"xxx' + assert pretty( 'xxx\"xxx' ) == 'xxx\"xxx' + assert pretty( "xxx'xxx" ) == 'xxx\'xxx' + assert pretty( "xxx\'xxx" ) == 'xxx\'xxx' + assert pretty( "xxx\"xxx" ) == 'xxx\"xxx' + assert pretty( "xxx\"xxx\'xxx" ) == 'xxx"xxx\'xxx' + assert pretty( "xxx\nxxx" ) == 'xxx\nxxx' + + +def test_upretty_greek(): + assert upretty( oo ) == '∞' + assert upretty( Symbol('alpha^+_1') ) == 'α⁺₁' + assert upretty( Symbol('beta') ) == 'β' + assert upretty(Symbol('lambda')) == 'λ' + + +def test_upretty_multiindex(): + assert upretty( Symbol('beta12') ) == 'β₁₂' + assert upretty( Symbol('Y00') ) == 'Y₀₀' + assert upretty( Symbol('Y_00') ) == 'Y₀₀' + assert upretty( Symbol('F^+-') ) == 'F⁺⁻' + + +def test_upretty_sub_super(): + assert upretty( Symbol('beta_1_2') ) == 'β₁ ₂' + assert upretty( Symbol('beta^1^2') ) == 'β¹ ²' + assert upretty( Symbol('beta_1^2') ) == 'β²₁' + assert upretty( Symbol('beta_10_20') ) == 'β₁₀ ₂₀' + assert upretty( Symbol('beta_ax_gamma^i') ) == 'βⁱₐₓ ᵧ' + assert upretty( Symbol("F^1^2_3_4") ) == 'F¹ ²₃ ₄' + assert upretty( Symbol("F_1_2^3^4") ) == 'F³ ⁴₁ ₂' + assert upretty( Symbol("F_1_2_3_4") ) == 'F₁ ₂ ₃ ₄' + assert upretty( Symbol("F^1^2^3^4") ) == 'F¹ ² ³ ⁴' + + +def test_upretty_subs_missing_in_24(): + assert upretty( Symbol('F_beta') ) == 'Fᵦ' + assert upretty( Symbol('F_gamma') ) == 'Fᵧ' + assert upretty( Symbol('F_rho') ) == 'Fᵨ' + assert upretty( Symbol('F_phi') ) == 'Fᵩ' + assert upretty( Symbol('F_chi') ) == 'Fᵪ' + + assert upretty( Symbol('F_a') ) == 'Fₐ' + assert upretty( Symbol('F_e') ) == 'Fₑ' + assert upretty( Symbol('F_i') ) == 'Fᵢ' + assert upretty( Symbol('F_o') ) == 'Fₒ' + assert upretty( Symbol('F_u') ) == 'Fᵤ' + assert upretty( Symbol('F_r') ) == 'Fᵣ' + assert upretty( Symbol('F_v') ) == 'Fᵥ' + assert upretty( Symbol('F_x') ) == 'Fₓ' + + +def test_missing_in_2X_issue_9047(): + assert upretty( Symbol('F_h') ) == 'Fₕ' + assert upretty( Symbol('F_k') ) == 'Fₖ' + assert upretty( Symbol('F_l') ) == 'Fₗ' + assert upretty( Symbol('F_m') ) == 'Fₘ' + assert upretty( Symbol('F_n') ) == 'Fₙ' + assert upretty( Symbol('F_p') ) == 'Fₚ' + assert upretty( Symbol('F_s') ) == 'Fₛ' + assert upretty( Symbol('F_t') ) == 'Fₜ' + + +def test_upretty_modifiers(): + # Accents + assert upretty( Symbol('Fmathring') ) == 'F̊' + assert upretty( Symbol('Fddddot') ) == 'F⃜' + assert upretty( Symbol('Fdddot') ) == 'F⃛' + assert upretty( Symbol('Fddot') ) == 'F̈' + assert upretty( Symbol('Fdot') ) == 'Ḟ' + assert upretty( Symbol('Fcheck') ) == 'F̌' + assert upretty( Symbol('Fbreve') ) == 'F̆' + assert upretty( Symbol('Facute') ) == 'F́' + assert upretty( Symbol('Fgrave') ) == 'F̀' + assert upretty( Symbol('Ftilde') ) == 'F̃' + assert upretty( Symbol('Fhat') ) == 'F̂' + assert upretty( Symbol('Fbar') ) == 'F̅' + assert upretty( Symbol('Fvec') ) == 'F⃗' + assert upretty( Symbol('Fprime') ) == 'F′' + assert upretty( Symbol('Fprm') ) == 'F′' + # No faces are actually implemented, but test to make sure the modifiers are stripped + assert upretty( Symbol('Fbold') ) == 'Fbold' + assert upretty( Symbol('Fbm') ) == 'Fbm' + assert upretty( Symbol('Fcal') ) == 'Fcal' + assert upretty( Symbol('Fscr') ) == 'Fscr' + assert upretty( Symbol('Ffrak') ) == 'Ffrak' + # Brackets + assert upretty( Symbol('Fnorm') ) == '‖F‖' + assert upretty( Symbol('Favg') ) == '⟨F⟩' + assert upretty( Symbol('Fabs') ) == '|F|' + assert upretty( Symbol('Fmag') ) == '|F|' + # Combinations + assert upretty( Symbol('xvecdot') ) == 'x⃗̇' + assert upretty( Symbol('xDotVec') ) == 'ẋ⃗' + assert upretty( Symbol('xHATNorm') ) == '‖x̂‖' + assert upretty( Symbol('xMathring_yCheckPRM__zbreveAbs') ) == 'x̊_y̌′__|z̆|' + assert upretty( Symbol('alphadothat_nVECDOT__tTildePrime') ) == 'α̇̂_n⃗̇__t̃′' + assert upretty( Symbol('x_dot') ) == 'x_dot' + assert upretty( Symbol('x__dot') ) == 'x__dot' + + +def test_pretty_Cycle(): + from sympy.combinatorics.permutations import Cycle + assert pretty(Cycle(1, 2)) == '(1 2)' + assert pretty(Cycle(2)) == '(2)' + assert pretty(Cycle(1, 3)(4, 5)) == '(1 3)(4 5)' + assert pretty(Cycle()) == '()' + + +def test_pretty_Permutation(): + from sympy.combinatorics.permutations import Permutation + p1 = Permutation(1, 2)(3, 4) + assert xpretty(p1, perm_cyclic=True, use_unicode=True) == "(1 2)(3 4)" + assert xpretty(p1, perm_cyclic=True, use_unicode=False) == "(1 2)(3 4)" + assert xpretty(p1, perm_cyclic=False, use_unicode=True) == \ + '⎛0 1 2 3 4⎞\n'\ + '⎝0 2 1 4 3⎠' + assert xpretty(p1, perm_cyclic=False, use_unicode=False) == \ + "/0 1 2 3 4\\\n"\ + "\\0 2 1 4 3/" + + with warns_deprecated_sympy(): + old_print_cyclic = Permutation.print_cyclic + Permutation.print_cyclic = False + assert xpretty(p1, use_unicode=True) == \ + '⎛0 1 2 3 4⎞\n'\ + '⎝0 2 1 4 3⎠' + assert xpretty(p1, use_unicode=False) == \ + "/0 1 2 3 4\\\n"\ + "\\0 2 1 4 3/" + Permutation.print_cyclic = old_print_cyclic + + +def test_pretty_basic(): + assert pretty( -Rational(1)/2 ) == '-1/2' + assert pretty( -Rational(13)/22 ) == \ +"""\ +-13 \n\ +----\n\ + 22 \ +""" + expr = oo + ascii_str = \ +"""\ +oo\ +""" + ucode_str = \ +"""\ +∞\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = (x**2) + ascii_str = \ +"""\ + 2\n\ +x \ +""" + ucode_str = \ +"""\ + 2\n\ +x \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = 1/x + ascii_str = \ +"""\ +1\n\ +-\n\ +x\ +""" + ucode_str = \ +"""\ +1\n\ +─\n\ +x\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + # not the same as 1/x + expr = x**-1.0 + ascii_str = \ +"""\ + -1.0\n\ +x \ +""" + ucode_str = \ +"""\ + -1.0\n\ +x \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + # see issue #2860 + expr = Pow(S(2), -1.0, evaluate=False) + ascii_str = \ +"""\ + -1.0\n\ +2 \ +""" + ucode_str = \ +"""\ + -1.0\n\ +2 \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = y*x**-2 + ascii_str = \ +"""\ +y \n\ +--\n\ + 2\n\ +x \ +""" + ucode_str = \ +"""\ +y \n\ +──\n\ + 2\n\ +x \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + #see issue #14033 + expr = x**Rational(1, 3) + ascii_str = \ +"""\ + 1/3\n\ +x \ +""" + ucode_str = \ +"""\ + 1/3\n\ +x \ +""" + assert xpretty(expr, use_unicode=False, wrap_line=False,\ + root_notation = False) == ascii_str + assert xpretty(expr, use_unicode=True, wrap_line=False,\ + root_notation = False) == ucode_str + + expr = x**Rational(-5, 2) + ascii_str = \ +"""\ + 1 \n\ +----\n\ + 5/2\n\ +x \ +""" + ucode_str = \ +"""\ + 1 \n\ +────\n\ + 5/2\n\ +x \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = (-2)**x + ascii_str = \ +"""\ + x\n\ +(-2) \ +""" + ucode_str = \ +"""\ + x\n\ +(-2) \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + # See issue 4923 + expr = Pow(3, 1, evaluate=False) + ascii_str = \ +"""\ + 1\n\ +3 \ +""" + ucode_str = \ +"""\ + 1\n\ +3 \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = (x**2 + x + 1) + ascii_str_1 = \ +"""\ + 2\n\ +1 + x + x \ +""" + ascii_str_2 = \ +"""\ + 2 \n\ +x + x + 1\ +""" + ascii_str_3 = \ +"""\ + 2 \n\ +x + 1 + x\ +""" + ucode_str_1 = \ +"""\ + 2\n\ +1 + x + x \ +""" + ucode_str_2 = \ +"""\ + 2 \n\ +x + x + 1\ +""" + ucode_str_3 = \ +"""\ + 2 \n\ +x + 1 + x\ +""" + assert pretty(expr) in [ascii_str_1, ascii_str_2, ascii_str_3] + assert upretty(expr) in [ucode_str_1, ucode_str_2, ucode_str_3] + + expr = 1 - x + ascii_str_1 = \ +"""\ +1 - x\ +""" + ascii_str_2 = \ +"""\ +-x + 1\ +""" + ucode_str_1 = \ +"""\ +1 - x\ +""" + ucode_str_2 = \ +"""\ +-x + 1\ +""" + assert pretty(expr) in [ascii_str_1, ascii_str_2] + assert upretty(expr) in [ucode_str_1, ucode_str_2] + + expr = 1 - 2*x + ascii_str_1 = \ +"""\ +1 - 2*x\ +""" + ascii_str_2 = \ +"""\ +-2*x + 1\ +""" + ucode_str_1 = \ +"""\ +1 - 2⋅x\ +""" + ucode_str_2 = \ +"""\ +-2⋅x + 1\ +""" + assert pretty(expr) in [ascii_str_1, ascii_str_2] + assert upretty(expr) in [ucode_str_1, ucode_str_2] + + expr = x/y + ascii_str = \ +"""\ +x\n\ +-\n\ +y\ +""" + ucode_str = \ +"""\ +x\n\ +─\n\ +y\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = -x/y + ascii_str = \ +"""\ +-x \n\ +---\n\ + y \ +""" + ucode_str = \ +"""\ +-x \n\ +───\n\ + y \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = (x + 2)/y + ascii_str_1 = \ +"""\ +2 + x\n\ +-----\n\ + y \ +""" + ascii_str_2 = \ +"""\ +x + 2\n\ +-----\n\ + y \ +""" + ucode_str_1 = \ +"""\ +2 + x\n\ +─────\n\ + y \ +""" + ucode_str_2 = \ +"""\ +x + 2\n\ +─────\n\ + y \ +""" + assert pretty(expr) in [ascii_str_1, ascii_str_2] + assert upretty(expr) in [ucode_str_1, ucode_str_2] + + expr = (1 + x)*y + ascii_str_1 = \ +"""\ +y*(1 + x)\ +""" + ascii_str_2 = \ +"""\ +(1 + x)*y\ +""" + ascii_str_3 = \ +"""\ +y*(x + 1)\ +""" + ucode_str_1 = \ +"""\ +y⋅(1 + x)\ +""" + ucode_str_2 = \ +"""\ +(1 + x)⋅y\ +""" + ucode_str_3 = \ +"""\ +y⋅(x + 1)\ +""" + assert pretty(expr) in [ascii_str_1, ascii_str_2, ascii_str_3] + assert upretty(expr) in [ucode_str_1, ucode_str_2, ucode_str_3] + + # Test for correct placement of the negative sign + expr = -5*x/(x + 10) + ascii_str_1 = \ +"""\ +-5*x \n\ +------\n\ +10 + x\ +""" + ascii_str_2 = \ +"""\ +-5*x \n\ +------\n\ +x + 10\ +""" + ucode_str_1 = \ +"""\ +-5⋅x \n\ +──────\n\ +10 + x\ +""" + ucode_str_2 = \ +"""\ +-5⋅x \n\ +──────\n\ +x + 10\ +""" + assert pretty(expr) in [ascii_str_1, ascii_str_2] + assert upretty(expr) in [ucode_str_1, ucode_str_2] + + expr = -S.Half - 3*x + ascii_str = \ +"""\ +-3*x - 1/2\ +""" + ucode_str = \ +"""\ +-3⋅x - 1/2\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = S.Half - 3*x + ascii_str = \ +"""\ +1/2 - 3*x\ +""" + ucode_str = \ +"""\ +1/2 - 3⋅x\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = -S.Half - 3*x/2 + ascii_str = \ +"""\ + 3*x 1\n\ +- --- - -\n\ + 2 2\ +""" + ucode_str = \ +"""\ + 3⋅x 1\n\ +- ─── - ─\n\ + 2 2\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = S.Half - 3*x/2 + ascii_str = \ +"""\ +1 3*x\n\ +- - ---\n\ +2 2 \ +""" + ucode_str = \ +"""\ +1 3⋅x\n\ +─ - ───\n\ +2 2 \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + +def test_negative_fractions(): + expr = -x/y + ascii_str =\ +"""\ +-x \n\ +---\n\ + y \ +""" + ucode_str =\ +"""\ +-x \n\ +───\n\ + y \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + expr = -x*z/y + ascii_str =\ +"""\ +-x*z \n\ +-----\n\ + y \ +""" + ucode_str =\ +"""\ +-x⋅z \n\ +─────\n\ + y \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + expr = x**2/y + ascii_str =\ +"""\ + 2\n\ +x \n\ +--\n\ +y \ +""" + ucode_str =\ +"""\ + 2\n\ +x \n\ +──\n\ +y \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + expr = -x**2/y + ascii_str =\ +"""\ + 2 \n\ +-x \n\ +----\n\ + y \ +""" + ucode_str =\ +"""\ + 2 \n\ +-x \n\ +────\n\ + y \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + expr = -x/(y*z) + ascii_str =\ +"""\ +-x \n\ +---\n\ +y*z\ +""" + ucode_str =\ +"""\ +-x \n\ +───\n\ +y⋅z\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + expr = -a/y**2 + ascii_str =\ +"""\ +-a \n\ +---\n\ + 2 \n\ +y \ +""" + ucode_str =\ +"""\ +-a \n\ +───\n\ + 2 \n\ +y \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + expr = y**(-a/b) + ascii_str =\ +"""\ + -a \n\ + ---\n\ + b \n\ +y \ +""" + ucode_str =\ +"""\ + -a \n\ + ───\n\ + b \n\ +y \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + expr = -1/y**2 + ascii_str =\ +"""\ +-1 \n\ +---\n\ + 2 \n\ +y \ +""" + ucode_str =\ +"""\ +-1 \n\ +───\n\ + 2 \n\ +y \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + expr = -10/b**2 + ascii_str =\ +"""\ +-10 \n\ +----\n\ + 2 \n\ + b \ +""" + ucode_str =\ +"""\ +-10 \n\ +────\n\ + 2 \n\ + b \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + expr = Rational(-200, 37) + ascii_str =\ +"""\ +-200 \n\ +-----\n\ + 37 \ +""" + ucode_str =\ +"""\ +-200 \n\ +─────\n\ + 37 \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + +def test_Mul(): + expr = Mul(0, 1, evaluate=False) + assert pretty(expr) == "0*1" + assert upretty(expr) == "0⋅1" + expr = Mul(1, 0, evaluate=False) + assert pretty(expr) == "1*0" + assert upretty(expr) == "1⋅0" + expr = Mul(1, 1, evaluate=False) + assert pretty(expr) == "1*1" + assert upretty(expr) == "1⋅1" + expr = Mul(1, 1, 1, evaluate=False) + assert pretty(expr) == "1*1*1" + assert upretty(expr) == "1⋅1⋅1" + expr = Mul(1, 2, evaluate=False) + assert pretty(expr) == "1*2" + assert upretty(expr) == "1⋅2" + expr = Add(0, 1, evaluate=False) + assert pretty(expr) == "0 + 1" + assert upretty(expr) == "0 + 1" + expr = Mul(1, 1, 2, evaluate=False) + assert pretty(expr) == "1*1*2" + assert upretty(expr) == "1⋅1⋅2" + expr = Add(0, 0, 1, evaluate=False) + assert pretty(expr) == "0 + 0 + 1" + assert upretty(expr) == "0 + 0 + 1" + expr = Mul(1, -1, evaluate=False) + assert pretty(expr) == "1*-1" + assert upretty(expr) == "1⋅-1" + expr = Mul(1.0, x, evaluate=False) + assert pretty(expr) == "1.0*x" + assert upretty(expr) == "1.0⋅x" + expr = Mul(1, 1, 2, 3, x, evaluate=False) + assert pretty(expr) == "1*1*2*3*x" + assert upretty(expr) == "1⋅1⋅2⋅3⋅x" + expr = Mul(-1, 1, evaluate=False) + assert pretty(expr) == "-1*1" + assert upretty(expr) == "-1⋅1" + expr = Mul(4, 3, 2, 1, 0, y, x, evaluate=False) + assert pretty(expr) == "4*3*2*1*0*y*x" + assert upretty(expr) == "4⋅3⋅2⋅1⋅0⋅y⋅x" + expr = Mul(4, 3, 2, 1+z, 0, y, x, evaluate=False) + assert pretty(expr) == "4*3*2*(z + 1)*0*y*x" + assert upretty(expr) == "4⋅3⋅2⋅(z + 1)⋅0⋅y⋅x" + expr = Mul(Rational(2, 3), Rational(5, 7), evaluate=False) + assert pretty(expr) == "2/3*5/7" + assert upretty(expr) == "2/3⋅5/7" + expr = Mul(x + y, Rational(1, 2), evaluate=False) + assert pretty(expr) == "(x + y)*1/2" + assert upretty(expr) == "(x + y)⋅1/2" + expr = Mul(Rational(1, 2), x + y, evaluate=False) + assert pretty(expr) == "x + y\n-----\n 2 " + assert upretty(expr) == "x + y\n─────\n 2 " + expr = Mul(S.One, x + y, evaluate=False) + assert pretty(expr) == "1*(x + y)" + assert upretty(expr) == "1⋅(x + y)" + expr = Mul(x - y, S.One, evaluate=False) + assert pretty(expr) == "(x - y)*1" + assert upretty(expr) == "(x - y)⋅1" + expr = Mul(Rational(1, 2), x - y, S.One, x + y, evaluate=False) + assert pretty(expr) == "1/2*(x - y)*1*(x + y)" + assert upretty(expr) == "1/2⋅(x - y)⋅1⋅(x + y)" + expr = Mul(x + y, Rational(3, 4), S.One, y - z, evaluate=False) + assert pretty(expr) == "(x + y)*3/4*1*(y - z)" + assert upretty(expr) == "(x + y)⋅3/4⋅1⋅(y - z)" + expr = Mul(x + y, Rational(1, 1), Rational(3, 4), Rational(5, 6),evaluate=False) + assert pretty(expr) == "(x + y)*1*3/4*5/6" + assert upretty(expr) == "(x + y)⋅1⋅3/4⋅5/6" + expr = Mul(Rational(3, 4), x + y, S.One, y - z, evaluate=False) + assert pretty(expr) == "3/4*(x + y)*1*(y - z)" + assert upretty(expr) == "3/4⋅(x + y)⋅1⋅(y - z)" + + +def test_issue_5524(): + assert pretty(-(-x + 5)*(-x - 2*sqrt(2) + 5) - (-y + 5)*(-y + 5)) == \ +"""\ + 2 / ___ \\\n\ +- (5 - y) + (x - 5)*\\-x - 2*\\/ 2 + 5/\ +""" + + assert upretty(-(-x + 5)*(-x - 2*sqrt(2) + 5) - (-y + 5)*(-y + 5)) == \ +"""\ + 2 \n\ +- (5 - y) + (x - 5)⋅(-x - 2⋅√2 + 5)\ +""" + + +def test_pretty_ordering(): + assert pretty(x**2 + x + 1, order='lex') == \ +"""\ + 2 \n\ +x + x + 1\ +""" + assert pretty(x**2 + x + 1, order='rev-lex') == \ +"""\ + 2\n\ +1 + x + x \ +""" + assert pretty(1 - x, order='lex') == '-x + 1' + assert pretty(1 - x, order='rev-lex') == '1 - x' + + assert pretty(1 - 2*x, order='lex') == '-2*x + 1' + assert pretty(1 - 2*x, order='rev-lex') == '1 - 2*x' + + f = 2*x**4 + y**2 - x**2 + y**3 + assert pretty(f, order=None) == \ +"""\ + 4 2 3 2\n\ +2*x - x + y + y \ +""" + assert pretty(f, order='lex') == \ +"""\ + 4 2 3 2\n\ +2*x - x + y + y \ +""" + assert pretty(f, order='rev-lex') == \ +"""\ + 2 3 2 4\n\ +y + y - x + 2*x \ +""" + + expr = x - x**3/6 + x**5/120 + O(x**6) + ascii_str = \ +"""\ + 3 5 \n\ + x x / 6\\\n\ +x - -- + --- + O\\x /\n\ + 6 120 \ +""" + ucode_str = \ +"""\ + 3 5 \n\ + x x ⎛ 6⎞\n\ +x - ── + ─── + O⎝x ⎠\n\ + 6 120 \ +""" + assert pretty(expr, order=None) == ascii_str + assert upretty(expr, order=None) == ucode_str + + assert pretty(expr, order='lex') == ascii_str + assert upretty(expr, order='lex') == ucode_str + + assert pretty(expr, order='rev-lex') == ascii_str + assert upretty(expr, order='rev-lex') == ucode_str + + +def test_EulerGamma(): + assert pretty(EulerGamma) == str(EulerGamma) == "EulerGamma" + assert upretty(EulerGamma) == "γ" + + +def test_GoldenRatio(): + assert pretty(GoldenRatio) == str(GoldenRatio) == "GoldenRatio" + assert upretty(GoldenRatio) == "φ" + + +def test_Catalan(): + assert pretty(Catalan) == upretty(Catalan) == "G" + + +def test_pretty_relational(): + expr = Eq(x, y) + ascii_str = \ +"""\ +x = y\ +""" + ucode_str = \ +"""\ +x = y\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Lt(x, y) + ascii_str = \ +"""\ +x < y\ +""" + ucode_str = \ +"""\ +x < y\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Gt(x, y) + ascii_str = \ +"""\ +x > y\ +""" + ucode_str = \ +"""\ +x > y\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Le(x, y) + ascii_str = \ +"""\ +x <= y\ +""" + ucode_str = \ +"""\ +x ≤ y\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Ge(x, y) + ascii_str = \ +"""\ +x >= y\ +""" + ucode_str = \ +"""\ +x ≥ y\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Ne(x/(y + 1), y**2) + ascii_str_1 = \ +"""\ + x 2\n\ +----- != y \n\ +1 + y \ +""" + ascii_str_2 = \ +"""\ + x 2\n\ +----- != y \n\ +y + 1 \ +""" + ucode_str_1 = \ +"""\ + x 2\n\ +───── ≠ y \n\ +1 + y \ +""" + ucode_str_2 = \ +"""\ + x 2\n\ +───── ≠ y \n\ +y + 1 \ +""" + assert pretty(expr) in [ascii_str_1, ascii_str_2] + assert upretty(expr) in [ucode_str_1, ucode_str_2] + + +def test_Assignment(): + expr = Assignment(x, y) + ascii_str = \ +"""\ +x := y\ +""" + ucode_str = \ +"""\ +x := y\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + +def test_AugmentedAssignment(): + expr = AddAugmentedAssignment(x, y) + ascii_str = \ +"""\ +x += y\ +""" + ucode_str = \ +"""\ +x += y\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = SubAugmentedAssignment(x, y) + ascii_str = \ +"""\ +x -= y\ +""" + ucode_str = \ +"""\ +x -= y\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = MulAugmentedAssignment(x, y) + ascii_str = \ +"""\ +x *= y\ +""" + ucode_str = \ +"""\ +x *= y\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = DivAugmentedAssignment(x, y) + ascii_str = \ +"""\ +x /= y\ +""" + ucode_str = \ +"""\ +x /= y\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = ModAugmentedAssignment(x, y) + ascii_str = \ +"""\ +x %= y\ +""" + ucode_str = \ +"""\ +x %= y\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + +def test_pretty_rational(): + expr = y*x**-2 + ascii_str = \ +"""\ +y \n\ +--\n\ + 2\n\ +x \ +""" + ucode_str = \ +"""\ +y \n\ +──\n\ + 2\n\ +x \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = y**Rational(3, 2) * x**Rational(-5, 2) + ascii_str = \ +"""\ + 3/2\n\ +y \n\ +----\n\ + 5/2\n\ +x \ +""" + ucode_str = \ +"""\ + 3/2\n\ +y \n\ +────\n\ + 5/2\n\ +x \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = sin(x)**3/tan(x)**2 + ascii_str = \ +"""\ + 3 \n\ +sin (x)\n\ +-------\n\ + 2 \n\ +tan (x)\ +""" + ucode_str = \ +"""\ + 3 \n\ +sin (x)\n\ +───────\n\ + 2 \n\ +tan (x)\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + +@_both_exp_pow +def test_pretty_functions(): + """Tests for Abs, conjugate, exp, function braces, and factorial.""" + expr = (2*x + exp(x)) + ascii_str_1 = \ +"""\ + x\n\ +2*x + e \ +""" + ascii_str_2 = \ +"""\ + x \n\ +e + 2*x\ +""" + ucode_str_1 = \ +"""\ + x\n\ +2⋅x + ℯ \ +""" + ucode_str_2 = \ +"""\ + x \n\ +ℯ + 2⋅x\ +""" + ucode_str_3 = \ +"""\ + x \n\ +ℯ + 2⋅x\ +""" + assert pretty(expr) in [ascii_str_1, ascii_str_2] + assert upretty(expr) in [ucode_str_1, ucode_str_2, ucode_str_3] + + expr = Abs(x) + ascii_str = \ +"""\ +|x|\ +""" + ucode_str = \ +"""\ +│x│\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Abs(x/(x**2 + 1)) + ascii_str_1 = \ +"""\ +| x |\n\ +|------|\n\ +| 2|\n\ +|1 + x |\ +""" + ascii_str_2 = \ +"""\ +| x |\n\ +|------|\n\ +| 2 |\n\ +|x + 1|\ +""" + ucode_str_1 = \ +"""\ +│ x │\n\ +│──────│\n\ +│ 2│\n\ +│1 + x │\ +""" + ucode_str_2 = \ +"""\ +│ x │\n\ +│──────│\n\ +│ 2 │\n\ +│x + 1│\ +""" + assert pretty(expr) in [ascii_str_1, ascii_str_2] + assert upretty(expr) in [ucode_str_1, ucode_str_2] + + expr = Abs(1 / (y - Abs(x))) + ascii_str = \ +"""\ + 1 \n\ +---------\n\ +|y - |x||\ +""" + ucode_str = \ +"""\ + 1 \n\ +─────────\n\ +│y - │x││\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + n = Symbol('n', integer=True) + expr = factorial(n) + ascii_str = \ +"""\ +n!\ +""" + ucode_str = \ +"""\ +n!\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = factorial(2*n) + ascii_str = \ +"""\ +(2*n)!\ +""" + ucode_str = \ +"""\ +(2⋅n)!\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = factorial(factorial(factorial(n))) + ascii_str = \ +"""\ +((n!)!)!\ +""" + ucode_str = \ +"""\ +((n!)!)!\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = factorial(n + 1) + ascii_str_1 = \ +"""\ +(1 + n)!\ +""" + ascii_str_2 = \ +"""\ +(n + 1)!\ +""" + ucode_str_1 = \ +"""\ +(1 + n)!\ +""" + ucode_str_2 = \ +"""\ +(n + 1)!\ +""" + + assert pretty(expr) in [ascii_str_1, ascii_str_2] + assert upretty(expr) in [ucode_str_1, ucode_str_2] + + expr = subfactorial(n) + ascii_str = \ +"""\ +!n\ +""" + ucode_str = \ +"""\ +!n\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = subfactorial(2*n) + ascii_str = \ +"""\ +!(2*n)\ +""" + ucode_str = \ +"""\ +!(2⋅n)\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + n = Symbol('n', integer=True) + expr = factorial2(n) + ascii_str = \ +"""\ +n!!\ +""" + ucode_str = \ +"""\ +n!!\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = factorial2(2*n) + ascii_str = \ +"""\ +(2*n)!!\ +""" + ucode_str = \ +"""\ +(2⋅n)!!\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = factorial2(factorial2(factorial2(n))) + ascii_str = \ +"""\ +((n!!)!!)!!\ +""" + ucode_str = \ +"""\ +((n!!)!!)!!\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = factorial2(n + 1) + ascii_str_1 = \ +"""\ +(1 + n)!!\ +""" + ascii_str_2 = \ +"""\ +(n + 1)!!\ +""" + ucode_str_1 = \ +"""\ +(1 + n)!!\ +""" + ucode_str_2 = \ +"""\ +(n + 1)!!\ +""" + + assert pretty(expr) in [ascii_str_1, ascii_str_2] + assert upretty(expr) in [ucode_str_1, ucode_str_2] + + expr = 2*binomial(n, k) + ascii_str = \ +"""\ + /n\\\n\ +2*| |\n\ + \\k/\ +""" + ucode_str = \ +"""\ + ⎛n⎞\n\ +2⋅⎜ ⎟\n\ + ⎝k⎠\ +""" + + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = 2*binomial(2*n, k) + ascii_str = \ +"""\ + /2*n\\\n\ +2*| |\n\ + \\ k /\ +""" + ucode_str = \ +"""\ + ⎛2⋅n⎞\n\ +2⋅⎜ ⎟\n\ + ⎝ k ⎠\ +""" + + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = 2*binomial(n**2, k) + ascii_str = \ +"""\ + / 2\\\n\ + |n |\n\ +2*| |\n\ + \\k /\ +""" + ucode_str = \ +"""\ + ⎛ 2⎞\n\ + ⎜n ⎟\n\ +2⋅⎜ ⎟\n\ + ⎝k ⎠\ +""" + + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = catalan(n) + ascii_str = \ +"""\ +C \n\ + n\ +""" + ucode_str = \ +"""\ +C \n\ + n\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = catalan(n) + ascii_str = \ +"""\ +C \n\ + n\ +""" + ucode_str = \ +"""\ +C \n\ + n\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = bell(n) + ascii_str = \ +"""\ +B \n\ + n\ +""" + ucode_str = \ +"""\ +B \n\ + n\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = bernoulli(n) + ascii_str = \ +"""\ +B \n\ + n\ +""" + ucode_str = \ +"""\ +B \n\ + n\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = bernoulli(n, x) + ascii_str = \ +"""\ +B (x)\n\ + n \ +""" + ucode_str = \ +"""\ +B (x)\n\ + n \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = fibonacci(n) + ascii_str = \ +"""\ +F \n\ + n\ +""" + ucode_str = \ +"""\ +F \n\ + n\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = lucas(n) + ascii_str = \ +"""\ +L \n\ + n\ +""" + ucode_str = \ +"""\ +L \n\ + n\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = tribonacci(n) + ascii_str = \ +"""\ +T \n\ + n\ +""" + ucode_str = \ +"""\ +T \n\ + n\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = stieltjes(n) + ascii_str = \ +"""\ +stieltjes \n\ + n\ +""" + ucode_str = \ +"""\ +γ \n\ + n\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = stieltjes(n, x) + ascii_str = \ +"""\ +stieltjes (x)\n\ + n \ +""" + ucode_str = \ +"""\ +γ (x)\n\ + n \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = mathieuc(x, y, z) + ascii_str = 'C(x, y, z)' + ucode_str = 'C(x, y, z)' + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = mathieus(x, y, z) + ascii_str = 'S(x, y, z)' + ucode_str = 'S(x, y, z)' + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = mathieucprime(x, y, z) + ascii_str = "C'(x, y, z)" + ucode_str = "C'(x, y, z)" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = mathieusprime(x, y, z) + ascii_str = "S'(x, y, z)" + ucode_str = "S'(x, y, z)" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = conjugate(x) + ascii_str = \ +"""\ +_\n\ +x\ +""" + ucode_str = \ +"""\ +_\n\ +x\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + f = Function('f') + expr = conjugate(f(x + 1)) + ascii_str_1 = \ +"""\ +________\n\ +f(1 + x)\ +""" + ascii_str_2 = \ +"""\ +________\n\ +f(x + 1)\ +""" + ucode_str_1 = \ +"""\ +________\n\ +f(1 + x)\ +""" + ucode_str_2 = \ +"""\ +________\n\ +f(x + 1)\ +""" + assert pretty(expr) in [ascii_str_1, ascii_str_2] + assert upretty(expr) in [ucode_str_1, ucode_str_2] + + expr = f(x) + ascii_str = \ +"""\ +f(x)\ +""" + ucode_str = \ +"""\ +f(x)\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = f(x, y) + ascii_str = \ +"""\ +f(x, y)\ +""" + ucode_str = \ +"""\ +f(x, y)\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = f(x/(y + 1), y) + ascii_str_1 = \ +"""\ + / x \\\n\ +f|-----, y|\n\ + \\1 + y /\ +""" + ascii_str_2 = \ +"""\ + / x \\\n\ +f|-----, y|\n\ + \\y + 1 /\ +""" + ucode_str_1 = \ +"""\ + ⎛ x ⎞\n\ +f⎜─────, y⎟\n\ + ⎝1 + y ⎠\ +""" + ucode_str_2 = \ +"""\ + ⎛ x ⎞\n\ +f⎜─────, y⎟\n\ + ⎝y + 1 ⎠\ +""" + assert pretty(expr) in [ascii_str_1, ascii_str_2] + assert upretty(expr) in [ucode_str_1, ucode_str_2] + + expr = f(x**x**x**x**x**x) + ascii_str = \ +"""\ + / / / / / x\\\\\\\\\\ + | | | | \\x /|||| + | | | \\x /||| + | | \\x /|| + | \\x /| +f\\x /\ +""" + ucode_str = \ +"""\ + ⎛ ⎛ ⎛ ⎛ ⎛ x⎞⎞⎞⎞⎞ + ⎜ ⎜ ⎜ ⎜ ⎝x ⎠⎟⎟⎟⎟ + ⎜ ⎜ ⎜ ⎝x ⎠⎟⎟⎟ + ⎜ ⎜ ⎝x ⎠⎟⎟ + ⎜ ⎝x ⎠⎟ +f⎝x ⎠\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = sin(x)**2 + ascii_str = \ +"""\ + 2 \n\ +sin (x)\ +""" + ucode_str = \ +"""\ + 2 \n\ +sin (x)\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = conjugate(a + b*I) + ascii_str = \ +"""\ +_ _\n\ +a - I*b\ +""" + ucode_str = \ +"""\ +_ _\n\ +a - ⅈ⋅b\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = conjugate(exp(a + b*I)) + ascii_str = \ +"""\ + _ _\n\ + a - I*b\n\ +e \ +""" + ucode_str = \ +"""\ + _ _\n\ + a - ⅈ⋅b\n\ +ℯ \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = conjugate( f(1 + conjugate(f(x))) ) + ascii_str_1 = \ +"""\ +___________\n\ + / ____\\\n\ +f\\1 + f(x)/\ +""" + ascii_str_2 = \ +"""\ +___________\n\ + /____ \\\n\ +f\\f(x) + 1/\ +""" + ucode_str_1 = \ +"""\ +___________\n\ + ⎛ ____⎞\n\ +f⎝1 + f(x)⎠\ +""" + ucode_str_2 = \ +"""\ +___________\n\ + ⎛____ ⎞\n\ +f⎝f(x) + 1⎠\ +""" + assert pretty(expr) in [ascii_str_1, ascii_str_2] + assert upretty(expr) in [ucode_str_1, ucode_str_2] + + expr = f(x/(y + 1), y) + ascii_str_1 = \ +"""\ + / x \\\n\ +f|-----, y|\n\ + \\1 + y /\ +""" + ascii_str_2 = \ +"""\ + / x \\\n\ +f|-----, y|\n\ + \\y + 1 /\ +""" + ucode_str_1 = \ +"""\ + ⎛ x ⎞\n\ +f⎜─────, y⎟\n\ + ⎝1 + y ⎠\ +""" + ucode_str_2 = \ +"""\ + ⎛ x ⎞\n\ +f⎜─────, y⎟\n\ + ⎝y + 1 ⎠\ +""" + assert pretty(expr) in [ascii_str_1, ascii_str_2] + assert upretty(expr) in [ucode_str_1, ucode_str_2] + + expr = floor(1 / (y - floor(x))) + ascii_str = \ +"""\ + / 1 \\\n\ +floor|------------|\n\ + \\y - floor(x)/\ +""" + ucode_str = \ +"""\ +⎢ 1 ⎥\n\ +⎢───────⎥\n\ +⎣y - ⌊x⌋⎦\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = ceiling(1 / (y - ceiling(x))) + ascii_str = \ +"""\ + / 1 \\\n\ +ceiling|--------------|\n\ + \\y - ceiling(x)/\ +""" + ucode_str = \ +"""\ +⎡ 1 ⎤\n\ +⎢───────⎥\n\ +⎢y - ⌈x⌉⎥\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = euler(n) + ascii_str = \ +"""\ +E \n\ + n\ +""" + ucode_str = \ +"""\ +E \n\ + n\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = euler(1/(1 + 1/(1 + 1/n))) + ascii_str = \ +"""\ +E \n\ + 1 \n\ + ---------\n\ + 1 \n\ + 1 + -----\n\ + 1\n\ + 1 + -\n\ + n\ +""" + + ucode_str = \ +"""\ +E \n\ + 1 \n\ + ─────────\n\ + 1 \n\ + 1 + ─────\n\ + 1\n\ + 1 + ─\n\ + n\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = euler(n, x) + ascii_str = \ +"""\ +E (x)\n\ + n \ +""" + ucode_str = \ +"""\ +E (x)\n\ + n \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = euler(n, x/2) + ascii_str = \ +"""\ + /x\\\n\ +E |-|\n\ + n\\2/\ +""" + ucode_str = \ +"""\ + ⎛x⎞\n\ +E ⎜─⎟\n\ + n⎝2⎠\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + +def test_pretty_sqrt(): + expr = sqrt(2) + ascii_str = \ +"""\ + ___\n\ +\\/ 2 \ +""" + ucode_str = \ +"√2" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = 2**Rational(1, 3) + ascii_str = \ +"""\ +3 ___\n\ +\\/ 2 \ +""" + ucode_str = \ +"""\ +3 ___\n\ +╲╱ 2 \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = 2**Rational(1, 1000) + ascii_str = \ +"""\ +1000___\n\ + \\/ 2 \ +""" + ucode_str = \ +"""\ +1000___\n\ + ╲╱ 2 \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = sqrt(x**2 + 1) + ascii_str = \ +"""\ + ________\n\ + / 2 \n\ +\\/ x + 1 \ +""" + ucode_str = \ +"""\ + ________\n\ + ╱ 2 \n\ +╲╱ x + 1 \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = (1 + sqrt(5))**Rational(1, 3) + ascii_str = \ +"""\ + ___________\n\ +3 / ___ \n\ +\\/ 1 + \\/ 5 \ +""" + ucode_str = \ +"""\ +3 ________\n\ +╲╱ 1 + √5 \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = 2**(1/x) + ascii_str = \ +"""\ +x ___\n\ +\\/ 2 \ +""" + ucode_str = \ +"""\ +x ___\n\ +╲╱ 2 \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = sqrt(2 + pi) + ascii_str = \ +"""\ + ________\n\ +\\/ 2 + pi \ +""" + ucode_str = \ +"""\ + _______\n\ +╲╱ 2 + π \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = (2 + ( + 1 + x**2)/(2 + x))**Rational(1, 4) + (1 + x**Rational(1, 1000))/sqrt(3 + x**2) + ascii_str = \ +"""\ + ____________ \n\ + / 2 1000___ \n\ + / x + 1 \\/ x + 1\n\ +4 / 2 + ------ + -----------\n\ +\\/ x + 2 ________\n\ + / 2 \n\ + \\/ x + 3 \ +""" + ucode_str = \ +"""\ + ____________ \n\ + ╱ 2 1000___ \n\ + ╱ x + 1 ╲╱ x + 1\n\ +4 ╱ 2 + ────── + ───────────\n\ +╲╱ x + 2 ________\n\ + ╱ 2 \n\ + ╲╱ x + 3 \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + +def test_pretty_sqrt_char_knob(): + # See PR #9234. + expr = sqrt(2) + ucode_str1 = \ +"""\ + ___\n\ +╲╱ 2 \ +""" + ucode_str2 = \ +"√2" + assert xpretty(expr, use_unicode=True, + use_unicode_sqrt_char=False) == ucode_str1 + assert xpretty(expr, use_unicode=True, + use_unicode_sqrt_char=True) == ucode_str2 + + +def test_pretty_sqrt_longsymbol_no_sqrt_char(): + # Do not use unicode sqrt char for long symbols (see PR #9234). + expr = sqrt(Symbol('C1')) + ucode_str = \ +"""\ + ____\n\ +╲╱ C₁ \ +""" + assert upretty(expr) == ucode_str + + +def test_pretty_KroneckerDelta(): + x, y = symbols("x, y") + expr = KroneckerDelta(x, y) + ascii_str = \ +"""\ +d \n\ + x,y\ +""" + ucode_str = \ +"""\ +δ \n\ + x,y\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + +def test_pretty_product(): + n, m, k, l = symbols('n m k l') + f = symbols('f', cls=Function) + expr = Product(f((n/3)**2), (n, k**2, l)) + + unicode_str = \ +"""\ + l \n\ +─┬──────┬─ \n\ + │ │ ⎛ 2⎞\n\ + │ │ ⎜n ⎟\n\ + │ │ f⎜──⎟\n\ + │ │ ⎝9 ⎠\n\ + │ │ \n\ + 2 \n\ + n = k """ + ascii_str = \ +"""\ + l \n\ +__________ \n\ + | | / 2\\\n\ + | | |n |\n\ + | | f|--|\n\ + | | \\9 /\n\ + | | \n\ + 2 \n\ + n = k """ + + expr = Product(f((n/3)**2), (n, k**2, l), (l, 1, m)) + + unicode_str = \ +"""\ + m l \n\ +─┬──────┬─ ─┬──────┬─ \n\ + │ │ │ │ ⎛ 2⎞\n\ + │ │ │ │ ⎜n ⎟\n\ + │ │ │ │ f⎜──⎟\n\ + │ │ │ │ ⎝9 ⎠\n\ + │ │ │ │ \n\ + l = 1 2 \n\ + n = k """ + ascii_str = \ +"""\ + m l \n\ +__________ __________ \n\ + | | | | / 2\\\n\ + | | | | |n |\n\ + | | | | f|--|\n\ + | | | | \\9 /\n\ + | | | | \n\ + l = 1 2 \n\ + n = k """ + + assert pretty(expr) == ascii_str + assert upretty(expr) == unicode_str + + +def test_pretty_Lambda(): + # S.IdentityFunction is a special case + expr = Lambda(y, y) + assert pretty(expr) == "x -> x" + assert upretty(expr) == "x ↦ x" + + expr = Lambda(x, x+1) + assert pretty(expr) == "x -> x + 1" + assert upretty(expr) == "x ↦ x + 1" + + expr = Lambda(x, x**2) + ascii_str = \ +"""\ + 2\n\ +x -> x \ +""" + ucode_str = \ +"""\ + 2\n\ +x ↦ x \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Lambda(x, x**2)**2 + ascii_str = \ +"""\ + 2 +/ 2\\ \n\ +\\x -> x / \ +""" + ucode_str = \ +"""\ + 2 +⎛ 2⎞ \n\ +⎝x ↦ x ⎠ \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Lambda((x, y), x) + ascii_str = "(x, y) -> x" + ucode_str = "(x, y) ↦ x" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Lambda((x, y), x**2) + ascii_str = \ +"""\ + 2\n\ +(x, y) -> x \ +""" + ucode_str = \ +"""\ + 2\n\ +(x, y) ↦ x \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Lambda(((x, y),), x**2) + ascii_str = \ +"""\ + 2\n\ +((x, y),) -> x \ +""" + ucode_str = \ +"""\ + 2\n\ +((x, y),) ↦ x \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + +def test_pretty_TransferFunction(): + tf1 = TransferFunction(s - 1, s + 1, s) + assert upretty(tf1) == "s - 1\n─────\ns + 1" + tf2 = TransferFunction(2*s + 1, 3 - p, s) + assert upretty(tf2) == "2⋅s + 1\n───────\n 3 - p " + tf3 = TransferFunction(p, p + 1, p) + assert upretty(tf3) == " p \n─────\np + 1" + + +def test_pretty_Series(): + tf1 = TransferFunction(x + y, x - 2*y, y) + tf2 = TransferFunction(x - y, x + y, y) + tf3 = TransferFunction(x**2 + y, y - x, y) + tf4 = TransferFunction(2, 3, y) + + tfm1 = TransferFunctionMatrix([[tf1, tf2], [tf3, tf4]]) + tfm2 = TransferFunctionMatrix([[tf3], [-tf4]]) + tfm3 = TransferFunctionMatrix([[tf1, -tf2, -tf3], [tf3, -tf4, tf2]]) + tfm4 = TransferFunctionMatrix([[tf1, tf2], [tf3, -tf4], [-tf2, -tf1]]) + tfm5 = TransferFunctionMatrix([[-tf2, -tf1], [tf4, -tf3], [tf1, tf2]]) + + expected1 = \ +"""\ + ⎛ 2 ⎞\n\ +⎛ x + y ⎞ ⎜x + y⎟\n\ +⎜───────⎟⋅⎜──────⎟\n\ +⎝x - 2⋅y⎠ ⎝-x + y⎠\ +""" + expected2 = \ +"""\ +⎛-x + y⎞ ⎛-x - y ⎞\n\ +⎜──────⎟⋅⎜───────⎟\n\ +⎝x + y ⎠ ⎝x - 2⋅y⎠\ +""" + expected3 = \ +"""\ +⎛ 2 ⎞ \n\ +⎜x + y⎟ ⎛ x + y ⎞ ⎛-x - y x - y⎞\n\ +⎜──────⎟⋅⎜───────⎟⋅⎜─────── + ─────⎟\n\ +⎝-x + y⎠ ⎝x - 2⋅y⎠ ⎝x - 2⋅y x + y⎠\ +""" + expected4 = \ +"""\ + ⎛ 2 ⎞\n\ +⎛ x + y x - y⎞ ⎜x - y x + y⎟\n\ +⎜─────── + ─────⎟⋅⎜───── + ──────⎟\n\ +⎝x - 2⋅y x + y⎠ ⎝x + y -x + y⎠\ +""" + expected5 = \ +"""\ +⎡ x + y x - y⎤ ⎡ 2 ⎤ \n\ +⎢─────── ─────⎥ ⎢x + y⎥ \n\ +⎢x - 2⋅y x + y⎥ ⎢──────⎥ \n\ +⎢ ⎥ ⎢-x + y⎥ \n\ +⎢ 2 ⎥ ⋅⎢ ⎥ \n\ +⎢x + y 2 ⎥ ⎢ -2 ⎥ \n\ +⎢────── ─ ⎥ ⎢ ─── ⎥ \n\ +⎣-x + y 3 ⎦τ ⎣ 3 ⎦τ\ +""" + expected6 = \ +"""\ + ⎛⎡ x + y x - y ⎤ ⎡ x - y x + y ⎤ ⎞\n\ + ⎜⎢─────── ───── ⎥ ⎢ ───── ───────⎥ ⎟\n\ +⎡ x + y x - y⎤ ⎡ 2 ⎤ ⎜⎢x - 2⋅y x + y ⎥ ⎢ x + y x - 2⋅y⎥ ⎟\n\ +⎢─────── ─────⎥ ⎢ x + y -x + y - x - y⎥ ⎜⎢ ⎥ ⎢ ⎥ ⎟\n\ +⎢x - 2⋅y x + y⎥ ⎢─────── ────── ────────⎥ ⎜⎢ 2 ⎥ ⎢ 2 ⎥ ⎟\n\ +⎢ ⎥ ⎢x - 2⋅y x + y -x + y ⎥ ⎜⎢x + y -2 ⎥ ⎢ -2 x + y ⎥ ⎟\n\ +⎢ 2 ⎥ ⋅⎢ ⎥ ⋅⎜⎢────── ─── ⎥ + ⎢ ─── ────── ⎥ ⎟\n\ +⎢x + y 2 ⎥ ⎢ 2 ⎥ ⎜⎢-x + y 3 ⎥ ⎢ 3 -x + y ⎥ ⎟\n\ +⎢────── ─ ⎥ ⎢x + y -2 x - y ⎥ ⎜⎢ ⎥ ⎢ ⎥ ⎟\n\ +⎣-x + y 3 ⎦τ ⎢────── ─── ───── ⎥ ⎜⎢-x + y -x - y ⎥ ⎢-x - y -x + y ⎥ ⎟\n\ + ⎣-x + y 3 x + y ⎦τ ⎜⎢────── ───────⎥ ⎢─────── ────── ⎥ ⎟\n\ + ⎝⎣x + y x - 2⋅y⎦τ ⎣x - 2⋅y x + y ⎦τ⎠\ +""" + + assert upretty(Series(tf1, tf3)) == expected1 + assert upretty(Series(-tf2, -tf1)) == expected2 + assert upretty(Series(tf3, tf1, Parallel(-tf1, tf2))) == expected3 + assert upretty(Series(Parallel(tf1, tf2), Parallel(tf2, tf3))) == expected4 + assert upretty(MIMOSeries(tfm2, tfm1)) == expected5 + assert upretty(MIMOSeries(MIMOParallel(tfm4, -tfm5), tfm3, tfm1)) == expected6 + + +def test_pretty_Parallel(): + tf1 = TransferFunction(x + y, x - 2*y, y) + tf2 = TransferFunction(x - y, x + y, y) + tf3 = TransferFunction(x**2 + y, y - x, y) + tf4 = TransferFunction(y**2 - x, x**3 + x, y) + + tfm1 = TransferFunctionMatrix([[tf1, tf2], [tf3, -tf4], [-tf2, -tf1]]) + tfm2 = TransferFunctionMatrix([[-tf2, -tf1], [tf4, -tf3], [tf1, tf2]]) + tfm3 = TransferFunctionMatrix([[-tf1, tf2], [-tf3, tf4], [tf2, tf1]]) + tfm4 = TransferFunctionMatrix([[-tf1, -tf2], [-tf3, -tf4]]) + + expected1 = \ +"""\ + x + y x - y\n\ +─────── + ─────\n\ +x - 2⋅y x + y\ +""" + expected2 = \ +"""\ +-x + y -x - y \n\ +────── + ─────── +x + y x - 2⋅y\ +""" + expected3 = \ +"""\ + 2 \n\ +x + y x + y ⎛-x - y ⎞ ⎛x - y⎞ +────── + ─────── + ⎜───────⎟⋅⎜─────⎟ +-x + y x - 2⋅y ⎝x - 2⋅y⎠ ⎝x + y⎠\ +""" + + expected4 = \ +"""\ + ⎛ 2 ⎞\n\ +⎛ x + y ⎞ ⎛x - y⎞ ⎛x - y⎞ ⎜x + y⎟\n\ +⎜───────⎟⋅⎜─────⎟ + ⎜─────⎟⋅⎜──────⎟\n\ +⎝x - 2⋅y⎠ ⎝x + y⎠ ⎝x + y⎠ ⎝-x + y⎠\ +""" + expected5 = \ +"""\ +⎡ x + y -x + y ⎤ ⎡ x - y x + y ⎤ ⎡ x + y x - y ⎤ \n\ +⎢─────── ────── ⎥ ⎢ ───── ───────⎥ ⎢─────── ───── ⎥ \n\ +⎢x - 2⋅y x + y ⎥ ⎢ x + y x - 2⋅y⎥ ⎢x - 2⋅y x + y ⎥ \n\ +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ \n\ +⎢ 2 2 ⎥ ⎢ 2 2 ⎥ ⎢ 2 2 ⎥ \n\ +⎢x + y x - y ⎥ ⎢x - y x + y ⎥ ⎢x + y x - y ⎥ \n\ +⎢────── ────── ⎥ + ⎢────── ────── ⎥ + ⎢────── ────── ⎥ \n\ +⎢-x + y 3 ⎥ ⎢ 3 -x + y ⎥ ⎢-x + y 3 ⎥ \n\ +⎢ x + x ⎥ ⎢x + x ⎥ ⎢ x + x ⎥ \n\ +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ \n\ +⎢-x + y -x - y ⎥ ⎢-x - y -x + y ⎥ ⎢-x + y -x - y ⎥ \n\ +⎢────── ───────⎥ ⎢─────── ────── ⎥ ⎢────── ───────⎥ \n\ +⎣x + y x - 2⋅y⎦τ ⎣x - 2⋅y x + y ⎦τ ⎣x + y x - 2⋅y⎦τ\ +""" + expected6 = \ +"""\ +⎡ x - y x + y ⎤ ⎡-x + y -x - y ⎤ \n\ +⎢ ───── ───────⎥ ⎢────── ─────── ⎥ \n\ +⎢ x + y x - 2⋅y⎥ ⎡-x - y -x + y⎤ ⎢x + y x - 2⋅y ⎥ \n\ +⎢ ⎥ ⎢─────── ──────⎥ ⎢ ⎥ \n\ +⎢ 2 2 ⎥ ⎢x - 2⋅y x + y ⎥ ⎢ 2 2 ⎥ \n\ +⎢x - y x + y ⎥ ⎢ ⎥ ⎢-x + y - x - y⎥ \n\ +⎢────── ────── ⎥ ⋅⎢ 2 2⎥ + ⎢─────── ────────⎥ \n\ +⎢ 3 -x + y ⎥ ⎢- x - y x - y ⎥ ⎢ 3 -x + y ⎥ \n\ +⎢x + x ⎥ ⎢──────── ──────⎥ ⎢x + x ⎥ \n\ +⎢ ⎥ ⎢ -x + y 3 ⎥ ⎢ ⎥ \n\ +⎢-x - y -x + y ⎥ ⎣ x + x⎦τ ⎢ x + y x - y ⎥ \n\ +⎢─────── ────── ⎥ ⎢─────── ───── ⎥ \n\ +⎣x - 2⋅y x + y ⎦τ ⎣x - 2⋅y x + y ⎦τ\ +""" + assert upretty(Parallel(tf1, tf2)) == expected1 + assert upretty(Parallel(-tf2, -tf1)) == expected2 + assert upretty(Parallel(tf3, tf1, Series(-tf1, tf2))) == expected3 + assert upretty(Parallel(Series(tf1, tf2), Series(tf2, tf3))) == expected4 + assert upretty(MIMOParallel(-tfm3, -tfm2, tfm1)) == expected5 + assert upretty(MIMOParallel(MIMOSeries(tfm4, -tfm2), tfm2)) == expected6 + + +def test_pretty_Feedback(): + tf = TransferFunction(1, 1, y) + tf1 = TransferFunction(x + y, x - 2*y, y) + tf2 = TransferFunction(x - y, x + y, y) + tf3 = TransferFunction(y**2 - 2*y + 1, y + 5, y) + tf4 = TransferFunction(x - 2*y**3, x + y, x) + tf5 = TransferFunction(1 - x, x - y, y) + tf6 = TransferFunction(2, 2, x) + expected1 = \ +"""\ + ⎛1⎞ \n\ + ⎜─⎟ \n\ + ⎝1⎠ \n\ +─────────────\n\ +1 ⎛ x + y ⎞\n\ +─ + ⎜───────⎟\n\ +1 ⎝x - 2⋅y⎠\ +""" + expected2 = \ +"""\ + ⎛1⎞ \n\ + ⎜─⎟ \n\ + ⎝1⎠ \n\ +────────────────────────────────────\n\ + ⎛ 2 ⎞\n\ +1 ⎛x - y⎞ ⎛ x + y ⎞ ⎜y - 2⋅y + 1⎟\n\ +─ + ⎜─────⎟⋅⎜───────⎟⋅⎜────────────⎟\n\ +1 ⎝x + y⎠ ⎝x - 2⋅y⎠ ⎝ y + 5 ⎠\ +""" + expected3 = \ +"""\ + ⎛ x + y ⎞ \n\ + ⎜───────⎟ \n\ + ⎝x - 2⋅y⎠ \n\ +────────────────────────────────────────────\n\ + ⎛ 2 ⎞ \n\ +1 ⎛ x + y ⎞ ⎛x - y⎞ ⎜y - 2⋅y + 1⎟ ⎛1 - x⎞\n\ +─ + ⎜───────⎟⋅⎜─────⎟⋅⎜────────────⎟⋅⎜─────⎟\n\ +1 ⎝x - 2⋅y⎠ ⎝x + y⎠ ⎝ y + 5 ⎠ ⎝x - y⎠\ +""" + expected4 = \ +"""\ + ⎛ x + y ⎞ ⎛x - y⎞ \n\ + ⎜───────⎟⋅⎜─────⎟ \n\ + ⎝x - 2⋅y⎠ ⎝x + y⎠ \n\ +─────────────────────\n\ +1 ⎛ x + y ⎞ ⎛x - y⎞\n\ +─ + ⎜───────⎟⋅⎜─────⎟\n\ +1 ⎝x - 2⋅y⎠ ⎝x + y⎠\ +""" + expected5 = \ +"""\ + ⎛ x + y ⎞ ⎛x - y⎞ \n\ + ⎜───────⎟⋅⎜─────⎟ \n\ + ⎝x - 2⋅y⎠ ⎝x + y⎠ \n\ +─────────────────────────────\n\ +1 ⎛ x + y ⎞ ⎛x - y⎞ ⎛1 - x⎞\n\ +─ + ⎜───────⎟⋅⎜─────⎟⋅⎜─────⎟\n\ +1 ⎝x - 2⋅y⎠ ⎝x + y⎠ ⎝x - y⎠\ +""" + expected6 = \ +"""\ + ⎛ 2 ⎞ \n\ + ⎜y - 2⋅y + 1⎟ ⎛1 - x⎞ \n\ + ⎜────────────⎟⋅⎜─────⎟ \n\ + ⎝ y + 5 ⎠ ⎝x - y⎠ \n\ +────────────────────────────────────────────\n\ + ⎛ 2 ⎞ \n\ +1 ⎜y - 2⋅y + 1⎟ ⎛1 - x⎞ ⎛x - y⎞ ⎛ x + y ⎞\n\ +─ + ⎜────────────⎟⋅⎜─────⎟⋅⎜─────⎟⋅⎜───────⎟\n\ +1 ⎝ y + 5 ⎠ ⎝x - y⎠ ⎝x + y⎠ ⎝x - 2⋅y⎠\ +""" + expected7 = \ +"""\ + ⎛ 3⎞ \n\ + ⎜x - 2⋅y ⎟ \n\ + ⎜────────⎟ \n\ + ⎝ x + y ⎠ \n\ +──────────────────\n\ + ⎛ 3⎞ \n\ +1 ⎜x - 2⋅y ⎟ ⎛2⎞\n\ +─ + ⎜────────⎟⋅⎜─⎟\n\ +1 ⎝ x + y ⎠ ⎝2⎠\ +""" + expected8 = \ +"""\ + ⎛1 - x⎞ \n\ + ⎜─────⎟ \n\ + ⎝x - y⎠ \n\ +───────────\n\ +1 ⎛1 - x⎞\n\ +─ + ⎜─────⎟\n\ +1 ⎝x - y⎠\ +""" + expected9 = \ +"""\ + ⎛ x + y ⎞ ⎛x - y⎞ \n\ + ⎜───────⎟⋅⎜─────⎟ \n\ + ⎝x - 2⋅y⎠ ⎝x + y⎠ \n\ +─────────────────────────────\n\ +1 ⎛ x + y ⎞ ⎛x - y⎞ ⎛1 - x⎞\n\ +─ - ⎜───────⎟⋅⎜─────⎟⋅⎜─────⎟\n\ +1 ⎝x - 2⋅y⎠ ⎝x + y⎠ ⎝x - y⎠\ +""" + expected10 = \ +"""\ + ⎛1 - x⎞ \n\ + ⎜─────⎟ \n\ + ⎝x - y⎠ \n\ +───────────\n\ +1 ⎛1 - x⎞\n\ +─ - ⎜─────⎟\n\ +1 ⎝x - y⎠\ +""" + assert upretty(Feedback(tf, tf1)) == expected1 + assert upretty(Feedback(tf, tf2*tf1*tf3)) == expected2 + assert upretty(Feedback(tf1, tf2*tf3*tf5)) == expected3 + assert upretty(Feedback(tf1*tf2, tf)) == expected4 + assert upretty(Feedback(tf1*tf2, tf5)) == expected5 + assert upretty(Feedback(tf3*tf5, tf2*tf1)) == expected6 + assert upretty(Feedback(tf4, tf6)) == expected7 + assert upretty(Feedback(tf5, tf)) == expected8 + + assert upretty(Feedback(tf1*tf2, tf5, 1)) == expected9 + assert upretty(Feedback(tf5, tf, 1)) == expected10 + + +def test_pretty_MIMOFeedback(): + tf1 = TransferFunction(x + y, x - 2*y, y) + tf2 = TransferFunction(x - y, x + y, y) + tfm_1 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf1]]) + tfm_2 = TransferFunctionMatrix([[tf2, tf1], [tf1, tf2]]) + tfm_3 = TransferFunctionMatrix([[tf1, tf1], [tf2, tf2]]) + + expected1 = \ +"""\ +⎛ ⎡ x + y x - y ⎤ ⎡ x - y x + y ⎤ ⎞-1 ⎡ x + y x - y ⎤ \n\ +⎜ ⎢─────── ───── ⎥ ⎢ ───── ───────⎥ ⎟ ⎢─────── ───── ⎥ \n\ +⎜ ⎢x - 2⋅y x + y ⎥ ⎢ x + y x - 2⋅y⎥ ⎟ ⎢x - 2⋅y x + y ⎥ \n\ +⎜I - ⎢ ⎥ ⋅⎢ ⎥ ⎟ ⋅ ⎢ ⎥ \n\ +⎜ ⎢ x - y x + y ⎥ ⎢ x + y x - y ⎥ ⎟ ⎢ x - y x + y ⎥ \n\ +⎜ ⎢ ───── ───────⎥ ⎢─────── ───── ⎥ ⎟ ⎢ ───── ───────⎥ \n\ +⎝ ⎣ x + y x - 2⋅y⎦τ ⎣x - 2⋅y x + y ⎦τ⎠ ⎣ x + y x - 2⋅y⎦τ\ +""" + expected2 = \ +"""\ +⎛ ⎡ x + y x - y ⎤ ⎡ x - y x + y ⎤ ⎡ x + y x + y ⎤ ⎞-1 ⎡ x + y x - y ⎤ ⎡ x - y x + y ⎤ \n\ +⎜ ⎢─────── ───── ⎥ ⎢ ───── ───────⎥ ⎢─────── ───────⎥ ⎟ ⎢─────── ───── ⎥ ⎢ ───── ───────⎥ \n\ +⎜ ⎢x - 2⋅y x + y ⎥ ⎢ x + y x - 2⋅y⎥ ⎢x - 2⋅y x - 2⋅y⎥ ⎟ ⎢x - 2⋅y x + y ⎥ ⎢ x + y x - 2⋅y⎥ \n\ +⎜I + ⎢ ⎥ ⋅⎢ ⎥ ⋅⎢ ⎥ ⎟ ⋅ ⎢ ⎥ ⋅⎢ ⎥ \n\ +⎜ ⎢ x - y x + y ⎥ ⎢ x + y x - y ⎥ ⎢ x - y x - y ⎥ ⎟ ⎢ x - y x + y ⎥ ⎢ x + y x - y ⎥ \n\ +⎜ ⎢ ───── ───────⎥ ⎢─────── ───── ⎥ ⎢ ───── ───── ⎥ ⎟ ⎢ ───── ───────⎥ ⎢─────── ───── ⎥ \n\ +⎝ ⎣ x + y x - 2⋅y⎦τ ⎣x - 2⋅y x + y ⎦τ ⎣ x + y x + y ⎦τ⎠ ⎣ x + y x - 2⋅y⎦τ ⎣x - 2⋅y x + y ⎦τ\ +""" + + assert upretty(MIMOFeedback(tfm_1, tfm_2, 1)) == \ + expected1 # Positive MIMOFeedback + assert upretty(MIMOFeedback(tfm_1*tfm_2, tfm_3)) == \ + expected2 # Negative MIMOFeedback (Default) + + +def test_pretty_TransferFunctionMatrix(): + tf1 = TransferFunction(x + y, x - 2*y, y) + tf2 = TransferFunction(x - y, x + y, y) + tf3 = TransferFunction(y**2 - 2*y + 1, y + 5, y) + tf4 = TransferFunction(y, x**2 + x + 1, y) + tf5 = TransferFunction(1 - x, x - y, y) + tf6 = TransferFunction(2, 2, y) + expected1 = \ +"""\ +⎡ x + y ⎤ \n\ +⎢───────⎥ \n\ +⎢x - 2⋅y⎥ \n\ +⎢ ⎥ \n\ +⎢ x - y ⎥ \n\ +⎢ ───── ⎥ \n\ +⎣ x + y ⎦τ\ +""" + expected2 = \ +"""\ +⎡ x + y ⎤ \n\ +⎢ ─────── ⎥ \n\ +⎢ x - 2⋅y ⎥ \n\ +⎢ ⎥ \n\ +⎢ x - y ⎥ \n\ +⎢ ───── ⎥ \n\ +⎢ x + y ⎥ \n\ +⎢ ⎥ \n\ +⎢ 2 ⎥ \n\ +⎢- y + 2⋅y - 1⎥ \n\ +⎢──────────────⎥ \n\ +⎣ y + 5 ⎦τ\ +""" + expected3 = \ +"""\ +⎡ x + y x - y ⎤ \n\ +⎢ ─────── ───── ⎥ \n\ +⎢ x - 2⋅y x + y ⎥ \n\ +⎢ ⎥ \n\ +⎢ 2 ⎥ \n\ +⎢y - 2⋅y + 1 y ⎥ \n\ +⎢──────────── ──────────⎥ \n\ +⎢ y + 5 2 ⎥ \n\ +⎢ x + x + 1⎥ \n\ +⎢ ⎥ \n\ +⎢ 1 - x 2 ⎥ \n\ +⎢ ───── ─ ⎥ \n\ +⎣ x - y 2 ⎦τ\ +""" + expected4 = \ +"""\ +⎡ x - y x + y y ⎤ \n\ +⎢ ───── ─────── ──────────⎥ \n\ +⎢ x + y x - 2⋅y 2 ⎥ \n\ +⎢ x + x + 1⎥ \n\ +⎢ ⎥ \n\ +⎢ 2 ⎥ \n\ +⎢- y + 2⋅y - 1 x - 1 -2 ⎥ \n\ +⎢────────────── ───── ─── ⎥ \n\ +⎣ y + 5 x - y 2 ⎦τ\ +""" + expected5 = \ +"""\ +⎡ x + y x - y x + y y ⎤ \n\ +⎢───────⋅───── ─────── ──────────⎥ \n\ +⎢x - 2⋅y x + y x - 2⋅y 2 ⎥ \n\ +⎢ x + x + 1⎥ \n\ +⎢ ⎥ \n\ +⎢ 1 - x 2 x + y -2 ⎥ \n\ +⎢ ───── + ─ ─────── ─── ⎥ \n\ +⎣ x - y 2 x - 2⋅y 2 ⎦τ\ +""" + + assert upretty(TransferFunctionMatrix([[tf1], [tf2]])) == expected1 + assert upretty(TransferFunctionMatrix([[tf1], [tf2], [-tf3]])) == expected2 + assert upretty(TransferFunctionMatrix([[tf1, tf2], [tf3, tf4], [tf5, tf6]])) == expected3 + assert upretty(TransferFunctionMatrix([[tf2, tf1, tf4], [-tf3, -tf5, -tf6]])) == expected4 + assert upretty(TransferFunctionMatrix([[Series(tf2, tf1), tf1, tf4], [Parallel(tf6, tf5), tf1, -tf6]])) == \ + expected5 + + +def test_pretty_StateSpace(): + ss1 = StateSpace(Matrix([a]), Matrix([b]), Matrix([c]), Matrix([d])) + A = Matrix([[0, 1], [1, 0]]) + B = Matrix([1, 0]) + C = Matrix([[0, 1]]) + D = Matrix([0]) + ss2 = StateSpace(A, B, C, D) + ss3 = StateSpace(Matrix([[-1.5, -2], [1, 0]]), + Matrix([[0.5, 0], [0, 1]]), + Matrix([[0, 1], [0, 2]]), + Matrix([[2, 2], [1, 1]])) + + expected1 = \ +"""\ +⎡[a] [b]⎤\n\ +⎢ ⎥\n\ +⎣[c] [d]⎦\ +""" + expected2 = \ +"""\ +⎡⎡0 1⎤ ⎡1⎤⎤\n\ +⎢⎢ ⎥ ⎢ ⎥⎥\n\ +⎢⎣1 0⎦ ⎣0⎦⎥\n\ +⎢ ⎥\n\ +⎣[0 1] [0]⎦\ +""" + expected3 = \ +"""\ +⎡⎡-1.5 -2⎤ ⎡0.5 0⎤⎤\n\ +⎢⎢ ⎥ ⎢ ⎥⎥\n\ +⎢⎣ 1 0 ⎦ ⎣ 0 1⎦⎥\n\ +⎢ ⎥\n\ +⎢ ⎡0 1⎤ ⎡2 2⎤ ⎥\n\ +⎢ ⎢ ⎥ ⎢ ⎥ ⎥\n\ +⎣ ⎣0 2⎦ ⎣1 1⎦ ⎦\ +""" + + assert upretty(ss1) == expected1 + assert upretty(ss2) == expected2 + assert upretty(ss3) == expected3 + +def test_pretty_order(): + expr = O(1) + ascii_str = \ +"""\ +O(1)\ +""" + ucode_str = \ +"""\ +O(1)\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = O(1/x) + ascii_str = \ +"""\ + /1\\\n\ +O|-|\n\ + \\x/\ +""" + ucode_str = \ +"""\ + ⎛1⎞\n\ +O⎜─⎟\n\ + ⎝x⎠\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = O(x**2 + y**2) + ascii_str = \ +"""\ + / 2 2 \\\n\ +O\\x + y ; (x, y) -> (0, 0)/\ +""" + ucode_str = \ +"""\ + ⎛ 2 2 ⎞\n\ +O⎝x + y ; (x, y) → (0, 0)⎠\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = O(1, (x, oo)) + ascii_str = \ +"""\ +O(1; x -> oo)\ +""" + ucode_str = \ +"""\ +O(1; x → ∞)\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = O(1/x, (x, oo)) + ascii_str = \ +"""\ + /1 \\\n\ +O|-; x -> oo|\n\ + \\x /\ +""" + ucode_str = \ +"""\ + ⎛1 ⎞\n\ +O⎜─; x → ∞⎟\n\ + ⎝x ⎠\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = O(x**2 + y**2, (x, oo), (y, oo)) + ascii_str = \ +"""\ + / 2 2 \\\n\ +O\\x + y ; (x, y) -> (oo, oo)/\ +""" + ucode_str = \ +"""\ + ⎛ 2 2 ⎞\n\ +O⎝x + y ; (x, y) → (∞, ∞)⎠\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + +def test_pretty_derivatives(): + # Simple + expr = Derivative(log(x), x, evaluate=False) + ascii_str = \ +"""\ +d \n\ +--(log(x))\n\ +dx \ +""" + ucode_str = \ +"""\ +d \n\ +──(log(x))\n\ +dx \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Derivative(log(x), x, evaluate=False) + x + ascii_str_1 = \ +"""\ + d \n\ +x + --(log(x))\n\ + dx \ +""" + ascii_str_2 = \ +"""\ +d \n\ +--(log(x)) + x\n\ +dx \ +""" + ucode_str_1 = \ +"""\ + d \n\ +x + ──(log(x))\n\ + dx \ +""" + ucode_str_2 = \ +"""\ +d \n\ +──(log(x)) + x\n\ +dx \ +""" + assert pretty(expr) in [ascii_str_1, ascii_str_2] + assert upretty(expr) in [ucode_str_1, ucode_str_2] + + # basic partial derivatives + expr = Derivative(log(x + y) + x, x) + ascii_str_1 = \ +"""\ +d \n\ +--(log(x + y) + x)\n\ +dx \ +""" + ascii_str_2 = \ +"""\ +d \n\ +--(x + log(x + y))\n\ +dx \ +""" + ucode_str_1 = \ +"""\ +∂ \n\ +──(log(x + y) + x)\n\ +∂x \ +""" + ucode_str_2 = \ +"""\ +∂ \n\ +──(x + log(x + y))\n\ +∂x \ +""" + assert pretty(expr) in [ascii_str_1, ascii_str_2] + assert upretty(expr) in [ucode_str_1, ucode_str_2], upretty(expr) + + # Multiple symbols + expr = Derivative(log(x) + x**2, x, y) + ascii_str_1 = \ +"""\ + 2 \n\ + d / 2\\\n\ +-----\\log(x) + x /\n\ +dy dx \ +""" + ascii_str_2 = \ +"""\ + 2 \n\ + d / 2 \\\n\ +-----\\x + log(x)/\n\ +dy dx \ +""" + ascii_str_3 = \ +"""\ + 2 \n\ + d / 2 \\\n\ +-----\\x + log(x)/\n\ +dy dx \ +""" + ucode_str_1 = \ +"""\ + 2 \n\ + d ⎛ 2⎞\n\ +─────⎝log(x) + x ⎠\n\ +dy dx \ +""" + ucode_str_2 = \ +"""\ + 2 \n\ + d ⎛ 2 ⎞\n\ +─────⎝x + log(x)⎠\n\ +dy dx \ +""" + ucode_str_3 = \ +"""\ + 2 \n\ + d ⎛ 2 ⎞\n\ +─────⎝x + log(x)⎠\n\ +dy dx \ +""" + assert pretty(expr) in [ascii_str_1, ascii_str_2, ascii_str_3] + assert upretty(expr) in [ucode_str_1, ucode_str_2, ucode_str_3] + + expr = Derivative(2*x*y, y, x) + x**2 + ascii_str_1 = \ +"""\ + 2 \n\ + d 2\n\ +-----(2*x*y) + x \n\ +dx dy \ +""" + ascii_str_2 = \ +"""\ + 2 \n\ + 2 d \n\ +x + -----(2*x*y)\n\ + dx dy \ +""" + ascii_str_3 = \ +"""\ + 2 \n\ + 2 d \n\ +x + -----(2*x*y)\n\ + dx dy \ +""" + ucode_str_1 = \ +"""\ + 2 \n\ + ∂ 2\n\ +─────(2⋅x⋅y) + x \n\ +∂x ∂y \ +""" + ucode_str_2 = \ +"""\ + 2 \n\ + 2 ∂ \n\ +x + ─────(2⋅x⋅y)\n\ + ∂x ∂y \ +""" + ucode_str_3 = \ +"""\ + 2 \n\ + 2 ∂ \n\ +x + ─────(2⋅x⋅y)\n\ + ∂x ∂y \ +""" + assert pretty(expr) in [ascii_str_1, ascii_str_2, ascii_str_3] + assert upretty(expr) in [ucode_str_1, ucode_str_2, ucode_str_3] + + expr = Derivative(2*x*y, x, x) + ascii_str = \ +"""\ + 2 \n\ +d \n\ +---(2*x*y)\n\ + 2 \n\ +dx \ +""" + ucode_str = \ +"""\ + 2 \n\ +∂ \n\ +───(2⋅x⋅y)\n\ + 2 \n\ +∂x \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Derivative(2*x*y, x, 17) + ascii_str = \ +"""\ + 17 \n\ +d \n\ +----(2*x*y)\n\ + 17 \n\ +dx \ +""" + ucode_str = \ +"""\ + 17 \n\ +∂ \n\ +────(2⋅x⋅y)\n\ + 17 \n\ +∂x \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Derivative(2*x*y, x, x, y) + ascii_str = \ +"""\ + 3 \n\ + d \n\ +------(2*x*y)\n\ + 2 \n\ +dy dx \ +""" + ucode_str = \ +"""\ + 3 \n\ + ∂ \n\ +──────(2⋅x⋅y)\n\ + 2 \n\ +∂y ∂x \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + # Greek letters + alpha = Symbol('alpha') + beta = Function('beta') + expr = beta(alpha).diff(alpha) + ascii_str = \ +"""\ + d \n\ +------(beta(alpha))\n\ +dalpha \ +""" + ucode_str = \ +"""\ +d \n\ +──(β(α))\n\ +dα \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Derivative(f(x), (x, n)) + + ascii_str = \ +"""\ + n \n\ +d \n\ +---(f(x))\n\ + n \n\ +dx \ +""" + ucode_str = \ +"""\ + n \n\ +d \n\ +───(f(x))\n\ + n \n\ +dx \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + +def test_pretty_integrals(): + expr = Integral(log(x), x) + ascii_str = \ +"""\ + / \n\ + | \n\ + | log(x) dx\n\ + | \n\ +/ \ +""" + ucode_str = \ +"""\ +⌠ \n\ +⎮ log(x) dx\n\ +⌡ \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Integral(x**2, x) + ascii_str = \ +"""\ + / \n\ + | \n\ + | 2 \n\ + | x dx\n\ + | \n\ +/ \ +""" + ucode_str = \ +"""\ +⌠ \n\ +⎮ 2 \n\ +⎮ x dx\n\ +⌡ \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Integral((sin(x))**2 / (tan(x))**2) + ascii_str = \ +"""\ + / \n\ + | \n\ + | 2 \n\ + | sin (x) \n\ + | ------- dx\n\ + | 2 \n\ + | tan (x) \n\ + | \n\ +/ \ +""" + ucode_str = \ +"""\ +⌠ \n\ +⎮ 2 \n\ +⎮ sin (x) \n\ +⎮ ─────── dx\n\ +⎮ 2 \n\ +⎮ tan (x) \n\ +⌡ \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Integral(x**(2**x), x) + ascii_str = \ +"""\ + / \n\ + | \n\ + | / x\\ \n\ + | \\2 / \n\ + | x dx\n\ + | \n\ +/ \ +""" + ucode_str = \ +"""\ +⌠ \n\ +⎮ ⎛ x⎞ \n\ +⎮ ⎝2 ⎠ \n\ +⎮ x dx\n\ +⌡ \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Integral(x**2, (x, 1, 2)) + ascii_str = \ +"""\ + 2 \n\ + / \n\ + | \n\ + | 2 \n\ + | x dx\n\ + | \n\ +/ \n\ +1 \ +""" + ucode_str = \ +"""\ +2 \n\ +⌠ \n\ +⎮ 2 \n\ +⎮ x dx\n\ +⌡ \n\ +1 \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Integral(x**2, (x, Rational(1, 2), 10)) + ascii_str = \ +"""\ + 10 \n\ + / \n\ + | \n\ + | 2 \n\ + | x dx\n\ + | \n\ +/ \n\ +1/2 \ +""" + ucode_str = \ +"""\ +10 \n\ +⌠ \n\ +⎮ 2 \n\ +⎮ x dx\n\ +⌡ \n\ +1/2 \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Integral(x**2*y**2, x, y) + ascii_str = \ +"""\ + / / \n\ + | | \n\ + | | 2 2 \n\ + | | x *y dx dy\n\ + | | \n\ +/ / \ +""" + ucode_str = \ +"""\ +⌠ ⌠ \n\ +⎮ ⎮ 2 2 \n\ +⎮ ⎮ x ⋅y dx dy\n\ +⌡ ⌡ \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Integral(sin(th)/cos(ph), (th, 0, pi), (ph, 0, 2*pi)) + ascii_str = \ +"""\ + 2*pi pi \n\ + / / \n\ + | | \n\ + | | sin(theta) \n\ + | | ---------- d(theta) d(phi)\n\ + | | cos(phi) \n\ + | | \n\ + / / \n\ +0 0 \ +""" + ucode_str = \ +"""\ +2⋅π π \n\ + ⌠ ⌠ \n\ + ⎮ ⎮ sin(θ) \n\ + ⎮ ⎮ ────── dθ dφ\n\ + ⎮ ⎮ cos(φ) \n\ + ⌡ ⌡ \n\ + 0 0 \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + +def test_pretty_matrix(): + # Empty Matrix + expr = Matrix() + ascii_str = "[]" + unicode_str = "[]" + assert pretty(expr) == ascii_str + assert upretty(expr) == unicode_str + expr = Matrix(2, 0, lambda i, j: 0) + ascii_str = "[]" + unicode_str = "[]" + assert pretty(expr) == ascii_str + assert upretty(expr) == unicode_str + expr = Matrix(0, 2, lambda i, j: 0) + ascii_str = "[]" + unicode_str = "[]" + assert pretty(expr) == ascii_str + assert upretty(expr) == unicode_str + expr = Matrix([[x**2 + 1, 1], [y, x + y]]) + ascii_str_1 = \ +"""\ +[ 2 ] +[1 + x 1 ] +[ ] +[ y x + y]\ +""" + ascii_str_2 = \ +"""\ +[ 2 ] +[x + 1 1 ] +[ ] +[ y x + y]\ +""" + ucode_str_1 = \ +"""\ +⎡ 2 ⎤ +⎢1 + x 1 ⎥ +⎢ ⎥ +⎣ y x + y⎦\ +""" + ucode_str_2 = \ +"""\ +⎡ 2 ⎤ +⎢x + 1 1 ⎥ +⎢ ⎥ +⎣ y x + y⎦\ +""" + assert pretty(expr) in [ascii_str_1, ascii_str_2] + assert upretty(expr) in [ucode_str_1, ucode_str_2] + + expr = Matrix([[x/y, y, th], [0, exp(I*k*ph), 1]]) + ascii_str = \ +"""\ +[x ] +[- y theta] +[y ] +[ ] +[ I*k*phi ] +[0 e 1 ]\ +""" + ucode_str = \ +"""\ +⎡x ⎤ +⎢─ y θ⎥ +⎢y ⎥ +⎢ ⎥ +⎢ ⅈ⋅k⋅φ ⎥ +⎣0 ℯ 1⎦\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + unicode_str = \ +"""\ +⎡v̇_msc_00 0 0 ⎤ +⎢ ⎥ +⎢ 0 v̇_msc_01 0 ⎥ +⎢ ⎥ +⎣ 0 0 v̇_msc_02⎦\ +""" + + expr = diag(*MatrixSymbol('vdot_msc',1,3)) + assert upretty(expr) == unicode_str + + +def test_pretty_ndim_arrays(): + x, y, z, w = symbols("x y z w") + + for ArrayType in (ImmutableDenseNDimArray, ImmutableSparseNDimArray, MutableDenseNDimArray, MutableSparseNDimArray): + # Basic: scalar array + M = ArrayType(x) + + assert pretty(M) == "x" + assert upretty(M) == "x" + + M = ArrayType([[1/x, y], [z, w]]) + M1 = ArrayType([1/x, y, z]) + + M2 = tensorproduct(M1, M) + M3 = tensorproduct(M, M) + + ascii_str = \ +"""\ +[1 ]\n\ +[- y]\n\ +[x ]\n\ +[ ]\n\ +[z w]\ +""" + ucode_str = \ +"""\ +⎡1 ⎤\n\ +⎢─ y⎥\n\ +⎢x ⎥\n\ +⎢ ⎥\n\ +⎣z w⎦\ +""" + assert pretty(M) == ascii_str + assert upretty(M) == ucode_str + + ascii_str = \ +"""\ +[1 ]\n\ +[- y z]\n\ +[x ]\ +""" + ucode_str = \ +"""\ +⎡1 ⎤\n\ +⎢─ y z⎥\n\ +⎣x ⎦\ +""" + assert pretty(M1) == ascii_str + assert upretty(M1) == ucode_str + + ascii_str = \ +"""\ +[[1 y] ]\n\ +[[-- -] [z ]]\n\ +[[ 2 x] [ y 2 ] [- y*z]]\n\ +[[x ] [ - y ] [x ]]\n\ +[[ ] [ x ] [ ]]\n\ +[[z w] [ ] [ 2 ]]\n\ +[[- -] [y*z w*y] [z w*z]]\n\ +[[x x] ]\ +""" + ucode_str = \ +"""\ +⎡⎡1 y⎤ ⎤\n\ +⎢⎢── ─⎥ ⎡z ⎤⎥\n\ +⎢⎢ 2 x⎥ ⎡ y 2 ⎤ ⎢─ y⋅z⎥⎥\n\ +⎢⎢x ⎥ ⎢ ─ y ⎥ ⎢x ⎥⎥\n\ +⎢⎢ ⎥ ⎢ x ⎥ ⎢ ⎥⎥\n\ +⎢⎢z w⎥ ⎢ ⎥ ⎢ 2 ⎥⎥\n\ +⎢⎢─ ─⎥ ⎣y⋅z w⋅y⎦ ⎣z w⋅z⎦⎥\n\ +⎣⎣x x⎦ ⎦\ +""" + assert pretty(M2) == ascii_str + assert upretty(M2) == ucode_str + + ascii_str = \ +"""\ +[ [1 y] ]\n\ +[ [-- -] ]\n\ +[ [ 2 x] [ y 2 ]]\n\ +[ [x ] [ - y ]]\n\ +[ [ ] [ x ]]\n\ +[ [z w] [ ]]\n\ +[ [- -] [y*z w*y]]\n\ +[ [x x] ]\n\ +[ ]\n\ +[[z ] [ w ]]\n\ +[[- y*z] [ - w*y]]\n\ +[[x ] [ x ]]\n\ +[[ ] [ ]]\n\ +[[ 2 ] [ 2 ]]\n\ +[[z w*z] [w*z w ]]\ +""" + ucode_str = \ +"""\ +⎡ ⎡1 y⎤ ⎤\n\ +⎢ ⎢── ─⎥ ⎥\n\ +⎢ ⎢ 2 x⎥ ⎡ y 2 ⎤⎥\n\ +⎢ ⎢x ⎥ ⎢ ─ y ⎥⎥\n\ +⎢ ⎢ ⎥ ⎢ x ⎥⎥\n\ +⎢ ⎢z w⎥ ⎢ ⎥⎥\n\ +⎢ ⎢─ ─⎥ ⎣y⋅z w⋅y⎦⎥\n\ +⎢ ⎣x x⎦ ⎥\n\ +⎢ ⎥\n\ +⎢⎡z ⎤ ⎡ w ⎤⎥\n\ +⎢⎢─ y⋅z⎥ ⎢ ─ w⋅y⎥⎥\n\ +⎢⎢x ⎥ ⎢ x ⎥⎥\n\ +⎢⎢ ⎥ ⎢ ⎥⎥\n\ +⎢⎢ 2 ⎥ ⎢ 2 ⎥⎥\n\ +⎣⎣z w⋅z⎦ ⎣w⋅z w ⎦⎦\ +""" + assert pretty(M3) == ascii_str + assert upretty(M3) == ucode_str + + Mrow = ArrayType([[x, y, 1 / z]]) + Mcolumn = ArrayType([[x], [y], [1 / z]]) + Mcol2 = ArrayType([Mcolumn.tolist()]) + + ascii_str = \ +"""\ +[[ 1]]\n\ +[[x y -]]\n\ +[[ z]]\ +""" + ucode_str = \ +"""\ +⎡⎡ 1⎤⎤\n\ +⎢⎢x y ─⎥⎥\n\ +⎣⎣ z⎦⎦\ +""" + assert pretty(Mrow) == ascii_str + assert upretty(Mrow) == ucode_str + + ascii_str = \ +"""\ +[x]\n\ +[ ]\n\ +[y]\n\ +[ ]\n\ +[1]\n\ +[-]\n\ +[z]\ +""" + ucode_str = \ +"""\ +⎡x⎤\n\ +⎢ ⎥\n\ +⎢y⎥\n\ +⎢ ⎥\n\ +⎢1⎥\n\ +⎢─⎥\n\ +⎣z⎦\ +""" + assert pretty(Mcolumn) == ascii_str + assert upretty(Mcolumn) == ucode_str + + ascii_str = \ +"""\ +[[x]]\n\ +[[ ]]\n\ +[[y]]\n\ +[[ ]]\n\ +[[1]]\n\ +[[-]]\n\ +[[z]]\ +""" + ucode_str = \ +"""\ +⎡⎡x⎤⎤\n\ +⎢⎢ ⎥⎥\n\ +⎢⎢y⎥⎥\n\ +⎢⎢ ⎥⎥\n\ +⎢⎢1⎥⎥\n\ +⎢⎢─⎥⎥\n\ +⎣⎣z⎦⎦\ +""" + assert pretty(Mcol2) == ascii_str + assert upretty(Mcol2) == ucode_str + + +def test_tensor_TensorProduct(): + A = MatrixSymbol("A", 3, 3) + B = MatrixSymbol("B", 3, 3) + assert upretty(TensorProduct(A, B)) == "A\u2297B" + assert upretty(TensorProduct(A, B, A)) == "A\u2297B\u2297A" + + +def test_diffgeom_print_WedgeProduct(): + from sympy.diffgeom.rn import R2 + from sympy.diffgeom import WedgeProduct + wp = WedgeProduct(R2.dx, R2.dy) + assert upretty(wp) == "ⅆ x∧ⅆ y" + assert pretty(wp) == r"d x/\d y" + + +def test_Adjoint(): + X = MatrixSymbol('X', 2, 2) + Y = MatrixSymbol('Y', 2, 2) + assert pretty(Adjoint(X)) == " +\nX " + assert pretty(Adjoint(X + Y)) == " +\n(X + Y) " + assert pretty(Adjoint(X) + Adjoint(Y)) == " + +\nX + Y " + assert pretty(Adjoint(X*Y)) == " +\n(X*Y) " + assert pretty(Adjoint(Y)*Adjoint(X)) == " + +\nY *X " + assert pretty(Adjoint(X**2)) == " +\n/ 2\\ \n\\X / " + assert pretty(Adjoint(X)**2) == " 2\n/ +\\ \n\\X / " + assert pretty(Adjoint(Inverse(X))) == " +\n/ -1\\ \n\\X / " + assert pretty(Inverse(Adjoint(X))) == " -1\n/ +\\ \n\\X / " + assert pretty(Adjoint(Transpose(X))) == " +\n/ T\\ \n\\X / " + assert pretty(Transpose(Adjoint(X))) == " T\n/ +\\ \n\\X / " + assert upretty(Adjoint(X)) == " †\nX " + assert upretty(Adjoint(X + Y)) == " †\n(X + Y) " + assert upretty(Adjoint(X) + Adjoint(Y)) == " † †\nX + Y " + assert upretty(Adjoint(X*Y)) == " †\n(X⋅Y) " + assert upretty(Adjoint(Y)*Adjoint(X)) == " † †\nY ⋅X " + assert upretty(Adjoint(X**2)) == \ + " †\n⎛ 2⎞ \n⎝X ⎠ " + assert upretty(Adjoint(X)**2) == \ + " 2\n⎛ †⎞ \n⎝X ⎠ " + assert upretty(Adjoint(Inverse(X))) == \ + " †\n⎛ -1⎞ \n⎝X ⎠ " + assert upretty(Inverse(Adjoint(X))) == \ + " -1\n⎛ †⎞ \n⎝X ⎠ " + assert upretty(Adjoint(Transpose(X))) == \ + " †\n⎛ T⎞ \n⎝X ⎠ " + assert upretty(Transpose(Adjoint(X))) == \ + " T\n⎛ †⎞ \n⎝X ⎠ " + m = Matrix(((1, 2), (3, 4))) + assert upretty(Adjoint(m)) == \ + ' †\n'\ + '⎡1 2⎤ \n'\ + '⎢ ⎥ \n'\ + '⎣3 4⎦ ' + assert upretty(Adjoint(m+X)) == \ + ' †\n'\ + '⎛⎡1 2⎤ ⎞ \n'\ + '⎜⎢ ⎥ + X⎟ \n'\ + '⎝⎣3 4⎦ ⎠ ' + assert upretty(Adjoint(BlockMatrix(((OneMatrix(2, 2), X), + (m, ZeroMatrix(2, 2)))))) == \ + ' †\n'\ + '⎡ 𝟙 X⎤ \n'\ + '⎢ ⎥ \n'\ + '⎢⎡1 2⎤ ⎥ \n'\ + '⎢⎢ ⎥ 𝟘⎥ \n'\ + '⎣⎣3 4⎦ ⎦ ' + + +def test_Transpose(): + X = MatrixSymbol('X', 2, 2) + Y = MatrixSymbol('Y', 2, 2) + assert pretty(Transpose(X)) == " T\nX " + assert pretty(Transpose(X + Y)) == " T\n(X + Y) " + assert pretty(Transpose(X) + Transpose(Y)) == " T T\nX + Y " + assert pretty(Transpose(X*Y)) == " T\n(X*Y) " + assert pretty(Transpose(Y)*Transpose(X)) == " T T\nY *X " + assert pretty(Transpose(X**2)) == " T\n/ 2\\ \n\\X / " + assert pretty(Transpose(X)**2) == " 2\n/ T\\ \n\\X / " + assert pretty(Transpose(Inverse(X))) == " T\n/ -1\\ \n\\X / " + assert pretty(Inverse(Transpose(X))) == " -1\n/ T\\ \n\\X / " + assert upretty(Transpose(X)) == " T\nX " + assert upretty(Transpose(X + Y)) == " T\n(X + Y) " + assert upretty(Transpose(X) + Transpose(Y)) == " T T\nX + Y " + assert upretty(Transpose(X*Y)) == " T\n(X⋅Y) " + assert upretty(Transpose(Y)*Transpose(X)) == " T T\nY ⋅X " + assert upretty(Transpose(X**2)) == \ + " T\n⎛ 2⎞ \n⎝X ⎠ " + assert upretty(Transpose(X)**2) == \ + " 2\n⎛ T⎞ \n⎝X ⎠ " + assert upretty(Transpose(Inverse(X))) == \ + " T\n⎛ -1⎞ \n⎝X ⎠ " + assert upretty(Inverse(Transpose(X))) == \ + " -1\n⎛ T⎞ \n⎝X ⎠ " + m = Matrix(((1, 2), (3, 4))) + assert upretty(Transpose(m)) == \ + ' T\n'\ + '⎡1 2⎤ \n'\ + '⎢ ⎥ \n'\ + '⎣3 4⎦ ' + assert upretty(Transpose(m+X)) == \ + ' T\n'\ + '⎛⎡1 2⎤ ⎞ \n'\ + '⎜⎢ ⎥ + X⎟ \n'\ + '⎝⎣3 4⎦ ⎠ ' + assert upretty(Transpose(BlockMatrix(((OneMatrix(2, 2), X), + (m, ZeroMatrix(2, 2)))))) == \ + ' T\n'\ + '⎡ 𝟙 X⎤ \n'\ + '⎢ ⎥ \n'\ + '⎢⎡1 2⎤ ⎥ \n'\ + '⎢⎢ ⎥ 𝟘⎥ \n'\ + '⎣⎣3 4⎦ ⎦ ' + + +def test_pretty_Trace_issue_9044(): + X = Matrix([[1, 2], [3, 4]]) + Y = Matrix([[2, 4], [6, 8]]) + ascii_str_1 = \ +"""\ + /[1 2]\\ +tr|[ ]| + \\[3 4]/\ +""" + ucode_str_1 = \ +"""\ + ⎛⎡1 2⎤⎞ +tr⎜⎢ ⎥⎟ + ⎝⎣3 4⎦⎠\ +""" + ascii_str_2 = \ +"""\ + /[1 2]\\ /[2 4]\\ +tr|[ ]| + tr|[ ]| + \\[3 4]/ \\[6 8]/\ +""" + ucode_str_2 = \ +"""\ + ⎛⎡1 2⎤⎞ ⎛⎡2 4⎤⎞ +tr⎜⎢ ⎥⎟ + tr⎜⎢ ⎥⎟ + ⎝⎣3 4⎦⎠ ⎝⎣6 8⎦⎠\ +""" + assert pretty(Trace(X)) == ascii_str_1 + assert upretty(Trace(X)) == ucode_str_1 + + assert pretty(Trace(X) + Trace(Y)) == ascii_str_2 + assert upretty(Trace(X) + Trace(Y)) == ucode_str_2 + + +def test_MatrixSlice(): + n = Symbol('n', integer=True) + x, y, z, w, t, = symbols('x y z w t') + X = MatrixSymbol('X', n, n) + Y = MatrixSymbol('Y', 10, 10) + Z = MatrixSymbol('Z', 10, 10) + + expr = MatrixSlice(X, (None, None, None), (None, None, None)) + assert pretty(expr) == upretty(expr) == 'X[:, :]' + expr = X[x:x + 1, y:y + 1] + assert pretty(expr) == upretty(expr) == 'X[x:x + 1, y:y + 1]' + expr = X[x:x + 1:2, y:y + 1:2] + assert pretty(expr) == upretty(expr) == 'X[x:x + 1:2, y:y + 1:2]' + expr = X[:x, y:] + assert pretty(expr) == upretty(expr) == 'X[:x, y:]' + expr = X[:x, y:] + assert pretty(expr) == upretty(expr) == 'X[:x, y:]' + expr = X[x:, :y] + assert pretty(expr) == upretty(expr) == 'X[x:, :y]' + expr = X[x:y, z:w] + assert pretty(expr) == upretty(expr) == 'X[x:y, z:w]' + expr = X[x:y:t, w:t:x] + assert pretty(expr) == upretty(expr) == 'X[x:y:t, w:t:x]' + expr = X[x::y, t::w] + assert pretty(expr) == upretty(expr) == 'X[x::y, t::w]' + expr = X[:x:y, :t:w] + assert pretty(expr) == upretty(expr) == 'X[:x:y, :t:w]' + expr = X[::x, ::y] + assert pretty(expr) == upretty(expr) == 'X[::x, ::y]' + expr = MatrixSlice(X, (0, None, None), (0, None, None)) + assert pretty(expr) == upretty(expr) == 'X[:, :]' + expr = MatrixSlice(X, (None, n, None), (None, n, None)) + assert pretty(expr) == upretty(expr) == 'X[:, :]' + expr = MatrixSlice(X, (0, n, None), (0, n, None)) + assert pretty(expr) == upretty(expr) == 'X[:, :]' + expr = MatrixSlice(X, (0, n, 2), (0, n, 2)) + assert pretty(expr) == upretty(expr) == 'X[::2, ::2]' + expr = X[1:2:3, 4:5:6] + assert pretty(expr) == upretty(expr) == 'X[1:2:3, 4:5:6]' + expr = X[1:3:5, 4:6:8] + assert pretty(expr) == upretty(expr) == 'X[1:3:5, 4:6:8]' + expr = X[1:10:2] + assert pretty(expr) == upretty(expr) == 'X[1:10:2, :]' + expr = Y[:5, 1:9:2] + assert pretty(expr) == upretty(expr) == 'Y[:5, 1:9:2]' + expr = Y[:5, 1:10:2] + assert pretty(expr) == upretty(expr) == 'Y[:5, 1::2]' + expr = Y[5, :5:2] + assert pretty(expr) == upretty(expr) == 'Y[5:6, :5:2]' + expr = X[0:1, 0:1] + assert pretty(expr) == upretty(expr) == 'X[:1, :1]' + expr = X[0:1:2, 0:1:2] + assert pretty(expr) == upretty(expr) == 'X[:1:2, :1:2]' + expr = (Y + Z)[2:, 2:] + assert pretty(expr) == upretty(expr) == '(Y + Z)[2:, 2:]' + + +def test_MatrixExpressions(): + n = Symbol('n', integer=True) + X = MatrixSymbol('X', n, n) + + assert pretty(X) == upretty(X) == "X" + + # Apply function elementwise (`ElementwiseApplyFunc`): + + expr = (X.T*X).applyfunc(sin) + + ascii_str = """\ + / T \\\n\ +(d -> sin(d)).\\X *X/\ +""" + ucode_str = """\ + ⎛ T ⎞\n\ +(d ↦ sin(d))˳⎝X ⋅X⎠\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + lamda = Lambda(x, 1/x) + expr = (n*X).applyfunc(lamda) + ascii_str = """\ +/ 1\\ \n\ +|x -> -|.(n*X)\n\ +\\ x/ \ +""" + ucode_str = """\ +⎛ 1⎞ \n\ +⎜x ↦ ─⎟˳(n⋅X)\n\ +⎝ x⎠ \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + +def test_pretty_dotproduct(): + from sympy.matrices.expressions.dotproduct import DotProduct + n = symbols("n", integer=True) + A = MatrixSymbol('A', n, 1) + B = MatrixSymbol('B', n, 1) + C = Matrix(1, 3, [1, 2, 3]) + D = Matrix(1, 3, [1, 3, 4]) + + assert pretty(DotProduct(A, B)) == "A*B" + assert pretty(DotProduct(C, D)) == "[1 2 3]*[1 3 4]" + assert upretty(DotProduct(A, B)) == "A⋅B" + assert upretty(DotProduct(C, D)) == "[1 2 3]⋅[1 3 4]" + + +def test_pretty_Determinant(): + from sympy.matrices import Determinant, Inverse, BlockMatrix, OneMatrix, ZeroMatrix + m = Matrix(((1, 2), (3, 4))) + assert upretty(Determinant(m)) == '│1 2│\n│ │\n│3 4│' + assert upretty(Determinant(Inverse(m))) == \ + '│ -1│\n'\ + '│⎡1 2⎤ │\n'\ + '│⎢ ⎥ │\n'\ + '│⎣3 4⎦ │' + X = MatrixSymbol('X', 2, 2) + assert upretty(Determinant(X)) == '│X│' + assert upretty(Determinant(X + m)) == \ + '│⎡1 2⎤ │\n'\ + '│⎢ ⎥ + X│\n'\ + '│⎣3 4⎦ │' + assert upretty(Determinant(BlockMatrix(((OneMatrix(2, 2), X), + (m, ZeroMatrix(2, 2)))))) == \ + '│ 𝟙 X│\n'\ + '│ │\n'\ + '│⎡1 2⎤ │\n'\ + '│⎢ ⎥ 𝟘│\n'\ + '│⎣3 4⎦ │' + + +def test_pretty_piecewise(): + expr = Piecewise((x, x < 1), (x**2, True)) + ascii_str = \ +"""\ +/x for x < 1\n\ +| \n\ +< 2 \n\ +|x otherwise\n\ +\\ \ +""" + ucode_str = \ +"""\ +⎧x for x < 1\n\ +⎪ \n\ +⎨ 2 \n\ +⎪x otherwise\n\ +⎩ \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = -Piecewise((x, x < 1), (x**2, True)) + ascii_str = \ +"""\ + //x for x < 1\\\n\ + || |\n\ +-|< 2 |\n\ + ||x otherwise|\n\ + \\\\ /\ +""" + ucode_str = \ +"""\ + ⎛⎧x for x < 1⎞\n\ + ⎜⎪ ⎟\n\ +-⎜⎨ 2 ⎟\n\ + ⎜⎪x otherwise⎟\n\ + ⎝⎩ ⎠\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = x + Piecewise((x, x > 0), (y, True)) + Piecewise((x/y, x < 2), + (y**2, x > 2), (1, True)) + 1 + ascii_str = \ +"""\ + //x \\ \n\ + ||- for x < 2| \n\ + ||y | \n\ + //x for x > 0\\ || | \n\ +x + |< | + |< 2 | + 1\n\ + \\\\y otherwise/ ||y for x > 2| \n\ + || | \n\ + ||1 otherwise| \n\ + \\\\ / \ +""" + ucode_str = \ +"""\ + ⎛⎧x ⎞ \n\ + ⎜⎪─ for x < 2⎟ \n\ + ⎜⎪y ⎟ \n\ + ⎛⎧x for x > 0⎞ ⎜⎪ ⎟ \n\ +x + ⎜⎨ ⎟ + ⎜⎨ 2 ⎟ + 1\n\ + ⎝⎩y otherwise⎠ ⎜⎪y for x > 2⎟ \n\ + ⎜⎪ ⎟ \n\ + ⎜⎪1 otherwise⎟ \n\ + ⎝⎩ ⎠ \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = x - Piecewise((x, x > 0), (y, True)) + Piecewise((x/y, x < 2), + (y**2, x > 2), (1, True)) + 1 + ascii_str = \ +"""\ + //x \\ \n\ + ||- for x < 2| \n\ + ||y | \n\ + //x for x > 0\\ || | \n\ +x - |< | + |< 2 | + 1\n\ + \\\\y otherwise/ ||y for x > 2| \n\ + || | \n\ + ||1 otherwise| \n\ + \\\\ / \ +""" + ucode_str = \ +"""\ + ⎛⎧x ⎞ \n\ + ⎜⎪─ for x < 2⎟ \n\ + ⎜⎪y ⎟ \n\ + ⎛⎧x for x > 0⎞ ⎜⎪ ⎟ \n\ +x - ⎜⎨ ⎟ + ⎜⎨ 2 ⎟ + 1\n\ + ⎝⎩y otherwise⎠ ⎜⎪y for x > 2⎟ \n\ + ⎜⎪ ⎟ \n\ + ⎜⎪1 otherwise⎟ \n\ + ⎝⎩ ⎠ \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = x*Piecewise((x, x > 0), (y, True)) + ascii_str = \ +"""\ + //x for x > 0\\\n\ +x*|< |\n\ + \\\\y otherwise/\ +""" + ucode_str = \ +"""\ + ⎛⎧x for x > 0⎞\n\ +x⋅⎜⎨ ⎟\n\ + ⎝⎩y otherwise⎠\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Piecewise((x, x > 0), (y, True))*Piecewise((x/y, x < 2), (y**2, x > + 2), (1, True)) + ascii_str = \ +"""\ + //x \\\n\ + ||- for x < 2|\n\ + ||y |\n\ +//x for x > 0\\ || |\n\ +|< |*|< 2 |\n\ +\\\\y otherwise/ ||y for x > 2|\n\ + || |\n\ + ||1 otherwise|\n\ + \\\\ /\ +""" + ucode_str = \ +"""\ + ⎛⎧x ⎞\n\ + ⎜⎪─ for x < 2⎟\n\ + ⎜⎪y ⎟\n\ +⎛⎧x for x > 0⎞ ⎜⎪ ⎟\n\ +⎜⎨ ⎟⋅⎜⎨ 2 ⎟\n\ +⎝⎩y otherwise⎠ ⎜⎪y for x > 2⎟\n\ + ⎜⎪ ⎟\n\ + ⎜⎪1 otherwise⎟\n\ + ⎝⎩ ⎠\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = -Piecewise((x, x > 0), (y, True))*Piecewise((x/y, x < 2), (y**2, x + > 2), (1, True)) + ascii_str = \ +"""\ + //x \\\n\ + ||- for x < 2|\n\ + ||y |\n\ + //x for x > 0\\ || |\n\ +-|< |*|< 2 |\n\ + \\\\y otherwise/ ||y for x > 2|\n\ + || |\n\ + ||1 otherwise|\n\ + \\\\ /\ +""" + ucode_str = \ +"""\ + ⎛⎧x ⎞\n\ + ⎜⎪─ for x < 2⎟\n\ + ⎜⎪y ⎟\n\ + ⎛⎧x for x > 0⎞ ⎜⎪ ⎟\n\ +-⎜⎨ ⎟⋅⎜⎨ 2 ⎟\n\ + ⎝⎩y otherwise⎠ ⎜⎪y for x > 2⎟\n\ + ⎜⎪ ⎟\n\ + ⎜⎪1 otherwise⎟\n\ + ⎝⎩ ⎠\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Piecewise((0, Abs(1/y) < 1), (1, Abs(y) < 1), (y*meijerg(((2, 1), + ()), ((), (1, 0)), 1/y), True)) + ascii_str = \ +"""\ +/ 1 \n\ +| 0 for --- < 1\n\ +| |y| \n\ +| \n\ +< 1 for |y| < 1\n\ +| \n\ +| __0, 2 /1, 2 | 1\\ \n\ +|y*/__ | | -| otherwise \n\ +\\ \\_|2, 2 \\ 0, 1 | y/ \ +""" + ucode_str = \ +"""\ +⎧ 1 \n\ +⎪ 0 for ─── < 1\n\ +⎪ │y│ \n\ +⎪ \n\ +⎨ 1 for │y│ < 1\n\ +⎪ \n\ +⎪ ╭─╮0, 2 ⎛1, 2 │ 1⎞ \n\ +⎪y⋅│╶┐ ⎜ │ ─⎟ otherwise \n\ +⎩ ╰─╯2, 2 ⎝ 0, 1 │ y⎠ \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + # XXX: We have to use evaluate=False here because Piecewise._eval_power + # denests the power. + expr = Pow(Piecewise((x, x > 0), (y, True)), 2, evaluate=False) + ascii_str = \ +"""\ + 2\n\ +//x for x > 0\\ \n\ +|< | \n\ +\\\\y otherwise/ \ +""" + ucode_str = \ +"""\ + 2\n\ +⎛⎧x for x > 0⎞ \n\ +⎜⎨ ⎟ \n\ +⎝⎩y otherwise⎠ \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + +def test_pretty_ITE(): + expr = ITE(x, y, z) + assert pretty(expr) == ( + '/y for x \n' + '< \n' + '\\z otherwise' + ) + assert upretty(expr) == """\ +⎧y for x \n\ +⎨ \n\ +⎩z otherwise\ +""" + + +def test_pretty_seq(): + expr = () + ascii_str = \ +"""\ +()\ +""" + ucode_str = \ +"""\ +()\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = [] + ascii_str = \ +"""\ +[]\ +""" + ucode_str = \ +"""\ +[]\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = {} + expr_2 = {} + ascii_str = \ +"""\ +{}\ +""" + ucode_str = \ +"""\ +{}\ +""" + assert pretty(expr) == ascii_str + assert pretty(expr_2) == ascii_str + assert upretty(expr) == ucode_str + assert upretty(expr_2) == ucode_str + + expr = (1/x,) + ascii_str = \ +"""\ + 1 \n\ +(-,)\n\ + x \ +""" + ucode_str = \ +"""\ +⎛1 ⎞\n\ +⎜─,⎟\n\ +⎝x ⎠\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = [x**2, 1/x, x, y, sin(th)**2/cos(ph)**2] + ascii_str = \ +"""\ + 2 \n\ + 2 1 sin (theta) \n\ +[x , -, x, y, -----------]\n\ + x 2 \n\ + cos (phi) \ +""" + ucode_str = \ +"""\ +⎡ 2 ⎤\n\ +⎢ 2 1 sin (θ)⎥\n\ +⎢x , ─, x, y, ───────⎥\n\ +⎢ x 2 ⎥\n\ +⎣ cos (φ)⎦\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = (x**2, 1/x, x, y, sin(th)**2/cos(ph)**2) + ascii_str = \ +"""\ + 2 \n\ + 2 1 sin (theta) \n\ +(x , -, x, y, -----------)\n\ + x 2 \n\ + cos (phi) \ +""" + ucode_str = \ +"""\ +⎛ 2 ⎞\n\ +⎜ 2 1 sin (θ)⎟\n\ +⎜x , ─, x, y, ───────⎟\n\ +⎜ x 2 ⎟\n\ +⎝ cos (φ)⎠\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Tuple(x**2, 1/x, x, y, sin(th)**2/cos(ph)**2) + ascii_str = \ +"""\ + 2 \n\ + 2 1 sin (theta) \n\ +(x , -, x, y, -----------)\n\ + x 2 \n\ + cos (phi) \ +""" + ucode_str = \ +"""\ +⎛ 2 ⎞\n\ +⎜ 2 1 sin (θ)⎟\n\ +⎜x , ─, x, y, ───────⎟\n\ +⎜ x 2 ⎟\n\ +⎝ cos (φ)⎠\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = {x: sin(x)} + expr_2 = Dict({x: sin(x)}) + ascii_str = \ +"""\ +{x: sin(x)}\ +""" + ucode_str = \ +"""\ +{x: sin(x)}\ +""" + assert pretty(expr) == ascii_str + assert pretty(expr_2) == ascii_str + assert upretty(expr) == ucode_str + assert upretty(expr_2) == ucode_str + + expr = {1/x: 1/y, x: sin(x)**2} + expr_2 = Dict({1/x: 1/y, x: sin(x)**2}) + ascii_str = \ +"""\ + 1 1 2 \n\ +{-: -, x: sin (x)}\n\ + x y \ +""" + ucode_str = \ +"""\ +⎧1 1 2 ⎫\n\ +⎨─: ─, x: sin (x)⎬\n\ +⎩x y ⎭\ +""" + assert pretty(expr) == ascii_str + assert pretty(expr_2) == ascii_str + assert upretty(expr) == ucode_str + assert upretty(expr_2) == ucode_str + + # There used to be a bug with pretty-printing sequences of even height. + expr = [x**2] + ascii_str = \ +"""\ + 2 \n\ +[x ]\ +""" + ucode_str = \ +"""\ +⎡ 2⎤\n\ +⎣x ⎦\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = (x**2,) + ascii_str = \ +"""\ + 2 \n\ +(x ,)\ +""" + ucode_str = \ +"""\ +⎛ 2 ⎞\n\ +⎝x ,⎠\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Tuple(x**2) + ascii_str = \ +"""\ + 2 \n\ +(x ,)\ +""" + ucode_str = \ +"""\ +⎛ 2 ⎞\n\ +⎝x ,⎠\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = {x**2: 1} + expr_2 = Dict({x**2: 1}) + ascii_str = \ +"""\ + 2 \n\ +{x : 1}\ +""" + ucode_str = \ +"""\ +⎧ 2 ⎫\n\ +⎨x : 1⎬\n\ +⎩ ⎭\ +""" + assert pretty(expr) == ascii_str + assert pretty(expr_2) == ascii_str + assert upretty(expr) == ucode_str + assert upretty(expr_2) == ucode_str + + +def test_any_object_in_sequence(): + # Cf. issue 5306 + b1 = Basic() + b2 = Basic(Basic()) + + expr = [b2, b1] + assert pretty(expr) == "[Basic(Basic()), Basic()]" + assert upretty(expr) == "[Basic(Basic()), Basic()]" + + expr = {b2, b1} + assert pretty(expr) == "{Basic(), Basic(Basic())}" + assert upretty(expr) == "{Basic(), Basic(Basic())}" + + expr = {b2: b1, b1: b2} + expr2 = Dict({b2: b1, b1: b2}) + assert pretty(expr) == "{Basic(): Basic(Basic()), Basic(Basic()): Basic()}" + assert pretty( + expr2) == "{Basic(): Basic(Basic()), Basic(Basic()): Basic()}" + assert upretty( + expr) == "{Basic(): Basic(Basic()), Basic(Basic()): Basic()}" + assert upretty( + expr2) == "{Basic(): Basic(Basic()), Basic(Basic()): Basic()}" + + +def test_print_builtin_set(): + assert pretty(set()) == 'set()' + assert upretty(set()) == 'set()' + + assert pretty(frozenset()) == 'frozenset()' + assert upretty(frozenset()) == 'frozenset()' + + s1 = {1/x, x} + s2 = frozenset(s1) + + assert pretty(s1) == \ +"""\ + 1 \n\ +{-, x} + x \ +""" + assert upretty(s1) == \ +"""\ +⎧1 ⎫ +⎨─, x⎬ +⎩x ⎭\ +""" + + assert pretty(s2) == \ +"""\ + 1 \n\ +frozenset({-, x}) + x \ +""" + assert upretty(s2) == \ +"""\ + ⎛⎧1 ⎫⎞ +frozenset⎜⎨─, x⎬⎟ + ⎝⎩x ⎭⎠\ +""" + + +def test_pretty_sets(): + s = FiniteSet + assert pretty(s(*[x*y, x**2])) == \ +"""\ + 2 \n\ +{x , x*y}\ +""" + assert pretty(s(*range(1, 6))) == "{1, 2, 3, 4, 5}" + assert pretty(s(*range(1, 13))) == "{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}" + + assert pretty({x*y, x**2}) == \ +"""\ + 2 \n\ +{x , x*y}\ +""" + assert pretty(set(range(1, 6))) == "{1, 2, 3, 4, 5}" + assert pretty(set(range(1, 13))) == \ + "{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}" + + assert pretty(frozenset([x*y, x**2])) == \ +"""\ + 2 \n\ +frozenset({x , x*y})\ +""" + assert pretty(frozenset(range(1, 6))) == "frozenset({1, 2, 3, 4, 5})" + assert pretty(frozenset(range(1, 13))) == \ + "frozenset({1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12})" + + assert pretty(Range(0, 3, 1)) == '{0, 1, 2}' + + ascii_str = '{0, 1, ..., 29}' + ucode_str = '{0, 1, …, 29}' + assert pretty(Range(0, 30, 1)) == ascii_str + assert upretty(Range(0, 30, 1)) == ucode_str + + ascii_str = '{30, 29, ..., 2}' + ucode_str = '{30, 29, …, 2}' + assert pretty(Range(30, 1, -1)) == ascii_str + assert upretty(Range(30, 1, -1)) == ucode_str + + ascii_str = '{0, 2, ...}' + ucode_str = '{0, 2, …}' + assert pretty(Range(0, oo, 2)) == ascii_str + assert upretty(Range(0, oo, 2)) == ucode_str + + ascii_str = '{..., 2, 0}' + ucode_str = '{…, 2, 0}' + assert pretty(Range(oo, -2, -2)) == ascii_str + assert upretty(Range(oo, -2, -2)) == ucode_str + + ascii_str = '{-2, -3, ...}' + ucode_str = '{-2, -3, …}' + assert pretty(Range(-2, -oo, -1)) == ascii_str + assert upretty(Range(-2, -oo, -1)) == ucode_str + + +def test_pretty_SetExpr(): + iv = Interval(1, 3) + se = SetExpr(iv) + ascii_str = "SetExpr([1, 3])" + ucode_str = "SetExpr([1, 3])" + assert pretty(se) == ascii_str + assert upretty(se) == ucode_str + + +def test_pretty_ImageSet(): + imgset = ImageSet(Lambda((x, y), x + y), {1, 2, 3}, {3, 4}) + ascii_str = '{x + y | x in {1, 2, 3}, y in {3, 4}}' + ucode_str = '{x + y │ x ∊ {1, 2, 3}, y ∊ {3, 4}}' + assert pretty(imgset) == ascii_str + assert upretty(imgset) == ucode_str + + imgset = ImageSet(Lambda(((x, y),), x + y), ProductSet({1, 2, 3}, {3, 4})) + ascii_str = '{x + y | (x, y) in {1, 2, 3} x {3, 4}}' + ucode_str = '{x + y │ (x, y) ∊ {1, 2, 3} × {3, 4}}' + assert pretty(imgset) == ascii_str + assert upretty(imgset) == ucode_str + + imgset = ImageSet(Lambda(x, x**2), S.Naturals) + ascii_str = '''\ + 2 \n\ +{x | x in Naturals}''' + ucode_str = '''\ +⎧ 2 │ ⎫\n\ +⎨x │ x ∊ ℕ⎬\n\ +⎩ │ ⎭''' + assert pretty(imgset) == ascii_str + assert upretty(imgset) == ucode_str + + # TODO: The "x in N" parts below should be centered independently of the + # 1/x**2 fraction + imgset = ImageSet(Lambda(x, 1/x**2), S.Naturals) + ascii_str = '''\ + 1 \n\ +{-- | x in Naturals} + 2 \n\ + x ''' + ucode_str = '''\ +⎧1 │ ⎫\n\ +⎪── │ x ∊ ℕ⎪\n\ +⎨ 2 │ ⎬\n\ +⎪x │ ⎪\n\ +⎩ │ ⎭''' + assert pretty(imgset) == ascii_str + assert upretty(imgset) == ucode_str + + imgset = ImageSet(Lambda((x, y), 1/(x + y)**2), S.Naturals, S.Naturals) + ascii_str = '''\ + 1 \n\ +{-------- | x in Naturals, y in Naturals} + 2 \n\ + (x + y) ''' + ucode_str = '''\ +⎧ 1 │ ⎫ +⎪──────── │ x ∊ ℕ, y ∊ ℕ⎪ +⎨ 2 │ ⎬ +⎪(x + y) │ ⎪ +⎩ │ ⎭''' + assert pretty(imgset) == ascii_str + assert upretty(imgset) == ucode_str + + # issue 23449 centering issue + assert upretty([Symbol("ihat") / (Symbol("i") + 1)]) == '''\ +⎡ î ⎤ +⎢─────⎥ +⎣i + 1⎦\ +''' + assert upretty(Matrix([Symbol("ihat"), Symbol("i") + 1])) == '''\ +⎡ î ⎤ +⎢ ⎥ +⎣i + 1⎦\ +''' + + +def test_pretty_ConditionSet(): + ascii_str = '{x | x in (-oo, oo) and sin(x) = 0}' + ucode_str = '{x │ x ∊ ℝ ∧ (sin(x) = 0)}' + assert pretty(ConditionSet(x, Eq(sin(x), 0), S.Reals)) == ascii_str + assert upretty(ConditionSet(x, Eq(sin(x), 0), S.Reals)) == ucode_str + + assert pretty(ConditionSet(x, Contains(x, S.Reals, evaluate=False), FiniteSet(1))) == '{1}' + assert upretty(ConditionSet(x, Contains(x, S.Reals, evaluate=False), FiniteSet(1))) == '{1}' + + assert pretty(ConditionSet(x, And(x > 1, x < -1), FiniteSet(1, 2, 3))) == "EmptySet" + assert upretty(ConditionSet(x, And(x > 1, x < -1), FiniteSet(1, 2, 3))) == "∅" + + assert pretty(ConditionSet(x, Or(x > 1, x < -1), FiniteSet(1, 2))) == '{2}' + assert upretty(ConditionSet(x, Or(x > 1, x < -1), FiniteSet(1, 2))) == '{2}' + + condset = ConditionSet(x, 1/x**2 > 0) + ascii_str = '''\ + 1 \n\ +{x | -- > 0} + 2 \n\ + x ''' + ucode_str = '''\ +⎧ │ ⎛1 ⎞⎫ +⎪x │ ⎜── > 0⎟⎪ +⎨ │ ⎜ 2 ⎟⎬ +⎪ │ ⎝x ⎠⎪ +⎩ │ ⎭''' + assert pretty(condset) == ascii_str + assert upretty(condset) == ucode_str + + condset = ConditionSet(x, 1/x**2 > 0, S.Reals) + ascii_str = '''\ + 1 \n\ +{x | x in (-oo, oo) and -- > 0} + 2 \n\ + x ''' + ucode_str = '''\ +⎧ │ ⎛1 ⎞⎫ +⎪x │ x ∊ ℝ ∧ ⎜── > 0⎟⎪ +⎨ │ ⎜ 2 ⎟⎬ +⎪ │ ⎝x ⎠⎪ +⎩ │ ⎭''' + assert pretty(condset) == ascii_str + assert upretty(condset) == ucode_str + + +def test_pretty_ComplexRegion(): + from sympy.sets.fancysets import ComplexRegion + cregion = ComplexRegion(Interval(3, 5)*Interval(4, 6)) + ascii_str = '{x + y*I | x, y in [3, 5] x [4, 6]}' + ucode_str = '{x + y⋅ⅈ │ x, y ∊ [3, 5] × [4, 6]}' + assert pretty(cregion) == ascii_str + assert upretty(cregion) == ucode_str + + cregion = ComplexRegion(Interval(0, 1)*Interval(0, 2*pi), polar=True) + ascii_str = '{r*(I*sin(theta) + cos(theta)) | r, theta in [0, 1] x [0, 2*pi)}' + ucode_str = '{r⋅(ⅈ⋅sin(θ) + cos(θ)) │ r, θ ∊ [0, 1] × [0, 2⋅π)}' + assert pretty(cregion) == ascii_str + assert upretty(cregion) == ucode_str + + cregion = ComplexRegion(Interval(3, 1/a**2)*Interval(4, 6)) + ascii_str = '''\ + 1 \n\ +{x + y*I | x, y in [3, --] x [4, 6]} + 2 \n\ + a ''' + ucode_str = '''\ +⎧ │ ⎡ 1 ⎤ ⎫ +⎪x + y⋅ⅈ │ x, y ∊ ⎢3, ──⎥ × [4, 6]⎪ +⎨ │ ⎢ 2⎥ ⎬ +⎪ │ ⎣ a ⎦ ⎪ +⎩ │ ⎭''' + assert pretty(cregion) == ascii_str + assert upretty(cregion) == ucode_str + + cregion = ComplexRegion(Interval(0, 1/a**2)*Interval(0, 2*pi), polar=True) + ascii_str = '''\ + 1 \n\ +{r*(I*sin(theta) + cos(theta)) | r, theta in [0, --] x [0, 2*pi)} + 2 \n\ + a ''' + ucode_str = '''\ +⎧ │ ⎡ 1 ⎤ ⎫ +⎪r⋅(ⅈ⋅sin(θ) + cos(θ)) │ r, θ ∊ ⎢0, ──⎥ × [0, 2⋅π)⎪ +⎨ │ ⎢ 2⎥ ⎬ +⎪ │ ⎣ a ⎦ ⎪ +⎩ │ ⎭''' + assert pretty(cregion) == ascii_str + assert upretty(cregion) == ucode_str + + +def test_pretty_Union_issue_10414(): + a, b = Interval(2, 3), Interval(4, 7) + ucode_str = '[2, 3] ∪ [4, 7]' + ascii_str = '[2, 3] U [4, 7]' + assert upretty(Union(a, b)) == ucode_str + assert pretty(Union(a, b)) == ascii_str + + +def test_pretty_Intersection_issue_10414(): + x, y, z, w = symbols('x, y, z, w') + a, b = Interval(x, y), Interval(z, w) + ucode_str = '[x, y] ∩ [z, w]' + ascii_str = '[x, y] n [z, w]' + assert upretty(Intersection(a, b)) == ucode_str + assert pretty(Intersection(a, b)) == ascii_str + + +def test_ProductSet_exponent(): + ucode_str = ' 1\n[0, 1] ' + assert upretty(Interval(0, 1)**1) == ucode_str + ucode_str = ' 2\n[0, 1] ' + assert upretty(Interval(0, 1)**2) == ucode_str + + +def test_ProductSet_parenthesis(): + ucode_str = '([4, 7] × {1, 2}) ∪ ([2, 3] × [4, 7])' + + a, b = Interval(2, 3), Interval(4, 7) + assert upretty(Union(a*b, b*FiniteSet(1, 2))) == ucode_str + + +def test_ProductSet_prod_char_issue_10413(): + ascii_str = '[2, 3] x [4, 7]' + ucode_str = '[2, 3] × [4, 7]' + + a, b = Interval(2, 3), Interval(4, 7) + assert pretty(a*b) == ascii_str + assert upretty(a*b) == ucode_str + + +def test_pretty_sequences(): + s1 = SeqFormula(a**2, (0, oo)) + s2 = SeqPer((1, 2)) + + ascii_str = '[0, 1, 4, 9, ...]' + ucode_str = '[0, 1, 4, 9, …]' + + assert pretty(s1) == ascii_str + assert upretty(s1) == ucode_str + + ascii_str = '[1, 2, 1, 2, ...]' + ucode_str = '[1, 2, 1, 2, …]' + assert pretty(s2) == ascii_str + assert upretty(s2) == ucode_str + + s3 = SeqFormula(a**2, (0, 2)) + s4 = SeqPer((1, 2), (0, 2)) + + ascii_str = '[0, 1, 4]' + ucode_str = '[0, 1, 4]' + + assert pretty(s3) == ascii_str + assert upretty(s3) == ucode_str + + ascii_str = '[1, 2, 1]' + ucode_str = '[1, 2, 1]' + assert pretty(s4) == ascii_str + assert upretty(s4) == ucode_str + + s5 = SeqFormula(a**2, (-oo, 0)) + s6 = SeqPer((1, 2), (-oo, 0)) + + ascii_str = '[..., 9, 4, 1, 0]' + ucode_str = '[…, 9, 4, 1, 0]' + + assert pretty(s5) == ascii_str + assert upretty(s5) == ucode_str + + ascii_str = '[..., 2, 1, 2, 1]' + ucode_str = '[…, 2, 1, 2, 1]' + assert pretty(s6) == ascii_str + assert upretty(s6) == ucode_str + + ascii_str = '[1, 3, 5, 11, ...]' + ucode_str = '[1, 3, 5, 11, …]' + + assert pretty(SeqAdd(s1, s2)) == ascii_str + assert upretty(SeqAdd(s1, s2)) == ucode_str + + ascii_str = '[1, 3, 5]' + ucode_str = '[1, 3, 5]' + + assert pretty(SeqAdd(s3, s4)) == ascii_str + assert upretty(SeqAdd(s3, s4)) == ucode_str + + ascii_str = '[..., 11, 5, 3, 1]' + ucode_str = '[…, 11, 5, 3, 1]' + + assert pretty(SeqAdd(s5, s6)) == ascii_str + assert upretty(SeqAdd(s5, s6)) == ucode_str + + ascii_str = '[0, 2, 4, 18, ...]' + ucode_str = '[0, 2, 4, 18, …]' + + assert pretty(SeqMul(s1, s2)) == ascii_str + assert upretty(SeqMul(s1, s2)) == ucode_str + + ascii_str = '[0, 2, 4]' + ucode_str = '[0, 2, 4]' + + assert pretty(SeqMul(s3, s4)) == ascii_str + assert upretty(SeqMul(s3, s4)) == ucode_str + + ascii_str = '[..., 18, 4, 2, 0]' + ucode_str = '[…, 18, 4, 2, 0]' + + assert pretty(SeqMul(s5, s6)) == ascii_str + assert upretty(SeqMul(s5, s6)) == ucode_str + + # Sequences with symbolic limits, issue 12629 + s7 = SeqFormula(a**2, (a, 0, x)) + raises(NotImplementedError, lambda: pretty(s7)) + raises(NotImplementedError, lambda: upretty(s7)) + + b = Symbol('b') + s8 = SeqFormula(b*a**2, (a, 0, 2)) + ascii_str = '[0, b, 4*b]' + ucode_str = '[0, b, 4⋅b]' + assert pretty(s8) == ascii_str + assert upretty(s8) == ucode_str + + +def test_pretty_FourierSeries(): + f = fourier_series(x, (x, -pi, pi)) + + ascii_str = \ +"""\ + 2*sin(3*x) \n\ +2*sin(x) - sin(2*x) + ---------- + ...\n\ + 3 \ +""" + + ucode_str = \ +"""\ + 2⋅sin(3⋅x) \n\ +2⋅sin(x) - sin(2⋅x) + ────────── + …\n\ + 3 \ +""" + + assert pretty(f) == ascii_str + assert upretty(f) == ucode_str + + +def test_pretty_FormalPowerSeries(): + f = fps(log(1 + x)) + + + ascii_str = \ +"""\ + oo \n\ +____ \n\ +\\ ` \n\ + \\ -k k \n\ + \\ -(-1) *x \n\ + / -----------\n\ + / k \n\ +/___, \n\ +k = 1 \ +""" + + ucode_str = \ +"""\ + ∞ \n\ +____ \n\ +╲ \n\ + ╲ -k k \n\ + ╲ -(-1) ⋅x \n\ + ╱ ───────────\n\ + ╱ k \n\ +╱ \n\ +‾‾‾‾ \n\ +k = 1 \ +""" + + assert pretty(f) == ascii_str + assert upretty(f) == ucode_str + + +def test_pretty_limits(): + expr = Limit(x, x, oo) + ascii_str = \ +"""\ + lim x\n\ +x->oo \ +""" + ucode_str = \ +"""\ +lim x\n\ +x─→∞ \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Limit(x**2, x, 0) + ascii_str = \ +"""\ + 2\n\ + lim x \n\ +x->0+ \ +""" + ucode_str = \ +"""\ + 2\n\ + lim x \n\ +x─→0⁺ \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Limit(1/x, x, 0) + ascii_str = \ +"""\ + 1\n\ + lim -\n\ +x->0+x\ +""" + ucode_str = \ +"""\ + 1\n\ + lim ─\n\ +x─→0⁺x\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Limit(sin(x)/x, x, 0) + ascii_str = \ +"""\ + /sin(x)\\\n\ + lim |------|\n\ +x->0+\\ x /\ +""" + ucode_str = \ +"""\ + ⎛sin(x)⎞\n\ + lim ⎜──────⎟\n\ +x─→0⁺⎝ x ⎠\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Limit(sin(x)/x, x, 0, "-") + ascii_str = \ +"""\ + /sin(x)\\\n\ + lim |------|\n\ +x->0-\\ x /\ +""" + ucode_str = \ +"""\ + ⎛sin(x)⎞\n\ + lim ⎜──────⎟\n\ +x─→0⁻⎝ x ⎠\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Limit(x + sin(x), x, 0) + ascii_str = \ +"""\ + lim (x + sin(x))\n\ +x->0+ \ +""" + ucode_str = \ +"""\ + lim (x + sin(x))\n\ +x─→0⁺ \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Limit(x, x, 0)**2 + ascii_str = \ +"""\ + 2\n\ +/ lim x\\ \n\ +\\x->0+ / \ +""" + ucode_str = \ +"""\ + 2\n\ +⎛ lim x⎞ \n\ +⎝x─→0⁺ ⎠ \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Limit(x*Limit(y/2,y,0), x, 0) + ascii_str = \ +"""\ + / /y\\\\\n\ + lim |x* lim |-||\n\ +x->0+\\ y->0+\\2//\ +""" + ucode_str = \ +"""\ + ⎛ ⎛y⎞⎞\n\ + lim ⎜x⋅ lim ⎜─⎟⎟\n\ +x─→0⁺⎝ y─→0⁺⎝2⎠⎠\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = 2*Limit(x*Limit(y/2,y,0), x, 0) + ascii_str = \ +"""\ + / /y\\\\\n\ +2* lim |x* lim |-||\n\ + x->0+\\ y->0+\\2//\ +""" + ucode_str = \ +"""\ + ⎛ ⎛y⎞⎞\n\ +2⋅ lim ⎜x⋅ lim ⎜─⎟⎟\n\ + x─→0⁺⎝ y─→0⁺⎝2⎠⎠\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Limit(sin(x), x, 0, dir='+-') + ascii_str = \ +"""\ +lim sin(x)\n\ +x->0 \ +""" + ucode_str = \ +"""\ +lim sin(x)\n\ +x─→0 \ +""" + + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + +def test_pretty_ComplexRootOf(): + expr = rootof(x**5 + 11*x - 2, 0) + ascii_str = \ +"""\ + / 5 \\\n\ +CRootOf\\x + 11*x - 2, 0/\ +""" + ucode_str = \ +"""\ + ⎛ 5 ⎞\n\ +CRootOf⎝x + 11⋅x - 2, 0⎠\ +""" + + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + +def test_pretty_RootSum(): + expr = RootSum(x**5 + 11*x - 2, auto=False) + ascii_str = \ +"""\ + / 5 \\\n\ +RootSum\\x + 11*x - 2/\ +""" + ucode_str = \ +"""\ + ⎛ 5 ⎞\n\ +RootSum⎝x + 11⋅x - 2⎠\ +""" + + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = RootSum(x**5 + 11*x - 2, Lambda(z, exp(z))) + ascii_str = \ +"""\ + / 5 z\\\n\ +RootSum\\x + 11*x - 2, z -> e /\ +""" + ucode_str = \ +"""\ + ⎛ 5 z⎞\n\ +RootSum⎝x + 11⋅x - 2, z ↦ ℯ ⎠\ +""" + + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + +def test_GroebnerBasis(): + expr = groebner([], x, y) + + ascii_str = \ +"""\ +GroebnerBasis([], x, y, domain=ZZ, order=lex)\ +""" + ucode_str = \ +"""\ +GroebnerBasis([], x, y, domain=ℤ, order=lex)\ +""" + + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + F = [x**2 - 3*y - x + 1, y**2 - 2*x + y - 1] + expr = groebner(F, x, y, order='grlex') + + ascii_str = \ +"""\ + /[ 2 2 ] \\\n\ +GroebnerBasis\\[x - x - 3*y + 1, y - 2*x + y - 1], x, y, domain=ZZ, order=grlex/\ +""" + ucode_str = \ +"""\ + ⎛⎡ 2 2 ⎤ ⎞\n\ +GroebnerBasis⎝⎣x - x - 3⋅y + 1, y - 2⋅x + y - 1⎦, x, y, domain=ℤ, order=grlex⎠\ +""" + + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = expr.fglm('lex') + + ascii_str = \ +"""\ + /[ 2 4 3 2 ] \\\n\ +GroebnerBasis\\[2*x - y - y + 1, y + 2*y - 3*y - 16*y + 7], x, y, domain=ZZ, order=lex/\ +""" + ucode_str = \ +"""\ + ⎛⎡ 2 4 3 2 ⎤ ⎞\n\ +GroebnerBasis⎝⎣2⋅x - y - y + 1, y + 2⋅y - 3⋅y - 16⋅y + 7⎦, x, y, domain=ℤ, order=lex⎠\ +""" + + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + +def test_pretty_UniversalSet(): + assert pretty(S.UniversalSet) == "UniversalSet" + assert upretty(S.UniversalSet) == '𝕌' + + +def test_pretty_Boolean(): + expr = Not(x, evaluate=False) + + assert pretty(expr) == "Not(x)" + assert upretty(expr) == "¬x" + + expr = And(x, y) + + assert pretty(expr) == "And(x, y)" + assert upretty(expr) == "x ∧ y" + + expr = Or(x, y) + + assert pretty(expr) == "Or(x, y)" + assert upretty(expr) == "x ∨ y" + + syms = symbols('a:f') + expr = And(*syms) + + assert pretty(expr) == "And(a, b, c, d, e, f)" + assert upretty(expr) == "a ∧ b ∧ c ∧ d ∧ e ∧ f" + + expr = Or(*syms) + + assert pretty(expr) == "Or(a, b, c, d, e, f)" + assert upretty(expr) == "a ∨ b ∨ c ∨ d ∨ e ∨ f" + + expr = Xor(x, y, evaluate=False) + + assert pretty(expr) == "Xor(x, y)" + assert upretty(expr) == "x ⊻ y" + + expr = Nand(x, y, evaluate=False) + + assert pretty(expr) == "Nand(x, y)" + assert upretty(expr) == "x ⊼ y" + + expr = Nor(x, y, evaluate=False) + + assert pretty(expr) == "Nor(x, y)" + assert upretty(expr) == "x ⊽ y" + + expr = Implies(x, y, evaluate=False) + + assert pretty(expr) == "Implies(x, y)" + assert upretty(expr) == "x → y" + + # don't sort args + expr = Implies(y, x, evaluate=False) + + assert pretty(expr) == "Implies(y, x)" + assert upretty(expr) == "y → x" + + expr = Equivalent(x, y, evaluate=False) + + assert pretty(expr) == "Equivalent(x, y)" + assert upretty(expr) == "x ⇔ y" + + expr = Equivalent(y, x, evaluate=False) + + assert pretty(expr) == "Equivalent(x, y)" + assert upretty(expr) == "x ⇔ y" + + +def test_pretty_Domain(): + expr = FF(23) + + assert pretty(expr) == "GF(23)" + assert upretty(expr) == "ℤ₂₃" + + expr = ZZ + + assert pretty(expr) == "ZZ" + assert upretty(expr) == "ℤ" + + expr = QQ + + assert pretty(expr) == "QQ" + assert upretty(expr) == "ℚ" + + expr = RR + + assert pretty(expr) == "RR" + assert upretty(expr) == "ℝ" + + expr = QQ[x] + + assert pretty(expr) == "QQ[x]" + assert upretty(expr) == "ℚ[x]" + + expr = QQ[x, y] + + assert pretty(expr) == "QQ[x, y]" + assert upretty(expr) == "ℚ[x, y]" + + expr = ZZ.frac_field(x) + + assert pretty(expr) == "ZZ(x)" + assert upretty(expr) == "ℤ(x)" + + expr = ZZ.frac_field(x, y) + + assert pretty(expr) == "ZZ(x, y)" + assert upretty(expr) == "ℤ(x, y)" + + expr = QQ.poly_ring(x, y, order=grlex) + + assert pretty(expr) == "QQ[x, y, order=grlex]" + assert upretty(expr) == "ℚ[x, y, order=grlex]" + + expr = QQ.poly_ring(x, y, order=ilex) + + assert pretty(expr) == "QQ[x, y, order=ilex]" + assert upretty(expr) == "ℚ[x, y, order=ilex]" + + +def test_pretty_prec(): + assert xpretty(S("0.3"), full_prec=True, wrap_line=False) == "0.300000000000000" + assert xpretty(S("0.3"), full_prec="auto", wrap_line=False) == "0.300000000000000" + assert xpretty(S("0.3"), full_prec=False, wrap_line=False) == "0.3" + assert xpretty(S("0.3")*x, full_prec=True, use_unicode=False, wrap_line=False) in [ + "0.300000000000000*x", + "x*0.300000000000000" + ] + assert xpretty(S("0.3")*x, full_prec="auto", use_unicode=False, wrap_line=False) in [ + "0.3*x", + "x*0.3" + ] + assert xpretty(S("0.3")*x, full_prec=False, use_unicode=False, wrap_line=False) in [ + "0.3*x", + "x*0.3" + ] + + +def test_pprint(): + import sys + from io import StringIO + fd = StringIO() + sso = sys.stdout + sys.stdout = fd + try: + pprint(pi, use_unicode=False, wrap_line=False) + finally: + sys.stdout = sso + assert fd.getvalue() == 'pi\n' + + +def test_pretty_class(): + """Test that the printer dispatcher correctly handles classes.""" + class C: + pass # C has no .__class__ and this was causing problems + + class D: + pass + + assert pretty( C ) == str( C ) + assert pretty( D ) == str( D ) + + +def test_pretty_no_wrap_line(): + huge_expr = 0 + for i in range(20): + huge_expr += i*sin(i + x) + assert xpretty(huge_expr ).find('\n') != -1 + assert xpretty(huge_expr, wrap_line=False).find('\n') == -1 + + +def test_settings(): + raises(TypeError, lambda: pretty(S(4), method="garbage")) + + +def test_pretty_sum(): + from sympy.abc import x, a, b, k, m, n + + expr = Sum(k**k, (k, 0, n)) + ascii_str = \ +"""\ + n \n\ +___ \n\ +\\ ` \n\ + \\ k\n\ + / k \n\ +/__, \n\ +k = 0 \ +""" + ucode_str = \ +"""\ + n \n\ + ___ \n\ + ╲ \n\ + ╲ k\n\ + ╱ k \n\ + ╱ \n\ + ‾‾‾ \n\ +k = 0 \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Sum(k**k, (k, oo, n)) + ascii_str = \ +"""\ + n \n\ + ___ \n\ + \\ ` \n\ + \\ k\n\ + / k \n\ + /__, \n\ +k = oo \ +""" + ucode_str = \ +"""\ + n \n\ + ___ \n\ + ╲ \n\ + ╲ k\n\ + ╱ k \n\ + ╱ \n\ + ‾‾‾ \n\ +k = ∞ \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Sum(k**(Integral(x**n, (x, -oo, oo))), (k, 0, n**n)) + ascii_str = \ +"""\ + n \n\ + n \n\ +______ \n\ +\\ ` \n\ + \\ oo \n\ + \\ / \n\ + \\ | \n\ + \\ | n \n\ + ) | x dx\n\ + / | \n\ + / / \n\ + / -oo \n\ + / k \n\ +/_____, \n\ + k = 0 \ +""" + ucode_str = \ +"""\ + n \n\ + n \n\ +______ \n\ +╲ \n\ + ╲ \n\ + ╲ ∞ \n\ + ╲ ⌠ \n\ + ╲ ⎮ n \n\ + ╱ ⎮ x dx\n\ + ╱ ⌡ \n\ + ╱ -∞ \n\ + ╱ k \n\ +╱ \n\ +‾‾‾‾‾‾ \n\ +k = 0 \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Sum(k**( + Integral(x**n, (x, -oo, oo))), (k, 0, Integral(x**x, (x, -oo, oo)))) + ascii_str = \ +"""\ + oo \n\ + / \n\ + | \n\ + | x \n\ + | x dx \n\ + | \n\ +/ \n\ +-oo \n\ + ______ \n\ + \\ ` \n\ + \\ oo \n\ + \\ / \n\ + \\ | \n\ + \\ | n \n\ + ) | x dx\n\ + / | \n\ + / / \n\ + / -oo \n\ + / k \n\ + /_____, \n\ + k = 0 \ +""" + ucode_str = \ +"""\ +∞ \n\ +⌠ \n\ +⎮ x \n\ +⎮ x dx \n\ +⌡ \n\ +-∞ \n\ + ______ \n\ + ╲ \n\ + ╲ \n\ + ╲ ∞ \n\ + ╲ ⌠ \n\ + ╲ ⎮ n \n\ + ╱ ⎮ x dx\n\ + ╱ ⌡ \n\ + ╱ -∞ \n\ + ╱ k \n\ + ╱ \n\ + ‾‾‾‾‾‾ \n\ + k = 0 \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Sum(k**(Integral(x**n, (x, -oo, oo))), ( + k, x + n + x**2 + n**2 + (x/n) + (1/x), Integral(x**x, (x, -oo, oo)))) + ascii_str = \ +"""\ + oo \n\ + / \n\ + | \n\ + | x \n\ + | x dx \n\ + | \n\ + / \n\ + -oo \n\ + ______ \n\ + \\ ` \n\ + \\ oo \n\ + \\ / \n\ + \\ | \n\ + \\ | n \n\ + ) | x dx\n\ + / | \n\ + / / \n\ + / -oo \n\ + / k \n\ + /_____, \n\ + 2 2 1 x \n\ +k = n + n + x + x + - + - \n\ + x n \ +""" + ucode_str = \ +"""\ + ∞ \n\ + ⌠ \n\ + ⎮ x \n\ + ⎮ x dx \n\ + ⌡ \n\ + -∞ \n\ + ______ \n\ + ╲ \n\ + ╲ \n\ + ╲ ∞ \n\ + ╲ ⌠ \n\ + ╲ ⎮ n \n\ + ╱ ⎮ x dx\n\ + ╱ ⌡ \n\ + ╱ -∞ \n\ + ╱ k \n\ + ╱ \n\ + ‾‾‾‾‾‾ \n\ + 2 2 1 x \n\ +k = n + n + x + x + ─ + ─ \n\ + x n \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Sum(k**( + Integral(x**n, (x, -oo, oo))), (k, 0, x + n + x**2 + n**2 + (x/n) + (1/x))) + ascii_str = \ +"""\ + 2 2 1 x \n\ +n + n + x + x + - + - \n\ + x n \n\ + ______ \n\ + \\ ` \n\ + \\ oo \n\ + \\ / \n\ + \\ | \n\ + \\ | n \n\ + ) | x dx\n\ + / | \n\ + / / \n\ + / -oo \n\ + / k \n\ + /_____, \n\ + k = 0 \ +""" + ucode_str = \ +"""\ + 2 2 1 x \n\ +n + n + x + x + ─ + ─ \n\ + x n \n\ + ______ \n\ + ╲ \n\ + ╲ \n\ + ╲ ∞ \n\ + ╲ ⌠ \n\ + ╲ ⎮ n \n\ + ╱ ⎮ x dx\n\ + ╱ ⌡ \n\ + ╱ -∞ \n\ + ╱ k \n\ + ╱ \n\ + ‾‾‾‾‾‾ \n\ + k = 0 \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Sum(x, (x, 0, oo)) + ascii_str = \ +"""\ + oo \n\ + __ \n\ + \\ ` \n\ + ) x\n\ + /_, \n\ +x = 0 \ +""" + ucode_str = \ +"""\ + ∞ \n\ + ___ \n\ + ╲ \n\ + ╲ \n\ + ╱ x\n\ + ╱ \n\ + ‾‾‾ \n\ +x = 0 \ +""" + + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Sum(x**2, (x, 0, oo)) + ascii_str = \ +"""\ + oo \n\ +___ \n\ +\\ ` \n\ + \\ 2\n\ + / x \n\ +/__, \n\ +x = 0 \ +""" + ucode_str = \ +"""\ + ∞ \n\ + ___ \n\ + ╲ \n\ + ╲ 2\n\ + ╱ x \n\ + ╱ \n\ + ‾‾‾ \n\ +x = 0 \ +""" + + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Sum(x/2, (x, 0, oo)) + ascii_str = \ +"""\ + oo \n\ +___ \n\ +\\ ` \n\ + \\ x\n\ + ) -\n\ + / 2\n\ +/__, \n\ +x = 0 \ +""" + ucode_str = \ +"""\ + ∞ \n\ +____ \n\ +╲ \n\ + ╲ \n\ + ╲ x\n\ + ╱ ─\n\ + ╱ 2\n\ +╱ \n\ +‾‾‾‾ \n\ +x = 0 \ +""" + + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Sum(x**3/2, (x, 0, oo)) + ascii_str = \ +"""\ + oo \n\ +____ \n\ +\\ ` \n\ + \\ 3\n\ + \\ x \n\ + / --\n\ + / 2 \n\ +/___, \n\ +x = 0 \ +""" + ucode_str = \ +"""\ + ∞ \n\ +____ \n\ +╲ \n\ + ╲ 3\n\ + ╲ x \n\ + ╱ ──\n\ + ╱ 2 \n\ +╱ \n\ +‾‾‾‾ \n\ +x = 0 \ +""" + + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Sum((x**3*y**(x/2))**n, (x, 0, oo)) + ascii_str = \ +"""\ + oo \n\ +____ \n\ +\\ ` \n\ + \\ n\n\ + \\ / x\\ \n\ + ) | -| \n\ + / | 3 2| \n\ + / \\x *y / \n\ +/___, \n\ +x = 0 \ +""" + ucode_str = \ +"""\ + ∞ \n\ +_____ \n\ +╲ \n\ + ╲ \n\ + ╲ n\n\ + ╲ ⎛ x⎞ \n\ + ╱ ⎜ ─⎟ \n\ + ╱ ⎜ 3 2⎟ \n\ + ╱ ⎝x ⋅y ⎠ \n\ +╱ \n\ +‾‾‾‾‾ \n\ +x = 0 \ +""" + + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Sum(1/x**2, (x, 0, oo)) + ascii_str = \ +"""\ + oo \n\ +____ \n\ +\\ ` \n\ + \\ 1 \n\ + \\ --\n\ + / 2\n\ + / x \n\ +/___, \n\ +x = 0 \ +""" + ucode_str = \ +"""\ + ∞ \n\ +____ \n\ +╲ \n\ + ╲ 1 \n\ + ╲ ──\n\ + ╱ 2\n\ + ╱ x \n\ +╱ \n\ +‾‾‾‾ \n\ +x = 0 \ +""" + + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Sum(1/y**(a/b), (x, 0, oo)) + ascii_str = \ +"""\ + oo \n\ +____ \n\ +\\ ` \n\ + \\ -a \n\ + \\ ---\n\ + / b \n\ + / y \n\ +/___, \n\ +x = 0 \ +""" + ucode_str = \ +"""\ + ∞ \n\ +____ \n\ +╲ \n\ + ╲ -a \n\ + ╲ ───\n\ + ╱ b \n\ + ╱ y \n\ +╱ \n\ +‾‾‾‾ \n\ +x = 0 \ +""" + + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Sum(1/y**(a/b), (x, 0, oo), (y, 1, 2)) + ascii_str = \ +"""\ + 2 oo \n\ +____ ____ \n\ +\\ ` \\ ` \n\ + \\ \\ -a\n\ + \\ \\ --\n\ + / / b \n\ + / / y \n\ +/___, /___, \n\ +y = 1 x = 0 \ +""" + ucode_str = \ +"""\ + 2 ∞ \n\ +____ ____ \n\ +╲ ╲ \n\ + ╲ ╲ -a\n\ + ╲ ╲ ──\n\ + ╱ ╱ b \n\ + ╱ ╱ y \n\ +╱ ╱ \n\ +‾‾‾‾ ‾‾‾‾ \n\ +y = 1 x = 0 \ +""" + expr = Sum(1/(1 + 1/( + 1 + 1/k)) + 1, (k, 111, 1 + 1/n), (k, 1/(1 + m), oo)) + 1/(1 + 1/k) + ascii_str = \ +"""\ + 1 \n\ + 1 + - \n\ + oo n \n\ + _____ _____ \n\ + \\ ` \\ ` \n\ + \\ \\ / 1 \\ \n\ + \\ \\ |1 + ---------| \n\ + \\ \\ | 1 | 1 \n\ + ) ) | 1 + -----| + -----\n\ + / / | 1| 1\n\ + / / | 1 + -| 1 + -\n\ + / / \\ k/ k\n\ + /____, /____, \n\ + 1 k = 111 \n\ +k = ----- \n\ + m + 1 \ +""" + ucode_str = \ +"""\ + 1 \n\ + 1 + ─ \n\ + ∞ n \n\ + ______ ______ \n\ + ╲ ╲ \n\ + ╲ ╲ \n\ + ╲ ╲ ⎛ 1 ⎞ \n\ + ╲ ╲ ⎜1 + ─────────⎟ \n\ + ╲ ╲ ⎜ 1 ⎟ 1 \n\ + ╱ ╱ ⎜ 1 + ─────⎟ + ─────\n\ + ╱ ╱ ⎜ 1⎟ 1\n\ + ╱ ╱ ⎜ 1 + ─⎟ 1 + ─\n\ + ╱ ╱ ⎝ k⎠ k\n\ + ╱ ╱ \n\ + ‾‾‾‾‾‾ ‾‾‾‾‾‾ \n\ + 1 k = 111 \n\ +k = ───── \n\ + m + 1 \ +""" + + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + +def test_units(): + expr = joule + ascii_str1 = \ +"""\ + 2\n\ +kilogram*meter \n\ +---------------\n\ + 2 \n\ + second \ +""" + unicode_str1 = \ +"""\ + 2\n\ +kilogram⋅meter \n\ +───────────────\n\ + 2 \n\ + second \ +""" + + ascii_str2 = \ +"""\ + 2\n\ +3*x*y*kilogram*meter \n\ +---------------------\n\ + 2 \n\ + second \ +""" + unicode_str2 = \ +"""\ + 2\n\ +3⋅x⋅y⋅kilogram⋅meter \n\ +─────────────────────\n\ + 2 \n\ + second \ +""" + + from sympy.physics.units import kg, m, s + assert upretty(expr) == "joule" + assert pretty(expr) == "joule" + assert upretty(expr.convert_to(kg*m**2/s**2)) == unicode_str1 + assert pretty(expr.convert_to(kg*m**2/s**2)) == ascii_str1 + assert upretty(3*kg*x*m**2*y/s**2) == unicode_str2 + assert pretty(3*kg*x*m**2*y/s**2) == ascii_str2 + + +def test_pretty_Subs(): + f = Function('f') + expr = Subs(f(x), x, ph**2) + ascii_str = \ +"""\ +(f(x))| 2\n\ + |x=phi \ +""" + unicode_str = \ +"""\ +(f(x))│ 2\n\ + │x=φ \ +""" + + assert pretty(expr) == ascii_str + assert upretty(expr) == unicode_str + + expr = Subs(f(x).diff(x), x, 0) + ascii_str = \ +"""\ +/d \\| \n\ +|--(f(x))|| \n\ +\\dx /|x=0\ +""" + unicode_str = \ +"""\ +⎛d ⎞│ \n\ +⎜──(f(x))⎟│ \n\ +⎝dx ⎠│x=0\ +""" + + assert pretty(expr) == ascii_str + assert upretty(expr) == unicode_str + + expr = Subs(f(x).diff(x)/y, (x, y), (0, Rational(1, 2))) + ascii_str = \ +"""\ +/d \\| \n\ +|--(f(x))|| \n\ +|dx || \n\ +|--------|| \n\ +\\ y /|x=0, y=1/2\ +""" + unicode_str = \ +"""\ +⎛d ⎞│ \n\ +⎜──(f(x))⎟│ \n\ +⎜dx ⎟│ \n\ +⎜────────⎟│ \n\ +⎝ y ⎠│x=0, y=1/2\ +""" + + assert pretty(expr) == ascii_str + assert upretty(expr) == unicode_str + + +def test_gammas(): + assert upretty(lowergamma(x, y)) == "γ(x, y)" + assert upretty(uppergamma(x, y)) == "Γ(x, y)" + assert xpretty(gamma(x), use_unicode=True) == 'Γ(x)' + assert xpretty(gamma, use_unicode=True) == 'Γ' + assert xpretty(symbols('gamma', cls=Function)(x), use_unicode=True) == 'γ(x)' + assert xpretty(symbols('gamma', cls=Function), use_unicode=True) == 'γ' + + +def test_beta(): + assert xpretty(beta(x,y), use_unicode=True) == 'Β(x, y)' + assert xpretty(beta(x,y), use_unicode=False) == 'B(x, y)' + assert xpretty(beta, use_unicode=True) == 'Β' + assert xpretty(beta, use_unicode=False) == 'B' + mybeta = Function('beta') + assert xpretty(mybeta(x), use_unicode=True) == 'β(x)' + assert xpretty(mybeta(x, y, z), use_unicode=False) == 'beta(x, y, z)' + assert xpretty(mybeta, use_unicode=True) == 'β' + + +# test that notation passes to subclasses of the same name only +def test_function_subclass_different_name(): + class mygamma(gamma): + pass + assert xpretty(mygamma, use_unicode=True) == r"mygamma" + assert xpretty(mygamma(x), use_unicode=True) == r"mygamma(x)" + + +def test_SingularityFunction(): + assert xpretty(SingularityFunction(x, 0, n), use_unicode=True) == ( +"""\ + n\n\ + \ +""") + assert xpretty(SingularityFunction(x, 1, n), use_unicode=True) == ( +"""\ + n\n\ + \ +""") + assert xpretty(SingularityFunction(x, -1, n), use_unicode=True) == ( +"""\ + n\n\ + \ +""") + assert xpretty(SingularityFunction(x, a, n), use_unicode=True) == ( +"""\ + n\n\ +<-a + x> \ +""") + assert xpretty(SingularityFunction(x, y, n), use_unicode=True) == ( +"""\ + n\n\ + \ +""") + assert xpretty(SingularityFunction(x, 0, n), use_unicode=False) == ( +"""\ + n\n\ + \ +""") + assert xpretty(SingularityFunction(x, 1, n), use_unicode=False) == ( +"""\ + n\n\ + \ +""") + assert xpretty(SingularityFunction(x, -1, n), use_unicode=False) == ( +"""\ + n\n\ + \ +""") + assert xpretty(SingularityFunction(x, a, n), use_unicode=False) == ( +"""\ + n\n\ +<-a + x> \ +""") + assert xpretty(SingularityFunction(x, y, n), use_unicode=False) == ( +"""\ + n\n\ + \ +""") + + +def test_deltas(): + assert xpretty(DiracDelta(x), use_unicode=True) == 'δ(x)' + assert xpretty(DiracDelta(x, 1), use_unicode=True) == \ +"""\ + (1) \n\ +δ (x)\ +""" + assert xpretty(x*DiracDelta(x, 1), use_unicode=True) == \ +"""\ + (1) \n\ +x⋅δ (x)\ +""" + + +def test_hyper(): + expr = hyper((), (), z) + ucode_str = \ +"""\ + ┌─ ⎛ │ ⎞\n\ + ├─ ⎜ │ z⎟\n\ +0╵ 0 ⎝ │ ⎠\ +""" + ascii_str = \ +"""\ + _ \n\ + |_ / | \\\n\ + | | | z|\n\ +0 0 \\ | /\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = hyper((), (1,), x) + ucode_str = \ +"""\ + ┌─ ⎛ │ ⎞\n\ + ├─ ⎜ │ x⎟\n\ +0╵ 1 ⎝1 │ ⎠\ +""" + ascii_str = \ +"""\ + _ \n\ + |_ / | \\\n\ + | | | x|\n\ +0 1 \\1 | /\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = hyper([2], [1], x) + ucode_str = \ +"""\ + ┌─ ⎛2 │ ⎞\n\ + ├─ ⎜ │ x⎟\n\ +1╵ 1 ⎝1 │ ⎠\ +""" + ascii_str = \ +"""\ + _ \n\ + |_ /2 | \\\n\ + | | | x|\n\ +1 1 \\1 | /\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = hyper((pi/3, -2*k), (3, 4, 5, -3), x) + ucode_str = \ +"""\ + ⎛ π │ ⎞\n\ + ┌─ ⎜ ─, -2⋅k │ ⎟\n\ + ├─ ⎜ 3 │ x⎟\n\ +2╵ 4 ⎜ │ ⎟\n\ + ⎝-3, 3, 4, 5 │ ⎠\ +""" + ascii_str = \ +"""\ + \n\ + _ / pi | \\\n\ + |_ | --, -2*k | |\n\ + | | 3 | x|\n\ +2 4 | | |\n\ + \\-3, 3, 4, 5 | /\ +""" + + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = hyper((pi, S('2/3'), -2*k), (3, 4, 5, -3), x**2) + ucode_str = \ +"""\ + ┌─ ⎛2/3, π, -2⋅k │ 2⎞\n\ + ├─ ⎜ │ x ⎟\n\ +3╵ 4 ⎝-3, 3, 4, 5 │ ⎠\ +""" + ascii_str = \ +"""\ + _ \n\ + |_ /2/3, pi, -2*k | 2\\ + | | | x | +3 4 \\ -3, 3, 4, 5 | /""" + + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = hyper([1, 2], [3, 4], 1/(1/(1/(1/x + 1) + 1) + 1)) + ucode_str = \ +"""\ + ⎛ │ 1 ⎞\n\ + ⎜ │ ─────────────⎟\n\ + ⎜ │ 1 ⎟\n\ + ┌─ ⎜1, 2 │ 1 + ─────────⎟\n\ + ├─ ⎜ │ 1 ⎟\n\ +2╵ 2 ⎜3, 4 │ 1 + ─────⎟\n\ + ⎜ │ 1⎟\n\ + ⎜ │ 1 + ─⎟\n\ + ⎝ │ x⎠\ +""" + + ascii_str = \ +"""\ + \n\ + / | 1 \\\n\ + | | -------------|\n\ + _ | | 1 |\n\ + |_ |1, 2 | 1 + ---------|\n\ + | | | 1 |\n\ +2 2 |3, 4 | 1 + -----|\n\ + | | 1|\n\ + | | 1 + -|\n\ + \\ | x/\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + +def test_meijerg(): + expr = meijerg([pi, pi, x], [1], [0, 1], [1, 2, 3], z) + ucode_str = \ +"""\ +╭─╮2, 3 ⎛π, π, x 1 │ ⎞\n\ +│╶┐ ⎜ │ z⎟\n\ +╰─╯4, 5 ⎝ 0, 1 1, 2, 3 │ ⎠\ +""" + ascii_str = \ +"""\ + __2, 3 /pi, pi, x 1 | \\\n\ +/__ | | z|\n\ +\\_|4, 5 \\ 0, 1 1, 2, 3 | /\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = meijerg([1, pi/7], [2, pi, 5], [], [], z**2) + ucode_str = \ +"""\ + ⎛ π │ ⎞\n\ +╭─╮0, 2 ⎜1, ─ 2, 5, π │ 2⎟\n\ +│╶┐ ⎜ 7 │ z ⎟\n\ +╰─╯5, 0 ⎜ │ ⎟\n\ + ⎝ │ ⎠\ +""" + ascii_str = \ +"""\ + / pi | \\\n\ + __0, 2 |1, -- 2, 5, pi | 2|\n\ +/__ | 7 | z |\n\ +\\_|5, 0 | | |\n\ + \\ | /\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + ucode_str = \ +"""\ +╭─╮ 1, 10 ⎛1, 1, 1, 1, 1, 1, 1, 1, 1, 1 1 │ ⎞\n\ +│╶┐ ⎜ │ z⎟\n\ +╰─╯11, 2 ⎝ 1 1 │ ⎠\ +""" + ascii_str = \ +"""\ + __ 1, 10 /1, 1, 1, 1, 1, 1, 1, 1, 1, 1 1 | \\\n\ +/__ | | z|\n\ +\\_|11, 2 \\ 1 1 | /\ +""" + + expr = meijerg([1]*10, [1], [1], [1], z) + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = meijerg([1, 2, ], [4, 3], [3], [4, 5], 1/(1/(1/(1/x + 1) + 1) + 1)) + + ucode_str = \ +"""\ + ⎛ │ 1 ⎞\n\ + ⎜ │ ─────────────⎟\n\ + ⎜ │ 1 ⎟\n\ +╭─╮1, 2 ⎜1, 2 3, 4 │ 1 + ─────────⎟\n\ +│╶┐ ⎜ │ 1 ⎟\n\ +╰─╯4, 3 ⎜ 3 4, 5 │ 1 + ─────⎟\n\ + ⎜ │ 1⎟\n\ + ⎜ │ 1 + ─⎟\n\ + ⎝ │ x⎠\ +""" + + ascii_str = \ +"""\ + / | 1 \\\n\ + | | -------------|\n\ + | | 1 |\n\ + __1, 2 |1, 2 3, 4 | 1 + ---------|\n\ +/__ | | 1 |\n\ +\\_|4, 3 | 3 4, 5 | 1 + -----|\n\ + | | 1|\n\ + | | 1 + -|\n\ + \\ | x/\ +""" + + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = Integral(expr, x) + + ucode_str = \ +"""\ +⌠ \n\ +⎮ ⎛ │ 1 ⎞ \n\ +⎮ ⎜ │ ─────────────⎟ \n\ +⎮ ⎜ │ 1 ⎟ \n\ +⎮ ╭─╮1, 2 ⎜1, 2 3, 4 │ 1 + ─────────⎟ \n\ +⎮ │╶┐ ⎜ │ 1 ⎟ dx\n\ +⎮ ╰─╯4, 3 ⎜ 3 4, 5 │ 1 + ─────⎟ \n\ +⎮ ⎜ │ 1⎟ \n\ +⎮ ⎜ │ 1 + ─⎟ \n\ +⎮ ⎝ │ x⎠ \n\ +⌡ \ +""" + + ascii_str = \ +"""\ + / \n\ + | \n\ + | / | 1 \\ \n\ + | | | -------------| \n\ + | | | 1 | \n\ + | __1, 2 |1, 2 3, 4 | 1 + ---------| \n\ + | /__ | | 1 | dx\n\ + | \\_|4, 3 | 3 4, 5 | 1 + -----| \n\ + | | | 1| \n\ + | | | 1 + -| \n\ + | \\ | x/ \n\ + | \n\ +/ \ +""" + + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + +def test_noncommutative(): + A, B, C = symbols('A,B,C', commutative=False) + + expr = A*B*C**-1 + ascii_str = \ +"""\ + -1\n\ +A*B*C \ +""" + ucode_str = \ +"""\ + -1\n\ +A⋅B⋅C \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = C**-1*A*B + ascii_str = \ +"""\ + -1 \n\ +C *A*B\ +""" + ucode_str = \ +"""\ + -1 \n\ +C ⋅A⋅B\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = A*C**-1*B + ascii_str = \ +"""\ + -1 \n\ +A*C *B\ +""" + ucode_str = \ +"""\ + -1 \n\ +A⋅C ⋅B\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = A*C**-1*B/x + ascii_str = \ +"""\ + -1 \n\ +A*C *B\n\ +-------\n\ + x \ +""" + ucode_str = \ +"""\ + -1 \n\ +A⋅C ⋅B\n\ +───────\n\ + x \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + +def test_pretty_special_functions(): + x, y = symbols("x y") + + # atan2 + expr = atan2(y/sqrt(200), sqrt(x)) + ascii_str = \ +"""\ + / ___ \\\n\ + |\\/ 2 *y ___|\n\ +atan2|-------, \\/ x |\n\ + \\ 20 /\ +""" + ucode_str = \ +"""\ + ⎛√2⋅y ⎞\n\ +atan2⎜────, √x⎟\n\ + ⎝ 20 ⎠\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + +def test_pretty_geometry(): + e = Segment((0, 1), (0, 2)) + assert pretty(e) == 'Segment2D(Point2D(0, 1), Point2D(0, 2))' + e = Ray((1, 1), angle=4.02*pi) + assert pretty(e) == 'Ray2D(Point2D(1, 1), Point2D(2, tan(pi/50) + 1))' + + +def test_expint(): + expr = Ei(x) + string = 'Ei(x)' + assert pretty(expr) == string + assert upretty(expr) == string + + expr = expint(1, z) + ucode_str = "E₁(z)" + ascii_str = "expint(1, z)" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + assert pretty(Shi(x)) == 'Shi(x)' + assert pretty(Si(x)) == 'Si(x)' + assert pretty(Ci(x)) == 'Ci(x)' + assert pretty(Chi(x)) == 'Chi(x)' + assert upretty(Shi(x)) == 'Shi(x)' + assert upretty(Si(x)) == 'Si(x)' + assert upretty(Ci(x)) == 'Ci(x)' + assert upretty(Chi(x)) == 'Chi(x)' + + +def test_elliptic_functions(): + ascii_str = \ +"""\ + / 1 \\\n\ +K|-----|\n\ + \\z + 1/\ +""" + ucode_str = \ +"""\ + ⎛ 1 ⎞\n\ +K⎜─────⎟\n\ + ⎝z + 1⎠\ +""" + expr = elliptic_k(1/(z + 1)) + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + ascii_str = \ +"""\ + / | 1 \\\n\ +F|1|-----|\n\ + \\ |z + 1/\ +""" + ucode_str = \ +"""\ + ⎛ │ 1 ⎞\n\ +F⎜1│─────⎟\n\ + ⎝ │z + 1⎠\ +""" + expr = elliptic_f(1, 1/(1 + z)) + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + ascii_str = \ +"""\ + / 1 \\\n\ +E|-----|\n\ + \\z + 1/\ +""" + ucode_str = \ +"""\ + ⎛ 1 ⎞\n\ +E⎜─────⎟\n\ + ⎝z + 1⎠\ +""" + expr = elliptic_e(1/(z + 1)) + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + ascii_str = \ +"""\ + / | 1 \\\n\ +E|1|-----|\n\ + \\ |z + 1/\ +""" + ucode_str = \ +"""\ + ⎛ │ 1 ⎞\n\ +E⎜1│─────⎟\n\ + ⎝ │z + 1⎠\ +""" + expr = elliptic_e(1, 1/(1 + z)) + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + ascii_str = \ +"""\ + / |4\\\n\ +Pi|3|-|\n\ + \\ |x/\ +""" + ucode_str = \ +"""\ + ⎛ │4⎞\n\ +Π⎜3│─⎟\n\ + ⎝ │x⎠\ +""" + expr = elliptic_pi(3, 4/x) + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + ascii_str = \ +"""\ + / 4| \\\n\ +Pi|3; -|6|\n\ + \\ x| /\ +""" + ucode_str = \ +"""\ + ⎛ 4│ ⎞\n\ +Π⎜3; ─│6⎟\n\ + ⎝ x│ ⎠\ +""" + expr = elliptic_pi(3, 4/x, 6) + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + +def test_RandomDomain(): + from sympy.stats import Normal, Die, Exponential, pspace, where + X = Normal('x1', 0, 1) + assert upretty(where(X > 0)) == "Domain: 0 < x₁ ∧ x₁ < ∞" + + D = Die('d1', 6) + assert upretty(where(D > 4)) == 'Domain: d₁ = 5 ∨ d₁ = 6' + + A = Exponential('a', 1) + B = Exponential('b', 1) + assert upretty(pspace(Tuple(A, B)).domain) == \ + 'Domain: 0 ≤ a ∧ 0 ≤ b ∧ a < ∞ ∧ b < ∞' + + +def test_PrettyPoly(): + F = QQ.frac_field(x, y) + R = QQ.poly_ring(x, y) + + expr = F.convert(x/(x + y)) + assert pretty(expr) == "x/(x + y)" + assert upretty(expr) == "x/(x + y)" + + expr = R.convert(x + y) + assert pretty(expr) == "x + y" + assert upretty(expr) == "x + y" + + +def test_issue_6285(): + assert pretty(Pow(2, -5, evaluate=False)) == '1 \n--\n 5\n2 ' + assert pretty(Pow(x, (1/pi))) == \ + ' 1 \n'\ + ' --\n'\ + ' pi\n'\ + 'x ' + + +def test_issue_6359(): + assert pretty(Integral(x**2, x)**2) == \ +"""\ + 2 +/ / \\ \n\ +| | | \n\ +| | 2 | \n\ +| | x dx| \n\ +| | | \n\ +\\/ / \ +""" + assert upretty(Integral(x**2, x)**2) == \ +"""\ + 2 +⎛⌠ ⎞ \n\ +⎜⎮ 2 ⎟ \n\ +⎜⎮ x dx⎟ \n\ +⎝⌡ ⎠ \ +""" + + assert pretty(Sum(x**2, (x, 0, 1))**2) == \ +"""\ + 2\n\ +/ 1 \\ \n\ +|___ | \n\ +|\\ ` | \n\ +| \\ 2| \n\ +| / x | \n\ +|/__, | \n\ +\\x = 0 / \ +""" + assert upretty(Sum(x**2, (x, 0, 1))**2) == \ +"""\ + 2 +⎛ 1 ⎞ \n\ +⎜ ___ ⎟ \n\ +⎜ ╲ ⎟ \n\ +⎜ ╲ 2⎟ \n\ +⎜ ╱ x ⎟ \n\ +⎜ ╱ ⎟ \n\ +⎜ ‾‾‾ ⎟ \n\ +⎝x = 0 ⎠ \ +""" + + assert pretty(Product(x**2, (x, 1, 2))**2) == \ +"""\ + 2 +/ 2 \\ \n\ +|______ | \n\ +| | | 2| \n\ +| | | x | \n\ +| | | | \n\ +\\x = 1 / \ +""" + assert upretty(Product(x**2, (x, 1, 2))**2) == \ +"""\ + 2 +⎛ 2 ⎞ \n\ +⎜─┬──┬─ ⎟ \n\ +⎜ │ │ 2⎟ \n\ +⎜ │ │ x ⎟ \n\ +⎜ │ │ ⎟ \n\ +⎝x = 1 ⎠ \ +""" + + f = Function('f') + assert pretty(Derivative(f(x), x)**2) == \ +"""\ + 2 +/d \\ \n\ +|--(f(x))| \n\ +\\dx / \ +""" + assert upretty(Derivative(f(x), x)**2) == \ +"""\ + 2 +⎛d ⎞ \n\ +⎜──(f(x))⎟ \n\ +⎝dx ⎠ \ +""" + + +def test_issue_6739(): + ascii_str = \ +"""\ + 1 \n\ +-----\n\ + ___\n\ +\\/ x \ +""" + ucode_str = \ +"""\ +1 \n\ +──\n\ +√x\ +""" + assert pretty(1/sqrt(x)) == ascii_str + assert upretty(1/sqrt(x)) == ucode_str + + +def test_complicated_symbol_unchanged(): + for symb_name in ["dexpr2_d1tau", "dexpr2^d1tau"]: + assert pretty(Symbol(symb_name)) == symb_name + + +def test_categories(): + from sympy.categories import (Object, IdentityMorphism, + NamedMorphism, Category, Diagram, DiagramGrid) + + A1 = Object("A1") + A2 = Object("A2") + A3 = Object("A3") + + f1 = NamedMorphism(A1, A2, "f1") + f2 = NamedMorphism(A2, A3, "f2") + id_A1 = IdentityMorphism(A1) + + K1 = Category("K1") + + assert pretty(A1) == "A1" + assert upretty(A1) == "A₁" + + assert pretty(f1) == "f1:A1-->A2" + assert upretty(f1) == "f₁:A₁——▶A₂" + assert pretty(id_A1) == "id:A1-->A1" + assert upretty(id_A1) == "id:A₁——▶A₁" + + assert pretty(f2*f1) == "f2*f1:A1-->A3" + assert upretty(f2*f1) == "f₂∘f₁:A₁——▶A₃" + + assert pretty(K1) == "K1" + assert upretty(K1) == "K₁" + + # Test how diagrams are printed. + d = Diagram() + assert pretty(d) == "EmptySet" + assert upretty(d) == "∅" + + d = Diagram({f1: "unique", f2: S.EmptySet}) + assert pretty(d) == "{f2*f1:A1-->A3: EmptySet, id:A1-->A1: " \ + "EmptySet, id:A2-->A2: EmptySet, id:A3-->A3: " \ + "EmptySet, f1:A1-->A2: {unique}, f2:A2-->A3: EmptySet}" + + assert upretty(d) == "{f₂∘f₁:A₁——▶A₃: ∅, id:A₁——▶A₁: ∅, " \ + "id:A₂——▶A₂: ∅, id:A₃——▶A₃: ∅, f₁:A₁——▶A₂: {unique}, f₂:A₂——▶A₃: ∅}" + + d = Diagram({f1: "unique", f2: S.EmptySet}, {f2 * f1: "unique"}) + assert pretty(d) == "{f2*f1:A1-->A3: EmptySet, id:A1-->A1: " \ + "EmptySet, id:A2-->A2: EmptySet, id:A3-->A3: " \ + "EmptySet, f1:A1-->A2: {unique}, f2:A2-->A3: EmptySet}" \ + " ==> {f2*f1:A1-->A3: {unique}}" + assert upretty(d) == "{f₂∘f₁:A₁——▶A₃: ∅, id:A₁——▶A₁: ∅, id:A₂——▶A₂: " \ + "∅, id:A₃——▶A₃: ∅, f₁:A₁——▶A₂: {unique}, f₂:A₂——▶A₃: ∅}" \ + " ══▶ {f₂∘f₁:A₁——▶A₃: {unique}}" + + grid = DiagramGrid(d) + assert pretty(grid) == "A1 A2\n \nA3 " + assert upretty(grid) == "A₁ A₂\n \nA₃ " + + +def test_PrettyModules(): + R = QQ.old_poly_ring(x, y) + F = R.free_module(2) + M = F.submodule([x, y], [1, x**2]) + + ucode_str = \ +"""\ + 2\n\ +ℚ[x, y] \ +""" + ascii_str = \ +"""\ + 2\n\ +QQ[x, y] \ +""" + + assert upretty(F) == ucode_str + assert pretty(F) == ascii_str + + ucode_str = \ +"""\ +╱ ⎡ 2⎤╲\n\ +╲[x, y], ⎣1, x ⎦╱\ +""" + ascii_str = \ +"""\ + 2 \n\ +<[x, y], [1, x ]>\ +""" + + assert upretty(M) == ucode_str + assert pretty(M) == ascii_str + + I = R.ideal(x**2, y) + + ucode_str = \ +"""\ +╱ 2 ╲\n\ +╲x , y╱\ +""" + + ascii_str = \ +"""\ + 2 \n\ +\ +""" + + assert upretty(I) == ucode_str + assert pretty(I) == ascii_str + + Q = F / M + + ucode_str = \ +"""\ + 2 \n\ + ℚ[x, y] \n\ +─────────────────\n\ +╱ ⎡ 2⎤╲\n\ +╲[x, y], ⎣1, x ⎦╱\ +""" + + ascii_str = \ +"""\ + 2 \n\ + QQ[x, y] \n\ +-----------------\n\ + 2 \n\ +<[x, y], [1, x ]>\ +""" + + assert upretty(Q) == ucode_str + assert pretty(Q) == ascii_str + + ucode_str = \ +"""\ +╱⎡ 3⎤ ╲\n\ +│⎢ x ⎥ ╱ ⎡ 2⎤╲ ╱ ⎡ 2⎤╲│\n\ +│⎢1, ──⎥ + ╲[x, y], ⎣1, x ⎦╱, [2, y] + ╲[x, y], ⎣1, x ⎦╱│\n\ +╲⎣ 2 ⎦ ╱\ +""" + + ascii_str = \ +"""\ + 3 \n\ + x 2 2 \n\ +<[1, --] + <[x, y], [1, x ]>, [2, y] + <[x, y], [1, x ]>>\n\ + 2 \ +""" + + +def test_QuotientRing(): + R = QQ.old_poly_ring(x)/[x**2 + 1] + + ucode_str = \ +"""\ + ℚ[x] \n\ +────────\n\ +╱ 2 ╲\n\ +╲x + 1╱\ +""" + + ascii_str = \ +"""\ + QQ[x] \n\ +--------\n\ + 2 \n\ +\ +""" + + assert upretty(R) == ucode_str + assert pretty(R) == ascii_str + + ucode_str = \ +"""\ + ╱ 2 ╲\n\ +1 + ╲x + 1╱\ +""" + + ascii_str = \ +"""\ + 2 \n\ +1 + \ +""" + + assert upretty(R.one) == ucode_str + assert pretty(R.one) == ascii_str + + +def test_Homomorphism(): + from sympy.polys.agca import homomorphism + + R = QQ.old_poly_ring(x) + + expr = homomorphism(R.free_module(1), R.free_module(1), [0]) + + ucode_str = \ +"""\ + 1 1\n\ +[0] : ℚ[x] ──> ℚ[x] \ +""" + + ascii_str = \ +"""\ + 1 1\n\ +[0] : QQ[x] --> QQ[x] \ +""" + + assert upretty(expr) == ucode_str + assert pretty(expr) == ascii_str + + expr = homomorphism(R.free_module(2), R.free_module(2), [0, 0]) + + ucode_str = \ +"""\ +⎡0 0⎤ 2 2\n\ +⎢ ⎥ : ℚ[x] ──> ℚ[x] \n\ +⎣0 0⎦ \ +""" + + ascii_str = \ +"""\ +[0 0] 2 2\n\ +[ ] : QQ[x] --> QQ[x] \n\ +[0 0] \ +""" + + assert upretty(expr) == ucode_str + assert pretty(expr) == ascii_str + + expr = homomorphism(R.free_module(1), R.free_module(1) / [[x]], [0]) + + ucode_str = \ +"""\ + 1\n\ + 1 ℚ[x] \n\ +[0] : ℚ[x] ──> ─────\n\ + <[x]>\ +""" + + ascii_str = \ +"""\ + 1\n\ + 1 QQ[x] \n\ +[0] : QQ[x] --> ------\n\ + <[x]> \ +""" + + assert upretty(expr) == ucode_str + assert pretty(expr) == ascii_str + + +def test_Tr(): + A, B = symbols('A B', commutative=False) + t = Tr(A*B) + assert pretty(t) == r'Tr(A*B)' + assert upretty(t) == 'Tr(A⋅B)' + + +def test_pretty_Add(): + eq = Mul(-2, x - 2, evaluate=False) + 5 + assert pretty(eq) == '5 - 2*(x - 2)' + + +def test_issue_7179(): + assert upretty(Not(Equivalent(x, y))) == 'x ⇎ y' + assert upretty(Not(Implies(x, y))) == 'x ↛ y' + + +def test_issue_7180(): + assert upretty(Equivalent(x, y)) == 'x ⇔ y' + + +def test_pretty_Complement(): + assert pretty(S.Reals - S.Naturals) == '(-oo, oo) \\ Naturals' + assert upretty(S.Reals - S.Naturals) == 'ℝ \\ ℕ' + assert pretty(S.Reals - S.Naturals0) == '(-oo, oo) \\ Naturals0' + assert upretty(S.Reals - S.Naturals0) == 'ℝ \\ ℕ₀' + + +def test_pretty_SymmetricDifference(): + from sympy.sets.sets import SymmetricDifference + assert upretty(SymmetricDifference(Interval(2,3), Interval(3,5), \ + evaluate = False)) == '[2, 3] ∆ [3, 5]' + with raises(NotImplementedError): + pretty(SymmetricDifference(Interval(2,3), Interval(3,5), evaluate = False)) + + +def test_pretty_Contains(): + assert pretty(Contains(x, S.Integers)) == 'Contains(x, Integers)' + assert upretty(Contains(x, S.Integers)) == 'x ∈ ℤ' + + +def test_issue_8292(): + from sympy.core import sympify + e = sympify('((x+x**4)/(x-1))-(2*(x-1)**4/(x-1)**4)', evaluate=False) + ucode_str = \ +"""\ + 4 4 \n\ + 2⋅(x - 1) x + x\n\ +- ────────── + ──────\n\ + 4 x - 1 \n\ + (x - 1) \ +""" + ascii_str = \ +"""\ + 4 4 \n\ + 2*(x - 1) x + x\n\ +- ---------- + ------\n\ + 4 x - 1 \n\ + (x - 1) \ +""" + assert pretty(e) == ascii_str + assert upretty(e) == ucode_str + + +def test_issue_4335(): + y = Function('y') + expr = -y(x).diff(x) + ucode_str = \ +"""\ + d \n\ +-──(y(x))\n\ + dx \ +""" + ascii_str = \ +"""\ + d \n\ +- --(y(x))\n\ + dx \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + +def test_issue_8344(): + from sympy.core import sympify + e = sympify('2*x*y**2/1**2 + 1', evaluate=False) + ucode_str = \ +"""\ + 2 \n\ +2⋅x⋅y \n\ +────── + 1\n\ + 2 \n\ + 1 \ +""" + assert upretty(e) == ucode_str + + +def test_issue_6324(): + x = Pow(2, 3, evaluate=False) + y = Pow(10, -2, evaluate=False) + e = Mul(x, y, evaluate=False) + ucode_str = \ +"""\ + 3 \n\ +2 \n\ +───\n\ + 2\n\ +10 \ +""" + assert upretty(e) == ucode_str + + +def test_issue_7927(): + e = sin(x/2)**cos(x/2) + ucode_str = \ +"""\ + ⎛x⎞\n\ + cos⎜─⎟\n\ + ⎝2⎠\n\ +⎛ ⎛x⎞⎞ \n\ +⎜sin⎜─⎟⎟ \n\ +⎝ ⎝2⎠⎠ \ +""" + assert upretty(e) == ucode_str + e = sin(x)**(S(11)/13) + ucode_str = \ +"""\ + 11\n\ + ──\n\ + 13\n\ +(sin(x)) \ +""" + assert upretty(e) == ucode_str + + +def test_issue_6134(): + from sympy.abc import lamda, t + phi = Function('phi') + + e = lamda*x*Integral(phi(t)*pi*sin(pi*t), (t, 0, 1)) + lamda*x**2*Integral(phi(t)*2*pi*sin(2*pi*t), (t, 0, 1)) + ucode_str = \ +"""\ + 1 1 \n\ + 2 ⌠ ⌠ \n\ +λ⋅x ⋅⎮ 2⋅π⋅φ(t)⋅sin(2⋅π⋅t) dt + λ⋅x⋅⎮ π⋅φ(t)⋅sin(π⋅t) dt\n\ + ⌡ ⌡ \n\ + 0 0 \ +""" + assert upretty(e) == ucode_str + + +def test_issue_9877(): + ucode_str1 = '(2, 3) ∪ ([1, 2] \\ {x})' + a, b, c = Interval(2, 3, True, True), Interval(1, 2), FiniteSet(x) + assert upretty(Union(a, Complement(b, c))) == ucode_str1 + + ucode_str2 = '{x} ∩ {y} ∩ ({z} \\ [1, 2])' + d, e, f, g = FiniteSet(x), FiniteSet(y), FiniteSet(z), Interval(1, 2) + assert upretty(Intersection(d, e, Complement(f, g))) == ucode_str2 + + +def test_issue_13651(): + expr1 = c + Mul(-1, a + b, evaluate=False) + assert pretty(expr1) == 'c - (a + b)' + expr2 = c + Mul(-1, a - b + d, evaluate=False) + assert pretty(expr2) == 'c - (a - b + d)' + + +def test_pretty_primenu(): + from sympy.functions.combinatorial.numbers import primenu + + ascii_str1 = "nu(n)" + ucode_str1 = "ν(n)" + + n = symbols('n', integer=True) + assert pretty(primenu(n)) == ascii_str1 + assert upretty(primenu(n)) == ucode_str1 + + +def test_pretty_primeomega(): + from sympy.functions.combinatorial.numbers import primeomega + + ascii_str1 = "Omega(n)" + ucode_str1 = "Ω(n)" + + n = symbols('n', integer=True) + assert pretty(primeomega(n)) == ascii_str1 + assert upretty(primeomega(n)) == ucode_str1 + + +def test_pretty_Mod(): + from sympy.core import Mod + + ascii_str1 = "x mod 7" + ucode_str1 = "x mod 7" + + ascii_str2 = "(x + 1) mod 7" + ucode_str2 = "(x + 1) mod 7" + + ascii_str3 = "2*x mod 7" + ucode_str3 = "2⋅x mod 7" + + ascii_str4 = "(x mod 7) + 1" + ucode_str4 = "(x mod 7) + 1" + + ascii_str5 = "2*(x mod 7)" + ucode_str5 = "2⋅(x mod 7)" + + x = symbols('x', integer=True) + assert pretty(Mod(x, 7)) == ascii_str1 + assert upretty(Mod(x, 7)) == ucode_str1 + assert pretty(Mod(x + 1, 7)) == ascii_str2 + assert upretty(Mod(x + 1, 7)) == ucode_str2 + assert pretty(Mod(2 * x, 7)) == ascii_str3 + assert upretty(Mod(2 * x, 7)) == ucode_str3 + assert pretty(Mod(x, 7) + 1) == ascii_str4 + assert upretty(Mod(x, 7) + 1) == ucode_str4 + assert pretty(2 * Mod(x, 7)) == ascii_str5 + assert upretty(2 * Mod(x, 7)) == ucode_str5 + + +def test_issue_11801(): + assert pretty(Symbol("")) == "" + assert upretty(Symbol("")) == "" + + +def test_pretty_UnevaluatedExpr(): + x = symbols('x') + he = UnevaluatedExpr(1/x) + + ucode_str = \ +"""\ +1\n\ +─\n\ +x\ +""" + + assert upretty(he) == ucode_str + + ucode_str = \ +"""\ + 2\n\ +⎛1⎞ \n\ +⎜─⎟ \n\ +⎝x⎠ \ +""" + + assert upretty(he**2) == ucode_str + + ucode_str = \ +"""\ + 1\n\ +1 + ─\n\ + x\ +""" + + assert upretty(he + 1) == ucode_str + + ucode_str = \ +('''\ + 1\n\ +x⋅─\n\ + x\ +''') + assert upretty(x*he) == ucode_str + + +def test_issue_10472(): + M = (Matrix([[0, 0], [0, 0]]), Matrix([0, 0])) + + ucode_str = \ +"""\ +⎛⎡0 0⎤ ⎡0⎤⎞ +⎜⎢ ⎥, ⎢ ⎥⎟ +⎝⎣0 0⎦ ⎣0⎦⎠\ +""" + assert upretty(M) == ucode_str + + +def test_MatrixElement_printing(): + # test cases for issue #11821 + A = MatrixSymbol("A", 1, 3) + B = MatrixSymbol("B", 1, 3) + C = MatrixSymbol("C", 1, 3) + + ascii_str1 = "A_00" + ucode_str1 = "A₀₀" + assert pretty(A[0, 0]) == ascii_str1 + assert upretty(A[0, 0]) == ucode_str1 + + ascii_str1 = "3*A_00" + ucode_str1 = "3⋅A₀₀" + assert pretty(3*A[0, 0]) == ascii_str1 + assert upretty(3*A[0, 0]) == ucode_str1 + + ascii_str1 = "(-B + A)[0, 0]" + ucode_str1 = "(-B + A)[0, 0]" + F = C[0, 0].subs(C, A - B) + assert pretty(F) == ascii_str1 + assert upretty(F) == ucode_str1 + + +def test_issue_12675(): + x, y, t, j = symbols('x y t j') + e = CoordSys3D('e') + + ucode_str = \ +"""\ +⎛ t⎞ \n\ +⎜⎛x⎞ ⎟ j_e\n\ +⎜⎜─⎟ ⎟ \n\ +⎝⎝y⎠ ⎠ \ +""" + assert upretty((x/y)**t*e.j) == ucode_str + ucode_str = \ +"""\ +⎛1⎞ \n\ +⎜─⎟ j_e\n\ +⎝y⎠ \ +""" + assert upretty((1/y)*e.j) == ucode_str + + +def test_MatrixSymbol_printing(): + # test cases for issue #14237 + A = MatrixSymbol("A", 3, 3) + B = MatrixSymbol("B", 3, 3) + C = MatrixSymbol("C", 3, 3) + assert pretty(-A*B*C) == "-A*B*C" + assert pretty(A - B) == "-B + A" + assert pretty(A*B*C - A*B - B*C) == "-A*B -B*C + A*B*C" + + # issue #14814 + x = MatrixSymbol('x', n, n) + y = MatrixSymbol('y*', n, n) + assert pretty(x + y) == "x + y*" + ascii_str = \ +"""\ + 2 \n\ +-2*y* -a*x\ +""" + assert pretty(-a*x + -2*y*y) == ascii_str + + +def test_degree_printing(): + expr1 = 90*degree + assert pretty(expr1) == '90°' + expr2 = x*degree + assert pretty(expr2) == 'x°' + expr3 = cos(x*degree + 90*degree) + assert pretty(expr3) == 'cos(x° + 90°)' + + +def test_vector_expr_pretty_printing(): + A = CoordSys3D('A') + + assert upretty(Cross(A.i, A.x*A.i+3*A.y*A.j)) == "(i_A)×((x_A) i_A + (3⋅y_A) j_A)" + assert upretty(x*Cross(A.i, A.j)) == 'x⋅(i_A)×(j_A)' + + assert upretty(Curl(A.x*A.i + 3*A.y*A.j)) == "∇×((x_A) i_A + (3⋅y_A) j_A)" + + assert upretty(Divergence(A.x*A.i + 3*A.y*A.j)) == "∇⋅((x_A) i_A + (3⋅y_A) j_A)" + + assert upretty(Dot(A.i, A.x*A.i+3*A.y*A.j)) == "(i_A)⋅((x_A) i_A + (3⋅y_A) j_A)" + + assert upretty(Gradient(A.x+3*A.y)) == "∇(x_A + 3⋅y_A)" + assert upretty(Laplacian(A.x+3*A.y)) == "∆(x_A + 3⋅y_A)" + # TODO: add support for ASCII pretty. + + +def test_pretty_print_tensor_expr(): + L = TensorIndexType("L") + i, j, k = tensor_indices("i j k", L) + i0 = tensor_indices("i_0", L) + A, B, C, D = tensor_heads("A B C D", [L]) + H = TensorHead("H", [L, L]) + + expr = -i + ascii_str = \ +"""\ +-i\ +""" + ucode_str = \ +"""\ +-i\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = A(i) + ascii_str = \ +"""\ + i\n\ +A \n\ + \ +""" + ucode_str = \ +"""\ + i\n\ +A \n\ + \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = A(i0) + ascii_str = \ +"""\ + i_0\n\ +A \n\ + \ +""" + ucode_str = \ +"""\ + i₀\n\ +A \n\ + \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = A(-i) + ascii_str = \ +"""\ + \n\ +A \n\ + i\ +""" + ucode_str = \ +"""\ + \n\ +A \n\ + i\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = -3*A(-i) + ascii_str = \ +"""\ + \n\ +-3*A \n\ + i\ +""" + ucode_str = \ +"""\ + \n\ +-3⋅A \n\ + i\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = H(i, -j) + ascii_str = \ +"""\ + i \n\ +H \n\ + j\ +""" + ucode_str = \ +"""\ + i \n\ +H \n\ + j\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = H(i, -i) + ascii_str = \ +"""\ + L_0 \n\ +H \n\ + L_0\ +""" + ucode_str = \ +"""\ + L₀ \n\ +H \n\ + L₀\ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = H(i, -j)*A(j)*B(k) + ascii_str = \ +"""\ + i L_0 k\n\ +H *A *B \n\ + L_0 \ +""" + ucode_str = \ +"""\ + i L₀ k\n\ +H ⋅A ⋅B \n\ + L₀ \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = (1+x)*A(i) + ascii_str = \ +"""\ + i\n\ +(x + 1)*A \n\ + \ +""" + ucode_str = \ +"""\ + i\n\ +(x + 1)⋅A \n\ + \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = A(i) + 3*B(i) + ascii_str = \ +"""\ + i i\n\ +3*B + A \n\ + \ +""" + ucode_str = \ +"""\ + i i\n\ +3⋅B + A \n\ + \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + +def test_pretty_print_tensor_partial_deriv(): + from sympy.tensor.toperators import PartialDerivative + + L = TensorIndexType("L") + i, j, k = tensor_indices("i j k", L) + + A, B, C, D = tensor_heads("A B C D", [L]) + + H = TensorHead("H", [L, L]) + + expr = PartialDerivative(A(i), A(j)) + ascii_str = \ +"""\ + d / i\\\n\ +---|A |\n\ + j\\ /\n\ +dA \n\ + \ +""" + ucode_str = \ +"""\ + ∂ ⎛ i⎞\n\ +───⎜A ⎟\n\ + j⎝ ⎠\n\ +∂A \n\ + \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = A(i)*PartialDerivative(H(k, -i), A(j)) + ascii_str = \ +"""\ + L_0 d / k \\\n\ +A *---|H |\n\ + j\\ L_0/\n\ + dA \n\ + \ +""" + ucode_str = \ +"""\ + L₀ ∂ ⎛ k ⎞\n\ +A ⋅───⎜H ⎟\n\ + j⎝ L₀⎠\n\ + ∂A \n\ + \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = A(i)*PartialDerivative(B(k)*C(-i) + 3*H(k, -i), A(j)) + ascii_str = \ +"""\ + L_0 d / k k \\\n\ +A *---|3*H + B *C |\n\ + j\\ L_0 L_0/\n\ + dA \n\ + \ +""" + ucode_str = \ +"""\ + L₀ ∂ ⎛ k k ⎞\n\ +A ⋅───⎜3⋅H + B ⋅C ⎟\n\ + j⎝ L₀ L₀⎠\n\ + ∂A \n\ + \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = (A(i) + B(i))*PartialDerivative(C(j), D(j)) + ascii_str = \ +"""\ +/ i i\\ d / L_0\\\n\ +|A + B |*-----|C |\n\ +\\ / L_0\\ /\n\ + dD \n\ + \ +""" + ucode_str = \ +"""\ +⎛ i i⎞ ∂ ⎛ L₀⎞\n\ +⎜A + B ⎟⋅────⎜C ⎟\n\ +⎝ ⎠ L₀⎝ ⎠\n\ + ∂D \n\ + \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = (A(i) + B(i))*PartialDerivative(C(-i), D(j)) + ascii_str = \ +"""\ +/ L_0 L_0\\ d / \\\n\ +|A + B |*---|C |\n\ +\\ / j\\ L_0/\n\ + dD \n\ + \ +""" + ucode_str = \ +"""\ +⎛ L₀ L₀⎞ ∂ ⎛ ⎞\n\ +⎜A + B ⎟⋅───⎜C ⎟\n\ +⎝ ⎠ j⎝ L₀⎠\n\ + ∂D \n\ + \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = PartialDerivative(B(-i) + A(-i), A(-j), A(-n)) + ucode_str = """\ + 2 \n\ + ∂ ⎛ ⎞\n\ +───────⎜A + B ⎟\n\ + ⎝ i i⎠\n\ +∂A ∂A \n\ + n j \ +""" + assert upretty(expr) == ucode_str + + expr = PartialDerivative(3*A(-i), A(-j), A(-n)) + ucode_str = """\ + 2 \n\ + ∂ ⎛ ⎞\n\ +───────⎜3⋅A ⎟\n\ + ⎝ i⎠\n\ +∂A ∂A \n\ + n j \ +""" + assert upretty(expr) == ucode_str + + expr = TensorElement(H(i, j), {i:1}) + ascii_str = \ +"""\ + i=1,j\n\ +H \n\ + \ +""" + ucode_str = ascii_str + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = TensorElement(H(i, j), {i: 1, j: 1}) + ascii_str = \ +"""\ + i=1,j=1\n\ +H \n\ + \ +""" + ucode_str = ascii_str + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = TensorElement(H(i, j), {j: 1}) + ascii_str = \ +"""\ + i,j=1\n\ +H \n\ + \ +""" + ucode_str = ascii_str + + expr = TensorElement(H(-i, j), {-i: 1}) + ascii_str = \ +"""\ + j\n\ +H \n\ + i=1 \ +""" + ucode_str = ascii_str + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + +def test_issue_15560(): + a = MatrixSymbol('a', 1, 1) + e = pretty(a*(KroneckerProduct(a, a))) + result = 'a*(a x a)' + assert e == result + + +def test_print_polylog(): + # Part of issue 6013 + uresult = 'Li₂(3)' + aresult = 'polylog(2, 3)' + assert pretty(polylog(2, 3)) == aresult + assert upretty(polylog(2, 3)) == uresult + + +# Issue #25312 +def test_print_expint_polylog_symbolic_order(): + s, z = symbols("s, z") + uresult = 'Liₛ(z)' + aresult = 'polylog(s, z)' + assert pretty(polylog(s, z)) == aresult + assert upretty(polylog(s, z)) == uresult + # TODO: TBD polylog(s - 1, z) + uresult = 'Eₛ(z)' + aresult = 'expint(s, z)' + assert pretty(expint(s, z)) == aresult + assert upretty(expint(s, z)) == uresult + + + +def test_print_polylog_long_order_issue_25309(): + s, z = symbols("s, z") + ucode_str = \ +"""\ + ⎛ 2 ⎞\n\ +polylog⎝s , z⎠\ +""" + assert upretty(polylog(s**2, z)) == ucode_str + + +def test_print_lerchphi(): + # Part of issue 6013 + a = Symbol('a') + pretty(lerchphi(a, 1, 2)) + uresult = 'Φ(a, 1, 2)' + aresult = 'lerchphi(a, 1, 2)' + assert pretty(lerchphi(a, 1, 2)) == aresult + assert upretty(lerchphi(a, 1, 2)) == uresult + + +def test_issue_15583(): + + N = mechanics.ReferenceFrame('N') + result = '(n_x, n_y, n_z)' + e = pretty((N.x, N.y, N.z)) + assert e == result + + +def test_matrixSymbolBold(): + # Issue 15871 + def boldpretty(expr): + return xpretty(expr, use_unicode=True, wrap_line=False, mat_symbol_style="bold") + + from sympy.matrices.expressions.trace import trace + A = MatrixSymbol("A", 2, 2) + assert boldpretty(trace(A)) == 'tr(𝐀)' + + A = MatrixSymbol("A", 3, 3) + B = MatrixSymbol("B", 3, 3) + C = MatrixSymbol("C", 3, 3) + + assert boldpretty(-A) == '-𝐀' + assert boldpretty(A - A*B - B) == '-𝐁 -𝐀⋅𝐁 + 𝐀' + assert boldpretty(-A*B - A*B*C - B) == '-𝐁 -𝐀⋅𝐁 -𝐀⋅𝐁⋅𝐂' + + A = MatrixSymbol("Addot", 3, 3) + assert boldpretty(A) == '𝐀̈' + omega = MatrixSymbol("omega", 3, 3) + assert boldpretty(omega) == 'ω' + omega = MatrixSymbol("omeganorm", 3, 3) + assert boldpretty(omega) == '‖ω‖' + + a = Symbol('alpha') + b = Symbol('b') + c = MatrixSymbol("c", 3, 1) + d = MatrixSymbol("d", 3, 1) + + assert boldpretty(a*B*c+b*d) == 'b⋅𝐝 + α⋅𝐁⋅𝐜' + + d = MatrixSymbol("delta", 3, 1) + B = MatrixSymbol("Beta", 3, 3) + + assert boldpretty(a*B*c+b*d) == 'b⋅δ + α⋅Β⋅𝐜' + + A = MatrixSymbol("A_2", 3, 3) + assert boldpretty(A) == '𝐀₂' + + +def test_center_accent(): + assert center_accent('a', '\N{COMBINING TILDE}') == 'ã' + assert center_accent('aa', '\N{COMBINING TILDE}') == 'aã' + assert center_accent('aaa', '\N{COMBINING TILDE}') == 'aãa' + assert center_accent('aaaa', '\N{COMBINING TILDE}') == 'aaãa' + assert center_accent('aaaaa', '\N{COMBINING TILDE}') == 'aaãaa' + assert center_accent('abcdefg', '\N{COMBINING FOUR DOTS ABOVE}') == 'abcd⃜efg' + + +def test_imaginary_unit(): + from sympy.printing.pretty import pretty # b/c it was redefined above + assert pretty(1 + I, use_unicode=False) == '1 + I' + assert pretty(1 + I, use_unicode=True) == '1 + ⅈ' + assert pretty(1 + I, use_unicode=False, imaginary_unit='j') == '1 + I' + assert pretty(1 + I, use_unicode=True, imaginary_unit='j') == '1 + ⅉ' + + raises(TypeError, lambda: pretty(I, imaginary_unit=I)) + raises(ValueError, lambda: pretty(I, imaginary_unit="kkk")) + + +def test_str_special_matrices(): + from sympy.matrices import Identity, ZeroMatrix, OneMatrix + assert pretty(Identity(4)) == 'I' + assert upretty(Identity(4)) == '𝕀' + assert pretty(ZeroMatrix(2, 2)) == '0' + assert upretty(ZeroMatrix(2, 2)) == '𝟘' + assert pretty(OneMatrix(2, 2)) == '1' + assert upretty(OneMatrix(2, 2)) == '𝟙' + + +def test_pretty_misc_functions(): + assert pretty(LambertW(x)) == 'W(x)' + assert upretty(LambertW(x)) == 'W(x)' + assert pretty(LambertW(x, y)) == 'W(x, y)' + assert upretty(LambertW(x, y)) == 'W(x, y)' + assert pretty(airyai(x)) == 'Ai(x)' + assert upretty(airyai(x)) == 'Ai(x)' + assert pretty(airybi(x)) == 'Bi(x)' + assert upretty(airybi(x)) == 'Bi(x)' + assert pretty(airyaiprime(x)) == "Ai'(x)" + assert upretty(airyaiprime(x)) == "Ai'(x)" + assert pretty(airybiprime(x)) == "Bi'(x)" + assert upretty(airybiprime(x)) == "Bi'(x)" + assert pretty(fresnelc(x)) == 'C(x)' + assert upretty(fresnelc(x)) == 'C(x)' + assert pretty(fresnels(x)) == 'S(x)' + assert upretty(fresnels(x)) == 'S(x)' + assert pretty(Heaviside(x)) == 'Heaviside(x)' + assert upretty(Heaviside(x)) == 'θ(x)' + assert pretty(Heaviside(x, y)) == 'Heaviside(x, y)' + assert upretty(Heaviside(x, y)) == 'θ(x, y)' + assert pretty(dirichlet_eta(x)) == 'dirichlet_eta(x)' + assert upretty(dirichlet_eta(x)) == 'η(x)' + + +def test_hadamard_power(): + m, n, p = symbols('m, n, p', integer=True) + A = MatrixSymbol('A', m, n) + B = MatrixSymbol('B', m, n) + + # Testing printer: + expr = hadamard_power(A, n) + ascii_str = \ +"""\ + .n\n\ +A \ +""" + ucode_str = \ +"""\ + ∘n\n\ +A \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = hadamard_power(A, 1+n) + ascii_str = \ +"""\ + .(n + 1)\n\ +A \ +""" + ucode_str = \ +"""\ + ∘(n + 1)\n\ +A \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + expr = hadamard_power(A*B.T, 1+n) + ascii_str = \ +"""\ + .(n + 1)\n\ +/ T\\ \n\ +\\A*B / \ +""" + ucode_str = \ +"""\ + ∘(n + 1)\n\ +⎛ T⎞ \n\ +⎝A⋅B ⎠ \ +""" + assert pretty(expr) == ascii_str + assert upretty(expr) == ucode_str + + +def test_issue_17258(): + n = Symbol('n', integer=True) + assert pretty(Sum(n, (n, -oo, 1))) == \ + ' 1 \n'\ + ' __ \n'\ + ' \\ ` \n'\ + ' ) n\n'\ + ' /_, \n'\ + 'n = -oo ' + + assert upretty(Sum(n, (n, -oo, 1))) == \ +"""\ + 1 \n\ + ___ \n\ + ╲ \n\ + ╲ \n\ + ╱ n\n\ + ╱ \n\ + ‾‾‾ \n\ +n = -∞ \ +""" + + +def test_is_combining(): + line = "v̇_m" + assert [is_combining(sym) for sym in line] == \ + [False, True, False, False] + + +def test_issue_17616(): + assert pretty(pi**(1/exp(1))) == \ + ' / -1\\\n'\ + ' \\e /\n'\ + 'pi ' + + assert upretty(pi**(1/exp(1))) == \ + ' ⎛ -1⎞\n'\ + ' ⎝ℯ ⎠\n'\ + 'π ' + + assert pretty(pi**(1/pi)) == \ + ' 1 \n'\ + ' --\n'\ + ' pi\n'\ + 'pi ' + + assert upretty(pi**(1/pi)) == \ + ' 1\n'\ + ' ─\n'\ + ' π\n'\ + 'π ' + + assert pretty(pi**(1/EulerGamma)) == \ + ' 1 \n'\ + ' ----------\n'\ + ' EulerGamma\n'\ + 'pi ' + + assert upretty(pi**(1/EulerGamma)) == \ + ' 1\n'\ + ' ─\n'\ + ' γ\n'\ + 'π ' + + z = Symbol("x_17") + assert upretty(7**(1/z)) == \ + 'x₁₇___\n'\ + ' ╲╱ 7 ' + + assert pretty(7**(1/z)) == \ + 'x_17___\n'\ + ' \\/ 7 ' + + +def test_issue_17857(): + assert pretty(Range(-oo, oo)) == '{..., -1, 0, 1, ...}' + assert pretty(Range(oo, -oo, -1)) == '{..., 1, 0, -1, ...}' + + +def test_issue_18272(): + x = Symbol('x') + n = Symbol('n') + + assert upretty(ConditionSet(x, Eq(-x + exp(x), 0), S.Complexes)) == \ + '⎧ │ ⎛ x ⎞⎫\n'\ + '⎨x │ x ∊ ℂ ∧ ⎝-x + ℯ = 0⎠⎬\n'\ + '⎩ │ ⎭' + assert upretty(ConditionSet(x, Contains(n/2, Interval(0, oo)), FiniteSet(-n/2, n/2))) == \ + '⎧ │ ⎧-n n⎫ ⎛n ⎞⎫\n'\ + '⎨x │ x ∊ ⎨───, ─⎬ ∧ ⎜─ ∈ [0, ∞)⎟⎬\n'\ + '⎩ │ ⎩ 2 2⎭ ⎝2 ⎠⎭' + assert upretty(ConditionSet(x, Eq(Piecewise((1, x >= 3), (x/2 - 1/2, x >= 2), (1/2, x >= 1), + (x/2, True)) - 1/2, 0), Interval(0, 3))) == \ + '⎧ │ ⎛⎛⎧ 1 for x ≥ 3⎞ ⎞⎫\n'\ + '⎪ │ ⎜⎜⎪ ⎟ ⎟⎪\n'\ + '⎪ │ ⎜⎜⎪x ⎟ ⎟⎪\n'\ + '⎪ │ ⎜⎜⎪─ - 0.5 for x ≥ 2⎟ ⎟⎪\n'\ + '⎪ │ ⎜⎜⎪2 ⎟ ⎟⎪\n'\ + '⎨x │ x ∊ [0, 3] ∧ ⎜⎜⎨ ⎟ - 0.5 = 0⎟⎬\n'\ + '⎪ │ ⎜⎜⎪ 0.5 for x ≥ 1⎟ ⎟⎪\n'\ + '⎪ │ ⎜⎜⎪ ⎟ ⎟⎪\n'\ + '⎪ │ ⎜⎜⎪ x ⎟ ⎟⎪\n'\ + '⎪ │ ⎜⎜⎪ ─ otherwise⎟ ⎟⎪\n'\ + '⎩ │ ⎝⎝⎩ 2 ⎠ ⎠⎭' + + +def test_Str(): + from sympy.core.symbol import Str + assert pretty(Str('x')) == 'x' + + +def test_symbolic_probability(): + mu = symbols("mu") + sigma = symbols("sigma", positive=True) + X = Normal("X", mu, sigma) + assert pretty(Expectation(X)) == r'E[X]' + assert pretty(Variance(X)) == r'Var(X)' + assert pretty(Probability(X > 0)) == r'P(X > 0)' + Y = Normal("Y", mu, sigma) + assert pretty(Covariance(X, Y)) == 'Cov(X, Y)' + + +def test_issue_21758(): + from sympy.functions.elementary.piecewise import piecewise_fold + from sympy.series.fourier import FourierSeries + x = Symbol('x') + k, n = symbols('k n') + fo = FourierSeries(x, (x, -pi, pi), (0, SeqFormula(0, (k, 1, oo)), SeqFormula( + Piecewise((-2*pi*cos(n*pi)/n + 2*sin(n*pi)/n**2, (n > -oo) & (n < oo) & Ne(n, 0)), + (0, True))*sin(n*x)/pi, (n, 1, oo)))) + assert upretty(piecewise_fold(fo)) == \ + '⎧ 2⋅sin(3⋅x) \n'\ + '⎪2⋅sin(x) - sin(2⋅x) + ────────── + … for n > -∞ ∧ n < ∞ ∧ n ≠ 0\n'\ + '⎨ 3 \n'\ + '⎪ \n'\ + '⎩ 0 otherwise ' + assert pretty(FourierSeries(x, (x, -pi, pi), (0, SeqFormula(0, (k, 1, oo)), + SeqFormula(0, (n, 1, oo))))) == '0' + + +def test_diffgeom(): + from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField + x,y = symbols('x y', real=True) + m = Manifold('M', 2) + assert pretty(m) == 'M' + p = Patch('P', m) + assert pretty(p) == "P" + rect = CoordSystem('rect', p, [x, y]) + assert pretty(rect) == "rect" + b = BaseScalarField(rect, 0) + assert pretty(b) == "x" + + +def test_deprecated_prettyForm(): + with warns_deprecated_sympy(): + from sympy.printing.pretty.pretty_symbology import xstr + assert xstr(1) == '1' + + with warns_deprecated_sympy(): + from sympy.printing.pretty.stringpict import prettyForm + p = prettyForm('s', unicode='s') + + with warns_deprecated_sympy(): + assert p.unicode == p.s == 's' + + +def test_center(): + assert center('1', 2) == '1 ' + assert center('1', 3) == ' 1 ' + assert center('1', 3, '-') == '-1-' + assert center('1', 5, '-') == '--1--' diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/preview.py b/.venv/lib/python3.12/site-packages/sympy/printing/preview.py new file mode 100644 index 0000000000000000000000000000000000000000..b04a344b5b4acc086eb84ff068bc1c6a8b55d811 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/preview.py @@ -0,0 +1,390 @@ +import os +from os.path import join +import shutil +import tempfile +from pathlib import Path + +try: + from subprocess import STDOUT, CalledProcessError, check_output +except ImportError: + pass + +from sympy.utilities.decorator import doctest_depends_on +from sympy.utilities.misc import debug +from .latex import latex + +__doctest_requires__ = {('preview',): ['pyglet']} + + +def _check_output_no_window(*args, **kwargs): + # Avoid showing a cmd.exe window when running this + # on Windows + if os.name == 'nt': + creation_flag = 0x08000000 # CREATE_NO_WINDOW + else: + creation_flag = 0 # Default value + return check_output(*args, creationflags=creation_flag, **kwargs) + + +def system_default_viewer(fname, fmt): + """ Open fname with the default system viewer. + + In practice, it is impossible for python to know when the system viewer is + done. For this reason, we ensure the passed file will not be deleted under + it, and this function does not attempt to block. + """ + # copy to a new temporary file that will not be deleted + with tempfile.NamedTemporaryFile(prefix='sympy-preview-', + suffix=os.path.splitext(fname)[1], + delete=False) as temp_f: + with open(fname, 'rb') as f: + shutil.copyfileobj(f, temp_f) + + import platform + if platform.system() == 'Darwin': + import subprocess + subprocess.call(('open', temp_f.name)) + elif platform.system() == 'Windows': + os.startfile(temp_f.name) + else: + import subprocess + subprocess.call(('xdg-open', temp_f.name)) + + +def pyglet_viewer(fname, fmt): + try: + from pyglet import window, image, gl + from pyglet.window import key + from pyglet.image.codecs import ImageDecodeException + except ImportError: + raise ImportError("pyglet is required for preview.\n visit https://pyglet.org/") + + try: + img = image.load(fname) + except ImageDecodeException: + raise ValueError("pyglet preview does not work for '{}' files.".format(fmt)) + + offset = 25 + + config = gl.Config(double_buffer=False) + win = window.Window( + width=img.width + 2*offset, + height=img.height + 2*offset, + caption="SymPy", + resizable=False, + config=config + ) + + win.set_vsync(False) + + try: + def on_close(): + win.has_exit = True + + win.on_close = on_close + + def on_key_press(symbol, modifiers): + if symbol in [key.Q, key.ESCAPE]: + on_close() + + win.on_key_press = on_key_press + + def on_expose(): + gl.glClearColor(1.0, 1.0, 1.0, 1.0) + gl.glClear(gl.GL_COLOR_BUFFER_BIT) + + img.blit( + (win.width - img.width) / 2, + (win.height - img.height) / 2 + ) + + win.on_expose = on_expose + + while not win.has_exit: + win.dispatch_events() + win.flip() + except KeyboardInterrupt: + pass + + win.close() + + +def _get_latex_main(expr, *, preamble=None, packages=(), extra_preamble=None, + euler=True, fontsize=None, **latex_settings): + """ + Generate string of a LaTeX document rendering ``expr``. + """ + if preamble is None: + actual_packages = packages + ("amsmath", "amsfonts") + if euler: + actual_packages += ("euler",) + package_includes = "\n" + "\n".join(["\\usepackage{%s}" % p + for p in actual_packages]) + if extra_preamble: + package_includes += extra_preamble + + if not fontsize: + fontsize = "12pt" + elif isinstance(fontsize, int): + fontsize = "{}pt".format(fontsize) + preamble = r"""\documentclass[varwidth,%s]{standalone} +%s + +\begin{document} +""" % (fontsize, package_includes) + else: + if packages or extra_preamble: + raise ValueError("The \"packages\" or \"extra_preamble\" keywords" + "must not be set if a " + "custom LaTeX preamble was specified") + + if isinstance(expr, str): + latex_string = expr + else: + latex_string = ('$\\displaystyle ' + + latex(expr, mode='plain', **latex_settings) + + '$') + + return preamble + '\n' + latex_string + '\n\n' + r"\end{document}" + + +@doctest_depends_on(exe=('latex', 'dvipng'), modules=('pyglet',), + disable_viewers=('evince', 'gimp', 'superior-dvi-viewer')) +def preview(expr, output='png', viewer=None, euler=True, packages=(), + filename=None, outputbuffer=None, preamble=None, dvioptions=None, + outputTexFile=None, extra_preamble=None, fontsize=None, + **latex_settings): + r""" + View expression or LaTeX markup in PNG, DVI, PostScript or PDF form. + + If the expr argument is an expression, it will be exported to LaTeX and + then compiled using the available TeX distribution. The first argument, + 'expr', may also be a LaTeX string. The function will then run the + appropriate viewer for the given output format or use the user defined + one. By default png output is generated. + + By default pretty Euler fonts are used for typesetting (they were used to + typeset the well known "Concrete Mathematics" book). For that to work, you + need the 'eulervm.sty' LaTeX style (in Debian/Ubuntu, install the + texlive-fonts-extra package). If you prefer default AMS fonts or your + system lacks 'eulervm' LaTeX package then unset the 'euler' keyword + argument. + + To use viewer auto-detection, lets say for 'png' output, issue + + >>> from sympy import symbols, preview, Symbol + >>> x, y = symbols("x,y") + + >>> preview(x + y, output='png') + + This will choose 'pyglet' by default. To select a different one, do + + >>> preview(x + y, output='png', viewer='gimp') + + The 'png' format is considered special. For all other formats the rules + are slightly different. As an example we will take 'dvi' output format. If + you would run + + >>> preview(x + y, output='dvi') + + then 'view' will look for available 'dvi' viewers on your system + (predefined in the function, so it will try evince, first, then kdvi and + xdvi). If nothing is found, it will fall back to using a system file + association (via ``open`` and ``xdg-open``). To always use your system file + association without searching for the above readers, use + + >>> from sympy.printing.preview import system_default_viewer + >>> preview(x + y, output='dvi', viewer=system_default_viewer) + + If this still does not find the viewer you want, it can be set explicitly. + + >>> preview(x + y, output='dvi', viewer='superior-dvi-viewer') + + This will skip auto-detection and will run user specified + 'superior-dvi-viewer'. If ``view`` fails to find it on your system it will + gracefully raise an exception. + + You may also enter ``'file'`` for the viewer argument. Doing so will cause + this function to return a file object in read-only mode, if ``filename`` + is unset. However, if it was set, then 'preview' writes the generated + file to this filename instead. + + There is also support for writing to a ``io.BytesIO`` like object, which + needs to be passed to the ``outputbuffer`` argument. + + >>> from io import BytesIO + >>> obj = BytesIO() + >>> preview(x + y, output='png', viewer='BytesIO', + ... outputbuffer=obj) + + The LaTeX preamble can be customized by setting the 'preamble' keyword + argument. This can be used, e.g., to set a different font size, use a + custom documentclass or import certain set of LaTeX packages. + + >>> preamble = "\\documentclass[10pt]{article}\n" \ + ... "\\usepackage{amsmath,amsfonts}\\begin{document}" + >>> preview(x + y, output='png', preamble=preamble) + + It is also possible to use the standard preamble and provide additional + information to the preamble using the ``extra_preamble`` keyword argument. + + >>> from sympy import sin + >>> extra_preamble = "\\renewcommand{\\sin}{\\cos}" + >>> preview(sin(x), output='png', extra_preamble=extra_preamble) + + If the value of 'output' is different from 'dvi' then command line + options can be set ('dvioptions' argument) for the execution of the + 'dvi'+output conversion tool. These options have to be in the form of a + list of strings (see ``subprocess.Popen``). + + Additional keyword args will be passed to the :func:`~sympy.printing.latex.latex` call, + e.g., the ``symbol_names`` flag. + + >>> phidd = Symbol('phidd') + >>> preview(phidd, symbol_names={phidd: r'\ddot{\varphi}'}) + + For post-processing the generated TeX File can be written to a file by + passing the desired filename to the 'outputTexFile' keyword + argument. To write the TeX code to a file named + ``"sample.tex"`` and run the default png viewer to display the resulting + bitmap, do + + >>> preview(x + y, outputTexFile="sample.tex") + + + """ + # pyglet is the default for png + if viewer is None and output == "png": + try: + import pyglet # noqa: F401 + except ImportError: + pass + else: + viewer = pyglet_viewer + + # look up a known application + if viewer is None: + # sorted in order from most pretty to most ugly + # very discussable, but indeed 'gv' looks awful :) + candidates = { + "dvi": [ "evince", "okular", "kdvi", "xdvi" ], + "ps": [ "evince", "okular", "gsview", "gv" ], + "pdf": [ "evince", "okular", "kpdf", "acroread", "xpdf", "gv" ], + } + + for candidate in candidates.get(output, []): + path = shutil.which(candidate) + if path is not None: + viewer = path + break + + # otherwise, use the system default for file association + if viewer is None: + viewer = system_default_viewer + + if viewer == "file": + if filename is None: + raise ValueError("filename has to be specified if viewer=\"file\"") + elif viewer == "BytesIO": + if outputbuffer is None: + raise ValueError("outputbuffer has to be a BytesIO " + "compatible object if viewer=\"BytesIO\"") + elif not callable(viewer) and not shutil.which(viewer): + raise OSError("Unrecognized viewer: %s" % viewer) + + latex_main = _get_latex_main(expr, preamble=preamble, packages=packages, + euler=euler, extra_preamble=extra_preamble, + fontsize=fontsize, **latex_settings) + + debug("Latex code:") + debug(latex_main) + with tempfile.TemporaryDirectory() as workdir: + Path(join(workdir, 'texput.tex')).write_text(latex_main, encoding='utf-8') + + if outputTexFile is not None: + shutil.copyfile(join(workdir, 'texput.tex'), outputTexFile) + + if not shutil.which('latex'): + raise RuntimeError("latex program is not installed") + + try: + _check_output_no_window( + ['latex', '-halt-on-error', '-interaction=nonstopmode', + 'texput.tex'], + cwd=workdir, + stderr=STDOUT) + except CalledProcessError as e: + raise RuntimeError( + "'latex' exited abnormally with the following output:\n%s" % + e.output) + + src = "texput.%s" % (output) + + if output != "dvi": + # in order of preference + commandnames = { + "ps": ["dvips"], + "pdf": ["dvipdfmx", "dvipdfm", "dvipdf"], + "png": ["dvipng"], + "svg": ["dvisvgm"], + } + try: + cmd_variants = commandnames[output] + except KeyError: + raise ValueError("Invalid output format: %s" % output) from None + + # find an appropriate command + for cmd_variant in cmd_variants: + cmd_path = shutil.which(cmd_variant) + if cmd_path: + cmd = [cmd_path] + break + else: + if len(cmd_variants) > 1: + raise RuntimeError("None of %s are installed" % ", ".join(cmd_variants)) + else: + raise RuntimeError("%s is not installed" % cmd_variants[0]) + + defaultoptions = { + "dvipng": ["-T", "tight", "-z", "9", "--truecolor"], + "dvisvgm": ["--no-fonts"], + } + + commandend = { + "dvips": ["-o", src, "texput.dvi"], + "dvipdf": ["texput.dvi", src], + "dvipdfm": ["-o", src, "texput.dvi"], + "dvipdfmx": ["-o", src, "texput.dvi"], + "dvipng": ["-o", src, "texput.dvi"], + "dvisvgm": ["-o", src, "texput.dvi"], + } + + if dvioptions is not None: + cmd.extend(dvioptions) + else: + cmd.extend(defaultoptions.get(cmd_variant, [])) + cmd.extend(commandend[cmd_variant]) + + try: + _check_output_no_window(cmd, cwd=workdir, stderr=STDOUT) + except CalledProcessError as e: + raise RuntimeError( + "'%s' exited abnormally with the following output:\n%s" % + (' '.join(cmd), e.output)) + + + if viewer == "file": + shutil.move(join(workdir, src), filename) + elif viewer == "BytesIO": + s = Path(join(workdir, src)).read_bytes() + outputbuffer.write(s) + elif callable(viewer): + viewer(join(workdir, src), fmt=output) + else: + try: + _check_output_no_window( + [viewer, src], cwd=workdir, stderr=STDOUT) + except CalledProcessError as e: + raise RuntimeError( + "'%s %s' exited abnormally with the following output:\n%s" % + (viewer, src, e.output)) diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/printer.py b/.venv/lib/python3.12/site-packages/sympy/printing/printer.py new file mode 100644 index 0000000000000000000000000000000000000000..0c0a6970920cf0928ad330ed9a3ea4291107a29d --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/printer.py @@ -0,0 +1,432 @@ +"""Printing subsystem driver + +SymPy's printing system works the following way: Any expression can be +passed to a designated Printer who then is responsible to return an +adequate representation of that expression. + +**The basic concept is the following:** + +1. Let the object print itself if it knows how. +2. Take the best fitting method defined in the printer. +3. As fall-back use the emptyPrinter method for the printer. + +Which Method is Responsible for Printing? +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +The whole printing process is started by calling ``.doprint(expr)`` on the printer +which you want to use. This method looks for an appropriate method which can +print the given expression in the given style that the printer defines. +While looking for the method, it follows these steps: + +1. **Let the object print itself if it knows how.** + + The printer looks for a specific method in every object. The name of that method + depends on the specific printer and is defined under ``Printer.printmethod``. + For example, StrPrinter calls ``_sympystr`` and LatexPrinter calls ``_latex``. + Look at the documentation of the printer that you want to use. + The name of the method is specified there. + + This was the original way of doing printing in sympy. Every class had + its own latex, mathml, str and repr methods, but it turned out that it + is hard to produce a high quality printer, if all the methods are spread + out that far. Therefore all printing code was combined into the different + printers, which works great for built-in SymPy objects, but not that + good for user defined classes where it is inconvenient to patch the + printers. + +2. **Take the best fitting method defined in the printer.** + + The printer loops through expr classes (class + its bases), and tries + to dispatch the work to ``_print_`` + + e.g., suppose we have the following class hierarchy:: + + Basic + | + Atom + | + Number + | + Rational + + then, for ``expr=Rational(...)``, the Printer will try + to call printer methods in the order as shown in the figure below:: + + p._print(expr) + | + |-- p._print_Rational(expr) + | + |-- p._print_Number(expr) + | + |-- p._print_Atom(expr) + | + `-- p._print_Basic(expr) + + if ``._print_Rational`` method exists in the printer, then it is called, + and the result is returned back. Otherwise, the printer tries to call + ``._print_Number`` and so on. + +3. **As a fall-back use the emptyPrinter method for the printer.** + + As fall-back ``self.emptyPrinter`` will be called with the expression. If + not defined in the Printer subclass this will be the same as ``str(expr)``. + +.. _printer_example: + +Example of Custom Printer +^^^^^^^^^^^^^^^^^^^^^^^^^ + +In the example below, we have a printer which prints the derivative of a function +in a shorter form. + +.. code-block:: python + + from sympy.core.symbol import Symbol + from sympy.printing.latex import LatexPrinter, print_latex + from sympy.core.function import UndefinedFunction, Function + + + class MyLatexPrinter(LatexPrinter): + \"\"\"Print derivative of a function of symbols in a shorter form. + \"\"\" + def _print_Derivative(self, expr): + function, *vars = expr.args + if not isinstance(type(function), UndefinedFunction) or \\ + not all(isinstance(i, Symbol) for i in vars): + return super()._print_Derivative(expr) + + # If you want the printer to work correctly for nested + # expressions then use self._print() instead of str() or latex(). + # See the example of nested modulo below in the custom printing + # method section. + return "{}_{{{}}}".format( + self._print(Symbol(function.func.__name__)), + ''.join(self._print(i) for i in vars)) + + + def print_my_latex(expr): + \"\"\" Most of the printers define their own wrappers for print(). + These wrappers usually take printer settings. Our printer does not have + any settings. + \"\"\" + print(MyLatexPrinter().doprint(expr)) + + + y = Symbol("y") + x = Symbol("x") + f = Function("f") + expr = f(x, y).diff(x, y) + + # Print the expression using the normal latex printer and our custom + # printer. + print_latex(expr) + print_my_latex(expr) + +The output of the code above is:: + + \\frac{\\partial^{2}}{\\partial x\\partial y} f{\\left(x,y \\right)} + f_{xy} + +.. _printer_method_example: + +Example of Custom Printing Method +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +In the example below, the latex printing of the modulo operator is modified. +This is done by overriding the method ``_latex`` of ``Mod``. + +>>> from sympy import Symbol, Mod, Integer, print_latex + +>>> # Always use printer._print() +>>> class ModOp(Mod): +... def _latex(self, printer): +... a, b = [printer._print(i) for i in self.args] +... return r"\\operatorname{Mod}{\\left(%s, %s\\right)}" % (a, b) + +Comparing the output of our custom operator to the builtin one: + +>>> x = Symbol('x') +>>> m = Symbol('m') +>>> print_latex(Mod(x, m)) +x \\bmod m +>>> print_latex(ModOp(x, m)) +\\operatorname{Mod}{\\left(x, m\\right)} + +Common mistakes +~~~~~~~~~~~~~~~ +It's important to always use ``self._print(obj)`` to print subcomponents of +an expression when customizing a printer. Mistakes include: + +1. Using ``self.doprint(obj)`` instead: + + >>> # This example does not work properly, as only the outermost call may use + >>> # doprint. + >>> class ModOpModeWrong(Mod): + ... def _latex(self, printer): + ... a, b = [printer.doprint(i) for i in self.args] + ... return r"\\operatorname{Mod}{\\left(%s, %s\\right)}" % (a, b) + + This fails when the ``mode`` argument is passed to the printer: + + >>> print_latex(ModOp(x, m), mode='inline') # ok + $\\operatorname{Mod}{\\left(x, m\\right)}$ + >>> print_latex(ModOpModeWrong(x, m), mode='inline') # bad + $\\operatorname{Mod}{\\left($x$, $m$\\right)}$ + +2. Using ``str(obj)`` instead: + + >>> class ModOpNestedWrong(Mod): + ... def _latex(self, printer): + ... a, b = [str(i) for i in self.args] + ... return r"\\operatorname{Mod}{\\left(%s, %s\\right)}" % (a, b) + + This fails on nested objects: + + >>> # Nested modulo. + >>> print_latex(ModOp(ModOp(x, m), Integer(7))) # ok + \\operatorname{Mod}{\\left(\\operatorname{Mod}{\\left(x, m\\right)}, 7\\right)} + >>> print_latex(ModOpNestedWrong(ModOpNestedWrong(x, m), Integer(7))) # bad + \\operatorname{Mod}{\\left(ModOpNestedWrong(x, m), 7\\right)} + +3. Using ``LatexPrinter()._print(obj)`` instead. + + >>> from sympy.printing.latex import LatexPrinter + >>> class ModOpSettingsWrong(Mod): + ... def _latex(self, printer): + ... a, b = [LatexPrinter()._print(i) for i in self.args] + ... return r"\\operatorname{Mod}{\\left(%s, %s\\right)}" % (a, b) + + This causes all the settings to be discarded in the subobjects. As an + example, the ``full_prec`` setting which shows floats to full precision is + ignored: + + >>> from sympy import Float + >>> print_latex(ModOp(Float(1) * x, m), full_prec=True) # ok + \\operatorname{Mod}{\\left(1.00000000000000 x, m\\right)} + >>> print_latex(ModOpSettingsWrong(Float(1) * x, m), full_prec=True) # bad + \\operatorname{Mod}{\\left(1.0 x, m\\right)} + +""" + +from __future__ import annotations +import sys +from typing import Any, Type +import inspect +from contextlib import contextmanager +from functools import cmp_to_key, update_wrapper + +from sympy.core.add import Add +from sympy.core.basic import Basic + +from sympy.core.function import AppliedUndef, UndefinedFunction, Function + + + +@contextmanager +def printer_context(printer, **kwargs): + original = printer._context.copy() + try: + printer._context.update(kwargs) + yield + finally: + printer._context = original + + +class Printer: + """ Generic printer + + Its job is to provide infrastructure for implementing new printers easily. + + If you want to define your custom Printer or your custom printing method + for your custom class then see the example above: printer_example_ . + """ + + _global_settings: dict[str, Any] = {} + + _default_settings: dict[str, Any] = {} + + # must be initialized to pass tests and cannot be set to '| None' to pass mypy + printmethod = None # type: str + + @classmethod + def _get_initial_settings(cls): + settings = cls._default_settings.copy() + for key, val in cls._global_settings.items(): + if key in cls._default_settings: + settings[key] = val + return settings + + def __init__(self, settings=None): + self._str = str + + self._settings = self._get_initial_settings() + self._context = {} # mutable during printing + + if settings is not None: + self._settings.update(settings) + + if len(self._settings) > len(self._default_settings): + for key in self._settings: + if key not in self._default_settings: + raise TypeError("Unknown setting '%s'." % key) + + # _print_level is the number of times self._print() was recursively + # called. See StrPrinter._print_Float() for an example of usage + self._print_level = 0 + + @classmethod + def set_global_settings(cls, **settings): + """Set system-wide printing settings. """ + for key, val in settings.items(): + if val is not None: + cls._global_settings[key] = val + + @property + def order(self): + if 'order' in self._settings: + return self._settings['order'] + else: + raise AttributeError("No order defined.") + + def doprint(self, expr): + """Returns printer's representation for expr (as a string)""" + return self._str(self._print(expr)) + + def _print(self, expr, **kwargs) -> str: + """Internal dispatcher + + Tries the following concepts to print an expression: + 1. Let the object print itself if it knows how. + 2. Take the best fitting method defined in the printer. + 3. As fall-back use the emptyPrinter method for the printer. + """ + self._print_level += 1 + try: + # If the printer defines a name for a printing method + # (Printer.printmethod) and the object knows for itself how it + # should be printed, use that method. + if self.printmethod and hasattr(expr, self.printmethod): + if not (isinstance(expr, type) and issubclass(expr, Basic)): + return getattr(expr, self.printmethod)(self, **kwargs) + + # See if the class of expr is known, or if one of its super + # classes is known, and use that print function + # Exception: ignore the subclasses of Undefined, so that, e.g., + # Function('gamma') does not get dispatched to _print_gamma + classes = type(expr).__mro__ + if AppliedUndef in classes: + classes = classes[classes.index(AppliedUndef):] + if UndefinedFunction in classes: + classes = classes[classes.index(UndefinedFunction):] + # Another exception: if someone subclasses a known function, e.g., + # gamma, and changes the name, then ignore _print_gamma + if Function in classes: + i = classes.index(Function) + classes = tuple(c for c in classes[:i] if \ + c.__name__ == classes[0].__name__ or \ + c.__name__.endswith("Base")) + classes[i:] + for cls in classes: + printmethodname = '_print_' + cls.__name__ + printmethod = getattr(self, printmethodname, None) + if printmethod is not None: + return printmethod(expr, **kwargs) + # Unknown object, fall back to the emptyPrinter. + return self.emptyPrinter(expr) + finally: + self._print_level -= 1 + + def emptyPrinter(self, expr): + return str(expr) + + def _as_ordered_terms(self, expr, order=None): + """A compatibility function for ordering terms in Add. """ + order = order or self.order + + if order == 'old': + return sorted(Add.make_args(expr), key=cmp_to_key(self._compare_pretty)) + elif order == 'none': + return list(expr.args) + else: + return expr.as_ordered_terms(order=order) + + def _compare_pretty(self, a, b): + """return -1, 0, 1 if a is canonically less, equal or + greater than b. This is used when 'order=old' is selected + for printing. This puts Order last, orders Rationals + according to value, puts terms in order wrt the power of + the last power appearing in a term. Ties are broken using + Basic.compare. + """ + from sympy.core.numbers import Rational + from sympy.core.symbol import Wild + from sympy.series.order import Order + if isinstance(a, Order) and not isinstance(b, Order): + return 1 + if not isinstance(a, Order) and isinstance(b, Order): + return -1 + + if isinstance(a, Rational) and isinstance(b, Rational): + l = a.p * b.q + r = b.p * a.q + return (l > r) - (l < r) + else: + p1, p2, p3 = Wild("p1"), Wild("p2"), Wild("p3") + r_a = a.match(p1 * p2**p3) + if r_a and p3 in r_a: + a3 = r_a[p3] + r_b = b.match(p1 * p2**p3) + if r_b and p3 in r_b: + b3 = r_b[p3] + c = Basic.compare(a3, b3) + if c != 0: + return c + + # break ties + return Basic.compare(a, b) + + +class _PrintFunction: + """ + Function wrapper to replace ``**settings`` in the signature with printer defaults + """ + def __init__(self, f, print_cls: Type[Printer]): + # find all the non-setting arguments + params = list(inspect.signature(f).parameters.values()) + assert params.pop(-1).kind == inspect.Parameter.VAR_KEYWORD + self.__other_params = params + + self.__print_cls = print_cls + update_wrapper(self, f) + + def __reduce__(self): + # Since this is used as a decorator, it replaces the original function. + # The default pickling will try to pickle self.__wrapped__ and fail + # because the wrapped function can't be retrieved by name. + return self.__wrapped__.__qualname__ + + def __call__(self, *args, **kwargs): + return self.__wrapped__(*args, **kwargs) + + @property + def __signature__(self) -> inspect.Signature: + settings = self.__print_cls._get_initial_settings() + return inspect.Signature( + parameters=self.__other_params + [ + inspect.Parameter(k, inspect.Parameter.KEYWORD_ONLY, default=v) + for k, v in settings.items() + ], + return_annotation=self.__wrapped__.__annotations__.get('return', inspect.Signature.empty) # type:ignore + ) + + +def print_function(print_cls): + """ A decorator to replace kwargs with the printer settings in __signature__ """ + def decorator(f): + if sys.version_info < (3, 9): + # We have to create a subclass so that `help` actually shows the docstring in older Python versions. + # IPython and Sphinx do not need this, only a raw Python console. + cls = type(f'{f.__qualname__}_PrintFunction', (_PrintFunction,), {"__doc__": f.__doc__}) + else: + cls = _PrintFunction + return cls(f, print_cls) + return decorator diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/pycode.py b/.venv/lib/python3.12/site-packages/sympy/printing/pycode.py new file mode 100644 index 0000000000000000000000000000000000000000..09bdc6788775d409c06bdaae0a43c54544894602 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/pycode.py @@ -0,0 +1,852 @@ +""" +Python code printers + +This module contains Python code printers for plain Python as well as NumPy & SciPy enabled code. +""" +from collections import defaultdict +from itertools import chain +from sympy.core import S +from sympy.core.mod import Mod +from .precedence import precedence +from .codeprinter import CodePrinter + +_kw = { + 'and', 'as', 'assert', 'break', 'class', 'continue', 'def', 'del', 'elif', + 'else', 'except', 'finally', 'for', 'from', 'global', 'if', 'import', 'in', + 'is', 'lambda', 'not', 'or', 'pass', 'raise', 'return', 'try', 'while', + 'with', 'yield', 'None', 'False', 'nonlocal', 'True' +} + +_known_functions = { + 'Abs': 'abs', + 'Min': 'min', + 'Max': 'max', +} +_known_functions_math = { + 'acos': 'acos', + 'acosh': 'acosh', + 'asin': 'asin', + 'asinh': 'asinh', + 'atan': 'atan', + 'atan2': 'atan2', + 'atanh': 'atanh', + 'ceiling': 'ceil', + 'cos': 'cos', + 'cosh': 'cosh', + 'erf': 'erf', + 'erfc': 'erfc', + 'exp': 'exp', + 'expm1': 'expm1', + 'factorial': 'factorial', + 'floor': 'floor', + 'gamma': 'gamma', + 'hypot': 'hypot', + 'isinf': 'isinf', + 'isnan': 'isnan', + 'loggamma': 'lgamma', + 'log': 'log', + 'ln': 'log', + 'log10': 'log10', + 'log1p': 'log1p', + 'log2': 'log2', + 'sin': 'sin', + 'sinh': 'sinh', + 'Sqrt': 'sqrt', + 'tan': 'tan', + 'tanh': 'tanh' +} # Not used from ``math``: [copysign isclose isfinite isinf ldexp frexp pow modf +# radians trunc fmod fsum gcd degrees fabs] +_known_constants_math = { + 'Exp1': 'e', + 'Pi': 'pi', + 'E': 'e', + 'Infinity': 'inf', + 'NaN': 'nan', + 'ComplexInfinity': 'nan' +} + +def _print_known_func(self, expr): + known = self.known_functions[expr.__class__.__name__] + return '{name}({args})'.format(name=self._module_format(known), + args=', '.join((self._print(arg) for arg in expr.args))) + + +def _print_known_const(self, expr): + known = self.known_constants[expr.__class__.__name__] + return self._module_format(known) + + +class AbstractPythonCodePrinter(CodePrinter): + printmethod = "_pythoncode" + language = "Python" + reserved_words = _kw + modules = None # initialized to a set in __init__ + tab = ' ' + _kf = dict(chain( + _known_functions.items(), + [(k, 'math.' + v) for k, v in _known_functions_math.items()] + )) + _kc = {k: 'math.'+v for k, v in _known_constants_math.items()} + _operators = {'and': 'and', 'or': 'or', 'not': 'not'} + _default_settings = dict( + CodePrinter._default_settings, + user_functions={}, + precision=17, + inline=True, + fully_qualified_modules=True, + contract=False, + standard='python3', + ) + + def __init__(self, settings=None): + super().__init__(settings) + + # Python standard handler + std = self._settings['standard'] + if std is None: + import sys + std = 'python{}'.format(sys.version_info.major) + if std != 'python3': + raise ValueError('Only Python 3 is supported.') + self.standard = std + + self.module_imports = defaultdict(set) + + # Known functions and constants handler + self.known_functions = dict(self._kf, **(settings or {}).get( + 'user_functions', {})) + self.known_constants = dict(self._kc, **(settings or {}).get( + 'user_constants', {})) + + def _declare_number_const(self, name, value): + return "%s = %s" % (name, value) + + def _module_format(self, fqn, register=True): + parts = fqn.split('.') + if register and len(parts) > 1: + self.module_imports['.'.join(parts[:-1])].add(parts[-1]) + + if self._settings['fully_qualified_modules']: + return fqn + else: + return fqn.split('(')[0].split('[')[0].split('.')[-1] + + def _format_code(self, lines): + return lines + + def _get_statement(self, codestring): + return "{}".format(codestring) + + def _get_comment(self, text): + return " # {}".format(text) + + def _expand_fold_binary_op(self, op, args): + """ + This method expands a fold on binary operations. + + ``functools.reduce`` is an example of a folded operation. + + For example, the expression + + `A + B + C + D` + + is folded into + + `((A + B) + C) + D` + """ + if len(args) == 1: + return self._print(args[0]) + else: + return "%s(%s, %s)" % ( + self._module_format(op), + self._expand_fold_binary_op(op, args[:-1]), + self._print(args[-1]), + ) + + def _expand_reduce_binary_op(self, op, args): + """ + This method expands a reduction on binary operations. + + Notice: this is NOT the same as ``functools.reduce``. + + For example, the expression + + `A + B + C + D` + + is reduced into: + + `(A + B) + (C + D)` + """ + if len(args) == 1: + return self._print(args[0]) + else: + N = len(args) + Nhalf = N // 2 + return "%s(%s, %s)" % ( + self._module_format(op), + self._expand_reduce_binary_op(args[:Nhalf]), + self._expand_reduce_binary_op(args[Nhalf:]), + ) + + def _print_NaN(self, expr): + return "float('nan')" + + def _print_Infinity(self, expr): + return "float('inf')" + + def _print_NegativeInfinity(self, expr): + return "float('-inf')" + + def _print_ComplexInfinity(self, expr): + return self._print_NaN(expr) + + def _print_Mod(self, expr): + PREC = precedence(expr) + return ('{} % {}'.format(*(self.parenthesize(x, PREC) for x in expr.args))) + + def _print_Piecewise(self, expr): + result = [] + i = 0 + for arg in expr.args: + e = arg.expr + c = arg.cond + if i == 0: + result.append('(') + result.append('(') + result.append(self._print(e)) + result.append(')') + result.append(' if ') + result.append(self._print(c)) + result.append(' else ') + i += 1 + result = result[:-1] + if result[-1] == 'True': + result = result[:-2] + result.append(')') + else: + result.append(' else None)') + return ''.join(result) + + def _print_Relational(self, expr): + "Relational printer for Equality and Unequality" + op = { + '==' :'equal', + '!=' :'not_equal', + '<' :'less', + '<=' :'less_equal', + '>' :'greater', + '>=' :'greater_equal', + } + if expr.rel_op in op: + lhs = self._print(expr.lhs) + rhs = self._print(expr.rhs) + return '({lhs} {op} {rhs})'.format(op=expr.rel_op, lhs=lhs, rhs=rhs) + return super()._print_Relational(expr) + + def _print_ITE(self, expr): + from sympy.functions.elementary.piecewise import Piecewise + return self._print(expr.rewrite(Piecewise)) + + def _print_Sum(self, expr): + loops = ( + 'for {i} in range({a}, {b}+1)'.format( + i=self._print(i), + a=self._print(a), + b=self._print(b)) + for i, a, b in expr.limits[::-1]) + return '(builtins.sum({function} {loops}))'.format( + function=self._print(expr.function), + loops=' '.join(loops)) + + def _print_ImaginaryUnit(self, expr): + return '1j' + + def _print_KroneckerDelta(self, expr): + a, b = expr.args + + return '(1 if {a} == {b} else 0)'.format( + a = self._print(a), + b = self._print(b) + ) + + def _print_MatrixBase(self, expr): + name = expr.__class__.__name__ + func = self.known_functions.get(name, name) + return "%s(%s)" % (func, self._print(expr.tolist())) + + _print_SparseRepMatrix = \ + _print_MutableSparseMatrix = \ + _print_ImmutableSparseMatrix = \ + _print_Matrix = \ + _print_DenseMatrix = \ + _print_MutableDenseMatrix = \ + _print_ImmutableMatrix = \ + _print_ImmutableDenseMatrix = \ + lambda self, expr: self._print_MatrixBase(expr) + + def _indent_codestring(self, codestring): + return '\n'.join([self.tab + line for line in codestring.split('\n')]) + + def _print_FunctionDefinition(self, fd): + body = '\n'.join((self._print(arg) for arg in fd.body)) + return "def {name}({parameters}):\n{body}".format( + name=self._print(fd.name), + parameters=', '.join([self._print(var.symbol) for var in fd.parameters]), + body=self._indent_codestring(body) + ) + + def _print_While(self, whl): + body = '\n'.join((self._print(arg) for arg in whl.body)) + return "while {cond}:\n{body}".format( + cond=self._print(whl.condition), + body=self._indent_codestring(body) + ) + + def _print_Declaration(self, decl): + return '%s = %s' % ( + self._print(decl.variable.symbol), + self._print(decl.variable.value) + ) + + def _print_BreakToken(self, bt): + return 'break' + + def _print_Return(self, ret): + arg, = ret.args + return 'return %s' % self._print(arg) + + def _print_Raise(self, rs): + arg, = rs.args + return 'raise %s' % self._print(arg) + + def _print_RuntimeError_(self, re): + message, = re.args + return "RuntimeError(%s)" % self._print(message) + + def _print_Print(self, prnt): + print_args = ', '.join((self._print(arg) for arg in prnt.print_args)) + from sympy.codegen.ast import none + if prnt.format_string != none: + print_args = '{} % ({}), end=""'.format( + self._print(prnt.format_string), + print_args + ) + if prnt.file != None: # Must be '!= None', cannot be 'is not None' + print_args += ', file=%s' % self._print(prnt.file) + return 'print(%s)' % print_args + + def _print_Stream(self, strm): + if str(strm.name) == 'stdout': + return self._module_format('sys.stdout') + elif str(strm.name) == 'stderr': + return self._module_format('sys.stderr') + else: + return self._print(strm.name) + + def _print_NoneToken(self, arg): + return 'None' + + def _hprint_Pow(self, expr, rational=False, sqrt='math.sqrt'): + """Printing helper function for ``Pow`` + + Notes + ===== + + This preprocesses the ``sqrt`` as math formatter and prints division + + Examples + ======== + + >>> from sympy import sqrt + >>> from sympy.printing.pycode import PythonCodePrinter + >>> from sympy.abc import x + + Python code printer automatically looks up ``math.sqrt``. + + >>> printer = PythonCodePrinter() + >>> printer._hprint_Pow(sqrt(x), rational=True) + 'x**(1/2)' + >>> printer._hprint_Pow(sqrt(x), rational=False) + 'math.sqrt(x)' + >>> printer._hprint_Pow(1/sqrt(x), rational=True) + 'x**(-1/2)' + >>> printer._hprint_Pow(1/sqrt(x), rational=False) + '1/math.sqrt(x)' + >>> printer._hprint_Pow(1/x, rational=False) + '1/x' + >>> printer._hprint_Pow(1/x, rational=True) + 'x**(-1)' + + Using sqrt from numpy or mpmath + + >>> printer._hprint_Pow(sqrt(x), sqrt='numpy.sqrt') + 'numpy.sqrt(x)' + >>> printer._hprint_Pow(sqrt(x), sqrt='mpmath.sqrt') + 'mpmath.sqrt(x)' + + See Also + ======== + + sympy.printing.str.StrPrinter._print_Pow + """ + PREC = precedence(expr) + + if expr.exp == S.Half and not rational: + func = self._module_format(sqrt) + arg = self._print(expr.base) + return '{func}({arg})'.format(func=func, arg=arg) + + if expr.is_commutative and not rational: + if -expr.exp is S.Half: + func = self._module_format(sqrt) + num = self._print(S.One) + arg = self._print(expr.base) + return f"{num}/{func}({arg})" + if expr.exp is S.NegativeOne: + num = self._print(S.One) + arg = self.parenthesize(expr.base, PREC, strict=False) + return f"{num}/{arg}" + + + base_str = self.parenthesize(expr.base, PREC, strict=False) + exp_str = self.parenthesize(expr.exp, PREC, strict=False) + return "{}**{}".format(base_str, exp_str) + + +class ArrayPrinter: + + def _arrayify(self, indexed): + from sympy.tensor.array.expressions.from_indexed_to_array import convert_indexed_to_array + try: + return convert_indexed_to_array(indexed) + except Exception: + return indexed + + def _get_einsum_string(self, subranks, contraction_indices): + letters = self._get_letter_generator_for_einsum() + contraction_string = "" + counter = 0 + d = {j: min(i) for i in contraction_indices for j in i} + indices = [] + for rank_arg in subranks: + lindices = [] + for i in range(rank_arg): + if counter in d: + lindices.append(d[counter]) + else: + lindices.append(counter) + counter += 1 + indices.append(lindices) + mapping = {} + letters_free = [] + letters_dum = [] + for i in indices: + for j in i: + if j not in mapping: + l = next(letters) + mapping[j] = l + else: + l = mapping[j] + contraction_string += l + if j in d: + if l not in letters_dum: + letters_dum.append(l) + else: + letters_free.append(l) + contraction_string += "," + contraction_string = contraction_string[:-1] + return contraction_string, letters_free, letters_dum + + def _get_letter_generator_for_einsum(self): + for i in range(97, 123): + yield chr(i) + for i in range(65, 91): + yield chr(i) + raise ValueError("out of letters") + + def _print_ArrayTensorProduct(self, expr): + letters = self._get_letter_generator_for_einsum() + contraction_string = ",".join(["".join([next(letters) for j in range(i)]) for i in expr.subranks]) + return '%s("%s", %s)' % ( + self._module_format(self._module + "." + self._einsum), + contraction_string, + ", ".join([self._print(arg) for arg in expr.args]) + ) + + def _print_ArrayContraction(self, expr): + from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct + base = expr.expr + contraction_indices = expr.contraction_indices + + if isinstance(base, ArrayTensorProduct): + elems = ",".join(["%s" % (self._print(arg)) for arg in base.args]) + ranks = base.subranks + else: + elems = self._print(base) + ranks = [len(base.shape)] + + contraction_string, letters_free, letters_dum = self._get_einsum_string(ranks, contraction_indices) + + if not contraction_indices: + return self._print(base) + if isinstance(base, ArrayTensorProduct): + elems = ",".join(["%s" % (self._print(arg)) for arg in base.args]) + else: + elems = self._print(base) + return "%s(\"%s\", %s)" % ( + self._module_format(self._module + "." + self._einsum), + "{}->{}".format(contraction_string, "".join(sorted(letters_free))), + elems, + ) + + def _print_ArrayDiagonal(self, expr): + from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct + diagonal_indices = list(expr.diagonal_indices) + if isinstance(expr.expr, ArrayTensorProduct): + subranks = expr.expr.subranks + elems = expr.expr.args + else: + subranks = expr.subranks + elems = [expr.expr] + diagonal_string, letters_free, letters_dum = self._get_einsum_string(subranks, diagonal_indices) + elems = [self._print(i) for i in elems] + return '%s("%s", %s)' % ( + self._module_format(self._module + "." + self._einsum), + "{}->{}".format(diagonal_string, "".join(letters_free+letters_dum)), + ", ".join(elems) + ) + + def _print_PermuteDims(self, expr): + return "%s(%s, %s)" % ( + self._module_format(self._module + "." + self._transpose), + self._print(expr.expr), + self._print(expr.permutation.array_form), + ) + + def _print_ArrayAdd(self, expr): + return self._expand_fold_binary_op(self._module + "." + self._add, expr.args) + + def _print_OneArray(self, expr): + return "%s((%s,))" % ( + self._module_format(self._module+ "." + self._ones), + ','.join(map(self._print,expr.args)) + ) + + def _print_ZeroArray(self, expr): + return "%s((%s,))" % ( + self._module_format(self._module+ "." + self._zeros), + ','.join(map(self._print,expr.args)) + ) + + def _print_Assignment(self, expr): + #XXX: maybe this needs to happen at a higher level e.g. at _print or + #doprint? + lhs = self._print(self._arrayify(expr.lhs)) + rhs = self._print(self._arrayify(expr.rhs)) + return "%s = %s" % ( lhs, rhs ) + + def _print_IndexedBase(self, expr): + return self._print_ArraySymbol(expr) + + +class PythonCodePrinter(AbstractPythonCodePrinter): + + def _print_sign(self, e): + return '(0.0 if {e} == 0 else {f}(1, {e}))'.format( + f=self._module_format('math.copysign'), e=self._print(e.args[0])) + + def _print_Not(self, expr): + PREC = precedence(expr) + return self._operators['not'] + ' ' + self.parenthesize(expr.args[0], PREC) + + def _print_IndexedBase(self, expr): + return expr.name + + def _print_Indexed(self, expr): + base = expr.args[0] + index = expr.args[1:] + return "{}[{}]".format(str(base), ", ".join([self._print(ind) for ind in index])) + + def _print_Pow(self, expr, rational=False): + return self._hprint_Pow(expr, rational=rational) + + def _print_Rational(self, expr): + return '{}/{}'.format(expr.p, expr.q) + + def _print_Half(self, expr): + return self._print_Rational(expr) + + def _print_frac(self, expr): + return self._print_Mod(Mod(expr.args[0], 1)) + + def _print_Symbol(self, expr): + + name = super()._print_Symbol(expr) + + if name in self.reserved_words: + if self._settings['error_on_reserved']: + msg = ('This expression includes the symbol "{}" which is a ' + 'reserved keyword in this language.') + raise ValueError(msg.format(name)) + return name + self._settings['reserved_word_suffix'] + elif '{' in name: # Remove curly braces from subscripted variables + return name.replace('{', '').replace('}', '') + else: + return name + + _print_lowergamma = CodePrinter._print_not_supported + _print_uppergamma = CodePrinter._print_not_supported + _print_fresnelc = CodePrinter._print_not_supported + _print_fresnels = CodePrinter._print_not_supported + + +for k in PythonCodePrinter._kf: + setattr(PythonCodePrinter, '_print_%s' % k, _print_known_func) + +for k in _known_constants_math: + setattr(PythonCodePrinter, '_print_%s' % k, _print_known_const) + + +def pycode(expr, **settings): + """ Converts an expr to a string of Python code + + Parameters + ========== + + expr : Expr + A SymPy expression. + fully_qualified_modules : bool + Whether or not to write out full module names of functions + (``math.sin`` vs. ``sin``). default: ``True``. + standard : str or None, optional + Only 'python3' (default) is supported. + This parameter may be removed in the future. + + Examples + ======== + + >>> from sympy import pycode, tan, Symbol + >>> pycode(tan(Symbol('x')) + 1) + 'math.tan(x) + 1' + + """ + return PythonCodePrinter(settings).doprint(expr) + + +from itertools import chain +from sympy.printing.pycode import PythonCodePrinter + +_known_functions_cmath = { + 'exp': 'exp', + 'sqrt': 'sqrt', + 'log': 'log', + 'cos': 'cos', + 'sin': 'sin', + 'tan': 'tan', + 'acos': 'acos', + 'asin': 'asin', + 'atan': 'atan', + 'cosh': 'cosh', + 'sinh': 'sinh', + 'tanh': 'tanh', + 'acosh': 'acosh', + 'asinh': 'asinh', + 'atanh': 'atanh', +} + +_known_constants_cmath = { + 'Pi': 'pi', + 'E': 'e', + 'Infinity': 'inf', + 'NegativeInfinity': '-inf', +} + +class CmathPrinter(PythonCodePrinter): + """ Printer for Python's cmath module """ + printmethod = "_cmathcode" + language = "Python with cmath" + + _kf = dict(chain( + _known_functions_cmath.items() + )) + + _kc = {k: 'cmath.' + v for k, v in _known_constants_cmath.items()} + + def _print_Pow(self, expr, rational=False): + return self._hprint_Pow(expr, rational=rational, sqrt='cmath.sqrt') + + def _print_Float(self, e): + return '{func}({val})'.format(func=self._module_format('cmath.mpf'), val=self._print(e)) + + def _print_known_func(self, expr): + func_name = expr.func.__name__ + if func_name in self._kf: + return f"cmath.{self._kf[func_name]}({', '.join(map(self._print, expr.args))})" + return super()._print_Function(expr) + + def _print_known_const(self, expr): + return self._kc[expr.__class__.__name__] + + def _print_re(self, expr): + """Prints `re(z)` as `z.real`""" + return f"({self._print(expr.args[0])}).real" + + def _print_im(self, expr): + """Prints `im(z)` as `z.imag`""" + return f"({self._print(expr.args[0])}).imag" + + +for k in CmathPrinter._kf: + setattr(CmathPrinter, '_print_%s' % k, CmathPrinter._print_known_func) + +for k in _known_constants_cmath: + setattr(CmathPrinter, '_print_%s' % k, CmathPrinter._print_known_const) + + +_not_in_mpmath = 'log1p log2'.split() +_in_mpmath = [(k, v) for k, v in _known_functions_math.items() if k not in _not_in_mpmath] +_known_functions_mpmath = dict(_in_mpmath, **{ + 'beta': 'beta', + 'frac': 'frac', + 'fresnelc': 'fresnelc', + 'fresnels': 'fresnels', + 'sign': 'sign', + 'loggamma': 'loggamma', + 'hyper': 'hyper', + 'meijerg': 'meijerg', + 'besselj': 'besselj', + 'bessely': 'bessely', + 'besseli': 'besseli', + 'besselk': 'besselk', +}) +_known_constants_mpmath = { + 'Exp1': 'e', + 'Pi': 'pi', + 'GoldenRatio': 'phi', + 'EulerGamma': 'euler', + 'Catalan': 'catalan', + 'NaN': 'nan', + 'Infinity': 'inf', + 'NegativeInfinity': 'ninf' +} + + +def _unpack_integral_limits(integral_expr): + """ helper function for _print_Integral that + - accepts an Integral expression + - returns a tuple of + - a list variables of integration + - a list of tuples of the upper and lower limits of integration + """ + integration_vars = [] + limits = [] + for integration_range in integral_expr.limits: + if len(integration_range) == 3: + integration_var, lower_limit, upper_limit = integration_range + else: + raise NotImplementedError("Only definite integrals are supported") + integration_vars.append(integration_var) + limits.append((lower_limit, upper_limit)) + return integration_vars, limits + + +class MpmathPrinter(PythonCodePrinter): + """ + Lambda printer for mpmath which maintains precision for floats + """ + printmethod = "_mpmathcode" + + language = "Python with mpmath" + + _kf = dict(chain( + _known_functions.items(), + [(k, 'mpmath.' + v) for k, v in _known_functions_mpmath.items()] + )) + _kc = {k: 'mpmath.'+v for k, v in _known_constants_mpmath.items()} + + def _print_Float(self, e): + # XXX: This does not handle setting mpmath.mp.dps. It is assumed that + # the caller of the lambdified function will have set it to sufficient + # precision to match the Floats in the expression. + + # Remove 'mpz' if gmpy is installed. + args = str(tuple(map(int, e._mpf_))) + return '{func}({args})'.format(func=self._module_format('mpmath.mpf'), args=args) + + + def _print_Rational(self, e): + return "{func}({p})/{func}({q})".format( + func=self._module_format('mpmath.mpf'), + q=self._print(e.q), + p=self._print(e.p) + ) + + def _print_Half(self, e): + return self._print_Rational(e) + + def _print_uppergamma(self, e): + return "{}({}, {}, {})".format( + self._module_format('mpmath.gammainc'), + self._print(e.args[0]), + self._print(e.args[1]), + self._module_format('mpmath.inf')) + + def _print_lowergamma(self, e): + return "{}({}, 0, {})".format( + self._module_format('mpmath.gammainc'), + self._print(e.args[0]), + self._print(e.args[1])) + + def _print_log2(self, e): + return '{0}({1})/{0}(2)'.format( + self._module_format('mpmath.log'), self._print(e.args[0])) + + def _print_log1p(self, e): + return '{}({})'.format( + self._module_format('mpmath.log1p'), self._print(e.args[0])) + + def _print_Pow(self, expr, rational=False): + return self._hprint_Pow(expr, rational=rational, sqrt='mpmath.sqrt') + + def _print_Integral(self, e): + integration_vars, limits = _unpack_integral_limits(e) + + return "{}(lambda {}: {}, {})".format( + self._module_format("mpmath.quad"), + ", ".join(map(self._print, integration_vars)), + self._print(e.args[0]), + ", ".join("(%s, %s)" % tuple(map(self._print, l)) for l in limits)) + + + def _print_Derivative_zeta(self, args, seq_orders): + arg, = args + deriv_order, = seq_orders + return '{}({}, derivative={})'.format( + self._module_format('mpmath.zeta'), + self._print(arg), deriv_order + ) + + +for k in MpmathPrinter._kf: + setattr(MpmathPrinter, '_print_%s' % k, _print_known_func) + +for k in _known_constants_mpmath: + setattr(MpmathPrinter, '_print_%s' % k, _print_known_const) + + +class SymPyPrinter(AbstractPythonCodePrinter): + + language = "Python with SymPy" + + _default_settings = dict( + AbstractPythonCodePrinter._default_settings, + strict=False # any class name will per definition be what we target in SymPyPrinter. + ) + + def _print_Function(self, expr): + mod = expr.func.__module__ or '' + return '%s(%s)' % (self._module_format(mod + ('.' if mod else '') + expr.func.__name__), + ', '.join((self._print(arg) for arg in expr.args))) + + def _print_Pow(self, expr, rational=False): + return self._hprint_Pow(expr, rational=rational, sqrt='sympy.sqrt') diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/python.py b/.venv/lib/python3.12/site-packages/sympy/printing/python.py new file mode 100644 index 0000000000000000000000000000000000000000..2f6862574d99db90f289de65144c7122ed2d731a --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/python.py @@ -0,0 +1,92 @@ +import keyword as kw +import sympy +from .repr import ReprPrinter +from .str import StrPrinter + +# A list of classes that should be printed using StrPrinter +STRPRINT = ("Add", "Infinity", "Integer", "Mul", "NegativeInfinity", "Pow") + + +class PythonPrinter(ReprPrinter, StrPrinter): + """A printer which converts an expression into its Python interpretation.""" + + def __init__(self, settings=None): + super().__init__(settings) + self.symbols = [] + self.functions = [] + + # Create print methods for classes that should use StrPrinter instead + # of ReprPrinter. + for name in STRPRINT: + f_name = "_print_%s" % name + f = getattr(StrPrinter, f_name) + setattr(PythonPrinter, f_name, f) + + def _print_Function(self, expr): + func = expr.func.__name__ + if not hasattr(sympy, func) and func not in self.functions: + self.functions.append(func) + return StrPrinter._print_Function(self, expr) + + # procedure (!) for defining symbols which have be defined in print_python() + def _print_Symbol(self, expr): + symbol = self._str(expr) + if symbol not in self.symbols: + self.symbols.append(symbol) + return StrPrinter._print_Symbol(self, expr) + + def _print_module(self, expr): + raise ValueError('Modules in the expression are unacceptable') + + +def python(expr, **settings): + """Return Python interpretation of passed expression + (can be passed to the exec() function without any modifications)""" + + printer = PythonPrinter(settings) + exprp = printer.doprint(expr) + + result = '' + # Returning found symbols and functions + renamings = {} + for symbolname in printer.symbols: + # Remove curly braces from subscripted variables + if '{' in symbolname: + newsymbolname = symbolname.replace('{', '').replace('}', '') + renamings[sympy.Symbol(symbolname)] = newsymbolname + else: + newsymbolname = symbolname + + # Escape symbol names that are reserved Python keywords + if kw.iskeyword(newsymbolname): + while True: + newsymbolname += "_" + if (newsymbolname not in printer.symbols and + newsymbolname not in printer.functions): + renamings[sympy.Symbol( + symbolname)] = sympy.Symbol(newsymbolname) + break + result += newsymbolname + ' = Symbol(\'' + symbolname + '\')\n' + + for functionname in printer.functions: + newfunctionname = functionname + # Escape function names that are reserved Python keywords + if kw.iskeyword(newfunctionname): + while True: + newfunctionname += "_" + if (newfunctionname not in printer.symbols and + newfunctionname not in printer.functions): + renamings[sympy.Function( + functionname)] = sympy.Function(newfunctionname) + break + result += newfunctionname + ' = Function(\'' + functionname + '\')\n' + + if renamings: + exprp = expr.subs(renamings) + result += 'e = ' + printer._str(exprp) + return result + + +def print_python(expr, **settings): + """Print output of python() function""" + print(python(expr, **settings)) diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/pytorch.py b/.venv/lib/python3.12/site-packages/sympy/printing/pytorch.py new file mode 100644 index 0000000000000000000000000000000000000000..0e8ff01856fa1dce7f8e786065a32bb74d987254 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/pytorch.py @@ -0,0 +1,297 @@ + +from sympy.printing.pycode import AbstractPythonCodePrinter, ArrayPrinter +from sympy.matrices.expressions import MatrixExpr +from sympy.core.mul import Mul +from sympy.printing.precedence import PRECEDENCE +from sympy.external import import_module +from sympy.codegen.cfunctions import Sqrt +from sympy import S +from sympy import Integer + +import sympy + +torch = import_module('torch') + + +class TorchPrinter(ArrayPrinter, AbstractPythonCodePrinter): + + printmethod = "_torchcode" + + mapping = { + sympy.Abs: "torch.abs", + sympy.sign: "torch.sign", + + # XXX May raise error for ints. + sympy.ceiling: "torch.ceil", + sympy.floor: "torch.floor", + sympy.log: "torch.log", + sympy.exp: "torch.exp", + Sqrt: "torch.sqrt", + sympy.cos: "torch.cos", + sympy.acos: "torch.acos", + sympy.sin: "torch.sin", + sympy.asin: "torch.asin", + sympy.tan: "torch.tan", + sympy.atan: "torch.atan", + sympy.atan2: "torch.atan2", + # XXX Also may give NaN for complex results. + sympy.cosh: "torch.cosh", + sympy.acosh: "torch.acosh", + sympy.sinh: "torch.sinh", + sympy.asinh: "torch.asinh", + sympy.tanh: "torch.tanh", + sympy.atanh: "torch.atanh", + sympy.Pow: "torch.pow", + + sympy.re: "torch.real", + sympy.im: "torch.imag", + sympy.arg: "torch.angle", + + # XXX May raise error for ints and complexes + sympy.erf: "torch.erf", + sympy.loggamma: "torch.lgamma", + + sympy.Eq: "torch.eq", + sympy.Ne: "torch.ne", + sympy.StrictGreaterThan: "torch.gt", + sympy.StrictLessThan: "torch.lt", + sympy.LessThan: "torch.le", + sympy.GreaterThan: "torch.ge", + + sympy.And: "torch.logical_and", + sympy.Or: "torch.logical_or", + sympy.Not: "torch.logical_not", + sympy.Max: "torch.max", + sympy.Min: "torch.min", + + # Matrices + sympy.MatAdd: "torch.add", + sympy.HadamardProduct: "torch.mul", + sympy.Trace: "torch.trace", + + # XXX May raise error for integer matrices. + sympy.Determinant: "torch.det", + } + + _default_settings = dict( + AbstractPythonCodePrinter._default_settings, + torch_version=None, + requires_grad=False, + dtype="torch.float64", + ) + + def __init__(self, settings=None): + super().__init__(settings) + + version = self._settings['torch_version'] + self.requires_grad = self._settings['requires_grad'] + self.dtype = self._settings['dtype'] + if version is None and torch: + version = torch.__version__ + self.torch_version = version + + def _print_Function(self, expr): + + op = self.mapping.get(type(expr), None) + if op is None: + return super()._print_Basic(expr) + children = [self._print(arg) for arg in expr.args] + if len(children) == 1: + return "%s(%s)" % ( + self._module_format(op), + children[0] + ) + else: + return self._expand_fold_binary_op(op, children) + + # mirrors the tensorflow version + _print_Expr = _print_Function + _print_Application = _print_Function + _print_MatrixExpr = _print_Function + _print_Relational = _print_Function + _print_Not = _print_Function + _print_And = _print_Function + _print_Or = _print_Function + _print_HadamardProduct = _print_Function + _print_Trace = _print_Function + _print_Determinant = _print_Function + + def _print_Inverse(self, expr): + return '{}({})'.format(self._module_format("torch.linalg.inv"), + self._print(expr.args[0])) + + def _print_Transpose(self, expr): + if expr.arg.is_Matrix and expr.arg.shape[0] == expr.arg.shape[1]: + # For square matrices, we can use the .t() method + return "{}({}).t()".format("torch.transpose", self._print(expr.arg)) + else: + # For non-square matrices or more general cases + # transpose first and second dimensions (typical matrix transpose) + return "{}.permute({})".format( + self._print(expr.arg), + ", ".join([str(i) for i in range(len(expr.arg.shape))])[::-1] + ) + + def _print_PermuteDims(self, expr): + return "%s.permute(%s)" % ( + self._print(expr.expr), + ", ".join(str(i) for i in expr.permutation.array_form) + ) + + def _print_Derivative(self, expr): + # this version handles multi-variable and mixed partial derivatives. The tensorflow version does not. + variables = expr.variables + expr_arg = expr.expr + + # Handle multi-variable or repeated derivatives + if len(variables) > 1 or ( + len(variables) == 1 and not isinstance(variables[0], tuple) and variables.count(variables[0]) > 1): + result = self._print(expr_arg) + var_groups = {} + + # Group variables by base symbol + for var in variables: + if isinstance(var, tuple): + base_var, order = var + var_groups[base_var] = var_groups.get(base_var, 0) + order + else: + var_groups[var] = var_groups.get(var, 0) + 1 + + # Apply gradients in sequence + for var, order in var_groups.items(): + for _ in range(order): + result = "torch.autograd.grad({}, {}, create_graph=True)[0]".format(result, self._print(var)) + return result + + # Handle single variable case + if len(variables) == 1: + variable = variables[0] + if isinstance(variable, tuple) and len(variable) == 2: + base_var, order = variable + if not isinstance(order, Integer): raise NotImplementedError("Only integer orders are supported") + result = self._print(expr_arg) + for _ in range(order): + result = "torch.autograd.grad({}, {}, create_graph=True)[0]".format(result, self._print(base_var)) + return result + return "torch.autograd.grad({}, {})[0]".format(self._print(expr_arg), self._print(variable)) + + return self._print(expr_arg) # Empty variables case + + def _print_Piecewise(self, expr): + from sympy import Piecewise + e, cond = expr.args[0].args + if len(expr.args) == 1: + return '{}({}, {}, {})'.format( + self._module_format("torch.where"), + self._print(cond), + self._print(e), + 0) + + return '{}({}, {}, {})'.format( + self._module_format("torch.where"), + self._print(cond), + self._print(e), + self._print(Piecewise(*expr.args[1:]))) + + def _print_Pow(self, expr): + # XXX May raise error for + # int**float or int**complex or float**complex + base, exp = expr.args + if expr.exp == S.Half: + return "{}({})".format( + self._module_format("torch.sqrt"), self._print(base)) + return "{}({}, {})".format( + self._module_format("torch.pow"), + self._print(base), self._print(exp)) + + def _print_MatMul(self, expr): + # Separate matrix and scalar arguments + mat_args = [arg for arg in expr.args if isinstance(arg, MatrixExpr)] + args = [arg for arg in expr.args if arg not in mat_args] + # Handle scalar multipliers if present + if args: + return "%s*%s" % ( + self.parenthesize(Mul.fromiter(args), PRECEDENCE["Mul"]), + self._expand_fold_binary_op("torch.matmul", mat_args) + ) + else: + return self._expand_fold_binary_op("torch.matmul", mat_args) + + def _print_MatPow(self, expr): + return self._expand_fold_binary_op("torch.mm", [expr.base]*expr.exp) + + def _print_MatrixBase(self, expr): + data = "[" + ", ".join(["[" + ", ".join([self._print(j) for j in i]) + "]" for i in expr.tolist()]) + "]" + params = [str(data)] + params.append(f"dtype={self.dtype}") + if self.requires_grad: + params.append("requires_grad=True") + + return "{}({})".format( + self._module_format("torch.tensor"), + ", ".join(params) + ) + + def _print_isnan(self, expr): + return f'torch.isnan({self._print(expr.args[0])})' + + def _print_isinf(self, expr): + return f'torch.isinf({self._print(expr.args[0])})' + + def _print_Identity(self, expr): + if all(dim.is_Integer for dim in expr.shape): + return "{}({})".format( + self._module_format("torch.eye"), + self._print(expr.shape[0]) + ) + else: + # For symbolic dimensions, fall back to a more general approach + return "{}({}, {})".format( + self._module_format("torch.eye"), + self._print(expr.shape[0]), + self._print(expr.shape[1]) + ) + + def _print_ZeroMatrix(self, expr): + return "{}({})".format( + self._module_format("torch.zeros"), + self._print(expr.shape) + ) + + def _print_OneMatrix(self, expr): + return "{}({})".format( + self._module_format("torch.ones"), + self._print(expr.shape) + ) + + def _print_conjugate(self, expr): + return f"{self._module_format('torch.conj')}({self._print(expr.args[0])})" + + def _print_ImaginaryUnit(self, expr): + return "1j" # uses the Python built-in 1j notation for the imaginary unit + + def _print_Heaviside(self, expr): + args = [self._print(expr.args[0]), "0.5"] + if len(expr.args) > 1: + args[1] = self._print(expr.args[1]) + return f"{self._module_format('torch.heaviside')}({args[0]}, {args[1]})" + + def _print_gamma(self, expr): + return f"{self._module_format('torch.special.gamma')}({self._print(expr.args[0])})" + + def _print_polygamma(self, expr): + if expr.args[0] == S.Zero: + return f"{self._module_format('torch.special.digamma')}({self._print(expr.args[1])})" + else: + raise NotImplementedError("PyTorch only supports digamma (0th order polygamma)") + + _module = "torch" + _einsum = "einsum" + _add = "add" + _transpose = "t" + _ones = "ones" + _zeros = "zeros" + +def torch_code(expr, requires_grad=False, dtype="torch.float64", **settings): + printer = TorchPrinter(settings={'requires_grad': requires_grad, 'dtype': dtype}) + return printer.doprint(expr, **settings) diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/rcode.py b/.venv/lib/python3.12/site-packages/sympy/printing/rcode.py new file mode 100644 index 0000000000000000000000000000000000000000..3107e6e94d5c5acf0b2dc063e4a83af6970f6576 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/rcode.py @@ -0,0 +1,402 @@ +""" +R code printer + +The RCodePrinter converts single SymPy expressions into single R expressions, +using the functions defined in math.h where possible. + + + +""" + +from __future__ import annotations +from typing import Any + +from sympy.core.numbers import equal_valued +from sympy.printing.codeprinter import CodePrinter +from sympy.printing.precedence import precedence, PRECEDENCE +from sympy.sets.fancysets import Range + +# dictionary mapping SymPy function to (argument_conditions, C_function). +# Used in RCodePrinter._print_Function(self) +known_functions = { + #"Abs": [(lambda x: not x.is_integer, "fabs")], + "Abs": "abs", + "sin": "sin", + "cos": "cos", + "tan": "tan", + "asin": "asin", + "acos": "acos", + "atan": "atan", + "atan2": "atan2", + "exp": "exp", + "log": "log", + "erf": "erf", + "sinh": "sinh", + "cosh": "cosh", + "tanh": "tanh", + "asinh": "asinh", + "acosh": "acosh", + "atanh": "atanh", + "floor": "floor", + "ceiling": "ceiling", + "sign": "sign", + "Max": "max", + "Min": "min", + "factorial": "factorial", + "gamma": "gamma", + "digamma": "digamma", + "trigamma": "trigamma", + "beta": "beta", + "sqrt": "sqrt", # To enable automatic rewrite +} + +# These are the core reserved words in the R language. Taken from: +# https://cran.r-project.org/doc/manuals/r-release/R-lang.html#Reserved-words + +reserved_words = ['if', + 'else', + 'repeat', + 'while', + 'function', + 'for', + 'in', + 'next', + 'break', + 'TRUE', + 'FALSE', + 'NULL', + 'Inf', + 'NaN', + 'NA', + 'NA_integer_', + 'NA_real_', + 'NA_complex_', + 'NA_character_', + 'volatile'] + + +class RCodePrinter(CodePrinter): + """A printer to convert SymPy expressions to strings of R code""" + printmethod = "_rcode" + language = "R" + + _default_settings: dict[str, Any] = dict(CodePrinter._default_settings, **{ + 'precision': 15, + 'user_functions': {}, + 'contract': True, + 'dereference': set(), + }) + _operators = { + 'and': '&', + 'or': '|', + 'not': '!', + } + + _relationals: dict[str, str] = {} + + def __init__(self, settings={}): + CodePrinter.__init__(self, settings) + self.known_functions = dict(known_functions) + userfuncs = settings.get('user_functions', {}) + self.known_functions.update(userfuncs) + self._dereference = set(settings.get('dereference', [])) + self.reserved_words = set(reserved_words) + + def _rate_index_position(self, p): + return p*5 + + def _get_statement(self, codestring): + return "%s;" % codestring + + def _get_comment(self, text): + return "// {}".format(text) + + def _declare_number_const(self, name, value): + return "{} = {};".format(name, value) + + def _format_code(self, lines): + return self.indent_code(lines) + + def _traverse_matrix_indices(self, mat): + rows, cols = mat.shape + return ((i, j) for i in range(rows) for j in range(cols)) + + def _get_loop_opening_ending(self, indices): + """Returns a tuple (open_lines, close_lines) containing lists of codelines + """ + open_lines = [] + close_lines = [] + loopstart = "for (%(var)s in %(start)s:%(end)s){" + for i in indices: + # R arrays start at 1 and end at dimension + open_lines.append(loopstart % { + 'var': self._print(i.label), + 'start': self._print(i.lower+1), + 'end': self._print(i.upper + 1)}) + close_lines.append("}") + return open_lines, close_lines + + def _print_Pow(self, expr): + if "Pow" in self.known_functions: + return self._print_Function(expr) + PREC = precedence(expr) + if equal_valued(expr.exp, -1): + return '1.0/%s' % (self.parenthesize(expr.base, PREC)) + elif equal_valued(expr.exp, 0.5): + return 'sqrt(%s)' % self._print(expr.base) + else: + return '%s^%s' % (self.parenthesize(expr.base, PREC), + self.parenthesize(expr.exp, PREC)) + + + def _print_Rational(self, expr): + p, q = int(expr.p), int(expr.q) + return '%d.0/%d.0' % (p, q) + + def _print_Indexed(self, expr): + inds = [ self._print(i) for i in expr.indices ] + return "%s[%s]" % (self._print(expr.base.label), ", ".join(inds)) + + def _print_Exp1(self, expr): + return "exp(1)" + + def _print_Pi(self, expr): + return 'pi' + + def _print_Infinity(self, expr): + return 'Inf' + + def _print_NegativeInfinity(self, expr): + return '-Inf' + + def _print_Assignment(self, expr): + from sympy.codegen.ast import Assignment + + from sympy.matrices.expressions.matexpr import MatrixSymbol + from sympy.tensor.indexed import IndexedBase + lhs = expr.lhs + rhs = expr.rhs + # We special case assignments that take multiple lines + #if isinstance(expr.rhs, Piecewise): + # from sympy.functions.elementary.piecewise import Piecewise + # # Here we modify Piecewise so each expression is now + # # an Assignment, and then continue on the print. + # expressions = [] + # conditions = [] + # for (e, c) in rhs.args: + # expressions.append(Assignment(lhs, e)) + # conditions.append(c) + # temp = Piecewise(*zip(expressions, conditions)) + # return self._print(temp) + #elif isinstance(lhs, MatrixSymbol): + if isinstance(lhs, MatrixSymbol): + # Here we form an Assignment for each element in the array, + # printing each one. + lines = [] + for (i, j) in self._traverse_matrix_indices(lhs): + temp = Assignment(lhs[i, j], rhs[i, j]) + code0 = self._print(temp) + lines.append(code0) + return "\n".join(lines) + elif self._settings["contract"] and (lhs.has(IndexedBase) or + rhs.has(IndexedBase)): + # Here we check if there is looping to be done, and if so + # print the required loops. + return self._doprint_loops(rhs, lhs) + else: + lhs_code = self._print(lhs) + rhs_code = self._print(rhs) + return self._get_statement("%s = %s" % (lhs_code, rhs_code)) + + def _print_Piecewise(self, expr): + # This method is called only for inline if constructs + # Top level piecewise is handled in doprint() + if expr.args[-1].cond == True: + last_line = "%s" % self._print(expr.args[-1].expr) + else: + last_line = "ifelse(%s,%s,NA)" % (self._print(expr.args[-1].cond), self._print(expr.args[-1].expr)) + code=last_line + for e, c in reversed(expr.args[:-1]): + code= "ifelse(%s,%s," % (self._print(c), self._print(e))+code+")" + return(code) + + def _print_ITE(self, expr): + from sympy.functions import Piecewise + return self._print(expr.rewrite(Piecewise)) + + def _print_MatrixElement(self, expr): + return "{}[{}]".format(self.parenthesize(expr.parent, PRECEDENCE["Atom"], + strict=True), expr.j + expr.i*expr.parent.shape[1]) + + def _print_Symbol(self, expr): + name = super()._print_Symbol(expr) + if expr in self._dereference: + return '(*{})'.format(name) + else: + return name + + def _print_Relational(self, expr): + lhs_code = self._print(expr.lhs) + rhs_code = self._print(expr.rhs) + op = expr.rel_op + return "{} {} {}".format(lhs_code, op, rhs_code) + + def _print_AugmentedAssignment(self, expr): + lhs_code = self._print(expr.lhs) + op = expr.op + rhs_code = self._print(expr.rhs) + return "{} {} {};".format(lhs_code, op, rhs_code) + + def _print_For(self, expr): + target = self._print(expr.target) + if isinstance(expr.iterable, Range): + start, stop, step = expr.iterable.args + else: + raise NotImplementedError("Only iterable currently supported is Range") + body = self._print(expr.body) + return 'for({target} in seq(from={start}, to={stop}, by={step}){{\n{body}\n}}'.format(target=target, start=start, + stop=stop-1, step=step, body=body) + + + def indent_code(self, code): + """Accepts a string of code or a list of code lines""" + + if isinstance(code, str): + code_lines = self.indent_code(code.splitlines(True)) + return ''.join(code_lines) + + tab = " " + inc_token = ('{', '(', '{\n', '(\n') + dec_token = ('}', ')') + + code = [ line.lstrip(' \t') for line in code ] + + increase = [ int(any(map(line.endswith, inc_token))) for line in code ] + decrease = [ int(any(map(line.startswith, dec_token))) + for line in code ] + + pretty = [] + level = 0 + for n, line in enumerate(code): + if line in ('', '\n'): + pretty.append(line) + continue + level -= decrease[n] + pretty.append("%s%s" % (tab*level, line)) + level += increase[n] + return pretty + + +def rcode(expr, assign_to=None, **settings): + """Converts an expr to a string of r code + + Parameters + ========== + + expr : Expr + A SymPy expression to be converted. + assign_to : optional + When given, the argument is used as the name of the variable to which + the expression is assigned. Can be a string, ``Symbol``, + ``MatrixSymbol``, or ``Indexed`` type. This is helpful in case of + line-wrapping, or for expressions that generate multi-line statements. + precision : integer, optional + The precision for numbers such as pi [default=15]. + user_functions : dict, optional + A dictionary where the keys are string representations of either + ``FunctionClass`` or ``UndefinedFunction`` instances and the values + are their desired R string representations. Alternatively, the + dictionary value can be a list of tuples i.e. [(argument_test, + rfunction_string)] or [(argument_test, rfunction_formater)]. See below + for examples. + human : bool, optional + If True, the result is a single string that may contain some constant + declarations for the number symbols. If False, the same information is + returned in a tuple of (symbols_to_declare, not_supported_functions, + code_text). [default=True]. + contract: bool, optional + If True, ``Indexed`` instances are assumed to obey tensor contraction + rules and the corresponding nested loops over indices are generated. + Setting contract=False will not generate loops, instead the user is + responsible to provide values for the indices in the code. + [default=True]. + + Examples + ======== + + >>> from sympy import rcode, symbols, Rational, sin, ceiling, Abs, Function + >>> x, tau = symbols("x, tau") + >>> rcode((2*tau)**Rational(7, 2)) + '8*sqrt(2)*tau^(7.0/2.0)' + >>> rcode(sin(x), assign_to="s") + 's = sin(x);' + + Simple custom printing can be defined for certain types by passing a + dictionary of {"type" : "function"} to the ``user_functions`` kwarg. + Alternatively, the dictionary value can be a list of tuples i.e. + [(argument_test, cfunction_string)]. + + >>> custom_functions = { + ... "ceiling": "CEIL", + ... "Abs": [(lambda x: not x.is_integer, "fabs"), + ... (lambda x: x.is_integer, "ABS")], + ... "func": "f" + ... } + >>> func = Function('func') + >>> rcode(func(Abs(x) + ceiling(x)), user_functions=custom_functions) + 'f(fabs(x) + CEIL(x))' + + or if the R-function takes a subset of the original arguments: + + >>> rcode(2**x + 3**x, user_functions={'Pow': [ + ... (lambda b, e: b == 2, lambda b, e: 'exp2(%s)' % e), + ... (lambda b, e: b != 2, 'pow')]}) + 'exp2(x) + pow(3, x)' + + ``Piecewise`` expressions are converted into conditionals. If an + ``assign_to`` variable is provided an if statement is created, otherwise + the ternary operator is used. Note that if the ``Piecewise`` lacks a + default term, represented by ``(expr, True)`` then an error will be thrown. + This is to prevent generating an expression that may not evaluate to + anything. + + >>> from sympy import Piecewise + >>> expr = Piecewise((x + 1, x > 0), (x, True)) + >>> print(rcode(expr, assign_to=tau)) + tau = ifelse(x > 0,x + 1,x); + + Support for loops is provided through ``Indexed`` types. With + ``contract=True`` these expressions will be turned into loops, whereas + ``contract=False`` will just print the assignment expression that should be + looped over: + + >>> from sympy import Eq, IndexedBase, Idx + >>> len_y = 5 + >>> y = IndexedBase('y', shape=(len_y,)) + >>> t = IndexedBase('t', shape=(len_y,)) + >>> Dy = IndexedBase('Dy', shape=(len_y-1,)) + >>> i = Idx('i', len_y-1) + >>> e=Eq(Dy[i], (y[i+1]-y[i])/(t[i+1]-t[i])) + >>> rcode(e.rhs, assign_to=e.lhs, contract=False) + 'Dy[i] = (y[i + 1] - y[i])/(t[i + 1] - t[i]);' + + Matrices are also supported, but a ``MatrixSymbol`` of the same dimensions + must be provided to ``assign_to``. Note that any expression that can be + generated normally can also exist inside a Matrix: + + >>> from sympy import Matrix, MatrixSymbol + >>> mat = Matrix([x**2, Piecewise((x + 1, x > 0), (x, True)), sin(x)]) + >>> A = MatrixSymbol('A', 3, 1) + >>> print(rcode(mat, A)) + A[0] = x^2; + A[1] = ifelse(x > 0,x + 1,x); + A[2] = sin(x); + + """ + + return RCodePrinter(settings).doprint(expr, assign_to) + + +def print_rcode(expr, **settings): + """Prints R representation of the given expression.""" + print(rcode(expr, **settings)) diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/repr.py b/.venv/lib/python3.12/site-packages/sympy/printing/repr.py new file mode 100644 index 0000000000000000000000000000000000000000..0a4b756abbab77c3eb0fd77ee1f0bd97382c36fb --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/repr.py @@ -0,0 +1,339 @@ +""" +A Printer for generating executable code. + +The most important function here is srepr that returns a string so that the +relation eval(srepr(expr))=expr holds in an appropriate environment. +""" + +from __future__ import annotations +from typing import Any + +from sympy.core.function import AppliedUndef +from sympy.core.mul import Mul +from mpmath.libmp import repr_dps, to_str as mlib_to_str + +from .printer import Printer, print_function + + +class ReprPrinter(Printer): + printmethod = "_sympyrepr" + + _default_settings: dict[str, Any] = { + "order": None, + "perm_cyclic" : True, + } + + def reprify(self, args, sep): + """ + Prints each item in `args` and joins them with `sep`. + """ + return sep.join([self.doprint(item) for item in args]) + + def emptyPrinter(self, expr): + """ + The fallback printer. + """ + if isinstance(expr, str): + return expr + elif hasattr(expr, "__srepr__"): + return expr.__srepr__() + elif hasattr(expr, "args") and hasattr(expr.args, "__iter__"): + l = [] + for o in expr.args: + l.append(self._print(o)) + return expr.__class__.__name__ + '(%s)' % ', '.join(l) + elif hasattr(expr, "__module__") and hasattr(expr, "__name__"): + return "<'%s.%s'>" % (expr.__module__, expr.__name__) + else: + return str(expr) + + def _print_Add(self, expr, order=None): + args = self._as_ordered_terms(expr, order=order) + args = map(self._print, args) + clsname = type(expr).__name__ + return clsname + "(%s)" % ", ".join(args) + + def _print_Cycle(self, expr): + return expr.__repr__() + + def _print_Permutation(self, expr): + from sympy.combinatorics.permutations import Permutation, Cycle + from sympy.utilities.exceptions import sympy_deprecation_warning + + perm_cyclic = Permutation.print_cyclic + if perm_cyclic is not None: + sympy_deprecation_warning( + f""" + Setting Permutation.print_cyclic is deprecated. Instead use + init_printing(perm_cyclic={perm_cyclic}). + """, + deprecated_since_version="1.6", + active_deprecations_target="deprecated-permutation-print_cyclic", + stacklevel=7, + ) + else: + perm_cyclic = self._settings.get("perm_cyclic", True) + + if perm_cyclic: + if not expr.size: + return 'Permutation()' + # before taking Cycle notation, see if the last element is + # a singleton and move it to the head of the string + s = Cycle(expr)(expr.size - 1).__repr__()[len('Cycle'):] + last = s.rfind('(') + if not last == 0 and ',' not in s[last:]: + s = s[last:] + s[:last] + return 'Permutation%s' %s + else: + s = expr.support() + if not s: + if expr.size < 5: + return 'Permutation(%s)' % str(expr.array_form) + return 'Permutation([], size=%s)' % expr.size + trim = str(expr.array_form[:s[-1] + 1]) + ', size=%s' % expr.size + use = full = str(expr.array_form) + if len(trim) < len(full): + use = trim + return 'Permutation(%s)' % use + + def _print_Function(self, expr): + r = self._print(expr.func) + r += '(%s)' % ', '.join([self._print(a) for a in expr.args]) + return r + + def _print_Heaviside(self, expr): + # Same as _print_Function but uses pargs to suppress default value for + # 2nd arg. + r = self._print(expr.func) + r += '(%s)' % ', '.join([self._print(a) for a in expr.pargs]) + return r + + def _print_FunctionClass(self, expr): + if issubclass(expr, AppliedUndef): + return 'Function(%r)' % (expr.__name__) + else: + return expr.__name__ + + def _print_Half(self, expr): + return 'Rational(1, 2)' + + def _print_RationalConstant(self, expr): + return str(expr) + + def _print_AtomicExpr(self, expr): + return str(expr) + + def _print_NumberSymbol(self, expr): + return str(expr) + + def _print_Integer(self, expr): + return 'Integer(%i)' % expr.p + + def _print_Complexes(self, expr): + return 'Complexes' + + def _print_Integers(self, expr): + return 'Integers' + + def _print_Naturals(self, expr): + return 'Naturals' + + def _print_Naturals0(self, expr): + return 'Naturals0' + + def _print_Rationals(self, expr): + return 'Rationals' + + def _print_Reals(self, expr): + return 'Reals' + + def _print_EmptySet(self, expr): + return 'EmptySet' + + def _print_UniversalSet(self, expr): + return 'UniversalSet' + + def _print_EmptySequence(self, expr): + return 'EmptySequence' + + def _print_list(self, expr): + return "[%s]" % self.reprify(expr, ", ") + + def _print_dict(self, expr): + sep = ", " + dict_kvs = ["%s: %s" % (self.doprint(key), self.doprint(value)) for key, value in expr.items()] + return "{%s}" % sep.join(dict_kvs) + + def _print_set(self, expr): + if not expr: + return "set()" + return "{%s}" % self.reprify(expr, ", ") + + def _print_MatrixBase(self, expr): + # special case for some empty matrices + if (expr.rows == 0) ^ (expr.cols == 0): + return '%s(%s, %s, %s)' % (expr.__class__.__name__, + self._print(expr.rows), + self._print(expr.cols), + self._print([])) + l = [] + for i in range(expr.rows): + l.append([]) + for j in range(expr.cols): + l[-1].append(expr[i, j]) + return '%s(%s)' % (expr.__class__.__name__, self._print(l)) + + def _print_BooleanTrue(self, expr): + return "true" + + def _print_BooleanFalse(self, expr): + return "false" + + def _print_NaN(self, expr): + return "nan" + + def _print_Mul(self, expr, order=None): + if self.order not in ('old', 'none'): + args = expr.as_ordered_factors() + else: + # use make_args in case expr was something like -x -> x + args = Mul.make_args(expr) + + args = map(self._print, args) + clsname = type(expr).__name__ + return clsname + "(%s)" % ", ".join(args) + + def _print_Rational(self, expr): + return 'Rational(%s, %s)' % (self._print(expr.p), self._print(expr.q)) + + def _print_PythonRational(self, expr): + return "%s(%d, %d)" % (expr.__class__.__name__, expr.p, expr.q) + + def _print_Fraction(self, expr): + return 'Fraction(%s, %s)' % (self._print(expr.numerator), self._print(expr.denominator)) + + def _print_Float(self, expr): + r = mlib_to_str(expr._mpf_, repr_dps(expr._prec)) + return "%s('%s', precision=%i)" % (expr.__class__.__name__, r, expr._prec) + + def _print_Sum2(self, expr): + return "Sum2(%s, (%s, %s, %s))" % (self._print(expr.f), self._print(expr.i), + self._print(expr.a), self._print(expr.b)) + + def _print_Str(self, s): + return "%s(%s)" % (s.__class__.__name__, self._print(s.name)) + + def _print_Symbol(self, expr): + d = expr._assumptions_orig + # print the dummy_index like it was an assumption + if expr.is_Dummy: + d = d.copy() + d['dummy_index'] = expr.dummy_index + + if d == {}: + return "%s(%s)" % (expr.__class__.__name__, self._print(expr.name)) + else: + attr = ['%s=%s' % (k, v) for k, v in d.items()] + return "%s(%s, %s)" % (expr.__class__.__name__, + self._print(expr.name), ', '.join(attr)) + + def _print_CoordinateSymbol(self, expr): + d = expr._assumptions.generator + + if d == {}: + return "%s(%s, %s)" % ( + expr.__class__.__name__, + self._print(expr.coord_sys), + self._print(expr.index) + ) + else: + attr = ['%s=%s' % (k, v) for k, v in d.items()] + return "%s(%s, %s, %s)" % ( + expr.__class__.__name__, + self._print(expr.coord_sys), + self._print(expr.index), + ', '.join(attr) + ) + + def _print_Predicate(self, expr): + return "Q.%s" % expr.name + + def _print_AppliedPredicate(self, expr): + # will be changed to just expr.args when args overriding is removed + args = expr._args + return "%s(%s)" % (expr.__class__.__name__, self.reprify(args, ", ")) + + def _print_str(self, expr): + return repr(expr) + + def _print_tuple(self, expr): + if len(expr) == 1: + return "(%s,)" % self._print(expr[0]) + else: + return "(%s)" % self.reprify(expr, ", ") + + def _print_WildFunction(self, expr): + return "%s('%s')" % (expr.__class__.__name__, expr.name) + + def _print_AlgebraicNumber(self, expr): + return "%s(%s, %s)" % (expr.__class__.__name__, + self._print(expr.root), self._print(expr.coeffs())) + + def _print_PolyRing(self, ring): + return "%s(%s, %s, %s)" % (ring.__class__.__name__, + self._print(ring.symbols), self._print(ring.domain), self._print(ring.order)) + + def _print_FracField(self, field): + return "%s(%s, %s, %s)" % (field.__class__.__name__, + self._print(field.symbols), self._print(field.domain), self._print(field.order)) + + def _print_PolyElement(self, poly): + terms = list(poly.terms()) + terms.sort(key=poly.ring.order, reverse=True) + return "%s(%s, %s)" % (poly.__class__.__name__, self._print(poly.ring), self._print(terms)) + + def _print_FracElement(self, frac): + numer_terms = list(frac.numer.terms()) + numer_terms.sort(key=frac.field.order, reverse=True) + denom_terms = list(frac.denom.terms()) + denom_terms.sort(key=frac.field.order, reverse=True) + numer = self._print(numer_terms) + denom = self._print(denom_terms) + return "%s(%s, %s, %s)" % (frac.__class__.__name__, self._print(frac.field), numer, denom) + + def _print_FractionField(self, domain): + cls = domain.__class__.__name__ + field = self._print(domain.field) + return "%s(%s)" % (cls, field) + + def _print_PolynomialRingBase(self, ring): + cls = ring.__class__.__name__ + dom = self._print(ring.domain) + gens = ', '.join(map(self._print, ring.gens)) + order = str(ring.order) + if order != ring.default_order: + orderstr = ", order=" + order + else: + orderstr = "" + return "%s(%s, %s%s)" % (cls, dom, gens, orderstr) + + def _print_DMP(self, p): + cls = p.__class__.__name__ + rep = self._print(p.to_list()) + dom = self._print(p.dom) + return "%s(%s, %s)" % (cls, rep, dom) + + def _print_MonogenicFiniteExtension(self, ext): + # The expanded tree shown by srepr(ext.modulus) + # is not practical. + return "FiniteExtension(%s)" % str(ext.modulus) + + def _print_ExtensionElement(self, f): + rep = self._print(f.rep) + ext = self._print(f.ext) + return "ExtElem(%s, %s)" % (rep, ext) + +@print_function(ReprPrinter) +def srepr(expr, **settings): + """return expr in repr form""" + return ReprPrinter(settings).doprint(expr) diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/rust.py b/.venv/lib/python3.12/site-packages/sympy/printing/rust.py new file mode 100644 index 0000000000000000000000000000000000000000..5bfd481bec6b7350df281accfb9b7a598cf05baa --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/rust.py @@ -0,0 +1,637 @@ +""" +Rust code printer + +The `RustCodePrinter` converts SymPy expressions into Rust expressions. + +A complete code generator, which uses `rust_code` extensively, can be found +in `sympy.utilities.codegen`. The `codegen` module can be used to generate +complete source code files. + +""" + +# Possible Improvement +# +# * make sure we follow Rust Style Guidelines_ +# * make use of pattern matching +# * better support for reference +# * generate generic code and use trait to make sure they have specific methods +# * use crates_ to get more math support +# - num_ +# + BigInt_, BigUint_ +# + Complex_ +# + Rational64_, Rational32_, BigRational_ +# +# .. _crates: https://crates.io/ +# .. _Guidelines: https://github.com/rust-lang/rust/tree/master/src/doc/style +# .. _num: http://rust-num.github.io/num/num/ +# .. _BigInt: http://rust-num.github.io/num/num/bigint/struct.BigInt.html +# .. _BigUint: http://rust-num.github.io/num/num/bigint/struct.BigUint.html +# .. _Complex: http://rust-num.github.io/num/num/complex/struct.Complex.html +# .. _Rational32: http://rust-num.github.io/num/num/rational/type.Rational32.html +# .. _Rational64: http://rust-num.github.io/num/num/rational/type.Rational64.html +# .. _BigRational: http://rust-num.github.io/num/num/rational/type.BigRational.html + +from __future__ import annotations +from functools import reduce +import operator +from typing import Any + +from sympy.codegen.ast import ( + float32, float64, int32, + real, integer, bool_ +) +from sympy.core import S, Rational, Float, Lambda +from sympy.core.expr import Expr +from sympy.core.numbers import equal_valued +from sympy.functions.elementary.integers import ceiling, floor +from sympy.printing.codeprinter import CodePrinter +from sympy.printing.precedence import PRECEDENCE + +# Rust's methods for integer and float can be found at here : +# +# * `Rust - Primitive Type f64 `_ +# * `Rust - Primitive Type i64 `_ +# +# Function Style : +# +# 1. args[0].func(args[1:]), method with arguments +# 2. args[0].func(), method without arguments +# 3. args[1].func(), method without arguments (e.g. (e, x) => x.exp()) +# 4. func(args), function with arguments + +# dictionary mapping SymPy function to (argument_conditions, Rust_function). +# Used in RustCodePrinter._print_Function(self) + +class float_floor(floor): + """ + Same as `sympy.floor`, but mimics the Rust behavior of returning a float rather than an integer + """ + def _eval_is_integer(self): + return False + +class float_ceiling(ceiling): + """ + Same as `sympy.ceiling`, but mimics the Rust behavior of returning a float rather than an integer + """ + def _eval_is_integer(self): + return False + + +function_overrides = { + "floor": (floor, float_floor), + "ceiling": (ceiling, float_ceiling), +} + +# f64 method in Rust +known_functions = { + # "": "is_nan", + # "": "is_infinite", + # "": "is_finite", + # "": "is_normal", + # "": "classify", + "float_floor": "floor", + "float_ceiling": "ceil", + # "": "round", + # "": "trunc", + # "": "fract", + "Abs": "abs", + # "": "signum", + # "": "is_sign_positive", + # "": "is_sign_negative", + # "": "mul_add", + "Pow": [(lambda base, exp: equal_valued(exp, -1), "recip", 2), # 1.0/x + (lambda base, exp: equal_valued(exp, 0.5), "sqrt", 2), # x ** 0.5 + (lambda base, exp: equal_valued(exp, -0.5), "sqrt().recip", 2), # 1/(x ** 0.5) + (lambda base, exp: exp == Rational(1, 3), "cbrt", 2), # x ** (1/3) + (lambda base, exp: equal_valued(base, 2), "exp2", 3), # 2 ** x + (lambda base, exp: exp.is_integer, "powi", 1), # x ** y, for i32 + (lambda base, exp: not exp.is_integer, "powf", 1)], # x ** y, for f64 + "exp": [(lambda exp: True, "exp", 2)], # e ** x + "log": "ln", + # "": "log", # number.log(base) + # "": "log2", + # "": "log10", + # "": "to_degrees", + # "": "to_radians", + "Max": "max", + "Min": "min", + # "": "hypot", # (x**2 + y**2) ** 0.5 + "sin": "sin", + "cos": "cos", + "tan": "tan", + "asin": "asin", + "acos": "acos", + "atan": "atan", + "atan2": "atan2", + # "": "sin_cos", + # "": "exp_m1", # e ** x - 1 + # "": "ln_1p", # ln(1 + x) + "sinh": "sinh", + "cosh": "cosh", + "tanh": "tanh", + "asinh": "asinh", + "acosh": "acosh", + "atanh": "atanh", + "sqrt": "sqrt", # To enable automatic rewrites +} + +# i64 method in Rust +# known_functions_i64 = { +# "": "min_value", +# "": "max_value", +# "": "from_str_radix", +# "": "count_ones", +# "": "count_zeros", +# "": "leading_zeros", +# "": "trainling_zeros", +# "": "rotate_left", +# "": "rotate_right", +# "": "swap_bytes", +# "": "from_be", +# "": "from_le", +# "": "to_be", # to big endian +# "": "to_le", # to little endian +# "": "checked_add", +# "": "checked_sub", +# "": "checked_mul", +# "": "checked_div", +# "": "checked_rem", +# "": "checked_neg", +# "": "checked_shl", +# "": "checked_shr", +# "": "checked_abs", +# "": "saturating_add", +# "": "saturating_sub", +# "": "saturating_mul", +# "": "wrapping_add", +# "": "wrapping_sub", +# "": "wrapping_mul", +# "": "wrapping_div", +# "": "wrapping_rem", +# "": "wrapping_neg", +# "": "wrapping_shl", +# "": "wrapping_shr", +# "": "wrapping_abs", +# "": "overflowing_add", +# "": "overflowing_sub", +# "": "overflowing_mul", +# "": "overflowing_div", +# "": "overflowing_rem", +# "": "overflowing_neg", +# "": "overflowing_shl", +# "": "overflowing_shr", +# "": "overflowing_abs", +# "Pow": "pow", +# "Abs": "abs", +# "sign": "signum", +# "": "is_positive", +# "": "is_negnative", +# } + +# These are the core reserved words in the Rust language. Taken from: +# https://doc.rust-lang.org/reference/keywords.html + +reserved_words = ['abstract', + 'as', + 'async', + 'await', + 'become', + 'box', + 'break', + 'const', + 'continue', + 'crate', + 'do', + 'dyn', + 'else', + 'enum', + 'extern', + 'false', + 'final', + 'fn', + 'for', + 'gen', + 'if', + 'impl', + 'in', + 'let', + 'loop', + 'macro', + 'match', + 'mod', + 'move', + 'mut', + 'override', + 'priv', + 'pub', + 'ref', + 'return', + 'Self', + 'self', + 'static', + 'struct', + 'super', + 'trait', + 'true', + 'try', + 'type', + 'typeof', + 'unsafe', + 'unsized', + 'use', + 'virtual', + 'where', + 'while', + 'yield'] + + +class TypeCast(Expr): + """ + The type casting operator of the Rust language. + """ + + def __init__(self, expr, type_) -> None: + super().__init__() + self.explicit = expr.is_integer and type_ is not integer + self._assumptions = expr._assumptions + if self.explicit: + setattr(self, 'precedence', PRECEDENCE["Func"] + 10) + + @property + def expr(self): + return self.args[0] + + @property + def type_(self): + return self.args[1] + + def sort_key(self, order=None): + return self.args[0].sort_key(order=order) + + +class RustCodePrinter(CodePrinter): + """A printer to convert SymPy expressions to strings of Rust code""" + printmethod = "_rust_code" + language = "Rust" + + type_aliases = { + integer: int32, + real: float64, + } + + type_mappings = { + int32: 'i32', + float32: 'f32', + float64: 'f64', + bool_: 'bool' + } + + _default_settings: dict[str, Any] = dict(CodePrinter._default_settings, **{ + 'precision': 17, + 'user_functions': {}, + 'contract': True, + 'dereference': set(), + }) + + def __init__(self, settings={}): + CodePrinter.__init__(self, settings) + self.known_functions = dict(known_functions) + userfuncs = settings.get('user_functions', {}) + self.known_functions.update(userfuncs) + self._dereference = set(settings.get('dereference', [])) + self.reserved_words = set(reserved_words) + self.function_overrides = function_overrides + + def _rate_index_position(self, p): + return p*5 + + def _get_statement(self, codestring): + return "%s;" % codestring + + def _get_comment(self, text): + return "// %s" % text + + def _declare_number_const(self, name, value): + type_ = self.type_mappings[self.type_aliases[real]] + return "const %s: %s = %s;" % (name, type_, value) + + def _format_code(self, lines): + return self.indent_code(lines) + + def _traverse_matrix_indices(self, mat): + rows, cols = mat.shape + return ((i, j) for i in range(rows) for j in range(cols)) + + def _get_loop_opening_ending(self, indices): + open_lines = [] + close_lines = [] + loopstart = "for %(var)s in %(start)s..%(end)s {" + for i in indices: + # Rust arrays start at 0 and end at dimension-1 + open_lines.append(loopstart % { + 'var': self._print(i), + 'start': self._print(i.lower), + 'end': self._print(i.upper + 1)}) + close_lines.append("}") + return open_lines, close_lines + + def _print_caller_var(self, expr): + if len(expr.args) > 1: + # for something like `sin(x + y + z)`, + # make sure we can get '(x + y + z).sin()' + # instead of 'x + y + z.sin()' + return '(' + self._print(expr) + ')' + elif expr.is_number: + return self._print(expr, _type=True) + else: + return self._print(expr) + + def _print_Function(self, expr): + """ + basic function for printing `Function` + + Function Style : + + 1. args[0].func(args[1:]), method with arguments + 2. args[0].func(), method without arguments + 3. args[1].func(), method without arguments (e.g. (e, x) => x.exp()) + 4. func(args), function with arguments + """ + + if expr.func.__name__ in self.known_functions: + cond_func = self.known_functions[expr.func.__name__] + func = None + style = 1 + if isinstance(cond_func, str): + func = cond_func + else: + for cond, func, style in cond_func: + if cond(*expr.args): + break + if func is not None: + if style == 1: + ret = "%(var)s.%(method)s(%(args)s)" % { + 'var': self._print_caller_var(expr.args[0]), + 'method': func, + 'args': self.stringify(expr.args[1:], ", ") if len(expr.args) > 1 else '' + } + elif style == 2: + ret = "%(var)s.%(method)s()" % { + 'var': self._print_caller_var(expr.args[0]), + 'method': func, + } + elif style == 3: + ret = "%(var)s.%(method)s()" % { + 'var': self._print_caller_var(expr.args[1]), + 'method': func, + } + else: + ret = "%(func)s(%(args)s)" % { + 'func': func, + 'args': self.stringify(expr.args, ", "), + } + return ret + elif hasattr(expr, '_imp_') and isinstance(expr._imp_, Lambda): + # inlined function + return self._print(expr._imp_(*expr.args)) + else: + return self._print_not_supported(expr) + + def _print_Mul(self, expr): + contains_floats = any(arg.is_real and not arg.is_integer for arg in expr.args) + if contains_floats: + expr = reduce(operator.mul,(self._cast_to_float(arg) if arg != -1 else arg for arg in expr.args)) + + return super()._print_Mul(expr) + + def _print_Add(self, expr, order=None): + contains_floats = any(arg.is_real and not arg.is_integer for arg in expr.args) + if contains_floats: + expr = reduce(operator.add, (self._cast_to_float(arg) for arg in expr.args)) + + return super()._print_Add(expr, order) + + def _print_Pow(self, expr): + if expr.base.is_integer and not expr.exp.is_integer: + expr = type(expr)(Float(expr.base), expr.exp) + return self._print(expr) + return self._print_Function(expr) + + def _print_TypeCast(self, expr): + if not expr.explicit: + return self._print(expr.expr) + else: + return self._print(expr.expr) + ' as %s' % self.type_mappings[self.type_aliases[expr.type_]] + + def _print_Float(self, expr, _type=False): + ret = super()._print_Float(expr) + if _type: + return ret + '_%s' % self.type_mappings[self.type_aliases[real]] + else: + return ret + + def _print_Integer(self, expr, _type=False): + ret = super()._print_Integer(expr) + if _type: + return ret + '_%s' % self.type_mappings[self.type_aliases[integer]] + else: + return ret + + def _print_Rational(self, expr): + p, q = int(expr.p), int(expr.q) + float_suffix = self.type_mappings[self.type_aliases[real]] + return '%d_%s/%d.0' % (p, float_suffix, q) + + def _print_Relational(self, expr): + if (expr.lhs.is_integer and not expr.rhs.is_integer) or (expr.rhs.is_integer and not expr.lhs.is_integer): + lhs = self._cast_to_float(expr.lhs) + rhs = self._cast_to_float(expr.rhs) + else: + lhs = expr.lhs + rhs = expr.rhs + lhs_code = self._print(lhs) + rhs_code = self._print(rhs) + op = expr.rel_op + return "{} {} {}".format(lhs_code, op, rhs_code) + + def _print_Indexed(self, expr): + # calculate index for 1d array + dims = expr.shape + elem = S.Zero + offset = S.One + for i in reversed(range(expr.rank)): + elem += expr.indices[i]*offset + offset *= dims[i] + return "%s[%s]" % (self._print(expr.base.label), self._print(elem)) + + def _print_Idx(self, expr): + return expr.label.name + + def _print_Dummy(self, expr): + return expr.name + + def _print_Exp1(self, expr, _type=False): + return "E" + + def _print_Pi(self, expr, _type=False): + return 'PI' + + def _print_Infinity(self, expr, _type=False): + return 'INFINITY' + + def _print_NegativeInfinity(self, expr, _type=False): + return 'NEG_INFINITY' + + def _print_BooleanTrue(self, expr, _type=False): + return "true" + + def _print_BooleanFalse(self, expr, _type=False): + return "false" + + def _print_bool(self, expr, _type=False): + return str(expr).lower() + + def _print_NaN(self, expr, _type=False): + return "NAN" + + def _print_Piecewise(self, expr): + if expr.args[-1].cond != True: + # We need the last conditional to be a True, otherwise the resulting + # function may not return a result. + raise ValueError("All Piecewise expressions must contain an " + "(expr, True) statement to be used as a default " + "condition. Without one, the generated " + "expression may not evaluate to anything under " + "some condition.") + lines = [] + + for i, (e, c) in enumerate(expr.args): + if i == 0: + lines.append("if (%s) {" % self._print(c)) + elif i == len(expr.args) - 1 and c == True: + lines[-1] += " else {" + else: + lines[-1] += " else if (%s) {" % self._print(c) + code0 = self._print(e) + lines.append(code0) + lines.append("}") + + if self._settings['inline']: + return " ".join(lines) + else: + return "\n".join(lines) + + def _print_ITE(self, expr): + from sympy.functions import Piecewise + return self._print(expr.rewrite(Piecewise, deep=False)) + + def _print_MatrixBase(self, A): + if A.cols == 1: + return "[%s]" % ", ".join(self._print(a) for a in A) + else: + raise ValueError("Full Matrix Support in Rust need Crates (https://crates.io/keywords/matrix).") + + def _print_SparseRepMatrix(self, mat): + # do not allow sparse matrices to be made dense + return self._print_not_supported(mat) + + def _print_MatrixElement(self, expr): + return "%s[%s]" % (expr.parent, + expr.j + expr.i*expr.parent.shape[1]) + + def _print_Symbol(self, expr): + + name = super()._print_Symbol(expr) + + if expr in self._dereference: + return '(*%s)' % name + else: + return name + + def _print_Assignment(self, expr): + from sympy.tensor.indexed import IndexedBase + lhs = expr.lhs + rhs = expr.rhs + if self._settings["contract"] and (lhs.has(IndexedBase) or + rhs.has(IndexedBase)): + # Here we check if there is looping to be done, and if so + # print the required loops. + return self._doprint_loops(rhs, lhs) + else: + lhs_code = self._print(lhs) + rhs_code = self._print(rhs) + return self._get_statement("%s = %s" % (lhs_code, rhs_code)) + + def _print_sign(self, expr): + arg = self._print(expr.args[0]) + return "(if (%s == 0.0) { 0.0 } else { (%s).signum() })" % (arg, arg) + + def _cast_to_float(self, expr): + if not expr.is_number: + return TypeCast(expr, real) + elif expr.is_integer: + return Float(expr) + return expr + + def _can_print(self, name): + """ Check if function ``name`` is either a known function or has its own + printing method. Used to check if rewriting is possible.""" + + # since the whole point of function_overrides is to enable proper printing, + # we presume they all are printable + + return name in self.known_functions or name in function_overrides or getattr(self, '_print_{}'.format(name), False) + + def _collect_functions(self, expr): + functions = set() + if isinstance(expr, Expr): + if expr.is_Function: + functions.add(expr.func) + for arg in expr.args: + functions = functions.union(self._collect_functions(arg)) + return functions + + def _rewrite_known_functions(self, expr): + if not isinstance(expr, Expr): + return expr + + expression_functions = self._collect_functions(expr) + rewriteable_functions = { + name: (target_f, required_fs) + for name, (target_f, required_fs) in self._rewriteable_functions.items() + if self._can_print(target_f) + and all(self._can_print(f) for f in required_fs) + } + for func in expression_functions: + target_f, _ = rewriteable_functions.get(func.__name__, (None, None)) + if target_f: + expr = expr.rewrite(target_f) + return expr + + def indent_code(self, code): + """Accepts a string of code or a list of code lines""" + + if isinstance(code, str): + code_lines = self.indent_code(code.splitlines(True)) + return ''.join(code_lines) + + tab = " " + inc_token = ('{', '(', '{\n', '(\n') + dec_token = ('}', ')') + + code = [ line.lstrip(' \t') for line in code ] + + increase = [ int(any(map(line.endswith, inc_token))) for line in code ] + decrease = [ int(any(map(line.startswith, dec_token))) + for line in code ] + + pretty = [] + level = 0 + for n, line in enumerate(code): + if line in ('', '\n'): + pretty.append(line) + continue + level -= decrease[n] + pretty.append("%s%s" % (tab*level, line)) + level += increase[n] + return pretty diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/smtlib.py b/.venv/lib/python3.12/site-packages/sympy/printing/smtlib.py new file mode 100644 index 0000000000000000000000000000000000000000..8fa015c6198cb32837eb3a0d7fe9d61352da25ad --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/smtlib.py @@ -0,0 +1,583 @@ +import typing + +import sympy +from sympy.core import Add, Mul +from sympy.core import Symbol, Expr, Float, Rational, Integer, Basic +from sympy.core.function import UndefinedFunction, Function +from sympy.core.relational import Relational, Unequality, Equality, LessThan, GreaterThan, StrictLessThan, StrictGreaterThan +from sympy.functions.elementary.complexes import Abs +from sympy.functions.elementary.exponential import exp, log, Pow +from sympy.functions.elementary.hyperbolic import sinh, cosh, tanh +from sympy.functions.elementary.miscellaneous import Min, Max +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.elementary.trigonometric import sin, cos, tan, asin, acos, atan, atan2 +from sympy.logic.boolalg import And, Or, Xor, Implies, Boolean +from sympy.logic.boolalg import BooleanTrue, BooleanFalse, BooleanFunction, Not, ITE +from sympy.printing.printer import Printer +from sympy.sets import Interval +from mpmath.libmp.libmpf import prec_to_dps, to_str as mlib_to_str +from sympy.assumptions.assume import AppliedPredicate +from sympy.assumptions.relation.binrel import AppliedBinaryRelation +from sympy.assumptions.ask import Q +from sympy.assumptions.relation.equality import StrictGreaterThanPredicate, StrictLessThanPredicate, GreaterThanPredicate, LessThanPredicate, EqualityPredicate + + +class SMTLibPrinter(Printer): + printmethod = "_smtlib" + + # based on dReal, an automated reasoning tool for solving problems that can be encoded as first-order logic formulas over the real numbers. + # dReal's special strength is in handling problems that involve a wide range of nonlinear real functions. + _default_settings: dict = { + 'precision': None, + 'known_types': { + bool: 'Bool', + int: 'Int', + float: 'Real' + }, + 'known_constants': { + # pi: 'MY_VARIABLE_PI_DECLARED_ELSEWHERE', + }, + 'known_functions': { + Add: '+', + Mul: '*', + + Equality: '=', + LessThan: '<=', + GreaterThan: '>=', + StrictLessThan: '<', + StrictGreaterThan: '>', + + EqualityPredicate(): '=', + LessThanPredicate(): '<=', + GreaterThanPredicate(): '>=', + StrictLessThanPredicate(): '<', + StrictGreaterThanPredicate(): '>', + + exp: 'exp', + log: 'log', + Abs: 'abs', + sin: 'sin', + cos: 'cos', + tan: 'tan', + asin: 'arcsin', + acos: 'arccos', + atan: 'arctan', + atan2: 'arctan2', + sinh: 'sinh', + cosh: 'cosh', + tanh: 'tanh', + Min: 'min', + Max: 'max', + Pow: 'pow', + + And: 'and', + Or: 'or', + Xor: 'xor', + Not: 'not', + ITE: 'ite', + Implies: '=>', + } + } + + symbol_table: dict + + def __init__(self, settings: typing.Optional[dict] = None, + symbol_table=None): + settings = settings or {} + self.symbol_table = symbol_table or {} + Printer.__init__(self, settings) + self._precision = self._settings['precision'] + self._known_types = dict(self._settings['known_types']) + self._known_constants = dict(self._settings['known_constants']) + self._known_functions = dict(self._settings['known_functions']) + + for _ in self._known_types.values(): assert self._is_legal_name(_) + for _ in self._known_constants.values(): assert self._is_legal_name(_) + # for _ in self._known_functions.values(): assert self._is_legal_name(_) # +, *, <, >, etc. + + def _is_legal_name(self, s: str): + if not s: return False + if s[0].isnumeric(): return False + return all(_.isalnum() or _ == '_' for _ in s) + + def _s_expr(self, op: str, args: typing.Union[list, tuple]) -> str: + args_str = ' '.join( + a if isinstance(a, str) + else self._print(a) + for a in args + ) + return f'({op} {args_str})' + + def _print_Function(self, e): + if e in self._known_functions: + op = self._known_functions[e] + elif type(e) in self._known_functions: + op = self._known_functions[type(e)] + elif type(type(e)) == UndefinedFunction: + op = e.name + elif isinstance(e, AppliedBinaryRelation) and e.function in self._known_functions: + op = self._known_functions[e.function] + return self._s_expr(op, e.arguments) + else: + op = self._known_functions[e] # throw KeyError + + return self._s_expr(op, e.args) + + def _print_Relational(self, e: Relational): + return self._print_Function(e) + + def _print_BooleanFunction(self, e: BooleanFunction): + return self._print_Function(e) + + def _print_Expr(self, e: Expr): + return self._print_Function(e) + + def _print_Unequality(self, e: Unequality): + if type(e) in self._known_functions: + return self._print_Relational(e) # default + else: + eq_op = self._known_functions[Equality] + not_op = self._known_functions[Not] + return self._s_expr(not_op, [self._s_expr(eq_op, e.args)]) + + def _print_Piecewise(self, e: Piecewise): + def _print_Piecewise_recursive(args: typing.Union[list, tuple]): + e, c = args[0] + if len(args) == 1: + assert (c is True) or isinstance(c, BooleanTrue) + return self._print(e) + else: + ite = self._known_functions[ITE] + return self._s_expr(ite, [ + c, e, _print_Piecewise_recursive(args[1:]) + ]) + + return _print_Piecewise_recursive(e.args) + + def _print_Interval(self, e: Interval): + if e.start.is_infinite and e.end.is_infinite: + return '' + elif e.start.is_infinite != e.end.is_infinite: + raise ValueError(f'One-sided intervals (`{e}`) are not supported in SMT.') + else: + return f'[{e.start}, {e.end}]' + + def _print_AppliedPredicate(self, e: AppliedPredicate): + if e.function == Q.positive: + rel = Q.gt(e.arguments[0],0) + elif e.function == Q.negative: + rel = Q.lt(e.arguments[0], 0) + elif e.function == Q.zero: + rel = Q.eq(e.arguments[0], 0) + elif e.function == Q.nonpositive: + rel = Q.le(e.arguments[0], 0) + elif e.function == Q.nonnegative: + rel = Q.ge(e.arguments[0], 0) + elif e.function == Q.nonzero: + rel = Q.ne(e.arguments[0], 0) + else: + raise ValueError(f"Predicate (`{e}`) is not handled.") + + return self._print_AppliedBinaryRelation(rel) + + def _print_AppliedBinaryRelation(self, e: AppliedPredicate): + if e.function == Q.ne: + return self._print_Unequality(Unequality(*e.arguments)) + else: + return self._print_Function(e) + + # todo: Sympy does not support quantifiers yet as of 2022, but quantifiers can be handy in SMT. + # For now, users can extend this class and build in their own quantifier support. + # See `test_quantifier_extensions()` in test_smtlib.py for an example of how this might look. + + # def _print_ForAll(self, e: ForAll): + # return self._s('forall', [ + # self._s('', [ + # self._s(sym.name, [self._type_name(sym), Interval(start, end)]) + # for sym, start, end in e.limits + # ]), + # e.function + # ]) + + def _print_BooleanTrue(self, x: BooleanTrue): + return 'true' + + def _print_BooleanFalse(self, x: BooleanFalse): + return 'false' + + def _print_Float(self, x: Float): + dps = prec_to_dps(x._prec) + str_real = mlib_to_str(x._mpf_, dps, strip_zeros=True, min_fixed=None, max_fixed=None) + + if 'e' in str_real: + (mant, exp) = str_real.split('e') + + if exp[0] == '+': + exp = exp[1:] + + mul = self._known_functions[Mul] + pow = self._known_functions[Pow] + + return r"(%s %s (%s 10 %s))" % (mul, mant, pow, exp) + elif str_real in ["+inf", "-inf"]: + raise ValueError("Infinite values are not supported in SMT.") + else: + return str_real + + def _print_float(self, x: float): + return self._print(Float(x)) + + def _print_Rational(self, x: Rational): + return self._s_expr('/', [x.p, x.q]) + + def _print_Integer(self, x: Integer): + assert x.q == 1 + return str(x.p) + + def _print_int(self, x: int): + return str(x) + + def _print_Symbol(self, x: Symbol): + assert self._is_legal_name(x.name) + return x.name + + def _print_NumberSymbol(self, x): + name = self._known_constants.get(x) + if name: + return name + else: + f = x.evalf(self._precision) if self._precision else x.evalf() + return self._print_Float(f) + + def _print_UndefinedFunction(self, x): + assert self._is_legal_name(x.name) + return x.name + + def _print_Exp1(self, x): + return ( + self._print_Function(exp(1, evaluate=False)) + if exp in self._known_functions else + self._print_NumberSymbol(x) + ) + + def emptyPrinter(self, expr): + raise NotImplementedError(f'Cannot convert `{repr(expr)}` of type `{type(expr)}` to SMT.') + + +def smtlib_code( + expr, + auto_assert=True, auto_declare=True, + precision=None, + symbol_table=None, + known_types=None, known_constants=None, known_functions=None, + prefix_expressions=None, suffix_expressions=None, + log_warn=None +): + r"""Converts ``expr`` to a string of smtlib code. + + Parameters + ========== + + expr : Expr | List[Expr] + A SymPy expression or system to be converted. + auto_assert : bool, optional + If false, do not modify expr and produce only the S-Expression equivalent of expr. + If true, assume expr is a system and assert each boolean element. + auto_declare : bool, optional + If false, do not produce declarations for the symbols used in expr. + If true, prepend all necessary declarations for variables used in expr based on symbol_table. + precision : integer, optional + The ``evalf(..)`` precision for numbers such as pi. + symbol_table : dict, optional + A dictionary where keys are ``Symbol`` or ``Function`` instances and values are their Python type i.e. ``bool``, ``int``, ``float``, or ``Callable[...]``. + If incomplete, an attempt will be made to infer types from ``expr``. + known_types: dict, optional + A dictionary where keys are ``bool``, ``int``, ``float`` etc. and values are their corresponding SMT type names. + If not given, a partial listing compatible with several solvers will be used. + known_functions : dict, optional + A dictionary where keys are ``Function``, ``Relational``, ``BooleanFunction``, or ``Expr`` instances and values are their SMT string representations. + If not given, a partial listing optimized for dReal solver (but compatible with others) will be used. + known_constants: dict, optional + A dictionary where keys are ``NumberSymbol`` instances and values are their SMT variable names. + When using this feature, extra caution must be taken to avoid naming collisions between user symbols and listed constants. + If not given, constants will be expanded inline i.e. ``3.14159`` instead of ``MY_SMT_VARIABLE_FOR_PI``. + prefix_expressions: list, optional + A list of lists of ``str`` and/or expressions to convert into SMTLib and prefix to the output. + suffix_expressions: list, optional + A list of lists of ``str`` and/or expressions to convert into SMTLib and postfix to the output. + log_warn: lambda function, optional + A function to record all warnings during potentially risky operations. + Soundness is a core value in SMT solving, so it is good to log all assumptions made. + + Examples + ======== + >>> from sympy import smtlib_code, symbols, sin, Eq + >>> x = symbols('x') + >>> smtlib_code(sin(x).series(x).removeO(), log_warn=print) + Could not infer type of `x`. Defaulting to float. + Non-Boolean expression `x**5/120 - x**3/6 + x` will not be asserted. Converting to SMTLib verbatim. + '(declare-const x Real)\n(+ x (* (/ -1 6) (pow x 3)) (* (/ 1 120) (pow x 5)))' + + >>> from sympy import Rational + >>> x, y, tau = symbols("x, y, tau") + >>> smtlib_code((2*tau)**Rational(7, 2), log_warn=print) + Could not infer type of `tau`. Defaulting to float. + Non-Boolean expression `8*sqrt(2)*tau**(7/2)` will not be asserted. Converting to SMTLib verbatim. + '(declare-const tau Real)\n(* 8 (pow 2 (/ 1 2)) (pow tau (/ 7 2)))' + + ``Piecewise`` expressions are implemented with ``ite`` expressions by default. + Note that if the ``Piecewise`` lacks a default term, represented by + ``(expr, True)`` then an error will be thrown. This is to prevent + generating an expression that may not evaluate to anything. + + >>> from sympy import Piecewise + >>> pw = Piecewise((x + 1, x > 0), (x, True)) + >>> smtlib_code(Eq(pw, 3), symbol_table={x: float}, log_warn=print) + '(declare-const x Real)\n(assert (= (ite (> x 0) (+ 1 x) x) 3))' + + Custom printing can be defined for certain types by passing a dictionary of + PythonType : "SMT Name" to the ``known_types``, ``known_constants``, and ``known_functions`` kwargs. + + >>> from typing import Callable + >>> from sympy import Function, Add + >>> f = Function('f') + >>> g = Function('g') + >>> smt_builtin_funcs = { # functions our SMT solver will understand + ... f: "existing_smtlib_fcn", + ... Add: "sum", + ... } + >>> user_def_funcs = { # functions defined by the user must have their types specified explicitly + ... g: Callable[[int], float], + ... } + >>> smtlib_code(f(x) + g(x), symbol_table=user_def_funcs, known_functions=smt_builtin_funcs, log_warn=print) + Non-Boolean expression `f(x) + g(x)` will not be asserted. Converting to SMTLib verbatim. + '(declare-const x Int)\n(declare-fun g (Int) Real)\n(sum (existing_smtlib_fcn x) (g x))' + """ + log_warn = log_warn or (lambda _: None) + + if not isinstance(expr, list): expr = [expr] + expr = [ + sympy.sympify(_, strict=True, evaluate=False, convert_xor=False) + for _ in expr + ] + + if not symbol_table: symbol_table = {} + symbol_table = _auto_infer_smtlib_types( + *expr, symbol_table=symbol_table + ) + # See [FALLBACK RULES] + # Need SMTLibPrinter to populate known_functions and known_constants first. + + settings = {} + if precision: settings['precision'] = precision + del precision + + if known_types: settings['known_types'] = known_types + del known_types + + if known_functions: settings['known_functions'] = known_functions + del known_functions + + if known_constants: settings['known_constants'] = known_constants + del known_constants + + if not prefix_expressions: prefix_expressions = [] + if not suffix_expressions: suffix_expressions = [] + + p = SMTLibPrinter(settings, symbol_table) + del symbol_table + + # [FALLBACK RULES] + for e in expr: + for sym in e.atoms(Symbol, Function): + if ( + sym.is_Symbol and + sym not in p._known_constants and + sym not in p.symbol_table + ): + log_warn(f"Could not infer type of `{sym}`. Defaulting to float.") + p.symbol_table[sym] = float + if ( + sym.is_Function and + type(sym) not in p._known_functions and + type(sym) not in p.symbol_table and + not sym.is_Piecewise + ): raise TypeError( + f"Unknown type of undefined function `{sym}`. " + f"Must be mapped to ``str`` in known_functions or mapped to ``Callable[..]`` in symbol_table." + ) + + declarations = [] + if auto_declare: + constants = {sym.name: sym for e in expr for sym in e.free_symbols + if sym not in p._known_constants} + functions = {fnc.name: fnc for e in expr for fnc in e.atoms(Function) + if type(fnc) not in p._known_functions and not fnc.is_Piecewise} + declarations = \ + [ + _auto_declare_smtlib(sym, p, log_warn) + for sym in constants.values() + ] + [ + _auto_declare_smtlib(fnc, p, log_warn) + for fnc in functions.values() + ] + declarations = [decl for decl in declarations if decl] + + if auto_assert: + expr = [_auto_assert_smtlib(e, p, log_warn) for e in expr] + + # return SMTLibPrinter().doprint(expr) + return '\n'.join([ + # ';; PREFIX EXPRESSIONS', + *[ + e if isinstance(e, str) else p.doprint(e) + for e in prefix_expressions + ], + + # ';; DECLARATIONS', + *sorted(e for e in declarations), + + # ';; EXPRESSIONS', + *[ + e if isinstance(e, str) else p.doprint(e) + for e in expr + ], + + # ';; SUFFIX EXPRESSIONS', + *[ + e if isinstance(e, str) else p.doprint(e) + for e in suffix_expressions + ], + ]) + + +def _auto_declare_smtlib(sym: typing.Union[Symbol, Function], p: SMTLibPrinter, log_warn: typing.Callable[[str], None]): + if sym.is_Symbol: + type_signature = p.symbol_table[sym] + assert isinstance(type_signature, type) + type_signature = p._known_types[type_signature] + return p._s_expr('declare-const', [sym, type_signature]) + + elif sym.is_Function: + type_signature = p.symbol_table[type(sym)] + assert callable(type_signature) + type_signature = [p._known_types[_] for _ in type_signature.__args__] + assert len(type_signature) > 0 + params_signature = f"({' '.join(type_signature[:-1])})" + return_signature = type_signature[-1] + return p._s_expr('declare-fun', [type(sym), params_signature, return_signature]) + + else: + log_warn(f"Non-Symbol/Function `{sym}` will not be declared.") + return None + + +def _auto_assert_smtlib(e: Expr, p: SMTLibPrinter, log_warn: typing.Callable[[str], None]): + if isinstance(e, Boolean) or ( + e in p.symbol_table and p.symbol_table[e] == bool + ) or ( + e.is_Function and + type(e) in p.symbol_table and + p.symbol_table[type(e)].__args__[-1] == bool + ): + return p._s_expr('assert', [e]) + else: + log_warn(f"Non-Boolean expression `{e}` will not be asserted. Converting to SMTLib verbatim.") + return e + + +def _auto_infer_smtlib_types( + *exprs: Basic, + symbol_table: typing.Optional[dict] = None +) -> dict: + # [TYPE INFERENCE RULES] + # X is alone in an expr => X is bool + # X in BooleanFunction.args => X is bool + # X matches to a bool param of a symbol_table function => X is bool + # X matches to an int param of a symbol_table function => X is int + # X.is_integer => X is int + # X == Y, where X is T => Y is T + + # [FALLBACK RULES] + # see _auto_declare_smtlib(..) + # X is not bool and X is not int and X is Symbol => X is float + # else (e.g. X is Function) => error. must be specified explicitly. + + _symbols = dict(symbol_table) if symbol_table else {} + + def safe_update(syms: set, inf): + for s in syms: + assert s.is_Symbol + if (old_type := _symbols.setdefault(s, inf)) != inf: + raise TypeError(f"Could not infer type of `{s}`. Apparently both `{old_type}` and `{inf}`?") + + # EXPLICIT TYPES + safe_update({ + e + for e in exprs + if e.is_Symbol + }, bool) + + safe_update({ + symbol + for e in exprs + for boolfunc in e.atoms(BooleanFunction) + for symbol in boolfunc.args + if symbol.is_Symbol + }, bool) + + safe_update({ + symbol + for e in exprs + for boolfunc in e.atoms(Function) + if type(boolfunc) in _symbols + for symbol, param in zip(boolfunc.args, _symbols[type(boolfunc)].__args__) + if symbol.is_Symbol and param == bool + }, bool) + + safe_update({ + symbol + for e in exprs + for intfunc in e.atoms(Function) + if type(intfunc) in _symbols + for symbol, param in zip(intfunc.args, _symbols[type(intfunc)].__args__) + if symbol.is_Symbol and param == int + }, int) + + safe_update({ + symbol + for e in exprs + for symbol in e.atoms(Symbol) + if symbol.is_integer + }, int) + + safe_update({ + symbol + for e in exprs + for symbol in e.atoms(Symbol) + if symbol.is_real and not symbol.is_integer + }, float) + + # EQUALITY RELATION RULE + rels_eq = [rel for expr in exprs for rel in expr.atoms(Equality)] + rels = [ + (rel.lhs, rel.rhs) for rel in rels_eq if rel.lhs.is_Symbol + ] + [ + (rel.rhs, rel.lhs) for rel in rels_eq if rel.rhs.is_Symbol + ] + for infer, reltd in rels: + inference = ( + _symbols[infer] if infer in _symbols else + _symbols[reltd] if reltd in _symbols else + + _symbols[type(reltd)].__args__[-1] + if reltd.is_Function and type(reltd) in _symbols else + + bool if reltd.is_Boolean else + int if reltd.is_integer or reltd.is_Integer else + float if reltd.is_real else + None + ) + if inference: safe_update({infer}, inference) + + return _symbols diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/str.py b/.venv/lib/python3.12/site-packages/sympy/printing/str.py new file mode 100644 index 0000000000000000000000000000000000000000..9975776fbb73bec9c956fe359387fa8036995795 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/str.py @@ -0,0 +1,1021 @@ +""" +A Printer for generating readable representation of most SymPy classes. +""" + +from __future__ import annotations +from typing import Any + +from sympy.core import S, Rational, Pow, Basic, Mul, Number +from sympy.core.mul import _keep_coeff +from sympy.core.numbers import Integer +from sympy.core.relational import Relational +from sympy.core.sorting import default_sort_key +from sympy.utilities.iterables import sift +from .precedence import precedence, PRECEDENCE +from .printer import Printer, print_function + +from mpmath.libmp import prec_to_dps, to_str as mlib_to_str + + +class StrPrinter(Printer): + printmethod = "_sympystr" + _default_settings: dict[str, Any] = { + "order": None, + "full_prec": "auto", + "sympy_integers": False, + "abbrev": False, + "perm_cyclic": True, + "min": None, + "max": None, + "dps" : None + } + + _relationals: dict[str, str] = {} + + def parenthesize(self, item, level, strict=False): + if (precedence(item) < level) or ((not strict) and precedence(item) <= level): + return "(%s)" % self._print(item) + else: + return self._print(item) + + def stringify(self, args, sep, level=0): + return sep.join([self.parenthesize(item, level) for item in args]) + + def emptyPrinter(self, expr): + if isinstance(expr, str): + return expr + elif isinstance(expr, Basic): + return repr(expr) + else: + return str(expr) + + def _print_Add(self, expr, order=None): + terms = self._as_ordered_terms(expr, order=order) + + prec = precedence(expr) + l = [] + for term in terms: + t = self._print(term) + if t.startswith('-') and not term.is_Add: + sign = "-" + t = t[1:] + else: + sign = "+" + if precedence(term) < prec or term.is_Add: + l.extend([sign, "(%s)" % t]) + else: + l.extend([sign, t]) + sign = l.pop(0) + if sign == '+': + sign = "" + return sign + ' '.join(l) + + def _print_BooleanTrue(self, expr): + return "True" + + def _print_BooleanFalse(self, expr): + return "False" + + def _print_Not(self, expr): + return '~%s' %(self.parenthesize(expr.args[0],PRECEDENCE["Not"])) + + def _print_And(self, expr): + args = list(expr.args) + for j, i in enumerate(args): + if isinstance(i, Relational) and ( + i.canonical.rhs is S.NegativeInfinity): + args.insert(0, args.pop(j)) + return self.stringify(args, " & ", PRECEDENCE["BitwiseAnd"]) + + def _print_Or(self, expr): + return self.stringify(expr.args, " | ", PRECEDENCE["BitwiseOr"]) + + def _print_Xor(self, expr): + return self.stringify(expr.args, " ^ ", PRECEDENCE["BitwiseXor"]) + + def _print_AppliedPredicate(self, expr): + return '%s(%s)' % ( + self._print(expr.function), self.stringify(expr.arguments, ", ")) + + def _print_Basic(self, expr): + l = [self._print(o) for o in expr.args] + return expr.__class__.__name__ + "(%s)" % ", ".join(l) + + def _print_BlockMatrix(self, B): + if B.blocks.shape == (1, 1): + self._print(B.blocks[0, 0]) + return self._print(B.blocks) + + def _print_Catalan(self, expr): + return 'Catalan' + + def _print_ComplexInfinity(self, expr): + return 'zoo' + + def _print_ConditionSet(self, s): + args = tuple([self._print(i) for i in (s.sym, s.condition)]) + if s.base_set is S.UniversalSet: + return 'ConditionSet(%s, %s)' % args + args += (self._print(s.base_set),) + return 'ConditionSet(%s, %s, %s)' % args + + def _print_Derivative(self, expr): + dexpr = expr.expr + dvars = [i[0] if i[1] == 1 else i for i in expr.variable_count] + return 'Derivative(%s)' % ", ".join((self._print(arg) for arg in [dexpr] + dvars)) + + def _print_dict(self, d): + keys = sorted(d.keys(), key=default_sort_key) + items = [] + + for key in keys: + item = "%s: %s" % (self._print(key), self._print(d[key])) + items.append(item) + + return "{%s}" % ", ".join(items) + + def _print_Dict(self, expr): + return self._print_dict(expr) + + def _print_RandomDomain(self, d): + if hasattr(d, 'as_boolean'): + return 'Domain: ' + self._print(d.as_boolean()) + elif hasattr(d, 'set'): + return ('Domain: ' + self._print(d.symbols) + ' in ' + + self._print(d.set)) + else: + return 'Domain on ' + self._print(d.symbols) + + def _print_Dummy(self, expr): + return '_' + expr.name + + def _print_EulerGamma(self, expr): + return 'EulerGamma' + + def _print_Exp1(self, expr): + return 'E' + + def _print_ExprCondPair(self, expr): + return '(%s, %s)' % (self._print(expr.expr), self._print(expr.cond)) + + def _print_Function(self, expr): + return expr.func.__name__ + "(%s)" % self.stringify(expr.args, ", ") + + def _print_GoldenRatio(self, expr): + return 'GoldenRatio' + + def _print_Heaviside(self, expr): + # Same as _print_Function but uses pargs to suppress default 1/2 for + # 2nd args + return expr.func.__name__ + "(%s)" % self.stringify(expr.pargs, ", ") + + def _print_TribonacciConstant(self, expr): + return 'TribonacciConstant' + + def _print_ImaginaryUnit(self, expr): + return 'I' + + def _print_Infinity(self, expr): + return 'oo' + + def _print_Integral(self, expr): + def _xab_tostr(xab): + if len(xab) == 1: + return self._print(xab[0]) + else: + return self._print((xab[0],) + tuple(xab[1:])) + L = ', '.join([_xab_tostr(l) for l in expr.limits]) + return 'Integral(%s, %s)' % (self._print(expr.function), L) + + def _print_Interval(self, i): + fin = 'Interval{m}({a}, {b})' + a, b, l, r = i.args + if a.is_infinite and b.is_infinite: + m = '' + elif a.is_infinite and not r: + m = '' + elif b.is_infinite and not l: + m = '' + elif not l and not r: + m = '' + elif l and r: + m = '.open' + elif l: + m = '.Lopen' + else: + m = '.Ropen' + return fin.format(**{'a': a, 'b': b, 'm': m}) + + def _print_AccumulationBounds(self, i): + return "AccumBounds(%s, %s)" % (self._print(i.min), + self._print(i.max)) + + def _print_Inverse(self, I): + return "%s**(-1)" % self.parenthesize(I.arg, PRECEDENCE["Pow"]) + + def _print_Lambda(self, obj): + expr = obj.expr + sig = obj.signature + if len(sig) == 1 and sig[0].is_symbol: + sig = sig[0] + return "Lambda(%s, %s)" % (self._print(sig), self._print(expr)) + + def _print_LatticeOp(self, expr): + args = sorted(expr.args, key=default_sort_key) + return expr.func.__name__ + "(%s)" % ", ".join(self._print(arg) for arg in args) + + def _print_Limit(self, expr): + e, z, z0, dir = expr.args + return "Limit(%s, %s, %s, dir='%s')" % tuple(map(self._print, (e, z, z0, dir))) + + + def _print_list(self, expr): + return "[%s]" % self.stringify(expr, ", ") + + def _print_List(self, expr): + return self._print_list(expr) + + def _print_MatrixBase(self, expr): + return expr._format_str(self) + + def _print_MatrixElement(self, expr): + return self.parenthesize(expr.parent, PRECEDENCE["Atom"], strict=True) \ + + '[%s, %s]' % (self._print(expr.i), self._print(expr.j)) + + def _print_MatrixSlice(self, expr): + def strslice(x, dim): + x = list(x) + if x[2] == 1: + del x[2] + if x[0] == 0: + x[0] = '' + if x[1] == dim: + x[1] = '' + return ':'.join((self._print(arg) for arg in x)) + return (self.parenthesize(expr.parent, PRECEDENCE["Atom"], strict=True) + '[' + + strslice(expr.rowslice, expr.parent.rows) + ', ' + + strslice(expr.colslice, expr.parent.cols) + ']') + + def _print_DeferredVector(self, expr): + return expr.name + + def _print_Mul(self, expr): + + prec = precedence(expr) + + # Check for unevaluated Mul. In this case we need to make sure the + # identities are visible, multiple Rational factors are not combined + # etc so we display in a straight-forward form that fully preserves all + # args and their order. + args = expr.args + if args[0] is S.One or any( + isinstance(a, Number) or + a.is_Pow and all(ai.is_Integer for ai in a.args) + for a in args[1:]): + d, n = sift(args, lambda x: + isinstance(x, Pow) and bool(x.exp.as_coeff_Mul()[0] < 0), + binary=True) + for i, di in enumerate(d): + if di.exp.is_Number: + e = -di.exp + else: + dargs = list(di.exp.args) + dargs[0] = -dargs[0] + e = Mul._from_args(dargs) + d[i] = Pow(di.base, e, evaluate=False) if e - 1 else di.base + + pre = [] + # don't parenthesize first factor if negative + if n and not n[0].is_Add and n[0].could_extract_minus_sign(): + pre = [self._print(n.pop(0))] + + nfactors = pre + [self.parenthesize(a, prec, strict=False) + for a in n] + if not nfactors: + nfactors = ['1'] + + # don't parenthesize first of denominator unless singleton + if len(d) > 1 and d[0].could_extract_minus_sign(): + pre = [self._print(d.pop(0))] + else: + pre = [] + dfactors = pre + [self.parenthesize(a, prec, strict=False) + for a in d] + + n = '*'.join(nfactors) + d = '*'.join(dfactors) + if len(dfactors) > 1: + return '%s/(%s)' % (n, d) + elif dfactors: + return '%s/%s' % (n, d) + return n + + c, e = expr.as_coeff_Mul() + if c < 0: + expr = _keep_coeff(-c, e) + sign = "-" + else: + sign = "" + + a = [] # items in the numerator + b = [] # items that are in the denominator (if any) + + pow_paren = [] # Will collect all pow with more than one base element and exp = -1 + + if self.order not in ('old', 'none'): + args = expr.as_ordered_factors() + else: + # use make_args in case expr was something like -x -> x + args = Mul.make_args(expr) + + # Gather args for numerator/denominator + def apow(i): + b, e = i.as_base_exp() + eargs = list(Mul.make_args(e)) + if eargs[0] is S.NegativeOne: + eargs = eargs[1:] + else: + eargs[0] = -eargs[0] + e = Mul._from_args(eargs) + if isinstance(i, Pow): + return i.func(b, e, evaluate=False) + return i.func(e, evaluate=False) + for item in args: + if (item.is_commutative and + isinstance(item, Pow) and + bool(item.exp.as_coeff_Mul()[0] < 0)): + if item.exp is not S.NegativeOne: + b.append(apow(item)) + else: + if (len(item.args[0].args) != 1 and + isinstance(item.base, (Mul, Pow))): + # To avoid situations like #14160 + pow_paren.append(item) + b.append(item.base) + elif item.is_Rational and item is not S.Infinity: + if item.p != 1: + a.append(Rational(item.p)) + if item.q != 1: + b.append(Rational(item.q)) + else: + a.append(item) + + a = a or [S.One] + + a_str = [self.parenthesize(x, prec, strict=False) for x in a] + b_str = [self.parenthesize(x, prec, strict=False) for x in b] + + # To parenthesize Pow with exp = -1 and having more than one Symbol + for item in pow_paren: + if item.base in b: + b_str[b.index(item.base)] = "(%s)" % b_str[b.index(item.base)] + + if not b: + return sign + '*'.join(a_str) + elif len(b) == 1: + return sign + '*'.join(a_str) + "/" + b_str[0] + else: + return sign + '*'.join(a_str) + "/(%s)" % '*'.join(b_str) + + def _print_MatMul(self, expr): + c, m = expr.as_coeff_mmul() + + sign = "" + if c.is_number: + re, im = c.as_real_imag() + if im.is_zero and re.is_negative: + expr = _keep_coeff(-c, m) + sign = "-" + elif re.is_zero and im.is_negative: + expr = _keep_coeff(-c, m) + sign = "-" + + return sign + '*'.join( + [self.parenthesize(arg, precedence(expr)) for arg in expr.args] + ) + + def _print_ElementwiseApplyFunction(self, expr): + return "{}.({})".format( + expr.function, + self._print(expr.expr), + ) + + def _print_NaN(self, expr): + return 'nan' + + def _print_NegativeInfinity(self, expr): + return '-oo' + + def _print_Order(self, expr): + if not expr.variables or all(p is S.Zero for p in expr.point): + if len(expr.variables) <= 1: + return 'O(%s)' % self._print(expr.expr) + else: + return 'O(%s)' % self.stringify((expr.expr,) + expr.variables, ', ', 0) + else: + return 'O(%s)' % self.stringify(expr.args, ', ', 0) + + def _print_Ordinal(self, expr): + return expr.__str__() + + def _print_Cycle(self, expr): + return expr.__str__() + + def _print_Permutation(self, expr): + from sympy.combinatorics.permutations import Permutation, Cycle + from sympy.utilities.exceptions import sympy_deprecation_warning + + perm_cyclic = Permutation.print_cyclic + if perm_cyclic is not None: + sympy_deprecation_warning( + f""" + Setting Permutation.print_cyclic is deprecated. Instead use + init_printing(perm_cyclic={perm_cyclic}). + """, + deprecated_since_version="1.6", + active_deprecations_target="deprecated-permutation-print_cyclic", + stacklevel=7, + ) + else: + perm_cyclic = self._settings.get("perm_cyclic", True) + + if perm_cyclic: + if not expr.size: + return '()' + # before taking Cycle notation, see if the last element is + # a singleton and move it to the head of the string + s = Cycle(expr)(expr.size - 1).__repr__()[len('Cycle'):] + last = s.rfind('(') + if not last == 0 and ',' not in s[last:]: + s = s[last:] + s[:last] + s = s.replace(',', '') + return s + else: + s = expr.support() + if not s: + if expr.size < 5: + return 'Permutation(%s)' % self._print(expr.array_form) + return 'Permutation([], size=%s)' % self._print(expr.size) + trim = self._print(expr.array_form[:s[-1] + 1]) + ', size=%s' % self._print(expr.size) + use = full = self._print(expr.array_form) + if len(trim) < len(full): + use = trim + return 'Permutation(%s)' % use + + def _print_Subs(self, obj): + expr, old, new = obj.args + if len(obj.point) == 1: + old = old[0] + new = new[0] + return "Subs(%s, %s, %s)" % ( + self._print(expr), self._print(old), self._print(new)) + + def _print_TensorIndex(self, expr): + return expr._print() + + def _print_TensorHead(self, expr): + return expr._print() + + def _print_Tensor(self, expr): + return expr._print() + + def _print_TensMul(self, expr): + # prints expressions like "A(a)", "3*A(a)", "(1+x)*A(a)" + sign, args = expr._get_args_for_traditional_printer() + return sign + "*".join( + [self.parenthesize(arg, precedence(expr)) for arg in args] + ) + + def _print_TensAdd(self, expr): + return expr._print() + + def _print_ArraySymbol(self, expr): + return self._print(expr.name) + + def _print_ArrayElement(self, expr): + return "%s[%s]" % ( + self.parenthesize(expr.name, PRECEDENCE["Func"], True), ", ".join([self._print(i) for i in expr.indices])) + + def _print_PermutationGroup(self, expr): + p = [' %s' % self._print(a) for a in expr.args] + return 'PermutationGroup([\n%s])' % ',\n'.join(p) + + def _print_Pi(self, expr): + return 'pi' + + def _print_PolyRing(self, ring): + return "Polynomial ring in %s over %s with %s order" % \ + (", ".join((self._print(rs) for rs in ring.symbols)), + self._print(ring.domain), self._print(ring.order)) + + def _print_FracField(self, field): + return "Rational function field in %s over %s with %s order" % \ + (", ".join((self._print(fs) for fs in field.symbols)), + self._print(field.domain), self._print(field.order)) + + def _print_FreeGroupElement(self, elm): + return elm.__str__() + + def _print_GaussianElement(self, poly): + return "(%s + %s*I)" % (poly.x, poly.y) + + def _print_PolyElement(self, poly): + return poly.str(self, PRECEDENCE, "%s**%s", "*") + + def _print_FracElement(self, frac): + if frac.denom == 1: + return self._print(frac.numer) + else: + numer = self.parenthesize(frac.numer, PRECEDENCE["Mul"], strict=True) + denom = self.parenthesize(frac.denom, PRECEDENCE["Atom"], strict=True) + return numer + "/" + denom + + def _print_Poly(self, expr): + ATOM_PREC = PRECEDENCE["Atom"] - 1 + terms, gens = [], [ self.parenthesize(s, ATOM_PREC) for s in expr.gens ] + + for monom, coeff in expr.terms(): + s_monom = [] + + for i, e in enumerate(monom): + if e > 0: + if e == 1: + s_monom.append(gens[i]) + else: + s_monom.append(gens[i] + "**%d" % e) + + s_monom = "*".join(s_monom) + + if coeff.is_Add: + if s_monom: + s_coeff = "(" + self._print(coeff) + ")" + else: + s_coeff = self._print(coeff) + else: + if s_monom: + if coeff is S.One: + terms.extend(['+', s_monom]) + continue + + if coeff is S.NegativeOne: + terms.extend(['-', s_monom]) + continue + + s_coeff = self._print(coeff) + + if not s_monom: + s_term = s_coeff + else: + s_term = s_coeff + "*" + s_monom + + if s_term.startswith('-'): + terms.extend(['-', s_term[1:]]) + else: + terms.extend(['+', s_term]) + + if terms[0] in ('-', '+'): + modifier = terms.pop(0) + + if modifier == '-': + terms[0] = '-' + terms[0] + + format = expr.__class__.__name__ + "(%s, %s" + + from sympy.polys.polyerrors import PolynomialError + + try: + format += ", modulus=%s" % expr.get_modulus() + except PolynomialError: + format += ", domain='%s'" % expr.get_domain() + + format += ")" + + for index, item in enumerate(gens): + if len(item) > 2 and (item[:1] == "(" and item[len(item) - 1:] == ")"): + gens[index] = item[1:len(item) - 1] + + return format % (' '.join(terms), ', '.join(gens)) + + def _print_UniversalSet(self, p): + return 'UniversalSet' + + def _print_AlgebraicNumber(self, expr): + if expr.is_aliased: + return self._print(expr.as_poly().as_expr()) + else: + return self._print(expr.as_expr()) + + def _print_Pow(self, expr, rational=False): + """Printing helper function for ``Pow`` + + Parameters + ========== + + rational : bool, optional + If ``True``, it will not attempt printing ``sqrt(x)`` or + ``x**S.Half`` as ``sqrt``, and will use ``x**(1/2)`` + instead. + + See examples for additional details + + Examples + ======== + + >>> from sympy import sqrt, StrPrinter + >>> from sympy.abc import x + + How ``rational`` keyword works with ``sqrt``: + + >>> printer = StrPrinter() + >>> printer._print_Pow(sqrt(x), rational=True) + 'x**(1/2)' + >>> printer._print_Pow(sqrt(x), rational=False) + 'sqrt(x)' + >>> printer._print_Pow(1/sqrt(x), rational=True) + 'x**(-1/2)' + >>> printer._print_Pow(1/sqrt(x), rational=False) + '1/sqrt(x)' + + Notes + ===== + + ``sqrt(x)`` is canonicalized as ``Pow(x, S.Half)`` in SymPy, + so there is no need of defining a separate printer for ``sqrt``. + Instead, it should be handled here as well. + """ + PREC = precedence(expr) + + if expr.exp is S.Half and not rational: + return "sqrt(%s)" % self._print(expr.base) + + if expr.is_commutative: + if -expr.exp is S.Half and not rational: + # Note: Don't test "expr.exp == -S.Half" here, because that will + # match -0.5, which we don't want. + return "%s/sqrt(%s)" % tuple((self._print(arg) for arg in (S.One, expr.base))) + if expr.exp is -S.One: + # Similarly to the S.Half case, don't test with "==" here. + return '%s/%s' % (self._print(S.One), + self.parenthesize(expr.base, PREC, strict=False)) + + e = self.parenthesize(expr.exp, PREC, strict=False) + if self.printmethod == '_sympyrepr' and expr.exp.is_Rational and expr.exp.q != 1: + # the parenthesized exp should be '(Rational(a, b))' so strip parens, + # but just check to be sure. + if e.startswith('(Rational'): + return '%s**%s' % (self.parenthesize(expr.base, PREC, strict=False), e[1:-1]) + return '%s**%s' % (self.parenthesize(expr.base, PREC, strict=False), e) + + def _print_UnevaluatedExpr(self, expr): + return self._print(expr.args[0]) + + def _print_MatPow(self, expr): + PREC = precedence(expr) + return '%s**%s' % (self.parenthesize(expr.base, PREC, strict=False), + self.parenthesize(expr.exp, PREC, strict=False)) + + def _print_Integer(self, expr): + if self._settings.get("sympy_integers", False): + return "S(%s)" % (expr) + return str(expr.p) + + def _print_Integers(self, expr): + return 'Integers' + + def _print_Naturals(self, expr): + return 'Naturals' + + def _print_Naturals0(self, expr): + return 'Naturals0' + + def _print_Rationals(self, expr): + return 'Rationals' + + def _print_Reals(self, expr): + return 'Reals' + + def _print_Complexes(self, expr): + return 'Complexes' + + def _print_EmptySet(self, expr): + return 'EmptySet' + + def _print_EmptySequence(self, expr): + return 'EmptySequence' + + def _print_int(self, expr): + return str(expr) + + def _print_mpz(self, expr): + return str(expr) + + def _print_Rational(self, expr): + if expr.q == 1: + return str(expr.p) + else: + if self._settings.get("sympy_integers", False): + return "S(%s)/%s" % (expr.p, expr.q) + return "%s/%s" % (expr.p, expr.q) + + def _print_PythonRational(self, expr): + if expr.q == 1: + return str(expr.p) + else: + return "%d/%d" % (expr.p, expr.q) + + def _print_Fraction(self, expr): + if expr.denominator == 1: + return str(expr.numerator) + else: + return "%s/%s" % (expr.numerator, expr.denominator) + + def _print_mpq(self, expr): + if expr.denominator == 1: + return str(expr.numerator) + else: + return "%s/%s" % (expr.numerator, expr.denominator) + + def _print_Float(self, expr): + prec = expr._prec + dps = self._settings.get('dps', None) + if dps is None: + dps = 0 if prec < 5 else prec_to_dps(expr._prec) + if self._settings["full_prec"] is True: + strip = False + elif self._settings["full_prec"] is False: + strip = True + elif self._settings["full_prec"] == "auto": + strip = self._print_level > 1 + low = self._settings["min"] if "min" in self._settings else None + high = self._settings["max"] if "max" in self._settings else None + rv = mlib_to_str(expr._mpf_, dps, strip_zeros=strip, min_fixed=low, max_fixed=high) + if rv.startswith('-.0'): + rv = '-0.' + rv[3:] + elif rv.startswith('.0'): + rv = '0.' + rv[2:] + rv = rv.removeprefix('+') # e.g., +inf -> inf + return rv + + def _print_Relational(self, expr): + + charmap = { + "==": "Eq", + "!=": "Ne", + ":=": "Assignment", + '+=': "AddAugmentedAssignment", + "-=": "SubAugmentedAssignment", + "*=": "MulAugmentedAssignment", + "/=": "DivAugmentedAssignment", + "%=": "ModAugmentedAssignment", + } + + if expr.rel_op in charmap: + return '%s(%s, %s)' % (charmap[expr.rel_op], self._print(expr.lhs), + self._print(expr.rhs)) + + return '%s %s %s' % (self.parenthesize(expr.lhs, precedence(expr)), + self._relationals.get(expr.rel_op) or expr.rel_op, + self.parenthesize(expr.rhs, precedence(expr))) + + def _print_ComplexRootOf(self, expr): + return "CRootOf(%s, %d)" % (self._print_Add(expr.expr, order='lex'), + expr.index) + + def _print_RootSum(self, expr): + args = [self._print_Add(expr.expr, order='lex')] + + if expr.fun is not S.IdentityFunction: + args.append(self._print(expr.fun)) + + return "RootSum(%s)" % ", ".join(args) + + def _print_GroebnerBasis(self, basis): + cls = basis.__class__.__name__ + + exprs = [self._print_Add(arg, order=basis.order) for arg in basis.exprs] + exprs = "[%s]" % ", ".join(exprs) + + gens = [ self._print(gen) for gen in basis.gens ] + domain = "domain='%s'" % self._print(basis.domain) + order = "order='%s'" % self._print(basis.order) + + args = [exprs] + gens + [domain, order] + + return "%s(%s)" % (cls, ", ".join(args)) + + def _print_set(self, s): + items = sorted(s, key=default_sort_key) + + args = ', '.join(self._print(item) for item in items) + if not args: + return "set()" + return '{%s}' % args + + def _print_FiniteSet(self, s): + from sympy.sets.sets import FiniteSet + items = sorted(s, key=default_sort_key) + + args = ', '.join(self._print(item) for item in items) + if any(item.has(FiniteSet) for item in items): + return 'FiniteSet({})'.format(args) + return '{{{}}}'.format(args) + + def _print_Partition(self, s): + items = sorted(s, key=default_sort_key) + + args = ', '.join(self._print(arg) for arg in items) + return 'Partition({})'.format(args) + + def _print_frozenset(self, s): + if not s: + return "frozenset()" + return "frozenset(%s)" % self._print_set(s) + + def _print_Sum(self, expr): + def _xab_tostr(xab): + if len(xab) == 1: + return self._print(xab[0]) + else: + return self._print((xab[0],) + tuple(xab[1:])) + L = ', '.join([_xab_tostr(l) for l in expr.limits]) + return 'Sum(%s, %s)' % (self._print(expr.function), L) + + def _print_Symbol(self, expr): + return expr.name + _print_MatrixSymbol = _print_Symbol + _print_RandomSymbol = _print_Symbol + + def _print_Identity(self, expr): + return "I" + + def _print_ZeroMatrix(self, expr): + return "0" + + def _print_OneMatrix(self, expr): + return "1" + + def _print_Predicate(self, expr): + return "Q.%s" % expr.name + + def _print_str(self, expr): + return str(expr) + + def _print_tuple(self, expr): + if len(expr) == 1: + return "(%s,)" % self._print(expr[0]) + else: + return "(%s)" % self.stringify(expr, ", ") + + def _print_Tuple(self, expr): + return self._print_tuple(expr) + + def _print_Transpose(self, T): + return "%s.T" % self.parenthesize(T.arg, PRECEDENCE["Pow"]) + + def _print_Uniform(self, expr): + return "Uniform(%s, %s)" % (self._print(expr.a), self._print(expr.b)) + + def _print_Quantity(self, expr): + if self._settings.get("abbrev", False): + return "%s" % expr.abbrev + return "%s" % expr.name + + def _print_Quaternion(self, expr): + s = [self.parenthesize(i, PRECEDENCE["Mul"], strict=True) for i in expr.args] + a = [s[0]] + [i+"*"+j for i, j in zip(s[1:], "ijk")] + return " + ".join(a) + + def _print_Dimension(self, expr): + return str(expr) + + def _print_Wild(self, expr): + return expr.name + '_' + + def _print_WildFunction(self, expr): + return expr.name + '_' + + def _print_WildDot(self, expr): + return expr.name + + def _print_WildPlus(self, expr): + return expr.name + + def _print_WildStar(self, expr): + return expr.name + + def _print_Zero(self, expr): + if self._settings.get("sympy_integers", False): + return "S(0)" + return self._print_Integer(Integer(0)) + + def _print_DMP(self, p): + cls = p.__class__.__name__ + rep = self._print(p.to_list()) + dom = self._print(p.dom) + + return "%s(%s, %s)" % (cls, rep, dom) + + def _print_DMF(self, expr): + cls = expr.__class__.__name__ + num = self._print(expr.num) + den = self._print(expr.den) + dom = self._print(expr.dom) + + return "%s(%s, %s, %s)" % (cls, num, den, dom) + + def _print_Object(self, obj): + return 'Object("%s")' % obj.name + + def _print_IdentityMorphism(self, morphism): + return 'IdentityMorphism(%s)' % morphism.domain + + def _print_NamedMorphism(self, morphism): + return 'NamedMorphism(%s, %s, "%s")' % \ + (morphism.domain, morphism.codomain, morphism.name) + + def _print_Category(self, category): + return 'Category("%s")' % category.name + + def _print_Manifold(self, manifold): + return manifold.name.name + + def _print_Patch(self, patch): + return patch.name.name + + def _print_CoordSystem(self, coords): + return coords.name.name + + def _print_BaseScalarField(self, field): + return field._coord_sys.symbols[field._index].name + + def _print_BaseVectorField(self, field): + return 'e_%s' % field._coord_sys.symbols[field._index].name + + def _print_Differential(self, diff): + field = diff._form_field + if hasattr(field, '_coord_sys'): + return 'd%s' % field._coord_sys.symbols[field._index].name + else: + return 'd(%s)' % self._print(field) + + def _print_Tr(self, expr): + #TODO : Handle indices + return "%s(%s)" % ("Tr", self._print(expr.args[0])) + + def _print_Str(self, s): + return self._print(s.name) + + def _print_AppliedBinaryRelation(self, expr): + rel = expr.function + return '%s(%s, %s)' % (self._print(rel), + self._print(expr.lhs), + self._print(expr.rhs)) + + +@print_function(StrPrinter) +def sstr(expr, **settings): + """Returns the expression as a string. + + For large expressions where speed is a concern, use the setting + order='none'. If abbrev=True setting is used then units are printed in + abbreviated form. + + Examples + ======== + + >>> from sympy import symbols, Eq, sstr + >>> a, b = symbols('a b') + >>> sstr(Eq(a + b, 0)) + 'Eq(a + b, 0)' + """ + + p = StrPrinter(settings) + s = p.doprint(expr) + + return s + + +class StrReprPrinter(StrPrinter): + """(internal) -- see sstrrepr""" + + def _print_str(self, s): + return repr(s) + + def _print_Str(self, s): + # Str does not to be printed same as str here + return "%s(%s)" % (s.__class__.__name__, self._print(s.name)) + +@print_function(StrReprPrinter) +def sstrrepr(expr, **settings): + """return expr in mixed str/repr form + + i.e. strings are returned in repr form with quotes, and everything else + is returned in str form. + + This function could be useful for hooking into sys.displayhook + """ + + p = StrReprPrinter(settings) + s = p.doprint(expr) + + return s diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tableform.py b/.venv/lib/python3.12/site-packages/sympy/printing/tableform.py new file mode 100644 index 0000000000000000000000000000000000000000..4a84ef96ae92517a6ec01ca9db1a13e9afa67093 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tableform.py @@ -0,0 +1,366 @@ +from sympy.core.containers import Tuple +from sympy.core.singleton import S +from sympy.core.symbol import Symbol +from sympy.core.sympify import SympifyError + +from types import FunctionType + + +class TableForm: + r""" + Create a nice table representation of data. + + Examples + ======== + + >>> from sympy import TableForm + >>> t = TableForm([[5, 7], [4, 2], [10, 3]]) + >>> print(t) + 5 7 + 4 2 + 10 3 + + You can use the SymPy's printing system to produce tables in any + format (ascii, latex, html, ...). + + >>> print(t.as_latex()) + \begin{tabular}{l l} + $5$ & $7$ \\ + $4$ & $2$ \\ + $10$ & $3$ \\ + \end{tabular} + + """ + + def __init__(self, data, **kwarg): + """ + Creates a TableForm. + + Parameters: + + data ... + 2D data to be put into the table; data can be + given as a Matrix + + headings ... + gives the labels for rows and columns: + + Can be a single argument that applies to both + dimensions: + + - None ... no labels + - "automatic" ... labels are 1, 2, 3, ... + + Can be a list of labels for rows and columns: + The labels for each dimension can be given + as None, "automatic", or [l1, l2, ...] e.g. + ["automatic", None] will number the rows + + [default: None] + + alignments ... + alignment of the columns with: + + - "left" or "<" + - "center" or "^" + - "right" or ">" + + When given as a single value, the value is used for + all columns. The row headings (if given) will be + right justified unless an explicit alignment is + given for it and all other columns. + + [default: "left"] + + formats ... + a list of format strings or functions that accept + 3 arguments (entry, row number, col number) and + return a string for the table entry. (If a function + returns None then the _print method will be used.) + + wipe_zeros ... + Do not show zeros in the table. + + [default: True] + + pad ... + the string to use to indicate a missing value (e.g. + elements that are None or those that are missing + from the end of a row (i.e. any row that is shorter + than the rest is assumed to have missing values). + When None, nothing will be shown for values that + are missing from the end of a row; values that are + None, however, will be shown. + + [default: None] + + Examples + ======== + + >>> from sympy import TableForm, Symbol + >>> TableForm([[5, 7], [4, 2], [10, 3]]) + 5 7 + 4 2 + 10 3 + >>> TableForm([list('.'*i) for i in range(1, 4)], headings='automatic') + | 1 2 3 + --------- + 1 | . + 2 | . . + 3 | . . . + >>> TableForm([[Symbol('.'*(j if not i%2 else 1)) for i in range(3)] + ... for j in range(4)], alignments='rcl') + . + . . . + .. . .. + ... . ... + """ + from sympy.matrices.dense import Matrix + + # We only support 2D data. Check the consistency: + if isinstance(data, Matrix): + data = data.tolist() + _h = len(data) + + # fill out any short lines + pad = kwarg.get('pad', None) + ok_None = False + if pad is None: + pad = " " + ok_None = True + pad = Symbol(pad) + _w = max(len(line) for line in data) + for i, line in enumerate(data): + if len(line) != _w: + line.extend([pad]*(_w - len(line))) + for j, lj in enumerate(line): + if lj is None: + if not ok_None: + lj = pad + else: + try: + lj = S(lj) + except SympifyError: + lj = Symbol(str(lj)) + line[j] = lj + data[i] = line + _lines = Tuple(*[Tuple(*d) for d in data]) + + headings = kwarg.get("headings", [None, None]) + if headings == "automatic": + _headings = [range(1, _h + 1), range(1, _w + 1)] + else: + h1, h2 = headings + if h1 == "automatic": + h1 = range(1, _h + 1) + if h2 == "automatic": + h2 = range(1, _w + 1) + _headings = [h1, h2] + + allow = ('l', 'r', 'c') + alignments = kwarg.get("alignments", "l") + + def _std_align(a): + a = a.strip().lower() + if len(a) > 1: + return {'left': 'l', 'right': 'r', 'center': 'c'}.get(a, a) + else: + return {'<': 'l', '>': 'r', '^': 'c'}.get(a, a) + std_align = _std_align(alignments) + if std_align in allow: + _alignments = [std_align]*_w + else: + _alignments = [] + for a in alignments: + std_align = _std_align(a) + _alignments.append(std_align) + if std_align not in ('l', 'r', 'c'): + raise ValueError('alignment "%s" unrecognized' % + alignments) + if _headings[0] and len(_alignments) == _w + 1: + _head_align = _alignments[0] + _alignments = _alignments[1:] + else: + _head_align = 'r' + if len(_alignments) != _w: + raise ValueError( + 'wrong number of alignments: expected %s but got %s' % + (_w, len(_alignments))) + + _column_formats = kwarg.get("formats", [None]*_w) + + _wipe_zeros = kwarg.get("wipe_zeros", True) + + self._w = _w + self._h = _h + self._lines = _lines + self._headings = _headings + self._head_align = _head_align + self._alignments = _alignments + self._column_formats = _column_formats + self._wipe_zeros = _wipe_zeros + + def __repr__(self): + from .str import sstr + return sstr(self, order=None) + + def __str__(self): + from .str import sstr + return sstr(self, order=None) + + def as_matrix(self): + """Returns the data of the table in Matrix form. + + Examples + ======== + + >>> from sympy import TableForm + >>> t = TableForm([[5, 7], [4, 2], [10, 3]], headings='automatic') + >>> t + | 1 2 + -------- + 1 | 5 7 + 2 | 4 2 + 3 | 10 3 + >>> t.as_matrix() + Matrix([ + [ 5, 7], + [ 4, 2], + [10, 3]]) + """ + from sympy.matrices.dense import Matrix + return Matrix(self._lines) + + def as_str(self): + # XXX obsolete ? + return str(self) + + def as_latex(self): + from .latex import latex + return latex(self) + + def _sympystr(self, p): + """ + Returns the string representation of 'self'. + + Examples + ======== + + >>> from sympy import TableForm + >>> t = TableForm([[5, 7], [4, 2], [10, 3]]) + >>> s = t.as_str() + + """ + column_widths = [0] * self._w + lines = [] + for line in self._lines: + new_line = [] + for i in range(self._w): + # Format the item somehow if needed: + s = str(line[i]) + if self._wipe_zeros and (s == "0"): + s = " " + w = len(s) + if w > column_widths[i]: + column_widths[i] = w + new_line.append(s) + lines.append(new_line) + + # Check heading: + if self._headings[0]: + self._headings[0] = [str(x) for x in self._headings[0]] + _head_width = max(len(x) for x in self._headings[0]) + + if self._headings[1]: + new_line = [] + for i in range(self._w): + # Format the item somehow if needed: + s = str(self._headings[1][i]) + w = len(s) + if w > column_widths[i]: + column_widths[i] = w + new_line.append(s) + self._headings[1] = new_line + + format_str = [] + + def _align(align, w): + return '%%%s%ss' % ( + ("-" if align == "l" else ""), + str(w)) + format_str = [_align(align, w) for align, w in + zip(self._alignments, column_widths)] + if self._headings[0]: + format_str.insert(0, _align(self._head_align, _head_width)) + format_str.insert(1, '|') + format_str = ' '.join(format_str) + '\n' + + s = [] + if self._headings[1]: + d = self._headings[1] + if self._headings[0]: + d = [""] + d + first_line = format_str % tuple(d) + s.append(first_line) + s.append("-" * (len(first_line) - 1) + "\n") + for i, line in enumerate(lines): + d = [l if self._alignments[j] != 'c' else + l.center(column_widths[j]) for j, l in enumerate(line)] + if self._headings[0]: + l = self._headings[0][i] + l = (l if self._head_align != 'c' else + l.center(_head_width)) + d = [l] + d + s.append(format_str % tuple(d)) + return ''.join(s)[:-1] # don't include trailing newline + + def _latex(self, printer): + """ + Returns the string representation of 'self'. + """ + # Check heading: + if self._headings[1]: + new_line = [] + for i in range(self._w): + # Format the item somehow if needed: + new_line.append(str(self._headings[1][i])) + self._headings[1] = new_line + + alignments = [] + if self._headings[0]: + self._headings[0] = [str(x) for x in self._headings[0]] + alignments = [self._head_align] + alignments.extend(self._alignments) + + s = r"\begin{tabular}{" + " ".join(alignments) + "}\n" + + if self._headings[1]: + d = self._headings[1] + if self._headings[0]: + d = [""] + d + first_line = " & ".join(d) + r" \\" + "\n" + s += first_line + s += r"\hline" + "\n" + for i, line in enumerate(self._lines): + d = [] + for j, x in enumerate(line): + if self._wipe_zeros and (x in (0, "0")): + d.append(" ") + continue + f = self._column_formats[j] + if f: + if isinstance(f, FunctionType): + v = f(x, i, j) + if v is None: + v = printer._print(x) + else: + v = f % x + d.append(v) + else: + v = printer._print(x) + d.append("$%s$" % v) + if self._headings[0]: + d = [self._headings[0][i]] + d + s += " & ".join(d) + r" \\" + "\n" + s += r"\end{tabular}" + return s diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tensorflow.py b/.venv/lib/python3.12/site-packages/sympy/printing/tensorflow.py new file mode 100644 index 0000000000000000000000000000000000000000..78b0df62b611f336468769e4cee1695bc068eee9 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tensorflow.py @@ -0,0 +1,224 @@ +import sympy.codegen +import sympy.codegen.cfunctions +from sympy.external.importtools import version_tuple +from collections.abc import Iterable + +from sympy.core.mul import Mul +from sympy.core.singleton import S +from sympy.codegen.cfunctions import Sqrt +from sympy.external import import_module +from sympy.printing.precedence import PRECEDENCE +from sympy.printing.pycode import AbstractPythonCodePrinter, ArrayPrinter +import sympy + +tensorflow = import_module('tensorflow') + +class TensorflowPrinter(ArrayPrinter, AbstractPythonCodePrinter): + """ + Tensorflow printer which handles vectorized piecewise functions, + logical operators, max/min, and relational operators. + """ + printmethod = "_tensorflowcode" + + mapping = { + sympy.Abs: "tensorflow.math.abs", + sympy.sign: "tensorflow.math.sign", + + # XXX May raise error for ints. + sympy.ceiling: "tensorflow.math.ceil", + sympy.floor: "tensorflow.math.floor", + sympy.log: "tensorflow.math.log", + sympy.exp: "tensorflow.math.exp", + Sqrt: "tensorflow.math.sqrt", + sympy.cos: "tensorflow.math.cos", + sympy.acos: "tensorflow.math.acos", + sympy.sin: "tensorflow.math.sin", + sympy.asin: "tensorflow.math.asin", + sympy.tan: "tensorflow.math.tan", + sympy.atan: "tensorflow.math.atan", + sympy.atan2: "tensorflow.math.atan2", + # XXX Also may give NaN for complex results. + sympy.cosh: "tensorflow.math.cosh", + sympy.acosh: "tensorflow.math.acosh", + sympy.sinh: "tensorflow.math.sinh", + sympy.asinh: "tensorflow.math.asinh", + sympy.tanh: "tensorflow.math.tanh", + sympy.atanh: "tensorflow.math.atanh", + + sympy.re: "tensorflow.math.real", + sympy.im: "tensorflow.math.imag", + sympy.arg: "tensorflow.math.angle", + + # XXX May raise error for ints and complexes + sympy.erf: "tensorflow.math.erf", + sympy.loggamma: "tensorflow.math.lgamma", + + sympy.Eq: "tensorflow.math.equal", + sympy.Ne: "tensorflow.math.not_equal", + sympy.StrictGreaterThan: "tensorflow.math.greater", + sympy.StrictLessThan: "tensorflow.math.less", + sympy.LessThan: "tensorflow.math.less_equal", + sympy.GreaterThan: "tensorflow.math.greater_equal", + + sympy.And: "tensorflow.math.logical_and", + sympy.Or: "tensorflow.math.logical_or", + sympy.Not: "tensorflow.math.logical_not", + sympy.Max: "tensorflow.math.maximum", + sympy.Min: "tensorflow.math.minimum", + + # Matrices + sympy.MatAdd: "tensorflow.math.add", + sympy.HadamardProduct: "tensorflow.math.multiply", + sympy.Trace: "tensorflow.linalg.trace", + + # XXX May raise error for integer matrices. + sympy.Determinant : "tensorflow.linalg.det", + } + + _default_settings = dict( + AbstractPythonCodePrinter._default_settings, + tensorflow_version=None + ) + + def __init__(self, settings=None): + super().__init__(settings) + + version = self._settings['tensorflow_version'] + if version is None and tensorflow: + version = tensorflow.__version__ + self.tensorflow_version = version + + def _print_Function(self, expr): + op = self.mapping.get(type(expr), None) + if op is None: + return super()._print_Basic(expr) + children = [self._print(arg) for arg in expr.args] + if len(children) == 1: + return "%s(%s)" % ( + self._module_format(op), + children[0] + ) + else: + return self._expand_fold_binary_op(op, children) + + _print_Expr = _print_Function + _print_Application = _print_Function + _print_MatrixExpr = _print_Function + # TODO: a better class structure would avoid this mess: + _print_Relational = _print_Function + _print_Not = _print_Function + _print_And = _print_Function + _print_Or = _print_Function + _print_HadamardProduct = _print_Function + _print_Trace = _print_Function + _print_Determinant = _print_Function + + def _print_Inverse(self, expr): + op = self._module_format('tensorflow.linalg.inv') + return "{}({})".format(op, self._print(expr.arg)) + + def _print_Transpose(self, expr): + version = self.tensorflow_version + if version and version_tuple(version) < version_tuple('1.14'): + op = self._module_format('tensorflow.matrix_transpose') + else: + op = self._module_format('tensorflow.linalg.matrix_transpose') + return "{}({})".format(op, self._print(expr.arg)) + + def _print_Derivative(self, expr): + variables = expr.variables + if any(isinstance(i, Iterable) for i in variables): + raise NotImplementedError("derivation by multiple variables is not supported") + def unfold(expr, args): + if not args: + return self._print(expr) + return "%s(%s, %s)[0]" % ( + self._module_format("tensorflow.gradients"), + unfold(expr, args[:-1]), + self._print(args[-1]), + ) + return unfold(expr.expr, variables) + + def _print_Piecewise(self, expr): + version = self.tensorflow_version + if version and version_tuple(version) < version_tuple('1.0'): + tensorflow_piecewise = "tensorflow.select" + else: + tensorflow_piecewise = "tensorflow.where" + + from sympy.functions.elementary.piecewise import Piecewise + e, cond = expr.args[0].args + if len(expr.args) == 1: + return '{}({}, {}, {})'.format( + self._module_format(tensorflow_piecewise), + self._print(cond), + self._print(e), + 0) + + return '{}({}, {}, {})'.format( + self._module_format(tensorflow_piecewise), + self._print(cond), + self._print(e), + self._print(Piecewise(*expr.args[1:]))) + + def _print_Pow(self, expr): + # XXX May raise error for + # int**float or int**complex or float**complex + base, exp = expr.args + if expr.exp == S.Half: + return "{}({})".format( + self._module_format("tensorflow.math.sqrt"), self._print(base)) + return "{}({}, {})".format( + self._module_format("tensorflow.math.pow"), + self._print(base), self._print(exp)) + + def _print_MatrixBase(self, expr): + tensorflow_f = "tensorflow.Variable" if expr.free_symbols else "tensorflow.constant" + data = "["+", ".join(["["+", ".join([self._print(j) for j in i])+"]" for i in expr.tolist()])+"]" + return "%s(%s)" % ( + self._module_format(tensorflow_f), + data, + ) + + def _print_MatMul(self, expr): + from sympy.matrices.expressions import MatrixExpr + mat_args = [arg for arg in expr.args if isinstance(arg, MatrixExpr)] + args = [arg for arg in expr.args if arg not in mat_args] + if args: + return "%s*%s" % ( + self.parenthesize(Mul.fromiter(args), PRECEDENCE["Mul"]), + self._expand_fold_binary_op( + "tensorflow.linalg.matmul", mat_args) + ) + else: + return self._expand_fold_binary_op( + "tensorflow.linalg.matmul", mat_args) + + def _print_MatPow(self, expr): + return self._expand_fold_binary_op( + "tensorflow.linalg.matmul", [expr.base]*expr.exp) + + def _print_CodeBlock(self, expr): + # TODO: is this necessary? + ret = [] + for subexpr in expr.args: + ret.append(self._print(subexpr)) + return "\n".join(ret) + + def _print_isnan(self, exp): + return f'tensorflow.math.is_nan({self._print(*exp.args)})' + + def _print_isinf(self, exp): + 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0000000000000000000000000000000000000000..13308af65b382e77de33302bcd75344d2b00adbf --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_aesaracode.py @@ -0,0 +1,633 @@ +""" +Important note on tests in this module - the Aesara printing functions use a +global cache by default, which means that tests using it will modify global +state and thus not be independent from each other. Instead of using the "cache" +keyword argument each time, this module uses the aesara_code_ and +aesara_function_ functions defined below which default to using a new, empty +cache instead. +""" + +import logging + +from sympy.external import import_module +from sympy.testing.pytest import raises, SKIP, warns_deprecated_sympy + +from sympy.utilities.exceptions import ignore_warnings + + +aesaralogger = logging.getLogger('aesara.configdefaults') +aesaralogger.setLevel(logging.CRITICAL) +aesara = import_module('aesara') +aesaralogger.setLevel(logging.WARNING) + + +if aesara: + import numpy as np + aet = aesara.tensor + from aesara.scalar.basic import ScalarType + from aesara.graph.basic import Variable + from aesara.tensor.var import TensorVariable + from aesara.tensor.elemwise import Elemwise, DimShuffle + from aesara.tensor.math import Dot + + from sympy.printing.aesaracode import true_divide + + xt, yt, zt = [aet.scalar(name, 'floatX') for name in 'xyz'] + Xt, Yt, Zt = [aet.tensor('floatX', (False, False), name=n) for n in 'XYZ'] +else: + #bin/test will not execute any tests now + disabled = True + +import sympy as sy +from sympy.core.singleton import S +from sympy.abc import x, y, z, t +from sympy.printing.aesaracode import (aesara_code, dim_handling, + aesara_function) + + +# Default set of matrix symbols for testing - make square so we can both +# multiply and perform elementwise operations between them. +X, Y, Z = [sy.MatrixSymbol(n, 4, 4) for n in 'XYZ'] + +# For testing AppliedUndef +f_t = sy.Function('f')(t) + + +def aesara_code_(expr, **kwargs): + """ Wrapper for aesara_code that uses a new, empty cache by default. """ + kwargs.setdefault('cache', {}) + with warns_deprecated_sympy(): + return aesara_code(expr, **kwargs) + +def aesara_function_(inputs, outputs, **kwargs): + """ Wrapper for aesara_function that uses a new, empty cache by default. """ + kwargs.setdefault('cache', {}) + with warns_deprecated_sympy(): + return aesara_function(inputs, outputs, **kwargs) + + +def fgraph_of(*exprs): + """ Transform SymPy expressions into Aesara Computation. + + Parameters + ========== + exprs + SymPy expressions + + Returns + ======= + aesara.graph.fg.FunctionGraph + """ + outs = list(map(aesara_code_, exprs)) + ins = list(aesara.graph.basic.graph_inputs(outs)) + ins, outs = aesara.graph.basic.clone(ins, outs) + return aesara.graph.fg.FunctionGraph(ins, outs) + + +def aesara_simplify(fgraph): + """ Simplify a Aesara Computation. + + Parameters + ========== + fgraph : aesara.graph.fg.FunctionGraph + + Returns + ======= + aesara.graph.fg.FunctionGraph + """ + mode = aesara.compile.get_default_mode().excluding("fusion") + fgraph = fgraph.clone() + mode.optimizer.rewrite(fgraph) + return fgraph + + +def theq(a, b): + """ Test two Aesara objects for equality. + + Also accepts numeric types and lists/tuples of supported types. + + Note - debugprint() has a bug where it will accept numeric types but does + not respect the "file" argument and in this case and instead prints the number + to stdout and returns an empty string. This can lead to tests passing where + they should fail because any two numbers will always compare as equal. To + prevent this we treat numbers as a separate case. + """ + numeric_types = (int, float, np.number) + a_is_num = isinstance(a, numeric_types) + b_is_num = isinstance(b, numeric_types) + + # Compare numeric types using regular equality + if a_is_num or b_is_num: + if not (a_is_num and b_is_num): + return False + + return a == b + + # Compare sequences element-wise + a_is_seq = isinstance(a, (tuple, list)) + b_is_seq = isinstance(b, (tuple, list)) + + if a_is_seq or b_is_seq: + if not (a_is_seq and b_is_seq) or type(a) != type(b): + return False + + return list(map(theq, a)) == list(map(theq, b)) + + # Otherwise, assume debugprint() can handle it + astr = aesara.printing.debugprint(a, file='str') + bstr = aesara.printing.debugprint(b, file='str') + + # Check for bug mentioned above + for argname, argval, argstr in [('a', a, astr), ('b', b, bstr)]: + if argstr == '': + raise TypeError( + 'aesara.printing.debugprint(%s) returned empty string ' + '(%s is instance of %r)' + % (argname, argname, type(argval)) + ) + + return astr == bstr + + +def test_example_symbols(): + """ + Check that the example symbols in this module print to their Aesara + equivalents, as many of the other tests depend on this. + """ + assert theq(xt, aesara_code_(x)) + assert theq(yt, aesara_code_(y)) + assert theq(zt, aesara_code_(z)) + assert theq(Xt, aesara_code_(X)) + assert theq(Yt, aesara_code_(Y)) + assert theq(Zt, aesara_code_(Z)) + + +def test_Symbol(): + """ Test printing a Symbol to a aesara variable. """ + xx = aesara_code_(x) + assert isinstance(xx, Variable) + assert xx.broadcastable == () + assert xx.name == x.name + + xx2 = aesara_code_(x, broadcastables={x: (False,)}) + assert xx2.broadcastable == (False,) + assert xx2.name == x.name + +def test_MatrixSymbol(): + """ Test printing a MatrixSymbol to a aesara variable. """ + XX = aesara_code_(X) + assert isinstance(XX, TensorVariable) + assert XX.broadcastable == (False, False) + +@SKIP # TODO - this is currently not checked but should be implemented +def test_MatrixSymbol_wrong_dims(): + """ Test MatrixSymbol with invalid broadcastable. """ + bcs = [(), (False,), (True,), (True, False), (False, True,), (True, True)] + for bc in bcs: + with raises(ValueError): + aesara_code_(X, broadcastables={X: bc}) + +def test_AppliedUndef(): + """ Test printing AppliedUndef instance, which works similarly to Symbol. """ + ftt = aesara_code_(f_t) + assert isinstance(ftt, TensorVariable) + assert ftt.broadcastable == () + assert ftt.name == 'f_t' + + +def test_add(): + expr = x + y + comp = aesara_code_(expr) + assert comp.owner.op == aesara.tensor.add + +def test_trig(): + assert theq(aesara_code_(sy.sin(x)), aet.sin(xt)) + assert theq(aesara_code_(sy.tan(x)), aet.tan(xt)) + +def test_many(): + """ Test printing a complex expression with multiple symbols. """ + expr = sy.exp(x**2 + sy.cos(y)) * sy.log(2*z) + comp = aesara_code_(expr) + expected = aet.exp(xt**2 + aet.cos(yt)) * aet.log(2*zt) + assert theq(comp, expected) + + +def test_dtype(): + """ Test specifying specific data types through the dtype argument. """ + for dtype in ['float32', 'float64', 'int8', 'int16', 'int32', 'int64']: + assert aesara_code_(x, dtypes={x: dtype}).type.dtype == dtype + + # "floatX" type + assert aesara_code_(x, dtypes={x: 'floatX'}).type.dtype in ('float32', 'float64') + + # Type promotion + assert aesara_code_(x + 1, dtypes={x: 'float32'}).type.dtype == 'float32' + assert aesara_code_(x + y, dtypes={x: 'float64', y: 'float32'}).type.dtype == 'float64' + + +def test_broadcastables(): + """ Test the "broadcastables" argument when printing symbol-like objects. """ + + # No restrictions on shape + for s in [x, f_t]: + for bc in [(), (False,), (True,), (False, False), (True, False)]: + assert aesara_code_(s, broadcastables={s: bc}).broadcastable == bc + + # TODO - matrix broadcasting? + +def test_broadcasting(): + """ Test "broadcastable" attribute after applying element-wise binary op. """ + + expr = x + y + + cases = [ + [(), (), ()], + [(False,), (False,), (False,)], + [(True,), (False,), (False,)], + [(False, True), (False, False), (False, False)], + [(True, False), (False, False), (False, False)], + ] + + for bc1, bc2, bc3 in cases: + comp = aesara_code_(expr, broadcastables={x: bc1, y: bc2}) + assert comp.broadcastable == bc3 + + +def test_MatMul(): + expr = X*Y*Z + expr_t = aesara_code_(expr) + assert isinstance(expr_t.owner.op, Dot) + assert theq(expr_t, Xt.dot(Yt).dot(Zt)) + +def test_Transpose(): + assert isinstance(aesara_code_(X.T).owner.op, DimShuffle) + +def test_MatAdd(): + expr = X+Y+Z + assert isinstance(aesara_code_(expr).owner.op, Elemwise) + + +def test_Rationals(): + assert theq(aesara_code_(sy.Integer(2) / 3), true_divide(2, 3)) + assert theq(aesara_code_(S.Half), true_divide(1, 2)) + +def test_Integers(): + assert aesara_code_(sy.Integer(3)) == 3 + +def test_factorial(): + n = sy.Symbol('n') + assert aesara_code_(sy.factorial(n)) + +def test_Derivative(): + with ignore_warnings(UserWarning): + simp = lambda expr: aesara_simplify(fgraph_of(expr)) + assert theq(simp(aesara_code_(sy.Derivative(sy.sin(x), x, evaluate=False))), + simp(aesara.grad(aet.sin(xt), xt))) + + +def test_aesara_function_simple(): + """ Test aesara_function() with single output. """ + f = aesara_function_([x, y], [x+y]) + assert f(2, 3) == 5 + +def test_aesara_function_multi(): + """ Test aesara_function() with multiple outputs. """ + f = aesara_function_([x, y], [x+y, x-y]) + o1, o2 = f(2, 3) + assert o1 == 5 + assert o2 == -1 + +def test_aesara_function_numpy(): + """ Test aesara_function() vs Numpy implementation. """ + f = aesara_function_([x, y], [x+y], dim=1, + dtypes={x: 'float64', y: 'float64'}) + assert np.linalg.norm(f([1, 2], [3, 4]) - np.asarray([4, 6])) < 1e-9 + + f = aesara_function_([x, y], [x+y], dtypes={x: 'float64', y: 'float64'}, + dim=1) + xx = np.arange(3).astype('float64') + yy = 2*np.arange(3).astype('float64') + assert np.linalg.norm(f(xx, yy) - 3*np.arange(3)) < 1e-9 + + +def test_aesara_function_matrix(): + m = sy.Matrix([[x, y], [z, x + y + z]]) + expected = np.array([[1.0, 2.0], [3.0, 1.0 + 2.0 + 3.0]]) + f = aesara_function_([x, y, z], [m]) + np.testing.assert_allclose(f(1.0, 2.0, 3.0), expected) + f = aesara_function_([x, y, z], [m], scalar=True) + np.testing.assert_allclose(f(1.0, 2.0, 3.0), expected) + f = aesara_function_([x, y, z], [m, m]) + assert isinstance(f(1.0, 2.0, 3.0), type([])) + np.testing.assert_allclose(f(1.0, 2.0, 3.0)[0], expected) + np.testing.assert_allclose(f(1.0, 2.0, 3.0)[1], expected) + +def test_dim_handling(): + assert dim_handling([x], dim=2) == {x: (False, False)} + assert dim_handling([x, y], dims={x: 1, y: 2}) == {x: (False, True), + y: (False, False)} + assert dim_handling([x], broadcastables={x: (False,)}) == {x: (False,)} + +def test_aesara_function_kwargs(): + """ + Test passing additional kwargs from aesara_function() to aesara.function(). + """ + import numpy as np + f = aesara_function_([x, y, z], [x+y], dim=1, on_unused_input='ignore', + dtypes={x: 'float64', y: 'float64', z: 'float64'}) + assert np.linalg.norm(f([1, 2], [3, 4], [0, 0]) - np.asarray([4, 6])) < 1e-9 + + f = aesara_function_([x, y, z], [x+y], + dtypes={x: 'float64', y: 'float64', z: 'float64'}, + dim=1, on_unused_input='ignore') + xx = np.arange(3).astype('float64') + yy = 2*np.arange(3).astype('float64') + zz = 2*np.arange(3).astype('float64') + assert np.linalg.norm(f(xx, yy, zz) - 3*np.arange(3)) < 1e-9 + +def test_aesara_function_scalar(): + """ Test the "scalar" argument to aesara_function(). """ + from aesara.compile.function.types import Function + + args = [ + ([x, y], [x + y], None, [0]), # Single 0d output + ([X, Y], [X + Y], None, [2]), # Single 2d output + ([x, y], [x + y], {x: 0, y: 1}, [1]), # Single 1d output + ([x, y], [x + y, x - y], None, [0, 0]), # Two 0d outputs + ([x, y, X, Y], [x + y, X + Y], None, [0, 2]), # One 0d output, one 2d + ] + + # Create and test functions with and without the scalar setting + for inputs, outputs, in_dims, out_dims in args: + for scalar in [False, True]: + + f = aesara_function_(inputs, outputs, dims=in_dims, scalar=scalar) + + # Check the aesara_function attribute is set whether wrapped or not + assert isinstance(f.aesara_function, Function) + + # Feed in inputs of the appropriate size and get outputs + in_values = [ + np.ones([1 if bc else 5 for bc in i.type.broadcastable]) + for i in f.aesara_function.input_storage + ] + out_values = f(*in_values) + if not isinstance(out_values, list): + out_values = [out_values] + + # Check output types and shapes + assert len(out_dims) == len(out_values) + for d, value in zip(out_dims, out_values): + + if scalar and d == 0: + # Should have been converted to a scalar value + assert isinstance(value, np.number) + + else: + # Otherwise should be an array + assert isinstance(value, np.ndarray) + assert value.ndim == d + +def test_aesara_function_bad_kwarg(): + """ + Passing an unknown keyword argument to aesara_function() should raise an + exception. + """ + raises(Exception, lambda : aesara_function_([x], [x+1], foobar=3)) + + +def test_slice(): + assert aesara_code_(slice(1, 2, 3)) == slice(1, 2, 3) + + def theq_slice(s1, s2): + for attr in ['start', 'stop', 'step']: + a1 = getattr(s1, attr) + a2 = getattr(s2, attr) + if a1 is None or a2 is None: + if not (a1 is None or a2 is None): + return False + elif not theq(a1, a2): + return False + return True + + dtypes = {x: 'int32', y: 'int32'} + assert theq_slice(aesara_code_(slice(x, y), dtypes=dtypes), slice(xt, yt)) + assert theq_slice(aesara_code_(slice(1, x, 3), dtypes=dtypes), slice(1, xt, 3)) + +def test_MatrixSlice(): + cache = {} + + n = sy.Symbol('n', integer=True) + X = sy.MatrixSymbol('X', n, n) + + Y = X[1:2:3, 4:5:6] + Yt = aesara_code_(Y, cache=cache) + + s = ScalarType('int64') + assert tuple(Yt.owner.op.idx_list) == (slice(s, s, s), slice(s, s, s)) + assert Yt.owner.inputs[0] == aesara_code_(X, cache=cache) + # == doesn't work in Aesara like it does in SymPy. You have to use + # equals. + assert all(Yt.owner.inputs[i].data == i for i in range(1, 7)) + + k = sy.Symbol('k') + aesara_code_(k, dtypes={k: 'int32'}) + start, stop, step = 4, k, 2 + Y = X[start:stop:step] + Yt = aesara_code_(Y, dtypes={n: 'int32', k: 'int32'}) + # assert Yt.owner.op.idx_list[0].stop == kt + +def test_BlockMatrix(): + n = sy.Symbol('n', integer=True) + A, B, C, D = [sy.MatrixSymbol(name, n, n) for name in 'ABCD'] + At, Bt, Ct, Dt = map(aesara_code_, (A, B, C, D)) + Block = sy.BlockMatrix([[A, B], [C, D]]) + Blockt = aesara_code_(Block) + solutions = [aet.join(0, aet.join(1, At, Bt), aet.join(1, Ct, Dt)), + aet.join(1, aet.join(0, At, Ct), aet.join(0, Bt, Dt))] + assert any(theq(Blockt, solution) for solution in solutions) + +@SKIP +def test_BlockMatrix_Inverse_execution(): + k, n = 2, 4 + dtype = 'float32' + A = sy.MatrixSymbol('A', n, k) + B = sy.MatrixSymbol('B', n, n) + inputs = A, B + output = B.I*A + + cutsizes = {A: [(n//2, n//2), (k//2, k//2)], + B: [(n//2, n//2), (n//2, n//2)]} + cutinputs = [sy.blockcut(i, *cutsizes[i]) for i in inputs] + cutoutput = output.subs(dict(zip(inputs, cutinputs))) + + dtypes = dict(zip(inputs, [dtype]*len(inputs))) + f = aesara_function_(inputs, [output], dtypes=dtypes, cache={}) + fblocked = aesara_function_(inputs, [sy.block_collapse(cutoutput)], + dtypes=dtypes, cache={}) + + ninputs = [np.random.rand(*x.shape).astype(dtype) for x in inputs] + ninputs = [np.arange(n*k).reshape(A.shape).astype(dtype), + np.eye(n).astype(dtype)] + ninputs[1] += np.ones(B.shape)*1e-5 + + assert np.allclose(f(*ninputs), fblocked(*ninputs), rtol=1e-5) + +def test_DenseMatrix(): + from aesara.tensor.basic import Join + + t = sy.Symbol('theta') + for MatrixType in [sy.Matrix, sy.ImmutableMatrix]: + X = MatrixType([[sy.cos(t), -sy.sin(t)], [sy.sin(t), sy.cos(t)]]) + tX = aesara_code_(X) + assert isinstance(tX, TensorVariable) + assert isinstance(tX.owner.op, Join) + + +def test_cache_basic(): + """ Test single symbol-like objects are cached when printed by themselves. """ + + # Pairs of objects which should be considered equivalent with respect to caching + pairs = [ + (x, sy.Symbol('x')), + (X, sy.MatrixSymbol('X', *X.shape)), + (f_t, sy.Function('f')(sy.Symbol('t'))), + ] + + for s1, s2 in pairs: + cache = {} + st = aesara_code_(s1, cache=cache) + + # Test hit with same instance + assert aesara_code_(s1, cache=cache) is st + + # Test miss with same instance but new cache + assert aesara_code_(s1, cache={}) is not st + + # Test hit with different but equivalent instance + assert aesara_code_(s2, cache=cache) is st + +def test_global_cache(): + """ Test use of the global cache. """ + from sympy.printing.aesaracode import global_cache + + backup = dict(global_cache) + try: + # Temporarily empty global cache + global_cache.clear() + + for s in [x, X, f_t]: + with warns_deprecated_sympy(): + st = aesara_code(s) + assert aesara_code(s) is st + + finally: + # Restore global cache + global_cache.update(backup) + +def test_cache_types_distinct(): + """ + Test that symbol-like objects of different types (Symbol, MatrixSymbol, + AppliedUndef) are distinguished by the cache even if they have the same + name. + """ + symbols = [sy.Symbol('f_t'), sy.MatrixSymbol('f_t', 4, 4), f_t] + + cache = {} # Single shared cache + printed = {} + + for s in symbols: + st = aesara_code_(s, cache=cache) + assert st not in printed.values() + printed[s] = st + + # Check all printed objects are distinct + assert len(set(map(id, printed.values()))) == len(symbols) + + # Check retrieving + for s, st in printed.items(): + with warns_deprecated_sympy(): + assert aesara_code(s, cache=cache) is st + +def test_symbols_are_created_once(): + """ + Test that a symbol is cached and reused when it appears in an expression + more than once. + """ + expr = sy.Add(x, x, evaluate=False) + comp = aesara_code_(expr) + + assert theq(comp, xt + xt) + assert not theq(comp, xt + aesara_code_(x)) + +def test_cache_complex(): + """ + Test caching on a complicated expression with multiple symbols appearing + multiple times. + """ + expr = x ** 2 + (y - sy.exp(x)) * sy.sin(z - x * y) + symbol_names = {s.name for s in expr.free_symbols} + expr_t = aesara_code_(expr) + + # Iterate through variables in the Aesara computational graph that the + # printed expression depends on + seen = set() + for v in aesara.graph.basic.ancestors([expr_t]): + # Owner-less, non-constant variables should be our symbols + if v.owner is None and not isinstance(v, aesara.graph.basic.Constant): + # Check it corresponds to a symbol and appears only once + assert v.name in symbol_names + assert v.name not in seen + seen.add(v.name) + + # Check all were present + assert seen == symbol_names + + +def test_Piecewise(): + # A piecewise linear + expr = sy.Piecewise((0, x<0), (x, x<2), (1, True)) # ___/III + result = aesara_code_(expr) + assert result.owner.op == aet.switch + + expected = aet.switch(xt<0, 0, aet.switch(xt<2, xt, 1)) + assert theq(result, expected) + + expr = sy.Piecewise((x, x < 0)) + result = aesara_code_(expr) + expected = aet.switch(xt < 0, xt, np.nan) + assert theq(result, expected) + + expr = sy.Piecewise((0, sy.And(x>0, x<2)), \ + (x, sy.Or(x>2, x<0))) + result = aesara_code_(expr) + expected = aet.switch(aet.and_(xt>0,xt<2), 0, \ + aet.switch(aet.or_(xt>2, xt<0), xt, np.nan)) + assert theq(result, expected) + + +def test_Relationals(): + assert theq(aesara_code_(sy.Eq(x, y)), aet.eq(xt, yt)) + # assert theq(aesara_code_(sy.Ne(x, y)), aet.neq(xt, yt)) # TODO - implement + assert theq(aesara_code_(x > y), xt > yt) + assert theq(aesara_code_(x < y), xt < yt) + assert theq(aesara_code_(x >= y), xt >= yt) + assert theq(aesara_code_(x <= y), xt <= yt) + + +def test_complexfunctions(): + dtypes = {x:'complex128', y:'complex128'} + with warns_deprecated_sympy(): + xt, yt = aesara_code(x, dtypes=dtypes), aesara_code(y, dtypes=dtypes) + from sympy.functions.elementary.complexes import conjugate + from aesara.tensor import as_tensor_variable as atv + from aesara.tensor import complex as cplx + with warns_deprecated_sympy(): + assert theq(aesara_code(y*conjugate(x), dtypes=dtypes), yt*(xt.conj())) + assert theq(aesara_code((1+2j)*x), xt*(atv(1.0)+atv(2.0)*cplx(0,1))) + + +def test_constantfunctions(): + with warns_deprecated_sympy(): + tf = aesara_function([],[1+1j]) + assert(tf()==1+1j) diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_c.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_c.py new file mode 100644 index 0000000000000000000000000000000000000000..626e7b6f244ea3227b886cd897d327f5d7bf66ec --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_c.py @@ -0,0 +1,888 @@ +from sympy.core import ( + S, pi, oo, Symbol, symbols, Rational, Integer, Float, Function, Mod, GoldenRatio, EulerGamma, Catalan, + Lambda, Dummy, nan, Mul, Pow, UnevaluatedExpr +) +from sympy.core.relational import (Eq, Ge, Gt, Le, Lt, Ne) +from sympy.functions import ( + Abs, acos, acosh, asin, asinh, atan, atanh, atan2, ceiling, cos, cosh, erf, + erfc, exp, floor, gamma, log, loggamma, Max, Min, Piecewise, sign, sin, sinh, + sqrt, tan, tanh, fibonacci, lucas +) +from sympy.sets import Range +from sympy.logic import ITE, Implies, Equivalent +from sympy.codegen import For, aug_assign, Assignment +from sympy.testing.pytest import raises, XFAIL +from sympy.printing.codeprinter import PrintMethodNotImplementedError +from sympy.printing.c import C89CodePrinter, C99CodePrinter, get_math_macros +from sympy.codegen.ast import ( + AddAugmentedAssignment, Element, Type, FloatType, Declaration, Pointer, Variable, value_const, pointer_const, + While, Scope, Print, FunctionPrototype, FunctionDefinition, FunctionCall, Return, + real, float32, float64, float80, float128, intc, Comment, CodeBlock, stderr, QuotedString +) +from sympy.codegen.cfunctions import expm1, log1p, exp2, log2, fma, log10, Cbrt, hypot, Sqrt, isnan, isinf +from sympy.codegen.cnodes import restrict +from sympy.utilities.lambdify import implemented_function +from sympy.tensor import IndexedBase, Idx +from sympy.matrices import Matrix, MatrixSymbol, SparseMatrix + +from sympy.printing.codeprinter import ccode + +x, y, z = symbols('x,y,z') + + +def test_printmethod(): + class fabs(Abs): + def _ccode(self, printer): + return "fabs(%s)" % printer._print(self.args[0]) + + assert ccode(fabs(x)) == "fabs(x)" + + +def test_ccode_sqrt(): + assert ccode(sqrt(x)) == "sqrt(x)" + assert ccode(x**0.5) == "sqrt(x)" + assert ccode(sqrt(x)) == "sqrt(x)" + + +def test_ccode_Pow(): + assert ccode(x**3) == "pow(x, 3)" + assert ccode(x**(y**3)) == "pow(x, pow(y, 3))" + g = implemented_function('g', Lambda(x, 2*x)) + assert ccode(1/(g(x)*3.5)**(x - y**x)/(x**2 + y)) == \ + "pow(3.5*2*x, -x + pow(y, x))/(pow(x, 2) + y)" + assert ccode(x**-1.0) == '1.0/x' + assert ccode(x**Rational(2, 3)) == 'pow(x, 2.0/3.0)' + assert ccode(x**Rational(2, 3), type_aliases={real: float80}) == 'powl(x, 2.0L/3.0L)' + _cond_cfunc = [(lambda base, exp: exp.is_integer, "dpowi"), + (lambda base, exp: not exp.is_integer, "pow")] + assert ccode(x**3, user_functions={'Pow': _cond_cfunc}) == 'dpowi(x, 3)' + assert ccode(x**0.5, user_functions={'Pow': _cond_cfunc}) == 'pow(x, 0.5)' + assert ccode(x**Rational(16, 5), user_functions={'Pow': _cond_cfunc}) == 'pow(x, 16.0/5.0)' + _cond_cfunc2 = [(lambda base, exp: base == 2, lambda base, exp: 'exp2(%s)' % exp), + (lambda base, exp: base != 2, 'pow')] + # Related to gh-11353 + assert ccode(2**x, user_functions={'Pow': _cond_cfunc2}) == 'exp2(x)' + assert ccode(x**2, user_functions={'Pow': _cond_cfunc2}) == 'pow(x, 2)' + # For issue 14160 + assert ccode(Mul(-2, x, Pow(Mul(y,y,evaluate=False), -1, evaluate=False), + evaluate=False)) == '-2*x/(y*y)' + + +def test_ccode_Max(): + # Test for gh-11926 + assert ccode(Max(x,x*x),user_functions={"Max":"my_max", "Pow":"my_pow"}) == 'my_max(x, my_pow(x, 2))' + + +def test_ccode_Min_performance(): + #Shouldn't take more than a few seconds + big_min = Min(*symbols('a[0:50]')) + for curr_standard in ('c89', 'c99', 'c11'): + output = ccode(big_min, standard=curr_standard) + assert output.count('(') == output.count(')') + + +def test_ccode_constants_mathh(): + assert ccode(exp(1)) == "M_E" + assert ccode(pi) == "M_PI" + assert ccode(oo, standard='c89') == "HUGE_VAL" + assert ccode(-oo, standard='c89') == "-HUGE_VAL" + assert ccode(oo) == "INFINITY" + assert ccode(-oo, standard='c99') == "-INFINITY" + assert ccode(pi, type_aliases={real: float80}) == "M_PIl" + + +def test_ccode_constants_other(): + assert ccode(2*GoldenRatio) == "const double GoldenRatio = %s;\n2*GoldenRatio" % GoldenRatio.evalf(17) + assert ccode( + 2*Catalan) == "const double Catalan = %s;\n2*Catalan" % Catalan.evalf(17) + assert ccode(2*EulerGamma) == "const double EulerGamma = %s;\n2*EulerGamma" % EulerGamma.evalf(17) + + +def test_ccode_Rational(): + assert ccode(Rational(3, 7)) == "3.0/7.0" + assert ccode(Rational(3, 7), type_aliases={real: float80}) == "3.0L/7.0L" + assert ccode(Rational(18, 9)) == "2" + assert ccode(Rational(3, -7)) == "-3.0/7.0" + assert ccode(Rational(3, -7), type_aliases={real: float80}) == "-3.0L/7.0L" + assert ccode(Rational(-3, -7)) == "3.0/7.0" + assert ccode(Rational(-3, -7), type_aliases={real: float80}) == "3.0L/7.0L" + assert ccode(x + Rational(3, 7)) == "x + 3.0/7.0" + assert ccode(x + Rational(3, 7), type_aliases={real: float80}) == "x + 3.0L/7.0L" + assert ccode(Rational(3, 7)*x) == "(3.0/7.0)*x" + assert ccode(Rational(3, 7)*x, type_aliases={real: float80}) == "(3.0L/7.0L)*x" + + +def test_ccode_Integer(): + assert ccode(Integer(67)) == "67" + assert ccode(Integer(-1)) == "-1" + + +def test_ccode_functions(): + assert ccode(sin(x) ** cos(x)) == "pow(sin(x), cos(x))" + + +def test_ccode_inline_function(): + x = symbols('x') + g = implemented_function('g', Lambda(x, 2*x)) + assert ccode(g(x)) == "2*x" + g = implemented_function('g', Lambda(x, 2*x/Catalan)) + assert ccode( + g(x)) == "const double Catalan = %s;\n2*x/Catalan" % Catalan.evalf(17) + A = IndexedBase('A') + i = Idx('i', symbols('n', integer=True)) + g = implemented_function('g', Lambda(x, x*(1 + x)*(2 + x))) + assert ccode(g(A[i]), assign_to=A[i]) == ( + "for (int i=0; i y" + assert ccode(Ge(x, y)) == "x >= y" + + +def test_ccode_Piecewise(): + expr = Piecewise((x, x < 1), (x**2, True)) + assert ccode(expr) == ( + "((x < 1) ? (\n" + " x\n" + ")\n" + ": (\n" + " pow(x, 2)\n" + "))") + assert ccode(expr, assign_to="c") == ( + "if (x < 1) {\n" + " c = x;\n" + "}\n" + "else {\n" + " c = pow(x, 2);\n" + "}") + expr = Piecewise((x, x < 1), (x + 1, x < 2), (x**2, True)) + assert ccode(expr) == ( + "((x < 1) ? (\n" + " x\n" + ")\n" + ": ((x < 2) ? (\n" + " x + 1\n" + ")\n" + ": (\n" + " pow(x, 2)\n" + ")))") + assert ccode(expr, assign_to='c') == ( + "if (x < 1) {\n" + " c = x;\n" + "}\n" + "else if (x < 2) {\n" + " c = x + 1;\n" + "}\n" + "else {\n" + " c = pow(x, 2);\n" + "}") + # Check that Piecewise without a True (default) condition error + expr = Piecewise((x, x < 1), (x**2, x > 1), (sin(x), x > 0)) + raises(ValueError, lambda: ccode(expr)) + + +def test_ccode_sinc(): + from sympy.functions.elementary.trigonometric import sinc + expr = sinc(x) + assert ccode(expr) == ( + "(((x != 0) ? (\n" + " sin(x)/x\n" + ")\n" + ": (\n" + " 1\n" + ")))") + + +def test_ccode_Piecewise_deep(): + p = ccode(2*Piecewise((x, x < 1), (x + 1, x < 2), (x**2, True))) + assert p == ( + "2*((x < 1) ? (\n" + " x\n" + ")\n" + ": ((x < 2) ? (\n" + " x + 1\n" + ")\n" + ": (\n" + " pow(x, 2)\n" + ")))") + expr = x*y*z + x**2 + y**2 + Piecewise((0, x < 0.5), (1, True)) + cos(z) - 1 + assert ccode(expr) == ( + "pow(x, 2) + x*y*z + pow(y, 2) + ((x < 0.5) ? (\n" + " 0\n" + ")\n" + ": (\n" + " 1\n" + ")) + cos(z) - 1") + assert ccode(expr, assign_to='c') == ( + "c = pow(x, 2) + x*y*z + pow(y, 2) + ((x < 0.5) ? (\n" + " 0\n" + ")\n" + ": (\n" + " 1\n" + ")) + cos(z) - 1;") + + +def test_ccode_ITE(): + expr = ITE(x < 1, y, z) + assert ccode(expr) == ( + "((x < 1) ? (\n" + " y\n" + ")\n" + ": (\n" + " z\n" + "))") + + +def test_ccode_settings(): + raises(TypeError, lambda: ccode(sin(x), method="garbage")) + + +def test_ccode_Indexed(): + s, n, m, o = symbols('s n m o', integer=True) + i, j, k = Idx('i', n), Idx('j', m), Idx('k', o) + + x = IndexedBase('x')[j] + A = IndexedBase('A')[i, j] + B = IndexedBase('B')[i, j, k] + + p = C99CodePrinter() + + assert p._print_Indexed(x) == 'x[j]' + assert p._print_Indexed(A) == 'A[%s]' % (m*i+j) + assert p._print_Indexed(B) == 'B[%s]' % (i*o*m+j*o+k) + + A = IndexedBase('A', shape=(5,3))[i, j] + assert p._print_Indexed(A) == 'A[%s]' % (3*i + j) + + A = IndexedBase('A', shape=(5,3), strides='F')[i, j] + assert ccode(A) == 'A[%s]' % (i + 5*j) + + A = IndexedBase('A', shape=(29,29), strides=(1, s), offset=o)[i, j] + assert ccode(A) == 'A[o + s*j + i]' + + Abase = IndexedBase('A', strides=(s, m, n), offset=o) + assert ccode(Abase[i, j, k]) == 'A[m*j + n*k + o + s*i]' + assert ccode(Abase[2, 3, k]) == 'A[3*m + n*k + o + 2*s]' + + +def test_Element(): + assert ccode(Element('x', 'ij')) == 'x[i][j]' + assert ccode(Element('x', 'ij', strides='kl', offset='o')) == 'x[i*k + j*l + o]' + assert ccode(Element('x', (3,))) == 'x[3]' + assert ccode(Element('x', (3,4,5))) == 'x[3][4][5]' + + +def test_ccode_Indexed_without_looking_for_contraction(): + len_y = 5 + y = IndexedBase('y', shape=(len_y,)) + x = IndexedBase('x', shape=(len_y,)) + Dy = IndexedBase('Dy', shape=(len_y-1,)) + i = Idx('i', len_y-1) + e = Eq(Dy[i], (y[i+1]-y[i])/(x[i+1]-x[i])) + code0 = ccode(e.rhs, assign_to=e.lhs, contract=False) + assert code0 == 'Dy[i] = (y[%s] - y[i])/(x[%s] - x[i]);' % (i + 1, i + 1) + + +def test_ccode_loops_matrix_vector(): + n, m = symbols('n m', integer=True) + A = IndexedBase('A') + x = IndexedBase('x') + y = IndexedBase('y') + i = Idx('i', m) + j = Idx('j', n) + + s = ( + 'for (int i=0; i0), (y, True)), sin(z)]) + A = MatrixSymbol('A', 3, 1) + assert ccode(mat, A) == ( + "A[0] = x*y;\n" + "if (y > 0) {\n" + " A[1] = x + 2;\n" + "}\n" + "else {\n" + " A[1] = y;\n" + "}\n" + "A[2] = sin(z);") + # Test using MatrixElements in expressions + expr = Piecewise((2*A[2, 0], x > 0), (A[2, 0], True)) + sin(A[1, 0]) + A[0, 0] + assert ccode(expr) == ( + "((x > 0) ? (\n" + " 2*A[2]\n" + ")\n" + ": (\n" + " A[2]\n" + ")) + sin(A[1]) + A[0]") + # Test using MatrixElements in a Matrix + q = MatrixSymbol('q', 5, 1) + M = MatrixSymbol('M', 3, 3) + m = Matrix([[sin(q[1,0]), 0, cos(q[2,0])], + [q[1,0] + q[2,0], q[3, 0], 5], + [2*q[4, 0]/q[1,0], sqrt(q[0,0]) + 4, 0]]) + assert ccode(m, M) == ( + "M[0] = sin(q[1]);\n" + "M[1] = 0;\n" + "M[2] = cos(q[2]);\n" + "M[3] = q[1] + q[2];\n" + "M[4] = q[3];\n" + "M[5] = 5;\n" + "M[6] = 2*q[4]/q[1];\n" + "M[7] = sqrt(q[0]) + 4;\n" + "M[8] = 0;") + + +def test_sparse_matrix(): + # gh-15791 + with raises(PrintMethodNotImplementedError): + ccode(SparseMatrix([[1, 2, 3]])) + + assert 'Not supported in C' in C89CodePrinter({'strict': False}).doprint(SparseMatrix([[1, 2, 3]])) + + + +def test_ccode_reserved_words(): + x, y = symbols('x, if') + with raises(ValueError): + ccode(y**2, error_on_reserved=True, standard='C99') + assert ccode(y**2) == 'pow(if_, 2)' + assert ccode(x * y**2, dereference=[y]) == 'pow((*if_), 2)*x' + assert ccode(y**2, reserved_word_suffix='_unreserved') == 'pow(if_unreserved, 2)' + + +def test_ccode_sign(): + expr1, ref1 = sign(x) * y, 'y*(((x) > 0) - ((x) < 0))' + expr2, ref2 = sign(cos(x)), '(((cos(x)) > 0) - ((cos(x)) < 0))' + expr3, ref3 = sign(2 * x + x**2) * x + x**2, 'pow(x, 2) + x*(((pow(x, 2) + 2*x) > 0) - ((pow(x, 2) + 2*x) < 0))' + assert ccode(expr1) == ref1 + assert ccode(expr1, 'z') == 'z = %s;' % ref1 + assert ccode(expr2) == ref2 + assert ccode(expr3) == ref3 + +def test_ccode_Assignment(): + assert ccode(Assignment(x, y + z)) == 'x = y + z;' + assert ccode(aug_assign(x, '+', y + z)) == 'x += y + z;' + + +def test_ccode_For(): + f = For(x, Range(0, 10, 2), [aug_assign(y, '*', x)]) + assert ccode(f) == ("for (x = 0; x < 10; x += 2) {\n" + " y *= x;\n" + "}") + +def test_ccode_Max_Min(): + assert ccode(Max(x, 0), standard='C89') == '((0 > x) ? 0 : x)' + assert ccode(Max(x, 0), standard='C99') == 'fmax(0, x)' + assert ccode(Min(x, 0, sqrt(x)), standard='c89') == ( + '((0 < ((x < sqrt(x)) ? x : sqrt(x))) ? 0 : ((x < sqrt(x)) ? x : sqrt(x)))' + ) + +def test_ccode_standard(): + assert ccode(expm1(x), standard='c99') == 'expm1(x)' + assert ccode(nan, standard='c99') == 'NAN' + assert ccode(float('nan'), standard='c99') == 'NAN' + + +def test_C89CodePrinter(): + c89printer = C89CodePrinter() + assert c89printer.language == 'C' + assert c89printer.standard == 'C89' + assert 'void' in c89printer.reserved_words + assert 'template' not in c89printer.reserved_words + assert c89printer.doprint(log10(x)) == 'log10(x)' + + +def test_C99CodePrinter(): + assert C99CodePrinter().doprint(expm1(x)) == 'expm1(x)' + assert C99CodePrinter().doprint(log1p(x)) == 'log1p(x)' + assert C99CodePrinter().doprint(exp2(x)) == 'exp2(x)' + assert C99CodePrinter().doprint(log2(x)) == 'log2(x)' + assert C99CodePrinter().doprint(fma(x, y, -z)) == 'fma(x, y, -z)' + assert C99CodePrinter().doprint(log10(x)) == 'log10(x)' + assert C99CodePrinter().doprint(Cbrt(x)) == 'cbrt(x)' # note Cbrt due to cbrt already taken. + assert C99CodePrinter().doprint(hypot(x, y)) == 'hypot(x, y)' + assert C99CodePrinter().doprint(loggamma(x)) == 'lgamma(x)' + assert C99CodePrinter().doprint(Max(x, 3, x**2)) == 'fmax(3, fmax(x, pow(x, 2)))' + assert C99CodePrinter().doprint(Min(x, 3)) == 'fmin(3, x)' + c99printer = C99CodePrinter() + assert c99printer.language == 'C' + assert c99printer.standard == 'C99' + assert 'restrict' in c99printer.reserved_words + assert 'using' not in c99printer.reserved_words + + +@XFAIL +def test_C99CodePrinter__precision_f80(): + f80_printer = C99CodePrinter({"type_aliases": {real: float80}}) + assert f80_printer.doprint(sin(x + Float('2.1'))) == 'sinl(x + 2.1L)' + + +def test_C99CodePrinter__precision(): + n = symbols('n', integer=True) + p = symbols('p', integer=True, positive=True) + f32_printer = C99CodePrinter({"type_aliases": {real: float32}}) + f64_printer = C99CodePrinter({"type_aliases": {real: float64}}) + f80_printer = C99CodePrinter({"type_aliases": {real: float80}}) + assert f32_printer.doprint(sin(x+2.1)) == 'sinf(x + 2.1F)' + assert f64_printer.doprint(sin(x+2.1)) == 'sin(x + 2.1000000000000001)' + assert f80_printer.doprint(sin(x+Float('2.0'))) == 'sinl(x + 2.0L)' + + for printer, suffix in zip([f32_printer, f64_printer, f80_printer], ['f', '', 'l']): + def check(expr, ref): + assert printer.doprint(expr) == ref.format(s=suffix, S=suffix.upper()) + check(Abs(n), 'abs(n)') + check(Abs(x + 2.0), 'fabs{s}(x + 2.0{S})') + check(sin(x + 4.0)**cos(x - 2.0), 'pow{s}(sin{s}(x + 4.0{S}), cos{s}(x - 2.0{S}))') + check(exp(x*8.0), 'exp{s}(8.0{S}*x)') + check(exp2(x), 'exp2{s}(x)') + check(expm1(x*4.0), 'expm1{s}(4.0{S}*x)') + check(Mod(p, 2), 'p % 2') + check(Mod(2*p + 3, 3*p + 5, evaluate=False), '(2*p + 3) % (3*p + 5)') + check(Mod(x + 2.0, 3.0), 'fmod{s}(1.0{S}*x + 2.0{S}, 3.0{S})') + check(Mod(x, 2.0*x + 3.0), 'fmod{s}(1.0{S}*x, 2.0{S}*x + 3.0{S})') + check(log(x/2), 'log{s}((1.0{S}/2.0{S})*x)') + check(log10(3*x/2), 'log10{s}((3.0{S}/2.0{S})*x)') + check(log2(x*8.0), 'log2{s}(8.0{S}*x)') + check(log1p(x), 'log1p{s}(x)') + check(2**x, 'pow{s}(2, x)') + check(2.0**x, 'pow{s}(2.0{S}, x)') + check(x**3, 'pow{s}(x, 3)') + check(x**4.0, 'pow{s}(x, 4.0{S})') + check(sqrt(3+x), 'sqrt{s}(x + 3)') + check(Cbrt(x-2.0), 'cbrt{s}(x - 2.0{S})') + check(hypot(x, y), 'hypot{s}(x, y)') + check(sin(3.*x + 2.), 'sin{s}(3.0{S}*x + 2.0{S})') + check(cos(3.*x - 1.), 'cos{s}(3.0{S}*x - 1.0{S})') + check(tan(4.*y + 2.), 'tan{s}(4.0{S}*y + 2.0{S})') + check(asin(3.*x + 2.), 'asin{s}(3.0{S}*x + 2.0{S})') + check(acos(3.*x + 2.), 'acos{s}(3.0{S}*x + 2.0{S})') + check(atan(3.*x + 2.), 'atan{s}(3.0{S}*x + 2.0{S})') + check(atan2(3.*x, 2.*y), 'atan2{s}(3.0{S}*x, 2.0{S}*y)') + + check(sinh(3.*x + 2.), 'sinh{s}(3.0{S}*x + 2.0{S})') + check(cosh(3.*x - 1.), 'cosh{s}(3.0{S}*x - 1.0{S})') + check(tanh(4.0*y + 2.), 'tanh{s}(4.0{S}*y + 2.0{S})') + check(asinh(3.*x + 2.), 'asinh{s}(3.0{S}*x + 2.0{S})') + check(acosh(3.*x + 2.), 'acosh{s}(3.0{S}*x + 2.0{S})') + check(atanh(3.*x + 2.), 'atanh{s}(3.0{S}*x + 2.0{S})') + check(erf(42.*x), 'erf{s}(42.0{S}*x)') + check(erfc(42.*x), 'erfc{s}(42.0{S}*x)') + check(gamma(x), 'tgamma{s}(x)') + check(loggamma(x), 'lgamma{s}(x)') + + check(ceiling(x + 2.), "ceil{s}(x) + 2") + check(floor(x + 2.), "floor{s}(x) + 2") + check(fma(x, y, -z), 'fma{s}(x, y, -z)') + check(Max(x, 8.0, x**4.0), 'fmax{s}(8.0{S}, fmax{s}(x, pow{s}(x, 4.0{S})))') + check(Min(x, 2.0), 'fmin{s}(2.0{S}, x)') + + +def test_get_math_macros(): + macros = get_math_macros() + assert macros[exp(1)] == 'M_E' + assert macros[1/Sqrt(2)] == 'M_SQRT1_2' + + +def test_ccode_Declaration(): + i = symbols('i', integer=True) + var1 = Variable(i, type=Type.from_expr(i)) + dcl1 = Declaration(var1) + assert ccode(dcl1) == 'int i' + + var2 = Variable(x, type=float32, attrs={value_const}) + dcl2a = Declaration(var2) + assert ccode(dcl2a) == 'const float x' + dcl2b = var2.as_Declaration(value=pi) + assert ccode(dcl2b) == 'const float x = M_PI' + + var3 = Variable(y, type=Type('bool')) + dcl3 = Declaration(var3) + printer = C89CodePrinter() + assert 'stdbool.h' not in printer.headers + assert printer.doprint(dcl3) == 'bool y' + assert 'stdbool.h' in printer.headers + + u = symbols('u', real=True) + ptr4 = Pointer.deduced(u, attrs={pointer_const, restrict}) + dcl4 = Declaration(ptr4) + assert ccode(dcl4) == 'double * const restrict u' + + var5 = Variable(x, Type('__float128'), attrs={value_const}) + dcl5a = Declaration(var5) + assert ccode(dcl5a) == 'const __float128 x' + var5b = Variable(var5.symbol, var5.type, pi, attrs=var5.attrs) + dcl5b = Declaration(var5b) + assert ccode(dcl5b) == 'const __float128 x = M_PI' + + +def test_C99CodePrinter_custom_type(): + # We will look at __float128 (new in glibc 2.26) + f128 = FloatType('_Float128', float128.nbits, float128.nmant, float128.nexp) + p128 = C99CodePrinter({ + "type_aliases": {real: f128}, + "type_literal_suffixes": {f128: 'Q'}, + "type_func_suffixes": {f128: 'f128'}, + "type_math_macro_suffixes": { + real: 'f128', + f128: 'f128' + }, + "type_macros": { + f128: ('__STDC_WANT_IEC_60559_TYPES_EXT__',) + } + }) + assert p128.doprint(x) == 'x' + assert not p128.headers + assert not p128.libraries + assert not p128.macros + assert p128.doprint(2.0) == '2.0Q' + assert not p128.headers + assert not p128.libraries + assert p128.macros == {'__STDC_WANT_IEC_60559_TYPES_EXT__'} + + assert p128.doprint(Rational(1, 2)) == '1.0Q/2.0Q' + assert p128.doprint(sin(x)) == 'sinf128(x)' + assert p128.doprint(cos(2., evaluate=False)) == 'cosf128(2.0Q)' + assert p128.doprint(x**-1.0) == '1.0Q/x' + + var5 = Variable(x, f128, attrs={value_const}) + + dcl5a = Declaration(var5) + assert ccode(dcl5a) == 'const _Float128 x' + var5b = Variable(x, f128, pi, attrs={value_const}) + dcl5b = Declaration(var5b) + assert p128.doprint(dcl5b) == 'const _Float128 x = M_PIf128' + var5b = Variable(x, f128, value=Catalan.evalf(38), attrs={value_const}) + dcl5c = Declaration(var5b) + assert p128.doprint(dcl5c) == 'const _Float128 x = %sQ' % Catalan.evalf(f128.decimal_dig) + + +def test_MatrixElement_printing(): + # test cases for issue #11821 + A = MatrixSymbol("A", 1, 3) + B = MatrixSymbol("B", 1, 3) + C = MatrixSymbol("C", 1, 3) + + assert(ccode(A[0, 0]) == "A[0]") + assert(ccode(3 * A[0, 0]) == "3*A[0]") + + F = C[0, 0].subs(C, A - B) + assert(ccode(F) == "(A - B)[0]") + +def test_ccode_math_macros(): + assert ccode(z + exp(1)) == 'z + M_E' + assert ccode(z + log2(exp(1))) == 'z + M_LOG2E' + assert ccode(z + 1/log(2)) == 'z + M_LOG2E' + assert ccode(z + log(2)) == 'z + M_LN2' + assert ccode(z + log(10)) == 'z + M_LN10' + assert ccode(z + pi) == 'z + M_PI' + assert ccode(z + pi/2) == 'z + M_PI_2' + assert ccode(z + pi/4) == 'z + M_PI_4' + assert ccode(z + 1/pi) == 'z + M_1_PI' + assert ccode(z + 2/pi) == 'z + M_2_PI' + assert ccode(z + 2/sqrt(pi)) == 'z + M_2_SQRTPI' + assert ccode(z + 2/Sqrt(pi)) == 'z + M_2_SQRTPI' + assert ccode(z + sqrt(2)) == 'z + M_SQRT2' + assert ccode(z + Sqrt(2)) == 'z + M_SQRT2' + assert ccode(z + 1/sqrt(2)) == 'z + M_SQRT1_2' + assert ccode(z + 1/Sqrt(2)) == 'z + M_SQRT1_2' + + +def test_ccode_Type(): + assert ccode(Type('float')) == 'float' + assert ccode(intc) == 'int' + + +def test_ccode_codegen_ast(): + # Note that C only allows comments of the form /* ... */, double forward + # slash is not standard C, and some C compilers will grind to a halt upon + # encountering them. + assert ccode(Comment("this is a comment")) == "/* this is a comment */" # not // + assert ccode(While(abs(x) > 1, [aug_assign(x, '-', 1)])) == ( + 'while (fabs(x) > 1) {\n' + ' x -= 1;\n' + '}' + ) + assert ccode(Scope([AddAugmentedAssignment(x, 1)])) == ( + '{\n' + ' x += 1;\n' + '}' + ) + inp_x = Declaration(Variable(x, type=real)) + assert ccode(FunctionPrototype(real, 'pwer', [inp_x])) == 'double pwer(double x)' + assert ccode(FunctionDefinition(real, 'pwer', [inp_x], [Assignment(x, x**2)])) == ( + 'double pwer(double x){\n' + ' x = pow(x, 2);\n' + '}' + ) + + # Elements of CodeBlock are formatted as statements: + block = CodeBlock( + x, + Print([x, y], "%d %d"), + Print([QuotedString('hello'), y], "%s %d", file=stderr), + FunctionCall('pwer', [x]), + Return(x), + ) + assert ccode(block) == '\n'.join([ + 'x;', + 'printf("%d %d", x, y);', + 'fprintf(stderr, "%s %d", "hello", y);', + 'pwer(x);', + 'return x;', + ]) + +def test_ccode_UnevaluatedExpr(): + assert ccode(UnevaluatedExpr(y * x) + z) == "z + x*y" + assert ccode(UnevaluatedExpr(y + x) + z) == "z + (x + y)" # gh-21955 + w = symbols('w') + assert ccode(UnevaluatedExpr(y + x) + UnevaluatedExpr(z + w)) == "(w + z) + (x + y)" + + p, q, r = symbols("p q r", real=True) + q_r = UnevaluatedExpr(q + r) + expr = abs(exp(p+q_r)) + assert ccode(expr) == "exp(p + (q + r))" + + +def test_ccode_array_like_containers(): + assert ccode([2,3,4]) == "{2, 3, 4}" + assert ccode((2,3,4)) == "{2, 3, 4}" + +def test_ccode__isinf_isnan(): + assert ccode(isinf(x)) == 'isinf(x)' + assert ccode(isnan(x)) == 'isnan(x)' diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_codeprinter.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_codeprinter.py new file mode 100644 index 0000000000000000000000000000000000000000..4b077037eb84e218fcfd4a05fc03e40b211e45b9 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_codeprinter.py @@ -0,0 +1,77 @@ +from sympy.printing.codeprinter import CodePrinter, PrintMethodNotImplementedError +from sympy.core import symbols +from sympy.core.symbol import Dummy +from sympy.testing.pytest import raises +from sympy import cos +from sympy.utilities.lambdify import lambdify +from math import cos as math_cos +from sympy.printing.lambdarepr import LambdaPrinter + + +def setup_test_printer(**kwargs): + p = CodePrinter(settings=kwargs) + p._not_supported = set() + p._number_symbols = set() + return p + + +def test_print_Dummy(): + d = Dummy('d') + p = setup_test_printer() + assert p._print_Dummy(d) == "d_%i" % d.dummy_index + +def test_print_Symbol(): + + x, y = symbols('x, if') + + p = setup_test_printer() + assert p._print(x) == 'x' + assert p._print(y) == 'if' + + p.reserved_words.update(['if']) + assert p._print(y) == 'if_' + + p = setup_test_printer(error_on_reserved=True) + p.reserved_words.update(['if']) + with raises(ValueError): + p._print(y) + + p = setup_test_printer(reserved_word_suffix='_He_Man') + p.reserved_words.update(['if']) + assert p._print(y) == 'if_He_Man' + + +def test_lambdify_LaTeX_symbols_issue_23374(): + # Create symbols with Latex style names + x1, x2 = symbols("x_{1} x_2") + + # Lambdify the function + f1 = lambdify([x1, x2], cos(x1 ** 2 + x2 ** 2)) + + # Test that the function works correctly (numerically) + assert f1(1, 2) == math_cos(1 ** 2 + 2 ** 2) + + # Explicitly generate a custom printer to verify the naming convention + p = LambdaPrinter() + expr_str = p.doprint(cos(x1 ** 2 + x2 ** 2)) + assert 'x_1' in expr_str + assert 'x_2' in expr_str + + +def test_issue_15791(): + class CrashingCodePrinter(CodePrinter): + def emptyPrinter(self, obj): + raise NotImplementedError + + from sympy.matrices import ( + MutableSparseMatrix, + ImmutableSparseMatrix, + ) + + c = CrashingCodePrinter() + + # these should not silently succeed + with raises(PrintMethodNotImplementedError): + c.doprint(ImmutableSparseMatrix(2, 2, {})) + with raises(PrintMethodNotImplementedError): + c.doprint(MutableSparseMatrix(2, 2, {})) diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_conventions.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_conventions.py new file mode 100644 index 0000000000000000000000000000000000000000..e8f1fa8532f96130828b89d1ba5ba11fd5bed7a4 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_conventions.py @@ -0,0 +1,116 @@ +# -*- coding: utf-8 -*- + +from sympy.core.function import (Derivative, Function) +from sympy.core.numbers import oo +from sympy.core.symbol import symbols +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.trigonometric import cos +from sympy.integrals.integrals import Integral +from sympy.functions.special.bessel import besselj +from sympy.functions.special.polynomials import legendre +from sympy.functions.combinatorial.numbers import bell +from sympy.printing.conventions import split_super_sub, requires_partial +from sympy.testing.pytest import XFAIL + +def test_super_sub(): + assert split_super_sub("beta_13_2") == ("beta", [], ["13", "2"]) + assert split_super_sub("beta_132_20") == ("beta", [], ["132", "20"]) + assert split_super_sub("beta_13") == ("beta", [], ["13"]) + assert split_super_sub("x_a_b") == ("x", [], ["a", "b"]) + assert split_super_sub("x_1_2_3") == ("x", [], ["1", "2", "3"]) + assert split_super_sub("x_a_b1") == ("x", [], ["a", "b1"]) + assert split_super_sub("x_a_1") == ("x", [], ["a", "1"]) + assert split_super_sub("x_1_a") == ("x", [], ["1", "a"]) + assert split_super_sub("x_1^aa") == ("x", ["aa"], ["1"]) + assert split_super_sub("x_1__aa") == ("x", ["aa"], ["1"]) + assert split_super_sub("x_11^a") == ("x", ["a"], ["11"]) + assert split_super_sub("x_11__a") == ("x", ["a"], ["11"]) + assert split_super_sub("x_a_b_c_d") == ("x", [], ["a", "b", "c", "d"]) + assert split_super_sub("x_a_b^c^d") == ("x", ["c", "d"], ["a", "b"]) + assert split_super_sub("x_a_b__c__d") == ("x", ["c", "d"], ["a", "b"]) + assert split_super_sub("x_a^b_c^d") == ("x", ["b", "d"], ["a", "c"]) + assert split_super_sub("x_a__b_c__d") == ("x", ["b", "d"], ["a", "c"]) + assert split_super_sub("x^a^b_c_d") == ("x", ["a", "b"], ["c", "d"]) + assert split_super_sub("x__a__b_c_d") == ("x", ["a", "b"], ["c", "d"]) + assert split_super_sub("x^a^b^c^d") == ("x", ["a", "b", "c", "d"], []) + assert split_super_sub("x__a__b__c__d") == ("x", ["a", "b", "c", "d"], []) + assert split_super_sub("alpha_11") == ("alpha", [], ["11"]) + assert split_super_sub("alpha_11_11") == ("alpha", [], ["11", "11"]) + assert split_super_sub("w1") == ("w", [], ["1"]) + assert split_super_sub("w𝟙") == ("w", [], ["𝟙"]) + assert split_super_sub("w11") == ("w", [], ["11"]) + assert split_super_sub("w𝟙𝟙") == ("w", [], ["𝟙𝟙"]) + assert split_super_sub("w𝟙2𝟙") == ("w", [], ["𝟙2𝟙"]) + assert split_super_sub("w1^a") == ("w", ["a"], ["1"]) + assert split_super_sub("ω1") == ("ω", [], ["1"]) + assert split_super_sub("ω11") == ("ω", [], ["11"]) + assert split_super_sub("ω1^a") == ("ω", ["a"], ["1"]) + assert split_super_sub("ω𝟙^α") == ("ω", ["α"], ["𝟙"]) + assert split_super_sub("ω𝟙2^3α") == ("ω", ["3α"], ["𝟙2"]) + assert split_super_sub("") == ("", [], []) + + +def test_requires_partial(): + x, y, z, t, nu = symbols('x y z t nu') + n = symbols('n', integer=True) + + f = x * y + assert requires_partial(Derivative(f, x)) is True + assert requires_partial(Derivative(f, y)) is True + + ## integrating out one of the variables + assert requires_partial(Derivative(Integral(exp(-x * y), (x, 0, oo)), y, evaluate=False)) is False + + ## bessel function with smooth parameter + f = besselj(nu, x) + assert requires_partial(Derivative(f, x)) is True + assert requires_partial(Derivative(f, nu)) is True + + ## bessel function with integer parameter + f = besselj(n, x) + assert requires_partial(Derivative(f, x)) is False + # this is not really valid (differentiating with respect to an integer) + # but there's no reason to use the partial derivative symbol there. make + # sure we don't throw an exception here, though + assert requires_partial(Derivative(f, n)) is False + + ## bell polynomial + f = bell(n, x) + assert requires_partial(Derivative(f, x)) is False + # again, invalid + assert requires_partial(Derivative(f, n)) is False + + ## legendre polynomial + f = legendre(0, x) + assert requires_partial(Derivative(f, x)) is False + + f = legendre(n, x) + assert requires_partial(Derivative(f, x)) is False + # again, invalid + assert requires_partial(Derivative(f, n)) is False + + f = x ** n + assert requires_partial(Derivative(f, x)) is False + + assert requires_partial(Derivative(Integral((x*y) ** n * exp(-x * y), (x, 0, oo)), y, evaluate=False)) is False + + # parametric equation + f = (exp(t), cos(t)) + g = sum(f) + assert requires_partial(Derivative(g, t)) is False + + f = symbols('f', cls=Function) + assert requires_partial(Derivative(f(x), x)) is False + assert requires_partial(Derivative(f(x), y)) is False + assert requires_partial(Derivative(f(x, y), x)) is True + assert requires_partial(Derivative(f(x, y), y)) is True + assert requires_partial(Derivative(f(x, y), z)) is True + assert requires_partial(Derivative(f(x, y), x, y)) is True + +@XFAIL +def test_requires_partial_unspecified_variables(): + x, y = symbols('x y') + # function of unspecified variables + f = symbols('f', cls=Function) + assert requires_partial(Derivative(f, x)) is False + assert requires_partial(Derivative(f, x, y)) is True diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_cupy.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_cupy.py new file mode 100644 index 0000000000000000000000000000000000000000..cf111ec1623390a3dbbf489235d2ed387624a36c --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_cupy.py @@ -0,0 +1,56 @@ +from sympy.concrete.summations import Sum +from sympy.functions.elementary.exponential import log +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.utilities.lambdify import lambdify +from sympy.abc import x, i, a, b +from sympy.codegen.numpy_nodes import logaddexp +from sympy.printing.numpy import CuPyPrinter, _cupy_known_constants, _cupy_known_functions + +from sympy.testing.pytest import skip, raises +from sympy.external import import_module + +cp = import_module('cupy') + +def test_cupy_print(): + prntr = CuPyPrinter() + assert prntr.doprint(logaddexp(a, b)) == 'cupy.logaddexp(a, b)' + assert prntr.doprint(sqrt(x)) == 'cupy.sqrt(x)' + assert prntr.doprint(log(x)) == 'cupy.log(x)' + assert prntr.doprint("acos(x)") == 'cupy.arccos(x)' + assert prntr.doprint("exp(x)") == 'cupy.exp(x)' + assert prntr.doprint("Abs(x)") == 'abs(x)' + +def test_not_cupy_print(): + prntr = CuPyPrinter() + with raises(NotImplementedError): + prntr.doprint("abcd(x)") + +def test_cupy_sum(): + if not cp: + skip("CuPy not installed") + + s = Sum(x ** i, (i, a, b)) + f = lambdify((a, b, x), s, 'cupy') + + a_, b_ = 0, 10 + x_ = cp.linspace(-1, +1, 10) + assert cp.allclose(f(a_, b_, x_), sum(x_ ** i_ for i_ in range(a_, b_ + 1))) + + s = Sum(i * x, (i, a, b)) + f = lambdify((a, b, x), s, 'numpy') + + a_, b_ = 0, 10 + x_ = cp.linspace(-1, +1, 10) + assert cp.allclose(f(a_, b_, x_), sum(i_ * x_ for i_ in range(a_, b_ + 1))) + +def test_cupy_known_funcs_consts(): + assert _cupy_known_constants['NaN'] == 'cupy.nan' + assert _cupy_known_constants['EulerGamma'] == 'cupy.euler_gamma' + + assert _cupy_known_functions['acos'] == 'cupy.arccos' + assert _cupy_known_functions['log'] == 'cupy.log' + +def test_cupy_print_methods(): + prntr = CuPyPrinter() + assert hasattr(prntr, '_print_acos') + assert hasattr(prntr, '_print_log') diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_cxx.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_cxx.py new file mode 100644 index 0000000000000000000000000000000000000000..d84ec75cbf0eeb60a1176b9cb3b401a3384454e7 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_cxx.py @@ -0,0 +1,86 @@ +from sympy.core.numbers import Float, Integer, Rational +from sympy.core.symbol import symbols +from sympy.functions import beta, Ei, zeta, Max, Min, sqrt, riemann_xi, frac +from sympy.printing.cxx import CXX98CodePrinter, CXX11CodePrinter, CXX17CodePrinter, cxxcode +from sympy.codegen.cfunctions import log1p + + +x, y, u, v = symbols('x y u v') + + +def test_CXX98CodePrinter(): + assert CXX98CodePrinter().doprint(Max(x, 3)) in ('std::max(x, 3)', 'std::max(3, x)') + assert CXX98CodePrinter().doprint(Min(x, 3, sqrt(x))) == 'std::min(3, std::min(x, std::sqrt(x)))' + cxx98printer = CXX98CodePrinter() + assert cxx98printer.language == 'C++' + assert cxx98printer.standard == 'C++98' + assert 'template' in cxx98printer.reserved_words + assert 'alignas' not in cxx98printer.reserved_words + + +def test_CXX11CodePrinter(): + assert CXX11CodePrinter().doprint(log1p(x)) == 'std::log1p(x)' + + cxx11printer = CXX11CodePrinter() + assert cxx11printer.language == 'C++' + assert cxx11printer.standard == 'C++11' + assert 'operator' in cxx11printer.reserved_words + assert 'noexcept' in cxx11printer.reserved_words + assert 'concept' not in cxx11printer.reserved_words + + +def test_subclass_print_method(): + class MyPrinter(CXX11CodePrinter): + def _print_log1p(self, expr): + return 'my_library::log1p(%s)' % ', '.join(map(self._print, expr.args)) + + assert MyPrinter().doprint(log1p(x)) == 'my_library::log1p(x)' + + +def test_subclass_print_method__ns(): + class MyPrinter(CXX11CodePrinter): + _ns = 'my_library::' + + p = CXX11CodePrinter() + myp = MyPrinter() + + assert p.doprint(log1p(x)) == 'std::log1p(x)' + assert myp.doprint(log1p(x)) == 'my_library::log1p(x)' + + +def test_CXX17CodePrinter(): + assert CXX17CodePrinter().doprint(beta(x, y)) == 'std::beta(x, y)' + assert CXX17CodePrinter().doprint(Ei(x)) == 'std::expint(x)' + assert CXX17CodePrinter().doprint(zeta(x)) == 'std::riemann_zeta(x)' + + # Automatic rewrite + assert CXX17CodePrinter().doprint(frac(x)) == '(x - std::floor(x))' + assert CXX17CodePrinter().doprint(riemann_xi(x)) == '((1.0/2.0)*std::pow(M_PI, -1.0/2.0*x)*x*(x - 1)*std::tgamma((1.0/2.0)*x)*std::riemann_zeta(x))' + + +def test_cxxcode(): + assert sorted(cxxcode(sqrt(x)*.5).split('*')) == sorted(['0.5', 'std::sqrt(x)']) + +def test_cxxcode_nested_minmax(): + assert cxxcode(Max(Min(x, y), Min(u, v))) \ + == 'std::max(std::min(u, v), std::min(x, y))' + assert cxxcode(Min(Max(x, y), Max(u, v))) \ + == 'std::min(std::max(u, v), std::max(x, y))' + +def test_subclass_Integer_Float(): + class MyPrinter(CXX17CodePrinter): + def _print_Integer(self, arg): + return 'bigInt("%s")' % super()._print_Integer(arg) + + def _print_Float(self, arg): + rat = Rational(arg) + return 'bigFloat(%s, %s)' % ( + self._print(Integer(rat.p)), + self._print(Integer(rat.q)) + ) + + p = MyPrinter() + for i in range(13): + assert p.doprint(i) == 'bigInt("%d")' % i + assert p.doprint(Float(0.5)) == 'bigFloat(bigInt("1"), bigInt("2"))' + assert p.doprint(x**-1.0) == 'bigFloat(bigInt("1"), bigInt("1"))/x' diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_dot.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_dot.py new file mode 100644 index 0000000000000000000000000000000000000000..6213e237fb7aac6460a956b4c9fc1f7c8710fec6 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_dot.py @@ -0,0 +1,134 @@ +from sympy.printing.dot import (purestr, styleof, attrprint, dotnode, + dotedges, dotprint) +from sympy.core.basic import Basic +from sympy.core.expr import Expr +from sympy.core.numbers import (Float, Integer) +from sympy.core.singleton import S +from sympy.core.symbol import (Symbol, symbols) +from sympy.printing.repr import srepr +from sympy.abc import x + + +def test_purestr(): + assert purestr(Symbol('x')) == "Symbol('x')" + assert purestr(Basic(S(1), S(2))) == "Basic(Integer(1), Integer(2))" + assert purestr(Float(2)) == "Float('2.0', precision=53)" + + assert purestr(Symbol('x'), with_args=True) == ("Symbol('x')", ()) + assert purestr(Basic(S(1), S(2)), with_args=True) == \ + ('Basic(Integer(1), Integer(2))', ('Integer(1)', 'Integer(2)')) + assert purestr(Float(2), with_args=True) == \ + ("Float('2.0', precision=53)", ()) + + +def test_styleof(): + styles = [(Basic, {'color': 'blue', 'shape': 'ellipse'}), + (Expr, {'color': 'black'})] + assert styleof(Basic(S(1)), styles) == {'color': 'blue', 'shape': 'ellipse'} + + assert styleof(x + 1, styles) == {'color': 'black', 'shape': 'ellipse'} + + +def test_attrprint(): + assert attrprint({'color': 'blue', 'shape': 'ellipse'}) == \ + '"color"="blue", "shape"="ellipse"' + +def test_dotnode(): + + assert dotnode(x, repeat=False) == \ + '"Symbol(\'x\')" ["color"="black", "label"="x", "shape"="ellipse"];' + assert dotnode(x+2, repeat=False) == \ + '"Add(Integer(2), Symbol(\'x\'))" ' \ + '["color"="black", "label"="Add", "shape"="ellipse"];', \ + dotnode(x+2,repeat=0) + + assert dotnode(x + x**2, repeat=False) == \ + '"Add(Symbol(\'x\'), Pow(Symbol(\'x\'), Integer(2)))" ' \ + '["color"="black", "label"="Add", "shape"="ellipse"];' + assert dotnode(x + x**2, repeat=True) == \ + '"Add(Symbol(\'x\'), Pow(Symbol(\'x\'), Integer(2)))_()" ' \ + '["color"="black", "label"="Add", "shape"="ellipse"];' + +def test_dotedges(): + assert sorted(dotedges(x+2, repeat=False)) == [ + '"Add(Integer(2), Symbol(\'x\'))" -> "Integer(2)";', + '"Add(Integer(2), Symbol(\'x\'))" -> "Symbol(\'x\')";' + ] + assert sorted(dotedges(x + 2, repeat=True)) == [ + '"Add(Integer(2), Symbol(\'x\'))_()" -> "Integer(2)_(0,)";', + '"Add(Integer(2), Symbol(\'x\'))_()" -> "Symbol(\'x\')_(1,)";' + ] + +def test_dotprint(): + text = dotprint(x+2, repeat=False) + assert all(e in text for e in dotedges(x+2, repeat=False)) + assert all( + n in text for n in [dotnode(expr, repeat=False) + for expr in (x, Integer(2), x+2)]) + assert 'digraph' in text + + text = dotprint(x+x**2, repeat=False) + assert all(e in text for e in dotedges(x+x**2, repeat=False)) + assert all( + n in text for n in [dotnode(expr, repeat=False) + for expr in (x, Integer(2), x**2)]) + assert 'digraph' in text + + text = dotprint(x+x**2, repeat=True) + assert all(e in text for e in dotedges(x+x**2, repeat=True)) + assert all( + n in text for n in [dotnode(expr, pos=()) + for expr in [x + x**2]]) + + text = dotprint(x**x, repeat=True) + assert all(e in text for e in dotedges(x**x, repeat=True)) + assert all( + n in text for n in [dotnode(x, pos=(0,)), dotnode(x, pos=(1,))]) + assert 'digraph' in text + +def test_dotprint_depth(): + text = dotprint(3*x+2, depth=1) + assert dotnode(3*x+2) in text + assert dotnode(x) not in text + text = dotprint(3*x+2) + assert "depth" not in text + +def test_Matrix_and_non_basics(): + from sympy.matrices.expressions.matexpr import MatrixSymbol + n = Symbol('n') + assert dotprint(MatrixSymbol('X', n, n)) == \ +"""digraph{ + +# Graph style +"ordering"="out" +"rankdir"="TD" + +######### +# Nodes # +######### + +"MatrixSymbol(Str('X'), Symbol('n'), Symbol('n'))_()" ["color"="black", "label"="MatrixSymbol", "shape"="ellipse"]; +"Str('X')_(0,)" ["color"="blue", "label"="X", "shape"="ellipse"]; +"Symbol('n')_(1,)" ["color"="black", "label"="n", "shape"="ellipse"]; +"Symbol('n')_(2,)" ["color"="black", "label"="n", "shape"="ellipse"]; + +######### +# Edges # +######### + +"MatrixSymbol(Str('X'), Symbol('n'), Symbol('n'))_()" -> "Str('X')_(0,)"; +"MatrixSymbol(Str('X'), Symbol('n'), Symbol('n'))_()" -> "Symbol('n')_(1,)"; +"MatrixSymbol(Str('X'), Symbol('n'), Symbol('n'))_()" -> "Symbol('n')_(2,)"; +}""" + + +def test_labelfunc(): + text = dotprint(x + 2, labelfunc=srepr) + assert "Symbol('x')" in text + assert "Integer(2)" in text + + +def test_commutative(): + x, y = symbols('x y', commutative=False) + assert dotprint(x + y) == dotprint(y + x) + assert dotprint(x*y) != dotprint(y*x) diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_fortran.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_fortran.py new file mode 100644 index 0000000000000000000000000000000000000000..c28a1ea16dcf2157b58d763286428dccc1944b71 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_fortran.py @@ -0,0 +1,854 @@ +from sympy.core.add import Add +from sympy.core.expr import Expr +from sympy.core.function import (Function, Lambda, diff) +from sympy.core.mod import Mod +from sympy.core import (Catalan, EulerGamma, GoldenRatio) +from sympy.core.numbers import (E, Float, I, Integer, Rational, pi) +from sympy.core.relational import Eq +from sympy.core.singleton import S +from sympy.core.symbol import (Dummy, symbols) +from sympy.functions.combinatorial.factorials import factorial +from sympy.functions.elementary.complexes import (conjugate, sign) +from sympy.functions.elementary.exponential import (exp, log) +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.elementary.trigonometric import (atan2, cos, sin) +from sympy.functions.special.gamma_functions import gamma +from sympy.integrals.integrals import Integral +from sympy.sets.fancysets import Range + +from sympy.codegen import For, Assignment, aug_assign +from sympy.codegen.ast import Declaration, Variable, float32, float64, \ + value_const, real, bool_, While, FunctionPrototype, FunctionDefinition, \ + integer, Return, Element +from sympy.core.expr import UnevaluatedExpr +from sympy.core.relational import Relational +from sympy.logic.boolalg import And, Or, Not, Equivalent, Xor +from sympy.matrices import Matrix, MatrixSymbol +from sympy.printing.fortran import fcode, FCodePrinter +from sympy.tensor import IndexedBase, Idx +from sympy.tensor.array.expressions import ArraySymbol, ArrayElement +from sympy.utilities.lambdify import implemented_function +from sympy.testing.pytest import raises + + +def test_UnevaluatedExpr(): + p, q, r = symbols("p q r", real=True) + q_r = UnevaluatedExpr(q + r) + expr = abs(exp(p+q_r)) + assert fcode(expr, source_format="free") == "exp(p + (q + r))" + x, y, z = symbols("x y z") + y_z = UnevaluatedExpr(y + z) + expr2 = abs(exp(x+y_z)) + assert fcode(expr2, human=False)[2].lstrip() == "exp(re(x) + re(y + z))" + assert fcode(expr2, user_functions={"re": "realpart"}).lstrip() == "exp(realpart(x) + realpart(y + z))" + + +def test_printmethod(): + x = symbols('x') + + class nint(Function): + def _fcode(self, printer): + return "nint(%s)" % printer._print(self.args[0]) + assert fcode(nint(x)) == " nint(x)" + + +def test_fcode_sign(): #issue 12267 + x=symbols('x') + y=symbols('y', integer=True) + z=symbols('z', complex=True) + assert fcode(sign(x), standard=95, source_format='free') == "merge(0d0, dsign(1d0, x), x == 0d0)" + assert fcode(sign(y), standard=95, source_format='free') == "merge(0, isign(1, y), y == 0)" + assert fcode(sign(z), standard=95, source_format='free') == "merge(cmplx(0d0, 0d0), z/abs(z), abs(z) == 0d0)" + raises(NotImplementedError, lambda: fcode(sign(x))) + + +def test_fcode_Pow(): + x, y = symbols('x,y') + n = symbols('n', integer=True) + + assert fcode(x**3) == " x**3" + assert fcode(x**(y**3)) == " x**(y**3)" + assert fcode(1/(sin(x)*3.5)**(x - y**x)/(x**2 + y)) == \ + " (3.5d0*sin(x))**(-x + y**x)/(x**2 + y)" + assert fcode(sqrt(x)) == ' sqrt(x)' + assert fcode(sqrt(n)) == ' sqrt(dble(n))' + assert fcode(x**0.5) == ' sqrt(x)' + assert fcode(sqrt(x)) == ' sqrt(x)' + assert fcode(sqrt(10)) == ' sqrt(10.0d0)' + assert fcode(x**-1.0) == ' 1d0/x' + assert fcode(x**-2.0, 'y', source_format='free') == 'y = x**(-2.0d0)' # 2823 + assert fcode(x**Rational(3, 7)) == ' x**(3.0d0/7.0d0)' + + +def test_fcode_Rational(): + x = symbols('x') + assert fcode(Rational(3, 7)) == " 3.0d0/7.0d0" + assert fcode(Rational(18, 9)) == " 2" + assert fcode(Rational(3, -7)) == " -3.0d0/7.0d0" + assert fcode(Rational(-3, -7)) == " 3.0d0/7.0d0" + assert fcode(x + Rational(3, 7)) == " x + 3.0d0/7.0d0" + assert fcode(Rational(3, 7)*x) == " (3.0d0/7.0d0)*x" + + +def test_fcode_Integer(): + assert fcode(Integer(67)) == " 67" + assert fcode(Integer(-1)) == " -1" + + +def test_fcode_Float(): + assert fcode(Float(42.0)) == " 42.0000000000000d0" + assert fcode(Float(-1e20)) == " -1.00000000000000d+20" + + +def test_fcode_functions(): + x, y = symbols('x,y') + assert fcode(sin(x) ** cos(y)) == " sin(x)**cos(y)" + raises(NotImplementedError, lambda: fcode(Mod(x, y), standard=66)) + raises(NotImplementedError, lambda: fcode(x % y, standard=66)) + raises(NotImplementedError, lambda: fcode(Mod(x, y), standard=77)) + raises(NotImplementedError, lambda: fcode(x % y, standard=77)) + for standard in [90, 95, 2003, 2008]: + assert fcode(Mod(x, y), standard=standard) == " modulo(x, y)" + assert fcode(x % y, standard=standard) == " modulo(x, y)" + + +def test_case(): + ob = FCodePrinter() + x,x_,x__,y,X,X_,Y = symbols('x,x_,x__,y,X,X_,Y') + assert fcode(exp(x_) + sin(x*y) + cos(X*Y)) == \ + ' exp(x_) + sin(x*y) + cos(X__*Y_)' + assert fcode(exp(x__) + 2*x*Y*X_**Rational(7, 2)) == \ + ' 2*X_**(7.0d0/2.0d0)*Y*x + exp(x__)' + assert fcode(exp(x_) + sin(x*y) + cos(X*Y), name_mangling=False) == \ + ' exp(x_) + sin(x*y) + cos(X*Y)' + assert fcode(x - cos(X), name_mangling=False) == ' x - cos(X)' + assert ob.doprint(X*sin(x) + x_, assign_to='me') == ' me = X*sin(x_) + x__' + assert ob.doprint(X*sin(x), assign_to='mu') == ' mu = X*sin(x_)' + assert ob.doprint(x_, assign_to='ad') == ' ad = x__' + n, m = symbols('n,m', integer=True) + A = IndexedBase('A') + x = IndexedBase('x') + y = IndexedBase('y') + i = Idx('i', m) + I = Idx('I', n) + assert fcode(A[i, I]*x[I], assign_to=y[i], source_format='free') == ( + "do i = 1, m\n" + " y(i) = 0\n" + "end do\n" + "do i = 1, m\n" + " do I_ = 1, n\n" + " y(i) = A(i, I_)*x(I_) + y(i)\n" + " end do\n" + "end do" ) + + +#issue 6814 +def test_fcode_functions_with_integers(): + x= symbols('x') + log10_17 = log(10).evalf(17) + loglog10_17 = '0.8340324452479558d0' + assert fcode(x * log(10)) == " x*%sd0" % log10_17 + assert fcode(x * log(10)) == " x*%sd0" % log10_17 + assert fcode(x * log(S(10))) == " x*%sd0" % log10_17 + assert fcode(log(S(10))) == " %sd0" % log10_17 + assert fcode(exp(10)) == " %sd0" % exp(10).evalf(17) + assert fcode(x * log(log(10))) == " x*%s" % loglog10_17 + assert fcode(x * log(log(S(10)))) == " x*%s" % loglog10_17 + + +def test_fcode_NumberSymbol(): + prec = 17 + p = FCodePrinter() + assert fcode(Catalan) == ' parameter (Catalan = %sd0)\n Catalan' % Catalan.evalf(prec) + assert fcode(EulerGamma) == ' parameter (EulerGamma = %sd0)\n EulerGamma' % EulerGamma.evalf(prec) + assert fcode(E) == ' parameter (E = %sd0)\n E' % E.evalf(prec) + assert fcode(GoldenRatio) == ' parameter (GoldenRatio = %sd0)\n GoldenRatio' % GoldenRatio.evalf(prec) + assert fcode(pi) == ' parameter (pi = %sd0)\n pi' % pi.evalf(prec) + assert fcode( + pi, precision=5) == ' parameter (pi = %sd0)\n pi' % pi.evalf(5) + assert fcode(Catalan, human=False) == ({ + (Catalan, p._print(Catalan.evalf(prec)))}, set(), ' Catalan') + assert fcode(EulerGamma, human=False) == ({(EulerGamma, p._print( + EulerGamma.evalf(prec)))}, set(), ' EulerGamma') + assert fcode(E, human=False) == ( + {(E, p._print(E.evalf(prec)))}, set(), ' E') + assert fcode(GoldenRatio, human=False) == ({(GoldenRatio, p._print( + GoldenRatio.evalf(prec)))}, set(), ' GoldenRatio') + assert fcode(pi, human=False) == ( + {(pi, p._print(pi.evalf(prec)))}, set(), ' pi') + assert fcode(pi, precision=5, human=False) == ( + {(pi, p._print(pi.evalf(5)))}, set(), ' pi') + + +def test_fcode_complex(): + assert fcode(I) == " cmplx(0,1)" + x = symbols('x') + assert fcode(4*I) == " cmplx(0,4)" + assert fcode(3 + 4*I) == " cmplx(3,4)" + assert fcode(3 + 4*I + x) == " cmplx(3,4) + x" + assert fcode(I*x) == " cmplx(0,1)*x" + assert fcode(3 + 4*I - x) == " cmplx(3,4) - x" + x = symbols('x', imaginary=True) + assert fcode(5*x) == " 5*x" + assert fcode(I*x) == " cmplx(0,1)*x" + assert fcode(3 + x) == " x + 3" + + +def test_implicit(): + x, y = symbols('x,y') + assert fcode(sin(x)) == " sin(x)" + assert fcode(atan2(x, y)) == " atan2(x, y)" + assert fcode(conjugate(x)) == " conjg(x)" + + +def test_not_fortran(): + x = symbols('x') + g = Function('g') + with raises(NotImplementedError): + fcode(gamma(x)) + assert fcode(Integral(sin(x)), strict=False) == "C Not supported in Fortran:\nC Integral\n Integral(sin(x), x)" + with raises(NotImplementedError): + fcode(g(x)) + + +def test_user_functions(): + x = symbols('x') + assert fcode(sin(x), user_functions={"sin": "zsin"}) == " zsin(x)" + x = symbols('x') + assert fcode( + gamma(x), user_functions={"gamma": "mygamma"}) == " mygamma(x)" + g = Function('g') + assert fcode(g(x), user_functions={"g": "great"}) == " great(x)" + n = symbols('n', integer=True) + assert fcode( + factorial(n), user_functions={"factorial": "fct"}) == " fct(n)" + + +def test_inline_function(): + x = symbols('x') + g = implemented_function('g', Lambda(x, 2*x)) + assert fcode(g(x)) == " 2*x" + g = implemented_function('g', Lambda(x, 2*pi/x)) + assert fcode(g(x)) == ( + " parameter (pi = %sd0)\n" + " 2*pi/x" + ) % pi.evalf(17) + A = IndexedBase('A') + i = Idx('i', symbols('n', integer=True)) + g = implemented_function('g', Lambda(x, x*(1 + x)*(2 + x))) + assert fcode(g(A[i]), assign_to=A[i]) == ( + " do i = 1, n\n" + " A(i) = (A(i) + 1)*(A(i) + 2)*A(i)\n" + " end do" + ) + + +def test_assign_to(): + x = symbols('x') + assert fcode(sin(x), assign_to="s") == " s = sin(x)" + + +def test_line_wrapping(): + x, y = symbols('x,y') + assert fcode(((x + y)**10).expand(), assign_to="var") == ( + " var = x**10 + 10*x**9*y + 45*x**8*y**2 + 120*x**7*y**3 + 210*x**6*\n" + " @ y**4 + 252*x**5*y**5 + 210*x**4*y**6 + 120*x**3*y**7 + 45*x**2*y\n" + " @ **8 + 10*x*y**9 + y**10" + ) + e = [x**i for i in range(11)] + assert fcode(Add(*e)) == ( + " x**10 + x**9 + x**8 + x**7 + x**6 + x**5 + x**4 + x**3 + x**2 + x\n" + " @ + 1" + ) + + +def test_fcode_precedence(): + x, y = symbols("x y") + assert fcode(And(x < y, y < x + 1), source_format="free") == \ + "x < y .and. y < x + 1" + assert fcode(Or(x < y, y < x + 1), source_format="free") == \ + "x < y .or. y < x + 1" + assert fcode(Xor(x < y, y < x + 1, evaluate=False), + source_format="free") == "x < y .neqv. y < x + 1" + assert fcode(Equivalent(x < y, y < x + 1), source_format="free") == \ + "x < y .eqv. y < x + 1" + + +def test_fcode_Logical(): + x, y, z = symbols("x y z") + # unary Not + assert fcode(Not(x), source_format="free") == ".not. x" + # binary And + assert fcode(And(x, y), source_format="free") == "x .and. y" + assert fcode(And(x, Not(y)), source_format="free") == "x .and. .not. y" + assert fcode(And(Not(x), y), source_format="free") == "y .and. .not. x" + assert fcode(And(Not(x), Not(y)), source_format="free") == \ + ".not. x .and. .not. y" + assert fcode(Not(And(x, y), evaluate=False), source_format="free") == \ + ".not. (x .and. y)" + # binary Or + assert fcode(Or(x, y), source_format="free") == "x .or. y" + assert fcode(Or(x, Not(y)), source_format="free") == "x .or. .not. y" + assert fcode(Or(Not(x), y), source_format="free") == "y .or. .not. x" + assert fcode(Or(Not(x), Not(y)), source_format="free") == \ + ".not. x .or. .not. y" + assert fcode(Not(Or(x, y), evaluate=False), source_format="free") == \ + ".not. (x .or. y)" + # mixed And/Or + assert fcode(And(Or(y, z), x), source_format="free") == "x .and. (y .or. z)" + assert fcode(And(Or(z, x), y), source_format="free") == "y .and. (x .or. z)" + assert fcode(And(Or(x, y), z), source_format="free") == "z .and. (x .or. y)" + assert fcode(Or(And(y, z), x), source_format="free") == "x .or. y .and. z" + assert fcode(Or(And(z, x), y), source_format="free") == "y .or. x .and. z" + assert fcode(Or(And(x, y), z), source_format="free") == "z .or. x .and. y" + # trinary And + assert fcode(And(x, y, z), source_format="free") == "x .and. y .and. z" + assert fcode(And(x, y, Not(z)), source_format="free") == \ + "x .and. y .and. .not. z" + assert fcode(And(x, Not(y), z), source_format="free") == \ + "x .and. z .and. .not. y" + assert fcode(And(Not(x), y, z), source_format="free") == \ + "y .and. z .and. .not. x" + assert fcode(Not(And(x, y, z), evaluate=False), source_format="free") == \ + ".not. (x .and. y .and. z)" + # trinary Or + assert fcode(Or(x, y, z), source_format="free") == "x .or. y .or. z" + assert fcode(Or(x, y, Not(z)), source_format="free") == \ + "x .or. y .or. .not. z" + assert fcode(Or(x, Not(y), z), source_format="free") == \ + "x .or. z .or. .not. y" + assert fcode(Or(Not(x), y, z), source_format="free") == \ + "y .or. z .or. .not. x" + assert fcode(Not(Or(x, y, z), evaluate=False), source_format="free") == \ + ".not. (x .or. y .or. z)" + + +def test_fcode_Xlogical(): + x, y, z = symbols("x y z") + # binary Xor + assert fcode(Xor(x, y, evaluate=False), source_format="free") == \ + "x .neqv. y" + assert fcode(Xor(x, Not(y), evaluate=False), source_format="free") == \ + "x .neqv. .not. y" + assert fcode(Xor(Not(x), y, evaluate=False), source_format="free") == \ + "y .neqv. .not. x" + assert fcode(Xor(Not(x), Not(y), evaluate=False), + source_format="free") == ".not. x .neqv. .not. y" + assert fcode(Not(Xor(x, y, evaluate=False), evaluate=False), + source_format="free") == ".not. (x .neqv. y)" + # binary Equivalent + assert fcode(Equivalent(x, y), source_format="free") == "x .eqv. y" + assert fcode(Equivalent(x, Not(y)), source_format="free") == \ + "x .eqv. .not. y" + assert fcode(Equivalent(Not(x), y), source_format="free") == \ + "y .eqv. .not. x" + assert fcode(Equivalent(Not(x), Not(y)), source_format="free") == \ + ".not. x .eqv. .not. y" + assert fcode(Not(Equivalent(x, y), evaluate=False), + source_format="free") == ".not. (x .eqv. y)" + # mixed And/Equivalent + assert fcode(Equivalent(And(y, z), x), source_format="free") == \ + "x .eqv. y .and. z" + assert fcode(Equivalent(And(z, x), y), source_format="free") == \ + "y .eqv. x .and. z" + assert fcode(Equivalent(And(x, y), z), source_format="free") == \ + "z .eqv. x .and. y" + assert fcode(And(Equivalent(y, z), x), source_format="free") == \ + "x .and. (y .eqv. z)" + assert fcode(And(Equivalent(z, x), y), source_format="free") == \ + "y .and. (x .eqv. z)" + assert fcode(And(Equivalent(x, y), z), source_format="free") == \ + "z .and. (x .eqv. y)" + # mixed Or/Equivalent + assert fcode(Equivalent(Or(y, z), x), source_format="free") == \ + "x .eqv. y .or. z" + assert fcode(Equivalent(Or(z, x), y), source_format="free") == \ + "y .eqv. x .or. z" + assert fcode(Equivalent(Or(x, y), z), source_format="free") == \ + "z .eqv. x .or. y" + assert fcode(Or(Equivalent(y, z), x), source_format="free") == \ + "x .or. (y .eqv. z)" + assert fcode(Or(Equivalent(z, x), y), source_format="free") == \ + "y .or. (x .eqv. z)" + assert fcode(Or(Equivalent(x, y), z), source_format="free") == \ + "z .or. (x .eqv. y)" + # mixed Xor/Equivalent + assert fcode(Equivalent(Xor(y, z, evaluate=False), x), + source_format="free") == "x .eqv. (y .neqv. z)" + assert fcode(Equivalent(Xor(z, x, evaluate=False), y), + source_format="free") == "y .eqv. (x .neqv. z)" + assert fcode(Equivalent(Xor(x, y, evaluate=False), z), + source_format="free") == "z .eqv. (x .neqv. y)" + assert fcode(Xor(Equivalent(y, z), x, evaluate=False), + source_format="free") == "x .neqv. (y .eqv. z)" + assert fcode(Xor(Equivalent(z, x), y, evaluate=False), + source_format="free") == "y .neqv. (x .eqv. z)" + assert fcode(Xor(Equivalent(x, y), z, evaluate=False), + source_format="free") == "z .neqv. (x .eqv. y)" + # mixed And/Xor + assert fcode(Xor(And(y, z), x, evaluate=False), source_format="free") == \ + "x .neqv. y .and. z" + assert fcode(Xor(And(z, x), y, evaluate=False), source_format="free") == \ + "y .neqv. x .and. z" + assert fcode(Xor(And(x, y), z, evaluate=False), source_format="free") == \ + "z .neqv. x .and. y" + assert fcode(And(Xor(y, z, evaluate=False), x), source_format="free") == \ + "x .and. (y .neqv. z)" + assert fcode(And(Xor(z, x, evaluate=False), y), source_format="free") == \ + "y .and. (x .neqv. z)" + assert fcode(And(Xor(x, y, evaluate=False), z), source_format="free") == \ + "z .and. (x .neqv. y)" + # mixed Or/Xor + assert fcode(Xor(Or(y, z), x, evaluate=False), source_format="free") == \ + "x .neqv. y .or. z" + assert fcode(Xor(Or(z, x), y, evaluate=False), source_format="free") == \ + "y .neqv. x .or. z" + assert fcode(Xor(Or(x, y), z, evaluate=False), source_format="free") == \ + "z .neqv. x .or. y" + assert fcode(Or(Xor(y, z, evaluate=False), x), source_format="free") == \ + "x .or. (y .neqv. z)" + assert fcode(Or(Xor(z, x, evaluate=False), y), source_format="free") == \ + "y .or. (x .neqv. z)" + assert fcode(Or(Xor(x, y, evaluate=False), z), source_format="free") == \ + "z .or. (x .neqv. y)" + # trinary Xor + assert fcode(Xor(x, y, z, evaluate=False), source_format="free") == \ + "x .neqv. y .neqv. z" + assert fcode(Xor(x, y, Not(z), evaluate=False), source_format="free") == \ + "x .neqv. y .neqv. .not. z" + assert fcode(Xor(x, Not(y), z, evaluate=False), source_format="free") == \ + "x .neqv. z .neqv. .not. y" + assert fcode(Xor(Not(x), y, z, evaluate=False), source_format="free") == \ + "y .neqv. z .neqv. .not. x" + + +def test_fcode_Relational(): + x, y = symbols("x y") + assert fcode(Relational(x, y, "=="), source_format="free") == "x == y" + assert fcode(Relational(x, y, "!="), source_format="free") == "x /= y" + assert fcode(Relational(x, y, ">="), source_format="free") == "x >= y" + assert fcode(Relational(x, y, "<="), source_format="free") == "x <= y" + assert fcode(Relational(x, y, ">"), source_format="free") == "x > y" + assert fcode(Relational(x, y, "<"), source_format="free") == "x < y" + + +def test_fcode_Piecewise(): + x = symbols('x') + expr = Piecewise((x, x < 1), (x**2, True)) + # Check that inline conditional (merge) fails if standard isn't 95+ + raises(NotImplementedError, lambda: fcode(expr)) + code = fcode(expr, standard=95) + expected = " merge(x, x**2, x < 1)" + assert code == expected + assert fcode(Piecewise((x, x < 1), (x**2, True)), assign_to="var") == ( + " if (x < 1) then\n" + " var = x\n" + " else\n" + " var = x**2\n" + " end if" + ) + a = cos(x)/x + b = sin(x)/x + for i in range(10): + a = diff(a, x) + b = diff(b, x) + expected = ( + " if (x < 0) then\n" + " weird_name = -cos(x)/x + 10*sin(x)/x**2 + 90*cos(x)/x**3 - 720*\n" + " @ sin(x)/x**4 - 5040*cos(x)/x**5 + 30240*sin(x)/x**6 + 151200*cos(x\n" + " @ )/x**7 - 604800*sin(x)/x**8 - 1814400*cos(x)/x**9 + 3628800*sin(x\n" + " @ )/x**10 + 3628800*cos(x)/x**11\n" + " else\n" + " weird_name = -sin(x)/x - 10*cos(x)/x**2 + 90*sin(x)/x**3 + 720*\n" + " @ cos(x)/x**4 - 5040*sin(x)/x**5 - 30240*cos(x)/x**6 + 151200*sin(x\n" + " @ )/x**7 + 604800*cos(x)/x**8 - 1814400*sin(x)/x**9 - 3628800*cos(x\n" + " @ )/x**10 + 3628800*sin(x)/x**11\n" + " end if" + ) + code = fcode(Piecewise((a, x < 0), (b, True)), assign_to="weird_name") + assert code == expected + code = fcode(Piecewise((x, x < 1), (x**2, x > 1), (sin(x), True)), standard=95) + expected = " merge(x, merge(x**2, sin(x), x > 1), x < 1)" + assert code == expected + # Check that Piecewise without a True (default) condition error + expr = Piecewise((x, x < 1), (x**2, x > 1), (sin(x), x > 0)) + raises(ValueError, lambda: fcode(expr)) + + +def test_wrap_fortran(): + # "########################################################################" + printer = FCodePrinter() + lines = [ + "C This is a long comment on a single line that must be wrapped properly to produce nice output", + " this = is + a + long + and + nasty + fortran + statement + that * must + be + wrapped + properly", + " this = is + a + long + and + nasty + fortran + statement + that * must + be + wrapped + properly", + " this = is + a + long + and + nasty + fortran + statement + that * must + be + wrapped + properly", + " this = is + a + long + and + nasty + fortran + statement + that*must + be + wrapped + properly", + " this = is + a + long + and + nasty + fortran + statement + that*must + be + wrapped + properly", + " this = is + a + long + and + nasty + fortran + statement + that*must + be + wrapped + properly", + " this = is + a + long + and + nasty + fortran + statement + that*must + be + wrapped + properly", + " this = is + a + long + and + nasty + fortran + statement + that**must + be + wrapped + properly", + " this = is + a + long + and + nasty + fortran + statement + that**must + be + wrapped + properly", + " this = is + a + long + and + nasty + fortran + statement + that**must + be + wrapped + properly", + " this = is + a + long + and + nasty + fortran + statement + that**must + be + wrapped + properly", + " this = is + a + long + and + nasty + fortran + statement + that**must + be + wrapped + properly", + " this = is + a + long + and + nasty + fortran + statement(that)/must + be + wrapped + properly", + " this = is + a + long + and + nasty + fortran + statement(that)/must + be + wrapped + properly", + ] + wrapped_lines = printer._wrap_fortran(lines) + expected_lines = [ + "C This is a long comment on a single line that must be wrapped", + "C properly to produce nice output", + " this = is + a + long + and + nasty + fortran + statement + that *", + " @ must + be + wrapped + properly", + " this = is + a + long + and + nasty + fortran + statement + that *", + " @ must + be + wrapped + properly", + " this = is + a + long + and + nasty + fortran + statement + that", + " @ * must + be + wrapped + properly", + " this = is + a + long + and + nasty + fortran + statement + that*", + " @ must + be + wrapped + properly", + " this = is + a + long + and + nasty + fortran + statement + that*", + " @ must + be + wrapped + properly", + " this = is + a + long + and + nasty + fortran + statement + that", + " @ *must + be + wrapped + properly", + " this = is + a + long + and + nasty + fortran + statement +", + " @ that*must + be + wrapped + properly", + " this = is + a + long + and + nasty + fortran + statement + that**", + " @ must + be + wrapped + properly", + " this = is + a + long + and + nasty + fortran + statement + that**", + " @ must + be + wrapped + properly", + " this = is + a + long + and + nasty + fortran + statement + that", + " @ **must + be + wrapped + properly", + " this = is + a + long + and + nasty + fortran + statement + that", + " @ **must + be + wrapped + properly", + " this = is + a + long + and + nasty + fortran + statement +", + " @ that**must + be + wrapped + properly", + " this = is + a + long + and + nasty + fortran + statement(that)/", + " @ must + be + wrapped + properly", + " this = is + a + long + and + nasty + fortran + statement(that)", + " @ /must + be + wrapped + properly", + ] + for line in wrapped_lines: + assert len(line) <= 72 + for w, e in zip(wrapped_lines, expected_lines): + assert w == e + assert len(wrapped_lines) == len(expected_lines) + + +def test_wrap_fortran_keep_d0(): + printer = FCodePrinter() + lines = [ + ' this_variable_is_very_long_because_we_try_to_test_line_break=1.0d0', + ' this_variable_is_very_long_because_we_try_to_test_line_break =1.0d0', + ' this_variable_is_very_long_because_we_try_to_test_line_break = 1.0d0', + ' this_variable_is_very_long_because_we_try_to_test_line_break = 1.0d0', + ' this_variable_is_very_long_because_we_try_to_test_line_break = 1.0d0', + ' this_variable_is_very_long_because_we_try_to_test_line_break = 10.0d0' + ] + expected = [ + ' this_variable_is_very_long_because_we_try_to_test_line_break=1.0d0', + ' this_variable_is_very_long_because_we_try_to_test_line_break =', + ' @ 1.0d0', + ' this_variable_is_very_long_because_we_try_to_test_line_break =', + ' @ 1.0d0', + ' this_variable_is_very_long_because_we_try_to_test_line_break =', + ' @ 1.0d0', + ' this_variable_is_very_long_because_we_try_to_test_line_break =', + ' @ 1.0d0', + ' this_variable_is_very_long_because_we_try_to_test_line_break =', + ' @ 10.0d0' + ] + assert printer._wrap_fortran(lines) == expected + + +def test_settings(): + raises(TypeError, lambda: fcode(S(4), method="garbage")) + + +def test_free_form_code_line(): + x, y = symbols('x,y') + assert fcode(cos(x) + sin(y), source_format='free') == "sin(y) + cos(x)" + + +def test_free_form_continuation_line(): + x, y = symbols('x,y') + result = fcode(((cos(x) + sin(y))**(7)).expand(), source_format='free') + expected = ( + 'sin(y)**7 + 7*sin(y)**6*cos(x) + 21*sin(y)**5*cos(x)**2 + 35*sin(y)**4* &\n' + ' cos(x)**3 + 35*sin(y)**3*cos(x)**4 + 21*sin(y)**2*cos(x)**5 + 7* &\n' + ' sin(y)*cos(x)**6 + cos(x)**7' + ) + assert result == expected + + +def test_free_form_comment_line(): + printer = FCodePrinter({'source_format': 'free'}) + lines = [ "! This is a long comment on a single line that must be wrapped properly to produce nice output"] + expected = [ + '! This is a long comment on a single line that must be wrapped properly', + '! to produce nice output'] + assert printer._wrap_fortran(lines) == expected + + +def test_loops(): + n, m = symbols('n,m', integer=True) + A = IndexedBase('A') + x = IndexedBase('x') + y = IndexedBase('y') + i = Idx('i', m) + j = Idx('j', n) + + expected = ( + 'do i = 1, m\n' + ' y(i) = 0\n' + 'end do\n' + 'do i = 1, m\n' + ' do j = 1, n\n' + ' y(i) = %(rhs)s\n' + ' end do\n' + 'end do' + ) + + code = fcode(A[i, j]*x[j], assign_to=y[i], source_format='free') + assert (code == expected % {'rhs': 'y(i) + A(i, j)*x(j)'} or + code == expected % {'rhs': 'y(i) + x(j)*A(i, j)'} or + code == expected % {'rhs': 'x(j)*A(i, j) + y(i)'} or + code == expected % {'rhs': 'A(i, j)*x(j) + y(i)'}) + + +def test_dummy_loops(): + i, m = symbols('i m', integer=True, cls=Dummy) + x = IndexedBase('x') + y = IndexedBase('y') + i = Idx(i, m) + + expected = ( + 'do i_%(icount)i = 1, m_%(mcount)i\n' + ' y(i_%(icount)i) = x(i_%(icount)i)\n' + 'end do' + ) % {'icount': i.label.dummy_index, 'mcount': m.dummy_index} + code = fcode(x[i], assign_to=y[i], source_format='free') + assert code == expected + + +def test_fcode_Indexed_without_looking_for_contraction(): + len_y = 5 + y = IndexedBase('y', shape=(len_y,)) + x = IndexedBase('x', shape=(len_y,)) + Dy = IndexedBase('Dy', shape=(len_y-1,)) + i = Idx('i', len_y-1) + e=Eq(Dy[i], (y[i+1]-y[i])/(x[i+1]-x[i])) + code0 = fcode(e.rhs, assign_to=e.lhs, contract=False) + assert code0.endswith('Dy(i) = (y(i + 1) - y(i))/(x(i + 1) - x(i))') + + +def test_element_like_objects(): + len_y = 5 + y = ArraySymbol('y', shape=(len_y,)) + x = ArraySymbol('x', shape=(len_y,)) + Dy = ArraySymbol('Dy', shape=(len_y-1,)) + i = Idx('i', len_y-1) + e=Eq(Dy[i], (y[i+1]-y[i])/(x[i+1]-x[i])) + code0 = fcode(Assignment(e.lhs, e.rhs)) + assert code0.endswith('Dy(i) = (y(i + 1) - y(i))/(x(i + 1) - x(i))') + + class ElementExpr(Element, Expr): + pass + + e = e.subs((a, ElementExpr(a.name, a.indices)) for a in e.atoms(ArrayElement) ) + e=Eq(Dy[i], (y[i+1]-y[i])/(x[i+1]-x[i])) + code0 = fcode(Assignment(e.lhs, e.rhs)) + assert code0.endswith('Dy(i) = (y(i + 1) - y(i))/(x(i + 1) - x(i))') + + +def test_derived_classes(): + class MyFancyFCodePrinter(FCodePrinter): + _default_settings = FCodePrinter._default_settings.copy() + + printer = MyFancyFCodePrinter() + x = symbols('x') + assert printer.doprint(sin(x), "bork") == " bork = sin(x)" + + +def test_indent(): + codelines = ( + 'subroutine test(a)\n' + 'integer :: a, i, j\n' + '\n' + 'do\n' + 'do \n' + 'do j = 1, 5\n' + 'if (a>b) then\n' + 'if(b>0) then\n' + 'a = 3\n' + 'donot_indent_me = 2\n' + 'do_not_indent_me_either = 2\n' + 'ifIam_indented_something_went_wrong = 2\n' + 'if_I_am_indented_something_went_wrong = 2\n' + 'end should not be unindented here\n' + 'end if\n' + 'endif\n' + 'end do\n' + 'end do\n' + 'enddo\n' + 'end subroutine\n' + '\n' + 'subroutine test2(a)\n' + 'integer :: a\n' + 'do\n' + 'a = a + 1\n' + 'end do \n' + 'end subroutine\n' + ) + expected = ( + 'subroutine test(a)\n' + 'integer :: a, i, j\n' + '\n' + 'do\n' + ' do \n' + ' do j = 1, 5\n' + ' if (a>b) then\n' + ' if(b>0) then\n' + ' a = 3\n' + ' donot_indent_me = 2\n' + ' do_not_indent_me_either = 2\n' + ' ifIam_indented_something_went_wrong = 2\n' + ' if_I_am_indented_something_went_wrong = 2\n' + ' end should not be unindented here\n' + ' end if\n' + ' endif\n' + ' end do\n' + ' end do\n' + 'enddo\n' + 'end subroutine\n' + '\n' + 'subroutine test2(a)\n' + 'integer :: a\n' + 'do\n' + ' a = a + 1\n' + 'end do \n' + 'end subroutine\n' + ) + p = FCodePrinter({'source_format': 'free'}) + result = p.indent_code(codelines) + assert result == expected + +def test_Matrix_printing(): + x, y, z = symbols('x,y,z') + # Test returning a Matrix + mat = Matrix([x*y, Piecewise((2 + x, y>0), (y, True)), sin(z)]) + A = MatrixSymbol('A', 3, 1) + assert fcode(mat, A) == ( + " A(1, 1) = x*y\n" + " if (y > 0) then\n" + " A(2, 1) = x + 2\n" + " else\n" + " A(2, 1) = y\n" + " end if\n" + " A(3, 1) = sin(z)") + # Test using MatrixElements in expressions + expr = Piecewise((2*A[2, 0], x > 0), (A[2, 0], True)) + sin(A[1, 0]) + A[0, 0] + assert fcode(expr, standard=95) == ( + " merge(2*A(3, 1), A(3, 1), x > 0) + sin(A(2, 1)) + A(1, 1)") + # Test using MatrixElements in a Matrix + q = MatrixSymbol('q', 5, 1) + M = MatrixSymbol('M', 3, 3) + m = Matrix([[sin(q[1,0]), 0, cos(q[2,0])], + [q[1,0] + q[2,0], q[3, 0], 5], + [2*q[4, 0]/q[1,0], sqrt(q[0,0]) + 4, 0]]) + assert fcode(m, M) == ( + " M(1, 1) = sin(q(2, 1))\n" + " M(2, 1) = q(2, 1) + q(3, 1)\n" + " M(3, 1) = 2*q(5, 1)/q(2, 1)\n" + " M(1, 2) = 0\n" + " M(2, 2) = q(4, 1)\n" + " M(3, 2) = sqrt(q(1, 1)) + 4\n" + " M(1, 3) = cos(q(3, 1))\n" + " M(2, 3) = 5\n" + " M(3, 3) = 0") + + +def test_fcode_For(): + x, y = symbols('x y') + + f = For(x, Range(0, 10, 2), [Assignment(y, x * y)]) + sol = fcode(f) + assert sol == (" do x = 0, 9, 2\n" + " y = x*y\n" + " end do") + + +def test_fcode_Declaration(): + def check(expr, ref, **kwargs): + assert fcode(expr, standard=95, source_format='free', **kwargs) == ref + + i = symbols('i', integer=True) + var1 = Variable.deduced(i) + dcl1 = Declaration(var1) + check(dcl1, "integer*4 :: i") + + + x, y = symbols('x y') + var2 = Variable(x, float32, value=42, attrs={value_const}) + dcl2b = Declaration(var2) + check(dcl2b, 'real*4, parameter :: x = 42') + + var3 = Variable(y, type=bool_) + dcl3 = Declaration(var3) + check(dcl3, 'logical :: y') + + check(float32, "real*4") + check(float64, "real*8") + check(real, "real*4", type_aliases={real: float32}) + check(real, "real*8", type_aliases={real: float64}) + + +def test_MatrixElement_printing(): + # test cases for issue #11821 + A = MatrixSymbol("A", 1, 3) + B = MatrixSymbol("B", 1, 3) + C = MatrixSymbol("C", 1, 3) + + assert(fcode(A[0, 0]) == " A(1, 1)") + assert(fcode(3 * A[0, 0]) == " 3*A(1, 1)") + + F = C[0, 0].subs(C, A - B) + assert(fcode(F) == " (A - B)(1, 1)") + + +def test_aug_assign(): + x = symbols('x') + assert fcode(aug_assign(x, '+', 1), source_format='free') == 'x = x + 1' + + +def test_While(): + x = symbols('x') + assert fcode(While(abs(x) > 1, [aug_assign(x, '-', 1)]), source_format='free') == ( + 'do while (abs(x) > 1)\n' + ' x = x - 1\n' + 'end do' + ) + + +def test_FunctionPrototype_print(): + x = symbols('x') + n = symbols('n', integer=True) + vx = Variable(x, type=real) + vn = Variable(n, type=integer) + fp1 = FunctionPrototype(real, 'power', [vx, vn]) + # Should be changed to proper test once multi-line generation is working + # see https://github.com/sympy/sympy/issues/15824 + raises(NotImplementedError, lambda: fcode(fp1)) + + +def test_FunctionDefinition_print(): + x = symbols('x') + n = symbols('n', integer=True) + vx = Variable(x, type=real) + vn = Variable(n, type=integer) + body = [Assignment(x, x**n), Return(x)] + fd1 = FunctionDefinition(real, 'power', [vx, vn], body) + # Should be changed to proper test once multi-line generation is working + # see https://github.com/sympy/sympy/issues/15824 + raises(NotImplementedError, lambda: fcode(fd1)) diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_glsl.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_glsl.py new file mode 100644 index 0000000000000000000000000000000000000000..86ec1dfe4a37d141e8435c369cb692d3a9a3b7bc --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_glsl.py @@ -0,0 +1,998 @@ +from sympy.core import (pi, symbols, Rational, Integer, GoldenRatio, EulerGamma, + Catalan, Lambda, Dummy, Eq, Ne, Le, Lt, Gt, Ge) +from sympy.functions import Piecewise, sin, cos, Abs, exp, ceiling, sqrt +from sympy.testing.pytest import raises, warns_deprecated_sympy +from sympy.printing.glsl import GLSLPrinter +from sympy.printing.str import StrPrinter +from sympy.utilities.lambdify import implemented_function +from sympy.tensor import IndexedBase, Idx +from sympy.matrices import Matrix, MatrixSymbol +from sympy.core import Tuple +from sympy.printing.glsl import glsl_code +import textwrap + +x, y, z = symbols('x,y,z') + + +def test_printmethod(): + assert glsl_code(Abs(x)) == "abs(x)" + +def test_print_without_operators(): + assert glsl_code(x*y,use_operators = False) == 'mul(x, y)' + assert glsl_code(x**y+z,use_operators = False) == 'add(pow(x, y), z)' + assert glsl_code(x*(y+z),use_operators = False) == 'mul(x, add(y, z))' + assert glsl_code(x*(y+z),use_operators = False) == 'mul(x, add(y, z))' + assert glsl_code(x*(y+z**y**0.5),use_operators = False) == 'mul(x, add(y, pow(z, sqrt(y))))' + assert glsl_code(-x-y, use_operators=False, zero='zero()') == 'sub(zero(), add(x, y))' + assert glsl_code(-x-y, use_operators=False) == 'sub(0.0, add(x, y))' + +def test_glsl_code_sqrt(): + assert glsl_code(sqrt(x)) == "sqrt(x)" + assert glsl_code(x**0.5) == "sqrt(x)" + assert glsl_code(sqrt(x)) == "sqrt(x)" + + +def test_glsl_code_Pow(): + g = implemented_function('g', Lambda(x, 2*x)) + assert glsl_code(x**3) == "pow(x, 3.0)" + assert glsl_code(x**(y**3)) == "pow(x, pow(y, 3.0))" + assert glsl_code(1/(g(x)*3.5)**(x - y**x)/(x**2 + y)) == \ + "pow(3.5*2*x, -x + pow(y, x))/(pow(x, 2.0) + y)" + assert glsl_code(x**-1.0) == '1.0/x' + + +def test_glsl_code_Relational(): + assert glsl_code(Eq(x, y)) == "x == y" + assert glsl_code(Ne(x, y)) == "x != y" + assert glsl_code(Le(x, y)) == "x <= y" + assert glsl_code(Lt(x, y)) == "x < y" + assert glsl_code(Gt(x, y)) == "x > y" + assert glsl_code(Ge(x, y)) == "x >= y" + + +def test_glsl_code_constants_mathh(): + assert glsl_code(exp(1)) == "float E = 2.71828183;\nE" + assert glsl_code(pi) == "float pi = 3.14159265;\npi" + # assert glsl_code(oo) == "Number.POSITIVE_INFINITY" + # assert glsl_code(-oo) == "Number.NEGATIVE_INFINITY" + + +def test_glsl_code_constants_other(): + assert glsl_code(2*GoldenRatio) == "float GoldenRatio = 1.61803399;\n2*GoldenRatio" + assert glsl_code(2*Catalan) == "float Catalan = 0.915965594;\n2*Catalan" + assert glsl_code(2*EulerGamma) == "float EulerGamma = 0.577215665;\n2*EulerGamma" + + +def test_glsl_code_Rational(): + assert glsl_code(Rational(3, 7)) == "3.0/7.0" + assert glsl_code(Rational(18, 9)) == "2" + assert glsl_code(Rational(3, -7)) == "-3.0/7.0" + assert glsl_code(Rational(-3, -7)) == "3.0/7.0" + + +def test_glsl_code_Integer(): + assert glsl_code(Integer(67)) == "67" + assert glsl_code(Integer(-1)) == "-1" + + +def test_glsl_code_functions(): + assert glsl_code(sin(x) ** cos(x)) == "pow(sin(x), cos(x))" + + +def test_glsl_code_inline_function(): + x = symbols('x') + g = implemented_function('g', Lambda(x, 2*x)) + assert glsl_code(g(x)) == "2*x" + g = implemented_function('g', Lambda(x, 2*x/Catalan)) + assert glsl_code(g(x)) == "float Catalan = 0.915965594;\n2*x/Catalan" + A = IndexedBase('A') + i = Idx('i', symbols('n', integer=True)) + g = implemented_function('g', Lambda(x, x*(1 + x)*(2 + x))) + assert glsl_code(g(A[i]), assign_to=A[i]) == ( + "for (int i=0; i 1), (sin(x), x > 0)) + raises(ValueError, lambda: glsl_code(expr)) + + +def test_glsl_code_Piecewise_deep(): + p = glsl_code(2*Piecewise((x, x < 1), (x**2, True))) + s = \ +"""\ +2*((x < 1) ? ( + x +) +: ( + pow(x, 2.0) +))\ +""" + assert p == s + + +def test_glsl_code_settings(): + raises(TypeError, lambda: glsl_code(sin(x), method="garbage")) + + +def test_glsl_code_Indexed(): + n, m, o = symbols('n m o', integer=True) + i, j, k = Idx('i', n), Idx('j', m), Idx('k', o) + p = GLSLPrinter() + p._not_c = set() + + x = IndexedBase('x')[j] + assert p._print_Indexed(x) == 'x[j]' + A = IndexedBase('A')[i, j] + assert p._print_Indexed(A) == 'A[%s]' % (m*i+j) + B = IndexedBase('B')[i, j, k] + assert p._print_Indexed(B) == 'B[%s]' % (i*o*m+j*o+k) + + assert p._not_c == set() + +def test_glsl_code_list_tuple_Tuple(): + assert glsl_code([1,2,3,4]) == 'vec4(1, 2, 3, 4)' + assert glsl_code([1,2,3],glsl_types=False) == 'float[3](1, 2, 3)' + assert glsl_code([1,2,3]) == glsl_code((1,2,3)) + assert glsl_code([1,2,3]) == glsl_code(Tuple(1,2,3)) + + m = MatrixSymbol('A',3,4) + assert glsl_code([m[0],m[1]]) + +def test_glsl_code_loops_matrix_vector(): + n, m = symbols('n m', integer=True) + A = IndexedBase('A') + x = IndexedBase('x') + y = IndexedBase('y') + i = Idx('i', m) + j = Idx('j', n) + + s = ( + 'for (int i=0; i0), (y, True)), sin(z)]) + A = MatrixSymbol('A', 3, 1) + assert glsl_code(mat, assign_to=A) == ( +'''A[0][0] = x*y; +if (y > 0) { + A[1][0] = x + 2; +} +else { + A[1][0] = y; +} +A[2][0] = sin(z);''' ) + assert glsl_code(Matrix([A[0],A[1]])) + # Test using MatrixElements in expressions + expr = Piecewise((2*A[2, 0], x > 0), (A[2, 0], True)) + sin(A[1, 0]) + A[0, 0] + assert glsl_code(expr) == ( +'''((x > 0) ? ( + 2*A[2][0] +) +: ( + A[2][0] +)) + sin(A[1][0]) + A[0][0]''' ) + + # Test using MatrixElements in a Matrix + q = MatrixSymbol('q', 5, 1) + M = MatrixSymbol('M', 3, 3) + m = Matrix([[sin(q[1,0]), 0, cos(q[2,0])], + [q[1,0] + q[2,0], q[3, 0], 5], + [2*q[4, 0]/q[1,0], sqrt(q[0,0]) + 4, 0]]) + assert glsl_code(m,M) == ( +'''M[0][0] = sin(q[1]); +M[0][1] = 0; +M[0][2] = cos(q[2]); +M[1][0] = q[1] + q[2]; +M[1][1] = q[3]; +M[1][2] = 5; +M[2][0] = 2*q[4]/q[1]; +M[2][1] = sqrt(q[0]) + 4; +M[2][2] = 0;''' + ) + +def test_Matrices_1x7(): + gl = glsl_code + A = Matrix([1,2,3,4,5,6,7]) + assert gl(A) == 'float[7](1, 2, 3, 4, 5, 6, 7)' + assert gl(A.transpose()) == 'float[7](1, 2, 3, 4, 5, 6, 7)' + +def test_Matrices_1x7_array_type_int(): + gl = glsl_code + A = Matrix([1,2,3,4,5,6,7]) + assert gl(A, array_type='int') == 'int[7](1, 2, 3, 4, 5, 6, 7)' + +def test_Tuple_array_type_custom(): + gl = glsl_code + A = symbols('a b c') + assert gl(A, array_type='AbcType', glsl_types=False) == 'AbcType[3](a, b, c)' + +def test_Matrices_1x7_spread_assign_to_symbols(): + gl = glsl_code + A = Matrix([1,2,3,4,5,6,7]) + assign_to = symbols('x.a x.b x.c x.d x.e x.f x.g') + assert gl(A, assign_to=assign_to) == textwrap.dedent('''\ + x.a = 1; + x.b = 2; + x.c = 3; + x.d = 4; + x.e = 5; + x.f = 6; + x.g = 7;''' + ) + +def test_spread_assign_to_nested_symbols(): + gl = glsl_code + expr = ((1,2,3), (1,2,3)) + assign_to = (symbols('a b c'), symbols('x y z')) + assert gl(expr, assign_to=assign_to) == textwrap.dedent('''\ + a = 1; + b = 2; + c = 3; + x = 1; + y = 2; + z = 3;''' + ) + +def test_spread_assign_to_deeply_nested_symbols(): + gl = glsl_code + a, b, c, x, y, z = symbols('a b c x y z') + expr = (((1,2),3), ((1,2),3)) + assign_to = (((a, b), c), ((x, y), z)) + assert gl(expr, assign_to=assign_to) == textwrap.dedent('''\ + a = 1; + b = 2; + c = 3; + x = 1; + y = 2; + z = 3;''' + ) + +def test_matrix_of_tuples_spread_assign_to_symbols(): + gl = glsl_code + with warns_deprecated_sympy(): + expr = Matrix([[(1,2),(3,4)],[(5,6),(7,8)]]) + assign_to = (symbols('a b'), symbols('c d'), symbols('e f'), symbols('g h')) + assert gl(expr, assign_to) == textwrap.dedent('''\ + a = 1; + b = 2; + c = 3; + d = 4; + e = 5; + f = 6; + g = 7; + h = 8;''' + ) + +def test_cannot_assign_to_cause_mismatched_length(): + expr = (1, 2) + assign_to = symbols('x y z') + raises(ValueError, lambda: glsl_code(expr, assign_to)) + +def test_matrix_4x4_assign(): + gl = glsl_code + expr = MatrixSymbol('A',4,4) * MatrixSymbol('B',4,4) + MatrixSymbol('C',4,4) + assign_to = MatrixSymbol('X',4,4) + assert gl(expr, assign_to=assign_to) == textwrap.dedent('''\ + X[0][0] = A[0][0]*B[0][0] + A[0][1]*B[1][0] + A[0][2]*B[2][0] + A[0][3]*B[3][0] + C[0][0]; + X[0][1] = A[0][0]*B[0][1] + A[0][1]*B[1][1] + A[0][2]*B[2][1] + A[0][3]*B[3][1] + C[0][1]; + X[0][2] = A[0][0]*B[0][2] + A[0][1]*B[1][2] + A[0][2]*B[2][2] + A[0][3]*B[3][2] + C[0][2]; + X[0][3] = A[0][0]*B[0][3] + A[0][1]*B[1][3] + A[0][2]*B[2][3] + A[0][3]*B[3][3] + C[0][3]; + X[1][0] = A[1][0]*B[0][0] + A[1][1]*B[1][0] + A[1][2]*B[2][0] + A[1][3]*B[3][0] + C[1][0]; + X[1][1] = A[1][0]*B[0][1] + A[1][1]*B[1][1] + A[1][2]*B[2][1] + A[1][3]*B[3][1] + C[1][1]; + X[1][2] = A[1][0]*B[0][2] + A[1][1]*B[1][2] + A[1][2]*B[2][2] + A[1][3]*B[3][2] + C[1][2]; + X[1][3] = A[1][0]*B[0][3] + A[1][1]*B[1][3] + A[1][2]*B[2][3] + A[1][3]*B[3][3] + C[1][3]; + X[2][0] = A[2][0]*B[0][0] + A[2][1]*B[1][0] + A[2][2]*B[2][0] + A[2][3]*B[3][0] + C[2][0]; + X[2][1] = A[2][0]*B[0][1] + A[2][1]*B[1][1] + A[2][2]*B[2][1] + A[2][3]*B[3][1] + C[2][1]; + X[2][2] = A[2][0]*B[0][2] + A[2][1]*B[1][2] + A[2][2]*B[2][2] + A[2][3]*B[3][2] + C[2][2]; + X[2][3] = A[2][0]*B[0][3] + A[2][1]*B[1][3] + A[2][2]*B[2][3] + A[2][3]*B[3][3] + C[2][3]; + X[3][0] = A[3][0]*B[0][0] + A[3][1]*B[1][0] + A[3][2]*B[2][0] + A[3][3]*B[3][0] + C[3][0]; + X[3][1] = A[3][0]*B[0][1] + A[3][1]*B[1][1] + A[3][2]*B[2][1] + A[3][3]*B[3][1] + C[3][1]; + X[3][2] = A[3][0]*B[0][2] + A[3][1]*B[1][2] + A[3][2]*B[2][2] + A[3][3]*B[3][2] + C[3][2]; + X[3][3] = A[3][0]*B[0][3] + A[3][1]*B[1][3] + A[3][2]*B[2][3] + A[3][3]*B[3][3] + C[3][3];''' + ) + +def test_1xN_vecs(): + gl = glsl_code + for i in range(1,10): + A = Matrix(range(i)) + assert gl(A.transpose()) == gl(A) + assert gl(A,mat_transpose=True) == gl(A) + if i > 1: + if i <= 4: + assert gl(A) == 'vec%s(%s)' % (i,', '.join(str(s) for s in range(i))) + else: + assert gl(A) == 'float[%s](%s)' % (i,', '.join(str(s) for s in range(i))) + +def test_MxN_mats(): + generatedAssertions='def test_misc_mats():\n' + for i in range(1,6): + for j in range(1,6): + A = Matrix([[x + y*j for x in range(j)] for y in range(i)]) + gl = glsl_code(A) + glTransposed = glsl_code(A,mat_transpose=True) + generatedAssertions+=' mat = '+StrPrinter()._print(A)+'\n\n' + generatedAssertions+=' gl = \'\'\''+gl+'\'\'\'\n' + generatedAssertions+=' glTransposed = \'\'\''+glTransposed+'\'\'\'\n\n' + generatedAssertions+=' assert glsl_code(mat) == gl\n' + generatedAssertions+=' assert glsl_code(mat,mat_transpose=True) == glTransposed\n' + if i == 1 and j == 1: + assert gl == '0' + elif i <= 4 and j <= 4 and i>1 and j>1: + assert gl.startswith('mat%s' % j) + assert glTransposed.startswith('mat%s' % i) + elif i == 1 and j <= 4: + assert gl.startswith('vec') + elif j == 1 and i <= 4: + assert gl.startswith('vec') + elif i == 1: + assert gl.startswith('float[%s]('% j*i) + assert glTransposed.startswith('float[%s]('% j*i) + elif j == 1: + assert gl.startswith('float[%s]('% i*j) + assert glTransposed.startswith('float[%s]('% i*j) + else: + assert gl.startswith('float[%s](' % (i*j)) + assert glTransposed.startswith('float[%s](' % (i*j)) + glNested = glsl_code(A,mat_nested=True) + glNestedTransposed = glsl_code(A,mat_transpose=True,mat_nested=True) + assert glNested.startswith('float[%s][%s]' % (i,j)) + assert glNestedTransposed.startswith('float[%s][%s]' % (j,i)) + generatedAssertions+=' glNested = \'\'\''+glNested+'\'\'\'\n' + generatedAssertions+=' glNestedTransposed = \'\'\''+glNestedTransposed+'\'\'\'\n\n' + generatedAssertions+=' assert glsl_code(mat,mat_nested=True) == glNested\n' + generatedAssertions+=' assert glsl_code(mat,mat_nested=True,mat_transpose=True) == glNestedTransposed\n\n' + generateAssertions = False # set this to true to write bake these generated tests to a file + if generateAssertions: + gen = open('test_glsl_generated_matrices.py','w') + gen.write(generatedAssertions) + gen.close() + + +# these assertions were generated from the previous function +# glsl has complicated rules and this makes it easier to look over all the cases +def test_misc_mats(): + + mat = Matrix([[0]]) + + gl = '''0''' + glTransposed = '''0''' + + assert glsl_code(mat) == gl + assert glsl_code(mat,mat_transpose=True) == glTransposed + + mat = Matrix([[0, 1]]) + + gl = '''vec2(0, 1)''' + glTransposed = '''vec2(0, 1)''' + + assert glsl_code(mat) == gl + assert glsl_code(mat,mat_transpose=True) == glTransposed + + mat = Matrix([[0, 1, 2]]) + + gl = '''vec3(0, 1, 2)''' + glTransposed = '''vec3(0, 1, 2)''' + + assert glsl_code(mat) == gl + assert glsl_code(mat,mat_transpose=True) == glTransposed + + mat = Matrix([[0, 1, 2, 3]]) + + gl = '''vec4(0, 1, 2, 3)''' + glTransposed = '''vec4(0, 1, 2, 3)''' + + assert glsl_code(mat) == gl + assert glsl_code(mat,mat_transpose=True) == glTransposed + + mat = Matrix([[0, 1, 2, 3, 4]]) + + gl = '''float[5](0, 1, 2, 3, 4)''' + glTransposed = '''float[5](0, 1, 2, 3, 4)''' + + assert glsl_code(mat) == gl + assert glsl_code(mat,mat_transpose=True) == glTransposed + + mat = Matrix([ +[0], +[1]]) + + gl = '''vec2(0, 1)''' + glTransposed = '''vec2(0, 1)''' + + assert glsl_code(mat) == gl + assert glsl_code(mat,mat_transpose=True) == glTransposed + + mat = Matrix([ +[0, 1], +[2, 3]]) + + gl = '''mat2(0, 1, 2, 3)''' + glTransposed = '''mat2(0, 2, 1, 3)''' + + assert glsl_code(mat) == gl + assert glsl_code(mat,mat_transpose=True) == glTransposed + + mat = Matrix([ +[0, 1, 2], +[3, 4, 5]]) + + gl = '''mat3x2(0, 1, 2, 3, 4, 5)''' + glTransposed = '''mat2x3(0, 3, 1, 4, 2, 5)''' + + assert glsl_code(mat) == gl + assert glsl_code(mat,mat_transpose=True) == glTransposed + + mat = Matrix([ +[0, 1, 2, 3], +[4, 5, 6, 7]]) + + gl = '''mat4x2(0, 1, 2, 3, 4, 5, 6, 7)''' + glTransposed = '''mat2x4(0, 4, 1, 5, 2, 6, 3, 7)''' + + assert glsl_code(mat) == gl + assert glsl_code(mat,mat_transpose=True) == glTransposed + + mat = Matrix([ +[0, 1, 2, 3, 4], +[5, 6, 7, 8, 9]]) + + gl = '''float[10]( + 0, 1, 2, 3, 4, + 5, 6, 7, 8, 9 +) /* a 2x5 matrix */''' + glTransposed = '''float[10]( + 0, 5, + 1, 6, + 2, 7, + 3, 8, + 4, 9 +) /* a 5x2 matrix */''' + + assert glsl_code(mat) == gl + assert glsl_code(mat,mat_transpose=True) == glTransposed + glNested = '''float[2][5]( + float[](0, 1, 2, 3, 4), + float[](5, 6, 7, 8, 9) +)''' + glNestedTransposed = '''float[5][2]( + float[](0, 5), + float[](1, 6), + float[](2, 7), + float[](3, 8), + float[](4, 9) +)''' + + assert glsl_code(mat,mat_nested=True) == glNested + assert glsl_code(mat,mat_nested=True,mat_transpose=True) == glNestedTransposed + + mat = Matrix([ +[0], +[1], +[2]]) + + gl = '''vec3(0, 1, 2)''' + glTransposed = '''vec3(0, 1, 2)''' + + assert glsl_code(mat) == gl + assert glsl_code(mat,mat_transpose=True) == glTransposed + + mat = Matrix([ +[0, 1], +[2, 3], +[4, 5]]) + + gl = '''mat2x3(0, 1, 2, 3, 4, 5)''' + glTransposed = '''mat3x2(0, 2, 4, 1, 3, 5)''' + + assert glsl_code(mat) == gl + assert glsl_code(mat,mat_transpose=True) == glTransposed + + mat = Matrix([ +[0, 1, 2], +[3, 4, 5], +[6, 7, 8]]) + + gl = '''mat3(0, 1, 2, 3, 4, 5, 6, 7, 8)''' + glTransposed = '''mat3(0, 3, 6, 1, 4, 7, 2, 5, 8)''' + + assert glsl_code(mat) == gl + assert glsl_code(mat,mat_transpose=True) == glTransposed + + mat = Matrix([ +[0, 1, 2, 3], +[4, 5, 6, 7], +[8, 9, 10, 11]]) + + gl = '''mat4x3(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11)''' + glTransposed = '''mat3x4(0, 4, 8, 1, 5, 9, 2, 6, 10, 3, 7, 11)''' + + assert glsl_code(mat) == gl + assert glsl_code(mat,mat_transpose=True) == glTransposed + + mat = Matrix([ +[ 0, 1, 2, 3, 4], +[ 5, 6, 7, 8, 9], +[10, 11, 12, 13, 14]]) + + gl = '''float[15]( + 0, 1, 2, 3, 4, + 5, 6, 7, 8, 9, + 10, 11, 12, 13, 14 +) /* a 3x5 matrix */''' + glTransposed = '''float[15]( + 0, 5, 10, + 1, 6, 11, + 2, 7, 12, + 3, 8, 13, + 4, 9, 14 +) /* a 5x3 matrix */''' + + assert glsl_code(mat) == gl + assert glsl_code(mat,mat_transpose=True) == glTransposed + glNested = '''float[3][5]( + float[]( 0, 1, 2, 3, 4), + float[]( 5, 6, 7, 8, 9), + float[](10, 11, 12, 13, 14) +)''' + glNestedTransposed = '''float[5][3]( + float[](0, 5, 10), + float[](1, 6, 11), + float[](2, 7, 12), + float[](3, 8, 13), + float[](4, 9, 14) +)''' + + assert glsl_code(mat,mat_nested=True) == glNested + assert glsl_code(mat,mat_nested=True,mat_transpose=True) == glNestedTransposed + + mat = Matrix([ +[0], +[1], +[2], +[3]]) + + gl = '''vec4(0, 1, 2, 3)''' + glTransposed = '''vec4(0, 1, 2, 3)''' + + assert glsl_code(mat) == gl + assert glsl_code(mat,mat_transpose=True) == glTransposed + + mat = Matrix([ +[0, 1], +[2, 3], +[4, 5], +[6, 7]]) + + gl = '''mat2x4(0, 1, 2, 3, 4, 5, 6, 7)''' + glTransposed = '''mat4x2(0, 2, 4, 6, 1, 3, 5, 7)''' + + assert glsl_code(mat) == gl + assert glsl_code(mat,mat_transpose=True) == glTransposed + + mat = Matrix([ +[0, 1, 2], +[3, 4, 5], +[6, 7, 8], +[9, 10, 11]]) + + gl = '''mat3x4(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11)''' + glTransposed = '''mat4x3(0, 3, 6, 9, 1, 4, 7, 10, 2, 5, 8, 11)''' + + assert glsl_code(mat) == gl + assert glsl_code(mat,mat_transpose=True) == glTransposed + + mat = Matrix([ +[ 0, 1, 2, 3], +[ 4, 5, 6, 7], +[ 8, 9, 10, 11], +[12, 13, 14, 15]]) + + gl = '''mat4( 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15)''' + glTransposed = '''mat4(0, 4, 8, 12, 1, 5, 9, 13, 2, 6, 10, 14, 3, 7, 11, 15)''' + + assert glsl_code(mat) == gl + assert glsl_code(mat,mat_transpose=True) == glTransposed + + mat = Matrix([ +[ 0, 1, 2, 3, 4], +[ 5, 6, 7, 8, 9], +[10, 11, 12, 13, 14], +[15, 16, 17, 18, 19]]) + + gl = '''float[20]( + 0, 1, 2, 3, 4, + 5, 6, 7, 8, 9, + 10, 11, 12, 13, 14, + 15, 16, 17, 18, 19 +) /* a 4x5 matrix */''' + glTransposed = '''float[20]( + 0, 5, 10, 15, + 1, 6, 11, 16, + 2, 7, 12, 17, + 3, 8, 13, 18, + 4, 9, 14, 19 +) /* a 5x4 matrix */''' + + assert glsl_code(mat) == gl + assert glsl_code(mat,mat_transpose=True) == glTransposed + glNested = '''float[4][5]( + float[]( 0, 1, 2, 3, 4), + float[]( 5, 6, 7, 8, 9), + float[](10, 11, 12, 13, 14), + float[](15, 16, 17, 18, 19) +)''' + glNestedTransposed = '''float[5][4]( + float[](0, 5, 10, 15), + float[](1, 6, 11, 16), + float[](2, 7, 12, 17), + float[](3, 8, 13, 18), + float[](4, 9, 14, 19) +)''' + + assert glsl_code(mat,mat_nested=True) == glNested + assert glsl_code(mat,mat_nested=True,mat_transpose=True) == glNestedTransposed + + mat = Matrix([ +[0], +[1], +[2], +[3], +[4]]) + + gl = '''float[5](0, 1, 2, 3, 4)''' + glTransposed = '''float[5](0, 1, 2, 3, 4)''' + + assert glsl_code(mat) == gl + assert glsl_code(mat,mat_transpose=True) == glTransposed + + mat = Matrix([ +[0, 1], +[2, 3], +[4, 5], +[6, 7], +[8, 9]]) + + gl = '''float[10]( + 0, 1, + 2, 3, + 4, 5, + 6, 7, + 8, 9 +) /* a 5x2 matrix */''' + glTransposed = '''float[10]( + 0, 2, 4, 6, 8, + 1, 3, 5, 7, 9 +) /* a 2x5 matrix */''' + + assert glsl_code(mat) == gl + assert glsl_code(mat,mat_transpose=True) == glTransposed + glNested = '''float[5][2]( + float[](0, 1), + float[](2, 3), + float[](4, 5), + float[](6, 7), + float[](8, 9) +)''' + glNestedTransposed = '''float[2][5]( + float[](0, 2, 4, 6, 8), + float[](1, 3, 5, 7, 9) +)''' + + assert glsl_code(mat,mat_nested=True) == glNested + assert glsl_code(mat,mat_nested=True,mat_transpose=True) == glNestedTransposed + + mat = Matrix([ +[ 0, 1, 2], +[ 3, 4, 5], +[ 6, 7, 8], +[ 9, 10, 11], +[12, 13, 14]]) + + gl = '''float[15]( + 0, 1, 2, + 3, 4, 5, + 6, 7, 8, + 9, 10, 11, + 12, 13, 14 +) /* a 5x3 matrix */''' + glTransposed = '''float[15]( + 0, 3, 6, 9, 12, + 1, 4, 7, 10, 13, + 2, 5, 8, 11, 14 +) /* a 3x5 matrix */''' + + assert glsl_code(mat) == gl + assert glsl_code(mat,mat_transpose=True) == glTransposed + glNested = '''float[5][3]( + float[]( 0, 1, 2), + float[]( 3, 4, 5), + float[]( 6, 7, 8), + float[]( 9, 10, 11), + float[](12, 13, 14) +)''' + glNestedTransposed = '''float[3][5]( + float[](0, 3, 6, 9, 12), + float[](1, 4, 7, 10, 13), + float[](2, 5, 8, 11, 14) +)''' + + assert glsl_code(mat,mat_nested=True) == glNested + assert glsl_code(mat,mat_nested=True,mat_transpose=True) == glNestedTransposed + + mat = Matrix([ +[ 0, 1, 2, 3], +[ 4, 5, 6, 7], +[ 8, 9, 10, 11], +[12, 13, 14, 15], +[16, 17, 18, 19]]) + + gl = '''float[20]( + 0, 1, 2, 3, + 4, 5, 6, 7, + 8, 9, 10, 11, + 12, 13, 14, 15, + 16, 17, 18, 19 +) /* a 5x4 matrix */''' + glTransposed = '''float[20]( + 0, 4, 8, 12, 16, + 1, 5, 9, 13, 17, + 2, 6, 10, 14, 18, + 3, 7, 11, 15, 19 +) /* a 4x5 matrix */''' + + assert glsl_code(mat) == gl + assert glsl_code(mat,mat_transpose=True) == glTransposed + glNested = '''float[5][4]( + float[]( 0, 1, 2, 3), + float[]( 4, 5, 6, 7), + float[]( 8, 9, 10, 11), + float[](12, 13, 14, 15), + float[](16, 17, 18, 19) +)''' + glNestedTransposed = '''float[4][5]( + float[](0, 4, 8, 12, 16), + float[](1, 5, 9, 13, 17), + float[](2, 6, 10, 14, 18), + float[](3, 7, 11, 15, 19) +)''' + + assert glsl_code(mat,mat_nested=True) == glNested + assert glsl_code(mat,mat_nested=True,mat_transpose=True) == glNestedTransposed + + mat = Matrix([ +[ 0, 1, 2, 3, 4], +[ 5, 6, 7, 8, 9], +[10, 11, 12, 13, 14], +[15, 16, 17, 18, 19], +[20, 21, 22, 23, 24]]) + + gl = '''float[25]( + 0, 1, 2, 3, 4, + 5, 6, 7, 8, 9, + 10, 11, 12, 13, 14, + 15, 16, 17, 18, 19, + 20, 21, 22, 23, 24 +) /* a 5x5 matrix */''' + glTransposed = '''float[25]( + 0, 5, 10, 15, 20, + 1, 6, 11, 16, 21, + 2, 7, 12, 17, 22, + 3, 8, 13, 18, 23, + 4, 9, 14, 19, 24 +) /* a 5x5 matrix */''' + + assert glsl_code(mat) == gl + assert glsl_code(mat,mat_transpose=True) == glTransposed + glNested = '''float[5][5]( + float[]( 0, 1, 2, 3, 4), + float[]( 5, 6, 7, 8, 9), + float[](10, 11, 12, 13, 14), + float[](15, 16, 17, 18, 19), + float[](20, 21, 22, 23, 24) +)''' + glNestedTransposed = '''float[5][5]( + float[](0, 5, 10, 15, 20), + float[](1, 6, 11, 16, 21), + float[](2, 7, 12, 17, 22), + float[](3, 8, 13, 18, 23), + float[](4, 9, 14, 19, 24) +)''' + + assert glsl_code(mat,mat_nested=True) == glNested + assert glsl_code(mat,mat_nested=True,mat_transpose=True) == glNestedTransposed diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_gtk.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_gtk.py new file mode 100644 index 0000000000000000000000000000000000000000..5a595ab04d3a29d23e06ec12207bf917392aebce --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_gtk.py @@ -0,0 +1,18 @@ +from sympy.functions.elementary.trigonometric import sin +from sympy.printing.gtk import print_gtk +from sympy.testing.pytest import XFAIL, raises + +# this test fails if python-lxml isn't installed. We don't want to depend on +# anything with SymPy + + +@XFAIL +def test_1(): + from sympy.abc import x + print_gtk(x**2, start_viewer=False) + print_gtk(x**2 + sin(x)/4, start_viewer=False) + + +def test_settings(): + from sympy.abc import x + raises(TypeError, lambda: print_gtk(x, method="garbage")) diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_jax.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_jax.py new file mode 100644 index 0000000000000000000000000000000000000000..365d87c5b91fdd49a8e46cfde9c2b5792c23a03c --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_jax.py @@ -0,0 +1,370 @@ +from sympy.concrete.summations import Sum +from sympy.core.mod import Mod +from sympy.core.relational import (Equality, Unequality) +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.piecewise import Piecewise +from sympy.matrices.expressions.blockmatrix import BlockMatrix +from sympy.matrices.expressions.matexpr import MatrixSymbol +from sympy.matrices.expressions.special import Identity +from sympy.utilities.lambdify import lambdify + +from sympy.abc import x, i, j, a, b, c, d +from sympy.core import Function, Pow, Symbol +from sympy.codegen.matrix_nodes import MatrixSolve +from sympy.codegen.numpy_nodes import logaddexp, logaddexp2 +from sympy.codegen.cfunctions import log1p, expm1, hypot, log10, exp2, log2, Sqrt +from sympy.tensor.array import Array +from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct, ArrayAdd, \ + PermuteDims, ArrayDiagonal +from sympy.printing.numpy import JaxPrinter, _jax_known_constants, _jax_known_functions +from sympy.tensor.array.expressions.from_matrix_to_array import convert_matrix_to_array + +from sympy.testing.pytest import skip, raises +from sympy.external import import_module + +# Unlike NumPy which will aggressively promote operands to double precision, +# jax always uses single precision. Double precision in jax can be +# configured before the call to `import jax`, however this must be explicitly +# configured and is not fully supported. Thus, the tests here have been modified +# from the tests in test_numpy.py, only in the fact that they assert lambdify +# function accuracy to only single precision accuracy. +# https://jax.readthedocs.io/en/latest/notebooks/Common_Gotchas_in_JAX.html#double-64bit-precision + +jax = import_module('jax') + +if jax: + deafult_float_info = jax.numpy.finfo(jax.numpy.array([]).dtype) + JAX_DEFAULT_EPSILON = deafult_float_info.eps + + +def test_jax_piecewise_regression(): + """ + NumPyPrinter needs to print Piecewise()'s choicelist as a list to avoid + breaking compatibility with numpy 1.8. This is not necessary in numpy 1.9+. + See gh-9747 and gh-9749 for details. + """ + printer = JaxPrinter() + p = Piecewise((1, x < 0), (0, True)) + assert printer.doprint(p) == \ + 'jax.numpy.select([jax.numpy.less(x, 0),True], [1,0], default=jax.numpy.nan)' + assert printer.module_imports == {'jax.numpy': {'select', 'less', 'nan'}} + + +def test_jax_logaddexp(): + lae = logaddexp(a, b) + assert JaxPrinter().doprint(lae) == 'jax.numpy.logaddexp(a, b)' + lae2 = logaddexp2(a, b) + assert JaxPrinter().doprint(lae2) == 'jax.numpy.logaddexp2(a, b)' + + +def test_jax_sum(): + if not jax: + skip("JAX not installed") + + s = Sum(x ** i, (i, a, b)) + f = lambdify((a, b, x), s, 'jax') + + a_, b_ = 0, 10 + x_ = jax.numpy.linspace(-1, +1, 10) + assert jax.numpy.allclose(f(a_, b_, x_), sum(x_ ** i_ for i_ in range(a_, b_ + 1))) + + s = Sum(i * x, (i, a, b)) + f = lambdify((a, b, x), s, 'jax') + + a_, b_ = 0, 10 + x_ = jax.numpy.linspace(-1, +1, 10) + assert jax.numpy.allclose(f(a_, b_, x_), sum(i_ * x_ for i_ in range(a_, b_ + 1))) + + +def test_jax_multiple_sums(): + if not jax: + skip("JAX not installed") + + s = Sum((x + j) * i, (i, a, b), (j, c, d)) + f = lambdify((a, b, c, d, x), s, 'jax') + + a_, b_ = 0, 10 + c_, d_ = 11, 21 + x_ = jax.numpy.linspace(-1, +1, 10) + assert jax.numpy.allclose(f(a_, b_, c_, d_, x_), + sum((x_ + j_) * i_ for i_ in range(a_, b_ + 1) for j_ in range(c_, d_ + 1))) + + +def test_jax_codegen_einsum(): + if not jax: + skip("JAX not installed") + + M = MatrixSymbol("M", 2, 2) + N = MatrixSymbol("N", 2, 2) + + cg = convert_matrix_to_array(M * N) + f = lambdify((M, N), cg, 'jax') + + ma = jax.numpy.array([[1, 2], [3, 4]]) + mb = jax.numpy.array([[1,-2], [-1, 3]]) + assert (f(ma, mb) == jax.numpy.matmul(ma, mb)).all() + + +def test_jax_codegen_extra(): + if not jax: + skip("JAX not installed") + + M = MatrixSymbol("M", 2, 2) + N = MatrixSymbol("N", 2, 2) + P = MatrixSymbol("P", 2, 2) + Q = MatrixSymbol("Q", 2, 2) + ma = jax.numpy.array([[1, 2], [3, 4]]) + mb = jax.numpy.array([[1,-2], [-1, 3]]) + mc = jax.numpy.array([[2, 0], [1, 2]]) + md = jax.numpy.array([[1,-1], [4, 7]]) + + cg = ArrayTensorProduct(M, N) + f = lambdify((M, N), cg, 'jax') + assert (f(ma, mb) == jax.numpy.einsum(ma, [0, 1], mb, [2, 3])).all() + + cg = ArrayAdd(M, N) + f = lambdify((M, N), cg, 'jax') + assert (f(ma, mb) == ma+mb).all() + + cg = ArrayAdd(M, N, P) + f = lambdify((M, N, P), cg, 'jax') + assert (f(ma, mb, mc) == ma+mb+mc).all() + + cg = ArrayAdd(M, N, P, Q) + f = lambdify((M, N, P, Q), cg, 'jax') + assert (f(ma, mb, mc, md) == ma+mb+mc+md).all() + + cg = PermuteDims(M, [1, 0]) + f = lambdify((M,), cg, 'jax') + assert (f(ma) == ma.T).all() + + cg = PermuteDims(ArrayTensorProduct(M, N), [1, 2, 3, 0]) + f = lambdify((M, N), cg, 'jax') + assert (f(ma, mb) == jax.numpy.transpose(jax.numpy.einsum(ma, [0, 1], mb, [2, 3]), (1, 2, 3, 0))).all() + + cg = ArrayDiagonal(ArrayTensorProduct(M, N), (1, 2)) + f = lambdify((M, N), cg, 'jax') + assert (f(ma, mb) == jax.numpy.diagonal(jax.numpy.einsum(ma, [0, 1], mb, [2, 3]), axis1=1, axis2=2)).all() + + +def test_jax_relational(): + if not jax: + skip("JAX not installed") + + e = Equality(x, 1) + + f = lambdify((x,), e, 'jax') + x_ = jax.numpy.array([0, 1, 2]) + assert jax.numpy.array_equal(f(x_), [False, True, False]) + + e = Unequality(x, 1) + + f = lambdify((x,), e, 'jax') + x_ = jax.numpy.array([0, 1, 2]) + assert jax.numpy.array_equal(f(x_), [True, False, True]) + + e = (x < 1) + + f = lambdify((x,), e, 'jax') + x_ = jax.numpy.array([0, 1, 2]) + assert jax.numpy.array_equal(f(x_), [True, False, False]) + + e = (x <= 1) + + f = lambdify((x,), e, 'jax') + x_ = jax.numpy.array([0, 1, 2]) + assert jax.numpy.array_equal(f(x_), [True, True, False]) + + e = (x > 1) + + f = lambdify((x,), e, 'jax') + x_ = jax.numpy.array([0, 1, 2]) + assert jax.numpy.array_equal(f(x_), [False, False, True]) + + e = (x >= 1) + + f = lambdify((x,), e, 'jax') + x_ = jax.numpy.array([0, 1, 2]) + assert jax.numpy.array_equal(f(x_), [False, True, True]) + + # Multi-condition expressions + e = (x >= 1) & (x < 2) + f = lambdify((x,), e, 'jax') + x_ = jax.numpy.array([0, 1, 2]) + assert jax.numpy.array_equal(f(x_), [False, True, False]) + + e = (x >= 1) | (x < 2) + f = lambdify((x,), e, 'jax') + x_ = jax.numpy.array([0, 1, 2]) + assert jax.numpy.array_equal(f(x_), [True, True, True]) + +def test_jax_mod(): + if not jax: + skip("JAX not installed") + + e = Mod(a, b) + f = lambdify((a, b), e, 'jax') + + a_ = jax.numpy.array([0, 1, 2, 3]) + b_ = 2 + assert jax.numpy.array_equal(f(a_, b_), [0, 1, 0, 1]) + + a_ = jax.numpy.array([0, 1, 2, 3]) + b_ = jax.numpy.array([2, 2, 2, 2]) + assert jax.numpy.array_equal(f(a_, b_), [0, 1, 0, 1]) + + a_ = jax.numpy.array([2, 3, 4, 5]) + b_ = jax.numpy.array([2, 3, 4, 5]) + assert jax.numpy.array_equal(f(a_, b_), [0, 0, 0, 0]) + + +def test_jax_pow(): + if not jax: + skip('JAX not installed') + + expr = Pow(2, -1, evaluate=False) + f = lambdify([], expr, 'jax') + assert f() == 0.5 + + +def test_jax_expm1(): + if not jax: + skip("JAX not installed") + + f = lambdify((a,), expm1(a), 'jax') + assert abs(f(1e-10) - 1e-10 - 5e-21) <= 1e-10 * JAX_DEFAULT_EPSILON + + +def test_jax_log1p(): + if not jax: + skip("JAX not installed") + + f = lambdify((a,), log1p(a), 'jax') + assert abs(f(1e-99) - 1e-99) <= 1e-99 * JAX_DEFAULT_EPSILON + +def test_jax_hypot(): + if not jax: + skip("JAX not installed") + assert abs(lambdify((a, b), hypot(a, b), 'jax')(3, 4) - 5) <= JAX_DEFAULT_EPSILON + +def test_jax_log10(): + if not jax: + skip("JAX not installed") + + assert abs(lambdify((a,), log10(a), 'jax')(100) - 2) <= JAX_DEFAULT_EPSILON + + +def test_jax_exp2(): + if not jax: + skip("JAX not installed") + assert abs(lambdify((a,), exp2(a), 'jax')(5) - 32) <= JAX_DEFAULT_EPSILON + + +def test_jax_log2(): + if not jax: + skip("JAX not installed") + assert abs(lambdify((a,), log2(a), 'jax')(256) - 8) <= JAX_DEFAULT_EPSILON + + +def test_jax_Sqrt(): + if not jax: + skip("JAX not installed") + assert abs(lambdify((a,), Sqrt(a), 'jax')(4) - 2) <= JAX_DEFAULT_EPSILON + + +def test_jax_sqrt(): + if not jax: + skip("JAX not installed") + assert abs(lambdify((a,), sqrt(a), 'jax')(4) - 2) <= JAX_DEFAULT_EPSILON + + +def test_jax_matsolve(): + if not jax: + skip("JAX not installed") + + M = MatrixSymbol("M", 3, 3) + x = MatrixSymbol("x", 3, 1) + + expr = M**(-1) * x + x + matsolve_expr = MatrixSolve(M, x) + x + + f = lambdify((M, x), expr, 'jax') + f_matsolve = lambdify((M, x), matsolve_expr, 'jax') + + m0 = jax.numpy.array([[1, 2, 3], [3, 2, 5], [5, 6, 7]]) + assert jax.numpy.linalg.matrix_rank(m0) == 3 + + x0 = jax.numpy.array([3, 4, 5]) + + assert jax.numpy.allclose(f_matsolve(m0, x0), f(m0, x0)) + + +def test_16857(): + if not jax: + skip("JAX not installed") + + a_1 = MatrixSymbol('a_1', 10, 3) + a_2 = MatrixSymbol('a_2', 10, 3) + a_3 = MatrixSymbol('a_3', 10, 3) + a_4 = MatrixSymbol('a_4', 10, 3) + A = BlockMatrix([[a_1, a_2], [a_3, a_4]]) + assert A.shape == (20, 6) + + printer = JaxPrinter() + assert printer.doprint(A) == 'jax.numpy.block([[a_1, a_2], [a_3, a_4]])' + + +def test_issue_17006(): + if not jax: + skip("JAX not installed") + + M = MatrixSymbol("M", 2, 2) + + f = lambdify(M, M + Identity(2), 'jax') + ma = jax.numpy.array([[1, 2], [3, 4]]) + mr = jax.numpy.array([[2, 2], [3, 5]]) + + assert (f(ma) == mr).all() + + from sympy.core.symbol import symbols + n = symbols('n', integer=True) + N = MatrixSymbol("M", n, n) + raises(NotImplementedError, lambda: lambdify(N, N + Identity(n), 'jax')) + + +def test_jax_array(): + assert JaxPrinter().doprint(Array(((1, 2), (3, 5)))) == 'jax.numpy.array([[1, 2], [3, 5]])' + assert JaxPrinter().doprint(Array((1, 2))) == 'jax.numpy.array([1, 2])' + + +def test_jax_known_funcs_consts(): + assert _jax_known_constants['NaN'] == 'jax.numpy.nan' + assert _jax_known_constants['EulerGamma'] == 'jax.numpy.euler_gamma' + + assert _jax_known_functions['acos'] == 'jax.numpy.arccos' + assert _jax_known_functions['log'] == 'jax.numpy.log' + + +def test_jax_print_methods(): + prntr = JaxPrinter() + assert hasattr(prntr, '_print_acos') + assert hasattr(prntr, '_print_log') + + +def test_jax_printmethod(): + printer = JaxPrinter() + assert hasattr(printer, 'printmethod') + assert printer.printmethod == '_jaxcode' + + +def test_jax_custom_print_method(): + + class expm1(Function): + + def _jaxcode(self, printer): + x, = self.args + function = f'expm1({printer._print(x)})' + return printer._module_format(printer._module + '.' + function) + + printer = JaxPrinter() + assert printer.doprint(expm1(Symbol('x'))) == 'jax.numpy.expm1(x)' diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_jscode.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_jscode.py new file mode 100644 index 0000000000000000000000000000000000000000..9199a8e0d62e87f2e964cb1712726a21c894fd20 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_jscode.py @@ -0,0 +1,396 @@ +from sympy.core import (pi, oo, symbols, Rational, Integer, GoldenRatio, + EulerGamma, Catalan, Lambda, Dummy, S, Eq, Ne, Le, + Lt, Gt, Ge, Mod) +from sympy.functions import (Piecewise, sin, cos, Abs, exp, ceiling, sqrt, + sinh, cosh, tanh, asin, acos, acosh, Max, Min) +from sympy.testing.pytest import raises +from sympy.printing.jscode import JavascriptCodePrinter +from sympy.utilities.lambdify import implemented_function +from sympy.tensor import IndexedBase, Idx +from sympy.matrices import Matrix, MatrixSymbol + +from sympy.printing.jscode import jscode + +x, y, z = symbols('x,y,z') + + +def test_printmethod(): + assert jscode(Abs(x)) == "Math.abs(x)" + + +def test_jscode_sqrt(): + assert jscode(sqrt(x)) == "Math.sqrt(x)" + assert jscode(x**0.5) == "Math.sqrt(x)" + assert jscode(x**(S.One/3)) == "Math.cbrt(x)" + + +def test_jscode_Pow(): + g = implemented_function('g', Lambda(x, 2*x)) + assert jscode(x**3) == "Math.pow(x, 3)" + assert jscode(x**(y**3)) == "Math.pow(x, Math.pow(y, 3))" + assert jscode(1/(g(x)*3.5)**(x - y**x)/(x**2 + y)) == \ + "Math.pow(3.5*2*x, -x + Math.pow(y, x))/(Math.pow(x, 2) + y)" + assert jscode(x**-1.0) == '1/x' + + +def test_jscode_constants_mathh(): + assert jscode(exp(1)) == "Math.E" + assert jscode(pi) == "Math.PI" + assert jscode(oo) == "Number.POSITIVE_INFINITY" + assert jscode(-oo) == "Number.NEGATIVE_INFINITY" + + +def test_jscode_constants_other(): + assert jscode( + 2*GoldenRatio) == "var GoldenRatio = %s;\n2*GoldenRatio" % GoldenRatio.evalf(17) + assert jscode(2*Catalan) == "var Catalan = %s;\n2*Catalan" % Catalan.evalf(17) + assert jscode( + 2*EulerGamma) == "var EulerGamma = %s;\n2*EulerGamma" % EulerGamma.evalf(17) + + +def test_jscode_Rational(): + assert jscode(Rational(3, 7)) == "3/7" + assert jscode(Rational(18, 9)) == "2" + assert jscode(Rational(3, -7)) == "-3/7" + assert jscode(Rational(-3, -7)) == "3/7" + + +def test_Relational(): + assert jscode(Eq(x, y)) == "x == y" + assert jscode(Ne(x, y)) == "x != y" + assert jscode(Le(x, y)) == "x <= y" + assert jscode(Lt(x, y)) == "x < y" + assert jscode(Gt(x, y)) == "x > y" + assert jscode(Ge(x, y)) == "x >= y" + + +def test_Mod(): + assert jscode(Mod(x, y)) == '((x % y) + y) % y' + assert jscode(Mod(x, x + y)) == '((x % (x + y)) + (x + y)) % (x + y)' + p1, p2 = symbols('p1 p2', positive=True) + assert jscode(Mod(p1, p2)) == 'p1 % p2' + assert jscode(Mod(p1, p2 + 3)) == 'p1 % (p2 + 3)' + assert jscode(Mod(-3, -7, evaluate=False)) == '(-3) % (-7)' + assert jscode(-Mod(p1, p2)) == '-(p1 % p2)' + assert jscode(x*Mod(p1, p2)) == 'x*(p1 % p2)' + + +def test_jscode_Integer(): + assert jscode(Integer(67)) == "67" + assert jscode(Integer(-1)) == "-1" + + +def test_jscode_functions(): + assert jscode(sin(x) ** cos(x)) == "Math.pow(Math.sin(x), Math.cos(x))" + assert jscode(sinh(x) * cosh(x)) == "Math.sinh(x)*Math.cosh(x)" + assert jscode(Max(x, y) + Min(x, y)) == "Math.max(x, y) + Math.min(x, y)" + assert jscode(tanh(x)*acosh(y)) == "Math.tanh(x)*Math.acosh(y)" + assert jscode(asin(x)-acos(y)) == "-Math.acos(y) + Math.asin(x)" + + +def test_jscode_inline_function(): + x = symbols('x') + g = implemented_function('g', Lambda(x, 2*x)) + assert jscode(g(x)) == "2*x" + g = implemented_function('g', Lambda(x, 2*x/Catalan)) + assert jscode(g(x)) == "var Catalan = %s;\n2*x/Catalan" % Catalan.evalf(17) + A = IndexedBase('A') + i = Idx('i', symbols('n', integer=True)) + g = implemented_function('g', Lambda(x, x*(1 + x)*(2 + x))) + assert jscode(g(A[i]), assign_to=A[i]) == ( + "for (var i=0; i 1), (sin(x), x > 0)) + raises(ValueError, lambda: jscode(expr)) + + +def test_jscode_Piecewise_deep(): + p = jscode(2*Piecewise((x, x < 1), (x**2, True))) + s = \ +"""\ +2*((x < 1) ? ( + x +) +: ( + Math.pow(x, 2) +))\ +""" + assert p == s + + +def test_jscode_settings(): + raises(TypeError, lambda: jscode(sin(x), method="garbage")) + + +def test_jscode_Indexed(): + n, m, o = symbols('n m o', integer=True) + i, j, k = Idx('i', n), Idx('j', m), Idx('k', o) + p = JavascriptCodePrinter() + p._not_c = set() + + x = IndexedBase('x')[j] + assert p._print_Indexed(x) == 'x[j]' + A = IndexedBase('A')[i, j] + assert p._print_Indexed(A) == 'A[%s]' % (m*i+j) + B = IndexedBase('B')[i, j, k] + assert p._print_Indexed(B) == 'B[%s]' % (i*o*m+j*o+k) + + assert p._not_c == set() + + +def test_jscode_loops_matrix_vector(): + n, m = symbols('n m', integer=True) + A = IndexedBase('A') + x = IndexedBase('x') + y = IndexedBase('y') + i = Idx('i', m) + j = Idx('j', n) + + s = ( + 'for (var i=0; i0), (y, True)), sin(z)]) + A = MatrixSymbol('A', 3, 1) + assert jscode(mat, A) == ( + "A[0] = x*y;\n" + "if (y > 0) {\n" + " A[1] = x + 2;\n" + "}\n" + "else {\n" + " A[1] = y;\n" + "}\n" + "A[2] = Math.sin(z);") + # Test using MatrixElements in expressions + expr = Piecewise((2*A[2, 0], x > 0), (A[2, 0], True)) + sin(A[1, 0]) + A[0, 0] + assert jscode(expr) == ( + "((x > 0) ? (\n" + " 2*A[2]\n" + ")\n" + ": (\n" + " A[2]\n" + ")) + Math.sin(A[1]) + A[0]") + # Test using MatrixElements in a Matrix + q = MatrixSymbol('q', 5, 1) + M = MatrixSymbol('M', 3, 3) + m = Matrix([[sin(q[1,0]), 0, cos(q[2,0])], + [q[1,0] + q[2,0], q[3, 0], 5], + [2*q[4, 0]/q[1,0], sqrt(q[0,0]) + 4, 0]]) + assert jscode(m, M) == ( + "M[0] = Math.sin(q[1]);\n" + "M[1] = 0;\n" + "M[2] = Math.cos(q[2]);\n" + "M[3] = q[1] + q[2];\n" + "M[4] = q[3];\n" + "M[5] = 5;\n" + "M[6] = 2*q[4]/q[1];\n" + "M[7] = Math.sqrt(q[0]) + 4;\n" + "M[8] = 0;") + + +def test_MatrixElement_printing(): + # test cases for issue #11821 + A = MatrixSymbol("A", 1, 3) + B = MatrixSymbol("B", 1, 3) + C = MatrixSymbol("C", 1, 3) + + assert(jscode(A[0, 0]) == "A[0]") + assert(jscode(3 * A[0, 0]) == "3*A[0]") + + F = C[0, 0].subs(C, A - B) + assert(jscode(F) == "(A - B)[0]") diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_julia.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_julia.py new file mode 100644 index 0000000000000000000000000000000000000000..b19c7b4fd4f21d8402ca2f577605322b3ec10f5b --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_julia.py @@ -0,0 +1,390 @@ +from sympy.core import (S, pi, oo, symbols, Function, Rational, Integer, + Tuple, Symbol, Eq, Ne, Le, Lt, Gt, Ge) +from sympy.core import EulerGamma, GoldenRatio, Catalan, Lambda, Mul, Pow +from sympy.functions import Piecewise, sqrt, ceiling, exp, sin, cos, sinc +from sympy.testing.pytest import raises +from sympy.utilities.lambdify import implemented_function +from sympy.matrices import (eye, Matrix, MatrixSymbol, Identity, + HadamardProduct, SparseMatrix) +from sympy.functions.special.bessel import (jn, yn, besselj, bessely, besseli, + besselk, hankel1, hankel2, airyai, + airybi, airyaiprime, airybiprime) +from sympy.testing.pytest import XFAIL + +from sympy.printing.julia import julia_code + +x, y, z = symbols('x,y,z') + + +def test_Integer(): + assert julia_code(Integer(67)) == "67" + assert julia_code(Integer(-1)) == "-1" + + +def test_Rational(): + assert julia_code(Rational(3, 7)) == "3 // 7" + assert julia_code(Rational(18, 9)) == "2" + assert julia_code(Rational(3, -7)) == "-3 // 7" + assert julia_code(Rational(-3, -7)) == "3 // 7" + assert julia_code(x + Rational(3, 7)) == "x + 3 // 7" + assert julia_code(Rational(3, 7)*x) == "(3 // 7) * x" + + +def test_Relational(): + assert julia_code(Eq(x, y)) == "x == y" + assert julia_code(Ne(x, y)) == "x != y" + assert julia_code(Le(x, y)) == "x <= y" + assert julia_code(Lt(x, y)) == "x < y" + assert julia_code(Gt(x, y)) == "x > y" + assert julia_code(Ge(x, y)) == "x >= y" + + +def test_Function(): + assert julia_code(sin(x) ** cos(x)) == "sin(x) .^ cos(x)" + assert julia_code(abs(x)) == "abs(x)" + assert julia_code(ceiling(x)) == "ceil(x)" + + +def test_Pow(): + assert julia_code(x**3) == "x .^ 3" + assert julia_code(x**(y**3)) == "x .^ (y .^ 3)" + assert julia_code(x**Rational(2, 3)) == 'x .^ (2 // 3)' + g = implemented_function('g', Lambda(x, 2*x)) + assert julia_code(1/(g(x)*3.5)**(x - y**x)/(x**2 + y)) == \ + "(3.5 * 2 * x) .^ (-x + y .^ x) ./ (x .^ 2 + y)" + # For issue 14160 + assert julia_code(Mul(-2, x, Pow(Mul(y,y,evaluate=False), -1, evaluate=False), + evaluate=False)) == '-2 * x ./ (y .* y)' + + +def test_basic_ops(): + assert julia_code(x*y) == "x .* y" + assert julia_code(x + y) == "x + y" + assert julia_code(x - y) == "x - y" + assert julia_code(-x) == "-x" + + +def test_1_over_x_and_sqrt(): + # 1.0 and 0.5 would do something different in regular StrPrinter, + # but these are exact in IEEE floating point so no different here. + assert julia_code(1/x) == '1 ./ x' + assert julia_code(x**-1) == julia_code(x**-1.0) == '1 ./ x' + assert julia_code(1/sqrt(x)) == '1 ./ sqrt(x)' + assert julia_code(x**-S.Half) == julia_code(x**-0.5) == '1 ./ sqrt(x)' + assert julia_code(sqrt(x)) == 'sqrt(x)' + assert julia_code(x**S.Half) == julia_code(x**0.5) == 'sqrt(x)' + assert julia_code(1/pi) == '1 / pi' + assert julia_code(pi**-1) == julia_code(pi**-1.0) == '1 / pi' + assert julia_code(pi**-0.5) == '1 / sqrt(pi)' + + +def test_mix_number_mult_symbols(): + assert julia_code(3*x) == "3 * x" + assert julia_code(pi*x) == "pi * x" + assert julia_code(3/x) == "3 ./ x" + assert julia_code(pi/x) == "pi ./ x" + assert julia_code(x/3) == "x / 3" + assert julia_code(x/pi) == "x / pi" + assert julia_code(x*y) == "x .* y" + assert julia_code(3*x*y) == "3 * x .* y" + assert julia_code(3*pi*x*y) == "3 * pi * x .* y" + assert julia_code(x/y) == "x ./ y" + assert julia_code(3*x/y) == "3 * x ./ y" + assert julia_code(x*y/z) == "x .* y ./ z" + assert julia_code(x/y*z) == "x .* z ./ y" + assert julia_code(1/x/y) == "1 ./ (x .* y)" + assert julia_code(2*pi*x/y/z) == "2 * pi * x ./ (y .* z)" + assert julia_code(3*pi/x) == "3 * pi ./ x" + assert julia_code(S(3)/5) == "3 // 5" + assert julia_code(S(3)/5*x) == "(3 // 5) * x" + assert julia_code(x/y/z) == "x ./ (y .* z)" + assert julia_code((x+y)/z) == "(x + y) ./ z" + assert julia_code((x+y)/(z+x)) == "(x + y) ./ (x + z)" + assert julia_code((x+y)/EulerGamma) == "(x + y) / eulergamma" + assert julia_code(x/3/pi) == "x / (3 * pi)" + assert julia_code(S(3)/5*x*y/pi) == "(3 // 5) * x .* y / pi" + + +def test_mix_number_pow_symbols(): + assert julia_code(pi**3) == 'pi ^ 3' + assert julia_code(x**2) == 'x .^ 2' + assert julia_code(x**(pi**3)) == 'x .^ (pi ^ 3)' + assert julia_code(x**y) == 'x .^ y' + assert julia_code(x**(y**z)) == 'x .^ (y .^ z)' + assert julia_code((x**y)**z) == '(x .^ y) .^ z' + + +def test_imag(): + I = S('I') + assert julia_code(I) == "im" + assert julia_code(5*I) == "5im" + assert julia_code((S(3)/2)*I) == "(3 // 2) * im" + assert julia_code(3+4*I) == "3 + 4im" + + +def test_constants(): + assert julia_code(pi) == "pi" + assert julia_code(oo) == "Inf" + assert julia_code(-oo) == "-Inf" + assert julia_code(S.NegativeInfinity) == "-Inf" + assert julia_code(S.NaN) == "NaN" + assert julia_code(S.Exp1) == "e" + assert julia_code(exp(1)) == "e" + + +def test_constants_other(): + assert julia_code(2*GoldenRatio) == "2 * golden" + assert julia_code(2*Catalan) == "2 * catalan" + assert julia_code(2*EulerGamma) == "2 * eulergamma" + + +def test_boolean(): + assert julia_code(x & y) == "x && y" + assert julia_code(x | y) == "x || y" + assert julia_code(~x) == "!x" + assert julia_code(x & y & z) == "x && y && z" + assert julia_code(x | y | z) == "x || y || z" + assert julia_code((x & y) | z) == "z || x && y" + assert julia_code((x | y) & z) == "z && (x || y)" + +def test_sinc(): + assert julia_code(sinc(x)) == 'sinc(x / pi)' + assert julia_code(sinc(x + 3)) == 'sinc((x + 3) / pi)' + assert julia_code(sinc(pi * (x + 3))) == 'sinc(x + 3)' + +def test_Matrices(): + assert julia_code(Matrix(1, 1, [10])) == "[10]" + A = Matrix([[1, sin(x/2), abs(x)], + [0, 1, pi], + [0, exp(1), ceiling(x)]]) + expected = ("[1 sin(x / 2) abs(x);\n" + "0 1 pi;\n" + "0 e ceil(x)]") + assert julia_code(A) == expected + # row and columns + assert julia_code(A[:,0]) == "[1, 0, 0]" + assert julia_code(A[0,:]) == "[1 sin(x / 2) abs(x)]" + # empty matrices + assert julia_code(Matrix(0, 0, [])) == 'zeros(0, 0)' + assert julia_code(Matrix(0, 3, [])) == 'zeros(0, 3)' + # annoying to read but correct + assert julia_code(Matrix([[x, x - y, -y]])) == "[x x - y -y]" + + +def test_vector_entries_hadamard(): + # For a row or column, user might to use the other dimension + A = Matrix([[1, sin(2/x), 3*pi/x/5]]) + assert julia_code(A) == "[1 sin(2 ./ x) (3 // 5) * pi ./ x]" + assert julia_code(A.T) == "[1, sin(2 ./ x), (3 // 5) * pi ./ x]" + + +@XFAIL +def test_Matrices_entries_not_hadamard(): + # For Matrix with col >= 2, row >= 2, they need to be scalars + # FIXME: is it worth worrying about this? Its not wrong, just + # leave it user's responsibility to put scalar data for x. + A = Matrix([[1, sin(2/x), 3*pi/x/5], [1, 2, x*y]]) + expected = ("[1 sin(2/x) 3*pi/(5*x);\n" + "1 2 x*y]") # <- we give x.*y + assert julia_code(A) == expected + + +def test_MatrixSymbol(): + n = Symbol('n', integer=True) + A = MatrixSymbol('A', n, n) + B = MatrixSymbol('B', n, n) + assert julia_code(A*B) == "A * B" + assert julia_code(B*A) == "B * A" + assert julia_code(2*A*B) == "2 * A * B" + assert julia_code(B*2*A) == "2 * B * A" + assert julia_code(A*(B + 3*Identity(n))) == "A * (3 * eye(n) + B)" + assert julia_code(A**(x**2)) == "A ^ (x .^ 2)" + assert julia_code(A**3) == "A ^ 3" + assert julia_code(A**S.Half) == "A ^ (1 // 2)" + + +def test_special_matrices(): + assert julia_code(6*Identity(3)) == "6 * eye(3)" + + +def test_containers(): + assert julia_code([1, 2, 3, [4, 5, [6, 7]], 8, [9, 10], 11]) == \ + "Any[1, 2, 3, Any[4, 5, Any[6, 7]], 8, Any[9, 10], 11]" + assert julia_code((1, 2, (3, 4))) == "(1, 2, (3, 4))" + assert julia_code([1]) == "Any[1]" + assert julia_code((1,)) == "(1,)" + assert julia_code(Tuple(*[1, 2, 3])) == "(1, 2, 3)" + assert julia_code((1, x*y, (3, x**2))) == "(1, x .* y, (3, x .^ 2))" + # scalar, matrix, empty matrix and empty list + assert julia_code((1, eye(3), Matrix(0, 0, []), [])) == "(1, [1 0 0;\n0 1 0;\n0 0 1], zeros(0, 0), Any[])" + + +def test_julia_noninline(): + source = julia_code((x+y)/Catalan, assign_to='me', inline=False) + expected = ( + "const Catalan = %s\n" + "me = (x + y) / Catalan" + ) % Catalan.evalf(17) + assert source == expected + + +def test_julia_piecewise(): + expr = Piecewise((x, x < 1), (x**2, True)) + assert julia_code(expr) == "((x < 1) ? (x) : (x .^ 2))" + assert julia_code(expr, assign_to="r") == ( + "r = ((x < 1) ? (x) : (x .^ 2))") + assert julia_code(expr, assign_to="r", inline=False) == ( + "if (x < 1)\n" + " r = x\n" + "else\n" + " r = x .^ 2\n" + "end") + expr = Piecewise((x**2, x < 1), (x**3, x < 2), (x**4, x < 3), (x**5, True)) + expected = ("((x < 1) ? (x .^ 2) :\n" + "(x < 2) ? (x .^ 3) :\n" + "(x < 3) ? (x .^ 4) : (x .^ 5))") + assert julia_code(expr) == expected + assert julia_code(expr, assign_to="r") == "r = " + expected + assert julia_code(expr, assign_to="r", inline=False) == ( + "if (x < 1)\n" + " r = x .^ 2\n" + "elseif (x < 2)\n" + " r = x .^ 3\n" + "elseif (x < 3)\n" + " r = x .^ 4\n" + "else\n" + " r = x .^ 5\n" + "end") + # Check that Piecewise without a True (default) condition error + expr = Piecewise((x, x < 1), (x**2, x > 1), (sin(x), x > 0)) + raises(ValueError, lambda: julia_code(expr)) + + +def test_julia_piecewise_times_const(): + pw = Piecewise((x, x < 1), (x**2, True)) + assert julia_code(2*pw) == "2 * ((x < 1) ? (x) : (x .^ 2))" + assert julia_code(pw/x) == "((x < 1) ? (x) : (x .^ 2)) ./ x" + assert julia_code(pw/(x*y)) == "((x < 1) ? (x) : (x .^ 2)) ./ (x .* y)" + assert julia_code(pw/3) == "((x < 1) ? (x) : (x .^ 2)) / 3" + + +def test_julia_matrix_assign_to(): + A = Matrix([[1, 2, 3]]) + assert julia_code(A, assign_to='a') == "a = [1 2 3]" + A = Matrix([[1, 2], [3, 4]]) + assert julia_code(A, assign_to='A') == "A = [1 2;\n3 4]" + + +def test_julia_matrix_assign_to_more(): + # assigning to Symbol or MatrixSymbol requires lhs/rhs match + A = Matrix([[1, 2, 3]]) + B = MatrixSymbol('B', 1, 3) + C = MatrixSymbol('C', 2, 3) + assert julia_code(A, assign_to=B) == "B = [1 2 3]" + raises(ValueError, lambda: julia_code(A, assign_to=x)) + raises(ValueError, lambda: julia_code(A, assign_to=C)) + + +def test_julia_matrix_1x1(): + A = Matrix([[3]]) + B = MatrixSymbol('B', 1, 1) + C = MatrixSymbol('C', 1, 2) + assert julia_code(A, assign_to=B) == "B = [3]" + # FIXME? + #assert julia_code(A, assign_to=x) == "x = [3]" + raises(ValueError, lambda: julia_code(A, assign_to=C)) + + +def test_julia_matrix_elements(): + A = Matrix([[x, 2, x*y]]) + assert julia_code(A[0, 0]**2 + A[0, 1] + A[0, 2]) == "x .^ 2 + x .* y + 2" + A = MatrixSymbol('AA', 1, 3) + assert julia_code(A) == "AA" + assert julia_code(A[0, 0]**2 + sin(A[0,1]) + A[0,2]) == \ + "sin(AA[1,2]) + AA[1,1] .^ 2 + AA[1,3]" + assert julia_code(sum(A)) == "AA[1,1] + AA[1,2] + AA[1,3]" + + +def test_julia_boolean(): + assert julia_code(True) == "true" + assert julia_code(S.true) == "true" + assert julia_code(False) == "false" + assert julia_code(S.false) == "false" + + +def test_julia_not_supported(): + with raises(NotImplementedError): + julia_code(S.ComplexInfinity) + + f = Function('f') + assert julia_code(f(x).diff(x), strict=False) == ( + "# Not supported in Julia:\n" + "# Derivative\n" + "Derivative(f(x), x)" + ) + + +def test_trick_indent_with_end_else_words(): + # words starting with "end" or "else" do not confuse the indenter + t1 = S('endless') + t2 = S('elsewhere') + pw = Piecewise((t1, x < 0), (t2, x <= 1), (1, True)) + assert julia_code(pw, inline=False) == ( + "if (x < 0)\n" + " endless\n" + "elseif (x <= 1)\n" + " elsewhere\n" + "else\n" + " 1\n" + "end") + + +def test_haramard(): + A = MatrixSymbol('A', 3, 3) + B = MatrixSymbol('B', 3, 3) + v = MatrixSymbol('v', 3, 1) + h = MatrixSymbol('h', 1, 3) + C = HadamardProduct(A, B) + assert julia_code(C) == "A .* B" + assert julia_code(C*v) == "(A .* B) * v" + assert julia_code(h*C*v) == "h * (A .* B) * v" + assert julia_code(C*A) == "(A .* B) * A" + # mixing Hadamard and scalar strange b/c we vectorize scalars + assert julia_code(C*x*y) == "(x .* y) * (A .* B)" + + +def test_sparse(): + M = SparseMatrix(5, 6, {}) + M[2, 2] = 10 + M[1, 2] = 20 + M[1, 3] = 22 + M[0, 3] = 30 + M[3, 0] = x*y + assert julia_code(M) == ( + "sparse([4, 2, 3, 1, 2], [1, 3, 3, 4, 4], [x .* y, 20, 10, 30, 22], 5, 6)" + ) + + +def test_specfun(): + n = Symbol('n') + for f in [besselj, bessely, besseli, besselk]: + assert julia_code(f(n, x)) == f.__name__ + '(n, x)' + for f in [airyai, airyaiprime, airybi, airybiprime]: + assert julia_code(f(x)) == f.__name__ + '(x)' + assert julia_code(hankel1(n, x)) == 'hankelh1(n, x)' + assert julia_code(hankel2(n, x)) == 'hankelh2(n, x)' + assert julia_code(jn(n, x)) == 'sqrt(2) * sqrt(pi) * sqrt(1 ./ x) .* besselj(n + 1 // 2, x) / 2' + assert julia_code(yn(n, x)) == 'sqrt(2) * sqrt(pi) * sqrt(1 ./ x) .* bessely(n + 1 // 2, x) / 2' + + +def test_MatrixElement_printing(): + # test cases for issue #11821 + A = MatrixSymbol("A", 1, 3) + B = MatrixSymbol("B", 1, 3) + C = MatrixSymbol("C", 1, 3) + + assert(julia_code(A[0, 0]) == "A[1,1]") + assert(julia_code(3 * A[0, 0]) == "3 * A[1,1]") + + F = C[0, 0].subs(C, A - B) + assert(julia_code(F) == "(A - B)[1,1]") diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_lambdarepr.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_lambdarepr.py new file mode 100644 index 0000000000000000000000000000000000000000..94e09ada7a9ce7d01667edd8fc6ec35ebfbb9639 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_lambdarepr.py @@ -0,0 +1,246 @@ +from sympy.concrete.summations import Sum +from sympy.core.expr import Expr +from sympy.core.symbol import symbols +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.elementary.trigonometric import sin +from sympy.matrices.dense import MutableDenseMatrix as Matrix +from sympy.sets.sets import Interval +from sympy.utilities.lambdify import lambdify +from sympy.testing.pytest import raises + +from sympy.printing.tensorflow import TensorflowPrinter +from sympy.printing.lambdarepr import lambdarepr, LambdaPrinter, NumExprPrinter + + +x, y, z = symbols("x,y,z") +i, a, b = symbols("i,a,b") +j, c, d = symbols("j,c,d") + + +def test_basic(): + assert lambdarepr(x*y) == "x*y" + assert lambdarepr(x + y) in ["y + x", "x + y"] + assert lambdarepr(x**y) == "x**y" + + +def test_matrix(): + # Test printing a Matrix that has an element that is printed differently + # with the LambdaPrinter than with the StrPrinter. + e = x % 2 + assert lambdarepr(e) != str(e) + assert lambdarepr(Matrix([e])) == 'ImmutableDenseMatrix([[x % 2]])' + + +def test_piecewise(): + # In each case, test eval() the lambdarepr() to make sure there are a + # correct number of parentheses. It will give a SyntaxError if there aren't. + + h = "lambda x: " + + p = Piecewise((x, x < 0)) + l = lambdarepr(p) + eval(h + l) + assert l == "((x) if (x < 0) else None)" + + p = Piecewise( + (1, x < 1), + (2, x < 2), + (0, True) + ) + l = lambdarepr(p) + eval(h + l) + assert l == "((1) if (x < 1) else (2) if (x < 2) else (0))" + + p = Piecewise( + (1, x < 1), + (2, x < 2), + ) + l = lambdarepr(p) + eval(h + l) + assert l == "((1) if (x < 1) else (2) if (x < 2) else None)" + + p = Piecewise( + (x, x < 1), + (x**2, Interval(3, 4, True, False).contains(x)), + (0, True), + ) + l = lambdarepr(p) + eval(h + l) + assert l == "((x) if (x < 1) else (x**2) if (((x <= 4)) and ((x > 3))) else (0))" + + p = Piecewise( + (x**2, x < 0), + (x, x < 1), + (2 - x, x >= 1), + (0, True), evaluate=False + ) + l = lambdarepr(p) + eval(h + l) + assert l == "((x**2) if (x < 0) else (x) if (x < 1)"\ + " else (2 - x) if (x >= 1) else (0))" + + p = Piecewise( + (x**2, x < 0), + (x, x < 1), + (2 - x, x >= 1), evaluate=False + ) + l = lambdarepr(p) + eval(h + l) + assert l == "((x**2) if (x < 0) else (x) if (x < 1)"\ + " else (2 - x) if (x >= 1) else None)" + + p = Piecewise( + (1, x >= 1), + (2, x >= 2), + (3, x >= 3), + (4, x >= 4), + (5, x >= 5), + (6, True) + ) + l = lambdarepr(p) + eval(h + l) + assert l == "((1) if (x >= 1) else (2) if (x >= 2) else (3) if (x >= 3)"\ + " else (4) if (x >= 4) else (5) if (x >= 5) else (6))" + + p = Piecewise( + (1, x <= 1), + (2, x <= 2), + (3, x <= 3), + (4, x <= 4), + (5, x <= 5), + (6, True) + ) + l = lambdarepr(p) + eval(h + l) + assert l == "((1) if (x <= 1) else (2) if (x <= 2) else (3) if (x <= 3)"\ + " else (4) if (x <= 4) else (5) if (x <= 5) else (6))" + + p = Piecewise( + (1, x > 1), + (2, x > 2), + (3, x > 3), + (4, x > 4), + (5, x > 5), + (6, True) + ) + l = lambdarepr(p) + eval(h + l) + assert l =="((1) if (x > 1) else (2) if (x > 2) else (3) if (x > 3)"\ + " else (4) if (x > 4) else (5) if (x > 5) else (6))" + + p = Piecewise( + (1, x < 1), + (2, x < 2), + (3, x < 3), + (4, x < 4), + (5, x < 5), + (6, True) + ) + l = lambdarepr(p) + eval(h + l) + assert l == "((1) if (x < 1) else (2) if (x < 2) else (3) if (x < 3)"\ + " else (4) if (x < 4) else (5) if (x < 5) else (6))" + + p = Piecewise( + (Piecewise( + (1, x > 0), + (2, True) + ), y > 0), + (3, True) + ) + l = lambdarepr(p) + eval(h + l) + assert l == "((((1) if (x > 0) else (2))) if (y > 0) else (3))" + + +def test_sum__1(): + # In each case, test eval() the lambdarepr() to make sure that + # it evaluates to the same results as the symbolic expression + s = Sum(x ** i, (i, a, b)) + l = lambdarepr(s) + assert l == "(builtins.sum(x**i for i in range(a, b+1)))" + + args = x, a, b + f = lambdify(args, s) + v = 2, 3, 8 + assert f(*v) == s.subs(zip(args, v)).doit() + +def test_sum__2(): + s = Sum(i * x, (i, a, b)) + l = lambdarepr(s) + assert l == "(builtins.sum(i*x for i in range(a, b+1)))" + + args = x, a, b + f = lambdify(args, s) + v = 2, 3, 8 + assert f(*v) == s.subs(zip(args, v)).doit() + + +def test_multiple_sums(): + s = Sum(i * x + j, (i, a, b), (j, c, d)) + + l = lambdarepr(s) + assert l == "(builtins.sum(i*x + j for j in range(c, d+1) for i in range(a, b+1)))" + + args = x, a, b, c, d + f = lambdify(args, s) + vals = 2, 3, 4, 5, 6 + f_ref = s.subs(zip(args, vals)).doit() + f_res = f(*vals) + assert f_res == f_ref + + +def test_sqrt(): + prntr = LambdaPrinter({'standard' : 'python3'}) + assert prntr._print_Pow(sqrt(x), rational=False) == 'sqrt(x)' + assert prntr._print_Pow(sqrt(x), rational=True) == 'x**(1/2)' + + +def test_settings(): + raises(TypeError, lambda: lambdarepr(sin(x), method="garbage")) + + +def test_numexpr(): + # test ITE rewrite as Piecewise + from sympy.logic.boolalg import ITE + expr = ITE(x > 0, True, False, evaluate=False) + assert NumExprPrinter().doprint(expr) == \ + "numexpr.evaluate('where((x > 0), True, False)', truediv=True)" + + from sympy.codegen.ast import Return, FunctionDefinition, Variable, Assignment + func_def = FunctionDefinition(None, 'foo', [Variable(x)], [Assignment(y,x), Return(y**2)]) + expected = "def foo(x):\n"\ + " y = numexpr.evaluate('x', truediv=True)\n"\ + " return numexpr.evaluate('y**2', truediv=True)" + assert NumExprPrinter().doprint(func_def) == expected + + +class CustomPrintedObject(Expr): + def _lambdacode(self, printer): + return 'lambda' + + def _tensorflowcode(self, printer): + return 'tensorflow' + + def _numpycode(self, printer): + return 'numpy' + + def _numexprcode(self, printer): + return 'numexpr' + + def _mpmathcode(self, printer): + return 'mpmath' + + +def test_printmethod(): + # In each case, printmethod is called to test + # its working + + obj = CustomPrintedObject() + assert LambdaPrinter().doprint(obj) == 'lambda' + assert TensorflowPrinter().doprint(obj) == 'tensorflow' + assert NumExprPrinter().doprint(obj) == "numexpr.evaluate('numexpr', truediv=True)" + + assert NumExprPrinter().doprint(Piecewise((y, x >= 0), (z, x < 0))) == \ + "numexpr.evaluate('where((x >= 0), y, z)', truediv=True)" diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_latex.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_latex.py new file mode 100644 index 0000000000000000000000000000000000000000..063611d09a923881cd94bd693f3f3f721535fd0c --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_latex.py @@ -0,0 +1,3164 @@ +from sympy import MatAdd, MatMul, Array +from sympy.algebras.quaternion import Quaternion +from sympy.calculus.accumulationbounds import AccumBounds +from sympy.combinatorics.permutations import Cycle, Permutation, AppliedPermutation +from sympy.concrete.products import Product +from sympy.concrete.summations import Sum +from sympy.core.containers import Tuple, Dict +from sympy.core.expr import UnevaluatedExpr +from sympy.core.function import (Derivative, Function, Lambda, Subs, diff) +from sympy.core.mod import Mod +from sympy.core.mul import Mul +from sympy.core.numbers import (AlgebraicNumber, Float, I, Integer, Rational, oo, pi) +from sympy.core.parameters import evaluate +from sympy.core.power import Pow +from sympy.core.relational import Eq, Ne +from sympy.core.singleton import S +from sympy.core.symbol import (Symbol, Wild, symbols) +from sympy.functions.combinatorial.factorials import (FallingFactorial, RisingFactorial, binomial, factorial, factorial2, subfactorial) +from sympy.functions.combinatorial.numbers import (bernoulli, bell, catalan, euler, genocchi, + lucas, fibonacci, tribonacci, divisor_sigma, udivisor_sigma, + mobius, primenu, primeomega, + totient, reduced_totient) +from sympy.functions.elementary.complexes import (Abs, arg, conjugate, im, polar_lift, re) +from sympy.functions.elementary.exponential import (LambertW, exp, log) +from sympy.functions.elementary.hyperbolic import (asinh, coth) +from sympy.functions.elementary.integers import (ceiling, floor, frac) +from sympy.functions.elementary.miscellaneous import (Max, Min, root, sqrt) +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.elementary.trigonometric import (acsc, asin, cos, cot, sin, tan) +from sympy.functions.special.beta_functions import beta +from sympy.functions.special.delta_functions import (DiracDelta, Heaviside) +from sympy.functions.special.elliptic_integrals import (elliptic_e, elliptic_f, elliptic_k, elliptic_pi) +from sympy.functions.special.error_functions import (Chi, Ci, Ei, Shi, Si, expint) +from sympy.functions.special.gamma_functions import (gamma, uppergamma) +from sympy.functions.special.hyper import (hyper, meijerg) +from sympy.functions.special.mathieu_functions import (mathieuc, mathieucprime, mathieus, mathieusprime) +from sympy.functions.special.polynomials import (assoc_laguerre, assoc_legendre, chebyshevt, chebyshevu, gegenbauer, hermite, jacobi, laguerre, legendre) +from sympy.functions.special.singularity_functions import SingularityFunction +from sympy.functions.special.spherical_harmonics import (Ynm, Znm) +from sympy.functions.special.tensor_functions import (KroneckerDelta, LeviCivita) +from sympy.functions.special.zeta_functions import (dirichlet_eta, lerchphi, polylog, stieltjes, zeta) +from sympy.integrals.integrals import Integral +from sympy.integrals.transforms import (CosineTransform, FourierTransform, InverseCosineTransform, InverseFourierTransform, InverseLaplaceTransform, InverseMellinTransform, InverseSineTransform, LaplaceTransform, MellinTransform, SineTransform) +from sympy.logic import Implies +from sympy.logic.boolalg import (And, Or, Xor, Equivalent, false, Not, true) +from sympy.matrices.dense import Matrix +from sympy.matrices.expressions.kronecker import KroneckerProduct +from sympy.matrices.expressions.matexpr import MatrixSymbol +from sympy.matrices.expressions.permutation import PermutationMatrix +from sympy.matrices.expressions.slice import MatrixSlice +from sympy.matrices.expressions.dotproduct import DotProduct +from sympy.physics.control.lti import TransferFunction, Series, Parallel, Feedback, TransferFunctionMatrix, MIMOSeries, MIMOParallel, MIMOFeedback +from sympy.physics.quantum import Commutator, Operator +from sympy.physics.quantum.trace import Tr +from sympy.physics.units import meter, gibibyte, gram, microgram, second, milli, micro +from sympy.polys.domains.integerring import ZZ +from sympy.polys.fields import field +from sympy.polys.polytools import Poly +from sympy.polys.rings import ring +from sympy.polys.rootoftools import (RootSum, rootof) +from sympy.series.formal import fps +from sympy.series.fourier import fourier_series +from sympy.series.limits import Limit +from sympy.series.order import Order +from sympy.series.sequences import (SeqAdd, SeqFormula, SeqMul, SeqPer) +from sympy.sets.conditionset import ConditionSet +from sympy.sets.contains import Contains +from sympy.sets.fancysets import (ComplexRegion, ImageSet, Range) +from sympy.sets.ordinals import Ordinal, OrdinalOmega, OmegaPower +from sympy.sets.powerset import PowerSet +from sympy.sets.sets import (FiniteSet, Interval, Union, Intersection, Complement, SymmetricDifference, ProductSet) +from sympy.sets.setexpr import SetExpr +from sympy.stats.crv_types import Normal +from sympy.stats.symbolic_probability import (Covariance, Expectation, + Probability, Variance) +from sympy.tensor.array import (ImmutableDenseNDimArray, + ImmutableSparseNDimArray, + MutableSparseNDimArray, + MutableDenseNDimArray, + tensorproduct) +from sympy.tensor.array.expressions.array_expressions import ArraySymbol, ArrayElement +from sympy.tensor.indexed import (Idx, Indexed, IndexedBase) +from sympy.tensor.toperators import PartialDerivative +from sympy.vector import CoordSys3D, Cross, Curl, Dot, Divergence, Gradient, Laplacian + + +from sympy.testing.pytest import (XFAIL, raises, _both_exp_pow, + warns_deprecated_sympy) +from sympy.printing.latex import (latex, translate, greek_letters_set, + tex_greek_dictionary, multiline_latex, + latex_escape, LatexPrinter) + +import sympy as sym + +from sympy.abc import mu, tau + + +class lowergamma(sym.lowergamma): + pass # testing notation inheritance by a subclass with same name + + +x, y, z, t, w, a, b, c, s, p = symbols('x y z t w a b c s p') +k, m, n = symbols('k m n', integer=True) + + +def test_printmethod(): + class R(Abs): + def _latex(self, printer): + return "foo(%s)" % printer._print(self.args[0]) + assert latex(R(x)) == r"foo(x)" + + class R(Abs): + def _latex(self, printer): + return "foo" + assert latex(R(x)) == r"foo" + + +def test_latex_basic(): + assert latex(1 + x) == r"x + 1" + assert latex(x**2) == r"x^{2}" + assert latex(x**(1 + x)) == r"x^{x + 1}" + assert latex(x**3 + x + 1 + x**2) == r"x^{3} + x^{2} + x + 1" + + assert latex(2*x*y) == r"2 x y" + assert latex(2*x*y, mul_symbol='dot') == r"2 \cdot x \cdot y" + assert latex(3*x**2*y, mul_symbol='\\,') == r"3\,x^{2}\,y" + assert latex(1.5*3**x, mul_symbol='\\,') == r"1.5 \cdot 3^{x}" + + assert latex(x**S.Half**5) == r"\sqrt[32]{x}" + assert latex(Mul(S.Half, x**2, -5, evaluate=False)) == r"\frac{1}{2} x^{2} \left(-5\right)" + assert latex(Mul(S.Half, x**2, 5, evaluate=False)) == r"\frac{1}{2} x^{2} \cdot 5" + assert latex(Mul(-5, -5, evaluate=False)) == r"\left(-5\right) \left(-5\right)" + assert latex(Mul(5, -5, evaluate=False)) == r"5 \left(-5\right)" + assert latex(Mul(S.Half, -5, S.Half, evaluate=False)) == r"\frac{1}{2} \left(-5\right) \frac{1}{2}" + assert latex(Mul(5, I, 5, evaluate=False)) == r"5 i 5" + assert latex(Mul(5, I, -5, evaluate=False)) == r"5 i \left(-5\right)" + assert latex(Mul(Pow(x, 2), S.Half*x + 1)) == r"x^{2} \left(\frac{x}{2} + 1\right)" + assert latex(Mul(Pow(x, 3), Rational(2, 3)*x + 1)) == r"x^{3} \left(\frac{2 x}{3} + 1\right)" + assert latex(Mul(Pow(x, 11), 2*x + 1)) == r"x^{11} \left(2 x + 1\right)" + + assert latex(Mul(0, 1, evaluate=False)) == r'0 \cdot 1' + assert latex(Mul(1, 0, evaluate=False)) == r'1 \cdot 0' + assert latex(Mul(1, 1, evaluate=False)) == r'1 \cdot 1' + assert latex(Mul(-1, 1, evaluate=False)) == r'\left(-1\right) 1' + assert latex(Mul(1, 1, 1, evaluate=False)) == r'1 \cdot 1 \cdot 1' + assert latex(Mul(1, 2, evaluate=False)) == r'1 \cdot 2' + assert latex(Mul(1, S.Half, evaluate=False)) == r'1 \cdot \frac{1}{2}' + assert latex(Mul(1, 1, S.Half, evaluate=False)) == \ + r'1 \cdot 1 \cdot \frac{1}{2}' + assert latex(Mul(1, 1, 2, 3, x, evaluate=False)) == \ + r'1 \cdot 1 \cdot 2 \cdot 3 x' + assert latex(Mul(1, -1, evaluate=False)) == r'1 \left(-1\right)' + assert latex(Mul(4, 3, 2, 1, 0, y, x, evaluate=False)) == \ + r'4 \cdot 3 \cdot 2 \cdot 1 \cdot 0 y x' + assert latex(Mul(4, 3, 2, 1+z, 0, y, x, evaluate=False)) == \ + r'4 \cdot 3 \cdot 2 \left(z + 1\right) 0 y x' + assert latex(Mul(Rational(2, 3), Rational(5, 7), evaluate=False)) == \ + r'\frac{2}{3} \cdot \frac{5}{7}' + + assert latex(1/x) == r"\frac{1}{x}" + assert latex(1/x, fold_short_frac=True) == r"1 / x" + assert latex(-S(3)/2) == r"- \frac{3}{2}" + assert latex(-S(3)/2, fold_short_frac=True) == r"- 3 / 2" + assert latex(1/x**2) == r"\frac{1}{x^{2}}" + assert latex(1/(x + y)/2) == r"\frac{1}{2 \left(x + y\right)}" + assert latex(x/2) == r"\frac{x}{2}" + assert latex(x/2, fold_short_frac=True) == r"x / 2" + assert latex((x + y)/(2*x)) == r"\frac{x + y}{2 x}" + assert latex((x + y)/(2*x), fold_short_frac=True) == \ + r"\left(x + y\right) / 2 x" + assert latex((x + y)/(2*x), long_frac_ratio=0) == \ + r"\frac{1}{2 x} \left(x + y\right)" + assert latex((x + y)/x) == r"\frac{x + y}{x}" + assert latex((x + y)/x, long_frac_ratio=3) == r"\frac{x + y}{x}" + assert latex((2*sqrt(2)*x)/3) == r"\frac{2 \sqrt{2} x}{3}" + assert latex((2*sqrt(2)*x)/3, long_frac_ratio=2) == \ + r"\frac{2 x}{3} \sqrt{2}" + assert latex(binomial(x, y)) == r"{\binom{x}{y}}" + + x_star = Symbol('x^*') + f = Function('f') + assert latex(x_star**2) == r"\left(x^{*}\right)^{2}" + assert latex(x_star**2, parenthesize_super=False) == r"{x^{*}}^{2}" + assert latex(Derivative(f(x_star), x_star,2)) == r"\frac{d^{2}}{d \left(x^{*}\right)^{2}} f{\left(x^{*} \right)}" + assert latex(Derivative(f(x_star), x_star,2), parenthesize_super=False) == r"\frac{d^{2}}{d {x^{*}}^{2}} f{\left(x^{*} \right)}" + + assert latex(2*Integral(x, x)/3) == r"\frac{2 \int x\, dx}{3}" + assert latex(2*Integral(x, x)/3, fold_short_frac=True) == \ + r"\left(2 \int x\, dx\right) / 3" + + assert latex(sqrt(x)) == r"\sqrt{x}" + assert latex(x**Rational(1, 3)) == r"\sqrt[3]{x}" + assert latex(x**Rational(1, 3), root_notation=False) == r"x^{\frac{1}{3}}" + assert latex(sqrt(x)**3) == r"x^{\frac{3}{2}}" + assert latex(sqrt(x), itex=True) == r"\sqrt{x}" + assert latex(x**Rational(1, 3), itex=True) == r"\root{3}{x}" + assert latex(sqrt(x)**3, itex=True) == r"x^{\frac{3}{2}}" + assert latex(x**Rational(3, 4)) == r"x^{\frac{3}{4}}" + assert latex(x**Rational(3, 4), fold_frac_powers=True) == r"x^{3/4}" + assert latex((x + 1)**Rational(3, 4)) == \ + r"\left(x + 1\right)^{\frac{3}{4}}" + assert latex((x + 1)**Rational(3, 4), fold_frac_powers=True) == \ + r"\left(x + 1\right)^{3/4}" + assert latex(AlgebraicNumber(sqrt(2))) == r"\sqrt{2}" + assert latex(AlgebraicNumber(sqrt(2), [3, -7])) == r"-7 + 3 \sqrt{2}" + assert latex(AlgebraicNumber(sqrt(2), alias='alpha')) == r"\alpha" + assert latex(AlgebraicNumber(sqrt(2), [3, -7], alias='alpha')) == \ + r"3 \alpha - 7" + assert latex(AlgebraicNumber(2**(S(1)/3), [1, 3, -7], alias='beta')) == \ + r"\beta^{2} + 3 \beta - 7" + + k = ZZ.cyclotomic_field(5) + assert latex(k.ext.field_element([1, 2, 3, 4])) == \ + r"\zeta^{3} + 2 \zeta^{2} + 3 \zeta + 4" + assert latex(k.ext.field_element([1, 2, 3, 4]), order='old') == \ + r"4 + 3 \zeta + 2 \zeta^{2} + \zeta^{3}" + assert latex(k.primes_above(19)[0]) == \ + r"\left(19, \zeta^{2} + 5 \zeta + 1\right)" + assert latex(k.primes_above(19)[0], order='old') == \ + r"\left(19, 1 + 5 \zeta + \zeta^{2}\right)" + assert latex(k.primes_above(7)[0]) == r"\left(7\right)" + + assert latex(1.5e20*x) == r"1.5 \cdot 10^{20} x" + assert latex(1.5e20*x, mul_symbol='dot') == r"1.5 \cdot 10^{20} \cdot x" + assert latex(1.5e20*x, mul_symbol='times') == \ + r"1.5 \times 10^{20} \times x" + + assert latex(1/sin(x)) == r"\frac{1}{\sin{\left(x \right)}}" + assert latex(sin(x)**-1) == r"\frac{1}{\sin{\left(x \right)}}" + assert latex(sin(x)**Rational(3, 2)) == \ + r"\sin^{\frac{3}{2}}{\left(x \right)}" + assert latex(sin(x)**Rational(3, 2), fold_frac_powers=True) == \ + r"\sin^{3/2}{\left(x \right)}" + + assert latex(~x) == r"\neg x" + assert latex(x & y) == r"x \wedge y" + assert latex(x & y & z) == r"x \wedge y \wedge z" + assert latex(x | y) == r"x \vee y" + assert latex(x | y | z) == r"x \vee y \vee z" + assert latex((x & y) | z) == r"z \vee \left(x \wedge y\right)" + assert latex(Implies(x, y)) == r"x \Rightarrow y" + assert latex(~(x >> ~y)) == r"x \not\Rightarrow \neg y" + assert latex(Implies(Or(x,y), z)) == r"\left(x \vee y\right) \Rightarrow z" + assert latex(Implies(z, Or(x,y))) == r"z \Rightarrow \left(x \vee y\right)" + assert latex(~(x & y)) == r"\neg \left(x \wedge y\right)" + + assert latex(~x, symbol_names={x: "x_i"}) == r"\neg x_i" + assert latex(x & y, symbol_names={x: "x_i", y: "y_i"}) == \ + r"x_i \wedge y_i" + assert latex(x & y & z, symbol_names={x: "x_i", y: "y_i", z: "z_i"}) == \ + r"x_i \wedge y_i \wedge z_i" + assert latex(x | y, symbol_names={x: "x_i", y: "y_i"}) == r"x_i \vee y_i" + assert latex(x | y | z, symbol_names={x: "x_i", y: "y_i", z: "z_i"}) == \ + r"x_i \vee y_i \vee z_i" + assert latex((x & y) | z, symbol_names={x: "x_i", y: "y_i", z: "z_i"}) == \ + r"z_i \vee \left(x_i \wedge y_i\right)" + assert latex(Implies(x, y), symbol_names={x: "x_i", y: "y_i"}) == \ + r"x_i \Rightarrow y_i" + assert latex(Pow(Rational(1, 3), -1, evaluate=False)) == r"\frac{1}{\frac{1}{3}}" + assert latex(Pow(Rational(1, 3), -2, evaluate=False)) == r"\frac{1}{(\frac{1}{3})^{2}}" + assert latex(Pow(Integer(1)/100, -1, evaluate=False)) == r"\frac{1}{\frac{1}{100}}" + + p = Symbol('p', positive=True) + assert latex(exp(-p)*log(p)) == r"e^{- p} \log{\left(p \right)}" + + assert latex(Pow(Rational(2, 3), -1, evaluate=False)) == r'\frac{1}{\frac{2}{3}}' + assert latex(Pow(Rational(4, 3), -1, evaluate=False)) == r'\frac{1}{\frac{4}{3}}' + assert latex(Pow(Rational(-3, 4), -1, evaluate=False)) == r'\frac{1}{- \frac{3}{4}}' + assert latex(Pow(Rational(-4, 4), -1, evaluate=False)) == r'\frac{1}{-1}' + assert latex(Pow(Rational(1, 3), -1, evaluate=False)) == r'\frac{1}{\frac{1}{3}}' + assert latex(Pow(Rational(-1, 3), -1, evaluate=False)) == r'\frac{1}{- \frac{1}{3}}' + + +def test_latex_builtins(): + assert latex(True) == r"\text{True}" + assert latex(False) == r"\text{False}" + assert latex(None) == r"\text{None}" + assert latex(true) == r"\text{True}" + assert latex(false) == r'\text{False}' + + +def test_latex_SingularityFunction(): + assert latex(SingularityFunction(x, 4, 5)) == \ + r"{\left\langle x - 4 \right\rangle}^{5}" + assert latex(SingularityFunction(x, -3, 4)) == \ + r"{\left\langle x + 3 \right\rangle}^{4}" + assert latex(SingularityFunction(x, 0, 4)) == \ + r"{\left\langle x \right\rangle}^{4}" + assert latex(SingularityFunction(x, a, n)) == \ + r"{\left\langle - a + x \right\rangle}^{n}" + assert latex(SingularityFunction(x, 4, -2)) == \ + r"{\left\langle x - 4 \right\rangle}^{-2}" + assert latex(SingularityFunction(x, 4, -1)) == \ + r"{\left\langle x - 4 \right\rangle}^{-1}" + + assert latex(SingularityFunction(x, 4, 5)**3) == \ + r"{\left({\langle x - 4 \rangle}^{5}\right)}^{3}" + assert latex(SingularityFunction(x, -3, 4)**3) == \ + r"{\left({\langle x + 3 \rangle}^{4}\right)}^{3}" + assert latex(SingularityFunction(x, 0, 4)**3) == \ + r"{\left({\langle x \rangle}^{4}\right)}^{3}" + assert latex(SingularityFunction(x, a, n)**3) == \ + r"{\left({\langle - a + x \rangle}^{n}\right)}^{3}" + assert latex(SingularityFunction(x, 4, -2)**3) == \ + r"{\left({\langle x - 4 \rangle}^{-2}\right)}^{3}" + assert latex((SingularityFunction(x, 4, -1)**3)**3) == \ + r"{\left({\langle x - 4 \rangle}^{-1}\right)}^{9}" + + +def test_latex_cycle(): + assert latex(Cycle(1, 2, 4)) == r"\left( 1\; 2\; 4\right)" + assert latex(Cycle(1, 2)(4, 5, 6)) == \ + r"\left( 1\; 2\right)\left( 4\; 5\; 6\right)" + assert latex(Cycle()) == r"\left( \right)" + + +def test_latex_permutation(): + assert latex(Permutation(1, 2, 4)) == r"\left( 1\; 2\; 4\right)" + assert latex(Permutation(1, 2)(4, 5, 6)) == \ + r"\left( 1\; 2\right)\left( 4\; 5\; 6\right)" + assert latex(Permutation()) == r"\left( \right)" + assert latex(Permutation(2, 4)*Permutation(5)) == \ + r"\left( 2\; 4\right)\left( 5\right)" + assert latex(Permutation(5)) == r"\left( 5\right)" + + assert latex(Permutation(0, 1), perm_cyclic=False) == \ + r"\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}" + assert latex(Permutation(0, 1)(2, 3), perm_cyclic=False) == \ + r"\begin{pmatrix} 0 & 1 & 2 & 3 \\ 1 & 0 & 3 & 2 \end{pmatrix}" + assert latex(Permutation(), perm_cyclic=False) == \ + r"\left( \right)" + + with warns_deprecated_sympy(): + old_print_cyclic = Permutation.print_cyclic + Permutation.print_cyclic = False + assert latex(Permutation(0, 1)(2, 3)) == \ + r"\begin{pmatrix} 0 & 1 & 2 & 3 \\ 1 & 0 & 3 & 2 \end{pmatrix}" + Permutation.print_cyclic = old_print_cyclic + +def test_latex_Float(): + assert latex(Float(1.0e100)) == r"1.0 \cdot 10^{100}" + assert latex(Float(1.0e-100)) == r"1.0 \cdot 10^{-100}" + assert latex(Float(1.0e-100), mul_symbol="times") == \ + r"1.0 \times 10^{-100}" + assert latex(Float('10000.0'), full_prec=False, min=-2, max=2) == \ + r"1.0 \cdot 10^{4}" + assert latex(Float('10000.0'), full_prec=False, min=-2, max=4) == \ + r"1.0 \cdot 10^{4}" + assert latex(Float('10000.0'), full_prec=False, min=-2, max=5) == \ + r"10000.0" + assert latex(Float('0.099999'), full_prec=True, min=-2, max=5) == \ + r"9.99990000000000 \cdot 10^{-2}" + + +def test_latex_vector_expressions(): + A = CoordSys3D('A') + + assert latex(Cross(A.i, A.j*A.x*3+A.k)) == \ + r"\mathbf{\hat{i}_{A}} \times \left(\left(3 \mathbf{{x}_{A}}\right)\mathbf{\hat{j}_{A}} + \mathbf{\hat{k}_{A}}\right)" + assert latex(Cross(A.i, A.j)) == \ + r"\mathbf{\hat{i}_{A}} \times \mathbf{\hat{j}_{A}}" + assert latex(x*Cross(A.i, A.j)) == \ + r"x \left(\mathbf{\hat{i}_{A}} \times \mathbf{\hat{j}_{A}}\right)" + assert latex(Cross(x*A.i, A.j)) == \ + r'- \mathbf{\hat{j}_{A}} \times \left(\left(x\right)\mathbf{\hat{i}_{A}}\right)' + + assert latex(Curl(3*A.x*A.j)) == \ + r"\nabla\times \left(\left(3 \mathbf{{x}_{A}}\right)\mathbf{\hat{j}_{A}}\right)" + assert latex(Curl(3*A.x*A.j+A.i)) == \ + r"\nabla\times \left(\mathbf{\hat{i}_{A}} + \left(3 \mathbf{{x}_{A}}\right)\mathbf{\hat{j}_{A}}\right)" + assert latex(Curl(3*x*A.x*A.j)) == \ + r"\nabla\times \left(\left(3 \mathbf{{x}_{A}} x\right)\mathbf{\hat{j}_{A}}\right)" + assert latex(x*Curl(3*A.x*A.j)) == \ + r"x \left(\nabla\times \left(\left(3 \mathbf{{x}_{A}}\right)\mathbf{\hat{j}_{A}}\right)\right)" + + assert latex(Divergence(3*A.x*A.j+A.i)) == \ + r"\nabla\cdot \left(\mathbf{\hat{i}_{A}} + \left(3 \mathbf{{x}_{A}}\right)\mathbf{\hat{j}_{A}}\right)" + assert latex(Divergence(3*A.x*A.j)) == \ + r"\nabla\cdot \left(\left(3 \mathbf{{x}_{A}}\right)\mathbf{\hat{j}_{A}}\right)" + assert latex(x*Divergence(3*A.x*A.j)) == \ + r"x \left(\nabla\cdot \left(\left(3 \mathbf{{x}_{A}}\right)\mathbf{\hat{j}_{A}}\right)\right)" + + assert latex(Dot(A.i, A.j*A.x*3+A.k)) == \ + r"\mathbf{\hat{i}_{A}} \cdot \left(\left(3 \mathbf{{x}_{A}}\right)\mathbf{\hat{j}_{A}} + \mathbf{\hat{k}_{A}}\right)" + assert latex(Dot(A.i, A.j)) == \ + r"\mathbf{\hat{i}_{A}} \cdot \mathbf{\hat{j}_{A}}" + assert latex(Dot(x*A.i, A.j)) == \ + r"\mathbf{\hat{j}_{A}} \cdot \left(\left(x\right)\mathbf{\hat{i}_{A}}\right)" + assert latex(x*Dot(A.i, A.j)) == \ + r"x \left(\mathbf{\hat{i}_{A}} \cdot \mathbf{\hat{j}_{A}}\right)" + + assert latex(Gradient(A.x)) == r"\nabla \mathbf{{x}_{A}}" + assert latex(Gradient(A.x + 3*A.y)) == \ + r"\nabla \left(\mathbf{{x}_{A}} + 3 \mathbf{{y}_{A}}\right)" + assert latex(x*Gradient(A.x)) == r"x \left(\nabla \mathbf{{x}_{A}}\right)" + assert latex(Gradient(x*A.x)) == r"\nabla \left(\mathbf{{x}_{A}} x\right)" + + assert latex(Laplacian(A.x)) == r"\Delta \mathbf{{x}_{A}}" + assert latex(Laplacian(A.x + 3*A.y)) == \ + r"\Delta \left(\mathbf{{x}_{A}} + 3 \mathbf{{y}_{A}}\right)" + assert latex(x*Laplacian(A.x)) == r"x \left(\Delta \mathbf{{x}_{A}}\right)" + assert latex(Laplacian(x*A.x)) == r"\Delta \left(\mathbf{{x}_{A}} x\right)" + +def test_latex_symbols(): + Gamma, lmbda, rho = symbols('Gamma, lambda, rho') + tau, Tau, TAU, taU = symbols('tau, Tau, TAU, taU') + assert latex(tau) == r"\tau" + assert latex(Tau) == r"\mathrm{T}" + assert latex(TAU) == r"\tau" + assert latex(taU) == r"\tau" + # Check that all capitalized greek letters are handled explicitly + capitalized_letters = {l.capitalize() for l in greek_letters_set} + assert len(capitalized_letters - set(tex_greek_dictionary.keys())) == 0 + assert latex(Gamma + lmbda) == r"\Gamma + \lambda" + assert latex(Gamma * lmbda) == r"\Gamma \lambda" + assert latex(Symbol('q1')) == r"q_{1}" + assert latex(Symbol('q21')) == r"q_{21}" + assert latex(Symbol('epsilon0')) == r"\epsilon_{0}" + assert latex(Symbol('omega1')) == r"\omega_{1}" + assert latex(Symbol('91')) == r"91" + assert latex(Symbol('alpha_new')) == r"\alpha_{new}" + assert latex(Symbol('C^orig')) == r"C^{orig}" + assert latex(Symbol('x^alpha')) == r"x^{\alpha}" + assert latex(Symbol('beta^alpha')) == r"\beta^{\alpha}" + assert latex(Symbol('e^Alpha')) == r"e^{\mathrm{A}}" + assert latex(Symbol('omega_alpha^beta')) == r"\omega^{\beta}_{\alpha}" + assert latex(Symbol('omega') ** Symbol('beta')) == r"\omega^{\beta}" + + +@XFAIL +def test_latex_symbols_failing(): + rho, mass, volume = symbols('rho, mass, volume') + assert latex( + volume * rho == mass) == r"\rho \mathrm{volume} = \mathrm{mass}" + assert latex(volume / mass * rho == 1) == \ + r"\rho \mathrm{volume} {\mathrm{mass}}^{(-1)} = 1" + assert latex(mass**3 * volume**3) == \ + r"{\mathrm{mass}}^{3} \cdot {\mathrm{volume}}^{3}" + + +@_both_exp_pow +def test_latex_functions(): + assert latex(exp(x)) == r"e^{x}" + assert latex(exp(1) + exp(2)) == r"e + e^{2}" + + f = Function('f') + assert latex(f(x)) == r'f{\left(x \right)}' + assert latex(f) == r'f' + + g = Function('g') + assert latex(g(x, y)) == r'g{\left(x,y \right)}' + assert latex(g) == r'g' + + h = Function('h') + assert latex(h(x, y, z)) == r'h{\left(x,y,z \right)}' + assert latex(h) == r'h' + + Li = Function('Li') + assert latex(Li) == r'\operatorname{Li}' + assert latex(Li(x)) == r'\operatorname{Li}{\left(x \right)}' + + mybeta = Function('beta') + # not to be confused with the beta function + assert latex(mybeta(x, y, z)) == r"\beta{\left(x,y,z \right)}" + assert latex(beta(x, y)) == r'\operatorname{B}\left(x, y\right)' + assert latex(beta(x, evaluate=False)) == r'\operatorname{B}\left(x, x\right)' + assert latex(beta(x, y)**2) == r'\operatorname{B}^{2}\left(x, y\right)' + assert latex(mybeta(x)) == r"\beta{\left(x \right)}" + assert latex(mybeta) == r"\beta" + + g = Function('gamma') + # not to be confused with the gamma function + assert latex(g(x, y, z)) == r"\gamma{\left(x,y,z \right)}" + assert latex(g(x)) == r"\gamma{\left(x \right)}" + assert latex(g) == r"\gamma" + + a_1 = Function('a_1') + assert latex(a_1) == r"a_{1}" + assert latex(a_1(x)) == r"a_{1}{\left(x \right)}" + assert latex(Function('a_1')) == r"a_{1}" + + # Issue #16925 + # multi letter function names + # > simple + assert latex(Function('ab')) == r"\operatorname{ab}" + assert latex(Function('ab1')) == r"\operatorname{ab}_{1}" + assert latex(Function('ab12')) == r"\operatorname{ab}_{12}" + assert latex(Function('ab_1')) == r"\operatorname{ab}_{1}" + assert latex(Function('ab_12')) == r"\operatorname{ab}_{12}" + assert latex(Function('ab_c')) == r"\operatorname{ab}_{c}" + assert latex(Function('ab_cd')) == r"\operatorname{ab}_{cd}" + # > with argument + assert latex(Function('ab')(Symbol('x'))) == r"\operatorname{ab}{\left(x \right)}" + assert latex(Function('ab1')(Symbol('x'))) == r"\operatorname{ab}_{1}{\left(x \right)}" + assert latex(Function('ab12')(Symbol('x'))) == r"\operatorname{ab}_{12}{\left(x \right)}" + assert latex(Function('ab_1')(Symbol('x'))) == r"\operatorname{ab}_{1}{\left(x \right)}" + assert latex(Function('ab_c')(Symbol('x'))) == r"\operatorname{ab}_{c}{\left(x \right)}" + assert latex(Function('ab_cd')(Symbol('x'))) == r"\operatorname{ab}_{cd}{\left(x \right)}" + + # > with power + # does not work on functions without brackets + + # > with argument and power combined + assert latex(Function('ab')()**2) == r"\operatorname{ab}^{2}{\left( \right)}" + assert latex(Function('ab1')()**2) == r"\operatorname{ab}_{1}^{2}{\left( \right)}" + assert latex(Function('ab12')()**2) == r"\operatorname{ab}_{12}^{2}{\left( \right)}" + assert latex(Function('ab_1')()**2) == r"\operatorname{ab}_{1}^{2}{\left( \right)}" + assert latex(Function('ab_12')()**2) == r"\operatorname{ab}_{12}^{2}{\left( \right)}" + assert latex(Function('ab')(Symbol('x'))**2) == r"\operatorname{ab}^{2}{\left(x \right)}" + assert latex(Function('ab1')(Symbol('x'))**2) == r"\operatorname{ab}_{1}^{2}{\left(x \right)}" + assert latex(Function('ab12')(Symbol('x'))**2) == r"\operatorname{ab}_{12}^{2}{\left(x \right)}" + assert latex(Function('ab_1')(Symbol('x'))**2) == r"\operatorname{ab}_{1}^{2}{\left(x \right)}" + assert latex(Function('ab_12')(Symbol('x'))**2) == \ + r"\operatorname{ab}_{12}^{2}{\left(x \right)}" + + # single letter function names + # > simple + assert latex(Function('a')) == r"a" + assert latex(Function('a1')) == r"a_{1}" + assert latex(Function('a12')) == r"a_{12}" + assert latex(Function('a_1')) == r"a_{1}" + assert latex(Function('a_12')) == r"a_{12}" + + # > with argument + assert latex(Function('a')()) == r"a{\left( \right)}" + assert latex(Function('a1')()) == r"a_{1}{\left( \right)}" + assert latex(Function('a12')()) == r"a_{12}{\left( \right)}" + assert latex(Function('a_1')()) == r"a_{1}{\left( \right)}" + assert latex(Function('a_12')()) == r"a_{12}{\left( \right)}" + + # > with power + # does not work on functions without brackets + + # > with argument and power combined + assert latex(Function('a')()**2) == r"a^{2}{\left( \right)}" + assert latex(Function('a1')()**2) == r"a_{1}^{2}{\left( \right)}" + assert latex(Function('a12')()**2) == r"a_{12}^{2}{\left( \right)}" + assert latex(Function('a_1')()**2) == r"a_{1}^{2}{\left( \right)}" + assert latex(Function('a_12')()**2) == r"a_{12}^{2}{\left( \right)}" + assert latex(Function('a')(Symbol('x'))**2) == r"a^{2}{\left(x \right)}" + assert latex(Function('a1')(Symbol('x'))**2) == r"a_{1}^{2}{\left(x \right)}" + assert latex(Function('a12')(Symbol('x'))**2) == r"a_{12}^{2}{\left(x \right)}" + assert latex(Function('a_1')(Symbol('x'))**2) == r"a_{1}^{2}{\left(x \right)}" + assert latex(Function('a_12')(Symbol('x'))**2) == r"a_{12}^{2}{\left(x \right)}" + + assert latex(Function('a')()**32) == r"a^{32}{\left( \right)}" + assert latex(Function('a1')()**32) == r"a_{1}^{32}{\left( \right)}" + assert latex(Function('a12')()**32) == r"a_{12}^{32}{\left( \right)}" + assert latex(Function('a_1')()**32) == r"a_{1}^{32}{\left( \right)}" + assert latex(Function('a_12')()**32) == r"a_{12}^{32}{\left( \right)}" + assert latex(Function('a')(Symbol('x'))**32) == r"a^{32}{\left(x \right)}" + assert latex(Function('a1')(Symbol('x'))**32) == r"a_{1}^{32}{\left(x \right)}" + assert latex(Function('a12')(Symbol('x'))**32) == r"a_{12}^{32}{\left(x \right)}" + assert latex(Function('a_1')(Symbol('x'))**32) == r"a_{1}^{32}{\left(x \right)}" + assert latex(Function('a_12')(Symbol('x'))**32) == r"a_{12}^{32}{\left(x \right)}" + + assert latex(Function('a')()**a) == r"a^{a}{\left( \right)}" + assert latex(Function('a1')()**a) == r"a_{1}^{a}{\left( \right)}" + assert latex(Function('a12')()**a) == r"a_{12}^{a}{\left( \right)}" + assert latex(Function('a_1')()**a) == r"a_{1}^{a}{\left( \right)}" + assert latex(Function('a_12')()**a) == r"a_{12}^{a}{\left( \right)}" + assert latex(Function('a')(Symbol('x'))**a) == r"a^{a}{\left(x \right)}" + assert latex(Function('a1')(Symbol('x'))**a) == r"a_{1}^{a}{\left(x \right)}" + assert latex(Function('a12')(Symbol('x'))**a) == r"a_{12}^{a}{\left(x \right)}" + assert latex(Function('a_1')(Symbol('x'))**a) == r"a_{1}^{a}{\left(x \right)}" + assert latex(Function('a_12')(Symbol('x'))**a) == r"a_{12}^{a}{\left(x \right)}" + + ab = Symbol('ab') + assert latex(Function('a')()**ab) == r"a^{ab}{\left( \right)}" + assert latex(Function('a1')()**ab) == r"a_{1}^{ab}{\left( \right)}" + assert latex(Function('a12')()**ab) == r"a_{12}^{ab}{\left( \right)}" + assert latex(Function('a_1')()**ab) == r"a_{1}^{ab}{\left( \right)}" + assert latex(Function('a_12')()**ab) == r"a_{12}^{ab}{\left( \right)}" + assert latex(Function('a')(Symbol('x'))**ab) == r"a^{ab}{\left(x \right)}" + assert latex(Function('a1')(Symbol('x'))**ab) == r"a_{1}^{ab}{\left(x \right)}" + assert latex(Function('a12')(Symbol('x'))**ab) == r"a_{12}^{ab}{\left(x \right)}" + assert latex(Function('a_1')(Symbol('x'))**ab) == r"a_{1}^{ab}{\left(x \right)}" + assert latex(Function('a_12')(Symbol('x'))**ab) == r"a_{12}^{ab}{\left(x \right)}" + + assert latex(Function('a^12')(x)) == R"a^{12}{\left(x \right)}" + assert latex(Function('a^12')(x) ** ab) == R"\left(a^{12}\right)^{ab}{\left(x \right)}" + assert latex(Function('a__12')(x)) == R"a^{12}{\left(x \right)}" + assert latex(Function('a__12')(x) ** ab) == R"\left(a^{12}\right)^{ab}{\left(x \right)}" + assert latex(Function('a_1__1_2')(x)) == R"a^{1}_{1 2}{\left(x \right)}" + + # issue 5868 + omega1 = Function('omega1') + assert latex(omega1) == r"\omega_{1}" + assert latex(omega1(x)) == r"\omega_{1}{\left(x \right)}" + + assert latex(sin(x)) == r"\sin{\left(x \right)}" + assert latex(sin(x), fold_func_brackets=True) == r"\sin {x}" + assert latex(sin(2*x**2), fold_func_brackets=True) == \ + r"\sin {2 x^{2}}" + assert latex(sin(x**2), fold_func_brackets=True) == \ + r"\sin {x^{2}}" + + assert latex(asin(x)**2) == r"\operatorname{asin}^{2}{\left(x \right)}" + assert latex(asin(x)**2, inv_trig_style="full") == \ + r"\arcsin^{2}{\left(x \right)}" + assert latex(asin(x)**2, inv_trig_style="power") == \ + r"\sin^{-1}{\left(x \right)}^{2}" + assert latex(asin(x**2), inv_trig_style="power", + fold_func_brackets=True) == \ + r"\sin^{-1} {x^{2}}" + assert latex(acsc(x), inv_trig_style="full") == \ + r"\operatorname{arccsc}{\left(x \right)}" + assert latex(asinh(x), inv_trig_style="full") == \ + r"\operatorname{arsinh}{\left(x \right)}" + + assert latex(factorial(k)) == r"k!" + assert latex(factorial(-k)) == r"\left(- k\right)!" + assert latex(factorial(k)**2) == r"k!^{2}" + + assert latex(subfactorial(k)) == r"!k" + assert latex(subfactorial(-k)) == r"!\left(- k\right)" + assert latex(subfactorial(k)**2) == r"\left(!k\right)^{2}" + + assert latex(factorial2(k)) == r"k!!" + assert latex(factorial2(-k)) == r"\left(- k\right)!!" + assert latex(factorial2(k)**2) == r"k!!^{2}" + + assert latex(binomial(2, k)) == r"{\binom{2}{k}}" + assert latex(binomial(2, k)**2) == r"{\binom{2}{k}}^{2}" + + assert latex(FallingFactorial(3, k)) == r"{\left(3\right)}_{k}" + assert latex(RisingFactorial(3, k)) == r"{3}^{\left(k\right)}" + + assert latex(floor(x)) == r"\left\lfloor{x}\right\rfloor" + assert latex(ceiling(x)) == r"\left\lceil{x}\right\rceil" + assert latex(frac(x)) == r"\operatorname{frac}{\left(x\right)}" + assert latex(floor(x)**2) == r"\left\lfloor{x}\right\rfloor^{2}" + assert latex(ceiling(x)**2) == r"\left\lceil{x}\right\rceil^{2}" + assert latex(frac(x)**2) == r"\operatorname{frac}{\left(x\right)}^{2}" + + assert latex(Min(x, 2, x**3)) == r"\min\left(2, x, x^{3}\right)" + assert latex(Min(x, y)**2) == r"\min\left(x, y\right)^{2}" + assert latex(Max(x, 2, x**3)) == r"\max\left(2, x, x^{3}\right)" + assert latex(Max(x, y)**2) == r"\max\left(x, y\right)^{2}" + assert latex(Abs(x)) == r"\left|{x}\right|" + assert latex(Abs(x)**2) == r"\left|{x}\right|^{2}" + assert latex(re(x)) == r"\operatorname{re}{\left(x\right)}" + assert latex(re(x + y)) == \ + r"\operatorname{re}{\left(x\right)} + \operatorname{re}{\left(y\right)}" + assert latex(im(x)) == r"\operatorname{im}{\left(x\right)}" + assert latex(conjugate(x)) == r"\overline{x}" + assert latex(conjugate(x)**2) == r"\overline{x}^{2}" + assert latex(conjugate(x**2)) == r"\overline{x}^{2}" + assert latex(gamma(x)) == r"\Gamma\left(x\right)" + w = Wild('w') + assert latex(gamma(w)) == r"\Gamma\left(w\right)" + assert latex(Order(x)) == r"O\left(x\right)" + assert latex(Order(x, x)) == r"O\left(x\right)" + assert latex(Order(x, (x, 0))) == r"O\left(x\right)" + assert latex(Order(x, (x, oo))) == r"O\left(x; x\rightarrow \infty\right)" + assert latex(Order(x - y, (x, y))) == \ + r"O\left(x - y; x\rightarrow y\right)" + assert latex(Order(x, x, y)) == \ + r"O\left(x; \left( x, \ y\right)\rightarrow \left( 0, \ 0\right)\right)" + assert latex(Order(x, x, y)) == \ + r"O\left(x; \left( x, \ y\right)\rightarrow \left( 0, \ 0\right)\right)" + assert latex(Order(x, (x, oo), (y, oo))) == \ + r"O\left(x; \left( x, \ y\right)\rightarrow \left( \infty, \ \infty\right)\right)" + assert latex(lowergamma(x, y)) == r'\gamma\left(x, y\right)' + assert latex(lowergamma(x, y)**2) == r'\gamma^{2}\left(x, y\right)' + assert latex(uppergamma(x, y)) == r'\Gamma\left(x, y\right)' + assert latex(uppergamma(x, y)**2) == r'\Gamma^{2}\left(x, y\right)' + + assert latex(cot(x)) == r'\cot{\left(x \right)}' + assert latex(coth(x)) == r'\coth{\left(x \right)}' + assert latex(re(x)) == r'\operatorname{re}{\left(x\right)}' + assert latex(im(x)) == r'\operatorname{im}{\left(x\right)}' + assert latex(root(x, y)) == r'x^{\frac{1}{y}}' + assert latex(arg(x)) == r'\arg{\left(x \right)}' + + assert latex(zeta(x)) == r"\zeta\left(x\right)" + assert latex(zeta(x)**2) == r"\zeta^{2}\left(x\right)" + assert latex(zeta(x, y)) == r"\zeta\left(x, y\right)" + assert latex(zeta(x, y)**2) == r"\zeta^{2}\left(x, y\right)" + assert latex(dirichlet_eta(x)) == r"\eta\left(x\right)" + assert latex(dirichlet_eta(x)**2) == r"\eta^{2}\left(x\right)" + assert latex(polylog(x, y)) == r"\operatorname{Li}_{x}\left(y\right)" + assert latex( + polylog(x, y)**2) == r"\operatorname{Li}_{x}^{2}\left(y\right)" + assert latex(lerchphi(x, y, n)) == r"\Phi\left(x, y, n\right)" + assert latex(lerchphi(x, y, n)**2) == r"\Phi^{2}\left(x, y, n\right)" + assert latex(stieltjes(x)) == r"\gamma_{x}" + assert latex(stieltjes(x)**2) == r"\gamma_{x}^{2}" + assert latex(stieltjes(x, y)) == r"\gamma_{x}\left(y\right)" + assert latex(stieltjes(x, y)**2) == r"\gamma_{x}\left(y\right)^{2}" + + assert latex(elliptic_k(z)) == r"K\left(z\right)" + assert latex(elliptic_k(z)**2) == r"K^{2}\left(z\right)" + assert latex(elliptic_f(x, y)) == r"F\left(x\middle| y\right)" + assert latex(elliptic_f(x, y)**2) == r"F^{2}\left(x\middle| y\right)" + assert latex(elliptic_e(x, y)) == r"E\left(x\middle| y\right)" + assert latex(elliptic_e(x, y)**2) == r"E^{2}\left(x\middle| y\right)" + assert latex(elliptic_e(z)) == r"E\left(z\right)" + assert latex(elliptic_e(z)**2) == r"E^{2}\left(z\right)" + assert latex(elliptic_pi(x, y, z)) == r"\Pi\left(x; y\middle| z\right)" + assert latex(elliptic_pi(x, y, z)**2) == \ + r"\Pi^{2}\left(x; y\middle| z\right)" + assert latex(elliptic_pi(x, y)) == r"\Pi\left(x\middle| y\right)" + assert latex(elliptic_pi(x, y)**2) == r"\Pi^{2}\left(x\middle| y\right)" + + assert latex(Ei(x)) == r'\operatorname{Ei}{\left(x \right)}' + assert latex(Ei(x)**2) == r'\operatorname{Ei}^{2}{\left(x \right)}' + assert latex(expint(x, y)) == r'\operatorname{E}_{x}\left(y\right)' + assert latex(expint(x, y)**2) == r'\operatorname{E}_{x}^{2}\left(y\right)' + assert latex(Shi(x)**2) == r'\operatorname{Shi}^{2}{\left(x \right)}' + assert latex(Si(x)**2) == r'\operatorname{Si}^{2}{\left(x \right)}' + assert latex(Ci(x)**2) == r'\operatorname{Ci}^{2}{\left(x \right)}' + assert latex(Chi(x)**2) == r'\operatorname{Chi}^{2}\left(x\right)' + assert latex(Chi(x)) == r'\operatorname{Chi}\left(x\right)' + assert latex(jacobi(n, a, b, x)) == \ + r'P_{n}^{\left(a,b\right)}\left(x\right)' + assert latex(jacobi(n, a, b, x)**2) == \ + r'\left(P_{n}^{\left(a,b\right)}\left(x\right)\right)^{2}' + assert latex(gegenbauer(n, a, x)) == \ + r'C_{n}^{\left(a\right)}\left(x\right)' + assert latex(gegenbauer(n, a, x)**2) == \ + r'\left(C_{n}^{\left(a\right)}\left(x\right)\right)^{2}' + assert latex(chebyshevt(n, x)) == r'T_{n}\left(x\right)' + assert latex(chebyshevt(n, x)**2) == \ + r'\left(T_{n}\left(x\right)\right)^{2}' + assert latex(chebyshevu(n, x)) == r'U_{n}\left(x\right)' + assert latex(chebyshevu(n, x)**2) == \ + r'\left(U_{n}\left(x\right)\right)^{2}' + assert latex(legendre(n, x)) == r'P_{n}\left(x\right)' + assert latex(legendre(n, x)**2) == r'\left(P_{n}\left(x\right)\right)^{2}' + assert latex(assoc_legendre(n, a, x)) == \ + r'P_{n}^{\left(a\right)}\left(x\right)' + assert latex(assoc_legendre(n, a, x)**2) == \ + r'\left(P_{n}^{\left(a\right)}\left(x\right)\right)^{2}' + assert latex(laguerre(n, x)) == r'L_{n}\left(x\right)' + assert latex(laguerre(n, x)**2) == r'\left(L_{n}\left(x\right)\right)^{2}' + assert latex(assoc_laguerre(n, a, x)) == \ + r'L_{n}^{\left(a\right)}\left(x\right)' + assert latex(assoc_laguerre(n, a, x)**2) == \ + r'\left(L_{n}^{\left(a\right)}\left(x\right)\right)^{2}' + assert latex(hermite(n, x)) == r'H_{n}\left(x\right)' + assert latex(hermite(n, x)**2) == r'\left(H_{n}\left(x\right)\right)^{2}' + + theta = Symbol("theta", real=True) + phi = Symbol("phi", real=True) + assert latex(Ynm(n, m, theta, phi)) == r'Y_{n}^{m}\left(\theta,\phi\right)' + assert latex(Ynm(n, m, theta, phi)**3) == \ + r'\left(Y_{n}^{m}\left(\theta,\phi\right)\right)^{3}' + assert latex(Znm(n, m, theta, phi)) == r'Z_{n}^{m}\left(\theta,\phi\right)' + assert latex(Znm(n, m, theta, phi)**3) == \ + r'\left(Z_{n}^{m}\left(\theta,\phi\right)\right)^{3}' + + # Test latex printing of function names with "_" + assert latex(polar_lift(0)) == \ + r"\operatorname{polar\_lift}{\left(0 \right)}" + assert latex(polar_lift(0)**3) == \ + r"\operatorname{polar\_lift}^{3}{\left(0 \right)}" + + assert latex(totient(n)) == r'\phi\left(n\right)' + assert latex(totient(n) ** 2) == r'\left(\phi\left(n\right)\right)^{2}' + + assert latex(reduced_totient(n)) == r'\lambda\left(n\right)' + assert latex(reduced_totient(n) ** 2) == \ + r'\left(\lambda\left(n\right)\right)^{2}' + + assert latex(divisor_sigma(x)) == r"\sigma\left(x\right)" + assert latex(divisor_sigma(x)**2) == r"\sigma^{2}\left(x\right)" + assert latex(divisor_sigma(x, y)) == r"\sigma_y\left(x\right)" + assert latex(divisor_sigma(x, y)**2) == r"\sigma^{2}_y\left(x\right)" + + assert latex(udivisor_sigma(x)) == r"\sigma^*\left(x\right)" + assert latex(udivisor_sigma(x)**2) == r"\sigma^*^{2}\left(x\right)" + assert latex(udivisor_sigma(x, y)) == r"\sigma^*_y\left(x\right)" + assert latex(udivisor_sigma(x, y)**2) == r"\sigma^*^{2}_y\left(x\right)" + + assert latex(primenu(n)) == r'\nu\left(n\right)' + assert latex(primenu(n) ** 2) == r'\left(\nu\left(n\right)\right)^{2}' + + assert latex(primeomega(n)) == r'\Omega\left(n\right)' + assert latex(primeomega(n) ** 2) == \ + r'\left(\Omega\left(n\right)\right)^{2}' + + assert latex(LambertW(n)) == r'W\left(n\right)' + assert latex(LambertW(n, -1)) == r'W_{-1}\left(n\right)' + assert latex(LambertW(n, k)) == r'W_{k}\left(n\right)' + assert latex(LambertW(n) * LambertW(n)) == r"W^{2}\left(n\right)" + assert latex(Pow(LambertW(n), 2)) == r"W^{2}\left(n\right)" + assert latex(LambertW(n)**k) == r"W^{k}\left(n\right)" + assert latex(LambertW(n, k)**p) == r"W^{p}_{k}\left(n\right)" + + assert latex(Mod(x, 7)) == r'x \bmod 7' + assert latex(Mod(x + 1, 7)) == r'\left(x + 1\right) \bmod 7' + assert latex(Mod(7, x + 1)) == r'7 \bmod \left(x + 1\right)' + assert latex(Mod(2 * x, 7)) == r'2 x \bmod 7' + assert latex(Mod(7, 2 * x)) == r'7 \bmod 2 x' + assert latex(Mod(x, 7) + 1) == r'\left(x \bmod 7\right) + 1' + assert latex(2 * Mod(x, 7)) == r'2 \left(x \bmod 7\right)' + assert latex(Mod(7, 2 * x)**n) == r'\left(7 \bmod 2 x\right)^{n}' + + # some unknown function name should get rendered with \operatorname + fjlkd = Function('fjlkd') + assert latex(fjlkd(x)) == r'\operatorname{fjlkd}{\left(x \right)}' + # even when it is referred to without an argument + assert latex(fjlkd) == r'\operatorname{fjlkd}' + + +# test that notation passes to subclasses of the same name only +def test_function_subclass_different_name(): + class mygamma(gamma): + pass + assert latex(mygamma) == r"\operatorname{mygamma}" + assert latex(mygamma(x)) == r"\operatorname{mygamma}{\left(x \right)}" + + +def test_hyper_printing(): + from sympy.abc import x, z + + assert latex(meijerg(Tuple(pi, pi, x), Tuple(1), + (0, 1), Tuple(1, 2, 3/pi), z)) == \ + r'{G_{4, 5}^{2, 3}\left(\begin{matrix} \pi, \pi, x & 1 \\0, 1 & 1, 2, '\ + r'\frac{3}{\pi} \end{matrix} \middle| {z} \right)}' + assert latex(meijerg(Tuple(), Tuple(1), (0,), Tuple(), z)) == \ + r'{G_{1, 1}^{1, 0}\left(\begin{matrix} & 1 \\0 & \end{matrix} \middle| {z} \right)}' + assert latex(hyper((x, 2), (3,), z)) == \ + r'{{}_{2}F_{1}\left(\begin{matrix} 2, x ' \ + r'\\ 3 \end{matrix}\middle| {z} \right)}' + assert latex(hyper(Tuple(), Tuple(1), z)) == \ + r'{{}_{0}F_{1}\left(\begin{matrix} ' \ + r'\\ 1 \end{matrix}\middle| {z} \right)}' + + +def test_latex_bessel(): + from sympy.functions.special.bessel import (besselj, bessely, besseli, + besselk, hankel1, hankel2, + jn, yn, hn1, hn2) + from sympy.abc import z + assert latex(besselj(n, z**2)**k) == r'J^{k}_{n}\left(z^{2}\right)' + assert latex(bessely(n, z)) == r'Y_{n}\left(z\right)' + assert latex(besseli(n, z)) == r'I_{n}\left(z\right)' + assert latex(besselk(n, z)) == r'K_{n}\left(z\right)' + assert latex(hankel1(n, z**2)**2) == \ + r'\left(H^{(1)}_{n}\left(z^{2}\right)\right)^{2}' + assert latex(hankel2(n, z)) == r'H^{(2)}_{n}\left(z\right)' + assert latex(jn(n, z)) == r'j_{n}\left(z\right)' + assert latex(yn(n, z)) == r'y_{n}\left(z\right)' + assert latex(hn1(n, z)) == r'h^{(1)}_{n}\left(z\right)' + assert latex(hn2(n, z)) == r'h^{(2)}_{n}\left(z\right)' + + +def test_latex_fresnel(): + from sympy.functions.special.error_functions import (fresnels, fresnelc) + from sympy.abc import z + assert latex(fresnels(z)) == r'S\left(z\right)' + assert latex(fresnelc(z)) == r'C\left(z\right)' + assert latex(fresnels(z)**2) == r'S^{2}\left(z\right)' + assert latex(fresnelc(z)**2) == r'C^{2}\left(z\right)' + + +def test_latex_brackets(): + assert latex((-1)**x) == r"\left(-1\right)^{x}" + + +def test_latex_indexed(): + Psi_symbol = Symbol('Psi_0', complex=True, real=False) + Psi_indexed = IndexedBase(Symbol('Psi', complex=True, real=False)) + symbol_latex = latex(Psi_symbol * conjugate(Psi_symbol)) + indexed_latex = latex(Psi_indexed[0] * conjugate(Psi_indexed[0])) + # \\overline{{\\Psi}_{0}} {\\Psi}_{0} vs. \\Psi_{0} \\overline{\\Psi_{0}} + assert symbol_latex == r'\Psi_{0} \overline{\Psi_{0}}' + assert indexed_latex == r'\overline{{\Psi}_{0}} {\Psi}_{0}' + + # Symbol('gamma') gives r'\gamma' + interval = '\\mathrel{..}\\nobreak ' + assert latex(Indexed('x1', Symbol('i'))) == r'{x_{1}}_{i}' + assert latex(Indexed('x2', Idx('i'))) == r'{x_{2}}_{i}' + assert latex(Indexed('x3', Idx('i', Symbol('N')))) == r'{x_{3}}_{{i}_{0'+interval+'N - 1}}' + assert latex(Indexed('x3', Idx('i', Symbol('N')+1))) == r'{x_{3}}_{{i}_{0'+interval+'N}}' + assert latex(Indexed('x4', Idx('i', (Symbol('a'),Symbol('b'))))) == r'{x_{4}}_{{i}_{a'+interval+'b}}' + assert latex(IndexedBase('gamma')) == r'\gamma' + assert latex(IndexedBase('a b')) == r'a b' + assert latex(IndexedBase('a_b')) == r'a_{b}' + + +def test_latex_derivatives(): + # regular "d" for ordinary derivatives + assert latex(diff(x**3, x, evaluate=False)) == \ + r"\frac{d}{d x} x^{3}" + assert latex(diff(sin(x) + x**2, x, evaluate=False)) == \ + r"\frac{d}{d x} \left(x^{2} + \sin{\left(x \right)}\right)" + assert latex(diff(diff(sin(x) + x**2, x, evaluate=False), evaluate=False))\ + == \ + r"\frac{d^{2}}{d x^{2}} \left(x^{2} + \sin{\left(x \right)}\right)" + assert latex(diff(diff(diff(sin(x) + x**2, x, evaluate=False), evaluate=False), evaluate=False)) == \ + r"\frac{d^{3}}{d x^{3}} \left(x^{2} + \sin{\left(x \right)}\right)" + + # \partial for partial derivatives + assert latex(diff(sin(x * y), x, evaluate=False)) == \ + r"\frac{\partial}{\partial x} \sin{\left(x y \right)}" + assert latex(diff(sin(x * y) + x**2, x, evaluate=False)) == \ + r"\frac{\partial}{\partial x} \left(x^{2} + \sin{\left(x y \right)}\right)" + assert latex(diff(diff(sin(x*y) + x**2, x, evaluate=False), x, evaluate=False)) == \ + r"\frac{\partial^{2}}{\partial x^{2}} \left(x^{2} + \sin{\left(x y \right)}\right)" + assert latex(diff(diff(diff(sin(x*y) + x**2, x, evaluate=False), x, evaluate=False), x, evaluate=False)) == \ + r"\frac{\partial^{3}}{\partial x^{3}} \left(x^{2} + \sin{\left(x y \right)}\right)" + + # mixed partial derivatives + f = Function("f") + assert latex(diff(diff(f(x, y), x, evaluate=False), y, evaluate=False)) == \ + r"\frac{\partial^{2}}{\partial y\partial x} " + latex(f(x, y)) + + assert latex(diff(diff(diff(f(x, y), x, evaluate=False), x, evaluate=False), y, evaluate=False)) == \ + r"\frac{\partial^{3}}{\partial y\partial x^{2}} " + latex(f(x, y)) + + # for negative nested Derivative + assert latex(diff(-diff(y**2,x,evaluate=False),x,evaluate=False)) == r'\frac{d}{d x} \left(- \frac{d}{d x} y^{2}\right)' + assert latex(diff(diff(-diff(diff(y,x,evaluate=False),x,evaluate=False),x,evaluate=False),x,evaluate=False)) == \ + r'\frac{d^{2}}{d x^{2}} \left(- \frac{d^{2}}{d x^{2}} y\right)' + + # use ordinary d when one of the variables has been integrated out + assert latex(diff(Integral(exp(-x*y), (x, 0, oo)), y, evaluate=False)) == \ + r"\frac{d}{d y} \int\limits_{0}^{\infty} e^{- x y}\, dx" + + # Derivative wrapped in power: + assert latex(diff(x, x, evaluate=False)**2) == \ + r"\left(\frac{d}{d x} x\right)^{2}" + + assert latex(diff(f(x), x)**2) == \ + r"\left(\frac{d}{d x} f{\left(x \right)}\right)^{2}" + + assert latex(diff(f(x), (x, n))) == \ + r"\frac{d^{n}}{d x^{n}} f{\left(x \right)}" + + x1 = Symbol('x1') + x2 = Symbol('x2') + assert latex(diff(f(x1, x2), x1)) == r'\frac{\partial}{\partial x_{1}} f{\left(x_{1},x_{2} \right)}' + + n1 = Symbol('n1') + assert latex(diff(f(x), (x, n1))) == r'\frac{d^{n_{1}}}{d x^{n_{1}}} f{\left(x \right)}' + + n2 = Symbol('n2') + assert latex(diff(f(x), (x, Max(n1, n2)))) == \ + r'\frac{d^{\max\left(n_{1}, n_{2}\right)}}{d x^{\max\left(n_{1}, n_{2}\right)}} f{\left(x \right)}' + + # set diff operator + assert latex(diff(f(x), x), diff_operator="rd") == r'\frac{\mathrm{d}}{\mathrm{d} x} f{\left(x \right)}' + + +def test_latex_subs(): + assert latex(Subs(x*y, (x, y), (1, 2))) == r'\left. x y \right|_{\substack{ x=1\\ y=2 }}' + + +def test_latex_integrals(): + assert latex(Integral(log(x), x)) == r"\int \log{\left(x \right)}\, dx" + assert latex(Integral(x**2, (x, 0, 1))) == \ + r"\int\limits_{0}^{1} x^{2}\, dx" + assert latex(Integral(x**2, (x, 10, 20))) == \ + r"\int\limits_{10}^{20} x^{2}\, dx" + assert latex(Integral(y*x**2, (x, 0, 1), y)) == \ + r"\int\int\limits_{0}^{1} x^{2} y\, dx\, dy" + assert latex(Integral(y*x**2, (x, 0, 1), y), mode='equation*') == \ + r"\begin{equation*}\int\int\limits_{0}^{1} x^{2} y\, dx\, dy\end{equation*}" + assert latex(Integral(y*x**2, (x, 0, 1), y), mode='equation*', itex=True) \ + == r"$$\int\int_{0}^{1} x^{2} y\, dx\, dy$$" + assert latex(Integral(x, (x, 0))) == r"\int\limits^{0} x\, dx" + assert latex(Integral(x*y, x, y)) == r"\iint x y\, dx\, dy" + assert latex(Integral(x*y*z, x, y, z)) == r"\iiint x y z\, dx\, dy\, dz" + assert latex(Integral(x*y*z*t, x, y, z, t)) == \ + r"\iiiint t x y z\, dx\, dy\, dz\, dt" + assert latex(Integral(x, x, x, x, x, x, x)) == \ + r"\int\int\int\int\int\int x\, dx\, dx\, dx\, dx\, dx\, dx" + assert latex(Integral(x, x, y, (z, 0, 1))) == \ + r"\int\limits_{0}^{1}\int\int x\, dx\, dy\, dz" + + # for negative nested Integral + assert latex(Integral(-Integral(y**2,x),x)) == \ + r'\int \left(- \int y^{2}\, dx\right)\, dx' + assert latex(Integral(-Integral(-Integral(y,x),x),x)) == \ + r'\int \left(- \int \left(- \int y\, dx\right)\, dx\right)\, dx' + + # fix issue #10806 + assert latex(Integral(z, z)**2) == r"\left(\int z\, dz\right)^{2}" + assert latex(Integral(x + z, z)) == r"\int \left(x + z\right)\, dz" + assert latex(Integral(x+z/2, z)) == \ + r"\int \left(x + \frac{z}{2}\right)\, dz" + assert latex(Integral(x**y, z)) == r"\int x^{y}\, dz" + + # set diff operator + assert latex(Integral(x, x), diff_operator="rd") == r'\int x\, \mathrm{d}x' + assert latex(Integral(x, (x, 0, 1)), diff_operator="rd") == r'\int\limits_{0}^{1} x\, \mathrm{d}x' + + +def test_latex_sets(): + for s in (frozenset, set): + assert latex(s([x*y, x**2])) == r"\left\{x^{2}, x y\right\}" + assert latex(s(range(1, 6))) == r"\left\{1, 2, 3, 4, 5\right\}" + assert latex(s(range(1, 13))) == \ + r"\left\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\right\}" + + s = FiniteSet + assert latex(s(*[x*y, x**2])) == r"\left\{x^{2}, x y\right\}" + assert latex(s(*range(1, 6))) == r"\left\{1, 2, 3, 4, 5\right\}" + assert latex(s(*range(1, 13))) == \ + r"\left\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\right\}" + + +def test_latex_SetExpr(): + iv = Interval(1, 3) + se = SetExpr(iv) + assert latex(se) == r"SetExpr\left(\left[1, 3\right]\right)" + + +def test_latex_Range(): + assert latex(Range(1, 51)) == r'\left\{1, 2, \ldots, 50\right\}' + assert latex(Range(1, 4)) == r'\left\{1, 2, 3\right\}' + assert latex(Range(0, 3, 1)) == r'\left\{0, 1, 2\right\}' + assert latex(Range(0, 30, 1)) == r'\left\{0, 1, \ldots, 29\right\}' + assert latex(Range(30, 1, -1)) == r'\left\{30, 29, \ldots, 2\right\}' + assert latex(Range(0, oo, 2)) == r'\left\{0, 2, \ldots\right\}' + assert latex(Range(oo, -2, -2)) == r'\left\{\ldots, 2, 0\right\}' + assert latex(Range(-2, -oo, -1)) == r'\left\{-2, -3, \ldots\right\}' + assert latex(Range(-oo, oo)) == r'\left\{\ldots, -1, 0, 1, \ldots\right\}' + assert latex(Range(oo, -oo, -1)) == r'\left\{\ldots, 1, 0, -1, \ldots\right\}' + + a, b, c = symbols('a:c') + assert latex(Range(a, b, c)) == r'\text{Range}\left(a, b, c\right)' + assert latex(Range(a, 10, 1)) == r'\text{Range}\left(a, 10\right)' + assert latex(Range(0, b, 1)) == r'\text{Range}\left(b\right)' + assert latex(Range(0, 10, c)) == r'\text{Range}\left(0, 10, c\right)' + + i = Symbol('i', integer=True) + n = Symbol('n', negative=True, integer=True) + p = Symbol('p', positive=True, integer=True) + + assert latex(Range(i, i + 3)) == r'\left\{i, i + 1, i + 2\right\}' + assert latex(Range(-oo, n, 2)) == r'\left\{\ldots, n - 4, n - 2\right\}' + assert latex(Range(p, oo)) == r'\left\{p, p + 1, \ldots\right\}' + # The following will work if __iter__ is improved + # assert latex(Range(-3, p + 7)) == r'\left\{-3, -2, \ldots, p + 6\right\}' + # Must have integer assumptions + assert latex(Range(a, a + 3)) == r'\text{Range}\left(a, a + 3\right)' + + +def test_latex_sequences(): + s1 = SeqFormula(a**2, (0, oo)) + s2 = SeqPer((1, 2)) + + latex_str = r'\left[0, 1, 4, 9, \ldots\right]' + assert latex(s1) == latex_str + + latex_str = r'\left[1, 2, 1, 2, \ldots\right]' + assert latex(s2) == latex_str + + s3 = SeqFormula(a**2, (0, 2)) + s4 = SeqPer((1, 2), (0, 2)) + + latex_str = r'\left[0, 1, 4\right]' + assert latex(s3) == latex_str + + latex_str = r'\left[1, 2, 1\right]' + assert latex(s4) == latex_str + + s5 = SeqFormula(a**2, (-oo, 0)) + s6 = SeqPer((1, 2), (-oo, 0)) + + latex_str = r'\left[\ldots, 9, 4, 1, 0\right]' + assert latex(s5) == latex_str + + latex_str = r'\left[\ldots, 2, 1, 2, 1\right]' + assert latex(s6) == latex_str + + latex_str = r'\left[1, 3, 5, 11, \ldots\right]' + assert latex(SeqAdd(s1, s2)) == latex_str + + latex_str = r'\left[1, 3, 5\right]' + assert latex(SeqAdd(s3, s4)) == latex_str + + latex_str = r'\left[\ldots, 11, 5, 3, 1\right]' + assert latex(SeqAdd(s5, s6)) == latex_str + + latex_str = r'\left[0, 2, 4, 18, \ldots\right]' + assert latex(SeqMul(s1, s2)) == latex_str + + latex_str = r'\left[0, 2, 4\right]' + assert latex(SeqMul(s3, s4)) == latex_str + + latex_str = r'\left[\ldots, 18, 4, 2, 0\right]' + assert latex(SeqMul(s5, s6)) == latex_str + + # Sequences with symbolic limits, issue 12629 + s7 = SeqFormula(a**2, (a, 0, x)) + latex_str = r'\left\{a^{2}\right\}_{a=0}^{x}' + assert latex(s7) == latex_str + + b = Symbol('b') + s8 = SeqFormula(b*a**2, (a, 0, 2)) + latex_str = r'\left[0, b, 4 b\right]' + assert latex(s8) == latex_str + + +def test_latex_FourierSeries(): + latex_str = \ + r'2 \sin{\left(x \right)} - \sin{\left(2 x \right)} + \frac{2 \sin{\left(3 x \right)}}{3} + \ldots' + assert latex(fourier_series(x, (x, -pi, pi))) == latex_str + + +def test_latex_FormalPowerSeries(): + latex_str = r'\sum_{k=1}^{\infty} - \frac{\left(-1\right)^{- k} x^{k}}{k}' + assert latex(fps(log(1 + x))) == latex_str + + +def test_latex_intervals(): + a = Symbol('a', real=True) + assert latex(Interval(0, 0)) == r"\left\{0\right\}" + assert latex(Interval(0, a)) == r"\left[0, a\right]" + assert latex(Interval(0, a, False, False)) == r"\left[0, a\right]" + assert latex(Interval(0, a, True, False)) == r"\left(0, a\right]" + assert latex(Interval(0, a, False, True)) == r"\left[0, a\right)" + assert latex(Interval(0, a, True, True)) == r"\left(0, a\right)" + + +def test_latex_AccumuBounds(): + a = Symbol('a', real=True) + assert latex(AccumBounds(0, 1)) == r"\left\langle 0, 1\right\rangle" + assert latex(AccumBounds(0, a)) == r"\left\langle 0, a\right\rangle" + assert latex(AccumBounds(a + 1, a + 2)) == \ + r"\left\langle a + 1, a + 2\right\rangle" + + +def test_latex_emptyset(): + assert latex(S.EmptySet) == r"\emptyset" + + +def test_latex_universalset(): + assert latex(S.UniversalSet) == r"\mathbb{U}" + + +def test_latex_commutator(): + A = Operator('A') + B = Operator('B') + comm = Commutator(B, A) + assert latex(comm.doit()) == r"- (A B - B A)" + + +def test_latex_union(): + assert latex(Union(Interval(0, 1), Interval(2, 3))) == \ + r"\left[0, 1\right] \cup \left[2, 3\right]" + assert latex(Union(Interval(1, 1), Interval(2, 2), Interval(3, 4))) == \ + r"\left\{1, 2\right\} \cup \left[3, 4\right]" + + +def test_latex_intersection(): + assert latex(Intersection(Interval(0, 1), Interval(x, y))) == \ + r"\left[0, 1\right] \cap \left[x, y\right]" + + +def test_latex_symmetric_difference(): + assert latex(SymmetricDifference(Interval(2, 5), Interval(4, 7), + evaluate=False)) == \ + r'\left[2, 5\right] \triangle \left[4, 7\right]' + + +def test_latex_Complement(): + assert latex(Complement(S.Reals, S.Naturals)) == \ + r"\mathbb{R} \setminus \mathbb{N}" + + +def test_latex_productset(): + line = Interval(0, 1) + bigline = Interval(0, 10) + fset = FiniteSet(1, 2, 3) + assert latex(line**2) == r"%s^{2}" % latex(line) + assert latex(line**10) == r"%s^{10}" % latex(line) + assert latex((line * bigline * fset).flatten()) == r"%s \times %s \times %s" % ( + latex(line), latex(bigline), latex(fset)) + + +def test_latex_powerset(): + fset = FiniteSet(1, 2, 3) + assert latex(PowerSet(fset)) == r'\mathcal{P}\left(\left\{1, 2, 3\right\}\right)' + + +def test_latex_ordinals(): + w = OrdinalOmega() + assert latex(w) == r"\omega" + wp = OmegaPower(2, 3) + assert latex(wp) == r'3 \omega^{2}' + assert latex(Ordinal(wp, OmegaPower(1, 1))) == r'3 \omega^{2} + \omega' + assert latex(Ordinal(OmegaPower(2, 1), OmegaPower(1, 2))) == r'\omega^{2} + 2 \omega' + + +def test_set_operators_parenthesis(): + a, b, c, d = symbols('a:d') + A = FiniteSet(a) + B = FiniteSet(b) + C = FiniteSet(c) + D = FiniteSet(d) + + U1 = Union(A, B, evaluate=False) + U2 = Union(C, D, evaluate=False) + I1 = Intersection(A, B, evaluate=False) + I2 = Intersection(C, D, evaluate=False) + C1 = Complement(A, B, evaluate=False) + C2 = Complement(C, D, evaluate=False) + D1 = SymmetricDifference(A, B, evaluate=False) + D2 = SymmetricDifference(C, D, evaluate=False) + # XXX ProductSet does not support evaluate keyword + P1 = ProductSet(A, B) + P2 = ProductSet(C, D) + + assert latex(Intersection(A, U2, evaluate=False)) == \ + r'\left\{a\right\} \cap ' \ + r'\left(\left\{c\right\} \cup \left\{d\right\}\right)' + assert latex(Intersection(U1, U2, evaluate=False)) == \ + r'\left(\left\{a\right\} \cup \left\{b\right\}\right) ' \ + r'\cap \left(\left\{c\right\} \cup \left\{d\right\}\right)' + assert latex(Intersection(C1, C2, evaluate=False)) == \ + r'\left(\left\{a\right\} \setminus ' \ + r'\left\{b\right\}\right) \cap \left(\left\{c\right\} ' \ + r'\setminus \left\{d\right\}\right)' + assert latex(Intersection(D1, D2, evaluate=False)) == \ + r'\left(\left\{a\right\} \triangle ' \ + r'\left\{b\right\}\right) \cap \left(\left\{c\right\} ' \ + r'\triangle \left\{d\right\}\right)' + assert latex(Intersection(P1, P2, evaluate=False)) == \ + r'\left(\left\{a\right\} \times \left\{b\right\}\right) ' \ + r'\cap \left(\left\{c\right\} \times ' \ + r'\left\{d\right\}\right)' + + assert latex(Union(A, I2, evaluate=False)) == \ + r'\left\{a\right\} \cup ' \ + r'\left(\left\{c\right\} \cap \left\{d\right\}\right)' + assert latex(Union(I1, I2, evaluate=False)) == \ + r'\left(\left\{a\right\} \cap \left\{b\right\}\right) ' \ + r'\cup \left(\left\{c\right\} \cap \left\{d\right\}\right)' + assert latex(Union(C1, C2, evaluate=False)) == \ + r'\left(\left\{a\right\} \setminus ' \ + r'\left\{b\right\}\right) \cup \left(\left\{c\right\} ' \ + r'\setminus \left\{d\right\}\right)' + assert latex(Union(D1, D2, evaluate=False)) == \ + r'\left(\left\{a\right\} \triangle ' \ + r'\left\{b\right\}\right) \cup \left(\left\{c\right\} ' \ + r'\triangle \left\{d\right\}\right)' + assert latex(Union(P1, P2, evaluate=False)) == \ + r'\left(\left\{a\right\} \times \left\{b\right\}\right) ' \ + r'\cup \left(\left\{c\right\} \times ' \ + r'\left\{d\right\}\right)' + + assert latex(Complement(A, C2, evaluate=False)) == \ + r'\left\{a\right\} \setminus \left(\left\{c\right\} ' \ + r'\setminus \left\{d\right\}\right)' + assert latex(Complement(U1, U2, evaluate=False)) == \ + r'\left(\left\{a\right\} \cup \left\{b\right\}\right) ' \ + r'\setminus \left(\left\{c\right\} \cup ' \ + r'\left\{d\right\}\right)' + assert latex(Complement(I1, I2, evaluate=False)) == \ + r'\left(\left\{a\right\} \cap \left\{b\right\}\right) ' \ + r'\setminus \left(\left\{c\right\} \cap ' \ + r'\left\{d\right\}\right)' + assert latex(Complement(D1, D2, evaluate=False)) == \ + r'\left(\left\{a\right\} \triangle ' \ + r'\left\{b\right\}\right) \setminus ' \ + r'\left(\left\{c\right\} \triangle \left\{d\right\}\right)' + assert latex(Complement(P1, P2, evaluate=False)) == \ + r'\left(\left\{a\right\} \times \left\{b\right\}\right) '\ + r'\setminus \left(\left\{c\right\} \times '\ + r'\left\{d\right\}\right)' + + assert latex(SymmetricDifference(A, D2, evaluate=False)) == \ + r'\left\{a\right\} \triangle \left(\left\{c\right\} ' \ + r'\triangle \left\{d\right\}\right)' + assert latex(SymmetricDifference(U1, U2, evaluate=False)) == \ + r'\left(\left\{a\right\} \cup \left\{b\right\}\right) ' \ + r'\triangle \left(\left\{c\right\} \cup ' \ + r'\left\{d\right\}\right)' + assert latex(SymmetricDifference(I1, I2, evaluate=False)) == \ + r'\left(\left\{a\right\} \cap \left\{b\right\}\right) ' \ + r'\triangle \left(\left\{c\right\} \cap ' \ + r'\left\{d\right\}\right)' + assert latex(SymmetricDifference(C1, C2, evaluate=False)) == \ + r'\left(\left\{a\right\} \setminus ' \ + r'\left\{b\right\}\right) \triangle ' \ + r'\left(\left\{c\right\} \setminus \left\{d\right\}\right)' + assert latex(SymmetricDifference(P1, P2, evaluate=False)) == \ + r'\left(\left\{a\right\} \times \left\{b\right\}\right) ' \ + r'\triangle \left(\left\{c\right\} \times ' \ + r'\left\{d\right\}\right)' + + # XXX This can be incorrect since cartesian product is not associative + assert latex(ProductSet(A, P2).flatten()) == \ + r'\left\{a\right\} \times \left\{c\right\} \times ' \ + r'\left\{d\right\}' + assert latex(ProductSet(U1, U2)) == \ + r'\left(\left\{a\right\} \cup \left\{b\right\}\right) ' \ + r'\times \left(\left\{c\right\} \cup ' \ + r'\left\{d\right\}\right)' + assert latex(ProductSet(I1, I2)) == \ + r'\left(\left\{a\right\} \cap \left\{b\right\}\right) ' \ + r'\times \left(\left\{c\right\} \cap ' \ + r'\left\{d\right\}\right)' + assert latex(ProductSet(C1, C2)) == \ + r'\left(\left\{a\right\} \setminus ' \ + r'\left\{b\right\}\right) \times \left(\left\{c\right\} ' \ + r'\setminus \left\{d\right\}\right)' + assert latex(ProductSet(D1, D2)) == \ + r'\left(\left\{a\right\} \triangle ' \ + r'\left\{b\right\}\right) \times \left(\left\{c\right\} ' \ + r'\triangle \left\{d\right\}\right)' + + +def test_latex_Complexes(): + assert latex(S.Complexes) == r"\mathbb{C}" + + +def test_latex_Naturals(): + assert latex(S.Naturals) == r"\mathbb{N}" + + +def test_latex_Naturals0(): + assert latex(S.Naturals0) == r"\mathbb{N}_0" + + +def test_latex_Integers(): + assert latex(S.Integers) == r"\mathbb{Z}" + + +def test_latex_ImageSet(): + x = Symbol('x') + assert latex(ImageSet(Lambda(x, x**2), S.Naturals)) == \ + r"\left\{x^{2}\; \middle|\; x \in \mathbb{N}\right\}" + + y = Symbol('y') + imgset = ImageSet(Lambda((x, y), x + y), {1, 2, 3}, {3, 4}) + assert latex(imgset) == \ + r"\left\{x + y\; \middle|\; x \in \left\{1, 2, 3\right\}, y \in \left\{3, 4\right\}\right\}" + + imgset = ImageSet(Lambda(((x, y),), x + y), ProductSet({1, 2, 3}, {3, 4})) + assert latex(imgset) == \ + r"\left\{x + y\; \middle|\; \left( x, \ y\right) \in \left\{1, 2, 3\right\} \times \left\{3, 4\right\}\right\}" + + +def test_latex_ConditionSet(): + x = Symbol('x') + assert latex(ConditionSet(x, Eq(x**2, 1), S.Reals)) == \ + r"\left\{x\; \middle|\; x \in \mathbb{R} \wedge x^{2} = 1 \right\}" + assert latex(ConditionSet(x, Eq(x**2, 1), S.UniversalSet)) == \ + r"\left\{x\; \middle|\; x^{2} = 1 \right\}" + + +def test_latex_ComplexRegion(): + assert latex(ComplexRegion(Interval(3, 5)*Interval(4, 6))) == \ + r"\left\{x + y i\; \middle|\; x, y \in \left[3, 5\right] \times \left[4, 6\right] \right\}" + assert latex(ComplexRegion(Interval(0, 1)*Interval(0, 2*pi), polar=True)) == \ + r"\left\{r \left(i \sin{\left(\theta \right)} + \cos{\left(\theta "\ + r"\right)}\right)\; \middle|\; r, \theta \in \left[0, 1\right] \times \left[0, 2 \pi\right) \right\}" + + +def test_latex_Contains(): + x = Symbol('x') + assert latex(Contains(x, S.Naturals)) == r"x \in \mathbb{N}" + + +def test_latex_sum(): + assert latex(Sum(x*y**2, (x, -2, 2), (y, -5, 5))) == \ + r"\sum_{\substack{-2 \leq x \leq 2\\-5 \leq y \leq 5}} x y^{2}" + assert latex(Sum(x**2, (x, -2, 2))) == \ + r"\sum_{x=-2}^{2} x^{2}" + assert latex(Sum(x**2 + y, (x, -2, 2))) == \ + r"\sum_{x=-2}^{2} \left(x^{2} + y\right)" + assert latex(Sum(x**2 + y, (x, -2, 2))**2) == \ + r"\left(\sum_{x=-2}^{2} \left(x^{2} + y\right)\right)^{2}" + + +def test_latex_product(): + assert latex(Product(x*y**2, (x, -2, 2), (y, -5, 5))) == \ + r"\prod_{\substack{-2 \leq x \leq 2\\-5 \leq y \leq 5}} x y^{2}" + assert latex(Product(x**2, (x, -2, 2))) == \ + r"\prod_{x=-2}^{2} x^{2}" + assert latex(Product(x**2 + y, (x, -2, 2))) == \ + r"\prod_{x=-2}^{2} \left(x^{2} + y\right)" + + assert latex(Product(x, (x, -2, 2))**2) == \ + r"\left(\prod_{x=-2}^{2} x\right)^{2}" + + +def test_latex_limits(): + assert latex(Limit(x, x, oo)) == r"\lim_{x \to \infty} x" + + # issue 8175 + f = Function('f') + assert latex(Limit(f(x), x, 0)) == r"\lim_{x \to 0^+} f{\left(x \right)}" + assert latex(Limit(f(x), x, 0, "-")) == \ + r"\lim_{x \to 0^-} f{\left(x \right)}" + + # issue #10806 + assert latex(Limit(f(x), x, 0)**2) == \ + r"\left(\lim_{x \to 0^+} f{\left(x \right)}\right)^{2}" + # bi-directional limit + assert latex(Limit(f(x), x, 0, dir='+-')) == \ + r"\lim_{x \to 0} f{\left(x \right)}" + + +def test_latex_log(): + assert latex(log(x)) == r"\log{\left(x \right)}" + assert latex(log(x), ln_notation=True) == r"\ln{\left(x \right)}" + assert latex(log(x) + log(y)) == \ + r"\log{\left(x \right)} + \log{\left(y \right)}" + assert latex(log(x) + log(y), ln_notation=True) == \ + r"\ln{\left(x \right)} + \ln{\left(y \right)}" + assert latex(pow(log(x), x)) == r"\log{\left(x \right)}^{x}" + assert latex(pow(log(x), x), ln_notation=True) == \ + r"\ln{\left(x \right)}^{x}" + + +def test_issue_3568(): + beta = Symbol(r'\beta') + y = beta + x + assert latex(y) in [r'\beta + x', r'x + \beta'] + + beta = Symbol(r'beta') + y = beta + x + assert latex(y) in [r'\beta + x', r'x + \beta'] + + +def test_latex(): + assert latex((2*tau)**Rational(7, 2)) == r"8 \sqrt{2} \tau^{\frac{7}{2}}" + assert latex((2*mu)**Rational(7, 2), mode='equation*') == \ + r"\begin{equation*}8 \sqrt{2} \mu^{\frac{7}{2}}\end{equation*}" + assert latex((2*mu)**Rational(7, 2), mode='equation', itex=True) == \ + r"$$8 \sqrt{2} \mu^{\frac{7}{2}}$$" + assert latex([2/x, y]) == r"\left[ \frac{2}{x}, \ y\right]" + + +def test_latex_dict(): + d = {Rational(1): 1, x**2: 2, x: 3, x**3: 4} + assert latex(d) == \ + r'\left\{ 1 : 1, \ x : 3, \ x^{2} : 2, \ x^{3} : 4\right\}' + D = Dict(d) + assert latex(D) == \ + r'\left\{ 1 : 1, \ x : 3, \ x^{2} : 2, \ x^{3} : 4\right\}' + + +def test_latex_list(): + ll = [Symbol('omega1'), Symbol('a'), Symbol('alpha')] + assert latex(ll) == r'\left[ \omega_{1}, \ a, \ \alpha\right]' + + +def test_latex_NumberSymbols(): + assert latex(S.Catalan) == "G" + assert latex(S.EulerGamma) == r"\gamma" + assert latex(S.Exp1) == "e" + assert latex(S.GoldenRatio) == r"\phi" + assert latex(S.Pi) == r"\pi" + assert latex(S.TribonacciConstant) == r"\text{TribonacciConstant}" + + +def test_latex_rational(): + # tests issue 3973 + assert latex(-Rational(1, 2)) == r"- \frac{1}{2}" + assert latex(Rational(-1, 2)) == r"- \frac{1}{2}" + assert latex(Rational(1, -2)) == r"- \frac{1}{2}" + assert latex(-Rational(-1, 2)) == r"\frac{1}{2}" + assert latex(-Rational(1, 2)*x) == r"- \frac{x}{2}" + assert latex(-Rational(1, 2)*x + Rational(-2, 3)*y) == \ + r"- \frac{x}{2} - \frac{2 y}{3}" + + +def test_latex_inverse(): + # tests issue 4129 + assert latex(1/x) == r"\frac{1}{x}" + assert latex(1/(x + y)) == r"\frac{1}{x + y}" + + +def test_latex_DiracDelta(): + assert latex(DiracDelta(x)) == r"\delta\left(x\right)" + assert latex(DiracDelta(x)**2) == r"\left(\delta\left(x\right)\right)^{2}" + assert latex(DiracDelta(x, 0)) == r"\delta\left(x\right)" + assert latex(DiracDelta(x, 5)) == \ + r"\delta^{\left( 5 \right)}\left( x \right)" + assert latex(DiracDelta(x, 5)**2) == \ + r"\left(\delta^{\left( 5 \right)}\left( x \right)\right)^{2}" + + +def test_latex_Heaviside(): + assert latex(Heaviside(x)) == r"\theta\left(x\right)" + assert latex(Heaviside(x)**2) == r"\left(\theta\left(x\right)\right)^{2}" + + +def test_latex_KroneckerDelta(): + assert latex(KroneckerDelta(x, y)) == r"\delta_{x y}" + assert latex(KroneckerDelta(x, y + 1)) == r"\delta_{x, y + 1}" + # issue 6578 + assert latex(KroneckerDelta(x + 1, y)) == r"\delta_{y, x + 1}" + assert latex(Pow(KroneckerDelta(x, y), 2, evaluate=False)) == \ + r"\left(\delta_{x y}\right)^{2}" + + +def test_latex_LeviCivita(): + assert latex(LeviCivita(x, y, z)) == r"\varepsilon_{x y z}" + assert latex(LeviCivita(x, y, z)**2) == \ + r"\left(\varepsilon_{x y z}\right)^{2}" + assert latex(LeviCivita(x, y, z + 1)) == r"\varepsilon_{x, y, z + 1}" + assert latex(LeviCivita(x, y + 1, z)) == r"\varepsilon_{x, y + 1, z}" + assert latex(LeviCivita(x + 1, y, z)) == r"\varepsilon_{x + 1, y, z}" + + +def test_mode(): + expr = x + y + assert latex(expr) == r'x + y' + assert latex(expr, mode='plain') == r'x + y' + assert latex(expr, mode='inline') == r'$x + y$' + assert latex( + expr, mode='equation*') == r'\begin{equation*}x + y\end{equation*}' + assert latex( + expr, mode='equation') == r'\begin{equation}x + y\end{equation}' + raises(ValueError, lambda: latex(expr, mode='foo')) + + +def test_latex_mathieu(): + assert latex(mathieuc(x, y, z)) == r"C\left(x, y, z\right)" + assert latex(mathieus(x, y, z)) == r"S\left(x, y, z\right)" + assert latex(mathieuc(x, y, z)**2) == r"C\left(x, y, z\right)^{2}" + assert latex(mathieus(x, y, z)**2) == r"S\left(x, y, z\right)^{2}" + assert latex(mathieucprime(x, y, z)) == r"C^{\prime}\left(x, y, z\right)" + assert latex(mathieusprime(x, y, z)) == r"S^{\prime}\left(x, y, z\right)" + assert latex(mathieucprime(x, y, z)**2) == r"C^{\prime}\left(x, y, z\right)^{2}" + assert latex(mathieusprime(x, y, z)**2) == r"S^{\prime}\left(x, y, z\right)^{2}" + +def test_latex_Piecewise(): + p = Piecewise((x, x < 1), (x**2, True)) + assert latex(p) == r"\begin{cases} x & \text{for}\: x < 1 \\x^{2} &" \ + r" \text{otherwise} \end{cases}" + assert latex(p, itex=True) == \ + r"\begin{cases} x & \text{for}\: x \lt 1 \\x^{2} &" \ + r" \text{otherwise} \end{cases}" + p = Piecewise((x, x < 0), (0, x >= 0)) + assert latex(p) == r'\begin{cases} x & \text{for}\: x < 0 \\0 &' \ + r' \text{otherwise} \end{cases}' + A, B = symbols("A B", commutative=False) + p = Piecewise((A**2, Eq(A, B)), (A*B, True)) + s = r"\begin{cases} A^{2} & \text{for}\: A = B \\A B & \text{otherwise} \end{cases}" + assert latex(p) == s + assert latex(A*p) == r"A \left(%s\right)" % s + assert latex(p*A) == r"\left(%s\right) A" % s + assert latex(Piecewise((x, x < 1), (x**2, x < 2))) == \ + r'\begin{cases} x & ' \ + r'\text{for}\: x < 1 \\x^{2} & \text{for}\: x < 2 \end{cases}' + + +def test_latex_Matrix(): + M = Matrix([[1 + x, y], [y, x - 1]]) + assert latex(M) == \ + r'\left[\begin{matrix}x + 1 & y\\y & x - 1\end{matrix}\right]' + assert latex(M, mode='inline') == \ + r'$\left[\begin{smallmatrix}x + 1 & y\\' \ + r'y & x - 1\end{smallmatrix}\right]$' + assert latex(M, mat_str='array') == \ + r'\left[\begin{array}{cc}x + 1 & y\\y & x - 1\end{array}\right]' + assert latex(M, mat_str='bmatrix') == \ + r'\left[\begin{bmatrix}x + 1 & y\\y & x - 1\end{bmatrix}\right]' + assert latex(M, mat_delim=None, mat_str='bmatrix') == \ + r'\begin{bmatrix}x + 1 & y\\y & x - 1\end{bmatrix}' + + M2 = Matrix(1, 11, range(11)) + assert latex(M2) == \ + r'\left[\begin{array}{ccccccccccc}' \ + r'0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10\end{array}\right]' + + +def test_latex_matrix_with_functions(): + t = symbols('t') + theta1 = symbols('theta1', cls=Function) + + M = Matrix([[sin(theta1(t)), cos(theta1(t))], + [cos(theta1(t).diff(t)), sin(theta1(t).diff(t))]]) + + expected = (r'\left[\begin{matrix}\sin{\left(' + r'\theta_{1}{\left(t \right)} \right)} & ' + r'\cos{\left(\theta_{1}{\left(t \right)} \right)' + r'}\\\cos{\left(\frac{d}{d t} \theta_{1}{\left(t ' + r'\right)} \right)} & \sin{\left(\frac{d}{d t} ' + r'\theta_{1}{\left(t \right)} \right' + r')}\end{matrix}\right]') + + assert latex(M) == expected + + +def test_latex_NDimArray(): + x, y, z, w = symbols("x y z w") + + for ArrayType in (ImmutableDenseNDimArray, ImmutableSparseNDimArray, + MutableDenseNDimArray, MutableSparseNDimArray): + # Basic: scalar array + M = ArrayType(x) + + assert latex(M) == r"x" + + M = ArrayType([[1 / x, y], [z, w]]) + M1 = ArrayType([1 / x, y, z]) + + M2 = tensorproduct(M1, M) + M3 = tensorproduct(M, M) + + assert latex(M) == \ + r'\left[\begin{matrix}\frac{1}{x} & y\\z & w\end{matrix}\right]' + assert latex(M1) == \ + r"\left[\begin{matrix}\frac{1}{x} & y & z\end{matrix}\right]" + assert latex(M2) == \ + r"\left[\begin{matrix}" \ + r"\left[\begin{matrix}\frac{1}{x^{2}} & \frac{y}{x}\\\frac{z}{x} & \frac{w}{x}\end{matrix}\right] & " \ + r"\left[\begin{matrix}\frac{y}{x} & y^{2}\\y z & w y\end{matrix}\right] & " \ + r"\left[\begin{matrix}\frac{z}{x} & y z\\z^{2} & w z\end{matrix}\right]" \ + r"\end{matrix}\right]" + assert latex(M3) == \ + r"""\left[\begin{matrix}"""\ + r"""\left[\begin{matrix}\frac{1}{x^{2}} & \frac{y}{x}\\\frac{z}{x} & \frac{w}{x}\end{matrix}\right] & """\ + r"""\left[\begin{matrix}\frac{y}{x} & y^{2}\\y z & w y\end{matrix}\right]\\"""\ + r"""\left[\begin{matrix}\frac{z}{x} & y z\\z^{2} & w z\end{matrix}\right] & """\ + r"""\left[\begin{matrix}\frac{w}{x} & w y\\w z & w^{2}\end{matrix}\right]"""\ + r"""\end{matrix}\right]""" + + Mrow = ArrayType([[x, y, 1/z]]) + Mcolumn = ArrayType([[x], [y], [1/z]]) + Mcol2 = ArrayType([Mcolumn.tolist()]) + + assert latex(Mrow) == \ + r"\left[\left[\begin{matrix}x & y & \frac{1}{z}\end{matrix}\right]\right]" + assert latex(Mcolumn) == \ + r"\left[\begin{matrix}x\\y\\\frac{1}{z}\end{matrix}\right]" + assert latex(Mcol2) == \ + r'\left[\begin{matrix}\left[\begin{matrix}x\\y\\\frac{1}{z}\end{matrix}\right]\end{matrix}\right]' + + +def test_latex_mul_symbol(): + assert latex(4*4**x, mul_symbol='times') == r"4 \times 4^{x}" + assert latex(4*4**x, mul_symbol='dot') == r"4 \cdot 4^{x}" + assert latex(4*4**x, mul_symbol='ldot') == r"4 \,.\, 4^{x}" + + assert latex(4*x, mul_symbol='times') == r"4 \times x" + assert latex(4*x, mul_symbol='dot') == r"4 \cdot x" + assert latex(4*x, mul_symbol='ldot') == r"4 \,.\, x" + + +def test_latex_issue_4381(): + y = 4*4**log(2) + assert latex(y) == r'4 \cdot 4^{\log{\left(2 \right)}}' + assert latex(1/y) == r'\frac{1}{4 \cdot 4^{\log{\left(2 \right)}}}' + + +def test_latex_issue_4576(): + assert latex(Symbol("beta_13_2")) == r"\beta_{13 2}" + assert latex(Symbol("beta_132_20")) == r"\beta_{132 20}" + assert latex(Symbol("beta_13")) == r"\beta_{13}" + assert latex(Symbol("x_a_b")) == r"x_{a b}" + assert latex(Symbol("x_1_2_3")) == r"x_{1 2 3}" + assert latex(Symbol("x_a_b1")) == r"x_{a b1}" + assert latex(Symbol("x_a_1")) == r"x_{a 1}" + assert latex(Symbol("x_1_a")) == r"x_{1 a}" + assert latex(Symbol("x_1^aa")) == r"x^{aa}_{1}" + assert latex(Symbol("x_1__aa")) == r"x^{aa}_{1}" + assert latex(Symbol("x_11^a")) == r"x^{a}_{11}" + assert latex(Symbol("x_11__a")) == r"x^{a}_{11}" + assert latex(Symbol("x_a_a_a_a")) == r"x_{a a a a}" + assert latex(Symbol("x_a_a^a^a")) == r"x^{a a}_{a a}" + assert latex(Symbol("x_a_a__a__a")) == r"x^{a a}_{a a}" + assert latex(Symbol("alpha_11")) == r"\alpha_{11}" + assert latex(Symbol("alpha_11_11")) == r"\alpha_{11 11}" + assert latex(Symbol("alpha_alpha")) == r"\alpha_{\alpha}" + assert latex(Symbol("alpha^aleph")) == r"\alpha^{\aleph}" + assert latex(Symbol("alpha__aleph")) == r"\alpha^{\aleph}" + + +def test_latex_pow_fraction(): + x = Symbol('x') + # Testing exp + assert r'e^{-x}' in latex(exp(-x)/2).replace(' ', '') # Remove Whitespace + + # Testing e^{-x} in case future changes alter behavior of muls or fracs + # In particular current output is \frac{1}{2}e^{- x} but perhaps this will + # change to \frac{e^{-x}}{2} + + # Testing general, non-exp, power + assert r'3^{-x}' in latex(3**-x/2).replace(' ', '') + + +def test_noncommutative(): + A, B, C = symbols('A,B,C', commutative=False) + + assert latex(A*B*C**-1) == r"A B C^{-1}" + assert latex(C**-1*A*B) == r"C^{-1} A B" + assert latex(A*C**-1*B) == r"A C^{-1} B" + + +def test_latex_order(): + expr = x**3 + x**2*y + y**4 + 3*x*y**3 + + assert latex(expr, order='lex') == r"x^{3} + x^{2} y + 3 x y^{3} + y^{4}" + assert latex( + expr, order='rev-lex') == r"y^{4} + 3 x y^{3} + x^{2} y + x^{3}" + assert latex(expr, order='none') == r"x^{3} + y^{4} + y x^{2} + 3 x y^{3}" + + +def test_latex_Lambda(): + assert latex(Lambda(x, x + 1)) == r"\left( x \mapsto x + 1 \right)" + assert latex(Lambda((x, y), x + 1)) == r"\left( \left( x, \ y\right) \mapsto x + 1 \right)" + assert latex(Lambda(x, x)) == r"\left( x \mapsto x \right)" + +def test_latex_PolyElement(): + Ruv, u, v = ring("u,v", ZZ) + Rxyz, x, y, z = ring("x,y,z", Ruv) + + assert latex(x - x) == r"0" + assert latex(x - 1) == r"x - 1" + assert latex(x + 1) == r"x + 1" + + assert latex((u**2 + 3*u*v + 1)*x**2*y + u + 1) == \ + r"\left({u}^{2} + 3 u v + 1\right) {x}^{2} y + u + 1" + assert latex((u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x) == \ + r"\left({u}^{2} + 3 u v + 1\right) {x}^{2} y + \left(u + 1\right) x" + assert latex((u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x + 1) == \ + r"\left({u}^{2} + 3 u v + 1\right) {x}^{2} y + \left(u + 1\right) x + 1" + assert latex((-u**2 + 3*u*v - 1)*x**2*y - (u + 1)*x - 1) == \ + r"-\left({u}^{2} - 3 u v + 1\right) {x}^{2} y - \left(u + 1\right) x - 1" + + assert latex(-(v**2 + v + 1)*x + 3*u*v + 1) == \ + r"-\left({v}^{2} + v + 1\right) x + 3 u v + 1" + assert latex(-(v**2 + v + 1)*x - 3*u*v + 1) == \ + r"-\left({v}^{2} + v + 1\right) x - 3 u v + 1" + + +def test_latex_FracElement(): + Fuv, u, v = field("u,v", ZZ) + Fxyzt, x, y, z, t = field("x,y,z,t", Fuv) + + assert latex(x - x) == r"0" + assert latex(x - 1) == r"x - 1" + assert latex(x + 1) == r"x + 1" + + assert latex(x/3) == r"\frac{x}{3}" + assert latex(x/z) == r"\frac{x}{z}" + assert latex(x*y/z) == r"\frac{x y}{z}" + assert latex(x/(z*t)) == r"\frac{x}{z t}" + assert latex(x*y/(z*t)) == r"\frac{x y}{z t}" + + assert latex((x - 1)/y) == r"\frac{x - 1}{y}" + assert latex((x + 1)/y) == r"\frac{x + 1}{y}" + assert latex((-x - 1)/y) == r"\frac{-x - 1}{y}" + assert latex((x + 1)/(y*z)) == r"\frac{x + 1}{y z}" + assert latex(-y/(x + 1)) == r"\frac{-y}{x + 1}" + assert latex(y*z/(x + 1)) == r"\frac{y z}{x + 1}" + + assert latex(((u + 1)*x*y + 1)/((v - 1)*z - 1)) == \ + r"\frac{\left(u + 1\right) x y + 1}{\left(v - 1\right) z - 1}" + assert latex(((u + 1)*x*y + 1)/((v - 1)*z - t*u*v - 1)) == \ + r"\frac{\left(u + 1\right) x y + 1}{\left(v - 1\right) z - u v t - 1}" + + +def test_latex_Poly(): + assert latex(Poly(x**2 + 2 * x, x)) == \ + r"\operatorname{Poly}{\left( x^{2} + 2 x, x, domain=\mathbb{Z} \right)}" + assert latex(Poly(x/y, x)) == \ + r"\operatorname{Poly}{\left( \frac{1}{y} x, x, domain=\mathbb{Z}\left(y\right) \right)}" + assert latex(Poly(2.0*x + y)) == \ + r"\operatorname{Poly}{\left( 2.0 x + 1.0 y, x, y, domain=\mathbb{R} \right)}" + + +def test_latex_Poly_order(): + assert latex(Poly([a, 1, b, 2, c, 3], x)) == \ + r'\operatorname{Poly}{\left( a x^{5} + x^{4} + b x^{3} + 2 x^{2} + c'\ + r' x + 3, x, domain=\mathbb{Z}\left[a, b, c\right] \right)}' + assert latex(Poly([a, 1, b+c, 2, 3], x)) == \ + r'\operatorname{Poly}{\left( a x^{4} + x^{3} + \left(b + c\right) '\ + r'x^{2} + 2 x + 3, x, domain=\mathbb{Z}\left[a, b, c\right] \right)}' + assert latex(Poly(a*x**3 + x**2*y - x*y - c*y**3 - b*x*y**2 + y - a*x + b, + (x, y))) == \ + r'\operatorname{Poly}{\left( a x^{3} + x^{2}y - b xy^{2} - xy - '\ + r'a x - c y^{3} + y + b, x, y, domain=\mathbb{Z}\left[a, b, c\right] \right)}' + + +def test_latex_ComplexRootOf(): + assert latex(rootof(x**5 + x + 3, 0)) == \ + r"\operatorname{CRootOf} {\left(x^{5} + x + 3, 0\right)}" + + +def test_latex_RootSum(): + assert latex(RootSum(x**5 + x + 3, sin)) == \ + r"\operatorname{RootSum} {\left(x^{5} + x + 3, \left( x \mapsto \sin{\left(x \right)} \right)\right)}" + + +def test_settings(): + raises(TypeError, lambda: latex(x*y, method="garbage")) + + +def test_latex_numbers(): + assert latex(catalan(n)) == r"C_{n}" + assert latex(catalan(n)**2) == r"C_{n}^{2}" + assert latex(bernoulli(n)) == r"B_{n}" + assert latex(bernoulli(n, x)) == r"B_{n}\left(x\right)" + assert latex(bernoulli(n)**2) == r"B_{n}^{2}" + assert latex(bernoulli(n, x)**2) == r"B_{n}^{2}\left(x\right)" + assert latex(genocchi(n)) == r"G_{n}" + assert latex(genocchi(n, x)) == r"G_{n}\left(x\right)" + assert latex(genocchi(n)**2) == r"G_{n}^{2}" + assert latex(genocchi(n, x)**2) == r"G_{n}^{2}\left(x\right)" + assert latex(bell(n)) == r"B_{n}" + assert latex(bell(n, x)) == r"B_{n}\left(x\right)" + assert latex(bell(n, m, (x, y))) == r"B_{n, m}\left(x, y\right)" + assert latex(bell(n)**2) == r"B_{n}^{2}" + assert latex(bell(n, x)**2) == r"B_{n}^{2}\left(x\right)" + assert latex(bell(n, m, (x, y))**2) == r"B_{n, m}^{2}\left(x, y\right)" + assert latex(fibonacci(n)) == r"F_{n}" + assert latex(fibonacci(n, x)) == r"F_{n}\left(x\right)" + assert latex(fibonacci(n)**2) == r"F_{n}^{2}" + assert latex(fibonacci(n, x)**2) == r"F_{n}^{2}\left(x\right)" + assert latex(lucas(n)) == r"L_{n}" + assert latex(lucas(n)**2) == r"L_{n}^{2}" + assert latex(tribonacci(n)) == r"T_{n}" + assert latex(tribonacci(n, x)) == r"T_{n}\left(x\right)" + assert latex(tribonacci(n)**2) == r"T_{n}^{2}" + assert latex(tribonacci(n, x)**2) == r"T_{n}^{2}\left(x\right)" + assert latex(mobius(n)) == r"\mu\left(n\right)" + assert latex(mobius(n)**2) == r"\mu^{2}\left(n\right)" + + +def test_latex_euler(): + assert latex(euler(n)) == r"E_{n}" + assert latex(euler(n, x)) == r"E_{n}\left(x\right)" + assert latex(euler(n, x)**2) == r"E_{n}^{2}\left(x\right)" + + +def test_lamda(): + assert latex(Symbol('lamda')) == r"\lambda" + assert latex(Symbol('Lamda')) == r"\Lambda" + + +def test_custom_symbol_names(): + x = Symbol('x') + y = Symbol('y') + assert latex(x) == r"x" + assert latex(x, symbol_names={x: "x_i"}) == r"x_i" + assert latex(x + y, symbol_names={x: "x_i"}) == r"x_i + y" + assert latex(x**2, symbol_names={x: "x_i"}) == r"x_i^{2}" + assert latex(x + y, symbol_names={x: "x_i", y: "y_j"}) == r"x_i + y_j" + + +def test_matAdd(): + C = MatrixSymbol('C', 5, 5) + B = MatrixSymbol('B', 5, 5) + + n = symbols("n") + h = MatrixSymbol("h", 1, 1) + + assert latex(C - 2*B) in [r'- 2 B + C', r'C -2 B'] + assert latex(C + 2*B) in [r'2 B + C', r'C + 2 B'] + assert latex(B - 2*C) in [r'B - 2 C', r'- 2 C + B'] + assert latex(B + 2*C) in [r'B + 2 C', r'2 C + B'] + + assert latex(n * h - (-h + h.T) * (h + h.T)) == 'n h - \\left(- h + h^{T}\\right) \\left(h + h^{T}\\right)' + assert latex(MatAdd(MatAdd(h, h), MatAdd(h, h))) == '\\left(h + h\\right) + \\left(h + h\\right)' + assert latex(MatMul(MatMul(h, h), MatMul(h, h))) == '\\left(h h\\right) \\left(h h\\right)' + + +def test_matMul(): + A = MatrixSymbol('A', 5, 5) + B = MatrixSymbol('B', 5, 5) + x = Symbol('x') + assert latex(2*A) == r'2 A' + assert latex(2*x*A) == r'2 x A' + assert latex(-2*A) == r'- 2 A' + assert latex(1.5*A) == r'1.5 A' + assert latex(sqrt(2)*A) == r'\sqrt{2} A' + assert latex(-sqrt(2)*A) == r'- \sqrt{2} A' + assert latex(2*sqrt(2)*x*A) == r'2 \sqrt{2} x A' + assert latex(-2*A*(A + 2*B)) in [r'- 2 A \left(A + 2 B\right)', + r'- 2 A \left(2 B + A\right)'] + + +def test_latex_MatrixSlice(): + n = Symbol('n', integer=True) + x, y, z, w, t, = symbols('x y z w t') + X = MatrixSymbol('X', n, n) + Y = MatrixSymbol('Y', 10, 10) + Z = MatrixSymbol('Z', 10, 10) + + assert latex(MatrixSlice(X, (None, None, None), (None, None, None))) == r'X\left[:, :\right]' + assert latex(X[x:x + 1, y:y + 1]) == r'X\left[x:x + 1, y:y + 1\right]' + assert latex(X[x:x + 1:2, y:y + 1:2]) == r'X\left[x:x + 1:2, y:y + 1:2\right]' + assert latex(X[:x, y:]) == r'X\left[:x, y:\right]' + assert latex(X[:x, y:]) == r'X\left[:x, y:\right]' + assert latex(X[x:, :y]) == r'X\left[x:, :y\right]' + assert latex(X[x:y, z:w]) == r'X\left[x:y, z:w\right]' + assert latex(X[x:y:t, w:t:x]) == r'X\left[x:y:t, w:t:x\right]' + assert latex(X[x::y, t::w]) == r'X\left[x::y, t::w\right]' + assert latex(X[:x:y, :t:w]) == r'X\left[:x:y, :t:w\right]' + assert latex(X[::x, ::y]) == r'X\left[::x, ::y\right]' + assert latex(MatrixSlice(X, (0, None, None), (0, None, None))) == r'X\left[:, :\right]' + assert latex(MatrixSlice(X, (None, n, None), (None, n, None))) == r'X\left[:, :\right]' + assert latex(MatrixSlice(X, (0, n, None), (0, n, None))) == r'X\left[:, :\right]' + assert latex(MatrixSlice(X, (0, n, 2), (0, n, 2))) == r'X\left[::2, ::2\right]' + assert latex(X[1:2:3, 4:5:6]) == r'X\left[1:2:3, 4:5:6\right]' + assert latex(X[1:3:5, 4:6:8]) == r'X\left[1:3:5, 4:6:8\right]' + assert latex(X[1:10:2]) == r'X\left[1:10:2, :\right]' + assert latex(Y[:5, 1:9:2]) == r'Y\left[:5, 1:9:2\right]' + assert latex(Y[:5, 1:10:2]) == r'Y\left[:5, 1::2\right]' + assert latex(Y[5, :5:2]) == r'Y\left[5:6, :5:2\right]' + assert latex(X[0:1, 0:1]) == r'X\left[:1, :1\right]' + assert latex(X[0:1:2, 0:1:2]) == r'X\left[:1:2, :1:2\right]' + assert latex((Y + Z)[2:, 2:]) == r'\left(Y + Z\right)\left[2:, 2:\right]' + + +def test_latex_RandomDomain(): + from sympy.stats import Normal, Die, Exponential, pspace, where + from sympy.stats.rv import RandomDomain + + X = Normal('x1', 0, 1) + assert latex(where(X > 0)) == r"\text{Domain: }0 < x_{1} \wedge x_{1} < \infty" + + D = Die('d1', 6) + assert latex(where(D > 4)) == r"\text{Domain: }d_{1} = 5 \vee d_{1} = 6" + + A = Exponential('a', 1) + B = Exponential('b', 1) + assert latex( + pspace(Tuple(A, B)).domain) == \ + r"\text{Domain: }0 \leq a \wedge 0 \leq b \wedge a < \infty \wedge b < \infty" + + assert latex(RandomDomain(FiniteSet(x), FiniteSet(1, 2))) == \ + r'\text{Domain: }\left\{x\right\} \in \left\{1, 2\right\}' + +def test_PrettyPoly(): + from sympy.polys.domains import QQ + F = QQ.frac_field(x, y) + R = QQ[x, y] + + assert latex(F.convert(x/(x + y))) == latex(x/(x + y)) + assert latex(R.convert(x + y)) == latex(x + y) + + +def test_integral_transforms(): + x = Symbol("x") + k = Symbol("k") + f = Function("f") + a = Symbol("a") + b = Symbol("b") + + assert latex(MellinTransform(f(x), x, k)) == \ + r"\mathcal{M}_{x}\left[f{\left(x \right)}\right]\left(k\right)" + assert latex(InverseMellinTransform(f(k), k, x, a, b)) == \ + r"\mathcal{M}^{-1}_{k}\left[f{\left(k \right)}\right]\left(x\right)" + + assert latex(LaplaceTransform(f(x), x, k)) == \ + r"\mathcal{L}_{x}\left[f{\left(x \right)}\right]\left(k\right)" + assert latex(InverseLaplaceTransform(f(k), k, x, (a, b))) == \ + r"\mathcal{L}^{-1}_{k}\left[f{\left(k \right)}\right]\left(x\right)" + + assert latex(FourierTransform(f(x), x, k)) == \ + r"\mathcal{F}_{x}\left[f{\left(x \right)}\right]\left(k\right)" + assert latex(InverseFourierTransform(f(k), k, x)) == \ + r"\mathcal{F}^{-1}_{k}\left[f{\left(k \right)}\right]\left(x\right)" + + assert latex(CosineTransform(f(x), x, k)) == \ + r"\mathcal{COS}_{x}\left[f{\left(x \right)}\right]\left(k\right)" + assert latex(InverseCosineTransform(f(k), k, x)) == \ + r"\mathcal{COS}^{-1}_{k}\left[f{\left(k \right)}\right]\left(x\right)" + + assert latex(SineTransform(f(x), x, k)) == \ + r"\mathcal{SIN}_{x}\left[f{\left(x \right)}\right]\left(k\right)" + assert latex(InverseSineTransform(f(k), k, x)) == \ + r"\mathcal{SIN}^{-1}_{k}\left[f{\left(k \right)}\right]\left(x\right)" + + +def test_PolynomialRingBase(): + from sympy.polys.domains import QQ + assert latex(QQ.old_poly_ring(x, y)) == r"\mathbb{Q}\left[x, y\right]" + assert latex(QQ.old_poly_ring(x, y, order="ilex")) == \ + r"S_<^{-1}\mathbb{Q}\left[x, y\right]" + + +def test_categories(): + from sympy.categories import (Object, IdentityMorphism, + NamedMorphism, Category, Diagram, + DiagramGrid) + + A1 = Object("A1") + A2 = Object("A2") + A3 = Object("A3") + + f1 = NamedMorphism(A1, A2, "f1") + f2 = NamedMorphism(A2, A3, "f2") + id_A1 = IdentityMorphism(A1) + + K1 = Category("K1") + + assert latex(A1) == r"A_{1}" + assert latex(f1) == r"f_{1}:A_{1}\rightarrow A_{2}" + assert latex(id_A1) == r"id:A_{1}\rightarrow A_{1}" + assert latex(f2*f1) == r"f_{2}\circ f_{1}:A_{1}\rightarrow A_{3}" + + assert latex(K1) == r"\mathbf{K_{1}}" + + d = Diagram() + assert latex(d) == r"\emptyset" + + d = Diagram({f1: "unique", f2: S.EmptySet}) + assert latex(d) == r"\left\{ f_{2}\circ f_{1}:A_{1}" \ + r"\rightarrow A_{3} : \emptyset, \ id:A_{1}\rightarrow " \ + r"A_{1} : \emptyset, \ id:A_{2}\rightarrow A_{2} : " \ + r"\emptyset, \ id:A_{3}\rightarrow A_{3} : \emptyset, " \ + r"\ f_{1}:A_{1}\rightarrow A_{2} : \left\{unique\right\}, " \ + r"\ f_{2}:A_{2}\rightarrow A_{3} : \emptyset\right\}" + + d = Diagram({f1: "unique", f2: S.EmptySet}, {f2 * f1: "unique"}) + assert latex(d) == r"\left\{ f_{2}\circ f_{1}:A_{1}" \ + r"\rightarrow A_{3} : \emptyset, \ id:A_{1}\rightarrow " \ + r"A_{1} : \emptyset, \ id:A_{2}\rightarrow A_{2} : " \ + r"\emptyset, \ id:A_{3}\rightarrow A_{3} : \emptyset, " \ + r"\ f_{1}:A_{1}\rightarrow A_{2} : \left\{unique\right\}," \ + r" \ f_{2}:A_{2}\rightarrow A_{3} : \emptyset\right\}" \ + r"\Longrightarrow \left\{ f_{2}\circ f_{1}:A_{1}" \ + r"\rightarrow A_{3} : \left\{unique\right\}\right\}" + + # A linear diagram. + A = Object("A") + B = Object("B") + C = Object("C") + f = NamedMorphism(A, B, "f") + g = NamedMorphism(B, C, "g") + d = Diagram([f, g]) + grid = DiagramGrid(d) + + assert latex(grid) == r"\begin{array}{cc}" + "\n" \ + r"A & B \\" + "\n" \ + r" & C " + "\n" \ + r"\end{array}" + "\n" + + +def test_Modules(): + from sympy.polys.domains import QQ + from sympy.polys.agca import homomorphism + + R = QQ.old_poly_ring(x, y) + F = R.free_module(2) + M = F.submodule([x, y], [1, x**2]) + + assert latex(F) == r"{\mathbb{Q}\left[x, y\right]}^{2}" + assert latex(M) == \ + r"\left\langle {\left[ {x},{y} \right]},{\left[ {1},{x^{2}} \right]} \right\rangle" + + I = R.ideal(x**2, y) + assert latex(I) == r"\left\langle {x^{2}},{y} \right\rangle" + + Q = F / M + assert latex(Q) == \ + r"\frac{{\mathbb{Q}\left[x, y\right]}^{2}}{\left\langle {\left[ {x},"\ + r"{y} \right]},{\left[ {1},{x^{2}} \right]} \right\rangle}" + assert latex(Q.submodule([1, x**3/2], [2, y])) == \ + r"\left\langle {{\left[ {1},{\frac{x^{3}}{2}} \right]} + {\left"\ + r"\langle {\left[ {x},{y} \right]},{\left[ {1},{x^{2}} \right]} "\ + r"\right\rangle}},{{\left[ {2},{y} \right]} + {\left\langle {\left[ "\ + r"{x},{y} \right]},{\left[ {1},{x^{2}} \right]} \right\rangle}} \right\rangle" + + h = homomorphism(QQ.old_poly_ring(x).free_module(2), + QQ.old_poly_ring(x).free_module(2), [0, 0]) + + assert latex(h) == \ + r"{\left[\begin{matrix}0 & 0\\0 & 0\end{matrix}\right]} : "\ + r"{{\mathbb{Q}\left[x\right]}^{2}} \to {{\mathbb{Q}\left[x\right]}^{2}}" + + +def test_QuotientRing(): + from sympy.polys.domains import QQ + R = QQ.old_poly_ring(x)/[x**2 + 1] + + assert latex(R) == \ + r"\frac{\mathbb{Q}\left[x\right]}{\left\langle {x^{2} + 1} \right\rangle}" + assert latex(R.one) == r"{1} + {\left\langle {x^{2} + 1} \right\rangle}" + + +def test_Tr(): + #TODO: Handle indices + A, B = symbols('A B', commutative=False) + t = Tr(A*B) + assert latex(t) == r'\operatorname{tr}\left(A B\right)' + + +def test_Determinant(): + from sympy.matrices import Determinant, Inverse, BlockMatrix, OneMatrix, ZeroMatrix + m = Matrix(((1, 2), (3, 4))) + assert latex(Determinant(m)) == '\\left|{\\begin{matrix}1 & 2\\\\3 & 4\\end{matrix}}\\right|' + assert latex(Determinant(Inverse(m))) == \ + '\\left|{\\left[\\begin{matrix}1 & 2\\\\3 & 4\\end{matrix}\\right]^{-1}}\\right|' + X = MatrixSymbol('X', 2, 2) + assert latex(Determinant(X)) == '\\left|{X}\\right|' + assert latex(Determinant(X + m)) == \ + '\\left|{\\left[\\begin{matrix}1 & 2\\\\3 & 4\\end{matrix}\\right] + X}\\right|' + assert latex(Determinant(BlockMatrix(((OneMatrix(2, 2), X), + (m, ZeroMatrix(2, 2)))))) == \ + '\\left|{\\begin{matrix}1 & X\\\\\\left[\\begin{matrix}1 & 2\\\\3 & 4\\end{matrix}\\right] & 0\\end{matrix}}\\right|' + + +def test_Adjoint(): + from sympy.matrices import Adjoint, Inverse, Transpose + X = MatrixSymbol('X', 2, 2) + Y = MatrixSymbol('Y', 2, 2) + assert latex(Adjoint(X)) == r'X^{\dagger}' + assert latex(Adjoint(X + Y)) == r'\left(X + Y\right)^{\dagger}' + assert latex(Adjoint(X) + Adjoint(Y)) == r'X^{\dagger} + Y^{\dagger}' + assert latex(Adjoint(X*Y)) == r'\left(X Y\right)^{\dagger}' + assert latex(Adjoint(Y)*Adjoint(X)) == r'Y^{\dagger} X^{\dagger}' + assert latex(Adjoint(X**2)) == r'\left(X^{2}\right)^{\dagger}' + assert latex(Adjoint(X)**2) == r'\left(X^{\dagger}\right)^{2}' + assert latex(Adjoint(Inverse(X))) == r'\left(X^{-1}\right)^{\dagger}' + assert latex(Inverse(Adjoint(X))) == r'\left(X^{\dagger}\right)^{-1}' + assert latex(Adjoint(Transpose(X))) == r'\left(X^{T}\right)^{\dagger}' + assert latex(Transpose(Adjoint(X))) == r'\left(X^{\dagger}\right)^{T}' + assert latex(Transpose(Adjoint(X) + Y)) == r'\left(X^{\dagger} + Y\right)^{T}' + m = Matrix(((1, 2), (3, 4))) + assert latex(Adjoint(m)) == '\\left[\\begin{matrix}1 & 2\\\\3 & 4\\end{matrix}\\right]^{\\dagger}' + assert latex(Adjoint(m+X)) == \ + '\\left(\\left[\\begin{matrix}1 & 2\\\\3 & 4\\end{matrix}\\right] + X\\right)^{\\dagger}' + from sympy.matrices import BlockMatrix, OneMatrix, ZeroMatrix + assert latex(Adjoint(BlockMatrix(((OneMatrix(2, 2), X), + (m, ZeroMatrix(2, 2)))))) == \ + '\\left[\\begin{matrix}1 & X\\\\\\left[\\begin{matrix}1 & 2\\\\3 & 4\\end{matrix}\\right] & 0\\end{matrix}\\right]^{\\dagger}' + # Issue 20959 + Mx = MatrixSymbol('M^x', 2, 2) + assert latex(Adjoint(Mx)) == r'\left(M^{x}\right)^{\dagger}' + + # adjoint style + assert latex(Adjoint(X), adjoint_style="star") == r'X^{\ast}' + assert latex(Adjoint(X + Y), adjoint_style="hermitian") == r'\left(X + Y\right)^{\mathsf{H}}' + assert latex(Adjoint(X) + Adjoint(Y), adjoint_style="dagger") == r'X^{\dagger} + Y^{\dagger}' + assert latex(Adjoint(Y)*Adjoint(X)) == r'Y^{\dagger} X^{\dagger}' + assert latex(Adjoint(X**2), adjoint_style="star") == r'\left(X^{2}\right)^{\ast}' + assert latex(Adjoint(X)**2, adjoint_style="hermitian") == r'\left(X^{\mathsf{H}}\right)^{2}' + +def test_Transpose(): + from sympy.matrices import Transpose, MatPow, HadamardPower + X = MatrixSymbol('X', 2, 2) + Y = MatrixSymbol('Y', 2, 2) + assert latex(Transpose(X)) == r'X^{T}' + assert latex(Transpose(X + Y)) == r'\left(X + Y\right)^{T}' + + assert latex(Transpose(HadamardPower(X, 2))) == r'\left(X^{\circ {2}}\right)^{T}' + assert latex(HadamardPower(Transpose(X), 2)) == r'\left(X^{T}\right)^{\circ {2}}' + assert latex(Transpose(MatPow(X, 2))) == r'\left(X^{2}\right)^{T}' + assert latex(MatPow(Transpose(X), 2)) == r'\left(X^{T}\right)^{2}' + m = Matrix(((1, 2), (3, 4))) + assert latex(Transpose(m)) == '\\left[\\begin{matrix}1 & 2\\\\3 & 4\\end{matrix}\\right]^{T}' + assert latex(Transpose(m+X)) == \ + '\\left(\\left[\\begin{matrix}1 & 2\\\\3 & 4\\end{matrix}\\right] + X\\right)^{T}' + from sympy.matrices import BlockMatrix, OneMatrix, ZeroMatrix + assert latex(Transpose(BlockMatrix(((OneMatrix(2, 2), X), + (m, ZeroMatrix(2, 2)))))) == \ + '\\left[\\begin{matrix}1 & X\\\\\\left[\\begin{matrix}1 & 2\\\\3 & 4\\end{matrix}\\right] & 0\\end{matrix}\\right]^{T}' + # Issue 20959 + Mx = MatrixSymbol('M^x', 2, 2) + assert latex(Transpose(Mx)) == r'\left(M^{x}\right)^{T}' + + +def test_Hadamard(): + from sympy.matrices import HadamardProduct, HadamardPower + from sympy.matrices.expressions import MatAdd, MatMul, MatPow + X = MatrixSymbol('X', 2, 2) + Y = MatrixSymbol('Y', 2, 2) + assert latex(HadamardProduct(X, Y*Y)) == r'X \circ Y^{2}' + assert latex(HadamardProduct(X, Y)*Y) == r'\left(X \circ Y\right) Y' + + assert latex(HadamardPower(X, 2)) == r'X^{\circ {2}}' + assert latex(HadamardPower(X, -1)) == r'X^{\circ \left({-1}\right)}' + assert latex(HadamardPower(MatAdd(X, Y), 2)) == \ + r'\left(X + Y\right)^{\circ {2}}' + assert latex(HadamardPower(MatMul(X, Y), 2)) == \ + r'\left(X Y\right)^{\circ {2}}' + + assert latex(HadamardPower(MatPow(X, -1), -1)) == \ + r'\left(X^{-1}\right)^{\circ \left({-1}\right)}' + assert latex(MatPow(HadamardPower(X, -1), -1)) == \ + r'\left(X^{\circ \left({-1}\right)}\right)^{-1}' + + assert latex(HadamardPower(X, n+1)) == \ + r'X^{\circ \left({n + 1}\right)}' + + +def test_MatPow(): + from sympy.matrices.expressions import MatPow + X = MatrixSymbol('X', 2, 2) + Y = MatrixSymbol('Y', 2, 2) + assert latex(MatPow(X, 2)) == 'X^{2}' + assert latex(MatPow(X*X, 2)) == '\\left(X^{2}\\right)^{2}' + assert latex(MatPow(X*Y, 2)) == '\\left(X Y\\right)^{2}' + assert latex(MatPow(X + Y, 2)) == '\\left(X + Y\\right)^{2}' + assert latex(MatPow(X + X, 2)) == '\\left(2 X\\right)^{2}' + # Issue 20959 + Mx = MatrixSymbol('M^x', 2, 2) + assert latex(MatPow(Mx, 2)) == r'\left(M^{x}\right)^{2}' + + +def test_ElementwiseApplyFunction(): + X = MatrixSymbol('X', 2, 2) + expr = (X.T*X).applyfunc(sin) + assert latex(expr) == r"{\left( d \mapsto \sin{\left(d \right)} \right)}_{\circ}\left({X^{T} X}\right)" + expr = X.applyfunc(Lambda(x, 1/x)) + assert latex(expr) == r'{\left( x \mapsto \frac{1}{x} \right)}_{\circ}\left({X}\right)' + + +def test_ZeroMatrix(): + from sympy.matrices.expressions.special import ZeroMatrix + assert latex(ZeroMatrix(1, 1), mat_symbol_style='plain') == r"0" + assert latex(ZeroMatrix(1, 1), mat_symbol_style='bold') == r"\mathbf{0}" + + +def test_OneMatrix(): + from sympy.matrices.expressions.special import OneMatrix + assert latex(OneMatrix(3, 4), mat_symbol_style='plain') == r"1" + assert latex(OneMatrix(3, 4), mat_symbol_style='bold') == r"\mathbf{1}" + + +def test_Identity(): + from sympy.matrices.expressions.special import Identity + assert latex(Identity(1), mat_symbol_style='plain') == r"\mathbb{I}" + assert latex(Identity(1), mat_symbol_style='bold') == r"\mathbf{I}" + + +def test_latex_DFT_IDFT(): + from sympy.matrices.expressions.fourier import DFT, IDFT + assert latex(DFT(13)) == r"\text{DFT}_{13}" + assert latex(IDFT(x)) == r"\text{IDFT}_{x}" + + +def test_boolean_args_order(): + syms = symbols('a:f') + + expr = And(*syms) + assert latex(expr) == r'a \wedge b \wedge c \wedge d \wedge e \wedge f' + + expr = Or(*syms) + assert latex(expr) == r'a \vee b \vee c \vee d \vee e \vee f' + + expr = Equivalent(*syms) + assert latex(expr) == \ + r'a \Leftrightarrow b \Leftrightarrow c \Leftrightarrow d \Leftrightarrow e \Leftrightarrow f' + + expr = Xor(*syms) + assert latex(expr) == \ + r'a \veebar b \veebar c \veebar d \veebar e \veebar f' + + +def test_imaginary(): + i = sqrt(-1) + assert latex(i) == r'i' + + +def test_builtins_without_args(): + assert latex(sin) == r'\sin' + assert latex(cos) == r'\cos' + assert latex(tan) == r'\tan' + assert latex(log) == r'\log' + assert latex(Ei) == r'\operatorname{Ei}' + assert latex(zeta) == r'\zeta' + + +def test_latex_greek_functions(): + # bug because capital greeks that have roman equivalents should not use + # \Alpha, \Beta, \Eta, etc. + s = Function('Alpha') + assert latex(s) == r'\mathrm{A}' + assert latex(s(x)) == r'\mathrm{A}{\left(x \right)}' + s = Function('Beta') + assert latex(s) == r'\mathrm{B}' + s = Function('Eta') + assert latex(s) == r'\mathrm{H}' + assert latex(s(x)) == r'\mathrm{H}{\left(x \right)}' + + # bug because sympy.core.numbers.Pi is special + p = Function('Pi') + # assert latex(p(x)) == r'\Pi{\left(x \right)}' + assert latex(p) == r'\Pi' + + # bug because not all greeks are included + c = Function('chi') + assert latex(c(x)) == r'\chi{\left(x \right)}' + assert latex(c) == r'\chi' + + +def test_translate(): + s = 'Alpha' + assert translate(s) == r'\mathrm{A}' + s = 'Beta' + assert translate(s) == r'\mathrm{B}' + s = 'Eta' + assert translate(s) == r'\mathrm{H}' + s = 'omicron' + assert translate(s) == r'o' + s = 'Pi' + assert translate(s) == r'\Pi' + s = 'pi' + assert translate(s) == r'\pi' + s = 'LamdaHatDOT' + assert translate(s) == r'\dot{\hat{\Lambda}}' + + +def test_other_symbols(): + from sympy.printing.latex import other_symbols + for s in other_symbols: + assert latex(symbols(s)) == r"" "\\" + s + + +def test_modifiers(): + # Test each modifier individually in the simplest case + # (with funny capitalizations) + assert latex(symbols("xMathring")) == r"\mathring{x}" + assert latex(symbols("xCheck")) == r"\check{x}" + assert latex(symbols("xBreve")) == r"\breve{x}" + assert latex(symbols("xAcute")) == r"\acute{x}" + assert latex(symbols("xGrave")) == r"\grave{x}" + assert latex(symbols("xTilde")) == r"\tilde{x}" + assert latex(symbols("xPrime")) == r"{x}'" + assert latex(symbols("xddDDot")) == r"\ddddot{x}" + assert latex(symbols("xDdDot")) == r"\dddot{x}" + assert latex(symbols("xDDot")) == r"\ddot{x}" + assert latex(symbols("xBold")) == r"\boldsymbol{x}" + assert latex(symbols("xnOrM")) == r"\left\|{x}\right\|" + assert latex(symbols("xAVG")) == r"\left\langle{x}\right\rangle" + assert latex(symbols("xHat")) == r"\hat{x}" + assert latex(symbols("xDot")) == r"\dot{x}" + assert latex(symbols("xBar")) == r"\bar{x}" + assert latex(symbols("xVec")) == r"\vec{x}" + assert latex(symbols("xAbs")) == r"\left|{x}\right|" + assert latex(symbols("xMag")) == r"\left|{x}\right|" + assert latex(symbols("xPrM")) == r"{x}'" + assert latex(symbols("xBM")) == r"\boldsymbol{x}" + # Test strings that are *only* the names of modifiers + assert latex(symbols("Mathring")) == r"Mathring" + assert latex(symbols("Check")) == r"Check" + assert latex(symbols("Breve")) == r"Breve" + assert latex(symbols("Acute")) == r"Acute" + assert latex(symbols("Grave")) == r"Grave" + assert latex(symbols("Tilde")) == r"Tilde" + assert latex(symbols("Prime")) == r"Prime" + assert latex(symbols("DDot")) == r"\dot{D}" + assert latex(symbols("Bold")) == r"Bold" + assert latex(symbols("NORm")) == r"NORm" + assert latex(symbols("AVG")) == r"AVG" + assert latex(symbols("Hat")) == r"Hat" + assert latex(symbols("Dot")) == r"Dot" + assert latex(symbols("Bar")) == r"Bar" + assert latex(symbols("Vec")) == r"Vec" + assert latex(symbols("Abs")) == r"Abs" + assert latex(symbols("Mag")) == r"Mag" + assert latex(symbols("PrM")) == r"PrM" + assert latex(symbols("BM")) == r"BM" + assert latex(symbols("hbar")) == r"\hbar" + # Check a few combinations + assert latex(symbols("xvecdot")) == r"\dot{\vec{x}}" + assert latex(symbols("xDotVec")) == r"\vec{\dot{x}}" + assert latex(symbols("xHATNorm")) == r"\left\|{\hat{x}}\right\|" + # Check a couple big, ugly combinations + assert latex(symbols('xMathringBm_yCheckPRM__zbreveAbs')) == \ + r"\boldsymbol{\mathring{x}}^{\left|{\breve{z}}\right|}_{{\check{y}}'}" + assert latex(symbols('alphadothat_nVECDOT__tTildePrime')) == \ + r"\hat{\dot{\alpha}}^{{\tilde{t}}'}_{\dot{\vec{n}}}" + + +def test_greek_symbols(): + assert latex(Symbol('alpha')) == r'\alpha' + assert latex(Symbol('beta')) == r'\beta' + assert latex(Symbol('gamma')) == r'\gamma' + assert latex(Symbol('delta')) == r'\delta' + assert latex(Symbol('epsilon')) == r'\epsilon' + assert latex(Symbol('zeta')) == r'\zeta' + assert latex(Symbol('eta')) == r'\eta' + assert latex(Symbol('theta')) == r'\theta' + assert latex(Symbol('iota')) == r'\iota' + assert latex(Symbol('kappa')) == r'\kappa' + assert latex(Symbol('lambda')) == r'\lambda' + assert latex(Symbol('mu')) == r'\mu' + assert latex(Symbol('nu')) == r'\nu' + assert latex(Symbol('xi')) == r'\xi' + assert latex(Symbol('omicron')) == r'o' + assert latex(Symbol('pi')) == r'\pi' + assert latex(Symbol('rho')) == r'\rho' + assert latex(Symbol('sigma')) == r'\sigma' + assert latex(Symbol('tau')) == r'\tau' + assert latex(Symbol('upsilon')) == r'\upsilon' + assert latex(Symbol('phi')) == r'\phi' + assert latex(Symbol('chi')) == r'\chi' + assert latex(Symbol('psi')) == r'\psi' + assert latex(Symbol('omega')) == r'\omega' + + assert latex(Symbol('Alpha')) == r'\mathrm{A}' + assert latex(Symbol('Beta')) == r'\mathrm{B}' + assert latex(Symbol('Gamma')) == r'\Gamma' + assert latex(Symbol('Delta')) == r'\Delta' + assert latex(Symbol('Epsilon')) == r'\mathrm{E}' + assert latex(Symbol('Zeta')) == r'\mathrm{Z}' + assert latex(Symbol('Eta')) == r'\mathrm{H}' + assert latex(Symbol('Theta')) == r'\Theta' + assert latex(Symbol('Iota')) == r'\mathrm{I}' + assert latex(Symbol('Kappa')) == r'\mathrm{K}' + assert latex(Symbol('Lambda')) == r'\Lambda' + assert latex(Symbol('Mu')) == r'\mathrm{M}' + assert latex(Symbol('Nu')) == r'\mathrm{N}' + assert latex(Symbol('Xi')) == r'\Xi' + assert latex(Symbol('Omicron')) == r'\mathrm{O}' + assert latex(Symbol('Pi')) == r'\Pi' + assert latex(Symbol('Rho')) == r'\mathrm{P}' + assert latex(Symbol('Sigma')) == r'\Sigma' + assert latex(Symbol('Tau')) == r'\mathrm{T}' + assert latex(Symbol('Upsilon')) == r'\Upsilon' + assert latex(Symbol('Phi')) == r'\Phi' + assert latex(Symbol('Chi')) == r'\mathrm{X}' + assert latex(Symbol('Psi')) == r'\Psi' + assert latex(Symbol('Omega')) == r'\Omega' + + assert latex(Symbol('varepsilon')) == r'\varepsilon' + assert latex(Symbol('varkappa')) == r'\varkappa' + assert latex(Symbol('varphi')) == r'\varphi' + assert latex(Symbol('varpi')) == r'\varpi' + assert latex(Symbol('varrho')) == r'\varrho' + assert latex(Symbol('varsigma')) == r'\varsigma' + assert latex(Symbol('vartheta')) == r'\vartheta' + + +def test_fancyset_symbols(): + assert latex(S.Rationals) == r'\mathbb{Q}' + assert latex(S.Naturals) == r'\mathbb{N}' + assert latex(S.Naturals0) == r'\mathbb{N}_0' + assert latex(S.Integers) == r'\mathbb{Z}' + assert latex(S.Reals) == r'\mathbb{R}' + assert latex(S.Complexes) == r'\mathbb{C}' + + +@XFAIL +def test_builtin_without_args_mismatched_names(): + assert latex(CosineTransform) == r'\mathcal{COS}' + + +def test_builtin_no_args(): + assert latex(Chi) == r'\operatorname{Chi}' + assert latex(beta) == r'\operatorname{B}' + assert latex(gamma) == r'\Gamma' + assert latex(KroneckerDelta) == r'\delta' + assert latex(DiracDelta) == r'\delta' + assert latex(lowergamma) == r'\gamma' + + +def test_issue_6853(): + p = Function('Pi') + assert latex(p(x)) == r"\Pi{\left(x \right)}" + + +def test_Mul(): + e = Mul(-2, x + 1, evaluate=False) + assert latex(e) == r'- 2 \left(x + 1\right)' + e = Mul(2, x + 1, evaluate=False) + assert latex(e) == r'2 \left(x + 1\right)' + e = Mul(S.Half, x + 1, evaluate=False) + assert latex(e) == r'\frac{x + 1}{2}' + e = Mul(y, x + 1, evaluate=False) + assert latex(e) == r'y \left(x + 1\right)' + e = Mul(-y, x + 1, evaluate=False) + assert latex(e) == r'- y \left(x + 1\right)' + e = Mul(-2, x + 1) + assert latex(e) == r'- 2 x - 2' + e = Mul(2, x + 1) + assert latex(e) == r'2 x + 2' + + +def test_Pow(): + e = Pow(2, 2, evaluate=False) + assert latex(e) == r'2^{2}' + assert latex(x**(Rational(-1, 3))) == r'\frac{1}{\sqrt[3]{x}}' + x2 = Symbol(r'x^2') + assert latex(x2**2) == r'\left(x^{2}\right)^{2}' + # Issue 11011 + assert latex(S('1.453e4500')**x) == r'{1.453 \cdot 10^{4500}}^{x}' + + +def test_issue_7180(): + assert latex(Equivalent(x, y)) == r"x \Leftrightarrow y" + assert latex(Not(Equivalent(x, y))) == r"x \not\Leftrightarrow y" + + +def test_issue_8409(): + assert latex(S.Half**n) == r"\left(\frac{1}{2}\right)^{n}" + + +def test_issue_8470(): + from sympy.parsing.sympy_parser import parse_expr + e = parse_expr("-B*A", evaluate=False) + assert latex(e) == r"A \left(- B\right)" + + +def test_issue_15439(): + x = MatrixSymbol('x', 2, 2) + y = MatrixSymbol('y', 2, 2) + assert latex((x * y).subs(y, -y)) == r"x \left(- y\right)" + assert latex((x * y).subs(y, -2*y)) == r"x \left(- 2 y\right)" + assert latex((x * y).subs(x, -x)) == r"\left(- x\right) y" + + +def test_issue_2934(): + assert latex(Symbol(r'\frac{a_1}{b_1}')) == r'\frac{a_1}{b_1}' + + +def test_issue_10489(): + latexSymbolWithBrace = r'C_{x_{0}}' + s = Symbol(latexSymbolWithBrace) + assert latex(s) == latexSymbolWithBrace + assert latex(cos(s)) == r'\cos{\left(C_{x_{0}} \right)}' + + +def test_issue_12886(): + m__1, l__1 = symbols('m__1, l__1') + assert latex(m__1**2 + l__1**2) == \ + r'\left(l^{1}\right)^{2} + \left(m^{1}\right)^{2}' + + +def test_issue_13559(): + from sympy.parsing.sympy_parser import parse_expr + expr = parse_expr('5/1', evaluate=False) + assert latex(expr) == r"\frac{5}{1}" + + +def test_issue_13651(): + expr = c + Mul(-1, a + b, evaluate=False) + assert latex(expr) == r"c - \left(a + b\right)" + + +def test_latex_UnevaluatedExpr(): + x = symbols("x") + he = UnevaluatedExpr(1/x) + assert latex(he) == latex(1/x) == r"\frac{1}{x}" + assert latex(he**2) == r"\left(\frac{1}{x}\right)^{2}" + assert latex(he + 1) == r"1 + \frac{1}{x}" + assert latex(x*he) == r"x \frac{1}{x}" + + +def test_MatrixElement_printing(): + # test cases for issue #11821 + A = MatrixSymbol("A", 1, 3) + B = MatrixSymbol("B", 1, 3) + C = MatrixSymbol("C", 1, 3) + + assert latex(A[0, 0]) == r"{A}_{0,0}" + assert latex(3 * A[0, 0]) == r"3 {A}_{0,0}" + + F = C[0, 0].subs(C, A - B) + assert latex(F) == r"{\left(A - B\right)}_{0,0}" + + i, j, k = symbols("i j k") + M = MatrixSymbol("M", k, k) + N = MatrixSymbol("N", k, k) + assert latex((M*N)[i, j]) == \ + r'\sum_{i_{1}=0}^{k - 1} {M}_{i,i_{1}} {N}_{i_{1},j}' + + X_a = MatrixSymbol('X_a', 3, 3) + assert latex(X_a[0, 0]) == r"{X_{a}}_{0,0}" + + +def test_MatrixSymbol_printing(): + # test cases for issue #14237 + A = MatrixSymbol("A", 3, 3) + B = MatrixSymbol("B", 3, 3) + C = MatrixSymbol("C", 3, 3) + + assert latex(-A) == r"- A" + assert latex(A - A*B - B) == r"A - A B - B" + assert latex(-A*B - A*B*C - B) == r"- A B - A B C - B" + + +def test_DotProduct_printing(): + X = MatrixSymbol('X', 3, 1) + Y = MatrixSymbol('Y', 3, 1) + a = Symbol('a') + assert latex(DotProduct(X, Y)) == r"X \cdot Y" + assert latex(DotProduct(a * X, Y)) == r"a X \cdot Y" + assert latex(a * DotProduct(X, Y)) == r"a \left(X \cdot Y\right)" + + +def test_KroneckerProduct_printing(): + A = MatrixSymbol('A', 3, 3) + B = MatrixSymbol('B', 2, 2) + assert latex(KroneckerProduct(A, B)) == r'A \otimes B' + + +def test_Series_printing(): + tf1 = TransferFunction(x*y**2 - z, y**3 - t**3, y) + tf2 = TransferFunction(x - y, x + y, y) + tf3 = TransferFunction(t*x**2 - t**w*x + w, t - y, y) + assert latex(Series(tf1, tf2)) == \ + r'\left(\frac{x y^{2} - z}{- t^{3} + y^{3}}\right) \left(\frac{x - y}{x + y}\right)' + assert latex(Series(tf1, tf2, tf3)) == \ + r'\left(\frac{x y^{2} - z}{- t^{3} + y^{3}}\right) \left(\frac{x - y}{x + y}\right) \left(\frac{t x^{2} - t^{w} x + w}{t - y}\right)' + assert latex(Series(-tf2, tf1)) == \ + r'\left(\frac{- x + y}{x + y}\right) \left(\frac{x y^{2} - z}{- t^{3} + y^{3}}\right)' + + M_1 = Matrix([[5/s], [5/(2*s)]]) + T_1 = TransferFunctionMatrix.from_Matrix(M_1, s) + M_2 = Matrix([[5, 6*s**3]]) + T_2 = TransferFunctionMatrix.from_Matrix(M_2, s) + # Brackets + assert latex(T_1*(T_2 + T_2)) == \ + r'\left[\begin{matrix}\frac{5}{s}\\\frac{5}{2 s}\end{matrix}\right]_\tau\cdot\left(\left[\begin{matrix}\frac{5}{1} &' \ + r' \frac{6 s^{3}}{1}\end{matrix}\right]_\tau + \left[\begin{matrix}\frac{5}{1} & \frac{6 s^{3}}{1}\end{matrix}\right]_\tau\right)' \ + == latex(MIMOSeries(MIMOParallel(T_2, T_2), T_1)) + # No Brackets + M_3 = Matrix([[5, 6], [6, 5/s]]) + T_3 = TransferFunctionMatrix.from_Matrix(M_3, s) + assert latex(T_1*T_2 + T_3) == r'\left[\begin{matrix}\frac{5}{s}\\\frac{5}{2 s}\end{matrix}\right]_\tau\cdot\left[\begin{matrix}' \ + r'\frac{5}{1} & \frac{6 s^{3}}{1}\end{matrix}\right]_\tau + \left[\begin{matrix}\frac{5}{1} & \frac{6}{1}\\\frac{6}{1} & ' \ + r'\frac{5}{s}\end{matrix}\right]_\tau' == latex(MIMOParallel(MIMOSeries(T_2, T_1), T_3)) + + +def test_TransferFunction_printing(): + tf1 = TransferFunction(x - 1, x + 1, x) + assert latex(tf1) == r"\frac{x - 1}{x + 1}" + tf2 = TransferFunction(x + 1, 2 - y, x) + assert latex(tf2) == r"\frac{x + 1}{2 - y}" + tf3 = TransferFunction(y, y**2 + 2*y + 3, y) + assert latex(tf3) == r"\frac{y}{y^{2} + 2 y + 3}" + + +def test_Parallel_printing(): + tf1 = TransferFunction(x*y**2 - z, y**3 - t**3, y) + tf2 = TransferFunction(x - y, x + y, y) + assert latex(Parallel(tf1, tf2)) == \ + r'\frac{x y^{2} - z}{- t^{3} + y^{3}} + \frac{x - y}{x + y}' + assert latex(Parallel(-tf2, tf1)) == \ + r'\frac{- x + y}{x + y} + \frac{x y^{2} - z}{- t^{3} + y^{3}}' + + M_1 = Matrix([[5, 6], [6, 5/s]]) + T_1 = TransferFunctionMatrix.from_Matrix(M_1, s) + M_2 = Matrix([[5/s, 6], [6, 5/(s - 1)]]) + T_2 = TransferFunctionMatrix.from_Matrix(M_2, s) + M_3 = Matrix([[6, 5/(s*(s - 1))], [5, 6]]) + T_3 = TransferFunctionMatrix.from_Matrix(M_3, s) + assert latex(T_1 + T_2 + T_3) == r'\left[\begin{matrix}\frac{5}{1} & \frac{6}{1}\\\frac{6}{1} & \frac{5}{s}\end{matrix}\right]' \ + r'_\tau + \left[\begin{matrix}\frac{5}{s} & \frac{6}{1}\\\frac{6}{1} & \frac{5}{s - 1}\end{matrix}\right]_\tau + \left[\begin{matrix}' \ + r'\frac{6}{1} & \frac{5}{s \left(s - 1\right)}\\\frac{5}{1} & \frac{6}{1}\end{matrix}\right]_\tau' \ + == latex(MIMOParallel(T_1, T_2, T_3)) == latex(MIMOParallel(T_1, MIMOParallel(T_2, T_3))) == latex(MIMOParallel(MIMOParallel(T_1, T_2), T_3)) + + +def test_TransferFunctionMatrix_printing(): + tf1 = TransferFunction(p, p + x, p) + tf2 = TransferFunction(-s + p, p + s, p) + tf3 = TransferFunction(p, y**2 + 2*y + 3, p) + assert latex(TransferFunctionMatrix([[tf1], [tf2]])) == \ + r'\left[\begin{matrix}\frac{p}{p + x}\\\frac{p - s}{p + s}\end{matrix}\right]_\tau' + assert latex(TransferFunctionMatrix([[tf1, tf2], [tf3, -tf1]])) == \ + r'\left[\begin{matrix}\frac{p}{p + x} & \frac{p - s}{p + s}\\\frac{p}{y^{2} + 2 y + 3} & \frac{\left(-1\right) p}{p + x}\end{matrix}\right]_\tau' + + +def test_Feedback_printing(): + tf1 = TransferFunction(p, p + x, p) + tf2 = TransferFunction(-s + p, p + s, p) + # Negative Feedback (Default) + assert latex(Feedback(tf1, tf2)) == \ + r'\frac{\frac{p}{p + x}}{\frac{1}{1} + \left(\frac{p}{p + x}\right) \left(\frac{p - s}{p + s}\right)}' + assert latex(Feedback(tf1*tf2, TransferFunction(1, 1, p))) == \ + r'\frac{\left(\frac{p}{p + x}\right) \left(\frac{p - s}{p + s}\right)}{\frac{1}{1} + \left(\frac{p}{p + x}\right) \left(\frac{p - s}{p + s}\right)}' + # Positive Feedback + assert latex(Feedback(tf1, tf2, 1)) == \ + r'\frac{\frac{p}{p + x}}{\frac{1}{1} - \left(\frac{p}{p + x}\right) \left(\frac{p - s}{p + s}\right)}' + assert latex(Feedback(tf1*tf2, sign=1)) == \ + r'\frac{\left(\frac{p}{p + x}\right) \left(\frac{p - s}{p + s}\right)}{\frac{1}{1} - \left(\frac{p}{p + x}\right) \left(\frac{p - s}{p + s}\right)}' + + +def test_MIMOFeedback_printing(): + tf1 = TransferFunction(1, s, s) + tf2 = TransferFunction(s, s**2 - 1, s) + tf3 = TransferFunction(s, s - 1, s) + tf4 = TransferFunction(s**2, s**2 - 1, s) + + tfm_1 = TransferFunctionMatrix([[tf1, tf2], [tf3, tf4]]) + tfm_2 = TransferFunctionMatrix([[tf4, tf3], [tf2, tf1]]) + + # Negative Feedback (Default) + assert latex(MIMOFeedback(tfm_1, tfm_2)) == \ + r'\left(I_{\tau} + \left[\begin{matrix}\frac{1}{s} & \frac{s}{s^{2} - 1}\\\frac{s}{s - 1} & \frac{s^{2}}{s^{2} - 1}\end{matrix}\right]_\tau\cdot\left[' \ + r'\begin{matrix}\frac{s^{2}}{s^{2} - 1} & \frac{s}{s - 1}\\\frac{s}{s^{2} - 1} & \frac{1}{s}\end{matrix}\right]_\tau\right)^{-1} \cdot \left[\begin{matrix}' \ + r'\frac{1}{s} & \frac{s}{s^{2} - 1}\\\frac{s}{s - 1} & \frac{s^{2}}{s^{2} - 1}\end{matrix}\right]_\tau' + + # Positive Feedback + assert latex(MIMOFeedback(tfm_1*tfm_2, tfm_1, 1)) == \ + r'\left(I_{\tau} - \left[\begin{matrix}\frac{1}{s} & \frac{s}{s^{2} - 1}\\\frac{s}{s - 1} & \frac{s^{2}}{s^{2} - 1}\end{matrix}\right]_\tau\cdot\left' \ + r'[\begin{matrix}\frac{s^{2}}{s^{2} - 1} & \frac{s}{s - 1}\\\frac{s}{s^{2} - 1} & \frac{1}{s}\end{matrix}\right]_\tau\cdot\left[\begin{matrix}\frac{1}{s} & \frac{s}{s^{2} - 1}' \ + r'\\\frac{s}{s - 1} & \frac{s^{2}}{s^{2} - 1}\end{matrix}\right]_\tau\right)^{-1} \cdot \left[\begin{matrix}\frac{1}{s} & \frac{s}{s^{2} - 1}' \ + r'\\\frac{s}{s - 1} & \frac{s^{2}}{s^{2} - 1}\end{matrix}\right]_\tau\cdot\left[\begin{matrix}\frac{s^{2}}{s^{2} - 1} & \frac{s}{s - 1}\\\frac{s}{s^{2} - 1}' \ + r' & \frac{1}{s}\end{matrix}\right]_\tau' + + +def test_Quaternion_latex_printing(): + q = Quaternion(x, y, z, t) + assert latex(q) == r"x + y i + z j + t k" + q = Quaternion(x, y, z, x*t) + assert latex(q) == r"x + y i + z j + t x k" + q = Quaternion(x, y, z, x + t) + assert latex(q) == r"x + y i + z j + \left(t + x\right) k" + + +def test_TensorProduct_printing(): + from sympy.tensor.functions import TensorProduct + A = MatrixSymbol("A", 3, 3) + B = MatrixSymbol("B", 3, 3) + assert latex(TensorProduct(A, B)) == r"A \otimes B" + + +def test_WedgeProduct_printing(): + from sympy.diffgeom.rn import R2 + from sympy.diffgeom import WedgeProduct + wp = WedgeProduct(R2.dx, R2.dy) + assert latex(wp) == r"\operatorname{d}x \wedge \operatorname{d}y" + + +def test_issue_9216(): + expr_1 = Pow(1, -1, evaluate=False) + assert latex(expr_1) == r"1^{-1}" + + expr_2 = Pow(1, Pow(1, -1, evaluate=False), evaluate=False) + assert latex(expr_2) == r"1^{1^{-1}}" + + expr_3 = Pow(3, -2, evaluate=False) + assert latex(expr_3) == r"\frac{1}{9}" + + expr_4 = Pow(1, -2, evaluate=False) + assert latex(expr_4) == r"1^{-2}" + + +def test_latex_printer_tensor(): + from sympy.tensor.tensor import TensorIndexType, tensor_indices, TensorHead, tensor_heads + L = TensorIndexType("L") + i, j, k, l = tensor_indices("i j k l", L) + i0 = tensor_indices("i_0", L) + A, B, C, D = tensor_heads("A B C D", [L]) + H = TensorHead("H", [L, L]) + K = TensorHead("K", [L, L, L, L]) + + assert latex(i) == r"{}^{i}" + assert latex(-i) == r"{}_{i}" + + expr = A(i) + assert latex(expr) == r"A{}^{i}" + + expr = A(i0) + assert latex(expr) == r"A{}^{i_{0}}" + + expr = A(-i) + assert latex(expr) == r"A{}_{i}" + + expr = -3*A(i) + assert latex(expr) == r"-3A{}^{i}" + + expr = K(i, j, -k, -i0) + assert latex(expr) == r"K{}^{ij}{}_{ki_{0}}" + + expr = K(i, -j, -k, i0) + assert latex(expr) == r"K{}^{i}{}_{jk}{}^{i_{0}}" + + expr = K(i, -j, k, -i0) + assert latex(expr) == r"K{}^{i}{}_{j}{}^{k}{}_{i_{0}}" + + expr = H(i, -j) + assert latex(expr) == r"H{}^{i}{}_{j}" + + expr = H(i, j) + assert latex(expr) == r"H{}^{ij}" + + expr = H(-i, -j) + assert latex(expr) == r"H{}_{ij}" + + expr = (1+x)*A(i) + assert latex(expr) == r"\left(x + 1\right)A{}^{i}" + + expr = H(i, -i) + assert latex(expr) == r"H{}^{L_{0}}{}_{L_{0}}" + + expr = H(i, -j)*A(j)*B(k) + assert latex(expr) == r"H{}^{i}{}_{L_{0}}A{}^{L_{0}}B{}^{k}" + + expr = A(i) + 3*B(i) + assert latex(expr) == r"3B{}^{i} + A{}^{i}" + + # Test ``TensorElement``: + from sympy.tensor.tensor import TensorElement + + expr = TensorElement(K(i, j, k, l), {i: 3, k: 2}) + assert latex(expr) == r'K{}^{i=3,j,k=2,l}' + + expr = TensorElement(K(i, j, k, l), {i: 3}) + assert latex(expr) == r'K{}^{i=3,jkl}' + + expr = TensorElement(K(i, -j, k, l), {i: 3, k: 2}) + assert latex(expr) == r'K{}^{i=3}{}_{j}{}^{k=2,l}' + + expr = TensorElement(K(i, -j, k, -l), {i: 3, k: 2}) + assert latex(expr) == r'K{}^{i=3}{}_{j}{}^{k=2}{}_{l}' + + expr = TensorElement(K(i, j, -k, -l), {i: 3, -k: 2}) + assert latex(expr) == r'K{}^{i=3,j}{}_{k=2,l}' + + expr = TensorElement(K(i, j, -k, -l), {i: 3}) + assert latex(expr) == r'K{}^{i=3,j}{}_{kl}' + + expr = PartialDerivative(A(i), A(i)) + assert latex(expr) == r"\frac{\partial}{\partial {A{}^{L_{0}}}}{A{}^{L_{0}}}" + + expr = PartialDerivative(A(-i), A(-j)) + assert latex(expr) == r"\frac{\partial}{\partial {A{}_{j}}}{A{}_{i}}" + + expr = PartialDerivative(K(i, j, -k, -l), A(m), A(-n)) + assert latex(expr) == r"\frac{\partial^{2}}{\partial {A{}^{m}} \partial {A{}_{n}}}{K{}^{ij}{}_{kl}}" + + expr = PartialDerivative(B(-i) + A(-i), A(-j), A(-n)) + assert latex(expr) == r"\frac{\partial^{2}}{\partial {A{}_{j}} \partial {A{}_{n}}}{\left(A{}_{i} + B{}_{i}\right)}" + + expr = PartialDerivative(3*A(-i), A(-j), A(-n)) + assert latex(expr) == r"\frac{\partial^{2}}{\partial {A{}_{j}} \partial {A{}_{n}}}{\left(3A{}_{i}\right)}" + + +def test_multiline_latex(): + a, b, c, d, e, f = symbols('a b c d e f') + expr = -a + 2*b -3*c +4*d -5*e + expected = r"\begin{eqnarray}" + "\n"\ + r"f & = &- a \nonumber\\" + "\n"\ + r"& & + 2 b \nonumber\\" + "\n"\ + r"& & - 3 c \nonumber\\" + "\n"\ + r"& & + 4 d \nonumber\\" + "\n"\ + r"& & - 5 e " + "\n"\ + r"\end{eqnarray}" + assert multiline_latex(f, expr, environment="eqnarray") == expected + + expected2 = r'\begin{eqnarray}' + '\n'\ + r'f & = &- a + 2 b \nonumber\\' + '\n'\ + r'& & - 3 c + 4 d \nonumber\\' + '\n'\ + r'& & - 5 e ' + '\n'\ + r'\end{eqnarray}' + + assert multiline_latex(f, expr, 2, environment="eqnarray") == expected2 + + expected3 = r'\begin{eqnarray}' + '\n'\ + r'f & = &- a + 2 b - 3 c \nonumber\\'+ '\n'\ + r'& & + 4 d - 5 e ' + '\n'\ + r'\end{eqnarray}' + + assert multiline_latex(f, expr, 3, environment="eqnarray") == expected3 + + expected3dots = r'\begin{eqnarray}' + '\n'\ + r'f & = &- a + 2 b - 3 c \dots\nonumber\\'+ '\n'\ + r'& & + 4 d - 5 e ' + '\n'\ + r'\end{eqnarray}' + + assert multiline_latex(f, expr, 3, environment="eqnarray", use_dots=True) == expected3dots + + expected3align = r'\begin{align*}' + '\n'\ + r'f = &- a + 2 b - 3 c \\'+ '\n'\ + r'& + 4 d - 5 e ' + '\n'\ + r'\end{align*}' + + assert multiline_latex(f, expr, 3) == expected3align + assert multiline_latex(f, expr, 3, environment='align*') == expected3align + + expected2ieee = r'\begin{IEEEeqnarray}{rCl}' + '\n'\ + r'f & = &- a + 2 b \nonumber\\' + '\n'\ + r'& & - 3 c + 4 d \nonumber\\' + '\n'\ + r'& & - 5 e ' + '\n'\ + r'\end{IEEEeqnarray}' + + assert multiline_latex(f, expr, 2, environment="IEEEeqnarray") == expected2ieee + + raises(ValueError, lambda: multiline_latex(f, expr, environment="foo")) + +def test_issue_15353(): + a, x = symbols('a x') + # Obtained from nonlinsolve([(sin(a*x)),cos(a*x)],[x,a]) + sol = ConditionSet( + Tuple(x, a), Eq(sin(a*x), 0) & Eq(cos(a*x), 0), S.Complexes**2) + assert latex(sol) == \ + r'\left\{\left( x, \ a\right)\; \middle|\; \left( x, \ a\right) \in ' \ + r'\mathbb{C}^{2} \wedge \sin{\left(a x \right)} = 0 \wedge ' \ + r'\cos{\left(a x \right)} = 0 \right\}' + + +def test_latex_symbolic_probability(): + mu = symbols("mu") + sigma = symbols("sigma", positive=True) + X = Normal("X", mu, sigma) + assert latex(Expectation(X)) == r'\operatorname{E}\left[X\right]' + assert latex(Variance(X)) == r'\operatorname{Var}\left(X\right)' + assert latex(Probability(X > 0)) == r'\operatorname{P}\left(X > 0\right)' + Y = Normal("Y", mu, sigma) + assert latex(Covariance(X, Y)) == r'\operatorname{Cov}\left(X, Y\right)' + + +def test_trace(): + # Issue 15303 + from sympy.matrices.expressions.trace import trace + A = MatrixSymbol("A", 2, 2) + assert latex(trace(A)) == r"\operatorname{tr}\left(A \right)" + assert latex(trace(A**2)) == r"\operatorname{tr}\left(A^{2} \right)" + + +def test_print_basic(): + # Issue 15303 + from sympy.core.basic import Basic + from sympy.core.expr import Expr + + # dummy class for testing printing where the function is not + # implemented in latex.py + class UnimplementedExpr(Expr): + def __new__(cls, e): + return Basic.__new__(cls, e) + + # dummy function for testing + def unimplemented_expr(expr): + return UnimplementedExpr(expr).doit() + + # override class name to use superscript / subscript + def unimplemented_expr_sup_sub(expr): + result = UnimplementedExpr(expr) + result.__class__.__name__ = 'UnimplementedExpr_x^1' + return result + + assert latex(unimplemented_expr(x)) == r'\operatorname{UnimplementedExpr}\left(x\right)' + assert latex(unimplemented_expr(x**2)) == \ + r'\operatorname{UnimplementedExpr}\left(x^{2}\right)' + assert latex(unimplemented_expr_sup_sub(x)) == \ + r'\operatorname{UnimplementedExpr^{1}_{x}}\left(x\right)' + + +def test_MatrixSymbol_bold(): + # Issue #15871 + from sympy.matrices.expressions.trace import trace + A = MatrixSymbol("A", 2, 2) + assert latex(trace(A), mat_symbol_style='bold') == \ + r"\operatorname{tr}\left(\mathbf{A} \right)" + assert latex(trace(A), mat_symbol_style='plain') == \ + r"\operatorname{tr}\left(A \right)" + + A = MatrixSymbol("A", 3, 3) + B = MatrixSymbol("B", 3, 3) + C = MatrixSymbol("C", 3, 3) + + assert latex(-A, mat_symbol_style='bold') == r"- \mathbf{A}" + assert latex(A - A*B - B, mat_symbol_style='bold') == \ + r"\mathbf{A} - \mathbf{A} \mathbf{B} - \mathbf{B}" + assert latex(-A*B - A*B*C - B, mat_symbol_style='bold') == \ + r"- \mathbf{A} \mathbf{B} - \mathbf{A} \mathbf{B} \mathbf{C} - \mathbf{B}" + + A_k = MatrixSymbol("A_k", 3, 3) + assert latex(A_k, mat_symbol_style='bold') == r"\mathbf{A}_{k}" + + A = MatrixSymbol(r"\nabla_k", 3, 3) + assert latex(A, mat_symbol_style='bold') == r"\mathbf{\nabla}_{k}" + +def test_AppliedPermutation(): + p = Permutation(0, 1, 2) + x = Symbol('x') + assert latex(AppliedPermutation(p, x)) == \ + r'\sigma_{\left( 0\; 1\; 2\right)}(x)' + + +def test_PermutationMatrix(): + p = Permutation(0, 1, 2) + assert latex(PermutationMatrix(p)) == r'P_{\left( 0\; 1\; 2\right)}' + p = Permutation(0, 3)(1, 2) + assert latex(PermutationMatrix(p)) == \ + r'P_{\left( 0\; 3\right)\left( 1\; 2\right)}' + + +def test_issue_21758(): + from sympy.functions.elementary.piecewise import piecewise_fold + from sympy.series.fourier import FourierSeries + x = Symbol('x') + k, n = symbols('k n') + fo = FourierSeries(x, (x, -pi, pi), (0, SeqFormula(0, (k, 1, oo)), SeqFormula( + Piecewise((-2*pi*cos(n*pi)/n + 2*sin(n*pi)/n**2, (n > -oo) & (n < oo) & Ne(n, 0)), + (0, True))*sin(n*x)/pi, (n, 1, oo)))) + assert latex(piecewise_fold(fo)) == '\\begin{cases} 2 \\sin{\\left(x \\right)}' \ + ' - \\sin{\\left(2 x \\right)} + \\frac{2 \\sin{\\left(3 x \\right)}}{3} +' \ + ' \\ldots & \\text{for}\\: n > -\\infty \\wedge n < \\infty \\wedge ' \ + 'n \\neq 0 \\\\0 & \\text{otherwise} \\end{cases}' + assert latex(FourierSeries(x, (x, -pi, pi), (0, SeqFormula(0, (k, 1, oo)), + SeqFormula(0, (n, 1, oo))))) == '0' + + +def test_imaginary_unit(): + assert latex(1 + I) == r'1 + i' + assert latex(1 + I, imaginary_unit='i') == r'1 + i' + assert latex(1 + I, imaginary_unit='j') == r'1 + j' + assert latex(1 + I, imaginary_unit='foo') == r'1 + foo' + assert latex(I, imaginary_unit="ti") == r'\text{i}' + assert latex(I, imaginary_unit="tj") == r'\text{j}' + + +def test_text_re_im(): + assert latex(im(x), gothic_re_im=True) == r'\Im{\left(x\right)}' + assert latex(im(x), gothic_re_im=False) == r'\operatorname{im}{\left(x\right)}' + assert latex(re(x), gothic_re_im=True) == r'\Re{\left(x\right)}' + assert latex(re(x), gothic_re_im=False) == r'\operatorname{re}{\left(x\right)}' + + +def test_latex_diffgeom(): + from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField, Differential + from sympy.diffgeom.rn import R2 + x,y = symbols('x y', real=True) + m = Manifold('M', 2) + assert latex(m) == r'\text{M}' + p = Patch('P', m) + assert latex(p) == r'\text{P}_{\text{M}}' + rect = CoordSystem('rect', p, [x, y]) + assert latex(rect) == r'\text{rect}^{\text{P}}_{\text{M}}' + b = BaseScalarField(rect, 0) + assert latex(b) == r'\mathbf{x}' + + g = Function('g') + s_field = g(R2.x, R2.y) + assert latex(Differential(s_field)) == \ + r'\operatorname{d}\left(g{\left(\mathbf{x},\mathbf{y} \right)}\right)' + + +def test_unit_printing(): + assert latex(5*meter) == r'5 \text{m}' + assert latex(3*gibibyte) == r'3 \text{gibibyte}' + assert latex(4*microgram/second) == r'\frac{4 \mu\text{g}}{\text{s}}' + assert latex(4*micro*gram/second) == r'\frac{4 \mu \text{g}}{\text{s}}' + assert latex(5*milli*meter) == r'5 \text{m} \text{m}' + assert latex(milli) == r'\text{m}' + + +def test_issue_17092(): + x_star = Symbol('x^*') + assert latex(Derivative(x_star, x_star,2)) == r'\frac{d^{2}}{d \left(x^{*}\right)^{2}} x^{*}' + + +def test_latex_decimal_separator(): + + x, y, z, t = symbols('x y z t') + k, m, n = symbols('k m n', integer=True) + f, g, h = symbols('f g h', cls=Function) + + # comma decimal_separator + assert(latex([1, 2.3, 4.5], decimal_separator='comma') == r'\left[ 1; \ 2{,}3; \ 4{,}5\right]') + assert(latex(FiniteSet(1, 2.3, 4.5), decimal_separator='comma') == r'\left\{1; 2{,}3; 4{,}5\right\}') + assert(latex((1, 2.3, 4.6), decimal_separator = 'comma') == r'\left( 1; \ 2{,}3; \ 4{,}6\right)') + assert(latex((1,), decimal_separator='comma') == r'\left( 1;\right)') + + # period decimal_separator + assert(latex([1, 2.3, 4.5], decimal_separator='period') == r'\left[ 1, \ 2.3, \ 4.5\right]' ) + assert(latex(FiniteSet(1, 2.3, 4.5), decimal_separator='period') == r'\left\{1, 2.3, 4.5\right\}') + assert(latex((1, 2.3, 4.6), decimal_separator = 'period') == r'\left( 1, \ 2.3, \ 4.6\right)') + assert(latex((1,), decimal_separator='period') == r'\left( 1,\right)') + + # default decimal_separator + assert(latex([1, 2.3, 4.5]) == r'\left[ 1, \ 2.3, \ 4.5\right]') + assert(latex(FiniteSet(1, 2.3, 4.5)) == r'\left\{1, 2.3, 4.5\right\}') + assert(latex((1, 2.3, 4.6)) == r'\left( 1, \ 2.3, \ 4.6\right)') + assert(latex((1,)) == r'\left( 1,\right)') + + assert(latex(Mul(3.4,5.3), decimal_separator = 'comma') == r'18{,}02') + assert(latex(3.4*5.3, decimal_separator = 'comma') == r'18{,}02') + x = symbols('x') + y = symbols('y') + z = symbols('z') + assert(latex(x*5.3 + 2**y**3.4 + 4.5 + z, decimal_separator = 'comma') == r'2^{y^{3{,}4}} + 5{,}3 x + z + 4{,}5') + + assert(latex(0.987, decimal_separator='comma') == r'0{,}987') + assert(latex(S(0.987), decimal_separator='comma') == r'0{,}987') + assert(latex(.3, decimal_separator='comma') == r'0{,}3') + assert(latex(S(.3), decimal_separator='comma') == r'0{,}3') + + + assert(latex(5.8*10**(-7), decimal_separator='comma') == r'5{,}8 \cdot 10^{-7}') + assert(latex(S(5.7)*10**(-7), decimal_separator='comma') == r'5{,}7 \cdot 10^{-7}') + assert(latex(S(5.7*10**(-7)), decimal_separator='comma') == r'5{,}7 \cdot 10^{-7}') + + x = symbols('x') + assert(latex(1.2*x+3.4, decimal_separator='comma') == r'1{,}2 x + 3{,}4') + assert(latex(FiniteSet(1, 2.3, 4.5), decimal_separator='period') == r'\left\{1, 2.3, 4.5\right\}') + + # Error Handling tests + raises(ValueError, lambda: latex([1,2.3,4.5], decimal_separator='non_existing_decimal_separator_in_list')) + raises(ValueError, lambda: latex(FiniteSet(1,2.3,4.5), decimal_separator='non_existing_decimal_separator_in_set')) + raises(ValueError, lambda: latex((1,2.3,4.5), decimal_separator='non_existing_decimal_separator_in_tuple')) + +def test_Str(): + from sympy.core.symbol import Str + assert str(Str('x')) == r'x' + +def test_latex_escape(): + assert latex_escape(r"~^\&%$#_{}") == "".join([ + r'\textasciitilde', + r'\textasciicircum', + r'\textbackslash', + r'\&', + r'\%', + r'\$', + r'\#', + r'\_', + r'\{', + r'\}', + ]) + +def test_emptyPrinter(): + class MyObject: + def __repr__(self): + return "" + + # unknown objects are monospaced + assert latex(MyObject()) == r"\mathtt{\text{}}" + + # even if they are nested within other objects + assert latex((MyObject(),)) == r"\left( \mathtt{\text{}},\right)" + +def test_global_settings(): + import inspect + + # settings should be visible in the signature of `latex` + assert inspect.signature(latex).parameters['imaginary_unit'].default == r'i' + assert latex(I) == r'i' + try: + # but changing the defaults... + LatexPrinter.set_global_settings(imaginary_unit='j') + # ... should change the signature + assert inspect.signature(latex).parameters['imaginary_unit'].default == r'j' + assert latex(I) == r'j' + finally: + # there's no public API to undo this, but we need to make sure we do + # so as not to impact other tests + del LatexPrinter._global_settings['imaginary_unit'] + + # check we really did undo it + assert inspect.signature(latex).parameters['imaginary_unit'].default == r'i' + assert latex(I) == r'i' + +def test_pickleable(): + # this tests that the _PrintFunction instance is pickleable + import pickle + assert pickle.loads(pickle.dumps(latex)) is latex + +def test_printing_latex_array_expressions(): + assert latex(ArraySymbol("A", (2, 3, 4))) == "A" + assert latex(ArrayElement("A", (2, 1/(1-x), 0))) == "{{A}_{2, \\frac{1}{1 - x}, 0}}" + M = MatrixSymbol("M", 3, 3) + N = MatrixSymbol("N", 3, 3) + assert latex(ArrayElement(M*N, [x, 0])) == "{{\\left(M N\\right)}_{x, 0}}" + +def test_Array(): + arr = Array(range(10)) + assert latex(arr) == r'\left[\begin{matrix}0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\end{matrix}\right]' + + arr = Array(range(11)) + # fill the empty argument with a bunch of 'c' to avoid latex errors + assert latex(arr) == r'\left[\begin{array}{ccccccccccc}0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10\end{array}\right]' + +def test_latex_with_unevaluated(): + with evaluate(False): + assert latex(a * a) == r"a a" + + +def test_latex_disable_split_super_sub(): + assert latex(Symbol('u^a_b')) == 'u^{a}_{b}' + assert latex(Symbol('u^a_b'), disable_split_super_sub=False) == 'u^{a}_{b}' + assert latex(Symbol('u^a_b'), disable_split_super_sub=True) == 'u\\^a\\_b' diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_llvmjit.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_llvmjit.py new file mode 100644 index 0000000000000000000000000000000000000000..709476f1d7517dc629210341594a70dc6f41808f --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_llvmjit.py @@ -0,0 +1,224 @@ +from sympy.external import import_module +from sympy.testing.pytest import raises +import ctypes + + +if import_module('llvmlite'): + import sympy.printing.llvmjitcode as g +else: + disabled = True + +import sympy +from sympy.abc import a, b, n + + +# copied from numpy.isclose documentation +def isclose(a, b): + rtol = 1e-5 + atol = 1e-8 + return abs(a-b) <= atol + rtol*abs(b) + + +def test_simple_expr(): + e = a + 1.0 + f = g.llvm_callable([a], e) + res = float(e.subs({a: 4.0}).evalf()) + jit_res = f(4.0) + + assert isclose(jit_res, res) + + +def test_two_arg(): + e = 4.0*a + b + 3.0 + f = g.llvm_callable([a, b], e) + res = float(e.subs({a: 4.0, b: 3.0}).evalf()) + jit_res = f(4.0, 3.0) + + assert isclose(jit_res, res) + + +def test_func(): + e = 4.0*sympy.exp(-a) + f = g.llvm_callable([a], e) + res = float(e.subs({a: 1.5}).evalf()) + jit_res = f(1.5) + + assert isclose(jit_res, res) + + +def test_two_func(): + e = 4.0*sympy.exp(-a) + sympy.exp(b) + f = g.llvm_callable([a, b], e) + res = float(e.subs({a: 1.5, b: 2.0}).evalf()) + jit_res = f(1.5, 2.0) + + assert isclose(jit_res, res) + + +def test_two_sqrt(): + e = 4.0*sympy.sqrt(a) + sympy.sqrt(b) + f = g.llvm_callable([a, b], e) + res = float(e.subs({a: 1.5, b: 2.0}).evalf()) + jit_res = f(1.5, 2.0) + + assert isclose(jit_res, res) + + +def test_two_pow(): + e = a**1.5 + b**7 + f = g.llvm_callable([a, b], e) + res = float(e.subs({a: 1.5, b: 2.0}).evalf()) + jit_res = f(1.5, 2.0) + + assert isclose(jit_res, res) + + +def test_callback(): + e = a + 1.2 + f = g.llvm_callable([a], e, callback_type='scipy.integrate.test') + m = ctypes.c_int(1) + array_type = ctypes.c_double * 1 + inp = {a: 2.2} + array = array_type(inp[a]) + jit_res = f(m, array) + + res = float(e.subs(inp).evalf()) + + assert isclose(jit_res, res) + + +def test_callback_cubature(): + e = a + 1.2 + f = g.llvm_callable([a], e, callback_type='cubature') + m = ctypes.c_int(1) + array_type = ctypes.c_double * 1 + inp = {a: 2.2} + array = array_type(inp[a]) + out_array = array_type(0.0) + jit_ret = f(m, array, None, m, out_array) + + assert jit_ret == 0 + + res = float(e.subs(inp).evalf()) + + assert isclose(out_array[0], res) + + +def test_callback_two(): + e = 3*a*b + f = g.llvm_callable([a, b], e, callback_type='scipy.integrate.test') + m = ctypes.c_int(2) + array_type = ctypes.c_double * 2 + inp = {a: 0.2, b: 1.7} + array = array_type(inp[a], inp[b]) + jit_res = f(m, array) + + res = float(e.subs(inp).evalf()) + + assert isclose(jit_res, res) + + +def test_callback_alt_two(): + d = sympy.IndexedBase('d') + e = 3*d[0]*d[1] + f = g.llvm_callable([n, d], e, callback_type='scipy.integrate.test') + m = ctypes.c_int(2) + array_type = ctypes.c_double * 2 + inp = {d[0]: 0.2, d[1]: 1.7} + array = array_type(inp[d[0]], inp[d[1]]) + jit_res = f(m, array) + + res = float(e.subs(inp).evalf()) + + assert isclose(jit_res, res) + + +def test_multiple_statements(): + # Match return from CSE + e = [[(b, 4.0*a)], [b + 5]] + f = g.llvm_callable([a], e) + b_val = e[0][0][1].subs({a: 1.5}) + res = float(e[1][0].subs({b: b_val}).evalf()) + jit_res = f(1.5) + assert isclose(jit_res, res) + + f_callback = g.llvm_callable([a], e, callback_type='scipy.integrate.test') + m = ctypes.c_int(1) + array_type = ctypes.c_double * 1 + array = array_type(1.5) + jit_callback_res = f_callback(m, array) + assert isclose(jit_callback_res, res) + + +def test_cse(): + e = a*a + b*b + sympy.exp(-a*a - b*b) + e2 = sympy.cse(e) + f = g.llvm_callable([a, b], e2) + res = float(e.subs({a: 2.3, b: 0.1}).evalf()) + jit_res = f(2.3, 0.1) + + assert isclose(jit_res, res) + + +def eval_cse(e, sub_dict): + tmp_dict = {} + for tmp_name, tmp_expr in e[0]: + e2 = tmp_expr.subs(sub_dict) + e3 = e2.subs(tmp_dict) + tmp_dict[tmp_name] = e3 + return [e.subs(sub_dict).subs(tmp_dict) for e in e[1]] + + +def test_cse_multiple(): + e1 = a*a + e2 = a*a + b*b + e3 = sympy.cse([e1, e2]) + + raises(NotImplementedError, + lambda: g.llvm_callable([a, b], e3, callback_type='scipy.integrate')) + + f = g.llvm_callable([a, b], e3) + jit_res = f(0.1, 1.5) + assert len(jit_res) == 2 + res = eval_cse(e3, {a: 0.1, b: 1.5}) + assert isclose(res[0], jit_res[0]) + assert isclose(res[1], jit_res[1]) + + +def test_callback_cubature_multiple(): + e1 = a*a + e2 = a*a + b*b + e3 = sympy.cse([e1, e2, 4*e2]) + f = g.llvm_callable([a, b], e3, callback_type='cubature') + + # Number of input variables + ndim = 2 + # Number of output expression values + outdim = 3 + + m = ctypes.c_int(ndim) + fdim = ctypes.c_int(outdim) + array_type = ctypes.c_double * ndim + out_array_type = ctypes.c_double * outdim + inp = {a: 0.2, b: 1.5} + array = array_type(inp[a], inp[b]) + out_array = out_array_type() + jit_ret = f(m, array, None, fdim, out_array) + + assert jit_ret == 0 + + res = eval_cse(e3, inp) + + assert isclose(out_array[0], res[0]) + assert isclose(out_array[1], res[1]) + assert isclose(out_array[2], res[2]) + + +def test_symbol_not_found(): + e = a*a + b + raises(LookupError, lambda: g.llvm_callable([a], e)) + + +def test_bad_callback(): + e = a + raises(ValueError, lambda: g.llvm_callable([a], e, callback_type='bad_callback')) diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_maple.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_maple.py new file mode 100644 index 0000000000000000000000000000000000000000..9bb4c512ad3203bd64ae56b350e15734b3a6afb0 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_maple.py @@ -0,0 +1,381 @@ +from sympy.core import (S, pi, oo, symbols, Function, Rational, Integer, + Tuple, Symbol, Eq, Ne, Le, Lt, Gt, Ge) +from sympy.core import EulerGamma, GoldenRatio, Catalan, Lambda, Mul, Pow +from sympy.functions import Piecewise, sqrt, ceiling, exp, sin, cos, sinc, lucas +from sympy.testing.pytest import raises +from sympy.utilities.lambdify import implemented_function +from sympy.matrices import (eye, Matrix, MatrixSymbol, Identity, + HadamardProduct, SparseMatrix) +from sympy.functions.special.bessel import besseli + +from sympy.printing.maple import maple_code + +x, y, z = symbols('x,y,z') + + +def test_Integer(): + assert maple_code(Integer(67)) == "67" + assert maple_code(Integer(-1)) == "-1" + + +def test_Rational(): + assert maple_code(Rational(3, 7)) == "3/7" + assert maple_code(Rational(18, 9)) == "2" + assert maple_code(Rational(3, -7)) == "-3/7" + assert maple_code(Rational(-3, -7)) == "3/7" + assert maple_code(x + Rational(3, 7)) == "x + 3/7" + assert maple_code(Rational(3, 7) * x) == '(3/7)*x' + + +def test_Relational(): + assert maple_code(Eq(x, y)) == "x = y" + assert maple_code(Ne(x, y)) == "x <> y" + assert maple_code(Le(x, y)) == "x <= y" + assert maple_code(Lt(x, y)) == "x < y" + assert maple_code(Gt(x, y)) == "x > y" + assert maple_code(Ge(x, y)) == "x >= y" + + +def test_Function(): + assert maple_code(sin(x) ** cos(x)) == "sin(x)^cos(x)" + assert maple_code(abs(x)) == "abs(x)" + assert maple_code(ceiling(x)) == "ceil(x)" + + +def test_Pow(): + assert maple_code(x ** 3) == "x^3" + assert maple_code(x ** (y ** 3)) == "x^(y^3)" + + assert maple_code((x ** 3) ** y) == "(x^3)^y" + assert maple_code(x ** Rational(2, 3)) == 'x^(2/3)' + + g = implemented_function('g', Lambda(x, 2 * x)) + assert maple_code(1 / (g(x) * 3.5) ** (x - y ** x) / (x ** 2 + y)) == \ + "(3.5*2*x)^(-x + y^x)/(x^2 + y)" + # For issue 14160 + assert maple_code(Mul(-2, x, Pow(Mul(y, y, evaluate=False), -1, evaluate=False), + evaluate=False)) == '-2*x/(y*y)' + + +def test_basic_ops(): + assert maple_code(x * y) == "x*y" + assert maple_code(x + y) == "x + y" + assert maple_code(x - y) == "x - y" + assert maple_code(-x) == "-x" + + +def test_1_over_x_and_sqrt(): + # 1.0 and 0.5 would do something different in regular StrPrinter, + # but these are exact in IEEE floating point so no different here. + assert maple_code(1 / x) == '1/x' + assert maple_code(x ** -1) == maple_code(x ** -1.0) == '1/x' + assert maple_code(1 / sqrt(x)) == '1/sqrt(x)' + assert maple_code(x ** -S.Half) == maple_code(x ** -0.5) == '1/sqrt(x)' + assert maple_code(sqrt(x)) == 'sqrt(x)' + assert maple_code(x ** S.Half) == maple_code(x ** 0.5) == 'sqrt(x)' + assert maple_code(1 / pi) == '1/Pi' + assert maple_code(pi ** -1) == maple_code(pi ** -1.0) == '1/Pi' + assert maple_code(pi ** -0.5) == '1/sqrt(Pi)' + + +def test_mix_number_mult_symbols(): + assert maple_code(3 * x) == "3*x" + assert maple_code(pi * x) == "Pi*x" + assert maple_code(3 / x) == "3/x" + assert maple_code(pi / x) == "Pi/x" + assert maple_code(x / 3) == '(1/3)*x' + assert maple_code(x / pi) == "x/Pi" + assert maple_code(x * y) == "x*y" + assert maple_code(3 * x * y) == "3*x*y" + assert maple_code(3 * pi * x * y) == "3*Pi*x*y" + assert maple_code(x / y) == "x/y" + assert maple_code(3 * x / y) == "3*x/y" + assert maple_code(x * y / z) == "x*y/z" + assert maple_code(x / y * z) == "x*z/y" + assert maple_code(1 / x / y) == "1/(x*y)" + assert maple_code(2 * pi * x / y / z) == "2*Pi*x/(y*z)" + assert maple_code(3 * pi / x) == "3*Pi/x" + assert maple_code(S(3) / 5) == "3/5" + assert maple_code(S(3) / 5 * x) == '(3/5)*x' + assert maple_code(x / y / z) == "x/(y*z)" + assert maple_code((x + y) / z) == "(x + y)/z" + assert maple_code((x + y) / (z + x)) == "(x + y)/(x + z)" + assert maple_code((x + y) / EulerGamma) == '(x + y)/gamma' + assert maple_code(x / 3 / pi) == '(1/3)*x/Pi' + assert maple_code(S(3) / 5 * x * y / pi) == '(3/5)*x*y/Pi' + + +def test_mix_number_pow_symbols(): + assert maple_code(pi ** 3) == 'Pi^3' + assert maple_code(x ** 2) == 'x^2' + + assert maple_code(x ** (pi ** 3)) == 'x^(Pi^3)' + assert maple_code(x ** y) == 'x^y' + + assert maple_code(x ** (y ** z)) == 'x^(y^z)' + assert maple_code((x ** y) ** z) == '(x^y)^z' + + +def test_imag(): + I = S('I') + assert maple_code(I) == "I" + assert maple_code(5 * I) == "5*I" + + assert maple_code((S(3) / 2) * I) == "(3/2)*I" + assert maple_code(3 + 4 * I) == "3 + 4*I" + + +def test_constants(): + assert maple_code(pi) == "Pi" + assert maple_code(oo) == "infinity" + assert maple_code(-oo) == "-infinity" + assert maple_code(S.NegativeInfinity) == "-infinity" + assert maple_code(S.NaN) == "undefined" + assert maple_code(S.Exp1) == "exp(1)" + assert maple_code(exp(1)) == "exp(1)" + + +def test_constants_other(): + assert maple_code(2 * GoldenRatio) == '2*(1/2 + (1/2)*sqrt(5))' + assert maple_code(2 * Catalan) == '2*Catalan' + assert maple_code(2 * EulerGamma) == "2*gamma" + + +def test_boolean(): + assert maple_code(x & y) == "x and y" + assert maple_code(x | y) == "x or y" + assert maple_code(~x) == "not x" + assert maple_code(x & y & z) == "x and y and z" + assert maple_code(x | y | z) == "x or y or z" + assert maple_code((x & y) | z) == "z or x and y" + assert maple_code((x | y) & z) == "z and (x or y)" + + +def test_Matrices(): + assert maple_code(Matrix(1, 1, [10])) == \ + 'Matrix([[10]], storage = rectangular)' + + A = Matrix([[1, sin(x / 2), abs(x)], + [0, 1, pi], + [0, exp(1), ceiling(x)]]) + expected = \ + 'Matrix(' \ + '[[1, sin((1/2)*x), abs(x)],' \ + ' [0, 1, Pi],' \ + ' [0, exp(1), ceil(x)]], ' \ + 'storage = rectangular)' + assert maple_code(A) == expected + + # row and columns + assert maple_code(A[:, 0]) == \ + 'Matrix([[1], [0], [0]], storage = rectangular)' + assert maple_code(A[0, :]) == \ + 'Matrix([[1, sin((1/2)*x), abs(x)]], storage = rectangular)' + assert maple_code(Matrix([[x, x - y, -y]])) == \ + 'Matrix([[x, x - y, -y]], storage = rectangular)' + + # empty matrices + assert maple_code(Matrix(0, 0, [])) == \ + 'Matrix([], storage = rectangular)' + assert maple_code(Matrix(0, 3, [])) == \ + 'Matrix([], storage = rectangular)' + +def test_SparseMatrices(): + assert maple_code(SparseMatrix(Identity(2))) == 'Matrix([[1, 0], [0, 1]], storage = sparse)' + + +def test_vector_entries_hadamard(): + # For a row or column, user might to use the other dimension + A = Matrix([[1, sin(2 / x), 3 * pi / x / 5]]) + assert maple_code(A) == \ + 'Matrix([[1, sin(2/x), (3/5)*Pi/x]], storage = rectangular)' + assert maple_code(A.T) == \ + 'Matrix([[1], [sin(2/x)], [(3/5)*Pi/x]], storage = rectangular)' + + +def test_Matrices_entries_not_hadamard(): + A = Matrix([[1, sin(2 / x), 3 * pi / x / 5], [1, 2, x * y]]) + expected = \ + 'Matrix([[1, sin(2/x), (3/5)*Pi/x], [1, 2, x*y]], ' \ + 'storage = rectangular)' + assert maple_code(A) == expected + + +def test_MatrixSymbol(): + n = Symbol('n', integer=True) + A = MatrixSymbol('A', n, n) + B = MatrixSymbol('B', n, n) + assert maple_code(A * B) == "A.B" + assert maple_code(B * A) == "B.A" + assert maple_code(2 * A * B) == "2*A.B" + assert maple_code(B * 2 * A) == "2*B.A" + + assert maple_code( + A * (B + 3 * Identity(n))) == "A.(3*Matrix(n, shape = identity) + B)" + + assert maple_code(A ** (x ** 2)) == "MatrixPower(A, x^2)" + assert maple_code(A ** 3) == "MatrixPower(A, 3)" + assert maple_code(A ** (S.Half)) == "MatrixPower(A, 1/2)" + + +def test_special_matrices(): + assert maple_code(6 * Identity(3)) == "6*Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]], storage = sparse)" + assert maple_code(Identity(x)) == 'Matrix(x, shape = identity)' + + +def test_containers(): + assert maple_code([1, 2, 3, [4, 5, [6, 7]], 8, [9, 10], 11]) == \ + "[1, 2, 3, [4, 5, [6, 7]], 8, [9, 10], 11]" + + assert maple_code((1, 2, (3, 4))) == "[1, 2, [3, 4]]" + assert maple_code([1]) == "[1]" + assert maple_code((1,)) == "[1]" + assert maple_code(Tuple(*[1, 2, 3])) == "[1, 2, 3]" + assert maple_code((1, x * y, (3, x ** 2))) == "[1, x*y, [3, x^2]]" + # scalar, matrix, empty matrix and empty list + + assert maple_code((1, eye(3), Matrix(0, 0, []), [])) == \ + "[1, Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]], storage = rectangular), Matrix([], storage = rectangular), []]" + + +def test_maple_noninline(): + source = maple_code((x + y)/Catalan, assign_to='me', inline=False) + expected = "me := (x + y)/Catalan" + + assert source == expected + + +def test_maple_matrix_assign_to(): + A = Matrix([[1, 2, 3]]) + assert maple_code(A, assign_to='a') == "a := Matrix([[1, 2, 3]], storage = rectangular)" + A = Matrix([[1, 2], [3, 4]]) + assert maple_code(A, assign_to='A') == "A := Matrix([[1, 2], [3, 4]], storage = rectangular)" + + +def test_maple_matrix_assign_to_more(): + # assigning to Symbol or MatrixSymbol requires lhs/rhs match + A = Matrix([[1, 2, 3]]) + B = MatrixSymbol('B', 1, 3) + C = MatrixSymbol('C', 2, 3) + assert maple_code(A, assign_to=B) == "B := Matrix([[1, 2, 3]], storage = rectangular)" + raises(ValueError, lambda: maple_code(A, assign_to=x)) + raises(ValueError, lambda: maple_code(A, assign_to=C)) + + +def test_maple_matrix_1x1(): + A = Matrix([[3]]) + assert maple_code(A, assign_to='B') == "B := Matrix([[3]], storage = rectangular)" + + +def test_maple_matrix_elements(): + A = Matrix([[x, 2, x * y]]) + + assert maple_code(A[0, 0] ** 2 + A[0, 1] + A[0, 2]) == "x^2 + x*y + 2" + AA = MatrixSymbol('AA', 1, 3) + assert maple_code(AA) == "AA" + + assert maple_code(AA[0, 0] ** 2 + sin(AA[0, 1]) + AA[0, 2]) == \ + "sin(AA[1, 2]) + AA[1, 1]^2 + AA[1, 3]" + assert maple_code(sum(AA)) == "AA[1, 1] + AA[1, 2] + AA[1, 3]" + + +def test_maple_boolean(): + assert maple_code(True) == "true" + assert maple_code(S.true) == "true" + assert maple_code(False) == "false" + assert maple_code(S.false) == "false" + + +def test_sparse(): + M = SparseMatrix(5, 6, {}) + M[2, 2] = 10 + M[1, 2] = 20 + M[1, 3] = 22 + M[0, 3] = 30 + M[3, 0] = x * y + assert maple_code(M) == \ + 'Matrix([[0, 0, 0, 30, 0, 0],' \ + ' [0, 0, 20, 22, 0, 0],' \ + ' [0, 0, 10, 0, 0, 0],' \ + ' [x*y, 0, 0, 0, 0, 0],' \ + ' [0, 0, 0, 0, 0, 0]], ' \ + 'storage = sparse)' + +# Not an important point. +def test_maple_not_supported(): + with raises(NotImplementedError): + maple_code(S.ComplexInfinity) + + +def test_MatrixElement_printing(): + # test cases for issue #11821 + A = MatrixSymbol("A", 1, 3) + B = MatrixSymbol("B", 1, 3) + + assert (maple_code(A[0, 0]) == "A[1, 1]") + assert (maple_code(3 * A[0, 0]) == "3*A[1, 1]") + + F = A-B + + assert (maple_code(F[0,0]) == "A[1, 1] - B[1, 1]") + + +def test_hadamard(): + A = MatrixSymbol('A', 3, 3) + B = MatrixSymbol('B', 3, 3) + v = MatrixSymbol('v', 3, 1) + h = MatrixSymbol('h', 1, 3) + C = HadamardProduct(A, B) + assert maple_code(C) == "A*B" + + assert maple_code(C * v) == "(A*B).v" + # HadamardProduct is higher than dot product. + + assert maple_code(h * C * v) == "h.(A*B).v" + + assert maple_code(C * A) == "(A*B).A" + # mixing Hadamard and scalar strange b/c we vectorize scalars + + assert maple_code(C * x * y) == "x*y*(A*B)" + + +def test_maple_piecewise(): + expr = Piecewise((x, x < 1), (x ** 2, True)) + + assert maple_code(expr) == "piecewise(x < 1, x, x^2)" + assert maple_code(expr, assign_to="r") == ( + "r := piecewise(x < 1, x, x^2)") + + expr = Piecewise((x ** 2, x < 1), (x ** 3, x < 2), (x ** 4, x < 3), (x ** 5, True)) + expected = "piecewise(x < 1, x^2, x < 2, x^3, x < 3, x^4, x^5)" + assert maple_code(expr) == expected + assert maple_code(expr, assign_to="r") == "r := " + expected + + # Check that Piecewise without a True (default) condition error + expr = Piecewise((x, x < 1), (x ** 2, x > 1), (sin(x), x > 0)) + raises(ValueError, lambda: maple_code(expr)) + + +def test_maple_piecewise_times_const(): + pw = Piecewise((x, x < 1), (x ** 2, True)) + + assert maple_code(2 * pw) == "2*piecewise(x < 1, x, x^2)" + assert maple_code(pw / x) == "piecewise(x < 1, x, x^2)/x" + assert maple_code(pw / (x * y)) == "piecewise(x < 1, x, x^2)/(x*y)" + assert maple_code(pw / 3) == "(1/3)*piecewise(x < 1, x, x^2)" + + +def test_maple_derivatives(): + f = Function('f') + assert maple_code(f(x).diff(x)) == 'diff(f(x), x)' + assert maple_code(f(x).diff(x, 2)) == 'diff(f(x), x$2)' + + +def test_automatic_rewrites(): + assert maple_code(lucas(x)) == '(2^(-x)*((1 - sqrt(5))^x + (1 + sqrt(5))^x))' + assert maple_code(sinc(x)) == '(piecewise(x <> 0, sin(x)/x, 1))' + + +def test_specfun(): + assert maple_code('asin(x)') == 'arcsin(x)' + assert maple_code(besseli(x, y)) == 'BesselI(x, y)' diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_mathematica.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_mathematica.py new file mode 100644 index 0000000000000000000000000000000000000000..aaf6b537677442ae59a4f1bbd2b5774d6646f4e2 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_mathematica.py @@ -0,0 +1,287 @@ +from sympy.core import (S, pi, oo, symbols, Function, Rational, Integer, Tuple, + Derivative, Eq, Ne, Le, Lt, Gt, Ge) +from sympy.integrals import Integral +from sympy.concrete import Sum +from sympy.functions import (exp, sin, cos, fresnelc, fresnels, conjugate, Max, + Min, gamma, polygamma, loggamma, erf, erfi, erfc, + erf2, expint, erfinv, erfcinv, Ei, Si, Ci, li, + Shi, Chi, uppergamma, beta, subfactorial, erf2inv, + factorial, factorial2, catalan, RisingFactorial, + FallingFactorial, harmonic, atan2, sec, acsc, + hermite, laguerre, assoc_laguerre, jacobi, + gegenbauer, chebyshevt, chebyshevu, legendre, + assoc_legendre, Li, LambertW) + +from sympy.printing.mathematica import mathematica_code as mcode + +x, y, z, w = symbols('x,y,z,w') +f = Function('f') + + +def test_Integer(): + assert mcode(Integer(67)) == "67" + assert mcode(Integer(-1)) == "-1" + + +def test_Rational(): + assert mcode(Rational(3, 7)) == "3/7" + assert mcode(Rational(18, 9)) == "2" + assert mcode(Rational(3, -7)) == "-3/7" + assert mcode(Rational(-3, -7)) == "3/7" + assert mcode(x + Rational(3, 7)) == "x + 3/7" + assert mcode(Rational(3, 7)*x) == "(3/7)*x" + + +def test_Relational(): + assert mcode(Eq(x, y)) == "x == y" + assert mcode(Ne(x, y)) == "x != y" + assert mcode(Le(x, y)) == "x <= y" + assert mcode(Lt(x, y)) == "x < y" + assert mcode(Gt(x, y)) == "x > y" + assert mcode(Ge(x, y)) == "x >= y" + + +def test_Function(): + assert mcode(f(x, y, z)) == "f[x, y, z]" + assert mcode(sin(x) ** cos(x)) == "Sin[x]^Cos[x]" + assert mcode(sec(x) * acsc(x)) == "ArcCsc[x]*Sec[x]" + assert mcode(atan2(y, x)) == "ArcTan[x, y]" + assert mcode(conjugate(x)) == "Conjugate[x]" + assert mcode(Max(x, y, z)*Min(y, z)) == "Max[x, y, z]*Min[y, z]" + assert mcode(fresnelc(x)) == "FresnelC[x]" + assert mcode(fresnels(x)) == "FresnelS[x]" + assert mcode(gamma(x)) == "Gamma[x]" + assert mcode(uppergamma(x, y)) == "Gamma[x, y]" + assert mcode(polygamma(x, y)) == "PolyGamma[x, y]" + assert mcode(loggamma(x)) == "LogGamma[x]" + assert mcode(erf(x)) == "Erf[x]" + assert mcode(erfc(x)) == "Erfc[x]" + assert mcode(erfi(x)) == "Erfi[x]" + assert mcode(erf2(x, y)) == "Erf[x, y]" + assert mcode(expint(x, y)) == "ExpIntegralE[x, y]" + assert mcode(erfcinv(x)) == "InverseErfc[x]" + assert mcode(erfinv(x)) == "InverseErf[x]" + assert mcode(erf2inv(x, y)) == "InverseErf[x, y]" + assert mcode(Ei(x)) == "ExpIntegralEi[x]" + assert mcode(Ci(x)) == "CosIntegral[x]" + assert mcode(li(x)) == "LogIntegral[x]" + assert mcode(Si(x)) == "SinIntegral[x]" + assert mcode(Shi(x)) == "SinhIntegral[x]" + assert mcode(Chi(x)) == "CoshIntegral[x]" + assert mcode(beta(x, y)) == "Beta[x, y]" + assert mcode(factorial(x)) == "Factorial[x]" + assert mcode(factorial2(x)) == "Factorial2[x]" + assert mcode(subfactorial(x)) == "Subfactorial[x]" + assert mcode(FallingFactorial(x, y)) == "FactorialPower[x, y]" + assert mcode(RisingFactorial(x, y)) == "Pochhammer[x, y]" + assert mcode(catalan(x)) == "CatalanNumber[x]" + assert mcode(harmonic(x)) == "HarmonicNumber[x]" + assert mcode(harmonic(x, y)) == "HarmonicNumber[x, y]" + assert mcode(Li(x)) == "LogIntegral[x] - LogIntegral[2]" + assert mcode(LambertW(x)) == "ProductLog[x]" + assert mcode(LambertW(x, -1)) == "ProductLog[-1, x]" + assert mcode(LambertW(x, y)) == "ProductLog[y, x]" + + +def test_special_polynomials(): + assert mcode(hermite(x, y)) == "HermiteH[x, y]" + assert mcode(laguerre(x, y)) == "LaguerreL[x, y]" + assert mcode(assoc_laguerre(x, y, z)) == "LaguerreL[x, y, z]" + assert mcode(jacobi(x, y, z, w)) == "JacobiP[x, y, z, w]" + assert mcode(gegenbauer(x, y, z)) == "GegenbauerC[x, y, z]" + assert mcode(chebyshevt(x, y)) == "ChebyshevT[x, y]" + assert mcode(chebyshevu(x, y)) == "ChebyshevU[x, y]" + assert mcode(legendre(x, y)) == "LegendreP[x, y]" + assert mcode(assoc_legendre(x, y, z)) == "LegendreP[x, y, z]" + + +def test_Pow(): + assert mcode(x**3) == "x^3" + assert mcode(x**(y**3)) == "x^(y^3)" + assert mcode(1/(f(x)*3.5)**(x - y**x)/(x**2 + y)) == \ + "(3.5*f[x])^(-x + y^x)/(x^2 + y)" + assert mcode(x**-1.0) == 'x^(-1.0)' + assert mcode(x**Rational(2, 3)) == 'x^(2/3)' + + +def test_Mul(): + A, B, C, D = symbols('A B C D', commutative=False) + assert mcode(x*y*z) == "x*y*z" + assert mcode(x*y*A) == "x*y*A" + assert mcode(x*y*A*B) == "x*y*A**B" + assert mcode(x*y*A*B*C) == "x*y*A**B**C" + assert mcode(x*A*B*(C + D)*A*y) == "x*y*A**B**(C + D)**A" + + +def test_constants(): + assert mcode(S.Zero) == "0" + assert mcode(S.One) == "1" + assert mcode(S.NegativeOne) == "-1" + assert mcode(S.Half) == "1/2" + assert mcode(S.ImaginaryUnit) == "I" + + assert mcode(oo) == "Infinity" + assert mcode(S.NegativeInfinity) == "-Infinity" + assert mcode(S.ComplexInfinity) == "ComplexInfinity" + assert mcode(S.NaN) == "Indeterminate" + + assert mcode(S.Exp1) == "E" + assert mcode(pi) == "Pi" + assert mcode(S.GoldenRatio) == "GoldenRatio" + assert mcode(S.TribonacciConstant) == \ + "(1/3 + (1/3)*(19 - 3*33^(1/2))^(1/3) + " \ + "(1/3)*(3*33^(1/2) + 19)^(1/3))" + assert mcode(2*S.TribonacciConstant) == \ + "2*(1/3 + (1/3)*(19 - 3*33^(1/2))^(1/3) + " \ + "(1/3)*(3*33^(1/2) + 19)^(1/3))" + assert mcode(S.EulerGamma) == "EulerGamma" + assert mcode(S.Catalan) == "Catalan" + + +def test_containers(): + assert mcode([1, 2, 3, [4, 5, [6, 7]], 8, [9, 10], 11]) == \ + "{1, 2, 3, {4, 5, {6, 7}}, 8, {9, 10}, 11}" + assert mcode((1, 2, (3, 4))) == "{1, 2, {3, 4}}" + assert mcode([1]) == "{1}" + assert mcode((1,)) == "{1}" + assert mcode(Tuple(*[1, 2, 3])) == "{1, 2, 3}" + + +def test_matrices(): + from sympy.matrices import MutableDenseMatrix, MutableSparseMatrix, \ + ImmutableDenseMatrix, ImmutableSparseMatrix + A = MutableDenseMatrix( + [[1, -1, 0, 0], + [0, 1, -1, 0], + [0, 0, 1, -1], + [0, 0, 0, 1]] + ) + B = MutableSparseMatrix(A) + C = ImmutableDenseMatrix(A) + D = ImmutableSparseMatrix(A) + + assert mcode(C) == mcode(A) == \ + "{{1, -1, 0, 0}, " \ + "{0, 1, -1, 0}, " \ + "{0, 0, 1, -1}, " \ + "{0, 0, 0, 1}}" + + assert mcode(D) == mcode(B) == \ + "SparseArray[{" \ + "{1, 1} -> 1, {1, 2} -> -1, {2, 2} -> 1, {2, 3} -> -1, " \ + "{3, 3} -> 1, {3, 4} -> -1, {4, 4} -> 1" \ + "}, {4, 4}]" + + # Trivial cases of matrices + assert mcode(MutableDenseMatrix(0, 0, [])) == '{}' + assert mcode(MutableSparseMatrix(0, 0, [])) == 'SparseArray[{}, {0, 0}]' + assert mcode(MutableDenseMatrix(0, 3, [])) == '{}' + assert mcode(MutableSparseMatrix(0, 3, [])) == 'SparseArray[{}, {0, 3}]' + assert mcode(MutableDenseMatrix(3, 0, [])) == '{{}, {}, {}}' + assert mcode(MutableSparseMatrix(3, 0, [])) == 'SparseArray[{}, {3, 0}]' + +def test_NDArray(): + from sympy.tensor.array import ( + MutableDenseNDimArray, ImmutableDenseNDimArray, + MutableSparseNDimArray, ImmutableSparseNDimArray) + + example = MutableDenseNDimArray( + [[[1, 2, 3, 4], + [5, 6, 7, 8], + [9, 10, 11, 12]], + [[13, 14, 15, 16], + [17, 18, 19, 20], + [21, 22, 23, 24]]] + ) + + assert mcode(example) == \ + "{{{1, 2, 3, 4}, {5, 6, 7, 8}, {9, 10, 11, 12}}, " \ + "{{13, 14, 15, 16}, {17, 18, 19, 20}, {21, 22, 23, 24}}}" + + example = ImmutableDenseNDimArray(example) + + assert mcode(example) == \ + "{{{1, 2, 3, 4}, {5, 6, 7, 8}, {9, 10, 11, 12}}, " \ + "{{13, 14, 15, 16}, {17, 18, 19, 20}, {21, 22, 23, 24}}}" + + example = MutableSparseNDimArray(example) + + assert mcode(example) == \ + "SparseArray[{" \ + "{1, 1, 1} -> 1, {1, 1, 2} -> 2, {1, 1, 3} -> 3, " \ + "{1, 1, 4} -> 4, {1, 2, 1} -> 5, {1, 2, 2} -> 6, " \ + "{1, 2, 3} -> 7, {1, 2, 4} -> 8, {1, 3, 1} -> 9, " \ + "{1, 3, 2} -> 10, {1, 3, 3} -> 11, {1, 3, 4} -> 12, " \ + "{2, 1, 1} -> 13, {2, 1, 2} -> 14, {2, 1, 3} -> 15, " \ + "{2, 1, 4} -> 16, {2, 2, 1} -> 17, {2, 2, 2} -> 18, " \ + "{2, 2, 3} -> 19, {2, 2, 4} -> 20, {2, 3, 1} -> 21, " \ + "{2, 3, 2} -> 22, {2, 3, 3} -> 23, {2, 3, 4} -> 24" \ + "}, {2, 3, 4}]" + + example = ImmutableSparseNDimArray(example) + + assert mcode(example) == \ + "SparseArray[{" \ + "{1, 1, 1} -> 1, {1, 1, 2} -> 2, {1, 1, 3} -> 3, " \ + "{1, 1, 4} -> 4, {1, 2, 1} -> 5, {1, 2, 2} -> 6, " \ + "{1, 2, 3} -> 7, {1, 2, 4} -> 8, {1, 3, 1} -> 9, " \ + "{1, 3, 2} -> 10, {1, 3, 3} -> 11, {1, 3, 4} -> 12, " \ + "{2, 1, 1} -> 13, {2, 1, 2} -> 14, {2, 1, 3} -> 15, " \ + "{2, 1, 4} -> 16, {2, 2, 1} -> 17, {2, 2, 2} -> 18, " \ + "{2, 2, 3} -> 19, {2, 2, 4} -> 20, {2, 3, 1} -> 21, " \ + "{2, 3, 2} -> 22, {2, 3, 3} -> 23, {2, 3, 4} -> 24" \ + "}, {2, 3, 4}]" + + +def test_Integral(): + assert mcode(Integral(sin(sin(x)), x)) == "Hold[Integrate[Sin[Sin[x]], x]]" + assert mcode(Integral(exp(-x**2 - y**2), + (x, -oo, oo), + (y, -oo, oo))) == \ + "Hold[Integrate[Exp[-x^2 - y^2], {x, -Infinity, Infinity}, " \ + "{y, -Infinity, Infinity}]]" + + +def test_Derivative(): + assert mcode(Derivative(sin(x), x)) == "Hold[D[Sin[x], x]]" + assert mcode(Derivative(x, x)) == "Hold[D[x, x]]" + assert mcode(Derivative(sin(x)*y**4, x, 2)) == "Hold[D[y^4*Sin[x], {x, 2}]]" + assert mcode(Derivative(sin(x)*y**4, x, y, x)) == "Hold[D[y^4*Sin[x], x, y, x]]" + assert mcode(Derivative(sin(x)*y**4, x, y, 3, x)) == "Hold[D[y^4*Sin[x], x, {y, 3}, x]]" + + +def test_Sum(): + assert mcode(Sum(sin(x), (x, 0, 10))) == "Hold[Sum[Sin[x], {x, 0, 10}]]" + assert mcode(Sum(exp(-x**2 - y**2), + (x, -oo, oo), + (y, -oo, oo))) == \ + "Hold[Sum[Exp[-x^2 - y^2], {x, -Infinity, Infinity}, " \ + "{y, -Infinity, Infinity}]]" + + +def test_comment(): + from sympy.printing.mathematica import MCodePrinter + assert MCodePrinter()._get_comment("Hello World") == \ + "(* Hello World *)" + + +def test_userfuncs(): + # Dictionary mutation test + some_function = symbols("some_function", cls=Function) + my_user_functions = {"some_function": "SomeFunction"} + assert mcode( + some_function(z), + user_functions=my_user_functions) == \ + 'SomeFunction[z]' + assert mcode( + some_function(z), + user_functions=my_user_functions) == \ + 'SomeFunction[z]' + + # List argument test + my_user_functions = \ + {"some_function": [(lambda x: True, "SomeOtherFunction")]} + assert mcode( + some_function(z), + user_functions=my_user_functions) == \ + 'SomeOtherFunction[z]' diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_mathml.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_mathml.py new file mode 100644 index 0000000000000000000000000000000000000000..4e7c2253c98fb1a4e99375774ad158df9b80b439 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_mathml.py @@ -0,0 +1,2048 @@ +from sympy.calculus.accumulationbounds import AccumBounds +from sympy.concrete.summations import Sum +from sympy.core.basic import Basic +from sympy.core.containers import Tuple +from sympy.core.function import Derivative, Lambda, diff, Function +from sympy.core.numbers import (zoo, Float, Integer, I, oo, pi, E, + Rational) +from sympy.core.relational import Lt, Ge, Ne, Eq +from sympy.core.singleton import S +from sympy.core.symbol import symbols, Symbol +from sympy.core.sympify import sympify +from sympy.functions.combinatorial.factorials import (factorial2, + binomial, factorial) +from sympy.functions.combinatorial.numbers import (lucas, bell, + catalan, euler, tribonacci, fibonacci, bernoulli, primenu, primeomega, + totient, reduced_totient) +from sympy.functions.elementary.complexes import re, im, conjugate, Abs +from sympy.functions.elementary.exponential import exp, LambertW, log +from sympy.functions.elementary.hyperbolic import (tanh, acoth, atanh, + coth, asinh, acsch, asech, acosh, csch, sinh, cosh, sech) +from sympy.functions.elementary.integers import ceiling, floor +from sympy.functions.elementary.miscellaneous import Max, Min +from sympy.functions.elementary.trigonometric import (csc, sec, tan, + atan, sin, asec, cot, cos, acot, acsc, asin, acos) +from sympy.functions.special.delta_functions import Heaviside +from sympy.functions.special.elliptic_integrals import (elliptic_pi, + elliptic_f, elliptic_k, elliptic_e) +from sympy.functions.special.error_functions import (fresnelc, + fresnels, Ei, expint) +from sympy.functions.special.gamma_functions import (gamma, uppergamma, + lowergamma) +from sympy.functions.special.mathieu_functions import (mathieusprime, + mathieus, mathieucprime, mathieuc) +from sympy.functions.special.polynomials import (jacobi, chebyshevu, + chebyshevt, hermite, assoc_legendre, gegenbauer, assoc_laguerre, + legendre, laguerre) +from sympy.functions.special.singularity_functions import SingularityFunction +from sympy.functions.special.zeta_functions import (polylog, stieltjes, + lerchphi, dirichlet_eta, zeta) +from sympy.integrals.integrals import Integral +from sympy.logic.boolalg import (Xor, Or, false, true, And, Equivalent, + Implies, Not) +from sympy.matrices.dense import Matrix +from sympy.matrices.expressions.determinant import Determinant +from sympy.matrices.expressions.matexpr import MatrixSymbol +from sympy.physics.quantum import (ComplexSpace, FockSpace, hbar, + HilbertSpace, Dagger) +from sympy.printing.mathml import (MathMLPresentationPrinter, + MathMLPrinter, MathMLContentPrinter, mathml) +from sympy.series.limits import Limit +from sympy.sets.contains import Contains +from sympy.sets.fancysets import Range +from sympy.sets.sets import (Interval, Union, SymmetricDifference, + Complement, FiniteSet, Intersection, ProductSet) +from sympy.stats.rv import RandomSymbol +from sympy.tensor.indexed import IndexedBase +from sympy.vector import (Divergence, CoordSys3D, Cross, Curl, Dot, + Laplacian, Gradient) +from sympy.testing.pytest import raises, XFAIL + +x, y, z, a, b, c, d, e, n = symbols('x:z a:e n') +mp = MathMLContentPrinter() +mpp = MathMLPresentationPrinter() + + +def test_mathml_printer(): + m = MathMLPrinter() + assert m.doprint(1+x) == mp.doprint(1+x) + + +def test_content_printmethod(): + assert mp.doprint(1 + x) == 'x1' + + +def test_content_mathml_core(): + mml_1 = mp._print(1 + x) + assert mml_1.nodeName == 'apply' + nodes = mml_1.childNodes + assert len(nodes) == 3 + assert nodes[0].nodeName == 'plus' + assert nodes[0].hasChildNodes() is False + assert nodes[0].nodeValue is None + assert nodes[1].nodeName in ['cn', 'ci'] + if nodes[1].nodeName == 'cn': + assert nodes[1].childNodes[0].nodeValue == '1' + assert nodes[2].childNodes[0].nodeValue == 'x' + else: + assert nodes[1].childNodes[0].nodeValue == 'x' + assert nodes[2].childNodes[0].nodeValue == '1' + + mml_2 = mp._print(x**2) + assert mml_2.nodeName == 'apply' + nodes = mml_2.childNodes + assert nodes[1].childNodes[0].nodeValue == 'x' + assert nodes[2].childNodes[0].nodeValue == '2' + + mml_3 = mp._print(2*x) + assert mml_3.nodeName == 'apply' + nodes = mml_3.childNodes + assert nodes[0].nodeName == 'times' + assert nodes[1].childNodes[0].nodeValue == '2' + assert nodes[2].childNodes[0].nodeValue == 'x' + + mml = mp._print(Float(1.0, 2)*x) + assert mml.nodeName == 'apply' + nodes = mml.childNodes + assert nodes[0].nodeName == 'times' + assert nodes[1].childNodes[0].nodeValue == '1.0' + assert nodes[2].childNodes[0].nodeValue == 'x' + + +def test_content_mathml_functions(): + mml_1 = mp._print(sin(x)) + assert mml_1.nodeName == 'apply' + assert mml_1.childNodes[0].nodeName == 'sin' + assert mml_1.childNodes[1].nodeName == 'ci' + + mml_2 = mp._print(diff(sin(x), x, evaluate=False)) + assert mml_2.nodeName == 'apply' + assert mml_2.childNodes[0].nodeName == 'diff' + assert mml_2.childNodes[1].nodeName == 'bvar' + assert mml_2.childNodes[1].childNodes[ + 0].nodeName == 'ci' # below bvar there's x/ci> + + mml_3 = mp._print(diff(cos(x*y), x, evaluate=False)) + assert mml_3.nodeName == 'apply' + assert mml_3.childNodes[0].nodeName == 'partialdiff' + assert mml_3.childNodes[1].nodeName == 'bvar' + assert mml_3.childNodes[1].childNodes[ + 0].nodeName == 'ci' # below bvar there's x/ci> + + mml_4 = mp._print(Lambda((x, y), x * y)) + assert mml_4.nodeName == 'lambda' + assert mml_4.childNodes[0].nodeName == 'bvar' + assert mml_4.childNodes[0].childNodes[ + 0].nodeName == 'ci' # below bvar there's x/ci> + assert mml_4.childNodes[1].nodeName == 'bvar' + assert mml_4.childNodes[1].childNodes[ + 0].nodeName == 'ci' # below bvar there's y/ci> + assert mml_4.childNodes[2].nodeName == 'apply' + + +def test_content_mathml_limits(): + # XXX No unevaluated limits + lim_fun = sin(x)/x + mml_1 = mp._print(Limit(lim_fun, x, 0)) + assert mml_1.childNodes[0].nodeName == 'limit' + assert mml_1.childNodes[1].nodeName == 'bvar' + assert mml_1.childNodes[2].nodeName == 'lowlimit' + assert mml_1.childNodes[3].toxml() == mp._print(lim_fun).toxml() + + +def test_content_mathml_integrals(): + integrand = x + mml_1 = mp._print(Integral(integrand, (x, 0, 1))) + assert mml_1.childNodes[0].nodeName == 'int' + assert mml_1.childNodes[1].nodeName == 'bvar' + assert mml_1.childNodes[2].nodeName == 'lowlimit' + assert mml_1.childNodes[3].nodeName == 'uplimit' + assert mml_1.childNodes[4].toxml() == mp._print(integrand).toxml() + + +def test_content_mathml_matrices(): + A = Matrix([1, 2, 3]) + B = Matrix([[0, 5, 4], [2, 3, 1], [9, 7, 9]]) + mll_1 = mp._print(A) + assert mll_1.childNodes[0].nodeName == 'matrixrow' + assert mll_1.childNodes[0].childNodes[0].nodeName == 'cn' + assert mll_1.childNodes[0].childNodes[0].childNodes[0].nodeValue == '1' + assert mll_1.childNodes[1].nodeName == 'matrixrow' + assert mll_1.childNodes[1].childNodes[0].nodeName == 'cn' + assert mll_1.childNodes[1].childNodes[0].childNodes[0].nodeValue == '2' + assert mll_1.childNodes[2].nodeName == 'matrixrow' + assert mll_1.childNodes[2].childNodes[0].nodeName == 'cn' + assert mll_1.childNodes[2].childNodes[0].childNodes[0].nodeValue == '3' + mll_2 = mp._print(B) + assert mll_2.childNodes[0].nodeName == 'matrixrow' + assert mll_2.childNodes[0].childNodes[0].nodeName == 'cn' + assert mll_2.childNodes[0].childNodes[0].childNodes[0].nodeValue == '0' + assert mll_2.childNodes[0].childNodes[1].nodeName == 'cn' + assert mll_2.childNodes[0].childNodes[1].childNodes[0].nodeValue == '5' + assert mll_2.childNodes[0].childNodes[2].nodeName == 'cn' + assert mll_2.childNodes[0].childNodes[2].childNodes[0].nodeValue == '4' + assert mll_2.childNodes[1].nodeName == 'matrixrow' + assert mll_2.childNodes[1].childNodes[0].nodeName == 'cn' + assert mll_2.childNodes[1].childNodes[0].childNodes[0].nodeValue == '2' + assert mll_2.childNodes[1].childNodes[1].nodeName == 'cn' + assert mll_2.childNodes[1].childNodes[1].childNodes[0].nodeValue == '3' + assert mll_2.childNodes[1].childNodes[2].nodeName == 'cn' + assert mll_2.childNodes[1].childNodes[2].childNodes[0].nodeValue == '1' + assert mll_2.childNodes[2].nodeName == 'matrixrow' + assert mll_2.childNodes[2].childNodes[0].nodeName == 'cn' + assert mll_2.childNodes[2].childNodes[0].childNodes[0].nodeValue == '9' + assert mll_2.childNodes[2].childNodes[1].nodeName == 'cn' + assert mll_2.childNodes[2].childNodes[1].childNodes[0].nodeValue == '7' + assert mll_2.childNodes[2].childNodes[2].nodeName == 'cn' + assert mll_2.childNodes[2].childNodes[2].childNodes[0].nodeValue == '9' + + +def test_content_mathml_sums(): + summand = x + mml_1 = mp._print(Sum(summand, (x, 1, 10))) + assert mml_1.childNodes[0].nodeName == 'sum' + assert mml_1.childNodes[1].nodeName == 'bvar' + assert mml_1.childNodes[2].nodeName == 'lowlimit' + assert mml_1.childNodes[3].nodeName == 'uplimit' + assert mml_1.childNodes[4].toxml() == mp._print(summand).toxml() + + +def test_content_mathml_tuples(): + mml_1 = mp._print([2]) + assert mml_1.nodeName == 'list' + assert mml_1.childNodes[0].nodeName == 'cn' + assert len(mml_1.childNodes) == 1 + + mml_2 = mp._print([2, Integer(1)]) + assert mml_2.nodeName == 'list' + assert mml_2.childNodes[0].nodeName == 'cn' + assert mml_2.childNodes[1].nodeName == 'cn' + assert len(mml_2.childNodes) == 2 + + +def test_content_mathml_add(): + mml = mp._print(x**5 - x**4 + x) + assert mml.childNodes[0].nodeName == 'plus' + assert mml.childNodes[1].childNodes[0].nodeName == 'minus' + assert mml.childNodes[1].childNodes[1].nodeName == 'apply' + + +def test_content_mathml_Rational(): + mml_1 = mp._print(Rational(1, 1)) + """should just return a number""" + assert mml_1.nodeName == 'cn' + + mml_2 = mp._print(Rational(2, 5)) + assert mml_2.childNodes[0].nodeName == 'divide' + + +def test_content_mathml_constants(): + mml = mp._print(I) + assert mml.nodeName == 'imaginaryi' + + mml = mp._print(E) + assert mml.nodeName == 'exponentiale' + + mml = mp._print(oo) + assert mml.nodeName == 'infinity' + + mml = mp._print(pi) + assert mml.nodeName == 'pi' + + assert mathml(hbar) == '' + assert mathml(S.TribonacciConstant) == '' + assert mathml(S.GoldenRatio) == 'φ' + mml = mathml(S.EulerGamma) + assert mml == '' + + mml = mathml(S.EmptySet) + assert mml == '' + + mml = mathml(S.true) + assert mml == '' + + mml = mathml(S.false) + assert mml == '' + + mml = mathml(S.NaN) + assert mml == '' + + +def test_content_mathml_trig(): + mml = mp._print(sin(x)) + assert mml.childNodes[0].nodeName == 'sin' + + mml = mp._print(cos(x)) + assert mml.childNodes[0].nodeName == 'cos' + + mml = mp._print(tan(x)) + assert mml.childNodes[0].nodeName == 'tan' + + mml = mp._print(cot(x)) + assert mml.childNodes[0].nodeName == 'cot' + + mml = mp._print(csc(x)) + assert mml.childNodes[0].nodeName == 'csc' + + mml = mp._print(sec(x)) + assert mml.childNodes[0].nodeName == 'sec' + + mml = mp._print(asin(x)) + assert mml.childNodes[0].nodeName == 'arcsin' + + mml = mp._print(acos(x)) + assert mml.childNodes[0].nodeName == 'arccos' + + mml = mp._print(atan(x)) + assert mml.childNodes[0].nodeName == 'arctan' + + mml = mp._print(acot(x)) + assert mml.childNodes[0].nodeName == 'arccot' + + mml = mp._print(acsc(x)) + assert mml.childNodes[0].nodeName == 'arccsc' + + mml = mp._print(asec(x)) + assert mml.childNodes[0].nodeName == 'arcsec' + + mml = mp._print(sinh(x)) + assert mml.childNodes[0].nodeName == 'sinh' + + mml = mp._print(cosh(x)) + assert mml.childNodes[0].nodeName == 'cosh' + + mml = mp._print(tanh(x)) + assert mml.childNodes[0].nodeName == 'tanh' + + mml = mp._print(coth(x)) + assert mml.childNodes[0].nodeName == 'coth' + + mml = mp._print(csch(x)) + assert mml.childNodes[0].nodeName == 'csch' + + mml = mp._print(sech(x)) + assert mml.childNodes[0].nodeName == 'sech' + + mml = mp._print(asinh(x)) + assert mml.childNodes[0].nodeName == 'arcsinh' + + mml = mp._print(atanh(x)) + assert mml.childNodes[0].nodeName == 'arctanh' + + mml = mp._print(acosh(x)) + assert mml.childNodes[0].nodeName == 'arccosh' + + mml = mp._print(acoth(x)) + assert mml.childNodes[0].nodeName == 'arccoth' + + mml = mp._print(acsch(x)) + assert mml.childNodes[0].nodeName == 'arccsch' + + mml = mp._print(asech(x)) + assert mml.childNodes[0].nodeName == 'arcsech' + + +def test_content_mathml_relational(): + mml_1 = mp._print(Eq(x, 1)) + assert mml_1.nodeName == 'apply' + assert mml_1.childNodes[0].nodeName == 'eq' + assert mml_1.childNodes[1].nodeName == 'ci' + assert mml_1.childNodes[1].childNodes[0].nodeValue == 'x' + assert mml_1.childNodes[2].nodeName == 'cn' + assert mml_1.childNodes[2].childNodes[0].nodeValue == '1' + + mml_2 = mp._print(Ne(1, x)) + assert mml_2.nodeName == 'apply' + assert mml_2.childNodes[0].nodeName == 'neq' + assert mml_2.childNodes[1].nodeName == 'cn' + assert mml_2.childNodes[1].childNodes[0].nodeValue == '1' + assert mml_2.childNodes[2].nodeName == 'ci' + assert mml_2.childNodes[2].childNodes[0].nodeValue == 'x' + + mml_3 = mp._print(Ge(1, x)) + assert mml_3.nodeName == 'apply' + assert mml_3.childNodes[0].nodeName == 'geq' + assert mml_3.childNodes[1].nodeName == 'cn' + assert mml_3.childNodes[1].childNodes[0].nodeValue == '1' + assert mml_3.childNodes[2].nodeName == 'ci' + assert mml_3.childNodes[2].childNodes[0].nodeValue == 'x' + + mml_4 = mp._print(Lt(1, x)) + assert mml_4.nodeName == 'apply' + assert mml_4.childNodes[0].nodeName == 'lt' + assert mml_4.childNodes[1].nodeName == 'cn' + assert mml_4.childNodes[1].childNodes[0].nodeValue == '1' + assert mml_4.childNodes[2].nodeName == 'ci' + assert mml_4.childNodes[2].childNodes[0].nodeValue == 'x' + + +def test_content_symbol(): + mml = mp._print(x) + assert mml.nodeName == 'ci' + assert mml.childNodes[0].nodeValue == 'x' + del mml + + mml = mp._print(Symbol("x^2")) + assert mml.nodeName == 'ci' + assert mml.childNodes[0].nodeName == 'mml:msup' + assert mml.childNodes[0].childNodes[0].nodeName == 'mml:mi' + assert mml.childNodes[0].childNodes[0].childNodes[0].nodeValue == 'x' + assert mml.childNodes[0].childNodes[1].nodeName == 'mml:mi' + assert mml.childNodes[0].childNodes[1].childNodes[0].nodeValue == '2' + del mml + + mml = mp._print(Symbol("x__2")) + assert mml.nodeName == 'ci' + assert mml.childNodes[0].nodeName == 'mml:msup' + assert mml.childNodes[0].childNodes[0].nodeName == 'mml:mi' + assert mml.childNodes[0].childNodes[0].childNodes[0].nodeValue == 'x' + assert mml.childNodes[0].childNodes[1].nodeName == 'mml:mi' + assert mml.childNodes[0].childNodes[1].childNodes[0].nodeValue == '2' + del mml + + mml = mp._print(Symbol("x_2")) + assert mml.nodeName == 'ci' + assert mml.childNodes[0].nodeName == 'mml:msub' + assert mml.childNodes[0].childNodes[0].nodeName == 'mml:mi' + assert mml.childNodes[0].childNodes[0].childNodes[0].nodeValue == 'x' + assert mml.childNodes[0].childNodes[1].nodeName == 'mml:mi' + assert mml.childNodes[0].childNodes[1].childNodes[0].nodeValue == '2' + del mml + + mml = mp._print(Symbol("x^3_2")) + assert mml.nodeName == 'ci' + assert mml.childNodes[0].nodeName == 'mml:msubsup' + assert mml.childNodes[0].childNodes[0].nodeName == 'mml:mi' + assert mml.childNodes[0].childNodes[0].childNodes[0].nodeValue == 'x' + assert mml.childNodes[0].childNodes[1].nodeName == 'mml:mi' + assert mml.childNodes[0].childNodes[1].childNodes[0].nodeValue == '2' + assert mml.childNodes[0].childNodes[2].nodeName == 'mml:mi' + assert mml.childNodes[0].childNodes[2].childNodes[0].nodeValue == '3' + del mml + + mml = mp._print(Symbol("x__3_2")) + assert mml.nodeName == 'ci' + assert mml.childNodes[0].nodeName == 'mml:msubsup' + assert mml.childNodes[0].childNodes[0].nodeName == 'mml:mi' + assert mml.childNodes[0].childNodes[0].childNodes[0].nodeValue == 'x' + assert mml.childNodes[0].childNodes[1].nodeName == 'mml:mi' + assert mml.childNodes[0].childNodes[1].childNodes[0].nodeValue == '2' + assert mml.childNodes[0].childNodes[2].nodeName == 'mml:mi' + assert mml.childNodes[0].childNodes[2].childNodes[0].nodeValue == '3' + del mml + + mml = mp._print(Symbol("x_2_a")) + assert mml.nodeName == 'ci' + assert mml.childNodes[0].nodeName == 'mml:msub' + assert mml.childNodes[0].childNodes[0].nodeName == 'mml:mi' + assert mml.childNodes[0].childNodes[0].childNodes[0].nodeValue == 'x' + assert mml.childNodes[0].childNodes[1].nodeName == 'mml:mrow' + assert mml.childNodes[0].childNodes[1].childNodes[0].nodeName == 'mml:mi' + assert mml.childNodes[0].childNodes[1].childNodes[0].childNodes[ + 0].nodeValue == '2' + assert mml.childNodes[0].childNodes[1].childNodes[1].nodeName == 'mml:mo' + assert mml.childNodes[0].childNodes[1].childNodes[1].childNodes[ + 0].nodeValue == ' ' + assert mml.childNodes[0].childNodes[1].childNodes[2].nodeName == 'mml:mi' + assert mml.childNodes[0].childNodes[1].childNodes[2].childNodes[ + 0].nodeValue == 'a' + del mml + + mml = mp._print(Symbol("x^2^a")) + assert mml.nodeName == 'ci' + assert mml.childNodes[0].nodeName == 'mml:msup' + assert mml.childNodes[0].childNodes[0].nodeName == 'mml:mi' + assert mml.childNodes[0].childNodes[0].childNodes[0].nodeValue == 'x' + assert mml.childNodes[0].childNodes[1].nodeName == 'mml:mrow' + assert mml.childNodes[0].childNodes[1].childNodes[0].nodeName == 'mml:mi' + assert mml.childNodes[0].childNodes[1].childNodes[0].childNodes[ + 0].nodeValue == '2' + assert mml.childNodes[0].childNodes[1].childNodes[1].nodeName == 'mml:mo' + assert mml.childNodes[0].childNodes[1].childNodes[1].childNodes[ + 0].nodeValue == ' ' + assert mml.childNodes[0].childNodes[1].childNodes[2].nodeName == 'mml:mi' + assert mml.childNodes[0].childNodes[1].childNodes[2].childNodes[ + 0].nodeValue == 'a' + del mml + + mml = mp._print(Symbol("x__2__a")) + assert mml.nodeName == 'ci' + assert mml.childNodes[0].nodeName == 'mml:msup' + assert mml.childNodes[0].childNodes[0].nodeName == 'mml:mi' + assert mml.childNodes[0].childNodes[0].childNodes[0].nodeValue == 'x' + assert mml.childNodes[0].childNodes[1].nodeName == 'mml:mrow' + assert mml.childNodes[0].childNodes[1].childNodes[0].nodeName == 'mml:mi' + assert mml.childNodes[0].childNodes[1].childNodes[0].childNodes[ + 0].nodeValue == '2' + assert mml.childNodes[0].childNodes[1].childNodes[1].nodeName == 'mml:mo' + assert mml.childNodes[0].childNodes[1].childNodes[1].childNodes[ + 0].nodeValue == ' ' + assert mml.childNodes[0].childNodes[1].childNodes[2].nodeName == 'mml:mi' + assert mml.childNodes[0].childNodes[1].childNodes[2].childNodes[ + 0].nodeValue == 'a' + del mml + + +def test_content_mathml_greek(): + mml = mp._print(Symbol('alpha')) + assert mml.nodeName == 'ci' + assert mml.childNodes[0].nodeValue == '\N{GREEK SMALL LETTER ALPHA}' + + assert mp.doprint(Symbol('alpha')) == 'α' + assert mp.doprint(Symbol('beta')) == 'β' + assert mp.doprint(Symbol('gamma')) == 'γ' + assert mp.doprint(Symbol('delta')) == 'δ' + assert mp.doprint(Symbol('epsilon')) == 'ε' + assert mp.doprint(Symbol('zeta')) == 'ζ' + assert mp.doprint(Symbol('eta')) == 'η' + assert mp.doprint(Symbol('theta')) == 'θ' + assert mp.doprint(Symbol('iota')) == 'ι' + assert mp.doprint(Symbol('kappa')) == 'κ' + assert mp.doprint(Symbol('lambda')) == 'λ' + assert mp.doprint(Symbol('mu')) == 'μ' + assert mp.doprint(Symbol('nu')) == 'ν' + assert mp.doprint(Symbol('xi')) == 'ξ' + assert mp.doprint(Symbol('omicron')) == 'ο' + assert mp.doprint(Symbol('pi')) == 'π' + assert mp.doprint(Symbol('rho')) == 'ρ' + assert mp.doprint(Symbol('varsigma')) == 'ς' + assert mp.doprint(Symbol('sigma')) == 'σ' + assert mp.doprint(Symbol('tau')) == 'τ' + assert mp.doprint(Symbol('upsilon')) == 'υ' + assert mp.doprint(Symbol('phi')) == 'φ' + assert mp.doprint(Symbol('chi')) == 'χ' + assert mp.doprint(Symbol('psi')) == 'ψ' + assert mp.doprint(Symbol('omega')) == 'ω' + + assert mp.doprint(Symbol('Alpha')) == 'Α' + assert mp.doprint(Symbol('Beta')) == 'Β' + assert mp.doprint(Symbol('Gamma')) == 'Γ' + assert mp.doprint(Symbol('Delta')) == 'Δ' + assert mp.doprint(Symbol('Epsilon')) == 'Ε' + assert mp.doprint(Symbol('Zeta')) == 'Ζ' + assert mp.doprint(Symbol('Eta')) == 'Η' + assert mp.doprint(Symbol('Theta')) == 'Θ' + assert mp.doprint(Symbol('Iota')) == 'Ι' + assert mp.doprint(Symbol('Kappa')) == 'Κ' + assert mp.doprint(Symbol('Lambda')) == 'Λ' + assert mp.doprint(Symbol('Mu')) == 'Μ' + assert mp.doprint(Symbol('Nu')) == 'Ν' + assert mp.doprint(Symbol('Xi')) == 'Ξ' + assert mp.doprint(Symbol('Omicron')) == 'Ο' + assert mp.doprint(Symbol('Pi')) == 'Π' + assert mp.doprint(Symbol('Rho')) == 'Ρ' + assert mp.doprint(Symbol('Sigma')) == 'Σ' + assert mp.doprint(Symbol('Tau')) == 'Τ' + assert mp.doprint(Symbol('Upsilon')) == 'Υ' + assert mp.doprint(Symbol('Phi')) == 'Φ' + assert mp.doprint(Symbol('Chi')) == 'Χ' + assert mp.doprint(Symbol('Psi')) == 'Ψ' + assert mp.doprint(Symbol('Omega')) == 'Ω' + + +def test_content_mathml_order(): + expr = x**3 + x**2*y + 3*x*y**3 + y**4 + + mp = MathMLContentPrinter({'order': 'lex'}) + mml = mp._print(expr) + + assert mml.childNodes[1].childNodes[0].nodeName == 'power' + assert mml.childNodes[1].childNodes[1].childNodes[0].data == 'x' + assert mml.childNodes[1].childNodes[2].childNodes[0].data == '3' + + assert mml.childNodes[4].childNodes[0].nodeName == 'power' + assert mml.childNodes[4].childNodes[1].childNodes[0].data == 'y' + assert mml.childNodes[4].childNodes[2].childNodes[0].data == '4' + + mp = MathMLContentPrinter({'order': 'rev-lex'}) + mml = mp._print(expr) + + assert mml.childNodes[1].childNodes[0].nodeName == 'power' + assert mml.childNodes[1].childNodes[1].childNodes[0].data == 'y' + assert mml.childNodes[1].childNodes[2].childNodes[0].data == '4' + + assert mml.childNodes[4].childNodes[0].nodeName == 'power' + assert mml.childNodes[4].childNodes[1].childNodes[0].data == 'x' + assert mml.childNodes[4].childNodes[2].childNodes[0].data == '3' + + +def test_content_settings(): + raises(TypeError, lambda: mathml(x, method="garbage")) + + +def test_content_mathml_logic(): + assert mathml(And(x, y)) == 'xy' + assert mathml(Or(x, y)) == 'xy' + assert mathml(Xor(x, y)) == 'xy' + assert mathml(Implies(x, y)) == 'xy' + assert mathml(Not(x)) == 'x' + + +def test_content_finite_sets(): + assert mathml(FiniteSet(a)) == 'a' + assert mathml(FiniteSet(a, b)) == 'ab' + assert mathml(FiniteSet(FiniteSet(a, b), c)) == \ + 'cab' + + A = FiniteSet(a) + B = FiniteSet(b) + C = FiniteSet(c) + D = FiniteSet(d) + + U1 = Union(A, B, evaluate=False) + U2 = Union(C, D, evaluate=False) + I1 = Intersection(A, B, evaluate=False) + I2 = Intersection(C, D, evaluate=False) + C1 = Complement(A, B, evaluate=False) + C2 = Complement(C, D, evaluate=False) + # XXX ProductSet does not support evaluate keyword + P1 = ProductSet(A, B) + P2 = ProductSet(C, D) + + assert mathml(U1) == \ + 'ab' + assert mathml(I1) == \ + 'ab' \ + '' + assert mathml(C1) == \ + 'ab' + assert mathml(P1) == \ + 'ab' \ + '' + + assert mathml(Intersection(A, U2, evaluate=False)) == \ + 'a' \ + 'cd' + assert mathml(Intersection(U1, U2, evaluate=False)) == \ + 'a' \ + 'bc' \ + 'd' + + # XXX Does the parenthesis appear correctly for these examples in mathjax? + assert mathml(Intersection(C1, C2, evaluate=False)) == \ + 'a' \ + 'bc' \ + 'd' + assert mathml(Intersection(P1, P2, evaluate=False)) == \ + 'a' \ + 'b' \ + 'cd' + + assert mathml(Union(A, I2, evaluate=False)) == \ + 'a' \ + 'cd' + assert mathml(Union(I1, I2, evaluate=False)) == \ + 'a' \ + 'bc' \ + 'd' + assert mathml(Union(C1, C2, evaluate=False)) == \ + 'a' \ + 'bc' \ + 'd' + assert mathml(Union(P1, P2, evaluate=False)) == \ + 'a' \ + 'b' \ + 'cd' + + assert mathml(Complement(A, C2, evaluate=False)) == \ + 'a' \ + 'cd' + assert mathml(Complement(U1, U2, evaluate=False)) == \ + 'a' \ + 'bc' \ + 'd' + assert mathml(Complement(I1, I2, evaluate=False)) == \ + 'a' \ + 'bc' \ + 'd' + assert mathml(Complement(P1, P2, evaluate=False)) == \ + 'a' \ + 'b' \ + 'cd' + + assert mathml(ProductSet(A, P2)) == \ + 'a' \ + 'c' \ + 'd' + assert mathml(ProductSet(U1, U2)) == \ + 'a' \ + 'bc' \ + 'd' + assert mathml(ProductSet(I1, I2)) == \ + 'a' \ + 'b' \ + 'cd' + assert mathml(ProductSet(C1, C2)) == \ + 'a' \ + 'b' \ + 'cd' + + +def test_presentation_printmethod(): + assert mpp.doprint(1 + x) == 'x+1' + assert mpp.doprint(x**2) == 'x2' + assert mpp.doprint(x**-1) == '1x' + assert mpp.doprint(x**-2) == \ + '1x2' + assert mpp.doprint(2*x) == \ + '2⁢x' + + +def test_presentation_mathml_core(): + mml_1 = mpp._print(1 + x) + assert mml_1.nodeName == 'mrow' + nodes = mml_1.childNodes + assert len(nodes) == 3 + assert nodes[0].nodeName in ['mi', 'mn'] + assert nodes[1].nodeName == 'mo' + if nodes[0].nodeName == 'mn': + assert nodes[0].childNodes[0].nodeValue == '1' + assert nodes[2].childNodes[0].nodeValue == 'x' + else: + assert nodes[0].childNodes[0].nodeValue == 'x' + assert nodes[2].childNodes[0].nodeValue == '1' + + mml_2 = mpp._print(x**2) + assert mml_2.nodeName == 'msup' + nodes = mml_2.childNodes + assert nodes[0].childNodes[0].nodeValue == 'x' + assert nodes[1].childNodes[0].nodeValue == '2' + + mml_3 = mpp._print(2*x) + assert mml_3.nodeName == 'mrow' + nodes = mml_3.childNodes + assert nodes[0].childNodes[0].nodeValue == '2' + assert nodes[1].childNodes[0].nodeValue == '⁢' + assert nodes[2].childNodes[0].nodeValue == 'x' + + mml = mpp._print(Float(1.0, 2)*x) + assert mml.nodeName == 'mrow' + nodes = mml.childNodes + assert nodes[0].childNodes[0].nodeValue == '1.0' + assert nodes[1].childNodes[0].nodeValue == '⁢' + assert nodes[2].childNodes[0].nodeValue == 'x' + + +def test_presentation_mathml_functions(): + mml_1 = mpp._print(sin(x)) + assert mml_1.childNodes[0].childNodes[0 + ].nodeValue == 'sin' + assert mml_1.childNodes[1].childNodes[1 + ].childNodes[0].nodeValue == 'x' + + mml_2 = mpp._print(diff(sin(x), x, evaluate=False)) + assert mml_2.nodeName == 'mrow' + assert mml_2.childNodes[0].childNodes[0 + ].childNodes[0].childNodes[0].nodeValue == 'ⅆ' + assert mml_2.childNodes[1].childNodes[1 + ].nodeName == 'mrow' + assert mml_2.childNodes[0].childNodes[1 + ].childNodes[0].childNodes[0].nodeValue == 'ⅆ' + + mml_3 = mpp._print(diff(cos(x*y), x, evaluate=False)) + assert mml_3.childNodes[0].nodeName == 'mfrac' + assert mml_3.childNodes[0].childNodes[0 + ].childNodes[0].childNodes[0].nodeValue == '∂' + assert mml_3.childNodes[1].childNodes[0 + ].childNodes[0].nodeValue == 'cos' + + +def test_print_derivative(): + f = Function('f') + d = Derivative(f(x, y, z), x, z, x, z, z, y) + assert mathml(d) == \ + 'yz2xzxxyz' + assert mathml(d, printer='presentation') == \ + '∂6∂y∂2z∂x∂z∂xf(x,y,z)' + + +def test_presentation_mathml_limits(): + lim_fun = sin(x)/x + mml_1 = mpp._print(Limit(lim_fun, x, 0)) + assert mml_1.childNodes[0].nodeName == 'munder' + assert mml_1.childNodes[0].childNodes[0 + ].childNodes[0].nodeValue == 'lim' + assert mml_1.childNodes[0].childNodes[1 + ].childNodes[0].childNodes[0 + ].nodeValue == 'x' + assert mml_1.childNodes[0].childNodes[1 + ].childNodes[1].childNodes[0 + ].nodeValue == '→' + assert mml_1.childNodes[0].childNodes[1 + ].childNodes[2].childNodes[0 + ].nodeValue == '0' + + +def test_presentation_mathml_integrals(): + assert mpp.doprint(Integral(x, (x, 0, 1))) == \ + '∫01'\ + 'xⅆx' + assert mpp.doprint(Integral(log(x), x)) == \ + '∫log(x' \ + ')ⅆx' + assert mpp.doprint(Integral(x*y, x, y)) == \ + '∬x⁢'\ + 'yⅆyⅆx' + z, w = symbols('z w') + assert mpp.doprint(Integral(x*y*z, x, y, z)) == \ + '∭x⁢'\ + 'y⁢zⅆ'\ + 'zⅆyⅆx' + assert mpp.doprint(Integral(x*y*z*w, x, y, z, w)) == \ + '∫∫∫'\ + '∫w⁢'\ + 'x⁢y'\ + '⁢zⅆw'\ + 'ⅆzⅆyⅆx' + assert mpp.doprint(Integral(x, x, y, (z, 0, 1))) == \ + '∫01'\ + '∫∫xⅆz'\ + 'ⅆyⅆx' + assert mpp.doprint(Integral(x, (x, 0))) == \ + '∫0xⅆ'\ + 'x' + + +def test_presentation_mathml_matrices(): + A = Matrix([1, 2, 3]) + B = Matrix([[0, 5, 4], [2, 3, 1], [9, 7, 9]]) + mll_1 = mpp._print(A) + assert mll_1.childNodes[1].nodeName == 'mtable' + assert mll_1.childNodes[1].childNodes[0].nodeName == 'mtr' + assert len(mll_1.childNodes[1].childNodes) == 3 + assert mll_1.childNodes[1].childNodes[0].childNodes[0].nodeName == 'mtd' + assert len(mll_1.childNodes[1].childNodes[0].childNodes) == 1 + assert mll_1.childNodes[1].childNodes[0].childNodes[0 + ].childNodes[0].childNodes[0].nodeValue == '1' + assert mll_1.childNodes[1].childNodes[1].childNodes[0 + ].childNodes[0].childNodes[0].nodeValue == '2' + assert mll_1.childNodes[1].childNodes[2].childNodes[0 + ].childNodes[0].childNodes[0].nodeValue == '3' + mll_2 = mpp._print(B) + assert mll_2.childNodes[1].nodeName == 'mtable' + assert mll_2.childNodes[1].childNodes[0].nodeName == 'mtr' + assert len(mll_2.childNodes[1].childNodes) == 3 + assert mll_2.childNodes[1].childNodes[0].childNodes[0].nodeName == 'mtd' + assert len(mll_2.childNodes[1].childNodes[0].childNodes) == 3 + assert mll_2.childNodes[1].childNodes[0].childNodes[0 + ].childNodes[0].childNodes[0].nodeValue == '0' + assert mll_2.childNodes[1].childNodes[0].childNodes[1 + ].childNodes[0].childNodes[0].nodeValue == '5' + assert mll_2.childNodes[1].childNodes[0].childNodes[2 + ].childNodes[0].childNodes[0].nodeValue == '4' + assert mll_2.childNodes[1].childNodes[1].childNodes[0 + ].childNodes[0].childNodes[0].nodeValue == '2' + assert mll_2.childNodes[1].childNodes[1].childNodes[1 + ].childNodes[0].childNodes[0].nodeValue == '3' + assert mll_2.childNodes[1].childNodes[1].childNodes[2 + ].childNodes[0].childNodes[0].nodeValue == '1' + assert mll_2.childNodes[1].childNodes[2].childNodes[0 + ].childNodes[0].childNodes[0].nodeValue == '9' + assert mll_2.childNodes[1].childNodes[2].childNodes[1 + ].childNodes[0].childNodes[0].nodeValue == '7' + assert mll_2.childNodes[1].childNodes[2].childNodes[2 + ].childNodes[0].childNodes[0].nodeValue == '9' + + +def test_presentation_mathml_sums(): + mml_1 = mpp._print(Sum(x, (x, 1, 10))) + assert mml_1.childNodes[0].nodeName == 'munderover' + assert len(mml_1.childNodes[0].childNodes) == 3 + assert mml_1.childNodes[0].childNodes[0].childNodes[0 + ].nodeValue == '∑' + assert len(mml_1.childNodes[0].childNodes[1].childNodes) == 3 + assert mml_1.childNodes[0].childNodes[2].childNodes[0 + ].nodeValue == '10' + assert mml_1.childNodes[1].childNodes[0].nodeValue == 'x' + + assert mpp.doprint(Sum(x, (x, 1, 10))) == \ + '∑x=110x' + assert mpp.doprint(Sum(x + y, (x, 1, 10))) == \ + '∑x=110(x+y)' + + +def test_presentation_mathml_add(): + mml = mpp._print(x**5 - x**4 + x) + assert len(mml.childNodes) == 5 + assert mml.childNodes[0].childNodes[0].childNodes[0 + ].nodeValue == 'x' + assert mml.childNodes[0].childNodes[1].childNodes[0 + ].nodeValue == '5' + assert mml.childNodes[1].childNodes[0].nodeValue == '-' + assert mml.childNodes[2].childNodes[0].childNodes[0 + ].nodeValue == 'x' + assert mml.childNodes[2].childNodes[1].childNodes[0 + ].nodeValue == '4' + assert mml.childNodes[3].childNodes[0].nodeValue == '+' + assert mml.childNodes[4].childNodes[0].nodeValue == 'x' + + +def test_presentation_mathml_Rational(): + mml_1 = mpp._print(Rational(1, 1)) + assert mml_1.nodeName == 'mn' + + mml_2 = mpp._print(Rational(2, 5)) + assert mml_2.nodeName == 'mfrac' + assert mml_2.childNodes[0].childNodes[0].nodeValue == '2' + assert mml_2.childNodes[1].childNodes[0].nodeValue == '5' + + +def test_presentation_mathml_constants(): + mml = mpp._print(I) + assert mml.childNodes[0].nodeValue == 'ⅈ' + + mml = mpp._print(E) + assert mml.childNodes[0].nodeValue == 'ⅇ' + + mml = mpp._print(oo) + assert mml.childNodes[0].nodeValue == '∞' + + mml = mpp._print(pi) + assert mml.childNodes[0].nodeValue == 'π' + + assert mathml(hbar, printer='presentation') == 'ℏ' + assert mathml(S.TribonacciConstant, printer='presentation' + ) == 'TribonacciConstant' + assert mathml(S.EulerGamma, printer='presentation' + ) == 'γ' + assert mathml(S.GoldenRatio, printer='presentation' + ) == 'Φ' + + assert mathml(zoo, printer='presentation') == \ + '∞~' + + assert mathml(S.NaN, printer='presentation') == 'NaN' + +def test_presentation_mathml_trig(): + mml = mpp._print(sin(x)) + assert mml.childNodes[0].childNodes[0].nodeValue == 'sin' + + mml = mpp._print(cos(x)) + assert mml.childNodes[0].childNodes[0].nodeValue == 'cos' + + mml = mpp._print(tan(x)) + assert mml.childNodes[0].childNodes[0].nodeValue == 'tan' + + mml = mpp._print(asin(x)) + assert mml.childNodes[0].childNodes[0].nodeValue == 'arcsin' + + mml = mpp._print(acos(x)) + assert mml.childNodes[0].childNodes[0].nodeValue == 'arccos' + + mml = mpp._print(atan(x)) + assert mml.childNodes[0].childNodes[0].nodeValue == 'arctan' + + mml = mpp._print(sinh(x)) + assert mml.childNodes[0].childNodes[0].nodeValue == 'sinh' + + mml = mpp._print(cosh(x)) + assert mml.childNodes[0].childNodes[0].nodeValue == 'cosh' + + mml = mpp._print(tanh(x)) + assert mml.childNodes[0].childNodes[0].nodeValue == 'tanh' + + mml = mpp._print(asinh(x)) + assert mml.childNodes[0].childNodes[0].nodeValue == 'arcsinh' + + mml = mpp._print(atanh(x)) + assert mml.childNodes[0].childNodes[0].nodeValue == 'arctanh' + + mml = mpp._print(acosh(x)) + assert mml.childNodes[0].childNodes[0].nodeValue == 'arccosh' + + +def test_presentation_mathml_relational(): + mml_1 = mpp._print(Eq(x, 1)) + assert len(mml_1.childNodes) == 3 + assert mml_1.childNodes[0].nodeName == 'mi' + assert mml_1.childNodes[0].childNodes[0].nodeValue == 'x' + assert mml_1.childNodes[1].nodeName == 'mo' + assert mml_1.childNodes[1].childNodes[0].nodeValue == '=' + assert mml_1.childNodes[2].nodeName == 'mn' + assert mml_1.childNodes[2].childNodes[0].nodeValue == '1' + + mml_2 = mpp._print(Ne(1, x)) + assert len(mml_2.childNodes) == 3 + assert mml_2.childNodes[0].nodeName == 'mn' + assert mml_2.childNodes[0].childNodes[0].nodeValue == '1' + assert mml_2.childNodes[1].nodeName == 'mo' + assert mml_2.childNodes[1].childNodes[0].nodeValue == '≠' + assert mml_2.childNodes[2].nodeName == 'mi' + assert mml_2.childNodes[2].childNodes[0].nodeValue == 'x' + + mml_3 = mpp._print(Ge(1, x)) + assert len(mml_3.childNodes) == 3 + assert mml_3.childNodes[0].nodeName == 'mn' + assert mml_3.childNodes[0].childNodes[0].nodeValue == '1' + assert mml_3.childNodes[1].nodeName == 'mo' + assert mml_3.childNodes[1].childNodes[0].nodeValue == '≥' + assert mml_3.childNodes[2].nodeName == 'mi' + assert mml_3.childNodes[2].childNodes[0].nodeValue == 'x' + + mml_4 = mpp._print(Lt(1, x)) + assert len(mml_4.childNodes) == 3 + assert mml_4.childNodes[0].nodeName == 'mn' + assert mml_4.childNodes[0].childNodes[0].nodeValue == '1' + assert mml_4.childNodes[1].nodeName == 'mo' + assert mml_4.childNodes[1].childNodes[0].nodeValue == '<' + assert mml_4.childNodes[2].nodeName == 'mi' + assert mml_4.childNodes[2].childNodes[0].nodeValue == 'x' + + +def test_presentation_symbol(): + mml = mpp._print(x) + assert mml.nodeName == 'mi' + assert mml.childNodes[0].nodeValue == 'x' + del mml + + mml = mpp._print(Symbol("x^2")) + assert mml.nodeName == 'msup' + assert mml.childNodes[0].nodeName == 'mi' + assert mml.childNodes[0].childNodes[0].nodeValue == 'x' + assert mml.childNodes[1].nodeName == 'mi' + assert mml.childNodes[1].childNodes[0].nodeValue == '2' + del mml + + mml = mpp._print(Symbol("x__2")) + assert mml.nodeName == 'msup' + assert mml.childNodes[0].nodeName == 'mi' + assert mml.childNodes[0].childNodes[0].nodeValue == 'x' + assert mml.childNodes[1].nodeName == 'mi' + assert mml.childNodes[1].childNodes[0].nodeValue == '2' + del mml + + mml = mpp._print(Symbol("x_2")) + assert mml.nodeName == 'msub' + assert mml.childNodes[0].nodeName == 'mi' + assert mml.childNodes[0].childNodes[0].nodeValue == 'x' + assert mml.childNodes[1].nodeName == 'mi' + assert mml.childNodes[1].childNodes[0].nodeValue == '2' + del mml + + mml = mpp._print(Symbol("x^3_2")) + assert mml.nodeName == 'msubsup' + assert mml.childNodes[0].nodeName == 'mi' + assert mml.childNodes[0].childNodes[0].nodeValue == 'x' + assert mml.childNodes[1].nodeName == 'mi' + assert mml.childNodes[1].childNodes[0].nodeValue == '2' + assert mml.childNodes[2].nodeName == 'mi' + assert mml.childNodes[2].childNodes[0].nodeValue == '3' + del mml + + mml = mpp._print(Symbol("x__3_2")) + assert mml.nodeName == 'msubsup' + assert mml.childNodes[0].nodeName == 'mi' + assert mml.childNodes[0].childNodes[0].nodeValue == 'x' + assert mml.childNodes[1].nodeName == 'mi' + assert mml.childNodes[1].childNodes[0].nodeValue == '2' + assert mml.childNodes[2].nodeName == 'mi' + assert mml.childNodes[2].childNodes[0].nodeValue == '3' + del mml + + mml = mpp._print(Symbol("x_2_a")) + assert mml.nodeName == 'msub' + assert mml.childNodes[0].nodeName == 'mi' + assert mml.childNodes[0].childNodes[0].nodeValue == 'x' + assert mml.childNodes[1].nodeName == 'mrow' + assert mml.childNodes[1].childNodes[0].nodeName == 'mi' + assert mml.childNodes[1].childNodes[0].childNodes[0].nodeValue == '2' + assert mml.childNodes[1].childNodes[1].nodeName == 'mo' + assert mml.childNodes[1].childNodes[1].childNodes[0].nodeValue == ' ' + assert mml.childNodes[1].childNodes[2].nodeName == 'mi' + assert mml.childNodes[1].childNodes[2].childNodes[0].nodeValue == 'a' + del mml + + mml = mpp._print(Symbol("x^2^a")) + assert mml.nodeName == 'msup' + assert mml.childNodes[0].nodeName == 'mi' + assert mml.childNodes[0].childNodes[0].nodeValue == 'x' + assert mml.childNodes[1].nodeName == 'mrow' + assert mml.childNodes[1].childNodes[0].nodeName == 'mi' + assert mml.childNodes[1].childNodes[0].childNodes[0].nodeValue == '2' + assert mml.childNodes[1].childNodes[1].nodeName == 'mo' + assert mml.childNodes[1].childNodes[1].childNodes[0].nodeValue == ' ' + assert mml.childNodes[1].childNodes[2].nodeName == 'mi' + assert mml.childNodes[1].childNodes[2].childNodes[0].nodeValue == 'a' + del mml + + mml = mpp._print(Symbol("x__2__a")) + assert mml.nodeName == 'msup' + assert mml.childNodes[0].nodeName == 'mi' + assert mml.childNodes[0].childNodes[0].nodeValue == 'x' + assert mml.childNodes[1].nodeName == 'mrow' + assert mml.childNodes[1].childNodes[0].nodeName == 'mi' + assert mml.childNodes[1].childNodes[0].childNodes[0].nodeValue == '2' + assert mml.childNodes[1].childNodes[1].nodeName == 'mo' + assert mml.childNodes[1].childNodes[1].childNodes[0].nodeValue == ' ' + assert mml.childNodes[1].childNodes[2].nodeName == 'mi' + assert mml.childNodes[1].childNodes[2].childNodes[0].nodeValue == 'a' + del mml + + +def test_presentation_mathml_greek(): + mml = mpp._print(Symbol('alpha')) + assert mml.nodeName == 'mi' + assert mml.childNodes[0].nodeValue == '\N{GREEK SMALL LETTER ALPHA}' + + assert mpp.doprint(Symbol('alpha')) == 'α' + assert mpp.doprint(Symbol('beta')) == 'β' + assert mpp.doprint(Symbol('gamma')) == 'γ' + assert mpp.doprint(Symbol('delta')) == 'δ' + assert mpp.doprint(Symbol('epsilon')) == 'ε' + assert mpp.doprint(Symbol('zeta')) == 'ζ' + assert mpp.doprint(Symbol('eta')) == 'η' + assert mpp.doprint(Symbol('theta')) == 'θ' + assert mpp.doprint(Symbol('iota')) == 'ι' + assert mpp.doprint(Symbol('kappa')) == 'κ' + assert mpp.doprint(Symbol('lambda')) == 'λ' + assert mpp.doprint(Symbol('mu')) == 'μ' + assert mpp.doprint(Symbol('nu')) == 'ν' + assert mpp.doprint(Symbol('xi')) == 'ξ' + assert mpp.doprint(Symbol('omicron')) == 'ο' + assert mpp.doprint(Symbol('pi')) == 'π' + assert mpp.doprint(Symbol('rho')) == 'ρ' + assert mpp.doprint(Symbol('varsigma')) == 'ς' + assert mpp.doprint(Symbol('sigma')) == 'σ' + assert mpp.doprint(Symbol('tau')) == 'τ' + assert mpp.doprint(Symbol('upsilon')) == 'υ' + assert mpp.doprint(Symbol('phi')) == 'φ' + assert mpp.doprint(Symbol('chi')) == 'χ' + assert mpp.doprint(Symbol('psi')) == 'ψ' + assert mpp.doprint(Symbol('omega')) == 'ω' + + assert mpp.doprint(Symbol('Alpha')) == 'Α' + assert mpp.doprint(Symbol('Beta')) == 'Β' + assert mpp.doprint(Symbol('Gamma')) == 'Γ' + assert mpp.doprint(Symbol('Delta')) == 'Δ' + assert mpp.doprint(Symbol('Epsilon')) == 'Ε' + assert mpp.doprint(Symbol('Zeta')) == 'Ζ' + assert mpp.doprint(Symbol('Eta')) == 'Η' + assert mpp.doprint(Symbol('Theta')) == 'Θ' + assert mpp.doprint(Symbol('Iota')) == 'Ι' + assert mpp.doprint(Symbol('Kappa')) == 'Κ' + assert mpp.doprint(Symbol('Lambda')) == 'Λ' + assert mpp.doprint(Symbol('Mu')) == 'Μ' + assert mpp.doprint(Symbol('Nu')) == 'Ν' + assert mpp.doprint(Symbol('Xi')) == 'Ξ' + assert mpp.doprint(Symbol('Omicron')) == 'Ο' + assert mpp.doprint(Symbol('Pi')) == 'Π' + assert mpp.doprint(Symbol('Rho')) == 'Ρ' + assert mpp.doprint(Symbol('Sigma')) == 'Σ' + assert mpp.doprint(Symbol('Tau')) == 'Τ' + assert mpp.doprint(Symbol('Upsilon')) == 'Υ' + assert mpp.doprint(Symbol('Phi')) == 'Φ' + assert mpp.doprint(Symbol('Chi')) == 'Χ' + assert mpp.doprint(Symbol('Psi')) == 'Ψ' + assert mpp.doprint(Symbol('Omega')) == 'Ω' + + +def test_presentation_mathml_order(): + expr = x**3 + x**2*y + 3*x*y**3 + y**4 + + mp = MathMLPresentationPrinter({'order': 'lex'}) + mml = mp._print(expr) + assert mml.childNodes[0].nodeName == 'msup' + assert mml.childNodes[0].childNodes[0].childNodes[0].nodeValue == 'x' + assert mml.childNodes[0].childNodes[1].childNodes[0].nodeValue == '3' + + assert mml.childNodes[6].nodeName == 'msup' + assert mml.childNodes[6].childNodes[0].childNodes[0].nodeValue == 'y' + assert mml.childNodes[6].childNodes[1].childNodes[0].nodeValue == '4' + + mp = MathMLPresentationPrinter({'order': 'rev-lex'}) + mml = mp._print(expr) + + assert mml.childNodes[0].nodeName == 'msup' + assert mml.childNodes[0].childNodes[0].childNodes[0].nodeValue == 'y' + assert mml.childNodes[0].childNodes[1].childNodes[0].nodeValue == '4' + + assert mml.childNodes[6].nodeName == 'msup' + assert mml.childNodes[6].childNodes[0].childNodes[0].nodeValue == 'x' + assert mml.childNodes[6].childNodes[1].childNodes[0].nodeValue == '3' + + +def test_print_intervals(): + a = Symbol('a', real=True) + assert mpp.doprint(Interval(0, a)) == \ + '[0,a]' + assert mpp.doprint(Interval(0, a, False, False)) == \ + '[0,a]' + assert mpp.doprint(Interval(0, a, True, False)) == \ + '(0,a]' + assert mpp.doprint(Interval(0, a, False, True)) == \ + '[0,a)' + assert mpp.doprint(Interval(0, a, True, True)) == \ + '(0,a)' + + +def test_print_tuples(): + assert mpp.doprint(Tuple(0,)) == \ + '(0)' + assert mpp.doprint(Tuple(0, a)) == \ + '(0,a)' + assert mpp.doprint(Tuple(0, a, a)) == \ + '(0,a,a)' + assert mpp.doprint(Tuple(0, 1, 2, 3, 4)) == \ + '(0,1,2,3,4)' + assert mpp.doprint(Tuple(0, 1, Tuple(2, 3, 4))) == \ + '(0,1,(2,3'\ + ',4))' + + +def test_print_re_im(): + assert mpp.doprint(re(x)) == \ + 'ℜ(x)' + assert mpp.doprint(im(x)) == \ + 'ℑ(x)' + assert mpp.doprint(re(x + 1, evaluate=False)) == \ + 'ℜ(x+1)' + assert mpp.doprint(im(x + 1, evaluate=False)) == \ + 'ℑ(x+1)' + + +def test_print_Abs(): + assert mpp.doprint(Abs(x)) == \ + '|x|' + assert mpp.doprint(Abs(x + 1)) == \ + '|x+1|' + + +def test_print_Determinant(): + assert mpp.doprint(Determinant(Matrix([[1, 2], [3, 4]]))) == \ + '|[1234]|' + + +def test_presentation_settings(): + raises(TypeError, lambda: mathml(x, printer='presentation', + method="garbage")) + + +def test_print_domains(): + from sympy.sets import Integers, Naturals, Naturals0, Reals, Complexes + + assert mpp.doprint(Complexes) == 'ℂ' + assert mpp.doprint(Integers) == 'ℤ' + assert mpp.doprint(Naturals) == 'ℕ' + assert mpp.doprint(Naturals0) == \ + 'ℕ0' + assert mpp.doprint(Reals) == 'ℝ' + + +def test_print_expression_with_minus(): + assert mpp.doprint(-x) == '-x' + assert mpp.doprint(-x/y) == \ + '-xy' + assert mpp.doprint(-Rational(1, 2)) == \ + '-12' + + +def test_print_AssocOp(): + from sympy.core.operations import AssocOp + + class TestAssocOp(AssocOp): + identity = 0 + + expr = TestAssocOp(1, 2) + assert mpp.doprint(expr) == \ + 'testassocop12' + + +def test_print_basic(): + expr = Basic(S(1), S(2)) + assert mpp.doprint(expr) == \ + 'basic(1,2)' + assert mp.doprint(expr) == '12' + + +def test_mat_delim_print(): + expr = Matrix([[1, 2], [3, 4]]) + assert mathml(expr, printer='presentation', mat_delim='[') == \ + '[1'\ + '234'\ + ']' + assert mathml(expr, printer='presentation', mat_delim='(') == \ + '(12'\ + '34)' + assert mathml(expr, printer='presentation', mat_delim='') == \ + '12'\ + '34' + + +def test_ln_notation_print(): + expr = log(x) + assert mathml(expr, printer='presentation') == \ + 'log(x)' + assert mathml(expr, printer='presentation', ln_notation=False) == \ + 'log(x)' + assert mathml(expr, printer='presentation', ln_notation=True) == \ + 'ln(x)' + + +def test_mul_symbol_print(): + expr = x * y + assert mathml(expr, printer='presentation') == \ + 'x⁢y' + assert mathml(expr, printer='presentation', mul_symbol=None) == \ + 'x⁢y' + assert mathml(expr, printer='presentation', mul_symbol='dot') == \ + 'x·y' + assert mathml(expr, printer='presentation', mul_symbol='ldot') == \ + 'x․y' + assert mathml(expr, printer='presentation', mul_symbol='times') == \ + 'x×y' + + +def test_print_lerchphi(): + assert mpp.doprint(lerchphi(1, 2, 3)) == \ + 'Φ(1,2,3)' + + +def test_print_polylog(): + assert mp.doprint(polylog(x, y)) == \ + 'xy' + assert mpp.doprint(polylog(x, y)) == \ + 'Lix(y)' + + +def test_print_set_frozenset(): + f = frozenset({1, 5, 3}) + assert mpp.doprint(f) == \ + '{1,3,5}' + s = set({1, 2, 3}) + assert mpp.doprint(s) == \ + '{1,2,3}' + + +def test_print_FiniteSet(): + f1 = FiniteSet(x, 1, 3) + assert mpp.doprint(f1) == \ + '{1,3,x}' + + +def test_print_LambertW(): + assert mpp.doprint(LambertW(x)) == 'W(x)' + assert mpp.doprint(LambertW(x, y)) == 'W(x,y)' + + +def test_print_EmptySet(): + assert mpp.doprint(S.EmptySet) == '∅' + + +def test_print_UniversalSet(): + assert mpp.doprint(S.UniversalSet) == '𝕌' + + +def test_print_spaces(): + assert mpp.doprint(HilbertSpace()) == 'ℋ' + assert mpp.doprint(ComplexSpace(2)) == '𝒞2' + assert mpp.doprint(FockSpace()) == 'ℱ' + + +def test_print_constants(): + assert mpp.doprint(hbar) == 'ℏ' + assert mpp.doprint(S.TribonacciConstant) == 'TribonacciConstant' + assert mpp.doprint(S.GoldenRatio) == 'Φ' + assert mpp.doprint(S.EulerGamma) == 'γ' + + +def test_print_Contains(): + assert mpp.doprint(Contains(x, S.Naturals)) == \ + 'x∈ℕ' + + +def test_print_Dagger(): + x = symbols('x', commutative=False) + assert mpp.doprint(Dagger(x)) == 'x†' + + +def test_print_SetOp(): + f1 = FiniteSet(x, 1, 3) + f2 = FiniteSet(y, 2, 4) + + prntr = lambda x: mathml(x, printer='presentation') + + assert prntr(Union(f1, f2, evaluate=False)) == \ + '{1,3,x'\ + '}∪{2,'\ + '4,y}' + assert prntr(Intersection(f1, f2, evaluate=False)) == \ + '{1,3,x'\ + '}∩{2'\ + ',4,y}' + assert prntr(Complement(f1, f2, evaluate=False)) == \ + '{1,3,x'\ + '}∖{2'\ + ',4,y}' + assert prntr(SymmetricDifference(f1, f2, evaluate=False)) == \ + '{1,3,x'\ + '}∆{2'\ + ',4,y}' + + A = FiniteSet(a) + C = FiniteSet(c) + D = FiniteSet(d) + + U1 = Union(C, D, evaluate=False) + I1 = Intersection(C, D, evaluate=False) + C1 = Complement(C, D, evaluate=False) + D1 = SymmetricDifference(C, D, evaluate=False) + # XXX ProductSet does not support evaluate keyword + P1 = ProductSet(C, D) + + assert prntr(Union(A, I1, evaluate=False)) == \ + '{a}' \ + '∪({' \ + 'c}∩{' \ + 'd})' + assert prntr(Intersection(A, C1, evaluate=False)) == \ + '{a}' \ + '∩({' \ + 'c}∖{' \ + 'd})' + assert prntr(Complement(A, D1, evaluate=False)) == \ + '{a}' \ + '∖({' \ + 'c}∆{' \ + 'd})' + assert prntr(SymmetricDifference(A, P1, evaluate=False)) == \ + '{a}' \ + '∆({' \ + 'c}×{' \ + 'd})' + assert prntr(ProductSet(A, U1)) == \ + '{a}' \ + '×({' \ + 'c}∪{' \ + 'd})' + + +def test_print_logic(): + assert mpp.doprint(And(x, y)) == \ + 'x∧y' + assert mpp.doprint(Or(x, y)) == \ + 'x∨y' + assert mpp.doprint(Xor(x, y)) == \ + 'x⊻y' + assert mpp.doprint(Implies(x, y)) == \ + 'x⇒y' + assert mpp.doprint(Equivalent(x, y)) == \ + 'x⇔y' + + assert mpp.doprint(And(Eq(x, y), x > 4)) == \ + 'x=y∧'\ + 'x>4' + assert mpp.doprint(And(Eq(x, 3), y < 3, x > y + 1)) == \ + 'x=3∧'\ + 'x>y+1'\ + '∧y<3' + assert mpp.doprint(Or(Eq(x, y), x > 4)) == \ + 'x=y∨'\ + 'x>4' + assert mpp.doprint(And(Eq(x, 3), Or(y < 3, x > y + 1))) == \ + 'x=3'\ + '∧(x>'\ + 'y+1∨'\ + 'y<3)' + + assert mpp.doprint(Not(x)) == '¬x' + assert mpp.doprint(Not(And(x, y))) == \ + '¬(x∧y)' + + +def test_root_notation_print(): + assert mathml(x**(S.One/3), printer='presentation') == \ + 'x3' + assert mathml(x**(S.One/3), printer='presentation', root_notation=False) ==\ + 'x13' + assert mathml(x**(S.One/3), printer='content') == \ + '3x' + assert mathml(x**(S.One/3), printer='content', root_notation=False) == \ + 'x13' + assert mathml(x**(Rational(-1, 3)), printer='presentation') == \ + '1x3' + assert mathml(x**(Rational(-1, 3)), printer='presentation', root_notation=False) \ + == '1x13' + + +def test_fold_frac_powers_print(): + expr = x ** Rational(5, 2) + assert mathml(expr, printer='presentation') == \ + 'x52' + assert mathml(expr, printer='presentation', fold_frac_powers=True) == \ + 'x52' + assert mathml(expr, printer='presentation', fold_frac_powers=False) == \ + 'x52' + + +def test_fold_short_frac_print(): + expr = Rational(2, 5) + assert mathml(expr, printer='presentation') == \ + '25' + assert mathml(expr, printer='presentation', fold_short_frac=True) == \ + '25' + assert mathml(expr, printer='presentation', fold_short_frac=False) == \ + '25' + + +def test_print_factorials(): + assert mpp.doprint(factorial(x)) == 'x!' + assert mpp.doprint(factorial(x + 1)) == \ + '(x+1)!' + assert mpp.doprint(factorial2(x)) == 'x!!' + assert mpp.doprint(factorial2(x + 1)) == \ + '(x+1)!!' + assert mpp.doprint(binomial(x, y)) == \ + '(xy)' + assert mpp.doprint(binomial(4, x + y)) == \ + '(4x'\ + '+y)' + + +def test_print_floor(): + expr = floor(x) + assert mathml(expr, printer='presentation') == \ + '⌊x⌋' + + +def test_print_ceiling(): + expr = ceiling(x) + assert mathml(expr, printer='presentation') == \ + '⌈x⌉' + + +def test_print_Lambda(): + expr = Lambda(x, x+1) + assert mathml(expr, printer='presentation') == \ + '(x↦x+1)' + expr = Lambda((x, y), x + y) + assert mathml(expr, printer='presentation') == \ + '((x,y)↦x+y)' + + +def test_print_conjugate(): + assert mpp.doprint(conjugate(x)) == \ + 'x' + assert mpp.doprint(conjugate(x + 1)) == \ + 'x+1' + + +def test_print_AccumBounds(): + a = Symbol('a', real=True) + assert mpp.doprint(AccumBounds(0, 1)) == '⟨0,1⟩' + assert mpp.doprint(AccumBounds(0, a)) == '⟨0,a⟩' + assert mpp.doprint(AccumBounds(a + 1, a + 2)) == '⟨a+1,a+2⟩' + + +def test_print_Float(): + assert mpp.doprint(Float(1e100)) == '1.0·10100' + assert mpp.doprint(Float(1e-100)) == '1.0·10-100' + assert mpp.doprint(Float(-1e100)) == '-1.0·10100' + assert mpp.doprint(Float(1.0*oo)) == '∞' + assert mpp.doprint(Float(-1.0*oo)) == '-∞' + + +def test_print_different_functions(): + assert mpp.doprint(gamma(x)) == 'Γ(x)' + assert mpp.doprint(lowergamma(x, y)) == 'γ(x,y)' + assert mpp.doprint(uppergamma(x, y)) == 'Γ(x,y)' + assert mpp.doprint(zeta(x)) == 'ζ(x)' + assert mpp.doprint(zeta(x, y)) == 'ζ(x,y)' + assert mpp.doprint(dirichlet_eta(x)) == 'η(x)' + assert mpp.doprint(elliptic_k(x)) == 'Κ(x)' + assert mpp.doprint(totient(x)) == 'ϕ(x)' + assert mpp.doprint(reduced_totient(x)) == 'λ(x)' + assert mpp.doprint(primenu(x)) == 'ν(x)' + assert mpp.doprint(primeomega(x)) == 'Ω(x)' + assert mpp.doprint(fresnels(x)) == 'S(x)' + assert mpp.doprint(fresnelc(x)) == 'C(x)' + assert mpp.doprint(Heaviside(x)) == 'Θ(x,12)' + + +def test_mathml_builtins(): + assert mpp.doprint(None) == 'None' + assert mpp.doprint(true) == 'True' + assert mpp.doprint(false) == 'False' + + +def test_mathml_Range(): + assert mpp.doprint(Range(1, 51)) == \ + '{1,2,…,50}' + assert mpp.doprint(Range(1, 4)) == \ + '{1,2,3}' + assert mpp.doprint(Range(0, 3, 1)) == \ + '{0,1,2}' + assert mpp.doprint(Range(0, 30, 1)) == \ + '{0,1,…,29}' + assert mpp.doprint(Range(30, 1, -1)) == \ + '{30,29,…,2}' + assert mpp.doprint(Range(0, oo, 2)) == \ + '{0,2,…}' + assert mpp.doprint(Range(oo, -2, -2)) == \ + '{…,2,0}' + assert mpp.doprint(Range(-2, -oo, -1)) == \ + '{-2,-3,…}' + + +def test_print_exp(): + assert mpp.doprint(exp(x)) == \ + 'ⅇx' + assert mpp.doprint(exp(1) + exp(2)) == \ + 'ⅇ+ⅇ2' + + +def test_print_MinMax(): + assert mpp.doprint(Min(x, y)) == \ + 'min(x,y)' + assert mpp.doprint(Min(x, 2, x**3)) == \ + 'min(2,x,x3)' + assert mpp.doprint(Max(x, y)) == \ + 'max(x,y)' + assert mpp.doprint(Max(x, 2, x**3)) == \ + 'max(2,x,x3)' + + +def test_mathml_presentation_numbers(): + n = Symbol('n') + assert mathml(catalan(n), printer='presentation') == \ + 'Cn' + assert mathml(bernoulli(n), printer='presentation') == \ + 'Bn' + assert mathml(bell(n), printer='presentation') == \ + 'Bn' + assert mathml(euler(n), printer='presentation') == \ + 'En' + assert mathml(fibonacci(n), printer='presentation') == \ + 'Fn' + assert mathml(lucas(n), printer='presentation') == \ + 'Ln' + assert mathml(tribonacci(n), printer='presentation') == \ + 'Tn' + assert mathml(bernoulli(n, x), printer='presentation') == \ + mathml(bell(n, x), printer='presentation') == \ + 'Bn(x)' + assert mathml(euler(n, x), printer='presentation') == \ + 'En(x)' + assert mathml(fibonacci(n, x), printer='presentation') == \ + 'Fn(x)' + assert mathml(tribonacci(n, x), printer='presentation') == \ + 'Tn(x)' + + +def test_mathml_presentation_mathieu(): + assert mathml(mathieuc(x, y, z), printer='presentation') == \ + 'C(x,y,z)' + assert mathml(mathieus(x, y, z), printer='presentation') == \ + 'S(x,y,z)' + assert mathml(mathieucprime(x, y, z), printer='presentation') == \ + 'C′(x,y,z)' + assert mathml(mathieusprime(x, y, z), printer='presentation') == \ + 'S′(x,y,z)' + + +def test_mathml_presentation_stieltjes(): + assert mathml(stieltjes(n), printer='presentation') == \ + 'γn' + assert mathml(stieltjes(n, x), printer='presentation') == \ + 'γn(x)' + + +def test_print_matrix_symbol(): + A = MatrixSymbol('A', 1, 2) + assert mpp.doprint(A) == 'A' + assert mp.doprint(A) == 'A' + assert mathml(A, printer='presentation', mat_symbol_style="bold") == \ + 'A' + # No effect in content printer + assert mathml(A, mat_symbol_style="bold") == 'A' + + +def test_print_hadamard(): + from sympy.matrices.expressions import HadamardProduct + from sympy.matrices.expressions import Transpose + + X = MatrixSymbol('X', 2, 2) + Y = MatrixSymbol('Y', 2, 2) + + assert mathml(HadamardProduct(X, Y*Y), printer="presentation") == \ + '' \ + 'X' \ + '∘' \ + 'Y2' \ + '' + + assert mathml(HadamardProduct(X, Y)*Y, printer="presentation") == \ + '' \ + '(' \ + 'X∘Y' \ + ')' \ + '⁢Y' \ + '' + + assert mathml(HadamardProduct(X, Y, Y), printer="presentation") == \ + '' \ + 'X∘' \ + 'Y∘' \ + 'Y' \ + '' + + assert mathml( + Transpose(HadamardProduct(X, Y)), printer="presentation") == \ + '' \ + '(' \ + 'X∘Y' \ + ')' \ + 'T' \ + '' + + +def test_print_random_symbol(): + R = RandomSymbol(Symbol('R')) + assert mpp.doprint(R) == 'R' + assert mp.doprint(R) == 'R' + + +def test_print_IndexedBase(): + assert mathml(IndexedBase(a)[b], printer='presentation') == \ + 'ab' + assert mathml(IndexedBase(a)[b, c, d], printer='presentation') == \ + 'a(b,c,d)' + assert mathml(IndexedBase(a)[b]*IndexedBase(c)[d]*IndexedBase(e), + printer='presentation') == \ + 'ab⁢'\ + 'cd⁢e' + + +def test_print_Indexed(): + assert mathml(IndexedBase(a), printer='presentation') == 'a' + assert mathml(IndexedBase(a/b), printer='presentation') == \ + 'ab' + assert mathml(IndexedBase((a, b)), printer='presentation') == \ + '(a,b)' + +def test_print_MatrixElement(): + i, j = symbols('i j') + A = MatrixSymbol('A', i, j) + assert mathml(A[0,0],printer = 'presentation') == \ + 'A0,0' + assert mathml(A[i,j], printer = 'presentation') == \ + 'Ai,j' + assert mathml(A[i*j,0], printer = 'presentation') == \ + 'Ai⁢j,0' + + +def test_print_Vector(): + ACS = CoordSys3D('A') + assert mathml(Cross(ACS.i, ACS.j*ACS.x*3 + ACS.k), printer='presentation') == \ + 'i^'\ + 'A×('\ + '(3⁢'\ + 'xA'\ + ')⁢'\ + 'j^'\ + 'A+'\ + 'k^'\ + 'A)' + assert mathml(Cross(ACS.i, ACS.j), printer='presentation') == \ + 'i^'\ + 'A×'\ + 'j^'\ + 'A' + assert mathml(x*Cross(ACS.i, ACS.j), printer='presentation') == \ + 'x⁢('\ + 'i^'\ + 'A×'\ + 'j^'\ + 'A)' + assert mathml(Cross(x*ACS.i, ACS.j), printer='presentation') == \ + '-j'\ + '^A'\ + '×((x)'\ + '⁢i'\ + '^A'\ + ')' + assert mathml(Curl(3*ACS.x*ACS.j), printer='presentation') == \ + '∇×(('\ + '3⁢x'\ + 'A)⁢'\ + 'j^'\ + 'A)' + assert mathml(Curl(3*x*ACS.x*ACS.j), printer='presentation') == \ + '∇×(('\ + '3⁢x'\ + 'A⁢'\ + 'x)⁢'\ + 'j^'\ + 'A)' + assert mathml(x*Curl(3*ACS.x*ACS.j), printer='presentation') == \ + 'x⁢(∇'\ + '×((3'\ + '⁢x'\ + 'A)'\ + '⁢j'\ + '^A)'\ + ')' + assert mathml(Curl(3*x*ACS.x*ACS.j + ACS.i), printer='presentation') == \ + '∇×('\ + 'i^'\ + 'A+('\ + '3⁢x'\ + 'A⁢'\ + 'x)⁢'\ + 'j^'\ + 'A)' + assert mathml(Divergence(3*ACS.x*ACS.j), printer='presentation') == \ + '∇·(('\ + '3⁢x'\ + 'A)'\ + '⁢j'\ + '^A)' + assert mathml(x*Divergence(3*ACS.x*ACS.j), printer='presentation') == \ + 'x⁢(∇'\ + '·((3'\ + '⁢x'\ + 'A)'\ + '⁢j'\ + '^A'\ + '))' + assert mathml(Divergence(3*x*ACS.x*ACS.j + ACS.i), printer='presentation') == \ + '∇·('\ + 'i^'\ + 'A+('\ + '3⁢'\ + 'xA'\ + '⁢x)'\ + '⁢j'\ + '^A'\ + ')' + assert mathml(Dot(ACS.i, ACS.j*ACS.x*3+ACS.k), printer='presentation') == \ + 'i^'\ + 'A·('\ + '(3⁢'\ + 'xA'\ + ')⁢'\ + 'j^'\ + 'A+'\ + 'k^'\ + 'A)' + assert mathml(Dot(ACS.i, ACS.j), printer='presentation') == \ + 'i^'\ + 'A·'\ + 'j^'\ + 'A' + assert mathml(Dot(x*ACS.i, ACS.j), printer='presentation') == \ + 'j^'\ + 'A·('\ + '(x)⁢'\ + 'i^'\ + 'A)' + assert mathml(x*Dot(ACS.i, ACS.j), printer='presentation') == \ + 'x⁢('\ + 'i^'\ + 'A·'\ + 'j^'\ + 'A)' + assert mathml(Gradient(ACS.x), printer='presentation') == \ + '∇x'\ + 'A' + assert mathml(Gradient(ACS.x + 3*ACS.y), printer='presentation') == \ + '∇('\ + 'xA+3'\ + '⁢y'\ + 'A)' + assert mathml(x*Gradient(ACS.x), printer='presentation') == \ + 'x⁢(∇'\ + 'xA'\ + ')' + assert mathml(Gradient(x*ACS.x), printer='presentation') == \ + '∇('\ + 'xA⁢'\ + 'x)' + assert mathml(Cross(ACS.z, ACS.x), printer='presentation') == \ + '-x'\ + 'A×'\ + 'zA' + assert mathml(Laplacian(ACS.x), printer='presentation') == \ + '∆x'\ + 'A' + assert mathml(Laplacian(ACS.x + 3*ACS.y), printer='presentation') == \ + '∆('\ + 'xA+3'\ + '⁢y'\ + 'A)' + assert mathml(x*Laplacian(ACS.x), printer='presentation') == \ + 'x⁢(∆'\ + 'xA'\ + ')' + assert mathml(Laplacian(x*ACS.x), printer='presentation') == \ + '∆('\ + 'xA⁢'\ + 'x)' + +@XFAIL +def test_vector_cross_xfail(): + ACS = CoordSys3D('A') + assert mathml(Cross(ACS.x, ACS.z) + Cross(ACS.z, ACS.x), printer='presentation') == \ + '0^' + +def test_print_elliptic_f(): + assert mathml(elliptic_f(x, y), printer = 'presentation') == \ + '𝖥(x|y)' + assert mathml(elliptic_f(x/y, y), printer = 'presentation') == \ + '𝖥(xy|y)' + +def test_print_elliptic_e(): + assert mathml(elliptic_e(x), printer = 'presentation') == \ + '𝖤(x)' + assert mathml(elliptic_e(x, y), printer = 'presentation') == \ + '𝖤(x|y)' + +def test_print_elliptic_pi(): + assert mathml(elliptic_pi(x, y), printer = 'presentation') == \ + '𝛱(x|y)' + assert mathml(elliptic_pi(x, y, z), printer = 'presentation') == \ + '𝛱(x;y|z)' + +def test_print_Ei(): + assert mathml(Ei(x), printer = 'presentation') == \ + 'Ei(x)' + assert mathml(Ei(x**y), printer = 'presentation') == \ + 'Ei(xy)' + +def test_print_expint(): + assert mathml(expint(x, y), printer = 'presentation') == \ + 'Ex(y)' + assert mathml(expint(IndexedBase(x)[1], IndexedBase(x)[2]), printer = 'presentation') == \ + 'Ex1(x2)' + +def test_print_jacobi(): + assert mathml(jacobi(n, a, b, x), printer = 'presentation') == \ + 'Pn(a,b)(x)' + +def test_print_gegenbauer(): + assert mathml(gegenbauer(n, a, x), printer = 'presentation') == \ + 'Cn(a)(x)' + +def test_print_chebyshevt(): + assert mathml(chebyshevt(n, x), printer = 'presentation') == \ + 'Tn(x)' + +def test_print_chebyshevu(): + assert mathml(chebyshevu(n, x), printer = 'presentation') == \ + 'Un(x)' + +def test_print_legendre(): + assert mathml(legendre(n, x), printer = 'presentation') == \ + 'Pn(x)' + +def test_print_assoc_legendre(): + assert mathml(assoc_legendre(n, a, x), printer = 'presentation') == \ + 'Pn(a)(x)' + +def test_print_laguerre(): + assert mathml(laguerre(n, x), printer = 'presentation') == \ + 'Ln(x)' + +def test_print_assoc_laguerre(): + assert mathml(assoc_laguerre(n, a, x), printer = 'presentation') == \ + 'Ln(a)(x)' + +def test_print_hermite(): + assert mathml(hermite(n, x), printer = 'presentation') == \ + 'Hn(x)' + +def test_mathml_SingularityFunction(): + assert mathml(SingularityFunction(x, 4, 5), printer='presentation') == \ + '⟨x-4⟩5' + assert mathml(SingularityFunction(x, -3, 4), printer='presentation') == \ + '⟨x+3⟩4' + assert mathml(SingularityFunction(x, 0, 4), printer='presentation') == \ + '⟨x⟩4' + assert mathml(SingularityFunction(x, a, n), printer='presentation') == \ + '⟨-a+x⟩n' + assert mathml(SingularityFunction(x, 4, -2), printer='presentation') == \ + '⟨x-4⟩-2' + assert mathml(SingularityFunction(x, 4, -1), printer='presentation') == \ + '⟨x-4⟩-1' + + +def test_mathml_matrix_functions(): + from sympy.matrices import Adjoint, Inverse, Transpose + X = MatrixSymbol('X', 2, 2) + Y = MatrixSymbol('Y', 2, 2) + assert mathml(Adjoint(X), printer='presentation') == \ + 'X†' + assert mathml(Adjoint(X + Y), printer='presentation') == \ + '(X+Y)†' + assert mathml(Adjoint(X) + Adjoint(Y), printer='presentation') == \ + 'X†+' \ + 'Y†' + assert mathml(Adjoint(X*Y), printer='presentation') == \ + '(X⁢' \ + 'Y)†' + assert mathml(Adjoint(Y)*Adjoint(X), printer='presentation') == \ + 'Y†⁢' \ + 'X†' + assert mathml(Adjoint(X**2), printer='presentation') == \ + '(X2)†' + assert mathml(Adjoint(X)**2, printer='presentation') == \ + '(X†)2' + assert mathml(Adjoint(Inverse(X)), printer='presentation') == \ + '(X-1)†' + assert mathml(Inverse(Adjoint(X)), printer='presentation') == \ + '(X†)-1' + assert mathml(Adjoint(Transpose(X)), printer='presentation') == \ + '(XT)†' + assert mathml(Transpose(Adjoint(X)), printer='presentation') == \ + '(X†)T' + assert mathml(Transpose(Adjoint(X) + Y), printer='presentation') == \ + '(X†' \ + '+Y)T' + assert mathml(Transpose(X), printer='presentation') == \ + 'XT' + assert mathml(Transpose(X + Y), printer='presentation') == \ + '(X+Y)T' + + +def test_mathml_special_matrices(): + from sympy.matrices import Identity, ZeroMatrix, OneMatrix + assert mathml(Identity(4), printer='presentation') == '𝕀' + assert mathml(ZeroMatrix(2, 2), printer='presentation') == '𝟘' + assert mathml(OneMatrix(2, 2), printer='presentation') == '𝟙' + +def test_mathml_piecewise(): + from sympy.functions.elementary.piecewise import Piecewise + # Content MathML + assert mathml(Piecewise((x, x <= 1), (x**2, True))) == \ + 'xx1x2' + + raises(ValueError, lambda: mathml(Piecewise((x, x <= 1)))) + + +def test_issue_17857(): + assert mathml(Range(-oo, oo), printer='presentation') == \ + '{…,-1,0,1,…}' + assert mathml(Range(oo, -oo, -1), printer='presentation') == \ + '{…,1,0,-1,…}' + + +def test_float_roundtrip(): + x = sympify(0.8975979010256552) + y = float(mp.doprint(x).strip('')) + assert x == y + + +def test_content_mathml_disable_split_super_sub(): + mp = MathMLContentPrinter() + assert mp.doprint(Symbol('u_b')) == 'ub' + mp = MathMLContentPrinter({'disable_split_super_sub': False}) + assert mp.doprint(Symbol('u_b')) == 'ub' + mp = MathMLContentPrinter({'disable_split_super_sub': True}) + assert mp.doprint(Symbol('u_b')) == 'u_b' + +def test_presentation_mathml_disable_split_super_sub(): + mpp = MathMLPresentationPrinter() + assert mpp.doprint(Symbol('u_b')) == 'ub' + mpp = MathMLPresentationPrinter({'disable_split_super_sub': False}) + assert mpp.doprint(Symbol('u_b')) == 'ub' + mpp = MathMLPresentationPrinter({'disable_split_super_sub': True}) + assert mpp.doprint(Symbol('u_b')) == 'u_b' diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_numpy.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_numpy.py new file mode 100644 index 0000000000000000000000000000000000000000..fee1c6bd95e54790a048220f37b8e5de79017d2f --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_numpy.py @@ -0,0 +1,381 @@ +from sympy.concrete.summations import Sum +from sympy.core.mod import Mod +from sympy.core.relational import (Equality, Unequality) +from sympy.core.symbol import Symbol +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.special.gamma_functions import polygamma +from sympy.functions.special.error_functions import (Si, Ci) +from sympy.matrices import Matrix +from sympy.matrices.expressions.blockmatrix import BlockMatrix +from sympy.matrices.expressions.matexpr import MatrixSymbol +from sympy.matrices.expressions.special import Identity +from sympy.utilities.lambdify import lambdify +from sympy import symbols, Min, Max + +from sympy.abc import x, i, j, a, b, c, d +from sympy.core import Pow +from sympy.codegen.matrix_nodes import MatrixSolve +from sympy.codegen.numpy_nodes import logaddexp, logaddexp2 +from sympy.codegen.cfunctions import log1p, expm1, hypot, log10, exp2, log2, Sqrt +from sympy.tensor.array import Array +from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct, ArrayAdd, \ + PermuteDims, ArrayDiagonal +from sympy.printing.numpy import NumPyPrinter, SciPyPrinter, _numpy_known_constants, \ + _numpy_known_functions, _scipy_known_constants, _scipy_known_functions +from sympy.tensor.array.expressions.from_matrix_to_array import convert_matrix_to_array + +from sympy.testing.pytest import skip, raises +from sympy.external import import_module + +np = import_module('numpy') +jax = import_module('jax') + +if np: + deafult_float_info = np.finfo(np.array([]).dtype) + NUMPY_DEFAULT_EPSILON = deafult_float_info.eps + +def test_numpy_piecewise_regression(): + """ + NumPyPrinter needs to print Piecewise()'s choicelist as a list to avoid + breaking compatibility with numpy 1.8. This is not necessary in numpy 1.9+. + See gh-9747 and gh-9749 for details. + """ + printer = NumPyPrinter() + p = Piecewise((1, x < 0), (0, True)) + assert printer.doprint(p) == \ + 'numpy.select([numpy.less(x, 0),True], [1,0], default=numpy.nan)' + assert printer.module_imports == {'numpy': {'select', 'less', 'nan'}} + +def test_numpy_logaddexp(): + lae = logaddexp(a, b) + assert NumPyPrinter().doprint(lae) == 'numpy.logaddexp(a, b)' + lae2 = logaddexp2(a, b) + assert NumPyPrinter().doprint(lae2) == 'numpy.logaddexp2(a, b)' + + +def test_sum(): + if not np: + skip("NumPy not installed") + + s = Sum(x ** i, (i, a, b)) + f = lambdify((a, b, x), s, 'numpy') + + a_, b_ = 0, 10 + x_ = np.linspace(-1, +1, 10) + assert np.allclose(f(a_, b_, x_), sum(x_ ** i_ for i_ in range(a_, b_ + 1))) + + s = Sum(i * x, (i, a, b)) + f = lambdify((a, b, x), s, 'numpy') + + a_, b_ = 0, 10 + x_ = np.linspace(-1, +1, 10) + assert np.allclose(f(a_, b_, x_), sum(i_ * x_ for i_ in range(a_, b_ + 1))) + + +def test_multiple_sums(): + if not np: + skip("NumPy not installed") + + s = Sum((x + j) * i, (i, a, b), (j, c, d)) + f = lambdify((a, b, c, d, x), s, 'numpy') + + a_, b_ = 0, 10 + c_, d_ = 11, 21 + x_ = np.linspace(-1, +1, 10) + assert np.allclose(f(a_, b_, c_, d_, x_), + sum((x_ + j_) * i_ for i_ in range(a_, b_ + 1) for j_ in range(c_, d_ + 1))) + + +def test_codegen_einsum(): + if not np: + skip("NumPy not installed") + + M = MatrixSymbol("M", 2, 2) + N = MatrixSymbol("N", 2, 2) + + cg = convert_matrix_to_array(M * N) + f = lambdify((M, N), cg, 'numpy') + + ma = np.array([[1, 2], [3, 4]]) + mb = np.array([[1,-2], [-1, 3]]) + assert (f(ma, mb) == np.matmul(ma, mb)).all() + + +def test_codegen_extra(): + if not np: + skip("NumPy not installed") + + M = MatrixSymbol("M", 2, 2) + N = MatrixSymbol("N", 2, 2) + P = MatrixSymbol("P", 2, 2) + Q = MatrixSymbol("Q", 2, 2) + ma = np.array([[1, 2], [3, 4]]) + mb = np.array([[1,-2], [-1, 3]]) + mc = np.array([[2, 0], [1, 2]]) + md = np.array([[1,-1], [4, 7]]) + + cg = ArrayTensorProduct(M, N) + f = lambdify((M, N), cg, 'numpy') + assert (f(ma, mb) == np.einsum(ma, [0, 1], mb, [2, 3])).all() + + cg = ArrayAdd(M, N) + f = lambdify((M, N), cg, 'numpy') + assert (f(ma, mb) == ma+mb).all() + + cg = ArrayAdd(M, N, P) + f = lambdify((M, N, P), cg, 'numpy') + assert (f(ma, mb, mc) == ma+mb+mc).all() + + cg = ArrayAdd(M, N, P, Q) + f = lambdify((M, N, P, Q), cg, 'numpy') + assert (f(ma, mb, mc, md) == ma+mb+mc+md).all() + + cg = PermuteDims(M, [1, 0]) + f = lambdify((M,), cg, 'numpy') + assert (f(ma) == ma.T).all() + + cg = PermuteDims(ArrayTensorProduct(M, N), [1, 2, 3, 0]) + f = lambdify((M, N), cg, 'numpy') + assert (f(ma, mb) == np.transpose(np.einsum(ma, [0, 1], mb, [2, 3]), (1, 2, 3, 0))).all() + + cg = ArrayDiagonal(ArrayTensorProduct(M, N), (1, 2)) + f = lambdify((M, N), cg, 'numpy') + assert (f(ma, mb) == np.diagonal(np.einsum(ma, [0, 1], mb, [2, 3]), axis1=1, axis2=2)).all() + + +def test_relational(): + if not np: + skip("NumPy not installed") + + e = Equality(x, 1) + + f = lambdify((x,), e) + x_ = np.array([0, 1, 2]) + assert np.array_equal(f(x_), [False, True, False]) + + e = Unequality(x, 1) + + f = lambdify((x,), e) + x_ = np.array([0, 1, 2]) + assert np.array_equal(f(x_), [True, False, True]) + + e = (x < 1) + + f = lambdify((x,), e) + x_ = np.array([0, 1, 2]) + assert np.array_equal(f(x_), [True, False, False]) + + e = (x <= 1) + + f = lambdify((x,), e) + x_ = np.array([0, 1, 2]) + assert np.array_equal(f(x_), [True, True, False]) + + e = (x > 1) + + f = lambdify((x,), e) + x_ = np.array([0, 1, 2]) + assert np.array_equal(f(x_), [False, False, True]) + + e = (x >= 1) + + f = lambdify((x,), e) + x_ = np.array([0, 1, 2]) + assert np.array_equal(f(x_), [False, True, True]) + + +def test_mod(): + if not np: + skip("NumPy not installed") + + e = Mod(a, b) + f = lambdify((a, b), e) + + a_ = np.array([0, 1, 2, 3]) + b_ = 2 + assert np.array_equal(f(a_, b_), [0, 1, 0, 1]) + + a_ = np.array([0, 1, 2, 3]) + b_ = np.array([2, 2, 2, 2]) + assert np.array_equal(f(a_, b_), [0, 1, 0, 1]) + + a_ = np.array([2, 3, 4, 5]) + b_ = np.array([2, 3, 4, 5]) + assert np.array_equal(f(a_, b_), [0, 0, 0, 0]) + + +def test_pow(): + if not np: + skip('NumPy not installed') + + expr = Pow(2, -1, evaluate=False) + f = lambdify([], expr, 'numpy') + assert f() == 0.5 + + +def test_expm1(): + if not np: + skip("NumPy not installed") + + f = lambdify((a,), expm1(a), 'numpy') + assert abs(f(1e-10) - 1e-10 - 5e-21) <= 1e-10 * NUMPY_DEFAULT_EPSILON + + +def test_log1p(): + if not np: + skip("NumPy not installed") + + f = lambdify((a,), log1p(a), 'numpy') + assert abs(f(1e-99) - 1e-99) <= 1e-99 * NUMPY_DEFAULT_EPSILON + +def test_hypot(): + if not np: + skip("NumPy not installed") + assert abs(lambdify((a, b), hypot(a, b), 'numpy')(3, 4) - 5) <= NUMPY_DEFAULT_EPSILON + +def test_log10(): + if not np: + skip("NumPy not installed") + assert abs(lambdify((a,), log10(a), 'numpy')(100) - 2) <= NUMPY_DEFAULT_EPSILON + + +def test_exp2(): + if not np: + skip("NumPy not installed") + assert abs(lambdify((a,), exp2(a), 'numpy')(5) - 32) <= NUMPY_DEFAULT_EPSILON + + +def test_log2(): + if not np: + skip("NumPy not installed") + assert abs(lambdify((a,), log2(a), 'numpy')(256) - 8) <= NUMPY_DEFAULT_EPSILON + + +def test_Sqrt(): + if not np: + skip("NumPy not installed") + assert abs(lambdify((a,), Sqrt(a), 'numpy')(4) - 2) <= NUMPY_DEFAULT_EPSILON + + +def test_sqrt(): + if not np: + skip("NumPy not installed") + assert abs(lambdify((a,), sqrt(a), 'numpy')(4) - 2) <= NUMPY_DEFAULT_EPSILON + + +def test_matsolve(): + if not np: + skip("NumPy not installed") + + M = MatrixSymbol("M", 3, 3) + x = MatrixSymbol("x", 3, 1) + + expr = M**(-1) * x + x + matsolve_expr = MatrixSolve(M, x) + x + + f = lambdify((M, x), expr) + f_matsolve = lambdify((M, x), matsolve_expr) + + m0 = np.array([[1, 2, 3], [3, 2, 5], [5, 6, 7]]) + assert np.linalg.matrix_rank(m0) == 3 + + x0 = np.array([3, 4, 5]) + + assert np.allclose(f_matsolve(m0, x0), f(m0, x0)) + + +def test_16857(): + if not np: + skip("NumPy not installed") + + a_1 = MatrixSymbol('a_1', 10, 3) + a_2 = MatrixSymbol('a_2', 10, 3) + a_3 = MatrixSymbol('a_3', 10, 3) + a_4 = MatrixSymbol('a_4', 10, 3) + A = BlockMatrix([[a_1, a_2], [a_3, a_4]]) + assert A.shape == (20, 6) + + printer = NumPyPrinter() + assert printer.doprint(A) == 'numpy.block([[a_1, a_2], [a_3, a_4]])' + + +def test_issue_17006(): + if not np: + skip("NumPy not installed") + + M = MatrixSymbol("M", 2, 2) + + f = lambdify(M, M + Identity(2)) + ma = np.array([[1, 2], [3, 4]]) + mr = np.array([[2, 2], [3, 5]]) + + assert (f(ma) == mr).all() + + from sympy.core.symbol import symbols + n = symbols('n', integer=True) + N = MatrixSymbol("M", n, n) + raises(NotImplementedError, lambda: lambdify(N, N + Identity(n))) + +def test_jax_tuple_compatibility(): + if not jax: + skip("Jax not installed") + + x, y, z = symbols('x y z') + expr = Max(x, y, z) + Min(x, y, z) + func = lambdify((x, y, z), expr, 'jax') + input_tuple1, input_tuple2 = (1, 2, 3), (4, 5, 6) + input_array1, input_array2 = jax.numpy.asarray(input_tuple1), jax.numpy.asarray(input_tuple2) + assert np.allclose(func(*input_tuple1), func(*input_array1)) + assert np.allclose(func(*input_tuple2), func(*input_array2)) + +def test_numpy_array(): + p = NumPyPrinter() + assert p.doprint(Array([[1, 2], [3, 5]])) == 'numpy.array([[1, 2], [3, 5]])' + assert p.doprint(Array([1, 2])) == 'numpy.array([1, 2])' + assert p.doprint(Array([[[1, 2, 3]]])) == 'numpy.array([[[1, 2, 3]]])' + assert p.doprint(Array([], (0,))) == 'numpy.zeros((0,))' + assert p.doprint(Array([], (0, 0))) == 'numpy.zeros((0, 0))' + assert p.doprint(Array([], (0, 1))) == 'numpy.zeros((0, 1))' + assert p.doprint(Array([], (1, 0))) == 'numpy.zeros((1, 0))' + assert p.doprint(Array([1], ())) == 'numpy.array(1)' + +def test_numpy_matrix(): + p = NumPyPrinter() + assert p.doprint(Matrix([[1, 2], [3, 5]])) == 'numpy.array([[1, 2], [3, 5]])' + assert p.doprint(Matrix([1, 2])) == 'numpy.array([[1], [2]])' + assert p.doprint(Matrix(0, 0, [])) == 'numpy.zeros((0, 0))' + assert p.doprint(Matrix(0, 1, [])) == 'numpy.zeros((0, 1))' + assert p.doprint(Matrix(1, 0, [])) == 'numpy.zeros((1, 0))' + +def test_numpy_known_funcs_consts(): + assert _numpy_known_constants['NaN'] == 'numpy.nan' + assert _numpy_known_constants['EulerGamma'] == 'numpy.euler_gamma' + + assert _numpy_known_functions['acos'] == 'numpy.arccos' + assert _numpy_known_functions['log'] == 'numpy.log' + +def test_scipy_known_funcs_consts(): + assert _scipy_known_constants['GoldenRatio'] == 'scipy.constants.golden_ratio' + assert _scipy_known_constants['Pi'] == 'scipy.constants.pi' + + assert _scipy_known_functions['erf'] == 'scipy.special.erf' + assert _scipy_known_functions['factorial'] == 'scipy.special.factorial' + +def test_numpy_print_methods(): + prntr = NumPyPrinter() + assert hasattr(prntr, '_print_acos') + assert hasattr(prntr, '_print_log') + +def test_scipy_print_methods(): + prntr = SciPyPrinter() + assert hasattr(prntr, '_print_acos') + assert hasattr(prntr, '_print_log') + assert hasattr(prntr, '_print_erf') + assert hasattr(prntr, '_print_factorial') + assert hasattr(prntr, '_print_chebyshevt') + k = Symbol('k', integer=True, nonnegative=True) + x = Symbol('x', real=True) + assert prntr.doprint(polygamma(k, x)) == "scipy.special.polygamma(k, x)" + assert prntr.doprint(Si(x)) == "scipy.special.sici(x)[0]" + assert prntr.doprint(Ci(x)) == "scipy.special.sici(x)[1]" diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_octave.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_octave.py new file mode 100644 index 0000000000000000000000000000000000000000..1aba318f873c48ec702f1b4e3a6cc047f75d647d --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_octave.py @@ -0,0 +1,515 @@ +from sympy.core import (S, pi, oo, symbols, Function, Rational, Integer, + Tuple, Symbol, EulerGamma, GoldenRatio, Catalan, + Lambda, Mul, Pow, Mod, Eq, Ne, Le, Lt, Gt, Ge) +from sympy.codegen.matrix_nodes import MatrixSolve +from sympy.functions import (arg, atan2, bernoulli, beta, ceiling, chebyshevu, + chebyshevt, conjugate, DiracDelta, exp, expint, + factorial, floor, harmonic, Heaviside, im, + laguerre, LambertW, log, Max, Min, Piecewise, + polylog, re, RisingFactorial, sign, sinc, sqrt, + zeta, binomial, legendre, dirichlet_eta, + riemann_xi) +from sympy.functions import (sin, cos, tan, cot, sec, csc, asin, acos, acot, + atan, asec, acsc, sinh, cosh, tanh, coth, csch, + sech, asinh, acosh, atanh, acoth, asech, acsch) +from sympy.testing.pytest import raises, XFAIL +from sympy.utilities.lambdify import implemented_function +from sympy.matrices import (eye, Matrix, MatrixSymbol, Identity, + HadamardProduct, SparseMatrix, HadamardPower) +from sympy.functions.special.bessel import (jn, yn, besselj, bessely, besseli, + besselk, hankel1, hankel2, airyai, + airybi, airyaiprime, airybiprime) +from sympy.functions.special.gamma_functions import (gamma, lowergamma, + uppergamma, loggamma, + polygamma) +from sympy.functions.special.error_functions import (Chi, Ci, erf, erfc, erfi, + erfcinv, erfinv, fresnelc, + fresnels, li, Shi, Si, Li, + erf2, Ei) +from sympy.printing.octave import octave_code, octave_code as mcode + +x, y, z = symbols('x,y,z') + + +def test_Integer(): + assert mcode(Integer(67)) == "67" + assert mcode(Integer(-1)) == "-1" + + +def test_Rational(): + assert mcode(Rational(3, 7)) == "3/7" + assert mcode(Rational(18, 9)) == "2" + assert mcode(Rational(3, -7)) == "-3/7" + assert mcode(Rational(-3, -7)) == "3/7" + assert mcode(x + Rational(3, 7)) == "x + 3/7" + assert mcode(Rational(3, 7)*x) == "3*x/7" + + +def test_Relational(): + assert mcode(Eq(x, y)) == "x == y" + assert mcode(Ne(x, y)) == "x != y" + assert mcode(Le(x, y)) == "x <= y" + assert mcode(Lt(x, y)) == "x < y" + assert mcode(Gt(x, y)) == "x > y" + assert mcode(Ge(x, y)) == "x >= y" + + +def test_Function(): + assert mcode(sin(x) ** cos(x)) == "sin(x).^cos(x)" + assert mcode(sign(x)) == "sign(x)" + assert mcode(exp(x)) == "exp(x)" + assert mcode(log(x)) == "log(x)" + assert mcode(factorial(x)) == "factorial(x)" + assert mcode(floor(x)) == "floor(x)" + assert mcode(atan2(y, x)) == "atan2(y, x)" + assert mcode(beta(x, y)) == 'beta(x, y)' + assert mcode(polylog(x, y)) == 'polylog(x, y)' + assert mcode(harmonic(x)) == 'harmonic(x)' + assert mcode(bernoulli(x)) == "bernoulli(x)" + assert mcode(bernoulli(x, y)) == "bernoulli(x, y)" + assert mcode(legendre(x, y)) == "legendre(x, y)" + + +def test_Function_change_name(): + assert mcode(abs(x)) == "abs(x)" + assert mcode(ceiling(x)) == "ceil(x)" + assert mcode(arg(x)) == "angle(x)" + assert mcode(im(x)) == "imag(x)" + assert mcode(re(x)) == "real(x)" + assert mcode(conjugate(x)) == "conj(x)" + assert mcode(chebyshevt(y, x)) == "chebyshevT(y, x)" + assert mcode(chebyshevu(y, x)) == "chebyshevU(y, x)" + assert mcode(laguerre(x, y)) == "laguerreL(x, y)" + assert mcode(Chi(x)) == "coshint(x)" + assert mcode(Shi(x)) == "sinhint(x)" + assert mcode(Ci(x)) == "cosint(x)" + assert mcode(Si(x)) == "sinint(x)" + assert mcode(li(x)) == "logint(x)" + assert mcode(loggamma(x)) == "gammaln(x)" + assert mcode(polygamma(x, y)) == "psi(x, y)" + assert mcode(RisingFactorial(x, y)) == "pochhammer(x, y)" + assert mcode(DiracDelta(x)) == "dirac(x)" + assert mcode(DiracDelta(x, 3)) == "dirac(3, x)" + assert mcode(Heaviside(x)) == "heaviside(x, 1/2)" + assert mcode(Heaviside(x, y)) == "heaviside(x, y)" + assert mcode(binomial(x, y)) == "bincoeff(x, y)" + assert mcode(Mod(x, y)) == "mod(x, y)" + + +def test_minmax(): + assert mcode(Max(x, y) + Min(x, y)) == "max(x, y) + min(x, y)" + assert mcode(Max(x, y, z)) == "max(x, max(y, z))" + assert mcode(Min(x, y, z)) == "min(x, min(y, z))" + + +def test_Pow(): + assert mcode(x**3) == "x.^3" + assert mcode(x**(y**3)) == "x.^(y.^3)" + assert mcode(x**Rational(2, 3)) == 'x.^(2/3)' + g = implemented_function('g', Lambda(x, 2*x)) + assert mcode(1/(g(x)*3.5)**(x - y**x)/(x**2 + y)) == \ + "(3.5*2*x).^(-x + y.^x)./(x.^2 + y)" + # For issue 14160 + assert mcode(Mul(-2, x, Pow(Mul(y,y,evaluate=False), -1, evaluate=False), + evaluate=False)) == '-2*x./(y.*y)' + + +def test_basic_ops(): + assert mcode(x*y) == "x.*y" + assert mcode(x + y) == "x + y" + assert mcode(x - y) == "x - y" + assert mcode(-x) == "-x" + + +def test_1_over_x_and_sqrt(): + # 1.0 and 0.5 would do something different in regular StrPrinter, + # but these are exact in IEEE floating point so no different here. + assert mcode(1/x) == '1./x' + assert mcode(x**-1) == mcode(x**-1.0) == '1./x' + assert mcode(1/sqrt(x)) == '1./sqrt(x)' + assert mcode(x**-S.Half) == mcode(x**-0.5) == '1./sqrt(x)' + assert mcode(sqrt(x)) == 'sqrt(x)' + assert mcode(x**S.Half) == mcode(x**0.5) == 'sqrt(x)' + assert mcode(1/pi) == '1/pi' + assert mcode(pi**-1) == mcode(pi**-1.0) == '1/pi' + assert mcode(pi**-0.5) == '1/sqrt(pi)' + + +def test_mix_number_mult_symbols(): + assert mcode(3*x) == "3*x" + assert mcode(pi*x) == "pi*x" + assert mcode(3/x) == "3./x" + assert mcode(pi/x) == "pi./x" + assert mcode(x/3) == "x/3" + assert mcode(x/pi) == "x/pi" + assert mcode(x*y) == "x.*y" + assert mcode(3*x*y) == "3*x.*y" + assert mcode(3*pi*x*y) == "3*pi*x.*y" + assert mcode(x/y) == "x./y" + assert mcode(3*x/y) == "3*x./y" + assert mcode(x*y/z) == "x.*y./z" + assert mcode(x/y*z) == "x.*z./y" + assert mcode(1/x/y) == "1./(x.*y)" + assert mcode(2*pi*x/y/z) == "2*pi*x./(y.*z)" + assert mcode(3*pi/x) == "3*pi./x" + assert mcode(S(3)/5) == "3/5" + assert mcode(S(3)/5*x) == "3*x/5" + assert mcode(x/y/z) == "x./(y.*z)" + assert mcode((x+y)/z) == "(x + y)./z" + assert mcode((x+y)/(z+x)) == "(x + y)./(x + z)" + assert mcode((x+y)/EulerGamma) == "(x + y)/%s" % EulerGamma.evalf(17) + assert mcode(x/3/pi) == "x/(3*pi)" + assert mcode(S(3)/5*x*y/pi) == "3*x.*y/(5*pi)" + + +def test_mix_number_pow_symbols(): + assert mcode(pi**3) == 'pi^3' + assert mcode(x**2) == 'x.^2' + assert mcode(x**(pi**3)) == 'x.^(pi^3)' + assert mcode(x**y) == 'x.^y' + assert mcode(x**(y**z)) == 'x.^(y.^z)' + assert mcode((x**y)**z) == '(x.^y).^z' + + +def test_imag(): + I = S('I') + assert mcode(I) == "1i" + assert mcode(5*I) == "5i" + assert mcode((S(3)/2)*I) == "3*1i/2" + assert mcode(3+4*I) == "3 + 4i" + assert mcode(sqrt(3)*I) == "sqrt(3)*1i" + + +def test_constants(): + assert mcode(pi) == "pi" + assert mcode(oo) == "inf" + assert mcode(-oo) == "-inf" + assert mcode(S.NegativeInfinity) == "-inf" + assert mcode(S.NaN) == "NaN" + assert mcode(S.Exp1) == "exp(1)" + assert mcode(exp(1)) == "exp(1)" + + +def test_constants_other(): + assert mcode(2*GoldenRatio) == "2*(1+sqrt(5))/2" + assert mcode(2*Catalan) == "2*%s" % Catalan.evalf(17) + assert mcode(2*EulerGamma) == "2*%s" % EulerGamma.evalf(17) + + +def test_boolean(): + assert mcode(x & y) == "x & y" + assert mcode(x | y) == "x | y" + assert mcode(~x) == "~x" + assert mcode(x & y & z) == "x & y & z" + assert mcode(x | y | z) == "x | y | z" + assert mcode((x & y) | z) == "z | x & y" + assert mcode((x | y) & z) == "z & (x | y)" + + +def test_KroneckerDelta(): + from sympy.functions import KroneckerDelta + assert mcode(KroneckerDelta(x, y)) == "double(x == y)" + assert mcode(KroneckerDelta(x, y + 1)) == "double(x == (y + 1))" + assert mcode(KroneckerDelta(2**x, y)) == "double((2.^x) == y)" + + +def test_Matrices(): + assert mcode(Matrix(1, 1, [10])) == "10" + A = Matrix([[1, sin(x/2), abs(x)], + [0, 1, pi], + [0, exp(1), ceiling(x)]]) + expected = "[1 sin(x/2) abs(x); 0 1 pi; 0 exp(1) ceil(x)]" + assert mcode(A) == expected + # row and columns + assert mcode(A[:,0]) == "[1; 0; 0]" + assert mcode(A[0,:]) == "[1 sin(x/2) abs(x)]" + # empty matrices + assert mcode(Matrix(0, 0, [])) == '[]' + assert mcode(Matrix(0, 3, [])) == 'zeros(0, 3)' + # annoying to read but correct + assert mcode(Matrix([[x, x - y, -y]])) == "[x x - y -y]" + + +def test_vector_entries_hadamard(): + # For a row or column, user might to use the other dimension + A = Matrix([[1, sin(2/x), 3*pi/x/5]]) + assert mcode(A) == "[1 sin(2./x) 3*pi./(5*x)]" + assert mcode(A.T) == "[1; sin(2./x); 3*pi./(5*x)]" + + +@XFAIL +def test_Matrices_entries_not_hadamard(): + # For Matrix with col >= 2, row >= 2, they need to be scalars + # FIXME: is it worth worrying about this? Its not wrong, just + # leave it user's responsibility to put scalar data for x. + A = Matrix([[1, sin(2/x), 3*pi/x/5], [1, 2, x*y]]) + expected = ("[1 sin(2/x) 3*pi/(5*x);\n" + "1 2 x*y]") # <- we give x.*y + assert mcode(A) == expected + + +def test_MatrixSymbol(): + n = Symbol('n', integer=True) + A = MatrixSymbol('A', n, n) + B = MatrixSymbol('B', n, n) + assert mcode(A*B) == "A*B" + assert mcode(B*A) == "B*A" + assert mcode(2*A*B) == "2*A*B" + assert mcode(B*2*A) == "2*B*A" + assert mcode(A*(B + 3*Identity(n))) == "A*(3*eye(n) + B)" + assert mcode(A**(x**2)) == "A^(x.^2)" + assert mcode(A**3) == "A^3" + assert mcode(A**S.Half) == "A^(1/2)" + + +def test_MatrixSolve(): + n = Symbol('n', integer=True) + A = MatrixSymbol('A', n, n) + x = MatrixSymbol('x', n, 1) + assert mcode(MatrixSolve(A, x)) == "A \\ x" + +def test_special_matrices(): + assert mcode(6*Identity(3)) == "6*eye(3)" + + +def test_containers(): + assert mcode([1, 2, 3, [4, 5, [6, 7]], 8, [9, 10], 11]) == \ + "{1, 2, 3, {4, 5, {6, 7}}, 8, {9, 10}, 11}" + assert mcode((1, 2, (3, 4))) == "{1, 2, {3, 4}}" + assert mcode([1]) == "{1}" + assert mcode((1,)) == "{1}" + assert mcode(Tuple(*[1, 2, 3])) == "{1, 2, 3}" + assert mcode((1, x*y, (3, x**2))) == "{1, x.*y, {3, x.^2}}" + # scalar, matrix, empty matrix and empty list + assert mcode((1, eye(3), Matrix(0, 0, []), [])) == "{1, [1 0 0; 0 1 0; 0 0 1], [], {}}" + + +def test_octave_noninline(): + source = mcode((x+y)/Catalan, assign_to='me', inline=False) + expected = ( + "Catalan = %s;\n" + "me = (x + y)/Catalan;" + ) % Catalan.evalf(17) + assert source == expected + + +def test_octave_piecewise(): + expr = Piecewise((x, x < 1), (x**2, True)) + assert mcode(expr) == "((x < 1).*(x) + (~(x < 1)).*(x.^2))" + assert mcode(expr, assign_to="r") == ( + "r = ((x < 1).*(x) + (~(x < 1)).*(x.^2));") + assert mcode(expr, assign_to="r", inline=False) == ( + "if (x < 1)\n" + " r = x;\n" + "else\n" + " r = x.^2;\n" + "end") + expr = Piecewise((x**2, x < 1), (x**3, x < 2), (x**4, x < 3), (x**5, True)) + expected = ("((x < 1).*(x.^2) + (~(x < 1)).*( ...\n" + "(x < 2).*(x.^3) + (~(x < 2)).*( ...\n" + "(x < 3).*(x.^4) + (~(x < 3)).*(x.^5))))") + assert mcode(expr) == expected + assert mcode(expr, assign_to="r") == "r = " + expected + ";" + assert mcode(expr, assign_to="r", inline=False) == ( + "if (x < 1)\n" + " r = x.^2;\n" + "elseif (x < 2)\n" + " r = x.^3;\n" + "elseif (x < 3)\n" + " r = x.^4;\n" + "else\n" + " r = x.^5;\n" + "end") + # Check that Piecewise without a True (default) condition error + expr = Piecewise((x, x < 1), (x**2, x > 1), (sin(x), x > 0)) + raises(ValueError, lambda: mcode(expr)) + + +def test_octave_piecewise_times_const(): + pw = Piecewise((x, x < 1), (x**2, True)) + assert mcode(2*pw) == "2*((x < 1).*(x) + (~(x < 1)).*(x.^2))" + assert mcode(pw/x) == "((x < 1).*(x) + (~(x < 1)).*(x.^2))./x" + assert mcode(pw/(x*y)) == "((x < 1).*(x) + (~(x < 1)).*(x.^2))./(x.*y)" + assert mcode(pw/3) == "((x < 1).*(x) + (~(x < 1)).*(x.^2))/3" + + +def test_octave_matrix_assign_to(): + A = Matrix([[1, 2, 3]]) + assert mcode(A, assign_to='a') == "a = [1 2 3];" + A = Matrix([[1, 2], [3, 4]]) + assert mcode(A, assign_to='A') == "A = [1 2; 3 4];" + + +def test_octave_matrix_assign_to_more(): + # assigning to Symbol or MatrixSymbol requires lhs/rhs match + A = Matrix([[1, 2, 3]]) + B = MatrixSymbol('B', 1, 3) + C = MatrixSymbol('C', 2, 3) + assert mcode(A, assign_to=B) == "B = [1 2 3];" + raises(ValueError, lambda: mcode(A, assign_to=x)) + raises(ValueError, lambda: mcode(A, assign_to=C)) + + +def test_octave_matrix_1x1(): + A = Matrix([[3]]) + B = MatrixSymbol('B', 1, 1) + C = MatrixSymbol('C', 1, 2) + assert mcode(A, assign_to=B) == "B = 3;" + # FIXME? + #assert mcode(A, assign_to=x) == "x = 3;" + raises(ValueError, lambda: mcode(A, assign_to=C)) + + +def test_octave_matrix_elements(): + A = Matrix([[x, 2, x*y]]) + assert mcode(A[0, 0]**2 + A[0, 1] + A[0, 2]) == "x.^2 + x.*y + 2" + A = MatrixSymbol('AA', 1, 3) + assert mcode(A) == "AA" + assert mcode(A[0, 0]**2 + sin(A[0,1]) + A[0,2]) == \ + "sin(AA(1, 2)) + AA(1, 1).^2 + AA(1, 3)" + assert mcode(sum(A)) == "AA(1, 1) + AA(1, 2) + AA(1, 3)" + + +def test_octave_boolean(): + assert mcode(True) == "true" + assert mcode(S.true) == "true" + assert mcode(False) == "false" + assert mcode(S.false) == "false" + + +def test_octave_not_supported(): + with raises(NotImplementedError): + mcode(S.ComplexInfinity) + f = Function('f') + assert mcode(f(x).diff(x), strict=False) == ( + "% Not supported in Octave:\n" + "% Derivative\n" + "Derivative(f(x), x)" + ) + + +def test_octave_not_supported_not_on_whitelist(): + from sympy.functions.special.polynomials import assoc_laguerre + with raises(NotImplementedError): + mcode(assoc_laguerre(x, y, z)) + + +def test_octave_expint(): + assert mcode(expint(1, x)) == "expint(x)" + with raises(NotImplementedError): + mcode(expint(2, x)) + assert mcode(expint(y, x), strict=False) == ( + "% Not supported in Octave:\n" + "% expint\n" + "expint(y, x)" + ) + + +def test_trick_indent_with_end_else_words(): + # words starting with "end" or "else" do not confuse the indenter + t1 = S('endless') + t2 = S('elsewhere') + pw = Piecewise((t1, x < 0), (t2, x <= 1), (1, True)) + assert mcode(pw, inline=False) == ( + "if (x < 0)\n" + " endless\n" + "elseif (x <= 1)\n" + " elsewhere\n" + "else\n" + " 1\n" + "end") + + +def test_hadamard(): + A = MatrixSymbol('A', 3, 3) + B = MatrixSymbol('B', 3, 3) + v = MatrixSymbol('v', 3, 1) + h = MatrixSymbol('h', 1, 3) + C = HadamardProduct(A, B) + n = Symbol('n') + assert mcode(C) == "A.*B" + assert mcode(C*v) == "(A.*B)*v" + assert mcode(h*C*v) == "h*(A.*B)*v" + assert mcode(C*A) == "(A.*B)*A" + # mixing Hadamard and scalar strange b/c we vectorize scalars + assert mcode(C*x*y) == "(x.*y)*(A.*B)" + + # Testing HadamardPower: + assert mcode(HadamardPower(A, n)) == "A.**n" + assert mcode(HadamardPower(A, 1+n)) == "A.**(n + 1)" + assert mcode(HadamardPower(A*B.T, 1+n)) == "(A*B.T).**(n + 1)" + + +def test_sparse(): + M = SparseMatrix(5, 6, {}) + M[2, 2] = 10 + M[1, 2] = 20 + M[1, 3] = 22 + M[0, 3] = 30 + M[3, 0] = x*y + assert mcode(M) == ( + "sparse([4 2 3 1 2], [1 3 3 4 4], [x.*y 20 10 30 22], 5, 6)" + ) + + +def test_sinc(): + assert mcode(sinc(x)) == 'sinc(x/pi)' + assert mcode(sinc(x + 3)) == 'sinc((x + 3)/pi)' + assert mcode(sinc(pi*(x + 3))) == 'sinc(x + 3)' + + +def test_trigfun(): + for f in (sin, cos, tan, cot, sec, csc, asin, acos, acot, atan, asec, acsc, + sinh, cosh, tanh, coth, csch, sech, asinh, acosh, atanh, acoth, + asech, acsch): + assert octave_code(f(x) == f.__name__ + '(x)') + + +def test_specfun(): + n = Symbol('n') + for f in [besselj, bessely, besseli, besselk]: + assert octave_code(f(n, x)) == f.__name__ + '(n, x)' + for f in (erfc, erfi, erf, erfinv, erfcinv, fresnelc, fresnels, gamma): + assert octave_code(f(x)) == f.__name__ + '(x)' + assert octave_code(hankel1(n, x)) == 'besselh(n, 1, x)' + assert octave_code(hankel2(n, x)) == 'besselh(n, 2, x)' + assert octave_code(airyai(x)) == 'airy(0, x)' + assert octave_code(airyaiprime(x)) == 'airy(1, x)' + assert octave_code(airybi(x)) == 'airy(2, x)' + assert octave_code(airybiprime(x)) == 'airy(3, x)' + assert octave_code(uppergamma(n, x)) == '(gammainc(x, n, \'upper\').*gamma(n))' + assert octave_code(lowergamma(n, x)) == '(gammainc(x, n).*gamma(n))' + assert octave_code(z**lowergamma(n, x)) == 'z.^(gammainc(x, n).*gamma(n))' + assert octave_code(jn(n, x)) == 'sqrt(2)*sqrt(pi)*sqrt(1./x).*besselj(n + 1/2, x)/2' + assert octave_code(yn(n, x)) == 'sqrt(2)*sqrt(pi)*sqrt(1./x).*bessely(n + 1/2, x)/2' + assert octave_code(LambertW(x)) == 'lambertw(x)' + assert octave_code(LambertW(x, n)) == 'lambertw(n, x)' + + # Automatic rewrite + assert octave_code(Ei(x)) == '(logint(exp(x)))' + assert octave_code(dirichlet_eta(x)) == '(((x == 1).*(log(2)) + (~(x == 1)).*((1 - 2.^(1 - x)).*zeta(x))))' + assert octave_code(riemann_xi(x)) == '(pi.^(-x/2).*x.*(x - 1).*gamma(x/2).*zeta(x)/2)' + + +def test_MatrixElement_printing(): + # test cases for issue #11821 + A = MatrixSymbol("A", 1, 3) + B = MatrixSymbol("B", 1, 3) + C = MatrixSymbol("C", 1, 3) + + assert mcode(A[0, 0]) == "A(1, 1)" + assert mcode(3 * A[0, 0]) == "3*A(1, 1)" + + F = C[0, 0].subs(C, A - B) + assert mcode(F) == "(A - B)(1, 1)" + + +def test_zeta_printing_issue_14820(): + assert octave_code(zeta(x)) == 'zeta(x)' + with raises(NotImplementedError): + octave_code(zeta(x, y)) + + +def test_automatic_rewrite(): + assert octave_code(Li(x)) == '(logint(x) - logint(2))' + assert octave_code(erf2(x, y)) == '(-erf(x) + erf(y))' diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_precedence.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_precedence.py new file mode 100644 index 0000000000000000000000000000000000000000..d08ea07483857e8c2ee7f930aa53d2dacdc58193 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_precedence.py @@ -0,0 +1,128 @@ +from sympy.concrete.products import Product +from sympy.concrete.summations import Sum +from sympy.core.function import Derivative, Function +from sympy.core.numbers import Integer, Rational, Float, oo +from sympy.core.relational import Rel +from sympy.core.symbol import symbols +from sympy.functions import sin +from sympy.integrals.integrals import Integral +from sympy.series.order import Order + +from sympy.printing.precedence import precedence, PRECEDENCE + +x, y = symbols("x,y") + + +def test_Add(): + assert precedence(x + y) == PRECEDENCE["Add"] + assert precedence(x*y + 1) == PRECEDENCE["Add"] + + +def test_Function(): + assert precedence(sin(x)) == PRECEDENCE["Func"] + +def test_Derivative(): + assert precedence(Derivative(x, y)) == PRECEDENCE["Atom"] + +def test_Integral(): + assert precedence(Integral(x, y)) == PRECEDENCE["Atom"] + + +def test_Mul(): + assert precedence(x*y) == PRECEDENCE["Mul"] + assert precedence(-x*y) == PRECEDENCE["Add"] + + +def test_Number(): + assert precedence(Integer(0)) == PRECEDENCE["Atom"] + assert precedence(Integer(1)) == PRECEDENCE["Atom"] + assert precedence(Integer(-1)) == PRECEDENCE["Add"] + assert precedence(Integer(10)) == PRECEDENCE["Atom"] + assert precedence(Rational(5, 2)) == PRECEDENCE["Mul"] + assert precedence(Rational(-5, 2)) == PRECEDENCE["Add"] + assert precedence(Float(5)) == PRECEDENCE["Atom"] + assert precedence(Float(-5)) == PRECEDENCE["Add"] + assert precedence(oo) == PRECEDENCE["Atom"] + assert precedence(-oo) == PRECEDENCE["Add"] + + +def test_Order(): + assert precedence(Order(x)) == PRECEDENCE["Atom"] + + +def test_Pow(): + assert precedence(x**y) == PRECEDENCE["Pow"] + assert precedence(-x**y) == PRECEDENCE["Add"] + assert precedence(x**-y) == PRECEDENCE["Pow"] + + +def test_Product(): + assert precedence(Product(x, (x, y, y + 1))) == PRECEDENCE["Atom"] + + +def test_Relational(): + assert precedence(Rel(x + y, y, "<")) == PRECEDENCE["Relational"] + + +def test_Sum(): + assert precedence(Sum(x, (x, y, y + 1))) == PRECEDENCE["Atom"] + + +def test_Symbol(): + assert precedence(x) == PRECEDENCE["Atom"] + + +def test_And_Or(): + # precedence relations between logical operators, ... + assert precedence(x & y) > precedence(x | y) + assert precedence(~y) > precedence(x & y) + # ... and with other operators (cfr. other programming languages) + assert precedence(x + y) > precedence(x | y) + assert precedence(x + y) > precedence(x & y) + assert precedence(x*y) > precedence(x | y) + assert precedence(x*y) > precedence(x & y) + assert precedence(~y) > precedence(x*y) + assert precedence(~y) > precedence(x - y) + # double checks + assert precedence(x & y) == PRECEDENCE["And"] + assert precedence(x | y) == PRECEDENCE["Or"] + assert precedence(~y) == PRECEDENCE["Not"] + + +def test_custom_function_precedence_comparison(): + """ + Test cases for custom functions with different precedence values, + specifically handling: + 1. Functions with precedence < PRECEDENCE["Mul"] (50) + 2. Functions with precedence = Func (70) + + Key distinction: + 1. Lower precedence functions (45) need parentheses: -2*(x F y) + 2. Higher precedence functions (70) don't: -2*x F y + """ + class LowPrecedenceF(Function): + precedence = PRECEDENCE["Mul"] - 5 + def _sympystr(self, printer): + return f"{printer._print(self.args[0])} F {printer._print(self.args[1])}" + + class HighPrecedenceF(Function): + precedence = PRECEDENCE["Func"] + def _sympystr(self, printer): + return f"{printer._print(self.args[0])} F {printer._print(self.args[1])}" + + def test_low_precedence(): + expr1 = 2 * LowPrecedenceF(x, y) + assert str(expr1) == "2*(x F y)" + + expr2 = -2 * LowPrecedenceF(x, y) + assert str(expr2) == "-2*(x F y)" + + def test_high_precedence(): + expr1 = 2 * HighPrecedenceF(x, y) + assert str(expr1) == "2*x F y" + + expr2 = -2 * HighPrecedenceF(x, y) + assert str(expr2) == "-2*x F y" + + test_low_precedence() + test_high_precedence() diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_preview.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_preview.py new file mode 100644 index 0000000000000000000000000000000000000000..91771ceb0466d6b0fee00570426713d02da14872 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_preview.py @@ -0,0 +1,38 @@ +# -*- coding: utf-8 -*- + +from sympy.core.relational import Eq +from sympy.core.symbol import Symbol +from sympy.functions.elementary.piecewise import Piecewise +from sympy.printing.preview import preview + +from io import BytesIO + + +def test_preview(): + x = Symbol('x') + obj = BytesIO() + try: + preview(x, output='png', viewer='BytesIO', outputbuffer=obj) + except RuntimeError: + pass # latex not installed on CI server + + +def test_preview_unicode_symbol(): + # issue 9107 + a = Symbol('α') + obj = BytesIO() + try: + preview(a, output='png', viewer='BytesIO', outputbuffer=obj) + except RuntimeError: + pass # latex not installed on CI server + + +def test_preview_latex_construct_in_expr(): + # see PR 9801 + x = Symbol('x') + pw = Piecewise((1, Eq(x, 0)), (0, True)) + obj = BytesIO() + try: + preview(pw, output='png', viewer='BytesIO', outputbuffer=obj) + except RuntimeError: + pass # latex not installed on CI server diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_pycode.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_pycode.py new file mode 100644 index 0000000000000000000000000000000000000000..2c38fe81d830149cdce6b55f15e6e07513fdd146 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_pycode.py @@ -0,0 +1,493 @@ +from sympy import Not +from sympy.codegen import Assignment +from sympy.codegen.ast import none +from sympy.codegen.cfunctions import expm1, log1p +from sympy.codegen.scipy_nodes import cosm1 +from sympy.codegen.matrix_nodes import MatrixSolve +from sympy.core import Expr, Mod, symbols, Eq, Le, Gt, zoo, oo, Rational, Pow +from sympy.core.function import Derivative +from sympy.core.numbers import pi +from sympy.core.singleton import S +from sympy.functions import acos, KroneckerDelta, Piecewise, sign, sqrt, Min, Max, cot, acsch, asec, coth, sec, log, sin, cos, tan, asin, atan, sinh, cosh, tanh, asinh, acosh, atanh +from sympy.functions.elementary.trigonometric import atan2 +from sympy.logic import And, Or +from sympy.matrices import SparseMatrix, MatrixSymbol, Identity +from sympy.printing.codeprinter import PrintMethodNotImplementedError +from sympy.printing.pycode import ( + MpmathPrinter, CmathPrinter, PythonCodePrinter, pycode, SymPyPrinter +) +from sympy.printing.tensorflow import TensorflowPrinter +from sympy.printing.numpy import NumPyPrinter, SciPyPrinter +from sympy.testing.pytest import raises, skip +from sympy.tensor import IndexedBase, Idx +from sympy.tensor.array.expressions.array_expressions import ArraySymbol, ArrayDiagonal, ArrayContraction, ZeroArray, OneArray +from sympy.external import import_module +from sympy.functions.special.gamma_functions import loggamma + + + +x, y, z = symbols('x y z') +p = IndexedBase("p") + + +def test_PythonCodePrinter(): + prntr = PythonCodePrinter() + + assert not prntr.module_imports + + assert prntr.doprint(x**y) == 'x**y' + assert prntr.doprint(Mod(x, 2)) == 'x % 2' + assert prntr.doprint(-Mod(x, y)) == '-(x % y)' + assert prntr.doprint(Mod(-x, y)) == '(-x) % y' + assert prntr.doprint(And(x, y)) == 'x and y' + assert prntr.doprint(Or(x, y)) == 'x or y' + assert prntr.doprint(1/(x+y)) == '1/(x + y)' + assert prntr.doprint(Not(x)) == 'not x' + assert not prntr.module_imports + + assert prntr.doprint(pi) == 'math.pi' + assert prntr.module_imports == {'math': {'pi'}} + + assert prntr.doprint(x**Rational(1, 2)) == 'math.sqrt(x)' + assert prntr.doprint(sqrt(x)) == 'math.sqrt(x)' + assert prntr.module_imports == {'math': {'pi', 'sqrt'}} + + assert prntr.doprint(acos(x)) == 'math.acos(x)' + assert prntr.doprint(cot(x)) == '(1/math.tan(x))' + assert prntr.doprint(coth(x)) == '((math.exp(x) + math.exp(-x))/(math.exp(x) - math.exp(-x)))' + assert prntr.doprint(asec(x)) == '(math.acos(1/x))' + assert prntr.doprint(acsch(x)) == '(math.log(math.sqrt(1 + x**(-2)) + 1/x))' + + assert prntr.doprint(Assignment(x, 2)) == 'x = 2' + assert prntr.doprint(Piecewise((1, Eq(x, 0)), + (2, x>6))) == '((1) if (x == 0) else (2) if (x > 6) else None)' + assert prntr.doprint(Piecewise((2, Le(x, 0)), + (3, Gt(x, 0)), evaluate=False)) == '((2) if (x <= 0) else'\ + ' (3) if (x > 0) else None)' + assert prntr.doprint(sign(x)) == '(0.0 if x == 0 else math.copysign(1, x))' + assert prntr.doprint(p[0, 1]) == 'p[0, 1]' + assert prntr.doprint(KroneckerDelta(x,y)) == '(1 if x == y else 0)' + + assert prntr.doprint((2,3)) == "(2, 3)" + assert prntr.doprint([2,3]) == "[2, 3]" + + assert prntr.doprint(Min(x, y)) == "min(x, y)" + assert prntr.doprint(Max(x, y)) == "max(x, y)" + + +def test_PythonCodePrinter_standard(): + prntr = PythonCodePrinter() + + assert prntr.standard == 'python3' + + raises(ValueError, lambda: PythonCodePrinter({'standard':'python4'})) + + +def test_CmathPrinter(): + p = CmathPrinter() + + assert p.doprint(sqrt(x)) == 'cmath.sqrt(x)' + assert p.doprint(log(x)) == 'cmath.log(x)' + + assert p.doprint(sin(x)) == 'cmath.sin(x)' + assert p.doprint(cos(x)) == 'cmath.cos(x)' + assert p.doprint(tan(x)) == 'cmath.tan(x)' + + assert p.doprint(asin(x)) == 'cmath.asin(x)' + assert p.doprint(acos(x)) == 'cmath.acos(x)' + assert p.doprint(atan(x)) == 'cmath.atan(x)' + + assert p.doprint(sinh(x)) == 'cmath.sinh(x)' + assert p.doprint(cosh(x)) == 'cmath.cosh(x)' + assert p.doprint(tanh(x)) == 'cmath.tanh(x)' + + assert p.doprint(asinh(x)) == 'cmath.asinh(x)' + assert p.doprint(acosh(x)) == 'cmath.acosh(x)' + assert p.doprint(atanh(x)) == 'cmath.atanh(x)' + + +def test_MpmathPrinter(): + p = MpmathPrinter() + assert p.doprint(sign(x)) == 'mpmath.sign(x)' + assert p.doprint(Rational(1, 2)) == 'mpmath.mpf(1)/mpmath.mpf(2)' + + assert p.doprint(S.Exp1) == 'mpmath.e' + assert p.doprint(S.Pi) == 'mpmath.pi' + assert p.doprint(S.GoldenRatio) == 'mpmath.phi' + assert p.doprint(S.EulerGamma) == 'mpmath.euler' + assert p.doprint(S.NaN) == 'mpmath.nan' + assert p.doprint(S.Infinity) == 'mpmath.inf' + assert p.doprint(S.NegativeInfinity) == 'mpmath.ninf' + assert p.doprint(loggamma(x)) == 'mpmath.loggamma(x)' + + +def test_NumPyPrinter(): + from sympy.core.function import Lambda + from sympy.matrices.expressions.adjoint import Adjoint + from sympy.matrices.expressions.diagonal import (DiagMatrix, DiagonalMatrix, DiagonalOf) + from sympy.matrices.expressions.funcmatrix import FunctionMatrix + from sympy.matrices.expressions.hadamard import HadamardProduct + from sympy.matrices.expressions.kronecker import KroneckerProduct + from sympy.matrices.expressions.special import (OneMatrix, ZeroMatrix) + from sympy.abc import a, b + p = NumPyPrinter() + assert p.doprint(sign(x)) == 'numpy.sign(x)' + A = MatrixSymbol("A", 2, 2) + B = MatrixSymbol("B", 2, 2) + C = MatrixSymbol("C", 1, 5) + D = MatrixSymbol("D", 3, 4) + assert p.doprint(A**(-1)) == "numpy.linalg.inv(A)" + assert p.doprint(A**5) == "numpy.linalg.matrix_power(A, 5)" + assert p.doprint(Identity(3)) == "numpy.eye(3)" + + u = MatrixSymbol('x', 2, 1) + v = MatrixSymbol('y', 2, 1) + assert p.doprint(MatrixSolve(A, u)) == 'numpy.linalg.solve(A, x)' + assert p.doprint(MatrixSolve(A, u) + v) == 'numpy.linalg.solve(A, x) + y' + + assert p.doprint(ZeroMatrix(2, 3)) == "numpy.zeros((2, 3))" + assert p.doprint(OneMatrix(2, 3)) == "numpy.ones((2, 3))" + assert p.doprint(FunctionMatrix(4, 5, Lambda((a, b), a + b))) == \ + "numpy.fromfunction(lambda a, b: a + b, (4, 5))" + assert p.doprint(HadamardProduct(A, B)) == "numpy.multiply(A, B)" + assert p.doprint(KroneckerProduct(A, B)) == "numpy.kron(A, B)" + assert p.doprint(Adjoint(A)) == "numpy.conjugate(numpy.transpose(A))" + assert p.doprint(DiagonalOf(A)) == "numpy.reshape(numpy.diag(A), (-1, 1))" + assert p.doprint(DiagMatrix(C)) == "numpy.diagflat(C)" + assert p.doprint(DiagonalMatrix(D)) == "numpy.multiply(D, numpy.eye(3, 4))" + + # Workaround for numpy negative integer power errors + assert p.doprint(x**-1) == 'x**(-1.0)' + assert p.doprint(x**-2) == 'x**(-2.0)' + + expr = Pow(2, -1, evaluate=False) + assert p.doprint(expr) == "2**(-1.0)" + + assert p.doprint(S.Exp1) == 'numpy.e' + assert p.doprint(S.Pi) == 'numpy.pi' + assert p.doprint(S.EulerGamma) == 'numpy.euler_gamma' + assert p.doprint(S.NaN) == 'numpy.nan' + assert p.doprint(S.Infinity) == 'numpy.inf' + assert p.doprint(S.NegativeInfinity) == '-numpy.inf' + + # Function rewriting operator precedence fix + assert p.doprint(sec(x)**2) == '(numpy.cos(x)**(-1.0))**2' + + +def test_issue_18770(): + numpy = import_module('numpy') + if not numpy: + skip("numpy not installed.") + + from sympy.functions.elementary.miscellaneous import (Max, Min) + from sympy.utilities.lambdify import lambdify + + expr1 = Min(0.1*x + 3, x + 1, 0.5*x + 1) + func = lambdify(x, expr1, "numpy") + assert (func(numpy.linspace(0, 3, 3)) == [1.0, 1.75, 2.5 ]).all() + assert func(4) == 3 + + expr1 = Max(x**2, x**3) + func = lambdify(x,expr1, "numpy") + assert (func(numpy.linspace(-1, 2, 4)) == [1, 0, 1, 8] ).all() + assert func(4) == 64 + + +def test_SciPyPrinter(): + p = SciPyPrinter() + expr = acos(x) + assert 'numpy' not in p.module_imports + assert p.doprint(expr) == 'numpy.arccos(x)' + assert 'numpy' in p.module_imports + assert not any(m.startswith('scipy') for m in p.module_imports) + smat = SparseMatrix(2, 5, {(0, 1): 3}) + assert p.doprint(smat) == \ + 'scipy.sparse.coo_matrix(([3], ([0], [1])), shape=(2, 5))' + assert 'scipy.sparse' in p.module_imports + + assert p.doprint(S.GoldenRatio) == 'scipy.constants.golden_ratio' + assert p.doprint(S.Pi) == 'scipy.constants.pi' + assert p.doprint(S.Exp1) == 'numpy.e' + + +def test_pycode_reserved_words(): + s1, s2 = symbols('if else') + raises(ValueError, lambda: pycode(s1 + s2, error_on_reserved=True)) + py_str = pycode(s1 + s2) + assert py_str in ('else_ + if_', 'if_ + else_') + + +def test_issue_20762(): + # Make sure pycode removes curly braces from subscripted variables + a_b, b, a_11 = symbols('a_{b} b a_{11}') + expr = a_b*b + assert pycode(expr) == 'a_b*b' + expr = a_11*b + assert pycode(expr) == 'a_11*b' + + +def test_sqrt(): + prntr = PythonCodePrinter() + assert prntr._print_Pow(sqrt(x), rational=False) == 'math.sqrt(x)' + assert prntr._print_Pow(1/sqrt(x), rational=False) == '1/math.sqrt(x)' + + prntr = PythonCodePrinter({'standard' : 'python3'}) + assert prntr._print_Pow(sqrt(x), rational=True) == 'x**(1/2)' + assert prntr._print_Pow(1/sqrt(x), rational=True) == 'x**(-1/2)' + + prntr = MpmathPrinter() + assert prntr._print_Pow(sqrt(x), rational=False) == 'mpmath.sqrt(x)' + assert prntr._print_Pow(sqrt(x), rational=True) == \ + "x**(mpmath.mpf(1)/mpmath.mpf(2))" + + prntr = NumPyPrinter() + assert prntr._print_Pow(sqrt(x), rational=False) == 'numpy.sqrt(x)' + assert prntr._print_Pow(sqrt(x), rational=True) == 'x**(1/2)' + + prntr = SciPyPrinter() + assert prntr._print_Pow(sqrt(x), rational=False) == 'numpy.sqrt(x)' + assert prntr._print_Pow(sqrt(x), rational=True) == 'x**(1/2)' + + prntr = SymPyPrinter() + assert prntr._print_Pow(sqrt(x), rational=False) == 'sympy.sqrt(x)' + assert prntr._print_Pow(sqrt(x), rational=True) == 'x**(1/2)' + + +def test_frac(): + from sympy.functions.elementary.integers import frac + + expr = frac(x) + prntr = NumPyPrinter() + assert prntr.doprint(expr) == 'numpy.mod(x, 1)' + + prntr = SciPyPrinter() + assert prntr.doprint(expr) == 'numpy.mod(x, 1)' + + prntr = PythonCodePrinter() + assert prntr.doprint(expr) == 'x % 1' + + prntr = MpmathPrinter() + assert prntr.doprint(expr) == 'mpmath.frac(x)' + + prntr = SymPyPrinter() + assert prntr.doprint(expr) == 'sympy.functions.elementary.integers.frac(x)' + + +class CustomPrintedObject(Expr): + def _numpycode(self, printer): + return 'numpy' + + def _mpmathcode(self, printer): + return 'mpmath' + + +def test_printmethod(): + obj = CustomPrintedObject() + assert NumPyPrinter().doprint(obj) == 'numpy' + assert MpmathPrinter().doprint(obj) == 'mpmath' + + +def test_codegen_ast_nodes(): + assert pycode(none) == 'None' + + +def test_issue_14283(): + prntr = PythonCodePrinter() + + assert prntr.doprint(zoo) == "math.nan" + assert prntr.doprint(-oo) == "float('-inf')" + + +def test_NumPyPrinter_print_seq(): + n = NumPyPrinter() + + assert n._print_seq(range(2)) == '(0, 1,)' + + +def test_issue_16535_16536(): + from sympy.functions.special.gamma_functions import (lowergamma, uppergamma) + + a = symbols('a') + expr1 = lowergamma(a, x) + expr2 = uppergamma(a, x) + + prntr = SciPyPrinter() + assert prntr.doprint(expr1) == 'scipy.special.gamma(a)*scipy.special.gammainc(a, x)' + assert prntr.doprint(expr2) == 'scipy.special.gamma(a)*scipy.special.gammaincc(a, x)' + + p_numpy = NumPyPrinter() + p_pycode = PythonCodePrinter({'strict': False}) + + for expr in [expr1, expr2]: + with raises(NotImplementedError): + p_numpy.doprint(expr1) + assert "Not supported" in p_pycode.doprint(expr) + + +def test_Integral(): + from sympy.functions.elementary.exponential import exp + from sympy.integrals.integrals import Integral + + single = Integral(exp(-x), (x, 0, oo)) + double = Integral(x**2*exp(x*y), (x, -z, z), (y, 0, z)) + indefinite = Integral(x**2, x) + evaluateat = Integral(x**2, (x, 1)) + + prntr = SciPyPrinter() + assert prntr.doprint(single) == 'scipy.integrate.quad(lambda x: numpy.exp(-x), 0, numpy.inf)[0]' + assert prntr.doprint(double) == 'scipy.integrate.nquad(lambda x, y: x**2*numpy.exp(x*y), ((-z, z), (0, z)))[0]' + raises(NotImplementedError, lambda: prntr.doprint(indefinite)) + raises(NotImplementedError, lambda: prntr.doprint(evaluateat)) + + prntr = MpmathPrinter() + assert prntr.doprint(single) == 'mpmath.quad(lambda x: mpmath.exp(-x), (0, mpmath.inf))' + assert prntr.doprint(double) == 'mpmath.quad(lambda x, y: x**2*mpmath.exp(x*y), (-z, z), (0, z))' + raises(NotImplementedError, lambda: prntr.doprint(indefinite)) + raises(NotImplementedError, lambda: prntr.doprint(evaluateat)) + + +def test_fresnel_integrals(): + from sympy.functions.special.error_functions import (fresnelc, fresnels) + + expr1 = fresnelc(x) + expr2 = fresnels(x) + + prntr = SciPyPrinter() + assert prntr.doprint(expr1) == 'scipy.special.fresnel(x)[1]' + assert prntr.doprint(expr2) == 'scipy.special.fresnel(x)[0]' + + p_numpy = NumPyPrinter() + p_pycode = PythonCodePrinter() + p_mpmath = MpmathPrinter() + for expr in [expr1, expr2]: + with raises(NotImplementedError): + p_numpy.doprint(expr) + with raises(NotImplementedError): + p_pycode.doprint(expr) + + assert p_mpmath.doprint(expr1) == 'mpmath.fresnelc(x)' + assert p_mpmath.doprint(expr2) == 'mpmath.fresnels(x)' + + +def test_beta(): + from sympy.functions.special.beta_functions import beta + + expr = beta(x, y) + + prntr = SciPyPrinter() + assert prntr.doprint(expr) == 'scipy.special.beta(x, y)' + + prntr = NumPyPrinter() + assert prntr.doprint(expr) == '(math.gamma(x)*math.gamma(y)/math.gamma(x + y))' + + prntr = PythonCodePrinter() + assert prntr.doprint(expr) == '(math.gamma(x)*math.gamma(y)/math.gamma(x + y))' + + prntr = PythonCodePrinter({'allow_unknown_functions': True}) + assert prntr.doprint(expr) == '(math.gamma(x)*math.gamma(y)/math.gamma(x + y))' + + prntr = MpmathPrinter() + assert prntr.doprint(expr) == 'mpmath.beta(x, y)' + +def test_airy(): + from sympy.functions.special.bessel import (airyai, airybi) + + expr1 = airyai(x) + expr2 = airybi(x) + + prntr = SciPyPrinter() + assert prntr.doprint(expr1) == 'scipy.special.airy(x)[0]' + assert prntr.doprint(expr2) == 'scipy.special.airy(x)[2]' + + prntr = NumPyPrinter({'strict': False}) + assert "Not supported" in prntr.doprint(expr1) + assert "Not supported" in prntr.doprint(expr2) + + prntr = PythonCodePrinter({'strict': False}) + assert "Not supported" in prntr.doprint(expr1) + assert "Not supported" in prntr.doprint(expr2) + +def test_airy_prime(): + from sympy.functions.special.bessel import (airyaiprime, airybiprime) + + expr1 = airyaiprime(x) + expr2 = airybiprime(x) + + prntr = SciPyPrinter() + assert prntr.doprint(expr1) == 'scipy.special.airy(x)[1]' + assert prntr.doprint(expr2) == 'scipy.special.airy(x)[3]' + + prntr = NumPyPrinter({'strict': False}) + assert "Not supported" in prntr.doprint(expr1) + assert "Not supported" in prntr.doprint(expr2) + + prntr = PythonCodePrinter({'strict': False}) + assert "Not supported" in prntr.doprint(expr1) + assert "Not supported" in prntr.doprint(expr2) + + +def test_numerical_accuracy_functions(): + prntr = SciPyPrinter() + assert prntr.doprint(expm1(x)) == 'numpy.expm1(x)' + assert prntr.doprint(log1p(x)) == 'numpy.log1p(x)' + assert prntr.doprint(cosm1(x)) == 'scipy.special.cosm1(x)' + +def test_array_printer(): + A = ArraySymbol('A', (4,4,6,6,6)) + I = IndexedBase('I') + i,j,k = Idx('i', (0,1)), Idx('j', (2,3)), Idx('k', (4,5)) + + prntr = NumPyPrinter() + assert prntr.doprint(ZeroArray(5)) == 'numpy.zeros((5,))' + assert prntr.doprint(OneArray(5)) == 'numpy.ones((5,))' + assert prntr.doprint(ArrayContraction(A, [2,3])) == 'numpy.einsum("abccd->abd", A)' + assert prntr.doprint(I) == 'I' + assert prntr.doprint(ArrayDiagonal(A, [2,3,4])) == 'numpy.einsum("abccc->abc", A)' + assert prntr.doprint(ArrayDiagonal(A, [0,1], [2,3])) == 'numpy.einsum("aabbc->cab", A)' + assert prntr.doprint(ArrayContraction(A, [2], [3])) == 'numpy.einsum("abcde->abe", A)' + assert prntr.doprint(Assignment(I[i,j,k], I[i,j,k])) == 'I = I' + + prntr = TensorflowPrinter() + assert prntr.doprint(ZeroArray(5)) == 'tensorflow.zeros((5,))' + assert prntr.doprint(OneArray(5)) == 'tensorflow.ones((5,))' + assert prntr.doprint(ArrayContraction(A, [2,3])) == 'tensorflow.linalg.einsum("abccd->abd", A)' + assert prntr.doprint(I) == 'I' + assert prntr.doprint(ArrayDiagonal(A, [2,3,4])) == 'tensorflow.linalg.einsum("abccc->abc", A)' + assert prntr.doprint(ArrayDiagonal(A, [0,1], [2,3])) == 'tensorflow.linalg.einsum("aabbc->cab", A)' + assert prntr.doprint(ArrayContraction(A, [2], [3])) == 'tensorflow.linalg.einsum("abcde->abe", A)' + assert prntr.doprint(Assignment(I[i,j,k], I[i,j,k])) == 'I = I' + + +def test_custom_Derivative_methods(): + class MyPrinter(SciPyPrinter): + def _print_Derivative_cosm1(self, args, seq_orders): + arg, = args + order, = seq_orders + return 'my_custom_cosm1(%s, deriv_order=%d)' % (self._print(arg), order) + + def _print_Derivative_atan2(self, args, seq_orders): + arg1, arg2 = args + ord1, ord2 = seq_orders + return 'my_custom_atan2(%s, %s, deriv1=%d, deriv2=%d)' % ( + self._print(arg1), self._print(arg2), ord1, ord2 + ) + + p = MyPrinter() + cosm1_1 = cosm1(x).diff(x, evaluate=False) + assert p.doprint(cosm1_1) == 'my_custom_cosm1(x, deriv_order=1)' + atan2_2_3 = atan2(x, y).diff(x, 2, y, 3, evaluate=False) + assert p.doprint(atan2_2_3) == 'my_custom_atan2(x, y, deriv1=2, deriv2=3)' + + try: + p.doprint(expm1(x).diff(x, evaluate=False)) + except PrintMethodNotImplementedError as e: + assert '_print_Derivative_expm1' in repr(e) + else: + assert False # should have thrown + + try: + p.doprint(Derivative(cosm1(x**2),x)) + except ValueError as e: + assert '_print_Derivative(' in repr(e) + else: + assert False # should have thrown diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_python.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_python.py new file mode 100644 index 0000000000000000000000000000000000000000..fb94a662be90934a672d08b3de44a22e2580d8b6 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_python.py @@ -0,0 +1,203 @@ +from sympy.core.function import (Derivative, Function) +from sympy.core.numbers import (I, Rational, oo, pi) +from sympy.core.relational import (Eq, Ge, Gt, Le, Lt, Ne) +from sympy.core.symbol import (Symbol, symbols) +from sympy.functions.elementary.complexes import (Abs, conjugate) +from sympy.functions.elementary.exponential import (exp, log) +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import sin +from sympy.integrals.integrals import Integral +from sympy.matrices.dense import Matrix +from sympy.series.limits import limit + +from sympy.printing.python import python + +from sympy.testing.pytest import raises, XFAIL + +x, y = symbols('x,y') +th = Symbol('theta') +ph = Symbol('phi') + + +def test_python_basic(): + # Simple numbers/symbols + assert python(-Rational(1)/2) == "e = Rational(-1, 2)" + assert python(-Rational(13)/22) == "e = Rational(-13, 22)" + assert python(oo) == "e = oo" + + # Powers + assert python(x**2) == "x = Symbol(\'x\')\ne = x**2" + assert python(1/x) == "x = Symbol('x')\ne = 1/x" + assert python(y*x**-2) == "y = Symbol('y')\nx = Symbol('x')\ne = y/x**2" + assert python( + x**Rational(-5, 2)) == "x = Symbol('x')\ne = x**Rational(-5, 2)" + + # Sums of terms + assert python(x**2 + x + 1) in [ + "x = Symbol('x')\ne = 1 + x + x**2", + "x = Symbol('x')\ne = x + x**2 + 1", + "x = Symbol('x')\ne = x**2 + x + 1", ] + assert python(1 - x) in [ + "x = Symbol('x')\ne = 1 - x", + "x = Symbol('x')\ne = -x + 1"] + assert python(1 - 2*x) in [ + "x = Symbol('x')\ne = 1 - 2*x", + "x = Symbol('x')\ne = -2*x + 1"] + assert python(1 - Rational(3, 2)*y/x) in [ + "y = Symbol('y')\nx = Symbol('x')\ne = 1 - 3/2*y/x", + "y = Symbol('y')\nx = Symbol('x')\ne = -3/2*y/x + 1", + "y = Symbol('y')\nx = Symbol('x')\ne = 1 - 3*y/(2*x)"] + + # Multiplication + assert python(x/y) == "x = Symbol('x')\ny = Symbol('y')\ne = x/y" + assert python(-x/y) == "x = Symbol('x')\ny = Symbol('y')\ne = -x/y" + assert python((x + 2)/y) in [ + "y = Symbol('y')\nx = Symbol('x')\ne = 1/y*(2 + x)", + "y = Symbol('y')\nx = Symbol('x')\ne = 1/y*(x + 2)", + "x = Symbol('x')\ny = Symbol('y')\ne = 1/y*(2 + x)", + "x = Symbol('x')\ny = Symbol('y')\ne = (2 + x)/y", + "x = Symbol('x')\ny = Symbol('y')\ne = (x + 2)/y"] + assert python((1 + x)*y) in [ + "y = Symbol('y')\nx = Symbol('x')\ne = y*(1 + x)", + "y = Symbol('y')\nx = Symbol('x')\ne = y*(x + 1)", ] + + # Check for proper placement of negative sign + assert python(-5*x/(x + 10)) == "x = Symbol('x')\ne = -5*x/(x + 10)" + assert python(1 - Rational(3, 2)*(x + 1)) in [ + "x = Symbol('x')\ne = Rational(-3, 2)*x + Rational(-1, 2)", + "x = Symbol('x')\ne = -3*x/2 + Rational(-1, 2)", + "x = Symbol('x')\ne = -3*x/2 + Rational(-1, 2)" + ] + + +def test_python_keyword_symbol_name_escaping(): + # Check for escaping of keywords + assert python( + 5*Symbol("lambda")) == "lambda_ = Symbol('lambda')\ne = 5*lambda_" + assert (python(5*Symbol("lambda") + 7*Symbol("lambda_")) == + "lambda__ = Symbol('lambda')\nlambda_ = Symbol('lambda_')\ne = 7*lambda_ + 5*lambda__") + assert (python(5*Symbol("for") + Function("for_")(8)) == + "for__ = Symbol('for')\nfor_ = Function('for_')\ne = 5*for__ + for_(8)") + + +def test_python_keyword_function_name_escaping(): + assert python( + 5*Function("for")(8)) == "for_ = Function('for')\ne = 5*for_(8)" + + +def test_python_relational(): + assert python(Eq(x, y)) == "x = Symbol('x')\ny = Symbol('y')\ne = Eq(x, y)" + assert python(Ge(x, y)) == "x = Symbol('x')\ny = Symbol('y')\ne = x >= y" + assert python(Le(x, y)) == "x = Symbol('x')\ny = Symbol('y')\ne = x <= y" + assert python(Gt(x, y)) == "x = Symbol('x')\ny = Symbol('y')\ne = x > y" + assert python(Lt(x, y)) == "x = Symbol('x')\ny = Symbol('y')\ne = x < y" + assert python(Ne(x/(y + 1), y**2)) in [ + "x = Symbol('x')\ny = Symbol('y')\ne = Ne(x/(1 + y), y**2)", + "x = Symbol('x')\ny = Symbol('y')\ne = Ne(x/(y + 1), y**2)"] + + +def test_python_functions(): + # Simple + assert python(2*x + exp(x)) in "x = Symbol('x')\ne = 2*x + exp(x)" + assert python(sqrt(2)) == 'e = sqrt(2)' + assert python(2**Rational(1, 3)) == 'e = 2**Rational(1, 3)' + assert python(sqrt(2 + pi)) == 'e = sqrt(2 + pi)' + assert python((2 + pi)**Rational(1, 3)) == 'e = (2 + pi)**Rational(1, 3)' + assert python(2**Rational(1, 4)) == 'e = 2**Rational(1, 4)' + assert python(Abs(x)) == "x = Symbol('x')\ne = Abs(x)" + assert python( + Abs(x/(x**2 + 1))) in ["x = Symbol('x')\ne = Abs(x/(1 + x**2))", + "x = Symbol('x')\ne = Abs(x/(x**2 + 1))"] + + # Univariate/Multivariate functions + f = Function('f') + assert python(f(x)) == "x = Symbol('x')\nf = Function('f')\ne = f(x)" + assert python(f(x, y)) == "x = Symbol('x')\ny = Symbol('y')\nf = Function('f')\ne = f(x, y)" + assert python(f(x/(y + 1), y)) in [ + "x = Symbol('x')\ny = Symbol('y')\nf = Function('f')\ne = f(x/(1 + y), y)", + "x = Symbol('x')\ny = Symbol('y')\nf = Function('f')\ne = f(x/(y + 1), y)"] + + # Nesting of square roots + assert python(sqrt((sqrt(x + 1)) + 1)) in [ + "x = Symbol('x')\ne = sqrt(1 + sqrt(1 + x))", + "x = Symbol('x')\ne = sqrt(sqrt(x + 1) + 1)"] + + # Nesting of powers + assert python((((x + 1)**Rational(1, 3)) + 1)**Rational(1, 3)) in [ + "x = Symbol('x')\ne = (1 + (1 + x)**Rational(1, 3))**Rational(1, 3)", + "x = Symbol('x')\ne = ((x + 1)**Rational(1, 3) + 1)**Rational(1, 3)"] + + # Function powers + assert python(sin(x)**2) == "x = Symbol('x')\ne = sin(x)**2" + + +@XFAIL +def test_python_functions_conjugates(): + a, b = map(Symbol, 'ab') + assert python( conjugate(a + b*I) ) == '_ _\na - I*b' + assert python( conjugate(exp(a + b*I)) ) == ' _ _\n a - I*b\ne ' + + +def test_python_derivatives(): + # Simple + f_1 = Derivative(log(x), x, evaluate=False) + assert python(f_1) == "x = Symbol('x')\ne = Derivative(log(x), x)" + + f_2 = Derivative(log(x), x, evaluate=False) + x + assert python(f_2) == "x = Symbol('x')\ne = x + Derivative(log(x), x)" + + # Multiple symbols + f_3 = Derivative(log(x) + x**2, x, y, evaluate=False) + assert python(f_3) == \ + "x = Symbol('x')\ny = Symbol('y')\ne = Derivative(x**2 + log(x), x, y)" + + f_4 = Derivative(2*x*y, y, x, evaluate=False) + x**2 + assert python(f_4) in [ + "x = Symbol('x')\ny = Symbol('y')\ne = x**2 + Derivative(2*x*y, y, x)", + "x = Symbol('x')\ny = Symbol('y')\ne = Derivative(2*x*y, y, x) + x**2"] + + +def test_python_integrals(): + # Simple + f_1 = Integral(log(x), x) + assert python(f_1) == "x = Symbol('x')\ne = Integral(log(x), x)" + + f_2 = Integral(x**2, x) + assert python(f_2) == "x = Symbol('x')\ne = Integral(x**2, x)" + + # Double nesting of pow + f_3 = Integral(x**(2**x), x) + assert python(f_3) == "x = Symbol('x')\ne = Integral(x**(2**x), x)" + + # Definite integrals + f_4 = Integral(x**2, (x, 1, 2)) + assert python(f_4) == "x = Symbol('x')\ne = Integral(x**2, (x, 1, 2))" + + f_5 = Integral(x**2, (x, Rational(1, 2), 10)) + assert python( + f_5) == "x = Symbol('x')\ne = Integral(x**2, (x, Rational(1, 2), 10))" + + # Nested integrals + f_6 = Integral(x**2*y**2, x, y) + assert python(f_6) == "x = Symbol('x')\ny = Symbol('y')\ne = Integral(x**2*y**2, x, y)" + + +def test_python_matrix(): + p = python(Matrix([[x**2+1, 1], [y, x+y]])) + s = "x = Symbol('x')\ny = Symbol('y')\ne = MutableDenseMatrix([[x**2 + 1, 1], [y, x + y]])" + assert p == s + +def test_python_limits(): + assert python(limit(x, x, oo)) == 'e = oo' + assert python(limit(x**2, x, 0)) == 'e = 0' + +def test_issue_20762(): + # Make sure Python removes curly braces from subscripted variables + a_b = Symbol('a_{b}') + b = Symbol('b') + expr = a_b*b + assert python(expr) == "a_b = Symbol('a_{b}')\nb = Symbol('b')\ne = a_b*b" + + +def test_settings(): + raises(TypeError, lambda: python(x, method="garbage")) diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_rcode.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_rcode.py new file mode 100644 index 0000000000000000000000000000000000000000..a83235b0654c6bf24c30846dbf68678d29cd3c80 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_rcode.py @@ -0,0 +1,476 @@ +from sympy.core import (S, pi, oo, Symbol, symbols, Rational, Integer, + GoldenRatio, EulerGamma, Catalan, Lambda, Dummy) +from sympy.functions import (Piecewise, sin, cos, Abs, exp, ceiling, sqrt, + gamma, sign, Max, Min, factorial, beta) +from sympy.core.relational import (Eq, Ge, Gt, Le, Lt, Ne) +from sympy.sets import Range +from sympy.logic import ITE +from sympy.codegen import For, aug_assign, Assignment +from sympy.testing.pytest import raises +from sympy.printing.rcode import RCodePrinter +from sympy.utilities.lambdify import implemented_function +from sympy.tensor import IndexedBase, Idx +from sympy.matrices import Matrix, MatrixSymbol + +from sympy.printing.rcode import rcode + +x, y, z = symbols('x,y,z') + + +def test_printmethod(): + class fabs(Abs): + def _rcode(self, printer): + return "abs(%s)" % printer._print(self.args[0]) + + assert rcode(fabs(x)) == "abs(x)" + + +def test_rcode_sqrt(): + assert rcode(sqrt(x)) == "sqrt(x)" + assert rcode(x**0.5) == "sqrt(x)" + assert rcode(sqrt(x)) == "sqrt(x)" + + +def test_rcode_Pow(): + assert rcode(x**3) == "x^3" + assert rcode(x**(y**3)) == "x^(y^3)" + g = implemented_function('g', Lambda(x, 2*x)) + assert rcode(1/(g(x)*3.5)**(x - y**x)/(x**2 + y)) == \ + "(3.5*2*x)^(-x + y^x)/(x^2 + y)" + assert rcode(x**-1.0) == '1.0/x' + assert rcode(x**Rational(2, 3)) == 'x^(2.0/3.0)' + _cond_cfunc = [(lambda base, exp: exp.is_integer, "dpowi"), + (lambda base, exp: not exp.is_integer, "pow")] + assert rcode(x**3, user_functions={'Pow': _cond_cfunc}) == 'dpowi(x, 3)' + assert rcode(x**3.2, user_functions={'Pow': _cond_cfunc}) == 'pow(x, 3.2)' + + +def test_rcode_Max(): + # Test for gh-11926 + assert rcode(Max(x,x*x),user_functions={"Max":"my_max", "Pow":"my_pow"}) == 'my_max(x, my_pow(x, 2))' + + +def test_rcode_constants_mathh(): + assert rcode(exp(1)) == "exp(1)" + assert rcode(pi) == "pi" + assert rcode(oo) == "Inf" + assert rcode(-oo) == "-Inf" + + +def test_rcode_constants_other(): + assert rcode(2*GoldenRatio) == "GoldenRatio = 1.61803398874989;\n2*GoldenRatio" + assert rcode( + 2*Catalan) == "Catalan = 0.915965594177219;\n2*Catalan" + assert rcode(2*EulerGamma) == "EulerGamma = 0.577215664901533;\n2*EulerGamma" + + +def test_rcode_Rational(): + assert rcode(Rational(3, 7)) == "3.0/7.0" + assert rcode(Rational(18, 9)) == "2" + assert rcode(Rational(3, -7)) == "-3.0/7.0" + assert rcode(Rational(-3, -7)) == "3.0/7.0" + assert rcode(x + Rational(3, 7)) == "x + 3.0/7.0" + assert rcode(Rational(3, 7)*x) == "(3.0/7.0)*x" + + +def test_rcode_Integer(): + assert rcode(Integer(67)) == "67" + assert rcode(Integer(-1)) == "-1" + + +def test_rcode_functions(): + assert rcode(sin(x) ** cos(x)) == "sin(x)^cos(x)" + assert rcode(factorial(x) + gamma(y)) == "factorial(x) + gamma(y)" + assert rcode(beta(Min(x, y), Max(x, y))) == "beta(min(x, y), max(x, y))" + + +def test_rcode_inline_function(): + x = symbols('x') + g = implemented_function('g', Lambda(x, 2*x)) + assert rcode(g(x)) == "2*x" + g = implemented_function('g', Lambda(x, 2*x/Catalan)) + assert rcode( + g(x)) == "Catalan = %s;\n2*x/Catalan" % Catalan.n() + A = IndexedBase('A') + i = Idx('i', symbols('n', integer=True)) + g = implemented_function('g', Lambda(x, x*(1 + x)*(2 + x))) + res=rcode(g(A[i]), assign_to=A[i]) + ref=( + "for (i in 1:n){\n" + " A[i] = (A[i] + 1)*(A[i] + 2)*A[i];\n" + "}" + ) + assert res == ref + + +def test_rcode_exceptions(): + assert rcode(ceiling(x)) == "ceiling(x)" + assert rcode(Abs(x)) == "abs(x)" + assert rcode(gamma(x)) == "gamma(x)" + + +def test_rcode_user_functions(): + x = symbols('x', integer=False) + n = symbols('n', integer=True) + custom_functions = { + "ceiling": "myceil", + "Abs": [(lambda x: not x.is_integer, "fabs"), (lambda x: x.is_integer, "abs")], + } + assert rcode(ceiling(x), user_functions=custom_functions) == "myceil(x)" + assert rcode(Abs(x), user_functions=custom_functions) == "fabs(x)" + assert rcode(Abs(n), user_functions=custom_functions) == "abs(n)" + + +def test_rcode_boolean(): + assert rcode(True) == "True" + assert rcode(S.true) == "True" + assert rcode(False) == "False" + assert rcode(S.false) == "False" + assert rcode(x & y) == "x & y" + assert rcode(x | y) == "x | y" + assert rcode(~x) == "!x" + assert rcode(x & y & z) == "x & y & z" + assert rcode(x | y | z) == "x | y | z" + assert rcode((x & y) | z) == "z | x & y" + assert rcode((x | y) & z) == "z & (x | y)" + +def test_rcode_Relational(): + assert rcode(Eq(x, y)) == "x == y" + assert rcode(Ne(x, y)) == "x != y" + assert rcode(Le(x, y)) == "x <= y" + assert rcode(Lt(x, y)) == "x < y" + assert rcode(Gt(x, y)) == "x > y" + assert rcode(Ge(x, y)) == "x >= y" + + +def test_rcode_Piecewise(): + expr = Piecewise((x, x < 1), (x**2, True)) + res=rcode(expr) + ref="ifelse(x < 1,x,x^2)" + assert res == ref + tau=Symbol("tau") + res=rcode(expr,tau) + ref="tau = ifelse(x < 1,x,x^2);" + assert res == ref + + expr = 2*Piecewise((x, x < 1), (x**2, x<2), (x**3,True)) + assert rcode(expr) == "2*ifelse(x < 1,x,ifelse(x < 2,x^2,x^3))" + res = rcode(expr, assign_to='c') + assert res == "c = 2*ifelse(x < 1,x,ifelse(x < 2,x^2,x^3));" + + # Check that Piecewise without a True (default) condition error + #expr = Piecewise((x, x < 1), (x**2, x > 1), (sin(x), x > 0)) + #raises(ValueError, lambda: rcode(expr)) + expr = 2*Piecewise((x, x < 1), (x**2, x<2)) + assert(rcode(expr))== "2*ifelse(x < 1,x,ifelse(x < 2,x^2,NA))" + + +def test_rcode_sinc(): + from sympy.functions.elementary.trigonometric import sinc + expr = sinc(x) + res = rcode(expr) + ref = "(ifelse(x != 0,sin(x)/x,1))" + assert res == ref + + +def test_rcode_Piecewise_deep(): + p = rcode(2*Piecewise((x, x < 1), (x + 1, x < 2), (x**2, True))) + assert p == "2*ifelse(x < 1,x,ifelse(x < 2,x + 1,x^2))" + expr = x*y*z + x**2 + y**2 + Piecewise((0, x < 0.5), (1, True)) + cos(z) - 1 + p = rcode(expr) + ref="x^2 + x*y*z + y^2 + ifelse(x < 0.5,0,1) + cos(z) - 1" + assert p == ref + + ref="c = x^2 + x*y*z + y^2 + ifelse(x < 0.5,0,1) + cos(z) - 1;" + p = rcode(expr, assign_to='c') + assert p == ref + + +def test_rcode_ITE(): + expr = ITE(x < 1, y, z) + p = rcode(expr) + ref="ifelse(x < 1,y,z)" + assert p == ref + + +def test_rcode_settings(): + raises(TypeError, lambda: rcode(sin(x), method="garbage")) + + +def test_rcode_Indexed(): + n, m, o = symbols('n m o', integer=True) + i, j, k = Idx('i', n), Idx('j', m), Idx('k', o) + p = RCodePrinter() + p._not_r = set() + + x = IndexedBase('x')[j] + assert p._print_Indexed(x) == 'x[j]' + A = IndexedBase('A')[i, j] + assert p._print_Indexed(A) == 'A[i, j]' + B = IndexedBase('B')[i, j, k] + assert p._print_Indexed(B) == 'B[i, j, k]' + + assert p._not_r == set() + +def test_rcode_Indexed_without_looking_for_contraction(): + len_y = 5 + y = IndexedBase('y', shape=(len_y,)) + x = IndexedBase('x', shape=(len_y,)) + Dy = IndexedBase('Dy', shape=(len_y-1,)) + i = Idx('i', len_y-1) + e=Eq(Dy[i], (y[i+1]-y[i])/(x[i+1]-x[i])) + code0 = rcode(e.rhs, assign_to=e.lhs, contract=False) + assert code0 == 'Dy[i] = (y[%s] - y[i])/(x[%s] - x[i]);' % (i + 1, i + 1) + + +def test_rcode_loops_matrix_vector(): + n, m = symbols('n m', integer=True) + A = IndexedBase('A') + x = IndexedBase('x') + y = IndexedBase('y') + i = Idx('i', m) + j = Idx('j', n) + + s = ( + 'for (i in 1:m){\n' + ' y[i] = 0;\n' + '}\n' + 'for (i in 1:m){\n' + ' for (j in 1:n){\n' + ' y[i] = A[i, j]*x[j] + y[i];\n' + ' }\n' + '}' + ) + c = rcode(A[i, j]*x[j], assign_to=y[i]) + assert c == s + + +def test_dummy_loops(): + # the following line could also be + # [Dummy(s, integer=True) for s in 'im'] + # or [Dummy(integer=True) for s in 'im'] + i, m = symbols('i m', integer=True, cls=Dummy) + x = IndexedBase('x') + y = IndexedBase('y') + i = Idx(i, m) + + expected = ( + 'for (i_%(icount)i in 1:m_%(mcount)i){\n' + ' y[i_%(icount)i] = x[i_%(icount)i];\n' + '}' + ) % {'icount': i.label.dummy_index, 'mcount': m.dummy_index} + code = rcode(x[i], assign_to=y[i]) + assert code == expected + + +def test_rcode_loops_add(): + n, m = symbols('n m', integer=True) + A = IndexedBase('A') + x = IndexedBase('x') + y = IndexedBase('y') + z = IndexedBase('z') + i = Idx('i', m) + j = Idx('j', n) + + s = ( + 'for (i in 1:m){\n' + ' y[i] = x[i] + z[i];\n' + '}\n' + 'for (i in 1:m){\n' + ' for (j in 1:n){\n' + ' y[i] = A[i, j]*x[j] + y[i];\n' + ' }\n' + '}' + ) + c = rcode(A[i, j]*x[j] + x[i] + z[i], assign_to=y[i]) + assert c == s + + +def test_rcode_loops_multiple_contractions(): + n, m, o, p = symbols('n m o p', integer=True) + a = IndexedBase('a') + b = IndexedBase('b') + y = IndexedBase('y') + i = Idx('i', m) + j = Idx('j', n) + k = Idx('k', o) + l = Idx('l', p) + + s = ( + 'for (i in 1:m){\n' + ' y[i] = 0;\n' + '}\n' + 'for (i in 1:m){\n' + ' for (j in 1:n){\n' + ' for (k in 1:o){\n' + ' for (l in 1:p){\n' + ' y[i] = a[i, j, k, l]*b[j, k, l] + y[i];\n' + ' }\n' + ' }\n' + ' }\n' + '}' + ) + c = rcode(b[j, k, l]*a[i, j, k, l], assign_to=y[i]) + assert c == s + + +def test_rcode_loops_addfactor(): + n, m, o, p = symbols('n m o p', integer=True) + a = IndexedBase('a') + b = IndexedBase('b') + c = IndexedBase('c') + y = IndexedBase('y') + i = Idx('i', m) + j = Idx('j', n) + k = Idx('k', o) + l = Idx('l', p) + + s = ( + 'for (i in 1:m){\n' + ' y[i] = 0;\n' + '}\n' + 'for (i in 1:m){\n' + ' for (j in 1:n){\n' + ' for (k in 1:o){\n' + ' for (l in 1:p){\n' + ' y[i] = (a[i, j, k, l] + b[i, j, k, l])*c[j, k, l] + y[i];\n' + ' }\n' + ' }\n' + ' }\n' + '}' + ) + c = rcode((a[i, j, k, l] + b[i, j, k, l])*c[j, k, l], assign_to=y[i]) + assert c == s + + +def test_rcode_loops_multiple_terms(): + n, m, o, p = symbols('n m o p', integer=True) + a = IndexedBase('a') + b = IndexedBase('b') + c = IndexedBase('c') + y = IndexedBase('y') + i = Idx('i', m) + j = Idx('j', n) + k = Idx('k', o) + + s0 = ( + 'for (i in 1:m){\n' + ' y[i] = 0;\n' + '}\n' + ) + s1 = ( + 'for (i in 1:m){\n' + ' for (j in 1:n){\n' + ' for (k in 1:o){\n' + ' y[i] = b[j]*b[k]*c[i, j, k] + y[i];\n' + ' }\n' + ' }\n' + '}\n' + ) + s2 = ( + 'for (i in 1:m){\n' + ' for (k in 1:o){\n' + ' y[i] = a[i, k]*b[k] + y[i];\n' + ' }\n' + '}\n' + ) + s3 = ( + 'for (i in 1:m){\n' + ' for (j in 1:n){\n' + ' y[i] = a[i, j]*b[j] + y[i];\n' + ' }\n' + '}\n' + ) + c = rcode( + b[j]*a[i, j] + b[k]*a[i, k] + b[j]*b[k]*c[i, j, k], assign_to=y[i]) + + ref={} + ref[0] = s0 + s1 + s2 + s3[:-1] + ref[1] = s0 + s1 + s3 + s2[:-1] + ref[2] = s0 + s2 + s1 + s3[:-1] + ref[3] = s0 + s2 + s3 + s1[:-1] + ref[4] = s0 + s3 + s1 + s2[:-1] + ref[5] = s0 + s3 + s2 + s1[:-1] + + assert (c == ref[0] or + c == ref[1] or + c == ref[2] or + c == ref[3] or + c == ref[4] or + c == ref[5]) + + +def test_dereference_printing(): + expr = x + y + sin(z) + z + assert rcode(expr, dereference=[z]) == "x + y + (*z) + sin((*z))" + + +def test_Matrix_printing(): + # Test returning a Matrix + mat = Matrix([x*y, Piecewise((2 + x, y>0), (y, True)), sin(z)]) + A = MatrixSymbol('A', 3, 1) + p = rcode(mat, A) + assert p == ( + "A[0] = x*y;\n" + "A[1] = ifelse(y > 0,x + 2,y);\n" + "A[2] = sin(z);") + # Test using MatrixElements in expressions + expr = Piecewise((2*A[2, 0], x > 0), (A[2, 0], True)) + sin(A[1, 0]) + A[0, 0] + p = rcode(expr) + assert p == ("ifelse(x > 0,2*A[2],A[2]) + sin(A[1]) + A[0]") + # Test using MatrixElements in a Matrix + q = MatrixSymbol('q', 5, 1) + M = MatrixSymbol('M', 3, 3) + m = Matrix([[sin(q[1,0]), 0, cos(q[2,0])], + [q[1,0] + q[2,0], q[3, 0], 5], + [2*q[4, 0]/q[1,0], sqrt(q[0,0]) + 4, 0]]) + assert rcode(m, M) == ( + "M[0] = sin(q[1]);\n" + "M[1] = 0;\n" + "M[2] = cos(q[2]);\n" + "M[3] = q[1] + q[2];\n" + "M[4] = q[3];\n" + "M[5] = 5;\n" + "M[6] = 2*q[4]/q[1];\n" + "M[7] = sqrt(q[0]) + 4;\n" + "M[8] = 0;") + + +def test_rcode_sgn(): + + expr = sign(x) * y + assert rcode(expr) == 'y*sign(x)' + p = rcode(expr, 'z') + assert p == 'z = y*sign(x);' + + p = rcode(sign(2 * x + x**2) * x + x**2) + assert p == "x^2 + x*sign(x^2 + 2*x)" + + expr = sign(cos(x)) + p = rcode(expr) + assert p == 'sign(cos(x))' + +def test_rcode_Assignment(): + assert rcode(Assignment(x, y + z)) == 'x = y + z;' + assert rcode(aug_assign(x, '+', y + z)) == 'x += y + z;' + + +def test_rcode_For(): + f = For(x, Range(0, 10, 2), [aug_assign(y, '*', x)]) + sol = rcode(f) + assert sol == ("for(x in seq(from=0, to=9, by=2){\n" + " y *= x;\n" + "}") + + +def test_MatrixElement_printing(): + # test cases for issue #11821 + A = MatrixSymbol("A", 1, 3) + B = MatrixSymbol("B", 1, 3) + C = MatrixSymbol("C", 1, 3) + + assert(rcode(A[0, 0]) == "A[0]") + assert(rcode(3 * A[0, 0]) == "3*A[0]") + + F = C[0, 0].subs(C, A - B) + assert(rcode(F) == "(A - B)[0]") diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_repr.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_repr.py new file mode 100644 index 0000000000000000000000000000000000000000..da58883b4fb027ed82db842a0a1ce5f76a49a8bb --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_repr.py @@ -0,0 +1,382 @@ +from __future__ import annotations +from typing import Any + +from sympy.external.gmpy import GROUND_TYPES +from sympy.testing.pytest import raises, warns_deprecated_sympy +from sympy.assumptions.ask import Q +from sympy.core.function import (Function, WildFunction) +from sympy.core.numbers import (AlgebraicNumber, Float, Integer, Rational) +from sympy.core.singleton import S +from sympy.core.symbol import (Dummy, Symbol, Wild, symbols) +from sympy.core.sympify import sympify +from sympy.functions.elementary.complexes import Abs +from sympy.functions.elementary.miscellaneous import (root, sqrt) +from sympy.functions.elementary.trigonometric import sin +from sympy.functions.special.delta_functions import Heaviside +from sympy.logic.boolalg import (false, true) +from sympy.matrices.dense import (Matrix, ones) +from sympy.matrices.expressions.matexpr import MatrixSymbol +from sympy.matrices.immutable import ImmutableDenseMatrix +from sympy.combinatorics import Cycle, Permutation +from sympy.core.symbol import Str +from sympy.geometry import Point, Ellipse +from sympy.printing import srepr +from sympy.polys import ring, field, ZZ, QQ, lex, grlex, Poly +from sympy.polys.polyclasses import DMP +from sympy.polys.agca.extensions import FiniteExtension + +x, y = symbols('x,y') + +# eval(srepr(expr)) == expr has to succeed in the right environment. The right +# environment is the scope of "from sympy import *" for most cases. +ENV: dict[str, Any] = {"Str": Str} +exec("from sympy import *", ENV) + + +def sT(expr, string, import_stmt=None, **kwargs): + """ + sT := sreprTest + + Tests that srepr delivers the expected string and that + the condition eval(srepr(expr))==expr holds. + """ + if import_stmt is None: + ENV2 = ENV + else: + ENV2 = ENV.copy() + exec(import_stmt, ENV2) + + assert srepr(expr, **kwargs) == string + assert eval(string, ENV2) == expr + + +def test_printmethod(): + class R(Abs): + def _sympyrepr(self, printer): + return "foo(%s)" % printer._print(self.args[0]) + assert srepr(R(x)) == "foo(Symbol('x'))" + + +def test_Add(): + sT(x + y, "Add(Symbol('x'), Symbol('y'))") + assert srepr(x**2 + 1, order='lex') == "Add(Pow(Symbol('x'), Integer(2)), Integer(1))" + assert srepr(x**2 + 1, order='old') == "Add(Integer(1), Pow(Symbol('x'), Integer(2)))" + assert srepr(sympify('x + 3 - 2', evaluate=False), order='none') == "Add(Symbol('x'), Integer(3), Mul(Integer(-1), Integer(2)))" + + +def test_more_than_255_args_issue_10259(): + from sympy.core.add import Add + from sympy.core.mul import Mul + for op in (Add, Mul): + expr = op(*symbols('x:256')) + assert eval(srepr(expr)) == expr + + +def test_Function(): + sT(Function("f")(x), "Function('f')(Symbol('x'))") + # test unapplied Function + sT(Function('f'), "Function('f')") + + sT(sin(x), "sin(Symbol('x'))") + sT(sin, "sin") + + +def test_Heaviside(): + sT(Heaviside(x), "Heaviside(Symbol('x'))") + sT(Heaviside(x, 1), "Heaviside(Symbol('x'), Integer(1))") + + +def test_Geometry(): + sT(Point(0, 0), "Point2D(Integer(0), Integer(0))") + sT(Ellipse(Point(0, 0), 5, 1), + "Ellipse(Point2D(Integer(0), Integer(0)), Integer(5), Integer(1))") + # TODO more tests + + +def test_Singletons(): + sT(S.Catalan, 'Catalan') + sT(S.ComplexInfinity, 'zoo') + sT(S.EulerGamma, 'EulerGamma') + sT(S.Exp1, 'E') + sT(S.GoldenRatio, 'GoldenRatio') + sT(S.TribonacciConstant, 'TribonacciConstant') + sT(S.Half, 'Rational(1, 2)') + sT(S.ImaginaryUnit, 'I') + sT(S.Infinity, 'oo') + sT(S.NaN, 'nan') + sT(S.NegativeInfinity, '-oo') + sT(S.NegativeOne, 'Integer(-1)') + sT(S.One, 'Integer(1)') + sT(S.Pi, 'pi') + sT(S.Zero, 'Integer(0)') + sT(S.Complexes, 'Complexes') + sT(S.EmptySequence, 'EmptySequence') + sT(S.EmptySet, 'EmptySet') + # sT(S.IdentityFunction, 'Lambda(_x, _x)') + sT(S.Naturals, 'Naturals') + sT(S.Naturals0, 'Naturals0') + sT(S.Rationals, 'Rationals') + sT(S.Reals, 'Reals') + sT(S.UniversalSet, 'UniversalSet') + + +def test_Integer(): + sT(Integer(4), "Integer(4)") + + +def test_list(): + sT([x, Integer(4)], "[Symbol('x'), Integer(4)]") + + +def test_Matrix(): + for cls, name in [(Matrix, "MutableDenseMatrix"), (ImmutableDenseMatrix, "ImmutableDenseMatrix")]: + sT(cls([[x**+1, 1], [y, x + y]]), + "%s([[Symbol('x'), Integer(1)], [Symbol('y'), Add(Symbol('x'), Symbol('y'))]])" % name) + + sT(cls(), "%s([])" % name) + + sT(cls([[x**+1, 1], [y, x + y]]), "%s([[Symbol('x'), Integer(1)], [Symbol('y'), Add(Symbol('x'), Symbol('y'))]])" % name) + + +def test_empty_Matrix(): + sT(ones(0, 3), "MutableDenseMatrix(0, 3, [])") + sT(ones(4, 0), "MutableDenseMatrix(4, 0, [])") + sT(ones(0, 0), "MutableDenseMatrix([])") + + +def test_Rational(): + sT(Rational(1, 3), "Rational(1, 3)") + sT(Rational(-1, 3), "Rational(-1, 3)") + + +def test_Float(): + sT(Float('1.23', dps=3), "Float('1.22998', precision=13)") + sT(Float('1.23456789', dps=9), "Float('1.23456788994', precision=33)") + sT(Float('1.234567890123456789', dps=19), + "Float('1.234567890123456789013', precision=66)") + sT(Float('0.60038617995049726', dps=15), + "Float('0.60038617995049726', precision=53)") + + sT(Float('1.23', precision=13), "Float('1.22998', precision=13)") + sT(Float('1.23456789', precision=33), + "Float('1.23456788994', precision=33)") + sT(Float('1.234567890123456789', precision=66), + "Float('1.234567890123456789013', precision=66)") + sT(Float('0.60038617995049726', precision=53), + "Float('0.60038617995049726', precision=53)") + + sT(Float('0.60038617995049726', 15), + "Float('0.60038617995049726', precision=53)") + + +def test_Symbol(): + sT(x, "Symbol('x')") + sT(y, "Symbol('y')") + sT(Symbol('x', negative=True), "Symbol('x', negative=True)") + + +def test_Symbol_two_assumptions(): + x = Symbol('x', negative=0, integer=1) + # order could vary + s1 = "Symbol('x', integer=True, negative=False)" + s2 = "Symbol('x', negative=False, integer=True)" + assert srepr(x) in (s1, s2) + assert eval(srepr(x), ENV) == x + + +def test_Symbol_no_special_commutative_treatment(): + sT(Symbol('x'), "Symbol('x')") + sT(Symbol('x', commutative=False), "Symbol('x', commutative=False)") + sT(Symbol('x', commutative=0), "Symbol('x', commutative=False)") + sT(Symbol('x', commutative=True), "Symbol('x', commutative=True)") + sT(Symbol('x', commutative=1), "Symbol('x', commutative=True)") + + +def test_Wild(): + sT(Wild('x', even=True), "Wild('x', even=True)") + + +def test_Dummy(): + d = Dummy('d') + sT(d, "Dummy('d', dummy_index=%s)" % str(d.dummy_index)) + + +def test_Dummy_assumption(): + d = Dummy('d', nonzero=True) + assert d == eval(srepr(d)) + s1 = "Dummy('d', dummy_index=%s, nonzero=True)" % str(d.dummy_index) + s2 = "Dummy('d', nonzero=True, dummy_index=%s)" % str(d.dummy_index) + assert srepr(d) in (s1, s2) + + +def test_Dummy_from_Symbol(): + # should not get the full dictionary of assumptions + n = Symbol('n', integer=True) + d = n.as_dummy() + assert srepr(d + ) == "Dummy('n', dummy_index=%s)" % str(d.dummy_index) + + +def test_tuple(): + sT((x,), "(Symbol('x'),)") + sT((x, y), "(Symbol('x'), Symbol('y'))") + + +def test_WildFunction(): + sT(WildFunction('w'), "WildFunction('w')") + + +def test_settins(): + raises(TypeError, lambda: srepr(x, method="garbage")) + + +def test_Mul(): + sT(3*x**3*y, "Mul(Integer(3), Pow(Symbol('x'), Integer(3)), Symbol('y'))") + assert srepr(3*x**3*y, order='old') == "Mul(Integer(3), Symbol('y'), Pow(Symbol('x'), Integer(3)))" + assert srepr(sympify('(x+4)*2*x*7', evaluate=False), order='none') == "Mul(Add(Symbol('x'), Integer(4)), Integer(2), Symbol('x'), Integer(7))" + + +def test_AlgebraicNumber(): + a = AlgebraicNumber(sqrt(2)) + sT(a, "AlgebraicNumber(Pow(Integer(2), Rational(1, 2)), [Integer(1), Integer(0)])") + a = AlgebraicNumber(root(-2, 3)) + sT(a, "AlgebraicNumber(Pow(Integer(-2), Rational(1, 3)), [Integer(1), Integer(0)])") + + +def test_PolyRing(): + assert srepr(ring("x", ZZ, lex)[0]) == "PolyRing((Symbol('x'),), ZZ, lex)" + assert srepr(ring("x,y", QQ, grlex)[0]) == "PolyRing((Symbol('x'), Symbol('y')), QQ, grlex)" + assert srepr(ring("x,y,z", ZZ["t"], lex)[0]) == "PolyRing((Symbol('x'), Symbol('y'), Symbol('z')), ZZ[t], lex)" + + +def test_FracField(): + assert srepr(field("x", ZZ, lex)[0]) == "FracField((Symbol('x'),), ZZ, lex)" + assert srepr(field("x,y", QQ, grlex)[0]) == "FracField((Symbol('x'), Symbol('y')), QQ, grlex)" + assert srepr(field("x,y,z", ZZ["t"], lex)[0]) == "FracField((Symbol('x'), Symbol('y'), Symbol('z')), ZZ[t], lex)" + + +def test_PolyElement(): + R, x, y = ring("x,y", ZZ) + assert srepr(3*x**2*y + 1) == "PolyElement(PolyRing((Symbol('x'), Symbol('y')), ZZ, lex), [((2, 1), 3), ((0, 0), 1)])" + + +def test_FracElement(): + F, x, y = field("x,y", ZZ) + assert srepr((3*x**2*y + 1)/(x - y**2)) == "FracElement(FracField((Symbol('x'), Symbol('y')), ZZ, lex), [((2, 1), 3), ((0, 0), 1)], [((1, 0), 1), ((0, 2), -1)])" + + +def test_FractionField(): + assert srepr(QQ.frac_field(x)) == \ + "FractionField(FracField((Symbol('x'),), QQ, lex))" + assert srepr(QQ.frac_field(x, y, order=grlex)) == \ + "FractionField(FracField((Symbol('x'), Symbol('y')), QQ, grlex))" + + +def test_PolynomialRingBase(): + assert srepr(ZZ.old_poly_ring(x)) == \ + "GlobalPolynomialRing(ZZ, Symbol('x'))" + assert srepr(ZZ[x].old_poly_ring(y)) == \ + "GlobalPolynomialRing(ZZ[x], Symbol('y'))" + assert srepr(QQ.frac_field(x).old_poly_ring(y)) == \ + "GlobalPolynomialRing(FractionField(FracField((Symbol('x'),), QQ, lex)), Symbol('y'))" + + +def test_DMP(): + p1 = DMP([1, 2], ZZ) + p2 = ZZ.old_poly_ring(x)([1, 2]) + if GROUND_TYPES != 'flint': + assert srepr(p1) == "DMP_Python([1, 2], ZZ)" + assert srepr(p2) == "DMP_Python([1, 2], ZZ)" + else: + assert srepr(p1) == "DUP_Flint([1, 2], ZZ)" + assert srepr(p2) == "DUP_Flint([1, 2], ZZ)" + + +def test_FiniteExtension(): + assert srepr(FiniteExtension(Poly(x**2 + 1, x))) == \ + "FiniteExtension(Poly(x**2 + 1, x, domain='ZZ'))" + + +def test_ExtensionElement(): + A = FiniteExtension(Poly(x**2 + 1, x)) + if GROUND_TYPES != 'flint': + ans = "ExtElem(DMP_Python([1, 0], ZZ), FiniteExtension(Poly(x**2 + 1, x, domain='ZZ')))" + else: + ans = "ExtElem(DUP_Flint([1, 0], ZZ), FiniteExtension(Poly(x**2 + 1, x, domain='ZZ')))" + assert srepr(A.generator) == ans + +def test_BooleanAtom(): + assert srepr(true) == "true" + assert srepr(false) == "false" + + +def test_Integers(): + sT(S.Integers, "Integers") + + +def test_Naturals(): + sT(S.Naturals, "Naturals") + + +def test_Naturals0(): + sT(S.Naturals0, "Naturals0") + + +def test_Reals(): + sT(S.Reals, "Reals") + + +def test_matrix_expressions(): + n = symbols('n', integer=True) + A = MatrixSymbol("A", n, n) + B = MatrixSymbol("B", n, n) + sT(A, "MatrixSymbol(Str('A'), Symbol('n', integer=True), Symbol('n', integer=True))") + sT(A*B, "MatMul(MatrixSymbol(Str('A'), Symbol('n', integer=True), Symbol('n', integer=True)), MatrixSymbol(Str('B'), Symbol('n', integer=True), Symbol('n', integer=True)))") + sT(A + B, "MatAdd(MatrixSymbol(Str('A'), Symbol('n', integer=True), Symbol('n', integer=True)), MatrixSymbol(Str('B'), Symbol('n', integer=True), Symbol('n', integer=True)))") + + +def test_Cycle(): + # FIXME: sT fails because Cycle is not immutable and calling srepr(Cycle(1, 2)) + # adds keys to the Cycle dict (GH-17661) + #import_stmt = "from sympy.combinatorics import Cycle" + #sT(Cycle(1, 2), "Cycle(1, 2)", import_stmt) + assert srepr(Cycle(1, 2)) == "Cycle(1, 2)" + + +def test_Permutation(): + import_stmt = "from sympy.combinatorics import Permutation" + sT(Permutation(1, 2)(3, 4), "Permutation([0, 2, 1, 4, 3])", import_stmt, perm_cyclic=False) + sT(Permutation(1, 2)(3, 4), "Permutation(1, 2)(3, 4)", import_stmt, perm_cyclic=True) + + with warns_deprecated_sympy(): + old_print_cyclic = Permutation.print_cyclic + Permutation.print_cyclic = False + sT(Permutation(1, 2)(3, 4), "Permutation([0, 2, 1, 4, 3])", import_stmt) + Permutation.print_cyclic = old_print_cyclic + +def test_dict(): + from sympy.abc import x, y, z + d = {} + assert srepr(d) == "{}" + d = {x: y} + assert srepr(d) == "{Symbol('x'): Symbol('y')}" + d = {x: y, y: z} + assert srepr(d) in ( + "{Symbol('x'): Symbol('y'), Symbol('y'): Symbol('z')}", + "{Symbol('y'): Symbol('z'), Symbol('x'): Symbol('y')}", + ) + d = {x: {y: z}} + assert srepr(d) == "{Symbol('x'): {Symbol('y'): Symbol('z')}}" + +def test_set(): + from sympy.abc import x, y + s = set() + assert srepr(s) == "set()" + s = {x, y} + assert srepr(s) in ("{Symbol('x'), Symbol('y')}", "{Symbol('y'), Symbol('x')}") + +def test_Predicate(): + sT(Q.even, "Q.even") + +def test_AppliedPredicate(): + sT(Q.even(Symbol('z')), "AppliedPredicate(Q.even, Symbol('z'))") diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_rust.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_rust.py new file mode 100644 index 0000000000000000000000000000000000000000..c81d592faca0d4a31e5a9618a48d67cb19ca94d8 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_rust.py @@ -0,0 +1,363 @@ +from sympy.core import (S, pi, oo, symbols, Rational, Integer, + GoldenRatio, EulerGamma, Catalan, Lambda, Dummy, + Eq, Ne, Le, Lt, Gt, Ge, Mod) +from sympy.functions import (Piecewise, sin, cos, Abs, exp, ceiling, sqrt, + sign, floor) +from sympy.logic import ITE +from sympy.testing.pytest import raises +from sympy.utilities.lambdify import implemented_function +from sympy.tensor import IndexedBase, Idx +from sympy.matrices import MatrixSymbol, SparseMatrix, Matrix + +from sympy.printing.codeprinter import rust_code + +x, y, z = symbols('x,y,z', integer=False, real=True) +k, m, n = symbols('k,m,n', integer=True) + + +def test_Integer(): + assert rust_code(Integer(42)) == "42" + assert rust_code(Integer(-56)) == "-56" + + +def test_Relational(): + assert rust_code(Eq(x, y)) == "x == y" + assert rust_code(Ne(x, y)) == "x != y" + assert rust_code(Le(x, y)) == "x <= y" + assert rust_code(Lt(x, y)) == "x < y" + assert rust_code(Gt(x, y)) == "x > y" + assert rust_code(Ge(x, y)) == "x >= y" + + +def test_Rational(): + assert rust_code(Rational(3, 7)) == "3_f64/7.0" + assert rust_code(Rational(18, 9)) == "2" + assert rust_code(Rational(3, -7)) == "-3_f64/7.0" + assert rust_code(Rational(-3, -7)) == "3_f64/7.0" + assert rust_code(x + Rational(3, 7)) == "x + 3_f64/7.0" + assert rust_code(Rational(3, 7)*x) == "(3_f64/7.0)*x" + + +def test_basic_ops(): + assert rust_code(x + y) == "x + y" + assert rust_code(x - y) == "x - y" + assert rust_code(x * y) == "x*y" + assert rust_code(x / y) == "x*y.recip()" + assert rust_code(-x) == "-x" + assert rust_code(2 * x) == "2.0*x" + assert rust_code(y + 2) == "y + 2.0" + assert rust_code(x + n) == "n as f64 + x" + +def test_printmethod(): + class fabs(Abs): + def _rust_code(self, printer): + return "%s.fabs()" % printer._print(self.args[0]) + assert rust_code(fabs(x)) == "x.fabs()" + a = MatrixSymbol("a", 1, 3) + assert rust_code(a[0,0]) == 'a[0]' + + +def test_Functions(): + assert rust_code(sin(x) ** cos(x)) == "x.sin().powf(x.cos())" + assert rust_code(abs(x)) == "x.abs()" + assert rust_code(ceiling(x)) == "x.ceil()" + assert rust_code(floor(x)) == "x.floor()" + + # Automatic rewrite + assert rust_code(Mod(x, 3)) == 'x - 3.0*((1_f64/3.0)*x).floor()' + + +def test_Pow(): + assert rust_code(1/x) == "x.recip()" + assert rust_code(x**-1) == rust_code(x**-1.0) == "x.recip()" + assert rust_code(sqrt(x)) == "x.sqrt()" + assert rust_code(x**S.Half) == rust_code(x**0.5) == "x.sqrt()" + + assert rust_code(1/sqrt(x)) == "x.sqrt().recip()" + assert rust_code(x**-S.Half) == rust_code(x**-0.5) == "x.sqrt().recip()" + + assert rust_code(1/pi) == "PI.recip()" + assert rust_code(pi**-1) == rust_code(pi**-1.0) == "PI.recip()" + assert rust_code(pi**-0.5) == "PI.sqrt().recip()" + + assert rust_code(x**Rational(1, 3)) == "x.cbrt()" + assert rust_code(2**x) == "x.exp2()" + assert rust_code(exp(x)) == "x.exp()" + assert rust_code(x**3) == "x.powi(3)" + assert rust_code(x**(y**3)) == "x.powf(y.powi(3))" + assert rust_code(x**Rational(2, 3)) == "x.powf(2_f64/3.0)" + + g = implemented_function('g', Lambda(x, 2*x)) + assert rust_code(1/(g(x)*3.5)**(x - y**x)/(x**2 + y)) == \ + "(3.5*2.0*x).powf(-x + y.powf(x))/(x.powi(2) + y)" + _cond_cfunc = [(lambda base, exp: exp.is_integer, "dpowi", 1), + (lambda base, exp: not exp.is_integer, "pow", 1)] + assert rust_code(x**3, user_functions={'Pow': _cond_cfunc}) == 'x.dpowi(3)' + assert rust_code(x**3.2, user_functions={'Pow': _cond_cfunc}) == 'x.pow(3.2)' + + +def test_constants(): + assert rust_code(pi) == "PI" + assert rust_code(oo) == "INFINITY" + assert rust_code(S.Infinity) == "INFINITY" + assert rust_code(-oo) == "NEG_INFINITY" + assert rust_code(S.NegativeInfinity) == "NEG_INFINITY" + assert rust_code(S.NaN) == "NAN" + assert rust_code(exp(1)) == "E" + assert rust_code(S.Exp1) == "E" + + +def test_constants_other(): + assert rust_code(2*GoldenRatio) == "const GoldenRatio: f64 = %s;\n2.0*GoldenRatio" % GoldenRatio.evalf(17) + assert rust_code( + 2*Catalan) == "const Catalan: f64 = %s;\n2.0*Catalan" % Catalan.evalf(17) + assert rust_code(2*EulerGamma) == "const EulerGamma: f64 = %s;\n2.0*EulerGamma" % EulerGamma.evalf(17) + + +def test_boolean(): + assert rust_code(True) == "true" + assert rust_code(S.true) == "true" + assert rust_code(False) == "false" + assert rust_code(S.false) == "false" + assert rust_code(k & m) == "k && m" + assert rust_code(k | m) == "k || m" + assert rust_code(~k) == "!k" + assert rust_code(k & m & n) == "k && m && n" + assert rust_code(k | m | n) == "k || m || n" + assert rust_code((k & m) | n) == "n || k && m" + assert rust_code((k | m) & n) == "n && (k || m)" + + +def test_Piecewise(): + expr = Piecewise((x, x < 1), (x + 2, True)) + assert rust_code(expr) == ( + "if (x < 1.0) {\n" + " x\n" + "} else {\n" + " x + 2.0\n" + "}") + assert rust_code(expr, assign_to="r") == ( + "r = if (x < 1.0) {\n" + " x\n" + "} else {\n" + " x + 2.0\n" + "};") + assert rust_code(expr, assign_to="r", inline=True) == ( + "r = if (x < 1.0) { x } else { x + 2.0 };") + expr = Piecewise((x, x < 1), (x + 1, x < 5), (x + 2, True)) + assert rust_code(expr, inline=True) == ( + "if (x < 1.0) { x } else if (x < 5.0) { x + 1.0 } else { x + 2.0 }") + assert rust_code(expr, assign_to="r", inline=True) == ( + "r = if (x < 1.0) { x } else if (x < 5.0) { x + 1.0 } else { x + 2.0 };") + assert rust_code(expr, assign_to="r") == ( + "r = if (x < 1.0) {\n" + " x\n" + "} else if (x < 5.0) {\n" + " x + 1.0\n" + "} else {\n" + " x + 2.0\n" + "};") + expr = 2*Piecewise((x, x < 1), (x + 1, x < 5), (x + 2, True)) + assert rust_code(expr, inline=True) == ( + "2.0*if (x < 1.0) { x } else if (x < 5.0) { x + 1.0 } else { x + 2.0 }") + expr = 2*Piecewise((x, x < 1), (x + 1, x < 5), (x + 2, True)) - 42 + assert rust_code(expr, inline=True) == ( + "2.0*if (x < 1.0) { x } else if (x < 5.0) { x + 1.0 } else { x + 2.0 } - 42.0") + # Check that Piecewise without a True (default) condition error + expr = Piecewise((x, x < 1), (x**2, x > 1), (sin(x), x > 0)) + raises(ValueError, lambda: rust_code(expr)) + + +def test_dereference_printing(): + expr = x + y + sin(z) + z + assert rust_code(expr, dereference=[z]) == "x + y + (*z) + (*z).sin()" + + +def test_sign(): + expr = sign(x) * y + assert rust_code(expr) == "y*(if (x == 0.0) { 0.0 } else { (x).signum() }) as f64" + assert rust_code(expr, assign_to='r') == "r = y*(if (x == 0.0) { 0.0 } else { (x).signum() }) as f64;" + + expr = sign(x + y) + 42 + assert rust_code(expr) == "(if (x + y == 0.0) { 0.0 } else { (x + y).signum() }) + 42" + assert rust_code(expr, assign_to='r') == "r = (if (x + y == 0.0) { 0.0 } else { (x + y).signum() }) + 42;" + + expr = sign(cos(x)) + assert rust_code(expr) == "(if (x.cos() == 0.0) { 0.0 } else { (x.cos()).signum() })" + + +def test_reserved_words(): + + x, y = symbols("x if") + + expr = sin(y) + assert rust_code(expr) == "if_.sin()" + assert rust_code(expr, dereference=[y]) == "(*if_).sin()" + assert rust_code(expr, reserved_word_suffix='_unreserved') == "if_unreserved.sin()" + + with raises(ValueError): + rust_code(expr, error_on_reserved=True) + + +def test_ITE(): + ekpr = ITE(k < 1, m, n) + assert rust_code(ekpr) == ( + "if (k < 1) {\n" + " m\n" + "} else {\n" + " n\n" + "}") + + +def test_Indexed(): + n, m, o = symbols('n m o', integer=True) + i, j, k = Idx('i', n), Idx('j', m), Idx('k', o) + + x = IndexedBase('x')[j] + assert rust_code(x) == "x[j]" + + A = IndexedBase('A')[i, j] + assert rust_code(A) == "A[m*i + j]" + + B = IndexedBase('B')[i, j, k] + assert rust_code(B) == "B[m*o*i + o*j + k]" + + +def test_dummy_loops(): + i, m = symbols('i m', integer=True, cls=Dummy) + x = IndexedBase('x') + y = IndexedBase('y') + i = Idx(i, m) + + assert rust_code(x[i], assign_to=y[i]) == ( + "for i in 0..m {\n" + " y[i] = x[i];\n" + "}") + + +def test_loops(): + m, n = symbols('m n', integer=True) + A = IndexedBase('A') + x = IndexedBase('x') + y = IndexedBase('y') + z = IndexedBase('z') + i = Idx('i', m) + j = Idx('j', n) + + assert rust_code(A[i, j]*x[j], assign_to=y[i]) == ( + "for i in 0..m {\n" + " y[i] = 0;\n" + "}\n" + "for i in 0..m {\n" + " for j in 0..n {\n" + " y[i] = A[n*i + j]*x[j] + y[i];\n" + " }\n" + "}") + + assert rust_code(A[i, j]*x[j] + x[i] + z[i], assign_to=y[i]) == ( + "for i in 0..m {\n" + " y[i] = x[i] + z[i];\n" + "}\n" + "for i in 0..m {\n" + " for j in 0..n {\n" + " y[i] = A[n*i + j]*x[j] + y[i];\n" + " }\n" + "}") + + +def test_loops_multiple_contractions(): + n, m, o, p = symbols('n m o p', integer=True) + a = IndexedBase('a') + b = IndexedBase('b') + y = IndexedBase('y') + i = Idx('i', m) + j = Idx('j', n) + k = Idx('k', o) + l = Idx('l', p) + + assert rust_code(b[j, k, l]*a[i, j, k, l], assign_to=y[i]) == ( + "for i in 0..m {\n" + " y[i] = 0;\n" + "}\n" + "for i in 0..m {\n" + " for j in 0..n {\n" + " for k in 0..o {\n" + " for l in 0..p {\n" + " y[i] = a[%s]*b[%s] + y[i];\n" % (i*n*o*p + j*o*p + k*p + l, j*o*p + k*p + l) +\ + " }\n" + " }\n" + " }\n" + "}") + + +def test_loops_addfactor(): + m, n, o, p = symbols('m n o p', integer=True) + a = IndexedBase('a') + b = IndexedBase('b') + c = IndexedBase('c') + y = IndexedBase('y') + i = Idx('i', m) + j = Idx('j', n) + k = Idx('k', o) + l = Idx('l', p) + + code = rust_code((a[i, j, k, l] + b[i, j, k, l])*c[j, k, l], assign_to=y[i]) + assert code == ( + "for i in 0..m {\n" + " y[i] = 0;\n" + "}\n" + "for i in 0..m {\n" + " for j in 0..n {\n" + " for k in 0..o {\n" + " for l in 0..p {\n" + " y[i] = (a[%s] + b[%s])*c[%s] + y[i];\n" % (i*n*o*p + j*o*p + k*p + l, i*n*o*p + j*o*p + k*p + l, j*o*p + k*p + l) +\ + " }\n" + " }\n" + " }\n" + "}") + + +def test_settings(): + raises(TypeError, lambda: rust_code(sin(x), method="garbage")) + + +def test_inline_function(): + x = symbols('x') + g = implemented_function('g', Lambda(x, 2*x)) + assert rust_code(g(x)) == "2*x" + + g = implemented_function('g', Lambda(x, 2*x/Catalan)) + assert rust_code(g(x)) == ( + "const Catalan: f64 = %s;\n2.0*x/Catalan" % Catalan.evalf(17)) + + A = IndexedBase('A') + i = Idx('i', symbols('n', integer=True)) + g = implemented_function('g', Lambda(x, x*(1 + x)*(2 + x))) + assert rust_code(g(A[i]), assign_to=A[i]) == ( + "for i in 0..n {\n" + " A[i] = (A[i] + 1)*(A[i] + 2)*A[i];\n" + "}") + + +def test_user_functions(): + x = symbols('x', integer=False) + n = symbols('n', integer=True) + custom_functions = { + "ceiling": "ceil", + "Abs": [(lambda x: not x.is_integer, "fabs", 4), (lambda x: x.is_integer, "abs", 4)], + } + assert rust_code(ceiling(x), user_functions=custom_functions) == "x.ceil()" + assert rust_code(Abs(x), user_functions=custom_functions) == "fabs(x)" + assert rust_code(Abs(n), user_functions=custom_functions) == "abs(n)" + + +def test_matrix(): + assert rust_code(Matrix([1, 2, 3])) == '[1, 2, 3]' + with raises(ValueError): + rust_code(Matrix([[1, 2, 3]])) + + +def test_sparse_matrix(): + # gh-15791 + with raises(NotImplementedError): + rust_code(SparseMatrix([[1, 2, 3]])) diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_smtlib.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_smtlib.py new file mode 100644 index 0000000000000000000000000000000000000000..48ff3d432d9042bf178f4e52dc46c787059937a3 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_smtlib.py @@ -0,0 +1,553 @@ +import contextlib +import itertools +import re +import typing +from enum import Enum +from typing import Callable + +import sympy +from sympy import Add, Implies, sqrt +from sympy.core import Mul, Pow +from sympy.core import (S, pi, symbols, Function, Rational, Integer, + Symbol, Eq, Ne, Le, Lt, Gt, Ge) +from sympy.functions import Piecewise, exp, sin, cos +from sympy.assumptions.ask import Q +from sympy.printing.smtlib import smtlib_code +from sympy.testing.pytest import raises, Failed + +x, y, z = symbols('x,y,z') + + +class _W(Enum): + DEFAULTING_TO_FLOAT = re.compile("Could not infer type of `.+`. Defaulting to float.", re.IGNORECASE) + WILL_NOT_DECLARE = re.compile("Non-Symbol/Function `.+` will not be declared.", re.IGNORECASE) + WILL_NOT_ASSERT = re.compile("Non-Boolean expression `.+` will not be asserted. Converting to SMTLib verbatim.", re.IGNORECASE) + + +@contextlib.contextmanager +def _check_warns(expected: typing.Iterable[_W]): + warns: typing.List[str] = [] + log_warn = warns.append + yield log_warn + + errors = [] + for i, (w, e) in enumerate(itertools.zip_longest(warns, expected)): + if not e: + errors += [f"[{i}] Received unexpected warning `{w}`."] + elif not w: + errors += [f"[{i}] Did not receive expected warning `{e.name}`."] + elif not e.value.match(w): + errors += [f"[{i}] Warning `{w}` does not match expected {e.name}."] + + if errors: raise Failed('\n'.join(errors)) + + +def test_Integer(): + with _check_warns([_W.WILL_NOT_ASSERT] * 2) as w: + assert smtlib_code(Integer(67), log_warn=w) == "67" + assert smtlib_code(Integer(-1), log_warn=w) == "-1" + with _check_warns([]) as w: + assert smtlib_code(Integer(67)) == "67" + assert smtlib_code(Integer(-1)) == "-1" + + +def test_Rational(): + with _check_warns([_W.WILL_NOT_ASSERT] * 4) as w: + assert smtlib_code(Rational(3, 7), log_warn=w) == "(/ 3 7)" + assert smtlib_code(Rational(18, 9), log_warn=w) == "2" + assert smtlib_code(Rational(3, -7), log_warn=w) == "(/ -3 7)" + assert smtlib_code(Rational(-3, -7), log_warn=w) == "(/ 3 7)" + + with _check_warns([_W.DEFAULTING_TO_FLOAT, _W.WILL_NOT_ASSERT] * 2) as w: + assert smtlib_code(x + Rational(3, 7), auto_declare=False, log_warn=w) == "(+ (/ 3 7) x)" + assert smtlib_code(Rational(3, 7) * x, log_warn=w) == "(declare-const x Real)\n" \ + "(* (/ 3 7) x)" + + +def test_Relational(): + with _check_warns([_W.DEFAULTING_TO_FLOAT] * 12) as w: + assert smtlib_code(Eq(x, y), auto_declare=False, log_warn=w) == "(assert (= x y))" + assert smtlib_code(Ne(x, y), auto_declare=False, log_warn=w) == "(assert (not (= x y)))" + assert smtlib_code(Le(x, y), auto_declare=False, log_warn=w) == "(assert (<= x y))" + assert smtlib_code(Lt(x, y), auto_declare=False, log_warn=w) == "(assert (< x y))" + assert smtlib_code(Gt(x, y), auto_declare=False, log_warn=w) == "(assert (> x y))" + assert smtlib_code(Ge(x, y), auto_declare=False, log_warn=w) == "(assert (>= x y))" + + +def test_AppliedBinaryRelation(): + with _check_warns([_W.DEFAULTING_TO_FLOAT] * 12) as w: + assert smtlib_code(Q.eq(x, y), auto_declare=False, log_warn=w) == "(assert (= x y))" + assert smtlib_code(Q.ne(x, y), auto_declare=False, log_warn=w) == "(assert (not (= x y)))" + assert smtlib_code(Q.lt(x, y), auto_declare=False, log_warn=w) == "(assert (< x y))" + assert smtlib_code(Q.le(x, y), auto_declare=False, log_warn=w) == "(assert (<= x y))" + assert smtlib_code(Q.gt(x, y), auto_declare=False, log_warn=w) == "(assert (> x y))" + assert smtlib_code(Q.ge(x, y), auto_declare=False, log_warn=w) == "(assert (>= x y))" + + raises(ValueError, lambda: smtlib_code(Q.complex(x), log_warn=w)) + + +def test_AppliedPredicate(): + with _check_warns([_W.DEFAULTING_TO_FLOAT] * 6) as w: + assert smtlib_code(Q.positive(x), auto_declare=False, log_warn=w) == "(assert (> x 0))" + assert smtlib_code(Q.negative(x), auto_declare=False, log_warn=w) == "(assert (< x 0))" + assert smtlib_code(Q.zero(x), auto_declare=False, log_warn=w) == "(assert (= x 0))" + assert smtlib_code(Q.nonpositive(x), auto_declare=False, log_warn=w) == "(assert (<= x 0))" + assert smtlib_code(Q.nonnegative(x), auto_declare=False, log_warn=w) == "(assert (>= x 0))" + assert smtlib_code(Q.nonzero(x), auto_declare=False, log_warn=w) == "(assert (not (= x 0)))" + +def test_Function(): + with _check_warns([_W.DEFAULTING_TO_FLOAT, _W.WILL_NOT_ASSERT]) as w: + assert smtlib_code(sin(x) ** cos(x), auto_declare=False, log_warn=w) == "(pow (sin x) (cos x))" + + with _check_warns([_W.WILL_NOT_ASSERT]) as w: + assert smtlib_code( + abs(x), + symbol_table={x: int, y: bool}, + known_types={int: "INTEGER_TYPE"}, + known_functions={sympy.Abs: "ABSOLUTE_VALUE_OF"}, + log_warn=w + ) == "(declare-const x INTEGER_TYPE)\n" \ + "(ABSOLUTE_VALUE_OF x)" + + my_fun1 = Function('f1') + with _check_warns([_W.WILL_NOT_ASSERT]) as w: + assert smtlib_code( + my_fun1(x), + symbol_table={my_fun1: Callable[[bool], float]}, + log_warn=w + ) == "(declare-const x Bool)\n" \ + "(declare-fun f1 (Bool) Real)\n" \ + "(f1 x)" + + with _check_warns([]) as w: + assert smtlib_code( + my_fun1(x), + symbol_table={my_fun1: Callable[[bool], bool]}, + log_warn=w + ) == "(declare-const x Bool)\n" \ + "(declare-fun f1 (Bool) Bool)\n" \ + "(assert (f1 x))" + + assert smtlib_code( + Eq(my_fun1(x, z), y), + symbol_table={my_fun1: Callable[[int, bool], bool]}, + log_warn=w + ) == "(declare-const x Int)\n" \ + "(declare-const y Bool)\n" \ + "(declare-const z Bool)\n" \ + "(declare-fun f1 (Int Bool) Bool)\n" \ + "(assert (= (f1 x z) y))" + + assert smtlib_code( + Eq(my_fun1(x, z), y), + symbol_table={my_fun1: Callable[[int, bool], bool]}, + known_functions={my_fun1: "MY_KNOWN_FUN", Eq: '=='}, + log_warn=w + ) == "(declare-const x Int)\n" \ + "(declare-const y Bool)\n" \ + "(declare-const z Bool)\n" \ + "(assert (== (MY_KNOWN_FUN x z) y))" + + with _check_warns([_W.DEFAULTING_TO_FLOAT] * 3) as w: + assert smtlib_code( + Eq(my_fun1(x, z), y), + known_functions={my_fun1: "MY_KNOWN_FUN", Eq: '=='}, + log_warn=w + ) == "(declare-const x Real)\n" \ + "(declare-const y Real)\n" \ + "(declare-const z Real)\n" \ + "(assert (== (MY_KNOWN_FUN x z) y))" + + +def test_Pow(): + with _check_warns([_W.DEFAULTING_TO_FLOAT, _W.WILL_NOT_ASSERT]) as w: + assert smtlib_code(x ** 3, auto_declare=False, log_warn=w) == "(pow x 3)" + with _check_warns([_W.DEFAULTING_TO_FLOAT, _W.DEFAULTING_TO_FLOAT, _W.WILL_NOT_ASSERT]) as w: + assert smtlib_code(x ** (y ** 3), auto_declare=False, log_warn=w) == "(pow x (pow y 3))" + with _check_warns([_W.DEFAULTING_TO_FLOAT, _W.WILL_NOT_ASSERT]) as w: + assert smtlib_code(x ** Rational(2, 3), auto_declare=False, log_warn=w) == '(pow x (/ 2 3))' + + a = Symbol('a', integer=True) + b = Symbol('b', real=True) + c = Symbol('c') + + def g(x): return 2 * x + + # if x=1, y=2, then expr=2.333... + expr = 1 / (g(a) * 3.5) ** (a - b ** a) / (a ** 2 + b) + + with _check_warns([]) as w: + assert smtlib_code( + [ + Eq(a < 2, c), + Eq(b > a, c), + c & True, + Eq(expr, 2 + Rational(1, 3)) + ], + log_warn=w + ) == '(declare-const a Int)\n' \ + '(declare-const b Real)\n' \ + '(declare-const c Bool)\n' \ + '(assert (= (< a 2) c))\n' \ + '(assert (= (> b a) c))\n' \ + '(assert c)\n' \ + '(assert (= ' \ + '(* (pow (* 7.0 a) (+ (pow b a) (* -1 a))) (pow (+ b (pow a 2)) -1)) ' \ + '(/ 7 3)' \ + '))' + + with _check_warns([_W.DEFAULTING_TO_FLOAT, _W.WILL_NOT_ASSERT]) as w: + assert smtlib_code( + Mul(-2, c, Pow(Mul(b, b, evaluate=False), -1, evaluate=False), evaluate=False), + log_warn=w + ) == '(declare-const b Real)\n' \ + '(declare-const c Real)\n' \ + '(* -2 c (pow (* b b) -1))' + + +def test_basic_ops(): + with _check_warns([_W.DEFAULTING_TO_FLOAT, _W.DEFAULTING_TO_FLOAT, _W.WILL_NOT_ASSERT]) as w: + assert smtlib_code(x * y, auto_declare=False, log_warn=w) == "(* x y)" + + with _check_warns([_W.DEFAULTING_TO_FLOAT, _W.DEFAULTING_TO_FLOAT, _W.WILL_NOT_ASSERT]) as w: + assert smtlib_code(x + y, auto_declare=False, log_warn=w) == "(+ x y)" + + # with _check_warns([_SmtlibWarnings.DEFAULTING_TO_FLOAT, _SmtlibWarnings.DEFAULTING_TO_FLOAT, _SmtlibWarnings.WILL_NOT_ASSERT]) as w: + # todo: implement re-write, currently does '(+ x (* -1 y))' instead + # assert smtlib_code(x - y, auto_declare=False, log_warn=w) == "(- x y)" + + with _check_warns([_W.DEFAULTING_TO_FLOAT, _W.WILL_NOT_ASSERT]) as w: + assert smtlib_code(-x, auto_declare=False, log_warn=w) == "(* -1 x)" + + +def test_quantifier_extensions(): + from sympy.logic.boolalg import Boolean + from sympy import Interval, Tuple, sympify + + # start For-all quantifier class example + class ForAll(Boolean): + def _smtlib(self, printer): + bound_symbol_declarations = [ + printer._s_expr(sym.name, [ + printer._known_types[printer.symbol_table[sym]], + Interval(start, end) + ]) for sym, start, end in self.limits + ] + return printer._s_expr('forall', [ + printer._s_expr('', bound_symbol_declarations), + self.function + ]) + + @property + def bound_symbols(self): + return {s for s, _, _ in self.limits} + + @property + def free_symbols(self): + bound_symbol_names = {s.name for s in self.bound_symbols} + return { + s for s in self.function.free_symbols + if s.name not in bound_symbol_names + } + + def __new__(cls, *args): + limits = [sympify(a) for a in args if isinstance(a, (tuple, Tuple))] + function = [sympify(a) for a in args if isinstance(a, Boolean)] + assert len(limits) + len(function) == len(args) + assert len(function) == 1 + function = function[0] + + if isinstance(function, ForAll): return ForAll.__new__( + ForAll, *(limits + function.limits), function.function + ) + inst = Boolean.__new__(cls) + inst._args = tuple(limits + [function]) + inst.limits = limits + inst.function = function + return inst + + # end For-All Quantifier class example + + f = Function('f') + with _check_warns([_W.DEFAULTING_TO_FLOAT]) as w: + assert smtlib_code( + ForAll((x, -42, +21), Eq(f(x), f(x))), + symbol_table={f: Callable[[float], float]}, + log_warn=w + ) == '(assert (forall ( (x Real [-42, 21])) true))' + + with _check_warns([_W.DEFAULTING_TO_FLOAT] * 2) as w: + assert smtlib_code( + ForAll( + (x, -42, +21), (y, -100, 3), + Implies(Eq(x, y), Eq(f(x), f(y))) + ), + symbol_table={f: Callable[[float], float]}, + log_warn=w + ) == '(declare-fun f (Real) Real)\n' \ + '(assert (' \ + 'forall ( (x Real [-42, 21]) (y Real [-100, 3])) ' \ + '(=> (= x y) (= (f x) (f y)))' \ + '))' + + a = Symbol('a', integer=True) + b = Symbol('b', real=True) + c = Symbol('c') + + with _check_warns([]) as w: + assert smtlib_code( + ForAll( + (a, 2, 100), ForAll( + (b, 2, 100), + Implies(a < b, sqrt(a) < b) | c + )), + log_warn=w + ) == '(declare-const c Bool)\n' \ + '(assert (forall ( (a Int [2, 100]) (b Real [2, 100])) ' \ + '(or c (=> (< a b) (< (pow a (/ 1 2)) b)))' \ + '))' + + +def test_mix_number_mult_symbols(): + with _check_warns([_W.WILL_NOT_ASSERT]) as w: + assert smtlib_code( + 1 / pi, + known_constants={pi: "MY_PI"}, + log_warn=w + ) == '(pow MY_PI -1)' + + with _check_warns([_W.WILL_NOT_ASSERT]) as w: + assert smtlib_code( + [ + Eq(pi, 3.14, evaluate=False), + 1 / pi, + ], + known_constants={pi: "MY_PI"}, + log_warn=w + ) == '(assert (= MY_PI 3.14))\n' \ + '(pow MY_PI -1)' + + with _check_warns([_W.WILL_NOT_ASSERT]) as w: + assert smtlib_code( + Add(S.Zero, S.One, S.NegativeOne, S.Half, + S.Exp1, S.Pi, S.GoldenRatio, evaluate=False), + known_constants={ + S.Pi: 'p', S.GoldenRatio: 'g', + S.Exp1: 'e' + }, + known_functions={ + Add: 'plus', + exp: 'exp' + }, + precision=3, + log_warn=w + ) == '(plus 0 1 -1 (/ 1 2) (exp 1) p g)' + + with _check_warns([_W.WILL_NOT_ASSERT]) as w: + assert smtlib_code( + Add(S.Zero, S.One, S.NegativeOne, S.Half, + S.Exp1, S.Pi, S.GoldenRatio, evaluate=False), + known_constants={ + S.Pi: 'p' + }, + known_functions={ + Add: 'plus', + exp: 'exp' + }, + precision=3, + log_warn=w + ) == '(plus 0 1 -1 (/ 1 2) (exp 1) p 1.62)' + + with _check_warns([_W.WILL_NOT_ASSERT]) as w: + assert smtlib_code( + Add(S.Zero, S.One, S.NegativeOne, S.Half, + S.Exp1, S.Pi, S.GoldenRatio, evaluate=False), + known_functions={Add: 'plus'}, + precision=3, + log_warn=w + ) == '(plus 0 1 -1 (/ 1 2) 2.72 3.14 1.62)' + + with _check_warns([_W.WILL_NOT_ASSERT]) as w: + assert smtlib_code( + Add(S.Zero, S.One, S.NegativeOne, S.Half, + S.Exp1, S.Pi, S.GoldenRatio, evaluate=False), + known_constants={S.Exp1: 'e'}, + known_functions={Add: 'plus'}, + precision=3, + log_warn=w + ) == '(plus 0 1 -1 (/ 1 2) e 3.14 1.62)' + + +def test_boolean(): + with _check_warns([]) as w: + assert smtlib_code(x & y, log_warn=w) == '(declare-const x Bool)\n' \ + '(declare-const y Bool)\n' \ + '(assert (and x y))' + assert smtlib_code(x | y, log_warn=w) == '(declare-const x Bool)\n' \ + '(declare-const y Bool)\n' \ + '(assert (or x y))' + assert smtlib_code(~x, log_warn=w) == '(declare-const x Bool)\n' \ + '(assert (not x))' + assert smtlib_code(x & y & z, log_warn=w) == '(declare-const x Bool)\n' \ + '(declare-const y Bool)\n' \ + '(declare-const z Bool)\n' \ + '(assert (and x y z))' + + with _check_warns([_W.DEFAULTING_TO_FLOAT]) as w: + assert smtlib_code((x & ~y) | (z > 3), log_warn=w) == '(declare-const x Bool)\n' \ + '(declare-const y Bool)\n' \ + '(declare-const z Real)\n' \ + '(assert (or (> z 3) (and x (not y))))' + + f = Function('f') + g = Function('g') + h = Function('h') + with _check_warns([_W.DEFAULTING_TO_FLOAT]) as w: + assert smtlib_code( + [Gt(f(x), y), + Lt(y, g(z))], + symbol_table={ + f: Callable[[bool], int], g: Callable[[bool], int], + }, log_warn=w + ) == '(declare-const x Bool)\n' \ + '(declare-const y Real)\n' \ + '(declare-const z Bool)\n' \ + '(declare-fun f (Bool) Int)\n' \ + '(declare-fun g (Bool) Int)\n' \ + '(assert (> (f x) y))\n' \ + '(assert (< y (g z)))' + + with _check_warns([]) as w: + assert smtlib_code( + [Eq(f(x), y), + Lt(y, g(z))], + symbol_table={ + f: Callable[[bool], int], g: Callable[[bool], int], + }, log_warn=w + ) == '(declare-const x Bool)\n' \ + '(declare-const y Int)\n' \ + '(declare-const z Bool)\n' \ + '(declare-fun f (Bool) Int)\n' \ + '(declare-fun g (Bool) Int)\n' \ + '(assert (= (f x) y))\n' \ + '(assert (< y (g z)))' + + with _check_warns([]) as w: + assert smtlib_code( + [Eq(f(x), y), + Eq(g(f(x)), z), + Eq(h(g(f(x))), x)], + symbol_table={ + f: Callable[[float], int], + g: Callable[[int], bool], + h: Callable[[bool], float] + }, + log_warn=w + ) == '(declare-const x Real)\n' \ + '(declare-const y Int)\n' \ + '(declare-const z Bool)\n' \ + '(declare-fun f (Real) Int)\n' \ + '(declare-fun g (Int) Bool)\n' \ + '(declare-fun h (Bool) Real)\n' \ + '(assert (= (f x) y))\n' \ + '(assert (= (g (f x)) z))\n' \ + '(assert (= (h (g (f x))) x))' + + +# todo: make smtlib_code support arrays +# def test_containers(): +# assert julia_code([1, 2, 3, [4, 5, [6, 7]], 8, [9, 10], 11]) == \ +# "Any[1, 2, 3, Any[4, 5, Any[6, 7]], 8, Any[9, 10], 11]" +# assert julia_code((1, 2, (3, 4))) == "(1, 2, (3, 4))" +# assert julia_code([1]) == "Any[1]" +# assert julia_code((1,)) == "(1,)" +# assert julia_code(Tuple(*[1, 2, 3])) == "(1, 2, 3)" +# assert julia_code((1, x * y, (3, x ** 2))) == "(1, x .* y, (3, x .^ 2))" +# # scalar, matrix, empty matrix and empty list +# assert julia_code((1, eye(3), Matrix(0, 0, []), [])) == "(1, [1 0 0;\n0 1 0;\n0 0 1], zeros(0, 0), Any[])" + +def test_smtlib_piecewise(): + with _check_warns([_W.DEFAULTING_TO_FLOAT, _W.WILL_NOT_ASSERT]) as w: + assert smtlib_code( + Piecewise((x, x < 1), + (x ** 2, True)), + auto_declare=False, + log_warn=w + ) == '(ite (< x 1) x (pow x 2))' + + with _check_warns([_W.DEFAULTING_TO_FLOAT, _W.WILL_NOT_ASSERT]) as w: + assert smtlib_code( + Piecewise((x ** 2, x < 1), + (x ** 3, x < 2), + (x ** 4, x < 3), + (x ** 5, True)), + auto_declare=False, + log_warn=w + ) == '(ite (< x 1) (pow x 2) ' \ + '(ite (< x 2) (pow x 3) ' \ + '(ite (< x 3) (pow x 4) ' \ + '(pow x 5))))' + + # Check that Piecewise without a True (default) condition error + expr = Piecewise((x, x < 1), (x ** 2, x > 1), (sin(x), x > 0)) + with _check_warns([_W.DEFAULTING_TO_FLOAT, _W.WILL_NOT_ASSERT]) as w: + raises(AssertionError, lambda: smtlib_code(expr, log_warn=w)) + + +def test_smtlib_piecewise_times_const(): + pw = Piecewise((x, x < 1), (x ** 2, True)) + with _check_warns([_W.DEFAULTING_TO_FLOAT, _W.WILL_NOT_ASSERT]) as w: + assert smtlib_code(2 * pw, log_warn=w) == '(declare-const x Real)\n(* 2 (ite (< x 1) x (pow x 2)))' + with _check_warns([_W.DEFAULTING_TO_FLOAT, _W.WILL_NOT_ASSERT]) as w: + assert smtlib_code(pw / x, log_warn=w) == '(declare-const x Real)\n(* (pow x -1) (ite (< x 1) x (pow x 2)))' + with _check_warns([_W.DEFAULTING_TO_FLOAT, _W.DEFAULTING_TO_FLOAT, _W.WILL_NOT_ASSERT]) as w: + assert smtlib_code(pw / (x * y), log_warn=w) == '(declare-const x Real)\n(declare-const y Real)\n(* (pow x -1) (pow y -1) (ite (< x 1) x (pow x 2)))' + with _check_warns([_W.DEFAULTING_TO_FLOAT, _W.WILL_NOT_ASSERT]) as w: + assert smtlib_code(pw / 3, log_warn=w) == '(declare-const x Real)\n(* (/ 1 3) (ite (< x 1) x (pow x 2)))' + + +# todo: make smtlib_code support arrays / matrices ? +# def test_smtlib_matrix_assign_to(): +# A = Matrix([[1, 2, 3]]) +# assert smtlib_code(A, assign_to='a') == "a = [1 2 3]" +# A = Matrix([[1, 2], [3, 4]]) +# assert smtlib_code(A, assign_to='A') == "A = [1 2;\n3 4]" + +# def test_julia_matrix_1x1(): +# A = Matrix([[3]]) +# B = MatrixSymbol('B', 1, 1) +# C = MatrixSymbol('C', 1, 2) +# assert julia_code(A, assign_to=B) == "B = [3]" +# raises(ValueError, lambda: julia_code(A, assign_to=C)) + +# def test_julia_matrix_elements(): +# A = Matrix([[x, 2, x * y]]) +# assert julia_code(A[0, 0] ** 2 + A[0, 1] + A[0, 2]) == "x .^ 2 + x .* y + 2" +# A = MatrixSymbol('AA', 1, 3) +# assert julia_code(A) == "AA" +# assert julia_code(A[0, 0] ** 2 + sin(A[0, 1]) + A[0, 2]) == \ +# "sin(AA[1,2]) + AA[1,1] .^ 2 + AA[1,3]" +# assert julia_code(sum(A)) == "AA[1,1] + AA[1,2] + AA[1,3]" + +def test_smtlib_boolean(): + with _check_warns([]) as w: + assert smtlib_code(True, auto_assert=False, log_warn=w) == 'true' + assert smtlib_code(True, log_warn=w) == '(assert true)' + assert smtlib_code(S.true, log_warn=w) == '(assert true)' + assert smtlib_code(S.false, log_warn=w) == '(assert false)' + assert smtlib_code(False, log_warn=w) == '(assert false)' + assert smtlib_code(False, auto_assert=False, log_warn=w) == 'false' + + +def test_not_supported(): + f = Function('f') + with _check_warns([_W.DEFAULTING_TO_FLOAT, _W.WILL_NOT_ASSERT]) as w: + raises(KeyError, lambda: smtlib_code(f(x).diff(x), symbol_table={f: Callable[[float], float]}, log_warn=w)) + with _check_warns([_W.WILL_NOT_ASSERT]) as w: + raises(KeyError, lambda: smtlib_code(S.ComplexInfinity, log_warn=w)) + + +def test_Float(): + assert smtlib_code(0.0) == "0.0" + assert smtlib_code(0.000000000000000003) == '(* 3.0 (pow 10 -18))' + assert smtlib_code(5.3) == "5.3" diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_str.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_str.py new file mode 100644 index 0000000000000000000000000000000000000000..675212964b03bf9a9806088225c28d7f70971ca7 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_str.py @@ -0,0 +1,1206 @@ +from sympy import MatAdd +from sympy.algebras.quaternion import Quaternion +from sympy.assumptions.ask import Q +from sympy.calculus.accumulationbounds import AccumBounds +from sympy.combinatorics.partitions import Partition +from sympy.concrete.summations import (Sum, summation) +from sympy.core.add import Add +from sympy.core.containers import (Dict, Tuple) +from sympy.core.expr import UnevaluatedExpr, Expr +from sympy.core.function import (Derivative, Function, Lambda, Subs, WildFunction) +from sympy.core.mul import Mul +from sympy.core import (Catalan, EulerGamma, GoldenRatio, TribonacciConstant) +from sympy.core.numbers import (E, Float, I, Integer, Rational, nan, oo, pi, zoo) +from sympy.core.parameters import _exp_is_pow +from sympy.core.power import Pow +from sympy.core.relational import (Eq, Rel, Ne) +from sympy.core.singleton import S +from sympy.core.symbol import (Dummy, Symbol, Wild, symbols) +from sympy.functions.combinatorial.factorials import (factorial, factorial2, subfactorial) +from sympy.functions.elementary.complexes import Abs +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.functions.special.delta_functions import Heaviside +from sympy.functions.special.zeta_functions import zeta +from sympy.integrals.integrals import Integral +from sympy.logic.boolalg import (Equivalent, false, true, Xor) +from sympy.matrices.dense import Matrix +from sympy.matrices.expressions.matexpr import MatrixSymbol +from sympy.matrices.expressions import Identity +from sympy.matrices.expressions.slice import MatrixSlice +from sympy.matrices import SparseMatrix +from sympy.polys.polytools import factor +from sympy.series.limits import Limit +from sympy.series.order import O +from sympy.sets.sets import (Complement, FiniteSet, Interval, SymmetricDifference) +from sympy.stats import (Covariance, Expectation, Probability, Variance) +from sympy.stats.rv import RandomSymbol +from sympy.external import import_module +from sympy.physics.control.lti import TransferFunction, Series, Parallel, \ + Feedback, TransferFunctionMatrix, MIMOSeries, MIMOParallel, MIMOFeedback +from sympy.physics.units import second, joule +from sympy.polys import (Poly, rootof, RootSum, groebner, ring, field, ZZ, QQ, + ZZ_I, QQ_I, lex, grlex) +from sympy.geometry import Point, Circle, Polygon, Ellipse, Triangle +from sympy.tensor import NDimArray +from sympy.tensor.array.expressions.array_expressions import ArraySymbol, ArrayElement + +from sympy.testing.pytest import raises, warns_deprecated_sympy + +from sympy.printing import sstr, sstrrepr, StrPrinter +from sympy.physics.quantum.trace import Tr + +x, y, z, w, t = symbols('x,y,z,w,t') +d = Dummy('d') + + +def test_printmethod(): + class R(Abs): + def _sympystr(self, printer): + return "foo(%s)" % printer._print(self.args[0]) + assert sstr(R(x)) == "foo(x)" + + class R(Abs): + def _sympystr(self, printer): + return "foo" + assert sstr(R(x)) == "foo" + + +def test_Abs(): + assert str(Abs(x)) == "Abs(x)" + assert str(Abs(Rational(1, 6))) == "1/6" + assert str(Abs(Rational(-1, 6))) == "1/6" + + +def test_Add(): + assert str(x + y) == "x + y" + assert str(x + 1) == "x + 1" + assert str(x + x**2) == "x**2 + x" + assert str(Add(0, 1, evaluate=False)) == "0 + 1" + assert str(Add(0, 0, 1, evaluate=False)) == "0 + 0 + 1" + assert str(1.0*x) == "1.0*x" + assert str(5 + x + y + x*y + x**2 + y**2) == "x**2 + x*y + x + y**2 + y + 5" + assert str(1 + x + x**2/2 + x**3/3) == "x**3/3 + x**2/2 + x + 1" + assert str(2*x - 7*x**2 + 2 + 3*y) == "-7*x**2 + 2*x + 3*y + 2" + assert str(x - y) == "x - y" + assert str(2 - x) == "2 - x" + assert str(x - 2) == "x - 2" + assert str(x - y - z - w) == "-w + x - y - z" + assert str(x - z*y**2*z*w) == "-w*y**2*z**2 + x" + assert str(x - 1*y*x*y) == "-x*y**2 + x" + assert str(sin(x).series(x, 0, 15)) == "x - x**3/6 + x**5/120 - x**7/5040 + x**9/362880 - x**11/39916800 + x**13/6227020800 + O(x**15)" + assert str(Add(Add(-w, x, evaluate=False), Add(-y, z, evaluate=False), evaluate=False)) == "(-w + x) + (-y + z)" + assert str(Add(Add(-x, -y, evaluate=False), -z, evaluate=False)) == "-z + (-x - y)" + assert str(Add(Add(Add(-x, -y, evaluate=False), -z, evaluate=False), -t, evaluate=False)) == "-t + (-z + (-x - y))" + + +def test_Catalan(): + assert str(Catalan) == "Catalan" + + +def test_ComplexInfinity(): + assert str(zoo) == "zoo" + + +def test_Derivative(): + assert str(Derivative(x, y)) == "Derivative(x, y)" + assert str(Derivative(x**2, x, evaluate=False)) == "Derivative(x**2, x)" + assert str(Derivative( + x**2/y, x, y, evaluate=False)) == "Derivative(x**2/y, x, y)" + + +def test_dict(): + assert str({1: 1 + x}) == sstr({1: 1 + x}) == "{1: x + 1}" + assert str({1: x**2, 2: y*x}) in ("{1: x**2, 2: x*y}", "{2: x*y, 1: x**2}") + assert sstr({1: x**2, 2: y*x}) == "{1: x**2, 2: x*y}" + + +def test_Dict(): + assert str(Dict({1: 1 + x})) == sstr({1: 1 + x}) == "{1: x + 1}" + assert str(Dict({1: x**2, 2: y*x})) in ( + "{1: x**2, 2: x*y}", "{2: x*y, 1: x**2}") + assert sstr(Dict({1: x**2, 2: y*x})) == "{1: x**2, 2: x*y}" + + +def test_Dummy(): + assert str(d) == "_d" + assert str(d + x) == "_d + x" + + +def test_EulerGamma(): + assert str(EulerGamma) == "EulerGamma" + + +def test_Exp(): + assert str(E) == "E" + with _exp_is_pow(True): + assert str(exp(x)) == "E**x" + + +def test_factorial(): + n = Symbol('n', integer=True) + assert str(factorial(-2)) == "zoo" + assert str(factorial(0)) == "1" + assert str(factorial(7)) == "5040" + assert str(factorial(n)) == "factorial(n)" + assert str(factorial(2*n)) == "factorial(2*n)" + assert str(factorial(factorial(n))) == 'factorial(factorial(n))' + assert str(factorial(factorial2(n))) == 'factorial(factorial2(n))' + assert str(factorial2(factorial(n))) == 'factorial2(factorial(n))' + assert str(factorial2(factorial2(n))) == 'factorial2(factorial2(n))' + assert str(subfactorial(3)) == "2" + assert str(subfactorial(n)) == "subfactorial(n)" + assert str(subfactorial(2*n)) == "subfactorial(2*n)" + + +def test_Function(): + f = Function('f') + fx = f(x) + w = WildFunction('w') + assert str(f) == "f" + assert str(fx) == "f(x)" + assert str(w) == "w_" + + +def test_Geometry(): + assert sstr(Point(0, 0)) == 'Point2D(0, 0)' + assert sstr(Circle(Point(0, 0), 3)) == 'Circle(Point2D(0, 0), 3)' + assert sstr(Ellipse(Point(1, 2), 3, 4)) == 'Ellipse(Point2D(1, 2), 3, 4)' + assert sstr(Triangle(Point(1, 1), Point(7, 8), Point(0, -1))) == \ + 'Triangle(Point2D(1, 1), Point2D(7, 8), Point2D(0, -1))' + assert sstr(Polygon(Point(5, 6), Point(-2, -3), Point(0, 0), Point(4, 7))) == \ + 'Polygon(Point2D(5, 6), Point2D(-2, -3), Point2D(0, 0), Point2D(4, 7))' + assert sstr(Triangle(Point(0, 0), Point(1, 0), Point(0, 1)), sympy_integers=True) == \ + 'Triangle(Point2D(S(0), S(0)), Point2D(S(1), S(0)), Point2D(S(0), S(1)))' + assert sstr(Ellipse(Point(1, 2), 3, 4), sympy_integers=True) == \ + 'Ellipse(Point2D(S(1), S(2)), S(3), S(4))' + + +def test_GoldenRatio(): + assert str(GoldenRatio) == "GoldenRatio" + + +def test_Heaviside(): + assert str(Heaviside(x)) == str(Heaviside(x, S.Half)) == "Heaviside(x)" + assert str(Heaviside(x, 1)) == "Heaviside(x, 1)" + + +def test_TribonacciConstant(): + assert str(TribonacciConstant) == "TribonacciConstant" + + +def test_ImaginaryUnit(): + assert str(I) == "I" + + +def test_Infinity(): + assert str(oo) == "oo" + assert str(oo*I) == "oo*I" + + +def test_Integer(): + assert str(Integer(-1)) == "-1" + assert str(Integer(1)) == "1" + assert str(Integer(-3)) == "-3" + assert str(Integer(0)) == "0" + assert str(Integer(25)) == "25" + + +def test_Integral(): + assert str(Integral(sin(x), y)) == "Integral(sin(x), y)" + assert str(Integral(sin(x), (y, 0, 1))) == "Integral(sin(x), (y, 0, 1))" + + +def test_Interval(): + n = (S.NegativeInfinity, 1, 2, S.Infinity) + for i in range(len(n)): + for j in range(i + 1, len(n)): + for l in (True, False): + for r in (True, False): + ival = Interval(n[i], n[j], l, r) + assert S(str(ival)) == ival + + +def test_AccumBounds(): + a = Symbol('a', real=True) + assert str(AccumBounds(0, a)) == "AccumBounds(0, a)" + assert str(AccumBounds(0, 1)) == "AccumBounds(0, 1)" + + +def test_Lambda(): + assert str(Lambda(d, d**2)) == "Lambda(_d, _d**2)" + # issue 2908 + assert str(Lambda((), 1)) == "Lambda((), 1)" + assert str(Lambda((), x)) == "Lambda((), x)" + assert str(Lambda((x, y), x+y)) == "Lambda((x, y), x + y)" + assert str(Lambda(((x, y),), x+y)) == "Lambda(((x, y),), x + y)" + + +def test_Limit(): + assert str(Limit(sin(x)/x, x, y)) == "Limit(sin(x)/x, x, y, dir='+')" + assert str(Limit(1/x, x, 0)) == "Limit(1/x, x, 0, dir='+')" + assert str( + Limit(sin(x)/x, x, y, dir="-")) == "Limit(sin(x)/x, x, y, dir='-')" + + +def test_list(): + assert str([x]) == sstr([x]) == "[x]" + assert str([x**2, x*y + 1]) == sstr([x**2, x*y + 1]) == "[x**2, x*y + 1]" + assert str([x**2, [y + x]]) == sstr([x**2, [y + x]]) == "[x**2, [x + y]]" + + +def test_Matrix_str(): + M = Matrix([[x**+1, 1], [y, x + y]]) + assert str(M) == "Matrix([[x, 1], [y, x + y]])" + assert sstr(M) == "Matrix([\n[x, 1],\n[y, x + y]])" + M = Matrix([[1]]) + assert str(M) == sstr(M) == "Matrix([[1]])" + M = Matrix([[1, 2]]) + assert str(M) == sstr(M) == "Matrix([[1, 2]])" + M = Matrix() + assert str(M) == sstr(M) == "Matrix(0, 0, [])" + M = Matrix(0, 1, lambda i, j: 0) + assert str(M) == sstr(M) == "Matrix(0, 1, [])" + + +def test_Mul(): + assert str(x/y) == "x/y" + assert str(y/x) == "y/x" + assert str(x/y/z) == "x/(y*z)" + assert str((x + 1)/(y + 2)) == "(x + 1)/(y + 2)" + assert str(2*x/3) == '2*x/3' + assert str(-2*x/3) == '-2*x/3' + assert str(-1.0*x) == '-1.0*x' + assert str(1.0*x) == '1.0*x' + assert str(Mul(0, 1, evaluate=False)) == '0*1' + assert str(Mul(1, 0, evaluate=False)) == '1*0' + assert str(Mul(1, 1, evaluate=False)) == '1*1' + assert str(Mul(1, 1, 1, evaluate=False)) == '1*1*1' + assert str(Mul(1, 2, evaluate=False)) == '1*2' + assert str(Mul(1, S.Half, evaluate=False)) == '1*(1/2)' + assert str(Mul(1, 1, S.Half, evaluate=False)) == '1*1*(1/2)' + assert str(Mul(1, 1, 2, 3, x, evaluate=False)) == '1*1*2*3*x' + assert str(Mul(1, -1, evaluate=False)) == '1*(-1)' + assert str(Mul(-1, 1, evaluate=False)) == '-1*1' + assert str(Mul(4, 3, 2, 1, 0, y, x, evaluate=False)) == '4*3*2*1*0*y*x' + assert str(Mul(4, 3, 2, 1+z, 0, y, x, evaluate=False)) == '4*3*2*(z + 1)*0*y*x' + assert str(Mul(Rational(2, 3), Rational(5, 7), evaluate=False)) == '(2/3)*(5/7)' + # For issue 14160 + assert str(Mul(-2, x, Pow(Mul(y,y,evaluate=False), -1, evaluate=False), + evaluate=False)) == '-2*x/(y*y)' + # issue 21537 + assert str(Mul(x, Pow(1/y, -1, evaluate=False), evaluate=False)) == 'x/(1/y)' + + # Issue 24108 + from sympy.core.parameters import evaluate + with evaluate(False): + assert str(Mul(Pow(Integer(2), Integer(-1)), Add(Integer(-1), Mul(Integer(-1), Integer(1))))) == "(-1 - 1*1)/2" + + class CustomClass1(Expr): + is_commutative = True + + class CustomClass2(Expr): + is_commutative = True + cc1 = CustomClass1() + cc2 = CustomClass2() + assert str(Rational(2)*cc1) == '2*CustomClass1()' + assert str(cc1*Rational(2)) == '2*CustomClass1()' + assert str(cc1*Float("1.5")) == '1.5*CustomClass1()' + assert str(cc2*Rational(2)) == '2*CustomClass2()' + assert str(cc2*Rational(2)*cc1) == '2*CustomClass1()*CustomClass2()' + assert str(cc1*Rational(2)*cc2) == '2*CustomClass1()*CustomClass2()' + + +def test_NaN(): + assert str(nan) == "nan" + + +def test_NegativeInfinity(): + assert str(-oo) == "-oo" + +def test_Order(): + assert str(O(x)) == "O(x)" + assert str(O(x**2)) == "O(x**2)" + assert str(O(x*y)) == "O(x*y, x, y)" + assert str(O(x, x)) == "O(x)" + assert str(O(x, (x, 0))) == "O(x)" + assert str(O(x, (x, oo))) == "O(x, (x, oo))" + assert str(O(x, x, y)) == "O(x, x, y)" + assert str(O(x, x, y)) == "O(x, x, y)" + assert str(O(x, (x, oo), (y, oo))) == "O(x, (x, oo), (y, oo))" + + +def test_Permutation_Cycle(): + from sympy.combinatorics import Permutation, Cycle + + # general principle: economically, canonically show all moved elements + # and the size of the permutation. + + for p, s in [ + (Cycle(), + '()'), + (Cycle(2), + '(2)'), + (Cycle(2, 1), + '(1 2)'), + (Cycle(1, 2)(5)(6, 7)(10), + '(1 2)(6 7)(10)'), + (Cycle(3, 4)(1, 2)(3, 4), + '(1 2)(4)'), + ]: + assert sstr(p) == s + + for p, s in [ + (Permutation([]), + 'Permutation([])'), + (Permutation([], size=1), + 'Permutation([0])'), + (Permutation([], size=2), + 'Permutation([0, 1])'), + (Permutation([], size=10), + 'Permutation([], size=10)'), + (Permutation([1, 0, 2]), + 'Permutation([1, 0, 2])'), + (Permutation([1, 0, 2, 3, 4, 5]), + 'Permutation([1, 0], size=6)'), + (Permutation([1, 0, 2, 3, 4, 5], size=10), + 'Permutation([1, 0], size=10)'), + ]: + assert sstr(p, perm_cyclic=False) == s + + for p, s in [ + (Permutation([]), + '()'), + (Permutation([], size=1), + '(0)'), + (Permutation([], size=2), + '(1)'), + (Permutation([], size=10), + '(9)'), + (Permutation([1, 0, 2]), + '(2)(0 1)'), + (Permutation([1, 0, 2, 3, 4, 5]), + '(5)(0 1)'), + (Permutation([1, 0, 2, 3, 4, 5], size=10), + '(9)(0 1)'), + (Permutation([0, 1, 3, 2, 4, 5], size=10), + '(9)(2 3)'), + ]: + assert sstr(p) == s + + + with warns_deprecated_sympy(): + old_print_cyclic = Permutation.print_cyclic + Permutation.print_cyclic = False + assert sstr(Permutation([1, 0, 2])) == 'Permutation([1, 0, 2])' + Permutation.print_cyclic = old_print_cyclic + +def test_Pi(): + assert str(pi) == "pi" + + +def test_Poly(): + assert str(Poly(0, x)) == "Poly(0, x, domain='ZZ')" + assert str(Poly(1, x)) == "Poly(1, x, domain='ZZ')" + assert str(Poly(x, x)) == "Poly(x, x, domain='ZZ')" + + assert str(Poly(2*x + 1, x)) == "Poly(2*x + 1, x, domain='ZZ')" + assert str(Poly(2*x - 1, x)) == "Poly(2*x - 1, x, domain='ZZ')" + + assert str(Poly(-1, x)) == "Poly(-1, x, domain='ZZ')" + assert str(Poly(-x, x)) == "Poly(-x, x, domain='ZZ')" + + assert str(Poly(-2*x + 1, x)) == "Poly(-2*x + 1, x, domain='ZZ')" + assert str(Poly(-2*x - 1, x)) == "Poly(-2*x - 1, x, domain='ZZ')" + + assert str(Poly(x - 1, x)) == "Poly(x - 1, x, domain='ZZ')" + assert str(Poly(2*x + x**5, x)) == "Poly(x**5 + 2*x, x, domain='ZZ')" + + assert str(Poly(3**(2*x), 3**x)) == "Poly((3**x)**2, 3**x, domain='ZZ')" + assert str(Poly((x**2)**x)) == "Poly(((x**2)**x), (x**2)**x, domain='ZZ')" + + assert str(Poly((x + y)**3, (x + y), expand=False) + ) == "Poly((x + y)**3, x + y, domain='ZZ')" + assert str(Poly((x - 1)**2, (x - 1), expand=False) + ) == "Poly((x - 1)**2, x - 1, domain='ZZ')" + + assert str( + Poly(x**2 + 1 + y, x)) == "Poly(x**2 + y + 1, x, domain='ZZ[y]')" + assert str( + Poly(x**2 - 1 + y, x)) == "Poly(x**2 + y - 1, x, domain='ZZ[y]')" + + assert str(Poly(x**2 + I*x, x)) == "Poly(x**2 + I*x, x, domain='ZZ_I')" + assert str(Poly(x**2 - I*x, x)) == "Poly(x**2 - I*x, x, domain='ZZ_I')" + + assert str(Poly(-x*y*z + x*y - 1, x, y, z) + ) == "Poly(-x*y*z + x*y - 1, x, y, z, domain='ZZ')" + assert str(Poly(-w*x**21*y**7*z + (1 + w)*z**3 - 2*x*z + 1, x, y, z)) == \ + "Poly(-w*x**21*y**7*z - 2*x*z + (w + 1)*z**3 + 1, x, y, z, domain='ZZ[w]')" + + assert str(Poly(x**2 + 1, x, modulus=2)) == "Poly(x**2 + 1, x, modulus=2)" + assert str(Poly(2*x**2 + 3*x + 4, x, modulus=17)) == "Poly(2*x**2 + 3*x + 4, x, modulus=17)" + + +def test_PolyRing(): + assert str(ring("x", ZZ, lex)[0]) == "Polynomial ring in x over ZZ with lex order" + assert str(ring("x,y", QQ, grlex)[0]) == "Polynomial ring in x, y over QQ with grlex order" + assert str(ring("x,y,z", ZZ["t"], lex)[0]) == "Polynomial ring in x, y, z over ZZ[t] with lex order" + + +def test_FracField(): + assert str(field("x", ZZ, lex)[0]) == "Rational function field in x over ZZ with lex order" + assert str(field("x,y", QQ, grlex)[0]) == "Rational function field in x, y over QQ with grlex order" + assert str(field("x,y,z", ZZ["t"], lex)[0]) == "Rational function field in x, y, z over ZZ[t] with lex order" + + +def test_PolyElement(): + Ruv, u,v = ring("u,v", ZZ) + Rxyz, x,y,z = ring("x,y,z", Ruv) + Rx_zzi, xz = ring("x", ZZ_I) + + assert str(x - x) == "0" + assert str(x - 1) == "x - 1" + assert str(x + 1) == "x + 1" + assert str(x**2) == "x**2" + + assert str((u**2 + 3*u*v + 1)*x**2*y + u + 1) == "(u**2 + 3*u*v + 1)*x**2*y + u + 1" + assert str((u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x) == "(u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x" + assert str((u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x + 1) == "(u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x + 1" + assert str((-u**2 + 3*u*v - 1)*x**2*y - (u + 1)*x - 1) == "-(u**2 - 3*u*v + 1)*x**2*y - (u + 1)*x - 1" + + assert str(-(v**2 + v + 1)*x + 3*u*v + 1) == "-(v**2 + v + 1)*x + 3*u*v + 1" + assert str(-(v**2 + v + 1)*x - 3*u*v + 1) == "-(v**2 + v + 1)*x - 3*u*v + 1" + + assert str((1+I)*xz + 2) == "(1 + 1*I)*x + (2 + 0*I)" + + +def test_FracElement(): + Fuv, u,v = field("u,v", ZZ) + Fxyzt, x,y,z,t = field("x,y,z,t", Fuv) + Rx_zzi, xz = field("x", QQ_I) + i = QQ_I(0, 1) + + assert str(x - x) == "0" + assert str(x - 1) == "x - 1" + assert str(x + 1) == "x + 1" + + assert str(x/3) == "x/3" + assert str(x/z) == "x/z" + assert str(x*y/z) == "x*y/z" + assert str(x/(z*t)) == "x/(z*t)" + assert str(x*y/(z*t)) == "x*y/(z*t)" + + assert str((x - 1)/y) == "(x - 1)/y" + assert str((x + 1)/y) == "(x + 1)/y" + assert str((-x - 1)/y) == "(-x - 1)/y" + assert str((x + 1)/(y*z)) == "(x + 1)/(y*z)" + assert str(-y/(x + 1)) == "-y/(x + 1)" + assert str(y*z/(x + 1)) == "y*z/(x + 1)" + + assert str(((u + 1)*x*y + 1)/((v - 1)*z - 1)) == "((u + 1)*x*y + 1)/((v - 1)*z - 1)" + assert str(((u + 1)*x*y + 1)/((v - 1)*z - t*u*v - 1)) == "((u + 1)*x*y + 1)/((v - 1)*z - u*v*t - 1)" + + assert str((1+i)/xz) == "(1 + 1*I)/x" + assert str(((1+i)*xz - i)/xz) == "((1 + 1*I)*x + (0 + -1*I))/x" + + +def test_GaussianInteger(): + assert str(ZZ_I(1, 0)) == "1" + assert str(ZZ_I(-1, 0)) == "-1" + assert str(ZZ_I(0, 1)) == "I" + assert str(ZZ_I(0, -1)) == "-I" + assert str(ZZ_I(0, 2)) == "2*I" + assert str(ZZ_I(0, -2)) == "-2*I" + assert str(ZZ_I(1, 1)) == "1 + I" + assert str(ZZ_I(-1, -1)) == "-1 - I" + assert str(ZZ_I(-1, -2)) == "-1 - 2*I" + + +def test_GaussianRational(): + assert str(QQ_I(1, 0)) == "1" + assert str(QQ_I(QQ(2, 3), 0)) == "2/3" + assert str(QQ_I(0, QQ(2, 3))) == "2*I/3" + assert str(QQ_I(QQ(1, 2), QQ(-2, 3))) == "1/2 - 2*I/3" + + +def test_Pow(): + assert str(x**-1) == "1/x" + assert str(x**-2) == "x**(-2)" + assert str(x**2) == "x**2" + assert str((x + y)**-1) == "1/(x + y)" + assert str((x + y)**-2) == "(x + y)**(-2)" + assert str((x + y)**2) == "(x + y)**2" + assert str((x + y)**(1 + x)) == "(x + y)**(x + 1)" + assert str(x**Rational(1, 3)) == "x**(1/3)" + assert str(1/x**Rational(1, 3)) == "x**(-1/3)" + assert str(sqrt(sqrt(x))) == "x**(1/4)" + # not the same as x**-1 + assert str(x**-1.0) == 'x**(-1.0)' + # see issue #2860 + assert str(Pow(S(2), -1.0, evaluate=False)) == '2**(-1.0)' + + +def test_sqrt(): + assert str(sqrt(x)) == "sqrt(x)" + assert str(sqrt(x**2)) == "sqrt(x**2)" + assert str(1/sqrt(x)) == "1/sqrt(x)" + assert str(1/sqrt(x**2)) == "1/sqrt(x**2)" + assert str(y/sqrt(x)) == "y/sqrt(x)" + assert str(x**0.5) == "x**0.5" + assert str(1/x**0.5) == "x**(-0.5)" + + +def test_Rational(): + n1 = Rational(1, 4) + n2 = Rational(1, 3) + n3 = Rational(2, 4) + n4 = Rational(2, -4) + n5 = Rational(0) + n7 = Rational(3) + n8 = Rational(-3) + assert str(n1*n2) == "1/12" + assert str(n1*n2) == "1/12" + assert str(n3) == "1/2" + assert str(n1*n3) == "1/8" + assert str(n1 + n3) == "3/4" + assert str(n1 + n2) == "7/12" + assert str(n1 + n4) == "-1/4" + assert str(n4*n4) == "1/4" + assert str(n4 + n2) == "-1/6" + assert str(n4 + n5) == "-1/2" + assert str(n4*n5) == "0" + assert str(n3 + n4) == "0" + assert str(n1**n7) == "1/64" + assert str(n2**n7) == "1/27" + assert str(n2**n8) == "27" + assert str(n7**n8) == "1/27" + assert str(Rational("-25")) == "-25" + assert str(Rational("1.25")) == "5/4" + assert str(Rational("-2.6e-2")) == "-13/500" + assert str(S("25/7")) == "25/7" + assert str(S("-123/569")) == "-123/569" + assert str(S("0.1[23]", rational=1)) == "61/495" + assert str(S("5.1[666]", rational=1)) == "31/6" + assert str(S("-5.1[666]", rational=1)) == "-31/6" + assert str(S("0.[9]", rational=1)) == "1" + assert str(S("-0.[9]", rational=1)) == "-1" + + assert str(sqrt(Rational(1, 4))) == "1/2" + assert str(sqrt(Rational(1, 36))) == "1/6" + + assert str((123**25) ** Rational(1, 25)) == "123" + assert str((123**25 + 1)**Rational(1, 25)) != "123" + assert str((123**25 - 1)**Rational(1, 25)) != "123" + assert str((123**25 - 1)**Rational(1, 25)) != "122" + + assert str(sqrt(Rational(81, 36))**3) == "27/8" + assert str(1/sqrt(Rational(81, 36))**3) == "8/27" + + assert str(sqrt(-4)) == str(2*I) + assert str(2**Rational(1, 10**10)) == "2**(1/10000000000)" + + assert sstr(Rational(2, 3), sympy_integers=True) == "S(2)/3" + x = Symbol("x") + assert sstr(x**Rational(2, 3), sympy_integers=True) == "x**(S(2)/3)" + assert sstr(Eq(x, Rational(2, 3)), sympy_integers=True) == "Eq(x, S(2)/3)" + assert sstr(Limit(x, x, Rational(7, 2)), sympy_integers=True) == \ + "Limit(x, x, S(7)/2, dir='+')" + + +def test_Float(): + # NOTE dps is the whole number of decimal digits + assert str(Float('1.23', dps=1 + 2)) == '1.23' + assert str(Float('1.23456789', dps=1 + 8)) == '1.23456789' + assert str( + Float('1.234567890123456789', dps=1 + 18)) == '1.234567890123456789' + assert str(pi.evalf(1 + 2)) == '3.14' + assert str(pi.evalf(1 + 14)) == '3.14159265358979' + assert str(pi.evalf(1 + 64)) == ('3.141592653589793238462643383279' + '5028841971693993751058209749445923') + assert str(pi.round(-1)) == '0.0' + assert str((pi**400 - (pi**400).round(1)).n(2)) == '-0.e+88' + assert sstr(Float("100"), full_prec=False, min=-2, max=2) == '1.0e+2' + assert sstr(Float("100"), full_prec=False, min=-2, max=3) == '100.0' + assert sstr(Float("0.1"), full_prec=False, min=-2, max=3) == '0.1' + assert sstr(Float("0.099"), min=-2, max=3) == '9.90000000000000e-2' + + +def test_Relational(): + assert str(Rel(x, y, "<")) == "x < y" + assert str(Rel(x + y, y, "==")) == "Eq(x + y, y)" + assert str(Rel(x, y, "!=")) == "Ne(x, y)" + assert str(Eq(x, 1) | Eq(x, 2)) == "Eq(x, 1) | Eq(x, 2)" + assert str(Ne(x, 1) & Ne(x, 2)) == "Ne(x, 1) & Ne(x, 2)" + + +def test_AppliedBinaryRelation(): + assert str(Q.eq(x, y)) == "Q.eq(x, y)" + assert str(Q.ne(x, y)) == "Q.ne(x, y)" + + +def test_CRootOf(): + assert str(rootof(x**5 + 2*x - 1, 0)) == "CRootOf(x**5 + 2*x - 1, 0)" + + +def test_RootSum(): + f = x**5 + 2*x - 1 + + assert str( + RootSum(f, Lambda(z, z), auto=False)) == "RootSum(x**5 + 2*x - 1)" + assert str(RootSum(f, Lambda( + z, z**2), auto=False)) == "RootSum(x**5 + 2*x - 1, Lambda(z, z**2))" + + +def test_GroebnerBasis(): + assert str(groebner( + [], x, y)) == "GroebnerBasis([], x, y, domain='ZZ', order='lex')" + + F = [x**2 - 3*y - x + 1, y**2 - 2*x + y - 1] + + assert str(groebner(F, order='grlex')) == \ + "GroebnerBasis([x**2 - x - 3*y + 1, y**2 - 2*x + y - 1], x, y, domain='ZZ', order='grlex')" + assert str(groebner(F, order='lex')) == \ + "GroebnerBasis([2*x - y**2 - y + 1, y**4 + 2*y**3 - 3*y**2 - 16*y + 7], x, y, domain='ZZ', order='lex')" + +def test_set(): + assert sstr(set()) == 'set()' + assert sstr(frozenset()) == 'frozenset()' + + assert sstr({1}) == '{1}' + assert sstr(frozenset([1])) == 'frozenset({1})' + assert sstr({1, 2, 3}) == '{1, 2, 3}' + assert sstr(frozenset([1, 2, 3])) == 'frozenset({1, 2, 3})' + + assert sstr( + {1, x, x**2, x**3, x**4}) == '{1, x, x**2, x**3, x**4}' + assert sstr( + frozenset([1, x, x**2, x**3, x**4])) == 'frozenset({1, x, x**2, x**3, x**4})' + + +def test_SparseMatrix(): + M = SparseMatrix([[x**+1, 1], [y, x + y]]) + assert str(M) == "Matrix([[x, 1], [y, x + y]])" + assert sstr(M) == "Matrix([\n[x, 1],\n[y, x + y]])" + + +def test_Sum(): + assert str(summation(cos(3*z), (z, x, y))) == "Sum(cos(3*z), (z, x, y))" + assert str(Sum(x*y**2, (x, -2, 2), (y, -5, 5))) == \ + "Sum(x*y**2, (x, -2, 2), (y, -5, 5))" + + +def test_Symbol(): + assert str(y) == "y" + assert str(x) == "x" + e = x + assert str(e) == "x" + + +def test_tuple(): + assert str((x,)) == sstr((x,)) == "(x,)" + assert str((x + y, 1 + x)) == sstr((x + y, 1 + x)) == "(x + y, x + 1)" + assert str((x + y, ( + 1 + x, x**2))) == sstr((x + y, (1 + x, x**2))) == "(x + y, (x + 1, x**2))" + + +def test_Series_str(): + tf1 = TransferFunction(x*y**2 - z, y**3 - t**3, y) + tf2 = TransferFunction(x - y, x + y, y) + tf3 = TransferFunction(t*x**2 - t**w*x + w, t - y, y) + assert str(Series(tf1, tf2)) == \ + "Series(TransferFunction(x*y**2 - z, -t**3 + y**3, y), TransferFunction(x - y, x + y, y))" + assert str(Series(tf1, tf2, tf3)) == \ + "Series(TransferFunction(x*y**2 - z, -t**3 + y**3, y), TransferFunction(x - y, x + y, y), TransferFunction(t*x**2 - t**w*x + w, t - y, y))" + assert str(Series(-tf2, tf1)) == \ + "Series(TransferFunction(-x + y, x + y, y), TransferFunction(x*y**2 - z, -t**3 + y**3, y))" + + +def test_MIMOSeries_str(): + tf1 = TransferFunction(x*y**2 - z, y**3 - t**3, y) + tf2 = TransferFunction(x - y, x + y, y) + tfm_1 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf1]]) + tfm_2 = TransferFunctionMatrix([[tf2, tf1], [tf1, tf2]]) + assert str(MIMOSeries(tfm_1, tfm_2)) == \ + "MIMOSeries(TransferFunctionMatrix(((TransferFunction(x*y**2 - z, -t**3 + y**3, y), TransferFunction(x - y, x + y, y)), "\ + "(TransferFunction(x - y, x + y, y), TransferFunction(x*y**2 - z, -t**3 + y**3, y)))), "\ + "TransferFunctionMatrix(((TransferFunction(x - y, x + y, y), TransferFunction(x*y**2 - z, -t**3 + y**3, y)), "\ + "(TransferFunction(x*y**2 - z, -t**3 + y**3, y), TransferFunction(x - y, x + y, y)))))" + + +def test_TransferFunction_str(): + tf1 = TransferFunction(x - 1, x + 1, x) + assert str(tf1) == "TransferFunction(x - 1, x + 1, x)" + tf2 = TransferFunction(x + 1, 2 - y, x) + assert str(tf2) == "TransferFunction(x + 1, 2 - y, x)" + tf3 = TransferFunction(y, y**2 + 2*y + 3, y) + assert str(tf3) == "TransferFunction(y, y**2 + 2*y + 3, y)" + + +def test_Parallel_str(): + tf1 = TransferFunction(x*y**2 - z, y**3 - t**3, y) + tf2 = TransferFunction(x - y, x + y, y) + tf3 = TransferFunction(t*x**2 - t**w*x + w, t - y, y) + assert str(Parallel(tf1, tf2)) == \ + "Parallel(TransferFunction(x*y**2 - z, -t**3 + y**3, y), TransferFunction(x - y, x + y, y))" + assert str(Parallel(tf1, tf2, tf3)) == \ + "Parallel(TransferFunction(x*y**2 - z, -t**3 + y**3, y), TransferFunction(x - y, x + y, y), TransferFunction(t*x**2 - t**w*x + w, t - y, y))" + assert str(Parallel(-tf2, tf1)) == \ + "Parallel(TransferFunction(-x + y, x + y, y), TransferFunction(x*y**2 - z, -t**3 + y**3, y))" + + +def test_MIMOParallel_str(): + tf1 = TransferFunction(x*y**2 - z, y**3 - t**3, y) + tf2 = TransferFunction(x - y, x + y, y) + tfm_1 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf1]]) + tfm_2 = TransferFunctionMatrix([[tf2, tf1], [tf1, tf2]]) + assert str(MIMOParallel(tfm_1, tfm_2)) == \ + "MIMOParallel(TransferFunctionMatrix(((TransferFunction(x*y**2 - z, -t**3 + y**3, y), TransferFunction(x - y, x + y, y)), "\ + "(TransferFunction(x - y, x + y, y), TransferFunction(x*y**2 - z, -t**3 + y**3, y)))), "\ + "TransferFunctionMatrix(((TransferFunction(x - y, x + y, y), TransferFunction(x*y**2 - z, -t**3 + y**3, y)), "\ + "(TransferFunction(x*y**2 - z, -t**3 + y**3, y), TransferFunction(x - y, x + y, y)))))" + + +def test_Feedback_str(): + tf1 = TransferFunction(x*y**2 - z, y**3 - t**3, y) + tf2 = TransferFunction(x - y, x + y, y) + tf3 = TransferFunction(t*x**2 - t**w*x + w, t - y, y) + assert str(Feedback(tf1*tf2, tf3)) == \ + "Feedback(Series(TransferFunction(x*y**2 - z, -t**3 + y**3, y), TransferFunction(x - y, x + y, y)), " \ + "TransferFunction(t*x**2 - t**w*x + w, t - y, y), -1)" + assert str(Feedback(tf1, TransferFunction(1, 1, y), 1)) == \ + "Feedback(TransferFunction(x*y**2 - z, -t**3 + y**3, y), TransferFunction(1, 1, y), 1)" + + +def test_MIMOFeedback_str(): + tf1 = TransferFunction(x**2 - y**3, y - z, x) + tf2 = TransferFunction(y - x, z + y, x) + tfm_1 = TransferFunctionMatrix([[tf2, tf1], [tf1, tf2]]) + tfm_2 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf1]]) + assert (str(MIMOFeedback(tfm_1, tfm_2)) \ + == "MIMOFeedback(TransferFunctionMatrix(((TransferFunction(-x + y, y + z, x), TransferFunction(x**2 - y**3, y - z, x))," \ + " (TransferFunction(x**2 - y**3, y - z, x), TransferFunction(-x + y, y + z, x)))), " \ + "TransferFunctionMatrix(((TransferFunction(x**2 - y**3, y - z, x), " \ + "TransferFunction(-x + y, y + z, x)), (TransferFunction(-x + y, y + z, x), TransferFunction(x**2 - y**3, y - z, x)))), -1)") + assert (str(MIMOFeedback(tfm_1, tfm_2, 1)) \ + == "MIMOFeedback(TransferFunctionMatrix(((TransferFunction(-x + y, y + z, x), TransferFunction(x**2 - y**3, y - z, x)), " \ + "(TransferFunction(x**2 - y**3, y - z, x), TransferFunction(-x + y, y + z, x)))), " \ + "TransferFunctionMatrix(((TransferFunction(x**2 - y**3, y - z, x), TransferFunction(-x + y, y + z, x)), "\ + "(TransferFunction(-x + y, y + z, x), TransferFunction(x**2 - y**3, y - z, x)))), 1)") + + +def test_TransferFunctionMatrix_str(): + tf1 = TransferFunction(x*y**2 - z, y**3 - t**3, y) + tf2 = TransferFunction(x - y, x + y, y) + tf3 = TransferFunction(t*x**2 - t**w*x + w, t - y, y) + assert str(TransferFunctionMatrix([[tf1], [tf2]])) == \ + "TransferFunctionMatrix(((TransferFunction(x*y**2 - z, -t**3 + y**3, y),), (TransferFunction(x - y, x + y, y),)))" + assert str(TransferFunctionMatrix([[tf1, tf2], [tf3, tf2]])) == \ + "TransferFunctionMatrix(((TransferFunction(x*y**2 - z, -t**3 + y**3, y), TransferFunction(x - y, x + y, y)), (TransferFunction(t*x**2 - t**w*x + w, t - y, y), TransferFunction(x - y, x + y, y))))" + + +def test_Quaternion_str_printer(): + q = Quaternion(x, y, z, t) + assert str(q) == "x + y*i + z*j + t*k" + q = Quaternion(x,y,z,x*t) + assert str(q) == "x + y*i + z*j + t*x*k" + q = Quaternion(x,y,z,x+t) + assert str(q) == "x + y*i + z*j + (t + x)*k" + + +def test_Quantity_str(): + assert sstr(second, abbrev=True) == "s" + assert sstr(joule, abbrev=True) == "J" + assert str(second) == "second" + assert str(joule) == "joule" + + +def test_wild_str(): + # Check expressions containing Wild not causing infinite recursion + w = Wild('x') + assert str(w + 1) == 'x_ + 1' + assert str(exp(2**w) + 5) == 'exp(2**x_) + 5' + assert str(3*w + 1) == '3*x_ + 1' + assert str(1/w + 1) == '1 + 1/x_' + assert str(w**2 + 1) == 'x_**2 + 1' + assert str(1/(1 - w)) == '1/(1 - x_)' + + +def test_wild_matchpy(): + from sympy.utilities.matchpy_connector import WildDot, WildPlus, WildStar + + matchpy = import_module("matchpy") + + if matchpy is None: + return + + wd = WildDot('w_') + wp = WildPlus('w__') + ws = WildStar('w___') + + assert str(wd) == 'w_' + assert str(wp) == 'w__' + assert str(ws) == 'w___' + + assert str(wp/ws + 2**wd) == '2**w_ + w__/w___' + assert str(sin(wd)*cos(wp)*sqrt(ws)) == 'sqrt(w___)*sin(w_)*cos(w__)' + + +def test_zeta(): + assert str(zeta(3)) == "zeta(3)" + + +def test_issue_3101(): + e = x - y + a = str(e) + b = str(e) + assert a == b + + +def test_issue_3103(): + e = -2*sqrt(x) - y/sqrt(x)/2 + assert str(e) not in ["(-2)*x**1/2(-1/2)*x**(-1/2)*y", + "-2*x**1/2(-1/2)*x**(-1/2)*y", "-2*x**1/2-1/2*x**-1/2*w"] + assert str(e) == "-2*sqrt(x) - y/(2*sqrt(x))" + + +def test_issue_4021(): + e = Integral(x, x) + 1 + assert str(e) == 'Integral(x, x) + 1' + + +def test_sstrrepr(): + assert sstr('abc') == 'abc' + assert sstrrepr('abc') == "'abc'" + + e = ['a', 'b', 'c', x] + assert sstr(e) == "[a, b, c, x]" + assert sstrrepr(e) == "['a', 'b', 'c', x]" + + +def test_infinity(): + assert sstr(oo*I) == "oo*I" + + +def test_full_prec(): + assert sstr(S("0.3"), full_prec=True) == "0.300000000000000" + assert sstr(S("0.3"), full_prec="auto") == "0.300000000000000" + assert sstr(S("0.3"), full_prec=False) == "0.3" + assert sstr(S("0.3")*x, full_prec=True) in [ + "0.300000000000000*x", + "x*0.300000000000000" + ] + assert sstr(S("0.3")*x, full_prec="auto") in [ + "0.3*x", + "x*0.3" + ] + assert sstr(S("0.3")*x, full_prec=False) in [ + "0.3*x", + "x*0.3" + ] + + +def test_noncommutative(): + A, B, C = symbols('A,B,C', commutative=False) + + assert sstr(A*B*C**-1) == "A*B*C**(-1)" + assert sstr(C**-1*A*B) == "C**(-1)*A*B" + assert sstr(A*C**-1*B) == "A*C**(-1)*B" + assert sstr(sqrt(A)) == "sqrt(A)" + assert sstr(1/sqrt(A)) == "A**(-1/2)" + + +def test_empty_printer(): + str_printer = StrPrinter() + assert str_printer.emptyPrinter("foo") == "foo" + assert str_printer.emptyPrinter(x*y) == "x*y" + assert str_printer.emptyPrinter(32) == "32" + +def test_decimal_printer(): + dec_printer = StrPrinter(settings={"dps":3}) + f = Function('f') + assert dec_printer.doprint(f(1.329294)) == "f(1.33)" + + +def test_settings(): + raises(TypeError, lambda: sstr(S(4), method="garbage")) + + +def test_RandomDomain(): + from sympy.stats import Normal, Die, Exponential, pspace, where + X = Normal('x1', 0, 1) + assert str(where(X > 0)) == "Domain: (0 < x1) & (x1 < oo)" + + D = Die('d1', 6) + assert str(where(D > 4)) == "Domain: Eq(d1, 5) | Eq(d1, 6)" + + A = Exponential('a', 1) + B = Exponential('b', 1) + assert str(pspace(Tuple(A, B)).domain) == "Domain: (0 <= a) & (0 <= b) & (a < oo) & (b < oo)" + + +def test_FiniteSet(): + assert str(FiniteSet(*range(1, 51))) == ( + '{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,' + ' 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34,' + ' 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50}' + ) + assert str(FiniteSet(*range(1, 6))) == '{1, 2, 3, 4, 5}' + assert str(FiniteSet(*[x*y, x**2])) == '{x**2, x*y}' + assert str(FiniteSet(FiniteSet(FiniteSet(x, y), 5), FiniteSet(x,y), 5) + ) == 'FiniteSet(5, FiniteSet(5, {x, y}), {x, y})' + + +def test_Partition(): + assert str(Partition(FiniteSet(x, y), {z})) == 'Partition({z}, {x, y})' + +def test_UniversalSet(): + assert str(S.UniversalSet) == 'UniversalSet' + + +def test_PrettyPoly(): + F = QQ.frac_field(x, y) + R = QQ[x, y] + assert sstr(F.convert(x/(x + y))) == sstr(x/(x + y)) + assert sstr(R.convert(x + y)) == sstr(x + y) + + +def test_categories(): + from sympy.categories import (Object, NamedMorphism, + IdentityMorphism, Category) + + A = Object("A") + B = Object("B") + + f = NamedMorphism(A, B, "f") + id_A = IdentityMorphism(A) + + K = Category("K") + + assert str(A) == 'Object("A")' + assert str(f) == 'NamedMorphism(Object("A"), Object("B"), "f")' + assert str(id_A) == 'IdentityMorphism(Object("A"))' + + assert str(K) == 'Category("K")' + + +def test_Tr(): + A, B = symbols('A B', commutative=False) + t = Tr(A*B) + assert str(t) == 'Tr(A*B)' + + +def test_issue_6387(): + assert str(factor(-3.0*z + 3)) == '-3.0*(1.0*z - 1.0)' + + +def test_MatMul_MatAdd(): + X, Y = MatrixSymbol("X", 2, 2), MatrixSymbol("Y", 2, 2) + assert str(2*(X + Y)) == "2*X + 2*Y" + + assert str(I*X) == "I*X" + assert str(-I*X) == "-I*X" + assert str((1 + I)*X) == '(1 + I)*X' + assert str(-(1 + I)*X) == '(-1 - I)*X' + assert str(MatAdd(MatAdd(X, Y), MatAdd(X, Y))) == '(X + Y) + (X + Y)' + + +def test_MatrixSlice(): + n = Symbol('n', integer=True) + X = MatrixSymbol('X', n, n) + Y = MatrixSymbol('Y', 10, 10) + Z = MatrixSymbol('Z', 10, 10) + + assert str(MatrixSlice(X, (None, None, None), (None, None, None))) == 'X[:, :]' + assert str(X[x:x + 1, y:y + 1]) == 'X[x:x + 1, y:y + 1]' + assert str(X[x:x + 1:2, y:y + 1:2]) == 'X[x:x + 1:2, y:y + 1:2]' + assert str(X[:x, y:]) == 'X[:x, y:]' + assert str(X[:x, y:]) == 'X[:x, y:]' + assert str(X[x:, :y]) == 'X[x:, :y]' + assert str(X[x:y, z:w]) == 'X[x:y, z:w]' + assert str(X[x:y:t, w:t:x]) == 'X[x:y:t, w:t:x]' + assert str(X[x::y, t::w]) == 'X[x::y, t::w]' + assert str(X[:x:y, :t:w]) == 'X[:x:y, :t:w]' + assert str(X[::x, ::y]) == 'X[::x, ::y]' + assert str(MatrixSlice(X, (0, None, None), (0, None, None))) == 'X[:, :]' + assert str(MatrixSlice(X, (None, n, None), (None, n, None))) == 'X[:, :]' + assert str(MatrixSlice(X, (0, n, None), (0, n, None))) == 'X[:, :]' + assert str(MatrixSlice(X, (0, n, 2), (0, n, 2))) == 'X[::2, ::2]' + assert str(X[1:2:3, 4:5:6]) == 'X[1:2:3, 4:5:6]' + assert str(X[1:3:5, 4:6:8]) == 'X[1:3:5, 4:6:8]' + assert str(X[1:10:2]) == 'X[1:10:2, :]' + assert str(Y[:5, 1:9:2]) == 'Y[:5, 1:9:2]' + assert str(Y[:5, 1:10:2]) == 'Y[:5, 1::2]' + assert str(Y[5, :5:2]) == 'Y[5:6, :5:2]' + assert str(X[0:1, 0:1]) == 'X[:1, :1]' + assert str(X[0:1:2, 0:1:2]) == 'X[:1:2, :1:2]' + assert str((Y + Z)[2:, 2:]) == '(Y + Z)[2:, 2:]' + +def test_true_false(): + assert str(true) == repr(true) == sstr(true) == "True" + assert str(false) == repr(false) == sstr(false) == "False" + +def test_Equivalent(): + assert str(Equivalent(y, x)) == "Equivalent(x, y)" + +def test_Xor(): + assert str(Xor(y, x, evaluate=False)) == "x ^ y" + +def test_Complement(): + assert str(Complement(S.Reals, S.Naturals)) == 'Complement(Reals, Naturals)' + +def test_SymmetricDifference(): + assert str(SymmetricDifference(Interval(2, 3), Interval(3, 4),evaluate=False)) == \ + 'SymmetricDifference(Interval(2, 3), Interval(3, 4))' + + +def test_UnevaluatedExpr(): + a, b = symbols("a b") + expr1 = 2*UnevaluatedExpr(a+b) + assert str(expr1) == "2*(a + b)" + + +def test_MatrixElement_printing(): + # test cases for issue #11821 + A = MatrixSymbol("A", 1, 3) + B = MatrixSymbol("B", 1, 3) + C = MatrixSymbol("C", 1, 3) + + assert(str(A[0, 0]) == "A[0, 0]") + assert(str(3 * A[0, 0]) == "3*A[0, 0]") + + F = C[0, 0].subs(C, A - B) + assert str(F) == "(A - B)[0, 0]" + + +def test_MatrixSymbol_printing(): + A = MatrixSymbol("A", 3, 3) + B = MatrixSymbol("B", 3, 3) + + assert str(A - A*B - B) == "A - A*B - B" + assert str(A*B - (A+B)) == "-A + A*B - B" + assert str(A**(-1)) == "A**(-1)" + assert str(A**3) == "A**3" + + +def test_MatrixExpressions(): + n = Symbol('n', integer=True) + X = MatrixSymbol('X', n, n) + + assert str(X) == "X" + + # Apply function elementwise (`ElementwiseApplyFunc`): + + expr = (X.T*X).applyfunc(sin) + assert str(expr) == 'Lambda(_d, sin(_d)).(X.T*X)' + + lamda = Lambda(x, 1/x) + expr = (n*X).applyfunc(lamda) + assert str(expr) == 'Lambda(x, 1/x).(n*X)' + + +def test_Subs_printing(): + assert str(Subs(x, (x,), (1,))) == 'Subs(x, x, 1)' + assert str(Subs(x + y, (x, y), (1, 2))) == 'Subs(x + y, (x, y), (1, 2))' + + +def test_issue_15716(): + e = Integral(factorial(x), (x, -oo, oo)) + assert e.as_terms() == ([(e, ((1.0, 0.0), (1,), ()))], [e]) + + +def test_str_special_matrices(): + from sympy.matrices import Identity, ZeroMatrix, OneMatrix + assert str(Identity(4)) == 'I' + assert str(ZeroMatrix(2, 2)) == '0' + assert str(OneMatrix(2, 2)) == '1' + + +def test_issue_14567(): + assert factorial(Sum(-1, (x, 0, 0))) + y # doesn't raise an error + + +def test_issue_21823(): + assert str(Partition([1, 2])) == 'Partition({1, 2})' + assert str(Partition({1, 2})) == 'Partition({1, 2})' + + +def test_issue_22689(): + assert str(Mul(Pow(x,-2, evaluate=False), Pow(3,-1,evaluate=False), evaluate=False)) == "1/(x**2*3)" + + +def test_issue_21119_21460(): + ss = lambda x: str(S(x, evaluate=False)) + assert ss('4/2') == '4/2' + assert ss('4/-2') == '4/(-2)' + assert ss('-4/2') == '-4/2' + assert ss('-4/-2') == '-4/(-2)' + assert ss('-2*3/-1') == '-2*3/(-1)' + assert ss('-2*3/-1/2') == '-2*3/(-1*2)' + assert ss('4/2/1') == '4/(2*1)' + assert ss('-2/-1/2') == '-2/(-1*2)' + assert ss('2*3*4**(-2*3)') == '2*3/4**(2*3)' + assert ss('2*3*1*4**(-2*3)') == '2*3*1/4**(2*3)' + + +def test_Str(): + from sympy.core.symbol import Str + assert str(Str('x')) == 'x' + assert sstrrepr(Str('x')) == "Str('x')" + + +def test_diffgeom(): + from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField + x,y = symbols('x y', real=True) + m = Manifold('M', 2) + assert str(m) == "M" + p = Patch('P', m) + assert str(p) == "P" + rect = CoordSystem('rect', p, [x, y]) + assert str(rect) == "rect" + b = BaseScalarField(rect, 0) + assert str(b) == "x" + +def test_NDimArray(): + assert sstr(NDimArray(1.0), full_prec=True) == '1.00000000000000' + assert sstr(NDimArray(1.0), full_prec=False) == '1.0' + assert sstr(NDimArray([1.0, 2.0]), full_prec=True) == '[1.00000000000000, 2.00000000000000]' + assert sstr(NDimArray([1.0, 2.0]), full_prec=False) == '[1.0, 2.0]' + assert sstr(NDimArray([], (0,))) == 'ImmutableDenseNDimArray([], (0,))' + assert sstr(NDimArray([], (0, 0))) == 'ImmutableDenseNDimArray([], (0, 0))' + assert sstr(NDimArray([], (0, 1))) == 'ImmutableDenseNDimArray([], (0, 1))' + assert sstr(NDimArray([], (1, 0))) == 'ImmutableDenseNDimArray([], (1, 0))' + +def test_Predicate(): + assert sstr(Q.even) == 'Q.even' + +def test_AppliedPredicate(): + assert sstr(Q.even(x)) == 'Q.even(x)' + +def test_printing_str_array_expressions(): + assert sstr(ArraySymbol("A", (2, 3, 4))) == "A" + assert sstr(ArrayElement("A", (2, 1/(1-x), 0))) == "A[2, 1/(1 - x), 0]" + M = MatrixSymbol("M", 3, 3) + N = MatrixSymbol("N", 3, 3) + assert sstr(ArrayElement(M*N, [x, 0])) == "(M*N)[x, 0]" + +def test_printing_stats(): + # issue 24132 + x = RandomSymbol("x") + y = RandomSymbol("y") + z1 = Probability(x > 0)*Identity(2) + z2 = Expectation(x)*Identity(2) + z3 = Variance(x)*Identity(2) + z4 = Covariance(x, y) * Identity(2) + + assert str(z1) == "Probability(x > 0)*I" + assert str(z2) == "Expectation(x)*I" + assert str(z3) == "Variance(x)*I" + assert str(z4) == "Covariance(x, y)*I" + assert z1.is_commutative == False + assert z2.is_commutative == False + assert z3.is_commutative == False + assert z4.is_commutative == False + assert z2._eval_is_commutative() == False + assert z3._eval_is_commutative() == False + assert z4._eval_is_commutative() == False diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_tableform.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_tableform.py new file mode 100644 index 0000000000000000000000000000000000000000..05802dd104a12f2f53d137167ecf31d201ff8dfc --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_tableform.py @@ -0,0 +1,182 @@ +from sympy.core.singleton import S +from sympy.printing.tableform import TableForm +from sympy.printing.latex import latex +from sympy.abc import x +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import sin +from sympy.testing.pytest import raises + +from textwrap import dedent + + +def test_TableForm(): + s = str(TableForm([["a", "b"], ["c", "d"], ["e", 0]], + headings="automatic")) + assert s == ( + ' | 1 2\n' + '-------\n' + '1 | a b\n' + '2 | c d\n' + '3 | e ' + ) + s = str(TableForm([["a", "b"], ["c", "d"], ["e", 0]], + headings="automatic", wipe_zeros=False)) + assert s == dedent('''\ + | 1 2 + ------- + 1 | a b + 2 | c d + 3 | e 0''') + s = str(TableForm([[x**2, "b"], ["c", x**2], ["e", "f"]], + headings=("automatic", None))) + assert s == ( + '1 | x**2 b \n' + '2 | c x**2\n' + '3 | e f ' + ) + s = str(TableForm([["a", "b"], ["c", "d"], ["e", "f"]], + headings=(None, "automatic"))) + assert s == dedent('''\ + 1 2 + --- + a b + c d + e f''') + s = str(TableForm([[5, 7], [4, 2], [10, 3]], + headings=[["Group A", "Group B", "Group C"], ["y1", "y2"]])) + assert s == ( + ' | y1 y2\n' + '---------------\n' + 'Group A | 5 7 \n' + 'Group B | 4 2 \n' + 'Group C | 10 3 ' + ) + raises( + ValueError, + lambda: + TableForm( + [[5, 7], [4, 2], [10, 3]], + headings=[["Group A", "Group B", "Group C"], ["y1", "y2"]], + alignments="middle") + ) + s = str(TableForm([[5, 7], [4, 2], [10, 3]], + headings=[["Group A", "Group B", "Group C"], ["y1", "y2"]], + alignments="right")) + assert s == dedent('''\ + | y1 y2 + --------------- + Group A | 5 7 + Group B | 4 2 + Group C | 10 3''') + + # other alignment permutations + d = [[1, 100], [100, 1]] + s = TableForm(d, headings=(('xxx', 'x'), None), alignments='l') + assert str(s) == ( + 'xxx | 1 100\n' + ' x | 100 1 ' + ) + s = TableForm(d, headings=(('xxx', 'x'), None), alignments='lr') + assert str(s) == dedent('''\ + xxx | 1 100 + x | 100 1''') + s = TableForm(d, headings=(('xxx', 'x'), None), alignments='clr') + assert str(s) == dedent('''\ + xxx | 1 100 + x | 100 1''') + + s = TableForm(d, headings=(('xxx', 'x'), None)) + assert str(s) == ( + 'xxx | 1 100\n' + ' x | 100 1 ' + ) + + raises(ValueError, lambda: TableForm(d, alignments='clr')) + + #pad + s = str(TableForm([[None, "-", 2], [1]], pad='?')) + assert s == dedent('''\ + ? - 2 + 1 ? ?''') + + +def test_TableForm_latex(): + s = latex(TableForm([[0, x**3], ["c", S.One/4], [sqrt(x), sin(x**2)]], + wipe_zeros=True, headings=("automatic", "automatic"))) + assert s == ( + '\\begin{tabular}{r l l}\n' + ' & 1 & 2 \\\\\n' + '\\hline\n' + '1 & & $x^{3}$ \\\\\n' + '2 & $c$ & $\\frac{1}{4}$ \\\\\n' + '3 & $\\sqrt{x}$ & $\\sin{\\left(x^{2} \\right)}$ \\\\\n' + '\\end{tabular}' + ) + s = latex(TableForm([[0, x**3], ["c", S.One/4], [sqrt(x), sin(x**2)]], + wipe_zeros=True, headings=("automatic", "automatic"), alignments='l')) + assert s == ( + '\\begin{tabular}{r l l}\n' + ' & 1 & 2 \\\\\n' + '\\hline\n' + '1 & & $x^{3}$ \\\\\n' + '2 & $c$ & $\\frac{1}{4}$ \\\\\n' + '3 & $\\sqrt{x}$ & $\\sin{\\left(x^{2} \\right)}$ \\\\\n' + '\\end{tabular}' + ) + s = latex(TableForm([[0, x**3], ["c", S.One/4], [sqrt(x), sin(x**2)]], + wipe_zeros=True, headings=("automatic", "automatic"), alignments='l'*3)) + assert s == ( + '\\begin{tabular}{l l l}\n' + ' & 1 & 2 \\\\\n' + '\\hline\n' + '1 & & $x^{3}$ \\\\\n' + '2 & $c$ & $\\frac{1}{4}$ \\\\\n' + '3 & $\\sqrt{x}$ & $\\sin{\\left(x^{2} \\right)}$ \\\\\n' + '\\end{tabular}' + ) + s = latex(TableForm([["a", x**3], ["c", S.One/4], [sqrt(x), sin(x**2)]], + headings=("automatic", "automatic"))) + assert s == ( + '\\begin{tabular}{r l l}\n' + ' & 1 & 2 \\\\\n' + '\\hline\n' + '1 & $a$ & $x^{3}$ \\\\\n' + '2 & $c$ & $\\frac{1}{4}$ \\\\\n' + '3 & $\\sqrt{x}$ & $\\sin{\\left(x^{2} \\right)}$ \\\\\n' + '\\end{tabular}' + ) + s = latex(TableForm([["a", x**3], ["c", S.One/4], [sqrt(x), sin(x**2)]], + formats=['(%s)', None], headings=("automatic", "automatic"))) + assert s == ( + '\\begin{tabular}{r l l}\n' + ' & 1 & 2 \\\\\n' + '\\hline\n' + '1 & (a) & $x^{3}$ \\\\\n' + '2 & (c) & $\\frac{1}{4}$ \\\\\n' + '3 & (sqrt(x)) & $\\sin{\\left(x^{2} \\right)}$ \\\\\n' + '\\end{tabular}' + ) + + def neg_in_paren(x, i, j): + if i % 2: + return ('(%s)' if x < 0 else '%s') % x + else: + pass # use default print + s = latex(TableForm([[-1, 2], [-3, 4]], + formats=[neg_in_paren]*2, headings=("automatic", "automatic"))) + assert s == ( + '\\begin{tabular}{r l l}\n' + ' & 1 & 2 \\\\\n' + '\\hline\n' + '1 & -1 & 2 \\\\\n' + '2 & (-3) & 4 \\\\\n' + '\\end{tabular}' + ) + s = latex(TableForm([["a", x**3], ["c", S.One/4], [sqrt(x), sin(x**2)]])) + assert s == ( + '\\begin{tabular}{l l}\n' + '$a$ & $x^{3}$ \\\\\n' + '$c$ & $\\frac{1}{4}$ \\\\\n' + '$\\sqrt{x}$ & $\\sin{\\left(x^{2} \\right)}$ \\\\\n' + '\\end{tabular}' + ) diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_tensorflow.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_tensorflow.py new file mode 100644 index 0000000000000000000000000000000000000000..e9c92cd17b13e1148ebf83f13f66854b983491fe --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_tensorflow.py @@ -0,0 +1,493 @@ +import random +from sympy.core.function import Derivative +from sympy.core.symbol import symbols +from sympy import Piecewise +from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct, ArrayAdd, \ + PermuteDims, ArrayDiagonal +from sympy.core.relational import Eq, Ne, Ge, Gt, Le, Lt +from sympy.external import import_module +from sympy.functions import \ + Abs, ceiling, exp, floor, sign, sin, asin, sqrt, cos, \ + acos, tan, atan, atan2, cosh, acosh, sinh, asinh, tanh, atanh, \ + re, im, arg, erf, loggamma, log +from sympy.codegen.cfunctions import isnan, isinf +from sympy.matrices import Matrix, MatrixBase, eye, randMatrix +from sympy.matrices.expressions import \ + Determinant, HadamardProduct, Inverse, MatrixSymbol, Trace +from sympy.printing.tensorflow import tensorflow_code +from sympy.tensor.array.expressions.from_matrix_to_array import convert_matrix_to_array +from sympy.utilities.lambdify import lambdify +from sympy.testing.pytest import skip +from sympy.testing.pytest import XFAIL + + +tf = tensorflow = import_module("tensorflow") + +if tensorflow: + # Hide Tensorflow warnings + import os + os.environ['TF_CPP_MIN_LOG_LEVEL'] = '2' + + +M = MatrixSymbol("M", 3, 3) +N = MatrixSymbol("N", 3, 3) +P = MatrixSymbol("P", 3, 3) +Q = MatrixSymbol("Q", 3, 3) + +x, y, z, t = symbols("x y z t") + +if tf is not None: + llo = [list(range(i, i+3)) for i in range(0, 9, 3)] + m3x3 = tf.constant(llo) + m3x3sympy = Matrix(llo) + + +def _compare_tensorflow_matrix(variables, expr, use_float=False): + f = lambdify(variables, expr, 'tensorflow') + if not use_float: + random_matrices = [randMatrix(v.rows, v.cols) for v in variables] + else: + random_matrices = [randMatrix(v.rows, v.cols)/100. for v in variables] + + graph = tf.Graph() + r = None + with graph.as_default(): + random_variables = [eval(tensorflow_code(i)) for i in random_matrices] + session = tf.compat.v1.Session(graph=graph) + r = session.run(f(*random_variables)) + + e = expr.subs(dict(zip(variables, random_matrices))) + e = e.doit() + if e.is_Matrix: + if not isinstance(e, MatrixBase): + e = e.as_explicit() + e = e.tolist() + + if not use_float: + assert (r == e).all() + else: + r = [i for row in r for i in row] + e = [i for row in e for i in row] + assert all( + abs(a-b) < 10**-(4-int(log(abs(a), 10))) for a, b in zip(r, e)) + + +# Creating a custom inverse test. +# See https://github.com/sympy/sympy/issues/18469 +def _compare_tensorflow_matrix_inverse(variables, expr, use_float=False): + f = lambdify(variables, expr, 'tensorflow') + if not use_float: + random_matrices = [eye(v.rows, v.cols)*4 for v in variables] + else: + random_matrices = [eye(v.rows, v.cols)*3.14 for v in variables] + + graph = tf.Graph() + r = None + with graph.as_default(): + random_variables = [eval(tensorflow_code(i)) for i in random_matrices] + session = tf.compat.v1.Session(graph=graph) + r = session.run(f(*random_variables)) + + e = expr.subs(dict(zip(variables, random_matrices))) + e = e.doit() + if e.is_Matrix: + if not isinstance(e, MatrixBase): + e = e.as_explicit() + e = e.tolist() + + if not use_float: + assert (r == e).all() + else: + r = [i for row in r for i in row] + e = [i for row in e for i in row] + assert all( + abs(a-b) < 10**-(4-int(log(abs(a), 10))) for a, b in zip(r, e)) + + +def _compare_tensorflow_matrix_scalar(variables, expr): + f = lambdify(variables, expr, 'tensorflow') + random_matrices = [ + randMatrix(v.rows, v.cols).evalf() / 100 for v in variables] + + graph = tf.Graph() + r = None + with graph.as_default(): + random_variables = [eval(tensorflow_code(i)) for i in random_matrices] + session = tf.compat.v1.Session(graph=graph) + r = session.run(f(*random_variables)) + + e = expr.subs(dict(zip(variables, random_matrices))) + e = e.doit() + assert abs(r-e) < 10**-6 + + +def _compare_tensorflow_scalar( + variables, expr, rng=lambda: random.randint(0, 10)): + f = lambdify(variables, expr, 'tensorflow') + rvs = [rng() for v in variables] + + graph = tf.Graph() + r = None + with graph.as_default(): + tf_rvs = [eval(tensorflow_code(i)) for i in rvs] + session = tf.compat.v1.Session(graph=graph) + r = session.run(f(*tf_rvs)) + + e = expr.subs(dict(zip(variables, rvs))).evalf().doit() + assert abs(r-e) < 10**-6 + + +def _compare_tensorflow_relational( + variables, expr, rng=lambda: random.randint(0, 10)): + f = lambdify(variables, expr, 'tensorflow') + rvs = [rng() for v in variables] + + graph = tf.Graph() + r = None + with graph.as_default(): + tf_rvs = [eval(tensorflow_code(i)) for i in rvs] + session = tf.compat.v1.Session(graph=graph) + r = session.run(f(*tf_rvs)) + + e = expr.subs(dict(zip(variables, rvs))).doit() + assert r == e + + +def test_tensorflow_printing(): + assert tensorflow_code(eye(3)) == \ + "tensorflow.constant([[1, 0, 0], [0, 1, 0], [0, 0, 1]])" + + expr = Matrix([[x, sin(y)], [exp(z), -t]]) + assert tensorflow_code(expr) == \ + "tensorflow.Variable(" \ + "[[x, tensorflow.math.sin(y)]," \ + " [tensorflow.math.exp(z), -t]])" + + +# This (random) test is XFAIL because it fails occasionally +# See https://github.com/sympy/sympy/issues/18469 +@XFAIL +def test_tensorflow_math(): + if not tf: + skip("TensorFlow not installed") + + expr = Abs(x) + assert tensorflow_code(expr) == "tensorflow.math.abs(x)" + _compare_tensorflow_scalar((x,), expr) + + expr = sign(x) + assert tensorflow_code(expr) == "tensorflow.math.sign(x)" + _compare_tensorflow_scalar((x,), expr) + + expr = ceiling(x) + assert tensorflow_code(expr) == "tensorflow.math.ceil(x)" + _compare_tensorflow_scalar((x,), expr, rng=lambda: random.random()) + + expr = floor(x) + assert tensorflow_code(expr) == "tensorflow.math.floor(x)" + _compare_tensorflow_scalar((x,), expr, rng=lambda: random.random()) + + expr = exp(x) + assert tensorflow_code(expr) == "tensorflow.math.exp(x)" + _compare_tensorflow_scalar((x,), expr, rng=lambda: random.random()) + + expr = sqrt(x) + assert tensorflow_code(expr) == "tensorflow.math.sqrt(x)" + _compare_tensorflow_scalar((x,), expr, rng=lambda: random.random()) + + expr = x ** 4 + assert tensorflow_code(expr) == "tensorflow.math.pow(x, 4)" + _compare_tensorflow_scalar((x,), expr, rng=lambda: random.random()) + + expr = cos(x) + assert tensorflow_code(expr) == "tensorflow.math.cos(x)" + _compare_tensorflow_scalar((x,), expr, rng=lambda: random.random()) + + expr = acos(x) + assert tensorflow_code(expr) == "tensorflow.math.acos(x)" + _compare_tensorflow_scalar((x,), expr, rng=lambda: random.uniform(0, 0.95)) + + expr = sin(x) + assert tensorflow_code(expr) == "tensorflow.math.sin(x)" + _compare_tensorflow_scalar((x,), expr, rng=lambda: random.random()) + + expr = asin(x) + assert tensorflow_code(expr) == "tensorflow.math.asin(x)" + _compare_tensorflow_scalar((x,), expr, rng=lambda: random.random()) + + expr = tan(x) + assert tensorflow_code(expr) == "tensorflow.math.tan(x)" + _compare_tensorflow_scalar((x,), expr, rng=lambda: random.random()) + + expr = atan(x) + assert tensorflow_code(expr) == "tensorflow.math.atan(x)" + _compare_tensorflow_scalar((x,), expr, rng=lambda: random.random()) + + expr = atan2(y, x) + assert tensorflow_code(expr) == "tensorflow.math.atan2(y, x)" + _compare_tensorflow_scalar((y, x), expr, rng=lambda: random.random()) + + expr = cosh(x) + assert tensorflow_code(expr) == "tensorflow.math.cosh(x)" + _compare_tensorflow_scalar((x,), expr, rng=lambda: random.random()) + + expr = acosh(x) + assert tensorflow_code(expr) == "tensorflow.math.acosh(x)" + _compare_tensorflow_scalar((x,), expr, rng=lambda: random.uniform(1, 2)) + + expr = sinh(x) + assert tensorflow_code(expr) == "tensorflow.math.sinh(x)" + _compare_tensorflow_scalar((x,), expr, rng=lambda: random.uniform(1, 2)) + + expr = asinh(x) + assert tensorflow_code(expr) == "tensorflow.math.asinh(x)" + _compare_tensorflow_scalar((x,), expr, rng=lambda: random.uniform(1, 2)) + + expr = tanh(x) + assert tensorflow_code(expr) == "tensorflow.math.tanh(x)" + _compare_tensorflow_scalar((x,), expr, rng=lambda: random.uniform(1, 2)) + + expr = atanh(x) + assert tensorflow_code(expr) == "tensorflow.math.atanh(x)" + _compare_tensorflow_scalar( + (x,), expr, rng=lambda: random.uniform(-.5, .5)) + + expr = erf(x) + assert tensorflow_code(expr) == "tensorflow.math.erf(x)" + _compare_tensorflow_scalar( + (x,), expr, rng=lambda: random.random()) + + expr = loggamma(x) + assert tensorflow_code(expr) == "tensorflow.math.lgamma(x)" + _compare_tensorflow_scalar( + (x,), expr, rng=lambda: random.random()) + + +def test_tensorflow_complexes(): + assert tensorflow_code(re(x)) == "tensorflow.math.real(x)" + assert tensorflow_code(im(x)) == "tensorflow.math.imag(x)" + assert tensorflow_code(arg(x)) == "tensorflow.math.angle(x)" + + +def test_tensorflow_relational(): + if not tf: + skip("TensorFlow not installed") + + expr = Eq(x, y) + assert tensorflow_code(expr) == "tensorflow.math.equal(x, y)" + _compare_tensorflow_relational((x, y), expr) + + expr = Ne(x, y) + assert tensorflow_code(expr) == "tensorflow.math.not_equal(x, y)" + _compare_tensorflow_relational((x, y), expr) + + expr = Ge(x, y) + assert tensorflow_code(expr) == "tensorflow.math.greater_equal(x, y)" + _compare_tensorflow_relational((x, y), expr) + + expr = Gt(x, y) + assert tensorflow_code(expr) == "tensorflow.math.greater(x, y)" + _compare_tensorflow_relational((x, y), expr) + + expr = Le(x, y) + assert tensorflow_code(expr) == "tensorflow.math.less_equal(x, y)" + _compare_tensorflow_relational((x, y), expr) + + expr = Lt(x, y) + assert tensorflow_code(expr) == "tensorflow.math.less(x, y)" + _compare_tensorflow_relational((x, y), expr) + + +# This (random) test is XFAIL because it fails occasionally +# See https://github.com/sympy/sympy/issues/18469 +@XFAIL +def test_tensorflow_matrices(): + if not tf: + skip("TensorFlow not installed") + + expr = M + assert tensorflow_code(expr) == "M" + _compare_tensorflow_matrix((M,), expr) + + expr = M + N + assert tensorflow_code(expr) == "tensorflow.math.add(M, N)" + _compare_tensorflow_matrix((M, N), expr) + + expr = M * N + assert tensorflow_code(expr) == "tensorflow.linalg.matmul(M, N)" + _compare_tensorflow_matrix((M, N), expr) + + expr = HadamardProduct(M, N) + assert tensorflow_code(expr) == "tensorflow.math.multiply(M, N)" + _compare_tensorflow_matrix((M, N), expr) + + expr = M*N*P*Q + assert tensorflow_code(expr) == \ + "tensorflow.linalg.matmul(" \ + "tensorflow.linalg.matmul(" \ + "tensorflow.linalg.matmul(M, N), P), Q)" + _compare_tensorflow_matrix((M, N, P, Q), expr) + + expr = M**3 + assert tensorflow_code(expr) == \ + "tensorflow.linalg.matmul(tensorflow.linalg.matmul(M, M), M)" + _compare_tensorflow_matrix((M,), expr) + + expr = Trace(M) + assert tensorflow_code(expr) == "tensorflow.linalg.trace(M)" + _compare_tensorflow_matrix((M,), expr) + + expr = Determinant(M) + assert tensorflow_code(expr) == "tensorflow.linalg.det(M)" + _compare_tensorflow_matrix_scalar((M,), expr) + + expr = Inverse(M) + assert tensorflow_code(expr) == "tensorflow.linalg.inv(M)" + _compare_tensorflow_matrix_inverse((M,), expr, use_float=True) + + expr = M.T + assert tensorflow_code(expr, tensorflow_version='1.14') == \ + "tensorflow.linalg.matrix_transpose(M)" + assert tensorflow_code(expr, tensorflow_version='1.13') == \ + "tensorflow.matrix_transpose(M)" + + _compare_tensorflow_matrix((M,), expr) + + +def test_codegen_einsum(): + if not tf: + skip("TensorFlow not installed") + + graph = tf.Graph() + with graph.as_default(): + session = tf.compat.v1.Session(graph=graph) + + M = MatrixSymbol("M", 2, 2) + N = MatrixSymbol("N", 2, 2) + + cg = convert_matrix_to_array(M * N) + f = lambdify((M, N), cg, 'tensorflow') + + ma = tf.constant([[1, 2], [3, 4]]) + mb = tf.constant([[1,-2], [-1, 3]]) + y = session.run(f(ma, mb)) + c = session.run(tf.matmul(ma, mb)) + assert (y == c).all() + + +def test_codegen_extra(): + if not tf: + skip("TensorFlow not installed") + + graph = tf.Graph() + with graph.as_default(): + session = tf.compat.v1.Session() + + M = MatrixSymbol("M", 2, 2) + N = MatrixSymbol("N", 2, 2) + P = MatrixSymbol("P", 2, 2) + Q = MatrixSymbol("Q", 2, 2) + ma = tf.constant([[1, 2], [3, 4]]) + mb = tf.constant([[1,-2], [-1, 3]]) + mc = tf.constant([[2, 0], [1, 2]]) + md = tf.constant([[1,-1], [4, 7]]) + + cg = ArrayTensorProduct(M, N) + assert tensorflow_code(cg) == \ + 'tensorflow.linalg.einsum("ab,cd", M, N)' + f = lambdify((M, N), cg, 'tensorflow') + y = session.run(f(ma, mb)) + c = session.run(tf.einsum("ij,kl", ma, mb)) + assert (y == c).all() + + cg = ArrayAdd(M, N) + assert tensorflow_code(cg) == 'tensorflow.math.add(M, N)' + f = lambdify((M, N), cg, 'tensorflow') + y = session.run(f(ma, mb)) + c = session.run(ma + mb) + assert (y == c).all() + + cg = ArrayAdd(M, N, P) + assert tensorflow_code(cg) == \ + 'tensorflow.math.add(tensorflow.math.add(M, N), P)' + f = lambdify((M, N, P), cg, 'tensorflow') + y = session.run(f(ma, mb, mc)) + c = session.run(ma + mb + mc) + assert (y == c).all() + + cg = ArrayAdd(M, N, P, Q) + assert tensorflow_code(cg) == \ + 'tensorflow.math.add(' \ + 'tensorflow.math.add(tensorflow.math.add(M, N), P), Q)' + f = lambdify((M, N, P, Q), cg, 'tensorflow') + y = session.run(f(ma, mb, mc, md)) + c = session.run(ma + mb + mc + md) + assert (y == c).all() + + cg = PermuteDims(M, [1, 0]) + assert tensorflow_code(cg) == 'tensorflow.transpose(M, [1, 0])' + f = lambdify((M,), cg, 'tensorflow') + y = session.run(f(ma)) + c = session.run(tf.transpose(ma)) + assert (y == c).all() + + cg = PermuteDims(ArrayTensorProduct(M, N), [1, 2, 3, 0]) + assert tensorflow_code(cg) == \ + 'tensorflow.transpose(' \ + 'tensorflow.linalg.einsum("ab,cd", M, N), [1, 2, 3, 0])' + f = lambdify((M, N), cg, 'tensorflow') + y = session.run(f(ma, mb)) + c = session.run(tf.transpose(tf.einsum("ab,cd", ma, mb), [1, 2, 3, 0])) + assert (y == c).all() + + cg = ArrayDiagonal(ArrayTensorProduct(M, N), (1, 2)) + assert tensorflow_code(cg) == \ + 'tensorflow.linalg.einsum("ab,bc->acb", M, N)' + f = lambdify((M, N), cg, 'tensorflow') + y = session.run(f(ma, mb)) + c = session.run(tf.einsum("ab,bc->acb", ma, mb)) + assert (y == c).all() + + +def test_MatrixElement_printing(): + A = MatrixSymbol("A", 1, 3) + B = MatrixSymbol("B", 1, 3) + C = MatrixSymbol("C", 1, 3) + + assert tensorflow_code(A[0, 0]) == "A[0, 0]" + assert tensorflow_code(3 * A[0, 0]) == "3*A[0, 0]" + + F = C[0, 0].subs(C, A - B) + assert tensorflow_code(F) == "(tensorflow.math.add((-1)*B, A))[0, 0]" + + +def test_tensorflow_Derivative(): + expr = Derivative(sin(x), x) + assert tensorflow_code(expr) == \ + "tensorflow.gradients(tensorflow.math.sin(x), x)[0]" + +def test_tensorflow_isnan_isinf(): + if not tf: + skip("TensorFlow not installed") + + # Test for isnan + x = symbols("x") + # Return 0 if x is of nan value, and 1 otherwise + expression = Piecewise((0.0, isnan(x)), (1.0, True)) + printed_code = tensorflow_code(expression) + expected_printed_code = "tensorflow.where(tensorflow.math.is_nan(x), 0.0, 1.0)" + assert tensorflow_code(expression) == expected_printed_code, f"Incorrect printed result {printed_code}, expected {expected_printed_code}" + for _input, _expected in [(float('nan'), 0.0), (float('inf'), 1.0), (float('-inf'), 1.0), (1.0, 1.0)]: + _output = lambdify((x), expression, modules="tensorflow")(x=tf.constant([_input])) + assert (_output == _expected).numpy().all() + + # Test for isinf + x = symbols("x") + # Return 0 if x is of nan value, and 1 otherwise + expression = Piecewise((0.0, isinf(x)), (1.0, True)) + printed_code = tensorflow_code(expression) + expected_printed_code = "tensorflow.where(tensorflow.math.is_inf(x), 0.0, 1.0)" + assert tensorflow_code(expression) == expected_printed_code, f"Incorrect printed result {printed_code}, expected {expected_printed_code}" + for _input, _expected in [(float('inf'), 0.0), (float('-inf'), 0.0), (float('nan'), 1.0), (1.0, 1.0)]: + _output = lambdify((x), expression, modules="tensorflow")(x=tf.constant([_input])) + assert (_output == _expected).numpy().all() diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_theanocode.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_theanocode.py new file mode 100644 index 0000000000000000000000000000000000000000..6ff40f78cb4de16149cb5e780756b7e32b574b71 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_theanocode.py @@ -0,0 +1,639 @@ +""" +Important note on tests in this module - the Theano printing functions use a +global cache by default, which means that tests using it will modify global +state and thus not be independent from each other. Instead of using the "cache" +keyword argument each time, this module uses the theano_code_ and +theano_function_ functions defined below which default to using a new, empty +cache instead. +""" + +import logging + +from sympy.external import import_module +from sympy.testing.pytest import raises, SKIP, warns_deprecated_sympy + +theanologger = logging.getLogger('theano.configdefaults') +theanologger.setLevel(logging.CRITICAL) +theano = import_module('theano') +theanologger.setLevel(logging.WARNING) + + +if theano: + import numpy as np + ts = theano.scalar + tt = theano.tensor + xt, yt, zt = [tt.scalar(name, 'floatX') for name in 'xyz'] + Xt, Yt, Zt = [tt.tensor('floatX', (False, False), name=n) for n in 'XYZ'] +else: + #bin/test will not execute any tests now + disabled = True + +import sympy as sy +from sympy.core.singleton import S +from sympy.abc import x, y, z, t +from sympy.printing.theanocode import (theano_code, dim_handling, + theano_function) + + +# Default set of matrix symbols for testing - make square so we can both +# multiply and perform elementwise operations between them. +X, Y, Z = [sy.MatrixSymbol(n, 4, 4) for n in 'XYZ'] + +# For testing AppliedUndef +f_t = sy.Function('f')(t) + + +def theano_code_(expr, **kwargs): + """ Wrapper for theano_code that uses a new, empty cache by default. """ + kwargs.setdefault('cache', {}) + with warns_deprecated_sympy(): + return theano_code(expr, **kwargs) + +def theano_function_(inputs, outputs, **kwargs): + """ Wrapper for theano_function that uses a new, empty cache by default. """ + kwargs.setdefault('cache', {}) + with warns_deprecated_sympy(): + return theano_function(inputs, outputs, **kwargs) + + +def fgraph_of(*exprs): + """ Transform SymPy expressions into Theano Computation. + + Parameters + ========== + exprs + SymPy expressions + + Returns + ======= + theano.gof.FunctionGraph + """ + outs = list(map(theano_code_, exprs)) + ins = theano.gof.graph.inputs(outs) + ins, outs = theano.gof.graph.clone(ins, outs) + return theano.gof.FunctionGraph(ins, outs) + + +def theano_simplify(fgraph): + """ Simplify a Theano Computation. + + Parameters + ========== + fgraph : theano.gof.FunctionGraph + + Returns + ======= + theano.gof.FunctionGraph + """ + mode = theano.compile.get_default_mode().excluding("fusion") + fgraph = fgraph.clone() + mode.optimizer.optimize(fgraph) + return fgraph + + +def theq(a, b): + """ Test two Theano objects for equality. + + Also accepts numeric types and lists/tuples of supported types. + + Note - debugprint() has a bug where it will accept numeric types but does + not respect the "file" argument and in this case and instead prints the number + to stdout and returns an empty string. This can lead to tests passing where + they should fail because any two numbers will always compare as equal. To + prevent this we treat numbers as a separate case. + """ + numeric_types = (int, float, np.number) + a_is_num = isinstance(a, numeric_types) + b_is_num = isinstance(b, numeric_types) + + # Compare numeric types using regular equality + if a_is_num or b_is_num: + if not (a_is_num and b_is_num): + return False + + return a == b + + # Compare sequences element-wise + a_is_seq = isinstance(a, (tuple, list)) + b_is_seq = isinstance(b, (tuple, list)) + + if a_is_seq or b_is_seq: + if not (a_is_seq and b_is_seq) or type(a) != type(b): + return False + + return list(map(theq, a)) == list(map(theq, b)) + + # Otherwise, assume debugprint() can handle it + astr = theano.printing.debugprint(a, file='str') + bstr = theano.printing.debugprint(b, file='str') + + # Check for bug mentioned above + for argname, argval, argstr in [('a', a, astr), ('b', b, bstr)]: + if argstr == '': + raise TypeError( + 'theano.printing.debugprint(%s) returned empty string ' + '(%s is instance of %r)' + % (argname, argname, type(argval)) + ) + + return astr == bstr + + +def test_example_symbols(): + """ + Check that the example symbols in this module print to their Theano + equivalents, as many of the other tests depend on this. + """ + assert theq(xt, theano_code_(x)) + assert theq(yt, theano_code_(y)) + assert theq(zt, theano_code_(z)) + assert theq(Xt, theano_code_(X)) + assert theq(Yt, theano_code_(Y)) + assert theq(Zt, theano_code_(Z)) + + +def test_Symbol(): + """ Test printing a Symbol to a theano variable. """ + xx = theano_code_(x) + assert isinstance(xx, (tt.TensorVariable, ts.ScalarVariable)) + assert xx.broadcastable == () + assert xx.name == x.name + + xx2 = theano_code_(x, broadcastables={x: (False,)}) + assert xx2.broadcastable == (False,) + assert xx2.name == x.name + +def test_MatrixSymbol(): + """ Test printing a MatrixSymbol to a theano variable. """ + XX = theano_code_(X) + assert isinstance(XX, tt.TensorVariable) + assert XX.broadcastable == (False, False) + +@SKIP # TODO - this is currently not checked but should be implemented +def test_MatrixSymbol_wrong_dims(): + """ Test MatrixSymbol with invalid broadcastable. """ + bcs = [(), (False,), (True,), (True, False), (False, True,), (True, True)] + for bc in bcs: + with raises(ValueError): + theano_code_(X, broadcastables={X: bc}) + +def test_AppliedUndef(): + """ Test printing AppliedUndef instance, which works similarly to Symbol. """ + ftt = theano_code_(f_t) + assert isinstance(ftt, tt.TensorVariable) + assert ftt.broadcastable == () + assert ftt.name == 'f_t' + + +def test_add(): + expr = x + y + comp = theano_code_(expr) + assert comp.owner.op == theano.tensor.add + +def test_trig(): + assert theq(theano_code_(sy.sin(x)), tt.sin(xt)) + assert theq(theano_code_(sy.tan(x)), tt.tan(xt)) + +def test_many(): + """ Test printing a complex expression with multiple symbols. """ + expr = sy.exp(x**2 + sy.cos(y)) * sy.log(2*z) + comp = theano_code_(expr) + expected = tt.exp(xt**2 + tt.cos(yt)) * tt.log(2*zt) + assert theq(comp, expected) + + +def test_dtype(): + """ Test specifying specific data types through the dtype argument. """ + for dtype in ['float32', 'float64', 'int8', 'int16', 'int32', 'int64']: + assert theano_code_(x, dtypes={x: dtype}).type.dtype == dtype + + # "floatX" type + assert theano_code_(x, dtypes={x: 'floatX'}).type.dtype in ('float32', 'float64') + + # Type promotion + assert theano_code_(x + 1, dtypes={x: 'float32'}).type.dtype == 'float32' + assert theano_code_(x + y, dtypes={x: 'float64', y: 'float32'}).type.dtype == 'float64' + + +def test_broadcastables(): + """ Test the "broadcastables" argument when printing symbol-like objects. """ + + # No restrictions on shape + for s in [x, f_t]: + for bc in [(), (False,), (True,), (False, False), (True, False)]: + assert theano_code_(s, broadcastables={s: bc}).broadcastable == bc + + # TODO - matrix broadcasting? + +def test_broadcasting(): + """ Test "broadcastable" attribute after applying element-wise binary op. """ + + expr = x + y + + cases = [ + [(), (), ()], + [(False,), (False,), (False,)], + [(True,), (False,), (False,)], + [(False, True), (False, False), (False, False)], + [(True, False), (False, False), (False, False)], + ] + + for bc1, bc2, bc3 in cases: + comp = theano_code_(expr, broadcastables={x: bc1, y: bc2}) + assert comp.broadcastable == bc3 + + +def test_MatMul(): + expr = X*Y*Z + expr_t = theano_code_(expr) + assert isinstance(expr_t.owner.op, tt.Dot) + assert theq(expr_t, Xt.dot(Yt).dot(Zt)) + +def test_Transpose(): + assert isinstance(theano_code_(X.T).owner.op, tt.DimShuffle) + +def test_MatAdd(): + expr = X+Y+Z + assert isinstance(theano_code_(expr).owner.op, tt.Elemwise) + + +def test_Rationals(): + assert theq(theano_code_(sy.Integer(2) / 3), tt.true_div(2, 3)) + assert theq(theano_code_(S.Half), tt.true_div(1, 2)) + +def test_Integers(): + assert theano_code_(sy.Integer(3)) == 3 + +def test_factorial(): + n = sy.Symbol('n') + assert theano_code_(sy.factorial(n)) + +def test_Derivative(): + simp = lambda expr: theano_simplify(fgraph_of(expr)) + assert theq(simp(theano_code_(sy.Derivative(sy.sin(x), x, evaluate=False))), + simp(theano.grad(tt.sin(xt), xt))) + + +def test_theano_function_simple(): + """ Test theano_function() with single output. """ + f = theano_function_([x, y], [x+y]) + assert f(2, 3) == 5 + +def test_theano_function_multi(): + """ Test theano_function() with multiple outputs. """ + f = theano_function_([x, y], [x+y, x-y]) + o1, o2 = f(2, 3) + assert o1 == 5 + assert o2 == -1 + +def test_theano_function_numpy(): + """ Test theano_function() vs Numpy implementation. """ + f = theano_function_([x, y], [x+y], dim=1, + dtypes={x: 'float64', y: 'float64'}) + assert np.linalg.norm(f([1, 2], [3, 4]) - np.asarray([4, 6])) < 1e-9 + + f = theano_function_([x, y], [x+y], dtypes={x: 'float64', y: 'float64'}, + dim=1) + xx = np.arange(3).astype('float64') + yy = 2*np.arange(3).astype('float64') + assert np.linalg.norm(f(xx, yy) - 3*np.arange(3)) < 1e-9 + + +def test_theano_function_matrix(): + m = sy.Matrix([[x, y], [z, x + y + z]]) + expected = np.array([[1.0, 2.0], [3.0, 1.0 + 2.0 + 3.0]]) + f = theano_function_([x, y, z], [m]) + np.testing.assert_allclose(f(1.0, 2.0, 3.0), expected) + f = theano_function_([x, y, z], [m], scalar=True) + np.testing.assert_allclose(f(1.0, 2.0, 3.0), expected) + f = theano_function_([x, y, z], [m, m]) + assert isinstance(f(1.0, 2.0, 3.0), type([])) + np.testing.assert_allclose(f(1.0, 2.0, 3.0)[0], expected) + np.testing.assert_allclose(f(1.0, 2.0, 3.0)[1], expected) + +def test_dim_handling(): + assert dim_handling([x], dim=2) == {x: (False, False)} + assert dim_handling([x, y], dims={x: 1, y: 2}) == {x: (False, True), + y: (False, False)} + assert dim_handling([x], broadcastables={x: (False,)}) == {x: (False,)} + +def test_theano_function_kwargs(): + """ + Test passing additional kwargs from theano_function() to theano.function(). + """ + import numpy as np + f = theano_function_([x, y, z], [x+y], dim=1, on_unused_input='ignore', + dtypes={x: 'float64', y: 'float64', z: 'float64'}) + assert np.linalg.norm(f([1, 2], [3, 4], [0, 0]) - np.asarray([4, 6])) < 1e-9 + + f = theano_function_([x, y, z], [x+y], + dtypes={x: 'float64', y: 'float64', z: 'float64'}, + dim=1, on_unused_input='ignore') + xx = np.arange(3).astype('float64') + yy = 2*np.arange(3).astype('float64') + zz = 2*np.arange(3).astype('float64') + assert np.linalg.norm(f(xx, yy, zz) - 3*np.arange(3)) < 1e-9 + +def test_theano_function_scalar(): + """ Test the "scalar" argument to theano_function(). """ + + args = [ + ([x, y], [x + y], None, [0]), # Single 0d output + ([X, Y], [X + Y], None, [2]), # Single 2d output + ([x, y], [x + y], {x: 0, y: 1}, [1]), # Single 1d output + ([x, y], [x + y, x - y], None, [0, 0]), # Two 0d outputs + ([x, y, X, Y], [x + y, X + Y], None, [0, 2]), # One 0d output, one 2d + ] + + # Create and test functions with and without the scalar setting + for inputs, outputs, in_dims, out_dims in args: + for scalar in [False, True]: + + f = theano_function_(inputs, outputs, dims=in_dims, scalar=scalar) + + # Check the theano_function attribute is set whether wrapped or not + assert isinstance(f.theano_function, theano.compile.function_module.Function) + + # Feed in inputs of the appropriate size and get outputs + in_values = [ + np.ones([1 if bc else 5 for bc in i.type.broadcastable]) + for i in f.theano_function.input_storage + ] + out_values = f(*in_values) + if not isinstance(out_values, list): + out_values = [out_values] + + # Check output types and shapes + assert len(out_dims) == len(out_values) + for d, value in zip(out_dims, out_values): + + if scalar and d == 0: + # Should have been converted to a scalar value + assert isinstance(value, np.number) + + else: + # Otherwise should be an array + assert isinstance(value, np.ndarray) + assert value.ndim == d + +def test_theano_function_bad_kwarg(): + """ + Passing an unknown keyword argument to theano_function() should raise an + exception. + """ + raises(Exception, lambda : theano_function_([x], [x+1], foobar=3)) + + +def test_slice(): + assert theano_code_(slice(1, 2, 3)) == slice(1, 2, 3) + + def theq_slice(s1, s2): + for attr in ['start', 'stop', 'step']: + a1 = getattr(s1, attr) + a2 = getattr(s2, attr) + if a1 is None or a2 is None: + if not (a1 is None or a2 is None): + return False + elif not theq(a1, a2): + return False + return True + + dtypes = {x: 'int32', y: 'int32'} + assert theq_slice(theano_code_(slice(x, y), dtypes=dtypes), slice(xt, yt)) + assert theq_slice(theano_code_(slice(1, x, 3), dtypes=dtypes), slice(1, xt, 3)) + +def test_MatrixSlice(): + from theano import Constant + + cache = {} + + n = sy.Symbol('n', integer=True) + X = sy.MatrixSymbol('X', n, n) + + Y = X[1:2:3, 4:5:6] + Yt = theano_code_(Y, cache=cache) + + s = ts.Scalar('int64') + assert tuple(Yt.owner.op.idx_list) == (slice(s, s, s), slice(s, s, s)) + assert Yt.owner.inputs[0] == theano_code_(X, cache=cache) + # == doesn't work in theano like it does in SymPy. You have to use + # equals. + assert all(Yt.owner.inputs[i].equals(Constant(s, i)) for i in range(1, 7)) + + k = sy.Symbol('k') + theano_code_(k, dtypes={k: 'int32'}) + start, stop, step = 4, k, 2 + Y = X[start:stop:step] + Yt = theano_code_(Y, dtypes={n: 'int32', k: 'int32'}) + # assert Yt.owner.op.idx_list[0].stop == kt + +def test_BlockMatrix(): + n = sy.Symbol('n', integer=True) + A, B, C, D = [sy.MatrixSymbol(name, n, n) for name in 'ABCD'] + At, Bt, Ct, Dt = map(theano_code_, (A, B, C, D)) + Block = sy.BlockMatrix([[A, B], [C, D]]) + Blockt = theano_code_(Block) + solutions = [tt.join(0, tt.join(1, At, Bt), tt.join(1, Ct, Dt)), + tt.join(1, tt.join(0, At, Ct), tt.join(0, Bt, Dt))] + assert any(theq(Blockt, solution) for solution in solutions) + +@SKIP +def test_BlockMatrix_Inverse_execution(): + k, n = 2, 4 + dtype = 'float32' + A = sy.MatrixSymbol('A', n, k) + B = sy.MatrixSymbol('B', n, n) + inputs = A, B + output = B.I*A + + cutsizes = {A: [(n//2, n//2), (k//2, k//2)], + B: [(n//2, n//2), (n//2, n//2)]} + cutinputs = [sy.blockcut(i, *cutsizes[i]) for i in inputs] + cutoutput = output.subs(dict(zip(inputs, cutinputs))) + + dtypes = dict(zip(inputs, [dtype]*len(inputs))) + f = theano_function_(inputs, [output], dtypes=dtypes, cache={}) + fblocked = theano_function_(inputs, [sy.block_collapse(cutoutput)], + dtypes=dtypes, cache={}) + + ninputs = [np.random.rand(*x.shape).astype(dtype) for x in inputs] + ninputs = [np.arange(n*k).reshape(A.shape).astype(dtype), + np.eye(n).astype(dtype)] + ninputs[1] += np.ones(B.shape)*1e-5 + + assert np.allclose(f(*ninputs), fblocked(*ninputs), rtol=1e-5) + +def test_DenseMatrix(): + t = sy.Symbol('theta') + for MatrixType in [sy.Matrix, sy.ImmutableMatrix]: + X = MatrixType([[sy.cos(t), -sy.sin(t)], [sy.sin(t), sy.cos(t)]]) + tX = theano_code_(X) + assert isinstance(tX, tt.TensorVariable) + assert tX.owner.op == tt.join_ + + +def test_cache_basic(): + """ Test single symbol-like objects are cached when printed by themselves. """ + + # Pairs of objects which should be considered equivalent with respect to caching + pairs = [ + (x, sy.Symbol('x')), + (X, sy.MatrixSymbol('X', *X.shape)), + (f_t, sy.Function('f')(sy.Symbol('t'))), + ] + + for s1, s2 in pairs: + cache = {} + st = theano_code_(s1, cache=cache) + + # Test hit with same instance + assert theano_code_(s1, cache=cache) is st + + # Test miss with same instance but new cache + assert theano_code_(s1, cache={}) is not st + + # Test hit with different but equivalent instance + assert theano_code_(s2, cache=cache) is st + +def test_global_cache(): + """ Test use of the global cache. """ + from sympy.printing.theanocode import global_cache + + backup = dict(global_cache) + try: + # Temporarily empty global cache + global_cache.clear() + + for s in [x, X, f_t]: + with warns_deprecated_sympy(): + st = theano_code(s) + assert theano_code(s) is st + + finally: + # Restore global cache + global_cache.update(backup) + +def test_cache_types_distinct(): + """ + Test that symbol-like objects of different types (Symbol, MatrixSymbol, + AppliedUndef) are distinguished by the cache even if they have the same + name. + """ + symbols = [sy.Symbol('f_t'), sy.MatrixSymbol('f_t', 4, 4), f_t] + + cache = {} # Single shared cache + printed = {} + + for s in symbols: + st = theano_code_(s, cache=cache) + assert st not in printed.values() + printed[s] = st + + # Check all printed objects are distinct + assert len(set(map(id, printed.values()))) == len(symbols) + + # Check retrieving + for s, st in printed.items(): + with warns_deprecated_sympy(): + assert theano_code(s, cache=cache) is st + +def test_symbols_are_created_once(): + """ + Test that a symbol is cached and reused when it appears in an expression + more than once. + """ + expr = sy.Add(x, x, evaluate=False) + comp = theano_code_(expr) + + assert theq(comp, xt + xt) + assert not theq(comp, xt + theano_code_(x)) + +def test_cache_complex(): + """ + Test caching on a complicated expression with multiple symbols appearing + multiple times. + """ + expr = x ** 2 + (y - sy.exp(x)) * sy.sin(z - x * y) + symbol_names = {s.name for s in expr.free_symbols} + expr_t = theano_code_(expr) + + # Iterate through variables in the Theano computational graph that the + # printed expression depends on + seen = set() + for v in theano.gof.graph.ancestors([expr_t]): + # Owner-less, non-constant variables should be our symbols + if v.owner is None and not isinstance(v, theano.gof.graph.Constant): + # Check it corresponds to a symbol and appears only once + assert v.name in symbol_names + assert v.name not in seen + seen.add(v.name) + + # Check all were present + assert seen == symbol_names + + +def test_Piecewise(): + # A piecewise linear + expr = sy.Piecewise((0, x<0), (x, x<2), (1, True)) # ___/III + result = theano_code_(expr) + assert result.owner.op == tt.switch + + expected = tt.switch(xt<0, 0, tt.switch(xt<2, xt, 1)) + assert theq(result, expected) + + expr = sy.Piecewise((x, x < 0)) + result = theano_code_(expr) + expected = tt.switch(xt < 0, xt, np.nan) + assert theq(result, expected) + + expr = sy.Piecewise((0, sy.And(x>0, x<2)), \ + (x, sy.Or(x>2, x<0))) + result = theano_code_(expr) + expected = tt.switch(tt.and_(xt>0,xt<2), 0, \ + tt.switch(tt.or_(xt>2, xt<0), xt, np.nan)) + assert theq(result, expected) + + +def test_Relationals(): + assert theq(theano_code_(sy.Eq(x, y)), tt.eq(xt, yt)) + # assert theq(theano_code_(sy.Ne(x, y)), tt.neq(xt, yt)) # TODO - implement + assert theq(theano_code_(x > y), xt > yt) + assert theq(theano_code_(x < y), xt < yt) + assert theq(theano_code_(x >= y), xt >= yt) + assert theq(theano_code_(x <= y), xt <= yt) + + +def test_complexfunctions(): + with warns_deprecated_sympy(): + xt, yt = theano_code_(x, dtypes={x:'complex128'}), theano_code_(y, dtypes={y: 'complex128'}) + from sympy.functions.elementary.complexes import conjugate + from theano.tensor import as_tensor_variable as atv + from theano.tensor import complex as cplx + with warns_deprecated_sympy(): + assert theq(theano_code_(y*conjugate(x)), yt*(xt.conj())) + assert theq(theano_code_((1+2j)*x), xt*(atv(1.0)+atv(2.0)*cplx(0,1))) + + +def test_constantfunctions(): + with warns_deprecated_sympy(): + tf = theano_function_([],[1+1j]) + assert(tf()==1+1j) + + +def test_Exp1(): + """ + Test that exp(1) prints without error and evaluates close to SymPy's E + """ + # sy.exp(1) should yield same instance of E as sy.E (singleton), but extra + # check added for sanity + e_a = sy.exp(1) + e_b = sy.E + + np.testing.assert_allclose(float(e_a), np.e) + np.testing.assert_allclose(float(e_b), np.e) + + e = theano_code_(e_a) + np.testing.assert_allclose(float(e_a), e.eval()) + + e = theano_code_(e_b) + np.testing.assert_allclose(float(e_b), e.eval()) diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_torch.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_torch.py new file mode 100644 index 0000000000000000000000000000000000000000..8ce2c6cec75e03264f93b472a79eb073742e3486 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_torch.py @@ -0,0 +1,531 @@ +import random +import math + +from sympy import symbols, Derivative +from sympy.printing.pytorch import torch_code +from sympy import (eye, MatrixSymbol, Matrix) +from sympy.tensor.array import NDimArray +from sympy.tensor.array.expressions.array_expressions import ( + ArrayTensorProduct, ArrayAdd, + PermuteDims, ArrayDiagonal, _CodegenArrayAbstract) +from sympy.utilities.lambdify import lambdify +from sympy.core.relational import Eq, Ne, Ge, Gt, Le, Lt +from sympy.functions import \ + Abs, ceiling, exp, floor, sign, sin, asin, cos, \ + acos, tan, atan, atan2, cosh, acosh, sinh, asinh, tanh, atanh, \ + re, im, arg, erf, loggamma, sqrt +from sympy.testing.pytest import skip +from sympy.external import import_module +from sympy.matrices.expressions import \ + Determinant, HadamardProduct, Inverse, Trace +from sympy.matrices import randMatrix +from sympy.matrices import Identity, ZeroMatrix, OneMatrix +from sympy import conjugate, I +from sympy import Heaviside, gamma, polygamma + + + +torch = import_module("torch") + +M = MatrixSymbol("M", 3, 3) +N = MatrixSymbol("N", 3, 3) +P = MatrixSymbol("P", 3, 3) +Q = MatrixSymbol("Q", 3, 3) + +x, y, z, t = symbols("x y z t") + +if torch is not None: + llo = [list(range(i, i + 3)) for i in range(0, 9, 3)] + m3x3 = torch.tensor(llo, dtype=torch.float64) + m3x3sympy = Matrix(llo) + + +def _compare_torch_matrix(variables, expr): + f = lambdify(variables, expr, 'torch') + + random_matrices = [randMatrix(i.shape[0], i.shape[1]) for i in variables] + random_variables = [torch.tensor(i.tolist(), dtype=torch.float64) for i in random_matrices] + r = f(*random_variables) + e = expr.subs(dict(zip(variables, random_matrices))).doit() + + if isinstance(e, _CodegenArrayAbstract): + e = e.doit() + + if hasattr(e, 'is_number') and e.is_number: + if isinstance(r, torch.Tensor) and r.dim() == 0: + r = r.item() + e = float(e) + assert abs(r - e) < 1e-6 + return + + if e.is_Matrix or isinstance(e, NDimArray): + e = torch.tensor(e.tolist(), dtype=torch.float64) + assert torch.allclose(r, e, atol=1e-6) + else: + raise TypeError(f"Cannot compare {type(r)} with {type(e)}") + + +def _compare_torch_scalar(variables, expr, rng=lambda: random.uniform(-5, 5)): + f = lambdify(variables, expr, 'torch') + rvs = [rng() for v in variables] + t_rvs = [torch.tensor(i, dtype=torch.float64) for i in rvs] + r = f(*t_rvs) + if isinstance(r, torch.Tensor): + r = r.item() + e = expr.subs(dict(zip(variables, rvs))).doit() + assert abs(r - e) < 1e-6 + + +def _compare_torch_relational(variables, expr, rng=lambda: random.randint(0, 10)): + f = lambdify(variables, expr, 'torch') + rvs = [rng() for v in variables] + t_rvs = [torch.tensor(i, dtype=torch.float64) for i in rvs] + r = f(*t_rvs) + e = bool(expr.subs(dict(zip(variables, rvs))).doit()) + assert r.item() == e + + +def test_torch_math(): + if not torch: + skip("PyTorch not installed") + + expr = Abs(x) + assert torch_code(expr) == "torch.abs(x)" + f = lambdify(x, expr, 'torch') + ma = torch.tensor([[-1, 2, -3, -4]], dtype=torch.float64) + y_abs = f(ma) + c = torch.abs(ma) + assert torch.all(y_abs == c) + + expr = sign(x) + assert torch_code(expr) == "torch.sign(x)" + _compare_torch_scalar((x,), expr, rng=lambda: random.uniform(-10, 10)) + + expr = ceiling(x) + assert torch_code(expr) == "torch.ceil(x)" + _compare_torch_scalar((x,), expr, rng=lambda: random.random()) + + expr = floor(x) + assert torch_code(expr) == "torch.floor(x)" + _compare_torch_scalar((x,), expr, rng=lambda: random.random()) + + expr = exp(x) + assert torch_code(expr) == "torch.exp(x)" + _compare_torch_scalar((x,), expr, rng=lambda: random.uniform(-2, 2)) + + expr = sqrt(x) + assert torch_code(expr) == "torch.sqrt(x)" + _compare_torch_scalar((x,), expr, rng=lambda: random.random()) + + expr = x ** 4 + assert torch_code(expr) == "torch.pow(x, 4)" + _compare_torch_scalar((x,), expr, rng=lambda: random.random()) + + expr = cos(x) + assert torch_code(expr) == "torch.cos(x)" + _compare_torch_scalar((x,), expr, rng=lambda: random.random()) + + expr = acos(x) + assert torch_code(expr) == "torch.acos(x)" + _compare_torch_scalar((x,), expr, rng=lambda: random.uniform(-0.99, 0.99)) + + expr = sin(x) + assert torch_code(expr) == "torch.sin(x)" + _compare_torch_scalar((x,), expr, rng=lambda: random.random()) + + expr = asin(x) + assert torch_code(expr) == "torch.asin(x)" + _compare_torch_scalar((x,), expr, rng=lambda: random.uniform(-0.99, 0.99)) + + expr = tan(x) + assert torch_code(expr) == "torch.tan(x)" + _compare_torch_scalar((x,), expr, rng=lambda: random.uniform(-1.5, 1.5)) + + expr = atan(x) + assert torch_code(expr) == "torch.atan(x)" + _compare_torch_scalar((x,), expr, rng=lambda: random.uniform(-5, 5)) + + expr = atan2(y, x) + assert torch_code(expr) == "torch.atan2(y, x)" + _compare_torch_scalar((y, x), expr, rng=lambda: random.uniform(-5, 5)) + + expr = cosh(x) + assert torch_code(expr) == "torch.cosh(x)" + _compare_torch_scalar((x,), expr, rng=lambda: random.uniform(-2, 2)) + + expr = acosh(x) + assert torch_code(expr) == "torch.acosh(x)" + _compare_torch_scalar((x,), expr, rng=lambda: random.uniform(1.1, 5)) + + expr = sinh(x) + assert torch_code(expr) == "torch.sinh(x)" + _compare_torch_scalar((x,), expr, rng=lambda: random.uniform(-2, 2)) + + expr = asinh(x) + assert torch_code(expr) == "torch.asinh(x)" + _compare_torch_scalar((x,), expr, rng=lambda: random.uniform(-5, 5)) + + expr = tanh(x) + assert torch_code(expr) == "torch.tanh(x)" + _compare_torch_scalar((x,), expr, rng=lambda: random.uniform(-2, 2)) + + expr = atanh(x) + assert torch_code(expr) == "torch.atanh(x)" + _compare_torch_scalar((x,), expr, rng=lambda: random.uniform(-0.9, 0.9)) + + expr = erf(x) + assert torch_code(expr) == "torch.erf(x)" + _compare_torch_scalar((x,), expr, rng=lambda: random.uniform(-2, 2)) + + expr = loggamma(x) + assert torch_code(expr) == "torch.lgamma(x)" + _compare_torch_scalar((x,), expr, rng=lambda: random.uniform(0.5, 5)) + + +def test_torch_complexes(): + assert torch_code(re(x)) == "torch.real(x)" + assert torch_code(im(x)) == "torch.imag(x)" + assert torch_code(arg(x)) == "torch.angle(x)" + + +def test_torch_relational(): + if not torch: + skip("PyTorch not installed") + + expr = Eq(x, y) + assert torch_code(expr) == "torch.eq(x, y)" + _compare_torch_relational((x, y), expr) + + expr = Ne(x, y) + assert torch_code(expr) == "torch.ne(x, y)" + _compare_torch_relational((x, y), expr) + + expr = Ge(x, y) + assert torch_code(expr) == "torch.ge(x, y)" + _compare_torch_relational((x, y), expr) + + expr = Gt(x, y) + assert torch_code(expr) == "torch.gt(x, y)" + _compare_torch_relational((x, y), expr) + + expr = Le(x, y) + assert torch_code(expr) == "torch.le(x, y)" + _compare_torch_relational((x, y), expr) + + expr = Lt(x, y) + assert torch_code(expr) == "torch.lt(x, y)" + _compare_torch_relational((x, y), expr) + + +def test_torch_matrix(): + if torch is None: + skip("PyTorch not installed") + + expr = M + assert torch_code(expr) == "M" + f = lambdify((M,), expr, "torch") + eye_mat = eye(3) + eye_tensor = torch.tensor(eye_mat.tolist(), dtype=torch.float64) + assert torch.allclose(f(eye_tensor), eye_tensor) + + expr = M * N + assert torch_code(expr) == "torch.matmul(M, N)" + _compare_torch_matrix((M, N), expr) + + expr = M ** 3 + assert torch_code(expr) == "torch.mm(torch.mm(M, M), M)" + _compare_torch_matrix((M,), expr) + + expr = M * N * P * Q + assert torch_code(expr) == "torch.matmul(torch.matmul(torch.matmul(M, N), P), Q)" + _compare_torch_matrix((M, N, P, Q), expr) + + expr = Trace(M) + assert torch_code(expr) == "torch.trace(M)" + _compare_torch_matrix((M,), expr) + + expr = Determinant(M) + assert torch_code(expr) == "torch.det(M)" + _compare_torch_matrix((M,), expr) + + expr = HadamardProduct(M, N) + assert torch_code(expr) == "torch.mul(M, N)" + _compare_torch_matrix((M, N), expr) + + expr = Inverse(M) + assert torch_code(expr) == "torch.linalg.inv(M)" + + # For inverse, use a matrix that's guaranteed to be invertible + eye_mat = eye(3) + eye_tensor = torch.tensor(eye_mat.tolist(), dtype=torch.float64) + f = lambdify((M,), expr, "torch") + result = f(eye_tensor) + expected = torch.linalg.inv(eye_tensor) + assert torch.allclose(result, expected) + + +def test_torch_array_operations(): + if not torch: + skip("PyTorch not installed") + + M = MatrixSymbol("M", 2, 2) + N = MatrixSymbol("N", 2, 2) + P = MatrixSymbol("P", 2, 2) + Q = MatrixSymbol("Q", 2, 2) + + ma = torch.tensor([[1., 2.], [3., 4.]], dtype=torch.float64) + mb = torch.tensor([[1., -2.], [-1., 3.]], dtype=torch.float64) + mc = torch.tensor([[2., 0.], [1., 2.]], dtype=torch.float64) + md = torch.tensor([[1., -1.], [4., 7.]], dtype=torch.float64) + + cg = ArrayTensorProduct(M, N) + assert torch_code(cg) == 'torch.einsum("ab,cd", M, N)' + f = lambdify((M, N), cg, 'torch') + y = f(ma, mb) + c = torch.einsum("ij,kl", ma, mb) + assert torch.allclose(y, c) + + cg = ArrayAdd(M, N) + assert torch_code(cg) == 'torch.add(M, N)' + f = lambdify((M, N), cg, 'torch') + y = f(ma, mb) + c = ma + mb + assert torch.allclose(y, c) + + cg = ArrayAdd(M, N, P) + assert torch_code(cg) == 'torch.add(torch.add(M, N), P)' + f = lambdify((M, N, P), cg, 'torch') + y = f(ma, mb, mc) + c = ma + mb + mc + assert torch.allclose(y, c) + + cg = ArrayAdd(M, N, P, Q) + assert torch_code(cg) == 'torch.add(torch.add(torch.add(M, N), P), Q)' + f = lambdify((M, N, P, Q), cg, 'torch') + y = f(ma, mb, mc, md) + c = ma + mb + mc + md + assert torch.allclose(y, c) + + cg = PermuteDims(M, [1, 0]) + assert torch_code(cg) == 'M.permute(1, 0)' + f = lambdify((M,), cg, 'torch') + y = f(ma) + c = ma.T + assert torch.allclose(y, c) + + cg = PermuteDims(ArrayTensorProduct(M, N), [1, 2, 3, 0]) + assert torch_code(cg) == 'torch.einsum("ab,cd", M, N).permute(1, 2, 3, 0)' + f = lambdify((M, N), cg, 'torch') + y = f(ma, mb) + c = torch.einsum("ab,cd", ma, mb).permute(1, 2, 3, 0) + assert torch.allclose(y, c) + + cg = ArrayDiagonal(ArrayTensorProduct(M, N), (1, 2)) + assert torch_code(cg) == 'torch.einsum("ab,bc->acb", M, N)' + f = lambdify((M, N), cg, 'torch') + y = f(ma, mb) + c = torch.einsum("ab,bc->acb", ma, mb) + assert torch.allclose(y, c) + + +def test_torch_derivative(): + """Test derivative handling.""" + expr = Derivative(sin(x), x) + assert torch_code(expr) == 'torch.autograd.grad(torch.sin(x), x)[0]' + + +def test_torch_printing_dtype(): + if not torch: + skip("PyTorch not installed") + + # matrix printing with default dtype + expr = Matrix([[x, sin(y)], [exp(z), -t]]) + assert "dtype=torch.float64" in torch_code(expr) + + # explicit dtype + assert "dtype=torch.float32" in torch_code(expr, dtype="torch.float32") + + # with requires_grad + result = torch_code(expr, requires_grad=True) + assert "requires_grad=True" in result + assert "dtype=torch.float64" in result + + # both + result = torch_code(expr, requires_grad=True, dtype="torch.float32") + assert "requires_grad=True" in result + assert "dtype=torch.float32" in result + + +def test_requires_grad(): + if not torch: + skip("PyTorch not installed") + + expr = sin(x) + cos(y) + f = lambdify([x, y], expr, 'torch') + + # make sure the gradients flow + x_val = torch.tensor(1.0, requires_grad=True) + y_val = torch.tensor(2.0, requires_grad=True) + result = f(x_val, y_val) + assert result.requires_grad + result.backward() + + # x_val.grad should be cos(x_val) which is close to cos(1.0) + assert abs(x_val.grad.item() - float(cos(1.0).evalf())) < 1e-6 + + # y_val.grad should be -sin(y_val) which is close to -sin(2.0) + assert abs(y_val.grad.item() - float(-sin(2.0).evalf())) < 1e-6 + + +def test_torch_multi_variable_derivatives(): + if not torch: + skip("PyTorch not installed") + + x, y, z = symbols("x y z") + + expr = Derivative(sin(x), x) + assert torch_code(expr) == "torch.autograd.grad(torch.sin(x), x)[0]" + + expr = Derivative(sin(x), (x, 2)) + assert torch_code( + expr) == "torch.autograd.grad(torch.autograd.grad(torch.sin(x), x, create_graph=True)[0], x, create_graph=True)[0]" + + expr = Derivative(sin(x * y), x, y) + result = torch_code(expr) + expected = "torch.autograd.grad(torch.autograd.grad(torch.sin(x*y), x, create_graph=True)[0], y, create_graph=True)[0]" + normalized_result = result.replace(" ", "") + normalized_expected = expected.replace(" ", "") + assert normalized_result == normalized_expected + + expr = Derivative(sin(x), x, x) + result = torch_code(expr) + expected = "torch.autograd.grad(torch.autograd.grad(torch.sin(x), x, create_graph=True)[0], x, create_graph=True)[0]" + assert result == expected + + expr = Derivative(sin(x * y * z), x, (y, 2), z) + result = torch_code(expr) + expected = "torch.autograd.grad(torch.autograd.grad(torch.autograd.grad(torch.autograd.grad(torch.sin(x*y*z), x, create_graph=True)[0], y, create_graph=True)[0], y, create_graph=True)[0], z, create_graph=True)[0]" + normalized_result = result.replace(" ", "") + normalized_expected = expected.replace(" ", "") + assert normalized_result == normalized_expected + + +def test_torch_derivative_lambdify(): + if not torch: + skip("PyTorch not installed") + + x = symbols("x") + y = symbols("y") + + expr = Derivative(x ** 2, x) + f = lambdify(x, expr, 'torch') + x_val = torch.tensor(2.0, requires_grad=True) + result = f(x_val) + assert torch.isclose(result, torch.tensor(4.0)) + + expr = Derivative(sin(x), (x, 2)) + f = lambdify(x, expr, 'torch') + # Second derivative of sin(x) at x=0 is 0, not -1 + x_val = torch.tensor(0.0, requires_grad=True) + result = f(x_val) + assert torch.isclose(result, torch.tensor(0.0), atol=1e-5) + + x_val = torch.tensor(math.pi / 2, requires_grad=True) + result = f(x_val) + assert torch.isclose(result, torch.tensor(-1.0), atol=1e-5) + + expr = Derivative(x * y ** 2, x, y) + f = lambdify((x, y), expr, 'torch') + x_val = torch.tensor(2.0, requires_grad=True) + y_val = torch.tensor(3.0, requires_grad=True) + result = f(x_val, y_val) + assert torch.isclose(result, torch.tensor(6.0)) + + +def test_torch_special_matrices(): + if not torch: + skip("PyTorch not installed") + + expr = Identity(3) + assert torch_code(expr) == "torch.eye(3)" + + n = symbols("n") + expr = Identity(n) + assert torch_code(expr) == "torch.eye(n, n)" + + expr = ZeroMatrix(2, 3) + assert torch_code(expr) == "torch.zeros((2, 3))" + + m, n = symbols("m n") + expr = ZeroMatrix(m, n) + assert torch_code(expr) == "torch.zeros((m, n))" + + expr = OneMatrix(2, 3) + assert torch_code(expr) == "torch.ones((2, 3))" + + expr = OneMatrix(m, n) + assert torch_code(expr) == "torch.ones((m, n))" + + +def test_torch_special_matrices_lambdify(): + if not torch: + skip("PyTorch not installed") + + expr = Identity(3) + f = lambdify([], expr, 'torch') + result = f() + expected = torch.eye(3) + assert torch.allclose(result, expected) + + expr = ZeroMatrix(2, 3) + f = lambdify([], expr, 'torch') + result = f() + expected = torch.zeros((2, 3)) + assert torch.allclose(result, expected) + + expr = OneMatrix(2, 3) + f = lambdify([], expr, 'torch') + result = f() + expected = torch.ones((2, 3)) + assert torch.allclose(result, expected) + + +def test_torch_complex_operations(): + if not torch: + skip("PyTorch not installed") + + expr = conjugate(x) + assert torch_code(expr) == "torch.conj(x)" + + # SymPy distributes conjugate over addition and applies specific rules for each term + expr = conjugate(sin(x) + I * cos(y)) + assert torch_code(expr) == "torch.sin(torch.conj(x)) - 1j*torch.cos(torch.conj(y))" + + expr = I + assert torch_code(expr) == "1j" + + expr = 2 * I + x + assert torch_code(expr) == "x + 2*1j" + + expr = exp(I * x) + assert torch_code(expr) == "torch.exp(1j*x)" + + +def test_torch_special_functions(): + if not torch: + skip("PyTorch not installed") + + expr = Heaviside(x) + assert torch_code(expr) == "torch.heaviside(x, 1/2)" + + expr = Heaviside(x, 0) + assert torch_code(expr) == "torch.heaviside(x, 0)" + + expr = gamma(x) + assert torch_code(expr) == "torch.special.gamma(x)" + + expr = polygamma(0, x) # Use polygamma instead of digamma because sympy will default to that anyway + assert torch_code(expr) == "torch.special.digamma(x)" + + expr = gamma(sin(x)) + assert torch_code(expr) == "torch.special.gamma(torch.sin(x))" diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_tree.py b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_tree.py new file mode 100644 index 0000000000000000000000000000000000000000..cf116d0cac5d38f225815fcd2d4ac90cd0dd96d7 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tests/test_tree.py @@ -0,0 +1,196 @@ +from sympy.printing.tree import tree +from sympy.testing.pytest import XFAIL + + +# Remove this flag after making _assumptions cache deterministic. +@XFAIL +def test_print_tree_MatAdd(): + from sympy.matrices.expressions import MatrixSymbol + A = MatrixSymbol('A', 3, 3) + B = MatrixSymbol('B', 3, 3) + + test_str = [ + 'MatAdd: A + B\n', + 'algebraic: False\n', + 'commutative: False\n', + 'complex: False\n', + 'composite: False\n', + 'even: False\n', + 'extended_negative: False\n', + 'extended_nonnegative: False\n', + 'extended_nonpositive: False\n', + 'extended_nonzero: False\n', + 'extended_positive: False\n', + 'extended_real: False\n', + 'imaginary: False\n', + 'integer: False\n', + 'irrational: False\n', + 'negative: False\n', + 'noninteger: False\n', + 'nonnegative: False\n', + 'nonpositive: False\n', + 'nonzero: False\n', + 'odd: False\n', + 'positive: False\n', + 'prime: False\n', + 'rational: False\n', + 'real: False\n', + 'transcendental: False\n', + 'zero: False\n', + '+-MatrixSymbol: A\n', + '| algebraic: False\n', + '| commutative: False\n', + '| complex: False\n', + '| composite: False\n', + '| even: False\n', + '| extended_negative: False\n', + '| extended_nonnegative: False\n', + '| extended_nonpositive: False\n', + '| extended_nonzero: False\n', + '| extended_positive: False\n', + '| extended_real: False\n', + '| imaginary: False\n', + '| integer: False\n', + '| irrational: False\n', + '| negative: False\n', + '| noninteger: False\n', + '| nonnegative: False\n', + '| nonpositive: False\n', + '| nonzero: False\n', + '| odd: False\n', + '| positive: False\n', + '| prime: False\n', + '| rational: False\n', + '| real: False\n', + '| transcendental: False\n', + '| zero: False\n', + '| +-Symbol: A\n', + '| | commutative: True\n', + '| +-Integer: 3\n', + '| | algebraic: True\n', + '| | commutative: True\n', + '| | complex: True\n', + '| | extended_negative: False\n', + '| | extended_nonnegative: True\n', + '| | extended_real: True\n', + '| | finite: True\n', + '| | hermitian: True\n', + '| | imaginary: False\n', + '| | infinite: False\n', + '| | integer: True\n', + '| | irrational: False\n', + '| | negative: False\n', + '| | noninteger: False\n', + '| | nonnegative: True\n', + '| | rational: True\n', + '| | real: True\n', + '| | transcendental: False\n', + '| +-Integer: 3\n', + '| algebraic: True\n', + '| commutative: True\n', + '| complex: True\n', + '| extended_negative: False\n', + '| extended_nonnegative: True\n', + '| extended_real: True\n', + '| finite: True\n', + '| hermitian: True\n', + '| imaginary: False\n', + '| infinite: False\n', + '| integer: True\n', + '| irrational: False\n', + '| negative: False\n', + '| noninteger: False\n', + '| nonnegative: True\n', + '| rational: True\n', + '| real: True\n', + '| transcendental: False\n', + '+-MatrixSymbol: B\n', + ' algebraic: False\n', + ' commutative: False\n', + ' complex: False\n', + ' composite: False\n', + ' even: False\n', + ' extended_negative: False\n', + ' extended_nonnegative: False\n', + ' extended_nonpositive: False\n', + ' extended_nonzero: False\n', + ' extended_positive: False\n', + ' extended_real: False\n', + ' imaginary: False\n', + ' integer: False\n', + ' irrational: False\n', + ' negative: False\n', + ' noninteger: False\n', + ' nonnegative: False\n', + ' nonpositive: False\n', + ' nonzero: False\n', + ' odd: False\n', + ' positive: False\n', + ' prime: False\n', + ' rational: False\n', + ' real: False\n', + ' transcendental: False\n', + ' zero: False\n', + ' +-Symbol: B\n', + ' | commutative: True\n', + ' +-Integer: 3\n', + ' | algebraic: True\n', + ' | commutative: True\n', + ' | complex: True\n', + ' | extended_negative: False\n', + ' | extended_nonnegative: True\n', + ' | extended_real: True\n', + ' | finite: True\n', + ' | hermitian: True\n', + ' | imaginary: False\n', + ' | infinite: False\n', + ' | integer: True\n', + ' | irrational: False\n', + ' | negative: False\n', + ' | noninteger: False\n', + ' | nonnegative: True\n', + ' | rational: True\n', + ' | real: True\n', + ' | transcendental: False\n', + ' +-Integer: 3\n', + ' algebraic: True\n', + ' commutative: True\n', + ' complex: True\n', + ' extended_negative: False\n', + ' extended_nonnegative: True\n', + ' extended_real: True\n', + ' finite: True\n', + ' hermitian: True\n', + ' imaginary: False\n', + ' infinite: False\n', + ' integer: True\n', + ' irrational: False\n', + ' negative: False\n', + ' noninteger: False\n', + ' nonnegative: True\n', + ' rational: True\n', + ' real: True\n', + ' transcendental: False\n' + ] + + assert tree(A + B) == "".join(test_str) + + +def test_print_tree_MatAdd_noassumptions(): + from sympy.matrices.expressions import MatrixSymbol + A = MatrixSymbol('A', 3, 3) + B = MatrixSymbol('B', 3, 3) + + test_str = \ +"""MatAdd: A + B ++-MatrixSymbol: A +| +-Str: A +| +-Integer: 3 +| +-Integer: 3 ++-MatrixSymbol: B + +-Str: B + +-Integer: 3 + +-Integer: 3 +""" + + assert tree(A + B, assumptions=False) == test_str diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/theanocode.py b/.venv/lib/python3.12/site-packages/sympy/printing/theanocode.py new file mode 100644 index 0000000000000000000000000000000000000000..dce908865d426dabede2b6749ad944e5a420e4cf --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/theanocode.py @@ -0,0 +1,571 @@ +""" +.. deprecated:: 1.8 + + ``sympy.printing.theanocode`` is deprecated. Theano has been renamed to + Aesara. Use ``sympy.printing.aesaracode`` instead. See + :ref:`theanocode-deprecated` for more information. + +""" +from __future__ import annotations +import math +from typing import Any + +from sympy.external import import_module +from sympy.printing.printer import Printer +from sympy.utilities.iterables import is_sequence +import sympy +from functools import partial + +from sympy.utilities.decorator import doctest_depends_on +from sympy.utilities.exceptions import sympy_deprecation_warning + + +__doctest_requires__ = {('theano_function',): ['theano']} + + +theano = import_module('theano') + + +if theano: + ts = theano.scalar + tt = theano.tensor + from theano.sandbox import linalg as tlinalg + + mapping = { + sympy.Add: tt.add, + sympy.Mul: tt.mul, + sympy.Abs: tt.abs_, + sympy.sign: tt.sgn, + sympy.ceiling: tt.ceil, + sympy.floor: tt.floor, + sympy.log: tt.log, + sympy.exp: tt.exp, + sympy.sqrt: tt.sqrt, + sympy.cos: tt.cos, + sympy.acos: tt.arccos, + sympy.sin: tt.sin, + sympy.asin: tt.arcsin, + sympy.tan: tt.tan, + sympy.atan: tt.arctan, + sympy.atan2: tt.arctan2, + sympy.cosh: tt.cosh, + sympy.acosh: tt.arccosh, + sympy.sinh: tt.sinh, + sympy.asinh: tt.arcsinh, + sympy.tanh: tt.tanh, + sympy.atanh: tt.arctanh, + sympy.re: tt.real, + sympy.im: tt.imag, + sympy.arg: tt.angle, + sympy.erf: tt.erf, + sympy.gamma: tt.gamma, + sympy.loggamma: tt.gammaln, + sympy.Pow: tt.pow, + sympy.Eq: tt.eq, + sympy.StrictGreaterThan: tt.gt, + sympy.StrictLessThan: tt.lt, + sympy.LessThan: tt.le, + sympy.GreaterThan: tt.ge, + sympy.And: tt.and_, + sympy.Or: tt.or_, + sympy.Max: tt.maximum, # SymPy accept >2 inputs, Theano only 2 + sympy.Min: tt.minimum, # SymPy accept >2 inputs, Theano only 2 + sympy.conjugate: tt.conj, + sympy.core.numbers.ImaginaryUnit: lambda:tt.complex(0,1), + # Matrices + sympy.MatAdd: tt.Elemwise(ts.add), + sympy.HadamardProduct: tt.Elemwise(ts.mul), + sympy.Trace: tlinalg.trace, + sympy.Determinant : tlinalg.det, + sympy.Inverse: tlinalg.matrix_inverse, + sympy.Transpose: tt.DimShuffle((False, False), [1, 0]), + } + + +class TheanoPrinter(Printer): + """ Code printer which creates Theano symbolic expression graphs. + + Parameters + ========== + + cache : dict + Cache dictionary to use. If None (default) will use + the global cache. To create a printer which does not depend on or alter + global state pass an empty dictionary. Note: the dictionary is not + copied on initialization of the printer and will be updated in-place, + so using the same dict object when creating multiple printers or making + multiple calls to :func:`.theano_code` or :func:`.theano_function` means + the cache is shared between all these applications. + + Attributes + ========== + + cache : dict + A cache of Theano variables which have been created for SymPy + symbol-like objects (e.g. :class:`sympy.core.symbol.Symbol` or + :class:`sympy.matrices.expressions.MatrixSymbol`). This is used to + ensure that all references to a given symbol in an expression (or + multiple expressions) are printed as the same Theano variable, which is + created only once. Symbols are differentiated only by name and type. The + format of the cache's contents should be considered opaque to the user. + """ + printmethod = "_theano" + + def __init__(self, *args, **kwargs): + self.cache = kwargs.pop('cache', {}) + super().__init__(*args, **kwargs) + + def _get_key(self, s, name=None, dtype=None, broadcastable=None): + """ Get the cache key for a SymPy object. + + Parameters + ========== + + s : sympy.core.basic.Basic + SymPy object to get key for. + + name : str + Name of object, if it does not have a ``name`` attribute. + """ + + if name is None: + name = s.name + + return (name, type(s), s.args, dtype, broadcastable) + + def _get_or_create(self, s, name=None, dtype=None, broadcastable=None): + """ + Get the Theano variable for a SymPy symbol from the cache, or create it + if it does not exist. + """ + + # Defaults + if name is None: + name = s.name + if dtype is None: + dtype = 'floatX' + if broadcastable is None: + broadcastable = () + + key = self._get_key(s, name, dtype=dtype, broadcastable=broadcastable) + + if key in self.cache: + return self.cache[key] + + value = tt.tensor(name=name, dtype=dtype, broadcastable=broadcastable) + self.cache[key] = value + return value + + def _print_Symbol(self, s, **kwargs): + dtype = kwargs.get('dtypes', {}).get(s) + bc = kwargs.get('broadcastables', {}).get(s) + return self._get_or_create(s, dtype=dtype, broadcastable=bc) + + def _print_AppliedUndef(self, s, **kwargs): + name = str(type(s)) + '_' + str(s.args[0]) + dtype = kwargs.get('dtypes', {}).get(s) + bc = kwargs.get('broadcastables', {}).get(s) + return self._get_or_create(s, name=name, dtype=dtype, broadcastable=bc) + + def _print_Basic(self, expr, **kwargs): + op = mapping[type(expr)] + children = [self._print(arg, **kwargs) for arg in expr.args] + return op(*children) + + def _print_Number(self, n, **kwargs): + # Integers already taken care of below, interpret as float + return float(n.evalf()) + + def _print_MatrixSymbol(self, X, **kwargs): + dtype = kwargs.get('dtypes', {}).get(X) + return self._get_or_create(X, dtype=dtype, broadcastable=(None, None)) + + def _print_DenseMatrix(self, X, **kwargs): + if not hasattr(tt, 'stacklists'): + raise NotImplementedError( + "Matrix translation not yet supported in this version of Theano") + + return tt.stacklists([ + [self._print(arg, **kwargs) for arg in L] + for L in X.tolist() + ]) + + _print_ImmutableMatrix = _print_ImmutableDenseMatrix = _print_DenseMatrix + + def _print_MatMul(self, expr, **kwargs): + children = [self._print(arg, **kwargs) for arg in expr.args] + result = children[0] + for child in children[1:]: + result = tt.dot(result, child) + return result + + def _print_MatPow(self, expr, **kwargs): + children = [self._print(arg, **kwargs) for arg in expr.args] + result = 1 + if isinstance(children[1], int) and children[1] > 0: + for i in range(children[1]): + result = tt.dot(result, children[0]) + else: + raise NotImplementedError('''Only non-negative integer + powers of matrices can be handled by Theano at the moment''') + return result + + def _print_MatrixSlice(self, expr, **kwargs): + parent = self._print(expr.parent, **kwargs) + rowslice = self._print(slice(*expr.rowslice), **kwargs) + colslice = self._print(slice(*expr.colslice), **kwargs) + return parent[rowslice, colslice] + + def _print_BlockMatrix(self, expr, **kwargs): + nrows, ncols = expr.blocks.shape + blocks = [[self._print(expr.blocks[r, c], **kwargs) + for c in range(ncols)] + for r in range(nrows)] + return tt.join(0, *[tt.join(1, *row) for row in blocks]) + + + def _print_slice(self, expr, **kwargs): + return slice(*[self._print(i, **kwargs) + if isinstance(i, sympy.Basic) else i + for i in (expr.start, expr.stop, expr.step)]) + + def _print_Pi(self, expr, **kwargs): + return math.pi + + def _print_Exp1(self, expr, **kwargs): + return ts.exp(1) + + def _print_Piecewise(self, expr, **kwargs): + import numpy as np + e, cond = expr.args[0].args # First condition and corresponding value + + # Print conditional expression and value for first condition + p_cond = self._print(cond, **kwargs) + p_e = self._print(e, **kwargs) + + # One condition only + if len(expr.args) == 1: + # Return value if condition else NaN + return tt.switch(p_cond, p_e, np.nan) + + # Return value_1 if condition_1 else evaluate remaining conditions + p_remaining = self._print(sympy.Piecewise(*expr.args[1:]), **kwargs) + return tt.switch(p_cond, p_e, p_remaining) + + def _print_Rational(self, expr, **kwargs): + return tt.true_div(self._print(expr.p, **kwargs), + self._print(expr.q, **kwargs)) + + def _print_Integer(self, expr, **kwargs): + return expr.p + + def _print_factorial(self, expr, **kwargs): + return self._print(sympy.gamma(expr.args[0] + 1), **kwargs) + + def _print_Derivative(self, deriv, **kwargs): + rv = self._print(deriv.expr, **kwargs) + for var in deriv.variables: + var = self._print(var, **kwargs) + rv = tt.Rop(rv, var, tt.ones_like(var)) + return rv + + def emptyPrinter(self, expr): + return expr + + def doprint(self, expr, dtypes=None, broadcastables=None): + """ Convert a SymPy expression to a Theano graph variable. + + The ``dtypes`` and ``broadcastables`` arguments are used to specify the + data type, dimension, and broadcasting behavior of the Theano variables + corresponding to the free symbols in ``expr``. Each is a mapping from + SymPy symbols to the value of the corresponding argument to + ``theano.tensor.Tensor``. + + See the corresponding `documentation page`__ for more information on + broadcasting in Theano. + + .. __: http://deeplearning.net/software/theano/tutorial/broadcasting.html + + Parameters + ========== + + expr : sympy.core.expr.Expr + SymPy expression to print. + + dtypes : dict + Mapping from SymPy symbols to Theano datatypes to use when creating + new Theano variables for those symbols. Corresponds to the ``dtype`` + argument to ``theano.tensor.Tensor``. Defaults to ``'floatX'`` + for symbols not included in the mapping. + + broadcastables : dict + Mapping from SymPy symbols to the value of the ``broadcastable`` + argument to ``theano.tensor.Tensor`` to use when creating Theano + variables for those symbols. Defaults to the empty tuple for symbols + not included in the mapping (resulting in a scalar). + + Returns + ======= + + theano.gof.graph.Variable + A variable corresponding to the expression's value in a Theano + symbolic expression graph. + + """ + if dtypes is None: + dtypes = {} + if broadcastables is None: + broadcastables = {} + + return self._print(expr, dtypes=dtypes, broadcastables=broadcastables) + + +global_cache: dict[Any, Any] = {} + + +def theano_code(expr, cache=None, **kwargs): + """ + Convert a SymPy expression into a Theano graph variable. + + .. deprecated:: 1.8 + + ``sympy.printing.theanocode`` is deprecated. Theano has been renamed to + Aesara. Use ``sympy.printing.aesaracode`` instead. See + :ref:`theanocode-deprecated` for more information. + + Parameters + ========== + + expr : sympy.core.expr.Expr + SymPy expression object to convert. + + cache : dict + Cached Theano variables (see :class:`TheanoPrinter.cache + `). Defaults to the module-level global cache. + + dtypes : dict + Passed to :meth:`.TheanoPrinter.doprint`. + + broadcastables : dict + Passed to :meth:`.TheanoPrinter.doprint`. + + Returns + ======= + + theano.gof.graph.Variable + A variable corresponding to the expression's value in a Theano symbolic + expression graph. + + """ + sympy_deprecation_warning( + """ + sympy.printing.theanocode is deprecated. Theano has been renamed to + Aesara. Use sympy.printing.aesaracode instead.""", + deprecated_since_version="1.8", + active_deprecations_target='theanocode-deprecated') + + if not theano: + raise ImportError("theano is required for theano_code") + + if cache is None: + cache = global_cache + + return TheanoPrinter(cache=cache, settings={}).doprint(expr, **kwargs) + + +def dim_handling(inputs, dim=None, dims=None, broadcastables=None): + r""" + Get value of ``broadcastables`` argument to :func:`.theano_code` from + keyword arguments to :func:`.theano_function`. + + Included for backwards compatibility. + + Parameters + ========== + + inputs + Sequence of input symbols. + + dim : int + Common number of dimensions for all inputs. Overrides other arguments + if given. + + dims : dict + Mapping from input symbols to number of dimensions. Overrides + ``broadcastables`` argument if given. + + broadcastables : dict + Explicit value of ``broadcastables`` argument to + :meth:`.TheanoPrinter.doprint`. If not None function will return this value unchanged. + + Returns + ======= + dict + Dictionary mapping elements of ``inputs`` to their "broadcastable" + values (tuple of ``bool``\ s). + """ + if dim is not None: + return dict.fromkeys(inputs, (False,) * dim) + + if dims is not None: + maxdim = max(dims.values()) + return { + s: (False,) * d + (True,) * (maxdim - d) + for s, d in dims.items() + } + + if broadcastables is not None: + return broadcastables + + return {} + + +@doctest_depends_on(modules=('theano',)) +def theano_function(inputs, outputs, scalar=False, *, + dim=None, dims=None, broadcastables=None, **kwargs): + """ + Create a Theano function from SymPy expressions. + + .. deprecated:: 1.8 + + ``sympy.printing.theanocode`` is deprecated. Theano has been renamed to + Aesara. Use ``sympy.printing.aesaracode`` instead. See + :ref:`theanocode-deprecated` for more information. + + The inputs and outputs are converted to Theano variables using + :func:`.theano_code` and then passed to ``theano.function``. + + Parameters + ========== + + inputs + Sequence of symbols which constitute the inputs of the function. + + outputs + Sequence of expressions which constitute the outputs(s) of the + function. The free symbols of each expression must be a subset of + ``inputs``. + + scalar : bool + Convert 0-dimensional arrays in output to scalars. This will return a + Python wrapper function around the Theano function object. + + cache : dict + Cached Theano variables (see :class:`TheanoPrinter.cache + `). Defaults to the module-level global cache. + + dtypes : dict + Passed to :meth:`.TheanoPrinter.doprint`. + + broadcastables : dict + Passed to :meth:`.TheanoPrinter.doprint`. + + dims : dict + Alternative to ``broadcastables`` argument. Mapping from elements of + ``inputs`` to integers indicating the dimension of their associated + arrays/tensors. Overrides ``broadcastables`` argument if given. + + dim : int + Another alternative to the ``broadcastables`` argument. Common number of + dimensions to use for all arrays/tensors. + ``theano_function([x, y], [...], dim=2)`` is equivalent to using + ``broadcastables={x: (False, False), y: (False, False)}``. + + Returns + ======= + callable + A callable object which takes values of ``inputs`` as positional + arguments and returns an output array for each of the expressions + in ``outputs``. If ``outputs`` is a single expression the function will + return a Numpy array, if it is a list of multiple expressions the + function will return a list of arrays. See description of the ``squeeze`` + argument above for the behavior when a single output is passed in a list. + The returned object will either be an instance of + ``theano.compile.function_module.Function`` or a Python wrapper + function around one. In both cases, the returned value will have a + ``theano_function`` attribute which points to the return value of + ``theano.function``. + + Examples + ======== + + >>> from sympy.abc import x, y, z + >>> from sympy.printing.theanocode import theano_function + + A simple function with one input and one output: + + >>> f1 = theano_function([x], [x**2 - 1], scalar=True) + >>> f1(3) + 8.0 + + A function with multiple inputs and one output: + + >>> f2 = theano_function([x, y, z], [(x**z + y**z)**(1/z)], scalar=True) + >>> f2(3, 4, 2) + 5.0 + + A function with multiple inputs and multiple outputs: + + >>> f3 = theano_function([x, y], [x**2 + y**2, x**2 - y**2], scalar=True) + >>> f3(2, 3) + [13.0, -5.0] + + See also + ======== + + dim_handling + + """ + sympy_deprecation_warning( + """ + sympy.printing.theanocode is deprecated. Theano has been renamed to Aesara. Use sympy.printing.aesaracode instead""", + deprecated_since_version="1.8", + active_deprecations_target='theanocode-deprecated') + + if not theano: + raise ImportError("theano is required for theano_function") + + # Pop off non-theano keyword args + cache = kwargs.pop('cache', {}) + dtypes = kwargs.pop('dtypes', {}) + + broadcastables = dim_handling( + inputs, dim=dim, dims=dims, broadcastables=broadcastables, + ) + + # Print inputs/outputs + code = partial(theano_code, cache=cache, dtypes=dtypes, + broadcastables=broadcastables) + tinputs = list(map(code, inputs)) + toutputs = list(map(code, outputs)) + + #fix constant expressions as variables + toutputs = [output if isinstance(output, theano.Variable) else tt.as_tensor_variable(output) for output in toutputs] + + if len(toutputs) == 1: + toutputs = toutputs[0] + + # Compile theano func + func = theano.function(tinputs, toutputs, **kwargs) + + is_0d = [len(o.variable.broadcastable) == 0 for o in func.outputs] + + # No wrapper required + if not scalar or not any(is_0d): + func.theano_function = func + return func + + # Create wrapper to convert 0-dimensional outputs to scalars + def wrapper(*args): + out = func(*args) + # out can be array(1.0) or [array(1.0), array(2.0)] + + if is_sequence(out): + return [o[()] if is_0d[i] else o for i, o in enumerate(out)] + else: + return out[()] + + wrapper.__wrapped__ = func + wrapper.__doc__ = func.__doc__ + wrapper.theano_function = func + return wrapper diff --git a/.venv/lib/python3.12/site-packages/sympy/printing/tree.py b/.venv/lib/python3.12/site-packages/sympy/printing/tree.py new file mode 100644 index 0000000000000000000000000000000000000000..82dac013419fbe93f63dcf5b90b3a529d72a32bc --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/printing/tree.py @@ -0,0 +1,175 @@ +def pprint_nodes(subtrees): + """ + Prettyprints systems of nodes. + + Examples + ======== + + >>> from sympy.printing.tree import pprint_nodes + >>> print(pprint_nodes(["a", "b1\\nb2", "c"])) + +-a + +-b1 + | b2 + +-c + + """ + def indent(s, type=1): + x = s.split("\n") + r = "+-%s\n" % x[0] + for a in x[1:]: + if a == "": + continue + if type == 1: + r += "| %s\n" % a + else: + r += " %s\n" % a + return r + if not subtrees: + return "" + f = "" + for a in subtrees[:-1]: + f += indent(a) + f += indent(subtrees[-1], 2) + return f + + +def print_node(node, assumptions=True): + """ + Returns information about the "node". + + This includes class name, string representation and assumptions. + + Parameters + ========== + + assumptions : bool, optional + See the ``assumptions`` keyword in ``tree`` + """ + s = "%s: %s\n" % (node.__class__.__name__, str(node)) + + if assumptions: + d = node._assumptions + else: + d = None + + if d: + for a in sorted(d): + v = d[a] + if v is None: + continue + s += "%s: %s\n" % (a, v) + + return s + + +def tree(node, assumptions=True): + """ + Returns a tree representation of "node" as a string. + + It uses print_node() together with pprint_nodes() on node.args recursively. + + Parameters + ========== + + assumptions : bool, optional + The flag to decide whether to print out all the assumption data + (such as ``is_integer`, ``is_real``) associated with the + expression or not. + + Enabling the flag makes the result verbose, and the printed + result may not be deterministic because of the randomness used + in backtracing the assumptions. + + See Also + ======== + + print_tree + + """ + subtrees = [] + for arg in node.args: + subtrees.append(tree(arg, assumptions=assumptions)) + s = print_node(node, assumptions=assumptions) + pprint_nodes(subtrees) + return s + + +def print_tree(node, assumptions=True): + """ + Prints a tree representation of "node". + + Parameters + ========== + + assumptions : bool, optional + The flag to decide whether to print out all the assumption data + (such as ``is_integer`, ``is_real``) associated with the + expression or not. + + Enabling the flag makes the result verbose, and the printed + result may not be deterministic because of the randomness used + in backtracing the assumptions. + + Examples + ======== + + >>> from sympy.printing import print_tree + >>> from sympy import Symbol + >>> x = Symbol('x', odd=True) + >>> y = Symbol('y', even=True) + + Printing with full assumptions information: + + >>> print_tree(y**x) + Pow: y**x + +-Symbol: y + | algebraic: True + | commutative: True + | complex: True + | even: True + | extended_real: True + | finite: True + | hermitian: True + | imaginary: False + | infinite: False + | integer: True + | irrational: False + | noninteger: False + | odd: False + | rational: True + | real: True + | transcendental: False + +-Symbol: x + algebraic: True + commutative: True + complex: True + even: False + extended_nonzero: True + extended_real: True + finite: True + hermitian: True + imaginary: False + infinite: False + integer: True + irrational: False + noninteger: False + nonzero: True + odd: True + rational: True + real: True + transcendental: False + zero: False + + Hiding the assumptions: + + >>> print_tree(y**x, assumptions=False) + Pow: y**x + +-Symbol: y + +-Symbol: x + + See Also + ======== + + tree + + """ + print(tree(node, assumptions=assumptions)) diff --git a/.venv/lib/python3.12/site-packages/sympy/release.py b/.venv/lib/python3.12/site-packages/sympy/release.py new file mode 100644 index 0000000000000000000000000000000000000000..b9f68edb2f571e7bfaad9472d36a2d357b5574b2 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/release.py @@ -0,0 +1 @@ +__version__ = "1.14.0" diff --git a/.venv/lib/python3.12/site-packages/sympy/sandbox/__init__.py b/.venv/lib/python3.12/site-packages/sympy/sandbox/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..3a84b7517819bb2fc9886274e09d955a74cabca1 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/sandbox/__init__.py @@ -0,0 +1,8 @@ +""" +Sandbox module of SymPy. + +This module contains experimental code, use at your own risk! + +There is no warranty that this code will still be located here in future +versions of SymPy. +""" diff --git a/.venv/lib/python3.12/site-packages/sympy/sandbox/__pycache__/__init__.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/sandbox/__pycache__/__init__.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..92b8d67cbd68a558b873a0a4603866231a655e4c Binary files /dev/null and b/.venv/lib/python3.12/site-packages/sympy/sandbox/__pycache__/__init__.cpython-312.pyc differ diff --git a/.venv/lib/python3.12/site-packages/sympy/sandbox/__pycache__/indexed_integrals.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/sandbox/__pycache__/indexed_integrals.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..76eef2a9d1486dab2d6fc350568723defb9e916d Binary files /dev/null and b/.venv/lib/python3.12/site-packages/sympy/sandbox/__pycache__/indexed_integrals.cpython-312.pyc differ diff --git a/.venv/lib/python3.12/site-packages/sympy/sandbox/indexed_integrals.py b/.venv/lib/python3.12/site-packages/sympy/sandbox/indexed_integrals.py new file mode 100644 index 0000000000000000000000000000000000000000..c0c17d141448b5a71cb814bff76a710a5bd43f88 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/sandbox/indexed_integrals.py @@ -0,0 +1,72 @@ +from sympy.tensor import Indexed +from sympy.core.containers import Tuple +from sympy.core.symbol import Dummy +from sympy.core.sympify import sympify +from sympy.integrals.integrals import Integral + + +class IndexedIntegral(Integral): + """ + Experimental class to test integration by indexed variables. + + Usage is analogue to ``Integral``, it simply adds awareness of + integration over indices. + + Contraction of non-identical index symbols referring to the same + ``IndexedBase`` is not yet supported. + + Examples + ======== + + >>> from sympy.sandbox.indexed_integrals import IndexedIntegral + >>> from sympy import IndexedBase, symbols + >>> A = IndexedBase('A') + >>> i, j = symbols('i j', integer=True) + >>> ii = IndexedIntegral(A[i], A[i]) + >>> ii + Integral(_A[i], _A[i]) + >>> ii.doit() + A[i]**2/2 + + If the indices are different, indexed objects are considered to be + different variables: + + >>> i2 = IndexedIntegral(A[j], A[i]) + >>> i2 + Integral(A[j], _A[i]) + >>> i2.doit() + A[i]*A[j] + """ + + def __new__(cls, function, *limits, **assumptions): + repl, limits = IndexedIntegral._indexed_process_limits(limits) + function = sympify(function) + function = function.xreplace(repl) + obj = Integral.__new__(cls, function, *limits, **assumptions) + obj._indexed_repl = repl + obj._indexed_reverse_repl = {val: key for key, val in repl.items()} + return obj + + def doit(self): + res = super().doit() + return res.xreplace(self._indexed_reverse_repl) + + @staticmethod + def _indexed_process_limits(limits): + repl = {} + newlimits = [] + for i in limits: + if isinstance(i, (tuple, list, Tuple)): + v = i[0] + vrest = i[1:] + else: + v = i + vrest = () + if isinstance(v, Indexed): + if v not in repl: + r = Dummy(str(v)) + repl[v] = r + newlimits.append((r,)+vrest) + else: + newlimits.append(i) + return repl, newlimits diff --git a/.venv/lib/python3.12/site-packages/sympy/sandbox/tests/__init__.py b/.venv/lib/python3.12/site-packages/sympy/sandbox/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/.venv/lib/python3.12/site-packages/sympy/sandbox/tests/__pycache__/__init__.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/sandbox/tests/__pycache__/__init__.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..124b33a7f1a007821d13f639d39b7486f3ba0526 Binary files /dev/null and b/.venv/lib/python3.12/site-packages/sympy/sandbox/tests/__pycache__/__init__.cpython-312.pyc differ diff --git a/.venv/lib/python3.12/site-packages/sympy/sandbox/tests/__pycache__/test_indexed_integrals.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/sandbox/tests/__pycache__/test_indexed_integrals.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..841cfd4282270e89677ffc17e8788d1f12a8ff95 Binary files /dev/null and b/.venv/lib/python3.12/site-packages/sympy/sandbox/tests/__pycache__/test_indexed_integrals.cpython-312.pyc differ diff --git a/.venv/lib/python3.12/site-packages/sympy/sandbox/tests/test_indexed_integrals.py b/.venv/lib/python3.12/site-packages/sympy/sandbox/tests/test_indexed_integrals.py new file mode 100644 index 0000000000000000000000000000000000000000..61b98f0ffec29e026f6dfe8e16fde8b5818b0b09 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/sandbox/tests/test_indexed_integrals.py @@ -0,0 +1,25 @@ +from sympy.sandbox.indexed_integrals import IndexedIntegral +from sympy.core.symbol import symbols +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.tensor.indexed import (Idx, IndexedBase) + + +def test_indexed_integrals(): + A = IndexedBase('A') + i, j = symbols('i j', integer=True) + a1, a2 = symbols('a1:3', cls=Idx) + assert isinstance(a1, Idx) + + assert IndexedIntegral(1, A[i]).doit() == A[i] + assert IndexedIntegral(A[i], A[i]).doit() == A[i] ** 2 / 2 + assert IndexedIntegral(A[j], A[i]).doit() == A[i] * A[j] + assert IndexedIntegral(A[i] * A[j], A[i]).doit() == A[i] ** 2 * A[j] / 2 + assert IndexedIntegral(sin(A[i]), A[i]).doit() == -cos(A[i]) + assert IndexedIntegral(sin(A[j]), A[i]).doit() == sin(A[j]) * A[i] + + assert IndexedIntegral(1, A[a1]).doit() == A[a1] + assert IndexedIntegral(A[a1], A[a1]).doit() == A[a1] ** 2 / 2 + assert IndexedIntegral(A[a2], A[a1]).doit() == A[a1] * A[a2] + assert IndexedIntegral(A[a1] * A[a2], A[a1]).doit() == A[a1] ** 2 * A[a2] / 2 + assert IndexedIntegral(sin(A[a1]), A[a1]).doit() == -cos(A[a1]) + assert IndexedIntegral(sin(A[a2]), A[a1]).doit() == sin(A[a2]) * A[a1] diff --git a/.venv/lib/python3.12/site-packages/sympy/series/__init__.py b/.venv/lib/python3.12/site-packages/sympy/series/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..953653e21856b82bc0b708ccd922efb728a084ed --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/__init__.py @@ -0,0 +1,23 @@ +"""A module that handles series: find a limit, order the series etc. +""" +from .order import Order +from .limits import limit, Limit +from .gruntz import gruntz +from .series import series +from .approximants import approximants +from .residues import residue +from .sequences import SeqPer, SeqFormula, sequence, SeqAdd, SeqMul +from .fourier import fourier_series +from .formal import fps +from .limitseq import difference_delta, limit_seq + +from sympy.core.singleton import S +EmptySequence = S.EmptySequence + +O = Order + +__all__ = ['Order', 'O', 'limit', 'Limit', 'gruntz', 'series', 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/dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/acceleration.py @@ -0,0 +1,101 @@ +""" +Convergence acceleration / extrapolation methods for series and +sequences. + +References: +Carl M. Bender & Steven A. Orszag, "Advanced Mathematical Methods for +Scientists and Engineers: Asymptotic Methods and Perturbation Theory", +Springer 1999. (Shanks transformation: pp. 368-375, Richardson +extrapolation: pp. 375-377.) +""" + +from sympy.core.numbers import Integer +from sympy.core.singleton import S +from sympy.functions.combinatorial.factorials import factorial + + +def richardson(A, k, n, N): + """ + Calculate an approximation for lim k->oo A(k) using Richardson + extrapolation with the terms A(n), A(n+1), ..., A(n+N+1). + Choosing N ~= 2*n often gives good results. + + Examples + ======== + + A simple example is to calculate exp(1) using the limit definition. + This limit converges slowly; n = 100 only produces two accurate + digits: + + >>> from sympy.abc import n + >>> e = (1 + 1/n)**n + >>> print(round(e.subs(n, 100).evalf(), 10)) + 2.7048138294 + + Richardson extrapolation with 11 appropriately chosen terms gives + a value that is accurate to the indicated precision: + + >>> from sympy import E + >>> from sympy.series.acceleration import richardson + >>> print(round(richardson(e, n, 10, 20).evalf(), 10)) + 2.7182818285 + >>> print(round(E.evalf(), 10)) + 2.7182818285 + + Another useful application is to speed up convergence of series. + Computing 100 terms of the zeta(2) series 1/k**2 yields only + two accurate digits: + + >>> from sympy.abc import k, n + >>> from sympy import Sum + >>> A = Sum(k**-2, (k, 1, n)) + >>> print(round(A.subs(n, 100).evalf(), 10)) + 1.6349839002 + + Richardson extrapolation performs much better: + + >>> from sympy import pi + >>> print(round(richardson(A, n, 10, 20).evalf(), 10)) + 1.6449340668 + >>> print(round(((pi**2)/6).evalf(), 10)) # Exact value + 1.6449340668 + + """ + s = S.Zero + for j in range(0, N + 1): + s += (A.subs(k, Integer(n + j)).doit() * (n + j)**N * + S.NegativeOne**(j + N) / (factorial(j) * factorial(N - j))) + return s + + +def shanks(A, k, n, m=1): + """ + Calculate an approximation for lim k->oo A(k) using the n-term Shanks + transformation S(A)(n). With m > 1, calculate the m-fold recursive + Shanks transformation S(S(...S(A)...))(n). + + The Shanks transformation is useful for summing Taylor series that + converge slowly near a pole or singularity, e.g. for log(2): + + >>> from sympy.abc import k, n + >>> from sympy import Sum, Integer + >>> from sympy.series.acceleration import shanks + >>> A = Sum(Integer(-1)**(k+1) / k, (k, 1, n)) + >>> print(round(A.subs(n, 100).doit().evalf(), 10)) + 0.6881721793 + >>> print(round(shanks(A, n, 25).evalf(), 10)) + 0.6931396564 + >>> print(round(shanks(A, n, 25, 5).evalf(), 10)) + 0.6931471806 + + The correct value is 0.6931471805599453094172321215. + """ + table = [A.subs(k, Integer(j)).doit() for j in range(n + m + 2)] + table2 = table.copy() + + for i in range(1, m + 1): + for j in range(i, n + m + 1): + x, y, z = table[j - 1], table[j], table[j + 1] + table2[j] = (z*x - y**2) / (z + x - 2*y) + table = table2.copy() + return table[n] diff --git a/.venv/lib/python3.12/site-packages/sympy/series/approximants.py b/.venv/lib/python3.12/site-packages/sympy/series/approximants.py new file mode 100644 index 0000000000000000000000000000000000000000..3d54ce41bc7367606ae6260f8e9ac00149cedc0f --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/approximants.py @@ -0,0 +1,103 @@ +from sympy.core.singleton import S +from sympy.core.symbol import Symbol +from sympy.polys.polytools import lcm +from sympy.utilities import public + +@public +def approximants(l, X=Symbol('x'), simplify=False): + """ + Return a generator for consecutive Pade approximants for a series. + It can also be used for computing the rational generating function of a + series when possible, since the last approximant returned by the generator + will be the generating function (if any). + + Explanation + =========== + + The input list can contain more complex expressions than integer or rational + numbers; symbols may also be involved in the computation. An example below + show how to compute the generating function of the whole Pascal triangle. + + The generator can be asked to apply the sympy.simplify function on each + generated term, which will make the computation slower; however it may be + useful when symbols are involved in the expressions. + + Examples + ======== + + >>> from sympy.series import approximants + >>> from sympy import lucas, fibonacci, symbols, binomial + >>> g = [lucas(k) for k in range(16)] + >>> [e for e in approximants(g)] + [2, -4/(x - 2), (5*x - 2)/(3*x - 1), (x - 2)/(x**2 + x - 1)] + + >>> h = [fibonacci(k) for k in range(16)] + >>> [e for e in approximants(h)] + [x, -x/(x - 1), (x**2 - x)/(2*x - 1), -x/(x**2 + x - 1)] + + >>> x, t = symbols("x,t") + >>> p=[sum(binomial(k,i)*x**i for i in range(k+1)) for k in range(16)] + >>> y = approximants(p, t) + >>> for k in range(3): print(next(y)) + 1 + (x + 1)/((-x - 1)*(t*(x + 1) + (x + 1)/(-x - 1))) + nan + + >>> y = approximants(p, t, simplify=True) + >>> for k in range(3): print(next(y)) + 1 + -1/(t*(x + 1) - 1) + nan + + See Also + ======== + + sympy.concrete.guess.guess_generating_function_rational + mpmath.pade + """ + from sympy.simplify import simplify as simp + from sympy.simplify.radsimp import denom + p1, q1 = [S.One], [S.Zero] + p2, q2 = [S.Zero], [S.One] + while len(l): + b = 0 + while l[b]==0: + b += 1 + if b == len(l): + return + m = [S.One/l[b]] + for k in range(b+1, len(l)): + s = 0 + for j in range(b, k): + s -= l[j+1] * m[b-j-1] + m.append(s/l[b]) + l = m + a, l[0] = l[0], 0 + p = [0] * max(len(p2), b+len(p1)) + q = [0] * max(len(q2), b+len(q1)) + for k in range(len(p2)): + p[k] = a*p2[k] + for k in range(b, b+len(p1)): + p[k] += p1[k-b] + for k in range(len(q2)): + q[k] = a*q2[k] + for k in range(b, b+len(q1)): + q[k] += q1[k-b] + while p[-1]==0: p.pop() + while q[-1]==0: q.pop() + p1, p2 = p2, p + q1, q2 = q2, q + + # yield result + c = 1 + for x in p: + c = lcm(c, denom(x)) + for x in q: + c = lcm(c, denom(x)) + out = ( sum(c*e*X**k for k, e in enumerate(p)) + / sum(c*e*X**k for k, e in enumerate(q)) ) + if simplify: + yield(simp(out)) + else: + yield out + return diff --git a/.venv/lib/python3.12/site-packages/sympy/series/aseries.py b/.venv/lib/python3.12/site-packages/sympy/series/aseries.py new file mode 100644 index 0000000000000000000000000000000000000000..dbbe0664e6d43a9329f37789c16c48143eda5413 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/aseries.py @@ -0,0 +1,10 @@ +from sympy.core.sympify import sympify + + +def aseries(expr, x=None, n=6, bound=0, hir=False): + """ + See the docstring of Expr.aseries() for complete details of this wrapper. + + """ + expr = sympify(expr) + return expr.aseries(x, n, bound, hir) diff --git a/.venv/lib/python3.12/site-packages/sympy/series/benchmarks/__init__.py b/.venv/lib/python3.12/site-packages/sympy/series/benchmarks/__init__.py new file mode 100644 index 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a/.venv/lib/python3.12/site-packages/sympy/series/benchmarks/__pycache__/bench_order.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/series/benchmarks/__pycache__/bench_order.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..55fdef81e455a73175565705943c2bfd0331319b Binary files /dev/null and b/.venv/lib/python3.12/site-packages/sympy/series/benchmarks/__pycache__/bench_order.cpython-312.pyc differ diff --git a/.venv/lib/python3.12/site-packages/sympy/series/benchmarks/bench_limit.py b/.venv/lib/python3.12/site-packages/sympy/series/benchmarks/bench_limit.py new file mode 100644 index 0000000000000000000000000000000000000000..eafc28328848dad4b3ea433537971f5785253afe --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/benchmarks/bench_limit.py @@ -0,0 +1,9 @@ +from sympy.core.numbers import oo +from sympy.core.symbol import Symbol +from sympy.series.limits import limit + +x = Symbol('x') + + +def timeit_limit_1x(): + limit(1/x, x, oo) diff --git a/.venv/lib/python3.12/site-packages/sympy/series/benchmarks/bench_order.py b/.venv/lib/python3.12/site-packages/sympy/series/benchmarks/bench_order.py new file mode 100644 index 0000000000000000000000000000000000000000..1c85fa173dfc2a478792de8ab816c23ba9d408ef --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/benchmarks/bench_order.py @@ -0,0 +1,10 @@ +from sympy.core.add import Add +from sympy.core.symbol import Symbol +from sympy.series.order import O + +x = Symbol('x') +l = [x**i for i in range(1000)] +l.append(O(x**1001)) + +def timeit_order_1x(): + Add(*l) diff --git a/.venv/lib/python3.12/site-packages/sympy/series/formal.py b/.venv/lib/python3.12/site-packages/sympy/series/formal.py new file mode 100644 index 0000000000000000000000000000000000000000..ada591e03dd607daf8f8a5da1cf38cf283a3ed86 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/formal.py @@ -0,0 +1,1863 @@ +"""Formal Power Series""" + +from collections import defaultdict + +from sympy.core.numbers import (nan, oo, zoo) +from sympy.core.add import Add +from sympy.core.expr import Expr +from sympy.core.function import Derivative, Function, expand +from sympy.core.mul import Mul +from sympy.core.numbers import Rational +from sympy.core.relational import Eq +from sympy.sets.sets import Interval +from sympy.core.singleton import S +from sympy.core.symbol import Wild, Dummy, symbols, Symbol +from sympy.core.sympify import sympify +from sympy.discrete.convolutions import convolution +from sympy.functions.combinatorial.factorials import binomial, factorial, rf +from sympy.functions.combinatorial.numbers import bell +from sympy.functions.elementary.integers import floor, frac, ceiling +from sympy.functions.elementary.miscellaneous import Min, Max +from sympy.functions.elementary.piecewise import Piecewise +from sympy.series.limits import Limit +from sympy.series.order import Order +from sympy.series.sequences import sequence +from sympy.series.series_class import SeriesBase +from sympy.utilities.iterables import iterable + + + +def rational_algorithm(f, x, k, order=4, full=False): + """ + Rational algorithm for computing + formula of coefficients of Formal Power Series + of a function. + + Explanation + =========== + + Applicable when f(x) or some derivative of f(x) + is a rational function in x. + + :func:`rational_algorithm` uses :func:`~.apart` function for partial fraction + decomposition. :func:`~.apart` by default uses 'undetermined coefficients + method'. By setting ``full=True``, 'Bronstein's algorithm' can be used + instead. + + Looks for derivative of a function up to 4'th order (by default). + This can be overridden using order option. + + Parameters + ========== + + x : Symbol + order : int, optional + Order of the derivative of ``f``, Default is 4. + full : bool + + Returns + ======= + + formula : Expr + ind : Expr + Independent terms. + order : int + full : bool + + Examples + ======== + + >>> from sympy import log, atan + >>> from sympy.series.formal import rational_algorithm as ra + >>> from sympy.abc import x, k + + >>> ra(1 / (1 - x), x, k) + (1, 0, 0) + >>> ra(log(1 + x), x, k) + (-1/((-1)**k*k), 0, 1) + + >>> ra(atan(x), x, k, full=True) + ((-I/(2*(-I)**k) + I/(2*I**k))/k, 0, 1) + + Notes + ===== + + By setting ``full=True``, range of admissible functions to be solved using + ``rational_algorithm`` can be increased. This option should be used + carefully as it can significantly slow down the computation as ``doit`` is + performed on the :class:`~.RootSum` object returned by the :func:`~.apart` + function. Use ``full=False`` whenever possible. + + See Also + ======== + + sympy.polys.partfrac.apart + + References + ========== + + .. [1] Formal Power Series - Dominik Gruntz, Wolfram Koepf + .. [2] Power Series in Computer Algebra - Wolfram Koepf + + """ + from sympy.polys import RootSum, apart + from sympy.integrals import integrate + + diff = f + ds = [] # list of diff + + for i in range(order + 1): + if i: + diff = diff.diff(x) + + if diff.is_rational_function(x): + coeff, sep = S.Zero, S.Zero + + terms = apart(diff, x, full=full) + if terms.has(RootSum): + terms = terms.doit() + + for t in Add.make_args(terms): + num, den = t.as_numer_denom() + if not den.has(x): + sep += t + else: + if isinstance(den, Mul): + # m*(n*x - a)**j -> (n*x - a)**j + ind = den.as_independent(x) + den = ind[1] + num /= ind[0] + + # (n*x - a)**j -> (x - b) + den, j = den.as_base_exp() + a, xterm = den.as_coeff_add(x) + + # term -> m/x**n + if not a: + sep += t + continue + + xc = xterm[0].coeff(x) + a /= -xc + num /= xc**j + + ak = ((-1)**j * num * + binomial(j + k - 1, k).rewrite(factorial) / + a**(j + k)) + coeff += ak + + # Hacky, better way? + if coeff.is_zero: + return None + if (coeff.has(x) or coeff.has(zoo) or coeff.has(oo) or + coeff.has(nan)): + return None + + for j in range(i): + coeff = (coeff / (k + j + 1)) + sep = integrate(sep, x) + sep += (ds.pop() - sep).limit(x, 0) # constant of integration + return (coeff.subs(k, k - i), sep, i) + + else: + ds.append(diff) + + return None + + +def rational_independent(terms, x): + """ + Returns a list of all the rationally independent terms. + + Examples + ======== + + >>> from sympy import sin, cos + >>> from sympy.series.formal import rational_independent + >>> from sympy.abc import x + + >>> rational_independent([cos(x), sin(x)], x) + [cos(x), sin(x)] + >>> rational_independent([x**2, sin(x), x*sin(x), x**3], x) + [x**3 + x**2, x*sin(x) + sin(x)] + """ + if not terms: + return [] + + ind = terms[0:1] + + for t in terms[1:]: + n = t.as_independent(x)[1] + for i, term in enumerate(ind): + d = term.as_independent(x)[1] + q = (n / d).cancel() + if q.is_rational_function(x): + ind[i] += t + break + else: + ind.append(t) + return ind + + +def simpleDE(f, x, g, order=4): + r""" + Generates simple DE. + + Explanation + =========== + + DE is of the form + + .. math:: + f^k(x) + \sum\limits_{j=0}^{k-1} A_j f^j(x) = 0 + + where :math:`A_j` should be rational function in x. + + Generates DE's upto order 4 (default). DE's can also have free parameters. + + By increasing order, higher order DE's can be found. + + Yields a tuple of (DE, order). + """ + from sympy.solvers.solveset import linsolve + + a = symbols('a:%d' % (order)) + + def _makeDE(k): + eq = f.diff(x, k) + Add(*[a[i]*f.diff(x, i) for i in range(0, k)]) + DE = g(x).diff(x, k) + Add(*[a[i]*g(x).diff(x, i) for i in range(0, k)]) + return eq, DE + + found = False + for k in range(1, order + 1): + eq, DE = _makeDE(k) + eq = eq.expand() + terms = eq.as_ordered_terms() + ind = rational_independent(terms, x) + if found or len(ind) == k: + sol = dict(zip(a, (i for s in linsolve(ind, a[:k]) for i in s))) + if sol: + found = True + DE = DE.subs(sol) + DE = DE.as_numer_denom()[0] + DE = DE.factor().as_coeff_mul(Derivative)[1][0] + yield DE.collect(Derivative(g(x))), k + + +def exp_re(DE, r, k): + """Converts a DE with constant coefficients (explike) into a RE. + + Explanation + =========== + + Performs the substitution: + + .. math:: + f^j(x) \\to r(k + j) + + Normalises the terms so that lowest order of a term is always r(k). + + Examples + ======== + + >>> from sympy import Function, Derivative + >>> from sympy.series.formal import exp_re + >>> from sympy.abc import x, k + >>> f, r = Function('f'), Function('r') + + >>> exp_re(-f(x) + Derivative(f(x)), r, k) + -r(k) + r(k + 1) + >>> exp_re(Derivative(f(x), x) + Derivative(f(x), (x, 2)), r, k) + r(k) + r(k + 1) + + See Also + ======== + + sympy.series.formal.hyper_re + """ + RE = S.Zero + + g = DE.atoms(Function).pop() + + mini = None + for t in Add.make_args(DE): + coeff, d = t.as_independent(g) + if isinstance(d, Derivative): + j = d.derivative_count + else: + j = 0 + if mini is None or j < mini: + mini = j + RE += coeff * r(k + j) + if mini: + RE = RE.subs(k, k - mini) + return RE + + +def hyper_re(DE, r, k): + """ + Converts a DE into a RE. + + Explanation + =========== + + Performs the substitution: + + .. math:: + x^l f^j(x) \\to (k + 1 - l)_j . a_{k + j - l} + + Normalises the terms so that lowest order of a term is always r(k). + + Examples + ======== + + >>> from sympy import Function, Derivative + >>> from sympy.series.formal import hyper_re + >>> from sympy.abc import x, k + >>> f, r = Function('f'), Function('r') + + >>> hyper_re(-f(x) + Derivative(f(x)), r, k) + (k + 1)*r(k + 1) - r(k) + >>> hyper_re(-x*f(x) + Derivative(f(x), (x, 2)), r, k) + (k + 2)*(k + 3)*r(k + 3) - r(k) + + See Also + ======== + + sympy.series.formal.exp_re + """ + RE = S.Zero + + g = DE.atoms(Function).pop() + x = g.atoms(Symbol).pop() + + mini = None + for t in Add.make_args(DE.expand()): + coeff, d = t.as_independent(g) + c, v = coeff.as_independent(x) + l = v.as_coeff_exponent(x)[1] + if isinstance(d, Derivative): + j = d.derivative_count + else: + j = 0 + RE += c * rf(k + 1 - l, j) * r(k + j - l) + if mini is None or j - l < mini: + mini = j - l + + RE = RE.subs(k, k - mini) + + m = Wild('m') + return RE.collect(r(k + m)) + + +def _transformation_a(f, x, P, Q, k, m, shift): + f *= x**(-shift) + P = P.subs(k, k + shift) + Q = Q.subs(k, k + shift) + return f, P, Q, m + + +def _transformation_c(f, x, P, Q, k, m, scale): + f = f.subs(x, x**scale) + P = P.subs(k, k / scale) + Q = Q.subs(k, k / scale) + m *= scale + return f, P, Q, m + + +def _transformation_e(f, x, P, Q, k, m): + f = f.diff(x) + P = P.subs(k, k + 1) * (k + m + 1) + Q = Q.subs(k, k + 1) * (k + 1) + return f, P, Q, m + + +def _apply_shift(sol, shift): + return [(res, cond + shift) for res, cond in sol] + + +def _apply_scale(sol, scale): + return [(res, cond / scale) for res, cond in sol] + + +def _apply_integrate(sol, x, k): + return [(res / ((cond + 1)*(cond.as_coeff_Add()[1].coeff(k))), cond + 1) + for res, cond in sol] + + +def _compute_formula(f, x, P, Q, k, m, k_max): + """Computes the formula for f.""" + from sympy.polys import roots + + sol = [] + for i in range(k_max + 1, k_max + m + 1): + if (i < 0) == True: + continue + r = f.diff(x, i).limit(x, 0) / factorial(i) + if r.is_zero: + continue + + kterm = m*k + i + res = r + + p = P.subs(k, kterm) + q = Q.subs(k, kterm) + c1 = p.subs(k, 1/k).leadterm(k)[0] + c2 = q.subs(k, 1/k).leadterm(k)[0] + res *= (-c1 / c2)**k + + res *= Mul(*[rf(-r, k)**mul for r, mul in roots(p, k).items()]) + res /= Mul(*[rf(-r, k)**mul for r, mul in roots(q, k).items()]) + + sol.append((res, kterm)) + + return sol + + +def _rsolve_hypergeometric(f, x, P, Q, k, m): + """ + Recursive wrapper to rsolve_hypergeometric. + + Explanation + =========== + + Returns a Tuple of (formula, series independent terms, + maximum power of x in independent terms) if successful + otherwise ``None``. + + See :func:`rsolve_hypergeometric` for details. + """ + from sympy.polys import lcm, roots + from sympy.integrals import integrate + + # transformation - c + proots, qroots = roots(P, k), roots(Q, k) + all_roots = dict(proots) + all_roots.update(qroots) + scale = lcm([r.as_numer_denom()[1] for r, t in all_roots.items() + if r.is_rational]) + f, P, Q, m = _transformation_c(f, x, P, Q, k, m, scale) + + # transformation - a + qroots = roots(Q, k) + if qroots: + k_min = Min(*qroots.keys()) + else: + k_min = S.Zero + shift = k_min + m + f, P, Q, m = _transformation_a(f, x, P, Q, k, m, shift) + + l = (x*f).limit(x, 0) + if not isinstance(l, Limit) and l != 0: # Ideally should only be l != 0 + return None + + qroots = roots(Q, k) + if qroots: + k_max = Max(*qroots.keys()) + else: + k_max = S.Zero + + ind, mp = S.Zero, -oo + for i in range(k_max + m + 1): + r = f.diff(x, i).limit(x, 0) / factorial(i) + if r.is_finite is False: + old_f = f + f, P, Q, m = _transformation_a(f, x, P, Q, k, m, i) + f, P, Q, m = _transformation_e(f, x, P, Q, k, m) + sol, ind, mp = _rsolve_hypergeometric(f, x, P, Q, k, m) + sol = _apply_integrate(sol, x, k) + sol = _apply_shift(sol, i) + ind = integrate(ind, x) + ind += (old_f - ind).limit(x, 0) # constant of integration + mp += 1 + return sol, ind, mp + elif r: + ind += r*x**(i + shift) + pow_x = Rational((i + shift), scale) + if pow_x > mp: + mp = pow_x # maximum power of x + ind = ind.subs(x, x**(1/scale)) + + sol = _compute_formula(f, x, P, Q, k, m, k_max) + sol = _apply_shift(sol, shift) + sol = _apply_scale(sol, scale) + + return sol, ind, mp + + +def rsolve_hypergeometric(f, x, P, Q, k, m): + """ + Solves RE of hypergeometric type. + + Explanation + =========== + + Attempts to solve RE of the form + + Q(k)*a(k + m) - P(k)*a(k) + + Transformations that preserve Hypergeometric type: + + a. x**n*f(x): b(k + m) = R(k - n)*b(k) + b. f(A*x): b(k + m) = A**m*R(k)*b(k) + c. f(x**n): b(k + n*m) = R(k/n)*b(k) + d. f(x**(1/m)): b(k + 1) = R(k*m)*b(k) + e. f'(x): b(k + m) = ((k + m + 1)/(k + 1))*R(k + 1)*b(k) + + Some of these transformations have been used to solve the RE. + + Returns + ======= + + formula : Expr + ind : Expr + Independent terms. + order : int + + Examples + ======== + + >>> from sympy import exp, ln, S + >>> from sympy.series.formal import rsolve_hypergeometric as rh + >>> from sympy.abc import x, k + + >>> rh(exp(x), x, -S.One, (k + 1), k, 1) + (Piecewise((1/factorial(k), Eq(Mod(k, 1), 0)), (0, True)), 1, 1) + + >>> rh(ln(1 + x), x, k**2, k*(k + 1), k, 1) + (Piecewise(((-1)**(k - 1)*factorial(k - 1)/RisingFactorial(2, k - 1), + Eq(Mod(k, 1), 0)), (0, True)), x, 2) + + References + ========== + + .. [1] Formal Power Series - Dominik Gruntz, Wolfram Koepf + .. [2] Power Series in Computer Algebra - Wolfram Koepf + """ + result = _rsolve_hypergeometric(f, x, P, Q, k, m) + + if result is None: + return None + + sol_list, ind, mp = result + + sol_dict = defaultdict(lambda: S.Zero) + for res, cond in sol_list: + j, mk = cond.as_coeff_Add() + c = mk.coeff(k) + + if j.is_integer is False: + res *= x**frac(j) + j = floor(j) + + res = res.subs(k, (k - j) / c) + cond = Eq(k % c, j % c) + sol_dict[cond] += res # Group together formula for same conditions + + sol = [(res, cond) for cond, res in sol_dict.items()] + sol.append((S.Zero, True)) + sol = Piecewise(*sol) + + if mp is -oo: + s = S.Zero + elif mp.is_integer is False: + s = ceiling(mp) + else: + s = mp + 1 + + # save all the terms of + # form 1/x**k in ind + if s < 0: + ind += sum(sequence(sol * x**k, (k, s, -1))) + s = S.Zero + + return (sol, ind, s) + + +def _solve_hyper_RE(f, x, RE, g, k): + """See docstring of :func:`rsolve_hypergeometric` for details.""" + terms = Add.make_args(RE) + + if len(terms) == 2: + gs = list(RE.atoms(Function)) + P, Q = map(RE.coeff, gs) + m = gs[1].args[0] - gs[0].args[0] + if m < 0: + P, Q = Q, P + m = abs(m) + return rsolve_hypergeometric(f, x, P, Q, k, m) + + +def _solve_explike_DE(f, x, DE, g, k): + """Solves DE with constant coefficients.""" + from sympy.solvers import rsolve + + for t in Add.make_args(DE): + coeff, d = t.as_independent(g) + if coeff.free_symbols: + return + + RE = exp_re(DE, g, k) + + init = {} + for i in range(len(Add.make_args(RE))): + if i: + f = f.diff(x) + init[g(k).subs(k, i)] = f.limit(x, 0) + + sol = rsolve(RE, g(k), init) + + if sol: + return (sol / factorial(k), S.Zero, S.Zero) + + +def _solve_simple(f, x, DE, g, k): + """Converts DE into RE and solves using :func:`rsolve`.""" + from sympy.solvers import rsolve + + RE = hyper_re(DE, g, k) + + init = {} + for i in range(len(Add.make_args(RE))): + if i: + f = f.diff(x) + init[g(k).subs(k, i)] = f.limit(x, 0) / factorial(i) + + sol = rsolve(RE, g(k), init) + + if sol: + return (sol, S.Zero, S.Zero) + + +def _transform_explike_DE(DE, g, x, order, syms): + """Converts DE with free parameters into DE with constant coefficients.""" + from sympy.solvers.solveset import linsolve + + eq = [] + highest_coeff = DE.coeff(Derivative(g(x), x, order)) + for i in range(order): + coeff = DE.coeff(Derivative(g(x), x, i)) + coeff = (coeff / highest_coeff).expand().collect(x) + eq.extend(Add.make_args(coeff)) + temp = [] + for e in eq: + if e.has(x): + break + elif e.has(Symbol): + temp.append(e) + else: + eq = temp + if eq: + sol = dict(zip(syms, (i for s in linsolve(eq, list(syms)) for i in s))) + if sol: + DE = DE.subs(sol) + DE = DE.factor().as_coeff_mul(Derivative)[1][0] + DE = DE.collect(Derivative(g(x))) + return DE + + +def _transform_DE_RE(DE, g, k, order, syms): + """Converts DE with free parameters into RE of hypergeometric type.""" + from sympy.solvers.solveset import linsolve + + RE = hyper_re(DE, g, k) + + eq = [RE.coeff(g(k + i)) for i in range(1, order)] + sol = dict(zip(syms, (i for s in linsolve(eq, list(syms)) for i in s))) + if sol: + m = Wild('m') + RE = RE.subs(sol) + RE = RE.factor().as_numer_denom()[0].collect(g(k + m)) + RE = RE.as_coeff_mul(g)[1][0] + for i in range(order): # smallest order should be g(k) + if RE.coeff(g(k + i)) and i: + RE = RE.subs(k, k - i) + break + return RE + + +def solve_de(f, x, DE, order, g, k): + """ + Solves the DE. + + Explanation + =========== + + Tries to solve DE by either converting into a RE containing two terms or + converting into a DE having constant coefficients. + + Returns + ======= + + formula : Expr + ind : Expr + Independent terms. + order : int + + Examples + ======== + + >>> from sympy import Derivative as D, Function + >>> from sympy import exp, ln + >>> from sympy.series.formal import solve_de + >>> from sympy.abc import x, k + >>> f = Function('f') + + >>> solve_de(exp(x), x, D(f(x), x) - f(x), 1, f, k) + (Piecewise((1/factorial(k), Eq(Mod(k, 1), 0)), (0, True)), 1, 1) + + >>> solve_de(ln(1 + x), x, (x + 1)*D(f(x), x, 2) + D(f(x)), 2, f, k) + (Piecewise(((-1)**(k - 1)*factorial(k - 1)/RisingFactorial(2, k - 1), + Eq(Mod(k, 1), 0)), (0, True)), x, 2) + """ + sol = None + syms = DE.free_symbols.difference({g, x}) + + if syms: + RE = _transform_DE_RE(DE, g, k, order, syms) + else: + RE = hyper_re(DE, g, k) + if not RE.free_symbols.difference({k}): + sol = _solve_hyper_RE(f, x, RE, g, k) + + if sol: + return sol + + if syms: + DE = _transform_explike_DE(DE, g, x, order, syms) + if not DE.free_symbols.difference({x}): + sol = _solve_explike_DE(f, x, DE, g, k) + + if sol: + return sol + + +def hyper_algorithm(f, x, k, order=4): + """ + Hypergeometric algorithm for computing Formal Power Series. + + Explanation + =========== + + Steps: + * Generates DE + * Convert the DE into RE + * Solves the RE + + Examples + ======== + + >>> from sympy import exp, ln + >>> from sympy.series.formal import hyper_algorithm + + >>> from sympy.abc import x, k + + >>> hyper_algorithm(exp(x), x, k) + (Piecewise((1/factorial(k), Eq(Mod(k, 1), 0)), (0, True)), 1, 1) + + >>> hyper_algorithm(ln(1 + x), x, k) + (Piecewise(((-1)**(k - 1)*factorial(k - 1)/RisingFactorial(2, k - 1), + Eq(Mod(k, 1), 0)), (0, True)), x, 2) + + See Also + ======== + + sympy.series.formal.simpleDE + sympy.series.formal.solve_de + """ + g = Function('g') + + des = [] # list of DE's + sol = None + for DE, i in simpleDE(f, x, g, order): + if DE is not None: + sol = solve_de(f, x, DE, i, g, k) + if sol: + return sol + if not DE.free_symbols.difference({x}): + des.append(DE) + + # If nothing works + # Try plain rsolve + for DE in des: + sol = _solve_simple(f, x, DE, g, k) + if sol: + return sol + + +def _compute_fps(f, x, x0, dir, hyper, order, rational, full): + """Recursive wrapper to compute fps. + + See :func:`compute_fps` for details. + """ + if x0 in [S.Infinity, S.NegativeInfinity]: + dir = S.One if x0 is S.Infinity else -S.One + temp = f.subs(x, 1/x) + result = _compute_fps(temp, x, 0, dir, hyper, order, rational, full) + if result is None: + return None + return (result[0], result[1].subs(x, 1/x), result[2].subs(x, 1/x)) + elif x0 or dir == -S.One: + if dir == -S.One: + rep = -x + x0 + rep2 = -x + rep2b = x0 + else: + rep = x + x0 + rep2 = x + rep2b = -x0 + temp = f.subs(x, rep) + result = _compute_fps(temp, x, 0, S.One, hyper, order, rational, full) + if result is None: + return None + return (result[0], result[1].subs(x, rep2 + rep2b), + result[2].subs(x, rep2 + rep2b)) + + if f.is_polynomial(x): + k = Dummy('k') + ak = sequence(Coeff(f, x, k), (k, 1, oo)) + xk = sequence(x**k, (k, 0, oo)) + ind = f.coeff(x, 0) + return ak, xk, ind + + # Break instances of Add + # this allows application of different + # algorithms on different terms increasing the + # range of admissible functions. + if isinstance(f, Add): + result = False + ak = sequence(S.Zero, (0, oo)) + ind, xk = S.Zero, None + for t in Add.make_args(f): + res = _compute_fps(t, x, 0, S.One, hyper, order, rational, full) + if res: + if not result: + result = True + xk = res[1] + if res[0].start > ak.start: + seq = ak + s, f = ak.start, res[0].start + else: + seq = res[0] + s, f = res[0].start, ak.start + save = Add(*[z[0]*z[1] for z in zip(seq[0:(f - s)], xk[s:f])]) + ak += res[0] + ind += res[2] + save + else: + ind += t + if result: + return ak, xk, ind + return None + + # The symbolic term - symb, if present, is being separated from the function + # Otherwise symb is being set to S.One + syms = f.free_symbols.difference({x}) + (f, symb) = expand(f).as_independent(*syms) + + result = None + + # from here on it's x0=0 and dir=1 handling + k = Dummy('k') + if rational: + result = rational_algorithm(f, x, k, order, full) + + if result is None and hyper: + result = hyper_algorithm(f, x, k, order) + + if result is None: + return None + + from sympy.simplify.powsimp import powsimp + if symb.is_zero: + symb = S.One + else: + symb = powsimp(symb) + ak = sequence(result[0], (k, result[2], oo)) + xk_formula = powsimp(x**k * symb) + xk = sequence(xk_formula, (k, 0, oo)) + ind = powsimp(result[1] * symb) + + return ak, xk, ind + + +def compute_fps(f, x, x0=0, dir=1, hyper=True, order=4, rational=True, + full=False): + """ + Computes the formula for Formal Power Series of a function. + + Explanation + =========== + + Tries to compute the formula by applying the following techniques + (in order): + + * rational_algorithm + * Hypergeometric algorithm + + Parameters + ========== + + x : Symbol + x0 : number, optional + Point to perform series expansion about. Default is 0. + dir : {1, -1, '+', '-'}, optional + If dir is 1 or '+' the series is calculated from the right and + for -1 or '-' the series is calculated from the left. For smooth + functions this flag will not alter the results. Default is 1. + hyper : {True, False}, optional + Set hyper to False to skip the hypergeometric algorithm. + By default it is set to False. + order : int, optional + Order of the derivative of ``f``, Default is 4. + rational : {True, False}, optional + Set rational to False to skip rational algorithm. By default it is set + to True. + full : {True, False}, optional + Set full to True to increase the range of rational algorithm. + See :func:`rational_algorithm` for details. By default it is set to + False. + + Returns + ======= + + ak : sequence + Sequence of coefficients. + xk : sequence + Sequence of powers of x. + ind : Expr + Independent terms. + mul : Pow + Common terms. + + See Also + ======== + + sympy.series.formal.rational_algorithm + sympy.series.formal.hyper_algorithm + """ + f = sympify(f) + x = sympify(x) + + if not f.has(x): + return None + + x0 = sympify(x0) + + if dir == '+': + dir = S.One + elif dir == '-': + dir = -S.One + elif dir not in [S.One, -S.One]: + raise ValueError("Dir must be '+' or '-'") + else: + dir = sympify(dir) + + return _compute_fps(f, x, x0, dir, hyper, order, rational, full) + + +class Coeff(Function): + """ + Coeff(p, x, n) represents the nth coefficient of the polynomial p in x + """ + @classmethod + def eval(cls, p, x, n): + if p.is_polynomial(x) and n.is_integer: + return p.coeff(x, n) + + +class FormalPowerSeries(SeriesBase): + """ + Represents Formal Power Series of a function. + + Explanation + =========== + + No computation is performed. This class should only to be used to represent + a series. No checks are performed. + + For computing a series use :func:`fps`. + + See Also + ======== + + sympy.series.formal.fps + """ + def __new__(cls, *args): + args = map(sympify, args) + return Expr.__new__(cls, *args) + + def __init__(self, *args): + ak = args[4][0] + k = ak.variables[0] + self.ak_seq = sequence(ak.formula, (k, 1, oo)) + self.fact_seq = sequence(factorial(k), (k, 1, oo)) + self.bell_coeff_seq = self.ak_seq * self.fact_seq + self.sign_seq = sequence((-1, 1), (k, 1, oo)) + + @property + def function(self): + return self.args[0] + + @property + def x(self): + return self.args[1] + + @property + def x0(self): + return self.args[2] + + @property + def dir(self): + return self.args[3] + + @property + def ak(self): + return self.args[4][0] + + @property + def xk(self): + return self.args[4][1] + + @property + def ind(self): + return self.args[4][2] + + @property + def interval(self): + return Interval(0, oo) + + @property + def start(self): + return self.interval.inf + + @property + def stop(self): + return self.interval.sup + + @property + def length(self): + return oo + + @property + def infinite(self): + """Returns an infinite representation of the series""" + from sympy.concrete import Sum + ak, xk = self.ak, self.xk + k = ak.variables[0] + inf_sum = Sum(ak.formula * xk.formula, (k, ak.start, ak.stop)) + + return self.ind + inf_sum + + def _get_pow_x(self, term): + """Returns the power of x in a term.""" + xterm, pow_x = term.as_independent(self.x)[1].as_base_exp() + if not xterm.has(self.x): + return S.Zero + return pow_x + + def polynomial(self, n=6): + """ + Truncated series as polynomial. + + Explanation + =========== + + Returns series expansion of ``f`` upto order ``O(x**n)`` + as a polynomial(without ``O`` term). + """ + terms = [] + sym = self.free_symbols + for i, t in enumerate(self): + xp = self._get_pow_x(t) + if xp.has(*sym): + xp = xp.as_coeff_add(*sym)[0] + if xp >= n: + break + elif xp.is_integer is True and i == n + 1: + break + elif t is not S.Zero: + terms.append(t) + + return Add(*terms) + + def truncate(self, n=6): + """ + Truncated series. + + Explanation + =========== + + Returns truncated series expansion of f upto + order ``O(x**n)``. + + If n is ``None``, returns an infinite iterator. + """ + if n is None: + return iter(self) + + x, x0 = self.x, self.x0 + pt_xk = self.xk.coeff(n) + if x0 is S.NegativeInfinity: + x0 = S.Infinity + + return self.polynomial(n) + Order(pt_xk, (x, x0)) + + def zero_coeff(self): + return self._eval_term(0) + + def _eval_term(self, pt): + try: + pt_xk = self.xk.coeff(pt) + pt_ak = self.ak.coeff(pt).simplify() # Simplify the coefficients + except IndexError: + term = S.Zero + else: + term = (pt_ak * pt_xk) + + if self.ind: + ind = S.Zero + sym = self.free_symbols + for t in Add.make_args(self.ind): + pow_x = self._get_pow_x(t) + if pow_x.has(*sym): + pow_x = pow_x.as_coeff_add(*sym)[0] + if pt == 0 and pow_x < 1: + ind += t + elif pow_x >= pt and pow_x < pt + 1: + ind += t + term += ind + + return term.collect(self.x) + + def _eval_subs(self, old, new): + x = self.x + if old.has(x): + return self + + def _eval_as_leading_term(self, x, logx, cdir): + for t in self: + if t is not S.Zero: + return t + + def _eval_derivative(self, x): + f = self.function.diff(x) + ind = self.ind.diff(x) + + pow_xk = self._get_pow_x(self.xk.formula) + ak = self.ak + k = ak.variables[0] + if ak.formula.has(x): + form = [] + for e, c in ak.formula.args: + temp = S.Zero + for t in Add.make_args(e): + pow_x = self._get_pow_x(t) + temp += t * (pow_xk + pow_x) + form.append((temp, c)) + form = Piecewise(*form) + ak = sequence(form.subs(k, k + 1), (k, ak.start - 1, ak.stop)) + else: + ak = sequence((ak.formula * pow_xk).subs(k, k + 1), + (k, ak.start - 1, ak.stop)) + + return self.func(f, self.x, self.x0, self.dir, (ak, self.xk, ind)) + + def integrate(self, x=None, **kwargs): + """ + Integrate Formal Power Series. + + Examples + ======== + + >>> from sympy import fps, sin, integrate + >>> from sympy.abc import x + >>> f = fps(sin(x)) + >>> f.integrate(x).truncate() + -1 + x**2/2 - x**4/24 + O(x**6) + >>> integrate(f, (x, 0, 1)) + 1 - cos(1) + """ + from sympy.integrals import integrate + + if x is None: + x = self.x + elif iterable(x): + return integrate(self.function, x) + + f = integrate(self.function, x) + ind = integrate(self.ind, x) + ind += (f - ind).limit(x, 0) # constant of integration + + pow_xk = self._get_pow_x(self.xk.formula) + ak = self.ak + k = ak.variables[0] + if ak.formula.has(x): + form = [] + for e, c in ak.formula.args: + temp = S.Zero + for t in Add.make_args(e): + pow_x = self._get_pow_x(t) + temp += t / (pow_xk + pow_x + 1) + form.append((temp, c)) + form = Piecewise(*form) + ak = sequence(form.subs(k, k - 1), (k, ak.start + 1, ak.stop)) + else: + ak = sequence((ak.formula / (pow_xk + 1)).subs(k, k - 1), + (k, ak.start + 1, ak.stop)) + + return self.func(f, self.x, self.x0, self.dir, (ak, self.xk, ind)) + + def product(self, other, x=None, n=6): + """ + Multiplies two Formal Power Series, using discrete convolution and + return the truncated terms upto specified order. + + Parameters + ========== + + n : Number, optional + Specifies the order of the term up to which the polynomial should + be truncated. + + Examples + ======== + + >>> from sympy import fps, sin, exp + >>> from sympy.abc import x + >>> f1 = fps(sin(x)) + >>> f2 = fps(exp(x)) + + >>> f1.product(f2, x).truncate(4) + x + x**2 + x**3/3 + O(x**4) + + See Also + ======== + + sympy.discrete.convolutions + sympy.series.formal.FormalPowerSeriesProduct + + """ + + if n is None: + return iter(self) + + other = sympify(other) + + if not isinstance(other, FormalPowerSeries): + raise ValueError("Both series should be an instance of FormalPowerSeries" + " class.") + + if self.dir != other.dir: + raise ValueError("Both series should be calculated from the" + " same direction.") + elif self.x0 != other.x0: + raise ValueError("Both series should be calculated about the" + " same point.") + + elif self.x != other.x: + raise ValueError("Both series should have the same symbol.") + + return FormalPowerSeriesProduct(self, other) + + def coeff_bell(self, n): + r""" + self.coeff_bell(n) returns a sequence of Bell polynomials of the second kind. + Note that ``n`` should be a integer. + + The second kind of Bell polynomials (are sometimes called "partial" Bell + polynomials or incomplete Bell polynomials) are defined as + + .. math:: + B_{n,k}(x_1, x_2,\dotsc x_{n-k+1}) = + \sum_{j_1+j_2+j_2+\dotsb=k \atop j_1+2j_2+3j_2+\dotsb=n} + \frac{n!}{j_1!j_2!\dotsb j_{n-k+1}!} + \left(\frac{x_1}{1!} \right)^{j_1} + \left(\frac{x_2}{2!} \right)^{j_2} \dotsb + \left(\frac{x_{n-k+1}}{(n-k+1)!} \right) ^{j_{n-k+1}}. + + * ``bell(n, k, (x1, x2, ...))`` gives Bell polynomials of the second kind, + `B_{n,k}(x_1, x_2, \dotsc, x_{n-k+1})`. + + See Also + ======== + + sympy.functions.combinatorial.numbers.bell + + """ + + inner_coeffs = [bell(n, j, tuple(self.bell_coeff_seq[:n-j+1])) for j in range(1, n+1)] + + k = Dummy('k') + return sequence(tuple(inner_coeffs), (k, 1, oo)) + + def compose(self, other, x=None, n=6): + r""" + Returns the truncated terms of the formal power series of the composed function, + up to specified ``n``. + + Explanation + =========== + + If ``f`` and ``g`` are two formal power series of two different functions, + then the coefficient sequence ``ak`` of the composed formal power series `fp` + will be as follows. + + .. math:: + \sum\limits_{k=0}^{n} b_k B_{n,k}(x_1, x_2, \dotsc, x_{n-k+1}) + + Parameters + ========== + + n : Number, optional + Specifies the order of the term up to which the polynomial should + be truncated. + + Examples + ======== + + >>> from sympy import fps, sin, exp + >>> from sympy.abc import x + >>> f1 = fps(exp(x)) + >>> f2 = fps(sin(x)) + + >>> f1.compose(f2, x).truncate() + 1 + x + x**2/2 - x**4/8 - x**5/15 + O(x**6) + + >>> f1.compose(f2, x).truncate(8) + 1 + x + x**2/2 - x**4/8 - x**5/15 - x**6/240 + x**7/90 + O(x**8) + + See Also + ======== + + sympy.functions.combinatorial.numbers.bell + sympy.series.formal.FormalPowerSeriesCompose + + References + ========== + + .. [1] Comtet, Louis: Advanced combinatorics; the art of finite and infinite expansions. Reidel, 1974. + + """ + + if n is None: + return iter(self) + + other = sympify(other) + + if not isinstance(other, FormalPowerSeries): + raise ValueError("Both series should be an instance of FormalPowerSeries" + " class.") + + if self.dir != other.dir: + raise ValueError("Both series should be calculated from the" + " same direction.") + elif self.x0 != other.x0: + raise ValueError("Both series should be calculated about the" + " same point.") + + elif self.x != other.x: + raise ValueError("Both series should have the same symbol.") + + if other._eval_term(0).as_coeff_mul(other.x)[0] is not S.Zero: + raise ValueError("The formal power series of the inner function should not have any " + "constant coefficient term.") + + return FormalPowerSeriesCompose(self, other) + + def inverse(self, x=None, n=6): + r""" + Returns the truncated terms of the inverse of the formal power series, + up to specified ``n``. + + Explanation + =========== + + If ``f`` and ``g`` are two formal power series of two different functions, + then the coefficient sequence ``ak`` of the composed formal power series ``fp`` + will be as follows. + + .. math:: + \sum\limits_{k=0}^{n} (-1)^{k} x_0^{-k-1} B_{n,k}(x_1, x_2, \dotsc, x_{n-k+1}) + + Parameters + ========== + + n : Number, optional + Specifies the order of the term up to which the polynomial should + be truncated. + + Examples + ======== + + >>> from sympy import fps, exp, cos + >>> from sympy.abc import x + >>> f1 = fps(exp(x)) + >>> f2 = fps(cos(x)) + + >>> f1.inverse(x).truncate() + 1 - x + x**2/2 - x**3/6 + x**4/24 - x**5/120 + O(x**6) + + >>> f2.inverse(x).truncate(8) + 1 + x**2/2 + 5*x**4/24 + 61*x**6/720 + O(x**8) + + See Also + ======== + + sympy.functions.combinatorial.numbers.bell + sympy.series.formal.FormalPowerSeriesInverse + + References + ========== + + .. [1] Comtet, Louis: Advanced combinatorics; the art of finite and infinite expansions. Reidel, 1974. + + """ + + if n is None: + return iter(self) + + if self._eval_term(0).is_zero: + raise ValueError("Constant coefficient should exist for an inverse of a formal" + " power series to exist.") + + return FormalPowerSeriesInverse(self) + + def __add__(self, other): + other = sympify(other) + + if isinstance(other, FormalPowerSeries): + if self.dir != other.dir: + raise ValueError("Both series should be calculated from the" + " same direction.") + elif self.x0 != other.x0: + raise ValueError("Both series should be calculated about the" + " same point.") + + x, y = self.x, other.x + f = self.function + other.function.subs(y, x) + + if self.x not in f.free_symbols: + return f + + ak = self.ak + other.ak + if self.ak.start > other.ak.start: + seq = other.ak + s, e = other.ak.start, self.ak.start + else: + seq = self.ak + s, e = self.ak.start, other.ak.start + save = Add(*[z[0]*z[1] for z in zip(seq[0:(e - s)], self.xk[s:e])]) + ind = self.ind + other.ind + save + + return self.func(f, x, self.x0, self.dir, (ak, self.xk, ind)) + + elif not other.has(self.x): + f = self.function + other + ind = self.ind + other + + return self.func(f, self.x, self.x0, self.dir, + (self.ak, self.xk, ind)) + + return Add(self, other) + + def __radd__(self, other): + return self.__add__(other) + + def __neg__(self): + return self.func(-self.function, self.x, self.x0, self.dir, + (-self.ak, self.xk, -self.ind)) + + def __sub__(self, other): + return self.__add__(-other) + + def __rsub__(self, other): + return (-self).__add__(other) + + def __mul__(self, other): + other = sympify(other) + + if other.has(self.x): + return Mul(self, other) + + f = self.function * other + ak = self.ak.coeff_mul(other) + ind = self.ind * other + + return self.func(f, self.x, self.x0, self.dir, (ak, self.xk, ind)) + + def __rmul__(self, other): + return self.__mul__(other) + + +class FiniteFormalPowerSeries(FormalPowerSeries): + """Base Class for Product, Compose and Inverse classes""" + + def __init__(self, *args): + pass + + @property + def ffps(self): + return self.args[0] + + @property + def gfps(self): + return self.args[1] + + @property + def f(self): + return self.ffps.function + + @property + def g(self): + return self.gfps.function + + @property + def infinite(self): + raise NotImplementedError("No infinite version for an object of" + " FiniteFormalPowerSeries class.") + + def _eval_terms(self, n): + raise NotImplementedError("(%s)._eval_terms()" % self) + + def _eval_term(self, pt): + raise NotImplementedError("By the current logic, one can get terms" + "upto a certain order, instead of getting term by term.") + + def polynomial(self, n): + return self._eval_terms(n) + + def truncate(self, n=6): + ffps = self.ffps + pt_xk = ffps.xk.coeff(n) + x, x0 = ffps.x, ffps.x0 + + return self.polynomial(n) + Order(pt_xk, (x, x0)) + + def _eval_derivative(self, x): + raise NotImplementedError + + def integrate(self, x): + raise NotImplementedError + + +class FormalPowerSeriesProduct(FiniteFormalPowerSeries): + """Represents the product of two formal power series of two functions. + + Explanation + =========== + + No computation is performed. Terms are calculated using a term by term logic, + instead of a point by point logic. + + There are two differences between a :obj:`FormalPowerSeries` object and a + :obj:`FormalPowerSeriesProduct` object. The first argument contains the two + functions involved in the product. Also, the coefficient sequence contains + both the coefficient sequence of the formal power series of the involved functions. + + See Also + ======== + + sympy.series.formal.FormalPowerSeries + sympy.series.formal.FiniteFormalPowerSeries + + """ + + def __init__(self, *args): + ffps, gfps = self.ffps, self.gfps + + k = ffps.ak.variables[0] + self.coeff1 = sequence(ffps.ak.formula, (k, 0, oo)) + + k = gfps.ak.variables[0] + self.coeff2 = sequence(gfps.ak.formula, (k, 0, oo)) + + @property + def function(self): + """Function of the product of two formal power series.""" + return self.f * self.g + + def _eval_terms(self, n): + """ + Returns the first ``n`` terms of the product formal power series. + Term by term logic is implemented here. + + Examples + ======== + + >>> from sympy import fps, sin, exp + >>> from sympy.abc import x + >>> f1 = fps(sin(x)) + >>> f2 = fps(exp(x)) + >>> fprod = f1.product(f2, x) + + >>> fprod._eval_terms(4) + x**3/3 + x**2 + x + + See Also + ======== + + sympy.series.formal.FormalPowerSeries.product + + """ + coeff1, coeff2 = self.coeff1, self.coeff2 + + aks = convolution(coeff1[:n], coeff2[:n]) + + terms = [] + for i in range(0, n): + terms.append(aks[i] * self.ffps.xk.coeff(i)) + + return Add(*terms) + + +class FormalPowerSeriesCompose(FiniteFormalPowerSeries): + """ + Represents the composed formal power series of two functions. + + Explanation + =========== + + No computation is performed. Terms are calculated using a term by term logic, + instead of a point by point logic. + + There are two differences between a :obj:`FormalPowerSeries` object and a + :obj:`FormalPowerSeriesCompose` object. The first argument contains the outer + function and the inner function involved in the omposition. Also, the + coefficient sequence contains the generic sequence which is to be multiplied + by a custom ``bell_seq`` finite sequence. The finite terms will then be added up to + get the final terms. + + See Also + ======== + + sympy.series.formal.FormalPowerSeries + sympy.series.formal.FiniteFormalPowerSeries + + """ + + @property + def function(self): + """Function for the composed formal power series.""" + f, g, x = self.f, self.g, self.ffps.x + return f.subs(x, g) + + def _eval_terms(self, n): + """ + Returns the first `n` terms of the composed formal power series. + Term by term logic is implemented here. + + Explanation + =========== + + The coefficient sequence of the :obj:`FormalPowerSeriesCompose` object is the generic sequence. + It is multiplied by ``bell_seq`` to get a sequence, whose terms are added up to get + the final terms for the polynomial. + + Examples + ======== + + >>> from sympy import fps, sin, exp + >>> from sympy.abc import x + >>> f1 = fps(exp(x)) + >>> f2 = fps(sin(x)) + >>> fcomp = f1.compose(f2, x) + + >>> fcomp._eval_terms(6) + -x**5/15 - x**4/8 + x**2/2 + x + 1 + + >>> fcomp._eval_terms(8) + x**7/90 - x**6/240 - x**5/15 - x**4/8 + x**2/2 + x + 1 + + See Also + ======== + + sympy.series.formal.FormalPowerSeries.compose + sympy.series.formal.FormalPowerSeries.coeff_bell + + """ + + ffps, gfps = self.ffps, self.gfps + terms = [ffps.zero_coeff()] + + for i in range(1, n): + bell_seq = gfps.coeff_bell(i) + seq = (ffps.bell_coeff_seq * bell_seq) + terms.append(Add(*(seq[:i])) / ffps.fact_seq[i-1] * ffps.xk.coeff(i)) + + return Add(*terms) + + +class FormalPowerSeriesInverse(FiniteFormalPowerSeries): + """ + Represents the Inverse of a formal power series. + + Explanation + =========== + + No computation is performed. Terms are calculated using a term by term logic, + instead of a point by point logic. + + There is a single difference between a :obj:`FormalPowerSeries` object and a + :obj:`FormalPowerSeriesInverse` object. The coefficient sequence contains the + generic sequence which is to be multiplied by a custom ``bell_seq`` finite sequence. + The finite terms will then be added up to get the final terms. + + See Also + ======== + + sympy.series.formal.FormalPowerSeries + sympy.series.formal.FiniteFormalPowerSeries + + """ + def __init__(self, *args): + ffps = self.ffps + k = ffps.xk.variables[0] + + inv = ffps.zero_coeff() + inv_seq = sequence(inv ** (-(k + 1)), (k, 1, oo)) + self.aux_seq = ffps.sign_seq * ffps.fact_seq * inv_seq + + @property + def function(self): + """Function for the inverse of a formal power series.""" + f = self.f + return 1 / f + + @property + def g(self): + raise ValueError("Only one function is considered while performing" + "inverse of a formal power series.") + + @property + def gfps(self): + raise ValueError("Only one function is considered while performing" + "inverse of a formal power series.") + + def _eval_terms(self, n): + """ + Returns the first ``n`` terms of the composed formal power series. + Term by term logic is implemented here. + + Explanation + =========== + + The coefficient sequence of the `FormalPowerSeriesInverse` object is the generic sequence. + It is multiplied by ``bell_seq`` to get a sequence, whose terms are added up to get + the final terms for the polynomial. + + Examples + ======== + + >>> from sympy import fps, exp, cos + >>> from sympy.abc import x + >>> f1 = fps(exp(x)) + >>> f2 = fps(cos(x)) + >>> finv1, finv2 = f1.inverse(), f2.inverse() + + >>> finv1._eval_terms(6) + -x**5/120 + x**4/24 - x**3/6 + x**2/2 - x + 1 + + >>> finv2._eval_terms(8) + 61*x**6/720 + 5*x**4/24 + x**2/2 + 1 + + See Also + ======== + + sympy.series.formal.FormalPowerSeries.inverse + sympy.series.formal.FormalPowerSeries.coeff_bell + + """ + ffps = self.ffps + terms = [ffps.zero_coeff()] + + for i in range(1, n): + bell_seq = ffps.coeff_bell(i) + seq = (self.aux_seq * bell_seq) + terms.append(Add(*(seq[:i])) / ffps.fact_seq[i-1] * ffps.xk.coeff(i)) + + return Add(*terms) + + +def fps(f, x=None, x0=0, dir=1, hyper=True, order=4, rational=True, full=False): + """ + Generates Formal Power Series of ``f``. + + Explanation + =========== + + Returns the formal series expansion of ``f`` around ``x = x0`` + with respect to ``x`` in the form of a ``FormalPowerSeries`` object. + + Formal Power Series is represented using an explicit formula + computed using different algorithms. + + See :func:`compute_fps` for the more details regarding the computation + of formula. + + Parameters + ========== + + x : Symbol, optional + If x is None and ``f`` is univariate, the univariate symbols will be + supplied, otherwise an error will be raised. + x0 : number, optional + Point to perform series expansion about. Default is 0. + dir : {1, -1, '+', '-'}, optional + If dir is 1 or '+' the series is calculated from the right and + for -1 or '-' the series is calculated from the left. For smooth + functions this flag will not alter the results. Default is 1. + hyper : {True, False}, optional + Set hyper to False to skip the hypergeometric algorithm. + By default it is set to False. + order : int, optional + Order of the derivative of ``f``, Default is 4. + rational : {True, False}, optional + Set rational to False to skip rational algorithm. By default it is set + to True. + full : {True, False}, optional + Set full to True to increase the range of rational algorithm. + See :func:`rational_algorithm` for details. By default it is set to + False. + + Examples + ======== + + >>> from sympy import fps, ln, atan, sin + >>> from sympy.abc import x, n + + Rational Functions + + >>> fps(ln(1 + x)).truncate() + x - x**2/2 + x**3/3 - x**4/4 + x**5/5 + O(x**6) + + >>> fps(atan(x), full=True).truncate() + x - x**3/3 + x**5/5 + O(x**6) + + Symbolic Functions + + >>> fps(x**n*sin(x**2), x).truncate(8) + -x**(n + 6)/6 + x**(n + 2) + O(x**(n + 8)) + + See Also + ======== + + sympy.series.formal.FormalPowerSeries + sympy.series.formal.compute_fps + """ + f = sympify(f) + + if x is None: + free = f.free_symbols + if len(free) == 1: + x = free.pop() + elif not free: + return f + else: + raise NotImplementedError("multivariate formal power series") + + result = compute_fps(f, x, x0, dir, hyper, order, rational, full) + + if result is None: + return f + + return FormalPowerSeries(f, x, x0, dir, result) diff --git a/.venv/lib/python3.12/site-packages/sympy/series/fourier.py b/.venv/lib/python3.12/site-packages/sympy/series/fourier.py new file mode 100644 index 0000000000000000000000000000000000000000..c4ad43a75d0e1d24bfc591383f68965fd8b504c7 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/fourier.py @@ -0,0 +1,811 @@ +"""Fourier Series""" + +from sympy.core.numbers import (oo, pi) +from sympy.core.symbol import Wild +from sympy.core.expr import Expr +from sympy.core.add import Add +from sympy.core.containers import Tuple +from sympy.core.singleton import S +from sympy.core.symbol import Dummy, Symbol +from sympy.core.sympify import sympify +from sympy.functions.elementary.trigonometric import sin, cos, sinc +from sympy.series.series_class import SeriesBase +from sympy.series.sequences import SeqFormula +from sympy.sets.sets import Interval +from sympy.utilities.iterables import is_sequence + + +__doctest_requires__ = {('fourier_series',): ['matplotlib']} + + +def fourier_cos_seq(func, limits, n): + """Returns the cos sequence in a Fourier series""" + from sympy.integrals import integrate + x, L = limits[0], limits[2] - limits[1] + cos_term = cos(2*n*pi*x / L) + formula = 2 * cos_term * integrate(func * cos_term, limits) / L + a0 = formula.subs(n, S.Zero) / 2 + return a0, SeqFormula(2 * cos_term * integrate(func * cos_term, limits) + / L, (n, 1, oo)) + + +def fourier_sin_seq(func, limits, n): + """Returns the sin sequence in a Fourier series""" + from sympy.integrals import integrate + x, L = limits[0], limits[2] - limits[1] + sin_term = sin(2*n*pi*x / L) + return SeqFormula(2 * sin_term * integrate(func * sin_term, limits) + / L, (n, 1, oo)) + + +def _process_limits(func, limits): + """ + Limits should be of the form (x, start, stop). + x should be a symbol. Both start and stop should be bounded. + + Explanation + =========== + + * If x is not given, x is determined from func. + * If limits is None. Limit of the form (x, -pi, pi) is returned. + + Examples + ======== + + >>> from sympy.series.fourier import _process_limits as pari + >>> from sympy.abc import x + >>> pari(x**2, (x, -2, 2)) + (x, -2, 2) + >>> pari(x**2, (-2, 2)) + (x, -2, 2) + >>> pari(x**2, None) + (x, -pi, pi) + """ + def _find_x(func): + free = func.free_symbols + if len(free) == 1: + return free.pop() + elif not free: + return Dummy('k') + else: + raise ValueError( + " specify dummy variables for %s. If the function contains" + " more than one free symbol, a dummy variable should be" + " supplied explicitly e.g. FourierSeries(m*n**2, (n, -pi, pi))" + % func) + + x, start, stop = None, None, None + if limits is None: + x, start, stop = _find_x(func), -pi, pi + if is_sequence(limits, Tuple): + if len(limits) == 3: + x, start, stop = limits + elif len(limits) == 2: + x = _find_x(func) + start, stop = limits + + if not isinstance(x, Symbol) or start is None or stop is None: + raise ValueError('Invalid limits given: %s' % str(limits)) + + unbounded = [S.NegativeInfinity, S.Infinity] + if start in unbounded or stop in unbounded: + raise ValueError("Both the start and end value should be bounded") + + return sympify((x, start, stop)) + + +def finite_check(f, x, L): + + def check_fx(exprs, x): + return x not in exprs.free_symbols + + def check_sincos(_expr, x, L): + if isinstance(_expr, (sin, cos)): + sincos_args = _expr.args[0] + + if sincos_args.match(a*(pi/L)*x + b) is not None: + return True + else: + return False + + from sympy.simplify.fu import TR2, TR1, sincos_to_sum + _expr = sincos_to_sum(TR2(TR1(f))) + add_coeff = _expr.as_coeff_add() + + a = Wild('a', properties=[lambda k: k.is_Integer, lambda k: k != S.Zero, ]) + b = Wild('b', properties=[lambda k: x not in k.free_symbols, ]) + + for s in add_coeff[1]: + mul_coeffs = s.as_coeff_mul()[1] + for t in mul_coeffs: + if not (check_fx(t, x) or check_sincos(t, x, L)): + return False, f + + return True, _expr + + +class FourierSeries(SeriesBase): + r"""Represents Fourier sine/cosine series. + + Explanation + =========== + + This class only represents a fourier series. + No computation is performed. + + For how to compute Fourier series, see the :func:`fourier_series` + docstring. + + See Also + ======== + + sympy.series.fourier.fourier_series + """ + def __new__(cls, *args): + args = map(sympify, args) + return Expr.__new__(cls, *args) + + @property + def function(self): + return self.args[0] + + @property + def x(self): + return self.args[1][0] + + @property + def period(self): + return (self.args[1][1], self.args[1][2]) + + @property + def a0(self): + return self.args[2][0] + + @property + def an(self): + return self.args[2][1] + + @property + def bn(self): + return self.args[2][2] + + @property + def interval(self): + return Interval(0, oo) + + @property + def start(self): + return self.interval.inf + + @property + def stop(self): + return self.interval.sup + + @property + def length(self): + return oo + + @property + def L(self): + return abs(self.period[1] - self.period[0]) / 2 + + def _eval_subs(self, old, new): + x = self.x + if old.has(x): + return self + + def truncate(self, n=3): + """ + Return the first n nonzero terms of the series. + + If ``n`` is None return an iterator. + + Parameters + ========== + + n : int or None + Amount of non-zero terms in approximation or None. + + Returns + ======= + + Expr or iterator : + Approximation of function expanded into Fourier series. + + Examples + ======== + + >>> from sympy import fourier_series, pi + >>> from sympy.abc import x + >>> s = fourier_series(x, (x, -pi, pi)) + >>> s.truncate(4) + 2*sin(x) - sin(2*x) + 2*sin(3*x)/3 - sin(4*x)/2 + + See Also + ======== + + sympy.series.fourier.FourierSeries.sigma_approximation + """ + if n is None: + return iter(self) + + terms = [] + for t in self: + if len(terms) == n: + break + if t is not S.Zero: + terms.append(t) + + return Add(*terms) + + def sigma_approximation(self, n=3): + r""" + Return :math:`\sigma`-approximation of Fourier series with respect + to order n. + + Explanation + =========== + + Sigma approximation adjusts a Fourier summation to eliminate the Gibbs + phenomenon which would otherwise occur at discontinuities. + A sigma-approximated summation for a Fourier series of a T-periodical + function can be written as + + .. math:: + s(\theta) = \frac{1}{2} a_0 + \sum _{k=1}^{m-1} + \operatorname{sinc} \Bigl( \frac{k}{m} \Bigr) \cdot + \left[ a_k \cos \Bigl( \frac{2\pi k}{T} \theta \Bigr) + + b_k \sin \Bigl( \frac{2\pi k}{T} \theta \Bigr) \right], + + where :math:`a_0, a_k, b_k, k=1,\ldots,{m-1}` are standard Fourier + series coefficients and + :math:`\operatorname{sinc} \Bigl( \frac{k}{m} \Bigr)` is a Lanczos + :math:`\sigma` factor (expressed in terms of normalized + :math:`\operatorname{sinc}` function). + + Parameters + ========== + + n : int + Highest order of the terms taken into account in approximation. + + Returns + ======= + + Expr : + Sigma approximation of function expanded into Fourier series. + + Examples + ======== + + >>> from sympy import fourier_series, pi + >>> from sympy.abc import x + >>> s = fourier_series(x, (x, -pi, pi)) + >>> s.sigma_approximation(4) + 2*sin(x)*sinc(pi/4) - 2*sin(2*x)/pi + 2*sin(3*x)*sinc(3*pi/4)/3 + + See Also + ======== + + sympy.series.fourier.FourierSeries.truncate + + Notes + ===== + + The behaviour of + :meth:`~sympy.series.fourier.FourierSeries.sigma_approximation` + is different from :meth:`~sympy.series.fourier.FourierSeries.truncate` + - it takes all nonzero terms of degree smaller than n, rather than + first n nonzero ones. + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Gibbs_phenomenon + .. [2] https://en.wikipedia.org/wiki/Sigma_approximation + """ + terms = [sinc(pi * i / n) * t for i, t in enumerate(self[:n]) + if t is not S.Zero] + return Add(*terms) + + def shift(self, s): + """ + Shift the function by a term independent of x. + + Explanation + =========== + + f(x) -> f(x) + s + + This is fast, if Fourier series of f(x) is already + computed. + + Examples + ======== + + >>> from sympy import fourier_series, pi + >>> from sympy.abc import x + >>> s = fourier_series(x**2, (x, -pi, pi)) + >>> s.shift(1).truncate() + -4*cos(x) + cos(2*x) + 1 + pi**2/3 + """ + s, x = sympify(s), self.x + + if x in s.free_symbols: + raise ValueError("'%s' should be independent of %s" % (s, x)) + + a0 = self.a0 + s + sfunc = self.function + s + + return self.func(sfunc, self.args[1], (a0, self.an, self.bn)) + + def shiftx(self, s): + """ + Shift x by a term independent of x. + + Explanation + =========== + + f(x) -> f(x + s) + + This is fast, if Fourier series of f(x) is already + computed. + + Examples + ======== + + >>> from sympy import fourier_series, pi + >>> from sympy.abc import x + >>> s = fourier_series(x**2, (x, -pi, pi)) + >>> s.shiftx(1).truncate() + -4*cos(x + 1) + cos(2*x + 2) + pi**2/3 + """ + s, x = sympify(s), self.x + + if x in s.free_symbols: + raise ValueError("'%s' should be independent of %s" % (s, x)) + + an = self.an.subs(x, x + s) + bn = self.bn.subs(x, x + s) + sfunc = self.function.subs(x, x + s) + + return self.func(sfunc, self.args[1], (self.a0, an, bn)) + + def scale(self, s): + """ + Scale the function by a term independent of x. + + Explanation + =========== + + f(x) -> s * f(x) + + This is fast, if Fourier series of f(x) is already + computed. + + Examples + ======== + + >>> from sympy import fourier_series, pi + >>> from sympy.abc import x + >>> s = fourier_series(x**2, (x, -pi, pi)) + >>> s.scale(2).truncate() + -8*cos(x) + 2*cos(2*x) + 2*pi**2/3 + """ + s, x = sympify(s), self.x + + if x in s.free_symbols: + raise ValueError("'%s' should be independent of %s" % (s, x)) + + an = self.an.coeff_mul(s) + bn = self.bn.coeff_mul(s) + a0 = self.a0 * s + sfunc = self.args[0] * s + + return self.func(sfunc, self.args[1], (a0, an, bn)) + + def scalex(self, s): + """ + Scale x by a term independent of x. + + Explanation + =========== + + f(x) -> f(s*x) + + This is fast, if Fourier series of f(x) is already + computed. + + Examples + ======== + + >>> from sympy import fourier_series, pi + >>> from sympy.abc import x + >>> s = fourier_series(x**2, (x, -pi, pi)) + >>> s.scalex(2).truncate() + -4*cos(2*x) + cos(4*x) + pi**2/3 + """ + s, x = sympify(s), self.x + + if x in s.free_symbols: + raise ValueError("'%s' should be independent of %s" % (s, x)) + + an = self.an.subs(x, x * s) + bn = self.bn.subs(x, x * s) + sfunc = self.function.subs(x, x * s) + + return self.func(sfunc, self.args[1], (self.a0, an, bn)) + + def _eval_as_leading_term(self, x, logx, cdir): + for t in self: + if t is not S.Zero: + return t + + def _eval_term(self, pt): + if pt == 0: + return self.a0 + return self.an.coeff(pt) + self.bn.coeff(pt) + + def __neg__(self): + return self.scale(-1) + + def __add__(self, other): + if isinstance(other, FourierSeries): + if self.period != other.period: + raise ValueError("Both the series should have same periods") + + x, y = self.x, other.x + function = self.function + other.function.subs(y, x) + + if self.x not in function.free_symbols: + return function + + an = self.an + other.an + bn = self.bn + other.bn + a0 = self.a0 + other.a0 + + return self.func(function, self.args[1], (a0, an, bn)) + + return Add(self, other) + + def __sub__(self, other): + return self.__add__(-other) + + +class FiniteFourierSeries(FourierSeries): + r"""Represents Finite Fourier sine/cosine series. + + For how to compute Fourier series, see the :func:`fourier_series` + docstring. + + Parameters + ========== + + f : Expr + Expression for finding fourier_series + + limits : ( x, start, stop) + x is the independent variable for the expression f + (start, stop) is the period of the fourier series + + exprs: (a0, an, bn) or Expr + a0 is the constant term a0 of the fourier series + an is a dictionary of coefficients of cos terms + an[k] = coefficient of cos(pi*(k/L)*x) + bn is a dictionary of coefficients of sin terms + bn[k] = coefficient of sin(pi*(k/L)*x) + + or exprs can be an expression to be converted to fourier form + + Methods + ======= + + This class is an extension of FourierSeries class. + Please refer to sympy.series.fourier.FourierSeries for + further information. + + See Also + ======== + + sympy.series.fourier.FourierSeries + sympy.series.fourier.fourier_series + """ + + def __new__(cls, f, limits, exprs): + f = sympify(f) + limits = sympify(limits) + exprs = sympify(exprs) + + if not (isinstance(exprs, Tuple) and len(exprs) == 3): # exprs is not of form (a0, an, bn) + # Converts the expression to fourier form + c, e = exprs.as_coeff_add() + from sympy.simplify.fu import TR10 + rexpr = c + Add(*[TR10(i) for i in e]) + a0, exp_ls = rexpr.expand(trig=False, power_base=False, power_exp=False, log=False).as_coeff_add() + + x = limits[0] + L = abs(limits[2] - limits[1]) / 2 + + a = Wild('a', properties=[lambda k: k.is_Integer, lambda k: k is not S.Zero, ]) + b = Wild('b', properties=[lambda k: x not in k.free_symbols, ]) + + an = {} + bn = {} + + # separates the coefficients of sin and cos terms in dictionaries an, and bn + for p in exp_ls: + t = p.match(b * cos(a * (pi / L) * x)) + q = p.match(b * sin(a * (pi / L) * x)) + if t: + an[t[a]] = t[b] + an.get(t[a], S.Zero) + elif q: + bn[q[a]] = q[b] + bn.get(q[a], S.Zero) + else: + a0 += p + + exprs = Tuple(a0, an, bn) + + return Expr.__new__(cls, f, limits, exprs) + + @property + def interval(self): + _length = 1 if self.a0 else 0 + _length += max(set(self.an.keys()).union(set(self.bn.keys()))) + 1 + return Interval(0, _length) + + @property + def length(self): + return self.stop - self.start + + def shiftx(self, s): + s, x = sympify(s), self.x + + if x in s.free_symbols: + raise ValueError("'%s' should be independent of %s" % (s, x)) + + _expr = self.truncate().subs(x, x + s) + sfunc = self.function.subs(x, x + s) + + return self.func(sfunc, self.args[1], _expr) + + def scale(self, s): + s, x = sympify(s), self.x + + if x in s.free_symbols: + raise ValueError("'%s' should be independent of %s" % (s, x)) + + _expr = self.truncate() * s + sfunc = self.function * s + + return self.func(sfunc, self.args[1], _expr) + + def scalex(self, s): + s, x = sympify(s), self.x + + if x in s.free_symbols: + raise ValueError("'%s' should be independent of %s" % (s, x)) + + _expr = self.truncate().subs(x, x * s) + sfunc = self.function.subs(x, x * s) + + return self.func(sfunc, self.args[1], _expr) + + def _eval_term(self, pt): + if pt == 0: + return self.a0 + + _term = self.an.get(pt, S.Zero) * cos(pt * (pi / self.L) * self.x) \ + + self.bn.get(pt, S.Zero) * sin(pt * (pi / self.L) * self.x) + return _term + + def __add__(self, other): + if isinstance(other, FourierSeries): + return other.__add__(fourier_series(self.function, self.args[1],\ + finite=False)) + elif isinstance(other, FiniteFourierSeries): + if self.period != other.period: + raise ValueError("Both the series should have same periods") + + x, y = self.x, other.x + function = self.function + other.function.subs(y, x) + + if self.x not in function.free_symbols: + return function + + return fourier_series(function, limits=self.args[1]) + + +def fourier_series(f, limits=None, finite=True): + r"""Computes the Fourier trigonometric series expansion. + + Explanation + =========== + + Fourier trigonometric series of $f(x)$ over the interval $(a, b)$ + is defined as: + + .. math:: + \frac{a_0}{2} + \sum_{n=1}^{\infty} + (a_n \cos(\frac{2n \pi x}{L}) + b_n \sin(\frac{2n \pi x}{L})) + + where the coefficients are: + + .. math:: + L = b - a + + .. math:: + a_0 = \frac{2}{L} \int_{a}^{b}{f(x) dx} + + .. math:: + a_n = \frac{2}{L} \int_{a}^{b}{f(x) \cos(\frac{2n \pi x}{L}) dx} + + .. math:: + b_n = \frac{2}{L} \int_{a}^{b}{f(x) \sin(\frac{2n \pi x}{L}) dx} + + The condition whether the function $f(x)$ given should be periodic + or not is more than necessary, because it is sufficient to consider + the series to be converging to $f(x)$ only in the given interval, + not throughout the whole real line. + + This also brings a lot of ease for the computation because + you do not have to make $f(x)$ artificially periodic by + wrapping it with piecewise, modulo operations, + but you can shape the function to look like the desired periodic + function only in the interval $(a, b)$, and the computed series will + automatically become the series of the periodic version of $f(x)$. + + This property is illustrated in the examples section below. + + Parameters + ========== + + limits : (sym, start, end), optional + *sym* denotes the symbol the series is computed with respect to. + + *start* and *end* denotes the start and the end of the interval + where the fourier series converges to the given function. + + Default range is specified as $-\pi$ and $\pi$. + + Returns + ======= + + FourierSeries + A symbolic object representing the Fourier trigonometric series. + + Examples + ======== + + Computing the Fourier series of $f(x) = x^2$: + + >>> from sympy import fourier_series, pi + >>> from sympy.abc import x + >>> f = x**2 + >>> s = fourier_series(f, (x, -pi, pi)) + >>> s1 = s.truncate(n=3) + >>> s1 + -4*cos(x) + cos(2*x) + pi**2/3 + + Shifting of the Fourier series: + + >>> s.shift(1).truncate() + -4*cos(x) + cos(2*x) + 1 + pi**2/3 + >>> s.shiftx(1).truncate() + -4*cos(x + 1) + cos(2*x + 2) + pi**2/3 + + Scaling of the Fourier series: + + >>> s.scale(2).truncate() + -8*cos(x) + 2*cos(2*x) + 2*pi**2/3 + >>> s.scalex(2).truncate() + -4*cos(2*x) + cos(4*x) + pi**2/3 + + Computing the Fourier series of $f(x) = x$: + + This illustrates how truncating to the higher order gives better + convergence. + + .. plot:: + :context: reset + :format: doctest + :include-source: True + + >>> from sympy import fourier_series, pi, plot + >>> from sympy.abc import x + >>> f = x + >>> s = fourier_series(f, (x, -pi, pi)) + >>> s1 = s.truncate(n = 3) + >>> s2 = s.truncate(n = 5) + >>> s3 = s.truncate(n = 7) + >>> p = plot(f, s1, s2, s3, (x, -pi, pi), show=False, legend=True) + + >>> p[0].line_color = (0, 0, 0) + >>> p[0].label = 'x' + >>> p[1].line_color = (0.7, 0.7, 0.7) + >>> p[1].label = 'n=3' + >>> p[2].line_color = (0.5, 0.5, 0.5) + >>> p[2].label = 'n=5' + >>> p[3].line_color = (0.3, 0.3, 0.3) + >>> p[3].label = 'n=7' + + >>> p.show() + + This illustrates how the series converges to different sawtooth + waves if the different ranges are specified. + + .. plot:: + :context: close-figs + :format: doctest + :include-source: True + + >>> s1 = fourier_series(x, (x, -1, 1)).truncate(10) + >>> s2 = fourier_series(x, (x, -pi, pi)).truncate(10) + >>> s3 = fourier_series(x, (x, 0, 1)).truncate(10) + >>> p = plot(x, s1, s2, s3, (x, -5, 5), show=False, legend=True) + + >>> p[0].line_color = (0, 0, 0) + >>> p[0].label = 'x' + >>> p[1].line_color = (0.7, 0.7, 0.7) + >>> p[1].label = '[-1, 1]' + >>> p[2].line_color = (0.5, 0.5, 0.5) + >>> p[2].label = '[-pi, pi]' + >>> p[3].line_color = (0.3, 0.3, 0.3) + >>> p[3].label = '[0, 1]' + + >>> p.show() + + Notes + ===== + + Computing Fourier series can be slow + due to the integration required in computing + an, bn. + + It is faster to compute Fourier series of a function + by using shifting and scaling on an already + computed Fourier series rather than computing + again. + + e.g. If the Fourier series of ``x**2`` is known + the Fourier series of ``x**2 - 1`` can be found by shifting by ``-1``. + + See Also + ======== + + sympy.series.fourier.FourierSeries + + References + ========== + + .. [1] https://mathworld.wolfram.com/FourierSeries.html + """ + f = sympify(f) + + limits = _process_limits(f, limits) + x = limits[0] + + if x not in f.free_symbols: + return f + + if finite: + L = abs(limits[2] - limits[1]) / 2 + is_finite, res_f = finite_check(f, x, L) + if is_finite: + return FiniteFourierSeries(f, limits, res_f) + + n = Dummy('n') + center = (limits[1] + limits[2]) / 2 + if center.is_zero: + neg_f = f.subs(x, -x) + if f == neg_f: + a0, an = fourier_cos_seq(f, limits, n) + bn = SeqFormula(0, (1, oo)) + return FourierSeries(f, limits, (a0, an, bn)) + elif f == -neg_f: + a0 = S.Zero + an = SeqFormula(0, (1, oo)) + bn = fourier_sin_seq(f, limits, n) + return FourierSeries(f, limits, (a0, an, bn)) + a0, an = fourier_cos_seq(f, limits, n) + bn = fourier_sin_seq(f, limits, n) + return FourierSeries(f, limits, (a0, an, bn)) diff --git a/.venv/lib/python3.12/site-packages/sympy/series/gruntz.py b/.venv/lib/python3.12/site-packages/sympy/series/gruntz.py new file mode 100644 index 0000000000000000000000000000000000000000..20ba3150e9918384141e755a39f5bacb0e76db55 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/gruntz.py @@ -0,0 +1,701 @@ +""" +Limits +====== + +Implemented according to the PhD thesis +https://www.cybertester.com/data/gruntz.pdf, which contains very thorough +descriptions of the algorithm including many examples. We summarize here +the gist of it. + +All functions are sorted according to how rapidly varying they are at +infinity using the following rules. Any two functions f and g can be +compared using the properties of L: + +L=lim log|f(x)| / log|g(x)| (for x -> oo) + +We define >, < ~ according to:: + + 1. f > g .... L=+-oo + + we say that: + - f is greater than any power of g + - f is more rapidly varying than g + - f goes to infinity/zero faster than g + + 2. f < g .... L=0 + + we say that: + - f is lower than any power of g + + 3. f ~ g .... L!=0, +-oo + + we say that: + - both f and g are bounded from above and below by suitable integral + powers of the other + +Examples +======== +:: + 2 < x < exp(x) < exp(x**2) < exp(exp(x)) + 2 ~ 3 ~ -5 + x ~ x**2 ~ x**3 ~ 1/x ~ x**m ~ -x + exp(x) ~ exp(-x) ~ exp(2x) ~ exp(x)**2 ~ exp(x+exp(-x)) + f ~ 1/f + +So we can divide all the functions into comparability classes (x and x^2 +belong to one class, exp(x) and exp(-x) belong to some other class). In +principle, we could compare any two functions, but in our algorithm, we +do not compare anything below the class 2~3~-5 (for example log(x) is +below this), so we set 2~3~-5 as the lowest comparability class. + +Given the function f, we find the list of most rapidly varying (mrv set) +subexpressions of it. This list belongs to the same comparability class. +Let's say it is {exp(x), exp(2x)}. Using the rule f ~ 1/f we find an +element "w" (either from the list or a new one) from the same +comparability class which goes to zero at infinity. In our example we +set w=exp(-x) (but we could also set w=exp(-2x) or w=exp(-3x) ...). We +rewrite the mrv set using w, in our case {1/w, 1/w^2}, and substitute it +into f. Then we expand f into a series in w:: + + f = c0*w^e0 + c1*w^e1 + ... + O(w^en), where e0oo, lim f = lim c0*w^e0, because all the other terms go to zero, +because w goes to zero faster than the ci and ei. So:: + + for e0>0, lim f = 0 + for e0<0, lim f = +-oo (the sign depends on the sign of c0) + for e0=0, lim f = lim c0 + +We need to recursively compute limits at several places of the algorithm, but +as is shown in the PhD thesis, it always finishes. + +Important functions from the implementation: + +compare(a, b, x) compares "a" and "b" by computing the limit L. +mrv(e, x) returns list of most rapidly varying (mrv) subexpressions of "e" +rewrite(e, Omega, x, wsym) rewrites "e" in terms of w +leadterm(f, x) returns the lowest power term in the series of f +mrv_leadterm(e, x) returns the lead term (c0, e0) for e +limitinf(e, x) computes lim e (for x->oo) +limit(e, z, z0) computes any limit by converting it to the case x->oo + +All the functions are really simple and straightforward except +rewrite(), which is the most difficult/complex part of the algorithm. +When the algorithm fails, the bugs are usually in the series expansion +(i.e. in SymPy) or in rewrite. + +This code is almost exact rewrite of the Maple code inside the Gruntz +thesis. + +Debugging +--------- + +Because the gruntz algorithm is highly recursive, it's difficult to +figure out what went wrong inside a debugger. Instead, turn on nice +debug prints by defining the environment variable SYMPY_DEBUG. For +example: + +[user@localhost]: SYMPY_DEBUG=True ./bin/isympy + +In [1]: limit(sin(x)/x, x, 0) +limitinf(_x*sin(1/_x), _x) = 1 ++-mrv_leadterm(_x*sin(1/_x), _x) = (1, 0) +| +-mrv(_x*sin(1/_x), _x) = set([_x]) +| | +-mrv(_x, _x) = set([_x]) +| | +-mrv(sin(1/_x), _x) = set([_x]) +| | +-mrv(1/_x, _x) = set([_x]) +| | +-mrv(_x, _x) = set([_x]) +| +-mrv_leadterm(exp(_x)*sin(exp(-_x)), _x, set([exp(_x)])) = (1, 0) +| +-rewrite(exp(_x)*sin(exp(-_x)), set([exp(_x)]), _x, _w) = (1/_w*sin(_w), -_x) +| +-sign(_x, _x) = 1 +| +-mrv_leadterm(1, _x) = (1, 0) ++-sign(0, _x) = 0 ++-limitinf(1, _x) = 1 + +And check manually which line is wrong. Then go to the source code and +debug this function to figure out the exact problem. + +""" +from functools import reduce + +from sympy.core import Basic, S, Mul, PoleError +from sympy.core.cache import cacheit +from sympy.core.function import AppliedUndef +from sympy.core.intfunc import ilcm +from sympy.core.numbers import I, oo +from sympy.core.symbol import Dummy, Wild +from sympy.core.traversal import bottom_up + +from sympy.functions import log, exp, sign as _sign +from sympy.series.order import Order +from sympy.utilities.misc import debug_decorator as debug +from sympy.utilities.timeutils import timethis + +timeit = timethis('gruntz') + + +def compare(a, b, x): + """Returns "<" if a" for a>b""" + # log(exp(...)) must always be simplified here for termination + la, lb = log(a), log(b) + if isinstance(a, Basic) and (isinstance(a, exp) or (a.is_Pow and a.base == S.Exp1)): + la = a.exp + if isinstance(b, Basic) and (isinstance(b, exp) or (b.is_Pow and b.base == S.Exp1)): + lb = b.exp + + c = limitinf(la/lb, x) + if c == 0: + return "<" + elif c.is_infinite: + return ">" + else: + return "=" + + +class SubsSet(dict): + """ + Stores (expr, dummy) pairs, and how to rewrite expr-s. + + Explanation + =========== + + The gruntz algorithm needs to rewrite certain expressions in term of a new + variable w. We cannot use subs, because it is just too smart for us. For + example:: + + > Omega=[exp(exp(_p - exp(-_p))/(1 - 1/_p)), exp(exp(_p))] + > O2=[exp(-exp(_p) + exp(-exp(-_p))*exp(_p)/(1 - 1/_p))/_w, 1/_w] + > e = exp(exp(_p - exp(-_p))/(1 - 1/_p)) - exp(exp(_p)) + > e.subs(Omega[0],O2[0]).subs(Omega[1],O2[1]) + -1/w + exp(exp(p)*exp(-exp(-p))/(1 - 1/p)) + + is really not what we want! + + So we do it the hard way and keep track of all the things we potentially + want to substitute by dummy variables. Consider the expression:: + + exp(x - exp(-x)) + exp(x) + x. + + The mrv set is {exp(x), exp(-x), exp(x - exp(-x))}. + We introduce corresponding dummy variables d1, d2, d3 and rewrite:: + + d3 + d1 + x. + + This class first of all keeps track of the mapping expr->variable, i.e. + will at this stage be a dictionary:: + + {exp(x): d1, exp(-x): d2, exp(x - exp(-x)): d3}. + + [It turns out to be more convenient this way round.] + But sometimes expressions in the mrv set have other expressions from the + mrv set as subexpressions, and we need to keep track of that as well. In + this case, d3 is really exp(x - d2), so rewrites at this stage is:: + + {d3: exp(x-d2)}. + + The function rewrite uses all this information to correctly rewrite our + expression in terms of w. In this case w can be chosen to be exp(-x), + i.e. d2. The correct rewriting then is:: + + exp(-w)/w + 1/w + x. + """ + def __init__(self): + self.rewrites = {} + + def __repr__(self): + return super().__repr__() + ', ' + self.rewrites.__repr__() + + def __getitem__(self, key): + if key not in self: + self[key] = Dummy() + return dict.__getitem__(self, key) + + def do_subs(self, e): + """Substitute the variables with expressions""" + for expr, var in self.items(): + e = e.xreplace({var: expr}) + return e + + def meets(self, s2): + """Tell whether or not self and s2 have non-empty intersection""" + return set(self.keys()).intersection(list(s2.keys())) != set() + + def union(self, s2, exps=None): + """Compute the union of self and s2, adjusting exps""" + res = self.copy() + tr = {} + for expr, var in s2.items(): + if expr in self: + if exps: + exps = exps.xreplace({var: res[expr]}) + tr[var] = res[expr] + else: + res[expr] = var + for var, rewr in s2.rewrites.items(): + res.rewrites[var] = rewr.xreplace(tr) + return res, exps + + def copy(self): + """Create a shallow copy of SubsSet""" + r = SubsSet() + r.rewrites = self.rewrites.copy() + for expr, var in self.items(): + r[expr] = var + return r + + +@debug +def mrv(e, x): + """Returns a SubsSet of most rapidly varying (mrv) subexpressions of 'e', + and e rewritten in terms of these""" + from sympy.simplify.powsimp import powsimp + e = powsimp(e, deep=True, combine='exp') + if not isinstance(e, Basic): + raise TypeError("e should be an instance of Basic") + if not e.has(x): + return SubsSet(), e + elif e == x: + s = SubsSet() + return s, s[x] + elif e.is_Mul or e.is_Add: + i, d = e.as_independent(x) # throw away x-independent terms + if d.func != e.func: + s, expr = mrv(d, x) + return s, e.func(i, expr) + a, b = d.as_two_terms() + s1, e1 = mrv(a, x) + s2, e2 = mrv(b, x) + return mrv_max1(s1, s2, e.func(i, e1, e2), x) + elif e.is_Pow and e.base != S.Exp1: + e1 = S.One + while e.is_Pow: + b1 = e.base + e1 *= e.exp + e = b1 + if b1 == 1: + return SubsSet(), b1 + if e1.has(x): + return mrv(exp(e1*log(b1)), x) + else: + s, expr = mrv(b1, x) + return s, expr**e1 + elif isinstance(e, log): + s, expr = mrv(e.args[0], x) + return s, log(expr) + elif isinstance(e, exp) or (e.is_Pow and e.base == S.Exp1): + # We know from the theory of this algorithm that exp(log(...)) may always + # be simplified here, and doing so is vital for termination. + if isinstance(e.exp, log): + return mrv(e.exp.args[0], x) + # if a product has an infinite factor the result will be + # infinite if there is no zero, otherwise NaN; here, we + # consider the result infinite if any factor is infinite + li = limitinf(e.exp, x) + if any(_.is_infinite for _ in Mul.make_args(li)): + s1 = SubsSet() + e1 = s1[e] + s2, e2 = mrv(e.exp, x) + su = s1.union(s2)[0] + su.rewrites[e1] = exp(e2) + return mrv_max3(s1, e1, s2, exp(e2), su, e1, x) + else: + s, expr = mrv(e.exp, x) + return s, exp(expr) + elif isinstance(e, AppliedUndef): + raise ValueError("MRV set computation for UndefinedFunction is not allowed") + elif e.is_Function: + l = [mrv(a, x) for a in e.args] + l2 = [s for (s, _) in l if s != SubsSet()] + if len(l2) != 1: + # e.g. something like BesselJ(x, x) + raise NotImplementedError("MRV set computation for functions in" + " several variables not implemented.") + s, ss = l2[0], SubsSet() + args = [ss.do_subs(x[1]) for x in l] + return s, e.func(*args) + elif e.is_Derivative: + raise NotImplementedError("MRV set computation for derivatives" + " not implemented yet.") + raise NotImplementedError( + "Don't know how to calculate the mrv of '%s'" % e) + + +def mrv_max3(f, expsf, g, expsg, union, expsboth, x): + """ + Computes the maximum of two sets of expressions f and g, which + are in the same comparability class, i.e. max() compares (two elements of) + f and g and returns either (f, expsf) [if f is larger], (g, expsg) + [if g is larger] or (union, expsboth) [if f, g are of the same class]. + """ + if not isinstance(f, SubsSet): + raise TypeError("f should be an instance of SubsSet") + if not isinstance(g, SubsSet): + raise TypeError("g should be an instance of SubsSet") + if f == SubsSet(): + return g, expsg + elif g == SubsSet(): + return f, expsf + elif f.meets(g): + return union, expsboth + + c = compare(list(f.keys())[0], list(g.keys())[0], x) + if c == ">": + return f, expsf + elif c == "<": + return g, expsg + else: + if c != "=": + raise ValueError("c should be =") + return union, expsboth + + +def mrv_max1(f, g, exps, x): + """Computes the maximum of two sets of expressions f and g, which + are in the same comparability class, i.e. mrv_max1() compares (two elements of) + f and g and returns the set, which is in the higher comparability class + of the union of both, if they have the same order of variation. + Also returns exps, with the appropriate substitutions made. + """ + u, b = f.union(g, exps) + return mrv_max3(f, g.do_subs(exps), g, f.do_subs(exps), + u, b, x) + + +@debug +@cacheit +@timeit +def sign(e, x): + """ + Returns a sign of an expression e(x) for x->oo. + + :: + + e > 0 for x sufficiently large ... 1 + e == 0 for x sufficiently large ... 0 + e < 0 for x sufficiently large ... -1 + + The result of this function is currently undefined if e changes sign + arbitrarily often for arbitrarily large x (e.g. sin(x)). + + Note that this returns zero only if e is *constantly* zero + for x sufficiently large. [If e is constant, of course, this is just + the same thing as the sign of e.] + """ + if not isinstance(e, Basic): + raise TypeError("e should be an instance of Basic") + + if e.is_positive: + return 1 + elif e.is_negative: + return -1 + elif e.is_zero: + return 0 + + elif not e.has(x): + from sympy.simplify import logcombine + e = logcombine(e) + return _sign(e) + elif e == x: + return 1 + elif e.is_Mul: + a, b = e.as_two_terms() + sa = sign(a, x) + if not sa: + return 0 + return sa * sign(b, x) + elif isinstance(e, exp): + return 1 + elif e.is_Pow: + if e.base == S.Exp1: + return 1 + s = sign(e.base, x) + if s == 1: + return 1 + if e.exp.is_Integer: + return s**e.exp + elif isinstance(e, log) and e.args[0].is_positive: + return sign(e.args[0] - 1, x) + + # if all else fails, do it the hard way + c0, e0 = mrv_leadterm(e, x) + return sign(c0, x) + + +@debug +@timeit +@cacheit +def limitinf(e, x): + """Limit e(x) for x-> oo.""" + # rewrite e in terms of tractable functions only + + old = e + if not e.has(x): + return e # e is a constant + from sympy.simplify.powsimp import powdenest + from sympy.calculus.util import AccumBounds + if e.has(Order): + e = e.expand().removeO() + if not x.is_positive or x.is_integer: + # We make sure that x.is_positive is True and x.is_integer is None + # so we get all the correct mathematical behavior from the expression. + # We need a fresh variable. + p = Dummy('p', positive=True) + e = e.subs(x, p) + x = p + e = e.rewrite('tractable', deep=True, limitvar=x) + e = powdenest(e) + if isinstance(e, AccumBounds): + if mrv_leadterm(e.min, x) != mrv_leadterm(e.max, x): + raise NotImplementedError + c0, e0 = mrv_leadterm(e.min, x) + else: + c0, e0 = mrv_leadterm(e, x) + sig = sign(e0, x) + if sig == 1: + return S.Zero # e0>0: lim f = 0 + elif sig == -1: # e0<0: lim f = +-oo (the sign depends on the sign of c0) + if c0.match(I*Wild("a", exclude=[I])): + return c0*oo + s = sign(c0, x) + # the leading term shouldn't be 0: + if s == 0: + raise ValueError("Leading term should not be 0") + return s*oo + elif sig == 0: + if c0 == old: + c0 = c0.cancel() + return limitinf(c0, x) # e0=0: lim f = lim c0 + else: + raise ValueError("{} could not be evaluated".format(sig)) + + +def moveup2(s, x): + r = SubsSet() + for expr, var in s.items(): + r[expr.xreplace({x: exp(x)})] = var + for var, expr in s.rewrites.items(): + r.rewrites[var] = s.rewrites[var].xreplace({x: exp(x)}) + return r + + +def moveup(l, x): + return [e.xreplace({x: exp(x)}) for e in l] + + +@debug +@timeit +@cacheit +def mrv_leadterm(e, x): + """Returns (c0, e0) for e.""" + Omega = SubsSet() + if not e.has(x): + return (e, S.Zero) + if Omega == SubsSet(): + Omega, exps = mrv(e, x) + if not Omega: + # e really does not depend on x after simplification + return exps, S.Zero + if x in Omega: + # move the whole omega up (exponentiate each term): + Omega_up = moveup2(Omega, x) + exps_up = moveup([exps], x)[0] + # NOTE: there is no need to move this down! + Omega = Omega_up + exps = exps_up + # + # The positive dummy, w, is used here so log(w*2) etc. will expand; + # a unique dummy is needed in this algorithm + # + # For limits of complex functions, the algorithm would have to be + # improved, or just find limits of Re and Im components separately. + # + w = Dummy("w", positive=True) + f, logw = rewrite(exps, Omega, x, w) + + # Ensure expressions of the form exp(log(...)) don't get simplified automatically in the previous steps. + # see: https://github.com/sympy/sympy/issues/15323#issuecomment-478639399 + f = f.replace(lambda f: f.is_Pow and f.has(x), lambda f: exp(log(f.base)*f.exp)) + + try: + lt = f.leadterm(w, logx=logw) + except (NotImplementedError, PoleError, ValueError): + n0 = 1 + _series = Order(1) + incr = S.One + while _series.is_Order: + _series = f._eval_nseries(w, n=n0+incr, logx=logw) + incr *= 2 + series = _series.expand().removeO() + try: + lt = series.leadterm(w, logx=logw) + except (NotImplementedError, PoleError, ValueError): + lt = f.as_coeff_exponent(w) + if lt[0].has(w): + base = f.as_base_exp()[0].as_coeff_exponent(w) + ex = f.as_base_exp()[1] + lt = (base[0]**ex, base[1]*ex) + return (lt[0].subs(log(w), logw), lt[1]) + + +def build_expression_tree(Omega, rewrites): + r""" Helper function for rewrite. + + We need to sort Omega (mrv set) so that we replace an expression before + we replace any expression in terms of which it has to be rewritten:: + + e1 ---> e2 ---> e3 + \ + -> e4 + + Here we can do e1, e2, e3, e4 or e1, e2, e4, e3. + To do this we assemble the nodes into a tree, and sort them by height. + + This function builds the tree, rewrites then sorts the nodes. + """ + class Node: + def __init__(self): + self.before = [] + self.expr = None + self.var = None + def ht(self): + return reduce(lambda x, y: x + y, + [x.ht() for x in self.before], 1) + nodes = {} + for expr, v in Omega: + n = Node() + n.var = v + n.expr = expr + nodes[v] = n + for _, v in Omega: + if v in rewrites: + n = nodes[v] + r = rewrites[v] + for _, v2 in Omega: + if r.has(v2): + n.before.append(nodes[v2]) + + return nodes + + +@debug +@timeit +def rewrite(e, Omega, x, wsym): + """e(x) ... the function + Omega ... the mrv set + wsym ... the symbol which is going to be used for w + + Returns the rewritten e in terms of w and log(w). See test_rewrite1() + for examples and correct results. + """ + + from sympy import AccumBounds + if not isinstance(Omega, SubsSet): + raise TypeError("Omega should be an instance of SubsSet") + if len(Omega) == 0: + raise ValueError("Length cannot be 0") + # all items in Omega must be exponentials + for t in Omega.keys(): + if not isinstance(t, exp): + raise ValueError("Value should be exp") + rewrites = Omega.rewrites + Omega = list(Omega.items()) + + nodes = build_expression_tree(Omega, rewrites) + Omega.sort(key=lambda x: nodes[x[1]].ht(), reverse=True) + + # make sure we know the sign of each exp() term; after the loop, + # g is going to be the "w" - the simplest one in the mrv set + for g, _ in Omega: + sig = sign(g.exp, x) + if sig != 1 and sig != -1 and not sig.has(AccumBounds): + raise NotImplementedError('Result depends on the sign of %s' % sig) + if sig == 1: + wsym = 1/wsym # if g goes to oo, substitute 1/w + # O2 is a list, which results by rewriting each item in Omega using "w" + O2 = [] + denominators = [] + for f, var in Omega: + c = limitinf(f.exp/g.exp, x) + if c.is_Rational: + denominators.append(c.q) + arg = f.exp + if var in rewrites: + if not isinstance(rewrites[var], exp): + raise ValueError("Value should be exp") + arg = rewrites[var].args[0] + O2.append((var, exp((arg - c*g.exp))*wsym**c)) + + # Remember that Omega contains subexpressions of "e". So now we find + # them in "e" and substitute them for our rewriting, stored in O2 + + # the following powsimp is necessary to automatically combine exponentials, + # so that the .xreplace() below succeeds: + # TODO this should not be necessary + from sympy.simplify.powsimp import powsimp + f = powsimp(e, deep=True, combine='exp') + for a, b in O2: + f = f.xreplace({a: b}) + + for _, var in Omega: + assert not f.has(var) + + # finally compute the logarithm of w (logw). + logw = g.exp + if sig == 1: + logw = -logw # log(w)->log(1/w)=-log(w) + + # Some parts of SymPy have difficulty computing series expansions with + # non-integral exponents. The following heuristic improves the situation: + exponent = reduce(ilcm, denominators, 1) + f = f.subs({wsym: wsym**exponent}) + logw /= exponent + + # bottom_up function is required for a specific case - when f is + # -exp(p/(p + 1)) + exp(-p**2/(p + 1) + p). No current simplification + # methods reduce this to 0 while not expanding polynomials. + f = bottom_up(f, lambda w: getattr(w, 'normal', lambda: w)()) + + return f, logw + + +def gruntz(e, z, z0, dir="+"): + """ + Compute the limit of e(z) at the point z0 using the Gruntz algorithm. + + Explanation + =========== + + ``z0`` can be any expression, including oo and -oo. + + For ``dir="+"`` (default) it calculates the limit from the right + (z->z0+) and for ``dir="-"`` the limit from the left (z->z0-). For infinite z0 + (oo or -oo), the dir argument does not matter. + + This algorithm is fully described in the module docstring in the gruntz.py + file. It relies heavily on the series expansion. Most frequently, gruntz() + is only used if the faster limit() function (which uses heuristics) fails. + """ + if not z.is_symbol: + raise NotImplementedError("Second argument must be a Symbol") + + # convert all limits to the limit z->oo; sign of z is handled in limitinf + r = None + if z0 in (oo, I*oo): + e0 = e + elif z0 in (-oo, -I*oo): + e0 = e.subs(z, -z) + else: + if str(dir) == "-": + e0 = e.subs(z, z0 - 1/z) + elif str(dir) == "+": + e0 = e.subs(z, z0 + 1/z) + else: + raise NotImplementedError("dir must be '+' or '-'") + + r = limitinf(e0, z) + + # This is a bit of a heuristic for nice results... we always rewrite + # tractable functions in terms of familiar intractable ones. + # It might be nicer to rewrite the exactly to what they were initially, + # but that would take some work to implement. + return r.rewrite('intractable', deep=True) diff --git a/.venv/lib/python3.12/site-packages/sympy/series/kauers.py b/.venv/lib/python3.12/site-packages/sympy/series/kauers.py new file mode 100644 index 0000000000000000000000000000000000000000..9e9645ff15ee5ae3c1d1c8709f76aed1b366f50a --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/kauers.py @@ -0,0 +1,51 @@ +def finite_diff(expression, variable, increment=1): + """ + Takes as input a polynomial expression and the variable used to construct + it and returns the difference between function's value when the input is + incremented to 1 and the original function value. If you want an increment + other than one supply it as a third argument. + + Examples + ======== + + >>> from sympy.abc import x, y, z + >>> from sympy.series.kauers import finite_diff + >>> finite_diff(x**2, x) + 2*x + 1 + >>> finite_diff(y**3 + 2*y**2 + 3*y + 4, y) + 3*y**2 + 7*y + 6 + >>> finite_diff(x**2 + 3*x + 8, x, 2) + 4*x + 10 + >>> finite_diff(z**3 + 8*z, z, 3) + 9*z**2 + 27*z + 51 + """ + expression = expression.expand() + expression2 = expression.subs(variable, variable + increment) + expression2 = expression2.expand() + return expression2 - expression + +def finite_diff_kauers(sum): + """ + Takes as input a Sum instance and returns the difference between the sum + with the upper index incremented by 1 and the original sum. For example, + if S(n) is a sum, then finite_diff_kauers will return S(n + 1) - S(n). + + Examples + ======== + + >>> from sympy.series.kauers import finite_diff_kauers + >>> from sympy import Sum + >>> from sympy.abc import x, y, m, n, k + >>> finite_diff_kauers(Sum(k, (k, 1, n))) + n + 1 + >>> finite_diff_kauers(Sum(1/k, (k, 1, n))) + 1/(n + 1) + >>> finite_diff_kauers(Sum((x*y**2), (x, 1, n), (y, 1, m))) + (m + 1)**2*(n + 1) + >>> finite_diff_kauers(Sum((x*y), (x, 1, m), (y, 1, n))) + (m + 1)*(n + 1) + """ + function = sum.function + for l in sum.limits: + function = function.subs(l[0], l[- 1] + 1) + return function diff --git a/.venv/lib/python3.12/site-packages/sympy/series/limits.py b/.venv/lib/python3.12/site-packages/sympy/series/limits.py new file mode 100644 index 0000000000000000000000000000000000000000..e15f7a1243452075a76553903cabf60e43942d8c --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/limits.py @@ -0,0 +1,394 @@ +from sympy.calculus.accumulationbounds import AccumBounds +from sympy.core import S, Symbol, Add, sympify, Expr, PoleError, Mul +from sympy.core.exprtools import factor_terms +from sympy.core.numbers import Float, _illegal +from sympy.core.function import AppliedUndef +from sympy.core.symbol import Dummy +from sympy.functions.combinatorial.factorials import factorial +from sympy.functions.elementary.complexes import (Abs, sign, arg, re) +from sympy.functions.elementary.exponential import (exp, log) +from sympy.functions.special.gamma_functions import gamma +from sympy.polys import PolynomialError, factor +from sympy.series.order import Order +from .gruntz import gruntz + +def limit(e, z, z0, dir="+"): + """Computes the limit of ``e(z)`` at the point ``z0``. + + Parameters + ========== + + e : expression, the limit of which is to be taken + + z : symbol representing the variable in the limit. + Other symbols are treated as constants. Multivariate limits + are not supported. + + z0 : the value toward which ``z`` tends. Can be any expression, + including ``oo`` and ``-oo``. + + dir : string, optional (default: "+") + The limit is bi-directional if ``dir="+-"``, from the right + (z->z0+) if ``dir="+"``, and from the left (z->z0-) if + ``dir="-"``. For infinite ``z0`` (``oo`` or ``-oo``), the ``dir`` + argument is determined from the direction of the infinity + (i.e., ``dir="-"`` for ``oo``). + + Examples + ======== + + >>> from sympy import limit, sin, oo + >>> from sympy.abc import x + >>> limit(sin(x)/x, x, 0) + 1 + >>> limit(1/x, x, 0) # default dir='+' + oo + >>> limit(1/x, x, 0, dir="-") + -oo + >>> limit(1/x, x, 0, dir='+-') + zoo + >>> limit(1/x, x, oo) + 0 + + Notes + ===== + + First we try some heuristics for easy and frequent cases like "x", "1/x", + "x**2" and similar, so that it's fast. For all other cases, we use the + Gruntz algorithm (see the gruntz() function). + + See Also + ======== + + limit_seq : returns the limit of a sequence. + """ + + return Limit(e, z, z0, dir).doit(deep=False) + + +def heuristics(e, z, z0, dir): + """Computes the limit of an expression term-wise. + Parameters are the same as for the ``limit`` function. + Works with the arguments of expression ``e`` one by one, computing + the limit of each and then combining the results. This approach + works only for simple limits, but it is fast. + """ + + rv = None + if z0 is S.Infinity: + rv = limit(e.subs(z, 1/z), z, S.Zero, "+") + if isinstance(rv, Limit): + return + elif (e.is_Mul or e.is_Add or e.is_Pow or (e.is_Function and not isinstance(e, AppliedUndef))): + r = [] + from sympy.simplify.simplify import together + for a in e.args: + l = limit(a, z, z0, dir) + if l.has(S.Infinity) and l.is_finite is None: + if isinstance(e, Add): + m = factor_terms(e) + if not isinstance(m, Mul): # try together + m = together(m) + if not isinstance(m, Mul): # try factor if the previous methods failed + m = factor(e) + if isinstance(m, Mul): + return heuristics(m, z, z0, dir) + return + return + elif isinstance(l, Limit): + return + elif l is S.NaN: + return + else: + r.append(l) + if r: + rv = e.func(*r) + if rv is S.NaN and e.is_Mul and any(isinstance(rr, AccumBounds) for rr in r): + r2 = [] + e2 = [] + for ii, rval in enumerate(r): + if isinstance(rval, AccumBounds): + r2.append(rval) + else: + e2.append(e.args[ii]) + + if len(e2) > 0: + e3 = Mul(*e2).simplify() + l = limit(e3, z, z0, dir) + rv = l * Mul(*r2) + + if rv is S.NaN: + try: + from sympy.simplify.ratsimp import ratsimp + rat_e = ratsimp(e) + except PolynomialError: + return + if rat_e is S.NaN or rat_e == e: + return + return limit(rat_e, z, z0, dir) + return rv + + +class Limit(Expr): + """Represents an unevaluated limit. + + Examples + ======== + + >>> from sympy import Limit, sin + >>> from sympy.abc import x + >>> Limit(sin(x)/x, x, 0) + Limit(sin(x)/x, x, 0, dir='+') + >>> Limit(1/x, x, 0, dir="-") + Limit(1/x, x, 0, dir='-') + + """ + + def __new__(cls, e, z, z0, dir="+"): + e = sympify(e) + z = sympify(z) + z0 = sympify(z0) + + if z0 in (S.Infinity, S.ImaginaryUnit*S.Infinity): + dir = "-" + elif z0 in (S.NegativeInfinity, S.ImaginaryUnit*S.NegativeInfinity): + dir = "+" + + if(z0.has(z)): + raise NotImplementedError("Limits approaching a variable point are" + " not supported (%s -> %s)" % (z, z0)) + if isinstance(dir, str): + dir = Symbol(dir) + elif not isinstance(dir, Symbol): + raise TypeError("direction must be of type basestring or " + "Symbol, not %s" % type(dir)) + if str(dir) not in ('+', '-', '+-'): + raise ValueError("direction must be one of '+', '-' " + "or '+-', not %s" % dir) + + obj = Expr.__new__(cls) + obj._args = (e, z, z0, dir) + return obj + + + @property + def free_symbols(self): + e = self.args[0] + isyms = e.free_symbols + isyms.difference_update(self.args[1].free_symbols) + isyms.update(self.args[2].free_symbols) + return isyms + + + def pow_heuristics(self, e): + _, z, z0, _ = self.args + b1, e1 = e.base, e.exp + if not b1.has(z): + res = limit(e1*log(b1), z, z0) + return exp(res) + + ex_lim = limit(e1, z, z0) + base_lim = limit(b1, z, z0) + + if base_lim is S.One: + if ex_lim in (S.Infinity, S.NegativeInfinity): + res = limit(e1*(b1 - 1), z, z0) + return exp(res) + if base_lim is S.NegativeInfinity and ex_lim is S.Infinity: + return S.ComplexInfinity + + + def doit(self, **hints): + """Evaluates the limit. + + Parameters + ========== + + deep : bool, optional (default: True) + Invoke the ``doit`` method of the expressions involved before + taking the limit. + + hints : optional keyword arguments + To be passed to ``doit`` methods; only used if deep is True. + """ + + e, z, z0, dir = self.args + + if str(dir) == '+-': + r = limit(e, z, z0, dir='+') + l = limit(e, z, z0, dir='-') + if isinstance(r, Limit) and isinstance(l, Limit): + if r.args[0] == l.args[0]: + return self + if r == l: + return l + if r.is_infinite and l.is_infinite: + return S.ComplexInfinity + raise ValueError("The limit does not exist since " + "left hand limit = %s and right hand limit = %s" + % (l, r)) + + if z0 is S.ComplexInfinity: + raise NotImplementedError("Limits at complex " + "infinity are not implemented") + + if z0.is_infinite: + cdir = sign(z0) + cdir = cdir/abs(cdir) + e = e.subs(z, cdir*z) + dir = "-" + z0 = S.Infinity + + if hints.get('deep', True): + e = e.doit(**hints) + z = z.doit(**hints) + z0 = z0.doit(**hints) + + if e == z: + return z0 + + if not e.has(z): + return e + + if z0 is S.NaN: + return S.NaN + + if e.has(*_illegal): + return self + + if e.is_Order: + return Order(limit(e.expr, z, z0), *e.args[1:]) + + cdir = S.Zero + if str(dir) == "+": + cdir = S.One + elif str(dir) == "-": + cdir = S.NegativeOne + + def set_signs(expr): + if not expr.args: + return expr + newargs = tuple(set_signs(arg) for arg in expr.args) + if newargs != expr.args: + expr = expr.func(*newargs) + abs_flag = isinstance(expr, Abs) + arg_flag = isinstance(expr, arg) + sign_flag = isinstance(expr, sign) + if abs_flag or sign_flag or arg_flag: + try: + sig = limit(expr.args[0], z, z0, dir) + if sig.is_zero: + sig = limit(1/expr.args[0], z, z0, dir) + except NotImplementedError: + return expr + else: + if sig.is_extended_real: + if (sig < 0) == True: + return (-expr.args[0] if abs_flag else + S.NegativeOne if sign_flag else S.Pi) + elif (sig > 0) == True: + return (expr.args[0] if abs_flag else + S.One if sign_flag else S.Zero) + return expr + + if e.has(Float): + # Convert floats like 0.5 to exact SymPy numbers like S.Half, to + # prevent rounding errors which can lead to unexpected execution + # of conditional blocks that work on comparisons + # Also see comments in https://github.com/sympy/sympy/issues/19453 + from sympy.simplify.simplify import nsimplify + e = nsimplify(e) + e = set_signs(e) + + + if e.is_meromorphic(z, z0): + if z0 is S.Infinity: + newe = e.subs(z, 1/z) + # cdir changes sign as oo- should become 0+ + cdir = -cdir + else: + newe = e.subs(z, z + z0) + try: + coeff, ex = newe.leadterm(z, cdir=cdir) + except ValueError: + pass + else: + if ex > 0: + return S.Zero + elif ex == 0: + return coeff + if cdir == 1 or not(int(ex) & 1): + return S.Infinity*sign(coeff) + elif cdir == -1: + return S.NegativeInfinity*sign(coeff) + else: + return S.ComplexInfinity + + if z0 is S.Infinity: + if e.is_Mul: + e = factor_terms(e) + dummy = Dummy('z', positive=z.is_positive, negative=z.is_negative, real=z.is_real) + newe = e.subs(z, 1/dummy) + # cdir changes sign as oo- should become 0+ + cdir = -cdir + newz = dummy + else: + newe = e.subs(z, z + z0) + newz = z + try: + coeff, ex = newe.leadterm(newz, cdir=cdir) + except (ValueError, NotImplementedError, PoleError): + # The NotImplementedError catching is for custom functions + from sympy.simplify.powsimp import powsimp + e = powsimp(e) + if e.is_Pow: + r = self.pow_heuristics(e) + if r is not None: + return r + try: + coeff = newe.as_leading_term(newz, cdir=cdir) + if coeff != newe and (coeff.has(exp) or coeff.has(S.Exp1)): + return gruntz(coeff, newz, 0, "-" if re(cdir).is_negative else "+") + except (ValueError, NotImplementedError, PoleError): + pass + else: + if isinstance(coeff, AccumBounds) and ex == S.Zero: + return coeff + if coeff.has(S.Infinity, S.NegativeInfinity, S.ComplexInfinity, S.NaN): + return self + if not coeff.has(newz): + if ex.is_positive: + return S.Zero + elif ex == 0: + return coeff + elif ex.is_negative: + if cdir == 1: + return S.Infinity*sign(coeff) + elif cdir == -1: + return S.NegativeInfinity*sign(coeff)*S.NegativeOne**(S.One + ex) + else: + return S.ComplexInfinity + else: + raise NotImplementedError("Not sure of sign of %s" % ex) + + # gruntz fails on factorials but works with the gamma function + # If no factorial term is present, e should remain unchanged. + # factorial is defined to be zero for negative inputs (which + # differs from gamma) so only rewrite for non-negative z0. + if z0.is_extended_nonnegative: + e = e.rewrite(factorial, gamma) + + l = None + + try: + r = gruntz(e, z, z0, dir) + if r is S.NaN or l is S.NaN: + raise PoleError() + except (PoleError, ValueError): + if l is not None: + raise + r = heuristics(e, z, z0, dir) + if r is None: + return self + + return r diff --git a/.venv/lib/python3.12/site-packages/sympy/series/limitseq.py b/.venv/lib/python3.12/site-packages/sympy/series/limitseq.py new file mode 100644 index 0000000000000000000000000000000000000000..ceac4e7b63bfc09d9dfc26c12c7d2acc8b8d44da --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/limitseq.py @@ -0,0 +1,257 @@ +"""Limits of sequences""" + +from sympy.calculus.accumulationbounds import AccumulationBounds +from sympy.core.add import Add +from sympy.core.function import PoleError +from sympy.core.power import Pow +from sympy.core.singleton import S +from sympy.core.symbol import Dummy +from sympy.core.sympify import sympify +from sympy.functions.combinatorial.numbers import fibonacci +from sympy.functions.combinatorial.factorials import factorial, subfactorial +from sympy.functions.special.gamma_functions import gamma +from sympy.functions.elementary.complexes import Abs +from sympy.functions.elementary.miscellaneous import Max, Min +from sympy.functions.elementary.trigonometric import cos, sin +from sympy.series.limits import Limit + + +def difference_delta(expr, n=None, step=1): + """Difference Operator. + + Explanation + =========== + + Discrete analog of differential operator. Given a sequence x[n], + returns the sequence x[n + step] - x[n]. + + Examples + ======== + + >>> from sympy import difference_delta as dd + >>> from sympy.abc import n + >>> dd(n*(n + 1), n) + 2*n + 2 + >>> dd(n*(n + 1), n, 2) + 4*n + 6 + + References + ========== + + .. [1] https://reference.wolfram.com/language/ref/DifferenceDelta.html + """ + expr = sympify(expr) + + if n is None: + f = expr.free_symbols + if len(f) == 1: + n = f.pop() + elif len(f) == 0: + return S.Zero + else: + raise ValueError("Since there is more than one variable in the" + " expression, a variable must be supplied to" + " take the difference of %s" % expr) + step = sympify(step) + if step.is_number is False or step.is_finite is False: + raise ValueError("Step should be a finite number.") + + if hasattr(expr, '_eval_difference_delta'): + result = expr._eval_difference_delta(n, step) + if result: + return result + + return expr.subs(n, n + step) - expr + + +def dominant(expr, n): + """Finds the dominant term in a sum, that is a term that dominates + every other term. + + Explanation + =========== + + If limit(a/b, n, oo) is oo then a dominates b. + If limit(a/b, n, oo) is 0 then b dominates a. + Otherwise, a and b are comparable. + + If there is no unique dominant term, then returns ``None``. + + Examples + ======== + + >>> from sympy import Sum + >>> from sympy.series.limitseq import dominant + >>> from sympy.abc import n, k + >>> dominant(5*n**3 + 4*n**2 + n + 1, n) + 5*n**3 + >>> dominant(2**n + Sum(k, (k, 0, n)), n) + 2**n + + See Also + ======== + + sympy.series.limitseq.dominant + """ + terms = Add.make_args(expr.expand(func=True)) + term0 = terms[-1] + comp = [term0] # comparable terms + for t in terms[:-1]: + r = term0/t + e = r.gammasimp() + if e == r: + e = r.factor() + l = limit_seq(e, n) + if l is None: + return None + elif l.is_zero: + term0 = t + comp = [term0] + elif l not in [S.Infinity, S.NegativeInfinity]: + comp.append(t) + if len(comp) > 1: + return None + return term0 + + +def _limit_inf(expr, n): + try: + return Limit(expr, n, S.Infinity).doit(deep=False) + except (NotImplementedError, PoleError): + return None + + +def _limit_seq(expr, n, trials): + from sympy.concrete.summations import Sum + + for i in range(trials): + if not expr.has(Sum): + result = _limit_inf(expr, n) + if result is not None: + return result + + num, den = expr.as_numer_denom() + if not den.has(n) or not num.has(n): + result = _limit_inf(expr.doit(), n) + if result is not None: + return result + return None + + num, den = (difference_delta(t.expand(), n) for t in [num, den]) + expr = (num / den).gammasimp() + + if not expr.has(Sum): + result = _limit_inf(expr, n) + if result is not None: + return result + + num, den = expr.as_numer_denom() + + num = dominant(num, n) + if num is None: + return None + + den = dominant(den, n) + if den is None: + return None + + expr = (num / den).gammasimp() + + +def limit_seq(expr, n=None, trials=5): + """Finds the limit of a sequence as index ``n`` tends to infinity. + + Parameters + ========== + + expr : Expr + SymPy expression for the ``n-th`` term of the sequence + n : Symbol, optional + The index of the sequence, an integer that tends to positive + infinity. If None, inferred from the expression unless it has + multiple symbols. + trials: int, optional + The algorithm is highly recursive. ``trials`` is a safeguard from + infinite recursion in case the limit is not easily computed by the + algorithm. Try increasing ``trials`` if the algorithm returns ``None``. + + Admissible Terms + ================ + + The algorithm is designed for sequences built from rational functions, + indefinite sums, and indefinite products over an indeterminate n. Terms of + alternating sign are also allowed, but more complex oscillatory behavior is + not supported. + + Examples + ======== + + >>> from sympy import limit_seq, Sum, binomial + >>> from sympy.abc import n, k, m + >>> limit_seq((5*n**3 + 3*n**2 + 4) / (3*n**3 + 4*n - 5), n) + 5/3 + >>> limit_seq(binomial(2*n, n) / Sum(binomial(2*k, k), (k, 1, n)), n) + 3/4 + >>> limit_seq(Sum(k**2 * Sum(2**m/m, (m, 1, k)), (k, 1, n)) / (2**n*n), n) + 4 + + See Also + ======== + + sympy.series.limitseq.dominant + + References + ========== + + .. [1] Computing Limits of Sequences - Manuel Kauers + """ + + from sympy.concrete.summations import Sum + if n is None: + free = expr.free_symbols + if len(free) == 1: + n = free.pop() + elif not free: + return expr + else: + raise ValueError("Expression has more than one variable. " + "Please specify a variable.") + elif n not in expr.free_symbols: + return expr + + expr = expr.rewrite(fibonacci, S.GoldenRatio) + expr = expr.rewrite(factorial, subfactorial, gamma) + n_ = Dummy("n", integer=True, positive=True) + n1 = Dummy("n", odd=True, positive=True) + n2 = Dummy("n", even=True, positive=True) + + # If there is a negative term raised to a power involving n, or a + # trigonometric function, then consider even and odd n separately. + powers = (p.as_base_exp() for p in expr.atoms(Pow)) + if (any(b.is_negative and e.has(n) for b, e in powers) or + expr.has(cos, sin)): + L1 = _limit_seq(expr.xreplace({n: n1}), n1, trials) + if L1 is not None: + L2 = _limit_seq(expr.xreplace({n: n2}), n2, trials) + if L1 != L2: + if L1.is_comparable and L2.is_comparable: + return AccumulationBounds(Min(L1, L2), Max(L1, L2)) + else: + return None + else: + L1 = _limit_seq(expr.xreplace({n: n_}), n_, trials) + if L1 is not None: + return L1 + else: + if expr.is_Add: + limits = [limit_seq(term, n, trials) for term in expr.args] + if any(result is None for result in limits): + return None + else: + return Add(*limits) + # Maybe the absolute value is easier to deal with (though not if + # it has a Sum). If it tends to 0, the limit is 0. + elif not expr.has(Sum): + lim = _limit_seq(Abs(expr.xreplace({n: n_})), n_, trials) + if lim is not None and lim.is_zero: + return S.Zero diff --git a/.venv/lib/python3.12/site-packages/sympy/series/order.py b/.venv/lib/python3.12/site-packages/sympy/series/order.py new file mode 100644 index 0000000000000000000000000000000000000000..9cfd4309c2b7094ce02feab129e5f051c442d8cd --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/order.py @@ -0,0 +1,522 @@ +from sympy.core import S, sympify, Expr, Dummy, Add, Mul +from sympy.core.cache import cacheit +from sympy.core.containers import Tuple +from sympy.core.function import Function, PoleError, expand_power_base, expand_log +from sympy.core.sorting import default_sort_key +from sympy.functions.elementary.exponential import exp, log +from sympy.sets.sets import Complement +from sympy.utilities.iterables import uniq, is_sequence + + +class Order(Expr): + r""" Represents the limiting behavior of some function. + + Explanation + =========== + + The order of a function characterizes the function based on the limiting + behavior of the function as it goes to some limit. Only taking the limit + point to be a number is currently supported. This is expressed in + big O notation [1]_. + + The formal definition for the order of a function `g(x)` about a point `a` + is such that `g(x) = O(f(x))` as `x \rightarrow a` if and only if there + exists a `\delta > 0` and an `M > 0` such that `|g(x)| \leq M|f(x)|` for + `|x-a| < \delta`. This is equivalent to `\limsup_{x \rightarrow a} + |g(x)/f(x)| < \infty`. + + Let's illustrate it on the following example by taking the expansion of + `\sin(x)` about 0: + + .. math :: + \sin(x) = x - x^3/3! + O(x^5) + + where in this case `O(x^5) = x^5/5! - x^7/7! + \cdots`. By the definition + of `O`, there is a `\delta > 0` and an `M` such that: + + .. math :: + |x^5/5! - x^7/7! + ....| <= M|x^5| \text{ for } |x| < \delta + + or by the alternate definition: + + .. math :: + \lim_{x \rightarrow 0} | (x^5/5! - x^7/7! + ....) / x^5| < \infty + + which surely is true, because + + .. math :: + \lim_{x \rightarrow 0} | (x^5/5! - x^7/7! + ....) / x^5| = 1/5! + + + As it is usually used, the order of a function can be intuitively thought + of representing all terms of powers greater than the one specified. For + example, `O(x^3)` corresponds to any terms proportional to `x^3, + x^4,\ldots` and any higher power. For a polynomial, this leaves terms + proportional to `x^2`, `x` and constants. + + Examples + ======== + + >>> from sympy import O, oo, cos, pi + >>> from sympy.abc import x, y + + >>> O(x + x**2) + O(x) + >>> O(x + x**2, (x, 0)) + O(x) + >>> O(x + x**2, (x, oo)) + O(x**2, (x, oo)) + + >>> O(1 + x*y) + O(1, x, y) + >>> O(1 + x*y, (x, 0), (y, 0)) + O(1, x, y) + >>> O(1 + x*y, (x, oo), (y, oo)) + O(x*y, (x, oo), (y, oo)) + + >>> O(1) in O(1, x) + True + >>> O(1, x) in O(1) + False + >>> O(x) in O(1, x) + True + >>> O(x**2) in O(x) + True + + >>> O(x)*x + O(x**2) + >>> O(x) - O(x) + O(x) + >>> O(cos(x)) + O(1) + >>> O(cos(x), (x, pi/2)) + O(x - pi/2, (x, pi/2)) + + References + ========== + + .. [1] `Big O notation `_ + + Notes + ===== + + In ``O(f(x), x)`` the expression ``f(x)`` is assumed to have a leading + term. ``O(f(x), x)`` is automatically transformed to + ``O(f(x).as_leading_term(x),x)``. + + ``O(expr*f(x), x)`` is ``O(f(x), x)`` + + ``O(expr, x)`` is ``O(1)`` + + ``O(0, x)`` is 0. + + Multivariate O is also supported: + + ``O(f(x, y), x, y)`` is transformed to + ``O(f(x, y).as_leading_term(x,y).as_leading_term(y), x, y)`` + + In the multivariate case, it is assumed the limits w.r.t. the various + symbols commute. + + If no symbols are passed then all symbols in the expression are used + and the limit point is assumed to be zero. + + """ + + is_Order = True + + __slots__ = () + + @cacheit + def __new__(cls, expr, *args, **kwargs): + expr = sympify(expr) + + if not args: + if expr.is_Order: + variables = expr.variables + point = expr.point + else: + variables = list(expr.free_symbols) + point = [S.Zero]*len(variables) + else: + args = list(args if is_sequence(args) else [args]) + variables, point = [], [] + if is_sequence(args[0]): + for a in args: + v, p = list(map(sympify, a)) + variables.append(v) + point.append(p) + else: + variables = list(map(sympify, args)) + point = [S.Zero]*len(variables) + + if not all(v.is_symbol for v in variables): + raise TypeError('Variables are not symbols, got %s' % variables) + + if len(list(uniq(variables))) != len(variables): + raise ValueError('Variables are supposed to be unique symbols, got %s' % variables) + + if expr.is_Order: + expr_vp = dict(expr.args[1:]) + new_vp = dict(expr_vp) + vp = dict(zip(variables, point)) + for v, p in vp.items(): + if v in new_vp.keys(): + if p != new_vp[v]: + raise NotImplementedError( + "Mixing Order at different points is not supported.") + else: + new_vp[v] = p + if set(expr_vp.keys()) == set(new_vp.keys()): + return expr + else: + variables = list(new_vp.keys()) + point = [new_vp[v] for v in variables] + + if expr is S.NaN: + return S.NaN + + if any(x in p.free_symbols for x in variables for p in point): + raise ValueError('Got %s as a point.' % point) + + if variables: + if any(p != point[0] for p in point): + raise NotImplementedError( + "Multivariable orders at different points are not supported.") + if point[0] in (S.Infinity, S.Infinity*S.ImaginaryUnit): + s = {k: 1/Dummy() for k in variables} + rs = {1/v: 1/k for k, v in s.items()} + ps = [S.Zero for p in point] + elif point[0] in (S.NegativeInfinity, S.NegativeInfinity*S.ImaginaryUnit): + s = {k: -1/Dummy() for k in variables} + rs = {-1/v: -1/k for k, v in s.items()} + ps = [S.Zero for p in point] + elif point[0] is not S.Zero: + s = {k: Dummy() + point[0] for k in variables} + rs = {(v - point[0]).together(): k - point[0] for k, v in s.items()} + ps = [S.Zero for p in point] + else: + s = () + rs = () + ps = list(point) + + expr = expr.subs(s) + + if expr.is_Add: + expr = expr.factor() + + if s: + args = tuple([r[0] for r in rs.items()]) + else: + args = tuple(variables) + + if len(variables) > 1: + # XXX: better way? We need this expand() to + # workaround e.g: expr = x*(x + y). + # (x*(x + y)).as_leading_term(x, y) currently returns + # x*y (wrong order term!). That's why we want to deal with + # expand()'ed expr (handled in "if expr.is_Add" branch below). + expr = expr.expand() + + old_expr = None + while old_expr != expr: + old_expr = expr + if expr.is_Add: + lst = expr.extract_leading_order(args) + expr = Add(*[f.expr for (e, f) in lst]) + + elif expr: + try: + expr = expr.as_leading_term(*args) + except PoleError: + if isinstance(expr, Function) or\ + all(isinstance(arg, Function) for arg in expr.args): + # It is not possible to simplify an expression + # containing only functions (which raise error on + # call to leading term) further + pass + else: + orders = [] + pts = tuple(zip(args, ps)) + for arg in expr.args: + try: + lt = arg.as_leading_term(*args) + except PoleError: + lt = arg + if lt not in args: + order = Order(lt) + else: + order = Order(lt, *pts) + orders.append(order) + if expr.is_Add: + new_expr = Order(Add(*orders), *pts) + if new_expr.is_Add: + new_expr = Order(Add(*[a.expr for a in new_expr.args]), *pts) + expr = new_expr.expr + elif expr.is_Mul: + expr = Mul(*[a.expr for a in orders]) + elif expr.is_Pow: + e = expr.exp + b = expr.base + expr = exp(e * log(b)) + + # It would probably be better to handle this somewhere + # else. This is needed for a testcase in which there is a + # symbol with the assumptions zero=True. + if expr.is_zero: + expr = S.Zero + else: + expr = expr.as_independent(*args, as_Add=False)[1] + + expr = expand_power_base(expr) + expr = expand_log(expr) + + if len(args) == 1: + # The definition of O(f(x)) symbol explicitly stated that + # the argument of f(x) is irrelevant. That's why we can + # combine some power exponents (only "on top" of the + # expression tree for f(x)), e.g.: + # x**p * (-x)**q -> x**(p+q) for real p, q. + x = args[0] + margs = list(Mul.make_args( + expr.as_independent(x, as_Add=False)[1])) + + for i, t in enumerate(margs): + if t.is_Pow: + b, q = t.args + if b in (x, -x) and q.is_real and not q.has(x): + margs[i] = x**q + elif b.is_Pow and not b.exp.has(x): + b, r = b.args + if b in (x, -x) and r.is_real: + margs[i] = x**(r*q) + elif b.is_Mul and b.args[0] is S.NegativeOne: + b = -b + if b.is_Pow and not b.exp.has(x): + b, r = b.args + if b in (x, -x) and r.is_real: + margs[i] = x**(r*q) + + expr = Mul(*margs) + + expr = expr.subs(rs) + + if expr.is_Order: + expr = expr.expr + + if not expr.has(*variables) and not expr.is_zero: + expr = S.One + + # create Order instance: + vp = dict(zip(variables, point)) + variables.sort(key=default_sort_key) + point = [vp[v] for v in variables] + args = (expr,) + Tuple(*zip(variables, point)) + obj = Expr.__new__(cls, *args) + return obj + + def _eval_nseries(self, x, n, logx, cdir=0): + return self + + @property + def expr(self): + return self.args[0] + + @property + def variables(self): + if self.args[1:]: + return tuple(x[0] for x in self.args[1:]) + else: + return () + + @property + def point(self): + if self.args[1:]: + return tuple(x[1] for x in self.args[1:]) + else: + return () + + @property + def free_symbols(self): + return self.expr.free_symbols | set(self.variables) + + def _eval_power(b, e): + if e.is_Number and e.is_nonnegative: + return b.func(b.expr ** e, *b.args[1:]) + if e == O(1): + return b + return + + def as_expr_variables(self, order_symbols): + if order_symbols is None: + order_symbols = self.args[1:] + else: + if (not all(o[1] == order_symbols[0][1] for o in order_symbols) and + not all(p == self.point[0] for p in self.point)): # pragma: no cover + raise NotImplementedError('Order at points other than 0 ' + 'or oo not supported, got %s as a point.' % self.point) + if order_symbols and order_symbols[0][1] != self.point[0]: + raise NotImplementedError( + "Multiplying Order at different points is not supported.") + order_symbols = dict(order_symbols) + for s, p in dict(self.args[1:]).items(): + if s not in order_symbols.keys(): + order_symbols[s] = p + order_symbols = sorted(order_symbols.items(), key=lambda x: default_sort_key(x[0])) + return self.expr, tuple(order_symbols) + + def removeO(self): + return S.Zero + + def getO(self): + return self + + @cacheit + def contains(self, expr): + r""" + Return True if expr belongs to Order(self.expr, \*self.variables). + Return False if self belongs to expr. + Return None if the inclusion relation cannot be determined + (e.g. when self and expr have different symbols). + """ + expr = sympify(expr) + if expr.is_zero: + return True + if expr is S.NaN: + return False + point = self.point[0] if self.point else S.Zero + if expr.is_Order: + if (any(p != point for p in expr.point) or + any(p != point for p in self.point)): + return None + if expr.expr == self.expr: + # O(1) + O(1), O(1) + O(1, x), etc. + return all(x in self.args[1:] for x in expr.args[1:]) + if expr.expr.is_Add: + return all(self.contains(x) for x in expr.expr.args) + if self.expr.is_Add and point.is_zero: + return any(self.func(x, *self.args[1:]).contains(expr) + for x in self.expr.args) + if self.variables and expr.variables: + common_symbols = tuple( + [s for s in self.variables if s in expr.variables]) + elif self.variables: + common_symbols = self.variables + else: + common_symbols = expr.variables + if not common_symbols: + return None + if (self.expr.is_Pow and len(self.variables) == 1 + and self.variables == expr.variables): + symbol = self.variables[0] + other = expr.expr.as_independent(symbol, as_Add=False)[1] + if (other.is_Pow and other.base == symbol and + self.expr.base == symbol): + if point.is_zero: + rv = (self.expr.exp - other.exp).is_nonpositive + if point.is_infinite: + rv = (self.expr.exp - other.exp).is_nonnegative + if rv is not None: + return rv + + from sympy.simplify.powsimp import powsimp + r = None + ratio = self.expr/expr.expr + ratio = powsimp(ratio, deep=True, combine='exp') + for s in common_symbols: + from sympy.series.limits import Limit + l = Limit(ratio, s, point).doit(heuristics=False) + if not isinstance(l, Limit): + l = l != 0 + else: + l = None + if r is None: + r = l + else: + if r != l: + return + return r + + if self.expr.is_Pow and len(self.variables) == 1: + symbol = self.variables[0] + other = expr.as_independent(symbol, as_Add=False)[1] + if (other.is_Pow and other.base == symbol and + self.expr.base == symbol): + if point.is_zero: + rv = (self.expr.exp - other.exp).is_nonpositive + if point.is_infinite: + rv = (self.expr.exp - other.exp).is_nonnegative + if rv is not None: + return rv + + obj = self.func(expr, *self.args[1:]) + return self.contains(obj) + + def __contains__(self, other): + result = self.contains(other) + if result is None: + raise TypeError('contains did not evaluate to a bool') + return result + + def _eval_subs(self, old, new): + if old in self.variables: + newexpr = self.expr.subs(old, new) + i = self.variables.index(old) + newvars = list(self.variables) + newpt = list(self.point) + if new.is_symbol: + newvars[i] = new + else: + syms = new.free_symbols + if len(syms) == 1 or old in syms: + if old in syms: + var = self.variables[i] + else: + var = syms.pop() + # First, try to substitute self.point in the "new" + # expr to see if this is a fixed point. + # E.g. O(y).subs(y, sin(x)) + from sympy import limit + if new.has(Order) and limit(new.getO().expr, var, new.getO().point[0]) == self.point[i]: + point = new.getO().point[0] + return Order(newexpr, *zip([var], [point])) + else: + point = new.subs(var, self.point[i]) + if point != self.point[i]: + from sympy.solvers.solveset import solveset + d = Dummy() + sol = solveset(old - new.subs(var, d), d) + if isinstance(sol, Complement): + e1 = sol.args[0] + e2 = sol.args[1] + sol = set(e1) - set(e2) + res = [dict(zip((d, ), sol))] + point = d.subs(res[0]).limit(old, self.point[i]) + newvars[i] = var + newpt[i] = point + elif old not in syms: + del newvars[i], newpt[i] + if not syms and new == self.point[i]: + newvars.extend(syms) + newpt.extend([S.Zero]*len(syms)) + else: + return + return Order(newexpr, *zip(newvars, newpt)) + + def _eval_conjugate(self): + expr = self.expr._eval_conjugate() + if expr is not None: + return self.func(expr, *self.args[1:]) + + def _eval_derivative(self, x): + return self.func(self.expr.diff(x), *self.args[1:]) or self + + def _eval_transpose(self): + expr = self.expr._eval_transpose() + if expr is not None: + return self.func(expr, *self.args[1:]) + + def __neg__(self): + return self + +O = Order diff --git a/.venv/lib/python3.12/site-packages/sympy/series/residues.py b/.venv/lib/python3.12/site-packages/sympy/series/residues.py new file mode 100644 index 0000000000000000000000000000000000000000..a426f9e799bd040eea5124f718c2fa43e5de026b --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/residues.py @@ -0,0 +1,73 @@ +""" +This module implements the Residue function and related tools for working +with residues. +""" + +from sympy.core.mul import Mul +from sympy.core.singleton import S +from sympy.core.sympify import sympify +from sympy.utilities.timeutils import timethis + + +@timethis('residue') +def residue(expr, x, x0): + """ + Finds the residue of ``expr`` at the point x=x0. + + The residue is defined as the coefficient of ``1/(x-x0)`` in the power series + expansion about ``x=x0``. + + Examples + ======== + + >>> from sympy import Symbol, residue, sin + >>> x = Symbol("x") + >>> residue(1/x, x, 0) + 1 + >>> residue(1/x**2, x, 0) + 0 + >>> residue(2/sin(x), x, 0) + 2 + + This function is essential for the Residue Theorem [1]. + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Residue_theorem + """ + # The current implementation uses series expansion to + # calculate it. A more general implementation is explained in + # the section 5.6 of the Bronstein's book {M. Bronstein: + # Symbolic Integration I, Springer Verlag (2005)}. For purely + # rational functions, the algorithm is much easier. See + # sections 2.4, 2.5, and 2.7 (this section actually gives an + # algorithm for computing any Laurent series coefficient for + # a rational function). The theory in section 2.4 will help to + # understand why the resultant works in the general algorithm. + # For the definition of a resultant, see section 1.4 (and any + # previous sections for more review). + + from sympy.series.order import Order + from sympy.simplify.radsimp import collect + expr = sympify(expr) + if x0 != 0: + expr = expr.subs(x, x + x0) + for n in (0, 1, 2, 4, 8, 16, 32): + s = expr.nseries(x, n=n) + if not s.has(Order) or s.getn() >= 0: + break + s = collect(s.removeO(), x) + if s.is_Add: + args = s.args + else: + args = [s] + res = S.Zero + for arg in args: + c, m = arg.as_coeff_mul(x) + m = Mul(*m) + if not (m in (S.One, x) or (m.is_Pow and m.exp.is_Integer)): + raise NotImplementedError('term of unexpected form: %s' % m) + if m == 1/x: + res += c + return res diff --git a/.venv/lib/python3.12/site-packages/sympy/series/sequences.py b/.venv/lib/python3.12/site-packages/sympy/series/sequences.py new file mode 100644 index 0000000000000000000000000000000000000000..7787515ddb05afaf34751bf451544935723d0921 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/sequences.py @@ -0,0 +1,1239 @@ +from sympy.core.basic import Basic +from sympy.core.cache import cacheit +from sympy.core.containers import Tuple +from sympy.core.decorators import call_highest_priority +from sympy.core.parameters import global_parameters +from sympy.core.function import AppliedUndef, expand +from sympy.core.mul import Mul +from sympy.core.numbers import Integer +from sympy.core.relational import Eq +from sympy.core.singleton import S, Singleton +from sympy.core.sorting import ordered +from sympy.core.symbol import Dummy, Symbol, Wild +from sympy.core.sympify import sympify +from sympy.matrices import Matrix +from sympy.polys import lcm, factor +from sympy.sets.sets import Interval, Intersection +from sympy.tensor.indexed import Idx +from sympy.utilities.iterables import flatten, is_sequence, iterable + + +############################################################################### +# SEQUENCES # +############################################################################### + + +class SeqBase(Basic): + """Base class for sequences""" + + is_commutative = True + _op_priority = 15 + + @staticmethod + def _start_key(expr): + """Return start (if possible) else S.Infinity. + + adapted from Set._infimum_key + """ + try: + start = expr.start + except NotImplementedError: + start = S.Infinity + return start + + def _intersect_interval(self, other): + """Returns start and stop. + + Takes intersection over the two intervals. + """ + interval = Intersection(self.interval, other.interval) + return interval.inf, interval.sup + + @property + def gen(self): + """Returns the generator for the sequence""" + raise NotImplementedError("(%s).gen" % self) + + @property + def interval(self): + """The interval on which the sequence is defined""" + raise NotImplementedError("(%s).interval" % self) + + @property + def start(self): + """The starting point of the sequence. This point is included""" + raise NotImplementedError("(%s).start" % self) + + @property + def stop(self): + """The ending point of the sequence. This point is included""" + raise NotImplementedError("(%s).stop" % self) + + @property + def length(self): + """Length of the sequence""" + raise NotImplementedError("(%s).length" % self) + + @property + def variables(self): + """Returns a tuple of variables that are bounded""" + return () + + @property + def free_symbols(self): + """ + This method returns the symbols in the object, excluding those + that take on a specific value (i.e. the dummy symbols). + + Examples + ======== + + >>> from sympy import SeqFormula + >>> from sympy.abc import n, m + >>> SeqFormula(m*n**2, (n, 0, 5)).free_symbols + {m} + """ + return ({j for i in self.args for j in i.free_symbols + .difference(self.variables)}) + + @cacheit + def coeff(self, pt): + """Returns the coefficient at point pt""" + if pt < self.start or pt > self.stop: + raise IndexError("Index %s out of bounds %s" % (pt, self.interval)) + return self._eval_coeff(pt) + + def _eval_coeff(self, pt): + raise NotImplementedError("The _eval_coeff method should be added to" + "%s to return coefficient so it is available" + "when coeff calls it." + % self.func) + + def _ith_point(self, i): + """Returns the i'th point of a sequence. + + Explanation + =========== + + If start point is negative infinity, point is returned from the end. + Assumes the first point to be indexed zero. + + Examples + ========= + + >>> from sympy import oo + >>> from sympy.series.sequences import SeqPer + + bounded + + >>> SeqPer((1, 2, 3), (-10, 10))._ith_point(0) + -10 + >>> SeqPer((1, 2, 3), (-10, 10))._ith_point(5) + -5 + + End is at infinity + + >>> SeqPer((1, 2, 3), (0, oo))._ith_point(5) + 5 + + Starts at negative infinity + + >>> SeqPer((1, 2, 3), (-oo, 0))._ith_point(5) + -5 + """ + if self.start is S.NegativeInfinity: + initial = self.stop + else: + initial = self.start + + if self.start is S.NegativeInfinity: + step = -1 + else: + step = 1 + + return initial + i*step + + def _add(self, other): + """ + Should only be used internally. + + Explanation + =========== + + self._add(other) returns a new, term-wise added sequence if self + knows how to add with other, otherwise it returns ``None``. + + ``other`` should only be a sequence object. + + Used within :class:`SeqAdd` class. + """ + return None + + def _mul(self, other): + """ + Should only be used internally. + + Explanation + =========== + + self._mul(other) returns a new, term-wise multiplied sequence if self + knows how to multiply with other, otherwise it returns ``None``. + + ``other`` should only be a sequence object. + + Used within :class:`SeqMul` class. + """ + return None + + def coeff_mul(self, other): + """ + Should be used when ``other`` is not a sequence. Should be + defined to define custom behaviour. + + Examples + ======== + + >>> from sympy import SeqFormula + >>> from sympy.abc import n + >>> SeqFormula(n**2).coeff_mul(2) + SeqFormula(2*n**2, (n, 0, oo)) + + Notes + ===== + + '*' defines multiplication of sequences with sequences only. + """ + return Mul(self, other) + + def __add__(self, other): + """Returns the term-wise addition of 'self' and 'other'. + + ``other`` should be a sequence. + + Examples + ======== + + >>> from sympy import SeqFormula + >>> from sympy.abc import n + >>> SeqFormula(n**2) + SeqFormula(n**3) + SeqFormula(n**3 + n**2, (n, 0, oo)) + """ + if not isinstance(other, SeqBase): + raise TypeError('cannot add sequence and %s' % type(other)) + return SeqAdd(self, other) + + @call_highest_priority('__add__') + def __radd__(self, other): + return self + other + + def __sub__(self, other): + """Returns the term-wise subtraction of ``self`` and ``other``. + + ``other`` should be a sequence. + + Examples + ======== + + >>> from sympy import SeqFormula + >>> from sympy.abc import n + >>> SeqFormula(n**2) - (SeqFormula(n)) + SeqFormula(n**2 - n, (n, 0, oo)) + """ + if not isinstance(other, SeqBase): + raise TypeError('cannot subtract sequence and %s' % type(other)) + return SeqAdd(self, -other) + + @call_highest_priority('__sub__') + def __rsub__(self, other): + return (-self) + other + + def __neg__(self): + """Negates the sequence. + + Examples + ======== + + >>> from sympy import SeqFormula + >>> from sympy.abc import n + >>> -SeqFormula(n**2) + SeqFormula(-n**2, (n, 0, oo)) + """ + return self.coeff_mul(-1) + + def __mul__(self, other): + """Returns the term-wise multiplication of 'self' and 'other'. + + ``other`` should be a sequence. For ``other`` not being a + sequence see :func:`coeff_mul` method. + + Examples + ======== + + >>> from sympy import SeqFormula + >>> from sympy.abc import n + >>> SeqFormula(n**2) * (SeqFormula(n)) + SeqFormula(n**3, (n, 0, oo)) + """ + if not isinstance(other, SeqBase): + raise TypeError('cannot multiply sequence and %s' % type(other)) + return SeqMul(self, other) + + @call_highest_priority('__mul__') + def __rmul__(self, other): + return self * other + + def __iter__(self): + for i in range(self.length): + pt = self._ith_point(i) + yield self.coeff(pt) + + def __getitem__(self, index): + if isinstance(index, int): + index = self._ith_point(index) + return self.coeff(index) + elif isinstance(index, slice): + start, stop = index.start, index.stop + if start is None: + start = 0 + if stop is None: + stop = self.length + return [self.coeff(self._ith_point(i)) for i in + range(start, stop, index.step or 1)] + + def find_linear_recurrence(self,n,d=None,gfvar=None): + r""" + Finds the shortest linear recurrence that satisfies the first n + terms of sequence of order `\leq` ``n/2`` if possible. + If ``d`` is specified, find shortest linear recurrence of order + `\leq` min(d, n/2) if possible. + Returns list of coefficients ``[b(1), b(2), ...]`` corresponding to the + recurrence relation ``x(n) = b(1)*x(n-1) + b(2)*x(n-2) + ...`` + Returns ``[]`` if no recurrence is found. + If gfvar is specified, also returns ordinary generating function as a + function of gfvar. + + Examples + ======== + + >>> from sympy import sequence, sqrt, oo, lucas + >>> from sympy.abc import n, x, y + >>> sequence(n**2).find_linear_recurrence(10, 2) + [] + >>> sequence(n**2).find_linear_recurrence(10) + [3, -3, 1] + >>> sequence(2**n).find_linear_recurrence(10) + [2] + >>> sequence(23*n**4+91*n**2).find_linear_recurrence(10) + [5, -10, 10, -5, 1] + >>> sequence(sqrt(5)*(((1 + sqrt(5))/2)**n - (-(1 + sqrt(5))/2)**(-n))/5).find_linear_recurrence(10) + [1, 1] + >>> sequence(x+y*(-2)**(-n), (n, 0, oo)).find_linear_recurrence(30) + [1/2, 1/2] + >>> sequence(3*5**n + 12).find_linear_recurrence(20,gfvar=x) + ([6, -5], 3*(5 - 21*x)/((x - 1)*(5*x - 1))) + >>> sequence(lucas(n)).find_linear_recurrence(15,gfvar=x) + ([1, 1], (x - 2)/(x**2 + x - 1)) + """ + from sympy.simplify import simplify + x = [simplify(expand(t)) for t in self[:n]] + lx = len(x) + if d is None: + r = lx//2 + else: + r = min(d,lx//2) + coeffs = [] + for l in range(1, r+1): + l2 = 2*l + mlist = [] + for k in range(l): + mlist.append(x[k:k+l]) + m = Matrix(mlist) + if m.det() != 0: + y = simplify(m.LUsolve(Matrix(x[l:l2]))) + if lx == l2: + coeffs = flatten(y[::-1]) + break + mlist = [] + for k in range(l,lx-l): + mlist.append(x[k:k+l]) + m = Matrix(mlist) + if m*y == Matrix(x[l2:]): + coeffs = flatten(y[::-1]) + break + if gfvar is None: + return coeffs + else: + l = len(coeffs) + if l == 0: + return [], None + else: + n, d = x[l-1]*gfvar**(l-1), 1 - coeffs[l-1]*gfvar**l + for i in range(l-1): + n += x[i]*gfvar**i + for j in range(l-i-1): + n -= coeffs[i]*x[j]*gfvar**(i+j+1) + d -= coeffs[i]*gfvar**(i+1) + return coeffs, simplify(factor(n)/factor(d)) + +class EmptySequence(SeqBase, metaclass=Singleton): + """Represents an empty sequence. + + The empty sequence is also available as a singleton as + ``S.EmptySequence``. + + Examples + ======== + + >>> from sympy import EmptySequence, SeqPer + >>> from sympy.abc import x + >>> EmptySequence + EmptySequence + >>> SeqPer((1, 2), (x, 0, 10)) + EmptySequence + SeqPer((1, 2), (x, 0, 10)) + >>> SeqPer((1, 2)) * EmptySequence + EmptySequence + >>> EmptySequence.coeff_mul(-1) + EmptySequence + """ + + @property + def interval(self): + return S.EmptySet + + @property + def length(self): + return S.Zero + + def coeff_mul(self, coeff): + """See docstring of SeqBase.coeff_mul""" + return self + + def __iter__(self): + return iter([]) + + +class SeqExpr(SeqBase): + """Sequence expression class. + + Various sequences should inherit from this class. + + Examples + ======== + + >>> from sympy.series.sequences import SeqExpr + >>> from sympy.abc import x + >>> from sympy import Tuple + >>> s = SeqExpr(Tuple(1, 2, 3), Tuple(x, 0, 10)) + >>> s.gen + (1, 2, 3) + >>> s.interval + Interval(0, 10) + >>> s.length + 11 + + See Also + ======== + + sympy.series.sequences.SeqPer + sympy.series.sequences.SeqFormula + """ + + @property + def gen(self): + return self.args[0] + + @property + def interval(self): + return Interval(self.args[1][1], self.args[1][2]) + + @property + def start(self): + return self.interval.inf + + @property + def stop(self): + return self.interval.sup + + @property + def length(self): + return self.stop - self.start + 1 + + @property + def variables(self): + return (self.args[1][0],) + + +class SeqPer(SeqExpr): + """ + Represents a periodic sequence. + + The elements are repeated after a given period. + + Examples + ======== + + >>> from sympy import SeqPer, oo + >>> from sympy.abc import k + + >>> s = SeqPer((1, 2, 3), (0, 5)) + >>> s.periodical + (1, 2, 3) + >>> s.period + 3 + + For value at a particular point + + >>> s.coeff(3) + 1 + + supports slicing + + >>> s[:] + [1, 2, 3, 1, 2, 3] + + iterable + + >>> list(s) + [1, 2, 3, 1, 2, 3] + + sequence starts from negative infinity + + >>> SeqPer((1, 2, 3), (-oo, 0))[0:6] + [1, 2, 3, 1, 2, 3] + + Periodic formulas + + >>> SeqPer((k, k**2, k**3), (k, 0, oo))[0:6] + [0, 1, 8, 3, 16, 125] + + See Also + ======== + + sympy.series.sequences.SeqFormula + """ + + def __new__(cls, periodical, limits=None): + periodical = sympify(periodical) + + def _find_x(periodical): + free = periodical.free_symbols + if len(periodical.free_symbols) == 1: + return free.pop() + else: + return Dummy('k') + + x, start, stop = None, None, None + if limits is None: + x, start, stop = _find_x(periodical), 0, S.Infinity + if is_sequence(limits, Tuple): + if len(limits) == 3: + x, start, stop = limits + elif len(limits) == 2: + x = _find_x(periodical) + start, stop = limits + + if not isinstance(x, (Symbol, Idx)) or start is None or stop is None: + raise ValueError('Invalid limits given: %s' % str(limits)) + + if start is S.NegativeInfinity and stop is S.Infinity: + raise ValueError("Both the start and end value" + "cannot be unbounded") + + limits = sympify((x, start, stop)) + + if is_sequence(periodical, Tuple): + periodical = sympify(tuple(flatten(periodical))) + else: + raise ValueError("invalid period %s should be something " + "like e.g (1, 2) " % periodical) + + if Interval(limits[1], limits[2]) is S.EmptySet: + return S.EmptySequence + + return Basic.__new__(cls, periodical, limits) + + @property + def period(self): + return len(self.gen) + + @property + def periodical(self): + return self.gen + + def _eval_coeff(self, pt): + if self.start is S.NegativeInfinity: + idx = (self.stop - pt) % self.period + else: + idx = (pt - self.start) % self.period + return self.periodical[idx].subs(self.variables[0], pt) + + def _add(self, other): + """See docstring of SeqBase._add""" + if isinstance(other, SeqPer): + per1, lper1 = self.periodical, self.period + per2, lper2 = other.periodical, other.period + + per_length = lcm(lper1, lper2) + + new_per = [] + for x in range(per_length): + ele1 = per1[x % lper1] + ele2 = per2[x % lper2] + new_per.append(ele1 + ele2) + + start, stop = self._intersect_interval(other) + return SeqPer(new_per, (self.variables[0], start, stop)) + + def _mul(self, other): + """See docstring of SeqBase._mul""" + if isinstance(other, SeqPer): + per1, lper1 = self.periodical, self.period + per2, lper2 = other.periodical, other.period + + per_length = lcm(lper1, lper2) + + new_per = [] + for x in range(per_length): + ele1 = per1[x % lper1] + ele2 = per2[x % lper2] + new_per.append(ele1 * ele2) + + start, stop = self._intersect_interval(other) + return SeqPer(new_per, (self.variables[0], start, stop)) + + def coeff_mul(self, coeff): + """See docstring of SeqBase.coeff_mul""" + coeff = sympify(coeff) + per = [x * coeff for x in self.periodical] + return SeqPer(per, self.args[1]) + + +class SeqFormula(SeqExpr): + """ + Represents sequence based on a formula. + + Elements are generated using a formula. + + Examples + ======== + + >>> from sympy import SeqFormula, oo, Symbol + >>> n = Symbol('n') + >>> s = SeqFormula(n**2, (n, 0, 5)) + >>> s.formula + n**2 + + For value at a particular point + + >>> s.coeff(3) + 9 + + supports slicing + + >>> s[:] + [0, 1, 4, 9, 16, 25] + + iterable + + >>> list(s) + [0, 1, 4, 9, 16, 25] + + sequence starts from negative infinity + + >>> SeqFormula(n**2, (-oo, 0))[0:6] + [0, 1, 4, 9, 16, 25] + + See Also + ======== + + sympy.series.sequences.SeqPer + """ + + def __new__(cls, formula, limits=None): + formula = sympify(formula) + + def _find_x(formula): + free = formula.free_symbols + if len(free) == 1: + return free.pop() + elif not free: + return Dummy('k') + else: + raise ValueError( + " specify dummy variables for %s. If the formula contains" + " more than one free symbol, a dummy variable should be" + " supplied explicitly e.g., SeqFormula(m*n**2, (n, 0, 5))" + % formula) + + x, start, stop = None, None, None + if limits is None: + x, start, stop = _find_x(formula), 0, S.Infinity + if is_sequence(limits, Tuple): + if len(limits) == 3: + x, start, stop = limits + elif len(limits) == 2: + x = _find_x(formula) + start, stop = limits + + if not isinstance(x, (Symbol, Idx)) or start is None or stop is None: + raise ValueError('Invalid limits given: %s' % str(limits)) + + if start is S.NegativeInfinity and stop is S.Infinity: + raise ValueError("Both the start and end value " + "cannot be unbounded") + limits = sympify((x, start, stop)) + + if Interval(limits[1], limits[2]) is S.EmptySet: + return S.EmptySequence + + return Basic.__new__(cls, formula, limits) + + @property + def formula(self): + return self.gen + + def _eval_coeff(self, pt): + d = self.variables[0] + return self.formula.subs(d, pt) + + def _add(self, other): + """See docstring of SeqBase._add""" + if isinstance(other, SeqFormula): + form1, v1 = self.formula, self.variables[0] + form2, v2 = other.formula, other.variables[0] + formula = form1 + form2.subs(v2, v1) + start, stop = self._intersect_interval(other) + return SeqFormula(formula, (v1, start, stop)) + + def _mul(self, other): + """See docstring of SeqBase._mul""" + if isinstance(other, SeqFormula): + form1, v1 = self.formula, self.variables[0] + form2, v2 = other.formula, other.variables[0] + formula = form1 * form2.subs(v2, v1) + start, stop = self._intersect_interval(other) + return SeqFormula(formula, (v1, start, stop)) + + def coeff_mul(self, coeff): + """See docstring of SeqBase.coeff_mul""" + coeff = sympify(coeff) + formula = self.formula * coeff + return SeqFormula(formula, self.args[1]) + + def expand(self, *args, **kwargs): + return SeqFormula(expand(self.formula, *args, **kwargs), self.args[1]) + +class RecursiveSeq(SeqBase): + """ + A finite degree recursive sequence. + + Explanation + =========== + + That is, a sequence a(n) that depends on a fixed, finite number of its + previous values. The general form is + + a(n) = f(a(n - 1), a(n - 2), ..., a(n - d)) + + for some fixed, positive integer d, where f is some function defined by a + SymPy expression. + + Parameters + ========== + + recurrence : SymPy expression defining recurrence + This is *not* an equality, only the expression that the nth term is + equal to. For example, if :code:`a(n) = f(a(n - 1), ..., a(n - d))`, + then the expression should be :code:`f(a(n - 1), ..., a(n - d))`. + + yn : applied undefined function + Represents the nth term of the sequence as e.g. :code:`y(n)` where + :code:`y` is an undefined function and `n` is the sequence index. + + n : symbolic argument + The name of the variable that the recurrence is in, e.g., :code:`n` if + the recurrence function is :code:`y(n)`. + + initial : iterable with length equal to the degree of the recurrence + The initial values of the recurrence. + + start : start value of sequence (inclusive) + + Examples + ======== + + >>> from sympy import Function, symbols + >>> from sympy.series.sequences import RecursiveSeq + >>> y = Function("y") + >>> n = symbols("n") + >>> fib = RecursiveSeq(y(n - 1) + y(n - 2), y(n), n, [0, 1]) + + >>> fib.coeff(3) # Value at a particular point + 2 + + >>> fib[:6] # supports slicing + [0, 1, 1, 2, 3, 5] + + >>> fib.recurrence # inspect recurrence + Eq(y(n), y(n - 2) + y(n - 1)) + + >>> fib.degree # automatically determine degree + 2 + + >>> for x in zip(range(10), fib): # supports iteration + ... print(x) + (0, 0) + (1, 1) + (2, 1) + (3, 2) + (4, 3) + (5, 5) + (6, 8) + (7, 13) + (8, 21) + (9, 34) + + See Also + ======== + + sympy.series.sequences.SeqFormula + + """ + + def __new__(cls, recurrence, yn, n, initial=None, start=0): + if not isinstance(yn, AppliedUndef): + raise TypeError("recurrence sequence must be an applied undefined function" + ", found `{}`".format(yn)) + + if not isinstance(n, Basic) or not n.is_symbol: + raise TypeError("recurrence variable must be a symbol" + ", found `{}`".format(n)) + + if yn.args != (n,): + raise TypeError("recurrence sequence does not match symbol") + + y = yn.func + + k = Wild("k", exclude=(n,)) + degree = 0 + + # Find all applications of y in the recurrence and check that: + # 1. The function y is only being used with a single argument; and + # 2. All arguments are n + k for constant negative integers k. + + prev_ys = recurrence.find(y) + for prev_y in prev_ys: + if len(prev_y.args) != 1: + raise TypeError("Recurrence should be in a single variable") + + shift = prev_y.args[0].match(n + k)[k] + if not (shift.is_constant() and shift.is_integer and shift < 0): + raise TypeError("Recurrence should have constant," + " negative, integer shifts" + " (found {})".format(prev_y)) + + if -shift > degree: + degree = -shift + + if not initial: + initial = [Dummy("c_{}".format(k)) for k in range(degree)] + + if len(initial) != degree: + raise ValueError("Number of initial terms must equal degree") + + degree = Integer(degree) + start = sympify(start) + + initial = Tuple(*(sympify(x) for x in initial)) + + seq = Basic.__new__(cls, recurrence, yn, n, initial, start) + + seq.cache = {y(start + k): init for k, init in enumerate(initial)} + seq.degree = degree + + return seq + + @property + def _recurrence(self): + """Equation defining recurrence.""" + return self.args[0] + + @property + def recurrence(self): + """Equation defining recurrence.""" + return Eq(self.yn, self.args[0]) + + @property + def yn(self): + """Applied function representing the nth term""" + return self.args[1] + + @property + def y(self): + """Undefined function for the nth term of the sequence""" + return self.yn.func + + @property + def n(self): + """Sequence index symbol""" + return self.args[2] + + @property + def initial(self): + """The initial values of the sequence""" + return self.args[3] + + @property + def start(self): + """The starting point of the sequence. This point is included""" + return self.args[4] + + @property + def stop(self): + """The ending point of the sequence. (oo)""" + return S.Infinity + + @property + def interval(self): + """Interval on which sequence is defined.""" + return (self.start, S.Infinity) + + def _eval_coeff(self, index): + if index - self.start < len(self.cache): + return self.cache[self.y(index)] + + for current in range(len(self.cache), index + 1): + # Use xreplace over subs for performance. + # See issue #10697. + seq_index = self.start + current + current_recurrence = self._recurrence.xreplace({self.n: seq_index}) + new_term = current_recurrence.xreplace(self.cache) + + self.cache[self.y(seq_index)] = new_term + + return self.cache[self.y(self.start + current)] + + def __iter__(self): + index = self.start + while True: + yield self._eval_coeff(index) + index += 1 + + +def sequence(seq, limits=None): + """ + Returns appropriate sequence object. + + Explanation + =========== + + If ``seq`` is a SymPy sequence, returns :class:`SeqPer` object + otherwise returns :class:`SeqFormula` object. + + Examples + ======== + + >>> from sympy import sequence + >>> from sympy.abc import n + >>> sequence(n**2, (n, 0, 5)) + SeqFormula(n**2, (n, 0, 5)) + >>> sequence((1, 2, 3), (n, 0, 5)) + SeqPer((1, 2, 3), (n, 0, 5)) + + See Also + ======== + + sympy.series.sequences.SeqPer + sympy.series.sequences.SeqFormula + """ + seq = sympify(seq) + + if is_sequence(seq, Tuple): + return SeqPer(seq, limits) + else: + return SeqFormula(seq, limits) + + +############################################################################### +# OPERATIONS # +############################################################################### + + +class SeqExprOp(SeqBase): + """ + Base class for operations on sequences. + + Examples + ======== + + >>> from sympy.series.sequences import SeqExprOp, sequence + >>> from sympy.abc import n + >>> s1 = sequence(n**2, (n, 0, 10)) + >>> s2 = sequence((1, 2, 3), (n, 5, 10)) + >>> s = SeqExprOp(s1, s2) + >>> s.gen + (n**2, (1, 2, 3)) + >>> s.interval + Interval(5, 10) + >>> s.length + 6 + + See Also + ======== + + sympy.series.sequences.SeqAdd + sympy.series.sequences.SeqMul + """ + @property + def gen(self): + """Generator for the sequence. + + returns a tuple of generators of all the argument sequences. + """ + return tuple(a.gen for a in self.args) + + @property + def interval(self): + """Sequence is defined on the intersection + of all the intervals of respective sequences + """ + return Intersection(*(a.interval for a in self.args)) + + @property + def start(self): + return self.interval.inf + + @property + def stop(self): + return self.interval.sup + + @property + def variables(self): + """Cumulative of all the bound variables""" + return tuple(flatten([a.variables for a in self.args])) + + @property + def length(self): + return self.stop - self.start + 1 + + +class SeqAdd(SeqExprOp): + """Represents term-wise addition of sequences. + + Rules: + * The interval on which sequence is defined is the intersection + of respective intervals of sequences. + * Anything + :class:`EmptySequence` remains unchanged. + * Other rules are defined in ``_add`` methods of sequence classes. + + Examples + ======== + + >>> from sympy import EmptySequence, oo, SeqAdd, SeqPer, SeqFormula + >>> from sympy.abc import n + >>> SeqAdd(SeqPer((1, 2), (n, 0, oo)), EmptySequence) + SeqPer((1, 2), (n, 0, oo)) + >>> SeqAdd(SeqPer((1, 2), (n, 0, 5)), SeqPer((1, 2), (n, 6, 10))) + EmptySequence + >>> SeqAdd(SeqPer((1, 2), (n, 0, oo)), SeqFormula(n**2, (n, 0, oo))) + SeqAdd(SeqFormula(n**2, (n, 0, oo)), SeqPer((1, 2), (n, 0, oo))) + >>> SeqAdd(SeqFormula(n**3), SeqFormula(n**2)) + SeqFormula(n**3 + n**2, (n, 0, oo)) + + See Also + ======== + + sympy.series.sequences.SeqMul + """ + + def __new__(cls, *args, **kwargs): + evaluate = kwargs.get('evaluate', global_parameters.evaluate) + + # flatten inputs + args = list(args) + + # adapted from sympy.sets.sets.Union + def _flatten(arg): + if isinstance(arg, SeqBase): + if isinstance(arg, SeqAdd): + return sum(map(_flatten, arg.args), []) + else: + return [arg] + if iterable(arg): + return sum(map(_flatten, arg), []) + raise TypeError("Input must be Sequences or " + " iterables of Sequences") + args = _flatten(args) + + args = [a for a in args if a is not S.EmptySequence] + + # Addition of no sequences is EmptySequence + if not args: + return S.EmptySequence + + if Intersection(*(a.interval for a in args)) is S.EmptySet: + return S.EmptySequence + + # reduce using known rules + if evaluate: + return SeqAdd.reduce(args) + + args = list(ordered(args, SeqBase._start_key)) + + return Basic.__new__(cls, *args) + + @staticmethod + def reduce(args): + """Simplify :class:`SeqAdd` using known rules. + + Iterates through all pairs and ask the constituent + sequences if they can simplify themselves with any other constituent. + + Notes + ===== + + adapted from ``Union.reduce`` + + """ + new_args = True + while new_args: + for id1, s in enumerate(args): + new_args = False + for id2, t in enumerate(args): + if id1 == id2: + continue + new_seq = s._add(t) + # This returns None if s does not know how to add + # with t. Returns the newly added sequence otherwise + if new_seq is not None: + new_args = [a for a in args if a not in (s, t)] + new_args.append(new_seq) + break + if new_args: + args = new_args + break + + if len(args) == 1: + return args.pop() + else: + return SeqAdd(args, evaluate=False) + + def _eval_coeff(self, pt): + """adds up the coefficients of all the sequences at point pt""" + return sum(a.coeff(pt) for a in self.args) + + +class SeqMul(SeqExprOp): + r"""Represents term-wise multiplication of sequences. + + Explanation + =========== + + Handles multiplication of sequences only. For multiplication + with other objects see :func:`SeqBase.coeff_mul`. + + Rules: + * The interval on which sequence is defined is the intersection + of respective intervals of sequences. + * Anything \* :class:`EmptySequence` returns :class:`EmptySequence`. + * Other rules are defined in ``_mul`` methods of sequence classes. + + Examples + ======== + + >>> from sympy import EmptySequence, oo, SeqMul, SeqPer, SeqFormula + >>> from sympy.abc import n + >>> SeqMul(SeqPer((1, 2), (n, 0, oo)), EmptySequence) + EmptySequence + >>> SeqMul(SeqPer((1, 2), (n, 0, 5)), SeqPer((1, 2), (n, 6, 10))) + EmptySequence + >>> SeqMul(SeqPer((1, 2), (n, 0, oo)), SeqFormula(n**2)) + SeqMul(SeqFormula(n**2, (n, 0, oo)), SeqPer((1, 2), (n, 0, oo))) + >>> SeqMul(SeqFormula(n**3), SeqFormula(n**2)) + SeqFormula(n**5, (n, 0, oo)) + + See Also + ======== + + sympy.series.sequences.SeqAdd + """ + + def __new__(cls, *args, **kwargs): + evaluate = kwargs.get('evaluate', global_parameters.evaluate) + + # flatten inputs + args = list(args) + + # adapted from sympy.sets.sets.Union + def _flatten(arg): + if isinstance(arg, SeqBase): + if isinstance(arg, SeqMul): + return sum(map(_flatten, arg.args), []) + else: + return [arg] + elif iterable(arg): + return sum(map(_flatten, arg), []) + raise TypeError("Input must be Sequences or " + " iterables of Sequences") + args = _flatten(args) + + # Multiplication of no sequences is EmptySequence + if not args: + return S.EmptySequence + + if Intersection(*(a.interval for a in args)) is S.EmptySet: + return S.EmptySequence + + # reduce using known rules + if evaluate: + return SeqMul.reduce(args) + + args = list(ordered(args, SeqBase._start_key)) + + return Basic.__new__(cls, *args) + + @staticmethod + def reduce(args): + """Simplify a :class:`SeqMul` using known rules. + + Explanation + =========== + + Iterates through all pairs and ask the constituent + sequences if they can simplify themselves with any other constituent. + + Notes + ===== + + adapted from ``Union.reduce`` + + """ + new_args = True + while new_args: + for id1, s in enumerate(args): + new_args = False + for id2, t in enumerate(args): + if id1 == id2: + continue + new_seq = s._mul(t) + # This returns None if s does not know how to multiply + # with t. Returns the newly multiplied sequence otherwise + if new_seq is not None: + new_args = [a for a in args if a not in (s, t)] + new_args.append(new_seq) + break + if new_args: + args = new_args + break + + if len(args) == 1: + return args.pop() + else: + return SeqMul(args, evaluate=False) + + def _eval_coeff(self, pt): + """multiplies the coefficients of all the sequences at point pt""" + val = 1 + for a in self.args: + val *= a.coeff(pt) + return val diff --git a/.venv/lib/python3.12/site-packages/sympy/series/series.py b/.venv/lib/python3.12/site-packages/sympy/series/series.py new file mode 100644 index 0000000000000000000000000000000000000000..e9feec7d3b1987bfaa5238969f531e9f98b88b25 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/series.py @@ -0,0 +1,63 @@ +from sympy.core.sympify import sympify + + +def series(expr, x=None, x0=0, n=6, dir="+"): + """Series expansion of expr around point `x = x0`. + + Parameters + ========== + + expr : Expression + The expression whose series is to be expanded. + + x : Symbol + It is the variable of the expression to be calculated. + + x0 : Value + The value around which ``x`` is calculated. Can be any value + from ``-oo`` to ``oo``. + + n : Value + The number of terms upto which the series is to be expanded. + + dir : String, optional + The series-expansion can be bi-directional. If ``dir="+"``, + then (x->x0+). If ``dir="-"``, then (x->x0-). For infinite + ``x0`` (``oo`` or ``-oo``), the ``dir`` argument is determined + from the direction of the infinity (i.e., ``dir="-"`` for + ``oo``). + + Examples + ======== + + >>> from sympy import series, tan, oo + >>> from sympy.abc import x + >>> f = tan(x) + >>> series(f, x, 2, 6, "+") + tan(2) + (1 + tan(2)**2)*(x - 2) + (x - 2)**2*(tan(2)**3 + tan(2)) + + (x - 2)**3*(1/3 + 4*tan(2)**2/3 + tan(2)**4) + (x - 2)**4*(tan(2)**5 + + 5*tan(2)**3/3 + 2*tan(2)/3) + (x - 2)**5*(2/15 + 17*tan(2)**2/15 + + 2*tan(2)**4 + tan(2)**6) + O((x - 2)**6, (x, 2)) + + >>> series(f, x, 2, 3, "-") + tan(2) + (2 - x)*(-tan(2)**2 - 1) + (2 - x)**2*(tan(2)**3 + tan(2)) + + O((x - 2)**3, (x, 2)) + + >>> series(f, x, 2, oo, "+") + Traceback (most recent call last): + ... + TypeError: 'Infinity' object cannot be interpreted as an integer + + Returns + ======= + + Expr + Series expansion of the expression about x0 + + See Also + ======== + + sympy.core.expr.Expr.series: See the docstring of Expr.series() for complete details of this wrapper. + """ + expr = sympify(expr) + return expr.series(x, x0, n, dir) diff --git a/.venv/lib/python3.12/site-packages/sympy/series/series_class.py b/.venv/lib/python3.12/site-packages/sympy/series/series_class.py new file mode 100644 index 0000000000000000000000000000000000000000..ff04993b266a3cbd3f767042d4325fb11edb2168 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/series_class.py @@ -0,0 +1,99 @@ +""" +Contains the base class for series +Made using sequences in mind +""" + +from sympy.core.expr import Expr +from sympy.core.singleton import S +from sympy.core.cache import cacheit + + +class SeriesBase(Expr): + """Base Class for series""" + + @property + def interval(self): + """The interval on which the series is defined""" + raise NotImplementedError("(%s).interval" % self) + + @property + def start(self): + """The starting point of the series. This point is included""" + raise NotImplementedError("(%s).start" % self) + + @property + def stop(self): + """The ending point of the series. This point is included""" + raise NotImplementedError("(%s).stop" % self) + + @property + def length(self): + """Length of the series expansion""" + raise NotImplementedError("(%s).length" % self) + + @property + def variables(self): + """Returns a tuple of variables that are bounded""" + return () + + @property + def free_symbols(self): + """ + This method returns the symbols in the object, excluding those + that take on a specific value (i.e. the dummy symbols). + """ + return ({j for i in self.args for j in i.free_symbols} + .difference(self.variables)) + + @cacheit + def term(self, pt): + """Term at point pt of a series""" + if pt < self.start or pt > self.stop: + raise IndexError("Index %s out of bounds %s" % (pt, self.interval)) + return self._eval_term(pt) + + def _eval_term(self, pt): + raise NotImplementedError("The _eval_term method should be added to" + "%s to return series term so it is available" + "when 'term' calls it." + % self.func) + + def _ith_point(self, i): + """ + Returns the i'th point of a series + If start point is negative infinity, point is returned from the end. + Assumes the first point to be indexed zero. + + Examples + ======== + + TODO + """ + if self.start is S.NegativeInfinity: + initial = self.stop + step = -1 + else: + initial = self.start + step = 1 + + return initial + i*step + + def __iter__(self): + i = 0 + while i < self.length: + pt = self._ith_point(i) + yield self.term(pt) + i += 1 + + def __getitem__(self, index): + if isinstance(index, int): + index = self._ith_point(index) + return self.term(index) + elif isinstance(index, slice): + start, stop = index.start, index.stop + if start is None: + start = 0 + if stop is None: + stop = self.length + return [self.term(self._ith_point(i)) for i in + range(start, stop, index.step or 1)] diff --git a/.venv/lib/python3.12/site-packages/sympy/series/tests/__init__.py b/.venv/lib/python3.12/site-packages/sympy/series/tests/__init__.py new file mode 100644 index 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differ diff --git a/.venv/lib/python3.12/site-packages/sympy/series/tests/__pycache__/test_series.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/series/tests/__pycache__/test_series.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..d8e24e554611c93858284623afdef3bd8215ec42 Binary files /dev/null and b/.venv/lib/python3.12/site-packages/sympy/series/tests/__pycache__/test_series.cpython-312.pyc differ diff --git a/.venv/lib/python3.12/site-packages/sympy/series/tests/test_approximants.py b/.venv/lib/python3.12/site-packages/sympy/series/tests/test_approximants.py new file mode 100644 index 0000000000000000000000000000000000000000..9c03d2ce38add99b0dce8725b6c8d8844b31f76b --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/tests/test_approximants.py @@ -0,0 +1,23 @@ +from sympy.series import approximants +from sympy.core.symbol import symbols +from sympy.functions.combinatorial.factorials import binomial +from sympy.functions.combinatorial.numbers import (fibonacci, lucas) + + +def test_approximants(): + x, t = symbols("x,t") + g = [lucas(k) for k in range(16)] + assert list(approximants(g)) == ( + [2, -4/(x - 2), (5*x - 2)/(3*x - 1), (x - 2)/(x**2 + x - 1)] ) + g = [lucas(k)+fibonacci(k+2) for k in range(16)] + assert list(approximants(g)) == ( + [3, -3/(x - 1), (3*x - 3)/(2*x - 1), -3/(x**2 + x - 1)] ) + g = [lucas(k)**2 for k in range(16)] + assert list(approximants(g)) == ( + [4, -16/(x - 4), (35*x - 4)/(9*x - 1), (37*x - 28)/(13*x**2 + 11*x - 7), + (50*x**2 + 63*x - 52)/(37*x**2 + 19*x - 13), + (-x**2 - 7*x + 4)/(x**3 - 2*x**2 - 2*x + 1)] ) + p = [sum(binomial(k,i)*x**i for i in range(k+1)) for k in range(16)] + y = approximants(p, t, simplify=True) + assert next(y) == 1 + assert next(y) == -1/(t*(x + 1) - 1) diff --git a/.venv/lib/python3.12/site-packages/sympy/series/tests/test_aseries.py b/.venv/lib/python3.12/site-packages/sympy/series/tests/test_aseries.py new file mode 100644 index 0000000000000000000000000000000000000000..cae0ac0a43f2406dd96e45c6a31939ac6b4cdcaa --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/tests/test_aseries.py @@ -0,0 +1,55 @@ +from sympy.core.function import PoleError +from sympy.core.numbers import oo +from sympy.core.symbol import Symbol +from sympy.functions.elementary.exponential import (exp, log) +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.series.order import O +from sympy.abc import x + +from sympy.testing.pytest import raises + +def test_simple(): + # Gruntz' theses pp. 91 to 96 + # 6.6 + e = sin(1/x + exp(-x)) - sin(1/x) + assert e.aseries(x) == (1/(24*x**4) - 1/(2*x**2) + 1 + O(x**(-6), (x, oo)))*exp(-x) + + e = exp(x) * (exp(1/x + exp(-x)) - exp(1/x)) + assert e.aseries(x, n=4) == 1/(6*x**3) + 1/(2*x**2) + 1/x + 1 + O(x**(-4), (x, oo)) + + e = exp(exp(x) / (1 - 1/x)) + assert e.aseries(x) == exp(exp(x) / (1 - 1/x)) + + # The implementation of bound in aseries is incorrect currently. This test + # should be commented out when that is fixed. + # assert e.aseries(x, bound=3) == exp(exp(x) / x**2)*exp(exp(x) / x)*exp(-exp(x) + exp(x)/(1 - 1/x) - \ + # exp(x) / x - exp(x) / x**2) * exp(exp(x)) + + e = exp(sin(1/x + exp(-exp(x)))) - exp(sin(1/x)) + assert e.aseries(x, n=4) == (-1/(2*x**3) + 1/x + 1 + O(x**(-4), (x, oo)))*exp(-exp(x)) + + e3 = lambda x:exp(exp(exp(x))) + e = e3(x)/e3(x - 1/e3(x)) + assert e.aseries(x, n=3) == 1 + exp(2*x + 2*exp(x))*exp(-2*exp(exp(x)))/2\ + - exp(2*x + exp(x))*exp(-2*exp(exp(x)))/2 - exp(x + exp(x))*exp(-2*exp(exp(x)))/2\ + + exp(x + exp(x))*exp(-exp(exp(x))) + O(exp(-3*exp(exp(x))), (x, oo)) + + e = exp(exp(x)) * (exp(sin(1/x + 1/exp(exp(x)))) - exp(sin(1/x))) + assert e.aseries(x, n=4) == -1/(2*x**3) + 1/x + 1 + O(x**(-4), (x, oo)) + + n = Symbol('n', integer=True) + e = (sqrt(n)*log(n)**2*exp(sqrt(log(n))*log(log(n))**2*exp(sqrt(log(log(n)))*log(log(log(n)))**3)))/n + assert e.aseries(n) == \ + exp(exp(sqrt(log(log(n)))*log(log(log(n)))**3)*sqrt(log(n))*log(log(n))**2)*log(n)**2/sqrt(n) + + +def test_hierarchical(): + e = sin(1/x + exp(-x)) + assert e.aseries(x, n=3, hir=True) == -exp(-2*x)*sin(1/x)/2 + \ + exp(-x)*cos(1/x) + sin(1/x) + O(exp(-3*x), (x, oo)) + + e = sin(x) * cos(exp(-x)) + assert e.aseries(x, hir=True) == exp(-4*x)*sin(x)/24 - \ + exp(-2*x)*sin(x)/2 + sin(x) + O(exp(-6*x), (x, oo)) + raises(PoleError, lambda: e.aseries(x)) diff --git a/.venv/lib/python3.12/site-packages/sympy/series/tests/test_demidovich.py b/.venv/lib/python3.12/site-packages/sympy/series/tests/test_demidovich.py new file mode 100644 index 0000000000000000000000000000000000000000..98cafbae6f019dd3d97d306099d5780ed2f37f04 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/tests/test_demidovich.py @@ -0,0 +1,143 @@ +from sympy.core.numbers import (Rational, oo, pi) +from sympy.core.singleton import S +from sympy.core.symbol import Symbol +from sympy.functions.elementary.exponential import (exp, log) +from sympy.functions.elementary.miscellaneous import (root, sqrt) +from sympy.functions.elementary.trigonometric import (asin, cos, sin, tan) +from sympy.polys.rationaltools import together +from sympy.series.limits import limit + +# Numbers listed with the tests refer to problem numbers in the book +# "Anti-demidovich, problemas resueltos, Ed. URSS" + +x = Symbol("x") + + +def test_leadterm(): + assert (3 + 2*x**(log(3)/log(2) - 1)).leadterm(x) == (3, 0) + + +def root3(x): + return root(x, 3) + + +def root4(x): + return root(x, 4) + + +def test_Limits_simple_0(): + assert limit((2**(x + 1) + 3**(x + 1))/(2**x + 3**x), x, oo) == 3 # 175 + + +def test_Limits_simple_1(): + assert limit((x + 1)*(x + 2)*(x + 3)/x**3, x, oo) == 1 # 172 + assert limit(sqrt(x + 1) - sqrt(x), x, oo) == 0 # 179 + assert limit((2*x - 3)*(3*x + 5)*(4*x - 6)/(3*x**3 + x - 1), x, oo) == 8 # Primjer 1 + assert limit(x/root3(x**3 + 10), x, oo) == 1 # Primjer 2 + assert limit((x + 1)**2/(x**2 + 1), x, oo) == 1 # 181 + + +def test_Limits_simple_2(): + assert limit(1000*x/(x**2 - 1), x, oo) == 0 # 182 + assert limit((x**2 - 5*x + 1)/(3*x + 7), x, oo) is oo # 183 + assert limit((2*x**2 - x + 3)/(x**3 - 8*x + 5), x, oo) == 0 # 184 + assert limit((2*x**2 - 3*x - 4)/sqrt(x**4 + 1), x, oo) == 2 # 186 + assert limit((2*x + 3)/(x + root3(x)), x, oo) == 2 # 187 + assert limit(x**2/(10 + x*sqrt(x)), x, oo) is oo # 188 + assert limit(root3(x**2 + 1)/(x + 1), x, oo) == 0 # 189 + assert limit(sqrt(x)/sqrt(x + sqrt(x + sqrt(x))), x, oo) == 1 # 190 + + +def test_Limits_simple_3a(): + a = Symbol('a') + #issue 3513 + assert together(limit((x**2 - (a + 1)*x + a)/(x**3 - a**3), x, a)) == \ + (a - 1)/(3*a**2) # 196 + + +def test_Limits_simple_3b(): + h = Symbol("h") + assert limit(((x + h)**3 - x**3)/h, h, 0) == 3*x**2 # 197 + assert limit((1/(1 - x) - 3/(1 - x**3)), x, 1) == -1 # 198 + assert limit((sqrt(1 + x) - 1)/(root3(1 + x) - 1), x, 0) == Rational(3)/2 # Primer 4 + assert limit((sqrt(x) - 1)/(x - 1), x, 1) == Rational(1)/2 # 199 + assert limit((sqrt(x) - 8)/(root3(x) - 4), x, 64) == 3 # 200 + assert limit((root3(x) - 1)/(root4(x) - 1), x, 1) == Rational(4)/3 # 201 + assert limit( + (root3(x**2) - 2*root3(x) + 1)/(x - 1)**2, x, 1) == Rational(1)/9 # 202 + + +def test_Limits_simple_4a(): + a = Symbol('a') + assert limit((sqrt(x) - sqrt(a))/(x - a), x, a) == 1/(2*sqrt(a)) # Primer 5 + assert limit((sqrt(x) - 1)/(root3(x) - 1), x, 1) == Rational(3, 2) # 205 + assert limit((sqrt(1 + x) - sqrt(1 - x))/x, x, 0) == 1 # 207 + assert limit(sqrt(x**2 - 5*x + 6) - x, x, oo) == Rational(-5, 2) # 213 + + +def test_limits_simple_4aa(): + assert limit(x*(sqrt(x**2 + 1) - x), x, oo) == Rational(1)/2 # 214 + + +def test_Limits_simple_4b(): + #issue 3511 + assert limit(x - root3(x**3 - 1), x, oo) == 0 # 215 + + +def test_Limits_simple_4c(): + assert limit(log(1 + exp(x))/x, x, -oo) == 0 # 267a + assert limit(log(1 + exp(x))/x, x, oo) == 1 # 267b + + +def test_bounded(): + assert limit(sin(x)/x, x, oo) == 0 # 216b + assert limit(x*sin(1/x), x, 0) == 0 # 227a + + +def test_f1a(): + #issue 3508: + assert limit((sin(2*x)/x)**(1 + x), x, 0) == 2 # Primer 7 + + +def test_f1a2(): + #issue 3509: + assert limit(((x - 1)/(x + 1))**x, x, oo) == exp(-2) # Primer 9 + + +def test_f1b(): + m = Symbol("m") + n = Symbol("n") + h = Symbol("h") + a = Symbol("a") + assert limit(sin(x)/x, x, 2) == sin(2)/2 # 216a + assert limit(sin(3*x)/x, x, 0) == 3 # 217 + assert limit(sin(5*x)/sin(2*x), x, 0) == Rational(5, 2) # 218 + assert limit(sin(pi*x)/sin(3*pi*x), x, 0) == Rational(1, 3) # 219 + assert limit(x*sin(pi/x), x, oo) == pi # 220 + assert limit((1 - cos(x))/x**2, x, 0) == S.Half # 221 + assert limit(x*sin(1/x), x, oo) == 1 # 227b + assert limit((cos(m*x) - cos(n*x))/x**2, x, 0) == -m**2/2 + n**2/2 # 232 + assert limit((tan(x) - sin(x))/x**3, x, 0) == S.Half # 233 + assert limit((x - sin(2*x))/(x + sin(3*x)), x, 0) == -Rational(1, 4) # 237 + assert limit((1 - sqrt(cos(x)))/x**2, x, 0) == Rational(1, 4) # 239 + assert limit((sqrt(1 + sin(x)) - sqrt(1 - sin(x)))/x, x, 0) == 1 # 240 + + assert limit((1 + h/x)**x, x, oo) == exp(h) # Primer 9 + assert limit((sin(x) - sin(a))/(x - a), x, a) == cos(a) # 222, *176 + assert limit((cos(x) - cos(a))/(x - a), x, a) == -sin(a) # 223 + assert limit((sin(x + h) - sin(x))/h, h, 0) == cos(x) # 225 + + +def test_f2a(): + assert limit(((x + 1)/(2*x + 1))**(x**2), x, oo) == 0 # Primer 8 + + +def test_f2(): + assert limit((sqrt( + cos(x)) - root3(cos(x)))/(sin(x)**2), x, 0) == -Rational(1, 12) # *184 + + +def test_f3(): + a = Symbol('a') + #issue 3504 + assert limit(asin(a*x)/x, x, 0) == a diff --git a/.venv/lib/python3.12/site-packages/sympy/series/tests/test_formal.py b/.venv/lib/python3.12/site-packages/sympy/series/tests/test_formal.py new file mode 100644 index 0000000000000000000000000000000000000000..cac60b12534152a5783bb8f0faab2c06da6691fb --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/tests/test_formal.py @@ -0,0 +1,618 @@ +from sympy.concrete.summations import Sum +from sympy.core.add import Add +from sympy.core.function import (Derivative, Function) +from sympy.core.mul import Mul +from sympy.core.numbers import (I, Rational, oo, pi) +from sympy.core.singleton import S +from sympy.core.symbol import symbols +from sympy.functions.combinatorial.factorials import factorial +from sympy.functions.elementary.exponential import (exp, log) +from sympy.functions.elementary.hyperbolic import (acosh, asech) +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import (acos, asin, atan, cos, sin) +from sympy.functions.special.bessel import airyai +from sympy.functions.special.error_functions import erf +from sympy.functions.special.gamma_functions import gamma +from sympy.integrals.integrals import integrate +from sympy.series.formal import fps +from sympy.series.order import O +from sympy.series.formal import (rational_algorithm, FormalPowerSeries, + FormalPowerSeriesProduct, FormalPowerSeriesCompose, + FormalPowerSeriesInverse, simpleDE, + rational_independent, exp_re, hyper_re) +from sympy.testing.pytest import raises, XFAIL, slow + +x, y, z = symbols('x y z') +n, m, k = symbols('n m k', integer=True) +f, r = Function('f'), Function('r') + + +def test_rational_algorithm(): + f = 1 / ((x - 1)**2 * (x - 2)) + assert rational_algorithm(f, x, k) == \ + (-2**(-k - 1) + 1 - (factorial(k + 1) / factorial(k)), 0, 0) + + f = (1 + x + x**2 + x**3) / ((x - 1) * (x - 2)) + assert rational_algorithm(f, x, k) == \ + (-15*2**(-k - 1) + 4, x + 4, 0) + + f = z / (y*m - m*x - y*x + x**2) + assert rational_algorithm(f, x, k) == \ + (((-y**(-k - 1)*z) / (y - m)) + ((m**(-k - 1)*z) / (y - m)), 0, 0) + + f = x / (1 - x - x**2) + assert rational_algorithm(f, x, k) is None + assert rational_algorithm(f, x, k, full=True) == \ + (((Rational(-1, 2) + sqrt(5)/2)**(-k - 1) * + (-sqrt(5)/10 + S.Half)) + + ((-sqrt(5)/2 - S.Half)**(-k - 1) * + (sqrt(5)/10 + S.Half)), 0, 0) + + f = 1 / (x**2 + 2*x + 2) + assert rational_algorithm(f, x, k) is None + assert rational_algorithm(f, x, k, full=True) == \ + ((I*(-1 + I)**(-k - 1)) / 2 - (I*(-1 - I)**(-k - 1)) / 2, 0, 0) + + f = log(1 + x) + assert rational_algorithm(f, x, k) == \ + (-(-1)**(-k) / k, 0, 1) + + f = atan(x) + assert rational_algorithm(f, x, k) is None + assert rational_algorithm(f, x, k, full=True) == \ + (((I*I**(-k)) / 2 - (I*(-I)**(-k)) / 2) / k, 0, 1) + + f = x*atan(x) - log(1 + x**2) / 2 + assert rational_algorithm(f, x, k) is None + assert rational_algorithm(f, x, k, full=True) == \ + (((I*I**(-k + 1)) / 2 - (I*(-I)**(-k + 1)) / 2) / + (k*(k - 1)), 0, 2) + + f = log((1 + x) / (1 - x)) / 2 - atan(x) + assert rational_algorithm(f, x, k) is None + assert rational_algorithm(f, x, k, full=True) == \ + ((-(-1)**(-k) / 2 - (I*I**(-k)) / 2 + (I*(-I)**(-k)) / 2 + + S.Half) / k, 0, 1) + + assert rational_algorithm(cos(x), x, k) is None + + +def test_rational_independent(): + ri = rational_independent + assert ri([], x) == [] + assert ri([cos(x), sin(x)], x) == [cos(x), sin(x)] + assert ri([x**2, sin(x), x*sin(x), x**3], x) == \ + [x**3 + x**2, x*sin(x) + sin(x)] + assert ri([S.One, x*log(x), log(x), sin(x)/x, cos(x), sin(x), x], x) == \ + [x + 1, x*log(x) + log(x), sin(x)/x + sin(x), cos(x)] + + +def test_simpleDE(): + # Tests just the first valid DE + for DE in simpleDE(exp(x), x, f): + assert DE == (-f(x) + Derivative(f(x), x), 1) + break + for DE in simpleDE(sin(x), x, f): + assert DE == (f(x) + Derivative(f(x), x, x), 2) + break + for DE in simpleDE(log(1 + x), x, f): + assert DE == ((x + 1)*Derivative(f(x), x, 2) + Derivative(f(x), x), 2) + break + for DE in simpleDE(asin(x), x, f): + assert DE == (x*Derivative(f(x), x) + (x**2 - 1)*Derivative(f(x), x, x), + 2) + break + for DE in simpleDE(exp(x)*sin(x), x, f): + assert DE == (2*f(x) - 2*Derivative(f(x)) + Derivative(f(x), x, x), 2) + break + for DE in simpleDE(((1 + x)/(1 - x))**n, x, f): + assert DE == (2*n*f(x) + (x**2 - 1)*Derivative(f(x), x), 1) + break + for DE in simpleDE(airyai(x), x, f): + assert DE == (-x*f(x) + Derivative(f(x), x, x), 2) + break + + +def test_exp_re(): + d = -f(x) + Derivative(f(x), x) + assert exp_re(d, r, k) == -r(k) + r(k + 1) + + d = f(x) + Derivative(f(x), x, x) + assert exp_re(d, r, k) == r(k) + r(k + 2) + + d = f(x) + Derivative(f(x), x) + Derivative(f(x), x, x) + assert exp_re(d, r, k) == r(k) + r(k + 1) + r(k + 2) + + d = Derivative(f(x), x) + Derivative(f(x), x, x) + assert exp_re(d, r, k) == r(k) + r(k + 1) + + d = Derivative(f(x), x, 3) + Derivative(f(x), x, 4) + Derivative(f(x)) + assert exp_re(d, r, k) == r(k) + r(k + 2) + r(k + 3) + + +def test_hyper_re(): + d = f(x) + Derivative(f(x), x, x) + assert hyper_re(d, r, k) == r(k) + (k+1)*(k+2)*r(k + 2) + + d = -x*f(x) + Derivative(f(x), x, x) + assert hyper_re(d, r, k) == (k + 2)*(k + 3)*r(k + 3) - r(k) + + d = 2*f(x) - 2*Derivative(f(x), x) + Derivative(f(x), x, x) + assert hyper_re(d, r, k) == \ + (-2*k - 2)*r(k + 1) + (k + 1)*(k + 2)*r(k + 2) + 2*r(k) + + d = 2*n*f(x) + (x**2 - 1)*Derivative(f(x), x) + assert hyper_re(d, r, k) == \ + k*r(k) + 2*n*r(k + 1) + (-k - 2)*r(k + 2) + + d = (x**10 + 4)*Derivative(f(x), x) + x*(x**10 - 1)*Derivative(f(x), x, x) + assert hyper_re(d, r, k) == \ + (k*(k - 1) + k)*r(k) + (4*k - (k + 9)*(k + 10) + 40)*r(k + 10) + + d = ((x**2 - 1)*Derivative(f(x), x, 3) + 3*x*Derivative(f(x), x, x) + + Derivative(f(x), x)) + assert hyper_re(d, r, k) == \ + ((k*(k - 2)*(k - 1) + 3*k*(k - 1) + k)*r(k) + + (-k*(k + 1)*(k + 2))*r(k + 2)) + + +def test_fps(): + assert fps(1) == 1 + assert fps(2, x) == 2 + assert fps(2, x, dir='+') == 2 + assert fps(2, x, dir='-') == 2 + assert fps(1/x + 1/x**2) == 1/x + 1/x**2 + assert fps(log(1 + x), hyper=False, rational=False) == log(1 + x) + + f = fps(x**2 + x + 1) + assert isinstance(f, FormalPowerSeries) + assert f.function == x**2 + x + 1 + assert f[0] == 1 + assert f[2] == x**2 + assert f.truncate(4) == x**2 + x + 1 + O(x**4) + assert f.polynomial() == x**2 + x + 1 + + f = fps(log(1 + x)) + assert isinstance(f, FormalPowerSeries) + assert f.function == log(1 + x) + assert f.subs(x, y) == f + assert f[:5] == [0, x, -x**2/2, x**3/3, -x**4/4] + assert f.as_leading_term(x) == x + assert f.polynomial(6) == x - x**2/2 + x**3/3 - x**4/4 + x**5/5 + + k = f.ak.variables[0] + assert f.infinite == Sum((-(-1)**(-k)*x**k)/k, (k, 1, oo)) + + ft, s = f.truncate(n=None), f[:5] + for i, t in enumerate(ft): + if i == 5: + break + assert s[i] == t + + f = sin(x).fps(x) + assert isinstance(f, FormalPowerSeries) + assert f.truncate() == x - x**3/6 + x**5/120 + O(x**6) + + raises(NotImplementedError, lambda: fps(y*x)) + raises(ValueError, lambda: fps(x, dir=0)) + + +@slow +def test_fps__rational(): + assert fps(1/x) == (1/x) + assert fps((x**2 + x + 1) / x**3, dir=-1) == (x**2 + x + 1) / x**3 + + f = 1 / ((x - 1)**2 * (x - 2)) + assert fps(f, x).truncate() == \ + (Rational(-1, 2) - x*Rational(5, 4) - 17*x**2/8 - 49*x**3/16 - 129*x**4/32 - + 321*x**5/64 + O(x**6)) + + f = (1 + x + x**2 + x**3) / ((x - 1) * (x - 2)) + assert fps(f, x).truncate() == \ + (S.Half + x*Rational(5, 4) + 17*x**2/8 + 49*x**3/16 + 113*x**4/32 + + 241*x**5/64 + O(x**6)) + + f = x / (1 - x - x**2) + assert fps(f, x, full=True).truncate() == \ + x + x**2 + 2*x**3 + 3*x**4 + 5*x**5 + O(x**6) + + f = 1 / (x**2 + 2*x + 2) + assert fps(f, x, full=True).truncate() == \ + S.Half - x/2 + x**2/4 - x**4/8 + x**5/8 + O(x**6) + + f = log(1 + x) + assert fps(f, x).truncate() == \ + x - x**2/2 + x**3/3 - x**4/4 + x**5/5 + O(x**6) + assert fps(f, x, dir=1).truncate() == fps(f, x, dir=-1).truncate() + assert fps(f, x, 2).truncate() == \ + (log(3) - Rational(2, 3) - (x - 2)**2/18 + (x - 2)**3/81 - + (x - 2)**4/324 + (x - 2)**5/1215 + x/3 + O((x - 2)**6, (x, 2))) + assert fps(f, x, 2, dir=-1).truncate() == \ + (log(3) - Rational(2, 3) - (-x + 2)**2/18 - (-x + 2)**3/81 - + (-x + 2)**4/324 - (-x + 2)**5/1215 + x/3 + O((x - 2)**6, (x, 2))) + + f = atan(x) + assert fps(f, x, full=True).truncate() == x - x**3/3 + x**5/5 + O(x**6) + assert fps(f, x, full=True, dir=1).truncate() == \ + fps(f, x, full=True, dir=-1).truncate() + assert fps(f, x, 2, full=True).truncate() == \ + (atan(2) - Rational(2, 5) - 2*(x - 2)**2/25 + 11*(x - 2)**3/375 - + 6*(x - 2)**4/625 + 41*(x - 2)**5/15625 + x/5 + O((x - 2)**6, (x, 2))) + assert fps(f, x, 2, full=True, dir=-1).truncate() == \ + (atan(2) - Rational(2, 5) - 2*(-x + 2)**2/25 - 11*(-x + 2)**3/375 - + 6*(-x + 2)**4/625 - 41*(-x + 2)**5/15625 + x/5 + O((x - 2)**6, (x, 2))) + + f = x*atan(x) - log(1 + x**2) / 2 + assert fps(f, x, full=True).truncate() == x**2/2 - x**4/12 + O(x**6) + + f = log((1 + x) / (1 - x)) / 2 - atan(x) + assert fps(f, x, full=True).truncate(n=10) == 2*x**3/3 + 2*x**7/7 + O(x**10) + + +@slow +def test_fps__hyper(): + f = sin(x) + assert fps(f, x).truncate() == x - x**3/6 + x**5/120 + O(x**6) + + f = cos(x) + assert fps(f, x).truncate() == 1 - x**2/2 + x**4/24 + O(x**6) + + f = exp(x) + assert fps(f, x).truncate() == \ + 1 + x + x**2/2 + x**3/6 + x**4/24 + x**5/120 + O(x**6) + + f = atan(x) + assert fps(f, x).truncate() == x - x**3/3 + x**5/5 + O(x**6) + + f = exp(acos(x)) + assert fps(f, x).truncate() == \ + (exp(pi/2) - x*exp(pi/2) + x**2*exp(pi/2)/2 - x**3*exp(pi/2)/3 + + 5*x**4*exp(pi/2)/24 - x**5*exp(pi/2)/6 + O(x**6)) + + f = exp(acosh(x)) + assert fps(f, x).truncate() == I + x - I*x**2/2 - I*x**4/8 + O(x**6) + + f = atan(1/x) + assert fps(f, x).truncate() == pi/2 - x + x**3/3 - x**5/5 + O(x**6) + + f = x*atan(x) - log(1 + x**2) / 2 + assert fps(f, x, rational=False).truncate() == x**2/2 - x**4/12 + O(x**6) + + f = log(1 + x) + assert fps(f, x, rational=False).truncate() == \ + x - x**2/2 + x**3/3 - x**4/4 + x**5/5 + O(x**6) + + f = airyai(x**2) + assert fps(f, x).truncate() == \ + (3**Rational(5, 6)*gamma(Rational(1, 3))/(6*pi) - + 3**Rational(2, 3)*x**2/(3*gamma(Rational(1, 3))) + O(x**6)) + + f = exp(x)*sin(x) + assert fps(f, x).truncate() == x + x**2 + x**3/3 - x**5/30 + O(x**6) + + f = exp(x)*sin(x)/x + assert fps(f, x).truncate() == 1 + x + x**2/3 - x**4/30 - x**5/90 + O(x**6) + + f = sin(x) * cos(x) + assert fps(f, x).truncate() == x - 2*x**3/3 + 2*x**5/15 + O(x**6) + + +def test_fps_shift(): + f = x**-5*sin(x) + assert fps(f, x).truncate() == \ + 1/x**4 - 1/(6*x**2) + Rational(1, 120) - x**2/5040 + x**4/362880 + O(x**6) + + f = x**2*atan(x) + assert fps(f, x, rational=False).truncate() == \ + x**3 - x**5/3 + O(x**6) + + f = cos(sqrt(x))*x + assert fps(f, x).truncate() == \ + x - x**2/2 + x**3/24 - x**4/720 + x**5/40320 + O(x**6) + + f = x**2*cos(sqrt(x)) + assert fps(f, x).truncate() == \ + x**2 - x**3/2 + x**4/24 - x**5/720 + O(x**6) + + +def test_fps__Add_expr(): + f = x*atan(x) - log(1 + x**2) / 2 + assert fps(f, x).truncate() == x**2/2 - x**4/12 + O(x**6) + + f = sin(x) + cos(x) - exp(x) + log(1 + x) + assert fps(f, x).truncate() == x - 3*x**2/2 - x**4/4 + x**5/5 + O(x**6) + + f = 1/x + sin(x) + assert fps(f, x).truncate() == 1/x + x - x**3/6 + x**5/120 + O(x**6) + + f = sin(x) - cos(x) + 1/(x - 1) + assert fps(f, x).truncate() == \ + -2 - x**2/2 - 7*x**3/6 - 25*x**4/24 - 119*x**5/120 + O(x**6) + + +def test_fps__asymptotic(): + f = exp(x) + assert fps(f, x, oo) == f + assert fps(f, x, -oo).truncate() == O(1/x**6, (x, oo)) + + f = erf(x) + assert fps(f, x, oo).truncate() == 1 + O(1/x**6, (x, oo)) + assert fps(f, x, -oo).truncate() == -1 + O(1/x**6, (x, oo)) + + f = atan(x) + assert fps(f, x, oo, full=True).truncate() == \ + -1/(5*x**5) + 1/(3*x**3) - 1/x + pi/2 + O(1/x**6, (x, oo)) + assert fps(f, x, -oo, full=True).truncate() == \ + -1/(5*x**5) + 1/(3*x**3) - 1/x - pi/2 + O(1/x**6, (x, oo)) + + f = log(1 + x) + assert fps(f, x, oo) != \ + (-1/(5*x**5) - 1/(4*x**4) + 1/(3*x**3) - 1/(2*x**2) + 1/x - log(1/x) + + O(1/x**6, (x, oo))) + assert fps(f, x, -oo) != \ + (-1/(5*x**5) - 1/(4*x**4) + 1/(3*x**3) - 1/(2*x**2) + 1/x + I*pi - + log(-1/x) + O(1/x**6, (x, oo))) + + +def test_fps__fractional(): + f = sin(sqrt(x)) / x + assert fps(f, x).truncate() == \ + (1/sqrt(x) - sqrt(x)/6 + x**Rational(3, 2)/120 - + x**Rational(5, 2)/5040 + x**Rational(7, 2)/362880 - + x**Rational(9, 2)/39916800 + x**Rational(11, 2)/6227020800 + O(x**6)) + + f = sin(sqrt(x)) * x + assert fps(f, x).truncate() == \ + (x**Rational(3, 2) - x**Rational(5, 2)/6 + x**Rational(7, 2)/120 - + x**Rational(9, 2)/5040 + x**Rational(11, 2)/362880 + O(x**6)) + + f = atan(sqrt(x)) / x**2 + assert fps(f, x).truncate() == \ + (x**Rational(-3, 2) - x**Rational(-1, 2)/3 + x**S.Half/5 - + x**Rational(3, 2)/7 + x**Rational(5, 2)/9 - x**Rational(7, 2)/11 + + x**Rational(9, 2)/13 - x**Rational(11, 2)/15 + O(x**6)) + + f = exp(sqrt(x)) + assert fps(f, x).truncate().expand() == \ + (1 + x/2 + x**2/24 + x**3/720 + x**4/40320 + x**5/3628800 + sqrt(x) + + x**Rational(3, 2)/6 + x**Rational(5, 2)/120 + x**Rational(7, 2)/5040 + + x**Rational(9, 2)/362880 + x**Rational(11, 2)/39916800 + O(x**6)) + + f = exp(sqrt(x))*x + assert fps(f, x).truncate().expand() == \ + (x + x**2/2 + x**3/24 + x**4/720 + x**5/40320 + x**Rational(3, 2) + + x**Rational(5, 2)/6 + x**Rational(7, 2)/120 + x**Rational(9, 2)/5040 + + x**Rational(11, 2)/362880 + O(x**6)) + + +def test_fps__logarithmic_singularity(): + f = log(1 + 1/x) + assert fps(f, x) != \ + -log(x) + x - x**2/2 + x**3/3 - x**4/4 + x**5/5 + O(x**6) + assert fps(f, x, rational=False) != \ + -log(x) + x - x**2/2 + x**3/3 - x**4/4 + x**5/5 + O(x**6) + + +@XFAIL +def test_fps__logarithmic_singularity_fail(): + f = asech(x) # Algorithms for computing limits probably needs improvements + assert fps(f, x) == log(2) - log(x) - x**2/4 - 3*x**4/64 + O(x**6) + + +def test_fps_symbolic(): + f = x**n*sin(x**2) + assert fps(f, x).truncate(8) == x**(n + 2) - x**(n + 6)/6 + O(x**(n + 8), x) + + f = x**n*log(1 + x) + fp = fps(f, x) + k = fp.ak.variables[0] + assert fp.infinite == \ + Sum((-(-1)**(-k)*x**(k + n))/k, (k, 1, oo)) + + f = (x - 2)**n*log(1 + x) + assert fps(f, x, 2).truncate() == \ + ((x - 2)**n*log(3) + (x - 2)**(n + 1)/3 - (x - 2)**(n + 2)/18 + (x - 2)**(n + 3)/81 - + (x - 2)**(n + 4)/324 + (x - 2)**(n + 5)/1215 + O((x - 2)**(n + 6), (x, 2))) + + f = x**(n - 2)*cos(x) + assert fps(f, x).truncate() == \ + (x**(n - 2) - x**n/2 + x**(n + 2)/24 + O(x**(n + 4), x)) + + f = x**(n - 2)*sin(x) + x**n*exp(x) + assert fps(f, x).truncate() == \ + (x**(n - 1) + x**(n + 1) + x**(n + 2)/2 + x**n + + x**(n + 4)/24 + x**(n + 5)/60 + O(x**(n + 6), x)) + + f = x**n*atan(x) + assert fps(f, x, oo).truncate() == \ + (-x**(n - 5)/5 + x**(n - 3)/3 + x**n*(pi/2 - 1/x) + + O((1/x)**(-n)/x**6, (x, oo))) + + f = x**(n/2)*cos(x) + assert fps(f, x).truncate() == \ + x**(n/2) - x**(n/2 + 2)/2 + x**(n/2 + 4)/24 + O(x**(n/2 + 6), x) + + f = x**(n + m)*sin(x) + assert fps(f, x).truncate() == \ + x**(m + n + 1) - x**(m + n + 3)/6 + x**(m + n + 5)/120 + O(x**(m + n + 6), x) + + +def test_fps__slow(): + f = x*exp(x)*sin(2*x) # TODO: rsolve needs improvement + assert fps(f, x).truncate() == 2*x**2 + 2*x**3 - x**4/3 - x**5 + O(x**6) + + +def test_fps__operations(): + f1, f2 = fps(sin(x)), fps(cos(x)) + + fsum = f1 + f2 + assert fsum.function == sin(x) + cos(x) + assert fsum.truncate() == \ + 1 + x - x**2/2 - x**3/6 + x**4/24 + x**5/120 + O(x**6) + + fsum = f1 + 1 + assert fsum.function == sin(x) + 1 + assert fsum.truncate() == 1 + x - x**3/6 + x**5/120 + O(x**6) + + fsum = 1 + f2 + assert fsum.function == cos(x) + 1 + assert fsum.truncate() == 2 - x**2/2 + x**4/24 + O(x**6) + + assert (f1 + x) == Add(f1, x) + + assert -f2.truncate() == -1 + x**2/2 - x**4/24 + O(x**6) + assert (f1 - f1) is S.Zero + + fsub = f1 - f2 + assert fsub.function == sin(x) - cos(x) + assert fsub.truncate() == \ + -1 + x + x**2/2 - x**3/6 - x**4/24 + x**5/120 + O(x**6) + + fsub = f1 - 1 + assert fsub.function == sin(x) - 1 + assert fsub.truncate() == -1 + x - x**3/6 + x**5/120 + O(x**6) + + fsub = 1 - f2 + assert fsub.function == -cos(x) + 1 + assert fsub.truncate() == x**2/2 - x**4/24 + O(x**6) + + raises(ValueError, lambda: f1 + fps(exp(x), dir=-1)) + raises(ValueError, lambda: f1 + fps(exp(x), x0=1)) + + fm = f1 * 3 + + assert fm.function == 3*sin(x) + assert fm.truncate() == 3*x - x**3/2 + x**5/40 + O(x**6) + + fm = 3 * f2 + + assert fm.function == 3*cos(x) + assert fm.truncate() == 3 - 3*x**2/2 + x**4/8 + O(x**6) + + assert (f1 * f2) == Mul(f1, f2) + assert (f1 * x) == Mul(f1, x) + + fd = f1.diff() + assert fd.function == cos(x) + assert fd.truncate() == 1 - x**2/2 + x**4/24 + O(x**6) + + fd = f2.diff() + assert fd.function == -sin(x) + assert fd.truncate() == -x + x**3/6 - x**5/120 + O(x**6) + + fd = f2.diff().diff() + assert fd.function == -cos(x) + assert fd.truncate() == -1 + x**2/2 - x**4/24 + O(x**6) + + f3 = fps(exp(sqrt(x))) + fd = f3.diff() + assert fd.truncate().expand() == \ + (1/(2*sqrt(x)) + S.Half + x/12 + x**2/240 + x**3/10080 + x**4/725760 + + x**5/79833600 + sqrt(x)/4 + x**Rational(3, 2)/48 + x**Rational(5, 2)/1440 + + x**Rational(7, 2)/80640 + x**Rational(9, 2)/7257600 + x**Rational(11, 2)/958003200 + + O(x**6)) + + assert f1.integrate((x, 0, 1)) == -cos(1) + 1 + assert integrate(f1, (x, 0, 1)) == -cos(1) + 1 + + fi = integrate(f1, x) + assert fi.function == -cos(x) + assert fi.truncate() == -1 + x**2/2 - x**4/24 + O(x**6) + + fi = f2.integrate(x) + assert fi.function == sin(x) + assert fi.truncate() == x - x**3/6 + x**5/120 + O(x**6) + +def test_fps__product(): + f1, f2, f3 = fps(sin(x)), fps(exp(x)), fps(cos(x)) + + raises(ValueError, lambda: f1.product(exp(x), x)) + raises(ValueError, lambda: f1.product(fps(exp(x), dir=-1), x, 4)) + raises(ValueError, lambda: f1.product(fps(exp(x), x0=1), x, 4)) + raises(ValueError, lambda: f1.product(fps(exp(y)), x, 4)) + + fprod = f1.product(f2, x) + assert isinstance(fprod, FormalPowerSeriesProduct) + assert isinstance(fprod.ffps, FormalPowerSeries) + assert isinstance(fprod.gfps, FormalPowerSeries) + assert fprod.f == sin(x) + assert fprod.g == exp(x) + assert fprod.function == sin(x) * exp(x) + assert fprod._eval_terms(4) == x + x**2 + x**3/3 + assert fprod.truncate(4) == x + x**2 + x**3/3 + O(x**4) + assert fprod.polynomial(4) == x + x**2 + x**3/3 + + raises(NotImplementedError, lambda: fprod._eval_term(5)) + raises(NotImplementedError, lambda: fprod.infinite) + raises(NotImplementedError, lambda: fprod._eval_derivative(x)) + raises(NotImplementedError, lambda: fprod.integrate(x)) + + assert f1.product(f3, x)._eval_terms(4) == x - 2*x**3/3 + assert f1.product(f3, x).truncate(4) == x - 2*x**3/3 + O(x**4) + + +def test_fps__compose(): + f1, f2, f3 = fps(exp(x)), fps(sin(x)), fps(cos(x)) + + raises(ValueError, lambda: f1.compose(sin(x), x)) + raises(ValueError, lambda: f1.compose(fps(sin(x), dir=-1), x, 4)) + raises(ValueError, lambda: f1.compose(fps(sin(x), x0=1), x, 4)) + raises(ValueError, lambda: f1.compose(fps(sin(y)), x, 4)) + + raises(ValueError, lambda: f1.compose(f3, x)) + raises(ValueError, lambda: f2.compose(f3, x)) + + fcomp = f1.compose(f2, x) + assert isinstance(fcomp, FormalPowerSeriesCompose) + assert isinstance(fcomp.ffps, FormalPowerSeries) + assert isinstance(fcomp.gfps, FormalPowerSeries) + assert fcomp.f == exp(x) + assert fcomp.g == sin(x) + assert fcomp.function == exp(sin(x)) + assert fcomp._eval_terms(6) == 1 + x + x**2/2 - x**4/8 - x**5/15 + assert fcomp.truncate() == 1 + x + x**2/2 - x**4/8 - x**5/15 + O(x**6) + assert fcomp.truncate(5) == 1 + x + x**2/2 - x**4/8 + O(x**5) + + raises(NotImplementedError, lambda: fcomp._eval_term(5)) + raises(NotImplementedError, lambda: fcomp.infinite) + raises(NotImplementedError, lambda: fcomp._eval_derivative(x)) + raises(NotImplementedError, lambda: fcomp.integrate(x)) + + assert f1.compose(f2, x).truncate(4) == 1 + x + x**2/2 + O(x**4) + assert f1.compose(f2, x).truncate(8) == \ + 1 + x + x**2/2 - x**4/8 - x**5/15 - x**6/240 + x**7/90 + O(x**8) + assert f1.compose(f2, x).truncate(6) == \ + 1 + x + x**2/2 - x**4/8 - x**5/15 + O(x**6) + + assert f2.compose(f2, x).truncate(4) == x - x**3/3 + O(x**4) + assert f2.compose(f2, x).truncate(8) == x - x**3/3 + x**5/10 - 8*x**7/315 + O(x**8) + assert f2.compose(f2, x).truncate(6) == x - x**3/3 + x**5/10 + O(x**6) + + +def test_fps__inverse(): + f1, f2, f3 = fps(sin(x)), fps(exp(x)), fps(cos(x)) + + raises(ValueError, lambda: f1.inverse(x)) + + finv = f2.inverse(x) + assert isinstance(finv, FormalPowerSeriesInverse) + assert isinstance(finv.ffps, FormalPowerSeries) + raises(ValueError, lambda: finv.gfps) + + assert finv.f == exp(x) + assert finv.function == exp(-x) + assert finv._eval_terms(5) == 1 - x + x**2/2 - x**3/6 + x**4/24 + assert finv.truncate() == 1 - x + x**2/2 - x**3/6 + x**4/24 - x**5/120 + O(x**6) + assert finv.truncate(5) == 1 - x + x**2/2 - x**3/6 + x**4/24 + O(x**5) + + raises(NotImplementedError, lambda: finv._eval_term(5)) + raises(ValueError, lambda: finv.g) + raises(NotImplementedError, lambda: finv.infinite) + raises(NotImplementedError, lambda: finv._eval_derivative(x)) + raises(NotImplementedError, lambda: finv.integrate(x)) + + assert f2.inverse(x).truncate(8) == \ + 1 - x + x**2/2 - x**3/6 + x**4/24 - x**5/120 + x**6/720 - x**7/5040 + O(x**8) + + assert f3.inverse(x).truncate() == 1 + x**2/2 + 5*x**4/24 + O(x**6) + assert f3.inverse(x).truncate(8) == 1 + x**2/2 + 5*x**4/24 + 61*x**6/720 + O(x**8) diff --git a/.venv/lib/python3.12/site-packages/sympy/series/tests/test_fourier.py b/.venv/lib/python3.12/site-packages/sympy/series/tests/test_fourier.py new file mode 100644 index 0000000000000000000000000000000000000000..994f182088b09b038e0e1b3885fec1c27f69f2b0 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/tests/test_fourier.py @@ -0,0 +1,165 @@ +from sympy.core.add import Add +from sympy.core.numbers import (Rational, oo, pi) +from sympy.core.singleton import S +from sympy.core.symbol import symbols +from sympy.functions.elementary.exponential import (exp, log) +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.elementary.trigonometric import (cos, sin, sinc, tan) +from sympy.series.fourier import fourier_series +from sympy.series.fourier import FourierSeries +from sympy.testing.pytest import raises +from functools import lru_cache + +x, y, z = symbols('x y z') + +# Don't declare these during import because they are slow +@lru_cache() +def _get_examples(): + fo = fourier_series(x, (x, -pi, pi)) + fe = fourier_series(x**2, (-pi, pi)) + fp = fourier_series(Piecewise((0, x < 0), (pi, True)), (x, -pi, pi)) + return fo, fe, fp + + +def test_FourierSeries(): + fo, fe, fp = _get_examples() + + assert fourier_series(1, (-pi, pi)) == 1 + assert (Piecewise((0, x < 0), (pi, True)). + fourier_series((x, -pi, pi)).truncate()) == fp.truncate() + assert isinstance(fo, FourierSeries) + assert fo.function == x + assert fo.x == x + assert fo.period == (-pi, pi) + + assert fo.term(3) == 2*sin(3*x) / 3 + assert fe.term(3) == -4*cos(3*x) / 9 + assert fp.term(3) == 2*sin(3*x) / 3 + + assert fo.as_leading_term(x) == 2*sin(x) + assert fe.as_leading_term(x) == pi**2 / 3 + assert fp.as_leading_term(x) == pi / 2 + + assert fo.truncate() == 2*sin(x) - sin(2*x) + (2*sin(3*x) / 3) + assert fe.truncate() == -4*cos(x) + cos(2*x) + pi**2 / 3 + assert fp.truncate() == 2*sin(x) + (2*sin(3*x) / 3) + pi / 2 + + fot = fo.truncate(n=None) + s = [0, 2*sin(x), -sin(2*x)] + for i, t in enumerate(fot): + if i == 3: + break + assert s[i] == t + + def _check_iter(f, i): + for ind, t in enumerate(f): + assert t == f[ind] # noqa: PLR1736 + if ind == i: + break + + _check_iter(fo, 3) + _check_iter(fe, 3) + _check_iter(fp, 3) + + assert fo.subs(x, x**2) == fo + + raises(ValueError, lambda: fourier_series(x, (0, 1, 2))) + raises(ValueError, lambda: fourier_series(x, (x, 0, oo))) + raises(ValueError, lambda: fourier_series(x*y, (0, oo))) + + +def test_FourierSeries_2(): + p = Piecewise((0, x < 0), (x, True)) + f = fourier_series(p, (x, -2, 2)) + + assert f.term(3) == (2*sin(3*pi*x / 2) / (3*pi) - + 4*cos(3*pi*x / 2) / (9*pi**2)) + assert f.truncate() == (2*sin(pi*x / 2) / pi - sin(pi*x) / pi - + 4*cos(pi*x / 2) / pi**2 + S.Half) + + +def test_square_wave(): + """Test if fourier_series approximates discontinuous function correctly.""" + square_wave = Piecewise((1, x < pi), (-1, True)) + s = fourier_series(square_wave, (x, 0, 2*pi)) + + assert s.truncate(3) == 4 / pi * sin(x) + 4 / (3 * pi) * sin(3 * x) + \ + 4 / (5 * pi) * sin(5 * x) + assert s.sigma_approximation(4) == 4 / pi * sin(x) * sinc(pi / 4) + \ + 4 / (3 * pi) * sin(3 * x) * sinc(3 * pi / 4) + + +def test_sawtooth_wave(): + s = fourier_series(x, (x, 0, pi)) + assert s.truncate(4) == \ + pi/2 - sin(2*x) - sin(4*x)/2 - sin(6*x)/3 + s = fourier_series(x, (x, 0, 1)) + assert s.truncate(4) == \ + S.Half - sin(2*pi*x)/pi - sin(4*pi*x)/(2*pi) - sin(6*pi*x)/(3*pi) + + +def test_FourierSeries__operations(): + fo, fe, fp = _get_examples() + + fes = fe.scale(-1).shift(pi**2) + assert fes.truncate() == 4*cos(x) - cos(2*x) + 2*pi**2 / 3 + + assert fp.shift(-pi/2).truncate() == (2*sin(x) + (2*sin(3*x) / 3) + + (2*sin(5*x) / 5)) + + fos = fo.scale(3) + assert fos.truncate() == 6*sin(x) - 3*sin(2*x) + 2*sin(3*x) + + fx = fe.scalex(2).shiftx(1) + assert fx.truncate() == -4*cos(2*x + 2) + cos(4*x + 4) + pi**2 / 3 + + fl = fe.scalex(3).shift(-pi).scalex(2).shiftx(1).scale(4) + assert fl.truncate() == (-16*cos(6*x + 6) + 4*cos(12*x + 12) - + 4*pi + 4*pi**2 / 3) + + raises(ValueError, lambda: fo.shift(x)) + raises(ValueError, lambda: fo.shiftx(sin(x))) + raises(ValueError, lambda: fo.scale(x*y)) + raises(ValueError, lambda: fo.scalex(x**2)) + + +def test_FourierSeries__neg(): + fo, fe, fp = _get_examples() + + assert (-fo).truncate() == -2*sin(x) + sin(2*x) - (2*sin(3*x) / 3) + assert (-fe).truncate() == +4*cos(x) - cos(2*x) - pi**2 / 3 + + +def test_FourierSeries__add__sub(): + fo, fe, fp = _get_examples() + + assert fo + fo == fo.scale(2) + assert fo - fo == 0 + assert -fe - fe == fe.scale(-2) + + assert (fo + fe).truncate() == 2*sin(x) - sin(2*x) - 4*cos(x) + cos(2*x) \ + + pi**2 / 3 + assert (fo - fe).truncate() == 2*sin(x) - sin(2*x) + 4*cos(x) - cos(2*x) \ + - pi**2 / 3 + + assert isinstance(fo + 1, Add) + + raises(ValueError, lambda: fo + fourier_series(x, (x, 0, 2))) + + +def test_FourierSeries_finite(): + + assert fourier_series(sin(x)).truncate(1) == sin(x) + # assert type(fourier_series(sin(x)*log(x))).truncate() == FourierSeries + # assert type(fourier_series(sin(x**2+6))).truncate() == FourierSeries + assert fourier_series(sin(x)*log(y)*exp(z),(x,pi,-pi)).truncate() == sin(x)*log(y)*exp(z) + assert fourier_series(sin(x)**6).truncate(oo) == -15*cos(2*x)/32 + 3*cos(4*x)/16 - cos(6*x)/32 \ + + Rational(5, 16) + assert fourier_series(sin(x) ** 6).truncate() == -15 * cos(2 * x) / 32 + 3 * cos(4 * x) / 16 \ + + Rational(5, 16) + assert fourier_series(sin(4*x+3) + cos(3*x+4)).truncate(oo) == -sin(4)*sin(3*x) + sin(4*x)*cos(3) \ + + cos(4)*cos(3*x) + sin(3)*cos(4*x) + assert fourier_series(sin(x)+cos(x)*tan(x)).truncate(oo) == 2*sin(x) + assert fourier_series(cos(pi*x), (x, -1, 1)).truncate(oo) == cos(pi*x) + assert fourier_series(cos(3*pi*x + 4) - sin(4*pi*x)*log(pi*y), (x, -1, 1)).truncate(oo) == -log(pi*y)*sin(4*pi*x)\ + - sin(4)*sin(3*pi*x) + cos(4)*cos(3*pi*x) diff --git a/.venv/lib/python3.12/site-packages/sympy/series/tests/test_gruntz.py b/.venv/lib/python3.12/site-packages/sympy/series/tests/test_gruntz.py new file mode 100644 index 0000000000000000000000000000000000000000..4cae15297048bc52a69a3d9ca57a7614cfcdc61c --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/tests/test_gruntz.py @@ -0,0 +1,490 @@ +from sympy.core import EulerGamma +from sympy.core.function import Function +from sympy.core.numbers import (E, I, Integer, Rational, oo, pi) +from sympy.core.singleton import S +from sympy.core.symbol import Symbol +from sympy.functions.elementary.exponential import (exp, log) +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import (acot, atan, cos, sin) +from sympy.functions.special.error_functions import (Ei, erf) +from sympy.functions.special.gamma_functions import (digamma, gamma, loggamma) +from sympy.functions.special.zeta_functions import zeta +from sympy.polys.polytools import cancel +from sympy.functions.elementary.hyperbolic import cosh, coth, sinh, tanh +from sympy.series.gruntz import compare, mrv, rewrite, mrv_leadterm, gruntz, \ + sign +from sympy.testing.pytest import XFAIL, raises, skip, slow + +""" +This test suite is testing the limit algorithm using the bottom up approach. +See the documentation in limits2.py. The algorithm itself is highly recursive +by nature, so "compare" is logically the lowest part of the algorithm, yet in +some sense it's the most complex part, because it needs to calculate a limit +to return the result. + +Nevertheless, the rest of the algorithm depends on compare working correctly. +""" + +x = Symbol('x', real=True) +m = Symbol('m', real=True) + + +runslow = False + + +def _sskip(): + if not runslow: + skip("slow") + + +@slow +def test_gruntz_evaluation(): + # Gruntz' thesis pp. 122 to 123 + # 8.1 + assert gruntz(exp(x)*(exp(1/x - exp(-x)) - exp(1/x)), x, oo) == -1 + # 8.2 + assert gruntz(exp(x)*(exp(1/x + exp(-x) + exp(-x**2)) + - exp(1/x - exp(-exp(x)))), x, oo) == 1 + # 8.3 + assert gruntz(exp(exp(x - exp(-x))/(1 - 1/x)) - exp(exp(x)), x, oo) is oo + # 8.5 + assert gruntz(exp(exp(exp(x + exp(-x)))) / exp(exp(exp(x))), x, oo) is oo + # 8.6 + assert gruntz(exp(exp(exp(x))) / exp(exp(exp(x - exp(-exp(x))))), + x, oo) is oo + # 8.7 + assert gruntz(exp(exp(exp(x))) / exp(exp(exp(x - exp(-exp(exp(x)))))), + x, oo) == 1 + # 8.8 + assert gruntz(exp(exp(x)) / exp(exp(x - exp(-exp(exp(x))))), x, oo) == 1 + # 8.9 + assert gruntz(log(x)**2 * exp(sqrt(log(x))*(log(log(x)))**2 + * exp(sqrt(log(log(x))) * (log(log(log(x))))**3)) / sqrt(x), + x, oo) == 0 + # 8.10 + assert gruntz((x*log(x)*(log(x*exp(x) - x**2))**2) + / (log(log(x**2 + 2*exp(exp(3*x**3*log(x)))))), x, oo) == Rational(1, 3) + # 8.11 + assert gruntz((exp(x*exp(-x)/(exp(-x) + exp(-2*x**2/(x + 1)))) - exp(x))/x, + x, oo) == -exp(2) + # 8.12 + assert gruntz((3**x + 5**x)**(1/x), x, oo) == 5 + # 8.13 + assert gruntz(x/log(x**(log(x**(log(2)/log(x))))), x, oo) is oo + # 8.14 + assert gruntz(exp(exp(2*log(x**5 + x)*log(log(x)))) + / exp(exp(10*log(x)*log(log(x)))), x, oo) is oo + # 8.15 + assert gruntz(exp(exp(Rational(5, 2)*x**Rational(-5, 7) + Rational(21, 8)*x**Rational(6, 11) + + 2*x**(-8) + Rational(54, 17)*x**Rational(49, 45)))**8 + / log(log(-log(Rational(4, 3)*x**Rational(-5, 14))))**Rational(7, 6), x, oo) is oo + # 8.16 + assert gruntz((exp(4*x*exp(-x)/(1/exp(x) + 1/exp(2*x**2/(x + 1)))) - exp(x)) + / exp(x)**4, x, oo) == 1 + # 8.17 + assert gruntz(exp(x*exp(-x)/(exp(-x) + exp(-2*x**2/(x + 1))))/exp(x), x, oo) \ + == 1 + # 8.19 + assert gruntz(log(x)*(log(log(x) + log(log(x))) - log(log(x))) + / (log(log(x) + log(log(log(x))))), x, oo) == 1 + # 8.20 + assert gruntz(exp((log(log(x + exp(log(x)*log(log(x)))))) + / (log(log(log(exp(x) + x + log(x)))))), x, oo) == E + # Another + assert gruntz(exp(exp(exp(x + exp(-x)))) / exp(exp(x)), x, oo) is oo + + +def test_gruntz_evaluation_slow(): + _sskip() + # 8.4 + assert gruntz(exp(exp(exp(x)/(1 - 1/x))) + - exp(exp(exp(x)/(1 - 1/x - log(x)**(-log(x))))), x, oo) is -oo + # 8.18 + assert gruntz((exp(exp(-x/(1 + exp(-x))))*exp(-x/(1 + exp(-x/(1 + exp(-x))))) + *exp(exp(-x + exp(-x/(1 + exp(-x)))))) + / (exp(-x/(1 + exp(-x))))**2 - exp(x) + x, x, oo) == 2 + + +@slow +def test_gruntz_eval_special(): + # Gruntz, p. 126 + assert gruntz(exp(x)*(sin(1/x + exp(-x)) - sin(1/x + exp(-x**2))), x, oo) == 1 + assert gruntz((erf(x - exp(-exp(x))) - erf(x)) * exp(exp(x)) * exp(x**2), + x, oo) == -2/sqrt(pi) + assert gruntz(exp(exp(x)) * (exp(sin(1/x + exp(-exp(x)))) - exp(sin(1/x))), + x, oo) == 1 + assert gruntz(exp(x)*(gamma(x + exp(-x)) - gamma(x)), x, oo) is oo + assert gruntz(exp(exp(digamma(digamma(x))))/x, x, oo) == exp(Rational(-1, 2)) + assert gruntz(exp(exp(digamma(log(x))))/x, x, oo) == exp(Rational(-1, 2)) + assert gruntz(digamma(digamma(digamma(x))), x, oo) is oo + assert gruntz(loggamma(loggamma(x)), x, oo) is oo + assert gruntz(((gamma(x + 1/gamma(x)) - gamma(x))/log(x) - cos(1/x)) + * x*log(x), x, oo) == Rational(-1, 2) + assert gruntz(x * (gamma(x - 1/gamma(x)) - gamma(x) + log(x)), x, oo) \ + == S.Half + assert gruntz((gamma(x + 1/gamma(x)) - gamma(x)) / log(x), x, oo) == 1 + + +def test_gruntz_eval_special_slow(): + _sskip() + assert gruntz(gamma(x + 1)/sqrt(2*pi) + - exp(-x)*(x**(x + S.Half) + x**(x - S.Half)/12), x, oo) is oo + assert gruntz(exp(exp(exp(digamma(digamma(digamma(x))))))/x, x, oo) == 0 + + +@XFAIL +def test_grunts_eval_special_slow_sometimes_fail(): + _sskip() + # XXX This sometimes fails!!! + assert gruntz(exp(gamma(x - exp(-x))*exp(1/x)) - exp(gamma(x)), x, oo) is oo + + +def test_gruntz_Ei(): + assert gruntz((Ei(x - exp(-exp(x))) - Ei(x)) *exp(-x)*exp(exp(x))*x, x, oo) == -1 + + +@XFAIL +def test_gruntz_eval_special_fail(): + # TODO zeta function series + assert gruntz( + exp((log(2) + 1)*x) * (zeta(x + exp(-x)) - zeta(x)), x, oo) == -log(2) + + # TODO 8.35 - 8.37 (bessel, max-min) + + +def test_gruntz_hyperbolic(): + assert gruntz(cosh(x), x, oo) is oo + assert gruntz(cosh(x), x, -oo) is oo + assert gruntz(sinh(x), x, oo) is oo + assert gruntz(sinh(x), x, -oo) is -oo + assert gruntz(2*cosh(x)*exp(x), x, oo) is oo + assert gruntz(2*cosh(x)*exp(x), x, -oo) == 1 + assert gruntz(2*sinh(x)*exp(x), x, oo) is oo + assert gruntz(2*sinh(x)*exp(x), x, -oo) == -1 + assert gruntz(tanh(x), x, oo) == 1 + assert gruntz(tanh(x), x, -oo) == -1 + assert gruntz(coth(x), x, oo) == 1 + assert gruntz(coth(x), x, -oo) == -1 + + +def test_compare1(): + assert compare(2, x, x) == "<" + assert compare(x, exp(x), x) == "<" + assert compare(exp(x), exp(x**2), x) == "<" + assert compare(exp(x**2), exp(exp(x)), x) == "<" + assert compare(1, exp(exp(x)), x) == "<" + + assert compare(x, 2, x) == ">" + assert compare(exp(x), x, x) == ">" + assert compare(exp(x**2), exp(x), x) == ">" + assert compare(exp(exp(x)), exp(x**2), x) == ">" + assert compare(exp(exp(x)), 1, x) == ">" + + assert compare(2, 3, x) == "=" + assert compare(3, -5, x) == "=" + assert compare(2, -5, x) == "=" + + assert compare(x, x**2, x) == "=" + assert compare(x**2, x**3, x) == "=" + assert compare(x**3, 1/x, x) == "=" + assert compare(1/x, x**m, x) == "=" + assert compare(x**m, -x, x) == "=" + + assert compare(exp(x), exp(-x), x) == "=" + assert compare(exp(-x), exp(2*x), x) == "=" + assert compare(exp(2*x), exp(x)**2, x) == "=" + assert compare(exp(x)**2, exp(x + exp(-x)), x) == "=" + assert compare(exp(x), exp(x + exp(-x)), x) == "=" + + assert compare(exp(x**2), 1/exp(x**2), x) == "=" + + +def test_compare2(): + assert compare(exp(x), x**5, x) == ">" + assert compare(exp(x**2), exp(x)**2, x) == ">" + assert compare(exp(x), exp(x + exp(-x)), x) == "=" + assert compare(exp(x + exp(-x)), exp(x), x) == "=" + assert compare(exp(x + exp(-x)), exp(-x), x) == "=" + assert compare(exp(-x), x, x) == ">" + assert compare(x, exp(-x), x) == "<" + assert compare(exp(x + 1/x), x, x) == ">" + assert compare(exp(-exp(x)), exp(x), x) == ">" + assert compare(exp(exp(-exp(x)) + x), exp(-exp(x)), x) == "<" + + +def test_compare3(): + assert compare(exp(exp(x)), exp(x + exp(-exp(x))), x) == ">" + + +def test_sign1(): + assert sign(Rational(0), x) == 0 + assert sign(Rational(3), x) == 1 + assert sign(Rational(-5), x) == -1 + assert sign(log(x), x) == 1 + assert sign(exp(-x), x) == 1 + assert sign(exp(x), x) == 1 + assert sign(-exp(x), x) == -1 + assert sign(3 - 1/x, x) == 1 + assert sign(-3 - 1/x, x) == -1 + assert sign(sin(1/x), x) == 1 + assert sign((x**Integer(2)), x) == 1 + assert sign(x**2, x) == 1 + assert sign(x**5, x) == 1 + + +def test_sign2(): + assert sign(x, x) == 1 + assert sign(-x, x) == -1 + y = Symbol("y", positive=True) + assert sign(y, x) == 1 + assert sign(-y, x) == -1 + assert sign(y*x, x) == 1 + assert sign(-y*x, x) == -1 + + +def mmrv(a, b): + return set(mrv(a, b)[0].keys()) + + +def test_mrv1(): + assert mmrv(x, x) == {x} + assert mmrv(x + 1/x, x) == {x} + assert mmrv(x**2, x) == {x} + assert mmrv(log(x), x) == {x} + assert mmrv(exp(x), x) == {exp(x)} + assert mmrv(exp(-x), x) == {exp(-x)} + assert mmrv(exp(x**2), x) == {exp(x**2)} + assert mmrv(-exp(1/x), x) == {x} + assert mmrv(exp(x + 1/x), x) == {exp(x + 1/x)} + + +def test_mrv2a(): + assert mmrv(exp(x + exp(-exp(x))), x) == {exp(-exp(x))} + assert mmrv(exp(x + exp(-x)), x) == {exp(x + exp(-x)), exp(-x)} + assert mmrv(exp(1/x + exp(-x)), x) == {exp(-x)} + +#sometimes infinite recursion due to log(exp(x**2)) not simplifying + + +def test_mrv2b(): + assert mmrv(exp(x + exp(-x**2)), x) == {exp(-x**2)} + +#sometimes infinite recursion due to log(exp(x**2)) not simplifying + + +def test_mrv2c(): + assert mmrv( + exp(-x + 1/x**2) - exp(x + 1/x), x) == {exp(x + 1/x), exp(1/x**2 - x)} + +#sometimes infinite recursion due to log(exp(x**2)) not simplifying + + +def test_mrv3(): + assert mmrv(exp(x**2) + x*exp(x) + log(x)**x/x, x) == {exp(x**2)} + assert mmrv( + exp(x)*(exp(1/x + exp(-x)) - exp(1/x)), x) == {exp(x), exp(-x)} + assert mmrv(log( + x**2 + 2*exp(exp(3*x**3*log(x)))), x) == {exp(exp(3*x**3*log(x)))} + assert mmrv(log(x - log(x))/log(x), x) == {x} + assert mmrv( + (exp(1/x - exp(-x)) - exp(1/x))*exp(x), x) == {exp(x), exp(-x)} + assert mmrv( + 1/exp(-x + exp(-x)) - exp(x), x) == {exp(x), exp(-x), exp(x - exp(-x))} + assert mmrv(log(log(x*exp(x*exp(x)) + 1)), x) == {exp(x*exp(x))} + assert mmrv(exp(exp(log(log(x) + 1/x))), x) == {x} + + +def test_mrv4(): + ln = log + assert mmrv((ln(ln(x) + ln(ln(x))) - ln(ln(x)))/ln(ln(x) + ln(ln(ln(x))))*ln(x), + x) == {x} + assert mmrv(log(log(x*exp(x*exp(x)) + 1)) - exp(exp(log(log(x) + 1/x))), x) == \ + {exp(x*exp(x))} + + +def mrewrite(a, b, c): + return rewrite(a[1], a[0], b, c) + + +def test_rewrite1(): + e = exp(x) + assert mrewrite(mrv(e, x), x, m) == (1/m, -x) + e = exp(x**2) + assert mrewrite(mrv(e, x), x, m) == (1/m, -x**2) + e = exp(x + 1/x) + assert mrewrite(mrv(e, x), x, m) == (1/m, -x - 1/x) + e = 1/exp(-x + exp(-x)) - exp(x) + assert mrewrite(mrv(e, x), x, m) == ((-m*exp(m) + m)*exp(-m)/m**2, -x) + + +def test_rewrite2(): + e = exp(x)*log(log(exp(x))) + assert mmrv(e, x) == {exp(x)} + assert mrewrite(mrv(e, x), x, m) == (1/m*log(x), -x) + +#sometimes infinite recursion due to log(exp(x**2)) not simplifying + + +def test_rewrite3(): + e = exp(-x + 1/x**2) - exp(x + 1/x) + #both of these are correct and should be equivalent: + assert mrewrite(mrv(e, x), x, m) in [(-1/m + m*exp( + (x**2 + x)/x**3), -x - 1/x), ((m**2 - exp((x**2 + x)/x**3))/m, x**(-2) - x)] + + +def test_mrv_leadterm1(): + assert mrv_leadterm(-exp(1/x), x) == (-1, 0) + assert mrv_leadterm(1/exp(-x + exp(-x)) - exp(x), x) == (-1, 0) + assert mrv_leadterm( + (exp(1/x - exp(-x)) - exp(1/x))*exp(x), x) == (-exp(1/x), 0) + + +def test_mrv_leadterm2(): + #Gruntz: p51, 3.25 + assert mrv_leadterm((log(exp(x) + x) - x)/log(exp(x) + log(x))*exp(x), x) == \ + (1, 0) + + +def test_mrv_leadterm3(): + #Gruntz: p56, 3.27 + assert mmrv(exp(-x + exp(-x)*exp(-x*log(x))), x) == {exp(-x - x*log(x))} + assert mrv_leadterm(exp(-x + exp(-x)*exp(-x*log(x))), x) == (exp(-x), 0) + + +def test_limit1(): + assert gruntz(x, x, oo) is oo + assert gruntz(x, x, -oo) is -oo + assert gruntz(-x, x, oo) is -oo + assert gruntz(x**2, x, -oo) is oo + assert gruntz(-x**2, x, oo) is -oo + assert gruntz(x*log(x), x, 0, dir="+") == 0 + assert gruntz(1/x, x, oo) == 0 + assert gruntz(exp(x), x, oo) is oo + assert gruntz(-exp(x), x, oo) is -oo + assert gruntz(exp(x)/x, x, oo) is oo + assert gruntz(1/x - exp(-x), x, oo) == 0 + assert gruntz(x + 1/x, x, oo) is oo + + +def test_limit2(): + assert gruntz(x**x, x, 0, dir="+") == 1 + assert gruntz((exp(x) - 1)/x, x, 0) == 1 + assert gruntz(1 + 1/x, x, oo) == 1 + assert gruntz(-exp(1/x), x, oo) == -1 + assert gruntz(x + exp(-x), x, oo) is oo + assert gruntz(x + exp(-x**2), x, oo) is oo + assert gruntz(x + exp(-exp(x)), x, oo) is oo + assert gruntz(13 + 1/x - exp(-x), x, oo) == 13 + + +def test_limit3(): + a = Symbol('a') + assert gruntz(x - log(1 + exp(x)), x, oo) == 0 + assert gruntz(x - log(a + exp(x)), x, oo) == 0 + assert gruntz(exp(x)/(1 + exp(x)), x, oo) == 1 + assert gruntz(exp(x)/(a + exp(x)), x, oo) == 1 + + +def test_limit4(): + #issue 3463 + assert gruntz((3**x + 5**x)**(1/x), x, oo) == 5 + #issue 3463 + assert gruntz((3**(1/x) + 5**(1/x))**x, x, 0) == 5 + + +@XFAIL +def test_MrvTestCase_page47_ex3_21(): + h = exp(-x/(1 + exp(-x))) + expr = exp(h)*exp(-x/(1 + h))*exp(exp(-x + h))/h**2 - exp(x) + x + assert mmrv(expr, x) == {1/h, exp(-x), exp(x), exp(x - h), exp(x/(1 + h))} + + +def test_gruntz_I(): + y = Symbol("y") + assert gruntz(I*x, x, oo) == I*oo + assert gruntz(y*I*x, x, oo) == y*I*oo + assert gruntz(y*3*I*x, x, oo) == y*I*oo + assert gruntz(y*3*sin(I)*x, x, oo) == y*I*oo + + +def test_issue_4814(): + assert gruntz((x + 1)**(1/log(x + 1)), x, oo) == E + + +def test_intractable(): + assert gruntz(1/gamma(x), x, oo) == 0 + assert gruntz(1/loggamma(x), x, oo) == 0 + assert gruntz(gamma(x)/loggamma(x), x, oo) is oo + assert gruntz(exp(gamma(x))/gamma(x), x, oo) is oo + assert gruntz(gamma(x), x, 3) == 2 + assert gruntz(gamma(Rational(1, 7) + 1/x), x, oo) == gamma(Rational(1, 7)) + assert gruntz(log(x**x)/log(gamma(x)), x, oo) == 1 + assert gruntz(log(gamma(gamma(x)))/exp(x), x, oo) is oo + + +def test_aseries_trig(): + assert cancel(gruntz(1/log(atan(x)), x, oo) + - 1/(log(pi) + log(S.Half))) == 0 + assert gruntz(1/acot(x), x, -oo) is -oo + + +def test_exp_log_series(): + assert gruntz(x/log(log(x*exp(x))), x, oo) is oo + + +def test_issue_3644(): + assert gruntz(((x**7 + x + 1)/(2**x + x**2))**(-1/x), x, oo) == 2 + + +def test_issue_6843(): + n = Symbol('n', integer=True, positive=True) + r = (n + 1)*x**(n + 1)/(x**(n + 1) - 1) - x/(x - 1) + assert gruntz(r, x, 1).simplify() == n/2 + + +def test_issue_4190(): + assert gruntz(x - gamma(1/x), x, oo) == S.EulerGamma + + +@XFAIL +def test_issue_5172(): + n = Symbol('n') + r = Symbol('r', positive=True) + c = Symbol('c') + p = Symbol('p', positive=True) + m = Symbol('m', negative=True) + expr = ((2*n*(n - r + 1)/(n + r*(n - r + 1)))**c + \ + (r - 1)*(n*(n - r + 2)/(n + r*(n - r + 1)))**c - n)/(n**c - n) + expr = expr.subs(c, c + 1) + assert gruntz(expr.subs(c, m), n, oo) == 1 + # fail: + assert gruntz(expr.subs(c, p), n, oo).simplify() == \ + (2**(p + 1) + r - 1)/(r + 1)**(p + 1) + + +def test_issue_4109(): + assert gruntz(1/gamma(x), x, 0) == 0 + assert gruntz(x*gamma(x), x, 0) == 1 + + +def test_issue_6682(): + assert gruntz(exp(2*Ei(-x))/x**2, x, 0) == exp(2*EulerGamma) + + +def test_issue_7096(): + from sympy.functions import sign + assert gruntz(x**-pi, x, 0, dir='-') == oo*sign((-1)**(-pi)) + + +def test_issue_7391_8166(): + f = Function('f') + # limit should depend on the continuity of the expression at the point passed + raises(ValueError, lambda: gruntz(f(x), x, 4)) + raises(ValueError, lambda: gruntz(x*f(x)**2/(x**2 + f(x)**4), x, 0)) + + +def test_issue_24210_25885(): + eq = exp(x)/(1+1/x)**x**2 + ans = sqrt(E) + assert gruntz(eq, x, oo) == ans + assert gruntz(1/eq, x, oo) == 1/ans diff --git a/.venv/lib/python3.12/site-packages/sympy/series/tests/test_kauers.py b/.venv/lib/python3.12/site-packages/sympy/series/tests/test_kauers.py new file mode 100644 index 0000000000000000000000000000000000000000..bfb9044b33416bc38879649b258150ba2906250c --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/tests/test_kauers.py @@ -0,0 +1,23 @@ +from sympy.series.kauers import finite_diff +from sympy.series.kauers import finite_diff_kauers +from sympy.abc import x, y, z, m, n, w +from sympy.core.numbers import pi +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.concrete.summations import Sum + + +def test_finite_diff(): + assert finite_diff(x**2 + 2*x + 1, x) == 2*x + 3 + assert finite_diff(y**3 + 2*y**2 + 3*y + 5, y) == 3*y**2 + 7*y + 6 + assert finite_diff(z**2 - 2*z + 3, z) == 2*z - 1 + assert finite_diff(w**2 + 3*w - 2, w) == 2*w + 4 + assert finite_diff(sin(x), x, pi/6) == -sin(x) + sin(x + pi/6) + assert finite_diff(cos(y), y, pi/3) == -cos(y) + cos(y + pi/3) + assert finite_diff(x**2 - 2*x + 3, x, 2) == 4*x + assert finite_diff(n**2 - 2*n + 3, n, 3) == 6*n + 3 + +def test_finite_diff_kauers(): + assert finite_diff_kauers(Sum(x**2, (x, 1, n))) == (n + 1)**2 + assert finite_diff_kauers(Sum(y, (y, 1, m))) == (m + 1) + assert finite_diff_kauers(Sum((x*y), (x, 1, m), (y, 1, n))) == (m + 1)*(n + 1) + assert finite_diff_kauers(Sum((x*y**2), (x, 1, m), (y, 1, n))) == (n + 1)**2*(m + 1) diff --git a/.venv/lib/python3.12/site-packages/sympy/series/tests/test_limits.py b/.venv/lib/python3.12/site-packages/sympy/series/tests/test_limits.py new file mode 100644 index 0000000000000000000000000000000000000000..aa3ab7683f057424f1c3215a06381d27687710dc --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/tests/test_limits.py @@ -0,0 +1,1440 @@ +from itertools import product + +from sympy.concrete.summations import Sum +from sympy.core.function import (Function, diff) +from sympy.core import EulerGamma, GoldenRatio +from sympy.core.mod import Mod +from sympy.core.numbers import (E, I, Rational, oo, pi, zoo) +from sympy.core.singleton import S +from sympy.core.symbol import (Symbol, symbols) +from sympy.functions.combinatorial.numbers import fibonacci +from sympy.functions.combinatorial.factorials import (binomial, factorial, subfactorial) +from sympy.functions.elementary.complexes import (Abs, re, sign) +from sympy.functions.elementary.exponential import (LambertW, exp, log) +from sympy.functions.elementary.hyperbolic import (atanh, asinh, acosh, acoth, acsch, asech, tanh, sinh) +from sympy.functions.elementary.integers import (ceiling, floor, frac) +from sympy.functions.elementary.miscellaneous import (cbrt, real_root, sqrt) +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.elementary.trigonometric import (acos, acot, acsc, asec, asin, + atan, cos, cot, csc, sec, sin, tan) +from sympy.functions.special.bessel import (besseli, bessely, besselj, besselk) +from sympy.functions.special.error_functions import (Ei, erf, erfc, erfi, fresnelc, fresnels) +from sympy.functions.special.gamma_functions import (digamma, gamma, uppergamma) +from sympy.functions.special.hyper import meijerg +from sympy.integrals.integrals import (Integral, integrate) +from sympy.series.limits import (Limit, limit) +from sympy.simplify.simplify import (logcombine, simplify) +from sympy.simplify.hyperexpand import hyperexpand + +from sympy.calculus.accumulationbounds import AccumBounds +from sympy.core.mul import Mul +from sympy.series.limits import heuristics +from sympy.series.order import Order +from sympy.testing.pytest import XFAIL, raises + +from sympy import elliptic_e, elliptic_k + +from sympy.abc import x, y, z, k +n = Symbol('n', integer=True, positive=True) + + +def test_basic1(): + assert limit(x, x, oo) is oo + assert limit(x, x, -oo) is -oo + assert limit(-x, x, oo) is -oo + assert limit(x**2, x, -oo) is oo + assert limit(-x**2, x, oo) is -oo + assert limit(x*log(x), x, 0, dir="+") == 0 + assert limit(1/x, x, oo) == 0 + assert limit(exp(x), x, oo) is oo + assert limit(-exp(x), x, oo) is -oo + assert limit(exp(x)/x, x, oo) is oo + assert limit(1/x - exp(-x), x, oo) == 0 + assert limit(x + 1/x, x, oo) is oo + assert limit(x - x**2, x, oo) is -oo + assert limit((1 + x)**(1 + sqrt(2)), x, 0) == 1 + assert limit((1 + x)**oo, x, 0) == Limit((x + 1)**oo, x, 0) + assert limit((1 + x)**oo, x, 0, dir='-') == Limit((x + 1)**oo, x, 0, dir='-') + assert limit((1 + x + y)**oo, x, 0, dir='-') == Limit((1 + x + y)**oo, x, 0, dir='-') + assert limit(y/x/log(x), x, 0) == -oo*sign(y) + assert limit(cos(x + y)/x, x, 0) == sign(cos(y))*oo + assert limit(gamma(1/x + 3), x, oo) == 2 + assert limit(S.NaN, x, -oo) is S.NaN + assert limit(Order(2)*x, x, S.NaN) is S.NaN + assert limit(1/(x - 1), x, 1, dir="+") is oo + assert limit(1/(x - 1), x, 1, dir="-") is -oo + assert limit(1/(5 - x)**3, x, 5, dir="+") is -oo + assert limit(1/(5 - x)**3, x, 5, dir="-") is oo + assert limit(1/sin(x), x, pi, dir="+") is -oo + assert limit(1/sin(x), x, pi, dir="-") is oo + assert limit(1/cos(x), x, pi/2, dir="+") is -oo + assert limit(1/cos(x), x, pi/2, dir="-") is oo + assert limit(1/tan(x**3), x, (2*pi)**Rational(1, 3), dir="+") is oo + assert limit(1/tan(x**3), x, (2*pi)**Rational(1, 3), dir="-") is -oo + assert limit(1/cot(x)**3, x, (pi*Rational(3, 2)), dir="+") is -oo + assert limit(1/cot(x)**3, x, (pi*Rational(3, 2)), dir="-") is oo + assert limit(tan(x), x, oo) == AccumBounds(S.NegativeInfinity, S.Infinity) + assert limit(cot(x), x, oo) == AccumBounds(S.NegativeInfinity, S.Infinity) + assert limit(sec(x), x, oo) == AccumBounds(S.NegativeInfinity, S.Infinity) + assert limit(csc(x), x, oo) == AccumBounds(S.NegativeInfinity, S.Infinity) + + # test bi-directional limits + assert limit(sin(x)/x, x, 0, dir="+-") == 1 + assert limit(x**2, x, 0, dir="+-") == 0 + assert limit(1/x**2, x, 0, dir="+-") is oo + + # test failing bi-directional limits + assert limit(1/x, x, 0, dir="+-") is zoo + # approaching 0 + # from dir="+" + assert limit(1 + 1/x, x, 0) is oo + # from dir='-' + # Add + assert limit(1 + 1/x, x, 0, dir='-') is -oo + # Pow + assert limit(x**(-2), x, 0, dir='-') is oo + assert limit(x**(-3), x, 0, dir='-') is -oo + assert limit(1/sqrt(x), x, 0, dir='-') == (-oo)*I + assert limit(x**2, x, 0, dir='-') == 0 + assert limit(sqrt(x), x, 0, dir='-') == 0 + assert limit(x**-pi, x, 0, dir='-') == -oo*(-1)**(1 - pi) + assert limit((1 + cos(x))**oo, x, 0) == Limit((cos(x) + 1)**oo, x, 0) + + # test pull request 22491 + assert limit(1/asin(x), x, 0, dir = '+') == oo + assert limit(1/asin(x), x, 0, dir = '-') == -oo + assert limit(1/sinh(x), x, 0, dir = '+') == oo + assert limit(1/sinh(x), x, 0, dir = '-') == -oo + assert limit(log(1/x) + 1/sin(x), x, 0, dir = '+') == oo + assert limit(log(1/x) + 1/x, x, 0, dir = '+') == oo + + +def test_basic2(): + assert limit(x**x, x, 0, dir="+") == 1 + assert limit((exp(x) - 1)/x, x, 0) == 1 + assert limit(1 + 1/x, x, oo) == 1 + assert limit(-exp(1/x), x, oo) == -1 + assert limit(x + exp(-x), x, oo) is oo + assert limit(x + exp(-x**2), x, oo) is oo + assert limit(x + exp(-exp(x)), x, oo) is oo + assert limit(13 + 1/x - exp(-x), x, oo) == 13 + + +def test_basic3(): + assert limit(1/x, x, 0, dir="+") is oo + assert limit(1/x, x, 0, dir="-") is -oo + + +def test_basic4(): + assert limit(2*x + y*x, x, 0) == 0 + assert limit(2*x + y*x, x, 1) == 2 + y + assert limit(2*x**8 + y*x**(-3), x, -2) == 512 - y/8 + assert limit(sqrt(x + 1) - sqrt(x), x, oo) == 0 + assert integrate(1/(x**3 + 1), (x, 0, oo)) == 2*pi*sqrt(3)/9 + + +def test_log(): + # https://github.com/sympy/sympy/issues/21598 + a, b, c = symbols('a b c', positive=True) + A = log(a/b) - (log(a) - log(b)) + assert A.limit(a, oo) == 0 + assert (A * c).limit(a, oo) == 0 + + tau, x = symbols('tau x', positive=True) + # The value of manualintegrate in the issue + expr = tau**2*((tau - 1)*(tau + 1)*log(x + 1)/(tau**2 + 1)**2 + 1/((tau**2\ + + 1)*(x + 1)) - (-2*tau*atan(x/tau) + (tau**2/2 - 1/2)*log(tau**2\ + + x**2))/(tau**2 + 1)**2) + assert limit(expr, x, oo) == pi*tau**3/(tau**2 + 1)**2 + + +def test_piecewise(): + # https://github.com/sympy/sympy/issues/18363 + assert limit((real_root(x - 6, 3) + 2)/(x + 2), x, -2, '+') == Rational(1, 12) + + +def test_piecewise2(): + func1 = 2*sqrt(x)*Piecewise(((4*x - 2)/Abs(sqrt(4 - 4*(2*x - 1)**2)), 4*x - 2\ + >= 0), ((2 - 4*x)/Abs(sqrt(4 - 4*(2*x - 1)**2)), True)) + func2 = Piecewise((x**2/2, x <= 0.5), (x/2 - 0.125, True)) + func3 = Piecewise(((x - 9) / 5, x < -1), ((x - 9) / 5, x > 4), (sqrt(Abs(x - 3)), True)) + assert limit(func1, x, 0) == 1 + assert limit(func2, x, 0) == 0 + assert limit(func3, x, -1) == 2 + + +def test_basic5(): + class my(Function): + @classmethod + def eval(cls, arg): + if arg is S.Infinity: + return S.NaN + assert limit(my(x), x, oo) == Limit(my(x), x, oo) + + +def test_issue_3885(): + assert limit(x*y + x*z, z, 2) == x*y + 2*x + + +def test_Limit(): + assert Limit(sin(x)/x, x, 0) != 1 + assert Limit(sin(x)/x, x, 0).doit() == 1 + assert Limit(x, x, 0, dir='+-').args == (x, x, 0, Symbol('+-')) + + +def test_floor(): + assert limit(floor(x), x, -2, "+") == -2 + assert limit(floor(x), x, -2, "-") == -3 + assert limit(floor(x), x, -1, "+") == -1 + assert limit(floor(x), x, -1, "-") == -2 + assert limit(floor(x), x, 0, "+") == 0 + assert limit(floor(x), x, 0, "-") == -1 + assert limit(floor(x), x, 1, "+") == 1 + assert limit(floor(x), x, 1, "-") == 0 + assert limit(floor(x), x, 2, "+") == 2 + assert limit(floor(x), x, 2, "-") == 1 + assert limit(floor(x), x, 248, "+") == 248 + assert limit(floor(x), x, 248, "-") == 247 + + # https://github.com/sympy/sympy/issues/14478 + assert limit(x*floor(3/x)/2, x, 0, '+') == Rational(3, 2) + assert limit(floor(x + 1/2) - floor(x), x, oo) == AccumBounds(-S.Half, S(3)/2) + + # test issue 9158 + assert limit(floor(atan(x)), x, oo) == 1 + assert limit(floor(atan(x)), x, -oo) == -2 + assert limit(ceiling(atan(x)), x, oo) == 2 + assert limit(ceiling(atan(x)), x, -oo) == -1 + + +def test_floor_requires_robust_assumptions(): + assert limit(floor(sin(x)), x, 0, "+") == 0 + assert limit(floor(sin(x)), x, 0, "-") == -1 + assert limit(floor(cos(x)), x, 0, "+") == 0 + assert limit(floor(cos(x)), x, 0, "-") == 0 + assert limit(floor(5 + sin(x)), x, 0, "+") == 5 + assert limit(floor(5 + sin(x)), x, 0, "-") == 4 + assert limit(floor(5 + cos(x)), x, 0, "+") == 5 + assert limit(floor(5 + cos(x)), x, 0, "-") == 5 + + +def test_ceiling(): + assert limit(ceiling(x), x, -2, "+") == -1 + assert limit(ceiling(x), x, -2, "-") == -2 + assert limit(ceiling(x), x, -1, "+") == 0 + assert limit(ceiling(x), x, -1, "-") == -1 + assert limit(ceiling(x), x, 0, "+") == 1 + assert limit(ceiling(x), x, 0, "-") == 0 + assert limit(ceiling(x), x, 1, "+") == 2 + assert limit(ceiling(x), x, 1, "-") == 1 + assert limit(ceiling(x), x, 2, "+") == 3 + assert limit(ceiling(x), x, 2, "-") == 2 + assert limit(ceiling(x), x, 248, "+") == 249 + assert limit(ceiling(x), x, 248, "-") == 248 + + # https://github.com/sympy/sympy/issues/14478 + assert limit(x*ceiling(3/x)/2, x, 0, '+') == Rational(3, 2) + assert limit(ceiling(x + 1/2) - ceiling(x), x, oo) == AccumBounds(-S.Half, S(3)/2) + + +def test_ceiling_requires_robust_assumptions(): + assert limit(ceiling(sin(x)), x, 0, "+") == 1 + assert limit(ceiling(sin(x)), x, 0, "-") == 0 + assert limit(ceiling(cos(x)), x, 0, "+") == 1 + assert limit(ceiling(cos(x)), x, 0, "-") == 1 + assert limit(ceiling(5 + sin(x)), x, 0, "+") == 6 + assert limit(ceiling(5 + sin(x)), x, 0, "-") == 5 + assert limit(ceiling(5 + cos(x)), x, 0, "+") == 6 + assert limit(ceiling(5 + cos(x)), x, 0, "-") == 6 + + +def test_frac(): + assert limit(frac(x), x, oo) == AccumBounds(0, 1) + assert limit(frac(x)**(1/x), x, oo) == AccumBounds(0, 1) + assert limit(frac(x)**(1/x), x, -oo) == AccumBounds(1, oo) + assert limit(frac(x)**x, x, oo) == AccumBounds(0, oo) # wolfram gives (0, 1) + assert limit(frac(sin(x)), x, 0, "+") == 0 + assert limit(frac(sin(x)), x, 0, "-") == 1 + assert limit(frac(cos(x)), x, 0, "+-") == 1 + assert limit(frac(x**2), x, 0, "+-") == 0 + raises(ValueError, lambda: limit(frac(x), x, 0, '+-')) + assert limit(frac(-2*x + 1), x, 0, "+") == 1 + assert limit(frac(-2*x + 1), x, 0, "-") == 0 + assert limit(frac(x + S.Half), x, 0, "+-") == S(1)/2 + assert limit(frac(1/x), x, 0) == AccumBounds(0, 1) + + +def test_issue_14355(): + assert limit(floor(sin(x)/x), x, 0, '+') == 0 + assert limit(floor(sin(x)/x), x, 0, '-') == 0 + # test comment https://github.com/sympy/sympy/issues/14355#issuecomment-372121314 + assert limit(floor(-tan(x)/x), x, 0, '+') == -2 + assert limit(floor(-tan(x)/x), x, 0, '-') == -2 + + +def test_atan(): + x = Symbol("x", real=True) + assert limit(atan(x)*sin(1/x), x, 0) == 0 + assert limit(atan(x) + sqrt(x + 1) - sqrt(x), x, oo) == pi/2 + + +def test_set_signs(): + assert limit(abs(x), x, 0) == 0 + assert limit(abs(sin(x)), x, 0) == 0 + assert limit(abs(cos(x)), x, 0) == 1 + assert limit(abs(sin(x + 1)), x, 0) == sin(1) + + # https://github.com/sympy/sympy/issues/9449 + assert limit((Abs(x + y) - Abs(x - y))/(2*x), x, 0) == sign(y) + + # https://github.com/sympy/sympy/issues/12398 + assert limit(Abs(log(x)/x**3), x, oo) == 0 + assert limit(x*(Abs(log(x)/x**3)/Abs(log(x + 1)/(x + 1)**3) - 1), x, oo) == 3 + + # https://github.com/sympy/sympy/issues/18501 + assert limit(Abs(log(x - 1)**3 - 1), x, 1, '+') == oo + + # https://github.com/sympy/sympy/issues/18997 + assert limit(Abs(log(x)), x, 0) == oo + assert limit(Abs(log(Abs(x))), x, 0) == oo + + # https://github.com/sympy/sympy/issues/19026 + z = Symbol('z', positive=True) + assert limit(Abs(log(z) + 1)/log(z), z, oo) == 1 + + # https://github.com/sympy/sympy/issues/20704 + assert limit(z*(Abs(1/z + y) - Abs(y - 1/z))/2, z, 0) == 0 + + # https://github.com/sympy/sympy/issues/21606 + assert limit(cos(z)/sign(z), z, pi, '-') == -1 + + +def test_heuristic(): + x = Symbol("x", real=True) + assert heuristics(sin(1/x) + atan(x), x, 0, '+') == AccumBounds(-1, 1) + assert limit(log(2 + sqrt(atan(x))*sqrt(sin(1/x))), x, 0) == log(2) + + +def test_issue_3871(): + z = Symbol("z", positive=True) + f = -1/z*exp(-z*x) + assert limit(f, x, oo) == 0 + assert f.limit(x, oo) == 0 + + +def test_exponential(): + n = Symbol('n') + x = Symbol('x', real=True) + assert limit((1 + x/n)**n, n, oo) == exp(x) + assert limit((1 + x/(2*n))**n, n, oo) == exp(x/2) + assert limit((1 + x/(2*n + 1))**n, n, oo) == exp(x/2) + assert limit(((x - 1)/(x + 1))**x, x, oo) == exp(-2) + assert limit(1 + (1 + 1/x)**x, x, oo) == 1 + S.Exp1 + assert limit((2 + 6*x)**x/(6*x)**x, x, oo) == exp(S('1/3')) + + +def test_exponential2(): + n = Symbol('n') + assert limit((1 + x/(n + sin(n)))**n, n, oo) == exp(x) + + +def test_doit(): + f = Integral(2 * x, x) + l = Limit(f, x, oo) + assert l.doit() is oo + + +def test_series_AccumBounds(): + assert limit(sin(k) - sin(k + 1), k, oo) == AccumBounds(-2, 2) + assert limit(cos(k) - cos(k + 1) + 1, k, oo) == AccumBounds(-1, 3) + + # not the exact bound + assert limit(sin(k) - sin(k)*cos(k), k, oo) == AccumBounds(-2, 2) + + # test for issue #9934 + lo = (-3 + cos(1))/2 + hi = (1 + cos(1))/2 + t1 = Mul(AccumBounds(lo, hi), 1/(-1 + cos(1)), evaluate=False) + assert limit(simplify(Sum(cos(n).rewrite(exp), (n, 0, k)).doit().rewrite(sin)), k, oo) == t1 + + t2 = Mul(AccumBounds(-1 + sin(1)/2, sin(1)/2 + 1), 1/(1 - cos(1))) + assert limit(simplify(Sum(sin(n).rewrite(exp), (n, 0, k)).doit().rewrite(sin)), k, oo) == t2 + + assert limit(((sin(x) + 1)/2)**x, x, oo) == AccumBounds(0, oo) # wolfram says 0 + + # https://github.com/sympy/sympy/issues/12312 + e = 2**(-x)*(sin(x) + 1)**x + assert limit(e, x, oo) == AccumBounds(0, oo) + + +def test_bessel_functions_at_infinity(): + # Pull Request 23844 implements limits for all bessel and modified bessel + # functions approaching infinity along any direction i.e. abs(z0) tends to oo + + assert limit(besselj(1, x), x, oo) == 0 + assert limit(besselj(1, x), x, -oo) == 0 + assert limit(besselj(1, x), x, I*oo) == oo*I + assert limit(besselj(1, x), x, -I*oo) == -oo*I + assert limit(bessely(1, x), x, oo) == 0 + assert limit(bessely(1, x), x, -oo) == 0 + assert limit(bessely(1, x), x, I*oo) == -oo + assert limit(bessely(1, x), x, -I*oo) == -oo + assert limit(besseli(1, x), x, oo) == oo + assert limit(besseli(1, x), x, -oo) == -oo + assert limit(besseli(1, x), x, I*oo) == 0 + assert limit(besseli(1, x), x, -I*oo) == 0 + assert limit(besselk(1, x), x, oo) == 0 + assert limit(besselk(1, x), x, -oo) == -oo*I + assert limit(besselk(1, x), x, I*oo) == 0 + assert limit(besselk(1, x), x, -I*oo) == 0 + + # test issue 14874 + assert limit(besselk(0, x), x, oo) == 0 + + +@XFAIL +def test_doit2(): + f = Integral(2 * x, x) + l = Limit(f, x, oo) + # limit() breaks on the contained Integral. + assert l.doit(deep=False) == l + + +def test_issue_2929(): + assert limit((x * exp(x))/(exp(x) - 1), x, -oo) == 0 + + +def test_issue_3792(): + assert limit((1 - cos(x))/x**2, x, S.Half) == 4 - 4*cos(S.Half) + assert limit(sin(sin(x + 1) + 1), x, 0) == sin(1 + sin(1)) + assert limit(abs(sin(x + 1) + 1), x, 0) == 1 + sin(1) + + +def test_issue_4090(): + assert limit(1/(x + 3), x, 2) == Rational(1, 5) + assert limit(1/(x + pi), x, 2) == S.One/(2 + pi) + assert limit(log(x)/(x**2 + 3), x, 2) == log(2)/7 + assert limit(log(x)/(x**2 + pi), x, 2) == log(2)/(4 + pi) + + +def test_issue_4547(): + assert limit(cot(x), x, 0, dir='+') is oo + assert limit(cot(x), x, pi/2, dir='+') == 0 + + +def test_issue_5164(): + assert limit(x**0.5, x, oo) == oo**0.5 is oo + assert limit(x**0.5, x, 16) == 4 # Should this be a float? + assert limit(x**0.5, x, 0) == 0 + assert limit(x**(-0.5), x, oo) == 0 + assert limit(x**(-0.5), x, 4) == S.Half # Should this be a float? + + +def test_issue_5383(): + func = (1.0 * 1 + 1.0 * x)**(1.0 * 1 / x) + assert limit(func, x, 0) == E + + +def test_issue_14793(): + expr = ((x + S(1)/2) * log(x) - x + log(2*pi)/2 - \ + log(factorial(x)) + S(1)/(12*x))*x**3 + assert limit(expr, x, oo) == S(1)/360 + + +def test_issue_5183(): + # using list(...) so py.test can recalculate values + tests = list(product([x, -x], + [-1, 1], + [2, 3, S.Half, Rational(2, 3)], + ['-', '+'])) + results = (oo, oo, -oo, oo, -oo*I, oo, -oo*(-1)**Rational(1, 3), oo, + 0, 0, 0, 0, 0, 0, 0, 0, + oo, oo, oo, -oo, oo, -oo*I, oo, -oo*(-1)**Rational(1, 3), + 0, 0, 0, 0, 0, 0, 0, 0) + assert len(tests) == len(results) + for i, (args, res) in enumerate(zip(tests, results)): + y, s, e, d = args + eq = y**(s*e) + try: + assert limit(eq, x, 0, dir=d) == res + except AssertionError: + if 0: # change to 1 if you want to see the failing tests + print() + print(i, res, eq, d, limit(eq, x, 0, dir=d)) + else: + assert None + + +def test_issue_5184(): + assert limit(sin(x)/x, x, oo) == 0 + assert limit(atan(x), x, oo) == pi/2 + assert limit(gamma(x), x, oo) is oo + assert limit(cos(x)/x, x, oo) == 0 + assert limit(gamma(x), x, S.Half) == sqrt(pi) + + r = Symbol('r', real=True) + assert limit(r*sin(1/r), r, 0) == 0 + + +def test_issue_5229(): + assert limit((1 + y)**(1/y) - S.Exp1, y, 0) == 0 + + +def test_issue_4546(): + # using list(...) so py.test can recalculate values + tests = list(product([cot, tan], + [-pi/2, 0, pi/2, pi, pi*Rational(3, 2)], + ['-', '+'])) + results = (0, 0, -oo, oo, 0, 0, -oo, oo, 0, 0, + oo, -oo, 0, 0, oo, -oo, 0, 0, oo, -oo) + assert len(tests) == len(results) + for i, (args, res) in enumerate(zip(tests, results)): + f, l, d = args + eq = f(x) + try: + assert limit(eq, x, l, dir=d) == res + except AssertionError: + if 0: # change to 1 if you want to see the failing tests + print() + print(i, res, eq, l, d, limit(eq, x, l, dir=d)) + else: + assert None + + +def test_issue_3934(): + assert limit((1 + x**log(3))**(1/x), x, 0) == 1 + assert limit((5**(1/x) + 3**(1/x))**x, x, 0) == 5 + + +def test_issue_5955(): + assert limit((x**16)/(1 + x**16), x, oo) == 1 + assert limit((x**100)/(1 + x**100), x, oo) == 1 + assert limit((x**1885)/(1 + x**1885), x, oo) == 1 + assert limit((x**1000/((x + 1)**1000 + exp(-x))), x, oo) == 1 + + +def test_newissue(): + assert limit(exp(1/sin(x))/exp(cot(x)), x, 0) == 1 + + +def test_extended_real_line(): + assert limit(x - oo, x, oo) == Limit(x - oo, x, oo) + assert limit(1/(x + sin(x)) - oo, x, 0) == Limit(1/(x + sin(x)) - oo, x, 0) + assert limit(oo/x, x, oo) == Limit(oo/x, x, oo) + assert limit(x - oo + 1/x, x, oo) == Limit(x - oo + 1/x, x, oo) + + +@XFAIL +def test_order_oo(): + x = Symbol('x', positive=True) + assert Order(x)*oo != Order(1, x) + assert limit(oo/(x**2 - 4), x, oo) is oo + + +def test_issue_5436(): + raises(NotImplementedError, lambda: limit(exp(x*y), x, oo)) + raises(NotImplementedError, lambda: limit(exp(-x*y), x, oo)) + + +def test_Limit_dir(): + raises(TypeError, lambda: Limit(x, x, 0, dir=0)) + raises(ValueError, lambda: Limit(x, x, 0, dir='0')) + + +def test_polynomial(): + assert limit((x + 1)**1000/((x + 1)**1000 + 1), x, oo) == 1 + assert limit((x + 1)**1000/((x + 1)**1000 + 1), x, -oo) == 1 + assert limit(x ** Rational(77, 3) / (1 + x ** Rational(77, 3)), x, oo) == 1 + assert limit(x ** 101.1 / (1 + x ** 101.1), x, oo) == 1 + + +def test_rational(): + assert limit(1/y - (1/(y + x) + x/(y + x)/y)/z, x, oo) == (z - 1)/(y*z) + assert limit(1/y - (1/(y + x) + x/(y + x)/y)/z, x, -oo) == (z - 1)/(y*z) + + +def test_issue_5740(): + assert limit(log(x)*z - log(2*x)*y, x, 0) == oo*sign(y - z) + + +def test_issue_6366(): + n = Symbol('n', integer=True, positive=True) + r = (n + 1)*x**(n + 1)/(x**(n + 1) - 1) - x/(x - 1) + assert limit(r, x, 1).cancel() == n/2 + + +def test_factorial(): + f = factorial(x) + assert limit(f, x, oo) is oo + assert limit(x/f, x, oo) == 0 + # see Stirling's approximation: + # https://en.wikipedia.org/wiki/Stirling's_approximation + assert limit(f/(sqrt(2*pi*x)*(x/E)**x), x, oo) == 1 + assert limit(f, x, -oo) == gamma(-oo) + + +def test_issue_6560(): + e = (5*x**3/4 - x*Rational(3, 4) + (y*(3*x**2/2 - S.Half) + + 35*x**4/8 - 15*x**2/4 + Rational(3, 8))/(2*(y + 1))) + assert limit(e, y, oo) == 5*x**3/4 + 3*x**2/4 - 3*x/4 - Rational(1, 4) + +@XFAIL +def test_issue_5172(): + n = Symbol('n') + r = Symbol('r', positive=True) + c = Symbol('c') + p = Symbol('p', positive=True) + m = Symbol('m', negative=True) + expr = ((2*n*(n - r + 1)/(n + r*(n - r + 1)))**c + + (r - 1)*(n*(n - r + 2)/(n + r*(n - r + 1)))**c - n)/(n**c - n) + expr = expr.subs(c, c + 1) + raises(NotImplementedError, lambda: limit(expr, n, oo)) + assert limit(expr.subs(c, m), n, oo) == 1 + assert limit(expr.subs(c, p), n, oo).simplify() == \ + (2**(p + 1) + r - 1)/(r + 1)**(p + 1) + + +def test_issue_7088(): + a = Symbol('a') + assert limit(sqrt(x/(x + a)), x, oo) == 1 + + +def test_branch_cuts(): + assert limit(asin(I*x + 2), x, 0) == pi - asin(2) + assert limit(asin(I*x + 2), x, 0, '-') == asin(2) + assert limit(asin(I*x - 2), x, 0) == -asin(2) + assert limit(asin(I*x - 2), x, 0, '-') == -pi + asin(2) + assert limit(acos(I*x + 2), x, 0) == -acos(2) + assert limit(acos(I*x + 2), x, 0, '-') == acos(2) + assert limit(acos(I*x - 2), x, 0) == acos(-2) + assert limit(acos(I*x - 2), x, 0, '-') == 2*pi - acos(-2) + assert limit(atan(x + 2*I), x, 0) == I*atanh(2) + assert limit(atan(x + 2*I), x, 0, '-') == -pi + I*atanh(2) + assert limit(atan(x - 2*I), x, 0) == pi - I*atanh(2) + assert limit(atan(x - 2*I), x, 0, '-') == -I*atanh(2) + assert limit(atan(1/x), x, 0) == pi/2 + assert limit(atan(1/x), x, 0, '-') == -pi/2 + assert limit(atan(x), x, oo) == pi/2 + assert limit(atan(x), x, -oo) == -pi/2 + assert limit(acot(x + S(1)/2*I), x, 0) == pi - I*acoth(S(1)/2) + assert limit(acot(x + S(1)/2*I), x, 0, '-') == -I*acoth(S(1)/2) + assert limit(acot(x - S(1)/2*I), x, 0) == I*acoth(S(1)/2) + assert limit(acot(x - S(1)/2*I), x, 0, '-') == -pi + I*acoth(S(1)/2) + assert limit(acot(x), x, 0) == pi/2 + assert limit(acot(x), x, 0, '-') == -pi/2 + assert limit(asec(I*x + S(1)/2), x, 0) == asec(S(1)/2) + assert limit(asec(I*x + S(1)/2), x, 0, '-') == -asec(S(1)/2) + assert limit(asec(I*x - S(1)/2), x, 0) == 2*pi - asec(-S(1)/2) + assert limit(asec(I*x - S(1)/2), x, 0, '-') == asec(-S(1)/2) + assert limit(acsc(I*x + S(1)/2), x, 0) == acsc(S(1)/2) + assert limit(acsc(I*x + S(1)/2), x, 0, '-') == pi - acsc(S(1)/2) + assert limit(acsc(I*x - S(1)/2), x, 0) == -pi + acsc(S(1)/2) + assert limit(acsc(I*x - S(1)/2), x, 0, '-') == -acsc(S(1)/2) + + assert limit(log(I*x - 1), x, 0) == I*pi + assert limit(log(I*x - 1), x, 0, '-') == -I*pi + assert limit(log(-I*x - 1), x, 0) == -I*pi + assert limit(log(-I*x - 1), x, 0, '-') == I*pi + + assert limit(sqrt(I*x - 1), x, 0) == I + assert limit(sqrt(I*x - 1), x, 0, '-') == -I + assert limit(sqrt(-I*x - 1), x, 0) == -I + assert limit(sqrt(-I*x - 1), x, 0, '-') == I + + assert limit(cbrt(I*x - 1), x, 0) == (-1)**(S(1)/3) + assert limit(cbrt(I*x - 1), x, 0, '-') == -(-1)**(S(2)/3) + assert limit(cbrt(-I*x - 1), x, 0) == -(-1)**(S(2)/3) + assert limit(cbrt(-I*x - 1), x, 0, '-') == (-1)**(S(1)/3) + + +def test_issue_6364(): + a = Symbol('a') + e = z/(1 - sqrt(1 + z)*sin(a)**2 - sqrt(1 - z)*cos(a)**2) + assert limit(e, z, 0) == 1/(cos(a)**2 - S.Half) + + +def test_issue_6682(): + assert limit(exp(2*Ei(-x))/x**2, x, 0) == exp(2*EulerGamma) + + +def test_issue_4099(): + a = Symbol('a') + assert limit(a/x, x, 0) == oo*sign(a) + assert limit(-a/x, x, 0) == -oo*sign(a) + assert limit(-a*x, x, oo) == -oo*sign(a) + assert limit(a*x, x, oo) == oo*sign(a) + + +def test_issue_4503(): + dx = Symbol('dx') + assert limit((sqrt(1 + exp(x + dx)) - sqrt(1 + exp(x)))/dx, dx, 0) == \ + exp(x)/(2*sqrt(exp(x) + 1)) + + +def test_issue_6052(): + G = meijerg((), (), (1,), (0,), -x) + g = hyperexpand(G) + assert limit(g, x, 0, '+-') == 0 + assert limit(g, x, oo) == -oo + + +def test_issue_7224(): + expr = sqrt(x)*besseli(1,sqrt(8*x)) + assert limit(x*diff(expr, x, x)/expr, x, 0) == 2 + assert limit(x*diff(expr, x, x)/expr, x, 1).evalf() == 2.0 + + +def test_issue_7391_8166(): + f = Function('f') + # limit should depend on the continuity of the expression at the point passed + assert limit(f(x), x, 4) == Limit(f(x), x, 4, dir='+') + assert limit(x*f(x)**2/(x**2 + f(x)**4), x, 0) == Limit(x*f(x)**2/(x**2 + f(x)**4), x, 0, dir='+') + + +def test_issue_8208(): + assert limit(n**(Rational(1, 1e9) - 1), n, oo) == 0 + + +def test_issue_8229(): + assert limit((x**Rational(1, 4) - 2)/(sqrt(x) - 4)**Rational(2, 3), x, 16) == 0 + + +def test_issue_8433(): + d, t = symbols('d t', positive=True) + assert limit(erf(1 - t/d), t, oo) == -1 + + +def test_issue_8481(): + k = Symbol('k', integer=True, nonnegative=True) + lamda = Symbol('lamda', positive=True) + assert limit(lamda**k * exp(-lamda) / factorial(k), k, oo) == 0 + + +def test_issue_8462(): + assert limit(binomial(n, n/2), n, oo) == oo + assert limit(binomial(n, n/2) * 3 ** (-n), n, oo) == 0 + + +def test_issue_8634(): + n = Symbol('n', integer=True, positive=True) + x = Symbol('x') + assert limit(x**n, x, -oo) == oo*sign((-1)**n) + + +def test_issue_8635_18176(): + x = Symbol('x', real=True) + k = Symbol('k', positive=True) + assert limit(x**n - x**(n - 0), x, oo) == 0 + assert limit(x**n - x**(n - 5), x, oo) == oo + assert limit(x**n - x**(n - 2.5), x, oo) == oo + assert limit(x**n - x**(n - k - 1), x, oo) == oo + x = Symbol('x', positive=True) + assert limit(x**n - x**(n - 1), x, oo) == oo + assert limit(x**n - x**(n + 2), x, oo) == -oo + + +def test_issue_8730(): + assert limit(subfactorial(x), x, oo) is oo + + +def test_issue_9252(): + n = Symbol('n', integer=True) + c = Symbol('c', positive=True) + assert limit((log(n))**(n/log(n)) / (1 + c)**n, n, oo) == 0 + # limit should depend on the value of c + raises(NotImplementedError, lambda: limit((log(n))**(n/log(n)) / c**n, n, oo)) + + +def test_issue_9558(): + assert limit(sin(x)**15, x, 0, '-') == 0 + + +def test_issue_10801(): + # make sure limits work with binomial + assert limit(16**k / (k * binomial(2*k, k)**2), k, oo) == pi + + +def test_issue_10976(): + s, x = symbols('s x', real=True) + assert limit(erf(s*x)/erf(s), s, 0) == x + + +def test_issue_9041(): + assert limit(factorial(n) / ((n/exp(1))**n * sqrt(2*pi*n)), n, oo) == 1 + + +def test_issue_9205(): + x, y, a = symbols('x, y, a') + assert Limit(x, x, a).free_symbols == {a} + assert Limit(x, x, a, '-').free_symbols == {a} + assert Limit(x + y, x + y, a).free_symbols == {a} + assert Limit(-x**2 + y, x**2, a).free_symbols == {y, a} + + +def test_issue_9471(): + assert limit(((27**(log(n,3)))/n**3),n,oo) == 1 + assert limit(((27**(log(n,3)+1))/n**3),n,oo) == 27 + + +def test_issue_10382(): + assert limit(fibonacci(n + 1)/fibonacci(n), n, oo) == GoldenRatio + + +def test_issue_11496(): + assert limit(erfc(log(1/x)), x, oo) == 2 + + +def test_issue_11879(): + assert simplify(limit(((x+y)**n-x**n)/y, y, 0)) == n*x**(n-1) + + +def test_limit_with_Float(): + k = symbols("k") + assert limit(1.0 ** k, k, oo) == 1 + assert limit(0.3*1.0**k, k, oo) == Rational(3, 10) + + +def test_issue_10610(): + assert limit(3**x*3**(-x - 1)*(x + 1)**2/x**2, x, oo) == Rational(1, 3) + + +def test_issue_10868(): + assert limit(log(x) + asech(x), x, 0, '+') == log(2) + assert limit(log(x) + asech(x), x, 0, '-') == log(2) + 2*I*pi + raises(ValueError, lambda: limit(log(x) + asech(x), x, 0, '+-')) + assert limit(log(x) + asech(x), x, oo) == oo + assert limit(log(x) + acsch(x), x, 0, '+') == log(2) + assert limit(log(x) + acsch(x), x, 0, '-') == -oo + raises(ValueError, lambda: limit(log(x) + acsch(x), x, 0, '+-')) + assert limit(log(x) + acsch(x), x, oo) == oo + + +def test_issue_6599(): + assert limit((n + cos(n))/n, n, oo) == 1 + + +def test_issue_12555(): + assert limit((3**x + 2* x**10) / (x**10 + exp(x)), x, -oo) == 2 + assert limit((3**x + 2* x**10) / (x**10 + exp(x)), x, oo) is oo + + +def test_issue_12769(): + r, z, x = symbols('r z x', real=True) + a, b, s0, K, F0, s, T = symbols('a b s0 K F0 s T', positive=True, real=True) + fx = (F0**b*K**b*r*s0 - sqrt((F0**2*K**(2*b)*a**2*(b - 1) + \ + F0**(2*b)*K**2*a**2*(b - 1) + F0**(2*b)*K**(2*b)*s0**2*(b - 1)*(b**2 - 2*b + 1) - \ + 2*F0**(2*b)*K**(b + 1)*a*r*s0*(b**2 - 2*b + 1) + \ + 2*F0**(b + 1)*K**(2*b)*a*r*s0*(b**2 - 2*b + 1) - \ + 2*F0**(b + 1)*K**(b + 1)*a**2*(b - 1))/((b - 1)*(b**2 - 2*b + 1))))*(b*r - b - r + 1) + + assert fx.subs(K, F0).factor(deep=True) == limit(fx, K, F0).factor(deep=True) + + +def test_issue_13332(): + assert limit(sqrt(30)*5**(-5*x - 1)*(46656*x)**x*(5*x + 2)**(5*x + 5*S.Half) * + (6*x + 2)**(-6*x - 5*S.Half), x, oo) == Rational(25, 36) + + +def test_issue_12564(): + assert limit(x**2 + x*sin(x) + cos(x), x, -oo) is oo + assert limit(x**2 + x*sin(x) + cos(x), x, oo) is oo + assert limit(((x + cos(x))**2).expand(), x, oo) is oo + assert limit(((x + sin(x))**2).expand(), x, oo) is oo + assert limit(((x + cos(x))**2).expand(), x, -oo) is oo + assert limit(((x + sin(x))**2).expand(), x, -oo) is oo + + +def test_issue_14456(): + raises(NotImplementedError, lambda: Limit(exp(x), x, zoo).doit()) + raises(NotImplementedError, lambda: Limit(x**2/(x+1), x, zoo).doit()) + + +def test_issue_14411(): + assert limit(3*sec(4*pi*x - x/3), x, 3*pi/(24*pi - 2)) is -oo + + +def test_issue_13382(): + assert limit(x*(((x + 1)**2 + 1)/(x**2 + 1) - 1), x, oo) == 2 + + +def test_issue_13403(): + assert limit(x*(-1 + (x + log(x + 1) + 1)/(x + log(x))), x, oo) == 1 + + +def test_issue_13416(): + assert limit((-x**3*log(x)**3 + (x - 1)*(x + 1)**2*log(x + 1)**3)/(x**2*log(x)**3), x, oo) == 1 + + +def test_issue_13462(): + assert limit(n**2*(2*n*(-(1 - 1/(2*n))**x + 1) - x - (-x**2/4 + x/4)/n), n, oo) == x**3/24 - x**2/8 + x/12 + + +def test_issue_13750(): + a = Symbol('a') + assert limit(erf(a - x), x, oo) == -1 + assert limit(erf(sqrt(x) - x), x, oo) == -1 + + +def test_issue_14276(): + assert isinstance(limit(sin(x)**log(x), x, oo), Limit) + assert isinstance(limit(sin(x)**cos(x), x, oo), Limit) + assert isinstance(limit(sin(log(cos(x))), x, oo), Limit) + assert limit((1 + 1/(x**2 + cos(x)))**(x**2 + x), x, oo) == E + + +def test_issue_14514(): + assert limit((1/(log(x)**log(x)))**(1/x), x, oo) == 1 + + +def test_issues_14525(): + assert limit(sin(x)**2 - cos(x) + tan(x)*csc(x), x, oo) == AccumBounds(S.NegativeInfinity, S.Infinity) + assert limit(sin(x)**2 - cos(x) + sin(x)*cot(x), x, oo) == AccumBounds(S.NegativeInfinity, S.Infinity) + assert limit(cot(x) - tan(x)**2, x, oo) == AccumBounds(S.NegativeInfinity, S.Infinity) + assert limit(cos(x) - tan(x)**2, x, oo) == AccumBounds(S.NegativeInfinity, S.One) + assert limit(sin(x) - tan(x)**2, x, oo) == AccumBounds(S.NegativeInfinity, S.One) + assert limit(cos(x)**2 - tan(x)**2, x, oo) == AccumBounds(S.NegativeInfinity, S.One) + assert limit(tan(x)**2 + sin(x)**2 - cos(x), x, oo) == AccumBounds(-S.One, S.Infinity) + + +def test_issue_14574(): + assert limit(sqrt(x)*cos(x - x**2) / (x + 1), x, oo) == 0 + + +def test_issue_10102(): + assert limit(fresnels(x), x, oo) == S.Half + assert limit(3 + fresnels(x), x, oo) == 3 + S.Half + assert limit(5*fresnels(x), x, oo) == Rational(5, 2) + assert limit(fresnelc(x), x, oo) == S.Half + assert limit(fresnels(x), x, -oo) == Rational(-1, 2) + assert limit(4*fresnelc(x), x, -oo) == -2 + + +def test_issue_14377(): + raises(NotImplementedError, lambda: limit(exp(I*x)*sin(pi*x), x, oo)) + + +def test_issue_15146(): + e = (x/2) * (-2*x**3 - 2*(x**3 - 1) * x**2 * digamma(x**3 + 1) + \ + 2*(x**3 - 1) * x**2 * digamma(x**3 + x + 1) + x + 3) + assert limit(e, x, oo) == S(1)/3 + + +def test_issue_15202(): + e = (2**x*(2 + 2**(-x)*(-2*2**x + x + 2))/(x + 1))**(x + 1) + assert limit(e, x, oo) == exp(1) + + e = (log(x, 2)**7 + 10*x*factorial(x) + 5**x) / (factorial(x + 1) + 3*factorial(x) + 10**x) + assert limit(e, x, oo) == 10 + + +def test_issue_15282(): + assert limit((x**2000 - (x + 1)**2000) / x**1999, x, oo) == -2000 + + +def test_issue_15984(): + assert limit((-x + log(exp(x) + 1))/x, x, oo, dir='-') == 0 + + +def test_issue_13571(): + assert limit(uppergamma(x, 1) / gamma(x), x, oo) == 1 + + +def test_issue_13575(): + assert limit(acos(erfi(x)), x, 1) == acos(erfi(S.One)) + + +def test_issue_17325(): + assert Limit(sin(x)/x, x, 0, dir="+-").doit() == 1 + assert Limit(x**2, x, 0, dir="+-").doit() == 0 + assert Limit(1/x**2, x, 0, dir="+-").doit() is oo + assert Limit(1/x, x, 0, dir="+-").doit() is zoo + + +def test_issue_10978(): + assert LambertW(x).limit(x, 0) == 0 + + +def test_issue_14313_comment(): + assert limit(floor(n/2), n, oo) is oo + + +def test_issue_15323(): + d = ((1 - 1/x)**x).diff(x) + assert limit(d, x, 1, dir='+') == 1 + + +def test_issue_12571(): + assert limit(-LambertW(-log(x))/log(x), x, 1) == 1 + + +def test_issue_14590(): + assert limit((x**3*((x + 1)/x)**x)/((x + 1)*(x + 2)*(x + 3)), x, oo) == exp(1) + + +def test_issue_14393(): + a, b = symbols('a b') + assert limit((x**b - y**b)/(x**a - y**a), x, y) == b*y**(-a + b)/a + + +def test_issue_14556(): + assert limit(factorial(n + 1)**(1/(n + 1)) - factorial(n)**(1/n), n, oo) == exp(-1) + + +def test_issue_14811(): + assert limit(((1 + ((S(2)/3)**(x + 1)))**(2**x))/(2**((S(4)/3)**(x - 1))), x, oo) == oo + + +def test_issue_16222(): + assert limit(exp(x), x, 1000000000) == exp(1000000000) + + +def test_issue_16714(): + assert limit(((x**(x + 1) + (x + 1)**x) / x**(x + 1))**x, x, oo) == exp(exp(1)) + + +def test_issue_16722(): + z = symbols('z', positive=True) + assert limit(binomial(n + z, n)*n**-z, n, oo) == 1/gamma(z + 1) + z = symbols('z', positive=True, integer=True) + assert limit(binomial(n + z, n)*n**-z, n, oo) == 1/gamma(z + 1) + + +def test_issue_17431(): + assert limit(((n + 1) + 1) / (((n + 1) + 2) * factorial(n + 1)) * + (n + 2) * factorial(n) / (n + 1), n, oo) == 0 + assert limit((n + 2)**2*factorial(n)/((n + 1)*(n + 3)*factorial(n + 1)) + , n, oo) == 0 + assert limit((n + 1) * factorial(n) / (n * factorial(n + 1)), n, oo) == 0 + + +def test_issue_17671(): + assert limit(Ei(-log(x)) - log(log(x))/x, x, 1) == EulerGamma + + +def test_issue_17751(): + a, b, c, x = symbols('a b c x', positive=True) + assert limit((a + 1)*x - sqrt((a + 1)**2*x**2 + b*x + c), x, oo) == -b/(2*a + 2) + + +def test_issue_17792(): + assert limit(factorial(n)/sqrt(n)*(exp(1)/n)**n, n, oo) == sqrt(2)*sqrt(pi) + + +def test_issue_18118(): + assert limit(sign(sin(x)), x, 0, "-") == -1 + assert limit(sign(sin(x)), x, 0, "+") == 1 + + +def test_issue_18306(): + assert limit(sin(sqrt(x))/sqrt(sin(x)), x, 0, '+') == 1 + + +def test_issue_18378(): + assert limit(log(exp(3*x) + x)/log(exp(x) + x**100), x, oo) == 3 + + +def test_issue_18399(): + assert limit((1 - S(1)/2*x)**(3*x), x, oo) is zoo + assert limit((-x)**x, x, oo) is zoo + + +def test_issue_18442(): + assert limit(tan(x)**(2**(sqrt(pi))), x, oo, dir='-') == Limit(tan(x)**(2**(sqrt(pi))), x, oo, dir='-') + + +def test_issue_18452(): + assert limit(abs(log(x))**x, x, 0) == 1 + assert limit(abs(log(x))**x, x, 0, "-") == 1 + + +def test_issue_18473(): + assert limit(sin(x)**(1/x), x, oo) == Limit(sin(x)**(1/x), x, oo, dir='-') + assert limit(cos(x)**(1/x), x, oo) == Limit(cos(x)**(1/x), x, oo, dir='-') + assert limit(tan(x)**(1/x), x, oo) == Limit(tan(x)**(1/x), x, oo, dir='-') + assert limit((cos(x) + 2)**(1/x), x, oo) == 1 + assert limit((sin(x) + 10)**(1/x), x, oo) == 1 + assert limit((cos(x) - 2)**(1/x), x, oo) == Limit((cos(x) - 2)**(1/x), x, oo, dir='-') + assert limit((cos(x) + 1)**(1/x), x, oo) == AccumBounds(0, 1) + assert limit((tan(x)**2)**(2/x) , x, oo) == AccumBounds(0, oo) + assert limit((sin(x)**2)**(1/x), x, oo) == AccumBounds(0, 1) + # Tests for issue #23751 + assert limit((cos(x) + 1)**(1/x), x, -oo) == AccumBounds(1, oo) + assert limit((sin(x)**2)**(1/x), x, -oo) == AccumBounds(1, oo) + assert limit((tan(x)**2)**(2/x) , x, -oo) == AccumBounds(0, oo) + + +def test_issue_18482(): + assert limit((2*exp(3*x)/(exp(2*x) + 1))**(1/x), x, oo) == exp(1) + + +def test_issue_18508(): + assert limit(sin(x)/sqrt(1-cos(x)), x, 0) == sqrt(2) + assert limit(sin(x)/sqrt(1-cos(x)), x, 0, dir='+') == sqrt(2) + assert limit(sin(x)/sqrt(1-cos(x)), x, 0, dir='-') == -sqrt(2) + + +def test_issue_18521(): + raises(NotImplementedError, lambda: limit(exp((2 - n) * x), x, oo)) + + +def test_issue_18969(): + a, b = symbols('a b', positive=True) + assert limit(LambertW(a), a, b) == LambertW(b) + assert limit(exp(LambertW(a)), a, b) == exp(LambertW(b)) + + +def test_issue_18992(): + assert limit(n/(factorial(n)**(1/n)), n, oo) == exp(1) + + +def test_issue_19067(): + x = Symbol('x') + assert limit(gamma(x)/(gamma(x - 1)*gamma(x + 2)), x, 0) == -1 + + +def test_issue_19586(): + assert limit(x**(2**x*3**(-x)), x, oo) == 1 + + +def test_issue_13715(): + n = Symbol('n') + p = Symbol('p', zero=True) + assert limit(n + p, n, 0) == 0 + + +def test_issue_15055(): + assert limit(n**3*((-n - 1)*sin(1/n) + (n + 2)*sin(1/(n + 1)))/(-n + 1), n, oo) == 1 + + +def test_issue_16708(): + m, vi = symbols('m vi', positive=True) + B, ti, d = symbols('B ti d') + assert limit((B*ti*vi - sqrt(m)*sqrt(-2*B*d*vi + m*(vi)**2) + m*vi)/(B*vi), B, 0) == (d + ti*vi)/vi + + +def test_issue_19154(): + assert limit(besseli(1, 3 *x)/(x *besseli(1, x)**3), x , oo) == 2*sqrt(3)*pi/3 + assert limit(besseli(1, 3 *x)/(x *besseli(1, x)**3), x , -oo) == -2*sqrt(3)*pi/3 + + +def test_issue_19453(): + beta = Symbol("beta", positive=True) + h = Symbol("h", positive=True) + m = Symbol("m", positive=True) + w = Symbol("omega", positive=True) + g = Symbol("g", positive=True) + + e = exp(1) + q = 3*h**2*beta*g*e**(0.5*h*beta*w) + p = m**2*w**2 + s = e**(h*beta*w) - 1 + Z = -q/(4*p*s) - q/(2*p*s**2) - q*(e**(h*beta*w) + 1)/(2*p*s**3)\ + + e**(0.5*h*beta*w)/s + E = -diff(log(Z), beta) + + assert limit(E - 0.5*h*w, beta, oo) == 0 + assert limit(E.simplify() - 0.5*h*w, beta, oo) == 0 + + +def test_issue_19739(): + assert limit((-S(1)/4)**x, x, oo) == 0 + + +def test_issue_19766(): + assert limit(2**(-x)*sqrt(4**(x + 1) + 1), x, oo) == 2 + + +def test_issue_19770(): + m = Symbol('m') + # the result is not 0 for non-real m + assert limit(cos(m*x)/x, x, oo) == Limit(cos(m*x)/x, x, oo, dir='-') + m = Symbol('m', real=True) + # can be improved to give the correct result 0 + assert limit(cos(m*x)/x, x, oo) == Limit(cos(m*x)/x, x, oo, dir='-') + m = Symbol('m', nonzero=True) + assert limit(cos(m*x), x, oo) == AccumBounds(-1, 1) + assert limit(cos(m*x)/x, x, oo) == 0 + + +def test_issue_7535(): + assert limit(tan(x)/sin(tan(x)), x, pi/2) == Limit(tan(x)/sin(tan(x)), x, pi/2, dir='+') + assert limit(tan(x)/sin(tan(x)), x, pi/2, dir='-') == Limit(tan(x)/sin(tan(x)), x, pi/2, dir='-') + assert limit(tan(x)/sin(tan(x)), x, pi/2, dir='+-') == Limit(tan(x)/sin(tan(x)), x, pi/2, dir='+-') + assert limit(sin(tan(x)),x,pi/2) == AccumBounds(-1, 1) + assert -oo*(1/sin(-oo)) == AccumBounds(-oo, oo) + assert oo*(1/sin(oo)) == AccumBounds(-oo, oo) + assert oo*(1/sin(-oo)) == AccumBounds(-oo, oo) + assert -oo*(1/sin(oo)) == AccumBounds(-oo, oo) + + +def test_issue_20365(): + assert limit(((x + 1)**(1/x) - E)/x, x, 0) == -E/2 + + +def test_issue_21031(): + assert limit(((1 + x)**(1/x) - (1 + 2*x)**(1/(2*x)))/asin(x), x, 0) == E/2 + + +def test_issue_21038(): + assert limit(sin(pi*x)/(3*x - 12), x, 4) == pi/3 + + +def test_issue_20578(): + expr = abs(x) * sin(1/x) + assert limit(expr,x,0,'+') == 0 + assert limit(expr,x,0,'-') == 0 + assert limit(expr,x,0,'+-') == 0 + + +def test_issue_21227(): + f = log(x) + + assert f.nseries(x, logx=y) == y + assert f.nseries(x, logx=-x) == -x + + f = log(-log(x)) + + assert f.nseries(x, logx=y) == log(-y) + assert f.nseries(x, logx=-x) == log(x) + + f = log(log(x)) + + assert f.nseries(x, logx=y) == log(y) + assert f.nseries(x, logx=-x) == log(-x) + assert f.nseries(x, logx=x) == log(x) + + f = log(log(log(1/x))) + + assert f.nseries(x, logx=y) == log(log(-y)) + assert f.nseries(x, logx=-y) == log(log(y)) + assert f.nseries(x, logx=x) == log(log(-x)) + assert f.nseries(x, logx=-x) == log(log(x)) + + +def test_issue_21415(): + exp = (x-1)*cos(1/(x-1)) + assert exp.limit(x,1) == 0 + assert exp.expand().limit(x,1) == 0 + + +def test_issue_21530(): + assert limit(sinh(n + 1)/sinh(n), n, oo) == E + + +def test_issue_21550(): + r = (sqrt(5) - 1)/2 + assert limit((x - r)/(x**2 + x - 1), x, r) == sqrt(5)/5 + + +def test_issue_21661(): + out = limit((x**(x + 1) * (log(x) + 1) + 1) / x, x, 11) + assert out == S(3138428376722)/11 + 285311670611*log(11) + + +def test_issue_21701(): + assert limit((besselj(z, x)/x**z).subs(z, 7), x, 0) == S(1)/645120 + + +def test_issue_21721(): + a = Symbol('a', real=True) + I = integrate(1/(pi*(1 + (x - a)**2)), x) + assert I.limit(x, oo) == S.Half + + +def test_issue_21756(): + term = (1 - exp(-2*I*pi*z))/(1 - exp(-2*I*pi*z/5)) + assert term.limit(z, 0) == 5 + assert re(term).limit(z, 0) == 5 + + +def test_issue_21785(): + a = Symbol('a') + assert sqrt((-a**2 + x**2)/(1 - x**2)).limit(a, 1, '-') == I + + +def test_issue_22181(): + assert limit((-1)**x * 2**(-x), x, oo) == 0 + + +def test_issue_22220(): + e1 = sqrt(30)*atan(sqrt(30)*tan(x/2)/6)/30 + e2 = sqrt(30)*I*(-log(sqrt(2)*tan(x/2) - 2*sqrt(15)*I/5) + + +log(sqrt(2)*tan(x/2) + 2*sqrt(15)*I/5))/60 + + assert limit(e1, x, -pi) == -sqrt(30)*pi/60 + assert limit(e2, x, -pi) == -sqrt(30)*pi/30 + + assert limit(e1, x, -pi, '-') == sqrt(30)*pi/60 + assert limit(e2, x, -pi, '-') == 0 + + # test https://github.com/sympy/sympy/issues/22220#issuecomment-972727694 + expr = log(x - I) - log(-x - I) + expr2 = logcombine(expr, force=True) + assert limit(expr, x, oo) == limit(expr2, x, oo) == I*pi + + # test https://github.com/sympy/sympy/issues/22220#issuecomment-1077618340 + expr = expr = (-log(tan(x/2) - I) +log(tan(x/2) + I)) + assert limit(expr, x, pi, '+') == 2*I*pi + assert limit(expr, x, pi, '-') == 0 + + +def test_issue_22334(): + k, n = symbols('k, n', positive=True) + assert limit((n+1)**k/((n+1)**(k+1) - (n)**(k+1)), n, oo) == 1/(k + 1) + assert limit((n+1)**k/((n+1)**(k+1) - (n)**(k+1)).expand(), n, oo) == 1/(k + 1) + assert limit((n+1)**k/(n*(-n**k + (n + 1)**k) + (n + 1)**k), n, oo) == 1/(k + 1) + + +def test_issue_22836_limit(): + assert limit(2**(1/x)/factorial(1/(x)), x, 0) == S.Zero + + +def test_sympyissue_22986(): + assert limit(acosh(1 + 1/x)*sqrt(x), x, oo) == sqrt(2) + + +def test_issue_23231(): + f = (2**x - 2**(-x))/(2**x + 2**(-x)) + assert limit(f, x, -oo) == -1 + + +def test_issue_23596(): + assert integrate(((1 + x)/x**2)*exp(-1/x), (x, 0, oo)) == oo + + +def test_issue_23752(): + expr1 = sqrt(-I*x**2 + x - 3) + expr2 = sqrt(-I*x**2 + I*x - 3) + assert limit(expr1, x, 0, '+') == -sqrt(3)*I + assert limit(expr1, x, 0, '-') == -sqrt(3)*I + assert limit(expr2, x, 0, '+') == sqrt(3)*I + assert limit(expr2, x, 0, '-') == -sqrt(3)*I + + +def test_issue_24276(): + fx = log(tan(pi/2*tanh(x))).diff(x) + assert fx.limit(x, oo) == 2 + assert fx.simplify().limit(x, oo) == 2 + assert fx.rewrite(sin).limit(x, oo) == 2 + assert fx.rewrite(sin).simplify().limit(x, oo) == 2 + +def test_issue_25230(): + a = Symbol('a', real = True) + b = Symbol('b', positive = True) + c = Symbol('c', negative = True) + n = Symbol('n', integer = True) + raises(NotImplementedError, lambda: limit(Mod(x, a), x, a)) + assert limit(Mod(x, b), x, n*b, '+') == 0 + assert limit(Mod(x, b), x, n*b, '-') == b + assert limit(Mod(x, c), x, n*c, '+') == c + assert limit(Mod(x, c), x, n*c, '-') == 0 + + +def test_issue_25582(): + + assert limit(asin(exp(x)), x, oo, '-') == -oo*I + assert limit(acos(exp(x)), x, oo, '-') == oo*I + assert limit(atan(exp(x)), x, oo, '-') == pi/2 + assert limit(acot(exp(x)), x, oo, '-') == 0 + assert limit(asec(exp(x)), x, oo, '-') == pi/2 + assert limit(acsc(exp(x)), x, oo, '-') == 0 + + +def test_issue_25847(): + #atan + assert limit(atan(sin(x)/x), x, 0, '+-') == pi/4 + assert limit(atan(exp(1/x)), x, 0, '+') == pi/2 + assert limit(atan(exp(1/x)), x, 0, '-') == 0 + + #asin + assert limit(asin(sin(x)/x), x, 0, '+-') == pi/2 + assert limit(asin(exp(1/x)), x, 0, '+') == -oo*I + assert limit(asin(exp(1/x)), x, 0, '-') == 0 + + #acos + assert limit(acos(sin(x)/x), x, 0, '+-') == 0 + assert limit(acos(exp(1/x)), x, 0, '+') == oo*I + assert limit(acos(exp(1/x)), x, 0, '-') == pi/2 + + #acot + assert limit(acot(sin(x)/x), x, 0, '+-') == pi/4 + assert limit(acot(exp(1/x)), x, 0, '+') == 0 + assert limit(acot(exp(1/x)), x, 0, '-') == pi/2 + + #asec + assert limit(asec(sin(x)/x), x, 0, '+-') == 0 + assert limit(asec(exp(1/x)), x, 0, '+') == pi/2 + assert limit(asec(exp(1/x)), x, 0, '-') == oo*I + + #acsc + assert limit(acsc(sin(x)/x), x, 0, '+-') == pi/2 + assert limit(acsc(exp(1/x)), x, 0, '+') == 0 + assert limit(acsc(exp(1/x)), x, 0, '-') == -oo*I + + #atanh + assert limit(atanh(sin(x)/x), x, 0, '+-') == oo + assert limit(atanh(exp(1/x)), x, 0, '+') == -I*pi/2 + assert limit(atanh(exp(1/x)), x, 0, '-') == 0 + + #asinh + assert limit(asinh(sin(x)/x), x, 0, '+-') == log(1 + sqrt(2)) + assert limit(asinh(exp(1/x)), x, 0, '+') == oo + assert limit(asinh(exp(1/x)), x, 0, '-') == 0 + + #acosh + assert limit(acosh(sin(x)/x), x, 0, '+-') == 0 + assert limit(acosh(exp(1/x)), x, 0, '+') == oo + assert limit(acosh(exp(1/x)), x, 0, '-') == I*pi/2 + + #acoth + assert limit(acoth(sin(x)/x), x, 0, '+-') == oo + assert limit(acoth(exp(1/x)), x, 0, '+') == 0 + assert limit(acoth(exp(1/x)), x, 0, '-') == -I*pi/2 + + #asech + assert limit(asech(sin(x)/x), x, 0, '+-') == 0 + assert limit(asech(exp(1/x)), x, 0, '+') == I*pi/2 + assert limit(asech(exp(1/x)), x, 0, '-') == oo + + #acsch + assert limit(acsch(sin(x)/x), x, 0, '+-') == log(1 + sqrt(2)) + assert limit(acsch(exp(1/x)), x, 0, '+') == 0 + assert limit(acsch(exp(1/x)), x, 0, '-') == oo + + +def test_issue_26040(): + assert limit(besseli(0, x + 1)/besseli(0, x), x, oo) == S.Exp1 + + +def test_issue_26250(): + e = elliptic_e(4*x/(x**2 + 2*x + 1)) + k = elliptic_k(4*x/(x**2 + 2*x + 1)) + e1 = ((1-3*x**2)*e**2/2 - (x**2-2*x+1)*e*k/2) + e2 = pi**2*(x**8 - 2*x**7 - x**6 + 4*x**5 - x**4 - 2*x**3 + x**2) + assert limit(e1/e2, x, 0) == -S(1)/8 + + +def test_issue_26513(): + assert limit(abs((-x/(x+1))**x), x ,oo) == exp(-1) + assert limit((x/(x + 1))**x, x, oo) == exp(-1) + raises (NotImplementedError, lambda: limit((-x/(x+1))**x, x, oo)) + + +def test_issue_26916(): + assert limit(Ei(x)*exp(-x), x, +oo) == 0 + assert limit(Ei(x)*exp(-x), x, -oo) == 0 + + +def test_issue_22982_15323(): + assert limit((log(E + 1/x) - 1)**(1 - sqrt(E + 1/x)), x, oo) == oo + assert limit((1 - 1/x)**x*(log(1 - 1/x) + 1/(x*(1 - 1/x))), x, 1, dir='+') == 1 + assert limit((log(E + 1/x) )**(1 - sqrt(E + 1/x)), x, oo) == 1 + assert limit((log(E + 1/x) - 1)**(- sqrt(E + 1/x)), x, oo) == oo + + +def test_issue_26991(): + assert limit(x/((x - 6)*sinh(tanh(0.03*x)) + tanh(x) - 0.5), x, oo) == 1/sinh(1) + +def test_issue_27278(): + expr = (1/(x*log((x + 3)/x)))**x*((x + 1)*log((x + 4)/(x + 1)))**(x + 1)/3 + assert limit(expr, x, oo) == 1 diff --git a/.venv/lib/python3.12/site-packages/sympy/series/tests/test_limitseq.py b/.venv/lib/python3.12/site-packages/sympy/series/tests/test_limitseq.py new file mode 100644 index 0000000000000000000000000000000000000000..362bb0397feb0ec63929920855c81279eca0bd6a --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/tests/test_limitseq.py @@ -0,0 +1,177 @@ +from sympy.concrete.summations import Sum +from sympy.core.add import Add +from sympy.core.numbers import (I, Rational, oo, pi) +from sympy.core.singleton import S +from sympy.core.symbol import (Symbol, symbols) +from sympy.functions.combinatorial.factorials import (binomial, factorial, subfactorial) +from sympy.functions.combinatorial.numbers import (fibonacci, harmonic) +from sympy.functions.elementary.exponential import (exp, log) +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.functions.special.gamma_functions import gamma +from sympy.series.limitseq import limit_seq +from sympy.series.limitseq import difference_delta as dd +from sympy.testing.pytest import raises, XFAIL +from sympy.calculus.accumulationbounds import AccumulationBounds + +n, m, k = symbols('n m k', integer=True) + + +def test_difference_delta(): + e = n*(n + 1) + e2 = e * k + + assert dd(e) == 2*n + 2 + assert dd(e2, n, 2) == k*(4*n + 6) + + raises(ValueError, lambda: dd(e2)) + raises(ValueError, lambda: dd(e2, n, oo)) + + +def test_difference_delta__Sum(): + e = Sum(1/k, (k, 1, n)) + assert dd(e, n) == 1/(n + 1) + assert dd(e, n, 5) == Add(*[1/(i + n + 1) for i in range(5)]) + + e = Sum(1/k, (k, 1, 3*n)) + assert dd(e, n) == Add(*[1/(i + 3*n + 1) for i in range(3)]) + + e = n * Sum(1/k, (k, 1, n)) + assert dd(e, n) == 1 + Sum(1/k, (k, 1, n)) + + e = Sum(1/k, (k, 1, n), (m, 1, n)) + assert dd(e, n) == harmonic(n) + + +def test_difference_delta__Add(): + e = n + n*(n + 1) + assert dd(e, n) == 2*n + 3 + assert dd(e, n, 2) == 4*n + 8 + + e = n + Sum(1/k, (k, 1, n)) + assert dd(e, n) == 1 + 1/(n + 1) + assert dd(e, n, 5) == 5 + Add(*[1/(i + n + 1) for i in range(5)]) + + +def test_difference_delta__Pow(): + e = 4**n + assert dd(e, n) == 3*4**n + assert dd(e, n, 2) == 15*4**n + + e = 4**(2*n) + assert dd(e, n) == 15*4**(2*n) + assert dd(e, n, 2) == 255*4**(2*n) + + e = n**4 + assert dd(e, n) == (n + 1)**4 - n**4 + + e = n**n + assert dd(e, n) == (n + 1)**(n + 1) - n**n + + +def test_limit_seq(): + e = binomial(2*n, n) / Sum(binomial(2*k, k), (k, 1, n)) + assert limit_seq(e) == S(3) / 4 + assert limit_seq(e, m) == e + + e = (5*n**3 + 3*n**2 + 4) / (3*n**3 + 4*n - 5) + assert limit_seq(e, n) == S(5) / 3 + + e = (harmonic(n) * Sum(harmonic(k), (k, 1, n))) / (n * harmonic(2*n)**2) + assert limit_seq(e, n) == 1 + + e = Sum(k**2 * Sum(2**m/m, (m, 1, k)), (k, 1, n)) / (2**n*n) + assert limit_seq(e, n) == 4 + + e = (Sum(binomial(3*k, k) * binomial(5*k, k), (k, 1, n)) / + (binomial(3*n, n) * binomial(5*n, n))) + assert limit_seq(e, n) == S(84375) / 83351 + + e = Sum(harmonic(k)**2/k, (k, 1, 2*n)) / harmonic(n)**3 + assert limit_seq(e, n) == S.One / 3 + + raises(ValueError, lambda: limit_seq(e * m)) + + +def test_alternating_sign(): + assert limit_seq((-1)**n/n**2, n) == 0 + assert limit_seq((-2)**(n+1)/(n + 3**n), n) == 0 + assert limit_seq((2*n + (-1)**n)/(n + 1), n) == 2 + assert limit_seq(sin(pi*n), n) == 0 + assert limit_seq(cos(2*pi*n), n) == 1 + assert limit_seq((S.NegativeOne/5)**n, n) == 0 + assert limit_seq((Rational(-1, 5))**n, n) == 0 + assert limit_seq((I/3)**n, n) == 0 + assert limit_seq(sqrt(n)*(I/2)**n, n) == 0 + assert limit_seq(n**7*(I/3)**n, n) == 0 + assert limit_seq(n/(n + 1) + (I/2)**n, n) == 1 + + +def test_accum_bounds(): + assert limit_seq((-1)**n, n) == AccumulationBounds(-1, 1) + assert limit_seq(cos(pi*n), n) == AccumulationBounds(-1, 1) + assert limit_seq(sin(pi*n/2)**2, n) == AccumulationBounds(0, 1) + assert limit_seq(2*(-3)**n/(n + 3**n), n) == AccumulationBounds(-2, 2) + assert limit_seq(3*n/(n + 1) + 2*(-1)**n, n) == AccumulationBounds(1, 5) + + +def test_limitseq_sum(): + from sympy.abc import x, y, z + assert limit_seq(Sum(1/x, (x, 1, y)) - log(y), y) == S.EulerGamma + assert limit_seq(Sum(1/x, (x, 1, y)) - 1/y, y) is S.Infinity + assert (limit_seq(binomial(2*x, x) / Sum(binomial(2*y, y), (y, 1, x)), x) == + S(3) / 4) + assert (limit_seq(Sum(y**2 * Sum(2**z/z, (z, 1, y)), (y, 1, x)) / + (2**x*x), x) == 4) + + +def test_issue_9308(): + assert limit_seq(subfactorial(n)/factorial(n), n) == exp(-1) + + +def test_issue_10382(): + n = Symbol('n', integer=True) + assert limit_seq(fibonacci(n+1)/fibonacci(n), n).together() == S.GoldenRatio + + +def test_issue_11672(): + assert limit_seq(Rational(-1, 2)**n, n) == 0 + + +def test_issue_14196(): + k, n = symbols('k, n', positive=True) + m = Symbol('m') + assert limit_seq(Sum(m**k, (m, 1, n)).doit()/(n**(k + 1)), n) == 1/(k + 1) + + +def test_issue_16735(): + assert limit_seq(5**n/factorial(n), n) == 0 + + +def test_issue_19868(): + assert limit_seq(1/gamma(n + S.One/2), n) == 0 + + +@XFAIL +def test_limit_seq_fail(): + # improve Summation algorithm or add ad-hoc criteria + e = (harmonic(n)**3 * Sum(1/harmonic(k), (k, 1, n)) / + (n * Sum(harmonic(k)/k, (k, 1, n)))) + assert limit_seq(e, n) == 2 + + # No unique dominant term + e = (Sum(2**k * binomial(2*k, k) / k**2, (k, 1, n)) / + (Sum(2**k/k*2, (k, 1, n)) * Sum(binomial(2*k, k), (k, 1, n)))) + assert limit_seq(e, n) == S(3) / 7 + + # Simplifications of summations needs to be improved. + e = n**3*Sum(2**k/k**2, (k, 1, n))**2 / (2**n * Sum(2**k/k, (k, 1, n))) + assert limit_seq(e, n) == 2 + + e = (harmonic(n) * Sum(2**k/k, (k, 1, n)) / + (n * Sum(2**k*harmonic(k)/k**2, (k, 1, n)))) + assert limit_seq(e, n) == 1 + + e = (Sum(2**k*factorial(k) / k**2, (k, 1, 2*n)) / + (Sum(4**k/k**2, (k, 1, n)) * Sum(factorial(k), (k, 1, 2*n)))) + assert limit_seq(e, n) == S(3) / 16 diff --git a/.venv/lib/python3.12/site-packages/sympy/series/tests/test_lseries.py b/.venv/lib/python3.12/site-packages/sympy/series/tests/test_lseries.py new file mode 100644 index 0000000000000000000000000000000000000000..42d327bf60c76eebdc4570d631efef4bc84b58e3 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/tests/test_lseries.py @@ -0,0 +1,65 @@ +from sympy.core.numbers import E +from sympy.core.singleton import S +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.hyperbolic import tanh +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.series.order import Order +from sympy.abc import x, y + + +def test_sin(): + e = sin(x).lseries(x) + assert next(e) == x + assert next(e) == -x**3/6 + assert next(e) == x**5/120 + + +def test_cos(): + e = cos(x).lseries(x) + assert next(e) == 1 + assert next(e) == -x**2/2 + assert next(e) == x**4/24 + + +def test_exp(): + e = exp(x).lseries(x) + assert next(e) == 1 + assert next(e) == x + assert next(e) == x**2/2 + assert next(e) == x**3/6 + + +def test_exp2(): + e = exp(cos(x)).lseries(x) + assert next(e) == E + assert next(e) == -E*x**2/2 + assert next(e) == E*x**4/6 + assert next(e) == -31*E*x**6/720 + + +def test_simple(): + assert list(x.lseries()) == [x] + assert list(S.One.lseries(x)) == [1] + assert not next((x/(x + y)).lseries(y)).has(Order) + + +def test_issue_5183(): + s = (x + 1/x).lseries() + assert list(s) == [1/x, x] + assert next((x + x**2).lseries()) == x + assert next(((1 + x)**7).lseries(x)) == 1 + assert next((sin(x + y)).series(x, n=3).lseries(y)) == x + # it would be nice if all terms were grouped, but in the + # following case that would mean that all the terms would have + # to be known since, for example, every term has a constant in it. + s = ((1 + x)**7).series(x, 1, n=None) + assert [next(s) for i in range(2)] == [128, -448 + 448*x] + + +def test_issue_6999(): + s = tanh(x).lseries(x, 1) + assert next(s) == tanh(1) + assert next(s) == x - (x - 1)*tanh(1)**2 - 1 + assert next(s) == -(x - 1)**2*tanh(1) + (x - 1)**2*tanh(1)**3 + assert next(s) == -(x - 1)**3*tanh(1)**4 - (x - 1)**3/3 + \ + 4*(x - 1)**3*tanh(1)**2/3 diff --git a/.venv/lib/python3.12/site-packages/sympy/series/tests/test_nseries.py b/.venv/lib/python3.12/site-packages/sympy/series/tests/test_nseries.py new file mode 100644 index 0000000000000000000000000000000000000000..a2f20add82d3e858e2ce145fc9fcd4a6548a48cc --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/tests/test_nseries.py @@ -0,0 +1,557 @@ +from sympy.calculus.util import AccumBounds +from sympy.core.function import (Derivative, PoleError) +from sympy.core.numbers import (E, I, Integer, Rational, pi) +from sympy.core.singleton import S +from sympy.core.symbol import (Symbol, symbols) +from sympy.functions.elementary.complexes import sign +from sympy.functions.elementary.exponential import (exp, log) +from sympy.functions.elementary.hyperbolic import (acosh, acoth, asinh, atanh, cosh, coth, sinh, tanh) +from sympy.functions.elementary.integers import (ceiling, floor, frac) +from sympy.functions.elementary.miscellaneous import (cbrt, sqrt) +from sympy.functions.elementary.trigonometric import (asin, cos, cot, sin, tan) +from sympy.series.limits import limit +from sympy.series.order import O +from sympy.abc import x, y, z + +from sympy.testing.pytest import raises, XFAIL + + +def test_simple_1(): + assert x.nseries(x, n=5) == x + assert y.nseries(x, n=5) == y + assert (1/(x*y)).nseries(y, n=5) == 1/(x*y) + assert Rational(3, 4).nseries(x, n=5) == Rational(3, 4) + assert x.nseries() == x + + +def test_mul_0(): + assert (x*log(x)).nseries(x, n=5) == x*log(x) + + +def test_mul_1(): + assert (x*log(2 + x)).nseries(x, n=5) == x*log(2) + x**2/2 - x**3/8 + \ + x**4/24 + O(x**5) + assert (x*log(1 + x)).nseries( + x, n=5) == x**2 - x**3/2 + x**4/3 + O(x**5) + + +def test_pow_0(): + assert (x**2).nseries(x, n=5) == x**2 + assert (1/x).nseries(x, n=5) == 1/x + assert (1/x**2).nseries(x, n=5) == 1/x**2 + assert (x**Rational(2, 3)).nseries(x, n=5) == (x**Rational(2, 3)) + assert (sqrt(x)**3).nseries(x, n=5) == (sqrt(x)**3) + + +def test_pow_1(): + assert ((1 + x)**2).nseries(x, n=5) == x**2 + 2*x + 1 + + # https://github.com/sympy/sympy/issues/21075 + assert ((sqrt(x) + 1)**2).nseries(x) == 2*sqrt(x) + x + 1 + assert ((sqrt(x) + cbrt(x))**2).nseries(x) == 2*x**Rational(5, 6)\ + + x**Rational(2, 3) + x + + +def test_geometric_1(): + assert (1/(1 - x)).nseries(x, n=5) == 1 + x + x**2 + x**3 + x**4 + O(x**5) + assert (x/(1 - x)).nseries(x, n=6) == x + x**2 + x**3 + x**4 + x**5 + O(x**6) + assert (x**3/(1 - x)).nseries(x, n=8) == x**3 + x**4 + x**5 + x**6 + \ + x**7 + O(x**8) + + +def test_sqrt_1(): + assert sqrt(1 + x).nseries(x, n=5) == 1 + x/2 - x**2/8 + x**3/16 - 5*x**4/128 + O(x**5) + + +def test_exp_1(): + assert exp(x).nseries(x, n=5) == 1 + x + x**2/2 + x**3/6 + x**4/24 + O(x**5) + assert exp(x).nseries(x, n=12) == 1 + x + x**2/2 + x**3/6 + x**4/24 + x**5/120 + \ + x**6/720 + x**7/5040 + x**8/40320 + x**9/362880 + x**10/3628800 + \ + x**11/39916800 + O(x**12) + assert exp(1/x).nseries(x, n=5) == exp(1/x) + assert exp(1/(1 + x)).nseries(x, n=4) == \ + (E*(1 - x - 13*x**3/6 + 3*x**2/2)).expand() + O(x**4) + assert exp(2 + x).nseries(x, n=5) == \ + (exp(2)*(1 + x + x**2/2 + x**3/6 + x**4/24)).expand() + O(x**5) + + +def test_exp_sqrt_1(): + assert exp(1 + sqrt(x)).nseries(x, n=3) == \ + (exp(1)*(1 + sqrt(x) + x/2 + sqrt(x)*x/6)).expand() + O(sqrt(x)**3) + + +def test_power_x_x1(): + assert (exp(x*log(x))).nseries(x, n=4) == \ + 1 + x*log(x) + x**2*log(x)**2/2 + x**3*log(x)**3/6 + O(x**4*log(x)**4) + + +def test_power_x_x2(): + assert (x**x).nseries(x, n=4) == \ + 1 + x*log(x) + x**2*log(x)**2/2 + x**3*log(x)**3/6 + O(x**4*log(x)**4) + + +def test_log_singular1(): + assert log(1 + 1/x).nseries(x, n=5) == x - log(x) - x**2/2 + x**3/3 - \ + x**4/4 + O(x**5) + + +def test_log_power1(): + e = 1 / (1/x + x ** (log(3)/log(2))) + assert e.nseries(x, n=5) == -x**(log(3)/log(2) + 2) + x + O(x**5) + + +def test_log_series(): + l = Symbol('l') + e = 1/(1 - log(x)) + assert e.nseries(x, n=5, logx=l) == 1/(1 - l) + + +def test_log2(): + e = log(-1/x) + assert e.nseries(x, n=5) == -log(x) + log(-1) + + +def test_log3(): + l = Symbol('l') + e = 1/log(-1/x) + assert e.nseries(x, n=4, logx=l) == 1/(-l + log(-1)) + + +def test_series1(): + e = sin(x) + assert e.nseries(x, 0, 0) != 0 + assert e.nseries(x, 0, 0) == O(1, x) + assert e.nseries(x, 0, 1) == O(x, x) + assert e.nseries(x, 0, 2) == x + O(x**2, x) + assert e.nseries(x, 0, 3) == x + O(x**3, x) + assert e.nseries(x, 0, 4) == x - x**3/6 + O(x**4, x) + + e = (exp(x) - 1)/x + assert e.nseries(x, 0, 3) == 1 + x/2 + x**2/6 + O(x**3) + + assert x.nseries(x, 0, 2) == x + + +@XFAIL +def test_series1_failing(): + assert x.nseries(x, 0, 0) == O(1, x) + assert x.nseries(x, 0, 1) == O(x, x) + + +def test_seriesbug1(): + assert (1/x).nseries(x, 0, 3) == 1/x + assert (x + 1/x).nseries(x, 0, 3) == x + 1/x + + +def test_series2x(): + assert ((x + 1)**(-2)).nseries(x, 0, 4) == 1 - 2*x + 3*x**2 - 4*x**3 + O(x**4, x) + assert ((x + 1)**(-1)).nseries(x, 0, 4) == 1 - x + x**2 - x**3 + O(x**4, x) + assert ((x + 1)**0).nseries(x, 0, 3) == 1 + assert ((x + 1)**1).nseries(x, 0, 3) == 1 + x + assert ((x + 1)**2).nseries(x, 0, 3) == x**2 + 2*x + 1 + assert ((x + 1)**3).nseries(x, 0, 3) == 1 + 3*x + 3*x**2 + O(x**3) + + assert (1/(1 + x)).nseries(x, 0, 4) == 1 - x + x**2 - x**3 + O(x**4, x) + assert (x + 3/(1 + 2*x)).nseries(x, 0, 4) == 3 - 5*x + 12*x**2 - 24*x**3 + O(x**4, x) + + assert ((1/x + 1)**3).nseries(x, 0, 3) == 1 + 3/x + 3/x**2 + x**(-3) + assert (1/(1 + 1/x)).nseries(x, 0, 4) == x - x**2 + x**3 - O(x**4, x) + assert (1/(1 + 1/x**2)).nseries(x, 0, 6) == x**2 - x**4 + O(x**6, x) + + +def test_bug2(): # 1/log(0)*log(0) problem + w = Symbol("w") + e = (w**(-1) + w**( + -log(3)*log(2)**(-1)))**(-1)*(3*w**(-log(3)*log(2)**(-1)) + 2*w**(-1)) + e = e.expand() + assert e.nseries(w, 0, 4).subs(w, 0) == 3 + + +def test_exp(): + e = (1 + x)**(1/x) + assert e.nseries(x, n=3) == exp(1) - x*exp(1)/2 + 11*exp(1)*x**2/24 + O(x**3) + + +def test_exp2(): + w = Symbol("w") + e = w**(1 - log(x)/(log(2) + log(x))) + logw = Symbol("logw") + assert e.nseries( + w, 0, 1, logx=logw) == exp(logw*log(2)/(log(x) + log(2))) + + +def test_bug3(): + e = (2/x + 3/x**2)/(1/x + 1/x**2) + assert e.nseries(x, n=3) == 3 - x + x**2 + O(x**3) + + +def test_generalexponent(): + p = 2 + e = (2/x + 3/x**p)/(1/x + 1/x**p) + assert e.nseries(x, 0, 3) == 3 - x + x**2 + O(x**3) + p = S.Half + e = (2/x + 3/x**p)/(1/x + 1/x**p) + assert e.nseries(x, 0, 2) == 2 - x + sqrt(x) + x**(S(3)/2) + O(x**2) + + e = 1 + sqrt(x) + assert e.nseries(x, 0, 4) == 1 + sqrt(x) + +# more complicated example + + +def test_genexp_x(): + e = 1/(1 + sqrt(x)) + assert e.nseries(x, 0, 2) == \ + 1 + x - sqrt(x) - sqrt(x)**3 + O(x**2, x) + +# more complicated example + + +def test_genexp_x2(): + p = Rational(3, 2) + e = (2/x + 3/x**p)/(1/x + 1/x**p) + assert e.nseries(x, 0, 3) == 3 + x + x**2 - sqrt(x) - x**(S(3)/2) - x**(S(5)/2) + O(x**3) + + +def test_seriesbug2(): + w = Symbol("w") + #simple case (1): + e = ((2*w)/w)**(1 + w) + assert e.nseries(w, 0, 1) == 2 + O(w, w) + assert e.nseries(w, 0, 1).subs(w, 0) == 2 + + +def test_seriesbug2b(): + w = Symbol("w") + #test sin + e = sin(2*w)/w + assert e.nseries(w, 0, 3) == 2 - 4*w**2/3 + O(w**3) + + +def test_seriesbug2d(): + w = Symbol("w", real=True) + e = log(sin(2*w)/w) + assert e.series(w, n=5) == log(2) - 2*w**2/3 - 4*w**4/45 + O(w**5) + + +def test_seriesbug2c(): + w = Symbol("w", real=True) + #more complicated case, but sin(x)~x, so the result is the same as in (1) + e = (sin(2*w)/w)**(1 + w) + assert e.series(w, 0, 1) == 2 + O(w) + assert e.series(w, 0, 3) == 2 + 2*w*log(2) + \ + w**2*(Rational(-4, 3) + log(2)**2) + O(w**3) + assert e.series(w, 0, 2).subs(w, 0) == 2 + + +def test_expbug4(): + x = Symbol("x", real=True) + assert (log( + sin(2*x)/x)*(1 + x)).series(x, 0, 2) == log(2) + x*log(2) + O(x**2, x) + assert exp( + log(sin(2*x)/x)*(1 + x)).series(x, 0, 2) == 2 + 2*x*log(2) + O(x**2) + + assert exp(log(2) + O(x)).nseries(x, 0, 2) == 2 + O(x) + assert ((2 + O(x))**(1 + x)).nseries(x, 0, 2) == 2 + O(x) + + +def test_logbug4(): + assert log(2 + O(x)).nseries(x, 0, 2) == log(2) + O(x, x) + + +def test_expbug5(): + assert exp(log(1 + x)/x).nseries(x, n=3) == exp(1) + -exp(1)*x/2 + 11*exp(1)*x**2/24 + O(x**3) + + assert exp(O(x)).nseries(x, 0, 2) == 1 + O(x) + + +def test_sinsinbug(): + assert sin(sin(x)).nseries(x, 0, 8) == x - x**3/3 + x**5/10 - 8*x**7/315 + O(x**8) + + +def test_issue_3258(): + a = x/(exp(x) - 1) + assert a.nseries(x, 0, 5) == 1 - x/2 - x**4/720 + x**2/12 + O(x**5) + + +def test_issue_3204(): + x = Symbol("x", nonnegative=True) + f = sin(x**3)**Rational(1, 3) + assert f.nseries(x, 0, 17) == x - x**7/18 - x**13/3240 + O(x**17) + + +def test_issue_3224(): + f = sqrt(1 - sqrt(y)) + assert f.nseries(y, 0, 2) == 1 - sqrt(y)/2 - y/8 - sqrt(y)**3/16 + O(y**2) + + +def test_issue_3463(): + w, i = symbols('w,i') + r = log(5)/log(3) + p = w**(-1 + r) + e = 1/x*(-log(w**(1 + r)) + log(w + w**r)) + e_ser = -r*log(w)/x + p/x - p**2/(2*x) + O(w) + assert e.nseries(w, n=1) == e_ser + + +def test_sin(): + assert sin(8*x).nseries(x, n=4) == 8*x - 256*x**3/3 + O(x**4) + assert sin(x + y).nseries(x, n=1) == sin(y) + O(x) + assert sin(x + y).nseries(x, n=2) == sin(y) + cos(y)*x + O(x**2) + assert sin(x + y).nseries(x, n=5) == sin(y) + cos(y)*x - sin(y)*x**2/2 - \ + cos(y)*x**3/6 + sin(y)*x**4/24 + O(x**5) + + +def test_issue_3515(): + e = sin(8*x)/x + assert e.nseries(x, n=6) == 8 - 256*x**2/3 + 4096*x**4/15 + O(x**6) + + +def test_issue_3505(): + e = sin(x)**(-4)*(sqrt(cos(x))*sin(x)**2 - + cos(x)**Rational(1, 3)*sin(x)**2) + assert e.nseries(x, n=9) == Rational(-1, 12) - 7*x**2/288 - \ + 43*x**4/10368 - 1123*x**6/2488320 + 377*x**8/29859840 + O(x**9) + + +def test_issue_3501(): + a = Symbol("a") + e = x**(-2)*(x*sin(a + x) - x*sin(a)) + assert e.nseries(x, n=6) == cos(a) - sin(a)*x/2 - cos(a)*x**2/6 + \ + x**3*sin(a)/24 + x**4*cos(a)/120 - x**5*sin(a)/720 + O(x**6) + e = x**(-2)*(x*cos(a + x) - x*cos(a)) + assert e.nseries(x, n=6) == -sin(a) - cos(a)*x/2 + sin(a)*x**2/6 + \ + cos(a)*x**3/24 - x**4*sin(a)/120 - x**5*cos(a)/720 + O(x**6) + + +def test_issue_3502(): + e = sin(5*x)/sin(2*x) + assert e.nseries(x, n=2) == Rational(5, 2) + O(x**2) + assert e.nseries(x, n=6) == \ + Rational(5, 2) - 35*x**2/4 + 329*x**4/48 + O(x**6) + + +def test_issue_3503(): + e = sin(2 + x)/(2 + x) + assert e.nseries(x, n=2) == sin(2)/2 + x*cos(2)/2 - x*sin(2)/4 + O(x**2) + + +def test_issue_3506(): + e = (x + sin(3*x))**(-2)*(x*(x + sin(3*x)) - (x + sin(3*x))*sin(2*x)) + assert e.nseries(x, n=7) == \ + Rational(-1, 4) + 5*x**2/96 + 91*x**4/768 + 11117*x**6/129024 + O(x**7) + + +def test_issue_3508(): + x = Symbol("x", real=True) + assert log(sin(x)).series(x, n=5) == log(x) - x**2/6 - x**4/180 + O(x**5) + e = -log(x) + x*(-log(x) + log(sin(2*x))) + log(sin(2*x)) + assert e.series(x, n=5) == \ + log(2) + log(2)*x - 2*x**2/3 - 2*x**3/3 - 4*x**4/45 + O(x**5) + + +def test_issue_3507(): + e = x**(-4)*(x**2 - x**2*sqrt(cos(x))) + assert e.nseries(x, n=9) == \ + Rational(1, 4) + x**2/96 + 19*x**4/5760 + 559*x**6/645120 + 29161*x**8/116121600 + O(x**9) + + +def test_issue_3639(): + assert sin(cos(x)).nseries(x, n=5) == \ + sin(1) - x**2*cos(1)/2 - x**4*sin(1)/8 + x**4*cos(1)/24 + O(x**5) + + +def test_hyperbolic(): + assert sinh(x).nseries(x, n=6) == x + x**3/6 + x**5/120 + O(x**6) + assert cosh(x).nseries(x, n=5) == 1 + x**2/2 + x**4/24 + O(x**5) + assert tanh(x).nseries(x, n=6) == x - x**3/3 + 2*x**5/15 + O(x**6) + assert coth(x).nseries(x, n=6) == \ + 1/x - x**3/45 + x/3 + 2*x**5/945 + O(x**6) + assert asinh(x).nseries(x, n=6) == x - x**3/6 + 3*x**5/40 + O(x**6) + assert acosh(x).nseries(x, n=6) == \ + pi*I/2 - I*x - 3*I*x**5/40 - I*x**3/6 + O(x**6) + assert atanh(x).nseries(x, n=6) == x + x**3/3 + x**5/5 + O(x**6) + assert acoth(x).nseries(x, n=6) == -I*pi/2 + x + x**3/3 + x**5/5 + O(x**6) + + +def test_series2(): + w = Symbol("w", real=True) + x = Symbol("x", real=True) + e = w**(-2)*(w*exp(1/x - w) - w*exp(1/x)) + assert e.nseries(w, n=4) == -exp(1/x) + w*exp(1/x)/2 - w**2*exp(1/x)/6 + w**3*exp(1/x)/24 + O(w**4) + + +def test_series3(): + w = Symbol("w", real=True) + e = w**(-6)*(w**3*tan(w) - w**3*sin(w)) + assert e.nseries(w, n=8) == Integer(1)/2 + w**2/8 + 13*w**4/240 + 529*w**6/24192 + O(w**8) + + +def test_bug4(): + w = Symbol("w") + e = x/(w**4 + x**2*w**4 + 2*x*w**4)*w**4 + assert e.nseries(w, n=2).removeO().expand() in [x/(1 + 2*x + x**2), + 1/(1 + x/2 + 1/x/2)/2, 1/x/(1 + 2/x + x**(-2))] + + +def test_bug5(): + w = Symbol("w") + l = Symbol('l') + e = (-log(w) + log(1 + w*log(x)))**(-2)*w**(-2)*((-log(w) + + log(1 + x*w))*(-log(w) + log(1 + w*log(x)))*w - x*(-log(w) + + log(1 + w*log(x)))*w) + assert e.nseries(w, n=0, logx=l) == x/w/l + 1/w + O(1, w) + assert e.nseries(w, n=1, logx=l) == x/w/l + 1/w - x/l + 1/l*log(x) \ + + x*log(x)/l**2 + O(w) + + +def test_issue_4115(): + assert (sin(x)/(1 - cos(x))).nseries(x, n=1) == 2/x + O(x) + assert (sin(x)**2/(1 - cos(x))).nseries(x, n=1) == 2 + O(x) + + +def test_pole(): + raises(PoleError, lambda: sin(1/x).series(x, 0, 5)) + raises(PoleError, lambda: sin(1 + 1/x).series(x, 0, 5)) + raises(PoleError, lambda: (x*sin(1/x)).series(x, 0, 5)) + + +def test_expsinbug(): + assert exp(sin(x)).series(x, 0, 0) == O(1, x) + assert exp(sin(x)).series(x, 0, 1) == 1 + O(x) + assert exp(sin(x)).series(x, 0, 2) == 1 + x + O(x**2) + assert exp(sin(x)).series(x, 0, 3) == 1 + x + x**2/2 + O(x**3) + assert exp(sin(x)).series(x, 0, 4) == 1 + x + x**2/2 + O(x**4) + assert exp(sin(x)).series(x, 0, 5) == 1 + x + x**2/2 - x**4/8 + O(x**5) + + +def test_floor(): + x = Symbol('x') + assert floor(x).series(x) == 0 + assert floor(-x).series(x) == -1 + assert floor(sin(x)).series(x) == 0 + assert floor(sin(-x)).series(x) == -1 + assert floor(x**3).series(x) == 0 + assert floor(-x**3).series(x) == -1 + assert floor(cos(x)).series(x) == 0 + assert floor(cos(-x)).series(x) == 0 + assert floor(5 + sin(x)).series(x) == 5 + assert floor(5 + sin(-x)).series(x) == 4 + + assert floor(x).series(x, 2) == 2 + assert floor(-x).series(x, 2) == -3 + + x = Symbol('x', negative=True) + assert floor(x + 1.5).series(x) == 1 + + +def test_frac(): + assert frac(x).series(x, cdir=1) == x + assert frac(x).series(x, cdir=-1) == 1 + x + assert frac(2*x + 1).series(x, cdir=1) == 2*x + assert frac(2*x + 1).series(x, cdir=-1) == 1 + 2*x + assert frac(x**2).series(x, cdir=1) == x**2 + assert frac(x**2).series(x, cdir=-1) == x**2 + assert frac(sin(x) + 5).series(x, cdir=1) == x - x**3/6 + x**5/120 + O(x**6) + assert frac(sin(x) + 5).series(x, cdir=-1) == 1 + x - x**3/6 + x**5/120 + O(x**6) + assert frac(sin(x) + S.Half).series(x) == S.Half + x - x**3/6 + x**5/120 + O(x**6) + assert frac(x**8).series(x, cdir=1) == O(x**6) + assert frac(1/x).series(x) == AccumBounds(0, 1) + O(x**6) + + +def test_ceiling(): + assert ceiling(x).series(x) == 1 + assert ceiling(-x).series(x) == 0 + assert ceiling(sin(x)).series(x) == 1 + assert ceiling(sin(-x)).series(x) == 0 + assert ceiling(1 - cos(x)).series(x) == 1 + assert ceiling(1 - cos(-x)).series(x) == 1 + assert ceiling(x).series(x, 2) == 3 + assert ceiling(-x).series(x, 2) == -2 + + +def test_abs(): + a = Symbol('a') + assert abs(x).nseries(x, n=4) == x + assert abs(-x).nseries(x, n=4) == x + assert abs(x + 1).nseries(x, n=4) == x + 1 + assert abs(sin(x)).nseries(x, n=4) == x - Rational(1, 6)*x**3 + O(x**4) + assert abs(sin(-x)).nseries(x, n=4) == x - Rational(1, 6)*x**3 + O(x**4) + assert abs(x - a).nseries(x, 1) == -a*sign(1 - a) + (x - 1)*sign(1 - a) + sign(1 - a) + + +def test_dir(): + assert abs(x).series(x, 0, dir="+") == x + assert abs(x).series(x, 0, dir="-") == -x + assert floor(x + 2).series(x, 0, dir='+') == 2 + assert floor(x + 2).series(x, 0, dir='-') == 1 + assert floor(x + 2.2).series(x, 0, dir='-') == 2 + assert ceiling(x + 2.2).series(x, 0, dir='-') == 3 + assert sin(x + y).series(x, 0, dir='-') == sin(x + y).series(x, 0, dir='+') + + +def test_cdir(): + assert abs(x).series(x, 0, cdir=1) == x + assert abs(x).series(x, 0, cdir=-1) == -x + assert floor(x + 2).series(x, 0, cdir=1) == 2 + assert floor(x + 2).series(x, 0, cdir=-1) == 1 + assert floor(x + 2.2).series(x, 0, cdir=1) == 2 + assert ceiling(x + 2.2).series(x, 0, cdir=-1) == 3 + assert sin(x + y).series(x, 0, cdir=-1) == sin(x + y).series(x, 0, cdir=1) + + +def test_issue_3504(): + a = Symbol("a") + e = asin(a*x)/x + assert e.series(x, 4, n=2).removeO() == \ + (x - 4)*(a/(4*sqrt(-16*a**2 + 1)) - asin(4*a)/16) + asin(4*a)/4 + + +def test_issue_4441(): + a, b = symbols('a,b') + f = 1/(1 + a*x) + assert f.series(x, 0, 5) == 1 - a*x + a**2*x**2 - a**3*x**3 + \ + a**4*x**4 + O(x**5) + f = 1/(1 + (a + b)*x) + assert f.series(x, 0, 3) == 1 + x*(-a - b)\ + + x**2*(a + b)**2 + O(x**3) + + +def test_issue_4329(): + assert tan(x).series(x, pi/2, n=3).removeO() == \ + -pi/6 + x/3 - 1/(x - pi/2) + assert cot(x).series(x, pi, n=3).removeO() == \ + -x/3 + pi/3 + 1/(x - pi) + assert limit(tan(x)**tan(2*x), x, pi/4) == exp(-1) + + +def test_issue_5183(): + assert abs(x + x**2).series(n=1) == O(x) + assert abs(x + x**2).series(n=2) == x + O(x**2) + assert ((1 + x)**2).series(x, n=6) == x**2 + 2*x + 1 + assert (1 + 1/x).series() == 1 + 1/x + assert Derivative(exp(x).series(), x).doit() == \ + 1 + x + x**2/2 + x**3/6 + x**4/24 + O(x**5) + + +def test_issue_5654(): + a = Symbol('a') + assert (1/(x**2+a**2)**2).nseries(x, x0=I*a, n=0) == \ + -I/(4*a**3*(-I*a + x)) - 1/(4*a**2*(-I*a + x)**2) + O(1, (x, I*a)) + assert (1/(x**2+a**2)**2).nseries(x, x0=I*a, n=1) == 3/(16*a**4) \ + -I/(4*a**3*(-I*a + x)) - 1/(4*a**2*(-I*a + x)**2) + O(-I*a + x, (x, I*a)) + + +def test_issue_5925(): + sx = sqrt(x + z).series(z, 0, 1) + sxy = sqrt(x + y + z).series(z, 0, 1) + s1, s2 = sx.subs(x, x + y), sxy + assert (s1 - s2).expand().removeO().simplify() == 0 + + sx = sqrt(x + z).series(z, 0, 1) + sxy = sqrt(x + y + z).series(z, 0, 1) + assert sxy.subs({x:1, y:2}) == sx.subs(x, 3) + + +def test_exp_2(): + assert exp(x**3).nseries(x, 0, 14) == 1 + x**3 + x**6/2 + x**9/6 + x**12/24 + O(x**14) diff --git a/.venv/lib/python3.12/site-packages/sympy/series/tests/test_order.py b/.venv/lib/python3.12/site-packages/sympy/series/tests/test_order.py new file mode 100644 index 0000000000000000000000000000000000000000..50fcb861ee2a76c730baae6d26cc1e7a00347176 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/tests/test_order.py @@ -0,0 +1,503 @@ +from sympy.core.add import Add +from sympy.core.function import (Function, expand) +from sympy.core.numbers import (I, Rational, nan, oo, pi) +from sympy.core.singleton import S +from sympy.core.symbol import (Symbol, symbols) +from sympy.functions.combinatorial.factorials import factorial +from sympy.functions.elementary.complexes import (conjugate, transpose) +from sympy.functions.elementary.exponential import (exp, log) +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.integrals.integrals import Integral +from sympy.series.order import O, Order +from sympy.core.expr import unchanged +from sympy.testing.pytest import raises +from sympy.abc import w, x, y, z +from sympy.testing.pytest import XFAIL + + +def test_caching_bug(): + #needs to be a first test, so that all caches are clean + #cache it + O(w) + #and test that this won't raise an exception + O(w**(-1/x/log(3)*log(5)), w) + + +def test_free_symbols(): + assert Order(1).free_symbols == set() + assert Order(x).free_symbols == {x} + assert Order(1, x).free_symbols == {x} + assert Order(x*y).free_symbols == {x, y} + assert Order(x, x, y).free_symbols == {x, y} + + +def test_simple_1(): + o = Rational(0) + assert Order(2*x) == Order(x) + assert Order(x)*3 == Order(x) + assert -28*Order(x) == Order(x) + assert Order(Order(x)) == Order(x) + assert Order(Order(x), y) == Order(Order(x), x, y) + assert Order(-23) == Order(1) + assert Order(exp(x)) == Order(1, x) + assert Order(exp(1/x)).expr == exp(1/x) + assert Order(x*exp(1/x)).expr == x*exp(1/x) + assert Order(x**(o/3)).expr == x**(o/3) + assert Order(x**(o*Rational(5, 3))).expr == x**(o*Rational(5, 3)) + assert Order(x**2 + x + y, x) == O(1, x) + assert Order(x**2 + x + y, y) == O(1, y) + raises(ValueError, lambda: Order(exp(x), x, x)) + raises(TypeError, lambda: Order(x, 2 - x)) + + +def test_simple_2(): + assert Order(2*x)*x == Order(x**2) + assert Order(2*x)/x == Order(1, x) + assert Order(2*x)*x*exp(1/x) == Order(x**2*exp(1/x)) + assert (Order(2*x)*x*exp(1/x)/log(x)**3).expr == x**2*exp(1/x)*log(x)**-3 + + +def test_simple_3(): + assert Order(x) + x == Order(x) + assert Order(x) + 2 == 2 + Order(x) + assert Order(x) + x**2 == Order(x) + assert Order(x) + 1/x == 1/x + Order(x) + assert Order(1/x) + 1/x**2 == 1/x**2 + Order(1/x) + assert Order(x) + exp(1/x) == Order(x) + exp(1/x) + + +def test_simple_4(): + assert Order(x)**2 == Order(x**2) + + +def test_simple_5(): + assert Order(x) + Order(x**2) == Order(x) + assert Order(x) + Order(x**-2) == Order(x**-2) + assert Order(x) + Order(1/x) == Order(1/x) + + +def test_simple_6(): + assert Order(x) - Order(x) == Order(x) + assert Order(x) + Order(1) == Order(1) + assert Order(x) + Order(x**2) == Order(x) + assert Order(1/x) + Order(1) == Order(1/x) + assert Order(x) + Order(exp(1/x)) == Order(exp(1/x)) + assert Order(x**3) + Order(exp(2/x)) == Order(exp(2/x)) + assert Order(x**-3) + Order(exp(2/x)) == Order(exp(2/x)) + + +def test_simple_7(): + assert 1 + O(1) == O(1) + assert 2 + O(1) == O(1) + assert x + O(1) == O(1) + assert 1/x + O(1) == 1/x + O(1) + + +def test_simple_8(): + assert O(sqrt(-x)) == O(sqrt(x)) + assert O(x**2*sqrt(x)) == O(x**Rational(5, 2)) + assert O(x**3*sqrt(-(-x)**3)) == O(x**Rational(9, 2)) + assert O(x**Rational(3, 2)*sqrt((-x)**3)) == O(x**3) + assert O(x*(-2*x)**(I/2)) == O(x*(-x)**(I/2)) + + +def test_as_expr_variables(): + assert Order(x).as_expr_variables(None) == (x, ((x, 0),)) + assert Order(x).as_expr_variables(((x, 0),)) == (x, ((x, 0),)) + assert Order(y).as_expr_variables(((x, 0),)) == (y, ((x, 0), (y, 0))) + assert Order(y).as_expr_variables(((x, 0), (y, 0))) == (y, ((x, 0), (y, 0))) + + +def test_contains_0(): + assert Order(1, x).contains(Order(1, x)) + assert Order(1, x).contains(Order(1)) + assert Order(1).contains(Order(1, x)) is False + + +def test_contains_1(): + assert Order(x).contains(Order(x)) + assert Order(x).contains(Order(x**2)) + assert not Order(x**2).contains(Order(x)) + assert not Order(x).contains(Order(1/x)) + assert not Order(1/x).contains(Order(exp(1/x))) + assert not Order(x).contains(Order(exp(1/x))) + assert Order(1/x).contains(Order(x)) + assert Order(exp(1/x)).contains(Order(x)) + assert Order(exp(1/x)).contains(Order(1/x)) + assert Order(exp(1/x)).contains(Order(exp(1/x))) + assert Order(exp(2/x)).contains(Order(exp(1/x))) + assert not Order(exp(1/x)).contains(Order(exp(2/x))) + + +def test_contains_2(): + assert Order(x).contains(Order(y)) is None + assert Order(x).contains(Order(y*x)) + assert Order(y*x).contains(Order(x)) + assert Order(y).contains(Order(x*y)) + assert Order(x).contains(Order(y**2*x)) + + +def test_contains_3(): + assert Order(x*y**2).contains(Order(x**2*y)) is None + assert Order(x**2*y).contains(Order(x*y**2)) is None + + +def test_contains_4(): + assert Order(sin(1/x**2)).contains(Order(cos(1/x**2))) is True + assert Order(cos(1/x**2)).contains(Order(sin(1/x**2))) is True + + +def test_contains(): + assert Order(1, x) not in Order(1) + assert Order(1) in Order(1, x) + raises(TypeError, lambda: Order(x*y**2) in Order(x**2*y)) + + +def test_add_1(): + assert Order(x + x) == Order(x) + assert Order(3*x - 2*x**2) == Order(x) + assert Order(1 + x) == Order(1, x) + assert Order(1 + 1/x) == Order(1/x) + # TODO : A better output for Order(log(x) + 1/log(x)) + # could be Order(log(x)). Currently Order for expressions + # where all arguments would involve a log term would fall + # in this category and outputs for these should be improved. + assert Order(log(x) + 1/log(x)) == Order((log(x)**2 + 1)/log(x)) + assert Order(exp(1/x) + x) == Order(exp(1/x)) + assert Order(exp(1/x) + 1/x**20) == Order(exp(1/x)) + + +def test_ln_args(): + assert O(log(x)) + O(log(2*x)) == O(log(x)) + assert O(log(x)) + O(log(x**3)) == O(log(x)) + assert O(log(x*y)) + O(log(x) + log(y)) == O(log(x) + log(y), x, y) + + +def test_multivar_0(): + assert Order(x*y).expr == x*y + assert Order(x*y**2).expr == x*y**2 + assert Order(x*y, x).expr == x + assert Order(x*y**2, y).expr == y**2 + assert Order(x*y*z).expr == x*y*z + assert Order(x/y).expr == x/y + assert Order(x*exp(1/y)).expr == x*exp(1/y) + assert Order(exp(x)*exp(1/y)).expr == exp(x)*exp(1/y) + + +def test_multivar_0a(): + assert Order(exp(1/x)*exp(1/y)).expr == exp(1/x)*exp(1/y) + + +def test_multivar_1(): + assert Order(x + y).expr == x + y + assert Order(x + 2*y).expr == x + y + assert (Order(x + y) + x).expr == (x + y) + assert (Order(x + y) + x**2) == Order(x + y) + assert (Order(x + y) + 1/x) == 1/x + Order(x + y) + assert Order(x**2 + y*x).expr == x**2 + y*x + + +def test_multivar_2(): + assert Order(x**2*y + y**2*x, x, y).expr == x**2*y + y**2*x + + +def test_multivar_mul_1(): + assert Order(x + y)*x == Order(x**2 + y*x, x, y) + + +def test_multivar_3(): + assert (Order(x) + Order(y)).args in [ + (Order(x), Order(y)), + (Order(y), Order(x))] + assert Order(x) + Order(y) + Order(x + y) == Order(x + y) + assert (Order(x**2*y) + Order(y**2*x)).args in [ + (Order(x*y**2), Order(y*x**2)), + (Order(y*x**2), Order(x*y**2))] + assert (Order(x**2*y) + Order(y*x)) == Order(x*y) + + +def test_issue_3468(): + y = Symbol('y', negative=True) + z = Symbol('z', complex=True) + + # check that Order does not modify assumptions about symbols + Order(x) + Order(y) + Order(z) + + assert x.is_positive is None + assert y.is_positive is False + assert z.is_positive is None + + +def test_leading_order(): + assert (x + 1 + 1/x**5).extract_leading_order(x) == ((1/x**5, O(1/x**5)),) + assert (1 + 1/x).extract_leading_order(x) == ((1/x, O(1/x)),) + assert (1 + x).extract_leading_order(x) == ((1, O(1, x)),) + assert (1 + x**2).extract_leading_order(x) == ((1, O(1, x)),) + assert (2 + x**2).extract_leading_order(x) == ((2, O(1, x)),) + assert (x + x**2).extract_leading_order(x) == ((x, O(x)),) + + +def test_leading_order2(): + assert set((2 + pi + x**2).extract_leading_order(x)) == {(pi, O(1, x)), + (S(2), O(1, x))} + assert set((2*x + pi*x + x**2).extract_leading_order(x)) == {(2*x, O(x)), + (x*pi, O(x))} + + +def test_order_leadterm(): + assert O(x**2)._eval_as_leading_term(x, None, 1) == O(x**2) + + +def test_order_symbols(): + e = x*y*sin(x)*Integral(x, (x, 1, 2)) + assert O(e) == O(x**2*y, x, y) + assert O(e, x) == O(x**2) + + +def test_nan(): + assert O(nan) is nan + assert not O(x).contains(nan) + + +def test_O1(): + assert O(1, x) * x == O(x) + assert O(1, y) * x == O(1, y) + + +def test_getn(): + # other lines are tested incidentally by the suite + assert O(x).getn() == 1 + assert O(x/log(x)).getn() == 1 + assert O(x**2/log(x)**2).getn() == 2 + assert O(x*log(x)).getn() == 1 + raises(NotImplementedError, lambda: (O(x) + O(y)).getn()) + + +def test_diff(): + assert O(x**2).diff(x) == O(x) + + +def test_getO(): + assert (x).getO() is None + assert (x).removeO() == x + assert (O(x)).getO() == O(x) + assert (O(x)).removeO() == 0 + assert (z + O(x) + O(y)).getO() == O(x) + O(y) + assert (z + O(x) + O(y)).removeO() == z + raises(NotImplementedError, lambda: (O(x) + O(y)).getn()) + + +def test_leading_term(): + from sympy.functions.special.gamma_functions import digamma + assert O(1/digamma(1/x)) == O(1/log(x)) + + +def test_eval(): + assert Order(x).subs(Order(x), 1) == 1 + assert Order(x).subs(x, y) == Order(y) + assert Order(x).subs(y, x) == Order(x) + assert Order(x).subs(x, x + y) == Order(x + y, (x, -y)) + assert (O(1)**x).is_Pow + + +def test_issue_4279(): + a, b = symbols('a b') + assert O(a, a, b) + O(1, a, b) == O(1, a, b) + assert O(b, a, b) + O(1, a, b) == O(1, a, b) + assert O(a + b, a, b) + O(1, a, b) == O(1, a, b) + assert O(1, a, b) + O(a, a, b) == O(1, a, b) + assert O(1, a, b) + O(b, a, b) == O(1, a, b) + assert O(1, a, b) + O(a + b, a, b) == O(1, a, b) + + +def test_issue_4855(): + assert 1/O(1) != O(1) + assert 1/O(x) != O(1/x) + assert 1/O(x, (x, oo)) != O(1/x, (x, oo)) + + f = Function('f') + assert 1/O(f(x)) != O(1/x) + + +def test_order_conjugate_transpose(): + x = Symbol('x', real=True) + y = Symbol('y', imaginary=True) + assert conjugate(Order(x)) == Order(conjugate(x)) + assert conjugate(Order(y)) == Order(conjugate(y)) + assert conjugate(Order(x**2)) == Order(conjugate(x)**2) + assert conjugate(Order(y**2)) == Order(conjugate(y)**2) + assert transpose(Order(x)) == Order(transpose(x)) + assert transpose(Order(y)) == Order(transpose(y)) + assert transpose(Order(x**2)) == Order(transpose(x)**2) + assert transpose(Order(y**2)) == Order(transpose(y)**2) + + +def test_order_noncommutative(): + A = Symbol('A', commutative=False) + assert Order(A + A*x, x) == Order(1, x) + assert (A + A*x)*Order(x) == Order(x) + assert (A*x)*Order(x) == Order(x**2, x) + assert expand((1 + Order(x))*A*A*x) == A*A*x + Order(x**2, x) + assert expand((A*A + Order(x))*x) == A*A*x + Order(x**2, x) + assert expand((A + Order(x))*A*x) == A*A*x + Order(x**2, x) + + +def test_issue_6753(): + assert (1 + x**2)**10000*O(x) == O(x) + + +def test_order_at_infinity(): + assert Order(1 + x, (x, oo)) == Order(x, (x, oo)) + assert Order(3*x, (x, oo)) == Order(x, (x, oo)) + assert Order(x, (x, oo))*3 == Order(x, (x, oo)) + assert -28*Order(x, (x, oo)) == Order(x, (x, oo)) + assert Order(Order(x, (x, oo)), (x, oo)) == Order(x, (x, oo)) + assert Order(Order(x, (x, oo)), (y, oo)) == Order(x, (x, oo), (y, oo)) + assert Order(3, (x, oo)) == Order(1, (x, oo)) + assert Order(x**2 + x + y, (x, oo)) == O(x**2, (x, oo)) + assert Order(x**2 + x + y, (y, oo)) == O(y, (y, oo)) + + assert Order(2*x, (x, oo))*x == Order(x**2, (x, oo)) + assert Order(2*x, (x, oo))/x == Order(1, (x, oo)) + assert Order(2*x, (x, oo))*x*exp(1/x) == Order(x**2*exp(1/x), (x, oo)) + assert Order(2*x, (x, oo))*x*exp(1/x)/log(x)**3 == Order(x**2*exp(1/x)*log(x)**-3, (x, oo)) + + assert Order(x, (x, oo)) + 1/x == 1/x + Order(x, (x, oo)) == Order(x, (x, oo)) + assert Order(x, (x, oo)) + 1 == 1 + Order(x, (x, oo)) == Order(x, (x, oo)) + assert Order(x, (x, oo)) + x == x + Order(x, (x, oo)) == Order(x, (x, oo)) + assert Order(x, (x, oo)) + x**2 == x**2 + Order(x, (x, oo)) + assert Order(1/x, (x, oo)) + 1/x**2 == 1/x**2 + Order(1/x, (x, oo)) == Order(1/x, (x, oo)) + assert Order(x, (x, oo)) + exp(1/x) == exp(1/x) + Order(x, (x, oo)) + + assert Order(x, (x, oo))**2 == Order(x**2, (x, oo)) + + assert Order(x, (x, oo)) + Order(x**2, (x, oo)) == Order(x**2, (x, oo)) + assert Order(x, (x, oo)) + Order(x**-2, (x, oo)) == Order(x, (x, oo)) + assert Order(x, (x, oo)) + Order(1/x, (x, oo)) == Order(x, (x, oo)) + + assert Order(x, (x, oo)) - Order(x, (x, oo)) == Order(x, (x, oo)) + assert Order(x, (x, oo)) + Order(1, (x, oo)) == Order(x, (x, oo)) + assert Order(x, (x, oo)) + Order(x**2, (x, oo)) == Order(x**2, (x, oo)) + assert Order(1/x, (x, oo)) + Order(1, (x, oo)) == Order(1, (x, oo)) + assert Order(x, (x, oo)) + Order(exp(1/x), (x, oo)) == Order(x, (x, oo)) + assert Order(x**3, (x, oo)) + Order(exp(2/x), (x, oo)) == Order(x**3, (x, oo)) + assert Order(x**-3, (x, oo)) + Order(exp(2/x), (x, oo)) == Order(exp(2/x), (x, oo)) + + # issue 7207 + assert Order(exp(x), (x, oo)).expr == Order(2*exp(x), (x, oo)).expr == exp(x) + assert Order(y**x, (x, oo)).expr == Order(2*y**x, (x, oo)).expr == exp(x*log(y)) + + # issue 19545 + assert Order(1/x - 3/(3*x + 2), (x, oo)).expr == x**(-2) + +def test_mixing_order_at_zero_and_infinity(): + assert (Order(x, (x, 0)) + Order(x, (x, oo))).is_Add + assert Order(x, (x, 0)) + Order(x, (x, oo)) == Order(x, (x, oo)) + Order(x, (x, 0)) + assert Order(Order(x, (x, oo))) == Order(x, (x, oo)) + + # not supported (yet) + raises(NotImplementedError, lambda: Order(x, (x, 0))*Order(x, (x, oo))) + raises(NotImplementedError, lambda: Order(x, (x, oo))*Order(x, (x, 0))) + raises(NotImplementedError, lambda: Order(Order(x, (x, oo)), y)) + raises(NotImplementedError, lambda: Order(Order(x), (x, oo))) + + +def test_order_at_some_point(): + assert Order(x, (x, 1)) == Order(1, (x, 1)) + assert Order(2*x - 2, (x, 1)) == Order(x - 1, (x, 1)) + assert Order(-x + 1, (x, 1)) == Order(x - 1, (x, 1)) + assert Order(x - 1, (x, 1))**2 == Order((x - 1)**2, (x, 1)) + assert Order(x - 2, (x, 2)) - O(x - 2, (x, 2)) == Order(x - 2, (x, 2)) + + +def test_order_subs_limits(): + # issue 3333 + assert (1 + Order(x)).subs(x, 1/x) == 1 + Order(1/x, (x, oo)) + assert (1 + Order(x)).limit(x, 0) == 1 + # issue 5769 + assert ((x + Order(x**2))/x).limit(x, 0) == 1 + + assert Order(x**2).subs(x, y - 1) == Order((y - 1)**2, (y, 1)) + assert Order(10*x**2, (x, 2)).subs(x, y - 1) == Order(1, (y, 3)) + + #issue 19120 + assert O(x).subs(x, O(x)) == O(x) + assert O(x**2).subs(x, x + O(x)) == O(x**2) + assert O(x, (x, oo)).subs(x, O(x, (x, oo))) == O(x, (x, oo)) + assert O(x**2, (x, oo)).subs(x, x + O(x, (x, oo))) == O(x**2, (x, oo)) + assert (x + O(x**2)).subs(x, x + O(x**2)) == x + O(x**2) + assert (x**2 + O(x**2) + 1/x**2).subs(x, x + O(x**2)) == (x + O(x**2))**(-2) + O(x**2) + assert (x**2 + O(x**2) + 1).subs(x, x + O(x**2)) == 1 + O(x**2) + assert O(x, (x, oo)).subs(x, x + O(x**2, (x, oo))) == O(x**2, (x, oo)) + assert sin(x).series(n=8).subs(x,sin(x).series(n=8)).expand() == x - x**3/3 + x**5/10 - 8*x**7/315 + O(x**8) + assert cos(x).series(n=8).subs(x,sin(x).series(n=8)).expand() == 1 - x**2/2 + 5*x**4/24 - 37*x**6/720 + O(x**8) + assert O(x).subs(x, O(1/x, (x, oo))) == O(1/x, (x, oo)) + +@XFAIL +def test_order_failing_due_to_solveset(): + assert O(x**3).subs(x, exp(-x**2)) == O(exp(-3*x**2), (x, -oo)) + raises(NotImplementedError, lambda: O(x).subs(x, O(1/x))) # mixing of order at different points + + +def test_issue_9351(): + assert exp(x).series(x, 10, 1) == exp(10) + Order(x - 10, (x, 10)) + + +def test_issue_9192(): + assert O(1)*O(1) == O(1) + assert O(1)**O(1) == O(1) + + +def test_issue_9910(): + assert O(x*log(x) + sin(x), (x, oo)) == O(x*log(x), (x, oo)) + + +def test_performance_of_adding_order(): + l = [x**i for i in range(1000)] + l.append(O(x**1001)) + assert Add(*l).subs(x,1) == O(1) + +def test_issue_14622(): + assert (x**(-4) + x**(-3) + x**(-1) + O(x**(-6), (x, oo))).as_numer_denom() == ( + x**4 + x**5 + x**7 + O(x**2, (x, oo)), x**8) + assert (x**3 + O(x**2, (x, oo))).is_Add + assert O(x**2, (x, oo)).contains(x**3) is False + assert O(x, (x, oo)).contains(O(x, (x, 0))) is None + assert O(x, (x, 0)).contains(O(x, (x, oo))) is None + raises(NotImplementedError, lambda: O(x**3).contains(x**w)) + + +def test_issue_15539(): + assert O(1/x**2 + 1/x**4, (x, -oo)) == O(1/x**2, (x, -oo)) + assert O(1/x**4 + exp(x), (x, -oo)) == O(1/x**4, (x, -oo)) + assert O(1/x**4 + exp(-x), (x, -oo)) == O(exp(-x), (x, -oo)) + assert O(1/x, (x, oo)).subs(x, -x) == O(-1/x, (x, -oo)) + +def test_issue_18606(): + assert unchanged(Order, 0) + + +def test_issue_22165(): + assert O(log(x)).contains(2) + + +def test_issue_23231(): + # This test checks Order for expressions having + # arguments containing variables in exponents/powers. + assert O(x**x + 2**x, (x, oo)) == O(exp(x*log(x)), (x, oo)) + assert O(x**x + x**2, (x, oo)) == O(exp(x*log(x)), (x, oo)) + assert O(x**x + 1/x**2, (x, oo)) == O(exp(x*log(x)), (x, oo)) + assert O(2**x + 3**x , (x, oo)) == O(exp(x*log(3)), (x, oo)) + + +def test_issue_9917(): + assert O(x*sin(x) + 1, (x, oo)) == O(x, (x, oo)) + + +def test_issue_22836(): + assert O(2**x + factorial(x), (x, oo)) == O(factorial(x), (x, oo)) + assert O(2**x + factorial(x) + x**x, (x, oo)) == O(exp(x*log(x)), (x, oo)) + assert O(x + factorial(x), (x, oo)) == O(factorial(x), (x, oo)) diff --git a/.venv/lib/python3.12/site-packages/sympy/series/tests/test_residues.py b/.venv/lib/python3.12/site-packages/sympy/series/tests/test_residues.py new file mode 100644 index 0000000000000000000000000000000000000000..9f7d075a56500d008e3c8b46c1fda5db890fd76a --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/tests/test_residues.py @@ -0,0 +1,101 @@ +from sympy.core.function import Function +from sympy.core.numbers import (I, Rational, pi) +from sympy.core.singleton import S +from sympy.core.symbol import Symbol +from sympy.functions.combinatorial.factorials import factorial +from sympy.functions.elementary.exponential import (exp, log) +from sympy.functions.elementary.hyperbolic import tanh +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import (cot, sin, tan) +from sympy.series.residues import residue +from sympy.testing.pytest import XFAIL, raises +from sympy.abc import x, z, a, s, k + + +def test_basic1(): + assert residue(1/x, x, 0) == 1 + assert residue(-2/x, x, 0) == -2 + assert residue(81/x, x, 0) == 81 + assert residue(1/x**2, x, 0) == 0 + assert residue(0, x, 0) == 0 + assert residue(5, x, 0) == 0 + assert residue(x, x, 0) == 0 + assert residue(x**2, x, 0) == 0 + + +def test_basic2(): + assert residue(1/x, x, 1) == 0 + assert residue(-2/x, x, 1) == 0 + assert residue(81/x, x, -1) == 0 + assert residue(1/x**2, x, 1) == 0 + assert residue(0, x, 1) == 0 + assert residue(5, x, 1) == 0 + assert residue(x, x, 1) == 0 + assert residue(x**2, x, 5) == 0 + + +def test_f(): + f = Function("f") + assert residue(f(x)/x**5, x, 0) == f(x).diff(x, 4).subs(x, 0)/24 + + +def test_functions(): + assert residue(1/sin(x), x, 0) == 1 + assert residue(2/sin(x), x, 0) == 2 + assert residue(1/sin(x)**2, x, 0) == 0 + assert residue(1/sin(x)**5, x, 0) == Rational(3, 8) + + +def test_expressions(): + assert residue(1/(x + 1), x, 0) == 0 + assert residue(1/(x + 1), x, -1) == 1 + assert residue(1/(x**2 + 1), x, -1) == 0 + assert residue(1/(x**2 + 1), x, I) == -I/2 + assert residue(1/(x**2 + 1), x, -I) == I/2 + assert residue(1/(x**4 + 1), x, 0) == 0 + assert residue(1/(x**4 + 1), x, exp(I*pi/4)).equals(-(Rational(1, 4) + I/4)/sqrt(2)) + assert residue(1/(x**2 + a**2)**2, x, a*I) == -I/4/a**3 + + +@XFAIL +def test_expressions_failing(): + n = Symbol('n', integer=True, positive=True) + assert residue(exp(z)/(z - pi*I/4*a)**n, z, I*pi*a) == \ + exp(I*pi*a/4)/factorial(n - 1) + + +def test_NotImplemented(): + raises(NotImplementedError, lambda: residue(exp(1/z), z, 0)) + + +def test_bug(): + assert residue(2**(z)*(s + z)*(1 - s - z)/z**2, z, 0) == \ + 1 + s*log(2) - s**2*log(2) - 2*s + + +def test_issue_5654(): + assert residue(1/(x**2 + a**2)**2, x, a*I) == -I/(4*a**3) + assert residue(1/s*1/(z - exp(s)), s, 0) == 1/(z - 1) + assert residue((1 + k)/s*1/(z - exp(s)), s, 0) == k/(z - 1) + 1/(z - 1) + + +def test_issue_6499(): + assert residue(1/(exp(z) - 1), z, 0) == 1 + + +def test_issue_14037(): + assert residue(sin(x**50)/x**51, x, 0) == 1 + + +def test_issue_21176(): + f = x**2*cot(pi*x)/(x**4 + 1) + assert residue(f, x, -sqrt(2)/2 - sqrt(2)*I/2).cancel().together(deep=True)\ + == sqrt(2)*(1 - I)/(8*tan(sqrt(2)*pi*(1 + I)/2)) + + +def test_issue_21177(): + r = -sqrt(3)*tanh(sqrt(3)*pi/2)/3 + a = residue(cot(pi*x)/((x - 1)*(x - 2) + 1), x, S(3)/2 - sqrt(3)*I/2) + b = residue(cot(pi*x)/(x**2 - 3*x + 3), x, S(3)/2 - sqrt(3)*I/2) + assert a == r + assert (b - a).cancel() == 0 diff --git a/.venv/lib/python3.12/site-packages/sympy/series/tests/test_sequences.py b/.venv/lib/python3.12/site-packages/sympy/series/tests/test_sequences.py new file mode 100644 index 0000000000000000000000000000000000000000..61e276ad67982f0a9877de3548d70238976d28a5 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/tests/test_sequences.py @@ -0,0 +1,312 @@ +from sympy.core.containers import Tuple +from sympy.core.function import Function +from sympy.core.numbers import oo, Rational +from sympy.core.singleton import S +from sympy.core.symbol import symbols, Symbol +from sympy.functions.combinatorial.numbers import tribonacci, fibonacci +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import cos, sin +from sympy.series import EmptySequence +from sympy.series.sequences import (SeqMul, SeqAdd, SeqPer, SeqFormula, + sequence) +from sympy.sets.sets import Interval +from sympy.tensor.indexed import Indexed, Idx +from sympy.series.sequences import SeqExpr, SeqExprOp, RecursiveSeq +from sympy.testing.pytest import raises, slow + +x, y, z = symbols('x y z') +n, m = symbols('n m') + + +def test_EmptySequence(): + assert S.EmptySequence is EmptySequence + + assert S.EmptySequence.interval is S.EmptySet + assert S.EmptySequence.length is S.Zero + + assert list(S.EmptySequence) == [] + + +def test_SeqExpr(): + #SeqExpr is a baseclass and does not take care of + #ensuring all arguments are Basics hence the use of + #Tuple(...) here. + s = SeqExpr(Tuple(1, n, y), Tuple(x, 0, 10)) + + assert isinstance(s, SeqExpr) + assert s.gen == (1, n, y) + assert s.interval == Interval(0, 10) + assert s.start == 0 + assert s.stop == 10 + assert s.length == 11 + assert s.variables == (x,) + + assert SeqExpr(Tuple(1, 2, 3), Tuple(x, 0, oo)).length is oo + + +def test_SeqPer(): + s = SeqPer((1, n, 3), (x, 0, 5)) + + assert isinstance(s, SeqPer) + assert s.periodical == Tuple(1, n, 3) + assert s.period == 3 + assert s.coeff(3) == 1 + assert s.free_symbols == {n} + + assert list(s) == [1, n, 3, 1, n, 3] + assert s[:] == [1, n, 3, 1, n, 3] + assert SeqPer((1, n, 3), (x, -oo, 0))[0:6] == [1, n, 3, 1, n, 3] + + raises(ValueError, lambda: SeqPer((1, 2, 3), (0, 1, 2))) + raises(ValueError, lambda: SeqPer((1, 2, 3), (x, -oo, oo))) + raises(ValueError, lambda: SeqPer(n**2, (0, oo))) + + assert SeqPer((n, n**2, n**3), (m, 0, oo))[:6] == \ + [n, n**2, n**3, n, n**2, n**3] + assert SeqPer((n, n**2, n**3), (n, 0, oo))[:6] == [0, 1, 8, 3, 16, 125] + assert SeqPer((n, m), (n, 0, oo))[:6] == [0, m, 2, m, 4, m] + + +def test_SeqFormula(): + s = SeqFormula(n**2, (n, 0, 5)) + + assert isinstance(s, SeqFormula) + assert s.formula == n**2 + assert s.coeff(3) == 9 + + assert list(s) == [i**2 for i in range(6)] + assert s[:] == [i**2 for i in range(6)] + assert SeqFormula(n**2, (n, -oo, 0))[0:6] == [i**2 for i in range(6)] + + assert SeqFormula(n**2, (0, oo)) == SeqFormula(n**2, (n, 0, oo)) + + assert SeqFormula(n**2, (0, m)).subs(m, x) == SeqFormula(n**2, (0, x)) + assert SeqFormula(m*n**2, (n, 0, oo)).subs(m, x) == \ + SeqFormula(x*n**2, (n, 0, oo)) + + raises(ValueError, lambda: SeqFormula(n**2, (0, 1, 2))) + raises(ValueError, lambda: SeqFormula(n**2, (n, -oo, oo))) + raises(ValueError, lambda: SeqFormula(m*n**2, (0, oo))) + + seq = SeqFormula(x*(y**2 + z), (z, 1, 100)) + assert seq.expand() == SeqFormula(x*y**2 + x*z, (z, 1, 100)) + seq = SeqFormula(sin(x*(y**2 + z)),(z, 1, 100)) + assert seq.expand(trig=True) == SeqFormula(sin(x*y**2)*cos(x*z) + sin(x*z)*cos(x*y**2), (z, 1, 100)) + assert seq.expand() == SeqFormula(sin(x*y**2 + x*z), (z, 1, 100)) + assert seq.expand(trig=False) == SeqFormula(sin(x*y**2 + x*z), (z, 1, 100)) + seq = SeqFormula(exp(x*(y**2 + z)), (z, 1, 100)) + assert seq.expand() == SeqFormula(exp(x*y**2)*exp(x*z), (z, 1, 100)) + assert seq.expand(power_exp=False) == SeqFormula(exp(x*y**2 + x*z), (z, 1, 100)) + assert seq.expand(mul=False, power_exp=False) == SeqFormula(exp(x*(y**2 + z)), (z, 1, 100)) + +def test_sequence(): + form = SeqFormula(n**2, (n, 0, 5)) + per = SeqPer((1, 2, 3), (n, 0, 5)) + inter = SeqFormula(n**2) + + assert sequence(n**2, (n, 0, 5)) == form + assert sequence((1, 2, 3), (n, 0, 5)) == per + assert sequence(n**2) == inter + + +def test_SeqExprOp(): + form = SeqFormula(n**2, (n, 0, 10)) + per = SeqPer((1, 2, 3), (m, 5, 10)) + + s = SeqExprOp(form, per) + assert s.gen == (n**2, (1, 2, 3)) + assert s.interval == Interval(5, 10) + assert s.start == 5 + assert s.stop == 10 + assert s.length == 6 + assert s.variables == (n, m) + + +def test_SeqAdd(): + per = SeqPer((1, 2, 3), (n, 0, oo)) + form = SeqFormula(n**2) + + per_bou = SeqPer((1, 2), (n, 1, 5)) + form_bou = SeqFormula(n**2, (6, 10)) + form_bou2 = SeqFormula(n**2, (1, 5)) + + assert SeqAdd() == S.EmptySequence + assert SeqAdd(S.EmptySequence) == S.EmptySequence + assert SeqAdd(per) == per + assert SeqAdd(per, S.EmptySequence) == per + assert SeqAdd(per_bou, form_bou) == S.EmptySequence + + s = SeqAdd(per_bou, form_bou2, evaluate=False) + assert s.args == (form_bou2, per_bou) + assert s[:] == [2, 6, 10, 18, 26] + assert list(s) == [2, 6, 10, 18, 26] + + assert isinstance(SeqAdd(per, per_bou, evaluate=False), SeqAdd) + + s1 = SeqAdd(per, per_bou) + assert isinstance(s1, SeqPer) + assert s1 == SeqPer((2, 4, 4, 3, 3, 5), (n, 1, 5)) + s2 = SeqAdd(form, form_bou) + assert isinstance(s2, SeqFormula) + assert s2 == SeqFormula(2*n**2, (6, 10)) + + assert SeqAdd(form, form_bou, per) == \ + SeqAdd(per, SeqFormula(2*n**2, (6, 10))) + assert SeqAdd(form, SeqAdd(form_bou, per)) == \ + SeqAdd(per, SeqFormula(2*n**2, (6, 10))) + assert SeqAdd(per, SeqAdd(form, form_bou), evaluate=False) == \ + SeqAdd(per, SeqFormula(2*n**2, (6, 10))) + + assert SeqAdd(SeqPer((1, 2), (n, 0, oo)), SeqPer((1, 2), (m, 0, oo))) == \ + SeqPer((2, 4), (n, 0, oo)) + + +def test_SeqMul(): + per = SeqPer((1, 2, 3), (n, 0, oo)) + form = SeqFormula(n**2) + + per_bou = SeqPer((1, 2), (n, 1, 5)) + form_bou = SeqFormula(n**2, (n, 6, 10)) + form_bou2 = SeqFormula(n**2, (1, 5)) + + assert SeqMul() == S.EmptySequence + assert SeqMul(S.EmptySequence) == S.EmptySequence + assert SeqMul(per) == per + assert SeqMul(per, S.EmptySequence) == S.EmptySequence + assert SeqMul(per_bou, form_bou) == S.EmptySequence + + s = SeqMul(per_bou, form_bou2, evaluate=False) + assert s.args == (form_bou2, per_bou) + assert s[:] == [1, 8, 9, 32, 25] + assert list(s) == [1, 8, 9, 32, 25] + + assert isinstance(SeqMul(per, per_bou, evaluate=False), SeqMul) + + s1 = SeqMul(per, per_bou) + assert isinstance(s1, SeqPer) + assert s1 == SeqPer((1, 4, 3, 2, 2, 6), (n, 1, 5)) + s2 = SeqMul(form, form_bou) + assert isinstance(s2, SeqFormula) + assert s2 == SeqFormula(n**4, (6, 10)) + + assert SeqMul(form, form_bou, per) == \ + SeqMul(per, SeqFormula(n**4, (6, 10))) + assert SeqMul(form, SeqMul(form_bou, per)) == \ + SeqMul(per, SeqFormula(n**4, (6, 10))) + assert SeqMul(per, SeqMul(form, form_bou2, + evaluate=False), evaluate=False) == \ + SeqMul(form, per, form_bou2, evaluate=False) + + assert SeqMul(SeqPer((1, 2), (n, 0, oo)), SeqPer((1, 2), (n, 0, oo))) == \ + SeqPer((1, 4), (n, 0, oo)) + + +def test_add(): + per = SeqPer((1, 2), (n, 0, oo)) + form = SeqFormula(n**2) + + assert per + (SeqPer((2, 3))) == SeqPer((3, 5), (n, 0, oo)) + assert form + SeqFormula(n**3) == SeqFormula(n**2 + n**3) + + assert per + form == SeqAdd(per, form) + + raises(TypeError, lambda: per + n) + raises(TypeError, lambda: n + per) + + +def test_sub(): + per = SeqPer((1, 2), (n, 0, oo)) + form = SeqFormula(n**2) + + assert per - (SeqPer((2, 3))) == SeqPer((-1, -1), (n, 0, oo)) + assert form - (SeqFormula(n**3)) == SeqFormula(n**2 - n**3) + + assert per - form == SeqAdd(per, -form) + + raises(TypeError, lambda: per - n) + raises(TypeError, lambda: n - per) + + +def test_mul__coeff_mul(): + assert SeqPer((1, 2), (n, 0, oo)).coeff_mul(2) == SeqPer((2, 4), (n, 0, oo)) + assert SeqFormula(n**2).coeff_mul(2) == SeqFormula(2*n**2) + assert S.EmptySequence.coeff_mul(100) == S.EmptySequence + + assert SeqPer((1, 2), (n, 0, oo)) * (SeqPer((2, 3))) == \ + SeqPer((2, 6), (n, 0, oo)) + assert SeqFormula(n**2) * SeqFormula(n**3) == SeqFormula(n**5) + + assert S.EmptySequence * SeqFormula(n**2) == S.EmptySequence + assert SeqFormula(n**2) * S.EmptySequence == S.EmptySequence + + raises(TypeError, lambda: sequence(n**2) * n) + raises(TypeError, lambda: n * sequence(n**2)) + + +def test_neg(): + assert -SeqPer((1, -2), (n, 0, oo)) == SeqPer((-1, 2), (n, 0, oo)) + assert -SeqFormula(n**2) == SeqFormula(-n**2) + + +def test_operations(): + per = SeqPer((1, 2), (n, 0, oo)) + per2 = SeqPer((2, 4), (n, 0, oo)) + form = SeqFormula(n**2) + form2 = SeqFormula(n**3) + + assert per + form + form2 == SeqAdd(per, form, form2) + assert per + form - form2 == SeqAdd(per, form, -form2) + assert per + form - S.EmptySequence == SeqAdd(per, form) + assert per + per2 + form == SeqAdd(SeqPer((3, 6), (n, 0, oo)), form) + assert S.EmptySequence - per == -per + assert form + form == SeqFormula(2*n**2) + + assert per * form * form2 == SeqMul(per, form, form2) + assert form * form == SeqFormula(n**4) + assert form * -form == SeqFormula(-n**4) + + assert form * (per + form2) == SeqMul(form, SeqAdd(per, form2)) + assert form * (per + per) == SeqMul(form, per2) + + assert form.coeff_mul(m) == SeqFormula(m*n**2, (n, 0, oo)) + assert per.coeff_mul(m) == SeqPer((m, 2*m), (n, 0, oo)) + + +def test_Idx_limits(): + i = symbols('i', cls=Idx) + r = Indexed('r', i) + + assert SeqFormula(r, (i, 0, 5))[:] == [r.subs(i, j) for j in range(6)] + assert SeqPer((1, 2), (i, 0, 5))[:] == [1, 2, 1, 2, 1, 2] + + +@slow +def test_find_linear_recurrence(): + assert sequence((0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55), \ + (n, 0, 10)).find_linear_recurrence(11) == [1, 1] + assert sequence((1, 2, 4, 7, 28, 128, 582, 2745, 13021, 61699, 292521, \ + 1387138), (n, 0, 11)).find_linear_recurrence(12) == [5, -2, 6, -11] + assert sequence(x*n**3+y*n, (n, 0, oo)).find_linear_recurrence(10) \ + == [4, -6, 4, -1] + assert sequence(x**n, (n,0,20)).find_linear_recurrence(21) == [x] + assert sequence((1,2,3)).find_linear_recurrence(10, 5) == [0, 0, 1] + assert sequence(((1 + sqrt(5))/2)**n + \ + (-(1 + sqrt(5))/2)**(-n)).find_linear_recurrence(10) == [1, 1] + assert sequence(x*((1 + sqrt(5))/2)**n + y*(-(1 + sqrt(5))/2)**(-n), \ + (n,0,oo)).find_linear_recurrence(10) == [1, 1] + assert sequence((1,2,3,4,6),(n, 0, 4)).find_linear_recurrence(5) == [] + assert sequence((2,3,4,5,6,79),(n, 0, 5)).find_linear_recurrence(6,gfvar=x) \ + == ([], None) + assert sequence((2,3,4,5,8,30),(n, 0, 5)).find_linear_recurrence(6,gfvar=x) \ + == ([Rational(19, 2), -20, Rational(27, 2)], (-31*x**2 + 32*x - 4)/(27*x**3 - 40*x**2 + 19*x -2)) + assert sequence(fibonacci(n)).find_linear_recurrence(30,gfvar=x) \ + == ([1, 1], -x/(x**2 + x - 1)) + assert sequence(tribonacci(n)).find_linear_recurrence(30,gfvar=x) \ + == ([1, 1, 1], -x/(x**3 + x**2 + x - 1)) + +def test_RecursiveSeq(): + y = Function('y') + n = Symbol('n') + fib = RecursiveSeq(y(n - 1) + y(n - 2), y(n), n, [0, 1]) + assert fib.coeff(3) == 2 diff --git a/.venv/lib/python3.12/site-packages/sympy/series/tests/test_series.py b/.venv/lib/python3.12/site-packages/sympy/series/tests/test_series.py new file mode 100644 index 0000000000000000000000000000000000000000..e3f3c122b98c14a58c6d5c6636cbb53e1e66a75d --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/series/tests/test_series.py @@ -0,0 +1,421 @@ +from sympy.core.evalf import N +from sympy.core.function import (Derivative, Function, PoleError, Subs) +from sympy.core.numbers import (E, Float, Rational, oo, pi, I) +from sympy.core.singleton import S +from sympy.core.symbol import (Symbol, symbols) +from sympy.functions.elementary.exponential import (LambertW, exp, log) +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import (atan, cos, sin) +from sympy.functions.special.gamma_functions import gamma +from sympy.integrals.integrals import Integral, integrate +from sympy.series.order import O +from sympy.series.series import series +from sympy.abc import x, y, n, k +from sympy.testing.pytest import raises +from sympy.core import EulerGamma + + +def test_sin(): + e1 = sin(x).series(x, 0) + e2 = series(sin(x), x, 0) + assert e1 == e2 + + +def test_cos(): + e1 = cos(x).series(x, 0) + e2 = series(cos(x), x, 0) + assert e1 == e2 + + +def test_exp(): + e1 = exp(x).series(x, 0) + e2 = series(exp(x), x, 0) + assert e1 == e2 + + +def test_exp2(): + e1 = exp(cos(x)).series(x, 0) + e2 = series(exp(cos(x)), x, 0) + assert e1 == e2 + + +def test_issue_5223(): + assert series(1, x) == 1 + assert next(S.Zero.lseries(x)) == 0 + assert cos(x).series() == cos(x).series(x) + raises(ValueError, lambda: cos(x + y).series()) + raises(ValueError, lambda: x.series(dir="")) + + assert (cos(x).series(x, 1) - + cos(x + 1).series(x).subs(x, x - 1)).removeO() == 0 + e = cos(x).series(x, 1, n=None) + assert [next(e) for i in range(2)] == [cos(1), -((x - 1)*sin(1))] + e = cos(x).series(x, 1, n=None, dir='-') + assert [next(e) for i in range(2)] == [cos(1), (1 - x)*sin(1)] + # the following test is exact so no need for x -> x - 1 replacement + assert abs(x).series(x, 1, dir='-') == x + assert exp(x).series(x, 1, dir='-', n=3).removeO() == \ + E - E*(-x + 1) + E*(-x + 1)**2/2 + + D = Derivative + assert D(x**2 + x**3*y**2, x, 2, y, 1).series(x).doit() == 12*x*y + assert next(D(cos(x), x).lseries()) == D(1, x) + assert D( + exp(x), x).series(n=3) == D(1, x) + D(x, x) + D(x**2/2, x) + D(x**3/6, x) + O(x**3) + + assert Integral(x, (x, 1, 3), (y, 1, x)).series(x) == -4 + 4*x + + assert (1 + x + O(x**2)).getn() == 2 + assert (1 + x).getn() is None + + raises(PoleError, lambda: ((1/sin(x))**oo).series()) + logx = Symbol('logx') + assert ((sin(x))**y).nseries(x, n=1, logx=logx) == \ + exp(y*logx) + O(x*exp(y*logx), x) + + assert sin(1/x).series(x, oo, n=5) == 1/x - 1/(6*x**3) + O(x**(-5), (x, oo)) + assert abs(x).series(x, oo, n=5, dir='+') == x + assert abs(x).series(x, -oo, n=5, dir='-') == -x + assert abs(-x).series(x, oo, n=5, dir='+') == x + assert abs(-x).series(x, -oo, n=5, dir='-') == -x + + assert exp(x*log(x)).series(n=3) == \ + 1 + x*log(x) + x**2*log(x)**2/2 + O(x**3*log(x)**3) + # XXX is this right? If not, fix "ngot > n" handling in expr. + p = Symbol('p', positive=True) + assert exp(sqrt(p)**3*log(p)).series(n=3) == \ + 1 + p**S('3/2')*log(p) + O(p**3*log(p)**3) + + assert exp(sin(x)*log(x)).series(n=2) == 1 + x*log(x) + O(x**2*log(x)**2) + + +def test_issue_6350(): + expr = integrate(exp(k*(y**3 - 3*y)), (y, 0, oo), conds='none') + assert expr.series(k, 0, 3) == -(-1)**(S(2)/3)*sqrt(3)*gamma(S(1)/3)**2*gamma(S(2)/3)/(6*pi*k**(S(1)/3)) - \ + sqrt(3)*k*gamma(-S(2)/3)*gamma(-S(1)/3)/(6*pi) - \ + (-1)**(S(1)/3)*sqrt(3)*k**(S(1)/3)*gamma(-S(1)/3)*gamma(S(1)/3)*gamma(S(2)/3)/(6*pi) - \ + (-1)**(S(2)/3)*sqrt(3)*k**(S(5)/3)*gamma(S(1)/3)**2*gamma(S(2)/3)/(4*pi) - \ + (-1)**(S(1)/3)*sqrt(3)*k**(S(7)/3)*gamma(-S(1)/3)*gamma(S(1)/3)*gamma(S(2)/3)/(8*pi) + O(k**3) + + +def test_issue_11313(): + assert Integral(cos(x), x).series(x) == sin(x).series(x) + assert Derivative(sin(x), x).series(x, n=3).doit() == cos(x).series(x, n=3) + + assert Derivative(x**3, x).as_leading_term(x) == 3*x**2 + assert Derivative(x**3, y).as_leading_term(x) == 0 + assert Derivative(sin(x), x).as_leading_term(x) == 1 + assert Derivative(cos(x), x).as_leading_term(x) == -x + + # This result is equivalent to zero, zero is not return because + # `Expr.series` doesn't currently detect an `x` in its `free_symbol`s. + assert Derivative(1, x).as_leading_term(x) == Derivative(1, x) + + assert Derivative(exp(x), x).series(x).doit() == exp(x).series(x) + assert 1 + Integral(exp(x), x).series(x) == exp(x).series(x) + + assert Derivative(log(x), x).series(x).doit() == (1/x).series(x) + assert Integral(log(x), x).series(x) == Integral(log(x), x).doit().series(x).removeO() + + +def test_series_of_Subs(): + from sympy.abc import z + + subs1 = Subs(sin(x), x, y) + subs2 = Subs(sin(x) * cos(z), x, y) + subs3 = Subs(sin(x * z), (x, z), (y, x)) + + assert subs1.series(x) == subs1 + subs1_series = (Subs(x, x, y) + Subs(-x**3/6, x, y) + + Subs(x**5/120, x, y) + O(y**6)) + assert subs1.series() == subs1_series + assert subs1.series(y) == subs1_series + assert subs1.series(z) == subs1 + assert subs2.series(z) == (Subs(z**4*sin(x)/24, x, y) + + Subs(-z**2*sin(x)/2, x, y) + Subs(sin(x), x, y) + O(z**6)) + assert subs3.series(x).doit() == subs3.doit().series(x) + assert subs3.series(z).doit() == sin(x*y) + + raises(ValueError, lambda: Subs(x + 2*y, y, z).series()) + assert Subs(x + y, y, z).series(x).doit() == x + z + + +def test_issue_3978(): + f = Function('f') + assert f(x).series(x, 0, 3, dir='-') == \ + f(0) + x*Subs(Derivative(f(x), x), x, 0) + \ + x**2*Subs(Derivative(f(x), x, x), x, 0)/2 + O(x**3) + assert f(x).series(x, 0, 3) == \ + f(0) + x*Subs(Derivative(f(x), x), x, 0) + \ + x**2*Subs(Derivative(f(x), x, x), x, 0)/2 + O(x**3) + assert f(x**2).series(x, 0, 3) == \ + f(0) + x**2*Subs(Derivative(f(x), x), x, 0) + O(x**3) + assert f(x**2+1).series(x, 0, 3) == \ + f(1) + x**2*Subs(Derivative(f(x), x), x, 1) + O(x**3) + + class TestF(Function): + pass + + assert TestF(x).series(x, 0, 3) == TestF(0) + \ + x*Subs(Derivative(TestF(x), x), x, 0) + \ + x**2*Subs(Derivative(TestF(x), x, x), x, 0)/2 + O(x**3) + +from sympy.series.acceleration import richardson, shanks +from sympy.concrete.summations import Sum +from sympy.core.numbers import Integer + + +def test_acceleration(): + e = (1 + 1/n)**n + assert round(richardson(e, n, 10, 20).evalf(), 10) == round(E.evalf(), 10) + + A = Sum(Integer(-1)**(k + 1) / k, (k, 1, n)) + assert round(shanks(A, n, 25).evalf(), 4) == round(log(2).evalf(), 4) + assert round(shanks(A, n, 25, 5).evalf(), 10) == round(log(2).evalf(), 10) + + +def test_issue_5852(): + assert series(1/cos(x/log(x)), x, 0) == 1 + x**2/(2*log(x)**2) + \ + 5*x**4/(24*log(x)**4) + O(x**6) + + +def test_issue_4583(): + assert cos(1 + x + x**2).series(x, 0, 5) == cos(1) - x*sin(1) + \ + x**2*(-sin(1) - cos(1)/2) + x**3*(-cos(1) + sin(1)/6) + \ + x**4*(-11*cos(1)/24 + sin(1)/2) + O(x**5) + + +def test_issue_6318(): + eq = (1/x)**Rational(2, 3) + assert (eq + 1).as_leading_term(x) == eq + + +def test_x_is_base_detection(): + eq = (x**2)**Rational(2, 3) + assert eq.series() == x**Rational(4, 3) + + +def test_issue_7203(): + assert series(cos(x), x, pi, 3) == \ + -1 + (x - pi)**2/2 + O((x - pi)**3, (x, pi)) + + +def test_exp_product_positive_factors(): + a, b = symbols('a, b', positive=True) + x = a * b + assert series(exp(x), x, n=8) == 1 + a*b + a**2*b**2/2 + \ + a**3*b**3/6 + a**4*b**4/24 + a**5*b**5/120 + a**6*b**6/720 + \ + a**7*b**7/5040 + O(a**8*b**8, a, b) + + +def test_issue_8805(): + assert series(1, n=8) == 1 + + +def test_issue_9173(): + p0,p1,p2,p3,b0,b1,b2=symbols('p0 p1 p2 p3 b0 b1 b2') + Q=(p0+(p1+(p2+p3/y)/y)/y)/(1+((p3/(b0*y)+(b0*p2-b1*p3)/b0**2)/y+\ + (b0**2*p1-b0*b1*p2-p3*(b0*b2-b1**2))/b0**3)/y) + + series = Q.series(y,n=3) + + assert series == y*(b0*p2/p3+b0*(-p2/p3+b1/b0))+y**2*(b0*p1/p3+b0*p2*\ + (-p2/p3+b1/b0)/p3+b0*(-p1/p3+(p2/p3-b1/b0)**2+b1*p2/(b0*p3)+\ + b2/b0-b1**2/b0**2))+b0+O(y**3) + assert series.simplify() == b2*y**2 + b1*y + b0 + O(y**3) + + +def test_issue_9549(): + y = (x**2 + x + 1) / (x**3 + x**2) + assert series(y, x, oo) == x**(-5) - 1/x**4 + x**(-3) + 1/x + O(x**(-6), (x, oo)) + + +def test_issue_10761(): + assert series(1/(x**-2 + x**-3), x, 0) == x**3 - x**4 + x**5 + O(x**6) + + +def test_issue_12578(): + y = (1 - 1/(x/2 - 1/(2*x))**4)**(S(1)/8) + assert y.series(x, 0, n=17) == 1 - 2*x**4 - 8*x**6 - 34*x**8 - 152*x**10 - 714*x**12 - \ + 3472*x**14 - 17318*x**16 + O(x**17) + + +def test_issue_12791(): + beta = symbols('beta', positive=True) + theta, varphi = symbols('theta varphi', real=True) + + expr = (-beta**2*varphi*sin(theta) + beta**2*cos(theta) + \ + beta*varphi*sin(theta) - beta*cos(theta) - beta + 1)/(beta*cos(theta) - 1)**2 + + sol = (0.5/(0.5*cos(theta) - 1.0)**2 - 0.25*cos(theta)/(0.5*cos(theta) - 1.0)**2 + + (beta - 0.5)*(-0.25*varphi*sin(2*theta) - 1.5*cos(theta) + + 0.25*cos(2*theta) + 1.25)/((0.5*cos(theta) - 1.0)**2*(0.5*cos(theta) - 1.0)) + + 0.25*varphi*sin(theta)/(0.5*cos(theta) - 1.0)**2 + + O((beta - S.Half)**2, (beta, S.Half))) + + assert expr.series(beta, 0.5, 2).trigsimp() == sol + + +def test_issue_14384(): + x, a = symbols('x a') + assert series(x**a, x) == x**a + assert series(x**(-2*a), x) == x**(-2*a) + assert series(exp(a*log(x)), x) == exp(a*log(x)) + raises(PoleError, lambda: series(x**I, x)) + raises(PoleError, lambda: series(x**(I + 1), x)) + raises(PoleError, lambda: series(exp(I*log(x)), x)) + + +def test_issue_14885(): + assert series(x**Rational(-3, 2)*exp(x), x, 0) == (x**Rational(-3, 2) + 1/sqrt(x) + + sqrt(x)/2 + x**Rational(3, 2)/6 + x**Rational(5, 2)/24 + x**Rational(7, 2)/120 + + x**Rational(9, 2)/720 + x**Rational(11, 2)/5040 + O(x**6)) + + +def test_issue_15539(): + assert series(atan(x), x, -oo) == (-1/(5*x**5) + 1/(3*x**3) - 1/x - pi/2 + + O(x**(-6), (x, -oo))) + assert series(atan(x), x, oo) == (-1/(5*x**5) + 1/(3*x**3) - 1/x + pi/2 + + O(x**(-6), (x, oo))) + + +def test_issue_7259(): + assert series(LambertW(x), x) == x - x**2 + 3*x**3/2 - 8*x**4/3 + 125*x**5/24 + O(x**6) + assert series(LambertW(x**2), x, n=8) == x**2 - x**4 + 3*x**6/2 + O(x**8) + assert series(LambertW(sin(x)), x, n=4) == x - x**2 + 4*x**3/3 + O(x**4) + +def test_issue_11884(): + assert cos(x).series(x, 1, n=1) == cos(1) + O(x - 1, (x, 1)) + + +def test_issue_18008(): + y = x*(1 + x*(1 - x))/((1 + x*(1 - x)) - (1 - x)*(1 - x)) + assert y.series(x, oo, n=4) == -9/(32*x**3) - 3/(16*x**2) - 1/(8*x) + S(1)/4 + x/2 + \ + O(x**(-4), (x, oo)) + + +def test_issue_18842(): + f = log(x/(1 - x)) + assert f.series(x, 0.491, n=1).removeO().nsimplify() == \ + -S(180019443780011)/5000000000000000 + + +def test_issue_19534(): + dt = symbols('dt', real=True) + expr = 16*dt*(0.125*dt*(2.0*dt + 1.0) + 0.875*dt + 1.0)/45 + \ + 49*dt*(-0.049335189898860408029*dt*(2.0*dt + 1.0) + \ + 0.29601113939316244817*dt*(0.125*dt*(2.0*dt + 1.0) + 0.875*dt + 1.0) - \ + 0.12564355335492979587*dt*(0.074074074074074074074*dt*(2.0*dt + 1.0) + \ + 0.2962962962962962963*dt*(0.125*dt*(2.0*dt + 1.0) + 0.875*dt + 1.0) + \ + 0.96296296296296296296*dt + 1.0) + 0.051640768506639183825*dt + \ + dt*(1/2 - sqrt(21)/14) + 1.0)/180 + 49*dt*(-0.23637909581542530626*dt*(2.0*dt + 1.0) - \ + 0.74817562366625959291*dt*(0.125*dt*(2.0*dt + 1.0) + 0.875*dt + 1.0) + \ + 0.88085458023927036857*dt*(0.074074074074074074074*dt*(2.0*dt + 1.0) + \ + 0.2962962962962962963*dt*(0.125*dt*(2.0*dt + 1.0) + 0.875*dt + 1.0) + \ + 0.96296296296296296296*dt + 1.0) + \ + 2.1165151389911680013*dt*(-0.049335189898860408029*dt*(2.0*dt + 1.0) + \ + 0.29601113939316244817*dt*(0.125*dt*(2.0*dt + 1.0) + 0.875*dt + 1.0) - \ + 0.12564355335492979587*dt*(0.074074074074074074074*dt*(2.0*dt + 1.0) + \ + 0.2962962962962962963*dt*(0.125*dt*(2.0*dt + 1.0) + 0.875*dt + 1.0) + \ + 0.96296296296296296296*dt + 1.0) + 0.22431393315265061193*dt + 1.0) - \ + 1.1854881643947648988*dt + dt*(sqrt(21)/14 + 1/2) + 1.0)/180 + \ + dt*(0.66666666666666666667*dt*(2.0*dt + 1.0) + \ + 6.0173399699313066769*dt*(0.125*dt*(2.0*dt + 1.0) + 0.875*dt + 1.0) - \ + 4.1117044797036320069*dt*(0.074074074074074074074*dt*(2.0*dt + 1.0) + \ + 0.2962962962962962963*dt*(0.125*dt*(2.0*dt + 1.0) + 0.875*dt + 1.0) + \ + 0.96296296296296296296*dt + 1.0) - \ + 7.0189140975801991157*dt*(-0.049335189898860408029*dt*(2.0*dt + 1.0) + \ + 0.29601113939316244817*dt*(0.125*dt*(2.0*dt + 1.0) + 0.875*dt + 1.0) - \ + 0.12564355335492979587*dt*(0.074074074074074074074*dt*(2.0*dt + 1.0) + \ + 0.2962962962962962963*dt*(0.125*dt*(2.0*dt + 1.0) + 0.875*dt + 1.0) + \ + 0.96296296296296296296*dt + 1.0) + 0.22431393315265061193*dt + 1.0) + \ + 0.94010945196161777522*dt*(-0.23637909581542530626*dt*(2.0*dt + 1.0) - \ + 0.74817562366625959291*dt*(0.125*dt*(2.0*dt + 1.0) + 0.875*dt + 1.0) + \ + 0.88085458023927036857*dt*(0.074074074074074074074*dt*(2.0*dt + 1.0) + \ + 0.2962962962962962963*dt*(0.125*dt*(2.0*dt + 1.0) + 0.875*dt + 1.0) + \ + 0.96296296296296296296*dt + 1.0) + \ + 2.1165151389911680013*dt*(-0.049335189898860408029*dt*(2.0*dt + 1.0) + \ + 0.29601113939316244817*dt*(0.125*dt*(2.0*dt + 1.0) + 0.875*dt + 1.0) - \ + 0.12564355335492979587*dt*(0.074074074074074074074*dt*(2.0*dt + 1.0) + \ + 0.2962962962962962963*dt*(0.125*dt*(2.0*dt + 1.0) + 0.875*dt + 1.0) + \ + 0.96296296296296296296*dt + 1.0) + 0.22431393315265061193*dt + 1.0) - \ + 0.35816132904077632692*dt + 1.0) + 5.5065024887242400038*dt + 1.0)/20 + dt/20 + 1 + + assert N(expr.series(dt, 0, 8), 20) == ( + - Float('0.00092592592592592596126289', precision=70) * dt**7 + + Float('0.0027777777777777783174695', precision=70) * dt**6 + + Float('0.016666666666666656027029', precision=70) * dt**5 + + Float('0.083333333333333300951828', precision=70) * dt**4 + + Float('0.33333333333333337034077', precision=70) * dt**3 + + Float('1.0', precision=70) * dt**2 + + Float('1.0', precision=70) * dt + + Float('1.0', precision=70) + ) + + +def test_issue_11407(): + a, b, c, x = symbols('a b c x') + assert series(sqrt(a + b + c*x), x, 0, 1) == sqrt(a + b) + O(x) + assert series(sqrt(a + b + c + c*x), x, 0, 1) == sqrt(a + b + c) + O(x) + + +def test_issue_14037(): + assert (sin(x**50)/x**51).series(x, n=0) == 1/x + O(1, x) + + +def test_issue_20551(): + expr = (exp(x)/x).series(x, n=None) + terms = [ next(expr) for i in range(3) ] + assert terms == [1/x, 1, x/2] + + +def test_issue_20697(): + p_0, p_1, p_2, p_3, b_0, b_1, b_2 = symbols('p_0 p_1 p_2 p_3 b_0 b_1 b_2') + Q = (p_0 + (p_1 + (p_2 + p_3/y)/y)/y)/(1 + ((p_3/(b_0*y) + (b_0*p_2\ + - b_1*p_3)/b_0**2)/y + (b_0**2*p_1 - b_0*b_1*p_2 - p_3*(b_0*b_2\ + - b_1**2))/b_0**3)/y) + assert Q.series(y, n=3).ratsimp() == b_2*y**2 + b_1*y + b_0 + O(y**3) + + +def test_issue_21245(): + fi = (1 + sqrt(5))/2 + assert (1/(1 - x - x**2)).series(x, 1/fi, 1).factor() == \ + (-37*sqrt(5) - 83 + 13*sqrt(5)*x + 29*x + O((x - 2/(1 + sqrt(5)))**2, (x\ + , 2/(1 + sqrt(5)))))/((2*sqrt(5) + 5)**2*(x + sqrt(5)*x - 2)) + + + +def test_issue_21938(): + expr = sin(1/x + exp(-x)) - sin(1/x) + assert expr.series(x, oo) == (1/(24*x**4) - 1/(2*x**2) + 1 + O(x**(-6), (x, oo)))*exp(-x) + + +def test_issue_23432(): + expr = 1/sqrt(1 - x**2) + result = expr.series(x, 0.5) + assert result.is_Add and len(result.args) == 7 + + +def test_issue_23727(): + res = series(sqrt(1 - x**2), x, 0.1) + assert res.is_Add == True + + +def test_issue_24266(): + #type1: exp(f(x)) + assert (exp(-I*pi*(2*x+1))).series(x, 0, 3) == -1 + 2*I*pi*x + 2*pi**2*x**2 + O(x**3) + assert (exp(-I*pi*(2*x+1))*gamma(1+x)).series(x, 0, 3) == -1 + x*(EulerGamma + 2*I*pi) + \ + x**2*(-EulerGamma**2/2 + 23*pi**2/12 - 2*EulerGamma*I*pi) + O(x**3) + + #type2: c**f(x) + assert ((2*I)**(-I*pi*(2*x+1))).series(x, 0, 2) == exp(pi**2/2 - I*pi*log(2)) + \ + x*(pi**2*exp(pi**2/2 - I*pi*log(2)) - 2*I*pi*exp(pi**2/2 - I*pi*log(2))*log(2)) + O(x**2) + assert ((2)**(-I*pi*(2*x+1))).series(x, 0, 2) == exp(-I*pi*log(2)) - 2*I*pi*x*exp(-I*pi*log(2))*log(2) + O(x**2) + + #type3: f(y)**g(x) + assert ((y)**(I*pi*(2*x+1))).series(x, 0, 2) == exp(I*pi*log(y)) + 2*I*pi*x*exp(I*pi*log(y))*log(y) + O(x**2) + assert ((I*y)**(I*pi*(2*x+1))).series(x, 0, 2) == exp(I*pi*log(I*y)) + 2*I*pi*x*exp(I*pi*log(I*y))*log(I*y) + O(x**2) + + +def test_issue_26856(): + raises(ValueError, lambda: (2**x).series(x, oo, -1)) diff --git a/.venv/lib/python3.12/site-packages/sympy/sets/__init__.py b/.venv/lib/python3.12/site-packages/sympy/sets/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..8b909c0b5ef03b1e1e76dfbf4288f61860575da7 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/sets/__init__.py @@ -0,0 +1,36 @@ +from .sets import (Set, Interval, Union, FiniteSet, ProductSet, + Intersection, imageset, Complement, SymmetricDifference, + DisjointUnion) + +from .fancysets import ImageSet, Range, ComplexRegion +from .contains import Contains +from .conditionset import ConditionSet +from .ordinals import Ordinal, OmegaPower, ord0 +from .powerset import PowerSet +from ..core.singleton import S +from .handlers.comparison import _eval_is_eq # noqa:F401 +Complexes = S.Complexes +EmptySet = S.EmptySet +Integers = S.Integers +Naturals = S.Naturals +Naturals0 = S.Naturals0 +Rationals = S.Rationals +Reals = S.Reals +UniversalSet = S.UniversalSet + +__all__ = [ + 'Set', 'Interval', 'Union', 'EmptySet', 'FiniteSet', 'ProductSet', + 'Intersection', 'imageset', 'Complement', 'SymmetricDifference', 'DisjointUnion', + + 'ImageSet', 'Range', 'ComplexRegion', 'Reals', + + 'Contains', + + 'ConditionSet', + + 'Ordinal', 'OmegaPower', 'ord0', + + 'PowerSet', + + 'Reals', 'Naturals', 'Naturals0', 'UniversalSet', 'Integers', 'Rationals', +] diff --git a/.venv/lib/python3.12/site-packages/sympy/sets/__pycache__/__init__.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/sets/__pycache__/__init__.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..4132f9597aed4f18dba92521c91ed2998b32a7e8 Binary files /dev/null and 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b/.venv/lib/python3.12/site-packages/sympy/sets/__pycache__/sets.cpython-312.pyc @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9e03a9ebf2da3530658635ef10bfbddab3b4b6f3bd0bf77b2dd6efecd2273fde +size 115024 diff --git a/.venv/lib/python3.12/site-packages/sympy/sets/conditionset.py b/.venv/lib/python3.12/site-packages/sympy/sets/conditionset.py new file mode 100644 index 0000000000000000000000000000000000000000..e847e60ce97d7e9922ce907042ace941838b0ab1 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/sets/conditionset.py @@ -0,0 +1,246 @@ +from sympy.core.singleton import S +from sympy.core.basic import Basic +from sympy.core.containers import Tuple +from sympy.core.function import Lambda, BadSignatureError +from sympy.core.logic import fuzzy_bool +from sympy.core.relational import Eq +from sympy.core.symbol import Dummy +from sympy.core.sympify import _sympify +from sympy.logic.boolalg import And, as_Boolean +from sympy.utilities.iterables import sift, flatten, has_dups +from sympy.utilities.exceptions import sympy_deprecation_warning +from .contains import Contains +from .sets import Set, Union, FiniteSet, SetKind + + +adummy = Dummy('conditionset') + + +class ConditionSet(Set): + r""" + Set of elements which satisfies a given condition. + + .. math:: \{x \mid \textrm{condition}(x) = \texttt{True}, x \in S\} + + Examples + ======== + + >>> from sympy import Symbol, S, ConditionSet, pi, Eq, sin, Interval + >>> from sympy.abc import x, y, z + + >>> sin_sols = ConditionSet(x, Eq(sin(x), 0), Interval(0, 2*pi)) + >>> 2*pi in sin_sols + True + >>> pi/2 in sin_sols + False + >>> 3*pi in sin_sols + False + >>> 5 in ConditionSet(x, x**2 > 4, S.Reals) + True + + If the value is not in the base set, the result is false: + + >>> 5 in ConditionSet(x, x**2 > 4, Interval(2, 4)) + False + + Notes + ===== + + Symbols with assumptions should be avoided or else the + condition may evaluate without consideration of the set: + + >>> n = Symbol('n', negative=True) + >>> cond = (n > 0); cond + False + >>> ConditionSet(n, cond, S.Integers) + EmptySet + + Only free symbols can be changed by using `subs`: + + >>> c = ConditionSet(x, x < 1, {x, z}) + >>> c.subs(x, y) + ConditionSet(x, x < 1, {y, z}) + + To check if ``pi`` is in ``c`` use: + + >>> pi in c + False + + If no base set is specified, the universal set is implied: + + >>> ConditionSet(x, x < 1).base_set + UniversalSet + + Only symbols or symbol-like expressions can be used: + + >>> ConditionSet(x + 1, x + 1 < 1, S.Integers) + Traceback (most recent call last): + ... + ValueError: non-symbol dummy not recognized in condition + + When the base set is a ConditionSet, the symbols will be + unified if possible with preference for the outermost symbols: + + >>> ConditionSet(x, x < y, ConditionSet(z, z + y < 2, S.Integers)) + ConditionSet(x, (x < y) & (x + y < 2), Integers) + + """ + def __new__(cls, sym, condition, base_set=S.UniversalSet): + sym = _sympify(sym) + flat = flatten([sym]) + if has_dups(flat): + raise BadSignatureError("Duplicate symbols detected") + base_set = _sympify(base_set) + if not isinstance(base_set, Set): + raise TypeError( + 'base set should be a Set object, not %s' % base_set) + condition = _sympify(condition) + + if isinstance(condition, FiniteSet): + condition_orig = condition + temp = (Eq(lhs, 0) for lhs in condition) + condition = And(*temp) + sympy_deprecation_warning( + f""" +Using a set for the condition in ConditionSet is deprecated. Use a boolean +instead. + +In this case, replace + + {condition_orig} + +with + + {condition} +""", + deprecated_since_version='1.5', + active_deprecations_target="deprecated-conditionset-set", + ) + + condition = as_Boolean(condition) + + if condition is S.true: + return base_set + + if condition is S.false: + return S.EmptySet + + if base_set is S.EmptySet: + return S.EmptySet + + # no simple answers, so now check syms + for i in flat: + if not getattr(i, '_diff_wrt', False): + raise ValueError('`%s` is not symbol-like' % i) + + if base_set.contains(sym) is S.false: + raise TypeError('sym `%s` is not in base_set `%s`' % (sym, base_set)) + + know = None + if isinstance(base_set, FiniteSet): + sifted = sift( + base_set, lambda _: fuzzy_bool(condition.subs(sym, _))) + if sifted[None]: + know = FiniteSet(*sifted[True]) + base_set = FiniteSet(*sifted[None]) + else: + return FiniteSet(*sifted[True]) + + if isinstance(base_set, cls): + s, c, b = base_set.args + def sig(s): + return cls(s, Eq(adummy, 0)).as_dummy().sym + sa, sb = map(sig, (sym, s)) + if sa != sb: + raise BadSignatureError('sym does not match sym of base set') + reps = dict(zip(flatten([sym]), flatten([s]))) + if s == sym: + condition = And(condition, c) + base_set = b + elif not c.free_symbols & sym.free_symbols: + reps = {v: k for k, v in reps.items()} + condition = And(condition, c.xreplace(reps)) + base_set = b + elif not condition.free_symbols & s.free_symbols: + sym = sym.xreplace(reps) + condition = And(condition.xreplace(reps), c) + base_set = b + + # flatten ConditionSet(Contains(ConditionSet())) expressions + if isinstance(condition, Contains) and (sym == condition.args[0]): + if isinstance(condition.args[1], Set): + return condition.args[1].intersect(base_set) + + rv = Basic.__new__(cls, sym, condition, base_set) + return rv if know is None else Union(know, rv) + + sym = property(lambda self: self.args[0]) + condition = property(lambda self: self.args[1]) + base_set = property(lambda self: self.args[2]) + + @property + def free_symbols(self): + cond_syms = self.condition.free_symbols - self.sym.free_symbols + return cond_syms | self.base_set.free_symbols + + @property + def bound_symbols(self): + return flatten([self.sym]) + + def _contains(self, other): + def ok_sig(a, b): + tuples = [isinstance(i, Tuple) for i in (a, b)] + c = tuples.count(True) + if c == 1: + return False + if c == 0: + return True + return len(a) == len(b) and all( + ok_sig(i, j) for i, j in zip(a, b)) + if not ok_sig(self.sym, other): + return S.false + + # try doing base_cond first and return + # False immediately if it is False + base_cond = Contains(other, self.base_set) + if base_cond is S.false: + return S.false + + # Substitute other into condition. This could raise e.g. for + # ConditionSet(x, 1/x >= 0, Reals).contains(0) + lamda = Lambda((self.sym,), self.condition) + try: + lambda_cond = lamda(other) + except TypeError: + return None + else: + return And(base_cond, lambda_cond) + + def as_relational(self, other): + f = Lambda(self.sym, self.condition) + if isinstance(self.sym, Tuple): + f = f(*other) + else: + f = f(other) + return And(f, self.base_set.contains(other)) + + def _eval_subs(self, old, new): + sym, cond, base = self.args + dsym = sym.subs(old, adummy) + insym = dsym.has(adummy) + # prioritize changing a symbol in the base + newbase = base.subs(old, new) + if newbase != base: + if not insym: + cond = cond.subs(old, new) + return self.func(sym, cond, newbase) + if insym: + pass # no change of bound symbols via subs + elif getattr(new, '_diff_wrt', False): + cond = cond.subs(old, new) + else: + pass # let error about the symbol raise from __new__ + return self.func(sym, cond, base) + + def _kind(self): + return SetKind(self.sym.kind) diff --git a/.venv/lib/python3.12/site-packages/sympy/sets/contains.py b/.venv/lib/python3.12/site-packages/sympy/sets/contains.py new file mode 100644 index 0000000000000000000000000000000000000000..403d4875279d718724a898efa5cba41bc7bed6ea --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/sets/contains.py @@ -0,0 +1,63 @@ +from sympy.core import S +from sympy.core.sympify import sympify +from sympy.core.relational import Eq, Ne +from sympy.core.parameters import global_parameters +from sympy.logic.boolalg import Boolean +from sympy.utilities.misc import func_name +from .sets import Set + + +class Contains(Boolean): + """ + Asserts that x is an element of the set S. + + Examples + ======== + + >>> from sympy import Symbol, Integer, S, Contains + >>> Contains(Integer(2), S.Integers) + True + >>> Contains(Integer(-2), S.Naturals) + False + >>> i = Symbol('i', integer=True) + >>> Contains(i, S.Naturals) + Contains(i, Naturals) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Element_%28mathematics%29 + """ + def __new__(cls, x, s, evaluate=None): + x = sympify(x) + s = sympify(s) + + if evaluate is None: + evaluate = global_parameters.evaluate + + if not isinstance(s, Set): + raise TypeError('expecting Set, not %s' % func_name(s)) + + if evaluate: + # _contains can return symbolic booleans that would be returned by + # s.contains(x) but here for Contains(x, s) we only evaluate to + # true, false or return the unevaluated Contains. + result = s._contains(x) + + if isinstance(result, Boolean): + if result in (S.true, S.false): + return result + elif result is not None: + raise TypeError("_contains() should return Boolean or None") + + return super().__new__(cls, x, s) + + @property + def binary_symbols(self): + return set().union(*[i.binary_symbols + for i in self.args[1].args + if i.is_Boolean or i.is_Symbol or + isinstance(i, (Eq, Ne))]) + + def as_set(self): + return self.args[1] diff --git a/.venv/lib/python3.12/site-packages/sympy/sets/fancysets.py b/.venv/lib/python3.12/site-packages/sympy/sets/fancysets.py new file mode 100644 index 0000000000000000000000000000000000000000..e0e24a2a864222d16ba1a697558b5211127fb2ad --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/sets/fancysets.py @@ -0,0 +1,1523 @@ +from functools import reduce +from itertools import product + +from sympy.core.basic import Basic +from sympy.core.containers import Tuple +from sympy.core.expr import Expr +from sympy.core.function import Lambda +from sympy.core.logic import fuzzy_not, fuzzy_or, fuzzy_and +from sympy.core.mod import Mod +from sympy.core.intfunc import igcd +from sympy.core.numbers import oo, Rational, Integer +from sympy.core.relational import Eq, is_eq +from sympy.core.kind import NumberKind +from sympy.core.singleton import Singleton, S +from sympy.core.symbol import Dummy, symbols, Symbol +from sympy.core.sympify import _sympify, sympify, _sympy_converter +from sympy.functions.elementary.integers import ceiling, floor +from sympy.functions.elementary.trigonometric import sin, cos +from sympy.logic.boolalg import And, Or +from .sets import tfn, Set, Interval, Union, FiniteSet, ProductSet, SetKind +from sympy.utilities.misc import filldedent + + +class Rationals(Set, metaclass=Singleton): + """ + Represents the rational numbers. This set is also available as + the singleton ``S.Rationals``. + + Examples + ======== + + >>> from sympy import S + >>> S.Half in S.Rationals + True + >>> iterable = iter(S.Rationals) + >>> [next(iterable) for i in range(12)] + [0, 1, -1, 1/2, 2, -1/2, -2, 1/3, 3, -1/3, -3, 2/3] + """ + + is_iterable = True + _inf = S.NegativeInfinity + _sup = S.Infinity + is_empty = False + is_finite_set = False + + def _contains(self, other): + if not isinstance(other, Expr): + return S.false + return tfn[other.is_rational] + + def __iter__(self): + yield S.Zero + yield S.One + yield S.NegativeOne + d = 2 + while True: + for n in range(d): + if igcd(n, d) == 1: + yield Rational(n, d) + yield Rational(d, n) + yield Rational(-n, d) + yield Rational(-d, n) + d += 1 + + @property + def _boundary(self): + return S.Reals + + def _kind(self): + return SetKind(NumberKind) + + +class Naturals(Set, metaclass=Singleton): + """ + Represents the natural numbers (or counting numbers) which are all + positive integers starting from 1. This set is also available as + the singleton ``S.Naturals``. + + Examples + ======== + + >>> from sympy import S, Interval, pprint + >>> 5 in S.Naturals + True + >>> iterable = iter(S.Naturals) + >>> next(iterable) + 1 + >>> next(iterable) + 2 + >>> next(iterable) + 3 + >>> pprint(S.Naturals.intersect(Interval(0, 10))) + {1, 2, ..., 10} + + See Also + ======== + + Naturals0 : non-negative integers (i.e. includes 0, too) + Integers : also includes negative integers + """ + + is_iterable = True + _inf: Integer = S.One + _sup = S.Infinity + is_empty = False + is_finite_set = False + + def _contains(self, other): + if not isinstance(other, Expr): + return S.false + elif other.is_positive and other.is_integer: + return S.true + elif other.is_integer is False or other.is_positive is False: + return S.false + + def _eval_is_subset(self, other): + return Range(1, oo).is_subset(other) + + def _eval_is_superset(self, other): + return Range(1, oo).is_superset(other) + + def __iter__(self): + i = self._inf + while True: + yield i + i = i + 1 + + @property + def _boundary(self): + return self + + def as_relational(self, x): + return And(Eq(floor(x), x), x >= self.inf, x < oo) + + def _kind(self): + return SetKind(NumberKind) + + +class Naturals0(Naturals): + """Represents the whole numbers which are all the non-negative integers, + inclusive of zero. + + See Also + ======== + + Naturals : positive integers; does not include 0 + Integers : also includes the negative integers + """ + _inf = S.Zero + + def _contains(self, other): + if not isinstance(other, Expr): + return S.false + elif other.is_integer and other.is_nonnegative: + return S.true + elif other.is_integer is False or other.is_nonnegative is False: + return S.false + + def _eval_is_subset(self, other): + return Range(oo).is_subset(other) + + def _eval_is_superset(self, other): + return Range(oo).is_superset(other) + + +class Integers(Set, metaclass=Singleton): + """ + Represents all integers: positive, negative and zero. This set is also + available as the singleton ``S.Integers``. + + Examples + ======== + + >>> from sympy import S, Interval, pprint + >>> 5 in S.Naturals + True + >>> iterable = iter(S.Integers) + >>> next(iterable) + 0 + >>> next(iterable) + 1 + >>> next(iterable) + -1 + >>> next(iterable) + 2 + + >>> pprint(S.Integers.intersect(Interval(-4, 4))) + {-4, -3, ..., 4} + + See Also + ======== + + Naturals0 : non-negative integers + Integers : positive and negative integers and zero + """ + + is_iterable = True + is_empty = False + is_finite_set = False + + def _contains(self, other): + if not isinstance(other, Expr): + return S.false + return tfn[other.is_integer] + + def __iter__(self): + yield S.Zero + i = S.One + while True: + yield i + yield -i + i = i + 1 + + @property + def _inf(self): + return S.NegativeInfinity + + @property + def _sup(self): + return S.Infinity + + @property + def _boundary(self): + return self + + def _kind(self): + return SetKind(NumberKind) + + def as_relational(self, x): + return And(Eq(floor(x), x), -oo < x, x < oo) + + def _eval_is_subset(self, other): + return Range(-oo, oo).is_subset(other) + + def _eval_is_superset(self, other): + return Range(-oo, oo).is_superset(other) + + +class Reals(Interval, metaclass=Singleton): + """ + Represents all real numbers + from negative infinity to positive infinity, + including all integer, rational and irrational numbers. + This set is also available as the singleton ``S.Reals``. + + + Examples + ======== + + >>> from sympy import S, Rational, pi, I + >>> 5 in S.Reals + True + >>> Rational(-1, 2) in S.Reals + True + >>> pi in S.Reals + True + >>> 3*I in S.Reals + False + >>> S.Reals.contains(pi) + True + + + See Also + ======== + + ComplexRegion + """ + @property + def start(self): + return S.NegativeInfinity + + @property + def end(self): + return S.Infinity + + @property + def left_open(self): + return True + + @property + def right_open(self): + return True + + def __eq__(self, other): + return other == Interval(S.NegativeInfinity, S.Infinity) + + def __hash__(self): + return hash(Interval(S.NegativeInfinity, S.Infinity)) + + +class ImageSet(Set): + """ + Image of a set under a mathematical function. The transformation + must be given as a Lambda function which has as many arguments + as the elements of the set upon which it operates, e.g. 1 argument + when acting on the set of integers or 2 arguments when acting on + a complex region. + + This function is not normally called directly, but is called + from ``imageset``. + + + Examples + ======== + + >>> from sympy import Symbol, S, pi, Dummy, Lambda + >>> from sympy import FiniteSet, ImageSet, Interval + + >>> x = Symbol('x') + >>> N = S.Naturals + >>> squares = ImageSet(Lambda(x, x**2), N) # {x**2 for x in N} + >>> 4 in squares + True + >>> 5 in squares + False + + >>> FiniteSet(0, 1, 2, 3, 4, 5, 6, 7, 9, 10).intersect(squares) + {1, 4, 9} + + >>> square_iterable = iter(squares) + >>> for i in range(4): + ... next(square_iterable) + 1 + 4 + 9 + 16 + + If you want to get value for `x` = 2, 1/2 etc. (Please check whether the + `x` value is in ``base_set`` or not before passing it as args) + + >>> squares.lamda(2) + 4 + >>> squares.lamda(S(1)/2) + 1/4 + + >>> n = Dummy('n') + >>> solutions = ImageSet(Lambda(n, n*pi), S.Integers) # solutions of sin(x) = 0 + >>> dom = Interval(-1, 1) + >>> dom.intersect(solutions) + {0} + + See Also + ======== + + sympy.sets.sets.imageset + """ + def __new__(cls, flambda, *sets): + if not isinstance(flambda, Lambda): + raise ValueError('First argument must be a Lambda') + + signature = flambda.signature + + if len(signature) != len(sets): + raise ValueError('Incompatible signature') + + sets = [_sympify(s) for s in sets] + + if not all(isinstance(s, Set) for s in sets): + raise TypeError("Set arguments to ImageSet should of type Set") + + if not all(cls._check_sig(sg, st) for sg, st in zip(signature, sets)): + raise ValueError("Signature %s does not match sets %s" % (signature, sets)) + + if flambda is S.IdentityFunction and len(sets) == 1: + return sets[0] + + if not set(flambda.variables) & flambda.expr.free_symbols: + is_empty = fuzzy_or(s.is_empty for s in sets) + if is_empty == True: + return S.EmptySet + elif is_empty == False: + return FiniteSet(flambda.expr) + + return Basic.__new__(cls, flambda, *sets) + + lamda = property(lambda self: self.args[0]) + base_sets = property(lambda self: self.args[1:]) + + @property + def base_set(self): + # XXX: Maybe deprecate this? It is poorly defined in handling + # the multivariate case... + sets = self.base_sets + if len(sets) == 1: + return sets[0] + else: + return ProductSet(*sets).flatten() + + @property + def base_pset(self): + return ProductSet(*self.base_sets) + + @classmethod + def _check_sig(cls, sig_i, set_i): + if sig_i.is_symbol: + return True + elif isinstance(set_i, ProductSet): + sets = set_i.sets + if len(sig_i) != len(sets): + return False + # Recurse through the signature for nested tuples: + return all(cls._check_sig(ts, ps) for ts, ps in zip(sig_i, sets)) + else: + # XXX: Need a better way of checking whether a set is a set of + # Tuples or not. For example a FiniteSet can contain Tuples + # but so can an ImageSet or a ConditionSet. Others like + # Integers, Reals etc can not contain Tuples. We could just + # list the possibilities here... Current code for e.g. + # _contains probably only works for ProductSet. + return True # Give the benefit of the doubt + + def __iter__(self): + already_seen = set() + for i in self.base_pset: + val = self.lamda(*i) + if val in already_seen: + continue + else: + already_seen.add(val) + yield val + + def _is_multivariate(self): + return len(self.lamda.variables) > 1 + + def _contains(self, other): + from sympy.solvers.solveset import _solveset_multi + + def get_symsetmap(signature, base_sets): + '''Attempt to get a map of symbols to base_sets''' + queue = list(zip(signature, base_sets)) + symsetmap = {} + for sig, base_set in queue: + if sig.is_symbol: + symsetmap[sig] = base_set + elif base_set.is_ProductSet: + sets = base_set.sets + if len(sig) != len(sets): + raise ValueError("Incompatible signature") + # Recurse + queue.extend(zip(sig, sets)) + else: + # If we get here then we have something like sig = (x, y) and + # base_set = {(1, 2), (3, 4)}. For now we give up. + return None + + return symsetmap + + def get_equations(expr, candidate): + '''Find the equations relating symbols in expr and candidate.''' + queue = [(expr, candidate)] + for e, c in queue: + if not isinstance(e, Tuple): + yield Eq(e, c) + elif not isinstance(c, Tuple) or len(e) != len(c): + yield False + return + else: + queue.extend(zip(e, c)) + + # Get the basic objects together: + other = _sympify(other) + expr = self.lamda.expr + sig = self.lamda.signature + variables = self.lamda.variables + base_sets = self.base_sets + + # Use dummy symbols for ImageSet parameters so they don't match + # anything in other + rep = {v: Dummy(v.name) for v in variables} + variables = [v.subs(rep) for v in variables] + sig = sig.subs(rep) + expr = expr.subs(rep) + + # Map the parts of other to those in the Lambda expr + equations = [] + for eq in get_equations(expr, other): + # Unsatisfiable equation? + if eq is False: + return S.false + equations.append(eq) + + # Map the symbols in the signature to the corresponding domains + symsetmap = get_symsetmap(sig, base_sets) + if symsetmap is None: + # Can't factor the base sets to a ProductSet + return None + + # Which of the variables in the Lambda signature need to be solved for? + symss = (eq.free_symbols for eq in equations) + variables = set(variables) & reduce(set.union, symss, set()) + + # Use internal multivariate solveset + variables = tuple(variables) + base_sets = [symsetmap[v] for v in variables] + solnset = _solveset_multi(equations, variables, base_sets) + if solnset is None: + return None + return tfn[fuzzy_not(solnset.is_empty)] + + @property + def is_iterable(self): + return all(s.is_iterable for s in self.base_sets) + + def doit(self, **hints): + from sympy.sets.setexpr import SetExpr + f = self.lamda + sig = f.signature + if len(sig) == 1 and sig[0].is_symbol and isinstance(f.expr, Expr): + base_set = self.base_sets[0] + return SetExpr(base_set)._eval_func(f).set + if all(s.is_FiniteSet for s in self.base_sets): + return FiniteSet(*(f(*a) for a in product(*self.base_sets))) + return self + + def _kind(self): + return SetKind(self.lamda.expr.kind) + + +class Range(Set): + """ + Represents a range of integers. Can be called as ``Range(stop)``, + ``Range(start, stop)``, or ``Range(start, stop, step)``; when ``step`` is + not given it defaults to 1. + + ``Range(stop)`` is the same as ``Range(0, stop, 1)`` and the stop value + (just as for Python ranges) is not included in the Range values. + + >>> from sympy import Range + >>> list(Range(3)) + [0, 1, 2] + + The step can also be negative: + + >>> list(Range(10, 0, -2)) + [10, 8, 6, 4, 2] + + The stop value is made canonical so equivalent ranges always + have the same args: + + >>> Range(0, 10, 3) + Range(0, 12, 3) + + Infinite ranges are allowed. ``oo`` and ``-oo`` are never included in the + set (``Range`` is always a subset of ``Integers``). If the starting point + is infinite, then the final value is ``stop - step``. To iterate such a + range, it needs to be reversed: + + >>> from sympy import oo + >>> r = Range(-oo, 1) + >>> r[-1] + 0 + >>> next(iter(r)) + Traceback (most recent call last): + ... + TypeError: Cannot iterate over Range with infinite start + >>> next(iter(r.reversed)) + 0 + + Although ``Range`` is a :class:`Set` (and supports the normal set + operations) it maintains the order of the elements and can + be used in contexts where ``range`` would be used. + + >>> from sympy import Interval + >>> Range(0, 10, 2).intersect(Interval(3, 7)) + Range(4, 8, 2) + >>> list(_) + [4, 6] + + Although slicing of a Range will always return a Range -- possibly + empty -- an empty set will be returned from any intersection that + is empty: + + >>> Range(3)[:0] + Range(0, 0, 1) + >>> Range(3).intersect(Interval(4, oo)) + EmptySet + >>> Range(3).intersect(Range(4, oo)) + EmptySet + + Range will accept symbolic arguments but has very limited support + for doing anything other than displaying the Range: + + >>> from sympy import Symbol, pprint + >>> from sympy.abc import i, j, k + >>> Range(i, j, k).start + i + >>> Range(i, j, k).inf + Traceback (most recent call last): + ... + ValueError: invalid method for symbolic range + + Better success will be had when using integer symbols: + + >>> n = Symbol('n', integer=True) + >>> r = Range(n, n + 20, 3) + >>> r.inf + n + >>> pprint(r) + {n, n + 3, ..., n + 18} + """ + + def __new__(cls, *args): + if len(args) == 1: + if isinstance(args[0], range): + raise TypeError( + 'use sympify(%s) to convert range to Range' % args[0]) + + # expand range + slc = slice(*args) + + if slc.step == 0: + raise ValueError("step cannot be 0") + + start, stop, step = slc.start or 0, slc.stop, slc.step or 1 + try: + ok = [] + for w in (start, stop, step): + w = sympify(w) + if w in [S.NegativeInfinity, S.Infinity] or ( + w.has(Symbol) and w.is_integer != False): + ok.append(w) + elif not w.is_Integer: + if w.is_infinite: + raise ValueError('infinite symbols not allowed') + raise ValueError + else: + ok.append(w) + except ValueError: + raise ValueError(filldedent(''' + Finite arguments to Range must be integers; `imageset` can define + other cases, e.g. use `imageset(i, i/10, Range(3))` to give + [0, 1/10, 1/5].''')) + start, stop, step = ok + + null = False + if any(i.has(Symbol) for i in (start, stop, step)): + dif = stop - start + n = dif/step + if n.is_Rational: + if dif == 0: + null = True + else: # (x, x + 5, 2) or (x, 3*x, x) + n = floor(n) + end = start + n*step + if dif.is_Rational: # (x, x + 5, 2) + if (end - stop).is_negative: + end += step + else: # (x, 3*x, x) + if (end/stop - 1).is_negative: + end += step + elif n.is_extended_negative: + null = True + else: + end = stop # other methods like sup and reversed must fail + elif start.is_infinite: + span = step*(stop - start) + if span is S.NaN or span <= 0: + null = True + elif step.is_Integer and stop.is_infinite and abs(step) != 1: + raise ValueError(filldedent(''' + Step size must be %s in this case.''' % (1 if step > 0 else -1))) + else: + end = stop + else: + oostep = step.is_infinite + if oostep: + step = S.One if step > 0 else S.NegativeOne + n = ceiling((stop - start)/step) + if n <= 0: + null = True + elif oostep: + step = S.One # make it canonical + end = start + step + else: + end = start + n*step + if null: + start = end = S.Zero + step = S.One + return Basic.__new__(cls, start, end, step) + + start = property(lambda self: self.args[0]) + stop = property(lambda self: self.args[1]) + step = property(lambda self: self.args[2]) + + @property + def reversed(self): + """Return an equivalent Range in the opposite order. + + Examples + ======== + + >>> from sympy import Range + >>> Range(10).reversed + Range(9, -1, -1) + """ + if self.has(Symbol): + n = (self.stop - self.start)/self.step + if not n.is_extended_positive or not all( + i.is_integer or i.is_infinite for i in self.args): + raise ValueError('invalid method for symbolic range') + if self.start == self.stop: + return self + return self.func( + self.stop - self.step, self.start - self.step, -self.step) + + def _kind(self): + return SetKind(NumberKind) + + def _contains(self, other): + if self.start == self.stop: + return S.false + if other.is_infinite: + return S.false + if not other.is_integer: + return tfn[other.is_integer] + if self.has(Symbol): + n = (self.stop - self.start)/self.step + if not n.is_extended_positive or not all( + i.is_integer or i.is_infinite for i in self.args): + return + else: + n = self.size + if self.start.is_finite: + ref = self.start + elif self.stop.is_finite: + ref = self.stop + else: # both infinite; step is +/- 1 (enforced by __new__) + return S.true + if n == 1: + return Eq(other, self[0]) + res = (ref - other) % self.step + if res == S.Zero: + if self.has(Symbol): + d = Dummy('i') + return self.as_relational(d).subs(d, other) + return And(other >= self.inf, other <= self.sup) + elif res.is_Integer: # off sequence + return S.false + else: # symbolic/unsimplified residue modulo step + return None + + def __iter__(self): + n = self.size # validate + if not (n.has(S.Infinity) or n.has(S.NegativeInfinity) or n.is_Integer): + raise TypeError("Cannot iterate over symbolic Range") + if self.start in [S.NegativeInfinity, S.Infinity]: + raise TypeError("Cannot iterate over Range with infinite start") + elif self.start != self.stop: + i = self.start + if n.is_infinite: + while True: + yield i + i += self.step + else: + for _ in range(n): + yield i + i += self.step + + @property + def is_iterable(self): + # Check that size can be determined, used by __iter__ + dif = self.stop - self.start + n = dif/self.step + if not (n.has(S.Infinity) or n.has(S.NegativeInfinity) or n.is_Integer): + return False + if self.start in [S.NegativeInfinity, S.Infinity]: + return False + if not (n.is_extended_nonnegative and all(i.is_integer for i in self.args)): + return False + return True + + def __len__(self): + rv = self.size + if rv is S.Infinity: + raise ValueError('Use .size to get the length of an infinite Range') + return int(rv) + + @property + def size(self): + if self.start == self.stop: + return S.Zero + dif = self.stop - self.start + n = dif/self.step + if n.is_infinite: + return S.Infinity + if n.is_extended_nonnegative and all(i.is_integer for i in self.args): + return abs(floor(n)) + raise ValueError('Invalid method for symbolic Range') + + @property + def is_finite_set(self): + if self.start.is_integer and self.stop.is_integer: + return True + return self.size.is_finite + + @property + def is_empty(self): + try: + return self.size.is_zero + except ValueError: + return None + + def __bool__(self): + # this only distinguishes between definite null range + # and non-null/unknown null; getting True doesn't mean + # that it actually is not null + b = is_eq(self.start, self.stop) + if b is None: + raise ValueError('cannot tell if Range is null or not') + return not bool(b) + + def __getitem__(self, i): + ooslice = "cannot slice from the end with an infinite value" + zerostep = "slice step cannot be zero" + infinite = "slicing not possible on range with infinite start" + # if we had to take every other element in the following + # oo, ..., 6, 4, 2, 0 + # we might get oo, ..., 4, 0 or oo, ..., 6, 2 + ambiguous = "cannot unambiguously re-stride from the end " + \ + "with an infinite value" + if isinstance(i, slice): + if self.size.is_finite: # validates, too + if self.start == self.stop: + return Range(0) + start, stop, step = i.indices(self.size) + n = ceiling((stop - start)/step) + if n <= 0: + return Range(0) + canonical_stop = start + n*step + end = canonical_stop - step + ss = step*self.step + return Range(self[start], self[end] + ss, ss) + else: # infinite Range + start = i.start + stop = i.stop + if i.step == 0: + raise ValueError(zerostep) + step = i.step or 1 + ss = step*self.step + #--------------------- + # handle infinite Range + # i.e. Range(-oo, oo) or Range(oo, -oo, -1) + # -------------------- + if self.start.is_infinite and self.stop.is_infinite: + raise ValueError(infinite) + #--------------------- + # handle infinite on right + # e.g. Range(0, oo) or Range(0, -oo, -1) + # -------------------- + if self.stop.is_infinite: + # start and stop are not interdependent -- + # they only depend on step --so we use the + # equivalent reversed values + return self.reversed[ + stop if stop is None else -stop + 1: + start if start is None else -start: + step].reversed + #--------------------- + # handle infinite on the left + # e.g. Range(oo, 0, -1) or Range(-oo, 0) + # -------------------- + # consider combinations of + # start/stop {== None, < 0, == 0, > 0} and + # step {< 0, > 0} + if start is None: + if stop is None: + if step < 0: + return Range(self[-1], self.start, ss) + elif step > 1: + raise ValueError(ambiguous) + else: # == 1 + return self + elif stop < 0: + if step < 0: + return Range(self[-1], self[stop], ss) + else: # > 0 + return Range(self.start, self[stop], ss) + elif stop == 0: + if step > 0: + return Range(0) + else: # < 0 + raise ValueError(ooslice) + elif stop == 1: + if step > 0: + raise ValueError(ooslice) # infinite singleton + else: # < 0 + raise ValueError(ooslice) + else: # > 1 + raise ValueError(ooslice) + elif start < 0: + if stop is None: + if step < 0: + return Range(self[start], self.start, ss) + else: # > 0 + return Range(self[start], self.stop, ss) + elif stop < 0: + return Range(self[start], self[stop], ss) + elif stop == 0: + if step < 0: + raise ValueError(ooslice) + else: # > 0 + return Range(0) + elif stop > 0: + raise ValueError(ooslice) + elif start == 0: + if stop is None: + if step < 0: + raise ValueError(ooslice) # infinite singleton + elif step > 1: + raise ValueError(ambiguous) + else: # == 1 + return self + elif stop < 0: + if step > 1: + raise ValueError(ambiguous) + elif step == 1: + return Range(self.start, self[stop], ss) + else: # < 0 + return Range(0) + else: # >= 0 + raise ValueError(ooslice) + elif start > 0: + raise ValueError(ooslice) + else: + if self.start == self.stop: + raise IndexError('Range index out of range') + if not (all(i.is_integer or i.is_infinite + for i in self.args) and ((self.stop - self.start)/ + self.step).is_extended_positive): + raise ValueError('Invalid method for symbolic Range') + if i == 0: + if self.start.is_infinite: + raise ValueError(ooslice) + return self.start + if i == -1: + if self.stop.is_infinite: + raise ValueError(ooslice) + return self.stop - self.step + n = self.size # must be known for any other index + rv = (self.stop if i < 0 else self.start) + i*self.step + if rv.is_infinite: + raise ValueError(ooslice) + val = (rv - self.start)/self.step + rel = fuzzy_or([val.is_infinite, + fuzzy_and([val.is_nonnegative, (n-val).is_nonnegative])]) + if rel: + return rv + if rel is None: + raise ValueError('Invalid method for symbolic Range') + raise IndexError("Range index out of range") + + @property + def _inf(self): + if not self: + return S.EmptySet.inf + if self.has(Symbol): + if all(i.is_integer or i.is_infinite for i in self.args): + dif = self.stop - self.start + if self.step.is_positive and dif.is_positive: + return self.start + elif self.step.is_negative and dif.is_negative: + return self.stop - self.step + raise ValueError('invalid method for symbolic range') + if self.step > 0: + return self.start + else: + return self.stop - self.step + + @property + def _sup(self): + if not self: + return S.EmptySet.sup + if self.has(Symbol): + if all(i.is_integer or i.is_infinite for i in self.args): + dif = self.stop - self.start + if self.step.is_positive and dif.is_positive: + return self.stop - self.step + elif self.step.is_negative and dif.is_negative: + return self.start + raise ValueError('invalid method for symbolic range') + if self.step > 0: + return self.stop - self.step + else: + return self.start + + @property + def _boundary(self): + return self + + def as_relational(self, x): + """Rewrite a Range in terms of equalities and logic operators. """ + if self.start.is_infinite: + assert not self.stop.is_infinite # by instantiation + a = self.reversed.start + else: + a = self.start + step = self.step + in_seq = Eq(Mod(x - a, step), 0) + ints = And(Eq(Mod(a, 1), 0), Eq(Mod(step, 1), 0)) + n = (self.stop - self.start)/self.step + if n == 0: + return S.EmptySet.as_relational(x) + if n == 1: + return And(Eq(x, a), ints) + try: + a, b = self.inf, self.sup + except ValueError: + a = None + if a is not None: + range_cond = And( + x > a if a.is_infinite else x >= a, + x < b if b.is_infinite else x <= b) + else: + a, b = self.start, self.stop - self.step + range_cond = Or( + And(self.step >= 1, x > a if a.is_infinite else x >= a, + x < b if b.is_infinite else x <= b), + And(self.step <= -1, x < a if a.is_infinite else x <= a, + x > b if b.is_infinite else x >= b)) + return And(in_seq, ints, range_cond) + + +_sympy_converter[range] = lambda r: Range(r.start, r.stop, r.step) + +def normalize_theta_set(theta): + r""" + Normalize a Real Set `theta` in the interval `[0, 2\pi)`. It returns + a normalized value of theta in the Set. For Interval, a maximum of + one cycle $[0, 2\pi]$, is returned i.e. for theta equal to $[0, 10\pi]$, + returned normalized value would be $[0, 2\pi)$. As of now intervals + with end points as non-multiples of ``pi`` is not supported. + + Raises + ====== + + NotImplementedError + The algorithms for Normalizing theta Set are not yet + implemented. + ValueError + The input is not valid, i.e. the input is not a real set. + RuntimeError + It is a bug, please report to the github issue tracker. + + Examples + ======== + + >>> from sympy.sets.fancysets import normalize_theta_set + >>> from sympy import Interval, FiniteSet, pi + >>> normalize_theta_set(Interval(9*pi/2, 5*pi)) + Interval(pi/2, pi) + >>> normalize_theta_set(Interval(-3*pi/2, pi/2)) + Interval.Ropen(0, 2*pi) + >>> normalize_theta_set(Interval(-pi/2, pi/2)) + Union(Interval(0, pi/2), Interval.Ropen(3*pi/2, 2*pi)) + >>> normalize_theta_set(Interval(-4*pi, 3*pi)) + Interval.Ropen(0, 2*pi) + >>> normalize_theta_set(Interval(-3*pi/2, -pi/2)) + Interval(pi/2, 3*pi/2) + >>> normalize_theta_set(FiniteSet(0, pi, 3*pi)) + {0, pi} + + """ + from sympy.functions.elementary.trigonometric import _pi_coeff + + if theta.is_Interval: + interval_len = theta.measure + # one complete circle + if interval_len >= 2*S.Pi: + if interval_len == 2*S.Pi and theta.left_open and theta.right_open: + k = _pi_coeff(theta.start) + return Union(Interval(0, k*S.Pi, False, True), + Interval(k*S.Pi, 2*S.Pi, True, True)) + return Interval(0, 2*S.Pi, False, True) + + k_start, k_end = _pi_coeff(theta.start), _pi_coeff(theta.end) + + if k_start is None or k_end is None: + raise NotImplementedError("Normalizing theta without pi as coefficient is " + "not yet implemented") + new_start = k_start*S.Pi + new_end = k_end*S.Pi + + if new_start > new_end: + return Union(Interval(S.Zero, new_end, False, theta.right_open), + Interval(new_start, 2*S.Pi, theta.left_open, True)) + else: + return Interval(new_start, new_end, theta.left_open, theta.right_open) + + elif theta.is_FiniteSet: + new_theta = [] + for element in theta: + k = _pi_coeff(element) + if k is None: + raise NotImplementedError('Normalizing theta without pi as ' + 'coefficient, is not Implemented.') + else: + new_theta.append(k*S.Pi) + return FiniteSet(*new_theta) + + elif theta.is_Union: + return Union(*[normalize_theta_set(interval) for interval in theta.args]) + + elif theta.is_subset(S.Reals): + raise NotImplementedError("Normalizing theta when, it is of type %s is not " + "implemented" % type(theta)) + else: + raise ValueError(" %s is not a real set" % (theta)) + + +class ComplexRegion(Set): + r""" + Represents the Set of all Complex Numbers. It can represent a + region of Complex Plane in both the standard forms Polar and + Rectangular coordinates. + + * Polar Form + Input is in the form of the ProductSet or Union of ProductSets + of the intervals of ``r`` and ``theta``, and use the flag ``polar=True``. + + .. math:: Z = \{z \in \mathbb{C} \mid z = r\times (\cos(\theta) + I\sin(\theta)), r \in [\texttt{r}], \theta \in [\texttt{theta}]\} + + * Rectangular Form + Input is in the form of the ProductSet or Union of ProductSets + of interval of x and y, the real and imaginary parts of the Complex numbers in a plane. + Default input type is in rectangular form. + + .. math:: Z = \{z \in \mathbb{C} \mid z = x + Iy, x \in [\operatorname{re}(z)], y \in [\operatorname{im}(z)]\} + + Examples + ======== + + >>> from sympy import ComplexRegion, Interval, S, I, Union + >>> a = Interval(2, 3) + >>> b = Interval(4, 6) + >>> c1 = ComplexRegion(a*b) # Rectangular Form + >>> c1 + CartesianComplexRegion(ProductSet(Interval(2, 3), Interval(4, 6))) + + * c1 represents the rectangular region in complex plane + surrounded by the coordinates (2, 4), (3, 4), (3, 6) and + (2, 6), of the four vertices. + + >>> c = Interval(1, 8) + >>> c2 = ComplexRegion(Union(a*b, b*c)) + >>> c2 + CartesianComplexRegion(Union(ProductSet(Interval(2, 3), Interval(4, 6)), ProductSet(Interval(4, 6), Interval(1, 8)))) + + * c2 represents the Union of two rectangular regions in complex + plane. One of them surrounded by the coordinates of c1 and + other surrounded by the coordinates (4, 1), (6, 1), (6, 8) and + (4, 8). + + >>> 2.5 + 4.5*I in c1 + True + >>> 2.5 + 6.5*I in c1 + False + + >>> r = Interval(0, 1) + >>> theta = Interval(0, 2*S.Pi) + >>> c2 = ComplexRegion(r*theta, polar=True) # Polar Form + >>> c2 # unit Disk + PolarComplexRegion(ProductSet(Interval(0, 1), Interval.Ropen(0, 2*pi))) + + * c2 represents the region in complex plane inside the + Unit Disk centered at the origin. + + >>> 0.5 + 0.5*I in c2 + True + >>> 1 + 2*I in c2 + False + + >>> unit_disk = ComplexRegion(Interval(0, 1)*Interval(0, 2*S.Pi), polar=True) + >>> upper_half_unit_disk = ComplexRegion(Interval(0, 1)*Interval(0, S.Pi), polar=True) + >>> intersection = unit_disk.intersect(upper_half_unit_disk) + >>> intersection + PolarComplexRegion(ProductSet(Interval(0, 1), Interval(0, pi))) + >>> intersection == upper_half_unit_disk + True + + See Also + ======== + + CartesianComplexRegion + PolarComplexRegion + Complexes + + """ + is_ComplexRegion = True + + def __new__(cls, sets, polar=False): + if polar is False: + return CartesianComplexRegion(sets) + elif polar is True: + return PolarComplexRegion(sets) + else: + raise ValueError("polar should be either True or False") + + @property + def sets(self): + """ + Return raw input sets to the self. + + Examples + ======== + + >>> from sympy import Interval, ComplexRegion, Union + >>> a = Interval(2, 3) + >>> b = Interval(4, 5) + >>> c = Interval(1, 7) + >>> C1 = ComplexRegion(a*b) + >>> C1.sets + ProductSet(Interval(2, 3), Interval(4, 5)) + >>> C2 = ComplexRegion(Union(a*b, b*c)) + >>> C2.sets + Union(ProductSet(Interval(2, 3), Interval(4, 5)), ProductSet(Interval(4, 5), Interval(1, 7))) + + """ + return self.args[0] + + @property + def psets(self): + """ + Return a tuple of sets (ProductSets) input of the self. + + Examples + ======== + + >>> from sympy import Interval, ComplexRegion, Union + >>> a = Interval(2, 3) + >>> b = Interval(4, 5) + >>> c = Interval(1, 7) + >>> C1 = ComplexRegion(a*b) + >>> C1.psets + (ProductSet(Interval(2, 3), Interval(4, 5)),) + >>> C2 = ComplexRegion(Union(a*b, b*c)) + >>> C2.psets + (ProductSet(Interval(2, 3), Interval(4, 5)), ProductSet(Interval(4, 5), Interval(1, 7))) + + """ + if self.sets.is_ProductSet: + psets = () + psets = psets + (self.sets, ) + else: + psets = self.sets.args + return psets + + @property + def a_interval(self): + """ + Return the union of intervals of `x` when, self is in + rectangular form, or the union of intervals of `r` when + self is in polar form. + + Examples + ======== + + >>> from sympy import Interval, ComplexRegion, Union + >>> a = Interval(2, 3) + >>> b = Interval(4, 5) + >>> c = Interval(1, 7) + >>> C1 = ComplexRegion(a*b) + >>> C1.a_interval + Interval(2, 3) + >>> C2 = ComplexRegion(Union(a*b, b*c)) + >>> C2.a_interval + Union(Interval(2, 3), Interval(4, 5)) + + """ + a_interval = [] + for element in self.psets: + a_interval.append(element.args[0]) + + a_interval = Union(*a_interval) + return a_interval + + @property + def b_interval(self): + """ + Return the union of intervals of `y` when, self is in + rectangular form, or the union of intervals of `theta` + when self is in polar form. + + Examples + ======== + + >>> from sympy import Interval, ComplexRegion, Union + >>> a = Interval(2, 3) + >>> b = Interval(4, 5) + >>> c = Interval(1, 7) + >>> C1 = ComplexRegion(a*b) + >>> C1.b_interval + Interval(4, 5) + >>> C2 = ComplexRegion(Union(a*b, b*c)) + >>> C2.b_interval + Interval(1, 7) + + """ + b_interval = [] + for element in self.psets: + b_interval.append(element.args[1]) + + b_interval = Union(*b_interval) + return b_interval + + @property + def _measure(self): + """ + The measure of self.sets. + + Examples + ======== + + >>> from sympy import Interval, ComplexRegion, S + >>> a, b = Interval(2, 5), Interval(4, 8) + >>> c = Interval(0, 2*S.Pi) + >>> c1 = ComplexRegion(a*b) + >>> c1.measure + 12 + >>> c2 = ComplexRegion(a*c, polar=True) + >>> c2.measure + 6*pi + + """ + return self.sets._measure + + def _kind(self): + return self.args[0].kind + + @classmethod + def from_real(cls, sets): + """ + Converts given subset of real numbers to a complex region. + + Examples + ======== + + >>> from sympy import Interval, ComplexRegion + >>> unit = Interval(0,1) + >>> ComplexRegion.from_real(unit) + CartesianComplexRegion(ProductSet(Interval(0, 1), {0})) + + """ + if not sets.is_subset(S.Reals): + raise ValueError("sets must be a subset of the real line") + + return CartesianComplexRegion(sets * FiniteSet(0)) + + def _contains(self, other): + from sympy.functions import arg, Abs + + isTuple = isinstance(other, Tuple) + if isTuple and len(other) != 2: + raise ValueError('expecting Tuple of length 2') + + # If the other is not an Expression, and neither a Tuple + if not isinstance(other, (Expr, Tuple)): + return S.false + + # self in rectangular form + if not self.polar: + re, im = other if isTuple else other.as_real_imag() + return tfn[fuzzy_or(fuzzy_and([ + pset.args[0]._contains(re), + pset.args[1]._contains(im)]) + for pset in self.psets)] + + # self in polar form + elif self.polar: + if other.is_zero: + # ignore undefined complex argument + return tfn[fuzzy_or(pset.args[0]._contains(S.Zero) + for pset in self.psets)] + if isTuple: + r, theta = other + else: + r, theta = Abs(other), arg(other) + if theta.is_real and theta.is_number: + # angles in psets are normalized to [0, 2pi) + theta %= 2*S.Pi + return tfn[fuzzy_or(fuzzy_and([ + pset.args[0]._contains(r), + pset.args[1]._contains(theta)]) + for pset in self.psets)] + + +class CartesianComplexRegion(ComplexRegion): + r""" + Set representing a square region of the complex plane. + + .. math:: Z = \{z \in \mathbb{C} \mid z = x + Iy, x \in [\operatorname{re}(z)], y \in [\operatorname{im}(z)]\} + + Examples + ======== + + >>> from sympy import ComplexRegion, I, Interval + >>> region = ComplexRegion(Interval(1, 3) * Interval(4, 6)) + >>> 2 + 5*I in region + True + >>> 5*I in region + False + + See also + ======== + + ComplexRegion + PolarComplexRegion + Complexes + """ + + polar = False + variables = symbols('x, y', cls=Dummy) + + def __new__(cls, sets): + + if sets == S.Reals*S.Reals: + return S.Complexes + + if all(_a.is_FiniteSet for _a in sets.args) and (len(sets.args) == 2): + + # ** ProductSet of FiniteSets in the Complex Plane. ** + # For Cases like ComplexRegion({2, 4}*{3}), It + # would return {2 + 3*I, 4 + 3*I} + + # FIXME: This should probably be handled with something like: + # return ImageSet(Lambda((x, y), x+I*y), sets).rewrite(FiniteSet) + complex_num = [] + for x in sets.args[0]: + for y in sets.args[1]: + complex_num.append(x + S.ImaginaryUnit*y) + return FiniteSet(*complex_num) + else: + return Set.__new__(cls, sets) + + @property + def expr(self): + x, y = self.variables + return x + S.ImaginaryUnit*y + + +class PolarComplexRegion(ComplexRegion): + r""" + Set representing a polar region of the complex plane. + + .. math:: Z = \{z \in \mathbb{C} \mid z = r\times (\cos(\theta) + I\sin(\theta)), r \in [\texttt{r}], \theta \in [\texttt{theta}]\} + + Examples + ======== + + >>> from sympy import ComplexRegion, Interval, oo, pi, I + >>> rset = Interval(0, oo) + >>> thetaset = Interval(0, pi) + >>> upper_half_plane = ComplexRegion(rset * thetaset, polar=True) + >>> 1 + I in upper_half_plane + True + >>> 1 - I in upper_half_plane + False + + See also + ======== + + ComplexRegion + CartesianComplexRegion + Complexes + + """ + + polar = True + variables = symbols('r, theta', cls=Dummy) + + def __new__(cls, sets): + + new_sets = [] + # sets is Union of ProductSets + if not sets.is_ProductSet: + for k in sets.args: + new_sets.append(k) + # sets is ProductSets + else: + new_sets.append(sets) + # Normalize input theta + for k, v in enumerate(new_sets): + new_sets[k] = ProductSet(v.args[0], + normalize_theta_set(v.args[1])) + sets = Union(*new_sets) + return Set.__new__(cls, sets) + + @property + def expr(self): + r, theta = self.variables + return r*(cos(theta) + S.ImaginaryUnit*sin(theta)) + + +class Complexes(CartesianComplexRegion, metaclass=Singleton): + """ + The :class:`Set` of all complex numbers + + Examples + ======== + + >>> from sympy import S, I + >>> S.Complexes + Complexes + >>> 1 + I in S.Complexes + True + + See also + ======== + + Reals + ComplexRegion + + """ + + is_empty = False + is_finite_set = False + + # Override property from superclass since Complexes has no args + @property + def sets(self): + return ProductSet(S.Reals, S.Reals) + + def __new__(cls): + return Set.__new__(cls) diff --git a/.venv/lib/python3.12/site-packages/sympy/sets/handlers/__init__.py b/.venv/lib/python3.12/site-packages/sympy/sets/handlers/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git 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a/.venv/lib/python3.12/site-packages/sympy/sets/handlers/add.py b/.venv/lib/python3.12/site-packages/sympy/sets/handlers/add.py new file mode 100644 index 0000000000000000000000000000000000000000..8c07b25ed19d21febffd6b23a92b34b787179f44 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/sets/handlers/add.py @@ -0,0 +1,79 @@ +from sympy.core.numbers import oo, Infinity, NegativeInfinity +from sympy.core.singleton import S +from sympy.core import Basic, Expr +from sympy.multipledispatch import Dispatcher +from sympy.sets import Interval, FiniteSet + + + +# XXX: The functions in this module are clearly not tested and are broken in a +# number of ways. + +_set_add = Dispatcher('_set_add') +_set_sub = Dispatcher('_set_sub') + + +@_set_add.register(Basic, Basic) +def _(x, y): + return None + + +@_set_add.register(Expr, Expr) +def _(x, y): + return x+y + + +@_set_add.register(Interval, Interval) +def _(x, y): + """ + Additions in interval arithmetic + https://en.wikipedia.org/wiki/Interval_arithmetic + """ + return Interval(x.start + y.start, x.end + y.end, + x.left_open or y.left_open, x.right_open or y.right_open) + + +@_set_add.register(Interval, Infinity) +def _(x, y): + if x.start is S.NegativeInfinity: + return Interval(-oo, oo) + return FiniteSet({S.Infinity}) + +@_set_add.register(Interval, NegativeInfinity) +def _(x, y): + if x.end is S.Infinity: + return Interval(-oo, oo) + return FiniteSet({S.NegativeInfinity}) + + +@_set_sub.register(Basic, Basic) +def _(x, y): + return None + + +@_set_sub.register(Expr, Expr) +def _(x, y): + return x-y + + +@_set_sub.register(Interval, Interval) +def _(x, y): + """ + Subtractions in interval arithmetic + https://en.wikipedia.org/wiki/Interval_arithmetic + """ + return Interval(x.start - y.end, x.end - y.start, + x.left_open or y.right_open, x.right_open or y.left_open) + + +@_set_sub.register(Interval, Infinity) +def _(x, y): + if x.start is S.NegativeInfinity: + return Interval(-oo, oo) + return FiniteSet(-oo) + +@_set_sub.register(Interval, NegativeInfinity) +def _(x, y): + if x.start is S.NegativeInfinity: + return Interval(-oo, oo) + return FiniteSet(-oo) diff --git a/.venv/lib/python3.12/site-packages/sympy/sets/handlers/comparison.py b/.venv/lib/python3.12/site-packages/sympy/sets/handlers/comparison.py new file mode 100644 index 0000000000000000000000000000000000000000..b64d1a2a22e15d09f6f10fb4fef730163d468d45 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/sets/handlers/comparison.py @@ -0,0 +1,53 @@ +from sympy.core.relational import Eq, is_eq +from sympy.core.basic import Basic +from sympy.core.logic import fuzzy_and, fuzzy_bool +from sympy.logic.boolalg import And +from sympy.multipledispatch import dispatch +from sympy.sets.sets import tfn, ProductSet, Interval, FiniteSet, Set + + +@dispatch(Interval, FiniteSet) # type:ignore +def _eval_is_eq(lhs, rhs): # noqa: F811 + return False + + +@dispatch(FiniteSet, Interval) # type:ignore +def _eval_is_eq(lhs, rhs): # noqa: F811 + return False + + +@dispatch(Interval, Interval) # type:ignore +def _eval_is_eq(lhs, rhs): # noqa: F811 + return And(Eq(lhs.left, rhs.left), + Eq(lhs.right, rhs.right), + lhs.left_open == rhs.left_open, + lhs.right_open == rhs.right_open) + +@dispatch(FiniteSet, FiniteSet) # type:ignore +def _eval_is_eq(lhs, rhs): # noqa: F811 + def all_in_both(): + s_set = set(lhs.args) + o_set = set(rhs.args) + yield fuzzy_and(lhs._contains(e) for e in o_set - s_set) + yield fuzzy_and(rhs._contains(e) for e in s_set - o_set) + + return tfn[fuzzy_and(all_in_both())] + + +@dispatch(ProductSet, ProductSet) # type:ignore +def _eval_is_eq(lhs, rhs): # noqa: F811 + if len(lhs.sets) != len(rhs.sets): + return False + + eqs = (is_eq(x, y) for x, y in zip(lhs.sets, rhs.sets)) + return tfn[fuzzy_and(map(fuzzy_bool, eqs))] + + +@dispatch(Set, Basic) # type:ignore +def _eval_is_eq(lhs, rhs): # noqa: F811 + return False + + +@dispatch(Set, Set) # type:ignore +def _eval_is_eq(lhs, rhs): # noqa: F811 + return tfn[fuzzy_and(a.is_subset(b) for a, b in [(lhs, rhs), (rhs, lhs)])] diff --git a/.venv/lib/python3.12/site-packages/sympy/sets/handlers/functions.py b/.venv/lib/python3.12/site-packages/sympy/sets/handlers/functions.py new file mode 100644 index 0000000000000000000000000000000000000000..2529dbfd458451d7d09e91c717b170df77b1d9fe --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/sets/handlers/functions.py @@ -0,0 +1,262 @@ +from sympy.core.singleton import S +from sympy.sets.sets import Set +from sympy.calculus.singularities import singularities +from sympy.core import Expr, Add +from sympy.core.function import Lambda, FunctionClass, diff, expand_mul +from sympy.core.numbers import Float, oo +from sympy.core.symbol import Dummy, symbols, Wild +from sympy.functions.elementary.exponential import exp, log +from sympy.functions.elementary.miscellaneous import Min, Max +from sympy.logic.boolalg import true +from sympy.multipledispatch import Dispatcher +from sympy.sets import (imageset, Interval, FiniteSet, Union, ImageSet, + Intersection, Range, Complement) +from sympy.sets.sets import EmptySet, is_function_invertible_in_set +from sympy.sets.fancysets import Integers, Naturals, Reals +from sympy.functions.elementary.exponential import match_real_imag + + +_x, _y = symbols("x y") + +FunctionUnion = (FunctionClass, Lambda) + +_set_function = Dispatcher('_set_function') + + +@_set_function.register(FunctionClass, Set) +def _(f, x): + return None + +@_set_function.register(FunctionUnion, FiniteSet) +def _(f, x): + return FiniteSet(*map(f, x)) + +@_set_function.register(Lambda, Interval) +def _(f, x): + from sympy.solvers.solveset import solveset + from sympy.series import limit + # TODO: handle functions with infinitely many solutions (eg, sin, tan) + # TODO: handle multivariate functions + + expr = f.expr + if len(expr.free_symbols) > 1 or len(f.variables) != 1: + return + var = f.variables[0] + if not var.is_real: + if expr.subs(var, Dummy(real=True)).is_real is False: + return + + if expr.is_Piecewise: + result = S.EmptySet + domain_set = x + for (p_expr, p_cond) in expr.args: + if p_cond is true: + intrvl = domain_set + else: + intrvl = p_cond.as_set() + intrvl = Intersection(domain_set, intrvl) + + if p_expr.is_Number: + image = FiniteSet(p_expr) + else: + image = imageset(Lambda(var, p_expr), intrvl) + result = Union(result, image) + + # remove the part which has been `imaged` + domain_set = Complement(domain_set, intrvl) + if domain_set is S.EmptySet: + break + return result + + if not x.start.is_comparable or not x.end.is_comparable: + return + + try: + from sympy.polys.polyutils import _nsort + sing = list(singularities(expr, var, x)) + if len(sing) > 1: + sing = _nsort(sing) + except NotImplementedError: + return + + if x.left_open: + _start = limit(expr, var, x.start, dir="+") + elif x.start not in sing: + _start = f(x.start) + if x.right_open: + _end = limit(expr, var, x.end, dir="-") + elif x.end not in sing: + _end = f(x.end) + + if len(sing) == 0: + soln_expr = solveset(diff(expr, var), var) + if not (isinstance(soln_expr, FiniteSet) + or soln_expr is S.EmptySet): + return + solns = list(soln_expr) + + extr = [_start, _end] + [f(i) for i in solns + if i.is_real and i in x] + start, end = Min(*extr), Max(*extr) + + left_open, right_open = False, False + if _start <= _end: + # the minimum or maximum value can occur simultaneously + # on both the edge of the interval and in some interior + # point + if start == _start and start not in solns: + left_open = x.left_open + if end == _end and end not in solns: + right_open = x.right_open + else: + if start == _end and start not in solns: + left_open = x.right_open + if end == _start and end not in solns: + right_open = x.left_open + + return Interval(start, end, left_open, right_open) + else: + return imageset(f, Interval(x.start, sing[0], + x.left_open, True)) + \ + Union(*[imageset(f, Interval(sing[i], sing[i + 1], True, True)) + for i in range(0, len(sing) - 1)]) + \ + imageset(f, Interval(sing[-1], x.end, True, x.right_open)) + +@_set_function.register(FunctionClass, Interval) +def _(f, x): + if f == exp: + return Interval(exp(x.start), exp(x.end), x.left_open, x.right_open) + elif f == log: + return Interval(log(x.start), log(x.end), x.left_open, x.right_open) + return ImageSet(Lambda(_x, f(_x)), x) + +@_set_function.register(FunctionUnion, Union) +def _(f, x): + return Union(*(imageset(f, arg) for arg in x.args)) + +@_set_function.register(FunctionUnion, Intersection) +def _(f, x): + # If the function is invertible, intersect the maps of the sets. + if is_function_invertible_in_set(f, x): + return Intersection(*(imageset(f, arg) for arg in x.args)) + else: + return ImageSet(Lambda(_x, f(_x)), x) + +@_set_function.register(FunctionUnion, EmptySet) +def _(f, x): + return x + +@_set_function.register(FunctionUnion, Set) +def _(f, x): + return ImageSet(Lambda(_x, f(_x)), x) + +@_set_function.register(FunctionUnion, Range) +def _(f, self): + if not self: + return S.EmptySet + if not isinstance(f.expr, Expr): + return + if self.size == 1: + return FiniteSet(f(self[0])) + if f is S.IdentityFunction: + return self + + x = f.variables[0] + expr = f.expr + # handle f that is linear in f's variable + if x not in expr.free_symbols or x in expr.diff(x).free_symbols: + return + if self.start.is_finite: + F = f(self.step*x + self.start) # for i in range(len(self)) + else: + F = f(-self.step*x + self[-1]) + F = expand_mul(F) + if F != expr: + return imageset(x, F, Range(self.size)) + +@_set_function.register(FunctionUnion, Integers) +def _(f, self): + expr = f.expr + if not isinstance(expr, Expr): + return + + n = f.variables[0] + if expr == abs(n): + return S.Naturals0 + + # f(x) + c and f(-x) + c cover the same integers + # so choose the form that has the fewest negatives + c = f(0) + fx = f(n) - c + f_x = f(-n) - c + neg_count = lambda e: sum(_.could_extract_minus_sign() + for _ in Add.make_args(e)) + if neg_count(f_x) < neg_count(fx): + expr = f_x + c + + a = Wild('a', exclude=[n]) + b = Wild('b', exclude=[n]) + match = expr.match(a*n + b) + if match and match[a] and ( + not match[a].atoms(Float) and + not match[b].atoms(Float)): + # canonical shift + a, b = match[a], match[b] + if a in [1, -1]: + # drop integer addends in b + nonint = [] + for bi in Add.make_args(b): + if not bi.is_integer: + nonint.append(bi) + b = Add(*nonint) + if b.is_number and a.is_real: + # avoid Mod for complex numbers, #11391 + br, bi = match_real_imag(b) + if br and br.is_comparable and a.is_comparable: + br %= a + b = br + S.ImaginaryUnit*bi + elif b.is_number and a.is_imaginary: + br, bi = match_real_imag(b) + ai = a/S.ImaginaryUnit + if bi and bi.is_comparable and ai.is_comparable: + bi %= ai + b = br + S.ImaginaryUnit*bi + expr = a*n + b + + if expr != f.expr: + return ImageSet(Lambda(n, expr), S.Integers) + + +@_set_function.register(FunctionUnion, Naturals) +def _(f, self): + expr = f.expr + if not isinstance(expr, Expr): + return + + x = f.variables[0] + if not expr.free_symbols - {x}: + if expr == abs(x): + if self is S.Naturals: + return self + return S.Naturals0 + step = expr.coeff(x) + c = expr.subs(x, 0) + if c.is_Integer and step.is_Integer and expr == step*x + c: + if self is S.Naturals: + c += step + if step > 0: + if step == 1: + if c == 0: + return S.Naturals0 + elif c == 1: + return S.Naturals + return Range(c, oo, step) + return Range(c, -oo, step) + + +@_set_function.register(FunctionUnion, Reals) +def _(f, self): + expr = f.expr + if not isinstance(expr, Expr): + return + return _set_function(f, Interval(-oo, oo)) diff --git a/.venv/lib/python3.12/site-packages/sympy/sets/handlers/intersection.py b/.venv/lib/python3.12/site-packages/sympy/sets/handlers/intersection.py new file mode 100644 index 0000000000000000000000000000000000000000..fcb9309ef3e9d2722ab1bfe664f1d1644f17da5d --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/sets/handlers/intersection.py @@ -0,0 +1,533 @@ +from sympy.core.basic import _aresame +from sympy.core.function import Lambda, expand_complex +from sympy.core.mul import Mul +from sympy.core.numbers import ilcm, Float +from sympy.core.relational import Eq +from sympy.core.singleton import S +from sympy.core.symbol import (Dummy, symbols) +from sympy.core.sorting import ordered +from sympy.functions.elementary.complexes import sign +from sympy.functions.elementary.integers import floor, ceiling +from sympy.sets.fancysets import ComplexRegion +from sympy.sets.sets import (FiniteSet, Intersection, Interval, Set, Union) +from sympy.multipledispatch import Dispatcher +from sympy.sets.conditionset import ConditionSet +from sympy.sets.fancysets import (Integers, Naturals, Reals, Range, + ImageSet, Rationals) +from sympy.sets.sets import EmptySet, UniversalSet, imageset, ProductSet +from sympy.simplify.radsimp import numer + + +intersection_sets = Dispatcher('intersection_sets') + + +@intersection_sets.register(ConditionSet, ConditionSet) +def _(a, b): + return None + +@intersection_sets.register(ConditionSet, Set) +def _(a, b): + return ConditionSet(a.sym, a.condition, Intersection(a.base_set, b)) + +@intersection_sets.register(Naturals, Integers) +def _(a, b): + return a + +@intersection_sets.register(Naturals, Naturals) +def _(a, b): + return a if a is S.Naturals else b + +@intersection_sets.register(Interval, Naturals) +def _(a, b): + return intersection_sets(b, a) + +@intersection_sets.register(ComplexRegion, Set) +def _(self, other): + if other.is_ComplexRegion: + # self in rectangular form + if (not self.polar) and (not other.polar): + return ComplexRegion(Intersection(self.sets, other.sets)) + + # self in polar form + elif self.polar and other.polar: + r1, theta1 = self.a_interval, self.b_interval + r2, theta2 = other.a_interval, other.b_interval + new_r_interval = Intersection(r1, r2) + new_theta_interval = Intersection(theta1, theta2) + + # 0 and 2*Pi means the same + if ((2*S.Pi in theta1 and S.Zero in theta2) or + (2*S.Pi in theta2 and S.Zero in theta1)): + new_theta_interval = Union(new_theta_interval, + FiniteSet(0)) + return ComplexRegion(new_r_interval*new_theta_interval, + polar=True) + + + if other.is_subset(S.Reals): + new_interval = [] + x = symbols("x", cls=Dummy, real=True) + + # self in rectangular form + if not self.polar: + for element in self.psets: + if S.Zero in element.args[1]: + new_interval.append(element.args[0]) + new_interval = Union(*new_interval) + return Intersection(new_interval, other) + + # self in polar form + elif self.polar: + for element in self.psets: + if S.Zero in element.args[1]: + new_interval.append(element.args[0]) + if S.Pi in element.args[1]: + new_interval.append(ImageSet(Lambda(x, -x), element.args[0])) + if S.Zero in element.args[0]: + new_interval.append(FiniteSet(0)) + new_interval = Union(*new_interval) + return Intersection(new_interval, other) + +@intersection_sets.register(Integers, Reals) +def _(a, b): + return a + +@intersection_sets.register(Range, Interval) +def _(a, b): + # Check that there are no symbolic arguments + if not all(i.is_number for i in a.args + b.args[:2]): + return + + # In case of null Range, return an EmptySet. + if a.size == 0: + return S.EmptySet + + # trim down to self's size, and represent + # as a Range with step 1. + start = ceiling(max(b.inf, a.inf)) + if start not in b: + start += 1 + end = floor(min(b.sup, a.sup)) + if end not in b: + end -= 1 + return intersection_sets(a, Range(start, end + 1)) + +@intersection_sets.register(Range, Naturals) +def _(a, b): + return intersection_sets(a, Interval(b.inf, S.Infinity)) + +@intersection_sets.register(Range, Range) +def _(a, b): + # Check that there are no symbolic range arguments + if not all(all(v.is_number for v in r.args) for r in [a, b]): + return None + + # non-overlap quick exits + if not b: + return S.EmptySet + if not a: + return S.EmptySet + if b.sup < a.inf: + return S.EmptySet + if b.inf > a.sup: + return S.EmptySet + + # work with finite end at the start + r1 = a + if r1.start.is_infinite: + r1 = r1.reversed + r2 = b + if r2.start.is_infinite: + r2 = r2.reversed + + # If both ends are infinite then it means that one Range is just the set + # of all integers (the step must be 1). + if r1.start.is_infinite: + return b + if r2.start.is_infinite: + return a + + from sympy.solvers.diophantine.diophantine import diop_linear + + # this equation represents the values of the Range; + # it's a linear equation + eq = lambda r, i: r.start + i*r.step + + # we want to know when the two equations might + # have integer solutions so we use the diophantine + # solver + va, vb = diop_linear(eq(r1, Dummy('a')) - eq(r2, Dummy('b'))) + + # check for no solution + no_solution = va is None and vb is None + if no_solution: + return S.EmptySet + + # there is a solution + # ------------------- + + # find the coincident point, c + a0 = va.as_coeff_Add()[0] + c = eq(r1, a0) + + # find the first point, if possible, in each range + # since c may not be that point + def _first_finite_point(r1, c): + if c == r1.start: + return c + # st is the signed step we need to take to + # get from c to r1.start + st = sign(r1.start - c)*step + # use Range to calculate the first point: + # we want to get as close as possible to + # r1.start; the Range will not be null since + # it will at least contain c + s1 = Range(c, r1.start + st, st)[-1] + if s1 == r1.start: + pass + else: + # if we didn't hit r1.start then, if the + # sign of st didn't match the sign of r1.step + # we are off by one and s1 is not in r1 + if sign(r1.step) != sign(st): + s1 -= st + if s1 not in r1: + return + return s1 + + # calculate the step size of the new Range + step = abs(ilcm(r1.step, r2.step)) + s1 = _first_finite_point(r1, c) + if s1 is None: + return S.EmptySet + s2 = _first_finite_point(r2, c) + if s2 is None: + return S.EmptySet + + # replace the corresponding start or stop in + # the original Ranges with these points; the + # result must have at least one point since + # we know that s1 and s2 are in the Ranges + def _updated_range(r, first): + st = sign(r.step)*step + if r.start.is_finite: + rv = Range(first, r.stop, st) + else: + rv = Range(r.start, first + st, st) + return rv + r1 = _updated_range(a, s1) + r2 = _updated_range(b, s2) + + # work with them both in the increasing direction + if sign(r1.step) < 0: + r1 = r1.reversed + if sign(r2.step) < 0: + r2 = r2.reversed + + # return clipped Range with positive step; it + # can't be empty at this point + start = max(r1.start, r2.start) + stop = min(r1.stop, r2.stop) + return Range(start, stop, step) + + +@intersection_sets.register(Range, Integers) +def _(a, b): + return a + + +@intersection_sets.register(Range, Rationals) +def _(a, b): + return a + + +@intersection_sets.register(ImageSet, Set) +def _(self, other): + from sympy.solvers.diophantine import diophantine + + # Only handle the straight-forward univariate case + if (len(self.lamda.variables) > 1 + or self.lamda.signature != self.lamda.variables): + return None + base_set = self.base_sets[0] + + # Intersection between ImageSets with Integers as base set + # For {f(n) : n in Integers} & {g(m) : m in Integers} we solve the + # diophantine equations f(n)=g(m). + # If the solutions for n are {h(t) : t in Integers} then we return + # {f(h(t)) : t in integers}. + # If the solutions for n are {n_1, n_2, ..., n_k} then we return + # {f(n_i) : 1 <= i <= k}. + if base_set is S.Integers: + gm = None + if isinstance(other, ImageSet) and other.base_sets == (S.Integers,): + gm = other.lamda.expr + var = other.lamda.variables[0] + # Symbol of second ImageSet lambda must be distinct from first + m = Dummy('m') + gm = gm.subs(var, m) + elif other is S.Integers: + m = gm = Dummy('m') + if gm is not None: + fn = self.lamda.expr + n = self.lamda.variables[0] + try: + solns = list(diophantine(fn - gm, syms=(n, m), permute=True)) + except (TypeError, NotImplementedError): + # TypeError if equation not polynomial with rational coeff. + # NotImplementedError if correct format but no solver. + return + # 3 cases are possible for solns: + # - empty set, + # - one or more parametric (infinite) solutions, + # - a finite number of (non-parametric) solution couples. + # Among those, there is one type of solution set that is + # not helpful here: multiple parametric solutions. + if len(solns) == 0: + return S.EmptySet + elif any(s.free_symbols for tupl in solns for s in tupl): + if len(solns) == 1: + soln, solm = solns[0] + (t,) = soln.free_symbols + expr = fn.subs(n, soln.subs(t, n)).expand() + return imageset(Lambda(n, expr), S.Integers) + else: + return + else: + return FiniteSet(*(fn.subs(n, s[0]) for s in solns)) + + if other == S.Reals: + from sympy.solvers.solvers import denoms, solve_linear + + def _solution_union(exprs, sym): + # return a union of linear solutions to i in expr; + # if i cannot be solved, use a ConditionSet for solution + sols = [] + for i in exprs: + x, xis = solve_linear(i, 0, [sym]) + if x == sym: + sols.append(FiniteSet(xis)) + else: + sols.append(ConditionSet(sym, Eq(i, 0))) + return Union(*sols) + + f = self.lamda.expr + n = self.lamda.variables[0] + + n_ = Dummy(n.name, real=True) + f_ = f.subs(n, n_) + + re, im = f_.as_real_imag() + im = expand_complex(im) + + re = re.subs(n_, n) + im = im.subs(n_, n) + ifree = im.free_symbols + lam = Lambda(n, re) + if im.is_zero: + # allow re-evaluation + # of self in this case to make + # the result canonical + pass + elif im.is_zero is False: + return S.EmptySet + elif ifree != {n}: + return None + else: + # univarite imaginary part in same variable; + # use numer instead of as_numer_denom to keep + # this as fast as possible while still handling + # simple cases + base_set &= _solution_union( + Mul.make_args(numer(im)), n) + # exclude values that make denominators 0 + base_set -= _solution_union(denoms(f), n) + return imageset(lam, base_set) + + elif isinstance(other, Interval): + from sympy.solvers.solveset import (invert_real, invert_complex, + solveset) + + f = self.lamda.expr + n = self.lamda.variables[0] + new_inf, new_sup = None, None + new_lopen, new_ropen = other.left_open, other.right_open + + if f.is_real: + inverter = invert_real + else: + inverter = invert_complex + + g1, h1 = inverter(f, other.inf, n) + g2, h2 = inverter(f, other.sup, n) + + if all(isinstance(i, FiniteSet) for i in (h1, h2)): + if g1 == n: + if len(h1) == 1: + new_inf = h1.args[0] + if g2 == n: + if len(h2) == 1: + new_sup = h2.args[0] + # TODO: Design a technique to handle multiple-inverse + # functions + + # Any of the new boundary values cannot be determined + if any(i is None for i in (new_sup, new_inf)): + return + + + range_set = S.EmptySet + + if all(i.is_real for i in (new_sup, new_inf)): + # this assumes continuity of underlying function + # however fixes the case when it is decreasing + if new_inf > new_sup: + new_inf, new_sup = new_sup, new_inf + new_interval = Interval(new_inf, new_sup, new_lopen, new_ropen) + range_set = base_set.intersect(new_interval) + else: + if other.is_subset(S.Reals): + solutions = solveset(f, n, S.Reals) + if not isinstance(range_set, (ImageSet, ConditionSet)): + range_set = solutions.intersect(other) + else: + return + + if range_set is S.EmptySet: + return S.EmptySet + elif isinstance(range_set, Range) and range_set.size is not S.Infinity: + range_set = FiniteSet(*list(range_set)) + + if range_set is not None: + return imageset(Lambda(n, f), range_set) + return + else: + return + + +@intersection_sets.register(ProductSet, ProductSet) +def _(a, b): + if len(b.args) != len(a.args): + return S.EmptySet + return ProductSet(*(i.intersect(j) for i, j in zip(a.sets, b.sets))) + + +@intersection_sets.register(Interval, Interval) +def _(a, b): + # handle (-oo, oo) + infty = S.NegativeInfinity, S.Infinity + if a == Interval(*infty): + l, r = a.left, a.right + if l.is_real or l in infty or r.is_real or r in infty: + return b + + # We can't intersect [0,3] with [x,6] -- we don't know if x>0 or x<0 + if not a._is_comparable(b): + return None + + empty = False + + if a.start <= b.end and b.start <= a.end: + # Get topology right. + if a.start < b.start: + start = b.start + left_open = b.left_open + elif a.start > b.start: + start = a.start + left_open = a.left_open + else: + start = a.start + if not _aresame(a.start, b.start): + # For example Integer(2) != Float(2) + # Prefer the Float boundary because Floats should be + # contagious in calculations. + if b.start.has(Float) and not a.start.has(Float): + start = b.start + elif a.start.has(Float) and not b.start.has(Float): + start = a.start + else: + #this is to ensure that if Eq(a.start, b.start) but + #type(a.start) != type(b.start) the order of a and b + #does not matter for the result + start = list(ordered([a,b]))[0].start + left_open = a.left_open or b.left_open + + if a.end < b.end: + end = a.end + right_open = a.right_open + elif a.end > b.end: + end = b.end + right_open = b.right_open + else: + # see above for logic with start + end = a.end + if not _aresame(a.end, b.end): + if b.end.has(Float) and not a.end.has(Float): + end = b.end + elif a.end.has(Float) and not b.end.has(Float): + end = a.end + else: + end = list(ordered([a,b]))[0].end + right_open = a.right_open or b.right_open + + if end - start == 0 and (left_open or right_open): + empty = True + else: + empty = True + + if empty: + return S.EmptySet + + return Interval(start, end, left_open, right_open) + +@intersection_sets.register(EmptySet, Set) +def _(a, b): + return S.EmptySet + +@intersection_sets.register(UniversalSet, Set) +def _(a, b): + return b + +@intersection_sets.register(FiniteSet, FiniteSet) +def _(a, b): + return FiniteSet(*(a._elements & b._elements)) + +@intersection_sets.register(FiniteSet, Set) +def _(a, b): + try: + return FiniteSet(*[el for el in a if el in b]) + except TypeError: + return None # could not evaluate `el in b` due to symbolic ranges. + +@intersection_sets.register(Set, Set) +def _(a, b): + return None + +@intersection_sets.register(Integers, Rationals) +def _(a, b): + return a + +@intersection_sets.register(Naturals, Rationals) +def _(a, b): + return a + +@intersection_sets.register(Rationals, Reals) +def _(a, b): + return a + +def _intlike_interval(a, b): + try: + if b._inf is S.NegativeInfinity and b._sup is S.Infinity: + return a + s = Range(max(a.inf, ceiling(b.left)), floor(b.right) + 1) + return intersection_sets(s, b) # take out endpoints if open interval + except ValueError: + return None + +@intersection_sets.register(Integers, Interval) +def _(a, b): + return _intlike_interval(a, b) + +@intersection_sets.register(Naturals, Interval) +def _(a, b): + return _intlike_interval(a, b) diff --git a/.venv/lib/python3.12/site-packages/sympy/sets/handlers/issubset.py b/.venv/lib/python3.12/site-packages/sympy/sets/handlers/issubset.py new file mode 100644 index 0000000000000000000000000000000000000000..cc23e8bf56f1743cd7f08452dd09a0acf981f5da --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/sets/handlers/issubset.py @@ -0,0 +1,144 @@ +from sympy.core.singleton import S +from sympy.core.symbol import Symbol +from sympy.core.logic import fuzzy_and, fuzzy_bool, fuzzy_not, fuzzy_or +from sympy.core.relational import Eq +from sympy.sets.sets import FiniteSet, Interval, Set, Union, ProductSet +from sympy.sets.fancysets import Complexes, Reals, Range, Rationals +from sympy.multipledispatch import Dispatcher + + +_inf_sets = [S.Naturals, S.Naturals0, S.Integers, S.Rationals, S.Reals, S.Complexes] + + +is_subset_sets = Dispatcher('is_subset_sets') + + +@is_subset_sets.register(Set, Set) +def _(a, b): + return None + +@is_subset_sets.register(Interval, Interval) +def _(a, b): + # This is correct but can be made more comprehensive... + if fuzzy_bool(a.start < b.start): + return False + if fuzzy_bool(a.end > b.end): + return False + if (b.left_open and not a.left_open and fuzzy_bool(Eq(a.start, b.start))): + return False + if (b.right_open and not a.right_open and fuzzy_bool(Eq(a.end, b.end))): + return False + +@is_subset_sets.register(Interval, FiniteSet) +def _(a_interval, b_fs): + # An Interval can only be a subset of a finite set if it is finite + # which can only happen if it has zero measure. + if fuzzy_not(a_interval.measure.is_zero): + return False + +@is_subset_sets.register(Interval, Union) +def _(a_interval, b_u): + if all(isinstance(s, (Interval, FiniteSet)) for s in b_u.args): + intervals = [s for s in b_u.args if isinstance(s, Interval)] + if all(fuzzy_bool(a_interval.start < s.start) for s in intervals): + return False + if all(fuzzy_bool(a_interval.end > s.end) for s in intervals): + return False + if a_interval.measure.is_nonzero: + no_overlap = lambda s1, s2: fuzzy_or([ + fuzzy_bool(s1.end <= s2.start), + fuzzy_bool(s1.start >= s2.end), + ]) + if all(no_overlap(s, a_interval) for s in intervals): + return False + +@is_subset_sets.register(Range, Range) +def _(a, b): + if a.step == b.step == 1: + return fuzzy_and([fuzzy_bool(a.start >= b.start), + fuzzy_bool(a.stop <= b.stop)]) + +@is_subset_sets.register(Range, Interval) +def _(a_range, b_interval): + if a_range.step.is_positive: + if b_interval.left_open and a_range.inf.is_finite: + cond_left = a_range.inf > b_interval.left + else: + cond_left = a_range.inf >= b_interval.left + if b_interval.right_open and a_range.sup.is_finite: + cond_right = a_range.sup < b_interval.right + else: + cond_right = a_range.sup <= b_interval.right + return fuzzy_and([cond_left, cond_right]) + +@is_subset_sets.register(Range, FiniteSet) +def _(a_range, b_finiteset): + try: + a_size = a_range.size + except ValueError: + # symbolic Range of unknown size + return None + if a_size > len(b_finiteset): + return False + elif any(arg.has(Symbol) for arg in a_range.args): + return fuzzy_and(b_finiteset.contains(x) for x in a_range) + else: + # Checking A \ B == EmptySet is more efficient than repeated naive + # membership checks on an arbitrary FiniteSet. + a_set = set(a_range) + b_remaining = len(b_finiteset) + # Symbolic expressions and numbers of unknown type (integer or not) are + # all counted as "candidates", i.e. *potentially* matching some a in + # a_range. + cnt_candidate = 0 + for b in b_finiteset: + if b.is_Integer: + a_set.discard(b) + elif fuzzy_not(b.is_integer): + pass + else: + cnt_candidate += 1 + b_remaining -= 1 + if len(a_set) > b_remaining + cnt_candidate: + return False + if len(a_set) == 0: + return True + return None + +@is_subset_sets.register(Interval, Range) +def _(a_interval, b_range): + if a_interval.measure.is_extended_nonzero: + return False + +@is_subset_sets.register(Interval, Rationals) +def _(a_interval, b_rationals): + if a_interval.measure.is_extended_nonzero: + return False + +@is_subset_sets.register(Range, Complexes) +def _(a, b): + return True + +@is_subset_sets.register(Complexes, Interval) +def _(a, b): + return False + +@is_subset_sets.register(Complexes, Range) +def _(a, b): + return False + +@is_subset_sets.register(Complexes, Rationals) +def _(a, b): + return False + +@is_subset_sets.register(Rationals, Reals) +def _(a, b): + return True + +@is_subset_sets.register(Rationals, Range) +def _(a, b): + return False + +@is_subset_sets.register(ProductSet, FiniteSet) +def _(a_ps, b_fs): + return fuzzy_and(b_fs.contains(x) for x in a_ps) diff --git a/.venv/lib/python3.12/site-packages/sympy/sets/handlers/mul.py b/.venv/lib/python3.12/site-packages/sympy/sets/handlers/mul.py new file mode 100644 index 0000000000000000000000000000000000000000..0dedc8068b7973fd4cb6fbf2854e5fa671d188de --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/sets/handlers/mul.py @@ -0,0 +1,79 @@ +from sympy.core import Basic, Expr +from sympy.core.numbers import oo +from sympy.core.symbol import symbols +from sympy.multipledispatch import Dispatcher +from sympy.sets.setexpr import set_mul +from sympy.sets.sets import Interval, Set + + +_x, _y = symbols("x y") + + +_set_mul = Dispatcher('_set_mul') +_set_div = Dispatcher('_set_div') + + +@_set_mul.register(Basic, Basic) +def _(x, y): + return None + +@_set_mul.register(Set, Set) +def _(x, y): + return None + +@_set_mul.register(Expr, Expr) +def _(x, y): + return x*y + +@_set_mul.register(Interval, Interval) +def _(x, y): + """ + Multiplications in interval arithmetic + https://en.wikipedia.org/wiki/Interval_arithmetic + """ + # TODO: some intervals containing 0 and oo will fail as 0*oo returns nan. + comvals = ( + (x.start * y.start, bool(x.left_open or y.left_open)), + (x.start * y.end, bool(x.left_open or y.right_open)), + (x.end * y.start, bool(x.right_open or y.left_open)), + (x.end * y.end, bool(x.right_open or y.right_open)), + ) + # TODO: handle symbolic intervals + minval, minopen = min(comvals) + maxval, maxopen = max(comvals) + return Interval( + minval, + maxval, + minopen, + maxopen + ) + +@_set_div.register(Basic, Basic) +def _(x, y): + return None + +@_set_div.register(Expr, Expr) +def _(x, y): + return x/y + +@_set_div.register(Set, Set) +def _(x, y): + return None + +@_set_div.register(Interval, Interval) +def _(x, y): + """ + Divisions in interval arithmetic + https://en.wikipedia.org/wiki/Interval_arithmetic + """ + if (y.start*y.end).is_negative: + return Interval(-oo, oo) + if y.start == 0: + s2 = oo + else: + s2 = 1/y.start + if y.end == 0: + s1 = -oo + else: + s1 = 1/y.end + return set_mul(x, Interval(s1, s2, y.right_open, y.left_open)) diff --git a/.venv/lib/python3.12/site-packages/sympy/sets/handlers/power.py b/.venv/lib/python3.12/site-packages/sympy/sets/handlers/power.py new file mode 100644 index 0000000000000000000000000000000000000000..3cad4ee49ab27770143bc121d1fbcd024bf01548 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/sets/handlers/power.py @@ -0,0 +1,107 @@ +from sympy.core import Basic, Expr +from sympy.core.function import Lambda +from sympy.core.numbers import oo, Infinity, NegativeInfinity, Zero, Integer +from sympy.core.singleton import S +from sympy.core.symbol import symbols +from sympy.functions.elementary.miscellaneous import (Max, Min) +from sympy.sets.fancysets import ImageSet +from sympy.sets.setexpr import set_div +from sympy.sets.sets import Set, Interval, FiniteSet, Union +from sympy.multipledispatch import Dispatcher + + +_x, _y = symbols("x y") + + +_set_pow = Dispatcher('_set_pow') + + +@_set_pow.register(Basic, Basic) +def _(x, y): + return None + +@_set_pow.register(Set, Set) +def _(x, y): + return ImageSet(Lambda((_x, _y), (_x ** _y)), x, y) + +@_set_pow.register(Expr, Expr) +def _(x, y): + return x**y + +@_set_pow.register(Interval, Zero) +def _(x, z): + return FiniteSet(S.One) + +@_set_pow.register(Interval, Integer) +def _(x, exponent): + """ + Powers in interval arithmetic + https://en.wikipedia.org/wiki/Interval_arithmetic + """ + s1 = x.start**exponent + s2 = x.end**exponent + if ((s2 > s1) if exponent > 0 else (x.end > -x.start)) == True: + left_open = x.left_open + right_open = x.right_open + # TODO: handle unevaluated condition. + sleft = s2 + else: + # TODO: `s2 > s1` could be unevaluated. + left_open = x.right_open + right_open = x.left_open + sleft = s1 + + if x.start.is_positive: + return Interval( + Min(s1, s2), + Max(s1, s2), left_open, right_open) + elif x.end.is_negative: + return Interval( + Min(s1, s2), + Max(s1, s2), left_open, right_open) + + # Case where x.start < 0 and x.end > 0: + if exponent.is_odd: + if exponent.is_negative: + if x.start.is_zero: + return Interval(s2, oo, x.right_open) + if x.end.is_zero: + return Interval(-oo, s1, True, x.left_open) + return Union(Interval(-oo, s1, True, x.left_open), Interval(s2, oo, x.right_open)) + else: + return Interval(s1, s2, x.left_open, x.right_open) + elif exponent.is_even: + if exponent.is_negative: + if x.start.is_zero: + return Interval(s2, oo, x.right_open) + if x.end.is_zero: + return Interval(s1, oo, x.left_open) + return Interval(0, oo) + else: + return Interval(S.Zero, sleft, S.Zero not in x, left_open) + +@_set_pow.register(Interval, Infinity) +def _(b, e): + # TODO: add logic for open intervals? + if b.start.is_nonnegative: + if b.end < 1: + return FiniteSet(S.Zero) + if b.start > 1: + return FiniteSet(S.Infinity) + return Interval(0, oo) + elif b.end.is_negative: + if b.start > -1: + return FiniteSet(S.Zero) + if b.end < -1: + return FiniteSet(-oo, oo) + return Interval(-oo, oo) + else: + if b.start > -1: + if b.end < 1: + return FiniteSet(S.Zero) + return Interval(0, oo) + return Interval(-oo, oo) + +@_set_pow.register(Interval, NegativeInfinity) +def _(b, e): + return _set_pow(set_div(S.One, b), oo) diff --git a/.venv/lib/python3.12/site-packages/sympy/sets/handlers/union.py b/.venv/lib/python3.12/site-packages/sympy/sets/handlers/union.py new file mode 100644 index 0000000000000000000000000000000000000000..75d867b49969ae2aeea76155dbaae7e05c1a6847 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/sets/handlers/union.py @@ -0,0 +1,147 @@ +from sympy.core.singleton import S +from sympy.core.sympify import sympify +from sympy.functions.elementary.miscellaneous import Min, Max +from sympy.sets.sets import (EmptySet, FiniteSet, Intersection, + Interval, ProductSet, Set, Union, UniversalSet) +from sympy.sets.fancysets import (ComplexRegion, Naturals, Naturals0, + Integers, Rationals, Reals) +from sympy.multipledispatch import Dispatcher + + +union_sets = Dispatcher('union_sets') + + +@union_sets.register(Naturals0, Naturals) +def _(a, b): + return a + +@union_sets.register(Rationals, Naturals) +def _(a, b): + return a + +@union_sets.register(Rationals, Naturals0) +def _(a, b): + return a + +@union_sets.register(Reals, Naturals) +def _(a, b): + return a + +@union_sets.register(Reals, Naturals0) +def _(a, b): + return a + +@union_sets.register(Reals, Rationals) +def _(a, b): + return a + +@union_sets.register(Integers, Set) +def _(a, b): + intersect = Intersection(a, b) + if intersect == a: + return b + elif intersect == b: + return a + +@union_sets.register(ComplexRegion, Set) +def _(a, b): + if b.is_subset(S.Reals): + # treat a subset of reals as a complex region + b = ComplexRegion.from_real(b) + + if b.is_ComplexRegion: + # a in rectangular form + if (not a.polar) and (not b.polar): + return ComplexRegion(Union(a.sets, b.sets)) + # a in polar form + elif a.polar and b.polar: + return ComplexRegion(Union(a.sets, b.sets), polar=True) + return None + +@union_sets.register(EmptySet, Set) +def _(a, b): + return b + + +@union_sets.register(UniversalSet, Set) +def _(a, b): + return a + +@union_sets.register(ProductSet, ProductSet) +def _(a, b): + if b.is_subset(a): + return a + if len(b.sets) != len(a.sets): + return None + if len(a.sets) == 2: + a1, a2 = a.sets + b1, b2 = b.sets + if a1 == b1: + return a1 * Union(a2, b2) + if a2 == b2: + return Union(a1, b1) * a2 + return None + +@union_sets.register(ProductSet, Set) +def _(a, b): + if b.is_subset(a): + return a + return None + +@union_sets.register(Interval, Interval) +def _(a, b): + if a._is_comparable(b): + # Non-overlapping intervals + end = Min(a.end, b.end) + start = Max(a.start, b.start) + if (end < start or + (end == start and (end not in a and end not in b))): + return None + else: + start = Min(a.start, b.start) + end = Max(a.end, b.end) + + left_open = ((a.start != start or a.left_open) and + (b.start != start or b.left_open)) + right_open = ((a.end != end or a.right_open) and + (b.end != end or b.right_open)) + return Interval(start, end, left_open, right_open) + +@union_sets.register(Interval, UniversalSet) +def _(a, b): + return S.UniversalSet + +@union_sets.register(Interval, Set) +def _(a, b): + # If I have open end points and these endpoints are contained in b + # But only in case, when endpoints are finite. Because + # interval does not contain oo or -oo. + open_left_in_b_and_finite = (a.left_open and + sympify(b.contains(a.start)) is S.true and + a.start.is_finite) + open_right_in_b_and_finite = (a.right_open and + sympify(b.contains(a.end)) is S.true and + a.end.is_finite) + if open_left_in_b_and_finite or open_right_in_b_and_finite: + # Fill in my end points and return + open_left = a.left_open and a.start not in b + open_right = a.right_open and a.end not in b + new_a = Interval(a.start, a.end, open_left, open_right) + return {new_a, b} + return None + +@union_sets.register(FiniteSet, FiniteSet) +def _(a, b): + return FiniteSet(*(a._elements | b._elements)) + +@union_sets.register(FiniteSet, Set) +def _(a, b): + # If `b` set contains one of my elements, remove it from `a` + if any(b.contains(x) == True for x in a): + return { + FiniteSet(*[x for x in a if b.contains(x) != True]), b} + return None + +@union_sets.register(Set, Set) +def _(a, b): + return None diff --git a/.venv/lib/python3.12/site-packages/sympy/sets/ordinals.py b/.venv/lib/python3.12/site-packages/sympy/sets/ordinals.py new file mode 100644 index 0000000000000000000000000000000000000000..cfe062354cfe58a4747998e51fa0d261e67576cc --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/sets/ordinals.py @@ -0,0 +1,282 @@ +from sympy.core import Basic, Integer +import operator + + +class OmegaPower(Basic): + """ + Represents ordinal exponential and multiplication terms one of the + building blocks of the :class:`Ordinal` class. + In ``OmegaPower(a, b)``, ``a`` represents exponent and ``b`` represents multiplicity. + """ + def __new__(cls, a, b): + if isinstance(b, int): + b = Integer(b) + if not isinstance(b, Integer) or b <= 0: + raise TypeError("multiplicity must be a positive integer") + + if not isinstance(a, Ordinal): + a = Ordinal.convert(a) + + return Basic.__new__(cls, a, b) + + @property + def exp(self): + return self.args[0] + + @property + def mult(self): + return self.args[1] + + def _compare_term(self, other, op): + if self.exp == other.exp: + return op(self.mult, other.mult) + else: + return op(self.exp, other.exp) + + def __eq__(self, other): + if not isinstance(other, OmegaPower): + try: + other = OmegaPower(0, other) + except TypeError: + return NotImplemented + return self.args == other.args + + def __hash__(self): + return Basic.__hash__(self) + + def __lt__(self, other): + if not isinstance(other, OmegaPower): + try: + other = OmegaPower(0, other) + except TypeError: + return NotImplemented + return self._compare_term(other, operator.lt) + + +class Ordinal(Basic): + """ + Represents ordinals in Cantor normal form. + + Internally, this class is just a list of instances of OmegaPower. + + Examples + ======== + >>> from sympy import Ordinal, OmegaPower + >>> from sympy.sets.ordinals import omega + >>> w = omega + >>> w.is_limit_ordinal + True + >>> Ordinal(OmegaPower(w + 1, 1), OmegaPower(3, 2)) + w**(w + 1) + w**3*2 + >>> 3 + w + w + >>> (w + 1) * w + w**2 + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Ordinal_arithmetic + """ + def __new__(cls, *terms): + obj = super().__new__(cls, *terms) + powers = [i.exp for i in obj.args] + if not all(powers[i] >= powers[i+1] for i in range(len(powers) - 1)): + raise ValueError("powers must be in decreasing order") + return obj + + @property + def terms(self): + return self.args + + @property + def leading_term(self): + if self == ord0: + raise ValueError("ordinal zero has no leading term") + return self.terms[0] + + @property + def trailing_term(self): + if self == ord0: + raise ValueError("ordinal zero has no trailing term") + return self.terms[-1] + + @property + def is_successor_ordinal(self): + try: + return self.trailing_term.exp == ord0 + except ValueError: + return False + + @property + def is_limit_ordinal(self): + try: + return not self.trailing_term.exp == ord0 + except ValueError: + return False + + @property + def degree(self): + return self.leading_term.exp + + @classmethod + def convert(cls, integer_value): + if integer_value == 0: + return ord0 + return Ordinal(OmegaPower(0, integer_value)) + + def __eq__(self, other): + if not isinstance(other, Ordinal): + try: + other = Ordinal.convert(other) + except TypeError: + return NotImplemented + return self.terms == other.terms + + def __hash__(self): + return hash(self.args) + + def __lt__(self, other): + if not isinstance(other, Ordinal): + try: + other = Ordinal.convert(other) + except TypeError: + return NotImplemented + for term_self, term_other in zip(self.terms, other.terms): + if term_self != term_other: + return term_self < term_other + return len(self.terms) < len(other.terms) + + def __le__(self, other): + return (self == other or self < other) + + def __gt__(self, other): + return not self <= other + + def __ge__(self, other): + return not self < other + + def __str__(self): + net_str = "" + plus_count = 0 + if self == ord0: + return 'ord0' + for i in self.terms: + if plus_count: + net_str += " + " + + if i.exp == ord0: + net_str += str(i.mult) + elif i.exp == 1: + net_str += 'w' + elif len(i.exp.terms) > 1 or i.exp.is_limit_ordinal: + net_str += 'w**(%s)'%i.exp + else: + net_str += 'w**%s'%i.exp + + if not i.mult == 1 and not i.exp == ord0: + net_str += '*%s'%i.mult + + plus_count += 1 + return(net_str) + + __repr__ = __str__ + + def __add__(self, other): + if not isinstance(other, Ordinal): + try: + other = Ordinal.convert(other) + except TypeError: + return NotImplemented + if other == ord0: + return self + a_terms = list(self.terms) + b_terms = list(other.terms) + r = len(a_terms) - 1 + b_exp = other.degree + while r >= 0 and a_terms[r].exp < b_exp: + r -= 1 + if r < 0: + terms = b_terms + elif a_terms[r].exp == b_exp: + sum_term = OmegaPower(b_exp, a_terms[r].mult + other.leading_term.mult) + terms = a_terms[:r] + [sum_term] + b_terms[1:] + else: + terms = a_terms[:r+1] + b_terms + return Ordinal(*terms) + + def __radd__(self, other): + if not isinstance(other, Ordinal): + try: + other = Ordinal.convert(other) + except TypeError: + return NotImplemented + return other + self + + def __mul__(self, other): + if not isinstance(other, Ordinal): + try: + other = Ordinal.convert(other) + except TypeError: + return NotImplemented + if ord0 in (self, other): + return ord0 + a_exp = self.degree + a_mult = self.leading_term.mult + summation = [] + if other.is_limit_ordinal: + for arg in other.terms: + summation.append(OmegaPower(a_exp + arg.exp, arg.mult)) + + else: + for arg in other.terms[:-1]: + summation.append(OmegaPower(a_exp + arg.exp, arg.mult)) + b_mult = other.trailing_term.mult + summation.append(OmegaPower(a_exp, a_mult*b_mult)) + summation += list(self.terms[1:]) + return Ordinal(*summation) + + def __rmul__(self, other): + if not isinstance(other, Ordinal): + try: + other = Ordinal.convert(other) + except TypeError: + return NotImplemented + return other * self + + def __pow__(self, other): + if not self == omega: + return NotImplemented + return Ordinal(OmegaPower(other, 1)) + + +class OrdinalZero(Ordinal): + """The ordinal zero. + + OrdinalZero can be imported as ``ord0``. + """ + pass + + +class OrdinalOmega(Ordinal): + """The ordinal omega which forms the base of all ordinals in cantor normal form. + + OrdinalOmega can be imported as ``omega``. + + Examples + ======== + + >>> from sympy.sets.ordinals import omega + >>> omega + omega + w*2 + """ + def __new__(cls): + return Ordinal.__new__(cls) + + @property + def terms(self): + return (OmegaPower(1, 1),) + + +ord0 = OrdinalZero() +omega = OrdinalOmega() diff --git a/.venv/lib/python3.12/site-packages/sympy/sets/powerset.py b/.venv/lib/python3.12/site-packages/sympy/sets/powerset.py new file mode 100644 index 0000000000000000000000000000000000000000..2eb3b41b9859281480bc9517a1cad0abe7a5683f --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/sets/powerset.py @@ -0,0 +1,119 @@ +from sympy.core.decorators import _sympifyit +from sympy.core.parameters import global_parameters +from sympy.core.logic import fuzzy_bool +from sympy.core.singleton import S +from sympy.core.sympify import _sympify + +from .sets import Set, FiniteSet, SetKind + + +class PowerSet(Set): + r"""A symbolic object representing a power set. + + Parameters + ========== + + arg : Set + The set to take power of. + + evaluate : bool + The flag to control evaluation. + + If the evaluation is disabled for finite sets, it can take + advantage of using subset test as a membership test. + + Notes + ===== + + Power set `\mathcal{P}(S)` is defined as a set containing all the + subsets of `S`. + + If the set `S` is a finite set, its power set would have + `2^{\left| S \right|}` elements, where `\left| S \right|` denotes + the cardinality of `S`. + + Examples + ======== + + >>> from sympy import PowerSet, S, FiniteSet + + A power set of a finite set: + + >>> PowerSet(FiniteSet(1, 2, 3)) + PowerSet({1, 2, 3}) + + A power set of an empty set: + + >>> PowerSet(S.EmptySet) + PowerSet(EmptySet) + >>> PowerSet(PowerSet(S.EmptySet)) + PowerSet(PowerSet(EmptySet)) + + A power set of an infinite set: + + >>> PowerSet(S.Reals) + PowerSet(Reals) + + Evaluating the power set of a finite set to its explicit form: + + >>> PowerSet(FiniteSet(1, 2, 3)).rewrite(FiniteSet) + FiniteSet(EmptySet, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Power_set + + .. [2] https://en.wikipedia.org/wiki/Axiom_of_power_set + """ + def __new__(cls, arg, evaluate=None): + if evaluate is None: + evaluate=global_parameters.evaluate + + arg = _sympify(arg) + + if not isinstance(arg, Set): + raise ValueError('{} must be a set.'.format(arg)) + + return super().__new__(cls, arg) + + @property + def arg(self): + return self.args[0] + + def _eval_rewrite_as_FiniteSet(self, *args, **kwargs): + arg = self.arg + if arg.is_FiniteSet: + return arg.powerset() + return None + + @_sympifyit('other', NotImplemented) + def _contains(self, other): + if not isinstance(other, Set): + return None + + return fuzzy_bool(self.arg.is_superset(other)) + + def _eval_is_subset(self, other): + if isinstance(other, PowerSet): + return self.arg.is_subset(other.arg) + + def __len__(self): + return 2 ** len(self.arg) + + def __iter__(self): + found = [S.EmptySet] + yield S.EmptySet + + for x in self.arg: + temp = [] + x = FiniteSet(x) + for y in found: + new = x + y + yield new + temp.append(new) + found.extend(temp) + + @property + def kind(self): + return SetKind(self.arg.kind) diff --git a/.venv/lib/python3.12/site-packages/sympy/sets/setexpr.py b/.venv/lib/python3.12/site-packages/sympy/sets/setexpr.py new file mode 100644 index 0000000000000000000000000000000000000000..94d77d5293617a620b70a945888987ce6cc61157 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/sets/setexpr.py @@ -0,0 +1,97 @@ +from sympy.core import Expr +from sympy.core.decorators import call_highest_priority, _sympifyit +from .fancysets import ImageSet +from .sets import set_add, set_sub, set_mul, set_div, set_pow, set_function + + +class SetExpr(Expr): + """An expression that can take on values of a set. + + Examples + ======== + + >>> from sympy import Interval, FiniteSet + >>> from sympy.sets.setexpr import SetExpr + + >>> a = SetExpr(Interval(0, 5)) + >>> b = SetExpr(FiniteSet(1, 10)) + >>> (a + b).set + Union(Interval(1, 6), Interval(10, 15)) + >>> (2*a + b).set + Interval(1, 20) + """ + _op_priority = 11.0 + + def __new__(cls, setarg): + return Expr.__new__(cls, setarg) + + set = property(lambda self: self.args[0]) + + def _latex(self, printer): + return r"SetExpr\left({}\right)".format(printer._print(self.set)) + + @_sympifyit('other', NotImplemented) + @call_highest_priority('__radd__') + def __add__(self, other): + return _setexpr_apply_operation(set_add, self, other) + + @_sympifyit('other', NotImplemented) + @call_highest_priority('__add__') + def __radd__(self, other): + return _setexpr_apply_operation(set_add, other, self) + + @_sympifyit('other', NotImplemented) + @call_highest_priority('__rmul__') + def __mul__(self, other): + return _setexpr_apply_operation(set_mul, self, other) + + @_sympifyit('other', NotImplemented) + @call_highest_priority('__mul__') + def __rmul__(self, other): + return _setexpr_apply_operation(set_mul, other, self) + + @_sympifyit('other', NotImplemented) + @call_highest_priority('__rsub__') + def __sub__(self, other): + return _setexpr_apply_operation(set_sub, self, other) + + @_sympifyit('other', NotImplemented) + @call_highest_priority('__sub__') + def __rsub__(self, other): + return _setexpr_apply_operation(set_sub, other, self) + + @_sympifyit('other', NotImplemented) + @call_highest_priority('__rpow__') + def __pow__(self, other): + return _setexpr_apply_operation(set_pow, self, other) + + @_sympifyit('other', NotImplemented) + @call_highest_priority('__pow__') + def __rpow__(self, other): + return _setexpr_apply_operation(set_pow, other, self) + + @_sympifyit('other', NotImplemented) + @call_highest_priority('__rtruediv__') + def __truediv__(self, other): + return _setexpr_apply_operation(set_div, self, other) + + @_sympifyit('other', NotImplemented) + @call_highest_priority('__truediv__') + def __rtruediv__(self, other): + return _setexpr_apply_operation(set_div, other, self) + + def _eval_func(self, func): + # TODO: this could be implemented straight into `imageset`: + res = set_function(func, self.set) + if res is None: + return SetExpr(ImageSet(func, self.set)) + return SetExpr(res) + + +def _setexpr_apply_operation(op, x, y): + if isinstance(x, SetExpr): + x = x.set + if isinstance(y, SetExpr): + y = y.set + out = op(x, y) + return SetExpr(out) diff --git a/.venv/lib/python3.12/site-packages/sympy/sets/sets.py b/.venv/lib/python3.12/site-packages/sympy/sets/sets.py new file mode 100644 index 0000000000000000000000000000000000000000..3c85ce87c515cfd4520dcc6b9265fe76d8c6163f --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/sets/sets.py @@ -0,0 +1,2804 @@ +from __future__ import annotations + +from typing import Any, Callable, TYPE_CHECKING, overload +from functools import reduce +from collections import defaultdict +from collections.abc import Mapping, Iterable +import inspect + +from sympy.core.kind import Kind, UndefinedKind, NumberKind +from sympy.core.basic import Basic +from sympy.core.containers import Tuple, TupleKind +from sympy.core.decorators import sympify_method_args, sympify_return +from sympy.core.evalf import EvalfMixin +from sympy.core.expr import Expr +from sympy.core.function import Lambda +from sympy.core.logic import (FuzzyBool, fuzzy_bool, fuzzy_or, fuzzy_and, + fuzzy_not) +from sympy.core.numbers import Float, Integer +from sympy.core.operations import LatticeOp +from sympy.core.parameters import global_parameters +from sympy.core.relational import Eq, Ne, is_lt +from sympy.core.singleton import Singleton, S +from sympy.core.sorting import ordered +from sympy.core.symbol import symbols, Symbol, Dummy, uniquely_named_symbol +from sympy.core.sympify import _sympify, sympify, _sympy_converter +from sympy.functions.elementary.exponential import exp, log +from sympy.functions.elementary.miscellaneous import Max, Min +from sympy.logic.boolalg import And, Or, Not, Xor, true, false +from sympy.utilities.decorator import deprecated +from sympy.utilities.exceptions import sympy_deprecation_warning +from sympy.utilities.iterables import (iproduct, sift, roundrobin, iterable, + subsets) +from sympy.utilities.misc import func_name, filldedent + +from mpmath import mpi, mpf + +from mpmath.libmp.libmpf import prec_to_dps + + +tfn = defaultdict(lambda: None, { + True: S.true, + S.true: S.true, + False: S.false, + S.false: S.false}) + + +@sympify_method_args +class Set(Basic, EvalfMixin): + """ + The base class for any kind of set. + + Explanation + =========== + + This is not meant to be used directly as a container of items. It does not + behave like the builtin ``set``; see :class:`FiniteSet` for that. + + Real intervals are represented by the :class:`Interval` class and unions of + sets by the :class:`Union` class. The empty set is represented by the + :class:`EmptySet` class and available as a singleton as ``S.EmptySet``. + """ + + __slots__: tuple[()] = () + + is_number = False + is_iterable = False + is_interval = False + + is_FiniteSet = False + is_Interval = False + is_ProductSet = False + is_Union = False + is_Intersection: FuzzyBool = None + is_UniversalSet: FuzzyBool = None + is_Complement: FuzzyBool = None + is_ComplexRegion = False + + is_empty: FuzzyBool = None + is_finite_set: FuzzyBool = None + + @property # type: ignore + @deprecated( + """ + The is_EmptySet attribute of Set objects is deprecated. + Use 's is S.EmptySet" or 's.is_empty' instead. + """, + deprecated_since_version="1.5", + active_deprecations_target="deprecated-is-emptyset", + ) + def is_EmptySet(self): + return None + + if TYPE_CHECKING: + + def __new__(cls, *args: Basic | complex) -> Set: + ... + + @overload # type: ignore + def subs(self, arg1: Mapping[Basic | complex, Set | complex], arg2: None=None) -> Set: ... + @overload + def subs(self, arg1: Iterable[tuple[Basic | complex, Set | complex]], arg2: None=None, **kwargs: Any) -> Set: ... + @overload + def subs(self, arg1: Set | complex, arg2: Set | complex) -> Set: ... + @overload + def subs(self, arg1: Mapping[Basic | complex, Basic | complex], arg2: None=None, **kwargs: Any) -> Basic: ... + @overload + def subs(self, arg1: Iterable[tuple[Basic | complex, Basic | complex]], arg2: None=None, **kwargs: Any) -> Basic: ... + @overload + def subs(self, arg1: Basic | complex, arg2: Basic | complex, **kwargs: Any) -> Basic: ... + + def subs(self, arg1: Mapping[Basic | complex, Basic | complex] | Basic | complex, # type: ignore + arg2: Basic | complex | None = None, **kwargs: Any) -> Basic: + ... + + def simplify(self, **kwargs) -> Set: + assert False + + def evalf(self, n: int = 15, subs: dict[Basic, Basic | float] | None = None, + maxn: int = 100, chop: bool = False, strict: bool = False, + quad: str | None = None, verbose: bool = False) -> Set: + ... + + n = evalf + + @staticmethod + def _infimum_key(expr): + """ + Return infimum (if possible) else S.Infinity. + """ + try: + infimum = expr.inf + assert infimum.is_comparable + infimum = infimum.evalf() # issue #18505 + except (NotImplementedError, + AttributeError, AssertionError, ValueError): + infimum = S.Infinity + return infimum + + def union(self, other): + """ + Returns the union of ``self`` and ``other``. + + Examples + ======== + + As a shortcut it is possible to use the ``+`` operator: + + >>> from sympy import Interval, FiniteSet + >>> Interval(0, 1).union(Interval(2, 3)) + Union(Interval(0, 1), Interval(2, 3)) + >>> Interval(0, 1) + Interval(2, 3) + Union(Interval(0, 1), Interval(2, 3)) + >>> Interval(1, 2, True, True) + FiniteSet(2, 3) + Union({3}, Interval.Lopen(1, 2)) + + Similarly it is possible to use the ``-`` operator for set differences: + + >>> Interval(0, 2) - Interval(0, 1) + Interval.Lopen(1, 2) + >>> Interval(1, 3) - FiniteSet(2) + Union(Interval.Ropen(1, 2), Interval.Lopen(2, 3)) + + """ + return Union(self, other) + + def intersect(self, other): + """ + Returns the intersection of 'self' and 'other'. + + Examples + ======== + + >>> from sympy import Interval + + >>> Interval(1, 3).intersect(Interval(1, 2)) + Interval(1, 2) + + >>> from sympy import imageset, Lambda, symbols, S + >>> n, m = symbols('n m') + >>> a = imageset(Lambda(n, 2*n), S.Integers) + >>> a.intersect(imageset(Lambda(m, 2*m + 1), S.Integers)) + EmptySet + + """ + return Intersection(self, other) + + def intersection(self, other): + """ + Alias for :meth:`intersect()` + """ + return self.intersect(other) + + def is_disjoint(self, other): + """ + Returns True if ``self`` and ``other`` are disjoint. + + Examples + ======== + + >>> from sympy import Interval + >>> Interval(0, 2).is_disjoint(Interval(1, 2)) + False + >>> Interval(0, 2).is_disjoint(Interval(3, 4)) + True + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Disjoint_sets + """ + return self.intersect(other) == S.EmptySet + + def isdisjoint(self, other): + """ + Alias for :meth:`is_disjoint()` + """ + return self.is_disjoint(other) + + def complement(self, universe): + r""" + The complement of 'self' w.r.t the given universe. + + Examples + ======== + + >>> from sympy import Interval, S + >>> Interval(0, 1).complement(S.Reals) + Union(Interval.open(-oo, 0), Interval.open(1, oo)) + + >>> Interval(0, 1).complement(S.UniversalSet) + Complement(UniversalSet, Interval(0, 1)) + + """ + return Complement(universe, self) + + def _complement(self, other): + # this behaves as other - self + if isinstance(self, ProductSet) and isinstance(other, ProductSet): + # If self and other are disjoint then other - self == self + if len(self.sets) != len(other.sets): + return other + + # There can be other ways to represent this but this gives: + # (A x B) - (C x D) = ((A - C) x B) U (A x (B - D)) + overlaps = [] + pairs = list(zip(self.sets, other.sets)) + for n in range(len(pairs)): + sets = (o if i != n else o-s for i, (s, o) in enumerate(pairs)) + overlaps.append(ProductSet(*sets)) + return Union(*overlaps) + + elif isinstance(other, Interval): + if isinstance(self, (Interval, FiniteSet)): + return Intersection(other, self.complement(S.Reals)) + + elif isinstance(other, Union): + return Union(*(o - self for o in other.args)) + + elif isinstance(other, Complement): + return Complement(other.args[0], Union(other.args[1], self), evaluate=False) + + elif other is S.EmptySet: + return S.EmptySet + + elif isinstance(other, FiniteSet): + sifted = sift(other, lambda x: fuzzy_bool(self.contains(x))) + # ignore those that are contained in self + return Union(FiniteSet(*(sifted[False])), + Complement(FiniteSet(*(sifted[None])), self, evaluate=False) + if sifted[None] else S.EmptySet) + + def symmetric_difference(self, other): + """ + Returns symmetric difference of ``self`` and ``other``. + + Examples + ======== + + >>> from sympy import Interval, S + >>> Interval(1, 3).symmetric_difference(S.Reals) + Union(Interval.open(-oo, 1), Interval.open(3, oo)) + >>> Interval(1, 10).symmetric_difference(S.Reals) + Union(Interval.open(-oo, 1), Interval.open(10, oo)) + + >>> from sympy import S, EmptySet + >>> S.Reals.symmetric_difference(EmptySet) + Reals + + References + ========== + .. [1] https://en.wikipedia.org/wiki/Symmetric_difference + + """ + return SymmetricDifference(self, other) + + def _symmetric_difference(self, other): + return Union(Complement(self, other), Complement(other, self)) + + @property + def inf(self): + """ + The infimum of ``self``. + + Examples + ======== + + >>> from sympy import Interval, Union + >>> Interval(0, 1).inf + 0 + >>> Union(Interval(0, 1), Interval(2, 3)).inf + 0 + + """ + return self._inf + + @property + def _inf(self): + raise NotImplementedError("(%s)._inf" % self) + + @property + def sup(self): + """ + The supremum of ``self``. + + Examples + ======== + + >>> from sympy import Interval, Union + >>> Interval(0, 1).sup + 1 + >>> Union(Interval(0, 1), Interval(2, 3)).sup + 3 + + """ + return self._sup + + @property + def _sup(self): + raise NotImplementedError("(%s)._sup" % self) + + def contains(self, other): + """ + Returns a SymPy value indicating whether ``other`` is contained + in ``self``: ``true`` if it is, ``false`` if it is not, else + an unevaluated ``Contains`` expression (or, as in the case of + ConditionSet and a union of FiniteSet/Intervals, an expression + indicating the conditions for containment). + + Examples + ======== + + >>> from sympy import Interval, S + >>> from sympy.abc import x + + >>> Interval(0, 1).contains(0.5) + True + + As a shortcut it is possible to use the ``in`` operator, but that + will raise an error unless an affirmative true or false is not + obtained. + + >>> Interval(0, 1).contains(x) + (0 <= x) & (x <= 1) + >>> x in Interval(0, 1) + Traceback (most recent call last): + ... + TypeError: did not evaluate to a bool: None + + The result of 'in' is a bool, not a SymPy value + + >>> 1 in Interval(0, 2) + True + >>> _ is S.true + False + """ + from .contains import Contains + other = sympify(other, strict=True) + + c = self._contains(other) + if isinstance(c, Contains): + return c + if c is None: + return Contains(other, self, evaluate=False) + b = tfn[c] + if b is None: + return c + return b + + def _contains(self, other): + """Test if ``other`` is an element of the set ``self``. + + This is an internal method that is expected to be overridden by + subclasses of ``Set`` and will be called by the public + :func:`Set.contains` method or the :class:`Contains` expression. + + Parameters + ========== + + other: Sympified :class:`Basic` instance + The object whose membership in ``self`` is to be tested. + + Returns + ======= + + Symbolic :class:`Boolean` or ``None``. + + A return value of ``None`` indicates that it is unknown whether + ``other`` is contained in ``self``. Returning ``None`` from here + ensures that ``self.contains(other)`` or ``Contains(self, other)`` will + return an unevaluated :class:`Contains` expression. + + If not ``None`` then the returned value is a :class:`Boolean` that is + logically equivalent to the statement that ``other`` is an element of + ``self``. Usually this would be either ``S.true`` or ``S.false`` but + not always. + """ + raise NotImplementedError(f"{type(self).__name__}._contains") + + def is_subset(self, other): + """ + Returns True if ``self`` is a subset of ``other``. + + Examples + ======== + + >>> from sympy import Interval + >>> Interval(0, 0.5).is_subset(Interval(0, 1)) + True + >>> Interval(0, 1).is_subset(Interval(0, 1, left_open=True)) + False + + """ + if not isinstance(other, Set): + raise ValueError("Unknown argument '%s'" % other) + + # Handle the trivial cases + if self == other: + return True + is_empty = self.is_empty + if is_empty is True: + return True + elif fuzzy_not(is_empty) and other.is_empty: + return False + if self.is_finite_set is False and other.is_finite_set: + return False + + # Dispatch on subclass rules + ret = self._eval_is_subset(other) + if ret is not None: + return ret + ret = other._eval_is_superset(self) + if ret is not None: + return ret + + # Use pairwise rules from multiple dispatch + from sympy.sets.handlers.issubset import is_subset_sets + ret = is_subset_sets(self, other) + if ret is not None: + return ret + + # Fall back on computing the intersection + # XXX: We shouldn't do this. A query like this should be handled + # without evaluating new Set objects. It should be the other way round + # so that the intersect method uses is_subset for evaluation. + if self.intersect(other) == self: + return True + + def _eval_is_subset(self, other): + '''Returns a fuzzy bool for whether self is a subset of other.''' + return None + + def _eval_is_superset(self, other): + '''Returns a fuzzy bool for whether self is a subset of other.''' + return None + + # This should be deprecated: + def issubset(self, other): + """ + Alias for :meth:`is_subset()` + """ + return self.is_subset(other) + + def is_proper_subset(self, other): + """ + Returns True if ``self`` is a proper subset of ``other``. + + Examples + ======== + + >>> from sympy import Interval + >>> Interval(0, 0.5).is_proper_subset(Interval(0, 1)) + True + >>> Interval(0, 1).is_proper_subset(Interval(0, 1)) + False + + """ + if isinstance(other, Set): + return self != other and self.is_subset(other) + else: + raise ValueError("Unknown argument '%s'" % other) + + def is_superset(self, other): + """ + Returns True if ``self`` is a superset of ``other``. + + Examples + ======== + + >>> from sympy import Interval + >>> Interval(0, 0.5).is_superset(Interval(0, 1)) + False + >>> Interval(0, 1).is_superset(Interval(0, 1, left_open=True)) + True + + """ + if isinstance(other, Set): + return other.is_subset(self) + else: + raise ValueError("Unknown argument '%s'" % other) + + # This should be deprecated: + def issuperset(self, other): + """ + Alias for :meth:`is_superset()` + """ + return self.is_superset(other) + + def is_proper_superset(self, other): + """ + Returns True if ``self`` is a proper superset of ``other``. + + Examples + ======== + + >>> from sympy import Interval + >>> Interval(0, 1).is_proper_superset(Interval(0, 0.5)) + True + >>> Interval(0, 1).is_proper_superset(Interval(0, 1)) + False + + """ + if isinstance(other, Set): + return self != other and self.is_superset(other) + else: + raise ValueError("Unknown argument '%s'" % other) + + def _eval_powerset(self): + from .powerset import PowerSet + return PowerSet(self) + + def powerset(self): + """ + Find the Power set of ``self``. + + Examples + ======== + + >>> from sympy import EmptySet, FiniteSet, Interval + + A power set of an empty set: + + >>> A = EmptySet + >>> A.powerset() + {EmptySet} + + A power set of a finite set: + + >>> A = FiniteSet(1, 2) + >>> a, b, c = FiniteSet(1), FiniteSet(2), FiniteSet(1, 2) + >>> A.powerset() == FiniteSet(a, b, c, EmptySet) + True + + A power set of an interval: + + >>> Interval(1, 2).powerset() + PowerSet(Interval(1, 2)) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Power_set + + """ + return self._eval_powerset() + + @property + def measure(self): + """ + The (Lebesgue) measure of ``self``. + + Examples + ======== + + >>> from sympy import Interval, Union + >>> Interval(0, 1).measure + 1 + >>> Union(Interval(0, 1), Interval(2, 3)).measure + 2 + + """ + return self._measure + + @property + def kind(self): + """ + The kind of a Set + + Explanation + =========== + + Any :class:`Set` will have kind :class:`SetKind` which is + parametrised by the kind of the elements of the set. For example + most sets are sets of numbers and will have kind + ``SetKind(NumberKind)``. If elements of sets are different in kind than + their kind will ``SetKind(UndefinedKind)``. See + :class:`sympy.core.kind.Kind` for an explanation of the kind system. + + Examples + ======== + + >>> from sympy import Interval, Matrix, FiniteSet, EmptySet, ProductSet, PowerSet + + >>> FiniteSet(Matrix([1, 2])).kind + SetKind(MatrixKind(NumberKind)) + + >>> Interval(1, 2).kind + SetKind(NumberKind) + + >>> EmptySet.kind + SetKind() + + A :class:`sympy.sets.powerset.PowerSet` is a set of sets: + + >>> PowerSet({1, 2, 3}).kind + SetKind(SetKind(NumberKind)) + + A :class:`ProductSet` represents the set of tuples of elements of + other sets. Its kind is :class:`sympy.core.containers.TupleKind` + parametrised by the kinds of the elements of those sets: + + >>> p = ProductSet(FiniteSet(1, 2), FiniteSet(3, 4)) + >>> list(p) + [(1, 3), (2, 3), (1, 4), (2, 4)] + >>> p.kind + SetKind(TupleKind(NumberKind, NumberKind)) + + When all elements of the set do not have same kind, the kind + will be returned as ``SetKind(UndefinedKind)``: + + >>> FiniteSet(0, Matrix([1, 2])).kind + SetKind(UndefinedKind) + + The kind of the elements of a set are given by the ``element_kind`` + attribute of ``SetKind``: + + >>> Interval(1, 2).kind.element_kind + NumberKind + + See Also + ======== + + NumberKind + sympy.core.kind.UndefinedKind + sympy.core.containers.TupleKind + MatrixKind + sympy.matrices.expressions.sets.MatrixSet + sympy.sets.conditionset.ConditionSet + Rationals + Naturals + Integers + sympy.sets.fancysets.ImageSet + sympy.sets.fancysets.Range + sympy.sets.fancysets.ComplexRegion + sympy.sets.powerset.PowerSet + sympy.sets.sets.ProductSet + sympy.sets.sets.Interval + sympy.sets.sets.Union + sympy.sets.sets.Intersection + sympy.sets.sets.Complement + sympy.sets.sets.EmptySet + sympy.sets.sets.UniversalSet + sympy.sets.sets.FiniteSet + sympy.sets.sets.SymmetricDifference + sympy.sets.sets.DisjointUnion + """ + return self._kind() + + @property + def boundary(self): + """ + The boundary or frontier of a set. + + Explanation + =========== + + A point x is on the boundary of a set S if + + 1. x is in the closure of S. + I.e. Every neighborhood of x contains a point in S. + 2. x is not in the interior of S. + I.e. There does not exist an open set centered on x contained + entirely within S. + + There are the points on the outer rim of S. If S is open then these + points need not actually be contained within S. + + For example, the boundary of an interval is its start and end points. + This is true regardless of whether or not the interval is open. + + Examples + ======== + + >>> from sympy import Interval + >>> Interval(0, 1).boundary + {0, 1} + >>> Interval(0, 1, True, False).boundary + {0, 1} + """ + return self._boundary + + @property + def is_open(self): + """ + Property method to check whether a set is open. + + Explanation + =========== + + A set is open if and only if it has an empty intersection with its + boundary. In particular, a subset A of the reals is open if and only + if each one of its points is contained in an open interval that is a + subset of A. + + Examples + ======== + >>> from sympy import S + >>> S.Reals.is_open + True + >>> S.Rationals.is_open + False + """ + return Intersection(self, self.boundary).is_empty + + @property + def is_closed(self): + """ + A property method to check whether a set is closed. + + Explanation + =========== + + A set is closed if its complement is an open set. The closedness of a + subset of the reals is determined with respect to R and its standard + topology. + + Examples + ======== + >>> from sympy import Interval + >>> Interval(0, 1).is_closed + True + """ + return self.boundary.is_subset(self) + + @property + def closure(self): + """ + Property method which returns the closure of a set. + The closure is defined as the union of the set itself and its + boundary. + + Examples + ======== + >>> from sympy import S, Interval + >>> S.Reals.closure + Reals + >>> Interval(0, 1).closure + Interval(0, 1) + """ + return self + self.boundary + + @property + def interior(self): + """ + Property method which returns the interior of a set. + The interior of a set S consists all points of S that do not + belong to the boundary of S. + + Examples + ======== + >>> from sympy import Interval + >>> Interval(0, 1).interior + Interval.open(0, 1) + >>> Interval(0, 1).boundary.interior + EmptySet + """ + return self - self.boundary + + @property + def _boundary(self): + raise NotImplementedError() + + @property + def _measure(self): + raise NotImplementedError("(%s)._measure" % self) + + def _kind(self): + return SetKind(UndefinedKind) + + def _eval_evalf(self, prec): + dps = prec_to_dps(prec) + return self.func(*[arg.evalf(n=dps) for arg in self.args]) + + @sympify_return([('other', 'Set')], NotImplemented) + def __add__(self, other): + return self.union(other) + + @sympify_return([('other', 'Set')], NotImplemented) + def __or__(self, other): + return self.union(other) + + @sympify_return([('other', 'Set')], NotImplemented) + def __and__(self, other): + return self.intersect(other) + + @sympify_return([('other', 'Set')], NotImplemented) + def __mul__(self, other): + return ProductSet(self, other) + + @sympify_return([('other', 'Set')], NotImplemented) + def __xor__(self, other): + return SymmetricDifference(self, other) + + @sympify_return([('exp', Expr)], NotImplemented) + def __pow__(self, exp): + if not (exp.is_Integer and exp >= 0): + raise ValueError("%s: Exponent must be a positive Integer" % exp) + return ProductSet(*[self]*exp) + + @sympify_return([('other', 'Set')], NotImplemented) + def __sub__(self, other): + return Complement(self, other) + + def __contains__(self, other): + other = _sympify(other) + c = self._contains(other) + b = tfn[c] + if b is None: + # x in y must evaluate to T or F; to entertain a None + # result with Set use y.contains(x) + raise TypeError('did not evaluate to a bool: %r' % c) + return b + + +class ProductSet(Set): + """ + Represents a Cartesian Product of Sets. + + Explanation + =========== + + Returns a Cartesian product given several sets as either an iterable + or individual arguments. + + Can use ``*`` operator on any sets for convenient shorthand. + + Examples + ======== + + >>> from sympy import Interval, FiniteSet, ProductSet + >>> I = Interval(0, 5); S = FiniteSet(1, 2, 3) + >>> ProductSet(I, S) + ProductSet(Interval(0, 5), {1, 2, 3}) + + >>> (2, 2) in ProductSet(I, S) + True + + >>> Interval(0, 1) * Interval(0, 1) # The unit square + ProductSet(Interval(0, 1), Interval(0, 1)) + + >>> coin = FiniteSet('H', 'T') + >>> set(coin**2) + {(H, H), (H, T), (T, H), (T, T)} + + The Cartesian product is not commutative or associative e.g.: + + >>> I*S == S*I + False + >>> (I*I)*I == I*(I*I) + False + + Notes + ===== + + - Passes most operations down to the argument sets + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Cartesian_product + """ + is_ProductSet = True + + def __new__(cls, *sets, **assumptions): + if len(sets) == 1 and iterable(sets[0]) and not isinstance(sets[0], (Set, set)): + sympy_deprecation_warning( + """ +ProductSet(iterable) is deprecated. Use ProductSet(*iterable) instead. + """, + deprecated_since_version="1.5", + active_deprecations_target="deprecated-productset-iterable", + ) + sets = tuple(sets[0]) + + sets = [sympify(s) for s in sets] + + if not all(isinstance(s, Set) for s in sets): + raise TypeError("Arguments to ProductSet should be of type Set") + + # Nullary product of sets is *not* the empty set + if len(sets) == 0: + return FiniteSet(()) + + if S.EmptySet in sets: + return S.EmptySet + + return Basic.__new__(cls, *sets, **assumptions) + + @property + def sets(self): + return self.args + + def flatten(self): + def _flatten(sets): + for s in sets: + if s.is_ProductSet: + yield from _flatten(s.sets) + else: + yield s + return ProductSet(*_flatten(self.sets)) + + + + def _contains(self, element): + """ + ``in`` operator for ProductSets. + + Examples + ======== + + >>> from sympy import Interval + >>> (2, 3) in Interval(0, 5) * Interval(0, 5) + True + + >>> (10, 10) in Interval(0, 5) * Interval(0, 5) + False + + Passes operation on to constituent sets + """ + if element.is_Symbol: + return None + + if not isinstance(element, Tuple) or len(element) != len(self.sets): + return S.false + + return And(*[s.contains(e) for s, e in zip(self.sets, element)]) + + def as_relational(self, *symbols): + symbols = [_sympify(s) for s in symbols] + if len(symbols) != len(self.sets) or not all( + i.is_Symbol for i in symbols): + raise ValueError( + 'number of symbols must match the number of sets') + return And(*[s.as_relational(i) for s, i in zip(self.sets, symbols)]) + + @property + def _boundary(self): + return Union(*(ProductSet(*(b + b.boundary if i != j else b.boundary + for j, b in enumerate(self.sets))) + for i, a in enumerate(self.sets))) + + @property + def is_iterable(self): + """ + A property method which tests whether a set is iterable or not. + Returns True if set is iterable, otherwise returns False. + + Examples + ======== + + >>> from sympy import FiniteSet, Interval + >>> I = Interval(0, 1) + >>> A = FiniteSet(1, 2, 3, 4, 5) + >>> I.is_iterable + False + >>> A.is_iterable + True + + """ + return all(set.is_iterable for set in self.sets) + + def __iter__(self): + """ + A method which implements is_iterable property method. + If self.is_iterable returns True (both constituent sets are iterable), + then return the Cartesian Product. Otherwise, raise TypeError. + """ + return iproduct(*self.sets) + + @property + def is_empty(self): + return fuzzy_or(s.is_empty for s in self.sets) + + @property + def is_finite_set(self): + all_finite = fuzzy_and(s.is_finite_set for s in self.sets) + return fuzzy_or([self.is_empty, all_finite]) + + @property + def _measure(self): + measure = 1 + for s in self.sets: + measure *= s.measure + return measure + + def _kind(self): + return SetKind(TupleKind(*(i.kind.element_kind for i in self.args))) + + def __len__(self): + return reduce(lambda a, b: a*b, (len(s) for s in self.args)) + + def __bool__(self): + return all(self.sets) + + +class Interval(Set): + """ + Represents a real interval as a Set. + + Usage: + Returns an interval with end points ``start`` and ``end``. + + For ``left_open=True`` (default ``left_open`` is ``False``) the interval + will be open on the left. Similarly, for ``right_open=True`` the interval + will be open on the right. + + Examples + ======== + + >>> from sympy import Symbol, Interval + >>> Interval(0, 1) + Interval(0, 1) + >>> Interval.Ropen(0, 1) + Interval.Ropen(0, 1) + >>> Interval.Ropen(0, 1) + Interval.Ropen(0, 1) + >>> Interval.Lopen(0, 1) + Interval.Lopen(0, 1) + >>> Interval.open(0, 1) + Interval.open(0, 1) + + >>> a = Symbol('a', real=True) + >>> Interval(0, a) + Interval(0, a) + + Notes + ===== + - Only real end points are supported + - ``Interval(a, b)`` with $a > b$ will return the empty set + - Use the ``evalf()`` method to turn an Interval into an mpmath + ``mpi`` interval instance + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Interval_%28mathematics%29 + """ + is_Interval = True + + def __new__(cls, start, end, left_open=False, right_open=False): + + start = _sympify(start) + end = _sympify(end) + left_open = _sympify(left_open) + right_open = _sympify(right_open) + + if not all(isinstance(a, (type(true), type(false))) + for a in [left_open, right_open]): + raise NotImplementedError( + "left_open and right_open can have only true/false values, " + "got %s and %s" % (left_open, right_open)) + + # Only allow real intervals + if fuzzy_not(fuzzy_and(i.is_extended_real for i in (start, end, end-start))): + raise ValueError("Non-real intervals are not supported") + + # evaluate if possible + if is_lt(end, start): + return S.EmptySet + elif (end - start).is_negative: + return S.EmptySet + + if end == start and (left_open or right_open): + return S.EmptySet + if end == start and not (left_open or right_open): + if start is S.Infinity or start is S.NegativeInfinity: + return S.EmptySet + return FiniteSet(end) + + # Make sure infinite interval end points are open. + if start is S.NegativeInfinity: + left_open = true + if end is S.Infinity: + right_open = true + if start == S.Infinity or end == S.NegativeInfinity: + return S.EmptySet + + return Basic.__new__(cls, start, end, left_open, right_open) + + @property + def start(self): + """ + The left end point of the interval. + + This property takes the same value as the ``inf`` property. + + Examples + ======== + + >>> from sympy import Interval + >>> Interval(0, 1).start + 0 + + """ + return self._args[0] + + @property + def end(self): + """ + The right end point of the interval. + + This property takes the same value as the ``sup`` property. + + Examples + ======== + + >>> from sympy import Interval + >>> Interval(0, 1).end + 1 + + """ + return self._args[1] + + @property + def left_open(self): + """ + True if interval is left-open. + + Examples + ======== + + >>> from sympy import Interval + >>> Interval(0, 1, left_open=True).left_open + True + >>> Interval(0, 1, left_open=False).left_open + False + + """ + return self._args[2] + + @property + def right_open(self): + """ + True if interval is right-open. + + Examples + ======== + + >>> from sympy import Interval + >>> Interval(0, 1, right_open=True).right_open + True + >>> Interval(0, 1, right_open=False).right_open + False + + """ + return self._args[3] + + @classmethod + def open(cls, a, b): + """Return an interval including neither boundary.""" + return cls(a, b, True, True) + + @classmethod + def Lopen(cls, a, b): + """Return an interval not including the left boundary.""" + return cls(a, b, True, False) + + @classmethod + def Ropen(cls, a, b): + """Return an interval not including the right boundary.""" + return cls(a, b, False, True) + + @property + def _inf(self): + return self.start + + @property + def _sup(self): + return self.end + + @property + def left(self): + return self.start + + @property + def right(self): + return self.end + + @property + def is_empty(self): + if self.left_open or self.right_open: + cond = self.start >= self.end # One/both bounds open + else: + cond = self.start > self.end # Both bounds closed + return fuzzy_bool(cond) + + @property + def is_finite_set(self): + return self.measure.is_zero + + def _complement(self, other): + if other == S.Reals: + a = Interval(S.NegativeInfinity, self.start, + True, not self.left_open) + b = Interval(self.end, S.Infinity, not self.right_open, True) + return Union(a, b) + + if isinstance(other, FiniteSet): + nums = [m for m in other.args if m.is_number] + if nums == []: + return None + + return Set._complement(self, other) + + @property + def _boundary(self): + finite_points = [p for p in (self.start, self.end) + if abs(p) != S.Infinity] + return FiniteSet(*finite_points) + + def _contains(self, other): + if (not isinstance(other, Expr) or other is S.NaN + or other.is_real is False or other.has(S.ComplexInfinity)): + # if an expression has zoo it will be zoo or nan + # and neither of those is real + return false + + if self.start is S.NegativeInfinity and self.end is S.Infinity: + if other.is_real is not None: + return tfn[other.is_real] + + d = Dummy() + return self.as_relational(d).subs(d, other) + + def as_relational(self, x): + """Rewrite an interval in terms of inequalities and logic operators.""" + x = sympify(x) + if self.right_open: + right = x < self.end + else: + right = x <= self.end + if self.left_open: + left = self.start < x + else: + left = self.start <= x + return And(left, right) + + @property + def _measure(self): + return self.end - self.start + + def _kind(self): + return SetKind(NumberKind) + + def to_mpi(self, prec=53): + return mpi(mpf(self.start._eval_evalf(prec)), + mpf(self.end._eval_evalf(prec))) + + def _eval_evalf(self, prec): + return Interval(self.left._evalf(prec), self.right._evalf(prec), + left_open=self.left_open, right_open=self.right_open) + + def _is_comparable(self, other): + is_comparable = self.start.is_comparable + is_comparable &= self.end.is_comparable + is_comparable &= other.start.is_comparable + is_comparable &= other.end.is_comparable + + return is_comparable + + @property + def is_left_unbounded(self): + """Return ``True`` if the left endpoint is negative infinity. """ + return self.left is S.NegativeInfinity or self.left == Float("-inf") + + @property + def is_right_unbounded(self): + """Return ``True`` if the right endpoint is positive infinity. """ + return self.right is S.Infinity or self.right == Float("+inf") + + def _eval_Eq(self, other): + if not isinstance(other, Interval): + if isinstance(other, FiniteSet): + return false + elif isinstance(other, Set): + return None + return false + + +class Union(Set, LatticeOp): + """ + Represents a union of sets as a :class:`Set`. + + Examples + ======== + + >>> from sympy import Union, Interval + >>> Union(Interval(1, 2), Interval(3, 4)) + Union(Interval(1, 2), Interval(3, 4)) + + The Union constructor will always try to merge overlapping intervals, + if possible. For example: + + >>> Union(Interval(1, 2), Interval(2, 3)) + Interval(1, 3) + + See Also + ======== + + Intersection + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Union_%28set_theory%29 + """ + is_Union = True + + @property + def identity(self): + return S.EmptySet + + @property + def zero(self): + return S.UniversalSet + + def __new__(cls, *args, **kwargs): + evaluate = kwargs.get('evaluate', global_parameters.evaluate) + + # flatten inputs to merge intersections and iterables + args = _sympify(args) + + # Reduce sets using known rules + if evaluate: + args = list(cls._new_args_filter(args)) + return simplify_union(args) + + args = list(ordered(args, Set._infimum_key)) + + obj = Basic.__new__(cls, *args) + obj._argset = frozenset(args) + return obj + + @property + def args(self): + return self._args + + def _complement(self, universe): + # DeMorgan's Law + return Intersection(s.complement(universe) for s in self.args) + + @property + def _inf(self): + # We use Min so that sup is meaningful in combination with symbolic + # interval end points. + return Min(*[set.inf for set in self.args]) + + @property + def _sup(self): + # We use Max so that sup is meaningful in combination with symbolic + # end points. + return Max(*[set.sup for set in self.args]) + + @property + def is_empty(self): + return fuzzy_and(set.is_empty for set in self.args) + + @property + def is_finite_set(self): + return fuzzy_and(set.is_finite_set for set in self.args) + + @property + def _measure(self): + # Measure of a union is the sum of the measures of the sets minus + # the sum of their pairwise intersections plus the sum of their + # triple-wise intersections minus ... etc... + + # Sets is a collection of intersections and a set of elementary + # sets which made up those intersections (called "sos" for set of sets) + # An example element might of this list might be: + # ( {A,B,C}, A.intersect(B).intersect(C) ) + + # Start with just elementary sets ( ({A}, A), ({B}, B), ... ) + # Then get and subtract ( ({A,B}, (A int B), ... ) while non-zero + sets = [(FiniteSet(s), s) for s in self.args] + measure = 0 + parity = 1 + while sets: + # Add up the measure of these sets and add or subtract it to total + measure += parity * sum(inter.measure for sos, inter in sets) + + # For each intersection in sets, compute the intersection with every + # other set not already part of the intersection. + sets = ((sos + FiniteSet(newset), newset.intersect(intersection)) + for sos, intersection in sets for newset in self.args + if newset not in sos) + + # Clear out sets with no measure + sets = [(sos, inter) for sos, inter in sets if inter.measure != 0] + + # Clear out duplicates + sos_list = [] + sets_list = [] + for _set in sets: + if _set[0] in sos_list: + continue + else: + sos_list.append(_set[0]) + sets_list.append(_set) + sets = sets_list + + # Flip Parity - next time subtract/add if we added/subtracted here + parity *= -1 + return measure + + def _kind(self): + kinds = tuple(arg.kind for arg in self.args if arg is not S.EmptySet) + if not kinds: + return SetKind() + elif all(i == kinds[0] for i in kinds): + return kinds[0] + else: + return SetKind(UndefinedKind) + + @property + def _boundary(self): + def boundary_of_set(i): + """ The boundary of set i minus interior of all other sets """ + b = self.args[i].boundary + for j, a in enumerate(self.args): + if j != i: + b = b - a.interior + return b + return Union(*map(boundary_of_set, range(len(self.args)))) + + def _contains(self, other): + return Or(*[s.contains(other) for s in self.args]) + + def is_subset(self, other): + return fuzzy_and(s.is_subset(other) for s in self.args) + + def as_relational(self, symbol): + """Rewrite a Union in terms of equalities and logic operators. """ + if (len(self.args) == 2 and + all(isinstance(i, Interval) for i in self.args)): + # optimization to give 3 args as (x > 1) & (x < 5) & Ne(x, 3) + # instead of as 4, ((1 <= x) & (x < 3)) | ((x <= 5) & (3 < x)) + # XXX: This should be ideally be improved to handle any number of + # intervals and also not to assume that the intervals are in any + # particular sorted order. + a, b = self.args + if a.sup == b.inf and a.right_open and b.left_open: + mincond = symbol > a.inf if a.left_open else symbol >= a.inf + maxcond = symbol < b.sup if b.right_open else symbol <= b.sup + necond = Ne(symbol, a.sup) + return And(necond, mincond, maxcond) + return Or(*[i.as_relational(symbol) for i in self.args]) + + @property + def is_iterable(self): + return all(arg.is_iterable for arg in self.args) + + def __iter__(self): + return roundrobin(*(iter(arg) for arg in self.args)) + + +class Intersection(Set, LatticeOp): + """ + Represents an intersection of sets as a :class:`Set`. + + Examples + ======== + + >>> from sympy import Intersection, Interval + >>> Intersection(Interval(1, 3), Interval(2, 4)) + Interval(2, 3) + + We often use the .intersect method + + >>> Interval(1,3).intersect(Interval(2,4)) + Interval(2, 3) + + See Also + ======== + + Union + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Intersection_%28set_theory%29 + """ + is_Intersection = True + + @property + def identity(self): + return S.UniversalSet + + @property + def zero(self): + return S.EmptySet + + def __new__(cls, *args , evaluate=None): + if evaluate is None: + evaluate = global_parameters.evaluate + + # flatten inputs to merge intersections and iterables + args = list(ordered(set(_sympify(args)))) + + # Reduce sets using known rules + if evaluate: + args = list(cls._new_args_filter(args)) + return simplify_intersection(args) + + args = list(ordered(args, Set._infimum_key)) + + obj = Basic.__new__(cls, *args) + obj._argset = frozenset(args) + return obj + + @property + def args(self): + return self._args + + @property + def is_iterable(self): + return any(arg.is_iterable for arg in self.args) + + @property + def is_finite_set(self): + if fuzzy_or(arg.is_finite_set for arg in self.args): + return True + + def _kind(self): + kinds = tuple(arg.kind for arg in self.args if arg is not S.UniversalSet) + if not kinds: + return SetKind(UndefinedKind) + elif all(i == kinds[0] for i in kinds): + return kinds[0] + else: + return SetKind() + + @property + def _inf(self): + raise NotImplementedError() + + @property + def _sup(self): + raise NotImplementedError() + + def _contains(self, other): + return And(*[set.contains(other) for set in self.args]) + + def __iter__(self): + sets_sift = sift(self.args, lambda x: x.is_iterable) + + completed = False + candidates = sets_sift[True] + sets_sift[None] + + finite_candidates, others = [], [] + for candidate in candidates: + length = None + try: + length = len(candidate) + except TypeError: + others.append(candidate) + + if length is not None: + finite_candidates.append(candidate) + finite_candidates.sort(key=len) + + for s in finite_candidates + others: + other_sets = set(self.args) - {s} + other = Intersection(*other_sets, evaluate=False) + completed = True + for x in s: + try: + if x in other: + yield x + except TypeError: + completed = False + if completed: + return + + if not completed: + if not candidates: + raise TypeError("None of the constituent sets are iterable") + raise TypeError( + "The computation had not completed because of the " + "undecidable set membership is found in every candidates.") + + @staticmethod + def _handle_finite_sets(args): + '''Simplify intersection of one or more FiniteSets and other sets''' + + # First separate the FiniteSets from the others + fs_args, others = sift(args, lambda x: x.is_FiniteSet, binary=True) + + # Let the caller handle intersection of non-FiniteSets + if not fs_args: + return + + # Convert to Python sets and build the set of all elements + fs_sets = [set(fs) for fs in fs_args] + all_elements = reduce(lambda a, b: a | b, fs_sets, set()) + + # Extract elements that are definitely in or definitely not in the + # intersection. Here we check contains for all of args. + definite = set() + for e in all_elements: + inall = fuzzy_and(s.contains(e) for s in args) + if inall is True: + definite.add(e) + if inall is not None: + for s in fs_sets: + s.discard(e) + + # At this point all elements in all of fs_sets are possibly in the + # intersection. In some cases this is because they are definitely in + # the intersection of the finite sets but it's not clear if they are + # members of others. We might have {m, n}, {m}, and Reals where we + # don't know if m or n is real. We want to remove n here but it is + # possibly in because it might be equal to m. So what we do now is + # extract the elements that are definitely in the remaining finite + # sets iteratively until we end up with {n}, {}. At that point if we + # get any empty set all remaining elements are discarded. + + fs_elements = reduce(lambda a, b: a | b, fs_sets, set()) + + # Need fuzzy containment testing + fs_symsets = [FiniteSet(*s) for s in fs_sets] + + while fs_elements: + for e in fs_elements: + infs = fuzzy_and(s.contains(e) for s in fs_symsets) + if infs is True: + definite.add(e) + if infs is not None: + for n, s in enumerate(fs_sets): + # Update Python set and FiniteSet + if e in s: + s.remove(e) + fs_symsets[n] = FiniteSet(*s) + fs_elements.remove(e) + break + # If we completed the for loop without removing anything we are + # done so quit the outer while loop + else: + break + + # If any of the sets of remainder elements is empty then we discard + # all of them for the intersection. + if not all(fs_sets): + fs_sets = [set()] + + # Here we fold back the definitely included elements into each fs. + # Since they are definitely included they must have been members of + # each FiniteSet to begin with. We could instead fold these in with a + # Union at the end to get e.g. {3}|({x}&{y}) rather than {3,x}&{3,y}. + if definite: + fs_sets = [fs | definite for fs in fs_sets] + + if fs_sets == [set()]: + return S.EmptySet + + sets = [FiniteSet(*s) for s in fs_sets] + + # Any set in others is redundant if it contains all the elements that + # are in the finite sets so we don't need it in the Intersection + all_elements = reduce(lambda a, b: a | b, fs_sets, set()) + is_redundant = lambda o: all(fuzzy_bool(o.contains(e)) for e in all_elements) + others = [o for o in others if not is_redundant(o)] + + if others: + rest = Intersection(*others) + # XXX: Maybe this shortcut should be at the beginning. For large + # FiniteSets it could much more efficient to process the other + # sets first... + if rest is S.EmptySet: + return S.EmptySet + # Flatten the Intersection + if rest.is_Intersection: + sets.extend(rest.args) + else: + sets.append(rest) + + if len(sets) == 1: + return sets[0] + else: + return Intersection(*sets, evaluate=False) + + def as_relational(self, symbol): + """Rewrite an Intersection in terms of equalities and logic operators""" + return And(*[set.as_relational(symbol) for set in self.args]) + + +class Complement(Set): + r"""Represents the set difference or relative complement of a set with + another set. + + $$A - B = \{x \in A \mid x \notin B\}$$ + + + Examples + ======== + + >>> from sympy import Complement, FiniteSet + >>> Complement(FiniteSet(0, 1, 2), FiniteSet(1)) + {0, 2} + + See Also + ========= + + Intersection, Union + + References + ========== + + .. [1] https://mathworld.wolfram.com/ComplementSet.html + """ + + is_Complement = True + + def __new__(cls, a, b, evaluate=True): + a, b = map(_sympify, (a, b)) + if evaluate: + return Complement.reduce(a, b) + + return Basic.__new__(cls, a, b) + + @staticmethod + def reduce(A, B): + """ + Simplify a :class:`Complement`. + + """ + if B == S.UniversalSet or A.is_subset(B): + return S.EmptySet + + if isinstance(B, Union): + return Intersection(*(s.complement(A) for s in B.args)) + + result = B._complement(A) + if result is not None: + return result + else: + return Complement(A, B, evaluate=False) + + def _contains(self, other): + A = self.args[0] + B = self.args[1] + return And(A.contains(other), Not(B.contains(other))) + + def as_relational(self, symbol): + """Rewrite a complement in terms of equalities and logic + operators""" + A, B = self.args + + A_rel = A.as_relational(symbol) + B_rel = Not(B.as_relational(symbol)) + + return And(A_rel, B_rel) + + def _kind(self): + return self.args[0].kind + + @property + def is_iterable(self): + if self.args[0].is_iterable: + return True + + @property + def is_finite_set(self): + A, B = self.args + a_finite = A.is_finite_set + if a_finite is True: + return True + elif a_finite is False and B.is_finite_set: + return False + + def __iter__(self): + A, B = self.args + for a in A: + if a not in B: + yield a + else: + continue + + +class EmptySet(Set, metaclass=Singleton): + """ + Represents the empty set. The empty set is available as a singleton + as ``S.EmptySet``. + + Examples + ======== + + >>> from sympy import S, Interval + >>> S.EmptySet + EmptySet + + >>> Interval(1, 2).intersect(S.EmptySet) + EmptySet + + See Also + ======== + + UniversalSet + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Empty_set + """ + is_empty = True + is_finite_set = True + is_FiniteSet = True + + @property # type: ignore + @deprecated( + """ + The is_EmptySet attribute of Set objects is deprecated. + Use 's is S.EmptySet" or 's.is_empty' instead. + """, + deprecated_since_version="1.5", + active_deprecations_target="deprecated-is-emptyset", + ) + def is_EmptySet(self): + return True + + @property + def _measure(self): + return 0 + + def _contains(self, other): + return false + + def as_relational(self, symbol): + return false + + def __len__(self): + return 0 + + def __iter__(self): + return iter([]) + + def _eval_powerset(self): + return FiniteSet(self) + + @property + def _boundary(self): + return self + + def _complement(self, other): + return other + + def _kind(self): + return SetKind() + + def _symmetric_difference(self, other): + return other + + +class UniversalSet(Set, metaclass=Singleton): + """ + Represents the set of all things. + The universal set is available as a singleton as ``S.UniversalSet``. + + Examples + ======== + + >>> from sympy import S, Interval + >>> S.UniversalSet + UniversalSet + + >>> Interval(1, 2).intersect(S.UniversalSet) + Interval(1, 2) + + See Also + ======== + + EmptySet + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Universal_set + """ + + is_UniversalSet = True + is_empty = False + is_finite_set = False + + def _complement(self, other): + return S.EmptySet + + def _symmetric_difference(self, other): + return other + + @property + def _measure(self): + return S.Infinity + + def _kind(self): + return SetKind(UndefinedKind) + + def _contains(self, other): + return true + + def as_relational(self, symbol): + return true + + @property + def _boundary(self): + return S.EmptySet + + +class FiniteSet(Set): + """ + Represents a finite set of Sympy expressions. + + Examples + ======== + + >>> from sympy import FiniteSet, Symbol, Interval, Naturals0 + >>> FiniteSet(1, 2, 3, 4) + {1, 2, 3, 4} + >>> 3 in FiniteSet(1, 2, 3, 4) + True + >>> FiniteSet(1, (1, 2), Symbol('x')) + {1, x, (1, 2)} + >>> FiniteSet(Interval(1, 2), Naturals0, {1, 2}) + FiniteSet({1, 2}, Interval(1, 2), Naturals0) + >>> members = [1, 2, 3, 4] + >>> f = FiniteSet(*members) + >>> f + {1, 2, 3, 4} + >>> f - FiniteSet(2) + {1, 3, 4} + >>> f + FiniteSet(2, 5) + {1, 2, 3, 4, 5} + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Finite_set + """ + is_FiniteSet = True + is_iterable = True + is_empty = False + is_finite_set = True + + def __new__(cls, *args, **kwargs): + evaluate = kwargs.get('evaluate', global_parameters.evaluate) + if evaluate: + args = list(map(sympify, args)) + + if len(args) == 0: + return S.EmptySet + else: + args = list(map(sympify, args)) + + # keep the form of the first canonical arg + dargs = {} + for i in reversed(list(ordered(args))): + if i.is_Symbol: + dargs[i] = i + else: + try: + dargs[i.as_dummy()] = i + except TypeError: + # e.g. i = class without args like `Interval` + dargs[i] = i + _args_set = set(dargs.values()) + args = list(ordered(_args_set, Set._infimum_key)) + obj = Basic.__new__(cls, *args) + obj._args_set = _args_set + return obj + + + def __iter__(self): + return iter(self.args) + + def _complement(self, other): + if isinstance(other, Interval): + # Splitting in sub-intervals is only done for S.Reals; + # other cases that need splitting will first pass through + # Set._complement(). + nums, syms = [], [] + for m in self.args: + if m.is_number and m.is_real: + nums.append(m) + elif m.is_real == False: + pass # drop non-reals + else: + syms.append(m) # various symbolic expressions + if other == S.Reals and nums != []: + nums.sort() + intervals = [] # Build up a list of intervals between the elements + intervals += [Interval(S.NegativeInfinity, nums[0], True, True)] + for a, b in zip(nums[:-1], nums[1:]): + intervals.append(Interval(a, b, True, True)) # both open + intervals.append(Interval(nums[-1], S.Infinity, True, True)) + if syms != []: + return Complement(Union(*intervals, evaluate=False), + FiniteSet(*syms), evaluate=False) + else: + return Union(*intervals, evaluate=False) + elif nums == []: # no splitting necessary or possible: + if syms: + return Complement(other, FiniteSet(*syms), evaluate=False) + else: + return other + + elif isinstance(other, FiniteSet): + unk = [] + for i in self: + c = sympify(other.contains(i)) + if c is not S.true and c is not S.false: + unk.append(i) + unk = FiniteSet(*unk) + if unk == self: + return + not_true = [] + for i in other: + c = sympify(self.contains(i)) + if c is not S.true: + not_true.append(i) + return Complement(FiniteSet(*not_true), unk) + + return Set._complement(self, other) + + def _contains(self, other): + """ + Tests whether an element, other, is in the set. + + Explanation + =========== + + The actual test is for mathematical equality (as opposed to + syntactical equality). In the worst case all elements of the + set must be checked. + + Examples + ======== + + >>> from sympy import FiniteSet + >>> 1 in FiniteSet(1, 2) + True + >>> 5 in FiniteSet(1, 2) + False + + """ + if other in self._args_set: + return S.true + else: + # evaluate=True is needed to override evaluate=False context; + # we need Eq to do the evaluation + return Or(*[Eq(e, other, evaluate=True) for e in self.args]) + + def _eval_is_subset(self, other): + return fuzzy_and(other._contains(e) for e in self.args) + + @property + def _boundary(self): + return self + + @property + def _inf(self): + return Min(*self) + + @property + def _sup(self): + return Max(*self) + + @property + def measure(self): + return 0 + + def _kind(self): + if not self.args: + return SetKind() + elif all(i.kind == self.args[0].kind for i in self.args): + return SetKind(self.args[0].kind) + else: + return SetKind(UndefinedKind) + + def __len__(self): + return len(self.args) + + def as_relational(self, symbol): + """Rewrite a FiniteSet in terms of equalities and logic operators. """ + return Or(*[Eq(symbol, elem) for elem in self]) + + def compare(self, other): + return (hash(self) - hash(other)) + + def _eval_evalf(self, prec): + dps = prec_to_dps(prec) + return FiniteSet(*[elem.evalf(n=dps) for elem in self]) + + def _eval_simplify(self, **kwargs): + from sympy.simplify import simplify + return FiniteSet(*[simplify(elem, **kwargs) for elem in self]) + + @property + def _sorted_args(self): + return self.args + + def _eval_powerset(self): + return self.func(*[self.func(*s) for s in subsets(self.args)]) + + def _eval_rewrite_as_PowerSet(self, *args, **kwargs): + """Rewriting method for a finite set to a power set.""" + from .powerset import PowerSet + + is2pow = lambda n: bool(n and not n & (n - 1)) + if not is2pow(len(self)): + return None + + fs_test = lambda arg: isinstance(arg, Set) and arg.is_FiniteSet + if not all(fs_test(arg) for arg in args): + return None + + biggest = max(args, key=len) + for arg in subsets(biggest.args): + arg_set = FiniteSet(*arg) + if arg_set not in args: + return None + return PowerSet(biggest) + + def __ge__(self, other): + if not isinstance(other, Set): + raise TypeError("Invalid comparison of set with %s" % func_name(other)) + return other.is_subset(self) + + def __gt__(self, other): + if not isinstance(other, Set): + raise TypeError("Invalid comparison of set with %s" % func_name(other)) + return self.is_proper_superset(other) + + def __le__(self, other): + if not isinstance(other, Set): + raise TypeError("Invalid comparison of set with %s" % func_name(other)) + return self.is_subset(other) + + def __lt__(self, other): + if not isinstance(other, Set): + raise TypeError("Invalid comparison of set with %s" % func_name(other)) + return self.is_proper_subset(other) + + def __eq__(self, other): + if isinstance(other, (set, frozenset)): + return self._args_set == other + return super().__eq__(other) + + __hash__ : Callable[[Basic], Any] = Basic.__hash__ + +_sympy_converter[set] = lambda x: FiniteSet(*x) +_sympy_converter[frozenset] = lambda x: FiniteSet(*x) + + +class SymmetricDifference(Set): + """Represents the set of elements which are in either of the + sets and not in their intersection. + + Examples + ======== + + >>> from sympy import SymmetricDifference, FiniteSet + >>> SymmetricDifference(FiniteSet(1, 2, 3), FiniteSet(3, 4, 5)) + {1, 2, 4, 5} + + See Also + ======== + + Complement, Union + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Symmetric_difference + """ + + is_SymmetricDifference = True + + def __new__(cls, a, b, evaluate=True): + if evaluate: + return SymmetricDifference.reduce(a, b) + + return Basic.__new__(cls, a, b) + + @staticmethod + def reduce(A, B): + result = B._symmetric_difference(A) + if result is not None: + return result + else: + return SymmetricDifference(A, B, evaluate=False) + + def as_relational(self, symbol): + """Rewrite a symmetric_difference in terms of equalities and + logic operators""" + A, B = self.args + + A_rel = A.as_relational(symbol) + B_rel = B.as_relational(symbol) + + return Xor(A_rel, B_rel) + + @property + def is_iterable(self): + if all(arg.is_iterable for arg in self.args): + return True + + def __iter__(self): + + args = self.args + union = roundrobin(*(iter(arg) for arg in args)) + + for item in union: + count = 0 + for s in args: + if item in s: + count += 1 + + if count % 2 == 1: + yield item + + + +class DisjointUnion(Set): + """ Represents the disjoint union (also known as the external disjoint union) + of a finite number of sets. + + Examples + ======== + + >>> from sympy import DisjointUnion, FiniteSet, Interval, Union, Symbol + >>> A = FiniteSet(1, 2, 3) + >>> B = Interval(0, 5) + >>> DisjointUnion(A, B) + DisjointUnion({1, 2, 3}, Interval(0, 5)) + >>> DisjointUnion(A, B).rewrite(Union) + Union(ProductSet({1, 2, 3}, {0}), ProductSet(Interval(0, 5), {1})) + >>> C = FiniteSet(Symbol('x'), Symbol('y'), Symbol('z')) + >>> DisjointUnion(C, C) + DisjointUnion({x, y, z}, {x, y, z}) + >>> DisjointUnion(C, C).rewrite(Union) + ProductSet({x, y, z}, {0, 1}) + + References + ========== + + https://en.wikipedia.org/wiki/Disjoint_union + """ + + def __new__(cls, *sets): + dj_collection = [] + for set_i in sets: + if isinstance(set_i, Set): + dj_collection.append(set_i) + else: + raise TypeError("Invalid input: '%s', input args \ + to DisjointUnion must be Sets" % set_i) + obj = Basic.__new__(cls, *dj_collection) + return obj + + @property + def sets(self): + return self.args + + @property + def is_empty(self): + return fuzzy_and(s.is_empty for s in self.sets) + + @property + def is_finite_set(self): + all_finite = fuzzy_and(s.is_finite_set for s in self.sets) + return fuzzy_or([self.is_empty, all_finite]) + + @property + def is_iterable(self): + if self.is_empty: + return False + iter_flag = True + for set_i in self.sets: + if not set_i.is_empty: + iter_flag = iter_flag and set_i.is_iterable + return iter_flag + + def _eval_rewrite_as_Union(self, *sets, **kwargs): + """ + Rewrites the disjoint union as the union of (``set`` x {``i``}) + where ``set`` is the element in ``sets`` at index = ``i`` + """ + + dj_union = S.EmptySet + index = 0 + for set_i in sets: + if isinstance(set_i, Set): + cross = ProductSet(set_i, FiniteSet(index)) + dj_union = Union(dj_union, cross) + index = index + 1 + return dj_union + + def _contains(self, element): + """ + ``in`` operator for DisjointUnion + + Examples + ======== + + >>> from sympy import Interval, DisjointUnion + >>> D = DisjointUnion(Interval(0, 1), Interval(0, 2)) + >>> (0.5, 0) in D + True + >>> (0.5, 1) in D + True + >>> (1.5, 0) in D + False + >>> (1.5, 1) in D + True + + Passes operation on to constituent sets + """ + if not isinstance(element, Tuple) or len(element) != 2: + return S.false + + if not element[1].is_Integer: + return S.false + + if element[1] >= len(self.sets) or element[1] < 0: + return S.false + + return self.sets[element[1]]._contains(element[0]) + + def _kind(self): + if not self.args: + return SetKind() + elif all(i.kind == self.args[0].kind for i in self.args): + return self.args[0].kind + else: + return SetKind(UndefinedKind) + + def __iter__(self): + if self.is_iterable: + + iters = [] + for i, s in enumerate(self.sets): + iters.append(iproduct(s, {Integer(i)})) + + return iter(roundrobin(*iters)) + else: + raise ValueError("'%s' is not iterable." % self) + + def __len__(self): + """ + Returns the length of the disjoint union, i.e., the number of elements in the set. + + Examples + ======== + + >>> from sympy import FiniteSet, DisjointUnion, EmptySet + >>> D1 = DisjointUnion(FiniteSet(1, 2, 3, 4), EmptySet, FiniteSet(3, 4, 5)) + >>> len(D1) + 7 + >>> D2 = DisjointUnion(FiniteSet(3, 5, 7), EmptySet, FiniteSet(3, 5, 7)) + >>> len(D2) + 6 + >>> D3 = DisjointUnion(EmptySet, EmptySet) + >>> len(D3) + 0 + + Adds up the lengths of the constituent sets. + """ + + if self.is_finite_set: + size = 0 + for set in self.sets: + size += len(set) + return size + else: + raise ValueError("'%s' is not a finite set." % self) + + +def imageset(*args): + r""" + Return an image of the set under transformation ``f``. + + Explanation + =========== + + If this function cannot compute the image, it returns an + unevaluated ImageSet object. + + .. math:: + \{ f(x) \mid x \in \mathrm{self} \} + + Examples + ======== + + >>> from sympy import S, Interval, imageset, sin, Lambda + >>> from sympy.abc import x + + >>> imageset(x, 2*x, Interval(0, 2)) + Interval(0, 4) + + >>> imageset(lambda x: 2*x, Interval(0, 2)) + Interval(0, 4) + + >>> imageset(Lambda(x, sin(x)), Interval(-2, 1)) + ImageSet(Lambda(x, sin(x)), Interval(-2, 1)) + + >>> imageset(sin, Interval(-2, 1)) + ImageSet(Lambda(x, sin(x)), Interval(-2, 1)) + >>> imageset(lambda y: x + y, Interval(-2, 1)) + ImageSet(Lambda(y, x + y), Interval(-2, 1)) + + Expressions applied to the set of Integers are simplified + to show as few negatives as possible and linear expressions + are converted to a canonical form. If this is not desirable + then the unevaluated ImageSet should be used. + + >>> imageset(x, -2*x + 5, S.Integers) + ImageSet(Lambda(x, 2*x + 1), Integers) + + See Also + ======== + + sympy.sets.fancysets.ImageSet + + """ + from .fancysets import ImageSet + from .setexpr import set_function + + if len(args) < 2: + raise ValueError('imageset expects at least 2 args, got: %s' % len(args)) + + if isinstance(args[0], (Symbol, tuple)) and len(args) > 2: + f = Lambda(args[0], args[1]) + set_list = args[2:] + else: + f = args[0] + set_list = args[1:] + + if isinstance(f, Lambda): + pass + elif callable(f): + nargs = getattr(f, 'nargs', {}) + if nargs: + if len(nargs) != 1: + raise NotImplementedError(filldedent(''' + This function can take more than 1 arg + but the potentially complicated set input + has not been analyzed at this point to + know its dimensions. TODO + ''')) + N = nargs.args[0] + if N == 1: + s = 'x' + else: + s = [Symbol('x%i' % i) for i in range(1, N + 1)] + else: + s = inspect.signature(f).parameters + + dexpr = _sympify(f(*[Dummy() for i in s])) + var = tuple(uniquely_named_symbol( + Symbol(i), dexpr) for i in s) + f = Lambda(var, f(*var)) + else: + raise TypeError(filldedent(''' + expecting lambda, Lambda, or FunctionClass, + not \'%s\'.''' % func_name(f))) + + if any(not isinstance(s, Set) for s in set_list): + name = [func_name(s) for s in set_list] + raise ValueError( + 'arguments after mapping should be sets, not %s' % name) + + if len(set_list) == 1: + set = set_list[0] + try: + # TypeError if arg count != set dimensions + r = set_function(f, set) + if r is None: + raise TypeError + if not r: + return r + except TypeError: + r = ImageSet(f, set) + if isinstance(r, ImageSet): + f, set = r.args + + if f.variables[0] == f.expr: + return set + + if isinstance(set, ImageSet): + # XXX: Maybe this should just be: + # f2 = set.lambda + # fun = Lambda(f2.signature, f(*f2.expr)) + # return imageset(fun, *set.base_sets) + if len(set.lamda.variables) == 1 and len(f.variables) == 1: + x = set.lamda.variables[0] + y = f.variables[0] + return imageset( + Lambda(x, f.expr.subs(y, set.lamda.expr)), *set.base_sets) + + if r is not None: + return r + + return ImageSet(f, *set_list) + + +def is_function_invertible_in_set(func, setv): + """ + Checks whether function ``func`` is invertible when the domain is + restricted to set ``setv``. + """ + # Functions known to always be invertible: + if func in (exp, log): + return True + u = Dummy("u") + fdiff = func(u).diff(u) + # monotonous functions: + # TODO: check subsets (`func` in `setv`) + if (fdiff > 0) == True or (fdiff < 0) == True: + return True + # TODO: support more + return None + + +def simplify_union(args): + """ + Simplify a :class:`Union` using known rules. + + Explanation + =========== + + We first start with global rules like 'Merge all FiniteSets' + + Then we iterate through all pairs and ask the constituent sets if they + can simplify themselves with any other constituent. This process depends + on ``union_sets(a, b)`` functions. + """ + from sympy.sets.handlers.union import union_sets + + # ===== Global Rules ===== + if not args: + return S.EmptySet + + for arg in args: + if not isinstance(arg, Set): + raise TypeError("Input args to Union must be Sets") + + # Merge all finite sets + finite_sets = [x for x in args if x.is_FiniteSet] + if len(finite_sets) > 1: + a = (x for set in finite_sets for x in set) + finite_set = FiniteSet(*a) + args = [finite_set] + [x for x in args if not x.is_FiniteSet] + + # ===== Pair-wise Rules ===== + # Here we depend on rules built into the constituent sets + args = set(args) + new_args = True + while new_args: + for s in args: + new_args = False + for t in args - {s}: + new_set = union_sets(s, t) + # This returns None if s does not know how to intersect + # with t. Returns the newly intersected set otherwise + if new_set is not None: + if not isinstance(new_set, set): + new_set = {new_set} + new_args = (args - {s, t}).union(new_set) + break + if new_args: + args = new_args + break + + if len(args) == 1: + return args.pop() + else: + return Union(*args, evaluate=False) + + +def simplify_intersection(args): + """ + Simplify an intersection using known rules. + + Explanation + =========== + + We first start with global rules like + 'if any empty sets return empty set' and 'distribute any unions' + + Then we iterate through all pairs and ask the constituent sets if they + can simplify themselves with any other constituent + """ + + # ===== Global Rules ===== + if not args: + return S.UniversalSet + + for arg in args: + if not isinstance(arg, Set): + raise TypeError("Input args to Union must be Sets") + + # If any EmptySets return EmptySet + if S.EmptySet in args: + return S.EmptySet + + # Handle Finite sets + rv = Intersection._handle_finite_sets(args) + + if rv is not None: + return rv + + # If any of the sets are unions, return a Union of Intersections + for s in args: + if s.is_Union: + other_sets = set(args) - {s} + if len(other_sets) > 0: + other = Intersection(*other_sets) + return Union(*(Intersection(arg, other) for arg in s.args)) + else: + return Union(*s.args) + + for s in args: + if s.is_Complement: + args.remove(s) + other_sets = args + [s.args[0]] + return Complement(Intersection(*other_sets), s.args[1]) + + from sympy.sets.handlers.intersection import intersection_sets + + # At this stage we are guaranteed not to have any + # EmptySets, FiniteSets, or Unions in the intersection + + # ===== Pair-wise Rules ===== + # Here we depend on rules built into the constituent sets + args = set(args) + new_args = True + while new_args: + for s in args: + new_args = False + for t in args - {s}: + new_set = intersection_sets(s, t) + # This returns None if s does not know how to intersect + # with t. Returns the newly intersected set otherwise + + if new_set is not None: + new_args = (args - {s, t}).union({new_set}) + break + if new_args: + args = new_args + break + + if len(args) == 1: + return args.pop() + else: + return Intersection(*args, evaluate=False) + + +def _handle_finite_sets(op, x, y, commutative): + # Handle finite sets: + fs_args, other = sift([x, y], lambda x: isinstance(x, FiniteSet), binary=True) + if len(fs_args) == 2: + return FiniteSet(*[op(i, j) for i in fs_args[0] for j in fs_args[1]]) + elif len(fs_args) == 1: + sets = [_apply_operation(op, other[0], i, commutative) for i in fs_args[0]] + return Union(*sets) + else: + return None + + +def _apply_operation(op, x, y, commutative): + from .fancysets import ImageSet + d = Dummy('d') + + out = _handle_finite_sets(op, x, y, commutative) + if out is None: + out = op(x, y) + + if out is None and commutative: + out = op(y, x) + if out is None: + _x, _y = symbols("x y") + if isinstance(x, Set) and not isinstance(y, Set): + out = ImageSet(Lambda(d, op(d, y)), x).doit() + elif not isinstance(x, Set) and isinstance(y, Set): + out = ImageSet(Lambda(d, op(x, d)), y).doit() + else: + out = ImageSet(Lambda((_x, _y), op(_x, _y)), x, y) + return out + + +def set_add(x, y): + from sympy.sets.handlers.add import _set_add + return _apply_operation(_set_add, x, y, commutative=True) + + +def set_sub(x, y): + from sympy.sets.handlers.add import _set_sub + return _apply_operation(_set_sub, x, y, commutative=False) + + +def set_mul(x, y): + from sympy.sets.handlers.mul import _set_mul + return _apply_operation(_set_mul, x, y, commutative=True) + + +def set_div(x, y): + from sympy.sets.handlers.mul import _set_div + return _apply_operation(_set_div, x, y, commutative=False) + + +def set_pow(x, y): + from sympy.sets.handlers.power import _set_pow + return _apply_operation(_set_pow, x, y, commutative=False) + + +def set_function(f, x): + from sympy.sets.handlers.functions import _set_function + return _set_function(f, x) + + +class SetKind(Kind): + """ + SetKind is kind for all Sets + + Every instance of Set will have kind ``SetKind`` parametrised by the kind + of the elements of the ``Set``. The kind of the elements might be + ``NumberKind``, or ``TupleKind`` or something else. When not all elements + have the same kind then the kind of the elements will be given as + ``UndefinedKind``. + + Parameters + ========== + + element_kind: Kind (optional) + The kind of the elements of the set. In a well defined set all elements + will have the same kind. Otherwise the kind should + :class:`sympy.core.kind.UndefinedKind`. The ``element_kind`` argument is optional but + should only be omitted in the case of ``EmptySet`` whose kind is simply + ``SetKind()`` + + Examples + ======== + + >>> from sympy import Interval + >>> Interval(1, 2).kind + SetKind(NumberKind) + >>> Interval(1,2).kind.element_kind + NumberKind + + See Also + ======== + + sympy.core.kind.NumberKind + sympy.matrices.kind.MatrixKind + sympy.core.containers.TupleKind + """ + def __new__(cls, element_kind=None): + obj = super().__new__(cls, element_kind) + obj.element_kind = element_kind + return obj + + def __repr__(self): + if not self.element_kind: + return "SetKind()" + else: + return "SetKind(%s)" % self.element_kind diff --git a/.venv/lib/python3.12/site-packages/sympy/sets/tests/__init__.py b/.venv/lib/python3.12/site-packages/sympy/sets/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git 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b/.venv/lib/python3.12/site-packages/sympy/sets/tests/test_conditionset.py new file mode 100644 index 0000000000000000000000000000000000000000..4818246f306afd46a09a2cbea1faab858a9e7806 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/sets/tests/test_conditionset.py @@ -0,0 +1,294 @@ +from sympy.core.expr import unchanged +from sympy.sets import (ConditionSet, Intersection, FiniteSet, + EmptySet, Union, Contains, ImageSet) +from sympy.sets.sets import SetKind +from sympy.core.function import (Function, Lambda) +from sympy.core.mod import Mod +from sympy.core.kind import NumberKind +from sympy.core.numbers import (oo, pi) +from sympy.core.relational import (Eq, Ne) +from sympy.core.singleton import S +from sympy.core.symbol import (Symbol, symbols) +from sympy.functions.elementary.complexes import Abs +from sympy.functions.elementary.trigonometric import (asin, sin) +from sympy.logic.boolalg import And +from sympy.matrices.dense import Matrix +from sympy.matrices.expressions.matexpr import MatrixSymbol +from sympy.sets.sets import Interval +from sympy.testing.pytest import raises, warns_deprecated_sympy + + +w = Symbol('w') +x = Symbol('x') +y = Symbol('y') +z = Symbol('z') +f = Function('f') + + +def test_CondSet(): + sin_sols_principal = ConditionSet(x, Eq(sin(x), 0), + Interval(0, 2*pi, False, True)) + assert pi in sin_sols_principal + assert pi/2 not in sin_sols_principal + assert 3*pi not in sin_sols_principal + assert oo not in sin_sols_principal + assert 5 in ConditionSet(x, x**2 > 4, S.Reals) + assert 1 not in ConditionSet(x, x**2 > 4, S.Reals) + # in this case, 0 is not part of the base set so + # it can't be in any subset selected by the condition + assert 0 not in ConditionSet(x, y > 5, Interval(1, 7)) + # since 'in' requires a true/false, the following raises + # an error because the given value provides no information + # for the condition to evaluate (since the condition does + # not depend on the dummy symbol): the result is `y > 5`. + # In this case, ConditionSet is just acting like + # Piecewise((Interval(1, 7), y > 5), (S.EmptySet, True)). + raises(TypeError, lambda: 6 in ConditionSet(x, y > 5, + Interval(1, 7))) + + X = MatrixSymbol('X', 2, 2) + matrix_set = ConditionSet(X, Eq(X*Matrix([[1, 1], [1, 1]]), X)) + Y = Matrix([[0, 0], [0, 0]]) + assert matrix_set.contains(Y).doit() is S.true + Z = Matrix([[1, 2], [3, 4]]) + assert matrix_set.contains(Z).doit() is S.false + + assert isinstance(ConditionSet(x, x < 1, {x, y}).base_set, + FiniteSet) + raises(TypeError, lambda: ConditionSet(x, x + 1, {x, y})) + raises(TypeError, lambda: ConditionSet(x, x, 1)) + + I = S.Integers + U = S.UniversalSet + C = ConditionSet + assert C(x, False, I) is S.EmptySet + assert C(x, True, I) is I + assert C(x, x < 1, C(x, x < 2, I) + ) == C(x, (x < 1) & (x < 2), I) + assert C(y, y < 1, C(x, y < 2, I) + ) == C(x, (x < 1) & (y < 2), I), C(y, y < 1, C(x, y < 2, I)) + assert C(y, y < 1, C(x, x < 2, I) + ) == C(y, (y < 1) & (y < 2), I) + assert C(y, y < 1, C(x, y < x, I) + ) == C(x, (x < 1) & (y < x), I) + assert unchanged(C, y, x < 1, C(x, y < x, I)) + assert ConditionSet(x, x < 1).base_set is U + # arg checking is not done at instantiation but this + # will raise an error when containment is tested + assert ConditionSet((x,), x < 1).base_set is U + + c = ConditionSet((x, y), x < y, I**2) + assert (1, 2) in c + assert (1, pi) not in c + + raises(TypeError, lambda: C(x, x > 1, C((x, y), x > 1, I**2))) + # signature mismatch since only 3 args are accepted + raises(TypeError, lambda: C((x, y), x + y < 2, U, U)) + + +def test_CondSet_intersect(): + input_conditionset = ConditionSet(x, x**2 > 4, Interval(1, 4, False, + False)) + other_domain = Interval(0, 3, False, False) + output_conditionset = ConditionSet(x, x**2 > 4, Interval( + 1, 3, False, False)) + assert Intersection(input_conditionset, other_domain + ) == output_conditionset + + +def test_issue_9849(): + assert ConditionSet(x, Eq(x, x), S.Naturals + ) is S.Naturals + assert ConditionSet(x, Eq(Abs(sin(x)), -1), S.Naturals + ) == S.EmptySet + + +def test_simplified_FiniteSet_in_CondSet(): + assert ConditionSet(x, And(x < 1, x > -3), FiniteSet(0, 1, 2) + ) == FiniteSet(0) + assert ConditionSet(x, x < 0, FiniteSet(0, 1, 2)) == EmptySet + assert ConditionSet(x, And(x < -3), EmptySet) == EmptySet + y = Symbol('y') + assert (ConditionSet(x, And(x > 0), FiniteSet(-1, 0, 1, y)) == + Union(FiniteSet(1), ConditionSet(x, And(x > 0), FiniteSet(y)))) + assert (ConditionSet(x, Eq(Mod(x, 3), 1), FiniteSet(1, 4, 2, y)) == + Union(FiniteSet(1, 4), ConditionSet(x, Eq(Mod(x, 3), 1), + FiniteSet(y)))) + + +def test_free_symbols(): + assert ConditionSet(x, Eq(y, 0), FiniteSet(z) + ).free_symbols == {y, z} + assert ConditionSet(x, Eq(x, 0), FiniteSet(z) + ).free_symbols == {z} + assert ConditionSet(x, Eq(x, 0), FiniteSet(x, z) + ).free_symbols == {x, z} + assert ConditionSet(x, Eq(x, 0), ImageSet(Lambda(y, y**2), + S.Integers)).free_symbols == set() + + +def test_bound_symbols(): + assert ConditionSet(x, Eq(y, 0), FiniteSet(z) + ).bound_symbols == [x] + assert ConditionSet(x, Eq(x, 0), FiniteSet(x, y) + ).bound_symbols == [x] + assert ConditionSet(x, x < 10, ImageSet(Lambda(y, y**2), S.Integers) + ).bound_symbols == [x] + assert ConditionSet(x, x < 10, ConditionSet(y, y > 1, S.Integers) + ).bound_symbols == [x] + + +def test_as_dummy(): + _0, _1 = symbols('_0 _1') + assert ConditionSet(x, x < 1, Interval(y, oo) + ).as_dummy() == ConditionSet(_0, _0 < 1, Interval(y, oo)) + assert ConditionSet(x, x < 1, Interval(x, oo) + ).as_dummy() == ConditionSet(_0, _0 < 1, Interval(x, oo)) + assert ConditionSet(x, x < 1, ImageSet(Lambda(y, y**2), S.Integers) + ).as_dummy() == ConditionSet( + _0, _0 < 1, ImageSet(Lambda(_0, _0**2), S.Integers)) + e = ConditionSet((x, y), x <= y, S.Reals**2) + assert e.bound_symbols == [x, y] + assert e.as_dummy() == ConditionSet((_0, _1), _0 <= _1, S.Reals**2) + assert e.as_dummy() == ConditionSet((y, x), y <= x, S.Reals**2 + ).as_dummy() + + +def test_subs_CondSet(): + s = FiniteSet(z, y) + c = ConditionSet(x, x < 2, s) + assert c.subs(x, y) == c + assert c.subs(z, y) == ConditionSet(x, x < 2, FiniteSet(y)) + assert c.xreplace({x: y}) == ConditionSet(y, y < 2, s) + + assert ConditionSet(x, x < y, s + ).subs(y, w) == ConditionSet(x, x < w, s.subs(y, w)) + # if the user uses assumptions that cause the condition + # to evaluate, that can't be helped from SymPy's end + n = Symbol('n', negative=True) + assert ConditionSet(n, 0 < n, S.Integers) is S.EmptySet + p = Symbol('p', positive=True) + assert ConditionSet(n, n < y, S.Integers + ).subs(n, x) == ConditionSet(n, n < y, S.Integers) + raises(ValueError, lambda: ConditionSet( + x + 1, x < 1, S.Integers)) + assert ConditionSet( + p, n < x, Interval(-5, 5)).subs(x, p) == Interval(-5, 5), ConditionSet( + p, n < x, Interval(-5, 5)).subs(x, p) + assert ConditionSet( + n, n < x, Interval(-oo, 0)).subs(x, p + ) == Interval(-oo, 0) + + assert ConditionSet(f(x), f(x) < 1, {w, z} + ).subs(f(x), y) == ConditionSet(f(x), f(x) < 1, {w, z}) + + # issue 17341 + k = Symbol('k') + img1 = ImageSet(Lambda(k, 2*k*pi + asin(y)), S.Integers) + img2 = ImageSet(Lambda(k, 2*k*pi + asin(S.One/3)), S.Integers) + assert ConditionSet(x, Contains( + y, Interval(-1,1)), img1).subs(y, S.One/3).dummy_eq(img2) + + assert (0, 1) in ConditionSet((x, y), x + y < 3, S.Integers**2) + + raises(TypeError, lambda: ConditionSet(n, n < -10, Interval(0, 10))) + + +def test_subs_CondSet_tebr(): + with warns_deprecated_sympy(): + assert ConditionSet((x, y), {x + 1, x + y}, S.Reals**2) == \ + ConditionSet((x, y), Eq(x + 1, 0) & Eq(x + y, 0), S.Reals**2) + + +def test_dummy_eq(): + C = ConditionSet + I = S.Integers + c = C(x, x < 1, I) + assert c.dummy_eq(C(y, y < 1, I)) + assert c.dummy_eq(1) == False + assert c.dummy_eq(C(x, x < 1, S.Reals)) == False + + c1 = ConditionSet((x, y), Eq(x + 1, 0) & Eq(x + y, 0), S.Reals**2) + c2 = ConditionSet((x, y), Eq(x + 1, 0) & Eq(x + y, 0), S.Reals**2) + c3 = ConditionSet((x, y), Eq(x + 1, 0) & Eq(x + y, 0), S.Complexes**2) + assert c1.dummy_eq(c2) + assert c1.dummy_eq(c3) is False + assert c.dummy_eq(c1) is False + assert c1.dummy_eq(c) is False + + # issue 19496 + m = Symbol('m') + n = Symbol('n') + a = Symbol('a') + d1 = ImageSet(Lambda(m, m*pi), S.Integers) + d2 = ImageSet(Lambda(n, n*pi), S.Integers) + c1 = ConditionSet(x, Ne(a, 0), d1) + c2 = ConditionSet(x, Ne(a, 0), d2) + assert c1.dummy_eq(c2) + + +def test_contains(): + assert 6 in ConditionSet(x, x > 5, Interval(1, 7)) + assert (8 in ConditionSet(x, y > 5, Interval(1, 7))) is False + # `in` should give True or False; in this case there is not + # enough information for that result + raises(TypeError, + lambda: 6 in ConditionSet(x, y > 5, Interval(1, 7))) + # here, there is enough information but the comparison is + # not defined + raises(TypeError, lambda: 0 in ConditionSet(x, 1/x >= 0, S.Reals)) + assert ConditionSet(x, y > 5, Interval(1, 7) + ).contains(6) == (y > 5) + assert ConditionSet(x, y > 5, Interval(1, 7) + ).contains(8) is S.false + assert ConditionSet(x, y > 5, Interval(1, 7) + ).contains(w) == And(Contains(w, Interval(1, 7)), y > 5) + # This returns an unevaluated Contains object + # because 1/0 should not be defined for 1 and 0 in the context of + # reals. + assert ConditionSet(x, 1/x >= 0, S.Reals).contains(0) == \ + Contains(0, ConditionSet(x, 1/x >= 0, S.Reals), evaluate=False) + c = ConditionSet((x, y), x + y > 1, S.Integers**2) + assert not c.contains(1) + assert c.contains((2, 1)) + assert not c.contains((0, 1)) + c = ConditionSet((w, (x, y)), w + x + y > 1, S.Integers*S.Integers**2) + assert not c.contains(1) + assert not c.contains((1, 2)) + assert not c.contains(((1, 2), 3)) + assert not c.contains(((1, 2), (3, 4))) + assert c.contains((1, (3, 4))) + + +def test_as_relational(): + assert ConditionSet((x, y), x > 1, S.Integers**2).as_relational((x, y) + ) == (x > 1) & Contains(x, S.Integers) & Contains(y, S.Integers) + assert ConditionSet(x, x > 1, S.Integers).as_relational(x + ) == Contains(x, S.Integers) & (x > 1) + + +def test_flatten(): + """Tests whether there is basic denesting functionality""" + inner = ConditionSet(x, sin(x) + x > 0) + outer = ConditionSet(x, Contains(x, inner), S.Reals) + assert outer == ConditionSet(x, sin(x) + x > 0, S.Reals) + + inner = ConditionSet(y, sin(y) + y > 0) + outer = ConditionSet(x, Contains(y, inner), S.Reals) + assert outer != ConditionSet(x, sin(x) + x > 0, S.Reals) + + inner = ConditionSet(x, sin(x) + x > 0).intersect(Interval(-1, 1)) + outer = ConditionSet(x, Contains(x, inner), S.Reals) + assert outer == ConditionSet(x, sin(x) + x > 0, Interval(-1, 1)) + + +def test_duplicate(): + from sympy.core.function import BadSignatureError + # test coverage for line 95 in conditionset.py, check for duplicates in symbols + dup = symbols('a,a') + raises(BadSignatureError, lambda: ConditionSet(dup, x < 0)) + + +def test_SetKind_ConditionSet(): + assert ConditionSet(x, Eq(sin(x), 0), Interval(0, 2*pi)).kind is SetKind(NumberKind) + assert ConditionSet(x, x < 0).kind is SetKind(NumberKind) diff --git a/.venv/lib/python3.12/site-packages/sympy/sets/tests/test_contains.py b/.venv/lib/python3.12/site-packages/sympy/sets/tests/test_contains.py new file mode 100644 index 0000000000000000000000000000000000000000..bb6b98940946f98bf377aad6810f5b32eb6dd069 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/sets/tests/test_contains.py @@ -0,0 +1,52 @@ +from sympy.core.expr import unchanged +from sympy.core.numbers import oo +from sympy.core.relational import Eq +from sympy.core.singleton import S +from sympy.core.symbol import Symbol +from sympy.sets.contains import Contains +from sympy.sets.sets import (FiniteSet, Interval) +from sympy.testing.pytest import raises + + +def test_contains_basic(): + raises(TypeError, lambda: Contains(S.Integers, 1)) + assert Contains(2, S.Integers) is S.true + assert Contains(-2, S.Naturals) is S.false + + i = Symbol('i', integer=True) + assert Contains(i, S.Naturals) == Contains(i, S.Naturals, evaluate=False) + + +def test_issue_6194(): + x = Symbol('x') + assert unchanged(Contains, x, Interval(0, 1)) + assert Interval(0, 1).contains(x) == (S.Zero <= x) & (x <= 1) + assert Contains(x, FiniteSet(0)) != S.false + assert Contains(x, Interval(1, 1)) != S.false + assert Contains(x, S.Integers) != S.false + + +def test_issue_10326(): + assert Contains(oo, Interval(-oo, oo)) == False + assert Contains(-oo, Interval(-oo, oo)) == False + + +def test_binary_symbols(): + x = Symbol('x') + y = Symbol('y') + z = Symbol('z') + assert Contains(x, FiniteSet(y, Eq(z, True)) + ).binary_symbols == {y, z} + + +def test_as_set(): + x = Symbol('x') + y = Symbol('y') + assert Contains(x, FiniteSet(y)).as_set() == FiniteSet(y) + assert Contains(x, S.Integers).as_set() == S.Integers + assert Contains(x, S.Reals).as_set() == S.Reals + + +def test_type_error(): + # Pass in a parameter not of type "set" + raises(TypeError, lambda: Contains(2, None)) diff --git a/.venv/lib/python3.12/site-packages/sympy/sets/tests/test_fancysets.py b/.venv/lib/python3.12/site-packages/sympy/sets/tests/test_fancysets.py new file mode 100644 index 0000000000000000000000000000000000000000..b23c2a99fce0af5bfe7c667185465ee417de19ce --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/sets/tests/test_fancysets.py @@ -0,0 +1,1313 @@ + +from sympy.core.expr import unchanged +from sympy.sets.contains import Contains +from sympy.sets.fancysets import (ImageSet, Range, normalize_theta_set, + ComplexRegion) +from sympy.sets.sets import (FiniteSet, Interval, Union, imageset, + Intersection, ProductSet, SetKind) +from sympy.sets.conditionset import ConditionSet +from sympy.simplify.simplify import simplify +from sympy.core.basic import Basic +from sympy.core.containers import Tuple, TupleKind +from sympy.core.function import Lambda +from sympy.core.kind import NumberKind +from sympy.core.numbers import (I, Rational, oo, pi) +from sympy.core.relational import Eq +from sympy.core.singleton import S +from sympy.core.symbol import (Dummy, Symbol, symbols) +from sympy.functions.elementary.complexes import Abs +from sympy.functions.elementary.exponential import (exp, log) +from sympy.functions.elementary.integers import floor +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import (cos, sin, tan) +from sympy.logic.boolalg import And +from sympy.matrices.dense import eye +from sympy.testing.pytest import XFAIL, raises +from sympy.abc import x, y, t, z +from sympy.core.mod import Mod + +import itertools + + +def test_naturals(): + N = S.Naturals + assert 5 in N + assert -5 not in N + assert 5.5 not in N + ni = iter(N) + a, b, c, d = next(ni), next(ni), next(ni), next(ni) + assert (a, b, c, d) == (1, 2, 3, 4) + assert isinstance(a, Basic) + + assert N.intersect(Interval(-5, 5)) == Range(1, 6) + assert N.intersect(Interval(-5, 5, True, True)) == Range(1, 5) + + assert N.boundary == N + assert N.is_open == False + assert N.is_closed == True + + assert N.inf == 1 + assert N.sup is oo + assert not N.contains(oo) + for s in (S.Naturals0, S.Naturals): + assert s.intersection(S.Reals) is s + assert s.is_subset(S.Reals) + + assert N.as_relational(x) == And(Eq(floor(x), x), x >= 1, x < oo) + + +def test_naturals0(): + N = S.Naturals0 + assert 0 in N + assert -1 not in N + assert next(iter(N)) == 0 + assert not N.contains(oo) + assert N.contains(sin(x)) == Contains(sin(x), N) + + +def test_integers(): + Z = S.Integers + assert 5 in Z + assert -5 in Z + assert 5.5 not in Z + assert not Z.contains(oo) + assert not Z.contains(-oo) + + zi = iter(Z) + a, b, c, d = next(zi), next(zi), next(zi), next(zi) + assert (a, b, c, d) == (0, 1, -1, 2) + assert isinstance(a, Basic) + + assert Z.intersect(Interval(-5, 5)) == Range(-5, 6) + assert Z.intersect(Interval(-5, 5, True, True)) == Range(-4, 5) + assert Z.intersect(Interval(5, S.Infinity)) == Range(5, S.Infinity) + assert Z.intersect(Interval.Lopen(5, S.Infinity)) == Range(6, S.Infinity) + + assert Z.inf is -oo + assert Z.sup is oo + + assert Z.boundary == Z + assert Z.is_open == False + assert Z.is_closed == True + + assert Z.as_relational(x) == And(Eq(floor(x), x), -oo < x, x < oo) + + +def test_ImageSet(): + raises(ValueError, lambda: ImageSet(x, S.Integers)) + assert ImageSet(Lambda(x, 1), S.Integers) == FiniteSet(1) + assert ImageSet(Lambda(x, y), S.Integers) == {y} + assert ImageSet(Lambda(x, 1), S.EmptySet) == S.EmptySet + empty = Intersection(FiniteSet(log(2)/pi), S.Integers) + assert unchanged(ImageSet, Lambda(x, 1), empty) # issue #17471 + squares = ImageSet(Lambda(x, x**2), S.Naturals) + assert 4 in squares + assert 5 not in squares + assert FiniteSet(*range(10)).intersect(squares) == FiniteSet(1, 4, 9) + + assert 16 not in squares.intersect(Interval(0, 10)) + + si = iter(squares) + a, b, c, d = next(si), next(si), next(si), next(si) + assert (a, b, c, d) == (1, 4, 9, 16) + + harmonics = ImageSet(Lambda(x, 1/x), S.Naturals) + assert Rational(1, 5) in harmonics + assert Rational(.25) in harmonics + assert harmonics.contains(.25) == Contains( + 0.25, ImageSet(Lambda(x, 1/x), S.Naturals), evaluate=False) + assert Rational(.3) not in harmonics + assert (1, 2) not in harmonics + + assert harmonics.is_iterable + + assert imageset(x, -x, Interval(0, 1)) == Interval(-1, 0) + + assert ImageSet(Lambda(x, x**2), Interval(0, 2)).doit() == Interval(0, 4) + assert ImageSet(Lambda((x, y), 2*x), {4}, {3}).doit() == FiniteSet(8) + assert (ImageSet(Lambda((x, y), x+y), {1, 2, 3}, {10, 20, 30}).doit() == + FiniteSet(11, 12, 13, 21, 22, 23, 31, 32, 33)) + + c = Interval(1, 3) * Interval(1, 3) + assert Tuple(2, 6) in ImageSet(Lambda(((x, y),), (x, 2*y)), c) + assert Tuple(2, S.Half) in ImageSet(Lambda(((x, y),), (x, 1/y)), c) + assert Tuple(2, -2) not in ImageSet(Lambda(((x, y),), (x, y**2)), c) + assert Tuple(2, -2) in ImageSet(Lambda(((x, y),), (x, -2)), c) + c3 = ProductSet(Interval(3, 7), Interval(8, 11), Interval(5, 9)) + assert Tuple(8, 3, 9) in ImageSet(Lambda(((t, y, x),), (y, t, x)), c3) + assert Tuple(Rational(1, 8), 3, 9) in ImageSet(Lambda(((t, y, x),), (1/y, t, x)), c3) + assert 2/pi not in ImageSet(Lambda(((x, y),), 2/x), c) + assert 2/S(100) not in ImageSet(Lambda(((x, y),), 2/x), c) + assert Rational(2, 3) in ImageSet(Lambda(((x, y),), 2/x), c) + + S1 = imageset(lambda x, y: x + y, S.Integers, S.Naturals) + assert S1.base_pset == ProductSet(S.Integers, S.Naturals) + assert S1.base_sets == (S.Integers, S.Naturals) + + # Passing a set instead of a FiniteSet shouldn't raise + assert unchanged(ImageSet, Lambda(x, x**2), {1, 2, 3}) + + S2 = ImageSet(Lambda(((x, y),), x+y), {(1, 2), (3, 4)}) + assert 3 in S2.doit() + # FIXME: This doesn't yet work: + #assert 3 in S2 + assert S2._contains(3) is None + + raises(TypeError, lambda: ImageSet(Lambda(x, x**2), 1)) + + +def test_image_is_ImageSet(): + assert isinstance(imageset(x, sqrt(sin(x)), Range(5)), ImageSet) + + +def test_halfcircle(): + r, th = symbols('r, theta', real=True) + L = Lambda(((r, th),), (r*cos(th), r*sin(th))) + halfcircle = ImageSet(L, Interval(0, 1)*Interval(0, pi)) + + assert (1, 0) in halfcircle + assert (0, -1) not in halfcircle + assert (0, 0) in halfcircle + assert halfcircle._contains((r, 0)) is None + assert not halfcircle.is_iterable + + +@XFAIL +def test_halfcircle_fail(): + r, th = symbols('r, theta', real=True) + L = Lambda(((r, th),), (r*cos(th), r*sin(th))) + halfcircle = ImageSet(L, Interval(0, 1)*Interval(0, pi)) + assert (r, 2*pi) not in halfcircle + + +def test_ImageSet_iterator_not_injective(): + L = Lambda(x, x - x % 2) # produces 0, 2, 2, 4, 4, 6, 6, ... + evens = ImageSet(L, S.Naturals) + i = iter(evens) + # No repeats here + assert (next(i), next(i), next(i), next(i)) == (0, 2, 4, 6) + + +def test_inf_Range_len(): + raises(ValueError, lambda: len(Range(0, oo, 2))) + assert Range(0, oo, 2).size is S.Infinity + assert Range(0, -oo, -2).size is S.Infinity + assert Range(oo, 0, -2).size is S.Infinity + assert Range(-oo, 0, 2).size is S.Infinity + + +def test_Range_set(): + empty = Range(0) + + assert Range(5) == Range(0, 5) == Range(0, 5, 1) + + r = Range(10, 20, 2) + assert 12 in r + assert 8 not in r + assert 11 not in r + assert 30 not in r + + assert list(Range(0, 5)) == list(range(5)) + assert list(Range(5, 0, -1)) == list(range(5, 0, -1)) + + + assert Range(5, 15).sup == 14 + assert Range(5, 15).inf == 5 + assert Range(15, 5, -1).sup == 15 + assert Range(15, 5, -1).inf == 6 + assert Range(10, 67, 10).sup == 60 + assert Range(60, 7, -10).inf == 10 + + assert len(Range(10, 38, 10)) == 3 + + assert Range(0, 0, 5) == empty + assert Range(oo, oo, 1) == empty + assert Range(oo, 1, 1) == empty + assert Range(-oo, 1, -1) == empty + assert Range(1, oo, -1) == empty + assert Range(1, -oo, 1) == empty + assert Range(1, -4, oo) == empty + ip = symbols('ip', positive=True) + assert Range(0, ip, -1) == empty + assert Range(0, -ip, 1) == empty + assert Range(1, -4, -oo) == Range(1, 2) + assert Range(1, 4, oo) == Range(1, 2) + assert Range(-oo, oo).size == oo + assert Range(oo, -oo, -1).size == oo + raises(ValueError, lambda: Range(-oo, oo, 2)) + raises(ValueError, lambda: Range(x, pi, y)) + raises(ValueError, lambda: Range(x, y, 0)) + + assert 5 in Range(0, oo, 5) + assert -5 in Range(-oo, 0, 5) + assert oo not in Range(0, oo) + ni = symbols('ni', integer=False) + assert ni not in Range(oo) + u = symbols('u', integer=None) + assert Range(oo).contains(u) is not False + inf = symbols('inf', infinite=True) + assert inf not in Range(-oo, oo) + raises(ValueError, lambda: Range(0, oo, 2)[-1]) + raises(ValueError, lambda: Range(0, -oo, -2)[-1]) + assert Range(-oo, 1, 1)[-1] is S.Zero + assert Range(oo, 1, -1)[-1] == 2 + assert inf not in Range(oo) + assert Range(1, 10, 1)[-1] == 9 + assert all(i.is_Integer for i in Range(0, -1, 1)) + it = iter(Range(-oo, 0, 2)) + raises(TypeError, lambda: next(it)) + + assert empty.intersect(S.Integers) == empty + assert Range(-1, 10, 1).intersect(S.Complexes) == Range(-1, 10, 1) + assert Range(-1, 10, 1).intersect(S.Reals) == Range(-1, 10, 1) + assert Range(-1, 10, 1).intersect(S.Rationals) == Range(-1, 10, 1) + assert Range(-1, 10, 1).intersect(S.Integers) == Range(-1, 10, 1) + assert Range(-1, 10, 1).intersect(S.Naturals) == Range(1, 10, 1) + assert Range(-1, 10, 1).intersect(S.Naturals0) == Range(0, 10, 1) + + # test slicing + assert Range(1, 10, 1)[5] == 6 + assert Range(1, 12, 2)[5] == 11 + assert Range(1, 10, 1)[-1] == 9 + assert Range(1, 10, 3)[-1] == 7 + raises(ValueError, lambda: Range(oo,0,-1)[1:3:0]) + raises(ValueError, lambda: Range(oo,0,-1)[:1]) + raises(ValueError, lambda: Range(1, oo)[-2]) + raises(ValueError, lambda: Range(-oo, 1)[2]) + raises(IndexError, lambda: Range(10)[-20]) + raises(IndexError, lambda: Range(10)[20]) + raises(ValueError, lambda: Range(2, -oo, -2)[2:2:0]) + assert Range(2, -oo, -2)[2:2:2] == empty + assert Range(2, -oo, -2)[:2:2] == Range(2, -2, -4) + raises(ValueError, lambda: Range(-oo, 4, 2)[:2:2]) + assert Range(-oo, 4, 2)[::-2] == Range(2, -oo, -4) + raises(ValueError, lambda: Range(-oo, 4, 2)[::2]) + assert Range(oo, 2, -2)[::] == Range(oo, 2, -2) + assert Range(-oo, 4, 2)[:-2:-2] == Range(2, 0, -4) + assert Range(-oo, 4, 2)[:-2:2] == Range(-oo, 0, 4) + raises(ValueError, lambda: Range(-oo, 4, 2)[:0:-2]) + raises(ValueError, lambda: Range(-oo, 4, 2)[:2:-2]) + assert Range(-oo, 4, 2)[-2::-2] == Range(0, -oo, -4) + raises(ValueError, lambda: Range(-oo, 4, 2)[-2:0:-2]) + raises(ValueError, lambda: Range(-oo, 4, 2)[0::2]) + assert Range(oo, 2, -2)[0::] == Range(oo, 2, -2) + raises(ValueError, lambda: Range(-oo, 4, 2)[0:-2:2]) + assert Range(oo, 2, -2)[0:-2:] == Range(oo, 6, -2) + raises(ValueError, lambda: Range(oo, 2, -2)[0:2:]) + raises(ValueError, lambda: Range(-oo, 4, 2)[2::-1]) + assert Range(-oo, 4, 2)[-2::2] == Range(0, 4, 4) + assert Range(oo, 0, -2)[-10:0:2] == empty + raises(ValueError, lambda: Range(oo, 0, -2)[0]) + raises(ValueError, lambda: Range(oo, 0, -2)[-10:10:2]) + raises(ValueError, lambda: Range(oo, 0, -2)[0::-2]) + assert Range(oo, 0, -2)[0:-4:-2] == empty + assert Range(oo, 0, -2)[:0:2] == empty + raises(ValueError, lambda: Range(oo, 0, -2)[:1:-1]) + + # test empty Range + assert Range(x, x, y) == empty + assert empty.reversed == empty + assert 0 not in empty + assert list(empty) == [] + assert len(empty) == 0 + assert empty.size is S.Zero + assert empty.intersect(FiniteSet(0)) is S.EmptySet + assert bool(empty) is False + raises(IndexError, lambda: empty[0]) + assert empty[:0] == empty + raises(NotImplementedError, lambda: empty.inf) + raises(NotImplementedError, lambda: empty.sup) + assert empty.as_relational(x) is S.false + + AB = [None] + list(range(12)) + for R in [ + Range(1, 10), + Range(1, 10, 2), + ]: + r = list(R) + for a, b, c in itertools.product(AB, AB, [-3, -1, None, 1, 3]): + for reverse in range(2): + r = list(reversed(r)) + R = R.reversed + result = list(R[a:b:c]) + ans = r[a:b:c] + txt = ('\n%s[%s:%s:%s] = %s -> %s' % ( + R, a, b, c, result, ans)) + check = ans == result + assert check, txt + + assert Range(1, 10, 1).boundary == Range(1, 10, 1) + + for r in (Range(1, 10, 2), Range(1, oo, 2)): + rev = r.reversed + assert r.inf == rev.inf and r.sup == rev.sup + assert r.step == -rev.step + + builtin_range = range + + raises(TypeError, lambda: Range(builtin_range(1))) + assert S(builtin_range(10)) == Range(10) + assert S(builtin_range(1000000000000)) == Range(1000000000000) + + # test Range.as_relational + assert Range(1, 4).as_relational(x) == (x >= 1) & (x <= 3) & Eq(Mod(x, 1), 0) + assert Range(oo, 1, -2).as_relational(x) == (x >= 3) & (x < oo) & Eq(Mod(x + 1, -2), 0) + + +def test_Range_symbolic(): + # symbolic Range + xr = Range(x, x + 4, 5) + sr = Range(x, y, t) + i = Symbol('i', integer=True) + ip = Symbol('i', integer=True, positive=True) + ipr = Range(ip) + inr = Range(0, -ip, -1) + ir = Range(i, i + 19, 2) + ir2 = Range(i, i*8, 3*i) + i = Symbol('i', integer=True) + inf = symbols('inf', infinite=True) + raises(ValueError, lambda: Range(inf)) + raises(ValueError, lambda: Range(inf, 0, -1)) + raises(ValueError, lambda: Range(inf, inf, 1)) + raises(ValueError, lambda: Range(1, 1, inf)) + # args + assert xr.args == (x, x + 5, 5) + assert sr.args == (x, y, t) + assert ir.args == (i, i + 20, 2) + assert ir2.args == (i, 10*i, 3*i) + # reversed + raises(ValueError, lambda: xr.reversed) + raises(ValueError, lambda: sr.reversed) + assert ipr.reversed.args == (ip - 1, -1, -1) + assert inr.reversed.args == (-ip + 1, 1, 1) + assert ir.reversed.args == (i + 18, i - 2, -2) + assert ir2.reversed.args == (7*i, -2*i, -3*i) + # contains + assert inf not in sr + assert inf not in ir + assert 0 in ipr + assert 0 in inr + raises(TypeError, lambda: 1 in ipr) + raises(TypeError, lambda: -1 in inr) + assert .1 not in sr + assert .1 not in ir + assert i + 1 not in ir + assert i + 2 in ir + raises(TypeError, lambda: x in xr) # XXX is this what contains is supposed to do? + raises(TypeError, lambda: 1 in sr) # XXX is this what contains is supposed to do? + # iter + raises(ValueError, lambda: next(iter(xr))) + raises(ValueError, lambda: next(iter(sr))) + assert next(iter(ir)) == i + assert next(iter(ir2)) == i + assert sr.intersect(S.Integers) == sr + assert sr.intersect(FiniteSet(x)) == Intersection({x}, sr) + raises(ValueError, lambda: sr[:2]) + raises(ValueError, lambda: xr[0]) + raises(ValueError, lambda: sr[0]) + # len + assert len(ir) == ir.size == 10 + assert len(ir2) == ir2.size == 3 + raises(ValueError, lambda: len(xr)) + raises(ValueError, lambda: xr.size) + raises(ValueError, lambda: len(sr)) + raises(ValueError, lambda: sr.size) + # bool + assert bool(Range(0)) == False + assert bool(xr) + assert bool(ir) + assert bool(ipr) + assert bool(inr) + raises(ValueError, lambda: bool(sr)) + raises(ValueError, lambda: bool(ir2)) + # inf + raises(ValueError, lambda: xr.inf) + raises(ValueError, lambda: sr.inf) + assert ipr.inf == 0 + assert inr.inf == -ip + 1 + assert ir.inf == i + raises(ValueError, lambda: ir2.inf) + # sup + raises(ValueError, lambda: xr.sup) + raises(ValueError, lambda: sr.sup) + assert ipr.sup == ip - 1 + assert inr.sup == 0 + assert ir.inf == i + raises(ValueError, lambda: ir2.sup) + # getitem + raises(ValueError, lambda: xr[0]) + raises(ValueError, lambda: sr[0]) + raises(ValueError, lambda: sr[-1]) + raises(ValueError, lambda: sr[:2]) + assert ir[:2] == Range(i, i + 4, 2) + assert ir[0] == i + assert ir[-2] == i + 16 + assert ir[-1] == i + 18 + assert ir2[:2] == Range(i, 7*i, 3*i) + assert ir2[0] == i + assert ir2[-2] == 4*i + assert ir2[-1] == 7*i + raises(ValueError, lambda: Range(i)[-1]) + assert ipr[0] == ipr.inf == 0 + assert ipr[-1] == ipr.sup == ip - 1 + assert inr[0] == inr.sup == 0 + assert inr[-1] == inr.inf == -ip + 1 + raises(ValueError, lambda: ipr[-2]) + assert ir.inf == i + assert ir.sup == i + 18 + raises(ValueError, lambda: Range(i).inf) + # as_relational + assert ir.as_relational(x) == ((x >= i) & (x <= i + 18) & + Eq(Mod(-i + x, 2), 0)) + assert ir2.as_relational(x) == Eq( + Mod(-i + x, 3*i), 0) & (((x >= i) & (x <= 7*i) & (3*i >= 1)) | + ((x <= i) & (x >= 7*i) & (3*i <= -1))) + assert Range(i, i + 1).as_relational(x) == Eq(x, i) + assert sr.as_relational(z) == Eq( + Mod(t, 1), 0) & Eq(Mod(x, 1), 0) & Eq(Mod(-x + z, t), 0 + ) & (((z >= x) & (z <= -t + y) & (t >= 1)) | + ((z <= x) & (z >= -t + y) & (t <= -1))) + assert xr.as_relational(z) == Eq(z, x) & Eq(Mod(x, 1), 0) + # symbols can clash if user wants (but it must be integer) + assert xr.as_relational(x) == Eq(Mod(x, 1), 0) + # contains() for symbolic values (issue #18146) + e = Symbol('e', integer=True, even=True) + o = Symbol('o', integer=True, odd=True) + assert Range(5).contains(i) == And(i >= 0, i <= 4) + assert Range(1).contains(i) == Eq(i, 0) + assert Range(-oo, 5, 1).contains(i) == (i <= 4) + assert Range(-oo, oo).contains(i) == True + assert Range(0, 8, 2).contains(i) == Contains(i, Range(0, 8, 2)) + assert Range(0, 8, 2).contains(e) == And(e >= 0, e <= 6) + assert Range(0, 8, 2).contains(2*i) == And(2*i >= 0, 2*i <= 6) + assert Range(0, 8, 2).contains(o) == False + assert Range(1, 9, 2).contains(e) == False + assert Range(1, 9, 2).contains(o) == And(o >= 1, o <= 7) + assert Range(8, 0, -2).contains(o) == False + assert Range(9, 1, -2).contains(o) == And(o >= 3, o <= 9) + assert Range(-oo, 8, 2).contains(i) == Contains(i, Range(-oo, 8, 2)) + + +def test_range_range_intersection(): + for a, b, r in [ + (Range(0), Range(1), S.EmptySet), + (Range(3), Range(4, oo), S.EmptySet), + (Range(3), Range(-3, -1), S.EmptySet), + (Range(1, 3), Range(0, 3), Range(1, 3)), + (Range(1, 3), Range(1, 4), Range(1, 3)), + (Range(1, oo, 2), Range(2, oo, 2), S.EmptySet), + (Range(0, oo, 2), Range(oo), Range(0, oo, 2)), + (Range(0, oo, 2), Range(100), Range(0, 100, 2)), + (Range(2, oo, 2), Range(oo), Range(2, oo, 2)), + (Range(0, oo, 2), Range(5, 6), S.EmptySet), + (Range(2, 80, 1), Range(55, 71, 4), Range(55, 71, 4)), + (Range(0, 6, 3), Range(-oo, 5, 3), S.EmptySet), + (Range(0, oo, 2), Range(5, oo, 3), Range(8, oo, 6)), + (Range(4, 6, 2), Range(2, 16, 7), S.EmptySet),]: + assert a.intersect(b) == r + assert a.intersect(b.reversed) == r + assert a.reversed.intersect(b) == r + assert a.reversed.intersect(b.reversed) == r + a, b = b, a + assert a.intersect(b) == r + assert a.intersect(b.reversed) == r + assert a.reversed.intersect(b) == r + assert a.reversed.intersect(b.reversed) == r + + +def test_range_interval_intersection(): + p = symbols('p', positive=True) + assert isinstance(Range(3).intersect(Interval(p, p + 2)), Intersection) + assert Range(4).intersect(Interval(0, 3)) == Range(4) + assert Range(4).intersect(Interval(-oo, oo)) == Range(4) + assert Range(4).intersect(Interval(1, oo)) == Range(1, 4) + assert Range(4).intersect(Interval(1.1, oo)) == Range(2, 4) + assert Range(4).intersect(Interval(0.1, 3)) == Range(1, 4) + assert Range(4).intersect(Interval(0.1, 3.1)) == Range(1, 4) + assert Range(4).intersect(Interval.open(0, 3)) == Range(1, 3) + assert Range(4).intersect(Interval.open(0.1, 0.5)) is S.EmptySet + assert Interval(-1, 5).intersect(S.Complexes) == Interval(-1, 5) + assert Interval(-1, 5).intersect(S.Reals) == Interval(-1, 5) + assert Interval(-1, 5).intersect(S.Integers) == Range(-1, 6) + assert Interval(-1, 5).intersect(S.Naturals) == Range(1, 6) + assert Interval(-1, 5).intersect(S.Naturals0) == Range(0, 6) + + # Null Range intersections + assert Range(0).intersect(Interval(0.2, 0.8)) is S.EmptySet + assert Range(0).intersect(Interval(-oo, oo)) is S.EmptySet + + +def test_range_is_finite_set(): + assert Range(-100, 100).is_finite_set is True + assert Range(2, oo).is_finite_set is False + assert Range(-oo, 50).is_finite_set is False + assert Range(-oo, oo).is_finite_set is False + assert Range(oo, -oo).is_finite_set is True + assert Range(0, 0).is_finite_set is True + assert Range(oo, oo).is_finite_set is True + assert Range(-oo, -oo).is_finite_set is True + n = Symbol('n', integer=True) + m = Symbol('m', integer=True) + assert Range(n, n + 49).is_finite_set is True + assert Range(n, 0).is_finite_set is True + assert Range(-3, n + 7).is_finite_set is True + assert Range(n, m).is_finite_set is True + assert Range(n + m, m - n).is_finite_set is True + assert Range(n, n + m + n).is_finite_set is True + assert Range(n, oo).is_finite_set is False + assert Range(-oo, n).is_finite_set is False + assert Range(n, -oo).is_finite_set is True + assert Range(oo, n).is_finite_set is True + + +def test_Range_is_iterable(): + assert Range(-100, 100).is_iterable is True + assert Range(2, oo).is_iterable is False + assert Range(-oo, 50).is_iterable is False + assert Range(-oo, oo).is_iterable is False + assert Range(oo, -oo).is_iterable is True + assert Range(0, 0).is_iterable is True + assert Range(oo, oo).is_iterable is True + assert Range(-oo, -oo).is_iterable is True + n = Symbol('n', integer=True) + m = Symbol('m', integer=True) + p = Symbol('p', integer=True, positive=True) + assert Range(n, n + 49).is_iterable is True + assert Range(n, 0).is_iterable is False + assert Range(-3, n + 7).is_iterable is False + assert Range(-3, p + 7).is_iterable is False # Should work with better __iter__ + assert Range(n, m).is_iterable is False + assert Range(n + m, m - n).is_iterable is False + assert Range(n, n + m + n).is_iterable is False + assert Range(n, oo).is_iterable is False + assert Range(-oo, n).is_iterable is False + x = Symbol('x') + assert Range(x, x + 49).is_iterable is False + assert Range(x, 0).is_iterable is False + assert Range(-3, x + 7).is_iterable is False + assert Range(x, m).is_iterable is False + assert Range(x + m, m - x).is_iterable is False + assert Range(x, x + m + x).is_iterable is False + assert Range(x, oo).is_iterable is False + assert Range(-oo, x).is_iterable is False + + +def test_Integers_eval_imageset(): + ans = ImageSet(Lambda(x, 2*x + Rational(3, 7)), S.Integers) + im = imageset(Lambda(x, -2*x + Rational(3, 7)), S.Integers) + assert im == ans + im = imageset(Lambda(x, -2*x - Rational(11, 7)), S.Integers) + assert im == ans + y = Symbol('y') + L = imageset(x, 2*x + y, S.Integers) + assert y + 4 in L + a, b, c = 0.092, 0.433, 0.341 + assert a in imageset(x, a + c*x, S.Integers) + assert b in imageset(x, b + c*x, S.Integers) + + _x = symbols('x', negative=True) + eq = _x**2 - _x + 1 + assert imageset(_x, eq, S.Integers).lamda.expr == _x**2 + _x + 1 + eq = 3*_x - 1 + assert imageset(_x, eq, S.Integers).lamda.expr == 3*_x + 2 + + assert imageset(x, (x, 1/x), S.Integers) == \ + ImageSet(Lambda(x, (x, 1/x)), S.Integers) + + +def test_Range_eval_imageset(): + a, b, c = symbols('a b c') + assert imageset(x, a*(x + b) + c, Range(3)) == \ + imageset(x, a*x + a*b + c, Range(3)) + eq = (x + 1)**2 + assert imageset(x, eq, Range(3)).lamda.expr == eq + eq = a*(x + b) + c + r = Range(3, -3, -2) + imset = imageset(x, eq, r) + assert imset.lamda.expr != eq + assert list(imset) == [eq.subs(x, i).expand() for i in list(r)] + + +def test_fun(): + assert (FiniteSet(*ImageSet(Lambda(x, sin(pi*x/4)), + Range(-10, 11))) == FiniteSet(-1, -sqrt(2)/2, 0, sqrt(2)/2, 1)) + + +def test_Range_is_empty(): + i = Symbol('i', integer=True) + n = Symbol('n', negative=True, integer=True) + p = Symbol('p', positive=True, integer=True) + + assert Range(0).is_empty + assert not Range(1).is_empty + assert Range(1, 0).is_empty + assert not Range(-1, 0).is_empty + assert Range(i).is_empty is None + assert Range(n).is_empty + assert Range(p).is_empty is False + assert Range(n, 0).is_empty is False + assert Range(n, p).is_empty is False + assert Range(p, n).is_empty + assert Range(n, -1).is_empty is None + assert Range(p, n, -1).is_empty is False + + +def test_Reals(): + assert 5 in S.Reals + assert S.Pi in S.Reals + assert -sqrt(2) in S.Reals + assert (2, 5) not in S.Reals + assert sqrt(-1) not in S.Reals + assert S.Reals == Interval(-oo, oo) + assert S.Reals != Interval(0, oo) + assert S.Reals.is_subset(Interval(-oo, oo)) + assert S.Reals.intersect(Range(-oo, oo)) == Range(-oo, oo) + assert S.ComplexInfinity not in S.Reals + assert S.NaN not in S.Reals + assert x + S.ComplexInfinity not in S.Reals + + +def test_Complex(): + assert 5 in S.Complexes + assert 5 + 4*I in S.Complexes + assert S.Pi in S.Complexes + assert -sqrt(2) in S.Complexes + assert -I in S.Complexes + assert sqrt(-1) in S.Complexes + assert S.Complexes.intersect(S.Reals) == S.Reals + assert S.Complexes.union(S.Reals) == S.Complexes + assert S.Complexes == ComplexRegion(S.Reals*S.Reals) + assert (S.Complexes == ComplexRegion(Interval(1, 2)*Interval(3, 4))) == False + assert str(S.Complexes) == "Complexes" + assert repr(S.Complexes) == "Complexes" + + +def take(n, iterable): + "Return first n items of the iterable as a list" + return list(itertools.islice(iterable, n)) + + +def test_intersections(): + assert S.Integers.intersect(S.Reals) == S.Integers + assert 5 in S.Integers.intersect(S.Reals) + assert 5 in S.Integers.intersect(S.Reals) + assert -5 not in S.Naturals.intersect(S.Reals) + assert 5.5 not in S.Integers.intersect(S.Reals) + assert 5 in S.Integers.intersect(Interval(3, oo)) + assert -5 in S.Integers.intersect(Interval(-oo, 3)) + assert all(x.is_Integer + for x in take(10, S.Integers.intersect(Interval(3, oo)) )) + + +def test_infinitely_indexed_set_1(): + from sympy.abc import n, m + assert imageset(Lambda(n, n), S.Integers) == imageset(Lambda(m, m), S.Integers) + + assert imageset(Lambda(n, 2*n), S.Integers).intersect( + imageset(Lambda(m, 2*m + 1), S.Integers)) is S.EmptySet + + assert imageset(Lambda(n, 2*n), S.Integers).intersect( + imageset(Lambda(n, 2*n + 1), S.Integers)) is S.EmptySet + + assert imageset(Lambda(m, 2*m), S.Integers).intersect( + imageset(Lambda(n, 3*n), S.Integers)).dummy_eq( + ImageSet(Lambda(t, 6*t), S.Integers)) + + assert imageset(x, x/2 + Rational(1, 3), S.Integers).intersect(S.Integers) is S.EmptySet + assert imageset(x, x/2 + S.Half, S.Integers).intersect(S.Integers) is S.Integers + + # https://github.com/sympy/sympy/issues/17355 + S53 = ImageSet(Lambda(n, 5*n + 3), S.Integers) + assert S53.intersect(S.Integers) == S53 + + +def test_infinitely_indexed_set_2(): + from sympy.abc import n + a = Symbol('a', integer=True) + assert imageset(Lambda(n, n), S.Integers) == \ + imageset(Lambda(n, n + a), S.Integers) + assert imageset(Lambda(n, n + pi), S.Integers) == \ + imageset(Lambda(n, n + a + pi), S.Integers) + assert imageset(Lambda(n, n), S.Integers) == \ + imageset(Lambda(n, -n + a), S.Integers) + assert imageset(Lambda(n, -6*n), S.Integers) == \ + ImageSet(Lambda(n, 6*n), S.Integers) + assert imageset(Lambda(n, 2*n + pi), S.Integers) == \ + ImageSet(Lambda(n, 2*n + pi - 2), S.Integers) + + +def test_imageset_intersect_real(): + from sympy.abc import n + assert imageset(Lambda(n, n + (n - 1)*(n + 1)*I), S.Integers).intersect(S.Reals) == FiniteSet(-1, 1) + im = (n - 1)*(n + S.Half) + assert imageset(Lambda(n, n + im*I), S.Integers + ).intersect(S.Reals) == FiniteSet(1) + assert imageset(Lambda(n, n + im*(n + 1)*I), S.Naturals0 + ).intersect(S.Reals) == FiniteSet(1) + assert imageset(Lambda(n, n/2 + im.expand()*I), S.Integers + ).intersect(S.Reals) == ImageSet(Lambda(x, x/2), ConditionSet( + n, Eq(n**2 - n/2 - S(1)/2, 0), S.Integers)) + assert imageset(Lambda(n, n/(1/n - 1) + im*(n + 1)*I), S.Integers + ).intersect(S.Reals) == FiniteSet(S.Half) + assert imageset(Lambda(n, n/(n - 6) + + (n - 3)*(n + 1)*I/(2*n + 2)), S.Integers).intersect( + S.Reals) == FiniteSet(-1) + assert imageset(Lambda(n, n/(n**2 - 9) + + (n - 3)*(n + 1)*I/(2*n + 2)), S.Integers).intersect( + S.Reals) is S.EmptySet + s = ImageSet( + Lambda(n, -I*(I*(2*pi*n - pi/4) + log(Abs(sqrt(-I))))), + S.Integers) + # s is unevaluated, but after intersection the result + # should be canonical + assert s.intersect(S.Reals) == imageset( + Lambda(n, 2*n*pi - pi/4), S.Integers) == ImageSet( + Lambda(n, 2*pi*n + pi*Rational(7, 4)), S.Integers) + + +def test_imageset_intersect_interval(): + from sympy.abc import n + f1 = ImageSet(Lambda(n, n*pi), S.Integers) + f2 = ImageSet(Lambda(n, 2*n), Interval(0, pi)) + f3 = ImageSet(Lambda(n, 2*n*pi + pi/2), S.Integers) + # complex expressions + f4 = ImageSet(Lambda(n, n*I*pi), S.Integers) + f5 = ImageSet(Lambda(n, 2*I*n*pi + pi/2), S.Integers) + # non-linear expressions + f6 = ImageSet(Lambda(n, log(n)), S.Integers) + f7 = ImageSet(Lambda(n, n**2), S.Integers) + f8 = ImageSet(Lambda(n, Abs(n)), S.Integers) + f9 = ImageSet(Lambda(n, exp(n)), S.Naturals0) + + assert f1.intersect(Interval(-1, 1)) == FiniteSet(0) + assert f1.intersect(Interval(0, 2*pi, False, True)) == FiniteSet(0, pi) + assert f2.intersect(Interval(1, 2)) == Interval(1, 2) + assert f3.intersect(Interval(-1, 1)) == S.EmptySet + assert f3.intersect(Interval(-5, 5)) == FiniteSet(pi*Rational(-3, 2), pi/2) + assert f4.intersect(Interval(-1, 1)) == FiniteSet(0) + assert f4.intersect(Interval(1, 2)) == S.EmptySet + assert f5.intersect(Interval(0, 1)) == S.EmptySet + assert f6.intersect(Interval(0, 1)) == FiniteSet(S.Zero, log(2)) + assert f7.intersect(Interval(0, 10)) == Intersection(f7, Interval(0, 10)) + assert f8.intersect(Interval(0, 2)) == Intersection(f8, Interval(0, 2)) + assert f9.intersect(Interval(1, 2)) == Intersection(f9, Interval(1, 2)) + + +def test_imageset_intersect_diophantine(): + from sympy.abc import m, n + # Check that same lambda variable for both ImageSets is handled correctly + img1 = ImageSet(Lambda(n, 2*n + 1), S.Integers) + img2 = ImageSet(Lambda(n, 4*n + 1), S.Integers) + assert img1.intersect(img2) == img2 + # Empty solution set returned by diophantine: + assert ImageSet(Lambda(n, 2*n), S.Integers).intersect( + ImageSet(Lambda(n, 2*n + 1), S.Integers)) == S.EmptySet + # Check intersection with S.Integers: + assert ImageSet(Lambda(n, 9/n + 20*n/3), S.Integers).intersect( + S.Integers) == FiniteSet(-61, -23, 23, 61) + # Single solution (2, 3) for diophantine solution: + assert ImageSet(Lambda(n, (n - 2)**2), S.Integers).intersect( + ImageSet(Lambda(n, -(n - 3)**2), S.Integers)) == FiniteSet(0) + # Single parametric solution for diophantine solution: + assert ImageSet(Lambda(n, n**2 + 5), S.Integers).intersect( + ImageSet(Lambda(m, 2*m), S.Integers)).dummy_eq(ImageSet( + Lambda(n, 4*n**2 + 4*n + 6), S.Integers)) + # 4 non-parametric solution couples for dioph. equation: + assert ImageSet(Lambda(n, n**2 - 9), S.Integers).intersect( + ImageSet(Lambda(m, -m**2), S.Integers)) == FiniteSet(-9, 0) + # Double parametric solution for diophantine solution: + assert ImageSet(Lambda(m, m**2 + 40), S.Integers).intersect( + ImageSet(Lambda(n, 41*n), S.Integers)).dummy_eq(Intersection( + ImageSet(Lambda(m, m**2 + 40), S.Integers), + ImageSet(Lambda(n, 41*n), S.Integers))) + # Check that diophantine returns *all* (8) solutions (permute=True) + assert ImageSet(Lambda(n, n**4 - 2**4), S.Integers).intersect( + ImageSet(Lambda(m, -m**4 + 3**4), S.Integers)) == FiniteSet(0, 65) + assert ImageSet(Lambda(n, pi/12 + n*5*pi/12), S.Integers).intersect( + ImageSet(Lambda(n, 7*pi/12 + n*11*pi/12), S.Integers)).dummy_eq(ImageSet( + Lambda(n, 55*pi*n/12 + 17*pi/4), S.Integers)) + # TypeError raised by diophantine (#18081) + assert ImageSet(Lambda(n, n*log(2)), S.Integers).intersection( + S.Integers).dummy_eq(Intersection(ImageSet( + Lambda(n, n*log(2)), S.Integers), S.Integers)) + # NotImplementedError raised by diophantine (no solver for cubic_thue) + assert ImageSet(Lambda(n, n**3 + 1), S.Integers).intersect( + ImageSet(Lambda(n, n**3), S.Integers)).dummy_eq(Intersection( + ImageSet(Lambda(n, n**3 + 1), S.Integers), + ImageSet(Lambda(n, n**3), S.Integers))) + + +def test_infinitely_indexed_set_3(): + from sympy.abc import n, m + assert imageset(Lambda(m, 2*pi*m), S.Integers).intersect( + imageset(Lambda(n, 3*pi*n), S.Integers)).dummy_eq( + ImageSet(Lambda(t, 6*pi*t), S.Integers)) + assert imageset(Lambda(n, 2*n + 1), S.Integers) == \ + imageset(Lambda(n, 2*n - 1), S.Integers) + assert imageset(Lambda(n, 3*n + 2), S.Integers) == \ + imageset(Lambda(n, 3*n - 1), S.Integers) + + +def test_ImageSet_simplification(): + from sympy.abc import n, m + assert imageset(Lambda(n, n), S.Integers) == S.Integers + assert imageset(Lambda(n, sin(n)), + imageset(Lambda(m, tan(m)), S.Integers)) == \ + imageset(Lambda(m, sin(tan(m))), S.Integers) + assert imageset(n, 1 + 2*n, S.Naturals) == Range(3, oo, 2) + assert imageset(n, 1 + 2*n, S.Naturals0) == Range(1, oo, 2) + assert imageset(n, 1 - 2*n, S.Naturals) == Range(-1, -oo, -2) + + +def test_ImageSet_contains(): + assert (2, S.Half) in imageset(x, (x, 1/x), S.Integers) + assert imageset(x, x + I*3, S.Integers).intersection(S.Reals) is S.EmptySet + i = Dummy(integer=True) + q = imageset(x, x + I*y, S.Integers).intersection(S.Reals) + assert q.subs(y, I*i).intersection(S.Integers) is S.Integers + q = imageset(x, x + I*y/x, S.Integers).intersection(S.Reals) + assert q.subs(y, 0) is S.Integers + assert q.subs(y, I*i*x).intersection(S.Integers) is S.Integers + z = cos(1)**2 + sin(1)**2 - 1 + q = imageset(x, x + I*z, S.Integers).intersection(S.Reals) + assert q is not S.EmptySet + + +def test_ComplexRegion_contains(): + r = Symbol('r', real=True) + # contains in ComplexRegion + a = Interval(2, 3) + b = Interval(4, 6) + c = Interval(7, 9) + c1 = ComplexRegion(a*b) + c2 = ComplexRegion(Union(a*b, c*a)) + assert 2.5 + 4.5*I in c1 + assert 2 + 4*I in c1 + assert 3 + 4*I in c1 + assert 8 + 2.5*I in c2 + assert 2.5 + 6.1*I not in c1 + assert 4.5 + 3.2*I not in c1 + assert c1.contains(x) == Contains(x, c1, evaluate=False) + assert c1.contains(r) == False + assert c2.contains(x) == Contains(x, c2, evaluate=False) + assert c2.contains(r) == False + + r1 = Interval(0, 1) + theta1 = Interval(0, 2*S.Pi) + c3 = ComplexRegion(r1*theta1, polar=True) + assert (0.5 + I*6/10) in c3 + assert (S.Half + I*6/10) in c3 + assert (S.Half + .6*I) in c3 + assert (0.5 + .6*I) in c3 + assert I in c3 + assert 1 in c3 + assert 0 in c3 + assert 1 + I not in c3 + assert 1 - I not in c3 + assert c3.contains(x) == Contains(x, c3, evaluate=False) + assert c3.contains(r + 2*I) == Contains( + r + 2*I, c3, evaluate=False) # is in fact False + assert c3.contains(1/(1 + r**2)) == Contains( + 1/(1 + r**2), c3, evaluate=False) # is in fact True + + r2 = Interval(0, 3) + theta2 = Interval(pi, 2*pi, left_open=True) + c4 = ComplexRegion(r2*theta2, polar=True) + assert c4.contains(0) == True + assert c4.contains(2 + I) == False + assert c4.contains(-2 + I) == False + assert c4.contains(-2 - I) == True + assert c4.contains(2 - I) == True + assert c4.contains(-2) == False + assert c4.contains(2) == True + assert c4.contains(x) == Contains(x, c4, evaluate=False) + assert c4.contains(3/(1 + r**2)) == Contains( + 3/(1 + r**2), c4, evaluate=False) # is in fact True + + raises(ValueError, lambda: ComplexRegion(r1*theta1, polar=2)) + + +def test_symbolic_Range(): + n = Symbol('n') + raises(ValueError, lambda: Range(n)[0]) + raises(IndexError, lambda: Range(n, n)[0]) + raises(ValueError, lambda: Range(n, n+1)[0]) + raises(ValueError, lambda: Range(n).size) + + n = Symbol('n', integer=True) + raises(ValueError, lambda: Range(n)[0]) + raises(IndexError, lambda: Range(n, n)[0]) + assert Range(n, n+1)[0] == n + raises(ValueError, lambda: Range(n).size) + assert Range(n, n+1).size == 1 + + n = Symbol('n', integer=True, nonnegative=True) + raises(ValueError, lambda: Range(n)[0]) + raises(IndexError, lambda: Range(n, n)[0]) + assert Range(n+1)[0] == 0 + assert Range(n, n+1)[0] == n + assert Range(n).size == n + assert Range(n+1).size == n+1 + assert Range(n, n+1).size == 1 + + n = Symbol('n', integer=True, positive=True) + assert Range(n)[0] == 0 + assert Range(n, n+1)[0] == n + assert Range(n).size == n + assert Range(n, n+1).size == 1 + + m = Symbol('m', integer=True, positive=True) + + assert Range(n, n+m)[0] == n + assert Range(n, n+m).size == m + assert Range(n, n+1).size == 1 + assert Range(n, n+m, 2).size == floor(m/2) + + m = Symbol('m', integer=True, positive=True, even=True) + assert Range(n, n+m, 2).size == m/2 + + +def test_issue_18400(): + n = Symbol('n', integer=True) + raises(ValueError, lambda: imageset(lambda x: x*2, Range(n))) + + n = Symbol('n', integer=True, positive=True) + # No exception + assert imageset(lambda x: x*2, Range(n)) == imageset(lambda x: x*2, Range(n)) + + +def test_ComplexRegion_intersect(): + # Polar form + X_axis = ComplexRegion(Interval(0, oo)*FiniteSet(0, S.Pi), polar=True) + + unit_disk = ComplexRegion(Interval(0, 1)*Interval(0, 2*S.Pi), polar=True) + upper_half_unit_disk = ComplexRegion(Interval(0, 1)*Interval(0, S.Pi), polar=True) + upper_half_disk = ComplexRegion(Interval(0, oo)*Interval(0, S.Pi), polar=True) + lower_half_disk = ComplexRegion(Interval(0, oo)*Interval(S.Pi, 2*S.Pi), polar=True) + right_half_disk = ComplexRegion(Interval(0, oo)*Interval(-S.Pi/2, S.Pi/2), polar=True) + first_quad_disk = ComplexRegion(Interval(0, oo)*Interval(0, S.Pi/2), polar=True) + + assert upper_half_disk.intersect(unit_disk) == upper_half_unit_disk + assert right_half_disk.intersect(first_quad_disk) == first_quad_disk + assert upper_half_disk.intersect(right_half_disk) == first_quad_disk + assert upper_half_disk.intersect(lower_half_disk) == X_axis + + c1 = ComplexRegion(Interval(0, 4)*Interval(0, 2*S.Pi), polar=True) + assert c1.intersect(Interval(1, 5)) == Interval(1, 4) + assert c1.intersect(Interval(4, 9)) == FiniteSet(4) + assert c1.intersect(Interval(5, 12)) is S.EmptySet + + # Rectangular form + X_axis = ComplexRegion(Interval(-oo, oo)*FiniteSet(0)) + + unit_square = ComplexRegion(Interval(-1, 1)*Interval(-1, 1)) + upper_half_unit_square = ComplexRegion(Interval(-1, 1)*Interval(0, 1)) + upper_half_plane = ComplexRegion(Interval(-oo, oo)*Interval(0, oo)) + lower_half_plane = ComplexRegion(Interval(-oo, oo)*Interval(-oo, 0)) + right_half_plane = ComplexRegion(Interval(0, oo)*Interval(-oo, oo)) + first_quad_plane = ComplexRegion(Interval(0, oo)*Interval(0, oo)) + + assert upper_half_plane.intersect(unit_square) == upper_half_unit_square + assert right_half_plane.intersect(first_quad_plane) == first_quad_plane + assert upper_half_plane.intersect(right_half_plane) == first_quad_plane + assert upper_half_plane.intersect(lower_half_plane) == X_axis + + c1 = ComplexRegion(Interval(-5, 5)*Interval(-10, 10)) + assert c1.intersect(Interval(2, 7)) == Interval(2, 5) + assert c1.intersect(Interval(5, 7)) == FiniteSet(5) + assert c1.intersect(Interval(6, 9)) is S.EmptySet + + # unevaluated object + C1 = ComplexRegion(Interval(0, 1)*Interval(0, 2*S.Pi), polar=True) + C2 = ComplexRegion(Interval(-1, 1)*Interval(-1, 1)) + assert C1.intersect(C2) == Intersection(C1, C2, evaluate=False) + + +def test_ComplexRegion_union(): + # Polar form + c1 = ComplexRegion(Interval(0, 1)*Interval(0, 2*S.Pi), polar=True) + c2 = ComplexRegion(Interval(0, 1)*Interval(0, S.Pi), polar=True) + c3 = ComplexRegion(Interval(0, oo)*Interval(0, S.Pi), polar=True) + c4 = ComplexRegion(Interval(0, oo)*Interval(S.Pi, 2*S.Pi), polar=True) + + p1 = Union(Interval(0, 1)*Interval(0, 2*S.Pi), Interval(0, 1)*Interval(0, S.Pi)) + p2 = Union(Interval(0, oo)*Interval(0, S.Pi), Interval(0, oo)*Interval(S.Pi, 2*S.Pi)) + + assert c1.union(c2) == ComplexRegion(p1, polar=True) + assert c3.union(c4) == ComplexRegion(p2, polar=True) + + # Rectangular form + c5 = ComplexRegion(Interval(2, 5)*Interval(6, 9)) + c6 = ComplexRegion(Interval(4, 6)*Interval(10, 12)) + c7 = ComplexRegion(Interval(0, 10)*Interval(-10, 0)) + c8 = ComplexRegion(Interval(12, 16)*Interval(14, 20)) + + p3 = Union(Interval(2, 5)*Interval(6, 9), Interval(4, 6)*Interval(10, 12)) + p4 = Union(Interval(0, 10)*Interval(-10, 0), Interval(12, 16)*Interval(14, 20)) + + assert c5.union(c6) == ComplexRegion(p3) + assert c7.union(c8) == ComplexRegion(p4) + + assert c1.union(Interval(2, 4)) == Union(c1, Interval(2, 4), evaluate=False) + assert c5.union(Interval(2, 4)) == Union(c5, ComplexRegion.from_real(Interval(2, 4))) + + +def test_ComplexRegion_from_real(): + c1 = ComplexRegion(Interval(0, 1) * Interval(0, 2 * S.Pi), polar=True) + + raises(ValueError, lambda: c1.from_real(c1)) + assert c1.from_real(Interval(-1, 1)) == ComplexRegion(Interval(-1, 1) * FiniteSet(0), False) + + +def test_ComplexRegion_measure(): + a, b = Interval(2, 5), Interval(4, 8) + theta1, theta2 = Interval(0, 2*S.Pi), Interval(0, S.Pi) + c1 = ComplexRegion(a*b) + c2 = ComplexRegion(Union(a*theta1, b*theta2), polar=True) + + assert c1.measure == 12 + assert c2.measure == 9*pi + + +def test_normalize_theta_set(): + # Interval + assert normalize_theta_set(Interval(pi, 2*pi)) == \ + Union(FiniteSet(0), Interval.Ropen(pi, 2*pi)) + assert normalize_theta_set(Interval(pi*Rational(9, 2), 5*pi)) == Interval(pi/2, pi) + assert normalize_theta_set(Interval(pi*Rational(-3, 2), pi/2)) == Interval.Ropen(0, 2*pi) + assert normalize_theta_set(Interval.open(pi*Rational(-3, 2), pi/2)) == \ + Union(Interval.Ropen(0, pi/2), Interval.open(pi/2, 2*pi)) + assert normalize_theta_set(Interval.open(pi*Rational(-7, 2), pi*Rational(-3, 2))) == \ + Union(Interval.Ropen(0, pi/2), Interval.open(pi/2, 2*pi)) + assert normalize_theta_set(Interval(-pi/2, pi/2)) == \ + Union(Interval(0, pi/2), Interval.Ropen(pi*Rational(3, 2), 2*pi)) + assert normalize_theta_set(Interval.open(-pi/2, pi/2)) == \ + Union(Interval.Ropen(0, pi/2), Interval.open(pi*Rational(3, 2), 2*pi)) + assert normalize_theta_set(Interval(-4*pi, 3*pi)) == Interval.Ropen(0, 2*pi) + assert normalize_theta_set(Interval(pi*Rational(-3, 2), -pi/2)) == Interval(pi/2, pi*Rational(3, 2)) + assert normalize_theta_set(Interval.open(0, 2*pi)) == Interval.open(0, 2*pi) + assert normalize_theta_set(Interval.Ropen(-pi/2, pi/2)) == \ + Union(Interval.Ropen(0, pi/2), Interval.Ropen(pi*Rational(3, 2), 2*pi)) + assert normalize_theta_set(Interval.Lopen(-pi/2, pi/2)) == \ + Union(Interval(0, pi/2), Interval.open(pi*Rational(3, 2), 2*pi)) + assert normalize_theta_set(Interval(-pi/2, pi/2)) == \ + Union(Interval(0, pi/2), Interval.Ropen(pi*Rational(3, 2), 2*pi)) + assert normalize_theta_set(Interval.open(4*pi, pi*Rational(9, 2))) == Interval.open(0, pi/2) + assert normalize_theta_set(Interval.Lopen(4*pi, pi*Rational(9, 2))) == Interval.Lopen(0, pi/2) + assert normalize_theta_set(Interval.Ropen(4*pi, pi*Rational(9, 2))) == Interval.Ropen(0, pi/2) + assert normalize_theta_set(Interval.open(3*pi, 5*pi)) == \ + Union(Interval.Ropen(0, pi), Interval.open(pi, 2*pi)) + + # FiniteSet + assert normalize_theta_set(FiniteSet(0, pi, 3*pi)) == FiniteSet(0, pi) + assert normalize_theta_set(FiniteSet(0, pi/2, pi, 2*pi)) == FiniteSet(0, pi/2, pi) + assert normalize_theta_set(FiniteSet(0, -pi/2, -pi, -2*pi)) == FiniteSet(0, pi, pi*Rational(3, 2)) + assert normalize_theta_set(FiniteSet(pi*Rational(-3, 2), pi/2)) == \ + FiniteSet(pi/2) + assert normalize_theta_set(FiniteSet(2*pi)) == FiniteSet(0) + + # Unions + assert normalize_theta_set(Union(Interval(0, pi/3), Interval(pi/2, pi))) == \ + Union(Interval(0, pi/3), Interval(pi/2, pi)) + assert normalize_theta_set(Union(Interval(0, pi), Interval(2*pi, pi*Rational(7, 3)))) == \ + Interval(0, pi) + + # ValueError for non-real sets + raises(ValueError, lambda: normalize_theta_set(S.Complexes)) + + # NotImplementedError for subset of reals + raises(NotImplementedError, lambda: normalize_theta_set(Interval(0, 1))) + + # NotImplementedError without pi as coefficient + raises(NotImplementedError, lambda: normalize_theta_set(Interval(1, 2*pi))) + raises(NotImplementedError, lambda: normalize_theta_set(Interval(2*pi, 10))) + raises(NotImplementedError, lambda: normalize_theta_set(FiniteSet(0, 3, 3*pi))) + + +def test_ComplexRegion_FiniteSet(): + x, y, z, a, b, c = symbols('x y z a b c') + + # Issue #9669 + assert ComplexRegion(FiniteSet(a, b, c)*FiniteSet(x, y, z)) == \ + FiniteSet(a + I*x, a + I*y, a + I*z, b + I*x, b + I*y, + b + I*z, c + I*x, c + I*y, c + I*z) + assert ComplexRegion(FiniteSet(2)*FiniteSet(3)) == FiniteSet(2 + 3*I) + + +def test_union_RealSubSet(): + assert (S.Complexes).union(Interval(1, 2)) == S.Complexes + assert (S.Complexes).union(S.Integers) == S.Complexes + + +def test_SetKind_fancySet(): + G = lambda *args: ImageSet(Lambda(x, x ** 2), *args) + assert G(Interval(1, 4)).kind is SetKind(NumberKind) + assert G(FiniteSet(1, 4)).kind is SetKind(NumberKind) + assert S.Rationals.kind is SetKind(NumberKind) + assert S.Naturals.kind is SetKind(NumberKind) + assert S.Integers.kind is SetKind(NumberKind) + assert Range(3).kind is SetKind(NumberKind) + a = Interval(2, 3) + b = Interval(4, 6) + c1 = ComplexRegion(a*b) + assert c1.kind is SetKind(TupleKind(NumberKind, NumberKind)) + + +def test_issue_9980(): + c1 = ComplexRegion(Interval(1, 2)*Interval(2, 3)) + c2 = ComplexRegion(Interval(1, 5)*Interval(1, 3)) + R = Union(c1, c2) + assert simplify(R) == ComplexRegion(Union(Interval(1, 2)*Interval(2, 3), \ + Interval(1, 5)*Interval(1, 3)), False) + assert c1.func(*c1.args) == c1 + assert R.func(*R.args) == R + + +def test_issue_11732(): + interval12 = Interval(1, 2) + finiteset1234 = FiniteSet(1, 2, 3, 4) + pointComplex = Tuple(1, 5) + + assert (interval12 in S.Naturals) == False + assert (interval12 in S.Naturals0) == False + assert (interval12 in S.Integers) == False + assert (interval12 in S.Complexes) == False + + assert (finiteset1234 in S.Naturals) == False + assert (finiteset1234 in S.Naturals0) == False + assert (finiteset1234 in S.Integers) == False + assert (finiteset1234 in S.Complexes) == False + + assert (pointComplex in S.Naturals) == False + assert (pointComplex in S.Naturals0) == False + assert (pointComplex in S.Integers) == False + assert (pointComplex in S.Complexes) == True + + +def test_issue_11730(): + unit = Interval(0, 1) + square = ComplexRegion(unit ** 2) + + assert Union(S.Complexes, FiniteSet(oo)) != S.Complexes + assert Union(S.Complexes, FiniteSet(eye(4))) != S.Complexes + assert Union(unit, square) == square + assert Intersection(S.Reals, square) == unit + + +def test_issue_11938(): + unit = Interval(0, 1) + ival = Interval(1, 2) + cr1 = ComplexRegion(ival * unit) + + assert Intersection(cr1, S.Reals) == ival + assert Intersection(cr1, unit) == FiniteSet(1) + + arg1 = Interval(0, S.Pi) + arg2 = FiniteSet(S.Pi) + arg3 = Interval(S.Pi / 4, 3 * S.Pi / 4) + cp1 = ComplexRegion(unit * arg1, polar=True) + cp2 = ComplexRegion(unit * arg2, polar=True) + cp3 = ComplexRegion(unit * arg3, polar=True) + + assert Intersection(cp1, S.Reals) == Interval(-1, 1) + assert Intersection(cp2, S.Reals) == Interval(-1, 0) + assert Intersection(cp3, S.Reals) == FiniteSet(0) + + +def test_issue_11914(): + a, b = Interval(0, 1), Interval(0, pi) + c, d = Interval(2, 3), Interval(pi, 3 * pi / 2) + cp1 = ComplexRegion(a * b, polar=True) + cp2 = ComplexRegion(c * d, polar=True) + + assert -3 in cp1.union(cp2) + assert -3 in cp2.union(cp1) + assert -5 not in cp1.union(cp2) + + +def test_issue_9543(): + assert ImageSet(Lambda(x, x**2), S.Naturals).is_subset(S.Reals) + + +def test_issue_16871(): + assert ImageSet(Lambda(x, x), FiniteSet(1)) == {1} + assert ImageSet(Lambda(x, x - 3), S.Integers + ).intersection(S.Integers) is S.Integers + + +@XFAIL +def test_issue_16871b(): + assert ImageSet(Lambda(x, x - 3), S.Integers).is_subset(S.Integers) + + +def test_issue_18050(): + assert imageset(Lambda(x, I*x + 1), S.Integers + ) == ImageSet(Lambda(x, I*x + 1), S.Integers) + assert imageset(Lambda(x, 3*I*x + 4 + 8*I), S.Integers + ) == ImageSet(Lambda(x, 3*I*x + 4 + 2*I), S.Integers) + # no 'Mod' for next 2 tests: + assert imageset(Lambda(x, 2*x + 3*I), S.Integers + ) == ImageSet(Lambda(x, 2*x + 3*I), S.Integers) + r = Symbol('r', positive=True) + assert imageset(Lambda(x, r*x + 10), S.Integers + ) == ImageSet(Lambda(x, r*x + 10), S.Integers) + # reduce real part: + assert imageset(Lambda(x, 3*x + 8 + 5*I), S.Integers + ) == ImageSet(Lambda(x, 3*x + 2 + 5*I), S.Integers) + + +def test_Rationals(): + assert S.Integers.is_subset(S.Rationals) + assert S.Naturals.is_subset(S.Rationals) + assert S.Naturals0.is_subset(S.Rationals) + assert S.Rationals.is_subset(S.Reals) + assert S.Rationals.inf is -oo + assert S.Rationals.sup is oo + it = iter(S.Rationals) + assert [next(it) for i in range(12)] == [ + 0, 1, -1, S.Half, 2, Rational(-1, 2), -2, + Rational(1, 3), 3, Rational(-1, 3), -3, Rational(2, 3)] + assert Basic() not in S.Rationals + assert S.Half in S.Rationals + assert S.Rationals.contains(0.5) == Contains( + 0.5, S.Rationals, evaluate=False) + assert 2 in S.Rationals + r = symbols('r', rational=True) + assert r in S.Rationals + raises(TypeError, lambda: x in S.Rationals) + # issue #18134: + assert S.Rationals.boundary == S.Reals + assert S.Rationals.closure == S.Reals + assert S.Rationals.is_open == False + assert S.Rationals.is_closed == False + + +def test_NZQRC_unions(): + # check that all trivial number set unions are simplified: + nbrsets = (S.Naturals, S.Naturals0, S.Integers, S.Rationals, + S.Reals, S.Complexes) + unions = (Union(a, b) for a in nbrsets for b in nbrsets) + assert all(u.is_Union is False for u in unions) + + +def test_imageset_intersection(): + n = Dummy() + s = ImageSet(Lambda(n, -I*(I*(2*pi*n - pi/4) + + log(Abs(sqrt(-I))))), S.Integers) + assert s.intersect(S.Reals) == ImageSet( + Lambda(n, 2*pi*n + pi*Rational(7, 4)), S.Integers) + + +def test_issue_17858(): + assert 1 in Range(-oo, oo) + assert 0 in Range(oo, -oo, -1) + assert oo not in Range(-oo, oo) + assert -oo not in Range(-oo, oo) + +def test_issue_17859(): + r = Range(-oo,oo) + raises(ValueError,lambda: r[::2]) + raises(ValueError, lambda: r[::-2]) + r = Range(oo,-oo,-1) + raises(ValueError,lambda: r[::2]) + raises(ValueError, lambda: r[::-2]) diff --git a/.venv/lib/python3.12/site-packages/sympy/sets/tests/test_ordinals.py b/.venv/lib/python3.12/site-packages/sympy/sets/tests/test_ordinals.py new file mode 100644 index 0000000000000000000000000000000000000000..973ca329586f3e904f9377c44022c266f81c805c --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/sets/tests/test_ordinals.py @@ -0,0 +1,67 @@ +from sympy.sets.ordinals import Ordinal, OmegaPower, ord0, omega +from sympy.testing.pytest import raises + +def test_string_ordinals(): + assert str(omega) == 'w' + assert str(Ordinal(OmegaPower(5, 3), OmegaPower(3, 2))) == 'w**5*3 + w**3*2' + assert str(Ordinal(OmegaPower(5, 3), OmegaPower(0, 5))) == 'w**5*3 + 5' + assert str(Ordinal(OmegaPower(1, 3), OmegaPower(0, 5))) == 'w*3 + 5' + assert str(Ordinal(OmegaPower(omega + 1, 1), OmegaPower(3, 2))) == 'w**(w + 1) + w**3*2' + +def test_addition_with_integers(): + assert 3 + Ordinal(OmegaPower(5, 3)) == Ordinal(OmegaPower(5, 3)) + assert Ordinal(OmegaPower(5, 3))+3 == Ordinal(OmegaPower(5, 3), OmegaPower(0, 3)) + assert Ordinal(OmegaPower(5, 3), OmegaPower(0, 2))+3 == \ + Ordinal(OmegaPower(5, 3), OmegaPower(0, 5)) + + +def test_addition_with_ordinals(): + assert Ordinal(OmegaPower(5, 3), OmegaPower(3, 2)) + Ordinal(OmegaPower(3, 3)) == \ + Ordinal(OmegaPower(5, 3), OmegaPower(3, 5)) + assert Ordinal(OmegaPower(5, 3), OmegaPower(3, 2)) + Ordinal(OmegaPower(4, 2)) == \ + Ordinal(OmegaPower(5, 3), OmegaPower(4, 2)) + assert Ordinal(OmegaPower(omega, 2), OmegaPower(3, 2)) + Ordinal(OmegaPower(4, 2)) == \ + Ordinal(OmegaPower(omega, 2), OmegaPower(4, 2)) + +def test_comparison(): + assert Ordinal(OmegaPower(5, 3)) > Ordinal(OmegaPower(4, 3), OmegaPower(2, 1)) + assert Ordinal(OmegaPower(5, 3), OmegaPower(3, 2)) < Ordinal(OmegaPower(5, 4)) + assert Ordinal(OmegaPower(5, 4)) < Ordinal(OmegaPower(5, 5), OmegaPower(4, 1)) + + assert Ordinal(OmegaPower(5, 3), OmegaPower(3, 2)) == \ + Ordinal(OmegaPower(5, 3), OmegaPower(3, 2)) + assert not Ordinal(OmegaPower(5, 3), OmegaPower(3, 2)) == Ordinal(OmegaPower(5, 3)) + assert Ordinal(OmegaPower(omega, 3)) > Ordinal(OmegaPower(5, 3)) + +def test_multiplication_with_integers(): + w = omega + assert 3*w == w + assert w*9 == Ordinal(OmegaPower(1, 9)) + +def test_multiplication(): + w = omega + assert w*(w + 1) == w*w + w + assert (w + 1)*(w + 1) == w*w + w + 1 + assert w*1 == w + assert 1*w == w + assert w*ord0 == ord0 + assert ord0*w == ord0 + assert w**w == w * w**w + assert (w**w)*w*w == w**(w + 2) + +def test_exponentiation(): + w = omega + assert w**2 == w*w + assert w**3 == w*w*w + assert w**(w + 1) == Ordinal(OmegaPower(omega + 1, 1)) + assert (w**w)*(w**w) == w**(w*2) + +def test_comapre_not_instance(): + w = OmegaPower(omega + 1, 1) + assert(not (w == None)) + assert(not (w < 5)) + raises(TypeError, lambda: w < 6.66) + +def test_is_successort(): + w = Ordinal(OmegaPower(5, 1)) + assert not w.is_successor_ordinal diff --git a/.venv/lib/python3.12/site-packages/sympy/sets/tests/test_powerset.py b/.venv/lib/python3.12/site-packages/sympy/sets/tests/test_powerset.py new file mode 100644 index 0000000000000000000000000000000000000000..2e3a407d565f6b9537a296af103ec0a4e137cff9 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/sets/tests/test_powerset.py @@ -0,0 +1,141 @@ +from sympy.core.expr import unchanged +from sympy.core.singleton import S +from sympy.core.symbol import Symbol +from sympy.sets.contains import Contains +from sympy.sets.fancysets import Interval +from sympy.sets.powerset import PowerSet +from sympy.sets.sets import FiniteSet +from sympy.testing.pytest import raises, XFAIL + + +def test_powerset_creation(): + assert unchanged(PowerSet, FiniteSet(1, 2)) + assert unchanged(PowerSet, S.EmptySet) + raises(ValueError, lambda: PowerSet(123)) + assert unchanged(PowerSet, S.Reals) + assert unchanged(PowerSet, S.Integers) + + +def test_powerset_rewrite_FiniteSet(): + assert PowerSet(FiniteSet(1, 2)).rewrite(FiniteSet) == \ + FiniteSet(S.EmptySet, FiniteSet(1), FiniteSet(2), FiniteSet(1, 2)) + assert PowerSet(S.EmptySet).rewrite(FiniteSet) == FiniteSet(S.EmptySet) + assert PowerSet(S.Naturals).rewrite(FiniteSet) == PowerSet(S.Naturals) + + +def test_finiteset_rewrite_powerset(): + assert FiniteSet(S.EmptySet).rewrite(PowerSet) == PowerSet(S.EmptySet) + assert FiniteSet( + S.EmptySet, FiniteSet(1), + FiniteSet(2), FiniteSet(1, 2)).rewrite(PowerSet) == \ + PowerSet(FiniteSet(1, 2)) + assert FiniteSet(1, 2, 3).rewrite(PowerSet) == FiniteSet(1, 2, 3) + + +def test_powerset__contains__(): + subset_series = [ + S.EmptySet, + FiniteSet(1, 2), + S.Naturals, + S.Naturals0, + S.Integers, + S.Rationals, + S.Reals, + S.Complexes] + + l = len(subset_series) + for i in range(l): + for j in range(l): + if i <= j: + assert subset_series[i] in \ + PowerSet(subset_series[j], evaluate=False) + else: + assert subset_series[i] not in \ + PowerSet(subset_series[j], evaluate=False) + + +@XFAIL +def test_failing_powerset__contains__(): + # XXX These are failing when evaluate=True, + # but using unevaluated PowerSet works fine. + assert FiniteSet(1, 2) not in PowerSet(S.EmptySet).rewrite(FiniteSet) + assert S.Naturals not in PowerSet(S.EmptySet).rewrite(FiniteSet) + assert S.Naturals not in PowerSet(FiniteSet(1, 2)).rewrite(FiniteSet) + assert S.Naturals0 not in PowerSet(S.EmptySet).rewrite(FiniteSet) + assert S.Naturals0 not in PowerSet(FiniteSet(1, 2)).rewrite(FiniteSet) + assert S.Integers not in PowerSet(S.EmptySet).rewrite(FiniteSet) + assert S.Integers not in PowerSet(FiniteSet(1, 2)).rewrite(FiniteSet) + assert S.Rationals not in PowerSet(S.EmptySet).rewrite(FiniteSet) + assert S.Rationals not in PowerSet(FiniteSet(1, 2)).rewrite(FiniteSet) + assert S.Reals not in PowerSet(S.EmptySet).rewrite(FiniteSet) + assert S.Reals not in PowerSet(FiniteSet(1, 2)).rewrite(FiniteSet) + assert S.Complexes not in PowerSet(S.EmptySet).rewrite(FiniteSet) + assert S.Complexes not in PowerSet(FiniteSet(1, 2)).rewrite(FiniteSet) + + +def test_powerset__len__(): + A = PowerSet(S.EmptySet, evaluate=False) + assert len(A) == 1 + A = PowerSet(A, evaluate=False) + assert len(A) == 2 + A = PowerSet(A, evaluate=False) + assert len(A) == 4 + A = PowerSet(A, evaluate=False) + assert len(A) == 16 + + +def test_powerset__iter__(): + a = PowerSet(FiniteSet(1, 2)).__iter__() + assert next(a) == S.EmptySet + assert next(a) == FiniteSet(1) + assert next(a) == FiniteSet(2) + assert next(a) == FiniteSet(1, 2) + + a = PowerSet(S.Naturals).__iter__() + assert next(a) == S.EmptySet + assert next(a) == FiniteSet(1) + assert next(a) == FiniteSet(2) + assert next(a) == FiniteSet(1, 2) + assert next(a) == FiniteSet(3) + assert next(a) == FiniteSet(1, 3) + assert next(a) == FiniteSet(2, 3) + assert next(a) == FiniteSet(1, 2, 3) + + +def test_powerset_contains(): + A = PowerSet(FiniteSet(1), evaluate=False) + assert A.contains(2) == Contains(2, A) + + x = Symbol('x') + + A = PowerSet(FiniteSet(x), evaluate=False) + assert A.contains(FiniteSet(1)) == Contains(FiniteSet(1), A) + + +def test_powerset_method(): + # EmptySet + A = FiniteSet() + pset = A.powerset() + assert len(pset) == 1 + assert pset == FiniteSet(S.EmptySet) + + # FiniteSets + A = FiniteSet(1, 2) + pset = A.powerset() + assert len(pset) == 2**len(A) + assert pset == FiniteSet(FiniteSet(), FiniteSet(1), + FiniteSet(2), A) + # Not finite sets + A = Interval(0, 1) + assert A.powerset() == PowerSet(A) + +def test_is_subset(): + # covers line 101-102 + # initialize powerset(1), which is a subset of powerset(1,2) + subset = PowerSet(FiniteSet(1)) + pset = PowerSet(FiniteSet(1, 2)) + bad_set = PowerSet(FiniteSet(2, 3)) + # assert "subset" is subset of pset == True + assert subset.is_subset(pset) + # assert "bad_set" is subset of pset == False + assert not pset.is_subset(bad_set) diff --git a/.venv/lib/python3.12/site-packages/sympy/sets/tests/test_setexpr.py b/.venv/lib/python3.12/site-packages/sympy/sets/tests/test_setexpr.py new file mode 100644 index 0000000000000000000000000000000000000000..faab1261c8d3e86901b04d30e8bc94de31642b93 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/sets/tests/test_setexpr.py @@ -0,0 +1,317 @@ +from sympy.sets.setexpr import SetExpr +from sympy.sets import Interval, FiniteSet, Intersection, ImageSet, Union + +from sympy.core.expr import Expr +from sympy.core.function import Lambda +from sympy.core.numbers import (I, Rational, oo) +from sympy.core.singleton import S +from sympy.core.symbol import (Dummy, Symbol, symbols) +from sympy.functions.elementary.exponential import (exp, log) +from sympy.functions.elementary.miscellaneous import (Max, Min, sqrt) +from sympy.functions.elementary.trigonometric import cos +from sympy.sets.sets import Set + + +a, x = symbols("a, x") +_d = Dummy("d") + + +def test_setexpr(): + se = SetExpr(Interval(0, 1)) + assert isinstance(se.set, Set) + assert isinstance(se, Expr) + + +def test_scalar_funcs(): + assert SetExpr(Interval(0, 1)).set == Interval(0, 1) + a, b = Symbol('a', real=True), Symbol('b', real=True) + a, b = 1, 2 + # TODO: add support for more functions in the future: + for f in [exp, log]: + input_se = f(SetExpr(Interval(a, b))) + output = input_se.set + expected = Interval(Min(f(a), f(b)), Max(f(a), f(b))) + assert output == expected + + +def test_Add_Mul(): + assert (SetExpr(Interval(0, 1)) + 1).set == Interval(1, 2) + assert (SetExpr(Interval(0, 1))*2).set == Interval(0, 2) + + +def test_Pow(): + assert (SetExpr(Interval(0, 2))**2).set == Interval(0, 4) + + +def test_compound(): + assert (exp(SetExpr(Interval(0, 1))*2 + 1)).set == \ + Interval(exp(1), exp(3)) + + +def test_Interval_Interval(): + assert (SetExpr(Interval(1, 2)) + SetExpr(Interval(10, 20))).set == \ + Interval(11, 22) + assert (SetExpr(Interval(1, 2))*SetExpr(Interval(10, 20))).set == \ + Interval(10, 40) + + +def test_FiniteSet_FiniteSet(): + assert (SetExpr(FiniteSet(1, 2, 3)) + SetExpr(FiniteSet(1, 2))).set == \ + FiniteSet(2, 3, 4, 5) + assert (SetExpr(FiniteSet(1, 2, 3))*SetExpr(FiniteSet(1, 2))).set == \ + FiniteSet(1, 2, 3, 4, 6) + + +def test_Interval_FiniteSet(): + assert (SetExpr(FiniteSet(1, 2)) + SetExpr(Interval(0, 10))).set == \ + Interval(1, 12) + + +def test_Many_Sets(): + assert (SetExpr(Interval(0, 1)) + + SetExpr(Interval(2, 3)) + + SetExpr(FiniteSet(10, 11, 12))).set == Interval(12, 16) + + +def test_same_setexprs_are_not_identical(): + a = SetExpr(FiniteSet(0, 1)) + b = SetExpr(FiniteSet(0, 1)) + assert (a + b).set == FiniteSet(0, 1, 2) + + # Cannot detect the set being the same: + # assert (a + a).set == FiniteSet(0, 2) + + +def test_Interval_arithmetic(): + i12cc = SetExpr(Interval(1, 2)) + i12lo = SetExpr(Interval.Lopen(1, 2)) + i12ro = SetExpr(Interval.Ropen(1, 2)) + i12o = SetExpr(Interval.open(1, 2)) + + n23cc = SetExpr(Interval(-2, 3)) + n23lo = SetExpr(Interval.Lopen(-2, 3)) + n23ro = SetExpr(Interval.Ropen(-2, 3)) + n23o = SetExpr(Interval.open(-2, 3)) + + n3n2cc = SetExpr(Interval(-3, -2)) + + assert i12cc + i12cc == SetExpr(Interval(2, 4)) + assert i12cc - i12cc == SetExpr(Interval(-1, 1)) + assert i12cc*i12cc == SetExpr(Interval(1, 4)) + assert i12cc/i12cc == SetExpr(Interval(S.Half, 2)) + assert i12cc**2 == SetExpr(Interval(1, 4)) + assert i12cc**3 == SetExpr(Interval(1, 8)) + + assert i12lo + i12ro == SetExpr(Interval.open(2, 4)) + assert i12lo - i12ro == SetExpr(Interval.Lopen(-1, 1)) + assert i12lo*i12ro == SetExpr(Interval.open(1, 4)) + assert i12lo/i12ro == SetExpr(Interval.Lopen(S.Half, 2)) + assert i12lo + i12lo == SetExpr(Interval.Lopen(2, 4)) + assert i12lo - i12lo == SetExpr(Interval.open(-1, 1)) + assert i12lo*i12lo == SetExpr(Interval.Lopen(1, 4)) + assert i12lo/i12lo == SetExpr(Interval.open(S.Half, 2)) + assert i12lo + i12cc == SetExpr(Interval.Lopen(2, 4)) + assert i12lo - i12cc == SetExpr(Interval.Lopen(-1, 1)) + assert i12lo*i12cc == SetExpr(Interval.Lopen(1, 4)) + assert i12lo/i12cc == SetExpr(Interval.Lopen(S.Half, 2)) + assert i12lo + i12o == SetExpr(Interval.open(2, 4)) + assert i12lo - i12o == SetExpr(Interval.open(-1, 1)) + assert i12lo*i12o == SetExpr(Interval.open(1, 4)) + assert i12lo/i12o == SetExpr(Interval.open(S.Half, 2)) + assert i12lo**2 == SetExpr(Interval.Lopen(1, 4)) + assert i12lo**3 == SetExpr(Interval.Lopen(1, 8)) + + assert i12ro + i12ro == SetExpr(Interval.Ropen(2, 4)) + assert i12ro - i12ro == SetExpr(Interval.open(-1, 1)) + assert i12ro*i12ro == SetExpr(Interval.Ropen(1, 4)) + assert i12ro/i12ro == SetExpr(Interval.open(S.Half, 2)) + assert i12ro + i12cc == SetExpr(Interval.Ropen(2, 4)) + assert i12ro - i12cc == SetExpr(Interval.Ropen(-1, 1)) + assert i12ro*i12cc == SetExpr(Interval.Ropen(1, 4)) + assert i12ro/i12cc == SetExpr(Interval.Ropen(S.Half, 2)) + assert i12ro + i12o == SetExpr(Interval.open(2, 4)) + assert i12ro - i12o == SetExpr(Interval.open(-1, 1)) + assert i12ro*i12o == SetExpr(Interval.open(1, 4)) + assert i12ro/i12o == SetExpr(Interval.open(S.Half, 2)) + assert i12ro**2 == SetExpr(Interval.Ropen(1, 4)) + assert i12ro**3 == SetExpr(Interval.Ropen(1, 8)) + + assert i12o + i12lo == SetExpr(Interval.open(2, 4)) + assert i12o - i12lo == SetExpr(Interval.open(-1, 1)) + assert i12o*i12lo == SetExpr(Interval.open(1, 4)) + assert i12o/i12lo == SetExpr(Interval.open(S.Half, 2)) + assert i12o + i12ro == SetExpr(Interval.open(2, 4)) + assert i12o - i12ro == SetExpr(Interval.open(-1, 1)) + assert i12o*i12ro == SetExpr(Interval.open(1, 4)) + assert i12o/i12ro == SetExpr(Interval.open(S.Half, 2)) + assert i12o + i12cc == SetExpr(Interval.open(2, 4)) + assert i12o - i12cc == SetExpr(Interval.open(-1, 1)) + assert i12o*i12cc == SetExpr(Interval.open(1, 4)) + assert i12o/i12cc == SetExpr(Interval.open(S.Half, 2)) + assert i12o**2 == SetExpr(Interval.open(1, 4)) + assert i12o**3 == SetExpr(Interval.open(1, 8)) + + assert n23cc + n23cc == SetExpr(Interval(-4, 6)) + assert n23cc - n23cc == SetExpr(Interval(-5, 5)) + assert n23cc*n23cc == SetExpr(Interval(-6, 9)) + assert n23cc/n23cc == SetExpr(Interval.open(-oo, oo)) + assert n23cc + n23ro == SetExpr(Interval.Ropen(-4, 6)) + assert n23cc - n23ro == SetExpr(Interval.Lopen(-5, 5)) + assert n23cc*n23ro == SetExpr(Interval.Ropen(-6, 9)) + assert n23cc/n23ro == SetExpr(Interval.Lopen(-oo, oo)) + assert n23cc + n23lo == SetExpr(Interval.Lopen(-4, 6)) + assert n23cc - n23lo == SetExpr(Interval.Ropen(-5, 5)) + assert n23cc*n23lo == SetExpr(Interval(-6, 9)) + assert n23cc/n23lo == SetExpr(Interval.open(-oo, oo)) + assert n23cc + n23o == SetExpr(Interval.open(-4, 6)) + assert n23cc - n23o == SetExpr(Interval.open(-5, 5)) + assert n23cc*n23o == SetExpr(Interval.open(-6, 9)) + assert n23cc/n23o == SetExpr(Interval.open(-oo, oo)) + assert n23cc**2 == SetExpr(Interval(0, 9)) + assert n23cc**3 == SetExpr(Interval(-8, 27)) + + n32cc = SetExpr(Interval(-3, 2)) + n32lo = SetExpr(Interval.Lopen(-3, 2)) + n32ro = SetExpr(Interval.Ropen(-3, 2)) + assert n32cc*n32lo == SetExpr(Interval.Ropen(-6, 9)) + assert n32cc*n32cc == SetExpr(Interval(-6, 9)) + assert n32lo*n32cc == SetExpr(Interval.Ropen(-6, 9)) + assert n32cc*n32ro == SetExpr(Interval(-6, 9)) + assert n32lo*n32ro == SetExpr(Interval.Ropen(-6, 9)) + assert n32cc/n32lo == SetExpr(Interval.Ropen(-oo, oo)) + assert i12cc/n32lo == SetExpr(Interval.Ropen(-oo, oo)) + + assert n3n2cc**2 == SetExpr(Interval(4, 9)) + assert n3n2cc**3 == SetExpr(Interval(-27, -8)) + + assert n23cc + i12cc == SetExpr(Interval(-1, 5)) + assert n23cc - i12cc == SetExpr(Interval(-4, 2)) + assert n23cc*i12cc == SetExpr(Interval(-4, 6)) + assert n23cc/i12cc == SetExpr(Interval(-2, 3)) + + +def test_SetExpr_Intersection(): + x, y, z, w = symbols("x y z w") + set1 = Interval(x, y) + set2 = Interval(w, z) + inter = Intersection(set1, set2) + se = SetExpr(inter) + assert exp(se).set == Intersection( + ImageSet(Lambda(x, exp(x)), set1), + ImageSet(Lambda(x, exp(x)), set2)) + assert cos(se).set == ImageSet(Lambda(x, cos(x)), inter) + + +def test_SetExpr_Interval_div(): + # TODO: some expressions cannot be calculated due to bugs (currently + # commented): + assert SetExpr(Interval(-3, -2))/SetExpr(Interval(-2, 1)) == SetExpr(Interval(-oo, oo)) + assert SetExpr(Interval(2, 3))/SetExpr(Interval(-2, 2)) == SetExpr(Interval(-oo, oo)) + + assert SetExpr(Interval(-3, -2))/SetExpr(Interval(0, 4)) == SetExpr(Interval(-oo, Rational(-1, 2))) + assert SetExpr(Interval(2, 4))/SetExpr(Interval(-3, 0)) == SetExpr(Interval(-oo, Rational(-2, 3))) + assert SetExpr(Interval(2, 4))/SetExpr(Interval(0, 3)) == SetExpr(Interval(Rational(2, 3), oo)) + + # assert SetExpr(Interval(0, 1))/SetExpr(Interval(0, 1)) == SetExpr(Interval(0, oo)) + # assert SetExpr(Interval(-1, 0))/SetExpr(Interval(0, 1)) == SetExpr(Interval(-oo, 0)) + assert SetExpr(Interval(-1, 2))/SetExpr(Interval(-2, 2)) == SetExpr(Interval(-oo, oo)) + + assert 1/SetExpr(Interval(-1, 2)) == SetExpr(Union(Interval(-oo, -1), Interval(S.Half, oo))) + + assert 1/SetExpr(Interval(0, 2)) == SetExpr(Interval(S.Half, oo)) + assert (-1)/SetExpr(Interval(0, 2)) == SetExpr(Interval(-oo, Rational(-1, 2))) + assert 1/SetExpr(Interval(-oo, 0)) == SetExpr(Interval.open(-oo, 0)) + assert 1/SetExpr(Interval(-1, 0)) == SetExpr(Interval(-oo, -1)) + # assert (-2)/SetExpr(Interval(-oo, 0)) == SetExpr(Interval(0, oo)) + # assert 1/SetExpr(Interval(-oo, -1)) == SetExpr(Interval(-1, 0)) + + # assert SetExpr(Interval(1, 2))/a == Mul(SetExpr(Interval(1, 2)), 1/a, evaluate=False) + + # assert SetExpr(Interval(1, 2))/0 == SetExpr(Interval(1, 2))*zoo + # assert SetExpr(Interval(1, oo))/oo == SetExpr(Interval(0, oo)) + # assert SetExpr(Interval(1, oo))/(-oo) == SetExpr(Interval(-oo, 0)) + # assert SetExpr(Interval(-oo, -1))/oo == SetExpr(Interval(-oo, 0)) + # assert SetExpr(Interval(-oo, -1))/(-oo) == SetExpr(Interval(0, oo)) + # assert SetExpr(Interval(-oo, oo))/oo == SetExpr(Interval(-oo, oo)) + # assert SetExpr(Interval(-oo, oo))/(-oo) == SetExpr(Interval(-oo, oo)) + # assert SetExpr(Interval(-1, oo))/oo == SetExpr(Interval(0, oo)) + # assert SetExpr(Interval(-1, oo))/(-oo) == SetExpr(Interval(-oo, 0)) + # assert SetExpr(Interval(-oo, 1))/oo == SetExpr(Interval(-oo, 0)) + # assert SetExpr(Interval(-oo, 1))/(-oo) == SetExpr(Interval(0, oo)) + + +def test_SetExpr_Interval_pow(): + assert SetExpr(Interval(0, 2))**2 == SetExpr(Interval(0, 4)) + assert SetExpr(Interval(-1, 1))**2 == SetExpr(Interval(0, 1)) + assert SetExpr(Interval(1, 2))**2 == SetExpr(Interval(1, 4)) + assert SetExpr(Interval(-1, 2))**3 == SetExpr(Interval(-1, 8)) + assert SetExpr(Interval(-1, 1))**0 == SetExpr(FiniteSet(1)) + + + assert SetExpr(Interval(1, 2))**Rational(5, 2) == SetExpr(Interval(1, 4*sqrt(2))) + #assert SetExpr(Interval(-1, 2))**Rational(1, 3) == SetExpr(Interval(-1, 2**Rational(1, 3))) + #assert SetExpr(Interval(0, 2))**S.Half == SetExpr(Interval(0, sqrt(2))) + + #assert SetExpr(Interval(-4, 2))**Rational(2, 3) == SetExpr(Interval(0, 2*2**Rational(1, 3))) + + #assert SetExpr(Interval(-1, 5))**S.Half == SetExpr(Interval(0, sqrt(5))) + #assert SetExpr(Interval(-oo, 2))**S.Half == SetExpr(Interval(0, sqrt(2))) + #assert SetExpr(Interval(-2, 3))**(Rational(-1, 4)) == SetExpr(Interval(0, oo)) + + assert SetExpr(Interval(1, 5))**(-2) == SetExpr(Interval(Rational(1, 25), 1)) + assert SetExpr(Interval(-1, 3))**(-2) == SetExpr(Interval(0, oo)) + + assert SetExpr(Interval(0, 2))**(-2) == SetExpr(Interval(Rational(1, 4), oo)) + assert SetExpr(Interval(-1, 2))**(-3) == SetExpr(Union(Interval(-oo, -1), Interval(Rational(1, 8), oo))) + assert SetExpr(Interval(-3, -2))**(-3) == SetExpr(Interval(Rational(-1, 8), Rational(-1, 27))) + assert SetExpr(Interval(-3, -2))**(-2) == SetExpr(Interval(Rational(1, 9), Rational(1, 4))) + #assert SetExpr(Interval(0, oo))**S.Half == SetExpr(Interval(0, oo)) + #assert SetExpr(Interval(-oo, -1))**Rational(1, 3) == SetExpr(Interval(-oo, -1)) + #assert SetExpr(Interval(-2, 3))**(Rational(-1, 3)) == SetExpr(Interval(-oo, oo)) + + assert SetExpr(Interval(-oo, 0))**(-2) == SetExpr(Interval.open(0, oo)) + assert SetExpr(Interval(-2, 0))**(-2) == SetExpr(Interval(Rational(1, 4), oo)) + + assert SetExpr(Interval(Rational(1, 3), S.Half))**oo == SetExpr(FiniteSet(0)) + assert SetExpr(Interval(0, S.Half))**oo == SetExpr(FiniteSet(0)) + assert SetExpr(Interval(S.Half, 1))**oo == SetExpr(Interval(0, oo)) + assert SetExpr(Interval(0, 1))**oo == SetExpr(Interval(0, oo)) + assert SetExpr(Interval(2, 3))**oo == SetExpr(FiniteSet(oo)) + assert SetExpr(Interval(1, 2))**oo == SetExpr(Interval(0, oo)) + assert SetExpr(Interval(S.Half, 3))**oo == SetExpr(Interval(0, oo)) + assert SetExpr(Interval(Rational(-1, 3), Rational(-1, 4)))**oo == SetExpr(FiniteSet(0)) + assert SetExpr(Interval(-1, Rational(-1, 2)))**oo == SetExpr(Interval(-oo, oo)) + assert SetExpr(Interval(-3, -2))**oo == SetExpr(FiniteSet(-oo, oo)) + assert SetExpr(Interval(-2, -1))**oo == SetExpr(Interval(-oo, oo)) + assert SetExpr(Interval(-2, Rational(-1, 2)))**oo == SetExpr(Interval(-oo, oo)) + assert SetExpr(Interval(Rational(-1, 2), S.Half))**oo == SetExpr(FiniteSet(0)) + assert SetExpr(Interval(Rational(-1, 2), 1))**oo == SetExpr(Interval(0, oo)) + assert SetExpr(Interval(Rational(-2, 3), 2))**oo == SetExpr(Interval(0, oo)) + assert SetExpr(Interval(-1, 1))**oo == SetExpr(Interval(-oo, oo)) + assert SetExpr(Interval(-1, S.Half))**oo == SetExpr(Interval(-oo, oo)) + assert SetExpr(Interval(-1, 2))**oo == SetExpr(Interval(-oo, oo)) + assert SetExpr(Interval(-2, S.Half))**oo == SetExpr(Interval(-oo, oo)) + + assert (SetExpr(Interval(1, 2))**x).dummy_eq(SetExpr(ImageSet(Lambda(_d, _d**x), Interval(1, 2)))) + + assert SetExpr(Interval(2, 3))**(-oo) == SetExpr(FiniteSet(0)) + assert SetExpr(Interval(0, 2))**(-oo) == SetExpr(Interval(0, oo)) + assert (SetExpr(Interval(-1, 2))**(-oo)).dummy_eq(SetExpr(ImageSet(Lambda(_d, _d**(-oo)), Interval(-1, 2)))) + + +def test_SetExpr_Integers(): + assert SetExpr(S.Integers) + 1 == SetExpr(S.Integers) + assert (SetExpr(S.Integers) + I).dummy_eq( + SetExpr(ImageSet(Lambda(_d, _d + I), S.Integers))) + assert SetExpr(S.Integers)*(-1) == SetExpr(S.Integers) + assert (SetExpr(S.Integers)*2).dummy_eq( + SetExpr(ImageSet(Lambda(_d, 2*_d), S.Integers))) + assert (SetExpr(S.Integers)*I).dummy_eq( + SetExpr(ImageSet(Lambda(_d, I*_d), S.Integers))) + # issue #18050: + assert SetExpr(S.Integers)._eval_func(Lambda(x, I*x + 1)).dummy_eq( + SetExpr(ImageSet(Lambda(_d, I*_d + 1), S.Integers))) + # needs improvement: + assert (SetExpr(S.Integers)*I + 1).dummy_eq( + SetExpr(ImageSet(Lambda(x, x + 1), + ImageSet(Lambda(_d, _d*I), S.Integers)))) diff --git a/.venv/lib/python3.12/site-packages/sympy/sets/tests/test_sets.py b/.venv/lib/python3.12/site-packages/sympy/sets/tests/test_sets.py new file mode 100644 index 0000000000000000000000000000000000000000..657ab19a90eb88ca48f266f7a5cf050504caed43 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/sets/tests/test_sets.py @@ -0,0 +1,1753 @@ +from sympy.concrete.summations import Sum +from sympy.core.add import Add +from sympy.core.containers import TupleKind +from sympy.core.function import Lambda +from sympy.core.kind import NumberKind, UndefinedKind +from sympy.core.numbers import (Float, I, Rational, nan, oo, pi, zoo) +from sympy.core.power import Pow +from sympy.core.singleton import S +from sympy.core.symbol import (Symbol, symbols) +from sympy.core.sympify import sympify +from sympy.functions.elementary.miscellaneous import (Max, Min, sqrt) +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.logic.boolalg import (false, true) +from sympy.matrices.kind import MatrixKind +from sympy.matrices.dense import Matrix +from sympy.polys.rootoftools import rootof +from sympy.sets.contains import Contains +from sympy.sets.fancysets import (ImageSet, Range) +from sympy.sets.sets import (Complement, DisjointUnion, FiniteSet, Intersection, Interval, ProductSet, Set, SymmetricDifference, Union, imageset, SetKind) +from mpmath import mpi + +from sympy.core.expr import unchanged +from sympy.core.relational import Eq, Ne, Le, Lt, LessThan +from sympy.logic import And, Or, Xor +from sympy.testing.pytest import raises, XFAIL, warns_deprecated_sympy +from sympy.utilities.iterables import cartes + +from sympy.abc import x, y, z, m, n + +EmptySet = S.EmptySet + +def test_imageset(): + ints = S.Integers + assert imageset(x, x - 1, S.Naturals) is S.Naturals0 + assert imageset(x, x + 1, S.Naturals0) is S.Naturals + assert imageset(x, abs(x), S.Naturals0) is S.Naturals0 + assert imageset(x, abs(x), S.Naturals) is S.Naturals + assert imageset(x, abs(x), S.Integers) is S.Naturals0 + # issue 16878a + r = symbols('r', real=True) + assert imageset(x, (x, x), S.Reals)._contains((1, r)) == None + assert imageset(x, (x, x), S.Reals)._contains((1, 2)) == False + assert (r, r) in imageset(x, (x, x), S.Reals) + assert 1 + I in imageset(x, x + I, S.Reals) + assert {1} not in imageset(x, (x,), S.Reals) + assert (1, 1) not in imageset(x, (x,), S.Reals) + raises(TypeError, lambda: imageset(x, ints)) + raises(ValueError, lambda: imageset(x, y, z, ints)) + raises(ValueError, lambda: imageset(Lambda(x, cos(x)), y)) + assert (1, 2) in imageset(Lambda((x, y), (x, y)), ints, ints) + raises(ValueError, lambda: imageset(Lambda(x, x), ints, ints)) + assert imageset(cos, ints) == ImageSet(Lambda(x, cos(x)), ints) + def f(x): + return cos(x) + assert imageset(f, ints) == imageset(x, cos(x), ints) + f = lambda x: cos(x) + assert imageset(f, ints) == ImageSet(Lambda(x, cos(x)), ints) + assert imageset(x, 1, ints) == FiniteSet(1) + assert imageset(x, y, ints) == {y} + assert imageset((x, y), (1, z), ints, S.Reals) == {(1, z)} + clash = Symbol('x', integer=true) + assert (str(imageset(lambda x: x + clash, Interval(-2, 1)).lamda.expr) + in ('x0 + x', 'x + x0')) + x1, x2 = symbols("x1, x2") + assert imageset(lambda x, y: + Add(x, y), Interval(1, 2), Interval(2, 3)).dummy_eq( + ImageSet(Lambda((x1, x2), x1 + x2), + Interval(1, 2), Interval(2, 3))) + + +def test_is_empty(): + for s in [S.Naturals, S.Naturals0, S.Integers, S.Rationals, S.Reals, + S.UniversalSet]: + assert s.is_empty is False + + assert S.EmptySet.is_empty is True + + +def test_is_finiteset(): + for s in [S.Naturals, S.Naturals0, S.Integers, S.Rationals, S.Reals, + S.UniversalSet]: + assert s.is_finite_set is False + + assert S.EmptySet.is_finite_set is True + + assert FiniteSet(1, 2).is_finite_set is True + assert Interval(1, 2).is_finite_set is False + assert Interval(x, y).is_finite_set is None + assert ProductSet(FiniteSet(1), FiniteSet(2)).is_finite_set is True + assert ProductSet(FiniteSet(1), Interval(1, 2)).is_finite_set is False + assert ProductSet(FiniteSet(1), Interval(x, y)).is_finite_set is None + assert Union(Interval(0, 1), Interval(2, 3)).is_finite_set is False + assert Union(FiniteSet(1), Interval(2, 3)).is_finite_set is False + assert Union(FiniteSet(1), FiniteSet(2)).is_finite_set is True + assert Union(FiniteSet(1), Interval(x, y)).is_finite_set is None + assert Intersection(Interval(x, y), FiniteSet(1)).is_finite_set is True + assert Intersection(Interval(x, y), Interval(1, 2)).is_finite_set is None + assert Intersection(FiniteSet(x), FiniteSet(y)).is_finite_set is True + assert Complement(FiniteSet(1), Interval(x, y)).is_finite_set is True + assert Complement(Interval(x, y), FiniteSet(1)).is_finite_set is None + assert Complement(Interval(1, 2), FiniteSet(x)).is_finite_set is False + assert DisjointUnion(Interval(-5, 3), FiniteSet(x, y)).is_finite_set is False + assert DisjointUnion(S.EmptySet, FiniteSet(x, y), S.EmptySet).is_finite_set is True + + +def test_deprecated_is_EmptySet(): + with warns_deprecated_sympy(): + S.EmptySet.is_EmptySet + + with warns_deprecated_sympy(): + FiniteSet(1).is_EmptySet + + +def test_interval_arguments(): + assert Interval(0, oo) == Interval(0, oo, False, True) + assert Interval(0, oo).right_open is true + assert Interval(-oo, 0) == Interval(-oo, 0, True, False) + assert Interval(-oo, 0).left_open is true + assert Interval(oo, -oo) == S.EmptySet + assert Interval(oo, oo) == S.EmptySet + assert Interval(-oo, -oo) == S.EmptySet + assert Interval(oo, x) == S.EmptySet + assert Interval(oo, oo) == S.EmptySet + assert Interval(x, -oo) == S.EmptySet + assert Interval(x, x) == {x} + + assert isinstance(Interval(1, 1), FiniteSet) + e = Sum(x, (x, 1, 3)) + assert isinstance(Interval(e, e), FiniteSet) + + assert Interval(1, 0) == S.EmptySet + assert Interval(1, 1).measure == 0 + + assert Interval(1, 1, False, True) == S.EmptySet + assert Interval(1, 1, True, False) == S.EmptySet + assert Interval(1, 1, True, True) == S.EmptySet + + + assert isinstance(Interval(0, Symbol('a')), Interval) + assert Interval(Symbol('a', positive=True), 0) == S.EmptySet + raises(ValueError, lambda: Interval(0, S.ImaginaryUnit)) + raises(ValueError, lambda: Interval(0, Symbol('z', extended_real=False))) + raises(ValueError, lambda: Interval(x, x + S.ImaginaryUnit)) + + raises(NotImplementedError, lambda: Interval(0, 1, And(x, y))) + raises(NotImplementedError, lambda: Interval(0, 1, False, And(x, y))) + raises(NotImplementedError, lambda: Interval(0, 1, z, And(x, y))) + + +def test_interval_symbolic_end_points(): + a = Symbol('a', real=True) + + assert Union(Interval(0, a), Interval(0, 3)).sup == Max(a, 3) + assert Union(Interval(a, 0), Interval(-3, 0)).inf == Min(-3, a) + + assert Interval(0, a).contains(1) == LessThan(1, a) + + +def test_interval_is_empty(): + x, y = symbols('x, y') + r = Symbol('r', real=True) + p = Symbol('p', positive=True) + n = Symbol('n', negative=True) + nn = Symbol('nn', nonnegative=True) + assert Interval(1, 2).is_empty == False + assert Interval(3, 3).is_empty == False # FiniteSet + assert Interval(r, r).is_empty == False # FiniteSet + assert Interval(r, r + nn).is_empty == False + assert Interval(x, x).is_empty == False + assert Interval(1, oo).is_empty == False + assert Interval(-oo, oo).is_empty == False + assert Interval(-oo, 1).is_empty == False + assert Interval(x, y).is_empty == None + assert Interval(r, oo).is_empty == False # real implies finite + assert Interval(n, 0).is_empty == False + assert Interval(n, 0, left_open=True).is_empty == False + assert Interval(p, 0).is_empty == True # EmptySet + assert Interval(nn, 0).is_empty == None + assert Interval(n, p).is_empty == False + assert Interval(0, p, left_open=True).is_empty == False + assert Interval(0, p, right_open=True).is_empty == False + assert Interval(0, nn, left_open=True).is_empty == None + assert Interval(0, nn, right_open=True).is_empty == None + + +def test_union(): + assert Union(Interval(1, 2), Interval(2, 3)) == Interval(1, 3) + assert Union(Interval(1, 2), Interval(2, 3, True)) == Interval(1, 3) + assert Union(Interval(1, 3), Interval(2, 4)) == Interval(1, 4) + assert Union(Interval(1, 2), Interval(1, 3)) == Interval(1, 3) + assert Union(Interval(1, 3), Interval(1, 2)) == Interval(1, 3) + assert Union(Interval(1, 3, False, True), Interval(1, 2)) == \ + Interval(1, 3, False, True) + assert Union(Interval(1, 3), Interval(1, 2, False, True)) == Interval(1, 3) + assert Union(Interval(1, 2, True), Interval(1, 3)) == Interval(1, 3) + assert Union(Interval(1, 2, True), Interval(1, 3, True)) == \ + Interval(1, 3, True) + assert Union(Interval(1, 2, True), Interval(1, 3, True, True)) == \ + Interval(1, 3, True, True) + assert Union(Interval(1, 2, True, True), Interval(1, 3, True)) == \ + Interval(1, 3, True) + assert Union(Interval(1, 3), Interval(2, 3)) == Interval(1, 3) + assert Union(Interval(1, 3, False, True), Interval(2, 3)) == \ + Interval(1, 3) + assert Union(Interval(1, 2, False, True), Interval(2, 3, True)) != \ + Interval(1, 3) + assert Union(Interval(1, 2), S.EmptySet) == Interval(1, 2) + assert Union(S.EmptySet) == S.EmptySet + + assert Union(Interval(0, 1), *[FiniteSet(1.0/n) for n in range(1, 10)]) == \ + Interval(0, 1) + # issue #18241: + x = Symbol('x') + assert Union(Interval(0, 1), FiniteSet(1, x)) == Union( + Interval(0, 1), FiniteSet(x)) + assert unchanged(Union, Interval(0, 1), FiniteSet(2, x)) + + assert Interval(1, 2).union(Interval(2, 3)) == \ + Interval(1, 2) + Interval(2, 3) + + assert Interval(1, 2).union(Interval(2, 3)) == Interval(1, 3) + + assert Union(Set()) == Set() + + assert FiniteSet(1) + FiniteSet(2) + FiniteSet(3) == FiniteSet(1, 2, 3) + assert FiniteSet('ham') + FiniteSet('eggs') == FiniteSet('ham', 'eggs') + assert FiniteSet(1, 2, 3) + S.EmptySet == FiniteSet(1, 2, 3) + + assert FiniteSet(1, 2, 3) & FiniteSet(2, 3, 4) == FiniteSet(2, 3) + assert FiniteSet(1, 2, 3) | FiniteSet(2, 3, 4) == FiniteSet(1, 2, 3, 4) + + assert FiniteSet(1, 2, 3) & S.EmptySet == S.EmptySet + assert FiniteSet(1, 2, 3) | S.EmptySet == FiniteSet(1, 2, 3) + + x = Symbol("x") + y = Symbol("y") + z = Symbol("z") + assert S.EmptySet | FiniteSet(x, FiniteSet(y, z)) == \ + FiniteSet(x, FiniteSet(y, z)) + + # Test that Intervals and FiniteSets play nicely + assert Interval(1, 3) + FiniteSet(2) == Interval(1, 3) + assert Interval(1, 3, True, True) + FiniteSet(3) == \ + Interval(1, 3, True, False) + X = Interval(1, 3) + FiniteSet(5) + Y = Interval(1, 2) + FiniteSet(3) + XandY = X.intersect(Y) + assert 2 in X and 3 in X and 3 in XandY + assert XandY.is_subset(X) and XandY.is_subset(Y) + + raises(TypeError, lambda: Union(1, 2, 3)) + + assert X.is_iterable is False + + # issue 7843 + assert Union(S.EmptySet, FiniteSet(-sqrt(-I), sqrt(-I))) == \ + FiniteSet(-sqrt(-I), sqrt(-I)) + + assert Union(S.Reals, S.Integers) == S.Reals + + +def test_union_iter(): + # Use Range because it is ordered + u = Union(Range(3), Range(5), Range(4), evaluate=False) + + # Round robin + assert list(u) == [0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4] + + +def test_union_is_empty(): + assert (Interval(x, y) + FiniteSet(1)).is_empty == False + assert (Interval(x, y) + Interval(-x, y)).is_empty == None + + +def test_difference(): + assert Interval(1, 3) - Interval(1, 2) == Interval(2, 3, True) + assert Interval(1, 3) - Interval(2, 3) == Interval(1, 2, False, True) + assert Interval(1, 3, True) - Interval(2, 3) == Interval(1, 2, True, True) + assert Interval(1, 3, True) - Interval(2, 3, True) == \ + Interval(1, 2, True, False) + assert Interval(0, 2) - FiniteSet(1) == \ + Union(Interval(0, 1, False, True), Interval(1, 2, True, False)) + + # issue #18119 + assert S.Reals - FiniteSet(I) == S.Reals + assert S.Reals - FiniteSet(-I, I) == S.Reals + assert Interval(0, 10) - FiniteSet(-I, I) == Interval(0, 10) + assert Interval(0, 10) - FiniteSet(1, I) == Union( + Interval.Ropen(0, 1), Interval.Lopen(1, 10)) + assert S.Reals - FiniteSet(1, 2 + I, x, y**2) == Complement( + Union(Interval.open(-oo, 1), Interval.open(1, oo)), FiniteSet(x, y**2), + evaluate=False) + + assert FiniteSet(1, 2, 3) - FiniteSet(2) == FiniteSet(1, 3) + assert FiniteSet('ham', 'eggs') - FiniteSet('eggs') == FiniteSet('ham') + assert FiniteSet(1, 2, 3, 4) - Interval(2, 10, True, False) == \ + FiniteSet(1, 2) + assert FiniteSet(1, 2, 3, 4) - S.EmptySet == FiniteSet(1, 2, 3, 4) + assert Union(Interval(0, 2), FiniteSet(2, 3, 4)) - Interval(1, 3) == \ + Union(Interval(0, 1, False, True), FiniteSet(4)) + + assert -1 in S.Reals - S.Naturals + + +def test_Complement(): + A = FiniteSet(1, 3, 4) + B = FiniteSet(3, 4) + C = Interval(1, 3) + D = Interval(1, 2) + + assert Complement(A, B, evaluate=False).is_iterable is True + assert Complement(A, C, evaluate=False).is_iterable is True + assert Complement(C, D, evaluate=False).is_iterable is None + + assert FiniteSet(*Complement(A, B, evaluate=False)) == FiniteSet(1) + assert FiniteSet(*Complement(A, C, evaluate=False)) == FiniteSet(4) + raises(TypeError, lambda: FiniteSet(*Complement(C, A, evaluate=False))) + + assert Complement(Interval(1, 3), Interval(1, 2)) == Interval(2, 3, True) + assert Complement(FiniteSet(1, 3, 4), FiniteSet(3, 4)) == FiniteSet(1) + assert Complement(Union(Interval(0, 2), FiniteSet(2, 3, 4)), + Interval(1, 3)) == \ + Union(Interval(0, 1, False, True), FiniteSet(4)) + + assert 3 not in Complement(Interval(0, 5), Interval(1, 4), evaluate=False) + assert -1 in Complement(S.Reals, S.Naturals, evaluate=False) + assert 1 not in Complement(S.Reals, S.Naturals, evaluate=False) + + assert Complement(S.Integers, S.UniversalSet) == EmptySet + assert S.UniversalSet.complement(S.Integers) == EmptySet + + assert (0 not in S.Reals.intersect(S.Integers - FiniteSet(0))) + + assert S.EmptySet - S.Integers == S.EmptySet + + assert (S.Integers - FiniteSet(0)) - FiniteSet(1) == S.Integers - FiniteSet(0, 1) + + assert S.Reals - Union(S.Naturals, FiniteSet(pi)) == \ + Intersection(S.Reals - S.Naturals, S.Reals - FiniteSet(pi)) + # issue 12712 + assert Complement(FiniteSet(x, y, 2), Interval(-10, 10)) == \ + Complement(FiniteSet(x, y), Interval(-10, 10)) + + A = FiniteSet(*symbols('a:c')) + B = FiniteSet(*symbols('d:f')) + assert unchanged(Complement, ProductSet(A, A), B) + + A2 = ProductSet(A, A) + B3 = ProductSet(B, B, B) + assert A2 - B3 == A2 + assert B3 - A2 == B3 + + +def test_set_operations_nonsets(): + '''Tests that e.g. FiniteSet(1) * 2 raises TypeError''' + ops = [ + lambda a, b: a + b, + lambda a, b: a - b, + lambda a, b: a * b, + lambda a, b: a / b, + lambda a, b: a // b, + lambda a, b: a | b, + lambda a, b: a & b, + lambda a, b: a ^ b, + # FiniteSet(1) ** 2 gives a ProductSet + #lambda a, b: a ** b, + ] + Sx = FiniteSet(x) + Sy = FiniteSet(y) + sets = [ + {1}, + FiniteSet(1), + Interval(1, 2), + Union(Sx, Interval(1, 2)), + Intersection(Sx, Sy), + Complement(Sx, Sy), + ProductSet(Sx, Sy), + S.EmptySet, + ] + nums = [0, 1, 2, S(0), S(1), S(2)] + + for si in sets: + for ni in nums: + for op in ops: + raises(TypeError, lambda : op(si, ni)) + raises(TypeError, lambda : op(ni, si)) + raises(TypeError, lambda: si ** object()) + raises(TypeError, lambda: si ** {1}) + + +def test_complement(): + assert Complement({1, 2}, {1}) == {2} + assert Interval(0, 1).complement(S.Reals) == \ + Union(Interval(-oo, 0, True, True), Interval(1, oo, True, True)) + assert Interval(0, 1, True, False).complement(S.Reals) == \ + Union(Interval(-oo, 0, True, False), Interval(1, oo, True, True)) + assert Interval(0, 1, False, True).complement(S.Reals) == \ + Union(Interval(-oo, 0, True, True), Interval(1, oo, False, True)) + assert Interval(0, 1, True, True).complement(S.Reals) == \ + Union(Interval(-oo, 0, True, False), Interval(1, oo, False, True)) + + assert S.UniversalSet.complement(S.EmptySet) == S.EmptySet + assert S.UniversalSet.complement(S.Reals) == S.EmptySet + assert S.UniversalSet.complement(S.UniversalSet) == S.EmptySet + + assert S.EmptySet.complement(S.Reals) == S.Reals + + assert Union(Interval(0, 1), Interval(2, 3)).complement(S.Reals) == \ + Union(Interval(-oo, 0, True, True), Interval(1, 2, True, True), + Interval(3, oo, True, True)) + + assert FiniteSet(0).complement(S.Reals) == \ + Union(Interval(-oo, 0, True, True), Interval(0, oo, True, True)) + + assert (FiniteSet(5) + Interval(S.NegativeInfinity, + 0)).complement(S.Reals) == \ + Interval(0, 5, True, True) + Interval(5, S.Infinity, True, True) + + assert FiniteSet(1, 2, 3).complement(S.Reals) == \ + Interval(S.NegativeInfinity, 1, True, True) + \ + Interval(1, 2, True, True) + Interval(2, 3, True, True) +\ + Interval(3, S.Infinity, True, True) + + assert FiniteSet(x).complement(S.Reals) == Complement(S.Reals, FiniteSet(x)) + + assert FiniteSet(0, x).complement(S.Reals) == Complement(Interval(-oo, 0, True, True) + + Interval(0, oo, True, True) + , FiniteSet(x), evaluate=False) + + square = Interval(0, 1) * Interval(0, 1) + notsquare = square.complement(S.Reals*S.Reals) + + assert all(pt in square for pt in [(0, 0), (.5, .5), (1, 0), (1, 1)]) + assert not any( + pt in notsquare for pt in [(0, 0), (.5, .5), (1, 0), (1, 1)]) + assert not any(pt in square for pt in [(-1, 0), (1.5, .5), (10, 10)]) + assert all(pt in notsquare for pt in [(-1, 0), (1.5, .5), (10, 10)]) + + +def test_intersect1(): + assert all(S.Integers.intersection(i) is i for i in + (S.Naturals, S.Naturals0)) + assert all(i.intersection(S.Integers) is i for i in + (S.Naturals, S.Naturals0)) + s = S.Naturals0 + assert S.Naturals.intersection(s) is S.Naturals + assert s.intersection(S.Naturals) is S.Naturals + x = Symbol('x') + assert Interval(0, 2).intersect(Interval(1, 2)) == Interval(1, 2) + assert Interval(0, 2).intersect(Interval(1, 2, True)) == \ + Interval(1, 2, True) + assert Interval(0, 2, True).intersect(Interval(1, 2)) == \ + Interval(1, 2, False, False) + assert Interval(0, 2, True, True).intersect(Interval(1, 2)) == \ + Interval(1, 2, False, True) + assert Interval(0, 2).intersect(Union(Interval(0, 1), Interval(2, 3))) == \ + Union(Interval(0, 1), Interval(2, 2)) + + assert FiniteSet(1, 2).intersect(FiniteSet(1, 2, 3)) == FiniteSet(1, 2) + assert FiniteSet(1, 2, x).intersect(FiniteSet(x)) == FiniteSet(x) + assert FiniteSet('ham', 'eggs').intersect(FiniteSet('ham')) == \ + FiniteSet('ham') + assert FiniteSet(1, 2, 3, 4, 5).intersect(S.EmptySet) == S.EmptySet + + assert Interval(0, 5).intersect(FiniteSet(1, 3)) == FiniteSet(1, 3) + assert Interval(0, 1, True, True).intersect(FiniteSet(1)) == S.EmptySet + + assert Union(Interval(0, 1), Interval(2, 3)).intersect(Interval(1, 2)) == \ + Union(Interval(1, 1), Interval(2, 2)) + assert Union(Interval(0, 1), Interval(2, 3)).intersect(Interval(0, 2)) == \ + Union(Interval(0, 1), Interval(2, 2)) + assert Union(Interval(0, 1), Interval(2, 3)).intersect(Interval(1, 2, True, True)) == \ + S.EmptySet + assert Union(Interval(0, 1), Interval(2, 3)).intersect(S.EmptySet) == \ + S.EmptySet + assert Union(Interval(0, 5), FiniteSet('ham')).intersect(FiniteSet(2, 3, 4, 5, 6)) == \ + Intersection(FiniteSet(2, 3, 4, 5, 6), Union(FiniteSet('ham'), Interval(0, 5))) + assert Intersection(FiniteSet(1, 2, 3), Interval(2, x), Interval(3, y)) == \ + Intersection(FiniteSet(3), Interval(2, x), Interval(3, y), evaluate=False) + assert Intersection(FiniteSet(1, 2), Interval(0, 3), Interval(x, y)) == \ + Intersection({1, 2}, Interval(x, y), evaluate=False) + assert Intersection(FiniteSet(1, 2, 4), Interval(0, 3), Interval(x, y)) == \ + Intersection({1, 2}, Interval(x, y), evaluate=False) + # XXX: Is the real=True necessary here? + # https://github.com/sympy/sympy/issues/17532 + m, n = symbols('m, n', real=True) + assert Intersection(FiniteSet(m), FiniteSet(m, n), Interval(m, m+1)) == \ + FiniteSet(m) + + # issue 8217 + assert Intersection(FiniteSet(x), FiniteSet(y)) == \ + Intersection(FiniteSet(x), FiniteSet(y), evaluate=False) + assert FiniteSet(x).intersect(S.Reals) == \ + Intersection(S.Reals, FiniteSet(x), evaluate=False) + + # tests for the intersection alias + assert Interval(0, 5).intersection(FiniteSet(1, 3)) == FiniteSet(1, 3) + assert Interval(0, 1, True, True).intersection(FiniteSet(1)) == S.EmptySet + + assert Union(Interval(0, 1), Interval(2, 3)).intersection(Interval(1, 2)) == \ + Union(Interval(1, 1), Interval(2, 2)) + + # canonical boundary selected + a = sqrt(2*sqrt(6) + 5) + b = sqrt(2) + sqrt(3) + assert Interval(a, 4).intersection(Interval(b, 5)) == Interval(b, 4) + assert Interval(1, a).intersection(Interval(0, b)) == Interval(1, b) + + +def test_intersection_interval_float(): + # intersection of Intervals with mixed Rational/Float boundaries should + # lead to Float boundaries in all cases regardless of which Interval is + # open or closed. + typs = [ + (Interval, Interval, Interval), + (Interval, Interval.open, Interval.open), + (Interval, Interval.Lopen, Interval.Lopen), + (Interval, Interval.Ropen, Interval.Ropen), + (Interval.open, Interval.open, Interval.open), + (Interval.open, Interval.Lopen, Interval.open), + (Interval.open, Interval.Ropen, Interval.open), + (Interval.Lopen, Interval.Lopen, Interval.Lopen), + (Interval.Lopen, Interval.Ropen, Interval.open), + (Interval.Ropen, Interval.Ropen, Interval.Ropen), + ] + + as_float = lambda a1, a2: a2 if isinstance(a2, float) else a1 + + for t1, t2, t3 in typs: + for t1i, t2i in [(t1, t2), (t2, t1)]: + for a1, a2, b1, b2 in cartes([2, 2.0], [2, 2.0], [3, 3.0], [3, 3.0]): + I1 = t1(a1, b1) + I2 = t2(a2, b2) + I3 = t3(as_float(a1, a2), as_float(b1, b2)) + assert I1.intersect(I2) == I3 + + +def test_intersection(): + # iterable + i = Intersection(FiniteSet(1, 2, 3), Interval(2, 5), evaluate=False) + assert i.is_iterable + assert set(i) == {S(2), S(3)} + + # challenging intervals + x = Symbol('x', real=True) + i = Intersection(Interval(0, 3), Interval(x, 6)) + assert (5 in i) is False + raises(TypeError, lambda: 2 in i) + + # Singleton special cases + assert Intersection(Interval(0, 1), S.EmptySet) == S.EmptySet + assert Intersection(Interval(-oo, oo), Interval(-oo, x)) == Interval(-oo, x) + + # Products + line = Interval(0, 5) + i = Intersection(line**2, line**3, evaluate=False) + assert (2, 2) not in i + assert (2, 2, 2) not in i + raises(TypeError, lambda: list(i)) + + a = Intersection(Intersection(S.Integers, S.Naturals, evaluate=False), S.Reals, evaluate=False) + assert a._argset == frozenset([Intersection(S.Naturals, S.Integers, evaluate=False), S.Reals]) + + assert Intersection(S.Complexes, FiniteSet(S.ComplexInfinity)) == S.EmptySet + + # issue 12178 + assert Intersection() == S.UniversalSet + + # issue 16987 + assert Intersection({1}, {1}, {x}) == Intersection({1}, {x}) + + +def test_issue_9623(): + n = Symbol('n') + + a = S.Reals + b = Interval(0, oo) + c = FiniteSet(n) + + assert Intersection(a, b, c) == Intersection(b, c) + assert Intersection(Interval(1, 2), Interval(3, 4), FiniteSet(n)) == EmptySet + + +def test_is_disjoint(): + assert Interval(0, 2).is_disjoint(Interval(1, 2)) == False + assert Interval(0, 2).is_disjoint(Interval(3, 4)) == True + + +def test_ProductSet__len__(): + A = FiniteSet(1, 2) + B = FiniteSet(1, 2, 3) + assert ProductSet(A).__len__() == 2 + assert ProductSet(A).__len__() is not S(2) + assert ProductSet(A, B).__len__() == 6 + assert ProductSet(A, B).__len__() is not S(6) + + +def test_ProductSet(): + # ProductSet is always a set of Tuples + assert ProductSet(S.Reals) == S.Reals ** 1 + assert ProductSet(S.Reals, S.Reals) == S.Reals ** 2 + assert ProductSet(S.Reals, S.Reals, S.Reals) == S.Reals ** 3 + + assert ProductSet(S.Reals) != S.Reals + assert ProductSet(S.Reals, S.Reals) == S.Reals * S.Reals + assert ProductSet(S.Reals, S.Reals, S.Reals) != S.Reals * S.Reals * S.Reals + assert ProductSet(S.Reals, S.Reals, S.Reals) == (S.Reals * S.Reals * S.Reals).flatten() + + assert 1 not in ProductSet(S.Reals) + assert (1,) in ProductSet(S.Reals) + + assert 1 not in ProductSet(S.Reals, S.Reals) + assert (1, 2) in ProductSet(S.Reals, S.Reals) + assert (1, I) not in ProductSet(S.Reals, S.Reals) + + assert (1, 2, 3) in ProductSet(S.Reals, S.Reals, S.Reals) + assert (1, 2, 3) in S.Reals ** 3 + assert (1, 2, 3) not in S.Reals * S.Reals * S.Reals + assert ((1, 2), 3) in S.Reals * S.Reals * S.Reals + assert (1, (2, 3)) not in S.Reals * S.Reals * S.Reals + assert (1, (2, 3)) in S.Reals * (S.Reals * S.Reals) + + assert ProductSet() == FiniteSet(()) + assert ProductSet(S.Reals, S.EmptySet) == S.EmptySet + + # See GH-17458 + + for ni in range(5): + Rn = ProductSet(*(S.Reals,) * ni) + assert (1,) * ni in Rn + assert 1 not in Rn + + assert (S.Reals * S.Reals) * S.Reals != S.Reals * (S.Reals * S.Reals) + + S1 = S.Reals + S2 = S.Integers + x1 = pi + x2 = 3 + assert x1 in S1 + assert x2 in S2 + assert (x1, x2) in S1 * S2 + S3 = S1 * S2 + x3 = (x1, x2) + assert x3 in S3 + assert (x3, x3) in S3 * S3 + assert x3 + x3 not in S3 * S3 + + raises(ValueError, lambda: S.Reals**-1) + with warns_deprecated_sympy(): + ProductSet(FiniteSet(s) for s in range(2)) + raises(TypeError, lambda: ProductSet(None)) + + S1 = FiniteSet(1, 2) + S2 = FiniteSet(3, 4) + S3 = ProductSet(S1, S2) + assert (S3.as_relational(x, y) + == And(S1.as_relational(x), S2.as_relational(y)) + == And(Or(Eq(x, 1), Eq(x, 2)), Or(Eq(y, 3), Eq(y, 4)))) + raises(ValueError, lambda: S3.as_relational(x)) + raises(ValueError, lambda: S3.as_relational(x, 1)) + raises(ValueError, lambda: ProductSet(Interval(0, 1)).as_relational(x, y)) + + Z2 = ProductSet(S.Integers, S.Integers) + assert Z2.contains((1, 2)) is S.true + assert Z2.contains((1,)) is S.false + assert Z2.contains(x) == Contains(x, Z2, evaluate=False) + assert Z2.contains(x).subs(x, 1) is S.false + assert Z2.contains((x, 1)).subs(x, 2) is S.true + assert Z2.contains((x, y)) == Contains(x, S.Integers) & Contains(y, S.Integers) + assert unchanged(Contains, (x, y), Z2) + assert Contains((1, 2), Z2) is S.true + + +def test_ProductSet_of_single_arg_is_not_arg(): + assert unchanged(ProductSet, Interval(0, 1)) + assert unchanged(ProductSet, ProductSet(Interval(0, 1))) + + +def test_ProductSet_is_empty(): + assert ProductSet(S.Integers, S.Reals).is_empty == False + assert ProductSet(Interval(x, 1), S.Reals).is_empty == None + + +def test_interval_subs(): + a = Symbol('a', real=True) + + assert Interval(0, a).subs(a, 2) == Interval(0, 2) + assert Interval(a, 0).subs(a, 2) == S.EmptySet + + +def test_interval_to_mpi(): + assert Interval(0, 1).to_mpi() == mpi(0, 1) + assert Interval(0, 1, True, False).to_mpi() == mpi(0, 1) + assert type(Interval(0, 1).to_mpi()) == type(mpi(0, 1)) + + +def test_set_evalf(): + assert Interval(S(11)/64, S.Half).evalf() == Interval( + Float('0.171875'), Float('0.5')) + assert Interval(x, S.Half, right_open=True).evalf() == Interval( + x, Float('0.5'), right_open=True) + assert Interval(-oo, S.Half).evalf() == Interval(-oo, Float('0.5')) + assert FiniteSet(2, x).evalf() == FiniteSet(Float('2.0'), x) + + +def test_measure(): + a = Symbol('a', real=True) + + assert Interval(1, 3).measure == 2 + assert Interval(0, a).measure == a + assert Interval(1, a).measure == a - 1 + + assert Union(Interval(1, 2), Interval(3, 4)).measure == 2 + assert Union(Interval(1, 2), Interval(3, 4), FiniteSet(5, 6, 7)).measure \ + == 2 + + assert FiniteSet(1, 2, oo, a, -oo, -5).measure == 0 + + assert S.EmptySet.measure == 0 + + square = Interval(0, 10) * Interval(0, 10) + offsetsquare = Interval(5, 15) * Interval(5, 15) + band = Interval(-oo, oo) * Interval(2, 4) + + assert square.measure == offsetsquare.measure == 100 + assert (square + offsetsquare).measure == 175 # there is some overlap + assert (square - offsetsquare).measure == 75 + assert (square * FiniteSet(1, 2, 3)).measure == 0 + assert (square.intersect(band)).measure == 20 + assert (square + band).measure is oo + assert (band * FiniteSet(1, 2, 3)).measure is nan + + +def test_is_subset(): + assert Interval(0, 1).is_subset(Interval(0, 2)) is True + assert Interval(0, 3).is_subset(Interval(0, 2)) is False + assert Interval(0, 1).is_subset(FiniteSet(0, 1)) is False + + assert FiniteSet(1, 2).is_subset(FiniteSet(1, 2, 3, 4)) + assert FiniteSet(4, 5).is_subset(FiniteSet(1, 2, 3, 4)) is False + assert FiniteSet(1).is_subset(Interval(0, 2)) + assert FiniteSet(1, 2).is_subset(Interval(0, 2, True, True)) is False + assert (Interval(1, 2) + FiniteSet(3)).is_subset( + Interval(0, 2, False, True) + FiniteSet(2, 3)) + + assert Interval(3, 4).is_subset(Union(Interval(0, 1), Interval(2, 5))) is True + assert Interval(3, 6).is_subset(Union(Interval(0, 1), Interval(2, 5))) is False + + assert FiniteSet(1, 2, 3, 4).is_subset(Interval(0, 5)) is True + assert S.EmptySet.is_subset(FiniteSet(1, 2, 3)) is True + + assert Interval(0, 1).is_subset(S.EmptySet) is False + assert S.EmptySet.is_subset(S.EmptySet) is True + + raises(ValueError, lambda: S.EmptySet.is_subset(1)) + + # tests for the issubset alias + assert FiniteSet(1, 2, 3, 4).issubset(Interval(0, 5)) is True + assert S.EmptySet.issubset(FiniteSet(1, 2, 3)) is True + + assert S.Naturals.is_subset(S.Integers) + assert S.Naturals0.is_subset(S.Integers) + + assert FiniteSet(x).is_subset(FiniteSet(y)) is None + assert FiniteSet(x).is_subset(FiniteSet(y).subs(y, x)) is True + assert FiniteSet(x).is_subset(FiniteSet(y).subs(y, x+1)) is False + + assert Interval(0, 1).is_subset(Interval(0, 1, left_open=True)) is False + assert Interval(-2, 3).is_subset(Union(Interval(-oo, -2), Interval(3, oo))) is False + + n = Symbol('n', integer=True) + assert Range(-3, 4, 1).is_subset(FiniteSet(-10, 10)) is False + assert Range(S(10)**100).is_subset(FiniteSet(0, 1, 2)) is False + assert Range(6, 0, -2).is_subset(FiniteSet(2, 4, 6)) is True + assert Range(1, oo).is_subset(FiniteSet(1, 2)) is False + assert Range(-oo, 1).is_subset(FiniteSet(1)) is False + assert Range(3).is_subset(FiniteSet(0, 1, n)) is None + assert Range(n, n + 2).is_subset(FiniteSet(n, n + 1)) is True + assert Range(5).is_subset(Interval(0, 4, right_open=True)) is False + #issue 19513 + assert imageset(Lambda(n, 1/n), S.Integers).is_subset(S.Reals) is None + +def test_is_proper_subset(): + assert Interval(0, 1).is_proper_subset(Interval(0, 2)) is True + assert Interval(0, 3).is_proper_subset(Interval(0, 2)) is False + assert S.EmptySet.is_proper_subset(FiniteSet(1, 2, 3)) is True + + raises(ValueError, lambda: Interval(0, 1).is_proper_subset(0)) + + +def test_is_superset(): + assert Interval(0, 1).is_superset(Interval(0, 2)) == False + assert Interval(0, 3).is_superset(Interval(0, 2)) + + assert FiniteSet(1, 2).is_superset(FiniteSet(1, 2, 3, 4)) == False + assert FiniteSet(4, 5).is_superset(FiniteSet(1, 2, 3, 4)) == False + assert FiniteSet(1).is_superset(Interval(0, 2)) == False + assert FiniteSet(1, 2).is_superset(Interval(0, 2, True, True)) == False + assert (Interval(1, 2) + FiniteSet(3)).is_superset( + Interval(0, 2, False, True) + FiniteSet(2, 3)) == False + + assert Interval(3, 4).is_superset(Union(Interval(0, 1), Interval(2, 5))) == False + + assert FiniteSet(1, 2, 3, 4).is_superset(Interval(0, 5)) == False + assert S.EmptySet.is_superset(FiniteSet(1, 2, 3)) == False + + assert Interval(0, 1).is_superset(S.EmptySet) == True + assert S.EmptySet.is_superset(S.EmptySet) == True + + raises(ValueError, lambda: S.EmptySet.is_superset(1)) + + # tests for the issuperset alias + assert Interval(0, 1).issuperset(S.EmptySet) == True + assert S.EmptySet.issuperset(S.EmptySet) == True + + +def test_is_proper_superset(): + assert Interval(0, 1).is_proper_superset(Interval(0, 2)) is False + assert Interval(0, 3).is_proper_superset(Interval(0, 2)) is True + assert FiniteSet(1, 2, 3).is_proper_superset(S.EmptySet) is True + + raises(ValueError, lambda: Interval(0, 1).is_proper_superset(0)) + + +def test_contains(): + assert Interval(0, 2).contains(1) is S.true + assert Interval(0, 2).contains(3) is S.false + assert Interval(0, 2, True, False).contains(0) is S.false + assert Interval(0, 2, True, False).contains(2) is S.true + assert Interval(0, 2, False, True).contains(0) is S.true + assert Interval(0, 2, False, True).contains(2) is S.false + assert Interval(0, 2, True, True).contains(0) is S.false + assert Interval(0, 2, True, True).contains(2) is S.false + + assert (Interval(0, 2) in Interval(0, 2)) is False + + assert FiniteSet(1, 2, 3).contains(2) is S.true + assert FiniteSet(1, 2, Symbol('x')).contains(Symbol('x')) is S.true + + assert FiniteSet(y)._contains(x) == Eq(y, x, evaluate=False) + raises(TypeError, lambda: x in FiniteSet(y)) + assert FiniteSet({x, y})._contains({x}) == Eq({x, y}, {x}, evaluate=False) + assert FiniteSet({x, y}).subs(y, x)._contains({x}) is S.true + assert FiniteSet({x, y}).subs(y, x+1)._contains({x}) is S.false + + # issue 8197 + from sympy.abc import a, b + assert FiniteSet(b).contains(-a) == Eq(b, -a) + assert FiniteSet(b).contains(a) == Eq(b, a) + assert FiniteSet(a).contains(1) == Eq(a, 1) + raises(TypeError, lambda: 1 in FiniteSet(a)) + + # issue 8209 + rad1 = Pow(Pow(2, Rational(1, 3)) - 1, Rational(1, 3)) + rad2 = Pow(Rational(1, 9), Rational(1, 3)) - Pow(Rational(2, 9), Rational(1, 3)) + Pow(Rational(4, 9), Rational(1, 3)) + s1 = FiniteSet(rad1) + s2 = FiniteSet(rad2) + assert s1 - s2 == S.EmptySet + + items = [1, 2, S.Infinity, S('ham'), -1.1] + fset = FiniteSet(*items) + assert all(item in fset for item in items) + assert all(fset.contains(item) is S.true for item in items) + + assert Union(Interval(0, 1), Interval(2, 5)).contains(3) is S.true + assert Union(Interval(0, 1), Interval(2, 5)).contains(6) is S.false + assert Union(Interval(0, 1), FiniteSet(2, 5)).contains(3) is S.false + + assert S.EmptySet.contains(1) is S.false + assert FiniteSet(rootof(x**3 + x - 1, 0)).contains(S.Infinity) is S.false + + assert rootof(x**5 + x**3 + 1, 0) in S.Reals + assert not rootof(x**5 + x**3 + 1, 1) in S.Reals + + # non-bool results + assert Union(Interval(1, 2), Interval(3, 4)).contains(x) == \ + Or(And(S.One <= x, x <= 2), And(S(3) <= x, x <= 4)) + assert Intersection(Interval(1, x), Interval(2, 3)).contains(y) == \ + And(y <= 3, y <= x, S.One <= y, S(2) <= y) + + assert (S.Complexes).contains(S.ComplexInfinity) == S.false + + +def test_interval_symbolic(): + x = Symbol('x') + e = Interval(0, 1) + assert e.contains(x) == And(S.Zero <= x, x <= 1) + raises(TypeError, lambda: x in e) + e = Interval(0, 1, True, True) + assert e.contains(x) == And(S.Zero < x, x < 1) + c = Symbol('c', real=False) + assert Interval(x, x + 1).contains(c) == False + e = Symbol('e', extended_real=True) + assert Interval(-oo, oo).contains(e) == And( + S.NegativeInfinity < e, e < S.Infinity) + + +def test_union_contains(): + x = Symbol('x') + i1 = Interval(0, 1) + i2 = Interval(2, 3) + i3 = Union(i1, i2) + assert i3.as_relational(x) == Or(And(S.Zero <= x, x <= 1), And(S(2) <= x, x <= 3)) + raises(TypeError, lambda: x in i3) + e = i3.contains(x) + assert e == i3.as_relational(x) + assert e.subs(x, -0.5) is false + assert e.subs(x, 0.5) is true + assert e.subs(x, 1.5) is false + assert e.subs(x, 2.5) is true + assert e.subs(x, 3.5) is false + + U = Interval(0, 2, True, True) + Interval(10, oo) + FiniteSet(-1, 2, 5, 6) + assert all(el not in U for el in [0, 4, -oo]) + assert all(el in U for el in [2, 5, 10]) + + +def test_is_number(): + assert Interval(0, 1).is_number is False + assert Set().is_number is False + + +def test_Interval_is_left_unbounded(): + assert Interval(3, 4).is_left_unbounded is False + assert Interval(-oo, 3).is_left_unbounded is True + assert Interval(Float("-inf"), 3).is_left_unbounded is True + + +def test_Interval_is_right_unbounded(): + assert Interval(3, 4).is_right_unbounded is False + assert Interval(3, oo).is_right_unbounded is True + assert Interval(3, Float("+inf")).is_right_unbounded is True + + +def test_Interval_as_relational(): + x = Symbol('x') + + assert Interval(-1, 2, False, False).as_relational(x) == \ + And(Le(-1, x), Le(x, 2)) + assert Interval(-1, 2, True, False).as_relational(x) == \ + And(Lt(-1, x), Le(x, 2)) + assert Interval(-1, 2, False, True).as_relational(x) == \ + And(Le(-1, x), Lt(x, 2)) + assert Interval(-1, 2, True, True).as_relational(x) == \ + And(Lt(-1, x), Lt(x, 2)) + + assert Interval(-oo, 2, right_open=False).as_relational(x) == And(Lt(-oo, x), Le(x, 2)) + assert Interval(-oo, 2, right_open=True).as_relational(x) == And(Lt(-oo, x), Lt(x, 2)) + + assert Interval(-2, oo, left_open=False).as_relational(x) == And(Le(-2, x), Lt(x, oo)) + assert Interval(-2, oo, left_open=True).as_relational(x) == And(Lt(-2, x), Lt(x, oo)) + + assert Interval(-oo, oo).as_relational(x) == And(Lt(-oo, x), Lt(x, oo)) + x = Symbol('x', real=True) + y = Symbol('y', real=True) + assert Interval(x, y).as_relational(x) == (x <= y) + assert Interval(y, x).as_relational(x) == (y <= x) + + +def test_Finite_as_relational(): + x = Symbol('x') + y = Symbol('y') + + assert FiniteSet(1, 2).as_relational(x) == Or(Eq(x, 1), Eq(x, 2)) + assert FiniteSet(y, -5).as_relational(x) == Or(Eq(x, y), Eq(x, -5)) + + +def test_Union_as_relational(): + x = Symbol('x') + assert (Interval(0, 1) + FiniteSet(2)).as_relational(x) == \ + Or(And(Le(0, x), Le(x, 1)), Eq(x, 2)) + assert (Interval(0, 1, True, True) + FiniteSet(1)).as_relational(x) == \ + And(Lt(0, x), Le(x, 1)) + assert Or(x < 0, x > 0).as_set().as_relational(x) == \ + And((x > -oo), (x < oo), Ne(x, 0)) + assert (Interval.Ropen(1, 3) + Interval.Lopen(3, 5) + ).as_relational(x) == And(Ne(x,3),(x>=1),(x<=5)) + + +def test_Intersection_as_relational(): + x = Symbol('x') + assert (Intersection(Interval(0, 1), FiniteSet(2), + evaluate=False).as_relational(x) + == And(And(Le(0, x), Le(x, 1)), Eq(x, 2))) + + +def test_Complement_as_relational(): + x = Symbol('x') + expr = Complement(Interval(0, 1), FiniteSet(2), evaluate=False) + assert expr.as_relational(x) == \ + And(Le(0, x), Le(x, 1), Ne(x, 2)) + + +@XFAIL +def test_Complement_as_relational_fail(): + x = Symbol('x') + expr = Complement(Interval(0, 1), FiniteSet(2), evaluate=False) + # XXX This example fails because 0 <= x changes to x >= 0 + # during the evaluation. + assert expr.as_relational(x) == \ + (0 <= x) & (x <= 1) & Ne(x, 2) + + +def test_SymmetricDifference_as_relational(): + x = Symbol('x') + expr = SymmetricDifference(Interval(0, 1), FiniteSet(2), evaluate=False) + assert expr.as_relational(x) == Xor(Eq(x, 2), Le(0, x) & Le(x, 1)) + + +def test_EmptySet(): + assert S.EmptySet.as_relational(Symbol('x')) is S.false + assert S.EmptySet.intersect(S.UniversalSet) == S.EmptySet + assert S.EmptySet.boundary == S.EmptySet + + +def test_finite_basic(): + x = Symbol('x') + A = FiniteSet(1, 2, 3) + B = FiniteSet(3, 4, 5) + AorB = Union(A, B) + AandB = A.intersect(B) + assert A.is_subset(AorB) and B.is_subset(AorB) + assert AandB.is_subset(A) + assert AandB == FiniteSet(3) + + assert A.inf == 1 and A.sup == 3 + assert AorB.inf == 1 and AorB.sup == 5 + assert FiniteSet(x, 1, 5).sup == Max(x, 5) + assert FiniteSet(x, 1, 5).inf == Min(x, 1) + + # issue 7335 + assert FiniteSet(S.EmptySet) != S.EmptySet + assert FiniteSet(FiniteSet(1, 2, 3)) != FiniteSet(1, 2, 3) + assert FiniteSet((1, 2, 3)) != FiniteSet(1, 2, 3) + + # Ensure a variety of types can exist in a FiniteSet + assert FiniteSet((1, 2), A, -5, x, 'eggs', x**2) + + assert (A > B) is False + assert (A >= B) is False + assert (A < B) is False + assert (A <= B) is False + assert AorB > A and AorB > B + assert AorB >= A and AorB >= B + assert A >= A and A <= A + assert A >= AandB and B >= AandB + assert A > AandB and B > AandB + + +def test_product_basic(): + H, T = 'H', 'T' + unit_line = Interval(0, 1) + d6 = FiniteSet(1, 2, 3, 4, 5, 6) + d4 = FiniteSet(1, 2, 3, 4) + coin = FiniteSet(H, T) + + square = unit_line * unit_line + + assert (0, 0) in square + assert 0 not in square + assert (H, T) in coin ** 2 + assert (.5, .5, .5) in (square * unit_line).flatten() + assert ((.5, .5), .5) in square * unit_line + assert (H, 3, 3) in (coin * d6 * d6).flatten() + assert ((H, 3), 3) in coin * d6 * d6 + HH, TT = sympify(H), sympify(T) + assert set(coin**2) == {(HH, HH), (HH, TT), (TT, HH), (TT, TT)} + + assert (d4*d4).is_subset(d6*d6) + + assert square.complement(Interval(-oo, oo)*Interval(-oo, oo)) == Union( + (Interval(-oo, 0, True, True) + + Interval(1, oo, True, True))*Interval(-oo, oo), + Interval(-oo, oo)*(Interval(-oo, 0, True, True) + + Interval(1, oo, True, True))) + + assert (Interval(-5, 5)**3).is_subset(Interval(-10, 10)**3) + assert not (Interval(-10, 10)**3).is_subset(Interval(-5, 5)**3) + assert not (Interval(-5, 5)**2).is_subset(Interval(-10, 10)**3) + + assert (Interval(.2, .5)*FiniteSet(.5)).is_subset(square) # segment in square + + assert len(coin*coin*coin) == 8 + assert len(S.EmptySet*S.EmptySet) == 0 + assert len(S.EmptySet*coin) == 0 + raises(TypeError, lambda: len(coin*Interval(0, 2))) + + +def test_real(): + x = Symbol('x', real=True) + + I = Interval(0, 5) + J = Interval(10, 20) + A = FiniteSet(1, 2, 30, x, S.Pi) + B = FiniteSet(-4, 0) + C = FiniteSet(100) + D = FiniteSet('Ham', 'Eggs') + + assert all(s.is_subset(S.Reals) for s in [I, J, A, B, C]) + assert not D.is_subset(S.Reals) + assert all((a + b).is_subset(S.Reals) for a in [I, J, A, B, C] for b in [I, J, A, B, C]) + assert not any((a + D).is_subset(S.Reals) for a in [I, J, A, B, C, D]) + + assert not (I + A + D).is_subset(S.Reals) + + +def test_supinf(): + x = Symbol('x', real=True) + y = Symbol('y', real=True) + + assert (Interval(0, 1) + FiniteSet(2)).sup == 2 + assert (Interval(0, 1) + FiniteSet(2)).inf == 0 + assert (Interval(0, 1) + FiniteSet(x)).sup == Max(1, x) + assert (Interval(0, 1) + FiniteSet(x)).inf == Min(0, x) + assert FiniteSet(5, 1, x).sup == Max(5, x) + assert FiniteSet(5, 1, x).inf == Min(1, x) + assert FiniteSet(5, 1, x, y).sup == Max(5, x, y) + assert FiniteSet(5, 1, x, y).inf == Min(1, x, y) + assert FiniteSet(5, 1, x, y, S.Infinity, S.NegativeInfinity).sup == \ + S.Infinity + assert FiniteSet(5, 1, x, y, S.Infinity, S.NegativeInfinity).inf == \ + S.NegativeInfinity + assert FiniteSet('Ham', 'Eggs').sup == Max('Ham', 'Eggs') + + +def test_universalset(): + U = S.UniversalSet + x = Symbol('x') + assert U.as_relational(x) is S.true + assert U.union(Interval(2, 4)) == U + + assert U.intersect(Interval(2, 4)) == Interval(2, 4) + assert U.measure is S.Infinity + assert U.boundary == S.EmptySet + assert U.contains(0) is S.true + + +def test_Union_of_ProductSets_shares(): + line = Interval(0, 2) + points = FiniteSet(0, 1, 2) + assert Union(line * line, line * points) == line * line + + +def test_Interval_free_symbols(): + # issue 6211 + assert Interval(0, 1).free_symbols == set() + x = Symbol('x', real=True) + assert Interval(0, x).free_symbols == {x} + + +def test_image_interval(): + x = Symbol('x', real=True) + a = Symbol('a', real=True) + assert imageset(x, 2*x, Interval(-2, 1)) == Interval(-4, 2) + assert imageset(x, 2*x, Interval(-2, 1, True, False)) == \ + Interval(-4, 2, True, False) + assert imageset(x, x**2, Interval(-2, 1, True, False)) == \ + Interval(0, 4, False, True) + assert imageset(x, x**2, Interval(-2, 1)) == Interval(0, 4) + assert imageset(x, x**2, Interval(-2, 1, True, False)) == \ + Interval(0, 4, False, True) + assert imageset(x, x**2, Interval(-2, 1, True, True)) == \ + Interval(0, 4, False, True) + assert imageset(x, (x - 2)**2, Interval(1, 3)) == Interval(0, 1) + assert imageset(x, 3*x**4 - 26*x**3 + 78*x**2 - 90*x, Interval(0, 4)) == \ + Interval(-35, 0) # Multiple Maxima + assert imageset(x, x + 1/x, Interval(-oo, oo)) == Interval(-oo, -2) \ + + Interval(2, oo) # Single Infinite discontinuity + assert imageset(x, 1/x + 1/(x-1)**2, Interval(0, 2, True, False)) == \ + Interval(Rational(3, 2), oo, False) # Multiple Infinite discontinuities + + # Test for Python lambda + assert imageset(lambda x: 2*x, Interval(-2, 1)) == Interval(-4, 2) + + assert imageset(Lambda(x, a*x), Interval(0, 1)) == \ + ImageSet(Lambda(x, a*x), Interval(0, 1)) + + assert imageset(Lambda(x, sin(cos(x))), Interval(0, 1)) == \ + ImageSet(Lambda(x, sin(cos(x))), Interval(0, 1)) + + +def test_image_piecewise(): + f = Piecewise((x, x <= -1), (1/x**2, x <= 5), (x**3, True)) + f1 = Piecewise((0, x <= 1), (1, x <= 2), (2, True)) + assert imageset(x, f, Interval(-5, 5)) == Union(Interval(-5, -1), Interval(Rational(1, 25), oo)) + assert imageset(x, f1, Interval(1, 2)) == FiniteSet(0, 1) + + +@XFAIL # See: https://github.com/sympy/sympy/pull/2723#discussion_r8659826 +def test_image_Intersection(): + x = Symbol('x', real=True) + y = Symbol('y', real=True) + assert imageset(x, x**2, Interval(-2, 0).intersect(Interval(x, y))) == \ + Interval(0, 4).intersect(Interval(Min(x**2, y**2), Max(x**2, y**2))) + + +def test_image_FiniteSet(): + x = Symbol('x', real=True) + assert imageset(x, 2*x, FiniteSet(1, 2, 3)) == FiniteSet(2, 4, 6) + + +def test_image_Union(): + x = Symbol('x', real=True) + assert imageset(x, x**2, Interval(-2, 0) + FiniteSet(1, 2, 3)) == \ + (Interval(0, 4) + FiniteSet(9)) + + +def test_image_EmptySet(): + x = Symbol('x', real=True) + assert imageset(x, 2*x, S.EmptySet) == S.EmptySet + + +def test_issue_5724_7680(): + assert I not in S.Reals # issue 7680 + assert Interval(-oo, oo).contains(I) is S.false + + +def test_boundary(): + assert FiniteSet(1).boundary == FiniteSet(1) + assert all(Interval(0, 1, left_open, right_open).boundary == FiniteSet(0, 1) + for left_open in (true, false) for right_open in (true, false)) + + +def test_boundary_Union(): + assert (Interval(0, 1) + Interval(2, 3)).boundary == FiniteSet(0, 1, 2, 3) + assert ((Interval(0, 1, False, True) + + Interval(1, 2, True, False)).boundary == FiniteSet(0, 1, 2)) + + assert (Interval(0, 1) + FiniteSet(2)).boundary == FiniteSet(0, 1, 2) + assert Union(Interval(0, 10), Interval(5, 15), evaluate=False).boundary \ + == FiniteSet(0, 15) + + assert Union(Interval(0, 10), Interval(0, 1), evaluate=False).boundary \ + == FiniteSet(0, 10) + assert Union(Interval(0, 10, True, True), + Interval(10, 15, True, True), evaluate=False).boundary \ + == FiniteSet(0, 10, 15) + + +@XFAIL +def test_union_boundary_of_joining_sets(): + """ Testing the boundary of unions is a hard problem """ + assert Union(Interval(0, 10), Interval(10, 15), evaluate=False).boundary \ + == FiniteSet(0, 15) + + +def test_boundary_ProductSet(): + open_square = Interval(0, 1, True, True) ** 2 + assert open_square.boundary == (FiniteSet(0, 1) * Interval(0, 1) + + Interval(0, 1) * FiniteSet(0, 1)) + + second_square = Interval(1, 2, True, True) * Interval(0, 1, True, True) + assert (open_square + second_square).boundary == ( + FiniteSet(0, 1) * Interval(0, 1) + + FiniteSet(1, 2) * Interval(0, 1) + + Interval(0, 1) * FiniteSet(0, 1) + + Interval(1, 2) * FiniteSet(0, 1)) + + +def test_boundary_ProductSet_line(): + line_in_r2 = Interval(0, 1) * FiniteSet(0) + assert line_in_r2.boundary == line_in_r2 + + +def test_is_open(): + assert Interval(0, 1, False, False).is_open is False + assert Interval(0, 1, True, False).is_open is False + assert Interval(0, 1, True, True).is_open is True + assert FiniteSet(1, 2, 3).is_open is False + + +def test_is_closed(): + assert Interval(0, 1, False, False).is_closed is True + assert Interval(0, 1, True, False).is_closed is False + assert FiniteSet(1, 2, 3).is_closed is True + + +def test_closure(): + assert Interval(0, 1, False, True).closure == Interval(0, 1, False, False) + + +def test_interior(): + assert Interval(0, 1, False, True).interior == Interval(0, 1, True, True) + + +def test_issue_7841(): + raises(TypeError, lambda: x in S.Reals) + + +def test_Eq(): + assert Eq(Interval(0, 1), Interval(0, 1)) + assert Eq(Interval(0, 1), Interval(0, 2)) == False + + s1 = FiniteSet(0, 1) + s2 = FiniteSet(1, 2) + + assert Eq(s1, s1) + assert Eq(s1, s2) == False + + assert Eq(s1*s2, s1*s2) + assert Eq(s1*s2, s2*s1) == False + + assert unchanged(Eq, FiniteSet({x, y}), FiniteSet({x})) + assert Eq(FiniteSet({x, y}).subs(y, x), FiniteSet({x})) is S.true + assert Eq(FiniteSet({x, y}), FiniteSet({x})).subs(y, x) is S.true + assert Eq(FiniteSet({x, y}).subs(y, x+1), FiniteSet({x})) is S.false + assert Eq(FiniteSet({x, y}), FiniteSet({x})).subs(y, x+1) is S.false + + assert Eq(ProductSet({1}, {2}), Interval(1, 2)) is S.false + assert Eq(ProductSet({1}), ProductSet({1}, {2})) is S.false + + assert Eq(FiniteSet(()), FiniteSet(1)) is S.false + assert Eq(ProductSet(), FiniteSet(1)) is S.false + + i1 = Interval(0, 1) + i2 = Interval(x, y) + assert unchanged(Eq, ProductSet(i1, i1), ProductSet(i2, i2)) + + +def test_SymmetricDifference(): + A = FiniteSet(0, 1, 2, 3, 4, 5) + B = FiniteSet(2, 4, 6, 8, 10) + C = Interval(8, 10) + + assert SymmetricDifference(A, B, evaluate=False).is_iterable is True + assert SymmetricDifference(A, C, evaluate=False).is_iterable is None + assert FiniteSet(*SymmetricDifference(A, B, evaluate=False)) == \ + FiniteSet(0, 1, 3, 5, 6, 8, 10) + raises(TypeError, + lambda: FiniteSet(*SymmetricDifference(A, C, evaluate=False))) + + assert SymmetricDifference(FiniteSet(0, 1, 2, 3, 4, 5), \ + FiniteSet(2, 4, 6, 8, 10)) == FiniteSet(0, 1, 3, 5, 6, 8, 10) + assert SymmetricDifference(FiniteSet(2, 3, 4), FiniteSet(2, 3, 4 ,5)) \ + == FiniteSet(5) + assert FiniteSet(1, 2, 3, 4, 5) ^ FiniteSet(1, 2, 5, 6) == \ + FiniteSet(3, 4, 6) + assert Set(S(1), S(2), S(3)) ^ Set(S(2), S(3), S(4)) == Union(Set(S(1), S(2), S(3)) - Set(S(2), S(3), S(4)), \ + Set(S(2), S(3), S(4)) - Set(S(1), S(2), S(3))) + assert Interval(0, 4) ^ Interval(2, 5) == Union(Interval(0, 4) - \ + Interval(2, 5), Interval(2, 5) - Interval(0, 4)) + + +def test_issue_9536(): + from sympy.functions.elementary.exponential import log + a = Symbol('a', real=True) + assert FiniteSet(log(a)).intersect(S.Reals) == Intersection(S.Reals, FiniteSet(log(a))) + + +def test_issue_9637(): + n = Symbol('n') + a = FiniteSet(n) + b = FiniteSet(2, n) + assert Complement(S.Reals, a) == Complement(S.Reals, a, evaluate=False) + assert Complement(Interval(1, 3), a) == Complement(Interval(1, 3), a, evaluate=False) + assert Complement(Interval(1, 3), b) == \ + Complement(Union(Interval(1, 2, False, True), Interval(2, 3, True, False)), a) + assert Complement(a, S.Reals) == Complement(a, S.Reals, evaluate=False) + assert Complement(a, Interval(1, 3)) == Complement(a, Interval(1, 3), evaluate=False) + + +def test_issue_9808(): + # See https://github.com/sympy/sympy/issues/16342 + assert Complement(FiniteSet(y), FiniteSet(1)) == Complement(FiniteSet(y), FiniteSet(1), evaluate=False) + assert Complement(FiniteSet(1, 2, x), FiniteSet(x, y, 2, 3)) == \ + Complement(FiniteSet(1), FiniteSet(y), evaluate=False) + + +def test_issue_9956(): + assert Union(Interval(-oo, oo), FiniteSet(1)) == Interval(-oo, oo) + assert Interval(-oo, oo).contains(1) is S.true + + +def test_issue_Symbol_inter(): + i = Interval(0, oo) + r = S.Reals + mat = Matrix([0, 0, 0]) + assert Intersection(r, i, FiniteSet(m), FiniteSet(m, n)) == \ + Intersection(i, FiniteSet(m)) + assert Intersection(FiniteSet(1, m, n), FiniteSet(m, n, 2), i) == \ + Intersection(i, FiniteSet(m, n)) + assert Intersection(FiniteSet(m, n, x), FiniteSet(m, z), r) == \ + Intersection(Intersection({m, z}, {m, n, x}), r) + assert Intersection(FiniteSet(m, n, 3), FiniteSet(m, n, x), r) == \ + Intersection(FiniteSet(3, m, n), FiniteSet(m, n, x), r, evaluate=False) + assert Intersection(FiniteSet(m, n, 3), FiniteSet(m, n, 2, 3), r) == \ + Intersection(FiniteSet(3, m, n), r) + assert Intersection(r, FiniteSet(mat, 2, n), FiniteSet(0, mat, n)) == \ + Intersection(r, FiniteSet(n)) + assert Intersection(FiniteSet(sin(x), cos(x)), FiniteSet(sin(x), cos(x), 1), r) == \ + Intersection(r, FiniteSet(sin(x), cos(x))) + assert Intersection(FiniteSet(x**2, 1, sin(x)), FiniteSet(x**2, 2, sin(x)), r) == \ + Intersection(r, FiniteSet(x**2, sin(x))) + + +def test_issue_11827(): + assert S.Naturals0**4 + + +def test_issue_10113(): + f = x**2/(x**2 - 4) + assert imageset(x, f, S.Reals) == Union(Interval(-oo, 0), Interval(1, oo, True, True)) + assert imageset(x, f, Interval(-2, 2)) == Interval(-oo, 0) + assert imageset(x, f, Interval(-2, 3)) == Union(Interval(-oo, 0), Interval(Rational(9, 5), oo)) + + +def test_issue_10248(): + raises( + TypeError, lambda: list(Intersection(S.Reals, FiniteSet(x))) + ) + A = Symbol('A', real=True) + assert list(Intersection(S.Reals, FiniteSet(A))) == [A] + + +def test_issue_9447(): + a = Interval(0, 1) + Interval(2, 3) + assert Complement(S.UniversalSet, a) == Complement( + S.UniversalSet, Union(Interval(0, 1), Interval(2, 3)), evaluate=False) + assert Complement(S.Naturals, a) == Complement( + S.Naturals, Union(Interval(0, 1), Interval(2, 3)), evaluate=False) + + +def test_issue_10337(): + assert (FiniteSet(2) == 3) is False + assert (FiniteSet(2) != 3) is True + raises(TypeError, lambda: FiniteSet(2) < 3) + raises(TypeError, lambda: FiniteSet(2) <= 3) + raises(TypeError, lambda: FiniteSet(2) > 3) + raises(TypeError, lambda: FiniteSet(2) >= 3) + + +def test_issue_10326(): + bad = [ + EmptySet, + FiniteSet(1), + Interval(1, 2), + S.ComplexInfinity, + S.ImaginaryUnit, + S.Infinity, + S.NaN, + S.NegativeInfinity, + ] + interval = Interval(0, 5) + for i in bad: + assert i not in interval + + x = Symbol('x', real=True) + nr = Symbol('nr', extended_real=False) + assert x + 1 in Interval(x, x + 4) + assert nr not in Interval(x, x + 4) + assert Interval(1, 2) in FiniteSet(Interval(0, 5), Interval(1, 2)) + assert Interval(-oo, oo).contains(oo) is S.false + assert Interval(-oo, oo).contains(-oo) is S.false + + +def test_issue_2799(): + U = S.UniversalSet + a = Symbol('a', real=True) + inf_interval = Interval(a, oo) + R = S.Reals + + assert U + inf_interval == inf_interval + U + assert U + R == R + U + assert R + inf_interval == inf_interval + R + + +def test_issue_9706(): + assert Interval(-oo, 0).closure == Interval(-oo, 0, True, False) + assert Interval(0, oo).closure == Interval(0, oo, False, True) + assert Interval(-oo, oo).closure == Interval(-oo, oo) + + +def test_issue_8257(): + reals_plus_infinity = Union(Interval(-oo, oo), FiniteSet(oo)) + reals_plus_negativeinfinity = Union(Interval(-oo, oo), FiniteSet(-oo)) + assert Interval(-oo, oo) + FiniteSet(oo) == reals_plus_infinity + assert FiniteSet(oo) + Interval(-oo, oo) == reals_plus_infinity + assert Interval(-oo, oo) + FiniteSet(-oo) == reals_plus_negativeinfinity + assert FiniteSet(-oo) + Interval(-oo, oo) == reals_plus_negativeinfinity + + +def test_issue_10931(): + assert S.Integers - S.Integers == EmptySet + assert S.Integers - S.Reals == EmptySet + + +def test_issue_11174(): + soln = Intersection(Interval(-oo, oo), FiniteSet(-x), evaluate=False) + assert Intersection(FiniteSet(-x), S.Reals) == soln + + soln = Intersection(S.Reals, FiniteSet(x), evaluate=False) + assert Intersection(FiniteSet(x), S.Reals) == soln + + +def test_issue_18505(): + assert ImageSet(Lambda(n, sqrt(pi*n/2 - 1 + pi/2)), S.Integers).contains(0) == \ + Contains(0, ImageSet(Lambda(n, sqrt(pi*n/2 - 1 + pi/2)), S.Integers)) + + +def test_finite_set_intersection(): + # The following should not produce recursion errors + # Note: some of these are not completely correct. See + # https://github.com/sympy/sympy/issues/16342. + assert Intersection(FiniteSet(-oo, x), FiniteSet(x)) == FiniteSet(x) + assert Intersection._handle_finite_sets([FiniteSet(-oo, x), FiniteSet(0, x)]) == FiniteSet(x) + + assert Intersection._handle_finite_sets([FiniteSet(-oo, x), FiniteSet(x)]) == FiniteSet(x) + assert Intersection._handle_finite_sets([FiniteSet(2, 3, x, y), FiniteSet(1, 2, x)]) == \ + Intersection._handle_finite_sets([FiniteSet(1, 2, x), FiniteSet(2, 3, x, y)]) == \ + Intersection(FiniteSet(1, 2, x), FiniteSet(2, 3, x, y)) == \ + Intersection(FiniteSet(1, 2, x), FiniteSet(2, x, y)) + + assert FiniteSet(1+x-y) & FiniteSet(1) == \ + FiniteSet(1) & FiniteSet(1+x-y) == \ + Intersection(FiniteSet(1+x-y), FiniteSet(1), evaluate=False) + + assert FiniteSet(1) & FiniteSet(x) == FiniteSet(x) & FiniteSet(1) == \ + Intersection(FiniteSet(1), FiniteSet(x), evaluate=False) + + assert FiniteSet({x}) & FiniteSet({x, y}) == \ + Intersection(FiniteSet({x}), FiniteSet({x, y}), evaluate=False) + + +def test_union_intersection_constructor(): + # The actual exception does not matter here, so long as these fail + sets = [FiniteSet(1), FiniteSet(2)] + raises(Exception, lambda: Union(sets)) + raises(Exception, lambda: Intersection(sets)) + raises(Exception, lambda: Union(tuple(sets))) + raises(Exception, lambda: Intersection(tuple(sets))) + raises(Exception, lambda: Union(i for i in sets)) + raises(Exception, lambda: Intersection(i for i in sets)) + + # Python sets are treated the same as FiniteSet + # The union of a single set (of sets) is the set (of sets) itself + assert Union(set(sets)) == FiniteSet(*sets) + assert Intersection(set(sets)) == FiniteSet(*sets) + + assert Union({1}, {2}) == FiniteSet(1, 2) + assert Intersection({1, 2}, {2, 3}) == FiniteSet(2) + + +def test_Union_contains(): + assert zoo not in Union( + Interval.open(-oo, 0), Interval.open(0, oo)) + + +@XFAIL +def test_issue_16878b(): + # in intersection_sets for (ImageSet, Set) there is no code + # that handles the base_set of S.Reals like there is + # for Integers + assert imageset(x, (x, x), S.Reals).is_subset(S.Reals**2) is True + +def test_DisjointUnion(): + assert DisjointUnion(FiniteSet(1, 2, 3), FiniteSet(1, 2, 3), FiniteSet(1, 2, 3)).rewrite(Union) == (FiniteSet(1, 2, 3) * FiniteSet(0, 1, 2)) + assert DisjointUnion(Interval(1, 3), Interval(2, 4)).rewrite(Union) == Union(Interval(1, 3) * FiniteSet(0), Interval(2, 4) * FiniteSet(1)) + assert DisjointUnion(Interval(0, 5), Interval(0, 5)).rewrite(Union) == Union(Interval(0, 5) * FiniteSet(0), Interval(0, 5) * FiniteSet(1)) + assert DisjointUnion(Interval(-1, 2), S.EmptySet, S.EmptySet).rewrite(Union) == Interval(-1, 2) * FiniteSet(0) + assert DisjointUnion(Interval(-1, 2)).rewrite(Union) == Interval(-1, 2) * FiniteSet(0) + assert DisjointUnion(S.EmptySet, Interval(-1, 2), S.EmptySet).rewrite(Union) == Interval(-1, 2) * FiniteSet(1) + assert DisjointUnion(Interval(-oo, oo)).rewrite(Union) == Interval(-oo, oo) * FiniteSet(0) + assert DisjointUnion(S.EmptySet).rewrite(Union) == S.EmptySet + assert DisjointUnion().rewrite(Union) == S.EmptySet + raises(TypeError, lambda: DisjointUnion(Symbol('n'))) + + x = Symbol("x") + y = Symbol("y") + z = Symbol("z") + assert DisjointUnion(FiniteSet(x), FiniteSet(y, z)).rewrite(Union) == (FiniteSet(x) * FiniteSet(0)) + (FiniteSet(y, z) * FiniteSet(1)) + +def test_DisjointUnion_is_empty(): + assert DisjointUnion(S.EmptySet).is_empty is True + assert DisjointUnion(S.EmptySet, S.EmptySet).is_empty is True + assert DisjointUnion(S.EmptySet, FiniteSet(1, 2, 3)).is_empty is False + +def test_DisjointUnion_is_iterable(): + assert DisjointUnion(S.Integers, S.Naturals, S.Rationals).is_iterable is True + assert DisjointUnion(S.EmptySet, S.Reals).is_iterable is False + assert DisjointUnion(FiniteSet(1, 2, 3), S.EmptySet, FiniteSet(x, y)).is_iterable is True + assert DisjointUnion(S.EmptySet, S.EmptySet).is_iterable is False + +def test_DisjointUnion_contains(): + assert (0, 0) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2)) + assert (0, 1) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2)) + assert (0, 2) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2)) + assert (1, 0) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2)) + assert (1, 1) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2)) + assert (1, 2) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2)) + assert (2, 0) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2)) + assert (2, 1) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2)) + assert (2, 2) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2)) + assert (0, 1, 2) not in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2)) + assert (0, 0.5) not in DisjointUnion(FiniteSet(0.5)) + assert (0, 5) not in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2)) + assert (x, 0) in DisjointUnion(FiniteSet(x, y, z), S.EmptySet, FiniteSet(y)) + assert (y, 0) in DisjointUnion(FiniteSet(x, y, z), S.EmptySet, FiniteSet(y)) + assert (z, 0) in DisjointUnion(FiniteSet(x, y, z), S.EmptySet, FiniteSet(y)) + assert (y, 2) in DisjointUnion(FiniteSet(x, y, z), S.EmptySet, FiniteSet(y)) + assert (0.5, 0) in DisjointUnion(Interval(0, 1), Interval(0, 2)) + assert (0.5, 1) in DisjointUnion(Interval(0, 1), Interval(0, 2)) + assert (1.5, 0) not in DisjointUnion(Interval(0, 1), Interval(0, 2)) + assert (1.5, 1) in DisjointUnion(Interval(0, 1), Interval(0, 2)) + +def test_DisjointUnion_iter(): + D = DisjointUnion(FiniteSet(3, 5, 7, 9), FiniteSet(x, y, z)) + it = iter(D) + L1 = [(x, 1), (y, 1), (z, 1)] + L2 = [(3, 0), (5, 0), (7, 0), (9, 0)] + nxt = next(it) + assert nxt in L2 + L2.remove(nxt) + nxt = next(it) + assert nxt in L1 + L1.remove(nxt) + nxt = next(it) + assert nxt in L2 + L2.remove(nxt) + nxt = next(it) + assert nxt in L1 + L1.remove(nxt) + nxt = next(it) + assert nxt in L2 + L2.remove(nxt) + nxt = next(it) + assert nxt in L1 + L1.remove(nxt) + nxt = next(it) + assert nxt in L2 + L2.remove(nxt) + raises(StopIteration, lambda: next(it)) + + raises(ValueError, lambda: iter(DisjointUnion(Interval(0, 1), S.EmptySet))) + +def test_DisjointUnion_len(): + assert len(DisjointUnion(FiniteSet(3, 5, 7, 9), FiniteSet(x, y, z))) == 7 + assert len(DisjointUnion(S.EmptySet, S.EmptySet, FiniteSet(x, y, z), S.EmptySet)) == 3 + raises(ValueError, lambda: len(DisjointUnion(Interval(0, 1), S.EmptySet))) + +def test_SetKind_ProductSet(): + p = ProductSet(FiniteSet(Matrix([1, 2])), FiniteSet(Matrix([1, 2]))) + mk = MatrixKind(NumberKind) + k = SetKind(TupleKind(mk, mk)) + assert p.kind is k + assert ProductSet(Interval(1, 2), FiniteSet(Matrix([1, 2]))).kind is SetKind(TupleKind(NumberKind, mk)) + +def test_SetKind_Interval(): + assert Interval(1, 2).kind is SetKind(NumberKind) + +def test_SetKind_EmptySet_UniversalSet(): + assert S.UniversalSet.kind is SetKind(UndefinedKind) + assert EmptySet.kind is SetKind() + +def test_SetKind_FiniteSet(): + assert FiniteSet(1, Matrix([1, 2])).kind is SetKind(UndefinedKind) + assert FiniteSet(1, 2).kind is SetKind(NumberKind) + +def test_SetKind_Unions(): + assert Union(FiniteSet(Matrix([1, 2])), Interval(1, 2)).kind is SetKind(UndefinedKind) + assert Union(Interval(1, 2), Interval(1, 7)).kind is SetKind(NumberKind) + +def test_SetKind_DisjointUnion(): + A = FiniteSet(1, 2, 3) + B = Interval(0, 5) + assert DisjointUnion(A, B).kind is SetKind(NumberKind) + +def test_SetKind_evaluate_False(): + U = lambda *args: Union(*args, evaluate=False) + assert U({1}, EmptySet).kind is SetKind(NumberKind) + assert U(Interval(1, 2), EmptySet).kind is SetKind(NumberKind) + assert U({1}, S.UniversalSet).kind is SetKind(UndefinedKind) + assert U(Interval(1, 2), Interval(4, 5), + FiniteSet(1)).kind is SetKind(NumberKind) + I = lambda *args: Intersection(*args, evaluate=False) + assert I({1}, S.UniversalSet).kind is SetKind(NumberKind) + assert I({1}, EmptySet).kind is SetKind() + C = lambda *args: Complement(*args, evaluate=False) + assert C(S.UniversalSet, {1, 2, 4, 5}).kind is SetKind(UndefinedKind) + assert C({1, 2, 3, 4, 5}, EmptySet).kind is SetKind(NumberKind) + assert C(EmptySet, {1, 2, 3, 4, 5}).kind is SetKind() + +def test_SetKind_ImageSet_Special(): + f = ImageSet(Lambda(n, n ** 2), Interval(1, 4)) + assert (f - FiniteSet(3)).kind is SetKind(NumberKind) + assert (f + Interval(16, 17)).kind is SetKind(NumberKind) + assert (f + FiniteSet(17)).kind is SetKind(NumberKind) + +def test_issue_20089(): + B = FiniteSet(FiniteSet(1, 2), FiniteSet(1)) + assert 1 not in B + assert 1.0 not in B + assert not Eq(1, FiniteSet(1, 2)) + assert FiniteSet(1) in B + A = FiniteSet(1, 2) + assert A in B + assert B.issubset(B) + assert not A.issubset(B) + assert 1 in A + C = FiniteSet(FiniteSet(1, 2), FiniteSet(1), 1, 2) + assert A.issubset(C) + assert B.issubset(C) + +def test_issue_19378(): + a = FiniteSet(1, 2) + b = ProductSet(a, a) + c = FiniteSet((1, 1), (1, 2), (2, 1), (2, 2)) + assert b.is_subset(c) is True + d = FiniteSet(1) + assert b.is_subset(d) is False + assert Eq(c, b).simplify() is S.true + assert Eq(a, c).simplify() is S.false + assert Eq({1}, {x}).simplify() == Eq({1}, {x}) + +def test_intersection_symbolic(): + n = Symbol('n') + # These should not throw an error + assert isinstance(Intersection(Range(n), Range(100)), Intersection) + assert isinstance(Intersection(Range(n), Interval(1, 100)), Intersection) + assert isinstance(Intersection(Range(100), Interval(1, n)), Intersection) + + +@XFAIL +def test_intersection_symbolic_failing(): + n = Symbol('n', integer=True, positive=True) + assert Intersection(Range(10, n), Range(4, 500, 5)) == Intersection( + Range(14, n), Range(14, 500, 5)) + assert Intersection(Interval(10, n), Range(4, 500, 5)) == Intersection( + Interval(14, n), Range(14, 500, 5)) + + +def test_issue_20379(): + #https://github.com/sympy/sympy/issues/20379 + x = pi - 3.14159265358979 + assert FiniteSet(x).evalf(2) == FiniteSet(Float('3.23108914886517e-15', 2)) + +def test_finiteset_simplify(): + S = FiniteSet(1, cos(1)**2 + sin(1)**2) + assert S.simplify() == {1} + +def test_issue_14336(): + #https://github.com/sympy/sympy/issues/14336 + U = S.Complexes + x = Symbol("x") + U -= U.intersect(Ne(x, 1).as_set()) + U -= U.intersect(S.true.as_set()) + +def test_issue_9855(): + #https://github.com/sympy/sympy/issues/9855 + x, y, z = symbols('x, y, z', real=True) + s1 = Interval(1, x) & Interval(y, 2) + s2 = Interval(1, 2) + assert s1.is_subset(s2) == None diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/__init__.py b/.venv/lib/python3.12/site-packages/sympy/simplify/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..0619d1c3ebbd6c6a7d663093c7ed2202114148af --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/simplify/__init__.py @@ -0,0 +1,60 @@ +"""The module helps converting SymPy expressions into shorter forms of them. + +for example: +the expression E**(pi*I) will be converted into -1 +the expression (x+x)**2 will be converted into 4*x**2 +""" +from .simplify import (simplify, hypersimp, hypersimilar, + logcombine, separatevars, posify, besselsimp, kroneckersimp, + signsimp, nsimplify) + +from .fu import FU, fu + +from .sqrtdenest import sqrtdenest + +from .cse_main import cse + +from .epathtools import epath, EPath + +from .hyperexpand import hyperexpand + +from .radsimp import collect, rcollect, radsimp, collect_const, fraction, numer, denom + +from .trigsimp import trigsimp, exptrigsimp + +from .powsimp import powsimp, powdenest + +from .combsimp import combsimp + +from .gammasimp import gammasimp + +from .ratsimp import ratsimp, ratsimpmodprime + +__all__ = [ + 'simplify', 'hypersimp', 'hypersimilar', 'logcombine', 'separatevars', + 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b/.venv/lib/python3.12/site-packages/sympy/simplify/__pycache__/trigsimp.cpython-312.pyc differ diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/_cse_diff.py b/.venv/lib/python3.12/site-packages/sympy/simplify/_cse_diff.py new file mode 100644 index 0000000000000000000000000000000000000000..3496ad3b31a4f45312cac002429be40aa9aa0868 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/simplify/_cse_diff.py @@ -0,0 +1,291 @@ +"""Module for differentiation using CSE.""" + +from sympy import cse, Matrix, Derivative, MatrixBase +from sympy.utilities.iterables import iterable + + +def _remove_cse_from_derivative(replacements, reduced_expressions): + """ + This function is designed to postprocess the output of a common subexpression + elimination (CSE) operation. Specifically, it removes any CSE replacement + symbols from the arguments of ``Derivative`` terms in the expression. This + is necessary to ensure that the forward Jacobian function correctly handles + derivative terms. + + Parameters + ========== + + replacements : list of (Symbol, expression) pairs + Replacement symbols and relative common subexpressions that have been + replaced during a CSE operation. + + reduced_expressions : list of SymPy expressions + The reduced expressions with all the replacements from the + replacements list above. + + Returns + ======= + + processed_replacements : list of (Symbol, expression) pairs + Processed replacement list, in the same format of the + ``replacements`` input list. + + processed_reduced : list of SymPy expressions + Processed reduced list, in the same format of the + ``reduced_expressions`` input list. + """ + + def traverse(node, repl_dict): + if isinstance(node, Derivative): + return replace_all(node, repl_dict) + if not node.args: + return node + new_args = [traverse(arg, repl_dict) for arg in node.args] + return node.func(*new_args) + + def replace_all(node, repl_dict): + result = node + while True: + free_symbols = result.free_symbols + symbols_dict = {k: repl_dict[k] for k in free_symbols if k in repl_dict} + if not symbols_dict: + break + result = result.xreplace(symbols_dict) + return result + + repl_dict = dict(replacements) + processed_replacements = [ + (rep_sym, traverse(sub_exp, repl_dict)) + for rep_sym, sub_exp in replacements + ] + processed_reduced = [ + red_exp.__class__([traverse(exp, repl_dict) for exp in red_exp]) + for red_exp in reduced_expressions + ] + + return processed_replacements, processed_reduced + + +def _forward_jacobian_cse(replacements, reduced_expr, wrt): + """ + Core function to compute the Jacobian of an input Matrix of expressions + through forward accumulation. Takes directly the output of a CSE operation + (replacements and reduced_expr), and an iterable of variables (wrt) with + respect to which to differentiate the reduced expression and returns the + reduced Jacobian matrix and the ``replacements`` list. + + The function also returns a list of precomputed free symbols for each + subexpression, which are useful in the substitution process. + + Parameters + ========== + + replacements : list of (Symbol, expression) pairs + Replacement symbols and relative common subexpressions that have been + replaced during a CSE operation. + + reduced_expr : list of SymPy expressions + The reduced expressions with all the replacements from the + replacements list above. + + wrt : iterable + Iterable of expressions with respect to which to compute the + Jacobian matrix. + + Returns + ======= + + replacements : list of (Symbol, expression) pairs + Replacement symbols and relative common subexpressions that have been + replaced during a CSE operation. Compared to the input replacement list, + the output one doesn't contain replacement symbols inside + ``Derivative``'s arguments. + + jacobian : list of SymPy expressions + The list only contains one element, which is the Jacobian matrix with + elements in reduced form (replacement symbols are present). + + precomputed_fs: list + List of sets, which store the free symbols present in each sub-expression. + Useful in the substitution process. + """ + + if not isinstance(reduced_expr[0], MatrixBase): + raise TypeError("``expr`` must be of matrix type") + + if not (reduced_expr[0].shape[0] == 1 or reduced_expr[0].shape[1] == 1): + raise TypeError("``expr`` must be a row or a column matrix") + + if not iterable(wrt): + raise TypeError("``wrt`` must be an iterable of variables") + + elif not isinstance(wrt, MatrixBase): + wrt = Matrix(wrt) + + if not (wrt.shape[0] == 1 or wrt.shape[1] == 1): + raise TypeError("``wrt`` must be a row or a column matrix") + + replacements, reduced_expr = _remove_cse_from_derivative(replacements, reduced_expr) + + if replacements: + rep_sym, sub_expr = map(Matrix, zip(*replacements)) + else: + rep_sym, sub_expr = Matrix([]), Matrix([]) + + l_sub, l_wrt, l_red = len(sub_expr), len(wrt), len(reduced_expr[0]) + + f1 = reduced_expr[0].__class__.from_dok(l_red, l_wrt, + { + (i, j): diff_value + for i, r in enumerate(reduced_expr[0]) + for j, w in enumerate(wrt) + if (diff_value := r.diff(w)) != 0 + }, + ) + + if not replacements: + return [], [f1], [] + + f2 = Matrix.from_dok(l_red, l_sub, + { + (i, j): diff_value + for i, (r, fs) in enumerate([(r, r.free_symbols) for r in reduced_expr[0]]) + for j, s in enumerate(rep_sym) + if s in fs and (diff_value := r.diff(s)) != 0 + }, + ) + + rep_sym_set = set(rep_sym) + precomputed_fs = [s.free_symbols & rep_sym_set for s in sub_expr ] + + c_matrix = Matrix.from_dok(1, l_wrt, + {(0, j): diff_value for j, w in enumerate(wrt) + if (diff_value := sub_expr[0].diff(w)) != 0}) + + for i in range(1, l_sub): + + bi_matrix = Matrix.from_dok(1, i, + {(0, j): diff_value for j in range(i + 1) + if rep_sym[j] in precomputed_fs[i] + and (diff_value := sub_expr[i].diff(rep_sym[j])) != 0}) + + ai_matrix = Matrix.from_dok(1, l_wrt, + {(0, j): diff_value for j, w in enumerate(wrt) + if (diff_value := sub_expr[i].diff(w)) != 0}) + + if bi_matrix._rep.nnz(): + ci_matrix = bi_matrix.multiply(c_matrix).add(ai_matrix) + c_matrix = Matrix.vstack(c_matrix, ci_matrix) + else: + c_matrix = Matrix.vstack(c_matrix, ai_matrix) + + jacobian = f2.multiply(c_matrix).add(f1) + jacobian = [reduced_expr[0].__class__(jacobian)] + + return replacements, jacobian, precomputed_fs + + +def _forward_jacobian_norm_in_cse_out(expr, wrt): + """ + Function to compute the Jacobian of an input Matrix of expressions through + forward accumulation. Takes a sympy Matrix of expressions (expr) as input + and an iterable of variables (wrt) with respect to which to compute the + Jacobian matrix. The matrix is returned in reduced form (containing + replacement symbols) along with the ``replacements`` list. + + The function also returns a list of precomputed free symbols for each + subexpression, which are useful in the substitution process. + + Parameters + ========== + + expr : Matrix + The vector to be differentiated. + + wrt : iterable + The vector with respect to which to perform the differentiation. + Can be a matrix or an iterable of variables. + + Returns + ======= + + replacements : list of (Symbol, expression) pairs + Replacement symbols and relative common subexpressions that have been + replaced during a CSE operation. The output replacement list doesn't + contain replacement symbols inside ``Derivative``'s arguments. + + jacobian : list of SymPy expressions + The list only contains one element, which is the Jacobian matrix with + elements in reduced form (replacement symbols are present). + + precomputed_fs: list + List of sets, which store the free symbols present in each + sub-expression. Useful in the substitution process. + """ + + replacements, reduced_expr = cse(expr) + replacements, jacobian, precomputed_fs = _forward_jacobian_cse(replacements, reduced_expr, wrt) + + return replacements, jacobian, precomputed_fs + + +def _forward_jacobian(expr, wrt): + """ + Function to compute the Jacobian of an input Matrix of expressions through + forward accumulation. Takes a sympy Matrix of expressions (expr) as input + and an iterable of variables (wrt) with respect to which to compute the + Jacobian matrix. + + Explanation + =========== + + Expressions often contain repeated subexpressions. Using a tree structure, + these subexpressions are duplicated and differentiated multiple times, + leading to inefficiency. + + Instead, if a data structure called a directed acyclic graph (DAG) is used + then each of these repeated subexpressions will only exist a single time. + This function uses a combination of representing the expression as a DAG and + a forward accumulation algorithm (repeated application of the chain rule + symbolically) to more efficiently calculate the Jacobian matrix of a target + expression ``expr`` with respect to an expression or set of expressions + ``wrt``. + + Note that this function is intended to improve performance when + differentiating large expressions that contain many common subexpressions. + For small and simple expressions it is likely less performant than using + SymPy's standard differentiation functions and methods. + + Parameters + ========== + + expr : Matrix + The vector to be differentiated. + + wrt : iterable + The vector with respect to which to do the differentiation. + Can be a matrix or an iterable of variables. + + See Also + ======== + + Direct Acyclic Graph : https://en.wikipedia.org/wiki/Directed_acyclic_graph + """ + + replacements, reduced_expr = cse(expr) + + if replacements: + rep_sym, _ = map(Matrix, zip(*replacements)) + else: + rep_sym = Matrix([]) + + replacements, jacobian, precomputed_fs = _forward_jacobian_cse(replacements, reduced_expr, wrt) + + if not replacements: return jacobian[0] + + sub_rep = dict(replacements) + for i, ik in enumerate(precomputed_fs): + sub_dict = {j: sub_rep[j] for j in ik} + sub_rep[rep_sym[i]] = sub_rep[rep_sym[i]].xreplace(sub_dict) + + return jacobian[0].xreplace(sub_rep) diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/combsimp.py b/.venv/lib/python3.12/site-packages/sympy/simplify/combsimp.py new file mode 100644 index 0000000000000000000000000000000000000000..8b0b3cefcba11b4b7759b7d3ec3c2d4415cfd849 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/simplify/combsimp.py @@ -0,0 +1,114 @@ +from sympy.core import Mul +from sympy.core.function import count_ops +from sympy.core.traversal import preorder_traversal, bottom_up +from sympy.functions.combinatorial.factorials import binomial, factorial +from sympy.functions import gamma +from sympy.simplify.gammasimp import gammasimp, _gammasimp + +from sympy.utilities.timeutils import timethis + + +@timethis('combsimp') +def combsimp(expr): + r""" + Simplify combinatorial expressions. + + Explanation + =========== + + This function takes as input an expression containing factorials, + binomials, Pochhammer symbol and other "combinatorial" functions, + and tries to minimize the number of those functions and reduce + the size of their arguments. + + The algorithm works by rewriting all combinatorial functions as + gamma functions and applying gammasimp() except simplification + steps that may make an integer argument non-integer. See docstring + of gammasimp for more information. + + Then it rewrites expression in terms of factorials and binomials by + rewriting gammas as factorials and converting (a+b)!/a!b! into + binomials. + + If expression has gamma functions or combinatorial functions + with non-integer argument, it is automatically passed to gammasimp. + + Examples + ======== + + >>> from sympy.simplify import combsimp + >>> from sympy import factorial, binomial, symbols + >>> n, k = symbols('n k', integer = True) + + >>> combsimp(factorial(n)/factorial(n - 3)) + n*(n - 2)*(n - 1) + >>> combsimp(binomial(n+1, k+1)/binomial(n, k)) + (n + 1)/(k + 1) + + """ + + expr = expr.rewrite(gamma, piecewise=False) + if any(isinstance(node, gamma) and not node.args[0].is_integer + for node in preorder_traversal(expr)): + return gammasimp(expr) + + expr = _gammasimp(expr, as_comb = True) + expr = _gamma_as_comb(expr) + return expr + + +def _gamma_as_comb(expr): + """ + Helper function for combsimp. + + Rewrites expression in terms of factorials and binomials + """ + + expr = expr.rewrite(factorial) + + def f(rv): + if not rv.is_Mul: + return rv + rvd = rv.as_powers_dict() + nd_fact_args = [[], []] # numerator, denominator + + for k in rvd: + if isinstance(k, factorial) and rvd[k].is_Integer: + if rvd[k].is_positive: + nd_fact_args[0].extend([k.args[0]]*rvd[k]) + else: + nd_fact_args[1].extend([k.args[0]]*-rvd[k]) + rvd[k] = 0 + if not nd_fact_args[0] or not nd_fact_args[1]: + return rv + + hit = False + for m in range(2): + i = 0 + while i < len(nd_fact_args[m]): + ai = nd_fact_args[m][i] + for j in range(i + 1, len(nd_fact_args[m])): + aj = nd_fact_args[m][j] + + sum = ai + aj + if sum in nd_fact_args[1 - m]: + hit = True + + nd_fact_args[1 - m].remove(sum) + del nd_fact_args[m][j] + del nd_fact_args[m][i] + + rvd[binomial(sum, ai if count_ops(ai) < + count_ops(aj) else aj)] += ( + -1 if m == 0 else 1) + break + else: + i += 1 + + if hit: + return Mul(*([k**rvd[k] for k in rvd] + [factorial(k) + for k in nd_fact_args[0]]))/Mul(*[factorial(k) + for k in nd_fact_args[1]]) + return rv + + return bottom_up(expr, f) diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/cse_main.py b/.venv/lib/python3.12/site-packages/sympy/simplify/cse_main.py new file mode 100644 index 0000000000000000000000000000000000000000..bcd1b2e50adae8c3d3400d6c489e63a44ae1e59b --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/simplify/cse_main.py @@ -0,0 +1,945 @@ +""" Tools for doing common subexpression elimination. +""" +from collections import defaultdict + +from sympy.core import Basic, Mul, Add, Pow, sympify +from sympy.core.containers import Tuple, OrderedSet +from sympy.core.exprtools import factor_terms +from sympy.core.singleton import S +from sympy.core.sorting import ordered +from sympy.core.symbol import symbols, Symbol +from sympy.matrices import (MatrixBase, Matrix, ImmutableMatrix, + SparseMatrix, ImmutableSparseMatrix) +from sympy.matrices.expressions import (MatrixExpr, MatrixSymbol, MatMul, + MatAdd, MatPow, Inverse) +from sympy.matrices.expressions.matexpr import MatrixElement +from sympy.polys.rootoftools import RootOf +from sympy.utilities.iterables import numbered_symbols, sift, \ + topological_sort, iterable + +from . import cse_opts + +# (preprocessor, postprocessor) pairs which are commonly useful. They should +# each take a SymPy expression and return a possibly transformed expression. +# When used in the function ``cse()``, the target expressions will be transformed +# by each of the preprocessor functions in order. After the common +# subexpressions are eliminated, each resulting expression will have the +# postprocessor functions transform them in *reverse* order in order to undo the +# transformation if necessary. This allows the algorithm to operate on +# a representation of the expressions that allows for more optimization +# opportunities. +# ``None`` can be used to specify no transformation for either the preprocessor or +# postprocessor. + + +basic_optimizations = [(cse_opts.sub_pre, cse_opts.sub_post), + (factor_terms, None)] + +# sometimes we want the output in a different format; non-trivial +# transformations can be put here for users +# =============================================================== + + +def reps_toposort(r): + """Sort replacements ``r`` so (k1, v1) appears before (k2, v2) + if k2 is in v1's free symbols. This orders items in the + way that cse returns its results (hence, in order to use the + replacements in a substitution option it would make sense + to reverse the order). + + Examples + ======== + + >>> from sympy.simplify.cse_main import reps_toposort + >>> from sympy.abc import x, y + >>> from sympy import Eq + >>> for l, r in reps_toposort([(x, y + 1), (y, 2)]): + ... print(Eq(l, r)) + ... + Eq(y, 2) + Eq(x, y + 1) + + """ + r = sympify(r) + E = [] + for c1, (k1, v1) in enumerate(r): + for c2, (k2, v2) in enumerate(r): + if k1 in v2.free_symbols: + E.append((c1, c2)) + return [r[i] for i in topological_sort((range(len(r)), E))] + + +def cse_separate(r, e): + """Move expressions that are in the form (symbol, expr) out of the + expressions and sort them into the replacements using the reps_toposort. + + Examples + ======== + + >>> from sympy.simplify.cse_main import cse_separate + >>> from sympy.abc import x, y, z + >>> from sympy import cos, exp, cse, Eq, symbols + >>> x0, x1 = symbols('x:2') + >>> eq = (x + 1 + exp((x + 1)/(y + 1)) + cos(y + 1)) + >>> cse([eq, Eq(x, z + 1), z - 2], postprocess=cse_separate) in [ + ... [[(x0, y + 1), (x, z + 1), (x1, x + 1)], + ... [x1 + exp(x1/x0) + cos(x0), z - 2]], + ... [[(x1, y + 1), (x, z + 1), (x0, x + 1)], + ... [x0 + exp(x0/x1) + cos(x1), z - 2]]] + ... + True + """ + d = sift(e, lambda w: w.is_Equality and w.lhs.is_Symbol) + r = r + [w.args for w in d[True]] + e = d[False] + return [reps_toposort(r), e] + + +def cse_release_variables(r, e): + """ + Return tuples giving ``(a, b)`` where ``a`` is a symbol and ``b`` is + either an expression or None. The value of None is used when a + symbol is no longer needed for subsequent expressions. + + Use of such output can reduce the memory footprint of lambdified + expressions that contain large, repeated subexpressions. + + Examples + ======== + + >>> from sympy import cse + >>> from sympy.simplify.cse_main import cse_release_variables + >>> from sympy.abc import x, y + >>> eqs = [(x + y - 1)**2, x, x + y, (x + y)/(2*x + 1) + (x + y - 1)**2, (2*x + 1)**(x + y)] + >>> defs, rvs = cse_release_variables(*cse(eqs)) + >>> for i in defs: + ... print(i) + ... + (x0, x + y) + (x1, (x0 - 1)**2) + (x2, 2*x + 1) + (_3, x0/x2 + x1) + (_4, x2**x0) + (x2, None) + (_0, x1) + (x1, None) + (_2, x0) + (x0, None) + (_1, x) + >>> print(rvs) + (_0, _1, _2, _3, _4) + """ + if not r: + return r, e + + s, p = zip(*r) + esyms = symbols('_:%d' % len(e)) + syms = list(esyms) + s = list(s) + in_use = set(s) + p = list(p) + # sort e so those with most sub-expressions appear first + e = [(e[i], syms[i]) for i in range(len(e))] + e, syms = zip(*sorted(e, + key=lambda x: -sum(p[s.index(i)].count_ops() + for i in x[0].free_symbols & in_use))) + syms = list(syms) + p += e + rv = [] + i = len(p) - 1 + while i >= 0: + _p = p.pop() + c = in_use & _p.free_symbols + if c: # sorting for canonical results + rv.extend([(s, None) for s in sorted(c, key=str)]) + if i >= len(r): + rv.append((syms.pop(), _p)) + else: + rv.append((s[i], _p)) + in_use -= c + i -= 1 + rv.reverse() + return rv, esyms + + +# ====end of cse postprocess idioms=========================== + + +def preprocess_for_cse(expr, optimizations): + """ Preprocess an expression to optimize for common subexpression + elimination. + + Parameters + ========== + + expr : SymPy expression + The target expression to optimize. + optimizations : list of (callable, callable) pairs + The (preprocessor, postprocessor) pairs. + + Returns + ======= + + expr : SymPy expression + The transformed expression. + """ + for pre, post in optimizations: + if pre is not None: + expr = pre(expr) + return expr + + +def postprocess_for_cse(expr, optimizations): + """Postprocess an expression after common subexpression elimination to + return the expression to canonical SymPy form. + + Parameters + ========== + + expr : SymPy expression + The target expression to transform. + optimizations : list of (callable, callable) pairs, optional + The (preprocessor, postprocessor) pairs. The postprocessors will be + applied in reversed order to undo the effects of the preprocessors + correctly. + + Returns + ======= + + expr : SymPy expression + The transformed expression. + """ + for pre, post in reversed(optimizations): + if post is not None: + expr = post(expr) + return expr + + +class FuncArgTracker: + """ + A class which manages a mapping from functions to arguments and an inverse + mapping from arguments to functions. + """ + + def __init__(self, funcs): + # To minimize the number of symbolic comparisons, all function arguments + # get assigned a value number. + self.value_numbers = {} + self.value_number_to_value = [] + + # Both of these maps use integer indices for arguments / functions. + self.arg_to_funcset = [] + self.func_to_argset = [] + + for func_i, func in enumerate(funcs): + func_argset = OrderedSet() + + for func_arg in func.args: + arg_number = self.get_or_add_value_number(func_arg) + func_argset.add(arg_number) + self.arg_to_funcset[arg_number].add(func_i) + + self.func_to_argset.append(func_argset) + + def get_args_in_value_order(self, argset): + """ + Return the list of arguments in sorted order according to their value + numbers. + """ + return [self.value_number_to_value[argn] for argn in sorted(argset)] + + def get_or_add_value_number(self, value): + """ + Return the value number for the given argument. + """ + nvalues = len(self.value_numbers) + value_number = self.value_numbers.setdefault(value, nvalues) + if value_number == nvalues: + self.value_number_to_value.append(value) + self.arg_to_funcset.append(OrderedSet()) + return value_number + + def stop_arg_tracking(self, func_i): + """ + Remove the function func_i from the argument to function mapping. + """ + for arg in self.func_to_argset[func_i]: + self.arg_to_funcset[arg].remove(func_i) + + + def get_common_arg_candidates(self, argset, min_func_i=0): + """Return a dict whose keys are function numbers. The entries of the dict are + the number of arguments said function has in common with + ``argset``. Entries have at least 2 items in common. All keys have + value at least ``min_func_i``. + """ + count_map = defaultdict(lambda: 0) + if not argset: + return count_map + + funcsets = [self.arg_to_funcset[arg] for arg in argset] + # As an optimization below, we handle the largest funcset separately from + # the others. + largest_funcset = max(funcsets, key=len) + + for funcset in funcsets: + if largest_funcset is funcset: + continue + for func_i in funcset: + if func_i >= min_func_i: + count_map[func_i] += 1 + + # We pick the smaller of the two containers (count_map, largest_funcset) + # to iterate over to reduce the number of iterations needed. + (smaller_funcs_container, + larger_funcs_container) = sorted( + [largest_funcset, count_map], + key=len) + + for func_i in smaller_funcs_container: + # Not already in count_map? It can't possibly be in the output, so + # skip it. + if count_map[func_i] < 1: + continue + + if func_i in larger_funcs_container: + count_map[func_i] += 1 + + return {k: v for k, v in count_map.items() if v >= 2} + + def get_subset_candidates(self, argset, restrict_to_funcset=None): + """ + Return a set of functions each of which whose argument list contains + ``argset``, optionally filtered only to contain functions in + ``restrict_to_funcset``. + """ + iarg = iter(argset) + + indices = OrderedSet( + fi for fi in self.arg_to_funcset[next(iarg)]) + + if restrict_to_funcset is not None: + indices &= restrict_to_funcset + + for arg in iarg: + indices &= self.arg_to_funcset[arg] + + return indices + + def update_func_argset(self, func_i, new_argset): + """ + Update a function with a new set of arguments. + """ + new_args = OrderedSet(new_argset) + old_args = self.func_to_argset[func_i] + + for deleted_arg in old_args - new_args: + self.arg_to_funcset[deleted_arg].remove(func_i) + for added_arg in new_args - old_args: + self.arg_to_funcset[added_arg].add(func_i) + + self.func_to_argset[func_i].clear() + self.func_to_argset[func_i].update(new_args) + + +class Unevaluated: + + def __init__(self, func, args): + self.func = func + self.args = args + + def __str__(self): + return "Uneval<{}>({})".format( + self.func, ", ".join(str(a) for a in self.args)) + + def as_unevaluated_basic(self): + return self.func(*self.args, evaluate=False) + + @property + def free_symbols(self): + return set().union(*[a.free_symbols for a in self.args]) + + __repr__ = __str__ + + +def match_common_args(func_class, funcs, opt_subs): + """ + Recognize and extract common subexpressions of function arguments within a + set of function calls. For instance, for the following function calls:: + + x + z + y + sin(x + y) + + this will extract a common subexpression of `x + y`:: + + w = x + y + w + z + sin(w) + + The function we work with is assumed to be associative and commutative. + + Parameters + ========== + + func_class: class + The function class (e.g. Add, Mul) + funcs: list of functions + A list of function calls. + opt_subs: dict + A dictionary of substitutions which this function may update. + """ + + # Sort to ensure that whole-function subexpressions come before the items + # that use them. + funcs = sorted(funcs, key=lambda f: len(f.args)) + arg_tracker = FuncArgTracker(funcs) + + changed = OrderedSet() + + for i in range(len(funcs)): + common_arg_candidates_counts = arg_tracker.get_common_arg_candidates( + arg_tracker.func_to_argset[i], min_func_i=i + 1) + + # Sort the candidates in order of match size. + # This makes us try combining smaller matches first. + common_arg_candidates = OrderedSet(sorted( + common_arg_candidates_counts.keys(), + key=lambda k: (common_arg_candidates_counts[k], k))) + + while common_arg_candidates: + j = common_arg_candidates.pop(last=False) + + com_args = arg_tracker.func_to_argset[i].intersection( + arg_tracker.func_to_argset[j]) + + if len(com_args) <= 1: + # This may happen if a set of common arguments was already + # combined in a previous iteration. + continue + + # For all sets, replace the common symbols by the function + # over them, to allow recursive matches. + + diff_i = arg_tracker.func_to_argset[i].difference(com_args) + if diff_i: + # com_func needs to be unevaluated to allow for recursive matches. + com_func = Unevaluated( + func_class, arg_tracker.get_args_in_value_order(com_args)) + com_func_number = arg_tracker.get_or_add_value_number(com_func) + arg_tracker.update_func_argset(i, diff_i | OrderedSet([com_func_number])) + changed.add(i) + else: + # Treat the whole expression as a CSE. + # + # The reason this needs to be done is somewhat subtle. Within + # tree_cse(), to_eliminate only contains expressions that are + # seen more than once. The problem is unevaluated expressions + # do not compare equal to the evaluated equivalent. So + # tree_cse() won't mark funcs[i] as a CSE if we use an + # unevaluated version. + com_func_number = arg_tracker.get_or_add_value_number(funcs[i]) + + diff_j = arg_tracker.func_to_argset[j].difference(com_args) + arg_tracker.update_func_argset(j, diff_j | OrderedSet([com_func_number])) + changed.add(j) + + for k in arg_tracker.get_subset_candidates( + com_args, common_arg_candidates): + diff_k = arg_tracker.func_to_argset[k].difference(com_args) + arg_tracker.update_func_argset(k, diff_k | OrderedSet([com_func_number])) + changed.add(k) + + if i in changed: + opt_subs[funcs[i]] = Unevaluated(func_class, + arg_tracker.get_args_in_value_order(arg_tracker.func_to_argset[i])) + + arg_tracker.stop_arg_tracking(i) + + +def opt_cse(exprs, order='canonical'): + """Find optimization opportunities in Adds, Muls, Pows and negative + coefficient Muls. + + Parameters + ========== + + exprs : list of SymPy expressions + The expressions to optimize. + order : string, 'none' or 'canonical' + The order by which Mul and Add arguments are processed. For large + expressions where speed is a concern, use the setting order='none'. + + Returns + ======= + + opt_subs : dictionary of expression substitutions + The expression substitutions which can be useful to optimize CSE. + + Examples + ======== + + >>> from sympy.simplify.cse_main import opt_cse + >>> from sympy.abc import x + >>> opt_subs = opt_cse([x**-2]) + >>> k, v = list(opt_subs.keys())[0], list(opt_subs.values())[0] + >>> print((k, v.as_unevaluated_basic())) + (x**(-2), 1/(x**2)) + """ + opt_subs = {} + + adds = OrderedSet() + muls = OrderedSet() + + seen_subexp = set() + collapsible_subexp = set() + + def _find_opts(expr): + + if not isinstance(expr, (Basic, Unevaluated)): + return + + if expr.is_Atom or expr.is_Order: + return + + if iterable(expr): + list(map(_find_opts, expr)) + return + + if expr in seen_subexp: + return expr + seen_subexp.add(expr) + + list(map(_find_opts, expr.args)) + + if not isinstance(expr, MatrixExpr) and expr.could_extract_minus_sign(): + # XXX -expr does not always work rigorously for some expressions + # containing UnevaluatedExpr. + # https://github.com/sympy/sympy/issues/24818 + if isinstance(expr, Add): + neg_expr = Add(*(-i for i in expr.args)) + else: + neg_expr = -expr + + if not neg_expr.is_Atom: + opt_subs[expr] = Unevaluated(Mul, (S.NegativeOne, neg_expr)) + seen_subexp.add(neg_expr) + expr = neg_expr + + if isinstance(expr, (Mul, MatMul)): + if len(expr.args) == 1: + collapsible_subexp.add(expr) + else: + muls.add(expr) + + elif isinstance(expr, (Add, MatAdd)): + if len(expr.args) == 1: + collapsible_subexp.add(expr) + else: + adds.add(expr) + + elif isinstance(expr, Inverse): + # Do not want to treat `Inverse` as a `MatPow` + pass + + elif isinstance(expr, (Pow, MatPow)): + base, exp = expr.base, expr.exp + if exp.could_extract_minus_sign(): + opt_subs[expr] = Unevaluated(Pow, (Pow(base, -exp), -1)) + + for e in exprs: + if isinstance(e, (Basic, Unevaluated)): + _find_opts(e) + + # Handle collapsing of multinary operations with single arguments + edges = [(s, s.args[0]) for s in collapsible_subexp + if s.args[0] in collapsible_subexp] + for e in reversed(topological_sort((collapsible_subexp, edges))): + opt_subs[e] = opt_subs.get(e.args[0], e.args[0]) + + # split muls into commutative + commutative_muls = OrderedSet() + for m in muls: + c, nc = m.args_cnc(cset=False) + if c: + c_mul = m.func(*c) + if nc: + if c_mul == 1: + new_obj = m.func(*nc) + else: + if isinstance(m, MatMul): + new_obj = m.func(c_mul, *nc, evaluate=False) + else: + new_obj = m.func(c_mul, m.func(*nc), evaluate=False) + opt_subs[m] = new_obj + if len(c) > 1: + commutative_muls.add(c_mul) + + match_common_args(Add, adds, opt_subs) + match_common_args(Mul, commutative_muls, opt_subs) + + return opt_subs + + +def tree_cse(exprs, symbols, opt_subs=None, order='canonical', ignore=()): + """Perform raw CSE on expression tree, taking opt_subs into account. + + Parameters + ========== + + exprs : list of SymPy expressions + The expressions to reduce. + symbols : infinite iterator yielding unique Symbols + The symbols used to label the common subexpressions which are pulled + out. + opt_subs : dictionary of expression substitutions + The expressions to be substituted before any CSE action is performed. + order : string, 'none' or 'canonical' + The order by which Mul and Add arguments are processed. For large + expressions where speed is a concern, use the setting order='none'. + ignore : iterable of Symbols + Substitutions containing any Symbol from ``ignore`` will be ignored. + """ + if opt_subs is None: + opt_subs = {} + + ## Find repeated sub-expressions + + to_eliminate = set() + + seen_subexp = set() + excluded_symbols = set() + + def _find_repeated(expr): + if not isinstance(expr, (Basic, Unevaluated)): + return + + if isinstance(expr, RootOf): + return + + if isinstance(expr, Basic) and ( + expr.is_Atom or + expr.is_Order or + isinstance(expr, (MatrixSymbol, MatrixElement))): + if expr.is_Symbol: + excluded_symbols.add(expr.name) + return + + if iterable(expr): + args = expr + + else: + if expr in seen_subexp: + for ign in ignore: + if ign in expr.free_symbols: + break + else: + to_eliminate.add(expr) + return + + seen_subexp.add(expr) + + if expr in opt_subs: + expr = opt_subs[expr] + + args = expr.args + + list(map(_find_repeated, args)) + + for e in exprs: + if isinstance(e, Basic): + _find_repeated(e) + + ## Rebuild tree + + # Remove symbols from the generator that conflict with names in the expressions. + symbols = (_ for _ in symbols if _.name not in excluded_symbols) + + replacements = [] + + subs = {} + + def _rebuild(expr): + if not isinstance(expr, (Basic, Unevaluated)): + return expr + + if not expr.args: + return expr + + if iterable(expr): + new_args = [_rebuild(arg) for arg in expr.args] + return expr.func(*new_args) + + if expr in subs: + return subs[expr] + + orig_expr = expr + if expr in opt_subs: + expr = opt_subs[expr] + + # If enabled, parse Muls and Adds arguments by order to ensure + # replacement order independent from hashes + if order != 'none': + if isinstance(expr, (Mul, MatMul)): + c, nc = expr.args_cnc() + if c == [1]: + args = nc + else: + args = list(ordered(c)) + nc + elif isinstance(expr, (Add, MatAdd)): + args = list(ordered(expr.args)) + else: + args = expr.args + else: + args = expr.args + + new_args = list(map(_rebuild, args)) + if isinstance(expr, Unevaluated) or new_args != args: + new_expr = expr.func(*new_args) + else: + new_expr = expr + + if orig_expr in to_eliminate: + try: + sym = next(symbols) + except StopIteration: + raise ValueError("Symbols iterator ran out of symbols.") + + if isinstance(orig_expr, MatrixExpr): + sym = MatrixSymbol(sym.name, orig_expr.rows, + orig_expr.cols) + + subs[orig_expr] = sym + replacements.append((sym, new_expr)) + return sym + + else: + return new_expr + + reduced_exprs = [] + for e in exprs: + if isinstance(e, Basic): + reduced_e = _rebuild(e) + else: + reduced_e = e + reduced_exprs.append(reduced_e) + return replacements, reduced_exprs + + +def cse(exprs, symbols=None, optimizations=None, postprocess=None, + order='canonical', ignore=(), list=True): + """ Perform common subexpression elimination on an expression. + + Parameters + ========== + + exprs : list of SymPy expressions, or a single SymPy expression + The expressions to reduce. + symbols : infinite iterator yielding unique Symbols + The symbols used to label the common subexpressions which are pulled + out. The ``numbered_symbols`` generator is useful. The default is a + stream of symbols of the form "x0", "x1", etc. This must be an + infinite iterator. + optimizations : list of (callable, callable) pairs + The (preprocessor, postprocessor) pairs of external optimization + functions. Optionally 'basic' can be passed for a set of predefined + basic optimizations. Such 'basic' optimizations were used by default + in old implementation, however they can be really slow on larger + expressions. Now, no pre or post optimizations are made by default. + postprocess : a function which accepts the two return values of cse and + returns the desired form of output from cse, e.g. if you want the + replacements reversed the function might be the following lambda: + lambda r, e: return reversed(r), e + order : string, 'none' or 'canonical' + The order by which Mul and Add arguments are processed. If set to + 'canonical', arguments will be canonically ordered. If set to 'none', + ordering will be faster but dependent on expressions hashes, thus + machine dependent and variable. For large expressions where speed is a + concern, use the setting order='none'. + ignore : iterable of Symbols + Substitutions containing any Symbol from ``ignore`` will be ignored. + list : bool, (default True) + Returns expression in list or else with same type as input (when False). + + Returns + ======= + + replacements : list of (Symbol, expression) pairs + All of the common subexpressions that were replaced. Subexpressions + earlier in this list might show up in subexpressions later in this + list. + reduced_exprs : list of SymPy expressions + The reduced expressions with all of the replacements above. + + Examples + ======== + + >>> from sympy import cse, SparseMatrix + >>> from sympy.abc import x, y, z, w + >>> cse(((w + x + y + z)*(w + y + z))/(w + x)**3) + ([(x0, y + z), (x1, w + x)], [(w + x0)*(x0 + x1)/x1**3]) + + + List of expressions with recursive substitutions: + + >>> m = SparseMatrix([x + y, x + y + z]) + >>> cse([(x+y)**2, x + y + z, y + z, x + z + y, m]) + ([(x0, x + y), (x1, x0 + z)], [x0**2, x1, y + z, x1, Matrix([ + [x0], + [x1]])]) + + Note: the type and mutability of input matrices is retained. + + >>> isinstance(_[1][-1], SparseMatrix) + True + + The user may disallow substitutions containing certain symbols: + + >>> cse([y**2*(x + 1), 3*y**2*(x + 1)], ignore=(y,)) + ([(x0, x + 1)], [x0*y**2, 3*x0*y**2]) + + The default return value for the reduced expression(s) is a list, even if there is only + one expression. The `list` flag preserves the type of the input in the output: + + >>> cse(x) + ([], [x]) + >>> cse(x, list=False) + ([], x) + """ + if not list: + return _cse_homogeneous(exprs, + symbols=symbols, optimizations=optimizations, + postprocess=postprocess, order=order, ignore=ignore) + + if isinstance(exprs, (int, float)): + exprs = sympify(exprs) + + # Handle the case if just one expression was passed. + if isinstance(exprs, (Basic, MatrixBase)): + exprs = [exprs] + + copy = exprs + temp = [] + for e in exprs: + if isinstance(e, (Matrix, ImmutableMatrix)): + temp.append(Tuple(*e.flat())) + elif isinstance(e, (SparseMatrix, ImmutableSparseMatrix)): + temp.append(Tuple(*e.todok().items())) + else: + temp.append(e) + exprs = temp + del temp + + if optimizations is None: + optimizations = [] + elif optimizations == 'basic': + optimizations = basic_optimizations + + # Preprocess the expressions to give us better optimization opportunities. + reduced_exprs = [preprocess_for_cse(e, optimizations) for e in exprs] + + if symbols is None: + symbols = numbered_symbols(cls=Symbol) + else: + # In case we get passed an iterable with an __iter__ method instead of + # an actual iterator. + symbols = iter(symbols) + + # Find other optimization opportunities. + opt_subs = opt_cse(reduced_exprs, order) + + # Main CSE algorithm. + replacements, reduced_exprs = tree_cse(reduced_exprs, symbols, opt_subs, + order, ignore) + + # Postprocess the expressions to return the expressions to canonical form. + exprs = copy + replacements = [(sym, postprocess_for_cse(subtree, optimizations)) + for sym, subtree in replacements] + reduced_exprs = [postprocess_for_cse(e, optimizations) + for e in reduced_exprs] + + # Get the matrices back + for i, e in enumerate(exprs): + if isinstance(e, (Matrix, ImmutableMatrix)): + reduced_exprs[i] = Matrix(e.rows, e.cols, reduced_exprs[i]) + if isinstance(e, ImmutableMatrix): + reduced_exprs[i] = reduced_exprs[i].as_immutable() + elif isinstance(e, (SparseMatrix, ImmutableSparseMatrix)): + m = SparseMatrix(e.rows, e.cols, {}) + for k, v in reduced_exprs[i]: + m[k] = v + if isinstance(e, ImmutableSparseMatrix): + m = m.as_immutable() + reduced_exprs[i] = m + + if postprocess is None: + return replacements, reduced_exprs + + return postprocess(replacements, reduced_exprs) + + +def _cse_homogeneous(exprs, **kwargs): + """ + Same as ``cse`` but the ``reduced_exprs`` are returned + with the same type as ``exprs`` or a sympified version of the same. + + Parameters + ========== + + exprs : an Expr, iterable of Expr or dictionary with Expr values + the expressions in which repeated subexpressions will be identified + kwargs : additional arguments for the ``cse`` function + + Returns + ======= + + replacements : list of (Symbol, expression) pairs + All of the common subexpressions that were replaced. Subexpressions + earlier in this list might show up in subexpressions later in this + list. + reduced_exprs : list of SymPy expressions + The reduced expressions with all of the replacements above. + + Examples + ======== + + >>> from sympy.simplify.cse_main import cse + >>> from sympy import cos, Tuple, Matrix + >>> from sympy.abc import x + >>> output = lambda x: type(cse(x, list=False)[1]) + >>> output(1) + + >>> output('cos(x)') + + >>> output(cos(x)) + cos + >>> output(Tuple(1, x)) + + >>> output(Matrix([[1,0], [0,1]])) + + >>> output([1, x]) + + >>> output((1, x)) + + >>> output({1, x}) + + """ + if isinstance(exprs, str): + replacements, reduced_exprs = _cse_homogeneous( + sympify(exprs), **kwargs) + return replacements, repr(reduced_exprs) + if isinstance(exprs, (list, tuple, set)): + replacements, reduced_exprs = cse(exprs, **kwargs) + return replacements, type(exprs)(reduced_exprs) + if isinstance(exprs, dict): + keys = list(exprs.keys()) # In order to guarantee the order of the elements. + replacements, values = cse([exprs[k] for k in keys], **kwargs) + reduced_exprs = dict(zip(keys, values)) + return replacements, reduced_exprs + + try: + replacements, (reduced_exprs,) = cse(exprs, **kwargs) + except TypeError: # For example 'mpf' objects + return [], exprs + else: + return replacements, reduced_exprs diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/cse_opts.py b/.venv/lib/python3.12/site-packages/sympy/simplify/cse_opts.py new file mode 100644 index 0000000000000000000000000000000000000000..36a59857411de740ae47423442af88b118a3395d --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/simplify/cse_opts.py @@ -0,0 +1,52 @@ +""" Optimizations of the expression tree representation for better CSE +opportunities. +""" +from sympy.core import Add, Basic, Mul +from sympy.core.singleton import S +from sympy.core.sorting import default_sort_key +from sympy.core.traversal import preorder_traversal + + +def sub_pre(e): + """ Replace y - x with -(x - y) if -1 can be extracted from y - x. + """ + # replacing Add, A, from which -1 can be extracted with -1*-A + adds = [a for a in e.atoms(Add) if a.could_extract_minus_sign()] + reps = {} + ignore = set() + for a in adds: + na = -a + if na.is_Mul: # e.g. MatExpr + ignore.add(a) + continue + reps[a] = Mul._from_args([S.NegativeOne, na]) + + e = e.xreplace(reps) + + # repeat again for persisting Adds but mark these with a leading 1, -1 + # e.g. y - x -> 1*-1*(x - y) + if isinstance(e, Basic): + negs = {} + for a in sorted(e.atoms(Add), key=default_sort_key): + if a in ignore: + continue + if a in reps: + negs[a] = reps[a] + elif a.could_extract_minus_sign(): + negs[a] = Mul._from_args([S.One, S.NegativeOne, -a]) + e = e.xreplace(negs) + return e + + +def sub_post(e): + """ Replace 1*-1*x with -x. + """ + replacements = [] + for node in preorder_traversal(e): + if isinstance(node, Mul) and \ + node.args[0] is S.One and node.args[1] is S.NegativeOne: + replacements.append((node, -Mul._from_args(node.args[2:]))) + for node, replacement in replacements: + e = e.xreplace({node: replacement}) + + return e diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/epathtools.py b/.venv/lib/python3.12/site-packages/sympy/simplify/epathtools.py new file mode 100644 index 0000000000000000000000000000000000000000..7be983ada63fd39d7d467acf9afd62b3a41a2d85 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/simplify/epathtools.py @@ -0,0 +1,352 @@ +"""Tools for manipulation of expressions using paths. """ + +from sympy.core import Basic + + +class EPath: + r""" + Manipulate expressions using paths. + + EPath grammar in EBNF notation:: + + literal ::= /[A-Za-z_][A-Za-z_0-9]*/ + number ::= /-?\d+/ + type ::= literal + attribute ::= literal "?" + all ::= "*" + slice ::= "[" number? (":" number? (":" number?)?)? "]" + range ::= all | slice + query ::= (type | attribute) ("|" (type | attribute))* + selector ::= range | query range? + path ::= "/" selector ("/" selector)* + + See the docstring of the epath() function. + + """ + + __slots__ = ("_path", "_epath") + + def __new__(cls, path): + """Construct new EPath. """ + if isinstance(path, EPath): + return path + + if not path: + raise ValueError("empty EPath") + + _path = path + + if path[0] == '/': + path = path[1:] + else: + raise NotImplementedError("non-root EPath") + + epath = [] + + for selector in path.split('/'): + selector = selector.strip() + + if not selector: + raise ValueError("empty selector") + + index = 0 + + for c in selector: + if c.isalnum() or c in ('_', '|', '?'): + index += 1 + else: + break + + attrs = [] + types = [] + + if index: + elements = selector[:index] + selector = selector[index:] + + for element in elements.split('|'): + element = element.strip() + + if not element: + raise ValueError("empty element") + + if element.endswith('?'): + attrs.append(element[:-1]) + else: + types.append(element) + + span = None + + if selector == '*': + pass + else: + if selector.startswith('['): + try: + i = selector.index(']') + except ValueError: + raise ValueError("expected ']', got EOL") + + _span, span = selector[1:i], [] + + if ':' not in _span: + span = int(_span) + else: + for elt in _span.split(':', 3): + if not elt: + span.append(None) + else: + span.append(int(elt)) + + span = slice(*span) + + selector = selector[i + 1:] + + if selector: + raise ValueError("trailing characters in selector") + + epath.append((attrs, types, span)) + + obj = object.__new__(cls) + + obj._path = _path + obj._epath = epath + + return obj + + def __repr__(self): + return "%s(%r)" % (self.__class__.__name__, self._path) + + def _get_ordered_args(self, expr): + """Sort ``expr.args`` using printing order. """ + if expr.is_Add: + return expr.as_ordered_terms() + elif expr.is_Mul: + return expr.as_ordered_factors() + else: + return expr.args + + def _hasattrs(self, expr, attrs) -> bool: + """Check if ``expr`` has any of ``attrs``. """ + return all(hasattr(expr, attr) for attr in attrs) + + def _hastypes(self, expr, types): + """Check if ``expr`` is any of ``types``. """ + _types = [ cls.__name__ for cls in expr.__class__.mro() ] + return bool(set(_types).intersection(types)) + + def _has(self, expr, attrs, types): + """Apply ``_hasattrs`` and ``_hastypes`` to ``expr``. """ + if not (attrs or types): + return True + + if attrs and self._hasattrs(expr, attrs): + return True + + if types and self._hastypes(expr, types): + return True + + return False + + def apply(self, expr, func, args=None, kwargs=None): + """ + Modify parts of an expression selected by a path. + + Examples + ======== + + >>> from sympy.simplify.epathtools import EPath + >>> from sympy import sin, cos, E + >>> from sympy.abc import x, y, z, t + + >>> path = EPath("/*/[0]/Symbol") + >>> expr = [((x, 1), 2), ((3, y), z)] + + >>> path.apply(expr, lambda expr: expr**2) + [((x**2, 1), 2), ((3, y**2), z)] + + >>> path = EPath("/*/*/Symbol") + >>> expr = t + sin(x + 1) + cos(x + y + E) + + >>> path.apply(expr, lambda expr: 2*expr) + t + sin(2*x + 1) + cos(2*x + 2*y + E) + + """ + def _apply(path, expr, func): + if not path: + return func(expr) + else: + selector, path = path[0], path[1:] + attrs, types, span = selector + + if isinstance(expr, Basic): + if not expr.is_Atom: + args, basic = self._get_ordered_args(expr), True + else: + return expr + elif hasattr(expr, '__iter__'): + args, basic = expr, False + else: + return expr + + args = list(args) + + if span is not None: + if isinstance(span, slice): + indices = range(*span.indices(len(args))) + else: + indices = [span] + else: + indices = range(len(args)) + + for i in indices: + try: + arg = args[i] + except IndexError: + continue + + if self._has(arg, attrs, types): + args[i] = _apply(path, arg, func) + + if basic: + return expr.func(*args) + else: + return expr.__class__(args) + + _args, _kwargs = args or (), kwargs or {} + _func = lambda expr: func(expr, *_args, **_kwargs) + + return _apply(self._epath, expr, _func) + + def select(self, expr): + """ + Retrieve parts of an expression selected by a path. + + Examples + ======== + + >>> from sympy.simplify.epathtools import EPath + >>> from sympy import sin, cos, E + >>> from sympy.abc import x, y, z, t + + >>> path = EPath("/*/[0]/Symbol") + >>> expr = [((x, 1), 2), ((3, y), z)] + + >>> path.select(expr) + [x, y] + + >>> path = EPath("/*/*/Symbol") + >>> expr = t + sin(x + 1) + cos(x + y + E) + + >>> path.select(expr) + [x, x, y] + + """ + result = [] + + def _select(path, expr): + if not path: + result.append(expr) + else: + selector, path = path[0], path[1:] + attrs, types, span = selector + + if isinstance(expr, Basic): + args = self._get_ordered_args(expr) + elif hasattr(expr, '__iter__'): + args = expr + else: + return + + if span is not None: + if isinstance(span, slice): + args = args[span] + else: + try: + args = [args[span]] + except IndexError: + return + + for arg in args: + if self._has(arg, attrs, types): + _select(path, arg) + + _select(self._epath, expr) + return result + + +def epath(path, expr=None, func=None, args=None, kwargs=None): + r""" + Manipulate parts of an expression selected by a path. + + Explanation + =========== + + This function allows to manipulate large nested expressions in single + line of code, utilizing techniques to those applied in XML processing + standards (e.g. XPath). + + If ``func`` is ``None``, :func:`epath` retrieves elements selected by + the ``path``. Otherwise it applies ``func`` to each matching element. + + Note that it is more efficient to create an EPath object and use the select + and apply methods of that object, since this will compile the path string + only once. This function should only be used as a convenient shortcut for + interactive use. + + This is the supported syntax: + + * select all: ``/*`` + Equivalent of ``for arg in args:``. + * select slice: ``/[0]`` or ``/[1:5]`` or ``/[1:5:2]`` + Supports standard Python's slice syntax. + * select by type: ``/list`` or ``/list|tuple`` + Emulates ``isinstance()``. + * select by attribute: ``/__iter__?`` + Emulates ``hasattr()``. + + Parameters + ========== + + path : str | EPath + A path as a string or a compiled EPath. + expr : Basic | iterable + An expression or a container of expressions. + func : callable (optional) + A callable that will be applied to matching parts. + args : tuple (optional) + Additional positional arguments to ``func``. + kwargs : dict (optional) + Additional keyword arguments to ``func``. + + Examples + ======== + + >>> from sympy.simplify.epathtools import epath + >>> from sympy import sin, cos, E + >>> from sympy.abc import x, y, z, t + + >>> path = "/*/[0]/Symbol" + >>> expr = [((x, 1), 2), ((3, y), z)] + + >>> epath(path, expr) + [x, y] + >>> epath(path, expr, lambda expr: expr**2) + [((x**2, 1), 2), ((3, y**2), z)] + + >>> path = "/*/*/Symbol" + >>> expr = t + sin(x + 1) + cos(x + y + E) + + >>> epath(path, expr) + [x, x, y] + >>> epath(path, expr, lambda expr: 2*expr) + t + sin(2*x + 1) + cos(2*x + 2*y + E) + + """ + _epath = EPath(path) + + if expr is None: + return _epath + if func is None: + return _epath.select(expr) + else: + return _epath.apply(expr, func, args, kwargs) diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/fu.py b/.venv/lib/python3.12/site-packages/sympy/simplify/fu.py new file mode 100644 index 0000000000000000000000000000000000000000..a26706edca98385df0009a8ee41476a17d36420c --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/simplify/fu.py @@ -0,0 +1,2112 @@ +from collections import defaultdict + +from sympy.core.add import Add +from sympy.core.cache import cacheit +from sympy.core.expr import Expr +from sympy.core.exprtools import Factors, gcd_terms, factor_terms +from sympy.core.function import expand_mul +from sympy.core.mul import Mul +from sympy.core.numbers import pi, I +from sympy.core.power import Pow +from sympy.core.singleton import S +from sympy.core.sorting import ordered +from sympy.core.symbol import Dummy +from sympy.core.sympify import sympify +from sympy.core.traversal import bottom_up +from sympy.functions.combinatorial.factorials import binomial +from sympy.functions.elementary.hyperbolic import ( + cosh, sinh, tanh, coth, sech, csch, HyperbolicFunction) +from sympy.functions.elementary.trigonometric import ( + cos, sin, tan, cot, sec, csc, sqrt, TrigonometricFunction) +from sympy.ntheory.factor_ import perfect_power +from sympy.polys.polytools import factor +from sympy.strategies.tree import greedy +from sympy.strategies.core import identity, debug + +from sympy import SYMPY_DEBUG + + +# ================== Fu-like tools =========================== + + +def TR0(rv): + """Simplification of rational polynomials, trying to simplify + the expression, e.g. combine things like 3*x + 2*x, etc.... + """ + # although it would be nice to use cancel, it doesn't work + # with noncommutatives + return rv.normal().factor().expand() + + +def TR1(rv): + """Replace sec, csc with 1/cos, 1/sin + + Examples + ======== + + >>> from sympy.simplify.fu import TR1, sec, csc + >>> from sympy.abc import x + >>> TR1(2*csc(x) + sec(x)) + 1/cos(x) + 2/sin(x) + """ + + def f(rv): + if isinstance(rv, sec): + a = rv.args[0] + return S.One/cos(a) + elif isinstance(rv, csc): + a = rv.args[0] + return S.One/sin(a) + return rv + + return bottom_up(rv, f) + + +def TR2(rv): + """Replace tan and cot with sin/cos and cos/sin + + Examples + ======== + + >>> from sympy.simplify.fu import TR2 + >>> from sympy.abc import x + >>> from sympy import tan, cot, sin, cos + >>> TR2(tan(x)) + sin(x)/cos(x) + >>> TR2(cot(x)) + cos(x)/sin(x) + >>> TR2(tan(tan(x) - sin(x)/cos(x))) + 0 + + """ + + def f(rv): + if isinstance(rv, tan): + a = rv.args[0] + return sin(a)/cos(a) + elif isinstance(rv, cot): + a = rv.args[0] + return cos(a)/sin(a) + return rv + + return bottom_up(rv, f) + + +def TR2i(rv, half=False): + """Converts ratios involving sin and cos as follows:: + sin(x)/cos(x) -> tan(x) + sin(x)/(cos(x) + 1) -> tan(x/2) if half=True + + Examples + ======== + + >>> from sympy.simplify.fu import TR2i + >>> from sympy.abc import x, a + >>> from sympy import sin, cos + >>> TR2i(sin(x)/cos(x)) + tan(x) + + Powers of the numerator and denominator are also recognized + + >>> TR2i(sin(x)**2/(cos(x) + 1)**2, half=True) + tan(x/2)**2 + + The transformation does not take place unless assumptions allow + (i.e. the base must be positive or the exponent must be an integer + for both numerator and denominator) + + >>> TR2i(sin(x)**a/(cos(x) + 1)**a) + sin(x)**a/(cos(x) + 1)**a + + """ + + def f(rv): + if not rv.is_Mul: + return rv + + n, d = rv.as_numer_denom() + if n.is_Atom or d.is_Atom: + return rv + + def ok(k, e): + # initial filtering of factors + return ( + (e.is_integer or k.is_positive) and ( + k.func in (sin, cos) or (half and + k.is_Add and + len(k.args) >= 2 and + any(any(isinstance(ai, cos) or ai.is_Pow and ai.base is cos + for ai in Mul.make_args(a)) for a in k.args)))) + + n = n.as_powers_dict() + ndone = [(k, n.pop(k)) for k in list(n.keys()) if not ok(k, n[k])] + if not n: + return rv + + d = d.as_powers_dict() + ddone = [(k, d.pop(k)) for k in list(d.keys()) if not ok(k, d[k])] + if not d: + return rv + + # factoring if necessary + + def factorize(d, ddone): + newk = [] + for k in d: + if k.is_Add and len(k.args) > 1: + knew = factor(k) if half else factor_terms(k) + if knew != k: + newk.append((k, knew)) + if newk: + for i, (k, knew) in enumerate(newk): + del d[k] + newk[i] = knew + newk = Mul(*newk).as_powers_dict() + for k in newk: + v = d[k] + newk[k] + if ok(k, v): + d[k] = v + else: + ddone.append((k, v)) + del newk + factorize(n, ndone) + factorize(d, ddone) + + # joining + t = [] + for k in n: + if isinstance(k, sin): + a = cos(k.args[0], evaluate=False) + if a in d and d[a] == n[k]: + t.append(tan(k.args[0])**n[k]) + n[k] = d[a] = None + elif half: + a1 = 1 + a + if a1 in d and d[a1] == n[k]: + t.append((tan(k.args[0]/2))**n[k]) + n[k] = d[a1] = None + elif isinstance(k, cos): + a = sin(k.args[0], evaluate=False) + if a in d and d[a] == n[k]: + t.append(tan(k.args[0])**-n[k]) + n[k] = d[a] = None + elif half and k.is_Add and k.args[0] is S.One and \ + isinstance(k.args[1], cos): + a = sin(k.args[1].args[0], evaluate=False) + if a in d and d[a] == n[k] and (d[a].is_integer or \ + a.is_positive): + t.append(tan(a.args[0]/2)**-n[k]) + n[k] = d[a] = None + + if t: + rv = Mul(*(t + [b**e for b, e in n.items() if e]))/\ + Mul(*[b**e for b, e in d.items() if e]) + rv *= Mul(*[b**e for b, e in ndone])/Mul(*[b**e for b, e in ddone]) + + return rv + + return bottom_up(rv, f) + + +def TR3(rv): + """Induced formula: example sin(-a) = -sin(a) + + Examples + ======== + + >>> from sympy.simplify.fu import TR3 + >>> from sympy.abc import x, y + >>> from sympy import pi + >>> from sympy import cos + >>> TR3(cos(y - x*(y - x))) + cos(x*(x - y) + y) + >>> cos(pi/2 + x) + -sin(x) + >>> cos(30*pi/2 + x) + -cos(x) + + """ + from sympy.simplify.simplify import signsimp + + # Negative argument (already automatic for funcs like sin(-x) -> -sin(x) + # but more complicated expressions can use it, too). Also, trig angles + # between pi/4 and pi/2 are not reduced to an angle between 0 and pi/4. + # The following are automatically handled: + # Argument of type: pi/2 +/- angle + # Argument of type: pi +/- angle + # Argument of type : 2k*pi +/- angle + + def f(rv): + if not isinstance(rv, TrigonometricFunction): + return rv + rv = rv.func(signsimp(rv.args[0])) + if not isinstance(rv, TrigonometricFunction): + return rv + if (rv.args[0] - S.Pi/4).is_positive is (S.Pi/2 - rv.args[0]).is_positive is True: + fmap = {cos: sin, sin: cos, tan: cot, cot: tan, sec: csc, csc: sec} + rv = fmap[type(rv)](S.Pi/2 - rv.args[0]) + return rv + + # touch numbers iside of trig functions to let them automatically update + rv = rv.replace( + lambda x: isinstance(x, TrigonometricFunction), + lambda x: x.replace( + lambda n: n.is_number and n.is_Mul, + lambda n: n.func(*n.args))) + + return bottom_up(rv, f) + + +def TR4(rv): + """Identify values of special angles. + + a= 0 pi/6 pi/4 pi/3 pi/2 + ---------------------------------------------------- + sin(a) 0 1/2 sqrt(2)/2 sqrt(3)/2 1 + cos(a) 1 sqrt(3)/2 sqrt(2)/2 1/2 0 + tan(a) 0 sqt(3)/3 1 sqrt(3) -- + + Examples + ======== + + >>> from sympy import pi + >>> from sympy import cos, sin, tan, cot + >>> for s in (0, pi/6, pi/4, pi/3, pi/2): + ... print('%s %s %s %s' % (cos(s), sin(s), tan(s), cot(s))) + ... + 1 0 0 zoo + sqrt(3)/2 1/2 sqrt(3)/3 sqrt(3) + sqrt(2)/2 sqrt(2)/2 1 1 + 1/2 sqrt(3)/2 sqrt(3) sqrt(3)/3 + 0 1 zoo 0 + """ + # special values at 0, pi/6, pi/4, pi/3, pi/2 already handled + return rv.replace( + lambda x: + isinstance(x, TrigonometricFunction) and + (r:=x.args[0]/pi).is_Rational and r.q in (1, 2, 3, 4, 6), + lambda x: + x.func(x.args[0].func(*x.args[0].args))) + + +def _TR56(rv, f, g, h, max, pow): + """Helper for TR5 and TR6 to replace f**2 with h(g**2) + + Options + ======= + + max : controls size of exponent that can appear on f + e.g. if max=4 then f**4 will be changed to h(g**2)**2. + pow : controls whether the exponent must be a perfect power of 2 + e.g. if pow=True (and max >= 6) then f**6 will not be changed + but f**8 will be changed to h(g**2)**4 + + >>> from sympy.simplify.fu import _TR56 as T + >>> from sympy.abc import x + >>> from sympy import sin, cos + >>> h = lambda x: 1 - x + >>> T(sin(x)**3, sin, cos, h, 4, False) + (1 - cos(x)**2)*sin(x) + >>> T(sin(x)**6, sin, cos, h, 6, False) + (1 - cos(x)**2)**3 + >>> T(sin(x)**6, sin, cos, h, 6, True) + sin(x)**6 + >>> T(sin(x)**8, sin, cos, h, 10, True) + (1 - cos(x)**2)**4 + """ + + def _f(rv): + # I'm not sure if this transformation should target all even powers + # or only those expressible as powers of 2. Also, should it only + # make the changes in powers that appear in sums -- making an isolated + # change is not going to allow a simplification as far as I can tell. + if not (rv.is_Pow and rv.base.func == f): + return rv + if not rv.exp.is_real: + return rv + + if (rv.exp < 0) == True: + return rv + if (rv.exp > max) == True: + return rv + if rv.exp == 1: + return rv + if rv.exp == 2: + return h(g(rv.base.args[0])**2) + else: + if rv.exp % 2 == 1: + e = rv.exp//2 + return f(rv.base.args[0])*h(g(rv.base.args[0])**2)**e + elif rv.exp == 4: + e = 2 + elif not pow: + if rv.exp % 2: + return rv + e = rv.exp//2 + else: + p = perfect_power(rv.exp) + if not p: + return rv + e = rv.exp//2 + return h(g(rv.base.args[0])**2)**e + + return bottom_up(rv, _f) + + +def TR5(rv, max=4, pow=False): + """Replacement of sin**2 with 1 - cos(x)**2. + + See _TR56 docstring for advanced use of ``max`` and ``pow``. + + Examples + ======== + + >>> from sympy.simplify.fu import TR5 + >>> from sympy.abc import x + >>> from sympy import sin + >>> TR5(sin(x)**2) + 1 - cos(x)**2 + >>> TR5(sin(x)**-2) # unchanged + sin(x)**(-2) + >>> TR5(sin(x)**4) + (1 - cos(x)**2)**2 + """ + return _TR56(rv, sin, cos, lambda x: 1 - x, max=max, pow=pow) + + +def TR6(rv, max=4, pow=False): + """Replacement of cos**2 with 1 - sin(x)**2. + + See _TR56 docstring for advanced use of ``max`` and ``pow``. + + Examples + ======== + + >>> from sympy.simplify.fu import TR6 + >>> from sympy.abc import x + >>> from sympy import cos + >>> TR6(cos(x)**2) + 1 - sin(x)**2 + >>> TR6(cos(x)**-2) #unchanged + cos(x)**(-2) + >>> TR6(cos(x)**4) + (1 - sin(x)**2)**2 + """ + return _TR56(rv, cos, sin, lambda x: 1 - x, max=max, pow=pow) + + +def TR7(rv): + """Lowering the degree of cos(x)**2. + + Examples + ======== + + >>> from sympy.simplify.fu import TR7 + >>> from sympy.abc import x + >>> from sympy import cos + >>> TR7(cos(x)**2) + cos(2*x)/2 + 1/2 + >>> TR7(cos(x)**2 + 1) + cos(2*x)/2 + 3/2 + + """ + + def f(rv): + if not (rv.is_Pow and rv.base.func == cos and rv.exp == 2): + return rv + return (1 + cos(2*rv.base.args[0]))/2 + + return bottom_up(rv, f) + + +def TR8(rv, first=True): + """Converting products of ``cos`` and/or ``sin`` to a sum or + difference of ``cos`` and or ``sin`` terms. + + Examples + ======== + + >>> from sympy.simplify.fu import TR8 + >>> from sympy import cos, sin + >>> TR8(cos(2)*cos(3)) + cos(5)/2 + cos(1)/2 + >>> TR8(cos(2)*sin(3)) + sin(5)/2 + sin(1)/2 + >>> TR8(sin(2)*sin(3)) + -cos(5)/2 + cos(1)/2 + """ + + def f(rv): + if not ( + rv.is_Mul or + rv.is_Pow and + rv.base.func in (cos, sin) and + (rv.exp.is_integer or rv.base.is_positive)): + return rv + + if first: + n, d = [expand_mul(i) for i in rv.as_numer_denom()] + newn = TR8(n, first=False) + newd = TR8(d, first=False) + if newn != n or newd != d: + rv = gcd_terms(newn/newd) + if rv.is_Mul and rv.args[0].is_Rational and \ + len(rv.args) == 2 and rv.args[1].is_Add: + rv = Mul(*rv.as_coeff_Mul()) + return rv + + args = {cos: [], sin: [], None: []} + for a in Mul.make_args(rv): + if a.func in (cos, sin): + args[type(a)].append(a.args[0]) + elif (a.is_Pow and a.exp.is_Integer and a.exp > 0 and \ + a.base.func in (cos, sin)): + # XXX this is ok but pathological expression could be handled + # more efficiently as in TRmorrie + args[type(a.base)].extend([a.base.args[0]]*a.exp) + else: + args[None].append(a) + c = args[cos] + s = args[sin] + if not (c and s or len(c) > 1 or len(s) > 1): + return rv + + args = args[None] + n = min(len(c), len(s)) + for i in range(n): + a1 = s.pop() + a2 = c.pop() + args.append((sin(a1 + a2) + sin(a1 - a2))/2) + while len(c) > 1: + a1 = c.pop() + a2 = c.pop() + args.append((cos(a1 + a2) + cos(a1 - a2))/2) + if c: + args.append(cos(c.pop())) + while len(s) > 1: + a1 = s.pop() + a2 = s.pop() + args.append((-cos(a1 + a2) + cos(a1 - a2))/2) + if s: + args.append(sin(s.pop())) + return TR8(expand_mul(Mul(*args))) + + return bottom_up(rv, f) + + +def TR9(rv): + """Sum of ``cos`` or ``sin`` terms as a product of ``cos`` or ``sin``. + + Examples + ======== + + >>> from sympy.simplify.fu import TR9 + >>> from sympy import cos, sin + >>> TR9(cos(1) + cos(2)) + 2*cos(1/2)*cos(3/2) + >>> TR9(cos(1) + 2*sin(1) + 2*sin(2)) + cos(1) + 4*sin(3/2)*cos(1/2) + + If no change is made by TR9, no re-arrangement of the + expression will be made. For example, though factoring + of common term is attempted, if the factored expression + was not changed, the original expression will be returned: + + >>> TR9(cos(3) + cos(3)*cos(2)) + cos(3) + cos(2)*cos(3) + + """ + + def f(rv): + if not rv.is_Add: + return rv + + def do(rv, first=True): + # cos(a)+/-cos(b) can be combined into a product of cosines and + # sin(a)+/-sin(b) can be combined into a product of cosine and + # sine. + # + # If there are more than two args, the pairs which "work" will + # have a gcd extractable and the remaining two terms will have + # the above structure -- all pairs must be checked to find the + # ones that work. args that don't have a common set of symbols + # are skipped since this doesn't lead to a simpler formula and + # also has the arbitrariness of combining, for example, the x + # and y term instead of the y and z term in something like + # cos(x) + cos(y) + cos(z). + + if not rv.is_Add: + return rv + + args = list(ordered(rv.args)) + if len(args) != 2: + hit = False + for i in range(len(args)): + ai = args[i] + if ai is None: + continue + for j in range(i + 1, len(args)): + aj = args[j] + if aj is None: + continue + was = ai + aj + new = do(was) + if new != was: + args[i] = new # update in place + args[j] = None + hit = True + break # go to next i + if hit: + rv = Add(*[_f for _f in args if _f]) + if rv.is_Add: + rv = do(rv) + + return rv + + # two-arg Add + split = trig_split(*args) + if not split: + return rv + gcd, n1, n2, a, b, iscos = split + + # application of rule if possible + if iscos: + if n1 == n2: + return gcd*n1*2*cos((a + b)/2)*cos((a - b)/2) + if n1 < 0: + a, b = b, a + return -2*gcd*sin((a + b)/2)*sin((a - b)/2) + else: + if n1 == n2: + return gcd*n1*2*sin((a + b)/2)*cos((a - b)/2) + if n1 < 0: + a, b = b, a + return 2*gcd*cos((a + b)/2)*sin((a - b)/2) + + return process_common_addends(rv, do) # DON'T sift by free symbols + + return bottom_up(rv, f) + + +def TR10(rv, first=True): + """Separate sums in ``cos`` and ``sin``. + + Examples + ======== + + >>> from sympy.simplify.fu import TR10 + >>> from sympy.abc import a, b, c + >>> from sympy import cos, sin + >>> TR10(cos(a + b)) + -sin(a)*sin(b) + cos(a)*cos(b) + >>> TR10(sin(a + b)) + sin(a)*cos(b) + sin(b)*cos(a) + >>> TR10(sin(a + b + c)) + (-sin(a)*sin(b) + cos(a)*cos(b))*sin(c) + \ + (sin(a)*cos(b) + sin(b)*cos(a))*cos(c) + """ + + def f(rv): + if rv.func not in (cos, sin): + return rv + + f = rv.func + arg = rv.args[0] + if arg.is_Add: + if first: + args = list(ordered(arg.args)) + else: + args = list(arg.args) + a = args.pop() + b = Add._from_args(args) + if b.is_Add: + if f == sin: + return sin(a)*TR10(cos(b), first=False) + \ + cos(a)*TR10(sin(b), first=False) + else: + return cos(a)*TR10(cos(b), first=False) - \ + sin(a)*TR10(sin(b), first=False) + else: + if f == sin: + return sin(a)*cos(b) + cos(a)*sin(b) + else: + return cos(a)*cos(b) - sin(a)*sin(b) + return rv + + return bottom_up(rv, f) + + +def TR10i(rv): + """Sum of products to function of sum. + + Examples + ======== + + >>> from sympy.simplify.fu import TR10i + >>> from sympy import cos, sin, sqrt + >>> from sympy.abc import x + + >>> TR10i(cos(1)*cos(3) + sin(1)*sin(3)) + cos(2) + >>> TR10i(cos(1)*sin(3) + sin(1)*cos(3) + cos(3)) + cos(3) + sin(4) + >>> TR10i(sqrt(2)*cos(x)*x + sqrt(6)*sin(x)*x) + 2*sqrt(2)*x*sin(x + pi/6) + + """ + def f(rv): + if not rv.is_Add: + return rv + + def do(rv, first=True): + # args which can be expressed as A*(cos(a)*cos(b)+/-sin(a)*sin(b)) + # or B*(cos(a)*sin(b)+/-cos(b)*sin(a)) can be combined into + # A*f(a+/-b) where f is either sin or cos. + # + # If there are more than two args, the pairs which "work" will have + # a gcd extractable and the remaining two terms will have the above + # structure -- all pairs must be checked to find the ones that + # work. + + if not rv.is_Add: + return rv + + args = list(ordered(rv.args)) + if len(args) != 2: + hit = False + for i in range(len(args)): + ai = args[i] + if ai is None: + continue + for j in range(i + 1, len(args)): + aj = args[j] + if aj is None: + continue + was = ai + aj + new = do(was) + if new != was: + args[i] = new # update in place + args[j] = None + hit = True + break # go to next i + if hit: + rv = Add(*[_f for _f in args if _f]) + if rv.is_Add: + rv = do(rv) + + return rv + + # two-arg Add + split = trig_split(*args, two=True) + if not split: + return rv + gcd, n1, n2, a, b, same = split + + # identify and get c1 to be cos then apply rule if possible + if same: # coscos, sinsin + gcd = n1*gcd + if n1 == n2: + return gcd*cos(a - b) + return gcd*cos(a + b) + else: #cossin, cossin + gcd = n1*gcd + if n1 == n2: + return gcd*sin(a + b) + return gcd*sin(b - a) + + rv = process_common_addends( + rv, do, lambda x: tuple(ordered(x.free_symbols))) + + # need to check for inducible pairs in ratio of sqrt(3):1 that + # appeared in different lists when sorting by coefficient + while rv.is_Add: + byrad = defaultdict(list) + for a in rv.args: + hit = 0 + if a.is_Mul: + for ai in a.args: + if ai.is_Pow and ai.exp is S.Half and \ + ai.base.is_Integer: + byrad[ai].append(a) + hit = 1 + break + if not hit: + byrad[S.One].append(a) + + # no need to check all pairs -- just check for the onees + # that have the right ratio + args = [] + for a in byrad: + for b in [_ROOT3()*a, _invROOT3()]: + if b in byrad: + for i in range(len(byrad[a])): + if byrad[a][i] is None: + continue + for j in range(len(byrad[b])): + if byrad[b][j] is None: + continue + was = Add(byrad[a][i] + byrad[b][j]) + new = do(was) + if new != was: + args.append(new) + byrad[a][i] = None + byrad[b][j] = None + break + if args: + rv = Add(*(args + [Add(*[_f for _f in v if _f]) + for v in byrad.values()])) + else: + rv = do(rv) # final pass to resolve any new inducible pairs + break + + return rv + + return bottom_up(rv, f) + + +def TR11(rv, base=None): + """Function of double angle to product. The ``base`` argument can be used + to indicate what is the un-doubled argument, e.g. if 3*pi/7 is the base + then cosine and sine functions with argument 6*pi/7 will be replaced. + + Examples + ======== + + >>> from sympy.simplify.fu import TR11 + >>> from sympy import cos, sin, pi + >>> from sympy.abc import x + >>> TR11(sin(2*x)) + 2*sin(x)*cos(x) + >>> TR11(cos(2*x)) + -sin(x)**2 + cos(x)**2 + >>> TR11(sin(4*x)) + 4*(-sin(x)**2 + cos(x)**2)*sin(x)*cos(x) + >>> TR11(sin(4*x/3)) + 4*(-sin(x/3)**2 + cos(x/3)**2)*sin(x/3)*cos(x/3) + + If the arguments are simply integers, no change is made + unless a base is provided: + + >>> TR11(cos(2)) + cos(2) + >>> TR11(cos(4), 2) + -sin(2)**2 + cos(2)**2 + + There is a subtle issue here in that autosimplification will convert + some higher angles to lower angles + + >>> cos(6*pi/7) + cos(3*pi/7) + -cos(pi/7) + cos(3*pi/7) + + The 6*pi/7 angle is now pi/7 but can be targeted with TR11 by supplying + the 3*pi/7 base: + + >>> TR11(_, 3*pi/7) + -sin(3*pi/7)**2 + cos(3*pi/7)**2 + cos(3*pi/7) + + """ + + def f(rv): + if rv.func not in (cos, sin): + return rv + + if base: + f = rv.func + t = f(base*2) + co = S.One + if t.is_Mul: + co, t = t.as_coeff_Mul() + if t.func not in (cos, sin): + return rv + if rv.args[0] == t.args[0]: + c = cos(base) + s = sin(base) + if f is cos: + return (c**2 - s**2)/co + else: + return 2*c*s/co + return rv + + elif not rv.args[0].is_Number: + # make a change if the leading coefficient's numerator is + # divisible by 2 + c, m = rv.args[0].as_coeff_Mul(rational=True) + if c.p % 2 == 0: + arg = c.p//2*m/c.q + c = TR11(cos(arg)) + s = TR11(sin(arg)) + if rv.func == sin: + rv = 2*s*c + else: + rv = c**2 - s**2 + return rv + + return bottom_up(rv, f) + + +def _TR11(rv): + """ + Helper for TR11 to find half-arguments for sin in factors of + num/den that appear in cos or sin factors in the den/num. + + Examples + ======== + + >>> from sympy.simplify.fu import TR11, _TR11 + >>> from sympy import cos, sin + >>> from sympy.abc import x + >>> TR11(sin(x/3)/(cos(x/6))) + sin(x/3)/cos(x/6) + >>> _TR11(sin(x/3)/(cos(x/6))) + 2*sin(x/6) + >>> TR11(sin(x/6)/(sin(x/3))) + sin(x/6)/sin(x/3) + >>> _TR11(sin(x/6)/(sin(x/3))) + 1/(2*cos(x/6)) + + """ + def f(rv): + if not isinstance(rv, Expr): + return rv + + def sincos_args(flat): + # find arguments of sin and cos that + # appears as bases in args of flat + # and have Integer exponents + args = defaultdict(set) + for fi in Mul.make_args(flat): + b, e = fi.as_base_exp() + if e.is_Integer and e > 0: + if b.func in (cos, sin): + args[type(b)].add(b.args[0]) + return args + num_args, den_args = map(sincos_args, rv.as_numer_denom()) + def handle_match(rv, num_args, den_args): + # for arg in sin args of num_args, look for arg/2 + # in den_args and pass this half-angle to TR11 + # for handling in rv + for narg in num_args[sin]: + half = narg/2 + if half in den_args[cos]: + func = cos + elif half in den_args[sin]: + func = sin + else: + continue + rv = TR11(rv, half) + den_args[func].remove(half) + return rv + # sin in num, sin or cos in den + rv = handle_match(rv, num_args, den_args) + # sin in den, sin or cos in num + rv = handle_match(rv, den_args, num_args) + return rv + + return bottom_up(rv, f) + + +def TR12(rv, first=True): + """Separate sums in ``tan``. + + Examples + ======== + + >>> from sympy.abc import x, y + >>> from sympy import tan + >>> from sympy.simplify.fu import TR12 + >>> TR12(tan(x + y)) + (tan(x) + tan(y))/(-tan(x)*tan(y) + 1) + """ + + def f(rv): + if not rv.func == tan: + return rv + + arg = rv.args[0] + if arg.is_Add: + if first: + args = list(ordered(arg.args)) + else: + args = list(arg.args) + a = args.pop() + b = Add._from_args(args) + if b.is_Add: + tb = TR12(tan(b), first=False) + else: + tb = tan(b) + return (tan(a) + tb)/(1 - tan(a)*tb) + return rv + + return bottom_up(rv, f) + + +def TR12i(rv): + """Combine tan arguments as + (tan(y) + tan(x))/(tan(x)*tan(y) - 1) -> -tan(x + y). + + Examples + ======== + + >>> from sympy.simplify.fu import TR12i + >>> from sympy import tan + >>> from sympy.abc import a, b, c + >>> ta, tb, tc = [tan(i) for i in (a, b, c)] + >>> TR12i((ta + tb)/(-ta*tb + 1)) + tan(a + b) + >>> TR12i((ta + tb)/(ta*tb - 1)) + -tan(a + b) + >>> TR12i((-ta - tb)/(ta*tb - 1)) + tan(a + b) + >>> eq = (ta + tb)/(-ta*tb + 1)**2*(-3*ta - 3*tc)/(2*(ta*tc - 1)) + >>> TR12i(eq.expand()) + -3*tan(a + b)*tan(a + c)/(2*(tan(a) + tan(b) - 1)) + """ + def f(rv): + if not (rv.is_Add or rv.is_Mul or rv.is_Pow): + return rv + + n, d = rv.as_numer_denom() + if not d.args or not n.args: + return rv + + dok = {} + + def ok(di): + m = as_f_sign_1(di) + if m: + g, f, s = m + if s is S.NegativeOne and f.is_Mul and len(f.args) == 2 and \ + all(isinstance(fi, tan) for fi in f.args): + return g, f + + d_args = list(Mul.make_args(d)) + for i, di in enumerate(d_args): + m = ok(di) + if m: + g, t = m + s = Add(*[_.args[0] for _ in t.args]) + dok[s] = S.One + d_args[i] = g + continue + if di.is_Add: + di = factor(di) + if di.is_Mul: + d_args.extend(di.args) + d_args[i] = S.One + elif di.is_Pow and (di.exp.is_integer or di.base.is_positive): + m = ok(di.base) + if m: + g, t = m + s = Add(*[_.args[0] for _ in t.args]) + dok[s] = di.exp + d_args[i] = g**di.exp + else: + di = factor(di) + if di.is_Mul: + d_args.extend(di.args) + d_args[i] = S.One + if not dok: + return rv + + def ok(ni): + if ni.is_Add and len(ni.args) == 2: + a, b = ni.args + if isinstance(a, tan) and isinstance(b, tan): + return a, b + n_args = list(Mul.make_args(factor_terms(n))) + hit = False + for i, ni in enumerate(n_args): + m = ok(ni) + if not m: + m = ok(-ni) + if m: + n_args[i] = S.NegativeOne + else: + if ni.is_Add: + ni = factor(ni) + if ni.is_Mul: + n_args.extend(ni.args) + n_args[i] = S.One + continue + elif ni.is_Pow and ( + ni.exp.is_integer or ni.base.is_positive): + m = ok(ni.base) + if m: + n_args[i] = S.One + else: + ni = factor(ni) + if ni.is_Mul: + n_args.extend(ni.args) + n_args[i] = S.One + continue + else: + continue + else: + n_args[i] = S.One + hit = True + s = Add(*[_.args[0] for _ in m]) + ed = dok[s] + newed = ed.extract_additively(S.One) + if newed is not None: + if newed: + dok[s] = newed + else: + dok.pop(s) + n_args[i] *= -tan(s) + + if hit: + rv = Mul(*n_args)/Mul(*d_args)/Mul(*[(Add(*[ + tan(a) for a in i.args]) - 1)**e for i, e in dok.items()]) + + return rv + + return bottom_up(rv, f) + + +def TR13(rv): + """Change products of ``tan`` or ``cot``. + + Examples + ======== + + >>> from sympy.simplify.fu import TR13 + >>> from sympy import tan, cot + >>> TR13(tan(3)*tan(2)) + -tan(2)/tan(5) - tan(3)/tan(5) + 1 + >>> TR13(cot(3)*cot(2)) + cot(2)*cot(5) + 1 + cot(3)*cot(5) + """ + + def f(rv): + if not rv.is_Mul: + return rv + + # XXX handle products of powers? or let power-reducing handle it? + args = {tan: [], cot: [], None: []} + for a in Mul.make_args(rv): + if a.func in (tan, cot): + args[type(a)].append(a.args[0]) + else: + args[None].append(a) + t = args[tan] + c = args[cot] + if len(t) < 2 and len(c) < 2: + return rv + args = args[None] + while len(t) > 1: + t1 = t.pop() + t2 = t.pop() + args.append(1 - (tan(t1)/tan(t1 + t2) + tan(t2)/tan(t1 + t2))) + if t: + args.append(tan(t.pop())) + while len(c) > 1: + t1 = c.pop() + t2 = c.pop() + args.append(1 + cot(t1)*cot(t1 + t2) + cot(t2)*cot(t1 + t2)) + if c: + args.append(cot(c.pop())) + return Mul(*args) + + return bottom_up(rv, f) + + +def TRmorrie(rv): + """Returns cos(x)*cos(2*x)*...*cos(2**(k-1)*x) -> sin(2**k*x)/(2**k*sin(x)) + + Examples + ======== + + >>> from sympy.simplify.fu import TRmorrie, TR8, TR3 + >>> from sympy.abc import x + >>> from sympy import Mul, cos, pi + >>> TRmorrie(cos(x)*cos(2*x)) + sin(4*x)/(4*sin(x)) + >>> TRmorrie(7*Mul(*[cos(x) for x in range(10)])) + 7*sin(12)*sin(16)*cos(5)*cos(7)*cos(9)/(64*sin(1)*sin(3)) + + Sometimes autosimplification will cause a power to be + not recognized. e.g. in the following, cos(4*pi/7) automatically + simplifies to -cos(3*pi/7) so only 2 of the 3 terms are + recognized: + + >>> TRmorrie(cos(pi/7)*cos(2*pi/7)*cos(4*pi/7)) + -sin(3*pi/7)*cos(3*pi/7)/(4*sin(pi/7)) + + A touch by TR8 resolves the expression to a Rational + + >>> TR8(_) + -1/8 + + In this case, if eq is unsimplified, the answer is obtained + directly: + + >>> eq = cos(pi/9)*cos(2*pi/9)*cos(3*pi/9)*cos(4*pi/9) + >>> TRmorrie(eq) + 1/16 + + But if angles are made canonical with TR3 then the answer + is not simplified without further work: + + >>> TR3(eq) + sin(pi/18)*cos(pi/9)*cos(2*pi/9)/2 + >>> TRmorrie(_) + sin(pi/18)*sin(4*pi/9)/(8*sin(pi/9)) + >>> TR8(_) + cos(7*pi/18)/(16*sin(pi/9)) + >>> TR3(_) + 1/16 + + The original expression would have resolve to 1/16 directly with TR8, + however: + + >>> TR8(eq) + 1/16 + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Morrie%27s_law + + """ + + def f(rv, first=True): + if not rv.is_Mul: + return rv + if first: + n, d = rv.as_numer_denom() + return f(n, 0)/f(d, 0) + + args = defaultdict(list) + coss = {} + other = [] + for c in rv.args: + b, e = c.as_base_exp() + if e.is_Integer and isinstance(b, cos): + co, a = b.args[0].as_coeff_Mul() + args[a].append(co) + coss[b] = e + else: + other.append(c) + + new = [] + for a in args: + c = args[a] + c.sort() + while c: + k = 0 + cc = ci = c[0] + while cc in c: + k += 1 + cc *= 2 + if k > 1: + newarg = sin(2**k*ci*a)/2**k/sin(ci*a) + # see how many times this can be taken + take = None + ccs = [] + for i in range(k): + cc /= 2 + key = cos(a*cc, evaluate=False) + ccs.append(cc) + take = min(coss[key], take or coss[key]) + # update exponent counts + for i in range(k): + cc = ccs.pop() + key = cos(a*cc, evaluate=False) + coss[key] -= take + if not coss[key]: + c.remove(cc) + new.append(newarg**take) + else: + b = cos(c.pop(0)*a) + other.append(b**coss[b]) + + if new: + rv = Mul(*(new + other + [ + cos(k*a, evaluate=False) for a in args for k in args[a]])) + + return rv + + return bottom_up(rv, f) + + +def TR14(rv, first=True): + """Convert factored powers of sin and cos identities into simpler + expressions. + + Examples + ======== + + >>> from sympy.simplify.fu import TR14 + >>> from sympy.abc import x, y + >>> from sympy import cos, sin + >>> TR14((cos(x) - 1)*(cos(x) + 1)) + -sin(x)**2 + >>> TR14((sin(x) - 1)*(sin(x) + 1)) + -cos(x)**2 + >>> p1 = (cos(x) + 1)*(cos(x) - 1) + >>> p2 = (cos(y) - 1)*2*(cos(y) + 1) + >>> p3 = (3*(cos(y) - 1))*(3*(cos(y) + 1)) + >>> TR14(p1*p2*p3*(x - 1)) + -18*(x - 1)*sin(x)**2*sin(y)**4 + + """ + + def f(rv): + if not rv.is_Mul: + return rv + + if first: + # sort them by location in numerator and denominator + # so the code below can just deal with positive exponents + n, d = rv.as_numer_denom() + if d is not S.One: + newn = TR14(n, first=False) + newd = TR14(d, first=False) + if newn != n or newd != d: + rv = newn/newd + return rv + + other = [] + process = [] + for a in rv.args: + if a.is_Pow: + b, e = a.as_base_exp() + if not (e.is_integer or b.is_positive): + other.append(a) + continue + a = b + else: + e = S.One + m = as_f_sign_1(a) + if not m or m[1].func not in (cos, sin): + if e is S.One: + other.append(a) + else: + other.append(a**e) + continue + g, f, si = m + process.append((g, e.is_Number, e, f, si, a)) + + # sort them to get like terms next to each other + process = list(ordered(process)) + + # keep track of whether there was any change + nother = len(other) + + # access keys + keys = (g, t, e, f, si, a) = list(range(6)) + + while process: + A = process.pop(0) + if process: + B = process[0] + + if A[e].is_Number and B[e].is_Number: + # both exponents are numbers + if A[f] == B[f]: + if A[si] != B[si]: + B = process.pop(0) + take = min(A[e], B[e]) + + # reinsert any remainder + # the B will likely sort after A so check it first + if B[e] != take: + rem = [B[i] for i in keys] + rem[e] -= take + process.insert(0, rem) + elif A[e] != take: + rem = [A[i] for i in keys] + rem[e] -= take + process.insert(0, rem) + + if isinstance(A[f], cos): + t = sin + else: + t = cos + other.append((-A[g]*B[g]*t(A[f].args[0])**2)**take) + continue + + elif A[e] == B[e]: + # both exponents are equal symbols + if A[f] == B[f]: + if A[si] != B[si]: + B = process.pop(0) + take = A[e] + if isinstance(A[f], cos): + t = sin + else: + t = cos + other.append((-A[g]*B[g]*t(A[f].args[0])**2)**take) + continue + + # either we are done or neither condition above applied + other.append(A[a]**A[e]) + + if len(other) != nother: + rv = Mul(*other) + + return rv + + return bottom_up(rv, f) + + +def TR15(rv, max=4, pow=False): + """Convert sin(x)**-2 to 1 + cot(x)**2. + + See _TR56 docstring for advanced use of ``max`` and ``pow``. + + Examples + ======== + + >>> from sympy.simplify.fu import TR15 + >>> from sympy.abc import x + >>> from sympy import sin + >>> TR15(1 - 1/sin(x)**2) + -cot(x)**2 + + """ + + def f(rv): + if not (isinstance(rv, Pow) and isinstance(rv.base, sin)): + return rv + + e = rv.exp + if e % 2 == 1: + return TR15(rv.base**(e + 1))/rv.base + + ia = 1/rv + a = _TR56(ia, sin, cot, lambda x: 1 + x, max=max, pow=pow) + if a != ia: + rv = a + return rv + + return bottom_up(rv, f) + + +def TR16(rv, max=4, pow=False): + """Convert cos(x)**-2 to 1 + tan(x)**2. + + See _TR56 docstring for advanced use of ``max`` and ``pow``. + + Examples + ======== + + >>> from sympy.simplify.fu import TR16 + >>> from sympy.abc import x + >>> from sympy import cos + >>> TR16(1 - 1/cos(x)**2) + -tan(x)**2 + + """ + + def f(rv): + if not (isinstance(rv, Pow) and isinstance(rv.base, cos)): + return rv + + e = rv.exp + if e % 2 == 1: + return TR15(rv.base**(e + 1))/rv.base + + ia = 1/rv + a = _TR56(ia, cos, tan, lambda x: 1 + x, max=max, pow=pow) + if a != ia: + rv = a + return rv + + return bottom_up(rv, f) + + +def TR111(rv): + """Convert f(x)**-i to g(x)**i where either ``i`` is an integer + or the base is positive and f, g are: tan, cot; sin, csc; or cos, sec. + + Examples + ======== + + >>> from sympy.simplify.fu import TR111 + >>> from sympy.abc import x + >>> from sympy import tan + >>> TR111(1 - 1/tan(x)**2) + 1 - cot(x)**2 + + """ + + def f(rv): + if not ( + isinstance(rv, Pow) and + (rv.base.is_positive or rv.exp.is_integer and rv.exp.is_negative)): + return rv + + if isinstance(rv.base, tan): + return cot(rv.base.args[0])**-rv.exp + elif isinstance(rv.base, sin): + return csc(rv.base.args[0])**-rv.exp + elif isinstance(rv.base, cos): + return sec(rv.base.args[0])**-rv.exp + return rv + + return bottom_up(rv, f) + + +def TR22(rv, max=4, pow=False): + """Convert tan(x)**2 to sec(x)**2 - 1 and cot(x)**2 to csc(x)**2 - 1. + + See _TR56 docstring for advanced use of ``max`` and ``pow``. + + Examples + ======== + + >>> from sympy.simplify.fu import TR22 + >>> from sympy.abc import x + >>> from sympy import tan, cot + >>> TR22(1 + tan(x)**2) + sec(x)**2 + >>> TR22(1 + cot(x)**2) + csc(x)**2 + + """ + + def f(rv): + if not (isinstance(rv, Pow) and rv.base.func in (cot, tan)): + return rv + + rv = _TR56(rv, tan, sec, lambda x: x - 1, max=max, pow=pow) + rv = _TR56(rv, cot, csc, lambda x: x - 1, max=max, pow=pow) + return rv + + return bottom_up(rv, f) + + +def TRpower(rv): + """Convert sin(x)**n and cos(x)**n with positive n to sums. + + Examples + ======== + + >>> from sympy.simplify.fu import TRpower + >>> from sympy.abc import x + >>> from sympy import cos, sin + >>> TRpower(sin(x)**6) + -15*cos(2*x)/32 + 3*cos(4*x)/16 - cos(6*x)/32 + 5/16 + >>> TRpower(sin(x)**3*cos(2*x)**4) + (3*sin(x)/4 - sin(3*x)/4)*(cos(4*x)/2 + cos(8*x)/8 + 3/8) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Power-reduction_formulae + + """ + + def f(rv): + if not (isinstance(rv, Pow) and isinstance(rv.base, (sin, cos))): + return rv + b, n = rv.as_base_exp() + x = b.args[0] + if n.is_Integer and n.is_positive: + if n.is_odd and isinstance(b, cos): + rv = 2**(1-n)*Add(*[binomial(n, k)*cos((n - 2*k)*x) + for k in range((n + 1)/2)]) + elif n.is_odd and isinstance(b, sin): + rv = 2**(1-n)*S.NegativeOne**((n-1)/2)*Add(*[binomial(n, k)* + S.NegativeOne**k*sin((n - 2*k)*x) for k in range((n + 1)/2)]) + elif n.is_even and isinstance(b, cos): + rv = 2**(1-n)*Add(*[binomial(n, k)*cos((n - 2*k)*x) + for k in range(n/2)]) + elif n.is_even and isinstance(b, sin): + rv = 2**(1-n)*S.NegativeOne**(n/2)*Add(*[binomial(n, k)* + S.NegativeOne**k*cos((n - 2*k)*x) for k in range(n/2)]) + if n.is_even: + rv += 2**(-n)*binomial(n, n/2) + return rv + + return bottom_up(rv, f) + + +def L(rv): + """Return count of trigonometric functions in expression. + + Examples + ======== + + >>> from sympy.simplify.fu import L + >>> from sympy.abc import x + >>> from sympy import cos, sin + >>> L(cos(x)+sin(x)) + 2 + """ + return S(rv.count(TrigonometricFunction)) + + +# ============== end of basic Fu-like tools ===================== + +if SYMPY_DEBUG: + (TR0, TR1, TR2, TR3, TR4, TR5, TR6, TR7, TR8, TR9, TR10, TR11, TR12, TR13, + TR2i, TRmorrie, TR14, TR15, TR16, TR12i, TR111, TR22 + )= list(map(debug, + (TR0, TR1, TR2, TR3, TR4, TR5, TR6, TR7, TR8, TR9, TR10, TR11, TR12, TR13, + TR2i, TRmorrie, TR14, TR15, TR16, TR12i, TR111, TR22))) + + +# tuples are chains -- (f, g) -> lambda x: g(f(x)) +# lists are choices -- [f, g] -> lambda x: min(f(x), g(x), key=objective) + +CTR1 = [(TR5, TR0), (TR6, TR0), identity] + +CTR2 = (TR11, [(TR5, TR0), (TR6, TR0), TR0]) + +CTR3 = [(TRmorrie, TR8, TR0), (TRmorrie, TR8, TR10i, TR0), identity] + +CTR4 = [(TR4, TR10i), identity] + +RL1 = (TR4, TR3, TR4, TR12, TR4, TR13, TR4, TR0) + + +# XXX it's a little unclear how this one is to be implemented +# see Fu paper of reference, page 7. What is the Union symbol referring to? +# The diagram shows all these as one chain of transformations, but the +# text refers to them being applied independently. Also, a break +# if L starts to increase has not been implemented. +RL2 = [ + (TR4, TR3, TR10, TR4, TR3, TR11), + (TR5, TR7, TR11, TR4), + (CTR3, CTR1, TR9, CTR2, TR4, TR9, TR9, CTR4), + identity, + ] + + +def fu(rv, measure=lambda x: (L(x), x.count_ops())): + """Attempt to simplify expression by using transformation rules given + in the algorithm by Fu et al. + + :func:`fu` will try to minimize the objective function ``measure``. + By default this first minimizes the number of trig terms and then minimizes + the number of total operations. + + Examples + ======== + + >>> from sympy.simplify.fu import fu + >>> from sympy import cos, sin, tan, pi, S, sqrt + >>> from sympy.abc import x, y, a, b + + >>> fu(sin(50)**2 + cos(50)**2 + sin(pi/6)) + 3/2 + >>> fu(sqrt(6)*cos(x) + sqrt(2)*sin(x)) + 2*sqrt(2)*sin(x + pi/3) + + CTR1 example + + >>> eq = sin(x)**4 - cos(y)**2 + sin(y)**2 + 2*cos(x)**2 + >>> fu(eq) + cos(x)**4 - 2*cos(y)**2 + 2 + + CTR2 example + + >>> fu(S.Half - cos(2*x)/2) + sin(x)**2 + + CTR3 example + + >>> fu(sin(a)*(cos(b) - sin(b)) + cos(a)*(sin(b) + cos(b))) + sqrt(2)*sin(a + b + pi/4) + + CTR4 example + + >>> fu(sqrt(3)*cos(x)/2 + sin(x)/2) + sin(x + pi/3) + + Example 1 + + >>> fu(1-sin(2*x)**2/4-sin(y)**2-cos(x)**4) + -cos(x)**2 + cos(y)**2 + + Example 2 + + >>> fu(cos(4*pi/9)) + sin(pi/18) + >>> fu(cos(pi/9)*cos(2*pi/9)*cos(3*pi/9)*cos(4*pi/9)) + 1/16 + + Example 3 + + >>> fu(tan(7*pi/18)+tan(5*pi/18)-sqrt(3)*tan(5*pi/18)*tan(7*pi/18)) + -sqrt(3) + + Objective function example + + >>> fu(sin(x)/cos(x)) # default objective function + tan(x) + >>> fu(sin(x)/cos(x), measure=lambda x: -x.count_ops()) # maximize op count + sin(x)/cos(x) + + References + ========== + + .. [1] https://www.sciencedirect.com/science/article/pii/S0895717706001609 + """ + fRL1 = greedy(RL1, measure) + fRL2 = greedy(RL2, measure) + + was = rv + rv = sympify(rv) + if not isinstance(rv, Expr): + return rv.func(*[fu(a, measure=measure) for a in rv.args]) + rv = TR1(rv) + if rv.has(tan, cot): + rv1 = fRL1(rv) + if (measure(rv1) < measure(rv)): + rv = rv1 + if rv.has(tan, cot): + rv = TR2(rv) + if rv.has(sin, cos): + rv1 = fRL2(rv) + rv2 = TR8(TRmorrie(rv1)) + rv = min([was, rv, rv1, rv2], key=measure) + return min(TR2i(rv), rv, key=measure) + + +def process_common_addends(rv, do, key2=None, key1=True): + """Apply ``do`` to addends of ``rv`` that (if ``key1=True``) share at least + a common absolute value of their coefficient and the value of ``key2`` when + applied to the argument. If ``key1`` is False ``key2`` must be supplied and + will be the only key applied. + """ + + # collect by absolute value of coefficient and key2 + absc = defaultdict(list) + if key1: + for a in rv.args: + c, a = a.as_coeff_Mul() + if c < 0: + c = -c + a = -a # put the sign on `a` + absc[(c, key2(a) if key2 else 1)].append(a) + elif key2: + for a in rv.args: + absc[(S.One, key2(a))].append(a) + else: + raise ValueError('must have at least one key') + + args = [] + hit = False + for k in absc: + v = absc[k] + c, _ = k + if len(v) > 1: + e = Add(*v, evaluate=False) + new = do(e) + if new != e: + e = new + hit = True + args.append(c*e) + else: + args.append(c*v[0]) + if hit: + rv = Add(*args) + + return rv + + +fufuncs = ''' + TR0 TR1 TR2 TR3 TR4 TR5 TR6 TR7 TR8 TR9 TR10 TR10i TR11 + TR12 TR13 L TR2i TRmorrie TR12i + TR14 TR15 TR16 TR111 TR22'''.split() +FU = dict(list(zip(fufuncs, list(map(locals().get, fufuncs))))) + + +@cacheit +def _ROOT2(): + return sqrt(2) + + +@cacheit +def _ROOT3(): + return sqrt(3) + + +@cacheit +def _invROOT3(): + return 1/sqrt(3) + + +def trig_split(a, b, two=False): + """Return the gcd, s1, s2, a1, a2, bool where + + If two is False (default) then:: + a + b = gcd*(s1*f(a1) + s2*f(a2)) where f = cos if bool else sin + else: + if bool, a + b was +/- cos(a1)*cos(a2) +/- sin(a1)*sin(a2) and equals + n1*gcd*cos(a - b) if n1 == n2 else + n1*gcd*cos(a + b) + else a + b was +/- cos(a1)*sin(a2) +/- sin(a1)*cos(a2) and equals + n1*gcd*sin(a + b) if n1 = n2 else + n1*gcd*sin(b - a) + + Examples + ======== + + >>> from sympy.simplify.fu import trig_split + >>> from sympy.abc import x, y, z + >>> from sympy import cos, sin, sqrt + + >>> trig_split(cos(x), cos(y)) + (1, 1, 1, x, y, True) + >>> trig_split(2*cos(x), -2*cos(y)) + (2, 1, -1, x, y, True) + >>> trig_split(cos(x)*sin(y), cos(y)*sin(y)) + (sin(y), 1, 1, x, y, True) + + >>> trig_split(cos(x), -sqrt(3)*sin(x), two=True) + (2, 1, -1, x, pi/6, False) + >>> trig_split(cos(x), sin(x), two=True) + (sqrt(2), 1, 1, x, pi/4, False) + >>> trig_split(cos(x), -sin(x), two=True) + (sqrt(2), 1, -1, x, pi/4, False) + >>> trig_split(sqrt(2)*cos(x), -sqrt(6)*sin(x), two=True) + (2*sqrt(2), 1, -1, x, pi/6, False) + >>> trig_split(-sqrt(6)*cos(x), -sqrt(2)*sin(x), two=True) + (-2*sqrt(2), 1, 1, x, pi/3, False) + >>> trig_split(cos(x)/sqrt(6), sin(x)/sqrt(2), two=True) + (sqrt(6)/3, 1, 1, x, pi/6, False) + >>> trig_split(-sqrt(6)*cos(x)*sin(y), -sqrt(2)*sin(x)*sin(y), two=True) + (-2*sqrt(2)*sin(y), 1, 1, x, pi/3, False) + + >>> trig_split(cos(x), sin(x)) + >>> trig_split(cos(x), sin(z)) + >>> trig_split(2*cos(x), -sin(x)) + >>> trig_split(cos(x), -sqrt(3)*sin(x)) + >>> trig_split(cos(x)*cos(y), sin(x)*sin(z)) + >>> trig_split(cos(x)*cos(y), sin(x)*sin(y)) + >>> trig_split(-sqrt(6)*cos(x), sqrt(2)*sin(x)*sin(y), two=True) + """ + a, b = [Factors(i) for i in (a, b)] + ua, ub = a.normal(b) + gcd = a.gcd(b).as_expr() + n1 = n2 = 1 + if S.NegativeOne in ua.factors: + ua = ua.quo(S.NegativeOne) + n1 = -n1 + elif S.NegativeOne in ub.factors: + ub = ub.quo(S.NegativeOne) + n2 = -n2 + a, b = [i.as_expr() for i in (ua, ub)] + + def pow_cos_sin(a, two): + """Return ``a`` as a tuple (r, c, s) such that + ``a = (r or 1)*(c or 1)*(s or 1)``. + + Three arguments are returned (radical, c-factor, s-factor) as + long as the conditions set by ``two`` are met; otherwise None is + returned. If ``two`` is True there will be one or two non-None + values in the tuple: c and s or c and r or s and r or s or c with c + being a cosine function (if possible) else a sine, and s being a sine + function (if possible) else oosine. If ``two`` is False then there + will only be a c or s term in the tuple. + + ``two`` also require that either two cos and/or sin be present (with + the condition that if the functions are the same the arguments are + different or vice versa) or that a single cosine or a single sine + be present with an optional radical. + + If the above conditions dictated by ``two`` are not met then None + is returned. + """ + c = s = None + co = S.One + if a.is_Mul: + co, a = a.as_coeff_Mul() + if len(a.args) > 2 or not two: + return None + if a.is_Mul: + args = list(a.args) + else: + args = [a] + a = args.pop(0) + if isinstance(a, cos): + c = a + elif isinstance(a, sin): + s = a + elif a.is_Pow and a.exp is S.Half: # autoeval doesn't allow -1/2 + co *= a + else: + return None + if args: + b = args[0] + if isinstance(b, cos): + if c: + s = b + else: + c = b + elif isinstance(b, sin): + if s: + c = b + else: + s = b + elif b.is_Pow and b.exp is S.Half: + co *= b + else: + return None + return co if co is not S.One else None, c, s + elif isinstance(a, cos): + c = a + elif isinstance(a, sin): + s = a + if c is None and s is None: + return + co = co if co is not S.One else None + return co, c, s + + # get the parts + m = pow_cos_sin(a, two) + if m is None: + return + coa, ca, sa = m + m = pow_cos_sin(b, two) + if m is None: + return + cob, cb, sb = m + + # check them + if (not ca) and cb or ca and isinstance(ca, sin): + coa, ca, sa, cob, cb, sb = cob, cb, sb, coa, ca, sa + n1, n2 = n2, n1 + if not two: # need cos(x) and cos(y) or sin(x) and sin(y) + c = ca or sa + s = cb or sb + if not isinstance(c, s.func): + return None + return gcd, n1, n2, c.args[0], s.args[0], isinstance(c, cos) + else: + if not coa and not cob: + if (ca and cb and sa and sb): + if isinstance(ca, sa.func) is not isinstance(cb, sb.func): + return + args = {j.args for j in (ca, sa)} + if not all(i.args in args for i in (cb, sb)): + return + return gcd, n1, n2, ca.args[0], sa.args[0], isinstance(ca, sa.func) + if ca and sa or cb and sb or \ + two and (ca is None and sa is None or cb is None and sb is None): + return + c = ca or sa + s = cb or sb + if c.args != s.args: + return + if not coa: + coa = S.One + if not cob: + cob = S.One + if coa is cob: + gcd *= _ROOT2() + return gcd, n1, n2, c.args[0], pi/4, False + elif coa/cob == _ROOT3(): + gcd *= 2*cob + return gcd, n1, n2, c.args[0], pi/3, False + elif coa/cob == _invROOT3(): + gcd *= 2*coa + return gcd, n1, n2, c.args[0], pi/6, False + + +def as_f_sign_1(e): + """If ``e`` is a sum that can be written as ``g*(a + s)`` where + ``s`` is ``+/-1``, return ``g``, ``a``, and ``s`` where ``a`` does + not have a leading negative coefficient. + + Examples + ======== + + >>> from sympy.simplify.fu import as_f_sign_1 + >>> from sympy.abc import x + >>> as_f_sign_1(x + 1) + (1, x, 1) + >>> as_f_sign_1(x - 1) + (1, x, -1) + >>> as_f_sign_1(-x + 1) + (-1, x, -1) + >>> as_f_sign_1(-x - 1) + (-1, x, 1) + >>> as_f_sign_1(2*x + 2) + (2, x, 1) + """ + if not e.is_Add or len(e.args) != 2: + return + # exact match + a, b = e.args + if a in (S.NegativeOne, S.One): + g = S.One + if b.is_Mul and b.args[0].is_Number and b.args[0] < 0: + a, b = -a, -b + g = -g + return g, b, a + # gcd match + a, b = [Factors(i) for i in e.args] + ua, ub = a.normal(b) + gcd = a.gcd(b).as_expr() + if S.NegativeOne in ua.factors: + ua = ua.quo(S.NegativeOne) + n1 = -1 + n2 = 1 + elif S.NegativeOne in ub.factors: + ub = ub.quo(S.NegativeOne) + n1 = 1 + n2 = -1 + else: + n1 = n2 = 1 + a, b = [i.as_expr() for i in (ua, ub)] + if a is S.One: + a, b = b, a + n1, n2 = n2, n1 + if n1 == -1: + gcd = -gcd + n2 = -n2 + + if b is S.One: + return gcd, a, n2 + + +def _osborne(e, d): + """Replace all hyperbolic functions with trig functions using + the Osborne rule. + + Notes + ===== + + ``d`` is a dummy variable to prevent automatic evaluation + of trigonometric/hyperbolic functions. + + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Hyperbolic_function + """ + + def f(rv): + if not isinstance(rv, HyperbolicFunction): + return rv + a = rv.args[0] + a = a*d if not a.is_Add else Add._from_args([i*d for i in a.args]) + if isinstance(rv, sinh): + return I*sin(a) + elif isinstance(rv, cosh): + return cos(a) + elif isinstance(rv, tanh): + return I*tan(a) + elif isinstance(rv, coth): + return cot(a)/I + elif isinstance(rv, sech): + return sec(a) + elif isinstance(rv, csch): + return csc(a)/I + else: + raise NotImplementedError('unhandled %s' % rv.func) + + return bottom_up(e, f) + + +def _osbornei(e, d): + """Replace all trig functions with hyperbolic functions using + the Osborne rule. + + Notes + ===== + + ``d`` is a dummy variable to prevent automatic evaluation + of trigonometric/hyperbolic functions. + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Hyperbolic_function + """ + + def f(rv): + if not isinstance(rv, TrigonometricFunction): + return rv + const, x = rv.args[0].as_independent(d, as_Add=True) + a = x.xreplace({d: S.One}) + const*I + if isinstance(rv, sin): + return sinh(a)/I + elif isinstance(rv, cos): + return cosh(a) + elif isinstance(rv, tan): + return tanh(a)/I + elif isinstance(rv, cot): + return coth(a)*I + elif isinstance(rv, sec): + return sech(a) + elif isinstance(rv, csc): + return csch(a)*I + else: + raise NotImplementedError('unhandled %s' % rv.func) + + return bottom_up(e, f) + + +def hyper_as_trig(rv): + """Return an expression containing hyperbolic functions in terms + of trigonometric functions. Any trigonometric functions initially + present are replaced with Dummy symbols and the function to undo + the masking and the conversion back to hyperbolics is also returned. It + should always be true that:: + + t, f = hyper_as_trig(expr) + expr == f(t) + + Examples + ======== + + >>> from sympy.simplify.fu import hyper_as_trig, fu + >>> from sympy.abc import x + >>> from sympy import cosh, sinh + >>> eq = sinh(x)**2 + cosh(x)**2 + >>> t, f = hyper_as_trig(eq) + >>> f(fu(t)) + cosh(2*x) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Hyperbolic_function + """ + from sympy.simplify.simplify import signsimp + from sympy.simplify.radsimp import collect + + # mask off trig functions + trigs = rv.atoms(TrigonometricFunction) + reps = [(t, Dummy()) for t in trigs] + masked = rv.xreplace(dict(reps)) + + # get inversion substitutions in place + reps = [(v, k) for k, v in reps] + + d = Dummy() + + return _osborne(masked, d), lambda x: collect(signsimp( + _osbornei(x, d).xreplace(dict(reps))), S.ImaginaryUnit) + + +def sincos_to_sum(expr): + """Convert products and powers of sin and cos to sums. + + Explanation + =========== + + Applied power reduction TRpower first, then expands products, and + converts products to sums with TR8. + + Examples + ======== + + >>> from sympy.simplify.fu import sincos_to_sum + >>> from sympy.abc import x + >>> from sympy import cos, sin + >>> sincos_to_sum(16*sin(x)**3*cos(2*x)**2) + 7*sin(x) - 5*sin(3*x) + 3*sin(5*x) - sin(7*x) + """ + + if not expr.has(cos, sin): + return expr + else: + return TR8(expand_mul(TRpower(expr))) diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/gammasimp.py b/.venv/lib/python3.12/site-packages/sympy/simplify/gammasimp.py new file mode 100644 index 0000000000000000000000000000000000000000..aec20c56eb60efb8e1aadfb5bff3d1ba1ab51869 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/simplify/gammasimp.py @@ -0,0 +1,493 @@ +from sympy.core import Function, S, Mul, Pow, Add +from sympy.core.sorting import ordered, default_sort_key +from sympy.core.function import expand_func +from sympy.core.symbol import Dummy +from sympy.functions import gamma, sqrt, sin +from sympy.polys import factor, cancel +from sympy.utilities.iterables import sift, uniq + + +def gammasimp(expr): + r""" + Simplify expressions with gamma functions. + + Explanation + =========== + + This function takes as input an expression containing gamma + functions or functions that can be rewritten in terms of gamma + functions and tries to minimize the number of those functions and + reduce the size of their arguments. + + The algorithm works by rewriting all gamma functions as expressions + involving rising factorials (Pochhammer symbols) and applies + recurrence relations and other transformations applicable to rising + factorials, to reduce their arguments, possibly letting the resulting + rising factorial to cancel. Rising factorials with the second argument + being an integer are expanded into polynomial forms and finally all + other rising factorial are rewritten in terms of gamma functions. + + Then the following two steps are performed. + + 1. Reduce the number of gammas by applying the reflection theorem + gamma(x)*gamma(1-x) == pi/sin(pi*x). + 2. Reduce the number of gammas by applying the multiplication theorem + gamma(x)*gamma(x+1/n)*...*gamma(x+(n-1)/n) == C*gamma(n*x). + + It then reduces the number of prefactors by absorbing them into gammas + where possible and expands gammas with rational argument. + + All transformation rules can be found (or were derived from) here: + + .. [1] https://functions.wolfram.com/GammaBetaErf/Pochhammer/17/01/02/ + .. [2] https://functions.wolfram.com/GammaBetaErf/Pochhammer/27/01/0005/ + + Examples + ======== + + >>> from sympy.simplify import gammasimp + >>> from sympy import gamma, Symbol + >>> from sympy.abc import x + >>> n = Symbol('n', integer = True) + + >>> gammasimp(gamma(x)/gamma(x - 3)) + (x - 3)*(x - 2)*(x - 1) + >>> gammasimp(gamma(n + 3)) + gamma(n + 3) + + """ + + expr = expr.rewrite(gamma) + + # compute_ST will be looking for Functions and we don't want + # it looking for non-gamma functions: issue 22606 + # so we mask free, non-gamma functions + f = expr.atoms(Function) + # take out gammas + gammas = {i for i in f if isinstance(i, gamma)} + if not gammas: + return expr # avoid side effects like factoring + f -= gammas + # keep only those without bound symbols + f = f & expr.as_dummy().atoms(Function) + if f: + dum, fun, simp = zip(*[ + (Dummy(), fi, fi.func(*[ + _gammasimp(a, as_comb=False) for a in fi.args])) + for fi in ordered(f)]) + d = expr.xreplace(dict(zip(fun, dum))) + return _gammasimp(d, as_comb=False).xreplace(dict(zip(dum, simp))) + + return _gammasimp(expr, as_comb=False) + + +def _gammasimp(expr, as_comb): + """ + Helper function for gammasimp and combsimp. + + Explanation + =========== + + Simplifies expressions written in terms of gamma function. If + as_comb is True, it tries to preserve integer arguments. See + docstring of gammasimp for more information. This was part of + combsimp() in combsimp.py. + """ + expr = expr.replace(gamma, + lambda n: _rf(1, (n - 1).expand())) + + if as_comb: + expr = expr.replace(_rf, + lambda a, b: gamma(b + 1)) + else: + expr = expr.replace(_rf, + lambda a, b: gamma(a + b)/gamma(a)) + + def rule_gamma(expr, level=0): + """ Simplify products of gamma functions further. """ + + if expr.is_Atom: + return expr + + def gamma_rat(x): + # helper to simplify ratios of gammas + was = x.count(gamma) + xx = x.replace(gamma, lambda n: _rf(1, (n - 1).expand() + ).replace(_rf, lambda a, b: gamma(a + b)/gamma(a))) + if xx.count(gamma) < was: + x = xx + return x + + def gamma_factor(x): + # return True if there is a gamma factor in shallow args + if isinstance(x, gamma): + return True + if x.is_Add or x.is_Mul: + return any(gamma_factor(xi) for xi in x.args) + if x.is_Pow and (x.exp.is_integer or x.base.is_positive): + return gamma_factor(x.base) + return False + + # recursion step + if level == 0: + expr = expr.func(*[rule_gamma(x, level + 1) for x in expr.args]) + level += 1 + + if not expr.is_Mul: + return expr + + # non-commutative step + if level == 1: + args, nc = expr.args_cnc() + if not args: + return expr + if nc: + return rule_gamma(Mul._from_args(args), level + 1)*Mul._from_args(nc) + level += 1 + + # pure gamma handling, not factor absorption + if level == 2: + T, F = sift(expr.args, gamma_factor, binary=True) + gamma_ind = Mul(*F) + d = Mul(*T) + + nd, dd = d.as_numer_denom() + for ipass in range(2): + args = list(ordered(Mul.make_args(nd))) + for i, ni in enumerate(args): + if ni.is_Add: + ni, dd = Add(*[ + rule_gamma(gamma_rat(a/dd), level + 1) for a in ni.args] + ).as_numer_denom() + args[i] = ni + if not dd.has(gamma): + break + nd = Mul(*args) + if ipass == 0 and not gamma_factor(nd): + break + nd, dd = dd, nd # now process in reversed order + expr = gamma_ind*nd/dd + if not (expr.is_Mul and (gamma_factor(dd) or gamma_factor(nd))): + return expr + level += 1 + + # iteration until constant + if level == 3: + while True: + was = expr + expr = rule_gamma(expr, 4) + if expr == was: + return expr + + numer_gammas = [] + denom_gammas = [] + numer_others = [] + denom_others = [] + def explicate(p): + if p is S.One: + return None, [] + b, e = p.as_base_exp() + if e.is_Integer: + if isinstance(b, gamma): + return True, [b.args[0]]*e + else: + return False, [b]*e + else: + return False, [p] + + newargs = list(ordered(expr.args)) + while newargs: + n, d = newargs.pop().as_numer_denom() + isg, l = explicate(n) + if isg: + numer_gammas.extend(l) + elif isg is False: + numer_others.extend(l) + isg, l = explicate(d) + if isg: + denom_gammas.extend(l) + elif isg is False: + denom_others.extend(l) + + # =========== level 2 work: pure gamma manipulation ========= + + if not as_comb: + # Try to reduce the number of gamma factors by applying the + # reflection formula gamma(x)*gamma(1-x) = pi/sin(pi*x) + for gammas, numer, denom in [( + numer_gammas, numer_others, denom_others), + (denom_gammas, denom_others, numer_others)]: + new = [] + while gammas: + g1 = gammas.pop() + if g1.is_integer: + new.append(g1) + continue + for i, g2 in enumerate(gammas): + n = g1 + g2 - 1 + if not n.is_Integer: + continue + numer.append(S.Pi) + denom.append(sin(S.Pi*g1)) + gammas.pop(i) + if n > 0: + numer.extend(1 - g1 + k for k in range(n)) + elif n < 0: + denom.extend(-g1 - k for k in range(-n)) + break + else: + new.append(g1) + # /!\ updating IN PLACE + gammas[:] = new + + # Try to reduce the number of gammas by using the duplication + # theorem to cancel an upper and lower: gamma(2*s)/gamma(s) = + # 2**(2*s + 1)/(4*sqrt(pi))*gamma(s + 1/2). Although this could + # be done with higher argument ratios like gamma(3*x)/gamma(x), + # this would not reduce the number of gammas as in this case. + for ng, dg, no, do in [(numer_gammas, denom_gammas, numer_others, + denom_others), + (denom_gammas, numer_gammas, denom_others, + numer_others)]: + + while True: + for x in ng: + for y in dg: + n = x - 2*y + if n.is_Integer: + break + else: + continue + break + else: + break + ng.remove(x) + dg.remove(y) + if n > 0: + no.extend(2*y + k for k in range(n)) + elif n < 0: + do.extend(2*y - 1 - k for k in range(-n)) + ng.append(y + S.Half) + no.append(2**(2*y - 1)) + do.append(sqrt(S.Pi)) + + # Try to reduce the number of gamma factors by applying the + # multiplication theorem (used when n gammas with args differing + # by 1/n mod 1 are encountered). + # + # run of 2 with args differing by 1/2 + # + # >>> gammasimp(gamma(x)*gamma(x+S.Half)) + # 2*sqrt(2)*2**(-2*x - 1/2)*sqrt(pi)*gamma(2*x) + # + # run of 3 args differing by 1/3 (mod 1) + # + # >>> gammasimp(gamma(x)*gamma(x+S(1)/3)*gamma(x+S(2)/3)) + # 6*3**(-3*x - 1/2)*pi*gamma(3*x) + # >>> gammasimp(gamma(x)*gamma(x+S(1)/3)*gamma(x+S(5)/3)) + # 2*3**(-3*x - 1/2)*pi*(3*x + 2)*gamma(3*x) + # + def _run(coeffs): + # find runs in coeffs such that the difference in terms (mod 1) + # of t1, t2, ..., tn is 1/n + u = list(uniq(coeffs)) + for i in range(len(u)): + dj = ([((u[j] - u[i]) % 1, j) for j in range(i + 1, len(u))]) + for one, j in dj: + if one.p == 1 and one.q != 1: + n = one.q + got = [i] + get = list(range(1, n)) + for d, j in dj: + m = n*d + if m.is_Integer and m in get: + get.remove(m) + got.append(j) + if not get: + break + else: + continue + for i, j in enumerate(got): + c = u[j] + coeffs.remove(c) + got[i] = c + return one.q, got[0], got[1:] + + def _mult_thm(gammas, numer, denom): + # pull off and analyze the leading coefficient from each gamma arg + # looking for runs in those Rationals + + # expr -> coeff + resid -> rats[resid] = coeff + rats = {} + for g in gammas: + c, resid = g.as_coeff_Add() + rats.setdefault(resid, []).append(c) + + # look for runs in Rationals for each resid + keys = sorted(rats, key=default_sort_key) + for resid in keys: + coeffs = sorted(rats[resid]) + new = [] + while True: + run = _run(coeffs) + if run is None: + break + + # process the sequence that was found: + # 1) convert all the gamma functions to have the right + # argument (could be off by an integer) + # 2) append the factors corresponding to the theorem + # 3) append the new gamma function + + n, ui, other = run + + # (1) + for u in other: + con = resid + u - 1 + for k in range(int(u - ui)): + numer.append(con - k) + + con = n*(resid + ui) # for (2) and (3) + + # (2) + numer.append((2*S.Pi)**(S(n - 1)/2)* + n**(S.Half - con)) + # (3) + new.append(con) + + # restore resid to coeffs + rats[resid] = [resid + c for c in coeffs] + new + + # rebuild the gamma arguments + g = [] + for resid in keys: + g += rats[resid] + # /!\ updating IN PLACE + gammas[:] = g + + for l, numer, denom in [(numer_gammas, numer_others, denom_others), + (denom_gammas, denom_others, numer_others)]: + _mult_thm(l, numer, denom) + + # =========== level >= 2 work: factor absorption ========= + + if level >= 2: + # Try to absorb factors into the gammas: x*gamma(x) -> gamma(x + 1) + # and gamma(x)/(x - 1) -> gamma(x - 1) + # This code (in particular repeated calls to find_fuzzy) can be very + # slow. + def find_fuzzy(l, x): + if not l: + return + S1, T1 = compute_ST(x) + for y in l: + S2, T2 = inv[y] + if T1 != T2 or (not S1.intersection(S2) and + (S1 != set() or S2 != set())): + continue + # XXX we want some simplification (e.g. cancel or + # simplify) but no matter what it's slow. + a = len(cancel(x/y).free_symbols) + b = len(x.free_symbols) + c = len(y.free_symbols) + # TODO is there a better heuristic? + if a == 0 and (b > 0 or c > 0): + return y + + # We thus try to avoid expensive calls by building the following + # "invariants": For every factor or gamma function argument + # - the set of free symbols S + # - the set of functional components T + # We will only try to absorb if T1==T2 and (S1 intersect S2 != emptyset + # or S1 == S2 == emptyset) + inv = {} + + def compute_ST(expr): + if expr in inv: + return inv[expr] + return (expr.free_symbols, expr.atoms(Function).union( + {e.exp for e in expr.atoms(Pow)})) + + def update_ST(expr): + inv[expr] = compute_ST(expr) + for expr in numer_gammas + denom_gammas + numer_others + denom_others: + update_ST(expr) + + for gammas, numer, denom in [( + numer_gammas, numer_others, denom_others), + (denom_gammas, denom_others, numer_others)]: + new = [] + while gammas: + g = gammas.pop() + cont = True + while cont: + cont = False + y = find_fuzzy(numer, g) + if y is not None: + numer.remove(y) + if y != g: + numer.append(y/g) + update_ST(y/g) + g += 1 + cont = True + y = find_fuzzy(denom, g - 1) + if y is not None: + denom.remove(y) + if y != g - 1: + numer.append((g - 1)/y) + update_ST((g - 1)/y) + g -= 1 + cont = True + new.append(g) + # /!\ updating IN PLACE + gammas[:] = new + + # =========== rebuild expr ================================== + + return Mul(*[gamma(g) for g in numer_gammas]) \ + / Mul(*[gamma(g) for g in denom_gammas]) \ + * Mul(*numer_others) / Mul(*denom_others) + + was = factor(expr) + # (for some reason we cannot use Basic.replace in this case) + expr = rule_gamma(was) + if expr != was: + expr = factor(expr) + + expr = expr.replace(gamma, + lambda n: expand_func(gamma(n)) if n.is_Rational else gamma(n)) + + return expr + + +class _rf(Function): + @classmethod + def eval(cls, a, b): + if b.is_Integer: + if not b: + return S.One + + n = int(b) + + if n > 0: + return Mul(*[a + i for i in range(n)]) + elif n < 0: + return 1/Mul(*[a - i for i in range(1, -n + 1)]) + else: + if b.is_Add: + c, _b = b.as_coeff_Add() + + if c.is_Integer: + if c > 0: + return _rf(a, _b)*_rf(a + _b, c) + elif c < 0: + return _rf(a, _b)/_rf(a + _b + c, -c) + + if a.is_Add: + c, _a = a.as_coeff_Add() + + if c.is_Integer: + if c > 0: + return _rf(_a, b)*_rf(_a + b, c)/_rf(_a, c) + elif c < 0: + return _rf(_a, b)*_rf(_a + c, -c)/_rf(_a + b + c, -c) diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/hyperexpand.py b/.venv/lib/python3.12/site-packages/sympy/simplify/hyperexpand.py new file mode 100644 index 0000000000000000000000000000000000000000..c070aa2e44b92794107b3e33df897813a54307b9 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/simplify/hyperexpand.py @@ -0,0 +1,2494 @@ +""" +Expand Hypergeometric (and Meijer G) functions into named +special functions. + +The algorithm for doing this uses a collection of lookup tables of +hypergeometric functions, and various of their properties, to expand +many hypergeometric functions in terms of special functions. + +It is based on the following paper: + Kelly B. Roach. Meijer G Function Representations. + In: Proceedings of the 1997 International Symposium on Symbolic and + Algebraic Computation, pages 205-211, New York, 1997. ACM. + +It is described in great(er) detail in the Sphinx documentation. +""" +# SUMMARY OF EXTENSIONS FOR MEIJER G FUNCTIONS +# +# o z**rho G(ap, bq; z) = G(ap + rho, bq + rho; z) +# +# o denote z*d/dz by D +# +# o It is helpful to keep in mind that ap and bq play essentially symmetric +# roles: G(1/z) has slightly altered parameters, with ap and bq interchanged. +# +# o There are four shift operators: +# A_J = b_J - D, J = 1, ..., n +# B_J = 1 - a_j + D, J = 1, ..., m +# C_J = -b_J + D, J = m+1, ..., q +# D_J = a_J - 1 - D, J = n+1, ..., p +# +# A_J, C_J increment b_J +# B_J, D_J decrement a_J +# +# o The corresponding four inverse-shift operators are defined if there +# is no cancellation. Thus e.g. an index a_J (upper or lower) can be +# incremented if a_J != b_i for i = 1, ..., q. +# +# o Order reduction: if b_j - a_i is a non-negative integer, where +# j <= m and i > n, the corresponding quotient of gamma functions reduces +# to a polynomial. Hence the G function can be expressed using a G-function +# of lower order. +# Similarly if j > m and i <= n. +# +# Secondly, there are paired index theorems [Adamchik, The evaluation of +# integrals of Bessel functions via G-function identities]. Suppose there +# are three parameters a, b, c, where a is an a_i, i <= n, b is a b_j, +# j <= m and c is a denominator parameter (i.e. a_i, i > n or b_j, j > m). +# Suppose further all three differ by integers. +# Then the order can be reduced. +# TODO work this out in detail. +# +# o An index quadruple is called suitable if its order cannot be reduced. +# If there exists a sequence of shift operators transforming one index +# quadruple into another, we say one is reachable from the other. +# +# o Deciding if one index quadruple is reachable from another is tricky. For +# this reason, we use hand-built routines to match and instantiate formulas. +# +from collections import defaultdict +from itertools import product +from functools import reduce +from math import prod + +from sympy import SYMPY_DEBUG +from sympy.core import (S, Dummy, symbols, sympify, Tuple, expand, I, pi, Mul, + EulerGamma, oo, zoo, expand_func, Add, nan, Expr, Rational) +from sympy.core.mod import Mod +from sympy.core.sorting import default_sort_key +from sympy.functions import (exp, sqrt, root, log, lowergamma, cos, + besseli, gamma, uppergamma, expint, erf, sin, besselj, Ei, Ci, Si, Shi, + sinh, cosh, Chi, fresnels, fresnelc, polar_lift, exp_polar, floor, ceiling, + rf, factorial, lerchphi, Piecewise, re, elliptic_k, elliptic_e) +from sympy.functions.elementary.complexes import polarify, unpolarify +from sympy.functions.special.hyper import (hyper, HyperRep_atanh, + HyperRep_power1, HyperRep_power2, HyperRep_log1, HyperRep_asin1, + HyperRep_asin2, HyperRep_sqrts1, HyperRep_sqrts2, HyperRep_log2, + HyperRep_cosasin, HyperRep_sinasin, meijerg) +from sympy.matrices import Matrix, eye, zeros +from sympy.polys import apart, poly, Poly +from sympy.series import residue +from sympy.simplify.powsimp import powdenest +from sympy.utilities.iterables import sift + +# function to define "buckets" +def _mod1(x): + # TODO see if this can work as Mod(x, 1); this will require + # different handling of the "buckets" since these need to + # be sorted and that fails when there is a mixture of + # integers and expressions with parameters. With the current + # Mod behavior, Mod(k, 1) == Mod(1, 1) == 0 if k is an integer. + # Although the sorting can be done with Basic.compare, this may + # still require different handling of the sorted buckets. + if x.is_Number: + return Mod(x, 1) + c, x = x.as_coeff_Add() + return Mod(c, 1) + x + + +# leave add formulae at the top for easy reference +def add_formulae(formulae): + """ Create our knowledge base. """ + a, b, c, z = symbols('a b c, z', cls=Dummy) + + def add(ap, bq, res): + func = Hyper_Function(ap, bq) + formulae.append(Formula(func, z, res, (a, b, c))) + + def addb(ap, bq, B, C, M): + func = Hyper_Function(ap, bq) + formulae.append(Formula(func, z, None, (a, b, c), B, C, M)) + + # Luke, Y. L. (1969), The Special Functions and Their Approximations, + # Volume 1, section 6.2 + + # 0F0 + add((), (), exp(z)) + + # 1F0 + add((a, ), (), HyperRep_power1(-a, z)) + + # 2F1 + addb((a, a - S.Half), (2*a, ), + Matrix([HyperRep_power2(a, z), + HyperRep_power2(a + S.Half, z)/2]), + Matrix([[1, 0]]), + Matrix([[(a - S.Half)*z/(1 - z), (S.Half - a)*z/(1 - z)], + [a/(1 - z), a*(z - 2)/(1 - z)]])) + addb((1, 1), (2, ), + Matrix([HyperRep_log1(z), 1]), Matrix([[-1/z, 0]]), + Matrix([[0, z/(z - 1)], [0, 0]])) + addb((S.Half, 1), (S('3/2'), ), + Matrix([HyperRep_atanh(z), 1]), + Matrix([[1, 0]]), + Matrix([[Rational(-1, 2), 1/(1 - z)/2], [0, 0]])) + addb((S.Half, S.Half), (S('3/2'), ), + Matrix([HyperRep_asin1(z), HyperRep_power1(Rational(-1, 2), z)]), + Matrix([[1, 0]]), + Matrix([[Rational(-1, 2), S.Half], [0, z/(1 - z)/2]])) + addb((a, S.Half + a), (S.Half, ), + Matrix([HyperRep_sqrts1(-a, z), -HyperRep_sqrts2(-a - S.Half, z)]), + Matrix([[1, 0]]), + Matrix([[0, -a], + [z*(-2*a - 1)/2/(1 - z), S.Half - z*(-2*a - 1)/(1 - z)]])) + + # A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev (1990). + # Integrals and Series: More Special Functions, Vol. 3,. + # Gordon and Breach Science Publisher + addb([a, -a], [S.Half], + Matrix([HyperRep_cosasin(a, z), HyperRep_sinasin(a, z)]), + Matrix([[1, 0]]), + Matrix([[0, -a], [a*z/(1 - z), 1/(1 - z)/2]])) + addb([1, 1], [3*S.Half], + Matrix([HyperRep_asin2(z), 1]), Matrix([[1, 0]]), + Matrix([[(z - S.Half)/(1 - z), 1/(1 - z)/2], [0, 0]])) + + # Complete elliptic integrals K(z) and E(z), both a 2F1 function + addb([S.Half, S.Half], [S.One], + Matrix([elliptic_k(z), elliptic_e(z)]), + Matrix([[2/pi, 0]]), + Matrix([[Rational(-1, 2), -1/(2*z-2)], + [Rational(-1, 2), S.Half]])) + addb([Rational(-1, 2), S.Half], [S.One], + Matrix([elliptic_k(z), elliptic_e(z)]), + Matrix([[0, 2/pi]]), + Matrix([[Rational(-1, 2), -1/(2*z-2)], + [Rational(-1, 2), S.Half]])) + + # 3F2 + addb([Rational(-1, 2), 1, 1], [S.Half, 2], + Matrix([z*HyperRep_atanh(z), HyperRep_log1(z), 1]), + Matrix([[Rational(-2, 3), -S.One/(3*z), Rational(2, 3)]]), + Matrix([[S.Half, 0, z/(1 - z)/2], + [0, 0, z/(z - 1)], + [0, 0, 0]])) + # actually the formula for 3/2 is much nicer ... + addb([Rational(-1, 2), 1, 1], [2, 2], + Matrix([HyperRep_power1(S.Half, z), HyperRep_log2(z), 1]), + Matrix([[Rational(4, 9) - 16/(9*z), 4/(3*z), 16/(9*z)]]), + Matrix([[z/2/(z - 1), 0, 0], [1/(2*(z - 1)), 0, S.Half], [0, 0, 0]])) + + # 1F1 + addb([1], [b], Matrix([z**(1 - b) * exp(z) * lowergamma(b - 1, z), 1]), + Matrix([[b - 1, 0]]), Matrix([[1 - b + z, 1], [0, 0]])) + addb([a], [2*a], + Matrix([z**(S.Half - a)*exp(z/2)*besseli(a - S.Half, z/2) + * gamma(a + S.Half)/4**(S.Half - a), + z**(S.Half - a)*exp(z/2)*besseli(a + S.Half, z/2) + * gamma(a + S.Half)/4**(S.Half - a)]), + Matrix([[1, 0]]), + Matrix([[z/2, z/2], [z/2, (z/2 - 2*a)]])) + mz = polar_lift(-1)*z + addb([a], [a + 1], + Matrix([mz**(-a)*a*lowergamma(a, mz), a*exp(z)]), + Matrix([[1, 0]]), + Matrix([[-a, 1], [0, z]])) + # This one is redundant. + add([Rational(-1, 2)], [S.Half], exp(z) - sqrt(pi*z)*(-I)*erf(I*sqrt(z))) + + # Added to get nice results for Laplace transform of Fresnel functions + # https://functions.wolfram.com/07.22.03.6437.01 + # Basic rule + #add([1], [Rational(3, 4), Rational(5, 4)], + # sqrt(pi) * (cos(2*sqrt(polar_lift(-1)*z))*fresnelc(2*root(polar_lift(-1)*z,4)/sqrt(pi)) + + # sin(2*sqrt(polar_lift(-1)*z))*fresnels(2*root(polar_lift(-1)*z,4)/sqrt(pi))) + # / (2*root(polar_lift(-1)*z,4))) + # Manually tuned rule + addb([1], [Rational(3, 4), Rational(5, 4)], + Matrix([ sqrt(pi)*(I*sinh(2*sqrt(z))*fresnels(2*root(z, 4)*exp(I*pi/4)/sqrt(pi)) + + cosh(2*sqrt(z))*fresnelc(2*root(z, 4)*exp(I*pi/4)/sqrt(pi))) + * exp(-I*pi/4)/(2*root(z, 4)), + sqrt(pi)*root(z, 4)*(sinh(2*sqrt(z))*fresnelc(2*root(z, 4)*exp(I*pi/4)/sqrt(pi)) + + I*cosh(2*sqrt(z))*fresnels(2*root(z, 4)*exp(I*pi/4)/sqrt(pi))) + *exp(-I*pi/4)/2, + 1 ]), + Matrix([[1, 0, 0]]), + Matrix([[Rational(-1, 4), 1, Rational(1, 4)], + [ z, Rational(1, 4), 0], + [ 0, 0, 0]])) + + # 2F2 + addb([S.Half, a], [Rational(3, 2), a + 1], + Matrix([a/(2*a - 1)*(-I)*sqrt(pi/z)*erf(I*sqrt(z)), + a/(2*a - 1)*(polar_lift(-1)*z)**(-a)* + lowergamma(a, polar_lift(-1)*z), + a/(2*a - 1)*exp(z)]), + Matrix([[1, -1, 0]]), + Matrix([[Rational(-1, 2), 0, 1], [0, -a, 1], [0, 0, z]])) + # We make a "basis" of four functions instead of three, and give EulerGamma + # an extra slot (it could just be a coefficient to 1). The advantage is + # that this way Polys will not see multivariate polynomials (it treats + # EulerGamma as an indeterminate), which is *way* faster. + addb([1, 1], [2, 2], + Matrix([Ei(z) - log(z), exp(z), 1, EulerGamma]), + Matrix([[1/z, 0, 0, -1/z]]), + Matrix([[0, 1, -1, 0], [0, z, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]])) + + # 0F1 + add((), (S.Half, ), cosh(2*sqrt(z))) + addb([], [b], + Matrix([gamma(b)*z**((1 - b)/2)*besseli(b - 1, 2*sqrt(z)), + gamma(b)*z**(1 - b/2)*besseli(b, 2*sqrt(z))]), + Matrix([[1, 0]]), Matrix([[0, 1], [z, (1 - b)]])) + + # 0F3 + x = 4*z**Rational(1, 4) + + def fp(a, z): + return besseli(a, x) + besselj(a, x) + + def fm(a, z): + return besseli(a, x) - besselj(a, x) + + # TODO branching + addb([], [S.Half, a, a + S.Half], + Matrix([fp(2*a - 1, z), fm(2*a, z)*z**Rational(1, 4), + fm(2*a - 1, z)*sqrt(z), fp(2*a, z)*z**Rational(3, 4)]) + * 2**(-2*a)*gamma(2*a)*z**((1 - 2*a)/4), + Matrix([[1, 0, 0, 0]]), + Matrix([[0, 1, 0, 0], + [0, S.Half - a, 1, 0], + [0, 0, S.Half, 1], + [z, 0, 0, 1 - a]])) + x = 2*(4*z)**Rational(1, 4)*exp_polar(I*pi/4) + addb([], [a, a + S.Half, 2*a], + (2*sqrt(polar_lift(-1)*z))**(1 - 2*a)*gamma(2*a)**2 * + Matrix([besselj(2*a - 1, x)*besseli(2*a - 1, x), + x*(besseli(2*a, x)*besselj(2*a - 1, x) + - besseli(2*a - 1, x)*besselj(2*a, x)), + x**2*besseli(2*a, x)*besselj(2*a, x), + x**3*(besseli(2*a, x)*besselj(2*a - 1, x) + + besseli(2*a - 1, x)*besselj(2*a, x))]), + Matrix([[1, 0, 0, 0]]), + Matrix([[0, Rational(1, 4), 0, 0], + [0, (1 - 2*a)/2, Rational(-1, 2), 0], + [0, 0, 1 - 2*a, Rational(1, 4)], + [-32*z, 0, 0, 1 - a]])) + + # 1F2 + addb([a], [a - S.Half, 2*a], + Matrix([z**(S.Half - a)*besseli(a - S.Half, sqrt(z))**2, + z**(1 - a)*besseli(a - S.Half, sqrt(z)) + *besseli(a - Rational(3, 2), sqrt(z)), + z**(Rational(3, 2) - a)*besseli(a - Rational(3, 2), sqrt(z))**2]), + Matrix([[-gamma(a + S.Half)**2/4**(S.Half - a), + 2*gamma(a - S.Half)*gamma(a + S.Half)/4**(1 - a), + 0]]), + Matrix([[1 - 2*a, 1, 0], [z/2, S.Half - a, S.Half], [0, z, 0]])) + addb([S.Half], [b, 2 - b], + pi*(1 - b)/sin(pi*b)* + Matrix([besseli(1 - b, sqrt(z))*besseli(b - 1, sqrt(z)), + sqrt(z)*(besseli(-b, sqrt(z))*besseli(b - 1, sqrt(z)) + + besseli(1 - b, sqrt(z))*besseli(b, sqrt(z))), + besseli(-b, sqrt(z))*besseli(b, sqrt(z))]), + Matrix([[1, 0, 0]]), + Matrix([[b - 1, S.Half, 0], + [z, 0, z], + [0, S.Half, -b]])) + addb([S.Half], [Rational(3, 2), Rational(3, 2)], + Matrix([Shi(2*sqrt(z))/2/sqrt(z), sinh(2*sqrt(z))/2/sqrt(z), + cosh(2*sqrt(z))]), + Matrix([[1, 0, 0]]), + Matrix([[Rational(-1, 2), S.Half, 0], [0, Rational(-1, 2), S.Half], [0, 2*z, 0]])) + + # FresnelS + # Basic rule + #add([Rational(3, 4)], [Rational(3, 2),Rational(7, 4)], 6*fresnels( exp(pi*I/4)*root(z,4)*2/sqrt(pi) ) / ( pi * (exp(pi*I/4)*root(z,4)*2/sqrt(pi))**3 ) ) + # Manually tuned rule + addb([Rational(3, 4)], [Rational(3, 2), Rational(7, 4)], + Matrix( + [ fresnels( + exp( + pi*I/4)*root( + z, 4)*2/sqrt( + pi) ) / ( + pi * (exp(pi*I/4)*root(z, 4)*2/sqrt(pi))**3 ), + sinh(2*sqrt(z))/sqrt(z), + cosh(2*sqrt(z)) ]), + Matrix([[6, 0, 0]]), + Matrix([[Rational(-3, 4), Rational(1, 16), 0], + [ 0, Rational(-1, 2), 1], + [ 0, z, 0]])) + + # FresnelC + # Basic rule + #add([Rational(1, 4)], [S.Half,Rational(5, 4)], fresnelc( exp(pi*I/4)*root(z,4)*2/sqrt(pi) ) / ( exp(pi*I/4)*root(z,4)*2/sqrt(pi) ) ) + # Manually tuned rule + addb([Rational(1, 4)], [S.Half, Rational(5, 4)], + Matrix( + [ sqrt( + pi)*exp( + -I*pi/4)*fresnelc( + 2*root(z, 4)*exp(I*pi/4)/sqrt(pi))/(2*root(z, 4)), + cosh(2*sqrt(z)), + sinh(2*sqrt(z))*sqrt(z) ]), + Matrix([[1, 0, 0]]), + Matrix([[Rational(-1, 4), Rational(1, 4), 0 ], + [ 0, 0, 1 ], + [ 0, z, S.Half]])) + + # 2F3 + # XXX with this five-parameter formula is pretty slow with the current + # Formula.find_instantiations (creates 2!*3!*3**(2+3) ~ 3000 + # instantiations ... But it's not too bad. + addb([a, a + S.Half], [2*a, b, 2*a - b + 1], + gamma(b)*gamma(2*a - b + 1) * (sqrt(z)/2)**(1 - 2*a) * + Matrix([besseli(b - 1, sqrt(z))*besseli(2*a - b, sqrt(z)), + sqrt(z)*besseli(b, sqrt(z))*besseli(2*a - b, sqrt(z)), + sqrt(z)*besseli(b - 1, sqrt(z))*besseli(2*a - b + 1, sqrt(z)), + besseli(b, sqrt(z))*besseli(2*a - b + 1, sqrt(z))]), + Matrix([[1, 0, 0, 0]]), + Matrix([[0, S.Half, S.Half, 0], + [z/2, 1 - b, 0, z/2], + [z/2, 0, b - 2*a, z/2], + [0, S.Half, S.Half, -2*a]])) + # (C/f above comment about eulergamma in the basis). + addb([1, 1], [2, 2, Rational(3, 2)], + Matrix([Chi(2*sqrt(z)) - log(2*sqrt(z)), + cosh(2*sqrt(z)), sqrt(z)*sinh(2*sqrt(z)), 1, EulerGamma]), + Matrix([[1/z, 0, 0, 0, -1/z]]), + Matrix([[0, S.Half, 0, Rational(-1, 2), 0], + [0, 0, 1, 0, 0], + [0, z, S.Half, 0, 0], + [0, 0, 0, 0, 0], + [0, 0, 0, 0, 0]])) + + # 3F3 + # This is rule: https://functions.wolfram.com/07.31.03.0134.01 + # Initial reason to add it was a nice solution for + # integrate(erf(a*z)/z**2, z) and same for erfc and erfi. + # Basic rule + # add([1, 1, a], [2, 2, a+1], (a/(z*(a-1)**2)) * + # (1 - (-z)**(1-a) * (gamma(a) - uppergamma(a,-z)) + # - (a-1) * (EulerGamma + uppergamma(0,-z) + log(-z)) + # - exp(z))) + # Manually tuned rule + addb([1, 1, a], [2, 2, a+1], + Matrix([a*(log(-z) + expint(1, -z) + EulerGamma)/(z*(a**2 - 2*a + 1)), + a*(-z)**(-a)*(gamma(a) - uppergamma(a, -z))/(a - 1)**2, + a*exp(z)/(a**2 - 2*a + 1), + a/(z*(a**2 - 2*a + 1))]), + Matrix([[1-a, 1, -1/z, 1]]), + Matrix([[-1,0,-1/z,1], + [0,-a,1,0], + [0,0,z,0], + [0,0,0,-1]])) + + +def add_meijerg_formulae(formulae): + a, b, c, z = list(map(Dummy, 'abcz')) + rho = Dummy('rho') + + def add(an, ap, bm, bq, B, C, M, matcher): + formulae.append(MeijerFormula(an, ap, bm, bq, z, [a, b, c, rho], + B, C, M, matcher)) + + def detect_uppergamma(func): + x = func.an[0] + y, z = func.bm + swapped = False + if not _mod1((x - y).simplify()): + swapped = True + (y, z) = (z, y) + if _mod1((x - z).simplify()) or x - z > 0: + return None + l = [y, x] + if swapped: + l = [x, y] + return {rho: y, a: x - y}, G_Function([x], [], l, []) + + add([a + rho], [], [rho, a + rho], [], + Matrix([gamma(1 - a)*z**rho*exp(z)*uppergamma(a, z), + gamma(1 - a)*z**(a + rho)]), + Matrix([[1, 0]]), + Matrix([[rho + z, -1], [0, a + rho]]), + detect_uppergamma) + + def detect_3113(func): + """https://functions.wolfram.com/07.34.03.0984.01""" + x = func.an[0] + u, v, w = func.bm + if _mod1((u - v).simplify()) == 0: + if _mod1((v - w).simplify()) == 0: + return + sig = (S.Half, S.Half, S.Zero) + x1, x2, y = u, v, w + else: + if _mod1((x - u).simplify()) == 0: + sig = (S.Half, S.Zero, S.Half) + x1, y, x2 = u, v, w + else: + sig = (S.Zero, S.Half, S.Half) + y, x1, x2 = u, v, w + + if (_mod1((x - x1).simplify()) != 0 or + _mod1((x - x2).simplify()) != 0 or + _mod1((x - y).simplify()) != S.Half or + x - x1 > 0 or x - x2 > 0): + return + + return {a: x}, G_Function([x], [], [x - S.Half + t for t in sig], []) + + s = sin(2*sqrt(z)) + c_ = cos(2*sqrt(z)) + S_ = Si(2*sqrt(z)) - pi/2 + C = Ci(2*sqrt(z)) + add([a], [], [a, a, a - S.Half], [], + Matrix([sqrt(pi)*z**(a - S.Half)*(c_*S_ - s*C), + sqrt(pi)*z**a*(s*S_ + c_*C), + sqrt(pi)*z**a]), + Matrix([[-2, 0, 0]]), + Matrix([[a - S.Half, -1, 0], [z, a, S.Half], [0, 0, a]]), + detect_3113) + + +def make_simp(z): + """ Create a function that simplifies rational functions in ``z``. """ + + def simp(expr): + """ Efficiently simplify the rational function ``expr``. """ + numer, denom = expr.as_numer_denom() + numer = numer.expand() + # denom = denom.expand() # is this needed? + c, numer, denom = poly(numer, z).cancel(poly(denom, z)) + return c * numer.as_expr() / denom.as_expr() + + return simp + + +def debug(*args): + if SYMPY_DEBUG: + for a in args: + print(a, end="") + print() + + +class Hyper_Function(Expr): + """ A generalized hypergeometric function. """ + + def __new__(cls, ap, bq): + obj = super().__new__(cls) + obj.ap = Tuple(*list(map(expand, ap))) + obj.bq = Tuple(*list(map(expand, bq))) + return obj + + @property + def args(self): + return (self.ap, self.bq) + + @property + def sizes(self): + return (len(self.ap), len(self.bq)) + + @property + def gamma(self): + """ + Number of upper parameters that are negative integers + + This is a transformation invariant. + """ + return sum(bool(x.is_integer and x.is_negative) for x in self.ap) + + def _hashable_content(self): + return super()._hashable_content() + (self.ap, + self.bq) + + def __call__(self, arg): + return hyper(self.ap, self.bq, arg) + + def build_invariants(self): + """ + Compute the invariant vector. + + Explanation + =========== + + The invariant vector is: + (gamma, ((s1, n1), ..., (sk, nk)), ((t1, m1), ..., (tr, mr))) + where gamma is the number of integer a < 0, + s1 < ... < sk + nl is the number of parameters a_i congruent to sl mod 1 + t1 < ... < tr + ml is the number of parameters b_i congruent to tl mod 1 + + If the index pair contains parameters, then this is not truly an + invariant, since the parameters cannot be sorted uniquely mod1. + + Examples + ======== + + >>> from sympy.simplify.hyperexpand import Hyper_Function + >>> from sympy import S + >>> ap = (S.Half, S.One/3, S(-1)/2, -2) + >>> bq = (1, 2) + + Here gamma = 1, + k = 3, s1 = 0, s2 = 1/3, s3 = 1/2 + n1 = 1, n2 = 1, n2 = 2 + r = 1, t1 = 0 + m1 = 2: + + >>> Hyper_Function(ap, bq).build_invariants() + (1, ((0, 1), (1/3, 1), (1/2, 2)), ((0, 2),)) + """ + abuckets, bbuckets = sift(self.ap, _mod1), sift(self.bq, _mod1) + + def tr(bucket): + bucket = list(bucket.items()) + if not any(isinstance(x[0], Mod) for x in bucket): + bucket.sort(key=lambda x: default_sort_key(x[0])) + bucket = tuple([(mod, len(values)) for mod, values in bucket if + values]) + return bucket + + return (self.gamma, tr(abuckets), tr(bbuckets)) + + def difficulty(self, func): + """ Estimate how many steps it takes to reach ``func`` from self. + Return -1 if impossible. """ + if self.gamma != func.gamma: + return -1 + oabuckets, obbuckets, abuckets, bbuckets = [sift(params, _mod1) for + params in (self.ap, self.bq, func.ap, func.bq)] + + diff = 0 + for bucket, obucket in [(abuckets, oabuckets), (bbuckets, obbuckets)]: + for mod in set(list(bucket.keys()) + list(obucket.keys())): + if (mod not in bucket) or (mod not in obucket) \ + or len(bucket[mod]) != len(obucket[mod]): + return -1 + l1 = list(bucket[mod]) + l2 = list(obucket[mod]) + l1.sort() + l2.sort() + for i, j in zip(l1, l2): + diff += abs(i - j) + + return diff + + def _is_suitable_origin(self): + """ + Decide if ``self`` is a suitable origin. + + Explanation + =========== + + A function is a suitable origin iff: + * none of the ai equals bj + n, with n a non-negative integer + * none of the ai is zero + * none of the bj is a non-positive integer + + Note that this gives meaningful results only when none of the indices + are symbolic. + + """ + for a in self.ap: + for b in self.bq: + if (a - b).is_integer and (a - b).is_negative is False: + return False + for a in self.ap: + if a == 0: + return False + for b in self.bq: + if b.is_integer and b.is_nonpositive: + return False + return True + + +class G_Function(Expr): + """ A Meijer G-function. """ + + def __new__(cls, an, ap, bm, bq): + obj = super().__new__(cls) + obj.an = Tuple(*list(map(expand, an))) + obj.ap = Tuple(*list(map(expand, ap))) + obj.bm = Tuple(*list(map(expand, bm))) + obj.bq = Tuple(*list(map(expand, bq))) + return obj + + @property + def args(self): + return (self.an, self.ap, self.bm, self.bq) + + def _hashable_content(self): + return super()._hashable_content() + self.args + + def __call__(self, z): + return meijerg(self.an, self.ap, self.bm, self.bq, z) + + def compute_buckets(self): + """ + Compute buckets for the fours sets of parameters. + + Explanation + =========== + + We guarantee that any two equal Mod objects returned are actually the + same, and that the buckets are sorted by real part (an and bq + descendending, bm and ap ascending). + + Examples + ======== + + >>> from sympy.simplify.hyperexpand import G_Function + >>> from sympy.abc import y + >>> from sympy import S + + >>> a, b = [1, 3, 2, S(3)/2], [1 + y, y, 2, y + 3] + >>> G_Function(a, b, [2], [y]).compute_buckets() + ({0: [3, 2, 1], 1/2: [3/2]}, + {0: [2], y: [y, y + 1, y + 3]}, {0: [2]}, {y: [y]}) + + """ + dicts = pan, pap, pbm, pbq = [defaultdict(list) for i in range(4)] + for dic, lis in zip(dicts, (self.an, self.ap, self.bm, self.bq)): + for x in lis: + dic[_mod1(x)].append(x) + + for dic, flip in zip(dicts, (True, False, False, True)): + for m, items in dic.items(): + x0 = items[0] + items.sort(key=lambda x: x - x0, reverse=flip) + dic[m] = items + + return tuple([dict(w) for w in dicts]) + + @property + def signature(self): + return (len(self.an), len(self.ap), len(self.bm), len(self.bq)) + + +# Dummy variable. +_x = Dummy('x') + +class Formula: + """ + This class represents hypergeometric formulae. + + Explanation + =========== + + Its data members are: + - z, the argument + - closed_form, the closed form expression + - symbols, the free symbols (parameters) in the formula + - func, the function + - B, C, M (see _compute_basis) + + Examples + ======== + + >>> from sympy.abc import a, b, z + >>> from sympy.simplify.hyperexpand import Formula, Hyper_Function + >>> func = Hyper_Function((a/2, a/3 + b, (1+a)/2), (a, b, (a+b)/7)) + >>> f = Formula(func, z, None, [a, b]) + + """ + + def _compute_basis(self, closed_form): + """ + Compute a set of functions B=(f1, ..., fn), a nxn matrix M + and a 1xn matrix C such that: + closed_form = C B + z d/dz B = M B. + """ + afactors = [_x + a for a in self.func.ap] + bfactors = [_x + b - 1 for b in self.func.bq] + expr = _x*Mul(*bfactors) - self.z*Mul(*afactors) + poly = Poly(expr, _x) + + n = poly.degree() - 1 + b = [closed_form] + for _ in range(n): + b.append(self.z*b[-1].diff(self.z)) + + self.B = Matrix(b) + self.C = Matrix([[1] + [0]*n]) + + m = eye(n) + m = m.col_insert(0, zeros(n, 1)) + l = poly.all_coeffs()[1:] + l.reverse() + self.M = m.row_insert(n, -Matrix([l])/poly.all_coeffs()[0]) + + def __init__(self, func, z, res, symbols, B=None, C=None, M=None): + z = sympify(z) + res = sympify(res) + symbols = [x for x in sympify(symbols) if func.has(x)] + + self.z = z + self.symbols = symbols + self.B = B + self.C = C + self.M = M + self.func = func + + # TODO with symbolic parameters, it could be advantageous + # (for prettier answers) to compute a basis only *after* + # instantiation + if res is not None: + self._compute_basis(res) + + @property + def closed_form(self): + return reduce(lambda s,m: s+m[0]*m[1], zip(self.C, self.B), S.Zero) + + def find_instantiations(self, func): + """ + Find substitutions of the free symbols that match ``func``. + + Return the substitution dictionaries as a list. Note that the returned + instantiations need not actually match, or be valid! + + """ + from sympy.solvers import solve + ap = func.ap + bq = func.bq + if len(ap) != len(self.func.ap) or len(bq) != len(self.func.bq): + raise TypeError('Cannot instantiate other number of parameters') + symbol_values = [] + for a in self.symbols: + if a in self.func.ap.args: + symbol_values.append(ap) + elif a in self.func.bq.args: + symbol_values.append(bq) + else: + raise ValueError("At least one of the parameters of the " + "formula must be equal to %s" % (a,)) + base_repl = [dict(list(zip(self.symbols, values))) + for values in product(*symbol_values)] + abuckets, bbuckets = [sift(params, _mod1) for params in [ap, bq]] + a_inv, b_inv = [{a: len(vals) for a, vals in bucket.items()} + for bucket in [abuckets, bbuckets]] + critical_values = [[0] for _ in self.symbols] + result = [] + _n = Dummy() + for repl in base_repl: + symb_a, symb_b = [sift(params, lambda x: _mod1(x.xreplace(repl))) + for params in [self.func.ap, self.func.bq]] + for bucket, obucket in [(abuckets, symb_a), (bbuckets, symb_b)]: + for mod in set(list(bucket.keys()) + list(obucket.keys())): + if (mod not in bucket) or (mod not in obucket) \ + or len(bucket[mod]) != len(obucket[mod]): + break + for a, vals in zip(self.symbols, critical_values): + if repl[a].free_symbols: + continue + exprs = [expr for expr in obucket[mod] if expr.has(a)] + repl0 = repl.copy() + repl0[a] += _n + for expr in exprs: + for target in bucket[mod]: + n0, = solve(expr.xreplace(repl0) - target, _n) + if n0.free_symbols: + raise ValueError("Value should not be true") + vals.append(n0) + else: + values = [] + for a, vals in zip(self.symbols, critical_values): + a0 = repl[a] + min_ = floor(min(vals)) + max_ = ceiling(max(vals)) + values.append([a0 + n for n in range(min_, max_ + 1)]) + result.extend(dict(list(zip(self.symbols, l))) for l in product(*values)) + return result + + + + +class FormulaCollection: + """ A collection of formulae to use as origins. """ + + def __init__(self): + """ Doing this globally at module init time is a pain ... """ + self.symbolic_formulae = {} + self.concrete_formulae = {} + self.formulae = [] + + add_formulae(self.formulae) + + # Now process the formulae into a helpful form. + # These dicts are indexed by (p, q). + + for f in self.formulae: + sizes = f.func.sizes + if len(f.symbols) > 0: + self.symbolic_formulae.setdefault(sizes, []).append(f) + else: + inv = f.func.build_invariants() + self.concrete_formulae.setdefault(sizes, {})[inv] = f + + def lookup_origin(self, func): + """ + Given the suitable target ``func``, try to find an origin in our + knowledge base. + + Examples + ======== + + >>> from sympy.simplify.hyperexpand import (FormulaCollection, + ... Hyper_Function) + >>> f = FormulaCollection() + >>> f.lookup_origin(Hyper_Function((), ())).closed_form + exp(_z) + >>> f.lookup_origin(Hyper_Function([1], ())).closed_form + HyperRep_power1(-1, _z) + + >>> from sympy import S + >>> i = Hyper_Function([S('1/4'), S('3/4 + 4')], [S.Half]) + >>> f.lookup_origin(i).closed_form + HyperRep_sqrts1(-1/4, _z) + """ + inv = func.build_invariants() + sizes = func.sizes + if sizes in self.concrete_formulae and \ + inv in self.concrete_formulae[sizes]: + return self.concrete_formulae[sizes][inv] + + # We don't have a concrete formula. Try to instantiate. + if sizes not in self.symbolic_formulae: + return None # Too bad... + + possible = [] + for f in self.symbolic_formulae[sizes]: + repls = f.find_instantiations(func) + for repl in repls: + func2 = f.func.xreplace(repl) + if not func2._is_suitable_origin(): + continue + diff = func2.difficulty(func) + if diff == -1: + continue + possible.append((diff, repl, f, func2)) + + # find the nearest origin + possible.sort(key=lambda x: x[0]) + for _, repl, f, func2 in possible: + f2 = Formula(func2, f.z, None, [], f.B.subs(repl), + f.C.subs(repl), f.M.subs(repl)) + if not any(e.has(S.NaN, oo, -oo, zoo) for e in [f2.B, f2.M, f2.C]): + return f2 + + return None + + +class MeijerFormula: + """ + This class represents a Meijer G-function formula. + + Its data members are: + - z, the argument + - symbols, the free symbols (parameters) in the formula + - func, the function + - B, C, M (c/f ordinary Formula) + """ + + def __init__(self, an, ap, bm, bq, z, symbols, B, C, M, matcher): + an, ap, bm, bq = [Tuple(*list(map(expand, w))) for w in [an, ap, bm, bq]] + self.func = G_Function(an, ap, bm, bq) + self.z = z + self.symbols = symbols + self._matcher = matcher + self.B = B + self.C = C + self.M = M + + @property + def closed_form(self): + return reduce(lambda s,m: s+m[0]*m[1], zip(self.C, self.B), S.Zero) + + def try_instantiate(self, func): + """ + Try to instantiate the current formula to (almost) match func. + This uses the _matcher passed on init. + """ + if func.signature != self.func.signature: + return None + res = self._matcher(func) + if res is not None: + subs, newfunc = res + return MeijerFormula(newfunc.an, newfunc.ap, newfunc.bm, newfunc.bq, + self.z, [], + self.B.subs(subs), self.C.subs(subs), + self.M.subs(subs), None) + + +class MeijerFormulaCollection: + """ + This class holds a collection of meijer g formulae. + """ + + def __init__(self): + formulae = [] + add_meijerg_formulae(formulae) + self.formulae = defaultdict(list) + for formula in formulae: + self.formulae[formula.func.signature].append(formula) + self.formulae = dict(self.formulae) + + def lookup_origin(self, func): + """ Try to find a formula that matches func. """ + if func.signature not in self.formulae: + return None + for formula in self.formulae[func.signature]: + res = formula.try_instantiate(func) + if res is not None: + return res + + +class Operator: + """ + Base class for operators to be applied to our functions. + + Explanation + =========== + + These operators are differential operators. They are by convention + expressed in the variable D = z*d/dz (although this base class does + not actually care). + Note that when the operator is applied to an object, we typically do + *not* blindly differentiate but instead use a different representation + of the z*d/dz operator (see make_derivative_operator). + + To subclass from this, define a __init__ method that initializes a + self._poly variable. This variable stores a polynomial. By convention + the generator is z*d/dz, and acts to the right of all coefficients. + + Thus this poly + x**2 + 2*z*x + 1 + represents the differential operator + (z*d/dz)**2 + 2*z**2*d/dz. + + This class is used only in the implementation of the hypergeometric + function expansion algorithm. + """ + + def apply(self, obj, op): + """ + Apply ``self`` to the object ``obj``, where the generator is ``op``. + + Examples + ======== + + >>> from sympy.simplify.hyperexpand import Operator + >>> from sympy.polys.polytools import Poly + >>> from sympy.abc import x, y, z + >>> op = Operator() + >>> op._poly = Poly(x**2 + z*x + y, x) + >>> op.apply(z**7, lambda f: f.diff(z)) + y*z**7 + 7*z**7 + 42*z**5 + """ + coeffs = self._poly.all_coeffs() + coeffs.reverse() + diffs = [obj] + for c in coeffs[1:]: + diffs.append(op(diffs[-1])) + r = coeffs[0]*diffs[0] + for c, d in zip(coeffs[1:], diffs[1:]): + r += c*d + return r + + +class MultOperator(Operator): + """ Simply multiply by a "constant" """ + + def __init__(self, p): + self._poly = Poly(p, _x) + + +class ShiftA(Operator): + """ Increment an upper index. """ + + def __init__(self, ai): + ai = sympify(ai) + if ai == 0: + raise ValueError('Cannot increment zero upper index.') + self._poly = Poly(_x/ai + 1, _x) + + def __str__(self): + return '' % (1/self._poly.all_coeffs()[0]) + + +class ShiftB(Operator): + """ Decrement a lower index. """ + + def __init__(self, bi): + bi = sympify(bi) + if bi == 1: + raise ValueError('Cannot decrement unit lower index.') + self._poly = Poly(_x/(bi - 1) + 1, _x) + + def __str__(self): + return '' % (1/self._poly.all_coeffs()[0] + 1) + + +class UnShiftA(Operator): + """ Decrement an upper index. """ + + def __init__(self, ap, bq, i, z): + """ Note: i counts from zero! """ + ap, bq, i = list(map(sympify, [ap, bq, i])) + + self._ap = ap + self._bq = bq + self._i = i + + ap = list(ap) + bq = list(bq) + ai = ap.pop(i) - 1 + + if ai == 0: + raise ValueError('Cannot decrement unit upper index.') + + m = Poly(z*ai, _x) + for a in ap: + m *= Poly(_x + a, _x) + + A = Dummy('A') + n = D = Poly(ai*A - ai, A) + for b in bq: + n *= D + (b - 1).as_poly(A) + + b0 = -n.nth(0) + if b0 == 0: + raise ValueError('Cannot decrement upper index: ' + 'cancels with lower') + + n = Poly(Poly(n.all_coeffs()[:-1], A).as_expr().subs(A, _x/ai + 1), _x) + + self._poly = Poly((n - m)/b0, _x) + + def __str__(self): + return '' % (self._i, + self._ap, self._bq) + + +class UnShiftB(Operator): + """ Increment a lower index. """ + + def __init__(self, ap, bq, i, z): + """ Note: i counts from zero! """ + ap, bq, i = list(map(sympify, [ap, bq, i])) + + self._ap = ap + self._bq = bq + self._i = i + + ap = list(ap) + bq = list(bq) + bi = bq.pop(i) + 1 + + if bi == 0: + raise ValueError('Cannot increment -1 lower index.') + + m = Poly(_x*(bi - 1), _x) + for b in bq: + m *= Poly(_x + b - 1, _x) + + B = Dummy('B') + D = Poly((bi - 1)*B - bi + 1, B) + n = Poly(z, B) + for a in ap: + n *= (D + a.as_poly(B)) + + b0 = n.nth(0) + if b0 == 0: + raise ValueError('Cannot increment index: cancels with upper') + + n = Poly(Poly(n.all_coeffs()[:-1], B).as_expr().subs( + B, _x/(bi - 1) + 1), _x) + + self._poly = Poly((m - n)/b0, _x) + + def __str__(self): + return '' % (self._i, + self._ap, self._bq) + + +class MeijerShiftA(Operator): + """ Increment an upper b index. """ + + def __init__(self, bi): + bi = sympify(bi) + self._poly = Poly(bi - _x, _x) + + def __str__(self): + return '' % (self._poly.all_coeffs()[1]) + + +class MeijerShiftB(Operator): + """ Decrement an upper a index. """ + + def __init__(self, bi): + bi = sympify(bi) + self._poly = Poly(1 - bi + _x, _x) + + def __str__(self): + return '' % (1 - self._poly.all_coeffs()[1]) + + +class MeijerShiftC(Operator): + """ Increment a lower b index. """ + + def __init__(self, bi): + bi = sympify(bi) + self._poly = Poly(-bi + _x, _x) + + def __str__(self): + return '' % (-self._poly.all_coeffs()[1]) + + +class MeijerShiftD(Operator): + """ Decrement a lower a index. """ + + def __init__(self, bi): + bi = sympify(bi) + self._poly = Poly(bi - 1 - _x, _x) + + def __str__(self): + return '' % (self._poly.all_coeffs()[1] + 1) + + +class MeijerUnShiftA(Operator): + """ Decrement an upper b index. """ + + def __init__(self, an, ap, bm, bq, i, z): + """ Note: i counts from zero! """ + an, ap, bm, bq, i = list(map(sympify, [an, ap, bm, bq, i])) + + self._an = an + self._ap = ap + self._bm = bm + self._bq = bq + self._i = i + + an = list(an) + ap = list(ap) + bm = list(bm) + bq = list(bq) + bi = bm.pop(i) - 1 + + m = Poly(1, _x) * prod(Poly(b - _x, _x) for b in bm) * prod(Poly(_x - b, _x) for b in bq) + + A = Dummy('A') + D = Poly(bi - A, A) + n = Poly(z, A) * prod((D + 1 - a) for a in an) * prod((-D + a - 1) for a in ap) + + b0 = n.nth(0) + if b0 == 0: + raise ValueError('Cannot decrement upper b index (cancels)') + + n = Poly(Poly(n.all_coeffs()[:-1], A).as_expr().subs(A, bi - _x), _x) + + self._poly = Poly((m - n)/b0, _x) + + def __str__(self): + return '' % (self._i, + self._an, self._ap, self._bm, self._bq) + + +class MeijerUnShiftB(Operator): + """ Increment an upper a index. """ + + def __init__(self, an, ap, bm, bq, i, z): + """ Note: i counts from zero! """ + an, ap, bm, bq, i = list(map(sympify, [an, ap, bm, bq, i])) + + self._an = an + self._ap = ap + self._bm = bm + self._bq = bq + self._i = i + + an = list(an) + ap = list(ap) + bm = list(bm) + bq = list(bq) + ai = an.pop(i) + 1 + + m = Poly(z, _x) + for a in an: + m *= Poly(1 - a + _x, _x) + for a in ap: + m *= Poly(a - 1 - _x, _x) + + B = Dummy('B') + D = Poly(B + ai - 1, B) + n = Poly(1, B) + for b in bm: + n *= (-D + b) + for b in bq: + n *= (D - b) + + b0 = n.nth(0) + if b0 == 0: + raise ValueError('Cannot increment upper a index (cancels)') + + n = Poly(Poly(n.all_coeffs()[:-1], B).as_expr().subs( + B, 1 - ai + _x), _x) + + self._poly = Poly((m - n)/b0, _x) + + def __str__(self): + return '' % (self._i, + self._an, self._ap, self._bm, self._bq) + + +class MeijerUnShiftC(Operator): + """ Decrement a lower b index. """ + # XXX this is "essentially" the same as MeijerUnShiftA. This "essentially" + # can be made rigorous using the functional equation G(1/z) = G'(z), + # where G' denotes a G function of slightly altered parameters. + # However, sorting out the details seems harder than just coding it + # again. + + def __init__(self, an, ap, bm, bq, i, z): + """ Note: i counts from zero! """ + an, ap, bm, bq, i = list(map(sympify, [an, ap, bm, bq, i])) + + self._an = an + self._ap = ap + self._bm = bm + self._bq = bq + self._i = i + + an = list(an) + ap = list(ap) + bm = list(bm) + bq = list(bq) + bi = bq.pop(i) - 1 + + m = Poly(1, _x) + for b in bm: + m *= Poly(b - _x, _x) + for b in bq: + m *= Poly(_x - b, _x) + + C = Dummy('C') + D = Poly(bi + C, C) + n = Poly(z, C) + for a in an: + n *= (D + 1 - a) + for a in ap: + n *= (-D + a - 1) + + b0 = n.nth(0) + if b0 == 0: + raise ValueError('Cannot decrement lower b index (cancels)') + + n = Poly(Poly(n.all_coeffs()[:-1], C).as_expr().subs(C, _x - bi), _x) + + self._poly = Poly((m - n)/b0, _x) + + def __str__(self): + return '' % (self._i, + self._an, self._ap, self._bm, self._bq) + + +class MeijerUnShiftD(Operator): + """ Increment a lower a index. """ + # XXX This is essentially the same as MeijerUnShiftA. + # See comment at MeijerUnShiftC. + + def __init__(self, an, ap, bm, bq, i, z): + """ Note: i counts from zero! """ + an, ap, bm, bq, i = list(map(sympify, [an, ap, bm, bq, i])) + + self._an = an + self._ap = ap + self._bm = bm + self._bq = bq + self._i = i + + an = list(an) + ap = list(ap) + bm = list(bm) + bq = list(bq) + ai = ap.pop(i) + 1 + + m = Poly(z, _x) + for a in an: + m *= Poly(1 - a + _x, _x) + for a in ap: + m *= Poly(a - 1 - _x, _x) + + B = Dummy('B') # - this is the shift operator `D_I` + D = Poly(ai - 1 - B, B) + n = Poly(1, B) + for b in bm: + n *= (-D + b) + for b in bq: + n *= (D - b) + + b0 = n.nth(0) + if b0 == 0: + raise ValueError('Cannot increment lower a index (cancels)') + + n = Poly(Poly(n.all_coeffs()[:-1], B).as_expr().subs( + B, ai - 1 - _x), _x) + + self._poly = Poly((m - n)/b0, _x) + + def __str__(self): + return '' % (self._i, + self._an, self._ap, self._bm, self._bq) + + +class ReduceOrder(Operator): + """ Reduce Order by cancelling an upper and a lower index. """ + + def __new__(cls, ai, bj): + """ For convenience if reduction is not possible, return None. """ + ai = sympify(ai) + bj = sympify(bj) + n = ai - bj + if not n.is_Integer or n < 0: + return None + if bj.is_integer and bj.is_nonpositive: + return None + + expr = Operator.__new__(cls) + + p = S.One + for k in range(n): + p *= (_x + bj + k)/(bj + k) + + expr._poly = Poly(p, _x) + expr._a = ai + expr._b = bj + + return expr + + @classmethod + def _meijer(cls, b, a, sign): + """ Cancel b + sign*s and a + sign*s + This is for meijer G functions. """ + b = sympify(b) + a = sympify(a) + n = b - a + if n.is_negative or not n.is_Integer: + return None + + expr = Operator.__new__(cls) + + p = S.One + for k in range(n): + p *= (sign*_x + a + k) + + expr._poly = Poly(p, _x) + if sign == -1: + expr._a = b + expr._b = a + else: + expr._b = Add(1, a - 1, evaluate=False) + expr._a = Add(1, b - 1, evaluate=False) + + return expr + + @classmethod + def meijer_minus(cls, b, a): + return cls._meijer(b, a, -1) + + @classmethod + def meijer_plus(cls, a, b): + return cls._meijer(1 - a, 1 - b, 1) + + def __str__(self): + return '' % \ + (self._a, self._b) + + +def _reduce_order(ap, bq, gen, key): + """ Order reduction algorithm used in Hypergeometric and Meijer G """ + ap = list(ap) + bq = list(bq) + + ap.sort(key=key) + bq.sort(key=key) + + nap = [] + # we will edit bq in place + operators = [] + for a in ap: + op = None + for i in range(len(bq)): + op = gen(a, bq[i]) + if op is not None: + bq.pop(i) + break + if op is None: + nap.append(a) + else: + operators.append(op) + + return nap, bq, operators + + +def reduce_order(func): + """ + Given the hypergeometric function ``func``, find a sequence of operators to + reduces order as much as possible. + + Explanation + =========== + + Return (newfunc, [operators]), where applying the operators to the + hypergeometric function newfunc yields func. + + Examples + ======== + + >>> from sympy.simplify.hyperexpand import reduce_order, Hyper_Function + >>> reduce_order(Hyper_Function((1, 2), (3, 4))) + (Hyper_Function((1, 2), (3, 4)), []) + >>> reduce_order(Hyper_Function((1,), (1,))) + (Hyper_Function((), ()), []) + >>> reduce_order(Hyper_Function((2, 4), (3, 3))) + (Hyper_Function((2,), (3,)), []) + """ + nap, nbq, operators = _reduce_order(func.ap, func.bq, ReduceOrder, default_sort_key) + + return Hyper_Function(Tuple(*nap), Tuple(*nbq)), operators + + +def reduce_order_meijer(func): + """ + Given the Meijer G function parameters, ``func``, find a sequence of + operators that reduces order as much as possible. + + Return newfunc, [operators]. + + Examples + ======== + + >>> from sympy.simplify.hyperexpand import (reduce_order_meijer, + ... G_Function) + >>> reduce_order_meijer(G_Function([3, 4], [5, 6], [3, 4], [1, 2]))[0] + G_Function((4, 3), (5, 6), (3, 4), (2, 1)) + >>> reduce_order_meijer(G_Function([3, 4], [5, 6], [3, 4], [1, 8]))[0] + G_Function((3,), (5, 6), (3, 4), (1,)) + >>> reduce_order_meijer(G_Function([3, 4], [5, 6], [7, 5], [1, 5]))[0] + G_Function((3,), (), (), (1,)) + >>> reduce_order_meijer(G_Function([3, 4], [5, 6], [7, 5], [5, 3]))[0] + G_Function((), (), (), ()) + """ + + nan, nbq, ops1 = _reduce_order(func.an, func.bq, ReduceOrder.meijer_plus, + lambda x: default_sort_key(-x)) + nbm, nap, ops2 = _reduce_order(func.bm, func.ap, ReduceOrder.meijer_minus, + default_sort_key) + + return G_Function(nan, nap, nbm, nbq), ops1 + ops2 + + +def make_derivative_operator(M, z): + """ Create a derivative operator, to be passed to Operator.apply. """ + def doit(C): + r = z*C.diff(z) + C*M + r = r.applyfunc(make_simp(z)) + return r + return doit + + +def apply_operators(obj, ops, op): + """ + Apply the list of operators ``ops`` to object ``obj``, substituting + ``op`` for the generator. + """ + res = obj + for o in reversed(ops): + res = o.apply(res, op) + return res + + +def devise_plan(target, origin, z): + """ + Devise a plan (consisting of shift and un-shift operators) to be applied + to the hypergeometric function ``target`` to yield ``origin``. + Returns a list of operators. + + Examples + ======== + + >>> from sympy.simplify.hyperexpand import devise_plan, Hyper_Function + >>> from sympy.abc import z + + Nothing to do: + + >>> devise_plan(Hyper_Function((1, 2), ()), Hyper_Function((1, 2), ()), z) + [] + >>> devise_plan(Hyper_Function((), (1, 2)), Hyper_Function((), (1, 2)), z) + [] + + Very simple plans: + + >>> devise_plan(Hyper_Function((2,), ()), Hyper_Function((1,), ()), z) + [] + >>> devise_plan(Hyper_Function((), (2,)), Hyper_Function((), (1,)), z) + [] + + Several buckets: + + >>> from sympy import S + >>> devise_plan(Hyper_Function((1, S.Half), ()), + ... Hyper_Function((2, S('3/2')), ()), z) #doctest: +NORMALIZE_WHITESPACE + [, + ] + + A slightly more complicated plan: + + >>> devise_plan(Hyper_Function((1, 3), ()), Hyper_Function((2, 2), ()), z) + [, ] + + Another more complicated plan: (note that the ap have to be shifted first!) + + >>> devise_plan(Hyper_Function((1, -1), (2,)), Hyper_Function((3, -2), (4,)), z) + [, , + , + , ] + """ + abuckets, bbuckets, nabuckets, nbbuckets = [sift(params, _mod1) for + params in (target.ap, target.bq, origin.ap, origin.bq)] + + if len(list(abuckets.keys())) != len(list(nabuckets.keys())) or \ + len(list(bbuckets.keys())) != len(list(nbbuckets.keys())): + raise ValueError('%s not reachable from %s' % (target, origin)) + + ops = [] + + def do_shifts(fro, to, inc, dec): + ops = [] + for i in range(len(fro)): + if to[i] - fro[i] > 0: + sh = inc + ch = 1 + else: + sh = dec + ch = -1 + + while to[i] != fro[i]: + ops += [sh(fro, i)] + fro[i] += ch + + return ops + + def do_shifts_a(nal, nbk, al, aother, bother): + """ Shift us from (nal, nbk) to (al, nbk). """ + return do_shifts(nal, al, lambda p, i: ShiftA(p[i]), + lambda p, i: UnShiftA(p + aother, nbk + bother, i, z)) + + def do_shifts_b(nal, nbk, bk, aother, bother): + """ Shift us from (nal, nbk) to (nal, bk). """ + return do_shifts(nbk, bk, + lambda p, i: UnShiftB(nal + aother, p + bother, i, z), + lambda p, i: ShiftB(p[i])) + + for r in sorted(list(abuckets.keys()) + list(bbuckets.keys()), key=default_sort_key): + al = () + nal = () + bk = () + nbk = () + if r in abuckets: + al = abuckets[r] + nal = nabuckets[r] + if r in bbuckets: + bk = bbuckets[r] + nbk = nbbuckets[r] + if len(al) != len(nal) or len(bk) != len(nbk): + raise ValueError('%s not reachable from %s' % (target, origin)) + + al, nal, bk, nbk = [sorted(w, key=default_sort_key) + for w in [al, nal, bk, nbk]] + + def others(dic, key): + l = [] + for k in dic: + if k != key: + l.extend(dic[k]) + return l + aother = others(nabuckets, r) + bother = others(nbbuckets, r) + + if len(al) == 0: + # there can be no complications, just shift the bs as we please + ops += do_shifts_b([], nbk, bk, aother, bother) + elif len(bk) == 0: + # there can be no complications, just shift the as as we please + ops += do_shifts_a(nal, [], al, aother, bother) + else: + namax = nal[-1] + amax = al[-1] + + if nbk[0] - namax <= 0 or bk[0] - amax <= 0: + raise ValueError('Non-suitable parameters.') + + if namax - amax > 0: + # we are going to shift down - first do the as, then the bs + ops += do_shifts_a(nal, nbk, al, aother, bother) + ops += do_shifts_b(al, nbk, bk, aother, bother) + else: + # we are going to shift up - first do the bs, then the as + ops += do_shifts_b(nal, nbk, bk, aother, bother) + ops += do_shifts_a(nal, bk, al, aother, bother) + + nabuckets[r] = al + nbbuckets[r] = bk + + ops.reverse() + return ops + + +def try_shifted_sum(func, z): + """ Try to recognise a hypergeometric sum that starts from k > 0. """ + abuckets, bbuckets = sift(func.ap, _mod1), sift(func.bq, _mod1) + if len(abuckets[S.Zero]) != 1: + return None + r = abuckets[S.Zero][0] + if r <= 0: + return None + if S.Zero not in bbuckets: + return None + l = list(bbuckets[S.Zero]) + l.sort() + k = l[0] + if k <= 0: + return None + + nap = list(func.ap) + nap.remove(r) + nbq = list(func.bq) + nbq.remove(k) + k -= 1 + nap = [x - k for x in nap] + nbq = [x - k for x in nbq] + + ops = [] + for n in range(r - 1): + ops.append(ShiftA(n + 1)) + ops.reverse() + + fac = factorial(k)/z**k + fac *= Mul(*[rf(b, k) for b in nbq]) + fac /= Mul(*[rf(a, k) for a in nap]) + + ops += [MultOperator(fac)] + + p = 0 + for n in range(k): + m = z**n/factorial(n) + m *= Mul(*[rf(a, n) for a in nap]) + m /= Mul(*[rf(b, n) for b in nbq]) + p += m + + return Hyper_Function(nap, nbq), ops, -p + + +def try_polynomial(func, z): + """ Recognise polynomial cases. Returns None if not such a case. + Requires order to be fully reduced. """ + abuckets, bbuckets = sift(func.ap, _mod1), sift(func.bq, _mod1) + a0 = abuckets[S.Zero] + b0 = bbuckets[S.Zero] + a0.sort() + b0.sort() + al0 = [x for x in a0 if x <= 0] + bl0 = [x for x in b0 if x <= 0] + + if bl0 and all(a < bl0[-1] for a in al0): + return oo + if not al0: + return None + + a = al0[-1] + fac = 1 + res = S.One + for n in Tuple(*list(range(-a))): + fac *= z + fac /= n + 1 + fac *= Mul(*[a + n for a in func.ap]) + fac /= Mul(*[b + n for b in func.bq]) + res += fac + return res + + +def try_lerchphi(func): + """ + Try to find an expression for Hyper_Function ``func`` in terms of Lerch + Transcendents. + + Return None if no such expression can be found. + """ + # This is actually quite simple, and is described in Roach's paper, + # section 18. + # We don't need to implement the reduction to polylog here, this + # is handled by expand_func. + + # First we need to figure out if the summation coefficient is a rational + # function of the summation index, and construct that rational function. + abuckets, bbuckets = sift(func.ap, _mod1), sift(func.bq, _mod1) + + paired = {} + for key, value in abuckets.items(): + if key != 0 and key not in bbuckets: + return None + bvalue = bbuckets[key] + paired[key] = (list(value), list(bvalue)) + bbuckets.pop(key, None) + if bbuckets != {}: + return None + if S.Zero not in abuckets: + return None + aints, bints = paired[S.Zero] + # Account for the additional n! in denominator + paired[S.Zero] = (aints, bints + [1]) + + t = Dummy('t') + numer = S.One + denom = S.One + for key, (avalue, bvalue) in paired.items(): + if len(avalue) != len(bvalue): + return None + # Note that since order has been reduced fully, all the b are + # bigger than all the a they differ from by an integer. In particular + # if there are any negative b left, this function is not well-defined. + for a, b in zip(avalue, bvalue): + if (a - b).is_positive: + k = a - b + numer *= rf(b + t, k) + denom *= rf(b, k) + else: + k = b - a + numer *= rf(a, k) + denom *= rf(a + t, k) + + # Now do a partial fraction decomposition. + # We assemble two structures: a list monomials of pairs (a, b) representing + # a*t**b (b a non-negative integer), and a dict terms, where + # terms[a] = [(b, c)] means that there is a term b/(t-a)**c. + part = apart(numer/denom, t) + args = Add.make_args(part) + monomials = [] + terms = {} + for arg in args: + numer, denom = arg.as_numer_denom() + if not denom.has(t): + p = Poly(numer, t) + if not p.is_monomial: + raise TypeError("p should be monomial") + ((b, ), a) = p.LT() + monomials += [(a/denom, b)] + continue + if numer.has(t): + raise NotImplementedError('Need partial fraction decomposition' + ' with linear denominators') + indep, [dep] = denom.as_coeff_mul(t) + n = 1 + if dep.is_Pow: + n = dep.exp + dep = dep.base + if dep == t: + a = 0 + elif dep.is_Add: + a, tmp = dep.as_independent(t) + b = 1 + if tmp != t: + b, _ = tmp.as_independent(t) + if dep != b*t + a: + raise NotImplementedError('unrecognised form %s' % dep) + a /= b + indep *= b**n + else: + raise NotImplementedError('unrecognised form of partial fraction') + terms.setdefault(a, []).append((numer/indep, n)) + + # Now that we have this information, assemble our formula. All the + # monomials yield rational functions and go into one basis element. + # The terms[a] are related by differentiation. If the largest exponent is + # n, we need lerchphi(z, k, a) for k = 1, 2, ..., n. + # deriv maps a basis to its derivative, expressed as a C(z)-linear + # combination of other basis elements. + deriv = {} + coeffs = {} + z = Dummy('z') + monomials.sort(key=lambda x: x[1]) + mon = {0: 1/(1 - z)} + if monomials: + for k in range(monomials[-1][1]): + mon[k + 1] = z*mon[k].diff(z) + for a, n in monomials: + coeffs.setdefault(S.One, []).append(a*mon[n]) + for a, l in terms.items(): + for c, k in l: + coeffs.setdefault(lerchphi(z, k, a), []).append(c) + l.sort(key=lambda x: x[1]) + for k in range(2, l[-1][1] + 1): + deriv[lerchphi(z, k, a)] = [(-a, lerchphi(z, k, a)), + (1, lerchphi(z, k - 1, a))] + deriv[lerchphi(z, 1, a)] = [(-a, lerchphi(z, 1, a)), + (1/(1 - z), S.One)] + trans = {} + for n, b in enumerate([S.One] + list(deriv.keys())): + trans[b] = n + basis = [expand_func(b) for (b, _) in sorted(trans.items(), + key=lambda x:x[1])] + B = Matrix(basis) + C = Matrix([[0]*len(B)]) + for b, c in coeffs.items(): + C[trans[b]] = Add(*c) + M = zeros(len(B)) + for b, l in deriv.items(): + for c, b2 in l: + M[trans[b], trans[b2]] = c + return Formula(func, z, None, [], B, C, M) + + +def build_hypergeometric_formula(func): + """ + Create a formula object representing the hypergeometric function ``func``. + + """ + # We know that no `ap` are negative integers, otherwise "detect poly" + # would have kicked in. However, `ap` could be empty. In this case we can + # use a different basis. + # I'm not aware of a basis that works in all cases. + z = Dummy('z') + if func.ap: + afactors = [_x + a for a in func.ap] + bfactors = [_x + b - 1 for b in func.bq] + expr = _x*Mul(*bfactors) - z*Mul(*afactors) + poly = Poly(expr, _x) + n = poly.degree() + basis = [] + M = zeros(n) + for k in range(n): + a = func.ap[0] + k + basis += [hyper([a] + list(func.ap[1:]), func.bq, z)] + if k < n - 1: + M[k, k] = -a + M[k, k + 1] = a + B = Matrix(basis) + C = Matrix([[1] + [0]*(n - 1)]) + derivs = [eye(n)] + for k in range(n): + derivs.append(M*derivs[k]) + l = poly.all_coeffs() + l.reverse() + res = [0]*n + for k, c in enumerate(l): + for r, d in enumerate(C*derivs[k]): + res[r] += c*d + for k, c in enumerate(res): + M[n - 1, k] = -c/derivs[n - 1][0, n - 1]/poly.all_coeffs()[0] + return Formula(func, z, None, [], B, C, M) + else: + # Since there are no `ap`, none of the `bq` can be non-positive + # integers. + basis = [] + bq = list(func.bq[:]) + for i in range(len(bq)): + basis += [hyper([], bq, z)] + bq[i] += 1 + basis += [hyper([], bq, z)] + B = Matrix(basis) + n = len(B) + C = Matrix([[1] + [0]*(n - 1)]) + M = zeros(n) + M[0, n - 1] = z/Mul(*func.bq) + for k in range(1, n): + M[k, k - 1] = func.bq[k - 1] + M[k, k] = -func.bq[k - 1] + return Formula(func, z, None, [], B, C, M) + + +def hyperexpand_special(ap, bq, z): + """ + Try to find a closed-form expression for hyper(ap, bq, z), where ``z`` + is supposed to be a "special" value, e.g. 1. + + This function tries various of the classical summation formulae + (Gauss, Saalschuetz, etc). + """ + # This code is very ad-hoc. There are many clever algorithms + # (notably Zeilberger's) related to this problem. + # For now we just want a few simple cases to work. + p, q = len(ap), len(bq) + z_ = z + z = unpolarify(z) + if z == 0: + return S.One + from sympy.simplify.simplify import simplify + if p == 2 and q == 1: + # 2F1 + a, b, c = ap + bq + if z == 1: + # Gauss + return gamma(c - a - b)*gamma(c)/gamma(c - a)/gamma(c - b) + if z == -1 and simplify(b - a + c) == 1: + b, a = a, b + if z == -1 and simplify(a - b + c) == 1: + # Kummer + if b.is_integer and b.is_negative: + return 2*cos(pi*b/2)*gamma(-b)*gamma(b - a + 1) \ + /gamma(-b/2)/gamma(b/2 - a + 1) + else: + return gamma(b/2 + 1)*gamma(b - a + 1) \ + /gamma(b + 1)/gamma(b/2 - a + 1) + # TODO tons of more formulae + # investigate what algorithms exist + return hyper(ap, bq, z_) + +_collection = None + + +def _hyperexpand(func, z, ops0=[], z0=Dummy('z0'), premult=1, prem=0, + rewrite='default'): + """ + Try to find an expression for the hypergeometric function ``func``. + + Explanation + =========== + + The result is expressed in terms of a dummy variable ``z0``. Then it + is multiplied by ``premult``. Then ``ops0`` is applied. + ``premult`` must be a*z**prem for some a independent of ``z``. + """ + + if z.is_zero: + return S.One + + from sympy.simplify.simplify import simplify + + z = polarify(z, subs=False) + if rewrite == 'default': + rewrite = 'nonrepsmall' + + def carryout_plan(f, ops): + C = apply_operators(f.C.subs(f.z, z0), ops, + make_derivative_operator(f.M.subs(f.z, z0), z0)) + C = apply_operators(C, ops0, + make_derivative_operator(f.M.subs(f.z, z0) + + prem*eye(f.M.shape[0]), z0)) + + if premult == 1: + C = C.applyfunc(make_simp(z0)) + r = reduce(lambda s,m: s+m[0]*m[1], zip(C, f.B.subs(f.z, z0)), S.Zero)*premult + res = r.subs(z0, z) + if rewrite: + res = res.rewrite(rewrite) + return res + + # TODO + # The following would be possible: + # *) PFD Duplication (see Kelly Roach's paper) + # *) In a similar spirit, try_lerchphi() can be generalised considerably. + + global _collection + if _collection is None: + _collection = FormulaCollection() + + debug('Trying to expand hypergeometric function ', func) + + # First reduce order as much as possible. + func, ops = reduce_order(func) + if ops: + debug(' Reduced order to ', func) + else: + debug(' Could not reduce order.') + + # Now try polynomial cases + res = try_polynomial(func, z0) + if res is not None: + debug(' Recognised polynomial.') + p = apply_operators(res, ops, lambda f: z0*f.diff(z0)) + p = apply_operators(p*premult, ops0, lambda f: z0*f.diff(z0)) + return unpolarify(simplify(p).subs(z0, z)) + + # Try to recognise a shifted sum. + p = S.Zero + res = try_shifted_sum(func, z0) + if res is not None: + func, nops, p = res + debug(' Recognised shifted sum, reduced order to ', func) + ops += nops + + # apply the plan for poly + p = apply_operators(p, ops, lambda f: z0*f.diff(z0)) + p = apply_operators(p*premult, ops0, lambda f: z0*f.diff(z0)) + p = simplify(p).subs(z0, z) + + # Try special expansions early. + if unpolarify(z) in [1, -1] and (len(func.ap), len(func.bq)) == (2, 1): + f = build_hypergeometric_formula(func) + r = carryout_plan(f, ops).replace(hyper, hyperexpand_special) + if not r.has(hyper): + return r + p + + # Try to find a formula in our collection + formula = _collection.lookup_origin(func) + + # Now try a lerch phi formula + if formula is None: + formula = try_lerchphi(func) + + if formula is None: + debug(' Could not find an origin. ', + 'Will return answer in terms of ' + 'simpler hypergeometric functions.') + formula = build_hypergeometric_formula(func) + + debug(' Found an origin: ', formula.closed_form, ' ', formula.func) + + # We need to find the operators that convert formula into func. + ops += devise_plan(func, formula.func, z0) + + # Now carry out the plan. + r = carryout_plan(formula, ops) + p + + return powdenest(r, polar=True).replace(hyper, hyperexpand_special) + + +def devise_plan_meijer(fro, to, z): + """ + Find operators to convert G-function ``fro`` into G-function ``to``. + + Explanation + =========== + + It is assumed that ``fro`` and ``to`` have the same signatures, and that in fact + any corresponding pair of parameters differs by integers, and a direct path + is possible. I.e. if there are parameters a1 b1 c1 and a2 b2 c2 it is + assumed that a1 can be shifted to a2, etc. The only thing this routine + determines is the order of shifts to apply, nothing clever will be tried. + It is also assumed that ``fro`` is suitable. + + Examples + ======== + + >>> from sympy.simplify.hyperexpand import (devise_plan_meijer, + ... G_Function) + >>> from sympy.abc import z + + Empty plan: + + >>> devise_plan_meijer(G_Function([1], [2], [3], [4]), + ... G_Function([1], [2], [3], [4]), z) + [] + + Very simple plans: + + >>> devise_plan_meijer(G_Function([0], [], [], []), + ... G_Function([1], [], [], []), z) + [] + >>> devise_plan_meijer(G_Function([0], [], [], []), + ... G_Function([-1], [], [], []), z) + [] + >>> devise_plan_meijer(G_Function([], [1], [], []), + ... G_Function([], [2], [], []), z) + [] + + Slightly more complicated plans: + + >>> devise_plan_meijer(G_Function([0], [], [], []), + ... G_Function([2], [], [], []), z) + [, + ] + >>> devise_plan_meijer(G_Function([0], [], [0], []), + ... G_Function([-1], [], [1], []), z) + [, ] + + Order matters: + + >>> devise_plan_meijer(G_Function([0], [], [0], []), + ... G_Function([1], [], [1], []), z) + [, ] + """ + # TODO for now, we use the following simple heuristic: inverse-shift + # when possible, shift otherwise. Give up if we cannot make progress. + + def try_shift(f, t, shifter, diff, counter): + """ Try to apply ``shifter`` in order to bring some element in ``f`` + nearer to its counterpart in ``to``. ``diff`` is +/- 1 and + determines the effect of ``shifter``. Counter is a list of elements + blocking the shift. + + Return an operator if change was possible, else None. + """ + for idx, (a, b) in enumerate(zip(f, t)): + if ( + (a - b).is_integer and (b - a)/diff > 0 and + all(a != x for x in counter)): + sh = shifter(idx) + f[idx] += diff + return sh + fan = list(fro.an) + fap = list(fro.ap) + fbm = list(fro.bm) + fbq = list(fro.bq) + ops = [] + change = True + while change: + change = False + op = try_shift(fan, to.an, + lambda i: MeijerUnShiftB(fan, fap, fbm, fbq, i, z), + 1, fbm + fbq) + if op is not None: + ops += [op] + change = True + continue + op = try_shift(fap, to.ap, + lambda i: MeijerUnShiftD(fan, fap, fbm, fbq, i, z), + 1, fbm + fbq) + if op is not None: + ops += [op] + change = True + continue + op = try_shift(fbm, to.bm, + lambda i: MeijerUnShiftA(fan, fap, fbm, fbq, i, z), + -1, fan + fap) + if op is not None: + ops += [op] + change = True + continue + op = try_shift(fbq, to.bq, + lambda i: MeijerUnShiftC(fan, fap, fbm, fbq, i, z), + -1, fan + fap) + if op is not None: + ops += [op] + change = True + continue + op = try_shift(fan, to.an, lambda i: MeijerShiftB(fan[i]), -1, []) + if op is not None: + ops += [op] + change = True + continue + op = try_shift(fap, to.ap, lambda i: MeijerShiftD(fap[i]), -1, []) + if op is not None: + ops += [op] + change = True + continue + op = try_shift(fbm, to.bm, lambda i: MeijerShiftA(fbm[i]), 1, []) + if op is not None: + ops += [op] + change = True + continue + op = try_shift(fbq, to.bq, lambda i: MeijerShiftC(fbq[i]), 1, []) + if op is not None: + ops += [op] + change = True + continue + if fan != list(to.an) or fap != list(to.ap) or fbm != list(to.bm) or \ + fbq != list(to.bq): + raise NotImplementedError('Could not devise plan.') + ops.reverse() + return ops + +_meijercollection = None + + +def _meijergexpand(func, z0, allow_hyper=False, rewrite='default', + place=None): + """ + Try to find an expression for the Meijer G function specified + by the G_Function ``func``. If ``allow_hyper`` is True, then returning + an expression in terms of hypergeometric functions is allowed. + + Currently this just does Slater's theorem. + If expansions exist both at zero and at infinity, ``place`` + can be set to ``0`` or ``zoo`` for the preferred choice. + """ + global _meijercollection + if _meijercollection is None: + _meijercollection = MeijerFormulaCollection() + if rewrite == 'default': + rewrite = None + + func0 = func + debug('Try to expand Meijer G function corresponding to ', func) + + # We will play games with analytic continuation - rather use a fresh symbol + z = Dummy('z') + + func, ops = reduce_order_meijer(func) + if ops: + debug(' Reduced order to ', func) + else: + debug(' Could not reduce order.') + + # Try to find a direct formula + f = _meijercollection.lookup_origin(func) + if f is not None: + debug(' Found a Meijer G formula: ', f.func) + ops += devise_plan_meijer(f.func, func, z) + + # Now carry out the plan. + C = apply_operators(f.C.subs(f.z, z), ops, + make_derivative_operator(f.M.subs(f.z, z), z)) + + C = C.applyfunc(make_simp(z)) + r = C*f.B.subs(f.z, z) + r = r[0].subs(z, z0) + return powdenest(r, polar=True) + + debug(" Could not find a direct formula. Trying Slater's theorem.") + + # TODO the following would be possible: + # *) Paired Index Theorems + # *) PFD Duplication + # (See Kelly Roach's paper for details on either.) + # + # TODO Also, we tend to create combinations of gamma functions that can be + # simplified. + + def can_do(pbm, pap): + """ Test if slater applies. """ + for i in pbm: + if len(pbm[i]) > 1: + l = 0 + if i in pap: + l = len(pap[i]) + if l + 1 < len(pbm[i]): + return False + return True + + def do_slater(an, bm, ap, bq, z, zfinal): + # zfinal is the value that will eventually be substituted for z. + # We pass it to _hyperexpand to improve performance. + func = G_Function(an, bm, ap, bq) + _, pbm, pap, _ = func.compute_buckets() + if not can_do(pbm, pap): + return S.Zero, False + + cond = len(an) + len(ap) < len(bm) + len(bq) + if len(an) + len(ap) == len(bm) + len(bq): + cond = abs(z) < 1 + if cond is False: + return S.Zero, False + + res = S.Zero + for m in pbm: + if len(pbm[m]) == 1: + bh = pbm[m][0] + fac = 1 + bo = list(bm) + bo.remove(bh) + for bj in bo: + fac *= gamma(bj - bh) + for aj in an: + fac *= gamma(1 + bh - aj) + for bj in bq: + fac /= gamma(1 + bh - bj) + for aj in ap: + fac /= gamma(aj - bh) + nap = [1 + bh - a for a in list(an) + list(ap)] + nbq = [1 + bh - b for b in list(bo) + list(bq)] + + k = polar_lift(S.NegativeOne**(len(ap) - len(bm))) + harg = k*zfinal + # NOTE even though k "is" +-1, this has to be t/k instead of + # t*k ... we are using polar numbers for consistency! + premult = (t/k)**bh + hyp = _hyperexpand(Hyper_Function(nap, nbq), harg, ops, + t, premult, bh, rewrite=None) + res += fac * hyp + else: + b_ = pbm[m][0] + ki = [bi - b_ for bi in pbm[m][1:]] + u = len(ki) + li = [ai - b_ for ai in pap[m][:u + 1]] + bo = list(bm) + for b in pbm[m]: + bo.remove(b) + ao = list(ap) + for a in pap[m][:u]: + ao.remove(a) + lu = li[-1] + di = [l - k for (l, k) in zip(li, ki)] + + # We first work out the integrand: + s = Dummy('s') + integrand = z**s + for b in bm: + if not Mod(b, 1) and b.is_Number: + b = int(round(b)) + integrand *= gamma(b - s) + for a in an: + integrand *= gamma(1 - a + s) + for b in bq: + integrand /= gamma(1 - b + s) + for a in ap: + integrand /= gamma(a - s) + + # Now sum the finitely many residues: + # XXX This speeds up some cases - is it a good idea? + integrand = expand_func(integrand) + for r in range(int(round(lu))): + resid = residue(integrand, s, b_ + r) + resid = apply_operators(resid, ops, lambda f: z*f.diff(z)) + res -= resid + + # Now the hypergeometric term. + au = b_ + lu + k = polar_lift(S.NegativeOne**(len(ao) + len(bo) + 1)) + harg = k*zfinal + premult = (t/k)**au + nap = [1 + au - a for a in list(an) + list(ap)] + [1] + nbq = [1 + au - b for b in list(bm) + list(bq)] + + hyp = _hyperexpand(Hyper_Function(nap, nbq), harg, ops, + t, premult, au, rewrite=None) + + C = S.NegativeOne**(lu)/factorial(lu) + for i in range(u): + C *= S.NegativeOne**di[i]/rf(lu - li[i] + 1, di[i]) + for a in an: + C *= gamma(1 - a + au) + for b in bo: + C *= gamma(b - au) + for a in ao: + C /= gamma(a - au) + for b in bq: + C /= gamma(1 - b + au) + + res += C*hyp + + return res, cond + + t = Dummy('t') + slater1, cond1 = do_slater(func.an, func.bm, func.ap, func.bq, z, z0) + + def tr(l): + return [1 - x for x in l] + + for op in ops: + op._poly = Poly(op._poly.subs({z: 1/t, _x: -_x}), _x) + slater2, cond2 = do_slater(tr(func.bm), tr(func.an), tr(func.bq), tr(func.ap), + t, 1/z0) + + slater1 = powdenest(slater1.subs(z, z0), polar=True) + slater2 = powdenest(slater2.subs(t, 1/z0), polar=True) + if not isinstance(cond2, bool): + cond2 = cond2.subs(t, 1/z) + + m = func(z) + if m.delta > 0 or \ + (m.delta == 0 and len(m.ap) == len(m.bq) and + (re(m.nu) < -1) is not False and polar_lift(z0) == polar_lift(1)): + # The condition delta > 0 means that the convergence region is + # connected. Any expression we find can be continued analytically + # to the entire convergence region. + # The conditions delta==0, p==q, re(nu) < -1 imply that G is continuous + # on the positive reals, so the values at z=1 agree. + if cond1 is not False: + cond1 = True + if cond2 is not False: + cond2 = True + + if cond1 is True: + slater1 = slater1.rewrite(rewrite or 'nonrep') + else: + slater1 = slater1.rewrite(rewrite or 'nonrepsmall') + if cond2 is True: + slater2 = slater2.rewrite(rewrite or 'nonrep') + else: + slater2 = slater2.rewrite(rewrite or 'nonrepsmall') + + if cond1 is not False and cond2 is not False: + # If one condition is False, there is no choice. + if place == 0: + cond2 = False + if place == zoo: + cond1 = False + + if not isinstance(cond1, bool): + cond1 = cond1.subs(z, z0) + if not isinstance(cond2, bool): + cond2 = cond2.subs(z, z0) + + def weight(expr, cond): + if cond is True: + c0 = 0 + elif cond is False: + c0 = 1 + else: + c0 = 2 + if expr.has(oo, zoo, -oo, nan): + # XXX this actually should not happen, but consider + # S('meijerg(((0, -1/2, 0, -1/2, 1/2), ()), ((0,), + # (-1/2, -1/2, -1/2, -1)), exp_polar(I*pi))/4') + c0 = 3 + return (c0, expr.count(hyper), expr.count_ops()) + + w1 = weight(slater1, cond1) + w2 = weight(slater2, cond2) + if min(w1, w2) <= (0, 1, oo): + if w1 < w2: + return slater1 + else: + return slater2 + if max(w1[0], w2[0]) <= 1 and max(w1[1], w2[1]) <= 1: + return Piecewise((slater1, cond1), (slater2, cond2), (func0(z0), True)) + + # We couldn't find an expression without hypergeometric functions. + # TODO it would be helpful to give conditions under which the integral + # is known to diverge. + r = Piecewise((slater1, cond1), (slater2, cond2), (func0(z0), True)) + if r.has(hyper) and not allow_hyper: + debug(' Could express using hypergeometric functions, ' + 'but not allowed.') + if not r.has(hyper) or allow_hyper: + return r + + return func0(z0) + + +def hyperexpand(f, allow_hyper=False, rewrite='default', place=None): + """ + Expand hypergeometric functions. If allow_hyper is True, allow partial + simplification (that is a result different from input, + but still containing hypergeometric functions). + + If a G-function has expansions both at zero and at infinity, + ``place`` can be set to ``0`` or ``zoo`` to indicate the + preferred choice. + + Examples + ======== + + >>> from sympy.simplify.hyperexpand import hyperexpand + >>> from sympy.functions import hyper + >>> from sympy.abc import z + >>> hyperexpand(hyper([], [], z)) + exp(z) + + Non-hyperegeometric parts of the expression and hypergeometric expressions + that are not recognised are left unchanged: + + >>> hyperexpand(1 + hyper([1, 1, 1], [], z)) + hyper((1, 1, 1), (), z) + 1 + """ + f = sympify(f) + + def do_replace(ap, bq, z): + r = _hyperexpand(Hyper_Function(ap, bq), z, rewrite=rewrite) + if r is None: + return hyper(ap, bq, z) + else: + return r + + def do_meijer(ap, bq, z): + r = _meijergexpand(G_Function(ap[0], ap[1], bq[0], bq[1]), z, + allow_hyper, rewrite=rewrite, place=place) + if not r.has(nan, zoo, oo, -oo): + return r + return f.replace(hyper, do_replace).replace(meijerg, do_meijer) diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/hyperexpand_doc.py b/.venv/lib/python3.12/site-packages/sympy/simplify/hyperexpand_doc.py new file mode 100644 index 0000000000000000000000000000000000000000..a18377f3aede5214036fbf628825536611001584 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/simplify/hyperexpand_doc.py @@ -0,0 +1,18 @@ +""" This module cooks up a docstring when imported. Its only purpose is to + be displayed in the sphinx documentation. """ + +from sympy.core.relational import Eq +from sympy.functions.special.hyper import hyper +from sympy.printing.latex import latex +from sympy.simplify.hyperexpand import FormulaCollection + +c = FormulaCollection() + +doc = "" + +for f in c.formulae: + obj = Eq(hyper(f.func.ap, f.func.bq, f.z), + f.closed_form.rewrite('nonrepsmall')) + doc += ".. math::\n %s\n" % latex(obj) + +__doc__ = doc diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/powsimp.py b/.venv/lib/python3.12/site-packages/sympy/simplify/powsimp.py new file mode 100644 index 0000000000000000000000000000000000000000..f72dfeb072e0d0d4737ace310eda5c2a3a082c16 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/simplify/powsimp.py @@ -0,0 +1,718 @@ +from collections import defaultdict +from functools import reduce +from math import prod + +from sympy.core.function import expand_log, count_ops, _coeff_isneg +from sympy.core import sympify, Basic, Dummy, S, Add, Mul, Pow, expand_mul, factor_terms +from sympy.core.sorting import ordered, default_sort_key +from sympy.core.numbers import Integer, Rational, equal_valued +from sympy.core.mul import _keep_coeff +from sympy.core.rules import Transform +from sympy.functions import exp_polar, exp, log, root, polarify, unpolarify +from sympy.matrices.expressions.matexpr import MatrixSymbol +from sympy.polys import lcm, gcd +from sympy.ntheory.factor_ import multiplicity + + + +def powsimp(expr, deep=False, combine='all', force=False, measure=count_ops): + """ + Reduce expression by combining powers with similar bases and exponents. + + Explanation + =========== + + If ``deep`` is ``True`` then powsimp() will also simplify arguments of + functions. By default ``deep`` is set to ``False``. + + If ``force`` is ``True`` then bases will be combined without checking for + assumptions, e.g. sqrt(x)*sqrt(y) -> sqrt(x*y) which is not true + if x and y are both negative. + + You can make powsimp() only combine bases or only combine exponents by + changing combine='base' or combine='exp'. By default, combine='all', + which does both. combine='base' will only combine:: + + a a a 2x x + x * y => (x*y) as well as things like 2 => 4 + + and combine='exp' will only combine + :: + + a b (a + b) + x * x => x + + combine='exp' will strictly only combine exponents in the way that used + to be automatic. Also use deep=True if you need the old behavior. + + When combine='all', 'exp' is evaluated first. Consider the first + example below for when there could be an ambiguity relating to this. + This is done so things like the second example can be completely + combined. If you want 'base' combined first, do something like + powsimp(powsimp(expr, combine='base'), combine='exp'). + + Examples + ======== + + >>> from sympy import powsimp, exp, log, symbols + >>> from sympy.abc import x, y, z, n + >>> powsimp(x**y*x**z*y**z, combine='all') + x**(y + z)*y**z + >>> powsimp(x**y*x**z*y**z, combine='exp') + x**(y + z)*y**z + >>> powsimp(x**y*x**z*y**z, combine='base', force=True) + x**y*(x*y)**z + + >>> powsimp(x**z*x**y*n**z*n**y, combine='all', force=True) + (n*x)**(y + z) + >>> powsimp(x**z*x**y*n**z*n**y, combine='exp') + n**(y + z)*x**(y + z) + >>> powsimp(x**z*x**y*n**z*n**y, combine='base', force=True) + (n*x)**y*(n*x)**z + + >>> x, y = symbols('x y', positive=True) + >>> powsimp(log(exp(x)*exp(y))) + log(exp(x)*exp(y)) + >>> powsimp(log(exp(x)*exp(y)), deep=True) + x + y + + Radicals with Mul bases will be combined if combine='exp' + + >>> from sympy import sqrt + >>> x, y = symbols('x y') + + Two radicals are automatically joined through Mul: + + >>> a=sqrt(x*sqrt(y)) + >>> a*a**3 == a**4 + True + + But if an integer power of that radical has been + autoexpanded then Mul does not join the resulting factors: + + >>> a**4 # auto expands to a Mul, no longer a Pow + x**2*y + >>> _*a # so Mul doesn't combine them + x**2*y*sqrt(x*sqrt(y)) + >>> powsimp(_) # but powsimp will + (x*sqrt(y))**(5/2) + >>> powsimp(x*y*a) # but won't when doing so would violate assumptions + x*y*sqrt(x*sqrt(y)) + + """ + def recurse(arg, **kwargs): + _deep = kwargs.get('deep', deep) + _combine = kwargs.get('combine', combine) + _force = kwargs.get('force', force) + _measure = kwargs.get('measure', measure) + return powsimp(arg, _deep, _combine, _force, _measure) + + expr = sympify(expr) + + if (not isinstance(expr, Basic) or isinstance(expr, MatrixSymbol) or ( + expr.is_Atom or expr in (exp_polar(0), exp_polar(1)))): + return expr + + if deep or expr.is_Add or expr.is_Mul and _y not in expr.args: + expr = expr.func(*[recurse(w) for w in expr.args]) + + if expr.is_Pow: + return recurse(expr*_y, deep=False)/_y + + if not expr.is_Mul: + return expr + + # handle the Mul + if combine in ('exp', 'all'): + # Collect base/exp data, while maintaining order in the + # non-commutative parts of the product + c_powers = defaultdict(list) + nc_part = [] + newexpr = [] + coeff = S.One + for term in expr.args: + if term.is_Rational: + coeff *= term + continue + if term.is_Pow: + term = _denest_pow(term) + if term.is_commutative: + b, e = term.as_base_exp() + if deep: + b, e = [recurse(i) for i in [b, e]] + if b.is_Pow or isinstance(b, exp): + # don't let smthg like sqrt(x**a) split into x**a, 1/2 + # or else it will be joined as x**(a/2) later + b, e = b**e, S.One + c_powers[b].append(e) + else: + # This is the logic that combines exponents for equal, + # but non-commutative bases: A**x*A**y == A**(x+y). + if nc_part: + b1, e1 = nc_part[-1].as_base_exp() + b2, e2 = term.as_base_exp() + if (b1 == b2 and + e1.is_commutative and e2.is_commutative): + nc_part[-1] = Pow(b1, Add(e1, e2)) + continue + nc_part.append(term) + + # add up exponents of common bases + for b, e in ordered(iter(c_powers.items())): + # allow 2**x/4 -> 2**(x - 2); don't do this when b and e are + # Numbers since autoevaluation will undo it, e.g. + # 2**(1/3)/4 -> 2**(1/3 - 2) -> 2**(1/3)/4 + if (b and b.is_Rational and not all(ei.is_Number for ei in e) and \ + coeff is not S.One and + b not in (S.One, S.NegativeOne)): + m = multiplicity(abs(b), abs(coeff)) + if m: + e.append(m) + coeff /= b**m + c_powers[b] = Add(*e) + if coeff is not S.One: + if coeff in c_powers: + c_powers[coeff] += S.One + else: + c_powers[coeff] = S.One + + # convert to plain dictionary + c_powers = dict(c_powers) + + # check for base and inverted base pairs + be = list(c_powers.items()) + skip = set() # skip if we already saw them + for b, e in be: + if b in skip: + continue + bpos = b.is_positive or b.is_polar + if bpos: + binv = 1/b + #Special case for float 1 + if b.is_Float and equal_valued(b, 1): + c_powers[b] = S.One + continue + if b != binv and binv in c_powers: + if b.as_numer_denom()[0] is S.One: + c_powers.pop(b) + c_powers[binv] -= e + else: + skip.add(binv) + e = c_powers.pop(binv) + c_powers[b] -= e + + # check for base and negated base pairs + be = list(c_powers.items()) + _n = S.NegativeOne + for b, e in be: + if (b.is_Symbol or b.is_Add) and -b in c_powers and b in c_powers: + if (b.is_positive is not None or e.is_integer): + if e.is_integer or b.is_negative: + c_powers[-b] += c_powers.pop(b) + else: # (-b).is_positive so use its e + e = c_powers.pop(-b) + c_powers[b] += e + if _n in c_powers: + c_powers[_n] += e + else: + c_powers[_n] = e + + # filter c_powers and convert to a list + c_powers = [(b, e) for b, e in c_powers.items() if e] + + # ============================================================== + # check for Mul bases of Rational powers that can be combined with + # separated bases, e.g. x*sqrt(x*y)*sqrt(x*sqrt(x*y)) -> + # (x*sqrt(x*y))**(3/2) + # ---------------- helper functions + + def ratq(x): + '''Return Rational part of x's exponent as it appears in the bkey. + ''' + return bkey(x)[0][1] + + def bkey(b, e=None): + '''Return (b**s, c.q), c.p where e -> c*s. If e is not given then + it will be taken by using as_base_exp() on the input b. + e.g. + x**3/2 -> (x, 2), 3 + x**y -> (x**y, 1), 1 + x**(2*y/3) -> (x**y, 3), 2 + exp(x/2) -> (exp(a), 2), 1 + + ''' + if e is not None: # coming from c_powers or from below + if e.is_Integer: + return (b, S.One), e + elif e.is_Rational: + return (b, Integer(e.q)), Integer(e.p) + else: + c, m = e.as_coeff_Mul(rational=True) + if c is not S.One: + if m.is_integer: + return (b, Integer(c.q)), m*Integer(c.p) + return (b**m, Integer(c.q)), Integer(c.p) + else: + return (b**e, S.One), S.One + else: + return bkey(*b.as_base_exp()) + + def update(b): + '''Decide what to do with base, b. If its exponent is now an + integer multiple of the Rational denominator, then remove it + and put the factors of its base in the common_b dictionary or + update the existing bases if necessary. If it has been zeroed + out, simply remove the base. + ''' + newe, r = divmod(common_b[b], b[1]) + if not r: + common_b.pop(b) + if newe: + for m in Mul.make_args(b[0]**newe): + b, e = bkey(m) + if b not in common_b: + common_b[b] = 0 + common_b[b] += e + if b[1] != 1: + bases.append(b) + # ---------------- end of helper functions + + # assemble a dictionary of the factors having a Rational power + common_b = {} + done = [] + bases = [] + for b, e in c_powers: + b, e = bkey(b, e) + if b in common_b: + common_b[b] = common_b[b] + e + else: + common_b[b] = e + if b[1] != 1 and b[0].is_Mul: + bases.append(b) + bases.sort(key=default_sort_key) # this makes tie-breaking canonical + bases.sort(key=measure, reverse=True) # handle longest first + for base in bases: + if base not in common_b: # it may have been removed already + continue + b, exponent = base + last = False # True when no factor of base is a radical + qlcm = 1 # the lcm of the radical denominators + while True: + bstart = b + qstart = qlcm + + bb = [] # list of factors + ee = [] # (factor's expo. and it's current value in common_b) + for bi in Mul.make_args(b): + bib, bie = bkey(bi) + if bib not in common_b or common_b[bib] < bie: + ee = bb = [] # failed + break + ee.append([bie, common_b[bib]]) + bb.append(bib) + if ee: + # find the number of integral extractions possible + # e.g. [(1, 2), (2, 2)] -> min(2/1, 2/2) -> 1 + min1 = ee[0][1]//ee[0][0] + for i in range(1, len(ee)): + rat = ee[i][1]//ee[i][0] + if rat < 1: + break + min1 = min(min1, rat) + else: + # update base factor counts + # e.g. if ee = [(2, 5), (3, 6)] then min1 = 2 + # and the new base counts will be 5-2*2 and 6-2*3 + for i in range(len(bb)): + common_b[bb[i]] -= min1*ee[i][0] + update(bb[i]) + # update the count of the base + # e.g. x**2*y*sqrt(x*sqrt(y)) the count of x*sqrt(y) + # will increase by 4 to give bkey (x*sqrt(y), 2, 5) + common_b[base] += min1*qstart*exponent + if (last # no more radicals in base + or len(common_b) == 1 # nothing left to join with + or all(k[1] == 1 for k in common_b) # no rad's in common_b + ): + break + # see what we can exponentiate base by to remove any radicals + # so we know what to search for + # e.g. if base were x**(1/2)*y**(1/3) then we should + # exponentiate by 6 and look for powers of x and y in the ratio + # of 2 to 3 + qlcm = lcm([ratq(bi) for bi in Mul.make_args(bstart)]) + if qlcm == 1: + break # we are done + b = bstart**qlcm + qlcm *= qstart + if all(ratq(bi) == 1 for bi in Mul.make_args(b)): + last = True # we are going to be done after this next pass + # this base no longer can find anything to join with and + # since it was longer than any other we are done with it + b, q = base + done.append((b, common_b.pop(base)*Rational(1, q))) + + # update c_powers and get ready to continue with powsimp + c_powers = done + # there may be terms still in common_b that were bases that were + # identified as needing processing, so remove those, too + for (b, q), e in common_b.items(): + if (b.is_Pow or isinstance(b, exp)) and \ + q is not S.One and not b.exp.is_Rational: + b, be = b.as_base_exp() + b = b**(be/q) + else: + b = root(b, q) + c_powers.append((b, e)) + check = len(c_powers) + c_powers = dict(c_powers) + assert len(c_powers) == check # there should have been no duplicates + # ============================================================== + + # rebuild the expression + newexpr = expr.func(*(newexpr + [Pow(b, e) for b, e in c_powers.items()])) + if combine == 'exp': + return expr.func(newexpr, expr.func(*nc_part)) + else: + return recurse(expr.func(*nc_part), combine='base') * \ + recurse(newexpr, combine='base') + + elif combine == 'base': + + # Build c_powers and nc_part. These must both be lists not + # dicts because exp's are not combined. + c_powers = [] + nc_part = [] + for term in expr.args: + if term.is_commutative: + c_powers.append(list(term.as_base_exp())) + else: + nc_part.append(term) + + # Pull out numerical coefficients from exponent if assumptions allow + # e.g., 2**(2*x) => 4**x + for i in range(len(c_powers)): + b, e = c_powers[i] + if not (all(x.is_nonnegative for x in b.as_numer_denom()) or e.is_integer or force or b.is_polar): + continue + exp_c, exp_t = e.as_coeff_Mul(rational=True) + if exp_c is not S.One and exp_t is not S.One: + c_powers[i] = [Pow(b, exp_c), exp_t] + + # Combine bases whenever they have the same exponent and + # assumptions allow + # first gather the potential bases under the common exponent + c_exp = defaultdict(list) + for b, e in c_powers: + if deep: + e = recurse(e) + if e.is_Add and (b.is_positive or e.is_integer): + e = factor_terms(e) + if _coeff_isneg(e): + e = -e + b = 1/b + c_exp[e].append(b) + del c_powers + + # Merge back in the results of the above to form a new product + c_powers = defaultdict(list) + for e in c_exp: + bases = c_exp[e] + + # calculate the new base for e + + if len(bases) == 1: + new_base = bases[0] + elif e.is_integer or force: + new_base = expr.func(*bases) + else: + # see which ones can be joined + unk = [] + nonneg = [] + neg = [] + for bi in bases: + if bi.is_negative: + neg.append(bi) + elif bi.is_nonnegative: + nonneg.append(bi) + elif bi.is_polar: + nonneg.append( + bi) # polar can be treated like non-negative + else: + unk.append(bi) + if len(unk) == 1 and not neg or len(neg) == 1 and not unk: + # a single neg or a single unk can join the rest + nonneg.extend(unk + neg) + unk = neg = [] + elif neg: + # their negative signs cancel in groups of 2*q if we know + # that e = p/q else we have to treat them as unknown + israt = False + if e.is_Rational: + israt = True + else: + p, d = e.as_numer_denom() + if p.is_integer and d.is_integer: + israt = True + if israt: + neg = [-w for w in neg] + unk.extend([S.NegativeOne]*len(neg)) + else: + unk.extend(neg) + neg = [] + del israt + + # these shouldn't be joined + for b in unk: + c_powers[b].append(e) + # here is a new joined base + new_base = expr.func(*(nonneg + neg)) + # if there are positive parts they will just get separated + # again unless some change is made + + def _terms(e): + # return the number of terms of this expression + # when multiplied out -- assuming no joining of terms + if e.is_Add: + return sum(_terms(ai) for ai in e.args) + if e.is_Mul: + return prod([_terms(mi) for mi in e.args]) + return 1 + xnew_base = expand_mul(new_base, deep=False) + if len(Add.make_args(xnew_base)) < _terms(new_base): + new_base = factor_terms(xnew_base) + + c_powers[new_base].append(e) + + # break out the powers from c_powers now + c_part = [Pow(b, ei) for b, e in c_powers.items() for ei in e] + + # we're done + return expr.func(*(c_part + nc_part)) + + else: + raise ValueError("combine must be one of ('all', 'exp', 'base').") + + +def powdenest(eq, force=False, polar=False): + r""" + Collect exponents on powers as assumptions allow. + + Explanation + =========== + + Given ``(bb**be)**e``, this can be simplified as follows: + * if ``bb`` is positive, or + * ``e`` is an integer, or + * ``|be| < 1`` then this simplifies to ``bb**(be*e)`` + + Given a product of powers raised to a power, ``(bb1**be1 * + bb2**be2...)**e``, simplification can be done as follows: + + - if e is positive, the gcd of all bei can be joined with e; + - all non-negative bb can be separated from those that are negative + and their gcd can be joined with e; autosimplification already + handles this separation. + - integer factors from powers that have integers in the denominator + of the exponent can be removed from any term and the gcd of such + integers can be joined with e + + Setting ``force`` to ``True`` will make symbols that are not explicitly + negative behave as though they are positive, resulting in more + denesting. + + Setting ``polar`` to ``True`` will do simplifications on the Riemann surface of + the logarithm, also resulting in more denestings. + + When there are sums of logs in exp() then a product of powers may be + obtained e.g. ``exp(3*(log(a) + 2*log(b)))`` - > ``a**3*b**6``. + + Examples + ======== + + >>> from sympy.abc import a, b, x, y, z + >>> from sympy import Symbol, exp, log, sqrt, symbols, powdenest + + >>> powdenest((x**(2*a/3))**(3*x)) + (x**(2*a/3))**(3*x) + >>> powdenest(exp(3*x*log(2))) + 2**(3*x) + + Assumptions may prevent expansion: + + >>> powdenest(sqrt(x**2)) + sqrt(x**2) + + >>> p = symbols('p', positive=True) + >>> powdenest(sqrt(p**2)) + p + + No other expansion is done. + + >>> i, j = symbols('i,j', integer=True) + >>> powdenest((x**x)**(i + j)) # -X-> (x**x)**i*(x**x)**j + x**(x*(i + j)) + + But exp() will be denested by moving all non-log terms outside of + the function; this may result in the collapsing of the exp to a power + with a different base: + + >>> powdenest(exp(3*y*log(x))) + x**(3*y) + >>> powdenest(exp(y*(log(a) + log(b)))) + (a*b)**y + >>> powdenest(exp(3*(log(a) + log(b)))) + a**3*b**3 + + If assumptions allow, symbols can also be moved to the outermost exponent: + + >>> i = Symbol('i', integer=True) + >>> powdenest(((x**(2*i))**(3*y))**x) + ((x**(2*i))**(3*y))**x + >>> powdenest(((x**(2*i))**(3*y))**x, force=True) + x**(6*i*x*y) + + >>> powdenest(((x**(2*a/3))**(3*y/i))**x) + ((x**(2*a/3))**(3*y/i))**x + >>> powdenest((x**(2*i)*y**(4*i))**z, force=True) + (x*y**2)**(2*i*z) + + >>> n = Symbol('n', negative=True) + + >>> powdenest((x**i)**y, force=True) + x**(i*y) + >>> powdenest((n**i)**x, force=True) + (n**i)**x + + """ + from sympy.simplify.simplify import posify + + if force: + def _denest(b, e): + if not isinstance(b, (Pow, exp)): + return b.is_positive, Pow(b, e, evaluate=False) + return _denest(b.base, b.exp*e) + reps = [] + for p in eq.atoms(Pow, exp): + if isinstance(p.base, (Pow, exp)): + ok, dp = _denest(*p.args) + if ok is not False: + reps.append((p, dp)) + if reps: + eq = eq.subs(reps) + eq, reps = posify(eq) + return powdenest(eq, force=False, polar=polar).xreplace(reps) + + if polar: + eq, rep = polarify(eq) + return unpolarify(powdenest(unpolarify(eq, exponents_only=True)), rep) + + new = powsimp(eq) + return new.xreplace(Transform( + _denest_pow, filter=lambda m: m.is_Pow or isinstance(m, exp))) + +_y = Dummy('y') + + +def _denest_pow(eq): + """ + Denest powers. + + This is a helper function for powdenest that performs the actual + transformation. + """ + from sympy.simplify.simplify import logcombine + + b, e = eq.as_base_exp() + if b.is_Pow or isinstance(b, exp) and e != 1: + new = b._eval_power(e) + if new is not None: + eq = new + b, e = new.as_base_exp() + + # denest exp with log terms in exponent + if b is S.Exp1 and e.is_Mul: + logs = [] + other = [] + for ei in e.args: + if any(isinstance(ai, log) for ai in Add.make_args(ei)): + logs.append(ei) + else: + other.append(ei) + logs = logcombine(Mul(*logs)) + return Pow(exp(logs), Mul(*other)) + + _, be = b.as_base_exp() + if be is S.One and not (b.is_Mul or + b.is_Rational and b.q != 1 or + b.is_positive): + return eq + + # denest eq which is either pos**e or Pow**e or Mul**e or + # Mul(b1**e1, b2**e2) + + # handle polar numbers specially + polars, nonpolars = [], [] + for bb in Mul.make_args(b): + if bb.is_polar: + polars.append(bb.as_base_exp()) + else: + nonpolars.append(bb) + if len(polars) == 1 and not polars[0][0].is_Mul: + return Pow(polars[0][0], polars[0][1]*e)*powdenest(Mul(*nonpolars)**e) + elif polars: + return Mul(*[powdenest(bb**(ee*e)) for (bb, ee) in polars]) \ + *powdenest(Mul(*nonpolars)**e) + + if b.is_Integer: + # use log to see if there is a power here + logb = expand_log(log(b)) + if logb.is_Mul: + c, logb = logb.args + e *= c + base = logb.args[0] + return Pow(base, e) + + # if b is not a Mul or any factor is an atom then there is nothing to do + if not b.is_Mul or any(s.is_Atom for s in Mul.make_args(b)): + return eq + + # let log handle the case of the base of the argument being a Mul, e.g. + # sqrt(x**(2*i)*y**(6*i)) -> x**i*y**(3**i) if x and y are positive; we + # will take the log, expand it, and then factor out the common powers that + # now appear as coefficient. We do this manually since terms_gcd pulls out + # fractions, terms_gcd(x+x*y/2) -> x*(y + 2)/2 and we don't want the 1/2; + # gcd won't pull out numerators from a fraction: gcd(3*x, 9*x/2) -> x but + # we want 3*x. Neither work with noncommutatives. + + def nc_gcd(aa, bb): + a, b = [i.as_coeff_Mul() for i in [aa, bb]] + c = gcd(a[0], b[0]).as_numer_denom()[0] + g = Mul(*(a[1].args_cnc(cset=True)[0] & b[1].args_cnc(cset=True)[0])) + return _keep_coeff(c, g) + + glogb = expand_log(log(b)) + if glogb.is_Add: + args = glogb.args + g = reduce(nc_gcd, args) + if g != 1: + cg, rg = g.as_coeff_Mul() + glogb = _keep_coeff(cg, rg*Add(*[a/g for a in args])) + + # now put the log back together again + if isinstance(glogb, log) or not glogb.is_Mul: + if glogb.args[0].is_Pow or isinstance(glogb.args[0], exp): + glogb = _denest_pow(glogb.args[0]) + if (abs(glogb.exp) < 1) == True: + return Pow(glogb.base, glogb.exp*e) + return eq + + # the log(b) was a Mul so join any adds with logcombine + add = [] + other = [] + for a in glogb.args: + if a.is_Add: + add.append(a) + else: + other.append(a) + return Pow(exp(logcombine(Mul(*add))), e*Mul(*other)) diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/radsimp.py b/.venv/lib/python3.12/site-packages/sympy/simplify/radsimp.py new file mode 100644 index 0000000000000000000000000000000000000000..c878168ebfbc29fc632577d6325befc120c26f56 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/simplify/radsimp.py @@ -0,0 +1,1234 @@ +from collections import defaultdict + +from sympy.core import sympify, S, Mul, Derivative, Pow +from sympy.core.add import _unevaluated_Add, Add +from sympy.core.assumptions import assumptions +from sympy.core.exprtools import Factors, gcd_terms +from sympy.core.function import _mexpand, expand_mul, expand_power_base +from sympy.core.mul import _keep_coeff, _unevaluated_Mul, _mulsort +from sympy.core.numbers import Rational, zoo, nan +from sympy.core.parameters import global_parameters +from sympy.core.sorting import ordered, default_sort_key +from sympy.core.symbol import Dummy, Wild, symbols +from sympy.functions import exp, sqrt, log +from sympy.functions.elementary.complexes import Abs +from sympy.polys import gcd +from sympy.simplify.sqrtdenest import sqrtdenest +from sympy.utilities.iterables import iterable, sift + + + + +def collect(expr, syms, func=None, evaluate=None, exact=False, distribute_order_term=True): + """ + Collect additive terms of an expression. + + Explanation + =========== + + This function collects additive terms of an expression with respect + to a list of expression up to powers with rational exponents. By the + term symbol here are meant arbitrary expressions, which can contain + powers, products, sums etc. In other words symbol is a pattern which + will be searched for in the expression's terms. + + The input expression is not expanded by :func:`collect`, so user is + expected to provide an expression in an appropriate form. This makes + :func:`collect` more predictable as there is no magic happening behind the + scenes. However, it is important to note, that powers of products are + converted to products of powers using the :func:`~.expand_power_base` + function. + + There are two possible types of output. First, if ``evaluate`` flag is + set, this function will return an expression with collected terms or + else it will return a dictionary with expressions up to rational powers + as keys and collected coefficients as values. + + Examples + ======== + + >>> from sympy import S, collect, expand, factor, Wild + >>> from sympy.abc import a, b, c, x, y + + This function can collect symbolic coefficients in polynomials or + rational expressions. It will manage to find all integer or rational + powers of collection variable:: + + >>> collect(a*x**2 + b*x**2 + a*x - b*x + c, x) + c + x**2*(a + b) + x*(a - b) + + The same result can be achieved in dictionary form:: + + >>> d = collect(a*x**2 + b*x**2 + a*x - b*x + c, x, evaluate=False) + >>> d[x**2] + a + b + >>> d[x] + a - b + >>> d[S.One] + c + + You can also work with multivariate polynomials. However, remember that + this function is greedy so it will care only about a single symbol at time, + in specification order:: + + >>> collect(x**2 + y*x**2 + x*y + y + a*y, [x, y]) + x**2*(y + 1) + x*y + y*(a + 1) + + Also more complicated expressions can be used as patterns:: + + >>> from sympy import sin, log + >>> collect(a*sin(2*x) + b*sin(2*x), sin(2*x)) + (a + b)*sin(2*x) + + >>> collect(a*x*log(x) + b*(x*log(x)), x*log(x)) + x*(a + b)*log(x) + + You can use wildcards in the pattern:: + + >>> w = Wild('w1') + >>> collect(a*x**y - b*x**y, w**y) + x**y*(a - b) + + It is also possible to work with symbolic powers, although it has more + complicated behavior, because in this case power's base and symbolic part + of the exponent are treated as a single symbol:: + + >>> collect(a*x**c + b*x**c, x) + a*x**c + b*x**c + >>> collect(a*x**c + b*x**c, x**c) + x**c*(a + b) + + However if you incorporate rationals to the exponents, then you will get + well known behavior:: + + >>> collect(a*x**(2*c) + b*x**(2*c), x**c) + x**(2*c)*(a + b) + + Note also that all previously stated facts about :func:`collect` function + apply to the exponential function, so you can get:: + + >>> from sympy import exp + >>> collect(a*exp(2*x) + b*exp(2*x), exp(x)) + (a + b)*exp(2*x) + + If you are interested only in collecting specific powers of some symbols + then set ``exact`` flag to True:: + + >>> collect(a*x**7 + b*x**7, x, exact=True) + a*x**7 + b*x**7 + >>> collect(a*x**7 + b*x**7, x**7, exact=True) + x**7*(a + b) + + If you want to collect on any object containing symbols, set + ``exact`` to None: + + >>> collect(x*exp(x) + sin(x)*y + sin(x)*2 + 3*x, x, exact=None) + x*exp(x) + 3*x + (y + 2)*sin(x) + >>> collect(a*x*y + x*y + b*x + x, [x, y], exact=None) + x*y*(a + 1) + x*(b + 1) + + You can also apply this function to differential equations, where + derivatives of arbitrary order can be collected. Note that if you + collect with respect to a function or a derivative of a function, all + derivatives of that function will also be collected. Use + ``exact=True`` to prevent this from happening:: + + >>> from sympy import Derivative as D, collect, Function + >>> f = Function('f') (x) + + >>> collect(a*D(f,x) + b*D(f,x), D(f,x)) + (a + b)*Derivative(f(x), x) + + >>> collect(a*D(D(f,x),x) + b*D(D(f,x),x), f) + (a + b)*Derivative(f(x), (x, 2)) + + >>> collect(a*D(D(f,x),x) + b*D(D(f,x),x), D(f,x), exact=True) + a*Derivative(f(x), (x, 2)) + b*Derivative(f(x), (x, 2)) + + >>> collect(a*D(f,x) + b*D(f,x) + a*f + b*f, f) + (a + b)*f(x) + (a + b)*Derivative(f(x), x) + + Or you can even match both derivative order and exponent at the same time:: + + >>> collect(a*D(D(f,x),x)**2 + b*D(D(f,x),x)**2, D(f,x)) + (a + b)*Derivative(f(x), (x, 2))**2 + + Finally, you can apply a function to each of the collected coefficients. + For example you can factorize symbolic coefficients of polynomial:: + + >>> f = expand((x + a + 1)**3) + + >>> collect(f, x, factor) + x**3 + 3*x**2*(a + 1) + 3*x*(a + 1)**2 + (a + 1)**3 + + .. note:: Arguments are expected to be in expanded form, so you might have + to call :func:`~.expand` prior to calling this function. + + See Also + ======== + + collect_const, collect_sqrt, rcollect + """ + expr = sympify(expr) + syms = [sympify(i) for i in (syms if iterable(syms) else [syms])] + + # replace syms[i] if it is not x, -x or has Wild symbols + cond = lambda x: x.is_Symbol or (-x).is_Symbol or bool( + x.atoms(Wild)) + _, nonsyms = sift(syms, cond, binary=True) + if nonsyms: + reps = dict(zip(nonsyms, [Dummy(**assumptions(i)) for i in nonsyms])) + syms = [reps.get(s, s) for s in syms] + rv = collect(expr.subs(reps), syms, + func=func, evaluate=evaluate, exact=exact, + distribute_order_term=distribute_order_term) + urep = {v: k for k, v in reps.items()} + if not isinstance(rv, dict): + return rv.xreplace(urep) + else: + return {urep.get(k, k).xreplace(urep): v.xreplace(urep) + for k, v in rv.items()} + + # see if other expressions should be considered + if exact is None: + _syms = set() + for i in Add.make_args(expr): + if not i.has_free(*syms) or i in syms: + continue + if not i.is_Mul and i not in syms: + _syms.add(i) + else: + # identify compound generators + g = i._new_rawargs(*i.as_coeff_mul(*syms)[1]) + if g not in syms: + _syms.add(g) + simple = all(i.is_Pow and i.base in syms for i in _syms) + syms = syms + list(ordered(_syms)) + if not simple: + return collect(expr, syms, + func=func, evaluate=evaluate, exact=False, + distribute_order_term=distribute_order_term) + + if evaluate is None: + evaluate = global_parameters.evaluate + + def make_expression(terms): + product = [] + + for term, rat, sym, deriv in terms: + if deriv is not None: + var, order = deriv + for _ in range(order): + term = Derivative(term, var) + + if sym is None: + if rat is S.One: + product.append(term) + else: + product.append(Pow(term, rat)) + else: + product.append(Pow(term, rat*sym)) + + return Mul(*product) + + def parse_derivative(deriv): + # scan derivatives tower in the input expression and return + # underlying function and maximal differentiation order + expr, sym, order = deriv.expr, deriv.variables[0], 1 + + for s in deriv.variables[1:]: + if s == sym: + order += 1 + else: + raise NotImplementedError( + 'Improve MV Derivative support in collect') + + while isinstance(expr, Derivative): + s0 = expr.variables[0] + + if any(s != s0 for s in expr.variables): + raise NotImplementedError( + 'Improve MV Derivative support in collect') + + if s0 == sym: + expr, order = expr.expr, order + len(expr.variables) + else: + break + + return expr, (sym, Rational(order)) + + def parse_term(expr): + """Parses expression expr and outputs tuple (sexpr, rat_expo, + sym_expo, deriv) + where: + - sexpr is the base expression + - rat_expo is the rational exponent that sexpr is raised to + - sym_expo is the symbolic exponent that sexpr is raised to + - deriv contains the derivatives of the expression + + For example, the output of x would be (x, 1, None, None) + the output of 2**x would be (2, 1, x, None). + """ + rat_expo, sym_expo = S.One, None + sexpr, deriv = expr, None + + if expr.is_Pow: + if isinstance(expr.base, Derivative): + sexpr, deriv = parse_derivative(expr.base) + else: + sexpr = expr.base + + if expr.base == S.Exp1: + arg = expr.exp + if arg.is_Rational: + sexpr, rat_expo = S.Exp1, arg + elif arg.is_Mul: + coeff, tail = arg.as_coeff_Mul(rational=True) + sexpr, rat_expo = exp(tail), coeff + + elif expr.exp.is_Number: + rat_expo = expr.exp + else: + coeff, tail = expr.exp.as_coeff_Mul() + + if coeff.is_Number: + rat_expo, sym_expo = coeff, tail + else: + sym_expo = expr.exp + elif isinstance(expr, exp): + arg = expr.exp + if arg.is_Rational: + sexpr, rat_expo = S.Exp1, arg + elif arg.is_Mul: + coeff, tail = arg.as_coeff_Mul(rational=True) + sexpr, rat_expo = exp(tail), coeff + elif isinstance(expr, Derivative): + sexpr, deriv = parse_derivative(expr) + + return sexpr, rat_expo, sym_expo, deriv + + def parse_expression(terms, pattern): + """Parse terms searching for a pattern. + Terms is a list of tuples as returned by parse_terms; + Pattern is an expression treated as a product of factors. + """ + pattern = Mul.make_args(pattern) + + if len(terms) < len(pattern): + # pattern is longer than matched product + # so no chance for positive parsing result + return None + else: + pattern = [parse_term(elem) for elem in pattern] + + terms = terms[:] # need a copy + elems, common_expo, has_deriv = [], None, False + + for elem, e_rat, e_sym, e_ord in pattern: + + if elem.is_Number and e_rat == 1 and e_sym is None: + # a constant is a match for everything + continue + + for j in range(len(terms)): + if terms[j] is None: + continue + + term, t_rat, t_sym, t_ord = terms[j] + + # keeping track of whether one of the terms had + # a derivative or not as this will require rebuilding + # the expression later + if t_ord is not None: + has_deriv = True + + if (term.match(elem) is not None and + (t_sym == e_sym or t_sym is not None and + e_sym is not None and + t_sym.match(e_sym) is not None)): + if exact is False: + # we don't have to be exact so find common exponent + # for both expression's term and pattern's element + expo = t_rat / e_rat + + if common_expo is None: + # first time + common_expo = expo + else: + # common exponent was negotiated before so + # there is no chance for a pattern match unless + # common and current exponents are equal + if common_expo != expo: + common_expo = 1 + else: + # we ought to be exact so all fields of + # interest must match in every details + if e_rat != t_rat or e_ord != t_ord: + continue + + # found common term so remove it from the expression + # and try to match next element in the pattern + elems.append(terms[j]) + terms[j] = None + + break + + else: + # pattern element not found + return None + + return [_f for _f in terms if _f], elems, common_expo, has_deriv + + if evaluate: + if expr.is_Add: + o = expr.getO() or 0 + expr = expr.func(*[ + collect(a, syms, func, True, exact, distribute_order_term) + for a in expr.args if a != o]) + o + elif expr.is_Mul: + return expr.func(*[ + collect(term, syms, func, True, exact, distribute_order_term) + for term in expr.args]) + elif expr.is_Pow: + b = collect( + expr.base, syms, func, True, exact, distribute_order_term) + return Pow(b, expr.exp) + + syms = [expand_power_base(i, deep=False) for i in syms] + + order_term = None + + if distribute_order_term: + order_term = expr.getO() + + if order_term is not None: + if order_term.has(*syms): + order_term = None + else: + expr = expr.removeO() + + summa = [expand_power_base(i, deep=False) for i in Add.make_args(expr)] + + collected, disliked = defaultdict(list), S.Zero + for product in summa: + c, nc = product.args_cnc(split_1=False) + args = list(ordered(c)) + nc + terms = [parse_term(i) for i in args] + small_first = True + + for symbol in syms: + if isinstance(symbol, Derivative) and small_first: + terms = list(reversed(terms)) + small_first = not small_first + result = parse_expression(terms, symbol) + + if result is not None: + if not symbol.is_commutative: + raise AttributeError("Can not collect noncommutative symbol") + + terms, elems, common_expo, has_deriv = result + + # when there was derivative in current pattern we + # will need to rebuild its expression from scratch + if not has_deriv: + margs = [] + for elem in elems: + if elem[2] is None: + e = elem[1] + else: + e = elem[1]*elem[2] + margs.append(Pow(elem[0], e)) + index = Mul(*margs) + else: + index = make_expression(elems) + terms = expand_power_base(make_expression(terms), deep=False) + index = expand_power_base(index, deep=False) + collected[index].append(terms) + break + else: + # none of the patterns matched + disliked += product + # add terms now for each key + collected = {k: Add(*v) for k, v in collected.items()} + + if disliked is not S.Zero: + collected[S.One] = disliked + + if order_term is not None: + for key, val in collected.items(): + collected[key] = val + order_term + + if func is not None: + collected = { + key: func(val) for key, val in collected.items()} + + if evaluate: + return Add(*[key*val for key, val in collected.items()]) + else: + return collected + + +def rcollect(expr, *vars): + """ + Recursively collect sums in an expression. + + Examples + ======== + + >>> from sympy.simplify import rcollect + >>> from sympy.abc import x, y + + >>> expr = (x**2*y + x*y + x + y)/(x + y) + + >>> rcollect(expr, y) + (x + y*(x**2 + x + 1))/(x + y) + + See Also + ======== + + collect, collect_const, collect_sqrt + """ + if expr.is_Atom or not expr.has(*vars): + return expr + else: + expr = expr.__class__(*[rcollect(arg, *vars) for arg in expr.args]) + + if expr.is_Add: + return collect(expr, vars) + else: + return expr + + +def collect_sqrt(expr, evaluate=None): + """Return expr with terms having common square roots collected together. + If ``evaluate`` is False a count indicating the number of sqrt-containing + terms will be returned and, if non-zero, the terms of the Add will be + returned, else the expression itself will be returned as a single term. + If ``evaluate`` is True, the expression with any collected terms will be + returned. + + Note: since I = sqrt(-1), it is collected, too. + + Examples + ======== + + >>> from sympy import sqrt + >>> from sympy.simplify.radsimp import collect_sqrt + >>> from sympy.abc import a, b + + >>> r2, r3, r5 = [sqrt(i) for i in [2, 3, 5]] + >>> collect_sqrt(a*r2 + b*r2) + sqrt(2)*(a + b) + >>> collect_sqrt(a*r2 + b*r2 + a*r3 + b*r3) + sqrt(2)*(a + b) + sqrt(3)*(a + b) + >>> collect_sqrt(a*r2 + b*r2 + a*r3 + b*r5) + sqrt(3)*a + sqrt(5)*b + sqrt(2)*(a + b) + + If evaluate is False then the arguments will be sorted and + returned as a list and a count of the number of sqrt-containing + terms will be returned: + + >>> collect_sqrt(a*r2 + b*r2 + a*r3 + b*r5, evaluate=False) + ((sqrt(3)*a, sqrt(5)*b, sqrt(2)*(a + b)), 3) + >>> collect_sqrt(a*sqrt(2) + b, evaluate=False) + ((b, sqrt(2)*a), 1) + >>> collect_sqrt(a + b, evaluate=False) + ((a + b,), 0) + + See Also + ======== + + collect, collect_const, rcollect + """ + if evaluate is None: + evaluate = global_parameters.evaluate + # this step will help to standardize any complex arguments + # of sqrts + coeff, expr = expr.as_content_primitive() + vars = set() + for a in Add.make_args(expr): + for m in a.args_cnc()[0]: + if m.is_number and ( + m.is_Pow and m.exp.is_Rational and m.exp.q == 2 or + m is S.ImaginaryUnit): + vars.add(m) + + # we only want radicals, so exclude Number handling; in this case + # d will be evaluated + d = collect_const(expr, *vars, Numbers=False) + hit = expr != d + + if not evaluate: + nrad = 0 + # make the evaluated args canonical + args = list(ordered(Add.make_args(d))) + for i, m in enumerate(args): + c, nc = m.args_cnc() + for ci in c: + # XXX should this be restricted to ci.is_number as above? + if ci.is_Pow and ci.exp.is_Rational and ci.exp.q == 2 or \ + ci is S.ImaginaryUnit: + nrad += 1 + break + args[i] *= coeff + if not (hit or nrad): + args = [Add(*args)] + return tuple(args), nrad + + return coeff*d + + +def collect_abs(expr): + """Return ``expr`` with arguments of multiple Abs in a term collected + under a single instance. + + Examples + ======== + + >>> from sympy.simplify.radsimp import collect_abs + >>> from sympy.abc import x + >>> collect_abs(abs(x + 1)/abs(x**2 - 1)) + Abs((x + 1)/(x**2 - 1)) + >>> collect_abs(abs(1/x)) + Abs(1/x) + """ + def _abs(mul): + c, nc = mul.args_cnc() + a = [] + o = [] + for i in c: + if isinstance(i, Abs): + a.append(i.args[0]) + elif isinstance(i, Pow) and isinstance(i.base, Abs) and i.exp.is_real: + a.append(i.base.args[0]**i.exp) + else: + o.append(i) + if len(a) < 2 and not any(i.exp.is_negative for i in a if isinstance(i, Pow)): + return mul + absarg = Mul(*a) + A = Abs(absarg) + args = [A] + args.extend(o) + if not A.has(Abs): + args.extend(nc) + return Mul(*args) + if not isinstance(A, Abs): + # reevaluate and make it unevaluated + A = Abs(absarg, evaluate=False) + args[0] = A + _mulsort(args) + args.extend(nc) # nc always go last + return Mul._from_args(args, is_commutative=not nc) + + return expr.replace( + lambda x: isinstance(x, Mul), + lambda x: _abs(x)).replace( + lambda x: isinstance(x, Pow), + lambda x: _abs(x)) + + +def collect_const(expr, *vars, Numbers=True): + """A non-greedy collection of terms with similar number coefficients in + an Add expr. If ``vars`` is given then only those constants will be + targeted. Although any Number can also be targeted, if this is not + desired set ``Numbers=False`` and no Float or Rational will be collected. + + Parameters + ========== + + expr : SymPy expression + This parameter defines the expression the expression from which + terms with similar coefficients are to be collected. A non-Add + expression is returned as it is. + + vars : variable length collection of Numbers, optional + Specifies the constants to target for collection. Can be multiple in + number. + + Numbers : bool + Specifies to target all instance of + :class:`sympy.core.numbers.Number` class. If ``Numbers=False``, then + no Float or Rational will be collected. + + Returns + ======= + + expr : Expr + Returns an expression with similar coefficient terms collected. + + Examples + ======== + + >>> from sympy import sqrt + >>> from sympy.abc import s, x, y, z + >>> from sympy.simplify.radsimp import collect_const + >>> collect_const(sqrt(3) + sqrt(3)*(1 + sqrt(2))) + sqrt(3)*(sqrt(2) + 2) + >>> collect_const(sqrt(3)*s + sqrt(7)*s + sqrt(3) + sqrt(7)) + (sqrt(3) + sqrt(7))*(s + 1) + >>> s = sqrt(2) + 2 + >>> collect_const(sqrt(3)*s + sqrt(3) + sqrt(7)*s + sqrt(7)) + (sqrt(2) + 3)*(sqrt(3) + sqrt(7)) + >>> collect_const(sqrt(3)*s + sqrt(3) + sqrt(7)*s + sqrt(7), sqrt(3)) + sqrt(7) + sqrt(3)*(sqrt(2) + 3) + sqrt(7)*(sqrt(2) + 2) + + The collection is sign-sensitive, giving higher precedence to the + unsigned values: + + >>> collect_const(x - y - z) + x - (y + z) + >>> collect_const(-y - z) + -(y + z) + >>> collect_const(2*x - 2*y - 2*z, 2) + 2*(x - y - z) + >>> collect_const(2*x - 2*y - 2*z, -2) + 2*x - 2*(y + z) + + See Also + ======== + + collect, collect_sqrt, rcollect + """ + if not expr.is_Add: + return expr + + recurse = False + + if not vars: + recurse = True + vars = set() + for a in expr.args: + for m in Mul.make_args(a): + if m.is_number: + vars.add(m) + else: + vars = sympify(vars) + if not Numbers: + vars = [v for v in vars if not v.is_Number] + + vars = list(ordered(vars)) + for v in vars: + terms = defaultdict(list) + Fv = Factors(v) + for m in Add.make_args(expr): + f = Factors(m) + q, r = f.div(Fv) + if r.is_one: + # only accept this as a true factor if + # it didn't change an exponent from an Integer + # to a non-Integer, e.g. 2/sqrt(2) -> sqrt(2) + # -- we aren't looking for this sort of change + fwas = f.factors.copy() + fnow = q.factors + if not any(k in fwas and fwas[k].is_Integer and not + fnow[k].is_Integer for k in fnow): + terms[v].append(q.as_expr()) + continue + terms[S.One].append(m) + + args = [] + hit = False + uneval = False + for k in ordered(terms): + v = terms[k] + if k is S.One: + args.extend(v) + continue + + if len(v) > 1: + v = Add(*v) + hit = True + if recurse and v != expr: + vars.append(v) + else: + v = v[0] + + # be careful not to let uneval become True unless + # it must be because it's going to be more expensive + # to rebuild the expression as an unevaluated one + if Numbers and k.is_Number and v.is_Add: + args.append(_keep_coeff(k, v, sign=True)) + uneval = True + else: + args.append(k*v) + + if hit: + if uneval: + expr = _unevaluated_Add(*args) + else: + expr = Add(*args) + if not expr.is_Add: + break + + return expr + + +def radsimp(expr, symbolic=True, max_terms=4): + r""" + Rationalize the denominator by removing square roots. + + Explanation + =========== + + The expression returned from radsimp must be used with caution + since if the denominator contains symbols, it will be possible to make + substitutions that violate the assumptions of the simplification process: + that for a denominator matching a + b*sqrt(c), a != +/-b*sqrt(c). (If + there are no symbols, this assumptions is made valid by collecting terms + of sqrt(c) so the match variable ``a`` does not contain ``sqrt(c)``.) If + you do not want the simplification to occur for symbolic denominators, set + ``symbolic`` to False. + + If there are more than ``max_terms`` radical terms then the expression is + returned unchanged. + + Examples + ======== + + >>> from sympy import radsimp, sqrt, Symbol, pprint + >>> from sympy import factor_terms, fraction, signsimp + >>> from sympy.simplify.radsimp import collect_sqrt + >>> from sympy.abc import a, b, c + + >>> radsimp(1/(2 + sqrt(2))) + (2 - sqrt(2))/2 + >>> x,y = map(Symbol, 'xy') + >>> e = ((2 + 2*sqrt(2))*x + (2 + sqrt(8))*y)/(2 + sqrt(2)) + >>> radsimp(e) + sqrt(2)*(x + y) + + No simplification beyond removal of the gcd is done. One might + want to polish the result a little, however, by collecting + square root terms: + + >>> r2 = sqrt(2) + >>> r5 = sqrt(5) + >>> ans = radsimp(1/(y*r2 + x*r2 + a*r5 + b*r5)); pprint(ans) + ___ ___ ___ ___ + \/ 5 *a + \/ 5 *b - \/ 2 *x - \/ 2 *y + ------------------------------------------ + 2 2 2 2 + 5*a + 10*a*b + 5*b - 2*x - 4*x*y - 2*y + + >>> n, d = fraction(ans) + >>> pprint(factor_terms(signsimp(collect_sqrt(n))/d, radical=True)) + ___ ___ + \/ 5 *(a + b) - \/ 2 *(x + y) + ------------------------------------------ + 2 2 2 2 + 5*a + 10*a*b + 5*b - 2*x - 4*x*y - 2*y + + If radicals in the denominator cannot be removed or there is no denominator, + the original expression will be returned. + + >>> radsimp(sqrt(2)*x + sqrt(2)) + sqrt(2)*x + sqrt(2) + + Results with symbols will not always be valid for all substitutions: + + >>> eq = 1/(a + b*sqrt(c)) + >>> eq.subs(a, b*sqrt(c)) + 1/(2*b*sqrt(c)) + >>> radsimp(eq).subs(a, b*sqrt(c)) + nan + + If ``symbolic=False``, symbolic denominators will not be transformed (but + numeric denominators will still be processed): + + >>> radsimp(eq, symbolic=False) + 1/(a + b*sqrt(c)) + + """ + from sympy.core.expr import Expr + from sympy.simplify.simplify import signsimp + + syms = symbols("a:d A:D") + def _num(rterms): + # return the multiplier that will simplify the expression described + # by rterms [(sqrt arg, coeff), ... ] + a, b, c, d, A, B, C, D = syms + if len(rterms) == 2: + reps = dict(list(zip([A, a, B, b], [j for i in rterms for j in i]))) + return ( + sqrt(A)*a - sqrt(B)*b).xreplace(reps) + if len(rterms) == 3: + reps = dict(list(zip([A, a, B, b, C, c], [j for i in rterms for j in i]))) + return ( + (sqrt(A)*a + sqrt(B)*b - sqrt(C)*c)*(2*sqrt(A)*sqrt(B)*a*b - A*a**2 - + B*b**2 + C*c**2)).xreplace(reps) + elif len(rterms) == 4: + reps = dict(list(zip([A, a, B, b, C, c, D, d], [j for i in rterms for j in i]))) + return ((sqrt(A)*a + sqrt(B)*b - sqrt(C)*c - sqrt(D)*d)*(2*sqrt(A)*sqrt(B)*a*b + - A*a**2 - B*b**2 - 2*sqrt(C)*sqrt(D)*c*d + C*c**2 + + D*d**2)*(-8*sqrt(A)*sqrt(B)*sqrt(C)*sqrt(D)*a*b*c*d + A**2*a**4 - + 2*A*B*a**2*b**2 - 2*A*C*a**2*c**2 - 2*A*D*a**2*d**2 + B**2*b**4 - + 2*B*C*b**2*c**2 - 2*B*D*b**2*d**2 + C**2*c**4 - 2*C*D*c**2*d**2 + + D**2*d**4)).xreplace(reps) + elif len(rterms) == 1: + return sqrt(rterms[0][0]) + else: + raise NotImplementedError + + def ispow2(d, log2=False): + if not d.is_Pow: + return False + e = d.exp + if e.is_Rational and e.q == 2 or symbolic and denom(e) == 2: + return True + if log2: + q = 1 + if e.is_Rational: + q = e.q + elif symbolic: + d = denom(e) + if d.is_Integer: + q = d + if q != 1 and log(q, 2).is_Integer: + return True + return False + + def handle(expr): + # Handle first reduces to the case + # expr = 1/d, where d is an add, or d is base**p/2. + # We do this by recursively calling handle on each piece. + from sympy.simplify.simplify import nsimplify + + if expr.is_Atom: + return expr + elif not isinstance(expr, Expr): + return expr.func(*[handle(a) for a in expr.args]) + + n, d = fraction(expr) + + if d.is_Atom and n.is_Atom: + return expr + elif not n.is_Atom: + n = n.func(*[handle(a) for a in n.args]) + return _unevaluated_Mul(n, handle(1/d)) + elif n is not S.One: + return _unevaluated_Mul(n, handle(1/d)) + elif d.is_Mul: + return _unevaluated_Mul(*[handle(1/d) for d in d.args]) + + # By this step, expr is 1/d, and d is not a mul. + if not symbolic and d.free_symbols: + return expr + + if ispow2(d): + d2 = sqrtdenest(sqrt(d.base))**numer(d.exp) + if d2 != d: + return handle(1/d2) + elif d.is_Pow and (d.exp.is_integer or d.base.is_positive): + # (1/d**i) = (1/d)**i + return handle(1/d.base)**d.exp + + if not (d.is_Add or ispow2(d)): + return 1/d.func(*[handle(a) for a in d.args]) + + # handle 1/d treating d as an Add (though it may not be) + + keep = True # keep changes that are made + + # flatten it and collect radicals after checking for special + # conditions + d = _mexpand(d) + + # did it change? + if d.is_Atom: + return 1/d + + # is it a number that might be handled easily? + if d.is_number: + _d = nsimplify(d) + if _d.is_Number and _d.equals(d): + return 1/_d + + while True: + # collect similar terms + collected = defaultdict(list) + for m in Add.make_args(d): # d might have become non-Add + p2 = [] + other = [] + for i in Mul.make_args(m): + if ispow2(i, log2=True): + p2.append(i.base if i.exp is S.Half else i.base**(2*i.exp)) + elif i is S.ImaginaryUnit: + p2.append(S.NegativeOne) + else: + other.append(i) + collected[tuple(ordered(p2))].append(Mul(*other)) + rterms = list(ordered(list(collected.items()))) + rterms = [(Mul(*i), Add(*j)) for i, j in rterms] + nrad = len(rterms) - (1 if rterms[0][0] is S.One else 0) + if nrad < 1: + break + elif nrad > max_terms: + # there may have been invalid operations leading to this point + # so don't keep changes, e.g. this expression is troublesome + # in collecting terms so as not to raise the issue of 2834: + # r = sqrt(sqrt(5) + 5) + # eq = 1/(sqrt(5)*r + 2*sqrt(5)*sqrt(-sqrt(5) + 5) + 5*r) + keep = False + break + if len(rterms) > 4: + # in general, only 4 terms can be removed with repeated squaring + # but other considerations can guide selection of radical terms + # so that radicals are removed + if all(x.is_Integer and (y**2).is_Rational for x, y in rterms): + nd, d = rad_rationalize(S.One, Add._from_args( + [sqrt(x)*y for x, y in rterms])) + n *= nd + else: + # is there anything else that might be attempted? + keep = False + break + from sympy.simplify.powsimp import powsimp, powdenest + + num = powsimp(_num(rterms)) + n *= num + d *= num + d = powdenest(_mexpand(d), force=symbolic) + if d.has(S.Zero, nan, zoo): + return expr + if d.is_Atom: + break + + if not keep: + return expr + return _unevaluated_Mul(n, 1/d) + + if not isinstance(expr, Expr): + return expr.func(*[radsimp(a, symbolic=symbolic, max_terms=max_terms) for a in expr.args]) + + coeff, expr = expr.as_coeff_Add() + expr = expr.normal() + old = fraction(expr) + n, d = fraction(handle(expr)) + if old != (n, d): + if not d.is_Atom: + was = (n, d) + n = signsimp(n, evaluate=False) + d = signsimp(d, evaluate=False) + u = Factors(_unevaluated_Mul(n, 1/d)) + u = _unevaluated_Mul(*[k**v for k, v in u.factors.items()]) + n, d = fraction(u) + if old == (n, d): + n, d = was + n = expand_mul(n) + if d.is_Number or d.is_Add: + n2, d2 = fraction(gcd_terms(_unevaluated_Mul(n, 1/d))) + if d2.is_Number or (d2.count_ops() <= d.count_ops()): + n, d = [signsimp(i) for i in (n2, d2)] + if n.is_Mul and n.args[0].is_Number: + n = n.func(*n.args) + + return coeff + _unevaluated_Mul(n, 1/d) + + +def rad_rationalize(num, den): + """ + Rationalize ``num/den`` by removing square roots in the denominator; + num and den are sum of terms whose squares are positive rationals. + + Examples + ======== + + >>> from sympy import sqrt + >>> from sympy.simplify.radsimp import rad_rationalize + >>> rad_rationalize(sqrt(3), 1 + sqrt(2)/3) + (-sqrt(3) + sqrt(6)/3, -7/9) + """ + if not den.is_Add: + return num, den + g, a, b = split_surds(den) + a = a*sqrt(g) + num = _mexpand((a - b)*num) + den = _mexpand(a**2 - b**2) + return rad_rationalize(num, den) + + +def fraction(expr, exact=False): + """Returns a pair with expression's numerator and denominator. + If the given expression is not a fraction then this function + will return the tuple (expr, 1). + + This function will not make any attempt to simplify nested + fractions or to do any term rewriting at all. + + If only one of the numerator/denominator pair is needed then + use numer(expr) or denom(expr) functions respectively. + + >>> from sympy import fraction, Rational, Symbol + >>> from sympy.abc import x, y + + >>> fraction(x/y) + (x, y) + >>> fraction(x) + (x, 1) + + >>> fraction(1/y**2) + (1, y**2) + + >>> fraction(x*y/2) + (x*y, 2) + >>> fraction(Rational(1, 2)) + (1, 2) + + This function will also work fine with assumptions: + + >>> k = Symbol('k', negative=True) + >>> fraction(x * y**k) + (x, y**(-k)) + + If we know nothing about sign of some exponent and ``exact`` + flag is unset, then the exponent's structure will + be analyzed and pretty fraction will be returned: + + >>> from sympy import exp, Mul + >>> fraction(2*x**(-y)) + (2, x**y) + + >>> fraction(exp(-x)) + (1, exp(x)) + + >>> fraction(exp(-x), exact=True) + (exp(-x), 1) + + The ``exact`` flag will also keep any unevaluated Muls from + being evaluated: + + >>> u = Mul(2, x + 1, evaluate=False) + >>> fraction(u) + (2*x + 2, 1) + >>> fraction(u, exact=True) + (2*(x + 1), 1) + """ + expr = sympify(expr) + + numer, denom = [], [] + + for term in Mul.make_args(expr): + if term.is_commutative and (term.is_Pow or isinstance(term, exp)): + b, ex = term.as_base_exp() + if ex.is_negative: + if ex is S.NegativeOne: + denom.append(b) + elif exact: + if ex.is_constant(): + denom.append(Pow(b, -ex)) + else: + numer.append(term) + else: + denom.append(Pow(b, -ex)) + elif ex.is_positive: + numer.append(term) + elif not exact and ex.is_Mul: + n, d = term.as_numer_denom() # this will cause evaluation + if n != 1: + numer.append(n) + denom.append(d) + else: + numer.append(term) + elif term.is_Rational and not term.is_Integer: + if term.p != 1: + numer.append(term.p) + denom.append(term.q) + else: + numer.append(term) + return Mul(*numer, evaluate=not exact), Mul(*denom, evaluate=not exact) + + +def numer(expr, exact=False): # default matches fraction's default + return fraction(expr, exact=exact)[0] + + +def denom(expr, exact=False): # default matches fraction's default + return fraction(expr, exact=exact)[1] + + +def fraction_expand(expr, **hints): + return expr.expand(frac=True, **hints) + + +def numer_expand(expr, **hints): + # default matches fraction's default + a, b = fraction(expr, exact=hints.get('exact', False)) + return a.expand(numer=True, **hints) / b + + +def denom_expand(expr, **hints): + # default matches fraction's default + a, b = fraction(expr, exact=hints.get('exact', False)) + return a / b.expand(denom=True, **hints) + + +expand_numer = numer_expand +expand_denom = denom_expand +expand_fraction = fraction_expand + + +def split_surds(expr): + """ + Split an expression with terms whose squares are positive rationals + into a sum of terms whose surds squared have gcd equal to g + and a sum of terms with surds squared prime with g. + + Examples + ======== + + >>> from sympy import sqrt + >>> from sympy.simplify.radsimp import split_surds + >>> split_surds(3*sqrt(3) + sqrt(5)/7 + sqrt(6) + sqrt(10) + sqrt(15)) + (3, sqrt(2) + sqrt(5) + 3, sqrt(5)/7 + sqrt(10)) + """ + args = sorted(expr.args, key=default_sort_key) + coeff_muls = [x.as_coeff_Mul() for x in args] + surds = [x[1]**2 for x in coeff_muls if x[1].is_Pow] + surds.sort(key=default_sort_key) + g, b1, b2 = _split_gcd(*surds) + g2 = g + if not b2 and len(b1) >= 2: + b1n = [x/g for x in b1] + b1n = [x for x in b1n if x != 1] + # only a common factor has been factored; split again + g1, b1n, b2 = _split_gcd(*b1n) + g2 = g*g1 + a1v, a2v = [], [] + for c, s in coeff_muls: + if s.is_Pow and s.exp == S.Half: + s1 = s.base + if s1 in b1: + a1v.append(c*sqrt(s1/g2)) + else: + a2v.append(c*s) + else: + a2v.append(c*s) + a = Add(*a1v) + b = Add(*a2v) + return g2, a, b + + +def _split_gcd(*a): + """ + Split the list of integers ``a`` into a list of integers, ``a1`` having + ``g = gcd(a1)``, and a list ``a2`` whose elements are not divisible by + ``g``. Returns ``g, a1, a2``. + + Examples + ======== + + >>> from sympy.simplify.radsimp import _split_gcd + >>> _split_gcd(55, 35, 22, 14, 77, 10) + (5, [55, 35, 10], [22, 14, 77]) + """ + g = a[0] + b1 = [g] + b2 = [] + for x in a[1:]: + g1 = gcd(g, x) + if g1 == 1: + b2.append(x) + else: + g = g1 + b1.append(x) + return g, b1, b2 diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/ratsimp.py b/.venv/lib/python3.12/site-packages/sympy/simplify/ratsimp.py new file mode 100644 index 0000000000000000000000000000000000000000..95751fab47f585d3ae2e1289f014fba0f2708224 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/simplify/ratsimp.py @@ -0,0 +1,222 @@ +from itertools import combinations_with_replacement +from sympy.core import symbols, Add, Dummy +from sympy.core.numbers import Rational +from sympy.polys import cancel, ComputationFailed, parallel_poly_from_expr, reduced, Poly +from sympy.polys.monomials import Monomial, monomial_div +from sympy.polys.polyerrors import DomainError, PolificationFailed +from sympy.utilities.misc import debug, debugf + +def ratsimp(expr): + """ + Put an expression over a common denominator, cancel and reduce. + + Examples + ======== + + >>> from sympy import ratsimp + >>> from sympy.abc import x, y + >>> ratsimp(1/x + 1/y) + (x + y)/(x*y) + """ + + f, g = cancel(expr).as_numer_denom() + try: + Q, r = reduced(f, [g], field=True, expand=False) + except ComputationFailed: + return f/g + + return Add(*Q) + cancel(r/g) + + +def ratsimpmodprime(expr, G, *gens, quick=True, polynomial=False, **args): + """ + Simplifies a rational expression ``expr`` modulo the prime ideal + generated by ``G``. ``G`` should be a Groebner basis of the + ideal. + + Examples + ======== + + >>> from sympy.simplify.ratsimp import ratsimpmodprime + >>> from sympy.abc import x, y + >>> eq = (x + y**5 + y)/(x - y) + >>> ratsimpmodprime(eq, [x*y**5 - x - y], x, y, order='lex') + (-x**2 - x*y - x - y)/(-x**2 + x*y) + + If ``polynomial`` is ``False``, the algorithm computes a rational + simplification which minimizes the sum of the total degrees of + the numerator and the denominator. + + If ``polynomial`` is ``True``, this function just brings numerator and + denominator into a canonical form. This is much faster, but has + potentially worse results. + + References + ========== + + .. [1] M. Monagan, R. Pearce, Rational Simplification Modulo a Polynomial + Ideal, https://dl.acm.org/doi/pdf/10.1145/1145768.1145809 + (specifically, the second algorithm) + """ + from sympy.solvers.solvers import solve + + debug('ratsimpmodprime', expr) + + # usual preparation of polynomials: + + num, denom = cancel(expr).as_numer_denom() + + try: + polys, opt = parallel_poly_from_expr([num, denom] + G, *gens, **args) + except PolificationFailed: + return expr + + domain = opt.domain + + if domain.has_assoc_Field: + opt.domain = domain.get_field() + else: + raise DomainError( + "Cannot compute rational simplification over %s" % domain) + + # compute only once + leading_monomials = [g.LM(opt.order) for g in polys[2:]] + tested = set() + + def staircase(n): + """ + Compute all monomials with degree less than ``n`` that are + not divisible by any element of ``leading_monomials``. + """ + if n == 0: + return [1] + S = [] + for mi in combinations_with_replacement(range(len(opt.gens)), n): + m = [0]*len(opt.gens) + for i in mi: + m[i] += 1 + if all(monomial_div(m, lmg) is None for lmg in + leading_monomials): + S.append(m) + + return [Monomial(s).as_expr(*opt.gens) for s in S] + staircase(n - 1) + + def _ratsimpmodprime(a, b, allsol, N=0, D=0): + r""" + Computes a rational simplification of ``a/b`` which minimizes + the sum of the total degrees of the numerator and the denominator. + + Explanation + =========== + + The algorithm proceeds by looking at ``a * d - b * c`` modulo + the ideal generated by ``G`` for some ``c`` and ``d`` with degree + less than ``a`` and ``b`` respectively. + The coefficients of ``c`` and ``d`` are indeterminates and thus + the coefficients of the normalform of ``a * d - b * c`` are + linear polynomials in these indeterminates. + If these linear polynomials, considered as system of + equations, have a nontrivial solution, then `\frac{a}{b} + \equiv \frac{c}{d}` modulo the ideal generated by ``G``. So, + by construction, the degree of ``c`` and ``d`` is less than + the degree of ``a`` and ``b``, so a simpler representation + has been found. + After a simpler representation has been found, the algorithm + tries to reduce the degree of the numerator and denominator + and returns the result afterwards. + + As an extension, if quick=False, we look at all possible degrees such + that the total degree is less than *or equal to* the best current + solution. We retain a list of all solutions of minimal degree, and try + to find the best one at the end. + """ + c, d = a, b + steps = 0 + + maxdeg = a.total_degree() + b.total_degree() + if quick: + bound = maxdeg - 1 + else: + bound = maxdeg + while N + D <= bound: + if (N, D) in tested: + break + tested.add((N, D)) + + M1 = staircase(N) + M2 = staircase(D) + debugf('%s / %s: %s, %s', (N, D, M1, M2)) + + Cs = symbols("c:%d" % len(M1), cls=Dummy) + Ds = symbols("d:%d" % len(M2), cls=Dummy) + ng = Cs + Ds + + c_hat = Poly( + sum(Cs[i] * M1[i] for i in range(len(M1))), opt.gens + ng) + d_hat = Poly( + sum(Ds[i] * M2[i] for i in range(len(M2))), opt.gens + ng) + + r = reduced(a * d_hat - b * c_hat, G, opt.gens + ng, + order=opt.order, polys=True)[1] + + S = Poly(r, gens=opt.gens).coeffs() + sol = solve(S, Cs + Ds, particular=True, quick=True) + + if sol and not all(s == 0 for s in sol.values()): + c = c_hat.subs(sol) + d = d_hat.subs(sol) + + # The "free" variables occurring before as parameters + # might still be in the substituted c, d, so set them + # to the value chosen before: + c = c.subs(dict(list(zip(Cs + Ds, [1] * (len(Cs) + len(Ds)))))) + d = d.subs(dict(list(zip(Cs + Ds, [1] * (len(Cs) + len(Ds)))))) + + c = Poly(c, opt.gens) + d = Poly(d, opt.gens) + if d == 0: + raise ValueError('Ideal not prime?') + + allsol.append((c_hat, d_hat, S, Cs + Ds)) + if N + D != maxdeg: + allsol = [allsol[-1]] + + break + + steps += 1 + N += 1 + D += 1 + + if steps > 0: + c, d, allsol = _ratsimpmodprime(c, d, allsol, N, D - steps) + c, d, allsol = _ratsimpmodprime(c, d, allsol, N - steps, D) + + return c, d, allsol + + # preprocessing. this improves performance a bit when deg(num) + # and deg(denom) are large: + num = reduced(num, G, opt.gens, order=opt.order)[1] + denom = reduced(denom, G, opt.gens, order=opt.order)[1] + + if polynomial: + return (num/denom).cancel() + + c, d, allsol = _ratsimpmodprime( + Poly(num, opt.gens, domain=opt.domain), Poly(denom, opt.gens, domain=opt.domain), []) + if not quick and allsol: + debugf('Looking for best minimal solution. Got: %s', len(allsol)) + newsol = [] + for c_hat, d_hat, S, ng in allsol: + sol = solve(S, ng, particular=True, quick=False) + # all values of sol should be numbers; if not, solve is broken + newsol.append((c_hat.subs(sol), d_hat.subs(sol))) + c, d = min(newsol, key=lambda x: len(x[0].terms()) + len(x[1].terms())) + + if not domain.is_Field: + cn, c = c.clear_denoms(convert=True) + dn, d = d.clear_denoms(convert=True) + r = Rational(cn, dn) + else: + r = Rational(1) + + return (c*r.q)/(d*r.p) diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/simplify.py b/.venv/lib/python3.12/site-packages/sympy/simplify/simplify.py new file mode 100644 index 0000000000000000000000000000000000000000..8b315cc20c19fc10c37b903d16129a7f5579ecd3 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/simplify/simplify.py @@ -0,0 +1,2164 @@ +from __future__ import annotations + +from typing import overload + +from collections import defaultdict + +from sympy.concrete.products import Product +from sympy.concrete.summations import Sum +from sympy.core import (Basic, S, Add, Mul, Pow, Symbol, sympify, + expand_func, Function, Dummy, Expr, factor_terms, + expand_power_exp, Eq) +from sympy.core.exprtools import factor_nc +from sympy.core.parameters import global_parameters +from sympy.core.function import (expand_log, count_ops, _mexpand, + nfloat, expand_mul, expand) +from sympy.core.numbers import Float, I, pi, Rational, equal_valued +from sympy.core.relational import Relational +from sympy.core.rules import Transform +from sympy.core.sorting import ordered +from sympy.core.sympify import _sympify +from sympy.core.traversal import bottom_up as _bottom_up, walk as _walk +from sympy.functions import gamma, exp, sqrt, log, exp_polar, re +from sympy.functions.combinatorial.factorials import CombinatorialFunction +from sympy.functions.elementary.complexes import unpolarify, Abs, sign +from sympy.functions.elementary.exponential import ExpBase +from sympy.functions.elementary.hyperbolic import HyperbolicFunction +from sympy.functions.elementary.integers import ceiling +from sympy.functions.elementary.piecewise import (Piecewise, piecewise_fold, + piecewise_simplify) +from sympy.functions.elementary.trigonometric import TrigonometricFunction +from sympy.functions.special.bessel import (BesselBase, besselj, besseli, + besselk, bessely, jn) +from sympy.functions.special.tensor_functions import KroneckerDelta +from sympy.integrals.integrals import Integral +from sympy.logic.boolalg import Boolean +from sympy.matrices.expressions import (MatrixExpr, MatAdd, MatMul, + MatPow, MatrixSymbol) +from sympy.polys import together, cancel, factor +from sympy.polys.numberfields.minpoly import _is_sum_surds, _minimal_polynomial_sq +from sympy.sets.sets import Set +from sympy.simplify.combsimp import combsimp +from sympy.simplify.cse_opts import sub_pre, sub_post +from sympy.simplify.hyperexpand import hyperexpand +from sympy.simplify.powsimp import powsimp +from sympy.simplify.radsimp import radsimp, fraction, collect_abs +from sympy.simplify.sqrtdenest import sqrtdenest +from sympy.simplify.trigsimp import trigsimp, exptrigsimp +from sympy.utilities.decorator import deprecated +from sympy.utilities.iterables import has_variety, sift, subsets, iterable +from sympy.utilities.misc import as_int + +import mpmath + + +def separatevars(expr, symbols=[], dict=False, force=False): + """ + Separates variables in an expression, if possible. By + default, it separates with respect to all symbols in an + expression and collects constant coefficients that are + independent of symbols. + + Explanation + =========== + + If ``dict=True`` then the separated terms will be returned + in a dictionary keyed to their corresponding symbols. + By default, all symbols in the expression will appear as + keys; if symbols are provided, then all those symbols will + be used as keys, and any terms in the expression containing + other symbols or non-symbols will be returned keyed to the + string 'coeff'. (Passing None for symbols will return the + expression in a dictionary keyed to 'coeff'.) + + If ``force=True``, then bases of powers will be separated regardless + of assumptions on the symbols involved. + + Notes + ===== + + The order of the factors is determined by Mul, so that the + separated expressions may not necessarily be grouped together. + + Although factoring is necessary to separate variables in some + expressions, it is not necessary in all cases, so one should not + count on the returned factors being factored. + + Examples + ======== + + >>> from sympy.abc import x, y, z, alpha + >>> from sympy import separatevars, sin + >>> separatevars((x*y)**y) + (x*y)**y + >>> separatevars((x*y)**y, force=True) + x**y*y**y + + >>> e = 2*x**2*z*sin(y)+2*z*x**2 + >>> separatevars(e) + 2*x**2*z*(sin(y) + 1) + >>> separatevars(e, symbols=(x, y), dict=True) + {'coeff': 2*z, x: x**2, y: sin(y) + 1} + >>> separatevars(e, [x, y, alpha], dict=True) + {'coeff': 2*z, alpha: 1, x: x**2, y: sin(y) + 1} + + If the expression is not really separable, or is only partially + separable, separatevars will do the best it can to separate it + by using factoring. + + >>> separatevars(x + x*y - 3*x**2) + -x*(3*x - y - 1) + + If the expression is not separable then expr is returned unchanged + or (if dict=True) then None is returned. + + >>> eq = 2*x + y*sin(x) + >>> separatevars(eq) == eq + True + >>> separatevars(2*x + y*sin(x), symbols=(x, y), dict=True) is None + True + + """ + expr = sympify(expr) + if dict: + return _separatevars_dict(_separatevars(expr, force), symbols) + else: + return _separatevars(expr, force) + + +def _separatevars(expr, force): + if isinstance(expr, Abs): + arg = expr.args[0] + if arg.is_Mul and not arg.is_number: + s = separatevars(arg, dict=True, force=force) + if s is not None: + return Mul(*map(expr.func, s.values())) + else: + return expr + + if len(expr.free_symbols) < 2: + return expr + + # don't destroy a Mul since much of the work may already be done + if expr.is_Mul: + args = list(expr.args) + changed = False + for i, a in enumerate(args): + args[i] = separatevars(a, force) + changed = changed or args[i] != a + if changed: + expr = expr.func(*args) + return expr + + # get a Pow ready for expansion + if expr.is_Pow and expr.base != S.Exp1: + expr = Pow(separatevars(expr.base, force=force), expr.exp) + + # First try other expansion methods + expr = expr.expand(mul=False, multinomial=False, force=force) + + _expr, reps = posify(expr) if force else (expr, {}) + expr = factor(_expr).subs(reps) + + if not expr.is_Add: + return expr + + # Find any common coefficients to pull out + args = list(expr.args) + commonc = args[0].args_cnc(cset=True, warn=False)[0] + for i in args[1:]: + commonc &= i.args_cnc(cset=True, warn=False)[0] + commonc = Mul(*commonc) + commonc = commonc.as_coeff_Mul()[1] # ignore constants + commonc_set = commonc.args_cnc(cset=True, warn=False)[0] + + # remove them + for i, a in enumerate(args): + c, nc = a.args_cnc(cset=True, warn=False) + c = c - commonc_set + args[i] = Mul(*c)*Mul(*nc) + nonsepar = Add(*args) + + if len(nonsepar.free_symbols) > 1: + _expr = nonsepar + _expr, reps = posify(_expr) if force else (_expr, {}) + _expr = (factor(_expr)).subs(reps) + + if not _expr.is_Add: + nonsepar = _expr + + return commonc*nonsepar + + +def _separatevars_dict(expr, symbols): + if symbols: + if not all(t.is_Atom for t in symbols): + raise ValueError("symbols must be Atoms.") + symbols = list(symbols) + elif symbols is None: + return {'coeff': expr} + else: + symbols = list(expr.free_symbols) + if not symbols: + return None + + ret = {i: [] for i in symbols + ['coeff']} + + for i in Mul.make_args(expr): + expsym = i.free_symbols + intersection = set(symbols).intersection(expsym) + if len(intersection) > 1: + return None + if len(intersection) == 0: + # There are no symbols, so it is part of the coefficient + ret['coeff'].append(i) + else: + ret[intersection.pop()].append(i) + + # rebuild + for k, v in ret.items(): + ret[k] = Mul(*v) + + return ret + + +def posify(eq): + """Return ``eq`` (with generic symbols made positive) and a + dictionary containing the mapping between the old and new + symbols. + + Explanation + =========== + + Any symbol that has positive=None will be replaced with a positive dummy + symbol having the same name. This replacement will allow more symbolic + processing of expressions, especially those involving powers and + logarithms. + + A dictionary that can be sent to subs to restore ``eq`` to its original + symbols is also returned. + + >>> from sympy import posify, Symbol, log, solve + >>> from sympy.abc import x + >>> posify(x + Symbol('p', positive=True) + Symbol('n', negative=True)) + (_x + n + p, {_x: x}) + + >>> eq = 1/x + >>> log(eq).expand() + log(1/x) + >>> log(posify(eq)[0]).expand() + -log(_x) + >>> p, rep = posify(eq) + >>> log(p).expand().subs(rep) + -log(x) + + It is possible to apply the same transformations to an iterable + of expressions: + + >>> eq = x**2 - 4 + >>> solve(eq, x) + [-2, 2] + >>> eq_x, reps = posify([eq, x]); eq_x + [_x**2 - 4, _x] + >>> solve(*eq_x) + [2] + """ + eq = sympify(eq) + if not isinstance(eq, Basic) and iterable(eq): + f = type(eq) + eq = list(eq) + syms = set() + for e in eq: + syms = syms.union(e.atoms(Symbol)) + reps = {} + for s in syms: + reps.update({v: k for k, v in posify(s)[1].items()}) + for i, e in enumerate(eq): + eq[i] = e.subs(reps) + return f(eq), {r: s for s, r in reps.items()} + + reps = {s: Dummy(s.name, positive=True, **s.assumptions0) + for s in eq.free_symbols if s.is_positive is None} + eq = eq.subs(reps) + return eq, {r: s for s, r in reps.items()} + + +def hypersimp(f, k): + """Given combinatorial term f(k) simplify its consecutive term ratio + i.e. f(k+1)/f(k). The input term can be composed of functions and + integer sequences which have equivalent representation in terms + of gamma special function. + + Explanation + =========== + + The algorithm performs three basic steps: + + 1. Rewrite all functions in terms of gamma, if possible. + + 2. Rewrite all occurrences of gamma in terms of products + of gamma and rising factorial with integer, absolute + constant exponent. + + 3. Perform simplification of nested fractions, powers + and if the resulting expression is a quotient of + polynomials, reduce their total degree. + + If f(k) is hypergeometric then as result we arrive with a + quotient of polynomials of minimal degree. Otherwise None + is returned. + + For more information on the implemented algorithm refer to: + + 1. W. Koepf, Algorithms for m-fold Hypergeometric Summation, + Journal of Symbolic Computation (1995) 20, 399-417 + """ + f = sympify(f) + + g = f.subs(k, k + 1) / f + + g = g.rewrite(gamma) + if g.has(Piecewise): + g = piecewise_fold(g) + g = g.args[-1][0] + g = expand_func(g) + g = powsimp(g, deep=True, combine='exp') + + if g.is_rational_function(k): + return simplify(g, ratio=S.Infinity) + else: + return None + + +def hypersimilar(f, g, k): + """ + Returns True if ``f`` and ``g`` are hyper-similar. + + Explanation + =========== + + Similarity in hypergeometric sense means that a quotient of + f(k) and g(k) is a rational function in ``k``. This procedure + is useful in solving recurrence relations. + + For more information see hypersimp(). + + """ + f, g = list(map(sympify, (f, g))) + + h = (f/g).rewrite(gamma) + h = h.expand(func=True, basic=False) + + return h.is_rational_function(k) + + +def signsimp(expr, evaluate=None): + """Make all Add sub-expressions canonical wrt sign. + + Explanation + =========== + + If an Add subexpression, ``a``, can have a sign extracted, + as determined by could_extract_minus_sign, it is replaced + with Mul(-1, a, evaluate=False). This allows signs to be + extracted from powers and products. + + Examples + ======== + + >>> from sympy import signsimp, exp, symbols + >>> from sympy.abc import x, y + >>> i = symbols('i', odd=True) + >>> n = -1 + 1/x + >>> n/x/(-n)**2 - 1/n/x + (-1 + 1/x)/(x*(1 - 1/x)**2) - 1/(x*(-1 + 1/x)) + >>> signsimp(_) + 0 + >>> x*n + x*-n + x*(-1 + 1/x) + x*(1 - 1/x) + >>> signsimp(_) + 0 + + Since powers automatically handle leading signs + + >>> (-2)**i + -2**i + + signsimp can be used to put the base of a power with an integer + exponent into canonical form: + + >>> n**i + (-1 + 1/x)**i + + By default, signsimp does not leave behind any hollow simplification: + if making an Add canonical wrt sign didn't change the expression, the + original Add is restored. If this is not desired then the keyword + ``evaluate`` can be set to False: + + >>> e = exp(y - x) + >>> signsimp(e) == e + True + >>> signsimp(e, evaluate=False) + exp(-(x - y)) + + """ + if evaluate is None: + evaluate = global_parameters.evaluate + expr = sympify(expr) + if not isinstance(expr, (Expr, Relational)) or expr.is_Atom: + return expr + # get rid of an pre-existing unevaluation regarding sign + e = expr.replace(lambda x: x.is_Mul and -(-x) != x, lambda x: -(-x)) + e = sub_post(sub_pre(e)) + if not isinstance(e, (Expr, Relational)) or e.is_Atom: + return e + if e.is_Add: + rv = e.func(*[signsimp(a) for a in e.args]) + if not evaluate and isinstance(rv, Add + ) and rv.could_extract_minus_sign(): + return Mul(S.NegativeOne, -rv, evaluate=False) + return rv + if evaluate: + e = e.replace(lambda x: x.is_Mul and -(-x) != x, lambda x: -(-x)) + return e + + +@overload +def simplify(expr: Expr, **kwargs) -> Expr: ... +@overload +def simplify(expr: Boolean, **kwargs) -> Boolean: ... +@overload +def simplify(expr: Set, **kwargs) -> Set: ... +@overload +def simplify(expr: Basic, **kwargs) -> Basic: ... + +def simplify(expr, ratio=1.7, measure=count_ops, rational=False, inverse=False, doit=True, **kwargs): + """Simplifies the given expression. + + Explanation + =========== + + Simplification is not a well defined term and the exact strategies + this function tries can change in the future versions of SymPy. If + your algorithm relies on "simplification" (whatever it is), try to + determine what you need exactly - is it powsimp()?, radsimp()?, + together()?, logcombine()?, or something else? And use this particular + function directly, because those are well defined and thus your algorithm + will be robust. + + Nonetheless, especially for interactive use, or when you do not know + anything about the structure of the expression, simplify() tries to apply + intelligent heuristics to make the input expression "simpler". For + example: + + >>> from sympy import simplify, cos, sin + >>> from sympy.abc import x, y + >>> a = (x + x**2)/(x*sin(y)**2 + x*cos(y)**2) + >>> a + (x**2 + x)/(x*sin(y)**2 + x*cos(y)**2) + >>> simplify(a) + x + 1 + + Note that we could have obtained the same result by using specific + simplification functions: + + >>> from sympy import trigsimp, cancel + >>> trigsimp(a) + (x**2 + x)/x + >>> cancel(_) + x + 1 + + In some cases, applying :func:`simplify` may actually result in some more + complicated expression. The default ``ratio=1.7`` prevents more extreme + cases: if (result length)/(input length) > ratio, then input is returned + unmodified. The ``measure`` parameter lets you specify the function used + to determine how complex an expression is. The function should take a + single argument as an expression and return a number such that if + expression ``a`` is more complex than expression ``b``, then + ``measure(a) > measure(b)``. The default measure function is + :func:`~.count_ops`, which returns the total number of operations in the + expression. + + For example, if ``ratio=1``, ``simplify`` output cannot be longer + than input. + + :: + + >>> from sympy import sqrt, simplify, count_ops, oo + >>> root = 1/(sqrt(2)+3) + + Since ``simplify(root)`` would result in a slightly longer expression, + root is returned unchanged instead:: + + >>> simplify(root, ratio=1) == root + True + + If ``ratio=oo``, simplify will be applied anyway:: + + >>> count_ops(simplify(root, ratio=oo)) > count_ops(root) + True + + Note that the shortest expression is not necessary the simplest, so + setting ``ratio`` to 1 may not be a good idea. + Heuristically, the default value ``ratio=1.7`` seems like a reasonable + choice. + + You can easily define your own measure function based on what you feel + should represent the "size" or "complexity" of the input expression. Note + that some choices, such as ``lambda expr: len(str(expr))`` may appear to be + good metrics, but have other problems (in this case, the measure function + may slow down simplify too much for very large expressions). If you do not + know what a good metric would be, the default, ``count_ops``, is a good + one. + + For example: + + >>> from sympy import symbols, log + >>> a, b = symbols('a b', positive=True) + >>> g = log(a) + log(b) + log(a)*log(1/b) + >>> h = simplify(g) + >>> h + log(a*b**(1 - log(a))) + >>> count_ops(g) + 8 + >>> count_ops(h) + 5 + + So you can see that ``h`` is simpler than ``g`` using the count_ops metric. + However, we may not like how ``simplify`` (in this case, using + ``logcombine``) has created the ``b**(log(1/a) + 1)`` term. A simple way + to reduce this would be to give more weight to powers as operations in + ``count_ops``. We can do this by using the ``visual=True`` option: + + >>> print(count_ops(g, visual=True)) + 2*ADD + DIV + 4*LOG + MUL + >>> print(count_ops(h, visual=True)) + 2*LOG + MUL + POW + SUB + + >>> from sympy import Symbol, S + >>> def my_measure(expr): + ... POW = Symbol('POW') + ... # Discourage powers by giving POW a weight of 10 + ... count = count_ops(expr, visual=True).subs(POW, 10) + ... # Every other operation gets a weight of 1 (the default) + ... count = count.replace(Symbol, type(S.One)) + ... return count + >>> my_measure(g) + 8 + >>> my_measure(h) + 14 + >>> 15./8 > 1.7 # 1.7 is the default ratio + True + >>> simplify(g, measure=my_measure) + -log(a)*log(b) + log(a) + log(b) + + Note that because ``simplify()`` internally tries many different + simplification strategies and then compares them using the measure + function, we get a completely different result that is still different + from the input expression by doing this. + + If ``rational=True``, Floats will be recast as Rationals before simplification. + If ``rational=None``, Floats will be recast as Rationals but the result will + be recast as Floats. If rational=False(default) then nothing will be done + to the Floats. + + If ``inverse=True``, it will be assumed that a composition of inverse + functions, such as sin and asin, can be cancelled in any order. + For example, ``asin(sin(x))`` will yield ``x`` without checking whether + x belongs to the set where this relation is true. The default is + False. + + Note that ``simplify()`` automatically calls ``doit()`` on the final + expression. You can avoid this behavior by passing ``doit=False`` as + an argument. + + Also, it should be noted that simplifying a boolean expression is not + well defined. If the expression prefers automatic evaluation (such as + :obj:`~.Eq()` or :obj:`~.Or()`), simplification will return ``True`` or + ``False`` if truth value can be determined. If the expression is not + evaluated by default (such as :obj:`~.Predicate()`), simplification will + not reduce it and you should use :func:`~.refine` or :func:`~.ask` + function. This inconsistency will be resolved in future version. + + See Also + ======== + + sympy.assumptions.refine.refine : Simplification using assumptions. + sympy.assumptions.ask.ask : Query for boolean expressions using assumptions. + """ + + def shorter(*choices): + """ + Return the choice that has the fewest ops. In case of a tie, + the expression listed first is selected. + """ + if not has_variety(choices): + return choices[0] + return min(choices, key=measure) + + def done(e): + rv = e.doit() if doit else e + return shorter(rv, collect_abs(rv)) + + expr = sympify(expr, rational=rational) + kwargs = { + "ratio": kwargs.get('ratio', ratio), + "measure": kwargs.get('measure', measure), + "rational": kwargs.get('rational', rational), + "inverse": kwargs.get('inverse', inverse), + "doit": kwargs.get('doit', doit)} + # no routine for Expr needs to check for is_zero + if isinstance(expr, Expr) and expr.is_zero: + return S.Zero if not expr.is_Number else expr + + _eval_simplify = getattr(expr, '_eval_simplify', None) + if _eval_simplify is not None: + return _eval_simplify(**kwargs) + + original_expr = expr = collect_abs(signsimp(expr)) + + if not isinstance(expr, Basic) or not expr.args: # XXX: temporary hack + return expr + + if inverse and expr.has(Function): + expr = inversecombine(expr) + if not expr.args: # simplified to atomic + return expr + + # do deep simplification + handled = Add, Mul, Pow, ExpBase + expr = expr.replace( + # here, checking for x.args is not enough because Basic has + # args but Basic does not always play well with replace, e.g. + # when simultaneous is True found expressions will be masked + # off with a Dummy but not all Basic objects in an expression + # can be replaced with a Dummy + lambda x: isinstance(x, Expr) and x.args and not isinstance( + x, handled), + lambda x: x.func(*[simplify(i, **kwargs) for i in x.args]), + simultaneous=False) + if not isinstance(expr, handled): + return done(expr) + + if not expr.is_commutative: + expr = nc_simplify(expr) + + # TODO: Apply different strategies, considering expression pattern: + # is it a purely rational function? Is there any trigonometric function?... + # See also https://github.com/sympy/sympy/pull/185. + + # rationalize Floats + floats = False + if rational is not False and expr.has(Float): + floats = True + expr = nsimplify(expr, rational=True) + + expr = _bottom_up(expr, lambda w: getattr(w, 'normal', lambda: w)()) + expr = Mul(*powsimp(expr).as_content_primitive()) + _e = cancel(expr) + expr1 = shorter(_e, _mexpand(_e).cancel()) # issue 6829 + expr2 = shorter(together(expr, deep=True), together(expr1, deep=True)) + + if ratio is S.Infinity: + expr = expr2 + else: + expr = shorter(expr2, expr1, expr) + if not isinstance(expr, Basic): # XXX: temporary hack + return expr + + expr = factor_terms(expr, sign=False) + + # must come before `Piecewise` since this introduces more `Piecewise` terms + if expr.has(sign): + expr = expr.rewrite(Abs) + + # Deal with Piecewise separately to avoid recursive growth of expressions + if expr.has(Piecewise): + # Fold into a single Piecewise + expr = piecewise_fold(expr) + # Apply doit, if doit=True + expr = done(expr) + # Still a Piecewise? + if expr.has(Piecewise): + # Fold into a single Piecewise, in case doit lead to some + # expressions being Piecewise + expr = piecewise_fold(expr) + # kroneckersimp also affects Piecewise + if expr.has(KroneckerDelta): + expr = kroneckersimp(expr) + # Still a Piecewise? + if expr.has(Piecewise): + # Do not apply doit on the segments as it has already + # been done above, but simplify + expr = piecewise_simplify(expr, deep=True, doit=False) + # Still a Piecewise? + if expr.has(Piecewise): + # Try factor common terms + expr = shorter(expr, factor_terms(expr)) + # As all expressions have been simplified above with the + # complete simplify, nothing more needs to be done here + return expr + + # hyperexpand automatically only works on hypergeometric terms + # Do this after the Piecewise part to avoid recursive expansion + expr = hyperexpand(expr) + + if expr.has(KroneckerDelta): + expr = kroneckersimp(expr) + + if expr.has(BesselBase): + expr = besselsimp(expr) + + if expr.has(TrigonometricFunction, HyperbolicFunction): + expr = trigsimp(expr, deep=True) + + if expr.has(log): + expr = shorter(expand_log(expr, deep=True), logcombine(expr)) + + if expr.has(CombinatorialFunction, gamma): + # expression with gamma functions or non-integer arguments is + # automatically passed to gammasimp + expr = combsimp(expr) + + if expr.has(Sum): + expr = sum_simplify(expr, **kwargs) + + if expr.has(Integral): + expr = expr.xreplace({ + i: factor_terms(i) for i in expr.atoms(Integral)}) + + if expr.has(Product): + expr = product_simplify(expr, **kwargs) + + from sympy.physics.units import Quantity + + if expr.has(Quantity): + from sympy.physics.units.util import quantity_simplify + expr = quantity_simplify(expr) + + short = shorter(powsimp(expr, combine='exp', deep=True), powsimp(expr), expr) + short = shorter(short, cancel(short)) + short = shorter(short, factor_terms(short), expand_power_exp(expand_mul(short))) + if short.has(TrigonometricFunction, HyperbolicFunction, ExpBase, exp): + short = exptrigsimp(short) + + # get rid of hollow 2-arg Mul factorization + hollow_mul = Transform( + lambda x: Mul(*x.args), + lambda x: + x.is_Mul and + len(x.args) == 2 and + x.args[0].is_Number and + x.args[1].is_Add and + x.is_commutative) + expr = short.xreplace(hollow_mul) + + numer, denom = expr.as_numer_denom() + if denom.is_Add: + n, d = fraction(radsimp(1/denom, symbolic=False, max_terms=1)) + if n is not S.One: + expr = (numer*n).expand()/d + + if expr.could_extract_minus_sign(): + n, d = fraction(expr) + if d != 0: + expr = signsimp(-n/(-d)) + + if measure(expr) > ratio*measure(original_expr): + expr = original_expr + + # restore floats + if floats and rational is None: + expr = nfloat(expr, exponent=False) + + return done(expr) + + +def sum_simplify(s, **kwargs): + """Main function for Sum simplification""" + if not isinstance(s, Add): + s = s.xreplace({a: sum_simplify(a, **kwargs) + for a in s.atoms(Add) if a.has(Sum)}) + s = expand(s) + if not isinstance(s, Add): + return s + + terms = s.args + s_t = [] # Sum Terms + o_t = [] # Other Terms + + for term in terms: + sum_terms, other = sift(Mul.make_args(term), + lambda i: isinstance(i, Sum), binary=True) + if not sum_terms: + o_t.append(term) + continue + other = [Mul(*other)] + s_t.append(Mul(*(other + [s._eval_simplify(**kwargs) for s in sum_terms]))) + + result = Add(sum_combine(s_t), *o_t) + + return result + + +def sum_combine(s_t): + """Helper function for Sum simplification + + Attempts to simplify a list of sums, by combining limits / sum function's + returns the simplified sum + """ + used = [False] * len(s_t) + + for method in range(2): + for i, s_term1 in enumerate(s_t): + if not used[i]: + for j, s_term2 in enumerate(s_t): + if not used[j] and i != j: + temp = sum_add(s_term1, s_term2, method) + if isinstance(temp, (Sum, Mul)): + s_t[i] = temp + s_term1 = s_t[i] + used[j] = True + + result = S.Zero + for i, s_term in enumerate(s_t): + if not used[i]: + result = Add(result, s_term) + + return result + +def factor_sum(self, limits=None, radical=False, clear=False, fraction=False, sign=True): + """Return Sum with constant factors extracted. + + If ``limits`` is specified then ``self`` is the summand; the other + keywords are passed to ``factor_terms``. + + Examples + ======== + + >>> from sympy import Sum + >>> from sympy.abc import x, y + >>> from sympy.simplify.simplify import factor_sum + >>> s = Sum(x*y, (x, 1, 3)) + >>> factor_sum(s) + y*Sum(x, (x, 1, 3)) + >>> factor_sum(s.function, s.limits) + y*Sum(x, (x, 1, 3)) + """ + + # XXX deprecate in favor of direct call to factor_terms + kwargs = {"radical": radical, "clear": clear, + "fraction": fraction, "sign": sign} + expr = Sum(self, *limits) if limits else self + return factor_terms(expr, **kwargs) + + +def sum_add(self, other, method=0): + """Helper function for Sum simplification""" + #we know this is something in terms of a constant * a sum + #so we temporarily put the constants inside for simplification + #then simplify the result + def __refactor(val): + args = Mul.make_args(val) + sumv = next(x for x in args if isinstance(x, Sum)) + constant = Mul(*[x for x in args if x != sumv]) + return Sum(constant * sumv.function, *sumv.limits) + + if isinstance(self, Mul): + rself = __refactor(self) + else: + rself = self + + if isinstance(other, Mul): + rother = __refactor(other) + else: + rother = other + + if type(rself) is type(rother): + if method == 0: + if rself.limits == rother.limits: + return factor_sum(Sum(rself.function + rother.function, *rself.limits)) + elif method == 1: + if simplify(rself.function - rother.function) == 0: + if len(rself.limits) == len(rother.limits) == 1: + i = rself.limits[0][0] + x1 = rself.limits[0][1] + y1 = rself.limits[0][2] + j = rother.limits[0][0] + x2 = rother.limits[0][1] + y2 = rother.limits[0][2] + + if i == j: + if x2 == y1 + 1: + return factor_sum(Sum(rself.function, (i, x1, y2))) + elif x1 == y2 + 1: + return factor_sum(Sum(rself.function, (i, x2, y1))) + + return Add(self, other) + + +def product_simplify(s, **kwargs): + """Main function for Product simplification""" + terms = Mul.make_args(s) + p_t = [] # Product Terms + o_t = [] # Other Terms + + deep = kwargs.get('deep', True) + for term in terms: + if isinstance(term, Product): + if deep: + p_t.append(Product(term.function.simplify(**kwargs), + *term.limits)) + else: + p_t.append(term) + else: + o_t.append(term) + + used = [False] * len(p_t) + + for method in range(2): + for i, p_term1 in enumerate(p_t): + if not used[i]: + for j, p_term2 in enumerate(p_t): + if not used[j] and i != j: + tmp_prod = product_mul(p_term1, p_term2, method) + if isinstance(tmp_prod, Product): + p_t[i] = tmp_prod + used[j] = True + + result = Mul(*o_t) + + for i, p_term in enumerate(p_t): + if not used[i]: + result = Mul(result, p_term) + + return result + + +def product_mul(self, other, method=0): + """Helper function for Product simplification""" + if type(self) is type(other): + if method == 0: + if self.limits == other.limits: + return Product(self.function * other.function, *self.limits) + elif method == 1: + if simplify(self.function - other.function) == 0: + if len(self.limits) == len(other.limits) == 1: + i = self.limits[0][0] + x1 = self.limits[0][1] + y1 = self.limits[0][2] + j = other.limits[0][0] + x2 = other.limits[0][1] + y2 = other.limits[0][2] + + if i == j: + if x2 == y1 + 1: + return Product(self.function, (i, x1, y2)) + elif x1 == y2 + 1: + return Product(self.function, (i, x2, y1)) + + return Mul(self, other) + + +def _nthroot_solve(p, n, prec): + """ + helper function for ``nthroot`` + It denests ``p**Rational(1, n)`` using its minimal polynomial + """ + from sympy.solvers import solve + while n % 2 == 0: + p = sqrtdenest(sqrt(p)) + n = n // 2 + if n == 1: + return p + pn = p**Rational(1, n) + x = Symbol('x') + f = _minimal_polynomial_sq(p, n, x) + if f is None: + return None + sols = solve(f, x) + for sol in sols: + if abs(sol - pn).n() < 1./10**prec: + sol = sqrtdenest(sol) + if _mexpand(sol**n) == p: + return sol + + +def logcombine(expr, force=False): + """ + Takes logarithms and combines them using the following rules: + + - log(x) + log(y) == log(x*y) if both are positive + - a*log(x) == log(x**a) if x is positive and a is real + + If ``force`` is ``True`` then the assumptions above will be assumed to hold if + there is no assumption already in place on a quantity. For example, if + ``a`` is imaginary or the argument negative, force will not perform a + combination but if ``a`` is a symbol with no assumptions the change will + take place. + + Examples + ======== + + >>> from sympy import Symbol, symbols, log, logcombine, I + >>> from sympy.abc import a, x, y, z + >>> logcombine(a*log(x) + log(y) - log(z)) + a*log(x) + log(y) - log(z) + >>> logcombine(a*log(x) + log(y) - log(z), force=True) + log(x**a*y/z) + >>> x,y,z = symbols('x,y,z', positive=True) + >>> a = Symbol('a', real=True) + >>> logcombine(a*log(x) + log(y) - log(z)) + log(x**a*y/z) + + The transformation is limited to factors and/or terms that + contain logs, so the result depends on the initial state of + expansion: + + >>> eq = (2 + 3*I)*log(x) + >>> logcombine(eq, force=True) == eq + True + >>> logcombine(eq.expand(), force=True) + log(x**2) + I*log(x**3) + + See Also + ======== + + posify: replace all symbols with symbols having positive assumptions + sympy.core.function.expand_log: expand the logarithms of products + and powers; the opposite of logcombine + + """ + + def f(rv): + if not (rv.is_Add or rv.is_Mul): + return rv + + def gooda(a): + # bool to tell whether the leading ``a`` in ``a*log(x)`` + # could appear as log(x**a) + return (a is not S.NegativeOne and # -1 *could* go, but we disallow + (a.is_extended_real or force and a.is_extended_real is not False)) + + def goodlog(l): + # bool to tell whether log ``l``'s argument can combine with others + a = l.args[0] + return a.is_positive or force and a.is_nonpositive is not False + + other = [] + logs = [] + log1 = defaultdict(list) + for a in Add.make_args(rv): + if isinstance(a, log) and goodlog(a): + log1[()].append(([], a)) + elif not a.is_Mul: + other.append(a) + else: + ot = [] + co = [] + lo = [] + for ai in a.args: + if ai.is_Rational and ai < 0: + ot.append(S.NegativeOne) + co.append(-ai) + elif isinstance(ai, log) and goodlog(ai): + lo.append(ai) + elif gooda(ai): + co.append(ai) + else: + ot.append(ai) + if len(lo) > 1: + logs.append((ot, co, lo)) + elif lo: + log1[tuple(ot)].append((co, lo[0])) + else: + other.append(a) + + # if there is only one log in other, put it with the + # good logs + if len(other) == 1 and isinstance(other[0], log): + log1[()].append(([], other.pop())) + # if there is only one log at each coefficient and none have + # an exponent to place inside the log then there is nothing to do + if not logs and all(len(log1[k]) == 1 and log1[k][0] == [] for k in log1): + return rv + + # collapse multi-logs as far as possible in a canonical way + # TODO: see if x*log(a)+x*log(a)*log(b) -> x*log(a)*(1+log(b))? + # -- in this case, it's unambiguous, but if it were were a log(c) in + # each term then it's arbitrary whether they are grouped by log(a) or + # by log(c). So for now, just leave this alone; it's probably better to + # let the user decide + for o, e, l in logs: + l = list(ordered(l)) + e = log(l.pop(0).args[0]**Mul(*e)) + while l: + li = l.pop(0) + e = log(li.args[0]**e) + c, l = Mul(*o), e + if isinstance(l, log): # it should be, but check to be sure + log1[(c,)].append(([], l)) + else: + other.append(c*l) + + # logs that have the same coefficient can multiply + for k in list(log1.keys()): + log1[Mul(*k)] = log(logcombine(Mul(*[ + l.args[0]**Mul(*c) for c, l in log1.pop(k)]), + force=force), evaluate=False) + + # logs that have oppositely signed coefficients can divide + for k in ordered(list(log1.keys())): + if k not in log1: # already popped as -k + continue + if -k in log1: + # figure out which has the minus sign; the one with + # more op counts should be the one + num, den = k, -k + if num.count_ops() > den.count_ops(): + num, den = den, num + other.append( + num*log(log1.pop(num).args[0]/log1.pop(den).args[0], + evaluate=False)) + else: + other.append(k*log1.pop(k)) + + return Add(*other) + + return _bottom_up(expr, f) + + +def inversecombine(expr): + """Simplify the composition of a function and its inverse. + + Explanation + =========== + + No attention is paid to whether the inverse is a left inverse or a + right inverse; thus, the result will in general not be equivalent + to the original expression. + + Examples + ======== + + >>> from sympy.simplify.simplify import inversecombine + >>> from sympy import asin, sin, log, exp + >>> from sympy.abc import x + >>> inversecombine(asin(sin(x))) + x + >>> inversecombine(2*log(exp(3*x))) + 6*x + """ + + def f(rv): + if isinstance(rv, log): + if isinstance(rv.args[0], exp) or (rv.args[0].is_Pow and rv.args[0].base == S.Exp1): + rv = rv.args[0].exp + elif rv.is_Function and hasattr(rv, "inverse"): + if (len(rv.args) == 1 and len(rv.args[0].args) == 1 and + isinstance(rv.args[0], rv.inverse(argindex=1))): + rv = rv.args[0].args[0] + if rv.is_Pow and rv.base == S.Exp1: + if isinstance(rv.exp, log): + rv = rv.exp.args[0] + return rv + + return _bottom_up(expr, f) + + +def kroneckersimp(expr): + """ + Simplify expressions with KroneckerDelta. + + The only simplification currently attempted is to identify multiplicative cancellation: + + Examples + ======== + + >>> from sympy import KroneckerDelta, kroneckersimp + >>> from sympy.abc import i + >>> kroneckersimp(1 + KroneckerDelta(0, i) * KroneckerDelta(1, i)) + 1 + """ + def args_cancel(args1, args2): + for i1 in range(2): + for i2 in range(2): + a1 = args1[i1] + a2 = args2[i2] + a3 = args1[(i1 + 1) % 2] + a4 = args2[(i2 + 1) % 2] + if Eq(a1, a2) is S.true and Eq(a3, a4) is S.false: + return True + return False + + def cancel_kronecker_mul(m): + args = m.args + deltas = [a for a in args if isinstance(a, KroneckerDelta)] + for delta1, delta2 in subsets(deltas, 2): + args1 = delta1.args + args2 = delta2.args + if args_cancel(args1, args2): + return S.Zero * m # In case of oo etc + return m + + if not expr.has(KroneckerDelta): + return expr + + if expr.has(Piecewise): + expr = expr.rewrite(KroneckerDelta) + + newexpr = expr + expr = None + + while newexpr != expr: + expr = newexpr + newexpr = expr.replace(lambda e: isinstance(e, Mul), cancel_kronecker_mul) + + return expr + + +def besselsimp(expr): + """ + Simplify bessel-type functions. + + Explanation + =========== + + This routine tries to simplify bessel-type functions. Currently it only + works on the Bessel J and I functions, however. It works by looking at all + such functions in turn, and eliminating factors of "I" and "-1" (actually + their polar equivalents) in front of the argument. Then, functions of + half-integer order are rewritten using trigonometric functions and + functions of integer order (> 1) are rewritten using functions + of low order. Finally, if the expression was changed, compute + factorization of the result with factor(). + + >>> from sympy import besselj, besseli, besselsimp, polar_lift, I, S + >>> from sympy.abc import z, nu + >>> besselsimp(besselj(nu, z*polar_lift(-1))) + exp(I*pi*nu)*besselj(nu, z) + >>> besselsimp(besseli(nu, z*polar_lift(-I))) + exp(-I*pi*nu/2)*besselj(nu, z) + >>> besselsimp(besseli(S(-1)/2, z)) + sqrt(2)*cosh(z)/(sqrt(pi)*sqrt(z)) + >>> besselsimp(z*besseli(0, z) + z*(besseli(2, z))/2 + besseli(1, z)) + 3*z*besseli(0, z)/2 + """ + # TODO + # - better algorithm? + # - simplify (cos(pi*b)*besselj(b,z) - besselj(-b,z))/sin(pi*b) ... + # - use contiguity relations? + + def replacer(fro, to, factors): + factors = set(factors) + + def repl(nu, z): + if factors.intersection(Mul.make_args(z)): + return to(nu, z) + return fro(nu, z) + return repl + + def torewrite(fro, to): + def tofunc(nu, z): + return fro(nu, z).rewrite(to) + return tofunc + + def tominus(fro): + def tofunc(nu, z): + return exp(I*pi*nu)*fro(nu, exp_polar(-I*pi)*z) + return tofunc + + orig_expr = expr + + ifactors = [I, exp_polar(I*pi/2), exp_polar(-I*pi/2)] + expr = expr.replace( + besselj, replacer(besselj, + torewrite(besselj, besseli), ifactors)) + expr = expr.replace( + besseli, replacer(besseli, + torewrite(besseli, besselj), ifactors)) + + minusfactors = [-1, exp_polar(I*pi)] + expr = expr.replace( + besselj, replacer(besselj, tominus(besselj), minusfactors)) + expr = expr.replace( + besseli, replacer(besseli, tominus(besseli), minusfactors)) + + z0 = Dummy('z') + + def expander(fro): + def repl(nu, z): + if (nu % 1) == S.Half: + return simplify(trigsimp(unpolarify( + fro(nu, z0).rewrite(besselj).rewrite(jn).expand( + func=True)).subs(z0, z))) + elif nu.is_Integer and nu > 1: + return fro(nu, z).expand(func=True) + return fro(nu, z) + return repl + + expr = expr.replace(besselj, expander(besselj)) + expr = expr.replace(bessely, expander(bessely)) + expr = expr.replace(besseli, expander(besseli)) + expr = expr.replace(besselk, expander(besselk)) + + def _bessel_simp_recursion(expr): + + def _use_recursion(bessel, expr): + while True: + bessels = expr.find(lambda x: isinstance(x, bessel)) + try: + for ba in sorted(bessels, key=lambda x: re(x.args[0])): + a, x = ba.args + bap1 = bessel(a+1, x) + bap2 = bessel(a+2, x) + if expr.has(bap1) and expr.has(bap2): + expr = expr.subs(ba, 2*(a+1)/x*bap1 - bap2) + break + else: + return expr + except (ValueError, TypeError): + return expr + if expr.has(besselj): + expr = _use_recursion(besselj, expr) + if expr.has(bessely): + expr = _use_recursion(bessely, expr) + return expr + + expr = _bessel_simp_recursion(expr) + if expr != orig_expr: + expr = expr.factor() + + return expr + + +def nthroot(expr, n, max_len=4, prec=15): + """ + Compute a real nth-root of a sum of surds. + + Parameters + ========== + + expr : sum of surds + n : integer + max_len : maximum number of surds passed as constants to ``nsimplify`` + + Algorithm + ========= + + First ``nsimplify`` is used to get a candidate root; if it is not a + root the minimal polynomial is computed; the answer is one of its + roots. + + Examples + ======== + + >>> from sympy.simplify.simplify import nthroot + >>> from sympy import sqrt + >>> nthroot(90 + 34*sqrt(7), 3) + sqrt(7) + 3 + + """ + expr = sympify(expr) + n = sympify(n) + p = expr**Rational(1, n) + if not n.is_integer: + return p + if not _is_sum_surds(expr): + return p + surds = [] + coeff_muls = [x.as_coeff_Mul() for x in expr.args] + for x, y in coeff_muls: + if not x.is_rational: + return p + if y is S.One: + continue + if not (y.is_Pow and y.exp == S.Half and y.base.is_integer): + return p + surds.append(y) + surds.sort() + surds = surds[:max_len] + if expr < 0 and n % 2 == 1: + p = (-expr)**Rational(1, n) + a = nsimplify(p, constants=surds) + res = a if _mexpand(a**n) == _mexpand(-expr) else p + return -res + a = nsimplify(p, constants=surds) + if _mexpand(a) is not _mexpand(p) and _mexpand(a**n) == _mexpand(expr): + return _mexpand(a) + expr = _nthroot_solve(expr, n, prec) + if expr is None: + return p + return expr + + +def nsimplify(expr, constants=(), tolerance=None, full=False, rational=None, + rational_conversion='base10'): + """ + Find a simple representation for a number or, if there are free symbols or + if ``rational=True``, then replace Floats with their Rational equivalents. If + no change is made and rational is not False then Floats will at least be + converted to Rationals. + + Explanation + =========== + + For numerical expressions, a simple formula that numerically matches the + given numerical expression is sought (and the input should be possible + to evalf to a precision of at least 30 digits). + + Optionally, a list of (rationally independent) constants to + include in the formula may be given. + + A lower tolerance may be set to find less exact matches. If no tolerance + is given then the least precise value will set the tolerance (e.g. Floats + default to 15 digits of precision, so would be tolerance=10**-15). + + With ``full=True``, a more extensive search is performed + (this is useful to find simpler numbers when the tolerance + is set low). + + When converting to rational, if rational_conversion='base10' (the default), then + convert floats to rationals using their base-10 (string) representation. + When rational_conversion='exact' it uses the exact, base-2 representation. + + Examples + ======== + + >>> from sympy import nsimplify, sqrt, GoldenRatio, exp, I, pi + >>> nsimplify(4/(1+sqrt(5)), [GoldenRatio]) + -2 + 2*GoldenRatio + >>> nsimplify((1/(exp(3*pi*I/5)+1))) + 1/2 - I*sqrt(sqrt(5)/10 + 1/4) + >>> nsimplify(I**I, [pi]) + exp(-pi/2) + >>> nsimplify(pi, tolerance=0.01) + 22/7 + + >>> nsimplify(0.333333333333333, rational=True, rational_conversion='exact') + 6004799503160655/18014398509481984 + >>> nsimplify(0.333333333333333, rational=True) + 1/3 + + See Also + ======== + + sympy.core.function.nfloat + + """ + try: + return sympify(as_int(expr)) + except (TypeError, ValueError): + pass + expr = sympify(expr).xreplace({ + Float('inf'): S.Infinity, + Float('-inf'): S.NegativeInfinity, + }) + if expr is S.Infinity or expr is S.NegativeInfinity: + return expr + if rational or expr.free_symbols: + return _real_to_rational(expr, tolerance, rational_conversion) + + # SymPy's default tolerance for Rationals is 15; other numbers may have + # lower tolerances set, so use them to pick the largest tolerance if None + # was given + if tolerance is None: + tolerance = 10**-min([15] + + [mpmath.libmp.libmpf.prec_to_dps(n._prec) + for n in expr.atoms(Float)]) + # XXX should prec be set independent of tolerance or should it be computed + # from tolerance? + prec = 30 + bprec = int(prec*3.33) + + constants_dict = {} + for constant in constants: + constant = sympify(constant) + v = constant.evalf(prec) + if not v.is_Float: + raise ValueError("constants must be real-valued") + constants_dict[str(constant)] = v._to_mpmath(bprec) + + exprval = expr.evalf(prec, chop=True) + re, im = exprval.as_real_imag() + + # safety check to make sure that this evaluated to a number + if not (re.is_Number and im.is_Number): + return expr + + def nsimplify_real(x): + orig = mpmath.mp.dps + xv = x._to_mpmath(bprec) + try: + # We'll be happy with low precision if a simple fraction + if not (tolerance or full): + mpmath.mp.dps = 15 + rat = mpmath.pslq([xv, 1]) + if rat is not None: + return Rational(-int(rat[1]), int(rat[0])) + mpmath.mp.dps = prec + newexpr = mpmath.identify(xv, constants=constants_dict, + tol=tolerance, full=full) + if not newexpr: + raise ValueError + if full: + newexpr = newexpr[0] + expr = sympify(newexpr) + if x and not expr: # don't let x become 0 + raise ValueError + if expr.is_finite is False and xv not in [mpmath.inf, mpmath.ninf]: + raise ValueError + return expr + finally: + # even though there are returns above, this is executed + # before leaving + mpmath.mp.dps = orig + try: + if re: + re = nsimplify_real(re) + if im: + im = nsimplify_real(im) + except ValueError: + if rational is None: + return _real_to_rational(expr, rational_conversion=rational_conversion) + return expr + + rv = re + im*S.ImaginaryUnit + # if there was a change or rational is explicitly not wanted + # return the value, else return the Rational representation + if rv != expr or rational is False: + return rv + return _real_to_rational(expr, rational_conversion=rational_conversion) + + +def _real_to_rational(expr, tolerance=None, rational_conversion='base10'): + """ + Replace all reals in expr with rationals. + + Examples + ======== + + >>> from sympy.simplify.simplify import _real_to_rational + >>> from sympy.abc import x + + >>> _real_to_rational(.76 + .1*x**.5) + sqrt(x)/10 + 19/25 + + If rational_conversion='base10', this uses the base-10 string. If + rational_conversion='exact', the exact, base-2 representation is used. + + >>> _real_to_rational(0.333333333333333, rational_conversion='exact') + 6004799503160655/18014398509481984 + >>> _real_to_rational(0.333333333333333) + 1/3 + + """ + expr = _sympify(expr) + inf = Float('inf') + p = expr + reps = {} + reduce_num = None + if tolerance is not None and tolerance < 1: + reduce_num = ceiling(1/tolerance) + for fl in p.atoms(Float): + key = fl + if reduce_num is not None: + r = Rational(fl).limit_denominator(reduce_num) + elif (tolerance is not None and tolerance >= 1 and + fl.is_Integer is False): + r = Rational(tolerance*round(fl/tolerance) + ).limit_denominator(int(tolerance)) + else: + if rational_conversion == 'exact': + r = Rational(fl) + reps[key] = r + continue + elif rational_conversion != 'base10': + raise ValueError("rational_conversion must be 'base10' or 'exact'") + + r = nsimplify(fl, rational=False) + # e.g. log(3).n() -> log(3) instead of a Rational + if fl and not r: + r = Rational(fl) + elif not r.is_Rational: + if fl in (inf, -inf): + r = S.ComplexInfinity + elif fl < 0: + fl = -fl + d = Pow(10, int(mpmath.log(fl)/mpmath.log(10))) + r = -Rational(str(fl/d))*d + elif fl > 0: + d = Pow(10, int(mpmath.log(fl)/mpmath.log(10))) + r = Rational(str(fl/d))*d + else: + r = S.Zero + reps[key] = r + return p.subs(reps, simultaneous=True) + + +def clear_coefficients(expr, rhs=S.Zero): + """Return `p, r` where `p` is the expression obtained when Rational + additive and multiplicative coefficients of `expr` have been stripped + away in a naive fashion (i.e. without simplification). The operations + needed to remove the coefficients will be applied to `rhs` and returned + as `r`. + + Examples + ======== + + >>> from sympy.simplify.simplify import clear_coefficients + >>> from sympy.abc import x, y + >>> from sympy import Dummy + >>> expr = 4*y*(6*x + 3) + >>> clear_coefficients(expr - 2) + (y*(2*x + 1), 1/6) + + When solving 2 or more expressions like `expr = a`, + `expr = b`, etc..., it is advantageous to provide a Dummy symbol + for `rhs` and simply replace it with `a`, `b`, etc... in `r`. + + >>> rhs = Dummy('rhs') + >>> clear_coefficients(expr, rhs) + (y*(2*x + 1), _rhs/12) + >>> _[1].subs(rhs, 2) + 1/6 + """ + was = None + free = expr.free_symbols + if expr.is_Rational: + return (S.Zero, rhs - expr) + while expr and was != expr: + was = expr + m, expr = ( + expr.as_content_primitive() + if free else + factor_terms(expr).as_coeff_Mul(rational=True)) + rhs /= m + c, expr = expr.as_coeff_Add(rational=True) + rhs -= c + expr = signsimp(expr, evaluate = False) + if expr.could_extract_minus_sign(): + expr = -expr + rhs = -rhs + return expr, rhs + +def nc_simplify(expr, deep=True): + ''' + Simplify a non-commutative expression composed of multiplication + and raising to a power by grouping repeated subterms into one power. + Priority is given to simplifications that give the fewest number + of arguments in the end (for example, in a*b*a*b*c*a*b*c simplifying + to (a*b)**2*c*a*b*c gives 5 arguments while a*b*(a*b*c)**2 has 3). + If ``expr`` is a sum of such terms, the sum of the simplified terms + is returned. + + Keyword argument ``deep`` controls whether or not subexpressions + nested deeper inside the main expression are simplified. See examples + below. Setting `deep` to `False` can save time on nested expressions + that do not need simplifying on all levels. + + Examples + ======== + + >>> from sympy import symbols + >>> from sympy.simplify.simplify import nc_simplify + >>> a, b, c = symbols("a b c", commutative=False) + >>> nc_simplify(a*b*a*b*c*a*b*c) + a*b*(a*b*c)**2 + >>> expr = a**2*b*a**4*b*a**4 + >>> nc_simplify(expr) + a**2*(b*a**4)**2 + >>> nc_simplify(a*b*a*b*c**2*(a*b)**2*c**2) + ((a*b)**2*c**2)**2 + >>> nc_simplify(a*b*a*b + 2*a*c*a**2*c*a**2*c*a) + (a*b)**2 + 2*(a*c*a)**3 + >>> nc_simplify(b**-1*a**-1*(a*b)**2) + a*b + >>> nc_simplify(a**-1*b**-1*c*a) + (b*a)**(-1)*c*a + >>> expr = (a*b*a*b)**2*a*c*a*c + >>> nc_simplify(expr) + (a*b)**4*(a*c)**2 + >>> nc_simplify(expr, deep=False) + (a*b*a*b)**2*(a*c)**2 + + ''' + if isinstance(expr, MatrixExpr): + expr = expr.doit(inv_expand=False) + _Add, _Mul, _Pow, _Symbol = MatAdd, MatMul, MatPow, MatrixSymbol + else: + _Add, _Mul, _Pow, _Symbol = Add, Mul, Pow, Symbol + + # =========== Auxiliary functions ======================== + def _overlaps(args): + # Calculate a list of lists m such that m[i][j] contains the lengths + # of all possible overlaps between args[:i+1] and args[i+1+j:]. + # An overlap is a suffix of the prefix that matches a prefix + # of the suffix. + # For example, let expr=c*a*b*a*b*a*b*a*b. Then m[3][0] contains + # the lengths of overlaps of c*a*b*a*b with a*b*a*b. The overlaps + # are a*b*a*b, a*b and the empty word so that m[3][0]=[4,2,0]. + # All overlaps rather than only the longest one are recorded + # because this information helps calculate other overlap lengths. + m = [[([1, 0] if a == args[0] else [0]) for a in args[1:]]] + for i in range(1, len(args)): + overlaps = [] + j = 0 + for j in range(len(args) - i - 1): + overlap = [] + for v in m[i-1][j+1]: + if j + i + 1 + v < len(args) and args[i] == args[j+i+1+v]: + overlap.append(v + 1) + overlap += [0] + overlaps.append(overlap) + m.append(overlaps) + return m + + def _reduce_inverses(_args): + # replace consecutive negative powers by an inverse + # of a product of positive powers, e.g. a**-1*b**-1*c + # will simplify to (a*b)**-1*c; + # return that new args list and the number of negative + # powers in it (inv_tot) + inv_tot = 0 # total number of inverses + inverses = [] + args = [] + for arg in _args: + if isinstance(arg, _Pow) and arg.args[1].is_extended_negative: + inverses = [arg**-1] + inverses + inv_tot += 1 + else: + if len(inverses) == 1: + args.append(inverses[0]**-1) + elif len(inverses) > 1: + args.append(_Pow(_Mul(*inverses), -1)) + inv_tot -= len(inverses) - 1 + inverses = [] + args.append(arg) + if inverses: + args.append(_Pow(_Mul(*inverses), -1)) + inv_tot -= len(inverses) - 1 + return inv_tot, tuple(args) + + def get_score(s): + # compute the number of arguments of s + # (including in nested expressions) overall + # but ignore exponents + if isinstance(s, _Pow): + return get_score(s.args[0]) + elif isinstance(s, (_Add, _Mul)): + return sum(get_score(a) for a in s.args) + return 1 + + def compare(s, alt_s): + # compare two possible simplifications and return a + # "better" one + if s != alt_s and get_score(alt_s) < get_score(s): + return alt_s + return s + # ======================================================== + + if not isinstance(expr, (_Add, _Mul, _Pow)) or expr.is_commutative: + return expr + args = expr.args[:] + if isinstance(expr, _Pow): + if deep: + return _Pow(nc_simplify(args[0]), args[1]).doit() + else: + return expr + elif isinstance(expr, _Add): + return _Add(*[nc_simplify(a, deep=deep) for a in args]).doit() + else: + # get the non-commutative part + c_args, args = expr.args_cnc() + com_coeff = Mul(*c_args) + if not equal_valued(com_coeff, 1): + return com_coeff*nc_simplify(expr/com_coeff, deep=deep) + + inv_tot, args = _reduce_inverses(args) + # if most arguments are negative, work with the inverse + # of the expression, e.g. a**-1*b*a**-1*c**-1 will become + # (c*a*b**-1*a)**-1 at the end so can work with c*a*b**-1*a + invert = False + if inv_tot > len(args)/2: + invert = True + args = [a**-1 for a in args[::-1]] + + if deep: + args = tuple(nc_simplify(a) for a in args) + + m = _overlaps(args) + + # simps will be {subterm: end} where `end` is the ending + # index of a sequence of repetitions of subterm; + # this is for not wasting time with subterms that are part + # of longer, already considered sequences + simps = {} + + post = 1 + pre = 1 + + # the simplification coefficient is the number of + # arguments by which contracting a given sequence + # would reduce the word; e.g. in a*b*a*b*c*a*b*c, + # contracting a*b*a*b to (a*b)**2 removes 3 arguments + # while a*b*c*a*b*c to (a*b*c)**2 removes 6. It's + # better to contract the latter so simplification + # with a maximum simplification coefficient will be chosen + max_simp_coeff = 0 + simp = None # information about future simplification + + for i in range(1, len(args)): + simp_coeff = 0 + l = 0 # length of a subterm + p = 0 # the power of a subterm + if i < len(args) - 1: + rep = m[i][0] + start = i # starting index of the repeated sequence + end = i+1 # ending index of the repeated sequence + if i == len(args)-1 or rep == [0]: + # no subterm is repeated at this stage, at least as + # far as the arguments are concerned - there may be + # a repetition if powers are taken into account + if (isinstance(args[i], _Pow) and + not isinstance(args[i].args[0], _Symbol)): + subterm = args[i].args[0].args + l = len(subterm) + if args[i-l:i] == subterm: + # e.g. a*b in a*b*(a*b)**2 is not repeated + # in args (= [a, b, (a*b)**2]) but it + # can be matched here + p += 1 + start -= l + if args[i+1:i+1+l] == subterm: + # e.g. a*b in (a*b)**2*a*b + p += 1 + end += l + if p: + p += args[i].args[1] + else: + continue + else: + l = rep[0] # length of the longest repeated subterm at this point + start -= l - 1 + subterm = args[start:end] + p = 2 + end += l + + if subterm in simps and simps[subterm] >= start: + # the subterm is part of a sequence that + # has already been considered + continue + + # count how many times it's repeated + while end < len(args): + if l in m[end-1][0]: + p += 1 + end += l + elif isinstance(args[end], _Pow) and args[end].args[0].args == subterm: + # for cases like a*b*a*b*(a*b)**2*a*b + p += args[end].args[1] + end += 1 + else: + break + + # see if another match can be made, e.g. + # for b*a**2 in b*a**2*b*a**3 or a*b in + # a**2*b*a*b + + pre_exp = 0 + pre_arg = 1 + if start - l >= 0 and args[start-l+1:start] == subterm[1:]: + if isinstance(subterm[0], _Pow): + pre_arg = subterm[0].args[0] + exp = subterm[0].args[1] + else: + pre_arg = subterm[0] + exp = 1 + if isinstance(args[start-l], _Pow) and args[start-l].args[0] == pre_arg: + pre_exp = args[start-l].args[1] - exp + start -= l + p += 1 + elif args[start-l] == pre_arg: + pre_exp = 1 - exp + start -= l + p += 1 + + post_exp = 0 + post_arg = 1 + if end + l - 1 < len(args) and args[end:end+l-1] == subterm[:-1]: + if isinstance(subterm[-1], _Pow): + post_arg = subterm[-1].args[0] + exp = subterm[-1].args[1] + else: + post_arg = subterm[-1] + exp = 1 + if isinstance(args[end+l-1], _Pow) and args[end+l-1].args[0] == post_arg: + post_exp = args[end+l-1].args[1] - exp + end += l + p += 1 + elif args[end+l-1] == post_arg: + post_exp = 1 - exp + end += l + p += 1 + + # Consider a*b*a**2*b*a**2*b*a: + # b*a**2 is explicitly repeated, but note + # that in this case a*b*a is also repeated + # so there are two possible simplifications: + # a*(b*a**2)**3*a**-1 or (a*b*a)**3 + # The latter is obviously simpler. + # But in a*b*a**2*b**2*a**2 the simplifications are + # a*(b*a**2)**2 and (a*b*a)**3*a in which case + # it's better to stick with the shorter subterm + if post_exp and exp % 2 == 0 and start > 0: + exp = exp/2 + _pre_exp = 1 + _post_exp = 1 + if isinstance(args[start-1], _Pow) and args[start-1].args[0] == post_arg: + _post_exp = post_exp + exp + _pre_exp = args[start-1].args[1] - exp + elif args[start-1] == post_arg: + _post_exp = post_exp + exp + _pre_exp = 1 - exp + if _pre_exp == 0 or _post_exp == 0: + if not pre_exp: + start -= 1 + post_exp = _post_exp + pre_exp = _pre_exp + pre_arg = post_arg + subterm = (post_arg**exp,) + subterm[:-1] + (post_arg**exp,) + + simp_coeff += end-start + + if post_exp: + simp_coeff -= 1 + if pre_exp: + simp_coeff -= 1 + + simps[subterm] = end + + if simp_coeff > max_simp_coeff: + max_simp_coeff = simp_coeff + simp = (start, _Mul(*subterm), p, end, l) + pre = pre_arg**pre_exp + post = post_arg**post_exp + + if simp: + subterm = _Pow(nc_simplify(simp[1], deep=deep), simp[2]) + pre = nc_simplify(_Mul(*args[:simp[0]])*pre, deep=deep) + post = post*nc_simplify(_Mul(*args[simp[3]:]), deep=deep) + simp = pre*subterm*post + if pre != 1 or post != 1: + # new simplifications may be possible but no need + # to recurse over arguments + simp = nc_simplify(simp, deep=False) + else: + simp = _Mul(*args) + + if invert: + simp = _Pow(simp, -1) + + # see if factor_nc(expr) is simplified better + if not isinstance(expr, MatrixExpr): + f_expr = factor_nc(expr) + if f_expr != expr: + alt_simp = nc_simplify(f_expr, deep=deep) + simp = compare(simp, alt_simp) + else: + simp = simp.doit(inv_expand=False) + return simp + + +def dotprodsimp(expr, withsimp=False): + """Simplification for a sum of products targeted at the kind of blowup that + occurs during summation of products. Intended to reduce expression blowup + during matrix multiplication or other similar operations. Only works with + algebraic expressions and does not recurse into non. + + Parameters + ========== + + withsimp : bool, optional + Specifies whether a flag should be returned along with the expression + to indicate roughly whether simplification was successful. It is used + in ``MatrixArithmetic._eval_pow_by_recursion`` to avoid attempting to + simplify an expression repetitively which does not simplify. + """ + + def count_ops_alg(expr): + """Optimized count algebraic operations with no recursion into + non-algebraic args that ``core.function.count_ops`` does. Also returns + whether rational functions may be present according to negative + exponents of powers or non-number fractions. + + Returns + ======= + + ops, ratfunc : int, bool + ``ops`` is the number of algebraic operations starting at the top + level expression (not recursing into non-alg children). ``ratfunc`` + specifies whether the expression MAY contain rational functions + which ``cancel`` MIGHT optimize. + """ + + ops = 0 + args = [expr] + ratfunc = False + + while args: + a = args.pop() + + if not isinstance(a, Basic): + continue + + if a.is_Rational: + if a is not S.One: # -1/3 = NEG + DIV + ops += bool (a.p < 0) + bool (a.q != 1) + + elif a.is_Mul: + if a.could_extract_minus_sign(): + ops += 1 + if a.args[0] is S.NegativeOne: + a = a.as_two_terms()[1] + else: + a = -a + + n, d = fraction(a) + + if n.is_Integer: + ops += 1 + bool (n < 0) + args.append(d) # won't be -Mul but could be Add + + elif d is not S.One: + if not d.is_Integer: + args.append(d) + ratfunc=True + + ops += 1 + args.append(n) # could be -Mul + + else: + ops += len(a.args) - 1 + args.extend(a.args) + + elif a.is_Add: + laargs = len(a.args) + negs = 0 + + for ai in a.args: + if ai.could_extract_minus_sign(): + negs += 1 + ai = -ai + args.append(ai) + + ops += laargs - (negs != laargs) # -x - y = NEG + SUB + + elif a.is_Pow: + ops += 1 + args.append(a.base) + + if not ratfunc: + ratfunc = a.exp.is_negative is not False + + return ops, ratfunc + + def nonalg_subs_dummies(expr, dummies): + """Substitute dummy variables for non-algebraic expressions to avoid + evaluation of non-algebraic terms that ``polys.polytools.cancel`` does. + """ + + if not expr.args: + return expr + + if expr.is_Add or expr.is_Mul or expr.is_Pow: + args = None + + for i, a in enumerate(expr.args): + c = nonalg_subs_dummies(a, dummies) + + if c is a: + continue + + if args is None: + args = list(expr.args) + + args[i] = c + + if args is None: + return expr + + return expr.func(*args) + + return dummies.setdefault(expr, Dummy()) + + simplified = False # doesn't really mean simplified, rather "can simplify again" + + if isinstance(expr, Basic) and (expr.is_Add or expr.is_Mul or expr.is_Pow): + expr2 = expr.expand(deep=True, modulus=None, power_base=False, + power_exp=False, mul=True, log=False, multinomial=True, basic=False) + + if expr2 != expr: + expr = expr2 + simplified = True + + exprops, ratfunc = count_ops_alg(expr) + + if exprops >= 6: # empirically tested cutoff for expensive simplification + if ratfunc: + dummies = {} + expr2 = nonalg_subs_dummies(expr, dummies) + + if expr2 is expr or count_ops_alg(expr2)[0] >= 6: # check again after substitution + expr3 = cancel(expr2) + + if expr3 != expr2: + expr = expr3.subs([(d, e) for e, d in dummies.items()]) + simplified = True + + # very special case: x/(x-1) - 1/(x-1) -> 1 + elif (exprops == 5 and expr.is_Add and expr.args [0].is_Mul and + expr.args [1].is_Mul and expr.args [0].args [-1].is_Pow and + expr.args [1].args [-1].is_Pow and + expr.args [0].args [-1].exp is S.NegativeOne and + expr.args [1].args [-1].exp is S.NegativeOne): + + expr2 = together (expr) + expr2ops = count_ops_alg(expr2)[0] + + if expr2ops < exprops: + expr = expr2 + simplified = True + + else: + simplified = True + + return (expr, simplified) if withsimp else expr + + +bottom_up = deprecated( + """ + Using bottom_up from the sympy.simplify.simplify submodule is + deprecated. + + Instead, use bottom_up from the top-level sympy namespace, like + + sympy.bottom_up + """, + deprecated_since_version="1.10", + active_deprecations_target="deprecated-traversal-functions-moved", +)(_bottom_up) + + +# XXX: This function really should either be private API or exported in the +# top-level sympy/__init__.py +walk = deprecated( + """ + Using walk from the sympy.simplify.simplify submodule is + deprecated. + + Instead, use walk from sympy.core.traversal.walk + """, + deprecated_since_version="1.10", + active_deprecations_target="deprecated-traversal-functions-moved", +)(_walk) diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/sqrtdenest.py b/.venv/lib/python3.12/site-packages/sympy/simplify/sqrtdenest.py new file mode 100644 index 0000000000000000000000000000000000000000..d266de7e62a4b7d37a2109f7091ff91e4df7c79d --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/simplify/sqrtdenest.py @@ -0,0 +1,678 @@ +from sympy.core import Add, Expr, Mul, S, sympify +from sympy.core.function import _mexpand, count_ops, expand_mul +from sympy.core.sorting import default_sort_key +from sympy.core.symbol import Dummy +from sympy.functions import root, sign, sqrt +from sympy.polys import Poly, PolynomialError + + +def is_sqrt(expr): + """Return True if expr is a sqrt, otherwise False.""" + + return expr.is_Pow and expr.exp.is_Rational and abs(expr.exp) is S.Half + + +def sqrt_depth(p) -> int: + """Return the maximum depth of any square root argument of p. + + >>> from sympy.functions.elementary.miscellaneous import sqrt + >>> from sympy.simplify.sqrtdenest import sqrt_depth + + Neither of these square roots contains any other square roots + so the depth is 1: + + >>> sqrt_depth(1 + sqrt(2)*(1 + sqrt(3))) + 1 + + The sqrt(3) is contained within a square root so the depth is + 2: + + >>> sqrt_depth(1 + sqrt(2)*sqrt(1 + sqrt(3))) + 2 + """ + if p is S.ImaginaryUnit: + return 1 + if p.is_Atom: + return 0 + if p.is_Add or p.is_Mul: + return max(sqrt_depth(x) for x in p.args) + if is_sqrt(p): + return sqrt_depth(p.base) + 1 + return 0 + + +def is_algebraic(p): + """Return True if p is comprised of only Rationals or square roots + of Rationals and algebraic operations. + + Examples + ======== + + >>> from sympy.functions.elementary.miscellaneous import sqrt + >>> from sympy.simplify.sqrtdenest import is_algebraic + >>> from sympy import cos + >>> is_algebraic(sqrt(2)*(3/(sqrt(7) + sqrt(5)*sqrt(2)))) + True + >>> is_algebraic(sqrt(2)*(3/(sqrt(7) + sqrt(5)*cos(2)))) + False + """ + + if p.is_Rational: + return True + elif p.is_Atom: + return False + elif is_sqrt(p) or p.is_Pow and p.exp.is_Integer: + return is_algebraic(p.base) + elif p.is_Add or p.is_Mul: + return all(is_algebraic(x) for x in p.args) + else: + return False + + +def _subsets(n): + """ + Returns all possible subsets of the set (0, 1, ..., n-1) except the + empty set, listed in reversed lexicographical order according to binary + representation, so that the case of the fourth root is treated last. + + Examples + ======== + + >>> from sympy.simplify.sqrtdenest import _subsets + >>> _subsets(2) + [[1, 0], [0, 1], [1, 1]] + + """ + if n == 1: + a = [[1]] + elif n == 2: + a = [[1, 0], [0, 1], [1, 1]] + elif n == 3: + a = [[1, 0, 0], [0, 1, 0], [1, 1, 0], + [0, 0, 1], [1, 0, 1], [0, 1, 1], [1, 1, 1]] + else: + b = _subsets(n - 1) + a0 = [x + [0] for x in b] + a1 = [x + [1] for x in b] + a = a0 + [[0]*(n - 1) + [1]] + a1 + return a + + +def sqrtdenest(expr, max_iter=3): + """Denests sqrts in an expression that contain other square roots + if possible, otherwise returns the expr unchanged. This is based on the + algorithms of [1]. + + Examples + ======== + + >>> from sympy.simplify.sqrtdenest import sqrtdenest + >>> from sympy import sqrt + >>> sqrtdenest(sqrt(5 + 2 * sqrt(6))) + sqrt(2) + sqrt(3) + + See Also + ======== + + sympy.solvers.solvers.unrad + + References + ========== + + .. [1] https://web.archive.org/web/20210806201615/https://researcher.watson.ibm.com/researcher/files/us-fagin/symb85.pdf + + .. [2] D. J. Jeffrey and A. D. Rich, 'Symplifying Square Roots of Square Roots + by Denesting' (available at https://www.cybertester.com/data/denest.pdf) + + """ + expr = expand_mul(expr) + for i in range(max_iter): + z = _sqrtdenest0(expr) + if expr == z: + return expr + expr = z + return expr + + +def _sqrt_match(p): + """Return [a, b, r] for p.match(a + b*sqrt(r)) where, in addition to + matching, sqrt(r) also has then maximal sqrt_depth among addends of p. + + Examples + ======== + + >>> from sympy.functions.elementary.miscellaneous import sqrt + >>> from sympy.simplify.sqrtdenest import _sqrt_match + >>> _sqrt_match(1 + sqrt(2) + sqrt(2)*sqrt(3) + 2*sqrt(1+sqrt(5))) + [1 + sqrt(2) + sqrt(6), 2, 1 + sqrt(5)] + """ + from sympy.simplify.radsimp import split_surds + + p = _mexpand(p) + if p.is_Number: + res = (p, S.Zero, S.Zero) + elif p.is_Add: + pargs = sorted(p.args, key=default_sort_key) + sqargs = [x**2 for x in pargs] + if all(sq.is_Rational and sq.is_positive for sq in sqargs): + r, b, a = split_surds(p) + res = a, b, r + return list(res) + # to make the process canonical, the argument is included in the tuple + # so when the max is selected, it will be the largest arg having a + # given depth + v = [(sqrt_depth(x), x, i) for i, x in enumerate(pargs)] + nmax = max(v, key=default_sort_key) + if nmax[0] == 0: + res = [] + else: + # select r + depth, _, i = nmax + r = pargs.pop(i) + v.pop(i) + b = S.One + if r.is_Mul: + bv = [] + rv = [] + for x in r.args: + if sqrt_depth(x) < depth: + bv.append(x) + else: + rv.append(x) + b = Mul._from_args(bv) + r = Mul._from_args(rv) + # collect terms containing r + a1 = [] + b1 = [b] + for x in v: + if x[0] < depth: + a1.append(x[1]) + else: + x1 = x[1] + if x1 == r: + b1.append(1) + else: + if x1.is_Mul: + x1args = list(x1.args) + if r in x1args: + x1args.remove(r) + b1.append(Mul(*x1args)) + else: + a1.append(x[1]) + else: + a1.append(x[1]) + a = Add(*a1) + b = Add(*b1) + res = (a, b, r**2) + else: + b, r = p.as_coeff_Mul() + if is_sqrt(r): + res = (S.Zero, b, r**2) + else: + res = [] + return list(res) + + +class SqrtdenestStopIteration(StopIteration): + pass + + +def _sqrtdenest0(expr): + """Returns expr after denesting its arguments.""" + + if is_sqrt(expr): + n, d = expr.as_numer_denom() + if d is S.One: # n is a square root + if n.base.is_Add: + args = sorted(n.base.args, key=default_sort_key) + if len(args) > 2 and all((x**2).is_Integer for x in args): + try: + return _sqrtdenest_rec(n) + except SqrtdenestStopIteration: + pass + expr = sqrt(_mexpand(Add(*[_sqrtdenest0(x) for x in args]))) + return _sqrtdenest1(expr) + else: + n, d = [_sqrtdenest0(i) for i in (n, d)] + return n/d + + if isinstance(expr, Add): + cs = [] + args = [] + for arg in expr.args: + c, a = arg.as_coeff_Mul() + cs.append(c) + args.append(a) + + if all(c.is_Rational for c in cs) and all(is_sqrt(arg) for arg in args): + return _sqrt_ratcomb(cs, args) + + if isinstance(expr, Expr): + args = expr.args + if args: + return expr.func(*[_sqrtdenest0(a) for a in args]) + return expr + + +def _sqrtdenest_rec(expr): + """Helper that denests the square root of three or more surds. + + Explanation + =========== + + It returns the denested expression; if it cannot be denested it + throws SqrtdenestStopIteration + + Algorithm: expr.base is in the extension Q_m = Q(sqrt(r_1),..,sqrt(r_k)); + split expr.base = a + b*sqrt(r_k), where `a` and `b` are on + Q_(m-1) = Q(sqrt(r_1),..,sqrt(r_(k-1))); then a**2 - b**2*r_k is + on Q_(m-1); denest sqrt(a**2 - b**2*r_k) and so on. + See [1], section 6. + + Examples + ======== + + >>> from sympy import sqrt + >>> from sympy.simplify.sqrtdenest import _sqrtdenest_rec + >>> _sqrtdenest_rec(sqrt(-72*sqrt(2) + 158*sqrt(5) + 498)) + -sqrt(10) + sqrt(2) + 9 + 9*sqrt(5) + >>> w=-6*sqrt(55)-6*sqrt(35)-2*sqrt(22)-2*sqrt(14)+2*sqrt(77)+6*sqrt(10)+65 + >>> _sqrtdenest_rec(sqrt(w)) + -sqrt(11) - sqrt(7) + sqrt(2) + 3*sqrt(5) + """ + from sympy.simplify.radsimp import radsimp, rad_rationalize, split_surds + if not expr.is_Pow: + return sqrtdenest(expr) + if expr.base < 0: + return sqrt(-1)*_sqrtdenest_rec(sqrt(-expr.base)) + g, a, b = split_surds(expr.base) + a = a*sqrt(g) + if a < b: + a, b = b, a + c2 = _mexpand(a**2 - b**2) + if len(c2.args) > 2: + g, a1, b1 = split_surds(c2) + a1 = a1*sqrt(g) + if a1 < b1: + a1, b1 = b1, a1 + c2_1 = _mexpand(a1**2 - b1**2) + c_1 = _sqrtdenest_rec(sqrt(c2_1)) + d_1 = _sqrtdenest_rec(sqrt(a1 + c_1)) + num, den = rad_rationalize(b1, d_1) + c = _mexpand(d_1/sqrt(2) + num/(den*sqrt(2))) + else: + c = _sqrtdenest1(sqrt(c2)) + + if sqrt_depth(c) > 1: + raise SqrtdenestStopIteration + ac = a + c + if len(ac.args) >= len(expr.args): + if count_ops(ac) >= count_ops(expr.base): + raise SqrtdenestStopIteration + d = sqrtdenest(sqrt(ac)) + if sqrt_depth(d) > 1: + raise SqrtdenestStopIteration + num, den = rad_rationalize(b, d) + r = d/sqrt(2) + num/(den*sqrt(2)) + r = radsimp(r) + return _mexpand(r) + + +def _sqrtdenest1(expr, denester=True): + """Return denested expr after denesting with simpler methods or, that + failing, using the denester.""" + + from sympy.simplify.simplify import radsimp + + if not is_sqrt(expr): + return expr + + a = expr.base + if a.is_Atom: + return expr + val = _sqrt_match(a) + if not val: + return expr + + a, b, r = val + # try a quick numeric denesting + d2 = _mexpand(a**2 - b**2*r) + if d2.is_Rational: + if d2.is_positive: + z = _sqrt_numeric_denest(a, b, r, d2) + if z is not None: + return z + else: + # fourth root case + # sqrtdenest(sqrt(3 + 2*sqrt(3))) = + # sqrt(2)*3**(1/4)/2 + sqrt(2)*3**(3/4)/2 + dr2 = _mexpand(-d2*r) + dr = sqrt(dr2) + if dr.is_Rational: + z = _sqrt_numeric_denest(_mexpand(b*r), a, r, dr2) + if z is not None: + return z/root(r, 4) + + else: + z = _sqrt_symbolic_denest(a, b, r) + if z is not None: + return z + + if not denester or not is_algebraic(expr): + return expr + + res = sqrt_biquadratic_denest(expr, a, b, r, d2) + if res: + return res + + # now call to the denester + av0 = [a, b, r, d2] + z = _denester([radsimp(expr**2)], av0, 0, sqrt_depth(expr))[0] + if av0[1] is None: + return expr + if z is not None: + if sqrt_depth(z) == sqrt_depth(expr) and count_ops(z) > count_ops(expr): + return expr + return z + return expr + + +def _sqrt_symbolic_denest(a, b, r): + """Given an expression, sqrt(a + b*sqrt(b)), return the denested + expression or None. + + Explanation + =========== + + If r = ra + rb*sqrt(rr), try replacing sqrt(rr) in ``a`` with + (y**2 - ra)/rb, and if the result is a quadratic, ca*y**2 + cb*y + cc, and + (cb + b)**2 - 4*ca*cc is 0, then sqrt(a + b*sqrt(r)) can be rewritten as + sqrt(ca*(sqrt(r) + (cb + b)/(2*ca))**2). + + Examples + ======== + + >>> from sympy.simplify.sqrtdenest import _sqrt_symbolic_denest, sqrtdenest + >>> from sympy import sqrt, Symbol + >>> from sympy.abc import x + + >>> a, b, r = 16 - 2*sqrt(29), 2, -10*sqrt(29) + 55 + >>> _sqrt_symbolic_denest(a, b, r) + sqrt(11 - 2*sqrt(29)) + sqrt(5) + + If the expression is numeric, it will be simplified: + + >>> w = sqrt(sqrt(sqrt(3) + 1) + 1) + 1 + sqrt(2) + >>> sqrtdenest(sqrt((w**2).expand())) + 1 + sqrt(2) + sqrt(1 + sqrt(1 + sqrt(3))) + + Otherwise, it will only be simplified if assumptions allow: + + >>> w = w.subs(sqrt(3), sqrt(x + 3)) + >>> sqrtdenest(sqrt((w**2).expand())) + sqrt((sqrt(sqrt(sqrt(x + 3) + 1) + 1) + 1 + sqrt(2))**2) + + Notice that the argument of the sqrt is a square. If x is made positive + then the sqrt of the square is resolved: + + >>> _.subs(x, Symbol('x', positive=True)) + sqrt(sqrt(sqrt(x + 3) + 1) + 1) + 1 + sqrt(2) + """ + + a, b, r = map(sympify, (a, b, r)) + rval = _sqrt_match(r) + if not rval: + return None + ra, rb, rr = rval + if rb: + y = Dummy('y', positive=True) + try: + newa = Poly(a.subs(sqrt(rr), (y**2 - ra)/rb), y) + except PolynomialError: + return None + if newa.degree() == 2: + ca, cb, cc = newa.all_coeffs() + cb += b + if _mexpand(cb**2 - 4*ca*cc).equals(0): + z = sqrt(ca*(sqrt(r) + cb/(2*ca))**2) + if z.is_number: + z = _mexpand(Mul._from_args(z.as_content_primitive())) + return z + + +def _sqrt_numeric_denest(a, b, r, d2): + r"""Helper that denest + $\sqrt{a + b \sqrt{r}}, d^2 = a^2 - b^2 r > 0$ + + If it cannot be denested, it returns ``None``. + """ + d = sqrt(d2) + s = a + d + # sqrt_depth(res) <= sqrt_depth(s) + 1 + # sqrt_depth(expr) = sqrt_depth(r) + 2 + # there is denesting if sqrt_depth(s) + 1 < sqrt_depth(r) + 2 + # if s**2 is Number there is a fourth root + if sqrt_depth(s) < sqrt_depth(r) + 1 or (s**2).is_Rational: + s1, s2 = sign(s), sign(b) + if s1 == s2 == -1: + s1 = s2 = 1 + res = (s1 * sqrt(a + d) + s2 * sqrt(a - d)) * sqrt(2) / 2 + return res.expand() + + +def sqrt_biquadratic_denest(expr, a, b, r, d2): + """denest expr = sqrt(a + b*sqrt(r)) + where a, b, r are linear combinations of square roots of + positive rationals on the rationals (SQRR) and r > 0, b != 0, + d2 = a**2 - b**2*r > 0 + + If it cannot denest it returns None. + + Explanation + =========== + + Search for a solution A of type SQRR of the biquadratic equation + 4*A**4 - 4*a*A**2 + b**2*r = 0 (1) + sqd = sqrt(a**2 - b**2*r) + Choosing the sqrt to be positive, the possible solutions are + A = sqrt(a/2 +/- sqd/2) + Since a, b, r are SQRR, then a**2 - b**2*r is a SQRR, + so if sqd can be denested, it is done by + _sqrtdenest_rec, and the result is a SQRR. + Similarly for A. + Examples of solutions (in both cases a and sqd are positive): + + Example of expr with solution sqrt(a/2 + sqd/2) but not + solution sqrt(a/2 - sqd/2): + expr = sqrt(-sqrt(15) - sqrt(2)*sqrt(-sqrt(5) + 5) - sqrt(3) + 8) + a = -sqrt(15) - sqrt(3) + 8; sqd = -2*sqrt(5) - 2 + 4*sqrt(3) + + Example of expr with solution sqrt(a/2 - sqd/2) but not + solution sqrt(a/2 + sqd/2): + w = 2 + r2 + r3 + (1 + r3)*sqrt(2 + r2 + 5*r3) + expr = sqrt((w**2).expand()) + a = 4*sqrt(6) + 8*sqrt(2) + 47 + 28*sqrt(3) + sqd = 29 + 20*sqrt(3) + + Define B = b/2*A; eq.(1) implies a = A**2 + B**2*r; then + expr**2 = a + b*sqrt(r) = (A + B*sqrt(r))**2 + + Examples + ======== + + >>> from sympy import sqrt + >>> from sympy.simplify.sqrtdenest import _sqrt_match, sqrt_biquadratic_denest + >>> z = sqrt((2*sqrt(2) + 4)*sqrt(2 + sqrt(2)) + 5*sqrt(2) + 8) + >>> a, b, r = _sqrt_match(z**2) + >>> d2 = a**2 - b**2*r + >>> sqrt_biquadratic_denest(z, a, b, r, d2) + sqrt(2) + sqrt(sqrt(2) + 2) + 2 + """ + from sympy.simplify.radsimp import radsimp, rad_rationalize + if r <= 0 or d2 < 0 or not b or sqrt_depth(expr.base) < 2: + return None + for x in (a, b, r): + for y in x.args: + y2 = y**2 + if not y2.is_Integer or not y2.is_positive: + return None + sqd = _mexpand(sqrtdenest(sqrt(radsimp(d2)))) + if sqrt_depth(sqd) > 1: + return None + x1, x2 = [a/2 + sqd/2, a/2 - sqd/2] + # look for a solution A with depth 1 + for x in (x1, x2): + A = sqrtdenest(sqrt(x)) + if sqrt_depth(A) > 1: + continue + Bn, Bd = rad_rationalize(b, _mexpand(2*A)) + B = Bn/Bd + z = A + B*sqrt(r) + if z < 0: + z = -z + return _mexpand(z) + return None + + +def _denester(nested, av0, h, max_depth_level): + """Denests a list of expressions that contain nested square roots. + + Explanation + =========== + + Algorithm based on . + + It is assumed that all of the elements of 'nested' share the same + bottom-level radicand. (This is stated in the paper, on page 177, in + the paragraph immediately preceding the algorithm.) + + When evaluating all of the arguments in parallel, the bottom-level + radicand only needs to be denested once. This means that calling + _denester with x arguments results in a recursive invocation with x+1 + arguments; hence _denester has polynomial complexity. + + However, if the arguments were evaluated separately, each call would + result in two recursive invocations, and the algorithm would have + exponential complexity. + + This is discussed in the paper in the middle paragraph of page 179. + """ + from sympy.simplify.simplify import radsimp + if h > max_depth_level: + return None, None + if av0[1] is None: + return None, None + if (av0[0] is None and + all(n.is_Number for n in nested)): # no arguments are nested + for f in _subsets(len(nested)): # test subset 'f' of nested + p = _mexpand(Mul(*[nested[i] for i in range(len(f)) if f[i]])) + if f.count(1) > 1 and f[-1]: + p = -p + sqp = sqrt(p) + if sqp.is_Rational: + return sqp, f # got a perfect square so return its square root. + # Otherwise, return the radicand from the previous invocation. + return sqrt(nested[-1]), [0]*len(nested) + else: + R = None + if av0[0] is not None: + values = [av0[:2]] + R = av0[2] + nested2 = [av0[3], R] + av0[0] = None + else: + values = list(filter(None, [_sqrt_match(expr) for expr in nested])) + for v in values: + if v[2]: # Since if b=0, r is not defined + if R is not None: + if R != v[2]: + av0[1] = None + return None, None + else: + R = v[2] + if R is None: + # return the radicand from the previous invocation + return sqrt(nested[-1]), [0]*len(nested) + nested2 = [_mexpand(v[0]**2) - + _mexpand(R*v[1]**2) for v in values] + [R] + d, f = _denester(nested2, av0, h + 1, max_depth_level) + if not f: + return None, None + if not any(f[i] for i in range(len(nested))): + v = values[-1] + return sqrt(v[0] + _mexpand(v[1]*d)), f + else: + p = Mul(*[nested[i] for i in range(len(nested)) if f[i]]) + v = _sqrt_match(p) + if 1 in f and f.index(1) < len(nested) - 1 and f[len(nested) - 1]: + v[0] = -v[0] + v[1] = -v[1] + if not f[len(nested)]: # Solution denests with square roots + vad = _mexpand(v[0] + d) + if vad <= 0: + # return the radicand from the previous invocation. + return sqrt(nested[-1]), [0]*len(nested) + if not(sqrt_depth(vad) <= sqrt_depth(R) + 1 or + (vad**2).is_Number): + av0[1] = None + return None, None + + sqvad = _sqrtdenest1(sqrt(vad), denester=False) + if not (sqrt_depth(sqvad) <= sqrt_depth(R) + 1): + av0[1] = None + return None, None + sqvad1 = radsimp(1/sqvad) + res = _mexpand(sqvad/sqrt(2) + (v[1]*sqrt(R)*sqvad1/sqrt(2))) + return res, f + + # sign(v[1])*sqrt(_mexpand(v[1]**2*R*vad1/2))), f + else: # Solution requires a fourth root + s2 = _mexpand(v[1]*R) + d + if s2 <= 0: + return sqrt(nested[-1]), [0]*len(nested) + FR, s = root(_mexpand(R), 4), sqrt(s2) + return _mexpand(s/(sqrt(2)*FR) + v[0]*FR/(sqrt(2)*s)), f + + +def _sqrt_ratcomb(cs, args): + """Denest rational combinations of radicals. + + Based on section 5 of [1]. + + Examples + ======== + + >>> from sympy import sqrt + >>> from sympy.simplify.sqrtdenest import sqrtdenest + >>> z = sqrt(1+sqrt(3)) + sqrt(3+3*sqrt(3)) - sqrt(10+6*sqrt(3)) + >>> sqrtdenest(z) + 0 + """ + from sympy.simplify.radsimp import radsimp + + # check if there exists a pair of sqrt that can be denested + def find(a): + n = len(a) + for i in range(n - 1): + for j in range(i + 1, n): + s1 = a[i].base + s2 = a[j].base + p = _mexpand(s1 * s2) + s = sqrtdenest(sqrt(p)) + if s != sqrt(p): + return s, i, j + + indices = find(args) + if indices is None: + return Add(*[c * arg for c, arg in zip(cs, args)]) + + s, i1, i2 = indices + + c2 = cs.pop(i2) + args.pop(i2) + a1 = args[i1] + + # replace a2 by s/a1 + cs[i1] += radsimp(c2 * s / a1.base) + + return _sqrt_ratcomb(cs, args) diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/tests/__init__.py b/.venv/lib/python3.12/site-packages/sympy/simplify/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git 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sympy.core.symbol import symbols +from sympy.functions.combinatorial.factorials import (FallingFactorial, RisingFactorial, binomial, factorial) +from sympy.functions.special.gamma_functions import gamma +from sympy.simplify.combsimp import combsimp +from sympy.abc import x + + +def test_combsimp(): + k, m, n = symbols('k m n', integer = True) + + assert combsimp(factorial(n)) == factorial(n) + assert combsimp(binomial(n, k)) == binomial(n, k) + + assert combsimp(factorial(n)/factorial(n - 3)) == n*(-1 + n)*(-2 + n) + assert combsimp(binomial(n + 1, k + 1)/binomial(n, k)) == (1 + n)/(1 + k) + + assert combsimp(binomial(3*n + 4, n + 1)/binomial(3*n + 1, n)) == \ + Rational(3, 2)*((3*n + 2)*(3*n + 4)/((n + 1)*(2*n + 3))) + + assert combsimp(factorial(n)**2/factorial(n - 3)) == \ + factorial(n)*n*(-1 + n)*(-2 + n) + assert combsimp(factorial(n)*binomial(n + 1, k + 1)/binomial(n, k)) == \ + factorial(n + 1)/(1 + k) + + assert combsimp(gamma(n + 3)) == factorial(n + 2) + + assert combsimp(factorial(x)) == gamma(x + 1) + + # issue 9699 + assert combsimp((n + 1)*factorial(n)) == factorial(n + 1) + assert combsimp(factorial(n)/n) == factorial(n-1) + + # issue 6658 + assert combsimp(binomial(n, n - k)) == binomial(n, k) + + # issue 6341, 7135 + assert combsimp(factorial(n)/(factorial(k)*factorial(n - k))) == \ + binomial(n, k) + assert combsimp(factorial(k)*factorial(n - k)/factorial(n)) == \ + 1/binomial(n, k) + assert combsimp(factorial(2*n)/factorial(n)**2) == binomial(2*n, n) + assert combsimp(factorial(2*n)*factorial(k)*factorial(n - k)/ + factorial(n)**3) == binomial(2*n, n)/binomial(n, k) + + assert combsimp(factorial(n*(1 + n) - n**2 - n)) == 1 + + assert combsimp(6*FallingFactorial(-4, n)/factorial(n)) == \ + (-1)**n*(n + 1)*(n + 2)*(n + 3) + assert combsimp(6*FallingFactorial(-4, n - 1)/factorial(n - 1)) == \ + (-1)**(n - 1)*n*(n + 1)*(n + 2) + assert combsimp(6*FallingFactorial(-4, n - 3)/factorial(n - 3)) == \ + (-1)**(n - 3)*n*(n - 1)*(n - 2) + assert combsimp(6*FallingFactorial(-4, -n - 1)/factorial(-n - 1)) == \ + -(-1)**(-n - 1)*n*(n - 1)*(n - 2) + + assert combsimp(6*RisingFactorial(4, n)/factorial(n)) == \ + (n + 1)*(n + 2)*(n + 3) + assert combsimp(6*RisingFactorial(4, n - 1)/factorial(n - 1)) == \ + n*(n + 1)*(n + 2) + assert combsimp(6*RisingFactorial(4, n - 3)/factorial(n - 3)) == \ + n*(n - 1)*(n - 2) + assert combsimp(6*RisingFactorial(4, -n - 1)/factorial(-n - 1)) == \ + -n*(n - 1)*(n - 2) + + +def test_issue_6878(): + n = symbols('n', integer=True) + assert combsimp(RisingFactorial(-10, n)) == 3628800*(-1)**n/factorial(10 - n) + + +def test_issue_14528(): + p = symbols("p", integer=True, positive=True) + assert combsimp(binomial(1,p)) == 1/(factorial(p)*factorial(1-p)) + assert combsimp(factorial(2-p)) == factorial(2-p) diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_cse.py b/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_cse.py new file mode 100644 index 0000000000000000000000000000000000000000..c2a34dfb0e227547bd41bed2491284fd7150d0b6 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_cse.py @@ -0,0 +1,761 @@ +from functools import reduce +import itertools +from operator import add + +from sympy.codegen.matrix_nodes import MatrixSolve +from sympy.core.add import Add +from sympy.core.containers import Tuple +from sympy.core.expr import UnevaluatedExpr +from sympy.core.function import Function +from sympy.core.mul import Mul +from sympy.core.power import Pow +from sympy.core.relational import Eq +from sympy.core.singleton import S +from sympy.core.symbol import (Symbol, symbols) +from sympy.core.sympify import sympify +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.matrices.dense import Matrix +from sympy.matrices.expressions import Inverse, MatAdd, MatMul, Transpose +from sympy.polys.rootoftools import CRootOf +from sympy.series.order import O +from sympy.simplify.cse_main import cse +from sympy.simplify.simplify import signsimp +from sympy.tensor.indexed import (Idx, IndexedBase) + +from sympy.core.function import count_ops +from sympy.simplify.cse_opts import sub_pre, sub_post +from sympy.functions.special.hyper import meijerg +from sympy.simplify import cse_main, cse_opts +from sympy.utilities.iterables import subsets +from sympy.testing.pytest import XFAIL, raises +from sympy.matrices import (MutableDenseMatrix, MutableSparseMatrix, + ImmutableDenseMatrix, ImmutableSparseMatrix) +from sympy.matrices.expressions import MatrixSymbol + + +w, x, y, z = symbols('w,x,y,z') +x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12 = symbols('x:13') + + +def test_numbered_symbols(): + ns = cse_main.numbered_symbols(prefix='y') + assert list(itertools.islice( + ns, 0, 10)) == [Symbol('y%s' % i) for i in range(0, 10)] + ns = cse_main.numbered_symbols(prefix='y') + assert list(itertools.islice( + ns, 10, 20)) == [Symbol('y%s' % i) for i in range(10, 20)] + ns = cse_main.numbered_symbols() + assert list(itertools.islice( + ns, 0, 10)) == [Symbol('x%s' % i) for i in range(0, 10)] + +# Dummy "optimization" functions for testing. + + +def opt1(expr): + return expr + y + + +def opt2(expr): + return expr*z + + +def test_preprocess_for_cse(): + assert cse_main.preprocess_for_cse(x, [(opt1, None)]) == x + y + assert cse_main.preprocess_for_cse(x, [(None, opt1)]) == x + assert cse_main.preprocess_for_cse(x, [(None, None)]) == x + assert cse_main.preprocess_for_cse(x, [(opt1, opt2)]) == x + y + assert cse_main.preprocess_for_cse( + x, [(opt1, None), (opt2, None)]) == (x + y)*z + + +def test_postprocess_for_cse(): + assert cse_main.postprocess_for_cse(x, [(opt1, None)]) == x + assert cse_main.postprocess_for_cse(x, [(None, opt1)]) == x + y + assert cse_main.postprocess_for_cse(x, [(None, None)]) == x + assert cse_main.postprocess_for_cse(x, [(opt1, opt2)]) == x*z + # Note the reverse order of application. + assert cse_main.postprocess_for_cse( + x, [(None, opt1), (None, opt2)]) == x*z + y + + +def test_cse_single(): + # Simple substitution. + e = Add(Pow(x + y, 2), sqrt(x + y)) + substs, reduced = cse([e]) + assert substs == [(x0, x + y)] + assert reduced == [sqrt(x0) + x0**2] + + subst42, (red42,) = cse([42]) # issue_15082 + assert len(subst42) == 0 and red42 == 42 + subst_half, (red_half,) = cse([0.5]) + assert len(subst_half) == 0 and red_half == 0.5 + + +def test_cse_single2(): + # Simple substitution, test for being able to pass the expression directly + e = Add(Pow(x + y, 2), sqrt(x + y)) + substs, reduced = cse(e) + assert substs == [(x0, x + y)] + assert reduced == [sqrt(x0) + x0**2] + substs, reduced = cse(Matrix([[1]])) + assert isinstance(reduced[0], Matrix) + + subst42, (red42,) = cse(42) # issue 15082 + assert len(subst42) == 0 and red42 == 42 + subst_half, (red_half,) = cse(0.5) # issue 15082 + assert len(subst_half) == 0 and red_half == 0.5 + + +def test_cse_not_possible(): + # No substitution possible. + e = Add(x, y) + substs, reduced = cse([e]) + assert substs == [] + assert reduced == [x + y] + # issue 6329 + eq = (meijerg((1, 2), (y, 4), (5,), [], x) + + meijerg((1, 3), (y, 4), (5,), [], x)) + assert cse(eq) == ([], [eq]) + + +def test_nested_substitution(): + # Substitution within a substitution. + e = Add(Pow(w*x + y, 2), sqrt(w*x + y)) + substs, reduced = cse([e]) + assert substs == [(x0, w*x + y)] + assert reduced == [sqrt(x0) + x0**2] + + +def test_subtraction_opt(): + # Make sure subtraction is optimized. + e = (x - y)*(z - y) + exp((x - y)*(z - y)) + substs, reduced = cse( + [e], optimizations=[(cse_opts.sub_pre, cse_opts.sub_post)]) + assert substs == [(x0, (x - y)*(y - z))] + assert reduced == [-x0 + exp(-x0)] + e = -(x - y)*(z - y) + exp(-(x - y)*(z - y)) + substs, reduced = cse( + [e], optimizations=[(cse_opts.sub_pre, cse_opts.sub_post)]) + assert substs == [(x0, (x - y)*(y - z))] + assert reduced == [x0 + exp(x0)] + # issue 4077 + n = -1 + 1/x + e = n/x/(-n)**2 - 1/n/x + assert cse(e, optimizations=[(cse_opts.sub_pre, cse_opts.sub_post)]) == \ + ([], [0]) + assert cse(((w + x + y + z)*(w - y - z))/(w + x)**3) == \ + ([(x0, w + x), (x1, y + z)], [(w - x1)*(x0 + x1)/x0**3]) + + +def test_multiple_expressions(): + e1 = (x + y)*z + e2 = (x + y)*w + substs, reduced = cse([e1, e2]) + assert substs == [(x0, x + y)] + assert reduced == [x0*z, x0*w] + l = [w*x*y + z, w*y] + substs, reduced = cse(l) + rsubsts, _ = cse(reversed(l)) + assert substs == rsubsts + assert reduced == [z + x*x0, x0] + l = [w*x*y, w*x*y + z, w*y] + substs, reduced = cse(l) + rsubsts, _ = cse(reversed(l)) + assert substs == rsubsts + assert reduced == [x1, x1 + z, x0] + l = [(x - z)*(y - z), x - z, y - z] + substs, reduced = cse(l) + rsubsts, _ = cse(reversed(l)) + assert substs == [(x0, -z), (x1, x + x0), (x2, x0 + y)] + assert rsubsts == [(x0, -z), (x1, x0 + y), (x2, x + x0)] + assert reduced == [x1*x2, x1, x2] + l = [w*y + w + x + y + z, w*x*y] + assert cse(l) == ([(x0, w*y)], [w + x + x0 + y + z, x*x0]) + assert cse([x + y, x + y + z]) == ([(x0, x + y)], [x0, z + x0]) + assert cse([x + y, x + z]) == ([], [x + y, x + z]) + assert cse([x*y, z + x*y, x*y*z + 3]) == \ + ([(x0, x*y)], [x0, z + x0, 3 + x0*z]) + + +@XFAIL # CSE of non-commutative Mul terms is disabled +def test_non_commutative_cse(): + A, B, C = symbols('A B C', commutative=False) + l = [A*B*C, A*C] + assert cse(l) == ([], l) + l = [A*B*C, A*B] + assert cse(l) == ([(x0, A*B)], [x0*C, x0]) + + +# Test if CSE of non-commutative Mul terms is disabled +def test_bypass_non_commutatives(): + A, B, C = symbols('A B C', commutative=False) + l = [A*B*C, A*C] + assert cse(l) == ([], l) + l = [A*B*C, A*B] + assert cse(l) == ([], l) + l = [B*C, A*B*C] + assert cse(l) == ([], l) + + +@XFAIL # CSE fails when replacing non-commutative sub-expressions +def test_non_commutative_order(): + A, B, C = symbols('A B C', commutative=False) + x0 = symbols('x0', commutative=False) + l = [B+C, A*(B+C)] + assert cse(l) == ([(x0, B+C)], [x0, A*x0]) + + +@XFAIL # Worked in gh-11232, but was reverted due to performance considerations +def test_issue_10228(): + assert cse([x*y**2 + x*y]) == ([(x0, x*y)], [x0*y + x0]) + assert cse([x + y, 2*x + y]) == ([(x0, x + y)], [x0, x + x0]) + assert cse((w + 2*x + y + z, w + x + 1)) == ( + [(x0, w + x)], [x0 + x + y + z, x0 + 1]) + assert cse(((w + x + y + z)*(w - x))/(w + x)) == ( + [(x0, w + x)], [(x0 + y + z)*(w - x)/x0]) + a, b, c, d, f, g, j, m = symbols('a, b, c, d, f, g, j, m') + exprs = (d*g**2*j*m, 4*a*f*g*m, a*b*c*f**2) + assert cse(exprs) == ( + [(x0, g*m), (x1, a*f)], [d*g*j*x0, 4*x0*x1, b*c*f*x1] +) + +@XFAIL +def test_powers(): + assert cse(x*y**2 + x*y) == ([(x0, x*y)], [x0*y + x0]) + + +def test_issue_4498(): + assert cse(w/(x - y) + z/(y - x), optimizations='basic') == \ + ([], [(w - z)/(x - y)]) + + +def test_issue_4020(): + assert cse(x**5 + x**4 + x**3 + x**2, optimizations='basic') \ + == ([(x0, x**2)], [x0*(x**3 + x + x0 + 1)]) + + +def test_issue_4203(): + assert cse(sin(x**x)/x**x) == ([(x0, x**x)], [sin(x0)/x0]) + + +def test_issue_6263(): + e = Eq(x*(-x + 1) + x*(x - 1), 0) + assert cse(e, optimizations='basic') == ([], [True]) + + +def test_issue_25043(): + c = symbols("c") + x = symbols("x0", real=True) + cse_expr = cse(c*x**2 + c*(x**4 - x**2))[-1][-1] + free = cse_expr.free_symbols + assert len(free) == len({i.name for i in free}) + + +def test_dont_cse_tuples(): + from sympy.core.function import Subs + f = Function("f") + g = Function("g") + + name_val, (expr,) = cse( + Subs(f(x, y), (x, y), (0, 1)) + + Subs(g(x, y), (x, y), (0, 1))) + + assert name_val == [] + assert expr == (Subs(f(x, y), (x, y), (0, 1)) + + Subs(g(x, y), (x, y), (0, 1))) + + name_val, (expr,) = cse( + Subs(f(x, y), (x, y), (0, x + y)) + + Subs(g(x, y), (x, y), (0, x + y))) + + assert name_val == [(x0, x + y)] + assert expr == Subs(f(x, y), (x, y), (0, x0)) + \ + Subs(g(x, y), (x, y), (0, x0)) + + +def test_pow_invpow(): + assert cse(1/x**2 + x**2) == \ + ([(x0, x**2)], [x0 + 1/x0]) + assert cse(x**2 + (1 + 1/x**2)/x**2) == \ + ([(x0, x**2), (x1, 1/x0)], [x0 + x1*(x1 + 1)]) + assert cse(1/x**2 + (1 + 1/x**2)*x**2) == \ + ([(x0, x**2), (x1, 1/x0)], [x0*(x1 + 1) + x1]) + assert cse(cos(1/x**2) + sin(1/x**2)) == \ + ([(x0, x**(-2))], [sin(x0) + cos(x0)]) + assert cse(cos(x**2) + sin(x**2)) == \ + ([(x0, x**2)], [sin(x0) + cos(x0)]) + assert cse(y/(2 + x**2) + z/x**2/y) == \ + ([(x0, x**2)], [y/(x0 + 2) + z/(x0*y)]) + assert cse(exp(x**2) + x**2*cos(1/x**2)) == \ + ([(x0, x**2)], [x0*cos(1/x0) + exp(x0)]) + assert cse((1 + 1/x**2)/x**2) == \ + ([(x0, x**(-2))], [x0*(x0 + 1)]) + assert cse(x**(2*y) + x**(-2*y)) == \ + ([(x0, x**(2*y))], [x0 + 1/x0]) + + +def test_postprocess(): + eq = (x + 1 + exp((x + 1)/(y + 1)) + cos(y + 1)) + assert cse([eq, Eq(x, z + 1), z - 2, (z + 1)*(x + 1)], + postprocess=cse_main.cse_separate) == \ + [[(x0, y + 1), (x2, z + 1), (x, x2), (x1, x + 1)], + [x1 + exp(x1/x0) + cos(x0), z - 2, x1*x2]] + + +def test_issue_4499(): + # previously, this gave 16 constants + from sympy.abc import a, b + B = Function('B') + G = Function('G') + t = Tuple(* + (a, a + S.Half, 2*a, b, 2*a - b + 1, (sqrt(z)/2)**(-2*a + 1)*B(2*a - + b, sqrt(z))*B(b - 1, sqrt(z))*G(b)*G(2*a - b + 1), + sqrt(z)*(sqrt(z)/2)**(-2*a + 1)*B(b, sqrt(z))*B(2*a - b, + sqrt(z))*G(b)*G(2*a - b + 1), sqrt(z)*(sqrt(z)/2)**(-2*a + 1)*B(b - 1, + sqrt(z))*B(2*a - b + 1, sqrt(z))*G(b)*G(2*a - b + 1), + (sqrt(z)/2)**(-2*a + 1)*B(b, sqrt(z))*B(2*a - b + 1, + sqrt(z))*G(b)*G(2*a - b + 1), 1, 0, S.Half, z/2, -b + 1, -2*a + b, + -2*a)) + c = cse(t) + ans = ( + [(x0, 2*a), (x1, -b + x0), (x2, x1 + 1), (x3, b - 1), (x4, sqrt(z)), + (x5, B(x3, x4)), (x6, (x4/2)**(1 - x0)*G(b)*G(x2)), (x7, x6*B(x1, x4)), + (x8, B(b, x4)), (x9, x6*B(x2, x4))], + [(a, a + S.Half, x0, b, x2, x5*x7, x4*x7*x8, x4*x5*x9, x8*x9, + 1, 0, S.Half, z/2, -x3, -x1, -x0)]) + assert ans == c + + +def test_issue_6169(): + r = CRootOf(x**6 - 4*x**5 - 2, 1) + assert cse(r) == ([], [r]) + # and a check that the right thing is done with the new + # mechanism + assert sub_post(sub_pre((-x - y)*z - x - y)) == -z*(x + y) - x - y + + +def test_cse_Indexed(): + len_y = 5 + y = IndexedBase('y', shape=(len_y,)) + x = IndexedBase('x', shape=(len_y,)) + i = Idx('i', len_y-1) + + expr1 = (y[i+1]-y[i])/(x[i+1]-x[i]) + expr2 = 1/(x[i+1]-x[i]) + replacements, reduced_exprs = cse([expr1, expr2]) + assert len(replacements) > 0 + + +def test_cse_MatrixSymbol(): + # MatrixSymbols have non-Basic args, so make sure that works + A = MatrixSymbol("A", 3, 3) + assert cse(A) == ([], [A]) + + n = symbols('n', integer=True) + B = MatrixSymbol("B", n, n) + assert cse(B) == ([], [B]) + + assert cse(A[0] * A[0]) == ([], [A[0]*A[0]]) + + assert cse(A[0,0]*A[0,1] + A[0,0]*A[0,1]*A[0,2]) == ([(x0, A[0, 0]*A[0, 1])], [x0*A[0, 2] + x0]) + +def test_cse_MatrixExpr(): + A = MatrixSymbol('A', 3, 3) + y = MatrixSymbol('y', 3, 1) + + expr1 = (A.T*A).I * A * y + expr2 = (A.T*A) * A * y + replacements, reduced_exprs = cse([expr1, expr2]) + assert len(replacements) > 0 + + replacements, reduced_exprs = cse([expr1 + expr2, expr1]) + assert replacements + + replacements, reduced_exprs = cse([A**2, A + A**2]) + assert replacements + + +def test_Piecewise(): + f = Piecewise((-z + x*y, Eq(y, 0)), (-z - x*y, True)) + ans = cse(f) + actual_ans = ([(x0, x*y)], + [Piecewise((x0 - z, Eq(y, 0)), (-z - x0, True))]) + assert ans == actual_ans + + +def test_ignore_order_terms(): + eq = exp(x).series(x,0,3) + sin(y+x**3) - 1 + assert cse(eq) == ([], [sin(x**3 + y) + x + x**2/2 + O(x**3)]) + + +def test_name_conflict(): + z1 = x0 + y + z2 = x2 + x3 + l = [cos(z1) + z1, cos(z2) + z2, x0 + x2] + substs, reduced = cse(l) + assert [e.subs(reversed(substs)) for e in reduced] == l + + +def test_name_conflict_cust_symbols(): + z1 = x0 + y + z2 = x2 + x3 + l = [cos(z1) + z1, cos(z2) + z2, x0 + x2] + substs, reduced = cse(l, symbols("x:10")) + assert [e.subs(reversed(substs)) for e in reduced] == l + + +def test_symbols_exhausted_error(): + l = cos(x+y)+x+y+cos(w+y)+sin(w+y) + sym = [x, y, z] + with raises(ValueError): + cse(l, symbols=sym) + + +def test_issue_7840(): + # daveknippers' example + C393 = sympify( \ + 'Piecewise((C391 - 1.65, C390 < 0.5), (Piecewise((C391 - 1.65, \ + C391 > 2.35), (C392, True)), True))' + ) + C391 = sympify( \ + 'Piecewise((2.05*C390**(-1.03), C390 < 0.5), (2.5*C390**(-0.625), True))' + ) + C393 = C393.subs('C391',C391) + # simple substitution + sub = {} + sub['C390'] = 0.703451854 + sub['C392'] = 1.01417794 + ss_answer = C393.subs(sub) + # cse + substitutions,new_eqn = cse(C393) + for pair in substitutions: + sub[pair[0].name] = pair[1].subs(sub) + cse_answer = new_eqn[0].subs(sub) + # both methods should be the same + assert ss_answer == cse_answer + + # GitRay's example + expr = sympify( + "Piecewise((Symbol('ON'), Equality(Symbol('mode'), Symbol('ON'))), \ + (Piecewise((Piecewise((Symbol('OFF'), StrictLessThan(Symbol('x'), \ + Symbol('threshold'))), (Symbol('ON'), true)), Equality(Symbol('mode'), \ + Symbol('AUTO'))), (Symbol('OFF'), true)), true))" + ) + substitutions, new_eqn = cse(expr) + # this Piecewise should be exactly the same + assert new_eqn[0] == expr + # there should not be any replacements + assert len(substitutions) < 1 + + +def test_issue_8891(): + for cls in (MutableDenseMatrix, MutableSparseMatrix, + ImmutableDenseMatrix, ImmutableSparseMatrix): + m = cls(2, 2, [x + y, 0, 0, 0]) + res = cse([x + y, m]) + ans = ([(x0, x + y)], [x0, cls([[x0, 0], [0, 0]])]) + assert res == ans + assert isinstance(res[1][-1], cls) + + +def test_issue_11230(): + # a specific test that always failed + a, b, f, k, l, i = symbols('a b f k l i') + p = [a*b*f*k*l, a*i*k**2*l, f*i*k**2*l] + R, C = cse(p) + assert not any(i.is_Mul for a in C for i in a.args) + + # random tests for the issue + from sympy.core.random import choice + from sympy.core.function import expand_mul + s = symbols('a:m') + # 35 Mul tests, none of which should ever fail + ex = [Mul(*[choice(s) for i in range(5)]) for i in range(7)] + for p in subsets(ex, 3): + p = list(p) + R, C = cse(p) + assert not any(i.is_Mul for a in C for i in a.args) + for ri in reversed(R): + for i in range(len(C)): + C[i] = C[i].subs(*ri) + assert p == C + # 35 Add tests, none of which should ever fail + ex = [Add(*[choice(s[:7]) for i in range(5)]) for i in range(7)] + for p in subsets(ex, 3): + p = list(p) + R, C = cse(p) + assert not any(i.is_Add for a in C for i in a.args) + for ri in reversed(R): + for i in range(len(C)): + C[i] = C[i].subs(*ri) + # use expand_mul to handle cases like this: + # p = [a + 2*b + 2*e, 2*b + c + 2*e, b + 2*c + 2*g] + # x0 = 2*(b + e) is identified giving a rebuilt p that + # is now `[a + 2*(b + e), c + 2*(b + e), b + 2*c + 2*g]` + assert p == [expand_mul(i) for i in C] + + +@XFAIL +def test_issue_11577(): + def check(eq): + r, c = cse(eq) + assert eq.count_ops() >= \ + len(r) + sum(i[1].count_ops() for i in r) + \ + count_ops(c) + + eq = x**5*y**2 + x**5*y + x**5 + assert cse(eq) == ( + [(x0, x**4), (x1, x*y)], [x**5 + x0*x1*y + x0*x1]) + # ([(x0, x**5*y)], [x0*y + x0 + x**5]) or + # ([(x0, x**5)], [x0*y**2 + x0*y + x0]) + check(eq) + + eq = x**2/(y + 1)**2 + x/(y + 1) + assert cse(eq) == ( + [(x0, y + 1)], [x**2/x0**2 + x/x0]) + # ([(x0, x/(y + 1))], [x0**2 + x0]) + check(eq) + + +def test_hollow_rejection(): + eq = [x + 3, x + 4] + assert cse(eq) == ([], eq) + + +def test_cse_ignore(): + exprs = [exp(y)*(3*y + 3*sqrt(x+1)), exp(y)*(5*y + 5*sqrt(x+1))] + subst1, red1 = cse(exprs) + assert any(y in sub.free_symbols for _, sub in subst1), "cse failed to identify any term with y" + + subst2, red2 = cse(exprs, ignore=(y,)) # y is not allowed in substitutions + assert not any(y in sub.free_symbols for _, sub in subst2), "Sub-expressions containing y must be ignored" + assert any(sub - sqrt(x + 1) == 0 for _, sub in subst2), "cse failed to identify sqrt(x + 1) as sub-expression" + + +def test_cse_ignore_issue_15002(): + l = [ + w*exp(x)*exp(-z), + exp(y)*exp(x)*exp(-z) + ] + substs, reduced = cse(l, ignore=(x,)) + rl = [e.subs(reversed(substs)) for e in reduced] + assert rl == l + + +def test_cse_unevaluated(): + xp1 = UnevaluatedExpr(x + 1) + # This used to cause RecursionError + [(x0, ue)], [red] = cse([(-1 - xp1) / (1 - xp1)]) + if ue == xp1: + assert red == (-1 - x0) / (1 - x0) + elif ue == -xp1: + assert red == (-1 + x0) / (1 + x0) + else: + msg = f'Expected common subexpression {xp1} or {-xp1}, instead got {ue}' + assert False, msg + + +def test_cse__performance(): + nexprs, nterms = 3, 20 + x = symbols('x:%d' % nterms) + exprs = [ + reduce(add, [x[j]*(-1)**(i+j) for j in range(nterms)]) + for i in range(nexprs) + ] + assert (exprs[0] + exprs[1]).simplify() == 0 + subst, red = cse(exprs) + assert len(subst) > 0, "exprs[0] == -exprs[2], i.e. a CSE" + for i, e in enumerate(red): + assert (e.subs(reversed(subst)) - exprs[i]).simplify() == 0 + + +def test_issue_12070(): + exprs = [x + y, 2 + x + y, x + y + z, 3 + x + y + z] + subst, red = cse(exprs) + assert 6 >= (len(subst) + sum(v.count_ops() for k, v in subst) + + count_ops(red)) + + +def test_issue_13000(): + eq = x/(-4*x**2 + y**2) + cse_eq = cse(eq)[1][0] + assert cse_eq == eq + + +def test_issue_18203(): + eq = CRootOf(x**5 + 11*x - 2, 0) + CRootOf(x**5 + 11*x - 2, 1) + assert cse(eq) == ([], [eq]) + + +def test_unevaluated_mul(): + eq = Mul(x + y, x + y, evaluate=False) + assert cse(eq) == ([(x0, x + y)], [x0**2]) + + +def test_cse_release_variables(): + from sympy.simplify.cse_main import cse_release_variables + _0, _1, _2, _3, _4 = symbols('_:5') + eqs = [(x + y - 1)**2, x, + x + y, (x + y)/(2*x + 1) + (x + y - 1)**2, + (2*x + 1)**(x + y)] + r, e = cse(eqs, postprocess=cse_release_variables) + # this can change in keeping with the intention of the function + assert r, e == ([ + (x0, x + y), (x1, (x0 - 1)**2), (x2, 2*x + 1), + (_3, x0/x2 + x1), (_4, x2**x0), (x2, None), (_0, x1), + (x1, None), (_2, x0), (x0, None), (_1, x)], (_0, _1, _2, _3, _4)) + r.reverse() + r = [(s, v) for s, v in r if v is not None] + assert eqs == [i.subs(r) for i in e] + + +def test_cse_list(): + _cse = lambda x: cse(x, list=False) + assert _cse(x) == ([], x) + assert _cse('x') == ([], 'x') + it = [x] + for c in (list, tuple, set): + assert _cse(c(it)) == ([], c(it)) + #Tuple works different from tuple: + assert _cse(Tuple(*it)) == ([], Tuple(*it)) + d = {x: 1} + assert _cse(d) == ([], d) + +def test_issue_18991(): + A = MatrixSymbol('A', 2, 2) + assert signsimp(-A * A - A) == -A * A - A + + +def test_unevaluated_Mul(): + m = [Mul(1, 2, evaluate=False)] + assert cse(m) == ([], m) + + +def test_cse_matrix_expression_inverse(): + A = ImmutableDenseMatrix(symbols('A:4')).reshape(2, 2) + x = Inverse(A) + cse_expr = cse(x) + assert cse_expr == ([], [Inverse(A)]) + + +def test_cse_matrix_expression_matmul_inverse(): + A = ImmutableDenseMatrix(symbols('A:4')).reshape(2, 2) + b = ImmutableDenseMatrix(symbols('b:2')) + x = MatMul(Inverse(A), b) + cse_expr = cse(x) + assert cse_expr == ([], [x]) + + +def test_cse_matrix_negate_matrix(): + A = ImmutableDenseMatrix(symbols('A:4')).reshape(2, 2) + x = MatMul(S.NegativeOne, A) + cse_expr = cse(x) + assert cse_expr == ([], [x]) + + +def test_cse_matrix_negate_matmul_not_extracted(): + A = ImmutableDenseMatrix(symbols('A:4')).reshape(2, 2) + B = ImmutableDenseMatrix(symbols('B:4')).reshape(2, 2) + x = MatMul(S.NegativeOne, A, B) + cse_expr = cse(x) + assert cse_expr == ([], [x]) + + +@XFAIL # No simplification rule for nested associative operations +def test_cse_matrix_nested_matmul_collapsed(): + A = ImmutableDenseMatrix(symbols('A:4')).reshape(2, 2) + B = ImmutableDenseMatrix(symbols('B:4')).reshape(2, 2) + x = MatMul(S.NegativeOne, MatMul(A, B)) + cse_expr = cse(x) + assert cse_expr == ([], [MatMul(S.NegativeOne, A, B)]) + + +def test_cse_matrix_optimize_out_single_argument_mul(): + A = ImmutableDenseMatrix(symbols('A:4')).reshape(2, 2) + x = MatMul(MatMul(MatMul(A))) + cse_expr = cse(x) + assert cse_expr == ([], [A]) + + +@XFAIL # Multiple simplification passed not supported in CSE +def test_cse_matrix_optimize_out_single_argument_mul_combined(): + A = ImmutableDenseMatrix(symbols('A:4')).reshape(2, 2) + x = MatAdd(MatMul(MatMul(MatMul(A))), MatMul(MatMul(A)), MatMul(A), A) + cse_expr = cse(x) + assert cse_expr == ([], [MatMul(4, A)]) + + +def test_cse_matrix_optimize_out_single_argument_add(): + A = ImmutableDenseMatrix(symbols('A:4')).reshape(2, 2) + x = MatAdd(MatAdd(MatAdd(MatAdd(A)))) + cse_expr = cse(x) + assert cse_expr == ([], [A]) + + +@XFAIL # Multiple simplification passed not supported in CSE +def test_cse_matrix_optimize_out_single_argument_add_combined(): + A = ImmutableDenseMatrix(symbols('A:4')).reshape(2, 2) + x = MatMul(MatAdd(MatAdd(MatAdd(A))), MatAdd(MatAdd(A)), MatAdd(A), A) + cse_expr = cse(x) + assert cse_expr == ([], [MatMul(4, A)]) + + +def test_cse_matrix_expression_matrix_solve(): + A = ImmutableDenseMatrix(symbols('A:4')).reshape(2, 2) + b = ImmutableDenseMatrix(symbols('b:2')) + x = MatrixSolve(A, b) + cse_expr = cse(x) + assert cse_expr == ([], [x]) + + +def test_cse_matrix_matrix_expression(): + X = ImmutableDenseMatrix(symbols('X:4')).reshape(2, 2) + y = ImmutableDenseMatrix(symbols('y:2')) + b = MatMul(Inverse(MatMul(Transpose(X), X)), Transpose(X), y) + cse_expr = cse(b) + x0 = MatrixSymbol('x0', 2, 2) + reduced_expr_expected = MatMul(Inverse(MatMul(x0, X)), x0, y) + assert cse_expr == ([(x0, Transpose(X))], [reduced_expr_expected]) + + +def test_cse_matrix_kalman_filter(): + """Kalman Filter example from Matthew Rocklin's SciPy 2013 talk. + + Talk titled: "Matrix Expressions and BLAS/LAPACK; SciPy 2013 Presentation" + + Video: https://pyvideo.org/scipy-2013/matrix-expressions-and-blaslapack-scipy-2013-pr.html + + Notes + ===== + + Equations are: + + new_mu = mu + Sigma*H.T * (R + H*Sigma*H.T).I * (H*mu - data) + = MatAdd(mu, MatMul(Sigma, Transpose(H), Inverse(MatAdd(R, MatMul(H, Sigma, Transpose(H)))), MatAdd(MatMul(H, mu), MatMul(S.NegativeOne, data)))) + new_Sigma = Sigma - Sigma*H.T * (R + H*Sigma*H.T).I * H * Sigma + = MatAdd(Sigma, MatMul(S.NegativeOne, Sigma, Transpose(H)), Inverse(MatAdd(R, MatMul(H*Sigma*Transpose(H)))), H, Sigma)) + + """ + N = 2 + mu = ImmutableDenseMatrix(symbols(f'mu:{N}')) + Sigma = ImmutableDenseMatrix(symbols(f'Sigma:{N * N}')).reshape(N, N) + H = ImmutableDenseMatrix(symbols(f'H:{N * N}')).reshape(N, N) + R = ImmutableDenseMatrix(symbols(f'R:{N * N}')).reshape(N, N) + data = ImmutableDenseMatrix(symbols(f'data:{N}')) + new_mu = MatAdd(mu, MatMul(Sigma, Transpose(H), Inverse(MatAdd(R, MatMul(H, Sigma, Transpose(H)))), MatAdd(MatMul(H, mu), MatMul(S.NegativeOne, data)))) + new_Sigma = MatAdd(Sigma, MatMul(S.NegativeOne, Sigma, Transpose(H), Inverse(MatAdd(R, MatMul(H, Sigma, Transpose(H)))), H, Sigma)) + cse_expr = cse([new_mu, new_Sigma]) + x0 = MatrixSymbol('x0', N, N) + x1 = MatrixSymbol('x1', N, N) + replacements_expected = [ + (x0, Transpose(H)), + (x1, Inverse(MatAdd(R, MatMul(H, Sigma, x0)))), + ] + reduced_exprs_expected = [ + MatAdd(mu, MatMul(Sigma, x0, x1, MatAdd(MatMul(H, mu), MatMul(S.NegativeOne, data)))), + MatAdd(Sigma, MatMul(S.NegativeOne, Sigma, x0, x1, H, Sigma)), + ] + assert cse_expr == (replacements_expected, reduced_exprs_expected) diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_cse_diff.py b/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_cse_diff.py new file mode 100644 index 0000000000000000000000000000000000000000..92b2d3d6bbaafb838a5e75f32a214511a1d39567 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_cse_diff.py @@ -0,0 +1,206 @@ +"""Tests for the ``sympy.simplify._cse_diff.py`` module.""" + +import pytest + +from sympy.core.symbol import (Symbol, symbols) +from sympy.core.numbers import Integer +from sympy.core.function import Function +from sympy.core import Derivative +from sympy.functions.elementary.exponential import exp +from sympy.matrices.immutable import ImmutableDenseMatrix +from sympy.physics.mechanics import dynamicsymbols +from sympy.simplify._cse_diff import (_forward_jacobian, + _remove_cse_from_derivative, + _forward_jacobian_cse, + _forward_jacobian_norm_in_cse_out) +from sympy.simplify.simplify import simplify +from sympy.matrices import Matrix, eye + +from sympy.testing.pytest import raises +from sympy.functions.elementary.trigonometric import (cos, sin, tan) +from sympy.simplify.trigsimp import trigsimp + +from sympy import cse + + +w = Symbol('w') +x = Symbol('x') +y = Symbol('y') +z = Symbol('z') + +q1, q2, q3 = dynamicsymbols('q1 q2 q3') + +# Define the custom functions +k = Function('k')(x, y) +f = Function('f')(k, z) + +zero = Integer(0) +one = Integer(1) +two = Integer(2) +neg_one = Integer(-1) + + +@pytest.mark.parametrize( + 'expr, wrt', + [ + ([zero], [x]), + ([one], [x]), + ([two], [x]), + ([neg_one], [x]), + ([x], [x]), + ([y], [x]), + ([x + y], [x]), + ([x*y], [x]), + ([x**2], [x]), + ([x**y], [x]), + ([exp(x)], [x]), + ([sin(x)], [x]), + ([tan(x)], [x]), + ([zero, one, x, y, x*y, x + y], [x, y]), + ([((x/y) + sin(x/y) - exp(y))*((x/y) - exp(y))], [x, y]), + ([w*tan(y*z)/(x - tan(y*z)), w*x*tan(y*z)/(x - tan(y*z))], [w, x, y, z]), + ([q1**2 + q2, q2**2 + q3, q3**2 + q1], [q1, q2, q3]), + ([f + Derivative(f, x) + k + 2*x], [x]) + ] +) + + +def test_forward_jacobian(expr, wrt): + expr = ImmutableDenseMatrix([expr]).T + wrt = ImmutableDenseMatrix([wrt]).T + jacobian = _forward_jacobian(expr, wrt) + zeros = ImmutableDenseMatrix.zeros(*jacobian.shape) + assert simplify(jacobian - expr.jacobian(wrt)) == zeros + + +def test_process_cse(): + x, y, z = symbols('x y z') + f = Function('f') + k = Function('k') + expr = Matrix([f(k(x,y), z) + Derivative(f(k(x,y), z), x) + k(x,y) + 2*x]) + repl, reduced = cse(expr) + p_repl, p_reduced = _remove_cse_from_derivative(repl, reduced) + + x0 = symbols('x0') + x1 = symbols('x1') + + expected_output = ( + [(x0, k(x, y)), (x1, f(x0, z))], + [Matrix([2 * x + x0 + x1 + Derivative(f(k(x, y), z), x)])] + ) + + assert p_repl == expected_output[0], f"Expected {expected_output[0]}, but got {p_repl}" + assert p_reduced == expected_output[1], f"Expected {expected_output[1]}, but got {p_reduced}" + + +def test_io_matrix_type(): + x, y, z = symbols('x y z') + expr = ImmutableDenseMatrix([ + x * y + y * z + x * y * z, + x ** 2 + y ** 2 + z ** 2, + x * y + x * z + y * z + ]) + wrt = ImmutableDenseMatrix([x, y, z]) + + replacements, reduced_expr = cse(expr) + + # Test _forward_jacobian_core + replacements_core, jacobian_core, precomputed_fs_core = _forward_jacobian_cse(replacements, reduced_expr, wrt) + assert isinstance(jacobian_core[0], type(reduced_expr[0])), "Jacobian should be a Matrix of the same type as the input" + + # Test _forward_jacobian_norm_in_dag_out + replacements_norm, jacobian_norm, precomputed_fs_norm = _forward_jacobian_norm_in_cse_out( + expr, wrt) + assert isinstance(jacobian_norm[0], type(reduced_expr[0])), "Jacobian should be a Matrix of the same type as the input" + + # Test _forward_jacobian + jacobian = _forward_jacobian(expr, wrt) + assert isinstance(jacobian, type(expr)), "Jacobian should be a Matrix of the same type as the input" + + +def test_forward_jacobian_input_output(): + x, y, z = symbols('x y z') + expr = Matrix([ + x * y + y * z + x * y * z, + x ** 2 + y ** 2 + z ** 2, + x * y + x * z + y * z + ]) + wrt = Matrix([x, y, z]) + + replacements, reduced_expr = cse(expr) + + # Test _forward_jacobian_core + replacements_core, jacobian_core, precomputed_fs_core = _forward_jacobian_cse(replacements, reduced_expr, wrt) + assert isinstance(replacements_core, type(replacements)), "Replacements should be a list" + assert isinstance(jacobian_core, type(reduced_expr)), "Jacobian should be a list" + assert isinstance(precomputed_fs_core, list), "Precomputed free symbols should be a list" + assert len(replacements_core) == len(replacements), "Length of replacements does not match" + assert len(jacobian_core) == 1, "Jacobian should have one element" + assert len(precomputed_fs_core) == len(replacements), "Length of precomputed free symbols does not match" + + # Test _forward_jacobian_norm_in_dag_out + replacements_norm, jacobian_norm, precomputed_fs_norm = _forward_jacobian_norm_in_cse_out(expr, wrt) + assert isinstance(replacements_norm, type(replacements)), "Replacements should be a list" + assert isinstance(jacobian_norm, type(reduced_expr)), "Jacobian should be a list" + assert isinstance(precomputed_fs_norm, list), "Precomputed free symbols should be a list" + assert len(replacements_norm) == len(replacements), "Length of replacements does not match" + assert len(jacobian_norm) == 1, "Jacobian should have one element" + assert len(precomputed_fs_norm) == len(replacements), "Length of precomputed free symbols does not match" + + +def test_jacobian_hessian(): + L = Matrix(1, 2, [x**2*y, 2*y**2 + x*y]) + syms = [x, y] + assert _forward_jacobian(L, syms) == Matrix([[2*x*y, x**2], [y, 4*y + x]]) + + L = Matrix(1, 2, [x, x**2*y**3]) + assert _forward_jacobian(L, syms) == Matrix([[1, 0], [2*x*y**3, x**2*3*y**2]]) + + +def test_jacobian_metrics(): + rho, phi = symbols("rho,phi") + X = Matrix([rho * cos(phi), rho * sin(phi)]) + Y = Matrix([rho, phi]) + J = _forward_jacobian(X, Y) + assert J == X.jacobian(Y.T) + assert J == (X.T).jacobian(Y) + assert J == (X.T).jacobian(Y.T) + g = J.T * eye(J.shape[0]) * J + g = g.applyfunc(trigsimp) + assert g == Matrix([[1, 0], [0, rho ** 2]]) + + +def test_jacobian2(): + rho, phi = symbols("rho,phi") + X = Matrix([rho * cos(phi), rho * sin(phi), rho ** 2]) + Y = Matrix([rho, phi]) + J = Matrix([ + [cos(phi), -rho * sin(phi)], + [sin(phi), rho * cos(phi)], + [2 * rho, 0], + ]) + assert _forward_jacobian(X, Y) == J + + +def test_issue_4564(): + X = Matrix([exp(x + y + z), exp(x + y + z), exp(x + y + z)]) + Y = Matrix([x, y, z]) + for i in range(1, 3): + for j in range(1, 3): + X_slice = X[:i, :] + Y_slice = Y[:j, :] + J = _forward_jacobian(X_slice, Y_slice) + assert J.rows == i + assert J.cols == j + for k in range(j): + assert J[:, k] == X_slice + + +def test_nonvectorJacobian(): + X = Matrix([[exp(x + y + z), exp(x + y + z)], + [exp(x + y + z), exp(x + y + z)]]) + raises(TypeError, lambda: _forward_jacobian(X, Matrix([x, y, z]))) + X = X[0, :] + Y = Matrix([[x, y], [x, z]]) + raises(TypeError, lambda: _forward_jacobian(X, Y)) + raises(TypeError, lambda: _forward_jacobian(X, Matrix([[x, y], [x, z]]))) diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_epathtools.py b/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_epathtools.py new file mode 100644 index 0000000000000000000000000000000000000000..a8bb47b2f2ff624077ab9905677b181c587ab5a7 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_epathtools.py @@ -0,0 +1,90 @@ +"""Tests for tools for manipulation of expressions using paths. """ + +from sympy.simplify.epathtools import epath, EPath +from sympy.testing.pytest import raises + +from sympy.core.numbers import E +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.abc import x, y, z, t + + +def test_epath_select(): + expr = [((x, 1, t), 2), ((3, y, 4), z)] + + assert epath("/*", expr) == [((x, 1, t), 2), ((3, y, 4), z)] + assert epath("/*/*", expr) == [(x, 1, t), 2, (3, y, 4), z] + assert epath("/*/*/*", expr) == [x, 1, t, 3, y, 4] + assert epath("/*/*/*/*", expr) == [] + + assert epath("/[:]", expr) == [((x, 1, t), 2), ((3, y, 4), z)] + assert epath("/[:]/[:]", expr) == [(x, 1, t), 2, (3, y, 4), z] + assert epath("/[:]/[:]/[:]", expr) == [x, 1, t, 3, y, 4] + assert epath("/[:]/[:]/[:]/[:]", expr) == [] + + assert epath("/*/[:]", expr) == [(x, 1, t), 2, (3, y, 4), z] + + assert epath("/*/[0]", expr) == [(x, 1, t), (3, y, 4)] + assert epath("/*/[1]", expr) == [2, z] + assert epath("/*/[2]", expr) == [] + + assert epath("/*/int", expr) == [2] + assert epath("/*/Symbol", expr) == [z] + assert epath("/*/tuple", expr) == [(x, 1, t), (3, y, 4)] + assert epath("/*/__iter__?", expr) == [(x, 1, t), (3, y, 4)] + + assert epath("/*/int|tuple", expr) == [(x, 1, t), 2, (3, y, 4)] + assert epath("/*/Symbol|tuple", expr) == [(x, 1, t), (3, y, 4), z] + assert epath("/*/int|Symbol|tuple", expr) == [(x, 1, t), 2, (3, y, 4), z] + + assert epath("/*/int|__iter__?", expr) == [(x, 1, t), 2, (3, y, 4)] + assert epath("/*/Symbol|__iter__?", expr) == [(x, 1, t), (3, y, 4), z] + assert epath( + "/*/int|Symbol|__iter__?", expr) == [(x, 1, t), 2, (3, y, 4), z] + + assert epath("/*/[0]/int", expr) == [1, 3, 4] + assert epath("/*/[0]/Symbol", expr) == [x, t, y] + + assert epath("/*/[0]/int[1:]", expr) == [1, 4] + assert epath("/*/[0]/Symbol[1:]", expr) == [t, y] + + assert epath("/Symbol", x + y + z + 1) == [x, y, z] + assert epath("/*/*/Symbol", t + sin(x + 1) + cos(x + y + E)) == [x, x, y] + + +def test_epath_apply(): + expr = [((x, 1, t), 2), ((3, y, 4), z)] + func = lambda expr: expr**2 + + assert epath("/*", expr, list) == [[(x, 1, t), 2], [(3, y, 4), z]] + + assert epath("/*/[0]", expr, list) == [([x, 1, t], 2), ([3, y, 4], z)] + assert epath("/*/[1]", expr, func) == [((x, 1, t), 4), ((3, y, 4), z**2)] + assert epath("/*/[2]", expr, list) == expr + + assert epath("/*/[0]/int", expr, func) == [((x, 1, t), 2), ((9, y, 16), z)] + assert epath("/*/[0]/Symbol", expr, func) == [((x**2, 1, t**2), 2), + ((3, y**2, 4), z)] + assert epath( + "/*/[0]/int[1:]", expr, func) == [((x, 1, t), 2), ((3, y, 16), z)] + assert epath("/*/[0]/Symbol[1:]", expr, func) == [((x, 1, t**2), + 2), ((3, y**2, 4), z)] + + assert epath("/Symbol", x + y + z + 1, func) == x**2 + y**2 + z**2 + 1 + assert epath("/*/*/Symbol", t + sin(x + 1) + cos(x + y + E), func) == \ + t + sin(x**2 + 1) + cos(x**2 + y**2 + E) + + +def test_EPath(): + assert EPath("/*/[0]")._path == "/*/[0]" + assert EPath(EPath("/*/[0]"))._path == "/*/[0]" + assert isinstance(epath("/*/[0]"), EPath) is True + + assert repr(EPath("/*/[0]")) == "EPath('/*/[0]')" + + raises(ValueError, lambda: EPath("")) + raises(ValueError, lambda: EPath("/")) + raises(ValueError, lambda: EPath("/|x")) + raises(ValueError, lambda: EPath("/[")) + raises(ValueError, lambda: EPath("/[0]%")) + + raises(NotImplementedError, lambda: EPath("Symbol")) diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_fu.py b/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_fu.py new file mode 100644 index 0000000000000000000000000000000000000000..2de2126b7333195fceeffe72dc9cb642e7eba9a9 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_fu.py @@ -0,0 +1,492 @@ +from sympy.core.add import Add +from sympy.core.mul import Mul +from sympy.core.numbers import (I, Rational, pi) +from sympy.core.parameters import evaluate +from sympy.core.singleton import S +from sympy.core.symbol import (Dummy, Symbol, symbols) +from sympy.functions.elementary.hyperbolic import (cosh, coth, csch, sech, sinh, tanh) +from sympy.functions.elementary.miscellaneous import (root, sqrt) +from sympy.functions.elementary.trigonometric import (cos, cot, csc, sec, sin, tan) +from sympy.simplify.powsimp import powsimp +from sympy.simplify.fu import ( + L, TR1, TR10, TR10i, TR11, _TR11, TR12, TR12i, TR13, TR14, TR15, TR16, + TR111, TR2, TR2i, TR3, TR4, TR5, TR6, TR7, TR8, TR9, TRmorrie, _TR56 as T, + TRpower, hyper_as_trig, fu, process_common_addends, trig_split, + as_f_sign_1) +from sympy.core.random import verify_numerically +from sympy.abc import a, b, c, x, y, z + + +def test_TR1(): + assert TR1(2*csc(x) + sec(x)) == 1/cos(x) + 2/sin(x) + + +def test_TR2(): + assert TR2(tan(x)) == sin(x)/cos(x) + assert TR2(cot(x)) == cos(x)/sin(x) + assert TR2(tan(tan(x) - sin(x)/cos(x))) == 0 + + +def test_TR2i(): + # just a reminder that ratios of powers only simplify if both + # numerator and denominator satisfy the condition that each + # has a positive base or an integer exponent; e.g. the following, + # at y=-1, x=1/2 gives sqrt(2)*I != -sqrt(2)*I + assert powsimp(2**x/y**x) != (2/y)**x + + assert TR2i(sin(x)/cos(x)) == tan(x) + assert TR2i(sin(x)*sin(y)/cos(x)) == tan(x)*sin(y) + assert TR2i(1/(sin(x)/cos(x))) == 1/tan(x) + assert TR2i(1/(sin(x)*sin(y)/cos(x))) == 1/tan(x)/sin(y) + assert TR2i(sin(x)/2/(cos(x) + 1)) == sin(x)/(cos(x) + 1)/2 + + assert TR2i(sin(x)/2/(cos(x) + 1), half=True) == tan(x/2)/2 + assert TR2i(sin(1)/(cos(1) + 1), half=True) == tan(S.Half) + assert TR2i(sin(2)/(cos(2) + 1), half=True) == tan(1) + assert TR2i(sin(4)/(cos(4) + 1), half=True) == tan(2) + assert TR2i(sin(5)/(cos(5) + 1), half=True) == tan(5*S.Half) + assert TR2i((cos(1) + 1)/sin(1), half=True) == 1/tan(S.Half) + assert TR2i((cos(2) + 1)/sin(2), half=True) == 1/tan(1) + assert TR2i((cos(4) + 1)/sin(4), half=True) == 1/tan(2) + assert TR2i((cos(5) + 1)/sin(5), half=True) == 1/tan(5*S.Half) + assert TR2i((cos(1) + 1)**(-a)*sin(1)**a, half=True) == tan(S.Half)**a + assert TR2i((cos(2) + 1)**(-a)*sin(2)**a, half=True) == tan(1)**a + assert TR2i((cos(4) + 1)**(-a)*sin(4)**a, half=True) == (cos(4) + 1)**(-a)*sin(4)**a + assert TR2i((cos(5) + 1)**(-a)*sin(5)**a, half=True) == (cos(5) + 1)**(-a)*sin(5)**a + assert TR2i((cos(1) + 1)**a*sin(1)**(-a), half=True) == tan(S.Half)**(-a) + assert TR2i((cos(2) + 1)**a*sin(2)**(-a), half=True) == tan(1)**(-a) + assert TR2i((cos(4) + 1)**a*sin(4)**(-a), half=True) == (cos(4) + 1)**a*sin(4)**(-a) + assert TR2i((cos(5) + 1)**a*sin(5)**(-a), half=True) == (cos(5) + 1)**a*sin(5)**(-a) + + i = symbols('i', integer=True) + assert TR2i(((cos(5) + 1)**i*sin(5)**(-i)), half=True) == tan(5*S.Half)**(-i) + assert TR2i(1/((cos(5) + 1)**i*sin(5)**(-i)), half=True) == tan(5*S.Half)**i + + +def test_TR3(): + assert TR3(cos(y - x*(y - x))) == cos(x*(x - y) + y) + assert cos(pi/2 + x) == -sin(x) + assert cos(30*pi/2 + x) == -cos(x) + + for f in (cos, sin, tan, cot, csc, sec): + i = f(pi*Rational(3, 7)) + j = TR3(i) + assert verify_numerically(i, j) and i.func != j.func + + with evaluate(False): + eq = cos(9*pi/22) + assert eq.has(9*pi) and TR3(eq) == sin(pi/11) + + +def test_TR4(): + for i in [0, pi/6, pi/4, pi/3, pi/2]: + with evaluate(False): + eq = cos(i) + assert isinstance(eq, cos) and TR4(eq) == cos(i) + + +def test__TR56(): + h = lambda x: 1 - x + assert T(sin(x)**3, sin, cos, h, 4, False) == sin(x)*(-cos(x)**2 + 1) + assert T(sin(x)**10, sin, cos, h, 4, False) == sin(x)**10 + assert T(sin(x)**6, sin, cos, h, 6, False) == (-cos(x)**2 + 1)**3 + assert T(sin(x)**6, sin, cos, h, 6, True) == sin(x)**6 + assert T(sin(x)**8, sin, cos, h, 10, True) == (-cos(x)**2 + 1)**4 + + # issue 17137 + assert T(sin(x)**I, sin, cos, h, 4, True) == sin(x)**I + assert T(sin(x)**(2*I + 1), sin, cos, h, 4, True) == sin(x)**(2*I + 1) + + +def test_TR5(): + assert TR5(sin(x)**2) == -cos(x)**2 + 1 + assert TR5(sin(x)**-2) == sin(x)**(-2) + assert TR5(sin(x)**4) == (-cos(x)**2 + 1)**2 + + +def test_TR6(): + assert TR6(cos(x)**2) == -sin(x)**2 + 1 + assert TR6(cos(x)**-2) == cos(x)**(-2) + assert TR6(cos(x)**4) == (-sin(x)**2 + 1)**2 + + +def test_TR7(): + assert TR7(cos(x)**2) == cos(2*x)/2 + S.Half + assert TR7(cos(x)**2 + 1) == cos(2*x)/2 + Rational(3, 2) + + +def test_TR8(): + assert TR8(cos(2)*cos(3)) == cos(5)/2 + cos(1)/2 + assert TR8(cos(2)*sin(3)) == sin(5)/2 + sin(1)/2 + assert TR8(sin(2)*sin(3)) == -cos(5)/2 + cos(1)/2 + assert TR8(sin(1)*sin(2)*sin(3)) == sin(4)/4 - sin(6)/4 + sin(2)/4 + assert TR8(cos(2)*cos(3)*cos(4)*cos(5)) == \ + cos(4)/4 + cos(10)/8 + cos(2)/8 + cos(8)/8 + cos(14)/8 + \ + cos(6)/8 + Rational(1, 8) + assert TR8(cos(2)*cos(3)*cos(4)*cos(5)*cos(6)) == \ + cos(10)/8 + cos(4)/8 + 3*cos(2)/16 + cos(16)/16 + cos(8)/8 + \ + cos(14)/16 + cos(20)/16 + cos(12)/16 + Rational(1, 16) + cos(6)/8 + assert TR8(sin(pi*Rational(3, 7))**2*cos(pi*Rational(3, 7))**2/(16*sin(pi/7)**2)) == Rational(1, 64) + +def test_TR9(): + a = S.Half + b = 3*a + assert TR9(a) == a + assert TR9(cos(1) + cos(2)) == 2*cos(a)*cos(b) + assert TR9(cos(1) - cos(2)) == 2*sin(a)*sin(b) + assert TR9(sin(1) - sin(2)) == -2*sin(a)*cos(b) + assert TR9(sin(1) + sin(2)) == 2*sin(b)*cos(a) + assert TR9(cos(1) + 2*sin(1) + 2*sin(2)) == cos(1) + 4*sin(b)*cos(a) + assert TR9(cos(4) + cos(2) + 2*cos(1)*cos(3)) == 4*cos(1)*cos(3) + assert TR9((cos(4) + cos(2))/cos(3)/2 + cos(3)) == 2*cos(1)*cos(2) + assert TR9(cos(3) + cos(4) + cos(5) + cos(6)) == \ + 4*cos(S.Half)*cos(1)*cos(Rational(9, 2)) + assert TR9(cos(3) + cos(3)*cos(2)) == cos(3) + cos(2)*cos(3) + assert TR9(-cos(y) + cos(x*y)) == -2*sin(x*y/2 - y/2)*sin(x*y/2 + y/2) + assert TR9(-sin(y) + sin(x*y)) == 2*sin(x*y/2 - y/2)*cos(x*y/2 + y/2) + c = cos(x) + s = sin(x) + for si in ((1, 1), (1, -1), (-1, 1), (-1, -1)): + for a in ((c, s), (s, c), (cos(x), cos(x*y)), (sin(x), sin(x*y))): + args = zip(si, a) + ex = Add(*[Mul(*ai) for ai in args]) + t = TR9(ex) + assert not (a[0].func == a[1].func and ( + not verify_numerically(ex, t.expand(trig=True)) or t.is_Add) + or a[1].func != a[0].func and ex != t) + + +def test_TR10(): + assert TR10(cos(a + b)) == -sin(a)*sin(b) + cos(a)*cos(b) + assert TR10(sin(a + b)) == sin(a)*cos(b) + sin(b)*cos(a) + assert TR10(sin(a + b + c)) == \ + (-sin(a)*sin(b) + cos(a)*cos(b))*sin(c) + \ + (sin(a)*cos(b) + sin(b)*cos(a))*cos(c) + assert TR10(cos(a + b + c)) == \ + (-sin(a)*sin(b) + cos(a)*cos(b))*cos(c) - \ + (sin(a)*cos(b) + sin(b)*cos(a))*sin(c) + + +def test_TR10i(): + assert TR10i(cos(1)*cos(3) + sin(1)*sin(3)) == cos(2) + assert TR10i(cos(1)*cos(3) - sin(1)*sin(3)) == cos(4) + assert TR10i(cos(1)*sin(3) - sin(1)*cos(3)) == sin(2) + assert TR10i(cos(1)*sin(3) + sin(1)*cos(3)) == sin(4) + assert TR10i(cos(1)*sin(3) + sin(1)*cos(3) + 7) == sin(4) + 7 + assert TR10i(cos(1)*sin(3) + sin(1)*cos(3) + cos(3)) == cos(3) + sin(4) + assert TR10i(2*cos(1)*sin(3) + 2*sin(1)*cos(3) + cos(3)) == \ + 2*sin(4) + cos(3) + assert TR10i(cos(2)*cos(3) + sin(2)*(cos(1)*sin(2) + cos(2)*sin(1))) == \ + cos(1) + eq = (cos(2)*cos(3) + sin(2)*( + cos(1)*sin(2) + cos(2)*sin(1)))*cos(5) + sin(1)*sin(5) + assert TR10i(eq) == TR10i(eq.expand()) == cos(4) + assert TR10i(sqrt(2)*cos(x)*x + sqrt(6)*sin(x)*x) == \ + 2*sqrt(2)*x*sin(x + pi/6) + assert TR10i(cos(x)/sqrt(6) + sin(x)/sqrt(2) + + cos(x)/sqrt(6)/3 + sin(x)/sqrt(2)/3) == 4*sqrt(6)*sin(x + pi/6)/9 + assert TR10i(cos(x)/sqrt(6) + sin(x)/sqrt(2) + + cos(y)/sqrt(6)/3 + sin(y)/sqrt(2)/3) == \ + sqrt(6)*sin(x + pi/6)/3 + sqrt(6)*sin(y + pi/6)/9 + assert TR10i(cos(x) + sqrt(3)*sin(x) + 2*sqrt(3)*cos(x + pi/6)) == 4*cos(x) + assert TR10i(cos(x) + sqrt(3)*sin(x) + + 2*sqrt(3)*cos(x + pi/6) + 4*sin(x)) == 4*sqrt(2)*sin(x + pi/4) + assert TR10i(cos(2)*sin(3) + sin(2)*cos(4)) == \ + sin(2)*cos(4) + sin(3)*cos(2) + + A = Symbol('A', commutative=False) + assert TR10i(sqrt(2)*cos(x)*A + sqrt(6)*sin(x)*A) == \ + 2*sqrt(2)*sin(x + pi/6)*A + + + c = cos(x) + s = sin(x) + h = sin(y) + r = cos(y) + for si in ((1, 1), (1, -1), (-1, 1), (-1, -1)): + for argsi in ((c*r, s*h), (c*h, s*r)): # explicit 2-args + args = zip(si, argsi) + ex = Add(*[Mul(*ai) for ai in args]) + t = TR10i(ex) + assert not (ex - t.expand(trig=True) or t.is_Add) + + c = cos(x) + s = sin(x) + h = sin(pi/6) + r = cos(pi/6) + for si in ((1, 1), (1, -1), (-1, 1), (-1, -1)): + for argsi in ((c*r, s*h), (c*h, s*r)): # induced + args = zip(si, argsi) + ex = Add(*[Mul(*ai) for ai in args]) + t = TR10i(ex) + assert not (ex - t.expand(trig=True) or t.is_Add) + + +def test_TR11(): + + assert TR11(sin(2*x)) == 2*sin(x)*cos(x) + assert TR11(sin(4*x)) == 4*((-sin(x)**2 + cos(x)**2)*sin(x)*cos(x)) + assert TR11(sin(x*Rational(4, 3))) == \ + 4*((-sin(x/3)**2 + cos(x/3)**2)*sin(x/3)*cos(x/3)) + + assert TR11(cos(2*x)) == -sin(x)**2 + cos(x)**2 + assert TR11(cos(4*x)) == \ + (-sin(x)**2 + cos(x)**2)**2 - 4*sin(x)**2*cos(x)**2 + + assert TR11(cos(2)) == cos(2) + + assert TR11(cos(pi*Rational(3, 7)), pi*Rational(2, 7)) == -cos(pi*Rational(2, 7))**2 + sin(pi*Rational(2, 7))**2 + assert TR11(cos(4), 2) == -sin(2)**2 + cos(2)**2 + assert TR11(cos(6), 2) == cos(6) + assert TR11(sin(x)/cos(x/2), x/2) == 2*sin(x/2) + +def test__TR11(): + + assert _TR11(sin(x/3)*sin(2*x)*sin(x/4)/(cos(x/6)*cos(x/8))) == \ + 4*sin(x/8)*sin(x/6)*sin(2*x),_TR11(sin(x/3)*sin(2*x)*sin(x/4)/(cos(x/6)*cos(x/8))) + assert _TR11(sin(x/3)/cos(x/6)) == 2*sin(x/6) + + assert _TR11(cos(x/6)/sin(x/3)) == 1/(2*sin(x/6)) + assert _TR11(sin(2*x)*cos(x/8)/sin(x/4)) == sin(2*x)/(2*sin(x/8)), _TR11(sin(2*x)*cos(x/8)/sin(x/4)) + assert _TR11(sin(x)/sin(x/2)) == 2*cos(x/2) + + +def test_TR12(): + assert TR12(tan(x + y)) == (tan(x) + tan(y))/(-tan(x)*tan(y) + 1) + assert TR12(tan(x + y + z)) ==\ + (tan(z) + (tan(x) + tan(y))/(-tan(x)*tan(y) + 1))/( + 1 - (tan(x) + tan(y))*tan(z)/(-tan(x)*tan(y) + 1)) + assert TR12(tan(x*y)) == tan(x*y) + + +def test_TR13(): + assert TR13(tan(3)*tan(2)) == -tan(2)/tan(5) - tan(3)/tan(5) + 1 + assert TR13(cot(3)*cot(2)) == 1 + cot(3)*cot(5) + cot(2)*cot(5) + assert TR13(tan(1)*tan(2)*tan(3)) == \ + (-tan(2)/tan(5) - tan(3)/tan(5) + 1)*tan(1) + assert TR13(tan(1)*tan(2)*cot(3)) == \ + (-tan(2)/tan(3) + 1 - tan(1)/tan(3))*cot(3) + + +def test_L(): + assert L(cos(x) + sin(x)) == 2 + + +def test_fu(): + + assert fu(sin(50)**2 + cos(50)**2 + sin(pi/6)) == Rational(3, 2) + assert fu(sqrt(6)*cos(x) + sqrt(2)*sin(x)) == 2*sqrt(2)*sin(x + pi/3) + + + eq = sin(x)**4 - cos(y)**2 + sin(y)**2 + 2*cos(x)**2 + assert fu(eq) == cos(x)**4 - 2*cos(y)**2 + 2 + + assert fu(S.Half - cos(2*x)/2) == sin(x)**2 + + assert fu(sin(a)*(cos(b) - sin(b)) + cos(a)*(sin(b) + cos(b))) == \ + sqrt(2)*sin(a + b + pi/4) + + assert fu(sqrt(3)*cos(x)/2 + sin(x)/2) == sin(x + pi/3) + + assert fu(1 - sin(2*x)**2/4 - sin(y)**2 - cos(x)**4) == \ + -cos(x)**2 + cos(y)**2 + + assert fu(cos(pi*Rational(4, 9))) == sin(pi/18) + assert fu(cos(pi/9)*cos(pi*Rational(2, 9))*cos(pi*Rational(3, 9))*cos(pi*Rational(4, 9))) == Rational(1, 16) + + assert fu( + tan(pi*Rational(7, 18)) + tan(pi*Rational(5, 18)) - sqrt(3)*tan(pi*Rational(5, 18))*tan(pi*Rational(7, 18))) == \ + -sqrt(3) + + assert fu(tan(1)*tan(2)) == tan(1)*tan(2) + + expr = Mul(*[cos(2**i) for i in range(10)]) + assert fu(expr) == sin(1024)/(1024*sin(1)) + + # issue #18059: + assert fu(cos(x) + sqrt(sin(x)**2)) == cos(x) + sqrt(sin(x)**2) + + assert fu((-14*sin(x)**3 + 35*sin(x) + 6*sqrt(3)*cos(x)**3 + 9*sqrt(3)*cos(x))/((cos(2*x) + 4))) == \ + 7*sin(x) + 3*sqrt(3)*cos(x) + + +def test_objective(): + assert fu(sin(x)/cos(x), measure=lambda x: x.count_ops()) == \ + tan(x) + assert fu(sin(x)/cos(x), measure=lambda x: -x.count_ops()) == \ + sin(x)/cos(x) + + +def test_process_common_addends(): + # this tests that the args are not evaluated as they are given to do + # and that key2 works when key1 is False + do = lambda x: Add(*[i**(i%2) for i in x.args]) + assert process_common_addends(Add(*[1, 2, 3, 4], evaluate=False), do, + key2=lambda x: x%2, key1=False) == 1**1 + 3**1 + 2**0 + 4**0 + + +def test_trig_split(): + assert trig_split(cos(x), cos(y)) == (1, 1, 1, x, y, True) + assert trig_split(2*cos(x), -2*cos(y)) == (2, 1, -1, x, y, True) + assert trig_split(cos(x)*sin(y), cos(y)*sin(y)) == \ + (sin(y), 1, 1, x, y, True) + + assert trig_split(cos(x), -sqrt(3)*sin(x), two=True) == \ + (2, 1, -1, x, pi/6, False) + assert trig_split(cos(x), sin(x), two=True) == \ + (sqrt(2), 1, 1, x, pi/4, False) + assert trig_split(cos(x), -sin(x), two=True) == \ + (sqrt(2), 1, -1, x, pi/4, False) + assert trig_split(sqrt(2)*cos(x), -sqrt(6)*sin(x), two=True) == \ + (2*sqrt(2), 1, -1, x, pi/6, False) + assert trig_split(-sqrt(6)*cos(x), -sqrt(2)*sin(x), two=True) == \ + (-2*sqrt(2), 1, 1, x, pi/3, False) + assert trig_split(cos(x)/sqrt(6), sin(x)/sqrt(2), two=True) == \ + (sqrt(6)/3, 1, 1, x, pi/6, False) + assert trig_split(-sqrt(6)*cos(x)*sin(y), + -sqrt(2)*sin(x)*sin(y), two=True) == \ + (-2*sqrt(2)*sin(y), 1, 1, x, pi/3, False) + + assert trig_split(cos(x), sin(x)) is None + assert trig_split(cos(x), sin(z)) is None + assert trig_split(2*cos(x), -sin(x)) is None + assert trig_split(cos(x), -sqrt(3)*sin(x)) is None + assert trig_split(cos(x)*cos(y), sin(x)*sin(z)) is None + assert trig_split(cos(x)*cos(y), sin(x)*sin(y)) is None + assert trig_split(-sqrt(6)*cos(x), sqrt(2)*sin(x)*sin(y), two=True) is \ + None + + assert trig_split(sqrt(3)*sqrt(x), cos(3), two=True) is None + assert trig_split(sqrt(3)*root(x, 3), sin(3)*cos(2), two=True) is None + assert trig_split(cos(5)*cos(6), cos(7)*sin(5), two=True) is None + + +def test_TRmorrie(): + assert TRmorrie(7*Mul(*[cos(i) for i in range(10)])) == \ + 7*sin(12)*sin(16)*cos(5)*cos(7)*cos(9)/(64*sin(1)*sin(3)) + assert TRmorrie(x) == x + assert TRmorrie(2*x) == 2*x + e = cos(pi/7)*cos(pi*Rational(2, 7))*cos(pi*Rational(4, 7)) + assert TR8(TRmorrie(e)) == Rational(-1, 8) + e = Mul(*[cos(2**i*pi/17) for i in range(1, 17)]) + assert TR8(TR3(TRmorrie(e))) == Rational(1, 65536) + # issue 17063 + eq = cos(x)/cos(x/2) + assert TRmorrie(eq) == eq + # issue #20430 + eq = cos(x/2)*sin(x/2)*cos(x)**3 + assert TRmorrie(eq) == sin(2*x)*cos(x)**2/4 + + +def test_TRpower(): + assert TRpower(1/sin(x)**2) == 1/sin(x)**2 + assert TRpower(cos(x)**3*sin(x/2)**4) == \ + (3*cos(x)/4 + cos(3*x)/4)*(-cos(x)/2 + cos(2*x)/8 + Rational(3, 8)) + for k in range(2, 8): + assert verify_numerically(sin(x)**k, TRpower(sin(x)**k)) + assert verify_numerically(cos(x)**k, TRpower(cos(x)**k)) + + +def test_hyper_as_trig(): + from sympy.simplify.fu import _osborne, _osbornei + + eq = sinh(x)**2 + cosh(x)**2 + t, f = hyper_as_trig(eq) + assert f(fu(t)) == cosh(2*x) + e, f = hyper_as_trig(tanh(x + y)) + assert f(TR12(e)) == (tanh(x) + tanh(y))/(tanh(x)*tanh(y) + 1) + + d = Dummy() + assert _osborne(sinh(x), d) == I*sin(x*d) + assert _osborne(tanh(x), d) == I*tan(x*d) + assert _osborne(coth(x), d) == cot(x*d)/I + assert _osborne(cosh(x), d) == cos(x*d) + assert _osborne(sech(x), d) == sec(x*d) + assert _osborne(csch(x), d) == csc(x*d)/I + for func in (sinh, cosh, tanh, coth, sech, csch): + h = func(pi) + assert _osbornei(_osborne(h, d), d) == h + # /!\ the _osborne functions are not meant to work + # in the o(i(trig, d), d) direction so we just check + # that they work as they are supposed to work + assert _osbornei(cos(x*y + z), y) == cosh(x + z*I) + assert _osbornei(sin(x*y + z), y) == sinh(x + z*I)/I + assert _osbornei(tan(x*y + z), y) == tanh(x + z*I)/I + assert _osbornei(cot(x*y + z), y) == coth(x + z*I)*I + assert _osbornei(sec(x*y + z), y) == sech(x + z*I) + assert _osbornei(csc(x*y + z), y) == csch(x + z*I)*I + + +def test_TR12i(): + ta, tb, tc = [tan(i) for i in (a, b, c)] + assert TR12i((ta + tb)/(-ta*tb + 1)) == tan(a + b) + assert TR12i((ta + tb)/(ta*tb - 1)) == -tan(a + b) + assert TR12i((-ta - tb)/(ta*tb - 1)) == tan(a + b) + eq = (ta + tb)/(-ta*tb + 1)**2*(-3*ta - 3*tc)/(2*(ta*tc - 1)) + assert TR12i(eq.expand()) == \ + -3*tan(a + b)*tan(a + c)/(tan(a) + tan(b) - 1)/2 + assert TR12i(tan(x)/sin(x)) == tan(x)/sin(x) + eq = (ta + cos(2))/(-ta*tb + 1) + assert TR12i(eq) == eq + eq = (ta + tb + 2)**2/(-ta*tb + 1) + assert TR12i(eq) == eq + eq = ta/(-ta*tb + 1) + assert TR12i(eq) == eq + eq = (((ta + tb)*(a + 1)).expand())**2/(ta*tb - 1) + assert TR12i(eq) == -(a + 1)**2*tan(a + b) + + +def test_TR14(): + eq = (cos(x) - 1)*(cos(x) + 1) + ans = -sin(x)**2 + assert TR14(eq) == ans + assert TR14(1/eq) == 1/ans + assert TR14((cos(x) - 1)**2*(cos(x) + 1)**2) == ans**2 + assert TR14((cos(x) - 1)**2*(cos(x) + 1)**3) == ans**2*(cos(x) + 1) + assert TR14((cos(x) - 1)**3*(cos(x) + 1)**2) == ans**2*(cos(x) - 1) + eq = (cos(x) - 1)**y*(cos(x) + 1)**y + assert TR14(eq) == eq + eq = (cos(x) - 2)**y*(cos(x) + 1) + assert TR14(eq) == eq + eq = (tan(x) - 2)**2*(cos(x) + 1) + assert TR14(eq) == eq + i = symbols('i', integer=True) + assert TR14((cos(x) - 1)**i*(cos(x) + 1)**i) == ans**i + assert TR14((sin(x) - 1)**i*(sin(x) + 1)**i) == (-cos(x)**2)**i + # could use extraction in this case + eq = (cos(x) - 1)**(i + 1)*(cos(x) + 1)**i + assert TR14(eq) in [(cos(x) - 1)*ans**i, eq] + + assert TR14((sin(x) - 1)*(sin(x) + 1)) == -cos(x)**2 + p1 = (cos(x) + 1)*(cos(x) - 1) + p2 = (cos(y) - 1)*2*(cos(y) + 1) + p3 = (3*(cos(y) - 1))*(3*(cos(y) + 1)) + assert TR14(p1*p2*p3*(x - 1)) == -18*((x - 1)*sin(x)**2*sin(y)**4) + + +def test_TR15_16_17(): + assert TR15(1 - 1/sin(x)**2) == -cot(x)**2 + assert TR16(1 - 1/cos(x)**2) == -tan(x)**2 + assert TR111(1 - 1/tan(x)**2) == 1 - cot(x)**2 + + +def test_as_f_sign_1(): + assert as_f_sign_1(x + 1) == (1, x, 1) + assert as_f_sign_1(x - 1) == (1, x, -1) + assert as_f_sign_1(-x + 1) == (-1, x, -1) + assert as_f_sign_1(-x - 1) == (-1, x, 1) + assert as_f_sign_1(2*x + 2) == (2, x, 1) + assert as_f_sign_1(x*y - y) == (y, x, -1) + assert as_f_sign_1(-x*y + y) == (-y, x, -1) + + +def test_issue_25590(): + A = Symbol('A', commutative=False) + B = Symbol('B', commutative=False) + + assert TR8(2*cos(x)*sin(x)*B*A) == sin(2*x)*B*A + assert TR13(tan(2)*tan(3)*B*A) == (-tan(2)/tan(5) - tan(3)/tan(5) + 1)*B*A + + # XXX The result may not be optimal than + # sin(2*x)*B*A + cos(x)**2 and may change in the future + assert (2*cos(x)*sin(x)*B*A + cos(x)**2).simplify() == sin(2*x)*B*A + cos(2*x)/2 + S.One/2 diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_function.py b/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_function.py new file mode 100644 index 0000000000000000000000000000000000000000..441b9faf1bb3c5e7f2279b2a61066d050e45f773 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_function.py @@ -0,0 +1,54 @@ +""" Unit tests for Hyper_Function""" +from sympy.core import symbols, Dummy, Tuple, S, Rational +from sympy.functions import hyper + +from sympy.simplify.hyperexpand import Hyper_Function + +def test_attrs(): + a, b = symbols('a, b', cls=Dummy) + f = Hyper_Function([2, a], [b]) + assert f.ap == Tuple(2, a) + assert f.bq == Tuple(b) + assert f.args == (Tuple(2, a), Tuple(b)) + assert f.sizes == (2, 1) + +def test_call(): + a, b, x = symbols('a, b, x', cls=Dummy) + f = Hyper_Function([2, a], [b]) + assert f(x) == hyper([2, a], [b], x) + +def test_has(): + a, b, c = symbols('a, b, c', cls=Dummy) + f = Hyper_Function([2, -a], [b]) + assert f.has(a) + assert f.has(Tuple(b)) + assert not f.has(c) + +def test_eq(): + assert Hyper_Function([1], []) == Hyper_Function([1], []) + assert (Hyper_Function([1], []) != Hyper_Function([1], [])) is False + assert Hyper_Function([1], []) != Hyper_Function([2], []) + assert Hyper_Function([1], []) != Hyper_Function([1, 2], []) + assert Hyper_Function([1], []) != Hyper_Function([1], [2]) + +def test_gamma(): + assert Hyper_Function([2, 3], [-1]).gamma == 0 + assert Hyper_Function([-2, -3], [-1]).gamma == 2 + n = Dummy(integer=True) + assert Hyper_Function([-1, n, 1], []).gamma == 1 + assert Hyper_Function([-1, -n, 1], []).gamma == 1 + p = Dummy(integer=True, positive=True) + assert Hyper_Function([-1, p, 1], []).gamma == 1 + assert Hyper_Function([-1, -p, 1], []).gamma == 2 + +def test_suitable_origin(): + assert Hyper_Function((S.Half,), (Rational(3, 2),))._is_suitable_origin() is True + assert Hyper_Function((S.Half,), (S.Half,))._is_suitable_origin() is False + assert Hyper_Function((S.Half,), (Rational(-1, 2),))._is_suitable_origin() is False + assert Hyper_Function((S.Half,), (0,))._is_suitable_origin() is False + assert Hyper_Function((S.Half,), (-1, 1,))._is_suitable_origin() is False + assert Hyper_Function((S.Half, 0), (1,))._is_suitable_origin() is False + assert Hyper_Function((S.Half, 1), + (2, Rational(-2, 3)))._is_suitable_origin() is True + assert Hyper_Function((S.Half, 1), + (2, Rational(-2, 3), Rational(3, 2)))._is_suitable_origin() is True diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_gammasimp.py b/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_gammasimp.py new file mode 100644 index 0000000000000000000000000000000000000000..e4c73093250b279510e3c2274db22818a9adffd8 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_gammasimp.py @@ -0,0 +1,127 @@ +from sympy.core.function import Function +from sympy.core.numbers import (Rational, pi) +from sympy.core.singleton import S +from sympy.core.symbol import symbols +from sympy.functions.combinatorial.factorials import (rf, binomial, factorial) +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.functions.special.gamma_functions import gamma +from sympy.simplify.gammasimp import gammasimp +from sympy.simplify.powsimp import powsimp +from sympy.simplify.simplify import simplify + +from sympy.abc import x, y, n, k + + +def test_gammasimp(): + R = Rational + + # was part of test_combsimp_gamma() in test_combsimp.py + assert gammasimp(gamma(x)) == gamma(x) + assert gammasimp(gamma(x + 1)/x) == gamma(x) + assert gammasimp(gamma(x)/(x - 1)) == gamma(x - 1) + assert gammasimp(x*gamma(x)) == gamma(x + 1) + assert gammasimp((x + 1)*gamma(x + 1)) == gamma(x + 2) + assert gammasimp(gamma(x + y)*(x + y)) == gamma(x + y + 1) + assert gammasimp(x/gamma(x + 1)) == 1/gamma(x) + assert gammasimp((x + 1)**2/gamma(x + 2)) == (x + 1)/gamma(x + 1) + assert gammasimp(x*gamma(x) + gamma(x + 3)/(x + 2)) == \ + (x + 2)*gamma(x + 1) + + assert gammasimp(gamma(2*x)*x) == gamma(2*x + 1)/2 + assert gammasimp(gamma(2*x)/(x - S.Half)) == 2*gamma(2*x - 1) + + assert gammasimp(gamma(x)*gamma(1 - x)) == pi/sin(pi*x) + assert gammasimp(gamma(x)*gamma(-x)) == -pi/(x*sin(pi*x)) + assert gammasimp(1/gamma(x + 3)/gamma(1 - x)) == \ + sin(pi*x)/(pi*x*(x + 1)*(x + 2)) + + assert gammasimp(factorial(n + 2)) == gamma(n + 3) + assert gammasimp(binomial(n, k)) == \ + gamma(n + 1)/(gamma(k + 1)*gamma(-k + n + 1)) + + assert powsimp(gammasimp( + gamma(x)*gamma(x + S.Half)*gamma(y)/gamma(x + y))) == \ + 2**(-2*x + 1)*sqrt(pi)*gamma(2*x)*gamma(y)/gamma(x + y) + assert gammasimp(1/gamma(x)/gamma(x - Rational(1, 3))/gamma(x + Rational(1, 3))) == \ + 3**(3*x - Rational(3, 2))/(2*pi*gamma(3*x - 1)) + assert simplify( + gamma(S.Half + x/2)*gamma(1 + x/2)/gamma(1 + x)/sqrt(pi)*2**x) == 1 + assert gammasimp(gamma(Rational(-1, 4))*gamma(Rational(-3, 4))) == 16*sqrt(2)*pi/3 + + assert powsimp(gammasimp(gamma(2*x)/gamma(x))) == \ + 2**(2*x - 1)*gamma(x + S.Half)/sqrt(pi) + + # issue 6792 + e = (-gamma(k)*gamma(k + 2) + gamma(k + 1)**2)/gamma(k)**2 + assert gammasimp(e) == -k + assert gammasimp(1/e) == -1/k + e = (gamma(x) + gamma(x + 1))/gamma(x) + assert gammasimp(e) == x + 1 + assert gammasimp(1/e) == 1/(x + 1) + e = (gamma(x) + gamma(x + 2))*(gamma(x - 1) + gamma(x))/gamma(x) + assert gammasimp(e) == (x**2 + x + 1)*gamma(x + 1)/(x - 1) + e = (-gamma(k)*gamma(k + 2) + gamma(k + 1)**2)/gamma(k)**2 + assert gammasimp(e**2) == k**2 + assert gammasimp(e**2/gamma(k + 1)) == k/gamma(k) + a = R(1, 2) + R(1, 3) + b = a + R(1, 3) + assert gammasimp(gamma(2*k)/gamma(k)*gamma(k + a)*gamma(k + b) + ) == 3*2**(2*k + 1)*3**(-3*k - 2)*sqrt(pi)*gamma(3*k + R(3, 2))/2 + + # issue 9699 + assert gammasimp((x + 1)*factorial(x)/gamma(y)) == gamma(x + 2)/gamma(y) + assert gammasimp(rf(x + n, k)*binomial(n, k)).simplify() == Piecewise( + (gamma(n + 1)*gamma(k + n + x)/(gamma(k + 1)*gamma(n + x)*gamma(-k + n + 1)), n > -x), + ((-1)**k*gamma(n + 1)*gamma(-n - x + 1)/(gamma(k + 1)*gamma(-k + n + 1)*gamma(-k - n - x + 1)), True)) + + A, B = symbols('A B', commutative=False) + assert gammasimp(e*B*A) == gammasimp(e)*B*A + + # check iteration + assert gammasimp(gamma(2*k)/gamma(k)*gamma(-k - R(1, 2))) == ( + -2**(2*k + 1)*sqrt(pi)/(2*((2*k + 1)*cos(pi*k)))) + assert gammasimp( + gamma(k)*gamma(k + R(1, 3))*gamma(k + R(2, 3))/gamma(k*R(3, 2))) == ( + 3*2**(3*k + 1)*3**(-3*k - S.Half)*sqrt(pi)*gamma(k*R(3, 2) + S.Half)/2) + + # issue 6153 + assert gammasimp(gamma(Rational(1, 4))/gamma(Rational(5, 4))) == 4 + + # was part of test_combsimp() in test_combsimp.py + assert gammasimp(binomial(n + 2, k + S.Half)) == gamma(n + 3)/ \ + (gamma(k + R(3, 2))*gamma(-k + n + R(5, 2))) + assert gammasimp(binomial(n + 2, k + 2.0)) == \ + gamma(n + 3)/(gamma(k + 3.0)*gamma(-k + n + 1)) + + # issue 11548 + assert gammasimp(binomial(0, x)) == sin(pi*x)/(pi*x) + + e = gamma(n + Rational(1, 3))*gamma(n + R(2, 3)) + assert gammasimp(e) == e + assert gammasimp(gamma(4*n + S.Half)/gamma(2*n - R(3, 4))) == \ + 2**(4*n - R(5, 2))*(8*n - 3)*gamma(2*n + R(3, 4))/sqrt(pi) + + i, m = symbols('i m', integer = True) + e = gamma(exp(i)) + assert gammasimp(e) == e + e = gamma(m + 3) + assert gammasimp(e) == e + e = gamma(m + 1)/(gamma(i + 1)*gamma(-i + m + 1)) + assert gammasimp(e) == e + + p = symbols("p", integer=True, positive=True) + assert gammasimp(gamma(-p + 4)) == gamma(-p + 4) + + +def test_issue_22606(): + fx = Function('f')(x) + eq = x + gamma(y) + # seems like ans should be `eq`, not `(x*y + gamma(y + 1))/y` + ans = gammasimp(eq) + assert gammasimp(eq.subs(x, fx)).subs(fx, x) == ans + assert gammasimp(eq.subs(x, cos(x))).subs(cos(x), x) == ans + assert 1/gammasimp(1/eq) == ans + assert gammasimp(fx.subs(x, eq)).args[0] == ans diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_hyperexpand.py b/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_hyperexpand.py new file mode 100644 index 0000000000000000000000000000000000000000..c703c228a13201de13cfd4c3413fc75a2cf5bdb6 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_hyperexpand.py @@ -0,0 +1,1063 @@ +from sympy.core.random import randrange + +from sympy.simplify.hyperexpand import (ShiftA, ShiftB, UnShiftA, UnShiftB, + MeijerShiftA, MeijerShiftB, MeijerShiftC, MeijerShiftD, + MeijerUnShiftA, MeijerUnShiftB, MeijerUnShiftC, + MeijerUnShiftD, + ReduceOrder, reduce_order, apply_operators, + devise_plan, make_derivative_operator, Formula, + hyperexpand, Hyper_Function, G_Function, + reduce_order_meijer, + build_hypergeometric_formula) +from sympy.concrete.summations import Sum +from sympy.core.containers import Tuple +from sympy.core.expr import Expr +from sympy.core.numbers import I +from sympy.core.singleton import S +from sympy.core.symbol import symbols +from sympy.functions.combinatorial.factorials import binomial +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.special.hyper import (hyper, meijerg) +from sympy.abc import z, a, b, c +from sympy.testing.pytest import XFAIL, raises, slow, tooslow +from sympy.core.random import verify_numerically as tn + +from sympy.core.numbers import (Rational, pi) +from sympy.functions.elementary.exponential import (exp, exp_polar, log) +from sympy.functions.elementary.hyperbolic import atanh +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import (asin, cos, sin) +from sympy.functions.special.bessel import besseli +from sympy.functions.special.error_functions import erf +from sympy.functions.special.gamma_functions import (gamma, lowergamma) + + +def test_branch_bug(): + assert hyperexpand(hyper((Rational(-1, 3), S.Half), (Rational(2, 3), Rational(3, 2)), -z)) == \ + -z**S('1/3')*lowergamma(exp_polar(I*pi)/3, z)/5 \ + + sqrt(pi)*erf(sqrt(z))/(5*sqrt(z)) + assert hyperexpand(meijerg([Rational(7, 6), 1], [], [Rational(2, 3)], [Rational(1, 6), 0], z)) == \ + 2*z**S('2/3')*(2*sqrt(pi)*erf(sqrt(z))/sqrt(z) - 2*lowergamma( + Rational(2, 3), z)/z**S('2/3'))*gamma(Rational(2, 3))/gamma(Rational(5, 3)) + + +def test_hyperexpand(): + # Luke, Y. L. (1969), The Special Functions and Their Approximations, + # Volume 1, section 6.2 + + assert hyperexpand(hyper([], [], z)) == exp(z) + assert hyperexpand(hyper([1, 1], [2], -z)*z) == log(1 + z) + assert hyperexpand(hyper([], [S.Half], -z**2/4)) == cos(z) + assert hyperexpand(z*hyper([], [S('3/2')], -z**2/4)) == sin(z) + assert hyperexpand(hyper([S('1/2'), S('1/2')], [S('3/2')], z**2)*z) \ + == asin(z) + assert isinstance(Sum(binomial(2, z)*z**2, (z, 0, a)).doit(), Expr) + + +def can_do(ap, bq, numerical=True, div=1, lowerplane=False): + r = hyperexpand(hyper(ap, bq, z)) + if r.has(hyper): + return False + if not numerical: + return True + repl = {} + randsyms = r.free_symbols - {z} + while randsyms: + # Only randomly generated parameters are checked. + for n, ai in enumerate(randsyms): + repl[ai] = randcplx(n)/div + if not any(b.is_Integer and b <= 0 for b in Tuple(*bq).subs(repl)): + break + [a, b, c, d] = [2, -1, 3, 1] + if lowerplane: + [a, b, c, d] = [2, -2, 3, -1] + return tn( + hyper(ap, bq, z).subs(repl), + r.replace(exp_polar, exp).subs(repl), + z, a=a, b=b, c=c, d=d) + + +def test_roach(): + # Kelly B. Roach. Meijer G Function Representations. + # Section "Gallery" + assert can_do([S.Half], [Rational(9, 2)]) + assert can_do([], [1, Rational(5, 2), 4]) + assert can_do([Rational(-1, 2), 1, 2], [3, 4]) + assert can_do([Rational(1, 3)], [Rational(-2, 3), Rational(-1, 2), S.Half, 1]) + assert can_do([Rational(-3, 2), Rational(-1, 2)], [Rational(-5, 2), 1]) + assert can_do([Rational(-3, 2), ], [Rational(-1, 2), S.Half]) # shine-integral + assert can_do([Rational(-3, 2), Rational(-1, 2)], [2]) # elliptic integrals + + +@XFAIL +def test_roach_fail(): + assert can_do([Rational(-1, 2), 1], [Rational(1, 4), S.Half, Rational(3, 4)]) # PFDD + assert can_do([Rational(3, 2)], [Rational(5, 2), 5]) # struve function + assert can_do([Rational(-1, 2), S.Half, 1], [Rational(3, 2), Rational(5, 2)]) # polylog, pfdd + assert can_do([1, 2, 3], [S.Half, 4]) # XXX ? + assert can_do([S.Half], [Rational(-1, 3), Rational(-1, 2), Rational(-2, 3)]) # PFDD ? + +# For the long table tests, see end of file + + +def test_polynomial(): + from sympy.core.numbers import oo + assert hyperexpand(hyper([], [-1], z)) is oo + assert hyperexpand(hyper([-2], [-1], z)) is oo + assert hyperexpand(hyper([0, 0], [-1], z)) == 1 + assert can_do([-5, -2, randcplx(), randcplx()], [-10, randcplx()]) + assert hyperexpand(hyper((-1, 1), (-2,), z)) == 1 + z/2 + + +def test_hyperexpand_bases(): + assert hyperexpand(hyper([2], [a], z)) == \ + a + z**(-a + 1)*(-a**2 + 3*a + z*(a - 1) - 2)*exp(z)* \ + lowergamma(a - 1, z) - 1 + # TODO [a+1, aRational(-1, 2)], [2*a] + assert hyperexpand(hyper([1, 2], [3], z)) == -2/z - 2*log(-z + 1)/z**2 + assert hyperexpand(hyper([S.Half, 2], [Rational(3, 2)], z)) == \ + -1/(2*z - 2) + atanh(sqrt(z))/sqrt(z)/2 + assert hyperexpand(hyper([S.Half, S.Half], [Rational(5, 2)], z)) == \ + (-3*z + 3)/4/(z*sqrt(-z + 1)) \ + + (6*z - 3)*asin(sqrt(z))/(4*z**Rational(3, 2)) + assert hyperexpand(hyper([1, 2], [Rational(3, 2)], z)) == -1/(2*z - 2) \ + - asin(sqrt(z))/(sqrt(z)*(2*z - 2)*sqrt(-z + 1)) + assert hyperexpand(hyper([Rational(-1, 2) - 1, 1, 2], [S.Half, 3], z)) == \ + sqrt(z)*(z*Rational(6, 7) - Rational(6, 5))*atanh(sqrt(z)) \ + + (-30*z**2 + 32*z - 6)/35/z - 6*log(-z + 1)/(35*z**2) + assert hyperexpand(hyper([1 + S.Half, 1, 1], [2, 2], z)) == \ + -4*log(sqrt(-z + 1)/2 + S.Half)/z + # TODO hyperexpand(hyper([a], [2*a + 1], z)) + # TODO [S.Half, a], [Rational(3, 2), a+1] + assert hyperexpand(hyper([2], [b, 1], z)) == \ + z**(-b/2 + S.Half)*besseli(b - 1, 2*sqrt(z))*gamma(b) \ + + z**(-b/2 + 1)*besseli(b, 2*sqrt(z))*gamma(b) + # TODO [a], [a - S.Half, 2*a] + + +def test_hyperexpand_parametric(): + assert hyperexpand(hyper([a, S.Half + a], [S.Half], z)) \ + == (1 + sqrt(z))**(-2*a)/2 + (1 - sqrt(z))**(-2*a)/2 + assert hyperexpand(hyper([a, Rational(-1, 2) + a], [2*a], z)) \ + == 2**(2*a - 1)*((-z + 1)**S.Half + 1)**(-2*a + 1) + + +def test_shifted_sum(): + from sympy.simplify.simplify import simplify + assert simplify(hyperexpand(z**4*hyper([2], [3, S('3/2')], -z**2))) \ + == z*sin(2*z) + (-z**2 + S.Half)*cos(2*z) - S.Half + + +def _randrat(): + """ Steer clear of integers. """ + return S(randrange(25) + 10)/50 + + +def randcplx(offset=-1): + """ Polys is not good with real coefficients. """ + return _randrat() + I*_randrat() + I*(1 + offset) + + +@slow +def test_formulae(): + from sympy.simplify.hyperexpand import FormulaCollection + formulae = FormulaCollection().formulae + for formula in formulae: + h = formula.func(formula.z) + rep = {} + for n, sym in enumerate(formula.symbols): + rep[sym] = randcplx(n) + + # NOTE hyperexpand returns truly branched functions. We know we are + # on the main sheet, but numerical evaluation can still go wrong + # (e.g. if exp_polar cannot be evalf'd). + # Just replace all exp_polar by exp, this usually works. + + # first test if the closed-form is actually correct + h = h.subs(rep) + closed_form = formula.closed_form.subs(rep).rewrite('nonrepsmall') + z = formula.z + assert tn(h, closed_form.replace(exp_polar, exp), z) + + # now test the computed matrix + cl = (formula.C * formula.B)[0].subs(rep).rewrite('nonrepsmall') + assert tn(closed_form.replace( + exp_polar, exp), cl.replace(exp_polar, exp), z) + deriv1 = z*formula.B.applyfunc(lambda t: t.rewrite( + 'nonrepsmall')).diff(z) + deriv2 = formula.M * formula.B + for d1, d2 in zip(deriv1, deriv2): + assert tn(d1.subs(rep).replace(exp_polar, exp), + d2.subs(rep).rewrite('nonrepsmall').replace(exp_polar, exp), z) + + +def test_meijerg_formulae(): + from sympy.simplify.hyperexpand import MeijerFormulaCollection + formulae = MeijerFormulaCollection().formulae + for sig in formulae: + for formula in formulae[sig]: + g = meijerg(formula.func.an, formula.func.ap, + formula.func.bm, formula.func.bq, + formula.z) + rep = {} + for sym in formula.symbols: + rep[sym] = randcplx() + + # first test if the closed-form is actually correct + g = g.subs(rep) + closed_form = formula.closed_form.subs(rep) + z = formula.z + assert tn(g, closed_form, z) + + # now test the computed matrix + cl = (formula.C * formula.B)[0].subs(rep) + assert tn(closed_form, cl, z) + deriv1 = z*formula.B.diff(z) + deriv2 = formula.M * formula.B + for d1, d2 in zip(deriv1, deriv2): + assert tn(d1.subs(rep), d2.subs(rep), z) + + +def op(f): + return z*f.diff(z) + + +def test_plan(): + assert devise_plan(Hyper_Function([0], ()), + Hyper_Function([0], ()), z) == [] + with raises(ValueError): + devise_plan(Hyper_Function([1], ()), Hyper_Function((), ()), z) + with raises(ValueError): + devise_plan(Hyper_Function([2], [1]), Hyper_Function([2], [2]), z) + with raises(ValueError): + devise_plan(Hyper_Function([2], []), Hyper_Function([S("1/2")], []), z) + + # We cannot use pi/(10000 + n) because polys is insanely slow. + a1, a2, b1 = (randcplx(n) for n in range(3)) + b1 += 2*I + h = hyper([a1, a2], [b1], z) + + h2 = hyper((a1 + 1, a2), [b1], z) + assert tn(apply_operators(h, + devise_plan(Hyper_Function((a1 + 1, a2), [b1]), + Hyper_Function((a1, a2), [b1]), z), op), + h2, z) + + h2 = hyper((a1 + 1, a2 - 1), [b1], z) + assert tn(apply_operators(h, + devise_plan(Hyper_Function((a1 + 1, a2 - 1), [b1]), + Hyper_Function((a1, a2), [b1]), z), op), + h2, z) + + +def test_plan_derivatives(): + a1, a2, a3 = 1, 2, S('1/2') + b1, b2 = 3, S('5/2') + h = Hyper_Function((a1, a2, a3), (b1, b2)) + h2 = Hyper_Function((a1 + 1, a2 + 1, a3 + 2), (b1 + 1, b2 + 1)) + ops = devise_plan(h2, h, z) + f = Formula(h, z, h(z), []) + deriv = make_derivative_operator(f.M, z) + assert tn((apply_operators(f.C, ops, deriv)*f.B)[0], h2(z), z) + + h2 = Hyper_Function((a1, a2 - 1, a3 - 2), (b1 - 1, b2 - 1)) + ops = devise_plan(h2, h, z) + assert tn((apply_operators(f.C, ops, deriv)*f.B)[0], h2(z), z) + + +def test_reduction_operators(): + a1, a2, b1 = (randcplx(n) for n in range(3)) + h = hyper([a1], [b1], z) + + assert ReduceOrder(2, 0) is None + assert ReduceOrder(2, -1) is None + assert ReduceOrder(1, S('1/2')) is None + + h2 = hyper((a1, a2), (b1, a2), z) + assert tn(ReduceOrder(a2, a2).apply(h, op), h2, z) + + h2 = hyper((a1, a2 + 1), (b1, a2), z) + assert tn(ReduceOrder(a2 + 1, a2).apply(h, op), h2, z) + + h2 = hyper((a2 + 4, a1), (b1, a2), z) + assert tn(ReduceOrder(a2 + 4, a2).apply(h, op), h2, z) + + # test several step order reduction + ap = (a2 + 4, a1, b1 + 1) + bq = (a2, b1, b1) + func, ops = reduce_order(Hyper_Function(ap, bq)) + assert func.ap == (a1,) + assert func.bq == (b1,) + assert tn(apply_operators(h, ops, op), hyper(ap, bq, z), z) + + +def test_shift_operators(): + a1, a2, b1, b2, b3 = (randcplx(n) for n in range(5)) + h = hyper((a1, a2), (b1, b2, b3), z) + + raises(ValueError, lambda: ShiftA(0)) + raises(ValueError, lambda: ShiftB(1)) + + assert tn(ShiftA(a1).apply(h, op), hyper((a1 + 1, a2), (b1, b2, b3), z), z) + assert tn(ShiftA(a2).apply(h, op), hyper((a1, a2 + 1), (b1, b2, b3), z), z) + assert tn(ShiftB(b1).apply(h, op), hyper((a1, a2), (b1 - 1, b2, b3), z), z) + assert tn(ShiftB(b2).apply(h, op), hyper((a1, a2), (b1, b2 - 1, b3), z), z) + assert tn(ShiftB(b3).apply(h, op), hyper((a1, a2), (b1, b2, b3 - 1), z), z) + + +def test_ushift_operators(): + a1, a2, b1, b2, b3 = (randcplx(n) for n in range(5)) + h = hyper((a1, a2), (b1, b2, b3), z) + + raises(ValueError, lambda: UnShiftA((1,), (), 0, z)) + raises(ValueError, lambda: UnShiftB((), (-1,), 0, z)) + raises(ValueError, lambda: UnShiftA((1,), (0, -1, 1), 0, z)) + raises(ValueError, lambda: UnShiftB((0, 1), (1,), 0, z)) + + s = UnShiftA((a1, a2), (b1, b2, b3), 0, z) + assert tn(s.apply(h, op), hyper((a1 - 1, a2), (b1, b2, b3), z), z) + s = UnShiftA((a1, a2), (b1, b2, b3), 1, z) + assert tn(s.apply(h, op), hyper((a1, a2 - 1), (b1, b2, b3), z), z) + + s = UnShiftB((a1, a2), (b1, b2, b3), 0, z) + assert tn(s.apply(h, op), hyper((a1, a2), (b1 + 1, b2, b3), z), z) + s = UnShiftB((a1, a2), (b1, b2, b3), 1, z) + assert tn(s.apply(h, op), hyper((a1, a2), (b1, b2 + 1, b3), z), z) + s = UnShiftB((a1, a2), (b1, b2, b3), 2, z) + assert tn(s.apply(h, op), hyper((a1, a2), (b1, b2, b3 + 1), z), z) + + +def can_do_meijer(a1, a2, b1, b2, numeric=True): + """ + This helper function tries to hyperexpand() the meijer g-function + corresponding to the parameters a1, a2, b1, b2. + It returns False if this expansion still contains g-functions. + If numeric is True, it also tests the so-obtained formula numerically + (at random values) and returns False if the test fails. + Else it returns True. + """ + from sympy.core.function import expand + from sympy.functions.elementary.complexes import unpolarify + r = hyperexpand(meijerg(a1, a2, b1, b2, z)) + if r.has(meijerg): + return False + # NOTE hyperexpand() returns a truly branched function, whereas numerical + # evaluation only works on the main branch. Since we are evaluating on + # the main branch, this should not be a problem, but expressions like + # exp_polar(I*pi/2*x)**a are evaluated incorrectly. We thus have to get + # rid of them. The expand heuristically does this... + r = unpolarify(expand(r, force=True, power_base=True, power_exp=False, + mul=False, log=False, multinomial=False, basic=False)) + + if not numeric: + return True + + repl = {} + for n, ai in enumerate(meijerg(a1, a2, b1, b2, z).free_symbols - {z}): + repl[ai] = randcplx(n) + return tn(meijerg(a1, a2, b1, b2, z).subs(repl), r.subs(repl), z) + + +@slow +def test_meijerg_expand(): + from sympy.simplify.gammasimp import gammasimp + from sympy.simplify.simplify import simplify + # from mpmath docs + assert hyperexpand(meijerg([[], []], [[0], []], -z)) == exp(z) + + assert hyperexpand(meijerg([[1, 1], []], [[1], [0]], z)) == \ + log(z + 1) + assert hyperexpand(meijerg([[1, 1], []], [[1], [1]], z)) == \ + z/(z + 1) + assert hyperexpand(meijerg([[], []], [[S.Half], [0]], (z/2)**2)) \ + == sin(z)/sqrt(pi) + assert hyperexpand(meijerg([[], []], [[0], [S.Half]], (z/2)**2)) \ + == cos(z)/sqrt(pi) + assert can_do_meijer([], [a], [a - 1, a - S.Half], []) + assert can_do_meijer([], [], [a/2], [-a/2], False) # branches... + assert can_do_meijer([a], [b], [a], [b, a - 1]) + + # wikipedia + assert hyperexpand(meijerg([1], [], [], [0], z)) == \ + Piecewise((0, abs(z) < 1), (1, abs(1/z) < 1), + (meijerg([1], [], [], [0], z), True)) + assert hyperexpand(meijerg([], [1], [0], [], z)) == \ + Piecewise((1, abs(z) < 1), (0, abs(1/z) < 1), + (meijerg([], [1], [0], [], z), True)) + + # The Special Functions and their Approximations + assert can_do_meijer([], [], [a + b/2], [a, a - b/2, a + S.Half]) + assert can_do_meijer( + [], [], [a], [b], False) # branches only agree for small z + assert can_do_meijer([], [S.Half], [a], [-a]) + assert can_do_meijer([], [], [a, b], []) + assert can_do_meijer([], [], [a, b], []) + assert can_do_meijer([], [], [a, a + S.Half], [b, b + S.Half]) + assert can_do_meijer([], [], [a, -a], [0, S.Half], False) # dito + assert can_do_meijer([], [], [a, a + S.Half, b, b + S.Half], []) + assert can_do_meijer([S.Half], [], [0], [a, -a]) + assert can_do_meijer([S.Half], [], [a], [0, -a], False) # dito + assert can_do_meijer([], [a - S.Half], [a, b], [a - S.Half], False) + assert can_do_meijer([], [a + S.Half], [a + b, a - b, a], [], False) + assert can_do_meijer([a + S.Half], [], [b, 2*a - b, a], [], False) + + # This for example is actually zero. + assert can_do_meijer([], [], [], [a, b]) + + # Testing a bug: + assert hyperexpand(meijerg([0, 2], [], [], [-1, 1], z)) == \ + Piecewise((0, abs(z) < 1), + (z*(1 - 1/z**2)/2, abs(1/z) < 1), + (meijerg([0, 2], [], [], [-1, 1], z), True)) + + # Test that the simplest possible answer is returned: + assert gammasimp(simplify(hyperexpand( + meijerg([1], [1 - a], [-a/2, -a/2 + S.Half], [], 1/z)))) == \ + -2*sqrt(pi)*(sqrt(z + 1) + 1)**a/a + + # Test that hyper is returned + assert hyperexpand(meijerg([1], [], [a], [0, 0], z)) == hyper( + (a,), (a + 1, a + 1), z*exp_polar(I*pi))*z**a*gamma(a)/gamma(a + 1)**2 + + # Test place option + f = meijerg(((0, 1), ()), ((S.Half,), (0,)), z**2) + assert hyperexpand(f) == sqrt(pi)/sqrt(1 + z**(-2)) + assert hyperexpand(f, place=0) == sqrt(pi)*z/sqrt(z**2 + 1) + + +def test_meijerg_lookup(): + from sympy.functions.special.error_functions import (Ci, Si) + from sympy.functions.special.gamma_functions import uppergamma + assert hyperexpand(meijerg([a], [], [b, a], [], z)) == \ + z**b*exp(z)*gamma(-a + b + 1)*uppergamma(a - b, z) + assert hyperexpand(meijerg([0], [], [0, 0], [], z)) == \ + exp(z)*uppergamma(0, z) + assert can_do_meijer([a], [], [b, a + 1], []) + assert can_do_meijer([a], [], [b + 2, a], []) + assert can_do_meijer([a], [], [b - 2, a], []) + + assert hyperexpand(meijerg([a], [], [a, a, a - S.Half], [], z)) == \ + -sqrt(pi)*z**(a - S.Half)*(2*cos(2*sqrt(z))*(Si(2*sqrt(z)) - pi/2) + - 2*sin(2*sqrt(z))*Ci(2*sqrt(z))) == \ + hyperexpand(meijerg([a], [], [a, a - S.Half, a], [], z)) == \ + hyperexpand(meijerg([a], [], [a - S.Half, a, a], [], z)) + assert can_do_meijer([a - 1], [], [a + 2, a - Rational(3, 2), a + 1], []) + + +@XFAIL +def test_meijerg_expand_fail(): + # These basically test hyper([], [1/2 - a, 1/2 + 1, 1/2], z), + # which is *very* messy. But since the meijer g actually yields a + # sum of bessel functions, things can sometimes be simplified a lot and + # are then put into tables... + assert can_do_meijer([], [], [a + S.Half], [a, a - b/2, a + b/2]) + assert can_do_meijer([], [], [0, S.Half], [a, -a]) + assert can_do_meijer([], [], [3*a - S.Half, a, -a - S.Half], [a - S.Half]) + assert can_do_meijer([], [], [0, a - S.Half, -a - S.Half], [S.Half]) + assert can_do_meijer([], [], [a, b + S.Half, b], [2*b - a]) + assert can_do_meijer([], [], [a, b + S.Half, b, 2*b - a]) + assert can_do_meijer([S.Half], [], [-a, a], [0]) + + +@slow +def test_meijerg(): + # carefully set up the parameters. + # NOTE: this used to fail sometimes. I believe it is fixed, but if you + # hit an inexplicable test failure here, please let me know the seed. + a1, a2 = (randcplx(n) - 5*I - n*I for n in range(2)) + b1, b2 = (randcplx(n) + 5*I + n*I for n in range(2)) + b3, b4, b5, a3, a4, a5 = (randcplx() for n in range(6)) + g = meijerg([a1], [a3, a4], [b1], [b3, b4], z) + + assert ReduceOrder.meijer_minus(3, 4) is None + assert ReduceOrder.meijer_plus(4, 3) is None + + g2 = meijerg([a1, a2], [a3, a4], [b1], [b3, b4, a2], z) + assert tn(ReduceOrder.meijer_plus(a2, a2).apply(g, op), g2, z) + + g2 = meijerg([a1, a2], [a3, a4], [b1], [b3, b4, a2 + 1], z) + assert tn(ReduceOrder.meijer_plus(a2, a2 + 1).apply(g, op), g2, z) + + g2 = meijerg([a1, a2 - 1], [a3, a4], [b1], [b3, b4, a2 + 2], z) + assert tn(ReduceOrder.meijer_plus(a2 - 1, a2 + 2).apply(g, op), g2, z) + + g2 = meijerg([a1], [a3, a4, b2 - 1], [b1, b2 + 2], [b3, b4], z) + assert tn(ReduceOrder.meijer_minus( + b2 + 2, b2 - 1).apply(g, op), g2, z, tol=1e-6) + + # test several-step reduction + an = [a1, a2] + bq = [b3, b4, a2 + 1] + ap = [a3, a4, b2 - 1] + bm = [b1, b2 + 1] + niq, ops = reduce_order_meijer(G_Function(an, ap, bm, bq)) + assert niq.an == (a1,) + assert set(niq.ap) == {a3, a4} + assert niq.bm == (b1,) + assert set(niq.bq) == {b3, b4} + assert tn(apply_operators(g, ops, op), meijerg(an, ap, bm, bq, z), z) + + +def test_meijerg_shift_operators(): + # carefully set up the parameters. XXX this still fails sometimes + a1, a2, a3, a4, a5, b1, b2, b3, b4, b5 = (randcplx(n) for n in range(10)) + g = meijerg([a1], [a3, a4], [b1], [b3, b4], z) + + assert tn(MeijerShiftA(b1).apply(g, op), + meijerg([a1], [a3, a4], [b1 + 1], [b3, b4], z), z) + assert tn(MeijerShiftB(a1).apply(g, op), + meijerg([a1 - 1], [a3, a4], [b1], [b3, b4], z), z) + assert tn(MeijerShiftC(b3).apply(g, op), + meijerg([a1], [a3, a4], [b1], [b3 + 1, b4], z), z) + assert tn(MeijerShiftD(a3).apply(g, op), + meijerg([a1], [a3 - 1, a4], [b1], [b3, b4], z), z) + + s = MeijerUnShiftA([a1], [a3, a4], [b1], [b3, b4], 0, z) + assert tn( + s.apply(g, op), meijerg([a1], [a3, a4], [b1 - 1], [b3, b4], z), z) + + s = MeijerUnShiftC([a1], [a3, a4], [b1], [b3, b4], 0, z) + assert tn( + s.apply(g, op), meijerg([a1], [a3, a4], [b1], [b3 - 1, b4], z), z) + + s = MeijerUnShiftB([a1], [a3, a4], [b1], [b3, b4], 0, z) + assert tn( + s.apply(g, op), meijerg([a1 + 1], [a3, a4], [b1], [b3, b4], z), z) + + s = MeijerUnShiftD([a1], [a3, a4], [b1], [b3, b4], 0, z) + assert tn( + s.apply(g, op), meijerg([a1], [a3 + 1, a4], [b1], [b3, b4], z), z) + + +@slow +def test_meijerg_confluence(): + def t(m, a, b): + from sympy.core.sympify import sympify + a, b = sympify([a, b]) + m_ = m + m = hyperexpand(m) + if not m == Piecewise((a, abs(z) < 1), (b, abs(1/z) < 1), (m_, True)): + return False + if not (m.args[0].args[0] == a and m.args[1].args[0] == b): + return False + z0 = randcplx()/10 + if abs(m.subs(z, z0).n() - a.subs(z, z0).n()).n() > 1e-10: + return False + if abs(m.subs(z, 1/z0).n() - b.subs(z, 1/z0).n()).n() > 1e-10: + return False + return True + + assert t(meijerg([], [1, 1], [0, 0], [], z), -log(z), 0) + assert t(meijerg( + [], [3, 1], [0, 0], [], z), -z**2/4 + z - log(z)/2 - Rational(3, 4), 0) + assert t(meijerg([], [3, 1], [-1, 0], [], z), + z**2/12 - z/2 + log(z)/2 + Rational(1, 4) + 1/(6*z), 0) + assert t(meijerg([], [1, 1, 1, 1], [0, 0, 0, 0], [], z), -log(z)**3/6, 0) + assert t(meijerg([1, 1], [], [], [0, 0], z), 0, -log(1/z)) + assert t(meijerg([1, 1], [2, 2], [1, 1], [0, 0], z), + -z*log(z) + 2*z, -log(1/z) + 2) + assert t(meijerg([S.Half], [1, 1], [0, 0], [Rational(3, 2)], z), log(z)/2 - 1, 0) + + def u(an, ap, bm, bq): + m = meijerg(an, ap, bm, bq, z) + m2 = hyperexpand(m, allow_hyper=True) + if m2.has(meijerg) and not (m2.is_Piecewise and len(m2.args) == 3): + return False + return tn(m, m2, z) + assert u([], [1], [0, 0], []) + assert u([1, 1], [], [], [0]) + assert u([1, 1], [2, 2, 5], [1, 1, 6], [0, 0]) + assert u([1, 1], [2, 2, 5], [1, 1, 6], [0]) + + +def test_meijerg_with_Floats(): + # see issue #10681 + from sympy.polys.domains.realfield import RR + f = meijerg(((3.0, 1), ()), ((Rational(3, 2),), (0,)), z) + a = -2.3632718012073 + g = a*z**Rational(3, 2)*hyper((-0.5, Rational(3, 2)), (Rational(5, 2),), z*exp_polar(I*pi)) + assert RR.almosteq((hyperexpand(f)/g).n(), 1.0, 1e-12) + + +def test_lerchphi(): + from sympy.functions.special.zeta_functions import (lerchphi, polylog) + from sympy.simplify.gammasimp import gammasimp + assert hyperexpand(hyper([1, a], [a + 1], z)/a) == lerchphi(z, 1, a) + assert hyperexpand( + hyper([1, a, a], [a + 1, a + 1], z)/a**2) == lerchphi(z, 2, a) + assert hyperexpand(hyper([1, a, a, a], [a + 1, a + 1, a + 1], z)/a**3) == \ + lerchphi(z, 3, a) + assert hyperexpand(hyper([1] + [a]*10, [a + 1]*10, z)/a**10) == \ + lerchphi(z, 10, a) + assert gammasimp(hyperexpand(meijerg([0, 1 - a], [], [0], + [-a], exp_polar(-I*pi)*z))) == lerchphi(z, 1, a) + assert gammasimp(hyperexpand(meijerg([0, 1 - a, 1 - a], [], [0], + [-a, -a], exp_polar(-I*pi)*z))) == lerchphi(z, 2, a) + assert gammasimp(hyperexpand(meijerg([0, 1 - a, 1 - a, 1 - a], [], [0], + [-a, -a, -a], exp_polar(-I*pi)*z))) == lerchphi(z, 3, a) + + assert hyperexpand(z*hyper([1, 1], [2], z)) == -log(1 + -z) + assert hyperexpand(z*hyper([1, 1, 1], [2, 2], z)) == polylog(2, z) + assert hyperexpand(z*hyper([1, 1, 1, 1], [2, 2, 2], z)) == polylog(3, z) + + assert hyperexpand(hyper([1, a, 1 + S.Half], [a + 1, S.Half], z)) == \ + -2*a/(z - 1) + (-2*a**2 + a)*lerchphi(z, 1, a) + + # Now numerical tests. These make sure reductions etc are carried out + # correctly + + # a rational function (polylog at negative integer order) + assert can_do([2, 2, 2], [1, 1]) + + # NOTE these contain log(1-x) etc ... better make sure we have |z| < 1 + # reduction of order for polylog + assert can_do([1, 1, 1, b + 5], [2, 2, b], div=10) + + # reduction of order for lerchphi + # XXX lerchphi in mpmath is flaky + assert can_do( + [1, a, a, a, b + 5], [a + 1, a + 1, a + 1, b], numerical=False) + + # test a bug + from sympy.functions.elementary.complexes import Abs + assert hyperexpand(hyper([S.Half, S.Half, S.Half, 1], + [Rational(3, 2), Rational(3, 2), Rational(3, 2)], Rational(1, 4))) == \ + Abs(-polylog(3, exp_polar(I*pi)/2) + polylog(3, S.Half)) + + +def test_partial_simp(): + # First test that hypergeometric function formulae work. + a, b, c, d, e = (randcplx() for _ in range(5)) + for func in [Hyper_Function([a, b, c], [d, e]), + Hyper_Function([], [a, b, c, d, e])]: + f = build_hypergeometric_formula(func) + z = f.z + assert f.closed_form == func(z) + deriv1 = f.B.diff(z)*z + deriv2 = f.M*f.B + for func1, func2 in zip(deriv1, deriv2): + assert tn(func1, func2, z) + + # Now test that formulae are partially simplified. + a, b, z = symbols('a b z') + assert hyperexpand(hyper([3, a], [1, b], z)) == \ + (-a*b/2 + a*z/2 + 2*a)*hyper([a + 1], [b], z) \ + + (a*b/2 - 2*a + 1)*hyper([a], [b], z) + assert tn( + hyperexpand(hyper([3, d], [1, e], z)), hyper([3, d], [1, e], z), z) + assert hyperexpand(hyper([3], [1, a, b], z)) == \ + hyper((), (a, b), z) \ + + z*hyper((), (a + 1, b), z)/(2*a) \ + - z*(b - 4)*hyper((), (a + 1, b + 1), z)/(2*a*b) + assert tn( + hyperexpand(hyper([3], [1, d, e], z)), hyper([3], [1, d, e], z), z) + + +def test_hyperexpand_special(): + assert hyperexpand(hyper([a, b], [c], 1)) == \ + gamma(c)*gamma(c - a - b)/gamma(c - a)/gamma(c - b) + assert hyperexpand(hyper([a, b], [1 + a - b], -1)) == \ + gamma(1 + a/2)*gamma(1 + a - b)/gamma(1 + a)/gamma(1 + a/2 - b) + assert hyperexpand(hyper([a, b], [1 + b - a], -1)) == \ + gamma(1 + b/2)*gamma(1 + b - a)/gamma(1 + b)/gamma(1 + b/2 - a) + assert hyperexpand(meijerg([1 - z - a/2], [1 - z + a/2], [b/2], [-b/2], 1)) == \ + gamma(1 - 2*z)*gamma(z + a/2 + b/2)/gamma(1 - z + a/2 - b/2) \ + /gamma(1 - z - a/2 + b/2)/gamma(1 - z + a/2 + b/2) + assert hyperexpand(hyper([a], [b], 0)) == 1 + assert hyper([a], [b], 0) != 0 + + +def test_Mod1_behavior(): + from sympy.core.symbol import Symbol + from sympy.simplify.simplify import simplify + n = Symbol('n', integer=True) + # Note: this should not hang. + assert simplify(hyperexpand(meijerg([1], [], [n + 1], [0], z))) == \ + lowergamma(n + 1, z) + + +@slow +def test_prudnikov_misc(): + assert can_do([1, (3 + I)/2, (3 - I)/2], [Rational(3, 2), 2]) + assert can_do([S.Half, a - 1], [Rational(3, 2), a + 1], lowerplane=True) + assert can_do([], [b + 1]) + assert can_do([a], [a - 1, b + 1]) + + assert can_do([a], [a - S.Half, 2*a]) + assert can_do([a], [a - S.Half, 2*a + 1]) + assert can_do([a], [a - S.Half, 2*a - 1]) + assert can_do([a], [a + S.Half, 2*a]) + assert can_do([a], [a + S.Half, 2*a + 1]) + assert can_do([a], [a + S.Half, 2*a - 1]) + assert can_do([S.Half], [b, 2 - b]) + assert can_do([S.Half], [b, 3 - b]) + assert can_do([1], [2, b]) + + assert can_do([a, a + S.Half], [2*a, b, 2*a - b + 1]) + assert can_do([a, a + S.Half], [S.Half, 2*a, 2*a + S.Half]) + assert can_do([a], [a + 1], lowerplane=True) # lowergamma + + +def test_prudnikov_1(): + # A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev (1990). + # Integrals and Series: More Special Functions, Vol. 3,. + # Gordon and Breach Science Publisher + + # 7.3.1 + assert can_do([a, -a], [S.Half]) + assert can_do([a, 1 - a], [S.Half]) + assert can_do([a, 1 - a], [Rational(3, 2)]) + assert can_do([a, 2 - a], [S.Half]) + assert can_do([a, 2 - a], [Rational(3, 2)]) + assert can_do([a, 2 - a], [Rational(3, 2)]) + assert can_do([a, a + S.Half], [2*a - 1]) + assert can_do([a, a + S.Half], [2*a]) + assert can_do([a, a + S.Half], [2*a + 1]) + assert can_do([a, a + S.Half], [S.Half]) + assert can_do([a, a + S.Half], [Rational(3, 2)]) + assert can_do([a, a/2 + 1], [a/2]) + assert can_do([1, b], [2]) + assert can_do([1, b], [b + 1], numerical=False) # Lerch Phi + # NOTE: branches are complicated for |z| > 1 + + assert can_do([a], [2*a]) + assert can_do([a], [2*a + 1]) + assert can_do([a], [2*a - 1]) + + +@slow +def test_prudnikov_2(): + h = S.Half + assert can_do([-h, -h], [h]) + assert can_do([-h, h], [3*h]) + assert can_do([-h, h], [5*h]) + assert can_do([-h, h], [7*h]) + assert can_do([-h, 1], [h]) + + for p in [-h, h]: + for n in [-h, h, 1, 3*h, 2, 5*h, 3, 7*h, 4]: + for m in [-h, h, 3*h, 5*h, 7*h]: + assert can_do([p, n], [m]) + for n in [1, 2, 3, 4]: + for m in [1, 2, 3, 4]: + assert can_do([p, n], [m]) + + +def test_prudnikov_3(): + h = S.Half + assert can_do([Rational(1, 4), Rational(3, 4)], [h]) + assert can_do([Rational(1, 4), Rational(3, 4)], [3*h]) + assert can_do([Rational(1, 3), Rational(2, 3)], [3*h]) + assert can_do([Rational(3, 4), Rational(5, 4)], [h]) + assert can_do([Rational(3, 4), Rational(5, 4)], [3*h]) + + +@tooslow +def test_prudnikov_3_slow(): + # XXX: This is marked as tooslow and hence skipped in CI. None of the + # individual cases below fails or hangs. Some cases are slow and the loops + # below generate 280 different cases. Is it really necessary to test all + # 280 cases here? + h = S.Half + for p in [1, 2, 3, 4]: + for n in [-h, h, 1, 3*h, 2, 5*h, 3, 7*h, 4, 9*h]: + for m in [1, 3*h, 2, 5*h, 3, 7*h, 4]: + assert can_do([p, m], [n]) + + +@slow +def test_prudnikov_4(): + h = S.Half + for p in [3*h, 5*h, 7*h]: + for n in [-h, h, 3*h, 5*h, 7*h]: + for m in [3*h, 2, 5*h, 3, 7*h, 4]: + assert can_do([p, m], [n]) + for n in [1, 2, 3, 4]: + for m in [2, 3, 4]: + assert can_do([p, m], [n]) + + +@slow +def test_prudnikov_5(): + h = S.Half + + for p in [1, 2, 3]: + for q in range(p, 4): + for r in [1, 2, 3]: + for s in range(r, 4): + assert can_do([-h, p, q], [r, s]) + + for p in [h, 1, 3*h, 2, 5*h, 3]: + for q in [h, 3*h, 5*h]: + for r in [h, 3*h, 5*h]: + for s in [h, 3*h, 5*h]: + if s <= q and s <= r: + assert can_do([-h, p, q], [r, s]) + + for p in [h, 1, 3*h, 2, 5*h, 3]: + for q in [1, 2, 3]: + for r in [h, 3*h, 5*h]: + for s in [1, 2, 3]: + assert can_do([-h, p, q], [r, s]) + + +@slow +def test_prudnikov_6(): + h = S.Half + + for m in [3*h, 5*h]: + for n in [1, 2, 3]: + for q in [h, 1, 2]: + for p in [1, 2, 3]: + assert can_do([h, q, p], [m, n]) + for q in [1, 2, 3]: + for p in [3*h, 5*h]: + assert can_do([h, q, p], [m, n]) + + for q in [1, 2]: + for p in [1, 2, 3]: + for m in [1, 2, 3]: + for n in [1, 2, 3]: + assert can_do([h, q, p], [m, n]) + + assert can_do([h, h, 5*h], [3*h, 3*h]) + assert can_do([h, 1, 5*h], [3*h, 3*h]) + assert can_do([h, 2, 2], [1, 3]) + + # pages 435 to 457 contain more PFDD and stuff like this + + +@slow +def test_prudnikov_7(): + assert can_do([3], [6]) + + h = S.Half + for n in [h, 3*h, 5*h, 7*h]: + assert can_do([-h], [n]) + for m in [-h, h, 1, 3*h, 2, 5*h, 3, 7*h, 4]: # HERE + for n in [-h, h, 3*h, 5*h, 7*h, 1, 2, 3, 4]: + assert can_do([m], [n]) + + +@slow +def test_prudnikov_8(): + h = S.Half + + # 7.12.2 + for ai in [1, 2, 3]: + for bi in [1, 2, 3]: + for ci in range(1, ai + 1): + for di in [h, 1, 3*h, 2, 5*h, 3]: + assert can_do([ai, bi], [ci, di]) + for bi in [3*h, 5*h]: + for ci in [h, 1, 3*h, 2, 5*h, 3]: + for di in [1, 2, 3]: + assert can_do([ai, bi], [ci, di]) + + for ai in [-h, h, 3*h, 5*h]: + for bi in [1, 2, 3]: + for ci in [h, 1, 3*h, 2, 5*h, 3]: + for di in [1, 2, 3]: + assert can_do([ai, bi], [ci, di]) + for bi in [h, 3*h, 5*h]: + for ci in [h, 3*h, 5*h, 3]: + for di in [h, 1, 3*h, 2, 5*h, 3]: + if ci <= bi: + assert can_do([ai, bi], [ci, di]) + + +def test_prudnikov_9(): + # 7.13.1 [we have a general formula ... so this is a bit pointless] + for i in range(9): + assert can_do([], [(S(i) + 1)/2]) + for i in range(5): + assert can_do([], [-(2*S(i) + 1)/2]) + + +@slow +def test_prudnikov_10(): + # 7.14.2 + h = S.Half + for p in [-h, h, 1, 3*h, 2, 5*h, 3, 7*h, 4]: + for m in [1, 2, 3, 4]: + for n in range(m, 5): + assert can_do([p], [m, n]) + + for p in [1, 2, 3, 4]: + for n in [h, 3*h, 5*h, 7*h]: + for m in [1, 2, 3, 4]: + assert can_do([p], [n, m]) + + for p in [3*h, 5*h, 7*h]: + for m in [h, 1, 2, 5*h, 3, 7*h, 4]: + assert can_do([p], [h, m]) + assert can_do([p], [3*h, m]) + + for m in [h, 1, 2, 5*h, 3, 7*h, 4]: + assert can_do([7*h], [5*h, m]) + + assert can_do([Rational(-1, 2)], [S.Half, S.Half]) # shine-integral shi + + +def test_prudnikov_11(): + # 7.15 + assert can_do([a, a + S.Half], [2*a, b, 2*a - b]) + assert can_do([a, a + S.Half], [Rational(3, 2), 2*a, 2*a - S.Half]) + + assert can_do([Rational(1, 4), Rational(3, 4)], [S.Half, S.Half, 1]) + assert can_do([Rational(5, 4), Rational(3, 4)], [Rational(3, 2), S.Half, 2]) + assert can_do([Rational(5, 4), Rational(3, 4)], [Rational(3, 2), Rational(3, 2), 1]) + assert can_do([Rational(5, 4), Rational(7, 4)], [Rational(3, 2), Rational(5, 2), 2]) + + assert can_do([1, 1], [Rational(3, 2), 2, 2]) # cosh-integral chi + + +def test_prudnikov_12(): + # 7.16 + assert can_do( + [], [a, a + S.Half, 2*a], False) # branches only agree for some z! + assert can_do([], [a, a + S.Half, 2*a + 1], False) # dito + assert can_do([], [S.Half, a, a + S.Half]) + assert can_do([], [Rational(3, 2), a, a + S.Half]) + + assert can_do([], [Rational(1, 4), S.Half, Rational(3, 4)]) + assert can_do([], [S.Half, S.Half, 1]) + assert can_do([], [S.Half, Rational(3, 2), 1]) + assert can_do([], [Rational(3, 4), Rational(3, 2), Rational(5, 4)]) + assert can_do([], [1, 1, Rational(3, 2)]) + assert can_do([], [1, 2, Rational(3, 2)]) + assert can_do([], [1, Rational(3, 2), Rational(3, 2)]) + assert can_do([], [Rational(5, 4), Rational(3, 2), Rational(7, 4)]) + assert can_do([], [2, Rational(3, 2), Rational(3, 2)]) + + +@slow +def test_prudnikov_2F1(): + h = S.Half + # Elliptic integrals + for p in [-h, h]: + for m in [h, 3*h, 5*h, 7*h]: + for n in [1, 2, 3, 4]: + assert can_do([p, m], [n]) + + +@XFAIL +def test_prudnikov_fail_2F1(): + assert can_do([a, b], [b + 1]) # incomplete beta function + assert can_do([-1, b], [c]) # Poly. also -2, -3 etc + + # TODO polys + + # Legendre functions: + assert can_do([a, b], [a + b + S.Half]) + assert can_do([a, b], [a + b - S.Half]) + assert can_do([a, b], [a + b + Rational(3, 2)]) + assert can_do([a, b], [(a + b + 1)/2]) + assert can_do([a, b], [(a + b)/2 + 1]) + assert can_do([a, b], [a - b + 1]) + assert can_do([a, b], [a - b + 2]) + assert can_do([a, b], [2*b]) + assert can_do([a, b], [S.Half]) + assert can_do([a, b], [Rational(3, 2)]) + assert can_do([a, 1 - a], [c]) + assert can_do([a, 2 - a], [c]) + assert can_do([a, 3 - a], [c]) + assert can_do([a, a + S.Half], [c]) + assert can_do([1, b], [c]) + assert can_do([1, b], [Rational(3, 2)]) + + assert can_do([Rational(1, 4), Rational(3, 4)], [1]) + + # PFDD + o = S.One + assert can_do([o/8, 1], [o/8*9]) + assert can_do([o/6, 1], [o/6*7]) + assert can_do([o/6, 1], [o/6*13]) + assert can_do([o/5, 1], [o/5*6]) + assert can_do([o/5, 1], [o/5*11]) + assert can_do([o/4, 1], [o/4*5]) + assert can_do([o/4, 1], [o/4*9]) + assert can_do([o/3, 1], [o/3*4]) + assert can_do([o/3, 1], [o/3*7]) + assert can_do([o/8*3, 1], [o/8*11]) + assert can_do([o/5*2, 1], [o/5*7]) + assert can_do([o/5*2, 1], [o/5*12]) + assert can_do([o/5*3, 1], [o/5*8]) + assert can_do([o/5*3, 1], [o/5*13]) + assert can_do([o/8*5, 1], [o/8*13]) + assert can_do([o/4*3, 1], [o/4*7]) + assert can_do([o/4*3, 1], [o/4*11]) + assert can_do([o/3*2, 1], [o/3*5]) + assert can_do([o/3*2, 1], [o/3*8]) + assert can_do([o/5*4, 1], [o/5*9]) + assert can_do([o/5*4, 1], [o/5*14]) + assert can_do([o/6*5, 1], [o/6*11]) + assert can_do([o/6*5, 1], [o/6*17]) + assert can_do([o/8*7, 1], [o/8*15]) + + +@XFAIL +def test_prudnikov_fail_3F2(): + assert can_do([a, a + Rational(1, 3), a + Rational(2, 3)], [Rational(1, 3), Rational(2, 3)]) + assert can_do([a, a + Rational(1, 3), a + Rational(2, 3)], [Rational(2, 3), Rational(4, 3)]) + assert can_do([a, a + Rational(1, 3), a + Rational(2, 3)], [Rational(4, 3), Rational(5, 3)]) + + # page 421 + assert can_do([a, a + Rational(1, 3), a + Rational(2, 3)], [a*Rational(3, 2), (3*a + 1)/2]) + + # pages 422 ... + assert can_do([Rational(-1, 2), S.Half, S.Half], [1, 1]) # elliptic integrals + assert can_do([Rational(-1, 2), S.Half, 1], [Rational(3, 2), Rational(3, 2)]) + # TODO LOTS more + + # PFDD + assert can_do([Rational(1, 8), Rational(3, 8), 1], [Rational(9, 8), Rational(11, 8)]) + assert can_do([Rational(1, 8), Rational(5, 8), 1], [Rational(9, 8), Rational(13, 8)]) + assert can_do([Rational(1, 8), Rational(7, 8), 1], [Rational(9, 8), Rational(15, 8)]) + assert can_do([Rational(1, 6), Rational(1, 3), 1], [Rational(7, 6), Rational(4, 3)]) + assert can_do([Rational(1, 6), Rational(2, 3), 1], [Rational(7, 6), Rational(5, 3)]) + assert can_do([Rational(1, 6), Rational(2, 3), 1], [Rational(5, 3), Rational(13, 6)]) + assert can_do([S.Half, 1, 1], [Rational(1, 4), Rational(3, 4)]) + # LOTS more + + +@XFAIL +def test_prudnikov_fail_other(): + # 7.11.2 + + # 7.12.1 + assert can_do([1, a], [b, 1 - 2*a + b]) # ??? + + # 7.14.2 + assert can_do([Rational(-1, 2)], [S.Half, 1]) # struve + assert can_do([1], [S.Half, S.Half]) # struve + assert can_do([Rational(1, 4)], [S.Half, Rational(5, 4)]) # PFDD + assert can_do([Rational(3, 4)], [Rational(3, 2), Rational(7, 4)]) # PFDD + assert can_do([1], [Rational(1, 4), Rational(3, 4)]) # PFDD + assert can_do([1], [Rational(3, 4), Rational(5, 4)]) # PFDD + assert can_do([1], [Rational(5, 4), Rational(7, 4)]) # PFDD + # TODO LOTS more + + # 7.15.2 + assert can_do([S.Half, 1], [Rational(3, 4), Rational(5, 4), Rational(3, 2)]) # PFDD + assert can_do([S.Half, 1], [Rational(7, 4), Rational(5, 4), Rational(3, 2)]) # PFDD + + # 7.16.1 + assert can_do([], [Rational(1, 3), S(2/3)]) # PFDD + assert can_do([], [Rational(2, 3), S(4/3)]) # PFDD + assert can_do([], [Rational(5, 3), S(4/3)]) # PFDD + + # XXX this does not *evaluate* right?? + assert can_do([], [a, a + S.Half, 2*a - 1]) + + +def test_bug(): + h = hyper([-1, 1], [z], -1) + assert hyperexpand(h) == (z + 1)/z + + +def test_omgissue_203(): + h = hyper((-5, -3, -4), (-6, -6), 1) + assert hyperexpand(h) == Rational(1, 30) + h = hyper((-6, -7, -5), (-6, -6), 1) + assert hyperexpand(h) == Rational(-1, 6) diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_powsimp.py b/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_powsimp.py new file mode 100644 index 0000000000000000000000000000000000000000..61bdc93d052baf4b1e80da8f5864cf22b8fa383e --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_powsimp.py @@ -0,0 +1,368 @@ +from sympy.core.function import Function +from sympy.core.mul import Mul +from sympy.core.numbers import (E, I, Rational, oo, pi) +from sympy.core.singleton import S +from sympy.core.symbol import (Dummy, Symbol, symbols) +from sympy.functions.elementary.exponential import (exp, log) +from sympy.functions.elementary.miscellaneous import (root, sqrt) +from sympy.functions.elementary.trigonometric import sin +from sympy.functions.special.gamma_functions import gamma +from sympy.functions.special.hyper import hyper +from sympy.matrices.expressions.matexpr import MatrixSymbol +from sympy.simplify.powsimp import (powdenest, powsimp) +from sympy.simplify.simplify import (signsimp, simplify) +from sympy.core.symbol import Str + +from sympy.abc import x, y, z, a, b + + +def test_powsimp(): + x, y, z, n = symbols('x,y,z,n') + f = Function('f') + assert powsimp( 4**x * 2**(-x) * 2**(-x) ) == 1 + assert powsimp( (-4)**x * (-2)**(-x) * 2**(-x) ) == 1 + + assert powsimp( + f(4**x * 2**(-x) * 2**(-x)) ) == f(4**x * 2**(-x) * 2**(-x)) + assert powsimp( f(4**x * 2**(-x) * 2**(-x)), deep=True ) == f(1) + assert exp(x)*exp(y) == exp(x)*exp(y) + assert powsimp(exp(x)*exp(y)) == exp(x + y) + assert powsimp(exp(x)*exp(y)*2**x*2**y) == (2*E)**(x + y) + assert powsimp(exp(x)*exp(y)*2**x*2**y, combine='exp') == \ + exp(x + y)*2**(x + y) + assert powsimp(exp(x)*exp(y)*exp(2)*sin(x) + sin(y) + 2**x*2**y) == \ + exp(2 + x + y)*sin(x) + sin(y) + 2**(x + y) + assert powsimp(sin(exp(x)*exp(y))) == sin(exp(x)*exp(y)) + assert powsimp(sin(exp(x)*exp(y)), deep=True) == sin(exp(x + y)) + assert powsimp(x**2*x**y) == x**(2 + y) + # This should remain factored, because 'exp' with deep=True is supposed + # to act like old automatic exponent combining. + assert powsimp((1 + E*exp(E))*exp(-E), combine='exp', deep=True) == \ + (1 + exp(1 + E))*exp(-E) + assert powsimp((1 + E*exp(E))*exp(-E), deep=True) == \ + (1 + exp(1 + E))*exp(-E) + assert powsimp((1 + E*exp(E))*exp(-E)) == (1 + exp(1 + E))*exp(-E) + assert powsimp((1 + E*exp(E))*exp(-E), combine='exp') == \ + (1 + exp(1 + E))*exp(-E) + assert powsimp((1 + E*exp(E))*exp(-E), combine='base') == \ + (1 + E*exp(E))*exp(-E) + x, y = symbols('x,y', nonnegative=True) + n = Symbol('n', real=True) + assert powsimp(y**n * (y/x)**(-n)) == x**n + assert powsimp(x**(x**(x*y)*y**(x*y))*y**(x**(x*y)*y**(x*y)), deep=True) \ + == (x*y)**(x*y)**(x*y) + assert powsimp(2**(2**(2*x)*x), deep=False) == 2**(2**(2*x)*x) + assert powsimp(2**(2**(2*x)*x), deep=True) == 2**(x*4**x) + assert powsimp( + exp(-x + exp(-x)*exp(-x*log(x))), deep=False, combine='exp') == \ + exp(-x + exp(-x)*exp(-x*log(x))) + assert powsimp( + exp(-x + exp(-x)*exp(-x*log(x))), deep=False, combine='exp') == \ + exp(-x + exp(-x)*exp(-x*log(x))) + assert powsimp((x + y)/(3*z), deep=False, combine='exp') == (x + y)/(3*z) + assert powsimp((x/3 + y/3)/z, deep=True, combine='exp') == (x/3 + y/3)/z + assert powsimp(exp(x)/(1 + exp(x)*exp(y)), deep=True) == \ + exp(x)/(1 + exp(x + y)) + assert powsimp(x*y**(z**x*z**y), deep=True) == x*y**(z**(x + y)) + assert powsimp((z**x*z**y)**x, deep=True) == (z**(x + y))**x + assert powsimp(x*(z**x*z**y)**x, deep=True) == x*(z**(x + y))**x + p = symbols('p', positive=True) + assert powsimp((1/x)**log(2)/x) == (1/x)**(1 + log(2)) + assert powsimp((1/p)**log(2)/p) == p**(-1 - log(2)) + + # coefficient of exponent can only be simplified for positive bases + assert powsimp(2**(2*x)) == 4**x + assert powsimp((-1)**(2*x)) == (-1)**(2*x) + i = symbols('i', integer=True) + assert powsimp((-1)**(2*i)) == 1 + assert powsimp((-1)**(-x)) != (-1)**x # could be 1/((-1)**x), but is not + # force=True overrides assumptions + assert powsimp((-1)**(2*x), force=True) == 1 + + # rational exponents allow combining of negative terms + w, n, m = symbols('w n m', negative=True) + e = i/a # not a rational exponent if `a` is unknown + ex = w**e*n**e*m**e + assert powsimp(ex) == m**(i/a)*n**(i/a)*w**(i/a) + e = i/3 + ex = w**e*n**e*m**e + assert powsimp(ex) == (-1)**i*(-m*n*w)**(i/3) + e = (3 + i)/i + ex = w**e*n**e*m**e + assert powsimp(ex) == (-1)**(3*e)*(-m*n*w)**e + + eq = x**(a*Rational(2, 3)) + # eq != (x**a)**(2/3) (try x = -1 and a = 3 to see) + assert powsimp(eq).exp == eq.exp == a*Rational(2, 3) + # powdenest goes the other direction + assert powsimp(2**(2*x)) == 4**x + + assert powsimp(exp(p/2)) == exp(p/2) + + # issue 6368 + eq = Mul(*[sqrt(Dummy(imaginary=True)) for i in range(3)]) + assert powsimp(eq) == eq and eq.is_Mul + + assert all(powsimp(e) == e for e in (sqrt(x**a), sqrt(x**2))) + + # issue 8836 + assert str( powsimp(exp(I*pi/3)*root(-1,3)) ) == '(-1)**(2/3)' + + # issue 9183 + assert powsimp(-0.1**x) == -0.1**x + + # issue 10095 + assert powsimp((1/(2*E))**oo) == (exp(-1)/2)**oo + + # PR 13131 + eq = sin(2*x)**2*sin(2.0*x)**2 + assert powsimp(eq) == eq + + # issue 14615 + assert powsimp(x**2*y**3*(x*y**2)**Rational(3, 2) + ) == x*y*(x*y**2)**Rational(5, 2) + + #issue 27380 + assert powsimp(1.0**(x+1)/1.0**x) == 1.0 + +def test_powsimp_negated_base(): + assert powsimp((-x + y)/sqrt(x - y)) == -sqrt(x - y) + assert powsimp((-x + y)*(-z + y)/sqrt(x - y)/sqrt(z - y)) == sqrt(x - y)*sqrt(z - y) + p = symbols('p', positive=True) + reps = {p: 2, a: S.Half} + assert powsimp((-p)**a/p**a).subs(reps) == ((-1)**a).subs(reps) + assert powsimp((-p)**a*p**a).subs(reps) == ((-p**2)**a).subs(reps) + n = symbols('n', negative=True) + reps = {p: -2, a: S.Half} + assert powsimp((-n)**a/n**a).subs(reps) == (-1)**(-a).subs(a, S.Half) + assert powsimp((-n)**a*n**a).subs(reps) == ((-n**2)**a).subs(reps) + # if x is 0 then the lhs is 0**a*oo**a which is not (-1)**a + eq = (-x)**a/x**a + assert powsimp(eq) == eq + + +def test_powsimp_nc(): + x, y, z = symbols('x,y,z') + A, B, C = symbols('A B C', commutative=False) + + assert powsimp(A**x*A**y, combine='all') == A**(x + y) + assert powsimp(A**x*A**y, combine='base') == A**x*A**y + assert powsimp(A**x*A**y, combine='exp') == A**(x + y) + + assert powsimp(A**x*B**x, combine='all') == A**x*B**x + assert powsimp(A**x*B**x, combine='base') == A**x*B**x + assert powsimp(A**x*B**x, combine='exp') == A**x*B**x + + assert powsimp(B**x*A**x, combine='all') == B**x*A**x + assert powsimp(B**x*A**x, combine='base') == B**x*A**x + assert powsimp(B**x*A**x, combine='exp') == B**x*A**x + + assert powsimp(A**x*A**y*A**z, combine='all') == A**(x + y + z) + assert powsimp(A**x*A**y*A**z, combine='base') == A**x*A**y*A**z + assert powsimp(A**x*A**y*A**z, combine='exp') == A**(x + y + z) + + assert powsimp(A**x*B**x*C**x, combine='all') == A**x*B**x*C**x + assert powsimp(A**x*B**x*C**x, combine='base') == A**x*B**x*C**x + assert powsimp(A**x*B**x*C**x, combine='exp') == A**x*B**x*C**x + + assert powsimp(B**x*A**x*C**x, combine='all') == B**x*A**x*C**x + assert powsimp(B**x*A**x*C**x, combine='base') == B**x*A**x*C**x + assert powsimp(B**x*A**x*C**x, combine='exp') == B**x*A**x*C**x + + +def test_issue_6440(): + assert powsimp(16*2**a*8**b) == 2**(a + 3*b + 4) + + +def test_powdenest(): + x, y = symbols('x,y') + p, q = symbols('p q', positive=True) + i, j = symbols('i,j', integer=True) + + assert powdenest(x) == x + assert powdenest(x + 2*(x**(a*Rational(2, 3)))**(3*x)) == (x + 2*(x**(a*Rational(2, 3)))**(3*x)) + assert powdenest((exp(a*Rational(2, 3)))**(3*x)) # -X-> (exp(a/3))**(6*x) + assert powdenest((x**(a*Rational(2, 3)))**(3*x)) == ((x**(a*Rational(2, 3)))**(3*x)) + assert powdenest(exp(3*x*log(2))) == 2**(3*x) + assert powdenest(sqrt(p**2)) == p + eq = p**(2*i)*q**(4*i) + assert powdenest(eq) == (p*q**2)**(2*i) + # -X-> (x**x)**i*(x**x)**j == x**(x*(i + j)) + assert powdenest((x**x)**(i + j)) + assert powdenest(exp(3*y*log(x))) == x**(3*y) + assert powdenest(exp(y*(log(a) + log(b)))) == (a*b)**y + assert powdenest(exp(3*(log(a) + log(b)))) == a**3*b**3 + assert powdenest(((x**(2*i))**(3*y))**x) == ((x**(2*i))**(3*y))**x + assert powdenest(((x**(2*i))**(3*y))**x, force=True) == x**(6*i*x*y) + assert powdenest(((x**(a*Rational(2, 3)))**(3*y/i))**x) == \ + (((x**(a*Rational(2, 3)))**(3*y/i))**x) + assert powdenest((x**(2*i)*y**(4*i))**z, force=True) == (x*y**2)**(2*i*z) + assert powdenest((p**(2*i)*q**(4*i))**j) == (p*q**2)**(2*i*j) + e = ((p**(2*a))**(3*y))**x + assert powdenest(e) == e + e = ((x**2*y**4)**a)**(x*y) + assert powdenest(e) == e + e = (((x**2*y**4)**a)**(x*y))**3 + assert powdenest(e) == ((x**2*y**4)**a)**(3*x*y) + assert powdenest((((x**2*y**4)**a)**(x*y)), force=True) == \ + (x*y**2)**(2*a*x*y) + assert powdenest((((x**2*y**4)**a)**(x*y))**3, force=True) == \ + (x*y**2)**(6*a*x*y) + assert powdenest((x**2*y**6)**i) != (x*y**3)**(2*i) + x, y = symbols('x,y', positive=True) + assert powdenest((x**2*y**6)**i) == (x*y**3)**(2*i) + + assert powdenest((x**(i*Rational(2, 3))*y**(i/2))**(2*i)) == (x**Rational(4, 3)*y)**(i**2) + assert powdenest(sqrt(x**(2*i)*y**(6*i))) == (x*y**3)**i + + assert powdenest(4**x) == 2**(2*x) + assert powdenest((4**x)**y) == 2**(2*x*y) + assert powdenest(4**x*y) == 2**(2*x)*y + + +def test_powdenest_polar(): + x, y, z = symbols('x y z', polar=True) + a, b, c = symbols('a b c') + assert powdenest((x*y*z)**a) == x**a*y**a*z**a + assert powdenest((x**a*y**b)**c) == x**(a*c)*y**(b*c) + assert powdenest(((x**a)**b*y**c)**c) == x**(a*b*c)*y**(c**2) + + +def test_issue_5805(): + arg = ((gamma(x)*hyper((), (), x))*pi)**2 + assert powdenest(arg) == (pi*gamma(x)*hyper((), (), x))**2 + assert arg.is_positive is None + + +def test_issue_9324_powsimp_on_matrix_symbol(): + M = MatrixSymbol('M', 10, 10) + expr = powsimp(M, deep=True) + assert expr == M + assert expr.args[0] == Str('M') + + +def test_issue_6367(): + z = -5*sqrt(2)/(2*sqrt(2*sqrt(29) + 29)) + sqrt(-sqrt(29)/29 + S.Half) + assert Mul(*[powsimp(a) for a in Mul.make_args(z.normal())]) == 0 + assert powsimp(z.normal()) == 0 + assert simplify(z) == 0 + assert powsimp(sqrt(2 + sqrt(3))*sqrt(2 - sqrt(3)) + 1) == 2 + assert powsimp(z) != 0 + + +def test_powsimp_polar(): + from sympy.functions.elementary.complexes import polar_lift + from sympy.functions.elementary.exponential import exp_polar + x, y, z = symbols('x y z') + p, q, r = symbols('p q r', polar=True) + + assert (polar_lift(-1))**(2*x) == exp_polar(2*pi*I*x) + assert powsimp(p**x * q**x) == (p*q)**x + assert p**x * (1/p)**x == 1 + assert (1/p)**x == p**(-x) + + assert exp_polar(x)*exp_polar(y) == exp_polar(x)*exp_polar(y) + assert powsimp(exp_polar(x)*exp_polar(y)) == exp_polar(x + y) + assert powsimp(exp_polar(x)*exp_polar(y)*p**x*p**y) == \ + (p*exp_polar(1))**(x + y) + assert powsimp(exp_polar(x)*exp_polar(y)*p**x*p**y, combine='exp') == \ + exp_polar(x + y)*p**(x + y) + assert powsimp( + exp_polar(x)*exp_polar(y)*exp_polar(2)*sin(x) + sin(y) + p**x*p**y) \ + == p**(x + y) + sin(x)*exp_polar(2 + x + y) + sin(y) + assert powsimp(sin(exp_polar(x)*exp_polar(y))) == \ + sin(exp_polar(x)*exp_polar(y)) + assert powsimp(sin(exp_polar(x)*exp_polar(y)), deep=True) == \ + sin(exp_polar(x + y)) + + +def test_issue_5728(): + b = x*sqrt(y) + a = sqrt(b) + c = sqrt(sqrt(x)*y) + assert powsimp(a*b) == sqrt(b)**3 + assert powsimp(a*b**2*sqrt(y)) == sqrt(y)*a**5 + assert powsimp(a*x**2*c**3*y) == c**3*a**5 + assert powsimp(a*x*c**3*y**2) == c**7*a + assert powsimp(x*c**3*y**2) == c**7 + assert powsimp(x*c**3*y) == x*y*c**3 + assert powsimp(sqrt(x)*c**3*y) == c**5 + assert powsimp(sqrt(x)*a**3*sqrt(y)) == sqrt(x)*sqrt(y)*a**3 + assert powsimp(Mul(sqrt(x)*c**3*sqrt(y), y, evaluate=False)) == \ + sqrt(x)*sqrt(y)**3*c**3 + assert powsimp(a**2*a*x**2*y) == a**7 + + # symbolic powers work, too + b = x**y*y + a = b*sqrt(b) + assert a.is_Mul is True + assert powsimp(a) == sqrt(b)**3 + + # as does exp + a = x*exp(y*Rational(2, 3)) + assert powsimp(a*sqrt(a)) == sqrt(a)**3 + assert powsimp(a**2*sqrt(a)) == sqrt(a)**5 + assert powsimp(a**2*sqrt(sqrt(a))) == sqrt(sqrt(a))**9 + + +def test_issue_from_PR1599(): + n1, n2, n3, n4 = symbols('n1 n2 n3 n4', negative=True) + assert (powsimp(sqrt(n1)*sqrt(n2)*sqrt(n3)) == + -I*sqrt(-n1)*sqrt(-n2)*sqrt(-n3)) + assert (powsimp(root(n1, 3)*root(n2, 3)*root(n3, 3)*root(n4, 3)) == + -(-1)**Rational(1, 3)* + (-n1)**Rational(1, 3)*(-n2)**Rational(1, 3)*(-n3)**Rational(1, 3)*(-n4)**Rational(1, 3)) + + +def test_issue_10195(): + a = Symbol('a', integer=True) + l = Symbol('l', even=True, nonzero=True) + n = Symbol('n', odd=True) + e_x = (-1)**(n/2 - S.Half) - (-1)**(n*Rational(3, 2) - S.Half) + assert powsimp((-1)**(l/2)) == I**l + assert powsimp((-1)**(n/2)) == I**n + assert powsimp((-1)**(n*Rational(3, 2))) == -I**n + assert powsimp(e_x) == (-1)**(n/2 - S.Half) + (-1)**(n*Rational(3, 2) + + S.Half) + assert powsimp((-1)**(a*Rational(3, 2))) == (-I)**a + +def test_issue_15709(): + assert powsimp(3**x*Rational(2, 3)) == 2*3**(x-1) + assert powsimp(2*3**x/3) == 2*3**(x-1) + + +def test_issue_11981(): + x, y = symbols('x y', commutative=False) + assert powsimp((x*y)**2 * (y*x)**2) == (x*y)**2 * (y*x)**2 + + +def test_issue_17524(): + a = symbols("a", real=True) + e = (-1 - a**2)*sqrt(1 + a**2) + assert signsimp(powsimp(e)) == signsimp(e) == -(a**2 + 1)**(S(3)/2) + + +def test_issue_19627(): + # if you use force the user must verify + assert powdenest(sqrt(sin(x)**2), force=True) == sin(x) + assert powdenest((x**(S.Half/y))**(2*y), force=True) == x + from sympy.core.function import expand_power_base + e = 1 - a + expr = (exp(z/e)*x**(b/e)*y**((1 - b)/e))**e + assert powdenest(expand_power_base(expr, force=True), force=True + ) == x**b*y**(1 - b)*exp(z) + + +def test_issue_22546(): + p1, p2 = symbols('p1, p2', positive=True) + ref = powsimp(p1**z/p2**z) + e = z + 1 + ans = ref.subs(z, e) + assert ans.is_Pow + assert powsimp(p1**e/p2**e) == ans + i = symbols('i', integer=True) + ref = powsimp(x**i/y**i) + e = i + 1 + ans = ref.subs(i, e) + assert ans.is_Pow + assert powsimp(x**e/y**e) == ans diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_radsimp.py b/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_radsimp.py new file mode 100644 index 0000000000000000000000000000000000000000..f8ff955e48a34536c1752c565c0864dedae6a214 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_radsimp.py @@ -0,0 +1,498 @@ +from sympy.core.add import Add +from sympy.core.function import (Derivative, Function, diff) +from sympy.core.mul import Mul +from sympy.core.numbers import (I, Rational) +from sympy.core.power import Pow +from sympy.core.singleton import S +from sympy.core.symbol import (Symbol, Wild, symbols) +from sympy.functions.elementary.complexes import Abs +from sympy.functions.elementary.exponential import (exp, log) +from sympy.functions.elementary.miscellaneous import (root, sqrt) +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.polys.polytools import factor +from sympy.series.order import O +from sympy.simplify.radsimp import (collect, collect_const, fraction, radsimp, rcollect) + +from sympy.core.expr import unchanged +from sympy.core.mul import _unevaluated_Mul as umul +from sympy.simplify.radsimp import (_unevaluated_Add, + collect_sqrt, fraction_expand, collect_abs) +from sympy.testing.pytest import raises + +from sympy.abc import x, y, z, a, b, c, d + + +def test_radsimp(): + r2 = sqrt(2) + r3 = sqrt(3) + r5 = sqrt(5) + r7 = sqrt(7) + assert fraction(radsimp(1/r2)) == (sqrt(2), 2) + assert radsimp(1/(1 + r2)) == \ + -1 + sqrt(2) + assert radsimp(1/(r2 + r3)) == \ + -sqrt(2) + sqrt(3) + assert fraction(radsimp(1/(1 + r2 + r3))) == \ + (-sqrt(6) + sqrt(2) + 2, 4) + assert fraction(radsimp(1/(r2 + r3 + r5))) == \ + (-sqrt(30) + 2*sqrt(3) + 3*sqrt(2), 12) + assert fraction(radsimp(1/(1 + r2 + r3 + r5))) == ( + (-34*sqrt(10) - 26*sqrt(15) - 55*sqrt(3) - 61*sqrt(2) + 14*sqrt(30) + + 93 + 46*sqrt(6) + 53*sqrt(5), 71)) + assert fraction(radsimp(1/(r2 + r3 + r5 + r7))) == ( + (-50*sqrt(42) - 133*sqrt(5) - 34*sqrt(70) - 145*sqrt(3) + 22*sqrt(105) + + 185*sqrt(2) + 62*sqrt(30) + 135*sqrt(7), 215)) + z = radsimp(1/(1 + r2/3 + r3/5 + r5 + r7)) + assert len((3616791619821680643598*z).args) == 16 + assert radsimp(1/z) == 1/z + assert radsimp(1/z, max_terms=20).expand() == 1 + r2/3 + r3/5 + r5 + r7 + assert radsimp(1/(r2*3)) == \ + sqrt(2)/6 + assert radsimp(1/(r2*a + r3 + r5 + r7)) == ( + (8*sqrt(2)*a**7 - 8*sqrt(7)*a**6 - 8*sqrt(5)*a**6 - 8*sqrt(3)*a**6 - + 180*sqrt(2)*a**5 + 8*sqrt(30)*a**5 + 8*sqrt(42)*a**5 + 8*sqrt(70)*a**5 + - 24*sqrt(105)*a**4 + 84*sqrt(3)*a**4 + 100*sqrt(5)*a**4 + + 116*sqrt(7)*a**4 - 72*sqrt(70)*a**3 - 40*sqrt(42)*a**3 - + 8*sqrt(30)*a**3 + 782*sqrt(2)*a**3 - 462*sqrt(3)*a**2 - + 302*sqrt(7)*a**2 - 254*sqrt(5)*a**2 + 120*sqrt(105)*a**2 - + 795*sqrt(2)*a - 62*sqrt(30)*a + 82*sqrt(42)*a + 98*sqrt(70)*a - + 118*sqrt(105) + 59*sqrt(7) + 295*sqrt(5) + 531*sqrt(3))/(16*a**8 - + 480*a**6 + 3128*a**4 - 6360*a**2 + 3481)) + assert radsimp(1/(r2*a + r2*b + r3 + r7)) == ( + (sqrt(2)*a*(a + b)**2 - 5*sqrt(2)*a + sqrt(42)*a + sqrt(2)*b*(a + + b)**2 - 5*sqrt(2)*b + sqrt(42)*b - sqrt(7)*(a + b)**2 - sqrt(3)*(a + + b)**2 - 2*sqrt(3) + 2*sqrt(7))/(2*a**4 + 8*a**3*b + 12*a**2*b**2 - + 20*a**2 + 8*a*b**3 - 40*a*b + 2*b**4 - 20*b**2 + 8)) + assert radsimp(1/(r2*a + r2*b + r2*c + r2*d)) == \ + sqrt(2)/(2*a + 2*b + 2*c + 2*d) + assert radsimp(1/(1 + r2*a + r2*b + r2*c + r2*d)) == ( + (sqrt(2)*a + sqrt(2)*b + sqrt(2)*c + sqrt(2)*d - 1)/(2*a**2 + 4*a*b + + 4*a*c + 4*a*d + 2*b**2 + 4*b*c + 4*b*d + 2*c**2 + 4*c*d + 2*d**2 - 1)) + assert radsimp((y**2 - x)/(y - sqrt(x))) == \ + sqrt(x) + y + assert radsimp(-(y**2 - x)/(y - sqrt(x))) == \ + -(sqrt(x) + y) + assert radsimp(1/(1 - I + a*I)) == \ + (-I*a + 1 + I)/(a**2 - 2*a + 2) + assert radsimp(1/((-x + y)*(x - sqrt(y)))) == \ + (-x - sqrt(y))/((x - y)*(x**2 - y)) + e = (3 + 3*sqrt(2))*x*(3*x - 3*sqrt(y)) + assert radsimp(e) == x*(3 + 3*sqrt(2))*(3*x - 3*sqrt(y)) + assert radsimp(1/e) == ( + (-9*x + 9*sqrt(2)*x - 9*sqrt(y) + 9*sqrt(2)*sqrt(y))/(9*x*(9*x**2 - + 9*y))) + assert radsimp(1 + 1/(1 + sqrt(3))) == \ + Mul(S.Half, -1 + sqrt(3), evaluate=False) + 1 + A = symbols("A", commutative=False) + assert radsimp(x**2 + sqrt(2)*x**2 - sqrt(2)*x*A) == \ + x**2 + sqrt(2)*x**2 - sqrt(2)*x*A + assert radsimp(1/sqrt(5 + 2 * sqrt(6))) == -sqrt(2) + sqrt(3) + assert radsimp(1/sqrt(5 + 2 * sqrt(6))**3) == -(-sqrt(3) + sqrt(2))**3 + + # issue 6532 + assert fraction(radsimp(1/sqrt(x))) == (sqrt(x), x) + assert fraction(radsimp(1/sqrt(2*x + 3))) == (sqrt(2*x + 3), 2*x + 3) + assert fraction(radsimp(1/sqrt(2*(x + 3)))) == (sqrt(2*x + 6), 2*x + 6) + + # issue 5994 + e = S('-(2 + 2*sqrt(2) + 4*2**(1/4))/' + '(1 + 2**(3/4) + 3*2**(1/4) + 3*sqrt(2))') + assert radsimp(e).expand() == -2*2**Rational(3, 4) - 2*2**Rational(1, 4) + 2 + 2*sqrt(2) + + # issue 5986 (modifications to radimp didn't initially recognize this so + # the test is included here) + assert radsimp(1/(-sqrt(5)/2 - S.Half + (-sqrt(5)/2 - S.Half)**2)) == 1 + + # from issue 5934 + eq = ( + (-240*sqrt(2)*sqrt(sqrt(5) + 5)*sqrt(8*sqrt(5) + 40) - + 360*sqrt(2)*sqrt(-8*sqrt(5) + 40)*sqrt(-sqrt(5) + 5) - + 120*sqrt(10)*sqrt(-8*sqrt(5) + 40)*sqrt(-sqrt(5) + 5) + + 120*sqrt(2)*sqrt(-sqrt(5) + 5)*sqrt(8*sqrt(5) + 40) + + 120*sqrt(2)*sqrt(-8*sqrt(5) + 40)*sqrt(sqrt(5) + 5) + + 120*sqrt(10)*sqrt(-sqrt(5) + 5)*sqrt(8*sqrt(5) + 40) + + 120*sqrt(10)*sqrt(-8*sqrt(5) + 40)*sqrt(sqrt(5) + 5))/(-36000 - + 7200*sqrt(5) + (12*sqrt(10)*sqrt(sqrt(5) + 5) + + 24*sqrt(10)*sqrt(-sqrt(5) + 5))**2)) + assert radsimp(eq) is S.NaN # it's 0/0 + + # work with normal form + e = 1/sqrt(sqrt(7)/7 + 2*sqrt(2) + 3*sqrt(3) + 5*sqrt(5)) + 3 + assert radsimp(e) == ( + -sqrt(sqrt(7) + 14*sqrt(2) + 21*sqrt(3) + + 35*sqrt(5))*(-11654899*sqrt(35) - 1577436*sqrt(210) - 1278438*sqrt(15) + - 1346996*sqrt(10) + 1635060*sqrt(6) + 5709765 + 7539830*sqrt(14) + + 8291415*sqrt(21))/1300423175 + 3) + + # obey power rules + base = sqrt(3) - sqrt(2) + assert radsimp(1/base**3) == (sqrt(3) + sqrt(2))**3 + assert radsimp(1/(-base)**3) == -(sqrt(2) + sqrt(3))**3 + assert radsimp(1/(-base)**x) == (-base)**(-x) + assert radsimp(1/base**x) == (sqrt(2) + sqrt(3))**x + assert radsimp(root(1/(-1 - sqrt(2)), -x)) == (-1)**(-1/x)*(1 + sqrt(2))**(1/x) + + # recurse + e = cos(1/(1 + sqrt(2))) + assert radsimp(e) == cos(-sqrt(2) + 1) + assert radsimp(e/2) == cos(-sqrt(2) + 1)/2 + assert radsimp(1/e) == 1/cos(-sqrt(2) + 1) + assert radsimp(2/e) == 2/cos(-sqrt(2) + 1) + assert fraction(radsimp(e/sqrt(x))) == (sqrt(x)*cos(-sqrt(2)+1), x) + + # test that symbolic denominators are not processed + r = 1 + sqrt(2) + assert radsimp(x/r, symbolic=False) == -x*(-sqrt(2) + 1) + assert radsimp(x/(y + r), symbolic=False) == x/(y + 1 + sqrt(2)) + assert radsimp(x/(y + r)/r, symbolic=False) == \ + -x*(-sqrt(2) + 1)/(y + 1 + sqrt(2)) + + # issue 7408 + eq = sqrt(x)/sqrt(y) + assert radsimp(eq) == umul(sqrt(x), sqrt(y), 1/y) + assert radsimp(eq, symbolic=False) == eq + + # issue 7498 + assert radsimp(sqrt(x)/sqrt(y)**3) == umul(sqrt(x), sqrt(y**3), 1/y**3) + + # for coverage + eq = sqrt(x)/y**2 + assert radsimp(eq) == eq + + # handle non-Expr args + from sympy.integrals.integrals import Integral + eq = Integral(x/(sqrt(2) - 1), (x, 0, 1/(sqrt(2) + 1))) + assert radsimp(eq) == Integral((sqrt(2) + 1)*x , (x, 0, sqrt(2) - 1)) + + from sympy.sets import FiniteSet + eq = FiniteSet(x/(sqrt(2) - 1)) + assert radsimp(eq) == FiniteSet((sqrt(2) + 1)*x) + +def test_radsimp_issue_3214(): + c, p = symbols('c p', positive=True) + s = sqrt(c**2 - p**2) + b = (c + I*p - s)/(c + I*p + s) + assert radsimp(b) == -I*(c + I*p - sqrt(c**2 - p**2))**2/(2*c*p) + + +def test_collect_1(): + """Collect with respect to Symbol""" + x, y, z, n = symbols('x,y,z,n') + assert collect(1, x) == 1 + assert collect( x + y*x, x ) == x * (1 + y) + assert collect( x + x**2, x ) == x + x**2 + assert collect( x**2 + y*x**2, x ) == (x**2)*(1 + y) + assert collect( x**2 + y*x, x ) == x*y + x**2 + assert collect( 2*x**2 + y*x**2 + 3*x*y, [x] ) == x**2*(2 + y) + 3*x*y + assert collect( 2*x**2 + y*x**2 + 3*x*y, [y] ) == 2*x**2 + y*(x**2 + 3*x) + + assert collect( ((1 + y + x)**4).expand(), x) == ((1 + y)**4).expand() + \ + x*(4*(1 + y)**3).expand() + x**2*(6*(1 + y)**2).expand() + \ + x**3*(4*(1 + y)).expand() + x**4 + # symbols can be given as any iterable + expr = x + y + assert collect(expr, expr.free_symbols) == expr + assert collect(x*exp(x) + sin(x)*y + sin(x)*2 + 3*x, x, exact=None + ) == x*exp(x) + 3*x + (y + 2)*sin(x) + assert collect(x*exp(x) + sin(x)*y + sin(x)*2 + 3*x + y*x + + y*x*exp(x), x, exact=None + ) == x*exp(x)*(y + 1) + (3 + y)*x + (y + 2)*sin(x) + + +def test_collect_2(): + """Collect with respect to a sum""" + a, b, x = symbols('a,b,x') + assert collect(a*(cos(x) + sin(x)) + b*(cos(x) + sin(x)), + sin(x) + cos(x)) == (a + b)*(cos(x) + sin(x)) + + +def test_collect_3(): + """Collect with respect to a product""" + a, b, c = symbols('a,b,c') + f = Function('f') + x, y, z, n = symbols('x,y,z,n') + + assert collect(-x/8 + x*y, -x) == x*(y - Rational(1, 8)) + + assert collect( 1 + x*(y**2), x*y ) == 1 + x*(y**2) + assert collect( x*y + a*x*y, x*y) == x*y*(1 + a) + assert collect( 1 + x*y + a*x*y, x*y) == 1 + x*y*(1 + a) + assert collect(a*x*f(x) + b*(x*f(x)), x*f(x)) == x*(a + b)*f(x) + + assert collect(a*x*log(x) + b*(x*log(x)), x*log(x)) == x*(a + b)*log(x) + assert collect(a*x**2*log(x)**2 + b*(x*log(x))**2, x*log(x)) == \ + x**2*log(x)**2*(a + b) + + # with respect to a product of three symbols + assert collect(y*x*z + a*x*y*z, x*y*z) == (1 + a)*x*y*z + + +def test_collect_4(): + """Collect with respect to a power""" + a, b, c, x = symbols('a,b,c,x') + + assert collect(a*x**c + b*x**c, x**c) == x**c*(a + b) + # issue 6096: 2 stays with c (unless c is integer or x is positive0 + assert collect(a*x**(2*c) + b*x**(2*c), x**c) == x**(2*c)*(a + b) + + +def test_collect_5(): + """Collect with respect to a tuple""" + a, x, y, z, n = symbols('a,x,y,z,n') + assert collect(x**2*y**4 + z*(x*y**2)**2 + z + a*z, [x*y**2, z]) in [ + z*(1 + a + x**2*y**4) + x**2*y**4, + z*(1 + a) + x**2*y**4*(1 + z) ] + assert collect((1 + (x + y) + (x + y)**2).expand(), + [x, y]) == 1 + y + x*(1 + 2*y) + x**2 + y**2 + + +def test_collect_pr19431(): + """Unevaluated collect with respect to a product""" + a = symbols('a') + assert collect(a**2*(a**2 + 1), a**2, evaluate=False)[a**2] == (a**2 + 1) + + +def test_collect_D(): + D = Derivative + f = Function('f') + x, a, b = symbols('x,a,b') + fx = D(f(x), x) + fxx = D(f(x), x, x) + + assert collect(a*fx + b*fx, fx) == (a + b)*fx + assert collect(a*D(fx, x) + b*D(fx, x), fx) == (a + b)*D(fx, x) + assert collect(a*fxx + b*fxx, fx) == (a + b)*D(fx, x) + # issue 4784 + assert collect(5*f(x) + 3*fx, fx) == 5*f(x) + 3*fx + assert collect(f(x) + f(x)*diff(f(x), x) + x*diff(f(x), x)*f(x), f(x).diff(x)) == \ + (x*f(x) + f(x))*D(f(x), x) + f(x) + assert collect(f(x) + f(x)*diff(f(x), x) + x*diff(f(x), x)*f(x), f(x).diff(x), exact=True) == \ + (x*f(x) + f(x))*D(f(x), x) + f(x) + assert collect(1/f(x) + 1/f(x)*diff(f(x), x) + x*diff(f(x), x)/f(x), f(x).diff(x), exact=True) == \ + (1/f(x) + x/f(x))*D(f(x), x) + 1/f(x) + e = (1 + x*fx + fx)/f(x) + assert collect(e.expand(), fx) == fx*(x/f(x) + 1/f(x)) + 1/f(x) + + +def test_collect_func(): + f = ((x + a + 1)**3).expand() + + assert collect(f, x) == a**3 + 3*a**2 + 3*a + x**3 + x**2*(3*a + 3) + \ + x*(3*a**2 + 6*a + 3) + 1 + assert collect(f, x, factor) == x**3 + 3*x**2*(a + 1) + 3*x*(a + 1)**2 + \ + (a + 1)**3 + + assert collect(f, x, evaluate=False) == { + S.One: a**3 + 3*a**2 + 3*a + 1, + x: 3*a**2 + 6*a + 3, x**2: 3*a + 3, + x**3: 1 + } + + assert collect(f, x, factor, evaluate=False) == { + S.One: (a + 1)**3, x: 3*(a + 1)**2, + x**2: umul(S(3), a + 1), x**3: 1} + + +def test_collect_order(): + a, b, x, t = symbols('a,b,x,t') + + assert collect(t + t*x + t*x**2 + O(x**3), t) == t*(1 + x + x**2 + O(x**3)) + assert collect(t + t*x + x**2 + O(x**3), t) == \ + t*(1 + x + O(x**3)) + x**2 + O(x**3) + + f = a*x + b*x + c*x**2 + d*x**2 + O(x**3) + g = x*(a + b) + x**2*(c + d) + O(x**3) + + assert collect(f, x) == g + assert collect(f, x, distribute_order_term=False) == g + + f = sin(a + b).series(b, 0, 10) + + assert collect(f, [sin(a), cos(a)]) == \ + sin(a)*cos(b).series(b, 0, 10) + cos(a)*sin(b).series(b, 0, 10) + assert collect(f, [sin(a), cos(a)], distribute_order_term=False) == \ + sin(a)*cos(b).series(b, 0, 10).removeO() + \ + cos(a)*sin(b).series(b, 0, 10).removeO() + O(b**10) + + +def test_rcollect(): + assert rcollect((x**2*y + x*y + x + y)/(x + y), y) == \ + (x + y*(1 + x + x**2))/(x + y) + assert rcollect(sqrt(-((x + 1)*(y + 1))), z) == sqrt(-((x + 1)*(y + 1))) + + +def test_collect_D_0(): + D = Derivative + f = Function('f') + x, a, b = symbols('x,a,b') + fxx = D(f(x), x, x) + + assert collect(a*fxx + b*fxx, fxx) == (a + b)*fxx + + +def test_collect_Wild(): + """Collect with respect to functions with Wild argument""" + a, b, x, y = symbols('a b x y') + f = Function('f') + w1 = Wild('.1') + w2 = Wild('.2') + assert collect(f(x) + a*f(x), f(w1)) == (1 + a)*f(x) + assert collect(f(x, y) + a*f(x, y), f(w1)) == f(x, y) + a*f(x, y) + assert collect(f(x, y) + a*f(x, y), f(w1, w2)) == (1 + a)*f(x, y) + assert collect(f(x, y) + a*f(x, y), f(w1, w1)) == f(x, y) + a*f(x, y) + assert collect(f(x, x) + a*f(x, x), f(w1, w1)) == (1 + a)*f(x, x) + assert collect(a*(x + 1)**y + (x + 1)**y, w1**y) == (1 + a)*(x + 1)**y + assert collect(a*(x + 1)**y + (x + 1)**y, w1**b) == \ + a*(x + 1)**y + (x + 1)**y + assert collect(a*(x + 1)**y + (x + 1)**y, (x + 1)**w2) == \ + (1 + a)*(x + 1)**y + assert collect(a*(x + 1)**y + (x + 1)**y, w1**w2) == (1 + a)*(x + 1)**y + + +def test_collect_const(): + # coverage not provided by above tests + assert collect_const(2*sqrt(3) + 4*a*sqrt(5)) == \ + 2*(2*sqrt(5)*a + sqrt(3)) # let the primitive reabsorb + assert collect_const(2*sqrt(3) + 4*a*sqrt(5), sqrt(3)) == \ + 2*sqrt(3) + 4*a*sqrt(5) + assert collect_const(sqrt(2)*(1 + sqrt(2)) + sqrt(3) + x*sqrt(2)) == \ + sqrt(2)*(x + 1 + sqrt(2)) + sqrt(3) + + # issue 5290 + assert collect_const(2*x + 2*y + 1, 2) == \ + collect_const(2*x + 2*y + 1) == \ + Add(S.One, Mul(2, x + y, evaluate=False), evaluate=False) + assert collect_const(-y - z) == Mul(-1, y + z, evaluate=False) + assert collect_const(2*x - 2*y - 2*z, 2) == \ + Mul(2, x - y - z, evaluate=False) + assert collect_const(2*x - 2*y - 2*z, -2) == \ + _unevaluated_Add(2*x, Mul(-2, y + z, evaluate=False)) + + # this is why the content_primitive is used + eq = (sqrt(15 + 5*sqrt(2))*x + sqrt(3 + sqrt(2))*y)*2 + assert collect_sqrt(eq + 2) == \ + 2*sqrt(sqrt(2) + 3)*(sqrt(5)*x + y) + 2 + + # issue 16296 + assert collect_const(a + b + x/2 + y/2) == a + b + Mul(S.Half, x + y, evaluate=False) + + +def test_issue_13143(): + f = Function('f') + fx = f(x).diff(x) + e = f(x) + fx + f(x)*fx + # collect function before derivative + assert collect(e, Wild('w')) == f(x)*(fx + 1) + fx + e = f(x) + f(x)*fx + x*fx*f(x) + assert collect(e, fx) == (x*f(x) + f(x))*fx + f(x) + assert collect(e, f(x)) == (x*fx + fx + 1)*f(x) + e = f(x) + fx + f(x)*fx + assert collect(e, [f(x), fx]) == f(x)*(1 + fx) + fx + assert collect(e, [fx, f(x)]) == fx*(1 + f(x)) + f(x) + + +def test_issue_6097(): + assert collect(a*y**(2.0*x) + b*y**(2.0*x), y**x) == (a + b)*(y**x)**2.0 + assert collect(a*2**(2.0*x) + b*2**(2.0*x), 2**x) == (a + b)*(2**x)**2.0 + + +def test_fraction_expand(): + eq = (x + y)*y/x + assert eq.expand(frac=True) == fraction_expand(eq) == (x*y + y**2)/x + assert eq.expand() == y + y**2/x + + +def test_fraction(): + x, y, z = map(Symbol, 'xyz') + A = Symbol('A', commutative=False) + + assert fraction(S.Half) == (1, 2) + + assert fraction(x) == (x, 1) + assert fraction(1/x) == (1, x) + assert fraction(x/y) == (x, y) + assert fraction(x/2) == (x, 2) + + assert fraction(x*y/z) == (x*y, z) + assert fraction(x/(y*z)) == (x, y*z) + + assert fraction(1/y**2) == (1, y**2) + assert fraction(x/y**2) == (x, y**2) + + assert fraction((x**2 + 1)/y) == (x**2 + 1, y) + assert fraction(x*(y + 1)/y**7) == (x*(y + 1), y**7) + + assert fraction(exp(-x), exact=True) == (exp(-x), 1) + assert fraction((1/(x + y))/2, exact=True) == (1, Mul(2,(x + y), evaluate=False)) + + assert fraction(x*A/y) == (x*A, y) + assert fraction(x*A**-1/y) == (x*A**-1, y) + + n = symbols('n', negative=True) + assert fraction(exp(n)) == (1, exp(-n)) + assert fraction(exp(-n)) == (exp(-n), 1) + + p = symbols('p', positive=True) + assert fraction(exp(-p)*log(p), exact=True) == (exp(-p)*log(p), 1) + + m = Mul(1, 1, S.Half, evaluate=False) + assert fraction(m) == (1, 2) + assert fraction(m, exact=True) == (Mul(1, 1, evaluate=False), 2) + + m = Mul(1, 1, S.Half, S.Half, Pow(1, -1, evaluate=False), evaluate=False) + assert fraction(m) == (1, 4) + assert fraction(m, exact=True) == \ + (Mul(1, 1, evaluate=False), Mul(2, 2, 1, evaluate=False)) + + +def test_issue_5615(): + aA, Re, a, b, D = symbols('aA Re a b D') + e = ((D**3*a + b*aA**3)/Re).expand() + assert collect(e, [aA**3/Re, a]) == e + + +def test_issue_5933(): + from sympy.geometry.polygon import (Polygon, RegularPolygon) + from sympy.simplify.radsimp import denom + x = Polygon(*RegularPolygon((0, 0), 1, 5).vertices).centroid.x + assert abs(denom(x).n()) > 1e-12 + assert abs(denom(radsimp(x))) > 1e-12 # in case simplify didn't handle it + + +def test_issue_14608(): + a, b = symbols('a b', commutative=False) + x, y = symbols('x y') + raises(AttributeError, lambda: collect(a*b + b*a, a)) + assert collect(x*y + y*(x+1), a) == x*y + y*(x+1) + assert collect(x*y + y*(x+1) + a*b + b*a, y) == y*(2*x + 1) + a*b + b*a + + +def test_collect_abs(): + s = abs(x) + abs(y) + assert collect_abs(s) == s + assert unchanged(Mul, abs(x), abs(y)) + ans = Abs(x*y) + assert isinstance(ans, Abs) + assert collect_abs(abs(x)*abs(y)) == ans + assert collect_abs(1 + exp(abs(x)*abs(y))) == 1 + exp(ans) + + # See https://github.com/sympy/sympy/issues/12910 + p = Symbol('p', positive=True) + assert collect_abs(p/abs(1-p)).is_commutative is True + + +def test_issue_19149(): + eq = exp(3*x/4) + assert collect(eq, exp(x)) == eq + +def test_issue_19719(): + a, b = symbols('a, b') + expr = a**2 * (b + 1) + (7 + 1/b)/a + collected = collect(expr, (a**2, 1/a), evaluate=False) + # Would return {_Dummy_20**(-2): b + 1, 1/a: 7 + 1/b} without xreplace + assert collected == {a**2: b + 1, 1/a: 7 + 1/b} + + +def test_issue_21355(): + assert radsimp(1/(x + sqrt(x**2))) == 1/(x + sqrt(x**2)) + assert radsimp(1/(x - sqrt(x**2))) == 1/(x - sqrt(x**2)) diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_ratsimp.py b/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_ratsimp.py new file mode 100644 index 0000000000000000000000000000000000000000..14e84fd2b227518baff1bda4e5b27ecc40a8bcdd --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_ratsimp.py @@ -0,0 +1,78 @@ +from sympy.core.numbers import (Rational, pi) +from sympy.functions.elementary.exponential import log +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.special.error_functions import erf +from sympy.polys.domains import GF +from sympy.simplify.ratsimp import (ratsimp, ratsimpmodprime) + +from sympy.abc import x, y, z, t, a, b, c, d, e + + +def test_ratsimp(): + f, g = 1/x + 1/y, (x + y)/(x*y) + + assert f != g and ratsimp(f) == g + + f, g = 1/(1 + 1/x), 1 - 1/(x + 1) + + assert f != g and ratsimp(f) == g + + f, g = x/(x + y) + y/(x + y), 1 + + assert f != g and ratsimp(f) == g + + f, g = -x - y - y**2/(x + y) + x**2/(x + y), -2*y + + assert f != g and ratsimp(f) == g + + f = (a*c*x*y + a*c*z - b*d*x*y - b*d*z - b*t*x*y - b*t*x - b*t*z + + e*x)/(x*y + z) + G = [a*c - b*d - b*t + (-b*t*x + e*x)/(x*y + z), + a*c - b*d - b*t - ( b*t*x - e*x)/(x*y + z)] + + assert f != g and ratsimp(f) in G + + A = sqrt(pi) + + B = log(erf(x) - 1) + C = log(erf(x) + 1) + + D = 8 - 8*erf(x) + + f = A*B/D - A*C/D + A*C*erf(x)/D - A*B*erf(x)/D + 2*A/D + + assert ratsimp(f) == A*B/8 - A*C/8 - A/(4*erf(x) - 4) + + +def test_ratsimpmodprime(): + a = y**5 + x + y + b = x - y + F = [x*y**5 - x - y] + assert ratsimpmodprime(a/b, F, x, y, order='lex') == \ + (-x**2 - x*y - x - y) / (-x**2 + x*y) + + a = x + y**2 - 2 + b = x + y**2 - y - 1 + F = [x*y - 1] + assert ratsimpmodprime(a/b, F, x, y, order='lex') == \ + (1 + y - x)/(y - x) + + a = 5*x**3 + 21*x**2 + 4*x*y + 23*x + 12*y + 15 + b = 7*x**3 - y*x**2 + 31*x**2 + 2*x*y + 15*y + 37*x + 21 + F = [x**2 + y**2 - 1] + assert ratsimpmodprime(a/b, F, x, y, order='lex') == \ + (1 + 5*y - 5*x)/(8*y - 6*x) + + a = x*y - x - 2*y + 4 + b = x + y**2 - 2*y + F = [x - 2, y - 3] + assert ratsimpmodprime(a/b, F, x, y, order='lex') == \ + Rational(2, 5) + + # Test a bug where denominators would be dropped + assert ratsimpmodprime(x, [y - 2*x], order='lex') == \ + y/2 + + a = (x**5 + 2*x**4 + 2*x**3 + 2*x**2 + x + 2/x + x**(-2)) + assert ratsimpmodprime(a, [x + 1], domain=GF(2)) == 1 + assert ratsimpmodprime(a, [x + 1], domain=GF(3)) == -1 diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_rewrite.py b/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_rewrite.py new file mode 100644 index 0000000000000000000000000000000000000000..56d2fb7a85bd959bd4accc2f36127429efbdbe70 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_rewrite.py @@ -0,0 +1,31 @@ +from sympy.core.numbers import I +from sympy.core.symbol import symbols +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.trigonometric import (cos, cot, sin) +from sympy.testing.pytest import _both_exp_pow + +x, y, z, n = symbols('x,y,z,n') + + +@_both_exp_pow +def test_has(): + assert cot(x).has(x) + assert cot(x).has(cot) + assert not cot(x).has(sin) + assert sin(x).has(x) + assert sin(x).has(sin) + assert not sin(x).has(cot) + assert exp(x).has(exp) + + +@_both_exp_pow +def test_sin_exp_rewrite(): + assert sin(x).rewrite(sin, exp) == -I/2*(exp(I*x) - exp(-I*x)) + assert sin(x).rewrite(sin, exp).rewrite(exp, sin) == sin(x) + assert cos(x).rewrite(cos, exp).rewrite(exp, cos) == cos(x) + assert (sin(5*y) - sin( + 2*x)).rewrite(sin, exp).rewrite(exp, sin) == sin(5*y) - sin(2*x) + assert sin(x + y).rewrite(sin, exp).rewrite(exp, sin) == sin(x + y) + assert cos(x + y).rewrite(cos, exp).rewrite(exp, cos) == cos(x + y) + # This next test currently passes... not clear whether it should or not? + assert cos(x).rewrite(cos, exp).rewrite(exp, sin) == cos(x) diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_simplify.py b/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_simplify.py new file mode 100644 index 0000000000000000000000000000000000000000..a5bf469f68adf5c5dfbdf7559414681e2fb28ba7 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_simplify.py @@ -0,0 +1,1093 @@ +from sympy.concrete.summations import Sum +from sympy.core.add import Add +from sympy.core.basic import Basic +from sympy.core.expr import unchanged +from sympy.core.function import (count_ops, diff, expand, expand_multinomial, Function, Derivative) +from sympy.core.mul import Mul, _keep_coeff +from sympy.core import GoldenRatio +from sympy.core.numbers import (E, Float, I, oo, pi, Rational, zoo) +from sympy.core.relational import (Eq, Lt, Gt, Ge, Le) +from sympy.core.singleton import S +from sympy.core.symbol import (Symbol, symbols) +from sympy.core.sympify import sympify +from sympy.functions.combinatorial.factorials import (binomial, factorial) +from sympy.functions.elementary.complexes import (Abs, sign) +from sympy.functions.elementary.exponential import (exp, exp_polar, log) +from sympy.functions.elementary.hyperbolic import (cosh, csch, sinh) +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.elementary.trigonometric import (acos, asin, atan, cos, sin, sinc, tan) +from sympy.functions.special.error_functions import erf +from sympy.functions.special.gamma_functions import gamma +from sympy.functions.special.hyper import hyper +from sympy.functions.special.tensor_functions import KroneckerDelta +from sympy.geometry.polygon import rad +from sympy.integrals.integrals import (Integral, integrate) +from sympy.logic.boolalg import (And, Or) +from sympy.matrices.dense import (Matrix, eye) +from sympy.matrices.expressions.matexpr import MatrixSymbol +from sympy.polys.polytools import (factor, Poly) +from sympy.simplify.simplify import (besselsimp, hypersimp, inversecombine, logcombine, nsimplify, nthroot, posify, separatevars, signsimp, simplify) +from sympy.solvers.solvers import solve + +from sympy.testing.pytest import XFAIL, slow, _both_exp_pow +from sympy.abc import x, y, z, t, a, b, c, d, e, f, g, h, i, n + + +def test_issue_7263(): + assert abs((simplify(30.8**2 - 82.5**2 * sin(rad(11.6))**2)).evalf() - \ + 673.447451402970) < 1e-12 + + +def test_factorial_simplify(): + # There are more tests in test_factorials.py. + x = Symbol('x') + assert simplify(factorial(x)/x) == gamma(x) + assert simplify(factorial(factorial(x))) == factorial(factorial(x)) + + +def test_simplify_expr(): + x, y, z, k, n, m, w, s, A = symbols('x,y,z,k,n,m,w,s,A') + f = Function('f') + + assert all(simplify(tmp) == tmp for tmp in [I, E, oo, x, -x, -oo, -E, -I]) + + e = 1/x + 1/y + assert e != (x + y)/(x*y) + assert simplify(e) == (x + y)/(x*y) + + e = A**2*s**4/(4*pi*k*m**3) + assert simplify(e) == e + + e = (4 + 4*x - 2*(2 + 2*x))/(2 + 2*x) + assert simplify(e) == 0 + + e = (-4*x*y**2 - 2*y**3 - 2*x**2*y)/(x + y)**2 + assert simplify(e) == -2*y + + e = -x - y - (x + y)**(-1)*y**2 + (x + y)**(-1)*x**2 + assert simplify(e) == -2*y + + e = (x + x*y)/x + assert simplify(e) == 1 + y + + e = (f(x) + y*f(x))/f(x) + assert simplify(e) == 1 + y + + e = (2 * (1/n - cos(n * pi)/n))/pi + assert simplify(e) == (-cos(pi*n) + 1)/(pi*n)*2 + + e = integrate(1/(x**3 + 1), x).diff(x) + assert simplify(e) == 1/(x**3 + 1) + + e = integrate(x/(x**2 + 3*x + 1), x).diff(x) + assert simplify(e) == x/(x**2 + 3*x + 1) + + f = Symbol('f') + A = Matrix([[2*k - m*w**2, -k], [-k, k - m*w**2]]).inv() + assert simplify((A*Matrix([0, f]))[1] - + (-f*(2*k - m*w**2)/(k**2 - (k - m*w**2)*(2*k - m*w**2)))) == 0 + + f = -x + y/(z + t) + z*x/(z + t) + z*a/(z + t) + t*x/(z + t) + assert simplify(f) == (y + a*z)/(z + t) + + # issue 10347 + expr = -x*(y**2 - 1)*(2*y**2*(x**2 - 1)/(a*(x**2 - y**2)**2) + (x**2 - 1) + /(a*(x**2 - y**2)))/(a*(x**2 - y**2)) + x*(-2*x**2*sqrt(-x**2*y**2 + x**2 + + y**2 - 1)*sin(z)/(a*(x**2 - y**2)**2) - x**2*sqrt(-x**2*y**2 + x**2 + + y**2 - 1)*sin(z)/(a*(x**2 - 1)*(x**2 - y**2)) + (x**2*sqrt((-x**2 + 1)* + (y**2 - 1))*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*sin(z)/(x**2 - 1) + sqrt( + (-x**2 + 1)*(y**2 - 1))*(x*(-x*y**2 + x)/sqrt(-x**2*y**2 + x**2 + y**2 - + 1) + sqrt(-x**2*y**2 + x**2 + y**2 - 1))*sin(z))/(a*sqrt((-x**2 + 1)*( + y**2 - 1))*(x**2 - y**2)))*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*sin(z)/(a* + (x**2 - y**2)) + x*(-2*x**2*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*cos(z)/(a* + (x**2 - y**2)**2) - x**2*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*cos(z)/(a* + (x**2 - 1)*(x**2 - y**2)) + (x**2*sqrt((-x**2 + 1)*(y**2 - 1))*sqrt(-x**2 + *y**2 + x**2 + y**2 - 1)*cos(z)/(x**2 - 1) + x*sqrt((-x**2 + 1)*(y**2 - + 1))*(-x*y**2 + x)*cos(z)/sqrt(-x**2*y**2 + x**2 + y**2 - 1) + sqrt((-x**2 + + 1)*(y**2 - 1))*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*cos(z))/(a*sqrt((-x**2 + + 1)*(y**2 - 1))*(x**2 - y**2)))*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*cos( + z)/(a*(x**2 - y**2)) - y*sqrt((-x**2 + 1)*(y**2 - 1))*(-x*y*sqrt(-x**2* + y**2 + x**2 + y**2 - 1)*sin(z)/(a*(x**2 - y**2)*(y**2 - 1)) + 2*x*y*sqrt( + -x**2*y**2 + x**2 + y**2 - 1)*sin(z)/(a*(x**2 - y**2)**2) + (x*y*sqrt(( + -x**2 + 1)*(y**2 - 1))*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*sin(z)/(y**2 - + 1) + x*sqrt((-x**2 + 1)*(y**2 - 1))*(-x**2*y + y)*sin(z)/sqrt(-x**2*y**2 + + x**2 + y**2 - 1))/(a*sqrt((-x**2 + 1)*(y**2 - 1))*(x**2 - y**2)))*sin( + z)/(a*(x**2 - y**2)) + y*(x**2 - 1)*(-2*x*y*(x**2 - 1)/(a*(x**2 - y**2) + **2) + 2*x*y/(a*(x**2 - y**2)))/(a*(x**2 - y**2)) + y*(x**2 - 1)*(y**2 - + 1)*(-x*y*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*cos(z)/(a*(x**2 - y**2)*(y**2 + - 1)) + 2*x*y*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*cos(z)/(a*(x**2 - y**2) + **2) + (x*y*sqrt((-x**2 + 1)*(y**2 - 1))*sqrt(-x**2*y**2 + x**2 + y**2 - + 1)*cos(z)/(y**2 - 1) + x*sqrt((-x**2 + 1)*(y**2 - 1))*(-x**2*y + y)*cos( + z)/sqrt(-x**2*y**2 + x**2 + y**2 - 1))/(a*sqrt((-x**2 + 1)*(y**2 - 1) + )*(x**2 - y**2)))*cos(z)/(a*sqrt((-x**2 + 1)*(y**2 - 1))*(x**2 - y**2) + ) - x*sqrt((-x**2 + 1)*(y**2 - 1))*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*sin( + z)**2/(a**2*(x**2 - 1)*(x**2 - y**2)*(y**2 - 1)) - x*sqrt((-x**2 + 1)*( + y**2 - 1))*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*cos(z)**2/(a**2*(x**2 - 1)*( + x**2 - y**2)*(y**2 - 1)) + assert simplify(expr) == 2*x/(a**2*(x**2 - y**2)) + + #issue 17631 + assert simplify('((-1/2)*Boole(True)*Boole(False)-1)*Boole(True)') == \ + Mul(sympify('(2 + Boole(True)*Boole(False))'), sympify('-Boole(True)/2')) + + A, B = symbols('A,B', commutative=False) + + assert simplify(A*B - B*A) == A*B - B*A + assert simplify(A/(1 + y/x)) == x*A/(x + y) + assert simplify(A*(1/x + 1/y)) == A/x + A/y #(x + y)*A/(x*y) + + assert simplify(log(2) + log(3)) == log(6) + assert simplify(log(2*x) - log(2)) == log(x) + + assert simplify(hyper([], [], x)) == exp(x) + + +def test_issue_3557(): + f_1 = x*a + y*b + z*c - 1 + f_2 = x*d + y*e + z*f - 1 + f_3 = x*g + y*h + z*i - 1 + + solutions = solve([f_1, f_2, f_3], x, y, z, simplify=False) + + assert simplify(solutions[y]) == \ + (a*i + c*d + f*g - a*f - c*g - d*i)/ \ + (a*e*i + b*f*g + c*d*h - a*f*h - b*d*i - c*e*g) + + +def test_simplify_other(): + assert simplify(sin(x)**2 + cos(x)**2) == 1 + assert simplify(gamma(x + 1)/gamma(x)) == x + assert simplify(sin(x)**2 + cos(x)**2 + factorial(x)/gamma(x)) == 1 + x + assert simplify( + Eq(sin(x)**2 + cos(x)**2, factorial(x)/gamma(x))) == Eq(x, 1) + nc = symbols('nc', commutative=False) + assert simplify(x + x*nc) == x*(1 + nc) + # issue 6123 + # f = exp(-I*(k*sqrt(t) + x/(2*sqrt(t)))**2) + # ans = integrate(f, (k, -oo, oo), conds='none') + ans = I*(-pi*x*exp(I*pi*Rational(-3, 4) + I*x**2/(4*t))*erf(x*exp(I*pi*Rational(-3, 4))/ + (2*sqrt(t)))/(2*sqrt(t)) + pi*x*exp(I*pi*Rational(-3, 4) + I*x**2/(4*t))/ + (2*sqrt(t)))*exp(-I*x**2/(4*t))/(sqrt(pi)*x) - I*sqrt(pi) * \ + (-erf(x*exp(I*pi/4)/(2*sqrt(t))) + 1)*exp(I*pi/4)/(2*sqrt(t)) + assert simplify(ans) == -(-1)**Rational(3, 4)*sqrt(pi)/sqrt(t) + # issue 6370 + assert simplify(2**(2 + x)/4) == 2**x + + +@_both_exp_pow +def test_simplify_complex(): + cosAsExp = cos(x)._eval_rewrite_as_exp(x) + tanAsExp = tan(x)._eval_rewrite_as_exp(x) + assert simplify(cosAsExp*tanAsExp) == sin(x) # issue 4341 + + # issue 10124 + assert simplify(exp(Matrix([[0, -1], [1, 0]]))) == Matrix([[cos(1), + -sin(1)], [sin(1), cos(1)]]) + + +def test_simplify_ratio(): + # roots of x**3-3*x+5 + roots = ['(1/2 - sqrt(3)*I/2)*(sqrt(21)/2 + 5/2)**(1/3) + 1/((1/2 - ' + 'sqrt(3)*I/2)*(sqrt(21)/2 + 5/2)**(1/3))', + '1/((1/2 + sqrt(3)*I/2)*(sqrt(21)/2 + 5/2)**(1/3)) + ' + '(1/2 + sqrt(3)*I/2)*(sqrt(21)/2 + 5/2)**(1/3)', + '-(sqrt(21)/2 + 5/2)**(1/3) - 1/(sqrt(21)/2 + 5/2)**(1/3)'] + + for r in roots: + r = S(r) + assert count_ops(simplify(r, ratio=1)) <= count_ops(r) + # If ratio=oo, simplify() is always applied: + assert simplify(r, ratio=oo) is not r + + +def test_simplify_measure(): + measure1 = lambda expr: len(str(expr)) + measure2 = lambda expr: -count_ops(expr) + # Return the most complicated result + expr = (x + 1)/(x + sin(x)**2 + cos(x)**2) + assert measure1(simplify(expr, measure=measure1)) <= measure1(expr) + assert measure2(simplify(expr, measure=measure2)) <= measure2(expr) + + expr2 = Eq(sin(x)**2 + cos(x)**2, 1) + assert measure1(simplify(expr2, measure=measure1)) <= measure1(expr2) + assert measure2(simplify(expr2, measure=measure2)) <= measure2(expr2) + + +def test_simplify_rational(): + expr = 2**x*2.**y + assert simplify(expr, rational = True) == 2**(x+y) + assert simplify(expr, rational = None) == 2.0**(x+y) + assert simplify(expr, rational = False) == expr + assert simplify('0.9 - 0.8 - 0.1', rational = True) == 0 + + +def test_simplify_issue_1308(): + assert simplify(exp(Rational(-1, 2)) + exp(Rational(-3, 2))) == \ + (1 + E)*exp(Rational(-3, 2)) + + +def test_issue_5652(): + assert simplify(E + exp(-E)) == exp(-E) + E + n = symbols('n', commutative=False) + assert simplify(n + n**(-n)) == n + n**(-n) + +def test_issue_27380(): + assert simplify(1.0**(x+1)/1.0**x) == 1.0 + +def test_simplify_fail1(): + x = Symbol('x') + y = Symbol('y') + e = (x + y)**2/(-4*x*y**2 - 2*y**3 - 2*x**2*y) + assert simplify(e) == 1 / (-2*y) + + +def test_nthroot(): + assert nthroot(90 + 34*sqrt(7), 3) == sqrt(7) + 3 + q = 1 + sqrt(2) - 2*sqrt(3) + sqrt(6) + sqrt(7) + assert nthroot(expand_multinomial(q**3), 3) == q + assert nthroot(41 + 29*sqrt(2), 5) == 1 + sqrt(2) + assert nthroot(-41 - 29*sqrt(2), 5) == -1 - sqrt(2) + expr = 1320*sqrt(10) + 4216 + 2576*sqrt(6) + 1640*sqrt(15) + assert nthroot(expr, 5) == 1 + sqrt(6) + sqrt(15) + q = 1 + sqrt(2) + sqrt(3) + sqrt(5) + assert expand_multinomial(nthroot(expand_multinomial(q**5), 5)) == q + q = 1 + sqrt(2) + 7*sqrt(6) + 2*sqrt(10) + assert nthroot(expand_multinomial(q**5), 5, 8) == q + q = 1 + sqrt(2) - 2*sqrt(3) + 1171*sqrt(6) + assert nthroot(expand_multinomial(q**3), 3) == q + assert nthroot(expand_multinomial(q**6), 6) == q + + +def test_nthroot1(): + q = 1 + sqrt(2) + sqrt(3) + S.One/10**20 + p = expand_multinomial(q**5) + assert nthroot(p, 5) == q + q = 1 + sqrt(2) + sqrt(3) + S.One/10**30 + p = expand_multinomial(q**5) + assert nthroot(p, 5) == q + + +@_both_exp_pow +def test_separatevars(): + x, y, z, n = symbols('x,y,z,n') + assert separatevars(2*n*x*z + 2*x*y*z) == 2*x*z*(n + y) + assert separatevars(x*z + x*y*z) == x*z*(1 + y) + assert separatevars(pi*x*z + pi*x*y*z) == pi*x*z*(1 + y) + assert separatevars(x*y**2*sin(x) + x*sin(x)*sin(y)) == \ + x*(sin(y) + y**2)*sin(x) + assert separatevars(x*exp(x + y) + x*exp(x)) == x*(1 + exp(y))*exp(x) + assert separatevars((x*(y + 1))**z).is_Pow # != x**z*(1 + y)**z + assert separatevars(1 + x + y + x*y) == (x + 1)*(y + 1) + assert separatevars(y/pi*exp(-(z - x)/cos(n))) == \ + y*exp(x/cos(n))*exp(-z/cos(n))/pi + assert separatevars((x + y)*(x - y) + y**2 + 2*x + 1) == (x + 1)**2 + # issue 4858 + p = Symbol('p', positive=True) + assert separatevars(sqrt(p**2 + x*p**2)) == p*sqrt(1 + x) + assert separatevars(sqrt(y*(p**2 + x*p**2))) == p*sqrt(y*(1 + x)) + assert separatevars(sqrt(y*(p**2 + x*p**2)), force=True) == \ + p*sqrt(y)*sqrt(1 + x) + # issue 4865 + assert separatevars(sqrt(x*y)).is_Pow + assert separatevars(sqrt(x*y), force=True) == sqrt(x)*sqrt(y) + # issue 4957 + # any type sequence for symbols is fine + assert separatevars(((2*x + 2)*y), dict=True, symbols=()) == \ + {'coeff': 1, x: 2*x + 2, y: y} + # separable + assert separatevars(((2*x + 2)*y), dict=True, symbols=[x]) == \ + {'coeff': y, x: 2*x + 2} + assert separatevars(((2*x + 2)*y), dict=True, symbols=[]) == \ + {'coeff': 1, x: 2*x + 2, y: y} + assert separatevars(((2*x + 2)*y), dict=True) == \ + {'coeff': 1, x: 2*x + 2, y: y} + assert separatevars(((2*x + 2)*y), dict=True, symbols=None) == \ + {'coeff': y*(2*x + 2)} + # not separable + assert separatevars(3, dict=True) is None + assert separatevars(2*x + y, dict=True, symbols=()) is None + assert separatevars(2*x + y, dict=True) is None + assert separatevars(2*x + y, dict=True, symbols=None) == {'coeff': 2*x + y} + # issue 4808 + n, m = symbols('n,m', commutative=False) + assert separatevars(m + n*m) == (1 + n)*m + assert separatevars(x + x*n) == x*(1 + n) + # issue 4910 + f = Function('f') + assert separatevars(f(x) + x*f(x)) == f(x) + x*f(x) + # a noncommutable object present + eq = x*(1 + hyper((), (), y*z)) + assert separatevars(eq) == eq + + s = separatevars(abs(x*y)) + assert s == abs(x)*abs(y) and s.is_Mul + z = cos(1)**2 + sin(1)**2 - 1 + a = abs(x*z) + s = separatevars(a) + assert not a.is_Mul and s.is_Mul and s == abs(x)*abs(z) + s = separatevars(abs(x*y*z)) + assert s == abs(x)*abs(y)*abs(z) + + # abs(x+y)/abs(z) would be better but we test this here to + # see that it doesn't raise + assert separatevars(abs((x+y)/z)) == abs((x+y)/z) + + +def test_separatevars_advanced_factor(): + x, y, z = symbols('x,y,z') + assert separatevars(1 + log(x)*log(y) + log(x) + log(y)) == \ + (log(x) + 1)*(log(y) + 1) + assert separatevars(1 + x - log(z) - x*log(z) - exp(y)*log(z) - + x*exp(y)*log(z) + x*exp(y) + exp(y)) == \ + -((x + 1)*(log(z) - 1)*(exp(y) + 1)) + x, y = symbols('x,y', positive=True) + assert separatevars(1 + log(x**log(y)) + log(x*y)) == \ + (log(x) + 1)*(log(y) + 1) + + +def test_hypersimp(): + n, k = symbols('n,k', integer=True) + + assert hypersimp(factorial(k), k) == k + 1 + assert hypersimp(factorial(k**2), k) is None + + assert hypersimp(1/factorial(k), k) == 1/(k + 1) + + assert hypersimp(2**k/factorial(k)**2, k) == 2/(k + 1)**2 + + assert hypersimp(binomial(n, k), k) == (n - k)/(k + 1) + assert hypersimp(binomial(n + 1, k), k) == (n - k + 1)/(k + 1) + + term = (4*k + 1)*factorial(k)/factorial(2*k + 1) + assert hypersimp(term, k) == S.Half*((4*k + 5)/(3 + 14*k + 8*k**2)) + + term = 1/((2*k - 1)*factorial(2*k + 1)) + assert hypersimp(term, k) == (k - S.Half)/((k + 1)*(2*k + 1)*(2*k + 3)) + + term = binomial(n, k)*(-1)**k/factorial(k) + assert hypersimp(term, k) == (k - n)/(k + 1)**2 + + +def test_nsimplify(): + x = Symbol("x") + assert nsimplify(0) == 0 + assert nsimplify(-1) == -1 + assert nsimplify(1) == 1 + assert nsimplify(1 + x) == 1 + x + assert nsimplify(2.7) == Rational(27, 10) + assert nsimplify(1 - GoldenRatio) == (1 - sqrt(5))/2 + assert nsimplify((1 + sqrt(5))/4, [GoldenRatio]) == GoldenRatio/2 + assert nsimplify(2/GoldenRatio, [GoldenRatio]) == 2*GoldenRatio - 2 + assert nsimplify(exp(pi*I*Rational(5, 3), evaluate=False)) == \ + sympify('1/2 - sqrt(3)*I/2') + assert nsimplify(sin(pi*Rational(3, 5), evaluate=False)) == \ + sympify('sqrt(sqrt(5)/8 + 5/8)') + assert nsimplify(sqrt(atan('1', evaluate=False))*(2 + I), [pi]) == \ + sqrt(pi) + sqrt(pi)/2*I + assert nsimplify(2 + exp(2*atan('1/4')*I)) == sympify('49/17 + 8*I/17') + assert nsimplify(pi, tolerance=0.01) == Rational(22, 7) + assert nsimplify(pi, tolerance=0.001) == Rational(355, 113) + assert nsimplify(0.33333, tolerance=1e-4) == Rational(1, 3) + assert nsimplify(2.0**(1/3.), tolerance=0.001) == Rational(635, 504) + assert nsimplify(2.0**(1/3.), tolerance=0.001, full=True) == \ + 2**Rational(1, 3) + assert nsimplify(x + .5, rational=True) == S.Half + x + assert nsimplify(1/.3 + x, rational=True) == Rational(10, 3) + x + assert nsimplify(log(3).n(), rational=True) == \ + sympify('109861228866811/100000000000000') + assert nsimplify(Float(0.272198261287950), [pi, log(2)]) == pi*log(2)/8 + assert nsimplify(Float(0.272198261287950).n(3), [pi, log(2)]) == \ + -pi/4 - log(2) + Rational(7, 4) + assert nsimplify(x/7.0) == x/7 + assert nsimplify(pi/1e2) == pi/100 + assert nsimplify(pi/1e2, rational=False) == pi/100.0 + assert nsimplify(pi/1e-7) == 10000000*pi + assert not nsimplify( + factor(-3.0*z**2*(z**2)**(-2.5) + 3*(z**2)**(-1.5))).atoms(Float) + e = x**0.0 + assert e.is_Pow and nsimplify(x**0.0) == 1 + assert nsimplify(3.333333, tolerance=0.1, rational=True) == Rational(10, 3) + assert nsimplify(3.333333, tolerance=0.01, rational=True) == Rational(10, 3) + assert nsimplify(3.666666, tolerance=0.1, rational=True) == Rational(11, 3) + assert nsimplify(3.666666, tolerance=0.01, rational=True) == Rational(11, 3) + assert nsimplify(33, tolerance=10, rational=True) == Rational(33) + assert nsimplify(33.33, tolerance=10, rational=True) == Rational(30) + assert nsimplify(37.76, tolerance=10, rational=True) == Rational(40) + assert nsimplify(-203.1) == Rational(-2031, 10) + assert nsimplify(.2, tolerance=0) == Rational(1, 5) + assert nsimplify(-.2, tolerance=0) == Rational(-1, 5) + assert nsimplify(.2222, tolerance=0) == Rational(1111, 5000) + assert nsimplify(-.2222, tolerance=0) == Rational(-1111, 5000) + # issue 7211, PR 4112 + assert nsimplify(S(2e-8)) == Rational(1, 50000000) + # issue 7322 direct test + assert nsimplify(1e-42, rational=True) != 0 + # issue 10336 + inf = Float('inf') + infs = (-oo, oo, inf, -inf) + for zi in infs: + ans = sign(zi)*oo + assert nsimplify(zi) == ans + assert nsimplify(zi + x) == x + ans + + assert nsimplify(0.33333333, rational=True, rational_conversion='exact') == Rational(0.33333333) + + # Make sure nsimplify on expressions uses full precision + assert nsimplify(pi.evalf(100)*x, rational_conversion='exact').evalf(100) == pi.evalf(100)*x + + +def test_issue_9448(): + tmp = sympify("1/(1 - (-1)**(2/3) - (-1)**(1/3)) + 1/(1 + (-1)**(2/3) + (-1)**(1/3))") + assert nsimplify(tmp) == S.Half + + +def test_extract_minus_sign(): + x = Symbol("x") + y = Symbol("y") + a = Symbol("a") + b = Symbol("b") + assert simplify(-x/-y) == x/y + assert simplify(-x/y) == -x/y + assert simplify(x/y) == x/y + assert simplify(x/-y) == -x/y + assert simplify(-x/0) == zoo*x + assert simplify(Rational(-5, 0)) is zoo + assert simplify(-a*x/(-y - b)) == a*x/(b + y) + + +def test_diff(): + x = Symbol("x") + y = Symbol("y") + f = Function("f") + g = Function("g") + assert simplify(g(x).diff(x)*f(x).diff(x) - f(x).diff(x)*g(x).diff(x)) == 0 + assert simplify(2*f(x)*f(x).diff(x) - diff(f(x)**2, x)) == 0 + assert simplify(diff(1/f(x), x) + f(x).diff(x)/f(x)**2) == 0 + assert simplify(f(x).diff(x, y) - f(x).diff(y, x)) == 0 + + +def test_logcombine_1(): + x, y = symbols("x,y") + a = Symbol("a") + z, w = symbols("z,w", positive=True) + b = Symbol("b", real=True) + assert logcombine(log(x) + 2*log(y)) == log(x) + 2*log(y) + assert logcombine(log(x) + 2*log(y), force=True) == log(x*y**2) + assert logcombine(a*log(w) + log(z)) == a*log(w) + log(z) + assert logcombine(b*log(z) + b*log(x)) == log(z**b) + b*log(x) + assert logcombine(b*log(z) - log(w)) == log(z**b/w) + assert logcombine(log(x)*log(z)) == log(x)*log(z) + assert logcombine(log(w)*log(x)) == log(w)*log(x) + assert logcombine(cos(-2*log(z) + b*log(w))) in [cos(log(w**b/z**2)), + cos(log(z**2/w**b))] + assert logcombine(log(log(x) - log(y)) - log(z), force=True) == \ + log(log(x/y)/z) + assert logcombine((2 + I)*log(x), force=True) == (2 + I)*log(x) + assert logcombine((x**2 + log(x) - log(y))/(x*y), force=True) == \ + (x**2 + log(x/y))/(x*y) + # the following could also give log(z*x**log(y**2)), what we + # are testing is that a canonical result is obtained + assert logcombine(log(x)*2*log(y) + log(z), force=True) == \ + log(z*y**log(x**2)) + assert logcombine((x*y + sqrt(x**4 + y**4) + log(x) - log(y))/(pi*x**Rational(2, 3)* + sqrt(y)**3), force=True) == ( + x*y + sqrt(x**4 + y**4) + log(x/y))/(pi*x**Rational(2, 3)*y**Rational(3, 2)) + assert logcombine(gamma(-log(x/y))*acos(-log(x/y)), force=True) == \ + acos(-log(x/y))*gamma(-log(x/y)) + + assert logcombine(2*log(z)*log(w)*log(x) + log(z) + log(w)) == \ + log(z**log(w**2))*log(x) + log(w*z) + assert logcombine(3*log(w) + 3*log(z)) == log(w**3*z**3) + assert logcombine(x*(y + 1) + log(2) + log(3)) == x*(y + 1) + log(6) + assert logcombine((x + y)*log(w) + (-x - y)*log(3)) == (x + y)*log(w/3) + # a single unknown can combine + assert logcombine(log(x) + log(2)) == log(2*x) + eq = log(abs(x)) + log(abs(y)) + assert logcombine(eq) == eq + reps = {x: 0, y: 0} + assert log(abs(x)*abs(y)).subs(reps) != eq.subs(reps) + + +def test_logcombine_complex_coeff(): + i = Integral((sin(x**2) + cos(x**3))/x, x) + assert logcombine(i, force=True) == i + assert logcombine(i + 2*log(x), force=True) == \ + i + log(x**2) + + +def test_issue_5950(): + x, y = symbols("x,y", positive=True) + assert logcombine(log(3) - log(2)) == log(Rational(3,2), evaluate=False) + assert logcombine(log(x) - log(y)) == log(x/y) + assert logcombine(log(Rational(3,2), evaluate=False) - log(2)) == \ + log(Rational(3,4), evaluate=False) + + +def test_posify(): + x = symbols('x') + + assert str(posify( + x + + Symbol('p', positive=True) + + Symbol('n', negative=True))) == '(_x + n + p, {_x: x})' + + eq, rep = posify(1/x) + assert log(eq).expand().subs(rep) == -log(x) + assert str(posify([x, 1 + x])) == '([_x, _x + 1], {_x: x})' + + p = symbols('p', positive=True) + n = symbols('n', negative=True) + orig = [x, n, p] + modified, reps = posify(orig) + assert str(modified) == '[_x, n, p]' + assert [w.subs(reps) for w in modified] == orig + + assert str(Integral(posify(1/x + y)[0], (y, 1, 3)).expand()) == \ + 'Integral(1/_x, (y, 1, 3)) + Integral(_y, (y, 1, 3))' + assert str(Sum(posify(1/x**n)[0], (n,1,3)).expand()) == \ + 'Sum(_x**(-n), (n, 1, 3))' + + A = Matrix([[1, 2, 3], [4, 5, 6 * Abs(x)]]) + Ap, rep = posify(A) + assert Ap == A.subs(*reversed(rep.popitem())) + + # issue 16438 + k = Symbol('k', finite=True) + eq, rep = posify(k) + assert eq.assumptions0 == {'positive': True, 'zero': False, 'imaginary': False, + 'nonpositive': False, 'commutative': True, 'hermitian': True, 'real': True, 'nonzero': True, + 'nonnegative': True, 'negative': False, 'complex': True, 'finite': True, + 'infinite': False, 'extended_real':True, 'extended_negative': False, + 'extended_nonnegative': True, 'extended_nonpositive': False, + 'extended_nonzero': True, 'extended_positive': True} + + +def test_issue_4194(): + # simplify should call cancel + f = Function('f') + assert simplify((4*x + 6*f(y))/(2*x + 3*f(y))) == 2 + + +@XFAIL +def test_simplify_float_vs_integer(): + # Test for issue 4473: + # https://github.com/sympy/sympy/issues/4473 + assert simplify(x**2.0 - x**2) == 0 + assert simplify(x**2 - x**2.0) == 0 + + +def test_as_content_primitive(): + assert (x/2 + y).as_content_primitive() == (S.Half, x + 2*y) + assert (x/2 + y).as_content_primitive(clear=False) == (S.One, x/2 + y) + assert (y*(x/2 + y)).as_content_primitive() == (S.Half, y*(x + 2*y)) + assert (y*(x/2 + y)).as_content_primitive(clear=False) == (S.One, y*(x/2 + y)) + + # although the _as_content_primitive methods do not alter the underlying structure, + # the as_content_primitive function will touch up the expression and join + # bases that would otherwise have not been joined. + assert (x*(2 + 2*x)*(3*x + 3)**2).as_content_primitive() == \ + (18, x*(x + 1)**3) + assert (2 + 2*x + 2*y*(3 + 3*y)).as_content_primitive() == \ + (2, x + 3*y*(y + 1) + 1) + assert ((2 + 6*x)**2).as_content_primitive() == \ + (4, (3*x + 1)**2) + assert ((2 + 6*x)**(2*y)).as_content_primitive() == \ + (1, (_keep_coeff(S(2), (3*x + 1)))**(2*y)) + assert (5 + 10*x + 2*y*(3 + 3*y)).as_content_primitive() == \ + (1, 10*x + 6*y*(y + 1) + 5) + assert (5*(x*(1 + y)) + 2*x*(3 + 3*y)).as_content_primitive() == \ + (11, x*(y + 1)) + assert ((5*(x*(1 + y)) + 2*x*(3 + 3*y))**2).as_content_primitive() == \ + (121, x**2*(y + 1)**2) + assert (y**2).as_content_primitive() == \ + (1, y**2) + assert (S.Infinity).as_content_primitive() == (1, oo) + eq = x**(2 + y) + assert (eq).as_content_primitive() == (1, eq) + assert (S.Half**(2 + x)).as_content_primitive() == (Rational(1, 4), 2**-x) + assert (Rational(-1, 2)**(2 + x)).as_content_primitive() == \ + (Rational(1, 4), (Rational(-1, 2))**x) + assert (Rational(-1, 2)**(2 + x)).as_content_primitive() == \ + (Rational(1, 4), Rational(-1, 2)**x) + assert (4**((1 + y)/2)).as_content_primitive() == (2, 4**(y/2)) + assert (3**((1 + y)/2)).as_content_primitive() == \ + (1, 3**(Mul(S.Half, 1 + y, evaluate=False))) + assert (5**Rational(3, 4)).as_content_primitive() == (1, 5**Rational(3, 4)) + assert (5**Rational(7, 4)).as_content_primitive() == (5, 5**Rational(3, 4)) + assert Add(z*Rational(5, 7), 0.5*x, y*Rational(3, 2), evaluate=False).as_content_primitive() == \ + (Rational(1, 14), 7.0*x + 21*y + 10*z) + assert (2**Rational(3, 4) + 2**Rational(1, 4)*sqrt(3)).as_content_primitive(radical=True) == \ + (1, 2**Rational(1, 4)*(sqrt(2) + sqrt(3))) + + +def test_signsimp(): + e = x*(-x + 1) + x*(x - 1) + assert signsimp(Eq(e, 0)) is S.true + assert Abs(x - 1) == Abs(1 - x) + assert signsimp(y - x) == y - x + assert signsimp(y - x, evaluate=False) == Mul(-1, x - y, evaluate=False) + + +def test_besselsimp(): + from sympy.functions.special.bessel import (besseli, besselj, bessely) + from sympy.integrals.transforms import cosine_transform + assert besselsimp(exp(-I*pi*y/2)*besseli(y, z*exp_polar(I*pi/2))) == \ + besselj(y, z) + assert besselsimp(exp(-I*pi*a/2)*besseli(a, 2*sqrt(x)*exp_polar(I*pi/2))) == \ + besselj(a, 2*sqrt(x)) + assert besselsimp(sqrt(2)*sqrt(pi)*x**Rational(1, 4)*exp(I*pi/4)*exp(-I*pi*a/2) * + besseli(Rational(-1, 2), sqrt(x)*exp_polar(I*pi/2)) * + besseli(a, sqrt(x)*exp_polar(I*pi/2))/2) == \ + besselj(a, sqrt(x)) * cos(sqrt(x)) + assert besselsimp(besseli(Rational(-1, 2), z)) == \ + sqrt(2)*cosh(z)/(sqrt(pi)*sqrt(z)) + assert besselsimp(besseli(a, z*exp_polar(-I*pi/2))) == \ + exp(-I*pi*a/2)*besselj(a, z) + assert cosine_transform(1/t*sin(a/t), t, y) == \ + sqrt(2)*sqrt(pi)*besselj(0, 2*sqrt(a)*sqrt(y))/2 + + assert besselsimp(x**2*(a*(-2*besselj(5*I, x) + besselj(-2 + 5*I, x) + + besselj(2 + 5*I, x)) + b*(-2*bessely(5*I, x) + bessely(-2 + 5*I, x) + + bessely(2 + 5*I, x)))/4 + x*(a*(besselj(-1 + 5*I, x)/2 - besselj(1 + 5*I, x)/2) + + b*(bessely(-1 + 5*I, x)/2 - bessely(1 + 5*I, x)/2)) + (x**2 + 25)*(a*besselj(5*I, x) + + b*bessely(5*I, x))) == 0 + + assert besselsimp(81*x**2*(a*(besselj(Rational(-5, 3), 9*x) - 2*besselj(Rational(1, 3), 9*x) + besselj(Rational(7, 3), 9*x)) + + b*(bessely(Rational(-5, 3), 9*x) - 2*bessely(Rational(1, 3), 9*x) + bessely(Rational(7, 3), 9*x)))/4 + x*(a*(9*besselj(Rational(-2, 3), 9*x)/2 + - 9*besselj(Rational(4, 3), 9*x)/2) + b*(9*bessely(Rational(-2, 3), 9*x)/2 - 9*bessely(Rational(4, 3), 9*x)/2)) + + (81*x**2 - Rational(1, 9))*(a*besselj(Rational(1, 3), 9*x) + b*bessely(Rational(1, 3), 9*x))) == 0 + + assert besselsimp(besselj(a-1,x) + besselj(a+1, x) - 2*a*besselj(a, x)/x) == 0 + + assert besselsimp(besselj(a-1,x) + besselj(a+1, x) + besselj(a, x)) == (2*a + x)*besselj(a, x)/x + + assert besselsimp(x**2* besselj(a,x) + x**3*besselj(a+1, x) + besselj(a+2, x)) == \ + 2*a*x*besselj(a + 1, x) + x**3*besselj(a + 1, x) - x**2*besselj(a + 2, x) + 2*x*besselj(a + 1, x) + besselj(a + 2, x) + +def test_Piecewise(): + e1 = x*(x + y) - y*(x + y) + e2 = sin(x)**2 + cos(x)**2 + e3 = expand((x + y)*y/x) + s1 = simplify(e1) + s2 = simplify(e2) + s3 = simplify(e3) + assert simplify(Piecewise((e1, x < e2), (e3, True))) == \ + Piecewise((s1, x < s2), (s3, True)) + + +def test_polymorphism(): + class A(Basic): + def _eval_simplify(x, **kwargs): + return S.One + + a = A(S(5), S(2)) + assert simplify(a) == 1 + + +def test_issue_from_PR1599(): + n1, n2, n3, n4 = symbols('n1 n2 n3 n4', negative=True) + assert simplify(I*sqrt(n1)) == -sqrt(-n1) + + +def test_issue_6811(): + eq = (x + 2*y)*(2*x + 2) + assert simplify(eq) == (x + 1)*(x + 2*y)*2 + # reject the 2-arg Mul -- these are a headache for test writing + assert simplify(eq.expand()) == \ + 2*x**2 + 4*x*y + 2*x + 4*y + + +def test_issue_6920(): + e = [cos(x) + I*sin(x), cos(x) - I*sin(x), + cosh(x) - sinh(x), cosh(x) + sinh(x)] + ok = [exp(I*x), exp(-I*x), exp(-x), exp(x)] + # wrap in f to show that the change happens wherever ei occurs + f = Function('f') + assert [simplify(f(ei)).args[0] for ei in e] == ok + + +def test_issue_7001(): + from sympy.abc import r, R + assert simplify(-(r*Piecewise((pi*Rational(4, 3), r <= R), + (-8*pi*R**3/(3*r**3), True)) + 2*Piecewise((pi*r*Rational(4, 3), r <= R), + (4*pi*R**3/(3*r**2), True)))/(4*pi*r)) == \ + Piecewise((-1, r <= R), (0, True)) + + +def test_inequality_no_auto_simplify(): + # no simplify on creation but can be simplified + lhs = cos(x)**2 + sin(x)**2 + rhs = 2 + e = Lt(lhs, rhs, evaluate=False) + assert e is not S.true + assert simplify(e) + + +def test_issue_9398(): + from sympy.core.numbers import Number + from sympy.polys.polytools import cancel + assert cancel(1e-14) != 0 + assert cancel(1e-14*I) != 0 + + assert simplify(1e-14) != 0 + assert simplify(1e-14*I) != 0 + + assert (I*Number(1.)*Number(10)**Number(-14)).simplify() != 0 + + assert cancel(1e-20) != 0 + assert cancel(1e-20*I) != 0 + + assert simplify(1e-20) != 0 + assert simplify(1e-20*I) != 0 + + assert cancel(1e-100) != 0 + assert cancel(1e-100*I) != 0 + + assert simplify(1e-100) != 0 + assert simplify(1e-100*I) != 0 + + f = Float("1e-1000") + assert cancel(f) != 0 + assert cancel(f*I) != 0 + + assert simplify(f) != 0 + assert simplify(f*I) != 0 + + +def test_issue_9324_simplify(): + M = MatrixSymbol('M', 10, 10) + e = M[0, 0] + M[5, 4] + 1304 + assert simplify(e) == e + + +def test_issue_9817_simplify(): + # simplify on trace of substituted explicit quadratic form of matrix + # expressions (a scalar) should return without errors (AttributeError) + # See issue #9817 and #9190 for the original bug more discussion on this + from sympy.matrices.expressions import Identity, trace + v = MatrixSymbol('v', 3, 1) + A = MatrixSymbol('A', 3, 3) + x = Matrix([i + 1 for i in range(3)]) + X = Identity(3) + quadratic = v.T * A * v + assert simplify((trace(quadratic.as_explicit())).xreplace({v:x, A:X})) == 14 + + +def test_issue_13474(): + x = Symbol('x') + assert simplify(x + csch(sinc(1))) == x + csch(sinc(1)) + + +@_both_exp_pow +def test_simplify_function_inverse(): + # "inverse" attribute does not guarantee that f(g(x)) is x + # so this simplification should not happen automatically. + # See issue #12140 + x, y = symbols('x, y') + g = Function('g') + + class f(Function): + def inverse(self, argindex=1): + return g + + assert simplify(f(g(x))) == f(g(x)) + assert inversecombine(f(g(x))) == x + assert simplify(f(g(x)), inverse=True) == x + assert simplify(f(g(sin(x)**2 + cos(x)**2)), inverse=True) == 1 + assert simplify(f(g(x, y)), inverse=True) == f(g(x, y)) + assert unchanged(asin, sin(x)) + assert simplify(asin(sin(x))) == asin(sin(x)) + assert simplify(2*asin(sin(3*x)), inverse=True) == 6*x + assert simplify(log(exp(x))) == log(exp(x)) + assert simplify(log(exp(x)), inverse=True) == x + assert simplify(exp(log(x)), inverse=True) == x + assert simplify(log(exp(x), 2), inverse=True) == x/log(2) + assert simplify(log(exp(x), 2, evaluate=False), inverse=True) == x/log(2) + + +def test_clear_coefficients(): + from sympy.simplify.simplify import clear_coefficients + assert clear_coefficients(4*y*(6*x + 3)) == (y*(2*x + 1), 0) + assert clear_coefficients(4*y*(6*x + 3) - 2) == (y*(2*x + 1), Rational(1, 6)) + assert clear_coefficients(4*y*(6*x + 3) - 2, x) == (y*(2*x + 1), x/12 + Rational(1, 6)) + assert clear_coefficients(sqrt(2) - 2) == (sqrt(2), 2) + assert clear_coefficients(4*sqrt(2) - 2) == (sqrt(2), S.Half) + assert clear_coefficients(S(3), x) == (0, x - 3) + assert clear_coefficients(S.Infinity, x) == (S.Infinity, x) + assert clear_coefficients(-S.Pi, x) == (S.Pi, -x) + assert clear_coefficients(2 - S.Pi/3, x) == (pi, -3*x + 6) + +def test_nc_simplify(): + from sympy.simplify.simplify import nc_simplify + from sympy.matrices.expressions import MatPow, Identity + from sympy.core import Pow + from functools import reduce + + a, b, c, d = symbols('a b c d', commutative = False) + x = Symbol('x') + A = MatrixSymbol("A", x, x) + B = MatrixSymbol("B", x, x) + C = MatrixSymbol("C", x, x) + D = MatrixSymbol("D", x, x) + subst = {a: A, b: B, c: C, d:D} + funcs = {Add: lambda x,y: x+y, Mul: lambda x,y: x*y } + + def _to_matrix(expr): + if expr in subst: + return subst[expr] + if isinstance(expr, Pow): + return MatPow(_to_matrix(expr.args[0]), expr.args[1]) + elif isinstance(expr, (Add, Mul)): + return reduce(funcs[expr.func],[_to_matrix(a) for a in expr.args]) + else: + return expr*Identity(x) + + def _check(expr, simplified, deep=True, matrix=True): + assert nc_simplify(expr, deep=deep) == simplified + assert expand(expr) == expand(simplified) + if matrix: + m_simp = _to_matrix(simplified).doit(inv_expand=False) + assert nc_simplify(_to_matrix(expr), deep=deep) == m_simp + + _check(a*b*a*b*a*b*c*(a*b)**3*c, ((a*b)**3*c)**2) + _check(a*b*(a*b)**-2*a*b, 1) + _check(a**2*b*a*b*a*b*(a*b)**-1, a*(a*b)**2, matrix=False) + _check(b*a*b**2*a*b**2*a*b**2, b*(a*b**2)**3) + _check(a*b*a**2*b*a**2*b*a**3, (a*b*a)**3*a**2) + _check(a**2*b*a**4*b*a**4*b*a**2, (a**2*b*a**2)**3) + _check(a**3*b*a**4*b*a**4*b*a, a**3*(b*a**4)**3*a**-3) + _check(a*b*a*b + a*b*c*x*a*b*c, (a*b)**2 + x*(a*b*c)**2) + _check(a*b*a*b*c*a*b*a*b*c, ((a*b)**2*c)**2) + _check(b**-1*a**-1*(a*b)**2, a*b) + _check(a**-1*b*c**-1, (c*b**-1*a)**-1) + expr = a**3*b*a**4*b*a**4*b*a**2*b*a**2*(b*a**2)**2*b*a**2*b*a**2 + for _ in range(10): + expr *= a*b + _check(expr, a**3*(b*a**4)**2*(b*a**2)**6*(a*b)**10) + _check((a*b*a*b)**2, (a*b*a*b)**2, deep=False) + _check(a*b*(c*d)**2, a*b*(c*d)**2) + expr = b**-1*(a**-1*b**-1 - a**-1*c*b**-1)**-1*a**-1 + assert nc_simplify(expr) == (1-c)**-1 + # commutative expressions should be returned without an error + assert nc_simplify(2*x**2) == 2*x**2 + +def test_issue_15965(): + A = Sum(z*x**y, (x, 1, a)) + anew = z*Sum(x**y, (x, 1, a)) + B = Integral(x*y, x) + bdo = x**2*y/2 + assert simplify(A + B) == anew + bdo + assert simplify(A) == anew + assert simplify(B) == bdo + assert simplify(B, doit=False) == y*Integral(x, x) + + +def test_issue_17137(): + assert simplify(cos(x)**I) == cos(x)**I + assert simplify(cos(x)**(2 + 3*I)) == cos(x)**(2 + 3*I) + + +def test_issue_21869(): + x = Symbol('x', real=True) + y = Symbol('y', real=True) + expr = And(Eq(x**2, 4), Le(x, y)) + assert expr.simplify() == expr + + expr = And(Eq(x**2, 4), Eq(x, 2)) + assert expr.simplify() == Eq(x, 2) + + expr = And(Eq(x**3, x**2), Eq(x, 1)) + assert expr.simplify() == Eq(x, 1) + + expr = And(Eq(sin(x), x**2), Eq(x, 0)) + assert expr.simplify() == Eq(x, 0) + + expr = And(Eq(x**3, x**2), Eq(x, 2)) + assert expr.simplify() == S.false + + expr = And(Eq(y, x**2), Eq(x, 1)) + assert expr.simplify() == And(Eq(y,1), Eq(x, 1)) + + expr = And(Eq(y**2, 1), Eq(y, x**2), Eq(x, 1)) + assert expr.simplify() == And(Eq(y,1), Eq(x, 1)) + + expr = And(Eq(y**2, 4), Eq(y, 2*x**2), Eq(x, 1)) + assert expr.simplify() == And(Eq(y,2), Eq(x, 1)) + + expr = And(Eq(y**2, 4), Eq(y, x**2), Eq(x, 1)) + assert expr.simplify() == S.false + + +def test_issue_7971_21740(): + z = Integral(x, (x, 1, 1)) + assert z != 0 + assert simplify(z) is S.Zero + assert simplify(S.Zero) is S.Zero + z = simplify(Float(0)) + assert z is not S.Zero and z == 0.0 + + +@slow +def test_issue_17141_slow(): + # Should not give RecursionError + assert simplify((2**acos(I+1)**2).rewrite('log')) == 2**((pi + 2*I*log(-1 + + sqrt(1 - 2*I) + I))**2/4) + + +def test_issue_17141(): + # Check that there is no RecursionError + assert simplify(x**(1 / acos(I))) == x**(2/(pi - 2*I*log(1 + sqrt(2)))) + assert simplify(acos(-I)**2*acos(I)**2) == \ + log(1 + sqrt(2))**4 + pi**2*log(1 + sqrt(2))**2/2 + pi**4/16 + assert simplify(2**acos(I)**2) == 2**((pi - 2*I*log(1 + sqrt(2)))**2/4) + p = 2**acos(I+1)**2 + assert simplify(p) == p + + +def test_simplify_kroneckerdelta(): + i, j = symbols("i j") + K = KroneckerDelta + + assert simplify(K(i, j)) == K(i, j) + assert simplify(K(0, j)) == K(0, j) + assert simplify(K(i, 0)) == K(i, 0) + + assert simplify(K(0, j).rewrite(Piecewise) * K(1, j)) == 0 + assert simplify(K(1, i) + Piecewise((1, Eq(j, 2)), (0, True))) == K(1, i) + K(2, j) + + # issue 17214 + assert simplify(K(0, j) * K(1, j)) == 0 + + n = Symbol('n', integer=True) + assert simplify(K(0, n) * K(1, n)) == 0 + + M = Matrix(4, 4, lambda i, j: K(j - i, n) if i <= j else 0) + assert simplify(M**2) == Matrix([[K(0, n), 0, K(1, n), 0], + [0, K(0, n), 0, K(1, n)], + [0, 0, K(0, n), 0], + [0, 0, 0, K(0, n)]]) + assert simplify(eye(1) * KroneckerDelta(0, n) * + KroneckerDelta(1, n)) == Matrix([[0]]) + + assert simplify(S.Infinity * KroneckerDelta(0, n) * + KroneckerDelta(1, n)) is S.NaN + + +def test_issue_17292(): + assert simplify(abs(x)/abs(x**2)) == 1/abs(x) + # this is bigger than the issue: check that deep processing works + assert simplify(5*abs((x**2 - 1)/(x - 1))) == 5*Abs(x + 1) + + +def test_issue_19822(): + expr = And(Gt(n-2, 1), Gt(n, 1)) + assert simplify(expr) == Gt(n, 3) + + +def test_issue_18645(): + expr = And(Ge(x, 3), Le(x, 3)) + assert simplify(expr) == Eq(x, 3) + expr = And(Eq(x, 3), Le(x, 3)) + assert simplify(expr) == Eq(x, 3) + + +@XFAIL +def test_issue_18642(): + i = Symbol("i", integer=True) + n = Symbol("n", integer=True) + expr = And(Eq(i, 2 * n), Le(i, 2*n -1)) + assert simplify(expr) == S.false + + +@XFAIL +def test_issue_18389(): + n = Symbol("n", integer=True) + expr = Eq(n, 0) | (n >= 1) + assert simplify(expr) == Ge(n, 0) + + +def test_issue_8373(): + x = Symbol('x', real=True) + assert simplify(Or(x < 1, x >= 1)) == S.true + + +def test_issue_7950(): + expr = And(Eq(x, 1), Eq(x, 2)) + assert simplify(expr) == S.false + + +def test_issue_22020(): + expr = I*pi/2 -oo + assert simplify(expr) == expr + # Used to throw an error + + +def test_issue_19484(): + assert simplify(sign(x) * Abs(x)) == x + + e = x + sign(x + x**3) + assert simplify(Abs(x + x**3)*e) == x**3 + x*Abs(x**3 + x) + x + + e = x**2 + sign(x**3 + 1) + assert simplify(Abs(x**3 + 1) * e) == x**3 + x**2*Abs(x**3 + 1) + 1 + + f = Function('f') + e = x + sign(x + f(x)**3) + assert simplify(Abs(x + f(x)**3) * e) == x*Abs(x + f(x)**3) + x + f(x)**3 + + +def test_issue_23543(): + # Used to give an error + x, y, z = symbols("x y z", commutative=False) + assert (x*(y + z/2)).simplify() == x*(2*y + z)/2 + + +def test_issue_11004(): + + def f(n): + return sqrt(2*pi*n) * (n/E)**n + + def m(n, k): + return f(n) / (f(n/k)**k) + + def p(n,k): + return m(n, k) / (k**n) + + N, k = symbols('N k') + half = Float('0.5', 4) + z = log(p(n, k) / p(n, k + 1)).expand(force=True) + r = simplify(z.subs(n, N).n(4)) + assert r == ( + half*k*log(k) + - half*k*log(k + 1) + + half*log(N) + - half*log(k + 1) + + Float(0.9189224, 4) + ) + + +def test_issue_19161(): + polynomial = Poly('x**2').simplify() + assert (polynomial-x**2).simplify() == 0 + + +def test_issue_22210(): + d = Symbol('d', integer=True) + expr = 2*Derivative(sin(x), (x, d)) + assert expr.simplify() == expr + + +def test_reduce_inverses_nc_pow(): + x, y = symbols("x y", commutative=True) + Z = symbols("Z", commutative=False) + assert simplify(2**Z * y**Z) == 2**Z * y**Z + assert simplify(x**Z * y**Z) == x**Z * y**Z + x, y = symbols("x y", positive=True) + assert expand((x*y)**Z) == x**Z * y**Z + assert simplify(x**Z * y**Z) == expand((x*y)**Z) + +def test_nc_recursion_coeff(): + X = symbols("X", commutative = False) + assert (2 * cos(pi/3) * X).simplify() == X + assert (2.0 * cos(pi/3) * X).simplify() == X diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_sqrtdenest.py b/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_sqrtdenest.py new file mode 100644 index 0000000000000000000000000000000000000000..41c771bb2055a1199d349ae3649f33927d79313a --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_sqrtdenest.py @@ -0,0 +1,204 @@ +from sympy.core.mul import Mul +from sympy.core.numbers import (I, Integer, Rational) +from sympy.core.symbol import Symbol +from sympy.functions.elementary.miscellaneous import (root, sqrt) +from sympy.functions.elementary.trigonometric import cos +from sympy.integrals.integrals import Integral +from sympy.simplify.sqrtdenest import sqrtdenest +from sympy.simplify.sqrtdenest import ( + _subsets as subsets, _sqrt_numeric_denest) + +r2, r3, r5, r6, r7, r10, r15, r29 = [sqrt(x) for x in (2, 3, 5, 6, 7, 10, + 15, 29)] + + +def test_sqrtdenest(): + d = {sqrt(5 + 2 * r6): r2 + r3, + sqrt(5. + 2 * r6): sqrt(5. + 2 * r6), + sqrt(5. + 4*sqrt(5 + 2 * r6)): sqrt(5.0 + 4*r2 + 4*r3), + sqrt(r2): sqrt(r2), + sqrt(5 + r7): sqrt(5 + r7), + sqrt(3 + sqrt(5 + 2*r7)): + 3*r2*(5 + 2*r7)**Rational(1, 4)/(2*sqrt(6 + 3*r7)) + + r2*sqrt(6 + 3*r7)/(2*(5 + 2*r7)**Rational(1, 4)), + sqrt(3 + 2*r3): 3**Rational(3, 4)*(r6/2 + 3*r2/2)/3} + for i in d: + assert sqrtdenest(i) == d[i], i + + +def test_sqrtdenest2(): + assert sqrtdenest(sqrt(16 - 2*r29 + 2*sqrt(55 - 10*r29))) == \ + r5 + sqrt(11 - 2*r29) + e = sqrt(-r5 + sqrt(-2*r29 + 2*sqrt(-10*r29 + 55) + 16)) + assert sqrtdenest(e) == root(-2*r29 + 11, 4) + r = sqrt(1 + r7) + assert sqrtdenest(sqrt(1 + r)) == sqrt(1 + r) + e = sqrt(((1 + sqrt(1 + 2*sqrt(3 + r2 + r5)))**2).expand()) + assert sqrtdenest(e) == 1 + sqrt(1 + 2*sqrt(r2 + r5 + 3)) + + assert sqrtdenest(sqrt(5*r3 + 6*r2)) == \ + sqrt(2)*root(3, 4) + root(3, 4)**3 + + assert sqrtdenest(sqrt(((1 + r5 + sqrt(1 + r3))**2).expand())) == \ + 1 + r5 + sqrt(1 + r3) + + assert sqrtdenest(sqrt(((1 + r5 + r7 + sqrt(1 + r3))**2).expand())) == \ + 1 + sqrt(1 + r3) + r5 + r7 + + e = sqrt(((1 + cos(2) + cos(3) + sqrt(1 + r3))**2).expand()) + assert sqrtdenest(e) == cos(3) + cos(2) + 1 + sqrt(1 + r3) + + e = sqrt(-2*r10 + 2*r2*sqrt(-2*r10 + 11) + 14) + assert sqrtdenest(e) == sqrt(-2*r10 - 2*r2 + 4*r5 + 14) + + # check that the result is not more complicated than the input + z = sqrt(-2*r29 + cos(2) + 2*sqrt(-10*r29 + 55) + 16) + assert sqrtdenest(z) == z + + assert sqrtdenest(sqrt(r6 + sqrt(15))) == sqrt(r6 + sqrt(15)) + + z = sqrt(15 - 2*sqrt(31) + 2*sqrt(55 - 10*r29)) + assert sqrtdenest(z) == z + + +def test_sqrtdenest_rec(): + assert sqrtdenest(sqrt(-4*sqrt(14) - 2*r6 + 4*sqrt(21) + 33)) == \ + -r2 + r3 + 2*r7 + assert sqrtdenest(sqrt(-28*r7 - 14*r5 + 4*sqrt(35) + 82)) == \ + -7 + r5 + 2*r7 + assert sqrtdenest(sqrt(6*r2/11 + 2*sqrt(22)/11 + 6*sqrt(11)/11 + 2)) == \ + sqrt(11)*(r2 + 3 + sqrt(11))/11 + assert sqrtdenest(sqrt(468*r3 + 3024*r2 + 2912*r6 + 19735)) == \ + 9*r3 + 26 + 56*r6 + z = sqrt(-490*r3 - 98*sqrt(115) - 98*sqrt(345) - 2107) + assert sqrtdenest(z) == sqrt(-1)*(7*r5 + 7*r15 + 7*sqrt(23)) + z = sqrt(-4*sqrt(14) - 2*r6 + 4*sqrt(21) + 34) + assert sqrtdenest(z) == z + assert sqrtdenest(sqrt(-8*r2 - 2*r5 + 18)) == -r10 + 1 + r2 + r5 + assert sqrtdenest(sqrt(8*r2 + 2*r5 - 18)) == \ + sqrt(-1)*(-r10 + 1 + r2 + r5) + assert sqrtdenest(sqrt(8*r2/3 + 14*r5/3 + Rational(154, 9))) == \ + -r10/3 + r2 + r5 + 3 + assert sqrtdenest(sqrt(sqrt(2*r6 + 5) + sqrt(2*r7 + 8))) == \ + sqrt(1 + r2 + r3 + r7) + assert sqrtdenest(sqrt(4*r15 + 8*r5 + 12*r3 + 24)) == 1 + r3 + r5 + r15 + + w = 1 + r2 + r3 + r5 + r7 + assert sqrtdenest(sqrt((w**2).expand())) == w + z = sqrt((w**2).expand() + 1) + assert sqrtdenest(z) == z + + z = sqrt(2*r10 + 6*r2 + 4*r5 + 12 + 10*r15 + 30*r3) + assert sqrtdenest(z) == z + + +def test_issue_6241(): + z = sqrt( -320 + 32*sqrt(5) + 64*r15) + assert sqrtdenest(z) == z + + +def test_sqrtdenest3(): + z = sqrt(13 - 2*r10 + 2*r2*sqrt(-2*r10 + 11)) + assert sqrtdenest(z) == -1 + r2 + r10 + assert sqrtdenest(z, max_iter=1) == -1 + sqrt(2) + sqrt(10) + z = sqrt(sqrt(r2 + 2) + 2) + assert sqrtdenest(z) == z + assert sqrtdenest(sqrt(-2*r10 + 4*r2*sqrt(-2*r10 + 11) + 20)) == \ + sqrt(-2*r10 - 4*r2 + 8*r5 + 20) + assert sqrtdenest(sqrt((112 + 70*r2) + (46 + 34*r2)*r5)) == \ + r10 + 5 + 4*r2 + 3*r5 + z = sqrt(5 + sqrt(2*r6 + 5)*sqrt(-2*r29 + 2*sqrt(-10*r29 + 55) + 16)) + r = sqrt(-2*r29 + 11) + assert sqrtdenest(z) == sqrt(r2*r + r3*r + r10 + r15 + 5) + + n = sqrt(2*r6/7 + 2*r7/7 + 2*sqrt(42)/7 + 2) + d = sqrt(16 - 2*r29 + 2*sqrt(55 - 10*r29)) + assert sqrtdenest(n/d) == r7*(1 + r6 + r7)/(Mul(7, (sqrt(-2*r29 + 11) + r5), + evaluate=False)) + + +def test_sqrtdenest4(): + # see Denest_en.pdf in https://github.com/sympy/sympy/issues/3192 + z = sqrt(8 - r2*sqrt(5 - r5) - sqrt(3)*(1 + r5)) + z1 = sqrtdenest(z) + c = sqrt(-r5 + 5) + z1 = ((-r15*c - r3*c + c + r5*c - r6 - r2 + r10 + sqrt(30))/4).expand() + assert sqrtdenest(z) == z1 + + z = sqrt(2*r2*sqrt(r2 + 2) + 5*r2 + 4*sqrt(r2 + 2) + 8) + assert sqrtdenest(z) == r2 + sqrt(r2 + 2) + 2 + + w = 2 + r2 + r3 + (1 + r3)*sqrt(2 + r2 + 5*r3) + z = sqrt((w**2).expand()) + assert sqrtdenest(z) == w.expand() + + +def test_sqrt_symbolic_denest(): + x = Symbol('x') + z = sqrt(((1 + sqrt(sqrt(2 + x) + 3))**2).expand()) + assert sqrtdenest(z) == sqrt((1 + sqrt(sqrt(2 + x) + 3))**2) + z = sqrt(((1 + sqrt(sqrt(2 + cos(1)) + 3))**2).expand()) + assert sqrtdenest(z) == 1 + sqrt(sqrt(2 + cos(1)) + 3) + z = ((1 + cos(2))**4 + 1).expand() + assert sqrtdenest(z) == z + z = sqrt(((1 + sqrt(sqrt(2 + cos(3*x)) + 3))**2 + 1).expand()) + assert sqrtdenest(z) == z + c = cos(3) + c2 = c**2 + assert sqrtdenest(sqrt(2*sqrt(1 + r3)*c + c2 + 1 + r3*c2)) == \ + -1 - sqrt(1 + r3)*c + ra = sqrt(1 + r3) + z = sqrt(20*ra*sqrt(3 + 3*r3) + 12*r3*ra*sqrt(3 + 3*r3) + 64*r3 + 112) + assert sqrtdenest(z) == z + + +def test_issue_5857(): + from sympy.abc import x, y + z = sqrt(1/(4*r3 + 7) + 1) + ans = (r2 + r6)/(r3 + 2) + assert sqrtdenest(z) == ans + assert sqrtdenest(1 + z) == 1 + ans + assert sqrtdenest(Integral(z + 1, (x, 1, 2))) == \ + Integral(1 + ans, (x, 1, 2)) + assert sqrtdenest(x + sqrt(y)) == x + sqrt(y) + ans = (r2 + r6)/(r3 + 2) + assert sqrtdenest(z) == ans + assert sqrtdenest(1 + z) == 1 + ans + assert sqrtdenest(Integral(z + 1, (x, 1, 2))) == \ + Integral(1 + ans, (x, 1, 2)) + assert sqrtdenest(x + sqrt(y)) == x + sqrt(y) + + +def test_subsets(): + assert subsets(1) == [[1]] + assert subsets(4) == [ + [1, 0, 0, 0], [0, 1, 0, 0], [1, 1, 0, 0], [0, 0, 1, 0], [1, 0, 1, 0], + [0, 1, 1, 0], [1, 1, 1, 0], [0, 0, 0, 1], [1, 0, 0, 1], [0, 1, 0, 1], + [1, 1, 0, 1], [0, 0, 1, 1], [1, 0, 1, 1], [0, 1, 1, 1], [1, 1, 1, 1]] + + +def test_issue_5653(): + assert sqrtdenest( + sqrt(2 + sqrt(2 + sqrt(2)))) == sqrt(2 + sqrt(2 + sqrt(2))) + +def test_issue_12420(): + assert sqrtdenest((3 - sqrt(2)*sqrt(4 + 3*I) + 3*I)/2) == I + e = 3 - sqrt(2)*sqrt(4 + I) + 3*I + assert sqrtdenest(e) == e + +def test_sqrt_ratcomb(): + assert sqrtdenest(sqrt(1 + r3) + sqrt(3 + 3*r3) - sqrt(10 + 6*r3)) == 0 + +def test_issue_18041(): + e = -sqrt(-2 + 2*sqrt(3)*I) + assert sqrtdenest(e) == -1 - sqrt(3)*I + +def test_issue_19914(): + a = Integer(-8) + b = Integer(-1) + r = Integer(63) + d2 = a*a - b*b*r + + assert _sqrt_numeric_denest(a, b, r, d2) == \ + sqrt(14)*I/2 + 3*sqrt(2)*I/2 + assert sqrtdenest(sqrt(-8-sqrt(63))) == sqrt(14)*I/2 + 3*sqrt(2)*I/2 diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_trigsimp.py b/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_trigsimp.py new file mode 100644 index 0000000000000000000000000000000000000000..ea091ec8a6c7d654405968e3d035c2bbe02ccdf7 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/simplify/tests/test_trigsimp.py @@ -0,0 +1,520 @@ +from itertools import product +from sympy.core.function import (Subs, count_ops, diff, expand) +from sympy.core.numbers import (E, I, Rational, pi) +from sympy.core.singleton import S +from sympy.core.symbol import (Symbol, symbols) +from sympy.functions.elementary.exponential import (exp, log) +from sympy.functions.elementary.hyperbolic import (cosh, coth, sinh, tanh) +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.elementary.trigonometric import (cos, cot, sin, tan) +from sympy.functions.elementary.trigonometric import (acos, asin, atan2) +from sympy.functions.elementary.trigonometric import (asec, acsc) +from sympy.functions.elementary.trigonometric import (acot, atan) +from sympy.integrals.integrals import integrate +from sympy.matrices.dense import Matrix +from sympy.simplify.simplify import simplify +from sympy.simplify.trigsimp import (exptrigsimp, trigsimp) + +from sympy.testing.pytest import XFAIL + +from sympy.abc import x, y + + + +def test_trigsimp1(): + x, y = symbols('x,y') + + assert trigsimp(1 - sin(x)**2) == cos(x)**2 + assert trigsimp(1 - cos(x)**2) == sin(x)**2 + assert trigsimp(sin(x)**2 + cos(x)**2) == 1 + assert trigsimp(1 + tan(x)**2) == 1/cos(x)**2 + assert trigsimp(1/cos(x)**2 - 1) == tan(x)**2 + assert trigsimp(1/cos(x)**2 - tan(x)**2) == 1 + assert trigsimp(1 + cot(x)**2) == 1/sin(x)**2 + assert trigsimp(1/sin(x)**2 - 1) == 1/tan(x)**2 + assert trigsimp(1/sin(x)**2 - cot(x)**2) == 1 + + assert trigsimp(5*cos(x)**2 + 5*sin(x)**2) == 5 + assert trigsimp(5*cos(x/2)**2 + 2*sin(x/2)**2) == 3*cos(x)/2 + Rational(7, 2) + + assert trigsimp(sin(x)/cos(x)) == tan(x) + assert trigsimp(2*tan(x)*cos(x)) == 2*sin(x) + assert trigsimp(cot(x)**3*sin(x)**3) == cos(x)**3 + assert trigsimp(y*tan(x)**2/sin(x)**2) == y/cos(x)**2 + assert trigsimp(cot(x)/cos(x)) == 1/sin(x) + + assert trigsimp(sin(x + y) + sin(x - y)) == 2*sin(x)*cos(y) + assert trigsimp(sin(x + y) - sin(x - y)) == 2*sin(y)*cos(x) + assert trigsimp(cos(x + y) + cos(x - y)) == 2*cos(x)*cos(y) + assert trigsimp(cos(x + y) - cos(x - y)) == -2*sin(x)*sin(y) + assert trigsimp(tan(x + y) - tan(x)/(1 - tan(x)*tan(y))) == \ + sin(y)/(-sin(y)*tan(x) + cos(y)) # -tan(y)/(tan(x)*tan(y) - 1) + + assert trigsimp(sinh(x + y) + sinh(x - y)) == 2*sinh(x)*cosh(y) + assert trigsimp(sinh(x + y) - sinh(x - y)) == 2*sinh(y)*cosh(x) + assert trigsimp(cosh(x + y) + cosh(x - y)) == 2*cosh(x)*cosh(y) + assert trigsimp(cosh(x + y) - cosh(x - y)) == 2*sinh(x)*sinh(y) + assert trigsimp(tanh(x + y) - tanh(x)/(1 + tanh(x)*tanh(y))) == \ + sinh(y)/(sinh(y)*tanh(x) + cosh(y)) + + assert trigsimp(cos(0.12345)**2 + sin(0.12345)**2) == 1.0 + e = 2*sin(x)**2 + 2*cos(x)**2 + assert trigsimp(log(e)) == log(2) + + +def test_trigsimp1a(): + assert trigsimp(sin(2)**2*cos(3)*exp(2)/cos(2)**2) == tan(2)**2*cos(3)*exp(2) + assert trigsimp(tan(2)**2*cos(3)*exp(2)*cos(2)**2) == sin(2)**2*cos(3)*exp(2) + assert trigsimp(cot(2)*cos(3)*exp(2)*sin(2)) == cos(3)*exp(2)*cos(2) + assert trigsimp(tan(2)*cos(3)*exp(2)/sin(2)) == cos(3)*exp(2)/cos(2) + assert trigsimp(cot(2)*cos(3)*exp(2)/cos(2)) == cos(3)*exp(2)/sin(2) + assert trigsimp(cot(2)*cos(3)*exp(2)*tan(2)) == cos(3)*exp(2) + assert trigsimp(sinh(2)*cos(3)*exp(2)/cosh(2)) == tanh(2)*cos(3)*exp(2) + assert trigsimp(tanh(2)*cos(3)*exp(2)*cosh(2)) == sinh(2)*cos(3)*exp(2) + assert trigsimp(coth(2)*cos(3)*exp(2)*sinh(2)) == cosh(2)*cos(3)*exp(2) + assert trigsimp(tanh(2)*cos(3)*exp(2)/sinh(2)) == cos(3)*exp(2)/cosh(2) + assert trigsimp(coth(2)*cos(3)*exp(2)/cosh(2)) == cos(3)*exp(2)/sinh(2) + assert trigsimp(coth(2)*cos(3)*exp(2)*tanh(2)) == cos(3)*exp(2) + + +def test_trigsimp2(): + x, y = symbols('x,y') + assert trigsimp(cos(x)**2*sin(y)**2 + cos(x)**2*cos(y)**2 + sin(x)**2, + recursive=True) == 1 + assert trigsimp(sin(x)**2*sin(y)**2 + sin(x)**2*cos(y)**2 + cos(x)**2, + recursive=True) == 1 + assert trigsimp( + Subs(x, x, sin(y)**2 + cos(y)**2)) == Subs(x, x, 1) + + +def test_issue_4373(): + x = Symbol("x") + assert abs(trigsimp(2.0*sin(x)**2 + 2.0*cos(x)**2) - 2.0) < 1e-10 + + +def test_trigsimp3(): + x, y = symbols('x,y') + assert trigsimp(sin(x)/cos(x)) == tan(x) + assert trigsimp(sin(x)**2/cos(x)**2) == tan(x)**2 + assert trigsimp(sin(x)**3/cos(x)**3) == tan(x)**3 + assert trigsimp(sin(x)**10/cos(x)**10) == tan(x)**10 + + assert trigsimp(cos(x)/sin(x)) == 1/tan(x) + assert trigsimp(cos(x)**2/sin(x)**2) == 1/tan(x)**2 + assert trigsimp(cos(x)**10/sin(x)**10) == 1/tan(x)**10 + + assert trigsimp(tan(x)) == trigsimp(sin(x)/cos(x)) + + +def test_issue_4661(): + a, x, y = symbols('a x y') + eq = -4*sin(x)**4 + 4*cos(x)**4 - 8*cos(x)**2 + assert trigsimp(eq) == -4 + n = sin(x)**6 + 4*sin(x)**4*cos(x)**2 + 5*sin(x)**2*cos(x)**4 + 2*cos(x)**6 + d = -sin(x)**2 - 2*cos(x)**2 + assert simplify(n/d) == -1 + assert trigsimp(-2*cos(x)**2 + cos(x)**4 - sin(x)**4) == -1 + eq = (- sin(x)**3/4)*cos(x) + (cos(x)**3/4)*sin(x) - sin(2*x)*cos(2*x)/8 + assert trigsimp(eq) == 0 + + +def test_issue_4494(): + a, b = symbols('a b') + eq = sin(a)**2*sin(b)**2 + cos(a)**2*cos(b)**2*tan(a)**2 + cos(a)**2 + assert trigsimp(eq) == 1 + + +def test_issue_5948(): + a, x, y = symbols('a x y') + assert trigsimp(diff(integrate(cos(x)/sin(x)**7, x), x)) == \ + cos(x)/sin(x)**7 + + +def test_issue_4775(): + a, x, y = symbols('a x y') + assert trigsimp(sin(x)*cos(y)+cos(x)*sin(y)) == sin(x + y) + assert trigsimp(sin(x)*cos(y)+cos(x)*sin(y)+3) == sin(x + y) + 3 + + +def test_issue_4280(): + a, x, y = symbols('a x y') + assert trigsimp(cos(x)**2 + cos(y)**2*sin(x)**2 + sin(y)**2*sin(x)**2) == 1 + assert trigsimp(a**2*sin(x)**2 + a**2*cos(y)**2*cos(x)**2 + a**2*cos(x)**2*sin(y)**2) == a**2 + assert trigsimp(a**2*cos(y)**2*sin(x)**2 + a**2*sin(y)**2*sin(x)**2) == a**2*sin(x)**2 + + +def test_issue_3210(): + eqs = (sin(2)*cos(3) + sin(3)*cos(2), + -sin(2)*sin(3) + cos(2)*cos(3), + sin(2)*cos(3) - sin(3)*cos(2), + sin(2)*sin(3) + cos(2)*cos(3), + sin(2)*sin(3) + cos(2)*cos(3) + cos(2), + sinh(2)*cosh(3) + sinh(3)*cosh(2), + sinh(2)*sinh(3) + cosh(2)*cosh(3), + ) + assert [trigsimp(e) for e in eqs] == [ + sin(5), + cos(5), + -sin(1), + cos(1), + cos(1) + cos(2), + sinh(5), + cosh(5), + ] + + +def test_trigsimp_issues(): + a, x, y = symbols('a x y') + + # issue 4625 - factor_terms works, too + assert trigsimp(sin(x)**3 + cos(x)**2*sin(x)) == sin(x) + + # issue 5948 + assert trigsimp(diff(integrate(cos(x)/sin(x)**3, x), x)) == \ + cos(x)/sin(x)**3 + assert trigsimp(diff(integrate(sin(x)/cos(x)**3, x), x)) == \ + sin(x)/cos(x)**3 + + # check integer exponents + e = sin(x)**y/cos(x)**y + assert trigsimp(e) == e + assert trigsimp(e.subs(y, 2)) == tan(x)**2 + assert trigsimp(e.subs(x, 1)) == tan(1)**y + + # check for multiple patterns + assert (cos(x)**2/sin(x)**2*cos(y)**2/sin(y)**2).trigsimp() == \ + 1/tan(x)**2/tan(y)**2 + assert trigsimp(cos(x)/sin(x)*cos(x+y)/sin(x+y)) == \ + 1/(tan(x)*tan(x + y)) + + eq = cos(2)*(cos(3) + 1)**2/(cos(3) - 1)**2 + assert trigsimp(eq) == eq.factor() # factor makes denom (-1 + cos(3))**2 + assert trigsimp(cos(2)*(cos(3) + 1)**2*(cos(3) - 1)**2) == \ + cos(2)*sin(3)**4 + + # issue 6789; this generates an expression that formerly caused + # trigsimp to hang + assert cot(x).equals(tan(x)) is False + + # nan or the unchanged expression is ok, but not sin(1) + z = cos(x)**2 + sin(x)**2 - 1 + z1 = tan(x)**2 - 1/cot(x)**2 + n = (1 + z1/z) + assert trigsimp(sin(n)) != sin(1) + eq = x*(n - 1) - x*n + assert trigsimp(eq) is S.NaN + assert trigsimp(eq, recursive=True) is S.NaN + assert trigsimp(1).is_Integer + + assert trigsimp(-sin(x)**4 - 2*sin(x)**2*cos(x)**2 - cos(x)**4) == -1 + + +def test_trigsimp_issue_2515(): + x = Symbol('x') + assert trigsimp(x*cos(x)*tan(x)) == x*sin(x) + assert trigsimp(-sin(x) + cos(x)*tan(x)) == 0 + + +def test_trigsimp_issue_3826(): + assert trigsimp(tan(2*x).expand(trig=True)) == tan(2*x) + + +def test_trigsimp_issue_4032(): + n = Symbol('n', integer=True, positive=True) + assert trigsimp(2**(n/2)*cos(pi*n/4)/2 + 2**(n - 1)/2) == \ + 2**(n/2)*cos(pi*n/4)/2 + 2**n/4 + + +def test_trigsimp_issue_7761(): + assert trigsimp(cosh(pi/4)) == cosh(pi/4) + + +def test_trigsimp_noncommutative(): + x, y = symbols('x,y') + A, B = symbols('A,B', commutative=False) + + assert trigsimp(A - A*sin(x)**2) == A*cos(x)**2 + assert trigsimp(A - A*cos(x)**2) == A*sin(x)**2 + assert trigsimp(A*sin(x)**2 + A*cos(x)**2) == A + assert trigsimp(A + A*tan(x)**2) == A/cos(x)**2 + assert trigsimp(A/cos(x)**2 - A) == A*tan(x)**2 + assert trigsimp(A/cos(x)**2 - A*tan(x)**2) == A + assert trigsimp(A + A*cot(x)**2) == A/sin(x)**2 + assert trigsimp(A/sin(x)**2 - A) == A/tan(x)**2 + assert trigsimp(A/sin(x)**2 - A*cot(x)**2) == A + + assert trigsimp(y*A*cos(x)**2 + y*A*sin(x)**2) == y*A + + assert trigsimp(A*sin(x)/cos(x)) == A*tan(x) + assert trigsimp(A*tan(x)*cos(x)) == A*sin(x) + assert trigsimp(A*cot(x)**3*sin(x)**3) == A*cos(x)**3 + assert trigsimp(y*A*tan(x)**2/sin(x)**2) == y*A/cos(x)**2 + assert trigsimp(A*cot(x)/cos(x)) == A/sin(x) + + assert trigsimp(A*sin(x + y) + A*sin(x - y)) == 2*A*sin(x)*cos(y) + assert trigsimp(A*sin(x + y) - A*sin(x - y)) == 2*A*sin(y)*cos(x) + assert trigsimp(A*cos(x + y) + A*cos(x - y)) == 2*A*cos(x)*cos(y) + assert trigsimp(A*cos(x + y) - A*cos(x - y)) == -2*A*sin(x)*sin(y) + + assert trigsimp(A*sinh(x + y) + A*sinh(x - y)) == 2*A*sinh(x)*cosh(y) + assert trigsimp(A*sinh(x + y) - A*sinh(x - y)) == 2*A*sinh(y)*cosh(x) + assert trigsimp(A*cosh(x + y) + A*cosh(x - y)) == 2*A*cosh(x)*cosh(y) + assert trigsimp(A*cosh(x + y) - A*cosh(x - y)) == 2*A*sinh(x)*sinh(y) + + assert trigsimp(A*cos(0.12345)**2 + A*sin(0.12345)**2) == 1.0*A + + +def test_hyperbolic_simp(): + x, y = symbols('x,y') + + assert trigsimp(sinh(x)**2 + 1) == cosh(x)**2 + assert trigsimp(cosh(x)**2 - 1) == sinh(x)**2 + assert trigsimp(cosh(x)**2 - sinh(x)**2) == 1 + assert trigsimp(1 - tanh(x)**2) == 1/cosh(x)**2 + assert trigsimp(1 - 1/cosh(x)**2) == tanh(x)**2 + assert trigsimp(tanh(x)**2 + 1/cosh(x)**2) == 1 + assert trigsimp(coth(x)**2 - 1) == 1/sinh(x)**2 + assert trigsimp(1/sinh(x)**2 + 1) == 1/tanh(x)**2 + assert trigsimp(coth(x)**2 - 1/sinh(x)**2) == 1 + + assert trigsimp(5*cosh(x)**2 - 5*sinh(x)**2) == 5 + assert trigsimp(5*cosh(x/2)**2 - 2*sinh(x/2)**2) == 3*cosh(x)/2 + Rational(7, 2) + + assert trigsimp(sinh(x)/cosh(x)) == tanh(x) + assert trigsimp(tanh(x)) == trigsimp(sinh(x)/cosh(x)) + assert trigsimp(cosh(x)/sinh(x)) == 1/tanh(x) + assert trigsimp(2*tanh(x)*cosh(x)) == 2*sinh(x) + assert trigsimp(coth(x)**3*sinh(x)**3) == cosh(x)**3 + assert trigsimp(y*tanh(x)**2/sinh(x)**2) == y/cosh(x)**2 + assert trigsimp(coth(x)/cosh(x)) == 1/sinh(x) + + for a in (pi/6*I, pi/4*I, pi/3*I): + assert trigsimp(sinh(a)*cosh(x) + cosh(a)*sinh(x)) == sinh(x + a) + assert trigsimp(-sinh(a)*cosh(x) + cosh(a)*sinh(x)) == sinh(x - a) + + e = 2*cosh(x)**2 - 2*sinh(x)**2 + assert trigsimp(log(e)) == log(2) + + # issue 19535: + assert trigsimp(sqrt(cosh(x)**2 - 1)) == sqrt(sinh(x)**2) + + assert trigsimp(cosh(x)**2*cosh(y)**2 - cosh(x)**2*sinh(y)**2 - sinh(x)**2, + recursive=True) == 1 + assert trigsimp(sinh(x)**2*sinh(y)**2 - sinh(x)**2*cosh(y)**2 + cosh(x)**2, + recursive=True) == 1 + + assert abs(trigsimp(2.0*cosh(x)**2 - 2.0*sinh(x)**2) - 2.0) < 1e-10 + + assert trigsimp(sinh(x)**2/cosh(x)**2) == tanh(x)**2 + assert trigsimp(sinh(x)**3/cosh(x)**3) == tanh(x)**3 + assert trigsimp(sinh(x)**10/cosh(x)**10) == tanh(x)**10 + assert trigsimp(cosh(x)**3/sinh(x)**3) == 1/tanh(x)**3 + + assert trigsimp(cosh(x)/sinh(x)) == 1/tanh(x) + assert trigsimp(cosh(x)**2/sinh(x)**2) == 1/tanh(x)**2 + assert trigsimp(cosh(x)**10/sinh(x)**10) == 1/tanh(x)**10 + + assert trigsimp(x*cosh(x)*tanh(x)) == x*sinh(x) + assert trigsimp(-sinh(x) + cosh(x)*tanh(x)) == 0 + + assert tan(x) != 1/cot(x) # cot doesn't auto-simplify + + assert trigsimp(tan(x) - 1/cot(x)) == 0 + assert trigsimp(3*tanh(x)**7 - 2/coth(x)**7) == tanh(x)**7 + + +def test_trigsimp_groebner(): + from sympy.simplify.trigsimp import trigsimp_groebner + + c = cos(x) + s = sin(x) + ex = (4*s*c + 12*s + 5*c**3 + 21*c**2 + 23*c + 15)/( + -s*c**2 + 2*s*c + 15*s + 7*c**3 + 31*c**2 + 37*c + 21) + resnum = (5*s - 5*c + 1) + resdenom = (8*s - 6*c) + results = [resnum/resdenom, (-resnum)/(-resdenom)] + assert trigsimp_groebner(ex) in results + assert trigsimp_groebner(s/c, hints=[tan]) == tan(x) + assert trigsimp_groebner(c*s) == c*s + assert trigsimp((-s + 1)/c + c/(-s + 1), + method='groebner') == 2/c + assert trigsimp((-s + 1)/c + c/(-s + 1), + method='groebner', polynomial=True) == 2/c + + # Test quick=False works + assert trigsimp_groebner(ex, hints=[2]) in results + assert trigsimp_groebner(ex, hints=[int(2)]) in results + + # test "I" + assert trigsimp_groebner(sin(I*x)/cos(I*x), hints=[tanh]) == I*tanh(x) + + # test hyperbolic / sums + assert trigsimp_groebner((tanh(x)+tanh(y))/(1+tanh(x)*tanh(y)), + hints=[(tanh, x, y)]) == tanh(x + y) + + +def test_issue_2827_trigsimp_methods(): + measure1 = lambda expr: len(str(expr)) + measure2 = lambda expr: -count_ops(expr) + # Return the most complicated result + expr = (x + 1)/(x + sin(x)**2 + cos(x)**2) + ans = Matrix([1]) + M = Matrix([expr]) + assert trigsimp(M, method='fu', measure=measure1) == ans + assert trigsimp(M, method='fu', measure=measure2) != ans + # all methods should work with Basic expressions even if they + # aren't Expr + M = Matrix.eye(1) + assert all(trigsimp(M, method=m) == M for m in + 'fu matching groebner old'.split()) + # watch for E in exptrigsimp, not only exp() + eq = 1/sqrt(E) + E + assert exptrigsimp(eq) == eq + +def test_issue_15129_trigsimp_methods(): + t1 = Matrix([sin(Rational(1, 50)), cos(Rational(1, 50)), 0]) + t2 = Matrix([sin(Rational(1, 25)), cos(Rational(1, 25)), 0]) + t3 = Matrix([cos(Rational(1, 25)), sin(Rational(1, 25)), 0]) + r1 = t1.dot(t2) + r2 = t1.dot(t3) + assert trigsimp(r1) == cos(Rational(1, 50)) + assert trigsimp(r2) == sin(Rational(3, 50)) + +def test_exptrigsimp(): + def valid(a, b): + from sympy.core.random import verify_numerically as tn + if not (tn(a, b) and a == b): + return False + return True + + assert exptrigsimp(exp(x) + exp(-x)) == 2*cosh(x) + assert exptrigsimp(exp(x) - exp(-x)) == 2*sinh(x) + assert exptrigsimp((2*exp(x)-2*exp(-x))/(exp(x)+exp(-x))) == 2*tanh(x) + assert exptrigsimp((2*exp(2*x)-2)/(exp(2*x)+1)) == 2*tanh(x) + e = [cos(x) + I*sin(x), cos(x) - I*sin(x), + cosh(x) - sinh(x), cosh(x) + sinh(x)] + ok = [exp(I*x), exp(-I*x), exp(-x), exp(x)] + assert all(valid(i, j) for i, j in zip( + [exptrigsimp(ei) for ei in e], ok)) + + ue = [cos(x) + sin(x), cos(x) - sin(x), + cosh(x) + I*sinh(x), cosh(x) - I*sinh(x)] + assert [exptrigsimp(ei) == ei for ei in ue] + + res = [] + ok = [y*tanh(1), 1/(y*tanh(1)), I*y*tan(1), -I/(y*tan(1)), + y*tanh(x), 1/(y*tanh(x)), I*y*tan(x), -I/(y*tan(x)), + y*tanh(1 + I), 1/(y*tanh(1 + I))] + for a in (1, I, x, I*x, 1 + I): + w = exp(a) + eq = y*(w - 1/w)/(w + 1/w) + res.append(simplify(eq)) + res.append(simplify(1/eq)) + assert all(valid(i, j) for i, j in zip(res, ok)) + + for a in range(1, 3): + w = exp(a) + e = w + 1/w + s = simplify(e) + assert s == exptrigsimp(e) + assert valid(s, 2*cosh(a)) + e = w - 1/w + s = simplify(e) + assert s == exptrigsimp(e) + assert valid(s, 2*sinh(a)) + +def test_exptrigsimp_noncommutative(): + a,b = symbols('a b', commutative=False) + x = Symbol('x', commutative=True) + assert exp(a + x) == exptrigsimp(exp(a)*exp(x)) + p = exp(a)*exp(b) - exp(b)*exp(a) + assert p == exptrigsimp(p) != 0 + +def test_powsimp_on_numbers(): + assert 2**(Rational(1, 3) - 2) == 2**Rational(1, 3)/4 + + +@XFAIL +def test_issue_6811_fail(): + # from doc/src/modules/physics/mechanics/examples.rst, the current `eq` + # at Line 576 (in different variables) was formerly the equivalent and + # shorter expression given below...it would be nice to get the short one + # back again + xp, y, x, z = symbols('xp, y, x, z') + eq = 4*(-19*sin(x)*y + 5*sin(3*x)*y + 15*cos(2*x)*z - 21*z)*xp/(9*cos(x) - 5*cos(3*x)) + assert trigsimp(eq) == -2*(2*cos(x)*tan(x)*y + 3*z)*xp/cos(x) + + +def test_Piecewise(): + e1 = x*(x + y) - y*(x + y) + e2 = sin(x)**2 + cos(x)**2 + e3 = expand((x + y)*y/x) + # s1 = simplify(e1) + s2 = simplify(e2) + # s3 = simplify(e3) + + # trigsimp tries not to touch non-trig containing args + assert trigsimp(Piecewise((e1, e3 < e2), (e3, True))) == \ + Piecewise((e1, e3 < s2), (e3, True)) + + +def test_issue_21594(): + assert simplify(exp(Rational(1,2)) + exp(Rational(-1,2))) == cosh(S.Half)*2 + + +def test_trigsimp_old(): + x, y = symbols('x,y') + + assert trigsimp(1 - sin(x)**2, old=True) == cos(x)**2 + assert trigsimp(1 - cos(x)**2, old=True) == sin(x)**2 + assert trigsimp(sin(x)**2 + cos(x)**2, old=True) == 1 + assert trigsimp(1 + tan(x)**2, old=True) == 1/cos(x)**2 + assert trigsimp(1/cos(x)**2 - 1, old=True) == tan(x)**2 + assert trigsimp(1/cos(x)**2 - tan(x)**2, old=True) == 1 + assert trigsimp(1 + cot(x)**2, old=True) == 1/sin(x)**2 + assert trigsimp(1/sin(x)**2 - cot(x)**2, old=True) == 1 + + assert trigsimp(5*cos(x)**2 + 5*sin(x)**2, old=True) == 5 + + assert trigsimp(sin(x)/cos(x), old=True) == tan(x) + assert trigsimp(2*tan(x)*cos(x), old=True) == 2*sin(x) + assert trigsimp(cot(x)**3*sin(x)**3, old=True) == cos(x)**3 + assert trigsimp(y*tan(x)**2/sin(x)**2, old=True) == y/cos(x)**2 + assert trigsimp(cot(x)/cos(x), old=True) == 1/sin(x) + + assert trigsimp(sin(x + y) + sin(x - y), old=True) == 2*sin(x)*cos(y) + assert trigsimp(sin(x + y) - sin(x - y), old=True) == 2*sin(y)*cos(x) + assert trigsimp(cos(x + y) + cos(x - y), old=True) == 2*cos(x)*cos(y) + assert trigsimp(cos(x + y) - cos(x - y), old=True) == -2*sin(x)*sin(y) + + assert trigsimp(sinh(x + y) + sinh(x - y), old=True) == 2*sinh(x)*cosh(y) + assert trigsimp(sinh(x + y) - sinh(x - y), old=True) == 2*sinh(y)*cosh(x) + assert trigsimp(cosh(x + y) + cosh(x - y), old=True) == 2*cosh(x)*cosh(y) + assert trigsimp(cosh(x + y) - cosh(x - y), old=True) == 2*sinh(x)*sinh(y) + + assert trigsimp(cos(0.12345)**2 + sin(0.12345)**2, old=True) == 1.0 + + assert trigsimp(sin(x)/cos(x), old=True, method='combined') == tan(x) + assert trigsimp(sin(x)/cos(x), old=True, method='groebner') == sin(x)/cos(x) + assert trigsimp(sin(x)/cos(x), old=True, method='groebner', hints=[tan]) == tan(x) + + assert trigsimp(1-sin(sin(x)**2+cos(x)**2)**2, old=True, deep=True) == cos(1)**2 + + +def test_trigsimp_inverse(): + alpha = symbols('alpha') + s, c = sin(alpha), cos(alpha) + + for finv in [asin, acos, asec, acsc, atan, acot]: + f = finv.inverse(None) + assert alpha == trigsimp(finv(f(alpha)), inverse=True) + + # test atan2(cos, sin), atan2(sin, cos), etc... + for a, b in [[c, s], [s, c]]: + for i, j in product([-1, 1], repeat=2): + angle = atan2(i*b, j*a) + angle_inverted = trigsimp(angle, inverse=True) + assert angle_inverted != angle # assures simplification happened + assert sin(angle_inverted) == trigsimp(sin(angle)) + assert cos(angle_inverted) == trigsimp(cos(angle)) diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/traversaltools.py b/.venv/lib/python3.12/site-packages/sympy/simplify/traversaltools.py new file mode 100644 index 0000000000000000000000000000000000000000..75b0bd0d8fd198cb12640ab8a0fe63a23c81ed8f --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/simplify/traversaltools.py @@ -0,0 +1,15 @@ +from sympy.core.traversal import use as _use +from sympy.utilities.decorator import deprecated + +use = deprecated( + """ + Using use from the sympy.simplify.traversaltools submodule is + deprecated. + + Instead, use use from the top-level sympy namespace, like + + sympy.use + """, + deprecated_since_version="1.10", + active_deprecations_target="deprecated-traversal-functions-moved" +)(_use) diff --git a/.venv/lib/python3.12/site-packages/sympy/simplify/trigsimp.py b/.venv/lib/python3.12/site-packages/sympy/simplify/trigsimp.py new file mode 100644 index 0000000000000000000000000000000000000000..fe5be1444a4625e4b63b339877e441d12cfbe8de --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/simplify/trigsimp.py @@ -0,0 +1,1252 @@ +from collections import defaultdict +from functools import reduce + +from sympy.core import (sympify, Basic, S, Expr, factor_terms, + Mul, Add, bottom_up) +from sympy.core.cache import cacheit +from sympy.core.function import (count_ops, _mexpand, FunctionClass, expand, + expand_mul, _coeff_isneg, Derivative) +from sympy.core.numbers import I, Integer +from sympy.core.intfunc import igcd +from sympy.core.sorting import _nodes +from sympy.core.symbol import Dummy, symbols, Wild +from sympy.external.gmpy import SYMPY_INTS +from sympy.functions import sin, cos, exp, cosh, tanh, sinh, tan, cot, coth +from sympy.functions import atan2 +from sympy.functions.elementary.hyperbolic import HyperbolicFunction +from sympy.functions.elementary.trigonometric import TrigonometricFunction +from sympy.polys import Poly, factor, cancel, parallel_poly_from_expr +from sympy.polys.domains import ZZ +from sympy.polys.polyerrors import PolificationFailed +from sympy.polys.polytools import groebner +from sympy.simplify.cse_main import cse +from sympy.strategies.core import identity +from sympy.strategies.tree import greedy +from sympy.utilities.iterables import iterable +from sympy.utilities.misc import debug + +def trigsimp_groebner(expr, hints=[], quick=False, order="grlex", + polynomial=False): + """ + Simplify trigonometric expressions using a groebner basis algorithm. + + Explanation + =========== + + This routine takes a fraction involving trigonometric or hyperbolic + expressions, and tries to simplify it. The primary metric is the + total degree. Some attempts are made to choose the simplest possible + expression of the minimal degree, but this is non-rigorous, and also + very slow (see the ``quick=True`` option). + + If ``polynomial`` is set to True, instead of simplifying numerator and + denominator together, this function just brings numerator and denominator + into a canonical form. This is much faster, but has potentially worse + results. However, if the input is a polynomial, then the result is + guaranteed to be an equivalent polynomial of minimal degree. + + The most important option is hints. Its entries can be any of the + following: + + - a natural number + - a function + - an iterable of the form (func, var1, var2, ...) + - anything else, interpreted as a generator + + A number is used to indicate that the search space should be increased. + A function is used to indicate that said function is likely to occur in a + simplified expression. + An iterable is used indicate that func(var1 + var2 + ...) is likely to + occur in a simplified . + An additional generator also indicates that it is likely to occur. + (See examples below). + + This routine carries out various computationally intensive algorithms. + The option ``quick=True`` can be used to suppress one particularly slow + step (at the expense of potentially more complicated results, but never at + the expense of increased total degree). + + Examples + ======== + + >>> from sympy.abc import x, y + >>> from sympy import sin, tan, cos, sinh, cosh, tanh + >>> from sympy.simplify.trigsimp import trigsimp_groebner + + Suppose you want to simplify ``sin(x)*cos(x)``. Naively, nothing happens: + + >>> ex = sin(x)*cos(x) + >>> trigsimp_groebner(ex) + sin(x)*cos(x) + + This is because ``trigsimp_groebner`` only looks for a simplification + involving just ``sin(x)`` and ``cos(x)``. You can tell it to also try + ``2*x`` by passing ``hints=[2]``: + + >>> trigsimp_groebner(ex, hints=[2]) + sin(2*x)/2 + >>> trigsimp_groebner(sin(x)**2 - cos(x)**2, hints=[2]) + -cos(2*x) + + Increasing the search space this way can quickly become expensive. A much + faster way is to give a specific expression that is likely to occur: + + >>> trigsimp_groebner(ex, hints=[sin(2*x)]) + sin(2*x)/2 + + Hyperbolic expressions are similarly supported: + + >>> trigsimp_groebner(sinh(2*x)/sinh(x)) + 2*cosh(x) + + Note how no hints had to be passed, since the expression already involved + ``2*x``. + + The tangent function is also supported. You can either pass ``tan`` in the + hints, to indicate that tan should be tried whenever cosine or sine are, + or you can pass a specific generator: + + >>> trigsimp_groebner(sin(x)/cos(x), hints=[tan]) + tan(x) + >>> trigsimp_groebner(sinh(x)/cosh(x), hints=[tanh(x)]) + tanh(x) + + Finally, you can use the iterable form to suggest that angle sum formulae + should be tried: + + >>> ex = (tan(x) + tan(y))/(1 - tan(x)*tan(y)) + >>> trigsimp_groebner(ex, hints=[(tan, x, y)]) + tan(x + y) + """ + # TODO + # - preprocess by replacing everything by funcs we can handle + # - optionally use cot instead of tan + # - more intelligent hinting. + # For example, if the ideal is small, and we have sin(x), sin(y), + # add sin(x + y) automatically... ? + # - algebraic numbers ... + # - expressions of lowest degree are not distinguished properly + # e.g. 1 - sin(x)**2 + # - we could try to order the generators intelligently, so as to influence + # which monomials appear in the quotient basis + + # THEORY + # ------ + # Ratsimpmodprime above can be used to "simplify" a rational function + # modulo a prime ideal. "Simplify" mainly means finding an equivalent + # expression of lower total degree. + # + # We intend to use this to simplify trigonometric functions. To do that, + # we need to decide (a) which ring to use, and (b) modulo which ideal to + # simplify. In practice, (a) means settling on a list of "generators" + # a, b, c, ..., such that the fraction we want to simplify is a rational + # function in a, b, c, ..., with coefficients in ZZ (integers). + # (2) means that we have to decide what relations to impose on the + # generators. There are two practical problems: + # (1) The ideal has to be *prime* (a technical term). + # (2) The relations have to be polynomials in the generators. + # + # We typically have two kinds of generators: + # - trigonometric expressions, like sin(x), cos(5*x), etc + # - "everything else", like gamma(x), pi, etc. + # + # Since this function is trigsimp, we will concentrate on what to do with + # trigonometric expressions. We can also simplify hyperbolic expressions, + # but the extensions should be clear. + # + # One crucial point is that all *other* generators really should behave + # like indeterminates. In particular if (say) "I" is one of them, then + # in fact I**2 + 1 = 0 and we may and will compute non-sensical + # expressions. However, we can work with a dummy and add the relation + # I**2 + 1 = 0 to our ideal, then substitute back in the end. + # + # Now regarding trigonometric generators. We split them into groups, + # according to the argument of the trigonometric functions. We want to + # organise this in such a way that most trigonometric identities apply in + # the same group. For example, given sin(x), cos(2*x) and cos(y), we would + # group as [sin(x), cos(2*x)] and [cos(y)]. + # + # Our prime ideal will be built in three steps: + # (1) For each group, compute a "geometrically prime" ideal of relations. + # Geometrically prime means that it generates a prime ideal in + # CC[gens], not just ZZ[gens]. + # (2) Take the union of all the generators of the ideals for all groups. + # By the geometric primality condition, this is still prime. + # (3) Add further inter-group relations which preserve primality. + # + # Step (1) works as follows. We will isolate common factors in the + # argument, so that all our generators are of the form sin(n*x), cos(n*x) + # or tan(n*x), with n an integer. Suppose first there are no tan terms. + # The ideal [sin(x)**2 + cos(x)**2 - 1] is geometrically prime, since + # X**2 + Y**2 - 1 is irreducible over CC. + # Now, if we have a generator sin(n*x), than we can, using trig identities, + # express sin(n*x) as a polynomial in sin(x) and cos(x). We can add this + # relation to the ideal, preserving geometric primality, since the quotient + # ring is unchanged. + # Thus we have treated all sin and cos terms. + # For tan(n*x), we add a relation tan(n*x)*cos(n*x) - sin(n*x) = 0. + # (This requires of course that we already have relations for cos(n*x) and + # sin(n*x).) It is not obvious, but it seems that this preserves geometric + # primality. + # XXX A real proof would be nice. HELP! + # Sketch that is a prime ideal of + # CC[S, C, T]: + # - it suffices to show that the projective closure in CP**3 is + # irreducible + # - using the half-angle substitutions, we can express sin(x), tan(x), + # cos(x) as rational functions in tan(x/2) + # - from this, we get a rational map from CP**1 to our curve + # - this is a morphism, hence the curve is prime + # + # Step (2) is trivial. + # + # Step (3) works by adding selected relations of the form + # sin(x + y) - sin(x)*cos(y) - sin(y)*cos(x), etc. Geometric primality is + # preserved by the same argument as before. + + def parse_hints(hints): + """Split hints into (n, funcs, iterables, gens).""" + n = 1 + funcs, iterables, gens = [], [], [] + for e in hints: + if isinstance(e, (SYMPY_INTS, Integer)): + n = e + elif isinstance(e, FunctionClass): + funcs.append(e) + elif iterable(e): + iterables.append((e[0], e[1:])) + # XXX sin(x+2y)? + # Note: we go through polys so e.g. + # sin(-x) -> -sin(x) -> sin(x) + gens.extend(parallel_poly_from_expr( + [e[0](x) for x in e[1:]] + [e[0](Add(*e[1:]))])[1].gens) + else: + gens.append(e) + return n, funcs, iterables, gens + + def build_ideal(x, terms): + """ + Build generators for our ideal. ``Terms`` is an iterable with elements of + the form (fn, coeff), indicating that we have a generator fn(coeff*x). + + If any of the terms is trigonometric, sin(x) and cos(x) are guaranteed + to appear in terms. Similarly for hyperbolic functions. For tan(n*x), + sin(n*x) and cos(n*x) are guaranteed. + """ + I = [] + y = Dummy('y') + for fn, coeff in terms: + for c, s, t, rel in ( + [cos, sin, tan, cos(x)**2 + sin(x)**2 - 1], + [cosh, sinh, tanh, cosh(x)**2 - sinh(x)**2 - 1]): + if coeff == 1 and fn in [c, s]: + I.append(rel) + elif fn == t: + I.append(t(coeff*x)*c(coeff*x) - s(coeff*x)) + elif fn in [c, s]: + cn = fn(coeff*y).expand(trig=True).subs(y, x) + I.append(fn(coeff*x) - cn) + return list(set(I)) + + def analyse_gens(gens, hints): + """ + Analyse the generators ``gens``, using the hints ``hints``. + + The meaning of ``hints`` is described in the main docstring. + Return a new list of generators, and also the ideal we should + work with. + """ + # First parse the hints + n, funcs, iterables, extragens = parse_hints(hints) + debug('n=%s funcs: %s iterables: %s extragens: %s', + (funcs, iterables, extragens)) + + # We just add the extragens to gens and analyse them as before + gens = list(gens) + gens.extend(extragens) + + # remove duplicates + funcs = list(set(funcs)) + iterables = list(set(iterables)) + gens = list(set(gens)) + + # all the functions we can do anything with + allfuncs = {sin, cos, tan, sinh, cosh, tanh} + # sin(3*x) -> ((3, x), sin) + trigterms = [(g.args[0].as_coeff_mul(), g.func) for g in gens + if g.func in allfuncs] + # Our list of new generators - start with anything that we cannot + # work with (i.e. is not a trigonometric term) + freegens = [g for g in gens if g.func not in allfuncs] + newgens = [] + trigdict = {} + for (coeff, var), fn in trigterms: + trigdict.setdefault(var, []).append((coeff, fn)) + res = [] # the ideal + + for key, val in trigdict.items(): + # We have now assembeled a dictionary. Its keys are common + # arguments in trigonometric expressions, and values are lists of + # pairs (fn, coeff). x0, (fn, coeff) in trigdict means that we + # need to deal with fn(coeff*x0). We take the rational gcd of the + # coeffs, call it ``gcd``. We then use x = x0/gcd as "base symbol", + # all other arguments are integral multiples thereof. + # We will build an ideal which works with sin(x), cos(x). + # If hint tan is provided, also work with tan(x). Moreover, if + # n > 1, also work with sin(k*x) for k <= n, and similarly for cos + # (and tan if the hint is provided). Finally, any generators which + # the ideal does not work with but we need to accommodate (either + # because it was in expr or because it was provided as a hint) + # we also build into the ideal. + # This selection process is expressed in the list ``terms``. + # build_ideal then generates the actual relations in our ideal, + # from this list. + fns = [x[1] for x in val] + val = [x[0] for x in val] + gcd = reduce(igcd, val) + terms = [(fn, v/gcd) for (fn, v) in zip(fns, val)] + fs = set(funcs + fns) + for c, s, t in ([cos, sin, tan], [cosh, sinh, tanh]): + if any(x in fs for x in (c, s, t)): + fs.add(c) + fs.add(s) + for fn in fs: + terms.extend((fn, k) for k in range(1, n + 1)) + extra = [] + for fn, v in terms: + if fn == tan: + extra.append((sin, v)) + extra.append((cos, v)) + if fn in [sin, cos] and tan in fs: + extra.append((tan, v)) + if fn == tanh: + extra.append((sinh, v)) + extra.append((cosh, v)) + if fn in [sinh, cosh] and tanh in fs: + extra.append((tanh, v)) + terms.extend(extra) + x = gcd*Mul(*key) + r = build_ideal(x, terms) + res.extend(r) + newgens.extend({fn(v*x) for fn, v in terms}) + + # Add generators for compound expressions from iterables + for fn, args in iterables: + if fn == tan: + # Tan expressions are recovered from sin and cos. + iterables.extend([(sin, args), (cos, args)]) + elif fn == tanh: + # Tanh expressions are recovered from sihn and cosh. + iterables.extend([(sinh, args), (cosh, args)]) + else: + dummys = symbols('d:%i' % len(args), cls=Dummy) + expr = fn( Add(*dummys)).expand(trig=True).subs(list(zip(dummys, args))) + res.append(fn(Add(*args)) - expr) + + if myI in gens: + res.append(myI**2 + 1) + freegens.remove(myI) + newgens.append(myI) + + return res, freegens, newgens + + myI = Dummy('I') + expr = expr.subs(S.ImaginaryUnit, myI) + subs = [(myI, S.ImaginaryUnit)] + + num, denom = cancel(expr).as_numer_denom() + try: + (pnum, pdenom), opt = parallel_poly_from_expr([num, denom]) + except PolificationFailed: + return expr + debug('initial gens:', opt.gens) + ideal, freegens, gens = analyse_gens(opt.gens, hints) + debug('ideal:', ideal) + debug('new gens:', gens, " -- len", len(gens)) + debug('free gens:', freegens, " -- len", len(gens)) + # NOTE we force the domain to be ZZ to stop polys from injecting generators + # (which is usually a sign of a bug in the way we build the ideal) + if not gens: + return expr + G = groebner(ideal, order=order, gens=gens, domain=ZZ) + debug('groebner basis:', list(G), " -- len", len(G)) + + # If our fraction is a polynomial in the free generators, simplify all + # coefficients separately: + + from sympy.simplify.ratsimp import ratsimpmodprime + + if freegens and pdenom.has_only_gens(*set(gens).intersection(pdenom.gens)): + num = Poly(num, gens=gens+freegens).eject(*gens) + res = [] + for monom, coeff in num.terms(): + ourgens = set(parallel_poly_from_expr([coeff, denom])[1].gens) + # We compute the transitive closure of all generators that can + # be reached from our generators through relations in the ideal. + changed = True + while changed: + changed = False + for p in ideal: + p = Poly(p) + if not ourgens.issuperset(p.gens) and \ + not p.has_only_gens(*set(p.gens).difference(ourgens)): + changed = True + ourgens.update(p.exclude().gens) + # NOTE preserve order! + realgens = [x for x in gens if x in ourgens] + # The generators of the ideal have now been (implicitly) split + # into two groups: those involving ourgens and those that don't. + # Since we took the transitive closure above, these two groups + # live in subgrings generated by a *disjoint* set of variables. + # Any sensible groebner basis algorithm will preserve this disjoint + # structure (i.e. the elements of the groebner basis can be split + # similarly), and and the two subsets of the groebner basis then + # form groebner bases by themselves. (For the smaller generating + # sets, of course.) + ourG = [g.as_expr() for g in G.polys if + g.has_only_gens(*ourgens.intersection(g.gens))] + res.append(Mul(*[a**b for a, b in zip(freegens, monom)]) * \ + ratsimpmodprime(coeff/denom, ourG, order=order, + gens=realgens, quick=quick, domain=ZZ, + polynomial=polynomial).subs(subs)) + return Add(*res) + # NOTE The following is simpler and has less assumptions on the + # groebner basis algorithm. If the above turns out to be broken, + # use this. + return Add(*[Mul(*[a**b for a, b in zip(freegens, monom)]) * \ + ratsimpmodprime(coeff/denom, list(G), order=order, + gens=gens, quick=quick, domain=ZZ) + for monom, coeff in num.terms()]) + else: + return ratsimpmodprime( + expr, list(G), order=order, gens=freegens+gens, + quick=quick, domain=ZZ, polynomial=polynomial).subs(subs) + + +_trigs = (TrigonometricFunction, HyperbolicFunction) + + +def _trigsimp_inverse(rv): + + def check_args(x, y): + try: + return x.args[0] == y.args[0] + except IndexError: + return False + + def f(rv): + # for simple functions + g = getattr(rv, 'inverse', None) + if (g is not None and isinstance(rv.args[0], g()) and + isinstance(g()(1), TrigonometricFunction)): + return rv.args[0].args[0] + + # for atan2 simplifications, harder because atan2 has 2 args + if isinstance(rv, atan2): + y, x = rv.args + if _coeff_isneg(y): + return -f(atan2(-y, x)) + elif _coeff_isneg(x): + return S.Pi - f(atan2(y, -x)) + + if check_args(x, y): + if isinstance(y, sin) and isinstance(x, cos): + return x.args[0] + if isinstance(y, cos) and isinstance(x, sin): + return S.Pi / 2 - x.args[0] + + return rv + + return bottom_up(rv, f) + + +def trigsimp(expr, inverse=False, **opts): + """Returns a reduced expression by using known trig identities. + + Parameters + ========== + + inverse : bool, optional + If ``inverse=True``, it will be assumed that a composition of inverse + functions, such as sin and asin, can be cancelled in any order. + For example, ``asin(sin(x))`` will yield ``x`` without checking whether + x belongs to the set where this relation is true. The default is False. + Default : True + + method : string, optional + Specifies the method to use. Valid choices are: + + - ``'matching'``, default + - ``'groebner'`` + - ``'combined'`` + - ``'fu'`` + - ``'old'`` + + If ``'matching'``, simplify the expression recursively by targeting + common patterns. If ``'groebner'``, apply an experimental groebner + basis algorithm. In this case further options are forwarded to + ``trigsimp_groebner``, please refer to + its docstring. If ``'combined'``, it first runs the groebner basis + algorithm with small default parameters, then runs the ``'matching'`` + algorithm. If ``'fu'``, run the collection of trigonometric + transformations described by Fu, et al. (see the + :py:func:`~sympy.simplify.fu.fu` docstring). If ``'old'``, the original + SymPy trig simplification function is run. + opts : + Optional keyword arguments passed to the method. See each method's + function docstring for details. + + Examples + ======== + + >>> from sympy import trigsimp, sin, cos, log + >>> from sympy.abc import x + >>> e = 2*sin(x)**2 + 2*cos(x)**2 + >>> trigsimp(e) + 2 + + Simplification occurs wherever trigonometric functions are located. + + >>> trigsimp(log(e)) + log(2) + + Using ``method='groebner'`` (or ``method='combined'``) might lead to + greater simplification. + + The old trigsimp routine can be accessed as with method ``method='old'``. + + >>> from sympy import coth, tanh + >>> t = 3*tanh(x)**7 - 2/coth(x)**7 + >>> trigsimp(t, method='old') == t + True + >>> trigsimp(t) + tanh(x)**7 + + """ + from sympy.simplify.fu import fu + + expr = sympify(expr) + + _eval_trigsimp = getattr(expr, '_eval_trigsimp', None) + if _eval_trigsimp is not None: + return _eval_trigsimp(**opts) + + old = opts.pop('old', False) + if not old: + opts.pop('deep', None) + opts.pop('recursive', None) + method = opts.pop('method', 'matching') + else: + method = 'old' + + def groebnersimp(ex, **opts): + def traverse(e): + if e.is_Atom: + return e + args = [traverse(x) for x in e.args] + if e.is_Function or e.is_Pow: + args = [trigsimp_groebner(x, **opts) for x in args] + return e.func(*args) + new = traverse(ex) + if not isinstance(new, Expr): + return new + return trigsimp_groebner(new, **opts) + + trigsimpfunc = { + 'fu': (lambda x: fu(x, **opts)), + 'matching': (lambda x: futrig(x)), + 'groebner': (lambda x: groebnersimp(x, **opts)), + 'combined': (lambda x: futrig(groebnersimp(x, + polynomial=True, hints=[2, tan]))), + 'old': lambda x: trigsimp_old(x, **opts), + }[method] + + expr_simplified = trigsimpfunc(expr) + if inverse: + expr_simplified = _trigsimp_inverse(expr_simplified) + + return expr_simplified + + +def exptrigsimp(expr): + """ + Simplifies exponential / trigonometric / hyperbolic functions. + + Examples + ======== + + >>> from sympy import exptrigsimp, exp, cosh, sinh + >>> from sympy.abc import z + + >>> exptrigsimp(exp(z) + exp(-z)) + 2*cosh(z) + >>> exptrigsimp(cosh(z) - sinh(z)) + exp(-z) + """ + from sympy.simplify.fu import hyper_as_trig, TR2i + + def exp_trig(e): + # select the better of e, and e rewritten in terms of exp or trig + # functions + choices = [e] + if e.has(*_trigs): + choices.append(e.rewrite(exp)) + choices.append(e.rewrite(cos)) + return min(*choices, key=count_ops) + newexpr = bottom_up(expr, exp_trig) + + def f(rv): + if not rv.is_Mul: + return rv + commutative_part, noncommutative_part = rv.args_cnc() + # Since as_powers_dict loses order information, + # if there is more than one noncommutative factor, + # it should only be used to simplify the commutative part. + if (len(noncommutative_part) > 1): + return f(Mul(*commutative_part))*Mul(*noncommutative_part) + rvd = rv.as_powers_dict() + newd = rvd.copy() + + def signlog(expr, sign=S.One): + if expr is S.Exp1: + return sign, S.One + elif isinstance(expr, exp) or (expr.is_Pow and expr.base == S.Exp1): + return sign, expr.exp + elif sign is S.One: + return signlog(-expr, sign=-S.One) + else: + return None, None + + ee = rvd[S.Exp1] + for k in rvd: + if k.is_Add and len(k.args) == 2: + # k == c*(1 + sign*E**x) + c = k.args[0] + sign, x = signlog(k.args[1]/c) + if not x: + continue + m = rvd[k] + newd[k] -= m + if ee == -x*m/2: + # sinh and cosh + newd[S.Exp1] -= ee + ee = 0 + if sign == 1: + newd[2*c*cosh(x/2)] += m + else: + newd[-2*c*sinh(x/2)] += m + elif newd[1 - sign*S.Exp1**x] == -m: + # tanh + del newd[1 - sign*S.Exp1**x] + if sign == 1: + newd[-c/tanh(x/2)] += m + else: + newd[-c*tanh(x/2)] += m + else: + newd[1 + sign*S.Exp1**x] += m + newd[c] += m + + return Mul(*[k**newd[k] for k in newd]) + newexpr = bottom_up(newexpr, f) + + # sin/cos and sinh/cosh ratios to tan and tanh, respectively + if newexpr.has(HyperbolicFunction): + e, f = hyper_as_trig(newexpr) + newexpr = f(TR2i(e)) + if newexpr.has(TrigonometricFunction): + newexpr = TR2i(newexpr) + + # can we ever generate an I where there was none previously? + if not (newexpr.has(I) and not expr.has(I)): + expr = newexpr + return expr + +#-------------------- the old trigsimp routines --------------------- + +def trigsimp_old(expr, *, first=True, **opts): + """ + Reduces expression by using known trig identities. + + Notes + ===== + + deep: + - Apply trigsimp inside all objects with arguments + + recursive: + - Use common subexpression elimination (cse()) and apply + trigsimp recursively (this is quite expensive if the + expression is large) + + method: + - Determine the method to use. Valid choices are 'matching' (default), + 'groebner', 'combined', 'fu' and 'futrig'. If 'matching', simplify the + expression recursively by pattern matching. If 'groebner', apply an + experimental groebner basis algorithm. In this case further options + are forwarded to ``trigsimp_groebner``, please refer to its docstring. + If 'combined', first run the groebner basis algorithm with small + default parameters, then run the 'matching' algorithm. 'fu' runs the + collection of trigonometric transformations described by Fu, et al. + (see the `fu` docstring) while `futrig` runs a subset of Fu-transforms + that mimic the behavior of `trigsimp`. + + compare: + - show input and output from `trigsimp` and `futrig` when different, + but returns the `trigsimp` value. + + Examples + ======== + + >>> from sympy import trigsimp, sin, cos, log, cot + >>> from sympy.abc import x + >>> e = 2*sin(x)**2 + 2*cos(x)**2 + >>> trigsimp(e, old=True) + 2 + >>> trigsimp(log(e), old=True) + log(2*sin(x)**2 + 2*cos(x)**2) + >>> trigsimp(log(e), deep=True, old=True) + log(2) + + Using `method="groebner"` (or `"combined"`) can sometimes lead to a lot + more simplification: + + >>> e = (-sin(x) + 1)/cos(x) + cos(x)/(-sin(x) + 1) + >>> trigsimp(e, old=True) + (1 - sin(x))/cos(x) + cos(x)/(1 - sin(x)) + >>> trigsimp(e, method="groebner", old=True) + 2/cos(x) + + >>> trigsimp(1/cot(x)**2, compare=True, old=True) + futrig: tan(x)**2 + cot(x)**(-2) + + """ + old = expr + if first: + if not expr.has(*_trigs): + return expr + + trigsyms = set().union(*[t.free_symbols for t in expr.atoms(*_trigs)]) + if len(trigsyms) > 1: + from sympy.simplify.simplify import separatevars + + d = separatevars(expr) + if d.is_Mul: + d = separatevars(d, dict=True) or d + if isinstance(d, dict): + expr = 1 + for v in d.values(): + # remove hollow factoring + was = v + v = expand_mul(v) + opts['first'] = False + vnew = trigsimp(v, **opts) + if vnew == v: + vnew = was + expr *= vnew + old = expr + else: + if d.is_Add: + for s in trigsyms: + r, e = expr.as_independent(s) + if r: + opts['first'] = False + expr = r + trigsimp(e, **opts) + if not expr.is_Add: + break + old = expr + + recursive = opts.pop('recursive', False) + deep = opts.pop('deep', False) + method = opts.pop('method', 'matching') + + def groebnersimp(ex, deep, **opts): + def traverse(e): + if e.is_Atom: + return e + args = [traverse(x) for x in e.args] + if e.is_Function or e.is_Pow: + args = [trigsimp_groebner(x, **opts) for x in args] + return e.func(*args) + if deep: + ex = traverse(ex) + return trigsimp_groebner(ex, **opts) + + trigsimpfunc = { + 'matching': (lambda x, d: _trigsimp(x, d)), + 'groebner': (lambda x, d: groebnersimp(x, d, **opts)), + 'combined': (lambda x, d: _trigsimp(groebnersimp(x, + d, polynomial=True, hints=[2, tan]), + d)) + }[method] + + if recursive: + w, g = cse(expr) + g = trigsimpfunc(g[0], deep) + + for sub in reversed(w): + g = g.subs(sub[0], sub[1]) + g = trigsimpfunc(g, deep) + result = g + else: + result = trigsimpfunc(expr, deep) + + if opts.get('compare', False): + f = futrig(old) + if f != result: + print('\tfutrig:', f) + + return result + + +def _dotrig(a, b): + """Helper to tell whether ``a`` and ``b`` have the same sorts + of symbols in them -- no need to test hyperbolic patterns against + expressions that have no hyperbolics in them.""" + return a.func == b.func and ( + a.has(TrigonometricFunction) and b.has(TrigonometricFunction) or + a.has(HyperbolicFunction) and b.has(HyperbolicFunction)) + + +_trigpat = None +def _trigpats(): + global _trigpat + a, b, c = symbols('a b c', cls=Wild) + d = Wild('d', commutative=False) + + # for the simplifications like sinh/cosh -> tanh: + # DO NOT REORDER THE FIRST 14 since these are assumed to be in this + # order in _match_div_rewrite. + matchers_division = ( + (a*sin(b)**c/cos(b)**c, a*tan(b)**c, sin(b), cos(b)), + (a*tan(b)**c*cos(b)**c, a*sin(b)**c, sin(b), cos(b)), + (a*cot(b)**c*sin(b)**c, a*cos(b)**c, sin(b), cos(b)), + (a*tan(b)**c/sin(b)**c, a/cos(b)**c, sin(b), cos(b)), + (a*cot(b)**c/cos(b)**c, a/sin(b)**c, sin(b), cos(b)), + (a*cot(b)**c*tan(b)**c, a, sin(b), cos(b)), + (a*(cos(b) + 1)**c*(cos(b) - 1)**c, + a*(-sin(b)**2)**c, cos(b) + 1, cos(b) - 1), + (a*(sin(b) + 1)**c*(sin(b) - 1)**c, + a*(-cos(b)**2)**c, sin(b) + 1, sin(b) - 1), + + (a*sinh(b)**c/cosh(b)**c, a*tanh(b)**c, S.One, S.One), + (a*tanh(b)**c*cosh(b)**c, a*sinh(b)**c, S.One, S.One), + (a*coth(b)**c*sinh(b)**c, a*cosh(b)**c, S.One, S.One), + (a*tanh(b)**c/sinh(b)**c, a/cosh(b)**c, S.One, S.One), + (a*coth(b)**c/cosh(b)**c, a/sinh(b)**c, S.One, S.One), + (a*coth(b)**c*tanh(b)**c, a, S.One, S.One), + + (c*(tanh(a) + tanh(b))/(1 + tanh(a)*tanh(b)), + tanh(a + b)*c, S.One, S.One), + ) + + matchers_add = ( + (c*sin(a)*cos(b) + c*cos(a)*sin(b) + d, sin(a + b)*c + d), + (c*cos(a)*cos(b) - c*sin(a)*sin(b) + d, cos(a + b)*c + d), + (c*sin(a)*cos(b) - c*cos(a)*sin(b) + d, sin(a - b)*c + d), + (c*cos(a)*cos(b) + c*sin(a)*sin(b) + d, cos(a - b)*c + d), + (c*sinh(a)*cosh(b) + c*sinh(b)*cosh(a) + d, sinh(a + b)*c + d), + (c*cosh(a)*cosh(b) + c*sinh(a)*sinh(b) + d, cosh(a + b)*c + d), + ) + + # for cos(x)**2 + sin(x)**2 -> 1 + matchers_identity = ( + (a*sin(b)**2, a - a*cos(b)**2), + (a*tan(b)**2, a*(1/cos(b))**2 - a), + (a*cot(b)**2, a*(1/sin(b))**2 - a), + (a*sin(b + c), a*(sin(b)*cos(c) + sin(c)*cos(b))), + (a*cos(b + c), a*(cos(b)*cos(c) - sin(b)*sin(c))), + (a*tan(b + c), a*((tan(b) + tan(c))/(1 - tan(b)*tan(c)))), + + (a*sinh(b)**2, a*cosh(b)**2 - a), + (a*tanh(b)**2, a - a*(1/cosh(b))**2), + (a*coth(b)**2, a + a*(1/sinh(b))**2), + (a*sinh(b + c), a*(sinh(b)*cosh(c) + sinh(c)*cosh(b))), + (a*cosh(b + c), a*(cosh(b)*cosh(c) + sinh(b)*sinh(c))), + (a*tanh(b + c), a*((tanh(b) + tanh(c))/(1 + tanh(b)*tanh(c)))), + + ) + + # Reduce any lingering artifacts, such as sin(x)**2 changing + # to 1-cos(x)**2 when sin(x)**2 was "simpler" + artifacts = ( + (a - a*cos(b)**2 + c, a*sin(b)**2 + c, cos), + (a - a*(1/cos(b))**2 + c, -a*tan(b)**2 + c, cos), + (a - a*(1/sin(b))**2 + c, -a*cot(b)**2 + c, sin), + + (a - a*cosh(b)**2 + c, -a*sinh(b)**2 + c, cosh), + (a - a*(1/cosh(b))**2 + c, a*tanh(b)**2 + c, cosh), + (a + a*(1/sinh(b))**2 + c, a*coth(b)**2 + c, sinh), + + # same as above but with noncommutative prefactor + (a*d - a*d*cos(b)**2 + c, a*d*sin(b)**2 + c, cos), + (a*d - a*d*(1/cos(b))**2 + c, -a*d*tan(b)**2 + c, cos), + (a*d - a*d*(1/sin(b))**2 + c, -a*d*cot(b)**2 + c, sin), + + (a*d - a*d*cosh(b)**2 + c, -a*d*sinh(b)**2 + c, cosh), + (a*d - a*d*(1/cosh(b))**2 + c, a*d*tanh(b)**2 + c, cosh), + (a*d + a*d*(1/sinh(b))**2 + c, a*d*coth(b)**2 + c, sinh), + ) + + _trigpat = (a, b, c, d, matchers_division, matchers_add, + matchers_identity, artifacts) + return _trigpat + + +def _replace_mul_fpowxgpow(expr, f, g, rexp, h, rexph): + """Helper for _match_div_rewrite. + + Replace f(b_)**c_*g(b_)**(rexp(c_)) with h(b)**rexph(c) if f(b_) + and g(b_) are both positive or if c_ is an integer. + """ + # assert expr.is_Mul and expr.is_commutative and f != g + fargs = defaultdict(int) + gargs = defaultdict(int) + args = [] + for x in expr.args: + if x.is_Pow or x.func in (f, g): + b, e = x.as_base_exp() + if b.is_positive or e.is_integer: + if b.func == f: + fargs[b.args[0]] += e + continue + elif b.func == g: + gargs[b.args[0]] += e + continue + args.append(x) + common = set(fargs) & set(gargs) + hit = False + while common: + key = common.pop() + fe = fargs.pop(key) + ge = gargs.pop(key) + if fe == rexp(ge): + args.append(h(key)**rexph(fe)) + hit = True + else: + fargs[key] = fe + gargs[key] = ge + if not hit: + return expr + while fargs: + key, e = fargs.popitem() + args.append(f(key)**e) + while gargs: + key, e = gargs.popitem() + args.append(g(key)**e) + return Mul(*args) + + +_idn = lambda x: x +_midn = lambda x: -x +_one = lambda x: S.One + +def _match_div_rewrite(expr, i): + """helper for __trigsimp""" + if i == 0: + expr = _replace_mul_fpowxgpow(expr, sin, cos, + _midn, tan, _idn) + elif i == 1: + expr = _replace_mul_fpowxgpow(expr, tan, cos, + _idn, sin, _idn) + elif i == 2: + expr = _replace_mul_fpowxgpow(expr, cot, sin, + _idn, cos, _idn) + elif i == 3: + expr = _replace_mul_fpowxgpow(expr, tan, sin, + _midn, cos, _midn) + elif i == 4: + expr = _replace_mul_fpowxgpow(expr, cot, cos, + _midn, sin, _midn) + elif i == 5: + expr = _replace_mul_fpowxgpow(expr, cot, tan, + _idn, _one, _idn) + # i in (6, 7) is skipped + elif i == 8: + expr = _replace_mul_fpowxgpow(expr, sinh, cosh, + _midn, tanh, _idn) + elif i == 9: + expr = _replace_mul_fpowxgpow(expr, tanh, cosh, + _idn, sinh, _idn) + elif i == 10: + expr = _replace_mul_fpowxgpow(expr, coth, sinh, + _idn, cosh, _idn) + elif i == 11: + expr = _replace_mul_fpowxgpow(expr, tanh, sinh, + _midn, cosh, _midn) + elif i == 12: + expr = _replace_mul_fpowxgpow(expr, coth, cosh, + _midn, sinh, _midn) + elif i == 13: + expr = _replace_mul_fpowxgpow(expr, coth, tanh, + _idn, _one, _idn) + else: + return None + return expr + + +def _trigsimp(expr, deep=False): + # protect the cache from non-trig patterns; we only allow + # trig patterns to enter the cache + if expr.has(*_trigs): + return __trigsimp(expr, deep) + return expr + + +@cacheit +def __trigsimp(expr, deep=False): + """recursive helper for trigsimp""" + from sympy.simplify.fu import TR10i + + if _trigpat is None: + _trigpats() + a, b, c, d, matchers_division, matchers_add, \ + matchers_identity, artifacts = _trigpat + + if expr.is_Mul: + # do some simplifications like sin/cos -> tan: + if not expr.is_commutative: + com, nc = expr.args_cnc() + expr = _trigsimp(Mul._from_args(com), deep)*Mul._from_args(nc) + else: + for i, (pattern, simp, ok1, ok2) in enumerate(matchers_division): + if not _dotrig(expr, pattern): + continue + + newexpr = _match_div_rewrite(expr, i) + if newexpr is not None: + if newexpr != expr: + expr = newexpr + break + else: + continue + + # use SymPy matching instead + res = expr.match(pattern) + if res and res.get(c, 0): + if not res[c].is_integer: + ok = ok1.subs(res) + if not ok.is_positive: + continue + ok = ok2.subs(res) + if not ok.is_positive: + continue + # if "a" contains any of trig or hyperbolic funcs with + # argument "b" then skip the simplification + if any(w.args[0] == res[b] for w in res[a].atoms( + TrigonometricFunction, HyperbolicFunction)): + continue + # simplify and finish: + expr = simp.subs(res) + break # process below + + if expr.is_Add: + args = [] + for term in expr.args: + if not term.is_commutative: + com, nc = term.args_cnc() + nc = Mul._from_args(nc) + term = Mul._from_args(com) + else: + nc = S.One + term = _trigsimp(term, deep) + for pattern, result in matchers_identity: + res = term.match(pattern) + if res is not None: + term = result.subs(res) + break + args.append(term*nc) + if args != expr.args: + expr = Add(*args) + expr = min(expr, expand(expr), key=count_ops) + if expr.is_Add: + for pattern, result in matchers_add: + if not _dotrig(expr, pattern): + continue + expr = TR10i(expr) + if expr.has(HyperbolicFunction): + res = expr.match(pattern) + # if "d" contains any trig or hyperbolic funcs with + # argument "a" or "b" then skip the simplification; + # this isn't perfect -- see tests + if res is None or not (a in res and b in res) or any( + w.args[0] in (res[a], res[b]) for w in res[d].atoms( + TrigonometricFunction, HyperbolicFunction)): + continue + expr = result.subs(res) + break + + # Reduce any lingering artifacts, such as sin(x)**2 changing + # to 1 - cos(x)**2 when sin(x)**2 was "simpler" + for pattern, result, ex in artifacts: + if not _dotrig(expr, pattern): + continue + # Substitute a new wild that excludes some function(s) + # to help influence a better match. This is because + # sometimes, for example, 'a' would match sec(x)**2 + a_t = Wild('a', exclude=[ex]) + pattern = pattern.subs(a, a_t) + result = result.subs(a, a_t) + + m = expr.match(pattern) + was = None + while m and was != expr: + was = expr + if m[a_t] == 0 or \ + -m[a_t] in m[c].args or m[a_t] + m[c] == 0: + break + if d in m and m[a_t]*m[d] + m[c] == 0: + break + expr = result.subs(m) + m = expr.match(pattern) + m.setdefault(c, S.Zero) + + elif expr.is_Mul or expr.is_Pow or deep and expr.args: + expr = expr.func(*[_trigsimp(a, deep) for a in expr.args]) + + try: + if not expr.has(*_trigs): + raise TypeError + e = expr.atoms(exp) + new = expr.rewrite(exp, deep=deep) + if new == e: + raise TypeError + fnew = factor(new) + if fnew != new: + new = min([new, factor(new)], key=count_ops) + # if all exp that were introduced disappeared then accept it + if not (new.atoms(exp) - e): + expr = new + except TypeError: + pass + + return expr +#------------------- end of old trigsimp routines -------------------- + + +def futrig(e, *, hyper=True, **kwargs): + """Return simplified ``e`` using Fu-like transformations. + This is not the "Fu" algorithm. This is called by default + from ``trigsimp``. By default, hyperbolics subexpressions + will be simplified, but this can be disabled by setting + ``hyper=False``. + + Examples + ======== + + >>> from sympy import trigsimp, tan, sinh, tanh + >>> from sympy.simplify.trigsimp import futrig + >>> from sympy.abc import x + >>> trigsimp(1/tan(x)**2) + tan(x)**(-2) + + >>> futrig(sinh(x)/tanh(x)) + cosh(x) + + """ + from sympy.simplify.fu import hyper_as_trig + + e = sympify(e) + + if not isinstance(e, Basic): + return e + + if not e.args: + return e + + old = e + e = bottom_up(e, _futrig) + + if hyper and e.has(HyperbolicFunction): + e, f = hyper_as_trig(e) + e = f(bottom_up(e, _futrig)) + + if e != old and e.is_Mul and e.args[0].is_Rational: + # redistribute leading coeff on 2-arg Add + e = Mul(*e.as_coeff_Mul()) + return e + + +def _futrig(e): + """Helper for futrig.""" + from sympy.simplify.fu import ( + TR1, TR2, TR3, TR2i, TR10, L, TR10i, + TR8, TR6, TR15, TR16, TR111, TR5, TRmorrie, TR11, _TR11, TR14, TR22, + TR12) + + if not e.has(TrigonometricFunction): + return e + + if e.is_Mul: + coeff, e = e.as_independent(TrigonometricFunction) + else: + coeff = None + + Lops = lambda x: (L(x), x.count_ops(), _nodes(x), len(x.args), x.is_Add) + trigs = lambda x: x.has(TrigonometricFunction) + + tree = [identity, + ( + TR3, # canonical angles + TR1, # sec-csc -> cos-sin + TR12, # expand tan of sum + lambda x: _eapply(factor, x, trigs), + TR2, # tan-cot -> sin-cos + [identity, lambda x: _eapply(_mexpand, x, trigs)], + TR2i, # sin-cos ratio -> tan + lambda x: _eapply(lambda i: factor(i.normal()), x, trigs), + TR14, # factored identities + TR5, # sin-pow -> cos_pow + TR10, # sin-cos of sums -> sin-cos prod + TR11, _TR11, TR6, # reduce double angles and rewrite cos pows + lambda x: _eapply(factor, x, trigs), + TR14, # factored powers of identities + [identity, lambda x: _eapply(_mexpand, x, trigs)], + TR10i, # sin-cos products > sin-cos of sums + TRmorrie, + [identity, TR8], # sin-cos products -> sin-cos of sums + [identity, lambda x: TR2i(TR2(x))], # tan -> sin-cos -> tan + [ + lambda x: _eapply(expand_mul, TR5(x), trigs), + lambda x: _eapply( + expand_mul, TR15(x), trigs)], # pos/neg powers of sin + [ + lambda x: _eapply(expand_mul, TR6(x), trigs), + lambda x: _eapply( + expand_mul, TR16(x), trigs)], # pos/neg powers of cos + TR111, # tan, sin, cos to neg power -> cot, csc, sec + [identity, TR2i], # sin-cos ratio to tan + [identity, lambda x: _eapply( + expand_mul, TR22(x), trigs)], # tan-cot to sec-csc + TR1, TR2, TR2i, + [identity, lambda x: _eapply( + factor_terms, TR12(x), trigs)], # expand tan of sum + )] + e = greedy(tree, objective=Lops)(e) + + if coeff is not None: + e = coeff * e + + return e + + +def _is_Expr(e): + """_eapply helper to tell whether ``e`` and all its args + are Exprs.""" + if isinstance(e, Derivative): + return _is_Expr(e.expr) + if not isinstance(e, Expr): + return False + return all(_is_Expr(i) for i in e.args) + + +def _eapply(func, e, cond=None): + """Apply ``func`` to ``e`` if all args are Exprs else only + apply it to those args that *are* Exprs.""" + if not isinstance(e, Expr): + return e + if _is_Expr(e) or not e.args: + return func(e) + return e.func(*[ + _eapply(func, ei) if (cond is None or cond(ei)) else ei + for ei in e.args]) diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/__init__.py b/.venv/lib/python3.12/site-packages/sympy/solvers/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..02cfb35a765c748e70ecc36dd78d8d4432118c64 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/__init__.py @@ -0,0 +1,75 @@ +"""A module for solving all kinds of equations. + + Examples + ======== + + >>> from sympy.solvers import solve + >>> from sympy.abc import x + >>> solve(((x + 1)**5).expand(), x) + [-1] +""" +from sympy.core.assumptions import check_assumptions, failing_assumptions + +from .solvers import solve, solve_linear_system, solve_linear_system_LU, \ + solve_undetermined_coeffs, nsolve, solve_linear, checksol, \ + det_quick, inv_quick + +from sympy.solvers.diophantine.diophantine import diophantine + +from .recurr import rsolve, rsolve_poly, rsolve_ratio, rsolve_hyper + +from .ode import checkodesol, classify_ode, dsolve, \ + homogeneous_order + +from .polysys import solve_poly_system, solve_triangulated, factor_system + +from .pde import pde_separate, pde_separate_add, pde_separate_mul, \ + pdsolve, classify_pde, checkpdesol + +from .deutils import ode_order + +from .inequalities import reduce_inequalities, reduce_abs_inequality, \ + reduce_abs_inequalities, solve_poly_inequality, solve_rational_inequalities, solve_univariate_inequality + +from .decompogen import decompogen + +from .solveset import solveset, linsolve, linear_eq_to_matrix, nonlinsolve, substitution + +from .simplex import lpmin, lpmax, linprog + +# This is here instead of sympy/sets/__init__.py to avoid circular import issues +from ..core.singleton import S +Complexes = S.Complexes + +__all__ = [ + 'solve', 'solve_linear_system', 'solve_linear_system_LU', + 'solve_undetermined_coeffs', 'nsolve', 'solve_linear', 'checksol', + 'det_quick', 'inv_quick', 'check_assumptions', 'failing_assumptions', + + 'diophantine', + + 'rsolve', 'rsolve_poly', 'rsolve_ratio', 'rsolve_hyper', + + 'checkodesol', 'classify_ode', 'dsolve', 'homogeneous_order', + + 'solve_poly_system', 'solve_triangulated', 'factor_system', + + 'pde_separate', 'pde_separate_add', 'pde_separate_mul', 'pdsolve', + 'classify_pde', 'checkpdesol', + + 'ode_order', + + 'reduce_inequalities', 'reduce_abs_inequality', 'reduce_abs_inequalities', + 'solve_poly_inequality', 'solve_rational_inequalities', + 'solve_univariate_inequality', + + 'decompogen', + + 'solveset', 'linsolve', 'linear_eq_to_matrix', 'nonlinsolve', + 'substitution', + + # This is here instead of sympy/sets/__init__.py to avoid circular import issues + 'Complexes', + + 'lpmin', 'lpmax', 'linprog' +] diff --git 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b/.venv/lib/python3.12/site-packages/sympy/solvers/benchmarks/__pycache__/bench_solvers.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..d3a194bccfd8da5076447957dd05520280fdd6b2 Binary files /dev/null and b/.venv/lib/python3.12/site-packages/sympy/solvers/benchmarks/__pycache__/bench_solvers.cpython-312.pyc differ diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/benchmarks/bench_solvers.py b/.venv/lib/python3.12/site-packages/sympy/solvers/benchmarks/bench_solvers.py new file mode 100644 index 0000000000000000000000000000000000000000..d18102873f7efcde1d111e0e8eca12e208f94663 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/benchmarks/bench_solvers.py @@ -0,0 +1,12 @@ +from sympy.core.symbol import Symbol +from sympy.matrices.dense import (eye, zeros) +from sympy.solvers.solvers import solve_linear_system + +N = 8 +M = zeros(N, N + 1) +M[:, :N] = eye(N) +S = [Symbol('A%i' % i) for i in range(N)] + + +def timeit_linsolve_trivial(): + solve_linear_system(M, *S) diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/bivariate.py b/.venv/lib/python3.12/site-packages/sympy/solvers/bivariate.py new file mode 100644 index 0000000000000000000000000000000000000000..72f8e266a16634fa65366e1058543dfe2171ba1c --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/bivariate.py @@ -0,0 +1,509 @@ +from sympy.core.add import Add +from sympy.core.exprtools import factor_terms +from sympy.core.function import expand_log, _mexpand +from sympy.core.power import Pow +from sympy.core.singleton import S +from sympy.core.sorting import ordered +from sympy.core.symbol import Dummy +from sympy.functions.elementary.exponential import (LambertW, exp, log) +from sympy.functions.elementary.miscellaneous import root +from sympy.polys.polyroots import roots +from sympy.polys.polytools import Poly, factor +from sympy.simplify.simplify import separatevars +from sympy.simplify.radsimp import collect +from sympy.simplify.simplify import powsimp +from sympy.solvers.solvers import solve, _invert +from sympy.utilities.iterables import uniq + + +def _filtered_gens(poly, symbol): + """process the generators of ``poly``, returning the set of generators that + have ``symbol``. If there are two generators that are inverses of each other, + prefer the one that has no denominator. + + Examples + ======== + + >>> from sympy.solvers.bivariate import _filtered_gens + >>> from sympy import Poly, exp + >>> from sympy.abc import x + >>> _filtered_gens(Poly(x + 1/x + exp(x)), x) + {x, exp(x)} + + """ + # TODO it would be good to pick the smallest divisible power + # instead of the base for something like x**4 + x**2 --> + # return x**2 not x + gens = {g for g in poly.gens if symbol in g.free_symbols} + for g in list(gens): + ag = 1/g + if g in gens and ag in gens: + if ag.as_numer_denom()[1] is not S.One: + g = ag + gens.remove(g) + return gens + + +def _mostfunc(lhs, func, X=None): + """Returns the term in lhs which contains the most of the + func-type things e.g. log(log(x)) wins over log(x) if both terms appear. + + ``func`` can be a function (exp, log, etc...) or any other SymPy object, + like Pow. + + If ``X`` is not ``None``, then the function returns the term composed with the + most ``func`` having the specified variable. + + Examples + ======== + + >>> from sympy.solvers.bivariate import _mostfunc + >>> from sympy import exp + >>> from sympy.abc import x, y + >>> _mostfunc(exp(x) + exp(exp(x) + 2), exp) + exp(exp(x) + 2) + >>> _mostfunc(exp(x) + exp(exp(y) + 2), exp) + exp(exp(y) + 2) + >>> _mostfunc(exp(x) + exp(exp(y) + 2), exp, x) + exp(x) + >>> _mostfunc(x, exp, x) is None + True + >>> _mostfunc(exp(x) + exp(x*y), exp, x) + exp(x) + """ + fterms = [tmp for tmp in lhs.atoms(func) if (not X or + X.is_Symbol and X in tmp.free_symbols or + not X.is_Symbol and tmp.has(X))] + if len(fterms) == 1: + return fterms[0] + elif fterms: + return max(list(ordered(fterms)), key=lambda x: x.count(func)) + return None + + +def _linab(arg, symbol): + """Return ``a, b, X`` assuming ``arg`` can be written as ``a*X + b`` + where ``X`` is a symbol-dependent factor and ``a`` and ``b`` are + independent of ``symbol``. + + Examples + ======== + + >>> from sympy.solvers.bivariate import _linab + >>> from sympy.abc import x, y + >>> from sympy import exp, S + >>> _linab(S(2), x) + (2, 0, 1) + >>> _linab(2*x, x) + (2, 0, x) + >>> _linab(y + y*x + 2*x, x) + (y + 2, y, x) + >>> _linab(3 + 2*exp(x), x) + (2, 3, exp(x)) + """ + arg = factor_terms(arg.expand()) + ind, dep = arg.as_independent(symbol) + if arg.is_Mul and dep.is_Add: + a, b, x = _linab(dep, symbol) + return ind*a, ind*b, x + if not arg.is_Add: + b = 0 + a, x = ind, dep + else: + b = ind + a, x = separatevars(dep).as_independent(symbol, as_Add=False) + if x.could_extract_minus_sign(): + a = -a + x = -x + return a, b, x + + +def _lambert(eq, x): + """ + Given an expression assumed to be in the form + ``F(X, a..f) = a*log(b*X + c) + d*X + f = 0`` + where X = g(x) and x = g^-1(X), return the Lambert solution, + ``x = g^-1(-c/b + (a/d)*W(d/(a*b)*exp(c*d/a/b)*exp(-f/a)))``. + """ + eq = _mexpand(expand_log(eq)) + mainlog = _mostfunc(eq, log, x) + if not mainlog: + return [] # violated assumptions + other = eq.subs(mainlog, 0) + if isinstance(-other, log): + eq = (eq - other).subs(mainlog, mainlog.args[0]) + mainlog = mainlog.args[0] + if not isinstance(mainlog, log): + return [] # violated assumptions + other = -(-other).args[0] + eq += other + if x not in other.free_symbols: + return [] # violated assumptions + d, f, X2 = _linab(other, x) + logterm = collect(eq - other, mainlog) + a = logterm.as_coefficient(mainlog) + if a is None or x in a.free_symbols: + return [] # violated assumptions + logarg = mainlog.args[0] + b, c, X1 = _linab(logarg, x) + if X1 != X2: + return [] # violated assumptions + + # invert the generator X1 so we have x(u) + u = Dummy('rhs') + xusolns = solve(X1 - u, x) + + # There are infinitely many branches for LambertW + # but only branches for k = -1 and 0 might be real. The k = 0 + # branch is real and the k = -1 branch is real if the LambertW argument + # in in range [-1/e, 0]. Since `solve` does not return infinite + # solutions we will only include the -1 branch if it tests as real. + # Otherwise, inclusion of any LambertW in the solution indicates to + # the user that there are imaginary solutions corresponding to + # different k values. + lambert_real_branches = [-1, 0] + sol = [] + + # solution of the given Lambert equation is like + # sol = -c/b + (a/d)*LambertW(arg, k), + # where arg = d/(a*b)*exp((c*d-b*f)/a/b) and k in lambert_real_branches. + # Instead of considering the single arg, `d/(a*b)*exp((c*d-b*f)/a/b)`, + # the individual `p` roots obtained when writing `exp((c*d-b*f)/a/b)` + # as `exp(A/p) = exp(A)**(1/p)`, where `p` is an Integer, are used. + + # calculating args for LambertW + num, den = ((c*d-b*f)/a/b).as_numer_denom() + p, den = den.as_coeff_Mul() + e = exp(num/den) + t = Dummy('t') + args = [d/(a*b)*t for t in roots(t**p - e, t).keys()] + + # calculating solutions from args + for arg in args: + for k in lambert_real_branches: + w = LambertW(arg, k) + if k and not w.is_real: + continue + rhs = -c/b + (a/d)*w + + sol.extend(xu.subs(u, rhs) for xu in xusolns) + return sol + + +def _solve_lambert(f, symbol, gens): + """Return solution to ``f`` if it is a Lambert-type expression + else raise NotImplementedError. + + For ``f(X, a..f) = a*log(b*X + c) + d*X - f = 0`` the solution + for ``X`` is ``X = -c/b + (a/d)*W(d/(a*b)*exp(c*d/a/b)*exp(f/a))``. + There are a variety of forms for `f(X, a..f)` as enumerated below: + + 1a1) + if B**B = R for R not in [0, 1] (since those cases would already + be solved before getting here) then log of both sides gives + log(B) + log(log(B)) = log(log(R)) and + X = log(B), a = 1, b = 1, c = 0, d = 1, f = log(log(R)) + 1a2) + if B*(b*log(B) + c)**a = R then log of both sides gives + log(B) + a*log(b*log(B) + c) = log(R) and + X = log(B), d=1, f=log(R) + 1b) + if a*log(b*B + c) + d*B = R and + X = B, f = R + 2a) + if (b*B + c)*exp(d*B + g) = R then log of both sides gives + log(b*B + c) + d*B + g = log(R) and + X = B, a = 1, f = log(R) - g + 2b) + if g*exp(d*B + h) - b*B = c then the log form is + log(g) + d*B + h - log(b*B + c) = 0 and + X = B, a = -1, f = -h - log(g) + 3) + if d*p**(a*B + g) - b*B = c then the log form is + log(d) + (a*B + g)*log(p) - log(b*B + c) = 0 and + X = B, a = -1, d = a*log(p), f = -log(d) - g*log(p) + """ + + def _solve_even_degree_expr(expr, t, symbol): + """Return the unique solutions of equations derived from + ``expr`` by replacing ``t`` with ``+/- symbol``. + + Parameters + ========== + + expr : Expr + The expression which includes a dummy variable t to be + replaced with +symbol and -symbol. + + symbol : Symbol + The symbol for which a solution is being sought. + + Returns + ======= + + List of unique solution of the two equations generated by + replacing ``t`` with positive and negative ``symbol``. + + Notes + ===== + + If ``expr = 2*log(t) + x/2` then solutions for + ``2*log(x) + x/2 = 0`` and ``2*log(-x) + x/2 = 0`` are + returned by this function. Though this may seem + counter-intuitive, one must note that the ``expr`` being + solved here has been derived from a different expression. For + an expression like ``eq = x**2*g(x) = 1``, if we take the + log of both sides we obtain ``log(x**2) + log(g(x)) = 0``. If + x is positive then this simplifies to + ``2*log(x) + log(g(x)) = 0``; the Lambert-solving routines will + return solutions for this, but we must also consider the + solutions for ``2*log(-x) + log(g(x))`` since those must also + be a solution of ``eq`` which has the same value when the ``x`` + in ``x**2`` is negated. If `g(x)` does not have even powers of + symbol then we do not want to replace the ``x`` there with + ``-x``. So the role of the ``t`` in the expression received by + this function is to mark where ``+/-x`` should be inserted + before obtaining the Lambert solutions. + + """ + nlhs, plhs = [ + expr.xreplace({t: sgn*symbol}) for sgn in (-1, 1)] + sols = _solve_lambert(nlhs, symbol, gens) + if plhs != nlhs: + sols.extend(_solve_lambert(plhs, symbol, gens)) + # uniq is needed for a case like + # 2*log(t) - log(-z**2) + log(z + log(x) + log(z)) + # where substituting t with +/-x gives all the same solution; + # uniq, rather than list(set()), is used to maintain canonical + # order + return list(uniq(sols)) + + nrhs, lhs = f.as_independent(symbol, as_Add=True) + rhs = -nrhs + + lamcheck = [tmp for tmp in gens + if (tmp.func in [exp, log] or + (tmp.is_Pow and symbol in tmp.exp.free_symbols))] + if not lamcheck: + raise NotImplementedError() + + if lhs.is_Add or lhs.is_Mul: + # replacing all even_degrees of symbol with dummy variable t + # since these will need special handling; non-Add/Mul do not + # need this handling + t = Dummy('t', **symbol.assumptions0) + lhs = lhs.replace( + lambda i: # find symbol**even + i.is_Pow and i.base == symbol and i.exp.is_even, + lambda i: # replace t**even + t**i.exp) + + if lhs.is_Add and lhs.has(t): + t_indep = lhs.subs(t, 0) + t_term = lhs - t_indep + _rhs = rhs - t_indep + if not t_term.is_Add and _rhs and not ( + t_term.has(S.ComplexInfinity, S.NaN)): + eq = expand_log(log(t_term) - log(_rhs)) + return _solve_even_degree_expr(eq, t, symbol) + elif lhs.is_Mul and rhs: + # this needs to happen whether t is present or not + lhs = expand_log(log(lhs), force=True) + rhs = log(rhs) + if lhs.has(t) and lhs.is_Add: + # it expanded from Mul to Add + eq = lhs - rhs + return _solve_even_degree_expr(eq, t, symbol) + + # restore symbol in lhs + lhs = lhs.xreplace({t: symbol}) + + lhs = powsimp(factor(lhs, deep=True)) + + # make sure we have inverted as completely as possible + r = Dummy() + i, lhs = _invert(lhs - r, symbol) + rhs = i.xreplace({r: rhs}) + + # For the first forms: + # + # 1a1) B**B = R will arrive here as B*log(B) = log(R) + # lhs is Mul so take log of both sides: + # log(B) + log(log(B)) = log(log(R)) + # 1a2) B*(b*log(B) + c)**a = R will arrive unchanged so + # lhs is Mul, so take log of both sides: + # log(B) + a*log(b*log(B) + c) = log(R) + # 1b) d*log(a*B + b) + c*B = R will arrive unchanged so + # lhs is Add, so isolate c*B and expand log of both sides: + # log(c) + log(B) = log(R - d*log(a*B + b)) + + soln = [] + if not soln: + mainlog = _mostfunc(lhs, log, symbol) + if mainlog: + if lhs.is_Mul and rhs != 0: + soln = _lambert(log(lhs) - log(rhs), symbol) + elif lhs.is_Add: + other = lhs.subs(mainlog, 0) + if other and not other.is_Add and [ + tmp for tmp in other.atoms(Pow) + if symbol in tmp.free_symbols]: + if not rhs: + diff = log(other) - log(other - lhs) + else: + diff = log(lhs - other) - log(rhs - other) + soln = _lambert(expand_log(diff), symbol) + else: + #it's ready to go + soln = _lambert(lhs - rhs, symbol) + + # For the next forms, + # + # collect on main exp + # 2a) (b*B + c)*exp(d*B + g) = R + # lhs is mul, so take log of both sides: + # log(b*B + c) + d*B = log(R) - g + # 2b) g*exp(d*B + h) - b*B = R + # lhs is add, so add b*B to both sides, + # take the log of both sides and rearrange to give + # log(R + b*B) - d*B = log(g) + h + + if not soln: + mainexp = _mostfunc(lhs, exp, symbol) + if mainexp: + lhs = collect(lhs, mainexp) + if lhs.is_Mul and rhs != 0: + soln = _lambert(expand_log(log(lhs) - log(rhs)), symbol) + elif lhs.is_Add: + # move all but mainexp-containing term to rhs + other = lhs.subs(mainexp, 0) + mainterm = lhs - other + rhs = rhs - other + if (mainterm.could_extract_minus_sign() and + rhs.could_extract_minus_sign()): + mainterm *= -1 + rhs *= -1 + diff = log(mainterm) - log(rhs) + soln = _lambert(expand_log(diff), symbol) + + # For the last form: + # + # 3) d*p**(a*B + g) - b*B = c + # collect on main pow, add b*B to both sides, + # take log of both sides and rearrange to give + # a*B*log(p) - log(b*B + c) = -log(d) - g*log(p) + if not soln: + mainpow = _mostfunc(lhs, Pow, symbol) + if mainpow and symbol in mainpow.exp.free_symbols: + lhs = collect(lhs, mainpow) + if lhs.is_Mul and rhs != 0: + # b*B = 0 + soln = _lambert(expand_log(log(lhs) - log(rhs)), symbol) + elif lhs.is_Add: + # move all but mainpow-containing term to rhs + other = lhs.subs(mainpow, 0) + mainterm = lhs - other + rhs = rhs - other + diff = log(mainterm) - log(rhs) + soln = _lambert(expand_log(diff), symbol) + + if not soln: + raise NotImplementedError('%s does not appear to have a solution in ' + 'terms of LambertW' % f) + + return list(ordered(soln)) + + +def bivariate_type(f, x, y, *, first=True): + """Given an expression, f, 3 tests will be done to see what type + of composite bivariate it might be, options for u(x, y) are:: + + x*y + x+y + x*y+x + x*y+y + + If it matches one of these types, ``u(x, y)``, ``P(u)`` and dummy + variable ``u`` will be returned. Solving ``P(u)`` for ``u`` and + equating the solutions to ``u(x, y)`` and then solving for ``x`` or + ``y`` is equivalent to solving the original expression for ``x`` or + ``y``. If ``x`` and ``y`` represent two functions in the same + variable, e.g. ``x = g(t)`` and ``y = h(t)``, then if ``u(x, y) - p`` + can be solved for ``t`` then these represent the solutions to + ``P(u) = 0`` when ``p`` are the solutions of ``P(u) = 0``. + + Only positive values of ``u`` are considered. + + Examples + ======== + + >>> from sympy import solve + >>> from sympy.solvers.bivariate import bivariate_type + >>> from sympy.abc import x, y + >>> eq = (x**2 - 3).subs(x, x + y) + >>> bivariate_type(eq, x, y) + (x + y, _u**2 - 3, _u) + >>> uxy, pu, u = _ + >>> usol = solve(pu, u); usol + [sqrt(3)] + >>> [solve(uxy - s) for s in solve(pu, u)] + [[{x: -y + sqrt(3)}]] + >>> all(eq.subs(s).equals(0) for sol in _ for s in sol) + True + + """ + + u = Dummy('u', positive=True) + + if first: + p = Poly(f, x, y) + f = p.as_expr() + _x = Dummy() + _y = Dummy() + rv = bivariate_type(Poly(f.subs({x: _x, y: _y}), _x, _y), _x, _y, first=False) + if rv: + reps = {_x: x, _y: y} + return rv[0].xreplace(reps), rv[1].xreplace(reps), rv[2] + return + + p = f + f = p.as_expr() + + # f(x*y) + args = Add.make_args(p.as_expr()) + new = [] + for a in args: + a = _mexpand(a.subs(x, u/y)) + free = a.free_symbols + if x in free or y in free: + break + new.append(a) + else: + return x*y, Add(*new), u + + def ok(f, v, c): + new = _mexpand(f.subs(v, c)) + free = new.free_symbols + return None if (x in free or y in free) else new + + # f(a*x + b*y) + new = [] + d = p.degree(x) + if p.degree(y) == d: + a = root(p.coeff_monomial(x**d), d) + b = root(p.coeff_monomial(y**d), d) + new = ok(f, x, (u - b*y)/a) + if new is not None: + return a*x + b*y, new, u + + # f(a*x*y + b*y) + new = [] + d = p.degree(x) + if p.degree(y) == d: + for itry in range(2): + a = root(p.coeff_monomial(x**d*y**d), d) + b = root(p.coeff_monomial(y**d), d) + new = ok(f, x, (u - b*y)/a/y) + if new is not None: + return a*x*y + b*y, new, u + x, y = y, x diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/decompogen.py b/.venv/lib/python3.12/site-packages/sympy/solvers/decompogen.py new file mode 100644 index 0000000000000000000000000000000000000000..ec1b3b683511a34e6f98b9839d112b87517390d8 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/decompogen.py @@ -0,0 +1,126 @@ +from sympy.core import (Function, Pow, sympify, Expr) +from sympy.core.relational import Relational +from sympy.core.singleton import S +from sympy.polys import Poly, decompose +from sympy.utilities.misc import func_name +from sympy.functions.elementary.miscellaneous import Min, Max + + +def decompogen(f, symbol): + """ + Computes General functional decomposition of ``f``. + Given an expression ``f``, returns a list ``[f_1, f_2, ..., f_n]``, + where:: + f = f_1 o f_2 o ... f_n = f_1(f_2(... f_n)) + + Note: This is a General decomposition function. It also decomposes + Polynomials. For only Polynomial decomposition see ``decompose`` in polys. + + Examples + ======== + + >>> from sympy.abc import x + >>> from sympy import decompogen, sqrt, sin, cos + >>> decompogen(sin(cos(x)), x) + [sin(x), cos(x)] + >>> decompogen(sin(x)**2 + sin(x) + 1, x) + [x**2 + x + 1, sin(x)] + >>> decompogen(sqrt(6*x**2 - 5), x) + [sqrt(x), 6*x**2 - 5] + >>> decompogen(sin(sqrt(cos(x**2 + 1))), x) + [sin(x), sqrt(x), cos(x), x**2 + 1] + >>> decompogen(x**4 + 2*x**3 - x - 1, x) + [x**2 - x - 1, x**2 + x] + + """ + f = sympify(f) + if not isinstance(f, Expr) or isinstance(f, Relational): + raise TypeError('expecting Expr but got: `%s`' % func_name(f)) + if symbol not in f.free_symbols: + return [f] + + + # ===== Simple Functions ===== # + if isinstance(f, (Function, Pow)): + if f.is_Pow and f.base == S.Exp1: + arg = f.exp + else: + arg = f.args[0] + if arg == symbol: + return [f] + return [f.subs(arg, symbol)] + decompogen(arg, symbol) + + # ===== Min/Max Functions ===== # + if isinstance(f, (Min, Max)): + args = list(f.args) + d0 = None + for i, a in enumerate(args): + if not a.has_free(symbol): + continue + d = decompogen(a, symbol) + if len(d) == 1: + d = [symbol] + d + if d0 is None: + d0 = d[1:] + elif d[1:] != d0: + # decomposition is not the same for each arg: + # mark as having no decomposition + d = [symbol] + break + args[i] = d[0] + if d[0] == symbol: + return [f] + return [f.func(*args)] + d0 + + # ===== Convert to Polynomial ===== # + fp = Poly(f) + gens = list(filter(lambda x: symbol in x.free_symbols, fp.gens)) + + if len(gens) == 1 and gens[0] != symbol: + f1 = f.subs(gens[0], symbol) + f2 = gens[0] + return [f1] + decompogen(f2, symbol) + + # ===== Polynomial decompose() ====== # + try: + return decompose(f) + except ValueError: + return [f] + + +def compogen(g_s, symbol): + """ + Returns the composition of functions. + Given a list of functions ``g_s``, returns their composition ``f``, + where: + f = g_1 o g_2 o .. o g_n + + Note: This is a General composition function. It also composes Polynomials. + For only Polynomial composition see ``compose`` in polys. + + Examples + ======== + + >>> from sympy.solvers.decompogen import compogen + >>> from sympy.abc import x + >>> from sympy import sqrt, sin, cos + >>> compogen([sin(x), cos(x)], x) + sin(cos(x)) + >>> compogen([x**2 + x + 1, sin(x)], x) + sin(x)**2 + sin(x) + 1 + >>> compogen([sqrt(x), 6*x**2 - 5], x) + sqrt(6*x**2 - 5) + >>> compogen([sin(x), sqrt(x), cos(x), x**2 + 1], x) + sin(sqrt(cos(x**2 + 1))) + >>> compogen([x**2 - x - 1, x**2 + x], x) + -x**2 - x + (x**2 + x)**2 - 1 + """ + if len(g_s) == 1: + return g_s[0] + + foo = g_s[0].subs(symbol, g_s[1]) + + if len(g_s) == 2: + return foo + + return compogen([foo] + g_s[2:], symbol) diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/deutils.py b/.venv/lib/python3.12/site-packages/sympy/solvers/deutils.py new file mode 100644 index 0000000000000000000000000000000000000000..b13f37b5004fe7bce2545ad2c788ec3feae56025 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/deutils.py @@ -0,0 +1,273 @@ +"""Utility functions for classifying and solving +ordinary and partial differential equations. + +Contains +======== +_preprocess +ode_order +_desolve + +""" +from sympy.core import Pow +from sympy.core.function import Derivative, AppliedUndef +from sympy.core.relational import Equality +from sympy.core.symbol import Wild + +def _preprocess(expr, func=None, hint='_Integral'): + """Prepare expr for solving by making sure that differentiation + is done so that only func remains in unevaluated derivatives and + (if hint does not end with _Integral) that doit is applied to all + other derivatives. If hint is None, do not do any differentiation. + (Currently this may cause some simple differential equations to + fail.) + + In case func is None, an attempt will be made to autodetect the + function to be solved for. + + >>> from sympy.solvers.deutils import _preprocess + >>> from sympy import Derivative, Function + >>> from sympy.abc import x, y, z + >>> f, g = map(Function, 'fg') + + If f(x)**p == 0 and p>0 then we can solve for f(x)=0 + >>> _preprocess((f(x).diff(x)-4)**5, f(x)) + (Derivative(f(x), x) - 4, f(x)) + + Apply doit to derivatives that contain more than the function + of interest: + + >>> _preprocess(Derivative(f(x) + x, x)) + (Derivative(f(x), x) + 1, f(x)) + + Do others if the differentiation variable(s) intersect with those + of the function of interest or contain the function of interest: + + >>> _preprocess(Derivative(g(x), y, z), f(y)) + (0, f(y)) + >>> _preprocess(Derivative(f(y), z), f(y)) + (0, f(y)) + + Do others if the hint does not end in '_Integral' (the default + assumes that it does): + + >>> _preprocess(Derivative(g(x), y), f(x)) + (Derivative(g(x), y), f(x)) + >>> _preprocess(Derivative(f(x), y), f(x), hint='') + (0, f(x)) + + Do not do any derivatives if hint is None: + + >>> eq = Derivative(f(x) + 1, x) + Derivative(f(x), y) + >>> _preprocess(eq, f(x), hint=None) + (Derivative(f(x) + 1, x) + Derivative(f(x), y), f(x)) + + If it's not clear what the function of interest is, it must be given: + + >>> eq = Derivative(f(x) + g(x), x) + >>> _preprocess(eq, g(x)) + (Derivative(f(x), x) + Derivative(g(x), x), g(x)) + >>> try: _preprocess(eq) + ... except ValueError: print("A ValueError was raised.") + A ValueError was raised. + + """ + if isinstance(expr, Pow): + # if f(x)**p=0 then f(x)=0 (p>0) + if (expr.exp).is_positive: + expr = expr.base + derivs = expr.atoms(Derivative) + if not func: + funcs = set().union(*[d.atoms(AppliedUndef) for d in derivs]) + if len(funcs) != 1: + raise ValueError('The function cannot be ' + 'automatically detected for %s.' % expr) + func = funcs.pop() + fvars = set(func.args) + if hint is None: + return expr, func + reps = [(d, d.doit()) for d in derivs if not hint.endswith('_Integral') or + d.has(func) or set(d.variables) & fvars] + eq = expr.subs(reps) + return eq, func + + +def ode_order(expr, func): + """ + Returns the order of a given differential + equation with respect to func. + + This function is implemented recursively. + + Examples + ======== + + >>> from sympy import Function + >>> from sympy.solvers.deutils import ode_order + >>> from sympy.abc import x + >>> f, g = map(Function, ['f', 'g']) + >>> ode_order(f(x).diff(x, 2) + f(x).diff(x)**2 + + ... f(x).diff(x), f(x)) + 2 + >>> ode_order(f(x).diff(x, 2) + g(x).diff(x, 3), f(x)) + 2 + >>> ode_order(f(x).diff(x, 2) + g(x).diff(x, 3), g(x)) + 3 + + """ + a = Wild('a', exclude=[func]) + if expr.match(a): + return 0 + + if isinstance(expr, Derivative): + if expr.args[0] == func: + return len(expr.variables) + else: + args = expr.args[0].args + rv = len(expr.variables) + if args: + rv += max(ode_order(_, func) for _ in args) + return rv + else: + return max(ode_order(_, func) for _ in expr.args) if expr.args else 0 + + +def _desolve(eq, func=None, hint="default", ics=None, simplify=True, *, prep=True, **kwargs): + """This is a helper function to dsolve and pdsolve in the ode + and pde modules. + + If the hint provided to the function is "default", then a dict with + the following keys are returned + + 'func' - It provides the function for which the differential equation + has to be solved. This is useful when the expression has + more than one function in it. + + 'default' - The default key as returned by classifier functions in ode + and pde.py + + 'hint' - The hint given by the user for which the differential equation + is to be solved. If the hint given by the user is 'default', + then the value of 'hint' and 'default' is the same. + + 'order' - The order of the function as returned by ode_order + + 'match' - It returns the match as given by the classifier functions, for + the default hint. + + If the hint provided to the function is not "default" and is not in + ('all', 'all_Integral', 'best'), then a dict with the above mentioned keys + is returned along with the keys which are returned when dict in + classify_ode or classify_pde is set True + + If the hint given is in ('all', 'all_Integral', 'best'), then this function + returns a nested dict, with the keys, being the set of classified hints + returned by classifier functions, and the values being the dict of form + as mentioned above. + + Key 'eq' is a common key to all the above mentioned hints which returns an + expression if eq given by user is an Equality. + + See Also + ======== + classify_ode(ode.py) + classify_pde(pde.py) + """ + if isinstance(eq, Equality): + eq = eq.lhs - eq.rhs + + # preprocess the equation and find func if not given + if prep or func is None: + eq, func = _preprocess(eq, func) + prep = False + + # type is an argument passed by the solve functions in ode and pde.py + # that identifies whether the function caller is an ordinary + # or partial differential equation. Accordingly corresponding + # changes are made in the function. + type = kwargs.get('type', None) + xi = kwargs.get('xi') + eta = kwargs.get('eta') + x0 = kwargs.get('x0', 0) + terms = kwargs.get('n') + + if type == 'ode': + from sympy.solvers.ode import classify_ode, allhints + classifier = classify_ode + string = 'ODE ' + dummy = '' + + elif type == 'pde': + from sympy.solvers.pde import classify_pde, allhints + classifier = classify_pde + string = 'PDE ' + dummy = 'p' + + # Magic that should only be used internally. Prevents classify_ode from + # being called more than it needs to be by passing its results through + # recursive calls. + if kwargs.get('classify', True): + hints = classifier(eq, func, dict=True, ics=ics, xi=xi, eta=eta, + n=terms, x0=x0, hint=hint, prep=prep) + + else: + # Here is what all this means: + # + # hint: The hint method given to _desolve() by the user. + # hints: The dictionary of hints that match the DE, along with other + # information (including the internal pass-through magic). + # default: The default hint to return, the first hint from allhints + # that matches the hint; obtained from classify_ode(). + # match: Dictionary containing the match dictionary for each hint + # (the parts of the DE for solving). When going through the + # hints in "all", this holds the match string for the current + # hint. + # order: The order of the DE, as determined by ode_order(). + hints = kwargs.get('hint', + {'default': hint, + hint: kwargs['match'], + 'order': kwargs['order']}) + if not hints['default']: + # classify_ode will set hints['default'] to None if no hints match. + if hint not in allhints and hint != 'default': + raise ValueError("Hint not recognized: " + hint) + elif hint not in hints['ordered_hints'] and hint != 'default': + raise ValueError(string + str(eq) + " does not match hint " + hint) + # If dsolve can't solve the purely algebraic equation then dsolve will raise + # ValueError + elif hints['order'] == 0: + raise ValueError( + str(eq) + " is not a solvable differential equation in " + str(func)) + else: + raise NotImplementedError(dummy + "solve" + ": Cannot solve " + str(eq)) + if hint == 'default': + return _desolve(eq, func, ics=ics, hint=hints['default'], simplify=simplify, + prep=prep, x0=x0, classify=False, order=hints['order'], + match=hints[hints['default']], xi=xi, eta=eta, n=terms, type=type) + elif hint in ('all', 'all_Integral', 'best'): + retdict = {} + gethints = set(hints) - {'order', 'default', 'ordered_hints'} + if hint == 'all_Integral': + for i in hints: + if i.endswith('_Integral'): + gethints.remove(i.removesuffix('_Integral')) + # special cases + for k in ["1st_homogeneous_coeff_best", "1st_power_series", + "lie_group", "2nd_power_series_ordinary", "2nd_power_series_regular"]: + if k in gethints: + gethints.remove(k) + for i in gethints: + sol = _desolve(eq, func, ics=ics, hint=i, x0=x0, simplify=simplify, prep=prep, + classify=False, n=terms, order=hints['order'], match=hints[i], type=type) + retdict[i] = sol + retdict['all'] = True + retdict['eq'] = eq + return retdict + elif hint not in allhints: # and hint not in ('default', 'ordered_hints'): + raise ValueError("Hint not recognized: " + hint) + elif hint not in hints: + raise ValueError(string + str(eq) + " does not match hint " + hint) + else: + # Key added to identify the hint needed to solve the equation + hints['hint'] = hint + hints.update({'func': func, 'eq': eq}) + return hints diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/diophantine/__init__.py b/.venv/lib/python3.12/site-packages/sympy/solvers/diophantine/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..23c21242208d6f520c130250ecdce43382b9d868 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/diophantine/__init__.py @@ -0,0 +1,5 @@ +from .diophantine import diophantine, classify_diop, diop_solve + +__all__ = [ + 'diophantine', 'classify_diop', 'diop_solve' +] diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/diophantine/__pycache__/__init__.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/solvers/diophantine/__pycache__/__init__.cpython-312.pyc new file mode 100644 index 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100644 index 0000000000000000000000000000000000000000..ffdef6344451c96ed48dff099cf8f02494f4b504 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/diophantine/diophantine.py @@ -0,0 +1,3980 @@ +from __future__ import annotations + +from sympy.core.add import Add +from sympy.core.assumptions import check_assumptions +from sympy.core.containers import Tuple +from sympy.core.exprtools import factor_terms +from sympy.core.function import _mexpand +from sympy.core.mul import Mul +from sympy.core.numbers import Rational, int_valued +from sympy.core.intfunc import igcdex, ilcm, igcd, integer_nthroot, isqrt +from sympy.core.relational import Eq +from sympy.core.singleton import S +from sympy.core.sorting import default_sort_key, ordered +from sympy.core.symbol import Symbol, symbols +from sympy.core.sympify import _sympify +from sympy.external.gmpy import jacobi, remove, invert, iroot +from sympy.functions.elementary.complexes import sign +from sympy.functions.elementary.integers import floor +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.matrices.dense import MutableDenseMatrix as Matrix +from sympy.ntheory.factor_ import divisors, factorint, perfect_power +from sympy.ntheory.generate import nextprime +from sympy.ntheory.primetest import is_square, isprime +from sympy.ntheory.modular import symmetric_residue +from sympy.ntheory.residue_ntheory import sqrt_mod, sqrt_mod_iter +from sympy.polys.polyerrors import GeneratorsNeeded +from sympy.polys.polytools import Poly, factor_list +from sympy.simplify.simplify import signsimp +from sympy.solvers.solveset import solveset_real +from sympy.utilities import numbered_symbols +from sympy.utilities.misc import as_int, filldedent +from sympy.utilities.iterables import (is_sequence, subsets, permute_signs, + signed_permutations, ordered_partitions) + + +# these are imported with 'from sympy.solvers.diophantine import * +__all__ = ['diophantine', 'classify_diop'] + + +class DiophantineSolutionSet(set): + """ + Container for a set of solutions to a particular diophantine equation. + + The base representation is a set of tuples representing each of the solutions. + + Parameters + ========== + + symbols : list + List of free symbols in the original equation. + parameters: list + List of parameters to be used in the solution. + + Examples + ======== + + Adding solutions: + + >>> from sympy.solvers.diophantine.diophantine import DiophantineSolutionSet + >>> from sympy.abc import x, y, t, u + >>> s1 = DiophantineSolutionSet([x, y], [t, u]) + >>> s1 + set() + >>> s1.add((2, 3)) + >>> s1.add((-1, u)) + >>> s1 + {(-1, u), (2, 3)} + >>> s2 = DiophantineSolutionSet([x, y], [t, u]) + >>> s2.add((3, 4)) + >>> s1.update(*s2) + >>> s1 + {(-1, u), (2, 3), (3, 4)} + + Conversion of solutions into dicts: + + >>> list(s1.dict_iterator()) + [{x: -1, y: u}, {x: 2, y: 3}, {x: 3, y: 4}] + + Substituting values: + + >>> s3 = DiophantineSolutionSet([x, y], [t, u]) + >>> s3.add((t**2, t + u)) + >>> s3 + {(t**2, t + u)} + >>> s3.subs({t: 2, u: 3}) + {(4, 5)} + >>> s3.subs(t, -1) + {(1, u - 1)} + >>> s3.subs(t, 3) + {(9, u + 3)} + + Evaluation at specific values. Positional arguments are given in the same order as the parameters: + + >>> s3(-2, 3) + {(4, 1)} + >>> s3(5) + {(25, u + 5)} + >>> s3(None, 2) + {(t**2, t + 2)} + """ + + def __init__(self, symbols_seq, parameters): + super().__init__() + + if not is_sequence(symbols_seq): + raise ValueError("Symbols must be given as a sequence.") + + if not is_sequence(parameters): + raise ValueError("Parameters must be given as a sequence.") + + self.symbols = tuple(symbols_seq) + self.parameters = tuple(parameters) + + def add(self, solution): + if len(solution) != len(self.symbols): + raise ValueError("Solution should have a length of %s, not %s" % (len(self.symbols), len(solution))) + # make solution canonical wrt sign (i.e. no -x unless x is also present as an arg) + args = set(solution) + for i in range(len(solution)): + x = solution[i] + if not type(x) is int and (-x).is_Symbol and -x not in args: + solution = [_.subs(-x, x) for _ in solution] + super().add(Tuple(*solution)) + + def update(self, *solutions): + for solution in solutions: + self.add(solution) + + def dict_iterator(self): + for solution in ordered(self): + yield dict(zip(self.symbols, solution)) + + def subs(self, *args, **kwargs): + result = DiophantineSolutionSet(self.symbols, self.parameters) + for solution in self: + result.add(solution.subs(*args, **kwargs)) + return result + + def __call__(self, *args): + if len(args) > len(self.parameters): + raise ValueError("Evaluation should have at most %s values, not %s" % (len(self.parameters), len(args))) + rep = {p: v for p, v in zip(self.parameters, args) if v is not None} + return self.subs(rep) + + +class DiophantineEquationType: + """ + Internal representation of a particular diophantine equation type. + + Parameters + ========== + + equation : + The diophantine equation that is being solved. + free_symbols : list (optional) + The symbols being solved for. + + Attributes + ========== + + total_degree : + The maximum of the degrees of all terms in the equation + homogeneous : + Does the equation contain a term of degree 0 + homogeneous_order : + Does the equation contain any coefficient that is in the symbols being solved for + dimension : + The number of symbols being solved for + """ + name: str + + def __init__(self, equation, free_symbols=None): + self.equation = _sympify(equation).expand(force=True) + + if free_symbols is not None: + self.free_symbols = free_symbols + else: + self.free_symbols = list(self.equation.free_symbols) + self.free_symbols.sort(key=default_sort_key) + + if not self.free_symbols: + raise ValueError('equation should have 1 or more free symbols') + + self.coeff = self.equation.as_coefficients_dict() + if not all(int_valued(c) for c in self.coeff.values()): + raise TypeError("Coefficients should be Integers") + + self.total_degree = Poly(self.equation).total_degree() + self.homogeneous = 1 not in self.coeff + self.homogeneous_order = not (set(self.coeff) & set(self.free_symbols)) + self.dimension = len(self.free_symbols) + self._parameters = None + + def matches(self): + """ + Determine whether the given equation can be matched to the particular equation type. + """ + return False + + @property + def n_parameters(self): + return self.dimension + + @property + def parameters(self): + if self._parameters is None: + self._parameters = symbols('t_:%i' % (self.n_parameters,), integer=True) + return self._parameters + + def solve(self, parameters=None, limit=None) -> DiophantineSolutionSet: + raise NotImplementedError('No solver has been written for %s.' % self.name) + + def pre_solve(self, parameters=None): + if not self.matches(): + raise ValueError("This equation does not match the %s equation type." % self.name) + + if parameters is not None: + if len(parameters) != self.n_parameters: + raise ValueError("Expected %s parameter(s) but got %s" % (self.n_parameters, len(parameters))) + + self._parameters = parameters + + +class Univariate(DiophantineEquationType): + """ + Representation of a univariate diophantine equation. + + A univariate diophantine equation is an equation of the form + `a_{0} + a_{1}x + a_{2}x^2 + .. + a_{n}x^n = 0` where `a_{1}, a_{2}, ..a_{n}` are + integer constants and `x` is an integer variable. + + Examples + ======== + + >>> from sympy.solvers.diophantine.diophantine import Univariate + >>> from sympy.abc import x + >>> Univariate((x - 2)*(x - 3)**2).solve() # solves equation (x - 2)*(x - 3)**2 == 0 + {(2,), (3,)} + + """ + + name = 'univariate' + + def matches(self): + return self.dimension == 1 + + def solve(self, parameters=None, limit=None): + self.pre_solve(parameters) + + result = DiophantineSolutionSet(self.free_symbols, parameters=self.parameters) + for i in solveset_real(self.equation, self.free_symbols[0]).intersect(S.Integers): + result.add((i,)) + return result + + +class Linear(DiophantineEquationType): + """ + Representation of a linear diophantine equation. + + A linear diophantine equation is an equation of the form `a_{1}x_{1} + + a_{2}x_{2} + .. + a_{n}x_{n} = 0` where `a_{1}, a_{2}, ..a_{n}` are + integer constants and `x_{1}, x_{2}, ..x_{n}` are integer variables. + + Examples + ======== + + >>> from sympy.solvers.diophantine.diophantine import Linear + >>> from sympy.abc import x, y, z + >>> l1 = Linear(2*x - 3*y - 5) + >>> l1.matches() # is this equation linear + True + >>> l1.solve() # solves equation 2*x - 3*y - 5 == 0 + {(3*t_0 - 5, 2*t_0 - 5)} + + Here x = -3*t_0 - 5 and y = -2*t_0 - 5 + + >>> Linear(2*x - 3*y - 4*z -3).solve() + {(t_0, 2*t_0 + 4*t_1 + 3, -t_0 - 3*t_1 - 3)} + + """ + + name = 'linear' + + def matches(self): + return self.total_degree == 1 + + def solve(self, parameters=None, limit=None): + self.pre_solve(parameters) + + coeff = self.coeff + var = self.free_symbols + + if 1 in coeff: + # negate coeff[] because input is of the form: ax + by + c == 0 + # but is used as: ax + by == -c + c = -coeff[1] + else: + c = 0 + + result = DiophantineSolutionSet(var, parameters=self.parameters) + params = result.parameters + + if len(var) == 1: + q, r = divmod(c, coeff[var[0]]) + if not r: + result.add((q,)) + return result + + ''' + base_solution_linear() can solve diophantine equations of the form: + + a*x + b*y == c + + We break down multivariate linear diophantine equations into a + series of bivariate linear diophantine equations which can then + be solved individually by base_solution_linear(). + + Consider the following: + + a_0*x_0 + a_1*x_1 + a_2*x_2 == c + + which can be re-written as: + + a_0*x_0 + g_0*y_0 == c + + where + + g_0 == gcd(a_1, a_2) + + and + + y == (a_1*x_1)/g_0 + (a_2*x_2)/g_0 + + This leaves us with two binary linear diophantine equations. + For the first equation: + + a == a_0 + b == g_0 + c == c + + For the second: + + a == a_1/g_0 + b == a_2/g_0 + c == the solution we find for y_0 in the first equation. + + The arrays A and B are the arrays of integers used for + 'a' and 'b' in each of the n-1 bivariate equations we solve. + ''' + + A = [coeff[v] for v in var] + B = [] + if len(var) > 2: + B.append(igcd(A[-2], A[-1])) + A[-2] = A[-2] // B[0] + A[-1] = A[-1] // B[0] + for i in range(len(A) - 3, 0, -1): + gcd = igcd(B[0], A[i]) + B[0] = B[0] // gcd + A[i] = A[i] // gcd + B.insert(0, gcd) + B.append(A[-1]) + + ''' + Consider the trivariate linear equation: + + 4*x_0 + 6*x_1 + 3*x_2 == 2 + + This can be re-written as: + + 4*x_0 + 3*y_0 == 2 + + where + + y_0 == 2*x_1 + x_2 + (Note that gcd(3, 6) == 3) + + The complete integral solution to this equation is: + + x_0 == 2 + 3*t_0 + y_0 == -2 - 4*t_0 + + where 't_0' is any integer. + + Now that we have a solution for 'x_0', find 'x_1' and 'x_2': + + 2*x_1 + x_2 == -2 - 4*t_0 + + We can then solve for '-2' and '-4' independently, + and combine the results: + + 2*x_1a + x_2a == -2 + x_1a == 0 + t_0 + x_2a == -2 - 2*t_0 + + 2*x_1b + x_2b == -4*t_0 + x_1b == 0*t_0 + t_1 + x_2b == -4*t_0 - 2*t_1 + + ==> + + x_1 == t_0 + t_1 + x_2 == -2 - 6*t_0 - 2*t_1 + + where 't_0' and 't_1' are any integers. + + Note that: + + 4*(2 + 3*t_0) + 6*(t_0 + t_1) + 3*(-2 - 6*t_0 - 2*t_1) == 2 + + for any integral values of 't_0', 't_1'; as required. + + This method is generalised for many variables, below. + + ''' + solutions = [] + for Ai, Bi in zip(A, B): + tot_x, tot_y = [], [] + + for arg in Add.make_args(c): + if arg.is_Integer: + # example: 5 -> k = 5 + k, p = arg, S.One + pnew = params[0] + else: # arg is a Mul or Symbol + # example: 3*t_1 -> k = 3 + # example: t_0 -> k = 1 + k, p = arg.as_coeff_Mul() + pnew = params[params.index(p) + 1] + + sol = sol_x, sol_y = base_solution_linear(k, Ai, Bi, pnew) + + if p is S.One: + if None in sol: + return result + else: + # convert a + b*pnew -> a*p + b*pnew + if isinstance(sol_x, Add): + sol_x = sol_x.args[0]*p + sol_x.args[1] + if isinstance(sol_y, Add): + sol_y = sol_y.args[0]*p + sol_y.args[1] + + tot_x.append(sol_x) + tot_y.append(sol_y) + + solutions.append(Add(*tot_x)) + c = Add(*tot_y) + + solutions.append(c) + result.add(solutions) + return result + + +class BinaryQuadratic(DiophantineEquationType): + """ + Representation of a binary quadratic diophantine equation. + + A binary quadratic diophantine equation is an equation of the + form `Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0`, where `A, B, C, D, E, + F` are integer constants and `x` and `y` are integer variables. + + Examples + ======== + + >>> from sympy.abc import x, y + >>> from sympy.solvers.diophantine.diophantine import BinaryQuadratic + >>> b1 = BinaryQuadratic(x**3 + y**2 + 1) + >>> b1.matches() + False + >>> b2 = BinaryQuadratic(x**2 + y**2 + 2*x + 2*y + 2) + >>> b2.matches() + True + >>> b2.solve() + {(-1, -1)} + + References + ========== + + .. [1] Methods to solve Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, [online], + Available: https://www.alpertron.com.ar/METHODS.HTM + .. [2] Solving the equation ax^2+ bxy + cy^2 + dx + ey + f= 0, [online], + Available: https://web.archive.org/web/20160323033111/http://www.jpr2718.org/ax2p.pdf + + """ + + name = 'binary_quadratic' + + def matches(self): + return self.total_degree == 2 and self.dimension == 2 + + def solve(self, parameters=None, limit=None) -> DiophantineSolutionSet: + self.pre_solve(parameters) + + var = self.free_symbols + coeff = self.coeff + + x, y = var + + A = coeff[x**2] + B = coeff[x*y] + C = coeff[y**2] + D = coeff[x] + E = coeff[y] + F = coeff[S.One] + + A, B, C, D, E, F = [as_int(i) for i in _remove_gcd(A, B, C, D, E, F)] + + # (1) Simple-Hyperbolic case: A = C = 0, B != 0 + # In this case equation can be converted to (Bx + E)(By + D) = DE - BF + # We consider two cases; DE - BF = 0 and DE - BF != 0 + # More details, https://www.alpertron.com.ar/METHODS.HTM#SHyperb + + result = DiophantineSolutionSet(var, self.parameters) + t, u = result.parameters + + discr = B**2 - 4*A*C + if A == 0 and C == 0 and B != 0: + + if D*E - B*F == 0: + q, r = divmod(E, B) + if not r: + result.add((-q, t)) + q, r = divmod(D, B) + if not r: + result.add((t, -q)) + else: + div = divisors(D*E - B*F) + div = div + [-term for term in div] + for d in div: + x0, r = divmod(d - E, B) + if not r: + q, r = divmod(D*E - B*F, d) + if not r: + y0, r = divmod(q - D, B) + if not r: + result.add((x0, y0)) + + # (2) Parabolic case: B**2 - 4*A*C = 0 + # There are two subcases to be considered in this case. + # sqrt(c)D - sqrt(a)E = 0 and sqrt(c)D - sqrt(a)E != 0 + # More Details, https://www.alpertron.com.ar/METHODS.HTM#Parabol + + elif discr == 0: + + if A == 0: + s = BinaryQuadratic(self.equation, free_symbols=[y, x]).solve(parameters=[t, u]) + for soln in s: + result.add((soln[1], soln[0])) + + else: + g = sign(A)*igcd(A, C) + a = A // g + c = C // g + e = sign(B / A) + + sqa = isqrt(a) + sqc = isqrt(c) + _c = e*sqc*D - sqa*E + if not _c: + z = Symbol("z", real=True) + eq = sqa*g*z**2 + D*z + sqa*F + roots = solveset_real(eq, z).intersect(S.Integers) + for root in roots: + ans = diop_solve(sqa*x + e*sqc*y - root) + result.add((ans[0], ans[1])) + + elif int_valued(c): + solve_x = lambda u: -e*sqc*g*_c*t**2 - (E + 2*e*sqc*g*u)*t \ + - (e*sqc*g*u**2 + E*u + e*sqc*F) // _c + + solve_y = lambda u: sqa*g*_c*t**2 + (D + 2*sqa*g*u)*t \ + + (sqa*g*u**2 + D*u + sqa*F) // _c + + for z0 in range(0, abs(_c)): + # Check if the coefficients of y and x obtained are integers or not + if (divisible(sqa*g*z0**2 + D*z0 + sqa*F, _c) and + divisible(e*sqc*g*z0**2 + E*z0 + e*sqc*F, _c)): + result.add((solve_x(z0), solve_y(z0))) + + # (3) Method used when B**2 - 4*A*C is a square, is described in p. 6 of the below paper + # by John P. Robertson. + # https://web.archive.org/web/20160323033111/http://www.jpr2718.org/ax2p.pdf + + elif is_square(discr): + if A != 0: + r = sqrt(discr) + u, v = symbols("u, v", integer=True) + eq = _mexpand( + 4*A*r*u*v + 4*A*D*(B*v + r*u + r*v - B*u) + + 2*A*4*A*E*(u - v) + 4*A*r*4*A*F) + + solution = diop_solve(eq, t) + + for s0, t0 in solution: + + num = B*t0 + r*s0 + r*t0 - B*s0 + x_0 = S(num) / (4*A*r) + y_0 = S(s0 - t0) / (2*r) + if isinstance(s0, Symbol) or isinstance(t0, Symbol): + if len(check_param(x_0, y_0, 4*A*r, parameters)) > 0: + ans = check_param(x_0, y_0, 4*A*r, parameters) + result.update(*ans) + elif x_0.is_Integer and y_0.is_Integer: + if is_solution_quad(var, coeff, x_0, y_0): + result.add((x_0, y_0)) + + else: + s = BinaryQuadratic(self.equation, free_symbols=var[::-1]).solve(parameters=[t, u]) # Interchange x and y + while s: + result.add(s.pop()[::-1]) # and solution <--------+ + + # (4) B**2 - 4*A*C > 0 and B**2 - 4*A*C not a square or B**2 - 4*A*C < 0 + + else: + + P, Q = _transformation_to_DN(var, coeff) + D, N = _find_DN(var, coeff) + solns_pell = diop_DN(D, N) + + if D < 0: + for x0, y0 in solns_pell: + for x in [-x0, x0]: + for y in [-y0, y0]: + s = P*Matrix([x, y]) + Q + try: + result.add([as_int(_) for _ in s]) + except ValueError: + pass + else: + # In this case equation can be transformed into a Pell equation + + solns_pell = set(solns_pell) + solns_pell.update((-X, -Y) for X, Y in list(solns_pell)) + + a = diop_DN(D, 1) + T = a[0][0] + U = a[0][1] + + if all(int_valued(_) for _ in P[:4] + Q[:2]): + for r, s in solns_pell: + _a = (r + s*sqrt(D))*(T + U*sqrt(D))**t + _b = (r - s*sqrt(D))*(T - U*sqrt(D))**t + x_n = _mexpand(S(_a + _b) / 2) + y_n = _mexpand(S(_a - _b) / (2*sqrt(D))) + s = P*Matrix([x_n, y_n]) + Q + result.add(s) + + else: + L = ilcm(*[_.q for _ in P[:4] + Q[:2]]) + + k = 1 + + T_k = T + U_k = U + + while (T_k - 1) % L != 0 or U_k % L != 0: + T_k, U_k = T_k*T + D*U_k*U, T_k*U + U_k*T + k += 1 + + for X, Y in solns_pell: + + for i in range(k): + if all(int_valued(_) for _ in P*Matrix([X, Y]) + Q): + _a = (X + sqrt(D)*Y)*(T_k + sqrt(D)*U_k)**t + _b = (X - sqrt(D)*Y)*(T_k - sqrt(D)*U_k)**t + Xt = S(_a + _b) / 2 + Yt = S(_a - _b) / (2*sqrt(D)) + s = P*Matrix([Xt, Yt]) + Q + result.add(s) + + X, Y = X*T + D*U*Y, X*U + Y*T + + return result + + +class InhomogeneousTernaryQuadratic(DiophantineEquationType): + """ + + Representation of an inhomogeneous ternary quadratic. + + No solver is currently implemented for this equation type. + + """ + + name = 'inhomogeneous_ternary_quadratic' + + def matches(self): + if not (self.total_degree == 2 and self.dimension == 3): + return False + if not self.homogeneous: + return False + return not self.homogeneous_order + + +class HomogeneousTernaryQuadraticNormal(DiophantineEquationType): + """ + Representation of a homogeneous ternary quadratic normal diophantine equation. + + Examples + ======== + + >>> from sympy.abc import x, y, z + >>> from sympy.solvers.diophantine.diophantine import HomogeneousTernaryQuadraticNormal + >>> HomogeneousTernaryQuadraticNormal(4*x**2 - 5*y**2 + z**2).solve() + {(1, 2, 4)} + + """ + + name = 'homogeneous_ternary_quadratic_normal' + + def matches(self): + if not (self.total_degree == 2 and self.dimension == 3): + return False + if not self.homogeneous: + return False + if not self.homogeneous_order: + return False + + nonzero = [k for k in self.coeff if self.coeff[k]] + return len(nonzero) == 3 and all(i**2 in nonzero for i in self.free_symbols) + + def solve(self, parameters=None, limit=None) -> DiophantineSolutionSet: + self.pre_solve(parameters) + + var = self.free_symbols + coeff = self.coeff + + x, y, z = var + + a = coeff[x**2] + b = coeff[y**2] + c = coeff[z**2] + + (sqf_of_a, sqf_of_b, sqf_of_c), (a_1, b_1, c_1), (a_2, b_2, c_2) = \ + sqf_normal(a, b, c, steps=True) + + A = -a_2*c_2 + B = -b_2*c_2 + + result = DiophantineSolutionSet(var, parameters=self.parameters) + + # If following two conditions are satisfied then there are no solutions + if A < 0 and B < 0: + return result + + if ( + sqrt_mod(-b_2*c_2, a_2) is None or + sqrt_mod(-c_2*a_2, b_2) is None or + sqrt_mod(-a_2*b_2, c_2) is None): + return result + + z_0, x_0, y_0 = descent(A, B) + + z_0, q = _rational_pq(z_0, abs(c_2)) + x_0 *= q + y_0 *= q + + x_0, y_0, z_0 = _remove_gcd(x_0, y_0, z_0) + + # Holzer reduction + if sign(a) == sign(b): + x_0, y_0, z_0 = holzer(x_0, y_0, z_0, abs(a_2), abs(b_2), abs(c_2)) + elif sign(a) == sign(c): + x_0, z_0, y_0 = holzer(x_0, z_0, y_0, abs(a_2), abs(c_2), abs(b_2)) + else: + y_0, z_0, x_0 = holzer(y_0, z_0, x_0, abs(b_2), abs(c_2), abs(a_2)) + + x_0 = reconstruct(b_1, c_1, x_0) + y_0 = reconstruct(a_1, c_1, y_0) + z_0 = reconstruct(a_1, b_1, z_0) + + sq_lcm = ilcm(sqf_of_a, sqf_of_b, sqf_of_c) + + x_0 = abs(x_0*sq_lcm // sqf_of_a) + y_0 = abs(y_0*sq_lcm // sqf_of_b) + z_0 = abs(z_0*sq_lcm // sqf_of_c) + + result.add(_remove_gcd(x_0, y_0, z_0)) + return result + + +class HomogeneousTernaryQuadratic(DiophantineEquationType): + """ + Representation of a homogeneous ternary quadratic diophantine equation. + + Examples + ======== + + >>> from sympy.abc import x, y, z + >>> from sympy.solvers.diophantine.diophantine import HomogeneousTernaryQuadratic + >>> HomogeneousTernaryQuadratic(x**2 + y**2 - 3*z**2 + x*y).solve() + {(-1, 2, 1)} + >>> HomogeneousTernaryQuadratic(3*x**2 + y**2 - 3*z**2 + 5*x*y + y*z).solve() + {(3, 12, 13)} + + """ + + name = 'homogeneous_ternary_quadratic' + + def matches(self): + if not (self.total_degree == 2 and self.dimension == 3): + return False + if not self.homogeneous: + return False + if not self.homogeneous_order: + return False + + nonzero = [k for k in self.coeff if self.coeff[k]] + return not (len(nonzero) == 3 and all(i**2 in nonzero for i in self.free_symbols)) + + def solve(self, parameters=None, limit=None): + self.pre_solve(parameters) + + _var = self.free_symbols + coeff = self.coeff + + x, y, z = _var + var = [x, y, z] + + # Equations of the form B*x*y + C*z*x + E*y*z = 0 and At least two of the + # coefficients A, B, C are non-zero. + # There are infinitely many solutions for the equation. + # Ex: (0, 0, t), (0, t, 0), (t, 0, 0) + # Equation can be re-written as y*(B*x + E*z) = -C*x*z and we can find rather + # unobvious solutions. Set y = -C and B*x + E*z = x*z. The latter can be solved by + # using methods for binary quadratic diophantine equations. Let's select the + # solution which minimizes |x| + |z| + + result = DiophantineSolutionSet(var, parameters=self.parameters) + + def unpack_sol(sol): + if len(sol) > 0: + return list(sol)[0] + return None, None, None + + if not any(coeff[i**2] for i in var): + if coeff[x*z]: + sols = diophantine(coeff[x*y]*x + coeff[y*z]*z - x*z) + s = min(sols, key=lambda r: abs(r[0]) + abs(r[1])) + result.add(_remove_gcd(s[0], -coeff[x*z], s[1])) + return result + + var[0], var[1] = _var[1], _var[0] + y_0, x_0, z_0 = unpack_sol(_diop_ternary_quadratic(var, coeff)) + if x_0 is not None: + result.add((x_0, y_0, z_0)) + return result + + if coeff[x**2] == 0: + # If the coefficient of x is zero change the variables + if coeff[y**2] == 0: + var[0], var[2] = _var[2], _var[0] + z_0, y_0, x_0 = unpack_sol(_diop_ternary_quadratic(var, coeff)) + + else: + var[0], var[1] = _var[1], _var[0] + y_0, x_0, z_0 = unpack_sol(_diop_ternary_quadratic(var, coeff)) + + else: + if coeff[x*y] or coeff[x*z]: + # Apply the transformation x --> X - (B*y + C*z)/(2*A) + A = coeff[x**2] + B = coeff[x*y] + C = coeff[x*z] + D = coeff[y**2] + E = coeff[y*z] + F = coeff[z**2] + + _coeff = {} + + _coeff[x**2] = 4*A**2 + _coeff[y**2] = 4*A*D - B**2 + _coeff[z**2] = 4*A*F - C**2 + _coeff[y*z] = 4*A*E - 2*B*C + _coeff[x*y] = 0 + _coeff[x*z] = 0 + + x_0, y_0, z_0 = unpack_sol(_diop_ternary_quadratic(var, _coeff)) + + if x_0 is None: + return result + + p, q = _rational_pq(B*y_0 + C*z_0, 2*A) + x_0, y_0, z_0 = x_0*q - p, y_0*q, z_0*q + + elif coeff[z*y] != 0: + if coeff[y**2] == 0: + if coeff[z**2] == 0: + # Equations of the form A*x**2 + E*yz = 0. + A = coeff[x**2] + E = coeff[y*z] + + b, a = _rational_pq(-E, A) + + x_0, y_0, z_0 = b, a, b + + else: + # Ax**2 + E*y*z + F*z**2 = 0 + var[0], var[2] = _var[2], _var[0] + z_0, y_0, x_0 = unpack_sol(_diop_ternary_quadratic(var, coeff)) + + else: + # A*x**2 + D*y**2 + E*y*z + F*z**2 = 0, C may be zero + var[0], var[1] = _var[1], _var[0] + y_0, x_0, z_0 = unpack_sol(_diop_ternary_quadratic(var, coeff)) + + else: + # Ax**2 + D*y**2 + F*z**2 = 0, C may be zero + x_0, y_0, z_0 = unpack_sol(_diop_ternary_quadratic_normal(var, coeff)) + + if x_0 is None: + return result + + result.add(_remove_gcd(x_0, y_0, z_0)) + return result + + +class InhomogeneousGeneralQuadratic(DiophantineEquationType): + """ + + Representation of an inhomogeneous general quadratic. + + No solver is currently implemented for this equation type. + + """ + + name = 'inhomogeneous_general_quadratic' + + def matches(self): + if not (self.total_degree == 2 and self.dimension >= 3): + return False + if not self.homogeneous_order: + return True + # there may be Pow keys like x**2 or Mul keys like x*y + return any(k.is_Mul for k in self.coeff) and not self.homogeneous + + +class HomogeneousGeneralQuadratic(DiophantineEquationType): + """ + + Representation of a homogeneous general quadratic. + + No solver is currently implemented for this equation type. + + """ + + name = 'homogeneous_general_quadratic' + + def matches(self): + if not (self.total_degree == 2 and self.dimension >= 3): + return False + if not self.homogeneous_order: + return False + # there may be Pow keys like x**2 or Mul keys like x*y + return any(k.is_Mul for k in self.coeff) and self.homogeneous + + +class GeneralSumOfSquares(DiophantineEquationType): + r""" + Representation of the diophantine equation + + `x_{1}^2 + x_{2}^2 + . . . + x_{n}^2 - k = 0`. + + Details + ======= + + When `n = 3` if `k = 4^a(8m + 7)` for some `a, m \in Z` then there will be + no solutions. Refer [1]_ for more details. + + Examples + ======== + + >>> from sympy.solvers.diophantine.diophantine import GeneralSumOfSquares + >>> from sympy.abc import a, b, c, d, e + >>> GeneralSumOfSquares(a**2 + b**2 + c**2 + d**2 + e**2 - 2345).solve() + {(15, 22, 22, 24, 24)} + + By default only 1 solution is returned. Use the `limit` keyword for more: + + >>> sorted(GeneralSumOfSquares(a**2 + b**2 + c**2 + d**2 + e**2 - 2345).solve(limit=3)) + [(15, 22, 22, 24, 24), (16, 19, 24, 24, 24), (16, 20, 22, 23, 26)] + + References + ========== + + .. [1] Representing an integer as a sum of three squares, [online], + Available: + https://proofwiki.org/wiki/Integer_as_Sum_of_Three_Squares + """ + + name = 'general_sum_of_squares' + + def matches(self): + if not (self.total_degree == 2 and self.dimension >= 3): + return False + if not self.homogeneous_order: + return False + if any(k.is_Mul for k in self.coeff): + return False + return all(self.coeff[k] == 1 for k in self.coeff if k != 1) + + def solve(self, parameters=None, limit=1): + self.pre_solve(parameters) + + var = self.free_symbols + k = -int(self.coeff[1]) + n = self.dimension + + result = DiophantineSolutionSet(var, parameters=self.parameters) + + if k < 0 or limit < 1: + return result + + signs = [-1 if x.is_nonpositive else 1 for x in var] + negs = signs.count(-1) != 0 + + took = 0 + for t in sum_of_squares(k, n, zeros=True): + if negs: + result.add([signs[i]*j for i, j in enumerate(t)]) + else: + result.add(t) + took += 1 + if took == limit: + break + return result + + +class GeneralPythagorean(DiophantineEquationType): + """ + Representation of the general pythagorean equation, + `a_{1}^2x_{1}^2 + a_{2}^2x_{2}^2 + . . . + a_{n}^2x_{n}^2 - a_{n + 1}^2x_{n + 1}^2 = 0`. + + Examples + ======== + + >>> from sympy.solvers.diophantine.diophantine import GeneralPythagorean + >>> from sympy.abc import a, b, c, d, e, x, y, z, t + >>> GeneralPythagorean(a**2 + b**2 + c**2 - d**2).solve() + {(t_0**2 + t_1**2 - t_2**2, 2*t_0*t_2, 2*t_1*t_2, t_0**2 + t_1**2 + t_2**2)} + >>> GeneralPythagorean(9*a**2 - 4*b**2 + 16*c**2 + 25*d**2 + e**2).solve(parameters=[x, y, z, t]) + {(-10*t**2 + 10*x**2 + 10*y**2 + 10*z**2, 15*t**2 + 15*x**2 + 15*y**2 + 15*z**2, 15*t*x, 12*t*y, 60*t*z)} + """ + + name = 'general_pythagorean' + + def matches(self): + if not (self.total_degree == 2 and self.dimension >= 3): + return False + if not self.homogeneous_order: + return False + if any(k.is_Mul for k in self.coeff): + return False + if all(self.coeff[k] == 1 for k in self.coeff if k != 1): + return False + if not all(is_square(abs(self.coeff[k])) for k in self.coeff): + return False + # all but one has the same sign + # e.g. 4*x**2 + y**2 - 4*z**2 + return abs(sum(sign(self.coeff[k]) for k in self.coeff)) == self.dimension - 2 + + @property + def n_parameters(self): + return self.dimension - 1 + + def solve(self, parameters=None, limit=1): + self.pre_solve(parameters) + + coeff = self.coeff + var = self.free_symbols + n = self.dimension + + if sign(coeff[var[0] ** 2]) + sign(coeff[var[1] ** 2]) + sign(coeff[var[2] ** 2]) < 0: + for key in coeff.keys(): + coeff[key] = -coeff[key] + + result = DiophantineSolutionSet(var, parameters=self.parameters) + + index = 0 + + for i, v in enumerate(var): + if sign(coeff[v ** 2]) == -1: + index = i + + m = result.parameters + + ith = sum(m_i ** 2 for m_i in m) + L = [ith - 2 * m[n - 2] ** 2] + L.extend([2 * m[i] * m[n - 2] for i in range(n - 2)]) + sol = L[:index] + [ith] + L[index:] + + lcm = 1 + for i, v in enumerate(var): + if i == index or (index > 0 and i == 0) or (index == 0 and i == 1): + lcm = ilcm(lcm, sqrt(abs(coeff[v ** 2]))) + else: + s = sqrt(coeff[v ** 2]) + lcm = ilcm(lcm, s if _odd(s) else s // 2) + + for i, v in enumerate(var): + sol[i] = (lcm * sol[i]) / sqrt(abs(coeff[v ** 2])) + + result.add(sol) + return result + + +class CubicThue(DiophantineEquationType): + """ + Representation of a cubic Thue diophantine equation. + + A cubic Thue diophantine equation is a polynomial of the form + `f(x, y) = r` of degree 3, where `x` and `y` are integers + and `r` is a rational number. + + No solver is currently implemented for this equation type. + + Examples + ======== + + >>> from sympy.abc import x, y + >>> from sympy.solvers.diophantine.diophantine import CubicThue + >>> c1 = CubicThue(x**3 + y**2 + 1) + >>> c1.matches() + True + + """ + + name = 'cubic_thue' + + def matches(self): + return self.total_degree == 3 and self.dimension == 2 + + +class GeneralSumOfEvenPowers(DiophantineEquationType): + """ + Representation of the diophantine equation + + `x_{1}^e + x_{2}^e + . . . + x_{n}^e - k = 0` + + where `e` is an even, integer power. + + Examples + ======== + + >>> from sympy.solvers.diophantine.diophantine import GeneralSumOfEvenPowers + >>> from sympy.abc import a, b + >>> GeneralSumOfEvenPowers(a**4 + b**4 - (2**4 + 3**4)).solve() + {(2, 3)} + + """ + + name = 'general_sum_of_even_powers' + + def matches(self): + if not self.total_degree > 3: + return False + if self.total_degree % 2 != 0: + return False + if not all(k.is_Pow and k.exp == self.total_degree for k in self.coeff if k != 1): + return False + return all(self.coeff[k] == 1 for k in self.coeff if k != 1) + + def solve(self, parameters=None, limit=1): + self.pre_solve(parameters) + + var = self.free_symbols + coeff = self.coeff + + p = None + for q in coeff.keys(): + if q.is_Pow and coeff[q]: + p = q.exp + + k = len(var) + n = -coeff[1] + + result = DiophantineSolutionSet(var, parameters=self.parameters) + + if n < 0 or limit < 1: + return result + + sign = [-1 if x.is_nonpositive else 1 for x in var] + negs = sign.count(-1) != 0 + + took = 0 + for t in power_representation(n, p, k): + if negs: + result.add([sign[i]*j for i, j in enumerate(t)]) + else: + result.add(t) + took += 1 + if took == limit: + break + return result + +# these types are known (but not necessarily handled) +# note that order is important here (in the current solver state) +all_diop_classes = [ + Linear, + Univariate, + BinaryQuadratic, + InhomogeneousTernaryQuadratic, + HomogeneousTernaryQuadraticNormal, + HomogeneousTernaryQuadratic, + InhomogeneousGeneralQuadratic, + HomogeneousGeneralQuadratic, + GeneralSumOfSquares, + GeneralPythagorean, + CubicThue, + GeneralSumOfEvenPowers, +] + +diop_known = {diop_class.name for diop_class in all_diop_classes} + + +def _remove_gcd(*x): + try: + g = igcd(*x) + except ValueError: + fx = list(filter(None, x)) + if len(fx) < 2: + return x + g = igcd(*[i.as_content_primitive()[0] for i in fx]) + except TypeError: + raise TypeError('_remove_gcd(a,b,c) or _remove_gcd(*container)') + if g == 1: + return x + return tuple([i//g for i in x]) + + +def _rational_pq(a, b): + # return `(numer, denom)` for a/b; sign in numer and gcd removed + return _remove_gcd(sign(b)*a, abs(b)) + + +def _nint_or_floor(p, q): + # return nearest int to p/q; in case of tie return floor(p/q) + w, r = divmod(p, q) + if abs(r) <= abs(q)//2: + return w + return w + 1 + + +def _odd(i): + return i % 2 != 0 + + +def _even(i): + return i % 2 == 0 + + +def diophantine(eq, param=symbols("t", integer=True), syms=None, + permute=False): + """ + Simplify the solution procedure of diophantine equation ``eq`` by + converting it into a product of terms which should equal zero. + + Explanation + =========== + + For example, when solving, `x^2 - y^2 = 0` this is treated as + `(x + y)(x - y) = 0` and `x + y = 0` and `x - y = 0` are solved + independently and combined. Each term is solved by calling + ``diop_solve()``. (Although it is possible to call ``diop_solve()`` + directly, one must be careful to pass an equation in the correct + form and to interpret the output correctly; ``diophantine()`` is + the public-facing function to use in general.) + + Output of ``diophantine()`` is a set of tuples. The elements of the + tuple are the solutions for each variable in the equation and + are arranged according to the alphabetic ordering of the variables. + e.g. For an equation with two variables, `a` and `b`, the first + element of the tuple is the solution for `a` and the second for `b`. + + Usage + ===== + + ``diophantine(eq, t, syms)``: Solve the diophantine + equation ``eq``. + ``t`` is the optional parameter to be used by ``diop_solve()``. + ``syms`` is an optional list of symbols which determines the + order of the elements in the returned tuple. + + By default, only the base solution is returned. If ``permute`` is set to + True then permutations of the base solution and/or permutations of the + signs of the values will be returned when applicable. + + Details + ======= + + ``eq`` should be an expression which is assumed to be zero. + ``t`` is the parameter to be used in the solution. + + Examples + ======== + + >>> from sympy import diophantine + >>> from sympy.abc import a, b + >>> eq = a**4 + b**4 - (2**4 + 3**4) + >>> diophantine(eq) + {(2, 3)} + >>> diophantine(eq, permute=True) + {(-3, -2), (-3, 2), (-2, -3), (-2, 3), (2, -3), (2, 3), (3, -2), (3, 2)} + + >>> from sympy.abc import x, y, z + >>> diophantine(x**2 - y**2) + {(t_0, -t_0), (t_0, t_0)} + + >>> diophantine(x*(2*x + 3*y - z)) + {(0, n1, n2), (t_0, t_1, 2*t_0 + 3*t_1)} + >>> diophantine(x**2 + 3*x*y + 4*x) + {(0, n1), (-3*t_0 - 4, t_0)} + + See Also + ======== + + diop_solve + sympy.utilities.iterables.permute_signs + sympy.utilities.iterables.signed_permutations + """ + + eq = _sympify(eq) + + if isinstance(eq, Eq): + eq = eq.lhs - eq.rhs + + try: + var = list(eq.expand(force=True).free_symbols) + var.sort(key=default_sort_key) + if syms: + if not is_sequence(syms): + raise TypeError( + 'syms should be given as a sequence, e.g. a list') + syms = [i for i in syms if i in var] + if syms != var: + dict_sym_index = dict(zip(syms, range(len(syms)))) + return {tuple([t[dict_sym_index[i]] for i in var]) + for t in diophantine(eq, param, permute=permute)} + n, d = eq.as_numer_denom() + if n.is_number: + return set() + if not d.is_number: + dsol = diophantine(d) + good = diophantine(n) - dsol + return {s for s in good if _mexpand(d.subs(zip(var, s)))} + eq = factor_terms(n) + assert not eq.is_number + eq = eq.as_independent(*var, as_Add=False)[1] + p = Poly(eq) + assert not any(g.is_number for g in p.gens) + eq = p.as_expr() + assert eq.is_polynomial() + except (GeneratorsNeeded, AssertionError): + raise TypeError(filldedent(''' + Equation should be a polynomial with Rational coefficients.''')) + + # permute only sign + do_permute_signs = False + # permute sign and values + do_permute_signs_var = False + # permute few signs + permute_few_signs = False + try: + # if we know that factoring should not be attempted, skip + # the factoring step + v, c, t = classify_diop(eq) + + # check for permute sign + if permute: + len_var = len(v) + permute_signs_for = [ + GeneralSumOfSquares.name, + GeneralSumOfEvenPowers.name] + permute_signs_check = [ + HomogeneousTernaryQuadratic.name, + HomogeneousTernaryQuadraticNormal.name, + BinaryQuadratic.name] + if t in permute_signs_for: + do_permute_signs_var = True + elif t in permute_signs_check: + # if all the variables in eq have even powers + # then do_permute_sign = True + if len_var == 3: + var_mul = list(subsets(v, 2)) + # here var_mul is like [(x, y), (x, z), (y, z)] + xy_coeff = True + x_coeff = True + var1_mul_var2 = (a[0]*a[1] for a in var_mul) + # if coeff(y*z), coeff(y*x), coeff(x*z) is not 0 then + # `xy_coeff` => True and do_permute_sign => False. + # Means no permuted solution. + for v1_mul_v2 in var1_mul_var2: + try: + coeff = c[v1_mul_v2] + except KeyError: + coeff = 0 + xy_coeff = bool(xy_coeff) and bool(coeff) + var_mul = list(subsets(v, 1)) + # here var_mul is like [(x,), (y, )] + for v1 in var_mul: + try: + coeff = c[v1[0]] + except KeyError: + coeff = 0 + x_coeff = bool(x_coeff) and bool(coeff) + if not any((xy_coeff, x_coeff)): + # means only x**2, y**2, z**2, const is present + do_permute_signs = True + elif not x_coeff: + permute_few_signs = True + elif len_var == 2: + var_mul = list(subsets(v, 2)) + # here var_mul is like [(x, y)] + xy_coeff = True + x_coeff = True + var1_mul_var2 = (x[0]*x[1] for x in var_mul) + for v1_mul_v2 in var1_mul_var2: + try: + coeff = c[v1_mul_v2] + except KeyError: + coeff = 0 + xy_coeff = bool(xy_coeff) and bool(coeff) + var_mul = list(subsets(v, 1)) + # here var_mul is like [(x,), (y, )] + for v1 in var_mul: + try: + coeff = c[v1[0]] + except KeyError: + coeff = 0 + x_coeff = bool(x_coeff) and bool(coeff) + if not any((xy_coeff, x_coeff)): + # means only x**2, y**2 and const is present + # so we can get more soln by permuting this soln. + do_permute_signs = True + elif not x_coeff: + # when coeff(x), coeff(y) is not present then signs of + # x, y can be permuted such that their sign are same + # as sign of x*y. + # e.g 1. (x_val,y_val)=> (x_val,y_val), (-x_val,-y_val) + # 2. (-x_vall, y_val)=> (-x_val,y_val), (x_val,-y_val) + permute_few_signs = True + if t == 'general_sum_of_squares': + # trying to factor such expressions will sometimes hang + terms = [(eq, 1)] + else: + raise TypeError + except (TypeError, NotImplementedError): + fl = factor_list(eq) + if fl[0].is_Rational and fl[0] != 1: + return diophantine(eq/fl[0], param=param, syms=syms, permute=permute) + terms = fl[1] + + sols = set() + + for term in terms: + + base, _ = term + var_t, _, eq_type = classify_diop(base, _dict=False) + _, base = signsimp(base, evaluate=False).as_coeff_Mul() + solution = diop_solve(base, param) + + if eq_type in [ + Linear.name, + HomogeneousTernaryQuadratic.name, + HomogeneousTernaryQuadraticNormal.name, + GeneralPythagorean.name]: + sols.add(merge_solution(var, var_t, solution)) + + elif eq_type in [ + BinaryQuadratic.name, + GeneralSumOfSquares.name, + GeneralSumOfEvenPowers.name, + Univariate.name]: + sols.update(merge_solution(var, var_t, sol) for sol in solution) + + else: + raise NotImplementedError('unhandled type: %s' % eq_type) + + sols.discard(()) + null = tuple([0]*len(var)) + # if there is no solution, return trivial solution + if not sols and eq.subs(zip(var, null)).is_zero: + if all(check_assumptions(val, **s.assumptions0) is not False for val, s in zip(null, var)): + sols.add(null) + + final_soln = set() + for sol in sols: + if all(int_valued(s) for s in sol): + if do_permute_signs: + permuted_sign = set(permute_signs(sol)) + final_soln.update(permuted_sign) + elif permute_few_signs: + lst = list(permute_signs(sol)) + lst = list(filter(lambda x: x[0]*x[1] == sol[1]*sol[0], lst)) + permuted_sign = set(lst) + final_soln.update(permuted_sign) + elif do_permute_signs_var: + permuted_sign_var = set(signed_permutations(sol)) + final_soln.update(permuted_sign_var) + else: + final_soln.add(sol) + else: + final_soln.add(sol) + return final_soln + + +def merge_solution(var, var_t, solution): + """ + This is used to construct the full solution from the solutions of sub + equations. + + Explanation + =========== + + For example when solving the equation `(x - y)(x^2 + y^2 - z^2) = 0`, + solutions for each of the equations `x - y = 0` and `x^2 + y^2 - z^2` are + found independently. Solutions for `x - y = 0` are `(x, y) = (t, t)`. But + we should introduce a value for z when we output the solution for the + original equation. This function converts `(t, t)` into `(t, t, n_{1})` + where `n_{1}` is an integer parameter. + """ + sol = [] + + if None in solution: + return () + + solution = iter(solution) + params = numbered_symbols("n", integer=True, start=1) + for v in var: + if v in var_t: + sol.append(next(solution)) + else: + sol.append(next(params)) + + for val, symb in zip(sol, var): + if check_assumptions(val, **symb.assumptions0) is False: + return () + + return tuple(sol) + + +def _diop_solve(eq, params=None): + for diop_type in all_diop_classes: + if diop_type(eq).matches(): + return diop_type(eq).solve(parameters=params) + + +def diop_solve(eq, param=symbols("t", integer=True)): + """ + Solves the diophantine equation ``eq``. + + Explanation + =========== + + Unlike ``diophantine()``, factoring of ``eq`` is not attempted. Uses + ``classify_diop()`` to determine the type of the equation and calls + the appropriate solver function. + + Use of ``diophantine()`` is recommended over other helper functions. + ``diop_solve()`` can return either a set or a tuple depending on the + nature of the equation. All non-trivial solutions are returned: assumptions + on symbols are ignored. + + Usage + ===== + + ``diop_solve(eq, t)``: Solve diophantine equation, ``eq`` using ``t`` + as a parameter if needed. + + Details + ======= + + ``eq`` should be an expression which is assumed to be zero. + ``t`` is a parameter to be used in the solution. + + Examples + ======== + + >>> from sympy.solvers.diophantine import diop_solve + >>> from sympy.abc import x, y, z, w + >>> diop_solve(2*x + 3*y - 5) + (3*t_0 - 5, 5 - 2*t_0) + >>> diop_solve(4*x + 3*y - 4*z + 5) + (t_0, 8*t_0 + 4*t_1 + 5, 7*t_0 + 3*t_1 + 5) + >>> diop_solve(x + 3*y - 4*z + w - 6) + (t_0, t_0 + t_1, 6*t_0 + 5*t_1 + 4*t_2 - 6, 5*t_0 + 4*t_1 + 3*t_2 - 6) + >>> diop_solve(x**2 + y**2 - 5) + {(-2, -1), (-2, 1), (-1, -2), (-1, 2), (1, -2), (1, 2), (2, -1), (2, 1)} + + + See Also + ======== + + diophantine() + """ + var, coeff, eq_type = classify_diop(eq, _dict=False) + + if eq_type == Linear.name: + return diop_linear(eq, param) + + elif eq_type == BinaryQuadratic.name: + return diop_quadratic(eq, param) + + elif eq_type == HomogeneousTernaryQuadratic.name: + return diop_ternary_quadratic(eq, parameterize=True) + + elif eq_type == HomogeneousTernaryQuadraticNormal.name: + return diop_ternary_quadratic_normal(eq, parameterize=True) + + elif eq_type == GeneralPythagorean.name: + return diop_general_pythagorean(eq, param) + + elif eq_type == Univariate.name: + return diop_univariate(eq) + + elif eq_type == GeneralSumOfSquares.name: + return diop_general_sum_of_squares(eq, limit=S.Infinity) + + elif eq_type == GeneralSumOfEvenPowers.name: + return diop_general_sum_of_even_powers(eq, limit=S.Infinity) + + if eq_type is not None and eq_type not in diop_known: + raise ValueError(filldedent(''' + Although this type of equation was identified, it is not yet + handled. It should, however, be listed in `diop_known` at the + top of this file. Developers should see comments at the end of + `classify_diop`. + ''')) # pragma: no cover + else: + raise NotImplementedError( + 'No solver has been written for %s.' % eq_type) + + +def classify_diop(eq, _dict=True): + # docstring supplied externally + + matched = False + diop_type = None + for diop_class in all_diop_classes: + diop_type = diop_class(eq) + if diop_type.matches(): + matched = True + break + + if matched: + return diop_type.free_symbols, dict(diop_type.coeff) if _dict else diop_type.coeff, diop_type.name + + # new diop type instructions + # -------------------------- + # if this error raises and the equation *can* be classified, + # * it should be identified in the if-block above + # * the type should be added to the diop_known + # if a solver can be written for it, + # * a dedicated handler should be written (e.g. diop_linear) + # * it should be passed to that handler in diop_solve + raise NotImplementedError(filldedent(''' + This equation is not yet recognized or else has not been + simplified sufficiently to put it in a form recognized by + diop_classify().''')) + + +classify_diop.func_doc = ( # type: ignore + ''' + Helper routine used by diop_solve() to find information about ``eq``. + + Explanation + =========== + + Returns a tuple containing the type of the diophantine equation + along with the variables (free symbols) and their coefficients. + Variables are returned as a list and coefficients are returned + as a dict with the key being the respective term and the constant + term is keyed to 1. The type is one of the following: + + * %s + + Usage + ===== + + ``classify_diop(eq)``: Return variables, coefficients and type of the + ``eq``. + + Details + ======= + + ``eq`` should be an expression which is assumed to be zero. + ``_dict`` is for internal use: when True (default) a dict is returned, + otherwise a defaultdict which supplies 0 for missing keys is returned. + + Examples + ======== + + >>> from sympy.solvers.diophantine import classify_diop + >>> from sympy.abc import x, y, z, w, t + >>> classify_diop(4*x + 6*y - 4) + ([x, y], {1: -4, x: 4, y: 6}, 'linear') + >>> classify_diop(x + 3*y -4*z + 5) + ([x, y, z], {1: 5, x: 1, y: 3, z: -4}, 'linear') + >>> classify_diop(x**2 + y**2 - x*y + x + 5) + ([x, y], {1: 5, x: 1, x**2: 1, y**2: 1, x*y: -1}, 'binary_quadratic') + ''' % ('\n * '.join(sorted(diop_known)))) + + +def diop_linear(eq, param=symbols("t", integer=True)): + """ + Solves linear diophantine equations. + + A linear diophantine equation is an equation of the form `a_{1}x_{1} + + a_{2}x_{2} + .. + a_{n}x_{n} = 0` where `a_{1}, a_{2}, ..a_{n}` are + integer constants and `x_{1}, x_{2}, ..x_{n}` are integer variables. + + Usage + ===== + + ``diop_linear(eq)``: Returns a tuple containing solutions to the + diophantine equation ``eq``. Values in the tuple is arranged in the same + order as the sorted variables. + + Details + ======= + + ``eq`` is a linear diophantine equation which is assumed to be zero. + ``param`` is the parameter to be used in the solution. + + Examples + ======== + + >>> from sympy.solvers.diophantine.diophantine import diop_linear + >>> from sympy.abc import x, y, z + >>> diop_linear(2*x - 3*y - 5) # solves equation 2*x - 3*y - 5 == 0 + (3*t_0 - 5, 2*t_0 - 5) + + Here x = -3*t_0 - 5 and y = -2*t_0 - 5 + + >>> diop_linear(2*x - 3*y - 4*z -3) + (t_0, 2*t_0 + 4*t_1 + 3, -t_0 - 3*t_1 - 3) + + See Also + ======== + + diop_quadratic(), diop_ternary_quadratic(), diop_general_pythagorean(), + diop_general_sum_of_squares() + """ + var, coeff, diop_type = classify_diop(eq, _dict=False) + + if diop_type == Linear.name: + parameters = None + if param is not None: + parameters = symbols('%s_0:%i' % (param, len(var)), integer=True) + + result = Linear(eq).solve(parameters=parameters) + + if param is None: + result = result(*[0]*len(result.parameters)) + + if len(result) > 0: + return list(result)[0] + else: + return tuple([None]*len(result.parameters)) + + +def base_solution_linear(c, a, b, t=None): + """ + Return the base solution for the linear equation, `ax + by = c`. + + Explanation + =========== + + Used by ``diop_linear()`` to find the base solution of a linear + Diophantine equation. If ``t`` is given then the parametrized solution is + returned. + + Usage + ===== + + ``base_solution_linear(c, a, b, t)``: ``a``, ``b``, ``c`` are coefficients + in `ax + by = c` and ``t`` is the parameter to be used in the solution. + + Examples + ======== + + >>> from sympy.solvers.diophantine.diophantine import base_solution_linear + >>> from sympy.abc import t + >>> base_solution_linear(5, 2, 3) # equation 2*x + 3*y = 5 + (-5, 5) + >>> base_solution_linear(0, 5, 7) # equation 5*x + 7*y = 0 + (0, 0) + >>> base_solution_linear(5, 2, 3, t) # equation 2*x + 3*y = 5 + (3*t - 5, 5 - 2*t) + >>> base_solution_linear(0, 5, 7, t) # equation 5*x + 7*y = 0 + (7*t, -5*t) + """ + a, b, c = _remove_gcd(a, b, c) + + if c == 0: + if t is None: + return (0, 0) + if b < 0: + t = -t + return (b*t, -a*t) + + x0, y0, d = igcdex(abs(a), abs(b)) + x0 *= sign(a) + y0 *= sign(b) + if c % d: + return (None, None) + if t is None: + return (c*x0, c*y0) + if b < 0: + t = -t + return (c*x0 + b*t, c*y0 - a*t) + + +def diop_univariate(eq): + """ + Solves a univariate diophantine equations. + + Explanation + =========== + + A univariate diophantine equation is an equation of the form + `a_{0} + a_{1}x + a_{2}x^2 + .. + a_{n}x^n = 0` where `a_{1}, a_{2}, ..a_{n}` are + integer constants and `x` is an integer variable. + + Usage + ===== + + ``diop_univariate(eq)``: Returns a set containing solutions to the + diophantine equation ``eq``. + + Details + ======= + + ``eq`` is a univariate diophantine equation which is assumed to be zero. + + Examples + ======== + + >>> from sympy.solvers.diophantine.diophantine import diop_univariate + >>> from sympy.abc import x + >>> diop_univariate((x - 2)*(x - 3)**2) # solves equation (x - 2)*(x - 3)**2 == 0 + {(2,), (3,)} + + """ + var, coeff, diop_type = classify_diop(eq, _dict=False) + + if diop_type == Univariate.name: + return {(int(i),) for i in solveset_real( + eq, var[0]).intersect(S.Integers)} + + +def divisible(a, b): + """ + Returns `True` if ``a`` is divisible by ``b`` and `False` otherwise. + """ + return not a % b + + +def diop_quadratic(eq, param=symbols("t", integer=True)): + """ + Solves quadratic diophantine equations. + + i.e. equations of the form `Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0`. Returns a + set containing the tuples `(x, y)` which contains the solutions. If there + are no solutions then `(None, None)` is returned. + + Usage + ===== + + ``diop_quadratic(eq, param)``: ``eq`` is a quadratic binary diophantine + equation. ``param`` is used to indicate the parameter to be used in the + solution. + + Details + ======= + + ``eq`` should be an expression which is assumed to be zero. + ``param`` is a parameter to be used in the solution. + + Examples + ======== + + >>> from sympy.abc import x, y, t + >>> from sympy.solvers.diophantine.diophantine import diop_quadratic + >>> diop_quadratic(x**2 + y**2 + 2*x + 2*y + 2, t) + {(-1, -1)} + + References + ========== + + .. [1] Methods to solve Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, [online], + Available: https://www.alpertron.com.ar/METHODS.HTM + .. [2] Solving the equation ax^2+ bxy + cy^2 + dx + ey + f= 0, [online], + Available: https://web.archive.org/web/20160323033111/http://www.jpr2718.org/ax2p.pdf + + See Also + ======== + + diop_linear(), diop_ternary_quadratic(), diop_general_sum_of_squares(), + diop_general_pythagorean() + """ + var, coeff, diop_type = classify_diop(eq, _dict=False) + + if diop_type == BinaryQuadratic.name: + if param is not None: + parameters = [param, Symbol("u", integer=True)] + else: + parameters = None + return set(BinaryQuadratic(eq).solve(parameters=parameters)) + + +def is_solution_quad(var, coeff, u, v): + """ + Check whether `(u, v)` is solution to the quadratic binary diophantine + equation with the variable list ``var`` and coefficient dictionary + ``coeff``. + + Not intended for use by normal users. + """ + reps = dict(zip(var, (u, v))) + eq = Add(*[j*i.xreplace(reps) for i, j in coeff.items()]) + return _mexpand(eq) == 0 + + +def diop_DN(D, N, t=symbols("t", integer=True)): + """ + Solves the equation `x^2 - Dy^2 = N`. + + Explanation + =========== + + Mainly concerned with the case `D > 0, D` is not a perfect square, + which is the same as the generalized Pell equation. The LMM + algorithm [1]_ is used to solve this equation. + + Returns one solution tuple, (`x, y)` for each class of the solutions. + Other solutions of the class can be constructed according to the + values of ``D`` and ``N``. + + Usage + ===== + + ``diop_DN(D, N, t)``: D and N are integers as in `x^2 - Dy^2 = N` and + ``t`` is the parameter to be used in the solutions. + + Details + ======= + + ``D`` and ``N`` correspond to D and N in the equation. + ``t`` is the parameter to be used in the solutions. + + Examples + ======== + + >>> from sympy.solvers.diophantine.diophantine import diop_DN + >>> diop_DN(13, -4) # Solves equation x**2 - 13*y**2 = -4 + [(3, 1), (393, 109), (36, 10)] + + The output can be interpreted as follows: There are three fundamental + solutions to the equation `x^2 - 13y^2 = -4` given by (3, 1), (393, 109) + and (36, 10). Each tuple is in the form (x, y), i.e. solution (3, 1) means + that `x = 3` and `y = 1`. + + >>> diop_DN(986, 1) # Solves equation x**2 - 986*y**2 = 1 + [(49299, 1570)] + + See Also + ======== + + find_DN(), diop_bf_DN() + + References + ========== + + .. [1] Solving the generalized Pell equation x**2 - D*y**2 = N, John P. + Robertson, July 31, 2004, Pages 16 - 17. [online], Available: + https://web.archive.org/web/20160323033128/http://www.jpr2718.org/pell.pdf + """ + if D < 0: + if N == 0: + return [(0, 0)] + if N < 0: + return [] + # N > 0: + sol = [] + for d in divisors(square_factor(N), generator=True): + for x, y in cornacchia(1, int(-D), int(N // d**2)): + sol.append((d*x, d*y)) + if D == -1: + sol.append((d*y, d*x)) + return sol + + if D == 0: + if N < 0: + return [] + if N == 0: + return [(0, t)] + sN, _exact = integer_nthroot(N, 2) + if _exact: + return [(sN, t)] + return [] + + # D > 0 + sD, _exact = integer_nthroot(D, 2) + if _exact: + if N == 0: + return [(sD*t, t)] + + sol = [] + for y in range(floor(sign(N)*(N - 1)/(2*sD)) + 1): + try: + sq, _exact = integer_nthroot(D*y**2 + N, 2) + except ValueError: + _exact = False + if _exact: + sol.append((sq, y)) + return sol + + if 1 < N**2 < D: + # It is much faster to call `_special_diop_DN`. + return _special_diop_DN(D, N) + + if N == 0: + return [(0, 0)] + + sol = [] + if abs(N) == 1: + pqa = PQa(0, 1, D) + *_, prev_B, prev_G = next(pqa) + for j, (*_, a, _, _B, _G) in enumerate(pqa): + if a == 2*sD: + break + prev_B, prev_G = _B, _G + if j % 2: + if N == 1: + sol.append((prev_G, prev_B)) + return sol + if N == -1: + return [(prev_G, prev_B)] + for _ in range(j): + *_, _B, _G = next(pqa) + return [(_G, _B)] + + for f in divisors(square_factor(N), generator=True): + m = N // f**2 + am = abs(m) + for sqm in sqrt_mod(D, am, all_roots=True): + z = symmetric_residue(sqm, am) + pqa = PQa(z, am, D) + *_, prev_B, prev_G = next(pqa) + for _ in range(length(z, am, D) - 1): + _, q, *_, _B, _G = next(pqa) + if abs(q) == 1: + if prev_G**2 - D*prev_B**2 == m: + sol.append((f*prev_G, f*prev_B)) + elif a := diop_DN(D, -1): + sol.append((f*(prev_G*a[0][0] + prev_B*D*a[0][1]), + f*(prev_G*a[0][1] + prev_B*a[0][0]))) + break + prev_B, prev_G = _B, _G + return sol + + +def _special_diop_DN(D, N): + """ + Solves the equation `x^2 - Dy^2 = N` for the special case where + `1 < N**2 < D` and `D` is not a perfect square. + It is better to call `diop_DN` rather than this function, as + the former checks the condition `1 < N**2 < D`, and calls the latter only + if appropriate. + + Usage + ===== + + WARNING: Internal method. Do not call directly! + + ``_special_diop_DN(D, N)``: D and N are integers as in `x^2 - Dy^2 = N`. + + Details + ======= + + ``D`` and ``N`` correspond to D and N in the equation. + + Examples + ======== + + >>> from sympy.solvers.diophantine.diophantine import _special_diop_DN + >>> _special_diop_DN(13, -3) # Solves equation x**2 - 13*y**2 = -3 + [(7, 2), (137, 38)] + + The output can be interpreted as follows: There are two fundamental + solutions to the equation `x^2 - 13y^2 = -3` given by (7, 2) and + (137, 38). Each tuple is in the form (x, y), i.e. solution (7, 2) means + that `x = 7` and `y = 2`. + + >>> _special_diop_DN(2445, -20) # Solves equation x**2 - 2445*y**2 = -20 + [(445, 9), (17625560, 356454), (698095554475, 14118073569)] + + See Also + ======== + + diop_DN() + + References + ========== + + .. [1] Section 4.4.4 of the following book: + Quadratic Diophantine Equations, T. Andreescu and D. Andrica, + Springer, 2015. + """ + + # The following assertion was removed for efficiency, with the understanding + # that this method is not called directly. The parent method, `diop_DN` + # is responsible for performing the appropriate checks. + # + # assert (1 < N**2 < D) and (not integer_nthroot(D, 2)[1]) + + sqrt_D = isqrt(D) + F = {N // f**2: f for f in divisors(square_factor(abs(N)), generator=True)} + P = 0 + Q = 1 + G0, G1 = 0, 1 + B0, B1 = 1, 0 + + solutions = [] + while True: + for _ in range(2): + a = (P + sqrt_D) // Q + P = a*Q - P + Q = (D - P**2) // Q + G0, G1 = G1, a*G1 + G0 + B0, B1 = B1, a*B1 + B0 + if (s := G1**2 - D*B1**2) in F: + f = F[s] + solutions.append((f*G1, f*B1)) + if Q == 1: + break + return solutions + + +def cornacchia(a:int, b:int, m:int) -> set[tuple[int, int]]: + r""" + Solves `ax^2 + by^2 = m` where `\gcd(a, b) = 1 = gcd(a, m)` and `a, b > 0`. + + Explanation + =========== + + Uses the algorithm due to Cornacchia. The method only finds primitive + solutions, i.e. ones with `\gcd(x, y) = 1`. So this method cannot be used to + find the solutions of `x^2 + y^2 = 20` since the only solution to former is + `(x, y) = (4, 2)` and it is not primitive. When `a = b`, only the + solutions with `x \leq y` are found. For more details, see the References. + + Examples + ======== + + >>> from sympy.solvers.diophantine.diophantine import cornacchia + >>> cornacchia(2, 3, 35) # equation 2x**2 + 3y**2 = 35 + {(2, 3), (4, 1)} + >>> cornacchia(1, 1, 25) # equation x**2 + y**2 = 25 + {(4, 3)} + + References + =========== + + .. [1] A. Nitaj, "L'algorithme de Cornacchia" + .. [2] Solving the diophantine equation ax**2 + by**2 = m by Cornacchia's + method, [online], Available: + http://www.numbertheory.org/php/cornacchia.html + + See Also + ======== + + sympy.utilities.iterables.signed_permutations + """ + # Assume gcd(a, b) = gcd(a, m) = 1 and a, b > 0 but no error checking + sols = set() + + if a + b > m: + # xy = 0 must hold if there exists a solution + if a == 1: + # y = 0 + s, _exact = iroot(m // a, 2) + if _exact: + sols.add((int(s), 0)) + if a == b: + # only keep one solution + return sols + if m % b == 0: + # x = 0 + s, _exact = iroot(m // b, 2) + if _exact: + sols.add((0, int(s))) + return sols + + # the original cornacchia + for t in sqrt_mod_iter(-b*invert(a, m), m): + if t < m // 2: + continue + u, r = m, t + while (m1 := m - a*r**2) <= 0: + u, r = r, u % r + m1, _r = divmod(m1, b) + if _r: + continue + s, _exact = iroot(m1, 2) + if _exact: + if a == b and r < s: + r, s = s, r + sols.add((int(r), int(s))) + return sols + + +def PQa(P_0, Q_0, D): + r""" + Returns useful information needed to solve the Pell equation. + + Explanation + =========== + + There are six sequences of integers defined related to the continued + fraction representation of `\\frac{P + \sqrt{D}}{Q}`, namely {`P_{i}`}, + {`Q_{i}`}, {`a_{i}`},{`A_{i}`}, {`B_{i}`}, {`G_{i}`}. ``PQa()`` Returns + these values as a 6-tuple in the same order as mentioned above. Refer [1]_ + for more detailed information. + + Usage + ===== + + ``PQa(P_0, Q_0, D)``: ``P_0``, ``Q_0`` and ``D`` are integers corresponding + to `P_{0}`, `Q_{0}` and `D` in the continued fraction + `\\frac{P_{0} + \sqrt{D}}{Q_{0}}`. + Also it's assumed that `P_{0}^2 == D mod(|Q_{0}|)` and `D` is square free. + + Examples + ======== + + >>> from sympy.solvers.diophantine.diophantine import PQa + >>> pqa = PQa(13, 4, 5) # (13 + sqrt(5))/4 + >>> next(pqa) # (P_0, Q_0, a_0, A_0, B_0, G_0) + (13, 4, 3, 3, 1, -1) + >>> next(pqa) # (P_1, Q_1, a_1, A_1, B_1, G_1) + (-1, 1, 1, 4, 1, 3) + + References + ========== + + .. [1] Solving the generalized Pell equation x^2 - Dy^2 = N, John P. + Robertson, July 31, 2004, Pages 4 - 8. https://web.archive.org/web/20160323033128/http://www.jpr2718.org/pell.pdf + """ + sqD = isqrt(D) + A2 = B1 = 0 + A1 = B2 = 1 + G1 = Q_0 + G2 = -P_0 + P_i = P_0 + Q_i = Q_0 + + while True: + a_i = (P_i + sqD) // Q_i + A1, A2 = a_i*A1 + A2, A1 + B1, B2 = a_i*B1 + B2, B1 + G1, G2 = a_i*G1 + G2, G1 + yield P_i, Q_i, a_i, A1, B1, G1 + + P_i = a_i*Q_i - P_i + Q_i = (D - P_i**2) // Q_i + + +def diop_bf_DN(D, N, t=symbols("t", integer=True)): + r""" + Uses brute force to solve the equation, `x^2 - Dy^2 = N`. + + Explanation + =========== + + Mainly concerned with the generalized Pell equation which is the case when + `D > 0, D` is not a perfect square. For more information on the case refer + [1]_. Let `(t, u)` be the minimal positive solution of the equation + `x^2 - Dy^2 = 1`. Then this method requires + `\sqrt{\\frac{\mid N \mid (t \pm 1)}{2D}}` to be small. + + Usage + ===== + + ``diop_bf_DN(D, N, t)``: ``D`` and ``N`` are coefficients in + `x^2 - Dy^2 = N` and ``t`` is the parameter to be used in the solutions. + + Details + ======= + + ``D`` and ``N`` correspond to D and N in the equation. + ``t`` is the parameter to be used in the solutions. + + Examples + ======== + + >>> from sympy.solvers.diophantine.diophantine import diop_bf_DN + >>> diop_bf_DN(13, -4) + [(3, 1), (-3, 1), (36, 10)] + >>> diop_bf_DN(986, 1) + [(49299, 1570)] + + See Also + ======== + + diop_DN() + + References + ========== + + .. [1] Solving the generalized Pell equation x**2 - D*y**2 = N, John P. + Robertson, July 31, 2004, Page 15. https://web.archive.org/web/20160323033128/http://www.jpr2718.org/pell.pdf + """ + D = as_int(D) + N = as_int(N) + + sol = [] + a = diop_DN(D, 1) + u = a[0][0] + + if N == 0: + if D < 0: + return [(0, 0)] + if D == 0: + return [(0, t)] + sD, _exact = integer_nthroot(D, 2) + if _exact: + return [(sD*t, t), (-sD*t, t)] + return [(0, 0)] + + if abs(N) == 1: + return diop_DN(D, N) + + if N > 1: + L1 = 0 + L2 = integer_nthroot(int(N*(u - 1)/(2*D)), 2)[0] + 1 + else: # N < -1 + L1, _exact = integer_nthroot(-int(N/D), 2) + if not _exact: + L1 += 1 + L2 = integer_nthroot(-int(N*(u + 1)/(2*D)), 2)[0] + 1 + + for y in range(L1, L2): + try: + x, _exact = integer_nthroot(N + D*y**2, 2) + except ValueError: + _exact = False + if _exact: + sol.append((x, y)) + if not equivalent(x, y, -x, y, D, N): + sol.append((-x, y)) + + return sol + + +def equivalent(u, v, r, s, D, N): + """ + Returns True if two solutions `(u, v)` and `(r, s)` of `x^2 - Dy^2 = N` + belongs to the same equivalence class and False otherwise. + + Explanation + =========== + + Two solutions `(u, v)` and `(r, s)` to the above equation fall to the same + equivalence class iff both `(ur - Dvs)` and `(us - vr)` are divisible by + `N`. See reference [1]_. No test is performed to test whether `(u, v)` and + `(r, s)` are actually solutions to the equation. User should take care of + this. + + Usage + ===== + + ``equivalent(u, v, r, s, D, N)``: `(u, v)` and `(r, s)` are two solutions + of the equation `x^2 - Dy^2 = N` and all parameters involved are integers. + + Examples + ======== + + >>> from sympy.solvers.diophantine.diophantine import equivalent + >>> equivalent(18, 5, -18, -5, 13, -1) + True + >>> equivalent(3, 1, -18, 393, 109, -4) + False + + References + ========== + + .. [1] Solving the generalized Pell equation x**2 - D*y**2 = N, John P. + Robertson, July 31, 2004, Page 12. https://web.archive.org/web/20160323033128/http://www.jpr2718.org/pell.pdf + + """ + return divisible(u*r - D*v*s, N) and divisible(u*s - v*r, N) + + +def length(P, Q, D): + r""" + Returns the (length of aperiodic part + length of periodic part) of + continued fraction representation of `\\frac{P + \sqrt{D}}{Q}`. + + It is important to remember that this does NOT return the length of the + periodic part but the sum of the lengths of the two parts as mentioned + above. + + Usage + ===== + + ``length(P, Q, D)``: ``P``, ``Q`` and ``D`` are integers corresponding to + the continued fraction `\\frac{P + \sqrt{D}}{Q}`. + + Details + ======= + + ``P``, ``D`` and ``Q`` corresponds to P, D and Q in the continued fraction, + `\\frac{P + \sqrt{D}}{Q}`. + + Examples + ======== + + >>> from sympy.solvers.diophantine.diophantine import length + >>> length(-2, 4, 5) # (-2 + sqrt(5))/4 + 3 + >>> length(-5, 4, 17) # (-5 + sqrt(17))/4 + 4 + + See Also + ======== + sympy.ntheory.continued_fraction.continued_fraction_periodic + """ + from sympy.ntheory.continued_fraction import continued_fraction_periodic + v = continued_fraction_periodic(P, Q, D) + if isinstance(v[-1], list): + rpt = len(v[-1]) + nonrpt = len(v) - 1 + else: + rpt = 0 + nonrpt = len(v) + return rpt + nonrpt + + +def transformation_to_DN(eq): + """ + This function transforms general quadratic, + `ax^2 + bxy + cy^2 + dx + ey + f = 0` + to more easy to deal with `X^2 - DY^2 = N` form. + + Explanation + =========== + + This is used to solve the general quadratic equation by transforming it to + the latter form. Refer to [1]_ for more detailed information on the + transformation. This function returns a tuple (A, B) where A is a 2 X 2 + matrix and B is a 2 X 1 matrix such that, + + Transpose([x y]) = A * Transpose([X Y]) + B + + Usage + ===== + + ``transformation_to_DN(eq)``: where ``eq`` is the quadratic to be + transformed. + + Examples + ======== + + >>> from sympy.abc import x, y + >>> from sympy.solvers.diophantine.diophantine import transformation_to_DN + >>> A, B = transformation_to_DN(x**2 - 3*x*y - y**2 - 2*y + 1) + >>> A + Matrix([ + [1/26, 3/26], + [ 0, 1/13]]) + >>> B + Matrix([ + [-6/13], + [-4/13]]) + + A, B returned are such that Transpose((x y)) = A * Transpose((X Y)) + B. + Substituting these values for `x` and `y` and a bit of simplifying work + will give an equation of the form `x^2 - Dy^2 = N`. + + >>> from sympy.abc import X, Y + >>> from sympy import Matrix, simplify + >>> u = (A*Matrix([X, Y]) + B)[0] # Transformation for x + >>> u + X/26 + 3*Y/26 - 6/13 + >>> v = (A*Matrix([X, Y]) + B)[1] # Transformation for y + >>> v + Y/13 - 4/13 + + Next we will substitute these formulas for `x` and `y` and do + ``simplify()``. + + >>> eq = simplify((x**2 - 3*x*y - y**2 - 2*y + 1).subs(zip((x, y), (u, v)))) + >>> eq + X**2/676 - Y**2/52 + 17/13 + + By multiplying the denominator appropriately, we can get a Pell equation + in the standard form. + + >>> eq * 676 + X**2 - 13*Y**2 + 884 + + If only the final equation is needed, ``find_DN()`` can be used. + + See Also + ======== + + find_DN() + + References + ========== + + .. [1] Solving the equation ax^2 + bxy + cy^2 + dx + ey + f = 0, + John P.Robertson, May 8, 2003, Page 7 - 11. + https://web.archive.org/web/20160323033111/http://www.jpr2718.org/ax2p.pdf + """ + + var, coeff, diop_type = classify_diop(eq, _dict=False) + if diop_type == BinaryQuadratic.name: + return _transformation_to_DN(var, coeff) + + +def _transformation_to_DN(var, coeff): + + x, y = var + + a = coeff[x**2] + b = coeff[x*y] + c = coeff[y**2] + d = coeff[x] + e = coeff[y] + f = coeff[1] + + a, b, c, d, e, f = [as_int(i) for i in _remove_gcd(a, b, c, d, e, f)] + + X, Y = symbols("X, Y", integer=True) + + if b: + B, C = _rational_pq(2*a, b) + A, T = _rational_pq(a, B**2) + + # eq_1 = A*B*X**2 + B*(c*T - A*C**2)*Y**2 + d*T*X + (B*e*T - d*T*C)*Y + f*T*B + coeff = {X**2: A*B, X*Y: 0, Y**2: B*(c*T - A*C**2), X: d*T, Y: B*e*T - d*T*C, 1: f*T*B} + A_0, B_0 = _transformation_to_DN([X, Y], coeff) + return Matrix(2, 2, [S.One/B, -S(C)/B, 0, 1])*A_0, Matrix(2, 2, [S.One/B, -S(C)/B, 0, 1])*B_0 + + if d: + B, C = _rational_pq(2*a, d) + A, T = _rational_pq(a, B**2) + + # eq_2 = A*X**2 + c*T*Y**2 + e*T*Y + f*T - A*C**2 + coeff = {X**2: A, X*Y: 0, Y**2: c*T, X: 0, Y: e*T, 1: f*T - A*C**2} + A_0, B_0 = _transformation_to_DN([X, Y], coeff) + return Matrix(2, 2, [S.One/B, 0, 0, 1])*A_0, Matrix(2, 2, [S.One/B, 0, 0, 1])*B_0 + Matrix([-S(C)/B, 0]) + + if e: + B, C = _rational_pq(2*c, e) + A, T = _rational_pq(c, B**2) + + # eq_3 = a*T*X**2 + A*Y**2 + f*T - A*C**2 + coeff = {X**2: a*T, X*Y: 0, Y**2: A, X: 0, Y: 0, 1: f*T - A*C**2} + A_0, B_0 = _transformation_to_DN([X, Y], coeff) + return Matrix(2, 2, [1, 0, 0, S.One/B])*A_0, Matrix(2, 2, [1, 0, 0, S.One/B])*B_0 + Matrix([0, -S(C)/B]) + + # TODO: pre-simplification: Not necessary but may simplify + # the equation. + return Matrix(2, 2, [S.One/a, 0, 0, 1]), Matrix([0, 0]) + + +def find_DN(eq): + """ + This function returns a tuple, `(D, N)` of the simplified form, + `x^2 - Dy^2 = N`, corresponding to the general quadratic, + `ax^2 + bxy + cy^2 + dx + ey + f = 0`. + + Solving the general quadratic is then equivalent to solving the equation + `X^2 - DY^2 = N` and transforming the solutions by using the transformation + matrices returned by ``transformation_to_DN()``. + + Usage + ===== + + ``find_DN(eq)``: where ``eq`` is the quadratic to be transformed. + + Examples + ======== + + >>> from sympy.abc import x, y + >>> from sympy.solvers.diophantine.diophantine import find_DN + >>> find_DN(x**2 - 3*x*y - y**2 - 2*y + 1) + (13, -884) + + Interpretation of the output is that we get `X^2 -13Y^2 = -884` after + transforming `x^2 - 3xy - y^2 - 2y + 1` using the transformation returned + by ``transformation_to_DN()``. + + See Also + ======== + + transformation_to_DN() + + References + ========== + + .. [1] Solving the equation ax^2 + bxy + cy^2 + dx + ey + f = 0, + John P.Robertson, May 8, 2003, Page 7 - 11. + https://web.archive.org/web/20160323033111/http://www.jpr2718.org/ax2p.pdf + """ + var, coeff, diop_type = classify_diop(eq, _dict=False) + if diop_type == BinaryQuadratic.name: + return _find_DN(var, coeff) + + +def _find_DN(var, coeff): + + x, y = var + X, Y = symbols("X, Y", integer=True) + A, B = _transformation_to_DN(var, coeff) + + u = (A*Matrix([X, Y]) + B)[0] + v = (A*Matrix([X, Y]) + B)[1] + eq = x**2*coeff[x**2] + x*y*coeff[x*y] + y**2*coeff[y**2] + x*coeff[x] + y*coeff[y] + coeff[1] + + simplified = _mexpand(eq.subs(zip((x, y), (u, v)))) + + coeff = simplified.as_coefficients_dict() + + return -coeff[Y**2]/coeff[X**2], -coeff[1]/coeff[X**2] + + +def check_param(x, y, a, params): + """ + If there is a number modulo ``a`` such that ``x`` and ``y`` are both + integers, then return a parametric representation for ``x`` and ``y`` + else return (None, None). + + Here ``x`` and ``y`` are functions of ``t``. + """ + from sympy.simplify.simplify import clear_coefficients + + if x.is_number and not x.is_Integer: + return DiophantineSolutionSet([x, y], parameters=params) + + if y.is_number and not y.is_Integer: + return DiophantineSolutionSet([x, y], parameters=params) + + m, n = symbols("m, n", integer=True) + c, p = (m*x + n*y).as_content_primitive() + if a % c.q: + return DiophantineSolutionSet([x, y], parameters=params) + + # clear_coefficients(mx + b, R)[1] -> (R - b)/m + eq = clear_coefficients(x, m)[1] - clear_coefficients(y, n)[1] + junk, eq = eq.as_content_primitive() + + return _diop_solve(eq, params=params) + + +def diop_ternary_quadratic(eq, parameterize=False): + """ + Solves the general quadratic ternary form, + `ax^2 + by^2 + cz^2 + fxy + gyz + hxz = 0`. + + Returns a tuple `(x, y, z)` which is a base solution for the above + equation. If there are no solutions, `(None, None, None)` is returned. + + Usage + ===== + + ``diop_ternary_quadratic(eq)``: Return a tuple containing a basic solution + to ``eq``. + + Details + ======= + + ``eq`` should be an homogeneous expression of degree two in three variables + and it is assumed to be zero. + + Examples + ======== + + >>> from sympy.abc import x, y, z + >>> from sympy.solvers.diophantine.diophantine import diop_ternary_quadratic + >>> diop_ternary_quadratic(x**2 + 3*y**2 - z**2) + (1, 0, 1) + >>> diop_ternary_quadratic(4*x**2 + 5*y**2 - z**2) + (1, 0, 2) + >>> diop_ternary_quadratic(45*x**2 - 7*y**2 - 8*x*y - z**2) + (28, 45, 105) + >>> diop_ternary_quadratic(x**2 - 49*y**2 - z**2 + 13*z*y -8*x*y) + (9, 1, 5) + """ + var, coeff, diop_type = classify_diop(eq, _dict=False) + + if diop_type in ( + HomogeneousTernaryQuadratic.name, + HomogeneousTernaryQuadraticNormal.name): + sol = _diop_ternary_quadratic(var, coeff) + if len(sol) > 0: + x_0, y_0, z_0 = list(sol)[0] + else: + x_0, y_0, z_0 = None, None, None + + if parameterize: + return _parametrize_ternary_quadratic( + (x_0, y_0, z_0), var, coeff) + return x_0, y_0, z_0 + + +def _diop_ternary_quadratic(_var, coeff): + eq = sum(i*coeff[i] for i in coeff) + if HomogeneousTernaryQuadratic(eq).matches(): + return HomogeneousTernaryQuadratic(eq, free_symbols=_var).solve() + elif HomogeneousTernaryQuadraticNormal(eq).matches(): + return HomogeneousTernaryQuadraticNormal(eq, free_symbols=_var).solve() + + +def transformation_to_normal(eq): + """ + Returns the transformation Matrix that converts a general ternary + quadratic equation ``eq`` (`ax^2 + by^2 + cz^2 + dxy + eyz + fxz`) + to a form without cross terms: `ax^2 + by^2 + cz^2 = 0`. This is + not used in solving ternary quadratics; it is only implemented for + the sake of completeness. + """ + var, coeff, diop_type = classify_diop(eq, _dict=False) + + if diop_type in ( + "homogeneous_ternary_quadratic", + "homogeneous_ternary_quadratic_normal"): + return _transformation_to_normal(var, coeff) + + +def _transformation_to_normal(var, coeff): + + _var = list(var) # copy + x, y, z = var + + if not any(coeff[i**2] for i in var): + # https://math.stackexchange.com/questions/448051/transform-quadratic-ternary-form-to-normal-form/448065#448065 + a = coeff[x*y] + b = coeff[y*z] + c = coeff[x*z] + swap = False + if not a: # b can't be 0 or else there aren't 3 vars + swap = True + a, b = b, a + T = Matrix(((1, 1, -b/a), (1, -1, -c/a), (0, 0, 1))) + if swap: + T.row_swap(0, 1) + T.col_swap(0, 1) + return T + + if coeff[x**2] == 0: + # If the coefficient of x is zero change the variables + if coeff[y**2] == 0: + _var[0], _var[2] = var[2], var[0] + T = _transformation_to_normal(_var, coeff) + T.row_swap(0, 2) + T.col_swap(0, 2) + return T + + _var[0], _var[1] = var[1], var[0] + T = _transformation_to_normal(_var, coeff) + T.row_swap(0, 1) + T.col_swap(0, 1) + return T + + # Apply the transformation x --> X - (B*Y + C*Z)/(2*A) + if coeff[x*y] != 0 or coeff[x*z] != 0: + A = coeff[x**2] + B = coeff[x*y] + C = coeff[x*z] + D = coeff[y**2] + E = coeff[y*z] + F = coeff[z**2] + + _coeff = {} + + _coeff[x**2] = 4*A**2 + _coeff[y**2] = 4*A*D - B**2 + _coeff[z**2] = 4*A*F - C**2 + _coeff[y*z] = 4*A*E - 2*B*C + _coeff[x*y] = 0 + _coeff[x*z] = 0 + + T_0 = _transformation_to_normal(_var, _coeff) + return Matrix(3, 3, [1, S(-B)/(2*A), S(-C)/(2*A), 0, 1, 0, 0, 0, 1])*T_0 + + elif coeff[y*z] != 0: + if coeff[y**2] == 0: + if coeff[z**2] == 0: + # Equations of the form A*x**2 + E*yz = 0. + # Apply transformation y -> Y + Z ans z -> Y - Z + return Matrix(3, 3, [1, 0, 0, 0, 1, 1, 0, 1, -1]) + + # Ax**2 + E*y*z + F*z**2 = 0 + _var[0], _var[2] = var[2], var[0] + T = _transformation_to_normal(_var, coeff) + T.row_swap(0, 2) + T.col_swap(0, 2) + return T + + # A*x**2 + D*y**2 + E*y*z + F*z**2 = 0, F may be zero + _var[0], _var[1] = var[1], var[0] + T = _transformation_to_normal(_var, coeff) + T.row_swap(0, 1) + T.col_swap(0, 1) + return T + + return Matrix.eye(3) + + +def parametrize_ternary_quadratic(eq): + """ + Returns the parametrized general solution for the ternary quadratic + equation ``eq`` which has the form + `ax^2 + by^2 + cz^2 + fxy + gyz + hxz = 0`. + + Examples + ======== + + >>> from sympy import Tuple, ordered + >>> from sympy.abc import x, y, z + >>> from sympy.solvers.diophantine.diophantine import parametrize_ternary_quadratic + + The parametrized solution may be returned with three parameters: + + >>> parametrize_ternary_quadratic(2*x**2 + y**2 - 2*z**2) + (p**2 - 2*q**2, -2*p**2 + 4*p*q - 4*p*r - 4*q**2, p**2 - 4*p*q + 2*q**2 - 4*q*r) + + There might also be only two parameters: + + >>> parametrize_ternary_quadratic(4*x**2 + 2*y**2 - 3*z**2) + (2*p**2 - 3*q**2, -4*p**2 + 12*p*q - 6*q**2, 4*p**2 - 8*p*q + 6*q**2) + + Notes + ===== + + Consider ``p`` and ``q`` in the previous 2-parameter + solution and observe that more than one solution can be represented + by a given pair of parameters. If `p` and ``q`` are not coprime, this is + trivially true since the common factor will also be a common factor of the + solution values. But it may also be true even when ``p`` and + ``q`` are coprime: + + >>> sol = Tuple(*_) + >>> p, q = ordered(sol.free_symbols) + >>> sol.subs([(p, 3), (q, 2)]) + (6, 12, 12) + >>> sol.subs([(q, 1), (p, 1)]) + (-1, 2, 2) + >>> sol.subs([(q, 0), (p, 1)]) + (2, -4, 4) + >>> sol.subs([(q, 1), (p, 0)]) + (-3, -6, 6) + + Except for sign and a common factor, these are equivalent to + the solution of (1, 2, 2). + + References + ========== + + .. [1] The algorithmic resolution of Diophantine equations, Nigel P. Smart, + London Mathematical Society Student Texts 41, Cambridge University + Press, Cambridge, 1998. + + """ + var, coeff, diop_type = classify_diop(eq, _dict=False) + + if diop_type in ( + "homogeneous_ternary_quadratic", + "homogeneous_ternary_quadratic_normal"): + x_0, y_0, z_0 = list(_diop_ternary_quadratic(var, coeff))[0] + return _parametrize_ternary_quadratic( + (x_0, y_0, z_0), var, coeff) + + +def _parametrize_ternary_quadratic(solution, _var, coeff): + # called for a*x**2 + b*y**2 + c*z**2 + d*x*y + e*y*z + f*x*z = 0 + assert 1 not in coeff + + x_0, y_0, z_0 = solution + + v = list(_var) # copy + + if x_0 is None: + return (None, None, None) + + if solution.count(0) >= 2: + # if there are 2 zeros the equation reduces + # to k*X**2 == 0 where X is x, y, or z so X must + # be zero, too. So there is only the trivial + # solution. + return (None, None, None) + + if x_0 == 0: + v[0], v[1] = v[1], v[0] + y_p, x_p, z_p = _parametrize_ternary_quadratic( + (y_0, x_0, z_0), v, coeff) + return x_p, y_p, z_p + + x, y, z = v + r, p, q = symbols("r, p, q", integer=True) + + eq = sum(k*v for k, v in coeff.items()) + eq_1 = _mexpand(eq.subs(zip( + (x, y, z), (r*x_0, r*y_0 + p, r*z_0 + q)))) + A, B = eq_1.as_independent(r, as_Add=True) + + + x = A*x_0 + y = (A*y_0 - _mexpand(B/r*p)) + z = (A*z_0 - _mexpand(B/r*q)) + + return _remove_gcd(x, y, z) + + +def diop_ternary_quadratic_normal(eq, parameterize=False): + """ + Solves the quadratic ternary diophantine equation, + `ax^2 + by^2 + cz^2 = 0`. + + Explanation + =========== + + Here the coefficients `a`, `b`, and `c` should be non zero. Otherwise the + equation will be a quadratic binary or univariate equation. If solvable, + returns a tuple `(x, y, z)` that satisfies the given equation. If the + equation does not have integer solutions, `(None, None, None)` is returned. + + Usage + ===== + + ``diop_ternary_quadratic_normal(eq)``: where ``eq`` is an equation of the form + `ax^2 + by^2 + cz^2 = 0`. + + Examples + ======== + + >>> from sympy.abc import x, y, z + >>> from sympy.solvers.diophantine.diophantine import diop_ternary_quadratic_normal + >>> diop_ternary_quadratic_normal(x**2 + 3*y**2 - z**2) + (1, 0, 1) + >>> diop_ternary_quadratic_normal(4*x**2 + 5*y**2 - z**2) + (1, 0, 2) + >>> diop_ternary_quadratic_normal(34*x**2 - 3*y**2 - 301*z**2) + (4, 9, 1) + """ + var, coeff, diop_type = classify_diop(eq, _dict=False) + if diop_type == HomogeneousTernaryQuadraticNormal.name: + sol = _diop_ternary_quadratic_normal(var, coeff) + if len(sol) > 0: + x_0, y_0, z_0 = list(sol)[0] + else: + x_0, y_0, z_0 = None, None, None + if parameterize: + return _parametrize_ternary_quadratic( + (x_0, y_0, z_0), var, coeff) + return x_0, y_0, z_0 + + +def _diop_ternary_quadratic_normal(var, coeff): + eq = sum(i * coeff[i] for i in coeff) + return HomogeneousTernaryQuadraticNormal(eq, free_symbols=var).solve() + + +def sqf_normal(a, b, c, steps=False): + """ + Return `a', b', c'`, the coefficients of the square-free normal + form of `ax^2 + by^2 + cz^2 = 0`, where `a', b', c'` are pairwise + prime. If `steps` is True then also return three tuples: + `sq`, `sqf`, and `(a', b', c')` where `sq` contains the square + factors of `a`, `b` and `c` after removing the `gcd(a, b, c)`; + `sqf` contains the values of `a`, `b` and `c` after removing + both the `gcd(a, b, c)` and the square factors. + + The solutions for `ax^2 + by^2 + cz^2 = 0` can be + recovered from the solutions of `a'x^2 + b'y^2 + c'z^2 = 0`. + + Examples + ======== + + >>> from sympy.solvers.diophantine.diophantine import sqf_normal + >>> sqf_normal(2 * 3**2 * 5, 2 * 5 * 11, 2 * 7**2 * 11) + (11, 1, 5) + >>> sqf_normal(2 * 3**2 * 5, 2 * 5 * 11, 2 * 7**2 * 11, True) + ((3, 1, 7), (5, 55, 11), (11, 1, 5)) + + References + ========== + + .. [1] Legendre's Theorem, Legrange's Descent, + https://public.csusm.edu/aitken_html/notes/legendre.pdf + + + See Also + ======== + + reconstruct() + """ + ABC = _remove_gcd(a, b, c) + sq = tuple(square_factor(i) for i in ABC) + sqf = A, B, C = tuple([i//j**2 for i,j in zip(ABC, sq)]) + pc = igcd(A, B) + A /= pc + B /= pc + pa = igcd(B, C) + B /= pa + C /= pa + pb = igcd(A, C) + A /= pb + B /= pb + + A *= pa + B *= pb + C *= pc + + if steps: + return (sq, sqf, (A, B, C)) + else: + return A, B, C + + +def square_factor(a): + r""" + Returns an integer `c` s.t. `a = c^2k, \ c,k \in Z`. Here `k` is square + free. `a` can be given as an integer or a dictionary of factors. + + Examples + ======== + + >>> from sympy.solvers.diophantine.diophantine import square_factor + >>> square_factor(24) + 2 + >>> square_factor(-36*3) + 6 + >>> square_factor(1) + 1 + >>> square_factor({3: 2, 2: 1, -1: 1}) # -18 + 3 + + See Also + ======== + sympy.ntheory.factor_.core + """ + f = a if isinstance(a, dict) else factorint(a) + return Mul(*[p**(e//2) for p, e in f.items()]) + + +def reconstruct(A, B, z): + """ + Reconstruct the `z` value of an equivalent solution of `ax^2 + by^2 + cz^2` + from the `z` value of a solution of the square-free normal form of the + equation, `a'*x^2 + b'*y^2 + c'*z^2`, where `a'`, `b'` and `c'` are square + free and `gcd(a', b', c') == 1`. + """ + f = factorint(igcd(A, B)) + for p, e in f.items(): + if e != 1: + raise ValueError('a and b should be square-free') + z *= p + return z + + +def ldescent(A, B): + """ + Return a non-trivial solution to `w^2 = Ax^2 + By^2` using + Lagrange's method; return None if there is no such solution. + + Parameters + ========== + + A : Integer + B : Integer + non-zero integer + + Returns + ======= + + (int, int, int) | None : a tuple `(w_0, x_0, y_0)` which is a solution to the above equation. + + Examples + ======== + + >>> from sympy.solvers.diophantine.diophantine import ldescent + >>> ldescent(1, 1) # w^2 = x^2 + y^2 + (1, 1, 0) + >>> ldescent(4, -7) # w^2 = 4x^2 - 7y^2 + (2, -1, 0) + + This means that `x = -1, y = 0` and `w = 2` is a solution to the equation + `w^2 = 4x^2 - 7y^2` + + >>> ldescent(5, -1) # w^2 = 5x^2 - y^2 + (2, 1, -1) + + References + ========== + + .. [1] The algorithmic resolution of Diophantine equations, Nigel P. Smart, + London Mathematical Society Student Texts 41, Cambridge University + Press, Cambridge, 1998. + .. [2] Cremona, J. E., Rusin, D. (2003). Efficient Solution of Rational Conics. + Mathematics of Computation, 72(243), 1417-1441. + https://doi.org/10.1090/S0025-5718-02-01480-1 + """ + if A == 0 or B == 0: + raise ValueError("A and B must be non-zero integers") + if abs(A) > abs(B): + w, y, x = ldescent(B, A) + return w, x, y + if A == 1: + return (1, 1, 0) + if B == 1: + return (1, 0, 1) + if B == -1: # and A == -1 + return + + r = sqrt_mod(A, B) + if r is None: + return + Q = (r**2 - A) // B + if Q == 0: + return r, -1, 0 + for i in divisors(Q): + d, _exact = integer_nthroot(abs(Q) // i, 2) + if _exact: + B_0 = sign(Q)*i + W, X, Y = ldescent(A, B_0) + return _remove_gcd(-A*X + r*W, r*X - W, Y*B_0*d) + + +def descent(A, B): + """ + Returns a non-trivial solution, (x, y, z), to `x^2 = Ay^2 + Bz^2` + using Lagrange's descent method with lattice-reduction. `A` and `B` + are assumed to be valid for such a solution to exist. + + This is faster than the normal Lagrange's descent algorithm because + the Gaussian reduction is used. + + Examples + ======== + + >>> from sympy.solvers.diophantine.diophantine import descent + >>> descent(3, 1) # x**2 = 3*y**2 + z**2 + (1, 0, 1) + + `(x, y, z) = (1, 0, 1)` is a solution to the above equation. + + >>> descent(41, -113) + (-16, -3, 1) + + References + ========== + + .. [1] Cremona, J. E., Rusin, D. (2003). Efficient Solution of Rational Conics. + Mathematics of Computation, 72(243), 1417-1441. + https://doi.org/10.1090/S0025-5718-02-01480-1 + """ + if abs(A) > abs(B): + x, y, z = descent(B, A) + return x, z, y + + if B == 1: + return (1, 0, 1) + if A == 1: + return (1, 1, 0) + if B == -A: + return (0, 1, 1) + if B == A: + x, z, y = descent(-1, A) + return (A*y, z, x) + + w = sqrt_mod(A, B) + x_0, z_0 = gaussian_reduce(w, A, B) + + t = (x_0**2 - A*z_0**2) // B + t_2 = square_factor(t) + t_1 = t // t_2**2 + + x_1, z_1, y_1 = descent(A, t_1) + + return _remove_gcd(x_0*x_1 + A*z_0*z_1, z_0*x_1 + x_0*z_1, t_1*t_2*y_1) + + +def gaussian_reduce(w:int, a:int, b:int) -> tuple[int, int]: + r""" + Returns a reduced solution `(x, z)` to the congruence + `X^2 - aZ^2 \equiv 0 \pmod{b}` so that `x^2 + |a|z^2` is as small as possible. + Here ``w`` is a solution of the congruence `x^2 \equiv a \pmod{b}`. + + This function is intended to be used only for ``descent()``. + + Explanation + =========== + + The Gaussian reduction can find the shortest vector for any norm. + So we define the special norm for the vectors `u = (u_1, u_2)` and `v = (v_1, v_2)` as follows. + + .. math :: + u \cdot v := (wu_1 + bu_2)(wv_1 + bv_2) + |a|u_1v_1 + + Note that, given the mapping `f: (u_1, u_2) \to (wu_1 + bu_2, u_1)`, + `f((u_1,u_2))` is the solution to `X^2 - aZ^2 \equiv 0 \pmod{b}`. + In other words, finding the shortest vector in this norm will yield a solution with smaller `X^2 + |a|Z^2`. + The algorithm starts from basis vectors `(0, 1)` and `(1, 0)` + (corresponding to solutions `(b, 0)` and `(w, 1)`, respectively) and finds the shortest vector. + The shortest vector does not necessarily correspond to the smallest solution, + but since ``descent()`` only wants the smallest possible solution, it is sufficient. + + Parameters + ========== + + w : int + ``w`` s.t. `w^2 \equiv a \pmod{b}` + a : int + square-free nonzero integer + b : int + square-free nonzero integer + + Examples + ======== + + >>> from sympy.solvers.diophantine.diophantine import gaussian_reduce + >>> from sympy.ntheory.residue_ntheory import sqrt_mod + >>> a, b = 19, 101 + >>> gaussian_reduce(sqrt_mod(a, b), a, b) # 1**2 - 19*(-4)**2 = -303 + (1, -4) + >>> a, b = 11, 14 + >>> x, z = gaussian_reduce(sqrt_mod(a, b), a, b) + >>> (x**2 - a*z**2) % b == 0 + True + + It does not always return the smallest solution. + + >>> a, b = 6, 95 + >>> min_x, min_z = 1, 4 + >>> x, z = gaussian_reduce(sqrt_mod(a, b), a, b) + >>> (x**2 - a*z**2) % b == 0 and (min_x**2 - a*min_z**2) % b == 0 + True + >>> min_x**2 + abs(a)*min_z**2 < x**2 + abs(a)*z**2 + True + + References + ========== + + .. [1] Gaussian lattice Reduction [online]. Available: + https://web.archive.org/web/20201021115213/http://home.ie.cuhk.edu.hk/~wkshum/wordpress/?p=404 + .. [2] Cremona, J. E., Rusin, D. (2003). Efficient Solution of Rational Conics. + Mathematics of Computation, 72(243), 1417-1441. + https://doi.org/10.1090/S0025-5718-02-01480-1 + """ + a = abs(a) + def _dot(u, v): + return u[0]*v[0] + a*u[1]*v[1] + + u = (b, 0) + v = (w, 1) if b*w >= 0 else (-w, -1) + # i.e., _dot(u, v) >= 0 + + if b**2 < w**2 + a: + u, v = v, u + # i.e., norm(u) >= norm(v), where norm(u) := sqrt(_dot(u, u)) + + while _dot(u, u) > (dv := _dot(v, v)): + k = _dot(u, v) // dv + u, v = v, (u[0] - k*v[0], u[1] - k*v[1]) + c = (v[0] - u[0], v[1] - u[1]) + if _dot(c, c) <= _dot(u, u) <= 2*_dot(u, v): + return c + return u + + +def holzer(x, y, z, a, b, c): + r""" + Simplify the solution `(x, y, z)` of the equation + `ax^2 + by^2 = cz^2` with `a, b, c > 0` and `z^2 \geq \mid ab \mid` to + a new reduced solution `(x', y', z')` such that `z'^2 \leq \mid ab \mid`. + + The algorithm is an interpretation of Mordell's reduction as described + on page 8 of Cremona and Rusin's paper [1]_ and the work of Mordell in + reference [2]_. + + References + ========== + + .. [1] Cremona, J. E., Rusin, D. (2003). Efficient Solution of Rational Conics. + Mathematics of Computation, 72(243), 1417-1441. + https://doi.org/10.1090/S0025-5718-02-01480-1 + .. [2] Diophantine Equations, L. J. Mordell, page 48. + + """ + + if _odd(c): + k = 2*c + else: + k = c//2 + + small = a*b*c + step = 0 + while True: + t1, t2, t3 = a*x**2, b*y**2, c*z**2 + # check that it's a solution + if t1 + t2 != t3: + if step == 0: + raise ValueError('bad starting solution') + break + x_0, y_0, z_0 = x, y, z + if max(t1, t2, t3) <= small: + # Holzer condition + break + + uv = u, v = base_solution_linear(k, y_0, -x_0) + if None in uv: + break + + p, q = -(a*u*x_0 + b*v*y_0), c*z_0 + r = Rational(p, q) + if _even(c): + w = _nint_or_floor(p, q) + assert abs(w - r) <= S.Half + else: + w = p//q # floor + if _odd(a*u + b*v + c*w): + w += 1 + assert abs(w - r) <= S.One + + A = (a*u**2 + b*v**2 + c*w**2) + B = (a*u*x_0 + b*v*y_0 + c*w*z_0) + x = Rational(x_0*A - 2*u*B, k) + y = Rational(y_0*A - 2*v*B, k) + z = Rational(z_0*A - 2*w*B, k) + assert all(i.is_Integer for i in (x, y, z)) + step += 1 + + return tuple([int(i) for i in (x_0, y_0, z_0)]) + + +def diop_general_pythagorean(eq, param=symbols("m", integer=True)): + """ + Solves the general pythagorean equation, + `a_{1}^2x_{1}^2 + a_{2}^2x_{2}^2 + . . . + a_{n}^2x_{n}^2 - a_{n + 1}^2x_{n + 1}^2 = 0`. + + Returns a tuple which contains a parametrized solution to the equation, + sorted in the same order as the input variables. + + Usage + ===== + + ``diop_general_pythagorean(eq, param)``: where ``eq`` is a general + pythagorean equation which is assumed to be zero and ``param`` is the base + parameter used to construct other parameters by subscripting. + + Examples + ======== + + >>> from sympy.solvers.diophantine.diophantine import diop_general_pythagorean + >>> from sympy.abc import a, b, c, d, e + >>> diop_general_pythagorean(a**2 + b**2 + c**2 - d**2) + (m1**2 + m2**2 - m3**2, 2*m1*m3, 2*m2*m3, m1**2 + m2**2 + m3**2) + >>> diop_general_pythagorean(9*a**2 - 4*b**2 + 16*c**2 + 25*d**2 + e**2) + (10*m1**2 + 10*m2**2 + 10*m3**2 - 10*m4**2, 15*m1**2 + 15*m2**2 + 15*m3**2 + 15*m4**2, 15*m1*m4, 12*m2*m4, 60*m3*m4) + """ + var, coeff, diop_type = classify_diop(eq, _dict=False) + + if diop_type == GeneralPythagorean.name: + if param is None: + params = None + else: + params = symbols('%s1:%i' % (param, len(var)), integer=True) + return list(GeneralPythagorean(eq).solve(parameters=params))[0] + + +def diop_general_sum_of_squares(eq, limit=1): + r""" + Solves the equation `x_{1}^2 + x_{2}^2 + . . . + x_{n}^2 - k = 0`. + + Returns at most ``limit`` number of solutions. + + Usage + ===== + + ``general_sum_of_squares(eq, limit)`` : Here ``eq`` is an expression which + is assumed to be zero. Also, ``eq`` should be in the form, + `x_{1}^2 + x_{2}^2 + . . . + x_{n}^2 - k = 0`. + + Details + ======= + + When `n = 3` if `k = 4^a(8m + 7)` for some `a, m \in Z` then there will be + no solutions. Refer to [1]_ for more details. + + Examples + ======== + + >>> from sympy.solvers.diophantine.diophantine import diop_general_sum_of_squares + >>> from sympy.abc import a, b, c, d, e + >>> diop_general_sum_of_squares(a**2 + b**2 + c**2 + d**2 + e**2 - 2345) + {(15, 22, 22, 24, 24)} + + Reference + ========= + + .. [1] Representing an integer as a sum of three squares, [online], + Available: + https://proofwiki.org/wiki/Integer_as_Sum_of_Three_Squares + """ + var, coeff, diop_type = classify_diop(eq, _dict=False) + + if diop_type == GeneralSumOfSquares.name: + return set(GeneralSumOfSquares(eq).solve(limit=limit)) + + +def diop_general_sum_of_even_powers(eq, limit=1): + """ + Solves the equation `x_{1}^e + x_{2}^e + . . . + x_{n}^e - k = 0` + where `e` is an even, integer power. + + Returns at most ``limit`` number of solutions. + + Usage + ===== + + ``general_sum_of_even_powers(eq, limit)`` : Here ``eq`` is an expression which + is assumed to be zero. Also, ``eq`` should be in the form, + `x_{1}^e + x_{2}^e + . . . + x_{n}^e - k = 0`. + + Examples + ======== + + >>> from sympy.solvers.diophantine.diophantine import diop_general_sum_of_even_powers + >>> from sympy.abc import a, b + >>> diop_general_sum_of_even_powers(a**4 + b**4 - (2**4 + 3**4)) + {(2, 3)} + + See Also + ======== + + power_representation + """ + var, coeff, diop_type = classify_diop(eq, _dict=False) + + if diop_type == GeneralSumOfEvenPowers.name: + return set(GeneralSumOfEvenPowers(eq).solve(limit=limit)) + + +## Functions below this comment can be more suitably grouped under +## an Additive number theory module rather than the Diophantine +## equation module. + + +def partition(n, k=None, zeros=False): + """ + Returns a generator that can be used to generate partitions of an integer + `n`. + + Explanation + =========== + + A partition of `n` is a set of positive integers which add up to `n`. For + example, partitions of 3 are 3, 1 + 2, 1 + 1 + 1. A partition is returned + as a tuple. If ``k`` equals None, then all possible partitions are returned + irrespective of their size, otherwise only the partitions of size ``k`` are + returned. If the ``zero`` parameter is set to True then a suitable + number of zeros are added at the end of every partition of size less than + ``k``. + + ``zero`` parameter is considered only if ``k`` is not None. When the + partitions are over, the last `next()` call throws the ``StopIteration`` + exception, so this function should always be used inside a try - except + block. + + Details + ======= + + ``partition(n, k)``: Here ``n`` is a positive integer and ``k`` is the size + of the partition which is also positive integer. + + Examples + ======== + + >>> from sympy.solvers.diophantine.diophantine import partition + >>> f = partition(5) + >>> next(f) + (1, 1, 1, 1, 1) + >>> next(f) + (1, 1, 1, 2) + >>> g = partition(5, 3) + >>> next(g) + (1, 1, 3) + >>> next(g) + (1, 2, 2) + >>> g = partition(5, 3, zeros=True) + >>> next(g) + (0, 0, 5) + + """ + if not zeros or k is None: + for i in ordered_partitions(n, k): + yield tuple(i) + else: + for m in range(1, k + 1): + for i in ordered_partitions(n, m): + i = tuple(i) + yield (0,)*(k - len(i)) + i + + +def prime_as_sum_of_two_squares(p): + """ + Represent a prime `p` as a unique sum of two squares; this can + only be done if the prime is congruent to 1 mod 4. + + Parameters + ========== + + p : Integer + A prime that is congruent to 1 mod 4 + + Returns + ======= + + (int, int) | None : Pair of positive integers ``(x, y)`` satisfying ``x**2 + y**2 = p``. + None if ``p`` is not congruent to 1 mod 4. + + Raises + ====== + + ValueError + If ``p`` is not prime number + + Examples + ======== + + >>> from sympy.solvers.diophantine.diophantine import prime_as_sum_of_two_squares + >>> prime_as_sum_of_two_squares(7) # can't be done + >>> prime_as_sum_of_two_squares(5) + (1, 2) + + Reference + ========= + + .. [1] Representing a number as a sum of four squares, [online], + Available: https://schorn.ch/lagrange.html + + See Also + ======== + + sum_of_squares + + """ + p = as_int(p) + if p % 4 != 1: + return + if not isprime(p): + raise ValueError("p should be a prime number") + + if p % 8 == 5: + # Legendre symbol (2/p) == -1 if p % 8 in [3, 5] + b = 2 + elif p % 12 == 5: + # Legendre symbol (3/p) == -1 if p % 12 in [5, 7] + b = 3 + elif p % 5 in [2, 3]: + # Legendre symbol (5/p) == -1 if p % 5 in [2, 3] + b = 5 + else: + b = 7 + while jacobi(b, p) == 1: + b = nextprime(b) + + b = pow(b, p >> 2, p) + a = p + while b**2 > p: + a, b = b, a % b + return (int(a % b), int(b)) # convert from long + + +def sum_of_three_squares(n): + r""" + Returns a 3-tuple $(a, b, c)$ such that $a^2 + b^2 + c^2 = n$ and + $a, b, c \geq 0$. + + Returns None if $n = 4^a(8m + 7)$ for some `a, m \in \mathbb{Z}`. See + [1]_ for more details. + + Parameters + ========== + + n : Integer + non-negative integer + + Returns + ======= + + (int, int, int) | None : 3-tuple non-negative integers ``(a, b, c)`` satisfying ``a**2 + b**2 + c**2 = n``. + a,b,c are sorted in ascending order. ``None`` if no such ``(a,b,c)``. + + Raises + ====== + + ValueError + If ``n`` is a negative integer + + Examples + ======== + + >>> from sympy.solvers.diophantine.diophantine import sum_of_three_squares + >>> sum_of_three_squares(44542) + (18, 37, 207) + + References + ========== + + .. [1] Representing a number as a sum of three squares, [online], + Available: https://schorn.ch/lagrange.html + + See Also + ======== + + power_representation : + ``sum_of_three_squares(n)`` is one of the solutions output by ``power_representation(n, 2, 3, zeros=True)`` + + """ + # https://math.stackexchange.com/questions/483101/rabin-and-shallit-algorithm/651425#651425 + # discusses these numbers (except for 1, 2, 3) as the exceptions of H&L's conjecture that + # Every sufficiently large number n is either a square or the sum of a prime and a square. + special = {1: (0, 0, 1), 2: (0, 1, 1), 3: (1, 1, 1), 10: (0, 1, 3), 34: (3, 3, 4), + 58: (0, 3, 7), 85: (0, 6, 7), 130: (0, 3, 11), 214: (3, 6, 13), 226: (8, 9, 9), + 370: (8, 9, 15), 526: (6, 7, 21), 706: (15, 15, 16), 730: (0, 1, 27), + 1414: (6, 17, 33), 1906: (13, 21, 36), 2986: (21, 32, 39), 9634: (56, 57, 57)} + n = as_int(n) + if n < 0: + raise ValueError("n should be a non-negative integer") + if n == 0: + return (0, 0, 0) + n, v = remove(n, 4) + v = 1 << v + if n % 8 == 7: + return + if n in special: + return tuple([v*i for i in special[n]]) + + s, _exact = integer_nthroot(n, 2) + if _exact: + return (0, 0, v*s) + if n % 8 == 3: + if not s % 2: + s -= 1 + for x in range(s, -1, -2): + N = (n - x**2) // 2 + if isprime(N): + # n % 8 == 3 and x % 2 == 1 => N % 4 == 1 + y, z = prime_as_sum_of_two_squares(N) + return tuple(sorted([v*x, v*(y + z), v*abs(y - z)])) + # We will never reach this point because there must be a solution. + assert False + + # assert n % 4 in [1, 2] + if not((n % 2) ^ (s % 2)): + s -= 1 + for x in range(s, -1, -2): + N = n - x**2 + if isprime(N): + # assert N % 4 == 1 + y, z = prime_as_sum_of_two_squares(N) + return tuple(sorted([v*x, v*y, v*z])) + # We will never reach this point because there must be a solution. + assert False + + +def sum_of_four_squares(n): + r""" + Returns a 4-tuple `(a, b, c, d)` such that `a^2 + b^2 + c^2 + d^2 = n`. + Here `a, b, c, d \geq 0`. + + Parameters + ========== + + n : Integer + non-negative integer + + Returns + ======= + + (int, int, int, int) : 4-tuple non-negative integers ``(a, b, c, d)`` satisfying ``a**2 + b**2 + c**2 + d**2 = n``. + a,b,c,d are sorted in ascending order. + + Raises + ====== + + ValueError + If ``n`` is a negative integer + + Examples + ======== + + >>> from sympy.solvers.diophantine.diophantine import sum_of_four_squares + >>> sum_of_four_squares(3456) + (8, 8, 32, 48) + >>> sum_of_four_squares(1294585930293) + (0, 1234, 2161, 1137796) + + References + ========== + + .. [1] Representing a number as a sum of four squares, [online], + Available: https://schorn.ch/lagrange.html + + See Also + ======== + + power_representation : + ``sum_of_four_squares(n)`` is one of the solutions output by ``power_representation(n, 2, 4, zeros=True)`` + + """ + n = as_int(n) + if n < 0: + raise ValueError("n should be a non-negative integer") + if n == 0: + return (0, 0, 0, 0) + # remove factors of 4 since a solution in terms of 3 squares is + # going to be returned; this is also done in sum_of_three_squares, + # but it needs to be done here to select d + n, v = remove(n, 4) + v = 1 << v + if n % 8 == 7: + d = 2 + n = n - 4 + elif n % 8 in (2, 6): + d = 1 + n = n - 1 + else: + d = 0 + x, y, z = sum_of_three_squares(n) # sorted + return tuple(sorted([v*d, v*x, v*y, v*z])) + + +def power_representation(n, p, k, zeros=False): + r""" + Returns a generator for finding k-tuples of integers, + `(n_{1}, n_{2}, . . . n_{k})`, such that + `n = n_{1}^p + n_{2}^p + . . . n_{k}^p`. + + Usage + ===== + + ``power_representation(n, p, k, zeros)``: Represent non-negative number + ``n`` as a sum of ``k`` ``p``\ th powers. If ``zeros`` is true, then the + solutions is allowed to contain zeros. + + Examples + ======== + + >>> from sympy.solvers.diophantine.diophantine import power_representation + + Represent 1729 as a sum of two cubes: + + >>> f = power_representation(1729, 3, 2) + >>> next(f) + (9, 10) + >>> next(f) + (1, 12) + + If the flag `zeros` is True, the solution may contain tuples with + zeros; any such solutions will be generated after the solutions + without zeros: + + >>> list(power_representation(125, 2, 3, zeros=True)) + [(5, 6, 8), (3, 4, 10), (0, 5, 10), (0, 2, 11)] + + For even `p` the `permute_sign` function can be used to get all + signed values: + + >>> from sympy.utilities.iterables import permute_signs + >>> list(permute_signs((1, 12))) + [(1, 12), (-1, 12), (1, -12), (-1, -12)] + + All possible signed permutations can also be obtained: + + >>> from sympy.utilities.iterables import signed_permutations + >>> list(signed_permutations((1, 12))) + [(1, 12), (-1, 12), (1, -12), (-1, -12), (12, 1), (-12, 1), (12, -1), (-12, -1)] + """ + n, p, k = [as_int(i) for i in (n, p, k)] + + if n < 0: + if p % 2: + for t in power_representation(-n, p, k, zeros): + yield tuple(-i for i in t) + return + + if p < 1 or k < 1: + raise ValueError(filldedent(''' + Expecting positive integers for `(p, k)`, but got `(%s, %s)`''' + % (p, k))) + + if n == 0: + if zeros: + yield (0,)*k + return + + if k == 1: + if p == 1: + yield (n,) + elif n == 1: + yield (1,) + else: + be = perfect_power(n) + if be: + b, e = be + d, r = divmod(e, p) + if not r: + yield (b**d,) + return + + if p == 1: + yield from partition(n, k, zeros=zeros) + return + + if p == 2: + if k == 3: + n, v = remove(n, 4) + if v: + v = 1 << v + for t in power_representation(n, p, k, zeros): + yield tuple(i*v for i in t) + return + feasible = _can_do_sum_of_squares(n, k) + if not feasible: + return + if not zeros: + if n > 33 and k >= 5 and k <= n and n - k in ( + 13, 10, 7, 5, 4, 2, 1): + '''Todd G. Will, "When Is n^2 a Sum of k Squares?", [online]. + Available: https://www.maa.org/sites/default/files/Will-MMz-201037918.pdf''' + return + # quick tests since feasibility includes the possibility of 0 + if k == 4 and (n in (1, 3, 5, 9, 11, 17, 29, 41) or remove(n, 4)[0] in (2, 6, 14)): + # A000534 + return + if k == 3 and n in (1, 2, 5, 10, 13, 25, 37, 58, 85, 130): # or n = some number >= 5*10**10 + # A051952 + return + if feasible is not True: # it's prime and k == 2 + yield prime_as_sum_of_two_squares(n) + return + + if k == 2 and p > 2: + be = perfect_power(n) + if be and be[1] % p == 0: + return # Fermat: a**n + b**n = c**n has no solution for n > 2 + + if n >= k: + a = integer_nthroot(n - (k - 1), p)[0] + for t in pow_rep_recursive(a, k, n, [], p): + yield tuple(reversed(t)) + + if zeros: + a = integer_nthroot(n, p)[0] + for i in range(1, k): + for t in pow_rep_recursive(a, i, n, [], p): + yield tuple(reversed(t + (0,)*(k - i))) + + +sum_of_powers = power_representation + + +def pow_rep_recursive(n_i, k, n_remaining, terms, p): + # Invalid arguments + if n_i <= 0 or k <= 0: + return + + # No solutions may exist + if n_remaining < k: + return + if k * pow(n_i, p) < n_remaining: + return + + if k == 0 and n_remaining == 0: + yield tuple(terms) + + elif k == 1: + # next_term^p must equal to n_remaining + next_term, exact = integer_nthroot(n_remaining, p) + if exact and next_term <= n_i: + yield tuple(terms + [next_term]) + return + + else: + # TODO: Fall back to diop_DN when k = 2 + if n_i >= 1 and k > 0: + for next_term in range(1, n_i + 1): + residual = n_remaining - pow(next_term, p) + if residual < 0: + break + yield from pow_rep_recursive(next_term, k - 1, residual, terms + [next_term], p) + + +def sum_of_squares(n, k, zeros=False): + """Return a generator that yields the k-tuples of nonnegative + values, the squares of which sum to n. If zeros is False (default) + then the solution will not contain zeros. The nonnegative + elements of a tuple are sorted. + + * If k == 1 and n is square, (n,) is returned. + + * If k == 2 then n can only be written as a sum of squares if + every prime in the factorization of n that has the form + 4*k + 3 has an even multiplicity. If n is prime then + it can only be written as a sum of two squares if it is + in the form 4*k + 1. + + * if k == 3 then n can be written as a sum of squares if it does + not have the form 4**m*(8*k + 7). + + * all integers can be written as the sum of 4 squares. + + * if k > 4 then n can be partitioned and each partition can + be written as a sum of 4 squares; if n is not evenly divisible + by 4 then n can be written as a sum of squares only if the + an additional partition can be written as sum of squares. + For example, if k = 6 then n is partitioned into two parts, + the first being written as a sum of 4 squares and the second + being written as a sum of 2 squares -- which can only be + done if the condition above for k = 2 can be met, so this will + automatically reject certain partitions of n. + + Examples + ======== + + >>> from sympy.solvers.diophantine.diophantine import sum_of_squares + >>> list(sum_of_squares(25, 2)) + [(3, 4)] + >>> list(sum_of_squares(25, 2, True)) + [(3, 4), (0, 5)] + >>> list(sum_of_squares(25, 4)) + [(1, 2, 2, 4)] + + See Also + ======== + + sympy.utilities.iterables.signed_permutations + """ + yield from power_representation(n, 2, k, zeros) + + +def _can_do_sum_of_squares(n, k): + """Return True if n can be written as the sum of k squares, + False if it cannot, or 1 if ``k == 2`` and ``n`` is prime (in which + case it *can* be written as a sum of two squares). A False + is returned only if it cannot be written as ``k``-squares, even + if 0s are allowed. + """ + if k < 1: + return False + if n < 0: + return False + if n == 0: + return True + if k == 1: + return is_square(n) + if k == 2: + if n in (1, 2): + return True + if isprime(n): + if n % 4 == 1: + return 1 # signal that it was prime + return False + # n is a composite number + # we can proceed iff no prime factor in the form 4*k + 3 + # has an odd multiplicity + return all(p % 4 !=3 or m % 2 == 0 for p, m in factorint(n).items()) + if k == 3: + return remove(n, 4)[0] % 8 != 7 + # every number can be written as a sum of 4 squares; for k > 4 partitions + # can be 0 + return True diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/diophantine/tests/__init__.py b/.venv/lib/python3.12/site-packages/sympy/solvers/diophantine/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/diophantine/tests/__pycache__/__init__.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/solvers/diophantine/tests/__pycache__/__init__.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..491bc67ed7ddaba8143cb420730e861285259501 Binary files /dev/null and b/.venv/lib/python3.12/site-packages/sympy/solvers/diophantine/tests/__pycache__/__init__.cpython-312.pyc differ diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/diophantine/tests/__pycache__/test_diophantine.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/solvers/diophantine/tests/__pycache__/test_diophantine.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..f3559bf48f79eb984f54a9085e899d8c6a66b5d2 Binary files /dev/null and b/.venv/lib/python3.12/site-packages/sympy/solvers/diophantine/tests/__pycache__/test_diophantine.cpython-312.pyc differ diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/diophantine/tests/test_diophantine.py b/.venv/lib/python3.12/site-packages/sympy/solvers/diophantine/tests/test_diophantine.py new file mode 100644 index 0000000000000000000000000000000000000000..b8b031a1e63fa445e3dbb0b425f84cfe88888667 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/diophantine/tests/test_diophantine.py @@ -0,0 +1,1071 @@ +from sympy.core.add import Add +from sympy.core.mul import Mul +from sympy.core.numbers import (Rational, oo, pi) +from sympy.core.relational import Eq +from sympy.core.singleton import S +from sympy.core.symbol import symbols +from sympy.matrices.dense import Matrix +from sympy.ntheory.factor_ import factorint +from sympy.simplify.powsimp import powsimp +from sympy.core.function import _mexpand +from sympy.core.sorting import default_sort_key, ordered +from sympy.functions.elementary.trigonometric import sin +from sympy.solvers.diophantine import diophantine +from sympy.solvers.diophantine.diophantine import (diop_DN, + diop_solve, diop_ternary_quadratic_normal, + diop_general_pythagorean, diop_ternary_quadratic, diop_linear, + diop_quadratic, diop_general_sum_of_squares, diop_general_sum_of_even_powers, + descent, diop_bf_DN, divisible, equivalent, find_DN, ldescent, length, + reconstruct, partition, power_representation, + prime_as_sum_of_two_squares, square_factor, sum_of_four_squares, + sum_of_three_squares, transformation_to_DN, transformation_to_normal, + classify_diop, base_solution_linear, cornacchia, sqf_normal, gaussian_reduce, holzer, + check_param, parametrize_ternary_quadratic, sum_of_powers, sum_of_squares, + _diop_ternary_quadratic_normal, _nint_or_floor, + _odd, _even, _remove_gcd, _can_do_sum_of_squares, DiophantineSolutionSet, GeneralPythagorean, + BinaryQuadratic) + +from sympy.testing.pytest import slow, raises, XFAIL +from sympy.utilities.iterables import ( + signed_permutations) + +a, b, c, d, p, q, x, y, z, w, t, u, v, X, Y, Z = symbols( + "a, b, c, d, p, q, x, y, z, w, t, u, v, X, Y, Z", integer=True) +t_0, t_1, t_2, t_3, t_4, t_5, t_6 = symbols("t_:7", integer=True) +m1, m2, m3 = symbols('m1:4', integer=True) +n1 = symbols('n1', integer=True) + + +def diop_simplify(eq): + return _mexpand(powsimp(_mexpand(eq))) + + +def test_input_format(): + raises(TypeError, lambda: diophantine(sin(x))) + raises(TypeError, lambda: diophantine(x/pi - 3)) + + +def test_nosols(): + # diophantine should sympify eq so that these are equivalent + assert diophantine(3) == set() + assert diophantine(S(3)) == set() + + +def test_univariate(): + assert diop_solve((x - 1)*(x - 2)**2) == {(1,), (2,)} + assert diop_solve((x - 1)*(x - 2)) == {(1,), (2,)} + + +def test_classify_diop(): + raises(TypeError, lambda: classify_diop(x**2/3 - 1)) + raises(ValueError, lambda: classify_diop(1)) + raises(NotImplementedError, lambda: classify_diop(w*x*y*z - 1)) + raises(NotImplementedError, lambda: classify_diop(x**3 + y**3 + z**4 - 90)) + assert classify_diop(14*x**2 + 15*x - 42) == ( + [x], {1: -42, x: 15, x**2: 14}, 'univariate') + assert classify_diop(x*y + z) == ( + [x, y, z], {x*y: 1, z: 1}, 'inhomogeneous_ternary_quadratic') + assert classify_diop(x*y + z + w + x**2) == ( + [w, x, y, z], {x*y: 1, w: 1, x**2: 1, z: 1}, 'inhomogeneous_general_quadratic') + assert classify_diop(x*y + x*z + x**2 + 1) == ( + [x, y, z], {x*y: 1, x*z: 1, x**2: 1, 1: 1}, 'inhomogeneous_general_quadratic') + assert classify_diop(x*y + z + w + 42) == ( + [w, x, y, z], {x*y: 1, w: 1, 1: 42, z: 1}, 'inhomogeneous_general_quadratic') + assert classify_diop(x*y + z*w) == ( + [w, x, y, z], {x*y: 1, w*z: 1}, 'homogeneous_general_quadratic') + assert classify_diop(x*y**2 + 1) == ( + [x, y], {x*y**2: 1, 1: 1}, 'cubic_thue') + assert classify_diop(x**4 + y**4 + z**4 - (1 + 16 + 81)) == ( + [x, y, z], {1: -98, x**4: 1, z**4: 1, y**4: 1}, 'general_sum_of_even_powers') + assert classify_diop(x**2 + y**2 + z**2) == ( + [x, y, z], {x**2: 1, y**2: 1, z**2: 1}, 'homogeneous_ternary_quadratic_normal') + + +def test_linear(): + assert diop_solve(x) == (0,) + assert diop_solve(1*x) == (0,) + assert diop_solve(3*x) == (0,) + assert diop_solve(x + 1) == (-1,) + assert diop_solve(2*x + 1) == (None,) + assert diop_solve(2*x + 4) == (-2,) + assert diop_solve(y + x) == (t_0, -t_0) + assert diop_solve(y + x + 0) == (t_0, -t_0) + assert diop_solve(y + x - 0) == (t_0, -t_0) + assert diop_solve(0*x - y - 5) == (-5,) + assert diop_solve(3*y + 2*x - 5) == (3*t_0 - 5, -2*t_0 + 5) + assert diop_solve(2*x - 3*y - 5) == (3*t_0 - 5, 2*t_0 - 5) + assert diop_solve(-2*x - 3*y - 5) == (3*t_0 + 5, -2*t_0 - 5) + assert diop_solve(7*x + 5*y) == (5*t_0, -7*t_0) + assert diop_solve(2*x + 4*y) == (-2*t_0, t_0) + assert diop_solve(4*x + 6*y - 4) == (3*t_0 - 2, -2*t_0 + 2) + assert diop_solve(4*x + 6*y - 3) == (None, None) + assert diop_solve(0*x + 3*y - 4*z + 5) == (4*t_0 + 5, 3*t_0 + 5) + assert diop_solve(4*x + 3*y - 4*z + 5) == (t_0, 8*t_0 + 4*t_1 + 5, 7*t_0 + 3*t_1 + 5) + assert diop_solve(4*x + 3*y - 4*z + 5, None) == (0, 5, 5) + assert diop_solve(4*x + 2*y + 8*z - 5) == (None, None, None) + assert diop_solve(5*x + 7*y - 2*z - 6) == (t_0, -3*t_0 + 2*t_1 + 6, -8*t_0 + 7*t_1 + 18) + assert diop_solve(3*x - 6*y + 12*z - 9) == (2*t_0 + 3, t_0 + 2*t_1, t_1) + assert diop_solve(6*w + 9*x + 20*y - z) == (t_0, t_1, t_1 + t_2, 6*t_0 + 29*t_1 + 20*t_2) + + # to ignore constant factors, use diophantine + raises(TypeError, lambda: diop_solve(x/2)) + + +def test_quadratic_simple_hyperbolic_case(): + # Simple Hyperbolic case: A = C = 0 and B != 0 + assert diop_solve(3*x*y + 34*x - 12*y + 1) == \ + {(-133, -11), (5, -57)} + assert diop_solve(6*x*y + 2*x + 3*y + 1) == set() + assert diop_solve(-13*x*y + 2*x - 4*y - 54) == {(27, 0)} + assert diop_solve(-27*x*y - 30*x - 12*y - 54) == {(-14, -1)} + assert diop_solve(2*x*y + 5*x + 56*y + 7) == {(-161, -3), (-47, -6), (-35, -12), + (-29, -69), (-27, 64), (-21, 7), + (-9, 1), (105, -2)} + assert diop_solve(6*x*y + 9*x + 2*y + 3) == set() + assert diop_solve(x*y + x + y + 1) == {(-1, t), (t, -1)} + assert diophantine(48*x*y) + + +def test_quadratic_elliptical_case(): + # Elliptical case: B**2 - 4AC < 0 + + assert diop_solve(42*x**2 + 8*x*y + 15*y**2 + 23*x + 17*y - 4915) == {(-11, -1)} + assert diop_solve(4*x**2 + 3*y**2 + 5*x - 11*y + 12) == set() + assert diop_solve(x**2 + y**2 + 2*x + 2*y + 2) == {(-1, -1)} + assert diop_solve(15*x**2 - 9*x*y + 14*y**2 - 23*x - 14*y - 4950) == {(-15, 6)} + assert diop_solve(10*x**2 + 12*x*y + 12*y**2 - 34) == \ + {(-1, -1), (-1, 2), (1, -2), (1, 1)} + + +def test_quadratic_parabolic_case(): + # Parabolic case: B**2 - 4AC = 0 + assert check_solutions(8*x**2 - 24*x*y + 18*y**2 + 5*x + 7*y + 16) + assert check_solutions(8*x**2 - 24*x*y + 18*y**2 + 6*x + 12*y - 6) + assert check_solutions(8*x**2 + 24*x*y + 18*y**2 + 4*x + 6*y - 7) + assert check_solutions(-4*x**2 + 4*x*y - y**2 + 2*x - 3) + assert check_solutions(x**2 + 2*x*y + y**2 + 2*x + 2*y + 1) + assert check_solutions(x**2 - 2*x*y + y**2 + 2*x + 2*y + 1) + assert check_solutions(y**2 - 41*x + 40) + + +def test_quadratic_perfect_square(): + # B**2 - 4*A*C > 0 + # B**2 - 4*A*C is a perfect square + assert check_solutions(48*x*y) + assert check_solutions(4*x**2 - 5*x*y + y**2 + 2) + assert check_solutions(-2*x**2 - 3*x*y + 2*y**2 -2*x - 17*y + 25) + assert check_solutions(12*x**2 + 13*x*y + 3*y**2 - 2*x + 3*y - 12) + assert check_solutions(8*x**2 + 10*x*y + 2*y**2 - 32*x - 13*y - 23) + assert check_solutions(4*x**2 - 4*x*y - 3*y- 8*x - 3) + assert check_solutions(- 4*x*y - 4*y**2 - 3*y- 5*x - 10) + assert check_solutions(x**2 - y**2 - 2*x - 2*y) + assert check_solutions(x**2 - 9*y**2 - 2*x - 6*y) + assert check_solutions(4*x**2 - 9*y**2 - 4*x - 12*y - 3) + + +def test_quadratic_non_perfect_square(): + # B**2 - 4*A*C is not a perfect square + # Used check_solutions() since the solutions are complex expressions involving + # square roots and exponents + assert check_solutions(x**2 - 2*x - 5*y**2) + assert check_solutions(3*x**2 - 2*y**2 - 2*x - 2*y) + assert check_solutions(x**2 - x*y - y**2 - 3*y) + assert check_solutions(x**2 - 9*y**2 - 2*x - 6*y) + assert BinaryQuadratic(x**2 + y**2 + 2*x + 2*y + 2).solve() == {(-1, -1)} + + +def test_issue_9106(): + eq = -48 - 2*x*(3*x - 1) + y*(3*y - 1) + v = (x, y) + for sol in diophantine(eq): + assert not diop_simplify(eq.xreplace(dict(zip(v, sol)))) + + +def test_issue_18138(): + eq = x**2 - x - y**2 + v = (x, y) + for sol in diophantine(eq): + assert not diop_simplify(eq.xreplace(dict(zip(v, sol)))) + + +@slow +def test_quadratic_non_perfect_slow(): + assert check_solutions(8*x**2 + 10*x*y - 2*y**2 - 32*x - 13*y - 23) + # This leads to very large numbers. + # assert check_solutions(5*x**2 - 13*x*y + y**2 - 4*x - 4*y - 15) + assert check_solutions(-3*x**2 - 2*x*y + 7*y**2 - 5*x - 7) + assert check_solutions(-4 - x + 4*x**2 - y - 3*x*y - 4*y**2) + assert check_solutions(1 + 2*x + 2*x**2 + 2*y + x*y - 2*y**2) + + +def test_DN(): + # Most of the test cases were adapted from, + # Solving the generalized Pell equation x**2 - D*y**2 = N, John P. Robertson, July 31, 2004. + # https://web.archive.org/web/20160323033128/http://www.jpr2718.org/pell.pdf + # others are verified using Wolfram Alpha. + + # Covers cases where D <= 0 or D > 0 and D is a square or N = 0 + # Solutions are straightforward in these cases. + assert diop_DN(3, 0) == [(0, 0)] + assert diop_DN(-17, -5) == [] + assert diop_DN(-19, 23) == [(2, 1)] + assert diop_DN(-13, 17) == [(2, 1)] + assert diop_DN(-15, 13) == [] + assert diop_DN(0, 5) == [] + assert diop_DN(0, 9) == [(3, t)] + assert diop_DN(9, 0) == [(3*t, t)] + assert diop_DN(16, 24) == [] + assert diop_DN(9, 180) == [(18, 4)] + assert diop_DN(9, -180) == [(12, 6)] + assert diop_DN(7, 0) == [(0, 0)] + + # When equation is x**2 + y**2 = N + # Solutions are interchangeable + assert diop_DN(-1, 5) == [(2, 1), (1, 2)] + assert diop_DN(-1, 169) == [(12, 5), (5, 12), (13, 0), (0, 13)] + + # D > 0 and D is not a square + + # N = 1 + assert diop_DN(13, 1) == [(649, 180)] + assert diop_DN(980, 1) == [(51841, 1656)] + assert diop_DN(981, 1) == [(158070671986249, 5046808151700)] + assert diop_DN(986, 1) == [(49299, 1570)] + assert diop_DN(991, 1) == [(379516400906811930638014896080, 12055735790331359447442538767)] + assert diop_DN(17, 1) == [(33, 8)] + assert diop_DN(19, 1) == [(170, 39)] + + # N = -1 + assert diop_DN(13, -1) == [(18, 5)] + assert diop_DN(991, -1) == [] + assert diop_DN(41, -1) == [(32, 5)] + assert diop_DN(290, -1) == [(17, 1)] + assert diop_DN(21257, -1) == [(13913102721304, 95427381109)] + assert diop_DN(32, -1) == [] + + # |N| > 1 + # Some tests were created using calculator at + # http://www.numbertheory.org/php/patz.html + + assert diop_DN(13, -4) == [(3, 1), (393, 109), (36, 10)] + # Source I referred returned (3, 1), (393, 109) and (-3, 1) as fundamental solutions + # So (-3, 1) and (393, 109) should be in the same equivalent class + assert equivalent(-3, 1, 393, 109, 13, -4) == True + + assert diop_DN(13, 27) == [(220, 61), (40, 11), (768, 213), (12, 3)] + assert set(diop_DN(157, 12)) == {(13, 1), (10663, 851), (579160, 46222), + (483790960, 38610722), (26277068347, 2097138361), + (21950079635497, 1751807067011)} + assert diop_DN(13, 25) == [(3245, 900)] + assert diop_DN(192, 18) == [] + assert diop_DN(23, 13) == [(-6, 1), (6, 1)] + assert diop_DN(167, 2) == [(13, 1)] + assert diop_DN(167, -2) == [] + + assert diop_DN(123, -2) == [(11, 1)] + # One calculator returned [(11, 1), (-11, 1)] but both of these are in + # the same equivalence class + assert equivalent(11, 1, -11, 1, 123, -2) + + assert diop_DN(123, -23) == [(-10, 1), (10, 1)] + + assert diop_DN(0, 0, t) == [(0, t)] + assert diop_DN(0, -1, t) == [] + + +def test_bf_pell(): + assert diop_bf_DN(13, -4) == [(3, 1), (-3, 1), (36, 10)] + assert diop_bf_DN(13, 27) == [(12, 3), (-12, 3), (40, 11), (-40, 11)] + assert diop_bf_DN(167, -2) == [] + assert diop_bf_DN(1729, 1) == [(44611924489705, 1072885712316)] + assert diop_bf_DN(89, -8) == [(9, 1), (-9, 1)] + assert diop_bf_DN(21257, -1) == [(13913102721304, 95427381109)] + assert diop_bf_DN(340, -4) == [(756, 41)] + assert diop_bf_DN(-1, 0, t) == [(0, 0)] + assert diop_bf_DN(0, 0, t) == [(0, t)] + assert diop_bf_DN(4, 0, t) == [(2*t, t), (-2*t, t)] + assert diop_bf_DN(3, 0, t) == [(0, 0)] + assert diop_bf_DN(1, -2, t) == [] + + +def test_length(): + assert length(2, 1, 0) == 1 + assert length(-2, 4, 5) == 3 + assert length(-5, 4, 17) == 4 + assert length(0, 4, 13) == 6 + assert length(7, 13, 11) == 23 + assert length(1, 6, 4) == 2 + + +def is_pell_transformation_ok(eq): + """ + Test whether X*Y, X, or Y terms are present in the equation + after transforming the equation using the transformation returned + by transformation_to_pell(). If they are not present we are good. + Moreover, coefficient of X**2 should be a divisor of coefficient of + Y**2 and the constant term. + """ + A, B = transformation_to_DN(eq) + u = (A*Matrix([X, Y]) + B)[0] + v = (A*Matrix([X, Y]) + B)[1] + simplified = diop_simplify(eq.subs(zip((x, y), (u, v)))) + + coeff = dict([reversed(t.as_independent(*[X, Y])) for t in simplified.args]) + + for term in [X*Y, X, Y]: + if term in coeff.keys(): + return False + + for term in [X**2, Y**2, 1]: + if term not in coeff.keys(): + coeff[term] = 0 + + if coeff[X**2] != 0: + return divisible(coeff[Y**2], coeff[X**2]) and \ + divisible(coeff[1], coeff[X**2]) + + return True + + +def test_transformation_to_pell(): + assert is_pell_transformation_ok(-13*x**2 - 7*x*y + y**2 + 2*x - 2*y - 14) + assert is_pell_transformation_ok(-17*x**2 + 19*x*y - 7*y**2 - 5*x - 13*y - 23) + assert is_pell_transformation_ok(x**2 - y**2 + 17) + assert is_pell_transformation_ok(-x**2 + 7*y**2 - 23) + assert is_pell_transformation_ok(25*x**2 - 45*x*y + 5*y**2 - 5*x - 10*y + 5) + assert is_pell_transformation_ok(190*x**2 + 30*x*y + y**2 - 3*y - 170*x - 130) + assert is_pell_transformation_ok(x**2 - 2*x*y -190*y**2 - 7*y - 23*x - 89) + assert is_pell_transformation_ok(15*x**2 - 9*x*y + 14*y**2 - 23*x - 14*y - 4950) + + +def test_find_DN(): + assert find_DN(x**2 - 2*x - y**2) == (1, 1) + assert find_DN(x**2 - 3*y**2 - 5) == (3, 5) + assert find_DN(x**2 - 2*x*y - 4*y**2 - 7) == (5, 7) + assert find_DN(4*x**2 - 8*x*y - y**2 - 9) == (20, 36) + assert find_DN(7*x**2 - 2*x*y - y**2 - 12) == (8, 84) + assert find_DN(-3*x**2 + 4*x*y -y**2) == (1, 0) + assert find_DN(-13*x**2 - 7*x*y + y**2 + 2*x - 2*y -14) == (101, -7825480) + + +def test_ldescent(): + # Equations which have solutions + u = ([(13, 23), (3, -11), (41, -113), (4, -7), (-7, 4), (91, -3), (1, 1), (1, -1), + (4, 32), (17, 13), (123689, 1), (19, -570)]) + for a, b in u: + w, x, y = ldescent(a, b) + assert a*x**2 + b*y**2 == w**2 + assert ldescent(-1, -1) is None + assert ldescent(2, 6) is None + + +def test_diop_ternary_quadratic_normal(): + assert check_solutions(234*x**2 - 65601*y**2 - z**2) + assert check_solutions(23*x**2 + 616*y**2 - z**2) + assert check_solutions(5*x**2 + 4*y**2 - z**2) + assert check_solutions(3*x**2 + 6*y**2 - 3*z**2) + assert check_solutions(x**2 + 3*y**2 - z**2) + assert check_solutions(4*x**2 + 5*y**2 - z**2) + assert check_solutions(x**2 + y**2 - z**2) + assert check_solutions(16*x**2 + y**2 - 25*z**2) + assert check_solutions(6*x**2 - y**2 + 10*z**2) + assert check_solutions(213*x**2 + 12*y**2 - 9*z**2) + assert check_solutions(34*x**2 - 3*y**2 - 301*z**2) + assert check_solutions(124*x**2 - 30*y**2 - 7729*z**2) + + +def is_normal_transformation_ok(eq): + A = transformation_to_normal(eq) + X, Y, Z = A*Matrix([x, y, z]) + simplified = diop_simplify(eq.subs(zip((x, y, z), (X, Y, Z)))) + + coeff = dict([reversed(t.as_independent(*[X, Y, Z])) for t in simplified.args]) + for term in [X*Y, Y*Z, X*Z]: + if term in coeff.keys(): + return False + + return True + + +def test_transformation_to_normal(): + assert is_normal_transformation_ok(x**2 + 3*y**2 + z**2 - 13*x*y - 16*y*z + 12*x*z) + assert is_normal_transformation_ok(x**2 + 3*y**2 - 100*z**2) + assert is_normal_transformation_ok(x**2 + 23*y*z) + assert is_normal_transformation_ok(3*y**2 - 100*z**2 - 12*x*y) + assert is_normal_transformation_ok(x**2 + 23*x*y - 34*y*z + 12*x*z) + assert is_normal_transformation_ok(z**2 + 34*x*y - 23*y*z + x*z) + assert is_normal_transformation_ok(x**2 + y**2 + z**2 - x*y - y*z - x*z) + assert is_normal_transformation_ok(x**2 + 2*y*z + 3*z**2) + assert is_normal_transformation_ok(x*y + 2*x*z + 3*y*z) + assert is_normal_transformation_ok(2*x*z + 3*y*z) + + +def test_diop_ternary_quadratic(): + assert check_solutions(2*x**2 + z**2 + y**2 - 4*x*y) + assert check_solutions(x**2 - y**2 - z**2 - x*y - y*z) + assert check_solutions(3*x**2 - x*y - y*z - x*z) + assert check_solutions(x**2 - y*z - x*z) + assert check_solutions(5*x**2 - 3*x*y - x*z) + assert check_solutions(4*x**2 - 5*y**2 - x*z) + assert check_solutions(3*x**2 + 2*y**2 - z**2 - 2*x*y + 5*y*z - 7*y*z) + assert check_solutions(8*x**2 - 12*y*z) + assert check_solutions(45*x**2 - 7*y**2 - 8*x*y - z**2) + assert check_solutions(x**2 - 49*y**2 - z**2 + 13*z*y -8*x*y) + assert check_solutions(90*x**2 + 3*y**2 + 5*x*y + 2*z*y + 5*x*z) + assert check_solutions(x**2 + 3*y**2 + z**2 - x*y - 17*y*z) + assert check_solutions(x**2 + 3*y**2 + z**2 - x*y - 16*y*z + 12*x*z) + assert check_solutions(x**2 + 3*y**2 + z**2 - 13*x*y - 16*y*z + 12*x*z) + assert check_solutions(x*y - 7*y*z + 13*x*z) + + assert diop_ternary_quadratic_normal(x**2 + y**2 + z**2) == (None, None, None) + assert diop_ternary_quadratic_normal(x**2 + y**2) is None + raises(ValueError, lambda: + _diop_ternary_quadratic_normal((x, y, z), + {x*y: 1, x**2: 2, y**2: 3, z**2: 0})) + eq = -2*x*y - 6*x*z + 7*y**2 - 3*y*z + 4*z**2 + assert diop_ternary_quadratic(eq) == (7, 2, 0) + assert diop_ternary_quadratic_normal(4*x**2 + 5*y**2 - z**2) == \ + (1, 0, 2) + assert diop_ternary_quadratic(x*y + 2*y*z) == \ + (-2, 0, n1) + eq = -5*x*y - 8*x*z - 3*y*z + 8*z**2 + assert parametrize_ternary_quadratic(eq) == \ + (8*p**2 - 3*p*q, -8*p*q + 8*q**2, 5*p*q) + # this cannot be tested with diophantine because it will + # factor into a product + assert diop_solve(x*y + 2*y*z) == (-2*p*q, -n1*p**2 + p**2, p*q) + + +def test_square_factor(): + assert square_factor(1) == square_factor(-1) == 1 + assert square_factor(0) == 1 + assert square_factor(5) == square_factor(-5) == 1 + assert square_factor(4) == square_factor(-4) == 2 + assert square_factor(12) == square_factor(-12) == 2 + assert square_factor(6) == 1 + assert square_factor(18) == 3 + assert square_factor(52) == 2 + assert square_factor(49) == 7 + assert square_factor(392) == 14 + assert square_factor(factorint(-12)) == 2 + + +def test_parametrize_ternary_quadratic(): + assert check_solutions(x**2 + y**2 - z**2) + assert check_solutions(x**2 + 2*x*y + z**2) + assert check_solutions(234*x**2 - 65601*y**2 - z**2) + assert check_solutions(3*x**2 + 2*y**2 - z**2 - 2*x*y + 5*y*z - 7*y*z) + assert check_solutions(x**2 - y**2 - z**2) + assert check_solutions(x**2 - 49*y**2 - z**2 + 13*z*y - 8*x*y) + assert check_solutions(8*x*y + z**2) + assert check_solutions(124*x**2 - 30*y**2 - 7729*z**2) + assert check_solutions(236*x**2 - 225*y**2 - 11*x*y - 13*y*z - 17*x*z) + assert check_solutions(90*x**2 + 3*y**2 + 5*x*y + 2*z*y + 5*x*z) + assert check_solutions(124*x**2 - 30*y**2 - 7729*z**2) + + +def test_no_square_ternary_quadratic(): + assert check_solutions(2*x*y + y*z - 3*x*z) + assert check_solutions(189*x*y - 345*y*z - 12*x*z) + assert check_solutions(23*x*y + 34*y*z) + assert check_solutions(x*y + y*z + z*x) + assert check_solutions(23*x*y + 23*y*z + 23*x*z) + + +def test_descent(): + + u = ([(13, 23), (3, -11), (41, -113), (91, -3), (1, 1), (1, -1), (17, 13), (123689, 1), (19, -570)]) + for a, b in u: + w, x, y = descent(a, b) + assert a*x**2 + b*y**2 == w**2 + # the docstring warns against bad input, so these are expected results + # - can't both be negative + raises(TypeError, lambda: descent(-1, -3)) + # A can't be zero unless B != 1 + raises(ZeroDivisionError, lambda: descent(0, 3)) + # supposed to be square-free + raises(TypeError, lambda: descent(4, 3)) + + +def test_diophantine(): + assert check_solutions((x - y)*(y - z)*(z - x)) + assert check_solutions((x - y)*(x**2 + y**2 - z**2)) + assert check_solutions((x - 3*y + 7*z)*(x**2 + y**2 - z**2)) + assert check_solutions(x**2 - 3*y**2 - 1) + assert check_solutions(y**2 + 7*x*y) + assert check_solutions(x**2 - 3*x*y + y**2) + assert check_solutions(z*(x**2 - y**2 - 15)) + assert check_solutions(x*(2*y - 2*z + 5)) + assert check_solutions((x**2 - 3*y**2 - 1)*(x**2 - y**2 - 15)) + assert check_solutions((x**2 - 3*y**2 - 1)*(y - 7*z)) + assert check_solutions((x**2 + y**2 - z**2)*(x - 7*y - 3*z + 4*w)) + # Following test case caused problems in parametric representation + # But this can be solved by factoring out y. + # No need to use methods for ternary quadratic equations. + assert check_solutions(y**2 - 7*x*y + 4*y*z) + assert check_solutions(x**2 - 2*x + 1) + + assert diophantine(x - y) == diophantine(Eq(x, y)) + # 18196 + eq = x**4 + y**4 - 97 + assert diophantine(eq, permute=True) == diophantine(-eq, permute=True) + assert diophantine(3*x*pi - 2*y*pi) == {(2*t_0, 3*t_0)} + eq = x**2 + y**2 + z**2 - 14 + base_sol = {(1, 2, 3)} + assert diophantine(eq) == base_sol + complete_soln = set(signed_permutations(base_sol.pop())) + assert diophantine(eq, permute=True) == complete_soln + + assert diophantine(x**2 + x*Rational(15, 14) - 3) == set() + # test issue 11049 + eq = 92*x**2 - 99*y**2 - z**2 + coeff = eq.as_coefficients_dict() + assert _diop_ternary_quadratic_normal((x, y, z), coeff) == \ + {(9, 7, 51)} + assert diophantine(eq) == {( + 891*p**2 + 9*q**2, -693*p**2 - 102*p*q + 7*q**2, + 5049*p**2 - 1386*p*q - 51*q**2)} + eq = 2*x**2 + 2*y**2 - z**2 + coeff = eq.as_coefficients_dict() + assert _diop_ternary_quadratic_normal((x, y, z), coeff) == \ + {(1, 1, 2)} + assert diophantine(eq) == {( + 2*p**2 - q**2, -2*p**2 + 4*p*q - q**2, + 4*p**2 - 4*p*q + 2*q**2)} + eq = 411*x**2+57*y**2-221*z**2 + coeff = eq.as_coefficients_dict() + assert _diop_ternary_quadratic_normal((x, y, z), coeff) == \ + {(2021, 2645, 3066)} + assert diophantine(eq) == \ + {(115197*p**2 - 446641*q**2, -150765*p**2 + 1355172*p*q - + 584545*q**2, 174762*p**2 - 301530*p*q + 677586*q**2)} + eq = 573*x**2+267*y**2-984*z**2 + coeff = eq.as_coefficients_dict() + assert _diop_ternary_quadratic_normal((x, y, z), coeff) == \ + {(49, 233, 127)} + assert diophantine(eq) == \ + {(4361*p**2 - 16072*q**2, -20737*p**2 + 83312*p*q - 76424*q**2, + 11303*p**2 - 41474*p*q + 41656*q**2)} + # this produces factors during reconstruction + eq = x**2 + 3*y**2 - 12*z**2 + coeff = eq.as_coefficients_dict() + assert _diop_ternary_quadratic_normal((x, y, z), coeff) == \ + {(0, 2, 1)} + assert diophantine(eq) == \ + {(24*p*q, 2*p**2 - 24*q**2, p**2 + 12*q**2)} + # solvers have not been written for every type + raises(NotImplementedError, lambda: diophantine(x*y**2 + 1)) + + # rational expressions + assert diophantine(1/x) == set() + assert diophantine(1/x + 1/y - S.Half) == {(6, 3), (-2, 1), (4, 4), (1, -2), (3, 6)} + assert diophantine(x**2 + y**2 +3*x- 5, permute=True) == \ + {(-1, 1), (-4, -1), (1, -1), (1, 1), (-4, 1), (-1, -1), (4, 1), (4, -1)} + + + #test issue 18186 + assert diophantine(y**4 + x**4 - 2**4 - 3**4, syms=(x, y), permute=True) == \ + {(-3, -2), (-3, 2), (-2, -3), (-2, 3), (2, -3), (2, 3), (3, -2), (3, 2)} + assert diophantine(y**4 + x**4 - 2**4 - 3**4, syms=(y, x), permute=True) == \ + {(-3, -2), (-3, 2), (-2, -3), (-2, 3), (2, -3), (2, 3), (3, -2), (3, 2)} + + # issue 18122 + assert check_solutions(x**2 - y) + assert check_solutions(y**2 - x) + assert diophantine((x**2 - y), t) == {(t, t**2)} + assert diophantine((y**2 - x), t) == {(t**2, t)} + + +def test_general_pythagorean(): + from sympy.abc import a, b, c, d, e + + assert check_solutions(a**2 + b**2 + c**2 - d**2) + assert check_solutions(a**2 + 4*b**2 + 4*c**2 - d**2) + assert check_solutions(9*a**2 + 4*b**2 + 4*c**2 - d**2) + assert check_solutions(9*a**2 + 4*b**2 - 25*d**2 + 4*c**2 ) + assert check_solutions(9*a**2 - 16*d**2 + 4*b**2 + 4*c**2) + assert check_solutions(-e**2 + 9*a**2 + 4*b**2 + 4*c**2 + 25*d**2) + assert check_solutions(16*a**2 - b**2 + 9*c**2 + d**2 + 25*e**2) + + assert GeneralPythagorean(a**2 + b**2 + c**2 - d**2).solve(parameters=[x, y, z]) == \ + {(x**2 + y**2 - z**2, 2*x*z, 2*y*z, x**2 + y**2 + z**2)} + + +def test_diop_general_sum_of_squares_quick(): + for i in range(3, 10): + assert check_solutions(sum(i**2 for i in symbols(':%i' % i)) - i) + + assert diop_general_sum_of_squares(x**2 + y**2 - 2) is None + assert diop_general_sum_of_squares(x**2 + y**2 + z**2 + 2) == set() + eq = x**2 + y**2 + z**2 - (1 + 4 + 9) + assert diop_general_sum_of_squares(eq) == \ + {(1, 2, 3)} + eq = u**2 + v**2 + x**2 + y**2 + z**2 - 1313 + assert len(diop_general_sum_of_squares(eq, 3)) == 3 + # issue 11016 + var = symbols(':5') + (symbols('6', negative=True),) + eq = Add(*[i**2 for i in var]) - 112 + + base_soln = {(0, 1, 1, 5, 6, -7), (1, 1, 1, 3, 6, -8), (2, 3, 3, 4, 5, -7), (0, 1, 1, 1, 3, -10), + (0, 0, 4, 4, 4, -8), (1, 2, 3, 3, 5, -8), (0, 1, 2, 3, 7, -7), (2, 2, 4, 4, 6, -6), + (1, 1, 3, 4, 6, -7), (0, 2, 3, 3, 3, -9), (0, 0, 2, 2, 2, -10), (1, 1, 2, 3, 4, -9), + (0, 1, 1, 2, 5, -9), (0, 0, 2, 6, 6, -6), (1, 3, 4, 5, 5, -6), (0, 2, 2, 2, 6, -8), + (0, 3, 3, 3, 6, -7), (0, 2, 3, 5, 5, -7), (0, 1, 5, 5, 5, -6)} + assert diophantine(eq) == base_soln + assert len(diophantine(eq, permute=True)) == 196800 + + # handle negated squares with signsimp + assert diophantine(12 - x**2 - y**2 - z**2) == {(2, 2, 2)} + # diophantine handles simplification, so classify_diop should + # not have to look for additional patterns that are removed + # by diophantine + eq = a**2 + b**2 + c**2 + d**2 - 4 + raises(NotImplementedError, lambda: classify_diop(-eq)) + + +def test_issue_23807(): + # fixes recursion error + eq = x**2 + y**2 + z**2 - 1000000 + base_soln = {(0, 0, 1000), (0, 352, 936), (480, 600, 640), (24, 640, 768), (192, 640, 744), + (192, 480, 856), (168, 224, 960), (0, 600, 800), (280, 576, 768), (152, 480, 864), + (0, 280, 960), (352, 360, 864), (424, 480, 768), (360, 480, 800), (224, 600, 768), + (96, 360, 928), (168, 576, 800), (96, 480, 872)} + + assert diophantine(eq) == base_soln + + +def test_diop_partition(): + for n in [8, 10]: + for k in range(1, 8): + for p in partition(n, k): + assert len(p) == k + assert list(partition(3, 5)) == [] + assert [list(p) for p in partition(3, 5, 1)] == [ + [0, 0, 0, 0, 3], [0, 0, 0, 1, 2], [0, 0, 1, 1, 1]] + assert list(partition(0)) == [()] + assert list(partition(1, 0)) == [()] + assert [list(i) for i in partition(3)] == [[1, 1, 1], [1, 2], [3]] + + +def test_prime_as_sum_of_two_squares(): + for i in [5, 13, 17, 29, 37, 41, 2341, 3557, 34841, 64601]: + a, b = prime_as_sum_of_two_squares(i) + assert a**2 + b**2 == i + assert prime_as_sum_of_two_squares(7) is None + ans = prime_as_sum_of_two_squares(800029) + assert ans == (450, 773) and type(ans[0]) is int + + +def test_sum_of_three_squares(): + for i in [0, 1, 2, 34, 123, 34304595905, 34304595905394941, 343045959052344, + 800, 801, 802, 803, 804, 805, 806]: + a, b, c = sum_of_three_squares(i) + assert a**2 + b**2 + c**2 == i + assert a >= 0 + + # error + raises(ValueError, lambda: sum_of_three_squares(-1)) + + assert sum_of_three_squares(7) is None + assert sum_of_three_squares((4**5)*15) is None + # if there are two zeros, there might be a solution + # with only one zero, e.g. 25 => (0, 3, 4) or + # with no zeros, e.g. 49 => (2, 3, 6) + assert sum_of_three_squares(25) == (0, 0, 5) + assert sum_of_three_squares(4) == (0, 0, 2) + + +def test_sum_of_four_squares(): + from sympy.core.random import randint + + # this should never fail + n = randint(1, 100000000000000) + assert sum(i**2 for i in sum_of_four_squares(n)) == n + + # error + raises(ValueError, lambda: sum_of_four_squares(-1)) + + for n in range(1000): + result = sum_of_four_squares(n) + assert len(result) == 4 + assert all(r >= 0 for r in result) + assert sum(r**2 for r in result) == n + assert list(result) == sorted(result) + + +def test_power_representation(): + tests = [(1729, 3, 2), (234, 2, 4), (2, 1, 2), (3, 1, 3), (5, 2, 2), (12352, 2, 4), + (32760, 2, 3)] + + for test in tests: + n, p, k = test + f = power_representation(n, p, k) + + while True: + try: + l = next(f) + assert len(l) == k + + chk_sum = 0 + for l_i in l: + chk_sum = chk_sum + l_i**p + assert chk_sum == n + + except StopIteration: + break + + assert list(power_representation(20, 2, 4, True)) == \ + [(1, 1, 3, 3), (0, 0, 2, 4)] + raises(ValueError, lambda: list(power_representation(1.2, 2, 2))) + raises(ValueError, lambda: list(power_representation(2, 0, 2))) + raises(ValueError, lambda: list(power_representation(2, 2, 0))) + assert list(power_representation(-1, 2, 2)) == [] + assert list(power_representation(1, 1, 1)) == [(1,)] + assert list(power_representation(3, 2, 1)) == [] + assert list(power_representation(4, 2, 1)) == [(2,)] + assert list(power_representation(3**4, 4, 6, zeros=True)) == \ + [(1, 2, 2, 2, 2, 2), (0, 0, 0, 0, 0, 3)] + assert list(power_representation(3**4, 4, 5, zeros=False)) == [] + assert list(power_representation(-2, 3, 2)) == [(-1, -1)] + assert list(power_representation(-2, 4, 2)) == [] + assert list(power_representation(0, 3, 2, True)) == [(0, 0)] + assert list(power_representation(0, 3, 2, False)) == [] + # when we are dealing with squares, do feasibility checks + assert len(list(power_representation(4**10*(8*10 + 7), 2, 3))) == 0 + # there will be a recursion error if these aren't recognized + big = 2**30 + for i in [13, 10, 7, 5, 4, 2, 1]: + assert list(sum_of_powers(big, 2, big - i)) == [] + + +def test_assumptions(): + """ + Test whether diophantine respects the assumptions. + """ + #Test case taken from the below so question regarding assumptions in diophantine module + #https://stackoverflow.com/questions/23301941/how-can-i-declare-natural-symbols-with-sympy + m, n = symbols('m n', integer=True, positive=True) + diof = diophantine(n**2 + m*n - 500) + assert diof == {(5, 20), (40, 10), (95, 5), (121, 4), (248, 2), (499, 1)} + + a, b = symbols('a b', integer=True, positive=False) + diof = diophantine(a*b + 2*a + 3*b - 6) + assert diof == {(-15, -3), (-9, -4), (-7, -5), (-6, -6), (-5, -8), (-4, -14)} + + x, y = symbols('x y', integer=True) + diof = diophantine(10*x**2 + 5*x*y - 3*y) + assert diof == {(1, -5), (-3, 5), (0, 0)} + + x, y = symbols('x y', integer=True, positive=True) + diof = diophantine(10*x**2 + 5*x*y - 3*y) + assert diof == set() + + x, y = symbols('x y', integer=True, negative=False) + diof = diophantine(10*x**2 + 5*x*y - 3*y) + assert diof == {(0, 0)} + + +def check_solutions(eq): + """ + Determines whether solutions returned by diophantine() satisfy the original + equation. Hope to generalize this so we can remove functions like check_ternay_quadratic, + check_solutions_normal, check_solutions() + """ + s = diophantine(eq) + + factors = Mul.make_args(eq) + + var = list(eq.free_symbols) + var.sort(key=default_sort_key) + + while s: + solution = s.pop() + for f in factors: + if diop_simplify(f.subs(zip(var, solution))) == 0: + break + else: + return False + return True + + +def test_diopcoverage(): + eq = (2*x + y + 1)**2 + assert diop_solve(eq) == {(t_0, -2*t_0 - 1)} + eq = 2*x**2 + 6*x*y + 12*x + 4*y**2 + 18*y + 18 + assert diop_solve(eq) == {(t, -t - 3), (-2*t - 3, t)} + assert diop_quadratic(x + y**2 - 3) == {(-t**2 + 3, t)} + + assert diop_linear(x + y - 3) == (t_0, 3 - t_0) + + assert base_solution_linear(0, 1, 2, t=None) == (0, 0) + ans = (3*t - 1, -2*t + 1) + assert base_solution_linear(4, 8, 12, t) == ans + assert base_solution_linear(4, 8, 12, t=None) == tuple(_.subs(t, 0) for _ in ans) + + assert cornacchia(1, 1, 20) == set() + assert cornacchia(1, 1, 5) == {(2, 1)} + assert cornacchia(1, 2, 17) == {(3, 2)} + + raises(ValueError, lambda: reconstruct(4, 20, 1)) + + assert gaussian_reduce(4, 1, 3) == (1, 1) + eq = -w**2 - x**2 - y**2 + z**2 + + assert diop_general_pythagorean(eq) == \ + diop_general_pythagorean(-eq) == \ + (m1**2 + m2**2 - m3**2, 2*m1*m3, + 2*m2*m3, m1**2 + m2**2 + m3**2) + + assert len(check_param(S(3) + x/3, S(4) + x/2, S(2), [x])) == 0 + assert len(check_param(Rational(3, 2), S(4) + x, S(2), [x])) == 0 + assert len(check_param(S(4) + x, Rational(3, 2), S(2), [x])) == 0 + + assert _nint_or_floor(16, 10) == 2 + assert _odd(1) == (not _even(1)) == True + assert _odd(0) == (not _even(0)) == False + assert _remove_gcd(2, 4, 6) == (1, 2, 3) + raises(TypeError, lambda: _remove_gcd((2, 4, 6))) + assert sqf_normal(2*3**2*5, 2*5*11, 2*7**2*11) == \ + (11, 1, 5) + + # it's ok if these pass some day when the solvers are implemented + raises(NotImplementedError, lambda: diophantine(x**2 + y**2 + x*y + 2*y*z - 12)) + raises(NotImplementedError, lambda: diophantine(x**3 + y**2)) + assert diop_quadratic(x**2 + y**2 - 1**2 - 3**4) == \ + {(-9, -1), (-9, 1), (-1, -9), (-1, 9), (1, -9), (1, 9), (9, -1), (9, 1)} + + +def test_holzer(): + # if the input is good, don't let it diverge in holzer() + # (but see test_fail_holzer below) + assert holzer(2, 7, 13, 4, 79, 23) == (2, 7, 13) + + # None in uv condition met; solution is not Holzer reduced + # so this will hopefully change but is here for coverage + assert holzer(2, 6, 2, 1, 1, 10) == (2, 6, 2) + + raises(ValueError, lambda: holzer(2, 7, 14, 4, 79, 23)) + + +@XFAIL +def test_fail_holzer(): + eq = lambda x, y, z: a*x**2 + b*y**2 - c*z**2 + a, b, c = 4, 79, 23 + x, y, z = xyz = 26, 1, 11 + X, Y, Z = ans = 2, 7, 13 + assert eq(*xyz) == 0 + assert eq(*ans) == 0 + assert max(a*x**2, b*y**2, c*z**2) <= a*b*c + assert max(a*X**2, b*Y**2, c*Z**2) <= a*b*c + h = holzer(x, y, z, a, b, c) + assert h == ans # it would be nice to get the smaller soln + + +def test_issue_9539(): + assert diophantine(6*w + 9*y + 20*x - z) == \ + {(t_0, t_1, t_1 + t_2, 6*t_0 + 29*t_1 + 9*t_2)} + + +def test_issue_8943(): + assert diophantine( + 3*(x**2 + y**2 + z**2) - 14*(x*y + y*z + z*x)) == \ + {(0, 0, 0)} + + +def test_diop_sum_of_even_powers(): + eq = x**4 + y**4 + z**4 - 2673 + assert diop_solve(eq) == {(3, 6, 6), (2, 4, 7)} + assert diop_general_sum_of_even_powers(eq, 2) == {(3, 6, 6), (2, 4, 7)} + raises(NotImplementedError, lambda: diop_general_sum_of_even_powers(-eq, 2)) + neg = symbols('neg', negative=True) + eq = x**4 + y**4 + neg**4 - 2673 + assert diop_general_sum_of_even_powers(eq) == {(-3, 6, 6)} + assert diophantine(x**4 + y**4 + 2) == set() + assert diop_general_sum_of_even_powers(x**4 + y**4 - 2, limit=0) == set() + + +def test_sum_of_squares_powers(): + tru = {(0, 0, 1, 1, 11), (0, 0, 5, 7, 7), (0, 1, 3, 7, 8), (0, 1, 4, 5, 9), (0, 3, 4, 7, 7), (0, 3, 5, 5, 8), + (1, 1, 2, 6, 9), (1, 1, 6, 6, 7), (1, 2, 3, 3, 10), (1, 3, 4, 4, 9), (1, 5, 5, 6, 6), (2, 2, 3, 5, 9), + (2, 3, 5, 6, 7), (3, 3, 4, 5, 8)} + eq = u**2 + v**2 + x**2 + y**2 + z**2 - 123 + ans = diop_general_sum_of_squares(eq, oo) # allow oo to be used + assert len(ans) == 14 + assert ans == tru + + raises(ValueError, lambda: list(sum_of_squares(10, -1))) + assert list(sum_of_squares(1, 1)) == [(1,)] + assert list(sum_of_squares(1, 2)) == [] + assert list(sum_of_squares(1, 2, True)) == [(0, 1)] + assert list(sum_of_squares(-10, 2)) == [] + assert list(sum_of_squares(2, 3)) == [] + assert list(sum_of_squares(0, 3, True)) == [(0, 0, 0)] + assert list(sum_of_squares(0, 3)) == [] + assert list(sum_of_squares(4, 1)) == [(2,)] + assert list(sum_of_squares(5, 1)) == [] + assert list(sum_of_squares(50, 2)) == [(5, 5), (1, 7)] + assert list(sum_of_squares(11, 5, True)) == [ + (1, 1, 1, 2, 2), (0, 0, 1, 1, 3)] + assert list(sum_of_squares(8, 8)) == [(1, 1, 1, 1, 1, 1, 1, 1)] + + assert [len(list(sum_of_squares(i, 5, True))) for i in range(30)] == [ + 1, 1, 1, 1, 2, + 2, 1, 1, 2, 2, + 2, 2, 2, 3, 2, + 1, 3, 3, 3, 3, + 4, 3, 3, 2, 2, + 4, 4, 4, 4, 5] + assert [len(list(sum_of_squares(i, 5))) for i in range(30)] == [ + 0, 0, 0, 0, 0, + 1, 0, 0, 1, 0, + 0, 1, 0, 1, 1, + 0, 1, 1, 0, 1, + 2, 1, 1, 1, 1, + 1, 1, 1, 1, 3] + for i in range(30): + s1 = set(sum_of_squares(i, 5, True)) + assert not s1 or all(sum(j**2 for j in t) == i for t in s1) + s2 = set(sum_of_squares(i, 5)) + assert all(sum(j**2 for j in t) == i for t in s2) + + raises(ValueError, lambda: list(sum_of_powers(2, -1, 1))) + raises(ValueError, lambda: list(sum_of_powers(2, 1, -1))) + assert list(sum_of_powers(-2, 3, 2)) == [(-1, -1)] + assert list(sum_of_powers(-2, 4, 2)) == [] + assert list(sum_of_powers(2, 1, 1)) == [(2,)] + assert list(sum_of_powers(2, 1, 3, True)) == [(0, 0, 2), (0, 1, 1)] + assert list(sum_of_powers(5, 1, 2, True)) == [(0, 5), (1, 4), (2, 3)] + assert list(sum_of_powers(6, 2, 2)) == [] + assert list(sum_of_powers(3**5, 3, 1)) == [] + assert list(sum_of_powers(3**6, 3, 1)) == [(9,)] and (9**3 == 3**6) + assert list(sum_of_powers(2**1000, 5, 2)) == [] + + +def test__can_do_sum_of_squares(): + assert _can_do_sum_of_squares(3, -1) is False + assert _can_do_sum_of_squares(-3, 1) is False + assert _can_do_sum_of_squares(0, 1) + assert _can_do_sum_of_squares(4, 1) + assert _can_do_sum_of_squares(1, 2) + assert _can_do_sum_of_squares(2, 2) + assert _can_do_sum_of_squares(3, 2) is False + + +def test_diophantine_permute_sign(): + from sympy.abc import a, b, c, d, e + eq = a**4 + b**4 - (2**4 + 3**4) + base_sol = {(2, 3)} + assert diophantine(eq) == base_sol + complete_soln = set(signed_permutations(base_sol.pop())) + assert diophantine(eq, permute=True) == complete_soln + + eq = a**2 + b**2 + c**2 + d**2 + e**2 - 234 + assert len(diophantine(eq)) == 35 + assert len(diophantine(eq, permute=True)) == 62000 + soln = {(-1, -1), (-1, 2), (1, -2), (1, 1)} + assert diophantine(10*x**2 + 12*x*y + 12*y**2 - 34, permute=True) == soln + + +@XFAIL +def test_not_implemented(): + eq = x**2 + y**4 - 1**2 - 3**4 + assert diophantine(eq, syms=[x, y]) == {(9, 1), (1, 3)} + + +def test_issue_9538(): + eq = x - 3*y + 2 + assert diophantine(eq, syms=[y,x]) == {(t_0, 3*t_0 - 2)} + raises(TypeError, lambda: diophantine(eq, syms={y, x})) + + +def test_ternary_quadratic(): + # solution with 3 parameters + s = diophantine(2*x**2 + y**2 - 2*z**2) + p, q, r = ordered(S(s).free_symbols) + assert s == {( + p**2 - 2*q**2, + -2*p**2 + 4*p*q - 4*p*r - 4*q**2, + p**2 - 4*p*q + 2*q**2 - 4*q*r)} + # solution with Mul in solution + s = diophantine(x**2 + 2*y**2 - 2*z**2) + assert s == {(4*p*q, p**2 - 2*q**2, p**2 + 2*q**2)} + # solution with no Mul in solution + s = diophantine(2*x**2 + 2*y**2 - z**2) + assert s == {(2*p**2 - q**2, -2*p**2 + 4*p*q - q**2, + 4*p**2 - 4*p*q + 2*q**2)} + # reduced form when parametrized + s = diophantine(3*x**2 + 72*y**2 - 27*z**2) + assert s == {(24*p**2 - 9*q**2, 6*p*q, 8*p**2 + 3*q**2)} + assert parametrize_ternary_quadratic( + 3*x**2 + 2*y**2 - z**2 - 2*x*y + 5*y*z - 7*y*z) == ( + 2*p**2 - 2*p*q - q**2, 2*p**2 + 2*p*q - q**2, 2*p**2 - + 2*p*q + 3*q**2) + assert parametrize_ternary_quadratic( + 124*x**2 - 30*y**2 - 7729*z**2) == ( + -1410*p**2 - 363263*q**2, 2700*p**2 + 30916*p*q - + 695610*q**2, -60*p**2 + 5400*p*q + 15458*q**2) + + +def test_diophantine_solution_set(): + s1 = DiophantineSolutionSet([], []) + assert set(s1) == set() + assert s1.symbols == () + assert s1.parameters == () + raises(ValueError, lambda: s1.add((x,))) + assert list(s1.dict_iterator()) == [] + + s2 = DiophantineSolutionSet([x, y], [t, u]) + assert s2.symbols == (x, y) + assert s2.parameters == (t, u) + raises(ValueError, lambda: s2.add((1,))) + s2.add((3, 4)) + assert set(s2) == {(3, 4)} + s2.update((3, 4), (-1, u)) + assert set(s2) == {(3, 4), (-1, u)} + raises(ValueError, lambda: s1.update(s2)) + assert list(s2.dict_iterator()) == [{x: -1, y: u}, {x: 3, y: 4}] + + s3 = DiophantineSolutionSet([x, y, z], [t, u]) + assert len(s3.parameters) == 2 + s3.add((t**2 + u, t - u, 1)) + assert set(s3) == {(t**2 + u, t - u, 1)} + assert s3.subs(t, 2) == {(u + 4, 2 - u, 1)} + assert s3(2) == {(u + 4, 2 - u, 1)} + assert s3.subs({t: 7, u: 8}) == {(57, -1, 1)} + assert s3(7, 8) == {(57, -1, 1)} + assert s3.subs({t: 5}) == {(u + 25, 5 - u, 1)} + assert s3(5) == {(u + 25, 5 - u, 1)} + assert s3.subs(u, -3) == {(t**2 - 3, t + 3, 1)} + assert s3(None, -3) == {(t**2 - 3, t + 3, 1)} + assert s3.subs({t: 2, u: 8}) == {(12, -6, 1)} + assert s3(2, 8) == {(12, -6, 1)} + assert s3.subs({t: 5, u: -3}) == {(22, 8, 1)} + assert s3(5, -3) == {(22, 8, 1)} + raises(TypeError, lambda: s3.subs(x=1)) + raises(TypeError, lambda: s3.subs(1, 2, 3)) + raises(ValueError, lambda: s3.add(())) + raises(ValueError, lambda: s3.add((1, 2, 3, 4))) + raises(ValueError, lambda: s3.add((1, 2))) + raises(ValueError, lambda: s3(1, 2, 3)) + raises(TypeError, lambda: s3(t=1)) + + s4 = DiophantineSolutionSet([x, y], [t, u]) + s4.add((t, 11*t)) + s4.add((-t, 22*t)) + assert s4(0, 0) == {(0, 0)} + + +def test_quadratic_parameter_passing(): + eq = -33*x*y + 3*y**2 + solution = BinaryQuadratic(eq).solve(parameters=[t, u]) + # test that parameters are passed all the way to the final solution + assert solution == {(t, 11*t), (t, -22*t)} + assert solution(0, 0) == {(0, 0)} + +def test_issue_18628(): + eq1 = x**2 - 15*x + y**2 - 8*y + sol = diophantine(eq1) + assert sol == {(15, 0), (15, 8), (-1, 4), (0, 0), (0, 8), (16, 4)} + eq2 = 2*x**2 - 9*x + 4*y**2 - 8*y + 14 + sol = diophantine(eq2) + assert sol == {(2, 1)} diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/inequalities.py b/.venv/lib/python3.12/site-packages/sympy/solvers/inequalities.py new file mode 100644 index 0000000000000000000000000000000000000000..f50f7a7572ff25e8f48a4214d7b0c5ec6b5b35f3 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/inequalities.py @@ -0,0 +1,986 @@ +"""Tools for solving inequalities and systems of inequalities. """ +import itertools + +from sympy.calculus.util import (continuous_domain, periodicity, + function_range) +from sympy.core import sympify +from sympy.core.exprtools import factor_terms +from sympy.core.relational import Relational, Lt, Ge, Eq +from sympy.core.symbol import Symbol, Dummy +from sympy.sets.sets import Interval, FiniteSet, Union, Intersection +from sympy.core.singleton import S +from sympy.core.function import expand_mul +from sympy.functions.elementary.complexes import Abs +from sympy.logic import And +from sympy.polys import Poly, PolynomialError, parallel_poly_from_expr +from sympy.polys.polyutils import _nsort +from sympy.solvers.solveset import solvify, solveset +from sympy.utilities.iterables import sift, iterable +from sympy.utilities.misc import filldedent + + +def solve_poly_inequality(poly, rel): + """Solve a polynomial inequality with rational coefficients. + + Examples + ======== + + >>> from sympy import solve_poly_inequality, Poly + >>> from sympy.abc import x + + >>> solve_poly_inequality(Poly(x, x, domain='ZZ'), '==') + [{0}] + + >>> solve_poly_inequality(Poly(x**2 - 1, x, domain='ZZ'), '!=') + [Interval.open(-oo, -1), Interval.open(-1, 1), Interval.open(1, oo)] + + >>> solve_poly_inequality(Poly(x**2 - 1, x, domain='ZZ'), '==') + [{-1}, {1}] + + See Also + ======== + solve_poly_inequalities + """ + if not isinstance(poly, Poly): + raise ValueError( + 'For efficiency reasons, `poly` should be a Poly instance') + if poly.as_expr().is_number: + t = Relational(poly.as_expr(), 0, rel) + if t is S.true: + return [S.Reals] + elif t is S.false: + return [S.EmptySet] + else: + raise NotImplementedError( + "could not determine truth value of %s" % t) + + reals, intervals = poly.real_roots(multiple=False), [] + + if rel == '==': + for root, _ in reals: + interval = Interval(root, root) + intervals.append(interval) + elif rel == '!=': + left = S.NegativeInfinity + + for right, _ in reals + [(S.Infinity, 1)]: + interval = Interval(left, right, True, True) + intervals.append(interval) + left = right + else: + if poly.LC() > 0: + sign = +1 + else: + sign = -1 + + eq_sign, equal = None, False + + if rel == '>': + eq_sign = +1 + elif rel == '<': + eq_sign = -1 + elif rel == '>=': + eq_sign, equal = +1, True + elif rel == '<=': + eq_sign, equal = -1, True + else: + raise ValueError("'%s' is not a valid relation" % rel) + + right, right_open = S.Infinity, True + + for left, multiplicity in reversed(reals): + if multiplicity % 2: + if sign == eq_sign: + intervals.insert( + 0, Interval(left, right, not equal, right_open)) + + sign, right, right_open = -sign, left, not equal + else: + if sign == eq_sign and not equal: + intervals.insert( + 0, Interval(left, right, True, right_open)) + right, right_open = left, True + elif sign != eq_sign and equal: + intervals.insert(0, Interval(left, left)) + + if sign == eq_sign: + intervals.insert( + 0, Interval(S.NegativeInfinity, right, True, right_open)) + + return intervals + + +def solve_poly_inequalities(polys): + """Solve polynomial inequalities with rational coefficients. + + Examples + ======== + + >>> from sympy import Poly + >>> from sympy.solvers.inequalities import solve_poly_inequalities + >>> from sympy.abc import x + >>> solve_poly_inequalities((( + ... Poly(x**2 - 3), ">"), ( + ... Poly(-x**2 + 1), ">"))) + Union(Interval.open(-oo, -sqrt(3)), Interval.open(-1, 1), Interval.open(sqrt(3), oo)) + """ + return Union(*[s for p in polys for s in solve_poly_inequality(*p)]) + + +def solve_rational_inequalities(eqs): + """Solve a system of rational inequalities with rational coefficients. + + Examples + ======== + + >>> from sympy.abc import x + >>> from sympy import solve_rational_inequalities, Poly + + >>> solve_rational_inequalities([[ + ... ((Poly(-x + 1), Poly(1, x)), '>='), + ... ((Poly(-x + 1), Poly(1, x)), '<=')]]) + {1} + + >>> solve_rational_inequalities([[ + ... ((Poly(x), Poly(1, x)), '!='), + ... ((Poly(-x + 1), Poly(1, x)), '>=')]]) + Union(Interval.open(-oo, 0), Interval.Lopen(0, 1)) + + See Also + ======== + solve_poly_inequality + """ + result = S.EmptySet + + for _eqs in eqs: + if not _eqs: + continue + + global_intervals = [Interval(S.NegativeInfinity, S.Infinity)] + + for (numer, denom), rel in _eqs: + numer_intervals = solve_poly_inequality(numer*denom, rel) + denom_intervals = solve_poly_inequality(denom, '==') + + intervals = [] + + for numer_interval, global_interval in itertools.product( + numer_intervals, global_intervals): + interval = numer_interval.intersect(global_interval) + + if interval is not S.EmptySet: + intervals.append(interval) + + global_intervals = intervals + + intervals = [] + + for global_interval in global_intervals: + for denom_interval in denom_intervals: + global_interval -= denom_interval + + if global_interval is not S.EmptySet: + intervals.append(global_interval) + + global_intervals = intervals + + if not global_intervals: + break + + for interval in global_intervals: + result = result.union(interval) + + return result + + +def reduce_rational_inequalities(exprs, gen, relational=True): + """Reduce a system of rational inequalities with rational coefficients. + + Examples + ======== + + >>> from sympy import Symbol + >>> from sympy.solvers.inequalities import reduce_rational_inequalities + + >>> x = Symbol('x', real=True) + + >>> reduce_rational_inequalities([[x**2 <= 0]], x) + Eq(x, 0) + + >>> reduce_rational_inequalities([[x + 2 > 0]], x) + -2 < x + >>> reduce_rational_inequalities([[(x + 2, ">")]], x) + -2 < x + >>> reduce_rational_inequalities([[x + 2]], x) + Eq(x, -2) + + This function find the non-infinite solution set so if the unknown symbol + is declared as extended real rather than real then the result may include + finiteness conditions: + + >>> y = Symbol('y', extended_real=True) + >>> reduce_rational_inequalities([[y + 2 > 0]], y) + (-2 < y) & (y < oo) + """ + exact = True + eqs = [] + solution = S.EmptySet # add pieces for each group + for _exprs in exprs: + if not _exprs: + continue + _eqs = [] + _sol = S.Reals + for expr in _exprs: + if isinstance(expr, tuple): + expr, rel = expr + else: + if expr.is_Relational: + expr, rel = expr.lhs - expr.rhs, expr.rel_op + else: + rel = '==' + + if expr is S.true: + numer, denom, rel = S.Zero, S.One, '==' + elif expr is S.false: + numer, denom, rel = S.One, S.One, '==' + else: + numer, denom = expr.together().as_numer_denom() + + try: + (numer, denom), opt = parallel_poly_from_expr( + (numer, denom), gen) + except PolynomialError: + raise PolynomialError(filldedent(''' + only polynomials and rational functions are + supported in this context. + ''')) + + if not opt.domain.is_Exact: + numer, denom, exact = numer.to_exact(), denom.to_exact(), False + + domain = opt.domain.get_exact() + + if not (domain.is_ZZ or domain.is_QQ): + expr = numer/denom + expr = Relational(expr, 0, rel) + _sol &= solve_univariate_inequality(expr, gen, relational=False) + else: + _eqs.append(((numer, denom), rel)) + + if _eqs: + _sol &= solve_rational_inequalities([_eqs]) + exclude = solve_rational_inequalities([[((d, d.one), '==') + for i in eqs for ((n, d), _) in i if d.has(gen)]]) + _sol -= exclude + + solution |= _sol + + if not exact and solution: + solution = solution.evalf() + + if relational: + solution = solution.as_relational(gen) + + return solution + + +def reduce_abs_inequality(expr, rel, gen): + """Reduce an inequality with nested absolute values. + + Examples + ======== + + >>> from sympy import reduce_abs_inequality, Abs, Symbol + >>> x = Symbol('x', real=True) + + >>> reduce_abs_inequality(Abs(x - 5) - 3, '<', x) + (2 < x) & (x < 8) + + >>> reduce_abs_inequality(Abs(x + 2)*3 - 13, '<', x) + (-19/3 < x) & (x < 7/3) + + See Also + ======== + + reduce_abs_inequalities + """ + if gen.is_extended_real is False: + raise TypeError(filldedent(''' + Cannot solve inequalities with absolute values containing + non-real variables. + ''')) + + def _bottom_up_scan(expr): + exprs = [] + + if expr.is_Add or expr.is_Mul: + op = expr.func + + for arg in expr.args: + _exprs = _bottom_up_scan(arg) + + if not exprs: + exprs = _exprs + else: + exprs = [(op(expr, _expr), conds + _conds) for (expr, conds), (_expr, _conds) in + itertools.product(exprs, _exprs)] + elif expr.is_Pow: + n = expr.exp + if not n.is_Integer: + raise ValueError("Only Integer Powers are allowed on Abs.") + + exprs.extend((expr**n, conds) for expr, conds in _bottom_up_scan(expr.base)) + elif isinstance(expr, Abs): + _exprs = _bottom_up_scan(expr.args[0]) + + for expr, conds in _exprs: + exprs.append(( expr, conds + [Ge(expr, 0)])) + exprs.append((-expr, conds + [Lt(expr, 0)])) + else: + exprs = [(expr, [])] + + return exprs + + mapping = {'<': '>', '<=': '>='} + inequalities = [] + + for expr, conds in _bottom_up_scan(expr): + if rel not in mapping.keys(): + expr = Relational( expr, 0, rel) + else: + expr = Relational(-expr, 0, mapping[rel]) + + inequalities.append([expr] + conds) + + return reduce_rational_inequalities(inequalities, gen) + + +def reduce_abs_inequalities(exprs, gen): + """Reduce a system of inequalities with nested absolute values. + + Examples + ======== + + >>> from sympy import reduce_abs_inequalities, Abs, Symbol + >>> x = Symbol('x', extended_real=True) + + >>> reduce_abs_inequalities([(Abs(3*x - 5) - 7, '<'), + ... (Abs(x + 25) - 13, '>')], x) + (-2/3 < x) & (x < 4) & (((-oo < x) & (x < -38)) | ((-12 < x) & (x < oo))) + + >>> reduce_abs_inequalities([(Abs(x - 4) + Abs(3*x - 5) - 7, '<')], x) + (1/2 < x) & (x < 4) + + See Also + ======== + + reduce_abs_inequality + """ + return And(*[ reduce_abs_inequality(expr, rel, gen) + for expr, rel in exprs ]) + + +def solve_univariate_inequality(expr, gen, relational=True, domain=S.Reals, continuous=False): + """Solves a real univariate inequality. + + Parameters + ========== + + expr : Relational + The target inequality + gen : Symbol + The variable for which the inequality is solved + relational : bool + A Relational type output is expected or not + domain : Set + The domain over which the equation is solved + continuous: bool + True if expr is known to be continuous over the given domain + (and so continuous_domain() does not need to be called on it) + + Raises + ====== + + NotImplementedError + The solution of the inequality cannot be determined due to limitation + in :func:`sympy.solvers.solveset.solvify`. + + Notes + ===== + + Currently, we cannot solve all the inequalities due to limitations in + :func:`sympy.solvers.solveset.solvify`. Also, the solution returned for trigonometric inequalities + are restricted in its periodic interval. + + See Also + ======== + + sympy.solvers.solveset.solvify: solver returning solveset solutions with solve's output API + + Examples + ======== + + >>> from sympy import solve_univariate_inequality, Symbol, sin, Interval, S + >>> x = Symbol('x') + + >>> solve_univariate_inequality(x**2 >= 4, x) + ((2 <= x) & (x < oo)) | ((-oo < x) & (x <= -2)) + + >>> solve_univariate_inequality(x**2 >= 4, x, relational=False) + Union(Interval(-oo, -2), Interval(2, oo)) + + >>> domain = Interval(0, S.Infinity) + >>> solve_univariate_inequality(x**2 >= 4, x, False, domain) + Interval(2, oo) + + >>> solve_univariate_inequality(sin(x) > 0, x, relational=False) + Interval.open(0, pi) + + """ + from sympy.solvers.solvers import denoms + + if domain.is_subset(S.Reals) is False: + raise NotImplementedError(filldedent(''' + Inequalities in the complex domain are + not supported. Try the real domain by + setting domain=S.Reals''')) + elif domain is not S.Reals: + rv = solve_univariate_inequality( + expr, gen, relational=False, continuous=continuous).intersection(domain) + if relational: + rv = rv.as_relational(gen) + return rv + else: + pass # continue with attempt to solve in Real domain + + # This keeps the function independent of the assumptions about `gen`. + # `solveset` makes sure this function is called only when the domain is + # real. + _gen = gen + _domain = domain + if gen.is_extended_real is False: + rv = S.EmptySet + return rv if not relational else rv.as_relational(_gen) + elif gen.is_extended_real is None: + gen = Dummy('gen', extended_real=True) + try: + expr = expr.xreplace({_gen: gen}) + except TypeError: + raise TypeError(filldedent(''' + When gen is real, the relational has a complex part + which leads to an invalid comparison like I < 0. + ''')) + + rv = None + + if expr is S.true: + rv = domain + + elif expr is S.false: + rv = S.EmptySet + + else: + e = expr.lhs - expr.rhs + period = periodicity(e, gen) + if period == S.Zero: + e = expand_mul(e) + const = expr.func(e, 0) + if const is S.true: + rv = domain + elif const is S.false: + rv = S.EmptySet + elif period is not None: + frange = function_range(e, gen, domain) + + rel = expr.rel_op + if rel in ('<', '<='): + if expr.func(frange.sup, 0): + rv = domain + elif not expr.func(frange.inf, 0): + rv = S.EmptySet + + elif rel in ('>', '>='): + if expr.func(frange.inf, 0): + rv = domain + elif not expr.func(frange.sup, 0): + rv = S.EmptySet + + inf, sup = domain.inf, domain.sup + if sup - inf is S.Infinity: + domain = Interval(0, period, False, True).intersect(_domain) + _domain = domain + + if rv is None: + n, d = e.as_numer_denom() + try: + if gen not in n.free_symbols and len(e.free_symbols) > 1: + raise ValueError + # this might raise ValueError on its own + # or it might give None... + solns = solvify(e, gen, domain) + if solns is None: + # in which case we raise ValueError + raise ValueError + except (ValueError, NotImplementedError): + # replace gen with generic x since it's + # univariate anyway + raise NotImplementedError(filldedent(''' + The inequality, %s, cannot be solved using + solve_univariate_inequality. + ''' % expr.subs(gen, Symbol('x')))) + + expanded_e = expand_mul(e) + def valid(x): + # this is used to see if gen=x satisfies the + # relational by substituting it into the + # expanded form and testing against 0, e.g. + # if expr = x*(x + 1) < 2 then e = x*(x + 1) - 2 + # and expanded_e = x**2 + x - 2; the test is + # whether a given value of x satisfies + # x**2 + x - 2 < 0 + # + # expanded_e, expr and gen used from enclosing scope + v = expanded_e.subs(gen, expand_mul(x)) + try: + r = expr.func(v, 0) + except TypeError: + r = S.false + if r in (S.true, S.false): + return r + if v.is_extended_real is False: + return S.false + else: + v = v.n(2) + if v.is_comparable: + return expr.func(v, 0) + # not comparable or couldn't be evaluated + raise NotImplementedError( + 'relationship did not evaluate: %s' % r) + + singularities = [] + for d in denoms(expr, gen): + singularities.extend(solvify(d, gen, domain)) + if not continuous: + domain = continuous_domain(expanded_e, gen, domain) + + include_x = '=' in expr.rel_op and expr.rel_op != '!=' + + try: + discontinuities = set(domain.boundary - + FiniteSet(domain.inf, domain.sup)) + # remove points that are not between inf and sup of domain + critical_points = FiniteSet(*(solns + singularities + list( + discontinuities))).intersection( + Interval(domain.inf, domain.sup, + domain.inf not in domain, domain.sup not in domain)) + if all(r.is_number for r in critical_points): + reals = _nsort(critical_points, separated=True)[0] + else: + sifted = sift(critical_points, lambda x: x.is_extended_real) + if sifted[None]: + # there were some roots that weren't known + # to be real + raise NotImplementedError + try: + reals = sifted[True] + if len(reals) > 1: + reals = sorted(reals) + except TypeError: + raise NotImplementedError + except NotImplementedError: + raise NotImplementedError('sorting of these roots is not supported') + + # If expr contains imaginary coefficients, only take real + # values of x for which the imaginary part is 0 + make_real = S.Reals + if (coeffI := expanded_e.coeff(S.ImaginaryUnit)) != S.Zero: + check = True + im_sol = FiniteSet() + try: + a = solveset(coeffI, gen, domain) + if not isinstance(a, Interval): + for z in a: + if z not in singularities and valid(z) and z.is_extended_real: + im_sol += FiniteSet(z) + else: + start, end = a.inf, a.sup + for z in _nsort(critical_points + FiniteSet(end)): + valid_start = valid(start) + if start != end: + valid_z = valid(z) + pt = _pt(start, z) + if pt not in singularities and pt.is_extended_real and valid(pt): + if valid_start and valid_z: + im_sol += Interval(start, z) + elif valid_start: + im_sol += Interval.Ropen(start, z) + elif valid_z: + im_sol += Interval.Lopen(start, z) + else: + im_sol += Interval.open(start, z) + start = z + for s in singularities: + im_sol -= FiniteSet(s) + except (TypeError): + im_sol = S.Reals + check = False + + if im_sol is S.EmptySet: + raise ValueError(filldedent(''' + %s contains imaginary parts which cannot be + made 0 for any value of %s satisfying the + inequality, leading to relations like I < 0. + ''' % (expr.subs(gen, _gen), _gen))) + + make_real = make_real.intersect(im_sol) + + sol_sets = [S.EmptySet] + + start = domain.inf + if start in domain and valid(start) and start.is_finite: + sol_sets.append(FiniteSet(start)) + + for x in reals: + end = x + + if valid(_pt(start, end)): + sol_sets.append(Interval(start, end, True, True)) + + if x in singularities: + singularities.remove(x) + else: + if x in discontinuities: + discontinuities.remove(x) + _valid = valid(x) + else: # it's a solution + _valid = include_x + if _valid: + sol_sets.append(FiniteSet(x)) + + start = end + + end = domain.sup + if end in domain and valid(end) and end.is_finite: + sol_sets.append(FiniteSet(end)) + + if valid(_pt(start, end)): + sol_sets.append(Interval.open(start, end)) + + if coeffI != S.Zero and check: + rv = (make_real).intersect(_domain) + else: + rv = Intersection( + (Union(*sol_sets)), make_real, _domain).subs(gen, _gen) + + return rv if not relational else rv.as_relational(_gen) + + +def _pt(start, end): + """Return a point between start and end""" + if not start.is_infinite and not end.is_infinite: + pt = (start + end)/2 + elif start.is_infinite and end.is_infinite: + pt = S.Zero + else: + if (start.is_infinite and start.is_extended_positive is None or + end.is_infinite and end.is_extended_positive is None): + raise ValueError('cannot proceed with unsigned infinite values') + if (end.is_infinite and end.is_extended_negative or + start.is_infinite and start.is_extended_positive): + start, end = end, start + # if possible, use a multiple of self which has + # better behavior when checking assumptions than + # an expression obtained by adding or subtracting 1 + if end.is_infinite: + if start.is_extended_positive: + pt = start*2 + elif start.is_extended_negative: + pt = start*S.Half + else: + pt = start + 1 + elif start.is_infinite: + if end.is_extended_positive: + pt = end*S.Half + elif end.is_extended_negative: + pt = end*2 + else: + pt = end - 1 + return pt + + +def _solve_inequality(ie, s, linear=False): + """Return the inequality with s isolated on the left, if possible. + If the relationship is non-linear, a solution involving And or Or + may be returned. False or True are returned if the relationship + is never True or always True, respectively. + + If `linear` is True (default is False) an `s`-dependent expression + will be isolated on the left, if possible + but it will not be solved for `s` unless the expression is linear + in `s`. Furthermore, only "safe" operations which do not change the + sense of the relationship are applied: no division by an unsigned + value is attempted unless the relationship involves Eq or Ne and + no division by a value not known to be nonzero is ever attempted. + + Examples + ======== + + >>> from sympy import Eq, Symbol + >>> from sympy.solvers.inequalities import _solve_inequality as f + >>> from sympy.abc import x, y + + For linear expressions, the symbol can be isolated: + + >>> f(x - 2 < 0, x) + x < 2 + >>> f(-x - 6 < x, x) + x > -3 + + Sometimes nonlinear relationships will be False + + >>> f(x**2 + 4 < 0, x) + False + + Or they may involve more than one region of values: + + >>> f(x**2 - 4 < 0, x) + (-2 < x) & (x < 2) + + To restrict the solution to a relational, set linear=True + and only the x-dependent portion will be isolated on the left: + + >>> f(x**2 - 4 < 0, x, linear=True) + x**2 < 4 + + Division of only nonzero quantities is allowed, so x cannot + be isolated by dividing by y: + + >>> y.is_nonzero is None # it is unknown whether it is 0 or not + True + >>> f(x*y < 1, x) + x*y < 1 + + And while an equality (or inequality) still holds after dividing by a + non-zero quantity + + >>> nz = Symbol('nz', nonzero=True) + >>> f(Eq(x*nz, 1), x) + Eq(x, 1/nz) + + the sign must be known for other inequalities involving > or <: + + >>> f(x*nz <= 1, x) + nz*x <= 1 + >>> p = Symbol('p', positive=True) + >>> f(x*p <= 1, x) + x <= 1/p + + When there are denominators in the original expression that + are removed by expansion, conditions for them will be returned + as part of the result: + + >>> f(x < x*(2/x - 1), x) + (x < 1) & Ne(x, 0) + """ + from sympy.solvers.solvers import denoms + if s not in ie.free_symbols: + return ie + if ie.rhs == s: + ie = ie.reversed + if ie.lhs == s and s not in ie.rhs.free_symbols: + return ie + + def classify(ie, s, i): + # return True or False if ie evaluates when substituting s with + # i else None (if unevaluated) or NaN (when there is an error + # in evaluating) + try: + v = ie.subs(s, i) + if v is S.NaN: + return v + elif v not in (True, False): + return + return v + except TypeError: + return S.NaN + + rv = None + oo = S.Infinity + expr = ie.lhs - ie.rhs + try: + p = Poly(expr, s) + if p.degree() == 0: + rv = ie.func(p.as_expr(), 0) + elif not linear and p.degree() > 1: + # handle in except clause + raise NotImplementedError + except (PolynomialError, NotImplementedError): + if not linear: + try: + rv = reduce_rational_inequalities([[ie]], s) + except PolynomialError: + rv = solve_univariate_inequality(ie, s) + # remove restrictions wrt +/-oo that may have been + # applied when using sets to simplify the relationship + okoo = classify(ie, s, oo) + if okoo is S.true and classify(rv, s, oo) is S.false: + rv = rv.subs(s < oo, True) + oknoo = classify(ie, s, -oo) + if (oknoo is S.true and + classify(rv, s, -oo) is S.false): + rv = rv.subs(-oo < s, True) + rv = rv.subs(s > -oo, True) + if rv is S.true: + rv = (s <= oo) if okoo is S.true else (s < oo) + if oknoo is not S.true: + rv = And(-oo < s, rv) + else: + p = Poly(expr) + + conds = [] + if rv is None: + e = p.as_expr() # this is in expanded form + # Do a safe inversion of e, moving non-s terms + # to the rhs and dividing by a nonzero factor if + # the relational is Eq/Ne; for other relationals + # the sign must also be positive or negative + rhs = 0 + b, ax = e.as_independent(s, as_Add=True) + e -= b + rhs -= b + ef = factor_terms(e) + a, e = ef.as_independent(s, as_Add=False) + if (a.is_zero != False or # don't divide by potential 0 + a.is_negative == + a.is_positive is None and # if sign is not known then + ie.rel_op not in ('!=', '==')): # reject if not Eq/Ne + e = ef + a = S.One + rhs /= a + if a.is_positive: + rv = ie.func(e, rhs) + else: + rv = ie.reversed.func(e, rhs) + + # return conditions under which the value is + # valid, too. + beginning_denoms = denoms(ie.lhs) | denoms(ie.rhs) + current_denoms = denoms(rv) + for d in beginning_denoms - current_denoms: + c = _solve_inequality(Eq(d, 0), s, linear=linear) + if isinstance(c, Eq) and c.lhs == s: + if classify(rv, s, c.rhs) is S.true: + # rv is permitting this value but it shouldn't + conds.append(~c) + for i in (-oo, oo): + if (classify(rv, s, i) is S.true and + classify(ie, s, i) is not S.true): + conds.append(s < i if i is oo else i < s) + + conds.append(rv) + return And(*conds) + + +def _reduce_inequalities(inequalities, symbols): + # helper for reduce_inequalities + + poly_part, abs_part = {}, {} + other = [] + + for inequality in inequalities: + + expr, rel = inequality.lhs, inequality.rel_op # rhs is 0 + + # check for gens using atoms which is more strict than free_symbols to + # guard against EX domain which won't be handled by + # reduce_rational_inequalities + gens = expr.atoms(Symbol) + + if len(gens) == 1: + gen = gens.pop() + else: + common = expr.free_symbols & symbols + if len(common) == 1: + gen = common.pop() + other.append(_solve_inequality(Relational(expr, 0, rel), gen)) + continue + else: + raise NotImplementedError(filldedent(''' + inequality has more than one symbol of interest. + ''')) + + if expr.is_polynomial(gen): + poly_part.setdefault(gen, []).append((expr, rel)) + else: + components = expr.find(lambda u: + u.has(gen) and ( + u.is_Function or u.is_Pow and not u.exp.is_Integer)) + if components and all(isinstance(i, Abs) for i in components): + abs_part.setdefault(gen, []).append((expr, rel)) + else: + other.append(_solve_inequality(Relational(expr, 0, rel), gen)) + + poly_reduced = [reduce_rational_inequalities([exprs], gen) for gen, exprs in poly_part.items()] + abs_reduced = [reduce_abs_inequalities(exprs, gen) for gen, exprs in abs_part.items()] + + return And(*(poly_reduced + abs_reduced + other)) + + +def reduce_inequalities(inequalities, symbols=[]): + """Reduce a system of inequalities with rational coefficients. + + Examples + ======== + + >>> from sympy.abc import x, y + >>> from sympy import reduce_inequalities + + >>> reduce_inequalities(0 <= x + 3, []) + (-3 <= x) & (x < oo) + + >>> reduce_inequalities(0 <= x + y*2 - 1, [x]) + (x < oo) & (x >= 1 - 2*y) + """ + if not iterable(inequalities): + inequalities = [inequalities] + inequalities = [sympify(i) for i in inequalities] + + gens = set().union(*[i.free_symbols for i in inequalities]) + + if not iterable(symbols): + symbols = [symbols] + symbols = (set(symbols) or gens) & gens + if any(i.is_extended_real is False for i in symbols): + raise TypeError(filldedent(''' + inequalities cannot contain symbols that are not real. + ''')) + + # make vanilla symbol real + recast = {i: Dummy(i.name, extended_real=True) + for i in gens if i.is_extended_real is None} + inequalities = [i.xreplace(recast) for i in inequalities] + symbols = {i.xreplace(recast) for i in symbols} + + # prefilter + keep = [] + for i in inequalities: + if isinstance(i, Relational): + i = i.func(i.lhs.as_expr() - i.rhs.as_expr(), 0) + elif i not in (True, False): + i = Eq(i, 0) + if i == True: + continue + elif i == False: + return S.false + if i.lhs.is_number: + raise NotImplementedError( + "could not determine truth value of %s" % i) + keep.append(i) + inequalities = keep + del keep + + # solve system + rv = _reduce_inequalities(inequalities, symbols) + + # restore original symbols and return + return rv.xreplace({v: k for k, v in recast.items()}) diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/ode/__init__.py b/.venv/lib/python3.12/site-packages/sympy/solvers/ode/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..2b543425251dea6380a1860279cb6d636f3dd629 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/ode/__init__.py @@ -0,0 +1,16 @@ +from .ode import (allhints, checkinfsol, classify_ode, + constantsimp, dsolve, homogeneous_order) + +from .lie_group import infinitesimals + +from .subscheck import checkodesol + +from .systems import (canonical_odes, linear_ode_to_matrix, + linodesolve) + + +__all__ = [ + 'allhints', 'checkinfsol', 'checkodesol', 'classify_ode', 'constantsimp', + 'dsolve', 'homogeneous_order', 'infinitesimals', 'canonical_odes', 'linear_ode_to_matrix', + 'linodesolve' +] diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/ode/__pycache__/__init__.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/solvers/ode/__pycache__/__init__.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..6cf62cfb7134d59e6653d35d4cb45c359915d418 Binary files /dev/null and b/.venv/lib/python3.12/site-packages/sympy/solvers/ode/__pycache__/__init__.cpython-312.pyc differ diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/ode/__pycache__/hypergeometric.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/solvers/ode/__pycache__/hypergeometric.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..c5f32dc6df15d01fba1b5af757525fe2bfe027b7 Binary 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This is an incomplete implementation of the algorithm described in [1]. +The algorithm solves 2nd order linear ODEs of the form + +.. math:: y'' + A(x) y' + B(x) y = 0\text{,} + +where `A` and `B` are rational functions. The algorithm should find any +solution of the form + +.. math:: y = P(x) _pF_q(..; ..;\frac{\alpha x^k + \beta}{\gamma x^k + \delta})\text{,} + +where pFq is any of 2F1, 1F1 or 0F1 and `P` is an "arbitrary function". +Currently only the 2F1 case is implemented in SymPy but the other cases are +described in the paper and could be implemented in future (contributions +welcome!). + +References +========== + +.. [1] L. Chan, E.S. Cheb-Terrab, Non-Liouvillian solutions for second order + linear ODEs, (2004). + https://arxiv.org/abs/math-ph/0402063 +''' + +from sympy.core import S, Pow +from sympy.core.function import expand +from sympy.core.relational import Eq +from sympy.core.symbol import Symbol, Wild +from sympy.functions import exp, sqrt, hyper +from sympy.integrals import Integral +from sympy.polys import roots, gcd +from sympy.polys.polytools import cancel, factor +from sympy.simplify import collect, simplify, logcombine # type: ignore +from sympy.simplify.powsimp import powdenest +from sympy.solvers.ode.ode import get_numbered_constants + + +def match_2nd_hypergeometric(eq, func): + x = func.args[0] + df = func.diff(x) + a3 = Wild('a3', exclude=[func, func.diff(x), func.diff(x, 2)]) + b3 = Wild('b3', exclude=[func, func.diff(x), func.diff(x, 2)]) + c3 = Wild('c3', exclude=[func, func.diff(x), func.diff(x, 2)]) + deq = a3*(func.diff(x, 2)) + b3*df + c3*func + r = collect(eq, + [func.diff(x, 2), func.diff(x), func]).match(deq) + if r: + if not all(val.is_polynomial() for val in r.values()): + n, d = eq.as_numer_denom() + eq = expand(n) + r = collect(eq, [func.diff(x, 2), func.diff(x), func]).match(deq) + + if r and r[a3]!=0: + A = cancel(r[b3]/r[a3]) + B = cancel(r[c3]/r[a3]) + return [A, B] + else: + return [] + + +def equivalence_hypergeometric(A, B, func): + # This method for finding the equivalence is only for 2F1 type. + # We can extend it for 1F1 and 0F1 type also. + x = func.args[0] + + # making given equation in normal form + I1 = factor(cancel(A.diff(x)/2 + A**2/4 - B)) + + # computing shifted invariant(J1) of the equation + J1 = factor(cancel(x**2*I1 + S(1)/4)) + num, dem = J1.as_numer_denom() + num = powdenest(expand(num)) + dem = powdenest(expand(dem)) + # this function will compute the different powers of variable(x) in J1. + # then it will help in finding value of k. k is power of x such that we can express + # J1 = x**k * J0(x**k) then all the powers in J0 become integers. + def _power_counting(num): + _pow = {0} + for val in num: + if val.has(x): + if isinstance(val, Pow) and val.as_base_exp()[0] == x: + _pow.add(val.as_base_exp()[1]) + elif val == x: + _pow.add(val.as_base_exp()[1]) + else: + _pow.update(_power_counting(val.args)) + return _pow + + pow_num = _power_counting((num, )) + pow_dem = _power_counting((dem, )) + pow_dem.update(pow_num) + + _pow = pow_dem + k = gcd(_pow) + + # computing I0 of the given equation + I0 = powdenest(simplify(factor(((J1/k**2) - S(1)/4)/((x**k)**2))), force=True) + I0 = factor(cancel(powdenest(I0.subs(x, x**(S(1)/k)), force=True))) + + # Before this point I0, J1 might be functions of e.g. sqrt(x) but replacing + # x with x**(1/k) should result in I0 being a rational function of x or + # otherwise the hypergeometric solver cannot be used. Note that k can be a + # non-integer rational such as 2/7. + if not I0.is_rational_function(x): + return None + + num, dem = I0.as_numer_denom() + + max_num_pow = max(_power_counting((num, ))) + dem_args = dem.args + sing_point = [] + dem_pow = [] + # calculating singular point of I0. + for arg in dem_args: + if arg.has(x): + if isinstance(arg, Pow): + # (x-a)**n + dem_pow.append(arg.as_base_exp()[1]) + sing_point.append(list(roots(arg.as_base_exp()[0], x).keys())[0]) + else: + # (x-a) type + dem_pow.append(arg.as_base_exp()[1]) + sing_point.append(list(roots(arg, x).keys())[0]) + + dem_pow.sort() + # checking if equivalence is exists or not. + + if equivalence(max_num_pow, dem_pow) == "2F1": + return {'I0':I0, 'k':k, 'sing_point':sing_point, 'type':"2F1"} + else: + return None + + +def match_2nd_2F1_hypergeometric(I, k, sing_point, func): + x = func.args[0] + a = Wild("a") + b = Wild("b") + c = Wild("c") + t = Wild("t") + s = Wild("s") + r = Wild("r") + alpha = Wild("alpha") + beta = Wild("beta") + gamma = Wild("gamma") + delta = Wild("delta") + # I0 of the standard 2F1 equation. + I0 = ((a-b+1)*(a-b-1)*x**2 + 2*((1-a-b)*c + 2*a*b)*x + c*(c-2))/(4*x**2*(x-1)**2) + if sing_point != [0, 1]: + # If singular point is [0, 1] then we have standard equation. + eqs = [] + sing_eqs = [-beta/alpha, -delta/gamma, (delta-beta)/(alpha-gamma)] + # making equations for the finding the mobius transformation + for i in range(3): + if i>> from sympy import Function, Eq, pprint + >>> from sympy.abc import x, y + >>> xi, eta, h = map(Function, ['xi', 'eta', 'h']) + >>> h = h(x, y) # dy/dx = h + >>> eta = eta(x, y) + >>> xi = xi(x, y) + >>> genform = Eq(eta.diff(x) + (eta.diff(y) - xi.diff(x))*h + ... - (xi.diff(y))*h**2 - xi*(h.diff(x)) - eta*(h.diff(y)), 0) + >>> pprint(genform) + /d d \ d 2 d d d + |--(eta(x, y)) - --(xi(x, y))|*h(x, y) - eta(x, y)*--(h(x, y)) - h (x, y)*--(xi(x, y)) - xi(x, y)*--(h(x, y)) + --(eta(x, y)) = 0 + \dy dx / dy dy dx dx + + Solving the above mentioned PDE is not trivial, and can be solved only by + making intelligent assumptions for `\xi` and `\eta` (heuristics). Once an + infinitesimal is found, the attempt to find more heuristics stops. This is done to + optimise the speed of solving the differential equation. If a list of all the + infinitesimals is needed, ``hint`` should be flagged as ``all``, which gives + the complete list of infinitesimals. If the infinitesimals for a particular + heuristic needs to be found, it can be passed as a flag to ``hint``. + + Examples + ======== + + >>> from sympy import Function + >>> from sympy.solvers.ode.lie_group import infinitesimals + >>> from sympy.abc import x + >>> f = Function('f') + >>> eq = f(x).diff(x) - x**2*f(x) + >>> infinitesimals(eq) + [{eta(x, f(x)): exp(x**3/3), xi(x, f(x)): 0}] + + References + ========== + + - Solving differential equations by Symmetry Groups, + John Starrett, pp. 1 - pp. 14 + + """ + + if isinstance(eq, Equality): + eq = eq.lhs - eq.rhs + if not func: + eq, func = _preprocess(eq) + variables = func.args + if len(variables) != 1: + raise ValueError("ODE's have only one independent variable") + else: + x = variables[0] + if not order: + order = ode_order(eq, func) + if order != 1: + raise NotImplementedError("Infinitesimals for only " + "first order ODE's have been implemented") + else: + df = func.diff(x) + # Matching differential equation of the form a*df + b + a = Wild('a', exclude = [df]) + b = Wild('b', exclude = [df]) + if match: # Used by lie_group hint + h = match['h'] + y = match['y'] + else: + match = collect(expand(eq), df).match(a*df + b) + if match: + h = -simplify(match[b]/match[a]) + else: + try: + sol = solve(eq, df) + except NotImplementedError: + raise NotImplementedError("Infinitesimals for the " + "first order ODE could not be found") + else: + h = sol[0] # Find infinitesimals for one solution + y = Dummy("y") + h = h.subs(func, y) + + u = Dummy("u") + hx = h.diff(x) + hy = h.diff(y) + hinv = ((1/h).subs([(x, u), (y, x)])).subs(u, y) # Inverse ODE + match = {'h': h, 'func': func, 'hx': hx, 'hy': hy, 'y': y, 'hinv': hinv} + if hint == 'all': + xieta = [] + for heuristic in lie_heuristics: + function = globals()['lie_heuristic_' + heuristic] + inflist = function(match, comp=True) + if inflist: + xieta.extend([inf for inf in inflist if inf not in xieta]) + if xieta: + return xieta + else: + raise NotImplementedError("Infinitesimals could not be found for " + "the given ODE") + + elif hint == 'default': + for heuristic in lie_heuristics: + function = globals()['lie_heuristic_' + heuristic] + xieta = function(match, comp=False) + if xieta: + return xieta + + raise NotImplementedError("Infinitesimals could not be found for" + " the given ODE") + + elif hint not in lie_heuristics: + raise ValueError("Heuristic not recognized: " + hint) + + else: + function = globals()['lie_heuristic_' + hint] + xieta = function(match, comp=True) + if xieta: + return xieta + else: + raise ValueError("Infinitesimals could not be found using the" + " given heuristic") + + +def lie_heuristic_abaco1_simple(match, comp=False): + r""" + The first heuristic uses the following four sets of + assumptions on `\xi` and `\eta` + + .. math:: \xi = 0, \eta = f(x) + + .. math:: \xi = 0, \eta = f(y) + + .. math:: \xi = f(x), \eta = 0 + + .. math:: \xi = f(y), \eta = 0 + + The success of this heuristic is determined by algebraic factorisation. + For the first assumption `\xi = 0` and `\eta` to be a function of `x`, the PDE + + .. math:: \frac{\partial \eta}{\partial x} + (\frac{\partial \eta}{\partial y} + - \frac{\partial \xi}{\partial x})*h + - \frac{\partial \xi}{\partial y}*h^{2} + - \xi*\frac{\partial h}{\partial x} - \eta*\frac{\partial h}{\partial y} = 0 + + reduces to `f'(x) - f\frac{\partial h}{\partial y} = 0` + If `\frac{\partial h}{\partial y}` is a function of `x`, then this can usually + be integrated easily. A similar idea is applied to the other 3 assumptions as well. + + + References + ========== + + - E.S Cheb-Terrab, L.G.S Duarte and L.A,C.P da Mota, Computer Algebra + Solving of First Order ODEs Using Symmetry Methods, pp. 8 + + + """ + + xieta = [] + y = match['y'] + h = match['h'] + func = match['func'] + x = func.args[0] + hx = match['hx'] + hy = match['hy'] + xi = Function('xi')(x, func) + eta = Function('eta')(x, func) + + hysym = hy.free_symbols + if y not in hysym: + try: + fx = exp(integrate(hy, x)) + except NotImplementedError: + pass + else: + inf = {xi: S.Zero, eta: fx} + if not comp: + return [inf] + if comp and inf not in xieta: + xieta.append(inf) + + factor = hy/h + facsym = factor.free_symbols + if x not in facsym: + try: + fy = exp(integrate(factor, y)) + except NotImplementedError: + pass + else: + inf = {xi: S.Zero, eta: fy.subs(y, func)} + if not comp: + return [inf] + if comp and inf not in xieta: + xieta.append(inf) + + factor = -hx/h + facsym = factor.free_symbols + if y not in facsym: + try: + fx = exp(integrate(factor, x)) + except NotImplementedError: + pass + else: + inf = {xi: fx, eta: S.Zero} + if not comp: + return [inf] + if comp and inf not in xieta: + xieta.append(inf) + + factor = -hx/(h**2) + facsym = factor.free_symbols + if x not in facsym: + try: + fy = exp(integrate(factor, y)) + except NotImplementedError: + pass + else: + inf = {xi: fy.subs(y, func), eta: S.Zero} + if not comp: + return [inf] + if comp and inf not in xieta: + xieta.append(inf) + + if xieta: + return xieta + +def lie_heuristic_abaco1_product(match, comp=False): + r""" + The second heuristic uses the following two assumptions on `\xi` and `\eta` + + .. math:: \eta = 0, \xi = f(x)*g(y) + + .. math:: \eta = f(x)*g(y), \xi = 0 + + The first assumption of this heuristic holds good if + `\frac{1}{h^{2}}\frac{\partial^2}{\partial x \partial y}\log(h)` is + separable in `x` and `y`, then the separated factors containing `x` + is `f(x)`, and `g(y)` is obtained by + + .. math:: e^{\int f\frac{\partial}{\partial x}\left(\frac{1}{f*h}\right)\,dy} + + provided `f\frac{\partial}{\partial x}\left(\frac{1}{f*h}\right)` is a function + of `y` only. + + The second assumption holds good if `\frac{dy}{dx} = h(x, y)` is rewritten as + `\frac{dy}{dx} = \frac{1}{h(y, x)}` and the same properties of the first assumption + satisfies. After obtaining `f(x)` and `g(y)`, the coordinates are again + interchanged, to get `\eta` as `f(x)*g(y)` + + + References + ========== + - E.S. Cheb-Terrab, A.D. Roche, Symmetries and First Order + ODE Patterns, pp. 7 - pp. 8 + + """ + + xieta = [] + y = match['y'] + h = match['h'] + hinv = match['hinv'] + func = match['func'] + x = func.args[0] + xi = Function('xi')(x, func) + eta = Function('eta')(x, func) + + + inf = separatevars(((log(h).diff(y)).diff(x))/h**2, dict=True, symbols=[x, y]) + if inf and inf['coeff']: + fx = inf[x] + gy = simplify(fx*((1/(fx*h)).diff(x))) + gysyms = gy.free_symbols + if x not in gysyms: + gy = exp(integrate(gy, y)) + inf = {eta: S.Zero, xi: (fx*gy).subs(y, func)} + if not comp: + return [inf] + if comp and inf not in xieta: + xieta.append(inf) + + u1 = Dummy("u1") + inf = separatevars(((log(hinv).diff(y)).diff(x))/hinv**2, dict=True, symbols=[x, y]) + if inf and inf['coeff']: + fx = inf[x] + gy = simplify(fx*((1/(fx*hinv)).diff(x))) + gysyms = gy.free_symbols + if x not in gysyms: + gy = exp(integrate(gy, y)) + etaval = fx*gy + etaval = (etaval.subs([(x, u1), (y, x)])).subs(u1, y) + inf = {eta: etaval.subs(y, func), xi: S.Zero} + if not comp: + return [inf] + if comp and inf not in xieta: + xieta.append(inf) + + if xieta: + return xieta + +def lie_heuristic_bivariate(match, comp=False): + r""" + The third heuristic assumes the infinitesimals `\xi` and `\eta` + to be bi-variate polynomials in `x` and `y`. The assumption made here + for the logic below is that `h` is a rational function in `x` and `y` + though that may not be necessary for the infinitesimals to be + bivariate polynomials. The coefficients of the infinitesimals + are found out by substituting them in the PDE and grouping similar terms + that are polynomials and since they form a linear system, solve and check + for non trivial solutions. The degree of the assumed bivariates + are increased till a certain maximum value. + + References + ========== + - Lie Groups and Differential Equations + pp. 327 - pp. 329 + + """ + + h = match['h'] + hx = match['hx'] + hy = match['hy'] + func = match['func'] + x = func.args[0] + y = match['y'] + xi = Function('xi')(x, func) + eta = Function('eta')(x, func) + + if h.is_rational_function(): + # The maximum degree that the infinitesimals can take is + # calculated by this technique. + etax, etay, etad, xix, xiy, xid = symbols("etax etay etad xix xiy xid") + ipde = etax + (etay - xix)*h - xiy*h**2 - xid*hx - etad*hy + num, denom = cancel(ipde).as_numer_denom() + deg = Poly(num, x, y).total_degree() + deta = Function('deta')(x, y) + dxi = Function('dxi')(x, y) + ipde = (deta.diff(x) + (deta.diff(y) - dxi.diff(x))*h - (dxi.diff(y))*h**2 + - dxi*hx - deta*hy) + xieq = Symbol("xi0") + etaeq = Symbol("eta0") + + for i in range(deg + 1): + if i: + xieq += Add(*[ + Symbol("xi_" + str(power) + "_" + str(i - power))*x**power*y**(i - power) + for power in range(i + 1)]) + etaeq += Add(*[ + Symbol("eta_" + str(power) + "_" + str(i - power))*x**power*y**(i - power) + for power in range(i + 1)]) + pden, denom = (ipde.subs({dxi: xieq, deta: etaeq}).doit()).as_numer_denom() + pden = expand(pden) + + # If the individual terms are monomials, the coefficients + # are grouped + if pden.is_polynomial(x, y) and pden.is_Add: + polyy = Poly(pden, x, y).as_dict() + if polyy: + symset = xieq.free_symbols.union(etaeq.free_symbols) - {x, y} + soldict = solve(polyy.values(), *symset) + if isinstance(soldict, list): + soldict = soldict[0] + if any(soldict.values()): + xired = xieq.subs(soldict) + etared = etaeq.subs(soldict) + # Scaling is done by substituting one for the parameters + # This can be any number except zero. + dict_ = dict.fromkeys(symset, 1) + inf = {eta: etared.subs(dict_).subs(y, func), + xi: xired.subs(dict_).subs(y, func)} + return [inf] + +def lie_heuristic_chi(match, comp=False): + r""" + The aim of the fourth heuristic is to find the function `\chi(x, y)` + that satisfies the PDE `\frac{d\chi}{dx} + h\frac{d\chi}{dx} + - \frac{\partial h}{\partial y}\chi = 0`. + + This assumes `\chi` to be a bivariate polynomial in `x` and `y`. By intuition, + `h` should be a rational function in `x` and `y`. The method used here is + to substitute a general binomial for `\chi` up to a certain maximum degree + is reached. The coefficients of the polynomials, are calculated by by collecting + terms of the same order in `x` and `y`. + + After finding `\chi`, the next step is to use `\eta = \xi*h + \chi`, to + determine `\xi` and `\eta`. This can be done by dividing `\chi` by `h` + which would give `-\xi` as the quotient and `\eta` as the remainder. + + + References + ========== + - E.S Cheb-Terrab, L.G.S Duarte and L.A,C.P da Mota, Computer Algebra + Solving of First Order ODEs Using Symmetry Methods, pp. 8 + + """ + + h = match['h'] + hy = match['hy'] + func = match['func'] + x = func.args[0] + y = match['y'] + xi = Function('xi')(x, func) + eta = Function('eta')(x, func) + + if h.is_rational_function(): + schi, schix, schiy = symbols("schi, schix, schiy") + cpde = schix + h*schiy - hy*schi + num, denom = cancel(cpde).as_numer_denom() + deg = Poly(num, x, y).total_degree() + + chi = Function('chi')(x, y) + chix = chi.diff(x) + chiy = chi.diff(y) + cpde = chix + h*chiy - hy*chi + chieq = Symbol("chi") + for i in range(1, deg + 1): + chieq += Add(*[ + Symbol("chi_" + str(power) + "_" + str(i - power))*x**power*y**(i - power) + for power in range(i + 1)]) + cnum, cden = cancel(cpde.subs({chi : chieq}).doit()).as_numer_denom() + cnum = expand(cnum) + if cnum.is_polynomial(x, y) and cnum.is_Add: + cpoly = Poly(cnum, x, y).as_dict() + if cpoly: + solsyms = chieq.free_symbols - {x, y} + soldict = solve(cpoly.values(), *solsyms) + if isinstance(soldict, list): + soldict = soldict[0] + if any(soldict.values()): + chieq = chieq.subs(soldict) + dict_ = dict.fromkeys(solsyms, 1) + chieq = chieq.subs(dict_) + # After finding chi, the main aim is to find out + # eta, xi by the equation eta = xi*h + chi + # One method to set xi, would be rearranging it to + # (eta/h) - xi = (chi/h). This would mean dividing + # chi by h would give -xi as the quotient and eta + # as the remainder. Thanks to Sean Vig for suggesting + # this method. + xic, etac = div(chieq, h) + inf = {eta: etac.subs(y, func), xi: -xic.subs(y, func)} + return [inf] + +def lie_heuristic_function_sum(match, comp=False): + r""" + This heuristic uses the following two assumptions on `\xi` and `\eta` + + .. math:: \eta = 0, \xi = f(x) + g(y) + + .. math:: \eta = f(x) + g(y), \xi = 0 + + The first assumption of this heuristic holds good if + + .. math:: \frac{\partial}{\partial y}[(h\frac{\partial^{2}}{ + \partial x^{2}}(h^{-1}))^{-1}] + + is separable in `x` and `y`, + + 1. The separated factors containing `y` is `\frac{\partial g}{\partial y}`. + From this `g(y)` can be determined. + 2. The separated factors containing `x` is `f''(x)`. + 3. `h\frac{\partial^{2}}{\partial x^{2}}(h^{-1})` equals + `\frac{f''(x)}{f(x) + g(y)}`. From this `f(x)` can be determined. + + The second assumption holds good if `\frac{dy}{dx} = h(x, y)` is rewritten as + `\frac{dy}{dx} = \frac{1}{h(y, x)}` and the same properties of the first + assumption satisfies. After obtaining `f(x)` and `g(y)`, the coordinates + are again interchanged, to get `\eta` as `f(x) + g(y)`. + + For both assumptions, the constant factors are separated among `g(y)` + and `f''(x)`, such that `f''(x)` obtained from 3] is the same as that + obtained from 2]. If not possible, then this heuristic fails. + + + References + ========== + - E.S. Cheb-Terrab, A.D. Roche, Symmetries and First Order + ODE Patterns, pp. 7 - pp. 8 + + """ + + xieta = [] + h = match['h'] + func = match['func'] + hinv = match['hinv'] + x = func.args[0] + y = match['y'] + xi = Function('xi')(x, func) + eta = Function('eta')(x, func) + + for odefac in [h, hinv]: + factor = odefac*((1/odefac).diff(x, 2)) + sep = separatevars((1/factor).diff(y), dict=True, symbols=[x, y]) + if sep and sep['coeff'] and sep[x].has(x) and sep[y].has(y): + k = Dummy("k") + try: + gy = k*integrate(sep[y], y) + except NotImplementedError: + pass + else: + fdd = 1/(k*sep[x]*sep['coeff']) + fx = simplify(fdd/factor - gy) + check = simplify(fx.diff(x, 2) - fdd) + if fx: + if not check: + fx = fx.subs(k, 1) + gy = (gy/k) + else: + sol = solve(check, k) + if sol: + sol = sol[0] + fx = fx.subs(k, sol) + gy = (gy/k)*sol + else: + continue + if odefac == hinv: # Inverse ODE + fx = fx.subs(x, y) + gy = gy.subs(y, x) + etaval = factor_terms(fx + gy) + if etaval.is_Mul: + etaval = Mul(*[arg for arg in etaval.args if arg.has(x, y)]) + if odefac == hinv: # Inverse ODE + inf = {eta: etaval.subs(y, func), xi : S.Zero} + else: + inf = {xi: etaval.subs(y, func), eta : S.Zero} + if not comp: + return [inf] + else: + xieta.append(inf) + + if xieta: + return xieta + +def lie_heuristic_abaco2_similar(match, comp=False): + r""" + This heuristic uses the following two assumptions on `\xi` and `\eta` + + .. math:: \eta = g(x), \xi = f(x) + + .. math:: \eta = f(y), \xi = g(y) + + For the first assumption, + + 1. First `\frac{\frac{\partial h}{\partial y}}{\frac{\partial^{2} h}{ + \partial yy}}` is calculated. Let us say this value is A + + 2. If this is constant, then `h` is matched to the form `A(x) + B(x)e^{ + \frac{y}{C}}` then, `\frac{e^{\int \frac{A(x)}{C} \,dx}}{B(x)}` gives `f(x)` + and `A(x)*f(x)` gives `g(x)` + + 3. Otherwise `\frac{\frac{\partial A}{\partial X}}{\frac{\partial A}{ + \partial Y}} = \gamma` is calculated. If + + a] `\gamma` is a function of `x` alone + + b] `\frac{\gamma\frac{\partial h}{\partial y} - \gamma'(x) - \frac{ + \partial h}{\partial x}}{h + \gamma} = G` is a function of `x` alone. + then, `e^{\int G \,dx}` gives `f(x)` and `-\gamma*f(x)` gives `g(x)` + + The second assumption holds good if `\frac{dy}{dx} = h(x, y)` is rewritten as + `\frac{dy}{dx} = \frac{1}{h(y, x)}` and the same properties of the first assumption + satisfies. After obtaining `f(x)` and `g(x)`, the coordinates are again + interchanged, to get `\xi` as `f(x^*)` and `\eta` as `g(y^*)` + + References + ========== + - E.S. Cheb-Terrab, A.D. Roche, Symmetries and First Order + ODE Patterns, pp. 10 - pp. 12 + + """ + + h = match['h'] + hx = match['hx'] + hy = match['hy'] + func = match['func'] + hinv = match['hinv'] + x = func.args[0] + y = match['y'] + xi = Function('xi')(x, func) + eta = Function('eta')(x, func) + + factor = cancel(h.diff(y)/h.diff(y, 2)) + factorx = factor.diff(x) + factory = factor.diff(y) + if not factor.has(x) and not factor.has(y): + A = Wild('A', exclude=[y]) + B = Wild('B', exclude=[y]) + C = Wild('C', exclude=[x, y]) + match = h.match(A + B*exp(y/C)) + try: + tau = exp(-integrate(match[A]/match[C]), x)/match[B] + except NotImplementedError: + pass + else: + gx = match[A]*tau + return [{xi: tau, eta: gx}] + + else: + gamma = cancel(factorx/factory) + if not gamma.has(y): + tauint = cancel((gamma*hy - gamma.diff(x) - hx)/(h + gamma)) + if not tauint.has(y): + try: + tau = exp(integrate(tauint, x)) + except NotImplementedError: + pass + else: + gx = -tau*gamma + return [{xi: tau, eta: gx}] + + factor = cancel(hinv.diff(y)/hinv.diff(y, 2)) + factorx = factor.diff(x) + factory = factor.diff(y) + if not factor.has(x) and not factor.has(y): + A = Wild('A', exclude=[y]) + B = Wild('B', exclude=[y]) + C = Wild('C', exclude=[x, y]) + match = h.match(A + B*exp(y/C)) + try: + tau = exp(-integrate(match[A]/match[C]), x)/match[B] + except NotImplementedError: + pass + else: + gx = match[A]*tau + return [{eta: tau.subs(x, func), xi: gx.subs(x, func)}] + + else: + gamma = cancel(factorx/factory) + if not gamma.has(y): + tauint = cancel((gamma*hinv.diff(y) - gamma.diff(x) - hinv.diff(x))/( + hinv + gamma)) + if not tauint.has(y): + try: + tau = exp(integrate(tauint, x)) + except NotImplementedError: + pass + else: + gx = -tau*gamma + return [{eta: tau.subs(x, func), xi: gx.subs(x, func)}] + + +def lie_heuristic_abaco2_unique_unknown(match, comp=False): + r""" + This heuristic assumes the presence of unknown functions or known functions + with non-integer powers. + + 1. A list of all functions and non-integer powers containing x and y + 2. Loop over each element `f` in the list, find `\frac{\frac{\partial f}{\partial x}}{ + \frac{\partial f}{\partial x}} = R` + + If it is separable in `x` and `y`, let `X` be the factors containing `x`. Then + + a] Check if `\xi = X` and `\eta = -\frac{X}{R}` satisfy the PDE. If yes, then return + `\xi` and `\eta` + b] Check if `\xi = \frac{-R}{X}` and `\eta = -\frac{1}{X}` satisfy the PDE. + If yes, then return `\xi` and `\eta` + + If not, then check if + + a] :math:`\xi = -R,\eta = 1` + + b] :math:`\xi = 1, \eta = -\frac{1}{R}` + + are solutions. + + References + ========== + - E.S. Cheb-Terrab, A.D. Roche, Symmetries and First Order + ODE Patterns, pp. 10 - pp. 12 + + """ + + h = match['h'] + hx = match['hx'] + hy = match['hy'] + func = match['func'] + x = func.args[0] + y = match['y'] + xi = Function('xi')(x, func) + eta = Function('eta')(x, func) + + funclist = [] + for atom in h.atoms(Pow): + base, exp = atom.as_base_exp() + if base.has(x) and base.has(y): + if not exp.is_Integer: + funclist.append(atom) + + for function in h.atoms(AppliedUndef): + syms = function.free_symbols + if x in syms and y in syms: + funclist.append(function) + + for f in funclist: + frac = cancel(f.diff(y)/f.diff(x)) + sep = separatevars(frac, dict=True, symbols=[x, y]) + if sep and sep['coeff']: + xitry1 = sep[x] + etatry1 = -1/(sep[y]*sep['coeff']) + pde1 = etatry1.diff(y)*h - xitry1.diff(x)*h - xitry1*hx - etatry1*hy + if not simplify(pde1): + return [{xi: xitry1, eta: etatry1.subs(y, func)}] + xitry2 = 1/etatry1 + etatry2 = 1/xitry1 + pde2 = etatry2.diff(x) - (xitry2.diff(y))*h**2 - xitry2*hx - etatry2*hy + if not simplify(expand(pde2)): + return [{xi: xitry2.subs(y, func), eta: etatry2}] + + else: + etatry = -1/frac + pde = etatry.diff(x) + etatry.diff(y)*h - hx - etatry*hy + if not simplify(pde): + return [{xi: S.One, eta: etatry.subs(y, func)}] + xitry = -frac + pde = -xitry.diff(x)*h -xitry.diff(y)*h**2 - xitry*hx -hy + if not simplify(expand(pde)): + return [{xi: xitry.subs(y, func), eta: S.One}] + + +def lie_heuristic_abaco2_unique_general(match, comp=False): + r""" + This heuristic finds if infinitesimals of the form `\eta = f(x)`, `\xi = g(y)` + without making any assumptions on `h`. + + The complete sequence of steps is given in the paper mentioned below. + + References + ========== + - E.S. Cheb-Terrab, A.D. Roche, Symmetries and First Order + ODE Patterns, pp. 10 - pp. 12 + + """ + hx = match['hx'] + hy = match['hy'] + func = match['func'] + x = func.args[0] + y = match['y'] + xi = Function('xi')(x, func) + eta = Function('eta')(x, func) + + A = hx.diff(y) + B = hy.diff(y) + hy**2 + C = hx.diff(x) - hx**2 + + if not (A and B and C): + return + + Ax = A.diff(x) + Ay = A.diff(y) + Axy = Ax.diff(y) + Axx = Ax.diff(x) + Ayy = Ay.diff(y) + D = simplify(2*Axy + hx*Ay - Ax*hy + (hx*hy + 2*A)*A)*A - 3*Ax*Ay + if not D: + E1 = simplify(3*Ax**2 + ((hx**2 + 2*C)*A - 2*Axx)*A) + if E1: + E2 = simplify((2*Ayy + (2*B - hy**2)*A)*A - 3*Ay**2) + if not E2: + E3 = simplify( + E1*((28*Ax + 4*hx*A)*A**3 - E1*(hy*A + Ay)) - E1.diff(x)*8*A**4) + if not E3: + etaval = cancel((4*A**3*(Ax - hx*A) + E1*(hy*A - Ay))/(S(2)*A*E1)) + if x not in etaval: + try: + etaval = exp(integrate(etaval, y)) + except NotImplementedError: + pass + else: + xival = -4*A**3*etaval/E1 + if y not in xival: + return [{xi: xival, eta: etaval.subs(y, func)}] + + else: + E1 = simplify((2*Ayy + (2*B - hy**2)*A)*A - 3*Ay**2) + if E1: + E2 = simplify( + 4*A**3*D - D**2 + E1*((2*Axx - (hx**2 + 2*C)*A)*A - 3*Ax**2)) + if not E2: + E3 = simplify( + -(A*D)*E1.diff(y) + ((E1.diff(x) - hy*D)*A + 3*Ay*D + + (A*hx - 3*Ax)*E1)*E1) + if not E3: + etaval = cancel(((A*hx - Ax)*E1 - (Ay + A*hy)*D)/(S(2)*A*D)) + if x not in etaval: + try: + etaval = exp(integrate(etaval, y)) + except NotImplementedError: + pass + else: + xival = -E1*etaval/D + if y not in xival: + return [{xi: xival, eta: etaval.subs(y, func)}] + + +def lie_heuristic_linear(match, comp=False): + r""" + This heuristic assumes + + 1. `\xi = ax + by + c` and + 2. `\eta = fx + gy + h` + + After substituting the following assumptions in the determining PDE, it + reduces to + + .. math:: f + (g - a)h - bh^{2} - (ax + by + c)\frac{\partial h}{\partial x} + - (fx + gy + c)\frac{\partial h}{\partial y} + + Solving the reduced PDE obtained, using the method of characteristics, becomes + impractical. The method followed is grouping similar terms and solving the system + of linear equations obtained. The difference between the bivariate heuristic is that + `h` need not be a rational function in this case. + + References + ========== + - E.S. Cheb-Terrab, A.D. Roche, Symmetries and First Order + ODE Patterns, pp. 10 - pp. 12 + + """ + h = match['h'] + hx = match['hx'] + hy = match['hy'] + func = match['func'] + x = func.args[0] + y = match['y'] + xi = Function('xi')(x, func) + eta = Function('eta')(x, func) + + coeffdict = {} + symbols = numbered_symbols("c", cls=Dummy) + symlist = [next(symbols) for _ in islice(symbols, 6)] + C0, C1, C2, C3, C4, C5 = symlist + pde = C3 + (C4 - C0)*h - (C0*x + C1*y + C2)*hx - (C3*x + C4*y + C5)*hy - C1*h**2 + pde, denom = pde.as_numer_denom() + pde = powsimp(expand(pde)) + if pde.is_Add: + terms = pde.args + for term in terms: + if term.is_Mul: + rem = Mul(*[m for m in term.args if not m.has(x, y)]) + xypart = term/rem + if xypart not in coeffdict: + coeffdict[xypart] = rem + else: + coeffdict[xypart] += rem + else: + if term not in coeffdict: + coeffdict[term] = S.One + else: + coeffdict[term] += S.One + + sollist = coeffdict.values() + soldict = solve(sollist, symlist) + if soldict: + if isinstance(soldict, list): + soldict = soldict[0] + subval = soldict.values() + if any(t for t in subval): + onedict = dict(zip(symlist, [1]*6)) + xival = C0*x + C1*func + C2 + etaval = C3*x + C4*func + C5 + xival = xival.subs(soldict) + etaval = etaval.subs(soldict) + xival = xival.subs(onedict) + etaval = etaval.subs(onedict) + return [{xi: xival, eta: etaval}] + + +def _lie_group_remove(coords): + r""" + This function is strictly meant for internal use by the Lie group ODE solving + method. It replaces arbitrary functions returned by pdsolve as follows: + + 1] If coords is an arbitrary function, then its argument is returned. + 2] An arbitrary function in an Add object is replaced by zero. + 3] An arbitrary function in a Mul object is replaced by one. + 4] If there is no arbitrary function coords is returned unchanged. + + Examples + ======== + + >>> from sympy.solvers.ode.lie_group import _lie_group_remove + >>> from sympy import Function + >>> from sympy.abc import x, y + >>> F = Function("F") + >>> eq = x**2*y + >>> _lie_group_remove(eq) + x**2*y + >>> eq = F(x**2*y) + >>> _lie_group_remove(eq) + x**2*y + >>> eq = x*y**2 + F(x**3) + >>> _lie_group_remove(eq) + x*y**2 + >>> eq = (F(x**3) + y)*x**4 + >>> _lie_group_remove(eq) + x**4*y + + """ + if isinstance(coords, AppliedUndef): + return coords.args[0] + elif coords.is_Add: + subfunc = coords.atoms(AppliedUndef) + if subfunc: + for func in subfunc: + coords = coords.subs(func, 0) + return coords + elif coords.is_Pow: + base, expr = coords.as_base_exp() + base = _lie_group_remove(base) + expr = _lie_group_remove(expr) + return base**expr + elif coords.is_Mul: + mulargs = [] + coordargs = coords.args + for arg in coordargs: + if not isinstance(coords, AppliedUndef): + mulargs.append(_lie_group_remove(arg)) + return Mul(*mulargs) + return coords diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/ode/nonhomogeneous.py b/.venv/lib/python3.12/site-packages/sympy/solvers/ode/nonhomogeneous.py new file mode 100644 index 0000000000000000000000000000000000000000..ae39d55664e4850168ca7d68f65cf02171979957 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/ode/nonhomogeneous.py @@ -0,0 +1,484 @@ +r""" +This File contains helper functions for nth_linear_constant_coeff_undetermined_coefficients, +nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients, +nth_linear_constant_coeff_variation_of_parameters, +and nth_linear_euler_eq_nonhomogeneous_variation_of_parameters. + +All the functions in this file are used by more than one solvers so, instead of creating +instances in other classes for using them it is better to keep it here as separate helpers. + +""" +from collections import Counter +from sympy.core import Add, S +from sympy.core.function import diff, expand, _mexpand, expand_mul +from sympy.core.relational import Eq +from sympy.core.sorting import default_sort_key +from sympy.core.symbol import Dummy, Wild +from sympy.functions import exp, cos, cosh, im, log, re, sin, sinh, \ + atan2, conjugate +from sympy.integrals import Integral +from sympy.polys import (Poly, RootOf, rootof, roots) +from sympy.simplify import collect, simplify, separatevars, powsimp, trigsimp # type: ignore +from sympy.utilities import numbered_symbols +from sympy.solvers.solvers import solve +from sympy.matrices import wronskian +from .subscheck import sub_func_doit +from sympy.solvers.ode.ode import get_numbered_constants + + +def _test_term(coeff, func, order): + r""" + Linear Euler ODEs have the form K*x**order*diff(y(x), x, order) = F(x), + where K is independent of x and y(x), order>= 0. + So we need to check that for each term, coeff == K*x**order from + some K. We have a few cases, since coeff may have several + different types. + """ + x = func.args[0] + f = func.func + if order < 0: + raise ValueError("order should be greater than 0") + if coeff == 0: + return True + if order == 0: + if x in coeff.free_symbols: + return False + return True + if coeff.is_Mul: + if coeff.has(f(x)): + return False + return x**order in coeff.args + elif coeff.is_Pow: + return coeff.as_base_exp() == (x, order) + elif order == 1: + return x == coeff + return False + + +def _get_euler_characteristic_eq_sols(eq, func, match_obj): + r""" + Returns the solution of homogeneous part of the linear euler ODE and + the list of roots of characteristic equation. + + The parameter ``match_obj`` is a dict of order:coeff terms, where order is the order + of the derivative on each term, and coeff is the coefficient of that derivative. + + """ + x = func.args[0] + f = func.func + + # First, set up characteristic equation. + chareq, symbol = S.Zero, Dummy('x') + + for i in match_obj: + if i >= 0: + chareq += (match_obj[i]*diff(x**symbol, x, i)*x**-symbol).expand() + + chareq = Poly(chareq, symbol) + chareqroots = [rootof(chareq, k) for k in range(chareq.degree())] + collectterms = [] + + # A generator of constants + constants = list(get_numbered_constants(eq, num=chareq.degree()*2)) + constants.reverse() + + # Create a dict root: multiplicity or charroots + charroots = Counter(chareqroots) + gsol = S.Zero + ln = log + for root, multiplicity in charroots.items(): + for i in range(multiplicity): + if isinstance(root, RootOf): + gsol += (x**root) * constants.pop() + if multiplicity != 1: + raise ValueError("Value should be 1") + collectterms = [(0, root, 0)] + collectterms + elif root.is_real: + gsol += ln(x)**i*(x**root) * constants.pop() + collectterms = [(i, root, 0)] + collectterms + else: + reroot = re(root) + imroot = im(root) + gsol += ln(x)**i * (x**reroot) * ( + constants.pop() * sin(abs(imroot)*ln(x)) + + constants.pop() * cos(imroot*ln(x))) + collectterms = [(i, reroot, imroot)] + collectterms + + gsol = Eq(f(x), gsol) + + gensols = [] + # Keep track of when to use sin or cos for nonzero imroot + for i, reroot, imroot in collectterms: + if imroot == 0: + gensols.append(ln(x)**i*x**reroot) + else: + sin_form = ln(x)**i*x**reroot*sin(abs(imroot)*ln(x)) + if sin_form in gensols: + cos_form = ln(x)**i*x**reroot*cos(imroot*ln(x)) + gensols.append(cos_form) + else: + gensols.append(sin_form) + return gsol, gensols + + +def _solve_variation_of_parameters(eq, func, roots, homogen_sol, order, match_obj, simplify_flag=True): + r""" + Helper function for the method of variation of parameters and nonhomogeneous euler eq. + + See the + :py:meth:`~sympy.solvers.ode.single.NthLinearConstantCoeffVariationOfParameters` + docstring for more information on this method. + + The parameter are ``match_obj`` should be a dictionary that has the following + keys: + + ``list`` + A list of solutions to the homogeneous equation. + + ``sol`` + The general solution. + + """ + f = func.func + x = func.args[0] + r = match_obj + psol = 0 + wr = wronskian(roots, x) + + if simplify_flag: + wr = simplify(wr) # We need much better simplification for + # some ODEs. See issue 4662, for example. + # To reduce commonly occurring sin(x)**2 + cos(x)**2 to 1 + wr = trigsimp(wr, deep=True, recursive=True) + if not wr: + # The wronskian will be 0 iff the solutions are not linearly + # independent. + raise NotImplementedError("Cannot find " + str(order) + + " solutions to the homogeneous equation necessary to apply " + + "variation of parameters to " + str(eq) + " (Wronskian == 0)") + if len(roots) != order: + raise NotImplementedError("Cannot find " + str(order) + + " solutions to the homogeneous equation necessary to apply " + + "variation of parameters to " + + str(eq) + " (number of terms != order)") + negoneterm = S.NegativeOne**(order) + for i in roots: + psol += negoneterm*Integral(wronskian([sol for sol in roots if sol != i], x)*r[-1]/wr, x)*i/r[order] + negoneterm *= -1 + + if simplify_flag: + psol = simplify(psol) + psol = trigsimp(psol, deep=True) + return Eq(f(x), homogen_sol.rhs + psol) + + +def _get_const_characteristic_eq_sols(r, func, order): + r""" + Returns the roots of characteristic equation of constant coefficient + linear ODE and list of collectterms which is later on used by simplification + to use collect on solution. + + The parameter `r` is a dict of order:coeff terms, where order is the order of the + derivative on each term, and coeff is the coefficient of that derivative. + + """ + x = func.args[0] + # First, set up characteristic equation. + chareq, symbol = S.Zero, Dummy('x') + + for i in r.keys(): + if isinstance(i, str) or i < 0: + pass + else: + chareq += r[i]*symbol**i + + chareq = Poly(chareq, symbol) + # Can't just call roots because it doesn't return rootof for unsolveable + # polynomials. + chareqroots = roots(chareq, multiple=True) + if len(chareqroots) != order: + chareqroots = [rootof(chareq, k) for k in range(chareq.degree())] + + chareq_is_complex = not all(i.is_real for i in chareq.all_coeffs()) + + # Create a dict root: multiplicity or charroots + charroots = Counter(chareqroots) + # We need to keep track of terms so we can run collect() at the end. + # This is necessary for constantsimp to work properly. + collectterms = [] + gensols = [] + conjugate_roots = [] # used to prevent double-use of conjugate roots + # Loop over roots in theorder provided by roots/rootof... + for root in chareqroots: + # but don't repoeat multiple roots. + if root not in charroots: + continue + multiplicity = charroots.pop(root) + for i in range(multiplicity): + if chareq_is_complex: + gensols.append(x**i*exp(root*x)) + collectterms = [(i, root, 0)] + collectterms + continue + reroot = re(root) + imroot = im(root) + if imroot.has(atan2) and reroot.has(atan2): + # Remove this condition when re and im stop returning + # circular atan2 usages. + gensols.append(x**i*exp(root*x)) + collectterms = [(i, root, 0)] + collectterms + else: + if root in conjugate_roots: + collectterms = [(i, reroot, imroot)] + collectterms + continue + if imroot == 0: + gensols.append(x**i*exp(reroot*x)) + collectterms = [(i, reroot, 0)] + collectterms + continue + conjugate_roots.append(conjugate(root)) + gensols.append(x**i*exp(reroot*x) * sin(abs(imroot) * x)) + gensols.append(x**i*exp(reroot*x) * cos( imroot * x)) + + # This ordering is important + collectterms = [(i, reroot, imroot)] + collectterms + return gensols, collectterms + + +# Ideally these kind of simplification functions shouldn't be part of solvers. +# odesimp should be improved to handle these kind of specific simplifications. +def _get_simplified_sol(sol, func, collectterms): + r""" + Helper function which collects the solution on + collectterms. Ideally this should be handled by odesimp.It is used + only when the simplify is set to True in dsolve. + + The parameter ``collectterms`` is a list of tuple (i, reroot, imroot) where `i` is + the multiplicity of the root, reroot is real part and imroot being the imaginary part. + + """ + f = func.func + x = func.args[0] + collectterms.sort(key=default_sort_key) + collectterms.reverse() + assert len(sol) == 1 and sol[0].lhs == f(x) + sol = sol[0].rhs + sol = expand_mul(sol) + for i, reroot, imroot in collectterms: + sol = collect(sol, x**i*exp(reroot*x)*sin(abs(imroot)*x)) + sol = collect(sol, x**i*exp(reroot*x)*cos(imroot*x)) + for i, reroot, imroot in collectterms: + sol = collect(sol, x**i*exp(reroot*x)) + sol = powsimp(sol) + return Eq(f(x), sol) + + +def _undetermined_coefficients_match(expr, x, func=None, eq_homogeneous=S.Zero): + r""" + Returns a trial function match if undetermined coefficients can be applied + to ``expr``, and ``None`` otherwise. + + A trial expression can be found for an expression for use with the method + of undetermined coefficients if the expression is an + additive/multiplicative combination of constants, polynomials in `x` (the + independent variable of expr), `\sin(a x + b)`, `\cos(a x + b)`, and + `e^{a x}` terms (in other words, it has a finite number of linearly + independent derivatives). + + Note that you may still need to multiply each term returned here by + sufficient `x` to make it linearly independent with the solutions to the + homogeneous equation. + + This is intended for internal use by ``undetermined_coefficients`` hints. + + SymPy currently has no way to convert `\sin^n(x) \cos^m(y)` into a sum of + only `\sin(a x)` and `\cos(b x)` terms, so these are not implemented. So, + for example, you will need to manually convert `\sin^2(x)` into `[1 + + \cos(2 x)]/2` to properly apply the method of undetermined coefficients on + it. + + Examples + ======== + + >>> from sympy import log, exp + >>> from sympy.solvers.ode.nonhomogeneous import _undetermined_coefficients_match + >>> from sympy.abc import x + >>> _undetermined_coefficients_match(9*x*exp(x) + exp(-x), x) + {'test': True, 'trialset': {x*exp(x), exp(-x), exp(x)}} + >>> _undetermined_coefficients_match(log(x), x) + {'test': False} + + """ + a = Wild('a', exclude=[x]) + b = Wild('b', exclude=[x]) + expr = powsimp(expr, combine='exp') # exp(x)*exp(2*x + 1) => exp(3*x + 1) + retdict = {} + + def _test_term(expr, x) -> bool: + r""" + Test if ``expr`` fits the proper form for undetermined coefficients. + """ + if not expr.has(x): + return True + if expr.is_Add: + return all(_test_term(i, x) for i in expr.args) + if expr.is_Mul: + if expr.has(sin, cos): + foundtrig = False + # Make sure that there is only one trig function in the args. + # See the docstring. + for i in expr.args: + if i.has(sin, cos): + if foundtrig: + return False + else: + foundtrig = True + return all(_test_term(i, x) for i in expr.args) + if expr.is_Function: + return expr.func in (sin, cos, exp, sinh, cosh) and \ + bool(expr.args[0].match(a*x + b)) + if expr.is_Pow and expr.base.is_Symbol and expr.exp.is_Integer and \ + expr.exp >= 0: + return True + if expr.is_Pow and expr.base.is_number: + return bool(expr.exp.match(a*x + b)) + return expr.is_Symbol or bool(expr.is_number) + + def _get_trial_set(expr, x, exprs=set()): + r""" + Returns a set of trial terms for undetermined coefficients. + + The idea behind undetermined coefficients is that the terms expression + repeat themselves after a finite number of derivatives, except for the + coefficients (they are linearly dependent). So if we collect these, + we should have the terms of our trial function. + """ + def _remove_coefficient(expr, x): + r""" + Returns the expression without a coefficient. + + Similar to expr.as_independent(x)[1], except it only works + multiplicatively. + """ + term = S.One + if expr.is_Mul: + for i in expr.args: + if i.has(x): + term *= i + elif expr.has(x): + term = expr + return term + + expr = expand_mul(expr) + if expr.is_Add: + for term in expr.args: + if _remove_coefficient(term, x) in exprs: + pass + else: + exprs.add(_remove_coefficient(term, x)) + exprs = exprs.union(_get_trial_set(term, x, exprs)) + else: + term = _remove_coefficient(expr, x) + tmpset = exprs.union({term}) + oldset = set() + while tmpset != oldset: + # If you get stuck in this loop, then _test_term is probably + # broken + oldset = tmpset.copy() + expr = expr.diff(x) + term = _remove_coefficient(expr, x) + if term.is_Add: + tmpset = tmpset.union(_get_trial_set(term, x, tmpset)) + else: + tmpset.add(term) + exprs = tmpset + return exprs + + def is_homogeneous_solution(term): + r""" This function checks whether the given trialset contains any root + of homogeneous equation""" + return expand(sub_func_doit(eq_homogeneous, func, term)).is_zero + + retdict['test'] = _test_term(expr, x) + if retdict['test']: + # Try to generate a list of trial solutions that will have the + # undetermined coefficients. Note that if any of these are not linearly + # independent with any of the solutions to the homogeneous equation, + # then they will need to be multiplied by sufficient x to make them so. + # This function DOES NOT do that (it doesn't even look at the + # homogeneous equation). + temp_set = set() + for i in Add.make_args(expr): + act = _get_trial_set(i, x) + if eq_homogeneous is not S.Zero: + while any(is_homogeneous_solution(ts) for ts in act): + act = {x*ts for ts in act} + temp_set = temp_set.union(act) + + retdict['trialset'] = temp_set + return retdict + + +def _solve_undetermined_coefficients(eq, func, order, match, trialset): + r""" + Helper function for the method of undetermined coefficients. + + See the + :py:meth:`~sympy.solvers.ode.single.NthLinearConstantCoeffUndeterminedCoefficients` + docstring for more information on this method. + + The parameter ``trialset`` is the set of trial functions as returned by + ``_undetermined_coefficients_match()['trialset']``. + + The parameter ``match`` should be a dictionary that has the following + keys: + + ``list`` + A list of solutions to the homogeneous equation. + + ``sol`` + The general solution. + + """ + r = match + coeffs = numbered_symbols('a', cls=Dummy) + coefflist = [] + gensols = r['list'] + gsol = r['sol'] + f = func.func + x = func.args[0] + + if len(gensols) != order: + raise NotImplementedError("Cannot find " + str(order) + + " solutions to the homogeneous equation necessary to apply" + + " undetermined coefficients to " + str(eq) + + " (number of terms != order)") + + trialfunc = 0 + for i in trialset: + c = next(coeffs) + coefflist.append(c) + trialfunc += c*i + + eqs = sub_func_doit(eq, f(x), trialfunc) + + coeffsdict = dict(list(zip(trialset, [0]*(len(trialset) + 1)))) + + eqs = _mexpand(eqs) + + for i in Add.make_args(eqs): + s = separatevars(i, dict=True, symbols=[x]) + if coeffsdict.get(s[x]): + coeffsdict[s[x]] += s['coeff'] + else: + coeffsdict[s[x]] = s['coeff'] + + coeffvals = solve(list(coeffsdict.values()), coefflist) + + if not coeffvals: + raise NotImplementedError( + "Could not solve `%s` using the " + "method of undetermined coefficients " + "(unable to solve for coefficients)." % eq) + + psol = trialfunc.subs(coeffvals) + + return Eq(f(x), gsol.rhs + psol) diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/ode/ode.py b/.venv/lib/python3.12/site-packages/sympy/solvers/ode/ode.py new file mode 100644 index 0000000000000000000000000000000000000000..6a28b2162b38a2dd8612c14e91fd588912f6756a --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/ode/ode.py @@ -0,0 +1,3572 @@ +r""" +This module contains :py:meth:`~sympy.solvers.ode.dsolve` and different helper +functions that it uses. + +:py:meth:`~sympy.solvers.ode.dsolve` solves ordinary differential equations. +See the docstring on the various functions for their uses. Note that partial +differential equations support is in ``pde.py``. Note that hint functions +have docstrings describing their various methods, but they are intended for +internal use. Use ``dsolve(ode, func, hint=hint)`` to solve an ODE using a +specific hint. See also the docstring on +:py:meth:`~sympy.solvers.ode.dsolve`. + +**Functions in this module** + + These are the user functions in this module: + + - :py:meth:`~sympy.solvers.ode.dsolve` - Solves ODEs. + - :py:meth:`~sympy.solvers.ode.classify_ode` - Classifies ODEs into + possible hints for :py:meth:`~sympy.solvers.ode.dsolve`. + - :py:meth:`~sympy.solvers.ode.checkodesol` - Checks if an equation is the + solution to an ODE. + - :py:meth:`~sympy.solvers.ode.homogeneous_order` - Returns the + homogeneous order of an expression. + - :py:meth:`~sympy.solvers.ode.infinitesimals` - Returns the infinitesimals + of the Lie group of point transformations of an ODE, such that it is + invariant. + - :py:meth:`~sympy.solvers.ode.checkinfsol` - Checks if the given infinitesimals + are the actual infinitesimals of a first order ODE. + + These are the non-solver helper functions that are for internal use. The + user should use the various options to + :py:meth:`~sympy.solvers.ode.dsolve` to obtain the functionality provided + by these functions: + + - :py:meth:`~sympy.solvers.ode.ode.odesimp` - Does all forms of ODE + simplification. + - :py:meth:`~sympy.solvers.ode.ode.ode_sol_simplicity` - A key function for + comparing solutions by simplicity. + - :py:meth:`~sympy.solvers.ode.constantsimp` - Simplifies arbitrary + constants. + - :py:meth:`~sympy.solvers.ode.ode.constant_renumber` - Renumber arbitrary + constants. + - :py:meth:`~sympy.solvers.ode.ode._handle_Integral` - Evaluate unevaluated + Integrals. + + See also the docstrings of these functions. + +**Currently implemented solver methods** + +The following methods are implemented for solving ordinary differential +equations. See the docstrings of the various hint functions for more +information on each (run ``help(ode)``): + + - 1st order separable differential equations. + - 1st order differential equations whose coefficients or `dx` and `dy` are + functions homogeneous of the same order. + - 1st order exact differential equations. + - 1st order linear differential equations. + - 1st order Bernoulli differential equations. + - Power series solutions for first order differential equations. + - Lie Group method of solving first order differential equations. + - 2nd order Liouville differential equations. + - Power series solutions for second order differential equations + at ordinary and regular singular points. + - `n`\th order differential equation that can be solved with algebraic + rearrangement and integration. + - `n`\th order linear homogeneous differential equation with constant + coefficients. + - `n`\th order linear inhomogeneous differential equation with constant + coefficients using the method of undetermined coefficients. + - `n`\th order linear inhomogeneous differential equation with constant + coefficients using the method of variation of parameters. + +**Philosophy behind this module** + +This module is designed to make it easy to add new ODE solving methods without +having to mess with the solving code for other methods. The idea is that +there is a :py:meth:`~sympy.solvers.ode.classify_ode` function, which takes in +an ODE and tells you what hints, if any, will solve the ODE. It does this +without attempting to solve the ODE, so it is fast. Each solving method is a +hint, and it has its own function, named ``ode_``. That function takes +in the ODE and any match expression gathered by +:py:meth:`~sympy.solvers.ode.classify_ode` and returns a solved result. If +this result has any integrals in it, the hint function will return an +unevaluated :py:class:`~sympy.integrals.integrals.Integral` class. +:py:meth:`~sympy.solvers.ode.dsolve`, which is the user wrapper function +around all of this, will then call :py:meth:`~sympy.solvers.ode.ode.odesimp` on +the result, which, among other things, will attempt to solve the equation for +the dependent variable (the function we are solving for), simplify the +arbitrary constants in the expression, and evaluate any integrals, if the hint +allows it. + +**How to add new solution methods** + +If you have an ODE that you want :py:meth:`~sympy.solvers.ode.dsolve` to be +able to solve, try to avoid adding special case code here. Instead, try +finding a general method that will solve your ODE, as well as others. This +way, the :py:mod:`~sympy.solvers.ode` module will become more robust, and +unhindered by special case hacks. WolphramAlpha and Maple's +DETools[odeadvisor] function are two resources you can use to classify a +specific ODE. It is also better for a method to work with an `n`\th order ODE +instead of only with specific orders, if possible. + +To add a new method, there are a few things that you need to do. First, you +need a hint name for your method. Try to name your hint so that it is +unambiguous with all other methods, including ones that may not be implemented +yet. If your method uses integrals, also include a ``hint_Integral`` hint. +If there is more than one way to solve ODEs with your method, include a hint +for each one, as well as a ``_best`` hint. Your ``ode__best()`` +function should choose the best using min with ``ode_sol_simplicity`` as the +key argument. See +:obj:`~sympy.solvers.ode.single.HomogeneousCoeffBest`, for example. +The function that uses your method will be called ``ode_()``, so the +hint must only use characters that are allowed in a Python function name +(alphanumeric characters and the underscore '``_``' character). Include a +function for every hint, except for ``_Integral`` hints +(:py:meth:`~sympy.solvers.ode.dsolve` takes care of those automatically). +Hint names should be all lowercase, unless a word is commonly capitalized +(such as Integral or Bernoulli). If you have a hint that you do not want to +run with ``all_Integral`` that does not have an ``_Integral`` counterpart (such +as a best hint that would defeat the purpose of ``all_Integral``), you will +need to remove it manually in the :py:meth:`~sympy.solvers.ode.dsolve` code. +See also the :py:meth:`~sympy.solvers.ode.classify_ode` docstring for +guidelines on writing a hint name. + +Determine *in general* how the solutions returned by your method compare with +other methods that can potentially solve the same ODEs. Then, put your hints +in the :py:data:`~sympy.solvers.ode.allhints` tuple in the order that they +should be called. The ordering of this tuple determines which hints are +default. Note that exceptions are ok, because it is easy for the user to +choose individual hints with :py:meth:`~sympy.solvers.ode.dsolve`. In +general, ``_Integral`` variants should go at the end of the list, and +``_best`` variants should go before the various hints they apply to. For +example, the ``undetermined_coefficients`` hint comes before the +``variation_of_parameters`` hint because, even though variation of parameters +is more general than undetermined coefficients, undetermined coefficients +generally returns cleaner results for the ODEs that it can solve than +variation of parameters does, and it does not require integration, so it is +much faster. + +Next, you need to have a match expression or a function that matches the type +of the ODE, which you should put in :py:meth:`~sympy.solvers.ode.classify_ode` +(if the match function is more than just a few lines. It should match the +ODE without solving for it as much as possible, so that +:py:meth:`~sympy.solvers.ode.classify_ode` remains fast and is not hindered by +bugs in solving code. Be sure to consider corner cases. For example, if your +solution method involves dividing by something, make sure you exclude the case +where that division will be 0. + +In most cases, the matching of the ODE will also give you the various parts +that you need to solve it. You should put that in a dictionary (``.match()`` +will do this for you), and add that as ``matching_hints['hint'] = matchdict`` +in the relevant part of :py:meth:`~sympy.solvers.ode.classify_ode`. +:py:meth:`~sympy.solvers.ode.classify_ode` will then send this to +:py:meth:`~sympy.solvers.ode.dsolve`, which will send it to your function as +the ``match`` argument. Your function should be named ``ode_(eq, func, +order, match)`. If you need to send more information, put it in the ``match`` +dictionary. For example, if you had to substitute in a dummy variable in +:py:meth:`~sympy.solvers.ode.classify_ode` to match the ODE, you will need to +pass it to your function using the `match` dict to access it. You can access +the independent variable using ``func.args[0]``, and the dependent variable +(the function you are trying to solve for) as ``func.func``. If, while trying +to solve the ODE, you find that you cannot, raise ``NotImplementedError``. +:py:meth:`~sympy.solvers.ode.dsolve` will catch this error with the ``all`` +meta-hint, rather than causing the whole routine to fail. + +Add a docstring to your function that describes the method employed. Like +with anything else in SymPy, you will need to add a doctest to the docstring, +in addition to real tests in ``test_ode.py``. Try to maintain consistency +with the other hint functions' docstrings. Add your method to the list at the +top of this docstring. Also, add your method to ``ode.rst`` in the +``docs/src`` directory, so that the Sphinx docs will pull its docstring into +the main SymPy documentation. Be sure to make the Sphinx documentation by +running ``make html`` from within the doc directory to verify that the +docstring formats correctly. + +If your solution method involves integrating, use :py:obj:`~.Integral` instead of +:py:meth:`~sympy.core.expr.Expr.integrate`. This allows the user to bypass +hard/slow integration by using the ``_Integral`` variant of your hint. In +most cases, calling :py:meth:`sympy.core.basic.Basic.doit` will integrate your +solution. If this is not the case, you will need to write special code in +:py:meth:`~sympy.solvers.ode.ode._handle_Integral`. Arbitrary constants should be +symbols named ``C1``, ``C2``, and so on. All solution methods should return +an equality instance. If you need an arbitrary number of arbitrary constants, +you can use ``constants = numbered_symbols(prefix='C', cls=Symbol, start=1)``. +If it is possible to solve for the dependent function in a general way, do so. +Otherwise, do as best as you can, but do not call solve in your +``ode_()`` function. :py:meth:`~sympy.solvers.ode.ode.odesimp` will attempt +to solve the solution for you, so you do not need to do that. Lastly, if your +ODE has a common simplification that can be applied to your solutions, you can +add a special case in :py:meth:`~sympy.solvers.ode.ode.odesimp` for it. For +example, solutions returned from the ``1st_homogeneous_coeff`` hints often +have many :obj:`~sympy.functions.elementary.exponential.log` terms, so +:py:meth:`~sympy.solvers.ode.ode.odesimp` calls +:py:meth:`~sympy.simplify.simplify.logcombine` on them (it also helps to write +the arbitrary constant as ``log(C1)`` instead of ``C1`` in this case). Also +consider common ways that you can rearrange your solution to have +:py:meth:`~sympy.solvers.ode.constantsimp` take better advantage of it. It is +better to put simplification in :py:meth:`~sympy.solvers.ode.ode.odesimp` than in +your method, because it can then be turned off with the simplify flag in +:py:meth:`~sympy.solvers.ode.dsolve`. If you have any extraneous +simplification in your function, be sure to only run it using ``if +match.get('simplify', True):``, especially if it can be slow or if it can +reduce the domain of the solution. + +Finally, as with every contribution to SymPy, your method will need to be +tested. Add a test for each method in ``test_ode.py``. Follow the +conventions there, i.e., test the solver using ``dsolve(eq, f(x), +hint=your_hint)``, and also test the solution using +:py:meth:`~sympy.solvers.ode.checkodesol` (you can put these in a separate +tests and skip/XFAIL if it runs too slow/does not work). Be sure to call your +hint specifically in :py:meth:`~sympy.solvers.ode.dsolve`, that way the test +will not be broken simply by the introduction of another matching hint. If your +method works for higher order (>1) ODEs, you will need to run ``sol = +constant_renumber(sol, 'C', 1, order)`` for each solution, where ``order`` is +the order of the ODE. This is because ``constant_renumber`` renumbers the +arbitrary constants by printing order, which is platform dependent. Try to +test every corner case of your solver, including a range of orders if it is a +`n`\th order solver, but if your solver is slow, such as if it involves hard +integration, try to keep the test run time down. + +Feel free to refactor existing hints to avoid duplicating code or creating +inconsistencies. If you can show that your method exactly duplicates an +existing method, including in the simplicity and speed of obtaining the +solutions, then you can remove the old, less general method. The existing +code is tested extensively in ``test_ode.py``, so if anything is broken, one +of those tests will surely fail. + +""" + +from sympy.core import Add, S, Mul, Pow, oo +from sympy.core.containers import Tuple +from sympy.core.expr import AtomicExpr, Expr +from sympy.core.function import (Function, Derivative, AppliedUndef, diff, + expand, expand_mul, Subs) +from sympy.core.multidimensional import vectorize +from sympy.core.numbers import nan, zoo, Number +from sympy.core.relational import Equality, Eq +from sympy.core.sorting import default_sort_key, ordered +from sympy.core.symbol import Symbol, Wild, Dummy, symbols +from sympy.core.sympify import sympify +from sympy.core.traversal import preorder_traversal + +from sympy.logic.boolalg import (BooleanAtom, BooleanTrue, + BooleanFalse) +from sympy.functions import exp, log, sqrt +from sympy.functions.combinatorial.factorials import factorial +from sympy.integrals.integrals import Integral +from sympy.polys import (Poly, terms_gcd, PolynomialError, lcm) +from sympy.polys.polytools import cancel +from sympy.series import Order +from sympy.series.series import series +from sympy.simplify import (collect, logcombine, powsimp, # type: ignore + separatevars, simplify, cse) +from sympy.simplify.radsimp import collect_const +from sympy.solvers import checksol, solve + +from sympy.utilities import numbered_symbols +from sympy.utilities.iterables import uniq, sift, iterable +from sympy.solvers.deutils import _preprocess, ode_order, _desolve + + +#: This is a list of hints in the order that they should be preferred by +#: :py:meth:`~sympy.solvers.ode.classify_ode`. In general, hints earlier in the +#: list should produce simpler solutions than those later in the list (for +#: ODEs that fit both). For now, the order of this list is based on empirical +#: observations by the developers of SymPy. +#: +#: The hint used by :py:meth:`~sympy.solvers.ode.dsolve` for a specific ODE +#: can be overridden (see the docstring). +#: +#: In general, ``_Integral`` hints are grouped at the end of the list, unless +#: there is a method that returns an unevaluable integral most of the time +#: (which go near the end of the list anyway). ``default``, ``all``, +#: ``best``, and ``all_Integral`` meta-hints should not be included in this +#: list, but ``_best`` and ``_Integral`` hints should be included. +allhints = ( + "factorable", + "nth_algebraic", + "separable", + "1st_exact", + "1st_linear", + "Bernoulli", + "1st_rational_riccati", + "Riccati_special_minus2", + "1st_homogeneous_coeff_best", + "1st_homogeneous_coeff_subs_indep_div_dep", + "1st_homogeneous_coeff_subs_dep_div_indep", + "almost_linear", + "linear_coefficients", + "separable_reduced", + "1st_power_series", + "lie_group", + "nth_linear_constant_coeff_homogeneous", + "nth_linear_euler_eq_homogeneous", + "nth_linear_constant_coeff_undetermined_coefficients", + "nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients", + "nth_linear_constant_coeff_variation_of_parameters", + "nth_linear_euler_eq_nonhomogeneous_variation_of_parameters", + "Liouville", + "2nd_linear_airy", + "2nd_linear_bessel", + "2nd_hypergeometric", + "2nd_hypergeometric_Integral", + "nth_order_reducible", + "2nd_power_series_ordinary", + "2nd_power_series_regular", + "nth_algebraic_Integral", + "separable_Integral", + "1st_exact_Integral", + "1st_linear_Integral", + "Bernoulli_Integral", + "1st_homogeneous_coeff_subs_indep_div_dep_Integral", + "1st_homogeneous_coeff_subs_dep_div_indep_Integral", + "almost_linear_Integral", + "linear_coefficients_Integral", + "separable_reduced_Integral", + "nth_linear_constant_coeff_variation_of_parameters_Integral", + "nth_linear_euler_eq_nonhomogeneous_variation_of_parameters_Integral", + "Liouville_Integral", + "2nd_nonlinear_autonomous_conserved", + "2nd_nonlinear_autonomous_conserved_Integral", + ) + + + +def get_numbered_constants(eq, num=1, start=1, prefix='C'): + """ + Returns a list of constants that do not occur + in eq already. + """ + + ncs = iter_numbered_constants(eq, start, prefix) + Cs = [next(ncs) for i in range(num)] + return (Cs[0] if num == 1 else tuple(Cs)) + + +def iter_numbered_constants(eq, start=1, prefix='C'): + """ + Returns an iterator of constants that do not occur + in eq already. + """ + + if isinstance(eq, (Expr, Eq)): + eq = [eq] + elif not iterable(eq): + raise ValueError("Expected Expr or iterable but got %s" % eq) + + atom_set = set().union(*[i.free_symbols for i in eq]) + func_set = set().union(*[i.atoms(Function) for i in eq]) + if func_set: + atom_set |= {Symbol(str(f.func)) for f in func_set} + return numbered_symbols(start=start, prefix=prefix, exclude=atom_set) + + +def dsolve(eq, func=None, hint="default", simplify=True, + ics= None, xi=None, eta=None, x0=0, n=6, **kwargs): + r""" + Solves any (supported) kind of ordinary differential equation and + system of ordinary differential equations. + + For single ordinary differential equation + ========================================= + + It is classified under this when number of equation in ``eq`` is one. + **Usage** + + ``dsolve(eq, f(x), hint)`` -> Solve ordinary differential equation + ``eq`` for function ``f(x)``, using method ``hint``. + + **Details** + + ``eq`` can be any supported ordinary differential equation (see the + :py:mod:`~sympy.solvers.ode` docstring for supported methods). + This can either be an :py:class:`~sympy.core.relational.Equality`, + or an expression, which is assumed to be equal to ``0``. + + ``f(x)`` is a function of one variable whose derivatives in that + variable make up the ordinary differential equation ``eq``. In + many cases it is not necessary to provide this; it will be + autodetected (and an error raised if it could not be detected). + + ``hint`` is the solving method that you want dsolve to use. Use + ``classify_ode(eq, f(x))`` to get all of the possible hints for an + ODE. The default hint, ``default``, will use whatever hint is + returned first by :py:meth:`~sympy.solvers.ode.classify_ode`. See + Hints below for more options that you can use for hint. + + ``simplify`` enables simplification by + :py:meth:`~sympy.solvers.ode.ode.odesimp`. See its docstring for more + information. Turn this off, for example, to disable solving of + solutions for ``func`` or simplification of arbitrary constants. + It will still integrate with this hint. Note that the solution may + contain more arbitrary constants than the order of the ODE with + this option enabled. + + ``xi`` and ``eta`` are the infinitesimal functions of an ordinary + differential equation. They are the infinitesimals of the Lie group + of point transformations for which the differential equation is + invariant. The user can specify values for the infinitesimals. If + nothing is specified, ``xi`` and ``eta`` are calculated using + :py:meth:`~sympy.solvers.ode.infinitesimals` with the help of various + heuristics. + + ``ics`` is the set of initial/boundary conditions for the differential equation. + It should be given in the form of ``{f(x0): x1, f(x).diff(x).subs(x, x2): + x3}`` and so on. For power series solutions, if no initial + conditions are specified ``f(0)`` is assumed to be ``C0`` and the power + series solution is calculated about 0. + + ``x0`` is the point about which the power series solution of a differential + equation is to be evaluated. + + ``n`` gives the exponent of the dependent variable up to which the power series + solution of a differential equation is to be evaluated. + + **Hints** + + Aside from the various solving methods, there are also some meta-hints + that you can pass to :py:meth:`~sympy.solvers.ode.dsolve`: + + ``default``: + This uses whatever hint is returned first by + :py:meth:`~sympy.solvers.ode.classify_ode`. This is the + default argument to :py:meth:`~sympy.solvers.ode.dsolve`. + + ``all``: + To make :py:meth:`~sympy.solvers.ode.dsolve` apply all + relevant classification hints, use ``dsolve(ODE, func, + hint="all")``. This will return a dictionary of + ``hint:solution`` terms. If a hint causes dsolve to raise the + ``NotImplementedError``, value of that hint's key will be the + exception object raised. The dictionary will also include + some special keys: + + - ``order``: The order of the ODE. See also + :py:meth:`~sympy.solvers.deutils.ode_order` in + ``deutils.py``. + - ``best``: The simplest hint; what would be returned by + ``best`` below. + - ``best_hint``: The hint that would produce the solution + given by ``best``. If more than one hint produces the best + solution, the first one in the tuple returned by + :py:meth:`~sympy.solvers.ode.classify_ode` is chosen. + - ``default``: The solution that would be returned by default. + This is the one produced by the hint that appears first in + the tuple returned by + :py:meth:`~sympy.solvers.ode.classify_ode`. + + ``all_Integral``: + This is the same as ``all``, except if a hint also has a + corresponding ``_Integral`` hint, it only returns the + ``_Integral`` hint. This is useful if ``all`` causes + :py:meth:`~sympy.solvers.ode.dsolve` to hang because of a + difficult or impossible integral. This meta-hint will also be + much faster than ``all``, because + :py:meth:`~sympy.core.expr.Expr.integrate` is an expensive + routine. + + ``best``: + To have :py:meth:`~sympy.solvers.ode.dsolve` try all methods + and return the simplest one. This takes into account whether + the solution is solvable in the function, whether it contains + any Integral classes (i.e. unevaluatable integrals), and + which one is the shortest in size. + + See also the :py:meth:`~sympy.solvers.ode.classify_ode` docstring for + more info on hints, and the :py:mod:`~sympy.solvers.ode` docstring for + a list of all supported hints. + + **Tips** + + - You can declare the derivative of an unknown function this way: + + >>> from sympy import Function, Derivative + >>> from sympy.abc import x # x is the independent variable + >>> f = Function("f")(x) # f is a function of x + >>> # f_ will be the derivative of f with respect to x + >>> f_ = Derivative(f, x) + + - See ``test_ode.py`` for many tests, which serves also as a set of + examples for how to use :py:meth:`~sympy.solvers.ode.dsolve`. + - :py:meth:`~sympy.solvers.ode.dsolve` always returns an + :py:class:`~sympy.core.relational.Equality` class (except for the + case when the hint is ``all`` or ``all_Integral``). If possible, it + solves the solution explicitly for the function being solved for. + Otherwise, it returns an implicit solution. + - Arbitrary constants are symbols named ``C1``, ``C2``, and so on. + - Because all solutions should be mathematically equivalent, some + hints may return the exact same result for an ODE. Often, though, + two different hints will return the same solution formatted + differently. The two should be equivalent. Also note that sometimes + the values of the arbitrary constants in two different solutions may + not be the same, because one constant may have "absorbed" other + constants into it. + - Do ``help(ode.ode_)`` to get help more information on a + specific hint, where ```` is the name of a hint without + ``_Integral``. + + For system of ordinary differential equations + ============================================= + + **Usage** + ``dsolve(eq, func)`` -> Solve a system of ordinary differential + equations ``eq`` for ``func`` being list of functions including + `x(t)`, `y(t)`, `z(t)` where number of functions in the list depends + upon the number of equations provided in ``eq``. + + **Details** + + ``eq`` can be any supported system of ordinary differential equations + This can either be an :py:class:`~sympy.core.relational.Equality`, + or an expression, which is assumed to be equal to ``0``. + + ``func`` holds ``x(t)`` and ``y(t)`` being functions of one variable which + together with some of their derivatives make up the system of ordinary + differential equation ``eq``. It is not necessary to provide this; it + will be autodetected (and an error raised if it could not be detected). + + **Hints** + + The hints are formed by parameters returned by classify_sysode, combining + them give hints name used later for forming method name. + + Examples + ======== + + >>> from sympy import Function, dsolve, Eq, Derivative, sin, cos, symbols + >>> from sympy.abc import x + >>> f = Function('f') + >>> dsolve(Derivative(f(x), x, x) + 9*f(x), f(x)) + Eq(f(x), C1*sin(3*x) + C2*cos(3*x)) + + >>> eq = sin(x)*cos(f(x)) + cos(x)*sin(f(x))*f(x).diff(x) + >>> dsolve(eq, hint='1st_exact') + [Eq(f(x), -acos(C1/cos(x)) + 2*pi), Eq(f(x), acos(C1/cos(x)))] + >>> dsolve(eq, hint='almost_linear') + [Eq(f(x), -acos(C1/cos(x)) + 2*pi), Eq(f(x), acos(C1/cos(x)))] + >>> t = symbols('t') + >>> x, y = symbols('x, y', cls=Function) + >>> eq = (Eq(Derivative(x(t),t), 12*t*x(t) + 8*y(t)), Eq(Derivative(y(t),t), 21*x(t) + 7*t*y(t))) + >>> dsolve(eq) + [Eq(x(t), C1*x0(t) + C2*x0(t)*Integral(8*exp(Integral(7*t, t))*exp(Integral(12*t, t))/x0(t)**2, t)), + Eq(y(t), C1*y0(t) + C2*(y0(t)*Integral(8*exp(Integral(7*t, t))*exp(Integral(12*t, t))/x0(t)**2, t) + + exp(Integral(7*t, t))*exp(Integral(12*t, t))/x0(t)))] + >>> eq = (Eq(Derivative(x(t),t),x(t)*y(t)*sin(t)), Eq(Derivative(y(t),t),y(t)**2*sin(t))) + >>> dsolve(eq) + {Eq(x(t), -exp(C1)/(C2*exp(C1) - cos(t))), Eq(y(t), -1/(C1 - cos(t)))} + """ + if iterable(eq): + from sympy.solvers.ode.systems import dsolve_system + + # This may have to be changed in future + # when we have weakly and strongly + # connected components. This have to + # changed to show the systems that haven't + # been solved. + try: + sol = dsolve_system(eq, funcs=func, ics=ics, doit=True) + return sol[0] if len(sol) == 1 else sol + except NotImplementedError: + pass + + match = classify_sysode(eq, func) + + eq = match['eq'] + order = match['order'] + func = match['func'] + t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0] + + # keep highest order term coefficient positive + for i in range(len(eq)): + for func_ in func: + if isinstance(func_, list): + pass + else: + if eq[i].coeff(diff(func[i],t,ode_order(eq[i], func[i]))).is_negative: + eq[i] = -eq[i] + match['eq'] = eq + if len(set(order.values()))!=1: + raise ValueError("It solves only those systems of equations whose orders are equal") + match['order'] = list(order.values())[0] + def recur_len(l): + return sum(recur_len(item) if isinstance(item,list) else 1 for item in l) + if recur_len(func) != len(eq): + raise ValueError("dsolve() and classify_sysode() work with " + "number of functions being equal to number of equations") + if match['type_of_equation'] is None: + raise NotImplementedError + else: + if match['is_linear'] == True: + solvefunc = globals()['sysode_linear_%(no_of_equation)seq_order%(order)s' % match] + else: + solvefunc = globals()['sysode_nonlinear_%(no_of_equation)seq_order%(order)s' % match] + sols = solvefunc(match) + if ics: + constants = Tuple(*sols).free_symbols - Tuple(*eq).free_symbols + solved_constants = solve_ics(sols, func, constants, ics) + return [sol.subs(solved_constants) for sol in sols] + return sols + else: + given_hint = hint # hint given by the user + + # See the docstring of _desolve for more details. + hints = _desolve(eq, func=func, + hint=hint, simplify=True, xi=xi, eta=eta, type='ode', ics=ics, + x0=x0, n=n, **kwargs) + eq = hints.pop('eq', eq) + all_ = hints.pop('all', False) + if all_: + retdict = {} + failed_hints = {} + gethints = classify_ode(eq, dict=True, hint='all') + orderedhints = gethints['ordered_hints'] + for hint in hints: + try: + rv = _helper_simplify(eq, hint, hints[hint], simplify) + except NotImplementedError as detail: + failed_hints[hint] = detail + else: + retdict[hint] = rv + func = hints[hint]['func'] + + retdict['best'] = min(list(retdict.values()), key=lambda x: + ode_sol_simplicity(x, func, trysolving=not simplify)) + if given_hint == 'best': + return retdict['best'] + for i in orderedhints: + if retdict['best'] == retdict.get(i, None): + retdict['best_hint'] = i + break + retdict['default'] = gethints['default'] + retdict['order'] = gethints['order'] + retdict.update(failed_hints) + return retdict + + else: + # The key 'hint' stores the hint needed to be solved for. + hint = hints['hint'] + return _helper_simplify(eq, hint, hints, simplify, ics=ics) + + +def _helper_simplify(eq, hint, match, simplify=True, ics=None, **kwargs): + r""" + Helper function of dsolve that calls the respective + :py:mod:`~sympy.solvers.ode` functions to solve for the ordinary + differential equations. This minimizes the computation in calling + :py:meth:`~sympy.solvers.deutils._desolve` multiple times. + """ + r = match + func = r['func'] + order = r['order'] + match = r[hint] + + if isinstance(match, SingleODESolver): + solvefunc = match + else: + solvefunc = globals()['ode_' + hint.removesuffix('_Integral')] + + free = eq.free_symbols + cons = lambda s: s.free_symbols.difference(free) + + if simplify: + # odesimp() will attempt to integrate, if necessary, apply constantsimp(), + # attempt to solve for func, and apply any other hint specific + # simplifications + if isinstance(solvefunc, SingleODESolver): + sols = solvefunc.get_general_solution() + else: + sols = solvefunc(eq, func, order, match) + if iterable(sols): + rv = [] + for s in sols: + simp = odesimp(eq, s, func, hint) + if iterable(simp): + rv.extend(simp) + else: + rv.append(simp) + else: + rv = odesimp(eq, sols, func, hint) + else: + # We still want to integrate (you can disable it separately with the hint) + if isinstance(solvefunc, SingleODESolver): + exprs = solvefunc.get_general_solution(simplify=False) + else: + match['simplify'] = False # Some hints can take advantage of this option + exprs = solvefunc(eq, func, order, match) + if isinstance(exprs, list): + rv = [_handle_Integral(expr, func, hint) for expr in exprs] + else: + rv = _handle_Integral(exprs, func, hint) + + if isinstance(rv, list): + assert all(isinstance(i, Eq) for i in rv), rv # if not => internal error + if simplify: + rv = _remove_redundant_solutions(eq, rv, order, func.args[0]) + if len(rv) == 1: + rv = rv[0] + if ics and 'power_series' not in hint: + if isinstance(rv, (Expr, Eq)): + solved_constants = solve_ics([rv], [r['func']], cons(rv), ics) + rv = rv.subs(solved_constants) + else: + rv1 = [] + for s in rv: + try: + solved_constants = solve_ics([s], [r['func']], cons(s), ics) + except ValueError: + continue + rv1.append(s.subs(solved_constants)) + if len(rv1) == 1: + return rv1[0] + rv = rv1 + return rv + + +def solve_ics(sols, funcs, constants, ics): + """ + Solve for the constants given initial conditions + + ``sols`` is a list of solutions. + + ``funcs`` is a list of functions. + + ``constants`` is a list of constants. + + ``ics`` is the set of initial/boundary conditions for the differential + equation. It should be given in the form of ``{f(x0): x1, + f(x).diff(x).subs(x, x2): x3}`` and so on. + + Returns a dictionary mapping constants to values. + ``solution.subs(constants)`` will replace the constants in ``solution``. + + Example + ======= + >>> # From dsolve(f(x).diff(x) - f(x), f(x)) + >>> from sympy import symbols, Eq, exp, Function + >>> from sympy.solvers.ode.ode import solve_ics + >>> f = Function('f') + >>> x, C1 = symbols('x C1') + >>> sols = [Eq(f(x), C1*exp(x))] + >>> funcs = [f(x)] + >>> constants = [C1] + >>> ics = {f(0): 2} + >>> solved_constants = solve_ics(sols, funcs, constants, ics) + >>> solved_constants + {C1: 2} + >>> sols[0].subs(solved_constants) + Eq(f(x), 2*exp(x)) + + """ + # Assume ics are of the form f(x0): value or Subs(diff(f(x), x, n), (x, + # x0)): value (currently checked by classify_ode). To solve, replace x + # with x0, f(x0) with value, then solve for constants. For f^(n)(x0), + # differentiate the solution n times, so that f^(n)(x) appears. + x = funcs[0].args[0] + diff_sols = [] + subs_sols = [] + diff_variables = set() + for funcarg, value in ics.items(): + if isinstance(funcarg, AppliedUndef): + x0 = funcarg.args[0] + matching_func = [f for f in funcs if f.func == funcarg.func][0] + S = sols + elif isinstance(funcarg, (Subs, Derivative)): + if isinstance(funcarg, Subs): + # Make sure it stays a subs. Otherwise subs below will produce + # a different looking term. + funcarg = funcarg.doit() + if isinstance(funcarg, Subs): + deriv = funcarg.expr + x0 = funcarg.point[0] + variables = funcarg.expr.variables + matching_func = deriv + elif isinstance(funcarg, Derivative): + deriv = funcarg + x0 = funcarg.variables[0] + variables = (x,)*len(funcarg.variables) + matching_func = deriv.subs(x0, x) + for sol in sols: + if sol.has(deriv.expr.func): + diff_sols.append(Eq(sol.lhs.diff(*variables), sol.rhs.diff(*variables))) + diff_variables.add(variables) + S = diff_sols + else: + raise NotImplementedError("Unrecognized initial condition") + + for sol in S: + if sol.has(matching_func): + sol2 = sol + sol2 = sol2.subs(x, x0) + sol2 = sol2.subs(funcarg, value) + # This check is necessary because of issue #15724 + if not isinstance(sol2, BooleanAtom) or not subs_sols: + subs_sols = [s for s in subs_sols if not isinstance(s, BooleanAtom)] + subs_sols.append(sol2) + + # TODO: Use solveset here + try: + solved_constants = solve(subs_sols, constants, dict=True) + except NotImplementedError: + solved_constants = [] + + # XXX: We can't differentiate between the solution not existing because of + # invalid initial conditions, and not existing because solve is not smart + # enough. If we could use solveset, this might be improvable, but for now, + # we use NotImplementedError in this case. + if not solved_constants: + raise ValueError("Couldn't solve for initial conditions") + + if solved_constants == True: + raise ValueError("Initial conditions did not produce any solutions for constants. Perhaps they are degenerate.") + + if len(solved_constants) > 1: + raise NotImplementedError("Initial conditions produced too many solutions for constants") + + return solved_constants[0] + +def classify_ode(eq, func=None, dict=False, ics=None, *, prep=True, xi=None, eta=None, n=None, **kwargs): + r""" + Returns a tuple of possible :py:meth:`~sympy.solvers.ode.dsolve` + classifications for an ODE. + + The tuple is ordered so that first item is the classification that + :py:meth:`~sympy.solvers.ode.dsolve` uses to solve the ODE by default. In + general, classifications at the near the beginning of the list will + produce better solutions faster than those near the end, thought there are + always exceptions. To make :py:meth:`~sympy.solvers.ode.dsolve` use a + different classification, use ``dsolve(ODE, func, + hint=)``. See also the + :py:meth:`~sympy.solvers.ode.dsolve` docstring for different meta-hints + you can use. + + If ``dict`` is true, :py:meth:`~sympy.solvers.ode.classify_ode` will + return a dictionary of ``hint:match`` expression terms. This is intended + for internal use by :py:meth:`~sympy.solvers.ode.dsolve`. Note that + because dictionaries are ordered arbitrarily, this will most likely not be + in the same order as the tuple. + + You can get help on different hints by executing + ``help(ode.ode_hintname)``, where ``hintname`` is the name of the hint + without ``_Integral``. + + See :py:data:`~sympy.solvers.ode.allhints` or the + :py:mod:`~sympy.solvers.ode` docstring for a list of all supported hints + that can be returned from :py:meth:`~sympy.solvers.ode.classify_ode`. + + Notes + ===== + + These are remarks on hint names. + + ``_Integral`` + + If a classification has ``_Integral`` at the end, it will return the + expression with an unevaluated :py:class:`~.Integral` + class in it. Note that a hint may do this anyway if + :py:meth:`~sympy.core.expr.Expr.integrate` cannot do the integral, + though just using an ``_Integral`` will do so much faster. Indeed, an + ``_Integral`` hint will always be faster than its corresponding hint + without ``_Integral`` because + :py:meth:`~sympy.core.expr.Expr.integrate` is an expensive routine. + If :py:meth:`~sympy.solvers.ode.dsolve` hangs, it is probably because + :py:meth:`~sympy.core.expr.Expr.integrate` is hanging on a tough or + impossible integral. Try using an ``_Integral`` hint or + ``all_Integral`` to get it return something. + + Note that some hints do not have ``_Integral`` counterparts. This is + because :py:func:`~sympy.integrals.integrals.integrate` is not used in + solving the ODE for those method. For example, `n`\th order linear + homogeneous ODEs with constant coefficients do not require integration + to solve, so there is no + ``nth_linear_homogeneous_constant_coeff_Integrate`` hint. You can + easily evaluate any unevaluated + :py:class:`~sympy.integrals.integrals.Integral`\s in an expression by + doing ``expr.doit()``. + + Ordinals + + Some hints contain an ordinal such as ``1st_linear``. This is to help + differentiate them from other hints, as well as from other methods + that may not be implemented yet. If a hint has ``nth`` in it, such as + the ``nth_linear`` hints, this means that the method used to applies + to ODEs of any order. + + ``indep`` and ``dep`` + + Some hints contain the words ``indep`` or ``dep``. These reference + the independent variable and the dependent function, respectively. For + example, if an ODE is in terms of `f(x)`, then ``indep`` will refer to + `x` and ``dep`` will refer to `f`. + + ``subs`` + + If a hints has the word ``subs`` in it, it means that the ODE is solved + by substituting the expression given after the word ``subs`` for a + single dummy variable. This is usually in terms of ``indep`` and + ``dep`` as above. The substituted expression will be written only in + characters allowed for names of Python objects, meaning operators will + be spelled out. For example, ``indep``/``dep`` will be written as + ``indep_div_dep``. + + ``coeff`` + + The word ``coeff`` in a hint refers to the coefficients of something + in the ODE, usually of the derivative terms. See the docstring for + the individual methods for more info (``help(ode)``). This is + contrast to ``coefficients``, as in ``undetermined_coefficients``, + which refers to the common name of a method. + + ``_best`` + + Methods that have more than one fundamental way to solve will have a + hint for each sub-method and a ``_best`` meta-classification. This + will evaluate all hints and return the best, using the same + considerations as the normal ``best`` meta-hint. + + + Examples + ======== + + >>> from sympy import Function, classify_ode, Eq + >>> from sympy.abc import x + >>> f = Function('f') + >>> classify_ode(Eq(f(x).diff(x), 0), f(x)) + ('nth_algebraic', + 'separable', + '1st_exact', + '1st_linear', + 'Bernoulli', + '1st_homogeneous_coeff_best', + '1st_homogeneous_coeff_subs_indep_div_dep', + '1st_homogeneous_coeff_subs_dep_div_indep', + '1st_power_series', 'lie_group', 'nth_linear_constant_coeff_homogeneous', + 'nth_linear_euler_eq_homogeneous', + 'nth_algebraic_Integral', 'separable_Integral', '1st_exact_Integral', + '1st_linear_Integral', 'Bernoulli_Integral', + '1st_homogeneous_coeff_subs_indep_div_dep_Integral', + '1st_homogeneous_coeff_subs_dep_div_indep_Integral') + >>> classify_ode(f(x).diff(x, 2) + 3*f(x).diff(x) + 2*f(x) - 4) + ('factorable', 'nth_linear_constant_coeff_undetermined_coefficients', + 'nth_linear_constant_coeff_variation_of_parameters', + 'nth_linear_constant_coeff_variation_of_parameters_Integral') + + """ + ics = sympify(ics) + + if func and len(func.args) != 1: + raise ValueError("dsolve() and classify_ode() only " + "work with functions of one variable, not %s" % func) + + if isinstance(eq, Equality): + eq = eq.lhs - eq.rhs + + # Some methods want the unprocessed equation + eq_orig = eq + + if prep or func is None: + eq, func_ = _preprocess(eq, func) + if func is None: + func = func_ + x = func.args[0] + f = func.func + y = Dummy('y') + terms = 5 if n is None else n + + order = ode_order(eq, f(x)) + # hint:matchdict or hint:(tuple of matchdicts) + # Also will contain "default": and "order":order items. + matching_hints = {"order": order} + + df = f(x).diff(x) + a = Wild('a', exclude=[f(x)]) + d = Wild('d', exclude=[df, f(x).diff(x, 2)]) + e = Wild('e', exclude=[df]) + n = Wild('n', exclude=[x, f(x), df]) + c1 = Wild('c1', exclude=[x]) + a3 = Wild('a3', exclude=[f(x), df, f(x).diff(x, 2)]) + b3 = Wild('b3', exclude=[f(x), df, f(x).diff(x, 2)]) + c3 = Wild('c3', exclude=[f(x), df, f(x).diff(x, 2)]) + boundary = {} # Used to extract initial conditions + C1 = Symbol("C1") + + # Preprocessing to get the initial conditions out + if ics is not None: + for funcarg in ics: + # Separating derivatives + if isinstance(funcarg, (Subs, Derivative)): + # f(x).diff(x).subs(x, 0) is a Subs, but f(x).diff(x).subs(x, + # y) is a Derivative + if isinstance(funcarg, Subs): + deriv = funcarg.expr + old = funcarg.variables[0] + new = funcarg.point[0] + elif isinstance(funcarg, Derivative): + deriv = funcarg + # No information on this. Just assume it was x + old = x + new = funcarg.variables[0] + + if (isinstance(deriv, Derivative) and isinstance(deriv.args[0], + AppliedUndef) and deriv.args[0].func == f and + len(deriv.args[0].args) == 1 and old == x and not + new.has(x) and all(i == deriv.variables[0] for i in + deriv.variables) and x not in ics[funcarg].free_symbols): + + dorder = ode_order(deriv, x) + temp = 'f' + str(dorder) + boundary.update({temp: new, temp + 'val': ics[funcarg]}) + else: + raise ValueError("Invalid boundary conditions for Derivatives") + + + # Separating functions + elif isinstance(funcarg, AppliedUndef): + if (funcarg.func == f and len(funcarg.args) == 1 and + not funcarg.args[0].has(x) and x not in ics[funcarg].free_symbols): + boundary.update({'f0': funcarg.args[0], 'f0val': ics[funcarg]}) + else: + raise ValueError("Invalid boundary conditions for Function") + + else: + raise ValueError("Enter boundary conditions of the form ics={f(point): value, f(x).diff(x, order).subs(x, point): value}") + + ode = SingleODEProblem(eq_orig, func, x, prep=prep, xi=xi, eta=eta) + user_hint = kwargs.get('hint', 'default') + # Used when dsolve is called without an explicit hint. + # We exit early to return the first valid match + early_exit = (user_hint=='default') + user_hint = user_hint.removesuffix('_Integral') + user_map = solver_map + # An explicit hint has been given to dsolve + # Skip matching code for other hints + if user_hint not in ['default', 'all', 'all_Integral', 'best'] and user_hint in solver_map: + user_map = {user_hint: solver_map[user_hint]} + + for hint in user_map: + solver = user_map[hint](ode) + if solver.matches(): + matching_hints[hint] = solver + if user_map[hint].has_integral: + matching_hints[hint + "_Integral"] = solver + if dict and early_exit: + matching_hints["default"] = hint + return matching_hints + + eq = expand(eq) + # Precondition to try remove f(x) from highest order derivative + reduced_eq = None + if eq.is_Add: + deriv_coef = eq.coeff(f(x).diff(x, order)) + if deriv_coef not in (1, 0): + r = deriv_coef.match(a*f(x)**c1) + if r and r[c1]: + den = f(x)**r[c1] + reduced_eq = Add(*[arg/den for arg in eq.args]) + if not reduced_eq: + reduced_eq = eq + + if order == 1: + + # NON-REDUCED FORM OF EQUATION matches + r = collect(eq, df, exact=True).match(d + e * df) + if r: + r['d'] = d + r['e'] = e + r['y'] = y + r[d] = r[d].subs(f(x), y) + r[e] = r[e].subs(f(x), y) + + # FIRST ORDER POWER SERIES WHICH NEEDS INITIAL CONDITIONS + # TODO: Hint first order series should match only if d/e is analytic. + # For now, only d/e and (d/e).diff(arg) is checked for existence at + # at a given point. + # This is currently done internally in ode_1st_power_series. + point = boundary.get('f0', 0) + value = boundary.get('f0val', C1) + check = cancel(r[d]/r[e]) + check1 = check.subs({x: point, y: value}) + if not check1.has(oo) and not check1.has(zoo) and \ + not check1.has(nan) and not check1.has(-oo): + check2 = (check1.diff(x)).subs({x: point, y: value}) + if not check2.has(oo) and not check2.has(zoo) and \ + not check2.has(nan) and not check2.has(-oo): + rseries = r.copy() + rseries.update({'terms': terms, 'f0': point, 'f0val': value}) + matching_hints["1st_power_series"] = rseries + + elif order == 2: + # Homogeneous second order differential equation of the form + # a3*f(x).diff(x, 2) + b3*f(x).diff(x) + c3 + # It has a definite power series solution at point x0 if, b3/a3 and c3/a3 + # are analytic at x0. + deq = a3*(f(x).diff(x, 2)) + b3*df + c3*f(x) + r = collect(reduced_eq, + [f(x).diff(x, 2), f(x).diff(x), f(x)]).match(deq) + ordinary = False + if r: + if not all(r[key].is_polynomial() for key in r): + n, d = reduced_eq.as_numer_denom() + reduced_eq = expand(n) + r = collect(reduced_eq, + [f(x).diff(x, 2), f(x).diff(x), f(x)]).match(deq) + if r and r[a3] != 0: + p = cancel(r[b3]/r[a3]) # Used below + q = cancel(r[c3]/r[a3]) # Used below + point = kwargs.get('x0', 0) + check = p.subs(x, point) + if not check.has(oo, nan, zoo, -oo): + check = q.subs(x, point) + if not check.has(oo, nan, zoo, -oo): + ordinary = True + r.update({'a3': a3, 'b3': b3, 'c3': c3, 'x0': point, 'terms': terms}) + matching_hints["2nd_power_series_ordinary"] = r + + # Checking if the differential equation has a regular singular point + # at x0. It has a regular singular point at x0, if (b3/a3)*(x - x0) + # and (c3/a3)*((x - x0)**2) are analytic at x0. + if not ordinary: + p = cancel((x - point)*p) + check = p.subs(x, point) + if not check.has(oo, nan, zoo, -oo): + q = cancel(((x - point)**2)*q) + check = q.subs(x, point) + if not check.has(oo, nan, zoo, -oo): + coeff_dict = {'p': p, 'q': q, 'x0': point, 'terms': terms} + matching_hints["2nd_power_series_regular"] = coeff_dict + + + # Order keys based on allhints. + retlist = [i for i in allhints if i in matching_hints] + if dict: + # Dictionaries are ordered arbitrarily, so make note of which + # hint would come first for dsolve(). Use an ordered dict in Py 3. + matching_hints["default"] = retlist[0] if retlist else None + matching_hints["ordered_hints"] = tuple(retlist) + return matching_hints + else: + return tuple(retlist) + + +def classify_sysode(eq, funcs=None, **kwargs): + r""" + Returns a dictionary of parameter names and values that define the system + of ordinary differential equations in ``eq``. + The parameters are further used in + :py:meth:`~sympy.solvers.ode.dsolve` for solving that system. + + Some parameter names and values are: + + 'is_linear' (boolean), which tells whether the given system is linear. + Note that "linear" here refers to the operator: terms such as ``x*diff(x,t)`` are + nonlinear, whereas terms like ``sin(t)*diff(x,t)`` are still linear operators. + + 'func' (list) contains the :py:class:`~sympy.core.function.Function`s that + appear with a derivative in the ODE, i.e. those that we are trying to solve + the ODE for. + + 'order' (dict) with the maximum derivative for each element of the 'func' + parameter. + + 'func_coeff' (dict or Matrix) with the coefficient for each triple ``(equation number, + function, order)```. The coefficients are those subexpressions that do not + appear in 'func', and hence can be considered constant for purposes of ODE + solving. The value of this parameter can also be a Matrix if the system of ODEs are + linear first order of the form X' = AX where X is the vector of dependent variables. + Here, this function returns the coefficient matrix A. + + 'eq' (list) with the equations from ``eq``, sympified and transformed into + expressions (we are solving for these expressions to be zero). + + 'no_of_equations' (int) is the number of equations (same as ``len(eq)``). + + 'type_of_equation' (string) is an internal classification of the type of + ODE. + + 'is_constant' (boolean), which tells if the system of ODEs is constant coefficient + or not. This key is temporary addition for now and is in the match dict only when + the system of ODEs is linear first order constant coefficient homogeneous. So, this + key's value is True for now if it is available else it does not exist. + + 'is_homogeneous' (boolean), which tells if the system of ODEs is homogeneous. Like the + key 'is_constant', this key is a temporary addition and it is True since this key value + is available only when the system is linear first order constant coefficient homogeneous. + + References + ========== + -https://eqworld.ipmnet.ru/en/solutions/sysode/sode-toc1.htm + -A. D. Polyanin and A. V. Manzhirov, Handbook of Mathematics for Engineers and Scientists + + Examples + ======== + + >>> from sympy import Function, Eq, symbols, diff + >>> from sympy.solvers.ode.ode import classify_sysode + >>> from sympy.abc import t + >>> f, x, y = symbols('f, x, y', cls=Function) + >>> k, l, m, n = symbols('k, l, m, n', Integer=True) + >>> x1 = diff(x(t), t) ; y1 = diff(y(t), t) + >>> x2 = diff(x(t), t, t) ; y2 = diff(y(t), t, t) + >>> eq = (Eq(x1, 12*x(t) - 6*y(t)), Eq(y1, 11*x(t) + 3*y(t))) + >>> classify_sysode(eq) + {'eq': [-12*x(t) + 6*y(t) + Derivative(x(t), t), -11*x(t) - 3*y(t) + Derivative(y(t), t)], 'func': [x(t), y(t)], + 'func_coeff': {(0, x(t), 0): -12, (0, x(t), 1): 1, (0, y(t), 0): 6, (0, y(t), 1): 0, (1, x(t), 0): -11, (1, x(t), 1): 0, (1, y(t), 0): -3, (1, y(t), 1): 1}, 'is_linear': True, 'no_of_equation': 2, 'order': {x(t): 1, y(t): 1}, 'type_of_equation': None} + >>> eq = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t) + 2), Eq(diff(y(t),t), -t**2*x(t) + 5*t*y(t))) + >>> classify_sysode(eq) + {'eq': [-t**2*y(t) - 5*t*x(t) + Derivative(x(t), t) - 2, t**2*x(t) - 5*t*y(t) + Derivative(y(t), t)], + 'func': [x(t), y(t)], 'func_coeff': {(0, x(t), 0): -5*t, (0, x(t), 1): 1, (0, y(t), 0): -t**2, (0, y(t), 1): 0, + (1, x(t), 0): t**2, (1, x(t), 1): 0, (1, y(t), 0): -5*t, (1, y(t), 1): 1}, 'is_linear': True, 'no_of_equation': 2, + 'order': {x(t): 1, y(t): 1}, 'type_of_equation': None} + + """ + + # Sympify equations and convert iterables of equations into + # a list of equations + def _sympify(eq): + return list(map(sympify, eq if iterable(eq) else [eq])) + + eq, funcs = (_sympify(w) for w in [eq, funcs]) + for i, fi in enumerate(eq): + if isinstance(fi, Equality): + eq[i] = fi.lhs - fi.rhs + + t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0] + matching_hints = {"no_of_equation":i+1} + matching_hints['eq'] = eq + if i==0: + raise ValueError("classify_sysode() works for systems of ODEs. " + "For scalar ODEs, classify_ode should be used") + + # find all the functions if not given + order = {} + if funcs==[None]: + funcs = _extract_funcs(eq) + + funcs = list(set(funcs)) + if len(funcs) != len(eq): + raise ValueError("Number of functions given is not equal to the number of equations %s" % funcs) + + # This logic of list of lists in funcs to + # be replaced later. + func_dict = {} + for func in funcs: + if not order.get(func, False): + max_order = 0 + for i, eqs_ in enumerate(eq): + order_ = ode_order(eqs_,func) + if max_order < order_: + max_order = order_ + eq_no = i + if eq_no in func_dict: + func_dict[eq_no] = [func_dict[eq_no], func] + else: + func_dict[eq_no] = func + order[func] = max_order + + funcs = [func_dict[i] for i in range(len(func_dict))] + matching_hints['func'] = funcs + for func in funcs: + if isinstance(func, list): + for func_elem in func: + if len(func_elem.args) != 1: + raise ValueError("dsolve() and classify_sysode() work with " + "functions of one variable only, not %s" % func) + else: + if func and len(func.args) != 1: + raise ValueError("dsolve() and classify_sysode() work with " + "functions of one variable only, not %s" % func) + + # find the order of all equation in system of odes + matching_hints["order"] = order + + # find coefficients of terms f(t), diff(f(t),t) and higher derivatives + # and similarly for other functions g(t), diff(g(t),t) in all equations. + # Here j denotes the equation number, funcs[l] denotes the function about + # which we are talking about and k denotes the order of function funcs[l] + # whose coefficient we are calculating. + def linearity_check(eqs, j, func, is_linear_): + for k in range(order[func] + 1): + func_coef[j, func, k] = collect(eqs.expand(), [diff(func, t, k)]).coeff(diff(func, t, k)) + if is_linear_ == True: + if func_coef[j, func, k] == 0: + if k == 0: + coef = eqs.as_independent(func, as_Add=True)[1] + for xr in range(1, ode_order(eqs,func) + 1): + coef -= eqs.as_independent(diff(func, t, xr), as_Add=True)[1] + if coef != 0: + is_linear_ = False + else: + if eqs.as_independent(diff(func, t, k), as_Add=True)[1]: + is_linear_ = False + else: + for func_ in funcs: + if isinstance(func_, list): + for elem_func_ in func_: + dep = func_coef[j, func, k].as_independent(elem_func_, as_Add=True)[1] + if dep != 0: + is_linear_ = False + else: + dep = func_coef[j, func, k].as_independent(func_, as_Add=True)[1] + if dep != 0: + is_linear_ = False + return is_linear_ + + func_coef = {} + is_linear = True + for j, eqs in enumerate(eq): + for func in funcs: + if isinstance(func, list): + for func_elem in func: + is_linear = linearity_check(eqs, j, func_elem, is_linear) + else: + is_linear = linearity_check(eqs, j, func, is_linear) + matching_hints['func_coeff'] = func_coef + matching_hints['is_linear'] = is_linear + + + if len(set(order.values())) == 1: + order_eq = list(matching_hints['order'].values())[0] + if matching_hints['is_linear'] == True: + if matching_hints['no_of_equation'] == 2: + if order_eq == 1: + type_of_equation = check_linear_2eq_order1(eq, funcs, func_coef) + else: + type_of_equation = None + # If the equation does not match up with any of the + # general case solvers in systems.py and the number + # of equations is greater than 2, then NotImplementedError + # should be raised. + else: + type_of_equation = None + + else: + if matching_hints['no_of_equation'] == 2: + if order_eq == 1: + type_of_equation = check_nonlinear_2eq_order1(eq, funcs, func_coef) + else: + type_of_equation = None + elif matching_hints['no_of_equation'] == 3: + if order_eq == 1: + type_of_equation = check_nonlinear_3eq_order1(eq, funcs, func_coef) + else: + type_of_equation = None + else: + type_of_equation = None + else: + type_of_equation = None + + matching_hints['type_of_equation'] = type_of_equation + + return matching_hints + + +def check_linear_2eq_order1(eq, func, func_coef): + x = func[0].func + y = func[1].func + fc = func_coef + t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0] + r = {} + # for equations Eq(a1*diff(x(t),t), b1*x(t) + c1*y(t) + d1) + # and Eq(a2*diff(y(t),t), b2*x(t) + c2*y(t) + d2) + r['a1'] = fc[0,x(t),1] ; r['a2'] = fc[1,y(t),1] + r['b1'] = -fc[0,x(t),0]/fc[0,x(t),1] ; r['b2'] = -fc[1,x(t),0]/fc[1,y(t),1] + r['c1'] = -fc[0,y(t),0]/fc[0,x(t),1] ; r['c2'] = -fc[1,y(t),0]/fc[1,y(t),1] + forcing = [S.Zero,S.Zero] + for i in range(2): + for j in Add.make_args(eq[i]): + if not j.has(x(t), y(t)): + forcing[i] += j + if not (forcing[0].has(t) or forcing[1].has(t)): + # We can handle homogeneous case and simple constant forcings + r['d1'] = forcing[0] + r['d2'] = forcing[1] + else: + # Issue #9244: nonhomogeneous linear systems are not supported + return None + + # Conditions to check for type 6 whose equations are Eq(diff(x(t),t), f(t)*x(t) + g(t)*y(t)) and + # Eq(diff(y(t),t), a*[f(t) + a*h(t)]x(t) + a*[g(t) - h(t)]*y(t)) + p = 0 + q = 0 + p1 = cancel(r['b2']/(cancel(r['b2']/r['c2']).as_numer_denom()[0])) + p2 = cancel(r['b1']/(cancel(r['b1']/r['c1']).as_numer_denom()[0])) + for n, i in enumerate([p1, p2]): + for j in Mul.make_args(collect_const(i)): + if not j.has(t): + q = j + if q and n==0: + if ((r['b2']/j - r['b1'])/(r['c1'] - r['c2']/j)) == j: + p = 1 + elif q and n==1: + if ((r['b1']/j - r['b2'])/(r['c2'] - r['c1']/j)) == j: + p = 2 + # End of condition for type 6 + + if r['d1']!=0 or r['d2']!=0: + return None + else: + if not any(r[k].has(t) for k in 'a1 a2 b1 b2 c1 c2'.split()): + return None + else: + r['b1'] = r['b1']/r['a1'] ; r['b2'] = r['b2']/r['a2'] + r['c1'] = r['c1']/r['a1'] ; r['c2'] = r['c2']/r['a2'] + if p: + return "type6" + else: + # Equations for type 7 are Eq(diff(x(t),t), f(t)*x(t) + g(t)*y(t)) and Eq(diff(y(t),t), h(t)*x(t) + p(t)*y(t)) + return "type7" +def check_nonlinear_2eq_order1(eq, func, func_coef): + t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0] + f = Wild('f') + g = Wild('g') + u, v = symbols('u, v', cls=Dummy) + def check_type(x, y): + r1 = eq[0].match(t*diff(x(t),t) - x(t) + f) + r2 = eq[1].match(t*diff(y(t),t) - y(t) + g) + if not (r1 and r2): + r1 = eq[0].match(diff(x(t),t) - x(t)/t + f/t) + r2 = eq[1].match(diff(y(t),t) - y(t)/t + g/t) + if not (r1 and r2): + r1 = (-eq[0]).match(t*diff(x(t),t) - x(t) + f) + r2 = (-eq[1]).match(t*diff(y(t),t) - y(t) + g) + if not (r1 and r2): + r1 = (-eq[0]).match(diff(x(t),t) - x(t)/t + f/t) + r2 = (-eq[1]).match(diff(y(t),t) - y(t)/t + g/t) + if r1 and r2 and not (r1[f].subs(diff(x(t),t),u).subs(diff(y(t),t),v).has(t) \ + or r2[g].subs(diff(x(t),t),u).subs(diff(y(t),t),v).has(t)): + return 'type5' + else: + return None + for func_ in func: + if isinstance(func_, list): + x = func[0][0].func + y = func[0][1].func + eq_type = check_type(x, y) + if not eq_type: + eq_type = check_type(y, x) + return eq_type + x = func[0].func + y = func[1].func + fc = func_coef + n = Wild('n', exclude=[x(t),y(t)]) + f1 = Wild('f1', exclude=[v,t]) + f2 = Wild('f2', exclude=[v,t]) + g1 = Wild('g1', exclude=[u,t]) + g2 = Wild('g2', exclude=[u,t]) + for i in range(2): + eqs = 0 + for terms in Add.make_args(eq[i]): + eqs += terms/fc[i,func[i],1] + eq[i] = eqs + r = eq[0].match(diff(x(t),t) - x(t)**n*f) + if r: + g = (diff(y(t),t) - eq[1])/r[f] + if r and not (g.has(x(t)) or g.subs(y(t),v).has(t) or r[f].subs(x(t),u).subs(y(t),v).has(t)): + return 'type1' + r = eq[0].match(diff(x(t),t) - exp(n*x(t))*f) + if r: + g = (diff(y(t),t) - eq[1])/r[f] + if r and not (g.has(x(t)) or g.subs(y(t),v).has(t) or r[f].subs(x(t),u).subs(y(t),v).has(t)): + return 'type2' + g = Wild('g') + r1 = eq[0].match(diff(x(t),t) - f) + r2 = eq[1].match(diff(y(t),t) - g) + if r1 and r2 and not (r1[f].subs(x(t),u).subs(y(t),v).has(t) or \ + r2[g].subs(x(t),u).subs(y(t),v).has(t)): + return 'type3' + r1 = eq[0].match(diff(x(t),t) - f) + r2 = eq[1].match(diff(y(t),t) - g) + num, den = ( + (r1[f].subs(x(t),u).subs(y(t),v))/ + (r2[g].subs(x(t),u).subs(y(t),v))).as_numer_denom() + R1 = num.match(f1*g1) + R2 = den.match(f2*g2) + # phi = (r1[f].subs(x(t),u).subs(y(t),v))/num + if R1 and R2: + return 'type4' + return None + + +def check_nonlinear_2eq_order2(eq, func, func_coef): + return None + +def check_nonlinear_3eq_order1(eq, func, func_coef): + x = func[0].func + y = func[1].func + z = func[2].func + fc = func_coef + t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0] + u, v, w = symbols('u, v, w', cls=Dummy) + a = Wild('a', exclude=[x(t), y(t), z(t), t]) + b = Wild('b', exclude=[x(t), y(t), z(t), t]) + c = Wild('c', exclude=[x(t), y(t), z(t), t]) + f = Wild('f') + F1 = Wild('F1') + F2 = Wild('F2') + F3 = Wild('F3') + for i in range(3): + eqs = 0 + for terms in Add.make_args(eq[i]): + eqs += terms/fc[i,func[i],1] + eq[i] = eqs + r1 = eq[0].match(diff(x(t),t) - a*y(t)*z(t)) + r2 = eq[1].match(diff(y(t),t) - b*z(t)*x(t)) + r3 = eq[2].match(diff(z(t),t) - c*x(t)*y(t)) + if r1 and r2 and r3: + num1, den1 = r1[a].as_numer_denom() + num2, den2 = r2[b].as_numer_denom() + num3, den3 = r3[c].as_numer_denom() + if solve([num1*u-den1*(v-w), num2*v-den2*(w-u), num3*w-den3*(u-v)],[u, v]): + return 'type1' + r = eq[0].match(diff(x(t),t) - y(t)*z(t)*f) + if r: + r1 = collect_const(r[f]).match(a*f) + r2 = ((diff(y(t),t) - eq[1])/r1[f]).match(b*z(t)*x(t)) + r3 = ((diff(z(t),t) - eq[2])/r1[f]).match(c*x(t)*y(t)) + if r1 and r2 and r3: + num1, den1 = r1[a].as_numer_denom() + num2, den2 = r2[b].as_numer_denom() + num3, den3 = r3[c].as_numer_denom() + if solve([num1*u-den1*(v-w), num2*v-den2*(w-u), num3*w-den3*(u-v)],[u, v]): + return 'type2' + r = eq[0].match(diff(x(t),t) - (F2-F3)) + if r: + r1 = collect_const(r[F2]).match(c*F2) + r1.update(collect_const(r[F3]).match(b*F3)) + if r1: + if eq[1].has(r1[F2]) and not eq[1].has(r1[F3]): + r1[F2], r1[F3] = r1[F3], r1[F2] + r1[c], r1[b] = -r1[b], -r1[c] + r2 = eq[1].match(diff(y(t),t) - a*r1[F3] + r1[c]*F1) + if r2: + r3 = (eq[2] == diff(z(t),t) - r1[b]*r2[F1] + r2[a]*r1[F2]) + if r1 and r2 and r3: + return 'type3' + r = eq[0].match(diff(x(t),t) - z(t)*F2 + y(t)*F3) + if r: + r1 = collect_const(r[F2]).match(c*F2) + r1.update(collect_const(r[F3]).match(b*F3)) + if r1: + if eq[1].has(r1[F2]) and not eq[1].has(r1[F3]): + r1[F2], r1[F3] = r1[F3], r1[F2] + r1[c], r1[b] = -r1[b], -r1[c] + r2 = (diff(y(t),t) - eq[1]).match(a*x(t)*r1[F3] - r1[c]*z(t)*F1) + if r2: + r3 = (diff(z(t),t) - eq[2] == r1[b]*y(t)*r2[F1] - r2[a]*x(t)*r1[F2]) + if r1 and r2 and r3: + return 'type4' + r = (diff(x(t),t) - eq[0]).match(x(t)*(F2 - F3)) + if r: + r1 = collect_const(r[F2]).match(c*F2) + r1.update(collect_const(r[F3]).match(b*F3)) + if r1: + if eq[1].has(r1[F2]) and not eq[1].has(r1[F3]): + r1[F2], r1[F3] = r1[F3], r1[F2] + r1[c], r1[b] = -r1[b], -r1[c] + r2 = (diff(y(t),t) - eq[1]).match(y(t)*(a*r1[F3] - r1[c]*F1)) + if r2: + r3 = (diff(z(t),t) - eq[2] == z(t)*(r1[b]*r2[F1] - r2[a]*r1[F2])) + if r1 and r2 and r3: + return 'type5' + return None + + +def check_nonlinear_3eq_order2(eq, func, func_coef): + return None + + +@vectorize(0) +def odesimp(ode, eq, func, hint): + r""" + Simplifies solutions of ODEs, including trying to solve for ``func`` and + running :py:meth:`~sympy.solvers.ode.constantsimp`. + + It may use knowledge of the type of solution that the hint returns to + apply additional simplifications. + + It also attempts to integrate any :py:class:`~sympy.integrals.integrals.Integral`\s + in the expression, if the hint is not an ``_Integral`` hint. + + This function should have no effect on expressions returned by + :py:meth:`~sympy.solvers.ode.dsolve`, as + :py:meth:`~sympy.solvers.ode.dsolve` already calls + :py:meth:`~sympy.solvers.ode.ode.odesimp`, but the individual hint functions + do not call :py:meth:`~sympy.solvers.ode.ode.odesimp` (because the + :py:meth:`~sympy.solvers.ode.dsolve` wrapper does). Therefore, this + function is designed for mainly internal use. + + Examples + ======== + + >>> from sympy import sin, symbols, dsolve, pprint, Function + >>> from sympy.solvers.ode.ode import odesimp + >>> x, u2, C1= symbols('x,u2,C1') + >>> f = Function('f') + + >>> eq = dsolve(x*f(x).diff(x) - f(x) - x*sin(f(x)/x), f(x), + ... hint='1st_homogeneous_coeff_subs_indep_div_dep_Integral', + ... simplify=False) + >>> pprint(eq, wrap_line=False) + x + ---- + f(x) + / + | + | / 1 \ + | -|u1 + -------| + | | /1 \| + | | sin|--|| + | \ \u1// + log(f(x)) = log(C1) + | ---------------- d(u1) + | 2 + | u1 + | + / + + >>> pprint(odesimp(eq, f(x), 1, {C1}, + ... hint='1st_homogeneous_coeff_subs_indep_div_dep' + ... )) #doctest: +SKIP + x + --------- = C1 + /f(x)\ + tan|----| + \2*x / + + """ + x = func.args[0] + f = func.func + C1 = get_numbered_constants(eq, num=1) + constants = eq.free_symbols - ode.free_symbols + + # First, integrate if the hint allows it. + eq = _handle_Integral(eq, func, hint) + if hint.startswith("nth_linear_euler_eq_nonhomogeneous"): + eq = simplify(eq) + if not isinstance(eq, Equality): + raise TypeError("eq should be an instance of Equality") + + # allow simplifications under assumption that symbols are nonzero + eq = eq.xreplace((_:={i: Dummy(nonzero=True) for i in constants})).xreplace({_[i]: i for i in _}) + + # Second, clean up the arbitrary constants. + # Right now, nth linear hints can put as many as 2*order constants in an + # expression. If that number grows with another hint, the third argument + # here should be raised accordingly, or constantsimp() rewritten to handle + # an arbitrary number of constants. + eq = constantsimp(eq, constants) + + # Lastly, now that we have cleaned up the expression, try solving for func. + # When CRootOf is implemented in solve(), we will want to return a CRootOf + # every time instead of an Equality. + + # Get the f(x) on the left if possible. + if eq.rhs == func and not eq.lhs.has(func): + eq = [Eq(eq.rhs, eq.lhs)] + + # make sure we are working with lists of solutions in simplified form. + if eq.lhs == func and not eq.rhs.has(func): + # The solution is already solved + eq = [eq] + + else: + # The solution is not solved, so try to solve it + try: + floats = any(i.is_Float for i in eq.atoms(Number)) + eqsol = solve(eq, func, force=True, rational=False if floats else None) + if not eqsol: + raise NotImplementedError + except (NotImplementedError, PolynomialError): + eq = [eq] + else: + def _expand(expr): + numer, denom = expr.as_numer_denom() + + if denom.is_Add: + return expr + else: + return powsimp(expr.expand(), combine='exp', deep=True) + + # XXX: the rest of odesimp() expects each ``t`` to be in a + # specific normal form: rational expression with numerator + # expanded, but with combined exponential functions (at + # least in this setup all tests pass). + eq = [Eq(f(x), _expand(t)) for t in eqsol] + + # special simplification of the lhs. + if hint.startswith("1st_homogeneous_coeff"): + for j, eqi in enumerate(eq): + newi = logcombine(eqi, force=True) + if isinstance(newi.lhs, log) and newi.rhs == 0: + newi = Eq(newi.lhs.args[0]/C1, C1) + eq[j] = newi + + # We cleaned up the constants before solving to help the solve engine with + # a simpler expression, but the solved expression could have introduced + # things like -C1, so rerun constantsimp() one last time before returning. + for i, eqi in enumerate(eq): + eq[i] = constantsimp(eqi, constants) + eq[i] = constant_renumber(eq[i], ode.free_symbols) + + # If there is only 1 solution, return it; + # otherwise return the list of solutions. + if len(eq) == 1: + eq = eq[0] + return eq + + +def ode_sol_simplicity(sol, func, trysolving=True): + r""" + Returns an extended integer representing how simple a solution to an ODE + is. + + The following things are considered, in order from most simple to least: + + - ``sol`` is solved for ``func``. + - ``sol`` is not solved for ``func``, but can be if passed to solve (e.g., + a solution returned by ``dsolve(ode, func, simplify=False``). + - If ``sol`` is not solved for ``func``, then base the result on the + length of ``sol``, as computed by ``len(str(sol))``. + - If ``sol`` has any unevaluated :py:class:`~sympy.integrals.integrals.Integral`\s, + this will automatically be considered less simple than any of the above. + + This function returns an integer such that if solution A is simpler than + solution B by above metric, then ``ode_sol_simplicity(sola, func) < + ode_sol_simplicity(solb, func)``. + + Currently, the following are the numbers returned, but if the heuristic is + ever improved, this may change. Only the ordering is guaranteed. + + +----------------------------------------------+-------------------+ + | Simplicity | Return | + +==============================================+===================+ + | ``sol`` solved for ``func`` | ``-2`` | + +----------------------------------------------+-------------------+ + | ``sol`` not solved for ``func`` but can be | ``-1`` | + +----------------------------------------------+-------------------+ + | ``sol`` is not solved nor solvable for | ``len(str(sol))`` | + | ``func`` | | + +----------------------------------------------+-------------------+ + | ``sol`` contains an | ``oo`` | + | :obj:`~sympy.integrals.integrals.Integral` | | + +----------------------------------------------+-------------------+ + + ``oo`` here means the SymPy infinity, which should compare greater than + any integer. + + If you already know :py:meth:`~sympy.solvers.solvers.solve` cannot solve + ``sol``, you can use ``trysolving=False`` to skip that step, which is the + only potentially slow step. For example, + :py:meth:`~sympy.solvers.ode.dsolve` with the ``simplify=False`` flag + should do this. + + If ``sol`` is a list of solutions, if the worst solution in the list + returns ``oo`` it returns that, otherwise it returns ``len(str(sol))``, + that is, the length of the string representation of the whole list. + + Examples + ======== + + This function is designed to be passed to ``min`` as the key argument, + such as ``min(listofsolutions, key=lambda i: ode_sol_simplicity(i, + f(x)))``. + + >>> from sympy import symbols, Function, Eq, tan, Integral + >>> from sympy.solvers.ode.ode import ode_sol_simplicity + >>> x, C1, C2 = symbols('x, C1, C2') + >>> f = Function('f') + + >>> ode_sol_simplicity(Eq(f(x), C1*x**2), f(x)) + -2 + >>> ode_sol_simplicity(Eq(x**2 + f(x), C1), f(x)) + -1 + >>> ode_sol_simplicity(Eq(f(x), C1*Integral(2*x, x)), f(x)) + oo + >>> eq1 = Eq(f(x)/tan(f(x)/(2*x)), C1) + >>> eq2 = Eq(f(x)/tan(f(x)/(2*x) + f(x)), C2) + >>> [ode_sol_simplicity(eq, f(x)) for eq in [eq1, eq2]] + [28, 35] + >>> min([eq1, eq2], key=lambda i: ode_sol_simplicity(i, f(x))) + Eq(f(x)/tan(f(x)/(2*x)), C1) + + """ + # TODO: if two solutions are solved for f(x), we still want to be + # able to get the simpler of the two + + # See the docstring for the coercion rules. We check easier (faster) + # things here first, to save time. + + if iterable(sol): + # See if there are Integrals + for i in sol: + if ode_sol_simplicity(i, func, trysolving=trysolving) == oo: + return oo + + return len(str(sol)) + + if sol.has(Integral): + return oo + + # Next, try to solve for func. This code will change slightly when CRootOf + # is implemented in solve(). Probably a CRootOf solution should fall + # somewhere between a normal solution and an unsolvable expression. + + # First, see if they are already solved + if sol.lhs == func and not sol.rhs.has(func) or \ + sol.rhs == func and not sol.lhs.has(func): + return -2 + # We are not so lucky, try solving manually + if trysolving: + try: + sols = solve(sol, func) + if not sols: + raise NotImplementedError + except NotImplementedError: + pass + else: + return -1 + + # Finally, a naive computation based on the length of the string version + # of the expression. This may favor combined fractions because they + # will not have duplicate denominators, and may slightly favor expressions + # with fewer additions and subtractions, as those are separated by spaces + # by the printer. + + # Additional ideas for simplicity heuristics are welcome, like maybe + # checking if a equation has a larger domain, or if constantsimp has + # introduced arbitrary constants numbered higher than the order of a + # given ODE that sol is a solution of. + return len(str(sol)) + + +def _extract_funcs(eqs): + funcs = [] + for eq in eqs: + derivs = [node for node in preorder_traversal(eq) if isinstance(node, Derivative)] + func = [] + for d in derivs: + func += list(d.atoms(AppliedUndef)) + for func_ in func: + funcs.append(func_) + funcs = list(uniq(funcs)) + + return funcs + + +def _get_constant_subexpressions(expr, Cs): + Cs = set(Cs) + Ces = [] + def _recursive_walk(expr): + expr_syms = expr.free_symbols + if expr_syms and expr_syms.issubset(Cs): + Ces.append(expr) + else: + if expr.func == exp: + expr = expr.expand(mul=True) + if expr.func in (Add, Mul): + d = sift(expr.args, lambda i : i.free_symbols.issubset(Cs)) + if len(d[True]) > 1: + x = expr.func(*d[True]) + if not x.is_number: + Ces.append(x) + elif isinstance(expr, Integral): + if expr.free_symbols.issubset(Cs) and \ + all(len(x) == 3 for x in expr.limits): + Ces.append(expr) + for i in expr.args: + _recursive_walk(i) + return + _recursive_walk(expr) + return Ces + +def __remove_linear_redundancies(expr, Cs): + cnts = {i: expr.count(i) for i in Cs} + Cs = [i for i in Cs if cnts[i] > 0] + + def _linear(expr): + if isinstance(expr, Add): + xs = [i for i in Cs if expr.count(i)==cnts[i] \ + and 0 == expr.diff(i, 2)] + d = {} + for x in xs: + y = expr.diff(x) + if y not in d: + d[y]=[] + d[y].append(x) + for y in d: + if len(d[y]) > 1: + d[y].sort(key=str) + for x in d[y][1:]: + expr = expr.subs(x, 0) + return expr + + def _recursive_walk(expr): + if len(expr.args) != 0: + expr = expr.func(*[_recursive_walk(i) for i in expr.args]) + expr = _linear(expr) + return expr + + if isinstance(expr, Equality): + lhs, rhs = [_recursive_walk(i) for i in expr.args] + f = lambda i: isinstance(i, Number) or i in Cs + if isinstance(lhs, Symbol) and lhs in Cs: + rhs, lhs = lhs, rhs + if lhs.func in (Add, Symbol) and rhs.func in (Add, Symbol): + dlhs = sift([lhs] if isinstance(lhs, AtomicExpr) else lhs.args, f) + drhs = sift([rhs] if isinstance(rhs, AtomicExpr) else rhs.args, f) + for i in [True, False]: + for hs in [dlhs, drhs]: + if i not in hs: + hs[i] = [0] + # this calculation can be simplified + lhs = Add(*dlhs[False]) - Add(*drhs[False]) + rhs = Add(*drhs[True]) - Add(*dlhs[True]) + elif lhs.func in (Mul, Symbol) and rhs.func in (Mul, Symbol): + dlhs = sift([lhs] if isinstance(lhs, AtomicExpr) else lhs.args, f) + if True in dlhs: + if False not in dlhs: + dlhs[False] = [1] + lhs = Mul(*dlhs[False]) + rhs = rhs/Mul(*dlhs[True]) + return Eq(lhs, rhs) + else: + return _recursive_walk(expr) + +@vectorize(0) +def constantsimp(expr, constants): + r""" + Simplifies an expression with arbitrary constants in it. + + This function is written specifically to work with + :py:meth:`~sympy.solvers.ode.dsolve`, and is not intended for general use. + + Simplification is done by "absorbing" the arbitrary constants into other + arbitrary constants, numbers, and symbols that they are not independent + of. + + The symbols must all have the same name with numbers after it, for + example, ``C1``, ``C2``, ``C3``. The ``symbolname`` here would be + '``C``', the ``startnumber`` would be 1, and the ``endnumber`` would be 3. + If the arbitrary constants are independent of the variable ``x``, then the + independent symbol would be ``x``. There is no need to specify the + dependent function, such as ``f(x)``, because it already has the + independent symbol, ``x``, in it. + + Because terms are "absorbed" into arbitrary constants and because + constants are renumbered after simplifying, the arbitrary constants in + expr are not necessarily equal to the ones of the same name in the + returned result. + + If two or more arbitrary constants are added, multiplied, or raised to the + power of each other, they are first absorbed together into a single + arbitrary constant. Then the new constant is combined into other terms if + necessary. + + Absorption of constants is done with limited assistance: + + 1. terms of :py:class:`~sympy.core.add.Add`\s are collected to try join + constants so `e^x (C_1 \cos(x) + C_2 \cos(x))` will simplify to `e^x + C_1 \cos(x)`; + + 2. powers with exponents that are :py:class:`~sympy.core.add.Add`\s are + expanded so `e^{C_1 + x}` will be simplified to `C_1 e^x`. + + Use :py:meth:`~sympy.solvers.ode.ode.constant_renumber` to renumber constants + after simplification or else arbitrary numbers on constants may appear, + e.g. `C_1 + C_3 x`. + + In rare cases, a single constant can be "simplified" into two constants. + Every differential equation solution should have as many arbitrary + constants as the order of the differential equation. The result here will + be technically correct, but it may, for example, have `C_1` and `C_2` in + an expression, when `C_1` is actually equal to `C_2`. Use your discretion + in such situations, and also take advantage of the ability to use hints in + :py:meth:`~sympy.solvers.ode.dsolve`. + + Examples + ======== + + >>> from sympy import symbols + >>> from sympy.solvers.ode.ode import constantsimp + >>> C1, C2, C3, x, y = symbols('C1, C2, C3, x, y') + >>> constantsimp(2*C1*x, {C1, C2, C3}) + C1*x + >>> constantsimp(C1 + 2 + x, {C1, C2, C3}) + C1 + x + >>> constantsimp(C1*C2 + 2 + C2 + C3*x, {C1, C2, C3}) + C1 + C3*x + + """ + # This function works recursively. The idea is that, for Mul, + # Add, Pow, and Function, if the class has a constant in it, then + # we can simplify it, which we do by recursing down and + # simplifying up. Otherwise, we can skip that part of the + # expression. + + Cs = constants + + orig_expr = expr + + constant_subexprs = _get_constant_subexpressions(expr, Cs) + for xe in constant_subexprs: + xes = list(xe.free_symbols) + if not xes: + continue + if all(expr.count(c) == xe.count(c) for c in xes): + xes.sort(key=str) + expr = expr.subs(xe, xes[0]) + + # try to perform common sub-expression elimination of constant terms + try: + commons, rexpr = cse(expr) + commons.reverse() + rexpr = rexpr[0] + for s in commons: + cs = list(s[1].atoms(Symbol)) + if len(cs) == 1 and cs[0] in Cs and \ + cs[0] not in rexpr.atoms(Symbol) and \ + not any(cs[0] in ex for ex in commons if ex != s): + rexpr = rexpr.subs(s[0], cs[0]) + else: + rexpr = rexpr.subs(*s) + expr = rexpr + except IndexError: + pass + expr = __remove_linear_redundancies(expr, Cs) + + def _conditional_term_factoring(expr): + new_expr = terms_gcd(expr, clear=False, deep=True, expand=False) + + # we do not want to factor exponentials, so handle this separately + if new_expr.is_Mul: + infac = False + asfac = False + for m in new_expr.args: + if isinstance(m, exp): + asfac = True + elif m.is_Add: + infac = any(isinstance(fi, exp) for t in m.args + for fi in Mul.make_args(t)) + if asfac and infac: + new_expr = expr + break + return new_expr + + expr = _conditional_term_factoring(expr) + + # call recursively if more simplification is possible + if orig_expr != expr: + return constantsimp(expr, Cs) + return expr + + +def constant_renumber(expr, variables=None, newconstants=None): + r""" + Renumber arbitrary constants in ``expr`` to use the symbol names as given + in ``newconstants``. In the process, this reorders expression terms in a + standard way. + + If ``newconstants`` is not provided then the new constant names will be + ``C1``, ``C2`` etc. Otherwise ``newconstants`` should be an iterable + giving the new symbols to use for the constants in order. + + The ``variables`` argument is a list of non-constant symbols. All other + free symbols found in ``expr`` are assumed to be constants and will be + renumbered. If ``variables`` is not given then any numbered symbol + beginning with ``C`` (e.g. ``C1``) is assumed to be a constant. + + Symbols are renumbered based on ``.sort_key()``, so they should be + numbered roughly in the order that they appear in the final, printed + expression. Note that this ordering is based in part on hashes, so it can + produce different results on different machines. + + The structure of this function is very similar to that of + :py:meth:`~sympy.solvers.ode.constantsimp`. + + Examples + ======== + + >>> from sympy import symbols + >>> from sympy.solvers.ode.ode import constant_renumber + >>> x, C1, C2, C3 = symbols('x,C1:4') + >>> expr = C3 + C2*x + C1*x**2 + >>> expr + C1*x**2 + C2*x + C3 + >>> constant_renumber(expr) + C1 + C2*x + C3*x**2 + + The ``variables`` argument specifies which are constants so that the + other symbols will not be renumbered: + + >>> constant_renumber(expr, [C1, x]) + C1*x**2 + C2 + C3*x + + The ``newconstants`` argument is used to specify what symbols to use when + replacing the constants: + + >>> constant_renumber(expr, [x], newconstants=symbols('E1:4')) + E1 + E2*x + E3*x**2 + + """ + + # System of expressions + if isinstance(expr, (set, list, tuple)): + return type(expr)(constant_renumber(Tuple(*expr), + variables=variables, newconstants=newconstants)) + + # Symbols in solution but not ODE are constants + if variables is not None: + variables = set(variables) + free_symbols = expr.free_symbols + constantsymbols = list(free_symbols - variables) + # Any Cn is a constant... + else: + variables = set() + isconstant = lambda s: s.startswith('C') and s[1:].isdigit() + constantsymbols = [sym for sym in expr.free_symbols if isconstant(sym.name)] + + # Find new constants checking that they aren't already in the ODE + if newconstants is None: + iter_constants = numbered_symbols(start=1, prefix='C', exclude=variables) + else: + iter_constants = (sym for sym in newconstants if sym not in variables) + + constants_found = [] + + # make a mapping to send all constantsymbols to S.One and use + # that to make sure that term ordering is not dependent on + # the indexed value of C + C_1 = [(ci, S.One) for ci in constantsymbols] + sort_key=lambda arg: default_sort_key(arg.subs(C_1)) + + def _constant_renumber(expr): + r""" + We need to have an internal recursive function + """ + + # For system of expressions + if isinstance(expr, Tuple): + renumbered = [_constant_renumber(e) for e in expr] + return Tuple(*renumbered) + + if isinstance(expr, Equality): + return Eq( + _constant_renumber(expr.lhs), + _constant_renumber(expr.rhs)) + + if type(expr) not in (Mul, Add, Pow) and not expr.is_Function and \ + not expr.has(*constantsymbols): + # Base case, as above. Hope there aren't constants inside + # of some other class, because they won't be renumbered. + return expr + elif expr.is_Piecewise: + return expr + elif expr in constantsymbols: + if expr not in constants_found: + constants_found.append(expr) + return expr + elif expr.is_Function or expr.is_Pow: + return expr.func( + *[_constant_renumber(x) for x in expr.args]) + else: + sortedargs = list(expr.args) + sortedargs.sort(key=sort_key) + return expr.func(*[_constant_renumber(x) for x in sortedargs]) + expr = _constant_renumber(expr) + + # Don't renumber symbols present in the ODE. + constants_found = [c for c in constants_found if c not in variables] + + # Renumbering happens here + subs_dict = dict(zip(constants_found, iter_constants)) + expr = expr.subs(subs_dict, simultaneous=True) + + return expr + + +def _handle_Integral(expr, func, hint): + r""" + Converts a solution with Integrals in it into an actual solution. + + For most hints, this simply runs ``expr.doit()``. + + """ + if hint == "nth_linear_constant_coeff_homogeneous": + sol = expr + elif not hint.endswith("_Integral"): + sol = expr.doit() + else: + sol = expr + return sol + + +# XXX: Should this function maybe go somewhere else? + + +def homogeneous_order(eq, *symbols): + r""" + Returns the order `n` if `g` is homogeneous and ``None`` if it is not + homogeneous. + + Determines if a function is homogeneous and if so of what order. A + function `f(x, y, \cdots)` is homogeneous of order `n` if `f(t x, t y, + \cdots) = t^n f(x, y, \cdots)`. + + If the function is of two variables, `F(x, y)`, then `f` being homogeneous + of any order is equivalent to being able to rewrite `F(x, y)` as `G(x/y)` + or `H(y/x)`. This fact is used to solve 1st order ordinary differential + equations whose coefficients are homogeneous of the same order (see the + docstrings of + :obj:`~sympy.solvers.ode.single.HomogeneousCoeffSubsDepDivIndep` and + :obj:`~sympy.solvers.ode.single.HomogeneousCoeffSubsIndepDivDep`). + + Symbols can be functions, but every argument of the function must be a + symbol, and the arguments of the function that appear in the expression + must match those given in the list of symbols. If a declared function + appears with different arguments than given in the list of symbols, + ``None`` is returned. + + Examples + ======== + + >>> from sympy import Function, homogeneous_order, sqrt + >>> from sympy.abc import x, y + >>> f = Function('f') + >>> homogeneous_order(f(x), f(x)) is None + True + >>> homogeneous_order(f(x,y), f(y, x), x, y) is None + True + >>> homogeneous_order(f(x), f(x), x) + 1 + >>> homogeneous_order(x**2*f(x)/sqrt(x**2+f(x)**2), x, f(x)) + 2 + >>> homogeneous_order(x**2+f(x), x, f(x)) is None + True + + """ + + if not symbols: + raise ValueError("homogeneous_order: no symbols were given.") + symset = set(symbols) + eq = sympify(eq) + + # The following are not supported + if eq.has(Order, Derivative): + return None + + # These are all constants + if (eq.is_Number or + eq.is_NumberSymbol or + eq.is_number + ): + return S.Zero + + # Replace all functions with dummy variables + dum = numbered_symbols(prefix='d', cls=Dummy) + newsyms = set() + for i in [j for j in symset if getattr(j, 'is_Function')]: + iargs = set(i.args) + if iargs.difference(symset): + return None + else: + dummyvar = next(dum) + eq = eq.subs(i, dummyvar) + symset.remove(i) + newsyms.add(dummyvar) + symset.update(newsyms) + + if not eq.free_symbols & symset: + return None + + # assuming order of a nested function can only be equal to zero + if isinstance(eq, Function): + return None if homogeneous_order( + eq.args[0], *tuple(symset)) != 0 else S.Zero + + # make the replacement of x with x*t and see if t can be factored out + t = Dummy('t', positive=True) # It is sufficient that t > 0 + eqs = separatevars(eq.subs([(i, t*i) for i in symset]), [t], dict=True)[t] + if eqs is S.One: + return S.Zero # there was no term with only t + i, d = eqs.as_independent(t, as_Add=False) + b, e = d.as_base_exp() + if b == t: + return e + + +def ode_2nd_power_series_ordinary(eq, func, order, match): + r""" + Gives a power series solution to a second order homogeneous differential + equation with polynomial coefficients at an ordinary point. A homogeneous + differential equation is of the form + + .. math :: P(x)\frac{d^2y}{dx^2} + Q(x)\frac{dy}{dx} + R(x) y(x) = 0 + + For simplicity it is assumed that `P(x)`, `Q(x)` and `R(x)` are polynomials, + it is sufficient that `\frac{Q(x)}{P(x)}` and `\frac{R(x)}{P(x)}` exists at + `x_{0}`. A recurrence relation is obtained by substituting `y` as `\sum_{n=0}^\infty a_{n}x^{n}`, + in the differential equation, and equating the nth term. Using this relation + various terms can be generated. + + + Examples + ======== + + >>> from sympy import dsolve, Function, pprint + >>> from sympy.abc import x + >>> f = Function("f") + >>> eq = f(x).diff(x, 2) + f(x) + >>> pprint(dsolve(eq, hint='2nd_power_series_ordinary')) + / 4 2 \ / 2\ + |x x | | x | / 6\ + f(x) = C2*|-- - -- + 1| + C1*x*|1 - --| + O\x / + \24 2 / \ 6 / + + + References + ========== + - https://tutorial.math.lamar.edu/Classes/DE/SeriesSolutions.aspx + - George E. Simmons, "Differential Equations with Applications and + Historical Notes", p.p 176 - 184 + + """ + x = func.args[0] + f = func.func + C0, C1 = get_numbered_constants(eq, num=2) + n = Dummy("n", integer=True) + s = Wild("s") + k = Wild("k", exclude=[x]) + x0 = match['x0'] + terms = match['terms'] + p = match[match['a3']] + q = match[match['b3']] + r = match[match['c3']] + seriesdict = {} + recurr = Function("r") + + # Generating the recurrence relation which works this way: + # for the second order term the summation begins at n = 2. The coefficients + # p is multiplied with an*(n - 1)*(n - 2)*x**n-2 and a substitution is made such that + # the exponent of x becomes n. + # For example, if p is x, then the second degree recurrence term is + # an*(n - 1)*(n - 2)*x**n-1, substituting (n - 1) as n, it transforms to + # an+1*n*(n - 1)*x**n. + # A similar process is done with the first order and zeroth order term. + + coefflist = [(recurr(n), r), (n*recurr(n), q), (n*(n - 1)*recurr(n), p)] + for index, coeff in enumerate(coefflist): + if coeff[1]: + f2 = powsimp(expand((coeff[1]*(x - x0)**(n - index)).subs(x, x + x0))) + if f2.is_Add: + addargs = f2.args + else: + addargs = [f2] + for arg in addargs: + powm = arg.match(s*x**k) + term = coeff[0]*powm[s] + if not powm[k].is_Symbol: + term = term.subs(n, n - powm[k].as_independent(n)[0]) + startind = powm[k].subs(n, index) + # Seeing if the startterm can be reduced further. + # If it vanishes for n lesser than startind, it is + # equal to summation from n. + if startind: + for i in reversed(range(startind)): + if not term.subs(n, i): + seriesdict[term] = i + else: + seriesdict[term] = i + 1 + break + else: + seriesdict[term] = S.Zero + + # Stripping of terms so that the sum starts with the same number. + teq = S.Zero + suminit = seriesdict.values() + rkeys = seriesdict.keys() + req = Add(*rkeys) + if any(suminit): + maxval = max(suminit) + for term in seriesdict: + val = seriesdict[term] + if val != maxval: + for i in range(val, maxval): + teq += term.subs(n, val) + + finaldict = {} + if teq: + fargs = teq.atoms(AppliedUndef) + if len(fargs) == 1: + finaldict[fargs.pop()] = 0 + else: + maxf = max(fargs, key = lambda x: x.args[0]) + sol = solve(teq, maxf) + if isinstance(sol, list): + sol = sol[0] + finaldict[maxf] = sol + + # Finding the recurrence relation in terms of the largest term. + fargs = req.atoms(AppliedUndef) + maxf = max(fargs, key = lambda x: x.args[0]) + minf = min(fargs, key = lambda x: x.args[0]) + if minf.args[0].is_Symbol: + startiter = 0 + else: + startiter = -minf.args[0].as_independent(n)[0] + lhs = maxf + rhs = solve(req, maxf) + if isinstance(rhs, list): + rhs = rhs[0] + + # Checking how many values are already present + tcounter = len([t for t in finaldict.values() if t]) + + for _ in range(tcounter, terms - 3): # Assuming c0 and c1 to be arbitrary + check = rhs.subs(n, startiter) + nlhs = lhs.subs(n, startiter) + nrhs = check.subs(finaldict) + finaldict[nlhs] = nrhs + startiter += 1 + + # Post processing + series = C0 + C1*(x - x0) + for term in finaldict: + if finaldict[term]: + fact = term.args[0] + series += (finaldict[term].subs([(recurr(0), C0), (recurr(1), C1)])*( + x - x0)**fact) + series = collect(expand_mul(series), [C0, C1]) + Order(x**terms) + return Eq(f(x), series) + + +def ode_2nd_power_series_regular(eq, func, order, match): + r""" + Gives a power series solution to a second order homogeneous differential + equation with polynomial coefficients at a regular point. A second order + homogeneous differential equation is of the form + + .. math :: P(x)\frac{d^2y}{dx^2} + Q(x)\frac{dy}{dx} + R(x) y(x) = 0 + + A point is said to regular singular at `x0` if `x - x0\frac{Q(x)}{P(x)}` + and `(x - x0)^{2}\frac{R(x)}{P(x)}` are analytic at `x0`. For simplicity + `P(x)`, `Q(x)` and `R(x)` are assumed to be polynomials. The algorithm for + finding the power series solutions is: + + 1. Try expressing `(x - x0)P(x)` and `((x - x0)^{2})Q(x)` as power series + solutions about x0. Find `p0` and `q0` which are the constants of the + power series expansions. + 2. Solve the indicial equation `f(m) = m(m - 1) + m*p0 + q0`, to obtain the + roots `m1` and `m2` of the indicial equation. + 3. If `m1 - m2` is a non integer there exists two series solutions. If + `m1 = m2`, there exists only one solution. If `m1 - m2` is an integer, + then the existence of one solution is confirmed. The other solution may + or may not exist. + + The power series solution is of the form `x^{m}\sum_{n=0}^\infty a_{n}x^{n}`. The + coefficients are determined by the following recurrence relation. + `a_{n} = -\frac{\sum_{k=0}^{n-1} q_{n-k} + (m + k)p_{n-k}}{f(m + n)}`. For the case + in which `m1 - m2` is an integer, it can be seen from the recurrence relation + that for the lower root `m`, when `n` equals the difference of both the + roots, the denominator becomes zero. So if the numerator is not equal to zero, + a second series solution exists. + + + Examples + ======== + + >>> from sympy import dsolve, Function, pprint + >>> from sympy.abc import x + >>> f = Function("f") + >>> eq = x*(f(x).diff(x, 2)) + 2*(f(x).diff(x)) + x*f(x) + >>> pprint(dsolve(eq, hint='2nd_power_series_regular')) + / 6 4 2 \ + | x x x | + / 4 2 \ C1*|- --- + -- - -- + 1| + |x x | \ 720 24 2 / / 6\ + f(x) = C2*|--- - -- + 1| + ------------------------ + O\x / + \120 6 / x + + + References + ========== + - George E. Simmons, "Differential Equations with Applications and + Historical Notes", p.p 176 - 184 + + """ + x = func.args[0] + f = func.func + C0, C1 = get_numbered_constants(eq, num=2) + m = Dummy("m") # for solving the indicial equation + x0 = match['x0'] + terms = match['terms'] + p = match['p'] + q = match['q'] + + # Generating the indicial equation + indicial = [] + for term in [p, q]: + if not term.has(x): + indicial.append(term) + else: + term = series(term, x=x, n=1, x0=x0) + if isinstance(term, Order): + indicial.append(S.Zero) + else: + for arg in term.args: + if not arg.has(x): + indicial.append(arg) + break + + p0, q0 = indicial + sollist = solve(m*(m - 1) + m*p0 + q0, m) + if sollist and isinstance(sollist, list) and all( + sol.is_real for sol in sollist): + serdict1 = {} + serdict2 = {} + if len(sollist) == 1: + # Only one series solution exists in this case. + m1 = m2 = sollist.pop() + if terms-m1-1 <= 0: + return Eq(f(x), Order(terms)) + serdict1 = _frobenius(terms-m1-1, m1, p0, q0, p, q, x0, x, C0) + + else: + m1 = sollist[0] + m2 = sollist[1] + if m1 < m2: + m1, m2 = m2, m1 + # Irrespective of whether m1 - m2 is an integer or not, one + # Frobenius series solution exists. + serdict1 = _frobenius(terms-m1-1, m1, p0, q0, p, q, x0, x, C0) + if not (m1 - m2).is_integer: + # Second frobenius series solution exists. + serdict2 = _frobenius(terms-m2-1, m2, p0, q0, p, q, x0, x, C1) + else: + # Check if second frobenius series solution exists. + serdict2 = _frobenius(terms-m2-1, m2, p0, q0, p, q, x0, x, C1, check=m1) + + if serdict1: + finalseries1 = C0 + for key in serdict1: + power = int(key.name[1:]) + finalseries1 += serdict1[key]*(x - x0)**power + finalseries1 = (x - x0)**m1*finalseries1 + finalseries2 = S.Zero + if serdict2: + for key in serdict2: + power = int(key.name[1:]) + finalseries2 += serdict2[key]*(x - x0)**power + finalseries2 += C1 + finalseries2 = (x - x0)**m2*finalseries2 + return Eq(f(x), collect(finalseries1 + finalseries2, + [C0, C1]) + Order(x**terms)) + + +def _frobenius(n, m, p0, q0, p, q, x0, x, c, check=None): + r""" + Returns a dict with keys as coefficients and values as their values in terms of C0 + """ + n = int(n) + # In cases where m1 - m2 is not an integer + m2 = check + + d = Dummy("d") + numsyms = numbered_symbols("C", start=0) + numsyms = [next(numsyms) for i in range(n + 1)] + serlist = [] + for ser in [p, q]: + # Order term not present + if ser.is_polynomial(x) and Poly(ser, x).degree() <= n: + if x0: + ser = ser.subs(x, x + x0) + dict_ = Poly(ser, x).as_dict() + # Order term present + else: + tseries = series(ser, x=x0, n=n+1) + # Removing order + dict_ = Poly(list(ordered(tseries.args))[: -1], x).as_dict() + # Fill in with zeros, if coefficients are zero. + for i in range(n + 1): + if (i,) not in dict_: + dict_[(i,)] = S.Zero + serlist.append(dict_) + + pseries = serlist[0] + qseries = serlist[1] + indicial = d*(d - 1) + d*p0 + q0 + frobdict = {} + for i in range(1, n + 1): + num = c*(m*pseries[(i,)] + qseries[(i,)]) + for j in range(1, i): + sym = Symbol("C" + str(j)) + num += frobdict[sym]*((m + j)*pseries[(i - j,)] + qseries[(i - j,)]) + + # Checking for cases when m1 - m2 is an integer. If num equals zero + # then a second Frobenius series solution cannot be found. If num is not zero + # then set constant as zero and proceed. + if m2 is not None and i == m2 - m: + if num: + return False + else: + frobdict[numsyms[i]] = S.Zero + else: + frobdict[numsyms[i]] = -num/(indicial.subs(d, m+i)) + + return frobdict + +def _remove_redundant_solutions(eq, solns, order, var): + r""" + Remove redundant solutions from the set of solutions. + + This function is needed because otherwise dsolve can return + redundant solutions. As an example consider: + + eq = Eq((f(x).diff(x, 2))*f(x).diff(x), 0) + + There are two ways to find solutions to eq. The first is to solve f(x).diff(x, 2) = 0 + leading to solution f(x)=C1 + C2*x. The second is to solve the equation f(x).diff(x) = 0 + leading to the solution f(x) = C1. In this particular case we then see + that the second solution is a special case of the first and we do not + want to return it. + + This does not always happen. If we have + + eq = Eq((f(x)**2-4)*(f(x).diff(x)-4), 0) + + then we get the algebraic solution f(x) = [-2, 2] and the integral solution + f(x) = x + C1 and in this case the two solutions are not equivalent wrt + initial conditions so both should be returned. + """ + def is_special_case_of(soln1, soln2): + return _is_special_case_of(soln1, soln2, eq, order, var) + + unique_solns = [] + for soln1 in solns: + for soln2 in unique_solns.copy(): + if is_special_case_of(soln1, soln2): + break + elif is_special_case_of(soln2, soln1): + unique_solns.remove(soln2) + else: + unique_solns.append(soln1) + + return unique_solns + +def _is_special_case_of(soln1, soln2, eq, order, var): + r""" + True if soln1 is found to be a special case of soln2 wrt some value of the + constants that appear in soln2. False otherwise. + """ + # The solutions returned by dsolve may be given explicitly or implicitly. + # We will equate the sol1=(soln1.rhs - soln1.lhs), sol2=(soln2.rhs - soln2.lhs) + # of the two solutions. + # + # Since this is supposed to hold for all x it also holds for derivatives. + # For an order n ode we should be able to differentiate + # each solution n times to get n+1 equations. + # + # We then try to solve those n+1 equations for the integrations constants + # in sol2. If we can find a solution that does not depend on x then it + # means that some value of the constants in sol1 is a special case of + # sol2 corresponding to a particular choice of the integration constants. + + # In case the solution is in implicit form we subtract the sides + soln1 = soln1.rhs - soln1.lhs + soln2 = soln2.rhs - soln2.lhs + + # Work for the series solution + if soln1.has(Order) and soln2.has(Order): + if soln1.getO() == soln2.getO(): + soln1 = soln1.removeO() + soln2 = soln2.removeO() + else: + return False + elif soln1.has(Order) or soln2.has(Order): + return False + + constants1 = soln1.free_symbols.difference(eq.free_symbols) + constants2 = soln2.free_symbols.difference(eq.free_symbols) + + constants1_new = get_numbered_constants(Tuple(soln1, soln2), len(constants1)) + if len(constants1) == 1: + constants1_new = {constants1_new} + for c_old, c_new in zip(constants1, constants1_new): + soln1 = soln1.subs(c_old, c_new) + + # n equations for sol1 = sol2, sol1'=sol2', ... + lhs = soln1 + rhs = soln2 + eqns = [Eq(lhs, rhs)] + for n in range(1, order): + lhs = lhs.diff(var) + rhs = rhs.diff(var) + eq = Eq(lhs, rhs) + eqns.append(eq) + + # BooleanTrue/False awkwardly show up for trivial equations + if any(isinstance(eq, BooleanFalse) for eq in eqns): + return False + eqns = [eq for eq in eqns if not isinstance(eq, BooleanTrue)] + + try: + constant_solns = solve(eqns, constants2) + except NotImplementedError: + return False + + # Sometimes returns a dict and sometimes a list of dicts + if isinstance(constant_solns, dict): + constant_solns = [constant_solns] + + # after solving the issue 17418, maybe we don't need the following checksol code. + for constant_soln in constant_solns: + for eq in eqns: + eq=eq.rhs-eq.lhs + if checksol(eq, constant_soln) is not True: + return False + + # If any solution gives all constants as expressions that don't depend on + # x then there exists constants for soln2 that give soln1 + for constant_soln in constant_solns: + if not any(c.has(var) for c in constant_soln.values()): + return True + + return False + + +def ode_1st_power_series(eq, func, order, match): + r""" + The power series solution is a method which gives the Taylor series expansion + to the solution of a differential equation. + + For a first order differential equation `\frac{dy}{dx} = h(x, y)`, a power + series solution exists at a point `x = x_{0}` if `h(x, y)` is analytic at `x_{0}`. + The solution is given by + + .. math:: y(x) = y(x_{0}) + \sum_{n = 1}^{\infty} \frac{F_{n}(x_{0},b)(x - x_{0})^n}{n!}, + + where `y(x_{0}) = b` is the value of y at the initial value of `x_{0}`. + To compute the values of the `F_{n}(x_{0},b)` the following algorithm is + followed, until the required number of terms are generated. + + 1. `F_1 = h(x_{0}, b)` + 2. `F_{n+1} = \frac{\partial F_{n}}{\partial x} + \frac{\partial F_{n}}{\partial y}F_{1}` + + Examples + ======== + + >>> from sympy import Function, pprint, exp, dsolve + >>> from sympy.abc import x + >>> f = Function('f') + >>> eq = exp(x)*(f(x).diff(x)) - f(x) + >>> pprint(dsolve(eq, hint='1st_power_series')) + 3 4 5 + C1*x C1*x C1*x / 6\ + f(x) = C1 + C1*x - ----- + ----- + ----- + O\x / + 6 24 60 + + + References + ========== + + - Travis W. Walker, Analytic power series technique for solving first-order + differential equations, p.p 17, 18 + + """ + x = func.args[0] + y = match['y'] + f = func.func + h = -match[match['d']]/match[match['e']] + point = match['f0'] + value = match['f0val'] + terms = match['terms'] + + # First term + F = h + if not h: + return Eq(f(x), value) + + # Initialization + series = value + if terms > 1: + hc = h.subs({x: point, y: value}) + if hc.has(oo) or hc.has(nan) or hc.has(zoo): + # Derivative does not exist, not analytic + return Eq(f(x), oo) + elif hc: + series += hc*(x - point) + + for factcount in range(2, terms): + Fnew = F.diff(x) + F.diff(y)*h + Fnewc = Fnew.subs({x: point, y: value}) + # Same logic as above + if Fnewc.has(oo) or Fnewc.has(nan) or Fnewc.has(-oo) or Fnewc.has(zoo): + return Eq(f(x), oo) + series += Fnewc*((x - point)**factcount)/factorial(factcount) + F = Fnew + series += Order(x**terms) + return Eq(f(x), series) + + +def checkinfsol(eq, infinitesimals, func=None, order=None): + r""" + This function is used to check if the given infinitesimals are the + actual infinitesimals of the given first order differential equation. + This method is specific to the Lie Group Solver of ODEs. + + As of now, it simply checks, by substituting the infinitesimals in the + partial differential equation. + + + .. math:: \frac{\partial \eta}{\partial x} + \left(\frac{\partial \eta}{\partial y} + - \frac{\partial \xi}{\partial x}\right)*h + - \frac{\partial \xi}{\partial y}*h^{2} + - \xi\frac{\partial h}{\partial x} - \eta\frac{\partial h}{\partial y} = 0 + + + where `\eta`, and `\xi` are the infinitesimals and `h(x,y) = \frac{dy}{dx}` + + The infinitesimals should be given in the form of a list of dicts + ``[{xi(x, y): inf, eta(x, y): inf}]``, corresponding to the + output of the function infinitesimals. It returns a list + of values of the form ``[(True/False, sol)]`` where ``sol`` is the value + obtained after substituting the infinitesimals in the PDE. If it + is ``True``, then ``sol`` would be 0. + + """ + if isinstance(eq, Equality): + eq = eq.lhs - eq.rhs + if not func: + eq, func = _preprocess(eq) + variables = func.args + if len(variables) != 1: + raise ValueError("ODE's have only one independent variable") + else: + x = variables[0] + if not order: + order = ode_order(eq, func) + if order != 1: + raise NotImplementedError("Lie groups solver has been implemented " + "only for first order differential equations") + else: + df = func.diff(x) + a = Wild('a', exclude = [df]) + b = Wild('b', exclude = [df]) + match = collect(expand(eq), df).match(a*df + b) + + if match: + h = -simplify(match[b]/match[a]) + else: + try: + sol = solve(eq, df) + except NotImplementedError: + raise NotImplementedError("Infinitesimals for the " + "first order ODE could not be found") + else: + h = sol[0] # Find infinitesimals for one solution + + y = Dummy('y') + h = h.subs(func, y) + xi = Function('xi')(x, y) + eta = Function('eta')(x, y) + dxi = Function('xi')(x, func) + deta = Function('eta')(x, func) + pde = (eta.diff(x) + (eta.diff(y) - xi.diff(x))*h - + (xi.diff(y))*h**2 - xi*(h.diff(x)) - eta*(h.diff(y))) + soltup = [] + for sol in infinitesimals: + tsol = {xi: S(sol[dxi]).subs(func, y), + eta: S(sol[deta]).subs(func, y)} + sol = simplify(pde.subs(tsol).doit()) + if sol: + soltup.append((False, sol.subs(y, func))) + else: + soltup.append((True, 0)) + return soltup + + +def sysode_linear_2eq_order1(match_): + x = match_['func'][0].func + y = match_['func'][1].func + func = match_['func'] + fc = match_['func_coeff'] + eq = match_['eq'] + r = {} + t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0] + for i in range(2): + eq[i] = Add(*[terms/fc[i,func[i],1] for terms in Add.make_args(eq[i])]) + + # for equations Eq(a1*diff(x(t),t), a*x(t) + b*y(t) + k1) + # and Eq(a2*diff(x(t),t), c*x(t) + d*y(t) + k2) + r['a'] = -fc[0,x(t),0]/fc[0,x(t),1] + r['c'] = -fc[1,x(t),0]/fc[1,y(t),1] + r['b'] = -fc[0,y(t),0]/fc[0,x(t),1] + r['d'] = -fc[1,y(t),0]/fc[1,y(t),1] + forcing = [S.Zero,S.Zero] + for i in range(2): + for j in Add.make_args(eq[i]): + if not j.has(x(t), y(t)): + forcing[i] += j + if not (forcing[0].has(t) or forcing[1].has(t)): + r['k1'] = forcing[0] + r['k2'] = forcing[1] + else: + raise NotImplementedError("Only homogeneous problems are supported" + + " (and constant inhomogeneity)") + + if match_['type_of_equation'] == 'type6': + sol = _linear_2eq_order1_type6(x, y, t, r, eq) + if match_['type_of_equation'] == 'type7': + sol = _linear_2eq_order1_type7(x, y, t, r, eq) + return sol + +def _linear_2eq_order1_type6(x, y, t, r, eq): + r""" + The equations of this type of ode are . + + .. math:: x' = f(t) x + g(t) y + + .. math:: y' = a [f(t) + a h(t)] x + a [g(t) - h(t)] y + + This is solved by first multiplying the first equation by `-a` and adding + it to the second equation to obtain + + .. math:: y' - a x' = -a h(t) (y - a x) + + Setting `U = y - ax` and integrating the equation we arrive at + + .. math:: y - ax = C_1 e^{-a \int h(t) \,dt} + + and on substituting the value of y in first equation give rise to first order ODEs. After solving for + `x`, we can obtain `y` by substituting the value of `x` in second equation. + + """ + C1, C2, C3, C4 = get_numbered_constants(eq, num=4) + p = 0 + q = 0 + p1 = cancel(r['c']/cancel(r['c']/r['d']).as_numer_denom()[0]) + p2 = cancel(r['a']/cancel(r['a']/r['b']).as_numer_denom()[0]) + for n, i in enumerate([p1, p2]): + for j in Mul.make_args(collect_const(i)): + if not j.has(t): + q = j + if q!=0 and n==0: + if ((r['c']/j - r['a'])/(r['b'] - r['d']/j)) == j: + p = 1 + s = j + break + if q!=0 and n==1: + if ((r['a']/j - r['c'])/(r['d'] - r['b']/j)) == j: + p = 2 + s = j + break + + if p == 1: + equ = diff(x(t),t) - r['a']*x(t) - r['b']*(s*x(t) + C1*exp(-s*Integral(r['b'] - r['d']/s, t))) + hint1 = classify_ode(equ)[1] + sol1 = dsolve(equ, hint=hint1+'_Integral').rhs + sol2 = s*sol1 + C1*exp(-s*Integral(r['b'] - r['d']/s, t)) + elif p ==2: + equ = diff(y(t),t) - r['c']*y(t) - r['d']*s*y(t) + C1*exp(-s*Integral(r['d'] - r['b']/s, t)) + hint1 = classify_ode(equ)[1] + sol2 = dsolve(equ, hint=hint1+'_Integral').rhs + sol1 = s*sol2 + C1*exp(-s*Integral(r['d'] - r['b']/s, t)) + return [Eq(x(t), sol1), Eq(y(t), sol2)] + +def _linear_2eq_order1_type7(x, y, t, r, eq): + r""" + The equations of this type of ode are . + + .. math:: x' = f(t) x + g(t) y + + .. math:: y' = h(t) x + p(t) y + + Differentiating the first equation and substituting the value of `y` + from second equation will give a second-order linear equation + + .. math:: g x'' - (fg + gp + g') x' + (fgp - g^{2} h + f g' - f' g) x = 0 + + This above equation can be easily integrated if following conditions are satisfied. + + 1. `fgp - g^{2} h + f g' - f' g = 0` + + 2. `fgp - g^{2} h + f g' - f' g = ag, fg + gp + g' = bg` + + If first condition is satisfied then it is solved by current dsolve solver and in second case it becomes + a constant coefficient differential equation which is also solved by current solver. + + Otherwise if the above condition fails then, + a particular solution is assumed as `x = x_0(t)` and `y = y_0(t)` + Then the general solution is expressed as + + .. math:: x = C_1 x_0(t) + C_2 x_0(t) \int \frac{g(t) F(t) P(t)}{x_0^{2}(t)} \,dt + + .. math:: y = C_1 y_0(t) + C_2 [\frac{F(t) P(t)}{x_0(t)} + y_0(t) \int \frac{g(t) F(t) P(t)}{x_0^{2}(t)} \,dt] + + where C1 and C2 are arbitrary constants and + + .. math:: F(t) = e^{\int f(t) \,dt}, P(t) = e^{\int p(t) \,dt} + + """ + C1, C2, C3, C4 = get_numbered_constants(eq, num=4) + e1 = r['a']*r['b']*r['c'] - r['b']**2*r['c'] + r['a']*diff(r['b'],t) - diff(r['a'],t)*r['b'] + e2 = r['a']*r['c']*r['d'] - r['b']*r['c']**2 + diff(r['c'],t)*r['d'] - r['c']*diff(r['d'],t) + m1 = r['a']*r['b'] + r['b']*r['d'] + diff(r['b'],t) + m2 = r['a']*r['c'] + r['c']*r['d'] + diff(r['c'],t) + if e1 == 0: + sol1 = dsolve(r['b']*diff(x(t),t,t) - m1*diff(x(t),t)).rhs + sol2 = dsolve(diff(y(t),t) - r['c']*sol1 - r['d']*y(t)).rhs + elif e2 == 0: + sol2 = dsolve(r['c']*diff(y(t),t,t) - m2*diff(y(t),t)).rhs + sol1 = dsolve(diff(x(t),t) - r['a']*x(t) - r['b']*sol2).rhs + elif not (e1/r['b']).has(t) and not (m1/r['b']).has(t): + sol1 = dsolve(diff(x(t),t,t) - (m1/r['b'])*diff(x(t),t) - (e1/r['b'])*x(t)).rhs + sol2 = dsolve(diff(y(t),t) - r['c']*sol1 - r['d']*y(t)).rhs + elif not (e2/r['c']).has(t) and not (m2/r['c']).has(t): + sol2 = dsolve(diff(y(t),t,t) - (m2/r['c'])*diff(y(t),t) - (e2/r['c'])*y(t)).rhs + sol1 = dsolve(diff(x(t),t) - r['a']*x(t) - r['b']*sol2).rhs + else: + x0 = Function('x0')(t) # x0 and y0 being particular solutions + y0 = Function('y0')(t) + F = exp(Integral(r['a'],t)) + P = exp(Integral(r['d'],t)) + sol1 = C1*x0 + C2*x0*Integral(r['b']*F*P/x0**2, t) + sol2 = C1*y0 + C2*(F*P/x0 + y0*Integral(r['b']*F*P/x0**2, t)) + return [Eq(x(t), sol1), Eq(y(t), sol2)] + + +def sysode_nonlinear_2eq_order1(match_): + func = match_['func'] + eq = match_['eq'] + fc = match_['func_coeff'] + t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0] + if match_['type_of_equation'] == 'type5': + sol = _nonlinear_2eq_order1_type5(func, t, eq) + return sol + x = func[0].func + y = func[1].func + for i in range(2): + eqs = 0 + for terms in Add.make_args(eq[i]): + eqs += terms/fc[i,func[i],1] + eq[i] = eqs + if match_['type_of_equation'] == 'type1': + sol = _nonlinear_2eq_order1_type1(x, y, t, eq) + elif match_['type_of_equation'] == 'type2': + sol = _nonlinear_2eq_order1_type2(x, y, t, eq) + elif match_['type_of_equation'] == 'type3': + sol = _nonlinear_2eq_order1_type3(x, y, t, eq) + elif match_['type_of_equation'] == 'type4': + sol = _nonlinear_2eq_order1_type4(x, y, t, eq) + return sol + + +def _nonlinear_2eq_order1_type1(x, y, t, eq): + r""" + Equations: + + .. math:: x' = x^n F(x,y) + + .. math:: y' = g(y) F(x,y) + + Solution: + + .. math:: x = \varphi(y), \int \frac{1}{g(y) F(\varphi(y),y)} \,dy = t + C_2 + + where + + if `n \neq 1` + + .. math:: \varphi = [C_1 + (1-n) \int \frac{1}{g(y)} \,dy]^{\frac{1}{1-n}} + + if `n = 1` + + .. math:: \varphi = C_1 e^{\int \frac{1}{g(y)} \,dy} + + where `C_1` and `C_2` are arbitrary constants. + + """ + C1, C2 = get_numbered_constants(eq, num=2) + n = Wild('n', exclude=[x(t),y(t)]) + f = Wild('f') + u, v = symbols('u, v') + r = eq[0].match(diff(x(t),t) - x(t)**n*f) + g = ((diff(y(t),t) - eq[1])/r[f]).subs(y(t),v) + F = r[f].subs(x(t),u).subs(y(t),v) + n = r[n] + if n!=1: + phi = (C1 + (1-n)*Integral(1/g, v))**(1/(1-n)) + else: + phi = C1*exp(Integral(1/g, v)) + phi = phi.doit() + sol2 = solve(Integral(1/(g*F.subs(u,phi)), v).doit() - t - C2, v) + sol = [] + for sols in sol2: + sol.append(Eq(x(t),phi.subs(v, sols))) + sol.append(Eq(y(t), sols)) + return sol + +def _nonlinear_2eq_order1_type2(x, y, t, eq): + r""" + Equations: + + .. math:: x' = e^{\lambda x} F(x,y) + + .. math:: y' = g(y) F(x,y) + + Solution: + + .. math:: x = \varphi(y), \int \frac{1}{g(y) F(\varphi(y),y)} \,dy = t + C_2 + + where + + if `\lambda \neq 0` + + .. math:: \varphi = -\frac{1}{\lambda} log(C_1 - \lambda \int \frac{1}{g(y)} \,dy) + + if `\lambda = 0` + + .. math:: \varphi = C_1 + \int \frac{1}{g(y)} \,dy + + where `C_1` and `C_2` are arbitrary constants. + + """ + C1, C2 = get_numbered_constants(eq, num=2) + n = Wild('n', exclude=[x(t),y(t)]) + f = Wild('f') + u, v = symbols('u, v') + r = eq[0].match(diff(x(t),t) - exp(n*x(t))*f) + g = ((diff(y(t),t) - eq[1])/r[f]).subs(y(t),v) + F = r[f].subs(x(t),u).subs(y(t),v) + n = r[n] + if n: + phi = -1/n*log(C1 - n*Integral(1/g, v)) + else: + phi = C1 + Integral(1/g, v) + phi = phi.doit() + sol2 = solve(Integral(1/(g*F.subs(u,phi)), v).doit() - t - C2, v) + sol = [] + for sols in sol2: + sol.append(Eq(x(t),phi.subs(v, sols))) + sol.append(Eq(y(t), sols)) + return sol + +def _nonlinear_2eq_order1_type3(x, y, t, eq): + r""" + Autonomous system of general form + + .. math:: x' = F(x,y) + + .. math:: y' = G(x,y) + + Assuming `y = y(x, C_1)` where `C_1` is an arbitrary constant is the general + solution of the first-order equation + + .. math:: F(x,y) y'_x = G(x,y) + + Then the general solution of the original system of equations has the form + + .. math:: \int \frac{1}{F(x,y(x,C_1))} \,dx = t + C_1 + + """ + C1, C2, C3, C4 = get_numbered_constants(eq, num=4) + v = Function('v') + u = Symbol('u') + f = Wild('f') + g = Wild('g') + r1 = eq[0].match(diff(x(t),t) - f) + r2 = eq[1].match(diff(y(t),t) - g) + F = r1[f].subs(x(t), u).subs(y(t), v(u)) + G = r2[g].subs(x(t), u).subs(y(t), v(u)) + sol2r = dsolve(Eq(diff(v(u), u), G/F)) + if isinstance(sol2r, Equality): + sol2r = [sol2r] + for sol2s in sol2r: + sol1 = solve(Integral(1/F.subs(v(u), sol2s.rhs), u).doit() - t - C2, u) + sol = [] + for sols in sol1: + sol.append(Eq(x(t), sols)) + sol.append(Eq(y(t), (sol2s.rhs).subs(u, sols))) + return sol + +def _nonlinear_2eq_order1_type4(x, y, t, eq): + r""" + Equation: + + .. math:: x' = f_1(x) g_1(y) \phi(x,y,t) + + .. math:: y' = f_2(x) g_2(y) \phi(x,y,t) + + First integral: + + .. math:: \int \frac{f_2(x)}{f_1(x)} \,dx - \int \frac{g_1(y)}{g_2(y)} \,dy = C + + where `C` is an arbitrary constant. + + On solving the first integral for `x` (resp., `y` ) and on substituting the + resulting expression into either equation of the original solution, one + arrives at a first-order equation for determining `y` (resp., `x` ). + + """ + C1, C2 = get_numbered_constants(eq, num=2) + u, v = symbols('u, v') + U, V = symbols('U, V', cls=Function) + f = Wild('f') + g = Wild('g') + f1 = Wild('f1', exclude=[v,t]) + f2 = Wild('f2', exclude=[v,t]) + g1 = Wild('g1', exclude=[u,t]) + g2 = Wild('g2', exclude=[u,t]) + r1 = eq[0].match(diff(x(t),t) - f) + r2 = eq[1].match(diff(y(t),t) - g) + num, den = ( + (r1[f].subs(x(t),u).subs(y(t),v))/ + (r2[g].subs(x(t),u).subs(y(t),v))).as_numer_denom() + R1 = num.match(f1*g1) + R2 = den.match(f2*g2) + phi = (r1[f].subs(x(t),u).subs(y(t),v))/num + F1 = R1[f1]; F2 = R2[f2] + G1 = R1[g1]; G2 = R2[g2] + sol1r = solve(Integral(F2/F1, u).doit() - Integral(G1/G2,v).doit() - C1, u) + sol2r = solve(Integral(F2/F1, u).doit() - Integral(G1/G2,v).doit() - C1, v) + sol = [] + for sols in sol1r: + sol.append(Eq(y(t), dsolve(diff(V(t),t) - F2.subs(u,sols).subs(v,V(t))*G2.subs(v,V(t))*phi.subs(u,sols).subs(v,V(t))).rhs)) + for sols in sol2r: + sol.append(Eq(x(t), dsolve(diff(U(t),t) - F1.subs(u,U(t))*G1.subs(v,sols).subs(u,U(t))*phi.subs(v,sols).subs(u,U(t))).rhs)) + return set(sol) + +def _nonlinear_2eq_order1_type5(func, t, eq): + r""" + Clairaut system of ODEs + + .. math:: x = t x' + F(x',y') + + .. math:: y = t y' + G(x',y') + + The following are solutions of the system + + `(i)` straight lines: + + .. math:: x = C_1 t + F(C_1, C_2), y = C_2 t + G(C_1, C_2) + + where `C_1` and `C_2` are arbitrary constants; + + `(ii)` envelopes of the above lines; + + `(iii)` continuously differentiable lines made up from segments of the lines + `(i)` and `(ii)`. + + """ + C1, C2 = get_numbered_constants(eq, num=2) + f = Wild('f') + g = Wild('g') + def check_type(x, y): + r1 = eq[0].match(t*diff(x(t),t) - x(t) + f) + r2 = eq[1].match(t*diff(y(t),t) - y(t) + g) + if not (r1 and r2): + r1 = eq[0].match(diff(x(t),t) - x(t)/t + f/t) + r2 = eq[1].match(diff(y(t),t) - y(t)/t + g/t) + if not (r1 and r2): + r1 = (-eq[0]).match(t*diff(x(t),t) - x(t) + f) + r2 = (-eq[1]).match(t*diff(y(t),t) - y(t) + g) + if not (r1 and r2): + r1 = (-eq[0]).match(diff(x(t),t) - x(t)/t + f/t) + r2 = (-eq[1]).match(diff(y(t),t) - y(t)/t + g/t) + return [r1, r2] + for func_ in func: + if isinstance(func_, list): + x = func[0][0].func + y = func[0][1].func + [r1, r2] = check_type(x, y) + if not (r1 and r2): + [r1, r2] = check_type(y, x) + x, y = y, x + x1 = diff(x(t),t); y1 = diff(y(t),t) + return {Eq(x(t), C1*t + r1[f].subs(x1,C1).subs(y1,C2)), Eq(y(t), C2*t + r2[g].subs(x1,C1).subs(y1,C2))} + +def sysode_nonlinear_3eq_order1(match_): + x = match_['func'][0].func + y = match_['func'][1].func + z = match_['func'][2].func + eq = match_['eq'] + t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0] + if match_['type_of_equation'] == 'type1': + sol = _nonlinear_3eq_order1_type1(x, y, z, t, eq) + if match_['type_of_equation'] == 'type2': + sol = _nonlinear_3eq_order1_type2(x, y, z, t, eq) + if match_['type_of_equation'] == 'type3': + sol = _nonlinear_3eq_order1_type3(x, y, z, t, eq) + if match_['type_of_equation'] == 'type4': + sol = _nonlinear_3eq_order1_type4(x, y, z, t, eq) + if match_['type_of_equation'] == 'type5': + sol = _nonlinear_3eq_order1_type5(x, y, z, t, eq) + return sol + +def _nonlinear_3eq_order1_type1(x, y, z, t, eq): + r""" + Equations: + + .. math:: a x' = (b - c) y z, \enspace b y' = (c - a) z x, \enspace c z' = (a - b) x y + + First Integrals: + + .. math:: a x^{2} + b y^{2} + c z^{2} = C_1 + + .. math:: a^{2} x^{2} + b^{2} y^{2} + c^{2} z^{2} = C_2 + + where `C_1` and `C_2` are arbitrary constants. On solving the integrals for `y` and + `z` and on substituting the resulting expressions into the first equation of the + system, we arrives at a separable first-order equation on `x`. Similarly doing that + for other two equations, we will arrive at first order equation on `y` and `z` too. + + References + ========== + -https://eqworld.ipmnet.ru/en/solutions/sysode/sode0401.pdf + + """ + C1, C2 = get_numbered_constants(eq, num=2) + u, v, w = symbols('u, v, w') + p = Wild('p', exclude=[x(t), y(t), z(t), t]) + q = Wild('q', exclude=[x(t), y(t), z(t), t]) + s = Wild('s', exclude=[x(t), y(t), z(t), t]) + r = (diff(x(t),t) - eq[0]).match(p*y(t)*z(t)) + r.update((diff(y(t),t) - eq[1]).match(q*z(t)*x(t))) + r.update((diff(z(t),t) - eq[2]).match(s*x(t)*y(t))) + n1, d1 = r[p].as_numer_denom() + n2, d2 = r[q].as_numer_denom() + n3, d3 = r[s].as_numer_denom() + val = solve([n1*u-d1*v+d1*w, d2*u+n2*v-d2*w, d3*u-d3*v-n3*w],[u,v]) + vals = [val[v], val[u]] + c = lcm(vals[0].as_numer_denom()[1], vals[1].as_numer_denom()[1]) + b = vals[0].subs(w, c) + a = vals[1].subs(w, c) + y_x = sqrt(((c*C1-C2) - a*(c-a)*x(t)**2)/(b*(c-b))) + z_x = sqrt(((b*C1-C2) - a*(b-a)*x(t)**2)/(c*(b-c))) + z_y = sqrt(((a*C1-C2) - b*(a-b)*y(t)**2)/(c*(a-c))) + x_y = sqrt(((c*C1-C2) - b*(c-b)*y(t)**2)/(a*(c-a))) + x_z = sqrt(((b*C1-C2) - c*(b-c)*z(t)**2)/(a*(b-a))) + y_z = sqrt(((a*C1-C2) - c*(a-c)*z(t)**2)/(b*(a-b))) + sol1 = dsolve(a*diff(x(t),t) - (b-c)*y_x*z_x) + sol2 = dsolve(b*diff(y(t),t) - (c-a)*z_y*x_y) + sol3 = dsolve(c*diff(z(t),t) - (a-b)*x_z*y_z) + return [sol1, sol2, sol3] + + +def _nonlinear_3eq_order1_type2(x, y, z, t, eq): + r""" + Equations: + + .. math:: a x' = (b - c) y z f(x, y, z, t) + + .. math:: b y' = (c - a) z x f(x, y, z, t) + + .. math:: c z' = (a - b) x y f(x, y, z, t) + + First Integrals: + + .. math:: a x^{2} + b y^{2} + c z^{2} = C_1 + + .. math:: a^{2} x^{2} + b^{2} y^{2} + c^{2} z^{2} = C_2 + + where `C_1` and `C_2` are arbitrary constants. On solving the integrals for `y` and + `z` and on substituting the resulting expressions into the first equation of the + system, we arrives at a first-order differential equations on `x`. Similarly doing + that for other two equations we will arrive at first order equation on `y` and `z`. + + References + ========== + -https://eqworld.ipmnet.ru/en/solutions/sysode/sode0402.pdf + + """ + C1, C2 = get_numbered_constants(eq, num=2) + u, v, w = symbols('u, v, w') + p = Wild('p', exclude=[x(t), y(t), z(t), t]) + q = Wild('q', exclude=[x(t), y(t), z(t), t]) + s = Wild('s', exclude=[x(t), y(t), z(t), t]) + f = Wild('f') + r1 = (diff(x(t),t) - eq[0]).match(y(t)*z(t)*f) + r = collect_const(r1[f]).match(p*f) + r.update(((diff(y(t),t) - eq[1])/r[f]).match(q*z(t)*x(t))) + r.update(((diff(z(t),t) - eq[2])/r[f]).match(s*x(t)*y(t))) + n1, d1 = r[p].as_numer_denom() + n2, d2 = r[q].as_numer_denom() + n3, d3 = r[s].as_numer_denom() + val = solve([n1*u-d1*v+d1*w, d2*u+n2*v-d2*w, -d3*u+d3*v+n3*w],[u,v]) + vals = [val[v], val[u]] + c = lcm(vals[0].as_numer_denom()[1], vals[1].as_numer_denom()[1]) + a = vals[0].subs(w, c) + b = vals[1].subs(w, c) + y_x = sqrt(((c*C1-C2) - a*(c-a)*x(t)**2)/(b*(c-b))) + z_x = sqrt(((b*C1-C2) - a*(b-a)*x(t)**2)/(c*(b-c))) + z_y = sqrt(((a*C1-C2) - b*(a-b)*y(t)**2)/(c*(a-c))) + x_y = sqrt(((c*C1-C2) - b*(c-b)*y(t)**2)/(a*(c-a))) + x_z = sqrt(((b*C1-C2) - c*(b-c)*z(t)**2)/(a*(b-a))) + y_z = sqrt(((a*C1-C2) - c*(a-c)*z(t)**2)/(b*(a-b))) + sol1 = dsolve(a*diff(x(t),t) - (b-c)*y_x*z_x*r[f]) + sol2 = dsolve(b*diff(y(t),t) - (c-a)*z_y*x_y*r[f]) + sol3 = dsolve(c*diff(z(t),t) - (a-b)*x_z*y_z*r[f]) + return [sol1, sol2, sol3] + +def _nonlinear_3eq_order1_type3(x, y, z, t, eq): + r""" + Equations: + + .. math:: x' = c F_2 - b F_3, \enspace y' = a F_3 - c F_1, \enspace z' = b F_1 - a F_2 + + where `F_n = F_n(x, y, z, t)`. + + 1. First Integral: + + .. math:: a x + b y + c z = C_1, + + where C is an arbitrary constant. + + 2. If we assume function `F_n` to be independent of `t`,i.e, `F_n` = `F_n (x, y, z)` + Then, on eliminating `t` and `z` from the first two equation of the system, one + arrives at the first-order equation + + .. math:: \frac{dy}{dx} = \frac{a F_3 (x, y, z) - c F_1 (x, y, z)}{c F_2 (x, y, z) - + b F_3 (x, y, z)} + + where `z = \frac{1}{c} (C_1 - a x - b y)` + + References + ========== + -https://eqworld.ipmnet.ru/en/solutions/sysode/sode0404.pdf + + """ + C1 = get_numbered_constants(eq, num=1) + u, v, w = symbols('u, v, w') + fu, fv, fw = symbols('u, v, w', cls=Function) + p = Wild('p', exclude=[x(t), y(t), z(t), t]) + q = Wild('q', exclude=[x(t), y(t), z(t), t]) + s = Wild('s', exclude=[x(t), y(t), z(t), t]) + F1, F2, F3 = symbols('F1, F2, F3', cls=Wild) + r1 = (diff(x(t), t) - eq[0]).match(F2-F3) + r = collect_const(r1[F2]).match(s*F2) + r.update(collect_const(r1[F3]).match(q*F3)) + if eq[1].has(r[F2]) and not eq[1].has(r[F3]): + r[F2], r[F3] = r[F3], r[F2] + r[s], r[q] = -r[q], -r[s] + r.update((diff(y(t), t) - eq[1]).match(p*r[F3] - r[s]*F1)) + a = r[p]; b = r[q]; c = r[s] + F1 = r[F1].subs(x(t), u).subs(y(t),v).subs(z(t), w) + F2 = r[F2].subs(x(t), u).subs(y(t),v).subs(z(t), w) + F3 = r[F3].subs(x(t), u).subs(y(t),v).subs(z(t), w) + z_xy = (C1-a*u-b*v)/c + y_zx = (C1-a*u-c*w)/b + x_yz = (C1-b*v-c*w)/a + y_x = dsolve(diff(fv(u),u) - ((a*F3-c*F1)/(c*F2-b*F3)).subs(w,z_xy).subs(v,fv(u))).rhs + z_x = dsolve(diff(fw(u),u) - ((b*F1-a*F2)/(c*F2-b*F3)).subs(v,y_zx).subs(w,fw(u))).rhs + z_y = dsolve(diff(fw(v),v) - ((b*F1-a*F2)/(a*F3-c*F1)).subs(u,x_yz).subs(w,fw(v))).rhs + x_y = dsolve(diff(fu(v),v) - ((c*F2-b*F3)/(a*F3-c*F1)).subs(w,z_xy).subs(u,fu(v))).rhs + y_z = dsolve(diff(fv(w),w) - ((a*F3-c*F1)/(b*F1-a*F2)).subs(u,x_yz).subs(v,fv(w))).rhs + x_z = dsolve(diff(fu(w),w) - ((c*F2-b*F3)/(b*F1-a*F2)).subs(v,y_zx).subs(u,fu(w))).rhs + sol1 = dsolve(diff(fu(t),t) - (c*F2 - b*F3).subs(v,y_x).subs(w,z_x).subs(u,fu(t))).rhs + sol2 = dsolve(diff(fv(t),t) - (a*F3 - c*F1).subs(u,x_y).subs(w,z_y).subs(v,fv(t))).rhs + sol3 = dsolve(diff(fw(t),t) - (b*F1 - a*F2).subs(u,x_z).subs(v,y_z).subs(w,fw(t))).rhs + return [sol1, sol2, sol3] + +def _nonlinear_3eq_order1_type4(x, y, z, t, eq): + r""" + Equations: + + .. math:: x' = c z F_2 - b y F_3, \enspace y' = a x F_3 - c z F_1, \enspace z' = b y F_1 - a x F_2 + + where `F_n = F_n (x, y, z, t)` + + 1. First integral: + + .. math:: a x^{2} + b y^{2} + c z^{2} = C_1 + + where `C` is an arbitrary constant. + + 2. Assuming the function `F_n` is independent of `t`: `F_n = F_n (x, y, z)`. Then on + eliminating `t` and `z` from the first two equations of the system, one arrives at + the first-order equation + + .. math:: \frac{dy}{dx} = \frac{a x F_3 (x, y, z) - c z F_1 (x, y, z)} + {c z F_2 (x, y, z) - b y F_3 (x, y, z)} + + where `z = \pm \sqrt{\frac{1}{c} (C_1 - a x^{2} - b y^{2})}` + + References + ========== + -https://eqworld.ipmnet.ru/en/solutions/sysode/sode0405.pdf + + """ + C1 = get_numbered_constants(eq, num=1) + u, v, w = symbols('u, v, w') + p = Wild('p', exclude=[x(t), y(t), z(t), t]) + q = Wild('q', exclude=[x(t), y(t), z(t), t]) + s = Wild('s', exclude=[x(t), y(t), z(t), t]) + F1, F2, F3 = symbols('F1, F2, F3', cls=Wild) + r1 = eq[0].match(diff(x(t),t) - z(t)*F2 + y(t)*F3) + r = collect_const(r1[F2]).match(s*F2) + r.update(collect_const(r1[F3]).match(q*F3)) + if eq[1].has(r[F2]) and not eq[1].has(r[F3]): + r[F2], r[F3] = r[F3], r[F2] + r[s], r[q] = -r[q], -r[s] + r.update((diff(y(t),t) - eq[1]).match(p*x(t)*r[F3] - r[s]*z(t)*F1)) + a = r[p]; b = r[q]; c = r[s] + F1 = r[F1].subs(x(t),u).subs(y(t),v).subs(z(t),w) + F2 = r[F2].subs(x(t),u).subs(y(t),v).subs(z(t),w) + F3 = r[F3].subs(x(t),u).subs(y(t),v).subs(z(t),w) + x_yz = sqrt((C1 - b*v**2 - c*w**2)/a) + y_zx = sqrt((C1 - c*w**2 - a*u**2)/b) + z_xy = sqrt((C1 - a*u**2 - b*v**2)/c) + y_x = dsolve(diff(v(u),u) - ((a*u*F3-c*w*F1)/(c*w*F2-b*v*F3)).subs(w,z_xy).subs(v,v(u))).rhs + z_x = dsolve(diff(w(u),u) - ((b*v*F1-a*u*F2)/(c*w*F2-b*v*F3)).subs(v,y_zx).subs(w,w(u))).rhs + z_y = dsolve(diff(w(v),v) - ((b*v*F1-a*u*F2)/(a*u*F3-c*w*F1)).subs(u,x_yz).subs(w,w(v))).rhs + x_y = dsolve(diff(u(v),v) - ((c*w*F2-b*v*F3)/(a*u*F3-c*w*F1)).subs(w,z_xy).subs(u,u(v))).rhs + y_z = dsolve(diff(v(w),w) - ((a*u*F3-c*w*F1)/(b*v*F1-a*u*F2)).subs(u,x_yz).subs(v,v(w))).rhs + x_z = dsolve(diff(u(w),w) - ((c*w*F2-b*v*F3)/(b*v*F1-a*u*F2)).subs(v,y_zx).subs(u,u(w))).rhs + sol1 = dsolve(diff(u(t),t) - (c*w*F2 - b*v*F3).subs(v,y_x).subs(w,z_x).subs(u,u(t))).rhs + sol2 = dsolve(diff(v(t),t) - (a*u*F3 - c*w*F1).subs(u,x_y).subs(w,z_y).subs(v,v(t))).rhs + sol3 = dsolve(diff(w(t),t) - (b*v*F1 - a*u*F2).subs(u,x_z).subs(v,y_z).subs(w,w(t))).rhs + return [sol1, sol2, sol3] + +def _nonlinear_3eq_order1_type5(x, y, z, t, eq): + r""" + .. math:: x' = x (c F_2 - b F_3), \enspace y' = y (a F_3 - c F_1), \enspace z' = z (b F_1 - a F_2) + + where `F_n = F_n (x, y, z, t)` and are arbitrary functions. + + First Integral: + + .. math:: \left|x\right|^{a} \left|y\right|^{b} \left|z\right|^{c} = C_1 + + where `C` is an arbitrary constant. If the function `F_n` is independent of `t`, + then, by eliminating `t` and `z` from the first two equations of the system, one + arrives at a first-order equation. + + References + ========== + -https://eqworld.ipmnet.ru/en/solutions/sysode/sode0406.pdf + + """ + C1 = get_numbered_constants(eq, num=1) + u, v, w = symbols('u, v, w') + fu, fv, fw = symbols('u, v, w', cls=Function) + p = Wild('p', exclude=[x(t), y(t), z(t), t]) + q = Wild('q', exclude=[x(t), y(t), z(t), t]) + s = Wild('s', exclude=[x(t), y(t), z(t), t]) + F1, F2, F3 = symbols('F1, F2, F3', cls=Wild) + r1 = eq[0].match(diff(x(t), t) - x(t)*F2 + x(t)*F3) + r = collect_const(r1[F2]).match(s*F2) + r.update(collect_const(r1[F3]).match(q*F3)) + if eq[1].has(r[F2]) and not eq[1].has(r[F3]): + r[F2], r[F3] = r[F3], r[F2] + r[s], r[q] = -r[q], -r[s] + r.update((diff(y(t), t) - eq[1]).match(y(t)*(p*r[F3] - r[s]*F1))) + a = r[p]; b = r[q]; c = r[s] + F1 = r[F1].subs(x(t), u).subs(y(t), v).subs(z(t), w) + F2 = r[F2].subs(x(t), u).subs(y(t), v).subs(z(t), w) + F3 = r[F3].subs(x(t), u).subs(y(t), v).subs(z(t), w) + x_yz = (C1*v**-b*w**-c)**-a + y_zx = (C1*w**-c*u**-a)**-b + z_xy = (C1*u**-a*v**-b)**-c + y_x = dsolve(diff(fv(u), u) - ((v*(a*F3 - c*F1))/(u*(c*F2 - b*F3))).subs(w, z_xy).subs(v, fv(u))).rhs + z_x = dsolve(diff(fw(u), u) - ((w*(b*F1 - a*F2))/(u*(c*F2 - b*F3))).subs(v, y_zx).subs(w, fw(u))).rhs + z_y = dsolve(diff(fw(v), v) - ((w*(b*F1 - a*F2))/(v*(a*F3 - c*F1))).subs(u, x_yz).subs(w, fw(v))).rhs + x_y = dsolve(diff(fu(v), v) - ((u*(c*F2 - b*F3))/(v*(a*F3 - c*F1))).subs(w, z_xy).subs(u, fu(v))).rhs + y_z = dsolve(diff(fv(w), w) - ((v*(a*F3 - c*F1))/(w*(b*F1 - a*F2))).subs(u, x_yz).subs(v, fv(w))).rhs + x_z = dsolve(diff(fu(w), w) - ((u*(c*F2 - b*F3))/(w*(b*F1 - a*F2))).subs(v, y_zx).subs(u, fu(w))).rhs + sol1 = dsolve(diff(fu(t), t) - (u*(c*F2 - b*F3)).subs(v, y_x).subs(w, z_x).subs(u, fu(t))).rhs + sol2 = dsolve(diff(fv(t), t) - (v*(a*F3 - c*F1)).subs(u, x_y).subs(w, z_y).subs(v, fv(t))).rhs + sol3 = dsolve(diff(fw(t), t) - (w*(b*F1 - a*F2)).subs(u, x_z).subs(v, y_z).subs(w, fw(t))).rhs + return [sol1, sol2, sol3] + + +#This import is written at the bottom to avoid circular imports. +from .single import SingleODEProblem, SingleODESolver, solver_map diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/ode/riccati.py b/.venv/lib/python3.12/site-packages/sympy/solvers/ode/riccati.py new file mode 100644 index 0000000000000000000000000000000000000000..2ef66ed0896d39bee8fba1b74a0c93734742fc1f --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/ode/riccati.py @@ -0,0 +1,893 @@ +r""" +This module contains :py:meth:`~sympy.solvers.ode.riccati.solve_riccati`, +a function which gives all rational particular solutions to first order +Riccati ODEs. A general first order Riccati ODE is given by - + +.. math:: y' = b_0(x) + b_1(x)w + b_2(x)w^2 + +where `b_0, b_1` and `b_2` can be arbitrary rational functions of `x` +with `b_2 \ne 0`. When `b_2 = 0`, the equation is not a Riccati ODE +anymore and becomes a Linear ODE. Similarly, when `b_0 = 0`, the equation +is a Bernoulli ODE. The algorithm presented below can find rational +solution(s) to all ODEs with `b_2 \ne 0` that have a rational solution, +or prove that no rational solution exists for the equation. + +Background +========== + +A Riccati equation can be transformed to its normal form + +.. math:: y' + y^2 = a(x) + +using the transformation + +.. math:: y = -b_2(x) - \frac{b'_2(x)}{2 b_2(x)} - \frac{b_1(x)}{2} + +where `a(x)` is given by + +.. math:: a(x) = \frac{1}{4}\left(\frac{b_2'}{b_2} + b_1\right)^2 - \frac{1}{2}\left(\frac{b_2'}{b_2} + b_1\right)' - b_0 b_2 + +Thus, we can develop an algorithm to solve for the Riccati equation +in its normal form, which would in turn give us the solution for +the original Riccati equation. + +Algorithm +========= + +The algorithm implemented here is presented in the Ph.D thesis +"Rational and Algebraic Solutions of First-Order Algebraic ODEs" +by N. Thieu Vo. The entire thesis can be found here - +https://www3.risc.jku.at/publications/download/risc_5387/PhDThesisThieu.pdf + +We have only implemented the Rational Riccati solver (Algorithm 11, +Pg 78-82 in Thesis). Before we proceed towards the implementation +of the algorithm, a few definitions to understand are - + +1. Valuation of a Rational Function at `\infty`: + The valuation of a rational function `p(x)` at `\infty` is equal + to the difference between the degree of the denominator and the + numerator of `p(x)`. + + NOTE: A general definition of valuation of a rational function + at any value of `x` can be found in Pg 63 of the thesis, but + is not of any interest for this algorithm. + +2. Zeros and Poles of a Rational Function: + Let `a(x) = \frac{S(x)}{T(x)}, T \ne 0` be a rational function + of `x`. Then - + + a. The Zeros of `a(x)` are the roots of `S(x)`. + b. The Poles of `a(x)` are the roots of `T(x)`. However, `\infty` + can also be a pole of a(x). We say that `a(x)` has a pole at + `\infty` if `a(\frac{1}{x})` has a pole at 0. + +Every pole is associated with an order that is equal to the multiplicity +of its appearance as a root of `T(x)`. A pole is called a simple pole if +it has an order 1. Similarly, a pole is called a multiple pole if it has +an order `\ge` 2. + +Necessary Conditions +==================== + +For a Riccati equation in its normal form, + +.. math:: y' + y^2 = a(x) + +we can define + +a. A pole is called a movable pole if it is a pole of `y(x)` and is not +a pole of `a(x)`. +b. Similarly, a pole is called a non-movable pole if it is a pole of both +`y(x)` and `a(x)`. + +Then, the algorithm states that a rational solution exists only if - + +a. Every pole of `a(x)` must be either a simple pole or a multiple pole +of even order. +b. The valuation of `a(x)` at `\infty` must be even or be `\ge` 2. + +This algorithm finds all possible rational solutions for the Riccati ODE. +If no rational solutions are found, it means that no rational solutions +exist. + +The algorithm works for Riccati ODEs where the coefficients are rational +functions in the independent variable `x` with rational number coefficients +i.e. in `Q(x)`. The coefficients in the rational function cannot be floats, +irrational numbers, symbols or any other kind of expression. The reasons +for this are - + +1. When using symbols, different symbols could take the same value and this +would affect the multiplicity of poles if symbols are present here. + +2. An integer degree bound is required to calculate a polynomial solution +to an auxiliary differential equation, which in turn gives the particular +solution for the original ODE. If symbols/floats/irrational numbers are +present, we cannot determine if the expression for the degree bound is an +integer or not. + +Solution +======== + +With these definitions, we can state a general form for the solution of +the equation. `y(x)` must have the form - + +.. math:: y(x) = \sum_{i=1}^{n} \sum_{j=1}^{r_i} \frac{c_{ij}}{(x - x_i)^j} + \sum_{i=1}^{m} \frac{1}{x - \chi_i} + \sum_{i=0}^{N} d_i x^i + +where `x_1, x_2, \dots, x_n` are non-movable poles of `a(x)`, +`\chi_1, \chi_2, \dots, \chi_m` are movable poles of `a(x)`, and the values +of `N, n, r_1, r_2, \dots, r_n` can be determined from `a(x)`. The +coefficient vectors `(d_0, d_1, \dots, d_N)` and `(c_{i1}, c_{i2}, \dots, c_{i r_i})` +can be determined from `a(x)`. We will have 2 choices each of these vectors +and part of the procedure is figuring out which of the 2 should be used +to get the solution correctly. + +Implementation +============== + +In this implementation, we use ``Poly`` to represent a rational function +rather than using ``Expr`` since ``Poly`` is much faster. Since we cannot +represent rational functions directly using ``Poly``, we instead represent +a rational function with 2 ``Poly`` objects - one for its numerator and +the other for its denominator. + +The code is written to match the steps given in the thesis (Pg 82) + +Step 0 : Match the equation - +Find `b_0, b_1` and `b_2`. If `b_2 = 0` or no such functions exist, raise +an error + +Step 1 : Transform the equation to its normal form as explained in the +theory section. + +Step 2 : Initialize an empty set of solutions, ``sol``. + +Step 3 : If `a(x) = 0`, append `\frac{1}/{(x - C1)}` to ``sol``. + +Step 4 : If `a(x)` is a rational non-zero number, append `\pm \sqrt{a}` +to ``sol``. + +Step 5 : Find the poles and their multiplicities of `a(x)`. Let +the number of poles be `n`. Also find the valuation of `a(x)` at +`\infty` using ``val_at_inf``. + +NOTE: Although the algorithm considers `\infty` as a pole, it is +not mentioned if it a part of the set of finite poles. `\infty` +is NOT a part of the set of finite poles. If a pole exists at +`\infty`, we use its multiplicity to find the laurent series of +`a(x)` about `\infty`. + +Step 6 : Find `n` c-vectors (one for each pole) and 1 d-vector using +``construct_c`` and ``construct_d``. Now, determine all the ``2**(n + 1)`` +combinations of choosing between 2 choices for each of the `n` c-vectors +and 1 d-vector. + +NOTE: The equation for `d_{-1}` in Case 4 (Pg 80) has a printinig +mistake. The term `- d_N` must be replaced with `-N d_N`. The same +has been explained in the code as well. + +For each of these above combinations, do + +Step 8 : Compute `m` in ``compute_m_ybar``. `m` is the degree bound of +the polynomial solution we must find for the auxiliary equation. + +Step 9 : In ``compute_m_ybar``, compute ybar as well where ``ybar`` is +one part of y(x) - + +.. math:: \overline{y}(x) = \sum_{i=1}^{n} \sum_{j=1}^{r_i} \frac{c_{ij}}{(x - x_i)^j} + \sum_{i=0}^{N} d_i x^i + +Step 10 : If `m` is a non-negative integer - + +Step 11: Find a polynomial solution of degree `m` for the auxiliary equation. + +There are 2 cases possible - + + a. `m` is a non-negative integer: We can solve for the coefficients + in `p(x)` using Undetermined Coefficients. + + b. `m` is not a non-negative integer: In this case, we cannot find + a polynomial solution to the auxiliary equation, and hence, we ignore + this value of `m`. + +Step 12 : For each `p(x)` that exists, append `ybar + \frac{p'(x)}{p(x)}` +to ``sol``. + +Step 13 : For each solution in ``sol``, apply an inverse transformation, +so that the solutions of the original equation are found using the +solutions of the equation in its normal form. +""" + + +from itertools import product +from sympy.core import S +from sympy.core.add import Add +from sympy.core.numbers import oo, Float +from sympy.core.function import count_ops +from sympy.core.relational import Eq +from sympy.core.symbol import symbols, Symbol, Dummy +from sympy.functions import sqrt, exp +from sympy.functions.elementary.complexes import sign +from sympy.integrals.integrals import Integral +from sympy.polys.domains import ZZ +from sympy.polys.polytools import Poly +from sympy.polys.polyroots import roots +from sympy.solvers.solveset import linsolve + + +def riccati_normal(w, x, b1, b2): + """ + Given a solution `w(x)` to the equation + + .. math:: w'(x) = b_0(x) + b_1(x)*w(x) + b_2(x)*w(x)^2 + + and rational function coefficients `b_1(x)` and + `b_2(x)`, this function transforms the solution to + give a solution `y(x)` for its corresponding normal + Riccati ODE + + .. math:: y'(x) + y(x)^2 = a(x) + + using the transformation + + .. math:: y(x) = -b_2(x)*w(x) - b'_2(x)/(2*b_2(x)) - b_1(x)/2 + """ + return -b2*w - b2.diff(x)/(2*b2) - b1/2 + + +def riccati_inverse_normal(y, x, b1, b2, bp=None): + """ + Inverse transforming the solution to the normal + Riccati ODE to get the solution to the Riccati ODE. + """ + # bp is the expression which is independent of the solution + # and hence, it need not be computed again + if bp is None: + bp = -b2.diff(x)/(2*b2**2) - b1/(2*b2) + # w(x) = -y(x)/b2(x) - b2'(x)/(2*b2(x)^2) - b1(x)/(2*b2(x)) + return -y/b2 + bp + + +def riccati_reduced(eq, f, x): + """ + Convert a Riccati ODE into its corresponding + normal Riccati ODE. + """ + match, funcs = match_riccati(eq, f, x) + # If equation is not a Riccati ODE, exit + if not match: + return False + # Using the rational functions, find the expression for a(x) + b0, b1, b2 = funcs + a = -b0*b2 + b1**2/4 - b1.diff(x)/2 + 3*b2.diff(x)**2/(4*b2**2) + b1*b2.diff(x)/(2*b2) - \ + b2.diff(x, 2)/(2*b2) + # Normal form of Riccati ODE is f'(x) + f(x)^2 = a(x) + return f(x).diff(x) + f(x)**2 - a + +def linsolve_dict(eq, syms): + """ + Get the output of linsolve as a dict + """ + # Convert tuple type return value of linsolve + # to a dictionary for ease of use + sol = linsolve(eq, syms) + if not sol: + return {} + return dict(zip(syms, list(sol)[0])) + + +def match_riccati(eq, f, x): + """ + A function that matches and returns the coefficients + if an equation is a Riccati ODE + + Parameters + ========== + + eq: Equation to be matched + f: Dependent variable + x: Independent variable + + Returns + ======= + + match: True if equation is a Riccati ODE, False otherwise + funcs: [b0, b1, b2] if match is True, [] otherwise. Here, + b0, b1 and b2 are rational functions which match the equation. + """ + # Group terms based on f(x) + if isinstance(eq, Eq): + eq = eq.lhs - eq.rhs + eq = eq.expand().collect(f(x)) + cf = eq.coeff(f(x).diff(x)) + + # There must be an f(x).diff(x) term. + # eq must be an Add object since we are using the expanded + # equation and it must have atleast 2 terms (b2 != 0) + if cf != 0 and isinstance(eq, Add): + + # Divide all coefficients by the coefficient of f(x).diff(x) + # and add the terms again to get the same equation + eq = Add(*((x/cf).cancel() for x in eq.args)).collect(f(x)) + + # Match the equation with the pattern + b1 = -eq.coeff(f(x)) + b2 = -eq.coeff(f(x)**2) + b0 = (f(x).diff(x) - b1*f(x) - b2*f(x)**2 - eq).expand() + funcs = [b0, b1, b2] + + # Check if coefficients are not symbols and floats + if any(len(x.atoms(Symbol)) > 1 or len(x.atoms(Float)) for x in funcs): + return False, [] + + # If b_0(x) contains f(x), it is not a Riccati ODE + if len(b0.atoms(f)) or not all((b2 != 0, b0.is_rational_function(x), + b1.is_rational_function(x), b2.is_rational_function(x))): + return False, [] + return True, funcs + return False, [] + + +def val_at_inf(num, den, x): + # Valuation of a rational function at oo = deg(denom) - deg(numer) + return den.degree(x) - num.degree(x) + + +def check_necessary_conds(val_inf, muls): + """ + The necessary conditions for a rational solution + to exist are as follows - + + i) Every pole of a(x) must be either a simple pole + or a multiple pole of even order. + + ii) The valuation of a(x) at infinity must be even + or be greater than or equal to 2. + + Here, a simple pole is a pole with multiplicity 1 + and a multiple pole is a pole with multiplicity + greater than 1. + """ + return (val_inf >= 2 or (val_inf <= 0 and val_inf%2 == 0)) and \ + all(mul == 1 or (mul%2 == 0 and mul >= 2) for mul in muls) + + +def inverse_transform_poly(num, den, x): + """ + A function to make the substitution + x -> 1/x in a rational function that + is represented using Poly objects for + numerator and denominator. + """ + # Declare for reuse + one = Poly(1, x) + xpoly = Poly(x, x) + + # Check if degree of numerator is same as denominator + pwr = val_at_inf(num, den, x) + if pwr >= 0: + # Denominator has greater degree. Substituting x with + # 1/x would make the extra power go to the numerator + if num.expr != 0: + num = num.transform(one, xpoly) * x**pwr + den = den.transform(one, xpoly) + else: + # Numerator has greater degree. Substituting x with + # 1/x would make the extra power go to the denominator + num = num.transform(one, xpoly) + den = den.transform(one, xpoly) * x**(-pwr) + return num.cancel(den, include=True) + + +def limit_at_inf(num, den, x): + """ + Find the limit of a rational function + at oo + """ + # pwr = degree(num) - degree(den) + pwr = -val_at_inf(num, den, x) + # Numerator has a greater degree than denominator + # Limit at infinity would depend on the sign of the + # leading coefficients of numerator and denominator + if pwr > 0: + return oo*sign(num.LC()/den.LC()) + # Degree of numerator is equal to that of denominator + # Limit at infinity is just the ratio of leading coeffs + elif pwr == 0: + return num.LC()/den.LC() + # Degree of numerator is less than that of denominator + # Limit at infinity is just 0 + else: + return 0 + + +def construct_c_case_1(num, den, x, pole): + # Find the coefficient of 1/(x - pole)**2 in the + # Laurent series expansion of a(x) about pole. + num1, den1 = (num*Poly((x - pole)**2, x, extension=True)).cancel(den, include=True) + r = (num1.subs(x, pole))/(den1.subs(x, pole)) + + # If multiplicity is 2, the coefficient to be added + # in the c-vector is c = (1 +- sqrt(1 + 4*r))/2 + if r != -S(1)/4: + return [[(1 + sqrt(1 + 4*r))/2], [(1 - sqrt(1 + 4*r))/2]] + return [[S.Half]] + + +def construct_c_case_2(num, den, x, pole, mul): + # Generate the coefficients using the recurrence + # relation mentioned in (5.14) in the thesis (Pg 80) + + # r_i = mul/2 + ri = mul//2 + + # Find the Laurent series coefficients about the pole + ser = rational_laurent_series(num, den, x, pole, mul, 6) + + # Start with an empty memo to store the coefficients + # This is for the plus case + cplus = [0 for i in range(ri)] + + # Base Case + cplus[ri-1] = sqrt(ser[2*ri]) + + # Iterate backwards to find all coefficients + s = ri - 1 + sm = 0 + for s in range(ri-1, 0, -1): + sm = 0 + for j in range(s+1, ri): + sm += cplus[j-1]*cplus[ri+s-j-1] + if s!= 1: + cplus[s-1] = (ser[ri+s] - sm)/(2*cplus[ri-1]) + + # Memo for the minus case + cminus = [-x for x in cplus] + + # Find the 0th coefficient in the recurrence + cplus[0] = (ser[ri+s] - sm - ri*cplus[ri-1])/(2*cplus[ri-1]) + cminus[0] = (ser[ri+s] - sm - ri*cminus[ri-1])/(2*cminus[ri-1]) + + # Add both the plus and minus cases' coefficients + if cplus != cminus: + return [cplus, cminus] + return cplus + + +def construct_c_case_3(): + # If multiplicity is 1, the coefficient to be added + # in the c-vector is 1 (no choice) + return [[1]] + + +def construct_c(num, den, x, poles, muls): + """ + Helper function to calculate the coefficients + in the c-vector for each pole. + """ + c = [] + for pole, mul in zip(poles, muls): + c.append([]) + + # Case 3 + if mul == 1: + # Add the coefficients from Case 3 + c[-1].extend(construct_c_case_3()) + + # Case 1 + elif mul == 2: + # Add the coefficients from Case 1 + c[-1].extend(construct_c_case_1(num, den, x, pole)) + + # Case 2 + else: + # Add the coefficients from Case 2 + c[-1].extend(construct_c_case_2(num, den, x, pole, mul)) + + return c + + +def construct_d_case_4(ser, N): + # Initialize an empty vector + dplus = [0 for i in range(N+2)] + # d_N = sqrt(a_{2*N}) + dplus[N] = sqrt(ser[2*N]) + + # Use the recurrence relations to find + # the value of d_s + for s in range(N-1, -2, -1): + sm = 0 + for j in range(s+1, N): + sm += dplus[j]*dplus[N+s-j] + if s != -1: + dplus[s] = (ser[N+s] - sm)/(2*dplus[N]) + + # Coefficients for the case of d_N = -sqrt(a_{2*N}) + dminus = [-x for x in dplus] + + # The third equation in Eq 5.15 of the thesis is WRONG! + # d_N must be replaced with N*d_N in that equation. + dplus[-1] = (ser[N+s] - N*dplus[N] - sm)/(2*dplus[N]) + dminus[-1] = (ser[N+s] - N*dminus[N] - sm)/(2*dminus[N]) + + if dplus != dminus: + return [dplus, dminus] + return dplus + + +def construct_d_case_5(ser): + # List to store coefficients for plus case + dplus = [0, 0] + + # d_0 = sqrt(a_0) + dplus[0] = sqrt(ser[0]) + + # d_(-1) = a_(-1)/(2*d_0) + dplus[-1] = ser[-1]/(2*dplus[0]) + + # Coefficients for the minus case are just the negative + # of the coefficients for the positive case. + dminus = [-x for x in dplus] + + if dplus != dminus: + return [dplus, dminus] + return dplus + + +def construct_d_case_6(num, den, x): + # s_oo = lim x->0 1/x**2 * a(1/x) which is equivalent to + # s_oo = lim x->oo x**2 * a(x) + s_inf = limit_at_inf(Poly(x**2, x)*num, den, x) + + # d_(-1) = (1 +- sqrt(1 + 4*s_oo))/2 + if s_inf != -S(1)/4: + return [[(1 + sqrt(1 + 4*s_inf))/2], [(1 - sqrt(1 + 4*s_inf))/2]] + return [[S.Half]] + + +def construct_d(num, den, x, val_inf): + """ + Helper function to calculate the coefficients + in the d-vector based on the valuation of the + function at oo. + """ + N = -val_inf//2 + # Multiplicity of oo as a pole + mul = -val_inf if val_inf < 0 else 0 + ser = rational_laurent_series(num, den, x, oo, mul, 1) + + # Case 4 + if val_inf < 0: + d = construct_d_case_4(ser, N) + + # Case 5 + elif val_inf == 0: + d = construct_d_case_5(ser) + + # Case 6 + else: + d = construct_d_case_6(num, den, x) + + return d + + +def rational_laurent_series(num, den, x, r, m, n): + r""" + The function computes the Laurent series coefficients + of a rational function. + + Parameters + ========== + + num: A Poly object that is the numerator of `f(x)`. + den: A Poly object that is the denominator of `f(x)`. + x: The variable of expansion of the series. + r: The point of expansion of the series. + m: Multiplicity of r if r is a pole of `f(x)`. Should + be zero otherwise. + n: Order of the term upto which the series is expanded. + + Returns + ======= + + series: A dictionary that has power of the term as key + and coefficient of that term as value. + + Below is a basic outline of how the Laurent series of a + rational function `f(x)` about `x_0` is being calculated - + + 1. Substitute `x + x_0` in place of `x`. If `x_0` + is a pole of `f(x)`, multiply the expression by `x^m` + where `m` is the multiplicity of `x_0`. Denote the + the resulting expression as g(x). We do this substitution + so that we can now find the Laurent series of g(x) about + `x = 0`. + + 2. We can then assume that the Laurent series of `g(x)` + takes the following form - + + .. math:: g(x) = \frac{num(x)}{den(x)} = \sum_{m = 0}^{\infty} a_m x^m + + where `a_m` denotes the Laurent series coefficients. + + 3. Multiply the denominator to the RHS of the equation + and form a recurrence relation for the coefficients `a_m`. + """ + one = Poly(1, x, extension=True) + + if r == oo: + # Series at x = oo is equal to first transforming + # the function from x -> 1/x and finding the + # series at x = 0 + num, den = inverse_transform_poly(num, den, x) + r = S(0) + + if r: + # For an expansion about a non-zero point, a + # transformation from x -> x + r must be made + num = num.transform(Poly(x + r, x, extension=True), one) + den = den.transform(Poly(x + r, x, extension=True), one) + + # Remove the pole from the denominator if the series + # expansion is about one of the poles + num, den = (num*x**m).cancel(den, include=True) + + # Equate coefficients for the first terms (base case) + maxdegree = 1 + max(num.degree(), den.degree()) + syms = symbols(f'a:{maxdegree}', cls=Dummy) + diff = num - den * Poly(syms[::-1], x) + coeff_diffs = diff.all_coeffs()[::-1][:maxdegree] + (coeffs, ) = linsolve(coeff_diffs, syms) + + # Use the recursion relation for the rest + recursion = den.all_coeffs()[::-1] + div, rec_rhs = recursion[0], recursion[1:] + series = list(coeffs) + while len(series) < n: + next_coeff = Add(*(c*series[-1-n] for n, c in enumerate(rec_rhs))) / div + series.append(-next_coeff) + series = {m - i: val for i, val in enumerate(series)} + return series + +def compute_m_ybar(x, poles, choice, N): + """ + Helper function to calculate - + + 1. m - The degree bound for the polynomial + solution that must be found for the auxiliary + differential equation. + + 2. ybar - Part of the solution which can be + computed using the poles, c and d vectors. + """ + ybar = 0 + m = Poly(choice[-1][-1], x, extension=True) + + # Calculate the first (nested) summation for ybar + # as given in Step 9 of the Thesis (Pg 82) + dybar = [] + for i, polei in enumerate(poles): + for j, cij in enumerate(choice[i]): + dybar.append(cij/(x - polei)**(j + 1)) + m -=Poly(choice[i][0], x, extension=True) # can't accumulate Poly and use with Add + ybar += Add(*dybar) + + # Calculate the second summation for ybar + for i in range(N+1): + ybar += choice[-1][i]*x**i + return (m.expr, ybar) + + +def solve_aux_eq(numa, dena, numy, deny, x, m): + """ + Helper function to find a polynomial solution + of degree m for the auxiliary differential + equation. + """ + # Assume that the solution is of the type + # p(x) = C_0 + C_1*x + ... + C_{m-1}*x**(m-1) + x**m + psyms = symbols(f'C0:{m}', cls=Dummy) + K = ZZ[psyms] + psol = Poly(K.gens, x, domain=K) + Poly(x**m, x, domain=K) + + # Eq (5.16) in Thesis - Pg 81 + auxeq = (dena*(numy.diff(x)*deny - numy*deny.diff(x) + numy**2) - numa*deny**2)*psol + if m >= 1: + px = psol.diff(x) + auxeq += px*(2*numy*deny*dena) + if m >= 2: + auxeq += px.diff(x)*(deny**2*dena) + if m != 0: + # m is a non-zero integer. Find the constant terms using undetermined coefficients + return psol, linsolve_dict(auxeq.all_coeffs(), psyms), True + else: + # m == 0 . Check if 1 (x**0) is a solution to the auxiliary equation + return S.One, auxeq, auxeq == 0 + + +def remove_redundant_sols(sol1, sol2, x): + """ + Helper function to remove redundant + solutions to the differential equation. + """ + # If y1 and y2 are redundant solutions, there is + # some value of the arbitrary constant for which + # they will be equal + + syms1 = sol1.atoms(Symbol, Dummy) + syms2 = sol2.atoms(Symbol, Dummy) + num1, den1 = [Poly(e, x, extension=True) for e in sol1.together().as_numer_denom()] + num2, den2 = [Poly(e, x, extension=True) for e in sol2.together().as_numer_denom()] + # Cross multiply + e = num1*den2 - den1*num2 + # Check if there are any constants + syms = list(e.atoms(Symbol, Dummy)) + if len(syms): + # Find values of constants for which solutions are equal + redn = linsolve(e.all_coeffs(), syms) + if len(redn): + # Return the general solution over a particular solution + if len(syms1) > len(syms2): + return sol2 + # If both have constants, return the lesser complex solution + elif len(syms1) == len(syms2): + return sol1 if count_ops(syms1) >= count_ops(syms2) else sol2 + else: + return sol1 + + +def get_gen_sol_from_part_sol(part_sols, a, x): + """" + Helper function which computes the general + solution for a Riccati ODE from its particular + solutions. + + There are 3 cases to find the general solution + from the particular solutions for a Riccati ODE + depending on the number of particular solution(s) + we have - 1, 2 or 3. + + For more information, see Section 6 of + "Methods of Solution of the Riccati Differential Equation" + by D. R. Haaheim and F. M. Stein + """ + + # If no particular solutions are found, a general + # solution cannot be found + if len(part_sols) == 0: + return [] + + # In case of a single particular solution, the general + # solution can be found by using the substitution + # y = y1 + 1/z and solving a Bernoulli ODE to find z. + elif len(part_sols) == 1: + y1 = part_sols[0] + i = exp(Integral(2*y1, x)) + z = i * Integral(a/i, x) + z = z.doit() + if a == 0 or z == 0: + return y1 + return y1 + 1/z + + # In case of 2 particular solutions, the general solution + # can be found by solving a separable equation. This is + # the most common case, i.e. most Riccati ODEs have 2 + # rational particular solutions. + elif len(part_sols) == 2: + y1, y2 = part_sols + # One of them already has a constant + if len(y1.atoms(Dummy)) + len(y2.atoms(Dummy)) > 0: + u = exp(Integral(y2 - y1, x)).doit() + # Introduce a constant + else: + C1 = Dummy('C1') + u = C1*exp(Integral(y2 - y1, x)).doit() + if u == 1: + return y2 + return (y2*u - y1)/(u - 1) + + # In case of 3 particular solutions, a closed form + # of the general solution can be obtained directly + else: + y1, y2, y3 = part_sols[:3] + C1 = Dummy('C1') + return (C1 + 1)*y2*(y1 - y3)/(C1*y1 + y2 - (C1 + 1)*y3) + + +def solve_riccati(fx, x, b0, b1, b2, gensol=False): + """ + The main function that gives particular/general + solutions to Riccati ODEs that have atleast 1 + rational particular solution. + """ + # Step 1 : Convert to Normal Form + a = -b0*b2 + b1**2/4 - b1.diff(x)/2 + 3*b2.diff(x)**2/(4*b2**2) + b1*b2.diff(x)/(2*b2) - \ + b2.diff(x, 2)/(2*b2) + a_t = a.together() + num, den = [Poly(e, x, extension=True) for e in a_t.as_numer_denom()] + num, den = num.cancel(den, include=True) + + # Step 2 + presol = [] + + # Step 3 : a(x) is 0 + if num == 0: + presol.append(1/(x + Dummy('C1'))) + + # Step 4 : a(x) is a non-zero constant + elif x not in num.free_symbols.union(den.free_symbols): + presol.extend([sqrt(a), -sqrt(a)]) + + # Step 5 : Find poles and valuation at infinity + poles = roots(den, x) + poles, muls = list(poles.keys()), list(poles.values()) + val_inf = val_at_inf(num, den, x) + + if len(poles): + # Check necessary conditions (outlined in the module docstring) + if not check_necessary_conds(val_inf, muls): + raise ValueError("Rational Solution doesn't exist") + + # Step 6 + # Construct c-vectors for each singular point + c = construct_c(num, den, x, poles, muls) + + # Construct d vectors for each singular point + d = construct_d(num, den, x, val_inf) + + # Step 7 : Iterate over all possible combinations and return solutions + # For each possible combination, generate an array of 0's and 1's + # where 0 means pick 1st choice and 1 means pick the second choice. + + # NOTE: We could exit from the loop if we find 3 particular solutions, + # but it is not implemented here as - + # a. Finding 3 particular solutions is very rare. Most of the time, + # only 2 particular solutions are found. + # b. In case we exit after finding 3 particular solutions, it might + # happen that 1 or 2 of them are redundant solutions. So, instead of + # spending some more time in computing the particular solutions, + # we will end up computing the general solution from a single + # particular solution which is usually slower than computing the + # general solution from 2 or 3 particular solutions. + c.append(d) + choices = product(*c) + for choice in choices: + m, ybar = compute_m_ybar(x, poles, choice, -val_inf//2) + numy, deny = [Poly(e, x, extension=True) for e in ybar.together().as_numer_denom()] + # Step 10 : Check if a valid solution exists. If yes, also check + # if m is a non-negative integer + if m.is_nonnegative == True and m.is_integer == True: + + # Step 11 : Find polynomial solutions of degree m for the auxiliary equation + psol, coeffs, exists = solve_aux_eq(num, den, numy, deny, x, m) + + # Step 12 : If valid polynomial solution exists, append solution. + if exists: + # m == 0 case + if psol == 1 and coeffs == 0: + # p(x) = 1, so p'(x)/p(x) term need not be added + presol.append(ybar) + # m is a positive integer and there are valid coefficients + elif len(coeffs): + # Substitute the valid coefficients to get p(x) + psol = psol.xreplace(coeffs) + # y(x) = ybar(x) + p'(x)/p(x) + presol.append(ybar + psol.diff(x)/psol) + + # Remove redundant solutions from the list of existing solutions + remove = set() + for i in range(len(presol)): + for j in range(i+1, len(presol)): + rem = remove_redundant_sols(presol[i], presol[j], x) + if rem is not None: + remove.add(rem) + sols = [x for x in presol if x not in remove] + + # Step 15 : Inverse transform the solutions of the equation in normal form + bp = -b2.diff(x)/(2*b2**2) - b1/(2*b2) + + # If general solution is required, compute it from the particular solutions + if gensol: + sols = [get_gen_sol_from_part_sol(sols, a, x)] + + # Inverse transform the particular solutions + presol = [Eq(fx, riccati_inverse_normal(y, x, b1, b2, bp).cancel(extension=True)) for y in sols] + return presol diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/ode/single.py b/.venv/lib/python3.12/site-packages/sympy/solvers/ode/single.py new file mode 100644 index 0000000000000000000000000000000000000000..c4829acf41293c2f6f20af3e9c45d37457802102 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/ode/single.py @@ -0,0 +1,2977 @@ +# +# This is the module for ODE solver classes for single ODEs. +# + +from __future__ import annotations +from typing import ClassVar, Iterator + +from .riccati import match_riccati, solve_riccati +from sympy.core import Add, S, Pow, Rational +from sympy.core.cache import cached_property +from sympy.core.exprtools import factor_terms +from sympy.core.expr import Expr +from sympy.core.function import AppliedUndef, Derivative, diff, Function, expand, Subs, _mexpand +from sympy.core.numbers import zoo +from sympy.core.relational import Equality, Eq +from sympy.core.symbol import Symbol, Dummy, Wild +from sympy.core.mul import Mul +from sympy.functions import exp, tan, log, sqrt, besselj, bessely, cbrt, airyai, airybi +from sympy.integrals import Integral +from sympy.polys import Poly +from sympy.polys.polytools import cancel, factor, degree +from sympy.simplify import collect, simplify, separatevars, logcombine, posify # type: ignore +from sympy.simplify.radsimp import fraction +from sympy.utilities import numbered_symbols +from sympy.solvers.solvers import solve +from sympy.solvers.deutils import ode_order, _preprocess +from sympy.polys.matrices.linsolve import _lin_eq2dict +from sympy.polys.solvers import PolyNonlinearError +from .hypergeometric import equivalence_hypergeometric, match_2nd_2F1_hypergeometric, \ + get_sol_2F1_hypergeometric, match_2nd_hypergeometric +from .nonhomogeneous import _get_euler_characteristic_eq_sols, _get_const_characteristic_eq_sols, \ + _solve_undetermined_coefficients, _solve_variation_of_parameters, _test_term, _undetermined_coefficients_match, \ + _get_simplified_sol +from .lie_group import _ode_lie_group + + +class ODEMatchError(NotImplementedError): + """Raised if a SingleODESolver is asked to solve an ODE it does not match""" + pass + + +class SingleODEProblem: + """Represents an ordinary differential equation (ODE) + + This class is used internally in the by dsolve and related + functions/classes so that properties of an ODE can be computed + efficiently. + + Examples + ======== + + This class is used internally by dsolve. To instantiate an instance + directly first define an ODE problem: + + >>> from sympy import Function, Symbol + >>> x = Symbol('x') + >>> f = Function('f') + >>> eq = f(x).diff(x, 2) + + Now you can create a SingleODEProblem instance and query its properties: + + >>> from sympy.solvers.ode.single import SingleODEProblem + >>> problem = SingleODEProblem(f(x).diff(x), f(x), x) + >>> problem.eq + Derivative(f(x), x) + >>> problem.func + f(x) + >>> problem.sym + x + """ + + # Instance attributes: + eq: Expr + func: AppliedUndef + sym: Symbol + _order: int + _eq_expanded: Expr + _eq_preprocessed: Expr + _eq_high_order_free = None + + def __init__(self, eq, func, sym, prep=True, **kwargs): + assert isinstance(eq, Expr) + assert isinstance(func, AppliedUndef) + assert isinstance(sym, Symbol) + assert isinstance(prep, bool) + self.eq = eq + self.func = func + self.sym = sym + self.prep = prep + self.params = kwargs + + @cached_property + def order(self) -> int: + return ode_order(self.eq, self.func) + + @cached_property + def eq_preprocessed(self) -> Expr: + return self._get_eq_preprocessed() + + @cached_property + def eq_high_order_free(self) -> Expr: + a = Wild('a', exclude=[self.func]) + c1 = Wild('c1', exclude=[self.sym]) + # Precondition to try remove f(x) from highest order derivative + reduced_eq = None + if self.eq.is_Add: + deriv_coef = self.eq.coeff(self.func.diff(self.sym, self.order)) + if deriv_coef not in (1, 0): + r = deriv_coef.match(a*self.func**c1) + if r and r[c1]: + den = self.func**r[c1] + reduced_eq = Add(*[arg/den for arg in self.eq.args]) + if reduced_eq is None: + reduced_eq = expand(self.eq) + return reduced_eq + + @cached_property + def eq_expanded(self) -> Expr: + return expand(self.eq_preprocessed) + + def _get_eq_preprocessed(self) -> Expr: + if self.prep: + process_eq, process_func = _preprocess(self.eq, self.func) + if process_func != self.func: + raise ValueError + else: + process_eq = self.eq + return process_eq + + def get_numbered_constants(self, num=1, start=1, prefix='C') -> list[Symbol]: + """ + Returns a list of constants that do not occur + in eq already. + """ + ncs = self.iter_numbered_constants(start, prefix) + Cs = [next(ncs) for i in range(num)] + return Cs + + def iter_numbered_constants(self, start=1, prefix='C') -> Iterator[Symbol]: + """ + Returns an iterator of constants that do not occur + in eq already. + """ + atom_set = self.eq.free_symbols + func_set = self.eq.atoms(Function) + if func_set: + atom_set |= {Symbol(str(f.func)) for f in func_set} + return numbered_symbols(start=start, prefix=prefix, exclude=atom_set) + + @cached_property + def is_autonomous(self): + u = Dummy('u') + x = self.sym + syms = self.eq.subs(self.func, u).free_symbols + return x not in syms + + def get_linear_coefficients(self, eq, func, order): + r""" + Matches a differential equation to the linear form: + + .. math:: a_n(x) y^{(n)} + \cdots + a_1(x)y' + a_0(x) y + B(x) = 0 + + Returns a dict of order:coeff terms, where order is the order of the + derivative on each term, and coeff is the coefficient of that derivative. + The key ``-1`` holds the function `B(x)`. Returns ``None`` if the ODE is + not linear. This function assumes that ``func`` has already been checked + to be good. + + Examples + ======== + + >>> from sympy import Function, cos, sin + >>> from sympy.abc import x + >>> from sympy.solvers.ode.single import SingleODEProblem + >>> f = Function('f') + >>> eq = f(x).diff(x, 3) + 2*f(x).diff(x) + \ + ... x*f(x).diff(x, 2) + cos(x)*f(x).diff(x) + x - f(x) - \ + ... sin(x) + >>> obj = SingleODEProblem(eq, f(x), x) + >>> obj.get_linear_coefficients(eq, f(x), 3) + {-1: x - sin(x), 0: -1, 1: cos(x) + 2, 2: x, 3: 1} + >>> eq = f(x).diff(x, 3) + 2*f(x).diff(x) + \ + ... x*f(x).diff(x, 2) + cos(x)*f(x).diff(x) + x - f(x) - \ + ... sin(f(x)) + >>> obj = SingleODEProblem(eq, f(x), x) + >>> obj.get_linear_coefficients(eq, f(x), 3) == None + True + + """ + f = func.func + x = func.args[0] + symset = {Derivative(f(x), x, i) for i in range(order+1)} + try: + rhs, lhs_terms = _lin_eq2dict(eq, symset) + except PolyNonlinearError: + return None + + if rhs.has(func) or any(c.has(func) for c in lhs_terms.values()): + return None + terms = {i: lhs_terms.get(f(x).diff(x, i), S.Zero) for i in range(order+1)} + terms[-1] = rhs + return terms + + # TODO: Add methods that can be used by many ODE solvers: + # order + # is_linear() + # get_linear_coefficients() + # eq_prepared (the ODE in prepared form) + + +class SingleODESolver: + """ + Base class for Single ODE solvers. + + Subclasses should implement the _matches and _get_general_solution + methods. This class is not intended to be instantiated directly but its + subclasses are as part of dsolve. + + Examples + ======== + + You can use a subclass of SingleODEProblem to solve a particular type of + ODE. We first define a particular ODE problem: + + >>> from sympy import Function, Symbol + >>> x = Symbol('x') + >>> f = Function('f') + >>> eq = f(x).diff(x, 2) + + Now we solve this problem using the NthAlgebraic solver which is a + subclass of SingleODESolver: + + >>> from sympy.solvers.ode.single import NthAlgebraic, SingleODEProblem + >>> problem = SingleODEProblem(eq, f(x), x) + >>> solver = NthAlgebraic(problem) + >>> solver.get_general_solution() + [Eq(f(x), _C*x + _C)] + + The normal way to solve an ODE is to use dsolve (which would use + NthAlgebraic and other solvers internally). When using dsolve a number of + other things are done such as evaluating integrals, simplifying the + solution and renumbering the constants: + + >>> from sympy import dsolve + >>> dsolve(eq, hint='nth_algebraic') + Eq(f(x), C1 + C2*x) + """ + + # Subclasses should store the hint name (the argument to dsolve) in this + # attribute + hint: ClassVar[str] + + # Subclasses should define this to indicate if they support an _Integral + # hint. + has_integral: ClassVar[bool] + + # The ODE to be solved + ode_problem: SingleODEProblem + + # Cache whether or not the equation has matched the method + _matched: bool | None = None + + # Subclasses should store in this attribute the list of order(s) of ODE + # that subclass can solve or leave it to None if not specific to any order + order: list | None = None + + def __init__(self, ode_problem): + self.ode_problem = ode_problem + + def matches(self) -> bool: + if self.order is not None and self.ode_problem.order not in self.order: + self._matched = False + return self._matched + + if self._matched is None: + self._matched = self._matches() + return self._matched + + def get_general_solution(self, *, simplify: bool = True) -> list[Equality]: + if not self.matches(): + msg = "%s solver cannot solve:\n%s" + raise ODEMatchError(msg % (self.hint, self.ode_problem.eq)) + return self._get_general_solution(simplify_flag=simplify) + + def _matches(self) -> bool: + msg = "Subclasses of SingleODESolver should implement matches." + raise NotImplementedError(msg) + + def _get_general_solution(self, *, simplify_flag: bool = True) -> list[Equality]: + msg = "Subclasses of SingleODESolver should implement get_general_solution." + raise NotImplementedError(msg) + + +class SinglePatternODESolver(SingleODESolver): + '''Superclass for ODE solvers based on pattern matching''' + + def wilds(self): + prob = self.ode_problem + f = prob.func.func + x = prob.sym + order = prob.order + return self._wilds(f, x, order) + + def wilds_match(self): + match = self._wilds_match + return [match.get(w, S.Zero) for w in self.wilds()] + + def _matches(self): + eq = self.ode_problem.eq_expanded + f = self.ode_problem.func.func + x = self.ode_problem.sym + order = self.ode_problem.order + df = f(x).diff(x, order) + + if order not in [1, 2]: + return False + + pattern = self._equation(f(x), x, order) + + if not pattern.coeff(df).has(Wild): + eq = expand(eq / eq.coeff(df)) + eq = eq.collect([f(x).diff(x), f(x)], func = cancel) + + self._wilds_match = match = eq.match(pattern) + if match is not None: + return self._verify(f(x)) + return False + + def _verify(self, fx) -> bool: + return True + + def _wilds(self, f, x, order): + msg = "Subclasses of SingleODESolver should implement _wilds" + raise NotImplementedError(msg) + + def _equation(self, fx, x, order): + msg = "Subclasses of SingleODESolver should implement _equation" + raise NotImplementedError(msg) + + +class NthAlgebraic(SingleODESolver): + r""" + Solves an `n`\th order ordinary differential equation using algebra and + integrals. + + There is no general form for the kind of equation that this can solve. The + the equation is solved algebraically treating differentiation as an + invertible algebraic function. + + Examples + ======== + + >>> from sympy import Function, dsolve, Eq + >>> from sympy.abc import x + >>> f = Function('f') + >>> eq = Eq(f(x) * (f(x).diff(x)**2 - 1), 0) + >>> dsolve(eq, f(x), hint='nth_algebraic') + [Eq(f(x), 0), Eq(f(x), C1 - x), Eq(f(x), C1 + x)] + + Note that this solver can return algebraic solutions that do not have any + integration constants (f(x) = 0 in the above example). + """ + + hint = 'nth_algebraic' + has_integral = True # nth_algebraic_Integral hint + + def _matches(self): + r""" + Matches any differential equation that nth_algebraic can solve. Uses + `sympy.solve` but teaches it how to integrate derivatives. + + This involves calling `sympy.solve` and does most of the work of finding a + solution (apart from evaluating the integrals). + """ + eq = self.ode_problem.eq + func = self.ode_problem.func + var = self.ode_problem.sym + + # Derivative that solve can handle: + diffx = self._get_diffx(var) + + # Replace derivatives wrt the independent variable with diffx + def replace(eq, var): + def expand_diffx(*args): + differand, diffs = args[0], args[1:] + toreplace = differand + for v, n in diffs: + for _ in range(n): + if v == var: + toreplace = diffx(toreplace) + else: + toreplace = Derivative(toreplace, v) + return toreplace + return eq.replace(Derivative, expand_diffx) + + # Restore derivatives in solution afterwards + def unreplace(eq, var): + return eq.replace(diffx, lambda e: Derivative(e, var)) + + subs_eqn = replace(eq, var) + try: + # turn off simplification to protect Integrals that have + # _t instead of fx in them and would otherwise factor + # as t_*Integral(1, x) + solns = solve(subs_eqn, func, simplify=False) + except NotImplementedError: + solns = [] + + solns = [simplify(unreplace(soln, var)) for soln in solns] + solns = [Equality(func, soln) for soln in solns] + + self.solutions = solns + return len(solns) != 0 + + def _get_general_solution(self, *, simplify_flag: bool = True): + return self.solutions + + # This needs to produce an invertible function but the inverse depends + # which variable we are integrating with respect to. Since the class can + # be stored in cached results we need to ensure that we always get the + # same class back for each particular integration variable so we store these + # classes in a global dict: + _diffx_stored: dict[Symbol, type[Function]] = {} + + @staticmethod + def _get_diffx(var): + diffcls = NthAlgebraic._diffx_stored.get(var, None) + + if diffcls is None: + # A class that behaves like Derivative wrt var but is "invertible". + class diffx(Function): + def inverse(self): + # don't use integrate here because fx has been replaced by _t + # in the equation; integrals will not be correct while solve + # is at work. + return lambda expr: Integral(expr, var) + Dummy('C') + + diffcls = NthAlgebraic._diffx_stored.setdefault(var, diffx) + + return diffcls + + +class FirstExact(SinglePatternODESolver): + r""" + Solves 1st order exact ordinary differential equations. + + A 1st order differential equation is called exact if it is the total + differential of a function. That is, the differential equation + + .. math:: P(x, y) \,\partial{}x + Q(x, y) \,\partial{}y = 0 + + is exact if there is some function `F(x, y)` such that `P(x, y) = + \partial{}F/\partial{}x` and `Q(x, y) = \partial{}F/\partial{}y`. It can + be shown that a necessary and sufficient condition for a first order ODE + to be exact is that `\partial{}P/\partial{}y = \partial{}Q/\partial{}x`. + Then, the solution will be as given below:: + + >>> from sympy import Function, Eq, Integral, symbols, pprint + >>> x, y, t, x0, y0, C1= symbols('x,y,t,x0,y0,C1') + >>> P, Q, F= map(Function, ['P', 'Q', 'F']) + >>> pprint(Eq(Eq(F(x, y), Integral(P(t, y), (t, x0, x)) + + ... Integral(Q(x0, t), (t, y0, y))), C1)) + x y + / / + | | + F(x, y) = | P(t, y) dt + | Q(x0, t) dt = C1 + | | + / / + x0 y0 + + Where the first partials of `P` and `Q` exist and are continuous in a + simply connected region. + + A note: SymPy currently has no way to represent inert substitution on an + expression, so the hint ``1st_exact_Integral`` will return an integral + with `dy`. This is supposed to represent the function that you are + solving for. + + Examples + ======== + + >>> from sympy import Function, dsolve, cos, sin + >>> from sympy.abc import x + >>> f = Function('f') + >>> dsolve(cos(f(x)) - (x*sin(f(x)) - f(x)**2)*f(x).diff(x), + ... f(x), hint='1st_exact') + Eq(x*cos(f(x)) + f(x)**3/3, C1) + + References + ========== + + - https://en.wikipedia.org/wiki/Exact_differential_equation + - M. Tenenbaum & H. Pollard, "Ordinary Differential Equations", + Dover 1963, pp. 73 + + # indirect doctest + + """ + hint = "1st_exact" + has_integral = True + order = [1] + + def _wilds(self, f, x, order): + P = Wild('P', exclude=[f(x).diff(x)]) + Q = Wild('Q', exclude=[f(x).diff(x)]) + return P, Q + + def _equation(self, fx, x, order): + P, Q = self.wilds() + return P + Q*fx.diff(x) + + def _verify(self, fx) -> bool: + P, Q = self.wilds() + x = self.ode_problem.sym + y = Dummy('y') + + m, n = self.wilds_match() + + m = m.subs(fx, y) + n = n.subs(fx, y) + numerator = cancel(m.diff(y) - n.diff(x)) + + if numerator.is_zero: + # Is exact + return True + else: + # The following few conditions try to convert a non-exact + # differential equation into an exact one. + # References: + # 1. Differential equations with applications + # and historical notes - George E. Simmons + # 2. https://math.okstate.edu/people/binegar/2233-S99/2233-l12.pdf + + factor_n = cancel(numerator/n) + factor_m = cancel(-numerator/m) + if y not in factor_n.free_symbols: + # If (dP/dy - dQ/dx) / Q = f(x) + # then exp(integral(f(x))*equation becomes exact + factor = factor_n + integration_variable = x + elif x not in factor_m.free_symbols: + # If (dP/dy - dQ/dx) / -P = f(y) + # then exp(integral(f(y))*equation becomes exact + factor = factor_m + integration_variable = y + else: + # Couldn't convert to exact + return False + + factor = exp(Integral(factor, integration_variable)) + m *= factor + n *= factor + self._wilds_match[P] = m.subs(y, fx) + self._wilds_match[Q] = n.subs(y, fx) + return True + + def _get_general_solution(self, *, simplify_flag: bool = True): + m, n = self.wilds_match() + fx = self.ode_problem.func + x = self.ode_problem.sym + (C1,) = self.ode_problem.get_numbered_constants(num=1) + y = Dummy('y') + + m = m.subs(fx, y) + n = n.subs(fx, y) + + gen_sol = Eq(Subs(Integral(m, x) + + Integral(n - Integral(m, x).diff(y), y), y, fx), C1) + return [gen_sol] + + +class FirstLinear(SinglePatternODESolver): + r""" + Solves 1st order linear differential equations. + + These are differential equations of the form + + .. math:: dy/dx + P(x) y = Q(x)\text{.} + + These kinds of differential equations can be solved in a general way. The + integrating factor `e^{\int P(x) \,dx}` will turn the equation into a + separable equation. The general solution is:: + + >>> from sympy import Function, dsolve, Eq, pprint, diff, sin + >>> from sympy.abc import x + >>> f, P, Q = map(Function, ['f', 'P', 'Q']) + >>> genform = Eq(f(x).diff(x) + P(x)*f(x), Q(x)) + >>> pprint(genform) + d + P(x)*f(x) + --(f(x)) = Q(x) + dx + >>> pprint(dsolve(genform, f(x), hint='1st_linear_Integral')) + / / \ + | | | + | | / | / + | | | | | + | | | P(x) dx | - | P(x) dx + | | | | | + | | / | / + f(x) = |C1 + | Q(x)*e dx|*e + | | | + \ / / + + + Examples + ======== + + >>> f = Function('f') + >>> pprint(dsolve(Eq(x*diff(f(x), x) - f(x), x**2*sin(x)), + ... f(x), '1st_linear')) + f(x) = x*(C1 - cos(x)) + + References + ========== + + - https://en.wikipedia.org/wiki/Linear_differential_equation#First-order_equation_with_variable_coefficients + - M. Tenenbaum & H. Pollard, "Ordinary Differential Equations", + Dover 1963, pp. 92 + + # indirect doctest + + """ + hint = '1st_linear' + has_integral = True + order = [1] + + def _wilds(self, f, x, order): + P = Wild('P', exclude=[f(x)]) + Q = Wild('Q', exclude=[f(x), f(x).diff(x)]) + return P, Q + + def _equation(self, fx, x, order): + P, Q = self.wilds() + return fx.diff(x) + P*fx - Q + + def _get_general_solution(self, *, simplify_flag: bool = True): + P, Q = self.wilds_match() + fx = self.ode_problem.func + x = self.ode_problem.sym + (C1,) = self.ode_problem.get_numbered_constants(num=1) + gensol = Eq(fx, ((C1 + Integral(Q*exp(Integral(P, x)), x)) + * exp(-Integral(P, x)))) + return [gensol] + + +class AlmostLinear(SinglePatternODESolver): + r""" + Solves an almost-linear differential equation. + + The general form of an almost linear differential equation is + + .. math:: a(x) g'(f(x)) f'(x) + b(x) g(f(x)) + c(x) + + Here `f(x)` is the function to be solved for (the dependent variable). + The substitution `g(f(x)) = u(x)` leads to a linear differential equation + for `u(x)` of the form `a(x) u' + b(x) u + c(x) = 0`. This can be solved + for `u(x)` by the `first_linear` hint and then `f(x)` is found by solving + `g(f(x)) = u(x)`. + + See Also + ======== + :obj:`sympy.solvers.ode.single.FirstLinear` + + Examples + ======== + + >>> from sympy import dsolve, Function, pprint, sin, cos + >>> from sympy.abc import x + >>> f = Function('f') + >>> d = f(x).diff(x) + >>> eq = x*d + x*f(x) + 1 + >>> dsolve(eq, f(x), hint='almost_linear') + Eq(f(x), (C1 - Ei(x))*exp(-x)) + >>> pprint(dsolve(eq, f(x), hint='almost_linear')) + -x + f(x) = (C1 - Ei(x))*e + >>> example = cos(f(x))*f(x).diff(x) + sin(f(x)) + 1 + >>> pprint(example) + d + sin(f(x)) + cos(f(x))*--(f(x)) + 1 + dx + >>> pprint(dsolve(example, f(x), hint='almost_linear')) + / -x \ / -x \ + [f(x) = pi - asin\C1*e - 1/, f(x) = asin\C1*e - 1/] + + + References + ========== + + - Joel Moses, "Symbolic Integration - The Stormy Decade", Communications + of the ACM, Volume 14, Number 8, August 1971, pp. 558 + """ + hint = "almost_linear" + has_integral = True + order = [1] + + def _wilds(self, f, x, order): + P = Wild('P', exclude=[f(x).diff(x)]) + Q = Wild('Q', exclude=[f(x).diff(x)]) + return P, Q + + def _equation(self, fx, x, order): + P, Q = self.wilds() + return P*fx.diff(x) + Q + + def _verify(self, fx): + a, b = self.wilds_match() + c, b = b.as_independent(fx) if b.is_Add else (S.Zero, b) + # a, b and c are the function a(x), b(x) and c(x) respectively. + # c(x) is obtained by separating out b as terms with and without fx i.e, l(y) + # The following conditions checks if the given equation is an almost-linear differential equation using the fact that + # a(x)*(l(y))' / l(y)' is independent of l(y) + + if b.diff(fx) != 0 and not simplify(b.diff(fx)/a).has(fx): + self.ly = factor_terms(b).as_independent(fx, as_Add=False)[1] # Gives the term containing fx i.e., l(y) + self.ax = a / self.ly.diff(fx) + self.cx = -c # cx is taken as -c(x) to simplify expression in the solution integral + self.bx = factor_terms(b) / self.ly + return True + + return False + + def _get_general_solution(self, *, simplify_flag: bool = True): + x = self.ode_problem.sym + (C1,) = self.ode_problem.get_numbered_constants(num=1) + gensol = Eq(self.ly, ((C1 + Integral((self.cx/self.ax)*exp(Integral(self.bx/self.ax, x)), x)) + * exp(-Integral(self.bx/self.ax, x)))) + + return [gensol] + + +class Bernoulli(SinglePatternODESolver): + r""" + Solves Bernoulli differential equations. + + These are equations of the form + + .. math:: dy/dx + P(x) y = Q(x) y^n\text{, }n \ne 1`\text{.} + + The substitution `w = 1/y^{1-n}` will transform an equation of this form + into one that is linear (see the docstring of + :obj:`~sympy.solvers.ode.single.FirstLinear`). The general solution is:: + + >>> from sympy import Function, dsolve, Eq, pprint + >>> from sympy.abc import x, n + >>> f, P, Q = map(Function, ['f', 'P', 'Q']) + >>> genform = Eq(f(x).diff(x) + P(x)*f(x), Q(x)*f(x)**n) + >>> pprint(genform) + d n + P(x)*f(x) + --(f(x)) = Q(x)*f (x) + dx + >>> pprint(dsolve(genform, f(x), hint='Bernoulli_Integral'), num_columns=110) + -1 + ----- + n - 1 + // / / \ \ + || | | | | + || | / | / | / | + || | | | | | | | + || | -(n - 1)* | P(x) dx | -(n - 1)* | P(x) dx | (n - 1)* | P(x) dx| + || | | | | | | | + || | / | / | / | + f(x) = ||C1 - n* | Q(x)*e dx + | Q(x)*e dx|*e | + || | | | | + \\ / / / / + + + Note that the equation is separable when `n = 1` (see the docstring of + :obj:`~sympy.solvers.ode.single.Separable`). + + >>> pprint(dsolve(Eq(f(x).diff(x) + P(x)*f(x), Q(x)*f(x)), f(x), + ... hint='separable_Integral')) + f(x) + / + | / + | 1 | + | - dy = C1 + | (-P(x) + Q(x)) dx + | y | + | / + / + + + Examples + ======== + + >>> from sympy import Function, dsolve, Eq, pprint, log + >>> from sympy.abc import x + >>> f = Function('f') + + >>> pprint(dsolve(Eq(x*f(x).diff(x) + f(x), log(x)*f(x)**2), + ... f(x), hint='Bernoulli')) + 1 + f(x) = ----------------- + C1*x + log(x) + 1 + + References + ========== + + - https://en.wikipedia.org/wiki/Bernoulli_differential_equation + + - M. Tenenbaum & H. Pollard, "Ordinary Differential Equations", + Dover 1963, pp. 95 + + # indirect doctest + + """ + hint = "Bernoulli" + has_integral = True + order = [1] + + def _wilds(self, f, x, order): + P = Wild('P', exclude=[f(x)]) + Q = Wild('Q', exclude=[f(x)]) + n = Wild('n', exclude=[x, f(x), f(x).diff(x)]) + return P, Q, n + + def _equation(self, fx, x, order): + P, Q, n = self.wilds() + return fx.diff(x) + P*fx - Q*fx**n + + def _get_general_solution(self, *, simplify_flag: bool = True): + P, Q, n = self.wilds_match() + fx = self.ode_problem.func + x = self.ode_problem.sym + (C1,) = self.ode_problem.get_numbered_constants(num=1) + if n==1: + gensol = Eq(log(fx), ( + C1 + Integral((-P + Q), x) + )) + else: + gensol = Eq(fx**(1-n), ( + (C1 - (n - 1) * Integral(Q*exp(-n*Integral(P, x)) + * exp(Integral(P, x)), x) + ) * exp(-(1 - n)*Integral(P, x))) + ) + return [gensol] + + +class Factorable(SingleODESolver): + r""" + Solves equations having a solvable factor. + + This function is used to solve the equation having factors. Factors may be of type algebraic or ode. It + will try to solve each factor independently. Factors will be solved by calling dsolve. We will return the + list of solutions. + + Examples + ======== + + >>> from sympy import Function, dsolve, pprint + >>> from sympy.abc import x + >>> f = Function('f') + >>> eq = (f(x)**2-4)*(f(x).diff(x)+f(x)) + >>> pprint(dsolve(eq, f(x))) + -x + [f(x) = 2, f(x) = -2, f(x) = C1*e ] + + + """ + hint = "factorable" + has_integral = False + + def _matches(self): + eq_orig = self.ode_problem.eq + f = self.ode_problem.func.func + x = self.ode_problem.sym + df = f(x).diff(x) + self.eqs = [] + eq = eq_orig.collect(f(x), func = cancel) + eq = fraction(factor(eq))[0] + factors = Mul.make_args(factor(eq)) + roots = [fac.as_base_exp() for fac in factors if len(fac.args)!=0] + if len(roots)>1 or roots[0][1]>1: + for base, expo in roots: + if base.has(f(x)): + self.eqs.append(base) + if len(self.eqs)>0: + return True + roots = solve(eq, df) + if len(roots)>0: + self.eqs = [(df - root) for root in roots] + # Avoid infinite recursion + matches = self.eqs != [eq_orig] + return matches + for i in factors: + if i.has(f(x)): + self.eqs.append(i) + return len(self.eqs)>0 and len(factors)>1 + + def _get_general_solution(self, *, simplify_flag: bool = True): + func = self.ode_problem.func.func + x = self.ode_problem.sym + eqns = self.eqs + sols = [] + for eq in eqns: + try: + sol = dsolve(eq, func(x)) + except NotImplementedError: + continue + else: + if isinstance(sol, list): + sols.extend(sol) + else: + sols.append(sol) + + if sols == []: + raise NotImplementedError("The given ODE " + str(eq) + " cannot be solved by" + + " the factorable group method") + return sols + + +class RiccatiSpecial(SinglePatternODESolver): + r""" + The general Riccati equation has the form + + .. math:: dy/dx = f(x) y^2 + g(x) y + h(x)\text{.} + + While it does not have a general solution [1], the "special" form, `dy/dx + = a y^2 - b x^c`, does have solutions in many cases [2]. This routine + returns a solution for `a(dy/dx) = b y^2 + c y/x + d/x^2` that is obtained + by using a suitable change of variables to reduce it to the special form + and is valid when neither `a` nor `b` are zero and either `c` or `d` is + zero. + + >>> from sympy.abc import x, a, b, c, d + >>> from sympy import dsolve, checkodesol, pprint, Function + >>> f = Function('f') + >>> y = f(x) + >>> genform = a*y.diff(x) - (b*y**2 + c*y/x + d/x**2) + >>> sol = dsolve(genform, y, hint="Riccati_special_minus2") + >>> pprint(sol, wrap_line=False) + / / __________________ \\ + | __________________ | / 2 || + | / 2 | \/ 4*b*d - (a + c) *log(x)|| + -|a + c - \/ 4*b*d - (a + c) *tan|C1 + ----------------------------|| + \ \ 2*a // + f(x) = ------------------------------------------------------------------------ + 2*b*x + + >>> checkodesol(genform, sol, order=1)[0] + True + + References + ========== + + - https://www.maplesoft.com/support/help/Maple/view.aspx?path=odeadvisor/Riccati + - https://eqworld.ipmnet.ru/en/solutions/ode/ode0106.pdf - + https://eqworld.ipmnet.ru/en/solutions/ode/ode0123.pdf + """ + hint = "Riccati_special_minus2" + has_integral = False + order = [1] + + def _wilds(self, f, x, order): + a = Wild('a', exclude=[x, f(x), f(x).diff(x), 0]) + b = Wild('b', exclude=[x, f(x), f(x).diff(x), 0]) + c = Wild('c', exclude=[x, f(x), f(x).diff(x)]) + d = Wild('d', exclude=[x, f(x), f(x).diff(x)]) + return a, b, c, d + + def _equation(self, fx, x, order): + a, b, c, d = self.wilds() + return a*fx.diff(x) + b*fx**2 + c*fx/x + d/x**2 + + def _get_general_solution(self, *, simplify_flag: bool = True): + a, b, c, d = self.wilds_match() + fx = self.ode_problem.func + x = self.ode_problem.sym + (C1,) = self.ode_problem.get_numbered_constants(num=1) + mu = sqrt(4*d*b - (a - c)**2) + + gensol = Eq(fx, (a - c - mu*tan(mu/(2*a)*log(x) + C1))/(2*b*x)) + return [gensol] + + +class RationalRiccati(SinglePatternODESolver): + r""" + Gives general solutions to the first order Riccati differential + equations that have atleast one rational particular solution. + + .. math :: y' = b_0(x) + b_1(x) y + b_2(x) y^2 + + where `b_0`, `b_1` and `b_2` are rational functions of `x` + with `b_2 \ne 0` (`b_2 = 0` would make it a Bernoulli equation). + + Examples + ======== + + >>> from sympy import Symbol, Function, dsolve, checkodesol + >>> f = Function('f') + >>> x = Symbol('x') + + >>> eq = -x**4*f(x)**2 + x**3*f(x).diff(x) + x**2*f(x) + 20 + >>> sol = dsolve(eq, hint="1st_rational_riccati") + >>> sol + Eq(f(x), (4*C1 - 5*x**9 - 4)/(x**2*(C1 + x**9 - 1))) + >>> checkodesol(eq, sol) + (True, 0) + + References + ========== + + - Riccati ODE: https://en.wikipedia.org/wiki/Riccati_equation + - N. Thieu Vo - Rational and Algebraic Solutions of First-Order Algebraic ODEs: + Algorithm 11, pp. 78 - https://www3.risc.jku.at/publications/download/risc_5387/PhDThesisThieu.pdf + """ + has_integral = False + hint = "1st_rational_riccati" + order = [1] + + def _wilds(self, f, x, order): + b0 = Wild('b0', exclude=[f(x), f(x).diff(x)]) + b1 = Wild('b1', exclude=[f(x), f(x).diff(x)]) + b2 = Wild('b2', exclude=[f(x), f(x).diff(x)]) + return (b0, b1, b2) + + def _equation(self, fx, x, order): + b0, b1, b2 = self.wilds() + return fx.diff(x) - b0 - b1*fx - b2*fx**2 + + def _matches(self): + eq = self.ode_problem.eq_expanded + f = self.ode_problem.func.func + x = self.ode_problem.sym + order = self.ode_problem.order + + if order != 1: + return False + + match, funcs = match_riccati(eq, f, x) + if not match: + return False + _b0, _b1, _b2 = funcs + b0, b1, b2 = self.wilds() + self._wilds_match = match = {b0: _b0, b1: _b1, b2: _b2} + return True + + def _get_general_solution(self, *, simplify_flag: bool = True): + # Match the equation + b0, b1, b2 = self.wilds_match() + fx = self.ode_problem.func + x = self.ode_problem.sym + return solve_riccati(fx, x, b0, b1, b2, gensol=True) + + +class SecondNonlinearAutonomousConserved(SinglePatternODESolver): + r""" + Gives solution for the autonomous second order nonlinear + differential equation of the form + + .. math :: f''(x) = g(f(x)) + + The solution for this differential equation can be computed + by multiplying by `f'(x)` and integrating on both sides, + converting it into a first order differential equation. + + Examples + ======== + + >>> from sympy import Function, symbols, dsolve + >>> f, g = symbols('f g', cls=Function) + >>> x = symbols('x') + + >>> eq = f(x).diff(x, 2) - g(f(x)) + >>> dsolve(eq, simplify=False) + [Eq(Integral(1/sqrt(C1 + 2*Integral(g(_u), _u)), (_u, f(x))), C2 + x), + Eq(Integral(1/sqrt(C1 + 2*Integral(g(_u), _u)), (_u, f(x))), C2 - x)] + + >>> from sympy import exp, log + >>> eq = f(x).diff(x, 2) - exp(f(x)) + log(f(x)) + >>> dsolve(eq, simplify=False) + [Eq(Integral(1/sqrt(-2*_u*log(_u) + 2*_u + C1 + 2*exp(_u)), (_u, f(x))), C2 + x), + Eq(Integral(1/sqrt(-2*_u*log(_u) + 2*_u + C1 + 2*exp(_u)), (_u, f(x))), C2 - x)] + + References + ========== + + - https://eqworld.ipmnet.ru/en/solutions/ode/ode0301.pdf + """ + hint = "2nd_nonlinear_autonomous_conserved" + has_integral = True + order = [2] + + def _wilds(self, f, x, order): + fy = Wild('fy', exclude=[0, f(x).diff(x), f(x).diff(x, 2)]) + return (fy, ) + + def _equation(self, fx, x, order): + fy = self.wilds()[0] + return fx.diff(x, 2) + fy + + def _verify(self, fx): + return self.ode_problem.is_autonomous + + def _get_general_solution(self, *, simplify_flag: bool = True): + g = self.wilds_match()[0] + fx = self.ode_problem.func + x = self.ode_problem.sym + u = Dummy('u') + g = g.subs(fx, u) + C1, C2 = self.ode_problem.get_numbered_constants(num=2) + inside = -2*Integral(g, u) + C1 + lhs = Integral(1/sqrt(inside), (u, fx)) + return [Eq(lhs, C2 + x), Eq(lhs, C2 - x)] + + +class Liouville(SinglePatternODESolver): + r""" + Solves 2nd order Liouville differential equations. + + The general form of a Liouville ODE is + + .. math:: \frac{d^2 y}{dx^2} + g(y) \left(\! + \frac{dy}{dx}\!\right)^2 + h(x) + \frac{dy}{dx}\text{.} + + The general solution is: + + >>> from sympy import Function, dsolve, Eq, pprint, diff + >>> from sympy.abc import x + >>> f, g, h = map(Function, ['f', 'g', 'h']) + >>> genform = Eq(diff(f(x),x,x) + g(f(x))*diff(f(x),x)**2 + + ... h(x)*diff(f(x),x), 0) + >>> pprint(genform) + 2 2 + /d \ d d + g(f(x))*|--(f(x))| + h(x)*--(f(x)) + ---(f(x)) = 0 + \dx / dx 2 + dx + >>> pprint(dsolve(genform, f(x), hint='Liouville_Integral')) + f(x) + / / + | | + | / | / + | | | | + | - | h(x) dx | | g(y) dy + | | | | + | / | / + C1 + C2* | e dx + | e dy = 0 + | | + / / + + Examples + ======== + + >>> from sympy import Function, dsolve, Eq, pprint + >>> from sympy.abc import x + >>> f = Function('f') + >>> pprint(dsolve(diff(f(x), x, x) + diff(f(x), x)**2/f(x) + + ... diff(f(x), x)/x, f(x), hint='Liouville')) + ________________ ________________ + [f(x) = -\/ C1 + C2*log(x) , f(x) = \/ C1 + C2*log(x) ] + + References + ========== + + - Goldstein and Braun, "Advanced Methods for the Solution of Differential + Equations", pp. 98 + - https://www.maplesoft.com/support/help/Maple/view.aspx?path=odeadvisor/Liouville + + # indirect doctest + + """ + hint = "Liouville" + has_integral = True + order = [2] + + def _wilds(self, f, x, order): + d = Wild('d', exclude=[f(x).diff(x), f(x).diff(x, 2)]) + e = Wild('e', exclude=[f(x).diff(x)]) + k = Wild('k', exclude=[f(x).diff(x)]) + return d, e, k + + def _equation(self, fx, x, order): + # Liouville ODE in the form + # f(x).diff(x, 2) + g(f(x))*(f(x).diff(x))**2 + h(x)*f(x).diff(x) + # See Goldstein and Braun, "Advanced Methods for the Solution of + # Differential Equations", pg. 98 + d, e, k = self.wilds() + return d*fx.diff(x, 2) + e*fx.diff(x)**2 + k*fx.diff(x) + + def _verify(self, fx): + d, e, k = self.wilds_match() + self.y = Dummy('y') + x = self.ode_problem.sym + self.g = simplify(e/d).subs(fx, self.y) + self.h = simplify(k/d).subs(fx, self.y) + if self.y in self.h.free_symbols or x in self.g.free_symbols: + return False + return True + + def _get_general_solution(self, *, simplify_flag: bool = True): + d, e, k = self.wilds_match() + fx = self.ode_problem.func + x = self.ode_problem.sym + C1, C2 = self.ode_problem.get_numbered_constants(num=2) + int = Integral(exp(Integral(self.g, self.y)), (self.y, None, fx)) + gen_sol = Eq(int + C1*Integral(exp(-Integral(self.h, x)), x) + C2, 0) + + return [gen_sol] + + +class Separable(SinglePatternODESolver): + r""" + Solves separable 1st order differential equations. + + This is any differential equation that can be written as `P(y) + \tfrac{dy}{dx} = Q(x)`. The solution can then just be found by + rearranging terms and integrating: `\int P(y) \,dy = \int Q(x) \,dx`. + This hint uses :py:meth:`sympy.simplify.simplify.separatevars` as its back + end, so if a separable equation is not caught by this solver, it is most + likely the fault of that function. + :py:meth:`~sympy.simplify.simplify.separatevars` is + smart enough to do most expansion and factoring necessary to convert a + separable equation `F(x, y)` into the proper form `P(x)\cdot{}Q(y)`. The + general solution is:: + + >>> from sympy import Function, dsolve, Eq, pprint + >>> from sympy.abc import x + >>> a, b, c, d, f = map(Function, ['a', 'b', 'c', 'd', 'f']) + >>> genform = Eq(a(x)*b(f(x))*f(x).diff(x), c(x)*d(f(x))) + >>> pprint(genform) + d + a(x)*b(f(x))*--(f(x)) = c(x)*d(f(x)) + dx + >>> pprint(dsolve(genform, f(x), hint='separable_Integral')) + f(x) + / / + | | + | b(y) | c(x) + | ---- dy = C1 + | ---- dx + | d(y) | a(x) + | | + / / + + Examples + ======== + + >>> from sympy import Function, dsolve, Eq + >>> from sympy.abc import x + >>> f = Function('f') + >>> pprint(dsolve(Eq(f(x)*f(x).diff(x) + x, 3*x*f(x)**2), f(x), + ... hint='separable', simplify=False)) + / 2 \ 2 + log\3*f (x) - 1/ x + ---------------- = C1 + -- + 6 2 + + References + ========== + + - M. Tenenbaum & H. Pollard, "Ordinary Differential Equations", + Dover 1963, pp. 52 + + # indirect doctest + + """ + hint = "separable" + has_integral = True + order = [1] + + def _wilds(self, f, x, order): + d = Wild('d', exclude=[f(x).diff(x), f(x).diff(x, 2)]) + e = Wild('e', exclude=[f(x).diff(x)]) + return d, e + + def _equation(self, fx, x, order): + d, e = self.wilds() + return d + e*fx.diff(x) + + def _verify(self, fx): + d, e = self.wilds_match() + self.y = Dummy('y') + x = self.ode_problem.sym + d = separatevars(d.subs(fx, self.y)) + e = separatevars(e.subs(fx, self.y)) + # m1[coeff]*m1[x]*m1[y] + m2[coeff]*m2[x]*m2[y]*y' + self.m1 = separatevars(d, dict=True, symbols=(x, self.y)) + self.m2 = separatevars(e, dict=True, symbols=(x, self.y)) + return bool(self.m1 and self.m2) + + def _get_match_object(self): + fx = self.ode_problem.func + x = self.ode_problem.sym + return self.m1, self.m2, x, fx + + def _get_general_solution(self, *, simplify_flag: bool = True): + m1, m2, x, fx = self._get_match_object() + (C1,) = self.ode_problem.get_numbered_constants(num=1) + int = Integral(m2['coeff']*m2[self.y]/m1[self.y], + (self.y, None, fx)) + gen_sol = Eq(int, Integral(-m1['coeff']*m1[x]/ + m2[x], x) + C1) + return [gen_sol] + + +class SeparableReduced(Separable): + r""" + Solves a differential equation that can be reduced to the separable form. + + The general form of this equation is + + .. math:: y' + (y/x) H(x^n y) = 0\text{}. + + This can be solved by substituting `u(y) = x^n y`. The equation then + reduces to the separable form `\frac{u'}{u (\mathrm{power} - H(u))} - + \frac{1}{x} = 0`. + + The general solution is: + + >>> from sympy import Function, dsolve, pprint + >>> from sympy.abc import x, n + >>> f, g = map(Function, ['f', 'g']) + >>> genform = f(x).diff(x) + (f(x)/x)*g(x**n*f(x)) + >>> pprint(genform) + / n \ + d f(x)*g\x *f(x)/ + --(f(x)) + --------------- + dx x + >>> pprint(dsolve(genform, hint='separable_reduced')) + n + x *f(x) + / + | + | 1 + | ------------ dy = C1 + log(x) + | y*(n - g(y)) + | + / + + See Also + ======== + :obj:`sympy.solvers.ode.single.Separable` + + Examples + ======== + + >>> from sympy import dsolve, Function, pprint + >>> from sympy.abc import x + >>> f = Function('f') + >>> d = f(x).diff(x) + >>> eq = (x - x**2*f(x))*d - f(x) + >>> dsolve(eq, hint='separable_reduced') + [Eq(f(x), (1 - sqrt(C1*x**2 + 1))/x), Eq(f(x), (sqrt(C1*x**2 + 1) + 1)/x)] + >>> pprint(dsolve(eq, hint='separable_reduced')) + ___________ ___________ + / 2 / 2 + 1 - \/ C1*x + 1 \/ C1*x + 1 + 1 + [f(x) = ------------------, f(x) = ------------------] + x x + + References + ========== + + - Joel Moses, "Symbolic Integration - The Stormy Decade", Communications + of the ACM, Volume 14, Number 8, August 1971, pp. 558 + """ + hint = "separable_reduced" + has_integral = True + order = [1] + + def _degree(self, expr, x): + # Made this function to calculate the degree of + # x in an expression. If expr will be of form + # x**p*y, (wheare p can be variables/rationals) then it + # will return p. + for val in expr: + if val.has(x): + if isinstance(val, Pow) and val.as_base_exp()[0] == x: + return (val.as_base_exp()[1]) + elif val == x: + return (val.as_base_exp()[1]) + else: + return self._degree(val.args, x) + return 0 + + def _powers(self, expr): + # this function will return all the different relative power of x w.r.t f(x). + # expr = x**p * f(x)**q then it will return {p/q}. + pows = set() + fx = self.ode_problem.func + x = self.ode_problem.sym + self.y = Dummy('y') + if isinstance(expr, Add): + exprs = expr.atoms(Add) + elif isinstance(expr, Mul): + exprs = expr.atoms(Mul) + elif isinstance(expr, Pow): + exprs = expr.atoms(Pow) + else: + exprs = {expr} + + for arg in exprs: + if arg.has(x): + _, u = arg.as_independent(x, fx) + pow = self._degree((u.subs(fx, self.y), ), x)/self._degree((u.subs(fx, self.y), ), self.y) + pows.add(pow) + return pows + + def _verify(self, fx): + num, den = self.wilds_match() + x = self.ode_problem.sym + factor = simplify(x/fx*num/den) + # Try representing factor in terms of x^n*y + # where n is lowest power of x in factor; + # first remove terms like sqrt(2)*3 from factor.atoms(Mul) + num, dem = factor.as_numer_denom() + num = expand(num) + dem = expand(dem) + pows = self._powers(num) + pows.update(self._powers(dem)) + pows = list(pows) + if(len(pows)==1) and pows[0]!=zoo: + self.t = Dummy('t') + self.r2 = {'t': self.t} + num = num.subs(x**pows[0]*fx, self.t) + dem = dem.subs(x**pows[0]*fx, self.t) + test = num/dem + free = test.free_symbols + if len(free) == 1 and free.pop() == self.t: + self.r2.update({'power' : pows[0], 'u' : test}) + return True + return False + return False + + def _get_match_object(self): + fx = self.ode_problem.func + x = self.ode_problem.sym + u = self.r2['u'].subs(self.r2['t'], self.y) + ycoeff = 1/(self.y*(self.r2['power'] - u)) + m1 = {self.y: 1, x: -1/x, 'coeff': 1} + m2 = {self.y: ycoeff, x: 1, 'coeff': 1} + return m1, m2, x, x**self.r2['power']*fx + + +class HomogeneousCoeffSubsDepDivIndep(SinglePatternODESolver): + r""" + Solves a 1st order differential equation with homogeneous coefficients + using the substitution `u_1 = \frac{\text{}}{\text{}}`. + + This is a differential equation + + .. math:: P(x, y) + Q(x, y) dy/dx = 0 + + such that `P` and `Q` are homogeneous and of the same order. A function + `F(x, y)` is homogeneous of order `n` if `F(x t, y t) = t^n F(x, y)`. + Equivalently, `F(x, y)` can be rewritten as `G(y/x)` or `H(x/y)`. See + also the docstring of :py:meth:`~sympy.solvers.ode.homogeneous_order`. + + If the coefficients `P` and `Q` in the differential equation above are + homogeneous functions of the same order, then it can be shown that the + substitution `y = u_1 x` (i.e. `u_1 = y/x`) will turn the differential + equation into an equation separable in the variables `x` and `u`. If + `h(u_1)` is the function that results from making the substitution `u_1 = + f(x)/x` on `P(x, f(x))` and `g(u_2)` is the function that results from the + substitution on `Q(x, f(x))` in the differential equation `P(x, f(x)) + + Q(x, f(x)) f'(x) = 0`, then the general solution is:: + + >>> from sympy import Function, dsolve, pprint + >>> from sympy.abc import x + >>> f, g, h = map(Function, ['f', 'g', 'h']) + >>> genform = g(f(x)/x) + h(f(x)/x)*f(x).diff(x) + >>> pprint(genform) + /f(x)\ /f(x)\ d + g|----| + h|----|*--(f(x)) + \ x / \ x / dx + >>> pprint(dsolve(genform, f(x), + ... hint='1st_homogeneous_coeff_subs_dep_div_indep_Integral')) + f(x) + ---- + x + / + | + | -h(u1) + log(x) = C1 + | ---------------- d(u1) + | u1*h(u1) + g(u1) + | + / + + Where `u_1 h(u_1) + g(u_1) \ne 0` and `x \ne 0`. + + See also the docstrings of + :obj:`~sympy.solvers.ode.single.HomogeneousCoeffBest` and + :obj:`~sympy.solvers.ode.single.HomogeneousCoeffSubsIndepDivDep`. + + Examples + ======== + + >>> from sympy import Function, dsolve + >>> from sympy.abc import x + >>> f = Function('f') + >>> pprint(dsolve(2*x*f(x) + (x**2 + f(x)**2)*f(x).diff(x), f(x), + ... hint='1st_homogeneous_coeff_subs_dep_div_indep', simplify=False)) + / 3 \ + |3*f(x) f (x)| + log|------ + -----| + | x 3 | + \ x / + log(x) = log(C1) - ------------------- + 3 + + References + ========== + + - https://en.wikipedia.org/wiki/Homogeneous_differential_equation + - M. Tenenbaum & H. Pollard, "Ordinary Differential Equations", + Dover 1963, pp. 59 + + # indirect doctest + + """ + hint = "1st_homogeneous_coeff_subs_dep_div_indep" + has_integral = True + order = [1] + + def _wilds(self, f, x, order): + d = Wild('d', exclude=[f(x).diff(x), f(x).diff(x, 2)]) + e = Wild('e', exclude=[f(x).diff(x)]) + return d, e + + def _equation(self, fx, x, order): + d, e = self.wilds() + return d + e*fx.diff(x) + + def _verify(self, fx): + self.d, self.e = self.wilds_match() + self.y = Dummy('y') + x = self.ode_problem.sym + self.d = separatevars(self.d.subs(fx, self.y)) + self.e = separatevars(self.e.subs(fx, self.y)) + ordera = homogeneous_order(self.d, x, self.y) + orderb = homogeneous_order(self.e, x, self.y) + if ordera == orderb and ordera is not None: + self.u = Dummy('u') + if simplify((self.d + self.u*self.e).subs({x: 1, self.y: self.u})) != 0: + return True + return False + return False + + def _get_match_object(self): + fx = self.ode_problem.func + x = self.ode_problem.sym + self.u1 = Dummy('u1') + xarg = 0 + yarg = 0 + return [self.d, self.e, fx, x, self.u, self.u1, self.y, xarg, yarg] + + def _get_general_solution(self, *, simplify_flag: bool = True): + d, e, fx, x, u, u1, y, xarg, yarg = self._get_match_object() + (C1,) = self.ode_problem.get_numbered_constants(num=1) + int = Integral( + (-e/(d + u1*e)).subs({x: 1, y: u1}), + (u1, None, fx/x)) + sol = logcombine(Eq(log(x), int + log(C1)), force=True) + gen_sol = sol.subs(fx, u).subs(((u, u - yarg), (x, x - xarg), (u, fx))) + return [gen_sol] + + +class HomogeneousCoeffSubsIndepDivDep(SinglePatternODESolver): + r""" + Solves a 1st order differential equation with homogeneous coefficients + using the substitution `u_2 = \frac{\text{}}{\text{}}`. + + This is a differential equation + + .. math:: P(x, y) + Q(x, y) dy/dx = 0 + + such that `P` and `Q` are homogeneous and of the same order. A function + `F(x, y)` is homogeneous of order `n` if `F(x t, y t) = t^n F(x, y)`. + Equivalently, `F(x, y)` can be rewritten as `G(y/x)` or `H(x/y)`. See + also the docstring of :py:meth:`~sympy.solvers.ode.homogeneous_order`. + + If the coefficients `P` and `Q` in the differential equation above are + homogeneous functions of the same order, then it can be shown that the + substitution `x = u_2 y` (i.e. `u_2 = x/y`) will turn the differential + equation into an equation separable in the variables `y` and `u_2`. If + `h(u_2)` is the function that results from making the substitution `u_2 = + x/f(x)` on `P(x, f(x))` and `g(u_2)` is the function that results from the + substitution on `Q(x, f(x))` in the differential equation `P(x, f(x)) + + Q(x, f(x)) f'(x) = 0`, then the general solution is: + + >>> from sympy import Function, dsolve, pprint + >>> from sympy.abc import x + >>> f, g, h = map(Function, ['f', 'g', 'h']) + >>> genform = g(x/f(x)) + h(x/f(x))*f(x).diff(x) + >>> pprint(genform) + / x \ / x \ d + g|----| + h|----|*--(f(x)) + \f(x)/ \f(x)/ dx + >>> pprint(dsolve(genform, f(x), + ... hint='1st_homogeneous_coeff_subs_indep_div_dep_Integral')) + x + ---- + f(x) + / + | + | -g(u1) + | ---------------- d(u1) + | u1*g(u1) + h(u1) + | + / + + f(x) = C1*e + + Where `u_1 g(u_1) + h(u_1) \ne 0` and `f(x) \ne 0`. + + See also the docstrings of + :obj:`~sympy.solvers.ode.single.HomogeneousCoeffBest` and + :obj:`~sympy.solvers.ode.single.HomogeneousCoeffSubsDepDivIndep`. + + Examples + ======== + + >>> from sympy import Function, pprint, dsolve + >>> from sympy.abc import x + >>> f = Function('f') + >>> pprint(dsolve(2*x*f(x) + (x**2 + f(x)**2)*f(x).diff(x), f(x), + ... hint='1st_homogeneous_coeff_subs_indep_div_dep', + ... simplify=False)) + / 2 \ + |3*x | + log|----- + 1| + | 2 | + \f (x) / + log(f(x)) = log(C1) - -------------- + 3 + + References + ========== + + - https://en.wikipedia.org/wiki/Homogeneous_differential_equation + - M. Tenenbaum & H. Pollard, "Ordinary Differential Equations", + Dover 1963, pp. 59 + + # indirect doctest + + """ + hint = "1st_homogeneous_coeff_subs_indep_div_dep" + has_integral = True + order = [1] + + def _wilds(self, f, x, order): + d = Wild('d', exclude=[f(x).diff(x), f(x).diff(x, 2)]) + e = Wild('e', exclude=[f(x).diff(x)]) + return d, e + + def _equation(self, fx, x, order): + d, e = self.wilds() + return d + e*fx.diff(x) + + def _verify(self, fx): + self.d, self.e = self.wilds_match() + self.y = Dummy('y') + x = self.ode_problem.sym + self.d = separatevars(self.d.subs(fx, self.y)) + self.e = separatevars(self.e.subs(fx, self.y)) + ordera = homogeneous_order(self.d, x, self.y) + orderb = homogeneous_order(self.e, x, self.y) + if ordera == orderb and ordera is not None: + self.u = Dummy('u') + if simplify((self.e + self.u*self.d).subs({x: self.u, self.y: 1})) != 0: + return True + return False + return False + + def _get_match_object(self): + fx = self.ode_problem.func + x = self.ode_problem.sym + self.u1 = Dummy('u1') + xarg = 0 + yarg = 0 + return [self.d, self.e, fx, x, self.u, self.u1, self.y, xarg, yarg] + + def _get_general_solution(self, *, simplify_flag: bool = True): + d, e, fx, x, u, u1, y, xarg, yarg = self._get_match_object() + (C1,) = self.ode_problem.get_numbered_constants(num=1) + int = Integral(simplify((-d/(e + u1*d)).subs({x: u1, y: 1})), (u1, None, x/fx)) # type: ignore + sol = logcombine(Eq(log(fx), int + log(C1)), force=True) + gen_sol = sol.subs(fx, u).subs(((u, u - yarg), (x, x - xarg), (u, fx))) + return [gen_sol] + + +class HomogeneousCoeffBest(HomogeneousCoeffSubsIndepDivDep, HomogeneousCoeffSubsDepDivIndep): + r""" + Returns the best solution to an ODE from the two hints + ``1st_homogeneous_coeff_subs_dep_div_indep`` and + ``1st_homogeneous_coeff_subs_indep_div_dep``. + + This is as determined by :py:meth:`~sympy.solvers.ode.ode.ode_sol_simplicity`. + + See the + :obj:`~sympy.solvers.ode.single.HomogeneousCoeffSubsIndepDivDep` + and + :obj:`~sympy.solvers.ode.single.HomogeneousCoeffSubsDepDivIndep` + docstrings for more information on these hints. Note that there is no + ``ode_1st_homogeneous_coeff_best_Integral`` hint. + + Examples + ======== + + >>> from sympy import Function, dsolve, pprint + >>> from sympy.abc import x + >>> f = Function('f') + >>> pprint(dsolve(2*x*f(x) + (x**2 + f(x)**2)*f(x).diff(x), f(x), + ... hint='1st_homogeneous_coeff_best', simplify=False)) + / 2 \ + |3*x | + log|----- + 1| + | 2 | + \f (x) / + log(f(x)) = log(C1) - -------------- + 3 + + References + ========== + + - https://en.wikipedia.org/wiki/Homogeneous_differential_equation + - M. Tenenbaum & H. Pollard, "Ordinary Differential Equations", + Dover 1963, pp. 59 + + # indirect doctest + + """ + hint = "1st_homogeneous_coeff_best" + has_integral = False + order = [1] + + def _verify(self, fx): + return HomogeneousCoeffSubsIndepDivDep._verify(self, fx) and \ + HomogeneousCoeffSubsDepDivIndep._verify(self, fx) + + def _get_general_solution(self, *, simplify_flag: bool = True): + # There are two substitutions that solve the equation, u1=y/x and u2=x/y + # # They produce different integrals, so try them both and see which + # # one is easier + sol1 = HomogeneousCoeffSubsIndepDivDep._get_general_solution(self) + sol2 = HomogeneousCoeffSubsDepDivIndep._get_general_solution(self) + fx = self.ode_problem.func + if simplify_flag: + sol1 = odesimp(self.ode_problem.eq, *sol1, fx, "1st_homogeneous_coeff_subs_indep_div_dep") + sol2 = odesimp(self.ode_problem.eq, *sol2, fx, "1st_homogeneous_coeff_subs_dep_div_indep") + # XXX: not simplify should be not simplify_flag. mypy correctly complains + return min([sol1, sol2], key=lambda x: ode_sol_simplicity(x, fx, trysolving=not simplify)) # type: ignore + + +class LinearCoefficients(HomogeneousCoeffBest): + r""" + Solves a differential equation with linear coefficients. + + The general form of a differential equation with linear coefficients is + + .. math:: y' + F\left(\!\frac{a_1 x + b_1 y + c_1}{a_2 x + b_2 y + + c_2}\!\right) = 0\text{,} + + where `a_1`, `b_1`, `c_1`, `a_2`, `b_2`, `c_2` are constants and `a_1 b_2 + - a_2 b_1 \ne 0`. + + This can be solved by substituting: + + .. math:: x = x' + \frac{b_2 c_1 - b_1 c_2}{a_2 b_1 - a_1 b_2} + + y = y' + \frac{a_1 c_2 - a_2 c_1}{a_2 b_1 - a_1 + b_2}\text{.} + + This substitution reduces the equation to a homogeneous differential + equation. + + See Also + ======== + :obj:`sympy.solvers.ode.single.HomogeneousCoeffBest` + :obj:`sympy.solvers.ode.single.HomogeneousCoeffSubsIndepDivDep` + :obj:`sympy.solvers.ode.single.HomogeneousCoeffSubsDepDivIndep` + + Examples + ======== + + >>> from sympy import dsolve, Function, pprint + >>> from sympy.abc import x + >>> f = Function('f') + >>> df = f(x).diff(x) + >>> eq = (x + f(x) + 1)*df + (f(x) - 6*x + 1) + >>> dsolve(eq, hint='linear_coefficients') + [Eq(f(x), -x - sqrt(C1 + 7*x**2) - 1), Eq(f(x), -x + sqrt(C1 + 7*x**2) - 1)] + >>> pprint(dsolve(eq, hint='linear_coefficients')) + ___________ ___________ + / 2 / 2 + [f(x) = -x - \/ C1 + 7*x - 1, f(x) = -x + \/ C1 + 7*x - 1] + + + References + ========== + + - Joel Moses, "Symbolic Integration - The Stormy Decade", Communications + of the ACM, Volume 14, Number 8, August 1971, pp. 558 + """ + hint = "linear_coefficients" + has_integral = True + order = [1] + + def _wilds(self, f, x, order): + d = Wild('d', exclude=[f(x).diff(x), f(x).diff(x, 2)]) + e = Wild('e', exclude=[f(x).diff(x)]) + return d, e + + def _equation(self, fx, x, order): + d, e = self.wilds() + return d + e*fx.diff(x) + + def _verify(self, fx): + self.d, self.e = self.wilds_match() + a, b = self.wilds() + F = self.d/self.e + x = self.ode_problem.sym + params = self._linear_coeff_match(F, fx) + if params: + self.xarg, self.yarg = params + u = Dummy('u') + t = Dummy('t') + self.y = Dummy('y') + # Dummy substitution for df and f(x). + dummy_eq = self.ode_problem.eq.subs(((fx.diff(x), t), (fx, u))) + reps = ((x, x + self.xarg), (u, u + self.yarg), (t, fx.diff(x)), (u, fx)) + dummy_eq = simplify(dummy_eq.subs(reps)) + # get the re-cast values for e and d + r2 = collect(expand(dummy_eq), [fx.diff(x), fx]).match(a*fx.diff(x) + b) + if r2: + self.d, self.e = r2[b], r2[a] + orderd = homogeneous_order(self.d, x, fx) + ordere = homogeneous_order(self.e, x, fx) + if orderd == ordere and orderd is not None: + self.d = self.d.subs(fx, self.y) + self.e = self.e.subs(fx, self.y) + return True + return False + return False + + def _linear_coeff_match(self, expr, func): + r""" + Helper function to match hint ``linear_coefficients``. + + Matches the expression to the form `(a_1 x + b_1 f(x) + c_1)/(a_2 x + b_2 + f(x) + c_2)` where the following conditions hold: + + 1. `a_1`, `b_1`, `c_1`, `a_2`, `b_2`, `c_2` are Rationals; + 2. `c_1` or `c_2` are not equal to zero; + 3. `a_2 b_1 - a_1 b_2` is not equal to zero. + + Return ``xarg``, ``yarg`` where + + 1. ``xarg`` = `(b_2 c_1 - b_1 c_2)/(a_2 b_1 - a_1 b_2)` + 2. ``yarg`` = `(a_1 c_2 - a_2 c_1)/(a_2 b_1 - a_1 b_2)` + + + Examples + ======== + + >>> from sympy import Function, sin + >>> from sympy.abc import x + >>> from sympy.solvers.ode.single import LinearCoefficients + >>> f = Function('f') + >>> eq = (-25*f(x) - 8*x + 62)/(4*f(x) + 11*x - 11) + >>> obj = LinearCoefficients(eq) + >>> obj._linear_coeff_match(eq, f(x)) + (1/9, 22/9) + >>> eq = sin((-5*f(x) - 8*x + 6)/(4*f(x) + x - 1)) + >>> obj = LinearCoefficients(eq) + >>> obj._linear_coeff_match(eq, f(x)) + (19/27, 2/27) + >>> eq = sin(f(x)/x) + >>> obj = LinearCoefficients(eq) + >>> obj._linear_coeff_match(eq, f(x)) + + """ + f = func.func + x = func.args[0] + def abc(eq): + r''' + Internal function of _linear_coeff_match + that returns Rationals a, b, c + if eq is a*x + b*f(x) + c, else None. + ''' + eq = _mexpand(eq) + c = eq.as_independent(x, f(x), as_Add=True)[0] + if not c.is_Rational: + return + a = eq.coeff(x) + if not a.is_Rational: + return + b = eq.coeff(f(x)) + if not b.is_Rational: + return + if eq == a*x + b*f(x) + c: + return a, b, c + + def match(arg): + r''' + Internal function of _linear_coeff_match that returns Rationals a1, + b1, c1, a2, b2, c2 and a2*b1 - a1*b2 of the expression (a1*x + b1*f(x) + + c1)/(a2*x + b2*f(x) + c2) if one of c1 or c2 and a2*b1 - a1*b2 is + non-zero, else None. + ''' + n, d = arg.together().as_numer_denom() + m = abc(n) + if m is not None: + a1, b1, c1 = m + m = abc(d) + if m is not None: + a2, b2, c2 = m + d = a2*b1 - a1*b2 + if (c1 or c2) and d: + return a1, b1, c1, a2, b2, c2, d + + m = [fi.args[0] for fi in expr.atoms(Function) if fi.func != f and + len(fi.args) == 1 and not fi.args[0].is_Function] or {expr} + m1 = match(m.pop()) + if m1 and all(match(mi) == m1 for mi in m): + a1, b1, c1, a2, b2, c2, denom = m1 + return (b2*c1 - b1*c2)/denom, (a1*c2 - a2*c1)/denom + + def _get_match_object(self): + fx = self.ode_problem.func + x = self.ode_problem.sym + self.u1 = Dummy('u1') + u = Dummy('u') + return [self.d, self.e, fx, x, u, self.u1, self.y, self.xarg, self.yarg] + + +class NthOrderReducible(SingleODESolver): + r""" + Solves ODEs that only involve derivatives of the dependent variable using + a substitution of the form `f^n(x) = g(x)`. + + For example any second order ODE of the form `f''(x) = h(f'(x), x)` can be + transformed into a pair of 1st order ODEs `g'(x) = h(g(x), x)` and + `f'(x) = g(x)`. Usually the 1st order ODE for `g` is easier to solve. If + that gives an explicit solution for `g` then `f` is found simply by + integration. + + + Examples + ======== + + >>> from sympy import Function, dsolve, Eq + >>> from sympy.abc import x + >>> f = Function('f') + >>> eq = Eq(x*f(x).diff(x)**2 + f(x).diff(x, 2), 0) + >>> dsolve(eq, f(x), hint='nth_order_reducible') + ... # doctest: +NORMALIZE_WHITESPACE + Eq(f(x), C1 - sqrt(-1/C2)*log(-C2*sqrt(-1/C2) + x) + sqrt(-1/C2)*log(C2*sqrt(-1/C2) + x)) + + """ + hint = "nth_order_reducible" + has_integral = False + + def _matches(self): + # Any ODE that can be solved with a substitution and + # repeated integration e.g.: + # `d^2/dx^2(y) + x*d/dx(y) = constant + #f'(x) must be finite for this to work + eq = self.ode_problem.eq_preprocessed + func = self.ode_problem.func + x = self.ode_problem.sym + r""" + Matches any differential equation that can be rewritten with a smaller + order. Only derivatives of ``func`` alone, wrt a single variable, + are considered, and only in them should ``func`` appear. + """ + # ODE only handles functions of 1 variable so this affirms that state + assert len(func.args) == 1 + vc = [d.variable_count[0] for d in eq.atoms(Derivative) + if d.expr == func and len(d.variable_count) == 1] + ords = [c for v, c in vc if v == x] + if len(ords) < 2: + return False + self.smallest = min(ords) + # make sure func does not appear outside of derivatives + D = Dummy() + if eq.subs(func.diff(x, self.smallest), D).has(func): + return False + return True + + def _get_general_solution(self, *, simplify_flag: bool = True): + eq = self.ode_problem.eq + f = self.ode_problem.func.func + x = self.ode_problem.sym + n = self.smallest + # get a unique function name for g + names = [a.name for a in eq.atoms(AppliedUndef)] + while True: + name = Dummy().name + if name not in names: + g = Function(name) + break + w = f(x).diff(x, n) + geq = eq.subs(w, g(x)) + gsol = dsolve(geq, g(x)) + + if not isinstance(gsol, list): + gsol = [gsol] + + # Might be multiple solutions to the reduced ODE: + fsol = [] + for gsoli in gsol: + fsoli = dsolve(gsoli.subs(g(x), w), f(x)) # or do integration n times + fsol.append(fsoli) + + return fsol + + +class SecondHypergeometric(SingleODESolver): + r""" + Solves 2nd order linear differential equations. + + It computes special function solutions which can be expressed using the + 2F1, 1F1 or 0F1 hypergeometric functions. + + .. math:: y'' + A(x) y' + B(x) y = 0\text{,} + + where `A` and `B` are rational functions. + + These kinds of differential equations have solution of non-Liouvillian form. + + Given linear ODE can be obtained from 2F1 given by + + .. math:: (x^2 - x) y'' + ((a + b + 1) x - c) y' + b a y = 0\text{,} + + where {a, b, c} are arbitrary constants. + + Notes + ===== + + The algorithm should find any solution of the form + + .. math:: y = P(x) _pF_q(..; ..;\frac{\alpha x^k + \beta}{\gamma x^k + \delta})\text{,} + + where pFq is any of 2F1, 1F1 or 0F1 and `P` is an "arbitrary function". + Currently only the 2F1 case is implemented in SymPy but the other cases are + described in the paper and could be implemented in future (contributions + welcome!). + + + Examples + ======== + + >>> from sympy import Function, dsolve, pprint + >>> from sympy.abc import x + >>> f = Function('f') + >>> eq = (x*x - x)*f(x).diff(x,2) + (5*x - 1)*f(x).diff(x) + 4*f(x) + >>> pprint(dsolve(eq, f(x), '2nd_hypergeometric')) + _ + / / 4 \\ |_ /-1, -1 | \ + |C1 + C2*|log(x) + -----||* | | | x| + \ \ x + 1// 2 1 \ 1 | / + f(x) = -------------------------------------------- + 3 + (x - 1) + + + References + ========== + + - "Non-Liouvillian solutions for second order linear ODEs" by L. Chan, E.S. Cheb-Terrab + + """ + hint = "2nd_hypergeometric" + has_integral = True + + def _matches(self): + eq = self.ode_problem.eq_preprocessed + func = self.ode_problem.func + r = match_2nd_hypergeometric(eq, func) + self.match_object = None + if r: + A, B = r + d = equivalence_hypergeometric(A, B, func) + if d: + if d['type'] == "2F1": + self.match_object = match_2nd_2F1_hypergeometric(d['I0'], d['k'], d['sing_point'], func) + if self.match_object is not None: + self.match_object.update({'A':A, 'B':B}) + # We can extend it for 1F1 and 0F1 type also. + return self.match_object is not None + + def _get_general_solution(self, *, simplify_flag: bool = True): + eq = self.ode_problem.eq + func = self.ode_problem.func + if self.match_object['type'] == "2F1": + sol = get_sol_2F1_hypergeometric(eq, func, self.match_object) + if sol is None: + raise NotImplementedError("The given ODE " + str(eq) + " cannot be solved by" + + " the hypergeometric method") + + return [sol] + + +class NthLinearConstantCoeffHomogeneous(SingleODESolver): + r""" + Solves an `n`\th order linear homogeneous differential equation with + constant coefficients. + + This is an equation of the form + + .. math:: a_n f^{(n)}(x) + a_{n-1} f^{(n-1)}(x) + \cdots + a_1 f'(x) + + a_0 f(x) = 0\text{.} + + These equations can be solved in a general manner, by taking the roots of + the characteristic equation `a_n m^n + a_{n-1} m^{n-1} + \cdots + a_1 m + + a_0 = 0`. The solution will then be the sum of `C_n x^i e^{r x}` terms, + for each where `C_n` is an arbitrary constant, `r` is a root of the + characteristic equation and `i` is one of each from 0 to the multiplicity + of the root - 1 (for example, a root 3 of multiplicity 2 would create the + terms `C_1 e^{3 x} + C_2 x e^{3 x}`). The exponential is usually expanded + for complex roots using Euler's equation `e^{I x} = \cos(x) + I \sin(x)`. + Complex roots always come in conjugate pairs in polynomials with real + coefficients, so the two roots will be represented (after simplifying the + constants) as `e^{a x} \left(C_1 \cos(b x) + C_2 \sin(b x)\right)`. + + If SymPy cannot find exact roots to the characteristic equation, a + :py:class:`~sympy.polys.rootoftools.ComplexRootOf` instance will be return + instead. + + >>> from sympy import Function, dsolve + >>> from sympy.abc import x + >>> f = Function('f') + >>> dsolve(f(x).diff(x, 5) + 10*f(x).diff(x) - 2*f(x), f(x), + ... hint='nth_linear_constant_coeff_homogeneous') + ... # doctest: +NORMALIZE_WHITESPACE + Eq(f(x), C5*exp(x*CRootOf(_x**5 + 10*_x - 2, 0)) + + (C1*sin(x*im(CRootOf(_x**5 + 10*_x - 2, 1))) + + C2*cos(x*im(CRootOf(_x**5 + 10*_x - 2, 1))))*exp(x*re(CRootOf(_x**5 + 10*_x - 2, 1))) + + (C3*sin(x*im(CRootOf(_x**5 + 10*_x - 2, 3))) + + C4*cos(x*im(CRootOf(_x**5 + 10*_x - 2, 3))))*exp(x*re(CRootOf(_x**5 + 10*_x - 2, 3)))) + + Note that because this method does not involve integration, there is no + ``nth_linear_constant_coeff_homogeneous_Integral`` hint. + + Examples + ======== + + >>> from sympy import Function, dsolve, pprint + >>> from sympy.abc import x + >>> f = Function('f') + >>> pprint(dsolve(f(x).diff(x, 4) + 2*f(x).diff(x, 3) - + ... 2*f(x).diff(x, 2) - 6*f(x).diff(x) + 5*f(x), f(x), + ... hint='nth_linear_constant_coeff_homogeneous')) + x -2*x + f(x) = (C1 + C2*x)*e + (C3*sin(x) + C4*cos(x))*e + + References + ========== + + - https://en.wikipedia.org/wiki/Linear_differential_equation section: + Nonhomogeneous_equation_with_constant_coefficients + - M. Tenenbaum & H. Pollard, "Ordinary Differential Equations", + Dover 1963, pp. 211 + + # indirect doctest + + """ + hint = "nth_linear_constant_coeff_homogeneous" + has_integral = False + + def _matches(self): + eq = self.ode_problem.eq_high_order_free + func = self.ode_problem.func + order = self.ode_problem.order + x = self.ode_problem.sym + self.r = self.ode_problem.get_linear_coefficients(eq, func, order) + if order and self.r and not any(self.r[i].has(x) for i in self.r if i >= 0): + if not self.r[-1]: + return True + else: + return False + return False + + def _get_general_solution(self, *, simplify_flag: bool = True): + fx = self.ode_problem.func + order = self.ode_problem.order + roots, collectterms = _get_const_characteristic_eq_sols(self.r, fx, order) + # A generator of constants + constants = self.ode_problem.get_numbered_constants(num=len(roots)) + gsol_rhs = Add(*[i*j for (i, j) in zip(constants, roots)]) + gsol = Eq(fx, gsol_rhs) + if simplify_flag: + gsol = _get_simplified_sol([gsol], fx, collectterms) + + return [gsol] + + +class NthLinearConstantCoeffVariationOfParameters(SingleODESolver): + r""" + Solves an `n`\th order linear differential equation with constant + coefficients using the method of variation of parameters. + + This method works on any differential equations of the form + + .. math:: f^{(n)}(x) + a_{n-1} f^{(n-1)}(x) + \cdots + a_1 f'(x) + a_0 + f(x) = P(x)\text{.} + + This method works by assuming that the particular solution takes the form + + .. math:: \sum_{x=1}^{n} c_i(x) y_i(x)\text{,} + + where `y_i` is the `i`\th solution to the homogeneous equation. The + solution is then solved using Wronskian's and Cramer's Rule. The + particular solution is given by + + .. math:: \sum_{x=1}^n \left( \int \frac{W_i(x)}{W(x)} \,dx + \right) y_i(x) \text{,} + + where `W(x)` is the Wronskian of the fundamental system (the system of `n` + linearly independent solutions to the homogeneous equation), and `W_i(x)` + is the Wronskian of the fundamental system with the `i`\th column replaced + with `[0, 0, \cdots, 0, P(x)]`. + + This method is general enough to solve any `n`\th order inhomogeneous + linear differential equation with constant coefficients, but sometimes + SymPy cannot simplify the Wronskian well enough to integrate it. If this + method hangs, try using the + ``nth_linear_constant_coeff_variation_of_parameters_Integral`` hint and + simplifying the integrals manually. Also, prefer using + ``nth_linear_constant_coeff_undetermined_coefficients`` when it + applies, because it does not use integration, making it faster and more + reliable. + + Warning, using simplify=False with + 'nth_linear_constant_coeff_variation_of_parameters' in + :py:meth:`~sympy.solvers.ode.dsolve` may cause it to hang, because it will + not attempt to simplify the Wronskian before integrating. It is + recommended that you only use simplify=False with + 'nth_linear_constant_coeff_variation_of_parameters_Integral' for this + method, especially if the solution to the homogeneous equation has + trigonometric functions in it. + + Examples + ======== + + >>> from sympy import Function, dsolve, pprint, exp, log + >>> from sympy.abc import x + >>> f = Function('f') + >>> pprint(dsolve(f(x).diff(x, 3) - 3*f(x).diff(x, 2) + + ... 3*f(x).diff(x) - f(x) - exp(x)*log(x), f(x), + ... hint='nth_linear_constant_coeff_variation_of_parameters')) + / / / x*log(x) 11*x\\\ x + f(x) = |C1 + x*|C2 + x*|C3 + -------- - ----|||*e + \ \ \ 6 36 /// + + References + ========== + + - https://en.wikipedia.org/wiki/Variation_of_parameters + - https://planetmath.org/VariationOfParameters + - M. Tenenbaum & H. Pollard, "Ordinary Differential Equations", + Dover 1963, pp. 233 + + # indirect doctest + + """ + hint = "nth_linear_constant_coeff_variation_of_parameters" + has_integral = True + + def _matches(self): + eq = self.ode_problem.eq_high_order_free + func = self.ode_problem.func + order = self.ode_problem.order + x = self.ode_problem.sym + self.r = self.ode_problem.get_linear_coefficients(eq, func, order) + + if order and self.r and not any(self.r[i].has(x) for i in self.r if i >= 0): + if self.r[-1]: + return True + else: + return False + return False + + def _get_general_solution(self, *, simplify_flag: bool = True): + eq = self.ode_problem.eq_high_order_free + f = self.ode_problem.func.func + x = self.ode_problem.sym + order = self.ode_problem.order + roots, collectterms = _get_const_characteristic_eq_sols(self.r, f(x), order) + # A generator of constants + constants = self.ode_problem.get_numbered_constants(num=len(roots)) + homogen_sol_rhs = Add(*[i*j for (i, j) in zip(constants, roots)]) + homogen_sol = Eq(f(x), homogen_sol_rhs) + homogen_sol = _solve_variation_of_parameters(eq, f(x), roots, homogen_sol, order, self.r, simplify_flag) + if simplify_flag: + homogen_sol = _get_simplified_sol([homogen_sol], f(x), collectterms) + return [homogen_sol] + + +class NthLinearConstantCoeffUndeterminedCoefficients(SingleODESolver): + r""" + Solves an `n`\th order linear differential equation with constant + coefficients using the method of undetermined coefficients. + + This method works on differential equations of the form + + .. math:: a_n f^{(n)}(x) + a_{n-1} f^{(n-1)}(x) + \cdots + a_1 f'(x) + + a_0 f(x) = P(x)\text{,} + + where `P(x)` is a function that has a finite number of linearly + independent derivatives. + + Functions that fit this requirement are finite sums functions of the form + `a x^i e^{b x} \sin(c x + d)` or `a x^i e^{b x} \cos(c x + d)`, where `i` + is a non-negative integer and `a`, `b`, `c`, and `d` are constants. For + example any polynomial in `x`, functions like `x^2 e^{2 x}`, `x \sin(x)`, + and `e^x \cos(x)` can all be used. Products of `\sin`'s and `\cos`'s have + a finite number of derivatives, because they can be expanded into `\sin(a + x)` and `\cos(b x)` terms. However, SymPy currently cannot do that + expansion, so you will need to manually rewrite the expression in terms of + the above to use this method. So, for example, you will need to manually + convert `\sin^2(x)` into `(1 + \cos(2 x))/2` to properly apply the method + of undetermined coefficients on it. + + This method works by creating a trial function from the expression and all + of its linear independent derivatives and substituting them into the + original ODE. The coefficients for each term will be a system of linear + equations, which are be solved for and substituted, giving the solution. + If any of the trial functions are linearly dependent on the solution to + the homogeneous equation, they are multiplied by sufficient `x` to make + them linearly independent. + + Examples + ======== + + >>> from sympy import Function, dsolve, pprint, exp, cos + >>> from sympy.abc import x + >>> f = Function('f') + >>> pprint(dsolve(f(x).diff(x, 2) + 2*f(x).diff(x) + f(x) - + ... 4*exp(-x)*x**2 + cos(2*x), f(x), + ... hint='nth_linear_constant_coeff_undetermined_coefficients')) + / / 3\\ + | | x || -x 4*sin(2*x) 3*cos(2*x) + f(x) = |C1 + x*|C2 + --||*e - ---------- + ---------- + \ \ 3 // 25 25 + + References + ========== + + - https://en.wikipedia.org/wiki/Method_of_undetermined_coefficients + - M. Tenenbaum & H. Pollard, "Ordinary Differential Equations", + Dover 1963, pp. 221 + + # indirect doctest + + """ + hint = "nth_linear_constant_coeff_undetermined_coefficients" + has_integral = False + + def _matches(self): + eq = self.ode_problem.eq_high_order_free + func = self.ode_problem.func + order = self.ode_problem.order + x = self.ode_problem.sym + self.r = self.ode_problem.get_linear_coefficients(eq, func, order) + does_match = False + if order and self.r and not any(self.r[i].has(x) for i in self.r if i >= 0): + if self.r[-1]: + eq_homogeneous = Add(eq, -self.r[-1]) + undetcoeff = _undetermined_coefficients_match(self.r[-1], x, func, eq_homogeneous) + if undetcoeff['test']: + self.trialset = undetcoeff['trialset'] + does_match = True + return does_match + + def _get_general_solution(self, *, simplify_flag: bool = True): + eq = self.ode_problem.eq + f = self.ode_problem.func.func + x = self.ode_problem.sym + order = self.ode_problem.order + roots, collectterms = _get_const_characteristic_eq_sols(self.r, f(x), order) + # A generator of constants + constants = self.ode_problem.get_numbered_constants(num=len(roots)) + homogen_sol_rhs = Add(*[i*j for (i, j) in zip(constants, roots)]) + homogen_sol = Eq(f(x), homogen_sol_rhs) + self.r.update({'list': roots, 'sol': homogen_sol, 'simpliy_flag': simplify_flag}) + gsol = _solve_undetermined_coefficients(eq, f(x), order, self.r, self.trialset) + if simplify_flag: + gsol = _get_simplified_sol([gsol], f(x), collectterms) + return [gsol] + + +class NthLinearEulerEqHomogeneous(SingleODESolver): + r""" + Solves an `n`\th order linear homogeneous variable-coefficient + Cauchy-Euler equidimensional ordinary differential equation. + + This is an equation with form `0 = a_0 f(x) + a_1 x f'(x) + a_2 x^2 f''(x) + \cdots`. + + These equations can be solved in a general manner, by substituting + solutions of the form `f(x) = x^r`, and deriving a characteristic equation + for `r`. When there are repeated roots, we include extra terms of the + form `C_{r k} \ln^k(x) x^r`, where `C_{r k}` is an arbitrary integration + constant, `r` is a root of the characteristic equation, and `k` ranges + over the multiplicity of `r`. In the cases where the roots are complex, + solutions of the form `C_1 x^a \sin(b \log(x)) + C_2 x^a \cos(b \log(x))` + are returned, based on expansions with Euler's formula. The general + solution is the sum of the terms found. If SymPy cannot find exact roots + to the characteristic equation, a + :py:obj:`~.ComplexRootOf` instance will be returned + instead. + + >>> from sympy import Function, dsolve + >>> from sympy.abc import x + >>> f = Function('f') + >>> dsolve(4*x**2*f(x).diff(x, 2) + f(x), f(x), + ... hint='nth_linear_euler_eq_homogeneous') + ... # doctest: +NORMALIZE_WHITESPACE + Eq(f(x), sqrt(x)*(C1 + C2*log(x))) + + Note that because this method does not involve integration, there is no + ``nth_linear_euler_eq_homogeneous_Integral`` hint. + + The following is for internal use: + + - ``returns = 'sol'`` returns the solution to the ODE. + - ``returns = 'list'`` returns a list of linearly independent solutions, + corresponding to the fundamental solution set, for use with non + homogeneous solution methods like variation of parameters and + undetermined coefficients. Note that, though the solutions should be + linearly independent, this function does not explicitly check that. You + can do ``assert simplify(wronskian(sollist)) != 0`` to check for linear + independence. Also, ``assert len(sollist) == order`` will need to pass. + - ``returns = 'both'``, return a dictionary ``{'sol': , + 'list': }``. + + Examples + ======== + + >>> from sympy import Function, dsolve, pprint + >>> from sympy.abc import x + >>> f = Function('f') + >>> eq = f(x).diff(x, 2)*x**2 - 4*f(x).diff(x)*x + 6*f(x) + >>> pprint(dsolve(eq, f(x), + ... hint='nth_linear_euler_eq_homogeneous')) + 2 + f(x) = x *(C1 + C2*x) + + References + ========== + + - https://en.wikipedia.org/wiki/Cauchy%E2%80%93Euler_equation + - C. Bender & S. Orszag, "Advanced Mathematical Methods for Scientists and + Engineers", Springer 1999, pp. 12 + + # indirect doctest + + """ + hint = "nth_linear_euler_eq_homogeneous" + has_integral = False + + def _matches(self): + eq = self.ode_problem.eq_preprocessed + f = self.ode_problem.func.func + order = self.ode_problem.order + x = self.ode_problem.sym + match = self.ode_problem.get_linear_coefficients(eq, f(x), order) + self.r = None + does_match = False + + if order and match: + coeff = match[order] + factor = x**order / coeff + self.r = {i: factor*match[i] for i in match} + if self.r and all(_test_term(self.r[i], f(x), i) for i in + self.r if i >= 0): + if not self.r[-1]: + does_match = True + return does_match + + def _get_general_solution(self, *, simplify_flag: bool = True): + fx = self.ode_problem.func + eq = self.ode_problem.eq + homogen_sol = _get_euler_characteristic_eq_sols(eq, fx, self.r)[0] + return [homogen_sol] + + +class NthLinearEulerEqNonhomogeneousVariationOfParameters(SingleODESolver): + r""" + Solves an `n`\th order linear non homogeneous Cauchy-Euler equidimensional + ordinary differential equation using variation of parameters. + + This is an equation with form `g(x) = a_0 f(x) + a_1 x f'(x) + a_2 x^2 f''(x) + \cdots`. + + This method works by assuming that the particular solution takes the form + + .. math:: \sum_{x=1}^{n} c_i(x) y_i(x) {a_n} {x^n} \text{, } + + where `y_i` is the `i`\th solution to the homogeneous equation. The + solution is then solved using Wronskian's and Cramer's Rule. The + particular solution is given by multiplying eq given below with `a_n x^{n}` + + .. math:: \sum_{x=1}^n \left( \int \frac{W_i(x)}{W(x)} \, dx + \right) y_i(x) \text{, } + + where `W(x)` is the Wronskian of the fundamental system (the system of `n` + linearly independent solutions to the homogeneous equation), and `W_i(x)` + is the Wronskian of the fundamental system with the `i`\th column replaced + with `[0, 0, \cdots, 0, \frac{x^{- n}}{a_n} g{\left(x \right)}]`. + + This method is general enough to solve any `n`\th order inhomogeneous + linear differential equation, but sometimes SymPy cannot simplify the + Wronskian well enough to integrate it. If this method hangs, try using the + ``nth_linear_constant_coeff_variation_of_parameters_Integral`` hint and + simplifying the integrals manually. Also, prefer using + ``nth_linear_constant_coeff_undetermined_coefficients`` when it + applies, because it does not use integration, making it faster and more + reliable. + + Warning, using simplify=False with + 'nth_linear_constant_coeff_variation_of_parameters' in + :py:meth:`~sympy.solvers.ode.dsolve` may cause it to hang, because it will + not attempt to simplify the Wronskian before integrating. It is + recommended that you only use simplify=False with + 'nth_linear_constant_coeff_variation_of_parameters_Integral' for this + method, especially if the solution to the homogeneous equation has + trigonometric functions in it. + + Examples + ======== + + >>> from sympy import Function, dsolve, Derivative + >>> from sympy.abc import x + >>> f = Function('f') + >>> eq = x**2*Derivative(f(x), x, x) - 2*x*Derivative(f(x), x) + 2*f(x) - x**4 + >>> dsolve(eq, f(x), + ... hint='nth_linear_euler_eq_nonhomogeneous_variation_of_parameters').expand() + Eq(f(x), C1*x + C2*x**2 + x**4/6) + + """ + hint = "nth_linear_euler_eq_nonhomogeneous_variation_of_parameters" + has_integral = True + + def _matches(self): + eq = self.ode_problem.eq_preprocessed + f = self.ode_problem.func.func + order = self.ode_problem.order + x = self.ode_problem.sym + match = self.ode_problem.get_linear_coefficients(eq, f(x), order) + self.r = None + does_match = False + + if order and match: + coeff = match[order] + factor = x**order / coeff + self.r = {i: factor*match[i] for i in match} + if self.r and all(_test_term(self.r[i], f(x), i) for i in + self.r if i >= 0): + if self.r[-1]: + does_match = True + + return does_match + + def _get_general_solution(self, *, simplify_flag: bool = True): + eq = self.ode_problem.eq + f = self.ode_problem.func.func + x = self.ode_problem.sym + order = self.ode_problem.order + homogen_sol, roots = _get_euler_characteristic_eq_sols(eq, f(x), self.r) + self.r[-1] = self.r[-1]/self.r[order] + sol = _solve_variation_of_parameters(eq, f(x), roots, homogen_sol, order, self.r, simplify_flag) + + return [Eq(f(x), homogen_sol.rhs + (sol.rhs - homogen_sol.rhs)*self.r[order])] + + +class NthLinearEulerEqNonhomogeneousUndeterminedCoefficients(SingleODESolver): + r""" + Solves an `n`\th order linear non homogeneous Cauchy-Euler equidimensional + ordinary differential equation using undetermined coefficients. + + This is an equation with form `g(x) = a_0 f(x) + a_1 x f'(x) + a_2 x^2 f''(x) + \cdots`. + + These equations can be solved in a general manner, by substituting + solutions of the form `x = exp(t)`, and deriving a characteristic equation + of form `g(exp(t)) = b_0 f(t) + b_1 f'(t) + b_2 f''(t) \cdots` which can + be then solved by nth_linear_constant_coeff_undetermined_coefficients if + g(exp(t)) has finite number of linearly independent derivatives. + + Functions that fit this requirement are finite sums functions of the form + `a x^i e^{b x} \sin(c x + d)` or `a x^i e^{b x} \cos(c x + d)`, where `i` + is a non-negative integer and `a`, `b`, `c`, and `d` are constants. For + example any polynomial in `x`, functions like `x^2 e^{2 x}`, `x \sin(x)`, + and `e^x \cos(x)` can all be used. Products of `\sin`'s and `\cos`'s have + a finite number of derivatives, because they can be expanded into `\sin(a + x)` and `\cos(b x)` terms. However, SymPy currently cannot do that + expansion, so you will need to manually rewrite the expression in terms of + the above to use this method. So, for example, you will need to manually + convert `\sin^2(x)` into `(1 + \cos(2 x))/2` to properly apply the method + of undetermined coefficients on it. + + After replacement of x by exp(t), this method works by creating a trial function + from the expression and all of its linear independent derivatives and + substituting them into the original ODE. The coefficients for each term + will be a system of linear equations, which are be solved for and + substituted, giving the solution. If any of the trial functions are linearly + dependent on the solution to the homogeneous equation, they are multiplied + by sufficient `x` to make them linearly independent. + + Examples + ======== + + >>> from sympy import dsolve, Function, Derivative, log + >>> from sympy.abc import x + >>> f = Function('f') + >>> eq = x**2*Derivative(f(x), x, x) - 2*x*Derivative(f(x), x) + 2*f(x) - log(x) + >>> dsolve(eq, f(x), + ... hint='nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients').expand() + Eq(f(x), C1*x + C2*x**2 + log(x)/2 + 3/4) + + """ + hint = "nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients" + has_integral = False + + def _matches(self): + eq = self.ode_problem.eq_high_order_free + f = self.ode_problem.func.func + order = self.ode_problem.order + x = self.ode_problem.sym + match = self.ode_problem.get_linear_coefficients(eq, f(x), order) + self.r = None + does_match = False + + if order and match: + coeff = match[order] + factor = x**order / coeff + self.r = {i: factor*match[i] for i in match} + if self.r and all(_test_term(self.r[i], f(x), i) for i in + self.r if i >= 0): + if self.r[-1]: + e, re = posify(self.r[-1].subs(x, exp(x))) + undetcoeff = _undetermined_coefficients_match(e.subs(re), x) + if undetcoeff['test']: + does_match = True + return does_match + + def _get_general_solution(self, *, simplify_flag: bool = True): + f = self.ode_problem.func.func + x = self.ode_problem.sym + chareq, eq, symbol = S.Zero, S.Zero, Dummy('x') + for i in self.r.keys(): + if i >= 0: + chareq += (self.r[i]*diff(x**symbol, x, i)*x**-symbol).expand() + + for i in range(1, degree(Poly(chareq, symbol))+1): + eq += chareq.coeff(symbol**i)*diff(f(x), x, i) + + if chareq.as_coeff_add(symbol)[0]: + eq += chareq.as_coeff_add(symbol)[0]*f(x) + e, re = posify(self.r[-1].subs(x, exp(x))) + eq += e.subs(re) + + self.const_undet_instance = NthLinearConstantCoeffUndeterminedCoefficients(SingleODEProblem(eq, f(x), x)) + sol = self.const_undet_instance.get_general_solution(simplify = simplify_flag)[0] + sol = sol.subs(x, log(x)) # type: ignore + sol = sol.subs(f(log(x)), f(x)).expand() # type: ignore + + return [sol] + + +class SecondLinearBessel(SingleODESolver): + r""" + Gives solution of the Bessel differential equation + + .. math :: x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} y(x) + (x^2-n^2) y(x) + + if `n` is integer then the solution is of the form ``Eq(f(x), C0 besselj(n,x) + + C1 bessely(n,x))`` as both the solutions are linearly independent else if + `n` is a fraction then the solution is of the form ``Eq(f(x), C0 besselj(n,x) + + C1 besselj(-n,x))`` which can also transform into ``Eq(f(x), C0 besselj(n,x) + + C1 bessely(n,x))``. + + Examples + ======== + + >>> from sympy.abc import x + >>> from sympy import Symbol + >>> v = Symbol('v', positive=True) + >>> from sympy import dsolve, Function + >>> f = Function('f') + >>> y = f(x) + >>> genform = x**2*y.diff(x, 2) + x*y.diff(x) + (x**2 - v**2)*y + >>> dsolve(genform) + Eq(f(x), C1*besselj(v, x) + C2*bessely(v, x)) + + References + ========== + + https://math24.net/bessel-differential-equation.html + + """ + hint = "2nd_linear_bessel" + has_integral = False + + def _matches(self): + eq = self.ode_problem.eq_high_order_free + f = self.ode_problem.func + order = self.ode_problem.order + x = self.ode_problem.sym + df = f.diff(x) + a = Wild('a', exclude=[f,df]) + b = Wild('b', exclude=[x, f,df]) + a4 = Wild('a4', exclude=[x,f,df]) + b4 = Wild('b4', exclude=[x,f,df]) + c4 = Wild('c4', exclude=[x,f,df]) + d4 = Wild('d4', exclude=[x,f,df]) + a3 = Wild('a3', exclude=[f, df, f.diff(x, 2)]) + b3 = Wild('b3', exclude=[f, df, f.diff(x, 2)]) + c3 = Wild('c3', exclude=[f, df, f.diff(x, 2)]) + deq = a3*(f.diff(x, 2)) + b3*df + c3*f + r = collect(eq, + [f.diff(x, 2), df, f]).match(deq) + if order == 2 and r: + if not all(r[key].is_polynomial() for key in r): + n, d = eq.as_numer_denom() + eq = expand(n) + r = collect(eq, + [f.diff(x, 2), df, f]).match(deq) + + if r and r[a3] != 0: + # leading coeff of f(x).diff(x, 2) + coeff = factor(r[a3]).match(a4*(x-b)**b4) + + if coeff: + # if coeff[b4] = 0 means constant coefficient + if coeff[b4] == 0: + return False + point = coeff[b] + else: + return False + + if point: + r[a3] = simplify(r[a3].subs(x, x+point)) + r[b3] = simplify(r[b3].subs(x, x+point)) + r[c3] = simplify(r[c3].subs(x, x+point)) + + # making a3 in the form of x**2 + r[a3] = cancel(r[a3]/(coeff[a4]*(x)**(-2+coeff[b4]))) + r[b3] = cancel(r[b3]/(coeff[a4]*(x)**(-2+coeff[b4]))) + r[c3] = cancel(r[c3]/(coeff[a4]*(x)**(-2+coeff[b4]))) + # checking if b3 is of form c*(x-b) + coeff1 = factor(r[b3]).match(a4*(x)) + if coeff1 is None: + return False + # c3 maybe of very complex form so I am simply checking (a - b) form + # if yes later I will match with the standard form of bessel in a and b + # a, b are wild variable defined above. + _coeff2 = expand(r[c3]).match(a - b) + if _coeff2 is None: + return False + # matching with standard form for c3 + coeff2 = factor(_coeff2[a]).match(c4**2*(x)**(2*a4)) + if coeff2 is None: + return False + + if _coeff2[b] == 0: + coeff2[d4] = 0 + else: + coeff2[d4] = factor(_coeff2[b]).match(d4**2)[d4] + + self.rn = {'n':coeff2[d4], 'a4':coeff2[c4], 'd4':coeff2[a4]} + self.rn['c4'] = coeff1[a4] + self.rn['b4'] = point + return True + return False + + def _get_general_solution(self, *, simplify_flag: bool = True): + f = self.ode_problem.func.func + x = self.ode_problem.sym + n = self.rn['n'] + a4 = self.rn['a4'] + c4 = self.rn['c4'] + d4 = self.rn['d4'] + b4 = self.rn['b4'] + n = sqrt(n**2 + Rational(1, 4)*(c4 - 1)**2) + (C1, C2) = self.ode_problem.get_numbered_constants(num=2) + return [Eq(f(x), ((x**(Rational(1-c4,2)))*(C1*besselj(n/d4,a4*x**d4/d4) + + C2*bessely(n/d4,a4*x**d4/d4))).subs(x, x-b4))] + + +class SecondLinearAiry(SingleODESolver): + r""" + Gives solution of the Airy differential equation + + .. math :: \frac{d^2y}{dx^2} + (a + b x) y(x) = 0 + + in terms of Airy special functions airyai and airybi. + + Examples + ======== + + >>> from sympy import dsolve, Function + >>> from sympy.abc import x + >>> f = Function("f") + >>> eq = f(x).diff(x, 2) - x*f(x) + >>> dsolve(eq) + Eq(f(x), C1*airyai(x) + C2*airybi(x)) + """ + hint = "2nd_linear_airy" + has_integral = False + + def _matches(self): + eq = self.ode_problem.eq_high_order_free + f = self.ode_problem.func + order = self.ode_problem.order + x = self.ode_problem.sym + df = f.diff(x) + a4 = Wild('a4', exclude=[x,f,df]) + b4 = Wild('b4', exclude=[x,f,df]) + match = self.ode_problem.get_linear_coefficients(eq, f, order) + does_match = False + if order == 2 and match and match[2] != 0: + if match[1].is_zero: + self.rn = cancel(match[0]/match[2]).match(a4+b4*x) + if self.rn and self.rn[b4] != 0: + self.rn = {'b':self.rn[a4],'m':self.rn[b4]} + does_match = True + return does_match + + def _get_general_solution(self, *, simplify_flag: bool = True): + f = self.ode_problem.func.func + x = self.ode_problem.sym + (C1, C2) = self.ode_problem.get_numbered_constants(num=2) + b = self.rn['b'] + m = self.rn['m'] + if m.is_positive: + arg = - b/cbrt(m)**2 - cbrt(m)*x + elif m.is_negative: + arg = - b/cbrt(-m)**2 + cbrt(-m)*x + else: + arg = - b/cbrt(-m)**2 + cbrt(-m)*x + + return [Eq(f(x), C1*airyai(arg) + C2*airybi(arg))] + + +class LieGroup(SingleODESolver): + r""" + This hint implements the Lie group method of solving first order differential + equations. The aim is to convert the given differential equation from the + given coordinate system into another coordinate system where it becomes + invariant under the one-parameter Lie group of translations. The converted + ODE can be easily solved by quadrature. It makes use of the + :py:meth:`sympy.solvers.ode.infinitesimals` function which returns the + infinitesimals of the transformation. + + The coordinates `r` and `s` can be found by solving the following Partial + Differential Equations. + + .. math :: \xi\frac{\partial r}{\partial x} + \eta\frac{\partial r}{\partial y} + = 0 + + .. math :: \xi\frac{\partial s}{\partial x} + \eta\frac{\partial s}{\partial y} + = 1 + + The differential equation becomes separable in the new coordinate system + + .. math :: \frac{ds}{dr} = \frac{\frac{\partial s}{\partial x} + + h(x, y)\frac{\partial s}{\partial y}}{ + \frac{\partial r}{\partial x} + h(x, y)\frac{\partial r}{\partial y}} + + After finding the solution by integration, it is then converted back to the original + coordinate system by substituting `r` and `s` in terms of `x` and `y` again. + + Examples + ======== + + >>> from sympy import Function, dsolve, exp, pprint + >>> from sympy.abc import x + >>> f = Function('f') + >>> pprint(dsolve(f(x).diff(x) + 2*x*f(x) - x*exp(-x**2), f(x), + ... hint='lie_group')) + / 2\ 2 + | x | -x + f(x) = |C1 + --|*e + \ 2 / + + + References + ========== + + - Solving differential equations by Symmetry Groups, + John Starrett, pp. 1 - pp. 14 + + """ + hint = "lie_group" + has_integral = False + + def _has_additional_params(self): + return 'xi' in self.ode_problem.params and 'eta' in self.ode_problem.params + + def _matches(self): + eq = self.ode_problem.eq + f = self.ode_problem.func.func + order = self.ode_problem.order + x = self.ode_problem.sym + df = f(x).diff(x) + y = Dummy('y') + d = Wild('d', exclude=[df, f(x).diff(x, 2)]) + e = Wild('e', exclude=[df]) + does_match = False + if self._has_additional_params() and order == 1: + xi = self.ode_problem.params['xi'] + eta = self.ode_problem.params['eta'] + self.r3 = {'xi': xi, 'eta': eta} + r = collect(eq, df, exact=True).match(d + e * df) + if r: + r['d'] = d + r['e'] = e + r['y'] = y + r[d] = r[d].subs(f(x), y) + r[e] = r[e].subs(f(x), y) + self.r3.update(r) + does_match = True + return does_match + + def _get_general_solution(self, *, simplify_flag: bool = True): + eq = self.ode_problem.eq + x = self.ode_problem.sym + func = self.ode_problem.func + order = self.ode_problem.order + df = func.diff(x) + + try: + eqsol = solve(eq, df) + except NotImplementedError: + eqsol = [] + + desols = [] + for s in eqsol: + sol = _ode_lie_group(s, func, order, match=self.r3) + if sol: + desols.extend(sol) + + if desols == []: + raise NotImplementedError("The given ODE " + str(eq) + " cannot be solved by" + + " the lie group method") + return desols + + +solver_map = { + 'factorable': Factorable, + 'nth_linear_constant_coeff_homogeneous': NthLinearConstantCoeffHomogeneous, + 'nth_linear_euler_eq_homogeneous': NthLinearEulerEqHomogeneous, + 'nth_linear_constant_coeff_undetermined_coefficients': NthLinearConstantCoeffUndeterminedCoefficients, + 'nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients': NthLinearEulerEqNonhomogeneousUndeterminedCoefficients, + 'separable': Separable, + '1st_exact': FirstExact, + '1st_linear': FirstLinear, + 'Bernoulli': Bernoulli, + 'Riccati_special_minus2': RiccatiSpecial, + '1st_rational_riccati': RationalRiccati, + '1st_homogeneous_coeff_best': HomogeneousCoeffBest, + '1st_homogeneous_coeff_subs_indep_div_dep': HomogeneousCoeffSubsIndepDivDep, + '1st_homogeneous_coeff_subs_dep_div_indep': HomogeneousCoeffSubsDepDivIndep, + 'almost_linear': AlmostLinear, + 'linear_coefficients': LinearCoefficients, + 'separable_reduced': SeparableReduced, + 'nth_linear_constant_coeff_variation_of_parameters': NthLinearConstantCoeffVariationOfParameters, + 'nth_linear_euler_eq_nonhomogeneous_variation_of_parameters': NthLinearEulerEqNonhomogeneousVariationOfParameters, + 'Liouville': Liouville, + '2nd_linear_airy': SecondLinearAiry, + '2nd_linear_bessel': SecondLinearBessel, + '2nd_hypergeometric': SecondHypergeometric, + 'nth_order_reducible': NthOrderReducible, + '2nd_nonlinear_autonomous_conserved': SecondNonlinearAutonomousConserved, + 'nth_algebraic': NthAlgebraic, + 'lie_group': LieGroup, + } + +# Avoid circular import: +from .ode import dsolve, ode_sol_simplicity, odesimp, homogeneous_order diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/ode/subscheck.py b/.venv/lib/python3.12/site-packages/sympy/solvers/ode/subscheck.py new file mode 100644 index 0000000000000000000000000000000000000000..6ac7fba7d364bf599e928ccf591b5bef096576d0 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/ode/subscheck.py @@ -0,0 +1,392 @@ +from sympy.core import S, Pow +from sympy.core.function import (Derivative, AppliedUndef, diff) +from sympy.core.relational import Equality, Eq +from sympy.core.symbol import Dummy +from sympy.core.sympify import sympify + +from sympy.logic.boolalg import BooleanAtom +from sympy.functions import exp +from sympy.series import Order +from sympy.simplify.simplify import simplify, posify, besselsimp +from sympy.simplify.trigsimp import trigsimp +from sympy.simplify.sqrtdenest import sqrtdenest +from sympy.solvers import solve +from sympy.solvers.deutils import _preprocess, ode_order +from sympy.utilities.iterables import iterable, is_sequence + + +def sub_func_doit(eq, func, new): + r""" + When replacing the func with something else, we usually want the + derivative evaluated, so this function helps in making that happen. + + Examples + ======== + + >>> from sympy import Derivative, symbols, Function + >>> from sympy.solvers.ode.subscheck import sub_func_doit + >>> x, z = symbols('x, z') + >>> y = Function('y') + + >>> sub_func_doit(3*Derivative(y(x), x) - 1, y(x), x) + 2 + + >>> sub_func_doit(x*Derivative(y(x), x) - y(x)**2 + y(x), y(x), + ... 1/(x*(z + 1/x))) + x*(-1/(x**2*(z + 1/x)) + 1/(x**3*(z + 1/x)**2)) + 1/(x*(z + 1/x)) + ...- 1/(x**2*(z + 1/x)**2) + """ + reps= {func: new} + for d in eq.atoms(Derivative): + if d.expr == func: + reps[d] = new.diff(*d.variable_count) + else: + reps[d] = d.xreplace({func: new}).doit(deep=False) + return eq.xreplace(reps) + + +def checkodesol(ode, sol, func=None, order='auto', solve_for_func=True): + r""" + Substitutes ``sol`` into ``ode`` and checks that the result is ``0``. + + This works when ``func`` is one function, like `f(x)` or a list of + functions like `[f(x), g(x)]` when `ode` is a system of ODEs. ``sol`` can + be a single solution or a list of solutions. Each solution may be an + :py:class:`~sympy.core.relational.Equality` that the solution satisfies, + e.g. ``Eq(f(x), C1), Eq(f(x) + C1, 0)``; or simply an + :py:class:`~sympy.core.expr.Expr`, e.g. ``f(x) - C1``. In most cases it + will not be necessary to explicitly identify the function, but if the + function cannot be inferred from the original equation it can be supplied + through the ``func`` argument. + + If a sequence of solutions is passed, the same sort of container will be + used to return the result for each solution. + + It tries the following methods, in order, until it finds zero equivalence: + + 1. Substitute the solution for `f` in the original equation. This only + works if ``ode`` is solved for `f`. It will attempt to solve it first + unless ``solve_for_func == False``. + 2. Take `n` derivatives of the solution, where `n` is the order of + ``ode``, and check to see if that is equal to the solution. This only + works on exact ODEs. + 3. Take the 1st, 2nd, ..., `n`\th derivatives of the solution, each time + solving for the derivative of `f` of that order (this will always be + possible because `f` is a linear operator). Then back substitute each + derivative into ``ode`` in reverse order. + + This function returns a tuple. The first item in the tuple is ``True`` if + the substitution results in ``0``, and ``False`` otherwise. The second + item in the tuple is what the substitution results in. It should always + be ``0`` if the first item is ``True``. Sometimes this function will + return ``False`` even when an expression is identically equal to ``0``. + This happens when :py:meth:`~sympy.simplify.simplify.simplify` does not + reduce the expression to ``0``. If an expression returned by this + function vanishes identically, then ``sol`` really is a solution to + the ``ode``. + + If this function seems to hang, it is probably because of a hard + simplification. + + To use this function to test, test the first item of the tuple. + + Examples + ======== + + >>> from sympy import (Eq, Function, checkodesol, symbols, + ... Derivative, exp) + >>> x, C1, C2 = symbols('x,C1,C2') + >>> f, g = symbols('f g', cls=Function) + >>> checkodesol(f(x).diff(x), Eq(f(x), C1)) + (True, 0) + >>> assert checkodesol(f(x).diff(x), C1)[0] + >>> assert not checkodesol(f(x).diff(x), x)[0] + >>> checkodesol(f(x).diff(x, 2), x**2) + (False, 2) + + >>> eqs = [Eq(Derivative(f(x), x), f(x)), Eq(Derivative(g(x), x), g(x))] + >>> sol = [Eq(f(x), C1*exp(x)), Eq(g(x), C2*exp(x))] + >>> checkodesol(eqs, sol) + (True, [0, 0]) + + """ + if iterable(ode): + return checksysodesol(ode, sol, func=func) + + if not isinstance(ode, Equality): + ode = Eq(ode, 0) + if func is None: + try: + _, func = _preprocess(ode.lhs) + except ValueError: + funcs = [s.atoms(AppliedUndef) for s in ( + sol if is_sequence(sol, set) else [sol])] + funcs = set().union(*funcs) + if len(funcs) != 1: + raise ValueError( + 'must pass func arg to checkodesol for this case.') + func = funcs.pop() + if not isinstance(func, AppliedUndef) or len(func.args) != 1: + raise ValueError( + "func must be a function of one variable, not %s" % func) + if is_sequence(sol, set): + return type(sol)([checkodesol(ode, i, order=order, solve_for_func=solve_for_func) for i in sol]) + + if not isinstance(sol, Equality): + sol = Eq(func, sol) + elif sol.rhs == func: + sol = sol.reversed + + if order == 'auto': + order = ode_order(ode, func) + solved = sol.lhs == func and not sol.rhs.has(func) + if solve_for_func and not solved: + rhs = solve(sol, func) + if rhs: + eqs = [Eq(func, t) for t in rhs] + if len(rhs) == 1: + eqs = eqs[0] + return checkodesol(ode, eqs, order=order, + solve_for_func=False) + + x = func.args[0] + + # Handle series solutions here + if sol.has(Order): + assert sol.lhs == func + Oterm = sol.rhs.getO() + solrhs = sol.rhs.removeO() + + Oexpr = Oterm.expr + assert isinstance(Oexpr, Pow) + sorder = Oexpr.exp + assert Oterm == Order(x**sorder) + + odesubs = (ode.lhs-ode.rhs).subs(func, solrhs).doit().expand() + + neworder = Order(x**(sorder - order)) + odesubs = odesubs + neworder + assert odesubs.getO() == neworder + residual = odesubs.removeO() + + return (residual == 0, residual) + + s = True + testnum = 0 + while s: + if testnum == 0: + # First pass, try substituting a solved solution directly into the + # ODE. This has the highest chance of succeeding. + ode_diff = ode.lhs - ode.rhs + + if sol.lhs == func: + s = sub_func_doit(ode_diff, func, sol.rhs) + s = besselsimp(s) + else: + testnum += 1 + continue + ss = simplify(s.rewrite(exp)) + if ss: + # with the new numer_denom in power.py, if we do a simple + # expansion then testnum == 0 verifies all solutions. + s = ss.expand(force=True) + else: + s = 0 + testnum += 1 + elif testnum == 1: + # Second pass. If we cannot substitute f, try seeing if the nth + # derivative is equal, this will only work for odes that are exact, + # by definition. + s = simplify( + trigsimp(diff(sol.lhs, x, order) - diff(sol.rhs, x, order)) - + trigsimp(ode.lhs) + trigsimp(ode.rhs)) + # s2 = simplify( + # diff(sol.lhs, x, order) - diff(sol.rhs, x, order) - \ + # ode.lhs + ode.rhs) + testnum += 1 + elif testnum == 2: + # Third pass. Try solving for df/dx and substituting that into the + # ODE. Thanks to Chris Smith for suggesting this method. Many of + # the comments below are his, too. + # The method: + # - Take each of 1..n derivatives of the solution. + # - Solve each nth derivative for d^(n)f/dx^(n) + # (the differential of that order) + # - Back substitute into the ODE in decreasing order + # (i.e., n, n-1, ...) + # - Check the result for zero equivalence + if sol.lhs == func and not sol.rhs.has(func): + diffsols = {0: sol.rhs} + elif sol.rhs == func and not sol.lhs.has(func): + diffsols = {0: sol.lhs} + else: + diffsols = {} + sol = sol.lhs - sol.rhs + for i in range(1, order + 1): + # Differentiation is a linear operator, so there should always + # be 1 solution. Nonetheless, we test just to make sure. + # We only need to solve once. After that, we automatically + # have the solution to the differential in the order we want. + if i == 1: + ds = sol.diff(x) + try: + sdf = solve(ds, func.diff(x, i)) + if not sdf: + raise NotImplementedError + except NotImplementedError: + testnum += 1 + break + else: + diffsols[i] = sdf[0] + else: + # This is what the solution says df/dx should be. + diffsols[i] = diffsols[i - 1].diff(x) + + # Make sure the above didn't fail. + if testnum > 2: + continue + else: + # Substitute it into ODE to check for self consistency. + lhs, rhs = ode.lhs, ode.rhs + for i in range(order, -1, -1): + if i == 0 and 0 not in diffsols: + # We can only substitute f(x) if the solution was + # solved for f(x). + break + lhs = sub_func_doit(lhs, func.diff(x, i), diffsols[i]) + rhs = sub_func_doit(rhs, func.diff(x, i), diffsols[i]) + ode_or_bool = Eq(lhs, rhs) + ode_or_bool = simplify(ode_or_bool) + + if isinstance(ode_or_bool, (bool, BooleanAtom)): + if ode_or_bool: + lhs = rhs = S.Zero + else: + lhs = ode_or_bool.lhs + rhs = ode_or_bool.rhs + # No sense in overworking simplify -- just prove that the + # numerator goes to zero + num = trigsimp((lhs - rhs).as_numer_denom()[0]) + # since solutions are obtained using force=True we test + # using the same level of assumptions + ## replace function with dummy so assumptions will work + _func = Dummy('func') + num = num.subs(func, _func) + ## posify the expression + num, reps = posify(num) + s = simplify(num).xreplace(reps).xreplace({_func: func}) + testnum += 1 + else: + break + + if not s: + return (True, s) + elif s is True: # The code above never was able to change s + raise NotImplementedError("Unable to test if " + str(sol) + + " is a solution to " + str(ode) + ".") + else: + return (False, s) + + +def checksysodesol(eqs, sols, func=None): + r""" + Substitutes corresponding ``sols`` for each functions into each ``eqs`` and + checks that the result of substitutions for each equation is ``0``. The + equations and solutions passed can be any iterable. + + This only works when each ``sols`` have one function only, like `x(t)` or `y(t)`. + For each function, ``sols`` can have a single solution or a list of solutions. + In most cases it will not be necessary to explicitly identify the function, + but if the function cannot be inferred from the original equation it + can be supplied through the ``func`` argument. + + When a sequence of equations is passed, the same sequence is used to return + the result for each equation with each function substituted with corresponding + solutions. + + It tries the following method to find zero equivalence for each equation: + + Substitute the solutions for functions, like `x(t)` and `y(t)` into the + original equations containing those functions. + This function returns a tuple. The first item in the tuple is ``True`` if + the substitution results for each equation is ``0``, and ``False`` otherwise. + The second item in the tuple is what the substitution results in. Each element + of the ``list`` should always be ``0`` corresponding to each equation if the + first item is ``True``. Note that sometimes this function may return ``False``, + but with an expression that is identically equal to ``0``, instead of returning + ``True``. This is because :py:meth:`~sympy.simplify.simplify.simplify` cannot + reduce the expression to ``0``. If an expression returned by each function + vanishes identically, then ``sols`` really is a solution to ``eqs``. + + If this function seems to hang, it is probably because of a difficult simplification. + + Examples + ======== + + >>> from sympy import Eq, diff, symbols, sin, cos, exp, sqrt, S, Function + >>> from sympy.solvers.ode.subscheck import checksysodesol + >>> C1, C2 = symbols('C1:3') + >>> t = symbols('t') + >>> x, y = symbols('x, y', cls=Function) + >>> eq = (Eq(diff(x(t),t), x(t) + y(t) + 17), Eq(diff(y(t),t), -2*x(t) + y(t) + 12)) + >>> sol = [Eq(x(t), (C1*sin(sqrt(2)*t) + C2*cos(sqrt(2)*t))*exp(t) - S(5)/3), + ... Eq(y(t), (sqrt(2)*C1*cos(sqrt(2)*t) - sqrt(2)*C2*sin(sqrt(2)*t))*exp(t) - S(46)/3)] + >>> checksysodesol(eq, sol) + (True, [0, 0]) + >>> eq = (Eq(diff(x(t),t),x(t)*y(t)**4), Eq(diff(y(t),t),y(t)**3)) + >>> sol = [Eq(x(t), C1*exp(-1/(4*(C2 + t)))), Eq(y(t), -sqrt(2)*sqrt(-1/(C2 + t))/2), + ... Eq(x(t), C1*exp(-1/(4*(C2 + t)))), Eq(y(t), sqrt(2)*sqrt(-1/(C2 + t))/2)] + >>> checksysodesol(eq, sol) + (True, [0, 0]) + + """ + def _sympify(eq): + return list(map(sympify, eq if iterable(eq) else [eq])) + eqs = _sympify(eqs) + for i in range(len(eqs)): + if isinstance(eqs[i], Equality): + eqs[i] = eqs[i].lhs - eqs[i].rhs + if func is None: + funcs = [] + for eq in eqs: + derivs = eq.atoms(Derivative) + func = set().union(*[d.atoms(AppliedUndef) for d in derivs]) + funcs.extend(func) + funcs = list(set(funcs)) + if not all(isinstance(func, AppliedUndef) and len(func.args) == 1 for func in funcs)\ + and len({func.args for func in funcs})!=1: + raise ValueError("func must be a function of one variable, not %s" % func) + for sol in sols: + if len(sol.atoms(AppliedUndef)) != 1: + raise ValueError("solutions should have one function only") + if len(funcs) != len({sol.lhs for sol in sols}): + raise ValueError("number of solutions provided does not match the number of equations") + dictsol = {} + for sol in sols: + func = list(sol.atoms(AppliedUndef))[0] + if sol.rhs == func: + sol = sol.reversed + solved = sol.lhs == func and not sol.rhs.has(func) + if not solved: + rhs = solve(sol, func) + if not rhs: + raise NotImplementedError + else: + rhs = sol.rhs + dictsol[func] = rhs + checkeq = [] + for eq in eqs: + for func in funcs: + eq = sub_func_doit(eq, func, dictsol[func]) + ss = simplify(eq) + if ss != 0: + eq = ss.expand(force=True) + if eq != 0: + eq = sqrtdenest(eq).simplify() + else: + eq = 0 + checkeq.append(eq) + if len(set(checkeq)) == 1 and list(set(checkeq))[0] == 0: + return (True, checkeq) + else: + return (False, checkeq) diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/ode/systems.py b/.venv/lib/python3.12/site-packages/sympy/solvers/ode/systems.py new file mode 100644 index 0000000000000000000000000000000000000000..2d2c9b57a969c7fb5c67c06ce952fa398e22a48d --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/ode/systems.py @@ -0,0 +1,2135 @@ +from sympy.core import Add, Mul, S +from sympy.core.containers import Tuple +from sympy.core.exprtools import factor_terms +from sympy.core.numbers import I +from sympy.core.relational import Eq, Equality +from sympy.core.sorting import default_sort_key, ordered +from sympy.core.symbol import Dummy, Symbol +from sympy.core.function import (expand_mul, expand, Derivative, + AppliedUndef, Function, Subs) +from sympy.functions import (exp, im, cos, sin, re, Piecewise, + piecewise_fold, sqrt, log) +from sympy.functions.combinatorial.factorials import factorial +from sympy.matrices import zeros, Matrix, NonSquareMatrixError, MatrixBase, eye +from sympy.polys import Poly, together +from sympy.simplify import collect, radsimp, signsimp # type: ignore +from sympy.simplify.powsimp import powdenest, powsimp +from sympy.simplify.ratsimp import ratsimp +from sympy.simplify.simplify import simplify +from sympy.sets.sets import FiniteSet +from sympy.solvers.deutils import ode_order +from sympy.solvers.solveset import NonlinearError, solveset +from sympy.utilities.iterables import (connected_components, iterable, + strongly_connected_components) +from sympy.utilities.misc import filldedent +from sympy.integrals.integrals import Integral, integrate + + +def _get_func_order(eqs, funcs): + return {func: max(ode_order(eq, func) for eq in eqs) for func in funcs} + + +class ODEOrderError(ValueError): + """Raised by linear_ode_to_matrix if the system has the wrong order""" + pass + + +class ODENonlinearError(NonlinearError): + """Raised by linear_ode_to_matrix if the system is nonlinear""" + pass + + +def _simpsol(soleq): + lhs = soleq.lhs + sol = soleq.rhs + sol = powsimp(sol) + gens = list(sol.atoms(exp)) + p = Poly(sol, *gens, expand=False) + gens = [factor_terms(g) for g in gens] + if not gens: + gens = p.gens + syms = [Symbol('C1'), Symbol('C2')] + terms = [] + for coeff, monom in zip(p.coeffs(), p.monoms()): + coeff = piecewise_fold(coeff) + if isinstance(coeff, Piecewise): + coeff = Piecewise(*((ratsimp(coef).collect(syms), cond) for coef, cond in coeff.args)) + else: + coeff = ratsimp(coeff).collect(syms) + monom = Mul(*(g ** i for g, i in zip(gens, monom))) + terms.append(coeff * monom) + return Eq(lhs, Add(*terms)) + + +def _solsimp(e, t): + no_t, has_t = powsimp(expand_mul(e)).as_independent(t) + + no_t = ratsimp(no_t) + has_t = has_t.replace(exp, lambda a: exp(factor_terms(a))) + + return no_t + has_t + + +def simpsol(sol, wrt1, wrt2, doit=True): + """Simplify solutions from dsolve_system.""" + + # The parameter sol is the solution as returned by dsolve (list of Eq). + # + # The parameters wrt1 and wrt2 are lists of symbols to be collected for + # with those in wrt1 being collected for first. This allows for collecting + # on any factors involving the independent variable before collecting on + # the integration constants or vice versa using e.g.: + # + # sol = simpsol(sol, [t], [C1, C2]) # t first, constants after + # sol = simpsol(sol, [C1, C2], [t]) # constants first, t after + # + # If doit=True (default) then simpsol will begin by evaluating any + # unevaluated integrals. Since many integrals will appear multiple times + # in the solutions this is done intelligently by computing each integral + # only once. + # + # The strategy is to first perform simple cancellation with factor_terms + # and then multiply out all brackets with expand_mul. This gives an Add + # with many terms. + # + # We split each term into two multiplicative factors dep and coeff where + # all factors that involve wrt1 are in dep and any constant factors are in + # coeff e.g. + # sqrt(2)*C1*exp(t) -> ( exp(t), sqrt(2)*C1 ) + # + # The dep factors are simplified using powsimp to combine expanded + # exponential factors e.g. + # exp(a*t)*exp(b*t) -> exp(t*(a+b)) + # + # We then collect coefficients for all terms having the same (simplified) + # dep. The coefficients are then simplified using together and ratsimp and + # lastly by recursively applying the same transformation to the + # coefficients to collect on wrt2. + # + # Finally the result is recombined into an Add and signsimp is used to + # normalise any minus signs. + + def simprhs(rhs, rep, wrt1, wrt2): + """Simplify the rhs of an ODE solution""" + if rep: + rhs = rhs.subs(rep) + rhs = factor_terms(rhs) + rhs = simp_coeff_dep(rhs, wrt1, wrt2) + rhs = signsimp(rhs) + return rhs + + def simp_coeff_dep(expr, wrt1, wrt2=None): + """Split rhs into terms, split terms into dep and coeff and collect on dep""" + add_dep_terms = lambda e: e.is_Add and e.has(*wrt1) + expandable = lambda e: e.is_Mul and any(map(add_dep_terms, e.args)) + expand_func = lambda e: expand_mul(e, deep=False) + expand_mul_mod = lambda e: e.replace(expandable, expand_func) + terms = Add.make_args(expand_mul_mod(expr)) + dc = {} + for term in terms: + coeff, dep = term.as_independent(*wrt1, as_Add=False) + # Collect together the coefficients for terms that have the same + # dependence on wrt1 (after dep is normalised using simpdep). + dep = simpdep(dep, wrt1) + + # See if the dependence on t cancels out... + if dep is not S.One: + dep2 = factor_terms(dep) + if not dep2.has(*wrt1): + coeff *= dep2 + dep = S.One + + if dep not in dc: + dc[dep] = coeff + else: + dc[dep] += coeff + # Apply the method recursively to the coefficients but this time + # collecting on wrt2 rather than wrt2. + termpairs = ((simpcoeff(c, wrt2), d) for d, c in dc.items()) + if wrt2 is not None: + termpairs = ((simp_coeff_dep(c, wrt2), d) for c, d in termpairs) + return Add(*(c * d for c, d in termpairs)) + + def simpdep(term, wrt1): + """Normalise factors involving t with powsimp and recombine exp""" + def canonicalise(a): + # Using factor_terms here isn't quite right because it leads to things + # like exp(t*(1+t)) that we don't want. We do want to cancel factors + # and pull out a common denominator but ideally the numerator would be + # expressed as a standard form polynomial in t so we expand_mul + # and collect afterwards. + a = factor_terms(a) + num, den = a.as_numer_denom() + num = expand_mul(num) + num = collect(num, wrt1) + return num / den + + term = powsimp(term) + rep = {e: exp(canonicalise(e.args[0])) for e in term.atoms(exp)} + term = term.subs(rep) + return term + + def simpcoeff(coeff, wrt2): + """Bring to a common fraction and cancel with ratsimp""" + coeff = together(coeff) + if coeff.is_polynomial(): + # Calling ratsimp can be expensive. The main reason is to simplify + # sums of terms with irrational denominators so we limit ourselves + # to the case where the expression is polynomial in any symbols. + # Maybe there's a better approach... + coeff = ratsimp(radsimp(coeff)) + # collect on secondary variables first and any remaining symbols after + if wrt2 is not None: + syms = list(wrt2) + list(ordered(coeff.free_symbols - set(wrt2))) + else: + syms = list(ordered(coeff.free_symbols)) + coeff = collect(coeff, syms) + coeff = together(coeff) + return coeff + + # There are often repeated integrals. Collect unique integrals and + # evaluate each once and then substitute into the final result to replace + # all occurrences in each of the solution equations. + if doit: + integrals = set().union(*(s.atoms(Integral) for s in sol)) + rep = {i: factor_terms(i).doit() for i in integrals} + else: + rep = {} + + sol = [Eq(s.lhs, simprhs(s.rhs, rep, wrt1, wrt2)) for s in sol] + return sol + + +def linodesolve_type(A, t, b=None): + r""" + Helper function that determines the type of the system of ODEs for solving with :obj:`sympy.solvers.ode.systems.linodesolve()` + + Explanation + =========== + + This function takes in the coefficient matrix and/or the non-homogeneous term + and returns the type of the equation that can be solved by :obj:`sympy.solvers.ode.systems.linodesolve()`. + + If the system is constant coefficient homogeneous, then "type1" is returned + + If the system is constant coefficient non-homogeneous, then "type2" is returned + + If the system is non-constant coefficient homogeneous, then "type3" is returned + + If the system is non-constant coefficient non-homogeneous, then "type4" is returned + + If the system has a non-constant coefficient matrix which can be factorized into constant + coefficient matrix, then "type5" or "type6" is returned for when the system is homogeneous or + non-homogeneous respectively. + + Note that, if the system of ODEs is of "type3" or "type4", then along with the type, + the commutative antiderivative of the coefficient matrix is also returned. + + If the system cannot be solved by :obj:`sympy.solvers.ode.systems.linodesolve()`, then + NotImplementedError is raised. + + Parameters + ========== + + A : Matrix + Coefficient matrix of the system of ODEs + b : Matrix or None + Non-homogeneous term of the system. The default value is None. + If this argument is None, then the system is assumed to be homogeneous. + + Examples + ======== + + >>> from sympy import symbols, Matrix + >>> from sympy.solvers.ode.systems import linodesolve_type + >>> t = symbols("t") + >>> A = Matrix([[1, 1], [2, 3]]) + >>> b = Matrix([t, 1]) + + >>> linodesolve_type(A, t) + {'antiderivative': None, 'type_of_equation': 'type1'} + + >>> linodesolve_type(A, t, b=b) + {'antiderivative': None, 'type_of_equation': 'type2'} + + >>> A_t = Matrix([[1, t], [-t, 1]]) + + >>> linodesolve_type(A_t, t) + {'antiderivative': Matrix([ + [ t, t**2/2], + [-t**2/2, t]]), 'type_of_equation': 'type3'} + + >>> linodesolve_type(A_t, t, b=b) + {'antiderivative': Matrix([ + [ t, t**2/2], + [-t**2/2, t]]), 'type_of_equation': 'type4'} + + >>> A_non_commutative = Matrix([[1, t], [t, -1]]) + >>> linodesolve_type(A_non_commutative, t) + Traceback (most recent call last): + ... + NotImplementedError: + The system does not have a commutative antiderivative, it cannot be + solved by linodesolve. + + Returns + ======= + + Dict + + Raises + ====== + + NotImplementedError + When the coefficient matrix does not have a commutative antiderivative + + See Also + ======== + + linodesolve: Function for which linodesolve_type gets the information + + """ + + match = {} + is_non_constant = not _matrix_is_constant(A, t) + is_non_homogeneous = not (b is None or b.is_zero_matrix) + type = "type{}".format(int("{}{}".format(int(is_non_constant), int(is_non_homogeneous)), 2) + 1) + + B = None + match.update({"type_of_equation": type, "antiderivative": B}) + + if is_non_constant: + B, is_commuting = _is_commutative_anti_derivative(A, t) + if not is_commuting: + raise NotImplementedError(filldedent(''' + The system does not have a commutative antiderivative, it cannot be solved + by linodesolve. + ''')) + + match['antiderivative'] = B + match.update(_first_order_type5_6_subs(A, t, b=b)) + + return match + + +def _first_order_type5_6_subs(A, t, b=None): + match = {} + + factor_terms = _factor_matrix(A, t) + is_homogeneous = b is None or b.is_zero_matrix + + if factor_terms is not None: + t_ = Symbol("{}_".format(t)) + F_t = integrate(factor_terms[0], t) + inverse = solveset(Eq(t_, F_t), t) + + # Note: A simple way to check if a function is invertible + # or not. + if isinstance(inverse, FiniteSet) and not inverse.has(Piecewise)\ + and len(inverse) == 1: + + A = factor_terms[1] + if not is_homogeneous: + b = b / factor_terms[0] + b = b.subs(t, list(inverse)[0]) + type = "type{}".format(5 + (not is_homogeneous)) + match.update({'func_coeff': A, 'tau': F_t, + 't_': t_, 'type_of_equation': type, 'rhs': b}) + + return match + + +def linear_ode_to_matrix(eqs, funcs, t, order): + r""" + Convert a linear system of ODEs to matrix form + + Explanation + =========== + + Express a system of linear ordinary differential equations as a single + matrix differential equation [1]. For example the system $x' = x + y + 1$ + and $y' = x - y$ can be represented as + + .. math:: A_1 X' = A_0 X + b + + where $A_1$ and $A_0$ are $2 \times 2$ matrices and $b$, $X$ and $X'$ are + $2 \times 1$ matrices with $X = [x, y]^T$. + + Higher-order systems are represented with additional matrices e.g. a + second-order system would look like + + .. math:: A_2 X'' = A_1 X' + A_0 X + b + + Examples + ======== + + >>> from sympy import Function, Symbol, Matrix, Eq + >>> from sympy.solvers.ode.systems import linear_ode_to_matrix + >>> t = Symbol('t') + >>> x = Function('x') + >>> y = Function('y') + + We can create a system of linear ODEs like + + >>> eqs = [ + ... Eq(x(t).diff(t), x(t) + y(t) + 1), + ... Eq(y(t).diff(t), x(t) - y(t)), + ... ] + >>> funcs = [x(t), y(t)] + >>> order = 1 # 1st order system + + Now ``linear_ode_to_matrix`` can represent this as a matrix + differential equation. + + >>> (A1, A0), b = linear_ode_to_matrix(eqs, funcs, t, order) + >>> A1 + Matrix([ + [1, 0], + [0, 1]]) + >>> A0 + Matrix([ + [1, 1], + [1, -1]]) + >>> b + Matrix([ + [1], + [0]]) + + The original equations can be recovered from these matrices: + + >>> eqs_mat = Matrix([eq.lhs - eq.rhs for eq in eqs]) + >>> X = Matrix(funcs) + >>> A1 * X.diff(t) - A0 * X - b == eqs_mat + True + + If the system of equations has a maximum order greater than the + order of the system specified, a ODEOrderError exception is raised. + + >>> eqs = [Eq(x(t).diff(t, 2), x(t).diff(t) + x(t)), Eq(y(t).diff(t), y(t) + x(t))] + >>> linear_ode_to_matrix(eqs, funcs, t, 1) + Traceback (most recent call last): + ... + ODEOrderError: Cannot represent system in 1-order form + + If the system of equations is nonlinear, then ODENonlinearError is + raised. + + >>> eqs = [Eq(x(t).diff(t), x(t) + y(t)), Eq(y(t).diff(t), y(t)**2 + x(t))] + >>> linear_ode_to_matrix(eqs, funcs, t, 1) + Traceback (most recent call last): + ... + ODENonlinearError: The system of ODEs is nonlinear. + + Parameters + ========== + + eqs : list of SymPy expressions or equalities + The equations as expressions (assumed equal to zero). + funcs : list of applied functions + The dependent variables of the system of ODEs. + t : symbol + The independent variable. + order : int + The order of the system of ODEs. + + Returns + ======= + + The tuple ``(As, b)`` where ``As`` is a tuple of matrices and ``b`` is the + the matrix representing the rhs of the matrix equation. + + Raises + ====== + + ODEOrderError + When the system of ODEs have an order greater than what was specified + ODENonlinearError + When the system of ODEs is nonlinear + + See Also + ======== + + linear_eq_to_matrix: for systems of linear algebraic equations. + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Matrix_differential_equation + + """ + from sympy.solvers.solveset import linear_eq_to_matrix + + if any(ode_order(eq, func) > order for eq in eqs for func in funcs): + msg = "Cannot represent system in {}-order form" + raise ODEOrderError(msg.format(order)) + + As = [] + + for o in range(order, -1, -1): + # Work from the highest derivative down + syms = [func.diff(t, o) for func in funcs] + + # Ai is the matrix for X(t).diff(t, o) + # eqs is minus the remainder of the equations. + try: + Ai, b = linear_eq_to_matrix(eqs, syms) + except NonlinearError: + raise ODENonlinearError("The system of ODEs is nonlinear.") + + Ai = Ai.applyfunc(expand_mul) + + As.append(Ai if o == order else -Ai) + + if o: + eqs = [-eq for eq in b] + else: + rhs = b + + return As, rhs + + +def matrix_exp(A, t): + r""" + Matrix exponential $\exp(A*t)$ for the matrix ``A`` and scalar ``t``. + + Explanation + =========== + + This functions returns the $\exp(A*t)$ by doing a simple + matrix multiplication: + + .. math:: \exp(A*t) = P * expJ * P^{-1} + + where $expJ$ is $\exp(J*t)$. $J$ is the Jordan normal + form of $A$ and $P$ is matrix such that: + + .. math:: A = P * J * P^{-1} + + The matrix exponential $\exp(A*t)$ appears in the solution of linear + differential equations. For example if $x$ is a vector and $A$ is a matrix + then the initial value problem + + .. math:: \frac{dx(t)}{dt} = A \times x(t), x(0) = x0 + + has the unique solution + + .. math:: x(t) = \exp(A t) x0 + + Examples + ======== + + >>> from sympy import Symbol, Matrix, pprint + >>> from sympy.solvers.ode.systems import matrix_exp + >>> t = Symbol('t') + + We will consider a 2x2 matrix for comupting the exponential + + >>> A = Matrix([[2, -5], [2, -4]]) + >>> pprint(A) + [2 -5] + [ ] + [2 -4] + + Now, exp(A*t) is given as follows: + + >>> pprint(matrix_exp(A, t)) + [ -t -t -t ] + [3*e *sin(t) + e *cos(t) -5*e *sin(t) ] + [ ] + [ -t -t -t ] + [ 2*e *sin(t) - 3*e *sin(t) + e *cos(t)] + + Parameters + ========== + + A : Matrix + The matrix $A$ in the expression $\exp(A*t)$ + t : Symbol + The independent variable + + See Also + ======== + + matrix_exp_jordan_form: For exponential of Jordan normal form + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Jordan_normal_form + .. [2] https://en.wikipedia.org/wiki/Matrix_exponential + + """ + P, expJ = matrix_exp_jordan_form(A, t) + return P * expJ * P.inv() + + +def matrix_exp_jordan_form(A, t): + r""" + Matrix exponential $\exp(A*t)$ for the matrix *A* and scalar *t*. + + Explanation + =========== + + Returns the Jordan form of the $\exp(A*t)$ along with the matrix $P$ such that: + + .. math:: + \exp(A*t) = P * expJ * P^{-1} + + Examples + ======== + + >>> from sympy import Matrix, Symbol + >>> from sympy.solvers.ode.systems import matrix_exp, matrix_exp_jordan_form + >>> t = Symbol('t') + + We will consider a 2x2 defective matrix. This shows that our method + works even for defective matrices. + + >>> A = Matrix([[1, 1], [0, 1]]) + + It can be observed that this function gives us the Jordan normal form + and the required invertible matrix P. + + >>> P, expJ = matrix_exp_jordan_form(A, t) + + Here, it is shown that P and expJ returned by this function is correct + as they satisfy the formula: P * expJ * P_inverse = exp(A*t). + + >>> P * expJ * P.inv() == matrix_exp(A, t) + True + + Parameters + ========== + + A : Matrix + The matrix $A$ in the expression $\exp(A*t)$ + t : Symbol + The independent variable + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Defective_matrix + .. [2] https://en.wikipedia.org/wiki/Jordan_matrix + .. [3] https://en.wikipedia.org/wiki/Jordan_normal_form + + """ + + N, M = A.shape + if N != M: + raise ValueError('Needed square matrix but got shape (%s, %s)' % (N, M)) + elif A.has(t): + raise ValueError('Matrix A should not depend on t') + + def jordan_chains(A): + '''Chains from Jordan normal form analogous to M.eigenvects(). + Returns a dict with eignevalues as keys like: + {e1: [[v111,v112,...], [v121, v122,...]], e2:...} + where vijk is the kth vector in the jth chain for eigenvalue i. + ''' + P, blocks = A.jordan_cells() + basis = [P[:,i] for i in range(P.shape[1])] + n = 0 + chains = {} + for b in blocks: + eigval = b[0, 0] + size = b.shape[0] + if eigval not in chains: + chains[eigval] = [] + chains[eigval].append(basis[n:n+size]) + n += size + return chains + + eigenchains = jordan_chains(A) + + # Needed for consistency across Python versions + eigenchains_iter = sorted(eigenchains.items(), key=default_sort_key) + isreal = not A.has(I) + + blocks = [] + vectors = [] + seen_conjugate = set() + for e, chains in eigenchains_iter: + for chain in chains: + n = len(chain) + if isreal and e != e.conjugate() and e.conjugate() in eigenchains: + if e in seen_conjugate: + continue + seen_conjugate.add(e.conjugate()) + exprt = exp(re(e) * t) + imrt = im(e) * t + imblock = Matrix([[cos(imrt), sin(imrt)], + [-sin(imrt), cos(imrt)]]) + expJblock2 = Matrix(n, n, lambda i,j: + imblock * t**(j-i) / factorial(j-i) if j >= i + else zeros(2, 2)) + expJblock = Matrix(2*n, 2*n, lambda i,j: expJblock2[i//2,j//2][i%2,j%2]) + + blocks.append(exprt * expJblock) + for i in range(n): + vectors.append(re(chain[i])) + vectors.append(im(chain[i])) + else: + vectors.extend(chain) + fun = lambda i,j: t**(j-i)/factorial(j-i) if j >= i else 0 + expJblock = Matrix(n, n, fun) + blocks.append(exp(e * t) * expJblock) + + expJ = Matrix.diag(*blocks) + P = Matrix(N, N, lambda i,j: vectors[j][i]) + + return P, expJ + + +# Note: To add a docstring example with tau +def linodesolve(A, t, b=None, B=None, type="auto", doit=False, + tau=None): + r""" + System of n equations linear first-order differential equations + + Explanation + =========== + + This solver solves the system of ODEs of the following form: + + .. math:: + X'(t) = A(t) X(t) + b(t) + + Here, $A(t)$ is the coefficient matrix, $X(t)$ is the vector of n independent variables, + $b(t)$ is the non-homogeneous term and $X'(t)$ is the derivative of $X(t)$ + + Depending on the properties of $A(t)$ and $b(t)$, this solver evaluates the solution + differently. + + When $A(t)$ is constant coefficient matrix and $b(t)$ is zero vector i.e. system is homogeneous, + the system is "type1". The solution is: + + .. math:: + X(t) = \exp(A t) C + + Here, $C$ is a vector of constants and $A$ is the constant coefficient matrix. + + When $A(t)$ is constant coefficient matrix and $b(t)$ is non-zero i.e. system is non-homogeneous, + the system is "type2". The solution is: + + .. math:: + X(t) = e^{A t} ( \int e^{- A t} b \,dt + C) + + When $A(t)$ is coefficient matrix such that its commutative with its antiderivative $B(t)$ and + $b(t)$ is a zero vector i.e. system is homogeneous, the system is "type3". The solution is: + + .. math:: + X(t) = \exp(B(t)) C + + When $A(t)$ is commutative with its antiderivative $B(t)$ and $b(t)$ is non-zero i.e. system is + non-homogeneous, the system is "type4". The solution is: + + .. math:: + X(t) = e^{B(t)} ( \int e^{-B(t)} b(t) \,dt + C) + + When $A(t)$ is a coefficient matrix such that it can be factorized into a scalar and a constant + coefficient matrix: + + .. math:: + A(t) = f(t) * A + + Where $f(t)$ is a scalar expression in the independent variable $t$ and $A$ is a constant matrix, + then we can do the following substitutions: + + .. math:: + tau = \int f(t) dt, X(t) = Y(tau), b(t) = b(f^{-1}(tau)) + + Here, the substitution for the non-homogeneous term is done only when its non-zero. + Using these substitutions, our original system becomes: + + .. math:: + Y'(tau) = A * Y(tau) + b(tau)/f(tau) + + The above system can be easily solved using the solution for "type1" or "type2" depending + on the homogeneity of the system. After we get the solution for $Y(tau)$, we substitute the + solution for $tau$ as $t$ to get back $X(t)$ + + .. math:: + X(t) = Y(tau) + + Systems of "type5" and "type6" have a commutative antiderivative but we use this solution + because its faster to compute. + + The final solution is the general solution for all the four equations since a constant coefficient + matrix is always commutative with its antidervative. + + An additional feature of this function is, if someone wants to substitute for value of the independent + variable, they can pass the substitution `tau` and the solution will have the independent variable + substituted with the passed expression(`tau`). + + Parameters + ========== + + A : Matrix + Coefficient matrix of the system of linear first order ODEs. + t : Symbol + Independent variable in the system of ODEs. + b : Matrix or None + Non-homogeneous term in the system of ODEs. If None is passed, + a homogeneous system of ODEs is assumed. + B : Matrix or None + Antiderivative of the coefficient matrix. If the antiderivative + is not passed and the solution requires the term, then the solver + would compute it internally. + type : String + Type of the system of ODEs passed. Depending on the type, the + solution is evaluated. The type values allowed and the corresponding + system it solves are: "type1" for constant coefficient homogeneous + "type2" for constant coefficient non-homogeneous, "type3" for non-constant + coefficient homogeneous, "type4" for non-constant coefficient non-homogeneous, + "type5" and "type6" for non-constant coefficient homogeneous and non-homogeneous + systems respectively where the coefficient matrix can be factorized to a constant + coefficient matrix. + The default value is "auto" which will let the solver decide the correct type of + the system passed. + doit : Boolean + Evaluate the solution if True, default value is False + tau: Expression + Used to substitute for the value of `t` after we get the solution of the system. + + Examples + ======== + + To solve the system of ODEs using this function directly, several things must be + done in the right order. Wrong inputs to the function will lead to incorrect results. + + >>> from sympy import symbols, Function, Eq + >>> from sympy.solvers.ode.systems import canonical_odes, linear_ode_to_matrix, linodesolve, linodesolve_type + >>> from sympy.solvers.ode.subscheck import checkodesol + >>> f, g = symbols("f, g", cls=Function) + >>> x, a = symbols("x, a") + >>> funcs = [f(x), g(x)] + >>> eqs = [Eq(f(x).diff(x) - f(x), a*g(x) + 1), Eq(g(x).diff(x) + g(x), a*f(x))] + + Here, it is important to note that before we derive the coefficient matrix, it is + important to get the system of ODEs into the desired form. For that we will use + :obj:`sympy.solvers.ode.systems.canonical_odes()`. + + >>> eqs = canonical_odes(eqs, funcs, x) + >>> eqs + [[Eq(Derivative(f(x), x), a*g(x) + f(x) + 1), Eq(Derivative(g(x), x), a*f(x) - g(x))]] + + Now, we will use :obj:`sympy.solvers.ode.systems.linear_ode_to_matrix()` to get the coefficient matrix and the + non-homogeneous term if it is there. + + >>> eqs = eqs[0] + >>> (A1, A0), b = linear_ode_to_matrix(eqs, funcs, x, 1) + >>> A = A0 + + We have the coefficient matrices and the non-homogeneous term ready. Now, we can use + :obj:`sympy.solvers.ode.systems.linodesolve_type()` to get the information for the system of ODEs + to finally pass it to the solver. + + >>> system_info = linodesolve_type(A, x, b=b) + >>> sol_vector = linodesolve(A, x, b=b, B=system_info['antiderivative'], type=system_info['type_of_equation']) + + Now, we can prove if the solution is correct or not by using :obj:`sympy.solvers.ode.checkodesol()` + + >>> sol = [Eq(f, s) for f, s in zip(funcs, sol_vector)] + >>> checkodesol(eqs, sol) + (True, [0, 0]) + + We can also use the doit method to evaluate the solutions passed by the function. + + >>> sol_vector_evaluated = linodesolve(A, x, b=b, type="type2", doit=True) + + Now, we will look at a system of ODEs which is non-constant. + + >>> eqs = [Eq(f(x).diff(x), f(x) + x*g(x)), Eq(g(x).diff(x), -x*f(x) + g(x))] + + The system defined above is already in the desired form, so we do not have to convert it. + + >>> (A1, A0), b = linear_ode_to_matrix(eqs, funcs, x, 1) + >>> A = A0 + + A user can also pass the commutative antiderivative required for type3 and type4 system of ODEs. + Passing an incorrect one will lead to incorrect results. If the coefficient matrix is not commutative + with its antiderivative, then :obj:`sympy.solvers.ode.systems.linodesolve_type()` raises a NotImplementedError. + If it does have a commutative antiderivative, then the function just returns the information about the system. + + >>> system_info = linodesolve_type(A, x, b=b) + + Now, we can pass the antiderivative as an argument to get the solution. If the system information is not + passed, then the solver will compute the required arguments internally. + + >>> sol_vector = linodesolve(A, x, b=b) + + Once again, we can verify the solution obtained. + + >>> sol = [Eq(f, s) for f, s in zip(funcs, sol_vector)] + >>> checkodesol(eqs, sol) + (True, [0, 0]) + + Returns + ======= + + List + + Raises + ====== + + ValueError + This error is raised when the coefficient matrix, non-homogeneous term + or the antiderivative, if passed, are not a matrix or + do not have correct dimensions + NonSquareMatrixError + When the coefficient matrix or its antiderivative, if passed is not a + square matrix + NotImplementedError + If the coefficient matrix does not have a commutative antiderivative + + See Also + ======== + + linear_ode_to_matrix: Coefficient matrix computation function + canonical_odes: System of ODEs representation change + linodesolve_type: Getting information about systems of ODEs to pass in this solver + + """ + + if not isinstance(A, MatrixBase): + raise ValueError(filldedent('''\ + The coefficients of the system of ODEs should be of type Matrix + ''')) + + if not A.is_square: + raise NonSquareMatrixError(filldedent('''\ + The coefficient matrix must be a square + ''')) + + if b is not None: + if not isinstance(b, MatrixBase): + raise ValueError(filldedent('''\ + The non-homogeneous terms of the system of ODEs should be of type Matrix + ''')) + + if A.rows != b.rows: + raise ValueError(filldedent('''\ + The system of ODEs should have the same number of non-homogeneous terms and the number of + equations + ''')) + + if B is not None: + if not isinstance(B, MatrixBase): + raise ValueError(filldedent('''\ + The antiderivative of coefficients of the system of ODEs should be of type Matrix + ''')) + + if not B.is_square: + raise NonSquareMatrixError(filldedent('''\ + The antiderivative of the coefficient matrix must be a square + ''')) + + if A.rows != B.rows: + raise ValueError(filldedent('''\ + The coefficient matrix and its antiderivative should have same dimensions + ''')) + + if not any(type == "type{}".format(i) for i in range(1, 7)) and not type == "auto": + raise ValueError(filldedent('''\ + The input type should be a valid one + ''')) + + n = A.rows + + # constants = numbered_symbols(prefix='C', cls=Dummy, start=const_idx+1) + Cvect = Matrix([Dummy() for _ in range(n)]) + + if b is None and any(type == typ for typ in ["type2", "type4", "type6"]): + b = zeros(n, 1) + + is_transformed = tau is not None + passed_type = type + + if type == "auto": + system_info = linodesolve_type(A, t, b=b) + type = system_info["type_of_equation"] + B = system_info["antiderivative"] + + if type in ("type5", "type6"): + is_transformed = True + if passed_type != "auto": + if tau is None: + system_info = _first_order_type5_6_subs(A, t, b=b) + if not system_info: + raise ValueError(filldedent(''' + The system passed isn't {}. + '''.format(type))) + + tau = system_info['tau'] + t = system_info['t_'] + A = system_info['A'] + b = system_info['b'] + + intx_wrtt = lambda x: Integral(x, t) if x else 0 + if type in ("type1", "type2", "type5", "type6"): + P, J = matrix_exp_jordan_form(A, t) + P = simplify(P) + + if type in ("type1", "type5"): + sol_vector = P * (J * Cvect) + else: + Jinv = J.subs(t, -t) + sol_vector = P * J * ((Jinv * P.inv() * b).applyfunc(intx_wrtt) + Cvect) + else: + if B is None: + B, _ = _is_commutative_anti_derivative(A, t) + + if type == "type3": + sol_vector = B.exp() * Cvect + else: + sol_vector = B.exp() * (((-B).exp() * b).applyfunc(intx_wrtt) + Cvect) + + if is_transformed: + sol_vector = sol_vector.subs(t, tau) + + gens = sol_vector.atoms(exp) + + if type != "type1": + sol_vector = [expand_mul(s) for s in sol_vector] + + sol_vector = [collect(s, ordered(gens), exact=True) for s in sol_vector] + + if doit: + sol_vector = [s.doit() for s in sol_vector] + + return sol_vector + + +def _matrix_is_constant(M, t): + """Checks if the matrix M is independent of t or not.""" + return all(coef.as_independent(t, as_Add=True)[1] == 0 for coef in M) + + +def canonical_odes(eqs, funcs, t): + r""" + Function that solves for highest order derivatives in a system + + Explanation + =========== + + This function inputs a system of ODEs and based on the system, + the dependent variables and their highest order, returns the system + in the following form: + + .. math:: + X'(t) = A(t) X(t) + b(t) + + Here, $X(t)$ is the vector of dependent variables of lower order, $A(t)$ is + the coefficient matrix, $b(t)$ is the non-homogeneous term and $X'(t)$ is the + vector of dependent variables in their respective highest order. We use the term + canonical form to imply the system of ODEs which is of the above form. + + If the system passed has a non-linear term with multiple solutions, then a list of + systems is returned in its canonical form. + + Parameters + ========== + + eqs : List + List of the ODEs + funcs : List + List of dependent variables + t : Symbol + Independent variable + + Examples + ======== + + >>> from sympy import symbols, Function, Eq, Derivative + >>> from sympy.solvers.ode.systems import canonical_odes + >>> f, g = symbols("f g", cls=Function) + >>> x, y = symbols("x y") + >>> funcs = [f(x), g(x)] + >>> eqs = [Eq(f(x).diff(x) - 7*f(x), 12*g(x)), Eq(g(x).diff(x) + g(x), 20*f(x))] + + >>> canonical_eqs = canonical_odes(eqs, funcs, x) + >>> canonical_eqs + [[Eq(Derivative(f(x), x), 7*f(x) + 12*g(x)), Eq(Derivative(g(x), x), 20*f(x) - g(x))]] + + >>> system = [Eq(Derivative(f(x), x)**2 - 2*Derivative(f(x), x) + 1, 4), Eq(-y*f(x) + Derivative(g(x), x), 0)] + + >>> canonical_system = canonical_odes(system, funcs, x) + >>> canonical_system + [[Eq(Derivative(f(x), x), -1), Eq(Derivative(g(x), x), y*f(x))], [Eq(Derivative(f(x), x), 3), Eq(Derivative(g(x), x), y*f(x))]] + + Returns + ======= + + List + + """ + from sympy.solvers.solvers import solve + + order = _get_func_order(eqs, funcs) + + canon_eqs = solve(eqs, *[func.diff(t, order[func]) for func in funcs], dict=True) + + systems = [] + for eq in canon_eqs: + system = [Eq(func.diff(t, order[func]), eq[func.diff(t, order[func])]) for func in funcs] + systems.append(system) + + return systems + + +def _is_commutative_anti_derivative(A, t): + r""" + Helper function for determining if the Matrix passed is commutative with its antiderivative + + Explanation + =========== + + This function checks if the Matrix $A$ passed is commutative with its antiderivative with respect + to the independent variable $t$. + + .. math:: + B(t) = \int A(t) dt + + The function outputs two values, first one being the antiderivative $B(t)$, second one being a + boolean value, if True, then the matrix $A(t)$ passed is commutative with $B(t)$, else the matrix + passed isn't commutative with $B(t)$. + + Parameters + ========== + + A : Matrix + The matrix which has to be checked + t : Symbol + Independent variable + + Examples + ======== + + >>> from sympy import symbols, Matrix + >>> from sympy.solvers.ode.systems import _is_commutative_anti_derivative + >>> t = symbols("t") + >>> A = Matrix([[1, t], [-t, 1]]) + + >>> B, is_commuting = _is_commutative_anti_derivative(A, t) + >>> is_commuting + True + + Returns + ======= + + Matrix, Boolean + + """ + B = integrate(A, t) + is_commuting = (B*A - A*B).applyfunc(expand).applyfunc(factor_terms).is_zero_matrix + + is_commuting = False if is_commuting is None else is_commuting + + return B, is_commuting + + +def _factor_matrix(A, t): + term = None + for element in A: + temp_term = element.as_independent(t)[1] + if temp_term.has(t): + term = temp_term + break + + if term is not None: + A_factored = (A/term).applyfunc(ratsimp) + can_factor = _matrix_is_constant(A_factored, t) + term = (term, A_factored) if can_factor else None + + return term + + +def _is_second_order_type2(A, t): + term = _factor_matrix(A, t) + is_type2 = False + + if term is not None: + term = 1/term[0] + is_type2 = term.is_polynomial() + + if is_type2: + poly = Poly(term.expand(), t) + monoms = poly.monoms() + + if monoms[0][0] in (2, 4): + cs = _get_poly_coeffs(poly, 4) + a, b, c, d, e = cs + + a1 = powdenest(sqrt(a), force=True) + c1 = powdenest(sqrt(e), force=True) + b1 = powdenest(sqrt(c - 2*a1*c1), force=True) + + is_type2 = (b == 2*a1*b1) and (d == 2*b1*c1) + term = a1*t**2 + b1*t + c1 + + else: + is_type2 = False + + return is_type2, term + + +def _get_poly_coeffs(poly, order): + cs = [0 for _ in range(order+1)] + for c, m in zip(poly.coeffs(), poly.monoms()): + cs[-1-m[0]] = c + return cs + + +def _match_second_order_type(A1, A0, t, b=None): + r""" + Works only for second order system in its canonical form. + + Type 0: Constant coefficient matrix, can be simply solved by + introducing dummy variables. + Type 1: When the substitution: $U = t*X' - X$ works for reducing + the second order system to first order system. + Type 2: When the system is of the form: $poly * X'' = A*X$ where + $poly$ is square of a quadratic polynomial with respect to + *t* and $A$ is a constant coefficient matrix. + + """ + match = {"type_of_equation": "type0"} + n = A1.shape[0] + + if _matrix_is_constant(A1, t) and _matrix_is_constant(A0, t): + return match + + if (A1 + A0*t).applyfunc(expand_mul).is_zero_matrix: + match.update({"type_of_equation": "type1", "A1": A1}) + + elif A1.is_zero_matrix and (b is None or b.is_zero_matrix): + is_type2, term = _is_second_order_type2(A0, t) + if is_type2: + a, b, c = _get_poly_coeffs(Poly(term, t), 2) + A = (A0*(term**2).expand()).applyfunc(ratsimp) + (b**2/4 - a*c)*eye(n, n) + tau = integrate(1/term, t) + t_ = Symbol("{}_".format(t)) + match.update({"type_of_equation": "type2", "A0": A, + "g(t)": sqrt(term), "tau": tau, "is_transformed": True, + "t_": t_}) + + return match + + +def _second_order_subs_type1(A, b, funcs, t): + r""" + For a linear, second order system of ODEs, a particular substitution. + + A system of the below form can be reduced to a linear first order system of + ODEs: + .. math:: + X'' = A(t) * (t*X' - X) + b(t) + + By substituting: + .. math:: U = t*X' - X + + To get the system: + .. math:: U' = t*(A(t)*U + b(t)) + + Where $U$ is the vector of dependent variables, $X$ is the vector of dependent + variables in `funcs` and $X'$ is the first order derivative of $X$ with respect to + $t$. It may or may not reduce the system into linear first order system of ODEs. + + Then a check is made to determine if the system passed can be reduced or not, if + this substitution works, then the system is reduced and its solved for the new + substitution. After we get the solution for $U$: + + .. math:: U = a(t) + + We substitute and return the reduced system: + + .. math:: + a(t) = t*X' - X + + Parameters + ========== + + A: Matrix + Coefficient matrix($A(t)*t$) of the second order system of this form. + b: Matrix + Non-homogeneous term($b(t)$) of the system of ODEs. + funcs: List + List of dependent variables + t: Symbol + Independent variable of the system of ODEs. + + Returns + ======= + + List + + """ + + U = Matrix([t*func.diff(t) - func for func in funcs]) + + sol = linodesolve(A, t, t*b) + reduced_eqs = [Eq(u, s) for s, u in zip(sol, U)] + reduced_eqs = canonical_odes(reduced_eqs, funcs, t)[0] + + return reduced_eqs + + +def _second_order_subs_type2(A, funcs, t_): + r""" + Returns a second order system based on the coefficient matrix passed. + + Explanation + =========== + + This function returns a system of second order ODE of the following form: + + .. math:: + X'' = A * X + + Here, $X$ is the vector of dependent variables, but a bit modified, $A$ is the + coefficient matrix passed. + + Along with returning the second order system, this function also returns the new + dependent variables with the new independent variable `t_` passed. + + Parameters + ========== + + A: Matrix + Coefficient matrix of the system + funcs: List + List of old dependent variables + t_: Symbol + New independent variable + + Returns + ======= + + List, List + + """ + func_names = [func.func.__name__ for func in funcs] + new_funcs = [Function(Dummy("{}_".format(name)))(t_) for name in func_names] + rhss = A * Matrix(new_funcs) + new_eqs = [Eq(func.diff(t_, 2), rhs) for func, rhs in zip(new_funcs, rhss)] + + return new_eqs, new_funcs + + +def _is_euler_system(As, t): + return all(_matrix_is_constant((A*t**i).applyfunc(ratsimp), t) for i, A in enumerate(As)) + + +def _classify_linear_system(eqs, funcs, t, is_canon=False): + r""" + Returns a dictionary with details of the eqs if the system passed is linear + and can be classified by this function else returns None + + Explanation + =========== + + This function takes the eqs, converts it into a form Ax = b where x is a vector of terms + containing dependent variables and their derivatives till their maximum order. If it is + possible to convert eqs into Ax = b, then all the equations in eqs are linear otherwise + they are non-linear. + + To check if the equations are constant coefficient, we need to check if all the terms in + A obtained above are constant or not. + + To check if the equations are homogeneous or not, we need to check if b is a zero matrix + or not. + + Parameters + ========== + + eqs: List + List of ODEs + funcs: List + List of dependent variables + t: Symbol + Independent variable of the equations in eqs + is_canon: Boolean + If True, then this function will not try to get the + system in canonical form. Default value is False + + Returns + ======= + + match = { + 'no_of_equation': len(eqs), + 'eq': eqs, + 'func': funcs, + 'order': order, + 'is_linear': is_linear, + 'is_constant': is_constant, + 'is_homogeneous': is_homogeneous, + } + + Dict or list of Dicts or None + Dict with values for keys: + 1. no_of_equation: Number of equations + 2. eq: The set of equations + 3. func: List of dependent variables + 4. order: A dictionary that gives the order of the + dependent variable in eqs + 5. is_linear: Boolean value indicating if the set of + equations are linear or not. + 6. is_constant: Boolean value indicating if the set of + equations have constant coefficients or not. + 7. is_homogeneous: Boolean value indicating if the set of + equations are homogeneous or not. + 8. commutative_antiderivative: Antiderivative of the coefficient + matrix if the coefficient matrix is non-constant + and commutative with its antiderivative. This key + may or may not exist. + 9. is_general: Boolean value indicating if the system of ODEs is + solvable using one of the general case solvers or not. + 10. rhs: rhs of the non-homogeneous system of ODEs in Matrix form. This + key may or may not exist. + 11. is_higher_order: True if the system passed has an order greater than 1. + This key may or may not exist. + 12. is_second_order: True if the system passed is a second order ODE. This + key may or may not exist. + This Dict is the answer returned if the eqs are linear and constant + coefficient. Otherwise, None is returned. + + """ + + # Error for i == 0 can be added but isn't for now + + # Check for len(funcs) == len(eqs) + if len(funcs) != len(eqs): + raise ValueError("Number of functions given is not equal to the number of equations %s" % funcs) + + # ValueError when functions have more than one arguments + for func in funcs: + if len(func.args) != 1: + raise ValueError("dsolve() and classify_sysode() work with " + "functions of one variable only, not %s" % func) + + # Getting the func_dict and order using the helper + # function + order = _get_func_order(eqs, funcs) + system_order = max(order[func] for func in funcs) + is_higher_order = system_order > 1 + is_second_order = system_order == 2 and all(order[func] == 2 for func in funcs) + + # Not adding the check if the len(func.args) for + # every func in funcs is 1 + + # Linearity check + try: + + canon_eqs = canonical_odes(eqs, funcs, t) if not is_canon else [eqs] + if len(canon_eqs) == 1: + As, b = linear_ode_to_matrix(canon_eqs[0], funcs, t, system_order) + else: + + match = { + 'is_implicit': True, + 'canon_eqs': canon_eqs + } + + return match + + # When the system of ODEs is non-linear, an ODENonlinearError is raised. + # This function catches the error and None is returned. + except ODENonlinearError: + return None + + is_linear = True + + # Homogeneous check + is_homogeneous = True if b.is_zero_matrix else False + + # Is general key is used to identify if the system of ODEs can be solved by + # one of the general case solvers or not. + match = { + 'no_of_equation': len(eqs), + 'eq': eqs, + 'func': funcs, + 'order': order, + 'is_linear': is_linear, + 'is_homogeneous': is_homogeneous, + 'is_general': True + } + + if not is_homogeneous: + match['rhs'] = b + + is_constant = all(_matrix_is_constant(A_, t) for A_ in As) + + # The match['is_linear'] check will be added in the future when this + # function becomes ready to deal with non-linear systems of ODEs + + if not is_higher_order: + A = As[1] + match['func_coeff'] = A + + # Constant coefficient check + is_constant = _matrix_is_constant(A, t) + match['is_constant'] = is_constant + + try: + system_info = linodesolve_type(A, t, b=b) + except NotImplementedError: + return None + + match.update(system_info) + antiderivative = match.pop("antiderivative") + + if not is_constant: + match['commutative_antiderivative'] = antiderivative + + return match + else: + match['type_of_equation'] = "type0" + + if is_second_order: + A1, A0 = As[1:] + + match_second_order = _match_second_order_type(A1, A0, t) + match.update(match_second_order) + + match['is_second_order'] = True + + # If system is constant, then no need to check if its in euler + # form or not. It will be easier and faster to directly proceed + # to solve it. + if match['type_of_equation'] == "type0" and not is_constant: + is_euler = _is_euler_system(As, t) + if is_euler: + t_ = Symbol('{}_'.format(t)) + match.update({'is_transformed': True, 'type_of_equation': 'type1', + 't_': t_}) + else: + is_jordan = lambda M: M == Matrix.jordan_block(M.shape[0], M[0, 0]) + terms = _factor_matrix(As[-1], t) + if all(A.is_zero_matrix for A in As[1:-1]) and terms is not None and not is_jordan(terms[1]): + P, J = terms[1].jordan_form() + match.update({'type_of_equation': 'type2', 'J': J, + 'f(t)': terms[0], 'P': P, 'is_transformed': True}) + + if match['type_of_equation'] != 'type0' and is_second_order: + match.pop('is_second_order', None) + + match['is_higher_order'] = is_higher_order + + return match + +def _preprocess_eqs(eqs): + processed_eqs = [] + for eq in eqs: + processed_eqs.append(eq if isinstance(eq, Equality) else Eq(eq, 0)) + + return processed_eqs + + +def _eqs2dict(eqs, funcs): + eqsorig = {} + eqsmap = {} + funcset = set(funcs) + for eq in eqs: + f1, = eq.lhs.atoms(AppliedUndef) + f2s = (eq.rhs.atoms(AppliedUndef) - {f1}) & funcset + eqsmap[f1] = f2s + eqsorig[f1] = eq + return eqsmap, eqsorig + + +def _dict2graph(d): + nodes = list(d) + edges = [(f1, f2) for f1, f2s in d.items() for f2 in f2s] + G = (nodes, edges) + return G + + +def _is_type1(scc, t): + eqs, funcs = scc + + try: + (A1, A0), b = linear_ode_to_matrix(eqs, funcs, t, 1) + except (ODENonlinearError, ODEOrderError): + return False + + if _matrix_is_constant(A0, t) and b.is_zero_matrix: + return True + + return False + + +def _combine_type1_subsystems(subsystem, funcs, t): + indices = [i for i, sys in enumerate(zip(subsystem, funcs)) if _is_type1(sys, t)] + remove = set() + for ip, i in enumerate(indices): + for j in indices[ip+1:]: + if any(eq2.has(funcs[i]) for eq2 in subsystem[j]): + subsystem[j] = subsystem[i] + subsystem[j] + remove.add(i) + subsystem = [sys for i, sys in enumerate(subsystem) if i not in remove] + return subsystem + + +def _component_division(eqs, funcs, t): + + # Assuming that each eq in eqs is in canonical form, + # that is, [f(x).diff(x) = .., g(x).diff(x) = .., etc] + # and that the system passed is in its first order + eqsmap, eqsorig = _eqs2dict(eqs, funcs) + + subsystems = [] + for cc in connected_components(_dict2graph(eqsmap)): + eqsmap_c = {f: eqsmap[f] for f in cc} + sccs = strongly_connected_components(_dict2graph(eqsmap_c)) + subsystem = [[eqsorig[f] for f in scc] for scc in sccs] + subsystem = _combine_type1_subsystems(subsystem, sccs, t) + subsystems.append(subsystem) + + return subsystems + + +# Returns: List of equations +def _linear_ode_solver(match): + t = match['t'] + funcs = match['func'] + + rhs = match.get('rhs', None) + tau = match.get('tau', None) + t = match['t_'] if 't_' in match else t + A = match['func_coeff'] + + # Note: To make B None when the matrix has constant + # coefficient + B = match.get('commutative_antiderivative', None) + type = match['type_of_equation'] + + sol_vector = linodesolve(A, t, b=rhs, B=B, + type=type, tau=tau) + + sol = [Eq(f, s) for f, s in zip(funcs, sol_vector)] + + return sol + + +def _select_equations(eqs, funcs, key=lambda x: x): + eq_dict = {e.lhs: e.rhs for e in eqs} + return [Eq(f, eq_dict[key(f)]) for f in funcs] + + +def _higher_order_ode_solver(match): + eqs = match["eq"] + funcs = match["func"] + t = match["t"] + sysorder = match['order'] + type = match.get('type_of_equation', "type0") + + is_second_order = match.get('is_second_order', False) + is_transformed = match.get('is_transformed', False) + is_euler = is_transformed and type == "type1" + is_higher_order_type2 = is_transformed and type == "type2" and 'P' in match + + if is_second_order: + new_eqs, new_funcs = _second_order_to_first_order(eqs, funcs, t, + A1=match.get("A1", None), A0=match.get("A0", None), + b=match.get("rhs", None), type=type, + t_=match.get("t_", None)) + else: + new_eqs, new_funcs = _higher_order_to_first_order(eqs, sysorder, t, funcs=funcs, + type=type, J=match.get('J', None), + f_t=match.get('f(t)', None), + P=match.get('P', None), b=match.get('rhs', None)) + + if is_transformed: + t = match.get('t_', t) + + if not is_higher_order_type2: + new_eqs = _select_equations(new_eqs, [f.diff(t) for f in new_funcs]) + + sol = None + + # NotImplementedError may be raised when the system may be actually + # solvable if it can be just divided into sub-systems + try: + if not is_higher_order_type2: + sol = _strong_component_solver(new_eqs, new_funcs, t) + except NotImplementedError: + sol = None + + # Dividing the system only when it becomes essential + if sol is None: + try: + sol = _component_solver(new_eqs, new_funcs, t) + except NotImplementedError: + sol = None + + if sol is None: + return sol + + is_second_order_type2 = is_second_order and type == "type2" + + underscores = '__' if is_transformed else '_' + + sol = _select_equations(sol, funcs, + key=lambda x: Function(Dummy('{}{}0'.format(x.func.__name__, underscores)))(t)) + + if match.get("is_transformed", False): + if is_second_order_type2: + g_t = match["g(t)"] + tau = match["tau"] + sol = [Eq(s.lhs, s.rhs.subs(t, tau) * g_t) for s in sol] + elif is_euler: + t = match['t'] + tau = match['t_'] + sol = [s.subs(tau, log(t)) for s in sol] + elif is_higher_order_type2: + P = match['P'] + sol_vector = P * Matrix([s.rhs for s in sol]) + sol = [Eq(f, s) for f, s in zip(funcs, sol_vector)] + + return sol + + +# Returns: List of equations or None +# If None is returned by this solver, then the system +# of ODEs cannot be solved directly by dsolve_system. +def _strong_component_solver(eqs, funcs, t): + from sympy.solvers.ode.ode import dsolve, constant_renumber + + match = _classify_linear_system(eqs, funcs, t, is_canon=True) + sol = None + + # Assuming that we can't get an implicit system + # since we are already canonical equations from + # dsolve_system + if match: + match['t'] = t + + if match.get('is_higher_order', False): + sol = _higher_order_ode_solver(match) + + elif match.get('is_linear', False): + sol = _linear_ode_solver(match) + + # Note: For now, only linear systems are handled by this function + # hence, the match condition is added. This can be removed later. + if sol is None and len(eqs) == 1: + sol = dsolve(eqs[0], func=funcs[0]) + variables = Tuple(eqs[0]).free_symbols + new_constants = [Dummy() for _ in range(ode_order(eqs[0], funcs[0]))] + sol = constant_renumber(sol, variables=variables, newconstants=new_constants) + sol = [sol] + + # To add non-linear case here in future + + return sol + + +def _get_funcs_from_canon(eqs): + return [eq.lhs.args[0] for eq in eqs] + + +# Returns: List of Equations(a solution) +def _weak_component_solver(wcc, t): + + # We will divide the systems into sccs + # only when the wcc cannot be solved as + # a whole + eqs = [] + for scc in wcc: + eqs += scc + funcs = _get_funcs_from_canon(eqs) + + sol = _strong_component_solver(eqs, funcs, t) + if sol: + return sol + + sol = [] + + for scc in wcc: + eqs = scc + funcs = _get_funcs_from_canon(eqs) + + # Substituting solutions for the dependent + # variables solved in previous SCC, if any solved. + comp_eqs = [eq.subs({s.lhs: s.rhs for s in sol}) for eq in eqs] + scc_sol = _strong_component_solver(comp_eqs, funcs, t) + + if scc_sol is None: + raise NotImplementedError(filldedent(''' + The system of ODEs passed cannot be solved by dsolve_system. + ''')) + + # scc_sol: List of equations + # scc_sol is a solution + sol += scc_sol + + return sol + + +# Returns: List of Equations(a solution) +def _component_solver(eqs, funcs, t): + components = _component_division(eqs, funcs, t) + sol = [] + + for wcc in components: + + # wcc_sol: List of Equations + sol += _weak_component_solver(wcc, t) + + # sol: List of Equations + return sol + + +def _second_order_to_first_order(eqs, funcs, t, type="auto", A1=None, + A0=None, b=None, t_=None): + r""" + Expects the system to be in second order and in canonical form + + Explanation + =========== + + Reduces a second order system into a first order one depending on the type of second + order system. + 1. "type0": If this is passed, then the system will be reduced to first order by + introducing dummy variables. + 2. "type1": If this is passed, then a particular substitution will be used to reduce the + the system into first order. + 3. "type2": If this is passed, then the system will be transformed with new dependent + variables and independent variables. This transformation is a part of solving + the corresponding system of ODEs. + + `A1` and `A0` are the coefficient matrices from the system and it is assumed that the + second order system has the form given below: + + .. math:: + A2 * X'' = A1 * X' + A0 * X + b + + Here, $A2$ is the coefficient matrix for the vector $X''$ and $b$ is the non-homogeneous + term. + + Default value for `b` is None but if `A1` and `A0` are passed and `b` is not passed, then the + system will be assumed homogeneous. + + """ + is_a1 = A1 is None + is_a0 = A0 is None + + if (type == "type1" and is_a1) or (type == "type2" and is_a0)\ + or (type == "auto" and (is_a1 or is_a0)): + (A2, A1, A0), b = linear_ode_to_matrix(eqs, funcs, t, 2) + + if not A2.is_Identity: + raise ValueError(filldedent(''' + The system must be in its canonical form. + ''')) + + if type == "auto": + match = _match_second_order_type(A1, A0, t) + type = match["type_of_equation"] + A1 = match.get("A1", None) + A0 = match.get("A0", None) + + sys_order = dict.fromkeys(funcs, 2) + + if type == "type1": + if b is None: + b = zeros(len(eqs)) + eqs = _second_order_subs_type1(A1, b, funcs, t) + sys_order = dict.fromkeys(funcs, 1) + + if type == "type2": + if t_ is None: + t_ = Symbol("{}_".format(t)) + t = t_ + eqs, funcs = _second_order_subs_type2(A0, funcs, t_) + sys_order = dict.fromkeys(funcs, 2) + + return _higher_order_to_first_order(eqs, sys_order, t, funcs=funcs) + + +def _higher_order_type2_to_sub_systems(J, f_t, funcs, t, max_order, b=None, P=None): + + # Note: To add a test for this ValueError + if J is None or f_t is None or not _matrix_is_constant(J, t): + raise ValueError(filldedent(''' + Correctly input for args 'A' and 'f_t' for Linear, Higher Order, + Type 2 + ''')) + + if P is None and b is not None and not b.is_zero_matrix: + raise ValueError(filldedent(''' + Provide the keyword 'P' for matrix P in A = P * J * P-1. + ''')) + + new_funcs = Matrix([Function(Dummy('{}__0'.format(f.func.__name__)))(t) for f in funcs]) + new_eqs = new_funcs.diff(t, max_order) - f_t * J * new_funcs + + if b is not None and not b.is_zero_matrix: + new_eqs -= P.inv() * b + + new_eqs = canonical_odes(new_eqs, new_funcs, t)[0] + + return new_eqs, new_funcs + + +def _higher_order_to_first_order(eqs, sys_order, t, funcs=None, type="type0", **kwargs): + if funcs is None: + funcs = sys_order.keys() + + # Standard Cauchy Euler system + if type == "type1": + t_ = Symbol('{}_'.format(t)) + new_funcs = [Function(Dummy('{}_'.format(f.func.__name__)))(t_) for f in funcs] + max_order = max(sys_order[func] for func in funcs) + subs_dict = dict(zip(funcs, new_funcs)) + subs_dict[t] = exp(t_) + + free_function = Function(Dummy()) + + def _get_coeffs_from_subs_expression(expr): + if isinstance(expr, Subs): + free_symbol = expr.args[1][0] + term = expr.args[0] + return {ode_order(term, free_symbol): 1} + + if isinstance(expr, Mul): + coeff = expr.args[0] + order = list(_get_coeffs_from_subs_expression(expr.args[1]).keys())[0] + return {order: coeff} + + if isinstance(expr, Add): + coeffs = {} + for arg in expr.args: + + if isinstance(arg, Mul): + coeffs.update(_get_coeffs_from_subs_expression(arg)) + + else: + order = list(_get_coeffs_from_subs_expression(arg).keys())[0] + coeffs[order] = 1 + + return coeffs + + for o in range(1, max_order + 1): + expr = free_function(log(t_)).diff(t_, o)*t_**o + coeff_dict = _get_coeffs_from_subs_expression(expr) + coeffs = [coeff_dict[order] if order in coeff_dict else 0 for order in range(o + 1)] + expr_to_subs = sum(free_function(t_).diff(t_, i) * c for i, c in + enumerate(coeffs)) / t**o + subs_dict.update({f.diff(t, o): expr_to_subs.subs(free_function(t_), nf) + for f, nf in zip(funcs, new_funcs)}) + + new_eqs = [eq.subs(subs_dict) for eq in eqs] + new_sys_order = {nf: sys_order[f] for f, nf in zip(funcs, new_funcs)} + + new_eqs = canonical_odes(new_eqs, new_funcs, t_)[0] + + return _higher_order_to_first_order(new_eqs, new_sys_order, t_, funcs=new_funcs) + + # Systems of the form: X(n)(t) = f(t)*A*X + b + # where X(n)(t) is the nth derivative of the vector of dependent variables + # with respect to the independent variable and A is a constant matrix. + if type == "type2": + J = kwargs.get('J', None) + f_t = kwargs.get('f_t', None) + b = kwargs.get('b', None) + P = kwargs.get('P', None) + max_order = max(sys_order[func] for func in funcs) + + return _higher_order_type2_to_sub_systems(J, f_t, funcs, t, max_order, P=P, b=b) + + # Note: To be changed to this after doit option is disabled for default cases + # new_sysorder = _get_func_order(new_eqs, new_funcs) + # + # return _higher_order_to_first_order(new_eqs, new_sysorder, t, funcs=new_funcs) + + new_funcs = [] + + for prev_func in funcs: + func_name = prev_func.func.__name__ + func = Function(Dummy('{}_0'.format(func_name)))(t) + new_funcs.append(func) + subs_dict = {prev_func: func} + new_eqs = [] + + for i in range(1, sys_order[prev_func]): + new_func = Function(Dummy('{}_{}'.format(func_name, i)))(t) + subs_dict[prev_func.diff(t, i)] = new_func + new_funcs.append(new_func) + + prev_f = subs_dict[prev_func.diff(t, i-1)] + new_eq = Eq(prev_f.diff(t), new_func) + new_eqs.append(new_eq) + + eqs = [eq.subs(subs_dict) for eq in eqs] + new_eqs + + return eqs, new_funcs + + +def dsolve_system(eqs, funcs=None, t=None, ics=None, doit=False, simplify=True): + r""" + Solves any(supported) system of Ordinary Differential Equations + + Explanation + =========== + + This function takes a system of ODEs as an input, determines if the + it is solvable by this function, and returns the solution if found any. + + This function can handle: + 1. Linear, First Order, Constant coefficient homogeneous system of ODEs + 2. Linear, First Order, Constant coefficient non-homogeneous system of ODEs + 3. Linear, First Order, non-constant coefficient homogeneous system of ODEs + 4. Linear, First Order, non-constant coefficient non-homogeneous system of ODEs + 5. Any implicit system which can be divided into system of ODEs which is of the above 4 forms + 6. Any higher order linear system of ODEs that can be reduced to one of the 5 forms of systems described above. + + The types of systems described above are not limited by the number of equations, i.e. this + function can solve the above types irrespective of the number of equations in the system passed. + But, the bigger the system, the more time it will take to solve the system. + + This function returns a list of solutions. Each solution is a list of equations where LHS is + the dependent variable and RHS is an expression in terms of the independent variable. + + Among the non constant coefficient types, not all the systems are solvable by this function. Only + those which have either a coefficient matrix with a commutative antiderivative or those systems which + may be divided further so that the divided systems may have coefficient matrix with commutative antiderivative. + + Parameters + ========== + + eqs : List + system of ODEs to be solved + funcs : List or None + List of dependent variables that make up the system of ODEs + t : Symbol or None + Independent variable in the system of ODEs + ics : Dict or None + Set of initial boundary/conditions for the system of ODEs + doit : Boolean + Evaluate the solutions if True. Default value is True. Can be + set to false if the integral evaluation takes too much time and/or + is not required. + simplify: Boolean + Simplify the solutions for the systems. Default value is True. + Can be set to false if simplification takes too much time and/or + is not required. + + Examples + ======== + + >>> from sympy import symbols, Eq, Function + >>> from sympy.solvers.ode.systems import dsolve_system + >>> f, g = symbols("f g", cls=Function) + >>> x = symbols("x") + + >>> eqs = [Eq(f(x).diff(x), g(x)), Eq(g(x).diff(x), f(x))] + >>> dsolve_system(eqs) + [[Eq(f(x), -C1*exp(-x) + C2*exp(x)), Eq(g(x), C1*exp(-x) + C2*exp(x))]] + + You can also pass the initial conditions for the system of ODEs: + + >>> dsolve_system(eqs, ics={f(0): 1, g(0): 0}) + [[Eq(f(x), exp(x)/2 + exp(-x)/2), Eq(g(x), exp(x)/2 - exp(-x)/2)]] + + Optionally, you can pass the dependent variables and the independent + variable for which the system is to be solved: + + >>> funcs = [f(x), g(x)] + >>> dsolve_system(eqs, funcs=funcs, t=x) + [[Eq(f(x), -C1*exp(-x) + C2*exp(x)), Eq(g(x), C1*exp(-x) + C2*exp(x))]] + + Lets look at an implicit system of ODEs: + + >>> eqs = [Eq(f(x).diff(x)**2, g(x)**2), Eq(g(x).diff(x), g(x))] + >>> dsolve_system(eqs) + [[Eq(f(x), C1 - C2*exp(x)), Eq(g(x), C2*exp(x))], [Eq(f(x), C1 + C2*exp(x)), Eq(g(x), C2*exp(x))]] + + Returns + ======= + + List of List of Equations + + Raises + ====== + + NotImplementedError + When the system of ODEs is not solvable by this function. + ValueError + When the parameters passed are not in the required form. + + """ + from sympy.solvers.ode.ode import solve_ics, _extract_funcs, constant_renumber + + if not iterable(eqs): + raise ValueError(filldedent(''' + List of equations should be passed. The input is not valid. + ''')) + + eqs = _preprocess_eqs(eqs) + + if funcs is not None and not isinstance(funcs, list): + raise ValueError(filldedent(''' + Input to the funcs should be a list of functions. + ''')) + + if funcs is None: + funcs = _extract_funcs(eqs) + + if any(len(func.args) != 1 for func in funcs): + raise ValueError(filldedent(''' + dsolve_system can solve a system of ODEs with only one independent + variable. + ''')) + + if len(eqs) != len(funcs): + raise ValueError(filldedent(''' + Number of equations and number of functions do not match + ''')) + + if t is not None and not isinstance(t, Symbol): + raise ValueError(filldedent(''' + The independent variable must be of type Symbol + ''')) + + if t is None: + t = list(list(eqs[0].atoms(Derivative))[0].atoms(Symbol))[0] + + sols = [] + canon_eqs = canonical_odes(eqs, funcs, t) + + for canon_eq in canon_eqs: + try: + sol = _strong_component_solver(canon_eq, funcs, t) + except NotImplementedError: + sol = None + + if sol 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+++ b/.venv/lib/python3.12/site-packages/sympy/solvers/ode/tests/test_lie_group.py @@ -0,0 +1,152 @@ +from sympy.core.function import Function +from sympy.core.numbers import Rational +from sympy.core.relational import Eq +from sympy.core.symbol import (Symbol, symbols) +from sympy.functions.elementary.exponential import (exp, log) +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import (atan, sin, tan) + +from sympy.solvers.ode import (classify_ode, checkinfsol, dsolve, infinitesimals) + +from sympy.solvers.ode.subscheck import checkodesol + +from sympy.testing.pytest import XFAIL + + +C1 = Symbol('C1') +x, y = symbols("x y") +f = Function('f') +xi = Function('xi') +eta = Function('eta') + + +def test_heuristic1(): + a, b, c, a4, a3, a2, a1, a0 = symbols("a b c a4 a3 a2 a1 a0") + df = f(x).diff(x) + eq = Eq(df, x**2*f(x)) + eq1 = f(x).diff(x) + a*f(x) - c*exp(b*x) + eq2 = f(x).diff(x) + 2*x*f(x) - x*exp(-x**2) + eq3 = (1 + 2*x)*df + 2 - 4*exp(-f(x)) + eq4 = f(x).diff(x) - (a4*x**4 + a3*x**3 + a2*x**2 + a1*x + a0)**Rational(-1, 2) + eq5 = x**2*df - f(x) + x**2*exp(x - (1/x)) + eqlist = [eq, eq1, eq2, eq3, eq4, eq5] + + i = infinitesimals(eq, hint='abaco1_simple') + assert i == [{eta(x, f(x)): exp(x**3/3), xi(x, f(x)): 0}, + {eta(x, f(x)): f(x), xi(x, f(x)): 0}, + {eta(x, f(x)): 0, xi(x, f(x)): x**(-2)}] + i1 = infinitesimals(eq1, hint='abaco1_simple') + assert i1 == [{eta(x, f(x)): exp(-a*x), xi(x, f(x)): 0}] + i2 = infinitesimals(eq2, hint='abaco1_simple') + assert i2 == [{eta(x, f(x)): exp(-x**2), xi(x, f(x)): 0}] + i3 = infinitesimals(eq3, hint='abaco1_simple') + assert i3 == [{eta(x, f(x)): 0, xi(x, f(x)): 2*x + 1}, + {eta(x, f(x)): 0, xi(x, f(x)): 1/(exp(f(x)) - 2)}] + i4 = infinitesimals(eq4, hint='abaco1_simple') + assert i4 == [{eta(x, f(x)): 1, xi(x, f(x)): 0}, + {eta(x, f(x)): 0, + xi(x, f(x)): sqrt(a0 + a1*x + a2*x**2 + a3*x**3 + a4*x**4)}] + i5 = infinitesimals(eq5, hint='abaco1_simple') + assert i5 == [{xi(x, f(x)): 0, eta(x, f(x)): exp(-1/x)}] + + ilist = [i, i1, i2, i3, i4, i5] + for eq, i in (zip(eqlist, ilist)): + check = checkinfsol(eq, i) + assert check[0] + + # This ODE can be solved by the Lie Group method, when there are + # better assumptions + eq6 = df - (f(x)/x)*(x*log(x**2/f(x)) + 2) + i = infinitesimals(eq6, hint='abaco1_product') + assert i == [{eta(x, f(x)): f(x)*exp(-x), xi(x, f(x)): 0}] + assert checkinfsol(eq6, i)[0] + + eq7 = x*(f(x).diff(x)) + 1 - f(x)**2 + i = infinitesimals(eq7, hint='chi') + assert checkinfsol(eq7, i)[0] + + +def test_heuristic3(): + a, b = symbols("a b") + df = f(x).diff(x) + + eq = x**2*df + x*f(x) + f(x)**2 + x**2 + i = infinitesimals(eq, hint='bivariate') + assert i == [{eta(x, f(x)): f(x), xi(x, f(x)): x}] + assert checkinfsol(eq, i)[0] + + eq = x**2*(-f(x)**2 + df)- a*x**2*f(x) + 2 - a*x + i = infinitesimals(eq, hint='bivariate') + assert checkinfsol(eq, i)[0] + + +def test_heuristic_function_sum(): + eq = f(x).diff(x) - (3*(1 + x**2/f(x)**2)*atan(f(x)/x) + (1 - 2*f(x))/x + + (1 - 3*f(x))*(x/f(x)**2)) + i = infinitesimals(eq, hint='function_sum') + assert i == [{eta(x, f(x)): f(x)**(-2) + x**(-2), xi(x, f(x)): 0}] + assert checkinfsol(eq, i)[0] + + +def test_heuristic_abaco2_similar(): + a, b = symbols("a b") + F = Function('F') + eq = f(x).diff(x) - F(a*x + b*f(x)) + i = infinitesimals(eq, hint='abaco2_similar') + assert i == [{eta(x, f(x)): -a/b, xi(x, f(x)): 1}] + assert checkinfsol(eq, i)[0] + + eq = f(x).diff(x) - (f(x)**2 / (sin(f(x) - x) - x**2 + 2*x*f(x))) + i = infinitesimals(eq, hint='abaco2_similar') + assert i == [{eta(x, f(x)): f(x)**2, xi(x, f(x)): f(x)**2}] + assert checkinfsol(eq, i)[0] + + +def test_heuristic_abaco2_unique_unknown(): + + a, b = symbols("a b") + F = Function('F') + eq = f(x).diff(x) - x**(a - 1)*(f(x)**(1 - b))*F(x**a/a + f(x)**b/b) + i = infinitesimals(eq, hint='abaco2_unique_unknown') + assert i == [{eta(x, f(x)): -f(x)*f(x)**(-b), xi(x, f(x)): x*x**(-a)}] + assert checkinfsol(eq, i)[0] + + eq = f(x).diff(x) + tan(F(x**2 + f(x)**2) + atan(x/f(x))) + i = infinitesimals(eq, hint='abaco2_unique_unknown') + assert i == [{eta(x, f(x)): x, xi(x, f(x)): -f(x)}] + assert checkinfsol(eq, i)[0] + + eq = (x*f(x).diff(x) + f(x) + 2*x)**2 -4*x*f(x) -4*x**2 -4*a + i = infinitesimals(eq, hint='abaco2_unique_unknown') + assert checkinfsol(eq, i)[0] + + +def test_heuristic_linear(): + a, b, m, n = symbols("a b m n") + + eq = x**(n*(m + 1) - m)*(f(x).diff(x)) - a*f(x)**n -b*x**(n*(m + 1)) + i = infinitesimals(eq, hint='linear') + assert checkinfsol(eq, i)[0] + + +@XFAIL +def test_kamke(): + a, b, alpha, c = symbols("a b alpha c") + eq = x**2*(a*f(x)**2+(f(x).diff(x))) + b*x**alpha + c + i = infinitesimals(eq, hint='sum_function') # XFAIL + assert checkinfsol(eq, i)[0] + + +def test_user_infinitesimals(): + x = Symbol("x") # assuming x is real generates an error + eq = x*(f(x).diff(x)) + 1 - f(x)**2 + sol = Eq(f(x), (C1 + x**2)/(C1 - x**2)) + infinitesimals = {'xi':sqrt(f(x) - 1)/sqrt(f(x) + 1), 'eta':0} + assert dsolve(eq, hint='lie_group', **infinitesimals) == sol + assert checkodesol(eq, sol) == (True, 0) + + +@XFAIL +def test_lie_group_issue15219(): + eqn = exp(f(x).diff(x)-f(x)) + assert 'lie_group' not in classify_ode(eqn, f(x)) diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/ode/tests/test_ode.py b/.venv/lib/python3.12/site-packages/sympy/solvers/ode/tests/test_ode.py new file mode 100644 index 0000000000000000000000000000000000000000..65e0fa62d52445a4669f3cdc5ef278dbf9c88ea4 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/ode/tests/test_ode.py @@ -0,0 +1,1105 @@ +from sympy.core.function import (Derivative, Function, Subs, diff) +from sympy.core.numbers import (E, I, Rational, pi) +from sympy.core.relational import Eq +from sympy.core.singleton import S +from sympy.core.symbol import (Symbol, symbols) +from sympy.functions.elementary.complexes import (im, re) +from sympy.functions.elementary.exponential import (exp, log) +from sympy.functions.elementary.hyperbolic import acosh +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import (atan2, cos, sin, tan) +from sympy.integrals.integrals import Integral +from sympy.polys.polytools import Poly +from sympy.series.order import O +from sympy.simplify.radsimp import collect + +from sympy.solvers.ode import (classify_ode, + homogeneous_order, dsolve) + +from sympy.solvers.ode.subscheck import checkodesol +from sympy.solvers.ode.ode import (classify_sysode, + constant_renumber, constantsimp, get_numbered_constants, solve_ics) + +from sympy.solvers.ode.nonhomogeneous import _undetermined_coefficients_match +from sympy.solvers.ode.single import LinearCoefficients +from sympy.solvers.deutils import ode_order +from sympy.testing.pytest import XFAIL, raises, slow, SKIP +from sympy.utilities.misc import filldedent + + +C0, C1, C2, C3, C4, C5, C6, C7, C8, C9, C10 = symbols('C0:11') +u, x, y, z = symbols('u,x:z', real=True) +f = Function('f') +g = Function('g') +h = Function('h') + +# Note: Examples which were specifically testing Single ODE solver are moved to test_single.py +# and all the system of ode examples are moved to test_systems.py +# Note: the tests below may fail (but still be correct) if ODE solver, +# the integral engine, solve(), or even simplify() changes. Also, in +# differently formatted solutions, the arbitrary constants might not be +# equal. Using specific hints in tests can help to avoid this. + +# Tests of order higher than 1 should run the solutions through +# constant_renumber because it will normalize it (constant_renumber causes +# dsolve() to return different results on different machines) + + +def test_get_numbered_constants(): + with raises(ValueError): + get_numbered_constants(None) + + +def test_dsolve_all_hint(): + eq = f(x).diff(x) + output = dsolve(eq, hint='all') + + # Match the Dummy variables: + sol1 = output['separable_Integral'] + _y = sol1.lhs.args[1][0] + sol1 = output['1st_homogeneous_coeff_subs_dep_div_indep_Integral'] + _u1 = sol1.rhs.args[1].args[1][0] + + expected = {'Bernoulli_Integral': Eq(f(x), C1 + Integral(0, x)), + '1st_homogeneous_coeff_best': Eq(f(x), C1), + 'Bernoulli': Eq(f(x), C1), + 'nth_algebraic': Eq(f(x), C1), + 'nth_linear_euler_eq_homogeneous': Eq(f(x), C1), + 'nth_linear_constant_coeff_homogeneous': Eq(f(x), C1), + 'separable': Eq(f(x), C1), + '1st_homogeneous_coeff_subs_indep_div_dep': Eq(f(x), C1), + 'nth_algebraic_Integral': Eq(f(x), C1), + '1st_linear': Eq(f(x), C1), + '1st_linear_Integral': Eq(f(x), C1 + Integral(0, x)), + '1st_exact': Eq(f(x), C1), + '1st_exact_Integral': Eq(Subs(Integral(0, x) + Integral(1, _y), _y, f(x)), C1), + 'lie_group': Eq(f(x), C1), + '1st_homogeneous_coeff_subs_dep_div_indep': Eq(f(x), C1), + '1st_homogeneous_coeff_subs_dep_div_indep_Integral': Eq(log(x), C1 + Integral(-1/_u1, (_u1, f(x)/x))), + '1st_power_series': Eq(f(x), C1), + 'separable_Integral': Eq(Integral(1, (_y, f(x))), C1 + Integral(0, x)), + '1st_homogeneous_coeff_subs_indep_div_dep_Integral': Eq(f(x), C1), + 'best': Eq(f(x), C1), + 'best_hint': 'nth_algebraic', + 'default': 'nth_algebraic', + 'order': 1} + assert output == expected + + assert dsolve(eq, hint='best') == Eq(f(x), C1) + + +def test_dsolve_ics(): + # Maybe this should just use one of the solutions instead of raising... + with raises(NotImplementedError): + dsolve(f(x).diff(x) - sqrt(f(x)), ics={f(1):1}) + + +@slow +def test_dsolve_options(): + eq = x*f(x).diff(x) + f(x) + a = dsolve(eq, hint='all') + b = dsolve(eq, hint='all', simplify=False) + c = dsolve(eq, hint='all_Integral') + keys = ['1st_exact', '1st_exact_Integral', '1st_homogeneous_coeff_best', + '1st_homogeneous_coeff_subs_dep_div_indep', + '1st_homogeneous_coeff_subs_dep_div_indep_Integral', + '1st_homogeneous_coeff_subs_indep_div_dep', + '1st_homogeneous_coeff_subs_indep_div_dep_Integral', '1st_linear', + '1st_linear_Integral', 'Bernoulli', 'Bernoulli_Integral', + 'almost_linear', 'almost_linear_Integral', 'best', 'best_hint', + 'default', 'factorable', 'lie_group', + 'nth_linear_euler_eq_homogeneous', 'order', + 'separable', 'separable_Integral'] + Integral_keys = ['1st_exact_Integral', + '1st_homogeneous_coeff_subs_dep_div_indep_Integral', + '1st_homogeneous_coeff_subs_indep_div_dep_Integral', '1st_linear_Integral', + 'Bernoulli_Integral', 'almost_linear_Integral', 'best', 'best_hint', 'default', + 'factorable', 'nth_linear_euler_eq_homogeneous', + 'order', 'separable_Integral'] + assert sorted(a.keys()) == keys + assert a['order'] == ode_order(eq, f(x)) + assert a['best'] == Eq(f(x), C1/x) + assert dsolve(eq, hint='best') == Eq(f(x), C1/x) + assert a['default'] == 'factorable' + assert a['best_hint'] == 'factorable' + assert not a['1st_exact'].has(Integral) + assert not a['separable'].has(Integral) + assert not a['1st_homogeneous_coeff_best'].has(Integral) + assert not a['1st_homogeneous_coeff_subs_dep_div_indep'].has(Integral) + assert not a['1st_homogeneous_coeff_subs_indep_div_dep'].has(Integral) + assert not a['1st_linear'].has(Integral) + assert a['1st_linear_Integral'].has(Integral) + assert a['1st_exact_Integral'].has(Integral) + assert a['1st_homogeneous_coeff_subs_dep_div_indep_Integral'].has(Integral) + assert a['1st_homogeneous_coeff_subs_indep_div_dep_Integral'].has(Integral) + assert a['separable_Integral'].has(Integral) + assert sorted(b.keys()) == keys + assert b['order'] == ode_order(eq, f(x)) + assert b['best'] == Eq(f(x), C1/x) + assert dsolve(eq, hint='best', simplify=False) == Eq(f(x), C1/x) + assert b['default'] == 'factorable' + assert b['best_hint'] == 'factorable' + assert a['separable'] != b['separable'] + assert a['1st_homogeneous_coeff_subs_dep_div_indep'] != \ + b['1st_homogeneous_coeff_subs_dep_div_indep'] + assert a['1st_homogeneous_coeff_subs_indep_div_dep'] != \ + b['1st_homogeneous_coeff_subs_indep_div_dep'] + assert not b['1st_exact'].has(Integral) + assert not b['separable'].has(Integral) + assert not b['1st_homogeneous_coeff_best'].has(Integral) + assert not b['1st_homogeneous_coeff_subs_dep_div_indep'].has(Integral) + assert not b['1st_homogeneous_coeff_subs_indep_div_dep'].has(Integral) + assert not b['1st_linear'].has(Integral) + assert b['1st_linear_Integral'].has(Integral) + assert b['1st_exact_Integral'].has(Integral) + assert b['1st_homogeneous_coeff_subs_dep_div_indep_Integral'].has(Integral) + assert b['1st_homogeneous_coeff_subs_indep_div_dep_Integral'].has(Integral) + assert b['separable_Integral'].has(Integral) + assert sorted(c.keys()) == Integral_keys + raises(ValueError, lambda: dsolve(eq, hint='notarealhint')) + raises(ValueError, lambda: dsolve(eq, hint='Liouville')) + assert dsolve(f(x).diff(x) - 1/f(x)**2, hint='all')['best'] == \ + dsolve(f(x).diff(x) - 1/f(x)**2, hint='best') + assert dsolve(f(x) + f(x).diff(x) + sin(x).diff(x) + 1, f(x), + hint="1st_linear_Integral") == \ + Eq(f(x), (C1 + Integral((-sin(x).diff(x) - 1)* + exp(Integral(1, x)), x))*exp(-Integral(1, x))) + + +def test_classify_ode(): + assert classify_ode(f(x).diff(x, 2), f(x)) == \ + ( + 'nth_algebraic', + 'nth_linear_constant_coeff_homogeneous', + 'nth_linear_euler_eq_homogeneous', + 'Liouville', + '2nd_power_series_ordinary', + 'nth_algebraic_Integral', + 'Liouville_Integral', + ) + assert classify_ode(f(x), f(x)) == ('nth_algebraic', 'nth_algebraic_Integral') + assert classify_ode(Eq(f(x).diff(x), 0), f(x)) == ( + 'nth_algebraic', + 'separable', + '1st_exact', + '1st_linear', + 'Bernoulli', + '1st_homogeneous_coeff_best', + '1st_homogeneous_coeff_subs_indep_div_dep', + '1st_homogeneous_coeff_subs_dep_div_indep', + '1st_power_series', 'lie_group', + 'nth_linear_constant_coeff_homogeneous', + 'nth_linear_euler_eq_homogeneous', + 'nth_algebraic_Integral', + 'separable_Integral', + '1st_exact_Integral', + '1st_linear_Integral', + 'Bernoulli_Integral', + '1st_homogeneous_coeff_subs_indep_div_dep_Integral', + '1st_homogeneous_coeff_subs_dep_div_indep_Integral') + assert classify_ode(f(x).diff(x)**2, f(x)) == ('factorable', + 'nth_algebraic', + 'separable', + '1st_exact', + '1st_linear', + 'Bernoulli', + '1st_homogeneous_coeff_best', + '1st_homogeneous_coeff_subs_indep_div_dep', + '1st_homogeneous_coeff_subs_dep_div_indep', + '1st_power_series', + 'lie_group', + 'nth_linear_euler_eq_homogeneous', + 'nth_algebraic_Integral', + 'separable_Integral', + '1st_exact_Integral', + '1st_linear_Integral', + 'Bernoulli_Integral', + '1st_homogeneous_coeff_subs_indep_div_dep_Integral', + '1st_homogeneous_coeff_subs_dep_div_indep_Integral') + # issue 4749: f(x) should be cleared from highest derivative before classifying + a = classify_ode(Eq(f(x).diff(x) + f(x), x), f(x)) + b = classify_ode(f(x).diff(x)*f(x) + f(x)*f(x) - x*f(x), f(x)) + c = classify_ode(f(x).diff(x)/f(x) + f(x)/f(x) - x/f(x), f(x)) + assert a == ('1st_exact', + '1st_linear', + 'Bernoulli', + 'almost_linear', + '1st_power_series', "lie_group", + 'nth_linear_constant_coeff_undetermined_coefficients', + 'nth_linear_constant_coeff_variation_of_parameters', + '1st_exact_Integral', + '1st_linear_Integral', + 'Bernoulli_Integral', + 'almost_linear_Integral', + 'nth_linear_constant_coeff_variation_of_parameters_Integral') + assert b == ('factorable', + '1st_linear', + 'Bernoulli', + '1st_power_series', + 'lie_group', + 'nth_linear_constant_coeff_undetermined_coefficients', + 'nth_linear_constant_coeff_variation_of_parameters', + '1st_linear_Integral', + 'Bernoulli_Integral', + 'nth_linear_constant_coeff_variation_of_parameters_Integral') + assert c == ('factorable', + '1st_linear', + 'Bernoulli', + '1st_power_series', + 'lie_group', + 'nth_linear_constant_coeff_undetermined_coefficients', + 'nth_linear_constant_coeff_variation_of_parameters', + '1st_linear_Integral', + 'Bernoulli_Integral', + 'nth_linear_constant_coeff_variation_of_parameters_Integral') + + assert classify_ode( + 2*x*f(x)*f(x).diff(x) + (1 + x)*f(x)**2 - exp(x), f(x) + ) == ('factorable', '1st_exact', 'Bernoulli', 'almost_linear', 'lie_group', + '1st_exact_Integral', 'Bernoulli_Integral', 'almost_linear_Integral') + assert 'Riccati_special_minus2' in \ + classify_ode(2*f(x).diff(x) + f(x)**2 - f(x)/x + 3*x**(-2), f(x)) + raises(ValueError, lambda: classify_ode(x + f(x, y).diff(x).diff( + y), f(x, y))) + # issue 5176 + k = Symbol('k') + assert classify_ode(f(x).diff(x)/(k*f(x) + k*x*f(x)) + 2*f(x)/(k*f(x) + + k*x*f(x)) + x*f(x).diff(x)/(k*f(x) + k*x*f(x)) + z, f(x)) == \ + ('factorable', 'separable', '1st_exact', '1st_linear', 'Bernoulli', + '1st_power_series', 'lie_group', 'separable_Integral', '1st_exact_Integral', + '1st_linear_Integral', 'Bernoulli_Integral') + # preprocessing + ans = ('factorable', 'nth_algebraic', 'separable', '1st_exact', '1st_linear', 'Bernoulli', + '1st_homogeneous_coeff_best', + '1st_homogeneous_coeff_subs_indep_div_dep', + '1st_homogeneous_coeff_subs_dep_div_indep', + '1st_power_series', 'lie_group', + 'nth_linear_constant_coeff_undetermined_coefficients', + 'nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients', + 'nth_linear_constant_coeff_variation_of_parameters', + 'nth_linear_euler_eq_nonhomogeneous_variation_of_parameters', + 'nth_algebraic_Integral', + 'separable_Integral', '1st_exact_Integral', + '1st_linear_Integral', + 'Bernoulli_Integral', + '1st_homogeneous_coeff_subs_indep_div_dep_Integral', + '1st_homogeneous_coeff_subs_dep_div_indep_Integral', + 'nth_linear_constant_coeff_variation_of_parameters_Integral', + 'nth_linear_euler_eq_nonhomogeneous_variation_of_parameters_Integral') + # w/o f(x) given + assert classify_ode(diff(f(x) + x, x) + diff(f(x), x)) == ans + # w/ f(x) and prep=True + assert classify_ode(diff(f(x) + x, x) + diff(f(x), x), f(x), + prep=True) == ans + + assert classify_ode(Eq(2*x**3*f(x).diff(x), 0), f(x)) == \ + ('factorable', 'nth_algebraic', 'separable', '1st_exact', + '1st_linear', 'Bernoulli', '1st_power_series', + 'lie_group', 'nth_linear_euler_eq_homogeneous', + 'nth_algebraic_Integral', 'separable_Integral', '1st_exact_Integral', + '1st_linear_Integral', 'Bernoulli_Integral') + + + assert classify_ode(Eq(2*f(x)**3*f(x).diff(x), 0), f(x)) == \ + ('factorable', 'nth_algebraic', 'separable', '1st_exact', '1st_linear', + 'Bernoulli', '1st_power_series', 'lie_group', 'nth_algebraic_Integral', + 'separable_Integral', '1st_exact_Integral', '1st_linear_Integral', + 'Bernoulli_Integral') + # test issue 13864 + assert classify_ode(Eq(diff(f(x), x) - f(x)**x, 0), f(x)) == \ + ('1st_power_series', 'lie_group') + assert isinstance(classify_ode(Eq(f(x), 5), f(x), dict=True), dict) + + #This is for new behavior of classify_ode when called internally with default, It should + # return the first hint which matches therefore, 'ordered_hints' key will not be there. + assert sorted(classify_ode(Eq(f(x).diff(x), 0), f(x), dict=True).keys()) == \ + ['default', 'nth_linear_constant_coeff_homogeneous', 'order'] + a = classify_ode(2*x*f(x)*f(x).diff(x) + (1 + x)*f(x)**2 - exp(x), f(x), dict=True, hint='Bernoulli') + assert sorted(a.keys()) == ['Bernoulli', 'Bernoulli_Integral', 'default', 'order', 'ordered_hints'] + + # test issue 22155 + a = classify_ode(f(x).diff(x) - exp(f(x) - x), f(x)) + assert a == ('separable', + '1st_exact', '1st_power_series', + 'lie_group', 'separable_Integral', + '1st_exact_Integral') + + +def test_classify_ode_ics(): + # Dummy + eq = f(x).diff(x, x) - f(x) + + # Not f(0) or f'(0) + ics = {x: 1} + raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics)) + + + ############################ + # f(0) type (AppliedUndef) # + ############################ + + + # Wrong function + ics = {g(0): 1} + raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics)) + + # Contains x + ics = {f(x): 1} + raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics)) + + # Too many args + ics = {f(0, 0): 1} + raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics)) + + # point contains x + ics = {f(0): f(x)} + raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics)) + + # Does not raise + ics = {f(0): f(0)} + classify_ode(eq, f(x), ics=ics) + + # Does not raise + ics = {f(0): 1} + classify_ode(eq, f(x), ics=ics) + + + ##################### + # f'(0) type (Subs) # + ##################### + + # Wrong function + ics = {g(x).diff(x).subs(x, 0): 1} + raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics)) + + # Contains x + ics = {f(y).diff(y).subs(y, x): 1} + raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics)) + + # Wrong variable + ics = {f(y).diff(y).subs(y, 0): 1} + raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics)) + + # Too many args + ics = {f(x, y).diff(x).subs(x, 0): 1} + raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics)) + + # Derivative wrt wrong vars + ics = {Derivative(f(x), x, y).subs(x, 0): 1} + raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics)) + + # point contains x + ics = {f(x).diff(x).subs(x, 0): f(x)} + raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics)) + + # Does not raise + ics = {f(x).diff(x).subs(x, 0): f(x).diff(x).subs(x, 0)} + classify_ode(eq, f(x), ics=ics) + + # Does not raise + ics = {f(x).diff(x).subs(x, 0): 1} + classify_ode(eq, f(x), ics=ics) + + ########################### + # f'(y) type (Derivative) # + ########################### + + # Wrong function + ics = {g(x).diff(x).subs(x, y): 1} + raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics)) + + # Contains x + ics = {f(y).diff(y).subs(y, x): 1} + raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics)) + + # Too many args + ics = {f(x, y).diff(x).subs(x, y): 1} + raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics)) + + # Derivative wrt wrong vars + ics = {Derivative(f(x), x, z).subs(x, y): 1} + raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics)) + + # point contains x + ics = {f(x).diff(x).subs(x, y): f(x)} + raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics)) + + # Does not raise + ics = {f(x).diff(x).subs(x, 0): f(0)} + classify_ode(eq, f(x), ics=ics) + + # Does not raise + ics = {f(x).diff(x).subs(x, y): 1} + classify_ode(eq, f(x), ics=ics) + +def test_classify_sysode(): + # Here x is assumed to be x(t) and y as y(t) for simplicity. + # Similarly diff(x,t) and diff(y,y) is assumed to be x1 and y1 respectively. + k, l, m, n = symbols('k, l, m, n', Integer=True) + k1, k2, k3, l1, l2, l3, m1, m2, m3 = symbols('k1, k2, k3, l1, l2, l3, m1, m2, m3', Integer=True) + P, Q, R, p, q, r = symbols('P, Q, R, p, q, r', cls=Function) + P1, P2, P3, Q1, Q2, R1, R2 = symbols('P1, P2, P3, Q1, Q2, R1, R2', cls=Function) + x, y, z = symbols('x, y, z', cls=Function) + t = symbols('t') + x1 = diff(x(t),t) + y1 = diff(y(t),t) + + eq6 = (Eq(x1, exp(k*x(t))*P(x(t),y(t))), Eq(y1,r(y(t))*P(x(t),y(t)))) + sol6 = {'no_of_equation': 2, 'func_coeff': {(0, x(t), 0): 0, (1, x(t), 1): 0, (0, x(t), 1): 1, (1, y(t), 0): 0, \ + (1, x(t), 0): 0, (0, y(t), 1): 0, (0, y(t), 0): 0, (1, y(t), 1): 1}, 'type_of_equation': 'type2', 'func': \ + [x(t), y(t)], 'is_linear': False, 'eq': [-P(x(t), y(t))*exp(k*x(t)) + Derivative(x(t), t), -P(x(t), \ + y(t))*r(y(t)) + Derivative(y(t), t)], 'order': {y(t): 1, x(t): 1}} + assert classify_sysode(eq6) == sol6 + + eq7 = (Eq(x1, x(t)**2+y(t)/x(t)), Eq(y1, x(t)/y(t))) + sol7 = {'no_of_equation': 2, 'func_coeff': {(0, x(t), 0): 0, (1, x(t), 1): 0, (0, x(t), 1): 1, (1, y(t), 0): 0, \ + (1, x(t), 0): -1/y(t), (0, y(t), 1): 0, (0, y(t), 0): -1/x(t), (1, y(t), 1): 1}, 'type_of_equation': 'type3', \ + 'func': [x(t), y(t)], 'is_linear': False, 'eq': [-x(t)**2 + Derivative(x(t), t) - y(t)/x(t), -x(t)/y(t) + \ + Derivative(y(t), t)], 'order': {y(t): 1, x(t): 1}} + assert classify_sysode(eq7) == sol7 + + eq8 = (Eq(x1, P1(x(t))*Q1(y(t))*R(x(t),y(t),t)), Eq(y1, P1(x(t))*Q1(y(t))*R(x(t),y(t),t))) + sol8 = {'func': [x(t), y(t)], 'is_linear': False, 'type_of_equation': 'type4', 'eq': \ + [-P1(x(t))*Q1(y(t))*R(x(t), y(t), t) + Derivative(x(t), t), -P1(x(t))*Q1(y(t))*R(x(t), y(t), t) + \ + Derivative(y(t), t)], 'func_coeff': {(0, y(t), 1): 0, (1, y(t), 1): 1, (1, x(t), 1): 0, (0, y(t), 0): 0, \ + (1, x(t), 0): 0, (0, x(t), 0): 0, (1, y(t), 0): 0, (0, x(t), 1): 1}, 'order': {y(t): 1, x(t): 1}, 'no_of_equation': 2} + assert classify_sysode(eq8) == sol8 + + eq11 = (Eq(x1,x(t)*y(t)**3), Eq(y1,y(t)**5)) + sol11 = {'no_of_equation': 2, 'func_coeff': {(0, x(t), 0): -y(t)**3, (1, x(t), 1): 0, (0, x(t), 1): 1, \ + (1, y(t), 0): 0, (1, x(t), 0): 0, (0, y(t), 1): 0, (0, y(t), 0): 0, (1, y(t), 1): 1}, 'type_of_equation': \ + 'type1', 'func': [x(t), y(t)], 'is_linear': False, 'eq': [-x(t)*y(t)**3 + Derivative(x(t), t), \ + -y(t)**5 + Derivative(y(t), t)], 'order': {y(t): 1, x(t): 1}} + assert classify_sysode(eq11) == sol11 + + eq13 = (Eq(x1,x(t)*y(t)*sin(t)**2), Eq(y1,y(t)**2*sin(t)**2)) + sol13 = {'no_of_equation': 2, 'func_coeff': {(0, x(t), 0): -y(t)*sin(t)**2, (1, x(t), 1): 0, (0, x(t), 1): 1, \ + (1, y(t), 0): 0, (1, x(t), 0): 0, (0, y(t), 1): 0, (0, y(t), 0): -x(t)*sin(t)**2, (1, y(t), 1): 1}, \ + 'type_of_equation': 'type4', 'func': [x(t), y(t)], 'is_linear': False, 'eq': [-x(t)*y(t)*sin(t)**2 + \ + Derivative(x(t), t), -y(t)**2*sin(t)**2 + Derivative(y(t), t)], 'order': {y(t): 1, x(t): 1}} + assert classify_sysode(eq13) == sol13 + + +def test_solve_ics(): + # Basic tests that things work from dsolve. + assert dsolve(f(x).diff(x) - 1/f(x), f(x), ics={f(1): 2}) == \ + Eq(f(x), sqrt(2 * x + 2)) + assert dsolve(f(x).diff(x) - f(x), f(x), ics={f(0): 1}) == Eq(f(x), exp(x)) + assert dsolve(f(x).diff(x) - f(x), f(x), ics={f(x).diff(x).subs(x, 0): 1}) == Eq(f(x), exp(x)) + assert dsolve(f(x).diff(x, x) + f(x), f(x), ics={f(0): 1, + f(x).diff(x).subs(x, 0): 1}) == Eq(f(x), sin(x) + cos(x)) + assert dsolve([f(x).diff(x) - f(x) + g(x), g(x).diff(x) - g(x) - f(x)], + [f(x), g(x)], ics={f(0): 1, g(0): 0}) == [Eq(f(x), exp(x)*cos(x)), Eq(g(x), exp(x)*sin(x))] + + # Test cases where dsolve returns two solutions. + eq = (x**2*f(x)**2 - x).diff(x) + assert dsolve(eq, f(x), ics={f(1): 0}) == [Eq(f(x), + -sqrt(x - 1)/x), Eq(f(x), sqrt(x - 1)/x)] + assert dsolve(eq, f(x), ics={f(x).diff(x).subs(x, 1): 0}) == [Eq(f(x), + -sqrt(x - S.Half)/x), Eq(f(x), sqrt(x - S.Half)/x)] + + eq = cos(f(x)) - (x*sin(f(x)) - f(x)**2)*f(x).diff(x) + assert dsolve(eq, f(x), + ics={f(0):1}, hint='1st_exact', simplify=False) == Eq(x*cos(f(x)) + f(x)**3/3, Rational(1, 3)) + assert dsolve(eq, f(x), + ics={f(0):1}, hint='1st_exact', simplify=True) == Eq(x*cos(f(x)) + f(x)**3/3, Rational(1, 3)) + + assert solve_ics([Eq(f(x), C1*exp(x))], [f(x)], [C1], {f(0): 1}) == {C1: 1} + assert solve_ics([Eq(f(x), C1*sin(x) + C2*cos(x))], [f(x)], [C1, C2], + {f(0): 1, f(pi/2): 1}) == {C1: 1, C2: 1} + + assert solve_ics([Eq(f(x), C1*sin(x) + C2*cos(x))], [f(x)], [C1, C2], + {f(0): 1, f(x).diff(x).subs(x, 0): 1}) == {C1: 1, C2: 1} + + assert solve_ics([Eq(f(x), C1*sin(x) + C2*cos(x))], [f(x)], [C1, C2], {f(0): 1}) == \ + {C2: 1} + + # Some more complicated tests Refer to PR #16098 + + assert set(dsolve(f(x).diff(x)*(f(x).diff(x, 2)-x), ics={f(0):0, f(x).diff(x).subs(x, 1):0})) == \ + {Eq(f(x), 0), Eq(f(x), x ** 3 / 6 - x / 2)} + assert set(dsolve(f(x).diff(x)*(f(x).diff(x, 2)-x), ics={f(0):0})) == \ + {Eq(f(x), 0), Eq(f(x), C2*x + x**3/6)} + + K, r, f0 = symbols('K r f0') + sol = Eq(f(x), K*f0*exp(r*x)/((-K + f0)*(f0*exp(r*x)/(-K + f0) - 1))) + assert (dsolve(Eq(f(x).diff(x), r * f(x) * (1 - f(x) / K)), f(x), ics={f(0): f0})) == sol + + + #Order dependent issues Refer to PR #16098 + assert set(dsolve(f(x).diff(x)*(f(x).diff(x, 2)-x), ics={f(x).diff(x).subs(x,0):0, f(0):0})) == \ + {Eq(f(x), 0), Eq(f(x), x ** 3 / 6)} + assert set(dsolve(f(x).diff(x)*(f(x).diff(x, 2)-x), ics={f(0):0, f(x).diff(x).subs(x,0):0})) == \ + {Eq(f(x), 0), Eq(f(x), x ** 3 / 6)} + + # XXX: Ought to be ValueError + raises(ValueError, lambda: solve_ics([Eq(f(x), C1*sin(x) + C2*cos(x))], [f(x)], [C1, C2], {f(0): 1, f(pi): 1})) + + # Degenerate case. f'(0) is identically 0. + raises(ValueError, lambda: solve_ics([Eq(f(x), sqrt(C1 - x**2))], [f(x)], [C1], {f(x).diff(x).subs(x, 0): 0})) + + EI, q, L = symbols('EI q L') + + # eq = Eq(EI*diff(f(x), x, 4), q) + sols = [Eq(f(x), C1 + C2*x + C3*x**2 + C4*x**3 + q*x**4/(24*EI))] + funcs = [f(x)] + constants = [C1, C2, C3, C4] + # Test both cases, Derivative (the default from f(x).diff(x).subs(x, L)), + # and Subs + ics1 = {f(0): 0, + f(x).diff(x).subs(x, 0): 0, + f(L).diff(L, 2): 0, + f(L).diff(L, 3): 0} + ics2 = {f(0): 0, + f(x).diff(x).subs(x, 0): 0, + Subs(f(x).diff(x, 2), x, L): 0, + Subs(f(x).diff(x, 3), x, L): 0} + + solved_constants1 = solve_ics(sols, funcs, constants, ics1) + solved_constants2 = solve_ics(sols, funcs, constants, ics2) + assert solved_constants1 == solved_constants2 == { + C1: 0, + C2: 0, + C3: L**2*q/(4*EI), + C4: -L*q/(6*EI)} + + # Allow the ics to refer to f + ics = {f(0): f(0)} + assert dsolve(f(x).diff(x) - f(x), f(x), ics=ics) == Eq(f(x), f(0)*exp(x)) + + ics = {f(x).diff(x).subs(x, 0): f(x).diff(x).subs(x, 0), f(0): f(0)} + assert dsolve(f(x).diff(x, x) + f(x), f(x), ics=ics) == \ + Eq(f(x), f(0)*cos(x) + f(x).diff(x).subs(x, 0)*sin(x)) + +def test_ode_order(): + f = Function('f') + g = Function('g') + x = Symbol('x') + assert ode_order(3*x*exp(f(x)), f(x)) == 0 + assert ode_order(x*diff(f(x), x) + 3*x*f(x) - sin(x)/x, f(x)) == 1 + assert ode_order(x**2*f(x).diff(x, x) + x*diff(f(x), x) - f(x), f(x)) == 2 + assert ode_order(diff(x*exp(f(x)), x, x), f(x)) == 2 + assert ode_order(diff(x*diff(x*exp(f(x)), x, x), x), f(x)) == 3 + assert ode_order(diff(f(x), x, x), g(x)) == 0 + assert ode_order(diff(f(x), x, x)*diff(g(x), x), f(x)) == 2 + assert ode_order(diff(f(x), x, x)*diff(g(x), x), g(x)) == 1 + assert ode_order(diff(x*diff(x*exp(f(x)), x, x), x), g(x)) == 0 + # issue 5835: ode_order has to also work for unevaluated derivatives + # (ie, without using doit()). + assert ode_order(Derivative(x*f(x), x), f(x)) == 1 + assert ode_order(x*sin(Derivative(x*f(x)**2, x, x)), f(x)) == 2 + assert ode_order(Derivative(x*Derivative(x*exp(f(x)), x, x), x), g(x)) == 0 + assert ode_order(Derivative(f(x), x, x), g(x)) == 0 + assert ode_order(Derivative(x*exp(f(x)), x, x), f(x)) == 2 + assert ode_order(Derivative(f(x), x, x)*Derivative(g(x), x), g(x)) == 1 + assert ode_order(Derivative(x*Derivative(f(x), x, x), x), f(x)) == 3 + assert ode_order( + x*sin(Derivative(x*Derivative(f(x), x)**2, x, x)), f(x)) == 3 + + +def test_homogeneous_order(): + assert homogeneous_order(exp(y/x) + tan(y/x), x, y) == 0 + assert homogeneous_order(x**2 + sin(x)*cos(y), x, y) is None + assert homogeneous_order(x - y - x*sin(y/x), x, y) == 1 + assert homogeneous_order((x*y + sqrt(x**4 + y**4) + x**2*(log(x) - log(y)))/ + (pi*x**Rational(2, 3)*sqrt(y)**3), x, y) == Rational(-1, 6) + assert homogeneous_order(y/x*cos(y/x) - x/y*sin(y/x) + cos(y/x), x, y) == 0 + assert homogeneous_order(f(x), x, f(x)) == 1 + assert homogeneous_order(f(x)**2, x, f(x)) == 2 + assert homogeneous_order(x*y*z, x, y) == 2 + assert homogeneous_order(x*y*z, x, y, z) == 3 + assert homogeneous_order(x**2*f(x)/sqrt(x**2 + f(x)**2), f(x)) is None + assert homogeneous_order(f(x, y)**2, x, f(x, y), y) == 2 + assert homogeneous_order(f(x, y)**2, x, f(x), y) is None + assert homogeneous_order(f(x, y)**2, x, f(x, y)) is None + assert homogeneous_order(f(y, x)**2, x, y, f(x, y)) is None + assert homogeneous_order(f(y), f(x), x) is None + assert homogeneous_order(-f(x)/x + 1/sin(f(x)/ x), f(x), x) == 0 + assert homogeneous_order(log(1/y) + log(x**2), x, y) is None + assert homogeneous_order(log(1/y) + log(x), x, y) == 0 + assert homogeneous_order(log(x/y), x, y) == 0 + assert homogeneous_order(2*log(1/y) + 2*log(x), x, y) == 0 + a = Symbol('a') + assert homogeneous_order(a*log(1/y) + a*log(x), x, y) == 0 + assert homogeneous_order(f(x).diff(x), x, y) is None + assert homogeneous_order(-f(x).diff(x) + x, x, y) is None + assert homogeneous_order(O(x), x, y) is None + assert homogeneous_order(x + O(x**2), x, y) is None + assert homogeneous_order(x**pi, x) == pi + assert homogeneous_order(x**x, x) is None + raises(ValueError, lambda: homogeneous_order(x*y)) + + +@XFAIL +def test_noncircularized_real_imaginary_parts(): + # If this passes, lines numbered 3878-3882 (at the time of this commit) + # of sympy/solvers/ode.py for nth_linear_constant_coeff_homogeneous + # should be removed. + y = sqrt(1+x) + i, r = im(y), re(y) + assert not (i.has(atan2) and r.has(atan2)) + + +def test_collect_respecting_exponentials(): + # If this test passes, lines 1306-1311 (at the time of this commit) + # of sympy/solvers/ode.py should be removed. + sol = 1 + exp(x/2) + assert sol == collect( sol, exp(x/3)) + + +def test_undetermined_coefficients_match(): + assert _undetermined_coefficients_match(g(x), x) == {'test': False} + assert _undetermined_coefficients_match(sin(2*x + sqrt(5)), x) == \ + {'test': True, 'trialset': + {cos(2*x + sqrt(5)), sin(2*x + sqrt(5))}} + assert _undetermined_coefficients_match(sin(x)*cos(x), x) == \ + {'test': False} + s = {cos(x), x*cos(x), x**2*cos(x), x**2*sin(x), x*sin(x), sin(x)} + assert _undetermined_coefficients_match(sin(x)*(x**2 + x + 1), x) == \ + {'test': True, 'trialset': s} + assert _undetermined_coefficients_match( + sin(x)*x**2 + sin(x)*x + sin(x), x) == {'test': True, 'trialset': s} + assert _undetermined_coefficients_match( + exp(2*x)*sin(x)*(x**2 + x + 1), x + ) == { + 'test': True, 'trialset': {exp(2*x)*sin(x), x**2*exp(2*x)*sin(x), + cos(x)*exp(2*x), x**2*cos(x)*exp(2*x), x*cos(x)*exp(2*x), + x*exp(2*x)*sin(x)}} + assert _undetermined_coefficients_match(1/sin(x), x) == {'test': False} + assert _undetermined_coefficients_match(log(x), x) == {'test': False} + assert _undetermined_coefficients_match(2**(x)*(x**2 + x + 1), x) == \ + {'test': True, 'trialset': {2**x, x*2**x, x**2*2**x}} + assert _undetermined_coefficients_match(x**y, x) == {'test': False} + assert _undetermined_coefficients_match(exp(x)*exp(2*x + 1), x) == \ + {'test': True, 'trialset': {exp(1 + 3*x)}} + assert _undetermined_coefficients_match(sin(x)*(x**2 + x + 1), x) == \ + {'test': True, 'trialset': {x*cos(x), x*sin(x), x**2*cos(x), + x**2*sin(x), cos(x), sin(x)}} + assert _undetermined_coefficients_match(sin(x)*(x + sin(x)), x) == \ + {'test': False} + assert _undetermined_coefficients_match(sin(x)*(x + sin(2*x)), x) == \ + {'test': False} + assert _undetermined_coefficients_match(sin(x)*tan(x), x) == \ + {'test': False} + assert _undetermined_coefficients_match( + x**2*sin(x)*exp(x) + x*sin(x) + x, x + ) == { + 'test': True, 'trialset': {x**2*cos(x)*exp(x), x, cos(x), S.One, + exp(x)*sin(x), sin(x), x*exp(x)*sin(x), x*cos(x), x*cos(x)*exp(x), + x*sin(x), cos(x)*exp(x), x**2*exp(x)*sin(x)}} + assert _undetermined_coefficients_match(4*x*sin(x - 2), x) == { + 'trialset': {x*cos(x - 2), x*sin(x - 2), cos(x - 2), sin(x - 2)}, + 'test': True, + } + assert _undetermined_coefficients_match(2**x*x, x) == \ + {'test': True, 'trialset': {2**x, x*2**x}} + assert _undetermined_coefficients_match(2**x*exp(2*x), x) == \ + {'test': True, 'trialset': {2**x*exp(2*x)}} + assert _undetermined_coefficients_match(exp(-x)/x, x) == \ + {'test': False} + # Below are from Ordinary Differential Equations, + # Tenenbaum and Pollard, pg. 231 + assert _undetermined_coefficients_match(S(4), x) == \ + {'test': True, 'trialset': {S.One}} + assert _undetermined_coefficients_match(12*exp(x), x) == \ + {'test': True, 'trialset': {exp(x)}} + assert _undetermined_coefficients_match(exp(I*x), x) == \ + {'test': True, 'trialset': {exp(I*x)}} + assert _undetermined_coefficients_match(sin(x), x) == \ + {'test': True, 'trialset': {cos(x), sin(x)}} + assert _undetermined_coefficients_match(cos(x), x) == \ + {'test': True, 'trialset': {cos(x), sin(x)}} + assert _undetermined_coefficients_match(8 + 6*exp(x) + 2*sin(x), x) == \ + {'test': True, 'trialset': {S.One, cos(x), sin(x), exp(x)}} + assert _undetermined_coefficients_match(x**2, x) == \ + {'test': True, 'trialset': {S.One, x, x**2}} + assert _undetermined_coefficients_match(9*x*exp(x) + exp(-x), x) == \ + {'test': True, 'trialset': {x*exp(x), exp(x), exp(-x)}} + assert _undetermined_coefficients_match(2*exp(2*x)*sin(x), x) == \ + {'test': True, 'trialset': {exp(2*x)*sin(x), cos(x)*exp(2*x)}} + assert _undetermined_coefficients_match(x - sin(x), x) == \ + {'test': True, 'trialset': {S.One, x, cos(x), sin(x)}} + assert _undetermined_coefficients_match(x**2 + 2*x, x) == \ + {'test': True, 'trialset': {S.One, x, x**2}} + assert _undetermined_coefficients_match(4*x*sin(x), x) == \ + {'test': True, 'trialset': {x*cos(x), x*sin(x), cos(x), sin(x)}} + assert _undetermined_coefficients_match(x*sin(2*x), x) == \ + {'test': True, 'trialset': + {x*cos(2*x), x*sin(2*x), cos(2*x), sin(2*x)}} + assert _undetermined_coefficients_match(x**2*exp(-x), x) == \ + {'test': True, 'trialset': {x*exp(-x), x**2*exp(-x), exp(-x)}} + assert _undetermined_coefficients_match(2*exp(-x) - x**2*exp(-x), x) == \ + {'test': True, 'trialset': {x*exp(-x), x**2*exp(-x), exp(-x)}} + assert _undetermined_coefficients_match(exp(-2*x) + x**2, x) == \ + {'test': True, 'trialset': {S.One, x, x**2, exp(-2*x)}} + assert _undetermined_coefficients_match(x*exp(-x), x) == \ + {'test': True, 'trialset': {x*exp(-x), exp(-x)}} + assert _undetermined_coefficients_match(x + exp(2*x), x) == \ + {'test': True, 'trialset': {S.One, x, exp(2*x)}} + assert _undetermined_coefficients_match(sin(x) + exp(-x), x) == \ + {'test': True, 'trialset': {cos(x), sin(x), exp(-x)}} + assert _undetermined_coefficients_match(exp(x), x) == \ + {'test': True, 'trialset': {exp(x)}} + # converted from sin(x)**2 + assert _undetermined_coefficients_match(S.Half - cos(2*x)/2, x) == \ + {'test': True, 'trialset': {S.One, cos(2*x), sin(2*x)}} + # converted from exp(2*x)*sin(x)**2 + assert _undetermined_coefficients_match( + exp(2*x)*(S.Half + cos(2*x)/2), x + ) == { + 'test': True, 'trialset': {exp(2*x)*sin(2*x), cos(2*x)*exp(2*x), + exp(2*x)}} + assert _undetermined_coefficients_match(2*x + sin(x) + cos(x), x) == \ + {'test': True, 'trialset': {S.One, x, cos(x), sin(x)}} + # converted from sin(2*x)*sin(x) + assert _undetermined_coefficients_match(cos(x)/2 - cos(3*x)/2, x) == \ + {'test': True, 'trialset': {cos(x), cos(3*x), sin(x), sin(3*x)}} + assert _undetermined_coefficients_match(cos(x**2), x) == {'test': False} + assert _undetermined_coefficients_match(2**(x**2), x) == {'test': False} + + +def test_issue_4785_22462(): + from sympy.abc import A + eq = x + A*(x + diff(f(x), x) + f(x)) + diff(f(x), x) + f(x) + 2 + assert classify_ode(eq, f(x)) == ('factorable', '1st_exact', '1st_linear', + 'Bernoulli', 'almost_linear', '1st_power_series', 'lie_group', + 'nth_linear_constant_coeff_undetermined_coefficients', + 'nth_linear_constant_coeff_variation_of_parameters', + '1st_exact_Integral', '1st_linear_Integral', 'Bernoulli_Integral', + 'almost_linear_Integral', + 'nth_linear_constant_coeff_variation_of_parameters_Integral') + # issue 4864 + eq = (x**2 + f(x)**2)*f(x).diff(x) - 2*x*f(x) + assert classify_ode(eq, f(x)) == ('factorable', '1st_exact', + '1st_homogeneous_coeff_best', + '1st_homogeneous_coeff_subs_indep_div_dep', + '1st_homogeneous_coeff_subs_dep_div_indep', + '1st_power_series', + 'lie_group', '1st_exact_Integral', + '1st_homogeneous_coeff_subs_indep_div_dep_Integral', + '1st_homogeneous_coeff_subs_dep_div_indep_Integral') + + +def test_issue_4825(): + raises(ValueError, lambda: dsolve(f(x, y).diff(x) - y*f(x, y), f(x))) + assert classify_ode(f(x, y).diff(x) - y*f(x, y), f(x), dict=True) == \ + {'order': 0, 'default': None, 'ordered_hints': ()} + # See also issue 3793, test Z13. + raises(ValueError, lambda: dsolve(f(x).diff(x), f(y))) + assert classify_ode(f(x).diff(x), f(y), dict=True) == \ + {'order': 0, 'default': None, 'ordered_hints': ()} + + +def test_constant_renumber_order_issue_5308(): + from sympy.utilities.iterables import variations + + assert constant_renumber(C1*x + C2*y) == \ + constant_renumber(C1*y + C2*x) == \ + C1*x + C2*y + e = C1*(C2 + x)*(C3 + y) + for a, b, c in variations([C1, C2, C3], 3): + assert constant_renumber(a*(b + x)*(c + y)) == e + + +def test_constant_renumber(): + e1, e2, x, y = symbols("e1:3 x y") + exprs = [e2*x, e1*x + e2*y] + + assert constant_renumber(exprs[0]) == e2*x + assert constant_renumber(exprs[0], variables=[x]) == C1*x + assert constant_renumber(exprs[0], variables=[x], newconstants=[C2]) == C2*x + assert constant_renumber(exprs, variables=[x, y]) == [C1*x, C1*y + C2*x] + assert constant_renumber(exprs, variables=[x, y], newconstants=symbols("C3:5")) == [C3*x, C3*y + C4*x] + + +def test_issue_5770(): + k = Symbol("k", real=True) + t = Symbol('t') + w = Function('w') + sol = dsolve(w(t).diff(t, 6) - k**6*w(t), w(t)) + assert len([s for s in sol.free_symbols if s.name.startswith('C')]) == 6 + assert constantsimp((C1*cos(x) + C2*cos(x))*exp(x), {C1, C2}) == \ + C1*cos(x)*exp(x) + assert constantsimp(C1*cos(x) + C2*cos(x) + C3*sin(x), {C1, C2, C3}) == \ + C1*cos(x) + C3*sin(x) + assert constantsimp(exp(C1 + x), {C1}) == C1*exp(x) + assert constantsimp(x + C1 + y, {C1, y}) == C1 + x + assert constantsimp(x + C1 + Integral(x, (x, 1, 2)), {C1}) == C1 + x + + +def test_issue_5112_5430(): + assert homogeneous_order(-log(x) + acosh(x), x) is None + assert homogeneous_order(y - log(x), x, y) is None + + +def test_issue_5095(): + f = Function('f') + raises(ValueError, lambda: dsolve(f(x).diff(x)**2, f(x), 'fdsjf')) + + +def test_homogeneous_function(): + f = Function('f') + eq1 = tan(x + f(x)) + eq2 = sin((3*x)/(4*f(x))) + eq3 = cos(x*f(x)*Rational(3, 4)) + eq4 = log((3*x + 4*f(x))/(5*f(x) + 7*x)) + eq5 = exp((2*x**2)/(3*f(x)**2)) + eq6 = log((3*x + 4*f(x))/(5*f(x) + 7*x) + exp((2*x**2)/(3*f(x)**2))) + eq7 = sin((3*x)/(5*f(x) + x**2)) + assert homogeneous_order(eq1, x, f(x)) == None + assert homogeneous_order(eq2, x, f(x)) == 0 + assert homogeneous_order(eq3, x, f(x)) == None + assert homogeneous_order(eq4, x, f(x)) == 0 + assert homogeneous_order(eq5, x, f(x)) == 0 + assert homogeneous_order(eq6, x, f(x)) == 0 + assert homogeneous_order(eq7, x, f(x)) == None + + +def test_linear_coeff_match(): + n, d = z*(2*x + 3*f(x) + 5), z*(7*x + 9*f(x) + 11) + rat = n/d + eq1 = sin(rat) + cos(rat.expand()) + obj1 = LinearCoefficients(eq1) + eq2 = rat + obj2 = LinearCoefficients(eq2) + eq3 = log(sin(rat)) + obj3 = LinearCoefficients(eq3) + ans = (4, Rational(-13, 3)) + assert obj1._linear_coeff_match(eq1, f(x)) == ans + assert obj2._linear_coeff_match(eq2, f(x)) == ans + assert obj3._linear_coeff_match(eq3, f(x)) == ans + + # no c + eq4 = (3*x)/f(x) + obj4 = LinearCoefficients(eq4) + # not x and f(x) + eq5 = (3*x + 2)/x + obj5 = LinearCoefficients(eq5) + # denom will be zero + eq6 = (3*x + 2*f(x) + 1)/(3*x + 2*f(x) + 5) + obj6 = LinearCoefficients(eq6) + # not rational coefficient + eq7 = (3*x + 2*f(x) + sqrt(2))/(3*x + 2*f(x) + 5) + obj7 = LinearCoefficients(eq7) + assert obj4._linear_coeff_match(eq4, f(x)) is None + assert obj5._linear_coeff_match(eq5, f(x)) is None + assert obj6._linear_coeff_match(eq6, f(x)) is None + assert obj7._linear_coeff_match(eq7, f(x)) is None + + +def test_constantsimp_take_problem(): + c = exp(C1) + 2 + assert len(Poly(constantsimp(exp(C1) + c + c*x, [C1])).gens) == 2 + + +def test_series(): + C1 = Symbol("C1") + eq = f(x).diff(x) - f(x) + sol = Eq(f(x), C1 + C1*x + C1*x**2/2 + C1*x**3/6 + C1*x**4/24 + + C1*x**5/120 + O(x**6)) + assert dsolve(eq, hint='1st_power_series') == sol + assert checkodesol(eq, sol, order=1)[0] + + eq = f(x).diff(x) - x*f(x) + sol = Eq(f(x), C1*x**4/8 + C1*x**2/2 + C1 + O(x**6)) + assert dsolve(eq, hint='1st_power_series') == sol + assert checkodesol(eq, sol, order=1)[0] + + eq = f(x).diff(x) - sin(x*f(x)) + sol = Eq(f(x), (x - 2)**2*(1+ sin(4))*cos(4) + (x - 2)*sin(4) + 2 + O(x**3)) + assert dsolve(eq, hint='1st_power_series', ics={f(2): 2}, n=3) == sol + # FIXME: The solution here should be O((x-2)**3) so is incorrect + #assert checkodesol(eq, sol, order=1)[0] + + +@slow +def test_2nd_power_series_ordinary(): + C1, C2 = symbols("C1 C2") + + eq = f(x).diff(x, 2) - x*f(x) + assert classify_ode(eq) == ('2nd_linear_airy', '2nd_power_series_ordinary') + sol = Eq(f(x), C2*(x**3/6 + 1) + C1*x*(x**3/12 + 1) + O(x**6)) + assert dsolve(eq, hint='2nd_power_series_ordinary') == sol + assert checkodesol(eq, sol) == (True, 0) + + sol = Eq(f(x), C2*((x + 2)**4/6 + (x + 2)**3/6 - (x + 2)**2 + 1) + + C1*(x + (x + 2)**4/12 - (x + 2)**3/3 + S(2)) + + O(x**6)) + assert dsolve(eq, hint='2nd_power_series_ordinary', x0=-2) == sol + # FIXME: Solution should be O((x+2)**6) + # assert checkodesol(eq, sol) == (True, 0) + + sol = Eq(f(x), C2*x + C1 + O(x**2)) + assert dsolve(eq, hint='2nd_power_series_ordinary', n=2) == sol + assert checkodesol(eq, sol) == (True, 0) + + eq = (1 + x**2)*(f(x).diff(x, 2)) + 2*x*(f(x).diff(x)) -2*f(x) + assert classify_ode(eq) == ('factorable', '2nd_hypergeometric', '2nd_hypergeometric_Integral', + '2nd_power_series_ordinary') + + sol = Eq(f(x), C2*(-x**4/3 + x**2 + 1) + C1*x + O(x**6)) + assert dsolve(eq, hint='2nd_power_series_ordinary') == sol + assert checkodesol(eq, sol) == (True, 0) + + eq = f(x).diff(x, 2) + x*(f(x).diff(x)) + f(x) + assert classify_ode(eq) == ('factorable', '2nd_power_series_ordinary',) + sol = Eq(f(x), C2*(x**4/8 - x**2/2 + 1) + C1*x*(-x**2/3 + 1) + O(x**6)) + assert dsolve(eq) == sol + # FIXME: checkodesol fails for this solution... + # assert checkodesol(eq, sol) == (True, 0) + + eq = f(x).diff(x, 2) + f(x).diff(x) - x*f(x) + assert classify_ode(eq) == ('2nd_power_series_ordinary',) + sol = Eq(f(x), C2*(-x**4/24 + x**3/6 + 1) + + C1*x*(x**3/24 + x**2/6 - x/2 + 1) + O(x**6)) + assert dsolve(eq) == sol + # FIXME: checkodesol fails for this solution... + # assert checkodesol(eq, sol) == (True, 0) + + eq = f(x).diff(x, 2) + x*f(x) + assert classify_ode(eq) == ('2nd_linear_airy', '2nd_power_series_ordinary') + sol = Eq(f(x), C2*(x**6/180 - x**3/6 + 1) + C1*x*(-x**3/12 + 1) + O(x**7)) + assert dsolve(eq, hint='2nd_power_series_ordinary', n=7) == sol + assert checkodesol(eq, sol) == (True, 0) + + +def test_2nd_power_series_regular(): + C1, C2, a = symbols("C1 C2 a") + eq = x**2*(f(x).diff(x, 2)) - 3*x*(f(x).diff(x)) + (4*x + 4)*f(x) + sol = Eq(f(x), C1*x**2*(-16*x**3/9 + 4*x**2 - 4*x + 1) + O(x**6)) + assert dsolve(eq, hint='2nd_power_series_regular') == sol + assert checkodesol(eq, sol) == (True, 0) + + eq = 4*x**2*(f(x).diff(x, 2)) -8*x**2*(f(x).diff(x)) + (4*x**2 + + 1)*f(x) + sol = Eq(f(x), C1*sqrt(x)*(x**4/24 + x**3/6 + x**2/2 + x + 1) + O(x**6)) + assert dsolve(eq, hint='2nd_power_series_regular') == sol + assert checkodesol(eq, sol) == (True, 0) + + eq = x**2*(f(x).diff(x, 2)) - x**2*(f(x).diff(x)) + ( + x**2 - 2)*f(x) + sol = Eq(f(x), C1*(-x**6/720 - 3*x**5/80 - x**4/8 + x**2/2 + x/2 + 1)/x + + C2*x**2*(-x**3/60 + x**2/20 + x/2 + 1) + O(x**6)) + assert dsolve(eq) == sol + assert checkodesol(eq, sol) == (True, 0) + + eq = x**2*(f(x).diff(x, 2)) + x*(f(x).diff(x)) + (x**2 - Rational(1, 4))*f(x) + sol = Eq(f(x), C1*(x**4/24 - x**2/2 + 1)/sqrt(x) + + C2*sqrt(x)*(x**4/120 - x**2/6 + 1) + O(x**6)) + assert dsolve(eq, hint='2nd_power_series_regular') == sol + assert checkodesol(eq, sol) == (True, 0) + + eq = x*f(x).diff(x, 2) + f(x).diff(x) - a*x*f(x) + sol = Eq(f(x), C1*(a**2*x**4/64 + a*x**2/4 + 1) + O(x**6)) + assert dsolve(eq, f(x), hint="2nd_power_series_regular") == sol + assert checkodesol(eq, sol) == (True, 0) + + eq = f(x).diff(x, 2) + ((1 - x)/x)*f(x).diff(x) + (a/x)*f(x) + sol = Eq(f(x), C1*(-a*x**5*(a - 4)*(a - 3)*(a - 2)*(a - 1)/14400 + \ + a*x**4*(a - 3)*(a - 2)*(a - 1)/576 - a*x**3*(a - 2)*(a - 1)/36 + \ + a*x**2*(a - 1)/4 - a*x + 1) + O(x**6)) + assert dsolve(eq, f(x), hint="2nd_power_series_regular") == sol + assert checkodesol(eq, sol) == (True, 0) + + +def test_issue_15056(): + t = Symbol('t') + C3 = Symbol('C3') + assert get_numbered_constants(Symbol('C1') * Function('C2')(t)) == C3 + + +def test_issue_15913(): + eq = -C1/x - 2*x*f(x) - f(x) + Derivative(f(x), x) + sol = C2*exp(x**2 + x) + exp(x**2 + x)*Integral(C1*exp(-x**2 - x)/x, x) + assert checkodesol(eq, sol) == (True, 0) + sol = C1 + C2*exp(-x*y) + eq = Derivative(y*f(x), x) + f(x).diff(x, 2) + assert checkodesol(eq, sol, f(x)) == (True, 0) + + +def test_issue_16146(): + raises(ValueError, lambda: dsolve([f(x).diff(x), g(x).diff(x)], [f(x), g(x), h(x)])) + raises(ValueError, lambda: dsolve([f(x).diff(x), g(x).diff(x)], [f(x)])) + + +def test_dsolve_remove_redundant_solutions(): + + eq = (f(x)-2)*f(x).diff(x) + sol = Eq(f(x), C1) + assert dsolve(eq) == sol + + eq = (f(x)-sin(x))*(f(x).diff(x, 2)) + sol = {Eq(f(x), C1 + C2*x), Eq(f(x), sin(x))} + assert set(dsolve(eq)) == sol + + eq = (f(x)**2-2*f(x)+1)*f(x).diff(x, 3) + sol = Eq(f(x), C1 + C2*x + C3*x**2) + assert dsolve(eq) == sol + + +def test_issue_13060(): + A, B = symbols("A B", cls=Function) + t = Symbol("t") + eq = [Eq(Derivative(A(t), t), A(t)*B(t)), Eq(Derivative(B(t), t), A(t)*B(t))] + sol = dsolve(eq) + assert checkodesol(eq, sol) == (True, [0, 0]) + + +def test_issue_22523(): + N, s = symbols('N s') + rho = Function('rho') + # intentionally use 4.0 to confirm issue with nfloat + # works here + eqn = 4.0*N*sqrt(N - 1)*rho(s) + (4*s**2*(N - 1) + (N - 2*s*(N - 1))**2 + )*Derivative(rho(s), (s, 2)) + match = classify_ode(eqn, dict=True, hint='all') + assert match['2nd_power_series_ordinary']['terms'] == 5 + C1, C2 = symbols('C1,C2') + sol = dsolve(eqn, hint='2nd_power_series_ordinary') + # there is no r(2.0) in this result + assert filldedent(sol) == filldedent(str(''' + Eq(rho(s), C2*(1 - 4.0*s**4*sqrt(N - 1.0)/N + 0.666666666666667*s**4/N + - 2.66666666666667*s**3*sqrt(N - 1.0)/N - 2.0*s**2*sqrt(N - 1.0)/N + + 9.33333333333333*s**4*sqrt(N - 1.0)/N**2 - 0.666666666666667*s**4/N**2 + + 2.66666666666667*s**3*sqrt(N - 1.0)/N**2 - + 5.33333333333333*s**4*sqrt(N - 1.0)/N**3) + C1*s*(1.0 - + 1.33333333333333*s**3*sqrt(N - 1.0)/N - 0.666666666666667*s**2*sqrt(N + - 1.0)/N + 1.33333333333333*s**3*sqrt(N - 1.0)/N**2) + O(s**6))''')) + + +def test_issue_22604(): + x1, x2 = symbols('x1, x2', cls = Function) + t, k1, k2, m1, m2 = symbols('t k1 k2 m1 m2', real = True) + k1, k2, m1, m2 = 1, 1, 1, 1 + eq1 = Eq(m1*diff(x1(t), t, 2) + k1*x1(t) - k2*(x2(t) - x1(t)), 0) + eq2 = Eq(m2*diff(x2(t), t, 2) + k2*(x2(t) - x1(t)), 0) + eqs = [eq1, eq2] + [x1sol, x2sol] = dsolve(eqs, [x1(t), x2(t)], ics = {x1(0):0, x1(t).diff().subs(t,0):0, \ + x2(0):1, x2(t).diff().subs(t,0):0}) + assert x1sol == Eq(x1(t), sqrt(3 - sqrt(5))*(sqrt(10) + 5*sqrt(2))*cos(sqrt(2)*t*sqrt(3 - sqrt(5))/2)/20 + \ + (-5*sqrt(2) + sqrt(10))*sqrt(sqrt(5) + 3)*cos(sqrt(2)*t*sqrt(sqrt(5) + 3)/2)/20) + assert x2sol == Eq(x2(t), (sqrt(5) + 5)*cos(sqrt(2)*t*sqrt(3 - sqrt(5))/2)/10 + (5 - sqrt(5))*cos(sqrt(2)*t*sqrt(sqrt(5) + 3)/2)/10) + + +def test_issue_22462(): + for de in [ + Eq(f(x).diff(x), -20*f(x)**2 - 500*f(x)/7200), + Eq(f(x).diff(x), -2*f(x)**2 - 5*f(x)/7)]: + assert 'Bernoulli' in classify_ode(de, f(x)) + + +def test_issue_23425(): + x = symbols('x') + y = Function('y') + eq = Eq(-E**x*y(x).diff().diff() + y(x).diff(), 0) + assert classify_ode(eq) == \ + ('Liouville', 'nth_order_reducible', \ + '2nd_power_series_ordinary', 'Liouville_Integral') + + +@SKIP("too slow for @slow") +def test_issue_25820(): + x = Symbol('x') + y = Function('y') + eq = y(x)**3*y(x).diff(x, 2) + 49 + assert dsolve(eq, y(x)) is not None # doesn't raise diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/ode/tests/test_riccati.py b/.venv/lib/python3.12/site-packages/sympy/solvers/ode/tests/test_riccati.py new file mode 100644 index 0000000000000000000000000000000000000000..548a1ee5b5e82d88f1b0aa319af09b8b9d1d9bfe --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/ode/tests/test_riccati.py @@ -0,0 +1,877 @@ +from sympy.core.random import randint +from sympy.core.function import Function +from sympy.core.mul import Mul +from sympy.core.numbers import (I, Rational, oo) +from sympy.core.relational import Eq +from sympy.core.singleton import S +from sympy.core.symbol import (Dummy, symbols) +from sympy.functions.elementary.exponential import (exp, log) +from sympy.functions.elementary.hyperbolic import tanh +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import sin +from sympy.polys.polytools import Poly +from sympy.simplify.ratsimp import ratsimp +from sympy.solvers.ode.subscheck import checkodesol +from sympy.testing.pytest import slow +from sympy.solvers.ode.riccati import (riccati_normal, riccati_inverse_normal, + riccati_reduced, match_riccati, inverse_transform_poly, limit_at_inf, + check_necessary_conds, val_at_inf, construct_c_case_1, + construct_c_case_2, construct_c_case_3, construct_d_case_4, + construct_d_case_5, construct_d_case_6, rational_laurent_series, + solve_riccati) + +f = Function('f') +x = symbols('x') + +# These are the functions used to generate the tests +# SHOULD NOT BE USED DIRECTLY IN TESTS + +def rand_rational(maxint): + return Rational(randint(-maxint, maxint), randint(1, maxint)) + + +def rand_poly(x, degree, maxint): + return Poly([rand_rational(maxint) for _ in range(degree+1)], x) + + +def rand_rational_function(x, degree, maxint): + degnum = randint(1, degree) + degden = randint(1, degree) + num = rand_poly(x, degnum, maxint) + den = rand_poly(x, degden, maxint) + while den == Poly(0, x): + den = rand_poly(x, degden, maxint) + return num / den + + +def find_riccati_ode(ratfunc, x, yf): + y = ratfunc + yp = y.diff(x) + q1 = rand_rational_function(x, 1, 3) + q2 = rand_rational_function(x, 1, 3) + while q2 == 0: + q2 = rand_rational_function(x, 1, 3) + q0 = ratsimp(yp - q1*y - q2*y**2) + eq = Eq(yf.diff(), q0 + q1*yf + q2*yf**2) + sol = Eq(yf, y) + assert checkodesol(eq, sol) == (True, 0) + return eq, q0, q1, q2 + + +# Testing functions start + +def test_riccati_transformation(): + """ + This function tests the transformation of the + solution of a Riccati ODE to the solution of + its corresponding normal Riccati ODE. + + Each test case 4 values - + + 1. w - The solution to be transformed + 2. b1 - The coefficient of f(x) in the ODE. + 3. b2 - The coefficient of f(x)**2 in the ODE. + 4. y - The solution to the normal Riccati ODE. + """ + tests = [ + ( + x/(x - 1), + (x**2 + 7)/3*x, + x, + -x**2/(x - 1) - x*(x**2/3 + S(7)/3)/2 - 1/(2*x) + ), + ( + (2*x + 3)/(2*x + 2), + (3 - 3*x)/(x + 1), + 5*x, + -5*x*(2*x + 3)/(2*x + 2) - (3 - 3*x)/(Mul(2, x + 1, evaluate=False)) - 1/(2*x) + ), + ( + -1/(2*x**2 - 1), + 0, + (2 - x)/(4*x - 2), + (2 - x)/((4*x - 2)*(2*x**2 - 1)) - (4*x - 2)*(Mul(-4, 2 - x, evaluate=False)/(4*x - \ + 2)**2 - 1/(4*x - 2))/(Mul(2, 2 - x, evaluate=False)) + ), + ( + x, + (8*x - 12)/(12*x + 9), + x**3/(6*x - 9), + -x**4/(6*x - 9) - (8*x - 12)/(Mul(2, 12*x + 9, evaluate=False)) - (6*x - 9)*(-6*x**3/(6*x \ + - 9)**2 + 3*x**2/(6*x - 9))/(2*x**3) + )] + for w, b1, b2, y in tests: + assert y == riccati_normal(w, x, b1, b2) + assert w == riccati_inverse_normal(y, x, b1, b2).cancel() + + # Test bp parameter in riccati_inverse_normal + tests = [ + ( + (-2*x - 1)/(2*x**2 + 2*x - 2), + -2/x, + (-x - 1)/(4*x), + 8*x**2*(1/(4*x) + (-x - 1)/(4*x**2))/(-x - 1)**2 + 4/(-x - 1), + -2*x*(-1/(4*x) - (-x - 1)/(4*x**2))/(-x - 1) - (-2*x - 1)*(-x - 1)/(4*x*(2*x**2 + 2*x \ + - 2)) + 1/x + ), + ( + 3/(2*x**2), + -2/x, + (-x - 1)/(4*x), + 8*x**2*(1/(4*x) + (-x - 1)/(4*x**2))/(-x - 1)**2 + 4/(-x - 1), + -2*x*(-1/(4*x) - (-x - 1)/(4*x**2))/(-x - 1) + 1/x - Mul(3, -x - 1, evaluate=False)/(8*x**3) + )] + for w, b1, b2, bp, y in tests: + assert y == riccati_normal(w, x, b1, b2) + assert w == riccati_inverse_normal(y, x, b1, b2, bp).cancel() + + +def test_riccati_reduced(): + """ + This function tests the transformation of a + Riccati ODE to its normal Riccati ODE. + + Each test case 2 values - + + 1. eq - A Riccati ODE. + 2. normal_eq - The normal Riccati ODE of eq. + """ + tests = [ + ( + f(x).diff(x) - x**2 - x*f(x) - x*f(x)**2, + + f(x).diff(x) + f(x)**2 + x**3 - x**2/4 - 3/(4*x**2) + ), + ( + 6*x/(2*x + 9) + f(x).diff(x) - (x + 1)*f(x)**2/x, + + -3*x**2*(1/x + (-x - 1)/x**2)**2/(4*(-x - 1)**2) + Mul(6, \ + -x - 1, evaluate=False)/(2*x + 9) + f(x)**2 + f(x).diff(x) \ + - (-1 + (x + 1)/x)/(x*(-x - 1)) + ), + ( + f(x)**2 + f(x).diff(x) - (x - 1)*f(x)/(-x - S(1)/2), + + -(2*x - 2)**2/(4*(2*x + 1)**2) + (2*x - 2)/(2*x + 1)**2 + \ + f(x)**2 + f(x).diff(x) - 1/(2*x + 1) + ), + ( + f(x).diff(x) - f(x)**2/x, + + f(x)**2 + f(x).diff(x) + 1/(4*x**2) + ), + ( + -3*(-x**2 - x + 1)/(x**2 + 6*x + 1) + f(x).diff(x) + f(x)**2/x, + + f(x)**2 + f(x).diff(x) + (3*x**2/(x**2 + 6*x + 1) + 3*x/(x**2 \ + + 6*x + 1) - 3/(x**2 + 6*x + 1))/x + 1/(4*x**2) + ), + ( + 6*x/(2*x + 9) + f(x).diff(x) - (x + 1)*f(x)/x, + + False + ), + ( + f(x)*f(x).diff(x) - 1/x + f(x)/3 + f(x)**2/(x**2 - 2), + + False + )] + for eq, normal_eq in tests: + assert normal_eq == riccati_reduced(eq, f, x) + + +def test_match_riccati(): + """ + This function tests if an ODE is Riccati or not. + + Each test case has 5 values - + + 1. eq - The Riccati ODE. + 2. match - Boolean indicating if eq is a Riccati ODE. + 3. b0 - + 4. b1 - Coefficient of f(x) in eq. + 5. b2 - Coefficient of f(x)**2 in eq. + """ + tests = [ + # Test Rational Riccati ODEs + ( + f(x).diff(x) - (405*x**3 - 882*x**2 - 78*x + 92)/(243*x**4 \ + - 945*x**3 + 846*x**2 + 180*x - 72) - 2 - f(x)**2/(3*x + 1) \ + - (S(1)/3 - x)*f(x)/(S(1)/3 - 3*x/2), + + True, + + 45*x**3/(27*x**4 - 105*x**3 + 94*x**2 + 20*x - 8) - 98*x**2/ \ + (27*x**4 - 105*x**3 + 94*x**2 + 20*x - 8) - 26*x/(81*x**4 - \ + 315*x**3 + 282*x**2 + 60*x - 24) + 2 + 92/(243*x**4 - 945*x**3 \ + + 846*x**2 + 180*x - 72), + + Mul(-1, 2 - 6*x, evaluate=False)/(9*x - 2), + + 1/(3*x + 1) + ), + ( + f(x).diff(x) + 4*x/27 - (x/3 - 1)*f(x)**2 - (2*x/3 + \ + 1)*f(x)/(3*x + 2) - S(10)/27 - (265*x**2 + 423*x + 162) \ + /(324*x**3 + 216*x**2), + + True, + + -4*x/27 + S(10)/27 + 3/(6*x**3 + 4*x**2) + 47/(36*x**2 \ + + 24*x) + 265/(324*x + 216), + + Mul(-1, -2*x - 3, evaluate=False)/(9*x + 6), + + x/3 - 1 + ), + ( + f(x).diff(x) - (304*x**5 - 745*x**4 + 631*x**3 - 876*x**2 \ + + 198*x - 108)/(36*x**6 - 216*x**5 + 477*x**4 - 567*x**3 + \ + 360*x**2 - 108*x) - S(17)/9 - (x - S(3)/2)*f(x)/(x/2 - \ + S(3)/2) - (x/3 - 3)*f(x)**2/(3*x), + + True, + + 304*x**4/(36*x**5 - 216*x**4 + 477*x**3 - 567*x**2 + 360*x - \ + 108) - 745*x**3/(36*x**5 - 216*x**4 + 477*x**3 - 567*x**2 + \ + 360*x - 108) + 631*x**2/(36*x**5 - 216*x**4 + 477*x**3 - 567* \ + x**2 + 360*x - 108) - 292*x/(12*x**5 - 72*x**4 + 159*x**3 - \ + 189*x**2 + 120*x - 36) + S(17)/9 - 12/(4*x**6 - 24*x**5 + \ + 53*x**4 - 63*x**3 + 40*x**2 - 12*x) + 22/(4*x**5 - 24*x**4 \ + + 53*x**3 - 63*x**2 + 40*x - 12), + + Mul(-1, 3 - 2*x, evaluate=False)/(x - 3), + + Mul(-1, 9 - x, evaluate=False)/(9*x) + ), + # Test Non-Rational Riccati ODEs + ( + f(x).diff(x) - x**(S(3)/2)/(x**(S(1)/2) - 2) + x**2*f(x) + \ + x*f(x)**2/(x**(S(3)/4)), + False, 0, 0, 0 + ), + ( + f(x).diff(x) - sin(x**2) + exp(x)*f(x) + log(x)*f(x)**2, + False, 0, 0, 0 + ), + ( + f(x).diff(x) - tanh(x + sqrt(x)) + f(x) + x**4*f(x)**2, + False, 0, 0, 0 + ), + # Test Non-Riccati ODEs + ( + (1 - x**2)*f(x).diff(x, 2) - 2*x*f(x).diff(x) + 20*f(x), + False, 0, 0, 0 + ), + ( + f(x).diff(x) - x**2 + x**3*f(x) + (x**2/(x + 1))*f(x)**3, + False, 0, 0, 0 + ), + ( + f(x).diff(x)*f(x)**2 + (x**2 - 1)/(x**3 + 1)*f(x) + 1/(2*x \ + + 3) + f(x)**2, + False, 0, 0, 0 + )] + for eq, res, b0, b1, b2 in tests: + match, funcs = match_riccati(eq, f, x) + assert match == res + if res: + assert [b0, b1, b2] == funcs + + +def test_val_at_inf(): + """ + This function tests the valuation of rational + function at oo. + + Each test case has 3 values - + + 1. num - Numerator of rational function. + 2. den - Denominator of rational function. + 3. val_inf - Valuation of rational function at oo + """ + tests = [ + # degree(denom) > degree(numer) + ( + Poly(10*x**3 + 8*x**2 - 13*x + 6, x), + Poly(-13*x**10 - x**9 + 5*x**8 + 7*x**7 + 10*x**6 + 6*x**5 - 7*x**4 + 11*x**3 - 8*x**2 + 5*x + 13, x), + 7 + ), + ( + Poly(1, x), + Poly(-9*x**4 + 3*x**3 + 15*x**2 - 6*x - 14, x), + 4 + ), + # degree(denom) == degree(numer) + ( + Poly(-6*x**3 - 8*x**2 + 8*x - 6, x), + Poly(-5*x**3 + 12*x**2 - 6*x - 9, x), + 0 + ), + # degree(denom) < degree(numer) + ( + Poly(12*x**8 - 12*x**7 - 11*x**6 + 8*x**5 + 3*x**4 - x**3 + x**2 - 11*x, x), + Poly(-14*x**2 + x, x), + -6 + ), + ( + Poly(5*x**6 + 9*x**5 - 11*x**4 - 9*x**3 + x**2 - 4*x + 4, x), + Poly(15*x**4 + 3*x**3 - 8*x**2 + 15*x + 12, x), + -2 + )] + for num, den, val in tests: + assert val_at_inf(num, den, x) == val + + +def test_necessary_conds(): + """ + This function tests the necessary conditions for + a Riccati ODE to have a rational particular solution. + """ + # Valuation at Infinity is an odd negative integer + assert check_necessary_conds(-3, [1, 2, 4]) == False + # Valuation at Infinity is a positive integer lesser than 2 + assert check_necessary_conds(1, [1, 2, 4]) == False + # Multiplicity of a pole is an odd integer greater than 1 + assert check_necessary_conds(2, [3, 1, 6]) == False + # All values are correct + assert check_necessary_conds(-10, [1, 2, 8, 12]) == True + + +def test_inverse_transform_poly(): + """ + This function tests the substitution x -> 1/x + in rational functions represented using Poly. + """ + fns = [ + (15*x**3 - 8*x**2 - 2*x - 6)/(18*x + 6), + + (180*x**5 + 40*x**4 + 80*x**3 + 30*x**2 - 60*x - 80)/(180*x**3 - 150*x**2 + 75*x + 12), + + (-15*x**5 - 36*x**4 + 75*x**3 - 60*x**2 - 80*x - 60)/(80*x**4 + 60*x**3 + 60*x**2 + 60*x - 80), + + (60*x**7 + 24*x**6 - 15*x**5 - 20*x**4 + 30*x**2 + 100*x - 60)/(240*x**2 - 20*x - 30), + + (30*x**6 - 12*x**5 + 15*x**4 - 15*x**2 + 10*x + 60)/(3*x**10 - 45*x**9 + 15*x**5 + 15*x**4 - 5*x**3 \ + + 15*x**2 + 45*x - 15) + ] + for f in fns: + num, den = [Poly(e, x) for e in f.as_numer_denom()] + num, den = inverse_transform_poly(num, den, x) + assert f.subs(x, 1/x).cancel() == num/den + + +def test_limit_at_inf(): + """ + This function tests the limit at oo of a + rational function. + + Each test case has 3 values - + + 1. num - Numerator of rational function. + 2. den - Denominator of rational function. + 3. limit_at_inf - Limit of rational function at oo + """ + tests = [ + # deg(denom) > deg(numer) + ( + Poly(-12*x**2 + 20*x + 32, x), + Poly(32*x**3 + 72*x**2 + 3*x - 32, x), + 0 + ), + # deg(denom) < deg(numer) + ( + Poly(1260*x**4 - 1260*x**3 - 700*x**2 - 1260*x + 1400, x), + Poly(6300*x**3 - 1575*x**2 + 756*x - 540, x), + oo + ), + # deg(denom) < deg(numer), one of the leading coefficients is negative + ( + Poly(-735*x**8 - 1400*x**7 + 1680*x**6 - 315*x**5 - 600*x**4 + 840*x**3 - 525*x**2 \ + + 630*x + 3780, x), + Poly(1008*x**7 - 2940*x**6 - 84*x**5 + 2940*x**4 - 420*x**3 + 1512*x**2 + 105*x + 168, x), + -oo + ), + # deg(denom) == deg(numer) + ( + Poly(105*x**7 - 960*x**6 + 60*x**5 + 60*x**4 - 80*x**3 + 45*x**2 + 120*x + 15, x), + Poly(735*x**7 + 525*x**6 + 720*x**5 + 720*x**4 - 8400*x**3 - 2520*x**2 + 2800*x + 280, x), + S(1)/7 + ), + ( + Poly(288*x**4 - 450*x**3 + 280*x**2 - 900*x - 90, x), + Poly(607*x**4 + 840*x**3 - 1050*x**2 + 420*x + 420, x), + S(288)/607 + )] + for num, den, lim in tests: + assert limit_at_inf(num, den, x) == lim + + +def test_construct_c_case_1(): + """ + This function tests the Case 1 in the step + to calculate coefficients of c-vectors. + + Each test case has 4 values - + + 1. num - Numerator of the rational function a(x). + 2. den - Denominator of the rational function a(x). + 3. pole - Pole of a(x) for which c-vector is being + calculated. + 4. c - The c-vector for the pole. + """ + tests = [ + ( + Poly(-3*x**3 + 3*x**2 + 4*x - 5, x, extension=True), + Poly(4*x**8 + 16*x**7 + 9*x**5 + 12*x**4 + 6*x**3 + 12*x**2, x, extension=True), + S(0), + [[S(1)/2 + sqrt(6)*I/6], [S(1)/2 - sqrt(6)*I/6]] + ), + ( + Poly(1200*x**3 + 1440*x**2 + 816*x + 560, x, extension=True), + Poly(128*x**5 - 656*x**4 + 1264*x**3 - 1125*x**2 + 385*x + 49, x, extension=True), + S(7)/4, + [[S(1)/2 + sqrt(16367978)/634], [S(1)/2 - sqrt(16367978)/634]] + ), + ( + Poly(4*x + 2, x, extension=True), + Poly(18*x**4 + (2 - 18*sqrt(3))*x**3 + (14 - 11*sqrt(3))*x**2 + (4 - 6*sqrt(3))*x \ + + 8*sqrt(3) + 16, x, domain='QQ'), + (S(1) + sqrt(3))/2, + [[S(1)/2 + sqrt(Mul(4, 2*sqrt(3) + 4, evaluate=False)/(19*sqrt(3) + 44) + 1)/2], \ + [S(1)/2 - sqrt(Mul(4, 2*sqrt(3) + 4, evaluate=False)/(19*sqrt(3) + 44) + 1)/2]] + )] + for num, den, pole, c in tests: + assert construct_c_case_1(num, den, x, pole) == c + + +def test_construct_c_case_2(): + """ + This function tests the Case 2 in the step + to calculate coefficients of c-vectors. + + Each test case has 5 values - + + 1. num - Numerator of the rational function a(x). + 2. den - Denominator of the rational function a(x). + 3. pole - Pole of a(x) for which c-vector is being + calculated. + 4. mul - The multiplicity of the pole. + 5. c - The c-vector for the pole. + """ + tests = [ + # Testing poles with multiplicity 2 + ( + Poly(1, x, extension=True), + Poly((x - 1)**2*(x - 2), x, extension=True), + 1, 2, + [[-I*(-1 - I)/2], [I*(-1 + I)/2]] + ), + ( + Poly(3*x**5 - 12*x**4 - 7*x**3 + 1, x, extension=True), + Poly((3*x - 1)**2*(x + 2)**2, x, extension=True), + S(1)/3, 2, + [[-S(89)/98], [-S(9)/98]] + ), + # Testing poles with multiplicity 4 + ( + Poly(x**3 - x**2 + 4*x, x, extension=True), + Poly((x - 2)**4*(x + 5)**2, x, extension=True), + 2, 4, + [[7*sqrt(3)*(S(60)/343 - 4*sqrt(3)/7)/12, 2*sqrt(3)/7], \ + [-7*sqrt(3)*(S(60)/343 + 4*sqrt(3)/7)/12, -2*sqrt(3)/7]] + ), + ( + Poly(3*x**5 + x**4 + 3, x, extension=True), + Poly((4*x + 1)**4*(x + 2), x, extension=True), + -S(1)/4, 4, + [[128*sqrt(439)*(-sqrt(439)/128 - S(55)/14336)/439, sqrt(439)/256], \ + [-128*sqrt(439)*(sqrt(439)/128 - S(55)/14336)/439, -sqrt(439)/256]] + ), + # Testing poles with multiplicity 6 + ( + Poly(x**3 + 2, x, extension=True), + Poly((3*x - 1)**6*(x**2 + 1), x, extension=True), + S(1)/3, 6, + [[27*sqrt(66)*(-sqrt(66)/54 - S(131)/267300)/22, -2*sqrt(66)/1485, sqrt(66)/162], \ + [-27*sqrt(66)*(sqrt(66)/54 - S(131)/267300)/22, 2*sqrt(66)/1485, -sqrt(66)/162]] + ), + ( + Poly(x**2 + 12, x, extension=True), + Poly((x - sqrt(2))**6, x, extension=True), + sqrt(2), 6, + [[sqrt(14)*(S(6)/7 - 3*sqrt(14))/28, sqrt(7)/7, sqrt(14)], \ + [-sqrt(14)*(S(6)/7 + 3*sqrt(14))/28, -sqrt(7)/7, -sqrt(14)]] + )] + for num, den, pole, mul, c in tests: + assert construct_c_case_2(num, den, x, pole, mul) == c + + +def test_construct_c_case_3(): + """ + This function tests the Case 3 in the step + to calculate coefficients of c-vectors. + """ + assert construct_c_case_3() == [[1]] + + +def test_construct_d_case_4(): + """ + This function tests the Case 4 in the step + to calculate coefficients of the d-vector. + + Each test case has 4 values - + + 1. num - Numerator of the rational function a(x). + 2. den - Denominator of the rational function a(x). + 3. mul - Multiplicity of oo as a pole. + 4. d - The d-vector. + """ + tests = [ + # Tests with multiplicity at oo = 2 + ( + Poly(-x**5 - 2*x**4 + 4*x**3 + 2*x + 5, x, extension=True), + Poly(9*x**3 - 2*x**2 + 10*x - 2, x, extension=True), + 2, + [[10*I/27, I/3, -3*I*(S(158)/243 - I/3)/2], \ + [-10*I/27, -I/3, 3*I*(S(158)/243 + I/3)/2]] + ), + ( + Poly(-x**6 + 9*x**5 + 5*x**4 + 6*x**3 + 5*x**2 + 6*x + 7, x, extension=True), + Poly(x**4 + 3*x**3 + 12*x**2 - x + 7, x, extension=True), + 2, + [[-6*I, I, -I*(17 - I)/2], [6*I, -I, I*(17 + I)/2]] + ), + # Tests with multiplicity at oo = 4 + ( + Poly(-2*x**6 - x**5 - x**4 - 2*x**3 - x**2 - 3*x - 3, x, extension=True), + Poly(3*x**2 + 10*x + 7, x, extension=True), + 4, + [[269*sqrt(6)*I/288, -17*sqrt(6)*I/36, sqrt(6)*I/3, -sqrt(6)*I*(S(16969)/2592 \ + - 2*sqrt(6)*I/3)/4], [-269*sqrt(6)*I/288, 17*sqrt(6)*I/36, -sqrt(6)*I/3, \ + sqrt(6)*I*(S(16969)/2592 + 2*sqrt(6)*I/3)/4]] + ), + ( + Poly(-3*x**5 - 3*x**4 - 3*x**3 - x**2 - 1, x, extension=True), + Poly(12*x - 2, x, extension=True), + 4, + [[41*I/192, 7*I/24, I/2, -I*(-S(59)/6912 - I)], \ + [-41*I/192, -7*I/24, -I/2, I*(-S(59)/6912 + I)]] + ), + # Tests with multiplicity at oo = 4 + ( + Poly(-x**7 - x**5 - x**4 - x**2 - x, x, extension=True), + Poly(x + 2, x, extension=True), + 6, + [[-5*I/2, 2*I, -I, I, -I*(-9 - 3*I)/2], [5*I/2, -2*I, I, -I, I*(-9 + 3*I)/2]] + ), + ( + Poly(-x**7 - x**6 - 2*x**5 - 2*x**4 - x**3 - x**2 + 2*x - 2, x, extension=True), + Poly(2*x - 2, x, extension=True), + 6, + [[3*sqrt(2)*I/4, 3*sqrt(2)*I/4, sqrt(2)*I/2, sqrt(2)*I/2, -sqrt(2)*I*(-S(7)/8 - \ + 3*sqrt(2)*I/2)/2], [-3*sqrt(2)*I/4, -3*sqrt(2)*I/4, -sqrt(2)*I/2, -sqrt(2)*I/2, \ + sqrt(2)*I*(-S(7)/8 + 3*sqrt(2)*I/2)/2]] + )] + for num, den, mul, d in tests: + ser = rational_laurent_series(num, den, x, oo, mul, 1) + assert construct_d_case_4(ser, mul//2) == d + + +def test_construct_d_case_5(): + """ + This function tests the Case 5 in the step + to calculate coefficients of the d-vector. + + Each test case has 3 values - + + 1. num - Numerator of the rational function a(x). + 2. den - Denominator of the rational function a(x). + 3. d - The d-vector. + """ + tests = [ + ( + Poly(2*x**3 + x**2 + x - 2, x, extension=True), + Poly(9*x**3 + 5*x**2 + 2*x - 1, x, extension=True), + [[sqrt(2)/3, -sqrt(2)/108], [-sqrt(2)/3, sqrt(2)/108]] + ), + ( + Poly(3*x**5 + x**4 - x**3 + x**2 - 2*x - 2, x, domain='ZZ'), + Poly(9*x**5 + 7*x**4 + 3*x**3 + 2*x**2 + 5*x + 7, x, domain='ZZ'), + [[sqrt(3)/3, -2*sqrt(3)/27], [-sqrt(3)/3, 2*sqrt(3)/27]] + ), + ( + Poly(x**2 - x + 1, x, domain='ZZ'), + Poly(3*x**2 + 7*x + 3, x, domain='ZZ'), + [[sqrt(3)/3, -5*sqrt(3)/9], [-sqrt(3)/3, 5*sqrt(3)/9]] + )] + for num, den, d in tests: + # Multiplicity of oo is 0 + ser = rational_laurent_series(num, den, x, oo, 0, 1) + assert construct_d_case_5(ser) == d + + +def test_construct_d_case_6(): + """ + This function tests the Case 6 in the step + to calculate coefficients of the d-vector. + + Each test case has 3 values - + + 1. num - Numerator of the rational function a(x). + 2. den - Denominator of the rational function a(x). + 3. d - The d-vector. + """ + tests = [ + ( + Poly(-2*x**2 - 5, x, domain='ZZ'), + Poly(4*x**4 + 2*x**2 + 10*x + 2, x, domain='ZZ'), + [[S(1)/2 + I/2], [S(1)/2 - I/2]] + ), + ( + Poly(-2*x**3 - 4*x**2 - 2*x - 5, x, domain='ZZ'), + Poly(x**6 - x**5 + 2*x**4 - 4*x**3 - 5*x**2 - 5*x + 9, x, domain='ZZ'), + [[1], [0]] + ), + ( + Poly(-5*x**3 + x**2 + 11*x + 12, x, domain='ZZ'), + Poly(6*x**8 - 26*x**7 - 27*x**6 - 10*x**5 - 44*x**4 - 46*x**3 - 34*x**2 \ + - 27*x - 42, x, domain='ZZ'), + [[1], [0]] + )] + for num, den, d in tests: + assert construct_d_case_6(num, den, x) == d + + +def test_rational_laurent_series(): + """ + This function tests the computation of coefficients + of Laurent series of a rational function. + + Each test case has 5 values - + + 1. num - Numerator of the rational function. + 2. den - Denominator of the rational function. + 3. x0 - Point about which Laurent series is to + be calculated. + 4. mul - Multiplicity of x0 if x0 is a pole of + the rational function (0 otherwise). + 5. n - Number of terms upto which the series + is to be calculated. + """ + tests = [ + # Laurent series about simple pole (Multiplicity = 1) + ( + Poly(x**2 - 3*x + 9, x, extension=True), + Poly(x**2 - x, x, extension=True), + S(1), 1, 6, + {1: 7, 0: -8, -1: 9, -2: -9, -3: 9, -4: -9} + ), + # Laurent series about multiple pole (Multiplicity > 1) + ( + Poly(64*x**3 - 1728*x + 1216, x, extension=True), + Poly(64*x**4 - 80*x**3 - 831*x**2 + 1809*x - 972, x, extension=True), + S(9)/8, 2, 3, + {0: S(32177152)/46521675, 2: S(1019)/984, -1: S(11947565056)/28610830125, \ + 1: S(209149)/75645} + ), + ( + Poly(1, x, extension=True), + Poly(x**5 + (-4*sqrt(2) - 1)*x**4 + (4*sqrt(2) + 12)*x**3 + (-12 - 8*sqrt(2))*x**2 \ + + (4 + 8*sqrt(2))*x - 4, x, extension=True), + sqrt(2), 4, 6, + {4: 1 + sqrt(2), 3: -3 - 2*sqrt(2), 2: Mul(-1, -3 - 2*sqrt(2), evaluate=False)/(-1 \ + + sqrt(2)), 1: (-3 - 2*sqrt(2))/(-1 + sqrt(2))**2, 0: Mul(-1, -3 - 2*sqrt(2), evaluate=False \ + )/(-1 + sqrt(2))**3, -1: (-3 - 2*sqrt(2))/(-1 + sqrt(2))**4} + ), + # Laurent series about oo + ( + Poly(x**5 - 4*x**3 + 6*x**2 + 10*x - 13, x, extension=True), + Poly(x**2 - 5, x, extension=True), + oo, 3, 6, + {3: 1, 2: 0, 1: 1, 0: 6, -1: 15, -2: 17} + ), + # Laurent series at x0 where x0 is not a pole of the function + # Using multiplicity as 0 (as x0 will not be a pole) + ( + Poly(3*x**3 + 6*x**2 - 2*x + 5, x, extension=True), + Poly(9*x**4 - x**3 - 3*x**2 + 4*x + 4, x, extension=True), + S(2)/5, 0, 1, + {0: S(3345)/3304, -1: S(399325)/2729104, -2: S(3926413375)/4508479808, \ + -3: S(-5000852751875)/1862002160704, -4: S(-6683640101653125)/6152055138966016} + ), + ( + Poly(-7*x**2 + 2*x - 4, x, extension=True), + Poly(7*x**5 + 9*x**4 + 8*x**3 + 3*x**2 + 6*x + 9, x, extension=True), + oo, 0, 6, + {0: 0, -2: 0, -5: -S(71)/49, -1: 0, -3: -1, -4: S(11)/7} + )] + for num, den, x0, mul, n, ser in tests: + assert ser == rational_laurent_series(num, den, x, x0, mul, n) + + +def check_dummy_sol(eq, solse, dummy_sym): + """ + Helper function to check if actual solution + matches expected solution if actual solution + contains dummy symbols. + """ + if isinstance(eq, Eq): + eq = eq.lhs - eq.rhs + _, funcs = match_riccati(eq, f, x) + + sols = solve_riccati(f(x), x, *funcs) + C1 = Dummy('C1') + sols = [sol.subs(C1, dummy_sym) for sol in sols] + + assert all(x[0] for x in checkodesol(eq, sols)) + assert all(s1.dummy_eq(s2, dummy_sym) for s1, s2 in zip(sols, solse)) + + +def test_solve_riccati(): + """ + This function tests the computation of rational + particular solutions for a Riccati ODE. + + Each test case has 2 values - + + 1. eq - Riccati ODE to be solved. + 2. sol - Expected solution to the equation. + + Some examples have been taken from the paper - "Statistical Investigation of + First-Order Algebraic ODEs and their Rational General Solutions" by + Georg Grasegger, N. Thieu Vo, Franz Winkler + + https://www3.risc.jku.at/publications/download/risc_5197/RISCReport15-19.pdf + """ + C0 = Dummy('C0') + # Type: 1st Order Rational Riccati, dy/dx = a + b*y + c*y**2, + # a, b, c are rational functions of x + + tests = [ + # a(x) is a constant + ( + Eq(f(x).diff(x) + f(x)**2 - 2, 0), + [Eq(f(x), sqrt(2)), Eq(f(x), -sqrt(2))] + ), + # a(x) is a constant + ( + f(x)**2 + f(x).diff(x) + 4*f(x)/x + 2/x**2, + [Eq(f(x), (-2*C0 - x)/(C0*x + x**2))] + ), + # a(x) is a constant + ( + 2*x**2*f(x).diff(x) - x*(4*f(x) + f(x).diff(x) - 4) + (f(x) - 1)*f(x), + [Eq(f(x), (C0 + 2*x**2)/(C0 + x))] + ), + # Pole with multiplicity 1 + ( + Eq(f(x).diff(x), -f(x)**2 - 2/(x**3 - x**2)), + [Eq(f(x), 1/(x**2 - x))] + ), + # One pole of multiplicity 2 + ( + x**2 - (2*x + 1/x)*f(x) + f(x)**2 + f(x).diff(x), + [Eq(f(x), (C0*x + x**3 + 2*x)/(C0 + x**2)), Eq(f(x), x)] + ), + ( + x**4*f(x).diff(x) + x**2 - x*(2*f(x)**2 + f(x).diff(x)) + f(x), + [Eq(f(x), (C0*x**2 + x)/(C0 + x**2)), Eq(f(x), x**2)] + ), + # Multiple poles of multiplicity 2 + ( + -f(x)**2 + f(x).diff(x) + (15*x**2 - 20*x + 7)/((x - 1)**2*(2*x \ + - 1)**2), + [Eq(f(x), (9*C0*x - 6*C0 - 15*x**5 + 60*x**4 - 94*x**3 + 72*x**2 \ + - 30*x + 6)/(6*C0*x**2 - 9*C0*x + 3*C0 + 6*x**6 - 29*x**5 + \ + 57*x**4 - 58*x**3 + 30*x**2 - 6*x)), Eq(f(x), (3*x - 2)/(2*x**2 \ + - 3*x + 1))] + ), + # Regression: Poles with even multiplicity > 2 fixed + ( + f(x)**2 + f(x).diff(x) - (4*x**6 - 8*x**5 + 12*x**4 + 4*x**3 + \ + 7*x**2 - 20*x + 4)/(4*x**4), + [Eq(f(x), (2*x**5 - 2*x**4 - x**3 + 4*x**2 + 3*x - 2)/(2*x**4 \ + - 2*x**2))] + ), + # Regression: Poles with even multiplicity > 2 fixed + ( + Eq(f(x).diff(x), (-x**6 + 15*x**4 - 40*x**3 + 45*x**2 - 24*x + 4)/\ + (x**12 - 12*x**11 + 66*x**10 - 220*x**9 + 495*x**8 - 792*x**7 + 924*x**6 - \ + 792*x**5 + 495*x**4 - 220*x**3 + 66*x**2 - 12*x + 1) + f(x)**2 + f(x)), + [Eq(f(x), 1/(x**6 - 6*x**5 + 15*x**4 - 20*x**3 + 15*x**2 - 6*x + 1))] + ), + # More than 2 poles with multiplicity 2 + # Regression: Fixed mistake in necessary conditions + ( + Eq(f(x).diff(x), x*f(x) + 2*x + (3*x - 2)*f(x)**2/(4*x + 2) + \ + (8*x**2 - 7*x + 26)/(16*x**3 - 24*x**2 + 8) - S(3)/2), + [Eq(f(x), (1 - 4*x)/(2*x - 2))] + ), + # Regression: Fixed mistake in necessary conditions + ( + Eq(f(x).diff(x), (-12*x**2 - 48*x - 15)/(24*x**3 - 40*x**2 + 8*x + 8) \ + + 3*f(x)**2/(6*x + 2)), + [Eq(f(x), (2*x + 1)/(2*x - 2))] + ), + # Imaginary poles + ( + f(x).diff(x) + (3*x**2 + 1)*f(x)**2/x + (6*x**2 - x + 3)*f(x)/(x*(x \ + - 1)) + (3*x**2 - 2*x + 2)/(x*(x - 1)**2), + [Eq(f(x), (-C0 - x**3 + x**2 - 2*x)/(C0*x - C0 + x**4 - x**3 + x**2 \ + - x)), Eq(f(x), -1/(x - 1))], + ), + # Imaginary coefficients in equation + ( + f(x).diff(x) - 2*I*(f(x)**2 + 1)/x, + [Eq(f(x), (-I*C0 + I*x**4)/(C0 + x**4)), Eq(f(x), -I)] + ), + # Regression: linsolve returning empty solution + # Large value of m (> 10) + ( + Eq(f(x).diff(x), x*f(x)/(S(3)/2 - 2*x) + (x/2 - S(1)/3)*f(x)**2/\ + (2*x/3 - S(1)/2) - S(5)/4 + (281*x**2 - 1260*x + 756)/(16*x**3 - 12*x**2)), + [Eq(f(x), (9 - x)/x), Eq(f(x), (40*x**14 + 28*x**13 + 420*x**12 + 2940*x**11 + \ + 18480*x**10 + 103950*x**9 + 519750*x**8 + 2286900*x**7 + 8731800*x**6 + 28378350*\ + x**5 + 76403250*x**4 + 163721250*x**3 + 261954000*x**2 + 278326125*x + 147349125)/\ + ((24*x**14 + 140*x**13 + 840*x**12 + 4620*x**11 + 23100*x**10 + 103950*x**9 + \ + 415800*x**8 + 1455300*x**7 + 4365900*x**6 + 10914750*x**5 + 21829500*x**4 + 32744250\ + *x**3 + 32744250*x**2 + 16372125*x)))] + ), + # Regression: Fixed bug due to a typo in paper + ( + Eq(f(x).diff(x), 18*x**3 + 18*x**2 + (-x/2 - S(1)/2)*f(x)**2 + 6), + [Eq(f(x), 6*x)] + ), + # Regression: Fixed bug due to a typo in paper + ( + Eq(f(x).diff(x), -3*x**3/4 + 15*x/2 + (x/3 - S(4)/3)*f(x)**2 \ + + 9 + (1 - x)*f(x)/x + 3/x), + [Eq(f(x), -3*x/2 - 3)] + )] + for eq, sol in tests: + check_dummy_sol(eq, sol, C0) + + +@slow +def test_solve_riccati_slow(): + """ + This function tests the computation of rational + particular solutions for a Riccati ODE. + + Each test case has 2 values - + + 1. eq - Riccati ODE to be solved. + 2. sol - Expected solution to the equation. + """ + C0 = Dummy('C0') + tests = [ + # Very large values of m (989 and 991) + ( + Eq(f(x).diff(x), (1 - x)*f(x)/(x - 3) + (2 - 12*x)*f(x)**2/(2*x - 9) + \ + (54924*x**3 - 405264*x**2 + 1084347*x - 1087533)/(8*x**4 - 132*x**3 + 810*x**2 - \ + 2187*x + 2187) + 495), + [Eq(f(x), (18*x + 6)/(2*x - 9))] + )] + for eq, sol in tests: + check_dummy_sol(eq, sol, C0) diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/ode/tests/test_single.py b/.venv/lib/python3.12/site-packages/sympy/solvers/ode/tests/test_single.py new file mode 100644 index 0000000000000000000000000000000000000000..45b38029e97b9e74236c45f2f3efb6aa87c26e5c --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/ode/tests/test_single.py @@ -0,0 +1,2902 @@ +# +# The main tests for the code in single.py are currently located in +# sympy/solvers/tests/test_ode.py +# +r""" +This File contains test functions for the individual hints used for solving ODEs. + +Examples of each solver will be returned by _get_examples_ode_sol_name_of_solver. + +Examples should have a key 'XFAIL' which stores the list of hints if they are +expected to fail for that hint. + +Functions that are for internal use: + +1) _ode_solver_test(ode_examples) - It takes a dictionary of examples returned by + _get_examples method and tests them with their respective hints. + +2) _test_particular_example(our_hint, example_name) - It tests the ODE example corresponding + to the hint provided. + +3) _test_all_hints(runxfail=False) - It is used to test all the examples with all the hints + currently implemented. It calls _test_all_examples_for_one_hint() which outputs whether the + given hint functions properly if it classifies the ODE example. + If runxfail flag is set to True then it will only test the examples which are expected to fail. + + Everytime the ODE of a particular solver is added, _test_all_hints() is to be executed to find + the possible failures of different solver hints. + +4) _test_all_examples_for_one_hint(our_hint, all_examples) - It takes hint as argument and checks + this hint against all the ODE examples and gives output as the number of ODEs matched, number + of ODEs which were solved correctly, list of ODEs which gives incorrect solution and list of + ODEs which raises exception. + +""" +from sympy.core.function import (Derivative, diff) +from sympy.core.mul import Mul +from sympy.core.numbers import (E, I, Rational, pi) +from sympy.core.relational import (Eq, Ne) +from sympy.core.singleton import S +from sympy.core.symbol import (Dummy, symbols) +from sympy.functions.elementary.complexes import (im, re) +from sympy.functions.elementary.exponential import (LambertW, exp, log) +from sympy.functions.elementary.hyperbolic import (asinh, cosh, sinh, tanh) +from sympy.functions.elementary.miscellaneous import (cbrt, sqrt) +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.elementary.trigonometric import (acos, asin, atan, cos, sec, sin, tan) +from sympy.functions.special.error_functions import (Ei, erfi) +from sympy.functions.special.hyper import hyper +from sympy.integrals.integrals import (Integral, integrate) +from sympy.polys.rootoftools import rootof + +from sympy.core import Function, Symbol +from sympy.functions import airyai, airybi, besselj, bessely, lowergamma +from sympy.integrals.risch import NonElementaryIntegral +from sympy.solvers.ode import classify_ode, dsolve +from sympy.solvers.ode.ode import allhints, _remove_redundant_solutions +from sympy.solvers.ode.single import (FirstLinear, ODEMatchError, + SingleODEProblem, SingleODESolver, NthOrderReducible) + +from sympy.solvers.ode.subscheck import checkodesol + +from sympy.testing.pytest import raises, slow +import traceback + + +x = Symbol('x') +u = Symbol('u') +_u = Dummy('u') +y = Symbol('y') +f = Function('f') +g = Function('g') +C1, C2, C3, C4, C5, C6, C7, C8, C9, C10 = symbols('C1:11') +a, b, c = symbols('a b c') + + +hint_message = """\ +Hint did not match the example {example}. + +The ODE is: +{eq}. + +The expected hint was +{our_hint}\ +""" + +expected_sol_message = """\ +Different solution found from dsolve for example {example}. + +The ODE is: +{eq} + +The expected solution was +{sol} + +What dsolve returned is: +{dsolve_sol}\ +""" + +checkodesol_msg = """\ +solution found is not correct for example {example}. + +The ODE is: +{eq}\ +""" + +dsol_incorrect_msg = """\ +solution returned by dsolve is incorrect when using {hint}. + +The ODE is: +{eq} + +The expected solution was +{sol} + +what dsolve returned is: +{dsolve_sol} + +You can test this with: + +eq = {eq} +sol = dsolve(eq, hint='{hint}') +print(sol) +print(checkodesol(eq, sol)) + +""" + +exception_msg = """\ +dsolve raised exception : {e} + +when using {hint} for the example {example} + +You can test this with: + +from sympy.solvers.ode.tests.test_single import _test_an_example + +_test_an_example('{hint}', example_name = '{example}') + +The ODE is: +{eq} + +\ +""" + +check_hint_msg = """\ +Tested hint was : {hint} + +Total of {matched} examples matched with this hint. + +Out of which {solve} gave correct results. + +Examples which gave incorrect results are {unsolve}. + +Examples which raised exceptions are {exceptions} +\ +""" + + +def _add_example_keys(func): + def inner(): + solver=func() + examples=[] + for example in solver['examples']: + temp={ + 'eq': solver['examples'][example]['eq'], + 'sol': solver['examples'][example]['sol'], + 'XFAIL': solver['examples'][example].get('XFAIL', []), + 'func': solver['examples'][example].get('func',solver['func']), + 'example_name': example, + 'slow': solver['examples'][example].get('slow', False), + 'simplify_flag':solver['examples'][example].get('simplify_flag',True), + 'checkodesol_XFAIL': solver['examples'][example].get('checkodesol_XFAIL', False), + 'dsolve_too_slow':solver['examples'][example].get('dsolve_too_slow',False), + 'checkodesol_too_slow':solver['examples'][example].get('checkodesol_too_slow',False), + 'hint': solver['hint'] + } + examples.append(temp) + return examples + return inner() + + +def _ode_solver_test(ode_examples, run_slow_test=False): + for example in ode_examples: + if ((not run_slow_test) and example['slow']) or (run_slow_test and (not example['slow'])): + continue + + result = _test_particular_example(example['hint'], example, solver_flag=True) + if result['xpass_msg'] != "": + print(result['xpass_msg']) + + +def _test_all_hints(runxfail=False): + all_hints = list(allhints)+["default"] + all_examples = _get_all_examples() + + for our_hint in all_hints: + if our_hint.endswith('_Integral') or 'series' in our_hint: + continue + _test_all_examples_for_one_hint(our_hint, all_examples, runxfail) + + +def _test_dummy_sol(expected_sol,dsolve_sol): + if type(dsolve_sol)==list: + return any(expected_sol.dummy_eq(sub_dsol) for sub_dsol in dsolve_sol) + else: + return expected_sol.dummy_eq(dsolve_sol) + + +def _test_an_example(our_hint, example_name): + all_examples = _get_all_examples() + for example in all_examples: + if example['example_name'] == example_name: + _test_particular_example(our_hint, example) + + +def _test_particular_example(our_hint, ode_example, solver_flag=False): + eq = ode_example['eq'] + expected_sol = ode_example['sol'] + example = ode_example['example_name'] + xfail = our_hint in ode_example['XFAIL'] + func = ode_example['func'] + result = {'msg': '', 'xpass_msg': ''} + simplify_flag=ode_example['simplify_flag'] + checkodesol_XFAIL = ode_example['checkodesol_XFAIL'] + dsolve_too_slow = ode_example['dsolve_too_slow'] + checkodesol_too_slow = ode_example['checkodesol_too_slow'] + xpass = True + if solver_flag: + if our_hint not in classify_ode(eq, func): + message = hint_message.format(example=example, eq=eq, our_hint=our_hint) + raise AssertionError(message) + + if our_hint in classify_ode(eq, func): + result['match_list'] = example + try: + if not (dsolve_too_slow): + dsolve_sol = dsolve(eq, func, simplify=simplify_flag,hint=our_hint) + else: + if len(expected_sol)==1: + dsolve_sol = expected_sol[0] + else: + dsolve_sol = expected_sol + + except Exception as e: + dsolve_sol = [] + result['exception_list'] = example + if not solver_flag: + traceback.print_exc() + result['msg'] = exception_msg.format(e=str(e), hint=our_hint, example=example, eq=eq) + if solver_flag and not xfail: + print(result['msg']) + raise + xpass = False + + if solver_flag and dsolve_sol!=[]: + expect_sol_check = False + if type(dsolve_sol)==list: + for sub_sol in expected_sol: + if sub_sol.has(Dummy): + expect_sol_check = not _test_dummy_sol(sub_sol, dsolve_sol) + else: + expect_sol_check = sub_sol not in dsolve_sol + if expect_sol_check: + break + else: + expect_sol_check = dsolve_sol not in expected_sol + for sub_sol in expected_sol: + if sub_sol.has(Dummy): + expect_sol_check = not _test_dummy_sol(sub_sol, dsolve_sol) + + if expect_sol_check: + message = expected_sol_message.format(example=example, eq=eq, sol=expected_sol, dsolve_sol=dsolve_sol) + raise AssertionError(message) + + expected_checkodesol = [(True, 0) for i in range(len(expected_sol))] + if len(expected_sol) == 1: + expected_checkodesol = (True, 0) + + if not checkodesol_too_slow: + if not checkodesol_XFAIL: + if checkodesol(eq, dsolve_sol, func, solve_for_func=False) != expected_checkodesol: + result['unsolve_list'] = example + xpass = False + message = dsol_incorrect_msg.format(hint=our_hint, eq=eq, sol=expected_sol,dsolve_sol=dsolve_sol) + if solver_flag: + message = checkodesol_msg.format(example=example, eq=eq) + raise AssertionError(message) + else: + result['msg'] = 'AssertionError: ' + message + + if xpass and xfail: + result['xpass_msg'] = example + "is now passing for the hint" + our_hint + return result + + +def _test_all_examples_for_one_hint(our_hint, all_examples=[], runxfail=None): + if all_examples == []: + all_examples = _get_all_examples() + match_list, unsolve_list, exception_list = [], [], [] + for ode_example in all_examples: + xfail = our_hint in ode_example['XFAIL'] + if runxfail and not xfail: + continue + if xfail: + continue + result = _test_particular_example(our_hint, ode_example) + match_list += result.get('match_list',[]) + unsolve_list += result.get('unsolve_list',[]) + exception_list += result.get('exception_list',[]) + if runxfail is not None: + msg = result['msg'] + if msg!='': + print(result['msg']) + # print(result.get('xpass_msg','')) + if runxfail is None: + match_count = len(match_list) + solved = len(match_list)-len(unsolve_list)-len(exception_list) + msg = check_hint_msg.format(hint=our_hint, matched=match_count, solve=solved, unsolve=unsolve_list, exceptions=exception_list) + print(msg) + + +def test_SingleODESolver(): + # Test that not implemented methods give NotImplementedError + # Subclasses should override these methods. + problem = SingleODEProblem(f(x).diff(x), f(x), x) + solver = SingleODESolver(problem) + raises(NotImplementedError, lambda: solver.matches()) + raises(NotImplementedError, lambda: solver.get_general_solution()) + raises(NotImplementedError, lambda: solver._matches()) + raises(NotImplementedError, lambda: solver._get_general_solution()) + + # This ODE can not be solved by the FirstLinear solver. Here we test that + # it does not match and the asking for a general solution gives + # ODEMatchError + + problem = SingleODEProblem(f(x).diff(x) + f(x)*f(x), f(x), x) + + solver = FirstLinear(problem) + raises(ODEMatchError, lambda: solver.get_general_solution()) + + solver = FirstLinear(problem) + assert solver.matches() is False + + #These are just test for order of ODE + + problem = SingleODEProblem(f(x).diff(x) + f(x), f(x), x) + assert problem.order == 1 + + problem = SingleODEProblem(f(x).diff(x,4) + f(x).diff(x,2) - f(x).diff(x,3), f(x), x) + assert problem.order == 4 + + problem = SingleODEProblem(f(x).diff(x, 3) + f(x).diff(x, 2) - f(x)**2, f(x), x) + assert problem.is_autonomous == True + + problem = SingleODEProblem(f(x).diff(x, 3) + x*f(x).diff(x, 2) - f(x)**2, f(x), x) + assert problem.is_autonomous == False + + +def test_linear_coefficients(): + _ode_solver_test(_get_examples_ode_sol_linear_coefficients) + + +@slow +def test_1st_homogeneous_coeff_ode(): + #These were marked as test_1st_homogeneous_coeff_corner_case + eq1 = f(x).diff(x) - f(x)/x + c1 = classify_ode(eq1, f(x)) + eq2 = x*f(x).diff(x) - f(x) + c2 = classify_ode(eq2, f(x)) + sdi = "1st_homogeneous_coeff_subs_dep_div_indep" + sid = "1st_homogeneous_coeff_subs_indep_div_dep" + assert sid not in c1 and sdi not in c1 + assert sid not in c2 and sdi not in c2 + _ode_solver_test(_get_examples_ode_sol_1st_homogeneous_coeff_subs_dep_div_indep) + _ode_solver_test(_get_examples_ode_sol_1st_homogeneous_coeff_best) + + +@slow +def test_slow_examples_1st_homogeneous_coeff_ode(): + _ode_solver_test(_get_examples_ode_sol_1st_homogeneous_coeff_subs_dep_div_indep, run_slow_test=True) + _ode_solver_test(_get_examples_ode_sol_1st_homogeneous_coeff_best, run_slow_test=True) + + +@slow +def test_nth_linear_constant_coeff_homogeneous(): + _ode_solver_test(_get_examples_ode_sol_nth_linear_constant_coeff_homogeneous) + + +@slow +def test_slow_examples_nth_linear_constant_coeff_homogeneous(): + _ode_solver_test(_get_examples_ode_sol_nth_linear_constant_coeff_homogeneous, run_slow_test=True) + + +def test_Airy_equation(): + _ode_solver_test(_get_examples_ode_sol_2nd_linear_airy) + + +@slow +def test_lie_group(): + _ode_solver_test(_get_examples_ode_sol_lie_group) + + +@slow +def test_separable_reduced(): + df = f(x).diff(x) + eq = (x / f(x))*df + tan(x**2*f(x) / (x**2*f(x) - 1)) + assert classify_ode(eq) == ('factorable', 'separable_reduced', 'lie_group', + 'separable_reduced_Integral') + _ode_solver_test(_get_examples_ode_sol_separable_reduced) + + +@slow +def test_slow_examples_separable_reduced(): + _ode_solver_test(_get_examples_ode_sol_separable_reduced, run_slow_test=True) + + +@slow +def test_2nd_2F1_hypergeometric(): + _ode_solver_test(_get_examples_ode_sol_2nd_2F1_hypergeometric) + + +def test_2nd_2F1_hypergeometric_integral(): + eq = x*(x-1)*f(x).diff(x, 2) + (-1+ S(7)/2*x)*f(x).diff(x) + f(x) + sol = Eq(f(x), (C1 + C2*Integral(exp(Integral((1 - x/2)/(x*(x - 1)), x))/(1 - + x/2)**2, x))*exp(Integral(1/(x - 1), x)/4)*exp(-Integral(7/(x - + 1), x)/4)*hyper((S(1)/2, -1), (1,), x)) + assert sol == dsolve(eq, hint='2nd_hypergeometric_Integral') + assert checkodesol(eq, sol) == (True, 0) + + +@slow +def test_2nd_nonlinear_autonomous_conserved(): + _ode_solver_test(_get_examples_ode_sol_2nd_nonlinear_autonomous_conserved) + + +def test_2nd_nonlinear_autonomous_conserved_integral(): + eq = f(x).diff(x, 2) + asin(f(x)) + actual = [Eq(Integral(1/sqrt(C1 - 2*Integral(asin(_u), _u)), (_u, f(x))), C2 + x), + Eq(Integral(1/sqrt(C1 - 2*Integral(asin(_u), _u)), (_u, f(x))), C2 - x)] + solved = dsolve(eq, hint='2nd_nonlinear_autonomous_conserved_Integral', simplify=False) + for a,s in zip(actual, solved): + assert a.dummy_eq(s) + # checkodesol unable to simplify solutions with f(x) in an integral equation + assert checkodesol(eq, [s.doit() for s in solved]) == [(True, 0), (True, 0)] + + +@slow +def test_2nd_linear_bessel_equation(): + _ode_solver_test(_get_examples_ode_sol_2nd_linear_bessel) + + +@slow +def test_nth_algebraic(): + eqn = f(x) + f(x)*f(x).diff(x) + solns = [Eq(f(x), exp(x)), + Eq(f(x), C1*exp(C2*x))] + solns_final = _remove_redundant_solutions(eqn, solns, 2, x) + assert solns_final == [Eq(f(x), C1*exp(C2*x))] + + _ode_solver_test(_get_examples_ode_sol_nth_algebraic) + + +@slow +def test_slow_examples_nth_linear_constant_coeff_var_of_parameters(): + _ode_solver_test(_get_examples_ode_sol_nth_linear_var_of_parameters, run_slow_test=True) + + +def test_nth_linear_constant_coeff_var_of_parameters(): + _ode_solver_test(_get_examples_ode_sol_nth_linear_var_of_parameters) + + +@slow +def test_nth_linear_constant_coeff_variation_of_parameters__integral(): + # solve_variation_of_parameters shouldn't attempt to simplify the + # Wronskian if simplify=False. If wronskian() ever gets good enough + # to simplify the result itself, this test might fail. + our_hint = 'nth_linear_constant_coeff_variation_of_parameters_Integral' + eq = f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x) - 2*x - exp(I*x) + sol_simp = dsolve(eq, f(x), hint=our_hint, simplify=True) + sol_nsimp = dsolve(eq, f(x), hint=our_hint, simplify=False) + assert sol_simp != sol_nsimp + assert checkodesol(eq, sol_simp, order=5, solve_for_func=False) == (True, 0) + assert checkodesol(eq, sol_simp, order=5, solve_for_func=False) == (True, 0) + + +@slow +def test_slow_examples_1st_exact(): + _ode_solver_test(_get_examples_ode_sol_1st_exact, run_slow_test=True) + + +@slow +def test_1st_exact(): + _ode_solver_test(_get_examples_ode_sol_1st_exact) + + +def test_1st_exact_integral(): + eq = cos(f(x)) - (x*sin(f(x)) - f(x)**2)*f(x).diff(x) + sol_1 = dsolve(eq, f(x), simplify=False, hint='1st_exact_Integral') + assert checkodesol(eq, sol_1, order=1, solve_for_func=False) + + +@slow +def test_slow_examples_nth_order_reducible(): + _ode_solver_test(_get_examples_ode_sol_nth_order_reducible, run_slow_test=True) + + +@slow +def test_slow_examples_nth_linear_constant_coeff_undetermined_coefficients(): + _ode_solver_test(_get_examples_ode_sol_nth_linear_undetermined_coefficients, run_slow_test=True) + + +@slow +def test_slow_examples_separable(): + _ode_solver_test(_get_examples_ode_sol_separable, run_slow_test=True) + + +@slow +def test_nth_linear_constant_coeff_undetermined_coefficients(): + #issue-https://github.com/sympy/sympy/issues/5787 + # This test case is to show the classification of imaginary constants under + # nth_linear_constant_coeff_undetermined_coefficients + eq = Eq(diff(f(x), x), I*f(x) + S.Half - I) + our_hint = 'nth_linear_constant_coeff_undetermined_coefficients' + assert our_hint in classify_ode(eq) + _ode_solver_test(_get_examples_ode_sol_nth_linear_undetermined_coefficients) + + +def test_nth_order_reducible(): + F = lambda eq: NthOrderReducible(SingleODEProblem(eq, f(x), x))._matches() + D = Derivative + assert F(D(y*f(x), x, y) + D(f(x), x)) == False + assert F(D(y*f(y), y, y) + D(f(y), y)) == False + assert F(f(x)*D(f(x), x) + D(f(x), x, 2))== False + assert F(D(x*f(y), y, 2) + D(u*y*f(x), x, 3)) == False # no simplification by design + assert F(D(f(y), y, 2) + D(f(y), y, 3) + D(f(x), x, 4)) == False + assert F(D(f(x), x, 2) + D(f(x), x, 3)) == True + _ode_solver_test(_get_examples_ode_sol_nth_order_reducible) + + +@slow +def test_separable(): + _ode_solver_test(_get_examples_ode_sol_separable) + + +@slow +def test_factorable(): + assert integrate(-asin(f(2*x)+pi), x) == -Integral(asin(pi + f(2*x)), x) + _ode_solver_test(_get_examples_ode_sol_factorable) + + +@slow +def test_slow_examples_factorable(): + _ode_solver_test(_get_examples_ode_sol_factorable, run_slow_test=True) + + +def test_Riccati_special_minus2(): + _ode_solver_test(_get_examples_ode_sol_riccati) + + +@slow +def test_1st_rational_riccati(): + _ode_solver_test(_get_examples_ode_sol_1st_rational_riccati) + + +def test_Bernoulli(): + _ode_solver_test(_get_examples_ode_sol_bernoulli) + + +def test_1st_linear(): + _ode_solver_test(_get_examples_ode_sol_1st_linear) + + +def test_almost_linear(): + _ode_solver_test(_get_examples_ode_sol_almost_linear) + + +@slow +def test_Liouville_ODE(): + hint = 'Liouville' + not_Liouville1 = classify_ode(diff(f(x), x)/x + f(x)*diff(f(x), x, x)/2 - + diff(f(x), x)**2/2, f(x)) + not_Liouville2 = classify_ode(diff(f(x), x)/x + diff(f(x), x, x)/2 - + x*diff(f(x), x)**2/2, f(x)) + assert hint not in not_Liouville1 + assert hint not in not_Liouville2 + assert hint + '_Integral' not in not_Liouville1 + assert hint + '_Integral' not in not_Liouville2 + + _ode_solver_test(_get_examples_ode_sol_liouville) + + +def test_nth_order_linear_euler_eq_homogeneous(): + x, t, a, b, c = symbols('x t a b c') + y = Function('y') + our_hint = "nth_linear_euler_eq_homogeneous" + + eq = diff(f(t), t, 4)*t**4 - 13*diff(f(t), t, 2)*t**2 + 36*f(t) + assert our_hint in classify_ode(eq) + + eq = a*y(t) + b*t*diff(y(t), t) + c*t**2*diff(y(t), t, 2) + assert our_hint in classify_ode(eq) + + _ode_solver_test(_get_examples_ode_sol_euler_homogeneous) + + +def test_nth_order_linear_euler_eq_nonhomogeneous_undetermined_coefficients(): + x, t = symbols('x t') + a, b, c, d = symbols('a b c d', integer=True) + our_hint = "nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients" + + eq = x**4*diff(f(x), x, 4) - 13*x**2*diff(f(x), x, 2) + 36*f(x) + x + assert our_hint in classify_ode(eq, f(x)) + + eq = a*x**2*diff(f(x), x, 2) + b*x*diff(f(x), x) + c*f(x) + d*log(x) + assert our_hint in classify_ode(eq, f(x)) + + _ode_solver_test(_get_examples_ode_sol_euler_undetermined_coeff) + + +@slow +def test_nth_order_linear_euler_eq_nonhomogeneous_variation_of_parameters(): + x, t = symbols('x, t') + a, b, c, d = symbols('a, b, c, d', integer=True) + our_hint = "nth_linear_euler_eq_nonhomogeneous_variation_of_parameters" + + eq = Eq(x**2*diff(f(x),x,2) - 8*x*diff(f(x),x) + 12*f(x), x**2) + assert our_hint in classify_ode(eq, f(x)) + + eq = Eq(a*x**3*diff(f(x),x,3) + b*x**2*diff(f(x),x,2) + c*x*diff(f(x),x) + d*f(x), x*log(x)) + assert our_hint in classify_ode(eq, f(x)) + + _ode_solver_test(_get_examples_ode_sol_euler_var_para) + + +@_add_example_keys +def _get_examples_ode_sol_euler_homogeneous(): + r1, r2, r3, r4, r5 = [rootof(x**5 - 14*x**4 + 71*x**3 - 154*x**2 + 120*x - 1, n) for n in range(5)] + return { + 'hint': "nth_linear_euler_eq_homogeneous", + 'func': f(x), + 'examples':{ + 'euler_hom_01': { + 'eq': Eq(-3*diff(f(x), x)*x + 2*x**2*diff(f(x), x, x), 0), + 'sol': [Eq(f(x), C1 + C2*x**Rational(5, 2))], + }, + + 'euler_hom_02': { + 'eq': Eq(3*f(x) - 5*diff(f(x), x)*x + 2*x**2*diff(f(x), x, x), 0), + 'sol': [Eq(f(x), C1*sqrt(x) + C2*x**3)] + }, + + 'euler_hom_03': { + 'eq': Eq(4*f(x) + 5*diff(f(x), x)*x + x**2*diff(f(x), x, x), 0), + 'sol': [Eq(f(x), (C1 + C2*log(x))/x**2)] + }, + + 'euler_hom_04': { + 'eq': Eq(6*f(x) - 6*diff(f(x), x)*x + 1*x**2*diff(f(x), x, x) + x**3*diff(f(x), x, x, x), 0), + 'sol': [Eq(f(x), C1/x**2 + C2*x + C3*x**3)] + }, + + 'euler_hom_05': { + 'eq': Eq(-125*f(x) + 61*diff(f(x), x)*x - 12*x**2*diff(f(x), x, x) + x**3*diff(f(x), x, x, x), 0), + 'sol': [Eq(f(x), x**5*(C1 + C2*log(x) + C3*log(x)**2))] + }, + + 'euler_hom_06': { + 'eq': x**2*diff(f(x), x, 2) + x*diff(f(x), x) - 9*f(x), + 'sol': [Eq(f(x), C1*x**-3 + C2*x**3)] + }, + + 'euler_hom_07': { + 'eq': sin(x)*x**2*f(x).diff(x, 2) + sin(x)*x*f(x).diff(x) + sin(x)*f(x), + 'sol': [Eq(f(x), C1*sin(log(x)) + C2*cos(log(x)))], + 'XFAIL': ['2nd_power_series_regular','nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients'] + }, + + 'euler_hom_08': { + 'eq': x**6 * f(x).diff(x, 6) - x*f(x).diff(x) + f(x), + 'sol': [Eq(f(x), C1*x + C2*x**r1 + C3*x**r2 + C4*x**r3 + C5*x**r4 + C6*x**r5)], + 'checkodesol_XFAIL':True + }, + + #This example is from issue: https://github.com/sympy/sympy/issues/15237 #This example is from issue: + # https://github.com/sympy/sympy/issues/15237 + 'euler_hom_09': { + 'eq': Derivative(x*f(x), x, x, x), + 'sol': [Eq(f(x), C1 + C2/x + C3*x)], + }, + } + } + + +@_add_example_keys +def _get_examples_ode_sol_euler_undetermined_coeff(): + return { + 'hint': "nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients", + 'func': f(x), + 'examples':{ + 'euler_undet_01': { + 'eq': Eq(x**2*diff(f(x), x, x) + x*diff(f(x), x), 1), + 'sol': [Eq(f(x), C1 + C2*log(x) + log(x)**2/2)] + }, + + 'euler_undet_02': { + 'eq': Eq(x**2*diff(f(x), x, x) - 2*x*diff(f(x), x) + 2*f(x), x**3), + 'sol': [Eq(f(x), x*(C1 + C2*x + Rational(1, 2)*x**2))] + }, + + 'euler_undet_03': { + 'eq': Eq(x**2*diff(f(x), x, x) - x*diff(f(x), x) - 3*f(x), log(x)/x), + 'sol': [Eq(f(x), (C1 + C2*x**4 - log(x)**2/8 - log(x)/16)/x)] + }, + + 'euler_undet_04': { + 'eq': Eq(x**2*diff(f(x), x, x) + 3*x*diff(f(x), x) - 8*f(x), log(x)**3 - log(x)), + 'sol': [Eq(f(x), C1/x**4 + C2*x**2 - Rational(1,8)*log(x)**3 - Rational(3,32)*log(x)**2 - Rational(1,64)*log(x) - Rational(7, 256))] + }, + + 'euler_undet_05': { + 'eq': Eq(x**3*diff(f(x), x, x, x) - 3*x**2*diff(f(x), x, x) + 6*x*diff(f(x), x) - 6*f(x), log(x)), + 'sol': [Eq(f(x), C1*x + C2*x**2 + C3*x**3 - Rational(1, 6)*log(x) - Rational(11, 36))] + }, + + #Below examples were added for the issue: https://github.com/sympy/sympy/issues/5096 + 'euler_undet_06': { + 'eq': 2*x**2*f(x).diff(x, 2) + f(x) + sqrt(2*x)*sin(log(2*x)/2), + 'sol': [Eq(f(x), sqrt(x)*(C1*sin(log(x)/2) + C2*cos(log(x)/2) + sqrt(2)*log(x)*cos(log(2*x)/2)/2))] + }, + + 'euler_undet_07': { + 'eq': 2*x**2*f(x).diff(x, 2) + f(x) + sin(log(2*x)/2), + 'sol': [Eq(f(x), C1*sqrt(x)*sin(log(x)/2) + C2*sqrt(x)*cos(log(x)/2) - 2*sin(log(2*x)/2)/5 - 4*cos(log(2*x)/2)/5)] + }, + } + } + + +@_add_example_keys +def _get_examples_ode_sol_euler_var_para(): + return { + 'hint': "nth_linear_euler_eq_nonhomogeneous_variation_of_parameters", + 'func': f(x), + 'examples':{ + 'euler_var_01': { + 'eq': Eq(x**2*Derivative(f(x), x, x) - 2*x*Derivative(f(x), x) + 2*f(x), x**4), + 'sol': [Eq(f(x), x*(C1 + C2*x + x**3/6))] + }, + + 'euler_var_02': { + 'eq': Eq(3*x**2*diff(f(x), x, x) + 6*x*diff(f(x), x) - 6*f(x), x**3*exp(x)), + 'sol': [Eq(f(x), C1/x**2 + C2*x + x*exp(x)/3 - 4*exp(x)/3 + 8*exp(x)/(3*x) - 8*exp(x)/(3*x**2))] + }, + + 'euler_var_03': { + 'eq': Eq(x**2*Derivative(f(x), x, x) - 2*x*Derivative(f(x), x) + 2*f(x), x**4*exp(x)), + 'sol': [Eq(f(x), x*(C1 + C2*x + x*exp(x) - 2*exp(x)))] + }, + + 'euler_var_04': { + 'eq': x**2*Derivative(f(x), x, x) - 2*x*Derivative(f(x), x) + 2*f(x) - log(x), + 'sol': [Eq(f(x), C1*x + C2*x**2 + log(x)/2 + Rational(3, 4))] + }, + + 'euler_var_05': { + 'eq': -exp(x) + (x*Derivative(f(x), (x, 2)) + Derivative(f(x), x))/x, + 'sol': [Eq(f(x), C1 + C2*log(x) + exp(x) - Ei(x))] + }, + + 'euler_var_06': { + 'eq': x**2 * f(x).diff(x, 2) + x * f(x).diff(x) + 4 * f(x) - 1/x, + 'sol': [Eq(f(x), C1*sin(2*log(x)) + C2*cos(2*log(x)) + 1/(5*x))] + }, + } + } + + +@_add_example_keys +def _get_examples_ode_sol_bernoulli(): + # Type: Bernoulli, f'(x) + p(x)*f(x) == q(x)*f(x)**n + return { + 'hint': "Bernoulli", + 'func': f(x), + 'examples':{ + 'bernoulli_01': { + 'eq': Eq(x*f(x).diff(x) + f(x) - f(x)**2, 0), + 'sol': [Eq(f(x), 1/(C1*x + 1))], + 'XFAIL': ['separable_reduced'] + }, + + 'bernoulli_02': { + 'eq': f(x).diff(x) - y*f(x), + 'sol': [Eq(f(x), C1*exp(x*y))] + }, + + 'bernoulli_03': { + 'eq': f(x)*f(x).diff(x) - 1, + 'sol': [Eq(f(x), -sqrt(C1 + 2*x)), Eq(f(x), sqrt(C1 + 2*x))] + }, + } + } + + +@_add_example_keys +def _get_examples_ode_sol_riccati(): + # Type: Riccati special alpha = -2, a*dy/dx + b*y**2 + c*y/x +d/x**2 + return { + 'hint': "Riccati_special_minus2", + 'func': f(x), + 'examples':{ + 'riccati_01': { + 'eq': 2*f(x).diff(x) + f(x)**2 - f(x)/x + 3*x**(-2), + 'sol': [Eq(f(x), (-sqrt(3)*tan(C1 + sqrt(3)*log(x)/4) + 3)/(2*x))], + }, + }, + } + + +@_add_example_keys +def _get_examples_ode_sol_1st_rational_riccati(): + # Type: 1st Order Rational Riccati, dy/dx = a + b*y + c*y**2, + # a, b, c are rational functions of x + return { + 'hint': "1st_rational_riccati", + 'func': f(x), + 'examples':{ + # a(x) is a constant + "rational_riccati_01": { + "eq": Eq(f(x).diff(x) + f(x)**2 - 2, 0), + "sol": [Eq(f(x), sqrt(2)*(-C1 - exp(2*sqrt(2)*x))/(C1 - exp(2*sqrt(2)*x)))] + }, + # a(x) is a constant + "rational_riccati_02": { + "eq": f(x)**2 + Derivative(f(x), x) + 4*f(x)/x + 2/x**2, + "sol": [Eq(f(x), (-2*C1 - x)/(x*(C1 + x)))] + }, + # a(x) is a constant + "rational_riccati_03": { + "eq": 2*x**2*Derivative(f(x), x) - x*(4*f(x) + Derivative(f(x), x) - 4) + (f(x) - 1)*f(x), + "sol": [Eq(f(x), (C1 + 2*x**2)/(C1 + x))] + }, + # Constant coefficients + "rational_riccati_04": { + "eq": f(x).diff(x) - 6 - 5*f(x) - f(x)**2, + "sol": [Eq(f(x), (-2*C1 + 3*exp(x))/(C1 - exp(x)))] + }, + # One pole of multiplicity 2 + "rational_riccati_05": { + "eq": x**2 - (2*x + 1/x)*f(x) + f(x)**2 + Derivative(f(x), x), + "sol": [Eq(f(x), x*(C1 + x**2 + 1)/(C1 + x**2 - 1))] + }, + # One pole of multiplicity 2 + "rational_riccati_06": { + "eq": x**4*Derivative(f(x), x) + x**2 - x*(2*f(x)**2 + Derivative(f(x), x)) + f(x), + "sol": [Eq(f(x), x*(C1*x - x + 1)/(C1 + x**2 - 1))] + }, + # Multiple poles of multiplicity 2 + "rational_riccati_07": { + "eq": -f(x)**2 + Derivative(f(x), x) + (15*x**2 - 20*x + 7)/((x - 1)**2*(2*x \ + - 1)**2), + "sol": [Eq(f(x), (9*C1*x - 6*C1 - 15*x**5 + 60*x**4 - 94*x**3 + 72*x**2 - \ + 33*x + 8)/(6*C1*x**2 - 9*C1*x + 3*C1 + 6*x**6 - 29*x**5 + 57*x**4 - \ + 58*x**3 + 28*x**2 - 3*x - 1))] + }, + # Imaginary poles + "rational_riccati_08": { + "eq": Derivative(f(x), x) + (3*x**2 + 1)*f(x)**2/x + (6*x**2 - x + 3)*f(x)/(x*(x \ + - 1)) + (3*x**2 - 2*x + 2)/(x*(x - 1)**2), + "sol": [Eq(f(x), (-C1 - x**3 + x**2 - 2*x + 1)/(C1*x - C1 + x**4 - x**3 + x**2 - \ + 2*x + 1))], + }, + # Imaginary coefficients in equation + "rational_riccati_09": { + "eq": Derivative(f(x), x) - 2*I*(f(x)**2 + 1)/x, + "sol": [Eq(f(x), (-I*C1 + I*x**4 + I)/(C1 + x**4 - 1))] + }, + # Regression: linsolve returning empty solution + # Large value of m (> 10) + "rational_riccati_10": { + "eq": Eq(Derivative(f(x), x), x*f(x)/(S(3)/2 - 2*x) + (x/2 - S(1)/3)*f(x)**2/\ + (2*x/3 - S(1)/2) - S(5)/4 + (281*x**2 - 1260*x + 756)/(16*x**3 - 12*x**2)), + "sol": [Eq(f(x), (40*C1*x**14 + 28*C1*x**13 + 420*C1*x**12 + 2940*C1*x**11 + \ + 18480*C1*x**10 + 103950*C1*x**9 + 519750*C1*x**8 + 2286900*C1*x**7 + \ + 8731800*C1*x**6 + 28378350*C1*x**5 + 76403250*C1*x**4 + 163721250*C1*x**3 \ + + 261954000*C1*x**2 + 278326125*C1*x + 147349125*C1 + x*exp(2*x) - 9*exp(2*x) \ + )/(x*(24*C1*x**13 + 140*C1*x**12 + 840*C1*x**11 + 4620*C1*x**10 + 23100*C1*x**9 \ + + 103950*C1*x**8 + 415800*C1*x**7 + 1455300*C1*x**6 + 4365900*C1*x**5 + \ + 10914750*C1*x**4 + 21829500*C1*x**3 + 32744250*C1*x**2 + 32744250*C1*x + \ + 16372125*C1 - exp(2*x))))] + } + } + } + + + +@_add_example_keys +def _get_examples_ode_sol_1st_linear(): + # Type: first order linear form f'(x)+p(x)f(x)=q(x) + return { + 'hint': "1st_linear", + 'func': f(x), + 'examples':{ + 'linear_01': { + 'eq': Eq(f(x).diff(x) + x*f(x), x**2), + 'sol': [Eq(f(x), (C1 + x*exp(x**2/2)- sqrt(2)*sqrt(pi)*erfi(sqrt(2)*x/2)/2)*exp(-x**2/2))], + }, + }, + } + + +@_add_example_keys +def _get_examples_ode_sol_factorable(): + """ some hints are marked as xfail for examples because they missed additional algebraic solution + which could be found by Factorable hint. Fact_01 raise exception for + nth_linear_constant_coeff_undetermined_coefficients""" + + y = Dummy('y') + a0,a1,a2,a3,a4 = symbols('a0, a1, a2, a3, a4') + return { + 'hint': "factorable", + 'func': f(x), + 'examples':{ + 'fact_01': { + 'eq': f(x) + f(x)*f(x).diff(x), + 'sol': [Eq(f(x), 0), Eq(f(x), C1 - x)], + 'XFAIL': ['separable', '1st_exact', '1st_linear', 'Bernoulli', '1st_homogeneous_coeff_best', + '1st_homogeneous_coeff_subs_indep_div_dep', '1st_homogeneous_coeff_subs_dep_div_indep', + 'lie_group', 'nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients', + 'nth_linear_constant_coeff_variation_of_parameters', + 'nth_linear_euler_eq_nonhomogeneous_variation_of_parameters', + 'nth_linear_constant_coeff_undetermined_coefficients'] + }, + + 'fact_02': { + 'eq': f(x)*(f(x).diff(x)+f(x)*x+2), + 'sol': [Eq(f(x), (C1 - sqrt(2)*sqrt(pi)*erfi(sqrt(2)*x/2))*exp(-x**2/2)), Eq(f(x), 0)], + 'XFAIL': ['Bernoulli', '1st_linear', 'lie_group'] + }, + + 'fact_03': { + 'eq': (f(x).diff(x)+f(x)*x**2)*(f(x).diff(x, 2) + x*f(x)), + 'sol': [Eq(f(x), C1*airyai(-x) + C2*airybi(-x)),Eq(f(x), C1*exp(-x**3/3))] + }, + + 'fact_04': { + 'eq': (f(x).diff(x)+f(x)*x**2)*(f(x).diff(x, 2) + f(x)), + 'sol': [Eq(f(x), C1*exp(-x**3/3)), Eq(f(x), C1*sin(x) + C2*cos(x))] + }, + + 'fact_05': { + 'eq': (f(x).diff(x)**2-1)*(f(x).diff(x)**2-4), + 'sol': [Eq(f(x), C1 - x), Eq(f(x), C1 + x), Eq(f(x), C1 + 2*x), Eq(f(x), C1 - 2*x)] + }, + + 'fact_06': { + 'eq': (f(x).diff(x, 2)-exp(f(x)))*f(x).diff(x), + 'sol': [ + Eq(f(x), log(-C1/(cos(sqrt(-C1)*(C2 + x)) + 1))), + Eq(f(x), log(-C1/(cos(sqrt(-C1)*(C2 - x)) + 1))), + Eq(f(x), C1) + ], + 'slow': True, + }, + + 'fact_07': { + 'eq': (f(x).diff(x)**2-1)*(f(x)*f(x).diff(x)-1), + 'sol': [Eq(f(x), C1 - x), Eq(f(x), -sqrt(C1 + 2*x)),Eq(f(x), sqrt(C1 + 2*x)), Eq(f(x), C1 + x)] + }, + + 'fact_08': { + 'eq': Derivative(f(x), x)**4 - 2*Derivative(f(x), x)**2 + 1, + 'sol': [Eq(f(x), C1 - x), Eq(f(x), C1 + x)] + }, + + 'fact_09': { + 'eq': f(x)**2*Derivative(f(x), x)**6 - 2*f(x)**2*Derivative(f(x), + x)**4 + f(x)**2*Derivative(f(x), x)**2 - 2*f(x)*Derivative(f(x), + x)**5 + 4*f(x)*Derivative(f(x), x)**3 - 2*f(x)*Derivative(f(x), + x) + Derivative(f(x), x)**4 - 2*Derivative(f(x), x)**2 + 1, + 'sol': [ + Eq(f(x), C1 - x), Eq(f(x), -sqrt(C1 + 2*x)), + Eq(f(x), sqrt(C1 + 2*x)), Eq(f(x), C1 + x) + ] + }, + + 'fact_10': { + 'eq': x**4*f(x)**2 + 2*x**4*f(x)*Derivative(f(x), (x, 2)) + x**4*Derivative(f(x), + (x, 2))**2 + 2*x**3*f(x)*Derivative(f(x), x) + 2*x**3*Derivative(f(x), + x)*Derivative(f(x), (x, 2)) - 7*x**2*f(x)**2 - 7*x**2*f(x)*Derivative(f(x), + (x, 2)) + x**2*Derivative(f(x), x)**2 - 7*x*f(x)*Derivative(f(x), x) + 12*f(x)**2, + 'sol': [ + Eq(f(x), C1*besselj(2, x) + C2*bessely(2, x)), + Eq(f(x), C1*besselj(sqrt(3), x) + C2*bessely(sqrt(3), x)) + ], + 'slow': True, + }, + + 'fact_11': { + 'eq': (f(x).diff(x, 2)-exp(f(x)))*(f(x).diff(x, 2)+exp(f(x))), + 'sol': [ + Eq(f(x), log(C1/(cos(C1*sqrt(-1/C1)*(C2 + x)) - 1))), + Eq(f(x), log(C1/(cos(C1*sqrt(-1/C1)*(C2 - x)) - 1))), + Eq(f(x), log(C1/(1 - cos(C1*sqrt(-1/C1)*(C2 + x))))), + Eq(f(x), log(C1/(1 - cos(C1*sqrt(-1/C1)*(C2 - x))))) + ], + 'dsolve_too_slow': True, + }, + + #Below examples were added for the issue: https://github.com/sympy/sympy/issues/15889 + 'fact_12': { + 'eq': exp(f(x).diff(x))-f(x)**2, + 'sol': [Eq(NonElementaryIntegral(1/log(y**2), (y, f(x))), C1 + x)], + 'XFAIL': ['lie_group'] #It shows not implemented error for lie_group. + }, + + 'fact_13': { + 'eq': f(x).diff(x)**2 - f(x)**3, + 'sol': [Eq(f(x), 4/(C1**2 - 2*C1*x + x**2))], + 'XFAIL': ['lie_group'] #It shows not implemented error for lie_group. + }, + + 'fact_14': { + 'eq': f(x).diff(x)**2 - f(x), + 'sol': [Eq(f(x), C1**2/4 - C1*x/2 + x**2/4)] + }, + + 'fact_15': { + 'eq': f(x).diff(x)**2 - f(x)**2, + 'sol': [Eq(f(x), C1*exp(x)), Eq(f(x), C1*exp(-x))] + }, + + 'fact_16': { + 'eq': f(x).diff(x)**2 - f(x)**3, + 'sol': [Eq(f(x), 4/(C1**2 - 2*C1*x + x**2))], + }, + + # kamke ode 1.1 + 'fact_17': { + 'eq': f(x).diff(x)-(a4*x**4 + a3*x**3 + a2*x**2 + a1*x + a0)**(-1/2), + 'sol': [Eq(f(x), C1 + Integral(1/sqrt(a0 + a1*x + a2*x**2 + a3*x**3 + a4*x**4), x))], + 'slow': True + }, + + # This is from issue: https://github.com/sympy/sympy/issues/9446 + 'fact_18':{ + 'eq': Eq(f(2 * x), sin(Derivative(f(x)))), + 'sol': [Eq(f(x), C1 + Integral(pi - asin(f(2*x)), x)), Eq(f(x), C1 + Integral(asin(f(2*x)), x))], + 'checkodesol_XFAIL':True + }, + + # This is from issue: https://github.com/sympy/sympy/issues/7093 + 'fact_19': { + 'eq': Derivative(f(x), x)**2 - x**3, + 'sol': [Eq(f(x), C1 - 2*x**Rational(5,2)/5), Eq(f(x), C1 + 2*x**Rational(5,2)/5)], + }, + + 'fact_20': { + 'eq': x*f(x).diff(x, 2) - x*f(x), + 'sol': [Eq(f(x), C1*exp(-x) + C2*exp(x))], + }, + } + } + + + +@_add_example_keys +def _get_examples_ode_sol_almost_linear(): + from sympy.functions.special.error_functions import Ei + A = Symbol('A', positive=True) + f = Function('f') + d = f(x).diff(x) + + return { + 'hint': "almost_linear", + 'func': f(x), + 'examples':{ + 'almost_lin_01': { + 'eq': x**2*f(x)**2*d + f(x)**3 + 1, + 'sol': [Eq(f(x), (C1*exp(3/x) - 1)**Rational(1, 3)), + Eq(f(x), (-1 - sqrt(3)*I)*(C1*exp(3/x) - 1)**Rational(1, 3)/2), + Eq(f(x), (-1 + sqrt(3)*I)*(C1*exp(3/x) - 1)**Rational(1, 3)/2)], + + }, + + 'almost_lin_02': { + 'eq': x*f(x)*d + 2*x*f(x)**2 + 1, + 'sol': [Eq(f(x), -sqrt((C1 - 2*Ei(4*x))*exp(-4*x))), Eq(f(x), sqrt((C1 - 2*Ei(4*x))*exp(-4*x)))] + }, + + 'almost_lin_03': { + 'eq': x*d + x*f(x) + 1, + 'sol': [Eq(f(x), (C1 - Ei(x))*exp(-x))] + }, + + 'almost_lin_04': { + 'eq': x*exp(f(x))*d + exp(f(x)) + 3*x, + 'sol': [Eq(f(x), log(C1/x - x*Rational(3, 2)))], + }, + + 'almost_lin_05': { + 'eq': x + A*(x + diff(f(x), x) + f(x)) + diff(f(x), x) + f(x) + 2, + 'sol': [Eq(f(x), (C1 + Piecewise( + (x, Eq(A + 1, 0)), ((-A*x + A - x - 1)*exp(x)/(A + 1), True)))*exp(-x))], + }, + } + } + + +@_add_example_keys +def _get_examples_ode_sol_liouville(): + n = Symbol('n') + _y = Dummy('y') + return { + 'hint': "Liouville", + 'func': f(x), + 'examples':{ + 'liouville_01': { + 'eq': diff(f(x), x)/x + diff(f(x), x, x)/2 - diff(f(x), x)**2/2, + 'sol': [Eq(f(x), log(x/(C1 + C2*x)))], + + }, + + 'liouville_02': { + 'eq': diff(x*exp(-f(x)), x, x), + 'sol': [Eq(f(x), log(x/(C1 + C2*x)))] + }, + + 'liouville_03': { + 'eq': ((diff(f(x), x)/x + diff(f(x), x, x)/2 - diff(f(x), x)**2/2)*exp(-f(x))/exp(f(x))).expand(), + 'sol': [Eq(f(x), log(x/(C1 + C2*x)))] + }, + + 'liouville_04': { + 'eq': diff(f(x), x, x) + 1/f(x)*(diff(f(x), x))**2 + 1/x*diff(f(x), x), + 'sol': [Eq(f(x), -sqrt(C1 + C2*log(x))), Eq(f(x), sqrt(C1 + C2*log(x)))], + }, + + 'liouville_05': { + 'eq': x*diff(f(x), x, x) + x/f(x)*diff(f(x), x)**2 + x*diff(f(x), x), + 'sol': [Eq(f(x), -sqrt(C1 + C2*exp(-x))), Eq(f(x), sqrt(C1 + C2*exp(-x)))], + }, + + 'liouville_06': { + 'eq': Eq((x*exp(f(x))).diff(x, x), 0), + 'sol': [Eq(f(x), log(C1 + C2/x))], + }, + + 'liouville_07': { + 'eq': (diff(f(x), x)/x + diff(f(x), x, x)/2 - diff(f(x), x)**2/2)*exp(-f(x))/exp(f(x)), + 'sol': [Eq(f(x), log(x/(C1 + C2*x)))], + }, + + 'liouville_08': { + 'eq': x**2*diff(f(x),x) + (n*f(x) + f(x)**2)*diff(f(x),x)**2 + diff(f(x), (x, 2)), + 'sol': [Eq(C1 + C2*lowergamma(Rational(1,3), x**3/3) + NonElementaryIntegral(exp(_y**3/3)*exp(_y**2*n/2), (_y, f(x))), 0)], + }, + } + } + + +@_add_example_keys +def _get_examples_ode_sol_nth_algebraic(): + M, m, r, t = symbols('M m r t') + phi = Function('phi') + k = Symbol('k') + # This one needs a substitution f' = g. + # 'algeb_12': { + # 'eq': -exp(x) + (x*Derivative(f(x), (x, 2)) + Derivative(f(x), x))/x, + # 'sol': [Eq(f(x), C1 + C2*log(x) + exp(x) - Ei(x))], + # }, + return { + 'hint': "nth_algebraic", + 'func': f(x), + 'examples':{ + 'algeb_01': { + 'eq': f(x) * f(x).diff(x) * f(x).diff(x, x) * (f(x) - 1) * (f(x).diff(x) - x), + 'sol': [Eq(f(x), C1 + x**2/2), Eq(f(x), C1 + C2*x)] + }, + + 'algeb_02': { + 'eq': f(x) * f(x).diff(x) * f(x).diff(x, x) * (f(x) - 1), + 'sol': [Eq(f(x), C1 + C2*x)] + }, + + 'algeb_03': { + 'eq': f(x) * f(x).diff(x) * f(x).diff(x, x), + 'sol': [Eq(f(x), C1 + C2*x)] + }, + + 'algeb_04': { + 'eq': Eq(-M * phi(t).diff(t), + Rational(3, 2) * m * r**2 * phi(t).diff(t) * phi(t).diff(t,t)), + 'sol': [Eq(phi(t), C1), Eq(phi(t), C1 + C2*t - M*t**2/(3*m*r**2))], + 'func': phi(t) + }, + + 'algeb_05': { + 'eq': (1 - sin(f(x))) * f(x).diff(x), + 'sol': [Eq(f(x), C1)], + 'XFAIL': ['separable'] #It raised exception. + }, + + 'algeb_06': { + 'eq': (diff(f(x)) - x)*(diff(f(x)) + x), + 'sol': [Eq(f(x), C1 - x**2/2), Eq(f(x), C1 + x**2/2)] + }, + + 'algeb_07': { + 'eq': Eq(Derivative(f(x), x), Derivative(g(x), x)), + 'sol': [Eq(f(x), C1 + g(x))], + }, + + 'algeb_08': { + 'eq': f(x).diff(x) - C1, #this example is from issue 15999 + 'sol': [Eq(f(x), C1*x + C2)], + }, + + 'algeb_09': { + 'eq': f(x)*f(x).diff(x), + 'sol': [Eq(f(x), C1)], + }, + + 'algeb_10': { + 'eq': (diff(f(x)) - x)*(diff(f(x)) + x), + 'sol': [Eq(f(x), C1 - x**2/2), Eq(f(x), C1 + x**2/2)], + }, + + 'algeb_11': { + 'eq': f(x) + f(x)*f(x).diff(x), + 'sol': [Eq(f(x), 0), Eq(f(x), C1 - x)], + 'XFAIL': ['separable', '1st_exact', '1st_linear', 'Bernoulli', '1st_homogeneous_coeff_best', + '1st_homogeneous_coeff_subs_indep_div_dep', '1st_homogeneous_coeff_subs_dep_div_indep', + 'lie_group', 'nth_linear_constant_coeff_undetermined_coefficients', + 'nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients', + 'nth_linear_constant_coeff_variation_of_parameters', + 'nth_linear_euler_eq_nonhomogeneous_variation_of_parameters'] + #nth_linear_constant_coeff_undetermined_coefficients raises exception rest all of them misses a solution. + }, + + 'algeb_12': { + 'eq': Derivative(x*f(x), x, x, x), + 'sol': [Eq(f(x), (C1 + C2*x + C3*x**2) / x)], + 'XFAIL': ['nth_algebraic'] # It passes only when prep=False is set in dsolve. + }, + + 'algeb_13': { + 'eq': Eq(Derivative(x*Derivative(f(x), x), x)/x, exp(x)), + 'sol': [Eq(f(x), C1 + C2*log(x) + exp(x) - Ei(x))], + 'XFAIL': ['nth_algebraic'] # It passes only when prep=False is set in dsolve. + }, + + # These are simple tests from the old ode module example 14-18 + 'algeb_14': { + 'eq': Eq(f(x).diff(x), 0), + 'sol': [Eq(f(x), C1)], + }, + + 'algeb_15': { + 'eq': Eq(3*f(x).diff(x) - 5, 0), + 'sol': [Eq(f(x), C1 + x*Rational(5, 3))], + }, + + 'algeb_16': { + 'eq': Eq(3*f(x).diff(x), 5), + 'sol': [Eq(f(x), C1 + x*Rational(5, 3))], + }, + + # Type: 2nd order, constant coefficients (two complex roots) + 'algeb_17': { + 'eq': Eq(3*f(x).diff(x) - 1, 0), + 'sol': [Eq(f(x), C1 + x/3)], + }, + + 'algeb_18': { + 'eq': Eq(x*f(x).diff(x) - 1, 0), + 'sol': [Eq(f(x), C1 + log(x))], + }, + + # https://github.com/sympy/sympy/issues/6989 + 'algeb_19': { + 'eq': f(x).diff(x) - x*exp(-k*x), + 'sol': [Eq(f(x), C1 + Piecewise(((-k*x - 1)*exp(-k*x)/k**2, Ne(k**2, 0)),(x**2/2, True)))], + }, + + 'algeb_20': { + 'eq': -f(x).diff(x) + x*exp(-k*x), + 'sol': [Eq(f(x), C1 + Piecewise(((-k*x - 1)*exp(-k*x)/k**2, Ne(k**2, 0)),(x**2/2, True)))], + }, + + # https://github.com/sympy/sympy/issues/10867 + 'algeb_21': { + 'eq': Eq(g(x).diff(x).diff(x), (x-2)**2 + (x-3)**3), + 'sol': [Eq(g(x), C1 + C2*x + x**5/20 - 2*x**4/3 + 23*x**3/6 - 23*x**2/2)], + 'func': g(x), + }, + + # https://github.com/sympy/sympy/issues/13691 + 'algeb_22': { + 'eq': f(x).diff(x) - C1*g(x).diff(x), + 'sol': [Eq(f(x), C2 + C1*g(x))], + 'func': f(x), + }, + + # https://github.com/sympy/sympy/issues/4838 + 'algeb_23': { + 'eq': f(x).diff(x) - 3*C1 - 3*x**2, + 'sol': [Eq(f(x), C2 + 3*C1*x + x**3)], + }, + } + } + + +@_add_example_keys +def _get_examples_ode_sol_nth_order_reducible(): + return { + 'hint': "nth_order_reducible", + 'func': f(x), + 'examples':{ + 'reducible_01': { + 'eq': Eq(x*Derivative(f(x), x)**2 + Derivative(f(x), x, 2), 0), + 'sol': [Eq(f(x),C1 - sqrt(-1/C2)*log(-C2*sqrt(-1/C2) + x) + + sqrt(-1/C2)*log(C2*sqrt(-1/C2) + x))], + 'slow': True, + }, + + 'reducible_02': { + 'eq': -exp(x) + (x*Derivative(f(x), (x, 2)) + Derivative(f(x), x))/x, + 'sol': [Eq(f(x), C1 + C2*log(x) + exp(x) - Ei(x))], + 'slow': True, + }, + + 'reducible_03': { + 'eq': Eq(sqrt(2) * f(x).diff(x,x,x) + f(x).diff(x), 0), + 'sol': [Eq(f(x), C1 + C2*sin(2**Rational(3, 4)*x/2) + C3*cos(2**Rational(3, 4)*x/2))], + 'slow': True, + }, + + 'reducible_04': { + 'eq': f(x).diff(x, 2) + 2*f(x).diff(x), + 'sol': [Eq(f(x), C1 + C2*exp(-2*x))], + }, + + 'reducible_05': { + 'eq': f(x).diff(x, 3) + f(x).diff(x, 2) - 6*f(x).diff(x), + 'sol': [Eq(f(x), C1 + C2*exp(-3*x) + C3*exp(2*x))], + 'slow': True, + }, + + 'reducible_06': { + 'eq': f(x).diff(x, 4) - f(x).diff(x, 3) - 4*f(x).diff(x, 2) + \ + 4*f(x).diff(x), + 'sol': [Eq(f(x), C1 + C2*exp(-2*x) + C3*exp(x) + C4*exp(2*x))], + 'slow': True, + }, + + 'reducible_07': { + 'eq': f(x).diff(x, 4) + 3*f(x).diff(x, 3), + 'sol': [Eq(f(x), C1 + C2*x + C3*x**2 + C4*exp(-3*x))], + 'slow': True, + }, + + 'reducible_08': { + 'eq': f(x).diff(x, 4) - 2*f(x).diff(x, 2), + 'sol': [Eq(f(x), C1 + C2*x + C3*exp(-sqrt(2)*x) + C4*exp(sqrt(2)*x))], + 'slow': True, + }, + + 'reducible_09': { + 'eq': f(x).diff(x, 4) + 4*f(x).diff(x, 2), + 'sol': [Eq(f(x), C1 + C2*x + C3*sin(2*x) + C4*cos(2*x))], + 'slow': True, + }, + + 'reducible_10': { + 'eq': f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x), + 'sol': [Eq(f(x), C1 + C2*x*sin(x) + C2*cos(x) - C3*x*cos(x) + C3*sin(x) + C4*sin(x) + C5*cos(x))], + 'slow': True, + }, + + 'reducible_11': { + 'eq': f(x).diff(x, 2) - f(x).diff(x)**3, + 'sol': [Eq(f(x), C1 - sqrt(2)*sqrt(-1/(C2 + x))*(C2 + x)), + Eq(f(x), C1 + sqrt(2)*sqrt(-1/(C2 + x))*(C2 + x))], + 'slow': True, + }, + + # Needs to be a way to know how to combine derivatives in the expression + 'reducible_12': { + 'eq': Derivative(x*f(x), x, x, x) + Derivative(f(x), x, x, x), + 'sol': [Eq(f(x), C1 + C3/Mul(2, (x**2 + 2*x + 1), evaluate=False) + + x*(C2 + C3/Mul(2, (x**2 + 2*x + 1), evaluate=False)))], # 2-arg Mul! + 'slow': True, + }, + } + } + + + +@_add_example_keys +def _get_examples_ode_sol_nth_linear_undetermined_coefficients(): + # examples 3-27 below are from Ordinary Differential Equations, + # Tenenbaum and Pollard, pg. 231 + g = exp(-x) + f2 = f(x).diff(x, 2) + c = 3*f(x).diff(x, 3) + 5*f2 + f(x).diff(x) - f(x) - x + t = symbols("t") + u = symbols("u",cls=Function) + R, L, C, E_0, alpha = symbols("R L C E_0 alpha",positive=True) + omega = Symbol('omega') + return { + 'hint': "nth_linear_constant_coeff_undetermined_coefficients", + 'func': f(x), + 'examples':{ + 'undet_01': { + 'eq': c - x*g, + 'sol': [Eq(f(x), C3*exp(x/3) - x + (C1 + x*(C2 - x**2/24 - 3*x/32))*exp(-x) - 1)], + 'slow': True, + }, + + 'undet_02': { + 'eq': c - g, + 'sol': [Eq(f(x), C3*exp(x/3) - x + (C1 + x*(C2 - x/8))*exp(-x) - 1)], + 'slow': True, + }, + + 'undet_03': { + 'eq': f2 + 3*f(x).diff(x) + 2*f(x) - 4, + 'sol': [Eq(f(x), C1*exp(-2*x) + C2*exp(-x) + 2)], + 'slow': True, + }, + + 'undet_04': { + 'eq': f2 + 3*f(x).diff(x) + 2*f(x) - 12*exp(x), + 'sol': [Eq(f(x), C1*exp(-2*x) + C2*exp(-x) + 2*exp(x))], + 'slow': True, + }, + + 'undet_05': { + 'eq': f2 + 3*f(x).diff(x) + 2*f(x) - exp(I*x), + 'sol': [Eq(f(x), (S(3)/10 + I/10)*(C1*exp(-2*x) + C2*exp(-x) - I*exp(I*x)))], + 'slow': True, + }, + + 'undet_06': { + 'eq': f2 + 3*f(x).diff(x) + 2*f(x) - sin(x), + 'sol': [Eq(f(x), C1*exp(-2*x) + C2*exp(-x) + sin(x)/10 - 3*cos(x)/10)], + 'slow': True, + }, + + 'undet_07': { + 'eq': f2 + 3*f(x).diff(x) + 2*f(x) - cos(x), + 'sol': [Eq(f(x), C1*exp(-2*x) + C2*exp(-x) + 3*sin(x)/10 + cos(x)/10)], + 'slow': True, + }, + + 'undet_08': { + 'eq': f2 + 3*f(x).diff(x) + 2*f(x) - (8 + 6*exp(x) + 2*sin(x)), + 'sol': [Eq(f(x), C1*exp(-2*x) + C2*exp(-x) + exp(x) + sin(x)/5 - 3*cos(x)/5 + 4)], + 'slow': True, + }, + + 'undet_09': { + 'eq': f2 + f(x).diff(x) + f(x) - x**2, + 'sol': [Eq(f(x), -2*x + x**2 + (C1*sin(x*sqrt(3)/2) + C2*cos(x*sqrt(3)/2))*exp(-x/2))], + 'slow': True, + }, + + 'undet_10': { + 'eq': f2 - 2*f(x).diff(x) - 8*f(x) - 9*x*exp(x) - 10*exp(-x), + 'sol': [Eq(f(x), -x*exp(x) - 2*exp(-x) + C1*exp(-2*x) + C2*exp(4*x))], + 'slow': True, + }, + + 'undet_11': { + 'eq': f2 - 3*f(x).diff(x) - 2*exp(2*x)*sin(x), + 'sol': [Eq(f(x), C1 + C2*exp(3*x) - 3*exp(2*x)*sin(x)/5 - exp(2*x)*cos(x)/5)], + 'slow': True, + }, + + 'undet_12': { + 'eq': f(x).diff(x, 4) - 2*f2 + f(x) - x + sin(x), + 'sol': [Eq(f(x), x - sin(x)/4 + (C1 + C2*x)*exp(-x) + (C3 + C4*x)*exp(x))], + 'slow': True, + }, + + 'undet_13': { + 'eq': f2 + f(x).diff(x) - x**2 - 2*x, + 'sol': [Eq(f(x), C1 + x**3/3 + C2*exp(-x))], + 'slow': True, + }, + + 'undet_14': { + 'eq': f2 + f(x).diff(x) - x - sin(2*x), + 'sol': [Eq(f(x), C1 - x - sin(2*x)/5 - cos(2*x)/10 + x**2/2 + C2*exp(-x))], + 'slow': True, + }, + + 'undet_15': { + 'eq': f2 + f(x) - 4*x*sin(x), + 'sol': [Eq(f(x), (C1 - x**2)*cos(x) + (C2 + x)*sin(x))], + 'slow': True, + }, + + 'undet_16': { + 'eq': f2 + 4*f(x) - x*sin(2*x), + 'sol': [Eq(f(x), (C1 - x**2/8)*cos(2*x) + (C2 + x/16)*sin(2*x))], + 'slow': True, + }, + + 'undet_17': { + 'eq': f2 + 2*f(x).diff(x) + f(x) - x**2*exp(-x), + 'sol': [Eq(f(x), (C1 + x*(C2 + x**3/12))*exp(-x))], + 'slow': True, + }, + + 'undet_18': { + 'eq': f(x).diff(x, 3) + 3*f2 + 3*f(x).diff(x) + f(x) - 2*exp(-x) + \ + x**2*exp(-x), + 'sol': [Eq(f(x), (C1 + x*(C2 + x*(C3 - x**3/60 + x/3)))*exp(-x))], + 'slow': True, + }, + + 'undet_19': { + 'eq': f2 + 3*f(x).diff(x) + 2*f(x) - exp(-2*x) - x**2, + 'sol': [Eq(f(x), C2*exp(-x) + x**2/2 - x*Rational(3,2) + (C1 - x)*exp(-2*x) + Rational(7,4))], + 'slow': True, + }, + + 'undet_20': { + 'eq': f2 - 3*f(x).diff(x) + 2*f(x) - x*exp(-x), + 'sol': [Eq(f(x), C1*exp(x) + C2*exp(2*x) + (6*x + 5)*exp(-x)/36)], + 'slow': True, + }, + + 'undet_21': { + 'eq': f2 + f(x).diff(x) - 6*f(x) - x - exp(2*x), + 'sol': [Eq(f(x), Rational(-1, 36) - x/6 + C2*exp(-3*x) + (C1 + x/5)*exp(2*x))], + 'slow': True, + }, + + 'undet_22': { + 'eq': f2 + f(x) - sin(x) - exp(-x), + 'sol': [Eq(f(x), C2*sin(x) + (C1 - x/2)*cos(x) + exp(-x)/2)], + 'slow': True, + }, + + 'undet_23': { + 'eq': f(x).diff(x, 3) - 3*f2 + 3*f(x).diff(x) - f(x) - exp(x), + 'sol': [Eq(f(x), (C1 + x*(C2 + x*(C3 + x/6)))*exp(x))], + 'slow': True, + }, + + 'undet_24': { + 'eq': f2 + f(x) - S.Half - cos(2*x)/2, + 'sol': [Eq(f(x), S.Half - cos(2*x)/6 + C1*sin(x) + C2*cos(x))], + 'slow': True, + }, + + 'undet_25': { + 'eq': f(x).diff(x, 3) - f(x).diff(x) - exp(2*x)*(S.Half - cos(2*x)/2), + 'sol': [Eq(f(x), C1 + C2*exp(-x) + C3*exp(x) + (-21*sin(2*x) + 27*cos(2*x) + 130)*exp(2*x)/1560)], + 'slow': True, + }, + + #Note: 'undet_26' is referred in 'undet_37' + 'undet_26': { + 'eq': (f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x) - 2*x - + sin(x) - cos(x)), + 'sol': [Eq(f(x), C1 + x**2 + (C2 + x*(C3 - x/8))*sin(x) + (C4 + x*(C5 + x/8))*cos(x))], + 'slow': True, + }, + + 'undet_27': { + 'eq': f2 + f(x) - cos(x)/2 + cos(3*x)/2, + 'sol': [Eq(f(x), cos(3*x)/16 + C2*cos(x) + (C1 + x/4)*sin(x))], + 'slow': True, + }, + + 'undet_28': { + 'eq': f(x).diff(x) - 1, + 'sol': [Eq(f(x), C1 + x)], + 'slow': True, + }, + + # https://github.com/sympy/sympy/issues/19358 + 'undet_29': { + 'eq': f2 + f(x).diff(x) + exp(x-C1), + 'sol': [Eq(f(x), C2 + C3*exp(-x) - exp(-C1 + x)/2)], + 'slow': True, + }, + + # https://github.com/sympy/sympy/issues/18408 + 'undet_30': { + 'eq': f(x).diff(x, 3) - f(x).diff(x) - sinh(x), + 'sol': [Eq(f(x), C1 + C2*exp(-x) + C3*exp(x) + x*sinh(x)/2)], + }, + + 'undet_31': { + 'eq': f(x).diff(x, 2) - 49*f(x) - sinh(3*x), + 'sol': [Eq(f(x), C1*exp(-7*x) + C2*exp(7*x) - sinh(3*x)/40)], + }, + + 'undet_32': { + 'eq': f(x).diff(x, 3) - f(x).diff(x) - sinh(x) - exp(x), + 'sol': [Eq(f(x), C1 + C3*exp(-x) + x*sinh(x)/2 + (C2 + x/2)*exp(x))], + }, + + # https://github.com/sympy/sympy/issues/5096 + 'undet_33': { + 'eq': f(x).diff(x, x) + f(x) - x*sin(x - 2), + 'sol': [Eq(f(x), C1*sin(x) + C2*cos(x) - x**2*cos(x - 2)/4 + x*sin(x - 2)/4)], + }, + + 'undet_34': { + 'eq': f(x).diff(x, 2) + f(x) - x**4*sin(x-1), + 'sol': [ Eq(f(x), C1*sin(x) + C2*cos(x) - x**5*cos(x - 1)/10 + x**4*sin(x - 1)/4 + x**3*cos(x - 1)/2 - 3*x**2*sin(x - 1)/4 - 3*x*cos(x - 1)/4)], + }, + + 'undet_35': { + 'eq': f(x).diff(x, 2) - f(x) - exp(x - 1), + 'sol': [Eq(f(x), C2*exp(-x) + (C1 + x*exp(-1)/2)*exp(x))], + }, + + 'undet_36': { + 'eq': f(x).diff(x, 2)+f(x)-(sin(x-2)+1), + 'sol': [Eq(f(x), C1*sin(x) + C2*cos(x) - x*cos(x - 2)/2 + 1)], + }, + + # Equivalent to example_name 'undet_26'. + # This previously failed because the algorithm for undetermined coefficients + # didn't know to multiply exp(I*x) by sufficient x because it is linearly + # dependent on sin(x) and cos(x). + 'undet_37': { + 'eq': f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x) - 2*x - exp(I*x), + 'sol': [Eq(f(x), C1 + x**2*(I*exp(I*x)/8 + 1) + (C2 + C3*x)*sin(x) + (C4 + C5*x)*cos(x))], + }, + + # https://github.com/sympy/sympy/issues/12623 + 'undet_38': { + 'eq': Eq( u(t).diff(t,t) + R /L*u(t).diff(t) + 1/(L*C)*u(t), alpha), + 'sol': [Eq(u(t), C*L*alpha + C2*exp(-t*(R + sqrt(C*R**2 - 4*L)/sqrt(C))/(2*L)) + + C1*exp(t*(-R + sqrt(C*R**2 - 4*L)/sqrt(C))/(2*L)))], + 'func': u(t) + }, + + 'undet_39': { + 'eq': Eq( L*C*u(t).diff(t,t) + R*C*u(t).diff(t) + u(t), E_0*exp(I*omega*t) ), + 'sol': [Eq(u(t), C2*exp(-t*(R + sqrt(C*R**2 - 4*L)/sqrt(C))/(2*L)) + + C1*exp(t*(-R + sqrt(C*R**2 - 4*L)/sqrt(C))/(2*L)) + - E_0*exp(I*omega*t)/(C*L*omega**2 - I*C*R*omega - 1))], + 'func': u(t), + }, + + # https://github.com/sympy/sympy/issues/6879 + 'undet_40': { + 'eq': Eq(Derivative(f(x), x, 2) - 2*Derivative(f(x), x) + f(x), sin(x)), + 'sol': [Eq(f(x), (C1 + C2*x)*exp(x) + cos(x)/2)], + }, + } + } + + +@_add_example_keys +def _get_examples_ode_sol_separable(): + # test_separable1-5 are from Ordinary Differential Equations, Tenenbaum and + # Pollard, pg. 55 + t,a = symbols('a,t') + m = 96 + g = 9.8 + k = .2 + f1 = g * m + v = Function('v') + return { + 'hint': "separable", + 'func': f(x), + 'examples':{ + 'separable_01': { + 'eq': f(x).diff(x) - f(x), + 'sol': [Eq(f(x), C1*exp(x))], + }, + + 'separable_02': { + 'eq': x*f(x).diff(x) - f(x), + 'sol': [Eq(f(x), C1*x)], + }, + + 'separable_03': { + 'eq': f(x).diff(x) + sin(x), + 'sol': [Eq(f(x), C1 + cos(x))], + }, + + 'separable_04': { + 'eq': f(x)**2 + 1 - (x**2 + 1)*f(x).diff(x), + 'sol': [Eq(f(x), tan(C1 + atan(x)))], + }, + + 'separable_05': { + 'eq': f(x).diff(x)/tan(x) - f(x) - 2, + 'sol': [Eq(f(x), C1/cos(x) - 2)], + }, + + 'separable_06': { + 'eq': f(x).diff(x) * (1 - sin(f(x))) - 1, + 'sol': [Eq(-x + f(x) + cos(f(x)), C1)], + }, + + 'separable_07': { + 'eq': f(x)*x**2*f(x).diff(x) - f(x)**3 - 2*x**2*f(x).diff(x), + 'sol': [Eq(f(x), (-x - sqrt(x*(4*C1*x + x - 4)))/(C1*x - 1)/2), + Eq(f(x), (-x + sqrt(x*(4*C1*x + x - 4)))/(C1*x - 1)/2)], + 'slow': True, + }, + + 'separable_08': { + 'eq': f(x)**2 - 1 - (2*f(x) + x*f(x))*f(x).diff(x), + 'sol': [Eq(f(x), -sqrt(C1*x**2 + 4*C1*x + 4*C1 + 1)), + Eq(f(x), sqrt(C1*x**2 + 4*C1*x + 4*C1 + 1))], + 'slow': True, + }, + + 'separable_09': { + 'eq': x*log(x)*f(x).diff(x) + sqrt(1 + f(x)**2), + 'sol': [Eq(f(x), sinh(C1 - log(log(x))))], #One more solution is f(x)=I + 'slow': True, + 'checkodesol_XFAIL': True, + }, + + 'separable_10': { + 'eq': exp(x + 1)*tan(f(x)) + cos(f(x))*f(x).diff(x), + 'sol': [Eq(E*exp(x) + log(cos(f(x)) - 1)/2 - log(cos(f(x)) + 1)/2 + cos(f(x)), C1)], + 'slow': True, + }, + + 'separable_11': { + 'eq': (x*cos(f(x)) + x**2*sin(f(x))*f(x).diff(x) - a**2*sin(f(x))*f(x).diff(x)), + 'sol': [ + Eq(f(x), -acos(C1*sqrt(-a**2 + x**2)) + 2*pi), + Eq(f(x), acos(C1*sqrt(-a**2 + x**2))) + ], + 'slow': True, + }, + + 'separable_12': { + 'eq': f(x).diff(x) - f(x)*tan(x), + 'sol': [Eq(f(x), C1/cos(x))], + }, + + 'separable_13': { + 'eq': (x - 1)*cos(f(x))*f(x).diff(x) - 2*x*sin(f(x)), + 'sol': [ + Eq(f(x), pi - asin(C1*(x**2 - 2*x + 1)*exp(2*x))), + Eq(f(x), asin(C1*(x**2 - 2*x + 1)*exp(2*x))) + ], + }, + + 'separable_14': { + 'eq': f(x).diff(x) - f(x)*log(f(x))/tan(x), + 'sol': [Eq(f(x), exp(C1*sin(x)))], + }, + + 'separable_15': { + 'eq': x*f(x).diff(x) + (1 + f(x)**2)*atan(f(x)), + 'sol': [Eq(f(x), tan(C1/x))], #Two more solutions are f(x)=0 and f(x)=I + 'slow': True, + 'checkodesol_XFAIL': True, + }, + + 'separable_16': { + 'eq': f(x).diff(x) + x*(f(x) + 1), + 'sol': [Eq(f(x), -1 + C1*exp(-x**2/2))], + }, + + 'separable_17': { + 'eq': exp(f(x)**2)*(x**2 + 2*x + 1) + (x*f(x) + f(x))*f(x).diff(x), + 'sol': [ + Eq(f(x), -sqrt(log(1/(C1 + x**2 + 2*x)))), + Eq(f(x), sqrt(log(1/(C1 + x**2 + 2*x)))) + ], + }, + + 'separable_18': { + 'eq': f(x).diff(x) + f(x), + 'sol': [Eq(f(x), C1*exp(-x))], + }, + + 'separable_19': { + 'eq': sin(x)*cos(2*f(x)) + cos(x)*sin(2*f(x))*f(x).diff(x), + 'sol': [Eq(f(x), pi - acos(C1/cos(x)**2)/2), Eq(f(x), acos(C1/cos(x)**2)/2)], + }, + + 'separable_20': { + 'eq': (1 - x)*f(x).diff(x) - x*(f(x) + 1), + 'sol': [Eq(f(x), (C1*exp(-x) - x + 1)/(x - 1))], + }, + + 'separable_21': { + 'eq': f(x)*diff(f(x), x) + x - 3*x*f(x)**2, + 'sol': [Eq(f(x), -sqrt(3)*sqrt(C1*exp(3*x**2) + 1)/3), + Eq(f(x), sqrt(3)*sqrt(C1*exp(3*x**2) + 1)/3)], + }, + + 'separable_22': { + 'eq': f(x).diff(x) - exp(x + f(x)), + 'sol': [Eq(f(x), log(-1/(C1 + exp(x))))], + 'XFAIL': ['lie_group'] #It shows 'NoneType' object is not subscriptable for lie_group. + }, + + # https://github.com/sympy/sympy/issues/7081 + 'separable_23': { + 'eq': x*(f(x).diff(x)) + 1 - f(x)**2, + 'sol': [Eq(f(x), (-C1 - x**2)/(-C1 + x**2))], + }, + + # https://github.com/sympy/sympy/issues/10379 + 'separable_24': { + 'eq': f(t).diff(t)-(1-51.05*y*f(t)), + 'sol': [Eq(f(t), (0.019588638589618023*exp(y*(C1 - 51.049999999999997*t)) + 0.019588638589618023)/y)], + 'func': f(t), + }, + + # https://github.com/sympy/sympy/issues/15999 + 'separable_25': { + 'eq': f(x).diff(x) - C1*f(x), + 'sol': [Eq(f(x), C2*exp(C1*x))], + }, + + 'separable_26': { + 'eq': f1 - k * (v(t) ** 2) - m * Derivative(v(t)), + 'sol': [Eq(v(t), -68.585712797928991/tanh(C1 - 0.14288690166235204*t))], + 'func': v(t), + 'checkodesol_XFAIL': True, + }, + + #https://github.com/sympy/sympy/issues/22155 + 'separable_27': { + 'eq': f(x).diff(x) - exp(f(x) - x), + 'sol': [Eq(f(x), log(-exp(x)/(C1*exp(x) - 1)))], + } + } + } + + +@_add_example_keys +def _get_examples_ode_sol_1st_exact(): + # Type: Exact differential equation, p(x,f) + q(x,f)*f' == 0, + # where dp/df == dq/dx + ''' + Example 7 is an exact equation that fails under the exact engine. It is caught + by first order homogeneous albeit with a much contorted solution. The + exact engine fails because of a poorly simplified integral of q(0,y)dy, + where q is the function multiplying f'. The solutions should be + Eq(sqrt(x**2+f(x)**2)**3+y**3, C1). The equation below is + equivalent, but it is so complex that checkodesol fails, and takes a long + time to do so. + ''' + return { + 'hint': "1st_exact", + 'func': f(x), + 'examples':{ + '1st_exact_01': { + 'eq': sin(x)*cos(f(x)) + cos(x)*sin(f(x))*f(x).diff(x), + 'sol': [Eq(f(x), -acos(C1/cos(x)) + 2*pi), Eq(f(x), acos(C1/cos(x)))], + 'slow': True, + }, + + '1st_exact_02': { + 'eq': (2*x*f(x) + 1)/f(x) + (f(x) - x)/f(x)**2*f(x).diff(x), + 'sol': [Eq(f(x), exp(C1 - x**2 + LambertW(-x*exp(-C1 + x**2))))], + 'XFAIL': ['lie_group'], #It shows dsolve raises an exception: List index out of range for lie_group + 'slow': True, + 'checkodesol_XFAIL':True + }, + + '1st_exact_03': { + 'eq': 2*x + f(x)*cos(x) + (2*f(x) + sin(x) - sin(f(x)))*f(x).diff(x), + 'sol': [Eq(f(x)*sin(x) + cos(f(x)) + x**2 + f(x)**2, C1)], + 'XFAIL': ['lie_group'], #It goes into infinite loop for lie_group. + 'slow': True, + }, + + '1st_exact_04': { + 'eq': cos(f(x)) - (x*sin(f(x)) - f(x)**2)*f(x).diff(x), + 'sol': [Eq(x*cos(f(x)) + f(x)**3/3, C1)], + 'slow': True, + }, + + '1st_exact_05': { + 'eq': 2*x*f(x) + (x**2 + f(x)**2)*f(x).diff(x), + 'sol': [Eq(x**2*f(x) + f(x)**3/3, C1)], + 'slow': True, + 'simplify_flag':False + }, + + # This was from issue: https://github.com/sympy/sympy/issues/11290 + '1st_exact_06': { + 'eq': cos(f(x)) - (x*sin(f(x)) - f(x)**2)*f(x).diff(x), + 'sol': [Eq(x*cos(f(x)) + f(x)**3/3, C1)], + 'simplify_flag':False + }, + + '1st_exact_07': { + 'eq': x*sqrt(x**2 + f(x)**2) - (x**2*f(x)/(f(x) - sqrt(x**2 + f(x)**2)))*f(x).diff(x), + 'sol': [Eq(log(x), + C1 - 9*sqrt(1 + f(x)**2/x**2)*asinh(f(x)/x)/(-27*f(x)/x + + 27*sqrt(1 + f(x)**2/x**2)) - 9*sqrt(1 + f(x)**2/x**2)* + log(1 - sqrt(1 + f(x)**2/x**2)*f(x)/x + 2*f(x)**2/x**2)/ + (-27*f(x)/x + 27*sqrt(1 + f(x)**2/x**2)) + + 9*asinh(f(x)/x)*f(x)/(x*(-27*f(x)/x + 27*sqrt(1 + f(x)**2/x**2))) + + 9*f(x)*log(1 - sqrt(1 + f(x)**2/x**2)*f(x)/x + 2*f(x)**2/x**2)/ + (x*(-27*f(x)/x + 27*sqrt(1 + f(x)**2/x**2))))], + 'slow': True, + 'dsolve_too_slow':True + }, + + # Type: a(x)f'(x)+b(x)*f(x)+c(x)=0 + '1st_exact_08': { + 'eq': Eq(x**2*f(x).diff(x) + 3*x*f(x) - sin(x)/x, 0), + 'sol': [Eq(f(x), (C1 - cos(x))/x**3)], + }, + + # these examples are from test_exact_enhancement + '1st_exact_09': { + 'eq': f(x)/x**2 + ((f(x)*x - 1)/x)*f(x).diff(x), + 'sol': [Eq(f(x), (i*sqrt(C1*x**2 + 1) + 1)/x) for i in (-1, 1)], + }, + + '1st_exact_10': { + 'eq': (x*f(x) - 1) + f(x).diff(x)*(x**2 - x*f(x)), + 'sol': [Eq(f(x), x - sqrt(C1 + x**2 - 2*log(x))), Eq(f(x), x + sqrt(C1 + x**2 - 2*log(x)))], + }, + + '1st_exact_11': { + 'eq': (x + 2)*sin(f(x)) + f(x).diff(x)*x*cos(f(x)), + 'sol': [Eq(f(x), -asin(C1*exp(-x)/x**2) + pi), Eq(f(x), asin(C1*exp(-x)/x**2))], + }, + } + } + + +@_add_example_keys +def _get_examples_ode_sol_nth_linear_var_of_parameters(): + g = exp(-x) + f2 = f(x).diff(x, 2) + c = 3*f(x).diff(x, 3) + 5*f2 + f(x).diff(x) - f(x) - x + return { + 'hint': "nth_linear_constant_coeff_variation_of_parameters", + 'func': f(x), + 'examples':{ + 'var_of_parameters_01': { + 'eq': c - x*g, + 'sol': [Eq(f(x), C3*exp(x/3) - x + (C1 + x*(C2 - x**2/24 - 3*x/32))*exp(-x) - 1)], + 'slow': True, + }, + + 'var_of_parameters_02': { + 'eq': c - g, + 'sol': [Eq(f(x), C3*exp(x/3) - x + (C1 + x*(C2 - x/8))*exp(-x) - 1)], + 'slow': True, + }, + + 'var_of_parameters_03': { + 'eq': f(x).diff(x) - 1, + 'sol': [Eq(f(x), C1 + x)], + 'slow': True, + }, + + 'var_of_parameters_04': { + 'eq': f2 + 3*f(x).diff(x) + 2*f(x) - 4, + 'sol': [Eq(f(x), C1*exp(-2*x) + C2*exp(-x) + 2)], + 'slow': True, + }, + + 'var_of_parameters_05': { + 'eq': f2 + 3*f(x).diff(x) + 2*f(x) - 12*exp(x), + 'sol': [Eq(f(x), C1*exp(-2*x) + C2*exp(-x) + 2*exp(x))], + 'slow': True, + }, + + 'var_of_parameters_06': { + 'eq': f2 - 2*f(x).diff(x) - 8*f(x) - 9*x*exp(x) - 10*exp(-x), + 'sol': [Eq(f(x), -x*exp(x) - 2*exp(-x) + C1*exp(-2*x) + C2*exp(4*x))], + 'slow': True, + }, + + 'var_of_parameters_07': { + 'eq': f2 + 2*f(x).diff(x) + f(x) - x**2*exp(-x), + 'sol': [Eq(f(x), (C1 + x*(C2 + x**3/12))*exp(-x))], + 'slow': True, + }, + + 'var_of_parameters_08': { + 'eq': f2 - 3*f(x).diff(x) + 2*f(x) - x*exp(-x), + 'sol': [Eq(f(x), C1*exp(x) + C2*exp(2*x) + (6*x + 5)*exp(-x)/36)], + 'slow': True, + }, + + 'var_of_parameters_09': { + 'eq': f(x).diff(x, 3) - 3*f2 + 3*f(x).diff(x) - f(x) - exp(x), + 'sol': [Eq(f(x), (C1 + x*(C2 + x*(C3 + x/6)))*exp(x))], + 'slow': True, + }, + + 'var_of_parameters_10': { + 'eq': f2 + 2*f(x).diff(x) + f(x) - exp(-x)/x, + 'sol': [Eq(f(x), (C1 + x*(C2 + log(x)))*exp(-x))], + 'slow': True, + }, + + 'var_of_parameters_11': { + 'eq': f2 + f(x) - 1/sin(x)*1/cos(x), + 'sol': [Eq(f(x), (C1 + log(sin(x) - 1)/2 - log(sin(x) + 1)/2 + )*cos(x) + (C2 + log(cos(x) - 1)/2 - log(cos(x) + 1)/2)*sin(x))], + 'slow': True, + }, + + 'var_of_parameters_12': { + 'eq': f(x).diff(x, 4) - 1/x, + 'sol': [Eq(f(x), C1 + C2*x + C3*x**2 + x**3*(C4 + log(x)/6))], + 'slow': True, + }, + + # These were from issue: https://github.com/sympy/sympy/issues/15996 + 'var_of_parameters_13': { + 'eq': f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x) - 2*x - exp(I*x), + 'sol': [Eq(f(x), C1 + x**2 + (C2 + x*(C3 - x/8 + 3*exp(I*x)/2 + 3*exp(-I*x)/2) + 5*exp(2*I*x)/16 + 2*I*exp(I*x) - 2*I*exp(-I*x))*sin(x) + (C4 + x*(C5 + I*x/8 + 3*I*exp(I*x)/2 - 3*I*exp(-I*x)/2) + + 5*I*exp(2*I*x)/16 - 2*exp(I*x) - 2*exp(-I*x))*cos(x) - I*exp(I*x))], + }, + + 'var_of_parameters_14': { + 'eq': f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x) - exp(I*x), + 'sol': [Eq(f(x), C1 + (C2 + x*(C3 - x/8) + 5*exp(2*I*x)/16)*sin(x) + (C4 + x*(C5 + I*x/8) + 5*I*exp(2*I*x)/16)*cos(x) - I*exp(I*x))], + }, + + # https://github.com/sympy/sympy/issues/14395 + 'var_of_parameters_15': { + 'eq': Derivative(f(x), x, x) + 9*f(x) - sec(x), + 'sol': [Eq(f(x), (C1 - x/3 + sin(2*x)/3)*sin(3*x) + (C2 + log(cos(x)) + - 2*log(cos(x)**2)/3 + 2*cos(x)**2/3)*cos(3*x))], + 'slow': True, + }, + } + } + + +@_add_example_keys +def _get_examples_ode_sol_2nd_linear_bessel(): + return { + 'hint': "2nd_linear_bessel", + 'func': f(x), + 'examples':{ + '2nd_lin_bessel_01': { + 'eq': x**2*(f(x).diff(x, 2)) + x*(f(x).diff(x)) + (x**2 - 4)*f(x), + 'sol': [Eq(f(x), C1*besselj(2, x) + C2*bessely(2, x))], + }, + + '2nd_lin_bessel_02': { + 'eq': x**2*(f(x).diff(x, 2)) + x*(f(x).diff(x)) + (x**2 +25)*f(x), + 'sol': [Eq(f(x), C1*besselj(5*I, x) + C2*bessely(5*I, x))], + }, + + '2nd_lin_bessel_03': { + 'eq': x**2*(f(x).diff(x, 2)) + x*(f(x).diff(x)) + (x**2)*f(x), + 'sol': [Eq(f(x), C1*besselj(0, x) + C2*bessely(0, x))], + }, + + '2nd_lin_bessel_04': { + 'eq': x**2*(f(x).diff(x, 2)) + x*(f(x).diff(x)) + (81*x**2 -S(1)/9)*f(x), + 'sol': [Eq(f(x), C1*besselj(S(1)/3, 9*x) + C2*bessely(S(1)/3, 9*x))], + }, + + '2nd_lin_bessel_05': { + 'eq': x**2*(f(x).diff(x, 2)) + x*(f(x).diff(x)) + (x**4 - 4)*f(x), + 'sol': [Eq(f(x), C1*besselj(1, x**2/2) + C2*bessely(1, x**2/2))], + }, + + '2nd_lin_bessel_06': { + 'eq': x**2*(f(x).diff(x, 2)) + 2*x*(f(x).diff(x)) + (x**4 - 4)*f(x), + 'sol': [Eq(f(x), (C1*besselj(sqrt(17)/4, x**2/2) + C2*bessely(sqrt(17)/4, x**2/2))/sqrt(x))], + }, + + '2nd_lin_bessel_07': { + 'eq': x**2*(f(x).diff(x, 2)) + x*(f(x).diff(x)) + (x**2 - S(1)/4)*f(x), + 'sol': [Eq(f(x), C1*besselj(S(1)/2, x) + C2*bessely(S(1)/2, x))], + }, + + '2nd_lin_bessel_08': { + 'eq': x**2*(f(x).diff(x, 2)) - 3*x*(f(x).diff(x)) + (4*x + 4)*f(x), + 'sol': [Eq(f(x), x**2*(C1*besselj(0, 4*sqrt(x)) + C2*bessely(0, 4*sqrt(x))))], + }, + + '2nd_lin_bessel_09': { + 'eq': x*(f(x).diff(x, 2)) - f(x).diff(x) + 4*x**3*f(x), + 'sol': [Eq(f(x), x*(C1*besselj(S(1)/2, x**2) + C2*bessely(S(1)/2, x**2)))], + }, + + '2nd_lin_bessel_10': { + 'eq': (x-2)**2*(f(x).diff(x, 2)) - (x-2)*f(x).diff(x) + 4*(x-2)**2*f(x), + 'sol': [Eq(f(x), (x - 2)*(C1*besselj(1, 2*x - 4) + C2*bessely(1, 2*x - 4)))], + }, + + # https://github.com/sympy/sympy/issues/4414 + '2nd_lin_bessel_11': { + 'eq': f(x).diff(x, x) + 2/x*f(x).diff(x) + f(x), + 'sol': [Eq(f(x), (C1*besselj(S(1)/2, x) + C2*bessely(S(1)/2, x))/sqrt(x))], + }, + '2nd_lin_bessel_12': { + 'eq': x**2*f(x).diff(x, 2) + x*f(x).diff(x) + (a**2*x**2/c**2 - b**2)*f(x), + 'sol': [Eq(f(x), C1*besselj(sqrt(b**2), x*sqrt(a**2/c**2)) + C2*bessely(sqrt(b**2), x*sqrt(a**2/c**2)))], + }, + } + } + + +@_add_example_keys +def _get_examples_ode_sol_2nd_2F1_hypergeometric(): + return { + 'hint': "2nd_hypergeometric", + 'func': f(x), + 'examples':{ + '2nd_2F1_hyper_01': { + 'eq': x*(x-1)*f(x).diff(x, 2) + (S(3)/2 -2*x)*f(x).diff(x) + 2*f(x), + 'sol': [Eq(f(x), C1*x**(S(5)/2)*hyper((S(3)/2, S(1)/2), (S(7)/2,), x) + C2*hyper((-1, -2), (-S(3)/2,), x))], + }, + + '2nd_2F1_hyper_02': { + 'eq': x*(x-1)*f(x).diff(x, 2) + (S(7)/2*x)*f(x).diff(x) + f(x), + 'sol': [Eq(f(x), (C1*(1 - x)**(S(5)/2)*hyper((S(1)/2, 2), (S(7)/2,), 1 - x) + + C2*hyper((-S(1)/2, -2), (-S(3)/2,), 1 - x))/(x - 1)**(S(5)/2))], + }, + + '2nd_2F1_hyper_03': { + 'eq': x*(x-1)*f(x).diff(x, 2) + (S(3)+ S(7)/2*x)*f(x).diff(x) + f(x), + 'sol': [Eq(f(x), (C1*(1 - x)**(S(11)/2)*hyper((S(1)/2, 2), (S(13)/2,), 1 - x) + + C2*hyper((-S(7)/2, -5), (-S(9)/2,), 1 - x))/(x - 1)**(S(11)/2))], + }, + + '2nd_2F1_hyper_04': { + 'eq': -x**(S(5)/7)*(-416*x**(S(9)/7)/9 - 2385*x**(S(5)/7)/49 + S(298)*x/3)*f(x)/(196*(-x**(S(6)/7) + + x)**2*(x**(S(6)/7) + x)**2) + Derivative(f(x), (x, 2)), + 'sol': [Eq(f(x), x**(S(45)/98)*(C1*x**(S(4)/49)*hyper((S(1)/3, -S(1)/2), (S(9)/7,), x**(S(2)/7)) + + C2*hyper((S(1)/21, -S(11)/14), (S(5)/7,), x**(S(2)/7)))/(x**(S(2)/7) - 1)**(S(19)/84))], + 'checkodesol_XFAIL':True, + }, + } + } + +@_add_example_keys +def _get_examples_ode_sol_2nd_nonlinear_autonomous_conserved(): + return { + 'hint': "2nd_nonlinear_autonomous_conserved", + 'func': f(x), + 'examples': { + '2nd_nonlinear_autonomous_conserved_01': { + 'eq': f(x).diff(x, 2) + exp(f(x)) + log(f(x)), + 'sol': [ + Eq(Integral(1/sqrt(C1 - 2*_u*log(_u) + 2*_u - 2*exp(_u)), (_u, f(x))), C2 + x), + Eq(Integral(1/sqrt(C1 - 2*_u*log(_u) + 2*_u - 2*exp(_u)), (_u, f(x))), C2 - x) + ], + 'simplify_flag': False, + }, + '2nd_nonlinear_autonomous_conserved_02': { + 'eq': f(x).diff(x, 2) + cbrt(f(x)) + 1/f(x), + 'sol': [ + Eq(sqrt(2)*Integral(1/sqrt(2*C1 - 3*_u**Rational(4, 3) - 4*log(_u)), (_u, f(x))), C2 + x), + Eq(sqrt(2)*Integral(1/sqrt(2*C1 - 3*_u**Rational(4, 3) - 4*log(_u)), (_u, f(x))), C2 - x) + ], + 'simplify_flag': False, + }, + '2nd_nonlinear_autonomous_conserved_03': { + 'eq': f(x).diff(x, 2) + sin(f(x)), + 'sol': [ + Eq(Integral(1/sqrt(C1 + 2*cos(_u)), (_u, f(x))), C2 + x), + Eq(Integral(1/sqrt(C1 + 2*cos(_u)), (_u, f(x))), C2 - x) + ], + 'simplify_flag': False, + }, + '2nd_nonlinear_autonomous_conserved_04': { + 'eq': f(x).diff(x, 2) + cosh(f(x)), + 'sol': [ + Eq(Integral(1/sqrt(C1 - 2*sinh(_u)), (_u, f(x))), C2 + x), + Eq(Integral(1/sqrt(C1 - 2*sinh(_u)), (_u, f(x))), C2 - x) + ], + 'simplify_flag': False, + }, + '2nd_nonlinear_autonomous_conserved_05': { + 'eq': f(x).diff(x, 2) + asin(f(x)), + 'sol': [ + Eq(Integral(1/sqrt(C1 - 2*_u*asin(_u) - 2*sqrt(1 - _u**2)), (_u, f(x))), C2 + x), + Eq(Integral(1/sqrt(C1 - 2*_u*asin(_u) - 2*sqrt(1 - _u**2)), (_u, f(x))), C2 - x) + ], + 'simplify_flag': False, + 'XFAIL': ['2nd_nonlinear_autonomous_conserved_Integral'] + } + } + } + + +@_add_example_keys +def _get_examples_ode_sol_separable_reduced(): + df = f(x).diff(x) + return { + 'hint': "separable_reduced", + 'func': f(x), + 'examples':{ + 'separable_reduced_01': { + 'eq': x* df + f(x)* (1 / (x**2*f(x) - 1)), + 'sol': [Eq(log(x**2*f(x))/3 + log(x**2*f(x) - Rational(3, 2))/6, C1 + log(x))], + 'simplify_flag': False, + 'XFAIL': ['lie_group'], #It hangs. + }, + + #Note: 'separable_reduced_02' is referred in 'separable_reduced_11' + 'separable_reduced_02': { + 'eq': f(x).diff(x) + (f(x) / (x**4*f(x) - x)), + 'sol': [Eq(log(x**3*f(x))/4 + log(x**3*f(x) - Rational(4,3))/12, C1 + log(x))], + 'simplify_flag': False, + 'checkodesol_XFAIL':True, #It hangs for this. + }, + + 'separable_reduced_03': { + 'eq': x*df + f(x)*(x**2*f(x)), + 'sol': [Eq(log(x**2*f(x))/2 - log(x**2*f(x) - 2)/2, C1 + log(x))], + 'simplify_flag': False, + }, + + 'separable_reduced_04': { + 'eq': Eq(f(x).diff(x) + f(x)/x * (1 + (x**(S(2)/3)*f(x))**2), 0), + 'sol': [Eq(-3*log(x**(S(2)/3)*f(x)) + 3*log(3*x**(S(4)/3)*f(x)**2 + 1)/2, C1 + log(x))], + 'simplify_flag': False, + }, + + 'separable_reduced_05': { + 'eq': Eq(f(x).diff(x) + f(x)/x * (1 + (x*f(x))**2), 0), + 'sol': [Eq(f(x), -sqrt(2)*sqrt(1/(C1 + log(x)))/(2*x)),\ + Eq(f(x), sqrt(2)*sqrt(1/(C1 + log(x)))/(2*x))], + }, + + 'separable_reduced_06': { + 'eq': Eq(f(x).diff(x) + (x**4*f(x)**2 + x**2*f(x))*f(x)/(x*(x**6*f(x)**3 + x**4*f(x)**2)), 0), + 'sol': [Eq(f(x), C1 + 1/(2*x**2))], + }, + + 'separable_reduced_07': { + 'eq': Eq(f(x).diff(x) + (f(x)**2)*f(x)/(x), 0), + 'sol': [ + Eq(f(x), -sqrt(2)*sqrt(1/(C1 + log(x)))/2), + Eq(f(x), sqrt(2)*sqrt(1/(C1 + log(x)))/2) + ], + }, + + 'separable_reduced_08': { + 'eq': Eq(f(x).diff(x) + (f(x)+3)*f(x)/(x*(f(x)+2)), 0), + 'sol': [Eq(-log(f(x) + 3)/3 - 2*log(f(x))/3, C1 + log(x))], + 'simplify_flag': False, + 'XFAIL': ['lie_group'], #It hangs. + }, + + 'separable_reduced_09': { + 'eq': Eq(f(x).diff(x) + (f(x)+3)*f(x)/x, 0), + 'sol': [Eq(f(x), 3/(C1*x**3 - 1))], + }, + + 'separable_reduced_10': { + 'eq': Eq(f(x).diff(x) + (f(x)**2+f(x))*f(x)/(x), 0), + 'sol': [Eq(- log(x) - log(f(x) + 1) + log(f(x)) + 1/f(x), C1)], + 'XFAIL': ['lie_group'],#No algorithms are implemented to solve equation -C1 + x*(_y + 1)*exp(-1/_y)/_y + + }, + + # Equivalent to example_name 'separable_reduced_02'. Only difference is testing with simplify=True + 'separable_reduced_11': { + 'eq': f(x).diff(x) + (f(x) / (x**4*f(x) - x)), + 'sol': [Eq(f(x), -sqrt(2)*sqrt(3*3**Rational(1,3)*(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) +- 3*3**Rational(2,3)*exp(12*C1)/(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) + 2/x**6)/6 +- sqrt(2)*sqrt(-3*3**Rational(1,3)*(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) ++ 3*3**Rational(2,3)*exp(12*C1)/(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) + 4/x**6 +- 4*sqrt(2)/(x**9*sqrt(3*3**Rational(1,3)*(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) +- 3*3**Rational(2,3)*exp(12*C1)/(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) + 2/x**6)))/6 + 1/(3*x**3)), +Eq(f(x), -sqrt(2)*sqrt(3*3**Rational(1,3)*(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) +- 3*3**Rational(2,3)*exp(12*C1)/(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) + 2/x**6)/6 ++ sqrt(2)*sqrt(-3*3**Rational(1,3)*(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) ++ 3*3**Rational(2,3)*exp(12*C1)/(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) + 4/x**6 +- 4*sqrt(2)/(x**9*sqrt(3*3**Rational(1,3)*(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) +- 3*3**Rational(2,3)*exp(12*C1)/(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) + 2/x**6)))/6 + 1/(3*x**3)), +Eq(f(x), sqrt(2)*sqrt(3*3**Rational(1,3)*(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) +- 3*3**Rational(2,3)*exp(12*C1)/(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) + 2/x**6)/6 +- sqrt(2)*sqrt(-3*3**Rational(1,3)*(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) ++ 3*3**Rational(2,3)*exp(12*C1)/(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) ++ 4/x**6 + 4*sqrt(2)/(x**9*sqrt(3*3**Rational(1,3)*(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) +- 3*3**Rational(2,3)*exp(12*C1)/(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) + 2/x**6)))/6 + 1/(3*x**3)), +Eq(f(x), sqrt(2)*sqrt(3*3**Rational(1,3)*(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) +- 3*3**Rational(2,3)*exp(12*C1)/(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) + 2/x**6)/6 ++ sqrt(2)*sqrt(-3*3**Rational(1,3)*(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) + 3*3**Rational(2,3)*exp(12*C1)/(sqrt((3*exp(12*C1) ++ x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) + 4/x**6 + 4*sqrt(2)/(x**9*sqrt(3*3**Rational(1,3)*(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) +- exp(12*C1)/x**6)**Rational(1,3) - 3*3**Rational(2,3)*exp(12*C1)/(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) + 2/x**6)))/6 + 1/(3*x**3))], + 'checkodesol_XFAIL':True, #It hangs for this. + 'slow': True, + }, + + #These were from issue: https://github.com/sympy/sympy/issues/6247 + 'separable_reduced_12': { + 'eq': x**2*f(x)**2 + x*Derivative(f(x), x), + 'sol': [Eq(f(x), 2*C1/(C1*x**2 - 1))], + }, + } + } + + +@_add_example_keys +def _get_examples_ode_sol_lie_group(): + a, b, c = symbols("a b c") + return { + 'hint': "lie_group", + 'func': f(x), + 'examples':{ + #Example 1-4 and 19-20 were from issue: https://github.com/sympy/sympy/issues/17322 + 'lie_group_01': { + 'eq': x*f(x).diff(x)*(f(x)+4) + (f(x)**2) -2*f(x)-2*x, + 'sol': [], + 'dsolve_too_slow': True, + 'checkodesol_too_slow': True, + }, + + 'lie_group_02': { + 'eq': x*f(x).diff(x)*(f(x)+4) + (f(x)**2) -2*f(x)-2*x, + 'sol': [], + 'dsolve_too_slow': True, + }, + + 'lie_group_03': { + 'eq': Eq(x**7*Derivative(f(x), x) + 5*x**3*f(x)**2 - (2*x**2 + 2)*f(x)**3, 0), + 'sol': [], + 'dsolve_too_slow': True, + }, + + 'lie_group_04': { + 'eq': f(x).diff(x) - (f(x) - x*log(x))**2/x**2 + log(x), + 'sol': [], + 'XFAIL': ['lie_group'], + }, + + 'lie_group_05': { + 'eq': f(x).diff(x)**2, + 'sol': [Eq(f(x), C1)], + 'XFAIL': ['factorable'], #It raises Not Implemented error + }, + + 'lie_group_06': { + 'eq': Eq(f(x).diff(x), x**2*f(x)), + 'sol': [Eq(f(x), C1*exp(x**3)**Rational(1, 3))], + }, + + 'lie_group_07': { + 'eq': f(x).diff(x) + a*f(x) - c*exp(b*x), + 'sol': [Eq(f(x), Piecewise(((-C1*(a + b) + c*exp(x*(a + b)))*exp(-a*x)/(a + b),\ + Ne(a, -b)), ((-C1 + c*x)*exp(-a*x), True)))], + }, + + 'lie_group_08': { + 'eq': f(x).diff(x) + 2*x*f(x) - x*exp(-x**2), + 'sol': [Eq(f(x), (C1 + x**2/2)*exp(-x**2))], + }, + + 'lie_group_09': { + 'eq': (1 + 2*x)*(f(x).diff(x)) + 2 - 4*exp(-f(x)), + 'sol': [Eq(f(x), log(C1/(2*x + 1) + 2))], + }, + + 'lie_group_10': { + 'eq': x**2*(f(x).diff(x)) - f(x) + x**2*exp(x - (1/x)), + 'sol': [Eq(f(x), (C1 - exp(x))*exp(-1/x))], + 'XFAIL': ['factorable'], #It raises Recursion Error (maixmum depth exceeded) + }, + + 'lie_group_11': { + 'eq': x**2*f(x)**2 + x*Derivative(f(x), x), + 'sol': [Eq(f(x), 2/(C1 + x**2))], + }, + + 'lie_group_12': { + 'eq': diff(f(x),x) + 2*x*f(x) - x*exp(-x**2), + 'sol': [Eq(f(x), exp(-x**2)*(C1 + x**2/2))], + }, + + 'lie_group_13': { + 'eq': diff(f(x),x) + f(x)*cos(x) - exp(2*x), + 'sol': [Eq(f(x), exp(-sin(x))*(C1 + Integral(exp(2*x)*exp(sin(x)), x)))], + }, + + 'lie_group_14': { + 'eq': diff(f(x),x) + f(x)*cos(x) - sin(2*x)/2, + 'sol': [Eq(f(x), C1*exp(-sin(x)) + sin(x) - 1)], + }, + + 'lie_group_15': { + 'eq': x*diff(f(x),x) + f(x) - x*sin(x), + 'sol': [Eq(f(x), (C1 - x*cos(x) + sin(x))/x)], + }, + + 'lie_group_16': { + 'eq': x*diff(f(x),x) - f(x) - x/log(x), + 'sol': [Eq(f(x), x*(C1 + log(log(x))))], + }, + + 'lie_group_17': { + 'eq': (f(x).diff(x)-f(x)) * (f(x).diff(x)+f(x)), + 'sol': [Eq(f(x), C1*exp(x)), Eq(f(x), C1*exp(-x))], + }, + + 'lie_group_18': { + 'eq': f(x).diff(x) * (f(x).diff(x) - f(x)), + 'sol': [Eq(f(x), C1*exp(x)), Eq(f(x), C1)], + }, + + 'lie_group_19': { + 'eq': (f(x).diff(x)-f(x)) * (f(x).diff(x)+f(x)), + 'sol': [Eq(f(x), C1*exp(-x)), Eq(f(x), C1*exp(x))], + }, + + 'lie_group_20': { + 'eq': f(x).diff(x)*(f(x).diff(x)+f(x)), + 'sol': [Eq(f(x), C1), Eq(f(x), C1*exp(-x))], + }, + } + } + + +@_add_example_keys +def _get_examples_ode_sol_2nd_linear_airy(): + return { + 'hint': "2nd_linear_airy", + 'func': f(x), + 'examples':{ + '2nd_lin_airy_01': { + 'eq': f(x).diff(x, 2) - x*f(x), + 'sol': [Eq(f(x), C1*airyai(x) + C2*airybi(x))], + }, + + '2nd_lin_airy_02': { + 'eq': f(x).diff(x, 2) + 2*x*f(x), + 'sol': [Eq(f(x), C1*airyai(-2**(S(1)/3)*x) + C2*airybi(-2**(S(1)/3)*x))], + }, + } + } + + +@_add_example_keys +def _get_examples_ode_sol_nth_linear_constant_coeff_homogeneous(): + # From Exercise 20, in Ordinary Differential Equations, + # Tenenbaum and Pollard, pg. 220 + a = Symbol('a', positive=True) + k = Symbol('k', real=True) + r1, r2, r3, r4, r5 = [rootof(x**5 + 11*x - 2, n) for n in range(5)] + r6, r7, r8, r9, r10 = [rootof(x**5 - 3*x + 1, n) for n in range(5)] + r11, r12, r13, r14, r15 = [rootof(x**5 - 100*x**3 + 1000*x + 1, n) for n in range(5)] + r16, r17, r18, r19, r20 = [rootof(x**5 - x**4 + 10, n) for n in range(5)] + r21, r22, r23, r24, r25 = [rootof(x**5 - x + 1, n) for n in range(5)] + E = exp(1) + return { + 'hint': "nth_linear_constant_coeff_homogeneous", + 'func': f(x), + 'examples':{ + 'lin_const_coeff_hom_01': { + 'eq': f(x).diff(x, 2) + 2*f(x).diff(x), + 'sol': [Eq(f(x), C1 + C2*exp(-2*x))], + }, + + 'lin_const_coeff_hom_02': { + 'eq': f(x).diff(x, 2) - 3*f(x).diff(x) + 2*f(x), + 'sol': [Eq(f(x), (C1 + C2*exp(x))*exp(x))], + }, + + 'lin_const_coeff_hom_03': { + 'eq': f(x).diff(x, 2) - f(x), + 'sol': [Eq(f(x), C1*exp(-x) + C2*exp(x))], + }, + + 'lin_const_coeff_hom_04': { + 'eq': f(x).diff(x, 3) + f(x).diff(x, 2) - 6*f(x).diff(x), + 'sol': [Eq(f(x), C1 + C2*exp(-3*x) + C3*exp(2*x))], + 'slow': True, + }, + + 'lin_const_coeff_hom_05': { + 'eq': 6*f(x).diff(x, 2) - 11*f(x).diff(x) + 4*f(x), + 'sol': [Eq(f(x), C1*exp(x/2) + C2*exp(x*Rational(4, 3)))], + 'slow': True, + }, + + 'lin_const_coeff_hom_06': { + 'eq': Eq(f(x).diff(x, 2) + 2*f(x).diff(x) - f(x), 0), + 'sol': [Eq(f(x), C1*exp(x*(-1 + sqrt(2))) + C2*exp(-x*(sqrt(2) + 1)))], + 'slow': True, + }, + + 'lin_const_coeff_hom_07': { + 'eq': diff(f(x), x, 3) + diff(f(x), x, 2) - 10*diff(f(x), x) - 6*f(x), + 'sol': [Eq(f(x), C1*exp(3*x) + C3*exp(-x*(2 + sqrt(2))) + C2*exp(x*(-2 + sqrt(2))))], + 'slow': True, + }, + + 'lin_const_coeff_hom_08': { + 'eq': f(x).diff(x, 4) - f(x).diff(x, 3) - 4*f(x).diff(x, 2) + \ + 4*f(x).diff(x), + 'sol': [Eq(f(x), C1 + C2*exp(-2*x) + C3*exp(x) + C4*exp(2*x))], + 'slow': True, + }, + + 'lin_const_coeff_hom_09': { + 'eq': f(x).diff(x, 4) + 4*f(x).diff(x, 3) + f(x).diff(x, 2) - \ + 4*f(x).diff(x) - 2*f(x), + 'sol': [Eq(f(x), C3*exp(-x) + C4*exp(x) + (C1*exp(-sqrt(2)*x) + C2*exp(sqrt(2)*x))*exp(-2*x))], + 'slow': True, + }, + + 'lin_const_coeff_hom_10': { + 'eq': f(x).diff(x, 4) - a**2*f(x), + 'sol': [Eq(f(x), C1*exp(-sqrt(a)*x) + C2*exp(sqrt(a)*x) + C3*sin(sqrt(a)*x) + C4*cos(sqrt(a)*x))], + 'slow': True, + }, + + 'lin_const_coeff_hom_11': { + 'eq': f(x).diff(x, 2) - 2*k*f(x).diff(x) - 2*f(x), + 'sol': [Eq(f(x), C1*exp(x*(k - sqrt(k**2 + 2))) + C2*exp(x*(k + sqrt(k**2 + 2))))], + 'slow': True, + }, + + 'lin_const_coeff_hom_12': { + 'eq': f(x).diff(x, 2) + 4*k*f(x).diff(x) - 12*k**2*f(x), + 'sol': [Eq(f(x), C1*exp(-6*k*x) + C2*exp(2*k*x))], + 'slow': True, + }, + + 'lin_const_coeff_hom_13': { + 'eq': f(x).diff(x, 4), + 'sol': [Eq(f(x), C1 + C2*x + C3*x**2 + C4*x**3)], + 'slow': True, + }, + + 'lin_const_coeff_hom_14': { + 'eq': f(x).diff(x, 2) + 4*f(x).diff(x) + 4*f(x), + 'sol': [Eq(f(x), (C1 + C2*x)*exp(-2*x))], + 'slow': True, + }, + + 'lin_const_coeff_hom_15': { + 'eq': 3*f(x).diff(x, 3) + 5*f(x).diff(x, 2) + f(x).diff(x) - f(x), + 'sol': [Eq(f(x), (C1 + C2*x)*exp(-x) + C3*exp(x/3))], + 'slow': True, + }, + + 'lin_const_coeff_hom_16': { + 'eq': f(x).diff(x, 3) - 6*f(x).diff(x, 2) + 12*f(x).diff(x) - 8*f(x), + 'sol': [Eq(f(x), (C1 + x*(C2 + C3*x))*exp(2*x))], + 'slow': True, + }, + + 'lin_const_coeff_hom_17': { + 'eq': f(x).diff(x, 2) - 2*a*f(x).diff(x) + a**2*f(x), + 'sol': [Eq(f(x), (C1 + C2*x)*exp(a*x))], + 'slow': True, + }, + + 'lin_const_coeff_hom_18': { + 'eq': f(x).diff(x, 4) + 3*f(x).diff(x, 3), + 'sol': [Eq(f(x), C1 + C2*x + C3*x**2 + C4*exp(-3*x))], + 'slow': True, + }, + + 'lin_const_coeff_hom_19': { + 'eq': f(x).diff(x, 4) - 2*f(x).diff(x, 2), + 'sol': [Eq(f(x), C1 + C2*x + C3*exp(-sqrt(2)*x) + C4*exp(sqrt(2)*x))], + 'slow': True, + }, + + 'lin_const_coeff_hom_20': { + 'eq': f(x).diff(x, 4) + 2*f(x).diff(x, 3) - 11*f(x).diff(x, 2) - \ + 12*f(x).diff(x) + 36*f(x), + 'sol': [Eq(f(x), (C1 + C2*x)*exp(-3*x) + (C3 + C4*x)*exp(2*x))], + 'slow': True, + }, + + 'lin_const_coeff_hom_21': { + 'eq': 36*f(x).diff(x, 4) - 37*f(x).diff(x, 2) + 4*f(x).diff(x) + 5*f(x), + 'sol': [Eq(f(x), C1*exp(-x) + C2*exp(-x/3) + C3*exp(x/2) + C4*exp(x*Rational(5, 6)))], + 'slow': True, + }, + + 'lin_const_coeff_hom_22': { + 'eq': f(x).diff(x, 4) - 8*f(x).diff(x, 2) + 16*f(x), + 'sol': [Eq(f(x), (C1 + C2*x)*exp(-2*x) + (C3 + C4*x)*exp(2*x))], + 'slow': True, + }, + + 'lin_const_coeff_hom_23': { + 'eq': f(x).diff(x, 2) - 2*f(x).diff(x) + 5*f(x), + 'sol': [Eq(f(x), (C1*sin(2*x) + C2*cos(2*x))*exp(x))], + 'slow': True, + }, + + 'lin_const_coeff_hom_24': { + 'eq': f(x).diff(x, 2) - f(x).diff(x) + f(x), + 'sol': [Eq(f(x), (C1*sin(x*sqrt(3)/2) + C2*cos(x*sqrt(3)/2))*exp(x/2))], + 'slow': True, + }, + + 'lin_const_coeff_hom_25': { + 'eq': f(x).diff(x, 4) + 5*f(x).diff(x, 2) + 6*f(x), + 'sol': [Eq(f(x), + C1*sin(sqrt(2)*x) + C2*sin(sqrt(3)*x) + C3*cos(sqrt(2)*x) + C4*cos(sqrt(3)*x))], + 'slow': True, + }, + + 'lin_const_coeff_hom_26': { + 'eq': f(x).diff(x, 2) - 4*f(x).diff(x) + 20*f(x), + 'sol': [Eq(f(x), (C1*sin(4*x) + C2*cos(4*x))*exp(2*x))], + 'slow': True, + }, + + 'lin_const_coeff_hom_27': { + 'eq': f(x).diff(x, 4) + 4*f(x).diff(x, 2) + 4*f(x), + 'sol': [Eq(f(x), (C1 + C2*x)*sin(x*sqrt(2)) + (C3 + C4*x)*cos(x*sqrt(2)))], + 'slow': True, + }, + + 'lin_const_coeff_hom_28': { + 'eq': f(x).diff(x, 3) + 8*f(x), + 'sol': [Eq(f(x), (C1*sin(x*sqrt(3)) + C2*cos(x*sqrt(3)))*exp(x) + C3*exp(-2*x))], + 'slow': True, + }, + + 'lin_const_coeff_hom_29': { + 'eq': f(x).diff(x, 4) + 4*f(x).diff(x, 2), + 'sol': [Eq(f(x), C1 + C2*x + C3*sin(2*x) + C4*cos(2*x))], + 'slow': True, + }, + + 'lin_const_coeff_hom_30': { + 'eq': f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x), + 'sol': [Eq(f(x), C1 + (C2 + C3*x)*sin(x) + (C4 + C5*x)*cos(x))], + 'slow': True, + }, + + 'lin_const_coeff_hom_31': { + 'eq': f(x).diff(x, 4) + f(x).diff(x, 2) + f(x), + 'sol': [Eq(f(x), (C1*sin(sqrt(3)*x/2) + C2*cos(sqrt(3)*x/2))*exp(-x/2) + + (C3*sin(sqrt(3)*x/2) + C4*cos(sqrt(3)*x/2))*exp(x/2))], + 'slow': True, + }, + + 'lin_const_coeff_hom_32': { + 'eq': f(x).diff(x, 4) + 4*f(x).diff(x, 2) + f(x), + 'sol': [Eq(f(x), C1*sin(x*sqrt(-sqrt(3) + 2)) + C2*sin(x*sqrt(sqrt(3) + 2)) + + C3*cos(x*sqrt(-sqrt(3) + 2)) + C4*cos(x*sqrt(sqrt(3) + 2)))], + 'slow': True, + }, + + # One real root, two complex conjugate pairs + 'lin_const_coeff_hom_33': { + 'eq': f(x).diff(x, 5) + 11*f(x).diff(x) - 2*f(x), + 'sol': [Eq(f(x), + C5*exp(r1*x) + exp(re(r2)*x) * (C1*sin(im(r2)*x) + C2*cos(im(r2)*x)) + + exp(re(r4)*x) * (C3*sin(im(r4)*x) + C4*cos(im(r4)*x)))], + 'checkodesol_XFAIL':True, #It Hangs + }, + + # Three real roots, one complex conjugate pair + 'lin_const_coeff_hom_34': { + 'eq': f(x).diff(x,5) - 3*f(x).diff(x) + f(x), + 'sol': [Eq(f(x), + C3*exp(r6*x) + C4*exp(r7*x) + C5*exp(r8*x) + + exp(re(r9)*x) * (C1*sin(im(r9)*x) + C2*cos(im(r9)*x)))], + 'checkodesol_XFAIL':True, #It Hangs + }, + + # Five distinct real roots + 'lin_const_coeff_hom_35': { + 'eq': f(x).diff(x,5) - 100*f(x).diff(x,3) + 1000*f(x).diff(x) + f(x), + 'sol': [Eq(f(x), C1*exp(r11*x) + C2*exp(r12*x) + C3*exp(r13*x) + C4*exp(r14*x) + C5*exp(r15*x))], + 'checkodesol_XFAIL':True, #It Hangs + }, + + # Rational root and unsolvable quintic + 'lin_const_coeff_hom_36': { + 'eq': f(x).diff(x, 6) - 6*f(x).diff(x, 5) + 5*f(x).diff(x, 4) + 10*f(x).diff(x) - 50 * f(x), + 'sol': [Eq(f(x), + C5*exp(5*x) + + C6*exp(x*r16) + + exp(re(r17)*x) * (C1*sin(im(r17)*x) + C2*cos(im(r17)*x)) + + exp(re(r19)*x) * (C3*sin(im(r19)*x) + C4*cos(im(r19)*x)))], + 'checkodesol_XFAIL':True, #It Hangs + }, + + # Five double roots (this is (x**5 - x + 1)**2) + 'lin_const_coeff_hom_37': { + 'eq': f(x).diff(x, 10) - 2*f(x).diff(x, 6) + 2*f(x).diff(x, 5) + + f(x).diff(x, 2) - 2*f(x).diff(x, 1) + f(x), + 'sol': [Eq(f(x), (C1 + C2*x)*exp(x*r21) + (-((C3 + C4*x)*sin(x*im(r22))) + + (C5 + C6*x)*cos(x*im(r22)))*exp(x*re(r22)) + (-((C7 + C8*x)*sin(x*im(r24))) + + (C10*x + C9)*cos(x*im(r24)))*exp(x*re(r24)))], + 'checkodesol_XFAIL':True, #It Hangs + }, + + 'lin_const_coeff_hom_38': { + 'eq': Eq(sqrt(2) * f(x).diff(x,x,x) + f(x).diff(x), 0), + 'sol': [Eq(f(x), C1 + C2*sin(2**Rational(3, 4)*x/2) + C3*cos(2**Rational(3, 4)*x/2))], + }, + + 'lin_const_coeff_hom_39': { + 'eq': Eq(E * f(x).diff(x,x,x) + f(x).diff(x), 0), + 'sol': [Eq(f(x), C1 + C2*sin(x/sqrt(E)) + C3*cos(x/sqrt(E)))], + }, + + 'lin_const_coeff_hom_40': { + 'eq': Eq(pi * f(x).diff(x,x,x) + f(x).diff(x), 0), + 'sol': [Eq(f(x), C1 + C2*sin(x/sqrt(pi)) + C3*cos(x/sqrt(pi)))], + }, + + 'lin_const_coeff_hom_41': { + 'eq': Eq(I * f(x).diff(x,x,x) + f(x).diff(x), 0), + 'sol': [Eq(f(x), C1 + C2*exp(-sqrt(I)*x) + C3*exp(sqrt(I)*x))], + }, + + 'lin_const_coeff_hom_42': { + 'eq': f(x).diff(x, x) + y*f(x), + 'sol': [Eq(f(x), C1*exp(-x*sqrt(-y)) + C2*exp(x*sqrt(-y)))], + }, + + 'lin_const_coeff_hom_43': { + 'eq': Eq(9*f(x).diff(x, x) + f(x), 0), + 'sol': [Eq(f(x), C1*sin(x/3) + C2*cos(x/3))], + }, + + 'lin_const_coeff_hom_44': { + 'eq': Eq(9*f(x).diff(x, x), f(x)), + 'sol': [Eq(f(x), C1*exp(-x/3) + C2*exp(x/3))], + }, + + 'lin_const_coeff_hom_45': { + 'eq': Eq(f(x).diff(x, x) - 3*diff(f(x), x) + 2*f(x), 0), + 'sol': [Eq(f(x), (C1 + C2*exp(x))*exp(x))], + }, + + 'lin_const_coeff_hom_46': { + 'eq': Eq(f(x).diff(x, x) - 4*diff(f(x), x) + 4*f(x), 0), + 'sol': [Eq(f(x), (C1 + C2*x)*exp(2*x))], + }, + + # Type: 2nd order, constant coefficients (two real equal roots) + 'lin_const_coeff_hom_47': { + 'eq': Eq(f(x).diff(x, x) + 2*diff(f(x), x) + 3*f(x), 0), + 'sol': [Eq(f(x), (C1*sin(x*sqrt(2)) + C2*cos(x*sqrt(2)))*exp(-x))], + }, + + #These were from issue: https://github.com/sympy/sympy/issues/6247 + 'lin_const_coeff_hom_48': { + 'eq': f(x).diff(x, x) + 4*f(x), + 'sol': [Eq(f(x), C1*sin(2*x) + C2*cos(2*x))], + }, + } + } + + +@_add_example_keys +def _get_examples_ode_sol_1st_homogeneous_coeff_subs_dep_div_indep(): + return { + 'hint': "1st_homogeneous_coeff_subs_dep_div_indep", + 'func': f(x), + 'examples':{ + 'dep_div_indep_01': { + 'eq': f(x)/x*cos(f(x)/x) - (x/f(x)*sin(f(x)/x) + cos(f(x)/x))*f(x).diff(x), + 'sol': [Eq(log(x), C1 - log(f(x)*sin(f(x)/x)/x))], + 'slow': True + }, + + #indep_div_dep actually has a simpler solution for example 2 but it runs too slow. + 'dep_div_indep_02': { + 'eq': x*f(x).diff(x) - f(x) - x*sin(f(x)/x), + 'sol': [Eq(log(x), log(C1) + log(cos(f(x)/x) - 1)/2 - log(cos(f(x)/x) + 1)/2)], + 'simplify_flag':False, + }, + + 'dep_div_indep_03': { + 'eq': x*exp(f(x)/x) - f(x)*sin(f(x)/x) + x*sin(f(x)/x)*f(x).diff(x), + 'sol': [Eq(log(x), C1 + exp(-f(x)/x)*sin(f(x)/x)/2 + exp(-f(x)/x)*cos(f(x)/x)/2)], + 'slow': True + }, + + 'dep_div_indep_04': { + 'eq': f(x).diff(x) - f(x)/x + 1/sin(f(x)/x), + 'sol': [Eq(f(x), x*(-acos(C1 + log(x)) + 2*pi)), Eq(f(x), x*acos(C1 + log(x)))], + 'slow': True + }, + + # previous code was testing with these other solution: + # example5_solb = Eq(f(x), log(log(C1/x)**(-x))) + 'dep_div_indep_05': { + 'eq': x*exp(f(x)/x) + f(x) - x*f(x).diff(x), + 'sol': [Eq(f(x), log((1/(C1 - log(x)))**x))], + 'checkodesol_XFAIL':True, #(because of **x?) + }, + } + } + +@_add_example_keys +def _get_examples_ode_sol_linear_coefficients(): + return { + 'hint': "linear_coefficients", + 'func': f(x), + 'examples':{ + 'linear_coeff_01': { + 'eq': f(x).diff(x) + (3 + 2*f(x))/(x + 3), + 'sol': [Eq(f(x), C1/(x**2 + 6*x + 9) - Rational(3, 2))], + }, + } + } + +@_add_example_keys +def _get_examples_ode_sol_1st_homogeneous_coeff_best(): + return { + 'hint': "1st_homogeneous_coeff_best", + 'func': f(x), + 'examples':{ + # previous code was testing this with other solution: + # example1_solb = Eq(-f(x)/(1 + log(x/f(x))), C1) + '1st_homogeneous_coeff_best_01': { + 'eq': f(x) + (x*log(f(x)/x) - 2*x)*diff(f(x), x), + 'sol': [Eq(f(x), -exp(C1)*LambertW(-x*exp(-C1 + 1)))], + 'checkodesol_XFAIL':True, #(because of LambertW?) + }, + + '1st_homogeneous_coeff_best_02': { + 'eq': 2*f(x)*exp(x/f(x)) + f(x)*f(x).diff(x) - 2*x*exp(x/f(x))*f(x).diff(x), + 'sol': [Eq(log(f(x)), C1 - 2*exp(x/f(x)))], + }, + + # previous code was testing this with other solution: + # example3_solb = Eq(log(C1*x*sqrt(1/x)*sqrt(f(x))) + x**2/(2*f(x)**2), 0) + '1st_homogeneous_coeff_best_03': { + 'eq': 2*x**2*f(x) + f(x)**3 + (x*f(x)**2 - 2*x**3)*f(x).diff(x), + 'sol': [Eq(f(x), exp(2*C1 + LambertW(-2*x**4*exp(-4*C1))/2)/x)], + 'checkodesol_XFAIL':True, #(because of LambertW?) + }, + + '1st_homogeneous_coeff_best_04': { + 'eq': (x + sqrt(f(x)**2 - x*f(x)))*f(x).diff(x) - f(x), + 'sol': [Eq(log(f(x)), C1 - 2*sqrt(-x/f(x) + 1))], + 'slow': True, + }, + + '1st_homogeneous_coeff_best_05': { + 'eq': x + f(x) - (x - f(x))*f(x).diff(x), + 'sol': [Eq(log(x), C1 - log(sqrt(1 + f(x)**2/x**2)) + atan(f(x)/x))], + }, + + '1st_homogeneous_coeff_best_06': { + 'eq': x*f(x).diff(x) - f(x) - x*sin(f(x)/x), + 'sol': [Eq(f(x), 2*x*atan(C1*x))], + }, + + '1st_homogeneous_coeff_best_07': { + 'eq': x**2 + f(x)**2 - 2*x*f(x)*f(x).diff(x), + 'sol': [Eq(f(x), -sqrt(x*(C1 + x))), Eq(f(x), sqrt(x*(C1 + x)))], + }, + + '1st_homogeneous_coeff_best_08': { + 'eq': f(x)**2 + (x*sqrt(f(x)**2 - x**2) - x*f(x))*f(x).diff(x), + 'sol': [Eq(f(x), -C1*sqrt(-x/(x - 2*C1))), Eq(f(x), C1*sqrt(-x/(x - 2*C1)))], + 'checkodesol_XFAIL': True # solutions are valid in a range + }, + } + } + + +def _get_all_examples(): + all_examples = _get_examples_ode_sol_euler_homogeneous + \ + _get_examples_ode_sol_euler_undetermined_coeff + \ + _get_examples_ode_sol_euler_var_para + \ + _get_examples_ode_sol_factorable + \ + _get_examples_ode_sol_bernoulli + \ + _get_examples_ode_sol_nth_algebraic + \ + _get_examples_ode_sol_riccati + \ + _get_examples_ode_sol_1st_linear + \ + _get_examples_ode_sol_1st_exact + \ + _get_examples_ode_sol_almost_linear + \ + _get_examples_ode_sol_nth_order_reducible + \ + _get_examples_ode_sol_nth_linear_undetermined_coefficients + \ + _get_examples_ode_sol_liouville + \ + _get_examples_ode_sol_separable + \ + _get_examples_ode_sol_1st_rational_riccati + \ + _get_examples_ode_sol_nth_linear_var_of_parameters + \ + _get_examples_ode_sol_2nd_linear_bessel + \ + _get_examples_ode_sol_2nd_2F1_hypergeometric + \ + _get_examples_ode_sol_2nd_nonlinear_autonomous_conserved + \ + _get_examples_ode_sol_separable_reduced + \ + _get_examples_ode_sol_lie_group + \ + _get_examples_ode_sol_2nd_linear_airy + \ + _get_examples_ode_sol_nth_linear_constant_coeff_homogeneous +\ + _get_examples_ode_sol_1st_homogeneous_coeff_best +\ + _get_examples_ode_sol_1st_homogeneous_coeff_subs_dep_div_indep +\ + _get_examples_ode_sol_linear_coefficients + + return all_examples diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/ode/tests/test_subscheck.py b/.venv/lib/python3.12/site-packages/sympy/solvers/ode/tests/test_subscheck.py new file mode 100644 index 0000000000000000000000000000000000000000..799c2854e878208721b600767de350cda08cd7e5 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/ode/tests/test_subscheck.py @@ -0,0 +1,203 @@ +from sympy.core.function import (Derivative, Function, diff) +from sympy.core.numbers import (I, Rational, pi) +from sympy.core.relational import Eq +from sympy.core.symbol import (Symbol, symbols) +from sympy.functions.elementary.exponential import (exp, log) +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.functions.special.error_functions import (Ei, erf, erfi) +from sympy.integrals.integrals import Integral + +from sympy.solvers.ode.subscheck import checkodesol, checksysodesol + +from sympy.functions import besselj, bessely + +from sympy.testing.pytest import raises, slow + + +C0, C1, C2, C3, C4 = symbols('C0:5') +u, x, y, z = symbols('u,x:z', real=True) +f = Function('f') +g = Function('g') +h = Function('h') + + +@slow +def test_checkodesol(): + # For the most part, checkodesol is well tested in the tests below. + # These tests only handle cases not checked below. + raises(ValueError, lambda: checkodesol(f(x, y).diff(x), Eq(f(x, y), x))) + raises(ValueError, lambda: checkodesol(f(x).diff(x), Eq(f(x, y), + x), f(x, y))) + assert checkodesol(f(x).diff(x), Eq(f(x, y), x)) == \ + (False, -f(x).diff(x) + f(x, y).diff(x) - 1) + assert checkodesol(f(x).diff(x), Eq(f(x), x)) is not True + assert checkodesol(f(x).diff(x), Eq(f(x), x)) == (False, 1) + sol1 = Eq(f(x)**5 + 11*f(x) - 2*f(x) + x, 0) + assert checkodesol(diff(sol1.lhs, x), sol1) == (True, 0) + assert checkodesol(diff(sol1.lhs, x)*exp(f(x)), sol1) == (True, 0) + assert checkodesol(diff(sol1.lhs, x, 2), sol1) == (True, 0) + assert checkodesol(diff(sol1.lhs, x, 2)*exp(f(x)), sol1) == (True, 0) + assert checkodesol(diff(sol1.lhs, x, 3), sol1) == (True, 0) + assert checkodesol(diff(sol1.lhs, x, 3)*exp(f(x)), sol1) == (True, 0) + assert checkodesol(diff(sol1.lhs, x, 3), Eq(f(x), x*log(x))) == \ + (False, 60*x**4*((log(x) + 1)**2 + log(x))*( + log(x) + 1)*log(x)**2 - 5*x**4*log(x)**4 - 9) + assert checkodesol(diff(exp(f(x)) + x, x)*x, Eq(exp(f(x)) + x, 0)) == \ + (True, 0) + assert checkodesol(diff(exp(f(x)) + x, x)*x, Eq(exp(f(x)) + x, 0), + solve_for_func=False) == (True, 0) + assert checkodesol(f(x).diff(x, 2), [Eq(f(x), C1 + C2*x), + Eq(f(x), C2 + C1*x), Eq(f(x), C1*x + C2*x**2)]) == \ + [(True, 0), (True, 0), (False, C2)] + assert checkodesol(f(x).diff(x, 2), {Eq(f(x), C1 + C2*x), + Eq(f(x), C2 + C1*x), Eq(f(x), C1*x + C2*x**2)}) == \ + {(True, 0), (True, 0), (False, C2)} + assert checkodesol(f(x).diff(x) - 1/f(x)/2, Eq(f(x)**2, x)) == \ + [(True, 0), (True, 0)] + assert checkodesol(f(x).diff(x) - f(x), Eq(C1*exp(x), f(x))) == (True, 0) + # Based on test_1st_homogeneous_coeff_ode2_eq3sol. Make sure that + # checkodesol tries back substituting f(x) when it can. + eq3 = x*exp(f(x)/x) + f(x) - x*f(x).diff(x) + sol3 = Eq(f(x), log(log(C1/x)**(-x))) + assert not checkodesol(eq3, sol3)[1].has(f(x)) + # This case was failing intermittently depending on hash-seed: + eqn = Eq(Derivative(x*Derivative(f(x), x), x)/x, exp(x)) + sol = Eq(f(x), C1 + C2*log(x) + exp(x) - Ei(x)) + assert checkodesol(eqn, sol, order=2, solve_for_func=False)[0] + eq = x**2*(f(x).diff(x, 2)) + x*(f(x).diff(x)) + (2*x**2 +25)*f(x) + sol = Eq(f(x), C1*besselj(5*I, sqrt(2)*x) + C2*bessely(5*I, sqrt(2)*x)) + assert checkodesol(eq, sol) == (True, 0) + + eqs = [Eq(f(x).diff(x), f(x) + g(x)), Eq(g(x).diff(x), f(x) + g(x))] + sol = [Eq(f(x), -C1 + C2*exp(2*x)), Eq(g(x), C1 + C2*exp(2*x))] + assert checkodesol(eqs, sol) == (True, [0, 0]) + + +def test_checksysodesol(): + x, y, z = symbols('x, y, z', cls=Function) + t = Symbol('t') + eq = (Eq(diff(x(t),t), 9*y(t)), Eq(diff(y(t),t), 12*x(t))) + sol = [Eq(x(t), 9*C1*exp(-6*sqrt(3)*t) + 9*C2*exp(6*sqrt(3)*t)), \ + Eq(y(t), -6*sqrt(3)*C1*exp(-6*sqrt(3)*t) + 6*sqrt(3)*C2*exp(6*sqrt(3)*t))] + assert checksysodesol(eq, sol) == (True, [0, 0]) + + eq = (Eq(diff(x(t),t), 2*x(t) + 4*y(t)), Eq(diff(y(t),t), 12*x(t) + 41*y(t))) + sol = [Eq(x(t), 4*C1*exp(t*(-sqrt(1713)/2 + Rational(43, 2))) + 4*C2*exp(t*(sqrt(1713)/2 + \ + Rational(43, 2)))), Eq(y(t), C1*(-sqrt(1713)/2 + Rational(39, 2))*exp(t*(-sqrt(1713)/2 + \ + Rational(43, 2))) + C2*(Rational(39, 2) + sqrt(1713)/2)*exp(t*(sqrt(1713)/2 + Rational(43, 2))))] + assert checksysodesol(eq, sol) == (True, [0, 0]) + + eq = (Eq(diff(x(t),t), x(t) + y(t)), Eq(diff(y(t),t), -2*x(t) + 2*y(t))) + sol = [Eq(x(t), (C1*sin(sqrt(7)*t/2) + C2*cos(sqrt(7)*t/2))*exp(t*Rational(3, 2))), \ + Eq(y(t), ((C1/2 - sqrt(7)*C2/2)*sin(sqrt(7)*t/2) + (sqrt(7)*C1/2 + \ + C2/2)*cos(sqrt(7)*t/2))*exp(t*Rational(3, 2)))] + assert checksysodesol(eq, sol) == (True, [0, 0]) + + eq = (Eq(diff(x(t),t), x(t) + y(t) + 9), Eq(diff(y(t),t), 2*x(t) + 5*y(t) + 23)) + sol = [Eq(x(t), C1*exp(t*(-sqrt(6) + 3)) + C2*exp(t*(sqrt(6) + 3)) - \ + Rational(22, 3)), Eq(y(t), C1*(-sqrt(6) + 2)*exp(t*(-sqrt(6) + 3)) + C2*(2 + \ + sqrt(6))*exp(t*(sqrt(6) + 3)) - Rational(5, 3))] + assert checksysodesol(eq, sol) == (True, [0, 0]) + + eq = (Eq(diff(x(t),t), x(t) + y(t) + 81), Eq(diff(y(t),t), -2*x(t) + y(t) + 23)) + sol = [Eq(x(t), (C1*sin(sqrt(2)*t) + C2*cos(sqrt(2)*t))*exp(t) - Rational(58, 3)), \ + Eq(y(t), (sqrt(2)*C1*cos(sqrt(2)*t) - sqrt(2)*C2*sin(sqrt(2)*t))*exp(t) - Rational(185, 3))] + assert checksysodesol(eq, sol) == (True, [0, 0]) + + eq = (Eq(diff(x(t),t), 5*t*x(t) + 2*y(t)), Eq(diff(y(t),t), 2*x(t) + 5*t*y(t))) + sol = [Eq(x(t), (C1*exp(Integral(2, t).doit()) + C2*exp(-(Integral(2, t)).doit()))*\ + exp((Integral(5*t, t)).doit())), Eq(y(t), (C1*exp((Integral(2, t)).doit()) - \ + C2*exp(-(Integral(2, t)).doit()))*exp((Integral(5*t, t)).doit()))] + assert checksysodesol(eq, sol) == (True, [0, 0]) + + eq = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t)), Eq(diff(y(t),t), -t**2*x(t) + 5*t*y(t))) + sol = [Eq(x(t), (C1*cos((Integral(t**2, t)).doit()) + C2*sin((Integral(t**2, t)).doit()))*\ + exp((Integral(5*t, t)).doit())), Eq(y(t), (-C1*sin((Integral(t**2, t)).doit()) + \ + C2*cos((Integral(t**2, t)).doit()))*exp((Integral(5*t, t)).doit()))] + assert checksysodesol(eq, sol) == (True, [0, 0]) + + eq = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t)), Eq(diff(y(t),t), -t**2*x(t) + (5*t+9*t**2)*y(t))) + sol = [Eq(x(t), (C1*exp((-sqrt(77)/2 + Rational(9, 2))*(Integral(t**2, t)).doit()) + \ + C2*exp((sqrt(77)/2 + Rational(9, 2))*(Integral(t**2, t)).doit()))*exp((Integral(5*t, t)).doit())), \ + Eq(y(t), (C1*(-sqrt(77)/2 + Rational(9, 2))*exp((-sqrt(77)/2 + Rational(9, 2))*(Integral(t**2, t)).doit()) + \ + C2*(sqrt(77)/2 + Rational(9, 2))*exp((sqrt(77)/2 + Rational(9, 2))*(Integral(t**2, t)).doit()))*exp((Integral(5*t, t)).doit()))] + assert checksysodesol(eq, sol) == (True, [0, 0]) + + eq = (Eq(diff(x(t),t,t), 5*x(t) + 43*y(t)), Eq(diff(y(t),t,t), x(t) + 9*y(t))) + root0 = -sqrt(-sqrt(47) + 7) + root1 = sqrt(-sqrt(47) + 7) + root2 = -sqrt(sqrt(47) + 7) + root3 = sqrt(sqrt(47) + 7) + sol = [Eq(x(t), 43*C1*exp(t*root0) + 43*C2*exp(t*root1) + 43*C3*exp(t*root2) + 43*C4*exp(t*root3)), \ + Eq(y(t), C1*(root0**2 - 5)*exp(t*root0) + C2*(root1**2 - 5)*exp(t*root1) + \ + C3*(root2**2 - 5)*exp(t*root2) + C4*(root3**2 - 5)*exp(t*root3))] + assert checksysodesol(eq, sol) == (True, [0, 0]) + + eq = (Eq(diff(x(t),t,t), 8*x(t)+3*y(t)+31), Eq(diff(y(t),t,t), 9*x(t)+7*y(t)+12)) + root0 = -sqrt(-sqrt(109)/2 + Rational(15, 2)) + root1 = sqrt(-sqrt(109)/2 + Rational(15, 2)) + root2 = -sqrt(sqrt(109)/2 + Rational(15, 2)) + root3 = sqrt(sqrt(109)/2 + Rational(15, 2)) + sol = [Eq(x(t), 3*C1*exp(t*root0) + 3*C2*exp(t*root1) + 3*C3*exp(t*root2) + 3*C4*exp(t*root3) - Rational(181, 29)), \ + Eq(y(t), C1*(root0**2 - 8)*exp(t*root0) + C2*(root1**2 - 8)*exp(t*root1) + \ + C3*(root2**2 - 8)*exp(t*root2) + C4*(root3**2 - 8)*exp(t*root3) + Rational(183, 29))] + assert checksysodesol(eq, sol) == (True, [0, 0]) + + eq = (Eq(diff(x(t),t,t) - 9*diff(y(t),t) + 7*x(t),0), Eq(diff(y(t),t,t) + 9*diff(x(t),t) + 7*y(t),0)) + sol = [Eq(x(t), C1*cos(t*(Rational(9, 2) + sqrt(109)/2)) + C2*sin(t*(Rational(9, 2) + sqrt(109)/2)) + \ + C3*cos(t*(-sqrt(109)/2 + Rational(9, 2))) + C4*sin(t*(-sqrt(109)/2 + Rational(9, 2)))), Eq(y(t), -C1*sin(t*(Rational(9, 2) + sqrt(109)/2)) \ + + C2*cos(t*(Rational(9, 2) + sqrt(109)/2)) - C3*sin(t*(-sqrt(109)/2 + Rational(9, 2))) + C4*cos(t*(-sqrt(109)/2 + Rational(9, 2))))] + assert checksysodesol(eq, sol) == (True, [0, 0]) + + eq = (Eq(diff(x(t),t,t), 9*t*diff(y(t),t)-9*y(t)), Eq(diff(y(t),t,t),7*t*diff(x(t),t)-7*x(t))) + I1 = sqrt(6)*7**Rational(1, 4)*sqrt(pi)*erfi(sqrt(6)*7**Rational(1, 4)*t/2)/2 - exp(3*sqrt(7)*t**2/2)/t + I2 = -sqrt(6)*7**Rational(1, 4)*sqrt(pi)*erf(sqrt(6)*7**Rational(1, 4)*t/2)/2 - exp(-3*sqrt(7)*t**2/2)/t + sol = [Eq(x(t), C3*t + t*(9*C1*I1 + 9*C2*I2)), Eq(y(t), C4*t + t*(3*sqrt(7)*C1*I1 - 3*sqrt(7)*C2*I2))] + assert checksysodesol(eq, sol) == (True, [0, 0]) + + eq = (Eq(diff(x(t),t), 21*x(t)), Eq(diff(y(t),t), 17*x(t)+3*y(t)), Eq(diff(z(t),t), 5*x(t)+7*y(t)+9*z(t))) + sol = [Eq(x(t), C1*exp(21*t)), Eq(y(t), 17*C1*exp(21*t)/18 + C2*exp(3*t)), \ + Eq(z(t), 209*C1*exp(21*t)/216 - 7*C2*exp(3*t)/6 + C3*exp(9*t))] + assert checksysodesol(eq, sol) == (True, [0, 0, 0]) + + eq = (Eq(diff(x(t),t),3*y(t)-11*z(t)),Eq(diff(y(t),t),7*z(t)-3*x(t)),Eq(diff(z(t),t),11*x(t)-7*y(t))) + sol = [Eq(x(t), 7*C0 + sqrt(179)*C1*cos(sqrt(179)*t) + (77*C1/3 + 130*C2/3)*sin(sqrt(179)*t)), \ + Eq(y(t), 11*C0 + sqrt(179)*C2*cos(sqrt(179)*t) + (-58*C1/3 - 77*C2/3)*sin(sqrt(179)*t)), \ + Eq(z(t), 3*C0 + sqrt(179)*(-7*C1/3 - 11*C2/3)*cos(sqrt(179)*t) + (11*C1 - 7*C2)*sin(sqrt(179)*t))] + assert checksysodesol(eq, sol) == (True, [0, 0, 0]) + + eq = (Eq(3*diff(x(t),t),4*5*(y(t)-z(t))),Eq(4*diff(y(t),t),3*5*(z(t)-x(t))),Eq(5*diff(z(t),t),3*4*(x(t)-y(t)))) + sol = [Eq(x(t), C0 + 5*sqrt(2)*C1*cos(5*sqrt(2)*t) + (12*C1/5 + 164*C2/15)*sin(5*sqrt(2)*t)), \ + Eq(y(t), C0 + 5*sqrt(2)*C2*cos(5*sqrt(2)*t) + (-51*C1/10 - 12*C2/5)*sin(5*sqrt(2)*t)), \ + Eq(z(t), C0 + 5*sqrt(2)*(-9*C1/25 - 16*C2/25)*cos(5*sqrt(2)*t) + (12*C1/5 - 12*C2/5)*sin(5*sqrt(2)*t))] + assert checksysodesol(eq, sol) == (True, [0, 0, 0]) + + eq = (Eq(diff(x(t),t),4*x(t) - z(t)),Eq(diff(y(t),t),2*x(t)+2*y(t)-z(t)),Eq(diff(z(t),t),3*x(t)+y(t))) + sol = [Eq(x(t), C1*exp(2*t) + C2*t*exp(2*t) + C2*exp(2*t) + C3*t**2*exp(2*t)/2 + C3*t*exp(2*t) + C3*exp(2*t)), \ + Eq(y(t), C1*exp(2*t) + C2*t*exp(2*t) + C2*exp(2*t) + C3*t**2*exp(2*t)/2 + C3*t*exp(2*t)), \ + Eq(z(t), 2*C1*exp(2*t) + 2*C2*t*exp(2*t) + C2*exp(2*t) + C3*t**2*exp(2*t) + C3*t*exp(2*t) + C3*exp(2*t))] + assert checksysodesol(eq, sol) == (True, [0, 0, 0]) + + eq = (Eq(diff(x(t),t),4*x(t) - y(t) - 2*z(t)),Eq(diff(y(t),t),2*x(t) + y(t)- 2*z(t)),Eq(diff(z(t),t),5*x(t)-3*z(t))) + sol = [Eq(x(t), C1*exp(2*t) + C2*(-sin(t) + 3*cos(t)) + C3*(3*sin(t) + cos(t))), \ + Eq(y(t), C2*(-sin(t) + 3*cos(t)) + C3*(3*sin(t) + cos(t))), Eq(z(t), C1*exp(2*t) + 5*C2*cos(t) + 5*C3*sin(t))] + assert checksysodesol(eq, sol) == (True, [0, 0, 0]) + + eq = (Eq(diff(x(t),t),x(t)*y(t)**3), Eq(diff(y(t),t),y(t)**5)) + sol = [Eq(x(t), C1*exp((-1/(4*C2 + 4*t))**(Rational(-1, 4)))), Eq(y(t), -(-1/(4*C2 + 4*t))**Rational(1, 4)), \ + Eq(x(t), C1*exp(-1/(-1/(4*C2 + 4*t))**Rational(1, 4))), Eq(y(t), (-1/(4*C2 + 4*t))**Rational(1, 4)), \ + Eq(x(t), C1*exp(-I/(-1/(4*C2 + 4*t))**Rational(1, 4))), Eq(y(t), -I*(-1/(4*C2 + 4*t))**Rational(1, 4)), \ + Eq(x(t), C1*exp(I/(-1/(4*C2 + 4*t))**Rational(1, 4))), Eq(y(t), I*(-1/(4*C2 + 4*t))**Rational(1, 4))] + assert checksysodesol(eq, sol) == (True, [0, 0]) + + eq = (Eq(diff(x(t),t), exp(3*x(t))*y(t)**3),Eq(diff(y(t),t), y(t)**5)) + sol = [Eq(x(t), -log(C1 - 3/(-1/(4*C2 + 4*t))**Rational(1, 4))/3), Eq(y(t), -(-1/(4*C2 + 4*t))**Rational(1, 4)), \ + Eq(x(t), -log(C1 + 3/(-1/(4*C2 + 4*t))**Rational(1, 4))/3), Eq(y(t), (-1/(4*C2 + 4*t))**Rational(1, 4)), \ + Eq(x(t), -log(C1 + 3*I/(-1/(4*C2 + 4*t))**Rational(1, 4))/3), Eq(y(t), -I*(-1/(4*C2 + 4*t))**Rational(1, 4)), \ + Eq(x(t), -log(C1 - 3*I/(-1/(4*C2 + 4*t))**Rational(1, 4))/3), Eq(y(t), I*(-1/(4*C2 + 4*t))**Rational(1, 4))] + assert checksysodesol(eq, sol) == (True, [0, 0]) + + eq = (Eq(x(t),t*diff(x(t),t)+diff(x(t),t)*diff(y(t),t)), Eq(y(t),t*diff(y(t),t)+diff(y(t),t)**2)) + sol = {Eq(x(t), C1*C2 + C1*t), Eq(y(t), C2**2 + C2*t)} + assert checksysodesol(eq, sol) == (True, [0, 0]) diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/ode/tests/test_systems.py b/.venv/lib/python3.12/site-packages/sympy/solvers/ode/tests/test_systems.py new file mode 100644 index 0000000000000000000000000000000000000000..9d206129dfcf38c7b8c2e0ab42bd875003253f35 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/ode/tests/test_systems.py @@ -0,0 +1,2544 @@ +from sympy.core.function import (Derivative, Function, diff) +from sympy.core.mul import Mul +from sympy.core.numbers import (I, Rational, pi) +from sympy.core.relational import Eq +from sympy.core.singleton import S +from sympy.core.symbol import (Symbol, symbols) +from sympy.functions.elementary.hyperbolic import sinh +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.matrices.dense import Matrix +from sympy.core.containers import Tuple +from sympy.functions import exp, cos, sin, log, Ci, Si, erf, erfi +from sympy.matrices import dotprodsimp, NonSquareMatrixError +from sympy.solvers.ode import dsolve +from sympy.solvers.ode.ode import constant_renumber +from sympy.solvers.ode.subscheck import checksysodesol +from sympy.solvers.ode.systems import (_classify_linear_system, linear_ode_to_matrix, + ODEOrderError, ODENonlinearError, _simpsol, + _is_commutative_anti_derivative, linodesolve, + canonical_odes, dsolve_system, _component_division, + _eqs2dict, _dict2graph) +from sympy.functions import airyai, airybi +from sympy.integrals.integrals import Integral +from sympy.simplify.ratsimp import ratsimp +from sympy.testing.pytest import raises, slow, tooslow, XFAIL + + +C0, C1, C2, C3, C4, C5, C6, C7, C8, C9, C10 = symbols('C0:11') +x = symbols('x') +f = Function('f') +g = Function('g') +h = Function('h') + + +def test_linear_ode_to_matrix(): + f, g, h = symbols("f, g, h", cls=Function) + t = Symbol("t") + funcs = [f(t), g(t), h(t)] + f1 = f(t).diff(t) + g1 = g(t).diff(t) + h1 = h(t).diff(t) + f2 = f(t).diff(t, 2) + g2 = g(t).diff(t, 2) + h2 = h(t).diff(t, 2) + + eqs_1 = [Eq(f1, g(t)), Eq(g1, f(t))] + sol_1 = ([Matrix([[1, 0], [0, 1]]), Matrix([[ 0, 1], [1, 0]])], Matrix([[0],[0]])) + assert linear_ode_to_matrix(eqs_1, funcs[:-1], t, 1) == sol_1 + + eqs_2 = [Eq(f1, f(t) + 2*g(t)), Eq(g1, h(t)), Eq(h1, g(t) + h(t) + f(t))] + sol_2 = ([Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]]), Matrix([[1, 2, 0], [ 0, 0, 1], [1, 1, 1]])], + Matrix([[0], [0], [0]])) + assert linear_ode_to_matrix(eqs_2, funcs, t, 1) == sol_2 + + eqs_3 = [Eq(2*f1 + 3*h1, f(t) + g(t)), Eq(4*h1 + 5*g1, f(t) + h(t)), Eq(5*f1 + 4*g1, g(t) + h(t))] + sol_3 = ([Matrix([[2, 0, 3], [0, 5, 4], [5, 4, 0]]), Matrix([[1, 1, 0], [1, 0, 1], [0, 1, 1]])], + Matrix([[0], [0], [0]])) + assert linear_ode_to_matrix(eqs_3, funcs, t, 1) == sol_3 + + eqs_4 = [Eq(f2 + h(t), f1 + g(t)), Eq(2*h2 + g2 + g1 + g(t), 0), Eq(3*h1, 4)] + sol_4 = ([Matrix([[1, 0, 0], [0, 1, 2], [0, 0, 0]]), Matrix([[1, 0, 0], [0, -1, 0], [0, 0, -3]]), + Matrix([[0, 1, -1], [0, -1, 0], [0, 0, 0]])], Matrix([[0], [0], [4]])) + assert linear_ode_to_matrix(eqs_4, funcs, t, 2) == sol_4 + + eqs_5 = [Eq(f2, g(t)), Eq(f1 + g1, f(t))] + raises(ODEOrderError, lambda: linear_ode_to_matrix(eqs_5, funcs[:-1], t, 1)) + + eqs_6 = [Eq(f1, f(t)**2), Eq(g1, f(t) + g(t))] + raises(ODENonlinearError, lambda: linear_ode_to_matrix(eqs_6, funcs[:-1], t, 1)) + + +def test__classify_linear_system(): + x, y, z, w = symbols('x, y, z, w', cls=Function) + t, k, l = symbols('t k l') + x1 = diff(x(t), t) + y1 = diff(y(t), t) + z1 = diff(z(t), t) + w1 = diff(w(t), t) + x2 = diff(x(t), t, t) + y2 = diff(y(t), t, t) + funcs = [x(t), y(t)] + funcs_2 = funcs + [z(t), w(t)] + + eqs_1 = (5 * x1 + 12 * x(t) - 6 * (y(t)), (2 * y1 - 11 * t * x(t) + 3 * y(t) + t)) + assert _classify_linear_system(eqs_1, funcs, t) is None + + eqs_2 = (5 * (x1**2) + 12 * x(t) - 6 * (y(t)), (2 * y1 - 11 * t * x(t) + 3 * y(t) + t)) + sol2 = {'is_implicit': True, + 'canon_eqs': [[Eq(Derivative(x(t), t), -sqrt(-12*x(t)/5 + 6*y(t)/5)), + Eq(Derivative(y(t), t), 11*t*x(t)/2 - t/2 - 3*y(t)/2)], + [Eq(Derivative(x(t), t), sqrt(-12*x(t)/5 + 6*y(t)/5)), + Eq(Derivative(y(t), t), 11*t*x(t)/2 - t/2 - 3*y(t)/2)]]} + assert _classify_linear_system(eqs_2, funcs, t) == sol2 + + eqs_2_1 = [Eq(Derivative(x(t), t), -sqrt(-12*x(t)/5 + 6*y(t)/5)), + Eq(Derivative(y(t), t), 11*t*x(t)/2 - t/2 - 3*y(t)/2)] + assert _classify_linear_system(eqs_2_1, funcs, t) is None + + eqs_2_2 = [Eq(Derivative(x(t), t), sqrt(-12*x(t)/5 + 6*y(t)/5)), + Eq(Derivative(y(t), t), 11*t*x(t)/2 - t/2 - 3*y(t)/2)] + assert _classify_linear_system(eqs_2_2, funcs, t) is None + + eqs_3 = (5 * x1 + 12 * x(t) - 6 * (y(t)), (2 * y1 - 11 * x(t) + 3 * y(t)), (5 * w1 + z(t)), (z1 + w(t))) + answer_3 = {'no_of_equation': 4, + 'eq': (12*x(t) - 6*y(t) + 5*Derivative(x(t), t), + -11*x(t) + 3*y(t) + 2*Derivative(y(t), t), + z(t) + 5*Derivative(w(t), t), + w(t) + Derivative(z(t), t)), + 'func': [x(t), y(t), z(t), w(t)], + 'order': {x(t): 1, y(t): 1, z(t): 1, w(t): 1}, + 'is_linear': True, + 'is_constant': True, + 'is_homogeneous': True, + 'func_coeff': -Matrix([ + [Rational(12, 5), Rational(-6, 5), 0, 0], + [Rational(-11, 2), Rational(3, 2), 0, 0], + [0, 0, 0, 1], + [0, 0, Rational(1, 5), 0]]), + 'type_of_equation': 'type1', + 'is_general': True} + assert _classify_linear_system(eqs_3, funcs_2, t) == answer_3 + + eqs_4 = (5 * x1 + 12 * x(t) - 6 * (y(t)), (2 * y1 - 11 * x(t) + 3 * y(t)), (z1 - w(t)), (w1 - z(t))) + answer_4 = {'no_of_equation': 4, + 'eq': (12 * x(t) - 6 * y(t) + 5 * Derivative(x(t), t), + -11 * x(t) + 3 * y(t) + 2 * Derivative(y(t), t), + -w(t) + Derivative(z(t), t), + -z(t) + Derivative(w(t), t)), + 'func': [x(t), y(t), z(t), w(t)], + 'order': {x(t): 1, y(t): 1, z(t): 1, w(t): 1}, + 'is_linear': True, + 'is_constant': True, + 'is_homogeneous': True, + 'func_coeff': -Matrix([ + [Rational(12, 5), Rational(-6, 5), 0, 0], + [Rational(-11, 2), Rational(3, 2), 0, 0], + [0, 0, 0, -1], + [0, 0, -1, 0]]), + 'type_of_equation': 'type1', + 'is_general': True} + assert _classify_linear_system(eqs_4, funcs_2, t) == answer_4 + + eqs_5 = (5*x1 + 12*x(t) - 6*(y(t)) + x2, (2*y1 - 11*x(t) + 3*y(t)), (z1 - w(t)), (w1 - z(t))) + answer_5 = {'no_of_equation': 4, 'eq': (12*x(t) - 6*y(t) + 5*Derivative(x(t), t) + Derivative(x(t), (t, 2)), + -11*x(t) + 3*y(t) + 2*Derivative(y(t), t), -w(t) + Derivative(z(t), t), -z(t) + Derivative(w(t), + t)), 'func': [x(t), y(t), z(t), w(t)], 'order': {x(t): 2, y(t): 1, z(t): 1, w(t): 1}, 'is_linear': + True, 'is_homogeneous': True, 'is_general': True, 'type_of_equation': 'type0', 'is_higher_order': True} + assert _classify_linear_system(eqs_5, funcs_2, t) == answer_5 + + eqs_6 = (Eq(x1, 3*y(t) - 11*z(t)), Eq(y1, 7*z(t) - 3*x(t)), Eq(z1, 11*x(t) - 7*y(t))) + answer_6 = {'no_of_equation': 3, 'eq': (Eq(Derivative(x(t), t), 3*y(t) - 11*z(t)), Eq(Derivative(y(t), t), -3*x(t) + 7*z(t)), + Eq(Derivative(z(t), t), 11*x(t) - 7*y(t))), 'func': [x(t), y(t), z(t)], 'order': {x(t): 1, y(t): 1, z(t): 1}, + 'is_linear': True, 'is_constant': True, 'is_homogeneous': True, + 'func_coeff': -Matrix([ + [ 0, -3, 11], + [ 3, 0, -7], + [-11, 7, 0]]), + 'type_of_equation': 'type1', 'is_general': True} + + assert _classify_linear_system(eqs_6, funcs_2[:-1], t) == answer_6 + + eqs_7 = (Eq(x1, y(t)), Eq(y1, x(t))) + answer_7 = {'no_of_equation': 2, 'eq': (Eq(Derivative(x(t), t), y(t)), Eq(Derivative(y(t), t), x(t))), + 'func': [x(t), y(t)], 'order': {x(t): 1, y(t): 1}, 'is_linear': True, 'is_constant': True, + 'is_homogeneous': True, 'func_coeff': -Matrix([ + [ 0, -1], + [-1, 0]]), + 'type_of_equation': 'type1', 'is_general': True} + assert _classify_linear_system(eqs_7, funcs, t) == answer_7 + + eqs_8 = (Eq(x1, 21*x(t)), Eq(y1, 17*x(t) + 3*y(t)), Eq(z1, 5*x(t) + 7*y(t) + 9*z(t))) + answer_8 = {'no_of_equation': 3, 'eq': (Eq(Derivative(x(t), t), 21*x(t)), Eq(Derivative(y(t), t), 17*x(t) + 3*y(t)), + Eq(Derivative(z(t), t), 5*x(t) + 7*y(t) + 9*z(t))), 'func': [x(t), y(t), z(t)], 'order': {x(t): 1, y(t): 1, z(t): 1}, + 'is_linear': True, 'is_constant': True, 'is_homogeneous': True, + 'func_coeff': -Matrix([ + [-21, 0, 0], + [-17, -3, 0], + [ -5, -7, -9]]), + 'type_of_equation': 'type1', 'is_general': True} + + assert _classify_linear_system(eqs_8, funcs_2[:-1], t) == answer_8 + + eqs_9 = (Eq(x1, 4*x(t) + 5*y(t) + 2*z(t)), Eq(y1, x(t) + 13*y(t) + 9*z(t)), Eq(z1, 32*x(t) + 41*y(t) + 11*z(t))) + answer_9 = {'no_of_equation': 3, 'eq': (Eq(Derivative(x(t), t), 4*x(t) + 5*y(t) + 2*z(t)), + Eq(Derivative(y(t), t), x(t) + 13*y(t) + 9*z(t)), Eq(Derivative(z(t), t), 32*x(t) + 41*y(t) + 11*z(t))), + 'func': [x(t), y(t), z(t)], 'order': {x(t): 1, y(t): 1, z(t): 1}, 'is_linear': True, + 'is_constant': True, 'is_homogeneous': True, + 'func_coeff': -Matrix([ + [ -4, -5, -2], + [ -1, -13, -9], + [-32, -41, -11]]), + 'type_of_equation': 'type1', 'is_general': True} + assert _classify_linear_system(eqs_9, funcs_2[:-1], t) == answer_9 + + eqs_10 = (Eq(3*x1, 4*5*(y(t) - z(t))), Eq(4*y1, 3*5*(z(t) - x(t))), Eq(5*z1, 3*4*(x(t) - y(t)))) + answer_10 = {'no_of_equation': 3, 'eq': (Eq(3*Derivative(x(t), t), 20*y(t) - 20*z(t)), + Eq(4*Derivative(y(t), t), -15*x(t) + 15*z(t)), Eq(5*Derivative(z(t), t), 12*x(t) - 12*y(t))), + 'func': [x(t), y(t), z(t)], 'order': {x(t): 1, y(t): 1, z(t): 1}, 'is_linear': True, + 'is_constant': True, 'is_homogeneous': True, + 'func_coeff': -Matrix([ + [ 0, Rational(-20, 3), Rational(20, 3)], + [Rational(15, 4), 0, Rational(-15, 4)], + [Rational(-12, 5), Rational(12, 5), 0]]), + 'type_of_equation': 'type1', 'is_general': True} + assert _classify_linear_system(eqs_10, funcs_2[:-1], t) == answer_10 + + eq11 = (Eq(x1, 3*y(t) - 11*z(t)), Eq(y1, 7*z(t) - 3*x(t)), Eq(z1, 11*x(t) - 7*y(t))) + sol11 = {'no_of_equation': 3, 'eq': (Eq(Derivative(x(t), t), 3*y(t) - 11*z(t)), Eq(Derivative(y(t), t), -3*x(t) + 7*z(t)), + Eq(Derivative(z(t), t), 11*x(t) - 7*y(t))), 'func': [x(t), y(t), z(t)], 'order': {x(t): 1, y(t): 1, z(t): 1}, + 'is_linear': True, 'is_constant': True, 'is_homogeneous': True, 'func_coeff': -Matrix([ + [ 0, -3, 11], [ 3, 0, -7], [-11, 7, 0]]), 'type_of_equation': 'type1', 'is_general': True} + assert _classify_linear_system(eq11, funcs_2[:-1], t) == sol11 + + eq12 = (Eq(Derivative(x(t), t), y(t)), Eq(Derivative(y(t), t), x(t))) + sol12 = {'no_of_equation': 2, 'eq': (Eq(Derivative(x(t), t), y(t)), Eq(Derivative(y(t), t), x(t))), + 'func': [x(t), y(t)], 'order': {x(t): 1, y(t): 1}, 'is_linear': True, 'is_constant': True, + 'is_homogeneous': True, 'func_coeff': -Matrix([ + [0, -1], + [-1, 0]]), 'type_of_equation': 'type1', 'is_general': True} + assert _classify_linear_system(eq12, [x(t), y(t)], t) == sol12 + + eq13 = (Eq(Derivative(x(t), t), 21*x(t)), Eq(Derivative(y(t), t), 17*x(t) + 3*y(t)), + Eq(Derivative(z(t), t), 5*x(t) + 7*y(t) + 9*z(t))) + sol13 = {'no_of_equation': 3, 'eq': ( + Eq(Derivative(x(t), t), 21 * x(t)), Eq(Derivative(y(t), t), 17 * x(t) + 3 * y(t)), + Eq(Derivative(z(t), t), 5 * x(t) + 7 * y(t) + 9 * z(t))), 'func': [x(t), y(t), z(t)], + 'order': {x(t): 1, y(t): 1, z(t): 1}, 'is_linear': True, 'is_constant': True, 'is_homogeneous': True, + 'func_coeff': -Matrix([ + [-21, 0, 0], + [-17, -3, 0], + [-5, -7, -9]]), 'type_of_equation': 'type1', 'is_general': True} + assert _classify_linear_system(eq13, [x(t), y(t), z(t)], t) == sol13 + + eq14 = ( + Eq(Derivative(x(t), t), 4*x(t) + 5*y(t) + 2*z(t)), Eq(Derivative(y(t), t), x(t) + 13*y(t) + 9*z(t)), + Eq(Derivative(z(t), t), 32*x(t) + 41*y(t) + 11*z(t))) + sol14 = {'no_of_equation': 3, 'eq': ( + Eq(Derivative(x(t), t), 4 * x(t) + 5 * y(t) + 2 * z(t)), Eq(Derivative(y(t), t), x(t) + 13 * y(t) + 9 * z(t)), + Eq(Derivative(z(t), t), 32 * x(t) + 41 * y(t) + 11 * z(t))), 'func': [x(t), y(t), z(t)], + 'order': {x(t): 1, y(t): 1, z(t): 1}, 'is_linear': True, 'is_constant': True, 'is_homogeneous': True, + 'func_coeff': -Matrix([ + [-4, -5, -2], + [-1, -13, -9], + [-32, -41, -11]]), 'type_of_equation': 'type1', 'is_general': True} + assert _classify_linear_system(eq14, [x(t), y(t), z(t)], t) == sol14 + + eq15 = (Eq(3*Derivative(x(t), t), 20*y(t) - 20*z(t)), Eq(4*Derivative(y(t), t), -15*x(t) + 15*z(t)), + Eq(5*Derivative(z(t), t), 12*x(t) - 12*y(t))) + sol15 = {'no_of_equation': 3, 'eq': ( + Eq(3 * Derivative(x(t), t), 20 * y(t) - 20 * z(t)), Eq(4 * Derivative(y(t), t), -15 * x(t) + 15 * z(t)), + Eq(5 * Derivative(z(t), t), 12 * x(t) - 12 * y(t))), 'func': [x(t), y(t), z(t)], + 'order': {x(t): 1, y(t): 1, z(t): 1}, 'is_linear': True, 'is_constant': True, 'is_homogeneous': True, + 'func_coeff': -Matrix([ + [0, Rational(-20, 3), Rational(20, 3)], + [Rational(15, 4), 0, Rational(-15, 4)], + [Rational(-12, 5), Rational(12, 5), 0]]), 'type_of_equation': 'type1', 'is_general': True} + assert _classify_linear_system(eq15, [x(t), y(t), z(t)], t) == sol15 + + # Constant coefficient homogeneous ODEs + eq1 = (Eq(diff(x(t), t), x(t) + y(t) + 9), Eq(diff(y(t), t), 2*x(t) + 5*y(t) + 23)) + sol1 = {'no_of_equation': 2, 'eq': (Eq(Derivative(x(t), t), x(t) + y(t) + 9), + Eq(Derivative(y(t), t), 2*x(t) + 5*y(t) + 23)), 'func': [x(t), y(t)], + 'order': {x(t): 1, y(t): 1}, 'is_linear': True, 'is_constant': True, 'is_homogeneous': False, 'is_general': True, + 'func_coeff': -Matrix([[-1, -1], [-2, -5]]), 'rhs': Matrix([[ 9], [23]]), 'type_of_equation': 'type2'} + assert _classify_linear_system(eq1, funcs, t) == sol1 + + # Non constant coefficient homogeneous ODEs + eq1 = (Eq(diff(x(t), t), 5*t*x(t) + 2*y(t)), Eq(diff(y(t), t), 2*x(t) + 5*t*y(t))) + sol1 = {'no_of_equation': 2, 'eq': (Eq(Derivative(x(t), t), 5*t*x(t) + 2*y(t)), Eq(Derivative(y(t), t), 5*t*y(t) + 2*x(t))), + 'func': [x(t), y(t)], 'order': {x(t): 1, y(t): 1}, 'is_linear': True, 'is_constant': False, + 'is_homogeneous': True, 'func_coeff': -Matrix([ [-5*t, -2], [ -2, -5*t]]), 'commutative_antiderivative': Matrix([ + [5*t**2/2, 2*t], [ 2*t, 5*t**2/2]]), 'type_of_equation': 'type3', 'is_general': True} + assert _classify_linear_system(eq1, funcs, t) == sol1 + + # Non constant coefficient non-homogeneous ODEs + eq1 = [Eq(x1, x(t) + t*y(t) + t), Eq(y1, t*x(t) + y(t))] + sol1 = {'no_of_equation': 2, 'eq': [Eq(Derivative(x(t), t), t*y(t) + t + x(t)), Eq(Derivative(y(t), t), + t*x(t) + y(t))], 'func': [x(t), y(t)], 'order': {x(t): 1, y(t): 1}, 'is_linear': True, + 'is_constant': False, 'is_homogeneous': False, 'is_general': True, 'func_coeff': -Matrix([ [-1, -t], + [-t, -1]]), 'commutative_antiderivative': Matrix([ [ t, t**2/2], [t**2/2, t]]), 'rhs': + Matrix([ [t], [0]]), 'type_of_equation': 'type4'} + assert _classify_linear_system(eq1, funcs, t) == sol1 + + eq2 = [Eq(x1, t*x(t) + t*y(t) + t), Eq(y1, t*x(t) + t*y(t) + cos(t))] + sol2 = {'no_of_equation': 2, 'eq': [Eq(Derivative(x(t), t), t*x(t) + t*y(t) + t), Eq(Derivative(y(t), t), + t*x(t) + t*y(t) + cos(t))], 'func': [x(t), y(t)], 'order': {x(t): 1, y(t): 1}, 'is_linear': True, + 'is_homogeneous': False, 'is_general': True, 'rhs': Matrix([ [ t], [cos(t)]]), 'func_coeff': + Matrix([ [t, t], [t, t]]), 'is_constant': False, 'type_of_equation': 'type4', + 'commutative_antiderivative': Matrix([ [t**2/2, t**2/2], [t**2/2, t**2/2]])} + assert _classify_linear_system(eq2, funcs, t) == sol2 + + eq3 = [Eq(x1, t*(x(t) + y(t) + z(t) + 1)), Eq(y1, t*(x(t) + y(t) + z(t))), Eq(z1, t*(x(t) + y(t) + z(t)))] + sol3 = {'no_of_equation': 3, 'eq': [Eq(Derivative(x(t), t), t*(x(t) + y(t) + z(t) + 1)), + Eq(Derivative(y(t), t), t*(x(t) + y(t) + z(t))), Eq(Derivative(z(t), t), t*(x(t) + y(t) + z(t)))], + 'func': [x(t), y(t), z(t)], 'order': {x(t): 1, y(t): 1, z(t): 1}, 'is_linear': True, 'is_constant': + False, 'is_homogeneous': False, 'is_general': True, 'func_coeff': -Matrix([ [-t, -t, -t], [-t, -t, + -t], [-t, -t, -t]]), 'commutative_antiderivative': Matrix([ [t**2/2, t**2/2, t**2/2], [t**2/2, + t**2/2, t**2/2], [t**2/2, t**2/2, t**2/2]]), 'rhs': Matrix([ [t], [0], [0]]), 'type_of_equation': + 'type4'} + assert _classify_linear_system(eq3, funcs_2[:-1], t) == sol3 + + eq4 = [Eq(x1, x(t) + y(t) + t*z(t) + 1), Eq(y1, x(t) + t*y(t) + z(t) + 10), Eq(z1, t*x(t) + y(t) + z(t) + t)] + sol4 = {'no_of_equation': 3, 'eq': [Eq(Derivative(x(t), t), t*z(t) + x(t) + y(t) + 1), Eq(Derivative(y(t), + t), t*y(t) + x(t) + z(t) + 10), Eq(Derivative(z(t), t), t*x(t) + t + y(t) + z(t))], 'func': [x(t), + y(t), z(t)], 'order': {x(t): 1, y(t): 1, z(t): 1}, 'is_linear': True, 'is_constant': False, + 'is_homogeneous': False, 'is_general': True, 'func_coeff': -Matrix([ [-1, -1, -t], [-1, -t, -1], [-t, + -1, -1]]), 'commutative_antiderivative': Matrix([ [ t, t, t**2/2], [ t, t**2/2, + t], [t**2/2, t, t]]), 'rhs': Matrix([ [ 1], [10], [ t]]), 'type_of_equation': 'type4'} + assert _classify_linear_system(eq4, funcs_2[:-1], t) == sol4 + + sum_terms = t*(x(t) + y(t) + z(t) + w(t)) + eq5 = [Eq(x1, sum_terms), Eq(y1, sum_terms), Eq(z1, sum_terms + 1), Eq(w1, sum_terms)] + sol5 = {'no_of_equation': 4, 'eq': [Eq(Derivative(x(t), t), t*(w(t) + x(t) + y(t) + z(t))), + Eq(Derivative(y(t), t), t*(w(t) + x(t) + y(t) + z(t))), Eq(Derivative(z(t), t), t*(w(t) + x(t) + + y(t) + z(t)) + 1), Eq(Derivative(w(t), t), t*(w(t) + x(t) + y(t) + z(t)))], 'func': [x(t), y(t), + z(t), w(t)], 'order': {x(t): 1, y(t): 1, z(t): 1, w(t): 1}, 'is_linear': True, 'is_constant': False, + 'is_homogeneous': False, 'is_general': True, 'func_coeff': -Matrix([ [-t, -t, -t, -t], [-t, -t, -t, + -t], [-t, -t, -t, -t], [-t, -t, -t, -t]]), 'commutative_antiderivative': Matrix([ [t**2/2, t**2/2, + t**2/2, t**2/2], [t**2/2, t**2/2, t**2/2, t**2/2], [t**2/2, t**2/2, t**2/2, t**2/2], [t**2/2, + t**2/2, t**2/2, t**2/2]]), 'rhs': Matrix([ [0], [0], [1], [0]]), 'type_of_equation': 'type4'} + assert _classify_linear_system(eq5, funcs_2, t) == sol5 + + # Second Order + t_ = symbols("t_") + + eq1 = (Eq(9*x(t) + 7*y(t) + 4*Derivative(x(t), t) + Derivative(x(t), (t, 2)) + 3*Derivative(y(t), t), 11*exp(I*t)), + Eq(3*x(t) + 12*y(t) + 5*Derivative(x(t), t) + 8*Derivative(y(t), t) + Derivative(y(t), (t, 2)), 2*exp(I*t))) + sol1 = {'no_of_equation': 2, 'eq': (Eq(9*x(t) + 7*y(t) + 4*Derivative(x(t), t) + Derivative(x(t), (t, 2)) + + 3*Derivative(y(t), t), 11*exp(I*t)), Eq(3*x(t) + 12*y(t) + 5*Derivative(x(t), t) + + 8*Derivative(y(t), t) + Derivative(y(t), (t, 2)), 2*exp(I*t))), 'func': [x(t), y(t)], 'order': + {x(t): 2, y(t): 2}, 'is_linear': True, 'is_homogeneous': False, 'is_general': True, 'rhs': Matrix([ + [11*exp(I*t)], [ 2*exp(I*t)]]), 'type_of_equation': 'type0', 'is_second_order': True, + 'is_higher_order': True} + assert _classify_linear_system(eq1, funcs, t) == sol1 + + eq2 = (Eq((4*t**2 + 7*t + 1)**2*Derivative(x(t), (t, 2)), 5*x(t) + 35*y(t)), + Eq((4*t**2 + 7*t + 1)**2*Derivative(y(t), (t, 2)), x(t) + 9*y(t))) + sol2 = {'no_of_equation': 2, 'eq': (Eq((4*t**2 + 7*t + 1)**2*Derivative(x(t), (t, 2)), 5*x(t) + 35*y(t)), + Eq((4*t**2 + 7*t + 1)**2*Derivative(y(t), (t, 2)), x(t) + 9*y(t))), 'func': [x(t), y(t)], 'order': + {x(t): 2, y(t): 2}, 'is_linear': True, 'is_homogeneous': True, 'is_general': True, + 'type_of_equation': 'type2', 'A0': Matrix([ [Rational(53, 4), 35], [ 1, Rational(69, 4)]]), 'g(t)': sqrt(4*t**2 + 7*t + + 1), 'tau': sqrt(33)*log(t - sqrt(33)/8 + Rational(7, 8))/33 - sqrt(33)*log(t + sqrt(33)/8 + Rational(7, 8))/33, + 'is_transformed': True, 't_': t_, 'is_second_order': True, 'is_higher_order': True} + assert _classify_linear_system(eq2, funcs, t) == sol2 + + eq3 = ((t*Derivative(x(t), t) - x(t))*log(t) + (t*Derivative(y(t), t) - y(t))*exp(t) + Derivative(x(t), (t, 2)), + t**2*(t*Derivative(x(t), t) - x(t)) + t*(t*Derivative(y(t), t) - y(t)) + Derivative(y(t), (t, 2))) + sol3 = {'no_of_equation': 2, 'eq': ((t*Derivative(x(t), t) - x(t))*log(t) + (t*Derivative(y(t), t) - + y(t))*exp(t) + Derivative(x(t), (t, 2)), t**2*(t*Derivative(x(t), t) - x(t)) + t*(t*Derivative(y(t), + t) - y(t)) + Derivative(y(t), (t, 2))), 'func': [x(t), y(t)], 'order': {x(t): 2, y(t): 2}, + 'is_linear': True, 'is_homogeneous': True, 'is_general': True, 'type_of_equation': 'type1', 'A1': + Matrix([ [-t*log(t), -t*exp(t)], [ -t**3, -t**2]]), 'is_second_order': True, + 'is_higher_order': True} + assert _classify_linear_system(eq3, funcs, t) == sol3 + + eq4 = (Eq(x2, k*x(t) - l*y1), Eq(y2, l*x1 + k*y(t))) + sol4 = {'no_of_equation': 2, 'eq': (Eq(Derivative(x(t), (t, 2)), k*x(t) - l*Derivative(y(t), t)), + Eq(Derivative(y(t), (t, 2)), k*y(t) + l*Derivative(x(t), t))), 'func': [x(t), y(t)], 'order': {x(t): + 2, y(t): 2}, 'is_linear': True, 'is_homogeneous': True, 'is_general': True, 'type_of_equation': + 'type0', 'is_second_order': True, 'is_higher_order': True} + assert _classify_linear_system(eq4, funcs, t) == sol4 + + + # Multiple matches + + f, g = symbols("f g", cls=Function) + y, t_ = symbols("y t_") + funcs = [f(t), g(t)] + + eq1 = [Eq(Derivative(f(t), t)**2 - 2*Derivative(f(t), t) + 1, 4), + Eq(-y*f(t) + Derivative(g(t), t), 0)] + sol1 = {'is_implicit': True, + 'canon_eqs': [[Eq(Derivative(f(t), t), -1), Eq(Derivative(g(t), t), y*f(t))], + [Eq(Derivative(f(t), t), 3), Eq(Derivative(g(t), t), y*f(t))]]} + assert _classify_linear_system(eq1, funcs, t) == sol1 + + raises(ValueError, lambda: _classify_linear_system(eq1, funcs[:1], t)) + + eq2 = [Eq(Derivative(f(t), t), (2*f(t) + g(t) + 1)/t), Eq(Derivative(g(t), t), (f(t) + 2*g(t))/t)] + sol2 = {'no_of_equation': 2, 'eq': [Eq(Derivative(f(t), t), (2*f(t) + g(t) + 1)/t), Eq(Derivative(g(t), t), + (f(t) + 2*g(t))/t)], 'func': [f(t), g(t)], 'order': {f(t): 1, g(t): 1}, 'is_linear': True, + 'is_homogeneous': False, 'is_general': True, 'rhs': Matrix([ [1], [0]]), 'func_coeff': Matrix([ [2, + 1], [1, 2]]), 'is_constant': False, 'type_of_equation': 'type6', 't_': t_, 'tau': log(t), + 'commutative_antiderivative': Matrix([ [2*log(t), log(t)], [ log(t), 2*log(t)]])} + assert _classify_linear_system(eq2, funcs, t) == sol2 + + eq3 = [Eq(Derivative(f(t), t), (2*f(t) + g(t))/t), Eq(Derivative(g(t), t), (f(t) + 2*g(t))/t)] + sol3 = {'no_of_equation': 2, 'eq': [Eq(Derivative(f(t), t), (2*f(t) + g(t))/t), Eq(Derivative(g(t), t), + (f(t) + 2*g(t))/t)], 'func': [f(t), g(t)], 'order': {f(t): 1, g(t): 1}, 'is_linear': True, + 'is_homogeneous': True, 'is_general': True, 'func_coeff': Matrix([ [2, 1], [1, 2]]), 'is_constant': + False, 'type_of_equation': 'type5', 't_': t_, 'rhs': Matrix([ [0], [0]]), 'tau': log(t), + 'commutative_antiderivative': Matrix([ [2*log(t), log(t)], [ log(t), 2*log(t)]])} + assert _classify_linear_system(eq3, funcs, t) == sol3 + + +def test_matrix_exp(): + from sympy.matrices.dense import Matrix, eye, zeros + from sympy.solvers.ode.systems import matrix_exp + t = Symbol('t') + + for n in range(1, 6+1): + assert matrix_exp(zeros(n), t) == eye(n) + + for n in range(1, 6+1): + A = eye(n) + expAt = exp(t) * eye(n) + assert matrix_exp(A, t) == expAt + + for n in range(1, 6+1): + A = Matrix(n, n, lambda i,j: i+1 if i==j else 0) + expAt = Matrix(n, n, lambda i,j: exp((i+1)*t) if i==j else 0) + assert matrix_exp(A, t) == expAt + + A = Matrix([[0, 1], [-1, 0]]) + expAt = Matrix([[cos(t), sin(t)], [-sin(t), cos(t)]]) + assert matrix_exp(A, t) == expAt + + A = Matrix([[2, -5], [2, -4]]) + expAt = Matrix([ + [3*exp(-t)*sin(t) + exp(-t)*cos(t), -5*exp(-t)*sin(t)], + [2*exp(-t)*sin(t), -3*exp(-t)*sin(t) + exp(-t)*cos(t)] + ]) + assert matrix_exp(A, t) == expAt + + A = Matrix([[21, 17, 6], [-5, -1, -6], [4, 4, 16]]) + # TO update this. + # expAt = Matrix([ + # [(8*t*exp(12*t) + 5*exp(12*t) - 1)*exp(4*t)/4, + # (8*t*exp(12*t) + 5*exp(12*t) - 5)*exp(4*t)/4, + # (exp(12*t) - 1)*exp(4*t)/2], + # [(-8*t*exp(12*t) - exp(12*t) + 1)*exp(4*t)/4, + # (-8*t*exp(12*t) - exp(12*t) + 5)*exp(4*t)/4, + # (-exp(12*t) + 1)*exp(4*t)/2], + # [4*t*exp(16*t), 4*t*exp(16*t), exp(16*t)]]) + expAt = Matrix([ + [2*t*exp(16*t) + 5*exp(16*t)/4 - exp(4*t)/4, 2*t*exp(16*t) + 5*exp(16*t)/4 - 5*exp(4*t)/4, exp(16*t)/2 - exp(4*t)/2], + [ -2*t*exp(16*t) - exp(16*t)/4 + exp(4*t)/4, -2*t*exp(16*t) - exp(16*t)/4 + 5*exp(4*t)/4, -exp(16*t)/2 + exp(4*t)/2], + [ 4*t*exp(16*t), 4*t*exp(16*t), exp(16*t)] + ]) + assert matrix_exp(A, t) == expAt + + A = Matrix([[1, 1, 0, 0], + [0, 1, 1, 0], + [0, 0, 1, -S(1)/8], + [0, 0, S(1)/2, S(1)/2]]) + expAt = Matrix([ + [exp(t), t*exp(t), 4*t*exp(3*t/4) + 8*t*exp(t) + 48*exp(3*t/4) - 48*exp(t), + -2*t*exp(3*t/4) - 2*t*exp(t) - 16*exp(3*t/4) + 16*exp(t)], + [0, exp(t), -t*exp(3*t/4) - 8*exp(3*t/4) + 8*exp(t), t*exp(3*t/4)/2 + 2*exp(3*t/4) - 2*exp(t)], + [0, 0, t*exp(3*t/4)/4 + exp(3*t/4), -t*exp(3*t/4)/8], + [0, 0, t*exp(3*t/4)/2, -t*exp(3*t/4)/4 + exp(3*t/4)] + ]) + assert matrix_exp(A, t) == expAt + + A = Matrix([ + [ 0, 1, 0, 0], + [-1, 0, 0, 0], + [ 0, 0, 0, 1], + [ 0, 0, -1, 0]]) + + expAt = Matrix([ + [ cos(t), sin(t), 0, 0], + [-sin(t), cos(t), 0, 0], + [ 0, 0, cos(t), sin(t)], + [ 0, 0, -sin(t), cos(t)]]) + assert matrix_exp(A, t) == expAt + + A = Matrix([ + [ 0, 1, 1, 0], + [-1, 0, 0, 1], + [ 0, 0, 0, 1], + [ 0, 0, -1, 0]]) + + expAt = Matrix([ + [ cos(t), sin(t), t*cos(t), t*sin(t)], + [-sin(t), cos(t), -t*sin(t), t*cos(t)], + [ 0, 0, cos(t), sin(t)], + [ 0, 0, -sin(t), cos(t)]]) + assert matrix_exp(A, t) == expAt + + # This case is unacceptably slow right now but should be solvable... + #a, b, c, d, e, f = symbols('a b c d e f') + #A = Matrix([ + #[-a, b, c, d], + #[ a, -b, e, 0], + #[ 0, 0, -c - e - f, 0], + #[ 0, 0, f, -d]]) + + A = Matrix([[0, I], [I, 0]]) + expAt = Matrix([ + [exp(I*t)/2 + exp(-I*t)/2, exp(I*t)/2 - exp(-I*t)/2], + [exp(I*t)/2 - exp(-I*t)/2, exp(I*t)/2 + exp(-I*t)/2]]) + assert matrix_exp(A, t) == expAt + + # Testing Errors + M = Matrix([[1, 2, 3], [4, 5, 6], [7, 7, 7]]) + M1 = Matrix([[t, 1], [1, 1]]) + + raises(ValueError, lambda: matrix_exp(M[:, :2], t)) + raises(ValueError, lambda: matrix_exp(M[:2, :], t)) + raises(ValueError, lambda: matrix_exp(M1, t)) + raises(ValueError, lambda: matrix_exp(M1[:1, :1], t)) + + +def test_canonical_odes(): + f, g, h = symbols('f g h', cls=Function) + x = symbols('x') + funcs = [f(x), g(x), h(x)] + + eqs1 = [Eq(f(x).diff(x, x), f(x) + 2*g(x)), Eq(g(x) + 1, g(x).diff(x) + f(x))] + sol1 = [[Eq(Derivative(f(x), (x, 2)), f(x) + 2*g(x)), Eq(Derivative(g(x), x), -f(x) + g(x) + 1)]] + assert canonical_odes(eqs1, funcs[:2], x) == sol1 + + eqs2 = [Eq(f(x).diff(x), h(x).diff(x) + f(x)), Eq(g(x).diff(x)**2, f(x) + h(x)), Eq(h(x).diff(x), f(x))] + sol2 = [[Eq(Derivative(f(x), x), 2*f(x)), Eq(Derivative(g(x), x), -sqrt(f(x) + h(x))), Eq(Derivative(h(x), x), f(x))], + [Eq(Derivative(f(x), x), 2*f(x)), Eq(Derivative(g(x), x), sqrt(f(x) + h(x))), Eq(Derivative(h(x), x), f(x))]] + assert canonical_odes(eqs2, funcs, x) == sol2 + + +def test_sysode_linear_neq_order1_type1(): + + f, g, x, y, h = symbols('f g x y h', cls=Function) + a, b, c, t = symbols('a b c t') + + eqs1 = [Eq(Derivative(x(t), t), x(t)), + Eq(Derivative(y(t), t), y(t))] + sol1 = [Eq(x(t), C1*exp(t)), + Eq(y(t), C2*exp(t))] + assert dsolve(eqs1) == sol1 + assert checksysodesol(eqs1, sol1) == (True, [0, 0]) + + eqs2 = [Eq(Derivative(x(t), t), 2*x(t)), + Eq(Derivative(y(t), t), 3*y(t))] + sol2 = [Eq(x(t), C1*exp(2*t)), + Eq(y(t), C2*exp(3*t))] + assert dsolve(eqs2) == sol2 + assert checksysodesol(eqs2, sol2) == (True, [0, 0]) + + eqs3 = [Eq(Derivative(x(t), t), a*x(t)), + Eq(Derivative(y(t), t), a*y(t))] + sol3 = [Eq(x(t), C1*exp(a*t)), + Eq(y(t), C2*exp(a*t))] + assert dsolve(eqs3) == sol3 + assert checksysodesol(eqs3, sol3) == (True, [0, 0]) + + # Regression test case for issue #15474 + # https://github.com/sympy/sympy/issues/15474 + eqs4 = [Eq(Derivative(x(t), t), a*x(t)), + Eq(Derivative(y(t), t), b*y(t))] + sol4 = [Eq(x(t), C1*exp(a*t)), + Eq(y(t), C2*exp(b*t))] + assert dsolve(eqs4) == sol4 + assert checksysodesol(eqs4, sol4) == (True, [0, 0]) + + eqs5 = [Eq(Derivative(x(t), t), -y(t)), + Eq(Derivative(y(t), t), x(t))] + sol5 = [Eq(x(t), -C1*sin(t) - C2*cos(t)), + Eq(y(t), C1*cos(t) - C2*sin(t))] + assert dsolve(eqs5) == sol5 + assert checksysodesol(eqs5, sol5) == (True, [0, 0]) + + eqs6 = [Eq(Derivative(x(t), t), -2*y(t)), + Eq(Derivative(y(t), t), 2*x(t))] + sol6 = [Eq(x(t), -C1*sin(2*t) - C2*cos(2*t)), + Eq(y(t), C1*cos(2*t) - C2*sin(2*t))] + assert dsolve(eqs6) == sol6 + assert checksysodesol(eqs6, sol6) == (True, [0, 0]) + + eqs7 = [Eq(Derivative(x(t), t), I*y(t)), + Eq(Derivative(y(t), t), I*x(t))] + sol7 = [Eq(x(t), -C1*exp(-I*t) + C2*exp(I*t)), + Eq(y(t), C1*exp(-I*t) + C2*exp(I*t))] + assert dsolve(eqs7) == sol7 + assert checksysodesol(eqs7, sol7) == (True, [0, 0]) + + eqs8 = [Eq(Derivative(x(t), t), -a*y(t)), + Eq(Derivative(y(t), t), a*x(t))] + sol8 = [Eq(x(t), -I*C1*exp(-I*a*t) + I*C2*exp(I*a*t)), + Eq(y(t), C1*exp(-I*a*t) + C2*exp(I*a*t))] + assert dsolve(eqs8) == sol8 + assert checksysodesol(eqs8, sol8) == (True, [0, 0]) + + eqs9 = [Eq(Derivative(x(t), t), x(t) + y(t)), + Eq(Derivative(y(t), t), x(t) - y(t))] + sol9 = [Eq(x(t), C1*(1 - sqrt(2))*exp(-sqrt(2)*t) + C2*(1 + sqrt(2))*exp(sqrt(2)*t)), + Eq(y(t), C1*exp(-sqrt(2)*t) + C2*exp(sqrt(2)*t))] + assert dsolve(eqs9) == sol9 + assert checksysodesol(eqs9, sol9) == (True, [0, 0]) + + eqs10 = [Eq(Derivative(x(t), t), x(t) + y(t)), + Eq(Derivative(y(t), t), x(t) + y(t))] + sol10 = [Eq(x(t), -C1 + C2*exp(2*t)), + Eq(y(t), C1 + C2*exp(2*t))] + assert dsolve(eqs10) == sol10 + assert checksysodesol(eqs10, sol10) == (True, [0, 0]) + + eqs11 = [Eq(Derivative(x(t), t), 2*x(t) + y(t)), + Eq(Derivative(y(t), t), -x(t) + 2*y(t))] + sol11 = [Eq(x(t), C1*exp(2*t)*sin(t) + C2*exp(2*t)*cos(t)), + Eq(y(t), C1*exp(2*t)*cos(t) - C2*exp(2*t)*sin(t))] + assert dsolve(eqs11) == sol11 + assert checksysodesol(eqs11, sol11) == (True, [0, 0]) + + eqs12 = [Eq(Derivative(x(t), t), x(t) + 2*y(t)), + Eq(Derivative(y(t), t), 2*x(t) + y(t))] + sol12 = [Eq(x(t), -C1*exp(-t) + C2*exp(3*t)), + Eq(y(t), C1*exp(-t) + C2*exp(3*t))] + assert dsolve(eqs12) == sol12 + assert checksysodesol(eqs12, sol12) == (True, [0, 0]) + + eqs13 = [Eq(Derivative(x(t), t), 4*x(t) + y(t)), + Eq(Derivative(y(t), t), -x(t) + 2*y(t))] + sol13 = [Eq(x(t), C2*t*exp(3*t) + (C1 + C2)*exp(3*t)), + Eq(y(t), -C1*exp(3*t) - C2*t*exp(3*t))] + assert dsolve(eqs13) == sol13 + assert checksysodesol(eqs13, sol13) == (True, [0, 0]) + + eqs14 = [Eq(Derivative(x(t), t), a*y(t)), + Eq(Derivative(y(t), t), a*x(t))] + sol14 = [Eq(x(t), -C1*exp(-a*t) + C2*exp(a*t)), + Eq(y(t), C1*exp(-a*t) + C2*exp(a*t))] + assert dsolve(eqs14) == sol14 + assert checksysodesol(eqs14, sol14) == (True, [0, 0]) + + eqs15 = [Eq(Derivative(x(t), t), a*y(t)), + Eq(Derivative(y(t), t), b*x(t))] + sol15 = [Eq(x(t), -C1*a*exp(-t*sqrt(a*b))/sqrt(a*b) + C2*a*exp(t*sqrt(a*b))/sqrt(a*b)), + Eq(y(t), C1*exp(-t*sqrt(a*b)) + C2*exp(t*sqrt(a*b)))] + assert dsolve(eqs15) == sol15 + assert checksysodesol(eqs15, sol15) == (True, [0, 0]) + + eqs16 = [Eq(Derivative(x(t), t), a*x(t) + b*y(t)), + Eq(Derivative(y(t), t), c*x(t))] + sol16 = [Eq(x(t), -2*C1*b*exp(t*(a + sqrt(a**2 + 4*b*c))/2)/(a - sqrt(a**2 + 4*b*c)) - 2*C2*b*exp(t*(a - + sqrt(a**2 + 4*b*c))/2)/(a + sqrt(a**2 + 4*b*c))), + Eq(y(t), C1*exp(t*(a + sqrt(a**2 + 4*b*c))/2) + C2*exp(t*(a - sqrt(a**2 + 4*b*c))/2))] + assert dsolve(eqs16) == sol16 + assert checksysodesol(eqs16, sol16) == (True, [0, 0]) + + # Regression test case for issue #18562 + # https://github.com/sympy/sympy/issues/18562 + eqs17 = [Eq(Derivative(x(t), t), a*y(t) + x(t)), + Eq(Derivative(y(t), t), a*x(t) - y(t))] + sol17 = [Eq(x(t), C1*a*exp(t*sqrt(a**2 + 1))/(sqrt(a**2 + 1) - 1) - C2*a*exp(-t*sqrt(a**2 + 1))/(sqrt(a**2 + + 1) + 1)), + Eq(y(t), C1*exp(t*sqrt(a**2 + 1)) + C2*exp(-t*sqrt(a**2 + 1)))] + assert dsolve(eqs17) == sol17 + assert checksysodesol(eqs17, sol17) == (True, [0, 0]) + + eqs18 = [Eq(Derivative(x(t), t), 0), + Eq(Derivative(y(t), t), 0)] + sol18 = [Eq(x(t), C1), + Eq(y(t), C2)] + assert dsolve(eqs18) == sol18 + assert checksysodesol(eqs18, sol18) == (True, [0, 0]) + + eqs19 = [Eq(Derivative(x(t), t), 2*x(t) - y(t)), + Eq(Derivative(y(t), t), x(t))] + sol19 = [Eq(x(t), C2*t*exp(t) + (C1 + C2)*exp(t)), + Eq(y(t), C1*exp(t) + C2*t*exp(t))] + assert dsolve(eqs19) == sol19 + assert checksysodesol(eqs19, sol19) == (True, [0, 0]) + + eqs20 = [Eq(Derivative(x(t), t), x(t)), + Eq(Derivative(y(t), t), x(t) + y(t))] + sol20 = [Eq(x(t), C1*exp(t)), + Eq(y(t), C1*t*exp(t) + C2*exp(t))] + assert dsolve(eqs20) == sol20 + assert checksysodesol(eqs20, sol20) == (True, [0, 0]) + + eqs21 = [Eq(Derivative(x(t), t), 3*x(t)), + Eq(Derivative(y(t), t), x(t) + y(t))] + sol21 = [Eq(x(t), 2*C1*exp(3*t)), + Eq(y(t), C1*exp(3*t) + C2*exp(t))] + assert dsolve(eqs21) == sol21 + assert checksysodesol(eqs21, sol21) == (True, [0, 0]) + + eqs22 = [Eq(Derivative(x(t), t), 3*x(t)), + Eq(Derivative(y(t), t), y(t))] + sol22 = [Eq(x(t), C1*exp(3*t)), + Eq(y(t), C2*exp(t))] + assert dsolve(eqs22) == sol22 + assert checksysodesol(eqs22, sol22) == (True, [0, 0]) + + +@slow +def test_sysode_linear_neq_order1_type1_slow(): + + t = Symbol('t') + Z0 = Function('Z0') + Z1 = Function('Z1') + Z2 = Function('Z2') + Z3 = Function('Z3') + + k01, k10, k20, k21, k23, k30 = symbols('k01 k10 k20 k21 k23 k30') + + eqs1 = [Eq(Derivative(Z0(t), t), -k01*Z0(t) + k10*Z1(t) + k20*Z2(t) + k30*Z3(t)), + Eq(Derivative(Z1(t), t), k01*Z0(t) - k10*Z1(t) + k21*Z2(t)), + Eq(Derivative(Z2(t), t), (-k20 - k21 - k23)*Z2(t)), + Eq(Derivative(Z3(t), t), k23*Z2(t) - k30*Z3(t))] + sol1 = [Eq(Z0(t), C1*k10/k01 - C2*(k10 - k30)*exp(-k30*t)/(k01 + k10 - k30) - C3*(k10*(k20 + k21 - k30) - + k20**2 - k20*(k21 + k23 - k30) + k23*k30)*exp(-t*(k20 + k21 + k23))/(k23*(-k01 - k10 + k20 + k21 + + k23)) - C4*exp(-t*(k01 + k10))), + Eq(Z1(t), C1 - C2*k01*exp(-k30*t)/(k01 + k10 - k30) + C3*(-k01*(k20 + k21 - k30) + k20*k21 + k21**2 + + k21*(k23 - k30))*exp(-t*(k20 + k21 + k23))/(k23*(-k01 - k10 + k20 + k21 + k23)) + C4*exp(-t*(k01 + + k10))), + Eq(Z2(t), -C3*(k20 + k21 + k23 - k30)*exp(-t*(k20 + k21 + k23))/k23), + Eq(Z3(t), C2*exp(-k30*t) + C3*exp(-t*(k20 + k21 + k23)))] + assert dsolve(eqs1) == sol1 + assert checksysodesol(eqs1, sol1) == (True, [0, 0, 0, 0]) + + x, y, z, u, v, w = symbols('x y z u v w', cls=Function) + k2, k3 = symbols('k2 k3') + a_b, a_c = symbols('a_b a_c', real=True) + + eqs2 = [Eq(Derivative(z(t), t), k2*y(t)), + Eq(Derivative(x(t), t), k3*y(t)), + Eq(Derivative(y(t), t), (-k2 - k3)*y(t))] + sol2 = [Eq(z(t), C1 - C2*k2*exp(-t*(k2 + k3))/(k2 + k3)), + Eq(x(t), -C2*k3*exp(-t*(k2 + k3))/(k2 + k3) + C3), + Eq(y(t), C2*exp(-t*(k2 + k3)))] + assert dsolve(eqs2) == sol2 + assert checksysodesol(eqs2, sol2) == (True, [0, 0, 0]) + + eqs3 = [4*u(t) - v(t) - 2*w(t) + Derivative(u(t), t), + 2*u(t) + v(t) - 2*w(t) + Derivative(v(t), t), + 5*u(t) + v(t) - 3*w(t) + Derivative(w(t), t)] + sol3 = [Eq(u(t), C3*exp(-2*t) + (C1/2 + sqrt(3)*C2/6)*cos(sqrt(3)*t) + sin(sqrt(3)*t)*(sqrt(3)*C1/6 + + C2*Rational(-1, 2))), + Eq(v(t), (C1/2 + sqrt(3)*C2/6)*cos(sqrt(3)*t) + sin(sqrt(3)*t)*(sqrt(3)*C1/6 + C2*Rational(-1, 2))), + Eq(w(t), C1*cos(sqrt(3)*t) - C2*sin(sqrt(3)*t) + C3*exp(-2*t))] + assert dsolve(eqs3) == sol3 + assert checksysodesol(eqs3, sol3) == (True, [0, 0, 0]) + + eqs4 = [Eq(Derivative(x(t), t), w(t)*Rational(-2, 9) + 2*x(t) + y(t) + z(t)*Rational(-8, 9)), + Eq(Derivative(y(t), t), w(t)*Rational(4, 9) + 2*y(t) + z(t)*Rational(16, 9)), + Eq(Derivative(z(t), t), w(t)*Rational(-2, 9) + z(t)*Rational(37, 9)), + Eq(Derivative(w(t), t), w(t)*Rational(44, 9) + z(t)*Rational(-4, 9))] + sol4 = [Eq(x(t), C1*exp(2*t) + C2*t*exp(2*t)), + Eq(y(t), C2*exp(2*t) + 2*C3*exp(4*t)), + Eq(z(t), 2*C3*exp(4*t) + C4*exp(5*t)*Rational(-1, 4)), + Eq(w(t), C3*exp(4*t) + C4*exp(5*t))] + assert dsolve(eqs4) == sol4 + assert checksysodesol(eqs4, sol4) == (True, [0, 0, 0, 0]) + + # Regression test case for issue #15574 + # https://github.com/sympy/sympy/issues/15574 + eq5 = [Eq(x(t).diff(t), x(t)), Eq(y(t).diff(t), y(t)), Eq(z(t).diff(t), z(t)), Eq(w(t).diff(t), w(t))] + sol5 = [Eq(x(t), C1*exp(t)), Eq(y(t), C2*exp(t)), Eq(z(t), C3*exp(t)), Eq(w(t), C4*exp(t))] + assert dsolve(eq5) == sol5 + assert checksysodesol(eq5, sol5) == (True, [0, 0, 0, 0]) + + eqs6 = [Eq(Derivative(x(t), t), x(t) + y(t)), + Eq(Derivative(y(t), t), y(t) + z(t)), + Eq(Derivative(z(t), t), w(t)*Rational(-1, 8) + z(t)), + Eq(Derivative(w(t), t), w(t)/2 + z(t)/2)] + sol6 = [Eq(x(t), C1*exp(t) + C2*t*exp(t) + 4*C4*t*exp(t*Rational(3, 4)) + (4*C3 + 48*C4)*exp(t*Rational(3, + 4))), + Eq(y(t), C2*exp(t) - C4*t*exp(t*Rational(3, 4)) - (C3 + 8*C4)*exp(t*Rational(3, 4))), + Eq(z(t), C4*t*exp(t*Rational(3, 4))/4 + (C3/4 + C4)*exp(t*Rational(3, 4))), + Eq(w(t), C3*exp(t*Rational(3, 4))/2 + C4*t*exp(t*Rational(3, 4))/2)] + assert dsolve(eqs6) == sol6 + assert checksysodesol(eqs6, sol6) == (True, [0, 0, 0, 0]) + + # Regression test case for issue #15574 + # https://github.com/sympy/sympy/issues/15574 + eq7 = [Eq(Derivative(x(t), t), x(t)), Eq(Derivative(y(t), t), y(t)), Eq(Derivative(z(t), t), z(t)), + Eq(Derivative(w(t), t), w(t)), Eq(Derivative(u(t), t), u(t))] + sol7 = [Eq(x(t), C1*exp(t)), Eq(y(t), C2*exp(t)), Eq(z(t), C3*exp(t)), Eq(w(t), C4*exp(t)), + Eq(u(t), C5*exp(t))] + assert dsolve(eq7) == sol7 + assert checksysodesol(eq7, sol7) == (True, [0, 0, 0, 0, 0]) + + eqs8 = [Eq(Derivative(x(t), t), 2*x(t) + y(t)), + Eq(Derivative(y(t), t), 2*y(t)), + Eq(Derivative(z(t), t), 4*z(t)), + Eq(Derivative(w(t), t), u(t) + 5*w(t)), + Eq(Derivative(u(t), t), 5*u(t))] + sol8 = [Eq(x(t), C1*exp(2*t) + C2*t*exp(2*t)), + Eq(y(t), C2*exp(2*t)), + Eq(z(t), C3*exp(4*t)), + Eq(w(t), C4*exp(5*t) + C5*t*exp(5*t)), + Eq(u(t), C5*exp(5*t))] + assert dsolve(eqs8) == sol8 + assert checksysodesol(eqs8, sol8) == (True, [0, 0, 0, 0, 0]) + + # Regression test case for issue #15574 + # https://github.com/sympy/sympy/issues/15574 + eq9 = [Eq(Derivative(x(t), t), x(t)), Eq(Derivative(y(t), t), y(t)), Eq(Derivative(z(t), t), z(t))] + sol9 = [Eq(x(t), C1*exp(t)), Eq(y(t), C2*exp(t)), Eq(z(t), C3*exp(t))] + assert dsolve(eq9) == sol9 + assert checksysodesol(eq9, sol9) == (True, [0, 0, 0]) + + # Regression test case for issue #15407 + # https://github.com/sympy/sympy/issues/15407 + eqs10 = [Eq(Derivative(x(t), t), (-a_b - a_c)*x(t)), + Eq(Derivative(y(t), t), a_b*y(t)), + Eq(Derivative(z(t), t), a_c*x(t))] + sol10 = [Eq(x(t), -C1*(a_b + a_c)*exp(-t*(a_b + a_c))/a_c), + Eq(y(t), C2*exp(a_b*t)), + Eq(z(t), C1*exp(-t*(a_b + a_c)) + C3)] + assert dsolve(eqs10) == sol10 + assert checksysodesol(eqs10, sol10) == (True, [0, 0, 0]) + + # Regression test case for issue #14312 + # https://github.com/sympy/sympy/issues/14312 + eqs11 = [Eq(Derivative(x(t), t), k3*y(t)), + Eq(Derivative(y(t), t), (-k2 - k3)*y(t)), + Eq(Derivative(z(t), t), k2*y(t))] + sol11 = [Eq(x(t), C1 + C2*k3*exp(-t*(k2 + k3))/k2), + Eq(y(t), -C2*(k2 + k3)*exp(-t*(k2 + k3))/k2), + Eq(z(t), C2*exp(-t*(k2 + k3)) + C3)] + assert dsolve(eqs11) == sol11 + assert checksysodesol(eqs11, sol11) == (True, [0, 0, 0]) + + # Regression test case for issue #14312 + # https://github.com/sympy/sympy/issues/14312 + eqs12 = [Eq(Derivative(z(t), t), k2*y(t)), + Eq(Derivative(x(t), t), k3*y(t)), + Eq(Derivative(y(t), t), (-k2 - k3)*y(t))] + sol12 = [Eq(z(t), C1 - C2*k2*exp(-t*(k2 + k3))/(k2 + k3)), + Eq(x(t), -C2*k3*exp(-t*(k2 + k3))/(k2 + k3) + C3), + Eq(y(t), C2*exp(-t*(k2 + k3)))] + assert dsolve(eqs12) == sol12 + assert checksysodesol(eqs12, sol12) == (True, [0, 0, 0]) + + f, g, h = symbols('f, g, h', cls=Function) + a, b, c = symbols('a, b, c') + + # Regression test case for issue #15474 + # https://github.com/sympy/sympy/issues/15474 + eqs13 = [Eq(Derivative(f(t), t), 2*f(t) + g(t)), + Eq(Derivative(g(t), t), a*f(t))] + sol13 = [Eq(f(t), C1*exp(t*(sqrt(a + 1) + 1))/(sqrt(a + 1) - 1) - C2*exp(-t*(sqrt(a + 1) - 1))/(sqrt(a + 1) + + 1)), + Eq(g(t), C1*exp(t*(sqrt(a + 1) + 1)) + C2*exp(-t*(sqrt(a + 1) - 1)))] + assert dsolve(eqs13) == sol13 + assert checksysodesol(eqs13, sol13) == (True, [0, 0]) + + eqs14 = [Eq(Derivative(f(t), t), 2*g(t) - 3*h(t)), + Eq(Derivative(g(t), t), -2*f(t) + 4*h(t)), + Eq(Derivative(h(t), t), 3*f(t) - 4*g(t))] + sol14 = [Eq(f(t), 2*C1 - sin(sqrt(29)*t)*(sqrt(29)*C2*Rational(3, 25) + C3*Rational(-8, 25)) - + cos(sqrt(29)*t)*(C2*Rational(8, 25) + sqrt(29)*C3*Rational(3, 25))), + Eq(g(t), C1*Rational(3, 2) + sin(sqrt(29)*t)*(sqrt(29)*C2*Rational(4, 25) + C3*Rational(6, 25)) - + cos(sqrt(29)*t)*(C2*Rational(6, 25) + sqrt(29)*C3*Rational(-4, 25))), + Eq(h(t), C1 + C2*cos(sqrt(29)*t) - C3*sin(sqrt(29)*t))] + assert dsolve(eqs14) == sol14 + assert checksysodesol(eqs14, sol14) == (True, [0, 0, 0]) + + eqs15 = [Eq(2*Derivative(f(t), t), 12*g(t) - 12*h(t)), + Eq(3*Derivative(g(t), t), -8*f(t) + 8*h(t)), + Eq(4*Derivative(h(t), t), 6*f(t) - 6*g(t))] + sol15 = [Eq(f(t), C1 - sin(sqrt(29)*t)*(sqrt(29)*C2*Rational(6, 13) + C3*Rational(-16, 13)) - + cos(sqrt(29)*t)*(C2*Rational(16, 13) + sqrt(29)*C3*Rational(6, 13))), + Eq(g(t), C1 + sin(sqrt(29)*t)*(sqrt(29)*C2*Rational(8, 39) + C3*Rational(16, 13)) - + cos(sqrt(29)*t)*(C2*Rational(16, 13) + sqrt(29)*C3*Rational(-8, 39))), + Eq(h(t), C1 + C2*cos(sqrt(29)*t) - C3*sin(sqrt(29)*t))] + assert dsolve(eqs15) == sol15 + assert checksysodesol(eqs15, sol15) == (True, [0, 0, 0]) + + eq16 = (Eq(diff(x(t), t), 21*x(t)), Eq(diff(y(t), t), 17*x(t) + 3*y(t)), + Eq(diff(z(t), t), 5*x(t) + 7*y(t) + 9*z(t))) + sol16 = [Eq(x(t), 216*C1*exp(21*t)/209), + Eq(y(t), 204*C1*exp(21*t)/209 - 6*C2*exp(3*t)/7), + Eq(z(t), C1*exp(21*t) + C2*exp(3*t) + C3*exp(9*t))] + assert dsolve(eq16) == sol16 + assert checksysodesol(eq16, sol16) == (True, [0, 0, 0]) + + eqs17 = [Eq(Derivative(x(t), t), 3*y(t) - 11*z(t)), + Eq(Derivative(y(t), t), -3*x(t) + 7*z(t)), + Eq(Derivative(z(t), t), 11*x(t) - 7*y(t))] + sol17 = [Eq(x(t), C1*Rational(7, 3) - sin(sqrt(179)*t)*(sqrt(179)*C2*Rational(11, 170) + C3*Rational(-21, + 170)) - cos(sqrt(179)*t)*(C2*Rational(21, 170) + sqrt(179)*C3*Rational(11, 170))), + Eq(y(t), C1*Rational(11, 3) + sin(sqrt(179)*t)*(sqrt(179)*C2*Rational(7, 170) + C3*Rational(33, + 170)) - cos(sqrt(179)*t)*(C2*Rational(33, 170) + sqrt(179)*C3*Rational(-7, 170))), + Eq(z(t), C1 + C2*cos(sqrt(179)*t) - C3*sin(sqrt(179)*t))] + assert dsolve(eqs17) == sol17 + assert checksysodesol(eqs17, sol17) == (True, [0, 0, 0]) + + eqs18 = [Eq(3*Derivative(x(t), t), 20*y(t) - 20*z(t)), + Eq(4*Derivative(y(t), t), -15*x(t) + 15*z(t)), + Eq(5*Derivative(z(t), t), 12*x(t) - 12*y(t))] + sol18 = [Eq(x(t), C1 - sin(5*sqrt(2)*t)*(sqrt(2)*C2*Rational(4, 3) - C3) - cos(5*sqrt(2)*t)*(C2 + + sqrt(2)*C3*Rational(4, 3))), + Eq(y(t), C1 + sin(5*sqrt(2)*t)*(sqrt(2)*C2*Rational(3, 4) + C3) - cos(5*sqrt(2)*t)*(C2 + + sqrt(2)*C3*Rational(-3, 4))), + Eq(z(t), C1 + C2*cos(5*sqrt(2)*t) - C3*sin(5*sqrt(2)*t))] + assert dsolve(eqs18) == sol18 + assert checksysodesol(eqs18, sol18) == (True, [0, 0, 0]) + + eqs19 = [Eq(Derivative(x(t), t), 4*x(t) - z(t)), + Eq(Derivative(y(t), t), 2*x(t) + 2*y(t) - z(t)), + Eq(Derivative(z(t), t), 3*x(t) + y(t))] + sol19 = [Eq(x(t), C2*t**2*exp(2*t)/2 + t*(2*C2 + C3)*exp(2*t) + (C1 + C2 + 2*C3)*exp(2*t)), + Eq(y(t), C2*t**2*exp(2*t)/2 + t*(2*C2 + C3)*exp(2*t) + (C1 + 2*C3)*exp(2*t)), + Eq(z(t), C2*t**2*exp(2*t) + t*(3*C2 + 2*C3)*exp(2*t) + (2*C1 + 3*C3)*exp(2*t))] + assert dsolve(eqs19) == sol19 + assert checksysodesol(eqs19, sol19) == (True, [0, 0, 0]) + + eqs20 = [Eq(Derivative(x(t), t), 4*x(t) - y(t) - 2*z(t)), + Eq(Derivative(y(t), t), 2*x(t) + y(t) - 2*z(t)), + Eq(Derivative(z(t), t), 5*x(t) - 3*z(t))] + sol20 = [Eq(x(t), C1*exp(2*t) - sin(t)*(C2*Rational(3, 5) + C3/5) - cos(t)*(C2/5 + C3*Rational(-3, 5))), + Eq(y(t), -sin(t)*(C2*Rational(3, 5) + C3/5) - cos(t)*(C2/5 + C3*Rational(-3, 5))), + Eq(z(t), C1*exp(2*t) - C2*sin(t) + C3*cos(t))] + assert dsolve(eqs20) == sol20 + assert checksysodesol(eqs20, sol20) == (True, [0, 0, 0]) + + eq21 = (Eq(diff(x(t), t), 9*y(t)), Eq(diff(y(t), t), 12*x(t))) + sol21 = [Eq(x(t), -sqrt(3)*C1*exp(-6*sqrt(3)*t)/2 + sqrt(3)*C2*exp(6*sqrt(3)*t)/2), + Eq(y(t), C1*exp(-6*sqrt(3)*t) + C2*exp(6*sqrt(3)*t))] + + assert dsolve(eq21) == sol21 + assert checksysodesol(eq21, sol21) == (True, [0, 0]) + + eqs22 = [Eq(Derivative(x(t), t), 2*x(t) + 4*y(t)), + Eq(Derivative(y(t), t), 12*x(t) + 41*y(t))] + sol22 = [Eq(x(t), C1*(39 - sqrt(1713))*exp(t*(sqrt(1713) + 43)/2)*Rational(-1, 24) + C2*(39 + + sqrt(1713))*exp(t*(43 - sqrt(1713))/2)*Rational(-1, 24)), + Eq(y(t), C1*exp(t*(sqrt(1713) + 43)/2) + C2*exp(t*(43 - sqrt(1713))/2))] + assert dsolve(eqs22) == sol22 + assert checksysodesol(eqs22, sol22) == (True, [0, 0]) + + eqs23 = [Eq(Derivative(x(t), t), x(t) + y(t)), + Eq(Derivative(y(t), t), -2*x(t) + 2*y(t))] + sol23 = [Eq(x(t), (C1/4 + sqrt(7)*C2/4)*cos(sqrt(7)*t/2)*exp(t*Rational(3, 2)) + + sin(sqrt(7)*t/2)*(sqrt(7)*C1/4 + C2*Rational(-1, 4))*exp(t*Rational(3, 2))), + Eq(y(t), C1*cos(sqrt(7)*t/2)*exp(t*Rational(3, 2)) - C2*sin(sqrt(7)*t/2)*exp(t*Rational(3, 2)))] + assert dsolve(eqs23) == sol23 + assert checksysodesol(eqs23, sol23) == (True, [0, 0]) + + # Regression test case for issue #15474 + # https://github.com/sympy/sympy/issues/15474 + a = Symbol("a", real=True) + eq24 = [x(t).diff(t) - a*y(t), y(t).diff(t) + a*x(t)] + sol24 = [Eq(x(t), C1*sin(a*t) + C2*cos(a*t)), Eq(y(t), C1*cos(a*t) - C2*sin(a*t))] + assert dsolve(eq24) == sol24 + assert checksysodesol(eq24, sol24) == (True, [0, 0]) + + # Regression test case for issue #19150 + # https://github.com/sympy/sympy/issues/19150 + eqs25 = [Eq(Derivative(f(t), t), 0), + Eq(Derivative(g(t), t), (f(t) - 2*g(t) + x(t))/(b*c)), + Eq(Derivative(x(t), t), (g(t) - 2*x(t) + y(t))/(b*c)), + Eq(Derivative(y(t), t), (h(t) + x(t) - 2*y(t))/(b*c)), + Eq(Derivative(h(t), t), 0)] + sol25 = [Eq(f(t), -3*C1 + 4*C2), + Eq(g(t), -2*C1 + 3*C2 - C3*exp(-2*t/(b*c)) + C4*exp(-t*(sqrt(2) + 2)/(b*c)) + C5*exp(-t*(2 - + sqrt(2))/(b*c))), + Eq(x(t), -C1 + 2*C2 - sqrt(2)*C4*exp(-t*(sqrt(2) + 2)/(b*c)) + sqrt(2)*C5*exp(-t*(2 - + sqrt(2))/(b*c))), + Eq(y(t), C2 + C3*exp(-2*t/(b*c)) + C4*exp(-t*(sqrt(2) + 2)/(b*c)) + C5*exp(-t*(2 - sqrt(2))/(b*c))), + Eq(h(t), C1)] + assert dsolve(eqs25) == sol25 + assert checksysodesol(eqs25, sol25) == (True, [0, 0, 0, 0, 0]) + + eq26 = [Eq(Derivative(f(t), t), 2*f(t)), Eq(Derivative(g(t), t), 3*f(t) + 7*g(t))] + sol26 = [Eq(f(t), -5*C1*exp(2*t)/3), Eq(g(t), C1*exp(2*t) + C2*exp(7*t))] + assert dsolve(eq26) == sol26 + assert checksysodesol(eq26, sol26) == (True, [0, 0]) + + eq27 = [Eq(Derivative(f(t), t), -9*I*f(t) - 4*g(t)), Eq(Derivative(g(t), t), -4*I*g(t))] + sol27 = [Eq(f(t), 4*I*C1*exp(-4*I*t)/5 + C2*exp(-9*I*t)), Eq(g(t), C1*exp(-4*I*t))] + assert dsolve(eq27) == sol27 + assert checksysodesol(eq27, sol27) == (True, [0, 0]) + + eq28 = [Eq(Derivative(f(t), t), -9*I*f(t)), Eq(Derivative(g(t), t), -4*I*g(t))] + sol28 = [Eq(f(t), C1*exp(-9*I*t)), Eq(g(t), C2*exp(-4*I*t))] + assert dsolve(eq28) == sol28 + assert checksysodesol(eq28, sol28) == (True, [0, 0]) + + eq29 = [Eq(Derivative(f(t), t), 0), Eq(Derivative(g(t), t), 0)] + sol29 = [Eq(f(t), C1), Eq(g(t), C2)] + assert dsolve(eq29) == sol29 + assert checksysodesol(eq29, sol29) == (True, [0, 0]) + + eq30 = [Eq(Derivative(f(t), t), f(t)), Eq(Derivative(g(t), t), 0)] + sol30 = [Eq(f(t), C1*exp(t)), Eq(g(t), C2)] + assert dsolve(eq30) == sol30 + assert checksysodesol(eq30, sol30) == (True, [0, 0]) + + eq31 = [Eq(Derivative(f(t), t), g(t)), Eq(Derivative(g(t), t), 0)] + sol31 = [Eq(f(t), C1 + C2*t), Eq(g(t), C2)] + assert dsolve(eq31) == sol31 + assert checksysodesol(eq31, sol31) == (True, [0, 0]) + + eq32 = [Eq(Derivative(f(t), t), 0), Eq(Derivative(g(t), t), f(t))] + sol32 = [Eq(f(t), C1), Eq(g(t), C1*t + C2)] + assert dsolve(eq32) == sol32 + assert checksysodesol(eq32, sol32) == (True, [0, 0]) + + eq33 = [Eq(Derivative(f(t), t), 0), Eq(Derivative(g(t), t), g(t))] + sol33 = [Eq(f(t), C1), Eq(g(t), C2*exp(t))] + assert dsolve(eq33) == sol33 + assert checksysodesol(eq33, sol33) == (True, [0, 0]) + + eq34 = [Eq(Derivative(f(t), t), f(t)), Eq(Derivative(g(t), t), I*g(t))] + sol34 = [Eq(f(t), C1*exp(t)), Eq(g(t), C2*exp(I*t))] + assert dsolve(eq34) == sol34 + assert checksysodesol(eq34, sol34) == (True, [0, 0]) + + eq35 = [Eq(Derivative(f(t), t), I*f(t)), Eq(Derivative(g(t), t), -I*g(t))] + sol35 = [Eq(f(t), C1*exp(I*t)), Eq(g(t), C2*exp(-I*t))] + assert dsolve(eq35) == sol35 + assert checksysodesol(eq35, sol35) == (True, [0, 0]) + + eq36 = [Eq(Derivative(f(t), t), I*g(t)), Eq(Derivative(g(t), t), 0)] + sol36 = [Eq(f(t), I*C1 + I*C2*t), Eq(g(t), C2)] + assert dsolve(eq36) == sol36 + assert checksysodesol(eq36, sol36) == (True, [0, 0]) + + eq37 = [Eq(Derivative(f(t), t), I*g(t)), Eq(Derivative(g(t), t), I*f(t))] + sol37 = [Eq(f(t), -C1*exp(-I*t) + C2*exp(I*t)), Eq(g(t), C1*exp(-I*t) + C2*exp(I*t))] + assert dsolve(eq37) == sol37 + assert checksysodesol(eq37, sol37) == (True, [0, 0]) + + # Multiple systems + eq1 = [Eq(Derivative(f(t), t)**2, g(t)**2), Eq(-f(t) + Derivative(g(t), t), 0)] + sol1 = [[Eq(f(t), -C1*sin(t) - C2*cos(t)), + Eq(g(t), C1*cos(t) - C2*sin(t))], + [Eq(f(t), -C1*exp(-t) + C2*exp(t)), + Eq(g(t), C1*exp(-t) + C2*exp(t))]] + assert dsolve(eq1) == sol1 + for sol in sol1: + assert checksysodesol(eq1, sol) == (True, [0, 0]) + + +def test_sysode_linear_neq_order1_type2(): + + f, g, h, k = symbols('f g h k', cls=Function) + x, t, a, b, c, d, y = symbols('x t a b c d y') + k1, k2 = symbols('k1 k2') + + + eqs1 = [Eq(Derivative(f(x), x), f(x) + g(x) + 5), + Eq(Derivative(g(x), x), -f(x) - g(x) + 7)] + sol1 = [Eq(f(x), C1 + C2 + 6*x**2 + x*(C2 + 5)), + Eq(g(x), -C1 - 6*x**2 - x*(C2 - 7))] + assert dsolve(eqs1) == sol1 + assert checksysodesol(eqs1, sol1) == (True, [0, 0]) + + eqs2 = [Eq(Derivative(f(x), x), f(x) + g(x) + 5), + Eq(Derivative(g(x), x), f(x) + g(x) + 7)] + sol2 = [Eq(f(x), -C1 + C2*exp(2*x) - x - 3), + Eq(g(x), C1 + C2*exp(2*x) + x - 3)] + assert dsolve(eqs2) == sol2 + assert checksysodesol(eqs2, sol2) == (True, [0, 0]) + + eqs3 = [Eq(Derivative(f(x), x), f(x) + 5), + Eq(Derivative(g(x), x), f(x) + 7)] + sol3 = [Eq(f(x), C1*exp(x) - 5), + Eq(g(x), C1*exp(x) + C2 + 2*x - 5)] + assert dsolve(eqs3) == sol3 + assert checksysodesol(eqs3, sol3) == (True, [0, 0]) + + eqs4 = [Eq(Derivative(f(x), x), f(x) + exp(x)), + Eq(Derivative(g(x), x), x*exp(x) + f(x) + g(x))] + sol4 = [Eq(f(x), C1*exp(x) + x*exp(x)), + Eq(g(x), C1*x*exp(x) + C2*exp(x) + x**2*exp(x))] + assert dsolve(eqs4) == sol4 + assert checksysodesol(eqs4, sol4) == (True, [0, 0]) + + eqs5 = [Eq(Derivative(f(x), x), 5*x + f(x) + g(x)), + Eq(Derivative(g(x), x), f(x) - g(x))] + sol5 = [Eq(f(x), C1*(1 + sqrt(2))*exp(sqrt(2)*x) + C2*(1 - sqrt(2))*exp(-sqrt(2)*x) + x*Rational(-5, 2) + + Rational(-5, 2)), + Eq(g(x), C1*exp(sqrt(2)*x) + C2*exp(-sqrt(2)*x) + x*Rational(-5, 2))] + assert dsolve(eqs5) == sol5 + assert checksysodesol(eqs5, sol5) == (True, [0, 0]) + + eqs6 = [Eq(Derivative(f(x), x), -9*f(x) - 4*g(x)), + Eq(Derivative(g(x), x), -4*g(x)), + Eq(Derivative(h(x), x), h(x) + exp(x))] + sol6 = [Eq(f(x), C2*exp(-4*x)*Rational(-4, 5) + C1*exp(-9*x)), + Eq(g(x), C2*exp(-4*x)), + Eq(h(x), C3*exp(x) + x*exp(x))] + assert dsolve(eqs6) == sol6 + assert checksysodesol(eqs6, sol6) == (True, [0, 0, 0]) + + # Regression test case for issue #8859 + # https://github.com/sympy/sympy/issues/8859 + eqs7 = [Eq(Derivative(f(t), t), 3*t + f(t)), + Eq(Derivative(g(t), t), g(t))] + sol7 = [Eq(f(t), C1*exp(t) - 3*t - 3), + Eq(g(t), C2*exp(t))] + assert dsolve(eqs7) == sol7 + assert checksysodesol(eqs7, sol7) == (True, [0, 0]) + + # Regression test case for issue #8567 + # https://github.com/sympy/sympy/issues/8567 + eqs8 = [Eq(Derivative(f(t), t), f(t) + 2*g(t)), + Eq(Derivative(g(t), t), -2*f(t) + g(t) + 2*exp(t))] + sol8 = [Eq(f(t), C1*exp(t)*sin(2*t) + C2*exp(t)*cos(2*t) + + exp(t)*sin(2*t)**2 + exp(t)*cos(2*t)**2), + Eq(g(t), C1*exp(t)*cos(2*t) - C2*exp(t)*sin(2*t))] + assert dsolve(eqs8) == sol8 + assert checksysodesol(eqs8, sol8) == (True, [0, 0]) + + # Regression test case for issue #19150 + # https://github.com/sympy/sympy/issues/19150 + eqs9 = [Eq(Derivative(f(t), t), (c - 2*f(t) + g(t))/(a*b)), + Eq(Derivative(g(t), t), (f(t) - 2*g(t) + h(t))/(a*b)), + Eq(Derivative(h(t), t), (d + g(t) - 2*h(t))/(a*b))] + sol9 = [Eq(f(t), -C1*exp(-2*t/(a*b)) + C2*exp(-t*(sqrt(2) + 2)/(a*b)) + C3*exp(-t*(2 - sqrt(2))/(a*b)) + + Mul(Rational(1, 4), 3*c + d, evaluate=False)), + Eq(g(t), -sqrt(2)*C2*exp(-t*(sqrt(2) + 2)/(a*b)) + sqrt(2)*C3*exp(-t*(2 - sqrt(2))/(a*b)) + + Mul(Rational(1, 2), c + d, evaluate=False)), + Eq(h(t), C1*exp(-2*t/(a*b)) + C2*exp(-t*(sqrt(2) + 2)/(a*b)) + C3*exp(-t*(2 - sqrt(2))/(a*b)) + + Mul(Rational(1, 4), c + 3*d, evaluate=False))] + assert dsolve(eqs9) == sol9 + assert checksysodesol(eqs9, sol9) == (True, [0, 0, 0]) + + # Regression test case for issue #16635 + # https://github.com/sympy/sympy/issues/16635 + eqs10 = [Eq(Derivative(f(t), t), 15*t + f(t) - g(t) - 10), + Eq(Derivative(g(t), t), -15*t + f(t) - g(t) - 5)] + sol10 = [Eq(f(t), C1 + C2 + 5*t**3 + 5*t**2 + t*(C2 - 10)), + Eq(g(t), C1 + 5*t**3 - 10*t**2 + t*(C2 - 5))] + assert dsolve(eqs10) == sol10 + assert checksysodesol(eqs10, sol10) == (True, [0, 0]) + + # Multiple solutions + eqs11 = [Eq(Derivative(f(t), t)**2 - 2*Derivative(f(t), t) + 1, 4), + Eq(-y*f(t) + Derivative(g(t), t), 0)] + sol11 = [[Eq(f(t), C1 - t), Eq(g(t), C1*t*y + C2*y + t**2*y*Rational(-1, 2))], + [Eq(f(t), C1 + 3*t), Eq(g(t), C1*t*y + C2*y + t**2*y*Rational(3, 2))]] + assert dsolve(eqs11) == sol11 + for s11 in sol11: + assert checksysodesol(eqs11, s11) == (True, [0, 0]) + + # test case for issue #19831 + # https://github.com/sympy/sympy/issues/19831 + n = symbols('n', positive=True) + x0 = symbols('x_0') + t0 = symbols('t_0') + x_0 = symbols('x_0') + t_0 = symbols('t_0') + t = symbols('t') + x = Function('x') + y = Function('y') + T = symbols('T') + + eqs12 = [Eq(Derivative(y(t), t), x(t)), + Eq(Derivative(x(t), t), n*(y(t) + 1))] + sol12 = [Eq(y(t), C1*exp(sqrt(n)*t)*n**Rational(-1, 2) - C2*exp(-sqrt(n)*t)*n**Rational(-1, 2) - 1), + Eq(x(t), C1*exp(sqrt(n)*t) + C2*exp(-sqrt(n)*t))] + assert dsolve(eqs12) == sol12 + assert checksysodesol(eqs12, sol12) == (True, [0, 0]) + + sol12b = [ + Eq(y(t), (T*exp(-sqrt(n)*t_0)/2 + exp(-sqrt(n)*t_0)/2 + + x_0*exp(-sqrt(n)*t_0)/(2*sqrt(n)))*exp(sqrt(n)*t) + + (T*exp(sqrt(n)*t_0)/2 + exp(sqrt(n)*t_0)/2 - + x_0*exp(sqrt(n)*t_0)/(2*sqrt(n)))*exp(-sqrt(n)*t) - 1), + Eq(x(t), (T*sqrt(n)*exp(-sqrt(n)*t_0)/2 + sqrt(n)*exp(-sqrt(n)*t_0)/2 + + x_0*exp(-sqrt(n)*t_0)/2)*exp(sqrt(n)*t) + - (T*sqrt(n)*exp(sqrt(n)*t_0)/2 + sqrt(n)*exp(sqrt(n)*t_0)/2 - + x_0*exp(sqrt(n)*t_0)/2)*exp(-sqrt(n)*t)) + ] + assert dsolve(eqs12, ics={y(t0): T, x(t0): x0}) == sol12b + assert checksysodesol(eqs12, sol12b) == (True, [0, 0]) + + #Test cases added for the issue 19763 + #https://github.com/sympy/sympy/issues/19763 + + eq13 = [Eq(Derivative(f(t), t), f(t) + g(t) + 9), + Eq(Derivative(g(t), t), 2*f(t) + 5*g(t) + 23)] + sol13 = [Eq(f(t), -C1*(2 + sqrt(6))*exp(t*(3 - sqrt(6)))/2 - C2*(2 - sqrt(6))*exp(t*(sqrt(6) + 3))/2 - + Rational(22,3)), + Eq(g(t), C1*exp(t*(3 - sqrt(6))) + C2*exp(t*(sqrt(6) + 3)) - Rational(5,3))] + assert dsolve(eq13) == sol13 + assert checksysodesol(eq13, sol13) == (True, [0, 0]) + + eq14 = [Eq(Derivative(f(t), t), f(t) + g(t) + 81), + Eq(Derivative(g(t), t), -2*f(t) + g(t) + 23)] + sol14 = [Eq(f(t), sqrt(2)*C1*exp(t)*sin(sqrt(2)*t)/2 + + sqrt(2)*C2*exp(t)*cos(sqrt(2)*t)/2 + - 58*sin(sqrt(2)*t)**2/3 - 58*cos(sqrt(2)*t)**2/3), + Eq(g(t), C1*exp(t)*cos(sqrt(2)*t) - C2*exp(t)*sin(sqrt(2)*t) + - 185*sin(sqrt(2)*t)**2/3 - 185*cos(sqrt(2)*t)**2/3)] + assert dsolve(eq14) == sol14 + assert checksysodesol(eq14, sol14) == (True, [0,0]) + + eq15 = [Eq(Derivative(f(t), t), f(t) + 2*g(t) + k1), + Eq(Derivative(g(t), t), 3*f(t) + 4*g(t) + k2)] + sol15 = [Eq(f(t), -C1*(3 - sqrt(33))*exp(t*(5 + sqrt(33))/2)/6 - + C2*(3 + sqrt(33))*exp(t*(5 - sqrt(33))/2)/6 + 2*k1 - k2), + Eq(g(t), C1*exp(t*(5 + sqrt(33))/2) + C2*exp(t*(5 - sqrt(33))/2) - + Mul(Rational(1,2), 3*k1 - k2, evaluate = False))] + assert dsolve(eq15) == sol15 + assert checksysodesol(eq15, sol15) == (True, [0,0]) + + eq16 = [Eq(Derivative(f(t), t), k1), + Eq(Derivative(g(t), t), k2)] + sol16 = [Eq(f(t), C1 + k1*t), + Eq(g(t), C2 + k2*t)] + assert dsolve(eq16) == sol16 + assert checksysodesol(eq16, sol16) == (True, [0,0]) + + eq17 = [Eq(Derivative(f(t), t), 0), + Eq(Derivative(g(t), t), c*f(t) + k2)] + sol17 = [Eq(f(t), C1), + Eq(g(t), C2*c + t*(C1*c + k2))] + assert dsolve(eq17) == sol17 + assert checksysodesol(eq17 , sol17) == (True , [0,0]) + + eq18 = [Eq(Derivative(f(t), t), k1), + Eq(Derivative(g(t), t), f(t) + k2)] + sol18 = [Eq(f(t), C1 + k1*t), + Eq(g(t), C2 + k1*t**2/2 + t*(C1 + k2))] + assert dsolve(eq18) == sol18 + assert checksysodesol(eq18 , sol18) == (True , [0,0]) + + eq19 = [Eq(Derivative(f(t), t), k1), + Eq(Derivative(g(t), t), f(t) + 2*g(t) + k2)] + sol19 = [Eq(f(t), -2*C1 + k1*t), + Eq(g(t), C1 + C2*exp(2*t) - k1*t/2 - Mul(Rational(1,4), k1 + 2*k2 , evaluate = False))] + assert dsolve(eq19) == sol19 + assert checksysodesol(eq19 , sol19) == (True , [0,0]) + + eq20 = [Eq(diff(f(t), t), f(t) + k1), + Eq(diff(g(t), t), k2)] + sol20 = [Eq(f(t), C1*exp(t) - k1), + Eq(g(t), C2 + k2*t)] + assert dsolve(eq20) == sol20 + assert checksysodesol(eq20 , sol20) == (True , [0,0]) + + eq21 = [Eq(diff(f(t), t), g(t) + k1), + Eq(diff(g(t), t), 0)] + sol21 = [Eq(f(t), C1 + t*(C2 + k1)), + Eq(g(t), C2)] + assert dsolve(eq21) == sol21 + assert checksysodesol(eq21 , sol21) == (True , [0,0]) + + eq22 = [Eq(Derivative(f(t), t), f(t) + 2*g(t) + k1), + Eq(Derivative(g(t), t), k2)] + sol22 = [Eq(f(t), -2*C1 + C2*exp(t) - k1 - 2*k2*t - 2*k2), + Eq(g(t), C1 + k2*t)] + assert dsolve(eq22) == sol22 + assert checksysodesol(eq22 , sol22) == (True , [0,0]) + + eq23 = [Eq(Derivative(f(t), t), g(t) + k1), + Eq(Derivative(g(t), t), 2*g(t) + k2)] + sol23 = [Eq(f(t), C1 + C2*exp(2*t)/2 - k2/4 + t*(2*k1 - k2)/2), + Eq(g(t), C2*exp(2*t) - k2/2)] + assert dsolve(eq23) == sol23 + assert checksysodesol(eq23 , sol23) == (True , [0,0]) + + eq24 = [Eq(Derivative(f(t), t), f(t) + k1), + Eq(Derivative(g(t), t), 2*f(t) + k2)] + sol24 = [Eq(f(t), C1*exp(t)/2 - k1), + Eq(g(t), C1*exp(t) + C2 - 2*k1 - t*(2*k1 - k2))] + assert dsolve(eq24) == sol24 + assert checksysodesol(eq24 , sol24) == (True , [0,0]) + + eq25 = [Eq(Derivative(f(t), t), f(t) + 2*g(t) + k1), + Eq(Derivative(g(t), t), 3*f(t) + 6*g(t) + k2)] + sol25 = [Eq(f(t), -2*C1 + C2*exp(7*t)/3 + 2*t*(3*k1 - k2)/7 - + Mul(Rational(1,49), k1 + 2*k2 , evaluate = False)), + Eq(g(t), C1 + C2*exp(7*t) - t*(3*k1 - k2)/7 - + Mul(Rational(3,49), k1 + 2*k2 , evaluate = False))] + assert dsolve(eq25) == sol25 + assert checksysodesol(eq25 , sol25) == (True , [0,0]) + + eq26 = [Eq(Derivative(f(t), t), 2*f(t) - g(t) + k1), + Eq(Derivative(g(t), t), 4*f(t) - 2*g(t) + 2*k1)] + sol26 = [Eq(f(t), C1 + 2*C2 + t*(2*C1 + k1)), + Eq(g(t), 4*C2 + t*(4*C1 + 2*k1))] + assert dsolve(eq26) == sol26 + assert checksysodesol(eq26 , sol26) == (True , [0,0]) + + # Test Case added for issue #22715 + # https://github.com/sympy/sympy/issues/22715 + + eq27 = [Eq(diff(x(t),t),-1*y(t)+10), Eq(diff(y(t),t),5*x(t)-2*y(t)+3)] + sol27 = [Eq(x(t), (C1/5 - 2*C2/5)*exp(-t)*cos(2*t) + - (2*C1/5 + C2/5)*exp(-t)*sin(2*t) + + 17*sin(2*t)**2/5 + 17*cos(2*t)**2/5), + Eq(y(t), C1*exp(-t)*cos(2*t) - C2*exp(-t)*sin(2*t) + + 10*sin(2*t)**2 + 10*cos(2*t)**2)] + assert dsolve(eq27) == sol27 + assert checksysodesol(eq27 , sol27) == (True , [0,0]) + + +def test_sysode_linear_neq_order1_type3(): + + f, g, h, k, x0 , y0 = symbols('f g h k x0 y0', cls=Function) + x, t, a = symbols('x t a') + r = symbols('r', real=True) + + eqs1 = [Eq(Derivative(f(r), r), r*g(r) + f(r)), + Eq(Derivative(g(r), r), -r*f(r) + g(r))] + sol1 = [Eq(f(r), C1*exp(r)*sin(r**2/2) + C2*exp(r)*cos(r**2/2)), + Eq(g(r), C1*exp(r)*cos(r**2/2) - C2*exp(r)*sin(r**2/2))] + assert dsolve(eqs1) == sol1 + assert checksysodesol(eqs1, sol1) == (True, [0, 0]) + + eqs2 = [Eq(Derivative(f(x), x), x**2*g(x) + x*f(x)), + Eq(Derivative(g(x), x), 2*x**2*f(x) + (3*x**2 + x)*g(x))] + sol2 = [Eq(f(x), (sqrt(17)*C1/17 + C2*(17 - 3*sqrt(17))/34)*exp(x**3*(3 + sqrt(17))/6 + x**2/2) - + exp(x**3*(3 - sqrt(17))/6 + x**2/2)*(sqrt(17)*C1/17 + C2*(3*sqrt(17) + 17)*Rational(-1, 34))), + Eq(g(x), exp(x**3*(3 - sqrt(17))/6 + x**2/2)*(C1*(17 - 3*sqrt(17))/34 + sqrt(17)*C2*Rational(-2, + 17)) + exp(x**3*(3 + sqrt(17))/6 + x**2/2)*(C1*(3*sqrt(17) + 17)/34 + sqrt(17)*C2*Rational(2, 17)))] + assert dsolve(eqs2) == sol2 + assert checksysodesol(eqs2, sol2) == (True, [0, 0]) + + eqs3 = [Eq(f(x).diff(x), x*f(x) + g(x)), + Eq(g(x).diff(x), -f(x) + x*g(x))] + sol3 = [Eq(f(x), (C1/2 + I*C2/2)*exp(x**2/2 - I*x) + exp(x**2/2 + I*x)*(C1/2 + I*C2*Rational(-1, 2))), + Eq(g(x), (I*C1/2 + C2/2)*exp(x**2/2 + I*x) - exp(x**2/2 - I*x)*(I*C1/2 + C2*Rational(-1, 2)))] + assert dsolve(eqs3) == sol3 + assert checksysodesol(eqs3, sol3) == (True, [0, 0]) + + eqs4 = [Eq(f(x).diff(x), x*(f(x) + g(x) + h(x))), Eq(g(x).diff(x), x*(f(x) + g(x) + h(x))), + Eq(h(x).diff(x), x*(f(x) + g(x) + h(x)))] + sol4 = [Eq(f(x), -C1/3 - C2/3 + 2*C3/3 + (C1/3 + C2/3 + C3/3)*exp(3*x**2/2)), + Eq(g(x), 2*C1/3 - C2/3 - C3/3 + (C1/3 + C2/3 + C3/3)*exp(3*x**2/2)), + Eq(h(x), -C1/3 + 2*C2/3 - C3/3 + (C1/3 + C2/3 + C3/3)*exp(3*x**2/2))] + assert dsolve(eqs4) == sol4 + assert checksysodesol(eqs4, sol4) == (True, [0, 0, 0]) + + eqs5 = [Eq(f(x).diff(x), x**2*(f(x) + g(x) + h(x))), Eq(g(x).diff(x), x**2*(f(x) + g(x) + h(x))), + Eq(h(x).diff(x), x**2*(f(x) + g(x) + h(x)))] + sol5 = [Eq(f(x), -C1/3 - C2/3 + 2*C3/3 + (C1/3 + C2/3 + C3/3)*exp(x**3)), + Eq(g(x), 2*C1/3 - C2/3 - C3/3 + (C1/3 + C2/3 + C3/3)*exp(x**3)), + Eq(h(x), -C1/3 + 2*C2/3 - C3/3 + (C1/3 + C2/3 + C3/3)*exp(x**3))] + assert dsolve(eqs5) == sol5 + assert checksysodesol(eqs5, sol5) == (True, [0, 0, 0]) + + eqs6 = [Eq(Derivative(f(x), x), x*(f(x) + g(x) + h(x) + k(x))), + Eq(Derivative(g(x), x), x*(f(x) + g(x) + h(x) + k(x))), + Eq(Derivative(h(x), x), x*(f(x) + g(x) + h(x) + k(x))), + Eq(Derivative(k(x), x), x*(f(x) + g(x) + h(x) + k(x)))] + sol6 = [Eq(f(x), -C1/4 - C2/4 - C3/4 + 3*C4/4 + (C1/4 + C2/4 + C3/4 + C4/4)*exp(2*x**2)), + Eq(g(x), 3*C1/4 - C2/4 - C3/4 - C4/4 + (C1/4 + C2/4 + C3/4 + C4/4)*exp(2*x**2)), + Eq(h(x), -C1/4 + 3*C2/4 - C3/4 - C4/4 + (C1/4 + C2/4 + C3/4 + C4/4)*exp(2*x**2)), + Eq(k(x), -C1/4 - C2/4 + 3*C3/4 - C4/4 + (C1/4 + C2/4 + C3/4 + C4/4)*exp(2*x**2))] + assert dsolve(eqs6) == sol6 + assert checksysodesol(eqs6, sol6) == (True, [0, 0, 0, 0]) + + y = symbols("y", real=True) + + eqs7 = [Eq(Derivative(f(y), y), y*f(y) + g(y)), + Eq(Derivative(g(y), y), y*g(y) - f(y))] + sol7 = [Eq(f(y), C1*exp(y**2/2)*sin(y) + C2*exp(y**2/2)*cos(y)), + Eq(g(y), C1*exp(y**2/2)*cos(y) - C2*exp(y**2/2)*sin(y))] + assert dsolve(eqs7) == sol7 + assert checksysodesol(eqs7, sol7) == (True, [0, 0]) + + #Test cases added for the issue 19763 + #https://github.com/sympy/sympy/issues/19763 + + eqs8 = [Eq(Derivative(f(t), t), 5*t*f(t) + 2*h(t)), + Eq(Derivative(h(t), t), 2*f(t) + 5*t*h(t))] + sol8 = [Eq(f(t), Mul(-1, (C1/2 - C2/2), evaluate = False)*exp(5*t**2/2 - 2*t) + (C1/2 + C2/2)*exp(5*t**2/2 + 2*t)), + Eq(h(t), (C1/2 - C2/2)*exp(5*t**2/2 - 2*t) + (C1/2 + C2/2)*exp(5*t**2/2 + 2*t))] + assert dsolve(eqs8) == sol8 + assert checksysodesol(eqs8, sol8) == (True, [0, 0]) + + eqs9 = [Eq(diff(f(t), t), 5*t*f(t) + t**2*g(t)), + Eq(diff(g(t), t), -t**2*f(t) + 5*t*g(t))] + sol9 = [Eq(f(t), (C1/2 - I*C2/2)*exp(I*t**3/3 + 5*t**2/2) + (C1/2 + I*C2/2)*exp(-I*t**3/3 + 5*t**2/2)), + Eq(g(t), Mul(-1, (I*C1/2 - C2/2) , evaluate = False)*exp(-I*t**3/3 + 5*t**2/2) + (I*C1/2 + C2/2)*exp(I*t**3/3 + 5*t**2/2))] + assert dsolve(eqs9) == sol9 + assert checksysodesol(eqs9 , sol9) == (True , [0,0]) + + eqs10 = [Eq(diff(f(t), t), t**2*g(t) + 5*t*f(t)), + Eq(diff(g(t), t), -t**2*f(t) + (9*t**2 + 5*t)*g(t))] + sol10 = [Eq(f(t), (C1*(77 - 9*sqrt(77))/154 + sqrt(77)*C2/77)*exp(t**3*(sqrt(77) + 9)/6 + 5*t**2/2) + (C1*(77 + 9*sqrt(77))/154 - sqrt(77)*C2/77)*exp(t**3*(9 - sqrt(77))/6 + 5*t**2/2)), + Eq(g(t), (sqrt(77)*C1/77 + C2*(77 - 9*sqrt(77))/154)*exp(t**3*(9 - sqrt(77))/6 + 5*t**2/2) - (sqrt(77)*C1/77 - C2*(77 + 9*sqrt(77))/154)*exp(t**3*(sqrt(77) + 9)/6 + 5*t**2/2))] + assert dsolve(eqs10) == sol10 + assert checksysodesol(eqs10 , sol10) == (True , [0,0]) + + eqs11 = [Eq(diff(f(t), t), 5*t*f(t) + t**2*g(t)), + Eq(diff(g(t), t), (1-t**2)*f(t) + (5*t + 9*t**2)*g(t))] + sol11 = [Eq(f(t), C1*x0(t) + C2*x0(t)*Integral(t**2*exp(Integral(5*t, t))*exp(Integral(9*t**2 + 5*t, t))/x0(t)**2, t)), + Eq(g(t), C1*y0(t) + C2*(y0(t)*Integral(t**2*exp(Integral(5*t, t))*exp(Integral(9*t**2 + 5*t, t))/x0(t)**2, t) + exp(Integral(5*t, t))*exp(Integral(9*t**2 + 5*t, t))/x0(t)))] + assert dsolve(eqs11) == sol11 + +@slow +def test_sysode_linear_neq_order1_type4(): + + f, g, h, k = symbols('f g h k', cls=Function) + x, t, a = symbols('x t a') + r = symbols('r', real=True) + + eqs1 = [Eq(diff(f(r), r), f(r) + r*g(r) + r**2), Eq(diff(g(r), r), -r*f(r) + g(r) + r)] + sol1 = [Eq(f(r), C1*exp(r)*sin(r**2/2) + C2*exp(r)*cos(r**2/2) + exp(r)*sin(r**2/2)*Integral(r**2*exp(-r)*sin(r**2/2) + + r*exp(-r)*cos(r**2/2), r) + exp(r)*cos(r**2/2)*Integral(r**2*exp(-r)*cos(r**2/2) - r*exp(-r)*sin(r**2/2), r)), + Eq(g(r), C1*exp(r)*cos(r**2/2) - C2*exp(r)*sin(r**2/2) - exp(r)*sin(r**2/2)*Integral(r**2*exp(-r)*cos(r**2/2) - + r*exp(-r)*sin(r**2/2), r) + exp(r)*cos(r**2/2)*Integral(r**2*exp(-r)*sin(r**2/2) + r*exp(-r)*cos(r**2/2), r))] + assert dsolve(eqs1) == sol1 + assert checksysodesol(eqs1, sol1) == (True, [0, 0]) + + eqs2 = [Eq(diff(f(r), r), f(r) + r*g(r) + r), Eq(diff(g(r), r), -r*f(r) + g(r) + log(r))] + sol2 = [Eq(f(r), C1*exp(r)*sin(r**2/2) + C2*exp(r)*cos(r**2/2) + exp(r)*sin(r**2/2)*Integral(r*exp(-r)*sin(r**2/2) + + exp(-r)*log(r)*cos(r**2/2), r) + exp(r)*cos(r**2/2)*Integral(r*exp(-r)*cos(r**2/2) - exp(-r)*log(r)*sin( + r**2/2), r)), + Eq(g(r), C1*exp(r)*cos(r**2/2) - C2*exp(r)*sin(r**2/2) - exp(r)*sin(r**2/2)*Integral(r*exp(-r)*cos(r**2/2) - + exp(-r)*log(r)*sin(r**2/2), r) + exp(r)*cos(r**2/2)*Integral(r*exp(-r)*sin(r**2/2) + exp(-r)*log(r)*cos( + r**2/2), r))] + # XXX: dsolve hangs for this in integration + assert dsolve_system(eqs2, simplify=False, doit=False) == [sol2] + assert checksysodesol(eqs2, sol2) == (True, [0, 0]) + + eqs3 = [Eq(Derivative(f(x), x), x*(f(x) + g(x) + h(x)) + x), + Eq(Derivative(g(x), x), x*(f(x) + g(x) + h(x)) + x), + Eq(Derivative(h(x), x), x*(f(x) + g(x) + h(x)) + 1)] + sol3 = [Eq(f(x), C1*Rational(-1, 3) + C2*Rational(-1, 3) + C3*Rational(2, 3) + x**2/6 + x*Rational(-1, 3) + + (C1/3 + C2/3 + C3/3)*exp(x**2*Rational(3, 2)) + + sqrt(6)*sqrt(pi)*erf(sqrt(6)*x/2)*exp(x**2*Rational(3, 2))/18 + Rational(-2, 9)), + Eq(g(x), C1*Rational(2, 3) + C2*Rational(-1, 3) + C3*Rational(-1, 3) + x**2/6 + x*Rational(-1, 3) + + (C1/3 + C2/3 + C3/3)*exp(x**2*Rational(3, 2)) + + sqrt(6)*sqrt(pi)*erf(sqrt(6)*x/2)*exp(x**2*Rational(3, 2))/18 + Rational(-2, 9)), + Eq(h(x), C1*Rational(-1, 3) + C2*Rational(2, 3) + C3*Rational(-1, 3) + x**2*Rational(-1, 3) + + x*Rational(2, 3) + (C1/3 + C2/3 + C3/3)*exp(x**2*Rational(3, 2)) + + sqrt(6)*sqrt(pi)*erf(sqrt(6)*x/2)*exp(x**2*Rational(3, 2))/18 + Rational(-2, 9))] + assert dsolve(eqs3) == sol3 + assert checksysodesol(eqs3, sol3) == (True, [0, 0, 0]) + + eqs4 = [Eq(Derivative(f(x), x), x*(f(x) + g(x) + h(x)) + sin(x)), + Eq(Derivative(g(x), x), x*(f(x) + g(x) + h(x)) + sin(x)), + Eq(Derivative(h(x), x), x*(f(x) + g(x) + h(x)) + sin(x))] + sol4 = [Eq(f(x), C1*Rational(-1, 3) + C2*Rational(-1, 3) + C3*Rational(2, 3) + (C1/3 + C2/3 + + C3/3)*exp(x**2*Rational(3, 2)) + Integral(sin(x)*exp(x**2*Rational(-3, 2)), x)*exp(x**2*Rational(3, + 2))), + Eq(g(x), C1*Rational(2, 3) + C2*Rational(-1, 3) + C3*Rational(-1, 3) + (C1/3 + C2/3 + + C3/3)*exp(x**2*Rational(3, 2)) + Integral(sin(x)*exp(x**2*Rational(-3, 2)), x)*exp(x**2*Rational(3, + 2))), + Eq(h(x), C1*Rational(-1, 3) + C2*Rational(2, 3) + C3*Rational(-1, 3) + (C1/3 + C2/3 + + C3/3)*exp(x**2*Rational(3, 2)) + Integral(sin(x)*exp(x**2*Rational(-3, 2)), x)*exp(x**2*Rational(3, + 2)))] + assert dsolve(eqs4) == sol4 + assert checksysodesol(eqs4, sol4) == (True, [0, 0, 0]) + + eqs5 = [Eq(Derivative(f(x), x), x*(f(x) + g(x) + h(x) + k(x) + 1)), + Eq(Derivative(g(x), x), x*(f(x) + g(x) + h(x) + k(x) + 1)), + Eq(Derivative(h(x), x), x*(f(x) + g(x) + h(x) + k(x) + 1)), + Eq(Derivative(k(x), x), x*(f(x) + g(x) + h(x) + k(x) + 1))] + sol5 = [Eq(f(x), C1*Rational(-1, 4) + C2*Rational(-1, 4) + C3*Rational(-1, 4) + C4*Rational(3, 4) + (C1/4 + + C2/4 + C3/4 + C4/4)*exp(2*x**2) + Rational(-1, 4)), + Eq(g(x), C1*Rational(3, 4) + C2*Rational(-1, 4) + C3*Rational(-1, 4) + C4*Rational(-1, 4) + (C1/4 + + C2/4 + C3/4 + C4/4)*exp(2*x**2) + Rational(-1, 4)), + Eq(h(x), C1*Rational(-1, 4) + C2*Rational(3, 4) + C3*Rational(-1, 4) + C4*Rational(-1, 4) + (C1/4 + + C2/4 + C3/4 + C4/4)*exp(2*x**2) + Rational(-1, 4)), + Eq(k(x), C1*Rational(-1, 4) + C2*Rational(-1, 4) + C3*Rational(3, 4) + C4*Rational(-1, 4) + (C1/4 + + C2/4 + C3/4 + C4/4)*exp(2*x**2) + Rational(-1, 4))] + assert dsolve(eqs5) == sol5 + assert checksysodesol(eqs5, sol5) == (True, [0, 0, 0, 0]) + + eqs6 = [Eq(Derivative(f(x), x), x**2*(f(x) + g(x) + h(x) + k(x) + 1)), + Eq(Derivative(g(x), x), x**2*(f(x) + g(x) + h(x) + k(x) + 1)), + Eq(Derivative(h(x), x), x**2*(f(x) + g(x) + h(x) + k(x) + 1)), + Eq(Derivative(k(x), x), x**2*(f(x) + g(x) + h(x) + k(x) + 1))] + sol6 = [Eq(f(x), C1*Rational(-1, 4) + C2*Rational(-1, 4) + C3*Rational(-1, 4) + C4*Rational(3, 4) + (C1/4 + + C2/4 + C3/4 + C4/4)*exp(x**3*Rational(4, 3)) + Rational(-1, 4)), + Eq(g(x), C1*Rational(3, 4) + C2*Rational(-1, 4) + C3*Rational(-1, 4) + C4*Rational(-1, 4) + (C1/4 + + C2/4 + C3/4 + C4/4)*exp(x**3*Rational(4, 3)) + Rational(-1, 4)), + Eq(h(x), C1*Rational(-1, 4) + C2*Rational(3, 4) + C3*Rational(-1, 4) + C4*Rational(-1, 4) + (C1/4 + + C2/4 + C3/4 + C4/4)*exp(x**3*Rational(4, 3)) + Rational(-1, 4)), + Eq(k(x), C1*Rational(-1, 4) + C2*Rational(-1, 4) + C3*Rational(3, 4) + C4*Rational(-1, 4) + (C1/4 + + C2/4 + C3/4 + C4/4)*exp(x**3*Rational(4, 3)) + Rational(-1, 4))] + assert dsolve(eqs6) == sol6 + assert checksysodesol(eqs6, sol6) == (True, [0, 0, 0, 0]) + + eqs7 = [Eq(Derivative(f(x), x), (f(x) + g(x) + h(x))*log(x) + sin(x)), Eq(Derivative(g(x), x), (f(x) + g(x) + + h(x))*log(x) + sin(x)), Eq(Derivative(h(x), x), (f(x) + g(x) + h(x))*log(x) + sin(x))] + sol7 = [Eq(f(x), -C1/3 - C2/3 + 2*C3/3 + (C1/3 + C2/3 + + C3/3)*exp(x*(3*log(x) - 3)) + exp(x*(3*log(x) - + 3))*Integral(exp(3*x)*exp(-3*x*log(x))*sin(x), x)), + Eq(g(x), 2*C1/3 - C2/3 - C3/3 + (C1/3 + C2/3 + + C3/3)*exp(x*(3*log(x) - 3)) + exp(x*(3*log(x) - + 3))*Integral(exp(3*x)*exp(-3*x*log(x))*sin(x), x)), + Eq(h(x), -C1/3 + 2*C2/3 - C3/3 + (C1/3 + C2/3 + + C3/3)*exp(x*(3*log(x) - 3)) + exp(x*(3*log(x) - + 3))*Integral(exp(3*x)*exp(-3*x*log(x))*sin(x), x))] + with dotprodsimp(True): + assert dsolve(eqs7, simplify=False, doit=False) == sol7 + assert checksysodesol(eqs7, sol7) == (True, [0, 0, 0]) + + eqs8 = [Eq(Derivative(f(x), x), (f(x) + g(x) + h(x) + k(x))*log(x) + sin(x)), Eq(Derivative(g(x), x), (f(x) + + g(x) + h(x) + k(x))*log(x) + sin(x)), Eq(Derivative(h(x), x), (f(x) + g(x) + h(x) + k(x))*log(x) + + sin(x)), Eq(Derivative(k(x), x), (f(x) + g(x) + h(x) + k(x))*log(x) + sin(x))] + sol8 = [Eq(f(x), -C1/4 - C2/4 - C3/4 + 3*C4/4 + (C1/4 + C2/4 + C3/4 + + C4/4)*exp(x*(4*log(x) - 4)) + exp(x*(4*log(x) - + 4))*Integral(exp(4*x)*exp(-4*x*log(x))*sin(x), x)), + Eq(g(x), 3*C1/4 - C2/4 - C3/4 - C4/4 + (C1/4 + C2/4 + C3/4 + + C4/4)*exp(x*(4*log(x) - 4)) + exp(x*(4*log(x) - + 4))*Integral(exp(4*x)*exp(-4*x*log(x))*sin(x), x)), + Eq(h(x), -C1/4 + 3*C2/4 - C3/4 - C4/4 + (C1/4 + C2/4 + C3/4 + + C4/4)*exp(x*(4*log(x) - 4)) + exp(x*(4*log(x) - + 4))*Integral(exp(4*x)*exp(-4*x*log(x))*sin(x), x)), + Eq(k(x), -C1/4 - C2/4 + 3*C3/4 - C4/4 + (C1/4 + C2/4 + C3/4 + + C4/4)*exp(x*(4*log(x) - 4)) + exp(x*(4*log(x) - + 4))*Integral(exp(4*x)*exp(-4*x*log(x))*sin(x), x))] + with dotprodsimp(True): + assert dsolve(eqs8) == sol8 + assert checksysodesol(eqs8, sol8) == (True, [0, 0, 0, 0]) + + +def test_sysode_linear_neq_order1_type5_type6(): + f, g = symbols("f g", cls=Function) + x, x_ = symbols("x x_") + + # Type 5 + eqs1 = [Eq(Derivative(f(x), x), (2*f(x) + g(x))/x), Eq(Derivative(g(x), x), (f(x) + 2*g(x))/x)] + sol1 = [Eq(f(x), -C1*x + C2*x**3), Eq(g(x), C1*x + C2*x**3)] + assert dsolve(eqs1) == sol1 + assert checksysodesol(eqs1, sol1) == (True, [0, 0]) + + # Type 6 + eqs2 = [Eq(Derivative(f(x), x), (2*f(x) + g(x) + 1)/x), + Eq(Derivative(g(x), x), (x + f(x) + 2*g(x))/x)] + sol2 = [Eq(f(x), C2*x**3 - x*(C1 + Rational(1, 4)) + x*log(x)*Rational(-1, 2) + Rational(-2, 3)), + Eq(g(x), C2*x**3 + x*log(x)/2 + x*(C1 + Rational(-1, 4)) + Rational(1, 3))] + assert dsolve(eqs2) == sol2 + assert checksysodesol(eqs2, sol2) == (True, [0, 0]) + + +def test_higher_order_to_first_order(): + f, g = symbols('f g', cls=Function) + x = symbols('x') + + eqs1 = [Eq(Derivative(f(x), (x, 2)), 2*f(x) + g(x)), + Eq(Derivative(g(x), (x, 2)), -f(x))] + sol1 = [Eq(f(x), -C2*x*exp(-x) + C3*x*exp(x) - (C1 - C2)*exp(-x) + (C3 + C4)*exp(x)), + Eq(g(x), C2*x*exp(-x) - C3*x*exp(x) + (C1 + C2)*exp(-x) + (C3 - C4)*exp(x))] + assert dsolve(eqs1) == sol1 + assert checksysodesol(eqs1, sol1) == (True, [0, 0]) + + eqs2 = [Eq(f(x).diff(x, 2), 0), Eq(g(x).diff(x, 2), f(x))] + sol2 = [Eq(f(x), C1 + C2*x), Eq(g(x), C1*x**2/2 + C2*x**3/6 + C3 + C4*x)] + assert dsolve(eqs2) == sol2 + assert checksysodesol(eqs2, sol2) == (True, [0, 0]) + + eqs3 = [Eq(Derivative(f(x), (x, 2)), 2*f(x)), + Eq(Derivative(g(x), (x, 2)), -f(x) + 2*g(x))] + sol3 = [Eq(f(x), 4*C1*exp(-sqrt(2)*x) + 4*C2*exp(sqrt(2)*x)), + Eq(g(x), sqrt(2)*C1*x*exp(-sqrt(2)*x) - sqrt(2)*C2*x*exp(sqrt(2)*x) + (C1 + + sqrt(2)*C4)*exp(-sqrt(2)*x) + (C2 - sqrt(2)*C3)*exp(sqrt(2)*x))] + assert dsolve(eqs3) == sol3 + assert checksysodesol(eqs3, sol3) == (True, [0, 0]) + + eqs4 = [Eq(Derivative(f(x), (x, 2)), 2*f(x) + g(x)), + Eq(Derivative(g(x), (x, 2)), 2*g(x))] + sol4 = [Eq(f(x), C1*x*exp(sqrt(2)*x)/4 + C3*x*exp(-sqrt(2)*x)/4 + (C2/4 + sqrt(2)*C3/8)*exp(-sqrt(2)*x) - + exp(sqrt(2)*x)*(sqrt(2)*C1/8 + C4*Rational(-1, 4))), + Eq(g(x), sqrt(2)*C1*exp(sqrt(2)*x)/2 + sqrt(2)*C3*exp(-sqrt(2)*x)*Rational(-1, 2))] + assert dsolve(eqs4) == sol4 + assert checksysodesol(eqs4, sol4) == (True, [0, 0]) + + eqs5 = [Eq(f(x).diff(x, 2), f(x)), Eq(g(x).diff(x, 2), f(x))] + sol5 = [Eq(f(x), -C1*exp(-x) + C2*exp(x)), Eq(g(x), -C1*exp(-x) + C2*exp(x) + C3 + C4*x)] + assert dsolve(eqs5) == sol5 + assert checksysodesol(eqs5, sol5) == (True, [0, 0]) + + eqs6 = [Eq(Derivative(f(x), (x, 2)), f(x) + g(x)), + Eq(Derivative(g(x), (x, 2)), -f(x) - g(x))] + sol6 = [Eq(f(x), C1 + C2*x**2/2 + C2 + C4*x**3/6 + x*(C3 + C4)), + Eq(g(x), -C1 + C2*x**2*Rational(-1, 2) - C3*x + C4*x**3*Rational(-1, 6))] + assert dsolve(eqs6) == sol6 + assert checksysodesol(eqs6, sol6) == (True, [0, 0]) + + eqs7 = [Eq(Derivative(f(x), (x, 2)), f(x) + g(x) + 1), + Eq(Derivative(g(x), (x, 2)), f(x) + g(x) + 1)] + sol7 = [Eq(f(x), -C1 - C2*x + sqrt(2)*C3*exp(sqrt(2)*x)/2 + sqrt(2)*C4*exp(-sqrt(2)*x)*Rational(-1, 2) + + Rational(-1, 2)), + Eq(g(x), C1 + C2*x + sqrt(2)*C3*exp(sqrt(2)*x)/2 + sqrt(2)*C4*exp(-sqrt(2)*x)*Rational(-1, 2) + + Rational(-1, 2))] + assert dsolve(eqs7) == sol7 + assert checksysodesol(eqs7, sol7) == (True, [0, 0]) + + eqs8 = [Eq(Derivative(f(x), (x, 2)), f(x) + g(x) + 1), + Eq(Derivative(g(x), (x, 2)), -f(x) - g(x) + 1)] + sol8 = [Eq(f(x), C1 + C2 + C4*x**3/6 + x**4/12 + x**2*(C2/2 + Rational(1, 2)) + x*(C3 + C4)), + Eq(g(x), -C1 - C3*x + C4*x**3*Rational(-1, 6) + x**4*Rational(-1, 12) - x**2*(C2/2 + Rational(-1, + 2)))] + assert dsolve(eqs8) == sol8 + assert checksysodesol(eqs8, sol8) == (True, [0, 0]) + + x, y = symbols('x, y', cls=Function) + t, l = symbols('t, l') + + eqs10 = [Eq(Derivative(x(t), (t, 2)), 5*x(t) + 43*y(t)), + Eq(Derivative(y(t), (t, 2)), x(t) + 9*y(t))] + sol10 = [Eq(x(t), C1*(61 - 9*sqrt(47))*sqrt(sqrt(47) + 7)*exp(-t*sqrt(sqrt(47) + 7))/2 + C2*sqrt(7 - + sqrt(47))*(61 + 9*sqrt(47))*exp(-t*sqrt(7 - sqrt(47)))/2 + C3*(61 - 9*sqrt(47))*sqrt(sqrt(47) + + 7)*exp(t*sqrt(sqrt(47) + 7))*Rational(-1, 2) + C4*sqrt(7 - sqrt(47))*(61 + 9*sqrt(47))*exp(t*sqrt(7 + - sqrt(47)))*Rational(-1, 2)), + Eq(y(t), C1*(7 - sqrt(47))*sqrt(sqrt(47) + 7)*exp(-t*sqrt(sqrt(47) + 7))*Rational(-1, 2) + C2*sqrt(7 + - sqrt(47))*(sqrt(47) + 7)*exp(-t*sqrt(7 - sqrt(47)))*Rational(-1, 2) + C3*(7 - + sqrt(47))*sqrt(sqrt(47) + 7)*exp(t*sqrt(sqrt(47) + 7))/2 + C4*sqrt(7 - sqrt(47))*(sqrt(47) + + 7)*exp(t*sqrt(7 - sqrt(47)))/2)] + assert dsolve(eqs10) == sol10 + assert checksysodesol(eqs10, sol10) == (True, [0, 0]) + + eqs11 = [Eq(7*x(t) + Derivative(x(t), (t, 2)) - 9*Derivative(y(t), t), 0), + Eq(7*y(t) + 9*Derivative(x(t), t) + Derivative(y(t), (t, 2)), 0)] + sol11 = [Eq(y(t), C1*(9 - sqrt(109))*sin(sqrt(2)*t*sqrt(9*sqrt(109) + 95)/2)/14 + C2*(9 - + sqrt(109))*cos(sqrt(2)*t*sqrt(9*sqrt(109) + 95)/2)*Rational(-1, 14) + C3*(9 + + sqrt(109))*sin(sqrt(2)*t*sqrt(95 - 9*sqrt(109))/2)/14 + C4*(9 + sqrt(109))*cos(sqrt(2)*t*sqrt(95 - + 9*sqrt(109))/2)*Rational(-1, 14)), + Eq(x(t), C1*(9 - sqrt(109))*cos(sqrt(2)*t*sqrt(9*sqrt(109) + 95)/2)*Rational(-1, 14) + C2*(9 - + sqrt(109))*sin(sqrt(2)*t*sqrt(9*sqrt(109) + 95)/2)*Rational(-1, 14) + C3*(9 + + sqrt(109))*cos(sqrt(2)*t*sqrt(95 - 9*sqrt(109))/2)/14 + C4*(9 + sqrt(109))*sin(sqrt(2)*t*sqrt(95 - + 9*sqrt(109))/2)/14)] + assert dsolve(eqs11) == sol11 + assert checksysodesol(eqs11, sol11) == (True, [0, 0]) + + # Euler Systems + # Note: To add examples of euler systems solver with non-homogeneous term. + eqs13 = [Eq(Derivative(f(t), (t, 2)), Derivative(f(t), t)/t + f(t)/t**2 + g(t)/t**2), + Eq(Derivative(g(t), (t, 2)), g(t)/t**2)] + sol13 = [Eq(f(t), C1*(sqrt(5) + 3)*Rational(-1, 2)*t**(Rational(1, 2) + + sqrt(5)*Rational(-1, 2)) + C2*t**(Rational(1, 2) + + sqrt(5)/2)*(3 - sqrt(5))*Rational(-1, 2) - C3*t**(1 - + sqrt(2))*(1 + sqrt(2)) - C4*t**(1 + sqrt(2))*(1 - sqrt(2))), + Eq(g(t), C1*(1 + sqrt(5))*Rational(-1, 2)*t**(Rational(1, 2) + + sqrt(5)*Rational(-1, 2)) + C2*t**(Rational(1, 2) + + sqrt(5)/2)*(1 - sqrt(5))*Rational(-1, 2))] + assert dsolve(eqs13) == sol13 + assert checksysodesol(eqs13, sol13) == (True, [0, 0]) + + # Solving systems using dsolve separately + eqs14 = [Eq(Derivative(f(t), (t, 2)), t*f(t)), + Eq(Derivative(g(t), (t, 2)), t*g(t))] + sol14 = [Eq(f(t), C1*airyai(t) + C2*airybi(t)), + Eq(g(t), C3*airyai(t) + C4*airybi(t))] + assert dsolve(eqs14) == sol14 + assert checksysodesol(eqs14, sol14) == (True, [0, 0]) + + + eqs15 = [Eq(Derivative(x(t), (t, 2)), t*(4*Derivative(x(t), t) + 8*Derivative(y(t), t))), + Eq(Derivative(y(t), (t, 2)), t*(12*Derivative(x(t), t) - 6*Derivative(y(t), t)))] + sol15 = [Eq(x(t), C1 - erf(sqrt(6)*t)*(sqrt(6)*sqrt(pi)*C2/33 + sqrt(6)*sqrt(pi)*C3*Rational(-1, 44)) + + erfi(sqrt(5)*t)*(sqrt(5)*sqrt(pi)*C2*Rational(2, 55) + sqrt(5)*sqrt(pi)*C3*Rational(4, 55))), + Eq(y(t), C4 + erf(sqrt(6)*t)*(sqrt(6)*sqrt(pi)*C2*Rational(2, 33) + sqrt(6)*sqrt(pi)*C3*Rational(-1, + 22)) + erfi(sqrt(5)*t)*(sqrt(5)*sqrt(pi)*C2*Rational(3, 110) + sqrt(5)*sqrt(pi)*C3*Rational(3, 55)))] + assert dsolve(eqs15) == sol15 + assert checksysodesol(eqs15, sol15) == (True, [0, 0]) + + +@slow +def test_higher_order_to_first_order_9(): + f, g = symbols('f g', cls=Function) + x = symbols('x') + + eqs9 = [f(x) + g(x) - 2*exp(I*x) + 2*Derivative(f(x), x) + Derivative(f(x), (x, 2)), + f(x) + g(x) - 2*exp(I*x) + 2*Derivative(g(x), x) + Derivative(g(x), (x, 2))] + sol9 = [Eq(f(x), -C1 + C4*exp(-2*x)/2 - (C2/2 - C3/2)*exp(-x)*cos(x) + + (C2/2 + C3/2)*exp(-x)*sin(x) + 2*((1 - 2*I)*exp(I*x)*sin(x)**2/5) + + 2*((1 - 2*I)*exp(I*x)*cos(x)**2/5)), + Eq(g(x), C1 - C4*exp(-2*x)/2 - (C2/2 - C3/2)*exp(-x)*cos(x) + + (C2/2 + C3/2)*exp(-x)*sin(x) + 2*((1 - 2*I)*exp(I*x)*sin(x)**2/5) + + 2*((1 - 2*I)*exp(I*x)*cos(x)**2/5))] + assert dsolve(eqs9) == sol9 + assert checksysodesol(eqs9, sol9) == (True, [0, 0]) + + +def test_higher_order_to_first_order_12(): + f, g = symbols('f g', cls=Function) + x = symbols('x') + + x, y = symbols('x, y', cls=Function) + t, l = symbols('t, l') + + eqs12 = [Eq(4*x(t) + Derivative(x(t), (t, 2)) + 8*Derivative(y(t), t), 0), + Eq(4*y(t) - 8*Derivative(x(t), t) + Derivative(y(t), (t, 2)), 0)] + sol12 = [Eq(y(t), C1*(2 - sqrt(5))*sin(2*t*sqrt(4*sqrt(5) + 9))*Rational(-1, 2) + C2*(2 - + sqrt(5))*cos(2*t*sqrt(4*sqrt(5) + 9))/2 + C3*(2 + sqrt(5))*sin(2*t*sqrt(9 - 4*sqrt(5)))*Rational(-1, + 2) + C4*(2 + sqrt(5))*cos(2*t*sqrt(9 - 4*sqrt(5)))/2), + Eq(x(t), C1*(2 - sqrt(5))*cos(2*t*sqrt(4*sqrt(5) + 9))*Rational(-1, 2) + C2*(2 - + sqrt(5))*sin(2*t*sqrt(4*sqrt(5) + 9))*Rational(-1, 2) + C3*(2 + sqrt(5))*cos(2*t*sqrt(9 - + 4*sqrt(5)))/2 + C4*(2 + sqrt(5))*sin(2*t*sqrt(9 - 4*sqrt(5)))/2)] + assert dsolve(eqs12) == sol12 + assert checksysodesol(eqs12, sol12) == (True, [0, 0]) + + +def test_second_order_to_first_order_2(): + f, g = symbols("f g", cls=Function) + x, t, x_, t_, d, a, m = symbols("x t x_ t_ d a m") + + eqs2 = [Eq(f(x).diff(x, 2), 2*(x*g(x).diff(x) - g(x))), + Eq(g(x).diff(x, 2),-2*(x*f(x).diff(x) - f(x)))] + sol2 = [Eq(f(x), C1*x + x*Integral(C2*exp(-x_)*exp(I*exp(2*x_))/2 + C2*exp(-x_)*exp(-I*exp(2*x_))/2 - + I*C3*exp(-x_)*exp(I*exp(2*x_))/2 + I*C3*exp(-x_)*exp(-I*exp(2*x_))/2, (x_, log(x)))), + Eq(g(x), C4*x + x*Integral(I*C2*exp(-x_)*exp(I*exp(2*x_))/2 - I*C2*exp(-x_)*exp(-I*exp(2*x_))/2 + + C3*exp(-x_)*exp(I*exp(2*x_))/2 + C3*exp(-x_)*exp(-I*exp(2*x_))/2, (x_, log(x))))] + # XXX: dsolve hangs for this in integration + assert dsolve_system(eqs2, simplify=False, doit=False) == [sol2] + assert checksysodesol(eqs2, sol2) == (True, [0, 0]) + + eqs3 = (Eq(diff(f(t),t,t), 9*t*diff(g(t),t)-9*g(t)), Eq(diff(g(t),t,t),7*t*diff(f(t),t)-7*f(t))) + sol3 = [Eq(f(t), C1*t + t*Integral(C2*exp(-t_)*exp(3*sqrt(7)*exp(2*t_)/2)/2 + C2*exp(-t_)* + exp(-3*sqrt(7)*exp(2*t_)/2)/2 + 3*sqrt(7)*C3*exp(-t_)*exp(3*sqrt(7)*exp(2*t_)/2)/14 - + 3*sqrt(7)*C3*exp(-t_)*exp(-3*sqrt(7)*exp(2*t_)/2)/14, (t_, log(t)))), + Eq(g(t), C4*t + t*Integral(sqrt(7)*C2*exp(-t_)*exp(3*sqrt(7)*exp(2*t_)/2)/6 - sqrt(7)*C2*exp(-t_)* + exp(-3*sqrt(7)*exp(2*t_)/2)/6 + C3*exp(-t_)*exp(3*sqrt(7)*exp(2*t_)/2)/2 + C3*exp(-t_)*exp(-3*sqrt(7)* + exp(2*t_)/2)/2, (t_, log(t))))] + # XXX: dsolve hangs for this in integration + assert dsolve_system(eqs3, simplify=False, doit=False) == [sol3] + assert checksysodesol(eqs3, sol3) == (True, [0, 0]) + + # Regression Test case for sympy#19238 + # https://github.com/sympy/sympy/issues/19238 + # Note: When the doit method is removed, these particular types of systems + # can be divided first so that we have lesser number of big matrices. + eqs5 = [Eq(Derivative(g(t), (t, 2)), a*m), + Eq(Derivative(f(t), (t, 2)), 0)] + sol5 = [Eq(g(t), C1 + C2*t + a*m*t**2/2), + Eq(f(t), C3 + C4*t)] + assert dsolve(eqs5) == sol5 + assert checksysodesol(eqs5, sol5) == (True, [0, 0]) + + # Type 2 + eqs6 = [Eq(Derivative(f(t), (t, 2)), f(t)/t**4), + Eq(Derivative(g(t), (t, 2)), d*g(t)/t**4)] + sol6 = [Eq(f(t), C1*sqrt(t**2)*exp(-1/t) - C2*sqrt(t**2)*exp(1/t)), + Eq(g(t), C3*sqrt(t**2)*exp(-sqrt(d)/t)*d**Rational(-1, 2) - + C4*sqrt(t**2)*exp(sqrt(d)/t)*d**Rational(-1, 2))] + assert dsolve(eqs6) == sol6 + assert checksysodesol(eqs6, sol6) == (True, [0, 0]) + + +@slow +def test_second_order_to_first_order_slow1(): + f, g = symbols("f g", cls=Function) + x, t, x_, t_, d, a, m = symbols("x t x_ t_ d a m") + + # Type 1 + + eqs1 = [Eq(f(x).diff(x, 2), 2/x *(x*g(x).diff(x) - g(x))), + Eq(g(x).diff(x, 2),-2/x *(x*f(x).diff(x) - f(x)))] + sol1 = [Eq(f(x), C1*x + 2*C2*x*Ci(2*x) - C2*sin(2*x) - 2*C3*x*Si(2*x) - C3*cos(2*x)), + Eq(g(x), -2*C2*x*Si(2*x) - C2*cos(2*x) - 2*C3*x*Ci(2*x) + C3*sin(2*x) + C4*x)] + assert dsolve(eqs1) == sol1 + assert checksysodesol(eqs1, sol1) == (True, [0, 0]) + + +def test_second_order_to_first_order_slow4(): + f, g = symbols("f g", cls=Function) + x, t, x_, t_, d, a, m = symbols("x t x_ t_ d a m") + + eqs4 = [Eq(Derivative(f(t), (t, 2)), t*sin(t)*Derivative(g(t), t) - g(t)*sin(t)), + Eq(Derivative(g(t), (t, 2)), t*sin(t)*Derivative(f(t), t) - f(t)*sin(t))] + sol4 = [Eq(f(t), C1*t + t*Integral(C2*exp(-t_)*exp(exp(t_)*cos(exp(t_)))*exp(-sin(exp(t_)))/2 + + C2*exp(-t_)*exp(-exp(t_)*cos(exp(t_)))*exp(sin(exp(t_)))/2 - C3*exp(-t_)*exp(exp(t_)*cos(exp(t_)))* + exp(-sin(exp(t_)))/2 + + C3*exp(-t_)*exp(-exp(t_)*cos(exp(t_)))*exp(sin(exp(t_)))/2, (t_, log(t)))), + Eq(g(t), C4*t + t*Integral(-C2*exp(-t_)*exp(exp(t_)*cos(exp(t_)))*exp(-sin(exp(t_)))/2 + + C2*exp(-t_)*exp(-exp(t_)*cos(exp(t_)))*exp(sin(exp(t_)))/2 + C3*exp(-t_)*exp(exp(t_)*cos(exp(t_)))* + exp(-sin(exp(t_)))/2 + C3*exp(-t_)*exp(-exp(t_)*cos(exp(t_)))*exp(sin(exp(t_)))/2, (t_, log(t))))] + # XXX: dsolve hangs for this in integration + assert dsolve_system(eqs4, simplify=False, doit=False) == [sol4] + assert checksysodesol(eqs4, sol4) == (True, [0, 0]) + + +def test_component_division(): + f, g, h, k = symbols('f g h k', cls=Function) + x = symbols("x") + funcs = [f(x), g(x), h(x), k(x)] + + eqs1 = [Eq(Derivative(f(x), x), 2*f(x)), + Eq(Derivative(g(x), x), f(x)), + Eq(Derivative(h(x), x), h(x)), + Eq(Derivative(k(x), x), h(x)**4 + k(x))] + sol1 = [Eq(f(x), 2*C1*exp(2*x)), + Eq(g(x), C1*exp(2*x) + C2), + Eq(h(x), C3*exp(x)), + Eq(k(x), C3**4*exp(4*x)/3 + C4*exp(x))] + assert dsolve(eqs1) == sol1 + assert checksysodesol(eqs1, sol1) == (True, [0, 0, 0, 0]) + + components1 = {((Eq(Derivative(f(x), x), 2*f(x)),), (Eq(Derivative(g(x), x), f(x)),)), + ((Eq(Derivative(h(x), x), h(x)),), (Eq(Derivative(k(x), x), h(x)**4 + k(x)),))} + eqsdict1 = ({f(x): set(), g(x): {f(x)}, h(x): set(), k(x): {h(x)}}, + {f(x): Eq(Derivative(f(x), x), 2*f(x)), + g(x): Eq(Derivative(g(x), x), f(x)), + h(x): Eq(Derivative(h(x), x), h(x)), + k(x): Eq(Derivative(k(x), x), h(x)**4 + k(x))}) + graph1 = [{f(x), g(x), h(x), k(x)}, {(g(x), f(x)), (k(x), h(x))}] + assert {tuple(tuple(scc) for scc in wcc) for wcc in _component_division(eqs1, funcs, x)} == components1 + assert _eqs2dict(eqs1, funcs) == eqsdict1 + assert [set(element) for element in _dict2graph(eqsdict1[0])] == graph1 + + eqs2 = [Eq(Derivative(f(x), x), 2*f(x)), + Eq(Derivative(g(x), x), f(x)), + Eq(Derivative(h(x), x), h(x)), + Eq(Derivative(k(x), x), f(x)**4 + k(x))] + sol2 = [Eq(f(x), C1*exp(2*x)), + Eq(g(x), C1*exp(2*x)/2 + C2), + Eq(h(x), C3*exp(x)), + Eq(k(x), C1**4*exp(8*x)/7 + C4*exp(x))] + assert dsolve(eqs2) == sol2 + assert checksysodesol(eqs2, sol2) == (True, [0, 0, 0, 0]) + + components2 = {frozenset([(Eq(Derivative(f(x), x), 2*f(x)),), + (Eq(Derivative(g(x), x), f(x)),), + (Eq(Derivative(k(x), x), f(x)**4 + k(x)),)]), + frozenset([(Eq(Derivative(h(x), x), h(x)),)])} + eqsdict2 = ({f(x): set(), g(x): {f(x)}, h(x): set(), k(x): {f(x)}}, + {f(x): Eq(Derivative(f(x), x), 2*f(x)), + g(x): Eq(Derivative(g(x), x), f(x)), + h(x): Eq(Derivative(h(x), x), h(x)), + k(x): Eq(Derivative(k(x), x), f(x)**4 + k(x))}) + graph2 = [{f(x), g(x), h(x), k(x)}, {(g(x), f(x)), (k(x), f(x))}] + assert {frozenset(tuple(scc) for scc in wcc) for wcc in _component_division(eqs2, funcs, x)} == components2 + assert _eqs2dict(eqs2, funcs) == eqsdict2 + assert [set(element) for element in _dict2graph(eqsdict2[0])] == graph2 + + eqs3 = [Eq(Derivative(f(x), x), 2*f(x)), + Eq(Derivative(g(x), x), x + f(x)), + Eq(Derivative(h(x), x), h(x)), + Eq(Derivative(k(x), x), f(x)**4 + k(x))] + sol3 = [Eq(f(x), C1*exp(2*x)), + Eq(g(x), C1*exp(2*x)/2 + C2 + x**2/2), + Eq(h(x), C3*exp(x)), + Eq(k(x), C1**4*exp(8*x)/7 + C4*exp(x))] + assert dsolve(eqs3) == sol3 + assert checksysodesol(eqs3, sol3) == (True, [0, 0, 0, 0]) + + components3 = {frozenset([(Eq(Derivative(f(x), x), 2*f(x)),), + (Eq(Derivative(g(x), x), x + f(x)),), + (Eq(Derivative(k(x), x), f(x)**4 + k(x)),)]), + frozenset([(Eq(Derivative(h(x), x), h(x)),),])} + eqsdict3 = ({f(x): set(), g(x): {f(x)}, h(x): set(), k(x): {f(x)}}, + {f(x): Eq(Derivative(f(x), x), 2*f(x)), + g(x): Eq(Derivative(g(x), x), x + f(x)), + h(x): Eq(Derivative(h(x), x), h(x)), + k(x): Eq(Derivative(k(x), x), f(x)**4 + k(x))}) + graph3 = [{f(x), g(x), h(x), k(x)}, {(g(x), f(x)), (k(x), f(x))}] + assert {frozenset(tuple(scc) for scc in wcc) for wcc in _component_division(eqs3, funcs, x)} == components3 + assert _eqs2dict(eqs3, funcs) == eqsdict3 + assert [set(l) for l in _dict2graph(eqsdict3[0])] == graph3 + + # Note: To be uncommented when the default option to call dsolve first for + # single ODE system can be rearranged. This can be done after the doit + # option in dsolve is made False by default. + + eqs4 = [Eq(Derivative(f(x), x), x*f(x) + 2*g(x)), + Eq(Derivative(g(x), x), f(x) + x*g(x) + x), + Eq(Derivative(h(x), x), h(x)), + Eq(Derivative(k(x), x), f(x)**4 + k(x))] + sol4 = [Eq(f(x), (C1/2 - sqrt(2)*C2/2 - sqrt(2)*Integral(x*exp(-x**2/2 - sqrt(2)*x)/2 + x*exp(-x**2/2 +\ + sqrt(2)*x)/2, x)/2 + Integral(sqrt(2)*x*exp(-x**2/2 - sqrt(2)*x)/2 - sqrt(2)*x*exp(-x**2/2 +\ + sqrt(2)*x)/2, x)/2)*exp(x**2/2 - sqrt(2)*x) + (C1/2 + sqrt(2)*C2/2 + sqrt(2)*Integral(x*exp(-x**2/2 + - sqrt(2)*x)/2 + x*exp(-x**2/2 + sqrt(2)*x)/2, x)/2 + Integral(sqrt(2)*x*exp(-x**2/2 - sqrt(2)*x)/2 + - sqrt(2)*x*exp(-x**2/2 + sqrt(2)*x)/2, x)/2)*exp(x**2/2 + sqrt(2)*x)), + Eq(g(x), (-sqrt(2)*C1/4 + C2/2 + Integral(x*exp(-x**2/2 - sqrt(2)*x)/2 + x*exp(-x**2/2 + sqrt(2)*x)/2, x)/2 -\ + sqrt(2)*Integral(sqrt(2)*x*exp(-x**2/2 - sqrt(2)*x)/2 - sqrt(2)*x*exp(-x**2/2 + sqrt(2)*x)/2, + x)/4)*exp(x**2/2 - sqrt(2)*x) + (sqrt(2)*C1/4 + C2/2 + Integral(x*exp(-x**2/2 - sqrt(2)*x)/2 + + x*exp(-x**2/2 + sqrt(2)*x)/2, x)/2 + sqrt(2)*Integral(sqrt(2)*x*exp(-x**2/2 - sqrt(2)*x)/2 - + sqrt(2)*x*exp(-x**2/2 + sqrt(2)*x)/2, x)/4)*exp(x**2/2 + sqrt(2)*x)), + Eq(h(x), C3*exp(x)), + Eq(k(x), C4*exp(x) + exp(x)*Integral((C1*exp(x**2/2 - sqrt(2)*x)/2 + C1*exp(x**2/2 + sqrt(2)*x)/2 - + sqrt(2)*C2*exp(x**2/2 - sqrt(2)*x)/2 + sqrt(2)*C2*exp(x**2/2 + sqrt(2)*x)/2 - sqrt(2)*exp(x**2/2 - + sqrt(2)*x)*Integral(x*exp(-x**2/2 - sqrt(2)*x)/2 + x*exp(-x**2/2 + sqrt(2)*x)/2, x)/2 + exp(x**2/2 - + sqrt(2)*x)*Integral(sqrt(2)*x*exp(-x**2/2 - sqrt(2)*x)/2 - sqrt(2)*x*exp(-x**2/2 + sqrt(2)*x)/2, + x)/2 + sqrt(2)*exp(x**2/2 + sqrt(2)*x)*Integral(x*exp(-x**2/2 - sqrt(2)*x)/2 + x*exp(-x**2/2 + + sqrt(2)*x)/2, x)/2 + exp(x**2/2 + sqrt(2)*x)*Integral(sqrt(2)*x*exp(-x**2/2 - sqrt(2)*x)/2 - + sqrt(2)*x*exp(-x**2/2 + sqrt(2)*x)/2, x)/2)**4*exp(-x), x))] + components4 = {(frozenset([Eq(Derivative(f(x), x), x*f(x) + 2*g(x)), + Eq(Derivative(g(x), x), x*g(x) + x + f(x))]), + frozenset([Eq(Derivative(k(x), x), f(x)**4 + k(x)),])), + (frozenset([Eq(Derivative(h(x), x), h(x)),]),)} + eqsdict4 = ({f(x): {g(x)}, g(x): {f(x)}, h(x): set(), k(x): {f(x)}}, + {f(x): Eq(Derivative(f(x), x), x*f(x) + 2*g(x)), + g(x): Eq(Derivative(g(x), x), x*g(x) + x + f(x)), + h(x): Eq(Derivative(h(x), x), h(x)), + k(x): Eq(Derivative(k(x), x), f(x)**4 + k(x))}) + graph4 = [{f(x), g(x), h(x), k(x)}, {(f(x), g(x)), (g(x), f(x)), (k(x), f(x))}] + assert {tuple(frozenset(scc) for scc in wcc) for wcc in _component_division(eqs4, funcs, x)} == components4 + assert _eqs2dict(eqs4, funcs) == eqsdict4 + assert [set(element) for element in _dict2graph(eqsdict4[0])] == graph4 + # XXX: dsolve hangs in integration here: + assert dsolve_system(eqs4, simplify=False, doit=False) == [sol4] + assert checksysodesol(eqs4, sol4) == (True, [0, 0, 0, 0]) + + eqs5 = [Eq(Derivative(f(x), x), x*f(x) + 2*g(x)), + Eq(Derivative(g(x), x), x*g(x) + f(x)), + Eq(Derivative(h(x), x), h(x)), + Eq(Derivative(k(x), x), f(x)**4 + k(x))] + sol5 = [Eq(f(x), (C1/2 - sqrt(2)*C2/2)*exp(x**2/2 - sqrt(2)*x) + (C1/2 + sqrt(2)*C2/2)*exp(x**2/2 + sqrt(2)*x)), + Eq(g(x), (-sqrt(2)*C1/4 + C2/2)*exp(x**2/2 - sqrt(2)*x) + (sqrt(2)*C1/4 + C2/2)*exp(x**2/2 + sqrt(2)*x)), + Eq(h(x), C3*exp(x)), + Eq(k(x), C4*exp(x) + exp(x)*Integral((C1*exp(x**2/2 - sqrt(2)*x)/2 + C1*exp(x**2/2 + sqrt(2)*x)/2 - + sqrt(2)*C2*exp(x**2/2 - sqrt(2)*x)/2 + sqrt(2)*C2*exp(x**2/2 + sqrt(2)*x)/2)**4*exp(-x), x))] + components5 = {(frozenset([Eq(Derivative(f(x), x), x*f(x) + 2*g(x)), + Eq(Derivative(g(x), x), x*g(x) + f(x))]), + frozenset([Eq(Derivative(k(x), x), f(x)**4 + k(x)),])), + (frozenset([Eq(Derivative(h(x), x), h(x)),]),)} + eqsdict5 = ({f(x): {g(x)}, g(x): {f(x)}, h(x): set(), k(x): {f(x)}}, + {f(x): Eq(Derivative(f(x), x), x*f(x) + 2*g(x)), + g(x): Eq(Derivative(g(x), x), x*g(x) + f(x)), + h(x): Eq(Derivative(h(x), x), h(x)), + k(x): Eq(Derivative(k(x), x), f(x)**4 + k(x))}) + graph5 = [{f(x), g(x), h(x), k(x)}, {(f(x), g(x)), (g(x), f(x)), (k(x), f(x))}] + assert {tuple(frozenset(scc) for scc in wcc) for wcc in _component_division(eqs5, funcs, x)} == components5 + assert _eqs2dict(eqs5, funcs) == eqsdict5 + assert [set(element) for element in _dict2graph(eqsdict5[0])] == graph5 + # XXX: dsolve hangs in integration here: + assert dsolve_system(eqs5, simplify=False, doit=False) == [sol5] + assert checksysodesol(eqs5, sol5) == (True, [0, 0, 0, 0]) + + +def test_linodesolve(): + t, x, a = symbols("t x a") + f, g, h = symbols("f g h", cls=Function) + + # Testing the Errors + raises(ValueError, lambda: linodesolve(1, t)) + raises(ValueError, lambda: linodesolve(a, t)) + + A1 = Matrix([[1, 2], [2, 4], [4, 6]]) + raises(NonSquareMatrixError, lambda: linodesolve(A1, t)) + + A2 = Matrix([[1, 2, 1], [3, 1, 2]]) + raises(NonSquareMatrixError, lambda: linodesolve(A2, t)) + + # Testing auto functionality + func = [f(t), g(t)] + eq = [Eq(f(t).diff(t) + g(t).diff(t), g(t)), Eq(g(t).diff(t), f(t))] + ceq = canonical_odes(eq, func, t) + (A1, A0), b = linear_ode_to_matrix(ceq[0], func, t, 1) + A = A0 + sol = [C1*(-Rational(1, 2) + sqrt(5)/2)*exp(t*(-Rational(1, 2) + sqrt(5)/2)) + C2*(-sqrt(5)/2 - Rational(1, 2))* + exp(t*(-sqrt(5)/2 - Rational(1, 2))), + C1*exp(t*(-Rational(1, 2) + sqrt(5)/2)) + C2*exp(t*(-sqrt(5)/2 - Rational(1, 2)))] + assert constant_renumber(linodesolve(A, t), variables=Tuple(*eq).free_symbols) == sol + + # Testing the Errors + raises(ValueError, lambda: linodesolve(1, t, b=Matrix([t+1]))) + raises(ValueError, lambda: linodesolve(a, t, b=Matrix([log(t) + sin(t)]))) + + raises(ValueError, lambda: linodesolve(Matrix([7]), t, b=t**2)) + raises(ValueError, lambda: linodesolve(Matrix([a+10]), t, b=log(t)*cos(t))) + + raises(ValueError, lambda: linodesolve(7, t, b=t**2)) + raises(ValueError, lambda: linodesolve(a, t, b=log(t) + sin(t))) + + A1 = Matrix([[1, 2], [2, 4], [4, 6]]) + b1 = Matrix([t, 1, t**2]) + raises(NonSquareMatrixError, lambda: linodesolve(A1, t, b=b1)) + + A2 = Matrix([[1, 2, 1], [3, 1, 2]]) + b2 = Matrix([t, t**2]) + raises(NonSquareMatrixError, lambda: linodesolve(A2, t, b=b2)) + + raises(ValueError, lambda: linodesolve(A1[:2, :], t, b=b1)) + raises(ValueError, lambda: linodesolve(A1[:2, :], t, b=b1[:1])) + + # DOIT check + A1 = Matrix([[1, -1], [1, -1]]) + b1 = Matrix([15*t - 10, -15*t - 5]) + sol1 = [C1 + C2*t + C2 - 10*t**3 + 10*t**2 + t*(15*t**2 - 5*t) - 10*t, + C1 + C2*t - 10*t**3 - 5*t**2 + t*(15*t**2 - 5*t) - 5*t] + assert constant_renumber(linodesolve(A1, t, b=b1, type="type2", doit=True), + variables=[t]) == sol1 + + # Testing auto functionality + func = [f(t), g(t)] + eq = [Eq(f(t).diff(t) + g(t).diff(t), g(t) + t), Eq(g(t).diff(t), f(t))] + ceq = canonical_odes(eq, func, t) + (A1, A0), b = linear_ode_to_matrix(ceq[0], func, t, 1) + A = A0 + sol = [-C1*exp(-t/2 + sqrt(5)*t/2)/2 + sqrt(5)*C1*exp(-t/2 + sqrt(5)*t/2)/2 - sqrt(5)*C2*exp(-sqrt(5)*t/2 - + t/2)/2 - C2*exp(-sqrt(5)*t/2 - t/2)/2 - exp(-t/2 + sqrt(5)*t/2)*Integral(t*exp(-sqrt(5)*t/2 + + t/2)/(-5 + sqrt(5)) - sqrt(5)*t*exp(-sqrt(5)*t/2 + t/2)/(-5 + sqrt(5)), t)/2 + sqrt(5)*exp(-t/2 + + sqrt(5)*t/2)*Integral(t*exp(-sqrt(5)*t/2 + t/2)/(-5 + sqrt(5)) - sqrt(5)*t*exp(-sqrt(5)*t/2 + + t/2)/(-5 + sqrt(5)), t)/2 - sqrt(5)*exp(-sqrt(5)*t/2 - t/2)*Integral(-sqrt(5)*t*exp(t/2 + + sqrt(5)*t/2)/5, t)/2 - exp(-sqrt(5)*t/2 - t/2)*Integral(-sqrt(5)*t*exp(t/2 + sqrt(5)*t/2)/5, t)/2, + C1*exp(-t/2 + sqrt(5)*t/2) + C2*exp(-sqrt(5)*t/2 - t/2) + exp(-t/2 + + sqrt(5)*t/2)*Integral(t*exp(-sqrt(5)*t/2 + t/2)/(-5 + sqrt(5)) - sqrt(5)*t*exp(-sqrt(5)*t/2 + + t/2)/(-5 + sqrt(5)), t) + exp(-sqrt(5)*t/2 - + t/2)*Integral(-sqrt(5)*t*exp(t/2 + sqrt(5)*t/2)/5, t)] + assert constant_renumber(linodesolve(A, t, b=b), variables=[t]) == sol + + # non-homogeneous term assumed to be 0 + sol1 = [-C1*exp(-t/2 + sqrt(5)*t/2)/2 + sqrt(5)*C1*exp(-t/2 + sqrt(5)*t/2)/2 - sqrt(5)*C2*exp(-sqrt(5)*t/2 + - t/2)/2 - C2*exp(-sqrt(5)*t/2 - t/2)/2, + C1*exp(-t/2 + sqrt(5)*t/2) + C2*exp(-sqrt(5)*t/2 - t/2)] + assert constant_renumber(linodesolve(A, t, type="type2"), variables=[t]) == sol1 + + # Testing the Errors + raises(ValueError, lambda: linodesolve(t+10, t)) + raises(ValueError, lambda: linodesolve(a*t, t)) + + A1 = Matrix([[1, t], [-t, 1]]) + B1, _ = _is_commutative_anti_derivative(A1, t) + raises(NonSquareMatrixError, lambda: linodesolve(A1[:, :1], t, B=B1)) + raises(ValueError, lambda: linodesolve(A1, t, B=1)) + + A2 = Matrix([[t, t, t], [t, t, t], [t, t, t]]) + B2, _ = _is_commutative_anti_derivative(A2, t) + raises(NonSquareMatrixError, lambda: linodesolve(A2, t, B=B2[:2, :])) + raises(ValueError, lambda: linodesolve(A2, t, B=2)) + raises(ValueError, lambda: linodesolve(A2, t, B=B2, type="type31")) + + raises(ValueError, lambda: linodesolve(A1, t, B=B2)) + raises(ValueError, lambda: linodesolve(A2, t, B=B1)) + + # Testing auto functionality + func = [f(t), g(t)] + eq = [Eq(f(t).diff(t), f(t) + t*g(t)), Eq(g(t).diff(t), -t*f(t) + g(t))] + ceq = canonical_odes(eq, func, t) + (A1, A0), b = linear_ode_to_matrix(ceq[0], func, t, 1) + A = A0 + sol = [(C1/2 - I*C2/2)*exp(I*t**2/2 + t) + (C1/2 + I*C2/2)*exp(-I*t**2/2 + t), + (-I*C1/2 + C2/2)*exp(-I*t**2/2 + t) + (I*C1/2 + C2/2)*exp(I*t**2/2 + t)] + assert constant_renumber(linodesolve(A, t), variables=Tuple(*eq).free_symbols) == sol + assert constant_renumber(linodesolve(A, t, type="type3"), variables=Tuple(*eq).free_symbols) == sol + + A1 = Matrix([[t, 1], [t, -1]]) + raises(NotImplementedError, lambda: linodesolve(A1, t)) + + # Testing the Errors + raises(ValueError, lambda: linodesolve(t+10, t, b=Matrix([t+1]))) + raises(ValueError, lambda: linodesolve(a*t, t, b=Matrix([log(t) + sin(t)]))) + + raises(ValueError, lambda: linodesolve(Matrix([7*t]), t, b=t**2)) + raises(ValueError, lambda: linodesolve(Matrix([a + 10*log(t)]), t, b=log(t)*cos(t))) + + raises(ValueError, lambda: linodesolve(7*t, t, b=t**2)) + raises(ValueError, lambda: linodesolve(a*t**2, t, b=log(t) + sin(t))) + + A1 = Matrix([[1, t], [-t, 1]]) + b1 = Matrix([t, t ** 2]) + B1, _ = _is_commutative_anti_derivative(A1, t) + raises(NonSquareMatrixError, lambda: linodesolve(A1[:, :1], t, b=b1)) + + A2 = Matrix([[t, t, t], [t, t, t], [t, t, t]]) + b2 = Matrix([t, 1, t**2]) + B2, _ = _is_commutative_anti_derivative(A2, t) + raises(NonSquareMatrixError, lambda: linodesolve(A2[:2, :], t, b=b2)) + + raises(ValueError, lambda: linodesolve(A1, t, b=b2)) + raises(ValueError, lambda: linodesolve(A2, t, b=b1)) + + raises(ValueError, lambda: linodesolve(A1, t, b=b1, B=B2)) + raises(ValueError, lambda: linodesolve(A2, t, b=b2, B=B1)) + + # Testing auto functionality + func = [f(x), g(x), h(x)] + eq = [Eq(f(x).diff(x), x*(f(x) + g(x) + h(x)) + x), + Eq(g(x).diff(x), x*(f(x) + g(x) + h(x)) + x), + Eq(h(x).diff(x), x*(f(x) + g(x) + h(x)) + 1)] + ceq = canonical_odes(eq, func, x) + (A1, A0), b = linear_ode_to_matrix(ceq[0], func, x, 1) + A = A0 + _x1 = exp(-3*x**2/2) + _x2 = exp(3*x**2/2) + _x3 = Integral(2*_x1*x/3 + _x1/3 + x/3 - Rational(1, 3), x) + _x4 = 2*_x2*_x3/3 + _x5 = Integral(2*_x1*x/3 + _x1/3 - 2*x/3 + Rational(2, 3), x) + sol = [ + C1*_x2/3 - C1/3 + C2*_x2/3 - C2/3 + C3*_x2/3 + 2*C3/3 + _x2*_x5/3 + _x3/3 + _x4 - _x5/3, + C1*_x2/3 + 2*C1/3 + C2*_x2/3 - C2/3 + C3*_x2/3 - C3/3 + _x2*_x5/3 + _x3/3 + _x4 - _x5/3, + C1*_x2/3 - C1/3 + C2*_x2/3 + 2*C2/3 + C3*_x2/3 - C3/3 + _x2*_x5/3 - 2*_x3/3 + _x4 + 2*_x5/3, + ] + assert constant_renumber(linodesolve(A, x, b=b), variables=Tuple(*eq).free_symbols) == sol + assert constant_renumber(linodesolve(A, x, b=b, type="type4"), + variables=Tuple(*eq).free_symbols) == sol + + A1 = Matrix([[t, 1], [t, -1]]) + raises(NotImplementedError, lambda: linodesolve(A1, t, b=b1)) + + # non-homogeneous term not passed + sol1 = [-C1/3 - C2/3 + 2*C3/3 + (C1/3 + C2/3 + C3/3)*exp(3*x**2/2), 2*C1/3 - C2/3 - C3/3 + (C1/3 + C2/3 + C3/3)*exp(3*x**2/2), + -C1/3 + 2*C2/3 - C3/3 + (C1/3 + C2/3 + C3/3)*exp(3*x**2/2)] + assert constant_renumber(linodesolve(A, x, type="type4", doit=True), variables=Tuple(*eq).free_symbols) == sol1 + + +@slow +def test_linear_3eq_order1_type4_slow(): + x, y, z = symbols('x, y, z', cls=Function) + t = Symbol('t') + + f = t ** 3 + log(t) + g = t ** 2 + sin(t) + eq1 = (Eq(diff(x(t), t), (4 * f + g) * x(t) - f * y(t) - 2 * f * z(t)), + Eq(diff(y(t), t), 2 * f * x(t) + (f + g) * y(t) - 2 * f * z(t)), Eq(diff(z(t), t), 5 * f * x(t) + f * y( + t) + (-3 * f + g) * z(t))) + with dotprodsimp(True): + dsolve(eq1) + + +@slow +def test_linear_neq_order1_type2_slow1(): + i, r1, c1, r2, c2, t = symbols('i, r1, c1, r2, c2, t') + x1 = Function('x1') + x2 = Function('x2') + + eq1 = r1*c1*Derivative(x1(t), t) + x1(t) - x2(t) - r1*i + eq2 = r2*c1*Derivative(x1(t), t) + r2*c2*Derivative(x2(t), t) + x2(t) - r2*i + eq = [eq1, eq2] + + # XXX: Solution is too complicated + [sol] = dsolve_system(eq, simplify=False, doit=False) + assert checksysodesol(eq, sol) == (True, [0, 0]) + + +# Regression test case for issue #9204 +# https://github.com/sympy/sympy/issues/9204 +@tooslow +def test_linear_new_order1_type2_de_lorentz_slow_check(): + m = Symbol("m", real=True) + q = Symbol("q", real=True) + t = Symbol("t", real=True) + + e1, e2, e3 = symbols("e1:4", real=True) + b1, b2, b3 = symbols("b1:4", real=True) + v1, v2, v3 = symbols("v1:4", cls=Function, real=True) + + eqs = [ + -e1*q + m*Derivative(v1(t), t) - q*(-b2*v3(t) + b3*v2(t)), + -e2*q + m*Derivative(v2(t), t) - q*(b1*v3(t) - b3*v1(t)), + -e3*q + m*Derivative(v3(t), t) - q*(-b1*v2(t) + b2*v1(t)) + ] + sol = dsolve(eqs) + assert checksysodesol(eqs, sol) == (True, [0, 0, 0]) + + +# Regression test case for issue #14001 +# https://github.com/sympy/sympy/issues/14001 +@slow +def test_linear_neq_order1_type2_slow_check(): + RC, t, C, Vs, L, R1, V0, I0 = symbols("RC t C Vs L R1 V0 I0") + V = Function("V") + I = Function("I") + system = [Eq(V(t).diff(t), -1/RC*V(t) + I(t)/C), Eq(I(t).diff(t), -R1/L*I(t) - 1/L*V(t) + Vs/L)] + [sol] = dsolve_system(system, simplify=False, doit=False) + + assert checksysodesol(system, sol) == (True, [0, 0]) + + +def _linear_3eq_order1_type4_long(): + x, y, z = symbols('x, y, z', cls=Function) + t = Symbol('t') + + f = t ** 3 + log(t) + g = t ** 2 + sin(t) + + eq1 = (Eq(diff(x(t), t), (4*f + g)*x(t) - f*y(t) - 2*f*z(t)), + Eq(diff(y(t), t), 2*f*x(t) + (f + g)*y(t) - 2*f*z(t)), Eq(diff(z(t), t), 5*f*x(t) + f*y( + t) + (-3*f + g)*z(t))) + + dsolve_sol = dsolve(eq1) + dsolve_sol1 = [_simpsol(sol) for sol in dsolve_sol] + + x_1 = sqrt(-t**6 - 8*t**3*log(t) + 8*t**3 - 16*log(t)**2 + 32*log(t) - 16) + x_2 = sqrt(3) + x_3 = 8324372644*C1*x_1*x_2 + 4162186322*C2*x_1*x_2 - 8324372644*C3*x_1*x_2 + x_4 = 1 / (1903457163*t**3 + 3825881643*x_1*x_2 + 7613828652*log(t) - 7613828652) + x_5 = exp(t**3/3 + t*x_1*x_2/4 - cos(t)) + x_6 = exp(t**3/3 - t*x_1*x_2/4 - cos(t)) + x_7 = exp(t**4/2 + t**3/3 + 2*t*log(t) - 2*t - cos(t)) + x_8 = 91238*C1*x_1*x_2 + 91238*C2*x_1*x_2 - 91238*C3*x_1*x_2 + x_9 = 1 / (66049*t**3 - 50629*x_1*x_2 + 264196*log(t) - 264196) + x_10 = 50629 * C1 / 25189 + 37909*C2/25189 - 50629*C3/25189 - x_3*x_4 + x_11 = -50629*C1/25189 - 12720*C2/25189 + 50629*C3/25189 + x_3*x_4 + sol = [Eq(x(t), x_10*x_5 + x_11*x_6 + x_7*(C1 - C2)), Eq(y(t), x_10*x_5 + x_11*x_6), Eq(z(t), x_5*( + -424*C1/257 - 167*C2/257 + 424*C3/257 - x_8*x_9) + x_6*(167*C1/257 + 424*C2/257 - + 167*C3/257 + x_8*x_9) + x_7*(C1 - C2))] + + assert dsolve_sol1 == sol + assert checksysodesol(eq1, dsolve_sol1) == (True, [0, 0, 0]) + + +@slow +def test_neq_order1_type4_slow_check1(): + f, g = symbols("f g", cls=Function) + x = symbols("x") + + eqs = [Eq(diff(f(x), x), x*f(x) + x**2*g(x) + x), + Eq(diff(g(x), x), 2*x**2*f(x) + (x + 3*x**2)*g(x) + 1)] + sol = dsolve(eqs) + assert checksysodesol(eqs, sol) == (True, [0, 0]) + + +@slow +def test_neq_order1_type4_slow_check2(): + f, g, h = symbols("f, g, h", cls=Function) + x = Symbol("x") + + eqs = [ + Eq(Derivative(f(x), x), x*h(x) + f(x) + g(x) + 1), + Eq(Derivative(g(x), x), x*g(x) + f(x) + h(x) + 10), + Eq(Derivative(h(x), x), x*f(x) + x + g(x) + h(x)) + ] + with dotprodsimp(True): + sol = dsolve(eqs) + assert checksysodesol(eqs, sol) == (True, [0, 0, 0]) + + +def _neq_order1_type4_slow3(): + f, g = symbols("f g", cls=Function) + x = symbols("x") + + eqs = [ + Eq(Derivative(f(x), x), x*f(x) + g(x) + sin(x)), + Eq(Derivative(g(x), x), x**2 + x*g(x) - f(x)) + ] + sol = [ + Eq(f(x), (C1/2 - I*C2/2 - I*Integral(x**2*exp(-x**2/2 - I*x)/2 + + x**2*exp(-x**2/2 + I*x)/2 + I*exp(-x**2/2 - I*x)*sin(x)/2 - + I*exp(-x**2/2 + I*x)*sin(x)/2, x)/2 + Integral(-I*x**2*exp(-x**2/2 + - I*x)/2 + I*x**2*exp(-x**2/2 + I*x)/2 + exp(-x**2/2 - + I*x)*sin(x)/2 + exp(-x**2/2 + I*x)*sin(x)/2, x)/2)*exp(x**2/2 + + I*x) + (C1/2 + I*C2/2 + I*Integral(x**2*exp(-x**2/2 - I*x)/2 + + x**2*exp(-x**2/2 + I*x)/2 + I*exp(-x**2/2 - I*x)*sin(x)/2 - + I*exp(-x**2/2 + I*x)*sin(x)/2, x)/2 + Integral(-I*x**2*exp(-x**2/2 + - I*x)/2 + I*x**2*exp(-x**2/2 + I*x)/2 + exp(-x**2/2 - + I*x)*sin(x)/2 + exp(-x**2/2 + I*x)*sin(x)/2, x)/2)*exp(x**2/2 - + I*x)), + Eq(g(x), (-I*C1/2 + C2/2 + Integral(x**2*exp(-x**2/2 - I*x)/2 + + x**2*exp(-x**2/2 + I*x)/2 + I*exp(-x**2/2 - I*x)*sin(x)/2 - + I*exp(-x**2/2 + I*x)*sin(x)/2, x)/2 - + I*Integral(-I*x**2*exp(-x**2/2 - I*x)/2 + I*x**2*exp(-x**2/2 + + I*x)/2 + exp(-x**2/2 - I*x)*sin(x)/2 + exp(-x**2/2 + + I*x)*sin(x)/2, x)/2)*exp(x**2/2 - I*x) + (I*C1/2 + C2/2 + + Integral(x**2*exp(-x**2/2 - I*x)/2 + x**2*exp(-x**2/2 + I*x)/2 + + I*exp(-x**2/2 - I*x)*sin(x)/2 - I*exp(-x**2/2 + I*x)*sin(x)/2, + x)/2 + I*Integral(-I*x**2*exp(-x**2/2 - I*x)/2 + + I*x**2*exp(-x**2/2 + I*x)/2 + exp(-x**2/2 - I*x)*sin(x)/2 + + exp(-x**2/2 + I*x)*sin(x)/2, x)/2)*exp(x**2/2 + I*x)) + ] + + return eqs, sol + + +def test_neq_order1_type4_slow3(): + eqs, sol = _neq_order1_type4_slow3() + assert dsolve_system(eqs, simplify=False, doit=False) == [sol] + # XXX: dsolve gives an error in integration: + # assert dsolve(eqs) == sol + # https://github.com/sympy/sympy/issues/20155 + + +@slow +def test_neq_order1_type4_slow_check3(): + eqs, sol = _neq_order1_type4_slow3() + assert checksysodesol(eqs, sol) == (True, [0, 0]) + + +@tooslow +@XFAIL +def test_linear_3eq_order1_type4_long_dsolve_slow_xfail(): + eq, sol = _linear_3eq_order1_type4_long() + + dsolve_sol = dsolve(eq) + dsolve_sol1 = [_simpsol(sol) for sol in dsolve_sol] + + assert dsolve_sol1 == sol + + +@tooslow +def test_linear_3eq_order1_type4_long_dsolve_dotprodsimp(): + eq, sol = _linear_3eq_order1_type4_long() + + # XXX: Only works with dotprodsimp see + # test_linear_3eq_order1_type4_long_dsolve_slow_xfail which is too slow + with dotprodsimp(True): + dsolve_sol = dsolve(eq) + + dsolve_sol1 = [_simpsol(sol) for sol in dsolve_sol] + assert dsolve_sol1 == sol + + +@tooslow +def test_linear_3eq_order1_type4_long_check(): + eq, sol = _linear_3eq_order1_type4_long() + assert checksysodesol(eq, sol) == (True, [0, 0, 0]) + + +def test_dsolve_system(): + f, g = symbols("f g", cls=Function) + x = symbols("x") + eqs = [Eq(f(x).diff(x), f(x) + g(x)), Eq(g(x).diff(x), f(x) + g(x))] + funcs = [f(x), g(x)] + + sol = [[Eq(f(x), -C1 + C2*exp(2*x)), Eq(g(x), C1 + C2*exp(2*x))]] + assert dsolve_system(eqs, funcs=funcs, t=x, doit=True) == sol + + raises(ValueError, lambda: dsolve_system(1)) + raises(ValueError, lambda: dsolve_system(eqs, 1)) + raises(ValueError, lambda: dsolve_system(eqs, funcs, 1)) + raises(ValueError, lambda: dsolve_system(eqs, funcs[:1], x)) + + eq = (Eq(f(x).diff(x), 12 * f(x) - 6 * g(x)), Eq(g(x).diff(x) ** 2, 11 * f(x) + 3 * g(x))) + raises(NotImplementedError, lambda: dsolve_system(eq) == ([], [])) + + raises(NotImplementedError, lambda: dsolve_system(eq, funcs=[f(x), g(x)]) == ([], [])) + raises(NotImplementedError, lambda: dsolve_system(eq, funcs=[f(x), g(x)], t=x) == ([], [])) + raises(NotImplementedError, lambda: dsolve_system(eq, funcs=[f(x), g(x)], t=x, ics={f(0): 1, g(0): 1}) == ([], [])) + raises(NotImplementedError, lambda: dsolve_system(eq, t=x, ics={f(0): 1, g(0): 1}) == ([], [])) + raises(NotImplementedError, lambda: dsolve_system(eq, ics={f(0): 1, g(0): 1}) == ([], [])) + raises(NotImplementedError, lambda: dsolve_system(eq, funcs=[f(x), g(x)], ics={f(0): 1, g(0): 1}) == ([], [])) + +def test_dsolve(): + + f, g = symbols('f g', cls=Function) + x, y = symbols('x y') + + eqs = [f(x).diff(x) - x, f(x).diff(x) + x] + with raises(ValueError): + dsolve(eqs) + + eqs = [f(x, y).diff(x)] + with raises(ValueError): + dsolve(eqs) + + eqs = [f(x, y).diff(x)+g(x).diff(x), g(x).diff(x)] + with raises(ValueError): + dsolve(eqs) + + +@slow +def test_higher_order1_slow1(): + x, y = symbols("x y", cls=Function) + t = symbols("t") + + eq = [ + Eq(diff(x(t),t,t), (log(t)+t**2)*diff(x(t),t)+(log(t)+t**2)*3*diff(y(t),t)), + Eq(diff(y(t),t,t), (log(t)+t**2)*2*diff(x(t),t)+(log(t)+t**2)*9*diff(y(t),t)) + ] + sol, = dsolve_system(eq, simplify=False, doit=False) + # The solution is too long to write out explicitly and checkodesol is too + # slow so we test for particular values of t: + for e in eq: + res = (e.lhs - e.rhs).subs({sol[0].lhs:sol[0].rhs, sol[1].lhs:sol[1].rhs}) + res = res.subs({d: d.doit(deep=False) for d in res.atoms(Derivative)}) + assert ratsimp(res.subs(t, 1)) == 0 + + +def test_second_order_type2_slow1(): + x, y, z = symbols('x, y, z', cls=Function) + t, l = symbols('t, l') + + eqs1 = [Eq(Derivative(x(t), (t, 2)), t*(2*x(t) + y(t))), + Eq(Derivative(y(t), (t, 2)), t*(-x(t) + 2*y(t)))] + sol1 = [Eq(x(t), I*C1*airyai(t*(2 - I)**(S(1)/3)) + I*C2*airybi(t*(2 - I)**(S(1)/3)) - I*C3*airyai(t*(2 + + I)**(S(1)/3)) - I*C4*airybi(t*(2 + I)**(S(1)/3))), + Eq(y(t), C1*airyai(t*(2 - I)**(S(1)/3)) + C2*airybi(t*(2 - I)**(S(1)/3)) + C3*airyai(t*(2 + I)**(S(1)/3)) + + C4*airybi(t*(2 + I)**(S(1)/3)))] + assert dsolve(eqs1) == sol1 + assert checksysodesol(eqs1, sol1) == (True, [0, 0]) + + +@tooslow +@XFAIL +def test_nonlinear_3eq_order1_type1(): + a, b, c = symbols('a b c') + + eqs = [ + a * f(x).diff(x) - (b - c) * g(x) * h(x), + b * g(x).diff(x) - (c - a) * h(x) * f(x), + c * h(x).diff(x) - (a - b) * f(x) * g(x), + ] + + assert dsolve(eqs) # NotImplementedError + + +@XFAIL +def test_nonlinear_3eq_order1_type4(): + eqs = [ + Eq(f(x).diff(x), (2*h(x)*g(x) - 3*g(x)*h(x))), + Eq(g(x).diff(x), (4*f(x)*h(x) - 2*h(x)*f(x))), + Eq(h(x).diff(x), (3*g(x)*f(x) - 4*f(x)*g(x))), + ] + dsolve(eqs) # KeyError when matching + # sol = ? + # assert dsolve_sol == sol + # assert checksysodesol(eqs, dsolve_sol) == (True, [0, 0, 0]) + + +@tooslow +@XFAIL +def test_nonlinear_3eq_order1_type3(): + eqs = [ + Eq(f(x).diff(x), (2*f(x)**2 - 3 )), + Eq(g(x).diff(x), (4 - 2*h(x) )), + Eq(h(x).diff(x), (3*h(x) - 4*f(x)**2)), + ] + dsolve(eqs) # Not sure if this finishes... + # sol = ? + # assert dsolve_sol == sol + # assert checksysodesol(eqs, dsolve_sol) == (True, [0, 0, 0]) + + +@XFAIL +def test_nonlinear_3eq_order1_type5(): + eqs = [ + Eq(f(x).diff(x), f(x)*(2*f(x) - 3*g(x))), + Eq(g(x).diff(x), g(x)*(4*g(x) - 2*h(x))), + Eq(h(x).diff(x), h(x)*(3*h(x) - 4*f(x))), + ] + dsolve(eqs) # KeyError + # sol = ? + # assert dsolve_sol == sol + # assert checksysodesol(eqs, dsolve_sol) == (True, [0, 0, 0]) + + +def test_linear_2eq_order1(): + x, y, z = symbols('x, y, z', cls=Function) + k, l, m, n = symbols('k, l, m, n', Integer=True) + t = Symbol('t') + x0, y0 = symbols('x0, y0', cls=Function) + + eq1 = (Eq(diff(x(t),t), x(t) + y(t) + 9), Eq(diff(y(t),t), 2*x(t) + 5*y(t) + 23)) + sol1 = [Eq(x(t), C1*exp(t*(sqrt(6) + 3)) + C2*exp(t*(-sqrt(6) + 3)) - Rational(22, 3)), \ + Eq(y(t), C1*(2 + sqrt(6))*exp(t*(sqrt(6) + 3)) + C2*(-sqrt(6) + 2)*exp(t*(-sqrt(6) + 3)) - Rational(5, 3))] + assert checksysodesol(eq1, sol1) == (True, [0, 0]) + + eq2 = (Eq(diff(x(t),t), x(t) + y(t) + 81), Eq(diff(y(t),t), -2*x(t) + y(t) + 23)) + sol2 = [Eq(x(t), (C1*cos(sqrt(2)*t) + C2*sin(sqrt(2)*t))*exp(t) - Rational(58, 3)), \ + Eq(y(t), (-sqrt(2)*C1*sin(sqrt(2)*t) + sqrt(2)*C2*cos(sqrt(2)*t))*exp(t) - Rational(185, 3))] + assert checksysodesol(eq2, sol2) == (True, [0, 0]) + + eq3 = (Eq(diff(x(t),t), 5*t*x(t) + 2*y(t)), Eq(diff(y(t),t), 2*x(t) + 5*t*y(t))) + sol3 = [Eq(x(t), (C1*exp(2*t) + C2*exp(-2*t))*exp(Rational(5, 2)*t**2)), \ + Eq(y(t), (C1*exp(2*t) - C2*exp(-2*t))*exp(Rational(5, 2)*t**2))] + assert checksysodesol(eq3, sol3) == (True, [0, 0]) + + eq4 = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t)), Eq(diff(y(t),t), -t**2*x(t) + 5*t*y(t))) + sol4 = [Eq(x(t), (C1*cos((t**3)/3) + C2*sin((t**3)/3))*exp(Rational(5, 2)*t**2)), \ + Eq(y(t), (-C1*sin((t**3)/3) + C2*cos((t**3)/3))*exp(Rational(5, 2)*t**2))] + assert checksysodesol(eq4, sol4) == (True, [0, 0]) + + eq5 = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t)), Eq(diff(y(t),t), -t**2*x(t) + (5*t+9*t**2)*y(t))) + sol5 = [Eq(x(t), (C1*exp((sqrt(77)/2 + Rational(9, 2))*(t**3)/3) + \ + C2*exp((-sqrt(77)/2 + Rational(9, 2))*(t**3)/3))*exp(Rational(5, 2)*t**2)), \ + Eq(y(t), (C1*(sqrt(77)/2 + Rational(9, 2))*exp((sqrt(77)/2 + Rational(9, 2))*(t**3)/3) + \ + C2*(-sqrt(77)/2 + Rational(9, 2))*exp((-sqrt(77)/2 + Rational(9, 2))*(t**3)/3))*exp(Rational(5, 2)*t**2))] + assert checksysodesol(eq5, sol5) == (True, [0, 0]) + + eq6 = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t)), Eq(diff(y(t),t), (1-t**2)*x(t) + (5*t+9*t**2)*y(t))) + sol6 = [Eq(x(t), C1*x0(t) + C2*x0(t)*Integral(t**2*exp(Integral(5*t, t))*exp(Integral(9*t**2 + 5*t, t))/x0(t)**2, t)), \ + Eq(y(t), C1*y0(t) + C2*(y0(t)*Integral(t**2*exp(Integral(5*t, t))*exp(Integral(9*t**2 + 5*t, t))/x0(t)**2, t) + \ + exp(Integral(5*t, t))*exp(Integral(9*t**2 + 5*t, t))/x0(t)))] + s = dsolve(eq6) + assert s == sol6 # too complicated to test with subs and simplify + # assert checksysodesol(eq10, sol10) == (True, [0, 0]) # this one fails + + +def test_nonlinear_2eq_order1(): + x, y, z = symbols('x, y, z', cls=Function) + t = Symbol('t') + eq1 = (Eq(diff(x(t),t),x(t)*y(t)**3), Eq(diff(y(t),t),y(t)**5)) + sol1 = [ + Eq(x(t), C1*exp((-1/(4*C2 + 4*t))**(Rational(-1, 4)))), + Eq(y(t), -(-1/(4*C2 + 4*t))**Rational(1, 4)), + Eq(x(t), C1*exp(-1/(-1/(4*C2 + 4*t))**Rational(1, 4))), + Eq(y(t), (-1/(4*C2 + 4*t))**Rational(1, 4)), + Eq(x(t), C1*exp(-I/(-1/(4*C2 + 4*t))**Rational(1, 4))), + Eq(y(t), -I*(-1/(4*C2 + 4*t))**Rational(1, 4)), + Eq(x(t), C1*exp(I/(-1/(4*C2 + 4*t))**Rational(1, 4))), + Eq(y(t), I*(-1/(4*C2 + 4*t))**Rational(1, 4))] + assert dsolve(eq1) == sol1 + assert checksysodesol(eq1, sol1) == (True, [0, 0]) + + eq2 = (Eq(diff(x(t),t), exp(3*x(t))*y(t)**3),Eq(diff(y(t),t), y(t)**5)) + sol2 = [ + Eq(x(t), -log(C1 - 3/(-1/(4*C2 + 4*t))**Rational(1, 4))/3), + Eq(y(t), -(-1/(4*C2 + 4*t))**Rational(1, 4)), + Eq(x(t), -log(C1 + 3/(-1/(4*C2 + 4*t))**Rational(1, 4))/3), + Eq(y(t), (-1/(4*C2 + 4*t))**Rational(1, 4)), + Eq(x(t), -log(C1 + 3*I/(-1/(4*C2 + 4*t))**Rational(1, 4))/3), + Eq(y(t), -I*(-1/(4*C2 + 4*t))**Rational(1, 4)), + Eq(x(t), -log(C1 - 3*I/(-1/(4*C2 + 4*t))**Rational(1, 4))/3), + Eq(y(t), I*(-1/(4*C2 + 4*t))**Rational(1, 4))] + assert dsolve(eq2) == sol2 + assert checksysodesol(eq2, sol2) == (True, [0, 0]) + + eq3 = (Eq(diff(x(t),t), y(t)*x(t)), Eq(diff(y(t),t), x(t)**3)) + tt = Rational(2, 3) + sol3 = [ + Eq(x(t), 6**tt/(6*(-sinh(sqrt(C1)*(C2 + t)/2)/sqrt(C1))**tt)), + Eq(y(t), sqrt(C1 + C1/sinh(sqrt(C1)*(C2 + t)/2)**2)/3)] + assert dsolve(eq3) == sol3 + # FIXME: assert checksysodesol(eq3, sol3) == (True, [0, 0]) + + eq4 = (Eq(diff(x(t),t),x(t)*y(t)*sin(t)**2), Eq(diff(y(t),t),y(t)**2*sin(t)**2)) + sol4 = {Eq(x(t), -2*exp(C1)/(C2*exp(C1) + t - sin(2*t)/2)), Eq(y(t), -2/(C1 + t - sin(2*t)/2))} + assert dsolve(eq4) == sol4 + # FIXME: assert checksysodesol(eq4, sol4) == (True, [0, 0]) + + eq5 = (Eq(x(t),t*diff(x(t),t)+diff(x(t),t)*diff(y(t),t)), Eq(y(t),t*diff(y(t),t)+diff(y(t),t)**2)) + sol5 = {Eq(x(t), C1*C2 + C1*t), Eq(y(t), C2**2 + C2*t)} + assert dsolve(eq5) == sol5 + assert checksysodesol(eq5, sol5) == (True, [0, 0]) + + eq6 = (Eq(diff(x(t),t),x(t)**2*y(t)**3), Eq(diff(y(t),t),y(t)**5)) + sol6 = [ + Eq(x(t), 1/(C1 - 1/(-1/(4*C2 + 4*t))**Rational(1, 4))), + Eq(y(t), -(-1/(4*C2 + 4*t))**Rational(1, 4)), + Eq(x(t), 1/(C1 + (-1/(4*C2 + 4*t))**(Rational(-1, 4)))), + Eq(y(t), (-1/(4*C2 + 4*t))**Rational(1, 4)), + Eq(x(t), 1/(C1 + I/(-1/(4*C2 + 4*t))**Rational(1, 4))), + Eq(y(t), -I*(-1/(4*C2 + 4*t))**Rational(1, 4)), + Eq(x(t), 1/(C1 - I/(-1/(4*C2 + 4*t))**Rational(1, 4))), + Eq(y(t), I*(-1/(4*C2 + 4*t))**Rational(1, 4))] + assert dsolve(eq6) == sol6 + assert checksysodesol(eq6, sol6) == (True, [0, 0]) + + +@slow +def test_nonlinear_3eq_order1(): + x, y, z = symbols('x, y, z', cls=Function) + t, u = symbols('t u') + eq1 = (4*diff(x(t),t) + 2*y(t)*z(t), 3*diff(y(t),t) - z(t)*x(t), 5*diff(z(t),t) - x(t)*y(t)) + sol1 = [Eq(4*Integral(1/(sqrt(-4*u**2 - 3*C1 + C2)*sqrt(-4*u**2 + 5*C1 - C2)), (u, x(t))), + C3 - sqrt(15)*t/15), Eq(3*Integral(1/(sqrt(-6*u**2 - C1 + 5*C2)*sqrt(3*u**2 + C1 - 4*C2)), + (u, y(t))), C3 + sqrt(5)*t/10), Eq(5*Integral(1/(sqrt(-10*u**2 - 3*C1 + C2)* + sqrt(5*u**2 + 4*C1 - C2)), (u, z(t))), C3 + sqrt(3)*t/6)] + assert [i.dummy_eq(j) for i, j in zip(dsolve(eq1), sol1)] + # FIXME: assert checksysodesol(eq1, sol1) == (True, [0, 0, 0]) + + eq2 = (4*diff(x(t),t) + 2*y(t)*z(t)*sin(t), 3*diff(y(t),t) - z(t)*x(t)*sin(t), 5*diff(z(t),t) - x(t)*y(t)*sin(t)) + sol2 = [Eq(3*Integral(1/(sqrt(-6*u**2 - C1 + 5*C2)*sqrt(3*u**2 + C1 - 4*C2)), (u, x(t))), C3 + + sqrt(5)*cos(t)/10), Eq(4*Integral(1/(sqrt(-4*u**2 - 3*C1 + C2)*sqrt(-4*u**2 + 5*C1 - C2)), + (u, y(t))), C3 - sqrt(15)*cos(t)/15), Eq(5*Integral(1/(sqrt(-10*u**2 - 3*C1 + C2)* + sqrt(5*u**2 + 4*C1 - C2)), (u, z(t))), C3 + sqrt(3)*cos(t)/6)] + assert [i.dummy_eq(j) for i, j in zip(dsolve(eq2), sol2)] + # FIXME: assert checksysodesol(eq2, sol2) == (True, [0, 0, 0]) + + +def test_C1_function_9239(): + t = Symbol('t') + C1 = Function('C1') + C2 = Function('C2') + C3 = Symbol('C3') + C4 = Symbol('C4') + eq = (Eq(diff(C1(t), t), 9*C2(t)), Eq(diff(C2(t), t), 12*C1(t))) + sol = [Eq(C1(t), 9*C3*exp(6*sqrt(3)*t) + 9*C4*exp(-6*sqrt(3)*t)), + Eq(C2(t), 6*sqrt(3)*C3*exp(6*sqrt(3)*t) - 6*sqrt(3)*C4*exp(-6*sqrt(3)*t))] + assert checksysodesol(eq, sol) == (True, [0, 0]) + + +def test_dsolve_linsystem_symbol(): + eps = Symbol('epsilon', positive=True) + eq1 = (Eq(diff(f(x), x), -eps*g(x)), Eq(diff(g(x), x), eps*f(x))) + sol1 = [Eq(f(x), -C1*eps*cos(eps*x) - C2*eps*sin(eps*x)), + Eq(g(x), -C1*eps*sin(eps*x) + C2*eps*cos(eps*x))] + assert checksysodesol(eq1, sol1) == (True, [0, 0]) diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/pde.py b/.venv/lib/python3.12/site-packages/sympy/solvers/pde.py new file mode 100644 index 0000000000000000000000000000000000000000..791ac67ae681ea952ea6e1dabacb7220d1843ebc --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/pde.py @@ -0,0 +1,966 @@ +""" +This module contains pdsolve() and different helper functions that it +uses. It is heavily inspired by the ode module and hence the basic +infrastructure remains the same. + +**Functions in this module** + + These are the user functions in this module: + + - pdsolve() - Solves PDE's + - classify_pde() - Classifies PDEs into possible hints for dsolve(). + - pde_separate() - Separate variables in partial differential equation either by + additive or multiplicative separation approach. + + These are the helper functions in this module: + + - pde_separate_add() - Helper function for searching additive separable solutions. + - pde_separate_mul() - Helper function for searching multiplicative + separable solutions. + +**Currently implemented solver methods** + +The following methods are implemented for solving partial differential +equations. See the docstrings of the various pde_hint() functions for +more information on each (run help(pde)): + + - 1st order linear homogeneous partial differential equations + with constant coefficients. + - 1st order linear general partial differential equations + with constant coefficients. + - 1st order linear partial differential equations with + variable coefficients. + +""" +from functools import reduce + +from itertools import combinations_with_replacement +from sympy.simplify import simplify # type: ignore +from sympy.core import Add, S +from sympy.core.function import Function, expand, AppliedUndef, Subs +from sympy.core.relational import Equality, Eq +from sympy.core.symbol import Symbol, Wild, symbols +from sympy.functions import exp +from sympy.integrals.integrals import Integral, integrate +from sympy.utilities.iterables import has_dups, is_sequence +from sympy.utilities.misc import filldedent + +from sympy.solvers.deutils import _preprocess, ode_order, _desolve +from sympy.solvers.solvers import solve +from sympy.simplify.radsimp import collect + +import operator + + +allhints = ( + "1st_linear_constant_coeff_homogeneous", + "1st_linear_constant_coeff", + "1st_linear_constant_coeff_Integral", + "1st_linear_variable_coeff" + ) + + +def pdsolve(eq, func=None, hint='default', dict=False, solvefun=None, **kwargs): + """ + Solves any (supported) kind of partial differential equation. + + **Usage** + + pdsolve(eq, f(x,y), hint) -> Solve partial differential equation + eq for function f(x,y), using method hint. + + **Details** + + ``eq`` can be any supported partial differential equation (see + the pde docstring for supported methods). This can either + be an Equality, or an expression, which is assumed to be + equal to 0. + + ``f(x,y)`` is a function of two variables whose derivatives in that + variable make up the partial differential equation. In many + cases it is not necessary to provide this; it will be autodetected + (and an error raised if it could not be detected). + + ``hint`` is the solving method that you want pdsolve to use. Use + classify_pde(eq, f(x,y)) to get all of the possible hints for + a PDE. The default hint, 'default', will use whatever hint + is returned first by classify_pde(). See Hints below for + more options that you can use for hint. + + ``solvefun`` is the convention used for arbitrary functions returned + by the PDE solver. If not set by the user, it is set by default + to be F. + + **Hints** + + Aside from the various solving methods, there are also some + meta-hints that you can pass to pdsolve(): + + "default": + This uses whatever hint is returned first by + classify_pde(). This is the default argument to + pdsolve(). + + "all": + To make pdsolve apply all relevant classification hints, + use pdsolve(PDE, func, hint="all"). This will return a + dictionary of hint:solution terms. If a hint causes + pdsolve to raise the NotImplementedError, value of that + hint's key will be the exception object raised. The + dictionary will also include some special keys: + + - order: The order of the PDE. See also ode_order() in + deutils.py + - default: The solution that would be returned by + default. This is the one produced by the hint that + appears first in the tuple returned by classify_pde(). + + "all_Integral": + This is the same as "all", except if a hint also has a + corresponding "_Integral" hint, it only returns the + "_Integral" hint. This is useful if "all" causes + pdsolve() to hang because of a difficult or impossible + integral. This meta-hint will also be much faster than + "all", because integrate() is an expensive routine. + + See also the classify_pde() docstring for more info on hints, + and the pde docstring for a list of all supported hints. + + **Tips** + - You can declare the derivative of an unknown function this way: + + >>> from sympy import Function, Derivative + >>> from sympy.abc import x, y # x and y are the independent variables + >>> f = Function("f")(x, y) # f is a function of x and y + >>> # fx will be the partial derivative of f with respect to x + >>> fx = Derivative(f, x) + >>> # fy will be the partial derivative of f with respect to y + >>> fy = Derivative(f, y) + + - See test_pde.py for many tests, which serves also as a set of + examples for how to use pdsolve(). + - pdsolve always returns an Equality class (except for the case + when the hint is "all" or "all_Integral"). Note that it is not possible + to get an explicit solution for f(x, y) as in the case of ODE's + - Do help(pde.pde_hintname) to get help more information on a + specific hint + + + Examples + ======== + + >>> from sympy.solvers.pde import pdsolve + >>> from sympy import Function, Eq + >>> from sympy.abc import x, y + >>> f = Function('f') + >>> u = f(x, y) + >>> ux = u.diff(x) + >>> uy = u.diff(y) + >>> eq = Eq(1 + (2*(ux/u)) + (3*(uy/u)), 0) + >>> pdsolve(eq) + Eq(f(x, y), F(3*x - 2*y)*exp(-2*x/13 - 3*y/13)) + + """ + + if not solvefun: + solvefun = Function('F') + + # See the docstring of _desolve for more details. + hints = _desolve(eq, func=func, hint=hint, simplify=True, + type='pde', **kwargs) + eq = hints.pop('eq', False) + all_ = hints.pop('all', False) + + if all_: + # TODO : 'best' hint should be implemented when adequate + # number of hints are added. + pdedict = {} + failed_hints = {} + gethints = classify_pde(eq, dict=True) + pdedict.update({'order': gethints['order'], + 'default': gethints['default']}) + for hint in hints: + try: + rv = _helper_simplify(eq, hint, hints[hint]['func'], + hints[hint]['order'], hints[hint][hint], solvefun) + except NotImplementedError as detail: + failed_hints[hint] = detail + else: + pdedict[hint] = rv + pdedict.update(failed_hints) + return pdedict + + else: + return _helper_simplify(eq, hints['hint'], hints['func'], + hints['order'], hints[hints['hint']], solvefun) + + +def _helper_simplify(eq, hint, func, order, match, solvefun): + """Helper function of pdsolve that calls the respective + pde functions to solve for the partial differential + equations. This minimizes the computation in + calling _desolve multiple times. + """ + solvefunc = globals()["pde_" + hint.removesuffix("_Integral")] + return _handle_Integral(solvefunc(eq, func, order, + match, solvefun), func, order, hint) + + +def _handle_Integral(expr, func, order, hint): + r""" + Converts a solution with integrals in it into an actual solution. + + Simplifies the integral mainly using doit() + """ + if hint.endswith("_Integral"): + return expr + + elif hint == "1st_linear_constant_coeff": + return simplify(expr.doit()) + + else: + return expr + + +def classify_pde(eq, func=None, dict=False, *, prep=True, **kwargs): + """ + Returns a tuple of possible pdsolve() classifications for a PDE. + + The tuple is ordered so that first item is the classification that + pdsolve() uses to solve the PDE by default. In general, + classifications near the beginning of the list will produce + better solutions faster than those near the end, though there are + always exceptions. To make pdsolve use a different classification, + use pdsolve(PDE, func, hint=). See also the pdsolve() + docstring for different meta-hints you can use. + + If ``dict`` is true, classify_pde() will return a dictionary of + hint:match expression terms. This is intended for internal use by + pdsolve(). Note that because dictionaries are ordered arbitrarily, + this will most likely not be in the same order as the tuple. + + You can get help on different hints by doing help(pde.pde_hintname), + where hintname is the name of the hint without "_Integral". + + See sympy.pde.allhints or the sympy.pde docstring for a list of all + supported hints that can be returned from classify_pde. + + + Examples + ======== + + >>> from sympy.solvers.pde import classify_pde + >>> from sympy import Function, Eq + >>> from sympy.abc import x, y + >>> f = Function('f') + >>> u = f(x, y) + >>> ux = u.diff(x) + >>> uy = u.diff(y) + >>> eq = Eq(1 + (2*(ux/u)) + (3*(uy/u)), 0) + >>> classify_pde(eq) + ('1st_linear_constant_coeff_homogeneous',) + """ + + if func and len(func.args) != 2: + raise NotImplementedError("Right now only partial " + "differential equations of two variables are supported") + + if prep or func is None: + prep, func_ = _preprocess(eq, func) + if func is None: + func = func_ + + if isinstance(eq, Equality): + if eq.rhs != 0: + return classify_pde(eq.lhs - eq.rhs, func) + eq = eq.lhs + + f = func.func + x = func.args[0] + y = func.args[1] + fx = f(x,y).diff(x) + fy = f(x,y).diff(y) + + # TODO : For now pde.py uses support offered by the ode_order function + # to find the order with respect to a multi-variable function. An + # improvement could be to classify the order of the PDE on the basis of + # individual variables. + order = ode_order(eq, f(x,y)) + + # hint:matchdict or hint:(tuple of matchdicts) + # Also will contain "default": and "order":order items. + matching_hints = {'order': order} + + if not order: + if dict: + matching_hints["default"] = None + return matching_hints + return () + + eq = expand(eq) + + a = Wild('a', exclude = [f(x,y)]) + b = Wild('b', exclude = [f(x,y), fx, fy, x, y]) + c = Wild('c', exclude = [f(x,y), fx, fy, x, y]) + d = Wild('d', exclude = [f(x,y), fx, fy, x, y]) + e = Wild('e', exclude = [f(x,y), fx, fy]) + n = Wild('n', exclude = [x, y]) + # Try removing the smallest power of f(x,y) + # from the highest partial derivatives of f(x,y) + reduced_eq = eq + if eq.is_Add: + power = None + for i in set(combinations_with_replacement((x,y), order)): + coeff = eq.coeff(f(x,y).diff(*i)) + if coeff == 1: + continue + match = coeff.match(a*f(x,y)**n) + if match and match[a]: + if power is None or match[n] < power: + power = match[n] + if power: + den = f(x,y)**power + reduced_eq = Add(*[arg/den for arg in eq.args]) + + if order == 1: + reduced_eq = collect(reduced_eq, f(x, y)) + r = reduced_eq.match(b*fx + c*fy + d*f(x,y) + e) + if r: + if not r[e]: + ## Linear first-order homogeneous partial-differential + ## equation with constant coefficients + r.update({'b': b, 'c': c, 'd': d}) + matching_hints["1st_linear_constant_coeff_homogeneous"] = r + elif r[b]**2 + r[c]**2 != 0: + ## Linear first-order general partial-differential + ## equation with constant coefficients + r.update({'b': b, 'c': c, 'd': d, 'e': e}) + matching_hints["1st_linear_constant_coeff"] = r + matching_hints["1st_linear_constant_coeff_Integral"] = r + + else: + b = Wild('b', exclude=[f(x, y), fx, fy]) + c = Wild('c', exclude=[f(x, y), fx, fy]) + d = Wild('d', exclude=[f(x, y), fx, fy]) + r = reduced_eq.match(b*fx + c*fy + d*f(x,y) + e) + if r: + r.update({'b': b, 'c': c, 'd': d, 'e': e}) + matching_hints["1st_linear_variable_coeff"] = r + + # Order keys based on allhints. + rettuple = tuple(i for i in allhints if i in matching_hints) + + if dict: + # Dictionaries are ordered arbitrarily, so make note of which + # hint would come first for pdsolve(). Use an ordered dict in Py 3. + matching_hints["default"] = None + matching_hints["ordered_hints"] = rettuple + for i in allhints: + if i in matching_hints: + matching_hints["default"] = i + break + return matching_hints + return rettuple + + +def checkpdesol(pde, sol, func=None, solve_for_func=True): + """ + Checks if the given solution satisfies the partial differential + equation. + + pde is the partial differential equation which can be given in the + form of an equation or an expression. sol is the solution for which + the pde is to be checked. This can also be given in an equation or + an expression form. If the function is not provided, the helper + function _preprocess from deutils is used to identify the function. + + If a sequence of solutions is passed, the same sort of container will be + used to return the result for each solution. + + The following methods are currently being implemented to check if the + solution satisfies the PDE: + + 1. Directly substitute the solution in the PDE and check. If the + solution has not been solved for f, then it will solve for f + provided solve_for_func has not been set to False. + + If the solution satisfies the PDE, then a tuple (True, 0) is returned. + Otherwise a tuple (False, expr) where expr is the value obtained + after substituting the solution in the PDE. However if a known solution + returns False, it may be due to the inability of doit() to simplify it to zero. + + Examples + ======== + + >>> from sympy import Function, symbols + >>> from sympy.solvers.pde import checkpdesol, pdsolve + >>> x, y = symbols('x y') + >>> f = Function('f') + >>> eq = 2*f(x,y) + 3*f(x,y).diff(x) + 4*f(x,y).diff(y) + >>> sol = pdsolve(eq) + >>> assert checkpdesol(eq, sol)[0] + >>> eq = x*f(x,y) + f(x,y).diff(x) + >>> checkpdesol(eq, sol) + (False, (x*F(4*x - 3*y) - 6*F(4*x - 3*y)/25 + 4*Subs(Derivative(F(_xi_1), _xi_1), _xi_1, 4*x - 3*y))*exp(-6*x/25 - 8*y/25)) + """ + + # Converting the pde into an equation + if not isinstance(pde, Equality): + pde = Eq(pde, 0) + + # If no function is given, try finding the function present. + if func is None: + try: + _, func = _preprocess(pde.lhs) + except ValueError: + funcs = [s.atoms(AppliedUndef) for s in ( + sol if is_sequence(sol, set) else [sol])] + funcs = set().union(funcs) + if len(funcs) != 1: + raise ValueError( + 'must pass func arg to checkpdesol for this case.') + func = funcs.pop() + + # If the given solution is in the form of a list or a set + # then return a list or set of tuples. + if is_sequence(sol, set): + return type(sol)([checkpdesol( + pde, i, func=func, + solve_for_func=solve_for_func) for i in sol]) + + # Convert solution into an equation + if not isinstance(sol, Equality): + sol = Eq(func, sol) + elif sol.rhs == func: + sol = sol.reversed + + # Try solving for the function + solved = sol.lhs == func and not sol.rhs.has(func) + if solve_for_func and not solved: + solved = solve(sol, func) + if solved: + if len(solved) == 1: + return checkpdesol(pde, Eq(func, solved[0]), + func=func, solve_for_func=False) + else: + return checkpdesol(pde, [Eq(func, t) for t in solved], + func=func, solve_for_func=False) + + # try direct substitution of the solution into the PDE and simplify + if sol.lhs == func: + pde = pde.lhs - pde.rhs + s = simplify(pde.subs(func, sol.rhs).doit()) + return s is S.Zero, s + + raise NotImplementedError(filldedent(''' + Unable to test if %s is a solution to %s.''' % (sol, pde))) + + + +def pde_1st_linear_constant_coeff_homogeneous(eq, func, order, match, solvefun): + r""" + Solves a first order linear homogeneous + partial differential equation with constant coefficients. + + The general form of this partial differential equation is + + .. math:: a \frac{\partial f(x,y)}{\partial x} + + b \frac{\partial f(x,y)}{\partial y} + c f(x,y) = 0 + + where `a`, `b` and `c` are constants. + + The general solution is of the form: + + .. math:: + f(x, y) = F(- a y + b x ) e^{- \frac{c (a x + b y)}{a^2 + b^2}} + + and can be found in SymPy with ``pdsolve``:: + + >>> from sympy.solvers import pdsolve + >>> from sympy.abc import x, y, a, b, c + >>> from sympy import Function, pprint + >>> f = Function('f') + >>> u = f(x,y) + >>> ux = u.diff(x) + >>> uy = u.diff(y) + >>> genform = a*ux + b*uy + c*u + >>> pprint(genform) + d d + a*--(f(x, y)) + b*--(f(x, y)) + c*f(x, y) + dx dy + + >>> pprint(pdsolve(genform)) + -c*(a*x + b*y) + --------------- + 2 2 + a + b + f(x, y) = F(-a*y + b*x)*e + + Examples + ======== + + >>> from sympy import pdsolve + >>> from sympy import Function, pprint + >>> from sympy.abc import x,y + >>> f = Function('f') + >>> pdsolve(f(x,y) + f(x,y).diff(x) + f(x,y).diff(y)) + Eq(f(x, y), F(x - y)*exp(-x/2 - y/2)) + >>> pprint(pdsolve(f(x,y) + f(x,y).diff(x) + f(x,y).diff(y))) + x y + - - - - + 2 2 + f(x, y) = F(x - y)*e + + References + ========== + + - Viktor Grigoryan, "Partial Differential Equations" + Math 124A - Fall 2010, pp.7 + + """ + # TODO : For now homogeneous first order linear PDE's having + # two variables are implemented. Once there is support for + # solving systems of ODE's, this can be extended to n variables. + + f = func.func + x = func.args[0] + y = func.args[1] + b = match[match['b']] + c = match[match['c']] + d = match[match['d']] + return Eq(f(x,y), exp(-S(d)/(b**2 + c**2)*(b*x + c*y))*solvefun(c*x - b*y)) + + +def pde_1st_linear_constant_coeff(eq, func, order, match, solvefun): + r""" + Solves a first order linear partial differential equation + with constant coefficients. + + The general form of this partial differential equation is + + .. math:: a \frac{\partial f(x,y)}{\partial x} + + b \frac{\partial f(x,y)}{\partial y} + + c f(x,y) = G(x,y) + + where `a`, `b` and `c` are constants and `G(x, y)` can be an arbitrary + function in `x` and `y`. + + The general solution of the PDE is: + + .. math:: + f(x, y) = \left. \left[F(\eta) + \frac{1}{a^2 + b^2} + \int\limits^{a x + b y} G\left(\frac{a \xi + b \eta}{a^2 + b^2}, + \frac{- a \eta + b \xi}{a^2 + b^2} \right) + e^{\frac{c \xi}{a^2 + b^2}}\, d\xi\right] + e^{- \frac{c \xi}{a^2 + b^2}} + \right|_{\substack{\eta=- a y + b x\\ \xi=a x + b y }}\, , + + where `F(\eta)` is an arbitrary single-valued function. The solution + can be found in SymPy with ``pdsolve``:: + + >>> from sympy.solvers import pdsolve + >>> from sympy.abc import x, y, a, b, c + >>> from sympy import Function, pprint + >>> f = Function('f') + >>> G = Function('G') + >>> u = f(x, y) + >>> ux = u.diff(x) + >>> uy = u.diff(y) + >>> genform = a*ux + b*uy + c*u - G(x,y) + >>> pprint(genform) + d d + a*--(f(x, y)) + b*--(f(x, y)) + c*f(x, y) - G(x, y) + dx dy + >>> pprint(pdsolve(genform, hint='1st_linear_constant_coeff_Integral')) + // a*x + b*y \ \| + || / | || + || | | || + || | c*xi | || + || | ------- | || + || | 2 2 | || + || | /a*xi + b*eta -a*eta + b*xi\ a + b | || + || | G|------------, -------------|*e d(xi)| || + || | | 2 2 2 2 | | || + || | \ a + b a + b / | -c*xi || + || | | -------|| + || / | 2 2|| + || | a + b || + f(x, y) = ||F(eta) + -------------------------------------------------------|*e || + || 2 2 | || + \\ a + b / /|eta=-a*y + b*x, xi=a*x + b*y + + Examples + ======== + + >>> from sympy.solvers.pde import pdsolve + >>> from sympy import Function, pprint, exp + >>> from sympy.abc import x,y + >>> f = Function('f') + >>> eq = -2*f(x,y).diff(x) + 4*f(x,y).diff(y) + 5*f(x,y) - exp(x + 3*y) + >>> pdsolve(eq) + Eq(f(x, y), (F(4*x + 2*y)*exp(x/2) + exp(x + 4*y)/15)*exp(-y)) + + References + ========== + + - Viktor Grigoryan, "Partial Differential Equations" + Math 124A - Fall 2010, pp.7 + + """ + + # TODO : For now homogeneous first order linear PDE's having + # two variables are implemented. Once there is support for + # solving systems of ODE's, this can be extended to n variables. + xi, eta = symbols("xi eta") + f = func.func + x = func.args[0] + y = func.args[1] + b = match[match['b']] + c = match[match['c']] + d = match[match['d']] + e = -match[match['e']] + expterm = exp(-S(d)/(b**2 + c**2)*xi) + functerm = solvefun(eta) + solvedict = solve((b*x + c*y - xi, c*x - b*y - eta), x, y) + # Integral should remain as it is in terms of xi, + # doit() should be done in _handle_Integral. + genterm = (1/S(b**2 + c**2))*Integral( + (1/expterm*e).subs(solvedict), (xi, b*x + c*y)) + return Eq(f(x,y), Subs(expterm*(functerm + genterm), + (eta, xi), (c*x - b*y, b*x + c*y))) + + +def pde_1st_linear_variable_coeff(eq, func, order, match, solvefun): + r""" + Solves a first order linear partial differential equation + with variable coefficients. The general form of this partial + differential equation is + + .. math:: a(x, y) \frac{\partial f(x, y)}{\partial x} + + b(x, y) \frac{\partial f(x, y)}{\partial y} + + c(x, y) f(x, y) = G(x, y) + + where `a(x, y)`, `b(x, y)`, `c(x, y)` and `G(x, y)` are arbitrary + functions in `x` and `y`. This PDE is converted into an ODE by + making the following transformation: + + 1. `\xi` as `x` + + 2. `\eta` as the constant in the solution to the differential + equation `\frac{dy}{dx} = -\frac{b}{a}` + + Making the previous substitutions reduces it to the linear ODE + + .. math:: a(\xi, \eta)\frac{du}{d\xi} + c(\xi, \eta)u - G(\xi, \eta) = 0 + + which can be solved using ``dsolve``. + + >>> from sympy.abc import x, y + >>> from sympy import Function, pprint + >>> a, b, c, G, f= [Function(i) for i in ['a', 'b', 'c', 'G', 'f']] + >>> u = f(x,y) + >>> ux = u.diff(x) + >>> uy = u.diff(y) + >>> genform = a(x, y)*u + b(x, y)*ux + c(x, y)*uy - G(x,y) + >>> pprint(genform) + d d + -G(x, y) + a(x, y)*f(x, y) + b(x, y)*--(f(x, y)) + c(x, y)*--(f(x, y)) + dx dy + + + Examples + ======== + + >>> from sympy.solvers.pde import pdsolve + >>> from sympy import Function, pprint + >>> from sympy.abc import x,y + >>> f = Function('f') + >>> eq = x*(u.diff(x)) - y*(u.diff(y)) + y**2*u - y**2 + >>> pdsolve(eq) + Eq(f(x, y), F(x*y)*exp(y**2/2) + 1) + + References + ========== + + - Viktor Grigoryan, "Partial Differential Equations" + Math 124A - Fall 2010, pp.7 + + """ + from sympy.solvers.ode import dsolve + + eta = symbols("eta") + f = func.func + x = func.args[0] + y = func.args[1] + b = match[match['b']] + c = match[match['c']] + d = match[match['d']] + e = -match[match['e']] + + + if not d: + # To deal with cases like b*ux = e or c*uy = e + if not (b and c): + if c: + try: + tsol = integrate(e/c, y) + except NotImplementedError: + raise NotImplementedError("Unable to find a solution" + " due to inability of integrate") + else: + return Eq(f(x,y), solvefun(x) + tsol) + if b: + try: + tsol = integrate(e/b, x) + except NotImplementedError: + raise NotImplementedError("Unable to find a solution" + " due to inability of integrate") + else: + return Eq(f(x,y), solvefun(y) + tsol) + + if not c: + # To deal with cases when c is 0, a simpler method is used. + # The PDE reduces to b*(u.diff(x)) + d*u = e, which is a linear ODE in x + plode = f(x).diff(x)*b + d*f(x) - e + sol = dsolve(plode, f(x)) + syms = sol.free_symbols - plode.free_symbols - {x, y} + rhs = _simplify_variable_coeff(sol.rhs, syms, solvefun, y) + return Eq(f(x, y), rhs) + + if not b: + # To deal with cases when b is 0, a simpler method is used. + # The PDE reduces to c*(u.diff(y)) + d*u = e, which is a linear ODE in y + plode = f(y).diff(y)*c + d*f(y) - e + sol = dsolve(plode, f(y)) + syms = sol.free_symbols - plode.free_symbols - {x, y} + rhs = _simplify_variable_coeff(sol.rhs, syms, solvefun, x) + return Eq(f(x, y), rhs) + + dummy = Function('d') + h = (c/b).subs(y, dummy(x)) + sol = dsolve(dummy(x).diff(x) - h, dummy(x)) + if isinstance(sol, list): + sol = sol[0] + solsym = sol.free_symbols - h.free_symbols - {x, y} + if len(solsym) == 1: + solsym = solsym.pop() + etat = (solve(sol, solsym)[0]).subs(dummy(x), y) + ysub = solve(eta - etat, y)[0] + deq = (b*(f(x).diff(x)) + d*f(x) - e).subs(y, ysub) + final = (dsolve(deq, f(x), hint='1st_linear')).rhs + if isinstance(final, list): + final = final[0] + finsyms = final.free_symbols - deq.free_symbols - {x, y} + rhs = _simplify_variable_coeff(final, finsyms, solvefun, etat) + return Eq(f(x, y), rhs) + + else: + raise NotImplementedError("Cannot solve the partial differential equation due" + " to inability of constantsimp") + + +def _simplify_variable_coeff(sol, syms, func, funcarg): + r""" + Helper function to replace constants by functions in 1st_linear_variable_coeff + """ + eta = Symbol("eta") + if len(syms) == 1: + sym = syms.pop() + final = sol.subs(sym, func(funcarg)) + + else: + for sym in syms: + final = sol.subs(sym, func(funcarg)) + + return simplify(final.subs(eta, funcarg)) + + +def pde_separate(eq, fun, sep, strategy='mul'): + """Separate variables in partial differential equation either by additive + or multiplicative separation approach. It tries to rewrite an equation so + that one of the specified variables occurs on a different side of the + equation than the others. + + :param eq: Partial differential equation + + :param fun: Original function F(x, y, z) + + :param sep: List of separated functions [X(x), u(y, z)] + + :param strategy: Separation strategy. You can choose between additive + separation ('add') and multiplicative separation ('mul') which is + default. + + Examples + ======== + + >>> from sympy import E, Eq, Function, pde_separate, Derivative as D + >>> from sympy.abc import x, t + >>> u, X, T = map(Function, 'uXT') + + >>> eq = Eq(D(u(x, t), x), E**(u(x, t))*D(u(x, t), t)) + >>> pde_separate(eq, u(x, t), [X(x), T(t)], strategy='add') + [exp(-X(x))*Derivative(X(x), x), exp(T(t))*Derivative(T(t), t)] + + >>> eq = Eq(D(u(x, t), x, 2), D(u(x, t), t, 2)) + >>> pde_separate(eq, u(x, t), [X(x), T(t)], strategy='mul') + [Derivative(X(x), (x, 2))/X(x), Derivative(T(t), (t, 2))/T(t)] + + See Also + ======== + pde_separate_add, pde_separate_mul + """ + + do_add = False + if strategy == 'add': + do_add = True + elif strategy == 'mul': + do_add = False + else: + raise ValueError('Unknown strategy: %s' % strategy) + + if isinstance(eq, Equality): + if eq.rhs != 0: + return pde_separate(Eq(eq.lhs - eq.rhs, 0), fun, sep, strategy) + else: + return pde_separate(Eq(eq, 0), fun, sep, strategy) + + if eq.rhs != 0: + raise ValueError("Value should be 0") + + # Handle arguments + orig_args = list(fun.args) + subs_args = [arg for s in sep for arg in s.args] + + if do_add: + functions = reduce(operator.add, sep) + else: + functions = reduce(operator.mul, sep) + + # Check whether variables match + if len(subs_args) != len(orig_args): + raise ValueError("Variable counts do not match") + # Check for duplicate arguments like [X(x), u(x, y)] + if has_dups(subs_args): + raise ValueError("Duplicate substitution arguments detected") + # Check whether the variables match + if set(orig_args) != set(subs_args): + raise ValueError("Arguments do not match") + + # Substitute original function with separated... + result = eq.lhs.subs(fun, functions).doit() + + # Divide by terms when doing multiplicative separation + if not do_add: + eq = 0 + for i in result.args: + eq += i/functions + result = eq + + svar = subs_args[0] + dvar = subs_args[1:] + return _separate(result, svar, dvar) + + +def pde_separate_add(eq, fun, sep): + """ + Helper function for searching additive separable solutions. + + Consider an equation of two independent variables x, y and a dependent + variable w, we look for the product of two functions depending on different + arguments: + + `w(x, y, z) = X(x) + y(y, z)` + + Examples + ======== + + >>> from sympy import E, Eq, Function, pde_separate_add, Derivative as D + >>> from sympy.abc import x, t + >>> u, X, T = map(Function, 'uXT') + + >>> eq = Eq(D(u(x, t), x), E**(u(x, t))*D(u(x, t), t)) + >>> pde_separate_add(eq, u(x, t), [X(x), T(t)]) + [exp(-X(x))*Derivative(X(x), x), exp(T(t))*Derivative(T(t), t)] + + """ + return pde_separate(eq, fun, sep, strategy='add') + + +def pde_separate_mul(eq, fun, sep): + """ + Helper function for searching multiplicative separable solutions. + + Consider an equation of two independent variables x, y and a dependent + variable w, we look for the product of two functions depending on different + arguments: + + `w(x, y, z) = X(x)*u(y, z)` + + Examples + ======== + + >>> from sympy import Function, Eq, pde_separate_mul, Derivative as D + >>> from sympy.abc import x, y + >>> u, X, Y = map(Function, 'uXY') + + >>> eq = Eq(D(u(x, y), x, 2), D(u(x, y), y, 2)) + >>> pde_separate_mul(eq, u(x, y), [X(x), Y(y)]) + [Derivative(X(x), (x, 2))/X(x), Derivative(Y(y), (y, 2))/Y(y)] + + """ + return pde_separate(eq, fun, sep, strategy='mul') + + +def _separate(eq, dep, others): + """Separate expression into two parts based on dependencies of variables.""" + + # FIRST PASS + # Extract derivatives depending our separable variable... + terms = set() + for term in eq.args: + if term.is_Mul: + for i in term.args: + if i.is_Derivative and not i.has(*others): + terms.add(term) + continue + elif term.is_Derivative and not term.has(*others): + terms.add(term) + # Find the factor that we need to divide by + div = set() + for term in terms: + ext, sep = term.expand().as_independent(dep) + # Failed? + if sep.has(*others): + return None + div.add(ext) + # FIXME: Find lcm() of all the divisors and divide with it, instead of + # current hack :( + # https://github.com/sympy/sympy/issues/4597 + if len(div) > 0: + # double sum required or some tests will fail + eq = Add(*[simplify(Add(*[term/i for i in div])) for term in eq.args]) + # SECOND PASS - separate the derivatives + div = set() + lhs = rhs = 0 + for term in eq.args: + # Check, whether we have already term with independent variable... + if not term.has(*others): + lhs += term + continue + # ...otherwise, try to separate + temp, sep = term.expand().as_independent(dep) + # Failed? + if sep.has(*others): + return None + # Extract the divisors + div.add(sep) + rhs -= term.expand() + # Do the division + fulldiv = reduce(operator.add, div) + lhs = simplify(lhs/fulldiv).expand() + rhs = simplify(rhs/fulldiv).expand() + # ...and check whether we were successful :) + if lhs.has(*others) or rhs.has(dep): + return None + return [lhs, rhs] diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/polysys.py b/.venv/lib/python3.12/site-packages/sympy/solvers/polysys.py new file mode 100644 index 0000000000000000000000000000000000000000..2edc70b36c25b986c975c33fcc57535ef0b31df2 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/polysys.py @@ -0,0 +1,872 @@ +"""Solvers of systems of polynomial equations. """ + +from __future__ import annotations + +from typing import Any +from collections.abc import Sequence, Iterable + +import itertools + +from sympy import Dummy +from sympy.core import S +from sympy.core.expr import Expr +from sympy.core.exprtools import factor_terms +from sympy.core.sorting import default_sort_key +from sympy.logic.boolalg import Boolean +from sympy.polys import Poly, groebner, roots +from sympy.polys.domains import ZZ +from sympy.polys.polyoptions import build_options +from sympy.polys.polytools import parallel_poly_from_expr, sqf_part +from sympy.polys.polyerrors import ( + ComputationFailed, + PolificationFailed, + CoercionFailed, + GeneratorsNeeded, + DomainError +) +from sympy.simplify import rcollect +from sympy.utilities import postfixes +from sympy.utilities.iterables import cartes +from sympy.utilities.misc import filldedent +from sympy.logic.boolalg import Or, And +from sympy.core.relational import Eq + + +class SolveFailed(Exception): + """Raised when solver's conditions were not met. """ + + +def solve_poly_system(seq, *gens, strict=False, **args): + """ + Return a list of solutions for the system of polynomial equations + or else None. + + Parameters + ========== + + seq: a list/tuple/set + Listing all the equations that are needed to be solved + gens: generators + generators of the equations in seq for which we want the + solutions + strict: a boolean (default is False) + if strict is True, NotImplementedError will be raised if + the solution is known to be incomplete (which can occur if + not all solutions are expressible in radicals) + args: Keyword arguments + Special options for solving the equations. + + + Returns + ======= + + List[Tuple] + a list of tuples with elements being solutions for the + symbols in the order they were passed as gens + None + None is returned when the computed basis contains only the ground. + + Examples + ======== + + >>> from sympy import solve_poly_system + >>> from sympy.abc import x, y + + >>> solve_poly_system([x*y - 2*y, 2*y**2 - x**2], x, y) + [(0, 0), (2, -sqrt(2)), (2, sqrt(2))] + + >>> solve_poly_system([x**5 - x + y**3, y**2 - 1], x, y, strict=True) + Traceback (most recent call last): + ... + UnsolvableFactorError + + """ + try: + polys, opt = parallel_poly_from_expr(seq, *gens, **args) + except PolificationFailed as exc: + raise ComputationFailed('solve_poly_system', len(seq), exc) + + if len(polys) == len(opt.gens) == 2: + f, g = polys + + if all(i <= 2 for i in f.degree_list() + g.degree_list()): + try: + return solve_biquadratic(f, g, opt) + except SolveFailed: + pass + + return solve_generic(polys, opt, strict=strict) + + +def solve_biquadratic(f, g, opt): + """Solve a system of two bivariate quadratic polynomial equations. + + Parameters + ========== + + f: a single Expr or Poly + First equation + g: a single Expr or Poly + Second Equation + opt: an Options object + For specifying keyword arguments and generators + + Returns + ======= + + List[Tuple] + a list of tuples with elements being solutions for the + symbols in the order they were passed as gens + None + None is returned when the computed basis contains only the ground. + + Examples + ======== + + >>> from sympy import Options, Poly + >>> from sympy.abc import x, y + >>> from sympy.solvers.polysys import solve_biquadratic + >>> NewOption = Options((x, y), {'domain': 'ZZ'}) + + >>> a = Poly(y**2 - 4 + x, y, x, domain='ZZ') + >>> b = Poly(y*2 + 3*x - 7, y, x, domain='ZZ') + >>> solve_biquadratic(a, b, NewOption) + [(1/3, 3), (41/27, 11/9)] + + >>> a = Poly(y + x**2 - 3, y, x, domain='ZZ') + >>> b = Poly(-y + x - 4, y, x, domain='ZZ') + >>> solve_biquadratic(a, b, NewOption) + [(7/2 - sqrt(29)/2, -sqrt(29)/2 - 1/2), (sqrt(29)/2 + 7/2, -1/2 + \ + sqrt(29)/2)] + """ + G = groebner([f, g]) + + if len(G) == 1 and G[0].is_ground: + return None + + if len(G) != 2: + raise SolveFailed + + x, y = opt.gens + p, q = G + if not p.gcd(q).is_ground: + # not 0-dimensional + raise SolveFailed + + p = Poly(p, x, expand=False) + p_roots = [rcollect(expr, y) for expr in roots(p).keys()] + + q = q.ltrim(-1) + q_roots = list(roots(q).keys()) + + solutions = [(p_root.subs(y, q_root), q_root) for q_root, p_root in + itertools.product(q_roots, p_roots)] + + return sorted(solutions, key=default_sort_key) + + +def solve_generic(polys, opt, strict=False): + """ + Solve a generic system of polynomial equations. + + Returns all possible solutions over C[x_1, x_2, ..., x_m] of a + set F = { f_1, f_2, ..., f_n } of polynomial equations, using + Groebner basis approach. For now only zero-dimensional systems + are supported, which means F can have at most a finite number + of solutions. If the basis contains only the ground, None is + returned. + + The algorithm works by the fact that, supposing G is the basis + of F with respect to an elimination order (here lexicographic + order is used), G and F generate the same ideal, they have the + same set of solutions. By the elimination property, if G is a + reduced, zero-dimensional Groebner basis, then there exists an + univariate polynomial in G (in its last variable). This can be + solved by computing its roots. Substituting all computed roots + for the last (eliminated) variable in other elements of G, new + polynomial system is generated. Applying the above procedure + recursively, a finite number of solutions can be found. + + The ability of finding all solutions by this procedure depends + on the root finding algorithms. If no solutions were found, it + means only that roots() failed, but the system is solvable. To + overcome this difficulty use numerical algorithms instead. + + Parameters + ========== + + polys: a list/tuple/set + Listing all the polynomial equations that are needed to be solved + opt: an Options object + For specifying keyword arguments and generators + strict: a boolean + If strict is True, NotImplementedError will be raised if the solution + is known to be incomplete + + Returns + ======= + + List[Tuple] + a list of tuples with elements being solutions for the + symbols in the order they were passed as gens + None + None is returned when the computed basis contains only the ground. + + References + ========== + + .. [Buchberger01] B. Buchberger, Groebner Bases: A Short + Introduction for Systems Theorists, In: R. Moreno-Diaz, + B. Buchberger, J.L. Freire, Proceedings of EUROCAST'01, + February, 2001 + + .. [Cox97] D. Cox, J. Little, D. O'Shea, Ideals, Varieties + and Algorithms, Springer, Second Edition, 1997, pp. 112 + + Raises + ======== + + NotImplementedError + If the system is not zero-dimensional (does not have a finite + number of solutions) + + UnsolvableFactorError + If ``strict`` is True and not all solution components are + expressible in radicals + + Examples + ======== + + >>> from sympy import Poly, Options + >>> from sympy.solvers.polysys import solve_generic + >>> from sympy.abc import x, y + >>> NewOption = Options((x, y), {'domain': 'ZZ'}) + + >>> a = Poly(x - y + 5, x, y, domain='ZZ') + >>> b = Poly(x + y - 3, x, y, domain='ZZ') + >>> solve_generic([a, b], NewOption) + [(-1, 4)] + + >>> a = Poly(x - 2*y + 5, x, y, domain='ZZ') + >>> b = Poly(2*x - y - 3, x, y, domain='ZZ') + >>> solve_generic([a, b], NewOption) + [(11/3, 13/3)] + + >>> a = Poly(x**2 + y, x, y, domain='ZZ') + >>> b = Poly(x + y*4, x, y, domain='ZZ') + >>> solve_generic([a, b], NewOption) + [(0, 0), (1/4, -1/16)] + + >>> a = Poly(x**5 - x + y**3, x, y, domain='ZZ') + >>> b = Poly(y**2 - 1, x, y, domain='ZZ') + >>> solve_generic([a, b], NewOption, strict=True) + Traceback (most recent call last): + ... + UnsolvableFactorError + + """ + def _is_univariate(f): + """Returns True if 'f' is univariate in its last variable. """ + for monom in f.monoms(): + if any(monom[:-1]): + return False + + return True + + def _subs_root(f, gen, zero): + """Replace generator with a root so that the result is nice. """ + p = f.as_expr({gen: zero}) + + if f.degree(gen) >= 2: + p = p.expand(deep=False) + + return p + + def _solve_reduced_system(system, gens, entry=False): + """Recursively solves reduced polynomial systems. """ + if len(system) == len(gens) == 1: + # the below line will produce UnsolvableFactorError if + # strict=True and the solution from `roots` is incomplete + zeros = list(roots(system[0], gens[-1], strict=strict).keys()) + return [(zero,) for zero in zeros] + + basis = groebner(system, gens, polys=True) + + if len(basis) == 1 and basis[0].is_ground: + if not entry: + return [] + else: + return None + + univariate = list(filter(_is_univariate, basis)) + + if len(basis) < len(gens): + raise NotImplementedError(filldedent(''' + only zero-dimensional systems supported + (finite number of solutions) + ''')) + + if len(univariate) == 1: + f = univariate.pop() + else: + raise NotImplementedError(filldedent(''' + only zero-dimensional systems supported + (finite number of solutions) + ''')) + + gens = f.gens + gen = gens[-1] + + # the below line will produce UnsolvableFactorError if + # strict=True and the solution from `roots` is incomplete + zeros = list(roots(f.ltrim(gen), strict=strict).keys()) + + if not zeros: + return [] + + if len(basis) == 1: + return [(zero,) for zero in zeros] + + solutions = [] + + for zero in zeros: + new_system = [] + new_gens = gens[:-1] + + for b in basis[:-1]: + eq = _subs_root(b, gen, zero) + + if eq is not S.Zero: + new_system.append(eq) + + for solution in _solve_reduced_system(new_system, new_gens): + solutions.append(solution + (zero,)) + + if solutions and len(solutions[0]) != len(gens): + raise NotImplementedError(filldedent(''' + only zero-dimensional systems supported + (finite number of solutions) + ''')) + return solutions + + try: + result = _solve_reduced_system(polys, opt.gens, entry=True) + except CoercionFailed: + raise NotImplementedError + + if result is not None: + return sorted(result, key=default_sort_key) + + +def solve_triangulated(polys, *gens, **args): + """ + Solve a polynomial system using Gianni-Kalkbrenner algorithm. + + The algorithm proceeds by computing one Groebner basis in the ground + domain and then by iteratively computing polynomial factorizations in + appropriately constructed algebraic extensions of the ground domain. + + Parameters + ========== + + polys: a list/tuple/set + Listing all the equations that are needed to be solved + gens: generators + generators of the equations in polys for which we want the + solutions + args: Keyword arguments + Special options for solving the equations + + Returns + ======= + + List[Tuple] + A List of tuples. Solutions for symbols that satisfy the + equations listed in polys + + Examples + ======== + + >>> from sympy import solve_triangulated + >>> from sympy.abc import x, y, z + + >>> F = [x**2 + y + z - 1, x + y**2 + z - 1, x + y + z**2 - 1] + + >>> solve_triangulated(F, x, y, z) + [(0, 0, 1), (0, 1, 0), (1, 0, 0)] + + Using extension for algebraic solutions. + + >>> solve_triangulated(F, x, y, z, extension=True) #doctest: +NORMALIZE_WHITESPACE + [(0, 0, 1), (0, 1, 0), (1, 0, 0), + (CRootOf(x**2 + 2*x - 1, 0), CRootOf(x**2 + 2*x - 1, 0), CRootOf(x**2 + 2*x - 1, 0)), + (CRootOf(x**2 + 2*x - 1, 1), CRootOf(x**2 + 2*x - 1, 1), CRootOf(x**2 + 2*x - 1, 1))] + + References + ========== + + 1. Patrizia Gianni, Teo Mora, Algebraic Solution of System of + Polynomial Equations using Groebner Bases, AAECC-5 on Applied Algebra, + Algebraic Algorithms and Error-Correcting Codes, LNCS 356 247--257, 1989 + + """ + opt = build_options(gens, args) + + G = groebner(polys, gens, polys=True) + G = list(reversed(G)) + + extension = opt.get('extension', False) + if extension: + def _solve_univariate(f): + return [r for r, _ in f.all_roots(multiple=False, radicals=False)] + else: + domain = opt.get('domain') + + if domain is not None: + for i, g in enumerate(G): + G[i] = g.set_domain(domain) + + def _solve_univariate(f): + return list(f.ground_roots().keys()) + + f, G = G[0].ltrim(-1), G[1:] + dom = f.get_domain() + + zeros = _solve_univariate(f) + + if extension: + solutions = {((zero,), dom.algebraic_field(zero)) for zero in zeros} + else: + solutions = {((zero,), dom) for zero in zeros} + + var_seq = reversed(gens[:-1]) + vars_seq = postfixes(gens[1:]) + + for var, vars in zip(var_seq, vars_seq): + _solutions = set() + + for values, dom in solutions: + H, mapping = [], list(zip(vars, values)) + + for g in G: + _vars = (var,) + vars + + if g.has_only_gens(*_vars) and g.degree(var) != 0: + if extension: + g = g.set_domain(g.domain.unify(dom)) + h = g.ltrim(var).eval(dict(mapping)) + + if g.degree(var) == h.degree(): + H.append(h) + + p = min(H, key=lambda h: h.degree()) + zeros = _solve_univariate(p) + + for zero in zeros: + if not (zero in dom): + dom_zero = dom.algebraic_field(zero) + else: + dom_zero = dom + + _solutions.add(((zero,) + values, dom_zero)) + + solutions = _solutions + return sorted((s for s, _ in solutions), key=default_sort_key) + + +def factor_system(eqs: Sequence[Expr | complex], gens: Sequence[Expr] = (), **kwargs: Any) -> list[list[Expr]]: + """ + Factorizes a system of polynomial equations into + irreducible subsystems. + + Parameters + ========== + + eqs : list + List of expressions to be factored. + Each expression is assumed to be equal to zero. + + gens : list, optional + Generator(s) of the polynomial ring. + If not provided, all free symbols will be used. + + **kwargs : dict, optional + Same optional arguments taken by ``factor`` + + Returns + ======= + + list[list[Expr]] + A list of lists of expressions, where each sublist represents + an irreducible subsystem. When solved, each subsystem gives + one component of the solution. Only generic solutions are + returned (cases not requiring parameters to be zero). + + Examples + ======== + + >>> from sympy.solvers.polysys import factor_system, factor_system_cond + >>> from sympy.abc import x, y, a, b, c + + A simple system with multiple solutions: + + >>> factor_system([x**2 - 1, y - 1]) + [[x + 1, y - 1], [x - 1, y - 1]] + + A system with no solution: + + >>> factor_system([x, 1]) + [] + + A system where any value of the symbol(s) is a solution: + + >>> factor_system([x - x, (x + 1)**2 - (x**2 + 2*x + 1)]) + [[]] + + A system with no generic solution: + + >>> factor_system([a*x*(x-1), b*y, c], [x, y]) + [] + + If c is added to the unknowns then the system has a generic solution: + + >>> factor_system([a*x*(x-1), b*y, c], [x, y, c]) + [[x - 1, y, c], [x, y, c]] + + Alternatively :func:`factor_system_cond` can be used to get degenerate + cases as well: + + >>> factor_system_cond([a*x*(x-1), b*y, c], [x, y]) + [[x - 1, y, c], [x, y, c], [x - 1, b, c], [x, b, c], [y, a, c], [a, b, c]] + + Each of the above cases is only satisfiable in the degenerate case `c = 0`. + + The solution set of the original system represented + by eqs is the union of the solution sets of the + factorized systems. + + An empty list [] means no generic solution exists. + A list containing an empty list [[]] means any value of + the symbol(s) is a solution. + + See Also + ======== + + factor_system_cond : Returns both generic and degenerate solutions + factor_system_bool : Returns a Boolean combination representing all solutions + sympy.polys.polytools.factor : Factors a polynomial into irreducible factors + over the rational numbers + """ + + systems = _factor_system_poly_from_expr(eqs, gens, **kwargs) + systems_generic = [sys for sys in systems if not _is_degenerate(sys)] + systems_expr = [[p.as_expr() for p in system] for system in systems_generic] + return systems_expr + + +def _is_degenerate(system: list[Poly]) -> bool: + """Helper function to check if a system is degenerate""" + return any(p.is_ground for p in system) + + +def factor_system_bool(eqs: Sequence[Expr | complex], gens: Sequence[Expr] = (), **kwargs: Any) -> Boolean: + """ + Factorizes a system of polynomial equations into irreducible DNF. + + The system of expressions(eqs) is taken and a Boolean combination + of equations is returned that represents the same solution set. + The result is in disjunctive normal form (OR of ANDs). + + Parameters + ========== + + eqs : list + List of expressions to be factored. + Each expression is assumed to be equal to zero. + + gens : list, optional + Generator(s) of the polynomial ring. + If not provided, all free symbols will be used. + + **kwargs : dict, optional + Optional keyword arguments + + + Returns + ======= + + Boolean: + A Boolean combination of equations. The result is typically in + the form of a conjunction (AND) of a disjunctive normal form + with additional conditions. + + Examples + ======== + + >>> from sympy.solvers.polysys import factor_system_bool + >>> from sympy.abc import x, y, a, b, c + >>> factor_system_bool([x**2 - 1]) + Eq(x - 1, 0) | Eq(x + 1, 0) + + >>> factor_system_bool([x**2 - 1, y - 1]) + (Eq(x - 1, 0) & Eq(y - 1, 0)) | (Eq(x + 1, 0) & Eq(y - 1, 0)) + + >>> eqs = [a * (x - 1), b] + >>> factor_system_bool([a*(x - 1), b]) + (Eq(a, 0) & Eq(b, 0)) | (Eq(b, 0) & Eq(x - 1, 0)) + + >>> factor_system_bool([a*x**2 - a, b*(x + 1), c], [x]) + (Eq(c, 0) & Eq(x + 1, 0)) | (Eq(a, 0) & Eq(b, 0) & Eq(c, 0)) | (Eq(b, 0) & Eq(c, 0) & Eq(x - 1, 0)) + + >>> factor_system_bool([x**2 + 2*x + 1 - (x + 1)**2]) + True + + The result is logically equivalent to the system of equations + i.e. eqs. The function returns ``True`` when all values of + the symbol(s) is a solution and ``False`` when the system + cannot be solved. + + See Also + ======== + + factor_system : Returns factors and solvability condition separately + factor_system_cond : Returns both factors and conditions + + """ + + systems = factor_system_cond(eqs, gens, **kwargs) + return Or(*[And(*[Eq(eq, 0) for eq in sys]) for sys in systems]) + + +def factor_system_cond(eqs: Sequence[Expr | complex], gens: Sequence[Expr] = (), **kwargs: Any) -> list[list[Expr]]: + """ + Factorizes a polynomial system into irreducible components and returns + both generic and degenerate solutions. + + Parameters + ========== + + eqs : list + List of expressions to be factored. + Each expression is assumed to be equal to zero. + + gens : list, optional + Generator(s) of the polynomial ring. + If not provided, all free symbols will be used. + + **kwargs : dict, optional + Optional keyword arguments. + + Returns + ======= + + list[list[Expr]] + A list of lists of expressions, where each sublist represents + an irreducible subsystem. Includes both generic solutions and + degenerate cases requiring equality conditions on parameters. + + Examples + ======== + + >>> from sympy.solvers.polysys import factor_system_cond + >>> from sympy.abc import x, y, a, b, c + + >>> factor_system_cond([x**2 - 4, a*y, b], [x, y]) + [[x + 2, y, b], [x - 2, y, b], [x + 2, a, b], [x - 2, a, b]] + + >>> factor_system_cond([a*x*(x-1), b*y, c], [x, y]) + [[x - 1, y, c], [x, y, c], [x - 1, b, c], [x, b, c], [y, a, c], [a, b, c]] + + An empty list [] means no solution exists. + A list containing an empty list [[]] means any value of + the symbol(s) is a solution. + + See Also + ======== + + factor_system : Returns only generic solutions + factor_system_bool : Returns a Boolean combination representing all solutions + sympy.polys.polytools.factor : Factors a polynomial into irreducible factors + over the rational numbers + """ + systems_poly = _factor_system_poly_from_expr(eqs, gens, **kwargs) + systems = [[p.as_expr() for p in system] for system in systems_poly] + return systems + + +def _factor_system_poly_from_expr( + eqs: Sequence[Expr | complex], gens: Sequence[Expr], **kwargs: Any +) -> list[list[Poly]]: + """ + Convert expressions to polynomials and factor the system. + + Takes a sequence of expressions, converts them to + polynomials, and factors the resulting system. Handles both regular + polynomial systems and purely numerical cases. + """ + try: + polys, opts = parallel_poly_from_expr(eqs, *gens, **kwargs) + only_numbers = False + except (GeneratorsNeeded, PolificationFailed): + _u = Dummy('u') + polys, opts = parallel_poly_from_expr(eqs, [_u], **kwargs) + assert opts['domain'].is_Numerical + only_numbers = True + + if only_numbers: + return [[]] if all(p == 0 for p in polys) else [] + + return factor_system_poly(polys) + + +def factor_system_poly(polys: list[Poly]) -> list[list[Poly]]: + """ + Factors a system of polynomial equations into irreducible subsystems + + Core implementation that works directly with Poly instances. + + Parameters + ========== + + polys : list[Poly] + A list of Poly instances to be factored. + + Returns + ======= + + list[list[Poly]] + A list of lists of polynomials, where each sublist represents + an irreducible component of the solution. Includes both + generic and degenerate cases. + + Examples + ======== + + >>> from sympy import symbols, Poly, ZZ + >>> from sympy.solvers.polysys import factor_system_poly + >>> a, b, c, x = symbols('a b c x') + >>> p1 = Poly((a - 1)*(x - 2), x, domain=ZZ[a,b,c]) + >>> p2 = Poly((b - 3)*(x - 2), x, domain=ZZ[a,b,c]) + >>> p3 = Poly(c, x, domain=ZZ[a,b,c]) + + The equation to be solved for x is ``x - 2 = 0`` provided either + of the two conditions on the parameters ``a`` and ``b`` is nonzero + and the constant parameter ``c`` should be zero. + + >>> sys1, sys2 = factor_system_poly([p1, p2, p3]) + >>> sys1 + [Poly(x - 2, x, domain='ZZ[a,b,c]'), + Poly(c, x, domain='ZZ[a,b,c]')] + >>> sys2 + [Poly(a - 1, x, domain='ZZ[a,b,c]'), + Poly(b - 3, x, domain='ZZ[a,b,c]'), + Poly(c, x, domain='ZZ[a,b,c]')] + + An empty list [] when returned means no solution exists. + Whereas a list containing an empty list [[]] means any value is a solution. + + See Also + ======== + + factor_system : Returns only generic solutions + factor_system_bool : Returns a Boolean combination representing the solutions + factor_system_cond : Returns both generic and degenerate solutions + sympy.polys.polytools.factor : Factors a polynomial into irreducible factors + over the rational numbers + """ + if not all(isinstance(poly, Poly) for poly in polys): + raise TypeError("polys should be a list of Poly instances") + if not polys: + return [[]] + + base_domain = polys[0].domain + base_gens = polys[0].gens + if not all(poly.domain == base_domain and poly.gens == base_gens for poly in polys[1:]): + raise DomainError("All polynomials must have the same domain and generators") + + factor_sets = [] + for poly in polys: + constant, factors_mult = poly.factor_list() + + if constant.is_zero is True: + continue + elif constant.is_zero is False: + if not factors_mult: + return [] + factor_sets.append([f for f, _ in factors_mult]) + else: + constant = sqf_part(factor_terms(constant).as_coeff_Mul()[1]) + constp = Poly(constant, base_gens, domain=base_domain) + factors = [f for f, _ in factors_mult] + factors.append(constp) + factor_sets.append(factors) + + if not factor_sets: + return [[]] + + result = _factor_sets(factor_sets) + return _sort_systems(result) + + +def _factor_sets_slow(eqs: list[list]) -> set[frozenset]: + """ + Helper to find the minimal set of factorised subsystems that is + equivalent to the original system. + + The result is in DNF. + """ + if not eqs: + return {frozenset()} + systems_set = {frozenset(sys) for sys in cartes(*eqs)} + return {s1 for s1 in systems_set if not any(s1 > s2 for s2 in systems_set)} + + +def _factor_sets(eqs: list[list]) -> set[frozenset]: + """ + Helper that builds factor combinations. + """ + if not eqs: + return {frozenset()} + + current_set = min(eqs, key=len) + other_sets = [s for s in eqs if s is not current_set] + + stack = [(factor, [s for s in other_sets if factor not in s], {factor}) + for factor in current_set] + + result = set() + + while stack: + factor, remaining_sets, current_solution = stack.pop() + + if not remaining_sets: + result.add(frozenset(current_solution)) + continue + + next_set = min(remaining_sets, key=len) + next_remaining = [s for s in remaining_sets if s is not next_set] + + for next_factor in next_set: + valid_remaining = [s for s in next_remaining if next_factor not in s] + new_solution = current_solution | {next_factor} + stack.append((next_factor, valid_remaining, new_solution)) + + return {s1 for s1 in result if not any(s1 > s2 for s2 in result)} + + +def _sort_systems(systems: Iterable[Iterable[Poly]]) -> list[list[Poly]]: + """Sorts a list of lists of polynomials""" + systems_list = [sorted(s, key=_poly_sort_key, reverse=True) for s in systems] + return sorted(systems_list, key=_sys_sort_key, reverse=True) + + +def _poly_sort_key(poly): + """Sort key for polynomials""" + if poly.domain.is_FF: + poly = poly.set_domain(ZZ) + return poly.degree_list(), poly.rep.to_list() + + +def _sys_sort_key(sys): + """Sort key for lists of polynomials""" + return list(zip(*map(_poly_sort_key, sys))) diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/recurr.py b/.venv/lib/python3.12/site-packages/sympy/solvers/recurr.py new file mode 100644 index 0000000000000000000000000000000000000000..ba627bbd4cb0844f11a8743634f5f10328aadca8 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/recurr.py @@ -0,0 +1,843 @@ +r""" +This module is intended for solving recurrences or, in other words, +difference equations. Currently supported are linear, inhomogeneous +equations with polynomial or rational coefficients. + +The solutions are obtained among polynomials, rational functions, +hypergeometric terms, or combinations of hypergeometric term which +are pairwise dissimilar. + +``rsolve_X`` functions were meant as a low level interface +for ``rsolve`` which would use Mathematica's syntax. + +Given a recurrence relation: + + .. math:: a_{k}(n) y(n+k) + a_{k-1}(n) y(n+k-1) + + ... + a_{0}(n) y(n) = f(n) + +where `k > 0` and `a_{i}(n)` are polynomials in `n`. To use +``rsolve_X`` we need to put all coefficients in to a list ``L`` of +`k+1` elements the following way: + + ``L = [a_{0}(n), ..., a_{k-1}(n), a_{k}(n)]`` + +where ``L[i]``, for `i=0, \ldots, k`, maps to +`a_{i}(n) y(n+i)` (`y(n+i)` is implicit). + +For example if we would like to compute `m`-th Bernoulli polynomial +up to a constant (example was taken from rsolve_poly docstring), +then we would use `b(n+1) - b(n) = m n^{m-1}` recurrence, which +has solution `b(n) = B_m + C`. + +Then ``L = [-1, 1]`` and `f(n) = m n^(m-1)` and finally for `m=4`: + +>>> from sympy import Symbol, bernoulli, rsolve_poly +>>> n = Symbol('n', integer=True) + +>>> rsolve_poly([-1, 1], 4*n**3, n) +C0 + n**4 - 2*n**3 + n**2 + +>>> bernoulli(4, n) +n**4 - 2*n**3 + n**2 - 1/30 + +For the sake of completeness, `f(n)` can be: + + [1] a polynomial -> rsolve_poly + [2] a rational function -> rsolve_ratio + [3] a hypergeometric function -> rsolve_hyper +""" +from collections import defaultdict + +from sympy.concrete import product +from sympy.core.singleton import S +from sympy.core.numbers import Rational, I +from sympy.core.symbol import Symbol, Wild, Dummy +from sympy.core.relational import Equality +from sympy.core.add import Add +from sympy.core.mul import Mul +from sympy.core.sorting import default_sort_key +from sympy.core.sympify import sympify + +from sympy.simplify import simplify, hypersimp, hypersimilar # type: ignore +from sympy.solvers import solve, solve_undetermined_coeffs +from sympy.polys import Poly, quo, gcd, lcm, roots, resultant +from sympy.functions import binomial, factorial, FallingFactorial, RisingFactorial +from sympy.matrices import Matrix, casoratian +from sympy.utilities.iterables import numbered_symbols + + +def rsolve_poly(coeffs, f, n, shift=0, **hints): + r""" + Given linear recurrence operator `\operatorname{L}` of order + `k` with polynomial coefficients and inhomogeneous equation + `\operatorname{L} y = f`, where `f` is a polynomial, we seek for + all polynomial solutions over field `K` of characteristic zero. + + The algorithm performs two basic steps: + + (1) Compute degree `N` of the general polynomial solution. + (2) Find all polynomials of degree `N` or less + of `\operatorname{L} y = f`. + + There are two methods for computing the polynomial solutions. + If the degree bound is relatively small, i.e. it's smaller than + or equal to the order of the recurrence, then naive method of + undetermined coefficients is being used. This gives a system + of algebraic equations with `N+1` unknowns. + + In the other case, the algorithm performs transformation of the + initial equation to an equivalent one for which the system of + algebraic equations has only `r` indeterminates. This method is + quite sophisticated (in comparison with the naive one) and was + invented together by Abramov, Bronstein and Petkovsek. + + It is possible to generalize the algorithm implemented here to + the case of linear q-difference and differential equations. + + Lets say that we would like to compute `m`-th Bernoulli polynomial + up to a constant. For this we can use `b(n+1) - b(n) = m n^{m-1}` + recurrence, which has solution `b(n) = B_m + C`. For example: + + >>> from sympy import Symbol, rsolve_poly + >>> n = Symbol('n', integer=True) + + >>> rsolve_poly([-1, 1], 4*n**3, n) + C0 + n**4 - 2*n**3 + n**2 + + References + ========== + + .. [1] S. A. Abramov, M. Bronstein and M. Petkovsek, On polynomial + solutions of linear operator equations, in: T. Levelt, ed., + Proc. ISSAC '95, ACM Press, New York, 1995, 290-296. + + .. [2] M. Petkovsek, Hypergeometric solutions of linear recurrences + with polynomial coefficients, J. Symbolic Computation, + 14 (1992), 243-264. + + .. [3] M. Petkovsek, H. S. Wilf, D. Zeilberger, A = B, 1996. + + """ + f = sympify(f) + + if not f.is_polynomial(n): + return None + + homogeneous = f.is_zero + + r = len(coeffs) - 1 + + coeffs = [Poly(coeff, n) for coeff in coeffs] + + polys = [Poly(0, n)]*(r + 1) + terms = [(S.Zero, S.NegativeInfinity)]*(r + 1) + + for i in range(r + 1): + for j in range(i, r + 1): + polys[i] += coeffs[j]*(binomial(j, i).as_poly(n)) + + if not polys[i].is_zero: + (exp,), coeff = polys[i].LT() + terms[i] = (coeff, exp) + + d = b = terms[0][1] + + for i in range(1, r + 1): + if terms[i][1] > d: + d = terms[i][1] + + if terms[i][1] - i > b: + b = terms[i][1] - i + + d, b = int(d), int(b) + + x = Dummy('x') + + degree_poly = S.Zero + + for i in range(r + 1): + if terms[i][1] - i == b: + degree_poly += terms[i][0]*FallingFactorial(x, i) + + nni_roots = list(roots(degree_poly, x, filter='Z', + predicate=lambda r: r >= 0).keys()) + + if nni_roots: + N = [max(nni_roots)] + else: + N = [] + + if homogeneous: + N += [-b - 1] + else: + N += [f.as_poly(n).degree() - b, -b - 1] + + N = int(max(N)) + + if N < 0: + if homogeneous: + if hints.get('symbols', False): + return (S.Zero, []) + else: + return S.Zero + else: + return None + + if N <= r: + C = [] + y = E = S.Zero + + for i in range(N + 1): + C.append(Symbol('C' + str(i + shift))) + y += C[i] * n**i + + for i in range(r + 1): + E += coeffs[i].as_expr()*y.subs(n, n + i) + + solutions = solve_undetermined_coeffs(E - f, C, n) + + if solutions is not None: + _C = C + C = [c for c in C if (c not in solutions)] + result = y.subs(solutions) + else: + return None # TBD + else: + A = r + U = N + A + b + 1 + + nni_roots = list(roots(polys[r], filter='Z', + predicate=lambda r: r >= 0).keys()) + + if nni_roots != []: + a = max(nni_roots) + 1 + else: + a = S.Zero + + def _zero_vector(k): + return [S.Zero] * k + + def _one_vector(k): + return [S.One] * k + + def _delta(p, k): + B = S.One + D = p.subs(n, a + k) + + for i in range(1, k + 1): + B *= Rational(i - k - 1, i) + D += B * p.subs(n, a + k - i) + + return D + + alpha = {} + + for i in range(-A, d + 1): + I = _one_vector(d + 1) + + for k in range(1, d + 1): + I[k] = I[k - 1] * (x + i - k + 1)/k + + alpha[i] = S.Zero + + for j in range(A + 1): + for k in range(d + 1): + B = binomial(k, i + j) + D = _delta(polys[j].as_expr(), k) + + alpha[i] += I[k]*B*D + + V = Matrix(U, A, lambda i, j: int(i == j)) + + if homogeneous: + for i in range(A, U): + v = _zero_vector(A) + + for k in range(1, A + b + 1): + if i - k < 0: + break + + B = alpha[k - A].subs(x, i - k) + + for j in range(A): + v[j] += B * V[i - k, j] + + denom = alpha[-A].subs(x, i) + + for j in range(A): + V[i, j] = -v[j] / denom + else: + G = _zero_vector(U) + + for i in range(A, U): + v = _zero_vector(A) + g = S.Zero + + for k in range(1, A + b + 1): + if i - k < 0: + break + + B = alpha[k - A].subs(x, i - k) + + for j in range(A): + v[j] += B * V[i - k, j] + + g += B * G[i - k] + + denom = alpha[-A].subs(x, i) + + for j in range(A): + V[i, j] = -v[j] / denom + + G[i] = (_delta(f, i - A) - g) / denom + + P, Q = _one_vector(U), _zero_vector(A) + + for i in range(1, U): + P[i] = (P[i - 1] * (n - a - i + 1)/i).expand() + + for i in range(A): + Q[i] = Add(*[(v*p).expand() for v, p in zip(V[:, i], P)]) + + if not homogeneous: + h = Add(*[(g*p).expand() for g, p in zip(G, P)]) + + C = [Symbol('C' + str(i + shift)) for i in range(A)] + + g = lambda i: Add(*[c*_delta(q, i) for c, q in zip(C, Q)]) + + if homogeneous: + E = [g(i) for i in range(N + 1, U)] + else: + E = [g(i) + _delta(h, i) for i in range(N + 1, U)] + + if E != []: + solutions = solve(E, *C) + + if not solutions: + if homogeneous: + if hints.get('symbols', False): + return (S.Zero, []) + else: + return S.Zero + else: + return None + else: + solutions = {} + + if homogeneous: + result = S.Zero + else: + result = h + + _C = C[:] + for c, q in list(zip(C, Q)): + if c in solutions: + s = solutions[c]*q + C.remove(c) + else: + s = c*q + + result += s.expand() + + if C != _C: + # renumber so they are contiguous + result = result.xreplace(dict(zip(C, _C))) + C = _C[:len(C)] + + if hints.get('symbols', False): + return (result, C) + else: + return result + + +def rsolve_ratio(coeffs, f, n, **hints): + r""" + Given linear recurrence operator `\operatorname{L}` of order `k` + with polynomial coefficients and inhomogeneous equation + `\operatorname{L} y = f`, where `f` is a polynomial, we seek + for all rational solutions over field `K` of characteristic zero. + + This procedure accepts only polynomials, however if you are + interested in solving recurrence with rational coefficients + then use ``rsolve`` which will pre-process the given equation + and run this procedure with polynomial arguments. + + The algorithm performs two basic steps: + + (1) Compute polynomial `v(n)` which can be used as universal + denominator of any rational solution of equation + `\operatorname{L} y = f`. + + (2) Construct new linear difference equation by substitution + `y(n) = u(n)/v(n)` and solve it for `u(n)` finding all its + polynomial solutions. Return ``None`` if none were found. + + The algorithm implemented here is a revised version of the original + Abramov's algorithm, developed in 1989. The new approach is much + simpler to implement and has better overall efficiency. This + method can be easily adapted to the q-difference equations case. + + Besides finding rational solutions alone, this functions is + an important part of Hyper algorithm where it is used to find + a particular solution for the inhomogeneous part of a recurrence. + + Examples + ======== + + >>> from sympy.abc import x + >>> from sympy.solvers.recurr import rsolve_ratio + >>> rsolve_ratio([-2*x**3 + x**2 + 2*x - 1, 2*x**3 + x**2 - 6*x, + ... - 2*x**3 - 11*x**2 - 18*x - 9, 2*x**3 + 13*x**2 + 22*x + 8], 0, x) + C0*(2*x - 3)/(2*(x**2 - 1)) + + References + ========== + + .. [1] S. A. Abramov, Rational solutions of linear difference + and q-difference equations with polynomial coefficients, + in: T. Levelt, ed., Proc. ISSAC '95, ACM Press, New York, + 1995, 285-289 + + See Also + ======== + + rsolve_hyper + """ + f = sympify(f) + + if not f.is_polynomial(n): + return None + + coeffs = list(map(sympify, coeffs)) + + r = len(coeffs) - 1 + + A, B = coeffs[r], coeffs[0] + A = A.subs(n, n - r).expand() + + h = Dummy('h') + + res = resultant(A, B.subs(n, n + h), n) + + if not res.is_polynomial(h): + p, q = res.as_numer_denom() + res = quo(p, q, h) + + nni_roots = list(roots(res, h, filter='Z', + predicate=lambda r: r >= 0).keys()) + + if not nni_roots: + return rsolve_poly(coeffs, f, n, **hints) + else: + C, numers = S.One, [S.Zero]*(r + 1) + + for i in range(int(max(nni_roots)), -1, -1): + d = gcd(A, B.subs(n, n + i), n) + + A = quo(A, d, n) + B = quo(B, d.subs(n, n - i), n) + + C *= Mul(*[d.subs(n, n - j) for j in range(i + 1)]) + + denoms = [C.subs(n, n + i) for i in range(r + 1)] + + for i in range(r + 1): + g = gcd(coeffs[i], denoms[i], n) + + numers[i] = quo(coeffs[i], g, n) + denoms[i] = quo(denoms[i], g, n) + + for i in range(r + 1): + numers[i] *= Mul(*(denoms[:i] + denoms[i + 1:])) + + result = rsolve_poly(numers, f * Mul(*denoms), n, **hints) + + if result is not None: + if hints.get('symbols', False): + return (simplify(result[0] / C), result[1]) + else: + return simplify(result / C) + else: + return None + + +def rsolve_hyper(coeffs, f, n, **hints): + r""" + Given linear recurrence operator `\operatorname{L}` of order `k` + with polynomial coefficients and inhomogeneous equation + `\operatorname{L} y = f` we seek for all hypergeometric solutions + over field `K` of characteristic zero. + + The inhomogeneous part can be either hypergeometric or a sum + of a fixed number of pairwise dissimilar hypergeometric terms. + + The algorithm performs three basic steps: + + (1) Group together similar hypergeometric terms in the + inhomogeneous part of `\operatorname{L} y = f`, and find + particular solution using Abramov's algorithm. + + (2) Compute generating set of `\operatorname{L}` and find basis + in it, so that all solutions are linearly independent. + + (3) Form final solution with the number of arbitrary + constants equal to dimension of basis of `\operatorname{L}`. + + Term `a(n)` is hypergeometric if it is annihilated by first order + linear difference equations with polynomial coefficients or, in + simpler words, if consecutive term ratio is a rational function. + + The output of this procedure is a linear combination of fixed + number of hypergeometric terms. However the underlying method + can generate larger class of solutions - D'Alembertian terms. + + Note also that this method not only computes the kernel of the + inhomogeneous equation, but also reduces in to a basis so that + solutions generated by this procedure are linearly independent + + Examples + ======== + + >>> from sympy.solvers import rsolve_hyper + >>> from sympy.abc import x + + >>> rsolve_hyper([-1, -1, 1], 0, x) + C0*(1/2 - sqrt(5)/2)**x + C1*(1/2 + sqrt(5)/2)**x + + >>> rsolve_hyper([-1, 1], 1 + x, x) + C0 + x*(x + 1)/2 + + References + ========== + + .. [1] M. Petkovsek, Hypergeometric solutions of linear recurrences + with polynomial coefficients, J. Symbolic Computation, + 14 (1992), 243-264. + + .. [2] M. Petkovsek, H. S. Wilf, D. Zeilberger, A = B, 1996. + """ + coeffs = list(map(sympify, coeffs)) + + f = sympify(f) + + r, kernel, symbols = len(coeffs) - 1, [], set() + + if not f.is_zero: + if f.is_Add: + similar = {} + + for g in f.expand().args: + if not g.is_hypergeometric(n): + return None + + for h in similar.keys(): + if hypersimilar(g, h, n): + similar[h] += g + break + else: + similar[g] = S.Zero + + inhomogeneous = [g + h for g, h in similar.items()] + elif f.is_hypergeometric(n): + inhomogeneous = [f] + else: + return None + + for i, g in enumerate(inhomogeneous): + coeff, polys = S.One, coeffs[:] + denoms = [S.One]*(r + 1) + + s = hypersimp(g, n) + + for j in range(1, r + 1): + coeff *= s.subs(n, n + j - 1) + + p, q = coeff.as_numer_denom() + + polys[j] *= p + denoms[j] = q + + for j in range(r + 1): + polys[j] *= Mul(*(denoms[:j] + denoms[j + 1:])) + + # FIXME: The call to rsolve_ratio below should suffice (rsolve_poly + # call can be removed) but the XFAIL test_rsolve_ratio_missed must + # be fixed first. + R = rsolve_ratio(polys, Mul(*denoms), n, symbols=True) + if R is not None: + R, syms = R + if syms: + R = R.subs(zip(syms, [0]*len(syms))) + else: + R = rsolve_poly(polys, Mul(*denoms), n) + + if R: + inhomogeneous[i] *= R + else: + return None + + result = Add(*inhomogeneous) + result = simplify(result) + else: + result = S.Zero + + Z = Dummy('Z') + + p, q = coeffs[0], coeffs[r].subs(n, n - r + 1) + + p_factors = list(roots(p, n).keys()) + q_factors = list(roots(q, n).keys()) + + factors = [(S.One, S.One)] + + for p in p_factors: + for q in q_factors: + if p.is_integer and q.is_integer and p <= q: + continue + else: + factors += [(n - p, n - q)] + + p = [(n - p, S.One) for p in p_factors] + q = [(S.One, n - q) for q in q_factors] + + factors = p + factors + q + + for A, B in factors: + polys, degrees = [], [] + D = A*B.subs(n, n + r - 1) + + for i in range(r + 1): + a = Mul(*[A.subs(n, n + j) for j in range(i)]) + b = Mul(*[B.subs(n, n + j) for j in range(i, r)]) + + poly = quo(coeffs[i]*a*b, D, n) + polys.append(poly.as_poly(n)) + + if not poly.is_zero: + degrees.append(polys[i].degree()) + + if degrees: + d, poly = max(degrees), S.Zero + else: + return None + + for i in range(r + 1): + coeff = polys[i].nth(d) + + if coeff is not S.Zero: + poly += coeff * Z**i + + for z in roots(poly, Z).keys(): + if z.is_zero: + continue + + recurr_coeffs = [polys[i].as_expr()*z**i for i in range(r + 1)] + if d == 0 and 0 != Add(*[recurr_coeffs[j]*j for j in range(1, r + 1)]): + # faster inline check (than calling rsolve_poly) for a + # constant solution to a constant coefficient recurrence. + sol = [Symbol("C" + str(len(symbols)))] + else: + sol, syms = rsolve_poly(recurr_coeffs, 0, n, len(symbols), symbols=True) + sol = sol.collect(syms) + sol = [sol.coeff(s) for s in syms] + + for C in sol: + ratio = z * A * C.subs(n, n + 1) / B / C + ratio = simplify(ratio) + # If there is a nonnegative root in the denominator of the ratio, + # this indicates that the term y(n_root) is zero, and one should + # start the product with the term y(n_root + 1). + n0 = 0 + for n_root in roots(ratio.as_numer_denom()[1], n).keys(): + if n_root.has(I): + return None + elif (n0 < (n_root + 1)) == True: + n0 = n_root + 1 + K = product(ratio, (n, n0, n - 1)) + if K.has(factorial, FallingFactorial, RisingFactorial): + K = simplify(K) + + if casoratian(kernel + [K], n, zero=False) != 0: + kernel.append(K) + + kernel.sort(key=default_sort_key) + sk = list(zip(numbered_symbols('C'), kernel)) + + for C, ker in sk: + result += C * ker + + if hints.get('symbols', False): + # XXX: This returns the symbols in a non-deterministic order + symbols |= {s for s, k in sk} + return (result, list(symbols)) + else: + return result + + +def rsolve(f, y, init=None): + r""" + Solve univariate recurrence with rational coefficients. + + Given `k`-th order linear recurrence `\operatorname{L} y = f`, + or equivalently: + + .. math:: a_{k}(n) y(n+k) + a_{k-1}(n) y(n+k-1) + + \cdots + a_{0}(n) y(n) = f(n) + + where `a_{i}(n)`, for `i=0, \ldots, k`, are polynomials or rational + functions in `n`, and `f` is a hypergeometric function or a sum + of a fixed number of pairwise dissimilar hypergeometric terms in + `n`, finds all solutions or returns ``None``, if none were found. + + Initial conditions can be given as a dictionary in two forms: + + (1) ``{ n_0 : v_0, n_1 : v_1, ..., n_m : v_m}`` + (2) ``{y(n_0) : v_0, y(n_1) : v_1, ..., y(n_m) : v_m}`` + + or as a list ``L`` of values: + + ``L = [v_0, v_1, ..., v_m]`` + + where ``L[i] = v_i``, for `i=0, \ldots, m`, maps to `y(n_i)`. + + Examples + ======== + + Lets consider the following recurrence: + + .. math:: (n - 1) y(n + 2) - (n^2 + 3 n - 2) y(n + 1) + + 2 n (n + 1) y(n) = 0 + + >>> from sympy import Function, rsolve + >>> from sympy.abc import n + >>> y = Function('y') + + >>> f = (n - 1)*y(n + 2) - (n**2 + 3*n - 2)*y(n + 1) + 2*n*(n + 1)*y(n) + + >>> rsolve(f, y(n)) + 2**n*C0 + C1*factorial(n) + + >>> rsolve(f, y(n), {y(0):0, y(1):3}) + 3*2**n - 3*factorial(n) + + See Also + ======== + + rsolve_poly, rsolve_ratio, rsolve_hyper + + """ + if isinstance(f, Equality): + f = f.lhs - f.rhs + + n = y.args[0] + k = Wild('k', exclude=(n,)) + + # Preprocess user input to allow things like + # y(n) + a*(y(n + 1) + y(n - 1))/2 + f = f.expand().collect(y.func(Wild('m', integer=True))) + + h_part = defaultdict(list) + i_part = [] + for g in Add.make_args(f): + coeff, dep = g.as_coeff_mul(y.func) + if not dep: + i_part.append(coeff) + continue + for h in dep: + if h.is_Function and h.func == y.func: + result = h.args[0].match(n + k) + if result is not None: + h_part[int(result[k])].append(coeff) + continue + raise ValueError( + "'%s(%s + k)' expected, got '%s'" % (y.func, n, h)) + for k in h_part: + h_part[k] = Add(*h_part[k]) + h_part.default_factory = lambda: 0 + i_part = Add(*i_part) + + for k, coeff in h_part.items(): + h_part[k] = simplify(coeff) + + common = S.One + + if not i_part.is_zero and not i_part.is_hypergeometric(n) and \ + not (i_part.is_Add and all((x.is_hypergeometric(n) for x in i_part.expand().args))): + raise ValueError("The independent term should be a sum of hypergeometric functions, got '%s'" % i_part) + + for coeff in h_part.values(): + if coeff.is_rational_function(n): + if not coeff.is_polynomial(n): + common = lcm(common, coeff.as_numer_denom()[1], n) + else: + raise ValueError( + "Polynomial or rational function expected, got '%s'" % coeff) + + i_numer, i_denom = i_part.as_numer_denom() + + if i_denom.is_polynomial(n): + common = lcm(common, i_denom, n) + + if common is not S.One: + for k, coeff in h_part.items(): + numer, denom = coeff.as_numer_denom() + h_part[k] = numer*quo(common, denom, n) + + i_part = i_numer*quo(common, i_denom, n) + + K_min = min(h_part.keys()) + + if K_min < 0: + K = abs(K_min) + + H_part = defaultdict(lambda: S.Zero) + i_part = i_part.subs(n, n + K).expand() + common = common.subs(n, n + K).expand() + + for k, coeff in h_part.items(): + H_part[k + K] = coeff.subs(n, n + K).expand() + else: + H_part = h_part + + K_max = max(H_part.keys()) + coeffs = [H_part[i] for i in range(K_max + 1)] + + result = rsolve_hyper(coeffs, -i_part, n, symbols=True) + + if result is None: + return None + + solution, symbols = result + + if init in ({}, []): + init = None + + if symbols and init is not None: + if isinstance(init, list): + init = {i: init[i] for i in range(len(init))} + + equations = [] + + for k, v in init.items(): + try: + i = int(k) + except TypeError: + if k.is_Function and k.func == y.func: + i = int(k.args[0]) + else: + raise ValueError("Integer or term expected, got '%s'" % k) + + eq = solution.subs(n, i) - v + if eq.has(S.NaN): + eq = solution.limit(n, i) - v + equations.append(eq) + + result = solve(equations, *symbols) + + if not result: + return None + else: + solution = solution.subs(result) + + return solution diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/simplex.py b/.venv/lib/python3.12/site-packages/sympy/solvers/simplex.py new file mode 100644 index 0000000000000000000000000000000000000000..c8e652cb626507d7829f9bc1c78fc6f49809865f --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/simplex.py @@ -0,0 +1,1104 @@ +"""Tools for optimizing a linear function for a given simplex. + +For the linear objective function ``f`` with linear constraints +expressed using `Le`, `Ge` or `Eq` can be found with ``lpmin`` or +``lpmax``. The symbols are **unbounded** unless specifically +constrained. + +As an alternative, the matrices describing the objective and the +constraints, and an optional list of bounds can be passed to +``linprog`` which will solve for the minimization of ``C*x`` +under constraints ``A*x <= b`` and/or ``Aeq*x = beq``, and +individual bounds for variables given as ``(lo, hi)``. The values +returned are **nonnegative** unless bounds are provided that +indicate otherwise. + +Errors that might be raised are UnboundedLPError when there is no +finite solution for the system or InfeasibleLPError when the +constraints represent impossible conditions (i.e. a non-existent + simplex). + +Here is a simple 1-D system: minimize `x` given that ``x >= 1``. + + >>> from sympy.solvers.simplex import lpmin, linprog + >>> from sympy.abc import x + + The function and a list with the constraint is passed directly + to `lpmin`: + + >>> lpmin(x, [x >= 1]) + (1, {x: 1}) + + For `linprog` the matrix for the objective is `[1]` and the + uivariate constraint can be passed as a bound with None acting + as infinity: + + >>> linprog([1], bounds=(1, None)) + (1, [1]) + + Or the matrices, corresponding to ``x >= 1`` expressed as + ``-x <= -1`` as required by the routine, can be passed: + + >>> linprog([1], [-1], [-1]) + (1, [1]) + + If there is no limit for the objective, an error is raised. + In this case there is a valid region of interest (simplex) + but no limit to how small ``x`` can be: + + >>> lpmin(x, []) + Traceback (most recent call last): + ... + sympy.solvers.simplex.UnboundedLPError: + Objective function can assume arbitrarily large values! + + An error is raised if there is no possible solution: + + >>> lpmin(x,[x<=1,x>=2]) + Traceback (most recent call last): + ... + sympy.solvers.simplex.InfeasibleLPError: + Inconsistent/False constraint +""" + +from sympy.core import sympify +from sympy.core.exprtools import factor_terms +from sympy.core.relational import Le, Ge, Eq +from sympy.core.singleton import S +from sympy.core.symbol import Dummy +from sympy.core.sorting import ordered +from sympy.functions.elementary.complexes import sign +from sympy.matrices.dense import Matrix, zeros +from sympy.solvers.solveset import linear_eq_to_matrix +from sympy.utilities.iterables import numbered_symbols +from sympy.utilities.misc import filldedent + + +class UnboundedLPError(Exception): + """ + A linear programming problem is said to be unbounded if its objective + function can assume arbitrarily large values. + + Example + ======= + + Suppose you want to maximize + 2x + subject to + x >= 0 + + There's no upper limit that 2x can take. + """ + + pass + + +class InfeasibleLPError(Exception): + """ + A linear programming problem is considered infeasible if its + constraint set is empty. That is, if the set of all vectors + satisfying the constraints is empty, then the problem is infeasible. + + Example + ======= + + Suppose you want to maximize + x + subject to + x >= 10 + x <= 9 + + No x can satisfy those constraints. + """ + + pass + + +def _pivot(M, i, j): + """ + The pivot element `M[i, j]` is inverted and the rest of the matrix + modified and returned as a new matrix; original is left unmodified. + + Example + ======= + + >>> from sympy.matrices.dense import Matrix + >>> from sympy.solvers.simplex import _pivot + >>> from sympy import var + >>> Matrix(3, 3, var('a:i')) + Matrix([ + [a, b, c], + [d, e, f], + [g, h, i]]) + >>> _pivot(_, 1, 0) + Matrix([ + [-a/d, -a*e/d + b, -a*f/d + c], + [ 1/d, e/d, f/d], + [-g/d, h - e*g/d, i - f*g/d]]) + """ + Mi, Mj, Mij = M[i, :], M[:, j], M[i, j] + if Mij == 0: + raise ZeroDivisionError( + "Tried to pivot about zero-valued entry.") + A = M - Mj * (Mi / Mij) + A[i, :] = Mi / Mij + A[:, j] = -Mj / Mij + A[i, j] = 1 / Mij + return A + + +def _choose_pivot_row(A, B, candidate_rows, pivot_col, Y): + # Choose row with smallest ratio + # If there are ties, pick using Bland's rule + return min(candidate_rows, key=lambda i: (B[i] / A[i, pivot_col], Y[i])) + + +def _simplex(A, B, C, D=None, dual=False): + """Return ``(o, x, y)`` obtained from the two-phase simplex method + using Bland's rule: ``o`` is the minimum value of primal, + ``Cx - D``, under constraints ``Ax <= B`` (with ``x >= 0``) and + the maximum of the dual, ``y^{T}B - D``, under constraints + ``A^{T}*y >= C^{T}`` (with ``y >= 0``). To compute the dual of + the system, pass `dual=True` and ``(o, y, x)`` will be returned. + + Note: the nonnegative constraints for ``x`` and ``y`` supercede + any values of ``A`` and ``B`` that are inconsistent with that + assumption, so if a constraint of ``x >= -1`` is represented + in ``A`` and ``B``, no value will be obtained that is negative; if + a constraint of ``x <= -1`` is represented, an error will be + raised since no solution is possible. + + This routine relies on the ability of determining whether an + expression is 0 or not. This is guaranteed if the input contains + only Float or Rational entries. It will raise a TypeError if + a relationship does not evaluate to True or False. + + Examples + ======== + + >>> from sympy.solvers.simplex import _simplex + >>> from sympy import Matrix + + Consider the simple minimization of ``f = x + y + 1`` under the + constraint that ``y + 2*x >= 4``. This is the "standard form" of + a minimization. + + In the nonnegative quadrant, this inequality describes a area above + a triangle with vertices at (0, 4), (0, 0) and (2, 0). The minimum + of ``f`` occurs at (2, 0). Define A, B, C, D for the standard + minimization: + + >>> A = Matrix([[2, 1]]) + >>> B = Matrix([4]) + >>> C = Matrix([[1, 1]]) + >>> D = Matrix([-1]) + + Confirm that this is the system of interest: + + >>> from sympy.abc import x, y + >>> X = Matrix([x, y]) + >>> (C*X - D)[0] + x + y + 1 + >>> [i >= j for i, j in zip(A*X, B)] + [2*x + y >= 4] + + Since `_simplex` will do a minimization for constraints given as + ``A*x <= B``, the signs of ``A`` and ``B`` must be negated since + the currently correspond to a greater-than inequality: + + >>> _simplex(-A, -B, C, D) + (3, [2, 0], [1/2]) + + The dual of minimizing ``f`` is maximizing ``F = c*y - d`` for + ``a*y <= b`` where ``a``, ``b``, ``c``, ``d`` are derived from the + transpose of the matrix representation of the standard minimization: + + >>> tr = lambda a, b, c, d: [i.T for i in (a, c, b, d)] + >>> a, b, c, d = tr(A, B, C, D) + + This time ``a*x <= b`` is the expected inequality for the `_simplex` + method, but to maximize ``F``, the sign of ``c`` and ``d`` must be + changed (so that minimizing the negative will give the negative of + the maximum of ``F``): + + >>> _simplex(a, b, -c, -d) + (-3, [1/2], [2, 0]) + + The negative of ``F`` and the min of ``f`` are the same. The dual + point `[1/2]` is the value of ``y`` that minimized ``F = c*y - d`` + under constraints a*x <= b``: + + >>> y = Matrix(['y']) + >>> (c*y - d)[0] + 4*y + 1 + >>> [i <= j for i, j in zip(a*y,b)] + [2*y <= 1, y <= 1] + + In this 1-dimensional dual system, the more restrictive constraint is + the first which limits ``y`` between 0 and 1/2 and the maximum of + ``F`` is attained at the nonzero value, hence is ``4*(1/2) + 1 = 3``. + + In this case the values for ``x`` and ``y`` were the same when the + dual representation was solved. This is not always the case (though + the value of the function will be the same). + + >>> l = [[1, 1], [-1, 1], [0, 1], [-1, 0]], [5, 1, 2, -1], [[1, 1]], [-1] + >>> A, B, C, D = [Matrix(i) for i in l] + >>> _simplex(A, B, -C, -D) + (-6, [3, 2], [1, 0, 0, 0]) + >>> _simplex(A, B, -C, -D, dual=True) # [5, 0] != [3, 2] + (-6, [1, 0, 0, 0], [5, 0]) + + In both cases the function has the same value: + + >>> Matrix(C)*Matrix([3, 2]) == Matrix(C)*Matrix([5, 0]) + True + + See Also + ======== + _lp - poses min/max problem in form compatible with _simplex + lpmin - minimization which calls _lp + lpmax - maximimzation which calls _lp + + References + ========== + + .. [1] Thomas S. Ferguson, LINEAR PROGRAMMING: A Concise Introduction + web.tecnico.ulisboa.pt/mcasquilho/acad/or/ftp/FergusonUCLA_lp.pdf + + """ + A, B, C, D = [Matrix(i) for i in (A, B, C, D or [0])] + if dual: + _o, d, p = _simplex(-A.T, C.T, B.T, -D) + return -_o, d, p + + if A and B: + M = Matrix([[A, B], [C, D]]) + else: + if A or B: + raise ValueError("must give A and B") + # no constraints given + M = Matrix([[C, D]]) + n = M.cols - 1 + m = M.rows - 1 + + if not all(i.is_Float or i.is_Rational for i in M): + # with literal Float and Rational we are guaranteed the + # ability of determining whether an expression is 0 or not + raise TypeError(filldedent(""" + Only rationals and floats are allowed. + """ + ) + ) + + # x variables have priority over y variables during Bland's rule + # since False < True + X = [(False, j) for j in range(n)] + Y = [(True, i) for i in range(m)] + + # Phase 1: find a feasible solution or determine none exist + + ## keep track of last pivot row and column + last = None + + while True: + B = M[:-1, -1] + A = M[:-1, :-1] + if all(B[i] >= 0 for i in range(B.rows)): + # We have found a feasible solution + break + + # Find k: first row with a negative rightmost entry + for k in range(B.rows): + if B[k] < 0: + break # use current value of k below + else: + pass # error will raise below + + # Choose pivot column, c + piv_cols = [_ for _ in range(A.cols) if A[k, _] < 0] + if not piv_cols: + raise InfeasibleLPError(filldedent(""" + The constraint set is empty!""")) + _, c = min((X[i], i) for i in piv_cols) # Bland's rule + + # Choose pivot row, r + piv_rows = [_ for _ in range(A.rows) if A[_, c] > 0 and B[_] > 0] + piv_rows.append(k) + r = _choose_pivot_row(A, B, piv_rows, c, Y) + + # check for oscillation + if (r, c) == last: + # Not sure what to do here; it looks like there will be + # oscillations; see o1 test added at this commit to + # see a system with no solution and the o2 for one + # with a solution. In the case of o2, the solution + # from linprog is the same as the one from lpmin, but + # the matrices created in the lpmin case are different + # than those created without replacements in linprog and + # the matrices in the linprog case lead to oscillations. + # If the matrices could be re-written in linprog like + # lpmin does, this behavior could be avoided and then + # perhaps the oscillating case would only occur when + # there is no solution. For now, the output is checked + # before exit if oscillations were detected and an + # error is raised there if the solution was invalid. + # + # cf section 6 of Ferguson for a non-cycling modification + last = True + break + last = r, c + + M = _pivot(M, r, c) + X[c], Y[r] = Y[r], X[c] + + # Phase 2: from a feasible solution, pivot to optimal + while True: + B = M[:-1, -1] + A = M[:-1, :-1] + C = M[-1, :-1] + + # Choose a pivot column, c + piv_cols = [_ for _ in range(n) if C[_] < 0] + if not piv_cols: + break + _, c = min((X[i], i) for i in piv_cols) # Bland's rule + + # Choose a pivot row, r + piv_rows = [_ for _ in range(m) if A[_, c] > 0] + if not piv_rows: + raise UnboundedLPError(filldedent(""" + Objective function can assume + arbitrarily large values!""")) + r = _choose_pivot_row(A, B, piv_rows, c, Y) + + M = _pivot(M, r, c) + X[c], Y[r] = Y[r], X[c] + + argmax = [None] * n + argmin_dual = [None] * m + + for i, (v, n) in enumerate(X): + if v == False: + argmax[n] = 0 + else: + argmin_dual[n] = M[-1, i] + + for i, (v, n) in enumerate(Y): + if v == True: + argmin_dual[n] = 0 + else: + argmax[n] = M[i, -1] + + if last and not all(i >= 0 for i in argmax + argmin_dual): + raise InfeasibleLPError(filldedent(""" + Oscillating system led to invalid solution. + If you believe there was a valid solution, please + report this as a bug.""")) + return -M[-1, -1], argmax, argmin_dual + + +## routines that use _simplex or support those that do + + +def _abcd(M, list=False): + """return parts of M as matrices or lists + + Examples + ======== + + >>> from sympy import Matrix + >>> from sympy.solvers.simplex import _abcd + + >>> m = Matrix(3, 3, range(9)); m + Matrix([ + [0, 1, 2], + [3, 4, 5], + [6, 7, 8]]) + >>> a, b, c, d = _abcd(m) + >>> a + Matrix([ + [0, 1], + [3, 4]]) + >>> b + Matrix([ + [2], + [5]]) + >>> c + Matrix([[6, 7]]) + >>> d + Matrix([[8]]) + + The matrices can be returned as compact lists, too: + + >>> L = a, b, c, d = _abcd(m, list=True); L + ([[0, 1], [3, 4]], [2, 5], [[6, 7]], [8]) + """ + + def aslist(i): + l = i.tolist() + if len(l[0]) == 1: # col vector + return [i[0] for i in l] + return l + + m = M[:-1, :-1], M[:-1, -1], M[-1, :-1], M[-1:, -1:] + if not list: + return m + return tuple([aslist(i) for i in m]) + + +def _m(a, b, c, d=None): + """return Matrix([[a, b], [c, d]]) from matrices + in Matrix or list form. + + Examples + ======== + + >>> from sympy import Matrix + >>> from sympy.solvers.simplex import _abcd, _m + >>> m = Matrix(3, 3, range(9)) + >>> L = _abcd(m, list=True); L + ([[0, 1], [3, 4]], [2, 5], [[6, 7]], [8]) + >>> _abcd(m) + (Matrix([ + [0, 1], + [3, 4]]), Matrix([ + [2], + [5]]), Matrix([[6, 7]]), Matrix([[8]])) + >>> assert m == _m(*L) == _m(*_) + """ + a, b, c, d = [Matrix(i) for i in (a, b, c, d or [0])] + return Matrix([[a, b], [c, d]]) + + +def _primal_dual(M, factor=True): + """return primal and dual function and constraints + assuming that ``M = Matrix([[A, b], [c, d]])`` and the + function ``c*x - d`` is being minimized with ``Ax >= b`` + for nonnegative values of ``x``. The dual and its + constraints will be for maximizing `b.T*y - d` subject + to ``A.T*y <= c.T``. + + Examples + ======== + + >>> from sympy.solvers.simplex import _primal_dual, lpmin, lpmax + >>> from sympy import Matrix + + The following matrix represents the primal task of + minimizing x + y + 7 for y >= x + 1 and y >= -2*x + 3. + The dual task seeks to maximize x + 3*y + 7 with + 2*y - x <= 1 and and x + y <= 1: + + >>> M = Matrix([ + ... [-1, 1, 1], + ... [ 2, 1, 3], + ... [ 1, 1, -7]]) + >>> p, d = _primal_dual(M) + + The minimum of the primal and maximum of the dual are the same + (though they occur at different points): + + >>> lpmin(*p) + (28/3, {x1: 2/3, x2: 5/3}) + >>> lpmax(*d) + (28/3, {y1: 1/3, y2: 2/3}) + + If the equivalent (but canonical) inequalities are + desired, leave `factor=True`, otherwise the unmodified + inequalities for M will be returned. + + >>> m = Matrix([ + ... [-3, -2, 4, -2], + ... [ 2, 0, 0, -2], + ... [ 0, 1, -3, 0]]) + + >>> _primal_dual(m, False) # last condition is 2*x1 >= -2 + ((x2 - 3*x3, + [-3*x1 - 2*x2 + 4*x3 >= -2, 2*x1 >= -2]), + (-2*y1 - 2*y2, + [-3*y1 + 2*y2 <= 0, -2*y1 <= 1, 4*y1 <= -3])) + + >>> _primal_dual(m) # condition now x1 >= -1 + ((x2 - 3*x3, + [-3*x1 - 2*x2 + 4*x3 >= -2, x1 >= -1]), + (-2*y1 - 2*y2, + [-3*y1 + 2*y2 <= 0, -2*y1 <= 1, 4*y1 <= -3])) + + If you pass the transpose of the matrix, the primal will be + identified as the standard minimization problem and the + dual as the standard maximization: + + >>> _primal_dual(m.T) + ((-2*x1 - 2*x2, + [-3*x1 + 2*x2 >= 0, -2*x1 >= 1, 4*x1 >= -3]), + (y2 - 3*y3, + [-3*y1 - 2*y2 + 4*y3 <= -2, y1 <= -1])) + + A matrix must have some size or else None will be returned for + the functions: + + >>> _primal_dual(Matrix([[1, 2]])) + ((x1 - 2, []), (-2, [])) + + >>> _primal_dual(Matrix([])) + ((None, []), (None, [])) + + References + ========== + + .. [1] David Galvin, Relations between Primal and Dual + www3.nd.edu/~dgalvin1/30210/30210_F07/presentations/dual_opt.pdf + """ + if not M: + return (None, []), (None, []) + if not hasattr(M, "shape"): + if len(M) not in (3, 4): + raise ValueError("expecting Matrix or 3 or 4 lists") + M = _m(*M) + m, n = [i - 1 for i in M.shape] + A, b, c, d = _abcd(M) + d = d[0] + _ = lambda x: numbered_symbols(x, start=1) + x = Matrix([i for i, j in zip(_("x"), range(n))]) + yT = Matrix([i for i, j in zip(_("y"), range(m))]).T + + def ineq(L, r, op): + rv = [] + for r in (op(i, j) for i, j in zip(L, r)): + if r == True: + continue + elif r == False: + return [False] + if factor: + f = factor_terms(r) + if f.lhs.is_Mul and f.rhs % f.lhs.args[0] == 0: + assert len(f.lhs.args) == 2, f.lhs + k = f.lhs.args[0] + r = r.func(sign(k) * f.lhs.args[1], f.rhs // abs(k)) + rv.append(r) + return rv + + eq = lambda x, d: x[0] - d if x else -d + F = eq(c * x, d) + f = eq(yT * b, d) + return (F, ineq(A * x, b, Ge)), (f, ineq(yT * A, c, Le)) + + +def _rel_as_nonpos(constr, syms): + """return `(np, d, aux)` where `np` is a list of nonpositive + expressions that represent the given constraints (possibly + rewritten in terms of auxilliary variables) expressible with + nonnegative symbols, and `d` is a dictionary mapping a given + symbols to an expression with an auxilliary variable. In some + cases a symbol will be used as part of the change of variables, + e.g. x: x - z1 instead of x: z1 - z2. + + If any constraint is False/empty, return None. All variables in + ``constr`` are assumed to be unbounded unless explicitly indicated + otherwise with a univariate constraint, e.g. ``x >= 0`` will + restrict ``x`` to nonnegative values. + + The ``syms`` must be included so all symbols can be given an + unbounded assumption if they are not otherwise bound with + univariate conditions like ``x <= 3``. + + Examples + ======== + + >>> from sympy.solvers.simplex import _rel_as_nonpos + >>> from sympy.abc import x, y + >>> _rel_as_nonpos([x >= y, x >= 0, y >= 0], (x, y)) + ([-x + y], {}, []) + >>> _rel_as_nonpos([x >= 3, x <= 5], [x]) + ([_z1 - 2], {x: _z1 + 3}, [_z1]) + >>> _rel_as_nonpos([x <= 5], [x]) + ([], {x: 5 - _z1}, [_z1]) + >>> _rel_as_nonpos([x >= 1], [x]) + ([], {x: _z1 + 1}, [_z1]) + """ + r = {} # replacements to handle change of variables + np = [] # nonpositive expressions + aux = [] # auxilliary symbols added + ui = numbered_symbols("z", start=1, cls=Dummy) # auxilliary symbols + univariate = {} # {x: interval} for univariate constraints + unbound = [] # symbols designated as unbound + syms = set(syms) # the expected syms of the system + + # separate out univariates + for i in constr: + if i == True: + continue # ignore + if i == False: + return # no solution + if i.has(S.Infinity, S.NegativeInfinity): + raise ValueError("only finite bounds are permitted") + if isinstance(i, (Le, Ge)): + i = i.lts - i.gts + freei = i.free_symbols + if freei - syms: + raise ValueError( + "unexpected symbol(s) in constraint: %s" % (freei - syms) + ) + if len(freei) > 1: + np.append(i) + elif freei: + x = freei.pop() + if x in unbound: + continue # will handle later + ivl = Le(i, 0, evaluate=False).as_set() + if x not in univariate: + univariate[x] = ivl + else: + univariate[x] &= ivl + elif i: + return False + else: + raise TypeError(filldedent(""" + only equalities like Eq(x, y) or non-strict + inequalities like x >= y are allowed in lp, not %s""" % i)) + + # introduce auxilliary variables as needed for univariate + # inequalities + for x in syms: + i = univariate.get(x, True) + if not i: + return None # no solution possible + if i == True: + unbound.append(x) + continue + a, b = i.inf, i.sup + if a.is_infinite: + u = next(ui) + r[x] = b - u + aux.append(u) + elif b.is_infinite: + if a: + u = next(ui) + r[x] = a + u + aux.append(u) + else: + # standard nonnegative relationship + pass + else: + u = next(ui) + aux.append(u) + # shift so u = x - a => x = u + a + r[x] = u + a + # add constraint for u <= b - a + # since when u = b-a then x = u + a = b - a + a = b: + # the upper limit for x + np.append(u - (b - a)) + + # make change of variables for unbound variables + for x in unbound: + u = next(ui) + r[x] = u - x # reusing x + aux.append(u) + + return np, r, aux + + +def _lp_matrices(objective, constraints): + """return A, B, C, D, r, x+X, X for maximizing + objective = Cx - D with constraints Ax <= B, introducing + introducing auxilliary variables, X, as necessary to make + replacements of symbols as given in r, {xi: expression with Xj}, + so all variables in x+X will take on nonnegative values. + + Every univariate condition creates a semi-infinite + condition, e.g. a single ``x <= 3`` creates the + interval ``[-oo, 3]`` while ``x <= 3`` and ``x >= 2`` + create an interval ``[2, 3]``. Variables not in a univariate + expression will take on nonnegative values. + """ + + # sympify input and collect free symbols + F = sympify(objective) + np = [sympify(i) for i in constraints] + syms = set.union(*[i.free_symbols for i in [F] + np], set()) + + # change Eq(x, y) to x - y <= 0 and y - x <= 0 + for i in range(len(np)): + if isinstance(np[i], Eq): + np[i] = np[i].lhs - np[i].rhs <= 0 + np.append(-np[i].lhs <= 0) + + # convert constraints to nonpositive expressions + _ = _rel_as_nonpos(np, syms) + if _ is None: + raise InfeasibleLPError(filldedent(""" + Inconsistent/False constraint""")) + np, r, aux = _ + + # do change of variables + F = F.xreplace(r) + np = [i.xreplace(r) for i in np] + + # convert to matrices + xx = list(ordered(syms)) + aux + A, B = linear_eq_to_matrix(np, xx) + C, D = linear_eq_to_matrix([F], xx) + return A, B, C, D, r, xx, aux + + +def _lp(min_max, f, constr): + """Return the optimization (min or max) of ``f`` with the given + constraints. All variables are unbounded unless constrained. + + If `min_max` is 'max' then the results corresponding to the + maximization of ``f`` will be returned, else the minimization. + The constraints can be given as Le, Ge or Eq expressions. + + Examples + ======== + + >>> from sympy.solvers.simplex import _lp as lp + >>> from sympy import Eq + >>> from sympy.abc import x, y, z + >>> f = x + y - 2*z + >>> c = [7*x + 4*y - 7*z <= 3, 3*x - y + 10*z <= 6] + >>> c += [i >= 0 for i in (x, y, z)] + >>> lp(min, f, c) + (-6/5, {x: 0, y: 0, z: 3/5}) + + By passing max, the maximum value for f under the constraints + is returned (if possible): + + >>> lp(max, f, c) + (3/4, {x: 0, y: 3/4, z: 0}) + + Constraints that are equalities will require that the solution + also satisfy them: + + >>> lp(max, f, c + [Eq(y - 9*x, 1)]) + (5/7, {x: 0, y: 1, z: 1/7}) + + All symbols are reported, even if they are not in the objective + function: + + >>> lp(min, x, [y + x >= 3, x >= 0]) + (0, {x: 0, y: 3}) + """ + # get the matrix components for the system expressed + # in terms of only nonnegative variables + A, B, C, D, r, xx, aux = _lp_matrices(f, constr) + + how = str(min_max).lower() + if "max" in how: + # _simplex minimizes for Ax <= B so we + # have to change the sign of the function + # and negate the optimal value returned + _o, p, d = _simplex(A, B, -C, -D) + o = -_o + elif "min" in how: + o, p, d = _simplex(A, B, C, D) + else: + raise ValueError("expecting min or max") + + # restore original variables and remove aux from p + p = dict(zip(xx, p)) + if r: # p has original symbols and auxilliary symbols + # if r has x: x - z1 use values from p to update + r = {k: v.xreplace(p) for k, v in r.items()} + # then use the actual value of x (= x - z1) in p + p.update(r) + # don't show aux + p = {k: p[k] for k in ordered(p) if k not in aux} + + # not returning dual since there may be extra constraints + # when a variable has finite bounds + return o, p + + +def lpmin(f, constr): + """return minimum of linear equation ``f`` under + linear constraints expressed using Ge, Le or Eq. + + All variables are unbounded unless constrained. + + Examples + ======== + + >>> from sympy.solvers.simplex import lpmin + >>> from sympy import Eq + >>> from sympy.abc import x, y + >>> lpmin(x, [2*x - 3*y >= -1, Eq(x + 3*y, 2), x <= 2*y]) + (1/3, {x: 1/3, y: 5/9}) + + Negative values for variables are permitted unless explicitly + excluding, so minimizing ``x`` for ``x <= 3`` is an + unbounded problem while the following has a bounded solution: + + >>> lpmin(x, [x >= 0, x <= 3]) + (0, {x: 0}) + + Without indicating that ``x`` is nonnegative, there + is no minimum for this objective: + + >>> lpmin(x, [x <= 3]) + Traceback (most recent call last): + ... + sympy.solvers.simplex.UnboundedLPError: + Objective function can assume arbitrarily large values! + + See Also + ======== + linprog, lpmax + """ + return _lp(min, f, constr) + + +def lpmax(f, constr): + """return maximum of linear equation ``f`` under + linear constraints expressed using Ge, Le or Eq. + + All variables are unbounded unless constrained. + + Examples + ======== + + >>> from sympy.solvers.simplex import lpmax + >>> from sympy import Eq + >>> from sympy.abc import x, y + >>> lpmax(x, [2*x - 3*y >= -1, Eq(x+ 3*y,2), x <= 2*y]) + (4/5, {x: 4/5, y: 2/5}) + + Negative values for variables are permitted unless explicitly + excluding: + + >>> lpmax(x, [x <= -1]) + (-1, {x: -1}) + + If a non-negative constraint is added for x, there is no + possible solution: + + >>> lpmax(x, [x <= -1, x >= 0]) + Traceback (most recent call last): + ... + sympy.solvers.simplex.InfeasibleLPError: inconsistent/False constraint + + See Also + ======== + linprog, lpmin + """ + return _lp(max, f, constr) + + +def _handle_bounds(bounds): + # introduce auxiliary variables as needed for univariate + # inequalities + + def _make_list(length: int, index_value_pairs): + li = [0] * length + for idx, val in index_value_pairs: + li[idx] = val + return li + + unbound = [] + row = [] + row2 = [] + b_len = len(bounds) + for x, (a, b) in enumerate(bounds): + if a is None and b is None: + unbound.append(x) + elif a is None: + # r[x] = b - u + b_len += 1 + row.append(_make_list(b_len, [(x, 1), (-1, 1)])) + row.append(_make_list(b_len, [(x, -1), (-1, -1)])) + row2.extend([[b], [-b]]) + elif b is None: + if a: + # r[x] = a + u + b_len += 1 + row.append(_make_list(b_len, [(x, 1), (-1, -1)])) + row.append(_make_list(b_len, [(x, -1), (-1, 1)])) + row2.extend([[a], [-a]]) + else: + # standard nonnegative relationship + pass + else: + # r[x] = u + a + b_len += 1 + row.append(_make_list(b_len, [(x, 1), (-1, -1)])) + row.append(_make_list(b_len, [(x, -1), (-1, 1)])) + # u <= b - a + row.append(_make_list(b_len, [(-1, 1)])) + row2.extend([[a], [-a], [b - a]]) + + # make change of variables for unbound variables + for x in unbound: + # r[x] = u - v + b_len += 2 + row.append(_make_list(b_len, [(x, 1), (-1, 1), (-2, -1)])) + row.append(_make_list(b_len, [(x, -1), (-1, -1), (-2, 1)])) + row2.extend([[0], [0]]) + + return Matrix([r + [0]*(b_len - len(r)) for r in row]), Matrix(row2) + + +def linprog(c, A=None, b=None, A_eq=None, b_eq=None, bounds=None): + """Return the minimization of ``c*x`` with the given + constraints ``A*x <= b`` and ``A_eq*x = b_eq``. Unless bounds + are given, variables will have nonnegative values in the solution. + + If ``A`` is not given, then the dimension of the system will + be determined by the length of ``C``. + + By default, all variables will be nonnegative. If ``bounds`` + is given as a single tuple, ``(lo, hi)``, then all variables + will be constrained to be between ``lo`` and ``hi``. Use + None for a ``lo`` or ``hi`` if it is unconstrained in the + negative or positive direction, respectively, e.g. + ``(None, 0)`` indicates nonpositive values. To set + individual ranges, pass a list with length equal to the + number of columns in ``A``, each element being a tuple; if + only a few variables take on non-default values they can be + passed as a dictionary with keys giving the corresponding + column to which the variable is assigned, e.g. ``bounds={2: + (1, 4)}`` would limit the 3rd variable to have a value in + range ``[1, 4]``. + + Examples + ======== + + >>> from sympy.solvers.simplex import linprog + >>> from sympy import symbols, Eq, linear_eq_to_matrix as M, Matrix + >>> x = x1, x2, x3, x4 = symbols('x1:5') + >>> X = Matrix(x) + >>> c, d = M(5*x2 + x3 + 4*x4 - x1, x) + >>> a, b = M([5*x2 + 2*x3 + 5*x4 - (x1 + 5)], x) + >>> aeq, beq = M([Eq(3*x2 + x4, 2), Eq(-x1 + x3 + 2*x4, 1)], x) + >>> constr = [i <= j for i,j in zip(a*X, b)] + >>> constr += [Eq(i, j) for i,j in zip(aeq*X, beq)] + >>> linprog(c, a, b, aeq, beq) + (9/2, [0, 1/2, 0, 1/2]) + >>> assert all(i.subs(dict(zip(x, _[1]))) for i in constr) + + See Also + ======== + lpmin, lpmax + """ + + ## the objective + C = Matrix(c) + if C.rows != 1 and C.cols == 1: + C = C.T + if C.rows != 1: + raise ValueError("C must be a single row.") + + ## the inequalities + if not A: + if b: + raise ValueError("A and b must both be given") + # the governing equations will be simple constraints + # on variables + A, b = zeros(0, C.cols), zeros(C.cols, 1) + else: + A, b = [Matrix(i) for i in (A, b)] + + if A.cols != C.cols: + raise ValueError("number of columns in A and C must match") + + ## the equalities + if A_eq is None: + if not b_eq is None: + raise ValueError("A_eq and b_eq must both be given") + else: + A_eq, b_eq = [Matrix(i) for i in (A_eq, b_eq)] + # if x == y then x <= y and x >= y (-x <= -y) + A = A.col_join(A_eq) + A = A.col_join(-A_eq) + b = b.col_join(b_eq) + b = b.col_join(-b_eq) + + if not (bounds is None or bounds == {} or bounds == (0, None)): + ## the bounds are interpreted + if type(bounds) is tuple and len(bounds) == 2: + bounds = [bounds] * A.cols + elif len(bounds) == A.cols and all( + type(i) is tuple and len(i) == 2 for i in bounds): + pass # individual bounds + elif type(bounds) is dict and all( + type(i) is tuple and len(i) == 2 + for i in bounds.values()): + # sparse bounds + db = bounds + bounds = [(0, None)] * A.cols + while db: + i, j = db.popitem() + bounds[i] = j # IndexError if out-of-bounds indices + else: + raise ValueError("unexpected bounds %s" % bounds) + A_, b_ = _handle_bounds(bounds) + aux = A_.cols - A.cols + if A: + A = Matrix([[A, zeros(A.rows, aux)], [A_]]) + b = b.col_join(b_) + else: + A = A_ + b = b_ + C = C.row_join(zeros(1, aux)) + else: + aux = -A.cols # set so -aux will give all cols below + + o, p, d = _simplex(A, b, C) + return o, p[:-aux] # don't include aux values + +def show_linprog(c, A=None, b=None, A_eq=None, b_eq=None, bounds=None): + from sympy import symbols + ## the objective + C = Matrix(c) + if C.rows != 1 and C.cols == 1: + C = C.T + if C.rows != 1: + raise ValueError("C must be a single row.") + + ## the inequalities + if not A: + if b: + raise ValueError("A and b must both be given") + # the governing equations will be simple constraints + # on variables + A, b = zeros(0, C.cols), zeros(C.cols, 1) + else: + A, b = [Matrix(i) for i in (A, b)] + + if A.cols != C.cols: + raise ValueError("number of columns in A and C must match") + + ## the equalities + if A_eq is None: + if not b_eq is None: + raise ValueError("A_eq and b_eq must both be given") + else: + A_eq, b_eq = [Matrix(i) for i in (A_eq, b_eq)] + + if not (bounds is None or bounds == {} or bounds == (0, None)): + ## the bounds are interpreted + if type(bounds) is tuple and len(bounds) == 2: + bounds = [bounds] * A.cols + elif len(bounds) == A.cols and all( + type(i) is tuple and len(i) == 2 for i in bounds): + pass # individual bounds + elif type(bounds) is dict and all( + type(i) is tuple and len(i) == 2 + for i in bounds.values()): + # sparse bounds + db = bounds + bounds = [(0, None)] * A.cols + while db: + i, j = db.popitem() + bounds[i] = j # IndexError if out-of-bounds indices + else: + raise ValueError("unexpected bounds %s" % bounds) + + x = Matrix(symbols('x1:%s' % (A.cols+1))) + f,c = (C*x)[0], [i<=j for i,j in zip(A*x, b)] + [Eq(i,j) for i,j in zip(A_eq*x,b_eq)] + for i, (lo, hi) in enumerate(bounds): + if lo is not None: + c.append(x[i]>=lo) + if hi is not None: + c.append(x[i]<=hi) + return f,c diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/solvers.py b/.venv/lib/python3.12/site-packages/sympy/solvers/solvers.py new file mode 100644 index 0000000000000000000000000000000000000000..ef621a84e34bde43a3181d2fd90e26fa7b05e968 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/solvers.py @@ -0,0 +1,3674 @@ +""" +This module contain solvers for all kinds of equations: + + - algebraic or transcendental, use solve() + + - recurrence, use rsolve() + + - differential, use dsolve() + + - nonlinear (numerically), use nsolve() + (you will need a good starting point) + +""" +from __future__ import annotations + +from sympy.core import (S, Add, Symbol, Dummy, Expr, Mul) +from sympy.core.assumptions import check_assumptions +from sympy.core.exprtools import factor_terms +from sympy.core.function import (expand_mul, expand_log, Derivative, + AppliedUndef, UndefinedFunction, nfloat, + Function, expand_power_exp, _mexpand, expand, + expand_func) +from sympy.core.logic import fuzzy_not, fuzzy_and +from sympy.core.numbers import Float, Rational, _illegal +from sympy.core.intfunc import integer_log, ilcm +from sympy.core.power import Pow +from sympy.core.relational import Eq, Ne +from sympy.core.sorting import ordered, default_sort_key +from sympy.core.sympify import sympify, _sympify +from sympy.core.traversal import preorder_traversal +from sympy.logic.boolalg import And, BooleanAtom + +from sympy.functions import (log, exp, LambertW, cos, sin, tan, acos, asin, atan, + Abs, re, im, arg, sqrt, atan2) +from sympy.functions.combinatorial.factorials import binomial +from sympy.functions.elementary.hyperbolic import HyperbolicFunction +from sympy.functions.elementary.piecewise import piecewise_fold, Piecewise +from sympy.functions.elementary.trigonometric import TrigonometricFunction +from sympy.integrals.integrals import Integral +from sympy.ntheory.factor_ import divisors +from sympy.simplify import (simplify, collect, powsimp, posify, # type: ignore + powdenest, nsimplify, denom, logcombine, sqrtdenest, fraction, + separatevars) +from sympy.simplify.sqrtdenest import sqrt_depth +from sympy.simplify.fu import TR1, TR2i, TR10, TR11 +from sympy.strategies.rl import rebuild +from sympy.matrices.exceptions import NonInvertibleMatrixError +from sympy.matrices import Matrix, zeros +from sympy.polys import roots, cancel, factor, Poly +from sympy.polys.solvers import sympy_eqs_to_ring, solve_lin_sys +from sympy.polys.polyerrors import GeneratorsNeeded, PolynomialError +from sympy.polys.polytools import gcd +from sympy.utilities.lambdify import lambdify +from sympy.utilities.misc import filldedent, debugf +from sympy.utilities.iterables import (connected_components, + generate_bell, uniq, iterable, is_sequence, subsets, flatten, sift) +from sympy.utilities.decorator import conserve_mpmath_dps + +from mpmath import findroot + +from sympy.solvers.polysys import solve_poly_system + +from types import GeneratorType +from collections import defaultdict +from itertools import combinations, product + +import warnings + + +def recast_to_symbols(eqs, symbols): + """ + Return (e, s, d) where e and s are versions of *eqs* and + *symbols* in which any non-Symbol objects in *symbols* have + been replaced with generic Dummy symbols and d is a dictionary + that can be used to restore the original expressions. + + Examples + ======== + + >>> from sympy.solvers.solvers import recast_to_symbols + >>> from sympy import symbols, Function + >>> x, y = symbols('x y') + >>> fx = Function('f')(x) + >>> eqs, syms = [fx + 1, x, y], [fx, y] + >>> e, s, d = recast_to_symbols(eqs, syms); (e, s, d) + ([_X0 + 1, x, y], [_X0, y], {_X0: f(x)}) + + The original equations and symbols can be restored using d: + + >>> assert [i.xreplace(d) for i in eqs] == eqs + >>> assert [d.get(i, i) for i in s] == syms + + """ + if not iterable(eqs) and iterable(symbols): + raise ValueError('Both eqs and symbols must be iterable') + orig = list(symbols) + symbols = list(ordered(symbols)) + swap_sym = {} + i = 0 + for s in symbols: + if not isinstance(s, Symbol) and s not in swap_sym: + swap_sym[s] = Dummy('X%d' % i) + i += 1 + new_f = [] + for i in eqs: + isubs = getattr(i, 'subs', None) + if isubs is not None: + new_f.append(isubs(swap_sym)) + else: + new_f.append(i) + restore = {v: k for k, v in swap_sym.items()} + return new_f, [swap_sym.get(i, i) for i in orig], restore + + +def _ispow(e): + """Return True if e is a Pow or is exp.""" + return isinstance(e, Expr) and (e.is_Pow or isinstance(e, exp)) + + +def _simple_dens(f, symbols): + # when checking if a denominator is zero, we can just check the + # base of powers with nonzero exponents since if the base is zero + # the power will be zero, too. To keep it simple and fast, we + # limit simplification to exponents that are Numbers + dens = set() + for d in denoms(f, symbols): + if d.is_Pow and d.exp.is_Number: + if d.exp.is_zero: + continue # foo**0 is never 0 + d = d.base + dens.add(d) + return dens + + +def denoms(eq, *symbols): + """ + Return (recursively) set of all denominators that appear in *eq* + that contain any symbol in *symbols*; if *symbols* are not + provided then all denominators will be returned. + + Examples + ======== + + >>> from sympy.solvers.solvers import denoms + >>> from sympy.abc import x, y, z + + >>> denoms(x/y) + {y} + + >>> denoms(x/(y*z)) + {y, z} + + >>> denoms(3/x + y/z) + {x, z} + + >>> denoms(x/2 + y/z) + {2, z} + + If *symbols* are provided then only denominators containing + those symbols will be returned: + + >>> denoms(1/x + 1/y + 1/z, y, z) + {y, z} + + """ + + pot = preorder_traversal(eq) + dens = set() + for p in pot: + # Here p might be Tuple or Relational + # Expr subtrees (e.g. lhs and rhs) will be traversed after by pot + if not isinstance(p, Expr): + continue + den = denom(p) + if den is S.One: + continue + dens.update(Mul.make_args(den)) + if not symbols: + return dens + elif len(symbols) == 1: + if iterable(symbols[0]): + symbols = symbols[0] + return {d for d in dens if any(s in d.free_symbols for s in symbols)} + + +def checksol(f, symbol, sol=None, **flags): + """ + Checks whether sol is a solution of equation f == 0. + + Explanation + =========== + + Input can be either a single symbol and corresponding value + or a dictionary of symbols and values. When given as a dictionary + and flag ``simplify=True``, the values in the dictionary will be + simplified. *f* can be a single equation or an iterable of equations. + A solution must satisfy all equations in *f* to be considered valid; + if a solution does not satisfy any equation, False is returned; if one or + more checks are inconclusive (and none are False) then None is returned. + + Examples + ======== + + >>> from sympy import checksol, symbols + >>> x, y = symbols('x,y') + >>> checksol(x**4 - 1, x, 1) + True + >>> checksol(x**4 - 1, x, 0) + False + >>> checksol(x**2 + y**2 - 5**2, {x: 3, y: 4}) + True + + To check if an expression is zero using ``checksol()``, pass it + as *f* and send an empty dictionary for *symbol*: + + >>> checksol(x**2 + x - x*(x + 1), {}) + True + + None is returned if ``checksol()`` could not conclude. + + flags: + 'numerical=True (default)' + do a fast numerical check if ``f`` has only one symbol. + 'minimal=True (default is False)' + a very fast, minimal testing. + 'warn=True (default is False)' + show a warning if checksol() could not conclude. + 'simplify=True (default)' + simplify solution before substituting into function and + simplify the function before trying specific simplifications + 'force=True (default is False)' + make positive all symbols without assumptions regarding sign. + + """ + from sympy.physics.units import Unit + + minimal = flags.get('minimal', False) + + if sol is not None: + sol = {symbol: sol} + elif isinstance(symbol, dict): + sol = symbol + else: + msg = 'Expecting (sym, val) or ({sym: val}, None) but got (%s, %s)' + raise ValueError(msg % (symbol, sol)) + + if iterable(f): + if not f: + raise ValueError('no functions to check') + return fuzzy_and(checksol(fi, sol, **flags) for fi in f) + + f = _sympify(f) + + if f.is_number: + return f.is_zero + + if isinstance(f, Poly): + f = f.as_expr() + elif isinstance(f, (Eq, Ne)): + if f.rhs in (S.true, S.false): + f = f.reversed + B, E = f.args + if isinstance(B, BooleanAtom): + f = f.subs(sol) + if not f.is_Boolean: + return + elif isinstance(f, Eq): + f = Add(f.lhs, -f.rhs, evaluate=False) + + if isinstance(f, BooleanAtom): + return bool(f) + elif not f.is_Relational and not f: + return True + + illegal = set(_illegal) + if any(sympify(v).atoms() & illegal for k, v in sol.items()): + return False + + attempt = -1 + numerical = flags.get('numerical', True) + while 1: + attempt += 1 + if attempt == 0: + val = f.subs(sol) + if isinstance(val, Mul): + val = val.as_independent(Unit)[0] + if val.atoms() & illegal: + return False + elif attempt == 1: + if not val.is_number: + if not val.is_constant(*list(sol.keys()), simplify=not minimal): + return False + # there are free symbols -- simple expansion might work + _, val = val.as_content_primitive() + val = _mexpand(val.as_numer_denom()[0], recursive=True) + elif attempt == 2: + if minimal: + return + if flags.get('simplify', True): + for k in sol: + sol[k] = simplify(sol[k]) + # start over without the failed expanded form, possibly + # with a simplified solution + val = simplify(f.subs(sol)) + if flags.get('force', True): + val, reps = posify(val) + # expansion may work now, so try again and check + exval = _mexpand(val, recursive=True) + if exval.is_number: + # we can decide now + val = exval + else: + # if there are no radicals and no functions then this can't be + # zero anymore -- can it? + pot = preorder_traversal(expand_mul(val)) + seen = set() + saw_pow_func = False + for p in pot: + if p in seen: + continue + seen.add(p) + if p.is_Pow and not p.exp.is_Integer: + saw_pow_func = True + elif p.is_Function: + saw_pow_func = True + elif isinstance(p, UndefinedFunction): + saw_pow_func = True + if saw_pow_func: + break + if saw_pow_func is False: + return False + if flags.get('force', True): + # don't do a zero check with the positive assumptions in place + val = val.subs(reps) + nz = fuzzy_not(val.is_zero) + if nz is not None: + # issue 5673: nz may be True even when False + # so these are just hacks to keep a false positive + # from being returned + + # HACK 1: LambertW (issue 5673) + if val.is_number and val.has(LambertW): + # don't eval this to verify solution since if we got here, + # numerical must be False + return None + + # add other HACKs here if necessary, otherwise we assume + # the nz value is correct + return not nz + break + if val.is_Rational: + return val == 0 + if numerical and val.is_number: + return (abs(val.n(18).n(12, chop=True)) < 1e-9) is S.true + + if flags.get('warn', False): + warnings.warn("\n\tWarning: could not verify solution %s." % sol) + # returns None if it can't conclude + # TODO: improve solution testing + + +def solve(f, *symbols, **flags): + r""" + Algebraically solves equations and systems of equations. + + Explanation + =========== + + Currently supported: + - polynomial + - transcendental + - piecewise combinations of the above + - systems of linear and polynomial equations + - systems containing relational expressions + - systems implied by undetermined coefficients + + Examples + ======== + + The default output varies according to the input and might + be a list (possibly empty), a dictionary, a list of + dictionaries or tuples, or an expression involving relationals. + For specifics regarding different forms of output that may appear, see :ref:`solve_output`. + Let it suffice here to say that to obtain a uniform output from + `solve` use ``dict=True`` or ``set=True`` (see below). + + >>> from sympy import solve, Poly, Eq, Matrix, Symbol + >>> from sympy.abc import x, y, z, a, b + + The expressions that are passed can be Expr, Equality, or Poly + classes (or lists of the same); a Matrix is considered to be a + list of all the elements of the matrix: + + >>> solve(x - 3, x) + [3] + >>> solve(Eq(x, 3), x) + [3] + >>> solve(Poly(x - 3), x) + [3] + >>> solve(Matrix([[x, x + y]]), x, y) == solve([x, x + y], x, y) + True + + If no symbols are indicated to be of interest and the equation is + univariate, a list of values is returned; otherwise, the keys in + a dictionary will indicate which (of all the variables used in + the expression(s)) variables and solutions were found: + + >>> solve(x**2 - 4) + [-2, 2] + >>> solve((x - a)*(y - b)) + [{a: x}, {b: y}] + >>> solve([x - 3, y - 1]) + {x: 3, y: 1} + >>> solve([x - 3, y**2 - 1]) + [{x: 3, y: -1}, {x: 3, y: 1}] + + If you pass symbols for which solutions are sought, the output will vary + depending on the number of symbols you passed, whether you are passing + a list of expressions or not, and whether a linear system was solved. + Uniform output is attained by using ``dict=True`` or ``set=True``. + + >>> #### *** feel free to skip to the stars below *** #### + >>> from sympy import TableForm + >>> h = [None, ';|;'.join(['e', 's', 'solve(e, s)', 'solve(e, s, dict=True)', + ... 'solve(e, s, set=True)']).split(';')] + >>> t = [] + >>> for e, s in [ + ... (x - y, y), + ... (x - y, [x, y]), + ... (x**2 - y, [x, y]), + ... ([x - 3, y -1], [x, y]), + ... ]: + ... how = [{}, dict(dict=True), dict(set=True)] + ... res = [solve(e, s, **f) for f in how] + ... t.append([e, '|', s, '|'] + [res[0], '|', res[1], '|', res[2]]) + ... + >>> # ******************************************************* # + >>> TableForm(t, headings=h, alignments="<") + e | s | solve(e, s) | solve(e, s, dict=True) | solve(e, s, set=True) + --------------------------------------------------------------------------------------- + x - y | y | [x] | [{y: x}] | ([y], {(x,)}) + x - y | [x, y] | [(y, y)] | [{x: y}] | ([x, y], {(y, y)}) + x**2 - y | [x, y] | [(x, x**2)] | [{y: x**2}] | ([x, y], {(x, x**2)}) + [x - 3, y - 1] | [x, y] | {x: 3, y: 1} | [{x: 3, y: 1}] | ([x, y], {(3, 1)}) + + * If any equation does not depend on the symbol(s) given, it will be + eliminated from the equation set and an answer may be given + implicitly in terms of variables that were not of interest: + + >>> solve([x - y, y - 3], x) + {x: y} + + When you pass all but one of the free symbols, an attempt + is made to find a single solution based on the method of + undetermined coefficients. If it succeeds, a dictionary of values + is returned. If you want an algebraic solutions for one + or more of the symbols, pass the expression to be solved in a list: + + >>> e = a*x + b - 2*x - 3 + >>> solve(e, [a, b]) + {a: 2, b: 3} + >>> solve([e], [a, b]) + {a: -b/x + (2*x + 3)/x} + + When there is no solution for any given symbol which will make all + expressions zero, the empty list is returned (or an empty set in + the tuple when ``set=True``): + + >>> from sympy import sqrt + >>> solve(3, x) + [] + >>> solve(x - 3, y) + [] + >>> solve(sqrt(x) + 1, x, set=True) + ([x], set()) + + When an object other than a Symbol is given as a symbol, it is + isolated algebraically and an implicit solution may be obtained. + This is mostly provided as a convenience to save you from replacing + the object with a Symbol and solving for that Symbol. It will only + work if the specified object can be replaced with a Symbol using the + subs method: + + >>> from sympy import exp, Function + >>> f = Function('f') + + >>> solve(f(x) - x, f(x)) + [x] + >>> solve(f(x).diff(x) - f(x) - x, f(x).diff(x)) + [x + f(x)] + >>> solve(f(x).diff(x) - f(x) - x, f(x)) + [-x + Derivative(f(x), x)] + >>> solve(x + exp(x)**2, exp(x), set=True) + ([exp(x)], {(-sqrt(-x),), (sqrt(-x),)}) + + >>> from sympy import Indexed, IndexedBase, Tuple + >>> A = IndexedBase('A') + >>> eqs = Tuple(A[1] + A[2] - 3, A[1] - A[2] + 1) + >>> solve(eqs, eqs.atoms(Indexed)) + {A[1]: 1, A[2]: 2} + + * To solve for a function within a derivative, use :func:`~.dsolve`. + + To solve for a symbol implicitly, use implicit=True: + + >>> solve(x + exp(x), x) + [-LambertW(1)] + >>> solve(x + exp(x), x, implicit=True) + [-exp(x)] + + It is possible to solve for anything in an expression that can be + replaced with a symbol using :obj:`~sympy.core.basic.Basic.subs`: + + >>> solve(x + 2 + sqrt(3), x + 2) + [-sqrt(3)] + >>> solve((x + 2 + sqrt(3), x + 4 + y), y, x + 2) + {y: -2 + sqrt(3), x + 2: -sqrt(3)} + + * Nothing heroic is done in this implicit solving so you may end up + with a symbol still in the solution: + + >>> eqs = (x*y + 3*y + sqrt(3), x + 4 + y) + >>> solve(eqs, y, x + 2) + {y: -sqrt(3)/(x + 3), x + 2: -2*x/(x + 3) - 6/(x + 3) + sqrt(3)/(x + 3)} + >>> solve(eqs, y*x, x) + {x: -y - 4, x*y: -3*y - sqrt(3)} + + * If you attempt to solve for a number, remember that the number + you have obtained does not necessarily mean that the value is + equivalent to the expression obtained: + + >>> solve(sqrt(2) - 1, 1) + [sqrt(2)] + >>> solve(x - y + 1, 1) # /!\ -1 is targeted, too + [x/(y - 1)] + >>> [_.subs(z, -1) for _ in solve((x - y + 1).subs(-1, z), 1)] + [-x + y] + + **Additional Examples** + + ``solve()`` with check=True (default) will run through the symbol tags to + eliminate unwanted solutions. If no assumptions are included, all possible + solutions will be returned: + + >>> x = Symbol("x") + >>> solve(x**2 - 1) + [-1, 1] + + By setting the ``positive`` flag, only one solution will be returned: + + >>> pos = Symbol("pos", positive=True) + >>> solve(pos**2 - 1) + [1] + + When the solutions are checked, those that make any denominator zero + are automatically excluded. If you do not want to exclude such solutions, + then use the check=False option: + + >>> from sympy import sin, limit + >>> solve(sin(x)/x) # 0 is excluded + [pi] + + If ``check=False``, then a solution to the numerator being zero is found + but the value of $x = 0$ is a spurious solution since $\sin(x)/x$ has the well + known limit (without discontinuity) of 1 at $x = 0$: + + >>> solve(sin(x)/x, check=False) + [0, pi] + + In the following case, however, the limit exists and is equal to the + value of $x = 0$ that is excluded when check=True: + + >>> eq = x**2*(1/x - z**2/x) + >>> solve(eq, x) + [] + >>> solve(eq, x, check=False) + [0] + >>> limit(eq, x, 0, '-') + 0 + >>> limit(eq, x, 0, '+') + 0 + + **Solving Relationships** + + When one or more expressions passed to ``solve`` is a relational, + a relational result is returned (and the ``dict`` and ``set`` flags + are ignored): + + >>> solve(x < 3) + (-oo < x) & (x < 3) + >>> solve([x < 3, x**2 > 4], x) + ((-oo < x) & (x < -2)) | ((2 < x) & (x < 3)) + >>> solve([x + y - 3, x > 3], x) + (3 < x) & (x < oo) & Eq(x, 3 - y) + + Although checking of assumptions on symbols in relationals + is not done, setting assumptions will affect how certain + relationals might automatically simplify: + + >>> solve(x**2 > 4) + ((-oo < x) & (x < -2)) | ((2 < x) & (x < oo)) + + >>> r = Symbol('r', real=True) + >>> solve(r**2 > 4) + (2 < r) | (r < -2) + + There is currently no algorithm in SymPy that allows you to use + relationships to resolve more than one variable. So the following + does not determine that ``q < 0`` (and trying to solve for ``r`` + and ``q`` will raise an error): + + >>> from sympy import symbols + >>> r, q = symbols('r, q', real=True) + >>> solve([r + q - 3, r > 3], r) + (3 < r) & Eq(r, 3 - q) + + You can directly call the routine that ``solve`` calls + when it encounters a relational: :func:`~.reduce_inequalities`. + It treats Expr like Equality. + + >>> from sympy import reduce_inequalities + >>> reduce_inequalities([x**2 - 4]) + Eq(x, -2) | Eq(x, 2) + + If each relationship contains only one symbol of interest, + the expressions can be processed for multiple symbols: + + >>> reduce_inequalities([0 <= x - 1, y < 3], [x, y]) + (-oo < y) & (1 <= x) & (x < oo) & (y < 3) + + But an error is raised if any relationship has more than one + symbol of interest: + + >>> reduce_inequalities([0 <= x*y - 1, y < 3], [x, y]) + Traceback (most recent call last): + ... + NotImplementedError: + inequality has more than one symbol of interest. + + **Disabling High-Order Explicit Solutions** + + When solving polynomial expressions, you might not want explicit solutions + (which can be quite long). If the expression is univariate, ``CRootOf`` + instances will be returned instead: + + >>> solve(x**3 - x + 1) + [-1/((-1/2 - sqrt(3)*I/2)*(3*sqrt(69)/2 + 27/2)**(1/3)) - + (-1/2 - sqrt(3)*I/2)*(3*sqrt(69)/2 + 27/2)**(1/3)/3, + -(-1/2 + sqrt(3)*I/2)*(3*sqrt(69)/2 + 27/2)**(1/3)/3 - + 1/((-1/2 + sqrt(3)*I/2)*(3*sqrt(69)/2 + 27/2)**(1/3)), + -(3*sqrt(69)/2 + 27/2)**(1/3)/3 - + 1/(3*sqrt(69)/2 + 27/2)**(1/3)] + >>> solve(x**3 - x + 1, cubics=False) + [CRootOf(x**3 - x + 1, 0), + CRootOf(x**3 - x + 1, 1), + CRootOf(x**3 - x + 1, 2)] + + If the expression is multivariate, no solution might be returned: + + >>> solve(x**3 - x + a, x, cubics=False) + [] + + Sometimes solutions will be obtained even when a flag is False because the + expression could be factored. In the following example, the equation can + be factored as the product of a linear and a quadratic factor so explicit + solutions (which did not require solving a cubic expression) are obtained: + + >>> eq = x**3 + 3*x**2 + x - 1 + >>> solve(eq, cubics=False) + [-1, -1 + sqrt(2), -sqrt(2) - 1] + + **Solving Equations Involving Radicals** + + Because of SymPy's use of the principle root, some solutions + to radical equations will be missed unless check=False: + + >>> from sympy import root + >>> eq = root(x**3 - 3*x**2, 3) + 1 - x + >>> solve(eq) + [] + >>> solve(eq, check=False) + [1/3] + + In the above example, there is only a single solution to the + equation. Other expressions will yield spurious roots which + must be checked manually; roots which give a negative argument + to odd-powered radicals will also need special checking: + + >>> from sympy import real_root, S + >>> eq = root(x, 3) - root(x, 5) + S(1)/7 + >>> solve(eq) # this gives 2 solutions but misses a 3rd + [CRootOf(7*x**5 - 7*x**3 + 1, 1)**15, + CRootOf(7*x**5 - 7*x**3 + 1, 2)**15] + >>> sol = solve(eq, check=False) + >>> [abs(eq.subs(x,i).n(2)) for i in sol] + [0.48, 0.e-110, 0.e-110, 0.052, 0.052] + + The first solution is negative so ``real_root`` must be used to see that it + satisfies the expression: + + >>> abs(real_root(eq.subs(x, sol[0])).n(2)) + 0.e-110 + + If the roots of the equation are not real then more care will be + necessary to find the roots, especially for higher order equations. + Consider the following expression: + + >>> expr = root(x, 3) - root(x, 5) + + We will construct a known value for this expression at x = 3 by selecting + the 1-th root for each radical: + + >>> expr1 = root(x, 3, 1) - root(x, 5, 1) + >>> v = expr1.subs(x, -3) + + The ``solve`` function is unable to find any exact roots to this equation: + + >>> eq = Eq(expr, v); eq1 = Eq(expr1, v) + >>> solve(eq, check=False), solve(eq1, check=False) + ([], []) + + The function ``unrad``, however, can be used to get a form of the equation + for which numerical roots can be found: + + >>> from sympy.solvers.solvers import unrad + >>> from sympy import nroots + >>> e, (p, cov) = unrad(eq) + >>> pvals = nroots(e) + >>> inversion = solve(cov, x)[0] + >>> xvals = [inversion.subs(p, i) for i in pvals] + + Although ``eq`` or ``eq1`` could have been used to find ``xvals``, the + solution can only be verified with ``expr1``: + + >>> z = expr - v + >>> [xi.n(chop=1e-9) for xi in xvals if abs(z.subs(x, xi).n()) < 1e-9] + [] + >>> z1 = expr1 - v + >>> [xi.n(chop=1e-9) for xi in xvals if abs(z1.subs(x, xi).n()) < 1e-9] + [-3.0] + + Parameters + ========== + + f : + - a single Expr or Poly that must be zero + - an Equality + - a Relational expression + - a Boolean + - iterable of one or more of the above + + symbols : (object(s) to solve for) specified as + - none given (other non-numeric objects will be used) + - single symbol + - denested list of symbols + (e.g., ``solve(f, x, y)``) + - ordered iterable of symbols + (e.g., ``solve(f, [x, y])``) + + flags : + dict=True (default is False) + Return list (perhaps empty) of solution mappings. + set=True (default is False) + Return list of symbols and set of tuple(s) of solution(s). + exclude=[] (default) + Do not try to solve for any of the free symbols in exclude; + if expressions are given, the free symbols in them will + be extracted automatically. + check=True (default) + If False, do not do any testing of solutions. This can be + useful if you want to include solutions that make any + denominator zero. + numerical=True (default) + Do a fast numerical check if *f* has only one symbol. + minimal=True (default is False) + A very fast, minimal testing. + warn=True (default is False) + Show a warning if ``checksol()`` could not conclude. + simplify=True (default) + Simplify all but polynomials of order 3 or greater before + returning them and (if check is not False) use the + general simplify function on the solutions and the + expression obtained when they are substituted into the + function which should be zero. + force=True (default is False) + Make positive all symbols without assumptions regarding sign. + rational=True (default) + Recast Floats as Rational; if this option is not used, the + system containing Floats may fail to solve because of issues + with polys. If rational=None, Floats will be recast as + rationals but the answer will be recast as Floats. If the + flag is False then nothing will be done to the Floats. + manual=True (default is False) + Do not use the polys/matrix method to solve a system of + equations, solve them one at a time as you might "manually." + implicit=True (default is False) + Allows ``solve`` to return a solution for a pattern in terms of + other functions that contain that pattern; this is only + needed if the pattern is inside of some invertible function + like cos, exp, etc. + particular=True (default is False) + Instructs ``solve`` to try to find a particular solution to + a linear system with as many zeros as possible; this is very + expensive. + quick=True (default is False; ``particular`` must be True) + Selects a fast heuristic to find a solution with many zeros + whereas a value of False uses the very slow method guaranteed + to find the largest number of zeros possible. + cubics=True (default) + Return explicit solutions when cubic expressions are encountered. + When False, quartics and quintics are disabled, too. + quartics=True (default) + Return explicit solutions when quartic expressions are encountered. + When False, quintics are disabled, too. + quintics=True (default) + Return explicit solutions (if possible) when quintic expressions + are encountered. + + See Also + ======== + + rsolve: For solving recurrence relationships + sympy.solvers.ode.dsolve: For solving differential equations + + """ + from .inequalities import reduce_inequalities + + # checking/recording flags + ########################################################################### + + # set solver types explicitly; as soon as one is False + # all the rest will be False + hints = ('cubics', 'quartics', 'quintics') + default = True + for k in hints: + default = flags.setdefault(k, bool(flags.get(k, default))) + + # allow solution to contain symbol if True: + implicit = flags.get('implicit', False) + + # record desire to see warnings + warn = flags.get('warn', False) + + # this flag will be needed for quick exits below, so record + # now -- but don't record `dict` yet since it might change + as_set = flags.get('set', False) + + # keeping track of how f was passed + bare_f = not iterable(f) + + # check flag usage for particular/quick which should only be used + # with systems of equations + if flags.get('quick', None) is not None: + if not flags.get('particular', None): + raise ValueError('when using `quick`, `particular` should be True') + if flags.get('particular', False) and bare_f: + raise ValueError(filldedent(""" + The 'particular/quick' flag is usually used with systems of + equations. Either pass your equation in a list or + consider using a solver like `diophantine` if you are + looking for a solution in integers.""")) + + # sympify everything, creating list of expressions and list of symbols + ########################################################################### + + def _sympified_list(w): + return list(map(sympify, w if iterable(w) else [w])) + f, symbols = (_sympified_list(w) for w in [f, symbols]) + + # preprocess symbol(s) + ########################################################################### + + ordered_symbols = None # were the symbols in a well defined order? + if not symbols: + # get symbols from equations + symbols = set().union(*[fi.free_symbols for fi in f]) + if len(symbols) < len(f): + for fi in f: + pot = preorder_traversal(fi) + for p in pot: + if isinstance(p, AppliedUndef): + if not as_set: + flags['dict'] = True # better show symbols + symbols.add(p) + pot.skip() # don't go any deeper + ordered_symbols = False + symbols = list(ordered(symbols)) # to make it canonical + else: + if len(symbols) == 1 and iterable(symbols[0]): + symbols = symbols[0] + ordered_symbols = symbols and is_sequence(symbols, + include=GeneratorType) + _symbols = list(uniq(symbols)) + if len(_symbols) != len(symbols): + ordered_symbols = False + symbols = list(ordered(symbols)) + else: + symbols = _symbols + + # check for duplicates + if len(symbols) != len(set(symbols)): + raise ValueError('duplicate symbols given') + # remove those not of interest + exclude = flags.pop('exclude', set()) + if exclude: + if isinstance(exclude, Expr): + exclude = [exclude] + exclude = set().union(*[e.free_symbols for e in sympify(exclude)]) + symbols = [s for s in symbols if s not in exclude] + + # preprocess equation(s) + ########################################################################### + + # automatically ignore True values + if isinstance(f, list): + f = [s for s in f if s is not S.true] + + # handle canonicalization of equation types + for i, fi in enumerate(f): + if isinstance(fi, (Eq, Ne)): + if 'ImmutableDenseMatrix' in [type(a).__name__ for a in fi.args]: + fi = fi.lhs - fi.rhs + else: + L, R = fi.args + if isinstance(R, BooleanAtom): + L, R = R, L + if isinstance(L, BooleanAtom): + if isinstance(fi, Ne): + L = ~L + if R.is_Relational: + fi = ~R if L is S.false else R + elif R.is_Symbol: + return L + elif R.is_Boolean and (~R).is_Symbol: + return ~L + else: + raise NotImplementedError(filldedent(''' + Unanticipated argument of Eq when other arg + is True or False. + ''')) + elif isinstance(fi, Eq): + fi = Add(fi.lhs, -fi.rhs, evaluate=False) + f[i] = fi + + # *** dispatch and handle as a system of relationals + # ************************************************** + if fi.is_Relational: + if len(symbols) != 1: + raise ValueError("can only solve for one symbol at a time") + if warn and symbols[0].assumptions0: + warnings.warn(filldedent(""" + \tWarning: assumptions about variable '%s' are + not handled currently.""" % symbols[0])) + return reduce_inequalities(f, symbols=symbols) + + # convert Poly to expression + if isinstance(fi, Poly): + f[i] = fi.as_expr() + + # rewrite hyperbolics in terms of exp if they have symbols of + # interest + f[i] = f[i].replace(lambda w: isinstance(w, HyperbolicFunction) and \ + w.has_free(*symbols), lambda w: w.rewrite(exp)) + + # if we have a Matrix, we need to iterate over its elements again + if f[i].is_Matrix: + try: + f[i] = f[i].as_explicit() + except ValueError: + raise ValueError( + "solve cannot handle matrices with symbolic shape." + ) + bare_f = False + f.extend(list(f[i])) + f[i] = S.Zero + + # if we can split it into real and imaginary parts then do so + freei = f[i].free_symbols + if freei and all(s.is_extended_real or s.is_imaginary for s in freei): + fr, fi = f[i].as_real_imag() + # accept as long as new re, im, arg or atan2 are not introduced + had = f[i].atoms(re, im, arg, atan2) + if fr and fi and fr != fi and not any( + i.atoms(re, im, arg, atan2) - had for i in (fr, fi)): + if bare_f: + bare_f = False + f[i: i + 1] = [fr, fi] + + # real/imag handling ----------------------------- + if any(isinstance(fi, (bool, BooleanAtom)) for fi in f): + if as_set: + return [], set() + return [] + + for i, fi in enumerate(f): + # Abs + while True: + was = fi + fi = fi.replace(Abs, lambda arg: + separatevars(Abs(arg)).rewrite(Piecewise) if arg.has(*symbols) + else Abs(arg)) + if was == fi: + break + + for e in fi.find(Abs): + if e.has(*symbols): + raise NotImplementedError('solving %s when the argument ' + 'is not real or imaginary.' % e) + + # arg + fi = fi.replace(arg, lambda a: arg(a).rewrite(atan2).rewrite(atan)) + + # save changes + f[i] = fi + + # see if re(s) or im(s) appear + freim = [fi for fi in f if fi.has(re, im)] + if freim: + irf = [] + for s in symbols: + if s.is_real or s.is_imaginary: + continue # neither re(x) nor im(x) will appear + # if re(s) or im(s) appear, the auxiliary equation must be present + if any(fi.has(re(s), im(s)) for fi in freim): + irf.append((s, re(s) + S.ImaginaryUnit*im(s))) + if irf: + for s, rhs in irf: + f = [fi.xreplace({s: rhs}) for fi in f] + [s - rhs] + symbols.extend([re(s), im(s)]) + if bare_f: + bare_f = False + flags['dict'] = True + # end of real/imag handling ----------------------------- + + # we can solve for non-symbol entities by replacing them with Dummy symbols + f, symbols, swap_sym = recast_to_symbols(f, symbols) + # this set of symbols (perhaps recast) is needed below + symset = set(symbols) + + # get rid of equations that have no symbols of interest; we don't + # try to solve them because the user didn't ask and they might be + # hard to solve; this means that solutions may be given in terms + # of the eliminated equations e.g. solve((x-y, y-3), x) -> {x: y} + newf = [] + for fi in f: + # let the solver handle equations that.. + # - have no symbols but are expressions + # - have symbols of interest + # - have no symbols of interest but are constant + # but when an expression is not constant and has no symbols of + # interest, it can't change what we obtain for a solution from + # the remaining equations so we don't include it; and if it's + # zero it can be removed and if it's not zero, there is no + # solution for the equation set as a whole + # + # The reason for doing this filtering is to allow an answer + # to be obtained to queries like solve((x - y, y), x); without + # this mod the return value is [] + ok = False + if fi.free_symbols & symset: + ok = True + else: + if fi.is_number: + if fi.is_Number: + if fi.is_zero: + continue + return [] + ok = True + else: + if fi.is_constant(): + ok = True + if ok: + newf.append(fi) + if not newf: + if as_set: + return symbols, set() + return [] + f = newf + del newf + + # mask off any Object that we aren't going to invert: Derivative, + # Integral, etc... so that solving for anything that they contain will + # give an implicit solution + seen = set() + non_inverts = set() + for fi in f: + pot = preorder_traversal(fi) + for p in pot: + if not isinstance(p, Expr) or isinstance(p, Piecewise): + pass + elif (isinstance(p, bool) or + not p.args or + p in symset or + p.is_Add or p.is_Mul or + p.is_Pow and not implicit or + p.is_Function and not implicit) and p.func not in (re, im): + continue + elif p not in seen: + seen.add(p) + if p.free_symbols & symset: + non_inverts.add(p) + else: + continue + pot.skip() + del seen + non_inverts = dict(list(zip(non_inverts, [Dummy() for _ in non_inverts]))) + f = [fi.subs(non_inverts) for fi in f] + + # Both xreplace and subs are needed below: xreplace to force substitution + # inside Derivative, subs to handle non-straightforward substitutions + non_inverts = [(v, k.xreplace(swap_sym).subs(swap_sym)) for k, v in non_inverts.items()] + + # rationalize Floats + floats = False + if flags.get('rational', True) is not False: + for i, fi in enumerate(f): + if fi.has(Float): + floats = True + f[i] = nsimplify(fi, rational=True) + + # capture any denominators before rewriting since + # they may disappear after the rewrite, e.g. issue 14779 + flags['_denominators'] = _simple_dens(f[0], symbols) + + # Any embedded piecewise functions need to be brought out to the + # top level so that the appropriate strategy gets selected. + # However, this is necessary only if one of the piecewise + # functions depends on one of the symbols we are solving for. + def _has_piecewise(e): + if e.is_Piecewise: + return e.has(*symbols) + return any(_has_piecewise(a) for a in e.args) + for i, fi in enumerate(f): + if _has_piecewise(fi): + f[i] = piecewise_fold(fi) + + # expand angles of sums; in general, expand_trig will allow + # more roots to be found but this is not a great solultion + # to not returning a parametric solution, otherwise + # many values can be returned that have a simple + # relationship between values + targs = {t for fi in f for t in fi.atoms(TrigonometricFunction)} + if len(targs) > 1: + add, other = sift(targs, lambda x: x.args[0].is_Add, binary=True) + add, other = [[i for i in l if i.has_free(*symbols)] for l in (add, other)] + trep = {} + for t in add: + a = t.args[0] + ind, dep = a.as_independent(*symbols) + if dep in symbols or -dep in symbols: + # don't let expansion expand wrt anything in ind + n = Dummy() if not ind.is_Number else ind + trep[t] = TR10(t.func(dep + n)).xreplace({n: ind}) + if other and len(other) <= 2: + base = gcd(*[i.args[0] for i in other]) if len(other) > 1 else other[0].args[0] + for i in other: + trep[i] = TR11(i, base) + f = [fi.xreplace(trep) for fi in f] + + # + # try to get a solution + ########################################################################### + if bare_f: + solution = None + if len(symbols) != 1: + solution = _solve_undetermined(f[0], symbols, flags) + if not solution: + solution = _solve(f[0], *symbols, **flags) + else: + linear, solution = _solve_system(f, symbols, **flags) + assert type(solution) is list + assert not solution or type(solution[0]) is dict, solution + # + # postprocessing + ########################################################################### + # capture as_dict flag now (as_set already captured) + as_dict = flags.get('dict', False) + + # define how solution will get unpacked + tuple_format = lambda s: [tuple([i.get(x, x) for x in symbols]) for i in s] + if as_dict or as_set: + unpack = None + elif bare_f: + if len(symbols) == 1: + unpack = lambda s: [i[symbols[0]] for i in s] + elif len(solution) == 1 and len(solution[0]) == len(symbols): + # undetermined linear coeffs solution + unpack = lambda s: s[0] + elif ordered_symbols: + unpack = tuple_format + else: + unpack = lambda s: s + else: + if solution: + if linear and len(solution) == 1: + # if you want the tuple solution for the linear + # case, use `set=True` + unpack = lambda s: s[0] + elif ordered_symbols: + unpack = tuple_format + else: + unpack = lambda s: s + else: + unpack = None + + # Restore masked-off objects + if non_inverts and type(solution) is list: + solution = [{k: v.subs(non_inverts) for k, v in s.items()} + for s in solution] + + # Restore original "symbols" if a dictionary is returned. + # This is not necessary for + # - the single univariate equation case + # since the symbol will have been removed from the solution; + # - the nonlinear poly_system since that only supports zero-dimensional + # systems and those results come back as a list + # + # ** unless there were Derivatives with the symbols, but those were handled + # above. + if swap_sym: + symbols = [swap_sym.get(k, k) for k in symbols] + for i, sol in enumerate(solution): + solution[i] = {swap_sym.get(k, k): v.subs(swap_sym) + for k, v in sol.items()} + + # Get assumptions about symbols, to filter solutions. + # Note that if assumptions about a solution can't be verified, it is still + # returned. + check = flags.get('check', True) + + # restore floats + if floats and solution and flags.get('rational', None) is None: + solution = nfloat(solution, exponent=False) + # nfloat might reveal more duplicates + solution = _remove_duplicate_solutions(solution) + + if check and solution: # assumption checking + warn = flags.get('warn', False) + got_None = [] # solutions for which one or more symbols gave None + no_False = [] # solutions for which no symbols gave False + for sol in solution: + v = fuzzy_and(check_assumptions(val, **symb.assumptions0) + for symb, val in sol.items()) + if v is False: + continue + no_False.append(sol) + if v is None: + got_None.append(sol) + + solution = no_False + if warn and got_None: + warnings.warn(filldedent(""" + \tWarning: assumptions concerning following solution(s) + cannot be checked:""" + '\n\t' + + ', '.join(str(s) for s in got_None))) + + # + # done + ########################################################################### + + if not solution: + if as_set: + return symbols, set() + return [] + + # make orderings canonical for list of dictionaries + if not as_set: # for set, no point in ordering + solution = [{k: s[k] for k in ordered(s)} for s in solution] + solution.sort(key=default_sort_key) + + if not (as_set or as_dict): + return unpack(solution) + + if as_dict: + return solution + + # set output: (symbols, {t1, t2, ...}) from list of dictionaries; + # include all symbols for those that like a verbose solution + # and to resolve any differences in dictionary keys. + # + # The set results can easily be used to make a verbose dict as + # k, v = solve(eqs, syms, set=True) + # sol = [dict(zip(k,i)) for i in v] + # + if ordered_symbols: + k = symbols # keep preferred order + else: + # just unify the symbols for which solutions were found + k = list(ordered(set(flatten(tuple(i.keys()) for i in solution)))) + return k, {tuple([s.get(ki, ki) for ki in k]) for s in solution} + + +def _solve_undetermined(g, symbols, flags): + """solve helper to return a list with one dict (solution) else None + + A direct call to solve_undetermined_coeffs is more flexible and + can return both multiple solutions and handle more than one independent + variable. Here, we have to be more cautious to keep from solving + something that does not look like an undetermined coeffs system -- + to minimize the surprise factor since singularities that cancel are not + prohibited in solve_undetermined_coeffs. + """ + if g.free_symbols - set(symbols): + sol = solve_undetermined_coeffs(g, symbols, **dict(flags, dict=True, set=None)) + if len(sol) == 1: + return sol + + +def _solve(f, *symbols, **flags): + """Return a checked solution for *f* in terms of one or more of the + symbols in the form of a list of dictionaries. + + If no method is implemented to solve the equation, a NotImplementedError + will be raised. In the case that conversion of an expression to a Poly + gives None a ValueError will be raised. + """ + + not_impl_msg = "No algorithms are implemented to solve equation %s" + + if len(symbols) != 1: + # look for solutions for desired symbols that are independent + # of symbols already solved for, e.g. if we solve for x = y + # then no symbol having x in its solution will be returned. + + # First solve for linear symbols (since that is easier and limits + # solution size) and then proceed with symbols appearing + # in a non-linear fashion. Ideally, if one is solving a single + # expression for several symbols, they would have to be + # appear in factors of an expression, but we do not here + # attempt factorization. XXX perhaps handling a Mul + # should come first in this routine whether there is + # one or several symbols. + nonlin_s = [] + got_s = set() + rhs_s = set() + result = [] + for s in symbols: + xi, v = solve_linear(f, symbols=[s]) + if xi == s: + # no need to check but we should simplify if desired + if flags.get('simplify', True): + v = simplify(v) + vfree = v.free_symbols + if vfree & got_s: + # was linear, but has redundant relationship + # e.g. x - y = 0 has y == x is redundant for x == y + # so ignore + continue + rhs_s |= vfree + got_s.add(xi) + result.append({xi: v}) + elif xi: # there might be a non-linear solution if xi is not 0 + nonlin_s.append(s) + if not nonlin_s: + return result + for s in nonlin_s: + try: + soln = _solve(f, s, **flags) + for sol in soln: + if sol[s].free_symbols & got_s: + # depends on previously solved symbols: ignore + continue + got_s.add(s) + result.append(sol) + except NotImplementedError: + continue + if got_s: + return result + else: + raise NotImplementedError(not_impl_msg % f) + + # solve f for a single variable + + symbol = symbols[0] + + # expand binomials only if it has the unknown symbol + f = f.replace(lambda e: isinstance(e, binomial) and e.has(symbol), + lambda e: expand_func(e)) + + # checking will be done unless it is turned off before making a + # recursive call; the variables `checkdens` and `check` are + # captured here (for reference below) in case flag value changes + flags['check'] = checkdens = check = flags.pop('check', True) + + # build up solutions if f is a Mul + if f.is_Mul: + result = set() + for m in f.args: + if m in {S.NegativeInfinity, S.ComplexInfinity, S.Infinity}: + result = set() + break + soln = _vsolve(m, symbol, **flags) + result.update(set(soln)) + result = [{symbol: v} for v in result] + if check: + # all solutions have been checked but now we must + # check that the solutions do not set denominators + # in any factor to zero + dens = flags.get('_denominators', _simple_dens(f, symbols)) + result = [s for s in result if + not any(checksol(den, s, **flags) for den in + dens)] + # set flags for quick exit at end; solutions for each + # factor were already checked and simplified + check = False + flags['simplify'] = False + + elif f.is_Piecewise: + result = set() + if any(e.is_zero for e, c in f.args): + f = f.simplify() # failure imminent w/o help + + cond = neg = True + for expr, cnd in f.args: + # the explicit condition for this expr is the current cond + # and none of the previous conditions + cond = And(neg, cnd) + neg = And(neg, ~cond) + + if expr.is_zero and cond.simplify() != False: + raise NotImplementedError(filldedent(''' + An expression is already zero when %s. + This means that in this *region* the solution + is zero but solve can only represent discrete, + not interval, solutions. If this is a spurious + interval it might be resolved with simplification + of the Piecewise conditions.''' % cond)) + candidates = _vsolve(expr, symbol, **flags) + + for candidate in candidates: + if candidate in result: + # an unconditional value was already there + continue + try: + v = cond.subs(symbol, candidate) + _eval_simplify = getattr(v, '_eval_simplify', None) + if _eval_simplify is not None: + # unconditionally take the simplification of v + v = _eval_simplify(ratio=2, measure=lambda x: 1) + except TypeError: + # incompatible type with condition(s) + continue + if v == False: + continue + if v == True: + result.add(candidate) + else: + result.add(Piecewise( + (candidate, v), + (S.NaN, True))) + # solutions already checked and simplified + # **************************************** + return [{symbol: r} for r in result] + else: + # first see if it really depends on symbol and whether there + # is only a linear solution + f_num, sol = solve_linear(f, symbols=symbols) + if f_num.is_zero or sol is S.NaN: + return [] + elif f_num.is_Symbol: + # no need to check but simplify if desired + if flags.get('simplify', True): + sol = simplify(sol) + return [{f_num: sol}] + + poly = None + # check for a single Add generator + if not f_num.is_Add: + add_args = [i for i in f_num.atoms(Add) + if symbol in i.free_symbols] + if len(add_args) == 1: + gen = add_args[0] + spart = gen.as_independent(symbol)[1].as_base_exp()[0] + if spart == symbol: + try: + poly = Poly(f_num, spart) + except PolynomialError: + pass + + result = False # no solution was obtained + msg = '' # there is no failure message + + # Poly is generally robust enough to convert anything to + # a polynomial and tell us the different generators that it + # contains, so we will inspect the generators identified by + # polys to figure out what to do. + + # try to identify a single generator that will allow us to solve this + # as a polynomial, followed (perhaps) by a change of variables if the + # generator is not a symbol + + try: + if poly is None: + poly = Poly(f_num) + if poly is None: + raise ValueError('could not convert %s to Poly' % f_num) + except GeneratorsNeeded: + simplified_f = simplify(f_num) + if simplified_f != f_num: + return _solve(simplified_f, symbol, **flags) + raise ValueError('expression appears to be a constant') + + gens = [g for g in poly.gens if g.has(symbol)] + + def _as_base_q(x): + """Return (b**e, q) for x = b**(p*e/q) where p/q is the leading + Rational of the exponent of x, e.g. exp(-2*x/3) -> (exp(x), 3) + """ + b, e = x.as_base_exp() + if e.is_Rational: + return b, e.q + if not e.is_Mul: + return x, 1 + c, ee = e.as_coeff_Mul() + if c.is_Rational and c is not S.One: # c could be a Float + return b**ee, c.q + return x, 1 + + if len(gens) > 1: + # If there is more than one generator, it could be that the + # generators have the same base but different powers, e.g. + # >>> Poly(exp(x) + 1/exp(x)) + # Poly(exp(-x) + exp(x), exp(-x), exp(x), domain='ZZ') + # + # If unrad was not disabled then there should be no rational + # exponents appearing as in + # >>> Poly(sqrt(x) + sqrt(sqrt(x))) + # Poly(sqrt(x) + x**(1/4), sqrt(x), x**(1/4), domain='ZZ') + + bases, qs = list(zip(*[_as_base_q(g) for g in gens])) + bases = set(bases) + + if len(bases) > 1 or not all(q == 1 for q in qs): + funcs = {b for b in bases if b.is_Function} + + trig = {_ for _ in funcs if + isinstance(_, TrigonometricFunction)} + other = funcs - trig + if not other and len(funcs.intersection(trig)) > 1: + newf = None + if f_num.is_Add and len(f_num.args) == 2: + # check for sin(x)**p = cos(x)**p + _args = f_num.args + t = a, b = [i.atoms(Function).intersection( + trig) for i in _args] + if all(len(i) == 1 for i in t): + a, b = [i.pop() for i in t] + if isinstance(a, cos): + a, b = b, a + _args = _args[::-1] + if isinstance(a, sin) and isinstance(b, cos + ) and a.args[0] == b.args[0]: + # sin(x) + cos(x) = 0 -> tan(x) + 1 = 0 + newf, _d = (TR2i(_args[0]/_args[1]) + 1 + ).as_numer_denom() + if not _d.is_Number: + newf = None + if newf is None: + newf = TR1(f_num).rewrite(tan) + if newf != f_num: + # don't check the rewritten form --check + # solutions in the un-rewritten form below + flags['check'] = False + result = _solve(newf, symbol, **flags) + flags['check'] = check + + # just a simple case - see if replacement of single function + # clears all symbol-dependent functions, e.g. + # log(x) - log(log(x) - 1) - 3 can be solved even though it has + # two generators. + + if result is False and funcs: + funcs = list(ordered(funcs)) # put shallowest function first + f1 = funcs[0] + t = Dummy('t') + # perform the substitution + ftry = f_num.subs(f1, t) + + # if no Functions left, we can proceed with usual solve + if not ftry.has(symbol): + cv_sols = _solve(ftry, t, **flags) + cv_inv = list(ordered(_vsolve(t - f1, symbol, **flags)))[0] + result = [{symbol: cv_inv.subs(sol)} for sol in cv_sols] + + if result is False: + msg = 'multiple generators %s' % gens + + else: + # e.g. case where gens are exp(x), exp(-x) + u = bases.pop() + t = Dummy('t') + inv = _vsolve(u - t, symbol, **flags) + if isinstance(u, (Pow, exp)): + # this will be resolved by factor in _tsolve but we might + # as well try a simple expansion here to get things in + # order so something like the following will work now without + # having to factor: + # + # >>> eq = (exp(I*(-x-2))+exp(I*(x+2))) + # >>> eq.subs(exp(x),y) # fails + # exp(I*(-x - 2)) + exp(I*(x + 2)) + # >>> eq.expand().subs(exp(x),y) # works + # y**I*exp(2*I) + y**(-I)*exp(-2*I) + def _expand(p): + b, e = p.as_base_exp() + e = expand_mul(e) + return expand_power_exp(b**e) + ftry = f_num.replace( + lambda w: w.is_Pow or isinstance(w, exp), + _expand).subs(u, t) + if not ftry.has(symbol): + soln = _solve(ftry, t, **flags) + result = [{symbol: i.subs(s)} for i in inv for s in soln] + + elif len(gens) == 1: + + # There is only one generator that we are interested in, but + # there may have been more than one generator identified by + # polys (e.g. for symbols other than the one we are interested + # in) so recast the poly in terms of our generator of interest. + # Also use composite=True with f_num since Poly won't update + # poly as documented in issue 8810. + + poly = Poly(f_num, gens[0], composite=True) + + # if we aren't on the tsolve-pass, use roots + if not flags.pop('tsolve', False): + soln = None + deg = poly.degree() + flags['tsolve'] = True + hints = ('cubics', 'quartics', 'quintics') + solvers = {h: flags.get(h) for h in hints} + soln = roots(poly, **solvers) + if sum(soln.values()) < deg: + # e.g. roots(32*x**5 + 400*x**4 + 2032*x**3 + + # 5000*x**2 + 6250*x + 3189) -> {} + # so all_roots is used and RootOf instances are + # returned *unless* the system is multivariate + # or high-order EX domain. + try: + soln = poly.all_roots() + except NotImplementedError: + if not flags.get('incomplete', True): + raise NotImplementedError( + filldedent(''' + Neither high-order multivariate polynomials + nor sorting of EX-domain polynomials is supported. + If you want to see any results, pass keyword incomplete=True to + solve; to see numerical values of roots + for univariate expressions, use nroots. + ''')) + else: + pass + else: + soln = list(soln.keys()) + + if soln is not None: + u = poly.gen + if u != symbol: + try: + t = Dummy('t') + inv = _vsolve(u - t, symbol, **flags) + soln = {i.subs(t, s) for i in inv for s in soln} + except NotImplementedError: + # perhaps _tsolve can handle f_num + soln = None + else: + check = False # only dens need to be checked + if soln is not None: + if len(soln) > 2: + # if the flag wasn't set then unset it since high-order + # results are quite long. Perhaps one could base this + # decision on a certain critical length of the + # roots. In addition, wester test M2 has an expression + # whose roots can be shown to be real with the + # unsimplified form of the solution whereas only one of + # the simplified forms appears to be real. + flags['simplify'] = flags.get('simplify', False) + if soln is not None: + result = [{symbol: v} for v in soln] + + # fallback if above fails + # ----------------------- + if result is False: + # try unrad + if flags.pop('_unrad', True): + try: + u = unrad(f_num, symbol) + except (ValueError, NotImplementedError): + u = False + if u: + eq, cov = u + if cov: + isym, ieq = cov + inv = _vsolve(ieq, symbol, **flags)[0] + rv = {inv.subs(xi) for xi in _solve(eq, isym, **flags)} + else: + try: + rv = set(_vsolve(eq, symbol, **flags)) + except NotImplementedError: + rv = None + if rv is not None: + result = [{symbol: v} for v in rv] + # if the flag wasn't set then unset it since unrad results + # can be quite long or of very high order + flags['simplify'] = flags.get('simplify', False) + else: + pass # for coverage + + # try _tsolve + if result is False: + flags.pop('tsolve', None) # allow tsolve to be used on next pass + try: + soln = _tsolve(f_num, symbol, **flags) + if soln is not None: + result = [{symbol: v} for v in soln] + except PolynomialError: + pass + # ----------- end of fallback ---------------------------- + + if result is False: + raise NotImplementedError('\n'.join([msg, not_impl_msg % f])) + + result = _remove_duplicate_solutions(result) + + if flags.get('simplify', True): + result = [{k: d[k].simplify() for k in d} for d in result] + # Simplification might reveal more duplicates + result = _remove_duplicate_solutions(result) + # we just simplified the solution so we now set the flag to + # False so the simplification doesn't happen again in checksol() + flags['simplify'] = False + + if checkdens: + # reject any result that makes any denom. affirmatively 0; + # if in doubt, keep it + dens = _simple_dens(f, symbols) + result = [r for r in result if + not any(checksol(d, r, **flags) + for d in dens)] + if check: + # keep only results if the check is not False + result = [r for r in result if + checksol(f_num, r, **flags) is not False] + return result + + +def _remove_duplicate_solutions(solutions: list[dict[Expr, Expr]] + ) -> list[dict[Expr, Expr]]: + """Remove duplicates from a list of dicts""" + solutions_set = set() + solutions_new = [] + + for sol in solutions: + solset = frozenset(sol.items()) + if solset not in solutions_set: + solutions_new.append(sol) + solutions_set.add(solset) + + return solutions_new + + +def _solve_system(exprs, symbols, **flags): + """return ``(linear, solution)`` where ``linear`` is True + if the system was linear, else False; ``solution`` + is a list of dictionaries giving solutions for the symbols + """ + if not exprs: + return False, [] + + if flags.pop('_split', True): + # Split the system into connected components + V = exprs + symsset = set(symbols) + exprsyms = {e: e.free_symbols & symsset for e in exprs} + E = [] + sym_indices = {sym: i for i, sym in enumerate(symbols)} + for n, e1 in enumerate(exprs): + for e2 in exprs[:n]: + # Equations are connected if they share a symbol + if exprsyms[e1] & exprsyms[e2]: + E.append((e1, e2)) + G = V, E + subexprs = connected_components(G) + if len(subexprs) > 1: + subsols = [] + linear = True + for subexpr in subexprs: + subsyms = set() + for e in subexpr: + subsyms |= exprsyms[e] + subsyms = sorted(subsyms, key = lambda x: sym_indices[x]) + flags['_split'] = False # skip split step + _linear, subsol = _solve_system(subexpr, subsyms, **flags) + if linear: + linear = linear and _linear + if not isinstance(subsol, list): + subsol = [subsol] + subsols.append(subsol) + # Full solution is cartesian product of subsystems + sols = [] + for soldicts in product(*subsols): + sols.append(dict(item for sd in soldicts + for item in sd.items())) + return linear, sols + + polys = [] + dens = set() + failed = [] + result = [] + solved_syms = [] + linear = True + manual = flags.get('manual', False) + checkdens = check = flags.get('check', True) + + for j, g in enumerate(exprs): + dens.update(_simple_dens(g, symbols)) + i, d = _invert(g, *symbols) + if d in symbols: + if linear: + linear = solve_linear(g, 0, [d])[0] == d + g = d - i + g = g.as_numer_denom()[0] + if manual: + failed.append(g) + continue + + poly = g.as_poly(*symbols, extension=True) + + if poly is not None: + polys.append(poly) + else: + failed.append(g) + + if polys: + if all(p.is_linear for p in polys): + n, m = len(polys), len(symbols) + matrix = zeros(n, m + 1) + + for i, poly in enumerate(polys): + for monom, coeff in poly.terms(): + try: + j = monom.index(1) + matrix[i, j] = coeff + except ValueError: + matrix[i, m] = -coeff + + # returns a dictionary ({symbols: values}) or None + if flags.pop('particular', False): + result = minsolve_linear_system(matrix, *symbols, **flags) + else: + result = solve_linear_system(matrix, *symbols, **flags) + result = [result] if result else [] + if failed: + if result: + solved_syms = list(result[0].keys()) # there is only one result dict + else: + solved_syms = [] + # linear doesn't change + else: + linear = False + if len(symbols) > len(polys): + + free = set().union(*[p.free_symbols for p in polys]) + free = list(ordered(free.intersection(symbols))) + got_s = set() + result = [] + for syms in subsets(free, min(len(free), len(polys))): + try: + # returns [], None or list of tuples + res = solve_poly_system(polys, *syms) + if res: + for r in set(res): + skip = False + for r1 in r: + if got_s and any(ss in r1.free_symbols + for ss in got_s): + # sol depends on previously + # solved symbols: discard it + skip = True + if not skip: + got_s.update(syms) + result.append(dict(list(zip(syms, r)))) + except NotImplementedError: + pass + if got_s: + solved_syms = list(got_s) + else: + failed.extend([g.as_expr() for g in polys]) + else: + try: + result = solve_poly_system(polys, *symbols) + if result: + solved_syms = symbols + result = [dict(list(zip(solved_syms, r))) for r in set(result)] + except NotImplementedError: + failed.extend([g.as_expr() for g in polys]) + solved_syms = [] + + # convert None or [] to [{}] + result = result or [{}] + + if failed: + linear = False + # For each failed equation, see if we can solve for one of the + # remaining symbols from that equation. If so, we update the + # solution set and continue with the next failed equation, + # repeating until we are done or we get an equation that can't + # be solved. + def _ok_syms(e, sort=False): + rv = e.free_symbols & legal + + # Solve first for symbols that have lower degree in the equation. + # Ideally we want to solve firstly for symbols that appear linearly + # with rational coefficients e.g. if e = x*y + z then we should + # solve for z first. + def key(sym): + ep = e.as_poly(sym) + if ep is None: + complexity = (S.Infinity, S.Infinity, S.Infinity) + else: + coeff_syms = ep.LC().free_symbols + complexity = (ep.degree(), len(coeff_syms & rv), len(coeff_syms)) + return complexity + (default_sort_key(sym),) + + if sort: + rv = sorted(rv, key=key) + return rv + + legal = set(symbols) # what we are interested in + # sort so equation with the fewest potential symbols is first + u = Dummy() # used in solution checking + for eq in ordered(failed, lambda _: len(_ok_syms(_))): + newresult = [] + bad_results = [] + hit = False + for r in result: + got_s = set() + # update eq with everything that is known so far + eq2 = eq.subs(r) + # if check is True then we see if it satisfies this + # equation, otherwise we just accept it + if check and r: + b = checksol(u, u, eq2, minimal=True) + if b is not None: + # this solution is sufficient to know whether + # it is valid or not so we either accept or + # reject it, then continue + if b: + newresult.append(r) + else: + bad_results.append(r) + continue + # search for a symbol amongst those available that + # can be solved for + ok_syms = _ok_syms(eq2, sort=True) + if not ok_syms: + if r: + newresult.append(r) + break # skip as it's independent of desired symbols + for s in ok_syms: + try: + soln = _vsolve(eq2, s, **flags) + except NotImplementedError: + continue + # put each solution in r and append the now-expanded + # result in the new result list; use copy since the + # solution for s is being added in-place + for sol in soln: + if got_s and any(ss in sol.free_symbols for ss in got_s): + # sol depends on previously solved symbols: discard it + continue + rnew = r.copy() + for k, v in r.items(): + rnew[k] = v.subs(s, sol) + # and add this new solution + rnew[s] = sol + # check that it is independent of previous solutions + iset = set(rnew.items()) + for i in newresult: + if len(i) < len(iset): + # update i with what is known + i_items_updated = {(k, v.xreplace(rnew)) for k, v in i.items()} + if not i_items_updated - iset: + # this is a superset of a known solution that + # is smaller + break + else: + # keep it + newresult.append(rnew) + hit = True + got_s.add(s) + if not hit: + raise NotImplementedError('could not solve %s' % eq2) + else: + result = newresult + for b in bad_results: + if b in result: + result.remove(b) + + if not result: + return False, [] + + # rely on linear/polynomial system solvers to simplify + # XXX the following tests show that the expressions + # returned are not the same as they would be if simplify + # were applied to this: + # sympy/solvers/ode/tests/test_systems/test__classify_linear_system + # sympy/solvers/tests/test_solvers/test_issue_4886 + # so the docs should be updated to reflect that or else + # the following should be `bool(failed) or not linear` + default_simplify = bool(failed) + if flags.get('simplify', default_simplify): + for r in result: + for k in r: + r[k] = simplify(r[k]) + flags['simplify'] = False # don't need to do so in checksol now + + if checkdens: + result = [r for r in result + if not any(checksol(d, r, **flags) for d in dens)] + + if check and not linear: + result = [r for r in result + if not any(checksol(e, r, **flags) is False for e in exprs)] + + result = [r for r in result if r] + return linear, result + + +def solve_linear(lhs, rhs=0, symbols=[], exclude=[]): + r""" + Return a tuple derived from ``f = lhs - rhs`` that is one of + the following: ``(0, 1)``, ``(0, 0)``, ``(symbol, solution)``, ``(n, d)``. + + Explanation + =========== + + ``(0, 1)`` meaning that ``f`` is independent of the symbols in *symbols* + that are not in *exclude*. + + ``(0, 0)`` meaning that there is no solution to the equation amongst the + symbols given. If the first element of the tuple is not zero, then the + function is guaranteed to be dependent on a symbol in *symbols*. + + ``(symbol, solution)`` where symbol appears linearly in the numerator of + ``f``, is in *symbols* (if given), and is not in *exclude* (if given). No + simplification is done to ``f`` other than a ``mul=True`` expansion, so the + solution will correspond strictly to a unique solution. + + ``(n, d)`` where ``n`` and ``d`` are the numerator and denominator of ``f`` + when the numerator was not linear in any symbol of interest; ``n`` will + never be a symbol unless a solution for that symbol was found (in which case + the second element is the solution, not the denominator). + + Examples + ======== + + >>> from sympy import cancel, Pow + + ``f`` is independent of the symbols in *symbols* that are not in + *exclude*: + + >>> from sympy import cos, sin, solve_linear + >>> from sympy.abc import x, y, z + >>> eq = y*cos(x)**2 + y*sin(x)**2 - y # = y*(1 - 1) = 0 + >>> solve_linear(eq) + (0, 1) + >>> eq = cos(x)**2 + sin(x)**2 # = 1 + >>> solve_linear(eq) + (0, 1) + >>> solve_linear(x, exclude=[x]) + (0, 1) + + The variable ``x`` appears as a linear variable in each of the + following: + + >>> solve_linear(x + y**2) + (x, -y**2) + >>> solve_linear(1/x - y**2) + (x, y**(-2)) + + When not linear in ``x`` or ``y`` then the numerator and denominator are + returned: + + >>> solve_linear(x**2/y**2 - 3) + (x**2 - 3*y**2, y**2) + + If the numerator of the expression is a symbol, then ``(0, 0)`` is + returned if the solution for that symbol would have set any + denominator to 0: + + >>> eq = 1/(1/x - 2) + >>> eq.as_numer_denom() + (x, 1 - 2*x) + >>> solve_linear(eq) + (0, 0) + + But automatic rewriting may cause a symbol in the denominator to + appear in the numerator so a solution will be returned: + + >>> (1/x)**-1 + x + >>> solve_linear((1/x)**-1) + (x, 0) + + Use an unevaluated expression to avoid this: + + >>> solve_linear(Pow(1/x, -1, evaluate=False)) + (0, 0) + + If ``x`` is allowed to cancel in the following expression, then it + appears to be linear in ``x``, but this sort of cancellation is not + done by ``solve_linear`` so the solution will always satisfy the + original expression without causing a division by zero error. + + >>> eq = x**2*(1/x - z**2/x) + >>> solve_linear(cancel(eq)) + (x, 0) + >>> solve_linear(eq) + (x**2*(1 - z**2), x) + + A list of symbols for which a solution is desired may be given: + + >>> solve_linear(x + y + z, symbols=[y]) + (y, -x - z) + + A list of symbols to ignore may also be given: + + >>> solve_linear(x + y + z, exclude=[x]) + (y, -x - z) + + (A solution for ``y`` is obtained because it is the first variable + from the canonically sorted list of symbols that had a linear + solution.) + + """ + if isinstance(lhs, Eq): + if rhs: + raise ValueError(filldedent(''' + If lhs is an Equality, rhs must be 0 but was %s''' % rhs)) + rhs = lhs.rhs + lhs = lhs.lhs + dens = None + eq = lhs - rhs + n, d = eq.as_numer_denom() + if not n: + return S.Zero, S.One + + free = n.free_symbols + if not symbols: + symbols = free + else: + bad = [s for s in symbols if not s.is_Symbol] + if bad: + if len(bad) == 1: + bad = bad[0] + if len(symbols) == 1: + eg = 'solve(%s, %s)' % (eq, symbols[0]) + else: + eg = 'solve(%s, *%s)' % (eq, list(symbols)) + raise ValueError(filldedent(''' + solve_linear only handles symbols, not %s. To isolate + non-symbols use solve, e.g. >>> %s <<<. + ''' % (bad, eg))) + symbols = free.intersection(symbols) + symbols = symbols.difference(exclude) + if not symbols: + return S.Zero, S.One + + # derivatives are easy to do but tricky to analyze to see if they + # are going to disallow a linear solution, so for simplicity we + # just evaluate the ones that have the symbols of interest + derivs = defaultdict(list) + for der in n.atoms(Derivative): + csym = der.free_symbols & symbols + for c in csym: + derivs[c].append(der) + + all_zero = True + for xi in sorted(symbols, key=default_sort_key): # canonical order + # if there are derivatives in this var, calculate them now + if isinstance(derivs[xi], list): + derivs[xi] = {der: der.doit() for der in derivs[xi]} + newn = n.subs(derivs[xi]) + dnewn_dxi = newn.diff(xi) + # dnewn_dxi can be nonzero if it survives differentation by any + # of its free symbols + free = dnewn_dxi.free_symbols + if dnewn_dxi and (not free or any(dnewn_dxi.diff(s) for s in free) or free == symbols): + all_zero = False + if dnewn_dxi is S.NaN: + break + if xi not in dnewn_dxi.free_symbols: + vi = -1/dnewn_dxi*(newn.subs(xi, 0)) + if dens is None: + dens = _simple_dens(eq, symbols) + if not any(checksol(di, {xi: vi}, minimal=True) is True + for di in dens): + # simplify any trivial integral + irep = [(i, i.doit()) for i in vi.atoms(Integral) if + i.function.is_number] + # do a slight bit of simplification + vi = expand_mul(vi.subs(irep)) + return xi, vi + if all_zero: + return S.Zero, S.One + if n.is_Symbol: # no solution for this symbol was found + return S.Zero, S.Zero + return n, d + + +def minsolve_linear_system(system, *symbols, **flags): + r""" + Find a particular solution to a linear system. + + Explanation + =========== + + In particular, try to find a solution with the minimal possible number + of non-zero variables using a naive algorithm with exponential complexity. + If ``quick=True``, a heuristic is used. + + """ + quick = flags.get('quick', False) + # Check if there are any non-zero solutions at all + s0 = solve_linear_system(system, *symbols, **flags) + if not s0 or all(v == 0 for v in s0.values()): + return s0 + if quick: + # We just solve the system and try to heuristically find a nice + # solution. + s = solve_linear_system(system, *symbols) + def update(determined, solution): + delete = [] + for k, v in solution.items(): + solution[k] = v.subs(determined) + if not solution[k].free_symbols: + delete.append(k) + determined[k] = solution[k] + for k in delete: + del solution[k] + determined = {} + update(determined, s) + while s: + # NOTE sort by default_sort_key to get deterministic result + k = max((k for k in s.values()), + key=lambda x: (len(x.free_symbols), default_sort_key(x))) + kfree = k.free_symbols + x = next(reversed(list(ordered(kfree)))) + if len(kfree) != 1: + determined[x] = S.Zero + else: + val = _vsolve(k, x, check=False)[0] + if not val and not any(v.subs(x, val) for v in s.values()): + determined[x] = S.One + else: + determined[x] = val + update(determined, s) + return determined + else: + # We try to select n variables which we want to be non-zero. + # All others will be assumed zero. We try to solve the modified system. + # If there is a non-trivial solution, just set the free variables to + # one. If we do this for increasing n, trying all combinations of + # variables, we will find an optimal solution. + # We speed up slightly by starting at one less than the number of + # variables the quick method manages. + N = len(symbols) + bestsol = minsolve_linear_system(system, *symbols, quick=True) + n0 = len([x for x in bestsol.values() if x != 0]) + for n in range(n0 - 1, 1, -1): + debugf('minsolve: %s', n) + thissol = None + for nonzeros in combinations(range(N), n): + subm = Matrix([system.col(i).T for i in nonzeros] + [system.col(-1).T]).T + s = solve_linear_system(subm, *[symbols[i] for i in nonzeros]) + if s and not all(v == 0 for v in s.values()): + subs = [(symbols[v], S.One) for v in nonzeros] + for k, v in s.items(): + s[k] = v.subs(subs) + for sym in symbols: + if sym not in s: + if symbols.index(sym) in nonzeros: + s[sym] = S.One + else: + s[sym] = S.Zero + thissol = s + break + if thissol is None: + break + bestsol = thissol + return bestsol + + +def solve_linear_system(system, *symbols, **flags): + r""" + Solve system of $N$ linear equations with $M$ variables, which means + both under- and overdetermined systems are supported. + + Explanation + =========== + + The possible number of solutions is zero, one, or infinite. Respectively, + this procedure will return None or a dictionary with solutions. In the + case of underdetermined systems, all arbitrary parameters are skipped. + This may cause a situation in which an empty dictionary is returned. + In that case, all symbols can be assigned arbitrary values. + + Input to this function is a $N\times M + 1$ matrix, which means it has + to be in augmented form. If you prefer to enter $N$ equations and $M$ + unknowns then use ``solve(Neqs, *Msymbols)`` instead. Note: a local + copy of the matrix is made by this routine so the matrix that is + passed will not be modified. + + The algorithm used here is fraction-free Gaussian elimination, + which results, after elimination, in an upper-triangular matrix. + Then solutions are found using back-substitution. This approach + is more efficient and compact than the Gauss-Jordan method. + + Examples + ======== + + >>> from sympy import Matrix, solve_linear_system + >>> from sympy.abc import x, y + + Solve the following system:: + + x + 4 y == 2 + -2 x + y == 14 + + >>> system = Matrix(( (1, 4, 2), (-2, 1, 14))) + >>> solve_linear_system(system, x, y) + {x: -6, y: 2} + + A degenerate system returns an empty dictionary: + + >>> system = Matrix(( (0,0,0), (0,0,0) )) + >>> solve_linear_system(system, x, y) + {} + + """ + assert system.shape[1] == len(symbols) + 1 + + # This is just a wrapper for solve_lin_sys + eqs = list(system * Matrix(symbols + (-1,))) + eqs, ring = sympy_eqs_to_ring(eqs, symbols) + sol = solve_lin_sys(eqs, ring, _raw=False) + if sol is not None: + sol = {sym:val for sym, val in sol.items() if sym != val} + return sol + + +def solve_undetermined_coeffs(equ, coeffs, *syms, **flags): + r""" + Solve a system of equations in $k$ parameters that is formed by + matching coefficients in variables ``coeffs`` that are on + factors dependent on the remaining variables (or those given + explicitly by ``syms``. + + Explanation + =========== + + The result of this function is a dictionary with symbolic values of those + parameters with respect to coefficients in $q$ -- empty if there + is no solution or coefficients do not appear in the equation -- else + None (if the system was not recognized). If there is more than one + solution, the solutions are passed as a list. The output can be modified using + the same semantics as for `solve` since the flags that are passed are sent + directly to `solve` so, for example the flag ``dict=True`` will always return a list + of solutions as dictionaries. + + This function accepts both Equality and Expr class instances. + The solving process is most efficient when symbols are specified + in addition to parameters to be determined, but an attempt to + determine them (if absent) will be made. If an expected solution is not + obtained (and symbols were not specified) try specifying them. + + Examples + ======== + + >>> from sympy import Eq, solve_undetermined_coeffs + >>> from sympy.abc import a, b, c, h, p, k, x, y + + >>> solve_undetermined_coeffs(Eq(a*x + a + b, x/2), [a, b], x) + {a: 1/2, b: -1/2} + >>> solve_undetermined_coeffs(a - 2, [a]) + {a: 2} + + The equation can be nonlinear in the symbols: + + >>> X, Y, Z = y, x**y, y*x**y + >>> eq = a*X + b*Y + c*Z - X - 2*Y - 3*Z + >>> coeffs = a, b, c + >>> syms = x, y + >>> solve_undetermined_coeffs(eq, coeffs, syms) + {a: 1, b: 2, c: 3} + + And the system can be nonlinear in coefficients, too, but if + there is only a single solution, it will be returned as a + dictionary: + + >>> eq = a*x**2 + b*x + c - ((x - h)**2 + 4*p*k)/4/p + >>> solve_undetermined_coeffs(eq, (h, p, k), x) + {h: -b/(2*a), k: (4*a*c - b**2)/(4*a), p: 1/(4*a)} + + Multiple solutions are always returned in a list: + + >>> solve_undetermined_coeffs(a**2*x + b - x, [a, b], x) + [{a: -1, b: 0}, {a: 1, b: 0}] + + Using flag ``dict=True`` (in keeping with semantics in :func:`~.solve`) + will force the result to always be a list with any solutions + as elements in that list. + + >>> solve_undetermined_coeffs(a*x - 2*x, [a], dict=True) + [{a: 2}] + """ + if not (coeffs and all(i.is_Symbol for i in coeffs)): + raise ValueError('must provide symbols for coeffs') + + if isinstance(equ, Eq): + eq = equ.lhs - equ.rhs + else: + eq = equ + + ceq = cancel(eq) + xeq = _mexpand(ceq.as_numer_denom()[0], recursive=True) + + free = xeq.free_symbols + coeffs = free & set(coeffs) + if not coeffs: + return ([], {}) if flags.get('set', None) else [] # solve(0, x) -> [] + + if not syms: + # e.g. A*exp(x) + B - (exp(x) + y) separated into parts that + # don't/do depend on coeffs gives + # -(exp(x) + y), A*exp(x) + B + # then see what symbols are common to both + # {x} = {x, A, B} - {x, y} + ind, dep = xeq.as_independent(*coeffs, as_Add=True) + dfree = dep.free_symbols + syms = dfree & ind.free_symbols + if not syms: + # but if the system looks like (a + b)*x + b - c + # then {} = {a, b, x} - c + # so calculate {x} = {a, b, x} - {a, b} + syms = dfree - set(coeffs) + if not syms: + syms = [Dummy()] + else: + if len(syms) == 1 and iterable(syms[0]): + syms = syms[0] + e, s, _ = recast_to_symbols([xeq], syms) + xeq = e[0] + syms = s + + # find the functional forms in which symbols appear + + gens = set(xeq.as_coefficients_dict(*syms).keys()) - {1} + cset = set(coeffs) + if any(g.has_xfree(cset) for g in gens): + return # a generator contained a coefficient symbol + + # make sure we are working with symbols for generators + + e, gens, _ = recast_to_symbols([xeq], list(gens)) + xeq = e[0] + + # collect coefficients in front of generators + + system = list(collect(xeq, gens, evaluate=False).values()) + + # get a solution + + soln = solve(system, coeffs, **flags) + + # unpack unless told otherwise if length is 1 + + settings = flags.get('dict', None) or flags.get('set', None) + if type(soln) is dict or settings or len(soln) != 1: + return soln + return soln[0] + + +def solve_linear_system_LU(matrix, syms): + """ + Solves the augmented matrix system using ``LUsolve`` and returns a + dictionary in which solutions are keyed to the symbols of *syms* as ordered. + + Explanation + =========== + + The matrix must be invertible. + + Examples + ======== + + >>> from sympy import Matrix, solve_linear_system_LU + >>> from sympy.abc import x, y, z + + >>> solve_linear_system_LU(Matrix([ + ... [1, 2, 0, 1], + ... [3, 2, 2, 1], + ... [2, 0, 0, 1]]), [x, y, z]) + {x: 1/2, y: 1/4, z: -1/2} + + See Also + ======== + + LUsolve + + """ + if matrix.rows != matrix.cols - 1: + raise ValueError("Rows should be equal to columns - 1") + A = matrix[:matrix.rows, :matrix.rows] + b = matrix[:, matrix.cols - 1:] + soln = A.LUsolve(b) + solutions = {} + for i in range(soln.rows): + solutions[syms[i]] = soln[i, 0] + return solutions + + +def det_perm(M): + """ + Return the determinant of *M* by using permutations to select factors. + + Explanation + =========== + + For sizes larger than 8 the number of permutations becomes prohibitively + large, or if there are no symbols in the matrix, it is better to use the + standard determinant routines (e.g., ``M.det()``.) + + See Also + ======== + + det_minor + det_quick + + """ + args = [] + s = True + n = M.rows + list_ = M.flat() + for perm in generate_bell(n): + fac = [] + idx = 0 + for j in perm: + fac.append(list_[idx + j]) + idx += n + term = Mul(*fac) # disaster with unevaluated Mul -- takes forever for n=7 + args.append(term if s else -term) + s = not s + return Add(*args) + + +def det_minor(M): + """ + Return the ``det(M)`` computed from minors without + introducing new nesting in products. + + See Also + ======== + + det_perm + det_quick + + """ + n = M.rows + if n == 2: + return M[0, 0]*M[1, 1] - M[1, 0]*M[0, 1] + else: + return sum((1, -1)[i % 2]*Add(*[M[0, i]*d for d in + Add.make_args(det_minor(M.minor_submatrix(0, i)))]) + if M[0, i] else S.Zero for i in range(n)) + + +def det_quick(M, method=None): + """ + Return ``det(M)`` assuming that either + there are lots of zeros or the size of the matrix + is small. If this assumption is not met, then the normal + Matrix.det function will be used with method = ``method``. + + See Also + ======== + + det_minor + det_perm + + """ + if any(i.has(Symbol) for i in M): + if M.rows < 8 and all(i.has(Symbol) for i in M): + return det_perm(M) + return det_minor(M) + else: + return M.det(method=method) if method else M.det() + + +def inv_quick(M): + """Return the inverse of ``M``, assuming that either + there are lots of zeros or the size of the matrix + is small. + """ + if not all(i.is_Number for i in M): + if not any(i.is_Number for i in M): + det = lambda _: det_perm(_) + else: + det = lambda _: det_minor(_) + else: + return M.inv() + n = M.rows + d = det(M) + if d == S.Zero: + raise NonInvertibleMatrixError("Matrix det == 0; not invertible") + ret = zeros(n) + s1 = -1 + for i in range(n): + s = s1 = -s1 + for j in range(n): + di = det(M.minor_submatrix(i, j)) + ret[j, i] = s*di/d + s = -s + return ret + + +# these are functions that have multiple inverse values per period +multi_inverses = { + sin: lambda x: (asin(x), S.Pi - asin(x)), + cos: lambda x: (acos(x), 2*S.Pi - acos(x)), +} + + +def _vsolve(e, s, **flags): + """return list of scalar values for the solution of e for symbol s""" + return [i[s] for i in _solve(e, s, **flags)] + + +def _tsolve(eq, sym, **flags): + """ + Helper for ``_solve`` that solves a transcendental equation with respect + to the given symbol. Various equations containing powers and logarithms, + can be solved. + + There is currently no guarantee that all solutions will be returned or + that a real solution will be favored over a complex one. + + Either a list of potential solutions will be returned or None will be + returned (in the case that no method was known to get a solution + for the equation). All other errors (like the inability to cast an + expression as a Poly) are unhandled. + + Examples + ======== + + >>> from sympy import log, ordered + >>> from sympy.solvers.solvers import _tsolve as tsolve + >>> from sympy.abc import x + + >>> list(ordered(tsolve(3**(2*x + 5) - 4, x))) + [-5/2 + log(2)/log(3), (-5*log(3)/2 + log(2) + I*pi)/log(3)] + + >>> tsolve(log(x) + 2*x, x) + [LambertW(2)/2] + + """ + if 'tsolve_saw' not in flags: + flags['tsolve_saw'] = [] + if eq in flags['tsolve_saw']: + return None + else: + flags['tsolve_saw'].append(eq) + + rhs, lhs = _invert(eq, sym) + + if lhs == sym: + return [rhs] + try: + if lhs.is_Add: + # it's time to try factoring; powdenest is used + # to try get powers in standard form for better factoring + f = factor(powdenest(lhs - rhs)) + if f.is_Mul: + return _vsolve(f, sym, **flags) + if rhs: + f = logcombine(lhs, force=flags.get('force', True)) + if f.count(log) != lhs.count(log): + if isinstance(f, log): + return _vsolve(f.args[0] - exp(rhs), sym, **flags) + return _tsolve(f - rhs, sym, **flags) + + elif lhs.is_Pow: + if lhs.exp.is_Integer: + if lhs - rhs != eq: + return _vsolve(lhs - rhs, sym, **flags) + + if sym not in lhs.exp.free_symbols: + return _vsolve(lhs.base - rhs**(1/lhs.exp), sym, **flags) + + # _tsolve calls this with Dummy before passing the actual number in. + if any(t.is_Dummy for t in rhs.free_symbols): + raise NotImplementedError # _tsolve will call here again... + + # a ** g(x) == 0 + if not rhs: + # f(x)**g(x) only has solutions where f(x) == 0 and g(x) != 0 at + # the same place + sol_base = _vsolve(lhs.base, sym, **flags) + return [s for s in sol_base if lhs.exp.subs(sym, s) != 0] # XXX use checksol here? + + # a ** g(x) == b + if not lhs.base.has(sym): + if lhs.base == 0: + return _vsolve(lhs.exp, sym, **flags) if rhs != 0 else [] + + # Gets most solutions... + if lhs.base == rhs.as_base_exp()[0]: + # handles case when bases are equal + sol = _vsolve(lhs.exp - rhs.as_base_exp()[1], sym, **flags) + else: + # handles cases when bases are not equal and exp + # may or may not be equal + f = exp(log(lhs.base)*lhs.exp) - exp(log(rhs)) + sol = _vsolve(f, sym, **flags) + + # Check for duplicate solutions + def equal(expr1, expr2): + _ = Dummy() + eq = checksol(expr1 - _, _, expr2) + if eq is None: + if nsimplify(expr1) != nsimplify(expr2): + return False + # they might be coincidentally the same + # so check more rigorously + eq = expr1.equals(expr2) # XXX expensive but necessary? + return eq + + # Guess a rational exponent + e_rat = nsimplify(log(abs(rhs))/log(abs(lhs.base))) + e_rat = simplify(posify(e_rat)[0]) + n, d = fraction(e_rat) + if expand(lhs.base**n - rhs**d) == 0: + sol = [s for s in sol if not equal(lhs.exp.subs(sym, s), e_rat)] + sol.extend(_vsolve(lhs.exp - e_rat, sym, **flags)) + + return list(set(sol)) + + # f(x) ** g(x) == c + else: + sol = [] + logform = lhs.exp*log(lhs.base) - log(rhs) + if logform != lhs - rhs: + try: + sol.extend(_vsolve(logform, sym, **flags)) + except NotImplementedError: + pass + + # Collect possible solutions and check with substitution later. + check = [] + if rhs == 1: + # f(x) ** g(x) = 1 -- g(x)=0 or f(x)=+-1 + check.extend(_vsolve(lhs.exp, sym, **flags)) + check.extend(_vsolve(lhs.base - 1, sym, **flags)) + check.extend(_vsolve(lhs.base + 1, sym, **flags)) + elif rhs.is_Rational: + for d in (i for i in divisors(abs(rhs.p)) if i != 1): + e, t = integer_log(rhs.p, d) + if not t: + continue # rhs.p != d**b + for s in divisors(abs(rhs.q)): + if s**e== rhs.q: + r = Rational(d, s) + check.extend(_vsolve(lhs.base - r, sym, **flags)) + check.extend(_vsolve(lhs.base + r, sym, **flags)) + check.extend(_vsolve(lhs.exp - e, sym, **flags)) + elif rhs.is_irrational: + b_l, e_l = lhs.base.as_base_exp() + n, d = (e_l*lhs.exp).as_numer_denom() + b, e = sqrtdenest(rhs).as_base_exp() + check = [sqrtdenest(i) for i in (_vsolve(lhs.base - b, sym, **flags))] + check.extend([sqrtdenest(i) for i in (_vsolve(lhs.exp - e, sym, **flags))]) + if e_l*d != 1: + check.extend(_vsolve(b_l**n - rhs**(e_l*d), sym, **flags)) + for s in check: + ok = checksol(eq, sym, s) + if ok is None: + ok = eq.subs(sym, s).equals(0) + if ok: + sol.append(s) + return list(set(sol)) + + elif lhs.is_Function and len(lhs.args) == 1: + if lhs.func in multi_inverses: + # sin(x) = 1/3 -> x - asin(1/3) & x - (pi - asin(1/3)) + soln = [] + for i in multi_inverses[type(lhs)](rhs): + soln.extend(_vsolve(lhs.args[0] - i, sym, **flags)) + return list(set(soln)) + elif lhs.func == LambertW: + return _vsolve(lhs.args[0] - rhs*exp(rhs), sym, **flags) + + rewrite = lhs.rewrite(exp) + rewrite = rebuild(rewrite) # avoid rewrites involving evaluate=False + if rewrite != lhs: + return _vsolve(rewrite - rhs, sym, **flags) + except NotImplementedError: + pass + + # maybe it is a lambert pattern + if flags.pop('bivariate', True): + # lambert forms may need some help being recognized, e.g. changing + # 2**(3*x) + x**3*log(2)**3 + 3*x**2*log(2)**2 + 3*x*log(2) + 1 + # to 2**(3*x) + (x*log(2) + 1)**3 + + # make generator in log have exponent of 1 + logs = eq.atoms(log) + spow = min( + {i.exp for j in logs for i in j.atoms(Pow) + if i.base == sym} or {1}) + if spow != 1: + p = sym**spow + u = Dummy('bivariate-cov') + ueq = eq.subs(p, u) + if not ueq.has_free(sym): + sol = _vsolve(ueq, u, **flags) + inv = _vsolve(p - u, sym) + return [i.subs(u, s) for i in inv for s in sol] + + g = _filtered_gens(eq.as_poly(), sym) + up_or_log = set() + for gi in g: + if isinstance(gi, (exp, log)) or (gi.is_Pow and gi.base == S.Exp1): + up_or_log.add(gi) + elif gi.is_Pow: + gisimp = powdenest(expand_power_exp(gi)) + if gisimp.is_Pow and sym in gisimp.exp.free_symbols: + up_or_log.add(gi) + eq_down = expand_log(expand_power_exp(eq)).subs( + dict(list(zip(up_or_log, [0]*len(up_or_log))))) + eq = expand_power_exp(factor(eq_down, deep=True) + (eq - eq_down)) + rhs, lhs = _invert(eq, sym) + if lhs.has(sym): + try: + poly = lhs.as_poly() + g = _filtered_gens(poly, sym) + _eq = lhs - rhs + sols = _solve_lambert(_eq, sym, g) + # use a simplified form if it satisfies eq + # and has fewer operations + for n, s in enumerate(sols): + ns = nsimplify(s) + if ns != s and ns.count_ops() <= s.count_ops(): + ok = checksol(_eq, sym, ns) + if ok is None: + ok = _eq.subs(sym, ns).equals(0) + if ok: + sols[n] = ns + return sols + except NotImplementedError: + # maybe it's a convoluted function + if len(g) == 2: + try: + gpu = bivariate_type(lhs - rhs, *g) + if gpu is None: + raise NotImplementedError + g, p, u = gpu + flags['bivariate'] = False + inversion = _tsolve(g - u, sym, **flags) + if inversion: + sol = _vsolve(p, u, **flags) + return list({i.subs(u, s) + for i in inversion for s in sol}) + except NotImplementedError: + pass + else: + pass + + if flags.pop('force', True): + flags['force'] = False + pos, reps = posify(lhs - rhs) + if rhs == S.ComplexInfinity: + return [] + for u, s in reps.items(): + if s == sym: + break + else: + u = sym + if pos.has(u): + try: + soln = _vsolve(pos, u, **flags) + return [s.subs(reps) for s in soln] + except NotImplementedError: + pass + else: + pass # here for coverage + + return # here for coverage + + +# TODO: option for calculating J numerically + +@conserve_mpmath_dps +def nsolve(*args, dict=False, **kwargs): + r""" + Solve a nonlinear equation system numerically: ``nsolve(f, [args,] x0, + modules=['mpmath'], **kwargs)``. + + Explanation + =========== + + ``f`` is a vector function of symbolic expressions representing the system. + *args* are the variables. If there is only one variable, this argument can + be omitted. ``x0`` is a starting vector close to a solution. + + Use the modules keyword to specify which modules should be used to + evaluate the function and the Jacobian matrix. Make sure to use a module + that supports matrices. For more information on the syntax, please see the + docstring of ``lambdify``. + + If the keyword arguments contain ``dict=True`` (default is False) ``nsolve`` + will return a list (perhaps empty) of solution mappings. This might be + especially useful if you want to use ``nsolve`` as a fallback to solve since + using the dict argument for both methods produces return values of + consistent type structure. Please note: to keep this consistent with + ``solve``, the solution will be returned in a list even though ``nsolve`` + (currently at least) only finds one solution at a time. + + Overdetermined systems are supported. + + Examples + ======== + + >>> from sympy import Symbol, nsolve + >>> import mpmath + >>> mpmath.mp.dps = 15 + >>> x1 = Symbol('x1') + >>> x2 = Symbol('x2') + >>> f1 = 3 * x1**2 - 2 * x2**2 - 1 + >>> f2 = x1**2 - 2 * x1 + x2**2 + 2 * x2 - 8 + >>> print(nsolve((f1, f2), (x1, x2), (-1, 1))) + Matrix([[-1.19287309935246], [1.27844411169911]]) + + For one-dimensional functions the syntax is simplified: + + >>> from sympy import sin, nsolve + >>> from sympy.abc import x + >>> nsolve(sin(x), x, 2) + 3.14159265358979 + >>> nsolve(sin(x), 2) + 3.14159265358979 + + To solve with higher precision than the default, use the prec argument: + + >>> from sympy import cos + >>> nsolve(cos(x) - x, 1) + 0.739085133215161 + >>> nsolve(cos(x) - x, 1, prec=50) + 0.73908513321516064165531208767387340401341175890076 + >>> cos(_) + 0.73908513321516064165531208767387340401341175890076 + + To solve for complex roots of real functions, a nonreal initial point + must be specified: + + >>> from sympy import I + >>> nsolve(x**2 + 2, I) + 1.4142135623731*I + + ``mpmath.findroot`` is used and you can find their more extensive + documentation, especially concerning keyword parameters and + available solvers. Note, however, that functions which are very + steep near the root, the verification of the solution may fail. In + this case you should use the flag ``verify=False`` and + independently verify the solution. + + >>> from sympy import cos, cosh + >>> f = cos(x)*cosh(x) - 1 + >>> nsolve(f, 3.14*100) + Traceback (most recent call last): + ... + ValueError: Could not find root within given tolerance. (1.39267e+230 > 2.1684e-19) + >>> ans = nsolve(f, 3.14*100, verify=False); ans + 312.588469032184 + >>> f.subs(x, ans).n(2) + 2.1e+121 + >>> (f/f.diff(x)).subs(x, ans).n(2) + 7.4e-15 + + One might safely skip the verification if bounds of the root are known + and a bisection method is used: + + >>> bounds = lambda i: (3.14*i, 3.14*(i + 1)) + >>> nsolve(f, bounds(100), solver='bisect', verify=False) + 315.730061685774 + + Alternatively, a function may be better behaved when the + denominator is ignored. Since this is not always the case, however, + the decision of what function to use is left to the discretion of + the user. + + >>> eq = x**2/(1 - x)/(1 - 2*x)**2 - 100 + >>> nsolve(eq, 0.46) + Traceback (most recent call last): + ... + ValueError: Could not find root within given tolerance. (10000 > 2.1684e-19) + Try another starting point or tweak arguments. + >>> nsolve(eq.as_numer_denom()[0], 0.46) + 0.46792545969349058 + + """ + # there are several other SymPy functions that use method= so + # guard against that here + if 'method' in kwargs: + raise ValueError(filldedent(''' + Keyword "method" should not be used in this context. When using + some mpmath solvers directly, the keyword "method" is + used, but when using nsolve (and findroot) the keyword to use is + "solver".''')) + + if 'prec' in kwargs: + import mpmath + mpmath.mp.dps = kwargs.pop('prec') + + # keyword argument to return result as a dictionary + as_dict = dict + from builtins import dict # to unhide the builtin + + # interpret arguments + if len(args) == 3: + f = args[0] + fargs = args[1] + x0 = args[2] + if iterable(fargs) and iterable(x0): + if len(x0) != len(fargs): + raise TypeError('nsolve expected exactly %i guess vectors, got %i' + % (len(fargs), len(x0))) + elif len(args) == 2: + f = args[0] + fargs = None + x0 = args[1] + if iterable(f): + raise TypeError('nsolve expected 3 arguments, got 2') + elif len(args) < 2: + raise TypeError('nsolve expected at least 2 arguments, got %i' + % len(args)) + else: + raise TypeError('nsolve expected at most 3 arguments, got %i' + % len(args)) + modules = kwargs.get('modules', ['mpmath']) + if iterable(f): + f = list(f) + for i, fi in enumerate(f): + if isinstance(fi, Eq): + f[i] = fi.lhs - fi.rhs + f = Matrix(f).T + if iterable(x0): + x0 = list(x0) + if not isinstance(f, Matrix): + # assume it's a SymPy expression + if isinstance(f, Eq): + f = f.lhs - f.rhs + elif f.is_Relational: + raise TypeError('nsolve cannot accept inequalities') + syms = f.free_symbols + if fargs is None: + fargs = syms.copy().pop() + if not (len(syms) == 1 and (fargs in syms or fargs[0] in syms)): + raise ValueError(filldedent(''' + expected a one-dimensional and numerical function''')) + + # the function is much better behaved if there is no denominator + # but sending the numerator is left to the user since sometimes + # the function is better behaved when the denominator is present + # e.g., issue 11768 + + f = lambdify(fargs, f, modules) + x = sympify(findroot(f, x0, **kwargs)) + if as_dict: + return [{fargs: x}] + return x + + if len(fargs) > f.cols: + raise NotImplementedError(filldedent(''' + need at least as many equations as variables''')) + verbose = kwargs.get('verbose', False) + if verbose: + print('f(x):') + print(f) + # derive Jacobian + J = f.jacobian(fargs) + if verbose: + print('J(x):') + print(J) + # create functions + f = lambdify(fargs, f.T, modules) + J = lambdify(fargs, J, modules) + # solve the system numerically + x = findroot(f, x0, J=J, **kwargs) + if as_dict: + return [dict(zip(fargs, [sympify(xi) for xi in x]))] + return Matrix(x) + + +def _invert(eq, *symbols, **kwargs): + """ + Return tuple (i, d) where ``i`` is independent of *symbols* and ``d`` + contains symbols. + + Explanation + =========== + + ``i`` and ``d`` are obtained after recursively using algebraic inversion + until an uninvertible ``d`` remains. If there are no free symbols then + ``d`` will be zero. Some (but not necessarily all) solutions to the + expression ``i - d`` will be related to the solutions of the original + expression. + + Examples + ======== + + >>> from sympy.solvers.solvers import _invert as invert + >>> from sympy import sqrt, cos + >>> from sympy.abc import x, y + >>> invert(x - 3) + (3, x) + >>> invert(3) + (3, 0) + >>> invert(2*cos(x) - 1) + (1/2, cos(x)) + >>> invert(sqrt(x) - 3) + (3, sqrt(x)) + >>> invert(sqrt(x) + y, x) + (-y, sqrt(x)) + >>> invert(sqrt(x) + y, y) + (-sqrt(x), y) + >>> invert(sqrt(x) + y, x, y) + (0, sqrt(x) + y) + + If there is more than one symbol in a power's base and the exponent + is not an Integer, then the principal root will be used for the + inversion: + + >>> invert(sqrt(x + y) - 2) + (4, x + y) + >>> invert(sqrt(x + y) + 2) # note +2 instead of -2 + (4, x + y) + + If the exponent is an Integer, setting ``integer_power`` to True + will force the principal root to be selected: + + >>> invert(x**2 - 4, integer_power=True) + (2, x) + + """ + eq = sympify(eq) + if eq.args: + # make sure we are working with flat eq + eq = eq.func(*eq.args) + free = eq.free_symbols + if not symbols: + symbols = free + if not free & set(symbols): + return eq, S.Zero + + dointpow = bool(kwargs.get('integer_power', False)) + + lhs = eq + rhs = S.Zero + while True: + was = lhs + while True: + indep, dep = lhs.as_independent(*symbols) + + # dep + indep == rhs + if lhs.is_Add: + # this indicates we have done it all + if indep.is_zero: + break + + lhs = dep + rhs -= indep + + # dep * indep == rhs + else: + # this indicates we have done it all + if indep is S.One: + break + + lhs = dep + rhs /= indep + + # collect like-terms in symbols + if lhs.is_Add: + terms = {} + for a in lhs.args: + i, d = a.as_independent(*symbols) + terms.setdefault(d, []).append(i) + if any(len(v) > 1 for v in terms.values()): + args = [] + for d, i in terms.items(): + if len(i) > 1: + args.append(Add(*i)*d) + else: + args.append(i[0]*d) + lhs = Add(*args) + + # if it's a two-term Add with rhs = 0 and two powers we can get the + # dependent terms together, e.g. 3*f(x) + 2*g(x) -> f(x)/g(x) = -2/3 + if lhs.is_Add and not rhs and len(lhs.args) == 2 and \ + not lhs.is_polynomial(*symbols): + a, b = ordered(lhs.args) + ai, ad = a.as_independent(*symbols) + bi, bd = b.as_independent(*symbols) + if any(_ispow(i) for i in (ad, bd)): + a_base, a_exp = ad.as_base_exp() + b_base, b_exp = bd.as_base_exp() + if a_base == b_base and a_exp.extract_additively(b_exp) is None: + # a = -b and exponents do not have canceling terms/factors + # e.g. if exponents were 3*x and x then the ratio would have + # an exponent of 2*x: one of the roots would be lost + rat = powsimp(powdenest(ad/bd)) + lhs = rat + rhs = -bi/ai + else: + rat = ad/bd + _lhs = powsimp(ad/bd) + if _lhs != rat: + lhs = _lhs + rhs = -bi/ai + elif ai == -bi: + if isinstance(ad, Function) and ad.func == bd.func: + if len(ad.args) == len(bd.args) == 1: + lhs = ad.args[0] - bd.args[0] + elif len(ad.args) == len(bd.args): + # should be able to solve + # f(x, y) - f(2 - x, 0) == 0 -> x == 1 + raise NotImplementedError( + 'equal function with more than 1 argument') + else: + raise ValueError( + 'function with different numbers of args') + + elif lhs.is_Mul and any(_ispow(a) for a in lhs.args): + lhs = powsimp(powdenest(lhs)) + + if lhs.is_Function: + if hasattr(lhs, 'inverse') and lhs.inverse() is not None and len(lhs.args) == 1: + # -1 + # f(x) = g -> x = f (g) + # + # /!\ inverse should not be defined if there are multiple values + # for the function -- these are handled in _tsolve + # + rhs = lhs.inverse()(rhs) + lhs = lhs.args[0] + elif isinstance(lhs, atan2): + y, x = lhs.args + lhs = 2*atan(y/(sqrt(x**2 + y**2) + x)) + elif lhs.func == rhs.func: + if len(lhs.args) == len(rhs.args) == 1: + lhs = lhs.args[0] + rhs = rhs.args[0] + elif len(lhs.args) == len(rhs.args): + # should be able to solve + # f(x, y) == f(2, 3) -> x == 2 + # f(x, x + y) == f(2, 3) -> x == 2 + raise NotImplementedError( + 'equal function with more than 1 argument') + else: + raise ValueError( + 'function with different numbers of args') + + + if rhs and lhs.is_Pow and lhs.exp.is_Integer and lhs.exp < 0: + lhs = 1/lhs + rhs = 1/rhs + + # base**a = b -> base = b**(1/a) if + # a is an Integer and dointpow=True (this gives real branch of root) + # a is not an Integer and the equation is multivariate and the + # base has more than 1 symbol in it + # The rationale for this is that right now the multi-system solvers + # doesn't try to resolve generators to see, for example, if the whole + # system is written in terms of sqrt(x + y) so it will just fail, so we + # do that step here. + if lhs.is_Pow and ( + lhs.exp.is_Integer and dointpow or not lhs.exp.is_Integer and + len(symbols) > 1 and len(lhs.base.free_symbols & set(symbols)) > 1): + rhs = rhs**(1/lhs.exp) + lhs = lhs.base + + if lhs == was: + break + return rhs, lhs + + +def unrad(eq, *syms, **flags): + """ + Remove radicals with symbolic arguments and return (eq, cov), + None, or raise an error. + + Explanation + =========== + + None is returned if there are no radicals to remove. + + NotImplementedError is raised if there are radicals and they cannot be + removed or if the relationship between the original symbols and the + change of variable needed to rewrite the system as a polynomial cannot + be solved. + + Otherwise the tuple, ``(eq, cov)``, is returned where: + + *eq*, ``cov`` + *eq* is an equation without radicals (in the symbol(s) of + interest) whose solutions are a superset of the solutions to the + original expression. *eq* might be rewritten in terms of a new + variable; the relationship to the original variables is given by + ``cov`` which is a list containing ``v`` and ``v**p - b`` where + ``p`` is the power needed to clear the radical and ``b`` is the + radical now expressed as a polynomial in the symbols of interest. + For example, for sqrt(2 - x) the tuple would be + ``(c, c**2 - 2 + x)``. The solutions of *eq* will contain + solutions to the original equation (if there are any). + + *syms* + An iterable of symbols which, if provided, will limit the focus of + radical removal: only radicals with one or more of the symbols of + interest will be cleared. All free symbols are used if *syms* is not + set. + + *flags* are used internally for communication during recursive calls. + Two options are also recognized: + + ``take``, when defined, is interpreted as a single-argument function + that returns True if a given Pow should be handled. + + Radicals can be removed from an expression if: + + * All bases of the radicals are the same; a change of variables is + done in this case. + * If all radicals appear in one term of the expression. + * There are only four terms with sqrt() factors or there are less than + four terms having sqrt() factors. + * There are only two terms with radicals. + + Examples + ======== + + >>> from sympy.solvers.solvers import unrad + >>> from sympy.abc import x + >>> from sympy import sqrt, Rational, root + + >>> unrad(sqrt(x)*x**Rational(1, 3) + 2) + (x**5 - 64, []) + >>> unrad(sqrt(x) + root(x + 1, 3)) + (-x**3 + x**2 + 2*x + 1, []) + >>> eq = sqrt(x) + root(x, 3) - 2 + >>> unrad(eq) + (_p**3 + _p**2 - 2, [_p, _p**6 - x]) + + """ + + uflags = {"check": False, "simplify": False} + + def _cov(p, e): + if cov: + # XXX - uncovered + oldp, olde = cov + if Poly(e, p).degree(p) in (1, 2): + cov[:] = [p, olde.subs(oldp, _vsolve(e, p, **uflags)[0])] + else: + raise NotImplementedError + else: + cov[:] = [p, e] + + def _canonical(eq, cov): + if cov: + # change symbol to vanilla so no solutions are eliminated + p, e = cov + rep = {p: Dummy(p.name)} + eq = eq.xreplace(rep) + cov = [p.xreplace(rep), e.xreplace(rep)] + + # remove constants and powers of factors since these don't change + # the location of the root; XXX should factor or factor_terms be used? + eq = factor_terms(_mexpand(eq.as_numer_denom()[0], recursive=True), clear=True) + if eq.is_Mul: + args = [] + for f in eq.args: + if f.is_number: + continue + if f.is_Pow: + args.append(f.base) + else: + args.append(f) + eq = Mul(*args) # leave as Mul for more efficient solving + + # make the sign canonical + margs = list(Mul.make_args(eq)) + changed = False + for i, m in enumerate(margs): + if m.could_extract_minus_sign(): + margs[i] = -m + changed = True + if changed: + eq = Mul(*margs, evaluate=False) + + return eq, cov + + def _Q(pow): + # return leading Rational of denominator of Pow's exponent + c = pow.as_base_exp()[1].as_coeff_Mul()[0] + if not c.is_Rational: + return S.One + return c.q + + # define the _take method that will determine whether a term is of interest + def _take(d): + # return True if coefficient of any factor's exponent's den is not 1 + for pow in Mul.make_args(d): + if not pow.is_Pow: + continue + if _Q(pow) == 1: + continue + if pow.free_symbols & syms: + return True + return False + _take = flags.setdefault('_take', _take) + + if isinstance(eq, Eq): + eq = eq.lhs - eq.rhs # XXX legacy Eq as Eqn support + elif not isinstance(eq, Expr): + return + + cov, nwas, rpt = [flags.setdefault(k, v) for k, v in + sorted({"cov": [], "n": None, "rpt": 0}.items())] + + # preconditioning + eq = powdenest(factor_terms(eq, radical=True, clear=True)) + eq = eq.as_numer_denom()[0] + eq = _mexpand(eq, recursive=True) + if eq.is_number: + return + + # see if there are radicals in symbols of interest + syms = set(syms) or eq.free_symbols # _take uses this + poly = eq.as_poly() + gens = [g for g in poly.gens if _take(g)] + if not gens: + return + + # recast poly in terms of eigen-gens + poly = eq.as_poly(*gens) + + # not a polynomial e.g. 1 + sqrt(x)*exp(sqrt(x)) with gen sqrt(x) + if poly is None: + return + + # - an exponent has a symbol of interest (don't handle) + if any(g.exp.has(*syms) for g in gens): + return + + def _rads_bases_lcm(poly): + # if all the bases are the same or all the radicals are in one + # term, `lcm` will be the lcm of the denominators of the + # exponents of the radicals + lcm = 1 + rads = set() + bases = set() + for g in poly.gens: + q = _Q(g) + if q != 1: + rads.add(g) + lcm = ilcm(lcm, q) + bases.add(g.base) + return rads, bases, lcm + rads, bases, lcm = _rads_bases_lcm(poly) + + covsym = Dummy('p', nonnegative=True) + + # only keep in syms symbols that actually appear in radicals; + # and update gens + newsyms = set() + for r in rads: + newsyms.update(syms & r.free_symbols) + if newsyms != syms: + syms = newsyms + # get terms together that have common generators + drad = dict(zip(rads, range(len(rads)))) + rterms = {(): []} + args = Add.make_args(poly.as_expr()) + for t in args: + if _take(t): + common = set(t.as_poly().gens).intersection(rads) + key = tuple(sorted([drad[i] for i in common])) + else: + key = () + rterms.setdefault(key, []).append(t) + others = Add(*rterms.pop(())) + rterms = [Add(*rterms[k]) for k in rterms.keys()] + + # the output will depend on the order terms are processed, so + # make it canonical quickly + rterms = list(reversed(list(ordered(rterms)))) + + ok = False # we don't have a solution yet + depth = sqrt_depth(eq) + + if len(rterms) == 1 and not (rterms[0].is_Add and lcm > 2): + eq = rterms[0]**lcm - ((-others)**lcm) + ok = True + else: + if len(rterms) == 1 and rterms[0].is_Add: + rterms = list(rterms[0].args) + if len(bases) == 1: + b = bases.pop() + if len(syms) > 1: + x = b.free_symbols + else: + x = syms + x = list(ordered(x))[0] + try: + inv = _vsolve(covsym**lcm - b, x, **uflags) + if not inv: + raise NotImplementedError + eq = poly.as_expr().subs(b, covsym**lcm).subs(x, inv[0]) + _cov(covsym, covsym**lcm - b) + return _canonical(eq, cov) + except NotImplementedError: + pass + + if len(rterms) == 2: + if not others: + eq = rterms[0]**lcm - (-rterms[1])**lcm + ok = True + elif not log(lcm, 2).is_Integer: + # the lcm-is-power-of-two case is handled below + r0, r1 = rterms + if flags.get('_reverse', False): + r1, r0 = r0, r1 + i0 = _rads0, _bases0, lcm0 = _rads_bases_lcm(r0.as_poly()) + i1 = _rads1, _bases1, lcm1 = _rads_bases_lcm(r1.as_poly()) + for reverse in range(2): + if reverse: + i0, i1 = i1, i0 + r0, r1 = r1, r0 + _rads1, _, lcm1 = i1 + _rads1 = Mul(*_rads1) + t1 = _rads1**lcm1 + c = covsym**lcm1 - t1 + for x in syms: + try: + sol = _vsolve(c, x, **uflags) + if not sol: + raise NotImplementedError + neweq = r0.subs(x, sol[0]) + covsym*r1/_rads1 + \ + others + tmp = unrad(neweq, covsym) + if tmp: + eq, newcov = tmp + if newcov: + newp, newc = newcov + _cov(newp, c.subs(covsym, + _vsolve(newc, covsym, **uflags)[0])) + else: + _cov(covsym, c) + else: + eq = neweq + _cov(covsym, c) + ok = True + break + except NotImplementedError: + if reverse: + raise NotImplementedError( + 'no successful change of variable found') + else: + pass + if ok: + break + elif len(rterms) == 3: + # two cube roots and another with order less than 5 + # (so an analytical solution can be found) or a base + # that matches one of the cube root bases + info = [_rads_bases_lcm(i.as_poly()) for i in rterms] + RAD = 0 + BASES = 1 + LCM = 2 + if info[0][LCM] != 3: + info.append(info.pop(0)) + rterms.append(rterms.pop(0)) + elif info[1][LCM] != 3: + info.append(info.pop(1)) + rterms.append(rterms.pop(1)) + if info[0][LCM] == info[1][LCM] == 3: + if info[1][BASES] != info[2][BASES]: + info[0], info[1] = info[1], info[0] + rterms[0], rterms[1] = rterms[1], rterms[0] + if info[1][BASES] == info[2][BASES]: + eq = rterms[0]**3 + (rterms[1] + rterms[2] + others)**3 + ok = True + elif info[2][LCM] < 5: + # a*root(A, 3) + b*root(B, 3) + others = c + a, b, c, d, A, B = [Dummy(i) for i in 'abcdAB'] + # zz represents the unraded expression into which the + # specifics for this case are substituted + zz = (c - d)*(A**3*a**9 + 3*A**2*B*a**6*b**3 - + 3*A**2*a**6*c**3 + 9*A**2*a**6*c**2*d - 9*A**2*a**6*c*d**2 + + 3*A**2*a**6*d**3 + 3*A*B**2*a**3*b**6 + 21*A*B*a**3*b**3*c**3 - + 63*A*B*a**3*b**3*c**2*d + 63*A*B*a**3*b**3*c*d**2 - + 21*A*B*a**3*b**3*d**3 + 3*A*a**3*c**6 - 18*A*a**3*c**5*d + + 45*A*a**3*c**4*d**2 - 60*A*a**3*c**3*d**3 + 45*A*a**3*c**2*d**4 - + 18*A*a**3*c*d**5 + 3*A*a**3*d**6 + B**3*b**9 - 3*B**2*b**6*c**3 + + 9*B**2*b**6*c**2*d - 9*B**2*b**6*c*d**2 + 3*B**2*b**6*d**3 + + 3*B*b**3*c**6 - 18*B*b**3*c**5*d + 45*B*b**3*c**4*d**2 - + 60*B*b**3*c**3*d**3 + 45*B*b**3*c**2*d**4 - 18*B*b**3*c*d**5 + + 3*B*b**3*d**6 - c**9 + 9*c**8*d - 36*c**7*d**2 + 84*c**6*d**3 - + 126*c**5*d**4 + 126*c**4*d**5 - 84*c**3*d**6 + 36*c**2*d**7 - + 9*c*d**8 + d**9) + def _t(i): + b = Mul(*info[i][RAD]) + return cancel(rterms[i]/b), Mul(*info[i][BASES]) + aa, AA = _t(0) + bb, BB = _t(1) + cc = -rterms[2] + dd = others + eq = zz.xreplace(dict(zip( + (a, A, b, B, c, d), + (aa, AA, bb, BB, cc, dd)))) + ok = True + # handle power-of-2 cases + if not ok: + if log(lcm, 2).is_Integer and (not others and + len(rterms) == 4 or len(rterms) < 4): + def _norm2(a, b): + return a**2 + b**2 + 2*a*b + + if len(rterms) == 4: + # (r0+r1)**2 - (r2+r3)**2 + r0, r1, r2, r3 = rterms + eq = _norm2(r0, r1) - _norm2(r2, r3) + ok = True + elif len(rterms) == 3: + # (r1+r2)**2 - (r0+others)**2 + r0, r1, r2 = rterms + eq = _norm2(r1, r2) - _norm2(r0, others) + ok = True + elif len(rterms) == 2: + # r0**2 - (r1+others)**2 + r0, r1 = rterms + eq = r0**2 - _norm2(r1, others) + ok = True + + new_depth = sqrt_depth(eq) if ok else depth + rpt += 1 # XXX how many repeats with others unchanging is enough? + if not ok or ( + nwas is not None and len(rterms) == nwas and + new_depth is not None and new_depth == depth and + rpt > 3): + raise NotImplementedError('Cannot remove all radicals') + + flags.update({"cov": cov, "n": len(rterms), "rpt": rpt}) + neq = unrad(eq, *syms, **flags) + if neq: + eq, cov = neq + eq, cov = _canonical(eq, cov) + return eq, cov + + +# delayed imports +from sympy.solvers.bivariate import ( + bivariate_type, _solve_lambert, _filtered_gens) diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/solveset.py b/.venv/lib/python3.12/site-packages/sympy/solvers/solveset.py new file mode 100644 index 0000000000000000000000000000000000000000..0ae242d9c8c4c0d1c1c46cd968a0c5e547ff0f66 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/solveset.py @@ -0,0 +1,4131 @@ +""" +This module contains functions to: + + - solve a single equation for a single variable, in any domain either real or complex. + + - solve a single transcendental equation for a single variable in any domain either real or complex. + (currently supports solving in real domain only) + + - solve a system of linear equations with N variables and M equations. + + - solve a system of Non Linear Equations with N variables and M equations +""" +from sympy.core.sympify import sympify +from sympy.core import (S, Pow, Dummy, pi, Expr, Wild, Mul, + Add, Basic) +from sympy.core.containers import Tuple +from sympy.core.function import (Lambda, expand_complex, AppliedUndef, + expand_log, _mexpand, expand_trig, nfloat) +from sympy.core.mod import Mod +from sympy.core.numbers import I, Number, Rational, oo +from sympy.core.intfunc import integer_log +from sympy.core.relational import Eq, Ne, Relational +from sympy.core.sorting import default_sort_key, ordered +from sympy.core.symbol import Symbol, _uniquely_named_symbol +from sympy.core.sympify import _sympify +from sympy.core.traversal import preorder_traversal +from sympy.external.gmpy import gcd as number_gcd, lcm as number_lcm +from sympy.polys.matrices.linsolve import _linear_eq_to_dict +from sympy.polys.polyroots import UnsolvableFactorError +from sympy.simplify.simplify import simplify, fraction, trigsimp, nsimplify +from sympy.simplify import powdenest, logcombine +from sympy.functions import (log, tan, cot, sin, cos, sec, csc, exp, + acos, asin, atan, acot, acsc, asec, + piecewise_fold, Piecewise) +from sympy.functions.combinatorial.numbers import totient +from sympy.functions.elementary.complexes import Abs, arg, re, im +from sympy.functions.elementary.hyperbolic import (HyperbolicFunction, + sinh, cosh, tanh, coth, sech, csch, + asinh, acosh, atanh, acoth, asech, acsch) +from sympy.functions.elementary.miscellaneous import real_root +from sympy.functions.elementary.trigonometric import TrigonometricFunction +from sympy.logic.boolalg import And, BooleanTrue +from sympy.sets import (FiniteSet, imageset, Interval, Intersection, + Union, ConditionSet, ImageSet, Complement, Contains) +from sympy.sets.sets import Set, ProductSet +from sympy.matrices import zeros, Matrix, MatrixBase +from sympy.ntheory.factor_ import divisors +from sympy.ntheory.residue_ntheory import discrete_log, nthroot_mod +from sympy.polys import (roots, Poly, degree, together, PolynomialError, + RootOf, factor, lcm, gcd) +from sympy.polys.polyerrors import CoercionFailed +from sympy.polys.polytools import invert, groebner, poly +from sympy.polys.solvers import (sympy_eqs_to_ring, solve_lin_sys, + PolyNonlinearError) +from sympy.polys.matrices.linsolve import _linsolve +from sympy.solvers.solvers import (checksol, denoms, unrad, + _simple_dens, recast_to_symbols) +from sympy.solvers.polysys import solve_poly_system +from sympy.utilities import filldedent +from sympy.utilities.iterables import (numbered_symbols, has_dups, + is_sequence, iterable) +from sympy.calculus.util import periodicity, continuous_domain, function_range + + +from types import GeneratorType + + +class NonlinearError(ValueError): + """Raised when unexpectedly encountering nonlinear equations""" + pass + + +def _masked(f, *atoms): + """Return ``f``, with all objects given by ``atoms`` replaced with + Dummy symbols, ``d``, and the list of replacements, ``(d, e)``, + where ``e`` is an object of type given by ``atoms`` in which + any other instances of atoms have been recursively replaced with + Dummy symbols, too. The tuples are ordered so that if they are + applied in sequence, the origin ``f`` will be restored. + + Examples + ======== + + >>> from sympy import cos + >>> from sympy.abc import x + >>> from sympy.solvers.solveset import _masked + + >>> f = cos(cos(x) + 1) + >>> f, reps = _masked(cos(1 + cos(x)), cos) + >>> f + _a1 + >>> reps + [(_a1, cos(_a0 + 1)), (_a0, cos(x))] + >>> for d, e in reps: + ... f = f.xreplace({d: e}) + >>> f + cos(cos(x) + 1) + """ + sym = numbered_symbols('a', cls=Dummy, real=True) + mask = [] + for a in ordered(f.atoms(*atoms)): + for i in mask: + a = a.replace(*i) + mask.append((a, next(sym))) + for i, (o, n) in enumerate(mask): + f = f.replace(o, n) + mask[i] = (n, o) + mask = list(reversed(mask)) + return f, mask + + +def _invert(f_x, y, x, domain=S.Complexes): + r""" + Reduce the complex valued equation $f(x) = y$ to a set of equations + + $$\left\{g(x) = h_1(y),\ g(x) = h_2(y),\ \dots,\ g(x) = h_n(y) \right\}$$ + + where $g(x)$ is a simpler function than $f(x)$. The return value is a tuple + $(g(x), \mathrm{set}_h)$, where $g(x)$ is a function of $x$ and $\mathrm{set}_h$ is + the set of function $\left\{h_1(y), h_2(y), \dots, h_n(y)\right\}$. + Here, $y$ is not necessarily a symbol. + + $\mathrm{set}_h$ contains the functions, along with the information + about the domain in which they are valid, through set + operations. For instance, if :math:`y = |x| - n` is inverted + in the real domain, then $\mathrm{set}_h$ is not simply + $\{-n, n\}$ as the nature of `n` is unknown; rather, it is: + + $$ \left(\left[0, \infty\right) \cap \left\{n\right\}\right) \cup + \left(\left(-\infty, 0\right] \cap \left\{- n\right\}\right)$$ + + By default, the complex domain is used which means that inverting even + seemingly simple functions like $\exp(x)$ will give very different + results from those obtained in the real domain. + (In the case of $\exp(x)$, the inversion via $\log$ is multi-valued + in the complex domain, having infinitely many branches.) + + If you are working with real values only (or you are not sure which + function to use) you should probably set the domain to + ``S.Reals`` (or use ``invert_real`` which does that automatically). + + + Examples + ======== + + >>> from sympy.solvers.solveset import invert_complex, invert_real + >>> from sympy.abc import x, y + >>> from sympy import exp + + When does exp(x) == y? + + >>> invert_complex(exp(x), y, x) + (x, ImageSet(Lambda(_n, I*(2*_n*pi + arg(y)) + log(Abs(y))), Integers)) + >>> invert_real(exp(x), y, x) + (x, Intersection({log(y)}, Reals)) + + When does exp(x) == 1? + + >>> invert_complex(exp(x), 1, x) + (x, ImageSet(Lambda(_n, 2*_n*I*pi), Integers)) + >>> invert_real(exp(x), 1, x) + (x, {0}) + + See Also + ======== + invert_real, invert_complex + """ + x = sympify(x) + if not x.is_Symbol: + raise ValueError("x must be a symbol") + f_x = sympify(f_x) + if x not in f_x.free_symbols: + raise ValueError("Inverse of constant function doesn't exist") + y = sympify(y) + if x in y.free_symbols: + raise ValueError("y should be independent of x ") + + if domain.is_subset(S.Reals): + x1, s = _invert_real(f_x, FiniteSet(y), x) + else: + x1, s = _invert_complex(f_x, FiniteSet(y), x) + + # f couldn't be inverted completely; return unmodified. + if x1 != x: + return x1, s + + # Avoid adding gratuitous intersections with S.Complexes. Actual + # conditions should be handled by the respective inverters. + if domain is S.Complexes: + return x1, s + + if isinstance(s, FiniteSet): + return x1, s.intersect(domain) + + # "Fancier" solution sets like those obtained by inversion of trigonometric + # functions already include general validity conditions (i.e. conditions on + # the domain of the respective inverse functions), so we should avoid adding + # blanket intersections with S.Reals. But subsets of R (or C) must still be + # accounted for. + if domain is S.Reals: + return x1, s + else: + return x1, s.intersect(domain) + + +invert_complex = _invert + + +def invert_real(f_x, y, x): + """ + Inverts a real-valued function. Same as :func:`invert_complex`, but sets + the domain to ``S.Reals`` before inverting. + """ + return _invert(f_x, y, x, S.Reals) + + +def _invert_real(f, g_ys, symbol): + """Helper function for _invert.""" + + if f == symbol or g_ys is S.EmptySet: + return (symbol, g_ys) + + n = Dummy('n', real=True) + + if isinstance(f, exp) or (f.is_Pow and f.base == S.Exp1): + return _invert_real(f.exp, + imageset(Lambda(n, log(n)), g_ys), + symbol) + + if hasattr(f, 'inverse') and f.inverse() is not None and not isinstance(f, ( + TrigonometricFunction, + HyperbolicFunction, + )): + if len(f.args) > 1: + raise ValueError("Only functions with one argument are supported.") + return _invert_real(f.args[0], + imageset(Lambda(n, f.inverse()(n)), g_ys), + symbol) + + if isinstance(f, Abs): + return _invert_abs(f.args[0], g_ys, symbol) + + if f.is_Add: + # f = g + h + g, h = f.as_independent(symbol) + if g is not S.Zero: + return _invert_real(h, imageset(Lambda(n, n - g), g_ys), symbol) + + if f.is_Mul: + # f = g*h + g, h = f.as_independent(symbol) + + if g is not S.One: + return _invert_real(h, imageset(Lambda(n, n/g), g_ys), symbol) + + if f.is_Pow: + base, expo = f.args + base_has_sym = base.has(symbol) + expo_has_sym = expo.has(symbol) + + if not expo_has_sym: + + if expo.is_rational: + num, den = expo.as_numer_denom() + + if den % 2 == 0 and num % 2 == 1 and den.is_zero is False: + # Here we have f(x)**(num/den) = y + # where den is nonzero and even and y is an element + # of the set g_ys. + # den is even, so we are only interested in the cases + # where both f(x) and y are positive. + # Restricting y to be positive (using the set g_ys_pos) + # means that y**(den/num) is always positive. + # Therefore it isn't necessary to also constrain f(x) + # to be positive because we are only going to + # find solutions of f(x) = y**(d/n) + # where the rhs is already required to be positive. + root = Lambda(n, real_root(n, expo)) + g_ys_pos = g_ys & Interval(0, oo) + res = imageset(root, g_ys_pos) + _inv, _set = _invert_real(base, res, symbol) + return (_inv, _set) + + if den % 2 == 1: + root = Lambda(n, real_root(n, expo)) + res = imageset(root, g_ys) + if num % 2 == 0: + neg_res = imageset(Lambda(n, -n), res) + return _invert_real(base, res + neg_res, symbol) + if num % 2 == 1: + return _invert_real(base, res, symbol) + + elif expo.is_irrational: + root = Lambda(n, real_root(n, expo)) + g_ys_pos = g_ys & Interval(0, oo) + res = imageset(root, g_ys_pos) + return _invert_real(base, res, symbol) + + else: + # indeterminate exponent, e.g. Float or parity of + # num, den of rational could not be determined + pass # use default return + + if not base_has_sym: + rhs = g_ys.args[0] + if base.is_positive: + return _invert_real(expo, + imageset(Lambda(n, log(n, base, evaluate=False)), g_ys), symbol) + elif base.is_negative: + s, b = integer_log(rhs, base) + if b: + return _invert_real(expo, FiniteSet(s), symbol) + else: + return (expo, S.EmptySet) + elif base.is_zero: + one = Eq(rhs, 1) + if one == S.true: + # special case: 0**x - 1 + return _invert_real(expo, FiniteSet(0), symbol) + elif one == S.false: + return (expo, S.EmptySet) + + if isinstance(f, (TrigonometricFunction, HyperbolicFunction)): + return _invert_trig_hyp_real(f, g_ys, symbol) + + return (f, g_ys) + + +# Dictionaries of inverses will be cached after first use. +_trig_inverses = None +_hyp_inverses = None + +def _invert_trig_hyp_real(f, g_ys, symbol): + """Helper function for inverting trigonometric and hyperbolic functions. + + This helper only handles inversion over the reals. + + For trigonometric functions only finite `g_ys` sets are implemented. + + For hyperbolic functions the set `g_ys` is checked against the domain of the + respective inverse functions. Infinite `g_ys` sets are also supported. + """ + + if isinstance(f, HyperbolicFunction): + n = Dummy('n', real=True) + + if isinstance(f, sinh): + # asinh is defined over R. + return _invert_real(f.args[0], imageset(n, asinh(n), g_ys), symbol) + + if isinstance(f, cosh): + g_ys_dom = g_ys.intersect(Interval(1, oo)) + if isinstance(g_ys_dom, Intersection): + # could not properly resolve domain check + if isinstance(g_ys, FiniteSet): + # If g_ys is a `FiniteSet`` it should be sufficient to just + # let the calling `_invert_real()` add an intersection with + # `S.Reals` (or a subset `domain`) to ensure that only valid + # (real) solutions are returned. + # This avoids adding "too many" Intersections or + # ConditionSets in the returned set. + g_ys_dom = g_ys + else: + return (f, g_ys) + return _invert_real(f.args[0], Union( + imageset(n, acosh(n), g_ys_dom), + imageset(n, -acosh(n), g_ys_dom)), symbol) + + if isinstance(f, sech): + g_ys_dom = g_ys.intersect(Interval.Lopen(0, 1)) + if isinstance(g_ys_dom, Intersection): + if isinstance(g_ys, FiniteSet): + g_ys_dom = g_ys + else: + return (f, g_ys) + return _invert_real(f.args[0], Union( + imageset(n, asech(n), g_ys_dom), + imageset(n, -asech(n), g_ys_dom)), symbol) + + if isinstance(f, tanh): + g_ys_dom = g_ys.intersect(Interval.open(-1, 1)) + if isinstance(g_ys_dom, Intersection): + if isinstance(g_ys, FiniteSet): + g_ys_dom = g_ys + else: + return (f, g_ys) + return _invert_real(f.args[0], + imageset(n, atanh(n), g_ys_dom), symbol) + + if isinstance(f, coth): + g_ys_dom = g_ys - Interval(-1, 1) + if isinstance(g_ys_dom, Complement): + if isinstance(g_ys, FiniteSet): + g_ys_dom = g_ys + else: + return (f, g_ys) + return _invert_real(f.args[0], + imageset(n, acoth(n), g_ys_dom), symbol) + + if isinstance(f, csch): + g_ys_dom = g_ys - FiniteSet(0) + if isinstance(g_ys_dom, Complement): + if isinstance(g_ys, FiniteSet): + g_ys_dom = g_ys + else: + return (f, g_ys) + return _invert_real(f.args[0], + imageset(n, acsch(n), g_ys_dom), symbol) + + elif isinstance(f, TrigonometricFunction) and isinstance(g_ys, FiniteSet): + def _get_trig_inverses(func): + global _trig_inverses + if _trig_inverses is None: + _trig_inverses = { + sin : ((asin, lambda y: pi-asin(y)), 2*pi, Interval(-1, 1)), + cos : ((acos, lambda y: -acos(y)), 2*pi, Interval(-1, 1)), + tan : ((atan,), pi, S.Reals), + cot : ((acot,), pi, S.Reals), + sec : ((asec, lambda y: -asec(y)), 2*pi, + Union(Interval(-oo, -1), Interval(1, oo))), + csc : ((acsc, lambda y: pi-acsc(y)), 2*pi, + Union(Interval(-oo, -1), Interval(1, oo)))} + return _trig_inverses[func] + + invs, period, rng = _get_trig_inverses(f.func) + n = Dummy('n', integer=True) + def create_return_set(g): + # returns ConditionSet that will be part of the final (x, set) tuple + invsimg = Union(*[ + imageset(n, period*n + inv(g), S.Integers) for inv in invs]) + inv_f, inv_g_ys = _invert_real(f.args[0], invsimg, symbol) + if inv_f == symbol: # inversion successful + conds = rng.contains(g) + return ConditionSet(symbol, conds, inv_g_ys) + else: + return ConditionSet(symbol, Eq(f, g), S.Reals) + + retset = Union(*[create_return_set(g) for g in g_ys]) + return (symbol, retset) + + else: + return (f, g_ys) + + +def _invert_trig_hyp_complex(f, g_ys, symbol): + """Helper function for inverting trigonometric and hyperbolic functions. + + This helper only handles inversion over the complex numbers. + Only finite `g_ys` sets are implemented. + + Handling of singularities is only implemented for hyperbolic equations. + In case of a symbolic element g in g_ys a ConditionSet may be returned. + """ + + if isinstance(f, TrigonometricFunction) and isinstance(g_ys, FiniteSet): + def inv(trig): + if isinstance(trig, (sin, csc)): + F = asin if isinstance(trig, sin) else acsc + return ( + lambda a: 2*n*pi + F(a), + lambda a: 2*n*pi + pi - F(a)) + if isinstance(trig, (cos, sec)): + F = acos if isinstance(trig, cos) else asec + return ( + lambda a: 2*n*pi + F(a), + lambda a: 2*n*pi - F(a)) + if isinstance(trig, (tan, cot)): + return (lambda a: n*pi + trig.inverse()(a),) + + n = Dummy('n', integer=True) + invs = S.EmptySet + for L in inv(f): + invs += Union(*[imageset(Lambda(n, L(g)), S.Integers) for g in g_ys]) + return _invert_complex(f.args[0], invs, symbol) + + elif isinstance(f, HyperbolicFunction) and isinstance(g_ys, FiniteSet): + # There are two main options regarding singularities / domain checking + # for symbolic elements in g_ys: + # 1. Add a "catch-all" intersection with S.Complexes. + # 2. ConditionSets. + # At present ConditionSets seem to work better and have the additional + # benefit of representing the precise conditions that must be satisfied. + # The conditions are also rather straightforward. (At most two isolated + # points.) + def _get_hyp_inverses(func): + global _hyp_inverses + if _hyp_inverses is None: + _hyp_inverses = { + sinh : ((asinh, lambda y: I*pi-asinh(y)), 2*I*pi, ()), + cosh : ((acosh, lambda y: -acosh(y)), 2*I*pi, ()), + tanh : ((atanh,), I*pi, (-1, 1)), + coth : ((acoth,), I*pi, (-1, 1)), + sech : ((asech, lambda y: -asech(y)), 2*I*pi, (0, )), + csch : ((acsch, lambda y: I*pi-acsch(y)), 2*I*pi, (0, ))} + return _hyp_inverses[func] + + # invs: iterable of main inverses, e.g. (acosh, -acosh). + # excl: iterable of singularities to be checked for. + invs, period, excl = _get_hyp_inverses(f.func) + n = Dummy('n', integer=True) + def create_return_set(g): + # returns ConditionSet that will be part of the final (x, set) tuple + invsimg = Union(*[ + imageset(n, period*n + inv(g), S.Integers) for inv in invs]) + inv_f, inv_g_ys = _invert_complex(f.args[0], invsimg, symbol) + if inv_f == symbol: # inversion successful + conds = And(*[Ne(g, e) for e in excl]) + return ConditionSet(symbol, conds, inv_g_ys) + else: + return ConditionSet(symbol, Eq(f, g), S.Complexes) + + retset = Union(*[create_return_set(g) for g in g_ys]) + return (symbol, retset) + + else: + return (f, g_ys) + + +def _invert_complex(f, g_ys, symbol): + """Helper function for _invert.""" + + if f == symbol or g_ys is S.EmptySet: + return (symbol, g_ys) + + n = Dummy('n') + + if f.is_Add: + # f = g + h + g, h = f.as_independent(symbol) + if g is not S.Zero: + return _invert_complex(h, imageset(Lambda(n, n - g), g_ys), symbol) + + if f.is_Mul: + # f = g*h + g, h = f.as_independent(symbol) + + if g is not S.One: + if g in {S.NegativeInfinity, S.ComplexInfinity, S.Infinity}: + return (h, S.EmptySet) + return _invert_complex(h, imageset(Lambda(n, n/g), g_ys), symbol) + + if f.is_Pow: + base, expo = f.args + # special case: g**r = 0 + # Could be improved like `_invert_real` to handle more general cases. + if expo.is_Rational and g_ys == FiniteSet(0): + if expo.is_positive: + return _invert_complex(base, g_ys, symbol) + + if hasattr(f, 'inverse') and f.inverse() is not None and \ + not isinstance(f, TrigonometricFunction) and \ + not isinstance(f, HyperbolicFunction) and \ + not isinstance(f, exp): + if len(f.args) > 1: + raise ValueError("Only functions with one argument are supported.") + return _invert_complex(f.args[0], + imageset(Lambda(n, f.inverse()(n)), g_ys), symbol) + + if isinstance(f, exp) or (f.is_Pow and f.base == S.Exp1): + if isinstance(g_ys, ImageSet): + # can solve up to `(d*exp(exp(...(exp(a*x + b))...) + c)` format. + # Further can be improved to `(d*exp(exp(...(exp(a*x**n + b*x**(n-1) + ... + f))...) + c)`. + g_ys_expr = g_ys.lamda.expr + g_ys_vars = g_ys.lamda.variables + k = Dummy('k{}'.format(len(g_ys_vars))) + g_ys_vars_1 = (k,) + g_ys_vars + exp_invs = Union(*[imageset(Lambda((g_ys_vars_1,), (I*(2*k*pi + arg(g_ys_expr)) + + log(Abs(g_ys_expr)))), S.Integers**(len(g_ys_vars_1)))]) + return _invert_complex(f.exp, exp_invs, symbol) + + elif isinstance(g_ys, FiniteSet): + exp_invs = Union(*[imageset(Lambda(n, I*(2*n*pi + arg(g_y)) + + log(Abs(g_y))), S.Integers) + for g_y in g_ys if g_y != 0]) + return _invert_complex(f.exp, exp_invs, symbol) + + if isinstance(f, (TrigonometricFunction, HyperbolicFunction)): + return _invert_trig_hyp_complex(f, g_ys, symbol) + + return (f, g_ys) + + +def _invert_abs(f, g_ys, symbol): + """Helper function for inverting absolute value functions. + + Returns the complete result of inverting an absolute value + function along with the conditions which must also be satisfied. + + If it is certain that all these conditions are met, a :class:`~.FiniteSet` + of all possible solutions is returned. If any condition cannot be + satisfied, an :class:`~.EmptySet` is returned. Otherwise, a + :class:`~.ConditionSet` of the solutions, with all the required conditions + specified, is returned. + + """ + if not g_ys.is_FiniteSet: + # this could be used for FiniteSet, but the + # results are more compact if they aren't, e.g. + # ConditionSet(x, Contains(n, Interval(0, oo)), {-n, n}) vs + # Union(Intersection(Interval(0, oo), {n}), Intersection(Interval(-oo, 0), {-n})) + # for the solution of abs(x) - n + pos = Intersection(g_ys, Interval(0, S.Infinity)) + parg = _invert_real(f, pos, symbol) + narg = _invert_real(-f, pos, symbol) + if parg[0] != narg[0]: + raise NotImplementedError + return parg[0], Union(narg[1], parg[1]) + + # check conditions: all these must be true. If any are unknown + # then return them as conditions which must be satisfied + unknown = [] + for a in g_ys.args: + ok = a.is_nonnegative if a.is_Number else a.is_positive + if ok is None: + unknown.append(a) + elif not ok: + return symbol, S.EmptySet + if unknown: + conditions = And(*[Contains(i, Interval(0, oo)) + for i in unknown]) + else: + conditions = True + n = Dummy('n', real=True) + # this is slightly different than above: instead of solving + # +/-f on positive values, here we solve for f on +/- g_ys + g_x, values = _invert_real(f, Union( + imageset(Lambda(n, n), g_ys), + imageset(Lambda(n, -n), g_ys)), symbol) + return g_x, ConditionSet(g_x, conditions, values) + + +def domain_check(f, symbol, p): + """Returns False if point p is infinite or any subexpression of f + is infinite or becomes so after replacing symbol with p. If none of + these conditions is met then True will be returned. + + Examples + ======== + + >>> from sympy import Mul, oo + >>> from sympy.abc import x + >>> from sympy.solvers.solveset import domain_check + >>> g = 1/(1 + (1/(x + 1))**2) + >>> domain_check(g, x, -1) + False + >>> domain_check(x**2, x, 0) + True + >>> domain_check(1/x, x, oo) + False + + * The function relies on the assumption that the original form + of the equation has not been changed by automatic simplification. + + >>> domain_check(x/x, x, 0) # x/x is automatically simplified to 1 + True + + * To deal with automatic evaluations use evaluate=False: + + >>> domain_check(Mul(x, 1/x, evaluate=False), x, 0) + False + """ + f, p = sympify(f), sympify(p) + if p.is_infinite: + return False + return _domain_check(f, symbol, p) + + +def _domain_check(f, symbol, p): + # helper for domain check + if f.is_Atom and f.is_finite: + return True + elif f.subs(symbol, p).is_infinite: + return False + elif isinstance(f, Piecewise): + # Check the cases of the Piecewise in turn. There might be invalid + # expressions in later cases that don't apply e.g. + # solveset(Piecewise((0, Eq(x, 0)), (1/x, True)), x) + for expr, cond in f.args: + condsubs = cond.subs(symbol, p) + if condsubs is S.false: + continue + elif condsubs is S.true: + return _domain_check(expr, symbol, p) + else: + # We don't know which case of the Piecewise holds. On this + # basis we cannot decide whether any solution is in or out of + # the domain. Ideally this function would allow returning a + # symbolic condition for the validity of the solution that + # could be handled in the calling code. In the mean time we'll + # give this particular solution the benefit of the doubt and + # let it pass. + return True + else: + # TODO : We should not blindly recurse through all args of arbitrary expressions like this + return all(_domain_check(g, symbol, p) + for g in f.args) + + +def _is_finite_with_finite_vars(f, domain=S.Complexes): + """ + Return True if the given expression is finite. For symbols that + do not assign a value for `complex` and/or `real`, the domain will + be used to assign a value; symbols that do not assign a value + for `finite` will be made finite. All other assumptions are + left unmodified. + """ + def assumptions(s): + A = s.assumptions0 + A.setdefault('finite', A.get('finite', True)) + if domain.is_subset(S.Reals): + # if this gets set it will make complex=True, too + A.setdefault('real', True) + else: + # don't change 'real' because being complex implies + # nothing about being real + A.setdefault('complex', True) + return A + + reps = {s: Dummy(**assumptions(s)) for s in f.free_symbols} + return f.xreplace(reps).is_finite + + +def _is_function_class_equation(func_class, f, symbol): + """ Tests whether the equation is an equation of the given function class. + + The given equation belongs to the given function class if it is + comprised of functions of the function class which are multiplied by + or added to expressions independent of the symbol. In addition, the + arguments of all such functions must be linear in the symbol as well. + + Examples + ======== + + >>> from sympy.solvers.solveset import _is_function_class_equation + >>> from sympy import tan, sin, tanh, sinh, exp + >>> from sympy.abc import x + >>> from sympy.functions.elementary.trigonometric import TrigonometricFunction + >>> from sympy.functions.elementary.hyperbolic import HyperbolicFunction + >>> _is_function_class_equation(TrigonometricFunction, exp(x) + tan(x), x) + False + >>> _is_function_class_equation(TrigonometricFunction, tan(x) + sin(x), x) + True + >>> _is_function_class_equation(TrigonometricFunction, tan(x**2), x) + False + >>> _is_function_class_equation(TrigonometricFunction, tan(x + 2), x) + True + >>> _is_function_class_equation(HyperbolicFunction, tanh(x) + sinh(x), x) + True + """ + if f.is_Mul or f.is_Add: + return all(_is_function_class_equation(func_class, arg, symbol) + for arg in f.args) + + if f.is_Pow: + if not f.exp.has(symbol): + return _is_function_class_equation(func_class, f.base, symbol) + else: + return False + + if not f.has(symbol): + return True + + if isinstance(f, func_class): + try: + g = Poly(f.args[0], symbol) + return g.degree() <= 1 + except PolynomialError: + return False + else: + return False + + +def _solve_as_rational(f, symbol, domain): + """ solve rational functions""" + f = together(_mexpand(f, recursive=True), deep=True) + g, h = fraction(f) + if not h.has(symbol): + try: + return _solve_as_poly(g, symbol, domain) + except NotImplementedError: + # The polynomial formed from g could end up having + # coefficients in a ring over which finding roots + # isn't implemented yet, e.g. ZZ[a] for some symbol a + return ConditionSet(symbol, Eq(f, 0), domain) + except CoercionFailed: + # contained oo, zoo or nan + return S.EmptySet + else: + valid_solns = _solveset(g, symbol, domain) + invalid_solns = _solveset(h, symbol, domain) + return valid_solns - invalid_solns + + +class _SolveTrig1Error(Exception): + """Raised when _solve_trig1 heuristics do not apply""" + +def _solve_trig(f, symbol, domain): + """Function to call other helpers to solve trigonometric equations """ + # If f is composed of a single trig function (potentially appearing multiple + # times) we should solve by either inverting directly or inverting after a + # suitable change of variable. + # + # _solve_trig is currently only called by _solveset for trig/hyperbolic + # functions of an argument linear in x. Inverting a symbolic argument should + # include a guard against division by zero in order to have a result that is + # consistent with similar processing done by _solve_trig1. + # (Ideally _invert should add these conditions by itself.) + trig_expr, count = None, 0 + for expr in preorder_traversal(f): + if isinstance(expr, (TrigonometricFunction, + HyperbolicFunction)) and expr.has(symbol): + if not trig_expr: + trig_expr, count = expr, 1 + elif expr == trig_expr: + count += 1 + else: + trig_expr, count = False, 0 + break + if count == 1: + # direct inversion + x, sol = _invert(f, 0, symbol, domain) + if x == symbol: + cond = True + if trig_expr.free_symbols - {symbol}: + a, h = trig_expr.args[0].as_independent(symbol, as_Add=True) + m, h = h.as_independent(symbol, as_Add=False) + num, den = m.as_numer_denom() + cond = Ne(num, 0) & Ne(den, 0) + return ConditionSet(symbol, cond, sol) + else: + return ConditionSet(symbol, Eq(f, 0), domain) + elif count: + # solve by change of variable + y = Dummy('y') + f_cov = f.subs(trig_expr, y) + sol_cov = solveset(f_cov, y, domain) + if isinstance(sol_cov, FiniteSet): + return Union( + *[_solve_trig(trig_expr-s, symbol, domain) for s in sol_cov]) + + sol = None + try: + # multiple trig/hyp functions; solve by rewriting to exp + sol = _solve_trig1(f, symbol, domain) + except _SolveTrig1Error: + try: + # multiple trig/hyp functions; solve by rewriting to tan(x/2) + sol = _solve_trig2(f, symbol, domain) + except ValueError: + raise NotImplementedError(filldedent(''' + Solution to this kind of trigonometric equations + is yet to be implemented''')) + return sol + + +def _solve_trig1(f, symbol, domain): + """Primary solver for trigonometric and hyperbolic equations + + Returns either the solution set as a ConditionSet (auto-evaluated to a + union of ImageSets if no variables besides 'symbol' are involved) or + raises _SolveTrig1Error if f == 0 cannot be solved. + + Notes + ===== + Algorithm: + 1. Do a change of variable x -> mu*x in arguments to trigonometric and + hyperbolic functions, in order to reduce them to small integers. (This + step is crucial to keep the degrees of the polynomials of step 4 low.) + 2. Rewrite trigonometric/hyperbolic functions as exponentials. + 3. Proceed to a 2nd change of variable, replacing exp(I*x) or exp(x) by y. + 4. Solve the resulting rational equation. + 5. Use invert_complex or invert_real to return to the original variable. + 6. If the coefficients of 'symbol' were symbolic in nature, add the + necessary consistency conditions in a ConditionSet. + + """ + # Prepare change of variable + x = Dummy('x') + if _is_function_class_equation(HyperbolicFunction, f, symbol): + cov = exp(x) + inverter = invert_real if domain.is_subset(S.Reals) else invert_complex + else: + cov = exp(I*x) + inverter = invert_complex + + f = trigsimp(f) + f_original = f + trig_functions = f.atoms(TrigonometricFunction, HyperbolicFunction) + trig_arguments = [e.args[0] for e in trig_functions] + # trigsimp may have reduced the equation to an expression + # that is independent of 'symbol' (e.g. cos**2+sin**2) + if not any(a.has(symbol) for a in trig_arguments): + return solveset(f_original, symbol, domain) + + denominators = [] + numerators = [] + for ar in trig_arguments: + try: + poly_ar = Poly(ar, symbol) + except PolynomialError: + raise _SolveTrig1Error("trig argument is not a polynomial") + if poly_ar.degree() > 1: # degree >1 still bad + raise _SolveTrig1Error("degree of variable must not exceed one") + if poly_ar.degree() == 0: # degree 0, don't care + continue + c = poly_ar.all_coeffs()[0] # got the coefficient of 'symbol' + numerators.append(fraction(c)[0]) + denominators.append(fraction(c)[1]) + + mu = lcm(denominators)/gcd(numerators) + f = f.subs(symbol, mu*x) + f = f.rewrite(exp) + f = together(f) + g, h = fraction(f) + y = Dummy('y') + g, h = g.expand(), h.expand() + g, h = g.subs(cov, y), h.subs(cov, y) + if g.has(x) or h.has(x): + raise _SolveTrig1Error("change of variable not possible") + + solns = solveset_complex(g, y) - solveset_complex(h, y) + if isinstance(solns, ConditionSet): + raise _SolveTrig1Error("polynomial has ConditionSet solution") + + if isinstance(solns, FiniteSet): + if any(isinstance(s, RootOf) for s in solns): + raise _SolveTrig1Error("polynomial results in RootOf object") + # revert the change of variable + cov = cov.subs(x, symbol/mu) + result = Union(*[inverter(cov, s, symbol)[1] for s in solns]) + # In case of symbolic coefficients, the solution set is only valid + # if numerator and denominator of mu are non-zero. + if mu.has(Symbol): + syms = (mu).atoms(Symbol) + munum, muden = fraction(mu) + condnum = munum.as_independent(*syms, as_Add=False)[1] + condden = muden.as_independent(*syms, as_Add=False)[1] + cond = And(Ne(condnum, 0), Ne(condden, 0)) + else: + cond = True + # Actual conditions are returned as part of the ConditionSet. Adding an + # intersection with C would only complicate some solution sets due to + # current limitations of intersection code. (e.g. #19154) + if domain is S.Complexes: + # This is a slight abuse of ConditionSet. Ideally this should + # be some kind of "PiecewiseSet". (See #19507 discussion) + return ConditionSet(symbol, cond, result) + else: + return ConditionSet(symbol, cond, Intersection(result, domain)) + elif solns is S.EmptySet: + return S.EmptySet + else: + raise _SolveTrig1Error("polynomial solutions must form FiniteSet") + + +def _solve_trig2(f, symbol, domain): + """Secondary helper to solve trigonometric equations, + called when first helper fails """ + f = trigsimp(f) + f_original = f + trig_functions = f.atoms(sin, cos, tan, sec, cot, csc) + trig_arguments = [e.args[0] for e in trig_functions] + denominators = [] + numerators = [] + + # todo: This solver can be extended to hyperbolics if the + # analogous change of variable to tanh (instead of tan) + # is used. + if not trig_functions: + return ConditionSet(symbol, Eq(f_original, 0), domain) + + # todo: The pre-processing below (extraction of numerators, denominators, + # gcd, lcm, mu, etc.) should be updated to the enhanced version in + # _solve_trig1. (See #19507) + for ar in trig_arguments: + try: + poly_ar = Poly(ar, symbol) + except PolynomialError: + raise ValueError("give up, we cannot solve if this is not a polynomial in x") + if poly_ar.degree() > 1: # degree >1 still bad + raise ValueError("degree of variable inside polynomial should not exceed one") + if poly_ar.degree() == 0: # degree 0, don't care + continue + c = poly_ar.all_coeffs()[0] # got the coefficient of 'symbol' + try: + numerators.append(Rational(c).p) + denominators.append(Rational(c).q) + except TypeError: + return ConditionSet(symbol, Eq(f_original, 0), domain) + + x = Dummy('x') + + mu = Rational(2)*number_lcm(*denominators)/number_gcd(*numerators) + f = f.subs(symbol, mu*x) + f = f.rewrite(tan) + f = expand_trig(f) + f = together(f) + + g, h = fraction(f) + y = Dummy('y') + g, h = g.expand(), h.expand() + g, h = g.subs(tan(x), y), h.subs(tan(x), y) + + if g.has(x) or h.has(x): + return ConditionSet(symbol, Eq(f_original, 0), domain) + solns = solveset(g, y, S.Reals) - solveset(h, y, S.Reals) + + if isinstance(solns, FiniteSet): + result = Union(*[invert_real(tan(symbol/mu), s, symbol)[1] + for s in solns]) + dsol = invert_real(tan(symbol/mu), oo, symbol)[1] + if degree(h) > degree(g): # If degree(denom)>degree(num) then there + result = Union(result, dsol) # would be another sol at Lim(denom-->oo) + return Intersection(result, domain) + elif solns is S.EmptySet: + return S.EmptySet + else: + return ConditionSet(symbol, Eq(f_original, 0), S.Reals) + + +def _solve_as_poly(f, symbol, domain=S.Complexes): + """ + Solve the equation using polynomial techniques if it already is a + polynomial equation or, with a change of variables, can be made so. + """ + result = None + if f.is_polynomial(symbol): + solns = roots(f, symbol, cubics=True, quartics=True, + quintics=True, domain='EX') + num_roots = sum(solns.values()) + if degree(f, symbol) <= num_roots: + result = FiniteSet(*solns.keys()) + else: + poly = Poly(f, symbol) + solns = poly.all_roots() + if poly.degree() <= len(solns): + result = FiniteSet(*solns) + else: + result = ConditionSet(symbol, Eq(f, 0), domain) + else: + poly = Poly(f) + if poly is None: + result = ConditionSet(symbol, Eq(f, 0), domain) + gens = [g for g in poly.gens if g.has(symbol)] + + if len(gens) == 1: + poly = Poly(poly, gens[0]) + gen = poly.gen + deg = poly.degree() + poly = Poly(poly.as_expr(), poly.gen, composite=True) + poly_solns = FiniteSet(*roots(poly, cubics=True, quartics=True, + quintics=True).keys()) + + if len(poly_solns) < deg: + result = ConditionSet(symbol, Eq(f, 0), domain) + + if gen != symbol: + y = Dummy('y') + inverter = invert_real if domain.is_subset(S.Reals) else invert_complex + lhs, rhs_s = inverter(gen, y, symbol) + if lhs == symbol: + result = Union(*[rhs_s.subs(y, s) for s in poly_solns]) + if isinstance(result, FiniteSet) and isinstance(gen, Pow + ) and gen.base.is_Rational: + result = FiniteSet(*[expand_log(i) for i in result]) + else: + result = ConditionSet(symbol, Eq(f, 0), domain) + else: + result = ConditionSet(symbol, Eq(f, 0), domain) + + if result is not None: + if isinstance(result, FiniteSet): + # this is to simplify solutions like -sqrt(-I) to sqrt(2)/2 + # - sqrt(2)*I/2. We are not expanding for solution with symbols + # or undefined functions because that makes the solution more complicated. + # For example, expand_complex(a) returns re(a) + I*im(a) + if all(s.atoms(Symbol, AppliedUndef) == set() and not isinstance(s, RootOf) + for s in result): + s = Dummy('s') + result = imageset(Lambda(s, expand_complex(s)), result) + if isinstance(result, FiniteSet) and domain != S.Complexes: + # Avoid adding gratuitous intersections with S.Complexes. Actual + # conditions should be handled elsewhere. + result = result.intersection(domain) + return result + else: + return ConditionSet(symbol, Eq(f, 0), domain) + + +def _solve_radical(f, unradf, symbol, solveset_solver): + """ Helper function to solve equations with radicals """ + res = unradf + eq, cov = res if res else (f, []) + if not cov: + result = solveset_solver(eq, symbol) - \ + Union(*[solveset_solver(g, symbol) for g in denoms(f, symbol)]) + else: + y, yeq = cov + if not solveset_solver(y - I, y): + yreal = Dummy('yreal', real=True) + yeq = yeq.xreplace({y: yreal}) + eq = eq.xreplace({y: yreal}) + y = yreal + g_y_s = solveset_solver(yeq, symbol) + f_y_sols = solveset_solver(eq, y) + result = Union(*[imageset(Lambda(y, g_y), f_y_sols) + for g_y in g_y_s]) + + def check_finiteset(solutions): + f_set = [] # solutions for FiniteSet + c_set = [] # solutions for ConditionSet + for s in solutions: + if checksol(f, symbol, s): + f_set.append(s) + else: + c_set.append(s) + return FiniteSet(*f_set) + ConditionSet(symbol, Eq(f, 0), FiniteSet(*c_set)) + + def check_set(solutions): + if solutions is S.EmptySet: + return solutions + elif isinstance(solutions, ConditionSet): + # XXX: Maybe the base set should be checked? + return solutions + elif isinstance(solutions, FiniteSet): + return check_finiteset(solutions) + elif isinstance(solutions, Complement): + A, B = solutions.args + return Complement(check_set(A), B) + elif isinstance(solutions, Union): + return Union(*[check_set(s) for s in solutions.args]) + else: + # XXX: There should be more cases checked here. The cases above + # are all those that come up in the test suite for now. + return solutions + + solution_set = check_set(result) + + return solution_set + + +def _solve_abs(f, symbol, domain): + """ Helper function to solve equation involving absolute value function """ + if not domain.is_subset(S.Reals): + raise ValueError(filldedent(''' + Absolute values cannot be inverted in the + complex domain.''')) + p, q, r = Wild('p'), Wild('q'), Wild('r') + pattern_match = f.match(p*Abs(q) + r) or {} + f_p, f_q, f_r = [pattern_match.get(i, S.Zero) for i in (p, q, r)] + + if not (f_p.is_zero or f_q.is_zero): + domain = continuous_domain(f_q, symbol, domain) + from .inequalities import solve_univariate_inequality + q_pos_cond = solve_univariate_inequality(f_q >= 0, symbol, + relational=False, domain=domain, continuous=True) + q_neg_cond = q_pos_cond.complement(domain) + + sols_q_pos = solveset_real(f_p*f_q + f_r, + symbol).intersect(q_pos_cond) + sols_q_neg = solveset_real(f_p*(-f_q) + f_r, + symbol).intersect(q_neg_cond) + return Union(sols_q_pos, sols_q_neg) + else: + return ConditionSet(symbol, Eq(f, 0), domain) + + +def solve_decomposition(f, symbol, domain): + """ + Function to solve equations via the principle of "Decomposition + and Rewriting". + + Examples + ======== + >>> from sympy import exp, sin, Symbol, pprint, S + >>> from sympy.solvers.solveset import solve_decomposition as sd + >>> x = Symbol('x') + >>> f1 = exp(2*x) - 3*exp(x) + 2 + >>> sd(f1, x, S.Reals) + {0, log(2)} + >>> f2 = sin(x)**2 + 2*sin(x) + 1 + >>> pprint(sd(f2, x, S.Reals), use_unicode=False) + 3*pi + {2*n*pi + ---- | n in Integers} + 2 + >>> f3 = sin(x + 2) + >>> pprint(sd(f3, x, S.Reals), use_unicode=False) + {2*n*pi - 2 | n in Integers} U {2*n*pi - 2 + pi | n in Integers} + + """ + from sympy.solvers.decompogen import decompogen + # decompose the given function + g_s = decompogen(f, symbol) + # `y_s` represents the set of values for which the function `g` is to be + # solved. + # `solutions` represent the solutions of the equations `g = y_s` or + # `g = 0` depending on the type of `y_s`. + # As we are interested in solving the equation: f = 0 + y_s = FiniteSet(0) + for g in g_s: + frange = function_range(g, symbol, domain) + y_s = Intersection(frange, y_s) + result = S.EmptySet + if isinstance(y_s, FiniteSet): + for y in y_s: + solutions = solveset(Eq(g, y), symbol, domain) + if not isinstance(solutions, ConditionSet): + result += solutions + + else: + if isinstance(y_s, ImageSet): + iter_iset = (y_s,) + + elif isinstance(y_s, Union): + iter_iset = y_s.args + + elif y_s is S.EmptySet: + # y_s is not in the range of g in g_s, so no solution exists + #in the given domain + return S.EmptySet + + for iset in iter_iset: + new_solutions = solveset(Eq(iset.lamda.expr, g), symbol, domain) + dummy_var = tuple(iset.lamda.expr.free_symbols)[0] + (base_set,) = iset.base_sets + if isinstance(new_solutions, FiniteSet): + new_exprs = new_solutions + + elif isinstance(new_solutions, Intersection): + if isinstance(new_solutions.args[1], FiniteSet): + new_exprs = new_solutions.args[1] + + for new_expr in new_exprs: + result += ImageSet(Lambda(dummy_var, new_expr), base_set) + + if result is S.EmptySet: + return ConditionSet(symbol, Eq(f, 0), domain) + + y_s = result + + return y_s + + +def _solveset(f, symbol, domain, _check=False): + """Helper for solveset to return a result from an expression + that has already been sympify'ed and is known to contain the + given symbol.""" + # _check controls whether the answer is checked or not + from sympy.simplify.simplify import signsimp + + if isinstance(f, BooleanTrue): + return domain + + orig_f = f + if f.is_Mul: + coeff, f = f.as_independent(symbol, as_Add=False) + if coeff in {S.ComplexInfinity, S.NegativeInfinity, S.Infinity}: + f = together(orig_f) + elif f.is_Add: + a, h = f.as_independent(symbol) + m, h = h.as_independent(symbol, as_Add=False) + if m not in {S.ComplexInfinity, S.Zero, S.Infinity, + S.NegativeInfinity}: + f = a/m + h # XXX condition `m != 0` should be added to soln + + # assign the solvers to use + solver = lambda f, x, domain=domain: _solveset(f, x, domain) + inverter = lambda f, rhs, symbol: _invert(f, rhs, symbol, domain) + + result = S.EmptySet + + if f.expand().is_zero: + return domain + elif not f.has(symbol): + return S.EmptySet + elif f.is_Mul and all(_is_finite_with_finite_vars(m, domain) + for m in f.args): + # if f(x) and g(x) are both finite we can say that the solution of + # f(x)*g(x) == 0 is same as Union(f(x) == 0, g(x) == 0) is not true in + # general. g(x) can grow to infinitely large for the values where + # f(x) == 0. To be sure that we are not silently allowing any + # wrong solutions we are using this technique only if both f and g are + # finite for a finite input. + result = Union(*[solver(m, symbol) for m in f.args]) + elif (_is_function_class_equation(TrigonometricFunction, f, symbol) or \ + _is_function_class_equation(HyperbolicFunction, f, symbol)): + result = _solve_trig(f, symbol, domain) + elif isinstance(f, arg): + a = f.args[0] + result = Intersection(_solveset(re(a) > 0, symbol, domain), + _solveset(im(a), symbol, domain)) + elif f.is_Piecewise: + expr_set_pairs = f.as_expr_set_pairs(domain) + for (expr, in_set) in expr_set_pairs: + if in_set.is_Relational: + in_set = in_set.as_set() + solns = solver(expr, symbol, in_set) + result += solns + elif isinstance(f, Eq): + result = solver(Add(f.lhs, -f.rhs, evaluate=False), symbol, domain) + + elif f.is_Relational: + from .inequalities import solve_univariate_inequality + try: + result = solve_univariate_inequality( + f, symbol, domain=domain, relational=False) + except NotImplementedError: + result = ConditionSet(symbol, f, domain) + return result + elif _is_modular(f, symbol): + result = _solve_modular(f, symbol, domain) + else: + lhs, rhs_s = inverter(f, 0, symbol) + if lhs == symbol: + # do some very minimal simplification since + # repeated inversion may have left the result + # in a state that other solvers (e.g. poly) + # would have simplified; this is done here + # rather than in the inverter since here it + # is only done once whereas there it would + # be repeated for each step of the inversion + if isinstance(rhs_s, FiniteSet): + rhs_s = FiniteSet(*[Mul(* + signsimp(i).as_content_primitive()) + for i in rhs_s]) + result = rhs_s + + elif isinstance(rhs_s, FiniteSet): + for equation in [lhs - rhs for rhs in rhs_s]: + if equation == f: + u = unrad(f, symbol) + if u: + result += _solve_radical(equation, u, + symbol, + solver) + elif equation.has(Abs): + result += _solve_abs(f, symbol, domain) + else: + result_rational = _solve_as_rational(equation, symbol, domain) + if not isinstance(result_rational, ConditionSet): + result += result_rational + else: + # may be a transcendental type equation + t_result = _transolve(equation, symbol, domain) + if isinstance(t_result, ConditionSet): + # might need factoring; this is expensive so we + # have delayed until now. To avoid recursion + # errors look for a non-trivial factoring into + # a product of symbol dependent terms; I think + # that something that factors as a Pow would + # have already been recognized by now. + factored = equation.factor() + if factored.is_Mul and equation != factored: + _, dep = factored.as_independent(symbol) + if not dep.is_Add: + # non-trivial factoring of equation + # but use form with constants + # in case they need special handling + t_results = [] + for fac in Mul.make_args(factored): + if fac.has(symbol): + t_results.append(solver(fac, symbol)) + t_result = Union(*t_results) + result += t_result + else: + result += solver(equation, symbol) + + elif rhs_s is not S.EmptySet: + result = ConditionSet(symbol, Eq(f, 0), domain) + + if isinstance(result, ConditionSet): + if isinstance(f, Expr): + num, den = f.as_numer_denom() + if den.has(symbol): + _result = _solveset(num, symbol, domain) + if not isinstance(_result, ConditionSet): + singularities = _solveset(den, symbol, domain) + result = _result - singularities + + if _check: + if isinstance(result, ConditionSet): + # it wasn't solved or has enumerated all conditions + # -- leave it alone + return result + + # whittle away all but the symbol-containing core + # to use this for testing + if isinstance(orig_f, Expr): + fx = orig_f.as_independent(symbol, as_Add=True)[1] + fx = fx.as_independent(symbol, as_Add=False)[1] + else: + fx = orig_f + + if isinstance(result, FiniteSet): + # check the result for invalid solutions + result = FiniteSet(*[s for s in result + if isinstance(s, RootOf) + or domain_check(fx, symbol, s)]) + + return result + + +def _is_modular(f, symbol): + """ + Helper function to check below mentioned types of modular equations. + ``A - Mod(B, C) = 0`` + + A -> This can or cannot be a function of symbol. + B -> This is surely a function of symbol. + C -> It is an integer. + + Parameters + ========== + + f : Expr + The equation to be checked. + + symbol : Symbol + The concerned variable for which the equation is to be checked. + + Examples + ======== + + >>> from sympy import symbols, exp, Mod + >>> from sympy.solvers.solveset import _is_modular as check + >>> x, y = symbols('x y') + >>> check(Mod(x, 3) - 1, x) + True + >>> check(Mod(x, 3) - 1, y) + False + >>> check(Mod(x, 3)**2 - 5, x) + False + >>> check(Mod(x, 3)**2 - y, x) + False + >>> check(exp(Mod(x, 3)) - 1, x) + False + >>> check(Mod(3, y) - 1, y) + False + """ + + if not f.has(Mod): + return False + + # extract modterms from f. + modterms = list(f.atoms(Mod)) + + return (len(modterms) == 1 and # only one Mod should be present + modterms[0].args[0].has(symbol) and # B-> function of symbol + modterms[0].args[1].is_integer and # C-> to be an integer. + any(isinstance(term, Mod) + for term in list(_term_factors(f))) # free from other funcs + ) + + +def _invert_modular(modterm, rhs, n, symbol): + """ + Helper function to invert modular equation. + ``Mod(a, m) - rhs = 0`` + + Generally it is inverted as (a, ImageSet(Lambda(n, m*n + rhs), S.Integers)). + More simplified form will be returned if possible. + + If it is not invertible then (modterm, rhs) is returned. + + The following cases arise while inverting equation ``Mod(a, m) - rhs = 0``: + + 1. If a is symbol then m*n + rhs is the required solution. + + 2. If a is an instance of ``Add`` then we try to find two symbol independent + parts of a and the symbol independent part gets transferred to the other + side and again the ``_invert_modular`` is called on the symbol + dependent part. + + 3. If a is an instance of ``Mul`` then same as we done in ``Add`` we separate + out the symbol dependent and symbol independent parts and transfer the + symbol independent part to the rhs with the help of invert and again the + ``_invert_modular`` is called on the symbol dependent part. + + 4. If a is an instance of ``Pow`` then two cases arise as following: + + - If a is of type (symbol_indep)**(symbol_dep) then the remainder is + evaluated with the help of discrete_log function and then the least + period is being found out with the help of totient function. + period*n + remainder is the required solution in this case. + For reference: (https://en.wikipedia.org/wiki/Euler's_theorem) + + - If a is of type (symbol_dep)**(symbol_indep) then we try to find all + primitive solutions list with the help of nthroot_mod function. + m*n + rem is the general solution where rem belongs to solutions list + from nthroot_mod function. + + Parameters + ========== + + modterm, rhs : Expr + The modular equation to be inverted, ``modterm - rhs = 0`` + + symbol : Symbol + The variable in the equation to be inverted. + + n : Dummy + Dummy variable for output g_n. + + Returns + ======= + + A tuple (f_x, g_n) is being returned where f_x is modular independent function + of symbol and g_n being set of values f_x can have. + + Examples + ======== + + >>> from sympy import symbols, exp, Mod, Dummy, S + >>> from sympy.solvers.solveset import _invert_modular as invert_modular + >>> x, y = symbols('x y') + >>> n = Dummy('n') + >>> invert_modular(Mod(exp(x), 7), S(5), n, x) + (Mod(exp(x), 7), 5) + >>> invert_modular(Mod(x, 7), S(5), n, x) + (x, ImageSet(Lambda(_n, 7*_n + 5), Integers)) + >>> invert_modular(Mod(3*x + 8, 7), S(5), n, x) + (x, ImageSet(Lambda(_n, 7*_n + 6), Integers)) + >>> invert_modular(Mod(x**4, 7), S(5), n, x) + (x, EmptySet) + >>> invert_modular(Mod(2**(x**2 + x + 1), 7), S(2), n, x) + (x**2 + x + 1, ImageSet(Lambda(_n, 3*_n + 1), Naturals0)) + + """ + a, m = modterm.args + + if rhs.is_integer is False: + return symbol, S.EmptySet + + if rhs.is_real is False or any(term.is_real is False + for term in list(_term_factors(a))): + # Check for complex arguments + return modterm, rhs + + if abs(rhs) >= abs(m): + # if rhs has value greater than value of m. + return symbol, S.EmptySet + + if a == symbol: + return symbol, ImageSet(Lambda(n, m*n + rhs), S.Integers) + + if a.is_Add: + # g + h = a + g, h = a.as_independent(symbol) + if g is not S.Zero: + x_indep_term = rhs - Mod(g, m) + return _invert_modular(Mod(h, m), Mod(x_indep_term, m), n, symbol) + + if a.is_Mul: + # g*h = a + g, h = a.as_independent(symbol) + if g is not S.One: + x_indep_term = rhs*invert(g, m) + return _invert_modular(Mod(h, m), Mod(x_indep_term, m), n, symbol) + + if a.is_Pow: + # base**expo = a + base, expo = a.args + if expo.has(symbol) and not base.has(symbol): + # remainder -> solution independent of n of equation. + # m, rhs are made coprime by dividing number_gcd(m, rhs) + if not m.is_Integer and rhs.is_Integer and a.base.is_Integer: + return modterm, rhs + + mdiv = m.p // number_gcd(m.p, rhs.p) + try: + remainder = discrete_log(mdiv, rhs.p, a.base.p) + except ValueError: # log does not exist + return modterm, rhs + # period -> coefficient of n in the solution and also referred as + # the least period of expo in which it is repeats itself. + # (a**(totient(m)) - 1) divides m. Here is link of theorem: + # (https://en.wikipedia.org/wiki/Euler's_theorem) + period = totient(m) + for p in divisors(period): + # there might a lesser period exist than totient(m). + if pow(a.base, p, m / number_gcd(m.p, a.base.p)) == 1: + period = p + break + # recursion is not applied here since _invert_modular is currently + # not smart enough to handle infinite rhs as here expo has infinite + # rhs = ImageSet(Lambda(n, period*n + remainder), S.Naturals0). + return expo, ImageSet(Lambda(n, period*n + remainder), S.Naturals0) + elif base.has(symbol) and not expo.has(symbol): + try: + remainder_list = nthroot_mod(rhs, expo, m, all_roots=True) + if remainder_list == []: + return symbol, S.EmptySet + except (ValueError, NotImplementedError): + return modterm, rhs + g_n = S.EmptySet + for rem in remainder_list: + g_n += ImageSet(Lambda(n, m*n + rem), S.Integers) + return base, g_n + + return modterm, rhs + + +def _solve_modular(f, symbol, domain): + r""" + Helper function for solving modular equations of type ``A - Mod(B, C) = 0``, + where A can or cannot be a function of symbol, B is surely a function of + symbol and C is an integer. + + Currently ``_solve_modular`` is only able to solve cases + where A is not a function of symbol. + + Parameters + ========== + + f : Expr + The modular equation to be solved, ``f = 0`` + + symbol : Symbol + The variable in the equation to be solved. + + domain : Set + A set over which the equation is solved. It has to be a subset of + Integers. + + Returns + ======= + + A set of integer solutions satisfying the given modular equation. + A ``ConditionSet`` if the equation is unsolvable. + + Examples + ======== + + >>> from sympy.solvers.solveset import _solve_modular as solve_modulo + >>> from sympy import S, Symbol, sin, Intersection, Interval, Mod + >>> x = Symbol('x') + >>> solve_modulo(Mod(5*x - 8, 7) - 3, x, S.Integers) + ImageSet(Lambda(_n, 7*_n + 5), Integers) + >>> solve_modulo(Mod(5*x - 8, 7) - 3, x, S.Reals) # domain should be subset of integers. + ConditionSet(x, Eq(Mod(5*x + 6, 7) - 3, 0), Reals) + >>> solve_modulo(-7 + Mod(x, 5), x, S.Integers) + EmptySet + >>> solve_modulo(Mod(12**x, 21) - 18, x, S.Integers) + ImageSet(Lambda(_n, 6*_n + 2), Naturals0) + >>> solve_modulo(Mod(sin(x), 7) - 3, x, S.Integers) # not solvable + ConditionSet(x, Eq(Mod(sin(x), 7) - 3, 0), Integers) + >>> solve_modulo(3 - Mod(x, 5), x, Intersection(S.Integers, Interval(0, 100))) + Intersection(ImageSet(Lambda(_n, 5*_n + 3), Integers), Range(0, 101, 1)) + """ + # extract modterm and g_y from f + unsolved_result = ConditionSet(symbol, Eq(f, 0), domain) + modterm = list(f.atoms(Mod))[0] + rhs = -S.One*(f.subs(modterm, S.Zero)) + if f.as_coefficients_dict()[modterm].is_negative: + # checks if coefficient of modterm is negative in main equation. + rhs *= -S.One + + if not domain.is_subset(S.Integers): + return unsolved_result + + if rhs.has(symbol): + # TODO Case: A-> function of symbol, can be extended here + # in future. + return unsolved_result + + n = Dummy('n', integer=True) + f_x, g_n = _invert_modular(modterm, rhs, n, symbol) + + if f_x == modterm and g_n == rhs: + return unsolved_result + + if f_x == symbol: + if domain is not S.Integers: + return domain.intersect(g_n) + return g_n + + if isinstance(g_n, ImageSet): + lamda_expr = g_n.lamda.expr + lamda_vars = g_n.lamda.variables + base_sets = g_n.base_sets + sol_set = _solveset(f_x - lamda_expr, symbol, S.Integers) + if isinstance(sol_set, FiniteSet): + tmp_sol = S.EmptySet + for sol in sol_set: + tmp_sol += ImageSet(Lambda(lamda_vars, sol), *base_sets) + sol_set = tmp_sol + else: + sol_set = ImageSet(Lambda(lamda_vars, sol_set), *base_sets) + return domain.intersect(sol_set) + + return unsolved_result + + +def _term_factors(f): + """ + Iterator to get the factors of all terms present + in the given equation. + + Parameters + ========== + f : Expr + Equation that needs to be addressed + + Returns + ======= + Factors of all terms present in the equation. + + Examples + ======== + + >>> from sympy import symbols + >>> from sympy.solvers.solveset import _term_factors + >>> x = symbols('x') + >>> list(_term_factors(-2 - x**2 + x*(x + 1))) + [-2, -1, x**2, x, x + 1] + """ + for add_arg in Add.make_args(f): + yield from Mul.make_args(add_arg) + + +def _solve_exponential(lhs, rhs, symbol, domain): + r""" + Helper function for solving (supported) exponential equations. + + Exponential equations are the sum of (currently) at most + two terms with one or both of them having a power with a + symbol-dependent exponent. + + For example + + .. math:: 5^{2x + 3} - 5^{3x - 1} + + .. math:: 4^{5 - 9x} - e^{2 - x} + + Parameters + ========== + + lhs, rhs : Expr + The exponential equation to be solved, `lhs = rhs` + + symbol : Symbol + The variable in which the equation is solved + + domain : Set + A set over which the equation is solved. + + Returns + ======= + + A set of solutions satisfying the given equation. + A ``ConditionSet`` if the equation is unsolvable or + if the assumptions are not properly defined, in that case + a different style of ``ConditionSet`` is returned having the + solution(s) of the equation with the desired assumptions. + + Examples + ======== + + >>> from sympy.solvers.solveset import _solve_exponential as solve_expo + >>> from sympy import symbols, S + >>> x = symbols('x', real=True) + >>> a, b = symbols('a b') + >>> solve_expo(2**x + 3**x - 5**x, 0, x, S.Reals) # not solvable + ConditionSet(x, Eq(2**x + 3**x - 5**x, 0), Reals) + >>> solve_expo(a**x - b**x, 0, x, S.Reals) # solvable but incorrect assumptions + ConditionSet(x, (a > 0) & (b > 0), {0}) + >>> solve_expo(3**(2*x) - 2**(x + 3), 0, x, S.Reals) + {-3*log(2)/(-2*log(3) + log(2))} + >>> solve_expo(2**x - 4**x, 0, x, S.Reals) + {0} + + * Proof of correctness of the method + + The logarithm function is the inverse of the exponential function. + The defining relation between exponentiation and logarithm is: + + .. math:: {\log_b x} = y \enspace if \enspace b^y = x + + Therefore if we are given an equation with exponent terms, we can + convert every term to its corresponding logarithmic form. This is + achieved by taking logarithms and expanding the equation using + logarithmic identities so that it can easily be handled by ``solveset``. + + For example: + + .. math:: 3^{2x} = 2^{x + 3} + + Taking log both sides will reduce the equation to + + .. math:: (2x)\log(3) = (x + 3)\log(2) + + This form can be easily handed by ``solveset``. + """ + unsolved_result = ConditionSet(symbol, Eq(lhs - rhs, 0), domain) + newlhs = powdenest(lhs) + if lhs != newlhs: + # it may also be advantageous to factor the new expr + neweq = factor(newlhs - rhs) + if neweq != (lhs - rhs): + return _solveset(neweq, symbol, domain) # try again with _solveset + + if not (isinstance(lhs, Add) and len(lhs.args) == 2): + # solving for the sum of more than two powers is possible + # but not yet implemented + return unsolved_result + + if rhs != 0: + return unsolved_result + + a, b = list(ordered(lhs.args)) + a_term = a.as_independent(symbol)[1] + b_term = b.as_independent(symbol)[1] + + a_base, a_exp = a_term.as_base_exp() + b_base, b_exp = b_term.as_base_exp() + + if domain.is_subset(S.Reals): + conditions = And( + a_base > 0, + b_base > 0, + Eq(im(a_exp), 0), + Eq(im(b_exp), 0)) + else: + conditions = And( + Ne(a_base, 0), + Ne(b_base, 0)) + + L, R = (expand_log(log(i), force=True) for i in (a, -b)) + solutions = _solveset(L - R, symbol, domain) + + return ConditionSet(symbol, conditions, solutions) + + +def _is_exponential(f, symbol): + r""" + Return ``True`` if one or more terms contain ``symbol`` only in + exponents, else ``False``. + + Parameters + ========== + + f : Expr + The equation to be checked + + symbol : Symbol + The variable in which the equation is checked + + Examples + ======== + + >>> from sympy import symbols, cos, exp + >>> from sympy.solvers.solveset import _is_exponential as check + >>> x, y = symbols('x y') + >>> check(y, y) + False + >>> check(x**y - 1, y) + True + >>> check(x**y*2**y - 1, y) + True + >>> check(exp(x + 3) + 3**x, x) + True + >>> check(cos(2**x), x) + False + + * Philosophy behind the helper + + The function extracts each term of the equation and checks if it is + of exponential form w.r.t ``symbol``. + """ + rv = False + for expr_arg in _term_factors(f): + if symbol not in expr_arg.free_symbols: + continue + if (isinstance(expr_arg, Pow) and + symbol not in expr_arg.base.free_symbols or + isinstance(expr_arg, exp)): + rv = True # symbol in exponent + else: + return False # dependent on symbol in non-exponential way + return rv + + +def _solve_logarithm(lhs, rhs, symbol, domain): + r""" + Helper to solve logarithmic equations which are reducible + to a single instance of `\log`. + + Logarithmic equations are (currently) the equations that contains + `\log` terms which can be reduced to a single `\log` term or + a constant using various logarithmic identities. + + For example: + + .. math:: \log(x) + \log(x - 4) + + can be reduced to: + + .. math:: \log(x(x - 4)) + + Parameters + ========== + + lhs, rhs : Expr + The logarithmic equation to be solved, `lhs = rhs` + + symbol : Symbol + The variable in which the equation is solved + + domain : Set + A set over which the equation is solved. + + Returns + ======= + + A set of solutions satisfying the given equation. + A ``ConditionSet`` if the equation is unsolvable. + + Examples + ======== + + >>> from sympy import symbols, log, S + >>> from sympy.solvers.solveset import _solve_logarithm as solve_log + >>> x = symbols('x') + >>> f = log(x - 3) + log(x + 3) + >>> solve_log(f, 0, x, S.Reals) + {-sqrt(10), sqrt(10)} + + * Proof of correctness + + A logarithm is another way to write exponent and is defined by + + .. math:: {\log_b x} = y \enspace if \enspace b^y = x + + When one side of the equation contains a single logarithm, the + equation can be solved by rewriting the equation as an equivalent + exponential equation as defined above. But if one side contains + more than one logarithm, we need to use the properties of logarithm + to condense it into a single logarithm. + + Take for example + + .. math:: \log(2x) - 15 = 0 + + contains single logarithm, therefore we can directly rewrite it to + exponential form as + + .. math:: x = \frac{e^{15}}{2} + + But if the equation has more than one logarithm as + + .. math:: \log(x - 3) + \log(x + 3) = 0 + + we use logarithmic identities to convert it into a reduced form + + Using, + + .. math:: \log(a) + \log(b) = \log(ab) + + the equation becomes, + + .. math:: \log((x - 3)(x + 3)) + + This equation contains one logarithm and can be solved by rewriting + to exponents. + """ + new_lhs = logcombine(lhs, force=True) + new_f = new_lhs - rhs + + return _solveset(new_f, symbol, domain) + + +def _is_logarithmic(f, symbol): + r""" + Return ``True`` if the equation is in the form + `a\log(f(x)) + b\log(g(x)) + ... + c` else ``False``. + + Parameters + ========== + + f : Expr + The equation to be checked + + symbol : Symbol + The variable in which the equation is checked + + Returns + ======= + + ``True`` if the equation is logarithmic otherwise ``False``. + + Examples + ======== + + >>> from sympy import symbols, tan, log + >>> from sympy.solvers.solveset import _is_logarithmic as check + >>> x, y = symbols('x y') + >>> check(log(x + 2) - log(x + 3), x) + True + >>> check(tan(log(2*x)), x) + False + >>> check(x*log(x), x) + False + >>> check(x + log(x), x) + False + >>> check(y + log(x), x) + True + + * Philosophy behind the helper + + The function extracts each term and checks whether it is + logarithmic w.r.t ``symbol``. + """ + rv = False + for term in Add.make_args(f): + saw_log = False + for term_arg in Mul.make_args(term): + if symbol not in term_arg.free_symbols: + continue + if isinstance(term_arg, log): + if saw_log: + return False # more than one log in term + saw_log = True + else: + return False # dependent on symbol in non-log way + if saw_log: + rv = True + return rv + + +def _is_lambert(f, symbol): + r""" + If this returns ``False`` then the Lambert solver (``_solve_lambert``) will not be called. + + Explanation + =========== + + Quick check for cases that the Lambert solver might be able to handle. + + 1. Equations containing more than two operands and `symbol`s involving any of + `Pow`, `exp`, `HyperbolicFunction`,`TrigonometricFunction`, `log` terms. + + 2. In `Pow`, `exp` the exponent should have `symbol` whereas for + `HyperbolicFunction`,`TrigonometricFunction`, `log` should contain `symbol`. + + 3. For `HyperbolicFunction`,`TrigonometricFunction` the number of trigonometric functions in + equation should be less than number of symbols. (since `A*cos(x) + B*sin(x) - c` + is not the Lambert type). + + Some forms of lambert equations are: + 1. X**X = C + 2. X*(B*log(X) + D)**A = C + 3. A*log(B*X + A) + d*X = C + 4. (B*X + A)*exp(d*X + g) = C + 5. g*exp(B*X + h) - B*X = C + 6. A*D**(E*X + g) - B*X = C + 7. A*cos(X) + B*sin(X) - D*X = C + 8. A*cosh(X) + B*sinh(X) - D*X = C + + Where X is any variable, + A, B, C, D, E are any constants, + g, h are linear functions or log terms. + + Parameters + ========== + + f : Expr + The equation to be checked + + symbol : Symbol + The variable in which the equation is checked + + Returns + ======= + + If this returns ``False`` then the Lambert solver (``_solve_lambert``) will not be called. + + Examples + ======== + + >>> from sympy.solvers.solveset import _is_lambert + >>> from sympy import symbols, cosh, sinh, log + >>> x = symbols('x') + + >>> _is_lambert(3*log(x) - x*log(3), x) + True + >>> _is_lambert(log(log(x - 3)) + log(x-3), x) + True + >>> _is_lambert(cosh(x) - sinh(x), x) + False + >>> _is_lambert((x**2 - 2*x + 1).subs(x, (log(x) + 3*x)**2 - 1), x) + True + + See Also + ======== + + _solve_lambert + + """ + term_factors = list(_term_factors(f.expand())) + + # total number of symbols in equation + no_of_symbols = len([arg for arg in term_factors if arg.has(symbol)]) + # total number of trigonometric terms in equation + no_of_trig = len([arg for arg in term_factors \ + if arg.has(HyperbolicFunction, TrigonometricFunction)]) + + if f.is_Add and no_of_symbols >= 2: + # `log`, `HyperbolicFunction`, `TrigonometricFunction` should have symbols + # and no_of_trig < no_of_symbols + lambert_funcs = (log, HyperbolicFunction, TrigonometricFunction) + if any(isinstance(arg, lambert_funcs)\ + for arg in term_factors if arg.has(symbol)): + if no_of_trig < no_of_symbols: + return True + # here, `Pow`, `exp` exponent should have symbols + elif any(isinstance(arg, (Pow, exp)) \ + for arg in term_factors if (arg.as_base_exp()[1]).has(symbol)): + return True + return False + + +def _transolve(f, symbol, domain): + r""" + Function to solve transcendental equations. It is a helper to + ``solveset`` and should be used internally. ``_transolve`` + currently supports the following class of equations: + + - Exponential equations + - Logarithmic equations + + Parameters + ========== + + f : Any transcendental equation that needs to be solved. + This needs to be an expression, which is assumed + to be equal to ``0``. + + symbol : The variable for which the equation is solved. + This needs to be of class ``Symbol``. + + domain : A set over which the equation is solved. + This needs to be of class ``Set``. + + Returns + ======= + + Set + A set of values for ``symbol`` for which ``f`` is equal to + zero. An ``EmptySet`` is returned if ``f`` does not have solutions + in respective domain. A ``ConditionSet`` is returned as unsolved + object if algorithms to evaluate complete solution are not + yet implemented. + + How to use ``_transolve`` + ========================= + + ``_transolve`` should not be used as an independent function, because + it assumes that the equation (``f``) and the ``symbol`` comes from + ``solveset`` and might have undergone a few modification(s). + To use ``_transolve`` as an independent function the equation (``f``) + and the ``symbol`` should be passed as they would have been by + ``solveset``. + + Examples + ======== + + >>> from sympy.solvers.solveset import _transolve as transolve + >>> from sympy.solvers.solvers import _tsolve as tsolve + >>> from sympy import symbols, S, pprint + >>> x = symbols('x', real=True) # assumption added + >>> transolve(5**(x - 3) - 3**(2*x + 1), x, S.Reals) + {-(log(3) + 3*log(5))/(-log(5) + 2*log(3))} + + How ``_transolve`` works + ======================== + + ``_transolve`` uses two types of helper functions to solve equations + of a particular class: + + Identifying helpers: To determine whether a given equation + belongs to a certain class of equation or not. Returns either + ``True`` or ``False``. + + Solving helpers: Once an equation is identified, a corresponding + helper either solves the equation or returns a form of the equation + that ``solveset`` might better be able to handle. + + * Philosophy behind the module + + The purpose of ``_transolve`` is to take equations which are not + already polynomial in their generator(s) and to either recast them + as such through a valid transformation or to solve them outright. + A pair of helper functions for each class of supported + transcendental functions are employed for this purpose. One + identifies the transcendental form of an equation and the other + either solves it or recasts it into a tractable form that can be + solved by ``solveset``. + For example, an equation in the form `ab^{f(x)} - cd^{g(x)} = 0` + can be transformed to + `\log(a) + f(x)\log(b) - \log(c) - g(x)\log(d) = 0` + (under certain assumptions) and this can be solved with ``solveset`` + if `f(x)` and `g(x)` are in polynomial form. + + How ``_transolve`` is better than ``_tsolve`` + ============================================= + + 1) Better output + + ``_transolve`` provides expressions in a more simplified form. + + Consider a simple exponential equation + + >>> f = 3**(2*x) - 2**(x + 3) + >>> pprint(transolve(f, x, S.Reals), use_unicode=False) + -3*log(2) + {------------------} + -2*log(3) + log(2) + >>> pprint(tsolve(f, x), use_unicode=False) + / 3 \ + | --------| + | log(2/9)| + [-log\2 /] + + 2) Extensible + + The API of ``_transolve`` is designed such that it is easily + extensible, i.e. the code that solves a given class of + equations is encapsulated in a helper and not mixed in with + the code of ``_transolve`` itself. + + 3) Modular + + ``_transolve`` is designed to be modular i.e, for every class of + equation a separate helper for identification and solving is + implemented. This makes it easy to change or modify any of the + method implemented directly in the helpers without interfering + with the actual structure of the API. + + 4) Faster Computation + + Solving equation via ``_transolve`` is much faster as compared to + ``_tsolve``. In ``solve``, attempts are made computing every possibility + to get the solutions. This series of attempts makes solving a bit + slow. In ``_transolve``, computation begins only after a particular + type of equation is identified. + + How to add new class of equations + ================================= + + Adding a new class of equation solver is a three-step procedure: + + - Identify the type of the equations + + Determine the type of the class of equations to which they belong: + it could be of ``Add``, ``Pow``, etc. types. Separate internal functions + are used for each type. Write identification and solving helpers + and use them from within the routine for the given type of equation + (after adding it, if necessary). Something like: + + .. code-block:: python + + def add_type(lhs, rhs, x): + .... + if _is_exponential(lhs, x): + new_eq = _solve_exponential(lhs, rhs, x) + .... + rhs, lhs = eq.as_independent(x) + if lhs.is_Add: + result = add_type(lhs, rhs, x) + + - Define the identification helper. + + - Define the solving helper. + + Apart from this, a few other things needs to be taken care while + adding an equation solver: + + - Naming conventions: + Name of the identification helper should be as + ``_is_class`` where class will be the name or abbreviation + of the class of equation. The solving helper will be named as + ``_solve_class``. + For example: for exponential equations it becomes + ``_is_exponential`` and ``_solve_expo``. + - The identifying helpers should take two input parameters, + the equation to be checked and the variable for which a solution + is being sought, while solving helpers would require an additional + domain parameter. + - Be sure to consider corner cases. + - Add tests for each helper. + - Add a docstring to your helper that describes the method + implemented. + The documentation of the helpers should identify: + + - the purpose of the helper, + - the method used to identify and solve the equation, + - a proof of correctness + - the return values of the helpers + """ + + def add_type(lhs, rhs, symbol, domain): + """ + Helper for ``_transolve`` to handle equations of + ``Add`` type, i.e. equations taking the form as + ``a*f(x) + b*g(x) + .... = c``. + For example: 4**x + 8**x = 0 + """ + result = ConditionSet(symbol, Eq(lhs - rhs, 0), domain) + + # check if it is exponential type equation + if _is_exponential(lhs, symbol): + result = _solve_exponential(lhs, rhs, symbol, domain) + # check if it is logarithmic type equation + elif _is_logarithmic(lhs, symbol): + result = _solve_logarithm(lhs, rhs, symbol, domain) + + return result + + result = ConditionSet(symbol, Eq(f, 0), domain) + + # invert_complex handles the call to the desired inverter based + # on the domain specified. + lhs, rhs_s = invert_complex(f, 0, symbol, domain) + + if isinstance(rhs_s, FiniteSet): + assert (len(rhs_s.args)) == 1 + rhs = rhs_s.args[0] + + if lhs.is_Add: + result = add_type(lhs, rhs, symbol, domain) + else: + result = rhs_s + + return result + + +def solveset(f, symbol=None, domain=S.Complexes): + r"""Solves a given inequality or equation with set as output + + Parameters + ========== + + f : Expr or a relational. + The target equation or inequality + symbol : Symbol + The variable for which the equation is solved + domain : Set + The domain over which the equation is solved + + Returns + ======= + + Set + A set of values for `symbol` for which `f` is True or is equal to + zero. An :class:`~.EmptySet` is returned if `f` is False or nonzero. + A :class:`~.ConditionSet` is returned as unsolved object if algorithms + to evaluate complete solution are not yet implemented. + + ``solveset`` claims to be complete in the solution set that it returns. + + Raises + ====== + + NotImplementedError + The algorithms to solve inequalities in complex domain are + not yet implemented. + ValueError + The input is not valid. + RuntimeError + It is a bug, please report to the github issue tracker. + + + Notes + ===== + + Python interprets 0 and 1 as False and True, respectively, but + in this function they refer to solutions of an expression. So 0 and 1 + return the domain and EmptySet, respectively, while True and False + return the opposite (as they are assumed to be solutions of relational + expressions). + + + See Also + ======== + + solveset_real: solver for real domain + solveset_complex: solver for complex domain + + Examples + ======== + + >>> from sympy import exp, sin, Symbol, pprint, S, Eq + >>> from sympy.solvers.solveset import solveset, solveset_real + + * The default domain is complex. Not specifying a domain will lead + to the solving of the equation in the complex domain (and this + is not affected by the assumptions on the symbol): + + >>> x = Symbol('x') + >>> pprint(solveset(exp(x) - 1, x), use_unicode=False) + {2*n*I*pi | n in Integers} + + >>> x = Symbol('x', real=True) + >>> pprint(solveset(exp(x) - 1, x), use_unicode=False) + {2*n*I*pi | n in Integers} + + * If you want to use ``solveset`` to solve the equation in the + real domain, provide a real domain. (Using ``solveset_real`` + does this automatically.) + + >>> R = S.Reals + >>> x = Symbol('x') + >>> solveset(exp(x) - 1, x, R) + {0} + >>> solveset_real(exp(x) - 1, x) + {0} + + The solution is unaffected by assumptions on the symbol: + + >>> p = Symbol('p', positive=True) + >>> pprint(solveset(p**2 - 4)) + {-2, 2} + + When a :class:`~.ConditionSet` is returned, symbols with assumptions that + would alter the set are replaced with more generic symbols: + + >>> i = Symbol('i', imaginary=True) + >>> solveset(Eq(i**2 + i*sin(i), 1), i, domain=S.Reals) + ConditionSet(_R, Eq(_R**2 + _R*sin(_R) - 1, 0), Reals) + + * Inequalities can be solved over the real domain only. Use of a complex + domain leads to a NotImplementedError. + + >>> solveset(exp(x) > 1, x, R) + Interval.open(0, oo) + + """ + f = sympify(f) + symbol = sympify(symbol) + + if f is S.true: + return domain + + if f is S.false: + return S.EmptySet + + if not isinstance(f, (Expr, Relational, Number)): + raise ValueError("%s is not a valid SymPy expression" % f) + + if not isinstance(symbol, (Expr, Relational)) and symbol is not None: + raise ValueError("%s is not a valid SymPy symbol" % (symbol,)) + + if not isinstance(domain, Set): + raise ValueError("%s is not a valid domain" %(domain)) + + free_symbols = f.free_symbols + + if f.has(Piecewise): + f = piecewise_fold(f) + + if symbol is None and not free_symbols: + b = Eq(f, 0) + if b is S.true: + return domain + elif b is S.false: + return S.EmptySet + else: + raise NotImplementedError(filldedent(''' + relationship between value and 0 is unknown: %s''' % b)) + + if symbol is None: + if len(free_symbols) == 1: + symbol = free_symbols.pop() + elif free_symbols: + raise ValueError(filldedent(''' + The independent variable must be specified for a + multivariate equation.''')) + elif not isinstance(symbol, Symbol): + f, s, swap = recast_to_symbols([f], [symbol]) + # the xreplace will be needed if a ConditionSet is returned + return solveset(f[0], s[0], domain).xreplace(swap) + + # solveset should ignore assumptions on symbols + newsym = None + if domain.is_subset(S.Reals): + if symbol._assumptions_orig != {'real': True}: + newsym = Dummy('R', real=True) + elif domain.is_subset(S.Complexes): + if symbol._assumptions_orig != {'complex': True}: + newsym = Dummy('C', complex=True) + + if newsym is not None: + rv = solveset(f.xreplace({symbol: newsym}), newsym, domain) + # try to use the original symbol if possible + try: + _rv = rv.xreplace({newsym: symbol}) + except TypeError: + _rv = rv + if rv.dummy_eq(_rv): + rv = _rv + return rv + + # Abs has its own handling method which avoids the + # rewriting property that the first piece of abs(x) + # is for x >= 0 and the 2nd piece for x < 0 -- solutions + # can look better if the 2nd condition is x <= 0. Since + # the solution is a set, duplication of results is not + # an issue, e.g. {y, -y} when y is 0 will be {0} + f, mask = _masked(f, Abs) + f = f.rewrite(Piecewise) # everything that's not an Abs + for d, e in mask: + # everything *in* an Abs + e = e.func(e.args[0].rewrite(Piecewise)) + f = f.xreplace({d: e}) + f = piecewise_fold(f) + + return _solveset(f, symbol, domain, _check=True) + + +def solveset_real(f, symbol): + return solveset(f, symbol, S.Reals) + + +def solveset_complex(f, symbol): + return solveset(f, symbol, S.Complexes) + + +def _solveset_multi(eqs, syms, domains): + '''Basic implementation of a multivariate solveset. + + For internal use (not ready for public consumption)''' + + rep = {} + for sym, dom in zip(syms, domains): + if dom is S.Reals: + rep[sym] = Symbol(sym.name, real=True) + eqs = [eq.subs(rep) for eq in eqs] + syms = [sym.subs(rep) for sym in syms] + + syms = tuple(syms) + + if len(eqs) == 0: + return ProductSet(*domains) + + if len(syms) == 1: + sym = syms[0] + domain = domains[0] + solsets = [solveset(eq, sym, domain) for eq in eqs] + solset = Intersection(*solsets) + return ImageSet(Lambda((sym,), (sym,)), solset).doit() + + eqs = sorted(eqs, key=lambda eq: len(eq.free_symbols & set(syms))) + + for n, eq in enumerate(eqs): + sols = [] + all_handled = True + for sym in syms: + if sym not in eq.free_symbols: + continue + sol = solveset(eq, sym, domains[syms.index(sym)]) + + if isinstance(sol, FiniteSet): + i = syms.index(sym) + symsp = syms[:i] + syms[i+1:] + domainsp = domains[:i] + domains[i+1:] + eqsp = eqs[:n] + eqs[n+1:] + for s in sol: + eqsp_sub = [eq.subs(sym, s) for eq in eqsp] + sol_others = _solveset_multi(eqsp_sub, symsp, domainsp) + fun = Lambda((symsp,), symsp[:i] + (s,) + symsp[i:]) + sols.append(ImageSet(fun, sol_others).doit()) + else: + all_handled = False + if all_handled: + return Union(*sols) + + +def solvify(f, symbol, domain): + """Solves an equation using solveset and returns the solution in accordance + with the `solve` output API. + + Returns + ======= + + We classify the output based on the type of solution returned by `solveset`. + + Solution | Output + ---------------------------------------- + FiniteSet | list + + ImageSet, | list (if `f` is periodic) + Union | + + Union | list (with FiniteSet) + + EmptySet | empty list + + Others | None + + + Raises + ====== + + NotImplementedError + A ConditionSet is the input. + + Examples + ======== + + >>> from sympy.solvers.solveset import solvify + >>> from sympy.abc import x + >>> from sympy import S, tan, sin, exp + >>> solvify(x**2 - 9, x, S.Reals) + [-3, 3] + >>> solvify(sin(x) - 1, x, S.Reals) + [pi/2] + >>> solvify(tan(x), x, S.Reals) + [0] + >>> solvify(exp(x) - 1, x, S.Complexes) + + >>> solvify(exp(x) - 1, x, S.Reals) + [0] + + """ + solution_set = solveset(f, symbol, domain) + result = None + if solution_set is S.EmptySet: + result = [] + + elif isinstance(solution_set, ConditionSet): + raise NotImplementedError('solveset is unable to solve this equation.') + + elif isinstance(solution_set, FiniteSet): + result = list(solution_set) + + else: + period = periodicity(f, symbol) + if period is not None: + solutions = S.EmptySet + iter_solutions = () + if isinstance(solution_set, ImageSet): + iter_solutions = (solution_set,) + elif isinstance(solution_set, Union): + if all(isinstance(i, ImageSet) for i in solution_set.args): + iter_solutions = solution_set.args + + for solution in iter_solutions: + solutions += solution.intersect(Interval(0, period, False, True)) + + if isinstance(solutions, FiniteSet): + result = list(solutions) + + else: + solution = solution_set.intersect(domain) + if isinstance(solution, Union): + # concerned about only FiniteSet with Union but not about ImageSet + # if required could be extend + if any(isinstance(i, FiniteSet) for i in solution.args): + result = [sol for soln in solution.args \ + for sol in soln.args if isinstance(soln,FiniteSet)] + else: + return None + + elif isinstance(solution, FiniteSet): + result += solution + + return result + + +############################################################################### +################################ LINSOLVE ##################################### +############################################################################### + + +def linear_coeffs(eq, *syms, dict=False): + """Return a list whose elements are the coefficients of the + corresponding symbols in the sum of terms in ``eq``. + The additive constant is returned as the last element of the + list. + + Raises + ====== + + NonlinearError + The equation contains a nonlinear term + ValueError + duplicate or unordered symbols are passed + + Parameters + ========== + + dict - (default False) when True, return coefficients as a + dictionary with coefficients keyed to syms that were present; + key 1 gives the constant term + + Examples + ======== + + >>> from sympy.solvers.solveset import linear_coeffs + >>> from sympy.abc import x, y, z + >>> linear_coeffs(3*x + 2*y - 1, x, y) + [3, 2, -1] + + It is not necessary to expand the expression: + + >>> linear_coeffs(x + y*(z*(x*3 + 2) + 3), x) + [3*y*z + 1, y*(2*z + 3)] + + When nonlinear is detected, an error will be raised: + + * even if they would cancel after expansion (so the + situation does not pass silently past the caller's + attention) + + >>> eq = 1/x*(x - 1) + 1/x + >>> linear_coeffs(eq.expand(), x) + [0, 1] + >>> linear_coeffs(eq, x) + Traceback (most recent call last): + ... + NonlinearError: + nonlinear in given generators + + * when there are cross terms + + >>> linear_coeffs(x*(y + 1), x, y) + Traceback (most recent call last): + ... + NonlinearError: + symbol-dependent cross-terms encountered + + * when there are terms that contain an expression + dependent on the symbols that is not linear + + >>> linear_coeffs(x**2, x) + Traceback (most recent call last): + ... + NonlinearError: + nonlinear in given generators + """ + eq = _sympify(eq) + if len(syms) == 1 and iterable(syms[0]) and not isinstance(syms[0], Basic): + raise ValueError('expecting unpacked symbols, *syms') + symset = set(syms) + if len(symset) != len(syms): + raise ValueError('duplicate symbols given') + try: + d, c = _linear_eq_to_dict([eq], symset) + d = d[0] + c = c[0] + except PolyNonlinearError as err: + raise NonlinearError(str(err)) + if dict: + if c: + d[S.One] = c + return d + rv = [S.Zero]*(len(syms) + 1) + rv[-1] = c + for i, k in enumerate(syms): + if k not in d: + continue + rv[i] = d[k] + return rv + + +def linear_eq_to_matrix(equations, *symbols): + r""" + Converts a given System of Equations into Matrix form. Here ``equations`` + must be a linear system of equations in ``symbols``. Element ``M[i, j]`` + corresponds to the coefficient of the jth symbol in the ith equation. + + The Matrix form corresponds to the augmented matrix form. For example: + + .. math:: + + 4x + 2y + 3z & = 1 \\ + 3x + y + z & = -6 \\ + 2x + 4y + 9z & = 2 + + This system will return :math:`A` and :math:`b` as: + + .. math:: + + A = \left[\begin{array}{ccc} + 4 & 2 & 3 \\ + 3 & 1 & 1 \\ + 2 & 4 & 9 + \end{array}\right] \\ + + .. math:: + + b = \left[\begin{array}{c} + 1 \\ -6 \\ 2 + \end{array}\right] + + The only simplification performed is to convert + ``Eq(a, b)`` :math:`\Rightarrow a - b`. + + Raises + ====== + + NonlinearError + The equations contain a nonlinear term. + ValueError + The symbols are not given or are not unique. + + Examples + ======== + + >>> from sympy import linear_eq_to_matrix, symbols + >>> c, x, y, z = symbols('c, x, y, z') + + The coefficients (numerical or symbolic) of the symbols will + be returned as matrices: + + >>> eqns = [c*x + z - 1 - c, y + z, x - y] + >>> A, b = linear_eq_to_matrix(eqns, [x, y, z]) + >>> A + Matrix([ + [c, 0, 1], + [0, 1, 1], + [1, -1, 0]]) + >>> b + Matrix([ + [c + 1], + [ 0], + [ 0]]) + + This routine does not simplify expressions and will raise an error + if nonlinearity is encountered: + + >>> eqns = [ + ... (x**2 - 3*x)/(x - 3) - 3, + ... y**2 - 3*y - y*(y - 4) + x - 4] + >>> linear_eq_to_matrix(eqns, [x, y]) + Traceback (most recent call last): + ... + NonlinearError: + symbol-dependent term can be ignored using `strict=False` + + Simplifying these equations will discard the removable singularity in the + first and reveal the linear structure of the second: + + >>> [e.simplify() for e in eqns] + [x - 3, x + y - 4] + + Any such simplification needed to eliminate nonlinear terms must be done + *before* calling this routine. + + """ + if not symbols: + raise ValueError(filldedent(''' + Symbols must be given, for which coefficients + are to be found. + ''')) + + # Check if 'symbols' is a set and raise an error if it is + if isinstance(symbols[0], set): + raise TypeError( + "Unordered 'set' type is not supported as input for symbols.") + + if hasattr(symbols[0], '__iter__'): + symbols = symbols[0] + + if has_dups(symbols): + raise ValueError('Symbols must be unique') + + equations = sympify(equations) + if isinstance(equations, MatrixBase): + equations = list(equations) + elif isinstance(equations, (Expr, Eq)): + equations = [equations] + elif not is_sequence(equations): + raise ValueError(filldedent(''' + Equation(s) must be given as a sequence, Expr, + Eq or Matrix. + ''')) + + # construct the dictionaries + try: + eq, c = _linear_eq_to_dict(equations, symbols) + except PolyNonlinearError as err: + raise NonlinearError(str(err)) + # prepare output matrices + n, m = shape = len(eq), len(symbols) + ix = dict(zip(symbols, range(m))) + A = zeros(*shape) + for row, d in enumerate(eq): + for k in d: + col = ix[k] + A[row, col] = d[k] + b = Matrix(n, 1, [-i for i in c]) + return A, b + + +def linsolve(system, *symbols): + r""" + Solve system of $N$ linear equations with $M$ variables; both + underdetermined and overdetermined systems are supported. + The possible number of solutions is zero, one or infinite. + Zero solutions throws a ValueError, whereas infinite + solutions are represented parametrically in terms of the given + symbols. For unique solution a :class:`~.FiniteSet` of ordered tuples + is returned. + + All standard input formats are supported: + For the given set of equations, the respective input types + are given below: + + .. math:: 3x + 2y - z = 1 + .. math:: 2x - 2y + 4z = -2 + .. math:: 2x - y + 2z = 0 + + * Augmented matrix form, ``system`` given below: + + $$ \text{system} = \left[{array}{cccc} + 3 & 2 & -1 & 1\\ + 2 & -2 & 4 & -2\\ + 2 & -1 & 2 & 0 + \end{array}\right] $$ + + :: + + system = Matrix([[3, 2, -1, 1], [2, -2, 4, -2], [2, -1, 2, 0]]) + + * List of equations form + + :: + + system = [3x + 2y - z - 1, 2x - 2y + 4z + 2, 2x - y + 2z] + + * Input $A$ and $b$ in matrix form (from $Ax = b$) are given as: + + $$ A = \left[\begin{array}{ccc} + 3 & 2 & -1 \\ + 2 & -2 & 4 \\ + 2 & -1 & 2 + \end{array}\right] \ \ b = \left[\begin{array}{c} + 1 \\ -2 \\ 0 + \end{array}\right] $$ + + :: + + A = Matrix([[3, 2, -1], [2, -2, 4], [2, -1, 2]]) + b = Matrix([[1], [-2], [0]]) + system = (A, b) + + Symbols can always be passed but are actually only needed + when 1) a system of equations is being passed and 2) the + system is passed as an underdetermined matrix and one wants + to control the name of the free variables in the result. + An error is raised if no symbols are used for case 1, but if + no symbols are provided for case 2, internally generated symbols + will be provided. When providing symbols for case 2, there should + be at least as many symbols are there are columns in matrix A. + + The algorithm used here is Gauss-Jordan elimination, which + results, after elimination, in a row echelon form matrix. + + Returns + ======= + + A FiniteSet containing an ordered tuple of values for the + unknowns for which the `system` has a solution. (Wrapping + the tuple in FiniteSet is used to maintain a consistent + output format throughout solveset.) + + Returns EmptySet, if the linear system is inconsistent. + + Raises + ====== + + ValueError + The input is not valid. + The symbols are not given. + + Examples + ======== + + >>> from sympy import Matrix, linsolve, symbols + >>> x, y, z = symbols("x, y, z") + >>> A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 10]]) + >>> b = Matrix([3, 6, 9]) + >>> A + Matrix([ + [1, 2, 3], + [4, 5, 6], + [7, 8, 10]]) + >>> b + Matrix([ + [3], + [6], + [9]]) + >>> linsolve((A, b), [x, y, z]) + {(-1, 2, 0)} + + * Parametric Solution: In case the system is underdetermined, the + function will return a parametric solution in terms of the given + symbols. Those that are free will be returned unchanged. e.g. in + the system below, `z` is returned as the solution for variable z; + it can take on any value. + + >>> A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) + >>> b = Matrix([3, 6, 9]) + >>> linsolve((A, b), x, y, z) + {(z - 1, 2 - 2*z, z)} + + If no symbols are given, internally generated symbols will be used. + The ``tau0`` in the third position indicates (as before) that the third + variable -- whatever it is named -- can take on any value: + + >>> linsolve((A, b)) + {(tau0 - 1, 2 - 2*tau0, tau0)} + + * List of equations as input + + >>> Eqns = [3*x + 2*y - z - 1, 2*x - 2*y + 4*z + 2, - x + y/2 - z] + >>> linsolve(Eqns, x, y, z) + {(1, -2, -2)} + + * Augmented matrix as input + + >>> aug = Matrix([[2, 1, 3, 1], [2, 6, 8, 3], [6, 8, 18, 5]]) + >>> aug + Matrix([ + [2, 1, 3, 1], + [2, 6, 8, 3], + [6, 8, 18, 5]]) + >>> linsolve(aug, x, y, z) + {(3/10, 2/5, 0)} + + * Solve for symbolic coefficients + + >>> a, b, c, d, e, f = symbols('a, b, c, d, e, f') + >>> eqns = [a*x + b*y - c, d*x + e*y - f] + >>> linsolve(eqns, x, y) + {((-b*f + c*e)/(a*e - b*d), (a*f - c*d)/(a*e - b*d))} + + * A degenerate system returns solution as set of given + symbols. + + >>> system = Matrix(([0, 0, 0], [0, 0, 0], [0, 0, 0])) + >>> linsolve(system, x, y) + {(x, y)} + + * For an empty system linsolve returns empty set + + >>> linsolve([], x) + EmptySet + + * An error is raised if any nonlinearity is detected, even + if it could be removed with expansion + + >>> linsolve([x*(1/x - 1)], x) + Traceback (most recent call last): + ... + NonlinearError: nonlinear term: 1/x + + >>> linsolve([x*(y + 1)], x, y) + Traceback (most recent call last): + ... + NonlinearError: nonlinear cross-term: x*(y + 1) + + >>> linsolve([x**2 - 1], x) + Traceback (most recent call last): + ... + NonlinearError: nonlinear term: x**2 + """ + if not system: + return S.EmptySet + + # If second argument is an iterable + if symbols and hasattr(symbols[0], '__iter__'): + symbols = symbols[0] + sym_gen = isinstance(symbols, GeneratorType) + dup_msg = 'duplicate symbols given' + + + b = None # if we don't get b the input was bad + # unpack system + + if hasattr(system, '__iter__'): + + # 1). (A, b) + if len(system) == 2 and isinstance(system[0], MatrixBase): + A, b = system + + # 2). (eq1, eq2, ...) + if not isinstance(system[0], MatrixBase): + if sym_gen or not symbols: + raise ValueError(filldedent(''' + When passing a system of equations, the explicit + symbols for which a solution is being sought must + be given as a sequence, too. + ''')) + if len(set(symbols)) != len(symbols): + raise ValueError(dup_msg) + + # + # Pass to the sparse solver implemented in polys. It is important + # that we do not attempt to convert the equations to a matrix + # because that would be very inefficient for large sparse systems + # of equations. + # + eqs = system + eqs = [sympify(eq) for eq in eqs] + try: + sol = _linsolve(eqs, symbols) + except PolyNonlinearError as exc: + # e.g. cos(x) contains an element of the set of generators + raise NonlinearError(str(exc)) + + if sol is None: + return S.EmptySet + + sol = FiniteSet(Tuple(*(sol.get(sym, sym) for sym in symbols))) + return sol + + elif isinstance(system, MatrixBase) and not ( + symbols and not isinstance(symbols, GeneratorType) and + isinstance(symbols[0], MatrixBase)): + # 3). A augmented with b + A, b = system[:, :-1], system[:, -1:] + + if b is None: + raise ValueError("Invalid arguments") + if sym_gen: + symbols = [next(symbols) for i in range(A.cols)] + symset = set(symbols) + if any(symset & (A.free_symbols | b.free_symbols)): + raise ValueError(filldedent(''' + At least one of the symbols provided + already appears in the system to be solved. + One way to avoid this is to use Dummy symbols in + the generator, e.g. numbered_symbols('%s', cls=Dummy) + ''' % symbols[0].name.rstrip('1234567890'))) + elif len(symset) != len(symbols): + raise ValueError(dup_msg) + + if not symbols: + symbols = [Dummy() for _ in range(A.cols)] + name = _uniquely_named_symbol('tau', (A, b), + compare=lambda i: str(i).rstrip('1234567890')).name + gen = numbered_symbols(name) + else: + gen = None + + # This is just a wrapper for solve_lin_sys + eqs = [] + rows = A.tolist() + for rowi, bi in zip(rows, b): + terms = [elem * sym for elem, sym in zip(rowi, symbols) if elem] + terms.append(-bi) + eqs.append(Add(*terms)) + + eqs, ring = sympy_eqs_to_ring(eqs, symbols) + sol = solve_lin_sys(eqs, ring, _raw=False) + if sol is None: + return S.EmptySet + #sol = {sym:val for sym, val in sol.items() if sym != val} + sol = FiniteSet(Tuple(*(sol.get(sym, sym) for sym in symbols))) + + if gen is not None: + solsym = sol.free_symbols + rep = {sym: next(gen) for sym in symbols if sym in solsym} + sol = sol.subs(rep) + + return sol + + +############################################################################## +# ------------------------------nonlinsolve ---------------------------------# +############################################################################## + + +def _return_conditionset(eqs, symbols): + # return conditionset + eqs = (Eq(lhs, 0) for lhs in eqs) + condition_set = ConditionSet( + Tuple(*symbols), And(*eqs), S.Complexes**len(symbols)) + return condition_set + + +def substitution(system, symbols, result=[{}], known_symbols=[], + exclude=[], all_symbols=None): + r""" + Solves the `system` using substitution method. It is used in + :func:`~.nonlinsolve`. This will be called from :func:`~.nonlinsolve` when any + equation(s) is non polynomial equation. + + Parameters + ========== + + system : list of equations + The target system of equations + symbols : list of symbols to be solved. + The variable(s) for which the system is solved + known_symbols : list of solved symbols + Values are known for these variable(s) + result : An empty list or list of dict + If No symbol values is known then empty list otherwise + symbol as keys and corresponding value in dict. + exclude : Set of expression. + Mostly denominator expression(s) of the equations of the system. + Final solution should not satisfy these expressions. + all_symbols : known_symbols + symbols(unsolved). + + Returns + ======= + + A FiniteSet of ordered tuple of values of `all_symbols` for which the + `system` has solution. Order of values in the tuple is same as symbols + present in the parameter `all_symbols`. If parameter `all_symbols` is None + then same as symbols present in the parameter `symbols`. + + Please note that general FiniteSet is unordered, the solution returned + here is not simply a FiniteSet of solutions, rather it is a FiniteSet of + ordered tuple, i.e. the first & only argument to FiniteSet is a tuple of + solutions, which is ordered, & hence the returned solution is ordered. + + Also note that solution could also have been returned as an ordered tuple, + FiniteSet is just a wrapper `{}` around the tuple. It has no other + significance except for the fact it is just used to maintain a consistent + output format throughout the solveset. + + Raises + ====== + + ValueError + The input is not valid. + The symbols are not given. + AttributeError + The input symbols are not :class:`~.Symbol` type. + + Examples + ======== + + >>> from sympy import symbols, substitution + >>> x, y = symbols('x, y', real=True) + >>> substitution([x + y], [x], [{y: 1}], [y], set([]), [x, y]) + {(-1, 1)} + + * When you want a soln not satisfying $x + 1 = 0$ + + >>> substitution([x + y], [x], [{y: 1}], [y], set([x + 1]), [y, x]) + EmptySet + >>> substitution([x + y], [x], [{y: 1}], [y], set([x - 1]), [y, x]) + {(1, -1)} + >>> substitution([x + y - 1, y - x**2 + 5], [x, y]) + {(-3, 4), (2, -1)} + + * Returns both real and complex solution + + >>> x, y, z = symbols('x, y, z') + >>> from sympy import exp, sin + >>> substitution([exp(x) - sin(y), y**2 - 4], [x, y]) + {(ImageSet(Lambda(_n, I*(2*_n*pi + pi) + log(sin(2))), Integers), -2), + (ImageSet(Lambda(_n, 2*_n*I*pi + log(sin(2))), Integers), 2)} + + >>> eqs = [z**2 + exp(2*x) - sin(y), -3 + exp(-y)] + >>> substitution(eqs, [y, z]) + {(-log(3), -sqrt(-exp(2*x) - sin(log(3)))), + (-log(3), sqrt(-exp(2*x) - sin(log(3)))), + (ImageSet(Lambda(_n, 2*_n*I*pi - log(3)), Integers), + ImageSet(Lambda(_n, -sqrt(-exp(2*x) + sin(2*_n*I*pi - log(3)))), Integers)), + (ImageSet(Lambda(_n, 2*_n*I*pi - log(3)), Integers), + ImageSet(Lambda(_n, sqrt(-exp(2*x) + sin(2*_n*I*pi - log(3)))), Integers))} + + """ + + if not system: + return S.EmptySet + + for i, e in enumerate(system): + if isinstance(e, Eq): + system[i] = e.lhs - e.rhs + + if not symbols: + msg = ('Symbols must be given, for which solution of the ' + 'system is to be found.') + raise ValueError(filldedent(msg)) + + if not is_sequence(symbols): + msg = ('symbols should be given as a sequence, e.g. a list.' + 'Not type %s: %s') + raise TypeError(filldedent(msg % (type(symbols), symbols))) + + if not getattr(symbols[0], 'is_Symbol', False): + msg = ('Iterable of symbols must be given as ' + 'second argument, not type %s: %s') + raise ValueError(filldedent(msg % (type(symbols[0]), symbols[0]))) + + # By default `all_symbols` will be same as `symbols` + if all_symbols is None: + all_symbols = symbols + + old_result = result + # storing complements and intersection for particular symbol + complements = {} + intersections = {} + + # when total_solveset_call equals total_conditionset + # it means that solveset failed to solve all eqs. + total_conditionset = -1 + total_solveset_call = -1 + + def _unsolved_syms(eq, sort=False): + """Returns the unsolved symbol present + in the equation `eq`. + """ + free = eq.free_symbols + unsolved = (free - set(known_symbols)) & set(all_symbols) + if sort: + unsolved = list(unsolved) + unsolved.sort(key=default_sort_key) + return unsolved + + # sort such that equation with the fewest potential symbols is first. + # means eq with less number of variable first in the list. + eqs_in_better_order = list( + ordered(system, lambda _: len(_unsolved_syms(_)))) + + def add_intersection_complement(result, intersection_dict, complement_dict): + # If solveset has returned some intersection/complement + # for any symbol, it will be added in the final solution. + final_result = [] + for res in result: + res_copy = res + for key_res, value_res in res.items(): + intersect_set, complement_set = None, None + for key_sym, value_sym in intersection_dict.items(): + if key_sym == key_res: + intersect_set = value_sym + for key_sym, value_sym in complement_dict.items(): + if key_sym == key_res: + complement_set = value_sym + if intersect_set or complement_set: + new_value = FiniteSet(value_res) + if intersect_set and intersect_set != S.Complexes: + new_value = Intersection(new_value, intersect_set) + if complement_set: + new_value = Complement(new_value, complement_set) + if new_value is S.EmptySet: + res_copy = None + break + elif new_value.is_FiniteSet and len(new_value) == 1: + res_copy[key_res] = set(new_value).pop() + else: + res_copy[key_res] = new_value + + if res_copy is not None: + final_result.append(res_copy) + return final_result + + def _extract_main_soln(sym, sol, soln_imageset): + """Separate the Complements, Intersections, ImageSet lambda expr and + its base_set. This function returns the unmasked sol from different classes + of sets and also returns the appended ImageSet elements in a + soln_imageset dict: `{unmasked element: ImageSet}`. + """ + # if there is union, then need to check + # Complement, Intersection, Imageset. + # Order should not be changed. + if isinstance(sol, ConditionSet): + # extracts any solution in ConditionSet + sol = sol.base_set + + if isinstance(sol, Complement): + # extract solution and complement + complements[sym] = sol.args[1] + sol = sol.args[0] + # complement will be added at the end + # using `add_intersection_complement` method + + # if there is union of Imageset or other in soln. + # no testcase is written for this if block + if isinstance(sol, Union): + sol_args = sol.args + sol = S.EmptySet + # We need in sequence so append finteset elements + # and then imageset or other. + for sol_arg2 in sol_args: + if isinstance(sol_arg2, FiniteSet): + sol += sol_arg2 + else: + # ImageSet, Intersection, complement then + # append them directly + sol += FiniteSet(sol_arg2) + + if isinstance(sol, Intersection): + # Interval/Set will be at 0th index always + if sol.args[0] not in (S.Reals, S.Complexes): + # Sometimes solveset returns soln with intersection + # S.Reals or S.Complexes. We don't consider that + # intersection. + intersections[sym] = sol.args[0] + sol = sol.args[1] + # after intersection and complement Imageset should + # be checked. + if isinstance(sol, ImageSet): + soln_imagest = sol + expr2 = sol.lamda.expr + sol = FiniteSet(expr2) + soln_imageset[expr2] = soln_imagest + + if not isinstance(sol, FiniteSet): + sol = FiniteSet(sol) + return sol, soln_imageset + + def _check_exclude(rnew, imgset_yes): + rnew_ = rnew + if imgset_yes: + # replace all dummy variables (Imageset lambda variables) + # with zero before `checksol`. Considering fundamental soln + # for `checksol`. + rnew_copy = rnew.copy() + dummy_n = imgset_yes[0] + for key_res, value_res in rnew_copy.items(): + rnew_copy[key_res] = value_res.subs(dummy_n, 0) + rnew_ = rnew_copy + # satisfy_exclude == true if it satisfies the expr of `exclude` list. + try: + # something like : `Mod(-log(3), 2*I*pi)` can't be + # simplified right now, so `checksol` returns `TypeError`. + # when this issue is fixed this try block should be + # removed. Mod(-log(3), 2*I*pi) == -log(3) + satisfy_exclude = any( + checksol(d, rnew_) for d in exclude) + except TypeError: + satisfy_exclude = None + return satisfy_exclude + + def _restore_imgset(rnew, original_imageset, newresult): + restore_sym = set(rnew.keys()) & \ + set(original_imageset.keys()) + for key_sym in restore_sym: + img = original_imageset[key_sym] + rnew[key_sym] = img + if rnew not in newresult: + newresult.append(rnew) + + def _append_eq(eq, result, res, delete_soln, n=None): + u = Dummy('u') + if n: + eq = eq.subs(n, 0) + satisfy = eq if eq in (True, False) else checksol(u, u, eq, minimal=True) + if satisfy is False: + delete_soln = True + res = {} + else: + result.append(res) + return result, res, delete_soln + + def _append_new_soln(rnew, sym, sol, imgset_yes, soln_imageset, + original_imageset, newresult, eq=None): + """If `rnew` (A dict ) contains valid soln + append it to `newresult` list. + `imgset_yes` is (base, dummy_var) if there was imageset in previously + calculated result(otherwise empty tuple). `original_imageset` is dict + of imageset expr and imageset from this result. + `soln_imageset` dict of imageset expr and imageset of new soln. + """ + satisfy_exclude = _check_exclude(rnew, imgset_yes) + delete_soln = False + # soln should not satisfy expr present in `exclude` list. + if not satisfy_exclude: + local_n = None + # if it is imageset + if imgset_yes: + local_n = imgset_yes[0] + base = imgset_yes[1] + if sym and sol: + # when `sym` and `sol` is `None` means no new + # soln. In that case we will append rnew directly after + # substituting original imagesets in rnew values if present + # (second last line of this function using _restore_imgset) + dummy_list = list(sol.atoms(Dummy)) + # use one dummy `n` which is in + # previous imageset + local_n_list = [ + local_n for i in range( + 0, len(dummy_list))] + + dummy_zip = zip(dummy_list, local_n_list) + lam = Lambda(local_n, sol.subs(dummy_zip)) + rnew[sym] = ImageSet(lam, base) + if eq is not None: + newresult, rnew, delete_soln = _append_eq( + eq, newresult, rnew, delete_soln, local_n) + elif eq is not None: + newresult, rnew, delete_soln = _append_eq( + eq, newresult, rnew, delete_soln) + elif sol in soln_imageset.keys(): + rnew[sym] = soln_imageset[sol] + # restore original imageset + _restore_imgset(rnew, original_imageset, newresult) + else: + newresult.append(rnew) + elif satisfy_exclude: + delete_soln = True + rnew = {} + _restore_imgset(rnew, original_imageset, newresult) + return newresult, delete_soln + + def _new_order_result(result, eq): + # separate first, second priority. `res` that makes `eq` value equals + # to zero, should be used first then other result(second priority). + # If it is not done then we may miss some soln. + first_priority = [] + second_priority = [] + for res in result: + if not any(isinstance(val, ImageSet) for val in res.values()): + if eq.subs(res) == 0: + first_priority.append(res) + else: + second_priority.append(res) + if first_priority or second_priority: + return first_priority + second_priority + return result + + def _solve_using_known_values(result, solver): + """Solves the system using already known solution + (result contains the dict ). + solver is :func:`~.solveset_complex` or :func:`~.solveset_real`. + """ + # stores imageset . + soln_imageset = {} + total_solvest_call = 0 + total_conditionst = 0 + + # sort equations so the one with the fewest potential + # symbols appears first + for index, eq in enumerate(eqs_in_better_order): + newresult = [] + # if imageset, expr is used to solve for other symbol + imgset_yes = False + for res in result: + original_imageset = {} + got_symbol = set() # symbols solved in one iteration + # find the imageset and use its expr. + for k, v in res.items(): + if isinstance(v, ImageSet): + res[k] = v.lamda.expr + original_imageset[k] = v + dummy_n = v.lamda.expr.atoms(Dummy).pop() + (base,) = v.base_sets + imgset_yes = (dummy_n, base) + assert not isinstance(v, FiniteSet) # if so, internal error + # update eq with everything that is known so far + eq2 = eq.subs(res).expand() + if imgset_yes and not eq2.has(imgset_yes[0]): + # The substituted equation simplified in such a way that + # it's no longer necessary to encapsulate a potential new + # solution in an ImageSet. (E.g. at the previous step some + # {n*2*pi} was found as partial solution for one of the + # unknowns, but its main solution expression n*2*pi has now + # been substituted in a trigonometric function.) + imgset_yes = False + + unsolved_syms = _unsolved_syms(eq2, sort=True) + if not unsolved_syms: + if res: + newresult, delete_res = _append_new_soln( + res, None, None, imgset_yes, soln_imageset, + original_imageset, newresult, eq2) + if delete_res: + # `delete_res` is true, means substituting `res` in + # eq2 doesn't return `zero` or deleting the `res` + # (a soln) since it satisfies expr of `exclude` + # list. + result.remove(res) + continue # skip as it's independent of desired symbols + depen1, depen2 = eq2.as_independent(*unsolved_syms) + if (depen1.has(Abs) or depen2.has(Abs)) and solver == solveset_complex: + # Absolute values cannot be inverted in the + # complex domain + continue + soln_imageset = {} + for sym in unsolved_syms: + not_solvable = False + try: + soln = solver(eq2, sym) + total_solvest_call += 1 + soln_new = S.EmptySet + if isinstance(soln, Complement): + # separate solution and complement + complements[sym] = soln.args[1] + soln = soln.args[0] + # complement will be added at the end + if isinstance(soln, Intersection): + # Interval will be at 0th index always + if soln.args[0] != Interval(-oo, oo): + # sometimes solveset returns soln + # with intersection S.Reals, to confirm that + # soln is in domain=S.Reals + intersections[sym] = soln.args[0] + soln_new += soln.args[1] + soln = soln_new if soln_new else soln + if index > 0 and solver == solveset_real: + # one symbol's real soln, another symbol may have + # corresponding complex soln. + if not isinstance(soln, (ImageSet, ConditionSet)): + soln += solveset_complex(eq2, sym) # might give ValueError with Abs + except (NotImplementedError, ValueError): + # If solveset is not able to solve equation `eq2`. Next + # time we may get soln using next equation `eq2` + continue + if isinstance(soln, ConditionSet): + if soln.base_set in (S.Reals, S.Complexes): + soln = S.EmptySet + # don't do `continue` we may get soln + # in terms of other symbol(s) + not_solvable = True + total_conditionst += 1 + else: + soln = soln.base_set + + if soln is not S.EmptySet: + soln, soln_imageset = _extract_main_soln( + sym, soln, soln_imageset) + + for sol in soln: + # sol is not a `Union` since we checked it + # before this loop + sol, soln_imageset = _extract_main_soln( + sym, sol, soln_imageset) + sol = set(sol).pop() # XXX what if there are more solutions? + free = sol.free_symbols + if got_symbol and any( + ss in free for ss in got_symbol + ): + # sol depends on previously solved symbols + # then continue + continue + rnew = res.copy() + # put each solution in res and append the new result + # in the new result list (solution for symbol `s`) + # along with old results. + for k, v in res.items(): + if isinstance(v, Expr) and isinstance(sol, Expr): + # if any unsolved symbol is present + # Then subs known value + rnew[k] = v.subs(sym, sol) + # and add this new solution + if sol in soln_imageset.keys(): + # replace all lambda variables with 0. + imgst = soln_imageset[sol] + rnew[sym] = imgst.lamda( + *[0 for i in range(0, len( + imgst.lamda.variables))]) + else: + rnew[sym] = sol + newresult, delete_res = _append_new_soln( + rnew, sym, sol, imgset_yes, soln_imageset, + original_imageset, newresult) + if delete_res: + # deleting the `res` (a soln) since it satisfies + # eq of `exclude` list + result.remove(res) + # solution got for sym + if not not_solvable: + got_symbol.add(sym) + # next time use this new soln + if newresult: + result = newresult + return result, total_solvest_call, total_conditionst + + new_result_real, solve_call1, cnd_call1 = _solve_using_known_values( + old_result, solveset_real) + new_result_complex, solve_call2, cnd_call2 = _solve_using_known_values( + old_result, solveset_complex) + + # If total_solveset_call is equal to total_conditionset + # then solveset failed to solve all of the equations. + # In this case we return a ConditionSet here. + total_conditionset += (cnd_call1 + cnd_call2) + total_solveset_call += (solve_call1 + solve_call2) + + if total_conditionset == total_solveset_call and total_solveset_call != -1: + return _return_conditionset(eqs_in_better_order, all_symbols) + + # don't keep duplicate solutions + filtered_complex = [] + for i in list(new_result_complex): + for j in list(new_result_real): + if i.keys() != j.keys(): + continue + if all(a.dummy_eq(b) for a, b in zip(i.values(), j.values()) \ + if not (isinstance(a, int) and isinstance(b, int))): + break + else: + filtered_complex.append(i) + # overall result + result = new_result_real + filtered_complex + + result_all_variables = [] + result_infinite = [] + for res in result: + if not res: + # means {None : None} + continue + # If length < len(all_symbols) means infinite soln. + # Some or all the soln is dependent on 1 symbol. + # eg. {x: y+2} then final soln {x: y+2, y: y} + if len(res) < len(all_symbols): + solved_symbols = res.keys() + unsolved = list(filter( + lambda x: x not in solved_symbols, all_symbols)) + for unsolved_sym in unsolved: + res[unsolved_sym] = unsolved_sym + result_infinite.append(res) + if res not in result_all_variables: + result_all_variables.append(res) + + if result_infinite: + # we have general soln + # eg : [{x: -1, y : 1}, {x : -y, y: y}] then + # return [{x : -y, y : y}] + result_all_variables = result_infinite + if intersections or complements: + result_all_variables = add_intersection_complement( + result_all_variables, intersections, complements) + + # convert to ordered tuple + result = S.EmptySet + for r in result_all_variables: + temp = [r[symb] for symb in all_symbols] + result += FiniteSet(tuple(temp)) + return result + + +def _solveset_work(system, symbols): + soln = solveset(system[0], symbols[0]) + if isinstance(soln, FiniteSet): + _soln = FiniteSet(*[(s,) for s in soln]) + return _soln + else: + return FiniteSet(tuple(FiniteSet(soln))) + + +def _handle_positive_dimensional(polys, symbols, denominators): + from sympy.polys.polytools import groebner + # substitution method where new system is groebner basis of the system + _symbols = list(symbols) + _symbols.sort(key=default_sort_key) + basis = groebner(polys, _symbols, polys=True) + new_system = [] + for poly_eq in basis: + new_system.append(poly_eq.as_expr()) + result = [{}] + result = substitution( + new_system, symbols, result, [], + denominators) + return result + + +def _handle_zero_dimensional(polys, symbols, system): + # solve 0 dimensional poly system using `solve_poly_system` + result = solve_poly_system(polys, *symbols) + # May be some extra soln is added because + # we used `unrad` in `_separate_poly_nonpoly`, so + # need to check and remove if it is not a soln. + result_update = S.EmptySet + for res in result: + dict_sym_value = dict(list(zip(symbols, res))) + if all(checksol(eq, dict_sym_value) for eq in system): + result_update += FiniteSet(res) + return result_update + + +def _separate_poly_nonpoly(system, symbols): + polys = [] + polys_expr = [] + nonpolys = [] + # unrad_changed stores a list of expressions containing + # radicals that were processed using unrad + # this is useful if solutions need to be checked later. + unrad_changed = [] + denominators = set() + poly = None + for eq in system: + # Store denom expressions that contain symbols + denominators.update(_simple_dens(eq, symbols)) + # Convert equality to expression + if isinstance(eq, Eq): + eq = eq.lhs - eq.rhs + # try to remove sqrt and rational power + without_radicals = unrad(simplify(eq), *symbols) + if without_radicals: + unrad_changed.append(eq) + eq_unrad, cov = without_radicals + if not cov: + eq = eq_unrad + if isinstance(eq, Expr): + eq = eq.as_numer_denom()[0] + poly = eq.as_poly(*symbols, extension=True) + elif simplify(eq).is_number: + continue + if poly is not None: + polys.append(poly) + polys_expr.append(poly.as_expr()) + else: + nonpolys.append(eq) + return polys, polys_expr, nonpolys, denominators, unrad_changed + + +def _handle_poly(polys, symbols): + # _handle_poly(polys, symbols) -> (poly_sol, poly_eqs) + # + # We will return possible solution information to nonlinsolve as well as a + # new system of polynomial equations to be solved if we cannot solve + # everything directly here. The new system of polynomial equations will be + # a lex-order Groebner basis for the original system. The lex basis + # hopefully separate some of the variables and equations and give something + # easier for substitution to work with. + + # The format for representing solution sets in nonlinsolve and substitution + # is a list of dicts. These are the special cases: + no_information = [{}] # No equations solved yet + no_solutions = [] # The system is inconsistent and has no solutions. + + # If there is no need to attempt further solution of these equations then + # we return no equations: + no_equations = [] + + inexact = any(not p.domain.is_Exact for p in polys) + if inexact: + # The use of Groebner over RR is likely to result incorrectly in an + # inconsistent Groebner basis. So, convert any float coefficients to + # Rational before computing the Groebner basis. + polys = [poly(nsimplify(p, rational=True)) for p in polys] + + # Compute a Groebner basis in grevlex order wrt the ordering given. We will + # try to convert this to lex order later. Usually it seems to be more + # efficient to compute a lex order basis by computing a grevlex basis and + # converting to lex with fglm. + basis = groebner(polys, symbols, order='grevlex', polys=False) + + # + # No solutions (inconsistent equations)? + # + if 1 in basis: + + # No solutions: + poly_sol = no_solutions + poly_eqs = no_equations + + # + # Finite number of solutions (zero-dimensional case) + # + elif basis.is_zero_dimensional: + + # Convert Groebner basis to lex ordering + basis = basis.fglm('lex') + + # Convert polynomial coefficients back to float before calling + # solve_poly_system + if inexact: + basis = [nfloat(p) for p in basis] + + # Solve the zero-dimensional case using solve_poly_system if possible. + # If some polynomials have factors that cannot be solved in radicals + # then this will fail. Using solve_poly_system(..., strict=True) + # ensures that we either get a complete solution set in radicals or + # UnsolvableFactorError will be raised. + try: + result = solve_poly_system(basis, *symbols, strict=True) + except UnsolvableFactorError: + # Failure... not fully solvable in radicals. Return the lex-order + # basis for substitution to handle. + poly_sol = no_information + poly_eqs = list(basis) + else: + # Success! We have a finite solution set and solve_poly_system has + # succeeded in finding all solutions. Return the solutions and also + # an empty list of remaining equations to be solved. + poly_sol = [dict(zip(symbols, res)) for res in result] + poly_eqs = no_equations + + # + # Infinite families of solutions (positive-dimensional case) + # + else: + # In this case the grevlex basis cannot be converted to lex using the + # fglm method and also solve_poly_system cannot solve the equations. We + # would like to return a lex basis but since we can't use fglm we + # compute the lex basis directly here. The time required to recompute + # the basis is generally significantly less than the time required by + # substitution to solve the new system. + poly_sol = no_information + poly_eqs = list(groebner(polys, symbols, order='lex', polys=False)) + + if inexact: + poly_eqs = [nfloat(p) for p in poly_eqs] + + return poly_sol, poly_eqs + + +def nonlinsolve(system, *symbols): + r""" + Solve system of $N$ nonlinear equations with $M$ variables, which means both + under and overdetermined systems are supported. Positive dimensional + system is also supported (A system with infinitely many solutions is said + to be positive-dimensional). In a positive dimensional system the solution will + be dependent on at least one symbol. Returns both real solution + and complex solution (if they exist). + + Parameters + ========== + + system : list of equations + The target system of equations + symbols : list of Symbols + symbols should be given as a sequence eg. list + + Returns + ======= + + A :class:`~.FiniteSet` of ordered tuple of values of `symbols` for which the `system` + has solution. Order of values in the tuple is same as symbols present in + the parameter `symbols`. + + Please note that general :class:`~.FiniteSet` is unordered, the solution + returned here is not simply a :class:`~.FiniteSet` of solutions, rather it + is a :class:`~.FiniteSet` of ordered tuple, i.e. the first and only + argument to :class:`~.FiniteSet` is a tuple of solutions, which is + ordered, and, hence ,the returned solution is ordered. + + Also note that solution could also have been returned as an ordered tuple, + FiniteSet is just a wrapper ``{}`` around the tuple. It has no other + significance except for the fact it is just used to maintain a consistent + output format throughout the solveset. + + For the given set of equations, the respective input types + are given below: + + .. math:: xy - 1 = 0 + .. math:: 4x^2 + y^2 - 5 = 0 + + :: + + system = [x*y - 1, 4*x**2 + y**2 - 5] + symbols = [x, y] + + Raises + ====== + + ValueError + The input is not valid. + The symbols are not given. + AttributeError + The input symbols are not `Symbol` type. + + Examples + ======== + + >>> from sympy import symbols, nonlinsolve + >>> x, y, z = symbols('x, y, z', real=True) + >>> nonlinsolve([x*y - 1, 4*x**2 + y**2 - 5], [x, y]) + {(-1, -1), (-1/2, -2), (1/2, 2), (1, 1)} + + 1. Positive dimensional system and complements: + + >>> from sympy import pprint + >>> from sympy.polys.polytools import is_zero_dimensional + >>> a, b, c, d = symbols('a, b, c, d', extended_real=True) + >>> eq1 = a + b + c + d + >>> eq2 = a*b + b*c + c*d + d*a + >>> eq3 = a*b*c + b*c*d + c*d*a + d*a*b + >>> eq4 = a*b*c*d - 1 + >>> system = [eq1, eq2, eq3, eq4] + >>> is_zero_dimensional(system) + False + >>> pprint(nonlinsolve(system, [a, b, c, d]), use_unicode=False) + -1 1 1 -1 + {(---, -d, -, {d} \ {0}), (-, -d, ---, {d} \ {0})} + d d d d + >>> nonlinsolve([(x+y)**2 - 4, x + y - 2], [x, y]) + {(2 - y, y)} + + 2. If some of the equations are non-polynomial then `nonlinsolve` + will call the ``substitution`` function and return real and complex solutions, + if present. + + >>> from sympy import exp, sin + >>> nonlinsolve([exp(x) - sin(y), y**2 - 4], [x, y]) + {(ImageSet(Lambda(_n, I*(2*_n*pi + pi) + log(sin(2))), Integers), -2), + (ImageSet(Lambda(_n, 2*_n*I*pi + log(sin(2))), Integers), 2)} + + 3. If system is non-linear polynomial and zero-dimensional then it + returns both solution (real and complex solutions, if present) using + :func:`~.solve_poly_system`: + + >>> from sympy import sqrt + >>> nonlinsolve([x**2 - 2*y**2 -2, x*y - 2], [x, y]) + {(-2, -1), (2, 1), (-sqrt(2)*I, sqrt(2)*I), (sqrt(2)*I, -sqrt(2)*I)} + + 4. ``nonlinsolve`` can solve some linear (zero or positive dimensional) + system (because it uses the :func:`sympy.polys.polytools.groebner` function to get the + groebner basis and then uses the ``substitution`` function basis as the + new `system`). But it is not recommended to solve linear system using + ``nonlinsolve``, because :func:`~.linsolve` is better for general linear systems. + + >>> nonlinsolve([x + 2*y -z - 3, x - y - 4*z + 9, y + z - 4], [x, y, z]) + {(3*z - 5, 4 - z, z)} + + 5. System having polynomial equations and only real solution is + solved using :func:`~.solve_poly_system`: + + >>> e1 = sqrt(x**2 + y**2) - 10 + >>> e2 = sqrt(y**2 + (-x + 10)**2) - 3 + >>> nonlinsolve((e1, e2), (x, y)) + {(191/20, -3*sqrt(391)/20), (191/20, 3*sqrt(391)/20)} + >>> nonlinsolve([x**2 + 2/y - 2, x + y - 3], [x, y]) + {(1, 2), (1 - sqrt(5), 2 + sqrt(5)), (1 + sqrt(5), 2 - sqrt(5))} + >>> nonlinsolve([x**2 + 2/y - 2, x + y - 3], [y, x]) + {(2, 1), (2 - sqrt(5), 1 + sqrt(5)), (2 + sqrt(5), 1 - sqrt(5))} + + 6. It is better to use symbols instead of trigonometric functions or + :class:`~.Function`. For example, replace $\sin(x)$ with a symbol, replace + $f(x)$ with a symbol and so on. Get a solution from ``nonlinsolve`` and then + use :func:`~.solveset` to get the value of $x$. + + How nonlinsolve is better than old solver ``_solve_system`` : + ============================================================= + + 1. A positive dimensional system solver: nonlinsolve can return + solution for positive dimensional system. It finds the + Groebner Basis of the positive dimensional system(calling it as + basis) then we can start solving equation(having least number of + variable first in the basis) using solveset and substituting that + solved solutions into other equation(of basis) to get solution in + terms of minimum variables. Here the important thing is how we + are substituting the known values and in which equations. + + 2. Real and complex solutions: nonlinsolve returns both real + and complex solution. If all the equations in the system are polynomial + then using :func:`~.solve_poly_system` both real and complex solution is returned. + If all the equations in the system are not polynomial equation then goes to + ``substitution`` method with this polynomial and non polynomial equation(s), + to solve for unsolved variables. Here to solve for particular variable + solveset_real and solveset_complex is used. For both real and complex + solution ``_solve_using_known_values`` is used inside ``substitution`` + (``substitution`` will be called when any non-polynomial equation is present). + If a solution is valid its general solution is added to the final result. + + 3. :class:`~.Complement` and :class:`~.Intersection` will be added: + nonlinsolve maintains dict for complements and intersections. If solveset + find complements or/and intersections with any interval or set during the + execution of ``substitution`` function, then complement or/and + intersection for that variable is added before returning final solution. + + """ + if not system: + return S.EmptySet + + if not symbols: + msg = ('Symbols must be given, for which solution of the ' + 'system is to be found.') + raise ValueError(filldedent(msg)) + + if hasattr(symbols[0], '__iter__'): + symbols = symbols[0] + + if not is_sequence(symbols) or not symbols: + msg = ('Symbols must be given, for which solution of the ' + 'system is to be found.') + raise IndexError(filldedent(msg)) + + symbols = list(map(_sympify, symbols)) + system, symbols, swap = recast_to_symbols(system, symbols) + if swap: + soln = nonlinsolve(system, symbols) + return FiniteSet(*[tuple(i.xreplace(swap) for i in s) for s in soln]) + + if len(system) == 1 and len(symbols) == 1: + return _solveset_work(system, symbols) + + # main code of def nonlinsolve() starts from here + + polys, polys_expr, nonpolys, denominators, unrad_changed = \ + _separate_poly_nonpoly(system, symbols) + + poly_eqs = [] + poly_sol = [{}] + + if polys: + poly_sol, poly_eqs = _handle_poly(polys, symbols) + if poly_sol and poly_sol[0]: + poly_syms = set().union(*(eq.free_symbols for eq in polys)) + unrad_syms = set().union(*(eq.free_symbols for eq in unrad_changed)) + if unrad_syms == poly_syms and unrad_changed: + # if all the symbols have been solved by _handle_poly + # and unrad has been used then check solutions + poly_sol = [sol for sol in poly_sol if checksol(unrad_changed, sol)] + + # Collect together the unsolved polynomials with the non-polynomial + # equations. + remaining = poly_eqs + nonpolys + + # to_tuple converts a solution dictionary to a tuple containing the + # value for each symbol + to_tuple = lambda sol: tuple(sol[s] for s in symbols) + + if not remaining: + # If there is nothing left to solve then return the solution from + # solve_poly_system directly. + return FiniteSet(*map(to_tuple, poly_sol)) + else: + # Here we handle: + # + # 1. The Groebner basis if solve_poly_system failed. + # 2. The Groebner basis in the positive-dimensional case. + # 3. Any non-polynomial equations + # + # If solve_poly_system did succeed then we pass those solutions in as + # preliminary results. + subs_res = substitution(remaining, symbols, result=poly_sol, exclude=denominators) + + if not isinstance(subs_res, FiniteSet): + return subs_res + + # check solutions produced by substitution. 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sha256:3f11e2b66ce820b8b26fad4db38f018fc7c09e98e118d8d41a723221531bb190 +size 364682 diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/tests/test_constantsimp.py b/.venv/lib/python3.12/site-packages/sympy/solvers/tests/test_constantsimp.py new file mode 100644 index 0000000000000000000000000000000000000000..efb966a4c8c2f93558d05e7c330f06530e69180c --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/tests/test_constantsimp.py @@ -0,0 +1,179 @@ +""" +If the arbitrary constant class from issue 4435 is ever implemented, this +should serve as a set of test cases. +""" + +from sympy.core.function import Function +from sympy.core.numbers import I +from sympy.core.power import Pow +from sympy.core.relational import Eq +from sympy.core.singleton import S +from sympy.core.symbol import Symbol +from sympy.functions.elementary.exponential import (exp, log) +from sympy.functions.elementary.hyperbolic import (cosh, sinh) +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import (acos, cos, sin) +from sympy.integrals.integrals import Integral +from sympy.solvers.ode.ode import constantsimp, constant_renumber +from sympy.testing.pytest import XFAIL + + +x = Symbol('x') +y = Symbol('y') +z = Symbol('z') +u2 = Symbol('u2') +_a = Symbol('_a') +C1 = Symbol('C1') +C2 = Symbol('C2') +C3 = Symbol('C3') +f = Function('f') + + +def test_constant_mul(): + # We want C1 (Constant) below to absorb the y's, but not the x's + assert constant_renumber(constantsimp(y*C1, [C1])) == C1*y + assert constant_renumber(constantsimp(C1*y, [C1])) == C1*y + assert constant_renumber(constantsimp(x*C1, [C1])) == x*C1 + assert constant_renumber(constantsimp(C1*x, [C1])) == x*C1 + assert constant_renumber(constantsimp(2*C1, [C1])) == C1 + assert constant_renumber(constantsimp(C1*2, [C1])) == C1 + assert constant_renumber(constantsimp(y*C1*x, [C1, y])) == C1*x + assert constant_renumber(constantsimp(x*y*C1, [C1, y])) == x*C1 + assert constant_renumber(constantsimp(y*x*C1, [C1, y])) == x*C1 + assert constant_renumber(constantsimp(C1*x*y, [C1, y])) == C1*x + assert constant_renumber(constantsimp(x*C1*y, [C1, y])) == x*C1 + assert constant_renumber(constantsimp(C1*y*(y + 1), [C1])) == C1*y*(y+1) + assert constant_renumber(constantsimp(y*C1*(y + 1), [C1])) == C1*y*(y+1) + assert constant_renumber(constantsimp(x*(y*C1), [C1])) == x*y*C1 + assert constant_renumber(constantsimp(x*(C1*y), [C1])) == x*y*C1 + assert constant_renumber(constantsimp(C1*(x*y), [C1, y])) == C1*x + assert constant_renumber(constantsimp((x*y)*C1, [C1, y])) == x*C1 + assert constant_renumber(constantsimp((y*x)*C1, [C1, y])) == x*C1 + assert constant_renumber(constantsimp(y*(y + 1)*C1, [C1, y])) == C1 + assert constant_renumber(constantsimp((C1*x)*y, [C1, y])) == C1*x + assert constant_renumber(constantsimp(y*(x*C1), [C1, y])) == x*C1 + assert constant_renumber(constantsimp((x*C1)*y, [C1, y])) == x*C1 + assert constant_renumber(constantsimp(C1*x*y*x*y*2, [C1, y])) == C1*x**2 + assert constant_renumber(constantsimp(C1*x*y*z, [C1, y, z])) == C1*x + assert constant_renumber(constantsimp(C1*x*y**2*sin(z), [C1, y, z])) == C1*x + assert constant_renumber(constantsimp(C1*C1, [C1])) == C1 + assert constant_renumber(constantsimp(C1*C2, [C1, C2])) == C1 + assert constant_renumber(constantsimp(C2*C2, [C1, C2])) == C1 + assert constant_renumber(constantsimp(C1*C1*C2, [C1, C2])) == C1 + assert constant_renumber(constantsimp(C1*x*2**x, [C1])) == C1*x*2**x + +def test_constant_add(): + assert constant_renumber(constantsimp(C1 + C1, [C1])) == C1 + assert constant_renumber(constantsimp(C1 + 2, [C1])) == C1 + assert constant_renumber(constantsimp(2 + C1, [C1])) == C1 + assert constant_renumber(constantsimp(C1 + y, [C1, y])) == C1 + assert constant_renumber(constantsimp(C1 + x, [C1])) == C1 + x + assert constant_renumber(constantsimp(C1 + C1, [C1])) == C1 + assert constant_renumber(constantsimp(C1 + C2, [C1, C2])) == C1 + assert constant_renumber(constantsimp(C2 + C1, [C1, C2])) == C1 + assert constant_renumber(constantsimp(C1 + C2 + C1, [C1, C2])) == C1 + + +def test_constant_power_as_base(): + assert constant_renumber(constantsimp(C1**C1, [C1])) == C1 + assert constant_renumber(constantsimp(Pow(C1, C1), [C1])) == C1 + assert constant_renumber(constantsimp(C1**C1, [C1])) == C1 + assert constant_renumber(constantsimp(C1**C2, [C1, C2])) == C1 + assert constant_renumber(constantsimp(C2**C1, [C1, C2])) == C1 + assert constant_renumber(constantsimp(C2**C2, [C1, C2])) == C1 + assert constant_renumber(constantsimp(C1**y, [C1, y])) == C1 + assert constant_renumber(constantsimp(C1**x, [C1])) == C1**x + assert constant_renumber(constantsimp(C1**2, [C1])) == C1 + assert constant_renumber( + constantsimp(C1**(x*y), [C1])) == C1**(x*y) + + +def test_constant_power_as_exp(): + assert constant_renumber(constantsimp(x**C1, [C1])) == x**C1 + assert constant_renumber(constantsimp(y**C1, [C1, y])) == C1 + assert constant_renumber(constantsimp(x**y**C1, [C1, y])) == x**C1 + assert constant_renumber( + constantsimp((x**y)**C1, [C1])) == (x**y)**C1 + assert constant_renumber( + constantsimp(x**(y**C1), [C1, y])) == x**C1 + assert constant_renumber(constantsimp(x**C1**y, [C1, y])) == x**C1 + assert constant_renumber( + constantsimp(x**(C1**y), [C1, y])) == x**C1 + assert constant_renumber( + constantsimp((x**C1)**y, [C1])) == (x**C1)**y + assert constant_renumber(constantsimp(2**C1, [C1])) == C1 + assert constant_renumber(constantsimp(S(2)**C1, [C1])) == C1 + assert constant_renumber(constantsimp(exp(C1), [C1])) == C1 + assert constant_renumber( + constantsimp(exp(C1 + x), [C1])) == C1*exp(x) + assert constant_renumber(constantsimp(Pow(2, C1), [C1])) == C1 + + +def test_constant_function(): + assert constant_renumber(constantsimp(sin(C1), [C1])) == C1 + assert constant_renumber(constantsimp(f(C1), [C1])) == C1 + assert constant_renumber(constantsimp(f(C1, C1), [C1])) == C1 + assert constant_renumber(constantsimp(f(C1, C2), [C1, C2])) == C1 + assert constant_renumber(constantsimp(f(C2, C1), [C1, C2])) == C1 + assert constant_renumber(constantsimp(f(C2, C2), [C1, C2])) == C1 + assert constant_renumber( + constantsimp(f(C1, x), [C1])) == f(C1, x) + assert constant_renumber(constantsimp(f(C1, y), [C1, y])) == C1 + assert constant_renumber(constantsimp(f(y, C1), [C1, y])) == C1 + assert constant_renumber(constantsimp(f(C1, y, C2), [C1, C2, y])) == C1 + + +def test_constant_function_multiple(): + # The rules to not renumber in this case would be too complicated, and + # dsolve is not likely to ever encounter anything remotely like this. + assert constant_renumber( + constantsimp(f(C1, C1, x), [C1])) == f(C1, C1, x) + + +def test_constant_multiple(): + assert constant_renumber(constantsimp(C1*2 + 2, [C1])) == C1 + assert constant_renumber(constantsimp(x*2/C1, [C1])) == C1*x + assert constant_renumber(constantsimp(C1**2*2 + 2, [C1])) == C1 + assert constant_renumber( + constantsimp(sin(2*C1) + x + sqrt(2), [C1])) == C1 + x + assert constant_renumber(constantsimp(2*C1 + C2, [C1, C2])) == C1 + +def test_constant_repeated(): + assert C1 + C1*x == constant_renumber( C1 + C1*x) + +def test_ode_solutions(): + # only a few examples here, the rest will be tested in the actual dsolve tests + assert constant_renumber(constantsimp(C1*exp(2*x) + exp(x)*(C2 + C3), [C1, C2, C3])) == \ + constant_renumber(C1*exp(x) + C2*exp(2*x)) + assert constant_renumber( + constantsimp(Eq(f(x), I*C1*sinh(x/3) + C2*cosh(x/3)), [C1, C2]) + ) == constant_renumber(Eq(f(x), C1*sinh(x/3) + C2*cosh(x/3))) + assert constant_renumber(constantsimp(Eq(f(x), acos((-C1)/cos(x))), [C1])) == \ + Eq(f(x), acos(C1/cos(x))) + assert constant_renumber( + constantsimp(Eq(log(f(x)/C1) + 2*exp(x/f(x)), 0), [C1]) + ) == Eq(log(C1*f(x)) + 2*exp(x/f(x)), 0) + assert constant_renumber(constantsimp(Eq(log(x*sqrt(2)*sqrt(1/x)*sqrt(f(x)) + /C1) + x**2/(2*f(x)**2), 0), [C1])) == \ + Eq(log(C1*sqrt(x)*sqrt(f(x))) + x**2/(2*f(x)**2), 0) + assert constant_renumber(constantsimp(Eq(-exp(-f(x)/x)*sin(f(x)/x)/2 + log(x/C1) - + cos(f(x)/x)*exp(-f(x)/x)/2, 0), [C1])) == \ + Eq(-exp(-f(x)/x)*sin(f(x)/x)/2 + log(C1*x) - cos(f(x)/x)* + exp(-f(x)/x)/2, 0) + assert constant_renumber(constantsimp(Eq(-Integral(-1/(sqrt(1 - u2**2)*u2), + (u2, _a, x/f(x))) + log(f(x)/C1), 0), [C1])) == \ + Eq(-Integral(-1/(u2*sqrt(1 - u2**2)), (u2, _a, x/f(x))) + + log(C1*f(x)), 0) + assert [constantsimp(i, [C1]) for i in [Eq(f(x), sqrt(-C1*x + x**2)), Eq(f(x), -sqrt(-C1*x + x**2))]] == \ + [Eq(f(x), sqrt(x*(C1 + x))), Eq(f(x), -sqrt(x*(C1 + x)))] + + +@XFAIL +def test_nonlocal_simplification(): + assert constantsimp(C1 + C2+x*C2, [C1, C2]) == C1 + C2*x + + +def test_constant_Eq(): + # C1 on the rhs is well-tested, but the lhs is only tested here + assert constantsimp(Eq(C1, 3 + f(x)*x), [C1]) == Eq(x*f(x), C1) + assert constantsimp(Eq(C1, 3 * f(x)*x), [C1]) == Eq(f(x)*x, C1) diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/tests/test_decompogen.py b/.venv/lib/python3.12/site-packages/sympy/solvers/tests/test_decompogen.py new file mode 100644 index 0000000000000000000000000000000000000000..1ba03f4b42558231b626b6ed169f8b0a81a72bf9 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/tests/test_decompogen.py @@ -0,0 +1,59 @@ +from sympy.solvers.decompogen import decompogen, compogen +from sympy.core.symbol import symbols +from sympy.functions.elementary.complexes import Abs +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.miscellaneous import sqrt, Max +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.testing.pytest import XFAIL, raises + +x, y = symbols('x y') + + +def test_decompogen(): + assert decompogen(sin(cos(x)), x) == [sin(x), cos(x)] + assert decompogen(sin(x)**2 + sin(x) + 1, x) == [x**2 + x + 1, sin(x)] + assert decompogen(sqrt(6*x**2 - 5), x) == [sqrt(x), 6*x**2 - 5] + assert decompogen(sin(sqrt(cos(x**2 + 1))), x) == [sin(x), sqrt(x), cos(x), x**2 + 1] + assert decompogen(Abs(cos(x)**2 + 3*cos(x) - 4), x) == [Abs(x), x**2 + 3*x - 4, cos(x)] + assert decompogen(sin(x)**2 + sin(x) - sqrt(3)/2, x) == [x**2 + x - sqrt(3)/2, sin(x)] + assert decompogen(Abs(cos(y)**2 + 3*cos(x) - 4), x) == [Abs(x), 3*x + cos(y)**2 - 4, cos(x)] + assert decompogen(x, y) == [x] + assert decompogen(1, x) == [1] + assert decompogen(Max(3, x), x) == [Max(3, x)] + raises(TypeError, lambda: decompogen(x < 5, x)) + u = 2*x + 3 + assert decompogen(Max(sqrt(u),(u)**2), x) == [Max(sqrt(x), x**2), u] + assert decompogen(Max(u, u**2, y), x) == [Max(x, x**2, y), u] + assert decompogen(Max(sin(x), u), x) == [Max(2*x + 3, sin(x))] + + +def test_decompogen_poly(): + assert decompogen(x**4 + 2*x**2 + 1, x) == [x**2 + 2*x + 1, x**2] + assert decompogen(x**4 + 2*x**3 - x - 1, x) == [x**2 - x - 1, x**2 + x] + + +@XFAIL +def test_decompogen_fails(): + A = lambda x: x**2 + 2*x + 3 + B = lambda x: 4*x**2 + 5*x + 6 + assert decompogen(A(x*exp(x)), x) == [x**2 + 2*x + 3, x*exp(x)] + assert decompogen(A(B(x)), x) == [x**2 + 2*x + 3, 4*x**2 + 5*x + 6] + assert decompogen(A(1/x + 1/x**2), x) == [x**2 + 2*x + 3, 1/x + 1/x**2] + assert decompogen(A(1/x + 2/(x + 1)), x) == [x**2 + 2*x + 3, 1/x + 2/(x + 1)] + + +def test_compogen(): + assert compogen([sin(x), cos(x)], x) == sin(cos(x)) + assert compogen([x**2 + x + 1, sin(x)], x) == sin(x)**2 + sin(x) + 1 + assert compogen([sqrt(x), 6*x**2 - 5], x) == sqrt(6*x**2 - 5) + assert compogen([sin(x), sqrt(x), cos(x), x**2 + 1], x) == sin(sqrt( + cos(x**2 + 1))) + assert compogen([Abs(x), x**2 + 3*x - 4, cos(x)], x) == Abs(cos(x)**2 + + 3*cos(x) - 4) + assert compogen([x**2 + x - sqrt(3)/2, sin(x)], x) == (sin(x)**2 + sin(x) - + sqrt(3)/2) + assert compogen([Abs(x), 3*x + cos(y)**2 - 4, cos(x)], x) == \ + Abs(3*cos(x) + cos(y)**2 - 4) + assert compogen([x**2 + 2*x + 1, x**2], x) == x**4 + 2*x**2 + 1 + # the result is in unsimplified form + assert compogen([x**2 - x - 1, x**2 + x], x) == -x**2 - x + (x**2 + x)**2 - 1 diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/tests/test_inequalities.py b/.venv/lib/python3.12/site-packages/sympy/solvers/tests/test_inequalities.py new file mode 100644 index 0000000000000000000000000000000000000000..6ce6f4520b52d8714102c95457c90d44543c685c --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/tests/test_inequalities.py @@ -0,0 +1,500 @@ +"""Tests for tools for solving inequalities and systems of inequalities. """ + +from sympy.concrete.summations import Sum +from sympy.core.function import Function +from sympy.core.numbers import I, Rational, oo, pi +from sympy.core.relational import Eq, Ge, Gt, Le, Lt, Ne +from sympy.core.singleton import S +from sympy.core.symbol import (Dummy, Symbol) +from sympy.functions.elementary.complexes import Abs +from sympy.functions.elementary.exponential import exp, log +from sympy.functions.elementary.miscellaneous import root, sqrt +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.elementary.trigonometric import cos, sin, tan +from sympy.integrals.integrals import Integral +from sympy.logic.boolalg import And, Or +from sympy.polys.polytools import Poly, PurePoly +from sympy.sets.sets import FiniteSet, Interval, Union +from sympy.solvers.inequalities import (reduce_inequalities, + solve_poly_inequality as psolve, + reduce_rational_inequalities, + solve_univariate_inequality as isolve, + reduce_abs_inequality, + _solve_inequality) +from sympy.polys.rootoftools import rootof +from sympy.solvers.solvers import solve +from sympy.solvers.solveset import solveset +from sympy.core.mod import Mod +from sympy.abc import x, y + +from sympy.testing.pytest import raises, XFAIL + + +inf = oo.evalf() + + +def test_solve_poly_inequality(): + assert psolve(Poly(0, x), '==') == [S.Reals] + assert psolve(Poly(1, x), '==') == [S.EmptySet] + assert psolve(PurePoly(x + 1, x), ">") == [Interval(-1, oo, True, False)] + + +def test_reduce_poly_inequalities_real_interval(): + assert reduce_rational_inequalities( + [[Eq(x**2, 0)]], x, relational=False) == FiniteSet(0) + assert reduce_rational_inequalities( + [[Le(x**2, 0)]], x, relational=False) == FiniteSet(0) + assert reduce_rational_inequalities( + [[Lt(x**2, 0)]], x, relational=False) == S.EmptySet + assert reduce_rational_inequalities( + [[Ge(x**2, 0)]], x, relational=False) == \ + S.Reals if x.is_real else Interval(-oo, oo) + assert reduce_rational_inequalities( + [[Gt(x**2, 0)]], x, relational=False) == \ + FiniteSet(0).complement(S.Reals) + assert reduce_rational_inequalities( + [[Ne(x**2, 0)]], x, relational=False) == \ + FiniteSet(0).complement(S.Reals) + + assert reduce_rational_inequalities( + [[Eq(x**2, 1)]], x, relational=False) == FiniteSet(-1, 1) + assert reduce_rational_inequalities( + [[Le(x**2, 1)]], x, relational=False) == Interval(-1, 1) + assert reduce_rational_inequalities( + [[Lt(x**2, 1)]], x, relational=False) == Interval(-1, 1, True, True) + assert reduce_rational_inequalities( + [[Ge(x**2, 1)]], x, relational=False) == \ + Union(Interval(-oo, -1), Interval(1, oo)) + assert reduce_rational_inequalities( + [[Gt(x**2, 1)]], x, relational=False) == \ + Interval(-1, 1).complement(S.Reals) + assert reduce_rational_inequalities( + [[Ne(x**2, 1)]], x, relational=False) == \ + FiniteSet(-1, 1).complement(S.Reals) + assert reduce_rational_inequalities([[Eq( + x**2, 1.0)]], x, relational=False) == FiniteSet(-1.0, 1.0).evalf() + assert reduce_rational_inequalities( + [[Le(x**2, 1.0)]], x, relational=False) == Interval(-1.0, 1.0) + assert reduce_rational_inequalities([[Lt( + x**2, 1.0)]], x, relational=False) == Interval(-1.0, 1.0, True, True) + assert reduce_rational_inequalities( + [[Ge(x**2, 1.0)]], x, relational=False) == \ + Union(Interval(-inf, -1.0), Interval(1.0, inf)) + assert reduce_rational_inequalities( + [[Gt(x**2, 1.0)]], x, relational=False) == \ + Union(Interval(-inf, -1.0, right_open=True), + Interval(1.0, inf, left_open=True)) + assert reduce_rational_inequalities([[Ne( + x**2, 1.0)]], x, relational=False) == \ + FiniteSet(-1.0, 1.0).complement(S.Reals) + + s = sqrt(2) + + assert reduce_rational_inequalities([[Lt( + x**2 - 1, 0), Gt(x**2 - 1, 0)]], x, relational=False) == S.EmptySet + assert reduce_rational_inequalities([[Le(x**2 - 1, 0), Ge( + x**2 - 1, 0)]], x, relational=False) == FiniteSet(-1, 1) + assert reduce_rational_inequalities( + [[Le(x**2 - 2, 0), Ge(x**2 - 1, 0)]], x, relational=False + ) == Union(Interval(-s, -1, False, False), Interval(1, s, False, False)) + assert reduce_rational_inequalities( + [[Le(x**2 - 2, 0), Gt(x**2 - 1, 0)]], x, relational=False + ) == Union(Interval(-s, -1, False, True), Interval(1, s, True, False)) + assert reduce_rational_inequalities( + [[Lt(x**2 - 2, 0), Ge(x**2 - 1, 0)]], x, relational=False + ) == Union(Interval(-s, -1, True, False), Interval(1, s, False, True)) + assert reduce_rational_inequalities( + [[Lt(x**2 - 2, 0), Gt(x**2 - 1, 0)]], x, relational=False + ) == Union(Interval(-s, -1, True, True), Interval(1, s, True, True)) + assert reduce_rational_inequalities( + [[Lt(x**2 - 2, 0), Ne(x**2 - 1, 0)]], x, relational=False + ) == Union(Interval(-s, -1, True, True), Interval(-1, 1, True, True), + Interval(1, s, True, True)) + + assert reduce_rational_inequalities([[Lt(x**2, -1.)]], x) is S.false + + +def test_reduce_poly_inequalities_complex_relational(): + assert reduce_rational_inequalities( + [[Eq(x**2, 0)]], x, relational=True) == Eq(x, 0) + assert reduce_rational_inequalities( + [[Le(x**2, 0)]], x, relational=True) == Eq(x, 0) + assert reduce_rational_inequalities( + [[Lt(x**2, 0)]], x, relational=True) == False + assert reduce_rational_inequalities( + [[Ge(x**2, 0)]], x, relational=True) == And(Lt(-oo, x), Lt(x, oo)) + assert reduce_rational_inequalities( + [[Gt(x**2, 0)]], x, relational=True) == \ + And(Gt(x, -oo), Lt(x, oo), Ne(x, 0)) + assert reduce_rational_inequalities( + [[Ne(x**2, 0)]], x, relational=True) == \ + And(Gt(x, -oo), Lt(x, oo), Ne(x, 0)) + + for one in (S.One, S(1.0)): + inf = one*oo + assert reduce_rational_inequalities( + [[Eq(x**2, one)]], x, relational=True) == \ + Or(Eq(x, -one), Eq(x, one)) + assert reduce_rational_inequalities( + [[Le(x**2, one)]], x, relational=True) == \ + And(And(Le(-one, x), Le(x, one))) + assert reduce_rational_inequalities( + [[Lt(x**2, one)]], x, relational=True) == \ + And(And(Lt(-one, x), Lt(x, one))) + assert reduce_rational_inequalities( + [[Ge(x**2, one)]], x, relational=True) == \ + And(Or(And(Le(one, x), Lt(x, inf)), And(Le(x, -one), Lt(-inf, x)))) + assert reduce_rational_inequalities( + [[Gt(x**2, one)]], x, relational=True) == \ + And(Or(And(Lt(-inf, x), Lt(x, -one)), And(Lt(one, x), Lt(x, inf)))) + assert reduce_rational_inequalities( + [[Ne(x**2, one)]], x, relational=True) == \ + Or(And(Lt(-inf, x), Lt(x, -one)), + And(Lt(-one, x), Lt(x, one)), + And(Lt(one, x), Lt(x, inf))) + + +def test_reduce_rational_inequalities_real_relational(): + assert reduce_rational_inequalities([], x) == False + assert reduce_rational_inequalities( + [[(x**2 + 3*x + 2)/(x**2 - 16) >= 0]], x, relational=False) == \ + Union(Interval.open(-oo, -4), Interval(-2, -1), Interval.open(4, oo)) + + assert reduce_rational_inequalities( + [[((-2*x - 10)*(3 - x))/((x**2 + 5)*(x - 2)**2) < 0]], x, + relational=False) == \ + Union(Interval.open(-5, 2), Interval.open(2, 3)) + + assert reduce_rational_inequalities([[(x + 1)/(x - 5) <= 0]], x, + relational=False) == \ + Interval.Ropen(-1, 5) + + assert reduce_rational_inequalities([[(x**2 + 4*x + 3)/(x - 1) > 0]], x, + relational=False) == \ + Union(Interval.open(-3, -1), Interval.open(1, oo)) + + assert reduce_rational_inequalities([[(x**2 - 16)/(x - 1)**2 < 0]], x, + relational=False) == \ + Union(Interval.open(-4, 1), Interval.open(1, 4)) + + assert reduce_rational_inequalities([[(3*x + 1)/(x + 4) >= 1]], x, + relational=False) == \ + Union(Interval.open(-oo, -4), Interval.Ropen(Rational(3, 2), oo)) + + assert reduce_rational_inequalities([[(x - 8)/x <= 3 - x]], x, + relational=False) == \ + Union(Interval.Lopen(-oo, -2), Interval.Lopen(0, 4)) + + # issue sympy/sympy#10237 + assert reduce_rational_inequalities( + [[x < oo, x >= 0, -oo < x]], x, relational=False) == Interval(0, oo) + + +def test_reduce_abs_inequalities(): + e = abs(x - 5) < 3 + ans = And(Lt(2, x), Lt(x, 8)) + assert reduce_inequalities(e) == ans + assert reduce_inequalities(e, x) == ans + assert reduce_inequalities(abs(x - 5)) == Eq(x, 5) + assert reduce_inequalities( + abs(2*x + 3) >= 8) == Or(And(Le(Rational(5, 2), x), Lt(x, oo)), + And(Le(x, Rational(-11, 2)), Lt(-oo, x))) + assert reduce_inequalities(abs(x - 4) + abs( + 3*x - 5) < 7) == And(Lt(S.Half, x), Lt(x, 4)) + assert reduce_inequalities(abs(x - 4) + abs(3*abs(x) - 5) < 7) == \ + Or(And(S(-2) < x, x < -1), And(S.Half < x, x < 4)) + + nr = Symbol('nr', extended_real=False) + raises(TypeError, lambda: reduce_inequalities(abs(nr - 5) < 3)) + assert reduce_inequalities(x < 3, symbols=[x, nr]) == And(-oo < x, x < 3) + + +def test_reduce_inequalities_general(): + assert reduce_inequalities(Ge(sqrt(2)*x, 1)) == And(sqrt(2)/2 <= x, x < oo) + assert reduce_inequalities(x + 1 > 0) == And(S.NegativeOne < x, x < oo) + + +def test_reduce_inequalities_boolean(): + assert reduce_inequalities( + [Eq(x**2, 0), True]) == Eq(x, 0) + assert reduce_inequalities([Eq(x**2, 0), False]) == False + assert reduce_inequalities(x**2 >= 0) is S.true # issue 10196 + + +def test_reduce_inequalities_multivariate(): + assert reduce_inequalities([Ge(x**2, 1), Ge(y**2, 1)]) == And( + Or(And(Le(S.One, x), Lt(x, oo)), And(Le(x, -1), Lt(-oo, x))), + Or(And(Le(S.One, y), Lt(y, oo)), And(Le(y, -1), Lt(-oo, y)))) + + +def test_reduce_inequalities_errors(): + raises(NotImplementedError, lambda: reduce_inequalities(Ge(sin(x) + x, 1))) + raises(NotImplementedError, lambda: reduce_inequalities(Ge(x**2*y + y, 1))) + + +def test__solve_inequalities(): + assert reduce_inequalities(x + y < 1, symbols=[x]) == (x < 1 - y) + assert reduce_inequalities(x + y >= 1, symbols=[x]) == (x < oo) & (x >= -y + 1) + assert reduce_inequalities(Eq(0, x - y), symbols=[x]) == Eq(x, y) + assert reduce_inequalities(Ne(0, x - y), symbols=[x]) == Ne(x, y) + + +def test_issue_6343(): + eq = -3*x**2/2 - x*Rational(45, 4) + Rational(33, 2) > 0 + assert reduce_inequalities(eq) == \ + And(x < Rational(-15, 4) + sqrt(401)/4, -sqrt(401)/4 - Rational(15, 4) < x) + + +def test_issue_8235(): + assert reduce_inequalities(x**2 - 1 < 0) == \ + And(S.NegativeOne < x, x < 1) + assert reduce_inequalities(x**2 - 1 <= 0) == \ + And(S.NegativeOne <= x, x <= 1) + assert reduce_inequalities(x**2 - 1 > 0) == \ + Or(And(-oo < x, x < -1), And(x < oo, S.One < x)) + assert reduce_inequalities(x**2 - 1 >= 0) == \ + Or(And(-oo < x, x <= -1), And(S.One <= x, x < oo)) + + eq = x**8 + x - 9 # we want CRootOf solns here + sol = solve(eq >= 0) + tru = Or(And(rootof(eq, 1) <= x, x < oo), And(-oo < x, x <= rootof(eq, 0))) + assert sol == tru + + # recast vanilla as real + assert solve(sqrt((-x + 1)**2) < 1) == And(S.Zero < x, x < 2) + + +def test_issue_5526(): + assert reduce_inequalities(0 <= + x + Integral(y**2, (y, 1, 3)) - 1, [x]) == \ + (x >= -Integral(y**2, (y, 1, 3)) + 1) + f = Function('f') + e = Sum(f(x), (x, 1, 3)) + assert reduce_inequalities(0 <= x + e + y**2, [x]) == \ + (x >= -y**2 - Sum(f(x), (x, 1, 3))) + + +def test_solve_univariate_inequality(): + assert isolve(x**2 >= 4, x, relational=False) == Union(Interval(-oo, -2), + Interval(2, oo)) + assert isolve(x**2 >= 4, x) == Or(And(Le(2, x), Lt(x, oo)), And(Le(x, -2), + Lt(-oo, x))) + assert isolve((x - 1)*(x - 2)*(x - 3) >= 0, x, relational=False) == \ + Union(Interval(1, 2), Interval(3, oo)) + assert isolve((x - 1)*(x - 2)*(x - 3) >= 0, x) == \ + Or(And(Le(1, x), Le(x, 2)), And(Le(3, x), Lt(x, oo))) + assert isolve((x - 1)*(x - 2)*(x - 4) < 0, x, domain = FiniteSet(0, 3)) == \ + Or(Eq(x, 0), Eq(x, 3)) + # issue 2785: + assert isolve(x**3 - 2*x - 1 > 0, x, relational=False) == \ + Union(Interval(-1, -sqrt(5)/2 + S.Half, True, True), + Interval(S.Half + sqrt(5)/2, oo, True, True)) + # issue 2794: + assert isolve(x**3 - x**2 + x - 1 > 0, x, relational=False) == \ + Interval(1, oo, True) + #issue 13105 + assert isolve((x + I)*(x + 2*I) < 0, x) == Eq(x, 0) + assert isolve(((x - 1)*(x - 2) + I)*((x - 1)*(x - 2) + 2*I) < 0, x) == Or(Eq(x, 1), Eq(x, 2)) + assert isolve((((x - 1)*(x - 2) + I)*((x - 1)*(x - 2) + 2*I))/(x - 2) > 0, x) == Eq(x, 1) + raises (ValueError, lambda: isolve((x**2 - 3*x*I + 2)/x < 0, x)) + + # numerical testing in valid() is needed + assert isolve(x**7 - x - 2 > 0, x) == \ + And(rootof(x**7 - x - 2, 0) < x, x < oo) + + # handle numerator and denominator; although these would be handled as + # rational inequalities, these test confirm that the right thing is done + # when the domain is EX (e.g. when 2 is replaced with sqrt(2)) + assert isolve(1/(x - 2) > 0, x) == And(S(2) < x, x < oo) + den = ((x - 1)*(x - 2)).expand() + assert isolve((x - 1)/den <= 0, x) == \ + (x > -oo) & (x < 2) & Ne(x, 1) + + n = Dummy('n') + raises(NotImplementedError, lambda: isolve(Abs(x) <= n, x, relational=False)) + c1 = Dummy("c1", positive=True) + raises(NotImplementedError, lambda: isolve(n/c1 < 0, c1)) + n = Dummy('n', negative=True) + assert isolve(n/c1 > -2, c1) == (-n/2 < c1) + assert isolve(n/c1 < 0, c1) == True + assert isolve(n/c1 > 0, c1) == False + + zero = cos(1)**2 + sin(1)**2 - 1 + raises(NotImplementedError, lambda: isolve(x**2 < zero, x)) + raises(NotImplementedError, lambda: isolve( + x**2 < zero*I, x)) + raises(NotImplementedError, lambda: isolve(1/(x - y) < 2, x)) + raises(NotImplementedError, lambda: isolve(1/(x - y) < 0, x)) + raises(TypeError, lambda: isolve(x - I < 0, x)) + + zero = x**2 + x - x*(x + 1) + assert isolve(zero < 0, x, relational=False) is S.EmptySet + assert isolve(zero <= 0, x, relational=False) is S.Reals + + # make sure iter_solutions gets a default value + raises(NotImplementedError, lambda: isolve( + Eq(cos(x)**2 + sin(x)**2, 1), x)) + + +def test_trig_inequalities(): + # all the inequalities are solved in a periodic interval. + assert isolve(sin(x) < S.Half, x, relational=False) == \ + Union(Interval(0, pi/6, False, True), Interval.open(pi*Rational(5, 6), 2*pi)) + assert isolve(sin(x) > S.Half, x, relational=False) == \ + Interval(pi/6, pi*Rational(5, 6), True, True) + assert isolve(cos(x) < S.Zero, x, relational=False) == \ + Interval(pi/2, pi*Rational(3, 2), True, True) + assert isolve(cos(x) >= S.Zero, x, relational=False) == \ + Union(Interval(0, pi/2), Interval.Ropen(pi*Rational(3, 2), 2*pi)) + + assert isolve(tan(x) < S.One, x, relational=False) == \ + Union(Interval.Ropen(0, pi/4), Interval.open(pi/2, pi)) + + assert isolve(sin(x) <= S.Zero, x, relational=False) == \ + Union(FiniteSet(S.Zero), Interval.Ropen(pi, 2*pi)) + + assert isolve(sin(x) <= S.One, x, relational=False) == S.Reals + assert isolve(cos(x) < S(-2), x, relational=False) == S.EmptySet + assert isolve(sin(x) >= S.NegativeOne, x, relational=False) == S.Reals + assert isolve(cos(x) > S.One, x, relational=False) == S.EmptySet + + +def test_issue_9954(): + assert isolve(x**2 >= 0, x, relational=False) == S.Reals + assert isolve(x**2 >= 0, x, relational=True) == S.Reals.as_relational(x) + assert isolve(x**2 < 0, x, relational=False) == S.EmptySet + assert isolve(x**2 < 0, x, relational=True) == S.EmptySet.as_relational(x) + + +@XFAIL +def test_slow_general_univariate(): + r = rootof(x**5 - x**2 + 1, 0) + assert solve(sqrt(x) + 1/root(x, 3) > 1) == \ + Or(And(0 < x, x < r**6), And(r**6 < x, x < oo)) + + +def test_issue_8545(): + eq = 1 - x - abs(1 - x) + ans = And(Lt(1, x), Lt(x, oo)) + assert reduce_abs_inequality(eq, '<', x) == ans + eq = 1 - x - sqrt((1 - x)**2) + assert reduce_inequalities(eq < 0) == ans + + +def test_issue_8974(): + assert isolve(-oo < x, x) == And(-oo < x, x < oo) + assert isolve(oo > x, x) == And(-oo < x, x < oo) + + +def test_issue_10198(): + assert reduce_inequalities( + -1 + 1/abs(1/x - 1) < 0) == (x > -oo) & (x < S(1)/2) & Ne(x, 0) + + assert reduce_inequalities(abs(1/sqrt(x)) - 1, x) == Eq(x, 1) + assert reduce_abs_inequality(-3 + 1/abs(1 - 1/x), '<', x) == \ + Or(And(-oo < x, x < 0), + And(S.Zero < x, x < Rational(3, 4)), And(Rational(3, 2) < x, x < oo)) + raises(ValueError,lambda: reduce_abs_inequality(-3 + 1/abs( + 1 - 1/sqrt(x)), '<', x)) + + +def test_issue_10047(): + # issue 10047: this must remain an inequality, not True, since if x + # is not real the inequality is invalid + # assert solve(sin(x) < 2) == (x <= oo) + + # with PR 16956, (x <= oo) autoevaluates when x is extended_real + # which is assumed in the current implementation of inequality solvers + assert solve(sin(x) < 2) == True + assert solveset(sin(x) < 2, domain=S.Reals) == S.Reals + + +def test_issue_10268(): + assert solve(log(x) < 1000) == And(S.Zero < x, x < exp(1000)) + + +@XFAIL +def test_isolve_Sets(): + n = Dummy('n') + assert isolve(Abs(x) <= n, x, relational=False) == \ + Piecewise((S.EmptySet, n < 0), (Interval(-n, n), True)) + + +def test_integer_domain_relational_isolve(): + + dom = FiniteSet(0, 3) + x = Symbol('x',zero=False) + assert isolve((x - 1)*(x - 2)*(x - 4) < 0, x, domain=dom) == Eq(x, 3) + + x = Symbol('x') + assert isolve(x + 2 < 0, x, domain=S.Integers) == \ + (x <= -3) & (x > -oo) & Eq(Mod(x, 1), 0) + assert isolve(2 * x + 3 > 0, x, domain=S.Integers) == \ + (x >= -1) & (x < oo) & Eq(Mod(x, 1), 0) + assert isolve((x ** 2 + 3 * x - 2) < 0, x, domain=S.Integers) == \ + (x >= -3) & (x <= 0) & Eq(Mod(x, 1), 0) + assert isolve((x ** 2 + 3 * x - 2) > 0, x, domain=S.Integers) == \ + ((x >= 1) & (x < oo) & Eq(Mod(x, 1), 0)) | ( + (x <= -4) & (x > -oo) & Eq(Mod(x, 1), 0)) + + +def test_issue_10671_12466(): + assert solveset(sin(y), y, Interval(0, pi)) == FiniteSet(0, pi) + i = Interval(1, 10) + assert solveset((1/x).diff(x) < 0, x, i) == i + assert solveset((log(x - 6)/x) <= 0, x, S.Reals) == \ + Interval.Lopen(6, 7) + + +def test__solve_inequality(): + for op in (Gt, Lt, Le, Ge, Eq, Ne): + assert _solve_inequality(op(x, 1), x).lhs == x + assert _solve_inequality(op(S.One, x), x).lhs == x + # don't get tricked by symbol on right: solve it + assert _solve_inequality(Eq(2*x - 1, x), x) == Eq(x, 1) + ie = Eq(S.One, y) + assert _solve_inequality(ie, x) == ie + for fx in (x**2, exp(x), sin(x) + cos(x), x*(1 + x)): + for c in (0, 1): + e = 2*fx - c > 0 + assert _solve_inequality(e, x, linear=True) == ( + fx > c/S(2)) + assert _solve_inequality(2*x**2 + 2*x - 1 < 0, x, linear=True) == ( + x*(x + 1) < S.Half) + assert _solve_inequality(Eq(x*y, 1), x) == Eq(x*y, 1) + nz = Symbol('nz', nonzero=True) + assert _solve_inequality(Eq(x*nz, 1), x) == Eq(x, 1/nz) + assert _solve_inequality(x*nz < 1, x) == (x*nz < 1) + a = Symbol('a', positive=True) + assert _solve_inequality(a/x > 1, x) == (S.Zero < x) & (x < a) + assert _solve_inequality(a/x > 1, x, linear=True) == (1/x > 1/a) + # make sure to include conditions under which solution is valid + e = Eq(1 - x, x*(1/x - 1)) + assert _solve_inequality(e, x) == Ne(x, 0) + assert _solve_inequality(x < x*(1/x - 1), x) == (x < S.Half) & Ne(x, 0) + + +def test__pt(): + from sympy.solvers.inequalities import _pt + assert _pt(-oo, oo) == 0 + assert _pt(S.One, S(3)) == 2 + assert _pt(S.One, oo) == _pt(oo, S.One) == 2 + assert _pt(S.One, -oo) == _pt(-oo, S.One) == S.Half + assert _pt(S.NegativeOne, oo) == _pt(oo, S.NegativeOne) == Rational(-1, 2) + assert _pt(S.NegativeOne, -oo) == _pt(-oo, S.NegativeOne) == -2 + assert _pt(x, oo) == _pt(oo, x) == x + 1 + assert _pt(x, -oo) == _pt(-oo, x) == x - 1 + raises(ValueError, lambda: _pt(Dummy('i', infinite=True), S.One)) + + +def test_issue_25697(): + assert _solve_inequality(log(x, 3) <= 2, x) == (x <= 9) & (S.Zero < x) + + +def test_issue_25738(): + assert reduce_inequalities(3 < abs(x) + ) == reduce_inequalities(pi < abs(x)).subs(pi, 3) + + +def test_issue_25983(): + assert(reduce_inequalities(pi/Abs(x) <= 1) == ((pi <= x) & (x < oo)) | ((-oo < x) & (x <= -pi))) diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/tests/test_numeric.py b/.venv/lib/python3.12/site-packages/sympy/solvers/tests/test_numeric.py new file mode 100644 index 0000000000000000000000000000000000000000..12abd38c80f07279ed41aefc7952762da0f9f430 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/tests/test_numeric.py @@ -0,0 +1,139 @@ +from sympy.core.function import nfloat +from sympy.core.numbers import (Float, I, Rational, pi) +from sympy.core.relational import Eq +from sympy.core.symbol import (Symbol, symbols) +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.elementary.trigonometric import sin +from sympy.integrals.integrals import Integral +from sympy.matrices.dense import Matrix +from mpmath import mnorm, mpf +from sympy.solvers import nsolve +from sympy.utilities.lambdify import lambdify +from sympy.testing.pytest import raises, XFAIL +from sympy.utilities.decorator import conserve_mpmath_dps + +@XFAIL +def test_nsolve_fail(): + x = symbols('x') + # Sometimes it is better to use the numerator (issue 4829) + # but sometimes it is not (issue 11768) so leave this to + # the discretion of the user + ans = nsolve(x**2/(1 - x)/(1 - 2*x)**2 - 100, x, 0) + assert ans > 0.46 and ans < 0.47 + + +def test_nsolve_denominator(): + x = symbols('x') + # Test that nsolve uses the full expression (numerator and denominator). + ans = nsolve((x**2 + 3*x + 2)/(x + 2), -2.1) + # The root -2 was divided out, so make sure we don't find it. + assert ans == -1.0 + +def test_nsolve(): + # onedimensional + x = Symbol('x') + assert nsolve(sin(x), 2) - pi.evalf() < 1e-15 + assert nsolve(Eq(2*x, 2), x, -10) == nsolve(2*x - 2, -10) + # Testing checks on number of inputs + raises(TypeError, lambda: nsolve(Eq(2*x, 2))) + raises(TypeError, lambda: nsolve(Eq(2*x, 2), x, 1, 2)) + # multidimensional + x1 = Symbol('x1') + x2 = Symbol('x2') + f1 = 3 * x1**2 - 2 * x2**2 - 1 + f2 = x1**2 - 2 * x1 + x2**2 + 2 * x2 - 8 + f = Matrix((f1, f2)).T + F = lambdify((x1, x2), f.T, modules='mpmath') + for x0 in [(-1, 1), (1, -2), (4, 4), (-4, -4)]: + x = nsolve(f, (x1, x2), x0, tol=1.e-8) + assert mnorm(F(*x), 1) <= 1.e-10 + # The Chinese mathematician Zhu Shijie was the very first to solve this + # nonlinear system 700 years ago (z was added to make it 3-dimensional) + x = Symbol('x') + y = Symbol('y') + z = Symbol('z') + f1 = -x + 2*y + f2 = (x**2 + x*(y**2 - 2) - 4*y) / (x + 4) + f3 = sqrt(x**2 + y**2)*z + f = Matrix((f1, f2, f3)).T + F = lambdify((x, y, z), f.T, modules='mpmath') + + def getroot(x0): + root = nsolve(f, (x, y, z), x0) + assert mnorm(F(*root), 1) <= 1.e-8 + return root + assert list(map(round, getroot((1, 1, 1)))) == [2, 1, 0] + assert nsolve([Eq( + f1, 0), Eq(f2, 0), Eq(f3, 0)], [x, y, z], (1, 1, 1)) # just see that it works + a = Symbol('a') + assert abs(nsolve(1/(0.001 + a)**3 - 6/(0.9 - a)**3, a, 0.3) - + mpf('0.31883011387318591')) < 1e-15 + + +def test_issue_6408(): + x = Symbol('x') + assert nsolve(Piecewise((x, x < 1), (x**2, True)), x, 2) == 0 + + +def test_issue_6408_integral(): + x, y = symbols('x y') + assert nsolve(Integral(x*y, (x, 0, 5)), y, 2) == 0 + + +@conserve_mpmath_dps +def test_increased_dps(): + # Issue 8564 + import mpmath + mpmath.mp.dps = 128 + x = Symbol('x') + e1 = x**2 - pi + q = nsolve(e1, x, 3.0) + + assert abs(sqrt(pi).evalf(128) - q) < 1e-128 + +def test_nsolve_precision(): + x, y = symbols('x y') + sol = nsolve(x**2 - pi, x, 3, prec=128) + assert abs(sqrt(pi).evalf(128) - sol) < 1e-128 + assert isinstance(sol, Float) + + sols = nsolve((y**2 - x, x**2 - pi), (x, y), (3, 3), prec=128) + assert isinstance(sols, Matrix) + assert sols.shape == (2, 1) + assert abs(sqrt(pi).evalf(128) - sols[0]) < 1e-128 + assert abs(sqrt(sqrt(pi)).evalf(128) - sols[1]) < 1e-128 + assert all(isinstance(i, Float) for i in sols) + +def test_nsolve_complex(): + x, y = symbols('x y') + + assert nsolve(x**2 + 2, 1j) == sqrt(2.)*I + assert nsolve(x**2 + 2, I) == sqrt(2.)*I + + assert nsolve([x**2 + 2, y**2 + 2], [x, y], [I, I]) == Matrix([sqrt(2.)*I, sqrt(2.)*I]) + assert nsolve([x**2 + 2, y**2 + 2], [x, y], [I, I]) == Matrix([sqrt(2.)*I, sqrt(2.)*I]) + +def test_nsolve_dict_kwarg(): + x, y = symbols('x y') + # one variable + assert nsolve(x**2 - 2, 1, dict = True) == \ + [{x: sqrt(2.)}] + # one variable with complex solution + assert nsolve(x**2 + 2, I, dict = True) == \ + [{x: sqrt(2.)*I}] + # two variables + assert nsolve([x**2 + y**2 - 5, x**2 - y**2 + 1], [x, y], [1, 1], dict = True) == \ + [{x: sqrt(2.), y: sqrt(3.)}] + +def test_nsolve_rational(): + x = symbols('x') + assert nsolve(x - Rational(1, 3), 0, prec=100) == Rational(1, 3).evalf(100) + + +def test_issue_14950(): + x = Matrix(symbols('t s')) + x0 = Matrix([17, 23]) + eqn = x + x0 + assert nsolve(eqn, x, x0) == nfloat(-x0) + assert nsolve(eqn.T, x.T, x0.T) == nfloat(-x0) diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/tests/test_pde.py b/.venv/lib/python3.12/site-packages/sympy/solvers/tests/test_pde.py new file mode 100644 index 0000000000000000000000000000000000000000..948d90c7be21a9e0e03753e723ef04f1fb08a5d6 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/tests/test_pde.py @@ -0,0 +1,239 @@ +from sympy.core.function import (Derivative as D, Function) +from sympy.core.relational import Eq +from sympy.core.symbol import (Symbol, symbols) +from sympy.functions.elementary.exponential import (exp, log) +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.core import S +from sympy.solvers.pde import (pde_separate, pde_separate_add, pde_separate_mul, + pdsolve, classify_pde, checkpdesol) +from sympy.testing.pytest import raises + + +a, b, c, x, y = symbols('a b c x y') + +def test_pde_separate_add(): + x, y, z, t = symbols("x,y,z,t") + F, T, X, Y, Z, u = map(Function, 'FTXYZu') + + eq = Eq(D(u(x, t), x), D(u(x, t), t)*exp(u(x, t))) + res = pde_separate_add(eq, u(x, t), [X(x), T(t)]) + assert res == [D(X(x), x)*exp(-X(x)), D(T(t), t)*exp(T(t))] + + +def test_pde_separate(): + x, y, z, t = symbols("x,y,z,t") + F, T, X, Y, Z, u = map(Function, 'FTXYZu') + + eq = Eq(D(u(x, t), x), D(u(x, t), t)*exp(u(x, t))) + raises(ValueError, lambda: pde_separate(eq, u(x, t), [X(x), T(t)], 'div')) + + +def test_pde_separate_mul(): + x, y, z, t = symbols("x,y,z,t") + c = Symbol("C", real=True) + Phi = Function('Phi') + F, R, T, X, Y, Z, u = map(Function, 'FRTXYZu') + r, theta, z = symbols('r,theta,z') + + # Something simple :) + eq = Eq(D(F(x, y, z), x) + D(F(x, y, z), y) + D(F(x, y, z), z), 0) + + # Duplicate arguments in functions + raises( + ValueError, lambda: pde_separate_mul(eq, F(x, y, z), [X(x), u(z, z)])) + # Wrong number of arguments + raises(ValueError, lambda: pde_separate_mul(eq, F(x, y, z), [X(x), Y(y)])) + # Wrong variables: [x, y] -> [x, z] + raises( + ValueError, lambda: pde_separate_mul(eq, F(x, y, z), [X(t), Y(x, y)])) + + assert pde_separate_mul(eq, F(x, y, z), [Y(y), u(x, z)]) == \ + [D(Y(y), y)/Y(y), -D(u(x, z), x)/u(x, z) - D(u(x, z), z)/u(x, z)] + assert pde_separate_mul(eq, F(x, y, z), [X(x), Y(y), Z(z)]) == \ + [D(X(x), x)/X(x), -D(Z(z), z)/Z(z) - D(Y(y), y)/Y(y)] + + # wave equation + wave = Eq(D(u(x, t), t, t), c**2*D(u(x, t), x, x)) + res = pde_separate_mul(wave, u(x, t), [X(x), T(t)]) + assert res == [D(X(x), x, x)/X(x), D(T(t), t, t)/(c**2*T(t))] + + # Laplace equation in cylindrical coords + eq = Eq(1/r * D(Phi(r, theta, z), r) + D(Phi(r, theta, z), r, 2) + + 1/r**2 * D(Phi(r, theta, z), theta, 2) + D(Phi(r, theta, z), z, 2), 0) + # Separate z + res = pde_separate_mul(eq, Phi(r, theta, z), [Z(z), u(theta, r)]) + assert res == [D(Z(z), z, z)/Z(z), + -D(u(theta, r), r, r)/u(theta, r) - + D(u(theta, r), r)/(r*u(theta, r)) - + D(u(theta, r), theta, theta)/(r**2*u(theta, r))] + # Lets use the result to create a new equation... + eq = Eq(res[1], c) + # ...and separate theta... + res = pde_separate_mul(eq, u(theta, r), [T(theta), R(r)]) + assert res == [D(T(theta), theta, theta)/T(theta), + -r*D(R(r), r)/R(r) - r**2*D(R(r), r, r)/R(r) - c*r**2] + # ...or r... + res = pde_separate_mul(eq, u(theta, r), [R(r), T(theta)]) + assert res == [r*D(R(r), r)/R(r) + r**2*D(R(r), r, r)/R(r) + c*r**2, + -D(T(theta), theta, theta)/T(theta)] + + +def test_issue_11726(): + x, t = symbols("x t") + f = symbols("f", cls=Function) + X, T = symbols("X T", cls=Function) + + u = f(x, t) + eq = u.diff(x, 2) - u.diff(t, 2) + res = pde_separate(eq, u, [T(x), X(t)]) + assert res == [D(T(x), x, x)/T(x),D(X(t), t, t)/X(t)] + + +def test_pde_classify(): + # When more number of hints are added, add tests for classifying here. + f = Function('f') + eq1 = a*f(x,y) + b*f(x,y).diff(x) + c*f(x,y).diff(y) + eq2 = 3*f(x,y) + 2*f(x,y).diff(x) + f(x,y).diff(y) + eq3 = a*f(x,y) + b*f(x,y).diff(x) + 2*f(x,y).diff(y) + eq4 = x*f(x,y) + f(x,y).diff(x) + 3*f(x,y).diff(y) + eq5 = x**2*f(x,y) + x*f(x,y).diff(x) + x*y*f(x,y).diff(y) + eq6 = y*x**2*f(x,y) + y*f(x,y).diff(x) + f(x,y).diff(y) + for eq in [eq1, eq2, eq3]: + assert classify_pde(eq) == ('1st_linear_constant_coeff_homogeneous',) + for eq in [eq4, eq5, eq6]: + assert classify_pde(eq) == ('1st_linear_variable_coeff',) + + +def test_checkpdesol(): + f, F = map(Function, ['f', 'F']) + eq1 = a*f(x,y) + b*f(x,y).diff(x) + c*f(x,y).diff(y) + eq2 = 3*f(x,y) + 2*f(x,y).diff(x) + f(x,y).diff(y) + eq3 = a*f(x,y) + b*f(x,y).diff(x) + 2*f(x,y).diff(y) + for eq in [eq1, eq2, eq3]: + assert checkpdesol(eq, pdsolve(eq))[0] + eq4 = x*f(x,y) + f(x,y).diff(x) + 3*f(x,y).diff(y) + eq5 = 2*f(x,y) + 1*f(x,y).diff(x) + 3*f(x,y).diff(y) + eq6 = f(x,y) + 1*f(x,y).diff(x) + 3*f(x,y).diff(y) + assert checkpdesol(eq4, [pdsolve(eq5), pdsolve(eq6)]) == [ + (False, (x - 2)*F(3*x - y)*exp(-x/S(5) - 3*y/S(5))), + (False, (x - 1)*F(3*x - y)*exp(-x/S(10) - 3*y/S(10)))] + for eq in [eq4, eq5, eq6]: + assert checkpdesol(eq, pdsolve(eq))[0] + sol = pdsolve(eq4) + sol4 = Eq(sol.lhs - sol.rhs, 0) + raises(NotImplementedError, lambda: + checkpdesol(eq4, sol4, solve_for_func=False)) + + +def test_solvefun(): + f, F, G, H = map(Function, ['f', 'F', 'G', 'H']) + eq1 = f(x,y) + f(x,y).diff(x) + f(x,y).diff(y) + assert pdsolve(eq1) == Eq(f(x, y), F(x - y)*exp(-x/2 - y/2)) + assert pdsolve(eq1, solvefun=G) == Eq(f(x, y), G(x - y)*exp(-x/2 - y/2)) + assert pdsolve(eq1, solvefun=H) == Eq(f(x, y), H(x - y)*exp(-x/2 - y/2)) + + +def test_pde_1st_linear_constant_coeff_homogeneous(): + f, F = map(Function, ['f', 'F']) + u = f(x, y) + eq = 2*u + u.diff(x) + u.diff(y) + assert classify_pde(eq) == ('1st_linear_constant_coeff_homogeneous',) + sol = pdsolve(eq) + assert sol == Eq(u, F(x - y)*exp(-x - y)) + assert checkpdesol(eq, sol)[0] + + eq = 4 + (3*u.diff(x)/u) + (2*u.diff(y)/u) + assert classify_pde(eq) == ('1st_linear_constant_coeff_homogeneous',) + sol = pdsolve(eq) + assert sol == Eq(u, F(2*x - 3*y)*exp(-S(12)*x/13 - S(8)*y/13)) + assert checkpdesol(eq, sol)[0] + + eq = u + (6*u.diff(x)) + (7*u.diff(y)) + assert classify_pde(eq) == ('1st_linear_constant_coeff_homogeneous',) + sol = pdsolve(eq) + assert sol == Eq(u, F(7*x - 6*y)*exp(-6*x/S(85) - 7*y/S(85))) + assert checkpdesol(eq, sol)[0] + + eq = a*u + b*u.diff(x) + c*u.diff(y) + sol = pdsolve(eq) + assert checkpdesol(eq, sol)[0] + + +def test_pde_1st_linear_constant_coeff(): + f, F = map(Function, ['f', 'F']) + u = f(x,y) + eq = -2*u.diff(x) + 4*u.diff(y) + 5*u - exp(x + 3*y) + sol = pdsolve(eq) + assert sol == Eq(f(x,y), + (F(4*x + 2*y)*exp(x/2) + exp(x + 4*y)/15)*exp(-y)) + assert classify_pde(eq) == ('1st_linear_constant_coeff', + '1st_linear_constant_coeff_Integral') + assert checkpdesol(eq, sol)[0] + + eq = (u.diff(x)/u) + (u.diff(y)/u) + 1 - (exp(x + y)/u) + sol = pdsolve(eq) + assert sol == Eq(f(x, y), F(x - y)*exp(-x/2 - y/2) + exp(x + y)/3) + assert classify_pde(eq) == ('1st_linear_constant_coeff', + '1st_linear_constant_coeff_Integral') + assert checkpdesol(eq, sol)[0] + + eq = 2*u + -u.diff(x) + 3*u.diff(y) + sin(x) + sol = pdsolve(eq) + assert sol == Eq(f(x, y), + F(3*x + y)*exp(x/5 - 3*y/5) - 2*sin(x)/5 - cos(x)/5) + assert classify_pde(eq) == ('1st_linear_constant_coeff', + '1st_linear_constant_coeff_Integral') + assert checkpdesol(eq, sol)[0] + + eq = u + u.diff(x) + u.diff(y) + x*y + sol = pdsolve(eq) + assert sol.expand() == Eq(f(x, y), + x + y + (x - y)**2/4 - (x + y)**2/4 + F(x - y)*exp(-x/2 - y/2) - 2).expand() + assert classify_pde(eq) == ('1st_linear_constant_coeff', + '1st_linear_constant_coeff_Integral') + assert checkpdesol(eq, sol)[0] + eq = u + u.diff(x) + u.diff(y) + log(x) + assert classify_pde(eq) == ('1st_linear_constant_coeff', + '1st_linear_constant_coeff_Integral') + + +def test_pdsolve_all(): + f, F = map(Function, ['f', 'F']) + u = f(x,y) + eq = u + u.diff(x) + u.diff(y) + x**2*y + sol = pdsolve(eq, hint = 'all') + keys = ['1st_linear_constant_coeff', + '1st_linear_constant_coeff_Integral', 'default', 'order'] + assert sorted(sol.keys()) == keys + assert sol['order'] == 1 + assert sol['default'] == '1st_linear_constant_coeff' + assert sol['1st_linear_constant_coeff'].expand() == Eq(f(x, y), + -x**2*y + x**2 + 2*x*y - 4*x - 2*y + F(x - y)*exp(-x/2 - y/2) + 6).expand() + + +def test_pdsolve_variable_coeff(): + f, F = map(Function, ['f', 'F']) + u = f(x, y) + eq = x*(u.diff(x)) - y*(u.diff(y)) + y**2*u - y**2 + sol = pdsolve(eq, hint="1st_linear_variable_coeff") + assert sol == Eq(u, F(x*y)*exp(y**2/2) + 1) + assert checkpdesol(eq, sol)[0] + + eq = x**2*u + x*u.diff(x) + x*y*u.diff(y) + sol = pdsolve(eq, hint='1st_linear_variable_coeff') + assert sol == Eq(u, F(y*exp(-x))*exp(-x**2/2)) + assert checkpdesol(eq, sol)[0] + + eq = y*x**2*u + y*u.diff(x) + u.diff(y) + sol = pdsolve(eq, hint='1st_linear_variable_coeff') + assert sol == Eq(u, F(-2*x + y**2)*exp(-x**3/3)) + assert checkpdesol(eq, sol)[0] + + eq = exp(x)**2*(u.diff(x)) + y + sol = pdsolve(eq, hint='1st_linear_variable_coeff') + assert sol == Eq(u, y*exp(-2*x)/2 + F(y)) + assert checkpdesol(eq, sol)[0] + + eq = exp(2*x)*(u.diff(y)) + y*u - u + sol = pdsolve(eq, hint='1st_linear_variable_coeff') + assert sol == Eq(u, F(x)*exp(-y*(y - 2)*exp(-2*x)/2)) diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/tests/test_polysys.py b/.venv/lib/python3.12/site-packages/sympy/solvers/tests/test_polysys.py new file mode 100644 index 0000000000000000000000000000000000000000..a119591a0354ba377a18767eae6e8b7812810a0d --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/tests/test_polysys.py @@ -0,0 +1,462 @@ +"""Tests for solvers of systems of polynomial equations. """ +from sympy.polys.domains import ZZ, QQ_I +from sympy.core.numbers import (I, Integer, Rational) +from sympy.core.singleton import S +from sympy.core.symbol import symbols +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.polys.domains.rationalfield import QQ +from sympy.polys.polyerrors import UnsolvableFactorError +from sympy.polys.polyoptions import Options +from sympy.polys.polytools import Poly +from sympy.polys.rootoftools import CRootOf +from sympy.solvers.solvers import solve +from sympy.utilities.iterables import flatten +from sympy.abc import a, b, c, x, y, z +from sympy.polys import PolynomialError +from sympy.solvers.polysys import (solve_poly_system, + solve_triangulated, + solve_biquadratic, SolveFailed, + solve_generic, factor_system_bool, + factor_system_cond, factor_system_poly, + factor_system, _factor_sets, _factor_sets_slow) +from sympy.polys.polytools import parallel_poly_from_expr +from sympy.testing.pytest import raises +from sympy.core.relational import Eq +from sympy.functions.elementary.trigonometric import sin, cos + +from sympy.functions.elementary.exponential import exp + + +def test_solve_poly_system(): + assert solve_poly_system([x - 1], x) == [(S.One,)] + + assert solve_poly_system([y - x, y - x - 1], x, y) is None + + assert solve_poly_system([y - x**2, y + x**2], x, y) == [(S.Zero, S.Zero)] + + assert solve_poly_system([2*x - 3, y*Rational(3, 2) - 2*x, z - 5*y], x, y, z) == \ + [(Rational(3, 2), Integer(2), Integer(10))] + + assert solve_poly_system([x*y - 2*y, 2*y**2 - x**2], x, y) == \ + [(0, 0), (2, -sqrt(2)), (2, sqrt(2))] + + assert solve_poly_system([y - x**2, y + x**2 + 1], x, y) == \ + [(-I*sqrt(S.Half), Rational(-1, 2)), (I*sqrt(S.Half), Rational(-1, 2))] + + f_1 = x**2 + y + z - 1 + f_2 = x + y**2 + z - 1 + f_3 = x + y + z**2 - 1 + + a, b = sqrt(2) - 1, -sqrt(2) - 1 + + assert solve_poly_system([f_1, f_2, f_3], x, y, z) == \ + [(0, 0, 1), (0, 1, 0), (1, 0, 0), (a, a, a), (b, b, b)] + + solution = [(1, -1), (1, 1)] + + assert solve_poly_system([Poly(x**2 - y**2), Poly(x - 1)]) == solution + assert solve_poly_system([x**2 - y**2, x - 1], x, y) == solution + assert solve_poly_system([x**2 - y**2, x - 1]) == solution + + assert solve_poly_system( + [x + x*y - 3, y + x*y - 4], x, y) == [(-3, -2), (1, 2)] + + raises(NotImplementedError, lambda: solve_poly_system([x**3 - y**3], x, y)) + raises(NotImplementedError, lambda: solve_poly_system( + [z, -2*x*y**2 + x + y**2*z, y**2*(-z - 4) + 2])) + raises(PolynomialError, lambda: solve_poly_system([1/x], x)) + + raises(NotImplementedError, lambda: solve_poly_system( + [x-1,], (x, y))) + raises(NotImplementedError, lambda: solve_poly_system( + [y-1,], (x, y))) + + # solve_poly_system should ideally construct solutions using + # CRootOf for the following four tests + assert solve_poly_system([x**5 - x + 1], [x], strict=False) == [] + raises(UnsolvableFactorError, lambda: solve_poly_system( + [x**5 - x + 1], [x], strict=True)) + + assert solve_poly_system([(x - 1)*(x**5 - x + 1), y**2 - 1], [x, y], + strict=False) == [(1, -1), (1, 1)] + raises(UnsolvableFactorError, + lambda: solve_poly_system([(x - 1)*(x**5 - x + 1), y**2-1], + [x, y], strict=True)) + + +def test_solve_generic(): + NewOption = Options((x, y), {'domain': 'ZZ'}) + assert solve_generic([x**2 - 2*y**2, y**2 - y + 1], NewOption) == \ + [(-sqrt(-1 - sqrt(3)*I), Rational(1, 2) - sqrt(3)*I/2), + (sqrt(-1 - sqrt(3)*I), Rational(1, 2) - sqrt(3)*I/2), + (-sqrt(-1 + sqrt(3)*I), Rational(1, 2) + sqrt(3)*I/2), + (sqrt(-1 + sqrt(3)*I), Rational(1, 2) + sqrt(3)*I/2)] + + # solve_generic should ideally construct solutions using + # CRootOf for the following two tests + assert solve_generic( + [2*x - y, (y - 1)*(y**5 - y + 1)], NewOption, strict=False) == \ + [(Rational(1, 2), 1)] + raises(UnsolvableFactorError, lambda: solve_generic( + [2*x - y, (y - 1)*(y**5 - y + 1)], NewOption, strict=True)) + + +def test_solve_biquadratic(): + x0, y0, x1, y1, r = symbols('x0 y0 x1 y1 r') + + f_1 = (x - 1)**2 + (y - 1)**2 - r**2 + f_2 = (x - 2)**2 + (y - 2)**2 - r**2 + s = sqrt(2*r**2 - 1) + a = (3 - s)/2 + b = (3 + s)/2 + assert solve_poly_system([f_1, f_2], x, y) == [(a, b), (b, a)] + + f_1 = (x - 1)**2 + (y - 2)**2 - r**2 + f_2 = (x - 1)**2 + (y - 1)**2 - r**2 + + assert solve_poly_system([f_1, f_2], x, y) == \ + [(1 - sqrt((2*r - 1)*(2*r + 1))/2, Rational(3, 2)), + (1 + sqrt((2*r - 1)*(2*r + 1))/2, Rational(3, 2))] + + query = lambda expr: expr.is_Pow and expr.exp is S.Half + + f_1 = (x - 1 )**2 + (y - 2)**2 - r**2 + f_2 = (x - x1)**2 + (y - 1)**2 - r**2 + + result = solve_poly_system([f_1, f_2], x, y) + + assert len(result) == 2 and all(len(r) == 2 for r in result) + assert all(r.count(query) == 1 for r in flatten(result)) + + f_1 = (x - x0)**2 + (y - y0)**2 - r**2 + f_2 = (x - x1)**2 + (y - y1)**2 - r**2 + + result = solve_poly_system([f_1, f_2], x, y) + + assert len(result) == 2 and all(len(r) == 2 for r in result) + assert all(len(r.find(query)) == 1 for r in flatten(result)) + + s1 = (x*y - y, x**2 - x) + assert solve(s1) == [{x: 1}, {x: 0, y: 0}] + s2 = (x*y - x, y**2 - y) + assert solve(s2) == [{y: 1}, {x: 0, y: 0}] + gens = (x, y) + for seq in (s1, s2): + (f, g), opt = parallel_poly_from_expr(seq, *gens) + raises(SolveFailed, lambda: solve_biquadratic(f, g, opt)) + seq = (x**2 + y**2 - 2, y**2 - 1) + (f, g), opt = parallel_poly_from_expr(seq, *gens) + assert solve_biquadratic(f, g, opt) == [ + (-1, -1), (-1, 1), (1, -1), (1, 1)] + ans = [(0, -1), (0, 1)] + seq = (x**2 + y**2 - 1, y**2 - 1) + (f, g), opt = parallel_poly_from_expr(seq, *gens) + assert solve_biquadratic(f, g, opt) == ans + seq = (x**2 + y**2 - 1, x**2 - x + y**2 - 1) + (f, g), opt = parallel_poly_from_expr(seq, *gens) + assert solve_biquadratic(f, g, opt) == ans + + +def test_solve_triangulated(): + f_1 = x**2 + y + z - 1 + f_2 = x + y**2 + z - 1 + f_3 = x + y + z**2 - 1 + + a, b = sqrt(2) - 1, -sqrt(2) - 1 + + assert solve_triangulated([f_1, f_2, f_3], x, y, z) == \ + [(0, 0, 1), (0, 1, 0), (1, 0, 0)] + + dom = QQ.algebraic_field(sqrt(2)) + + assert solve_triangulated([f_1, f_2, f_3], x, y, z, domain=dom) == \ + [(0, 0, 1), (0, 1, 0), (1, 0, 0), (a, a, a), (b, b, b)] + + a, b = CRootOf(z**2 + 2*z - 1, 0), CRootOf(z**2 + 2*z - 1, 1) + assert solve_triangulated([f_1, f_2, f_3], x, y, z, extension=True) == \ + [(0, 0, 1), (0, 1, 0), (1, 0, 0), (a, a, a), (b, b, b)] + + +def test_solve_issue_3686(): + roots = solve_poly_system([((x - 5)**2/250000 + (y - Rational(5, 10))**2/250000) - 1, x], x, y) + assert roots == [(0, S.Half - 15*sqrt(1111)), (0, S.Half + 15*sqrt(1111))] + + roots = solve_poly_system([((x - 5)**2/250000 + (y - 5.0/10)**2/250000) - 1, x], x, y) + # TODO: does this really have to be so complicated?! + assert len(roots) == 2 + assert roots[0][0] == 0 + assert roots[0][1].epsilon_eq(-499.474999374969, 1e12) + assert roots[1][0] == 0 + assert roots[1][1].epsilon_eq(500.474999374969, 1e12) + + +def test_factor_system(): + + assert factor_system([x**2 + 2*x + 1]) == [[x + 1]] + assert factor_system([x**2 + 2*x + 1, y**2 + 2*y + 1]) == [[x + 1, y + 1]] + assert factor_system([x**2 + 1]) == [[x**2 + 1]] + assert factor_system([]) == [[]] + + assert factor_system([x**2 + y**2 + 2*x*y, x**2 - 2], extension=sqrt(2)) == [ + [x + y, x + sqrt(2)], + [x + y, x - sqrt(2)], + ] + + assert factor_system([x**2 + 1, y**2 + 1], gaussian=True) == [ + [x + I, y + I], + [x + I, y - I], + [x - I, y + I], + [x - I, y - I], + ] + + assert factor_system([x**2 + 1, y**2 + 1], domain=QQ_I) == [ + [x + I, y + I], + [x + I, y - I], + [x - I, y + I], + [x - I, y - I], + ] + + assert factor_system([0]) == [[]] + assert factor_system([1]) == [] + assert factor_system([0 , x]) == [[x]] + assert factor_system([1, 0, x]) == [] + + assert factor_system([x**4 - 1, y**6 - 1]) == [ + [x**2 + 1, y**2 + y + 1], + [x**2 + 1, y**2 - y + 1], + [x**2 + 1, y + 1], + [x**2 + 1, y - 1], + [x + 1, y**2 + y + 1], + [x + 1, y**2 - y + 1], + [x - 1, y**2 + y + 1], + [x - 1, y**2 - y + 1], + [x + 1, y + 1], + [x + 1, y - 1], + [x - 1, y + 1], + [x - 1, y - 1], + ] + + assert factor_system([(x - 1)*(y - 2), (y - 2)*(z - 3)]) == [ + [x - 1, z - 3], + [y - 2] + ] + + assert factor_system([sin(x)**2 + cos(x)**2 - 1, x]) == [ + [x, sin(x)**2 + cos(x)**2 - 1], + ] + + assert factor_system([sin(x)**2 + cos(x)**2 - 1]) == [ + [sin(x)**2 + cos(x)**2 - 1] + ] + + assert factor_system([sin(x)**2 + cos(x)**2]) == [ + [sin(x)**2 + cos(x)**2] + ] + + assert factor_system([a*x, y, a]) == [[y, a]] + + assert factor_system([a*x, y, a], [x, y]) == [] + + assert factor_system([a ** 2 * x, y], [x, y]) == [[x, y]] + + assert factor_system([a*x*(x - 1), b*y, c], [x, y]) == [] + + assert factor_system([a*x*(x - 1), b*y, c], [x, y, c]) == [ + [x - 1, y, c], + [x, y, c], + ] + + assert factor_system([a*x*(x - 1), b*y, c]) == [ + [x - 1, y, c], + [x, y, c], + [x - 1, b, c], + [x, b, c], + [y, a, c], + [a, b, c], + ] + + assert factor_system([x**2 - 2], [y]) == [] + + assert factor_system([x**2 - 2], [x]) == [[x**2 - 2]] + + assert factor_system([cos(x)**2 - sin(x)**2, cos(x)**2 + sin(x)**2 - 1]) == [ + [sin(x)**2 + cos(x)**2 - 1, sin(x) + cos(x)], + [sin(x)**2 + cos(x)**2 - 1, -sin(x) + cos(x)], + ] + + assert factor_system([(cos(x) + sin(x))**2 - 1, cos(x)**2 - sin(x)**2 - cos(2*x)]) == [ + [sin(x)**2 - cos(x)**2 + cos(2*x), sin(x) + cos(x) + 1], + [sin(x)**2 - cos(x)**2 + cos(2*x), sin(x) + cos(x) - 1], + ] + + assert factor_system([(cos(x) + sin(x))*exp(y) - 1, (cos(x) - sin(x))*exp(y) - 1]) == [ + [exp(y)*sin(x) + exp(y)*cos(x) - 1, -exp(y)*sin(x) + exp(y)*cos(x) - 1] + ] + + +def test_factor_system_poly(): + + px = lambda e: Poly(e, x) + pxab = lambda e: Poly(e, x, domain=ZZ[a, b]) + pxI = lambda e: Poly(e, x, domain=QQ_I) + pxyz = lambda e: Poly(e, (x, y, z)) + + assert factor_system_poly([px(x**2 - 1), px(x**2 - 4)]) == [ + [px(x + 2), px(x + 1)], + [px(x + 2), px(x - 1)], + [px(x + 1), px(x - 2)], + [px(x - 1), px(x - 2)], + ] + + assert factor_system_poly([px(x**2 - 1)]) == [[px(x + 1)], [px(x - 1)]] + + assert factor_system_poly([pxyz(x**2*y - y), pxyz(x**2*z - z)]) == [ + [pxyz(x + 1)], + [pxyz(x - 1)], + [pxyz(y), pxyz(z)], + ] + + assert factor_system_poly([px(x**2*(x - 1)**2), px(x*(x - 1))]) == [ + [px(x)], + [px(x - 1)], + ] + + assert factor_system_poly([pxyz(x**2 + y*x), pxyz(x**2 + z*x)]) == [ + [pxyz(x + y), pxyz(x + z)], + [pxyz(x)], + ] + + assert factor_system_poly([pxab((a - 1)*(x - 2)), pxab((b - 3)*(x - 2))]) == [ + [pxab(x - 2)], + [pxab(a - 1), pxab(b - 3)], + ] + + assert factor_system_poly([pxI(x**2 + 1)]) == [[pxI(x + I)], [pxI(x - I)]] + + assert factor_system_poly([]) == [[]] + + assert factor_system_poly([px(1)]) == [] + assert factor_system_poly([px(0), px(x)]) == [[px(x)]] + + +def test_factor_system_cond(): + + assert factor_system_cond([x ** 2 - 1, x ** 2 - 4]) == [ + [x + 2, x + 1], + [x + 2, x - 1], + [x + 1, x - 2], + [x - 1, x - 2], + ] + + assert factor_system_cond([1]) == [] + assert factor_system_cond([0]) == [[]] + assert factor_system_cond([1, x]) == [] + assert factor_system_cond([0, x]) == [[x]] + assert factor_system_cond([]) == [[]] + + assert factor_system_cond([x**2 + y*x]) == [[x + y], [x]] + + assert factor_system_cond([(a - 1)*(x - 2), (b - 3)*(x - 2)], [x]) == [ + [x - 2], + [a - 1, b - 3], + ] + + assert factor_system_cond([a * (x - 1), b], [x]) == [[x - 1, b], [a, b]] + + assert factor_system_cond([a*x*(x-1), b*y, c], [x, y]) == [ + [x - 1, y, c], + [x, y, c], + [x - 1, b, c], + [x, b, c], + [y, a, c], + [a, b, c], + ] + + assert factor_system_cond([x*(x-1), y], [x, y]) == [[x - 1, y], [x, y]] + + assert factor_system_cond([a*x, y, a], [x, y]) == [[y, a]] + + assert factor_system_cond([a*x, b*x], [x, y]) == [[x], [a, b]] + + assert factor_system_cond([a*b*x, y], [x, y]) == [[x, y], [y, a*b]] + + assert factor_system_cond([a*b*x, y]) == [[x, y], [y, a], [y, b]] + + assert factor_system_cond([a**2*x, y], [x, y]) == [[x, y], [y, a]] + +def test_factor_system_bool(): + + eqs = [a*(x - 1)*(y - 1), b*(x - 2)*(y - 1)*(y - 2)] + assert factor_system_bool(eqs, [x, y]) == ( + Eq(y - 1, 0) + | (Eq(a, 0) & Eq(b, 0)) + | (Eq(a, 0) & Eq(x - 2, 0)) + | (Eq(a, 0) & Eq(y - 2, 0)) + | (Eq(b, 0) & Eq(x - 1, 0)) + | (Eq(x - 2, 0) & Eq(x - 1, 0)) + | (Eq(x - 1, 0) & Eq(y - 2, 0)) + ) + + assert factor_system_bool([x - 1], [x]) == Eq(x - 1, 0) + + assert factor_system_bool([(x - 1)*(x - 2)], [x]) == Eq(x - 2, 0) | Eq(x - 1, 0) + + assert factor_system_bool([], [x]) == True + assert factor_system_bool([0], [x]) == True + assert factor_system_bool([1], [x]) == False + assert factor_system_bool([a], [x]) == Eq(a, 0) + + assert factor_system_bool([a * x, y, a], [x, y]) == Eq(a, 0) & Eq(y, 0) + + assert (factor_system_bool([a*x, b*y*x, a], [x, y]) == ( + Eq(a, 0) & Eq(b, 0)) + | (Eq(a, 0) & Eq(x, 0)) + | (Eq(a, 0) & Eq(y, 0))) + + assert (factor_system_bool([a*x, b*x], [x, y]) == Eq(x, 0) | + (Eq(a, 0) & Eq(b, 0))) + + assert (factor_system_bool([a*b*x, y], [x, y]) == ( + Eq(x, 0) & Eq(y, 0)) | + (Eq(y, 0) & Eq(a*b, 0))) + + assert (factor_system_bool([a**2*x, y], [x, y]) == ( + Eq(a, 0) & Eq(y, 0)) | + (Eq(x, 0) & Eq(y, 0))) + + assert factor_system_bool([a*x*y, b*y*z], [x, y, z]) == ( + Eq(y, 0) + | (Eq(a, 0) & Eq(b, 0)) + | (Eq(a, 0) & Eq(z, 0)) + | (Eq(b, 0) & Eq(x, 0)) + | (Eq(x, 0) & Eq(z, 0)) + ) + + assert factor_system_bool([a*(x - 1), b], [x]) == ( + (Eq(a, 0) & Eq(b, 0)) + | (Eq(x - 1, 0) & Eq(b, 0)) + ) + + +def test_factor_sets(): + # + from random import randint + + def generate_random_system(n_eqs=3, n_factors=2, max_val=10): + return [ + [randint(0, max_val) for _ in range(randint(1, n_factors))] + for _ in range(n_eqs) + ] + + test_cases = [ + [[1, 2], [1, 3]], + [[1, 2], [3, 4]], + [[1], [1, 2], [2]], + ] + + for case in test_cases: + assert _factor_sets(case) == _factor_sets_slow(case) + + for _ in range(100): + system = generate_random_system() + assert _factor_sets(system) == _factor_sets_slow(system) diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/tests/test_recurr.py b/.venv/lib/python3.12/site-packages/sympy/solvers/tests/test_recurr.py new file mode 100644 index 0000000000000000000000000000000000000000..5a6306b51a5cf33ccd9fae131430a24690d540a7 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/tests/test_recurr.py @@ -0,0 +1,295 @@ +from sympy.core.function import (Function, Lambda, expand) +from sympy.core.numbers import (I, Rational) +from sympy.core.relational import Eq +from sympy.core.singleton import S +from sympy.core.symbol import (Symbol, symbols) +from sympy.functions.combinatorial.factorials import (rf, binomial, factorial) +from sympy.functions.elementary.complexes import Abs +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.polys.polytools import factor +from sympy.solvers.recurr import rsolve, rsolve_hyper, rsolve_poly, rsolve_ratio +from sympy.testing.pytest import raises, slow, XFAIL +from sympy.abc import a, b + +y = Function('y') +n, k = symbols('n,k', integer=True) +C0, C1, C2 = symbols('C0,C1,C2') + + +def test_rsolve_poly(): + assert rsolve_poly([-1, -1, 1], 0, n) == 0 + assert rsolve_poly([-1, -1, 1], 1, n) == -1 + + assert rsolve_poly([-1, n + 1], n, n) == 1 + assert rsolve_poly([-1, 1], n, n) == C0 + (n**2 - n)/2 + assert rsolve_poly([-n - 1, n], 1, n) == C0*n - 1 + assert rsolve_poly([-4*n - 2, 1], 4*n + 1, n) == -1 + + assert rsolve_poly([-1, 1], n**5 + n**3, n) == \ + C0 - n**3 / 2 - n**5 / 2 + n**2 / 6 + n**6 / 6 + 2*n**4 / 3 + + +def test_rsolve_ratio(): + solution = rsolve_ratio([-2*n**3 + n**2 + 2*n - 1, 2*n**3 + n**2 - 6*n, + -2*n**3 - 11*n**2 - 18*n - 9, 2*n**3 + 13*n**2 + 22*n + 8], 0, n) + assert solution == C0*(2*n - 3)/(n**2 - 1)/2 + + +def test_rsolve_hyper(): + assert rsolve_hyper([-1, -1, 1], 0, n) in [ + C0*(S.Half - S.Half*sqrt(5))**n + C1*(S.Half + S.Half*sqrt(5))**n, + C1*(S.Half - S.Half*sqrt(5))**n + C0*(S.Half + S.Half*sqrt(5))**n, + ] + + assert rsolve_hyper([n**2 - 2, -2*n - 1, 1], 0, n) in [ + C0*rf(sqrt(2), n) + C1*rf(-sqrt(2), n), + C1*rf(sqrt(2), n) + C0*rf(-sqrt(2), n), + ] + + assert rsolve_hyper([n**2 - k, -2*n - 1, 1], 0, n) in [ + C0*rf(sqrt(k), n) + C1*rf(-sqrt(k), n), + C1*rf(sqrt(k), n) + C0*rf(-sqrt(k), n), + ] + + assert rsolve_hyper( + [2*n*(n + 1), -n**2 - 3*n + 2, n - 1], 0, n) == C1*factorial(n) + C0*2**n + + assert rsolve_hyper( + [n + 2, -(2*n + 3)*(17*n**2 + 51*n + 39), n + 1], 0, n) == 0 + + assert rsolve_hyper([-n - 1, -1, 1], 0, n) == 0 + + assert rsolve_hyper([-1, 1], n, n).expand() == C0 + n**2/2 - n/2 + + assert rsolve_hyper([-1, 1], 1 + n, n).expand() == C0 + n**2/2 + n/2 + + assert rsolve_hyper([-1, 1], 3*(n + n**2), n).expand() == C0 + n**3 - n + + assert rsolve_hyper([-a, 1],0,n).expand() == C0*a**n + + assert rsolve_hyper([-a, 0, 1], 0, n).expand() == (-1)**n*C1*a**(n/2) + C0*a**(n/2) + + assert rsolve_hyper([1, 1, 1], 0, n).expand() == \ + C0*(Rational(-1, 2) - sqrt(3)*I/2)**n + C1*(Rational(-1, 2) + sqrt(3)*I/2)**n + + assert rsolve_hyper([1, -2*n/a - 2/a, 1], 0, n) == 0 + + +@XFAIL +def test_rsolve_ratio_missed(): + # this arises during computation + # assert rsolve_hyper([-1, 1], 3*(n + n**2), n).expand() == C0 + n**3 - n + assert rsolve_ratio([-n, n + 2], n, n) is not None + + +def recurrence_term(c, f): + """Compute RHS of recurrence in f(n) with coefficients in c.""" + return sum(c[i]*f.subs(n, n + i) for i in range(len(c))) + + +def test_rsolve_bulk(): + """Some bulk-generated tests.""" + funcs = [ n, n + 1, n**2, n**3, n**4, n + n**2, 27*n + 52*n**2 - 3* + n**3 + 12*n**4 - 52*n**5 ] + coeffs = [ [-2, 1], [-2, -1, 1], [-1, 1, 1, -1, 1], [-n, 1], [n**2 - + n + 12, 1] ] + for p in funcs: + # compute difference + for c in coeffs: + q = recurrence_term(c, p) + if p.is_polynomial(n): + assert rsolve_poly(c, q, n) == p + # See issue 3956: + if p.is_hypergeometric(n) and len(c) <= 3: + assert rsolve_hyper(c, q, n).subs(zip(symbols('C:3'), [0, 0, 0])).expand() == p + + +def test_rsolve_0_sol_homogeneous(): + # fixed by cherry-pick from + # https://github.com/diofant/diofant/commit/e1d2e52125199eb3df59f12e8944f8a5f24b00a5 + assert rsolve_hyper([n**2 - n + 12, 1], n*(n**2 - n + 12) + n + 1, n) == n + + +def test_rsolve(): + f = y(n + 2) - y(n + 1) - y(n) + h = sqrt(5)*(S.Half + S.Half*sqrt(5))**n \ + - sqrt(5)*(S.Half - S.Half*sqrt(5))**n + + assert rsolve(f, y(n)) in [ + C0*(S.Half - S.Half*sqrt(5))**n + C1*(S.Half + S.Half*sqrt(5))**n, + C1*(S.Half - S.Half*sqrt(5))**n + C0*(S.Half + S.Half*sqrt(5))**n, + ] + + assert rsolve(f, y(n), [0, 5]) == h + assert rsolve(f, y(n), {0: 0, 1: 5}) == h + assert rsolve(f, y(n), {y(0): 0, y(1): 5}) == h + assert rsolve(y(n) - y(n - 1) - y(n - 2), y(n), [0, 5]) == h + assert rsolve(Eq(y(n), y(n - 1) + y(n - 2)), y(n), [0, 5]) == h + + assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 + + f = (n - 1)*y(n + 2) - (n**2 + 3*n - 2)*y(n + 1) + 2*n*(n + 1)*y(n) + g = C1*factorial(n) + C0*2**n + h = -3*factorial(n) + 3*2**n + + assert rsolve(f, y(n)) == g + assert rsolve(f, y(n), []) == g + assert rsolve(f, y(n), {}) == g + + assert rsolve(f, y(n), [0, 3]) == h + assert rsolve(f, y(n), {0: 0, 1: 3}) == h + assert rsolve(f, y(n), {y(0): 0, y(1): 3}) == h + + assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 + + f = y(n) - y(n - 1) - 2 + + assert rsolve(f, y(n), {y(0): 0}) == 2*n + assert rsolve(f, y(n), {y(0): 1}) == 2*n + 1 + assert rsolve(f, y(n), {y(0): 0, y(1): 1}) is None + + assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 + + f = 3*y(n - 1) - y(n) - 1 + + assert rsolve(f, y(n), {y(0): 0}) == -3**n/2 + S.Half + assert rsolve(f, y(n), {y(0): 1}) == 3**n/2 + S.Half + assert rsolve(f, y(n), {y(0): 2}) == 3*3**n/2 + S.Half + + assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 + + f = y(n) - 1/n*y(n - 1) + assert rsolve(f, y(n)) == C0/factorial(n) + assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 + + f = y(n) - 1/n*y(n - 1) - 1 + assert rsolve(f, y(n)) is None + + f = 2*y(n - 1) + (1 - n)*y(n)/n + + assert rsolve(f, y(n), {y(1): 1}) == 2**(n - 1)*n + assert rsolve(f, y(n), {y(1): 2}) == 2**(n - 1)*n*2 + assert rsolve(f, y(n), {y(1): 3}) == 2**(n - 1)*n*3 + + assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 + + f = (n - 1)*(n - 2)*y(n + 2) - (n + 1)*(n + 2)*y(n) + + assert rsolve(f, y(n), {y(3): 6, y(4): 24}) == n*(n - 1)*(n - 2) + assert rsolve( + f, y(n), {y(3): 6, y(4): -24}) == -n*(n - 1)*(n - 2)*(-1)**(n) + + assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0 + + assert rsolve(Eq(y(n + 1), a*y(n)), y(n), {y(1): a}).simplify() == a**n + + assert rsolve(y(n) - a*y(n-2),y(n), \ + {y(1): sqrt(a)*(a + b), y(2): a*(a - b)}).simplify() == \ + a**(n/2 + 1) - b*(-sqrt(a))**n + + f = (-16*n**2 + 32*n - 12)*y(n - 1) + (4*n**2 - 12*n + 9)*y(n) + + yn = rsolve(f, y(n), {y(1): binomial(2*n + 1, 3)}) + sol = 2**(2*n)*n*(2*n - 1)**2*(2*n + 1)/12 + assert factor(expand(yn, func=True)) == sol + + sol = rsolve(y(n) + a*(y(n + 1) + y(n - 1))/2, y(n)) + assert str(sol) == 'C0*((-sqrt(1 - a**2) - 1)/a)**n + C1*((sqrt(1 - a**2) - 1)/a)**n' + + assert rsolve((k + 1)*y(k), y(k)) is None + assert (rsolve((k + 1)*y(k) + (k + 3)*y(k + 1) + (k + 5)*y(k + 2), y(k)) + is None) + + assert rsolve(y(n) + y(n + 1) + 2**n + 3**n, y(n)) == (-1)**n*C0 - 2**n/3 - 3**n/4 + + +def test_rsolve_raises(): + x = Function('x') + raises(ValueError, lambda: rsolve(y(n) - y(k + 1), y(n))) + raises(ValueError, lambda: rsolve(y(n) - y(n + 1), x(n))) + raises(ValueError, lambda: rsolve(y(n) - x(n + 1), y(n))) + raises(ValueError, lambda: rsolve(y(n) - sqrt(n)*y(n + 1), y(n))) + raises(ValueError, lambda: rsolve(y(n) - y(n + 1), y(n), {x(0): 0})) + raises(ValueError, lambda: rsolve(y(n) + y(n + 1) + 2**n + cos(n), y(n))) + + +def test_issue_6844(): + f = y(n + 2) - y(n + 1) + y(n)/4 + assert rsolve(f, y(n)) == 2**(-n + 1)*C1*n + 2**(-n)*C0 + assert rsolve(f, y(n), {y(0): 0, y(1): 1}) == 2**(1 - n)*n + + +def test_issue_18751(): + r = Symbol('r', positive=True) + theta = Symbol('theta', real=True) + f = y(n) - 2 * r * cos(theta) * y(n - 1) + r**2 * y(n - 2) + assert rsolve(f, y(n)) == \ + C0*(r*(cos(theta) - I*Abs(sin(theta))))**n + C1*(r*(cos(theta) + I*Abs(sin(theta))))**n + + +def test_constant_naming(): + #issue 8697 + assert rsolve(y(n+3) - y(n+2) - y(n+1) + y(n), y(n)) == (-1)**n*C1 + C0 + C2*n + assert rsolve(y(n+3)+3*y(n+2)+3*y(n+1)+y(n), y(n)).expand() == (-1)**n*C0 - (-1)**n*C1*n - (-1)**n*C2*n**2 + assert rsolve(y(n) - 2*y(n - 3) + 5*y(n - 2) - 4*y(n - 1),y(n),[1,3,8]) == 3*2**n - n - 2 + + #issue 19630 + assert rsolve(y(n+3) - 3*y(n+1) + 2*y(n), y(n), {y(1):0, y(2):8, y(3):-2}) == (-2)**n + 2*n + + +@slow +def test_issue_15751(): + f = y(n) + 21*y(n + 1) - 273*y(n + 2) - 1092*y(n + 3) + 1820*y(n + 4) + 1092*y(n + 5) - 273*y(n + 6) - 21*y(n + 7) + y(n + 8) + assert rsolve(f, y(n)) is not None + + +def test_issue_17990(): + f = -10*y(n) + 4*y(n + 1) + 6*y(n + 2) + 46*y(n + 3) + sol = rsolve(f, y(n)) + expected = C0*((86*18**(S(1)/3)/69 + (-12 + (-1 + sqrt(3)*I)*(290412 + + 3036*sqrt(9165))**(S(1)/3))*(1 - sqrt(3)*I)*(24201 + 253*sqrt(9165))** + (S(1)/3)/276)/((1 - sqrt(3)*I)*(24201 + 253*sqrt(9165))**(S(1)/3)) + )**n + C1*((86*18**(S(1)/3)/69 + (-12 + (-1 - sqrt(3)*I)*(290412 + 3036 + *sqrt(9165))**(S(1)/3))*(1 + sqrt(3)*I)*(24201 + 253*sqrt(9165))** + (S(1)/3)/276)/((1 + sqrt(3)*I)*(24201 + 253*sqrt(9165))**(S(1)/3)) + )**n + C2*(-43*18**(S(1)/3)/(69*(24201 + 253*sqrt(9165))**(S(1)/3)) - + S(1)/23 + (290412 + 3036*sqrt(9165))**(S(1)/3)/138)**n + assert sol == expected + e = sol.subs({C0: 1, C1: 1, C2: 1, n: 1}).evalf() + assert abs(e + 0.130434782608696) < 1e-13 + + +def test_issue_8697(): + a = Function('a') + eq = a(n + 3) - a(n + 2) - a(n + 1) + a(n) + assert rsolve(eq, a(n)) == (-1)**n*C1 + C0 + C2*n + eq2 = a(n + 3) + 3*a(n + 2) + 3*a(n + 1) + a(n) + assert (rsolve(eq2, a(n)) == + (-1)**n*C0 + (-1)**(n + 1)*C1*n + (-1)**(n + 1)*C2*n**2) + + assert rsolve(a(n) - 2*a(n - 3) + 5*a(n - 2) - 4*a(n - 1), + a(n), {a(0): 1, a(1): 3, a(2): 8}) == 3*2**n - n - 2 + + # From issue thread (but fixed by https://github.com/diofant/diofant/commit/da9789c6cd7d0c2ceeea19fbf59645987125b289): + assert rsolve(a(n) - 2*a(n - 1) - n, a(n), {a(0): 1}) == 3*2**n - n - 2 + + +def test_diofantissue_294(): + f = y(n) - y(n - 1) - 2*y(n - 2) - 2*n + assert rsolve(f, y(n)) == (-1)**n*C0 + 2**n*C1 - n - Rational(5, 2) + # issue sympy/sympy#11261 + assert rsolve(f, y(n), {y(0): -1, y(1): 1}) == (-(-1)**n/2 + 2*2**n - + n - Rational(5, 2)) + # issue sympy/sympy#7055 + assert rsolve(-2*y(n) + y(n + 1) + n - 1, y(n)) == 2**n*C0 + n + + +def test_issue_15553(): + f = Function("f") + assert rsolve(Eq(f(n), 2*f(n - 1) + n), f(n)) == 2**n*C0 - n - 2 + assert rsolve(Eq(f(n + 1), 2*f(n) + n**2 + 1), f(n)) == 2**n*C0 - n**2 - 2*n - 4 + assert rsolve(Eq(f(n + 1), 2*f(n) + n**2 + 1), f(n), {f(1): 0}) == 7*2**n/2 - n**2 - 2*n - 4 + assert rsolve(Eq(f(n), 2*f(n - 1) + 3*n**2), f(n)) == 2**n*C0 - 3*n**2 - 12*n - 18 + assert rsolve(Eq(f(n), 2*f(n - 1) + n**2), f(n)) == 2**n*C0 - n**2 - 4*n - 6 + assert rsolve(Eq(f(n), 2*f(n - 1) + n), f(n), {f(0): 1}) == 3*2**n - n - 2 diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/tests/test_simplex.py b/.venv/lib/python3.12/site-packages/sympy/solvers/tests/test_simplex.py new file mode 100644 index 0000000000000000000000000000000000000000..611205f5df009a6d0de6e687501695b63bb932c9 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/tests/test_simplex.py @@ -0,0 +1,254 @@ +from sympy.core.numbers import Rational +from sympy.core.relational import Eq, Ne +from sympy.core.symbol import symbols +from sympy.core.sympify import sympify +from sympy.core.singleton import S +from sympy.core.random import random, choice +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.ntheory.generate import randprime +from sympy.matrices.dense import Matrix +from sympy.solvers.solveset import linear_eq_to_matrix +from sympy.solvers.simplex import (_lp as lp, _primal_dual, + UnboundedLPError, InfeasibleLPError, lpmin, lpmax, + _m, _abcd, _simplex, linprog) + +from sympy.external.importtools import import_module + +from sympy.testing.pytest import raises + +from sympy.abc import x, y, z + + +np = import_module("numpy") +scipy = import_module("scipy") + + +def test_lp(): + r1 = y + 2*z <= 3 + r2 = -x - 3*z <= -2 + r3 = 2*x + y + 7*z <= 5 + constraints = [r1, r2, r3, x >= 0, y >= 0, z >= 0] + objective = -x - y - 5 * z + ans = optimum, argmax = lp(max, objective, constraints) + assert ans == lpmax(objective, constraints) + assert objective.subs(argmax) == optimum + for constr in constraints: + assert constr.subs(argmax) == True + + r1 = x - y + 2*z <= 3 + r2 = -x + 2*y - 3*z <= -2 + r3 = 2*x + y - 7*z <= -5 + constraints = [r1, r2, r3, x >= 0, y >= 0, z >= 0] + objective = -x - y - 5*z + ans = optimum, argmax = lp(max, objective, constraints) + assert ans == lpmax(objective, constraints) + assert objective.subs(argmax) == optimum + for constr in constraints: + assert constr.subs(argmax) == True + + r1 = x - y + 2*z <= -4 + r2 = -x + 2*y - 3*z <= 8 + r3 = 2*x + y - 7*z <= 10 + constraints = [r1, r2, r3, x >= 0, y >= 0, z >= 0] + const = 2 + objective = -x-y-5*z+const # has constant term + ans = optimum, argmax = lp(max, objective, constraints) + assert ans == lpmax(objective, constraints) + assert objective.subs(argmax) == optimum + for constr in constraints: + assert constr.subs(argmax) == True + + # Section 4 Problem 1 from + # http://web.tecnico.ulisboa.pt/mcasquilho/acad/or/ftp/FergusonUCLA_LP.pdf + # answer on page 55 + v = x1, x2, x3, x4 = symbols('x1 x2 x3 x4') + r1 = x1 - x2 - 2*x3 - x4 <= 4 + r2 = 2*x1 + x3 -4*x4 <= 2 + r3 = -2*x1 + x2 + x4 <= 1 + objective, constraints = x1 - 2*x2 - 3*x3 - x4, [r1, r2, r3] + [ + i >= 0 for i in v] + ans = optimum, argmax = lp(max, objective, constraints) + assert ans == lpmax(objective, constraints) + assert ans == (4, {x1: 7, x2: 0, x3: 0, x4: 3}) + + # input contains Floats + r1 = x - y + 2.0*z <= -4 + r2 = -x + 2*y - 3.0*z <= 8 + r3 = 2*x + y - 7*z <= 10 + constraints = [r1, r2, r3] + [i >= 0 for i in (x, y, z)] + objective = -x-y-5*z + optimum, argmax = lp(max, objective, constraints) + assert objective.subs(argmax) == optimum + for constr in constraints: + assert constr.subs(argmax) == True + + # input contains non-float or non-Rational + r1 = x - y + sqrt(2) * z <= -4 + r2 = -x + 2*y - 3*z <= 8 + r3 = 2*x + y - 7*z <= 10 + raises(TypeError, lambda: lp(max, -x-y-5*z, [r1, r2, r3])) + + r1 = x >= 0 + raises(UnboundedLPError, lambda: lp(max, x, [r1])) + r2 = x <= -1 + raises(InfeasibleLPError, lambda: lp(max, x, [r1, r2])) + + # strict inequalities are not allowed + r1 = x > 0 + raises(TypeError, lambda: lp(max, x, [r1])) + + # not equals not allowed + r1 = Ne(x, 0) + raises(TypeError, lambda: lp(max, x, [r1])) + + def make_random_problem(nvar=2, num_constraints=2, sparsity=.1): + def rand(): + if random() < sparsity: + return sympify(0) + int1, int2 = [randprime(0, 200) for _ in range(2)] + return Rational(int1, int2)*choice([-1, 1]) + variables = symbols('x1:%s' % (nvar + 1)) + constraints = [(sum(rand()*x for x in variables) <= rand()) + for _ in range(num_constraints)] + objective = sum(rand() * x for x in variables) + return objective, constraints, variables + + # equality + r1 = Eq(x, y) + r2 = Eq(y, z) + r3 = z <= 3 + constraints = [r1, r2, r3] + objective = x + ans = optimum, argmax = lp(max, objective, constraints) + assert ans == lpmax(objective, constraints) + assert objective.subs(argmax) == optimum + for constr in constraints: + assert constr.subs(argmax) == True + + +def test_simplex(): + L = [ + [[1, 1], [-1, 1], [0, 1], [-1, 0]], + [5, 1, 2, -1], + [[1, 1]], + [-1]] + A, B, C, D = _abcd(_m(*L), list=False) + assert _simplex(A, B, -C, -D) == (-6, [3, 2], [1, 0, 0, 0]) + assert _simplex(A, B, -C, -D, dual=True) == (-6, + [1, 0, 0, 0], [5, 0]) + + assert _simplex([[]],[],[[1]],[0]) == (0, [0], []) + + # handling of Eq (or Eq-like x<=y, x>=y conditions) + assert lpmax(x - y, [x <= y + 2, x >= y + 2, x >= 0, y >= 0] + ) == (2, {x: 2, y: 0}) + assert lpmax(x - y, [x <= y + 2, Eq(x, y + 2), x >= 0, y >= 0] + ) == (2, {x: 2, y: 0}) + assert lpmax(x - y, [x <= y + 2, Eq(x, 2)]) == (2, {x: 2, y: 0}) + assert lpmax(y, [Eq(y, 2)]) == (2, {y: 2}) + + # the conditions are equivalent to Eq(x, y + 2) + assert lpmin(y, [x <= y + 2, x >= y + 2, y >= 0] + ) == (0, {x: 2, y: 0}) + # equivalent to Eq(y, -2) + assert lpmax(y, [0 <= y + 2, 0 >= y + 2]) == (-2, {y: -2}) + assert lpmax(y, [0 <= y + 2, 0 >= y + 2, y <= 0] + ) == (-2, {y: -2}) + + # extra symbols symbols + assert lpmin(x, [y >= 1, x >= y]) == (1, {x: 1, y: 1}) + assert lpmin(x, [y >= 1, x >= y + z, x >= 0, z >= 0] + ) == (1, {x: 1, y: 1, z: 0}) + + # detect oscillation + # o1 + v = x1, x2, x3, x4 = symbols('x1 x2 x3 x4') + raises(InfeasibleLPError, lambda: lpmin( + 9*x2 - 8*x3 + 3*x4 + 6, + [5*x2 - 2*x3 <= 0, + -x1 - 8*x2 + 9*x3 <= -3, + 10*x1 - x2+ 9*x4 <= -4] + [i >= 0 for i in v])) + # o2 - equations fed to lpmin are changed into a matrix + # system that doesn't oscillate and has the same solution + # as below + M = linear_eq_to_matrix + f = 5*x2 + x3 + 4*x4 - x1 + L = 5*x2 + 2*x3 + 5*x4 - (x1 + 5) + cond = [L <= 0] + [Eq(3*x2 + x4, 2), Eq(-x1 + x3 + 2*x4, 1)] + c, d = M(f, v) + a, b = M(L, v) + aeq, beq = M(cond[1:], v) + ans = (S(9)/2, [0, S(1)/2, 0, S(1)/2]) + assert linprog(c, a, b, aeq, beq, bounds=(0, 1)) == ans + lpans = lpmin(f, cond + [x1 >= 0, x1 <= 1, + x2 >= 0, x2 <= 1, x3 >= 0, x3 <= 1, x4 >= 0, x4 <= 1]) + assert (lpans[0], list(lpans[1].values())) == ans + + +def test_lpmin_lpmax(): + v = x1, x2, y1, y2 = symbols('x1 x2 y1 y2') + L = [[1, -1]], [1], [[1, 1]], [2] + a, b, c, d = [Matrix(i) for i in L] + m = Matrix([[a, b], [c, d]]) + f, constr = _primal_dual(m)[0] + ans = lpmin(f, constr + [i >= 0 for i in v[:2]]) + assert ans == (-1, {x1: 1, x2: 0}),ans + + L = [[1, -1], [1, 1]], [1, 1], [[1, 1]], [2] + a, b, c, d = [Matrix(i) for i in L] + m = Matrix([[a, b], [c, d]]) + f, constr = _primal_dual(m)[1] + ans = lpmax(f, constr + [i >= 0 for i in v[-2:]]) + assert ans == (-1, {y1: 1, y2: 0}) + + +def test_linprog(): + for do in range(2): + if not do: + M = lambda a, b: linear_eq_to_matrix(a, b) + else: + # check matrices as list + M = lambda a, b: tuple([ + i.tolist() for i in linear_eq_to_matrix(a, b)]) + + v = x, y, z = symbols('x1:4') + f = x + y - 2*z + c = M(f, v)[0] + ineq = [7*x + 4*y - 7*z <= 3, + 3*x - y + 10*z <= 6, + x >= 0, y >= 0, z >= 0] + ab = M([i.lts - i.gts for i in ineq], v) + ans = (-S(6)/5, [0, 0, S(3)/5]) + assert lpmin(f, ineq) == (ans[0], dict(zip(v, ans[1]))) + assert linprog(c, *ab) == ans + + f += 1 + c = M(f, v)[0] + eq = [Eq(y - 9*x, 1)] + abeq = M([i.lhs - i.rhs for i in eq], v) + ans = (1 - S(2)/5, [0, 1, S(7)/10]) + assert lpmin(f, ineq + eq) == (ans[0], dict(zip(v, ans[1]))) + assert linprog(c, *ab, *abeq) == (ans[0] - 1, ans[1]) + + eq = [z - y <= S.Half] + abeq = M([i.lhs - i.rhs for i in eq], v) + ans = (1 - S(10)/9, [0, S(1)/9, S(11)/18]) + assert lpmin(f, ineq + eq) == (ans[0], dict(zip(v, ans[1]))) + assert linprog(c, *ab, *abeq) == (ans[0] - 1, ans[1]) + + bounds = [(0, None), (0, None), (None, S.Half)] + ans = (0, [0, 0, S.Half]) + assert lpmin(f, ineq + [z <= S.Half]) == ( + ans[0], dict(zip(v, ans[1]))) + assert linprog(c, *ab, bounds=bounds) == (ans[0] - 1, ans[1]) + assert linprog(c, *ab, bounds={v.index(z): bounds[-1]} + ) == (ans[0] - 1, ans[1]) + eq = [z - y <= S.Half] + + assert linprog([[1]], [], [], bounds=(2, 3)) == (2, [2]) + assert linprog([1], [], [], bounds=(2, 3)) == (2, [2]) + assert linprog([1], bounds=(2, 3)) == (2, [2]) + assert linprog([1, -1], [[1, 1]], [2], bounds={1:(None, None)} + ) == (-2, [0, 2]) + assert linprog([1, -1], [[1, 1]], [5], bounds={1:(3, None)} + ) == (-5, [0, 5]) diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/tests/test_solvers.py b/.venv/lib/python3.12/site-packages/sympy/solvers/tests/test_solvers.py new file mode 100644 index 0000000000000000000000000000000000000000..ac9550ad404c2ec7592caf6afd2910f425138987 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/tests/test_solvers.py @@ -0,0 +1,2725 @@ +from sympy.assumptions.ask import (Q, ask) +from sympy.core.add import Add +from sympy.core.containers import Tuple +from sympy.core.function import (Derivative, Function, diff) +from sympy.core.mod import Mod +from sympy.core.mul import Mul +from sympy.core import (GoldenRatio, TribonacciConstant) +from sympy.core.numbers import (E, Float, I, Rational, oo, pi) +from sympy.core.relational import (Eq, Gt, Lt, Ne) +from sympy.core.singleton import S +from sympy.core.symbol import (Dummy, Symbol, Wild, symbols) +from sympy.core.sympify import sympify +from sympy.functions.combinatorial.factorials import binomial +from sympy.functions.elementary.complexes import (Abs, arg, conjugate, im, re) +from sympy.functions.elementary.exponential import (LambertW, exp, log) +from sympy.functions.elementary.hyperbolic import (atanh, cosh, sinh, tanh) +from sympy.functions.elementary.integers import floor +from sympy.functions.elementary.miscellaneous import (cbrt, root, sqrt) +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.elementary.trigonometric import (acos, asin, atan, atan2, cos, sec, sin, tan) +from sympy.functions.special.error_functions import (erf, erfc, erfcinv, erfinv) +from sympy.integrals.integrals import Integral +from sympy.logic.boolalg import (And, Or) +from sympy.matrices.dense import Matrix +from sympy.matrices import MatrixSymbol, SparseMatrix +from sympy.polys.polytools import Poly, groebner +from sympy.printing.str import sstr +from sympy.simplify.radsimp import denom +from sympy.solvers.solvers import (nsolve, solve, solve_linear) + +from sympy.core.function import nfloat +from sympy.solvers import solve_linear_system, solve_linear_system_LU, \ + solve_undetermined_coeffs +from sympy.solvers.bivariate import _filtered_gens, _solve_lambert, _lambert +from sympy.solvers.solvers import _invert, unrad, checksol, posify, _ispow, \ + det_quick, det_perm, det_minor, _simple_dens, denoms + +from sympy.physics.units import cm +from sympy.polys.rootoftools import CRootOf + +from sympy.testing.pytest import slow, XFAIL, SKIP, raises +from sympy.core.random import verify_numerically as tn + +from sympy.abc import a, b, c, d, e, k, h, p, x, y, z, t, q, m, R + + +def NS(e, n=15, **options): + return sstr(sympify(e).evalf(n, **options), full_prec=True) + + +def test_swap_back(): + f, g = map(Function, 'fg') + fx, gx = f(x), g(x) + assert solve([fx + y - 2, fx - gx - 5], fx, y, gx) == \ + {fx: gx + 5, y: -gx - 3} + assert solve(fx + gx*x - 2, [fx, gx], dict=True) == [{fx: 2, gx: 0}] + assert solve(fx + gx**2*x - y, [fx, gx], dict=True) == [{fx: y, gx: 0}] + assert solve([f(1) - 2, x + 2], dict=True) == [{x: -2, f(1): 2}] + + +def guess_solve_strategy(eq, symbol): + try: + solve(eq, symbol) + return True + except (TypeError, NotImplementedError): + return False + + +def test_guess_poly(): + # polynomial equations + assert guess_solve_strategy( S(4), x ) # == GS_POLY + assert guess_solve_strategy( x, x ) # == GS_POLY + assert guess_solve_strategy( x + a, x ) # == GS_POLY + assert guess_solve_strategy( 2*x, x ) # == GS_POLY + assert guess_solve_strategy( x + sqrt(2), x) # == GS_POLY + assert guess_solve_strategy( x + 2**Rational(1, 4), x) # == GS_POLY + assert guess_solve_strategy( x**2 + 1, x ) # == GS_POLY + assert guess_solve_strategy( x**2 - 1, x ) # == GS_POLY + assert guess_solve_strategy( x*y + y, x ) # == GS_POLY + assert guess_solve_strategy( x*exp(y) + y, x) # == GS_POLY + assert guess_solve_strategy( + (x - y**3)/(y**2*sqrt(1 - y**2)), x) # == GS_POLY + + +def test_guess_poly_cv(): + # polynomial equations via a change of variable + assert guess_solve_strategy( sqrt(x) + 1, x ) # == GS_POLY_CV_1 + assert guess_solve_strategy( + x**Rational(1, 3) + sqrt(x) + 1, x ) # == GS_POLY_CV_1 + assert guess_solve_strategy( 4*x*(1 - sqrt(x)), x ) # == GS_POLY_CV_1 + + # polynomial equation multiplying both sides by x**n + assert guess_solve_strategy( x + 1/x + y, x ) # == GS_POLY_CV_2 + + +def test_guess_rational_cv(): + # rational functions + assert guess_solve_strategy( (x + 1)/(x**2 + 2), x) # == GS_RATIONAL + assert guess_solve_strategy( + (x - y**3)/(y**2*sqrt(1 - y**2)), y) # == GS_RATIONAL_CV_1 + + # rational functions via the change of variable y -> x**n + assert guess_solve_strategy( (sqrt(x) + 1)/(x**Rational(1, 3) + sqrt(x) + 1), x ) \ + #== GS_RATIONAL_CV_1 + + +def test_guess_transcendental(): + #transcendental functions + assert guess_solve_strategy( exp(x) + 1, x ) # == GS_TRANSCENDENTAL + assert guess_solve_strategy( 2*cos(x) - y, x ) # == GS_TRANSCENDENTAL + assert guess_solve_strategy( + exp(x) + exp(-x) - y, x ) # == GS_TRANSCENDENTAL + assert guess_solve_strategy(3**x - 10, x) # == GS_TRANSCENDENTAL + assert guess_solve_strategy(-3**x + 10, x) # == GS_TRANSCENDENTAL + + assert guess_solve_strategy(a*x**b - y, x) # == GS_TRANSCENDENTAL + + +def test_solve_args(): + # equation container, issue 5113 + ans = {x: -3, y: 1} + eqs = (x + 5*y - 2, -3*x + 6*y - 15) + assert all(solve(container(eqs), x, y) == ans for container in + (tuple, list, set, frozenset)) + assert solve(Tuple(*eqs), x, y) == ans + # implicit symbol to solve for + assert set(solve(x**2 - 4)) == {S(2), -S(2)} + assert solve([x + y - 3, x - y - 5]) == {x: 4, y: -1} + assert solve(x - exp(x), x, implicit=True) == [exp(x)] + # no symbol to solve for + assert solve(42) == solve(42, x) == [] + assert solve([1, 2]) == [] + assert solve([sqrt(2)],[x]) == [] + # duplicate symbols raises + raises(ValueError, lambda: solve((x - 3, y + 2), x, y, x)) + raises(ValueError, lambda: solve(x, x, x)) + # no error in exclude + assert solve(x, x, exclude=[y, y]) == [0] + # duplicate symbols raises + raises(ValueError, lambda: solve((x - 3, y + 2), x, y, x)) + raises(ValueError, lambda: solve(x, x, x)) + # no error in exclude + assert solve(x, x, exclude=[y, y]) == [0] + # unordered symbols + # only 1 + assert solve(y - 3, {y}) == [3] + # more than 1 + assert solve(y - 3, {x, y}) == [{y: 3}] + # multiple symbols: take the first linear solution+ + # - return as tuple with values for all requested symbols + assert solve(x + y - 3, [x, y]) == [(3 - y, y)] + # - unless dict is True + assert solve(x + y - 3, [x, y], dict=True) == [{x: 3 - y}] + # - or no symbols are given + assert solve(x + y - 3) == [{x: 3 - y}] + # multiple symbols might represent an undetermined coefficients system + assert solve(a + b*x - 2, [a, b]) == {a: 2, b: 0} + assert solve((a + b)*x + b - c, [a, b]) == {a: -c, b: c} + eq = a*x**2 + b*x + c - ((x - h)**2 + 4*p*k)/4/p + # - check that flags are obeyed + sol = solve(eq, [h, p, k], exclude=[a, b, c]) + assert sol == {h: -b/(2*a), k: (4*a*c - b**2)/(4*a), p: 1/(4*a)} + assert solve(eq, [h, p, k], dict=True) == [sol] + assert solve(eq, [h, p, k], set=True) == \ + ([h, p, k], {(-b/(2*a), 1/(4*a), (4*a*c - b**2)/(4*a))}) + # issue 23889 - polysys not simplified + assert solve(eq, [h, p, k], exclude=[a, b, c], simplify=False) == \ + {h: -b/(2*a), k: (4*a*c - b**2)/(4*a), p: 1/(4*a)} + # but this only happens when system has a single solution + args = (a + b)*x - b**2 + 2, a, b + assert solve(*args) == [((b**2 - b*x - 2)/x, b)] + # and if the system has a solution; the following doesn't so + # an algebraic solution is returned + assert solve(a*x + b**2/(x + 4) - 3*x - 4/x, a, b, dict=True) == \ + [{a: (-b**2*x + 3*x**3 + 12*x**2 + 4*x + 16)/(x**2*(x + 4))}] + # failed single equation + assert solve(1/(1/x - y + exp(y))) == [] + raises( + NotImplementedError, lambda: solve(exp(x) + sin(x) + exp(y) + sin(y))) + # failed system + # -- when no symbols given, 1 fails + assert solve([y, exp(x) + x]) == [{x: -LambertW(1), y: 0}] + # both fail + assert solve( + (exp(x) - x, exp(y) - y)) == [{x: -LambertW(-1), y: -LambertW(-1)}] + # -- when symbols given + assert solve([y, exp(x) + x], x, y) == [(-LambertW(1), 0)] + # symbol is a number + assert solve(x**2 - pi, pi) == [x**2] + # no equations + assert solve([], [x]) == [] + # nonlinear system + assert solve((x**2 - 4, y - 2), x, y) == [(-2, 2), (2, 2)] + assert solve((x**2 - 4, y - 2), y, x) == [(2, -2), (2, 2)] + assert solve((x**2 - 4 + z, y - 2 - z), a, z, y, x, set=True + ) == ([a, z, y, x], { + (a, z, z + 2, -sqrt(4 - z)), + (a, z, z + 2, sqrt(4 - z))}) + # overdetermined system + # - nonlinear + assert solve([(x + y)**2 - 4, x + y - 2]) == [{x: -y + 2}] + # - linear + assert solve((x + y - 2, 2*x + 2*y - 4)) == {x: -y + 2} + # When one or more args are Boolean + assert solve(Eq(x**2, 0.0)) == [0.0] # issue 19048 + assert solve([True, Eq(x, 0)], [x], dict=True) == [{x: 0}] + assert solve([Eq(x, x), Eq(x, 0), Eq(x, x+1)], [x], dict=True) == [] + assert not solve([Eq(x, x+1), x < 2], x) + assert solve([Eq(x, 0), x+1<2]) == Eq(x, 0) + assert solve([Eq(x, x), Eq(x, x+1)], x) == [] + assert solve(True, x) == [] + assert solve([x - 1, False], [x], set=True) == ([], set()) + assert solve([-y*(x + y - 1)/2, (y - 1)/x/y + 1/y], + set=True, check=False) == ([x, y], {(1 - y, y), (x, 0)}) + # ordering should be canonical, fastest to order by keys instead + # of by size + assert list(solve((y - 1, x - sqrt(3)*z)).keys()) == [x, y] + # as set always returns as symbols, set even if no solution + assert solve([x - 1, x], (y, x), set=True) == ([y, x], set()) + assert solve([x - 1, x], {y, x}, set=True) == ([x, y], set()) + + +def test_solve_polynomial1(): + assert solve(3*x - 2, x) == [Rational(2, 3)] + assert solve(Eq(3*x, 2), x) == [Rational(2, 3)] + + assert set(solve(x**2 - 1, x)) == {-S.One, S.One} + assert set(solve(Eq(x**2, 1), x)) == {-S.One, S.One} + + assert solve(x - y**3, x) == [y**3] + rx = root(x, 3) + assert solve(x - y**3, y) == [ + rx, -rx/2 - sqrt(3)*I*rx/2, -rx/2 + sqrt(3)*I*rx/2] + a11, a12, a21, a22, b1, b2 = symbols('a11,a12,a21,a22,b1,b2') + + assert solve([a11*x + a12*y - b1, a21*x + a22*y - b2], x, y) == \ + { + x: (a22*b1 - a12*b2)/(a11*a22 - a12*a21), + y: (a11*b2 - a21*b1)/(a11*a22 - a12*a21), + } + + solution = {x: S.Zero, y: S.Zero} + + assert solve((x - y, x + y), x, y ) == solution + assert solve((x - y, x + y), (x, y)) == solution + assert solve((x - y, x + y), [x, y]) == solution + + assert set(solve(x**3 - 15*x - 4, x)) == { + -2 + 3**S.Half, + S(4), + -2 - 3**S.Half + } + + assert set(solve((x**2 - 1)**2 - a, x)) == \ + {sqrt(1 + sqrt(a)), -sqrt(1 + sqrt(a)), + sqrt(1 - sqrt(a)), -sqrt(1 - sqrt(a))} + + +def test_solve_polynomial2(): + assert solve(4, x) == [] + + +def test_solve_polynomial_cv_1a(): + """ + Test for solving on equations that can be converted to a polynomial equation + using the change of variable y -> x**Rational(p, q) + """ + assert solve( sqrt(x) - 1, x) == [1] + assert solve( sqrt(x) - 2, x) == [4] + assert solve( x**Rational(1, 4) - 2, x) == [16] + assert solve( x**Rational(1, 3) - 3, x) == [27] + assert solve(sqrt(x) + x**Rational(1, 3) + x**Rational(1, 4), x) == [0] + + +def test_solve_polynomial_cv_1b(): + assert set(solve(4*x*(1 - a*sqrt(x)), x)) == {S.Zero, 1/a**2} + assert set(solve(x*(root(x, 3) - 3), x)) == {S.Zero, S(27)} + + +def test_solve_polynomial_cv_2(): + """ + Test for solving on equations that can be converted to a polynomial equation + multiplying both sides of the equation by x**m + """ + assert solve(x + 1/x - 1, x) in \ + [[ S.Half + I*sqrt(3)/2, S.Half - I*sqrt(3)/2], + [ S.Half - I*sqrt(3)/2, S.Half + I*sqrt(3)/2]] + + +def test_quintics_1(): + f = x**5 - 110*x**3 - 55*x**2 + 2310*x + 979 + s = solve(f, check=False) + for r in s: + res = f.subs(x, r.n()).n() + assert tn(res, 0) + + f = x**5 - 15*x**3 - 5*x**2 + 10*x + 20 + s = solve(f) + for r in s: + assert r.func == CRootOf + + # if one uses solve to get the roots of a polynomial that has a CRootOf + # solution, make sure that the use of nfloat during the solve process + # doesn't fail. Note: if you want numerical solutions to a polynomial + # it is *much* faster to use nroots to get them than to solve the + # equation only to get RootOf solutions which are then numerically + # evaluated. So for eq = x**5 + 3*x + 7 do Poly(eq).nroots() rather + # than [i.n() for i in solve(eq)] to get the numerical roots of eq. + assert nfloat(solve(x**5 + 3*x**3 + 7)[0], exponent=False) == \ + CRootOf(x**5 + 3*x**3 + 7, 0).n() + + +def test_quintics_2(): + f = x**5 + 15*x + 12 + s = solve(f, check=False) + for r in s: + res = f.subs(x, r.n()).n() + assert tn(res, 0) + + f = x**5 - 15*x**3 - 5*x**2 + 10*x + 20 + s = solve(f) + for r in s: + assert r.func == CRootOf + + assert solve(x**5 - 6*x**3 - 6*x**2 + x - 6) == [ + CRootOf(x**5 - 6*x**3 - 6*x**2 + x - 6, 0), + CRootOf(x**5 - 6*x**3 - 6*x**2 + x - 6, 1), + CRootOf(x**5 - 6*x**3 - 6*x**2 + x - 6, 2), + CRootOf(x**5 - 6*x**3 - 6*x**2 + x - 6, 3), + CRootOf(x**5 - 6*x**3 - 6*x**2 + x - 6, 4)] + +def test_quintics_3(): + y = x**5 + x**3 - 2**Rational(1, 3) + assert solve(y) == solve(-y) == [] + + +def test_highorder_poly(): + # just testing that the uniq generator is unpacked + sol = solve(x**6 - 2*x + 2) + assert all(isinstance(i, CRootOf) for i in sol) and len(sol) == 6 + + +def test_solve_rational(): + """Test solve for rational functions""" + assert solve( ( x - y**3 )/( (y**2)*sqrt(1 - y**2) ), x) == [y**3] + + +def test_solve_conjugate(): + """Test solve for simple conjugate functions""" + assert solve(conjugate(x) -3 + I) == [3 + I] + + +def test_solve_nonlinear(): + assert solve(x**2 - y**2, x, y, dict=True) == [{x: -y}, {x: y}] + assert solve(x**2 - y**2/exp(x), y, x, dict=True) == [{y: -x*sqrt(exp(x))}, + {y: x*sqrt(exp(x))}] + + +def test_issue_8666(): + x = symbols('x') + assert solve(Eq(x**2 - 1/(x**2 - 4), 4 - 1/(x**2 - 4)), x) == [] + assert solve(Eq(x + 1/x, 1/x), x) == [] + + +def test_issue_7228(): + assert solve(4**(2*(x**2) + 2*x) - 8, x) == [Rational(-3, 2), S.Half] + + +def test_issue_7190(): + assert solve(log(x-3) + log(x+3), x) == [sqrt(10)] + + +def test_issue_21004(): + x = symbols('x') + f = x/sqrt(x**2+1) + f_diff = f.diff(x) + assert solve(f_diff, x) == [] + + +def test_issue_24650(): + x = symbols('x') + r = solve(Eq(Piecewise((x, Eq(x, 0) | (x > 1))), 0)) + assert r == [0] + r = checksol(Eq(Piecewise((x, Eq(x, 0) | (x > 1))), 0), x, sol=0) + assert r is True + + +def test_linear_system(): + x, y, z, t, n = symbols('x, y, z, t, n') + + assert solve([x - 1, x - y, x - 2*y, y - 1], [x, y]) == [] + + assert solve([x - 1, x - y, x - 2*y, x - 1], [x, y]) == [] + assert solve([x - 1, x - 1, x - y, x - 2*y], [x, y]) == [] + + assert solve([x + 5*y - 2, -3*x + 6*y - 15], x, y) == {x: -3, y: 1} + + M = Matrix([[0, 0, n*(n + 1), (n + 1)**2, 0], + [n + 1, n + 1, -2*n - 1, -(n + 1), 0], + [-1, 0, 1, 0, 0]]) + + assert solve_linear_system(M, x, y, z, t) == \ + {x: t*(-n-1)/n, y: 0, z: t*(-n-1)/n} + + assert solve([x + y + z + t, -z - t], x, y, z, t) == {x: -y, z: -t} + + +@XFAIL +def test_linear_system_xfail(): + # https://github.com/sympy/sympy/issues/6420 + M = Matrix([[0, 15.0, 10.0, 700.0], + [1, 1, 1, 100.0], + [0, 10.0, 5.0, 200.0], + [-5.0, 0, 0, 0 ]]) + + assert solve_linear_system(M, x, y, z) == {x: 0, y: -60.0, z: 160.0} + + +def test_linear_system_function(): + a = Function('a') + assert solve([a(0, 0) + a(0, 1) + a(1, 0) + a(1, 1), -a(1, 0) - a(1, 1)], + a(0, 0), a(0, 1), a(1, 0), a(1, 1)) == {a(1, 0): -a(1, 1), a(0, 0): -a(0, 1)} + + +def test_linear_system_symbols_doesnt_hang_1(): + + def _mk_eqs(wy): + # Equations for fitting a wy*2 - 1 degree polynomial between two points, + # at end points derivatives are known up to order: wy - 1 + order = 2*wy - 1 + x, x0, x1 = symbols('x, x0, x1', real=True) + y0s = symbols('y0_:{}'.format(wy), real=True) + y1s = symbols('y1_:{}'.format(wy), real=True) + c = symbols('c_:{}'.format(order+1), real=True) + + expr = sum(coeff*x**o for o, coeff in enumerate(c)) + eqs = [] + for i in range(wy): + eqs.append(expr.diff(x, i).subs({x: x0}) - y0s[i]) + eqs.append(expr.diff(x, i).subs({x: x1}) - y1s[i]) + return eqs, c + + # + # The purpose of this test is just to see that these calls don't hang. The + # expressions returned are complicated so are not included here. Testing + # their correctness takes longer than solving the system. + # + + for n in range(1, 7+1): + eqs, c = _mk_eqs(n) + solve(eqs, c) + + +def test_linear_system_symbols_doesnt_hang_2(): + + M = Matrix([ + [66, 24, 39, 50, 88, 40, 37, 96, 16, 65, 31, 11, 37, 72, 16, 19, 55, 37, 28, 76], + [10, 93, 34, 98, 59, 44, 67, 74, 74, 94, 71, 61, 60, 23, 6, 2, 57, 8, 29, 78], + [19, 91, 57, 13, 64, 65, 24, 53, 77, 34, 85, 58, 87, 39, 39, 7, 36, 67, 91, 3], + [74, 70, 15, 53, 68, 43, 86, 83, 81, 72, 25, 46, 67, 17, 59, 25, 78, 39, 63, 6], + [69, 40, 67, 21, 67, 40, 17, 13, 93, 44, 46, 89, 62, 31, 30, 38, 18, 20, 12, 81], + [50, 22, 74, 76, 34, 45, 19, 76, 28, 28, 11, 99, 97, 82, 8, 46, 99, 57, 68, 35], + [58, 18, 45, 88, 10, 64, 9, 34, 90, 82, 17, 41, 43, 81, 45, 83, 22, 88, 24, 39], + [42, 21, 70, 68, 6, 33, 64, 81, 83, 15, 86, 75, 86, 17, 77, 34, 62, 72, 20, 24], + [ 7, 8, 2, 72, 71, 52, 96, 5, 32, 51, 31, 36, 79, 88, 25, 77, 29, 26, 33, 13], + [19, 31, 30, 85, 81, 39, 63, 28, 19, 12, 16, 49, 37, 66, 38, 13, 3, 71, 61, 51], + [29, 82, 80, 49, 26, 85, 1, 37, 2, 74, 54, 82, 26, 47, 54, 9, 35, 0, 99, 40], + [15, 49, 82, 91, 93, 57, 45, 25, 45, 97, 15, 98, 48, 52, 66, 24, 62, 54, 97, 37], + [62, 23, 73, 53, 52, 86, 28, 38, 0, 74, 92, 38, 97, 70, 71, 29, 26, 90, 67, 45], + [ 2, 32, 23, 24, 71, 37, 25, 71, 5, 41, 97, 65, 93, 13, 65, 45, 25, 88, 69, 50], + [40, 56, 1, 29, 79, 98, 79, 62, 37, 28, 45, 47, 3, 1, 32, 74, 98, 35, 84, 32], + [33, 15, 87, 79, 65, 9, 14, 63, 24, 19, 46, 28, 74, 20, 29, 96, 84, 91, 93, 1], + [97, 18, 12, 52, 1, 2, 50, 14, 52, 76, 19, 82, 41, 73, 51, 79, 13, 3, 82, 96], + [40, 28, 52, 10, 10, 71, 56, 78, 82, 5, 29, 48, 1, 26, 16, 18, 50, 76, 86, 52], + [38, 89, 83, 43, 29, 52, 90, 77, 57, 0, 67, 20, 81, 88, 48, 96, 88, 58, 14, 3]]) + + syms = x0,x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18 = symbols('x:19') + + sol = { + x0: -S(1967374186044955317099186851240896179)/3166636564687820453598895768302256588, + x1: -S(84268280268757263347292368432053826)/791659141171955113399723942075564147, + x2: -S(229962957341664730974463872411844965)/1583318282343910226799447884151128294, + x3: S(990156781744251750886760432229180537)/6333273129375640907197791536604513176, + x4: -S(2169830351210066092046760299593096265)/18999819388126922721593374609813539528, + x5: S(4680868883477577389628494526618745355)/9499909694063461360796687304906769764, + x6: -S(1590820774344371990683178396480879213)/3166636564687820453598895768302256588, + x7: -S(54104723404825537735226491634383072)/339282489073695048599881689460956063, + x8: S(3182076494196560075964847771774733847)/6333273129375640907197791536604513176, + x9: -S(10870817431029210431989147852497539675)/18999819388126922721593374609813539528, + x10: -S(13118019242576506476316318268573312603)/18999819388126922721593374609813539528, + x11: -S(5173852969886775824855781403820641259)/4749954847031730680398343652453384882, + x12: S(4261112042731942783763341580651820563)/4749954847031730680398343652453384882, + x13: -S(821833082694661608993818117038209051)/6333273129375640907197791536604513176, + x14: S(906881575107250690508618713632090559)/904753304196520129599684505229216168, + x15: -S(732162528717458388995329317371283987)/6333273129375640907197791536604513176, + x16: S(4524215476705983545537087360959896817)/9499909694063461360796687304906769764, + x17: -S(3898571347562055611881270844646055217)/6333273129375640907197791536604513176, + x18: S(7513502486176995632751685137907442269)/18999819388126922721593374609813539528 + } + + eqs = list(M * Matrix(syms + (1,))) + assert solve(eqs, syms) == sol + + y = Symbol('y') + eqs = list(y * M * Matrix(syms + (1,))) + assert solve(eqs, syms) == sol + + +def test_linear_systemLU(): + n = Symbol('n') + + M = Matrix([[1, 2, 0, 1], [1, 3, 2*n, 1], [4, -1, n**2, 1]]) + + assert solve_linear_system_LU(M, [x, y, z]) == {z: -3/(n**2 + 18*n), + x: 1 - 12*n/(n**2 + 18*n), + y: 6*n/(n**2 + 18*n)} + +# Note: multiple solutions exist for some of these equations, so the tests +# should be expected to break if the implementation of the solver changes +# in such a way that a different branch is chosen + +@slow +def test_solve_transcendental(): + from sympy.abc import a, b + + assert solve(exp(x) - 3, x) == [log(3)] + assert set(solve((a*x + b)*(exp(x) - 3), x)) == {-b/a, log(3)} + assert solve(cos(x) - y, x) == [-acos(y) + 2*pi, acos(y)] + assert solve(2*cos(x) - y, x) == [-acos(y/2) + 2*pi, acos(y/2)] + assert solve(Eq(cos(x), sin(x)), x) == [pi/4] + + assert set(solve(exp(x) + exp(-x) - y, x)) in [{ + log(y/2 - sqrt(y**2 - 4)/2), + log(y/2 + sqrt(y**2 - 4)/2), + }, { + log(y - sqrt(y**2 - 4)) - log(2), + log(y + sqrt(y**2 - 4)) - log(2)}, + { + log(y/2 - sqrt((y - 2)*(y + 2))/2), + log(y/2 + sqrt((y - 2)*(y + 2))/2)}] + assert solve(exp(x) - 3, x) == [log(3)] + assert solve(Eq(exp(x), 3), x) == [log(3)] + assert solve(log(x) - 3, x) == [exp(3)] + assert solve(sqrt(3*x) - 4, x) == [Rational(16, 3)] + assert solve(3**(x + 2), x) == [] + assert solve(3**(2 - x), x) == [] + assert solve(x + 2**x, x) == [-LambertW(log(2))/log(2)] + assert solve(2*x + 5 + log(3*x - 2), x) == \ + [Rational(2, 3) + LambertW(2*exp(Rational(-19, 3))/3)/2] + assert solve(3*x + log(4*x), x) == [LambertW(Rational(3, 4))/3] + assert set(solve((2*x + 8)*(8 + exp(x)), x)) == {S(-4), log(8) + pi*I} + eq = 2*exp(3*x + 4) - 3 + ans = solve(eq, x) # this generated a failure in flatten + assert len(ans) == 3 and all(eq.subs(x, a).n(chop=True) == 0 for a in ans) + assert solve(2*log(3*x + 4) - 3, x) == [(exp(Rational(3, 2)) - 4)/3] + assert solve(exp(x) + 1, x) == [pi*I] + + eq = 2*(3*x + 4)**5 - 6*7**(3*x + 9) + result = solve(eq, x) + x0 = -log(2401) + x1 = 3**Rational(1, 5) + x2 = log(7**(7*x1/20)) + x3 = sqrt(2) + x4 = sqrt(5) + x5 = x3*sqrt(x4 - 5) + x6 = x4 + 1 + x7 = 1/(3*log(7)) + x8 = -x4 + x9 = x3*sqrt(x8 - 5) + x10 = x8 + 1 + ans = [x7*(x0 - 5*LambertW(x2*(-x5 + x6))), + x7*(x0 - 5*LambertW(x2*(x5 + x6))), + x7*(x0 - 5*LambertW(x2*(x10 - x9))), + x7*(x0 - 5*LambertW(x2*(x10 + x9))), + x7*(x0 - 5*LambertW(-log(7**(7*x1/5))))] + assert result == ans, result + # it works if expanded, too + assert solve(eq.expand(), x) == result + + assert solve(z*cos(x) - y, x) == [-acos(y/z) + 2*pi, acos(y/z)] + assert solve(z*cos(2*x) - y, x) == [-acos(y/z)/2 + pi, acos(y/z)/2] + assert solve(z*cos(sin(x)) - y, x) == [ + pi - asin(acos(y/z)), asin(acos(y/z) - 2*pi) + pi, + -asin(acos(y/z) - 2*pi), asin(acos(y/z))] + + assert solve(z*cos(x), x) == [pi/2, pi*Rational(3, 2)] + + # issue 4508 + assert solve(y - b*x/(a + x), x) in [[-a*y/(y - b)], [a*y/(b - y)]] + assert solve(y - b*exp(a/x), x) == [a/log(y/b)] + # issue 4507 + assert solve(y - b/(1 + a*x), x) in [[(b - y)/(a*y)], [-((y - b)/(a*y))]] + # issue 4506 + assert solve(y - a*x**b, x) == [(y/a)**(1/b)] + # issue 4505 + assert solve(z**x - y, x) == [log(y)/log(z)] + # issue 4504 + assert solve(2**x - 10, x) == [1 + log(5)/log(2)] + # issue 6744 + assert solve(x*y) == [{x: 0}, {y: 0}] + assert solve([x*y]) == [{x: 0}, {y: 0}] + assert solve(x**y - 1) == [{x: 1}, {y: 0}] + assert solve([x**y - 1]) == [{x: 1}, {y: 0}] + assert solve(x*y*(x**2 - y**2)) == [{x: 0}, {x: -y}, {x: y}, {y: 0}] + assert solve([x*y*(x**2 - y**2)]) == [{x: 0}, {x: -y}, {x: y}, {y: 0}] + # issue 4739 + assert solve(exp(log(5)*x) - 2**x, x) == [0] + # issue 14791 + assert solve(exp(log(5)*x) - exp(log(2)*x), x) == [0] + f = Function('f') + assert solve(y*f(log(5)*x) - y*f(log(2)*x), x) == [0] + assert solve(f(x) - f(0), x) == [0] + assert solve(f(x) - f(2 - x), x) == [1] + raises(NotImplementedError, lambda: solve(f(x, y) - f(1, 2), x)) + raises(NotImplementedError, lambda: solve(f(x, y) - f(2 - x, 2), x)) + raises(ValueError, lambda: solve(f(x, y) - f(1 - x), x)) + raises(ValueError, lambda: solve(f(x, y) - f(1), x)) + + # misc + # make sure that the right variables is picked up in tsolve + # shouldn't generate a GeneratorsNeeded error in _tsolve when the NaN is generated + # for eq_down. Actual answers, as determined numerically are approx. +/- 0.83 + raises(NotImplementedError, lambda: + solve(sinh(x)*sinh(sinh(x)) + cosh(x)*cosh(sinh(x)) - 3)) + + # watch out for recursive loop in tsolve + raises(NotImplementedError, lambda: solve((x + 2)**y*x - 3, x)) + + # issue 7245 + assert solve(sin(sqrt(x))) == [0, pi**2] + + # issue 7602 + a, b = symbols('a, b', real=True, negative=False) + assert str(solve(Eq(a, 0.5 - cos(pi*b)/2), b)) == \ + '[2.0 - 0.318309886183791*acos(1.0 - 2.0*a), 0.318309886183791*acos(1.0 - 2.0*a)]' + + # issue 15325 + assert solve(y**(1/x) - z, x) == [log(y)/log(z)] + + # issue 25685 (basic trig identities should give simple solutions) + for yi in [cos(2*x),sin(2*x),cos(x - pi/3)]: + sol = solve([cos(x) - S(3)/5, yi - y]) + assert (sol[0][y] + sol[1][y]).is_Rational, (yi,sol) + # don't allow massive expansion + assert solve(cos(1000*x) - S.Half) == [pi/3000, pi/600] + assert solve(cos(x - 1000*y) - 1, x) == [1000*y, 1000*y + 2*pi] + assert solve(cos(x + y + z) - 1, x) == [-y - z, -y - z + 2*pi] + + # issue 26008 + assert solve(sin(x + pi/6)) == [-pi/6, 5*pi/6] + + +def test_solve_for_functions_derivatives(): + t = Symbol('t') + x = Function('x')(t) + y = Function('y')(t) + a11, a12, a21, a22, b1, b2 = symbols('a11,a12,a21,a22,b1,b2') + + soln = solve([a11*x + a12*y - b1, a21*x + a22*y - b2], x, y) + assert soln == { + x: (a22*b1 - a12*b2)/(a11*a22 - a12*a21), + y: (a11*b2 - a21*b1)/(a11*a22 - a12*a21), + } + + assert solve(x - 1, x) == [1] + assert solve(3*x - 2, x) == [Rational(2, 3)] + + soln = solve([a11*x.diff(t) + a12*y.diff(t) - b1, a21*x.diff(t) + + a22*y.diff(t) - b2], x.diff(t), y.diff(t)) + assert soln == { y.diff(t): (a11*b2 - a21*b1)/(a11*a22 - a12*a21), + x.diff(t): (a22*b1 - a12*b2)/(a11*a22 - a12*a21) } + + assert solve(x.diff(t) - 1, x.diff(t)) == [1] + assert solve(3*x.diff(t) - 2, x.diff(t)) == [Rational(2, 3)] + + eqns = {3*x - 1, 2*y - 4} + assert solve(eqns, {x, y}) == { x: Rational(1, 3), y: 2 } + x = Symbol('x') + f = Function('f') + F = x**2 + f(x)**2 - 4*x - 1 + assert solve(F.diff(x), diff(f(x), x)) == [(-x + 2)/f(x)] + + # Mixed cased with a Symbol and a Function + x = Symbol('x') + y = Function('y')(t) + + soln = solve([a11*x + a12*y.diff(t) - b1, a21*x + + a22*y.diff(t) - b2], x, y.diff(t)) + assert soln == { y.diff(t): (a11*b2 - a21*b1)/(a11*a22 - a12*a21), + x: (a22*b1 - a12*b2)/(a11*a22 - a12*a21) } + + # issue 13263 + x = Symbol('x') + f = Function('f') + soln = solve([f(x).diff(x) + f(x).diff(x, 2) - 1, f(x).diff(x) - f(x).diff(x, 2)], + f(x).diff(x), f(x).diff(x, 2)) + assert soln == { f(x).diff(x, 2): S(1)/2, f(x).diff(x): S(1)/2 } + + soln = solve([f(x).diff(x, 2) + f(x).diff(x, 3) - 1, 1 - f(x).diff(x, 2) - + f(x).diff(x, 3), 1 - f(x).diff(x,3)], f(x).diff(x, 2), f(x).diff(x, 3)) + assert soln == { f(x).diff(x, 2): 0, f(x).diff(x, 3): 1 } + + +def test_issue_3725(): + f = Function('f') + F = x**2 + f(x)**2 - 4*x - 1 + e = F.diff(x) + assert solve(e, f(x).diff(x)) in [[(2 - x)/f(x)], [-((x - 2)/f(x))]] + + +def test_solve_Matrix(): + # https://github.com/sympy/sympy/issues/3870 + a, b, c, d = symbols('a b c d') + A = Matrix(2, 2, [a, b, c, d]) + B = Matrix(2, 2, [0, 2, -3, 0]) + C = Matrix(2, 2, [1, 2, 3, 4]) + + assert solve(A*B - C, [a, b, c, d]) == {a: 1, b: Rational(-1, 3), c: 2, d: -1} + assert solve([A*B - C], [a, b, c, d]) == {a: 1, b: Rational(-1, 3), c: 2, d: -1} + assert solve(Eq(A*B, C), [a, b, c, d]) == {a: 1, b: Rational(-1, 3), c: 2, d: -1} + + assert solve([A*B - B*A], [a, b, c, d]) == {a: d, b: Rational(-2, 3)*c} + assert solve([A*C - C*A], [a, b, c, d]) == {a: d - c, b: Rational(2, 3)*c} + assert solve([A*B - B*A, A*C - C*A], [a, b, c, d]) == {a: d, b: 0, c: 0} + + assert solve([Eq(A*B, B*A)], [a, b, c, d]) == {a: d, b: Rational(-2, 3)*c} + assert solve([Eq(A*C, C*A)], [a, b, c, d]) == {a: d - c, b: Rational(2, 3)*c} + assert solve([Eq(A*B, B*A), Eq(A*C, C*A)], [a, b, c, d]) == {a: d, b: 0, c: 0} + + # https://github.com/sympy/sympy/issues/27854 + m, n = symbols("m n") + A = MatrixSymbol("A", m, n) + x = MatrixSymbol("x", n, 1) + b = MatrixSymbol('b', m, 1) + r = A * x - b + f = r.T * r + grad_f = f.diff(x) + raises(ValueError, lambda: solve(grad_f, x)) + + +def test_solve_linear(): + w = Wild('w') + assert solve_linear(x, x) == (0, 1) + assert solve_linear(x, exclude=[x]) == (0, 1) + assert solve_linear(x, symbols=[w]) == (0, 1) + assert solve_linear(x, y - 2*x) in [(x, y/3), (y, 3*x)] + assert solve_linear(x, y - 2*x, exclude=[x]) == (y, 3*x) + assert solve_linear(3*x - y, 0) in [(x, y/3), (y, 3*x)] + assert solve_linear(3*x - y, 0, [x]) == (x, y/3) + assert solve_linear(3*x - y, 0, [y]) == (y, 3*x) + assert solve_linear(x**2/y, 1) == (y, x**2) + assert solve_linear(w, x) in [(w, x), (x, w)] + assert solve_linear(cos(x)**2 + sin(x)**2 + 2 + y) == \ + (y, -2 - cos(x)**2 - sin(x)**2) + assert solve_linear(cos(x)**2 + sin(x)**2 + 2 + y, symbols=[x]) == (0, 1) + assert solve_linear(Eq(x, 3)) == (x, 3) + assert solve_linear(1/(1/x - 2)) == (0, 0) + assert solve_linear((x + 1)*exp(-x), symbols=[x]) == (x, -1) + assert solve_linear((x + 1)*exp(x), symbols=[x]) == ((x + 1)*exp(x), 1) + assert solve_linear(x*exp(-x**2), symbols=[x]) == (x, 0) + assert solve_linear(0**x - 1) == (0**x - 1, 1) + assert solve_linear(1 + 1/(x - 1)) == (x, 0) + eq = y*cos(x)**2 + y*sin(x)**2 - y # = y*(1 - 1) = 0 + assert solve_linear(eq) == (0, 1) + eq = cos(x)**2 + sin(x)**2 # = 1 + assert solve_linear(eq) == (0, 1) + raises(ValueError, lambda: solve_linear(Eq(x, 3), 3)) + + +def test_solve_undetermined_coeffs(): + assert solve_undetermined_coeffs( + a*x**2 + b*x**2 + b*x + 2*c*x + c + 1, [a, b, c], x + ) == {a: -2, b: 2, c: -1} + # Test that rational functions work + assert solve_undetermined_coeffs(a/x + b/(x + 1) + - (2*x + 1)/(x**2 + x), [a, b], x) == {a: 1, b: 1} + # Test cancellation in rational functions + assert solve_undetermined_coeffs( + ((c + 1)*a*x**2 + (c + 1)*b*x**2 + + (c + 1)*b*x + (c + 1)*2*c*x + (c + 1)**2)/(c + 1), + [a, b, c], x) == \ + {a: -2, b: 2, c: -1} + # multivariate + X, Y, Z = y, x**y, y*x**y + eq = a*X + b*Y + c*Z - X - 2*Y - 3*Z + coeffs = a, b, c + syms = x, y + assert solve_undetermined_coeffs(eq, coeffs) == { + a: 1, b: 2, c: 3} + assert solve_undetermined_coeffs(eq, coeffs, syms) == { + a: 1, b: 2, c: 3} + assert solve_undetermined_coeffs(eq, coeffs, *syms) == { + a: 1, b: 2, c: 3} + # check output format + assert solve_undetermined_coeffs(a*x + a - 2, [a]) == [] + assert solve_undetermined_coeffs(a**2*x - 4*x, [a]) == [ + {a: -2}, {a: 2}] + assert solve_undetermined_coeffs(0, [a]) == [] + assert solve_undetermined_coeffs(0, [a], dict=True) == [] + assert solve_undetermined_coeffs(0, [a], set=True) == ([], {}) + assert solve_undetermined_coeffs(1, [a]) == [] + abeq = a*x - 2*x + b - 3 + s = {b, a} + assert solve_undetermined_coeffs(abeq, s, x) == {a: 2, b: 3} + assert solve_undetermined_coeffs(abeq, s, x, set=True) == ([a, b], {(2, 3)}) + assert solve_undetermined_coeffs(sin(a*x) - sin(2*x), (a,)) is None + assert solve_undetermined_coeffs(a*x + b*x - 2*x, (a, b)) == {a: 2 - b} + + +def test_solve_inequalities(): + x = Symbol('x') + sol = And(S.Zero < x, x < oo) + assert solve(x + 1 > 1) == sol + assert solve([x + 1 > 1]) == sol + assert solve([x + 1 > 1], x) == sol + assert solve([x + 1 > 1], [x]) == sol + + system = [Lt(x**2 - 2, 0), Gt(x**2 - 1, 0)] + assert solve(system) == \ + And(Or(And(Lt(-sqrt(2), x), Lt(x, -1)), + And(Lt(1, x), Lt(x, sqrt(2)))), Eq(0, 0)) + + x = Symbol('x', real=True) + system = [Lt(x**2 - 2, 0), Gt(x**2 - 1, 0)] + assert solve(system) == \ + Or(And(Lt(-sqrt(2), x), Lt(x, -1)), And(Lt(1, x), Lt(x, sqrt(2)))) + + # issues 6627, 3448 + assert solve((x - 3)/(x - 2) < 0, x) == And(Lt(2, x), Lt(x, 3)) + assert solve(x/(x + 1) > 1, x) == And(Lt(-oo, x), Lt(x, -1)) + + assert solve(sin(x) > S.Half) == And(pi/6 < x, x < pi*Rational(5, 6)) + + assert solve(Eq(False, x < 1)) == (S.One <= x) & (x < oo) + assert solve(Eq(True, x < 1)) == (-oo < x) & (x < 1) + assert solve(Eq(x < 1, False)) == (S.One <= x) & (x < oo) + assert solve(Eq(x < 1, True)) == (-oo < x) & (x < 1) + + assert solve(Eq(False, x)) == False + assert solve(Eq(0, x)) == [0] + assert solve(Eq(True, x)) == True + assert solve(Eq(1, x)) == [1] + assert solve(Eq(False, ~x)) == True + assert solve(Eq(True, ~x)) == False + assert solve(Ne(True, x)) == False + assert solve(Ne(1, x)) == (x > -oo) & (x < oo) & Ne(x, 1) + + +def test_issue_4793(): + assert solve(1/x) == [] + assert solve(x*(1 - 5/x)) == [5] + assert solve(x + sqrt(x) - 2) == [1] + assert solve(-(1 + x)/(2 + x)**2 + 1/(2 + x)) == [] + assert solve(-x**2 - 2*x + (x + 1)**2 - 1) == [] + assert solve((x/(x + 1) + 3)**(-2)) == [] + assert solve(x/sqrt(x**2 + 1), x) == [0] + assert solve(exp(x) - y, x) == [log(y)] + assert solve(exp(x)) == [] + assert solve(x**2 + x + sin(y)**2 + cos(y)**2 - 1, x) in [[0, -1], [-1, 0]] + eq = 4*3**(5*x + 2) - 7 + ans = solve(eq, x) + assert len(ans) == 5 and all(eq.subs(x, a).n(chop=True) == 0 for a in ans) + assert solve(log(x**2) - y**2/exp(x), x, y, set=True) == ( + [x, y], + {(x, sqrt(exp(x) * log(x ** 2))), (x, -sqrt(exp(x) * log(x ** 2)))}) + assert solve(x**2*z**2 - z**2*y**2) == [{x: -y}, {x: y}, {z: 0}] + assert solve((x - 1)/(1 + 1/(x - 1))) == [] + assert solve(x**(y*z) - x, x) == [1] + raises(NotImplementedError, lambda: solve(log(x) - exp(x), x)) + raises(NotImplementedError, lambda: solve(2**x - exp(x) - 3)) + + +def test_PR1964(): + # issue 5171 + assert solve(sqrt(x)) == solve(sqrt(x**3)) == [0] + assert solve(sqrt(x - 1)) == [1] + # issue 4462 + a = Symbol('a') + assert solve(-3*a/sqrt(x), x) == [] + # issue 4486 + assert solve(2*x/(x + 2) - 1, x) == [2] + # issue 4496 + assert set(solve((x**2/(7 - x)).diff(x))) == {S.Zero, S(14)} + # issue 4695 + f = Function('f') + assert solve((3 - 5*x/f(x))*f(x), f(x)) == [x*Rational(5, 3)] + # issue 4497 + assert solve(1/root(5 + x, 5) - 9, x) == [Rational(-295244, 59049)] + + assert solve(sqrt(x) + sqrt(sqrt(x)) - 4) == [(Rational(-1, 2) + sqrt(17)/2)**4] + assert set(solve(Poly(sqrt(exp(x)) + sqrt(exp(-x)) - 4))) in \ + [ + {log((-sqrt(3) + 2)**2), log((sqrt(3) + 2)**2)}, + {2*log(-sqrt(3) + 2), 2*log(sqrt(3) + 2)}, + {log(-4*sqrt(3) + 7), log(4*sqrt(3) + 7)}, + ] + assert set(solve(Poly(exp(x) + exp(-x) - 4))) == \ + {log(-sqrt(3) + 2), log(sqrt(3) + 2)} + assert set(solve(x**y + x**(2*y) - 1, x)) == \ + {(Rational(-1, 2) + sqrt(5)/2)**(1/y), (Rational(-1, 2) - sqrt(5)/2)**(1/y)} + + assert solve(exp(x/y)*exp(-z/y) - 2, y) == [(x - z)/log(2)] + assert solve( + x**z*y**z - 2, z) in [[log(2)/(log(x) + log(y))], [log(2)/(log(x*y))]] + # if you do inversion too soon then multiple roots (as for the following) + # will be missed, e.g. if exp(3*x) = exp(3) -> 3*x = 3 + E = S.Exp1 + assert solve(exp(3*x) - exp(3), x) in [ + [1, log(E*(Rational(-1, 2) - sqrt(3)*I/2)), log(E*(Rational(-1, 2) + sqrt(3)*I/2))], + [1, log(-E/2 - sqrt(3)*E*I/2), log(-E/2 + sqrt(3)*E*I/2)], + ] + + # coverage test + p = Symbol('p', positive=True) + assert solve((1/p + 1)**(p + 1)) == [] + + +def test_issue_5197(): + x = Symbol('x', real=True) + assert solve(x**2 + 1, x) == [] + n = Symbol('n', integer=True, positive=True) + assert solve((n - 1)*(n + 2)*(2*n - 1), n) == [1] + x = Symbol('x', positive=True) + y = Symbol('y') + assert solve([x + 5*y - 2, -3*x + 6*y - 15], x, y) == [] + # not {x: -3, y: 1} b/c x is positive + # The solution following should not contain (-sqrt(2), sqrt(2)) + assert solve([(x + y), 2 - y**2], x, y) == [(sqrt(2), -sqrt(2))] + y = Symbol('y', positive=True) + # The solution following should not contain {y: -x*exp(x/2)} + assert solve(x**2 - y**2/exp(x), y, x, dict=True) == [{y: x*exp(x/2)}] + x, y, z = symbols('x y z', positive=True) + assert solve(z**2*x**2 - z**2*y**2/exp(x), y, x, z, dict=True) == [{y: x*exp(x/2)}] + + +def test_checking(): + assert set( + solve(x*(x - y/x), x, check=False)) == {sqrt(y), S.Zero, -sqrt(y)} + assert set(solve(x*(x - y/x), x, check=True)) == {sqrt(y), -sqrt(y)} + # {x: 0, y: 4} sets denominator to 0 in the following so system should return None + assert solve((1/(1/x + 2), 1/(y - 3) - 1)) == [] + # 0 sets denominator of 1/x to zero so None is returned + assert solve(1/(1/x + 2)) == [] + + +def test_issue_4671_4463_4467(): + assert solve(sqrt(x**2 - 1) - 2) in ([sqrt(5), -sqrt(5)], + [-sqrt(5), sqrt(5)]) + assert solve((2**exp(y**2/x) + 2)/(x**2 + 15), y) == [ + -sqrt(x*log(1 + I*pi/log(2))), sqrt(x*log(1 + I*pi/log(2)))] + + C1, C2 = symbols('C1 C2') + f = Function('f') + assert solve(C1 + C2/x**2 - exp(-f(x)), f(x)) == [log(x**2/(C1*x**2 + C2))] + a = Symbol('a') + E = S.Exp1 + assert solve(1 - log(a + 4*x**2), x) in ( + [-sqrt(-a + E)/2, sqrt(-a + E)/2], + [sqrt(-a + E)/2, -sqrt(-a + E)/2] + ) + assert solve(log(a**(-3) - x**2)/a, x) in ( + [-sqrt(-1 + a**(-3)), sqrt(-1 + a**(-3))], + [sqrt(-1 + a**(-3)), -sqrt(-1 + a**(-3))],) + assert solve(1 - log(a + 4*x**2), x) in ( + [-sqrt(-a + E)/2, sqrt(-a + E)/2], + [sqrt(-a + E)/2, -sqrt(-a + E)/2],) + assert solve((a**2 + 1)*(sin(a*x) + cos(a*x)), x) == [-pi/(4*a)] + assert solve(3 - (sinh(a*x) + cosh(a*x)), x) == [log(3)/a] + assert set(solve(3 - (sinh(a*x) + cosh(a*x)**2), x)) == \ + {log(-2 + sqrt(5))/a, log(-sqrt(2) + 1)/a, + log(-sqrt(5) - 2)/a, log(1 + sqrt(2))/a} + assert solve(atan(x) - 1) == [tan(1)] + + +def test_issue_5132(): + r, t = symbols('r,t') + assert set(solve([r - x**2 - y**2, tan(t) - y/x], [x, y])) == \ + {( + -sqrt(r*cos(t)**2), -1*sqrt(r*cos(t)**2)*tan(t)), + (sqrt(r*cos(t)**2), sqrt(r*cos(t)**2)*tan(t))} + assert solve([exp(x) - sin(y), 1/y - 3], [x, y]) == \ + [(log(sin(Rational(1, 3))), Rational(1, 3))] + assert solve([exp(x) - sin(y), 1/exp(y) - 3], [x, y]) == \ + [(log(-sin(log(3))), -log(3))] + assert set(solve([exp(x) - sin(y), y**2 - 4], [x, y])) == \ + {(log(-sin(2)), -S(2)), (log(sin(2)), S(2))} + eqs = [exp(x)**2 - sin(y) + z**2, 1/exp(y) - 3] + assert solve(eqs, set=True) == \ + ([y, z], { + (-log(3), sqrt(-exp(2*x) - sin(log(3)))), + (-log(3), -sqrt(-exp(2*x) - sin(log(3))))}) + assert solve(eqs, x, z, set=True) == ( + [x, z], + {(x, sqrt(-exp(2*x) + sin(y))), (x, -sqrt(-exp(2*x) + sin(y)))}) + assert set(solve(eqs, x, y)) == \ + { + (log(-sqrt(-z**2 - sin(log(3)))), -log(3)), + (log(-z**2 - sin(log(3)))/2, -log(3))} + assert set(solve(eqs, y, z)) == \ + { + (-log(3), -sqrt(-exp(2*x) - sin(log(3)))), + (-log(3), sqrt(-exp(2*x) - sin(log(3))))} + eqs = [exp(x)**2 - sin(y) + z, 1/exp(y) - 3] + assert solve(eqs, set=True) == ([y, z], { + (-log(3), -exp(2*x) - sin(log(3)))}) + assert solve(eqs, x, z, set=True) == ( + [x, z], {(x, -exp(2*x) + sin(y))}) + assert set(solve(eqs, x, y)) == { + (log(-sqrt(-z - sin(log(3)))), -log(3)), + (log(-z - sin(log(3)))/2, -log(3))} + assert solve(eqs, z, y) == \ + [(-exp(2*x) - sin(log(3)), -log(3))] + assert solve((sqrt(x**2 + y**2) - sqrt(10), x + y - 4), set=True) == ( + [x, y], {(S.One, S(3)), (S(3), S.One)}) + assert set(solve((sqrt(x**2 + y**2) - sqrt(10), x + y - 4), x, y)) == \ + {(S.One, S(3)), (S(3), S.One)} + + +def test_issue_5335(): + lam, a0, conc = symbols('lam a0 conc') + a = 0.005 + b = 0.743436700916726 + eqs = [lam + 2*y - a0*(1 - x/2)*x - a*x/2*x, + a0*(1 - x/2)*x - 1*y - b*y, + x + y - conc] + sym = [x, y, a0] + # there are 4 solutions obtained manually but only two are valid + assert len(solve(eqs, sym, manual=True, minimal=True)) == 2 + assert len(solve(eqs, sym)) == 2 # cf below with rational=False + + +@SKIP("Hangs") +def _test_issue_5335_float(): + # gives ZeroDivisionError: polynomial division + lam, a0, conc = symbols('lam a0 conc') + a = 0.005 + b = 0.743436700916726 + eqs = [lam + 2*y - a0*(1 - x/2)*x - a*x/2*x, + a0*(1 - x/2)*x - 1*y - b*y, + x + y - conc] + sym = [x, y, a0] + assert len(solve(eqs, sym, rational=False)) == 2 + + +def test_issue_5767(): + assert set(solve([x**2 + y + 4], [x])) == \ + {(-sqrt(-y - 4),), (sqrt(-y - 4),)} + + +def _make_example_24609(): + D, R, H, B_g, V, D_c = symbols("D, R, H, B_g, V, D_c", real=True, positive=True) + Sigma_f, Sigma_a, nu = symbols("Sigma_f, Sigma_a, nu", real=True, positive=True) + x = symbols("x", real=True, positive=True) + eq = ( + 2**(S(2)/3)*pi**(S(2)/3)*D_c*(S(231361)/10000 + pi**2/x**2) + /(6*V**(S(2)/3)*x**(S(1)/3)) + - 2**(S(2)/3)*pi**(S(8)/3)*D_c/(2*V**(S(2)/3)*x**(S(7)/3)) + ) + expected = 100*sqrt(2)*pi/481 + return eq, expected, x + + +def test_issue_24609(): + # https://github.com/sympy/sympy/issues/24609 + eq, expected, x = _make_example_24609() + assert solve(eq, x, simplify=True) == [expected] + [solapprox] = solve(eq.n(), x) + assert abs(solapprox - expected.n()) < 1e-14 + + +@XFAIL +def test_issue_24609_xfail(): + # + # This returns 5 solutions when it should be 1 (with x positive). + # Simplification reveals all solutions to be equivalent. It is expected + # that solve without simplify=True returns duplicate solutions in some + # cases but the core of this equation is a simple quadratic that can easily + # be solved without introducing any redundant solutions: + # + # >>> print(factor_terms(eq.as_numer_denom()[0])) + # 2**(2/3)*pi**(2/3)*D_c*V**(2/3)*x**(7/3)*(231361*x**2 - 20000*pi**2) + # + eq, expected, x = _make_example_24609() + assert len(solve(eq, x)) == [expected] + # + # We do not want to pass this test just by using simplify so if the above + # passes then uncomment the additional test below: + # + # assert len(solve(eq, x, simplify=False)) == 1 + + +def test_polysys(): + assert set(solve([x**2 + 2/y - 2, x + y - 3], [x, y])) == \ + {(S.One, S(2)), (1 + sqrt(5), 2 - sqrt(5)), + (1 - sqrt(5), 2 + sqrt(5))} + assert solve([x**2 + y - 2, x**2 + y]) == [] + # the ordering should be whatever the user requested + assert solve([x**2 + y - 3, x - y - 4], (x, y)) != solve([x**2 + + y - 3, x - y - 4], (y, x)) + + +@slow +def test_unrad1(): + raises(NotImplementedError, lambda: + unrad(sqrt(x) + sqrt(x + 1) + sqrt(1 - sqrt(x)) + 3)) + raises(NotImplementedError, lambda: + unrad(sqrt(x) + (x + 1)**Rational(1, 3) + 2*sqrt(y))) + + s = symbols('s', cls=Dummy) + + # checkers to deal with possibility of answer coming + # back with a sign change (cf issue 5203) + def check(rv, ans): + assert bool(rv[1]) == bool(ans[1]) + if ans[1]: + return s_check(rv, ans) + e = rv[0].expand() + a = ans[0].expand() + return e in [a, -a] and rv[1] == ans[1] + + def s_check(rv, ans): + # get the dummy + rv = list(rv) + d = rv[0].atoms(Dummy) + reps = list(zip(d, [s]*len(d))) + # replace s with this dummy + rv = (rv[0].subs(reps).expand(), [rv[1][0].subs(reps), rv[1][1].subs(reps)]) + ans = (ans[0].subs(reps).expand(), [ans[1][0].subs(reps), ans[1][1].subs(reps)]) + return str(rv[0]) in [str(ans[0]), str(-ans[0])] and \ + str(rv[1]) == str(ans[1]) + + assert unrad(1) is None + assert check(unrad(sqrt(x)), + (x, [])) + assert check(unrad(sqrt(x) + 1), + (x - 1, [])) + assert check(unrad(sqrt(x) + root(x, 3) + 2), + (s**3 + s**2 + 2, [s, s**6 - x])) + assert check(unrad(sqrt(x)*root(x, 3) + 2), + (x**5 - 64, [])) + assert check(unrad(sqrt(x) + (x + 1)**Rational(1, 3)), + (x**3 - (x + 1)**2, [])) + assert check(unrad(sqrt(x) + sqrt(x + 1) + sqrt(2*x)), + (-2*sqrt(2)*x - 2*x + 1, [])) + assert check(unrad(sqrt(x) + sqrt(x + 1) + 2), + (16*x - 9, [])) + assert check(unrad(sqrt(x) + sqrt(x + 1) + sqrt(1 - x)), + (5*x**2 - 4*x, [])) + assert check(unrad(a*sqrt(x) + b*sqrt(x) + c*sqrt(y) + d*sqrt(y)), + ((a*sqrt(x) + b*sqrt(x))**2 - (c*sqrt(y) + d*sqrt(y))**2, [])) + assert check(unrad(sqrt(x) + sqrt(1 - x)), + (2*x - 1, [])) + assert check(unrad(sqrt(x) + sqrt(1 - x) - 3), + (x**2 - x + 16, [])) + assert check(unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x)), + (5*x**2 - 2*x + 1, [])) + assert unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x) - 3) in [ + (25*x**4 + 376*x**3 + 1256*x**2 - 2272*x + 784, []), + (25*x**8 - 476*x**6 + 2534*x**4 - 1468*x**2 + 169, [])] + assert unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x) - sqrt(1 - 2*x)) == \ + (41*x**4 + 40*x**3 + 232*x**2 - 160*x + 16, []) # orig root at 0.487 + assert check(unrad(sqrt(x) + sqrt(x + 1)), (S.One, [])) + + eq = sqrt(x) + sqrt(x + 1) + sqrt(1 - sqrt(x)) + assert check(unrad(eq), + (16*x**2 - 9*x, [])) + assert set(solve(eq, check=False)) == {S.Zero, Rational(9, 16)} + assert solve(eq) == [] + # but this one really does have those solutions + assert set(solve(sqrt(x) - sqrt(x + 1) + sqrt(1 - sqrt(x)))) == \ + {S.Zero, Rational(9, 16)} + + assert check(unrad(sqrt(x) + root(x + 1, 3) + 2*sqrt(y), y), + (S('2*sqrt(x)*(x + 1)**(1/3) + x - 4*y + (x + 1)**(2/3)'), [])) + assert check(unrad(sqrt(x/(1 - x)) + (x + 1)**Rational(1, 3)), + (x**5 - x**4 - x**3 + 2*x**2 + x - 1, [])) + assert check(unrad(sqrt(x/(1 - x)) + 2*sqrt(y), y), + (4*x*y + x - 4*y, [])) + assert check(unrad(sqrt(x)*sqrt(1 - x) + 2, x), + (x**2 - x + 4, [])) + + # http://tutorial.math.lamar.edu/ + # Classes/Alg/SolveRadicalEqns.aspx#Solve_Rad_Ex2_a + assert solve(Eq(x, sqrt(x + 6))) == [3] + assert solve(Eq(x + sqrt(x - 4), 4)) == [4] + assert solve(Eq(1, x + sqrt(2*x - 3))) == [] + assert set(solve(Eq(sqrt(5*x + 6) - 2, x))) == {-S.One, S(2)} + assert set(solve(Eq(sqrt(2*x - 1) - sqrt(x - 4), 2))) == {S(5), S(13)} + assert solve(Eq(sqrt(x + 7) + 2, sqrt(3 - x))) == [-6] + # http://www.purplemath.com/modules/solverad.htm + assert solve((2*x - 5)**Rational(1, 3) - 3) == [16] + assert set(solve(x + 1 - root(x**4 + 4*x**3 - x, 4))) == \ + {Rational(-1, 2), Rational(-1, 3)} + assert set(solve(sqrt(2*x**2 - 7) - (3 - x))) == {-S(8), S(2)} + assert solve(sqrt(2*x + 9) - sqrt(x + 1) - sqrt(x + 4)) == [0] + assert solve(sqrt(x + 4) + sqrt(2*x - 1) - 3*sqrt(x - 1)) == [5] + assert solve(sqrt(x)*sqrt(x - 7) - 12) == [16] + assert solve(sqrt(x - 3) + sqrt(x) - 3) == [4] + assert solve(sqrt(9*x**2 + 4) - (3*x + 2)) == [0] + assert solve(sqrt(x) - 2 - 5) == [49] + assert solve(sqrt(x - 3) - sqrt(x) - 3) == [] + assert solve(sqrt(x - 1) - x + 7) == [10] + assert solve(sqrt(x - 2) - 5) == [27] + assert solve(sqrt(17*x - sqrt(x**2 - 5)) - 7) == [3] + assert solve(sqrt(x) - sqrt(x - 1) + sqrt(sqrt(x))) == [] + + # don't posify the expression in unrad and do use _mexpand + z = sqrt(2*x + 1)/sqrt(x) - sqrt(2 + 1/x) + p = posify(z)[0] + assert solve(p) == [] + assert solve(z) == [] + assert solve(z + 6*I) == [Rational(-1, 11)] + assert solve(p + 6*I) == [] + # issue 8622 + assert unrad(root(x + 1, 5) - root(x, 3)) == ( + -(x**5 - x**3 - 3*x**2 - 3*x - 1), []) + # issue #8679 + assert check(unrad(x + root(x, 3) + root(x, 3)**2 + sqrt(y), x), + (s**3 + s**2 + s + sqrt(y), [s, s**3 - x])) + + # for coverage + assert check(unrad(sqrt(x) + root(x, 3) + y), + (s**3 + s**2 + y, [s, s**6 - x])) + assert solve(sqrt(x) + root(x, 3) - 2) == [1] + raises(NotImplementedError, lambda: + solve(sqrt(x) + root(x, 3) + root(x + 1, 5) - 2)) + # fails through a different code path + raises(NotImplementedError, lambda: solve(-sqrt(2) + cosh(x)/x)) + # unrad some + assert solve(sqrt(x + root(x, 3))+root(x - y, 5), y) == [ + x + (x**Rational(1, 3) + x)**Rational(5, 2)] + assert check(unrad(sqrt(x) - root(x + 1, 3)*sqrt(x + 2) + 2), + (s**10 + 8*s**8 + 24*s**6 - 12*s**5 - 22*s**4 - 160*s**3 - 212*s**2 - + 192*s - 56, [s, s**2 - x])) + e = root(x + 1, 3) + root(x, 3) + assert unrad(e) == (2*x + 1, []) + eq = (sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6*sqrt(5)/5) + assert check(unrad(eq), + (15625*x**4 + 173000*x**3 + 355600*x**2 - 817920*x + 331776, [])) + assert check(unrad(root(x, 4) + root(x, 4)**3 - 1), + (s**3 + s - 1, [s, s**4 - x])) + assert check(unrad(root(x, 2) + root(x, 2)**3 - 1), + (x**3 + 2*x**2 + x - 1, [])) + assert unrad(x**0.5) is None + assert check(unrad(t + root(x + y, 5) + root(x + y, 5)**3), + (s**3 + s + t, [s, s**5 - x - y])) + assert check(unrad(x + root(x + y, 5) + root(x + y, 5)**3, y), + (s**3 + s + x, [s, s**5 - x - y])) + assert check(unrad(x + root(x + y, 5) + root(x + y, 5)**3, x), + (s**5 + s**3 + s - y, [s, s**5 - x - y])) + assert check(unrad(root(x - 1, 3) + root(x + 1, 5) + root(2, 5)), + (s**5 + 5*2**Rational(1, 5)*s**4 + s**3 + 10*2**Rational(2, 5)*s**3 + + 10*2**Rational(3, 5)*s**2 + 5*2**Rational(4, 5)*s + 4, [s, s**3 - x + 1])) + raises(NotImplementedError, lambda: + unrad((root(x, 2) + root(x, 3) + root(x, 4)).subs(x, x**5 - x + 1))) + + # the simplify flag should be reset to False for unrad results; + # if it's not then this next test will take a long time + assert solve(root(x, 3) + root(x, 5) - 2) == [1] + eq = (sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6*sqrt(5)/5) + assert check(unrad(eq), + ((5*x - 4)*(3125*x**3 + 37100*x**2 + 100800*x - 82944), [])) + ans = S(''' + [4/5, -1484/375 + 172564/(140625*(114*sqrt(12657)/78125 + + 12459439/52734375)**(1/3)) + + 4*(114*sqrt(12657)/78125 + 12459439/52734375)**(1/3)]''') + assert solve(eq) == ans + # duplicate radical handling + assert check(unrad(sqrt(x + root(x + 1, 3)) - root(x + 1, 3) - 2), + (s**3 - s**2 - 3*s - 5, [s, s**3 - x - 1])) + # cov post-processing + e = root(x**2 + 1, 3) - root(x**2 - 1, 5) - 2 + assert check(unrad(e), + (s**5 - 10*s**4 + 39*s**3 - 80*s**2 + 80*s - 30, + [s, s**3 - x**2 - 1])) + + e = sqrt(x + root(x + 1, 2)) - root(x + 1, 3) - 2 + assert check(unrad(e), + (s**6 - 2*s**5 - 7*s**4 - 3*s**3 + 26*s**2 + 40*s + 25, + [s, s**3 - x - 1])) + assert check(unrad(e, _reverse=True), + (s**6 - 14*s**5 + 73*s**4 - 187*s**3 + 276*s**2 - 228*s + 89, + [s, s**2 - x - sqrt(x + 1)])) + # this one needs r0, r1 reversal to work + assert check(unrad(sqrt(x + sqrt(root(x, 3) - 1)) - root(x, 6) - 2), + (s**12 - 2*s**8 - 8*s**7 - 8*s**6 + s**4 + 8*s**3 + 23*s**2 + + 32*s + 17, [s, s**6 - x])) + + # why does this pass + assert unrad(root(cosh(x), 3)/x*root(x + 1, 5) - 1) == ( + -(x**15 - x**3*cosh(x)**5 - 3*x**2*cosh(x)**5 - 3*x*cosh(x)**5 + - cosh(x)**5), []) + # and this fail? + #assert unrad(sqrt(cosh(x)/x) + root(x + 1, 3)*sqrt(x) - 1) == ( + # -s**6 + 6*s**5 - 15*s**4 + 20*s**3 - 15*s**2 + 6*s + x**5 + + # 2*x**4 + x**3 - 1, [s, s**2 - cosh(x)/x]) + + # watch for symbols in exponents + assert unrad(S('(x+y)**(2*y/3) + (x+y)**(1/3) + 1')) is None + assert check(unrad(S('(x+y)**(2*y/3) + (x+y)**(1/3) + 1'), x), + (s**(2*y) + s + 1, [s, s**3 - x - y])) + # should _Q be so lenient? + assert unrad(x**(S.Half/y) + y, x) == (x**(1/y) - y**2, []) + + # This tests two things: that if full unrad is attempted and fails + # the solution should still be found; also it tests that the use of + # composite + assert len(solve(sqrt(y)*x + x**3 - 1, x)) == 3 + assert len(solve(-512*y**3 + 1344*(x + 2)**Rational(1, 3)*y**2 - + 1176*(x + 2)**Rational(2, 3)*y - 169*x + 686, y, _unrad=False)) == 3 + + # watch out for when the cov doesn't involve the symbol of interest + eq = S('-x + (7*y/8 - (27*x/2 + 27*sqrt(x**2)/2)**(1/3)/3)**3 - 1') + assert solve(eq, y) == [ + 2**(S(2)/3)*(27*x + 27*sqrt(x**2))**(S(1)/3)*S(4)/21 + (512*x/343 + + S(512)/343)**(S(1)/3)*(-S(1)/2 - sqrt(3)*I/2), 2**(S(2)/3)*(27*x + + 27*sqrt(x**2))**(S(1)/3)*S(4)/21 + (512*x/343 + + S(512)/343)**(S(1)/3)*(-S(1)/2 + sqrt(3)*I/2), 2**(S(2)/3)*(27*x + + 27*sqrt(x**2))**(S(1)/3)*S(4)/21 + (512*x/343 + S(512)/343)**(S(1)/3)] + + eq = root(x + 1, 3) - (root(x, 3) + root(x, 5)) + assert check(unrad(eq), + (3*s**13 + 3*s**11 + s**9 - 1, [s, s**15 - x])) + assert check(unrad(eq - 2), + (3*s**13 + 3*s**11 + 6*s**10 + s**9 + 12*s**8 + 6*s**6 + 12*s**5 + + 12*s**3 + 7, [s, s**15 - x])) + assert check(unrad(root(x, 3) - root(x + 1, 4)/2 + root(x + 2, 3)), + (s*(4096*s**9 + 960*s**8 + 48*s**7 - s**6 - 1728), + [s, s**4 - x - 1])) # orig expr has two real roots: -1, -.389 + assert check(unrad(root(x, 3) + root(x + 1, 4) - root(x + 2, 3)/2), + (343*s**13 + 2904*s**12 + 1344*s**11 + 512*s**10 - 1323*s**9 - + 3024*s**8 - 1728*s**7 + 1701*s**5 + 216*s**4 - 729*s, [s, s**4 - x - + 1])) # orig expr has one real root: -0.048 + assert check(unrad(root(x, 3)/2 - root(x + 1, 4) + root(x + 2, 3)), + (729*s**13 - 216*s**12 + 1728*s**11 - 512*s**10 + 1701*s**9 - + 3024*s**8 + 1344*s**7 + 1323*s**5 - 2904*s**4 + 343*s, [s, s**4 - x - + 1])) # orig expr has 2 real roots: -0.91, -0.15 + assert check(unrad(root(x, 3)/2 - root(x + 1, 4) + root(x + 2, 3) - 2), + (729*s**13 + 1242*s**12 + 18496*s**10 + 129701*s**9 + 388602*s**8 + + 453312*s**7 - 612864*s**6 - 3337173*s**5 - 6332418*s**4 - 7134912*s**3 + - 5064768*s**2 - 2111913*s - 398034, [s, s**4 - x - 1])) + # orig expr has 1 real root: 19.53 + + ans = solve(sqrt(x) + sqrt(x + 1) - + sqrt(1 - x) - sqrt(2 + x)) + assert len(ans) == 1 and NS(ans[0])[:4] == '0.73' + # the fence optimization problem + # https://github.com/sympy/sympy/issues/4793#issuecomment-36994519 + F = Symbol('F') + eq = F - (2*x + 2*y + sqrt(x**2 + y**2)) + ans = F*Rational(2, 7) - sqrt(2)*F/14 + X = solve(eq, x, check=False) + for xi in reversed(X): # reverse since currently, ans is the 2nd one + Y = solve((x*y).subs(x, xi).diff(y), y, simplify=False, check=False) + if any((a - ans).expand().is_zero for a in Y): + break + else: + assert None # no answer was found + assert solve(sqrt(x + 1) + root(x, 3) - 2) == S(''' + [(-11/(9*(47/54 + sqrt(93)/6)**(1/3)) + 1/3 + (47/54 + + sqrt(93)/6)**(1/3))**3]''') + assert solve(sqrt(sqrt(x + 1)) + x**Rational(1, 3) - 2) == S(''' + [(-sqrt(-2*(-1/16 + sqrt(6913)/16)**(1/3) + 6/(-1/16 + + sqrt(6913)/16)**(1/3) + 17/2 + 121/(4*sqrt(-6/(-1/16 + + sqrt(6913)/16)**(1/3) + 2*(-1/16 + sqrt(6913)/16)**(1/3) + 17/4)))/2 + + sqrt(-6/(-1/16 + sqrt(6913)/16)**(1/3) + 2*(-1/16 + + sqrt(6913)/16)**(1/3) + 17/4)/2 + 9/4)**3]''') + assert solve(sqrt(x) + root(sqrt(x) + 1, 3) - 2) == S(''' + [(-(81/2 + 3*sqrt(741)/2)**(1/3)/3 + (81/2 + 3*sqrt(741)/2)**(-1/3) + + 2)**2]''') + eq = S(''' + -x + (1/2 - sqrt(3)*I/2)*(3*x**3/2 - x*(3*x**2 - 34)/2 + sqrt((-3*x**3 + + x*(3*x**2 - 34) + 90)**2/4 - 39304/27) - 45)**(1/3) + 34/(3*(1/2 - + sqrt(3)*I/2)*(3*x**3/2 - x*(3*x**2 - 34)/2 + sqrt((-3*x**3 + x*(3*x**2 + - 34) + 90)**2/4 - 39304/27) - 45)**(1/3))''') + assert check(unrad(eq), + (s*-(-s**6 + sqrt(3)*s**6*I - 153*2**Rational(2, 3)*3**Rational(1, 3)*s**4 + + 51*12**Rational(1, 3)*s**4 - 102*2**Rational(2, 3)*3**Rational(5, 6)*s**4*I - 1620*s**3 + + 1620*sqrt(3)*s**3*I + 13872*18**Rational(1, 3)*s**2 - 471648 + + 471648*sqrt(3)*I), [s, s**3 - 306*x - sqrt(3)*sqrt(31212*x**2 - + 165240*x + 61484) + 810])) + + assert solve(eq) == [] # not other code errors + eq = root(x, 3) - root(y, 3) + root(x, 5) + assert check(unrad(eq), + (s**15 + 3*s**13 + 3*s**11 + s**9 - y, [s, s**15 - x])) + eq = root(x, 3) + root(y, 3) + root(x*y, 4) + assert check(unrad(eq), + (s*y*(-s**12 - 3*s**11*y - 3*s**10*y**2 - s**9*y**3 - + 3*s**8*y**2 + 21*s**7*y**3 - 3*s**6*y**4 - 3*s**4*y**4 - + 3*s**3*y**5 - y**6), [s, s**4 - x*y])) + raises(NotImplementedError, + lambda: unrad(root(x, 3) + root(y, 3) + root(x*y, 5))) + + # Test unrad with an Equality + eq = Eq(-x**(S(1)/5) + x**(S(1)/3), -3**(S(1)/3) - (-1)**(S(3)/5)*3**(S(1)/5)) + assert check(unrad(eq), + (-s**5 + s**3 - 3**(S(1)/3) - (-1)**(S(3)/5)*3**(S(1)/5), [s, s**15 - x])) + + # make sure buried radicals are exposed + s = sqrt(x) - 1 + assert unrad(s**2 - s**3) == (x**3 - 6*x**2 + 9*x - 4, []) + # make sure numerators which are already polynomial are rejected + assert unrad((x/(x + 1) + 3)**(-2), x) is None + + # https://github.com/sympy/sympy/issues/23707 + eq = sqrt(x - y)*exp(t*sqrt(x - y)) - exp(t*sqrt(x - y)) + assert solve(eq, y) == [x - 1] + assert unrad(eq) is None + + +@slow +def test_unrad_slow(): + # this has roots with multiplicity > 1; there should be no + # repeats in roots obtained, however + eq = (sqrt(1 + sqrt(1 - 4*x**2)) - x*(1 + sqrt(1 + 2*sqrt(1 - 4*x**2)))) + assert solve(eq) == [S.Half] + + +@XFAIL +def test_unrad_fail(): + # this only works if we check real_root(eq.subs(x, Rational(1, 3))) + # but checksol doesn't work like that + assert solve(root(x**3 - 3*x**2, 3) + 1 - x) == [Rational(1, 3)] + assert solve(root(x + 1, 3) + root(x**2 - 2, 5) + 1) == [ + -1, -1 + CRootOf(x**5 + x**4 + 5*x**3 + 8*x**2 + 10*x + 5, 0)**3] + + +def test_checksol(): + x, y, r, t = symbols('x, y, r, t') + eq = r - x**2 - y**2 + dict_var_soln = {y: - sqrt(r) / sqrt(tan(t)**2 + 1), + x: -sqrt(r)*tan(t)/sqrt(tan(t)**2 + 1)} + assert checksol(eq, dict_var_soln) == True + assert checksol(Eq(x, False), {x: False}) is True + assert checksol(Ne(x, False), {x: False}) is False + assert checksol(Eq(x < 1, True), {x: 0}) is True + assert checksol(Eq(x < 1, True), {x: 1}) is False + assert checksol(Eq(x < 1, False), {x: 1}) is True + assert checksol(Eq(x < 1, False), {x: 0}) is False + assert checksol(Eq(x + 1, x**2 + 1), {x: 1}) is True + assert checksol([x - 1, x**2 - 1], x, 1) is True + assert checksol([x - 1, x**2 - 2], x, 1) is False + assert checksol(Poly(x**2 - 1), x, 1) is True + assert checksol(0, {}) is True + assert checksol([1e-10, x - 2], x, 2) is False + assert checksol([0.5, 0, x], x, 0) is False + assert checksol(y, x, 2) is False + assert checksol(x+1e-10, x, 0, numerical=True) is True + assert checksol(x+1e-10, x, 0, numerical=False) is False + assert checksol(exp(92*x), {x: log(sqrt(2)/2)}) is False + assert checksol(exp(92*x), {x: log(sqrt(2)/2) + I*pi}) is False + assert checksol(1/x**5, x, 1000) is False + raises(ValueError, lambda: checksol(x, 1)) + raises(ValueError, lambda: checksol([], x, 1)) + + +def test__invert(): + assert _invert(x - 2) == (2, x) + assert _invert(2) == (2, 0) + assert _invert(exp(1/x) - 3, x) == (1/log(3), x) + assert _invert(exp(1/x + a/x) - 3, x) == ((a + 1)/log(3), x) + assert _invert(a, x) == (a, 0) + + +def test_issue_4463(): + assert solve(-a*x + 2*x*log(x), x) == [exp(a/2)] + assert solve(x**x) == [] + assert solve(x**x - 2) == [exp(LambertW(log(2)))] + assert solve(((x - 3)*(x - 2))**((x - 3)*(x - 4))) == [2] + +@slow +def test_issue_5114_solvers(): + a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r = symbols('a:r') + + # there is no 'a' in the equation set but this is how the + # problem was originally posed + syms = a, b, c, f, h, k, n + eqs = [b + r/d - c/d, + c*(1/d + 1/e + 1/g) - f/g - r/d, + f*(1/g + 1/i + 1/j) - c/g - h/i, + h*(1/i + 1/l + 1/m) - f/i - k/m, + k*(1/m + 1/o + 1/p) - h/m - n/p, + n*(1/p + 1/q) - k/p] + assert len(solve(eqs, syms, manual=True, check=False, simplify=False)) == 1 + + +def test_issue_5849(): + # + # XXX: This system does not have a solution for most values of the + # parameters. Generally solve returns the empty set for systems that are + # generically inconsistent. + # + I1, I2, I3, I4, I5, I6 = symbols('I1:7') + dI1, dI4, dQ2, dQ4, Q2, Q4 = symbols('dI1,dI4,dQ2,dQ4,Q2,Q4') + + e = ( + I1 - I2 - I3, + I3 - I4 - I5, + I4 + I5 - I6, + -I1 + I2 + I6, + -2*I1 - 2*I3 - 2*I5 - 3*I6 - dI1/2 + 12, + -I4 + dQ4, + -I2 + dQ2, + 2*I3 + 2*I5 + 3*I6 - Q2, + I4 - 2*I5 + 2*Q4 + dI4 + ) + + ans = [{ + I1: I2 + I3, + dI1: -4*I2 - 8*I3 - 4*I5 - 6*I6 + 24, + I4: I3 - I5, + dQ4: I3 - I5, + Q4: -I3/2 + 3*I5/2 - dI4/2, + dQ2: I2, + Q2: 2*I3 + 2*I5 + 3*I6}] + + v = I1, I4, Q2, Q4, dI1, dI4, dQ2, dQ4 + assert solve(e, *v, manual=True, check=False, dict=True) == ans + assert solve(e, *v, manual=True, check=False) == [ + tuple([a.get(i, i) for i in v]) for a in ans] + assert solve(e, *v, manual=True) == [] + assert solve(e, *v) == [] + + # the matrix solver (tested below) doesn't like this because it produces + # a zero row in the matrix. Is this related to issue 4551? + assert [ei.subs( + ans[0]) for ei in e] == [0, 0, I3 - I6, -I3 + I6, 0, 0, 0, 0, 0] + + +def test_issue_5849_matrix(): + '''Same as test_issue_5849 but solved with the matrix solver. + + A solution only exists if I3 == I6 which is not generically true, + but `solve` does not return conditions under which the solution is + valid, only a solution that is canonical and consistent with the input. + ''' + # a simple example with the same issue + # assert solve([x+y+z, x+y], [x, y]) == {x: y} + # the longer example + I1, I2, I3, I4, I5, I6 = symbols('I1:7') + dI1, dI4, dQ2, dQ4, Q2, Q4 = symbols('dI1,dI4,dQ2,dQ4,Q2,Q4') + + e = ( + I1 - I2 - I3, + I3 - I4 - I5, + I4 + I5 - I6, + -I1 + I2 + I6, + -2*I1 - 2*I3 - 2*I5 - 3*I6 - dI1/2 + 12, + -I4 + dQ4, + -I2 + dQ2, + 2*I3 + 2*I5 + 3*I6 - Q2, + I4 - 2*I5 + 2*Q4 + dI4 + ) + assert solve(e, I1, I4, Q2, Q4, dI1, dI4, dQ2, dQ4) == [] + + +def test_issue_21882(): + + a, b, c, d, f, g, k = unknowns = symbols('a, b, c, d, f, g, k') + + equations = [ + -k*a + b + 5*f/6 + 2*c/9 + 5*d/6 + 4*a/3, + -k*f + 4*f/3 + d/2, + -k*d + f/6 + d, + 13*b/18 + 13*c/18 + 13*a/18, + -k*c + b/2 + 20*c/9 + a, + -k*b + b + c/18 + a/6, + 5*b/3 + c/3 + a, + 2*b/3 + 2*c + 4*a/3, + -g, + ] + + answer = [ + {a: 0, f: 0, b: 0, d: 0, c: 0, g: 0}, + {a: 0, f: -d, b: 0, k: S(5)/6, c: 0, g: 0}, + {a: -2*c, f: 0, b: c, d: 0, k: S(13)/18, g: 0}] + # but not {a: 0, f: 0, b: 0, k: S(3)/2, c: 0, d: 0, g: 0} + # since this is already covered by the first solution + got = solve(equations, unknowns, dict=True) + assert got == answer, (got,answer) + + +def test_issue_5901(): + f, g, h = map(Function, 'fgh') + a = Symbol('a') + D = Derivative(f(x), x) + G = Derivative(g(a), a) + assert solve(f(x) + f(x).diff(x), f(x)) == \ + [-D] + assert solve(f(x) - 3, f(x)) == \ + [3] + assert solve(f(x) - 3*f(x).diff(x), f(x)) == \ + [3*D] + assert solve([f(x) - 3*f(x).diff(x)], f(x)) == \ + {f(x): 3*D} + assert solve([f(x) - 3*f(x).diff(x), f(x)**2 - y + 4], f(x), y) == \ + [(3*D, 9*D**2 + 4)] + assert solve(-f(a)**2*g(a)**2 + f(a)**2*h(a)**2 + g(a).diff(a), + h(a), g(a), set=True) == \ + ([h(a), g(a)], { + (-sqrt(f(a)**2*g(a)**2 - G)/f(a), g(a)), + (sqrt(f(a)**2*g(a)**2 - G)/f(a), g(a))}), solve(-f(a)**2*g(a)**2 + f(a)**2*h(a)**2 + g(a).diff(a), + h(a), g(a), set=True) + args = [[f(x).diff(x, 2)*(f(x) + g(x)), 2 - g(x)**2], f(x), g(x)] + assert solve(*args, set=True)[1] == \ + {(-sqrt(2), sqrt(2)), (sqrt(2), -sqrt(2))} + eqs = [f(x)**2 + g(x) - 2*f(x).diff(x), g(x)**2 - 4] + assert solve(eqs, f(x), g(x), set=True) == \ + ([f(x), g(x)], { + (-sqrt(2*D - 2), S(2)), + (sqrt(2*D - 2), S(2)), + (-sqrt(2*D + 2), -S(2)), + (sqrt(2*D + 2), -S(2))}) + + # the underlying problem was in solve_linear that was not masking off + # anything but a Mul or Add; it now raises an error if it gets anything + # but a symbol and solve handles the substitutions necessary so solve_linear + # won't make this error + raises( + ValueError, lambda: solve_linear(f(x) + f(x).diff(x), symbols=[f(x)])) + assert solve_linear(f(x) + f(x).diff(x), symbols=[x]) == \ + (f(x) + Derivative(f(x), x), 1) + assert solve_linear(f(x) + Integral(x, (x, y)), symbols=[x]) == \ + (f(x) + Integral(x, (x, y)), 1) + assert solve_linear(f(x) + Integral(x, (x, y)) + x, symbols=[x]) == \ + (x + f(x) + Integral(x, (x, y)), 1) + assert solve_linear(f(y) + Integral(x, (x, y)) + x, symbols=[x]) == \ + (x, -f(y) - Integral(x, (x, y))) + assert solve_linear(x - f(x)/a + (f(x) - 1)/a, symbols=[x]) == \ + (x, 1/a) + assert solve_linear(x + Derivative(2*x, x)) == \ + (x, -2) + assert solve_linear(x + Integral(x, y), symbols=[x]) == \ + (x, 0) + assert solve_linear(x + Integral(x, y) - 2, symbols=[x]) == \ + (x, 2/(y + 1)) + + assert set(solve(x + exp(x)**2, exp(x))) == \ + {-sqrt(-x), sqrt(-x)} + assert solve(x + exp(x), x, implicit=True) == \ + [-exp(x)] + assert solve(cos(x) - sin(x), x, implicit=True) == [] + assert solve(x - sin(x), x, implicit=True) == \ + [sin(x)] + assert solve(x**2 + x - 3, x, implicit=True) == \ + [-x**2 + 3] + assert solve(x**2 + x - 3, x**2, implicit=True) == \ + [-x + 3] + + +def test_issue_5912(): + assert set(solve(x**2 - x - 0.1, rational=True)) == \ + {S.Half + sqrt(35)/10, -sqrt(35)/10 + S.Half} + ans = solve(x**2 - x - 0.1, rational=False) + assert len(ans) == 2 and all(a.is_Number for a in ans) + ans = solve(x**2 - x - 0.1) + assert len(ans) == 2 and all(a.is_Number for a in ans) + + +def test_float_handling(): + def test(e1, e2): + return len(e1.atoms(Float)) == len(e2.atoms(Float)) + assert solve(x - 0.5, rational=True)[0].is_Rational + assert solve(x - 0.5, rational=False)[0].is_Float + assert solve(x - S.Half, rational=False)[0].is_Rational + assert solve(x - 0.5, rational=None)[0].is_Float + assert solve(x - S.Half, rational=None)[0].is_Rational + assert test(nfloat(1 + 2*x), 1.0 + 2.0*x) + for contain in [list, tuple, set]: + ans = nfloat(contain([1 + 2*x])) + assert type(ans) is contain and test(list(ans)[0], 1.0 + 2.0*x) + k, v = list(nfloat({2*x: [1 + 2*x]}).items())[0] + assert test(k, 2*x) and test(v[0], 1.0 + 2.0*x) + assert test(nfloat(cos(2*x)), cos(2.0*x)) + assert test(nfloat(3*x**2), 3.0*x**2) + assert test(nfloat(3*x**2, exponent=True), 3.0*x**2.0) + assert test(nfloat(exp(2*x)), exp(2.0*x)) + assert test(nfloat(x/3), x/3.0) + assert test(nfloat(x**4 + 2*x + cos(Rational(1, 3)) + 1), + x**4 + 2.0*x + 1.94495694631474) + # don't call nfloat if there is no solution + tot = 100 + c + z + t + assert solve(((.7 + c)/tot - .6, (.2 + z)/tot - .3, t/tot - .1)) == [] + + +def test_check_assumptions(): + x = symbols('x', positive=True) + assert solve(x**2 - 1) == [1] + + +def test_issue_6056(): + assert solve(tanh(x + 3)*tanh(x - 3) - 1) == [] + assert solve(tanh(x - 1)*tanh(x + 1) + 1) == \ + [I*pi*Rational(-3, 4), -I*pi/4, I*pi/4, I*pi*Rational(3, 4)] + assert solve((tanh(x + 3)*tanh(x - 3) + 1)**2) == \ + [I*pi*Rational(-3, 4), -I*pi/4, I*pi/4, I*pi*Rational(3, 4)] + + +def test_issue_5673(): + eq = -x + exp(exp(LambertW(log(x)))*LambertW(log(x))) + assert checksol(eq, x, 2) is True + assert checksol(eq, x, 2, numerical=False) is None + + +def test_exclude(): + R, C, Ri, Vout, V1, Vminus, Vplus, s = \ + symbols('R, C, Ri, Vout, V1, Vminus, Vplus, s') + Rf = symbols('Rf', positive=True) # to eliminate Rf = 0 soln + eqs = [C*V1*s + Vplus*(-2*C*s - 1/R), + Vminus*(-1/Ri - 1/Rf) + Vout/Rf, + C*Vplus*s + V1*(-C*s - 1/R) + Vout/R, + -Vminus + Vplus] + assert solve(eqs, exclude=s*C*R) == [ + { + Rf: Ri*(C*R*s + 1)**2/(C*R*s), + Vminus: Vplus, + V1: 2*Vplus + Vplus/(C*R*s), + Vout: C*R*Vplus*s + 3*Vplus + Vplus/(C*R*s)}, + { + Vplus: 0, + Vminus: 0, + V1: 0, + Vout: 0}, + ] + + # TODO: Investigate why currently solution [0] is preferred over [1]. + assert solve(eqs, exclude=[Vplus, s, C]) in [[{ + Vminus: Vplus, + V1: Vout/2 + Vplus/2 + sqrt((Vout - 5*Vplus)*(Vout - Vplus))/2, + R: (Vout - 3*Vplus - sqrt(Vout**2 - 6*Vout*Vplus + 5*Vplus**2))/(2*C*Vplus*s), + Rf: Ri*(Vout - Vplus)/Vplus, + }, { + Vminus: Vplus, + V1: Vout/2 + Vplus/2 - sqrt((Vout - 5*Vplus)*(Vout - Vplus))/2, + R: (Vout - 3*Vplus + sqrt(Vout**2 - 6*Vout*Vplus + 5*Vplus**2))/(2*C*Vplus*s), + Rf: Ri*(Vout - Vplus)/Vplus, + }], [{ + Vminus: Vplus, + Vout: (V1**2 - V1*Vplus - Vplus**2)/(V1 - 2*Vplus), + Rf: Ri*(V1 - Vplus)**2/(Vplus*(V1 - 2*Vplus)), + R: Vplus/(C*s*(V1 - 2*Vplus)), + }]] + + +def test_high_order_roots(): + s = x**5 + 4*x**3 + 3*x**2 + Rational(7, 4) + assert set(solve(s)) == set(Poly(s*4, domain='ZZ').all_roots()) + + +def test_minsolve_linear_system(): + pqt = {"quick": True, "particular": True} + pqf = {"quick": False, "particular": True} + assert solve([x + y - 5, 2*x - y - 1], **pqt) == {x: 2, y: 3} + assert solve([x + y - 5, 2*x - y - 1], **pqf) == {x: 2, y: 3} + def count(dic): + return len([x for x in dic.values() if x == 0]) + assert count(solve([x + y + z, y + z + a + t], **pqt)) == 3 + assert count(solve([x + y + z, y + z + a + t], **pqf)) == 3 + assert count(solve([x + y + z, y + z + a], **pqt)) == 1 + assert count(solve([x + y + z, y + z + a], **pqf)) == 2 + # issue 22718 + A = Matrix([ + [ 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0], + [ 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, -1, -1, 0, 0], + [-1, -1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 1, 0, 1], + [ 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, -1, 0, -1, 0], + [-1, 0, -1, 0, 0, -1, 0, 0, 0, 0, 1, 0, 1, 1], + [-1, 0, 0, -1, 0, 0, -1, 0, 0, 0, -1, 0, 0, -1], + [ 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, -1, -1, 0], + [ 0, -1, -1, 0, 0, 0, 0, -1, 0, 0, 0, 1, 1, 1], + [ 0, -1, 0, -1, 0, 0, 0, 0, -1, 0, 0, -1, 0, -1], + [ 0, 0, -1, -1, 0, 0, 0, 0, 0, -1, 0, 0, -1, -1], + [ 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0], + [ 0, 0, 0, 0, -1, -1, 0, -1, 0, 0, 0, 0, 0, 0]]) + v = Matrix(symbols("v:14", integer=True)) + B = Matrix([[2], [-2], [0], [0], [0], [0], [0], [0], [0], + [0], [0], [0]]) + eqs = A@v-B + assert solve(eqs) == [] + assert solve(eqs, particular=True) == [] # assumption violated + assert all(v for v in solve([x + y + z, y + z + a]).values()) + for _q in (True, False): + assert not all(v for v in solve( + [x + y + z, y + z + a], quick=_q, + particular=True).values()) + # raise error if quick used w/o particular=True + raises(ValueError, lambda: solve([x + 1], quick=_q)) + raises(ValueError, lambda: solve([x + 1], quick=_q, particular=False)) + # and give a good error message if someone tries to use + # particular with a single equation + raises(ValueError, lambda: solve(x + 1, particular=True)) + + +def test_real_roots(): + # cf. issue 6650 + x = Symbol('x', real=True) + assert len(solve(x**5 + x**3 + 1)) == 1 + + +def test_issue_6528(): + eqs = [ + 327600995*x**2 - 37869137*x + 1809975124*y**2 - 9998905626, + 895613949*x**2 - 273830224*x*y + 530506983*y**2 - 10000000000] + # two expressions encountered are > 1400 ops long so if this hangs + # it is likely because simplification is being done + assert len(solve(eqs, y, x, check=False)) == 4 + + +def test_overdetermined(): + x = symbols('x', real=True) + eqs = [Abs(4*x - 7) - 5, Abs(3 - 8*x) - 1] + assert solve(eqs, x) == [(S.Half,)] + assert solve(eqs, x, manual=True) == [(S.Half,)] + assert solve(eqs, x, manual=True, check=False) == [(S.Half,), (S(3),)] + + +def test_issue_6605(): + x = symbols('x') + assert solve(4**(x/2) - 2**(x/3)) == [0, 3*I*pi/log(2)] + # while the first one passed, this one failed + x = symbols('x', real=True) + assert solve(5**(x/2) - 2**(x/3)) == [0] + b = sqrt(6)*sqrt(log(2))/sqrt(log(5)) + assert solve(5**(x/2) - 2**(3/x)) == [-b, b] + + +def test__ispow(): + assert _ispow(x**2) + assert not _ispow(x) + assert not _ispow(True) + + +def test_issue_6644(): + eq = -sqrt((m - q)**2 + (-m/(2*q) + S.Half)**2) + sqrt((-m**2/2 - sqrt( + 4*m**4 - 4*m**2 + 8*m + 1)/4 - Rational(1, 4))**2 + (m**2/2 - m - sqrt( + 4*m**4 - 4*m**2 + 8*m + 1)/4 - Rational(1, 4))**2) + sol = solve(eq, q, simplify=False, check=False) + assert len(sol) == 5 + + +def test_issue_6752(): + assert solve([a**2 + a, a - b], [a, b]) == [(-1, -1), (0, 0)] + assert solve([a**2 + a*c, a - b], [a, b]) == [(0, 0), (-c, -c)] + + +def test_issue_6792(): + assert solve(x*(x - 1)**2*(x + 1)*(x**6 - x + 1)) == [ + -1, 0, 1, CRootOf(x**6 - x + 1, 0), CRootOf(x**6 - x + 1, 1), + CRootOf(x**6 - x + 1, 2), CRootOf(x**6 - x + 1, 3), + CRootOf(x**6 - x + 1, 4), CRootOf(x**6 - x + 1, 5)] + + +def test_issues_6819_6820_6821_6248_8692_25777_25779(): + # issue 6821 + x, y = symbols('x y', real=True) + assert solve(abs(x + 3) - 2*abs(x - 3)) == [1, 9] + assert solve([abs(x) - 2, arg(x) - pi], x) == [(-2,)] + assert set(solve(abs(x - 7) - 8)) == {-S.One, S(15)} + + # issue 8692 + assert solve(Eq(Abs(x + 1) + Abs(x**2 - 7), 9), x) == [ + Rational(-1, 2) + sqrt(61)/2, -sqrt(69)/2 + S.Half] + + # issue 7145 + assert solve(2*abs(x) - abs(x - 1)) == [-1, Rational(1, 3)] + + # 25777 + assert solve(abs(x**3 + x + 2)/(x + 1)) == [] + + # 25779 + assert solve(abs(x)) == [0] + assert solve(Eq(abs(x**2 - 2*x), 4), x) == [ + 1 - sqrt(5), 1 + sqrt(5)] + nn = symbols('nn', nonnegative=True) + assert solve(abs(sqrt(nn))) == [0] + nz = symbols('nz', nonzero=True) + assert solve(Eq(Abs(4 + 1 / (4*nz)), 0)) == [-Rational(1, 16)] + + x = symbols('x') + assert solve([re(x) - 1, im(x) - 2], x) == [ + {x: 1 + 2*I, re(x): 1, im(x): 2}] + + # check for 'dict' handling of solution + eq = sqrt(re(x)**2 + im(x)**2) - 3 + assert solve(eq) == solve(eq, x) + + i = symbols('i', imaginary=True) + assert solve(abs(i) - 3) == [-3*I, 3*I] + raises(NotImplementedError, lambda: solve(abs(x) - 3)) + + w = symbols('w', integer=True) + assert solve(2*x**w - 4*y**w, w) == solve((x/y)**w - 2, w) + + x, y = symbols('x y', real=True) + assert solve(x + y*I + 3) == {y: 0, x: -3} + # issue 2642 + assert solve(x*(1 + I)) == [0] + + x, y = symbols('x y', imaginary=True) + assert solve(x + y*I + 3 + 2*I) == {x: -2*I, y: 3*I} + + x = symbols('x', real=True) + assert solve(x + y + 3 + 2*I) == {x: -3, y: -2*I} + + # issue 6248 + f = Function('f') + assert solve(f(x + 1) - f(2*x - 1)) == [2] + assert solve(log(x + 1) - log(2*x - 1)) == [2] + + x = symbols('x') + assert solve(2**x + 4**x) == [I*pi/log(2)] + +def test_issue_17638(): + + assert solve(((2-exp(2*x))*exp(x))/(exp(2*x)+2)**2 > 0, x) == (-oo < x) & (x < log(2)/2) + assert solve(((2-exp(2*x)+2)*exp(x+2))/(exp(x)+2)**2 > 0, x) == (-oo < x) & (x < log(4)/2) + assert solve((exp(x)+2+x**2)*exp(2*x+2)/(exp(x)+2)**2 > 0, x) == (-oo < x) & (x < oo) + + + +def test_issue_14607(): + # issue 14607 + s, tau_c, tau_1, tau_2, phi, K = symbols( + 's, tau_c, tau_1, tau_2, phi, K') + + target = (s**2*tau_1*tau_2 + s*tau_1 + s*tau_2 + 1)/(K*s*(-phi + tau_c)) + + K_C, tau_I, tau_D = symbols('K_C, tau_I, tau_D', + positive=True, nonzero=True) + PID = K_C*(1 + 1/(tau_I*s) + tau_D*s) + + eq = (target - PID).together() + eq *= denom(eq).simplify() + eq = Poly(eq, s) + c = eq.coeffs() + + vars = [K_C, tau_I, tau_D] + s = solve(c, vars, dict=True) + + assert len(s) == 1 + + knownsolution = {K_C: -(tau_1 + tau_2)/(K*(phi - tau_c)), + tau_I: tau_1 + tau_2, + tau_D: tau_1*tau_2/(tau_1 + tau_2)} + + for var in vars: + assert s[0][var].simplify() == knownsolution[var].simplify() + + +def test_lambert_multivariate(): + from sympy.abc import x, y + assert _filtered_gens(Poly(x + 1/x + exp(x) + y), x) == {x, exp(x)} + assert _lambert(x, x) == [] + assert solve((x**2 - 2*x + 1).subs(x, log(x) + 3*x)) == [LambertW(3*S.Exp1)/3] + assert solve((x**2 - 2*x + 1).subs(x, (log(x) + 3*x)**2 - 1)) == \ + [LambertW(3*exp(-sqrt(2)))/3, LambertW(3*exp(sqrt(2)))/3] + assert solve((x**2 - 2*x - 2).subs(x, log(x) + 3*x)) == \ + [LambertW(3*exp(1 - sqrt(3)))/3, LambertW(3*exp(1 + sqrt(3)))/3] + eq = (x*exp(x) - 3).subs(x, x*exp(x)) + assert solve(eq) == [LambertW(3*exp(-LambertW(3)))] + # coverage test + raises(NotImplementedError, lambda: solve(x - sin(x)*log(y - x), x)) + ans = [3, -3*LambertW(-log(3)/3)/log(3)] # 3 and 2.478... + assert solve(x**3 - 3**x, x) == ans + assert set(solve(3*log(x) - x*log(3))) == set(ans) + assert solve(LambertW(2*x) - y, x) == [y*exp(y)/2] + + +@XFAIL +def test_other_lambert(): + assert solve(3*sin(x) - x*sin(3), x) == [3] + assert set(solve(x**a - a**x), x) == { + a, -a*LambertW(-log(a)/a)/log(a)} + + +@slow +def test_lambert_bivariate(): + # tests passing current implementation + assert solve((x**2 + x)*exp(x**2 + x) - 1) == [ + Rational(-1, 2) + sqrt(1 + 4*LambertW(1))/2, + Rational(-1, 2) - sqrt(1 + 4*LambertW(1))/2] + assert solve((x**2 + x)*exp((x**2 + x)*2) - 1) == [ + Rational(-1, 2) + sqrt(1 + 2*LambertW(2))/2, + Rational(-1, 2) - sqrt(1 + 2*LambertW(2))/2] + assert solve(a/x + exp(x/2), x) == [2*LambertW(-a/2)] + assert solve((a/x + exp(x/2)).diff(x), x) == \ + [4*LambertW(-sqrt(2)*sqrt(a)/4), 4*LambertW(sqrt(2)*sqrt(a)/4)] + assert solve((1/x + exp(x/2)).diff(x), x) == \ + [4*LambertW(-sqrt(2)/4), + 4*LambertW(sqrt(2)/4), # nsimplifies as 2*2**(141/299)*3**(206/299)*5**(205/299)*7**(37/299)/21 + 4*LambertW(-sqrt(2)/4, -1)] + assert solve(x*log(x) + 3*x + 1, x) == \ + [exp(-3 + LambertW(-exp(3)))] + assert solve(-x**2 + 2**x, x) == [2, 4, -2*LambertW(log(2)/2)/log(2)] + assert solve(x**2 - 2**x, x) == [2, 4, -2*LambertW(log(2)/2)/log(2)] + ans = solve(3*x + 5 + 2**(-5*x + 3), x) + assert len(ans) == 1 and ans[0].expand() == \ + Rational(-5, 3) + LambertW(-10240*root(2, 3)*log(2)/3)/(5*log(2)) + assert solve(5*x - 1 + 3*exp(2 - 7*x), x) == \ + [Rational(1, 5) + LambertW(-21*exp(Rational(3, 5))/5)/7] + assert solve((log(x) + x).subs(x, x**2 + 1)) == [ + -I*sqrt(-LambertW(1) + 1), sqrt(-1 + LambertW(1))] + # check collection + ax = a**(3*x + 5) + ans = solve(3*log(ax) + b*log(ax) + ax, x) + x0 = 1/log(a) + x1 = sqrt(3)*I + x2 = b + 3 + x3 = x2*LambertW(1/x2)/a**5 + x4 = x3**Rational(1, 3)/2 + assert ans == [ + x0*log(x4*(-x1 - 1)), + x0*log(x4*(x1 - 1)), + x0*log(x3)/3] + x1 = LambertW(Rational(1, 3)) + x2 = a**(-5) + x3 = -3**Rational(1, 3) + x4 = 3**Rational(5, 6)*I + x5 = x1**Rational(1, 3)*x2**Rational(1, 3)/2 + ans = solve(3*log(ax) + ax, x) + assert ans == [ + x0*log(3*x1*x2)/3, + x0*log(x5*(x3 - x4)), + x0*log(x5*(x3 + x4))] + # coverage + p = symbols('p', positive=True) + eq = 4*2**(2*p + 3) - 2*p - 3 + assert _solve_lambert(eq, p, _filtered_gens(Poly(eq), p)) == [ + Rational(-3, 2) - LambertW(-4*log(2))/(2*log(2))] + assert set(solve(3**cos(x) - cos(x)**3)) == { + acos(3), acos(-3*LambertW(-log(3)/3)/log(3))} + # should give only one solution after using `uniq` + assert solve(2*log(x) - 2*log(z) + log(z + log(x) + log(z)), x) == [ + exp(-z + LambertW(2*z**4*exp(2*z))/2)/z] + # cases when p != S.One + # issue 4271 + ans = solve((a/x + exp(x/2)).diff(x, 2), x) + x0 = (-a)**Rational(1, 3) + x1 = sqrt(3)*I + x2 = x0/6 + assert ans == [ + 6*LambertW(x0/3), + 6*LambertW(x2*(-x1 - 1)), + 6*LambertW(x2*(x1 - 1))] + assert solve((1/x + exp(x/2)).diff(x, 2), x) == \ + [6*LambertW(Rational(-1, 3)), 6*LambertW(Rational(1, 6) - sqrt(3)*I/6), \ + 6*LambertW(Rational(1, 6) + sqrt(3)*I/6), 6*LambertW(Rational(-1, 3), -1)] + assert solve(x**2 - y**2/exp(x), x, y, dict=True) == \ + [{x: 2*LambertW(-y/2)}, {x: 2*LambertW(y/2)}] + # this is slow but not exceedingly slow + assert solve((x**3)**(x/2) + pi/2, x) == [ + exp(LambertW(-2*log(2)/3 + 2*log(pi)/3 + I*pi*Rational(2, 3)))] + + # issue 23253 + assert solve((1/log(sqrt(x) + 2)**2 - 1/x)) == [ + (LambertW(-exp(-2), -1) + 2)**2] + assert solve((1/log(1/sqrt(x) + 2)**2 - x)) == [ + (LambertW(-exp(-2), -1) + 2)**-2] + assert solve((1/log(x**2 + 2)**2 - x**-4)) == [ + -I*sqrt(2 - LambertW(exp(2))), + -I*sqrt(LambertW(-exp(-2)) + 2), + sqrt(-2 - LambertW(-exp(-2))), + sqrt(-2 + LambertW(exp(2))), + -sqrt(-2 - LambertW(-exp(-2), -1)), + sqrt(-2 - LambertW(-exp(-2), -1))] + + +def test_rewrite_trig(): + assert solve(sin(x) + tan(x)) == [0, -pi, pi, 2*pi] + assert solve(sin(x) + sec(x)) == [ + -2*atan(Rational(-1, 2) + sqrt(2)*sqrt(1 - sqrt(3)*I)/2 + sqrt(3)*I/2), + 2*atan(S.Half - sqrt(2)*sqrt(1 + sqrt(3)*I)/2 + sqrt(3)*I/2), 2*atan(S.Half + + sqrt(2)*sqrt(1 + sqrt(3)*I)/2 + sqrt(3)*I/2), 2*atan(S.Half - + sqrt(3)*I/2 + sqrt(2)*sqrt(1 - sqrt(3)*I)/2)] + assert solve(sinh(x) + tanh(x)) == [0, I*pi] + + # issue 6157 + assert solve(2*sin(x) - cos(x), x) == [atan(S.Half)] + + +@XFAIL +def test_rewrite_trigh(): + # if this import passes then the test below should also pass + from sympy.functions.elementary.hyperbolic import sech + assert solve(sinh(x) + sech(x)) == [ + 2*atanh(Rational(-1, 2) + sqrt(5)/2 - sqrt(-2*sqrt(5) + 2)/2), + 2*atanh(Rational(-1, 2) + sqrt(5)/2 + sqrt(-2*sqrt(5) + 2)/2), + 2*atanh(-sqrt(5)/2 - S.Half + sqrt(2 + 2*sqrt(5))/2), + 2*atanh(-sqrt(2 + 2*sqrt(5))/2 - sqrt(5)/2 - S.Half)] + + +def test_uselogcombine(): + eq = z - log(x) + log(y/(x*(-1 + y**2/x**2))) + assert solve(eq, x, force=True) == [-sqrt(y*(y - exp(z))), sqrt(y*(y - exp(z)))] + assert solve(log(x + 3) + log(1 + 3/x) - 3) in [ + [-3 + sqrt(-12 + exp(3))*exp(Rational(3, 2))/2 + exp(3)/2, + -sqrt(-12 + exp(3))*exp(Rational(3, 2))/2 - 3 + exp(3)/2], + [-3 + sqrt(-36 + (-exp(3) + 6)**2)/2 + exp(3)/2, + -3 - sqrt(-36 + (-exp(3) + 6)**2)/2 + exp(3)/2], + ] + assert solve(log(exp(2*x) + 1) + log(-tanh(x) + 1) - log(2)) == [] + + +def test_atan2(): + assert solve(atan2(x, 2) - pi/3, x) == [2*sqrt(3)] + + +def test_errorinverses(): + assert solve(erf(x) - y, x) == [erfinv(y)] + assert solve(erfinv(x) - y, x) == [erf(y)] + assert solve(erfc(x) - y, x) == [erfcinv(y)] + assert solve(erfcinv(x) - y, x) == [erfc(y)] + + +def test_issue_2725(): + R = Symbol('R') + eq = sqrt(2)*R*sqrt(1/(R + 1)) + (R + 1)*(sqrt(2)*sqrt(1/(R + 1)) - 1) + sol = solve(eq, R, set=True)[1] + assert sol == {(Rational(5, 3) + (Rational(-1, 2) - sqrt(3)*I/2)*(Rational(251, 27) + + sqrt(111)*I/9)**Rational(1, 3) + 40/(9*((Rational(-1, 2) - sqrt(3)*I/2)*(Rational(251, 27) + + sqrt(111)*I/9)**Rational(1, 3))),), (Rational(5, 3) + 40/(9*(Rational(251, 27) + + sqrt(111)*I/9)**Rational(1, 3)) + (Rational(251, 27) + sqrt(111)*I/9)**Rational(1, 3),)} + + +def test_issue_5114_6611(): + # See that it doesn't hang; this solves in about 2 seconds. + # Also check that the solution is relatively small. + # Note: the system in issue 6611 solves in about 5 seconds and has + # an op-count of 138336 (with simplify=False). + b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r = symbols('b:r') + eqs = Matrix([ + [b - c/d + r/d], [c*(1/g + 1/e + 1/d) - f/g - r/d], + [-c/g + f*(1/j + 1/i + 1/g) - h/i], [-f/i + h*(1/m + 1/l + 1/i) - k/m], + [-h/m + k*(1/p + 1/o + 1/m) - n/p], [-k/p + n*(1/q + 1/p)]]) + v = Matrix([f, h, k, n, b, c]) + ans = solve(list(eqs), list(v), simplify=False) + # If time is taken to simplify then then 2617 below becomes + # 1168 and the time is about 50 seconds instead of 2. + assert sum(s.count_ops() for s in ans.values()) <= 3270 + + +def test_det_quick(): + m = Matrix(3, 3, symbols('a:9')) + assert m.det() == det_quick(m) # calls det_perm + m[0, 0] = 1 + assert m.det() == det_quick(m) # calls det_minor + m = Matrix(3, 3, list(range(9))) + assert m.det() == det_quick(m) # defaults to .det() + # make sure they work with Sparse + s = SparseMatrix(2, 2, (1, 2, 1, 4)) + assert det_perm(s) == det_minor(s) == s.det() + + +def test_real_imag_splitting(): + a, b = symbols('a b', real=True) + assert solve(sqrt(a**2 + b**2) - 3, a) == \ + [-sqrt(-b**2 + 9), sqrt(-b**2 + 9)] + a, b = symbols('a b', imaginary=True) + assert solve(sqrt(a**2 + b**2) - 3, a) == [] + + +def test_issue_7110(): + y = -2*x**3 + 4*x**2 - 2*x + 5 + assert any(ask(Q.real(i)) for i in solve(y)) + + +def test_units(): + assert solve(1/x - 1/(2*cm)) == [2*cm] + + +def test_issue_7547(): + A, B, V = symbols('A,B,V') + eq1 = Eq(630.26*(V - 39.0)*V*(V + 39) - A + B, 0) + eq2 = Eq(B, 1.36*10**8*(V - 39)) + eq3 = Eq(A, 5.75*10**5*V*(V + 39.0)) + sol = Matrix(nsolve(Tuple(eq1, eq2, eq3), [A, B, V], (0, 0, 0))) + assert str(sol) == str(Matrix( + [['4442890172.68209'], + ['4289299466.1432'], + ['70.5389666628177']])) + + +def test_issue_7895(): + r = symbols('r', real=True) + assert solve(sqrt(r) - 2) == [4] + + +def test_issue_2777(): + # the equations represent two circles + x, y = symbols('x y', real=True) + e1, e2 = sqrt(x**2 + y**2) - 10, sqrt(y**2 + (-x + 10)**2) - 3 + a, b = Rational(191, 20), 3*sqrt(391)/20 + ans = [(a, -b), (a, b)] + assert solve((e1, e2), (x, y)) == ans + assert solve((e1, e2/(x - a)), (x, y)) == [] + # make the 2nd circle's radius be -3 + e2 += 6 + assert solve((e1, e2), (x, y)) == [] + assert solve((e1, e2), (x, y), check=False) == ans + + +def test_issue_7322(): + number = 5.62527e-35 + assert solve(x - number, x)[0] == number + + +def test_nsolve(): + raises(ValueError, lambda: nsolve(x, (-1, 1), method='bisect')) + raises(TypeError, lambda: nsolve((x - y + 3,x + y,z - y),(x,y,z),(-50,50))) + raises(TypeError, lambda: nsolve((x + y, x - y), (0, 1))) + raises(TypeError, lambda: nsolve(x < 0.5, x, 1)) + + +@slow +def test_high_order_multivariate(): + assert len(solve(a*x**3 - x + 1, x)) == 3 + assert len(solve(a*x**4 - x + 1, x)) == 4 + assert solve(a*x**5 - x + 1, x) == [] # incomplete solution allowed + raises(NotImplementedError, lambda: + solve(a*x**5 - x + 1, x, incomplete=False)) + + # result checking must always consider the denominator and CRootOf + # must be checked, too + d = x**5 - x + 1 + assert solve(d*(1 + 1/d)) == [CRootOf(d + 1, i) for i in range(5)] + d = x - 1 + assert solve(d*(2 + 1/d)) == [S.Half] + + +def test_base_0_exp_0(): + assert solve(0**x - 1) == [0] + assert solve(0**(x - 2) - 1) == [2] + assert solve(S('x*(1/x**0 - x)', evaluate=False)) == \ + [0, 1] + + +def test__simple_dens(): + assert _simple_dens(1/x**0, [x]) == set() + assert _simple_dens(1/x**y, [x]) == {x**y} + assert _simple_dens(1/root(x, 3), [x]) == {x} + + +def test_issue_8755(): + # This tests two things: that if full unrad is attempted and fails + # the solution should still be found; also it tests the use of + # keyword `composite`. + assert len(solve(sqrt(y)*x + x**3 - 1, x)) == 3 + assert len(solve(-512*y**3 + 1344*(x + 2)**Rational(1, 3)*y**2 - + 1176*(x + 2)**Rational(2, 3)*y - 169*x + 686, y, _unrad=False)) == 3 + + +@slow +def test_issue_8828(): + x1 = 0 + y1 = -620 + r1 = 920 + x2 = 126 + y2 = 276 + x3 = 51 + y3 = 205 + r3 = 104 + v = x, y, z + + f1 = (x - x1)**2 + (y - y1)**2 - (r1 - z)**2 + f2 = (x - x2)**2 + (y - y2)**2 - z**2 + f3 = (x - x3)**2 + (y - y3)**2 - (r3 - z)**2 + F = f1,f2,f3 + + g1 = sqrt((x - x1)**2 + (y - y1)**2) + z - r1 + g2 = f2 + g3 = sqrt((x - x3)**2 + (y - y3)**2) + z - r3 + G = g1,g2,g3 + + A = solve(F, v) + B = solve(G, v) + C = solve(G, v, manual=True) + + p, q, r = [{tuple(i.evalf(2) for i in j) for j in R} for R in [A, B, C]] + assert p == q == r + + +def test_issue_2840_8155(): + # with parameter-free solutions (i.e. no `n`), we want to avoid + # excessive periodic solutions + assert solve(sin(3*x) + sin(6*x)) == [0, -2*pi/9, 2*pi/9] + assert solve(sin(300*x) + sin(600*x)) == [0, -pi/450, pi/450] + assert solve(2*sin(x) - 2*sin(2*x)) == [0, -pi/3, pi/3] + + +def test_issue_9567(): + assert solve(1 + 1/(x - 1)) == [0] + + +def test_issue_11538(): + assert solve(x + E) == [-E] + assert solve(x**2 + E) == [-I*sqrt(E), I*sqrt(E)] + assert solve(x**3 + 2*E) == [ + -cbrt(2 * E), + cbrt(2)*cbrt(E)/2 - cbrt(2)*sqrt(3)*I*cbrt(E)/2, + cbrt(2)*cbrt(E)/2 + cbrt(2)*sqrt(3)*I*cbrt(E)/2] + assert solve([x + 4, y + E], x, y) == {x: -4, y: -E} + assert solve([x**2 + 4, y + E], x, y) == [ + (-2*I, -E), (2*I, -E)] + + e1 = x - y**3 + 4 + e2 = x + y + 4 + 4 * E + assert len(solve([e1, e2], x, y)) == 3 + + +@slow +def test_issue_12114(): + a, b, c, d, e, f, g = symbols('a,b,c,d,e,f,g') + terms = [1 + a*b + d*e, 1 + a*c + d*f, 1 + b*c + e*f, + g - a**2 - d**2, g - b**2 - e**2, g - c**2 - f**2] + sol = solve(terms, [a, b, c, d, e, f, g], dict=True) + s = sqrt(-f**2 - 1) + s2 = sqrt(2 - f**2) + s3 = sqrt(6 - 3*f**2) + s4 = sqrt(3)*f + s5 = sqrt(3)*s2 + assert sol == [ + {a: -s, b: -s, c: -s, d: f, e: f, g: -1}, + {a: s, b: s, c: s, d: f, e: f, g: -1}, + {a: -s4/2 - s2/2, b: s4/2 - s2/2, c: s2, + d: -f/2 + s3/2, e: -f/2 - s5/2, g: 2}, + {a: -s4/2 + s2/2, b: s4/2 + s2/2, c: -s2, + d: -f/2 - s3/2, e: -f/2 + s5/2, g: 2}, + {a: s4/2 - s2/2, b: -s4/2 - s2/2, c: s2, + d: -f/2 - s3/2, e: -f/2 + s5/2, g: 2}, + {a: s4/2 + s2/2, b: -s4/2 + s2/2, c: -s2, + d: -f/2 + s3/2, e: -f/2 - s5/2, g: 2}] + + +def test_inf(): + assert solve(1 - oo*x) == [] + assert solve(oo*x, x) == [] + assert solve(oo*x - oo, x) == [] + + +def test_issue_12448(): + f = Function('f') + fun = [f(i) for i in range(15)] + sym = symbols('x:15') + reps = dict(zip(fun, sym)) + + (x, y, z), c = sym[:3], sym[3:] + ssym = solve([c[4*i]*x + c[4*i + 1]*y + c[4*i + 2]*z + c[4*i + 3] + for i in range(3)], (x, y, z)) + + (x, y, z), c = fun[:3], fun[3:] + sfun = solve([c[4*i]*x + c[4*i + 1]*y + c[4*i + 2]*z + c[4*i + 3] + for i in range(3)], (x, y, z)) + + assert sfun[fun[0]].xreplace(reps).count_ops() == \ + ssym[sym[0]].count_ops() + + +def test_denoms(): + assert denoms(x/2 + 1/y) == {2, y} + assert denoms(x/2 + 1/y, y) == {y} + assert denoms(x/2 + 1/y, [y]) == {y} + assert denoms(1/x + 1/y + 1/z, [x, y]) == {x, y} + assert denoms(1/x + 1/y + 1/z, x, y) == {x, y} + assert denoms(1/x + 1/y + 1/z, {x, y}) == {x, y} + + +def test_issue_12476(): + x0, x1, x2, x3, x4, x5 = symbols('x0 x1 x2 x3 x4 x5') + eqns = [x0**2 - x0, x0*x1 - x1, x0*x2 - x2, x0*x3 - x3, x0*x4 - x4, x0*x5 - x5, + x0*x1 - x1, -x0/3 + x1**2 - 2*x2/3, x1*x2 - x1/3 - x2/3 - x3/3, + x1*x3 - x2/3 - x3/3 - x4/3, x1*x4 - 2*x3/3 - x5/3, x1*x5 - x4, x0*x2 - x2, + x1*x2 - x1/3 - x2/3 - x3/3, -x0/6 - x1/6 + x2**2 - x2/6 - x3/3 - x4/6, + -x1/6 + x2*x3 - x2/3 - x3/6 - x4/6 - x5/6, x2*x4 - x2/3 - x3/3 - x4/3, + x2*x5 - x3, x0*x3 - x3, x1*x3 - x2/3 - x3/3 - x4/3, + -x1/6 + x2*x3 - x2/3 - x3/6 - x4/6 - x5/6, + -x0/6 - x1/6 - x2/6 + x3**2 - x3/3 - x4/6, -x1/3 - x2/3 + x3*x4 - x3/3, + -x2 + x3*x5, x0*x4 - x4, x1*x4 - 2*x3/3 - x5/3, x2*x4 - x2/3 - x3/3 - x4/3, + -x1/3 - x2/3 + x3*x4 - x3/3, -x0/3 - 2*x2/3 + x4**2, -x1 + x4*x5, x0*x5 - x5, + x1*x5 - x4, x2*x5 - x3, -x2 + x3*x5, -x1 + x4*x5, -x0 + x5**2, x0 - 1] + sols = [{x0: 1, x3: Rational(1, 6), x2: Rational(1, 6), x4: Rational(-2, 3), x1: Rational(-2, 3), x5: 1}, + {x0: 1, x3: S.Half, x2: Rational(-1, 2), x4: 0, x1: 0, x5: -1}, + {x0: 1, x3: Rational(-1, 3), x2: Rational(-1, 3), x4: Rational(1, 3), x1: Rational(1, 3), x5: 1}, + {x0: 1, x3: 1, x2: 1, x4: 1, x1: 1, x5: 1}, + {x0: 1, x3: Rational(-1, 3), x2: Rational(1, 3), x4: sqrt(5)/3, x1: -sqrt(5)/3, x5: -1}, + {x0: 1, x3: Rational(-1, 3), x2: Rational(1, 3), x4: -sqrt(5)/3, x1: sqrt(5)/3, x5: -1}] + + assert solve(eqns) == sols + + +def test_issue_13849(): + t = symbols('t') + assert solve((t*(sqrt(5) + sqrt(2)) - sqrt(2), t), t) == [] + + +def test_issue_14860(): + from sympy.physics.units import newton, kilo + assert solve(8*kilo*newton + x + y, x) == [-8000*newton - y] + + +def test_issue_14721(): + k, h, a, b = symbols(':4') + assert solve([ + -1 + (-k + 1)**2/b**2 + (-h - 1)**2/a**2, + -1 + (-k + 1)**2/b**2 + (-h + 1)**2/a**2, + h, k + 2], h, k, a, b) == [ + (0, -2, -b*sqrt(1/(b**2 - 9)), b), + (0, -2, b*sqrt(1/(b**2 - 9)), b)] + assert solve([ + h, h/a + 1/b**2 - 2, -h/2 + 1/b**2 - 2], a, h, b) == [ + (a, 0, -sqrt(2)/2), (a, 0, sqrt(2)/2)] + assert solve((a + b**2 - 1, a + b**2 - 2)) == [] + + +def test_issue_14779(): + x = symbols('x', real=True) + assert solve(sqrt(x**4 - 130*x**2 + 1089) + sqrt(x**4 - 130*x**2 + + 3969) - 96*Abs(x)/x,x) == [sqrt(130)] + + +def test_issue_15307(): + assert solve((y - 2, Mul(x + 3,x - 2, evaluate=False))) == \ + [{x: -3, y: 2}, {x: 2, y: 2}] + assert solve((y - 2, Mul(3, x - 2, evaluate=False))) == \ + {x: 2, y: 2} + assert solve((y - 2, Add(x + 4, x - 2, evaluate=False))) == \ + {x: -1, y: 2} + eq1 = Eq(12513*x + 2*y - 219093, -5726*x - y) + eq2 = Eq(-2*x + 8, 2*x - 40) + assert solve([eq1, eq2]) == {x:12, y:75} + + +def test_issue_15415(): + assert solve(x - 3, x) == [3] + assert solve([x - 3], x) == {x:3} + assert solve(Eq(y + 3*x**2/2, y + 3*x), y) == [] + assert solve([Eq(y + 3*x**2/2, y + 3*x)], y) == [] + assert solve([Eq(y + 3*x**2/2, y + 3*x), Eq(x, 1)], y) == [] + + +@slow +def test_issue_15731(): + # f(x)**g(x)=c + assert solve(Eq((x**2 - 7*x + 11)**(x**2 - 13*x + 42), 1)) == [2, 3, 4, 5, 6, 7] + assert solve((x)**(x + 4) - 4) == [-2] + assert solve((-x)**(-x + 4) - 4) == [2] + assert solve((x**2 - 6)**(x**2 - 2) - 4) == [-2, 2] + assert solve((x**2 - 2*x - 1)**(x**2 - 3) - 1/(1 - 2*sqrt(2))) == [sqrt(2)] + assert solve(x**(x + S.Half) - 4*sqrt(2)) == [S(2)] + assert solve((x**2 + 1)**x - 25) == [2] + assert solve(x**(2/x) - 2) == [2, 4] + assert solve((x/2)**(2/x) - sqrt(2)) == [4, 8] + assert solve(x**(x + S.Half) - Rational(9, 4)) == [Rational(3, 2)] + # a**g(x)=c + assert solve((-sqrt(sqrt(2)))**x - 2) == [4, log(2)/(log(2**Rational(1, 4)) + I*pi)] + assert solve((sqrt(2))**x - sqrt(sqrt(2))) == [S.Half] + assert solve((-sqrt(2))**x + 2*(sqrt(2))) == [3, + (3*log(2)**2 + 4*pi**2 - 4*I*pi*log(2))/(log(2)**2 + 4*pi**2)] + assert solve((sqrt(2))**x - 2*(sqrt(2))) == [3] + assert solve(I**x + 1) == [2] + assert solve((1 + I)**x - 2*I) == [2] + assert solve((sqrt(2) + sqrt(3))**x - (2*sqrt(6) + 5)**Rational(1, 3)) == [Rational(2, 3)] + # bases of both sides are equal + b = Symbol('b') + assert solve(b**x - b**2, x) == [2] + assert solve(b**x - 1/b, x) == [-1] + assert solve(b**x - b, x) == [1] + b = Symbol('b', positive=True) + assert solve(b**x - b**2, x) == [2] + assert solve(b**x - 1/b, x) == [-1] + + +def test_issue_10933(): + assert solve(x**4 + y*(x + 0.1), x) # doesn't fail + assert solve(I*x**4 + x**3 + x**2 + 1.) # doesn't fail + + +def test_Abs_handling(): + x = symbols('x', real=True) + assert solve(abs(x/y), x) == [0] + + +def test_issue_7982(): + x = Symbol('x') + # Test that no exception happens + assert solve([2*x**2 + 5*x + 20 <= 0, x >= 1.5], x) is S.false + # From #8040 + assert solve([x**3 - 8.08*x**2 - 56.48*x/5 - 106 >= 0, x - 1 <= 0], [x]) is S.false + + +def test_issue_14645(): + x, y = symbols('x y') + assert solve([x*y - x - y, x*y - x - y], [x, y]) == [(y/(y - 1), y)] + + +def test_issue_12024(): + x, y = symbols('x y') + assert solve(Piecewise((0.0, x < 0.1), (x, x >= 0.1)) - y) == \ + [{y: Piecewise((0.0, x < 0.1), (x, True))}] + + +def test_issue_17452(): + assert solve((7**x)**x + pi, x) == [-sqrt(log(pi) + I*pi)/sqrt(log(7)), + sqrt(log(pi) + I*pi)/sqrt(log(7))] + assert solve(x**(x/11) + pi/11, x) == [exp(LambertW(-11*log(11) + 11*log(pi) + 11*I*pi))] + + +def test_issue_17799(): + assert solve(-erf(x**(S(1)/3))**pi + I, x) == [] + + +def test_issue_17650(): + x = Symbol('x', real=True) + assert solve(abs(abs(x**2 - 1) - x) - x) == [1, -1 + sqrt(2), 1 + sqrt(2)] + + +def test_issue_17882(): + eq = -8*x**2/(9*(x**2 - 1)**(S(4)/3)) + 4/(3*(x**2 - 1)**(S(1)/3)) + assert unrad(eq) is None + + +def test_issue_17949(): + assert solve(exp(+x+x**2), x) == [] + assert solve(exp(-x+x**2), x) == [] + assert solve(exp(+x-x**2), x) == [] + assert solve(exp(-x-x**2), x) == [] + + +def test_issue_10993(): + assert solve(Eq(binomial(x, 2), 3)) == [-2, 3] + assert solve(Eq(pow(x, 2) + binomial(x, 3), x)) == [-4, 0, 1] + assert solve(Eq(binomial(x, 2), 0)) == [0, 1] + assert solve(a+binomial(x, 3), a) == [-binomial(x, 3)] + assert solve(x-binomial(a, 3) + binomial(y, 2) + sin(a), x) == [-sin(a) + binomial(a, 3) - binomial(y, 2)] + assert solve((x+1)-binomial(x+1, 3), x) == [-2, -1, 3] + + +def test_issue_11553(): + eq1 = x + y + 1 + eq2 = x + GoldenRatio + assert solve([eq1, eq2], x, y) == {x: -GoldenRatio, y: -1 + GoldenRatio} + eq3 = x + 2 + TribonacciConstant + assert solve([eq1, eq3], x, y) == {x: -2 - TribonacciConstant, y: 1 + TribonacciConstant} + + +def test_issue_19113_19102(): + t = S(1)/3 + solve(cos(x)**5-sin(x)**5) + assert solve(4*cos(x)**3 - 2*sin(x)**3) == [ + atan(2**(t)), -atan(2**(t)*(1 - sqrt(3)*I)/2), + -atan(2**(t)*(1 + sqrt(3)*I)/2)] + h = S.Half + assert solve(cos(x)**2 + sin(x)) == [ + 2*atan(-h + sqrt(5)/2 + sqrt(2)*sqrt(1 - sqrt(5))/2), + -2*atan(h + sqrt(5)/2 + sqrt(2)*sqrt(1 + sqrt(5))/2), + -2*atan(-sqrt(5)/2 + h + sqrt(2)*sqrt(1 - sqrt(5))/2), + -2*atan(-sqrt(2)*sqrt(1 + sqrt(5))/2 + h + sqrt(5)/2)] + assert solve(3*cos(x) - sin(x)) == [atan(3)] + + +def test_issue_19509(): + a = S(3)/4 + b = S(5)/8 + c = sqrt(5)/8 + d = sqrt(5)/4 + assert solve(1/(x -1)**5 - 1) == [2, + -d + a - sqrt(-b + c), + -d + a + sqrt(-b + c), + d + a - sqrt(-b - c), + d + a + sqrt(-b - c)] + +def test_issue_20747(): + THT, HT, DBH, dib, c0, c1, c2, c3, c4 = symbols('THT HT DBH dib c0 c1 c2 c3 c4') + f = DBH*c3 + THT*c4 + c2 + rhs = 1 - ((HT - 1)/(THT - 1))**c1*(1 - exp(c0/f)) + eq = dib - DBH*(c0 - f*log(rhs)) + term = ((1 - exp((DBH*c0 - dib)/(DBH*(DBH*c3 + THT*c4 + c2)))) + / (1 - exp(c0/(DBH*c3 + THT*c4 + c2)))) + sol = [THT*term**(1/c1) - term**(1/c1) + 1] + assert solve(eq, HT) == sol + + +def test_issue_27001(): + assert solve((x, x**2), (x, y, z), dict=True) == [{x: 0}] + s = a1, a2, a3, a4, a5 = symbols('a1:6') + eqs = [8*a1**4*a2 + 4*a1**2*a2**3 - 8*a1**2*a2*a4 + a2**5/2 - 2*a2**3*a4 + + 8*a2*a3**2 + 2*a2*a4**2 + 8*a2*a5, 12*a1**4 + 6*a1**2*a2**2 - + 8*a1**2*a4 + 3*a2**4/4 - 2*a2**2*a4 + 4*a3**2 + a4**2 + 4*a5, 16*a1**3 + + 4*a1*a2**2 - 8*a1*a4, -8*a1**2*a2 - 2*a2**3 + 4*a2*a4] + sol = [{a4: 2*a1**2 + a2**2/2, a5: -a3**2}, {a1: 0, a2: 0, a5: -a3**2 - a4**2/4}] + assert solve(eqs, s, dict=True) == sol + assert (g:=solve(groebner(eqs, s), dict=True)) == sol, g + + +def test_issue_20902(): + f = (t / ((1 + t) ** 2)) + assert solve(f.subs({t: 3 * x + 2}).diff(x) > 0, x) == (S(-1) < x) & (x < S(-1)/3) + assert solve(f.subs({t: 3 * x + 3}).diff(x) > 0, x) == (S(-4)/3 < x) & (x < S(-2)/3) + assert solve(f.subs({t: 3 * x + 4}).diff(x) > 0, x) == (S(-5)/3 < x) & (x < S(-1)) + assert solve(f.subs({t: 3 * x + 2}).diff(x) > 0, x) == (S(-1) < x) & (x < S(-1)/3) + + +def test_issue_21034(): + a = symbols('a', real=True) + system = [x - cosh(cos(4)), y - sinh(cos(a)), z - tanh(x)] + # constants inside hyperbolic functions should not be rewritten in terms of exp + assert solve(system, x, y, z) == [(cosh(cos(4)), sinh(cos(a)), tanh(cosh(cos(4))))] + # but if the variable of interest is present in a hyperbolic function, + # then it should be rewritten in terms of exp and solved further + newsystem = [(exp(x) - exp(-x)) - tanh(x)*(exp(x) + exp(-x)) + x - 5] + assert solve(newsystem, x) == {x: 5} + + +def test_issue_4886(): + z = a*sqrt(R**2*a**2 + R**2*b**2 - c**2)/(a**2 + b**2) + t = b*c/(a**2 + b**2) + sol = [((b*(t - z) - c)/(-a), t - z), ((b*(t + z) - c)/(-a), t + z)] + assert solve([x**2 + y**2 - R**2, a*x + b*y - c], x, y) == sol + + +def test_issue_6819(): + a, b, c, d = symbols('a b c d', positive=True) + assert solve(a*b**x - c*d**x, x) == [log(c/a)/log(b/d)] + + +def test_issue_17454(): + x = Symbol('x') + assert solve((1 - x - I)**4, x) == [1 - I] + + +def test_issue_21852(): + solution = [21 - 21*sqrt(2)/2] + assert solve(2*x + sqrt(2*x**2) - 21) == solution + + +def test_issue_21942(): + eq = -d + (a*c**(1 - e) + b**(1 - e)*(1 - a))**(1/(1 - e)) + sol = solve(eq, c, simplify=False, check=False) + assert sol == [((a*b**(1 - e) - b**(1 - e) + + d**(1 - e))/a)**(1/(1 - e))] + + +def test_solver_flags(): + root = solve(x**5 + x**2 - x - 1, cubics=False) + rad = solve(x**5 + x**2 - x - 1, cubics=True) + assert root != rad + + +def test_issue_22768(): + eq = 2*x**3 - 16*(y - 1)**6*z**3 + assert solve(eq.expand(), x, simplify=False + ) == [2*z*(y - 1)**2, z*(-1 + sqrt(3)*I)*(y - 1)**2, + -z*(1 + sqrt(3)*I)*(y - 1)**2] + + +def test_issue_22717(): + assert solve((-y**2 + log(y**2/x) + 2, -2*x*y + 2*x/y)) == [ + {y: -1, x: E}, {y: 1, x: E}] + + +def test_issue_25176(): + eq = (x - 5)**-8 - 3 + sol = solve(eq) + assert not any(eq.subs(x, i) for i in sol) + + +def test_issue_10169(): + eq = S(-8*a - x**5*(a + b + c + e) - x**4*(4*a - 2**Rational(3,4)*c + 4*c + + d + 2**Rational(3,4)*e + 4*e + k) - x**3*(-4*2**Rational(3,4)*c + sqrt(2)*c - + 2**Rational(3,4)*d + 4*d + sqrt(2)*e + 4*2**Rational(3,4)*e + 2**Rational(3,4)*k + 4*k) - + x**2*(4*sqrt(2)*c - 4*2**Rational(3,4)*d + sqrt(2)*d + 4*sqrt(2)*e + + sqrt(2)*k + 4*2**Rational(3,4)*k) - x*(2*a + 2*b + 4*sqrt(2)*d + + 4*sqrt(2)*k) + 5) + assert solve_undetermined_coeffs(eq, [a, b, c, d, e, k], x) == { + a: Rational(5,8), + b: Rational(-5,1032), + c: Rational(-40,129) - 5*2**Rational(3,4)/129 + 5*2**Rational(1,4)/1032, + d: -20*2**Rational(3,4)/129 - 10*sqrt(2)/129 - 5*2**Rational(1,4)/258, + e: Rational(-40,129) - 5*2**Rational(1,4)/1032 + 5*2**Rational(3,4)/129, + k: -10*sqrt(2)/129 + 5*2**Rational(1,4)/258 + 20*2**Rational(3,4)/129 + } + + +def test_solve_undetermined_coeffs_issue_23927(): + A, B, r, phi = symbols('A, B, r, phi') + e = Eq(A*sin(t) + B*cos(t), r*sin(t - phi)) + eq = (e.lhs - e.rhs).expand(trig=True) + soln = solve_undetermined_coeffs(eq, (r, phi), t) + assert soln == [{ + phi: 2*atan((A - sqrt(A**2 + B**2))/B), + r: (-A**2 + A*sqrt(A**2 + B**2) - B**2)/(A - sqrt(A**2 + B**2)) + }, { + phi: 2*atan((A + sqrt(A**2 + B**2))/B), + r: (A**2 + A*sqrt(A**2 + B**2) + B**2)/(A + sqrt(A**2 + B**2))/-1 + }] + +def test_issue_24368(): + # Ideally these would produce a solution, but for now just check that they + # don't fail with a RuntimeError + raises(NotImplementedError, lambda: solve(Mod(x**2, 49), x)) + s2 = Symbol('s2', integer=True, positive=True) + f = floor(s2/2 - S(1)/2) + raises(NotImplementedError, lambda: solve((Mod(f**2/(f + 1) + 2*f/(f + 1) + 1/(f + 1), 1))*f + Mod(f**2/(f + 1) + 2*f/(f + 1) + 1/(f + 1), 1), s2)) + + +def test_solve_Piecewise(): + assert [S(10)/3] == solve(3*Piecewise( + (S.NaN, x <= 0), + (20*x - 3*(x - 6)**2/2 - 176, (x >= 0) & (x >= 2) & (x>= 4) & (x >= 6) & (x < 10)), + (100 - 26*x, (x >= 0) & (x >= 2) & (x >= 4) & (x < 10)), + (16*x - 3*(x - 6)**2/2 - 176, (x >= 2) & (x >= 4) & (x >= 6) & (x < 10)), + (100 - 30*x, (x >= 2) & (x >= 4) & (x < 10)), + (30*x - 3*(x - 6)**2/2 - 196, (x>= 0) & (x >= 4) & (x >= 6) & (x < 10)), + (80 - 16*x, (x >= 0) & (x >= 4) & (x < 10)), + (26*x - 3*(x - 6)**2/2 - 196, (x >= 4) & (x >= 6) & (x < 10)), + (80 - 20*x, (x >= 4) & (x < 10)), + (40*x - 3*(x - 6)**2/2 - 256, (x >= 0) & (x >= 2) & (x >= 6) & (x < 10)), + (20 - 6*x, (x >= 0) & (x >= 2) & (x < 10)), + (36*x - 3*(x - 6)**2/2 - 256, (x >= 2) & (x >= 6) & (x < 10)), + (20 - 10*x, (x >= 2) & (x < 10)), + (50*x - 3*(x - 6)**2/2 - 276, (x >= 0) & (x >= 6) & (x < 10)), + (4*x, (x >= 0) & (x < 10)), + (46*x - 3*(x - 6)**2/2 - 276, (x >= 6) & (x < 10)), + (0, x < 10), # this will simplify away + (S.NaN,True))) diff --git a/.venv/lib/python3.12/site-packages/sympy/solvers/tests/test_solveset.py b/.venv/lib/python3.12/site-packages/sympy/solvers/tests/test_solveset.py new file mode 100644 index 0000000000000000000000000000000000000000..a1ba7a11e68ed518c4d83c050947b78756ade181 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/solvers/tests/test_solveset.py @@ -0,0 +1,3548 @@ +from math import isclose + +from sympy.calculus.util import stationary_points +from sympy.core.containers import Tuple +from sympy.core.function import (Function, Lambda, nfloat, diff) +from sympy.core.mod import Mod +from sympy.core.numbers import (E, I, Rational, oo, pi, Integer, all_close) +from sympy.core.relational import (Eq, Gt, Ne, Ge) +from sympy.core.singleton import S +from sympy.core.sorting import ordered +from sympy.core.symbol import (Dummy, Symbol, symbols) +from sympy.core.sympify import sympify +from sympy.functions.elementary.complexes import (Abs, arg, im, re, sign, conjugate) +from sympy.functions.elementary.exponential import (LambertW, exp, log) +from sympy.functions.elementary.hyperbolic import (HyperbolicFunction, + sinh, cosh, tanh, coth, sech, csch, asinh, acosh, atanh, acoth, asech, acsch) +from sympy.functions.elementary.miscellaneous import sqrt, Min, Max +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.elementary.trigonometric import ( + TrigonometricFunction, acos, acot, acsc, asec, asin, atan, atan2, + cos, cot, csc, sec, sin, tan) +from sympy.functions.special.error_functions import (erf, erfc, + erfcinv, erfinv) +from sympy.logic.boolalg import And +from sympy.matrices.dense import MutableDenseMatrix as Matrix +from sympy.matrices.immutable import ImmutableDenseMatrix +from sympy.polys.polytools import Poly +from sympy.polys.rootoftools import CRootOf +from sympy.sets.contains import Contains +from sympy.sets.conditionset import ConditionSet +from sympy.sets.fancysets import ImageSet, Range +from sympy.sets.sets import (Complement, FiniteSet, + Intersection, Interval, Union, imageset, ProductSet) +from sympy.simplify import simplify +from sympy.tensor.indexed import Indexed +from sympy.utilities.iterables import numbered_symbols + +from sympy.testing.pytest import (XFAIL, raises, skip, slow, SKIP, _both_exp_pow) +from sympy.core.random import verify_numerically as tn +from sympy.physics.units import cm + +from sympy.solvers import solve +from sympy.solvers.solveset import ( + solveset_real, domain_check, solveset_complex, linear_eq_to_matrix, + linsolve, _is_function_class_equation, invert_real, invert_complex, + _invert_trig_hyp_real, solveset, solve_decomposition, substitution, + nonlinsolve, solvify, + _is_finite_with_finite_vars, _transolve, _is_exponential, + _solve_exponential, _is_logarithmic, _is_lambert, + _solve_logarithm, _term_factors, _is_modular, NonlinearError) + +from sympy.abc import (a, b, c, d, e, f, g, h, i, j, k, l, m, n, q, r, + t, w, x, y, z) + + +def dumeq(i, j): + if type(i) in (list, tuple): + return all(dumeq(i, j) for i, j in zip(i, j)) + return i == j or i.dummy_eq(j) + + +def assert_close_ss(sol1, sol2): + """Test solutions with floats from solveset are close""" + sol1 = sympify(sol1) + sol2 = sympify(sol2) + assert isinstance(sol1, FiniteSet) + assert isinstance(sol2, FiniteSet) + assert len(sol1) == len(sol2) + assert all(isclose(v1, v2) for v1, v2 in zip(sol1, sol2)) + + +def assert_close_nl(sol1, sol2): + """Test solutions with floats from nonlinsolve are close""" + sol1 = sympify(sol1) + sol2 = sympify(sol2) + assert isinstance(sol1, FiniteSet) + assert isinstance(sol2, FiniteSet) + assert len(sol1) == len(sol2) + for s1, s2 in zip(sol1, sol2): + assert len(s1) == len(s2) + assert all(isclose(v1, v2) for v1, v2 in zip(s1, s2)) + + +@_both_exp_pow +def test_invert_real(): + x = Symbol('x', real=True) + + def ireal(x, s=S.Reals): + return Intersection(s, x) + + assert invert_real(exp(x), z, x) == (x, ireal(FiniteSet(log(z)))) + + y = Symbol('y', positive=True) + n = Symbol('n', real=True) + assert invert_real(x + 3, y, x) == (x, FiniteSet(y - 3)) + assert invert_real(x*3, y, x) == (x, FiniteSet(y / 3)) + + assert invert_real(exp(x), y, x) == (x, FiniteSet(log(y))) + assert invert_real(exp(3*x), y, x) == (x, FiniteSet(log(y) / 3)) + assert invert_real(exp(x + 3), y, x) == (x, FiniteSet(log(y) - 3)) + + assert invert_real(exp(x) + 3, y, x) == (x, ireal(FiniteSet(log(y - 3)))) + assert invert_real(exp(x)*3, y, x) == (x, FiniteSet(log(y / 3))) + + assert invert_real(log(x), y, x) == (x, FiniteSet(exp(y))) + assert invert_real(log(3*x), y, x) == (x, FiniteSet(exp(y) / 3)) + assert invert_real(log(x + 3), y, x) == (x, FiniteSet(exp(y) - 3)) + + assert invert_real(Abs(x), y, x) == (x, FiniteSet(y, -y)) + + assert invert_real(2**x, y, x) == (x, FiniteSet(log(y)/log(2))) + assert invert_real(2**exp(x), y, x) == (x, ireal(FiniteSet(log(log(y)/log(2))))) + + assert invert_real(x**2, y, x) == (x, FiniteSet(sqrt(y), -sqrt(y))) + assert invert_real(x**S.Half, y, x) == (x, FiniteSet(y**2)) + + raises(ValueError, lambda: invert_real(x, x, x)) + + # issue 21236 + assert invert_real(x**pi, y, x) == (x, FiniteSet(y**(1/pi))) + assert invert_real(x**pi, -E, x) == (x, S.EmptySet) + assert invert_real(x**Rational(3/2), 1000, x) == (x, FiniteSet(100)) + assert invert_real(x**1.0, 1, x) == (x**1.0, FiniteSet(1)) + + raises(ValueError, lambda: invert_real(S.One, y, x)) + + assert invert_real(x**31 + x, y, x) == (x**31 + x, FiniteSet(y)) + + lhs = x**31 + x + base_values = FiniteSet(y - 1, -y - 1) + assert invert_real(Abs(x**31 + x + 1), y, x) == (lhs, base_values) + + assert dumeq(invert_real(sin(x), y, x), (x, + ConditionSet(x, (S(-1) <= y) & (y <= S(1)), Union( + ImageSet(Lambda(n, 2*n*pi + asin(y)), S.Integers), + ImageSet(Lambda(n, pi*2*n + pi - asin(y)), S.Integers))))) + + assert dumeq(invert_real(sin(exp(x)), y, x), (x, + ConditionSet(x, (S(-1) <= y) & (y <= S(1)), Union( + ImageSet(Lambda(n, log(2*n*pi + asin(y))), S.Integers), + ImageSet(Lambda(n, log(pi*2*n + pi - asin(y))), S.Integers))))) + + assert dumeq(invert_real(csc(x), y, x), (x, + ConditionSet(x, ((S(1) <= y) & (y < oo)) | ((-oo < y) & (y <= S(-1))), + Union(ImageSet(Lambda(n, 2*n*pi + acsc(y)), S.Integers), + ImageSet(Lambda(n, 2*n*pi - acsc(y) + pi), S.Integers))))) + + assert dumeq(invert_real(csc(exp(x)), y, x), (x, + ConditionSet(x, ((S(1) <= y) & (y < oo)) | ((-oo < y) & (y <= S(-1))), + Union(ImageSet(Lambda(n, log(2*n*pi + acsc(y))), S.Integers), + ImageSet(Lambda(n, log(2*n*pi - acsc(y) + pi)), S.Integers))))) + + assert dumeq(invert_real(cos(x), y, x), (x, + ConditionSet(x, (S(-1) <= y) & (y <= S(1)), Union( + ImageSet(Lambda(n, 2*n*pi + acos(y)), S.Integers), + ImageSet(Lambda(n, 2*n*pi - acos(y)), S.Integers))))) + + assert dumeq(invert_real(cos(exp(x)), y, x), (x, + ConditionSet(x, (S(-1) <= y) & (y <= S(1)), Union( + ImageSet(Lambda(n, log(2*n*pi + acos(y))), S.Integers), + ImageSet(Lambda(n, log(2*n*pi - acos(y))), S.Integers))))) + + assert dumeq(invert_real(sec(x), y, x), (x, + ConditionSet(x, ((S(1) <= y) & (y < oo)) | ((-oo < y) & (y <= S(-1))), + Union(ImageSet(Lambda(n, 2*n*pi + asec(y)), S.Integers), \ + ImageSet(Lambda(n, 2*n*pi - asec(y)), S.Integers))))) + + assert dumeq(invert_real(sec(exp(x)), y, x), (x, + ConditionSet(x, ((S(1) <= y) & (y < oo)) | ((-oo < y) & (y <= S(-1))), + Union(ImageSet(Lambda(n, log(2*n*pi - asec(y))), S.Integers), + ImageSet(Lambda(n, log(2*n*pi + asec(y))), S.Integers))))) + + assert dumeq(invert_real(tan(x), y, x), (x, + ConditionSet(x, (-oo < y) & (y < oo), + ImageSet(Lambda(n, n*pi + atan(y)), S.Integers)))) + + assert dumeq(invert_real(tan(exp(x)), y, x), (x, + ConditionSet(x, (-oo < y) & (y < oo), + ImageSet(Lambda(n, log(n*pi + atan(y))), S.Integers)))) + + assert dumeq(invert_real(cot(x), y, x), (x, + ConditionSet(x, (-oo < y) & (y < oo), + ImageSet(Lambda(n, n*pi + acot(y)), S.Integers)))) + + assert dumeq(invert_real(cot(exp(x)), y, x), (x, + ConditionSet(x, (-oo < y) & (y < oo), + ImageSet(Lambda(n, log(n*pi + acot(y))), S.Integers)))) + + assert dumeq(invert_real(tan(tan(x)), y, x), + (x, ConditionSet(x, Eq(tan(tan(x)), y), S.Reals))) + # slight regression compared to previous result: + # (tan(x), imageset(Lambda(n, n*pi + atan(y)), S.Integers))) + + x = Symbol('x', positive=True) + assert invert_real(x**pi, y, x) == (x, FiniteSet(y**(1/pi))) + + r = Symbol('r', real=True) + p = Symbol('p', positive=True) + assert invert_real(sinh(x), r, x) == (x, FiniteSet(asinh(r))) + assert invert_real(sinh(log(x)), p, x) == (x, FiniteSet(exp(asinh(p)))) + + assert invert_real(cosh(x), r, x) == (x, Intersection( + FiniteSet(-acosh(r), acosh(r)), S.Reals)) + assert invert_real(cosh(x), p + 1, x) == (x, + FiniteSet(-acosh(p + 1), acosh(p + 1))) + + assert invert_real(tanh(x), r, x) == (x, Intersection(FiniteSet(atanh(r)), S.Reals)) + assert invert_real(coth(x), p+1, x) == (x, FiniteSet(acoth(p+1))) + assert invert_real(sech(x), r, x) == (x, Intersection( + FiniteSet(-asech(r), asech(r)), S.Reals)) + assert invert_real(csch(x), p, x) == (x, FiniteSet(acsch(p))) + + assert dumeq(invert_real(tanh(sin(x)), r, x), (x, + ConditionSet(x, (S(-1) <= atanh(r)) & (atanh(r) <= S(1)), Union( + ImageSet(Lambda(n, 2*n*pi + asin(atanh(r))), S.Integers), + ImageSet(Lambda(n, 2*n*pi - asin(atanh(r)) + pi), S.Integers))))) + + +def test_invert_trig_hyp_real(): + # check some codepaths that are not as easily reached otherwise + n = Dummy('n') + assert _invert_trig_hyp_real(cosh(x), Range(-5, 10, 1), x)[1].dummy_eq(Union( + ImageSet(Lambda(n, -acosh(n)), Range(1, 10, 1)), + ImageSet(Lambda(n, acosh(n)), Range(1, 10, 1)))) + assert _invert_trig_hyp_real(coth(x), Interval(-3, 2), x) == (x, Union( + Interval(-oo, -acoth(3)), Interval(acoth(2), oo))) + assert _invert_trig_hyp_real(tanh(x), Interval(-S.Half, 1), x) == (x, + Interval(-atanh(S.Half), oo)) + assert _invert_trig_hyp_real(sech(x), imageset(n, S.Half + n/3, S.Naturals0), x) == \ + (x, FiniteSet(-asech(S(1)/2), asech(S(1)/2), -asech(S(5)/6), asech(S(5)/6))) + assert _invert_trig_hyp_real(csch(x), S.Reals, x) == (x, + Union(Interval.open(-oo, 0), Interval.open(0, oo))) + + +def test_invert_complex(): + assert invert_complex(x + 3, y, x) == (x, FiniteSet(y - 3)) + assert invert_complex(x*3, y, x) == (x, FiniteSet(y / 3)) + assert invert_complex((x - 1)**3, 0, x) == (x, FiniteSet(1)) + + assert dumeq(invert_complex(exp(x), y, x), + (x, imageset(Lambda(n, I*(2*pi*n + arg(y)) + log(Abs(y))), S.Integers))) + + assert invert_complex(log(x), y, x) == (x, FiniteSet(exp(y))) + + raises(ValueError, lambda: invert_real(1, y, x)) + raises(ValueError, lambda: invert_complex(x, x, x)) + raises(ValueError, lambda: invert_complex(x, x, 1)) + + assert dumeq(invert_complex(sin(x), I, x), (x, Union( + ImageSet(Lambda(n, 2*n*pi + I*log(1 + sqrt(2))), S.Integers), + ImageSet(Lambda(n, 2*n*pi + pi - I*log(1 + sqrt(2))), S.Integers)))) + assert dumeq(invert_complex(cos(x), 1+I, x), (x, Union( + ImageSet(Lambda(n, 2*n*pi - acos(1 + I)), S.Integers), + ImageSet(Lambda(n, 2*n*pi + acos(1 + I)), S.Integers)))) + assert dumeq(invert_complex(tan(2*x), 1, x), (x, + ImageSet(Lambda(n, n*pi/2 + pi/8), S.Integers))) + assert dumeq(invert_complex(cot(x), 2*I, x), (x, + ImageSet(Lambda(n, n*pi - I*acoth(2)), S.Integers))) + + assert dumeq(invert_complex(sinh(x), 0, x), (x, Union( + ImageSet(Lambda(n, 2*n*I*pi), S.Integers), + ImageSet(Lambda(n, 2*n*I*pi + I*pi), S.Integers)))) + assert dumeq(invert_complex(cosh(x), 0, x), (x, Union( + ImageSet(Lambda(n, 2*n*I*pi + I*pi/2), S.Integers), + ImageSet(Lambda(n, 2*n*I*pi + 3*I*pi/2), S.Integers)))) + assert invert_complex(tanh(x), 1, x) == (x, S.EmptySet) + assert dumeq(invert_complex(tanh(x), a, x), (x, + ConditionSet(x, Ne(a, -1) & Ne(a, 1), + ImageSet(Lambda(n, n*I*pi + atanh(a)), S.Integers)))) + assert invert_complex(coth(x), 1, x) == (x, S.EmptySet) + assert dumeq(invert_complex(coth(x), a, x), (x, + ConditionSet(x, Ne(a, -1) & Ne(a, 1), + ImageSet(Lambda(n, n*I*pi + acoth(a)), S.Integers)))) + assert dumeq(invert_complex(sech(x), 2, x), (x, Union( + ImageSet(Lambda(n, 2*n*I*pi + I*pi/3), S.Integers), + ImageSet(Lambda(n, 2*n*I*pi + 5*I*pi/3), S.Integers)))) + + +def test_domain_check(): + assert domain_check(1/(1 + (1/(x+1))**2), x, -1) is False + assert domain_check(x**2, x, 0) is True + assert domain_check(x, x, oo) is False + assert domain_check(0, x, oo) is False + + +def test_issue_11536(): + assert solveset(0**x - 100, x, S.Reals) == S.EmptySet + assert solveset(0**x - 1, x, S.Reals) == FiniteSet(0) + + +def test_issue_17479(): + f = (x**2 + y**2)**2 + (x**2 + z**2)**2 - 2*(2*x**2 + y**2 + z**2) + fx = f.diff(x) + fy = f.diff(y) + fz = f.diff(z) + sol = nonlinsolve([fx, fy, fz], [x, y, z]) + assert len(sol) >= 4 and len(sol) <= 20 + # nonlinsolve has been giving a varying number of solutions + # (originally 18, then 20, now 19) due to various internal changes. + # Unfortunately not all the solutions are actually valid and some are + # redundant. Since the original issue was that an exception was raised, + # this first test only checks that nonlinsolve returns a "plausible" + # solution set. The next test checks the result for correctness. + + +@XFAIL +def test_issue_18449(): + x, y, z = symbols("x, y, z") + f = (x**2 + y**2)**2 + (x**2 + z**2)**2 - 2*(2*x**2 + y**2 + z**2) + fx = diff(f, x) + fy = diff(f, y) + fz = diff(f, z) + sol = nonlinsolve([fx, fy, fz], [x, y, z]) + for (xs, ys, zs) in sol: + d = {x: xs, y: ys, z: zs} + assert tuple(_.subs(d).simplify() for _ in (fx, fy, fz)) == (0, 0, 0) + # After simplification and removal of duplicate elements, there should + # only be 4 parametric solutions left: + # simplifiedsolutions = FiniteSet((sqrt(1 - z**2), z, z), + # (-sqrt(1 - z**2), z, z), + # (sqrt(1 - z**2), -z, z), + # (-sqrt(1 - z**2), -z, z)) + # TODO: Is the above solution set definitely complete? + + +def test_issue_21047(): + f = (2 - x)**2 + (sqrt(x - 1) - 1)**6 + assert solveset(f, x, S.Reals) == FiniteSet(2) + + f = (sqrt(x)-1)**2 + (sqrt(x)+1)**2 -2*x**2 + sqrt(2) + assert solveset(f, x, S.Reals) == FiniteSet( + S.Half - sqrt(2*sqrt(2) + 5)/2, S.Half + sqrt(2*sqrt(2) + 5)/2) + + +def test_is_function_class_equation(): + assert _is_function_class_equation(TrigonometricFunction, + tan(x), x) is True + assert _is_function_class_equation(TrigonometricFunction, + tan(x) - 1, x) is True + assert _is_function_class_equation(TrigonometricFunction, + tan(x) + sin(x), x) is True + assert _is_function_class_equation(TrigonometricFunction, + tan(x) + sin(x) - a, x) is True + assert _is_function_class_equation(TrigonometricFunction, + sin(x)*tan(x) + sin(x), x) is True + assert _is_function_class_equation(TrigonometricFunction, + sin(x)*tan(x + a) + sin(x), x) is True + assert _is_function_class_equation(TrigonometricFunction, + sin(x)*tan(x*a) + sin(x), x) is True + assert _is_function_class_equation(TrigonometricFunction, + a*tan(x) - 1, x) is True + assert _is_function_class_equation(TrigonometricFunction, + tan(x)**2 + sin(x) - 1, x) is True + assert _is_function_class_equation(TrigonometricFunction, + tan(x) + x, x) is False + assert _is_function_class_equation(TrigonometricFunction, + tan(x**2), x) is False + assert _is_function_class_equation(TrigonometricFunction, + tan(x**2) + sin(x), x) is False + assert _is_function_class_equation(TrigonometricFunction, + tan(x)**sin(x), x) is False + assert _is_function_class_equation(TrigonometricFunction, + tan(sin(x)) + sin(x), x) is False + assert _is_function_class_equation(HyperbolicFunction, + tanh(x), x) is True + assert _is_function_class_equation(HyperbolicFunction, + tanh(x) - 1, x) is True + assert _is_function_class_equation(HyperbolicFunction, + tanh(x) + sinh(x), x) is True + assert _is_function_class_equation(HyperbolicFunction, + tanh(x) + sinh(x) - a, x) is True + assert _is_function_class_equation(HyperbolicFunction, + sinh(x)*tanh(x) + sinh(x), x) is True + assert _is_function_class_equation(HyperbolicFunction, + sinh(x)*tanh(x + a) + sinh(x), x) is True + assert _is_function_class_equation(HyperbolicFunction, + sinh(x)*tanh(x*a) + sinh(x), x) is True + assert _is_function_class_equation(HyperbolicFunction, + a*tanh(x) - 1, x) is True + assert _is_function_class_equation(HyperbolicFunction, + tanh(x)**2 + sinh(x) - 1, x) is True + assert _is_function_class_equation(HyperbolicFunction, + tanh(x) + x, x) is False + assert _is_function_class_equation(HyperbolicFunction, + tanh(x**2), x) is False + assert _is_function_class_equation(HyperbolicFunction, + tanh(x**2) + sinh(x), x) is False + assert _is_function_class_equation(HyperbolicFunction, + tanh(x)**sinh(x), x) is False + assert _is_function_class_equation(HyperbolicFunction, + tanh(sinh(x)) + sinh(x), x) is False + + +def test_garbage_input(): + raises(ValueError, lambda: solveset_real([y], y)) + x = Symbol('x', real=True) + assert solveset_real(x, 1) == S.EmptySet + assert solveset_real(x - 1, 1) == FiniteSet(x) + assert solveset_real(x, pi) == S.EmptySet + assert solveset_real(x, x**2) == S.EmptySet + + raises(ValueError, lambda: solveset_complex([x], x)) + assert solveset_complex(x, pi) == S.EmptySet + + raises(ValueError, lambda: solveset((x, y), x)) + raises(ValueError, lambda: solveset(x + 1, S.Reals)) + raises(ValueError, lambda: solveset(x + 1, x, 2)) + + +def test_solve_mul(): + assert solveset_real((a*x + b)*(exp(x) - 3), x) == \ + Union({log(3)}, Intersection({-b/a}, S.Reals)) + anz = Symbol('anz', nonzero=True) + bb = Symbol('bb', real=True) + assert solveset_real((anz*x + bb)*(exp(x) - 3), x) == \ + FiniteSet(-bb/anz, log(3)) + assert solveset_real((2*x + 8)*(8 + exp(x)), x) == FiniteSet(S(-4)) + assert solveset_real(x/log(x), x) is S.EmptySet + + +def test_solve_invert(): + assert solveset_real(exp(x) - 3, x) == FiniteSet(log(3)) + assert solveset_real(log(x) - 3, x) == FiniteSet(exp(3)) + + assert solveset_real(3**(x + 2), x) == FiniteSet() + assert solveset_real(3**(2 - x), x) == FiniteSet() + + assert solveset_real(y - b*exp(a/x), x) == Intersection( + S.Reals, FiniteSet(a/log(y/b))) + + # issue 4504 + assert solveset_real(2**x - 10, x) == FiniteSet(1 + log(5)/log(2)) + + +def test_issue_25768(): + assert dumeq(solveset_real(sin(x) - S.Half, x), Union( + ImageSet(Lambda(n, pi*2*n + pi/6), S.Integers), + ImageSet(Lambda(n, pi*2*n + pi*5/6), S.Integers))) + n1 = solveset_real(sin(x) - 0.5, x).n(5) + n2 = solveset_real(sin(x) - S.Half, x).n(5) + # help pass despite fp differences + eq = [i.replace( + lambda x:x.is_Float, + lambda x:Rational(x).limit_denominator(1000)) for i in (n1, n2)] + assert dumeq(*eq),(n1,n2) + + +def test_errorinverses(): + assert solveset_real(erf(x) - S.Half, x) == \ + FiniteSet(erfinv(S.Half)) + assert solveset_real(erfinv(x) - 2, x) == \ + FiniteSet(erf(2)) + assert solveset_real(erfc(x) - S.One, x) == \ + FiniteSet(erfcinv(S.One)) + assert solveset_real(erfcinv(x) - 2, x) == FiniteSet(erfc(2)) + + +def test_solve_polynomial(): + x = Symbol('x', real=True) + y = Symbol('y', real=True) + assert solveset_real(3*x - 2, x) == FiniteSet(Rational(2, 3)) + + assert solveset_real(x**2 - 1, x) == FiniteSet(-S.One, S.One) + assert solveset_real(x - y**3, x) == FiniteSet(y ** 3) + + assert solveset_real(x**3 - 15*x - 4, x) == FiniteSet( + -2 + 3 ** S.Half, + S(4), + -2 - 3 ** S.Half) + + assert solveset_real(sqrt(x) - 1, x) == FiniteSet(1) + assert solveset_real(sqrt(x) - 2, x) == FiniteSet(4) + assert solveset_real(x**Rational(1, 4) - 2, x) == FiniteSet(16) + assert solveset_real(x**Rational(1, 3) - 3, x) == FiniteSet(27) + assert len(solveset_real(x**5 + x**3 + 1, x)) == 1 + assert len(solveset_real(-2*x**3 + 4*x**2 - 2*x + 6, x)) > 0 + assert solveset_real(x**6 + x**4 + I, x) is S.EmptySet + + +def test_return_root_of(): + f = x**5 - 15*x**3 - 5*x**2 + 10*x + 20 + s = list(solveset_complex(f, x)) + for root in s: + assert root.func == CRootOf + + # if one uses solve to get the roots of a polynomial that has a CRootOf + # solution, make sure that the use of nfloat during the solve process + # doesn't fail. Note: if you want numerical solutions to a polynomial + # it is *much* faster to use nroots to get them than to solve the + # equation only to get CRootOf solutions which are then numerically + # evaluated. So for eq = x**5 + 3*x + 7 do Poly(eq).nroots() rather + # than [i.n() for i in solve(eq)] to get the numerical roots of eq. + assert nfloat(list(solveset_complex(x**5 + 3*x**3 + 7, x))[0], + exponent=False) == CRootOf(x**5 + 3*x**3 + 7, 0).n() + + sol = list(solveset_complex(x**6 - 2*x + 2, x)) + assert all(isinstance(i, CRootOf) for i in sol) and len(sol) == 6 + + f = x**5 - 15*x**3 - 5*x**2 + 10*x + 20 + s = list(solveset_complex(f, x)) + for root in s: + assert root.func == CRootOf + + s = x**5 + 4*x**3 + 3*x**2 + Rational(7, 4) + assert solveset_complex(s, x) == \ + FiniteSet(*Poly(s*4, domain='ZZ').all_roots()) + + # Refer issue #7876 + eq = x*(x - 1)**2*(x + 1)*(x**6 - x + 1) + assert solveset_complex(eq, x) == \ + FiniteSet(-1, 0, 1, CRootOf(x**6 - x + 1, 0), + CRootOf(x**6 - x + 1, 1), + CRootOf(x**6 - x + 1, 2), + CRootOf(x**6 - x + 1, 3), + CRootOf(x**6 - x + 1, 4), + CRootOf(x**6 - x + 1, 5)) + + +def test_solveset_sqrt_1(): + assert solveset_real(sqrt(5*x + 6) - 2 - x, x) == \ + FiniteSet(-S.One, S(2)) + assert solveset_real(sqrt(x - 1) - x + 7, x) == FiniteSet(10) + assert solveset_real(sqrt(x - 2) - 5, x) == FiniteSet(27) + assert solveset_real(sqrt(x) - 2 - 5, x) == FiniteSet(49) + assert solveset_real(sqrt(x**3), x) == FiniteSet(0) + assert solveset_real(sqrt(x - 1), x) == FiniteSet(1) + assert solveset_real(sqrt((x-3)/x), x) == FiniteSet(3) + assert solveset_real(sqrt((x-3)/x)-Rational(1, 2), x) == \ + FiniteSet(4) + +def test_solveset_sqrt_2(): + x = Symbol('x', real=True) + y = Symbol('y', real=True) + # http://tutorial.math.lamar.edu/Classes/Alg/SolveRadicalEqns.aspx#Solve_Rad_Ex2_a + assert solveset_real(sqrt(2*x - 1) - sqrt(x - 4) - 2, x) == \ + FiniteSet(S(5), S(13)) + assert solveset_real(sqrt(x + 7) + 2 - sqrt(3 - x), x) == \ + FiniteSet(-6) + + # http://www.purplemath.com/modules/solverad.htm + assert solveset_real(sqrt(17*x - sqrt(x**2 - 5)) - 7, x) == \ + FiniteSet(3) + + eq = x + 1 - (x**4 + 4*x**3 - x)**Rational(1, 4) + assert solveset_real(eq, x) == FiniteSet(Rational(-1, 2), Rational(-1, 3)) + + eq = sqrt(2*x + 9) - sqrt(x + 1) - sqrt(x + 4) + assert solveset_real(eq, x) == FiniteSet(0) + + eq = sqrt(x + 4) + sqrt(2*x - 1) - 3*sqrt(x - 1) + assert solveset_real(eq, x) == FiniteSet(5) + + eq = sqrt(x)*sqrt(x - 7) - 12 + assert solveset_real(eq, x) == FiniteSet(16) + + eq = sqrt(x - 3) + sqrt(x) - 3 + assert solveset_real(eq, x) == FiniteSet(4) + + eq = sqrt(2*x**2 - 7) - (3 - x) + assert solveset_real(eq, x) == FiniteSet(-S(8), S(2)) + + # others + eq = sqrt(9*x**2 + 4) - (3*x + 2) + assert solveset_real(eq, x) == FiniteSet(0) + + assert solveset_real(sqrt(x - 3) - sqrt(x) - 3, x) == FiniteSet() + + eq = (2*x - 5)**Rational(1, 3) - 3 + assert solveset_real(eq, x) == FiniteSet(16) + + assert solveset_real(sqrt(x) + sqrt(sqrt(x)) - 4, x) == \ + FiniteSet((Rational(-1, 2) + sqrt(17)/2)**4) + + eq = sqrt(x) - sqrt(x - 1) + sqrt(sqrt(x)) + assert solveset_real(eq, x) == FiniteSet() + + eq = (x - 4)**2 + (sqrt(x) - 2)**4 + assert solveset_real(eq, x) == FiniteSet(-4, 4) + + eq = (sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6*sqrt(5)/5) + ans = solveset_real(eq, x) + ra = S('''-1484/375 - 4*(-S(1)/2 + sqrt(3)*I/2)*(-12459439/52734375 + + 114*sqrt(12657)/78125)**(S(1)/3) - 172564/(140625*(-S(1)/2 + + sqrt(3)*I/2)*(-12459439/52734375 + 114*sqrt(12657)/78125)**(S(1)/3))''') + rb = Rational(4, 5) + assert all(abs(eq.subs(x, i).n()) < 1e-10 for i in (ra, rb)) and \ + len(ans) == 2 and \ + {i.n(chop=True) for i in ans} == \ + {i.n(chop=True) for i in (ra, rb)} + + assert solveset_real(sqrt(x) + x**Rational(1, 3) + + x**Rational(1, 4), x) == FiniteSet(0) + + assert solveset_real(x/sqrt(x**2 + 1), x) == FiniteSet(0) + + eq = (x - y**3)/((y**2)*sqrt(1 - y**2)) + assert solveset_real(eq, x) == FiniteSet(y**3) + + # issue 4497 + assert solveset_real(1/(5 + x)**Rational(1, 5) - 9, x) == \ + FiniteSet(Rational(-295244, 59049)) + + +@XFAIL +def test_solve_sqrt_fail(): + # this only works if we check real_root(eq.subs(x, Rational(1, 3))) + # but checksol doesn't work like that + eq = (x**3 - 3*x**2)**Rational(1, 3) + 1 - x + assert solveset_real(eq, x) == FiniteSet(Rational(1, 3)) + + +@slow +def test_solve_sqrt_3(): + R = Symbol('R') + eq = sqrt(2)*R*sqrt(1/(R + 1)) + (R + 1)*(sqrt(2)*sqrt(1/(R + 1)) - 1) + sol = solveset_complex(eq, R) + fset = [Rational(5, 3) + 4*sqrt(10)*cos(atan(3*sqrt(111)/251)/3)/3, + -sqrt(10)*cos(atan(3*sqrt(111)/251)/3)/3 + + 40*re(1/((Rational(-1, 2) - sqrt(3)*I/2)*(Rational(251, 27) + sqrt(111)*I/9)**Rational(1, 3)))/9 + + sqrt(30)*sin(atan(3*sqrt(111)/251)/3)/3 + Rational(5, 3) + + I*(-sqrt(30)*cos(atan(3*sqrt(111)/251)/3)/3 - + sqrt(10)*sin(atan(3*sqrt(111)/251)/3)/3 + + 40*im(1/((Rational(-1, 2) - sqrt(3)*I/2)*(Rational(251, 27) + sqrt(111)*I/9)**Rational(1, 3)))/9)] + cset = [40*re(1/((Rational(-1, 2) + sqrt(3)*I/2)*(Rational(251, 27) + sqrt(111)*I/9)**Rational(1, 3)))/9 - + sqrt(10)*cos(atan(3*sqrt(111)/251)/3)/3 - sqrt(30)*sin(atan(3*sqrt(111)/251)/3)/3 + + Rational(5, 3) + + I*(40*im(1/((Rational(-1, 2) + sqrt(3)*I/2)*(Rational(251, 27) + sqrt(111)*I/9)**Rational(1, 3)))/9 - + sqrt(10)*sin(atan(3*sqrt(111)/251)/3)/3 + + sqrt(30)*cos(atan(3*sqrt(111)/251)/3)/3)] + + fs = FiniteSet(*fset) + cs = ConditionSet(R, Eq(eq, 0), FiniteSet(*cset)) + assert sol == (fs - {-1}) | (cs - {-1}) + + # the number of real roots will depend on the value of m: for m=1 there are 4 + # and for m=-1 there are none. + eq = -sqrt((m - q)**2 + (-m/(2*q) + S.Half)**2) + sqrt((-m**2/2 - sqrt( + 4*m**4 - 4*m**2 + 8*m + 1)/4 - Rational(1, 4))**2 + (m**2/2 - m - sqrt( + 4*m**4 - 4*m**2 + 8*m + 1)/4 - Rational(1, 4))**2) + unsolved_object = ConditionSet(q, Eq(sqrt((m - q)**2 + (-m/(2*q) + S.Half)**2) - + sqrt((-m**2/2 - sqrt(4*m**4 - 4*m**2 + 8*m + 1)/4 - Rational(1, 4))**2 + (m**2/2 - m - + sqrt(4*m**4 - 4*m**2 + 8*m + 1)/4 - Rational(1, 4))**2), 0), S.Reals) + assert solveset_real(eq, q) == unsolved_object + + +def test_solve_polynomial_symbolic_param(): + assert solveset_complex((x**2 - 1)**2 - a, x) == \ + FiniteSet(sqrt(1 + sqrt(a)), -sqrt(1 + sqrt(a)), + sqrt(1 - sqrt(a)), -sqrt(1 - sqrt(a))) + + # issue 4507 + assert solveset_complex(y - b/(1 + a*x), x) == \ + FiniteSet((b/y - 1)/a) - FiniteSet(-1/a) + + # issue 4508 + assert solveset_complex(y - b*x/(a + x), x) == \ + FiniteSet(-a*y/(y - b)) - FiniteSet(-a) + + +def test_solve_rational(): + assert solveset_real(1/x + 1, x) == FiniteSet(-S.One) + assert solveset_real(1/exp(x) - 1, x) == FiniteSet(0) + assert solveset_real(x*(1 - 5/x), x) == FiniteSet(5) + assert solveset_real(2*x/(x + 2) - 1, x) == FiniteSet(2) + assert solveset_real((x**2/(7 - x)).diff(x), x) == \ + FiniteSet(S.Zero, S(14)) + + +def test_solveset_real_gen_is_pow(): + assert solveset_real(sqrt(1) + 1, x) is S.EmptySet + + +def test_no_sol(): + assert solveset(1 - oo*x) is S.EmptySet + assert solveset(oo*x, x) is S.EmptySet + assert solveset(oo*x - oo, x) is S.EmptySet + assert solveset_real(4, x) is S.EmptySet + assert solveset_real(exp(x), x) is S.EmptySet + assert solveset_real(x**2 + 1, x) is S.EmptySet + assert solveset_real(-3*a/sqrt(x), x) is S.EmptySet + assert solveset_real(1/x, x) is S.EmptySet + assert solveset_real(-(1 + x)/(2 + x)**2 + 1/(2 + x), x + ) is S.EmptySet + + +def test_sol_zero_real(): + assert solveset_real(0, x) == S.Reals + assert solveset(0, x, Interval(1, 2)) == Interval(1, 2) + assert solveset_real(-x**2 - 2*x + (x + 1)**2 - 1, x) == S.Reals + + +def test_no_sol_rational_extragenous(): + assert solveset_real((x/(x + 1) + 3)**(-2), x) is S.EmptySet + assert solveset_real((x - 1)/(1 + 1/(x - 1)), x) is S.EmptySet + + +def test_solve_polynomial_cv_1a(): + """ + Test for solving on equations that can be converted to + a polynomial equation using the change of variable y -> x**Rational(p, q) + """ + assert solveset_real(sqrt(x) - 1, x) == FiniteSet(1) + assert solveset_real(sqrt(x) - 2, x) == FiniteSet(4) + assert solveset_real(x**Rational(1, 4) - 2, x) == FiniteSet(16) + assert solveset_real(x**Rational(1, 3) - 3, x) == FiniteSet(27) + assert solveset_real(x*(x**(S.One / 3) - 3), x) == \ + FiniteSet(S.Zero, S(27)) + + +def test_solveset_real_rational(): + """Test solveset_real for rational functions""" + x = Symbol('x', real=True) + y = Symbol('y', real=True) + assert solveset_real((x - y**3) / ((y**2)*sqrt(1 - y**2)), x) \ + == FiniteSet(y**3) + # issue 4486 + assert solveset_real(2*x/(x + 2) - 1, x) == FiniteSet(2) + + +def test_solveset_real_log(): + assert solveset_real(log((x-1)*(x+1)), x) == \ + FiniteSet(sqrt(2), -sqrt(2)) + + +def test_poly_gens(): + assert solveset_real(4**(2*(x**2) + 2*x) - 8, x) == \ + FiniteSet(Rational(-3, 2), S.Half) + + +def test_solve_abs(): + n = Dummy('n') + raises(ValueError, lambda: solveset(Abs(x) - 1, x)) + assert solveset(Abs(x) - n, x, S.Reals).dummy_eq( + ConditionSet(x, Contains(n, Interval(0, oo)), {-n, n})) + assert solveset_real(Abs(x) - 2, x) == FiniteSet(-2, 2) + assert solveset_real(Abs(x) + 2, x) is S.EmptySet + assert solveset_real(Abs(x + 3) - 2*Abs(x - 3), x) == \ + FiniteSet(1, 9) + assert solveset_real(2*Abs(x) - Abs(x - 1), x) == \ + FiniteSet(-1, Rational(1, 3)) + + sol = ConditionSet( + x, + And( + Contains(b, Interval(0, oo)), + Contains(a + b, Interval(0, oo)), + Contains(a - b, Interval(0, oo))), + FiniteSet(-a - b - 3, -a + b - 3, a - b - 3, a + b - 3)) + eq = Abs(Abs(x + 3) - a) - b + assert invert_real(eq, 0, x)[1] == sol + reps = {a: 3, b: 1} + eqab = eq.subs(reps) + for si in sol.subs(reps): + assert not eqab.subs(x, si) + assert dumeq(solveset(Eq(sin(Abs(x)), 1), x, domain=S.Reals), Union( + Intersection(Interval(0, oo), Union( + Intersection(ImageSet(Lambda(n, 2*n*pi + 3*pi/2), S.Integers), + Interval(-oo, 0)), + Intersection(ImageSet(Lambda(n, 2*n*pi + pi/2), S.Integers), + Interval(0, oo)))))) + + +def test_issue_9824(): + assert dumeq(solveset(sin(x)**2 - 2*sin(x) + 1, x), ImageSet(Lambda(n, 2*n*pi + pi/2), S.Integers)) + assert dumeq(solveset(cos(x)**2 - 2*cos(x) + 1, x), ImageSet(Lambda(n, 2*n*pi), S.Integers)) + + +def test_issue_9565(): + assert solveset_real(Abs((x - 1)/(x - 5)) <= Rational(1, 3), x) == Interval(-1, 2) + + +def test_issue_10069(): + eq = abs(1/(x - 1)) - 1 > 0 + assert solveset_real(eq, x) == Union( + Interval.open(0, 1), Interval.open(1, 2)) + + +def test_real_imag_splitting(): + a, b = symbols('a b', real=True) + assert solveset_real(sqrt(a**2 - b**2) - 3, a) == \ + FiniteSet(-sqrt(b**2 + 9), sqrt(b**2 + 9)) + assert solveset_real(sqrt(a**2 + b**2) - 3, a) != \ + S.EmptySet + + +def test_units(): + assert solveset_real(1/x - 1/(2*cm), x) == FiniteSet(2*cm) + + +def test_solve_only_exp_1(): + y = Symbol('y', positive=True) + assert solveset_real(exp(x) - y, x) == FiniteSet(log(y)) + assert solveset_real(exp(x) + exp(-x) - 4, x) == \ + FiniteSet(log(-sqrt(3) + 2), log(sqrt(3) + 2)) + assert solveset_real(exp(x) + exp(-x) - y, x) != S.EmptySet + + +def test_atan2(): + # The .inverse() method on atan2 works only if x.is_real is True and the + # second argument is a real constant + assert solveset_real(atan2(x, 2) - pi/3, x) == FiniteSet(2*sqrt(3)) + + +def test_piecewise_solveset(): + eq = Piecewise((x - 2, Gt(x, 2)), (2 - x, True)) - 3 + assert set(solveset_real(eq, x)) == set(FiniteSet(-1, 5)) + + absxm3 = Piecewise( + (x - 3, 0 <= x - 3), + (3 - x, 0 > x - 3)) + y = Symbol('y', positive=True) + assert solveset_real(absxm3 - y, x) == FiniteSet(-y + 3, y + 3) + + f = Piecewise(((x - 2)**2, x >= 0), (0, True)) + assert solveset(f, x, domain=S.Reals) == Union(FiniteSet(2), Interval(-oo, 0, True, True)) + + assert solveset( + Piecewise((x + 1, x > 0), (I, True)) - I, x, S.Reals + ) == Interval(-oo, 0) + + assert solveset(Piecewise((x - 1, Ne(x, I)), (x, True)), x) == FiniteSet(1) + + # issue 19718 + g = Piecewise((1, x > 10), (0, True)) + assert solveset(g > 0, x, S.Reals) == Interval.open(10, oo) + + from sympy.logic.boolalg import BooleanTrue + f = BooleanTrue() + assert solveset(f, x, domain=Interval(-3, 10)) == Interval(-3, 10) + + # issue 20552 + f = Piecewise((0, Eq(x, 0)), (x**2/Abs(x), True)) + g = Piecewise((0, Eq(x, pi)), ((x - pi)/sin(x), True)) + assert solveset(f, x, domain=S.Reals) == FiniteSet(0) + assert solveset(g) == FiniteSet(pi) + + +def test_solveset_complex_polynomial(): + assert solveset_complex(a*x**2 + b*x + c, x) == \ + FiniteSet(-b/(2*a) - sqrt(-4*a*c + b**2)/(2*a), + -b/(2*a) + sqrt(-4*a*c + b**2)/(2*a)) + + assert solveset_complex(x - y**3, y) == FiniteSet( + (-x**Rational(1, 3))/2 + I*sqrt(3)*x**Rational(1, 3)/2, + x**Rational(1, 3), + (-x**Rational(1, 3))/2 - I*sqrt(3)*x**Rational(1, 3)/2) + + assert solveset_complex(x + 1/x - 1, x) == \ + FiniteSet(S.Half + I*sqrt(3)/2, S.Half - I*sqrt(3)/2) + + +def test_sol_zero_complex(): + assert solveset_complex(0, x) is S.Complexes + + +def test_solveset_complex_rational(): + assert solveset_complex((x - 1)*(x - I)/(x - 3), x) == \ + FiniteSet(1, I) + + assert solveset_complex((x - y**3)/((y**2)*sqrt(1 - y**2)), x) == \ + FiniteSet(y**3) + assert solveset_complex(-x**2 - I, x) == \ + FiniteSet(-sqrt(2)/2 + sqrt(2)*I/2, sqrt(2)/2 - sqrt(2)*I/2) + + +def test_solve_quintics(): + skip("This test is too slow") + f = x**5 - 110*x**3 - 55*x**2 + 2310*x + 979 + s = solveset_complex(f, x) + for root in s: + res = f.subs(x, root.n()).n() + assert tn(res, 0) + + f = x**5 + 15*x + 12 + s = solveset_complex(f, x) + for root in s: + res = f.subs(x, root.n()).n() + assert tn(res, 0) + + +def test_solveset_complex_exp(): + assert dumeq(solveset_complex(exp(x) - 1, x), + imageset(Lambda(n, I*2*n*pi), S.Integers)) + assert dumeq(solveset_complex(exp(x) - I, x), + imageset(Lambda(n, I*(2*n*pi + pi/2)), S.Integers)) + assert solveset_complex(1/exp(x), x) == S.EmptySet + assert dumeq(solveset_complex(sinh(x).rewrite(exp), x), + imageset(Lambda(n, n*pi*I), S.Integers)) + + +def test_solveset_real_exp(): + assert solveset(Eq((-2)**x, 4), x, S.Reals) == FiniteSet(2) + assert solveset(Eq(-2**x, 4), x, S.Reals) == S.EmptySet + assert solveset(Eq((-3)**x, 27), x, S.Reals) == S.EmptySet + assert solveset(Eq((-5)**(x+1), 625), x, S.Reals) == FiniteSet(3) + assert solveset(Eq(2**(x-3), -16), x, S.Reals) == S.EmptySet + assert solveset(Eq((-3)**(x - 3), -3**39), x, S.Reals) == FiniteSet(42) + assert solveset(Eq(2**x, y), x, S.Reals) == Intersection(S.Reals, FiniteSet(log(y)/log(2))) + + assert invert_real((-2)**(2*x) - 16, 0, x) == (x, FiniteSet(2)) + + +def test_solve_complex_log(): + assert solveset_complex(log(x), x) == FiniteSet(1) + assert solveset_complex(1 - log(a + 4*x**2), x) == \ + FiniteSet(-sqrt(-a + E)/2, sqrt(-a + E)/2) + + +def test_solve_complex_sqrt(): + assert solveset_complex(sqrt(5*x + 6) - 2 - x, x) == \ + FiniteSet(-S.One, S(2)) + assert solveset_complex(sqrt(5*x + 6) - (2 + 2*I) - x, x) == \ + FiniteSet(-S(2), 3 - 4*I) + assert solveset_complex(4*x*(1 - a * sqrt(x)), x) == \ + FiniteSet(S.Zero, 1 / a ** 2) + + +def test_solveset_complex_tan(): + s = solveset_complex(tan(x).rewrite(exp), x) + assert dumeq(s, imageset(Lambda(n, pi*n), S.Integers) - \ + imageset(Lambda(n, pi*n + pi/2), S.Integers)) + + +@_both_exp_pow +def test_solve_trig(): + assert dumeq(solveset_real(sin(x), x), + Union(imageset(Lambda(n, 2*pi*n), S.Integers), + imageset(Lambda(n, 2*pi*n + pi), S.Integers))) + + assert dumeq(solveset_real(sin(x) - 1, x), + imageset(Lambda(n, 2*pi*n + pi/2), S.Integers)) + + assert dumeq(solveset_real(cos(x), x), + Union(imageset(Lambda(n, 2*pi*n + pi/2), S.Integers), + imageset(Lambda(n, 2*pi*n + pi*Rational(3, 2)), S.Integers))) + + assert dumeq(solveset_real(sin(x) + cos(x), x), + Union(imageset(Lambda(n, 2*n*pi + pi*Rational(3, 4)), S.Integers), + imageset(Lambda(n, 2*n*pi + pi*Rational(7, 4)), S.Integers))) + + assert solveset_real(sin(x)**2 + cos(x)**2, x) == S.EmptySet + + assert dumeq(solveset_complex(cos(x) - S.Half, x), + Union(imageset(Lambda(n, 2*n*pi + pi*Rational(5, 3)), S.Integers), + imageset(Lambda(n, 2*n*pi + pi/3), S.Integers))) + + assert dumeq(solveset(sin(y + a) - sin(y), a, domain=S.Reals), + ConditionSet(a, (S(-1) <= sin(y)) & (sin(y) <= S(1)), Union( + ImageSet(Lambda(n, 2*n*pi - y + asin(sin(y))), S.Integers), + ImageSet(Lambda(n, 2*n*pi - y - asin(sin(y)) + pi), S.Integers)))) + + assert dumeq(solveset_real(sin(2*x)*cos(x) + cos(2*x)*sin(x)-1, x), + ImageSet(Lambda(n, n*pi*Rational(2, 3) + pi/6), S.Integers)) + + assert dumeq(solveset_real(2*tan(x)*sin(x) + 1, x), Union( + ImageSet(Lambda(n, 2*n*pi + atan(sqrt(2)*sqrt(-1 + sqrt(17))/ + (1 - sqrt(17))) + pi), S.Integers), + ImageSet(Lambda(n, 2*n*pi - atan(sqrt(2)*sqrt(-1 + sqrt(17))/ + (1 - sqrt(17))) + pi), S.Integers))) + + assert dumeq(solveset_real(cos(2*x)*cos(4*x) - 1, x), + ImageSet(Lambda(n, n*pi), S.Integers)) + + assert dumeq(solveset(sin(x/10) + Rational(3, 4)), Union( + ImageSet(Lambda(n, 20*n*pi - 10*asin(S(3)/4) + 20*pi), S.Integers), + ImageSet(Lambda(n, 20*n*pi + 10*asin(S(3)/4) + 10*pi), S.Integers))) + + assert dumeq(solveset(cos(x/15) + cos(x/5)), Union( + ImageSet(Lambda(n, 30*n*pi + 15*pi/2), S.Integers), + ImageSet(Lambda(n, 30*n*pi + 45*pi/2), S.Integers), + ImageSet(Lambda(n, 30*n*pi + 75*pi/4), S.Integers), + ImageSet(Lambda(n, 30*n*pi + 45*pi/4), S.Integers), + ImageSet(Lambda(n, 30*n*pi + 105*pi/4), S.Integers), + ImageSet(Lambda(n, 30*n*pi + 15*pi/4), S.Integers))) + + assert dumeq(solveset(sec(sqrt(2)*x/3) + 5), Union( + ImageSet(Lambda(n, 3*sqrt(2)*(2*n*pi - asec(-5))/2), S.Integers), + ImageSet(Lambda(n, 3*sqrt(2)*(2*n*pi + asec(-5))/2), S.Integers))) + + assert dumeq(simplify(solveset(tan(pi*x) - cot(pi/2*x))), Union( + ImageSet(Lambda(n, 4*n + 1), S.Integers), + ImageSet(Lambda(n, 4*n + 3), S.Integers), + ImageSet(Lambda(n, 4*n + Rational(7, 3)), S.Integers), + ImageSet(Lambda(n, 4*n + Rational(5, 3)), S.Integers), + ImageSet(Lambda(n, 4*n + Rational(11, 3)), S.Integers), + ImageSet(Lambda(n, 4*n + Rational(1, 3)), S.Integers))) + + assert dumeq(solveset(cos(9*x)), Union( + ImageSet(Lambda(n, 2*n*pi/9 + pi/18), S.Integers), + ImageSet(Lambda(n, 2*n*pi/9 + pi/6), S.Integers))) + + assert dumeq(solveset(sin(8*x) + cot(12*x), x, S.Reals), Union( + ImageSet(Lambda(n, n*pi/2 + pi/8), S.Integers), + ImageSet(Lambda(n, n*pi/2 + 3*pi/8), S.Integers), + ImageSet(Lambda(n, n*pi/2 + 5*pi/16), S.Integers), + ImageSet(Lambda(n, n*pi/2 + 3*pi/16), S.Integers), + ImageSet(Lambda(n, n*pi/2 + 7*pi/16), S.Integers), + ImageSet(Lambda(n, n*pi/2 + pi/16), S.Integers))) + + # This is the only remaining solveset test that actually ends up being solved + # by _solve_trig2(). All others are handled by the improved _solve_trig1. + assert dumeq(solveset_real(2*cos(x)*cos(2*x) - 1, x), + Union(ImageSet(Lambda(n, 2*n*pi + 2*atan(sqrt(-2*2**Rational(1, 3)*(67 + + 9*sqrt(57))**Rational(2, 3) + 8*2**Rational(2, 3) + 11*(67 + + 9*sqrt(57))**Rational(1, 3))/(3*(67 + 9*sqrt(57))**Rational(1, 6)))), S.Integers), + ImageSet(Lambda(n, 2*n*pi - 2*atan(sqrt(-2*2**Rational(1, 3)*(67 + + 9*sqrt(57))**Rational(2, 3) + 8*2**Rational(2, 3) + 11*(67 + + 9*sqrt(57))**Rational(1, 3))/(3*(67 + 9*sqrt(57))**Rational(1, 6))) + + 2*pi), S.Integers))) + + # issue #16870 + assert dumeq(simplify(solveset(sin(x/180*pi) - S.Half, x, S.Reals)), Union( + ImageSet(Lambda(n, 360*n + 150), S.Integers), + ImageSet(Lambda(n, 360*n + 30), S.Integers))) + + +def test_solve_trig_hyp_by_inversion(): + n = Dummy('n') + assert solveset_real(sin(2*x + 3) - S(1)/2, x).dummy_eq(Union( + ImageSet(Lambda(n, n*pi - S(3)/2 + 13*pi/12), S.Integers), + ImageSet(Lambda(n, n*pi - S(3)/2 + 17*pi/12), S.Integers))) + assert solveset_complex(sin(2*x + 3) - S(1)/2, x).dummy_eq(Union( + ImageSet(Lambda(n, n*pi - S(3)/2 + 13*pi/12), S.Integers), + ImageSet(Lambda(n, n*pi - S(3)/2 + 17*pi/12), S.Integers))) + assert solveset_real(tan(x) - tan(pi/10), x).dummy_eq( + ImageSet(Lambda(n, n*pi + pi/10), S.Integers)) + assert solveset_complex(tan(x) - tan(pi/10), x).dummy_eq( + ImageSet(Lambda(n, n*pi + pi/10), S.Integers)) + + assert solveset_real(3*cosh(2*x) - 5, x) == FiniteSet( + -acosh(S(5)/3)/2, acosh(S(5)/3)/2) + assert solveset_complex(3*cosh(2*x) - 5, x).dummy_eq(Union( + ImageSet(Lambda(n, n*I*pi - acosh(S(5)/3)/2), S.Integers), + ImageSet(Lambda(n, n*I*pi + acosh(S(5)/3)/2), S.Integers))) + assert solveset_real(sinh(x - 3) - 2, x) == FiniteSet( + asinh(2) + 3) + assert solveset_complex(sinh(x - 3) - 2, x).dummy_eq(Union( + ImageSet(Lambda(n, 2*n*I*pi + asinh(2) + 3), S.Integers), + ImageSet(Lambda(n, 2*n*I*pi - asinh(2) + 3 + I*pi), S.Integers))) + + assert solveset_real(cos(sinh(x))-cos(pi/12), x).dummy_eq(Union( + ImageSet(Lambda(n, asinh(2*n*pi + pi/12)), S.Integers), + ImageSet(Lambda(n, asinh(2*n*pi + 23*pi/12)), S.Integers))) + assert solveset(cos(sinh(x))-cos(pi/12), x, Interval(2,3)) == \ + FiniteSet(asinh(23*pi/12), asinh(25*pi/12)) + assert solveset_real(cosh(x**2-1)-2, x) == FiniteSet( + -sqrt(1 + acosh(2)), sqrt(1 + acosh(2))) + + assert solveset_real(sin(x) - 2, x) == S.EmptySet # issue #17334 + assert solveset_real(cos(x) + 2, x) == S.EmptySet + assert solveset_real(sec(x), x) == S.EmptySet + assert solveset_real(csc(x), x) == S.EmptySet + assert solveset_real(cosh(x) + 1, x) == S.EmptySet + assert solveset_real(coth(x), x) == S.EmptySet + assert solveset_real(sech(x) - 2, x) == S.EmptySet + assert solveset_real(sech(x), x) == S.EmptySet + assert solveset_real(tanh(x) + 1, x) == S.EmptySet + assert solveset_complex(tanh(x), 1) == S.EmptySet + assert solveset_complex(coth(x), -1) == S.EmptySet + assert solveset_complex(sech(x), 0) == S.EmptySet + assert solveset_complex(csch(x), 0) == S.EmptySet + + assert solveset_real(abs(csch(x)) - 3, x) == FiniteSet(-acsch(3), acsch(3)) + + assert solveset_real(tanh(x**2 - 1) - exp(-9), x) == FiniteSet( + -sqrt(atanh(exp(-9)) + 1), sqrt(atanh(exp(-9)) + 1)) + + assert solveset_real(coth(log(x)) + 2, x) == FiniteSet(exp(-acoth(2))) + assert solveset_real(coth(exp(x)) + 2, x) == S.EmptySet + + assert solveset_complex(sinh(x) - I/2, x).dummy_eq(Union( + ImageSet(Lambda(n, 2*I*pi*n + 5*I*pi/6), S.Integers), + ImageSet(Lambda(n, 2*I*pi*n + I*pi/6), S.Integers))) + assert solveset_complex(sinh(x/10) + Rational(3, 4), x).dummy_eq(Union( + ImageSet(Lambda(n, 20*n*I*pi - 10*asinh(S(3)/4)), S.Integers), + ImageSet(Lambda(n, 20*n*I*pi + 10*asinh(S(3)/4) + 10*I*pi), S.Integers))) + assert solveset_complex(sech(sqrt(2)*x/3) + 5, x).dummy_eq(Union( + ImageSet(Lambda(n, 3*sqrt(2)*(2*n*I*pi - asech(-5))/2), S.Integers), + ImageSet(Lambda(n, 3*sqrt(2)*(2*n*I*pi + asech(-5))/2), S.Integers))) + assert solveset_complex(cosh(9*x), x).dummy_eq(Union( + ImageSet(Lambda(n, 2*n*I*pi/9 + I*pi/18), S.Integers), + ImageSet(Lambda(n, 2*n*I*pi/9 + I*pi/6), S.Integers))) + + eq = (x**5 -4*x + 1).subs(x, coth(z)) + assert solveset(eq, z, S.Complexes).dummy_eq(Union( + ImageSet(Lambda(n, n*I*pi + acoth(CRootOf(x**5 -4*x + 1, 0))), S.Integers), + ImageSet(Lambda(n, n*I*pi + acoth(CRootOf(x**5 -4*x + 1, 1))), S.Integers), + ImageSet(Lambda(n, n*I*pi + acoth(CRootOf(x**5 -4*x + 1, 2))), S.Integers), + ImageSet(Lambda(n, n*I*pi + acoth(CRootOf(x**5 -4*x + 1, 3))), S.Integers), + ImageSet(Lambda(n, n*I*pi + acoth(CRootOf(x**5 -4*x + 1, 4))), S.Integers))) + assert solveset(eq, z, S.Reals) == FiniteSet( + acoth(CRootOf(x**5 - 4*x + 1, 0)), acoth(CRootOf(x**5 - 4*x + 1, 2))) + + eq = ((x-sqrt(3)/2)*(x+2)).expand().subs(x, cos(x)) + assert solveset(eq, x, S.Complexes).dummy_eq(Union( + ImageSet(Lambda(n, 2*n*pi - acos(-2)), S.Integers), + ImageSet(Lambda(n, 2*n*pi + acos(-2)), S.Integers), + ImageSet(Lambda(n, 2*n*pi + pi/6), S.Integers), + ImageSet(Lambda(n, 2*n*pi + 11*pi/6), S.Integers))) + assert solveset(eq, x, S.Reals).dummy_eq(Union( + ImageSet(Lambda(n, 2*n*pi + pi/6), S.Integers), + ImageSet(Lambda(n, 2*n*pi + 11*pi/6), S.Integers))) + + assert solveset((1+sec(sqrt(3)*x+4)**2)/(1-sec(sqrt(3)*x+4))).dummy_eq(Union( + ImageSet(Lambda(n, sqrt(3)*(2*n*pi - 4 - asec(I))/3), S.Integers), + ImageSet(Lambda(n, sqrt(3)*(2*n*pi - 4 + asec(I))/3), S.Integers), + ImageSet(Lambda(n, sqrt(3)*(2*n*pi - 4 - asec(-I))/3), S.Integers), + ImageSet(Lambda(n, sqrt(3)*(2*n*pi - 4 + asec(-I))/3), S.Integers))) + + assert all_close(solveset(tan(3.14*x)**(S(3)/2)-5.678, x, Interval(0, 3)), + FiniteSet(0.403301114561067, 0.403301114561067 + 0.318471337579618*pi, + 0.403301114561067 + 0.636942675159236*pi)) + + +def test_old_trig_issues(): + # issues #9606 / #9531: + assert solveset(sinh(x), x, S.Reals) == FiniteSet(0) + assert solveset(sinh(x), x, S.Complexes).dummy_eq(Union( + ImageSet(Lambda(n, 2*n*I*pi), S.Integers), + ImageSet(Lambda(n, 2*n*I*pi + I*pi), S.Integers))) + + # issues #11218 / #18427 + assert solveset(sin(pi*x), x, S.Reals).dummy_eq(Union( + ImageSet(Lambda(n, (2*n*pi + pi)/pi), S.Integers), + ImageSet(Lambda(n, 2*n), S.Integers))) + assert solveset(sin(pi*x), x).dummy_eq(Union( + ImageSet(Lambda(n, (2*n*pi + pi)/pi), S.Integers), + ImageSet(Lambda(n, 2*n), S.Integers))) + + # issue #17543 + assert solveset(I*cot(8*x - 8*E), x).dummy_eq( + ImageSet(Lambda(n, pi*n/8 - 13*pi/16 + E), S.Integers)) + + # issue #20798 + assert all_close(solveset(cos(2*x) - 0.5, x, Interval(0, 2*pi)), FiniteSet( + 0.523598775598299, -0.523598775598299 + pi, + -0.523598775598299 + 2*pi, 0.523598775598299 + pi)) + sol = Union(ImageSet(Lambda(n, n*pi - 0.523598775598299), S.Integers), + ImageSet(Lambda(n, n*pi + 0.523598775598299), S.Integers)) + ret = solveset(cos(2*x) - 0.5, x, S.Reals) + # replace Dummy n by the regular Symbol n to allow all_close comparison. + ret = ret.subs(ret.atoms(Dummy).pop(), n) + assert all_close(ret, sol) + ret = solveset(cos(2*x) - 0.5, x, S.Complexes) + ret = ret.subs(ret.atoms(Dummy).pop(), n) + assert all_close(ret, sol) + + # issue #21296 / #17667 + assert solveset(tan(x)-sqrt(2), x, Interval(0, pi/2)) == FiniteSet(atan(sqrt(2))) + assert solveset(tan(x)-pi, x, Interval(0, pi/2)) == FiniteSet(atan(pi)) + + # issue #17667 + # not yet working properly: + # solveset(cos(x)-y, x, Interval(0, pi)) + assert solveset(cos(x)-y, x, S.Reals).dummy_eq( + ConditionSet(x,(S(-1) <= y) & (y <= S(1)), Union( + ImageSet(Lambda(n, 2*n*pi - acos(y)), S.Integers), + ImageSet(Lambda(n, 2*n*pi + acos(y)), S.Integers)))) + + # issue #17579 + # Valid result, but the intersection could potentially be simplified. + assert solveset(sin(log(x)), x, Interval(0,1, True, False)).dummy_eq( + Union(Intersection(ImageSet(Lambda(n, exp(2*n*pi)), S.Integers), Interval.Lopen(0, 1)), + Intersection(ImageSet(Lambda(n, exp(2*n*pi + pi)), S.Integers), Interval.Lopen(0, 1)))) + + # issue #17334 + assert solveset(sin(x) - sin(1), x, S.Reals).dummy_eq(Union( + ImageSet(Lambda(n, 2*n*pi + 1), S.Integers), + ImageSet(Lambda(n, 2*n*pi - 1 + pi), S.Integers))) + assert solveset(sin(x) - sqrt(5)/3, x, S.Reals).dummy_eq(Union( + ImageSet(Lambda(n, 2*n*pi + asin(sqrt(5)/3)), S.Integers), + ImageSet(Lambda(n, 2*n*pi - asin(sqrt(5)/3) + pi), S.Integers))) + assert solveset(sinh(x)-cosh(2), x, S.Reals) == FiniteSet(asinh(cosh(2))) + + # issue 9825 + assert solveset(Eq(tan(x), y), x, domain=S.Reals).dummy_eq( + ConditionSet(x, (-oo < y) & (y < oo), + ImageSet(Lambda(n, n*pi + atan(y)), S.Integers))) + r = Symbol('r', real=True) + assert solveset(Eq(tan(x), r), x, domain=S.Reals).dummy_eq( + ImageSet(Lambda(n, n*pi + atan(r)), S.Integers)) + + +def test_solve_hyperbolic(): + # actual solver: _solve_trig1 + n = Dummy('n') + assert solveset(sinh(x) + cosh(x), x) == S.EmptySet + assert solveset(sinh(x) + cos(x), x) == ConditionSet(x, + Eq(cos(x) + sinh(x), 0), S.Complexes) + assert solveset_real(sinh(x) + sech(x), x) == FiniteSet( + log(sqrt(sqrt(5) - 2))) + assert solveset_real(cosh(2*x) + 2*sinh(x) - 5, x) == FiniteSet( + log(-2 + sqrt(5)), log(1 + sqrt(2))) + assert solveset_real((coth(x) + sinh(2*x))/cosh(x) - 3, x) == FiniteSet( + log(S.Half + sqrt(5)/2), log(1 + sqrt(2))) + assert solveset_real(cosh(x)*sinh(x) - 2, x) == FiniteSet( + log(4 + sqrt(17))/2) + assert solveset_real(sinh(x) + tanh(x) - 1, x) == FiniteSet( + log(sqrt(2)/2 + sqrt(-S(1)/2 + sqrt(2)))) + + assert dumeq(solveset_complex(sinh(x) + sech(x), x), Union( + ImageSet(Lambda(n, 2*n*I*pi + log(sqrt(-2 + sqrt(5)))), S.Integers), + ImageSet(Lambda(n, I*(2*n*pi + pi/2) + log(sqrt(2 + sqrt(5)))), S.Integers), + ImageSet(Lambda(n, I*(2*n*pi + pi) + log(sqrt(-2 + sqrt(5)))), S.Integers), + ImageSet(Lambda(n, I*(2*n*pi - pi/2) + log(sqrt(2 + sqrt(5)))), S.Integers))) + + assert dumeq(solveset(cosh(x/15) + cosh(x/5)), Union( + ImageSet(Lambda(n, 15*I*(2*n*pi + pi/2)), S.Integers), + ImageSet(Lambda(n, 15*I*(2*n*pi - pi/2)), S.Integers), + ImageSet(Lambda(n, 15*I*(2*n*pi - 3*pi/4)), S.Integers), + ImageSet(Lambda(n, 15*I*(2*n*pi + 3*pi/4)), S.Integers), + ImageSet(Lambda(n, 15*I*(2*n*pi - pi/4)), S.Integers), + ImageSet(Lambda(n, 15*I*(2*n*pi + pi/4)), S.Integers))) + + assert dumeq(solveset(tanh(pi*x) - coth(pi/2*x)), Union( + ImageSet(Lambda(n, 2*I*(2*n*pi + pi/2)/pi), S.Integers), + ImageSet(Lambda(n, 2*I*(2*n*pi - pi/2)/pi), S.Integers))) + + # issues #18490 / #19489 + assert solveset(cosh(x) + cosh(3*x) - cosh(5*x), x, S.Reals + ).dummy_eq(ConditionSet(x, + Eq(cosh(x) + cosh(3*x) - cosh(5*x), 0), S.Reals)) + assert solveset(sinh(8*x) + coth(12*x)).dummy_eq( + ConditionSet(x, Eq(sinh(8*x) + coth(12*x), 0), S.Complexes)) + + +def test_solve_trig_hyp_symbolic(): + # actual solver: invert_trig_hyp + assert dumeq(solveset(sin(a*x), x), ConditionSet(x, Ne(a, 0), Union( + ImageSet(Lambda(n, (2*n*pi + pi)/a), S.Integers), + ImageSet(Lambda(n, 2*n*pi/a), S.Integers)))) + + assert dumeq(solveset(cosh(x/a), x), ConditionSet(x, Ne(a, 0), Union( + ImageSet(Lambda(n, a*(2*n*I*pi + I*pi/2)), S.Integers), + ImageSet(Lambda(n, a*(2*n*I*pi + 3*I*pi/2)), S.Integers)))) + + assert dumeq(solveset(sin(2*sqrt(3)/3*a**2/(b*pi)*x) + + cos(4*sqrt(3)/3*a**2/(b*pi)*x), x), + ConditionSet(x, Ne(b, 0) & Ne(a**2, 0), Union( + ImageSet(Lambda(n, sqrt(3)*pi*b*(2*n*pi + pi/2)/(2*a**2)), S.Integers), + ImageSet(Lambda(n, sqrt(3)*pi*b*(2*n*pi - 5*pi/6)/(2*a**2)), S.Integers), + ImageSet(Lambda(n, sqrt(3)*pi*b*(2*n*pi - pi/6)/(2*a**2)), S.Integers)))) + + assert dumeq(solveset(cosh((a**2 + 1)*x) - 3, x), ConditionSet( + x, Ne(a**2 + 1, 0), Union( + ImageSet(Lambda(n, (2*n*I*pi - acosh(3))/(a**2 + 1)), S.Integers), + ImageSet(Lambda(n, (2*n*I*pi + acosh(3))/(a**2 + 1)), S.Integers)))) + + ar = Symbol('ar', real=True) + assert solveset(cosh((ar**2 + 1)*x) - 2, x, S.Reals) == FiniteSet( + -acosh(2)/(ar**2 + 1), acosh(2)/(ar**2 + 1)) + + # actual solver: _solve_trig1 + assert dumeq(simplify(solveset(cot((1 + I)*x) - cot((3 + 3*I)*x), x)), Union( + ImageSet(Lambda(n, pi*(1 - I)*(4*n + 1)/4), S.Integers), + ImageSet(Lambda(n, pi*(1 - I)*(4*n - 1)/4), S.Integers))) + + +def test_issue_9616(): + assert dumeq(solveset(sinh(x) + tanh(x) - 1, x), Union( + ImageSet(Lambda(n, 2*n*I*pi + log(sqrt(2)/2 + sqrt(-S.Half + sqrt(2)))), S.Integers), + ImageSet(Lambda(n, I*(2*n*pi - atan(sqrt(2)*sqrt(S.Half + sqrt(2))) + pi) + + log(sqrt(1 + sqrt(2)))), S.Integers), + ImageSet(Lambda(n, I*(2*n*pi + pi) + log(-sqrt(2)/2 + sqrt(-S.Half + sqrt(2)))), S.Integers), + ImageSet(Lambda(n, I*(2*n*pi - pi + atan(sqrt(2)*sqrt(S.Half + sqrt(2)))) + + log(sqrt(1 + sqrt(2)))), S.Integers))) + f1 = (sinh(x)).rewrite(exp) + f2 = (tanh(x)).rewrite(exp) + assert dumeq(solveset(f1 + f2 - 1, x), Union( + Complement(ImageSet( + Lambda(n, I*(2*n*pi + pi) + log(-sqrt(2)/2 + sqrt(-S.Half + sqrt(2)))), S.Integers), + ImageSet(Lambda(n, I*(2*n*pi + pi)/2), S.Integers)), + Complement(ImageSet(Lambda(n, I*(2*n*pi - pi + atan(sqrt(2)*sqrt(S.Half + sqrt(2)))) + + log(sqrt(1 + sqrt(2)))), S.Integers), + ImageSet(Lambda(n, I*(2*n*pi + pi)/2), S.Integers)), + Complement(ImageSet(Lambda(n, I*(2*n*pi - atan(sqrt(2)*sqrt(S.Half + sqrt(2))) + pi) + + log(sqrt(1 + sqrt(2)))), S.Integers), + ImageSet(Lambda(n, I*(2*n*pi + pi)/2), S.Integers)), + Complement( + ImageSet(Lambda(n, 2*n*I*pi + log(sqrt(2)/2 + sqrt(-S.Half + sqrt(2)))), S.Integers), + ImageSet(Lambda(n, I*(2*n*pi + pi)/2), S.Integers)))) + + +def test_solve_invalid_sol(): + assert 0 not in solveset_real(sin(x)/x, x) + assert 0 not in solveset_complex((exp(x) - 1)/x, x) + + +@XFAIL +def test_solve_trig_simplified(): + n = Dummy('n') + assert dumeq(solveset_real(sin(x), x), + imageset(Lambda(n, n*pi), S.Integers)) + + assert dumeq(solveset_real(cos(x), x), + imageset(Lambda(n, n*pi + pi/2), S.Integers)) + + assert dumeq(solveset_real(cos(x) + sin(x), x), + imageset(Lambda(n, n*pi - pi/4), S.Integers)) + + +@XFAIL +def test_solve_lambert(): + assert solveset_real(x*exp(x) - 1, x) == FiniteSet(LambertW(1)) + assert solveset_real(exp(x) + x, x) == FiniteSet(-LambertW(1)) + assert solveset_real(x + 2**x, x) == \ + FiniteSet(-LambertW(log(2))/log(2)) + + # issue 4739 + ans = solveset_real(3*x + 5 + 2**(-5*x + 3), x) + assert ans == FiniteSet(Rational(-5, 3) + + LambertW(-10240*2**Rational(1, 3)*log(2)/3)/(5*log(2))) + + eq = 2*(3*x + 4)**5 - 6*7**(3*x + 9) + result = solveset_real(eq, x) + ans = FiniteSet((log(2401) + + 5*LambertW(-log(7**(7*3**Rational(1, 5)/5))))/(3*log(7))/-1) + assert result == ans + assert solveset_real(eq.expand(), x) == result + + assert solveset_real(5*x - 1 + 3*exp(2 - 7*x), x) == \ + FiniteSet(Rational(1, 5) + LambertW(-21*exp(Rational(3, 5))/5)/7) + + assert solveset_real(2*x + 5 + log(3*x - 2), x) == \ + FiniteSet(Rational(2, 3) + LambertW(2*exp(Rational(-19, 3))/3)/2) + + assert solveset_real(3*x + log(4*x), x) == \ + FiniteSet(LambertW(Rational(3, 4))/3) + + assert solveset_real(x**x - 2) == FiniteSet(exp(LambertW(log(2)))) + + a = Symbol('a') + assert solveset_real(-a*x + 2*x*log(x), x) == FiniteSet(exp(a/2)) + a = Symbol('a', real=True) + assert solveset_real(a/x + exp(x/2), x) == \ + FiniteSet(2*LambertW(-a/2)) + assert solveset_real((a/x + exp(x/2)).diff(x), x) == \ + FiniteSet(4*LambertW(sqrt(2)*sqrt(a)/4)) + + # coverage test + assert solveset_real(tanh(x + 3)*tanh(x - 3) - 1, x) is S.EmptySet + + assert solveset_real((x**2 - 2*x + 1).subs(x, log(x) + 3*x), x) == \ + FiniteSet(LambertW(3*S.Exp1)/3) + assert solveset_real((x**2 - 2*x + 1).subs(x, (log(x) + 3*x)**2 - 1), x) == \ + FiniteSet(LambertW(3*exp(-sqrt(2)))/3, LambertW(3*exp(sqrt(2)))/3) + assert solveset_real((x**2 - 2*x - 2).subs(x, log(x) + 3*x), x) == \ + FiniteSet(LambertW(3*exp(1 + sqrt(3)))/3, LambertW(3*exp(-sqrt(3) + 1))/3) + assert solveset_real(x*log(x) + 3*x + 1, x) == \ + FiniteSet(exp(-3 + LambertW(-exp(3)))) + eq = (x*exp(x) - 3).subs(x, x*exp(x)) + assert solveset_real(eq, x) == \ + FiniteSet(LambertW(3*exp(-LambertW(3)))) + + assert solveset_real(3*log(a**(3*x + 5)) + a**(3*x + 5), x) == \ + FiniteSet(-((log(a**5) + LambertW(Rational(1, 3)))/(3*log(a)))) + p = symbols('p', positive=True) + assert solveset_real(3*log(p**(3*x + 5)) + p**(3*x + 5), x) == \ + FiniteSet( + log((-3**Rational(1, 3) - 3**Rational(5, 6)*I)*LambertW(Rational(1, 3))**Rational(1, 3)/(2*p**Rational(5, 3)))/log(p), + log((-3**Rational(1, 3) + 3**Rational(5, 6)*I)*LambertW(Rational(1, 3))**Rational(1, 3)/(2*p**Rational(5, 3)))/log(p), + log((3*LambertW(Rational(1, 3))/p**5)**(1/(3*log(p)))),) # checked numerically + # check collection + b = Symbol('b') + eq = 3*log(a**(3*x + 5)) + b*log(a**(3*x + 5)) + a**(3*x + 5) + assert solveset_real(eq, x) == FiniteSet( + -((log(a**5) + LambertW(1/(b + 3)))/(3*log(a)))) + + # issue 4271 + assert solveset_real((a/x + exp(x/2)).diff(x, 2), x) == FiniteSet( + 6*LambertW((-1)**Rational(1, 3)*a**Rational(1, 3)/3)) + + assert solveset_real(x**3 - 3**x, x) == \ + FiniteSet(-3/log(3)*LambertW(-log(3)/3)) + assert solveset_real(3**cos(x) - cos(x)**3) == FiniteSet( + acos(-3*LambertW(-log(3)/3)/log(3))) + + assert solveset_real(x**2 - 2**x, x) == \ + solveset_real(-x**2 + 2**x, x) + + assert solveset_real(3*log(x) - x*log(3)) == FiniteSet( + -3*LambertW(-log(3)/3)/log(3), + -3*LambertW(-log(3)/3, -1)/log(3)) + + assert solveset_real(LambertW(2*x) - y) == FiniteSet( + y*exp(y)/2) + + +@XFAIL +def test_other_lambert(): + a = Rational(6, 5) + assert solveset_real(x**a - a**x, x) == FiniteSet( + a, -a*LambertW(-log(a)/a)/log(a)) + + +@_both_exp_pow +def test_solveset(): + f = Function('f') + raises(ValueError, lambda: solveset(x + y)) + assert solveset(x, 1) == S.EmptySet + assert solveset(f(1)**2 + y + 1, f(1) + ) == FiniteSet(-sqrt(-y - 1), sqrt(-y - 1)) + assert solveset(f(1)**2 - 1, f(1), S.Reals) == FiniteSet(-1, 1) + assert solveset(f(1)**2 + 1, f(1)) == FiniteSet(-I, I) + assert solveset(x - 1, 1) == FiniteSet(x) + assert solveset(sin(x) - cos(x), sin(x)) == FiniteSet(cos(x)) + + assert solveset(0, domain=S.Reals) == S.Reals + assert solveset(1) == S.EmptySet + assert solveset(True, domain=S.Reals) == S.Reals # issue 10197 + assert solveset(False, domain=S.Reals) == S.EmptySet + + assert solveset(exp(x) - 1, domain=S.Reals) == FiniteSet(0) + assert solveset(exp(x) - 1, x, S.Reals) == FiniteSet(0) + assert solveset(Eq(exp(x), 1), x, S.Reals) == FiniteSet(0) + assert solveset(exp(x) - 1, exp(x), S.Reals) == FiniteSet(1) + A = Indexed('A', x) + assert solveset(A - 1, A, S.Reals) == FiniteSet(1) + + assert solveset(x - 1 >= 0, x, S.Reals) == Interval(1, oo) + assert solveset(exp(x) - 1 >= 0, x, S.Reals) == Interval(0, oo) + + assert dumeq(solveset(exp(x) - 1, x), imageset(Lambda(n, 2*I*pi*n), S.Integers)) + assert dumeq(solveset(Eq(exp(x), 1), x), imageset(Lambda(n, 2*I*pi*n), + S.Integers)) + # issue 13825 + assert solveset(x**2 + f(0) + 1, x) == {-sqrt(-f(0) - 1), sqrt(-f(0) - 1)} + + # issue 19977 + assert solveset(atan(log(x)) > 0, x, domain=Interval.open(0, oo)) == Interval.open(1, oo) + + +@_both_exp_pow +def test_multi_exp(): + k1, k2, k3 = symbols('k1, k2, k3') + assert dumeq(solveset(exp(exp(x)) - 5, x),\ + imageset(Lambda(((k1, n),), I*(2*k1*pi + arg(2*n*I*pi + log(5))) + log(Abs(2*n*I*pi + log(5)))),\ + ProductSet(S.Integers, S.Integers))) + assert dumeq(solveset((d*exp(exp(a*x + b)) + c), x),\ + imageset(Lambda(x, (-b + x)/a), ImageSet(Lambda(((k1, n),), \ + I*(2*k1*pi + arg(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d)))) + log(Abs(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d))))), \ + ProductSet(S.Integers, S.Integers)))) + + assert dumeq(solveset((d*exp(exp(exp(a*x + b))) + c), x),\ + imageset(Lambda(x, (-b + x)/a), ImageSet(Lambda(((k2, k1, n),), \ + I*(2*k2*pi + arg(I*(2*k1*pi + arg(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d)))) + \ + log(Abs(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d)))))) + log(Abs(I*(2*k1*pi + arg(I*(2*n*pi + arg(-c/d)) + \ + log(Abs(c/d)))) + log(Abs(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d))))))), \ + ProductSet(S.Integers, S.Integers, S.Integers)))) + + assert dumeq(solveset((d*exp(exp(exp(exp(a*x + b)))) + c), x),\ + ImageSet(Lambda(x, (-b + x)/a), ImageSet(Lambda(((k3, k2, k1, n),), \ + I*(2*k3*pi + arg(I*(2*k2*pi + arg(I*(2*k1*pi + arg(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d)))) + \ + log(Abs(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d)))))) + log(Abs(I*(2*k1*pi + arg(I*(2*n*pi + arg(-c/d)) + \ + log(Abs(c/d)))) + log(Abs(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d)))))))) + log(Abs(I*(2*k2*pi + \ + arg(I*(2*k1*pi + arg(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d)))) + log(Abs(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d)))))) + \ + log(Abs(I*(2*k1*pi + arg(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d)))) + log(Abs(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d))))))))), \ + ProductSet(S.Integers, S.Integers, S.Integers, S.Integers)))) + + +def test__solveset_multi(): + from sympy.solvers.solveset import _solveset_multi + from sympy.sets import Reals + + # Basic univariate case: + assert _solveset_multi([x**2-1], [x], [S.Reals]) == FiniteSet((1,), (-1,)) + + # Linear systems of two equations + assert _solveset_multi([x+y, x+1], [x, y], [Reals, Reals]) == FiniteSet((-1, 1)) + assert _solveset_multi([x+y, x+1], [y, x], [Reals, Reals]) == FiniteSet((1, -1)) + assert _solveset_multi([x+y, x-y-1], [x, y], [Reals, Reals]) == FiniteSet((S(1)/2, -S(1)/2)) + assert _solveset_multi([x-1, y-2], [x, y], [Reals, Reals]) == FiniteSet((1, 2)) + # assert dumeq(_solveset_multi([x+y], [x, y], [Reals, Reals]), ImageSet(Lambda(x, (x, -x)), Reals)) + assert dumeq(_solveset_multi([x+y], [x, y], [Reals, Reals]), Union( + ImageSet(Lambda(((x,),), (x, -x)), ProductSet(Reals)), + ImageSet(Lambda(((y,),), (-y, y)), ProductSet(Reals)))) + assert _solveset_multi([x+y, x+y+1], [x, y], [Reals, Reals]) == S.EmptySet + assert _solveset_multi([x+y, x-y, x-1], [x, y], [Reals, Reals]) == S.EmptySet + assert _solveset_multi([x+y, x-y, x-1], [y, x], [Reals, Reals]) == S.EmptySet + + # Systems of three equations: + assert _solveset_multi([x+y+z-1, x+y-z-2, x-y-z-3], [x, y, z], [Reals, + Reals, Reals]) == FiniteSet((2, -S.Half, -S.Half)) + + # Nonlinear systems: + from sympy.abc import theta + assert _solveset_multi([x**2+y**2-2, x+y], [x, y], [Reals, Reals]) == FiniteSet((-1, 1), (1, -1)) + assert _solveset_multi([x**2-1, y], [x, y], [Reals, Reals]) == FiniteSet((1, 0), (-1, 0)) + #assert _solveset_multi([x**2-y**2], [x, y], [Reals, Reals]) == Union( + # ImageSet(Lambda(x, (x, -x)), Reals), ImageSet(Lambda(x, (x, x)), Reals)) + assert dumeq(_solveset_multi([x**2-y**2], [x, y], [Reals, Reals]), Union( + ImageSet(Lambda(((x,),), (x, -Abs(x))), ProductSet(Reals)), + ImageSet(Lambda(((x,),), (x, Abs(x))), ProductSet(Reals)), + ImageSet(Lambda(((y,),), (-Abs(y), y)), ProductSet(Reals)), + ImageSet(Lambda(((y,),), (Abs(y), y)), ProductSet(Reals)))) + assert _solveset_multi([r*cos(theta)-1, r*sin(theta)], [theta, r], + [Interval(0, pi), Interval(-1, 1)]) == FiniteSet((0, 1), (pi, -1)) + assert _solveset_multi([r*cos(theta)-1, r*sin(theta)], [r, theta], + [Interval(0, 1), Interval(0, pi)]) == FiniteSet((1, 0)) + assert _solveset_multi([r*cos(theta)-r, r*sin(theta)], [r, theta], + [Interval(0, 1), Interval(0, pi)]) == Union( + ImageSet(Lambda(((r,),), (r, 0)), + ImageSet(Lambda(r, (r,)), Interval(0, 1))), + ImageSet(Lambda(((theta,),), (0, theta)), + ImageSet(Lambda(theta, (theta,)), Interval(0, pi)))) + + +def test_conditionset(): + assert solveset(Eq(sin(x)**2 + cos(x)**2, 1), x, domain=S.Reals + ) is S.Reals + + assert solveset(Eq(x**2 + x*sin(x), 1), x, domain=S.Reals + ).dummy_eq(ConditionSet(x, Eq(x**2 + x*sin(x) - 1, 0), S.Reals)) + + assert dumeq(solveset(Eq(-I*(exp(I*x) - exp(-I*x))/2, 1), x + ), imageset(Lambda(n, 2*n*pi + pi/2), S.Integers)) + + assert solveset(x + sin(x) > 1, x, domain=S.Reals + ).dummy_eq(ConditionSet(x, x + sin(x) > 1, S.Reals)) + + assert solveset(Eq(sin(Abs(x)), x), x, domain=S.Reals + ).dummy_eq(ConditionSet(x, Eq(-x + sin(Abs(x)), 0), S.Reals)) + + assert solveset(y**x-z, x, S.Reals + ).dummy_eq(ConditionSet(x, Eq(y**x - z, 0), S.Reals)) + + +@XFAIL +def test_conditionset_equality(): + ''' Checking equality of different representations of ConditionSet''' + assert solveset(Eq(tan(x), y), x) == ConditionSet(x, Eq(tan(x), y), S.Complexes) + + +def test_solveset_domain(): + assert solveset(x**2 - x - 6, x, Interval(0, oo)) == FiniteSet(3) + assert solveset(x**2 - 1, x, Interval(0, oo)) == FiniteSet(1) + assert solveset(x**4 - 16, x, Interval(0, 10)) == FiniteSet(2) + + +def test_improve_coverage(): + solution = solveset(exp(x) + sin(x), x, S.Reals) + unsolved_object = ConditionSet(x, Eq(exp(x) + sin(x), 0), S.Reals) + assert solution.dummy_eq(unsolved_object) + + +def test_issue_9522(): + expr1 = Eq(1/(x**2 - 4) + x, 1/(x**2 - 4) + 2) + expr2 = Eq(1/x + x, 1/x) + + assert solveset(expr1, x, S.Reals) is S.EmptySet + assert solveset(expr2, x, S.Reals) is S.EmptySet + + +def test_solvify(): + assert solvify(x**2 + 10, x, S.Reals) == [] + assert solvify(x**3 + 1, x, S.Complexes) == [-1, S.Half - sqrt(3)*I/2, + S.Half + sqrt(3)*I/2] + assert solvify(log(x), x, S.Reals) == [1] + assert solvify(cos(x), x, S.Reals) == [pi/2, pi*Rational(3, 2)] + assert solvify(sin(x) + 1, x, S.Reals) == [pi*Rational(3, 2)] + raises(NotImplementedError, lambda: solvify(sin(exp(x)), x, S.Complexes)) + + +def test_solvify_piecewise(): + p1 = Piecewise((0, x < -1), (x**2, x <= 1), (log(x), True)) + p2 = Piecewise((0, x < -10), (x**2 + 5*x - 6, x >= -9)) + p3 = Piecewise((0, Eq(x, 0)), (x**2/Abs(x), True)) + p4 = Piecewise((0, Eq(x, pi)), ((x - pi)/sin(x), True)) + + # issue 21079 + assert solvify(p1, x, S.Reals) == [0] + assert solvify(p2, x, S.Reals) == [-6, 1] + assert solvify(p3, x, S.Reals) == [0] + assert solvify(p4, x, S.Reals) == [pi] + + +def test_abs_invert_solvify(): + + x = Symbol('x',positive=True) + assert solvify(sin(Abs(x)), x, S.Reals) == [0, pi] + x = Symbol('x') + assert solvify(sin(Abs(x)), x, S.Reals) is None + + +def test_linear_eq_to_matrix(): + assert linear_eq_to_matrix(0, x) == (Matrix([[0]]), Matrix([[0]])) + assert linear_eq_to_matrix(1, x) == (Matrix([[0]]), Matrix([[-1]])) + + # integer coefficients + eqns1 = [2*x + y - 2*z - 3, x - y - z, x + y + 3*z - 12] + eqns2 = [Eq(3*x + 2*y - z, 1), Eq(2*x - 2*y + 4*z, -2), -2*x + y - 2*z] + + A, B = linear_eq_to_matrix(eqns1, x, y, z) + assert A == Matrix([[2, 1, -2], [1, -1, -1], [1, 1, 3]]) + assert B == Matrix([[3], [0], [12]]) + + A, B = linear_eq_to_matrix(eqns2, x, y, z) + assert A == Matrix([[3, 2, -1], [2, -2, 4], [-2, 1, -2]]) + assert B == Matrix([[1], [-2], [0]]) + + # Pure symbolic coefficients + eqns3 = [a*b*x + b*y + c*z - d, e*x + d*x + f*y + g*z - h, i*x + j*y + k*z - l] + A, B = linear_eq_to_matrix(eqns3, x, y, z) + assert A == Matrix([[a*b, b, c], [d + e, f, g], [i, j, k]]) + assert B == Matrix([[d], [h], [l]]) + + # raise Errors if + # 1) no symbols are given + raises(ValueError, lambda: linear_eq_to_matrix(eqns3)) + # 2) there are duplicates + raises(ValueError, lambda: linear_eq_to_matrix(eqns3, [x, x, y])) + # 3) a nonlinear term is detected in the original expression + raises(NonlinearError, lambda: linear_eq_to_matrix(Eq(1/x + x, 1/x), [x])) + raises(NonlinearError, lambda: linear_eq_to_matrix([x**2], [x])) + raises(NonlinearError, lambda: linear_eq_to_matrix([x*y], [x, y])) + # 4) Eq being used to represent equations autoevaluates + # (use unevaluated Eq instead) + raises(ValueError, lambda: linear_eq_to_matrix(Eq(x, x), x)) + raises(ValueError, lambda: linear_eq_to_matrix(Eq(x, x + 1), x)) + + + # if non-symbols are passed, the user is responsible for interpreting + assert linear_eq_to_matrix([x], [1/x]) == (Matrix([[0]]), Matrix([[-x]])) + + # issue 15195 + assert linear_eq_to_matrix(x + y*(z*(3*x + 2) + 3), x) == ( + Matrix([[3*y*z + 1]]), Matrix([[-y*(2*z + 3)]])) + assert linear_eq_to_matrix(Matrix( + [[a*x + b*y - 7], [5*x + 6*y - c]]), x, y) == ( + Matrix([[a, b], [5, 6]]), Matrix([[7], [c]])) + + # issue 15312 + assert linear_eq_to_matrix(Eq(x + 2, 1), x) == ( + Matrix([[1]]), Matrix([[-1]])) + + # issue 25423 + raises(TypeError, lambda: linear_eq_to_matrix([], {x, y})) + raises(TypeError, lambda: linear_eq_to_matrix([x + y], {x, y})) + raises(ValueError, lambda: linear_eq_to_matrix({x + y}, (x, y))) + + +def test_issue_16577(): + assert linear_eq_to_matrix(Eq(a*(2*x + 3*y) + 4*y, 5), x, y) == ( + Matrix([[2*a, 3*a + 4]]), Matrix([[5]])) + + +def test_issue_10085(): + assert invert_real(exp(x),0,x) == (x, S.EmptySet) + + +def test_linsolve(): + x1, x2, x3, x4 = symbols('x1, x2, x3, x4') + + # Test for different input forms + + M = Matrix([[1, 2, 1, 1, 7], [1, 2, 2, -1, 12], [2, 4, 0, 6, 4]]) + system1 = A, B = M[:, :-1], M[:, -1] + Eqns = [x1 + 2*x2 + x3 + x4 - 7, x1 + 2*x2 + 2*x3 - x4 - 12, + 2*x1 + 4*x2 + 6*x4 - 4] + + sol = FiniteSet((-2*x2 - 3*x4 + 2, x2, 2*x4 + 5, x4)) + assert linsolve(Eqns, (x1, x2, x3, x4)) == sol + assert linsolve(Eqns, *(x1, x2, x3, x4)) == sol + assert linsolve(system1, (x1, x2, x3, x4)) == sol + assert linsolve(system1, *(x1, x2, x3, x4)) == sol + # issue 9667 - symbols can be Dummy symbols + x1, x2, x3, x4 = symbols('x:4', cls=Dummy) + assert linsolve(system1, x1, x2, x3, x4) == FiniteSet( + (-2*x2 - 3*x4 + 2, x2, 2*x4 + 5, x4)) + + # raise ValueError for garbage value + raises(ValueError, lambda: linsolve(Eqns)) + raises(ValueError, lambda: linsolve(x1)) + raises(ValueError, lambda: linsolve(x1, x2)) + raises(ValueError, lambda: linsolve((A,), x1, x2)) + raises(ValueError, lambda: linsolve(A, B, x1, x2)) + raises(ValueError, lambda: linsolve([x1], x1, x1)) + raises(ValueError, lambda: linsolve([x1], (i for i in (x1, x1)))) + + #raise ValueError if equations are non-linear in given variables + raises(NonlinearError, lambda: linsolve([x + y - 1, x ** 2 + y - 3], [x, y])) + raises(NonlinearError, lambda: linsolve([cos(x) + y, x + y], [x, y])) + assert linsolve([x + z - 1, x ** 2 + y - 3], [z, y]) == {(-x + 1, -x**2 + 3)} + + # Fully symbolic test + A = Matrix([[a, b], [c, d]]) + B = Matrix([[e], [g]]) + system2 = (A, B) + sol = FiniteSet(((-b*g + d*e)/(a*d - b*c), (a*g - c*e)/(a*d - b*c))) + assert linsolve(system2, [x, y]) == sol + + # No solution + A = Matrix([[1, 2, 3], [2, 4, 6], [3, 6, 9]]) + B = Matrix([0, 0, 1]) + assert linsolve((A, B), (x, y, z)) is S.EmptySet + + # Issue #10056 + A, B, J1, J2 = symbols('A B J1 J2') + Augmatrix = Matrix([ + [2*I*J1, 2*I*J2, -2/J1], + [-2*I*J2, -2*I*J1, 2/J2], + [0, 2, 2*I/(J1*J2)], + [2, 0, 0], + ]) + + assert linsolve(Augmatrix, A, B) == FiniteSet((0, I/(J1*J2))) + + # Issue #10121 - Assignment of free variables + Augmatrix = Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0]]) + assert linsolve(Augmatrix, a, b, c, d, e) == FiniteSet((a, 0, c, 0, e)) + #raises(IndexError, lambda: linsolve(Augmatrix, a, b, c)) + + x0, x1, x2, _x0 = symbols('tau0 tau1 tau2 _tau0') + assert linsolve(Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, _x0]]) + ) == FiniteSet((x0, 0, x1, _x0, x2)) + x0, x1, x2, _x0 = symbols('tau00 tau01 tau02 tau0') + assert linsolve(Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, _x0]]) + ) == FiniteSet((x0, 0, x1, _x0, x2)) + x0, x1, x2, _x0 = symbols('tau00 tau01 tau02 tau1') + assert linsolve(Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, _x0]]) + ) == FiniteSet((x0, 0, x1, _x0, x2)) + # symbols can be given as generators + x0, x2, x4 = symbols('x0, x2, x4') + assert linsolve(Augmatrix, numbered_symbols('x') + ) == FiniteSet((x0, 0, x2, 0, x4)) + Augmatrix[-1, -1] = x0 + # use Dummy to avoid clash; the names may clash but the symbols + # will not + Augmatrix[-1, -1] = symbols('_x0') + assert len(linsolve( + Augmatrix, numbered_symbols('x', cls=Dummy)).free_symbols) == 4 + + # Issue #12604 + f = Function('f') + assert linsolve([f(x) - 5], f(x)) == FiniteSet((5,)) + + # Issue #14860 + from sympy.physics.units import meter, newton, kilo + kN = kilo*newton + Eqns = [8*kN + x + y, 28*kN*meter + 3*x*meter] + assert linsolve(Eqns, x, y) == { + (kilo*newton*Rational(-28, 3), kN*Rational(4, 3))} + + # linsolve does not allow expansion (real or implemented) + # to remove singularities, but it will cancel linear terms + assert linsolve([Eq(x, x + y)], [x, y]) == {(x, 0)} + assert linsolve([Eq(x + x*y, 1 + y)], [x]) == {(1,)} + assert linsolve([Eq(1 + y, x + x*y)], [x]) == {(1,)} + raises(NonlinearError, lambda: + linsolve([Eq(x**2, x**2 + y)], [x, y])) + + # corner cases + # + # XXX: The case below should give the same as for [0] + # assert linsolve([], [x]) == {(x,)} + assert linsolve([], [x]) is S.EmptySet + assert linsolve([0], [x]) == {(x,)} + assert linsolve([x], [x, y]) == {(0, y)} + assert linsolve([x, 0], [x, y]) == {(0, y)} + + +def test_linsolve_large_sparse(): + # + # This is mainly a performance test + # + + def _mk_eqs_sol(n): + xs = symbols('x:{}'.format(n)) + ys = symbols('y:{}'.format(n)) + syms = xs + ys + eqs = [] + sol = (-S.Half,) * n + (S.Half,) * n + for xi, yi in zip(xs, ys): + eqs.extend([xi + yi, xi - yi + 1]) + return eqs, syms, FiniteSet(sol) + + n = 500 + eqs, syms, sol = _mk_eqs_sol(n) + assert linsolve(eqs, syms) == sol + + +def test_linsolve_immutable(): + A = ImmutableDenseMatrix([[1, 1, 2], [0, 1, 2], [0, 0, 1]]) + B = ImmutableDenseMatrix([2, 1, -1]) + assert linsolve([A, B], (x, y, z)) == FiniteSet((1, 3, -1)) + + A = ImmutableDenseMatrix([[1, 1, 7], [1, -1, 3]]) + assert linsolve(A) == FiniteSet((5, 2)) + + +def test_solve_decomposition(): + n = Dummy('n') + + f1 = exp(3*x) - 6*exp(2*x) + 11*exp(x) - 6 + f2 = sin(x)**2 - 2*sin(x) + 1 + f3 = sin(x)**2 - sin(x) + f4 = sin(x + 1) + f5 = exp(x + 2) - 1 + f6 = 1/log(x) + f7 = 1/x + + s1 = ImageSet(Lambda(n, 2*n*pi), S.Integers) + s2 = ImageSet(Lambda(n, 2*n*pi + pi), S.Integers) + s3 = ImageSet(Lambda(n, 2*n*pi + pi/2), S.Integers) + s4 = ImageSet(Lambda(n, 2*n*pi - 1), S.Integers) + s5 = ImageSet(Lambda(n, 2*n*pi - 1 + pi), S.Integers) + + assert solve_decomposition(f1, x, S.Reals) == FiniteSet(0, log(2), log(3)) + assert dumeq(solve_decomposition(f2, x, S.Reals), s3) + assert dumeq(solve_decomposition(f3, x, S.Reals), Union(s1, s2, s3)) + assert dumeq(solve_decomposition(f4, x, S.Reals), Union(s4, s5)) + assert solve_decomposition(f5, x, S.Reals) == FiniteSet(-2) + assert solve_decomposition(f6, x, S.Reals) == S.EmptySet + assert solve_decomposition(f7, x, S.Reals) == S.EmptySet + assert solve_decomposition(x, x, Interval(1, 2)) == S.EmptySet + + +# nonlinsolve testcases +def test_nonlinsolve_basic(): + assert nonlinsolve([],[]) == S.EmptySet + assert nonlinsolve([],[x, y]) == S.EmptySet + + system = [x, y - x - 5] + assert nonlinsolve([x],[x, y]) == FiniteSet((0, y)) + assert nonlinsolve(system, [y]) == S.EmptySet + soln = (ImageSet(Lambda(n, 2*n*pi + pi/2), S.Integers),) + assert dumeq(nonlinsolve([sin(x) - 1], [x]), FiniteSet(tuple(soln))) + soln = ((ImageSet(Lambda(n, 2*n*pi + pi), S.Integers), 1), + (ImageSet(Lambda(n, 2*n*pi), S.Integers), 1)) + assert dumeq(nonlinsolve([sin(x), y - 1], [x, y]), FiniteSet(*soln)) + assert nonlinsolve([x**2 - 1], [x]) == FiniteSet((-1,), (1,)) + + soln = FiniteSet((y, y)) + assert nonlinsolve([x - y, 0], x, y) == soln + assert nonlinsolve([0, x - y], x, y) == soln + assert nonlinsolve([x - y, x - y], x, y) == soln + assert nonlinsolve([x, 0], x, y) == FiniteSet((0, y)) + f = Function('f') + assert nonlinsolve([f(x), 0], f(x), y) == FiniteSet((0, y)) + assert nonlinsolve([f(x), 0], f(x), f(y)) == FiniteSet((0, f(y))) + A = Indexed('A', x) + assert nonlinsolve([A, 0], A, y) == FiniteSet((0, y)) + assert nonlinsolve([x**2 -1], [sin(x)]) == FiniteSet((S.EmptySet,)) + assert nonlinsolve([x**2 -1], sin(x)) == FiniteSet((S.EmptySet,)) + assert nonlinsolve([x**2 -1], 1) == FiniteSet((x**2,)) + assert nonlinsolve([x**2 -1], x + y) == FiniteSet((S.EmptySet,)) + assert nonlinsolve([Eq(1, x + y), Eq(1, -x + y - 1), Eq(1, -x + y - 1)], x, y) == FiniteSet( + (-S.Half, 3*S.Half)) + + +def test_nonlinsolve_abs(): + soln = FiniteSet((y, y), (-y, y)) + assert nonlinsolve([Abs(x) - y], x, y) == soln + + +def test_raise_exception_nonlinsolve(): + raises(IndexError, lambda: nonlinsolve([x**2 -1], [])) + raises(ValueError, lambda: nonlinsolve([x**2 -1])) + + +def test_trig_system(): + # TODO: add more simple testcases when solveset returns + # simplified soln for Trig eq + assert nonlinsolve([sin(x) - 1, cos(x) -1 ], x) == S.EmptySet + soln1 = (ImageSet(Lambda(n, 2*n*pi + pi/2), S.Integers),) + soln = FiniteSet(soln1) + assert dumeq(nonlinsolve([sin(x) - 1, cos(x)], x), soln) + + +@XFAIL +def test_trig_system_fail(): + # fails because solveset trig solver is not much smart. + sys = [x + y - pi/2, sin(x) + sin(y) - 1] + # solveset returns conditionset for sin(x) + sin(y) - 1 + soln_1 = (ImageSet(Lambda(n, n*pi + pi/2), S.Integers), + ImageSet(Lambda(n, n*pi), S.Integers)) + soln_1 = FiniteSet(soln_1) + soln_2 = (ImageSet(Lambda(n, n*pi), S.Integers), + ImageSet(Lambda(n, n*pi+ pi/2), S.Integers)) + soln_2 = FiniteSet(soln_2) + soln = soln_1 + soln_2 + assert dumeq(nonlinsolve(sys, [x, y]), soln) + + # Add more cases from here + # http://www.vitutor.com/geometry/trigonometry/equations_systems.html#uno + sys = [sin(x) + sin(y) - (sqrt(3)+1)/2, sin(x) - sin(y) - (sqrt(3) - 1)/2] + soln_x = Union(ImageSet(Lambda(n, 2*n*pi + pi/3), S.Integers), + ImageSet(Lambda(n, 2*n*pi + pi*Rational(2, 3)), S.Integers)) + soln_y = Union(ImageSet(Lambda(n, 2*n*pi + pi/6), S.Integers), + ImageSet(Lambda(n, 2*n*pi + pi*Rational(5, 6)), S.Integers)) + assert dumeq(nonlinsolve(sys, [x, y]), FiniteSet((soln_x, soln_y))) + + +def test_nonlinsolve_positive_dimensional(): + x, y, a, b, c, d = symbols('x, y, a, b, c, d', extended_real=True) + assert nonlinsolve([x*y, x*y - x], [x, y]) == FiniteSet((0, y)) + + system = [a**2 + a*c, a - b] + assert nonlinsolve(system, [a, b]) == FiniteSet((0, 0), (-c, -c)) + # here (a= 0, b = 0) is independent soln so both is printed. + # if symbols = [a, b, c] then only {a : -c ,b : -c} + + eq1 = a + b + c + d + eq2 = a*b + b*c + c*d + d*a + eq3 = a*b*c + b*c*d + c*d*a + d*a*b + eq4 = a*b*c*d - 1 + system = [eq1, eq2, eq3, eq4] + sol1 = (-1/d, -d, 1/d, FiniteSet(d) - FiniteSet(0)) + sol2 = (1/d, -d, -1/d, FiniteSet(d) - FiniteSet(0)) + soln = FiniteSet(sol1, sol2) + assert nonlinsolve(system, [a, b, c, d]) == soln + + assert nonlinsolve([x**4 - 3*x**2 + y*x, x*z**2, y*z - 1], [x, y, z]) == \ + {(0, 1/z, z)} + + +def test_nonlinsolve_polysys(): + x, y, z = symbols('x, y, z', real=True) + assert nonlinsolve([x**2 + y - 2, x**2 + y], [x, y]) == S.EmptySet + + s = (-y + 2, y) + assert nonlinsolve([(x + y)**2 - 4, x + y - 2], [x, y]) == FiniteSet(s) + + system = [x**2 - y**2] + soln_real = FiniteSet((-y, y), (y, y)) + soln_complex = FiniteSet((-Abs(y), y), (Abs(y), y)) + soln =soln_real + soln_complex + assert nonlinsolve(system, [x, y]) == soln + + system = [x**2 - y**2] + soln_real= FiniteSet((y, -y), (y, y)) + soln_complex = FiniteSet((y, -Abs(y)), (y, Abs(y))) + soln = soln_real + soln_complex + assert nonlinsolve(system, [y, x]) == soln + + system = [x**2 + y - 3, x - y - 4] + assert nonlinsolve(system, (x, y)) != nonlinsolve(system, (y, x)) + + assert nonlinsolve([-x**2 - y**2 + z, -2*x, -2*y, S.One], [x, y, z]) == S.EmptySet + assert nonlinsolve([x + y + z, S.One, S.One, S.One], [x, y, z]) == S.EmptySet + + system = [-x**2*z**2 + x*y*z + y**4, -2*x*z**2 + y*z, x*z + 4*y**3, -2*x**2*z + x*y] + assert nonlinsolve(system, [x, y, z]) == FiniteSet((0, 0, z), (x, 0, 0)) + + +def test_nonlinsolve_using_substitution(): + x, y, z, n = symbols('x, y, z, n', real = True) + system = [(x + y)*n - y**2 + 2] + s_x = (n*y - y**2 + 2)/n + soln = (-s_x, y) + assert nonlinsolve(system, [x, y]) == FiniteSet(soln) + + system = [z**2*x**2 - z**2*y**2/exp(x)] + soln_real_1 = (y, x, 0) + soln_real_2 = (-exp(x/2)*Abs(x), x, z) + soln_real_3 = (exp(x/2)*Abs(x), x, z) + soln_complex_1 = (-x*exp(x/2), x, z) + soln_complex_2 = (x*exp(x/2), x, z) + syms = [y, x, z] + soln = FiniteSet(soln_real_1, soln_complex_1, soln_complex_2,\ + soln_real_2, soln_real_3) + assert nonlinsolve(system,syms) == soln + + +def test_nonlinsolve_complex(): + n = Dummy('n') + assert dumeq(nonlinsolve([exp(x) - sin(y), 1/y - 3], [x, y]), { + (ImageSet(Lambda(n, 2*n*I*pi + log(sin(Rational(1, 3)))), S.Integers), Rational(1, 3))}) + + system = [exp(x) - sin(y), 1/exp(y) - 3] + assert dumeq(nonlinsolve(system, [x, y]), { + (ImageSet(Lambda(n, I*(2*n*pi + pi) + + log(sin(log(3)))), S.Integers), -log(3)), + (ImageSet(Lambda(n, I*(2*n*pi + arg(sin(2*n*I*pi - log(3)))) + + log(Abs(sin(2*n*I*pi - log(3))))), S.Integers), + ImageSet(Lambda(n, 2*n*I*pi - log(3)), S.Integers))}) + + system = [exp(x) - sin(y), y**2 - 4] + assert dumeq(nonlinsolve(system, [x, y]), { + (ImageSet(Lambda(n, I*(2*n*pi + pi) + log(sin(2))), S.Integers), -2), + (ImageSet(Lambda(n, 2*n*I*pi + log(sin(2))), S.Integers), 2)}) + + system = [exp(x) - 2, y ** 2 - 2] + assert dumeq(nonlinsolve(system, [x, y]), { + (log(2), -sqrt(2)), (log(2), sqrt(2)), + (ImageSet(Lambda(n, 2*n*I*pi + log(2)), S.Integers), -sqrt(2)), + (ImageSet(Lambda(n, 2 * n * I * pi + log(2)), S.Integers), sqrt(2))}) + + +def test_nonlinsolve_radical(): + assert nonlinsolve([sqrt(y) - x - z, y - 1], [x, y, z]) == {(1 - z, 1, z)} + + +def test_nonlinsolve_inexact(): + sol = [(-1.625, -1.375), (1.625, 1.375)] + res = nonlinsolve([(x + y)**2 - 9, x**2 - y**2 - 0.75], [x, y]) + assert all(abs(res.args[i][j]-sol[i][j]) < 1e-9 + for i in range(2) for j in range(2)) + + assert nonlinsolve([(x + y)**2 - 9, (x + y)**2 - 0.75], [x, y]) == S.EmptySet + + assert nonlinsolve([y**2 + (x - 0.5)**2 - 0.0625, 2*x - 1.0, 2*y], [x, y]) == \ + S.EmptySet + + res = nonlinsolve([x**2 + y - 0.5, (x + y)**2, log(z)], [x, y, z]) + sol = [(-0.366025403784439, 0.366025403784439, 1), + (-0.366025403784439, 0.366025403784439, 1), + (1.36602540378444, -1.36602540378444, 1)] + assert all(abs(res.args[i][j]-sol[i][j]) < 1e-9 + for i in range(3) for j in range(3)) + + res = nonlinsolve([y - x**2, x**5 - x + 1.0], [x, y]) + sol = [(-1.16730397826142, 1.36259857766493), + (-0.181232444469876 - 1.08395410131771*I, + -1.14211129483496 + 0.392895302949911*I), + (-0.181232444469876 + 1.08395410131771*I, + -1.14211129483496 - 0.392895302949911*I), + (0.764884433600585 - 0.352471546031726*I, + 0.460812006002492 - 0.539199997693599*I), + (0.764884433600585 + 0.352471546031726*I, + 0.460812006002492 + 0.539199997693599*I)] + assert all(abs(res.args[i][j] - sol[i][j]) < 1e-9 + for i in range(5) for j in range(2)) + +@XFAIL +def test_solve_nonlinear_trans(): + # After the transcendental equation solver these will work + x, y = symbols('x, y', real=True) + soln1 = FiniteSet((2*LambertW(y/2), y)) + soln2 = FiniteSet((-x*sqrt(exp(x)), y), (x*sqrt(exp(x)), y)) + soln3 = FiniteSet((x*exp(x/2), x)) + soln4 = FiniteSet(2*LambertW(y/2), y) + assert nonlinsolve([x**2 - y**2/exp(x)], [x, y]) == soln1 + assert nonlinsolve([x**2 - y**2/exp(x)], [y, x]) == soln2 + assert nonlinsolve([x**2 - y**2/exp(x)], [y, x]) == soln3 + assert nonlinsolve([x**2 - y**2/exp(x)], [x, y]) == soln4 + + +def test_nonlinsolve_issue_25182(): + a1, b1, c1, ca, cb, cg = symbols('a1, b1, c1, ca, cb, cg') + eq1 = a1*a1 + b1*b1 - 2.*a1*b1*cg - c1*c1 + eq2 = a1*a1 + c1*c1 - 2.*a1*c1*cb - b1*b1 + eq3 = b1*b1 + c1*c1 - 2.*b1*c1*ca - a1*a1 + assert nonlinsolve([eq1, eq2, eq3], [c1, cb, cg]) == FiniteSet( + (1.0*b1*ca - 1.0*sqrt(a1**2 + b1**2*ca**2 - b1**2), + -1.0*sqrt(a1**2 + b1**2*ca**2 - b1**2)/a1, + -1.0*b1*(ca - 1)*(ca + 1)/a1 + 1.0*ca*sqrt(a1**2 + b1**2*ca**2 - b1**2)/a1), + (1.0*b1*ca + 1.0*sqrt(a1**2 + b1**2*ca**2 - b1**2), + 1.0*sqrt(a1**2 + b1**2*ca**2 - b1**2)/a1, + -1.0*b1*(ca - 1)*(ca + 1)/a1 - 1.0*ca*sqrt(a1**2 + b1**2*ca**2 - b1**2)/a1)) + + +def test_issue_14642(): + x = Symbol('x') + n1 = 0.5*x**3+x**2+0.5+I #add I in the Polynomials + solution = solveset(n1, x) + assert abs(solution.args[0] - (-2.28267560928153 - 0.312325580497716*I)) <= 1e-9 + assert abs(solution.args[1] - (-0.297354141679308 + 1.01904778618762*I)) <= 1e-9 + assert abs(solution.args[2] - (0.580029750960839 - 0.706722205689907*I)) <= 1e-9 + + # Symbolic + n1 = S.Half*x**3+x**2+S.Half+I + res = FiniteSet(-((3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 + + S(43)/2)**2 + (27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(S(172)/49) + /2)/2)**2)**(S(1)/6)*cos(atan((27 + 3*sqrt(3)*31985**(S(1)/4)* + cos(atan(S(172)/49)/2)/2)/(3*sqrt(3)*31985**(S(1)/4)*sin(atan( + S(172)/49)/2)/2 + S(43)/2))/3)/3 - S(2)/3 - 4*cos(atan((27 + + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(S(172)/49)/2)/2)/(3*sqrt(3)* + 31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 + S(43)/2))/3)/(3*((3* + sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 + S(43)/2)**2 + + (27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(S(172)/49)/2)/2)**2)**(S(1)/ + 6)) + I*(-((3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 + + S(43)/2)**2 + (27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(S(172)/49)/ + 2)/2)**2)**(S(1)/6)*sin(atan((27 + 3*sqrt(3)*31985**(S(1)/4)*cos( + atan(S(172)/49)/2)/2)/(3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49) + /2)/2 + S(43)/2))/3)/3 + 4*sin(atan((27 + 3*sqrt(3)*31985**(S(1)/4)* + cos(atan(S(172)/49)/2)/2)/(3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172) + /49)/2)/2 + S(43)/2))/3)/(3*((3*sqrt(3)*31985**(S(1)/4)*sin(atan( + S(172)/49)/2)/2 + S(43)/2)**2 + (27 + 3*sqrt(3)*31985**(S(1)/4)* + cos(atan(S(172)/49)/2)/2)**2)**(S(1)/6))), -S(2)/3 - sqrt(3)*((3* + sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 + S(43)/2)**2 + + (27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(S(172)/49)/2)/2)**2)**(S(1) + /6)*sin(atan((27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(S(172)/49)/2) + /2)/(3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 + S(43)/2)) + /3)/6 - 4*re(1/((-S(1)/2 - sqrt(3)*I/2)*(S(43)/2 + 27*I + sqrt(-256 + + (43 + 54*I)**2)/2)**(S(1)/3)))/3 + ((3*sqrt(3)*31985**(S(1)/4)*sin( + atan(S(172)/49)/2)/2 + S(43)/2)**2 + (27 + 3*sqrt(3)*31985**(S(1)/4)* + cos(atan(S(172)/49)/2)/2)**2)**(S(1)/6)*cos(atan((27 + 3*sqrt(3)* + 31985**(S(1)/4)*cos(atan(S(172)/49)/2)/2)/(3*sqrt(3)*31985**(S(1)/4)* + sin(atan(S(172)/49)/2)/2 + S(43)/2))/3)/6 + I*(-4*im(1/((-S(1)/2 - + sqrt(3)*I/2)*(S(43)/2 + 27*I + sqrt(-256 + (43 + 54*I)**2)/2)**(S(1)/ + 3)))/3 + ((3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 + + S(43)/2)**2 + (27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(S(172)/49)/2) + /2)**2)**(S(1)/6)*sin(atan((27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan( + S(172)/49)/2)/2)/(3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 + + S(43)/2))/3)/6 + sqrt(3)*((3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/ + 49)/2)/2 + S(43)/2)**2 + (27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan( + S(172)/49)/2)/2)**2)**(S(1)/6)*cos(atan((27 + 3*sqrt(3)*31985**(S(1)/ + 4)*cos(atan(S(172)/49)/2)/2)/(3*sqrt(3)*31985**(S(1)/4)*sin(atan( + S(172)/49)/2)/2 + S(43)/2))/3)/6), -S(2)/3 - 4*re(1/((-S(1)/2 + + sqrt(3)*I/2)*(S(43)/2 + 27*I + sqrt(-256 + (43 + 54*I)**2)/2)**(S(1) + /3)))/3 + sqrt(3)*((3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 + + S(43)/2)**2 + (27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(S(172)/49)/2) + /2)**2)**(S(1)/6)*sin(atan((27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan( + S(172)/49)/2)/2)/(3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 + + S(43)/2))/3)/6 + ((3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 + + S(43)/2)**2 + (27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(S(172)/49)/2) + /2)**2)**(S(1)/6)*cos(atan((27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan( + S(172)/49)/2)/2)/(3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 + + S(43)/2))/3)/6 + I*(-sqrt(3)*((3*sqrt(3)*31985**(S(1)/4)*sin(atan( + S(172)/49)/2)/2 + S(43)/2)**2 + (27 + 3*sqrt(3)*31985**(S(1)/4)*cos( + atan(S(172)/49)/2)/2)**2)**(S(1)/6)*cos(atan((27 + 3*sqrt(3)*31985**( + S(1)/4)*cos(atan(S(172)/49)/2)/2)/(3*sqrt(3)*31985**(S(1)/4)*sin( + atan(S(172)/49)/2)/2 + S(43)/2))/3)/6 + ((3*sqrt(3)*31985**(S(1)/4)* + sin(atan(S(172)/49)/2)/2 + S(43)/2)**2 + (27 + 3*sqrt(3)*31985**(S(1)/4)* + cos(atan(S(172)/49)/2)/2)**2)**(S(1)/6)*sin(atan((27 + 3*sqrt(3)*31985**( + S(1)/4)*cos(atan(S(172)/49)/2)/2)/(3*sqrt(3)*31985**(S(1)/4)*sin( + atan(S(172)/49)/2)/2 + S(43)/2))/3)/6 - 4*im(1/((-S(1)/2 + sqrt(3)*I/2)* + (S(43)/2 + 27*I + sqrt(-256 + (43 + 54*I)**2)/2)**(S(1)/3)))/3)) + + assert solveset(n1, x) == res + + +def test_issue_13961(): + V = (ax, bx, cx, gx, jx, lx, mx, nx, q) = symbols('ax bx cx gx jx lx mx nx q') + S = (ax*q - lx*q - mx, ax - gx*q - lx, bx*q**2 + cx*q - jx*q - nx, q*(-ax*q + lx*q + mx), q*(-ax + gx*q + lx)) + + sol = FiniteSet((lx + mx/q, (-cx*q + jx*q + nx)/q**2, cx, mx/q**2, jx, lx, mx, nx, Complement({q}, {0})), + (lx + mx/q, (cx*q - jx*q - nx)/q**2*-1, cx, mx/q**2, jx, lx, mx, nx, Complement({q}, {0}))) + assert nonlinsolve(S, *V) == sol + # The two solutions are in fact identical, so even better if only one is returned + + +def test_issue_14541(): + solutions = solveset(sqrt(-x**2 - 2.0), x) + assert abs(solutions.args[0]+1.4142135623731*I) <= 1e-9 + assert abs(solutions.args[1]-1.4142135623731*I) <= 1e-9 + + +def test_issue_13396(): + expr = -2*y*exp(-x**2 - y**2)*Abs(x) + sol = FiniteSet(0) + + assert solveset(expr, y, domain=S.Reals) == sol + + # Related type of equation also solved here + assert solveset(atan(x**2 - y**2)-pi/2, y, S.Reals) is S.EmptySet + + +def test_issue_12032(): + sol = FiniteSet(-sqrt(-2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) + + 2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)))/2 + + sqrt(Abs(-2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)) + + 2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) + + 2/sqrt(-2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) + + 2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)))))/2, + -sqrt(Abs(-2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)) + + 2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) + + 2/sqrt(-2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) + + 2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)))))/2 - + sqrt(-2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) + + 2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)))/2, + sqrt(-2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) + + 2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)))/2 - + I*sqrt(Abs(-2/sqrt(-2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) + + 2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) - + 2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)) + + 2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)))))/2, + sqrt(-2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) + + 2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)))/2 + + I*sqrt(Abs(-2/sqrt(-2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) + + 2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) - + 2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)) + + 2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1,3)))))/2) + assert solveset(x**4 + x - 1, x) == sol + + +def test_issue_10876(): + assert solveset(1/sqrt(x), x) == S.EmptySet + + +def test_issue_19050(): + # test_issue_19050 --> TypeError removed + assert dumeq(nonlinsolve([x + y, sin(y)], [x, y]), + FiniteSet((ImageSet(Lambda(n, -2*n*pi), S.Integers), ImageSet(Lambda(n, 2*n*pi), S.Integers)),\ + (ImageSet(Lambda(n, -2*n*pi - pi), S.Integers), ImageSet(Lambda(n, 2*n*pi + pi), S.Integers)))) + assert dumeq(nonlinsolve([x + y, sin(y) + cos(y)], [x, y]), + FiniteSet((ImageSet(Lambda(n, -2*n*pi - 3*pi/4), S.Integers), ImageSet(Lambda(n, 2*n*pi + 3*pi/4), S.Integers)), \ + (ImageSet(Lambda(n, -2*n*pi - 7*pi/4), S.Integers), ImageSet(Lambda(n, 2*n*pi + 7*pi/4), S.Integers)))) + + +def test_issue_16618(): + eqn = [sin(x)*sin(y), cos(x)*cos(y) - 1] + # nonlinsolve's answer is still suspicious since it contains only three + # distinct Dummys instead of 4. (Both 'x' ImageSets share the same Dummy.) + ans = FiniteSet((ImageSet(Lambda(n, 2*n*pi), S.Integers), ImageSet(Lambda(n, 2*n*pi), S.Integers)), + (ImageSet(Lambda(n, 2*n*pi + pi), S.Integers), ImageSet(Lambda(n, 2*n*pi + pi), S.Integers))) + sol = nonlinsolve(eqn, [x, y]) + + for i0, j0 in zip(ordered(sol), ordered(ans)): + assert len(i0) == len(j0) == 2 + assert all(a.dummy_eq(b) for a, b in zip(i0, j0)) + assert len(sol) == len(ans) + + +def test_issue_17566(): + assert nonlinsolve([32*(2**x)/2**(-y) - 4**y, 27*(3**x) - S(1)/3**y], x, y) ==\ + FiniteSet((-log(81)/log(3), 1)) + + +def test_issue_16643(): + n = Dummy('n') + assert solveset(x**2*sin(x), x).dummy_eq(Union(ImageSet(Lambda(n, 2*n*pi + pi), S.Integers), + ImageSet(Lambda(n, 2*n*pi), S.Integers))) + + +def test_issue_19587(): + n,m = symbols('n m') + assert nonlinsolve([32*2**m*2**n - 4**n, 27*3**m - 3**(-n)], m, n) ==\ + FiniteSet((-log(81)/log(3), 1)) + + +def test_issue_5132_1(): + system = [sqrt(x**2 + y**2) - sqrt(10), x + y - 4] + assert nonlinsolve(system, [x, y]) == FiniteSet((1, 3), (3, 1)) + + n = Dummy('n') + eqs = [exp(x)**2 - sin(y) + z**2, 1/exp(y) - 3] + s_real_y = -log(3) + s_real_z = sqrt(-exp(2*x) - sin(log(3))) + soln_real = FiniteSet((s_real_y, s_real_z), (s_real_y, -s_real_z)) + lam = Lambda(n, 2*n*I*pi + -log(3)) + s_complex_y = ImageSet(lam, S.Integers) + lam = Lambda(n, sqrt(-exp(2*x) + sin(2*n*I*pi + -log(3)))) + s_complex_z_1 = ImageSet(lam, S.Integers) + lam = Lambda(n, -sqrt(-exp(2*x) + sin(2*n*I*pi + -log(3)))) + s_complex_z_2 = ImageSet(lam, S.Integers) + soln_complex = FiniteSet( + (s_complex_y, s_complex_z_1), + (s_complex_y, s_complex_z_2) + ) + soln = soln_real + soln_complex + assert dumeq(nonlinsolve(eqs, [y, z]), soln) + + +def test_issue_5132_2(): + x, y = symbols('x, y', real=True) + eqs = [exp(x)**2 - sin(y) + z**2] + n = Dummy('n') + soln_real = (log(-z**2 + sin(y))/2, z) + lam = Lambda( n, I*(2*n*pi + arg(-z**2 + sin(y)))/2 + log(Abs(z**2 - sin(y)))/2) + img = ImageSet(lam, S.Integers) + # not sure about the complex soln. But it looks correct. + soln_complex = (img, z) + soln = FiniteSet(soln_real, soln_complex) + assert dumeq(nonlinsolve(eqs, [x, z]), soln) + + system = [r - x**2 - y**2, tan(t) - y/x] + s_x = sqrt(r/(tan(t)**2 + 1)) + s_y = sqrt(r/(tan(t)**2 + 1))*tan(t) + soln = FiniteSet((s_x, s_y), (-s_x, -s_y)) + assert nonlinsolve(system, [x, y]) == soln + + +def test_issue_6752(): + a, b = symbols('a, b', real=True) + assert nonlinsolve([a**2 + a, a - b], [a, b]) == {(-1, -1), (0, 0)} + + +@SKIP("slow") +def test_issue_5114_solveset(): + # slow testcase + from sympy.abc import o, p + + # there is no 'a' in the equation set but this is how the + # problem was originally posed + syms = [a, b, c, f, h, k, n] + eqs = [b + r/d - c/d, + c*(1/d + 1/e + 1/g) - f/g - r/d, + f*(1/g + 1/i + 1/j) - c/g - h/i, + h*(1/i + 1/l + 1/m) - f/i - k/m, + k*(1/m + 1/o + 1/p) - h/m - n/p, + n*(1/p + 1/q) - k/p] + assert len(nonlinsolve(eqs, syms)) == 1 + + +@SKIP("Hangs") +def _test_issue_5335(): + # Not able to check zero dimensional system. + # is_zero_dimensional Hangs + lam, a0, conc = symbols('lam a0 conc') + eqs = [lam + 2*y - a0*(1 - x/2)*x - 0.005*x/2*x, + a0*(1 - x/2)*x - 1*y - 0.743436700916726*y, + x + y - conc] + sym = [x, y, a0] + # there are 4 solutions but only two are valid + assert len(nonlinsolve(eqs, sym)) == 2 + # float + eqs = [lam + 2*y - a0*(1 - x/2)*x - 0.005*x/2*x, + a0*(1 - x/2)*x - 1*y - 0.743436700916726*y, + x + y - conc] + sym = [x, y, a0] + assert len(nonlinsolve(eqs, sym)) == 2 + + +def test_issue_2777(): + # the equations represent two circles + x, y = symbols('x y', real=True) + e1, e2 = sqrt(x**2 + y**2) - 10, sqrt(y**2 + (-x + 10)**2) - 3 + a, b = Rational(191, 20), 3*sqrt(391)/20 + ans = {(a, -b), (a, b)} + assert nonlinsolve((e1, e2), (x, y)) == ans + assert nonlinsolve((e1, e2/(x - a)), (x, y)) == S.EmptySet + # make the 2nd circle's radius be -3 + e2 += 6 + assert nonlinsolve((e1, e2), (x, y)) == S.EmptySet + + +def test_issue_8828(): + x1 = 0 + y1 = -620 + r1 = 920 + x2 = 126 + y2 = 276 + x3 = 51 + y3 = 205 + r3 = 104 + v = [x, y, z] + + f1 = (x - x1)**2 + (y - y1)**2 - (r1 - z)**2 + f2 = (x2 - x)**2 + (y2 - y)**2 - z**2 + f3 = (x - x3)**2 + (y - y3)**2 - (r3 - z)**2 + F = [f1, f2, f3] + + g1 = sqrt((x - x1)**2 + (y - y1)**2) + z - r1 + g2 = f2 + g3 = sqrt((x - x3)**2 + (y - y3)**2) + z - r3 + G = [g1, g2, g3] + + # both soln same + A = nonlinsolve(F, v) + B = nonlinsolve(G, v) + assert A == B + + +def test_nonlinsolve_conditionset(): + # when solveset failed to solve all the eq + # return conditionset + f = Function('f') + f1 = f(x) - pi/2 + f2 = f(y) - pi*Rational(3, 2) + intermediate_system = Eq(2*f(x) - pi, 0) & Eq(2*f(y) - 3*pi, 0) + syms = Tuple(x, y) + soln = ConditionSet( + syms, + intermediate_system, + S.Complexes**2) + assert nonlinsolve([f1, f2], [x, y]) == soln + + +def test_substitution_basic(): + assert substitution([], [x, y]) == S.EmptySet + assert substitution([], []) == S.EmptySet + system = [2*x**2 + 3*y**2 - 30, 3*x**2 - 2*y**2 - 19] + soln = FiniteSet((-3, -2), (-3, 2), (3, -2), (3, 2)) + assert substitution(system, [x, y]) == soln + + soln = FiniteSet((-1, 1)) + assert substitution([x + y], [x], [{y: 1}], [y], set(), [x, y]) == soln + assert substitution( + [x + y], [x], [{y: 1}], [y], + {x + 1}, [y, x]) == S.EmptySet + + +def test_substitution_incorrect(): + # the solutions in the following two tests are incorrect. The + # correct result is EmptySet in both cases. + assert substitution([h - 1, k - 1, f - 2, f - 4, -2 * k], + [h, k, f]) == {(1, 1, f)} + assert substitution([x + y + z, S.One, S.One, S.One], [x, y, z]) == \ + {(-y - z, y, z)} + + # the correct result in the test below is {(-I, I, I, -I), + # (I, -I, -I, I)} + assert substitution([a - d, b + d, c + d, d**2 + 1], [a, b, c, d]) == \ + {(d, -d, -d, d)} + + # the result in the test below is incomplete. The complete result + # is {(0, b), (log(2), 2)} + assert substitution([a*(a - log(b)), a*(b - 2)], [a, b]) == \ + {(0, b)} + + # The system in the test below is zero-dimensional, so the result + # should have no free symbols + assert substitution([-k*y + 6*x - 4*y, -81*k + 49*y**2 - 270, + -3*k*z + k + z**3, k**2 - 2*k + 4], + [x, y, z, k]).free_symbols == {z} + + +def test_substitution_redundant(): + # the third and fourth solutions are redundant in the test below + assert substitution([x**2 - y**2, z - 1], [x, z]) == \ + {(-y, 1), (y, 1), (-sqrt(y**2), 1), (sqrt(y**2), 1)} + + # the system below has three solutions. Two of the solutions + # returned by substitution are redundant. + res = substitution([x - y, y**3 - 3*y**2 + 1], [x, y]) + assert len(res) == 5 + + +def test_issue_5132_substitution(): + x, y, z, r, t = symbols('x, y, z, r, t', real=True) + system = [r - x**2 - y**2, tan(t) - y/x] + s_x_1 = Complement(FiniteSet(-sqrt(r/(tan(t)**2 + 1))), FiniteSet(0)) + s_x_2 = Complement(FiniteSet(sqrt(r/(tan(t)**2 + 1))), FiniteSet(0)) + s_y = sqrt(r/(tan(t)**2 + 1))*tan(t) + soln = FiniteSet((s_x_2, s_y)) + FiniteSet((s_x_1, -s_y)) + assert substitution(system, [x, y]) == soln + + n = Dummy('n') + eqs = [exp(x)**2 - sin(y) + z**2, 1/exp(y) - 3] + s_real_y = -log(3) + s_real_z = sqrt(-exp(2*x) - sin(log(3))) + soln_real = FiniteSet((s_real_y, s_real_z), (s_real_y, -s_real_z)) + lam = Lambda(n, 2*n*I*pi + -log(3)) + s_complex_y = ImageSet(lam, S.Integers) + lam = Lambda(n, sqrt(-exp(2*x) + sin(2*n*I*pi + -log(3)))) + s_complex_z_1 = ImageSet(lam, S.Integers) + lam = Lambda(n, -sqrt(-exp(2*x) + sin(2*n*I*pi + -log(3)))) + s_complex_z_2 = ImageSet(lam, S.Integers) + soln_complex = FiniteSet( + (s_complex_y, s_complex_z_1), + (s_complex_y, s_complex_z_2)) + soln = soln_real + soln_complex + assert dumeq(substitution(eqs, [y, z]), soln) + + +def test_raises_substitution(): + raises(ValueError, lambda: substitution([x**2 -1], [])) + raises(TypeError, lambda: substitution([x**2 -1])) + raises(ValueError, lambda: substitution([x**2 -1], [sin(x)])) + raises(TypeError, lambda: substitution([x**2 -1], x)) + raises(TypeError, lambda: substitution([x**2 -1], 1)) + + +def test_issue_21022(): + from sympy.core.sympify import sympify + + eqs = [ + 'k-16', + 'p-8', + 'y*y+z*z-x*x', + 'd - x + p', + 'd*d+k*k-y*y', + 'z*z-p*p-k*k', + 'abc-efg', + ] + efg = Symbol('efg') + eqs = [sympify(x) for x in eqs] + + syb = list(ordered(set.union(*[x.free_symbols for x in eqs]))) + res = nonlinsolve(eqs, syb) + + ans = FiniteSet( + (efg, 32, efg, 16, 8, 40, -16*sqrt(5), -8*sqrt(5)), + (efg, 32, efg, 16, 8, 40, -16*sqrt(5), 8*sqrt(5)), + (efg, 32, efg, 16, 8, 40, 16*sqrt(5), -8*sqrt(5)), + (efg, 32, efg, 16, 8, 40, 16*sqrt(5), 8*sqrt(5)), + ) + assert len(res) == len(ans) == 4 + assert res == ans + for result in res.args: + assert len(result) == 8 + + +def test_issue_17940(): + n = Dummy('n') + k1 = Dummy('k1') + sol = ImageSet(Lambda(((k1, n),), I*(2*k1*pi + arg(2*n*I*pi + log(5))) + + log(Abs(2*n*I*pi + log(5)))), + ProductSet(S.Integers, S.Integers)) + assert solveset(exp(exp(x)) - 5, x).dummy_eq(sol) + + +def test_issue_17906(): + assert solveset(7**(x**2 - 80) - 49**x, x) == FiniteSet(-8, 10) + + +@XFAIL +def test_issue_17933(): + eq1 = x*sin(45) - y*cos(q) + eq2 = x*cos(45) - y*sin(q) + eq3 = 9*x*sin(45)/10 + y*cos(q) + eq4 = 9*x*cos(45)/10 + y*sin(z) - z + assert nonlinsolve([eq1, eq2, eq3, eq4], x, y, z, q) ==\ + FiniteSet((0, 0, 0, q)) + +def test_issue_17933_bis(): + # nonlinsolve's result depends on the 'default_sort_key' ordering of + # the unknowns. + eq1 = x*sin(45) - y*cos(q) + eq2 = x*cos(45) - y*sin(q) + eq3 = 9*x*sin(45)/10 + y*cos(q) + eq4 = 9*x*cos(45)/10 + y*sin(z) - z + zz = Symbol('zz') + eqs = [e.subs(q, zz) for e in (eq1, eq2, eq3, eq4)] + assert nonlinsolve(eqs, x, y, z, zz) == FiniteSet((0, 0, 0, zz)) + + +def test_issue_14565(): + # removed redundancy + assert dumeq(nonlinsolve([k + m, k + m*exp(-2*pi*k)], [k, m]) , + FiniteSet((-n*I, ImageSet(Lambda(n, n*I), S.Integers)))) + + +# end of tests for nonlinsolve + + +def test_issue_9556(): + b = Symbol('b', positive=True) + + assert solveset(Abs(x) + 1, x, S.Reals) is S.EmptySet + assert solveset(Abs(x) + b, x, S.Reals) is S.EmptySet + assert solveset(Eq(b, -1), b, S.Reals) is S.EmptySet + + +def test_issue_9611(): + assert solveset(Eq(x - x + a, a), x, S.Reals) == S.Reals + assert solveset(Eq(y - y + a, a), y) == S.Complexes + + +def test_issue_9557(): + assert solveset(x**2 + a, x, S.Reals) == Intersection(S.Reals, + FiniteSet(-sqrt(-a), sqrt(-a))) + + +def test_issue_9778(): + x = Symbol('x', real=True) + y = Symbol('y', real=True) + assert solveset(x**3 + 1, x, S.Reals) == FiniteSet(-1) + assert solveset(x**Rational(3, 5) + 1, x, S.Reals) == S.EmptySet + assert solveset(x**3 + y, x, S.Reals) == \ + FiniteSet(-Abs(y)**Rational(1, 3)*sign(y)) + + +def test_issue_10214(): + assert solveset(x**Rational(3, 2) + 4, x, S.Reals) == S.EmptySet + assert solveset(x**(Rational(-3, 2)) + 4, x, S.Reals) == S.EmptySet + + ans = FiniteSet(-2**Rational(2, 3)) + assert solveset(x**(S(3)) + 4, x, S.Reals) == ans + assert (x**(S(3)) + 4).subs(x,list(ans)[0]) == 0 # substituting ans and verifying the result. + assert (x**(S(3)) + 4).subs(x,-(-2)**Rational(2, 3)) == 0 + + +def test_issue_9849(): + assert solveset(Abs(sin(x)) + 1, x, S.Reals) == S.EmptySet + + +def test_issue_9953(): + assert linsolve([ ], x) == S.EmptySet + + +def test_issue_9913(): + assert solveset(2*x + 1/(x - 10)**2, x, S.Reals) == \ + FiniteSet(-(3*sqrt(24081)/4 + Rational(4027, 4))**Rational(1, 3)/3 - 100/ + (3*(3*sqrt(24081)/4 + Rational(4027, 4))**Rational(1, 3)) + Rational(20, 3)) + + +def test_issue_10397(): + assert solveset(sqrt(x), x, S.Complexes) == FiniteSet(0) + + +def test_issue_14987(): + raises(ValueError, lambda: linear_eq_to_matrix( + [x**2], x)) + raises(ValueError, lambda: linear_eq_to_matrix( + [x*(-3/x + 1) + 2*y - a], [x, y])) + raises(ValueError, lambda: linear_eq_to_matrix( + [(x**2 - 3*x)/(x - 3) - 3], x)) + raises(ValueError, lambda: linear_eq_to_matrix( + [(x + 1)**3 - x**3 - 3*x**2 + 7], x)) + raises(ValueError, lambda: linear_eq_to_matrix( + [x*(1/x + 1) + y], [x, y])) + raises(ValueError, lambda: linear_eq_to_matrix( + [(x + 1)*y], [x, y])) + raises(ValueError, lambda: linear_eq_to_matrix( + [Eq(1/x, 1/x + y)], [x, y])) + raises(ValueError, lambda: linear_eq_to_matrix( + [Eq(y/x, y/x + y)], [x, y])) + raises(ValueError, lambda: linear_eq_to_matrix( + [Eq(x*(x + 1), x**2 + y)], [x, y])) + + +def test_simplification(): + eq = x + (a - b)/(-2*a + 2*b) + assert solveset(eq, x) == FiniteSet(S.Half) + assert solveset(eq, x, S.Reals) == Intersection({-((a - b)/(-2*a + 2*b))}, S.Reals) + # So that ap - bn is not zero: + ap = Symbol('ap', positive=True) + bn = Symbol('bn', negative=True) + eq = x + (ap - bn)/(-2*ap + 2*bn) + assert solveset(eq, x) == FiniteSet(S.Half) + assert solveset(eq, x, S.Reals) == FiniteSet(S.Half) + + +def test_integer_domain_relational(): + eq1 = 2*x + 3 > 0 + eq2 = x**2 + 3*x - 2 >= 0 + eq3 = x + 1/x > -2 + 1/x + eq4 = x + sqrt(x**2 - 5) > 0 + eq = x + 1/x > -2 + 1/x + eq5 = eq.subs(x,log(x)) + eq6 = log(x)/x <= 0 + eq7 = log(x)/x < 0 + eq8 = x/(x-3) < 3 + eq9 = x/(x**2-3) < 3 + + assert solveset(eq1, x, S.Integers) == Range(-1, oo, 1) + assert solveset(eq2, x, S.Integers) == Union(Range(-oo, -3, 1), Range(1, oo, 1)) + assert solveset(eq3, x, S.Integers) == Union(Range(-1, 0, 1), Range(1, oo, 1)) + assert solveset(eq4, x, S.Integers) == Range(3, oo, 1) + assert solveset(eq5, x, S.Integers) == Range(2, oo, 1) + assert solveset(eq6, x, S.Integers) == Range(1, 2, 1) + assert solveset(eq7, x, S.Integers) == S.EmptySet + assert solveset(eq8, x, domain=Range(0,5)) == Range(0, 3, 1) + assert solveset(eq9, x, domain=Range(0,5)) == Union(Range(0, 2, 1), Range(2, 5, 1)) + + # test_issue_19794 + assert solveset(x + 2 < 0, x, S.Integers) == Range(-oo, -2, 1) + + +def test_issue_10555(): + f = Function('f') + g = Function('g') + assert solveset(f(x) - pi/2, x, S.Reals).dummy_eq( + ConditionSet(x, Eq(f(x) - pi/2, 0), S.Reals)) + assert solveset(f(g(x)) - pi/2, g(x), S.Reals).dummy_eq( + ConditionSet(g(x), Eq(f(g(x)) - pi/2, 0), S.Reals)) + + +def test_issue_8715(): + eq = x + 1/x > -2 + 1/x + assert solveset(eq, x, S.Reals) == \ + (Interval.open(-2, oo) - FiniteSet(0)) + assert solveset(eq.subs(x,log(x)), x, S.Reals) == \ + Interval.open(exp(-2), oo) - FiniteSet(1) + + +def test_issue_11174(): + eq = z**2 + exp(2*x) - sin(y) + soln = Intersection(S.Reals, FiniteSet(log(-z**2 + sin(y))/2)) + assert solveset(eq, x, S.Reals) == soln + + eq = sqrt(r)*Abs(tan(t))/sqrt(tan(t)**2 + 1) + x*tan(t) + s = -sqrt(r)*Abs(tan(t))/(sqrt(tan(t)**2 + 1)*tan(t)) + soln = Intersection(S.Reals, FiniteSet(s)) + assert solveset(eq, x, S.Reals) == soln + + +def test_issue_11534(): + # eq1 and eq2 should not have the same solutions because squaring both + # sides of the radical equation introduces a spurious solution branch. + # The equations have a symbolic parameter y and it is easy to see that for + # y != 0 the solution s1 will not be valid for eq1. + x = Symbol('x', real=True) + y = Symbol('y', real=True) + eq1 = -y + x/sqrt(-x**2 + 1) + eq2 = -y**2 + x**2/(-x**2 + 1) + + # We get a ConditionSet here because s1 works in eq1 if y is equal to zero + # although not for any other value of y. That case is redundant though + # because if y=0 then s1=s2 so the solution for eq1 could just be returned + # as s2 - {-1, 1}. In fact we have + # |y/sqrt(y**2 + 1)| < 1 + # So the complements are not needed either. The ideal output here would be + # sol1 = s2 + # sol2 = s1 | s2. + s1, s2 = FiniteSet(-y/sqrt(y**2 + 1)), FiniteSet(y/sqrt(y**2 + 1)) + cset = ConditionSet(x, Eq(eq1, 0), s1) + sol1 = (s2 - {-1, 1}) | (cset - {-1, 1}) + sol2 = (s1 | s2) - {-1, 1} + + assert solveset(eq1, x, S.Reals) == sol1 + assert solveset(eq2, x, S.Reals) == sol2 + + +def test_issue_10477(): + assert solveset((x**2 + 4*x - 3)/x < 2, x, S.Reals) == \ + Union(Interval.open(-oo, -3), Interval.open(0, 1)) + + +def test_issue_10671(): + assert solveset(sin(y), y, Interval(0, pi)) == FiniteSet(0, pi) + i = Interval(1, 10) + assert solveset((1/x).diff(x) < 0, x, i) == i + + +def test_issue_11064(): + eq = x + sqrt(x**2 - 5) + assert solveset(eq > 0, x, S.Reals) == \ + Interval(sqrt(5), oo) + assert solveset(eq < 0, x, S.Reals) == \ + Interval(-oo, -sqrt(5)) + assert solveset(eq > sqrt(5), x, S.Reals) == \ + Interval.Lopen(sqrt(5), oo) + + +def test_issue_12478(): + eq = sqrt(x - 2) + 2 + soln = solveset_real(eq, x) + assert soln is S.EmptySet + assert solveset(eq < 0, x, S.Reals) is S.EmptySet + assert solveset(eq > 0, x, S.Reals) == Interval(2, oo) + + +def test_issue_12429(): + eq = solveset(log(x)/x <= 0, x, S.Reals) + sol = Interval.Lopen(0, 1) + assert eq == sol + + +def test_issue_19506(): + eq = arg(x + I) + C = Dummy('C') + assert solveset(eq).dummy_eq(Intersection(ConditionSet(C, Eq(im(C) + 1, 0), S.Complexes), + ConditionSet(C, re(C) > 0, S.Complexes))) + + +def test_solveset_arg(): + assert solveset(arg(x), x, S.Reals) == Interval.open(0, oo) + assert solveset(arg(4*x -3), x, S.Reals) == Interval.open(Rational(3, 4), oo) + + +def test__is_finite_with_finite_vars(): + f = _is_finite_with_finite_vars + # issue 12482 + assert all(f(1/x) is None for x in ( + Dummy(), Dummy(real=True), Dummy(complex=True))) + assert f(1/Dummy(real=False)) is True # b/c it's finite but not 0 + + +def test_issue_13550(): + assert solveset(x**2 - 2*x - 15, symbol = x, domain = Interval(-oo, 0)) == FiniteSet(-3) + + +def test_issue_13849(): + assert nonlinsolve((t*(sqrt(5) + sqrt(2)) - sqrt(2), t), t) is S.EmptySet + + +def test_issue_14223(): + assert solveset((Abs(x + Min(x, 2)) - 2).rewrite(Piecewise), x, + S.Reals) == FiniteSet(-1, 1) + assert solveset((Abs(x + Min(x, 2)) - 2).rewrite(Piecewise), x, + Interval(0, 2)) == FiniteSet(1) + assert solveset(x, x, FiniteSet(1, 2)) is S.EmptySet + + +def test_issue_10158(): + dom = S.Reals + assert solveset(x*Max(x, 15) - 10, x, dom) == FiniteSet(Rational(2, 3)) + assert solveset(x*Min(x, 15) - 10, x, dom) == FiniteSet(-sqrt(10), sqrt(10)) + assert solveset(Max(Abs(x - 3) - 1, x + 2) - 3, x, dom) == FiniteSet(-1, 1) + assert solveset(Abs(x - 1) - Abs(y), x, dom) == FiniteSet(-Abs(y) + 1, Abs(y) + 1) + assert solveset(Abs(x + 4*Abs(x + 1)), x, dom) == FiniteSet(Rational(-4, 3), Rational(-4, 5)) + assert solveset(2*Abs(x + Abs(x + Max(3, x))) - 2, x, S.Reals) == FiniteSet(-1, -2) + dom = S.Complexes + raises(ValueError, lambda: solveset(x*Max(x, 15) - 10, x, dom)) + raises(ValueError, lambda: solveset(x*Min(x, 15) - 10, x, dom)) + raises(ValueError, lambda: solveset(Max(Abs(x - 3) - 1, x + 2) - 3, x, dom)) + raises(ValueError, lambda: solveset(Abs(x - 1) - Abs(y), x, dom)) + raises(ValueError, lambda: solveset(Abs(x + 4*Abs(x + 1)), x, dom)) + + +def test_issue_14300(): + f = 1 - exp(-18000000*x) - y + a1 = FiniteSet(-log(-y + 1)/18000000) + + assert solveset(f, x, S.Reals) == \ + Intersection(S.Reals, a1) + assert dumeq(solveset(f, x), + ImageSet(Lambda(n, -I*(2*n*pi + arg(-y + 1))/18000000 - + log(Abs(y - 1))/18000000), S.Integers)) + + +def test_issue_14454(): + number = CRootOf(x**4 + x - 1, 2) + raises(ValueError, lambda: invert_real(number, 0, x)) + assert invert_real(x**2, number, x) # no error + + +def test_issue_17882(): + assert solveset(-8*x**2/(9*(x**2 - 1)**(S(4)/3)) + 4/(3*(x**2 - 1)**(S(1)/3)), x, S.Complexes) == \ + FiniteSet(sqrt(3), -sqrt(3)) + + +def test_term_factors(): + assert list(_term_factors(3**x - 2)) == [-2, 3**x] + expr = 4**(x + 1) + 4**(x + 2) + 4**(x - 1) - 3**(x + 2) - 3**(x + 3) + assert set(_term_factors(expr)) == { + 3**(x + 2), 4**(x + 2), 3**(x + 3), 4**(x - 1), -1, 4**(x + 1)} + + +#################### tests for transolve and its helpers ############### + +def test_transolve(): + + assert _transolve(3**x, x, S.Reals) == S.EmptySet + assert _transolve(3**x - 9**(x + 5), x, S.Reals) == FiniteSet(-10) + + +def test_issue_21276(): + eq = (2*x*(y - z) - y*erf(y - z) - y + z*erf(y - z) + z)**2 + assert solveset(eq.expand(), y) == FiniteSet(z, z + erfinv(2*x - 1)) + + +# exponential tests +def test_exponential_real(): + from sympy.abc import y + + e1 = 3**(2*x) - 2**(x + 3) + e2 = 4**(5 - 9*x) - 8**(2 - x) + e3 = 2**x + 4**x + e4 = exp(log(5)*x) - 2**x + e5 = exp(x/y)*exp(-z/y) - 2 + e6 = 5**(x/2) - 2**(x/3) + e7 = 4**(x + 1) + 4**(x + 2) + 4**(x - 1) - 3**(x + 2) - 3**(x + 3) + e8 = -9*exp(-2*x + 5) + 4*exp(3*x + 1) + e9 = 2**x + 4**x + 8**x - 84 + e10 = 29*2**(x + 1)*615**(x) - 123*2726**(x) + + assert solveset(e1, x, S.Reals) == FiniteSet( + -3*log(2)/(-2*log(3) + log(2))) + assert solveset(e2, x, S.Reals) == FiniteSet(Rational(4, 15)) + assert solveset(e3, x, S.Reals) == S.EmptySet + assert solveset(e4, x, S.Reals) == FiniteSet(0) + assert solveset(e5, x, S.Reals) == Intersection( + S.Reals, FiniteSet(y*log(2*exp(z/y)))) + assert solveset(e6, x, S.Reals) == FiniteSet(0) + assert solveset(e7, x, S.Reals) == FiniteSet(2) + assert solveset(e8, x, S.Reals) == FiniteSet(-2*log(2)/5 + 2*log(3)/5 + Rational(4, 5)) + assert solveset(e9, x, S.Reals) == FiniteSet(2) + assert solveset(e10,x, S.Reals) == FiniteSet((-log(29) - log(2) + log(123))/(-log(2726) + log(2) + log(615))) + + assert solveset_real(-9*exp(-2*x + 5) + 2**(x + 1), x) == FiniteSet( + -((-5 - 2*log(3) + log(2))/(log(2) + 2))) + assert solveset_real(4**(x/2) - 2**(x/3), x) == FiniteSet(0) + b = sqrt(6)*sqrt(log(2))/sqrt(log(5)) + assert solveset_real(5**(x/2) - 2**(3/x), x) == FiniteSet(-b, b) + + # coverage test + C1, C2 = symbols('C1 C2') + f = Function('f') + assert solveset_real(C1 + C2/x**2 - exp(-f(x)), f(x)) == Intersection( + S.Reals, FiniteSet(-log(C1 + C2/x**2))) + y = symbols('y', positive=True) + assert solveset_real(x**2 - y**2/exp(x), y) == Intersection( + S.Reals, FiniteSet(-sqrt(x**2*exp(x)), sqrt(x**2*exp(x)))) + p = Symbol('p', positive=True) + assert solveset_real((1/p + 1)**(p + 1), p).dummy_eq( + ConditionSet(x, Eq((1 + 1/x)**(x + 1), 0), S.Reals)) + assert solveset(2**x - 4**x + 12, x, S.Reals) == {2} + assert solveset(2**x - 2**(2*x) + 12, x, S.Reals) == {2} + + +@XFAIL +def test_exponential_complex(): + n = Dummy('n') + + assert dumeq(solveset_complex(2**x + 4**x, x),imageset( + Lambda(n, I*(2*n*pi + pi)/log(2)), S.Integers)) + assert solveset_complex(x**z*y**z - 2, z) == FiniteSet( + log(2)/(log(x) + log(y))) + assert dumeq(solveset_complex(4**(x/2) - 2**(x/3), x), imageset( + Lambda(n, 3*n*I*pi/log(2)), S.Integers)) + assert dumeq(solveset(2**x + 32, x), imageset( + Lambda(n, (I*(2*n*pi + pi) + 5*log(2))/log(2)), S.Integers)) + + eq = (2**exp(y**2/x) + 2)/(x**2 + 15) + a = sqrt(x)*sqrt(-log(log(2)) + log(log(2) + 2*n*I*pi)) + assert solveset_complex(eq, y) == FiniteSet(-a, a) + + union1 = imageset(Lambda(n, I*(2*n*pi - pi*Rational(2, 3))/log(2)), S.Integers) + union2 = imageset(Lambda(n, I*(2*n*pi + pi*Rational(2, 3))/log(2)), S.Integers) + assert dumeq(solveset(2**x + 4**x + 8**x, x), Union(union1, union2)) + + eq = 4**(x + 1) + 4**(x + 2) + 4**(x - 1) - 3**(x + 2) - 3**(x + 3) + res = solveset(eq, x) + num = 2*n*I*pi - 4*log(2) + 2*log(3) + den = -2*log(2) + log(3) + ans = imageset(Lambda(n, num/den), S.Integers) + assert dumeq(res, ans) + + +def test_expo_conditionset(): + + f1 = (exp(x) + 1)**x - 2 + f2 = (x + 2)**y*x - 3 + f3 = 2**x - exp(x) - 3 + f4 = log(x) - exp(x) + f5 = 2**x + 3**x - 5**x + + assert solveset(f1, x, S.Reals).dummy_eq(ConditionSet( + x, Eq((exp(x) + 1)**x - 2, 0), S.Reals)) + assert solveset(f2, x, S.Reals).dummy_eq(ConditionSet( + x, Eq(x*(x + 2)**y - 3, 0), S.Reals)) + assert solveset(f3, x, S.Reals).dummy_eq(ConditionSet( + x, Eq(2**x - exp(x) - 3, 0), S.Reals)) + assert solveset(f4, x, S.Reals).dummy_eq(ConditionSet( + x, Eq(-exp(x) + log(x), 0), S.Reals)) + assert solveset(f5, x, S.Reals).dummy_eq(ConditionSet( + x, Eq(2**x + 3**x - 5**x, 0), S.Reals)) + + +def test_exponential_symbols(): + x, y, z = symbols('x y z', positive=True) + xr, zr = symbols('xr, zr', real=True) + + assert solveset(z**x - y, x, S.Reals) == Intersection( + S.Reals, FiniteSet(log(y)/log(z))) + + f1 = 2*x**w - 4*y**w + f2 = (x/y)**w - 2 + sol1 = Intersection({log(2)/(log(x) - log(y))}, S.Reals) + sol2 = Intersection({log(2)/log(x/y)}, S.Reals) + assert solveset(f1, w, S.Reals) == sol1, solveset(f1, w, S.Reals) + assert solveset(f2, w, S.Reals) == sol2, solveset(f2, w, S.Reals) + + assert solveset(x**x, x, Interval.Lopen(0,oo)).dummy_eq( + ConditionSet(w, Eq(w**w, 0), Interval.open(0, oo))) + assert solveset(x**y - 1, y, S.Reals) == FiniteSet(0) + assert solveset(exp(x/y)*exp(-z/y) - 2, y, S.Reals) == \ + Complement(ConditionSet(y, Eq(im(x)/y, 0) & Eq(im(z)/y, 0), \ + Complement(Intersection(FiniteSet((x - z)/log(2)), S.Reals), FiniteSet(0))), FiniteSet(0)) + assert solveset(exp(xr/y)*exp(-zr/y) - 2, y, S.Reals) == \ + Complement(FiniteSet((xr - zr)/log(2)), FiniteSet(0)) + + assert solveset(a**x - b**x, x).dummy_eq(ConditionSet( + w, Ne(a, 0) & Ne(b, 0), FiniteSet(0))) + + +def test_ignore_assumptions(): + # make sure assumptions are ignored + xpos = symbols('x', positive=True) + x = symbols('x') + assert solveset_complex(xpos**2 - 4, xpos + ) == solveset_complex(x**2 - 4, x) + + +@XFAIL +def test_issue_10864(): + assert solveset(x**(y*z) - x, x, S.Reals) == FiniteSet(1) + + +@XFAIL +def test_solve_only_exp_2(): + assert solveset_real(sqrt(exp(x)) + sqrt(exp(-x)) - 4, x) == \ + FiniteSet(2*log(-sqrt(3) + 2), 2*log(sqrt(3) + 2)) + + +def test_is_exponential(): + assert _is_exponential(y, x) is False + assert _is_exponential(3**x - 2, x) is True + assert _is_exponential(5**x - 7**(2 - x), x) is True + assert _is_exponential(sin(2**x) - 4*x, x) is False + assert _is_exponential(x**y - z, y) is True + assert _is_exponential(x**y - z, x) is False + assert _is_exponential(2**x + 4**x - 1, x) is True + assert _is_exponential(x**(y*z) - x, x) is False + assert _is_exponential(x**(2*x) - 3**x, x) is False + assert _is_exponential(x**y - y*z, y) is False + assert _is_exponential(x**y - x*z, y) is True + + +def test_solve_exponential(): + assert _solve_exponential(3**(2*x) - 2**(x + 3), 0, x, S.Reals) == \ + FiniteSet(-3*log(2)/(-2*log(3) + log(2))) + assert _solve_exponential(2**y + 4**y, 1, y, S.Reals) == \ + FiniteSet(log(Rational(-1, 2) + sqrt(5)/2)/log(2)) + assert _solve_exponential(2**y + 4**y, 0, y, S.Reals) == \ + S.EmptySet + assert _solve_exponential(2**x + 3**x - 5**x, 0, x, S.Reals) == \ + ConditionSet(x, Eq(2**x + 3**x - 5**x, 0), S.Reals) + +# end of exponential tests + + +# logarithmic tests +def test_logarithmic(): + assert solveset_real(log(x - 3) + log(x + 3), x) == FiniteSet( + -sqrt(10), sqrt(10)) + assert solveset_real(log(x + 1) - log(2*x - 1), x) == FiniteSet(2) + assert solveset_real(log(x + 3) + log(1 + 3/x) - 3, x) == FiniteSet( + -3 + sqrt(-12 + exp(3))*exp(Rational(3, 2))/2 + exp(3)/2, + -sqrt(-12 + exp(3))*exp(Rational(3, 2))/2 - 3 + exp(3)/2) + + eq = z - log(x) + log(y/(x*(-1 + y**2/x**2))) + assert solveset_real(eq, x) == \ + Intersection(S.Reals, FiniteSet(-sqrt(y**2 - y*exp(z)), + sqrt(y**2 - y*exp(z)))) - \ + Intersection(S.Reals, FiniteSet(-sqrt(y**2), sqrt(y**2))) + assert solveset_real( + log(3*x) - log(-x + 1) - log(4*x + 1), x) == FiniteSet(Rational(-1, 2), S.Half) + assert solveset(log(x**y) - y*log(x), x, S.Reals) == S.Reals + +@XFAIL +def test_uselogcombine_2(): + eq = log(exp(2*x) + 1) + log(-tanh(x) + 1) - log(2) + assert solveset_real(eq, x) is S.EmptySet + eq = log(8*x) - log(sqrt(x) + 1) - 2 + assert solveset_real(eq, x) is S.EmptySet + + +def test_is_logarithmic(): + assert _is_logarithmic(y, x) is False + assert _is_logarithmic(log(x), x) is True + assert _is_logarithmic(log(x) - 3, x) is True + assert _is_logarithmic(log(x)*log(y), x) is True + assert _is_logarithmic(log(x)**2, x) is False + assert _is_logarithmic(log(x - 3) + log(x + 3), x) is True + assert _is_logarithmic(log(x**y) - y*log(x), x) is True + assert _is_logarithmic(sin(log(x)), x) is False + assert _is_logarithmic(x + y, x) is False + assert _is_logarithmic(log(3*x) - log(1 - x) + 4, x) is True + assert _is_logarithmic(log(x) + log(y) + x, x) is False + assert _is_logarithmic(log(log(x - 3)) + log(x - 3), x) is True + assert _is_logarithmic(log(log(3) + x) + log(x), x) is True + assert _is_logarithmic(log(x)*(y + 3) + log(x), y) is False + + +def test_solve_logarithm(): + y = Symbol('y') + assert _solve_logarithm(log(x**y) - y*log(x), 0, x, S.Reals) == S.Reals + y = Symbol('y', positive=True) + assert _solve_logarithm(log(x)*log(y), 0, x, S.Reals) == FiniteSet(1) + +# end of logarithmic tests + + +# lambert tests +def test_is_lambert(): + a, b, c = symbols('a,b,c') + assert _is_lambert(x**2, x) is False + assert _is_lambert(a**x**2+b*x+c, x) is True + assert _is_lambert(E**2, x) is False + assert _is_lambert(x*E**2, x) is False + assert _is_lambert(3*log(x) - x*log(3), x) is True + assert _is_lambert(log(log(x - 3)) + log(x-3), x) is True + assert _is_lambert(5*x - 1 + 3*exp(2 - 7*x), x) is True + assert _is_lambert((a/x + exp(x/2)).diff(x, 2), x) is True + assert _is_lambert((x**2 - 2*x + 1).subs(x, (log(x) + 3*x)**2 - 1), x) is True + assert _is_lambert(x*sinh(x) - 1, x) is True + assert _is_lambert(x*cos(x) - 5, x) is True + assert _is_lambert(tanh(x) - 5*x, x) is True + assert _is_lambert(cosh(x) - sinh(x), x) is False + +# end of lambert tests + + +def test_linear_coeffs(): + from sympy.solvers.solveset import linear_coeffs + assert linear_coeffs(0, x) == [0, 0] + assert all(i is S.Zero for i in linear_coeffs(0, x)) + assert linear_coeffs(x + 2*y + 3, x, y) == [1, 2, 3] + assert linear_coeffs(x + 2*y + 3, y, x) == [2, 1, 3] + assert linear_coeffs(x + 2*x**2 + 3, x, x**2) == [1, 2, 3] + raises(ValueError, lambda: + linear_coeffs(x + 2*x**2 + x**3, x, x**2)) + raises(ValueError, lambda: + linear_coeffs(1/x*(x - 1) + 1/x, x)) + raises(ValueError, lambda: + linear_coeffs(x, x, x)) + assert linear_coeffs(a*(x + y), x, y) == [a, a, 0] + assert linear_coeffs(1.0, x, y) == [0, 0, 1.0] + # don't include coefficients of 0 + assert linear_coeffs(Eq(x, x + y), x, y, dict=True) == {y: -1} + assert linear_coeffs(0, x, y, dict=True) == {} + + +def test_is_modular(): + assert _is_modular(y, x) is False + assert _is_modular(Mod(x, 3) - 1, x) is True + assert _is_modular(Mod(x**3 - 3*x**2 - x + 1, 3) - 1, x) is True + assert _is_modular(Mod(exp(x + y), 3) - 2, x) is True + assert _is_modular(Mod(exp(x + y), 3) - log(x), x) is True + assert _is_modular(Mod(x, 3) - 1, y) is False + assert _is_modular(Mod(x, 3)**2 - 5, x) is False + assert _is_modular(Mod(x, 3)**2 - y, x) is False + assert _is_modular(exp(Mod(x, 3)) - 1, x) is False + assert _is_modular(Mod(3, y) - 1, y) is False + + +def test_invert_modular(): + n = Dummy('n', integer=True) + from sympy.solvers.solveset import _invert_modular as invert_modular + + # no solutions + assert invert_modular(Mod(x, 12), S(1)/2, n, x) == (x, S.EmptySet) + # non invertible cases + assert invert_modular(Mod(sin(x), 7), S(5), n, x) == (Mod(sin(x), 7), 5) + assert invert_modular(Mod(exp(x), 7), S(5), n, x) == (Mod(exp(x), 7), 5) + assert invert_modular(Mod(log(x), 7), S(5), n, x) == (Mod(log(x), 7), 5) + # a is symbol + assert dumeq(invert_modular(Mod(x, 7), S(5), n, x), + (x, ImageSet(Lambda(n, 7*n + 5), S.Integers))) + # a.is_Add + assert dumeq(invert_modular(Mod(x + 8, 7), S(5), n, x), + (x, ImageSet(Lambda(n, 7*n + 4), S.Integers))) + assert invert_modular(Mod(x**2 + x, 7), S(5), n, x) == \ + (Mod(x**2 + x, 7), 5) + # a.is_Mul + assert dumeq(invert_modular(Mod(3*x, 7), S(5), n, x), + (x, ImageSet(Lambda(n, 7*n + 4), S.Integers))) + assert invert_modular(Mod((x + 1)*(x + 2), 7), S(5), n, x) == \ + (Mod((x + 1)*(x + 2), 7), 5) + # a.is_Pow + assert invert_modular(Mod(x**4, 7), S(5), n, x) == \ + (x, S.EmptySet) + assert dumeq(invert_modular(Mod(3**x, 4), S(3), n, x), + (x, ImageSet(Lambda(n, 2*n + 1), S.Naturals0))) + assert dumeq(invert_modular(Mod(2**(x**2 + x + 1), 7), S(2), n, x), + (x**2 + x + 1, ImageSet(Lambda(n, 3*n + 1), S.Naturals0))) + assert invert_modular(Mod(sin(x)**4, 7), S(5), n, x) == (x, S.EmptySet) + + +def test_solve_modular(): + n = Dummy('n', integer=True) + # if rhs has symbol (need to be implemented in future). + assert solveset(Mod(x, 4) - x, x, S.Integers + ).dummy_eq( + ConditionSet(x, Eq(-x + Mod(x, 4), 0), + S.Integers)) + # when _invert_modular fails to invert + assert solveset(3 - Mod(sin(x), 7), x, S.Integers + ).dummy_eq( + ConditionSet(x, Eq(Mod(sin(x), 7) - 3, 0), S.Integers)) + assert solveset(3 - Mod(log(x), 7), x, S.Integers + ).dummy_eq( + ConditionSet(x, Eq(Mod(log(x), 7) - 3, 0), S.Integers)) + assert solveset(3 - Mod(exp(x), 7), x, S.Integers + ).dummy_eq(ConditionSet(x, Eq(Mod(exp(x), 7) - 3, 0), + S.Integers)) + # EmptySet solution definitely + assert solveset(7 - Mod(x, 5), x, S.Integers) is S.EmptySet + assert solveset(5 - Mod(x, 5), x, S.Integers) is S.EmptySet + # Negative m + assert dumeq(solveset(2 + Mod(x, -3), x, S.Integers), + ImageSet(Lambda(n, -3*n - 2), S.Integers)) + assert solveset(4 + Mod(x, -3), x, S.Integers) is S.EmptySet + # linear expression in Mod + assert dumeq(solveset(3 - Mod(x, 5), x, S.Integers), + ImageSet(Lambda(n, 5*n + 3), S.Integers)) + assert dumeq(solveset(3 - Mod(5*x - 8, 7), x, S.Integers), + ImageSet(Lambda(n, 7*n + 5), S.Integers)) + assert dumeq(solveset(3 - Mod(5*x, 7), x, S.Integers), + ImageSet(Lambda(n, 7*n + 2), S.Integers)) + # higher degree expression in Mod + assert dumeq(solveset(Mod(x**2, 160) - 9, x, S.Integers), + Union(ImageSet(Lambda(n, 160*n + 3), S.Integers), + ImageSet(Lambda(n, 160*n + 13), S.Integers), + ImageSet(Lambda(n, 160*n + 67), S.Integers), + ImageSet(Lambda(n, 160*n + 77), S.Integers), + ImageSet(Lambda(n, 160*n + 83), S.Integers), + ImageSet(Lambda(n, 160*n + 93), S.Integers), + ImageSet(Lambda(n, 160*n + 147), S.Integers), + ImageSet(Lambda(n, 160*n + 157), S.Integers))) + assert solveset(3 - Mod(x**4, 7), x, S.Integers) is S.EmptySet + assert dumeq(solveset(Mod(x**4, 17) - 13, x, S.Integers), + Union(ImageSet(Lambda(n, 17*n + 3), S.Integers), + ImageSet(Lambda(n, 17*n + 5), S.Integers), + ImageSet(Lambda(n, 17*n + 12), S.Integers), + ImageSet(Lambda(n, 17*n + 14), S.Integers))) + # a.is_Pow tests + assert dumeq(solveset(Mod(7**x, 41) - 15, x, S.Integers), + ImageSet(Lambda(n, 40*n + 3), S.Naturals0)) + assert dumeq(solveset(Mod(12**x, 21) - 18, x, S.Integers), + ImageSet(Lambda(n, 6*n + 2), S.Naturals0)) + assert dumeq(solveset(Mod(3**x, 4) - 3, x, S.Integers), + ImageSet(Lambda(n, 2*n + 1), S.Naturals0)) + assert dumeq(solveset(Mod(2**x, 7) - 2 , x, S.Integers), + ImageSet(Lambda(n, 3*n + 1), S.Naturals0)) + assert dumeq(solveset(Mod(3**(3**x), 4) - 3, x, S.Integers), + Intersection(ImageSet(Lambda(n, Intersection({log(2*n + 1)/log(3)}, + S.Integers)), S.Naturals0), S.Integers)) + # Implemented for m without primitive root + assert solveset(Mod(x**3, 7) - 2, x, S.Integers) is S.EmptySet + assert dumeq(solveset(Mod(x**3, 8) - 1, x, S.Integers), + ImageSet(Lambda(n, 8*n + 1), S.Integers)) + assert dumeq(solveset(Mod(x**4, 9) - 4, x, S.Integers), + Union(ImageSet(Lambda(n, 9*n + 4), S.Integers), + ImageSet(Lambda(n, 9*n + 5), S.Integers))) + # domain intersection + assert dumeq(solveset(3 - Mod(5*x - 8, 7), x, S.Naturals0), + Intersection(ImageSet(Lambda(n, 7*n + 5), S.Integers), S.Naturals0)) + # Complex args + assert solveset(Mod(x, 3) - I, x, S.Integers) == \ + S.EmptySet + assert solveset(Mod(I*x, 3) - 2, x, S.Integers + ).dummy_eq( + ConditionSet(x, Eq(Mod(I*x, 3) - 2, 0), S.Integers)) + assert solveset(Mod(I + x, 3) - 2, x, S.Integers + ).dummy_eq( + ConditionSet(x, Eq(Mod(x + I, 3) - 2, 0), S.Integers)) + + # issue 17373 (https://github.com/sympy/sympy/issues/17373) + assert dumeq(solveset(Mod(x**4, 14) - 11, x, S.Integers), + Union(ImageSet(Lambda(n, 14*n + 3), S.Integers), + ImageSet(Lambda(n, 14*n + 11), S.Integers))) + assert dumeq(solveset(Mod(x**31, 74) - 43, x, S.Integers), + ImageSet(Lambda(n, 74*n + 31), S.Integers)) + + # issue 13178 + n = symbols('n', integer=True) + a = 742938285 + b = 1898888478 + m = 2**31 - 1 + c = 20170816 + assert dumeq(solveset(c - Mod(a**n*b, m), n, S.Integers), + ImageSet(Lambda(n, 2147483646*n + 100), S.Naturals0)) + assert dumeq(solveset(c - Mod(a**n*b, m), n, S.Naturals0), + Intersection(ImageSet(Lambda(n, 2147483646*n + 100), S.Naturals0), + S.Naturals0)) + assert dumeq(solveset(c - Mod(a**(2*n)*b, m), n, S.Integers), + Intersection(ImageSet(Lambda(n, 1073741823*n + 50), S.Naturals0), + S.Integers)) + assert solveset(c - Mod(a**(2*n + 7)*b, m), n, S.Integers) is S.EmptySet + assert dumeq(solveset(c - Mod(a**(n - 4)*b, m), n, S.Integers), + Intersection(ImageSet(Lambda(n, 2147483646*n + 104), S.Naturals0), + S.Integers)) + +# end of modular tests + +def test_issue_17276(): + assert nonlinsolve([Eq(x, 5**(S(1)/5)), Eq(x*y, 25*sqrt(5))], x, y) == \ + FiniteSet((5**(S(1)/5), 25*5**(S(3)/10))) + + +def test_issue_10426(): + x = Dummy('x') + a = Symbol('a') + n = Dummy('n') + assert (solveset(sin(x + a) - sin(x), a)).dummy_eq(Dummy('x')) == (Union( + ImageSet(Lambda(n, 2*n*pi), S.Integers), + Intersection(S.Complexes, ImageSet(Lambda(n, -I*(I*(2*n*pi + arg(-exp(-2*I*x))) + 2*im(x))), + S.Integers)))).dummy_eq(Dummy('x,n')) + + +def test_solveset_conjugate(): + """Test solveset for simple conjugate functions""" + assert solveset(conjugate(x) -3 + I) == FiniteSet(3 + I) + + +def test_issue_18208(): + variables = symbols('x0:16') + symbols('y0:12') + x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15,\ + y0, y1, y2, y3, y4, y5, y6, y7, y8, y9, y10, y11 = variables + + eqs = [x0 + x1 + x2 + x3 - 51, + x0 + x1 + x4 + x5 - 46, + x2 + x3 + x6 + x7 - 39, + x0 + x3 + x4 + x7 - 50, + x1 + x2 + x5 + x6 - 35, + x4 + x5 + x6 + x7 - 34, + x4 + x5 + x8 + x9 - 46, + x10 + x11 + x6 + x7 - 23, + x11 + x4 + x7 + x8 - 25, + x10 + x5 + x6 + x9 - 44, + x10 + x11 + x8 + x9 - 35, + x12 + x13 + x8 + x9 - 35, + x10 + x11 + x14 + x15 - 29, + x11 + x12 + x15 + x8 - 35, + x10 + x13 + x14 + x9 - 29, + x12 + x13 + x14 + x15 - 29, + y0 + y1 + y2 + y3 - 55, + y0 + y1 + y4 + y5 - 53, + y2 + y3 + y6 + y7 - 56, + y0 + y3 + y4 + y7 - 57, + y1 + y2 + y5 + y6 - 52, + y4 + y5 + y6 + y7 - 54, + y4 + y5 + y8 + y9 - 48, + y10 + y11 + y6 + y7 - 60, + y11 + y4 + y7 + y8 - 51, + y10 + y5 + y6 + y9 - 57, + y10 + y11 + y8 + y9 - 54, + x10 - 2, + x11 - 5, + x12 - 1, + x13 - 6, + x14 - 1, + x15 - 21, + y0 - 12, + y1 - 20] + + expected = [38 - x3, x3 - 10, 23 - x3, x3, 12 - x7, x7 + 6, 16 - x7, x7, + 8, 20, 2, 5, 1, 6, 1, 21, 12, 20, -y11 + y9 + 2, y11 - y9 + 21, + -y11 - y7 + y9 + 24, y11 + y7 - y9 - 3, 33 - y7, y7, 27 - y9, y9, + 27 - y11, y11] + + A, b = linear_eq_to_matrix(eqs, variables) + + # solve + solve_expected = {v:eq for v, eq in zip(variables, expected) if v != eq} + + assert solve(eqs, variables) == solve_expected + + # linsolve + linsolve_expected = FiniteSet(Tuple(*expected)) + + assert linsolve(eqs, variables) == linsolve_expected + assert linsolve((A, b), variables) == linsolve_expected + + # gauss_jordan_solve + gj_solve, new_vars = A.gauss_jordan_solve(b) + gj_solve = list(gj_solve) + + gj_expected = linsolve_expected.subs(zip([x3, x7, y7, y9, y11], new_vars)) + + assert FiniteSet(Tuple(*gj_solve)) == gj_expected + + # nonlinsolve + # The solution set of nonlinsolve is currently equivalent to linsolve and is + # also correct. However, we would prefer to use the same symbols as parameters + # for the solution to the underdetermined system in all cases if possible. + # We want a solution that is not just equivalent but also given in the same form. + # This test may be changed should nonlinsolve be modified in this way. + + nonlinsolve_expected = FiniteSet((38 - x3, x3 - 10, 23 - x3, x3, 12 - x7, x7 + 6, + 16 - x7, x7, 8, 20, 2, 5, 1, 6, 1, 21, 12, 20, + -y5 + y7 - 1, y5 - y7 + 24, 21 - y5, y5, 33 - y7, + y7, 27 - y9, y9, -y5 + y7 - y9 + 24, y5 - y7 + y9 + 3)) + + assert nonlinsolve(eqs, variables) == nonlinsolve_expected + + +def test_substitution_with_infeasible_solution(): + a00, a01, a10, a11, l0, l1, l2, l3, m0, m1, m2, m3, m4, m5, m6, m7, c00, c01, c10, c11, p00, p01, p10, p11 = symbols( + 'a00, a01, a10, a11, l0, l1, l2, l3, m0, m1, m2, m3, m4, m5, m6, m7, c00, c01, c10, c11, p00, p01, p10, p11' + ) + solvefor = [p00, p01, p10, p11, c00, c01, c10, c11, m0, m1, m3, l0, l1, l2, l3] + system = [ + -l0 * c00 - l1 * c01 + m0 + c00 + c01, + -l0 * c10 - l1 * c11 + m1, + -l2 * c00 - l3 * c01 + c00 + c01, + -l2 * c10 - l3 * c11 + m3, + -l0 * p00 - l2 * p10 + p00 + p10, + -l1 * p00 - l3 * p10 + p00 + p10, + -l0 * p01 - l2 * p11, + -l1 * p01 - l3 * p11, + -a00 + c00 * p00 + c10 * p01, + -a01 + c01 * p00 + c11 * p01, + -a10 + c00 * p10 + c10 * p11, + -a11 + c01 * p10 + c11 * p11, + -m0 * p00, + -m1 * p01, + -m2 * p10, + -m3 * p11, + -m4 * c00, + -m5 * c01, + -m6 * c10, + -m7 * c11, + m2, + m4, + m5, + m6, + m7 + ] + sol = FiniteSet( + (0, Complement(FiniteSet(p01), FiniteSet(0)), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, l2, l3), + (p00, Complement(FiniteSet(p01), FiniteSet(0)), 0, p11, 0, 0, 0, 0, 0, 0, 0, 1, 1, -p01/p11, -p01/p11), + (0, Complement(FiniteSet(p01), FiniteSet(0)), 0, p11, 0, 0, 0, 0, 0, 0, 0, 1, -l3*p11/p01, -p01/p11, l3), + (0, Complement(FiniteSet(p01), FiniteSet(0)), 0, p11, 0, 0, 0, 0, 0, 0, 0, -l2*p11/p01, -l3*p11/p01, l2, l3), + ) + assert sol != nonlinsolve(system, solvefor) + + +def test_issue_20097(): + assert solveset(1/sqrt(x)) is S.EmptySet + + +def test_issue_15350(): + assert solveset(diff(sqrt(1/x+x))) == FiniteSet(-1, 1) + + +def test_issue_18359(): + c1 = Piecewise((0, x < 0), (Min(1, x)/2 - Min(2, x)/2 + Min(3, x)/2, True)) + c2 = Piecewise((Piecewise((0, x < 0), (Min(1, x)/2 - Min(2, x)/2 + Min(3, x)/2, True)), x >= 0), (0, True)) + correct_result = Interval(1, 2) + result1 = solveset(c1 - Rational(1, 2), x, Interval(0, 3)) + result2 = solveset(c2 - Rational(1, 2), x, Interval(0, 3)) + assert result1 == correct_result + assert result2 == correct_result + + +def test_issue_17604(): + lhs = -2**(3*x/11)*exp(x/11) + pi**(x/11) + assert _is_exponential(lhs, x) + assert _solve_exponential(lhs, 0, x, S.Complexes) == FiniteSet(0) + + +def test_issue_17580(): + assert solveset(1/(1 - x**3)**2, x, S.Reals) is S.EmptySet + + +def test_issue_17566_actual(): + sys = [2**x + 2**y - 3, 4**x + 9**y - 5] + # Not clear this is the correct result, but at least no recursion error + assert nonlinsolve(sys, x, y) == FiniteSet((log(3 - 2**y)/log(2), y)) + + +def test_issue_17565(): + eq = Ge(2*(x - 2)**2/(3*(x + 1)**(Integer(1)/3)) + 2*(x - 2)*(x + 1)**(Integer(2)/3), 0) + res = Union(Interval.Lopen(-1, -Rational(1, 4)), Interval(2, oo)) + assert solveset(eq, x, S.Reals) == res + + +def test_issue_15024(): + function = (x + 5)/sqrt(-x**2 - 10*x) + assert solveset(function, x, S.Reals) == FiniteSet(Integer(-5)) + + +def test_issue_16877(): + assert dumeq(nonlinsolve([x - 1, sin(y)], x, y), + FiniteSet((1, ImageSet(Lambda(n, 2*n*pi), S.Integers)), + (1, ImageSet(Lambda(n, 2*n*pi + pi), S.Integers)))) + # Even better if (1, ImageSet(Lambda(n, n*pi), S.Integers)) is obtained + + +def test_issue_16876(): + assert dumeq(nonlinsolve([sin(x), 2*x - 4*y], x, y), + FiniteSet((ImageSet(Lambda(n, 2*n*pi), S.Integers), + ImageSet(Lambda(n, n*pi), S.Integers)), + (ImageSet(Lambda(n, 2*n*pi + pi), S.Integers), + ImageSet(Lambda(n, n*pi + pi/2), S.Integers)))) + # Even better if (ImageSet(Lambda(n, n*pi), S.Integers), + # ImageSet(Lambda(n, n*pi/2), S.Integers)) is obtained + +def test_issue_21236(): + x, z = symbols("x z") + y = symbols('y', rational=True) + assert solveset(x**y - z, x, S.Reals) == ConditionSet(x, Eq(x**y - z, 0), S.Reals) + e1, e2 = symbols('e1 e2', even=True) + y = e1/e2 # don't know if num or den will be odd and the other even + assert solveset(x**y - z, x, S.Reals) == ConditionSet(x, Eq(x**y - z, 0), S.Reals) + + +def test_issue_21908(): + assert nonlinsolve([(x**2 + 2*x - y**2)*exp(x), -2*y*exp(x)], x, y + ) == {(-2, 0), (0, 0)} + + +def test_issue_19144(): + # test case 1 + expr1 = [x + y - 1, y**2 + 1] + eq1 = [Eq(i, 0) for i in expr1] + soln1 = {(1 - I, I), (1 + I, -I)} + soln_expr1 = nonlinsolve(expr1, [x, y]) + soln_eq1 = nonlinsolve(eq1, [x, y]) + assert soln_eq1 == soln_expr1 == soln1 + # test case 2 - with denoms + expr2 = [x/y - 1, y**2 + 1] + eq2 = [Eq(i, 0) for i in expr2] + soln2 = {(-I, -I), (I, I)} + soln_expr2 = nonlinsolve(expr2, [x, y]) + soln_eq2 = nonlinsolve(eq2, [x, y]) + assert soln_eq2 == soln_expr2 == soln2 + # denominators that cancel in expression + assert nonlinsolve([Eq(x + 1/x, 1/x)], [x]) == FiniteSet((S.EmptySet,)) + + +def test_issue_22413(): + res = nonlinsolve((4*y*(2*x + 2*exp(y) + 1)*exp(2*x), + 4*x*exp(2*x) + 4*y*exp(2*x + y) + 4*exp(2*x + y) + 1), + x, y) + # First solution is not correct, but the issue was an exception + sols = FiniteSet((x, S.Zero), (-exp(y) - S.Half, y)) + assert res == sols + + +def test_issue_23318(): + eqs_eq = [ + Eq(53.5780461486929, x * log(y / (5.0 - y) + 1) / y), + Eq(x, 0.0015 * z), + Eq(0.0015, 7845.32 * y / z), + ] + eqs_expr = [eq.lhs - eq.rhs for eq in eqs_eq] + + sol = {(266.97755814852, 0.0340301680681629, 177985.03876568)} + + assert_close_nl(nonlinsolve(eqs_eq, [x, y, z]), sol) + assert_close_nl(nonlinsolve(eqs_expr, [x, y, z]), sol) + + logterm = log(1.91196789933362e-7*z/(5.0 - 1.91196789933362e-7*z) + 1) + eq = -0.0015*z*logterm + 1.02439504345316e-5*z + assert_close_ss(solveset(eq, z), {0, 177985.038765679}) + + +def test_issue_19814(): + assert nonlinsolve([ 2**m - 2**(2*n), 4*2**m - 2**(4*n)], m, n + ) == FiniteSet((log(2**(2*n))/log(2), S.Complexes)) + + +def test_issue_22058(): + sol = solveset(-sqrt(t)*x**2 + 2*x + sqrt(t), x, S.Reals) + # doesn't fail (and following numerical check) + assert sol.xreplace({t: 1}) == {1 - sqrt(2), 1 + sqrt(2)}, sol.xreplace({t: 1}) + + +def test_issue_11184(): + assert solveset(20*sqrt(y**2 + (sqrt(-(y - 10)*(y + 10)) + 10)**2) - 60, y, S.Reals) is S.EmptySet + + +def test_issue_21890(): + e = S(2)/3 + assert nonlinsolve([4*x**3*y**4 - 2*y, 4*x**4*y**3 - 2*x], x, y) == { + (2**e/(2*y), y), ((-2**e/4 - 2**e*sqrt(3)*I/4)/y, y), + ((-2**e/4 + 2**e*sqrt(3)*I/4)/y, y)} + assert nonlinsolve([(1 - 4*x**2)*exp(-2*x**2 - 2*y**2), + -4*x*y*exp(-2*x**2)*exp(-2*y**2)], x, y) == {(-S(1)/2, 0), (S(1)/2, 0)} + rx, ry = symbols('x y', real=True) + sol = nonlinsolve([4*rx**3*ry**4 - 2*ry, 4*rx**4*ry**3 - 2*rx], rx, ry) + ans = {(2**(S(2)/3)/(2*ry), ry), + ((-2**(S(2)/3)/4 - 2**(S(2)/3)*sqrt(3)*I/4)/ry, ry), + ((-2**(S(2)/3)/4 + 2**(S(2)/3)*sqrt(3)*I/4)/ry, ry)} + assert sol == ans + + +def test_issue_22628(): + assert nonlinsolve([h - 1, k - 1, f - 2, f - 4, -2*k], h, k, f) == S.EmptySet + assert nonlinsolve([x**3 - 1, x + y, x**2 - 4], [x, y]) == S.EmptySet + + +def test_issue_25781(): + assert solve(sqrt(x/2) - x) == [0, S.Half] + + +def test_issue_26077(): + _n = Symbol('_n') + function = x*cot(5*x) + critical_points = stationary_points(function, x, S.Reals) + excluded_points = Union( + ImageSet(Lambda(_n, 2*_n*pi/5), S.Integers), + ImageSet(Lambda(_n, 2*_n*pi/5 + pi/5), S.Integers) + ) + solution = ConditionSet(x, + Eq(x*(-5*cot(5*x)**2 - 5) + cot(5*x), 0), + Complement(S.Reals, excluded_points) + ) + assert solution.as_dummy() == critical_points.as_dummy() diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/__init__.py b/.venv/lib/python3.12/site-packages/sympy/stats/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..adb79261954924305c1837555d7d47cd53b8430b --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/__init__.py @@ -0,0 +1,202 @@ +""" +SymPy statistics module + +Introduces a random variable type into the SymPy language. + +Random variables may be declared using prebuilt functions such as +Normal, Exponential, Coin, Die, etc... or built with functions like FiniteRV. + +Queries on random expressions can be made using the functions + +========================= ============================= + Expression Meaning +------------------------- ----------------------------- + ``P(condition)`` Probability + ``E(expression)`` Expected value + ``H(expression)`` Entropy + ``variance(expression)`` Variance + ``density(expression)`` Probability Density Function + ``sample(expression)`` Produce a realization + ``where(condition)`` Where the condition is true +========================= ============================= + +Examples +======== + +>>> from sympy.stats import P, E, variance, Die, Normal +>>> from sympy import simplify +>>> X, Y = Die('X', 6), Die('Y', 6) # Define two six sided dice +>>> Z = Normal('Z', 0, 1) # Declare a Normal random variable with mean 0, std 1 +>>> P(X>3) # Probability X is greater than 3 +1/2 +>>> E(X+Y) # Expectation of the sum of two dice +7 +>>> variance(X+Y) # Variance of the sum of two dice +35/6 +>>> simplify(P(Z>1)) # Probability of Z being greater than 1 +1/2 - erf(sqrt(2)/2)/2 + + +One could also create custom distribution and define custom random variables +as follows: + +1. If you want to create a Continuous Random Variable: + +>>> from sympy.stats import ContinuousRV, P, E +>>> from sympy import exp, Symbol, Interval, oo +>>> x = Symbol('x') +>>> pdf = exp(-x) # pdf of the Continuous Distribution +>>> Z = ContinuousRV(x, pdf, set=Interval(0, oo)) +>>> E(Z) +1 +>>> P(Z > 5) +exp(-5) + +1.1 To create an instance of Continuous Distribution: + +>>> from sympy.stats import ContinuousDistributionHandmade +>>> from sympy import Lambda +>>> dist = ContinuousDistributionHandmade(Lambda(x, pdf), set=Interval(0, oo)) +>>> dist.pdf(x) +exp(-x) + +2. If you want to create a Discrete Random Variable: + +>>> from sympy.stats import DiscreteRV, P, E +>>> from sympy import Symbol, S +>>> p = S(1)/2 +>>> x = Symbol('x', integer=True, positive=True) +>>> pdf = p*(1 - p)**(x - 1) +>>> D = DiscreteRV(x, pdf, set=S.Naturals) +>>> E(D) +2 +>>> P(D > 3) +1/8 + +2.1 To create an instance of Discrete Distribution: + +>>> from sympy.stats import DiscreteDistributionHandmade +>>> from sympy import Lambda +>>> dist = DiscreteDistributionHandmade(Lambda(x, pdf), set=S.Naturals) +>>> dist.pdf(x) +2**(1 - x)/2 + +3. If you want to create a Finite Random Variable: + +>>> from sympy.stats import FiniteRV, P, E +>>> from sympy import Rational, Eq +>>> pmf = {1: Rational(1, 3), 2: Rational(1, 6), 3: Rational(1, 4), 4: Rational(1, 4)} +>>> X = FiniteRV('X', pmf) +>>> E(X) +29/12 +>>> P(X > 3) +1/4 + +3.1 To create an instance of Finite Distribution: + +>>> from sympy.stats import FiniteDistributionHandmade +>>> dist = FiniteDistributionHandmade(pmf) +>>> dist.pmf(x) +Lambda(x, Piecewise((1/3, Eq(x, 1)), (1/6, Eq(x, 2)), (1/4, Eq(x, 3) | Eq(x, 4)), (0, True))) +""" + +__all__ = [ + 'P', 'E', 'H', 'density', 'where', 'given', 'sample', 'cdf','median', + 'characteristic_function', 'pspace', 'sample_iter', 'variance', 'std', + 'skewness', 'kurtosis', 'covariance', 'dependent', 'entropy', 'independent', + 'random_symbols', 'correlation', 'factorial_moment', 'moment', 'cmoment', + 'sampling_density', 'moment_generating_function', 'smoment', 'quantile', + 'coskewness', 'sample_stochastic_process', + + 'FiniteRV', 'DiscreteUniform', 'Die', 'Bernoulli', 'Coin', 'Binomial', + 'BetaBinomial', 'Hypergeometric', 'Rademacher', 'IdealSoliton', 'RobustSoliton', + 'FiniteDistributionHandmade', + + 'ContinuousRV', 'Arcsin', 'Benini', 'Beta', 'BetaNoncentral', 'BetaPrime', + 'BoundedPareto', 'Cauchy', 'Chi', 'ChiNoncentral', 'ChiSquared', 'Dagum', 'Davis', 'Erlang', + 'ExGaussian', 'Exponential', 'ExponentialPower', 'FDistribution', + 'FisherZ', 'Frechet', 'Gamma', 'GammaInverse', 'Gompertz', 'Gumbel', + 'Kumaraswamy', 'Laplace', 'Levy', 'Logistic','LogCauchy', 'LogLogistic', 'LogitNormal', 'LogNormal', 'Lomax', + 'Moyal', 'Maxwell', 'Nakagami', 'Normal', 'GaussianInverse', 'Pareto', 'PowerFunction', + 'QuadraticU', 'RaisedCosine', 'Rayleigh','Reciprocal', 'StudentT', 'ShiftedGompertz', + 'Trapezoidal', 'Triangular', 'Uniform', 'UniformSum', 'VonMises', 'Wald', + 'Weibull', 'WignerSemicircle', 'ContinuousDistributionHandmade', + + 'FlorySchulz', 'Geometric','Hermite', 'Logarithmic', 'NegativeBinomial', 'Poisson', 'Skellam', + 'YuleSimon', 'Zeta', 'DiscreteRV', 'DiscreteDistributionHandmade', + + 'JointRV', 'Dirichlet', 'GeneralizedMultivariateLogGamma', + 'GeneralizedMultivariateLogGammaOmega', 'Multinomial', 'MultivariateBeta', + 'MultivariateEwens', 'MultivariateT', 'NegativeMultinomial', + 'NormalGamma', 'MultivariateNormal', 'MultivariateLaplace', 'marginal_distribution', + + 'StochasticProcess', 'DiscreteTimeStochasticProcess', + 'DiscreteMarkovChain', 'TransitionMatrixOf', 'StochasticStateSpaceOf', + 'GeneratorMatrixOf', 'ContinuousMarkovChain', 'BernoulliProcess', + 'PoissonProcess', 'WienerProcess', 'GammaProcess', + + 'CircularEnsemble', 'CircularUnitaryEnsemble', + 'CircularOrthogonalEnsemble', 'CircularSymplecticEnsemble', + 'GaussianEnsemble', 'GaussianUnitaryEnsemble', + 'GaussianOrthogonalEnsemble', 'GaussianSymplecticEnsemble', + 'joint_eigen_distribution', 'JointEigenDistribution', + 'level_spacing_distribution', + + 'MatrixGamma', 'Wishart', 'MatrixNormal', 'MatrixStudentT', + + 'Probability', 'Expectation', 'Variance', 'Covariance', 'Moment', + 'CentralMoment', + + 'ExpectationMatrix', 'VarianceMatrix', 'CrossCovarianceMatrix' + +] +from .rv_interface import (P, E, H, density, where, given, sample, cdf, median, + characteristic_function, pspace, sample_iter, variance, std, skewness, + kurtosis, covariance, dependent, entropy, independent, random_symbols, + correlation, factorial_moment, moment, cmoment, sampling_density, + moment_generating_function, smoment, quantile, coskewness, + sample_stochastic_process) + +from .frv_types import (FiniteRV, DiscreteUniform, Die, Bernoulli, Coin, + Binomial, BetaBinomial, Hypergeometric, Rademacher, + FiniteDistributionHandmade, IdealSoliton, RobustSoliton) + +from .crv_types import (ContinuousRV, Arcsin, Benini, Beta, BetaNoncentral, + BetaPrime, BoundedPareto, Cauchy, Chi, ChiNoncentral, ChiSquared, + Dagum, Davis, Erlang, ExGaussian, Exponential, ExponentialPower, + FDistribution, FisherZ, Frechet, Gamma, GammaInverse, GaussianInverse, + Gompertz, Gumbel, Kumaraswamy, Laplace, Levy, Logistic, LogCauchy, + LogLogistic, LogitNormal, LogNormal, Lomax, Maxwell, Moyal, Nakagami, + Normal, Pareto, QuadraticU, RaisedCosine, Rayleigh, Reciprocal, + StudentT, PowerFunction, ShiftedGompertz, Trapezoidal, Triangular, + Uniform, UniformSum, VonMises, Wald, Weibull, WignerSemicircle, + ContinuousDistributionHandmade) + +from .drv_types import (FlorySchulz, Geometric, Hermite, Logarithmic, NegativeBinomial, Poisson, + Skellam, YuleSimon, Zeta, DiscreteRV, DiscreteDistributionHandmade) + +from .joint_rv_types import (JointRV, Dirichlet, + GeneralizedMultivariateLogGamma, GeneralizedMultivariateLogGammaOmega, + Multinomial, MultivariateBeta, MultivariateEwens, MultivariateT, + NegativeMultinomial, NormalGamma, MultivariateNormal, MultivariateLaplace, + marginal_distribution) + +from .stochastic_process_types import (StochasticProcess, + DiscreteTimeStochasticProcess, DiscreteMarkovChain, + TransitionMatrixOf, StochasticStateSpaceOf, GeneratorMatrixOf, + ContinuousMarkovChain, BernoulliProcess, PoissonProcess, WienerProcess, + GammaProcess) + +from .random_matrix_models import (CircularEnsemble, CircularUnitaryEnsemble, + CircularOrthogonalEnsemble, CircularSymplecticEnsemble, + GaussianEnsemble, GaussianUnitaryEnsemble, GaussianOrthogonalEnsemble, + GaussianSymplecticEnsemble, joint_eigen_distribution, + JointEigenDistribution, level_spacing_distribution) + +from .matrix_distributions import MatrixGamma, Wishart, MatrixNormal, MatrixStudentT + +from .symbolic_probability import (Probability, Expectation, Variance, + Covariance, Moment, CentralMoment) + +from .symbolic_multivariate_probability import (ExpectationMatrix, VarianceMatrix, + CrossCovarianceMatrix) diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/__pycache__/__init__.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/stats/__pycache__/__init__.cpython-312.pyc new file 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b/.venv/lib/python3.12/site-packages/sympy/stats/compound_rv.py new file mode 100644 index 0000000000000000000000000000000000000000..27555f4233fe691bac303800a87736205acbdee6 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/compound_rv.py @@ -0,0 +1,223 @@ +from sympy.concrete.summations import Sum +from sympy.core.basic import Basic +from sympy.core.function import Lambda +from sympy.core.symbol import Dummy +from sympy.integrals.integrals import Integral +from sympy.stats.rv import (NamedArgsMixin, random_symbols, _symbol_converter, + PSpace, RandomSymbol, is_random, Distribution) +from sympy.stats.crv import ContinuousDistribution, SingleContinuousPSpace +from sympy.stats.drv import DiscreteDistribution, SingleDiscretePSpace +from sympy.stats.frv import SingleFiniteDistribution, SingleFinitePSpace +from sympy.stats.crv_types import ContinuousDistributionHandmade +from sympy.stats.drv_types import DiscreteDistributionHandmade +from sympy.stats.frv_types import FiniteDistributionHandmade + + +class CompoundPSpace(PSpace): + """ + A temporary Probability Space for the Compound Distribution. After + Marginalization, this returns the corresponding Probability Space of the + parent distribution. + """ + + def __new__(cls, s, distribution): + s = _symbol_converter(s) + if isinstance(distribution, ContinuousDistribution): + return SingleContinuousPSpace(s, distribution) + if isinstance(distribution, DiscreteDistribution): + return SingleDiscretePSpace(s, distribution) + if isinstance(distribution, SingleFiniteDistribution): + return SingleFinitePSpace(s, distribution) + if not isinstance(distribution, CompoundDistribution): + raise ValueError("%s should be an isinstance of " + "CompoundDistribution"%(distribution)) + return Basic.__new__(cls, s, distribution) + + @property + def value(self): + return RandomSymbol(self.symbol, self) + + @property + def symbol(self): + return self.args[0] + + @property + def is_Continuous(self): + return self.distribution.is_Continuous + + @property + def is_Finite(self): + return self.distribution.is_Finite + + @property + def is_Discrete(self): + return self.distribution.is_Discrete + + @property + def distribution(self): + return self.args[1] + + @property + def pdf(self): + return self.distribution.pdf(self.symbol) + + @property + def set(self): + return self.distribution.set + + @property + def domain(self): + return self._get_newpspace().domain + + def _get_newpspace(self, evaluate=False): + x = Dummy('x') + parent_dist = self.distribution.args[0] + func = Lambda(x, self.distribution.pdf(x, evaluate)) + new_pspace = self._transform_pspace(self.symbol, parent_dist, func) + if new_pspace is not None: + return new_pspace + message = ("Compound Distribution for %s is not implemented yet" % str(parent_dist)) + raise NotImplementedError(message) + + def _transform_pspace(self, sym, dist, pdf): + """ + This function returns the new pspace of the distribution using handmade + Distributions and their corresponding pspace. + """ + pdf = Lambda(sym, pdf(sym)) + _set = dist.set + if isinstance(dist, ContinuousDistribution): + return SingleContinuousPSpace(sym, ContinuousDistributionHandmade(pdf, _set)) + elif isinstance(dist, DiscreteDistribution): + return SingleDiscretePSpace(sym, DiscreteDistributionHandmade(pdf, _set)) + elif isinstance(dist, SingleFiniteDistribution): + dens = {k: pdf(k) for k in _set} + return SingleFinitePSpace(sym, FiniteDistributionHandmade(dens)) + + def compute_density(self, expr, *, compound_evaluate=True, **kwargs): + new_pspace = self._get_newpspace(compound_evaluate) + expr = expr.subs({self.value: new_pspace.value}) + return new_pspace.compute_density(expr, **kwargs) + + def compute_cdf(self, expr, *, compound_evaluate=True, **kwargs): + new_pspace = self._get_newpspace(compound_evaluate) + expr = expr.subs({self.value: new_pspace.value}) + return new_pspace.compute_cdf(expr, **kwargs) + + def compute_expectation(self, expr, rvs=None, evaluate=False, **kwargs): + new_pspace = self._get_newpspace(evaluate) + expr = expr.subs({self.value: new_pspace.value}) + if rvs: + rvs = rvs.subs({self.value: new_pspace.value}) + if isinstance(new_pspace, SingleFinitePSpace): + return new_pspace.compute_expectation(expr, rvs, **kwargs) + return new_pspace.compute_expectation(expr, rvs, evaluate, **kwargs) + + def probability(self, condition, *, compound_evaluate=True, **kwargs): + new_pspace = self._get_newpspace(compound_evaluate) + condition = condition.subs({self.value: new_pspace.value}) + return new_pspace.probability(condition) + + def conditional_space(self, condition, *, compound_evaluate=True, **kwargs): + new_pspace = self._get_newpspace(compound_evaluate) + condition = condition.subs({self.value: new_pspace.value}) + return new_pspace.conditional_space(condition) + + +class CompoundDistribution(Distribution, NamedArgsMixin): + """ + Class for Compound Distributions. + + Parameters + ========== + + dist : Distribution + Distribution must contain a random parameter + + Examples + ======== + + >>> from sympy.stats.compound_rv import CompoundDistribution + >>> from sympy.stats.crv_types import NormalDistribution + >>> from sympy.stats import Normal + >>> from sympy.abc import x + >>> X = Normal('X', 2, 4) + >>> N = NormalDistribution(X, 4) + >>> C = CompoundDistribution(N) + >>> C.set + Interval(-oo, oo) + >>> C.pdf(x, evaluate=True).simplify() + exp(-x**2/64 + x/16 - 1/16)/(8*sqrt(pi)) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Compound_probability_distribution + + """ + + def __new__(cls, dist): + if not isinstance(dist, (ContinuousDistribution, + SingleFiniteDistribution, DiscreteDistribution)): + message = "Compound Distribution for %s is not implemented yet" % str(dist) + raise NotImplementedError(message) + if not cls._compound_check(dist): + return dist + return Basic.__new__(cls, dist) + + @property + def set(self): + return self.args[0].set + + @property + def is_Continuous(self): + return isinstance(self.args[0], ContinuousDistribution) + + @property + def is_Finite(self): + return isinstance(self.args[0], SingleFiniteDistribution) + + @property + def is_Discrete(self): + return isinstance(self.args[0], DiscreteDistribution) + + def pdf(self, x, evaluate=False): + dist = self.args[0] + randoms = [rv for rv in dist.args if is_random(rv)] + if isinstance(dist, SingleFiniteDistribution): + y = Dummy('y', integer=True, negative=False) + expr = dist.pmf(y) + else: + y = Dummy('y') + expr = dist.pdf(y) + for rv in randoms: + expr = self._marginalise(expr, rv, evaluate) + return Lambda(y, expr)(x) + + def _marginalise(self, expr, rv, evaluate): + if isinstance(rv.pspace.distribution, SingleFiniteDistribution): + rv_dens = rv.pspace.distribution.pmf(rv) + else: + rv_dens = rv.pspace.distribution.pdf(rv) + rv_dom = rv.pspace.domain.set + if rv.pspace.is_Discrete or rv.pspace.is_Finite: + expr = Sum(expr*rv_dens, (rv, rv_dom._inf, + rv_dom._sup)) + else: + expr = Integral(expr*rv_dens, (rv, rv_dom._inf, + rv_dom._sup)) + if evaluate: + return expr.doit() + return expr + + @classmethod + def _compound_check(self, dist): + """ + Checks if the given distribution contains random parameters. + """ + randoms = [] + for arg in dist.args: + randoms.extend(random_symbols(arg)) + if len(randoms) == 0: + return False + return True diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/crv.py b/.venv/lib/python3.12/site-packages/sympy/stats/crv.py new file mode 100644 index 0000000000000000000000000000000000000000..0a5184029679f663c83d81aa6c1b6ca4d948c70f --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/crv.py @@ -0,0 +1,570 @@ +""" +Continuous Random Variables Module + +See Also +======== +sympy.stats.crv_types +sympy.stats.rv +sympy.stats.frv +""" + + +from sympy.core.basic import Basic +from sympy.core.cache import cacheit +from sympy.core.function import Lambda, PoleError +from sympy.core.numbers import (I, nan, oo) +from sympy.core.relational import (Eq, Ne) +from sympy.core.singleton import S +from sympy.core.symbol import (Dummy, symbols) +from sympy.core.sympify import _sympify, sympify +from sympy.functions.combinatorial.factorials import factorial +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.special.delta_functions import DiracDelta +from sympy.integrals.integrals import (Integral, integrate) +from sympy.logic.boolalg import (And, Or) +from sympy.polys.polyerrors import PolynomialError +from sympy.polys.polytools import poly +from sympy.series.series import series +from sympy.sets.sets import (FiniteSet, Intersection, Interval, Union) +from sympy.solvers.solveset import solveset +from sympy.solvers.inequalities import reduce_rational_inequalities +from sympy.stats.rv import (RandomDomain, SingleDomain, ConditionalDomain, is_random, + ProductDomain, PSpace, SinglePSpace, random_symbols, NamedArgsMixin, Distribution) + + +class ContinuousDomain(RandomDomain): + """ + A domain with continuous support + + Represented using symbols and Intervals. + """ + is_Continuous = True + + def as_boolean(self): + raise NotImplementedError("Not Implemented for generic Domains") + + +class SingleContinuousDomain(ContinuousDomain, SingleDomain): + """ + A univariate domain with continuous support + + Represented using a single symbol and interval. + """ + def compute_expectation(self, expr, variables=None, **kwargs): + if variables is None: + variables = self.symbols + if not variables: + return expr + if frozenset(variables) != frozenset(self.symbols): + raise ValueError("Values should be equal") + # assumes only intervals + return Integral(expr, (self.symbol, self.set), **kwargs) + + def as_boolean(self): + return self.set.as_relational(self.symbol) + + +class ProductContinuousDomain(ProductDomain, ContinuousDomain): + """ + A collection of independent domains with continuous support + """ + + def compute_expectation(self, expr, variables=None, **kwargs): + if variables is None: + variables = self.symbols + for domain in self.domains: + domain_vars = frozenset(variables) & frozenset(domain.symbols) + if domain_vars: + expr = domain.compute_expectation(expr, domain_vars, **kwargs) + return expr + + def as_boolean(self): + return And(*[domain.as_boolean() for domain in self.domains]) + + +class ConditionalContinuousDomain(ContinuousDomain, ConditionalDomain): + """ + A domain with continuous support that has been further restricted by a + condition such as $x > 3$. + """ + + def compute_expectation(self, expr, variables=None, **kwargs): + if variables is None: + variables = self.symbols + if not variables: + return expr + # Extract the full integral + fullintgrl = self.fulldomain.compute_expectation(expr, variables) + # separate into integrand and limits + integrand, limits = fullintgrl.function, list(fullintgrl.limits) + + conditions = [self.condition] + while conditions: + cond = conditions.pop() + if cond.is_Boolean: + if isinstance(cond, And): + conditions.extend(cond.args) + elif isinstance(cond, Or): + raise NotImplementedError("Or not implemented here") + elif cond.is_Relational: + if cond.is_Equality: + # Add the appropriate Delta to the integrand + integrand *= DiracDelta(cond.lhs - cond.rhs) + else: + symbols = cond.free_symbols & set(self.symbols) + if len(symbols) != 1: # Can't handle x > y + raise NotImplementedError( + "Multivariate Inequalities not yet implemented") + # Can handle x > 0 + symbol = symbols.pop() + # Find the limit with x, such as (x, -oo, oo) + for i, limit in enumerate(limits): + if limit[0] == symbol: + # Make condition into an Interval like [0, oo] + cintvl = reduce_rational_inequalities_wrap( + cond, symbol) + # Make limit into an Interval like [-oo, oo] + lintvl = Interval(limit[1], limit[2]) + # Intersect them to get [0, oo] + intvl = cintvl.intersect(lintvl) + # Put back into limits list + limits[i] = (symbol, intvl.left, intvl.right) + else: + raise TypeError( + "Condition %s is not a relational or Boolean" % cond) + + return Integral(integrand, *limits, **kwargs) + + def as_boolean(self): + return And(self.fulldomain.as_boolean(), self.condition) + + @property + def set(self): + if len(self.symbols) == 1: + return (self.fulldomain.set & reduce_rational_inequalities_wrap( + self.condition, tuple(self.symbols)[0])) + else: + raise NotImplementedError( + "Set of Conditional Domain not Implemented") + + +class ContinuousDistribution(Distribution): + def __call__(self, *args): + return self.pdf(*args) + + +class SingleContinuousDistribution(ContinuousDistribution, NamedArgsMixin): + """ Continuous distribution of a single variable. + + Explanation + =========== + + Serves as superclass for Normal/Exponential/UniformDistribution etc.... + + Represented by parameters for each of the specific classes. E.g + NormalDistribution is represented by a mean and standard deviation. + + Provides methods for pdf, cdf, and sampling. + + See Also + ======== + + sympy.stats.crv_types.* + """ + + set = Interval(-oo, oo) + + def __new__(cls, *args): + args = list(map(sympify, args)) + return Basic.__new__(cls, *args) + + @staticmethod + def check(*args): + pass + + @cacheit + def compute_cdf(self, **kwargs): + """ Compute the CDF from the PDF. + + Returns a Lambda. + """ + x, z = symbols('x, z', real=True, cls=Dummy) + left_bound = self.set.start + + # CDF is integral of PDF from left bound to z + pdf = self.pdf(x) + cdf = integrate(pdf.doit(), (x, left_bound, z), **kwargs) + # CDF Ensure that CDF left of left_bound is zero + cdf = Piecewise((cdf, z >= left_bound), (0, True)) + return Lambda(z, cdf) + + def _cdf(self, x): + return None + + def cdf(self, x, **kwargs): + """ Cumulative density function """ + if len(kwargs) == 0: + cdf = self._cdf(x) + if cdf is not None: + return cdf + return self.compute_cdf(**kwargs)(x) + + @cacheit + def compute_characteristic_function(self, **kwargs): + """ Compute the characteristic function from the PDF. + + Returns a Lambda. + """ + x, t = symbols('x, t', real=True, cls=Dummy) + pdf = self.pdf(x) + cf = integrate(exp(I*t*x)*pdf, (x, self.set)) + return Lambda(t, cf) + + def _characteristic_function(self, t): + return None + + def characteristic_function(self, t, **kwargs): + """ Characteristic function """ + if len(kwargs) == 0: + cf = self._characteristic_function(t) + if cf is not None: + return cf + return self.compute_characteristic_function(**kwargs)(t) + + @cacheit + def compute_moment_generating_function(self, **kwargs): + """ Compute the moment generating function from the PDF. + + Returns a Lambda. + """ + x, t = symbols('x, t', real=True, cls=Dummy) + pdf = self.pdf(x) + mgf = integrate(exp(t * x) * pdf, (x, self.set)) + return Lambda(t, mgf) + + def _moment_generating_function(self, t): + return None + + def moment_generating_function(self, t, **kwargs): + """ Moment generating function """ + if not kwargs: + mgf = self._moment_generating_function(t) + if mgf is not None: + return mgf + return self.compute_moment_generating_function(**kwargs)(t) + + def expectation(self, expr, var, evaluate=True, **kwargs): + """ Expectation of expression over distribution """ + if evaluate: + try: + p = poly(expr, var) + if p.is_zero: + return S.Zero + t = Dummy('t', real=True) + mgf = self._moment_generating_function(t) + if mgf is None: + return integrate(expr * self.pdf(var), (var, self.set), **kwargs) + deg = p.degree() + taylor = poly(series(mgf, t, 0, deg + 1).removeO(), t) + result = 0 + for k in range(deg+1): + result += p.coeff_monomial(var ** k) * taylor.coeff_monomial(t ** k) * factorial(k) + return result + except PolynomialError: + return integrate(expr * self.pdf(var), (var, self.set), **kwargs) + else: + return Integral(expr * self.pdf(var), (var, self.set), **kwargs) + + @cacheit + def compute_quantile(self, **kwargs): + """ Compute the Quantile from the PDF. + + Returns a Lambda. + """ + x, p = symbols('x, p', real=True, cls=Dummy) + left_bound = self.set.start + + pdf = self.pdf(x) + cdf = integrate(pdf, (x, left_bound, x), **kwargs) + quantile = solveset(cdf - p, x, self.set) + return Lambda(p, Piecewise((quantile, (p >= 0) & (p <= 1) ), (nan, True))) + + def _quantile(self, x): + return None + + def quantile(self, x, **kwargs): + """ Cumulative density function """ + if len(kwargs) == 0: + quantile = self._quantile(x) + if quantile is not None: + return quantile + return self.compute_quantile(**kwargs)(x) + + +class ContinuousPSpace(PSpace): + """ Continuous Probability Space + + Represents the likelihood of an event space defined over a continuum. + + Represented with a ContinuousDomain and a PDF (Lambda-Like) + """ + + is_Continuous = True + is_real = True + + @property + def pdf(self): + return self.density(*self.domain.symbols) + + def compute_expectation(self, expr, rvs=None, evaluate=False, **kwargs): + if rvs is None: + rvs = self.values + else: + rvs = frozenset(rvs) + + expr = expr.xreplace({rv: rv.symbol for rv in rvs}) + + domain_symbols = frozenset(rv.symbol for rv in rvs) + + return self.domain.compute_expectation(self.pdf * expr, + domain_symbols, **kwargs) + + def compute_density(self, expr, **kwargs): + # Common case Density(X) where X in self.values + if expr in self.values: + # Marginalize all other random symbols out of the density + randomsymbols = tuple(set(self.values) - frozenset([expr])) + symbols = tuple(rs.symbol for rs in randomsymbols) + pdf = self.domain.compute_expectation(self.pdf, symbols, **kwargs) + return Lambda(expr.symbol, pdf) + + z = Dummy('z', real=True) + return Lambda(z, self.compute_expectation(DiracDelta(expr - z), **kwargs)) + + @cacheit + def compute_cdf(self, expr, **kwargs): + if not self.domain.set.is_Interval: + raise ValueError( + "CDF not well defined on multivariate expressions") + + d = self.compute_density(expr, **kwargs) + x, z = symbols('x, z', real=True, cls=Dummy) + left_bound = self.domain.set.start + + # CDF is integral of PDF from left bound to z + cdf = integrate(d(x), (x, left_bound, z), **kwargs) + # CDF Ensure that CDF left of left_bound is zero + cdf = Piecewise((cdf, z >= left_bound), (0, True)) + return Lambda(z, cdf) + + @cacheit + def compute_characteristic_function(self, expr, **kwargs): + if not self.domain.set.is_Interval: + raise NotImplementedError("Characteristic function of multivariate expressions not implemented") + + d = self.compute_density(expr, **kwargs) + x, t = symbols('x, t', real=True, cls=Dummy) + cf = integrate(exp(I*t*x)*d(x), (x, -oo, oo), **kwargs) + return Lambda(t, cf) + + @cacheit + def compute_moment_generating_function(self, expr, **kwargs): + if not self.domain.set.is_Interval: + raise NotImplementedError("Moment generating function of multivariate expressions not implemented") + + d = self.compute_density(expr, **kwargs) + x, t = symbols('x, t', real=True, cls=Dummy) + mgf = integrate(exp(t * x) * d(x), (x, -oo, oo), **kwargs) + return Lambda(t, mgf) + + @cacheit + def compute_quantile(self, expr, **kwargs): + if not self.domain.set.is_Interval: + raise ValueError( + "Quantile not well defined on multivariate expressions") + + d = self.compute_cdf(expr, **kwargs) + x = Dummy('x', real=True) + p = Dummy('p', positive=True) + + quantile = solveset(d(x) - p, x, self.set) + + return Lambda(p, quantile) + + def probability(self, condition, **kwargs): + z = Dummy('z', real=True) + cond_inv = False + if isinstance(condition, Ne): + condition = Eq(condition.args[0], condition.args[1]) + cond_inv = True + # Univariate case can be handled by where + try: + domain = self.where(condition) + rv = [rv for rv in self.values if rv.symbol == domain.symbol][0] + # Integrate out all other random variables + pdf = self.compute_density(rv, **kwargs) + # return S.Zero if `domain` is empty set + if domain.set is S.EmptySet or isinstance(domain.set, FiniteSet): + return S.Zero if not cond_inv else S.One + if isinstance(domain.set, Union): + return sum( + Integral(pdf(z), (z, subset), **kwargs) for subset in + domain.set.args if isinstance(subset, Interval)) + # Integrate out the last variable over the special domain + return Integral(pdf(z), (z, domain.set), **kwargs) + + # Other cases can be turned into univariate case + # by computing a density handled by density computation + except NotImplementedError: + from sympy.stats.rv import density + expr = condition.lhs - condition.rhs + if not is_random(expr): + dens = self.density + comp = condition.rhs + else: + dens = density(expr, **kwargs) + comp = 0 + if not isinstance(dens, ContinuousDistribution): + from sympy.stats.crv_types import ContinuousDistributionHandmade + dens = ContinuousDistributionHandmade(dens, set=self.domain.set) + # Turn problem into univariate case + space = SingleContinuousPSpace(z, dens) + result = space.probability(condition.__class__(space.value, comp)) + return result if not cond_inv else S.One - result + + def where(self, condition): + rvs = frozenset(random_symbols(condition)) + if not (len(rvs) == 1 and rvs.issubset(self.values)): + raise NotImplementedError( + "Multiple continuous random variables not supported") + rv = tuple(rvs)[0] + interval = reduce_rational_inequalities_wrap(condition, rv) + interval = interval.intersect(self.domain.set) + return SingleContinuousDomain(rv.symbol, interval) + + def conditional_space(self, condition, normalize=True, **kwargs): + condition = condition.xreplace({rv: rv.symbol for rv in self.values}) + domain = ConditionalContinuousDomain(self.domain, condition) + if normalize: + # create a clone of the variable to + # make sure that variables in nested integrals are different + # from the variables outside the integral + # this makes sure that they are evaluated separately + # and in the correct order + replacement = {rv: Dummy(str(rv)) for rv in self.symbols} + norm = domain.compute_expectation(self.pdf, **kwargs) + pdf = self.pdf / norm.xreplace(replacement) + # XXX: Converting set to tuple. The order matters to Lambda though + # so we shouldn't be starting with a set here... + density = Lambda(tuple(domain.symbols), pdf) + + return ContinuousPSpace(domain, density) + + +class SingleContinuousPSpace(ContinuousPSpace, SinglePSpace): + """ + A continuous probability space over a single univariate variable. + + These consist of a Symbol and a SingleContinuousDistribution + + This class is normally accessed through the various random variable + functions, Normal, Exponential, Uniform, etc.... + """ + + @property + def set(self): + return self.distribution.set + + @property + def domain(self): + return SingleContinuousDomain(sympify(self.symbol), self.set) + + def sample(self, size=(), library='scipy', seed=None): + """ + Internal sample method. + + Returns dictionary mapping RandomSymbol to realization value. + """ + return {self.value: self.distribution.sample(size, library=library, seed=seed)} + + def compute_expectation(self, expr, rvs=None, evaluate=False, **kwargs): + rvs = rvs or (self.value,) + if self.value not in rvs: + return expr + + expr = _sympify(expr) + expr = expr.xreplace({rv: rv.symbol for rv in rvs}) + + x = self.value.symbol + try: + return self.distribution.expectation(expr, x, evaluate=evaluate, **kwargs) + except PoleError: + return Integral(expr * self.pdf, (x, self.set), **kwargs) + + def compute_cdf(self, expr, **kwargs): + if expr == self.value: + z = Dummy("z", real=True) + return Lambda(z, self.distribution.cdf(z, **kwargs)) + else: + return ContinuousPSpace.compute_cdf(self, expr, **kwargs) + + def compute_characteristic_function(self, expr, **kwargs): + if expr == self.value: + t = Dummy("t", real=True) + return Lambda(t, self.distribution.characteristic_function(t, **kwargs)) + else: + return ContinuousPSpace.compute_characteristic_function(self, expr, **kwargs) + + def compute_moment_generating_function(self, expr, **kwargs): + if expr == self.value: + t = Dummy("t", real=True) + return Lambda(t, self.distribution.moment_generating_function(t, **kwargs)) + else: + return ContinuousPSpace.compute_moment_generating_function(self, expr, **kwargs) + + def compute_density(self, expr, **kwargs): + # https://en.wikipedia.org/wiki/Random_variable#Functions_of_random_variables + if expr == self.value: + return self.density + y = Dummy('y', real=True) + + gs = solveset(expr - y, self.value, S.Reals) + + if isinstance(gs, Intersection): + if len(gs.args) == 2 and gs.args[0] is S.Reals: + gs = gs.args[1] + if not gs.is_FiniteSet: + raise ValueError("Can not solve %s for %s" % (expr, self.value)) + fx = self.compute_density(self.value) + fy = sum(fx(g) * abs(g.diff(y)) for g in gs) + return Lambda(y, fy) + + def compute_quantile(self, expr, **kwargs): + + if expr == self.value: + p = Dummy("p", real=True) + return Lambda(p, self.distribution.quantile(p, **kwargs)) + else: + return ContinuousPSpace.compute_quantile(self, expr, **kwargs) + +def _reduce_inequalities(conditions, var, **kwargs): + try: + return reduce_rational_inequalities(conditions, var, **kwargs) + except PolynomialError: + raise ValueError("Reduction of condition failed %s\n" % conditions[0]) + + +def reduce_rational_inequalities_wrap(condition, var): + if condition.is_Relational: + return _reduce_inequalities([[condition]], var, relational=False) + if isinstance(condition, Or): + return Union(*[_reduce_inequalities([[arg]], var, relational=False) + for arg in condition.args]) + if isinstance(condition, And): + intervals = [_reduce_inequalities([[arg]], var, relational=False) + for arg in condition.args] + I = intervals[0] + for i in intervals: + I = I.intersect(i) + return I diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/crv_types.py b/.venv/lib/python3.12/site-packages/sympy/stats/crv_types.py new file mode 100644 index 0000000000000000000000000000000000000000..073e7350fdf80aac39ecd1dd607488a8b76187e3 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/crv_types.py @@ -0,0 +1,4732 @@ +""" +Continuous Random Variables - Prebuilt variables + +Contains +======== +Arcsin +Benini +Beta +BetaNoncentral +BetaPrime +BoundedPareto +Cauchy +Chi +ChiNoncentral +ChiSquared +Dagum +Davis +Erlang +ExGaussian +Exponential +ExponentialPower +FDistribution +FisherZ +Frechet +Gamma +GammaInverse +Gumbel +Gompertz +Kumaraswamy +Laplace +Levy +LogCauchy +Logistic +LogLogistic +LogitNormal +LogNormal +Lomax +Maxwell +Moyal +Nakagami +Normal +Pareto +PowerFunction +QuadraticU +RaisedCosine +Rayleigh +Reciprocal +ShiftedGompertz +StudentT +Trapezoidal +Triangular +Uniform +UniformSum +VonMises +Wald +Weibull +WignerSemicircle +""" + + + +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.trigonometric import (atan, cos, sin, tan) +from sympy.functions.special.bessel import (besseli, besselj, besselk) +from sympy.functions.special.beta_functions import beta as beta_fn +from sympy.concrete.summations import Sum +from sympy.core.basic import Basic +from sympy.core.function import Lambda +from sympy.core.numbers import (I, Rational, pi) +from sympy.core.relational import (Eq, Ne) +from sympy.core.singleton import S +from sympy.core.symbol import Dummy +from sympy.core.sympify import sympify +from sympy.functions.combinatorial.factorials import (binomial, factorial) +from sympy.functions.elementary.complexes import (Abs, sign) +from sympy.functions.elementary.exponential import log +from sympy.functions.elementary.hyperbolic import sinh +from sympy.functions.elementary.integers import floor +from sympy.functions.elementary.miscellaneous import sqrt, Max, Min +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.elementary.trigonometric import asin +from sympy.functions.special.error_functions import (erf, erfc, erfi, erfinv, expint) +from sympy.functions.special.gamma_functions import (gamma, lowergamma, uppergamma) +from sympy.functions.special.zeta_functions import zeta +from sympy.functions.special.hyper import hyper +from sympy.integrals.integrals import integrate +from sympy.logic.boolalg import And +from sympy.sets.sets import Interval +from sympy.matrices import MatrixBase +from sympy.stats.crv import SingleContinuousPSpace, SingleContinuousDistribution +from sympy.stats.rv import _value_check, is_random + +oo = S.Infinity + +__all__ = ['ContinuousRV', +'Arcsin', +'Benini', +'Beta', +'BetaNoncentral', +'BetaPrime', +'BoundedPareto', +'Cauchy', +'Chi', +'ChiNoncentral', +'ChiSquared', +'Dagum', +'Davis', +'Erlang', +'ExGaussian', +'Exponential', +'ExponentialPower', +'FDistribution', +'FisherZ', +'Frechet', +'Gamma', +'GammaInverse', +'Gompertz', +'Gumbel', +'Kumaraswamy', +'Laplace', +'Levy', +'LogCauchy', +'Logistic', +'LogLogistic', +'LogitNormal', +'LogNormal', +'Lomax', +'Maxwell', +'Moyal', +'Nakagami', +'Normal', +'GaussianInverse', +'Pareto', +'PowerFunction', +'QuadraticU', +'RaisedCosine', +'Rayleigh', +'Reciprocal', +'StudentT', +'ShiftedGompertz', +'Trapezoidal', +'Triangular', +'Uniform', +'UniformSum', +'VonMises', +'Wald', +'Weibull', +'WignerSemicircle', +] + + +@is_random.register(MatrixBase) +def _(x): + return any(is_random(i) for i in x) + +def rv(symbol, cls, args, **kwargs): + args = list(map(sympify, args)) + dist = cls(*args) + if kwargs.pop('check', True): + dist.check(*args) + pspace = SingleContinuousPSpace(symbol, dist) + if any(is_random(arg) for arg in args): + from sympy.stats.compound_rv import CompoundPSpace, CompoundDistribution + pspace = CompoundPSpace(symbol, CompoundDistribution(dist)) + return pspace.value + + +class ContinuousDistributionHandmade(SingleContinuousDistribution): + _argnames = ('pdf',) + + def __new__(cls, pdf, set=Interval(-oo, oo)): + return Basic.__new__(cls, pdf, set) + + @property + def set(self): + return self.args[1] + + @staticmethod + def check(pdf, set): + x = Dummy('x') + val = integrate(pdf(x), (x, set)) + _value_check(Eq(val, 1) != S.false, "The pdf on the given set is incorrect.") + + +def ContinuousRV(symbol, density, set=Interval(-oo, oo), **kwargs): + """ + Create a Continuous Random Variable given the following: + + Parameters + ========== + + symbol : Symbol + Represents name of the random variable. + density : Expression containing symbol + Represents probability density function. + set : set/Interval + Represents the region where the pdf is valid, by default is real line. + check : bool + If True, it will check whether the given density + integrates to 1 over the given set. If False, it + will not perform this check. Default is False. + + + Returns + ======= + + RandomSymbol + + Many common continuous random variable types are already implemented. + This function should be necessary only very rarely. + + + Examples + ======== + + >>> from sympy import Symbol, sqrt, exp, pi + >>> from sympy.stats import ContinuousRV, P, E + + >>> x = Symbol("x") + + >>> pdf = sqrt(2)*exp(-x**2/2)/(2*sqrt(pi)) # Normal distribution + >>> X = ContinuousRV(x, pdf) + + >>> E(X) + 0 + >>> P(X>0) + 1/2 + """ + pdf = Piecewise((density, set.as_relational(symbol)), (0, True)) + pdf = Lambda(symbol, pdf) + # have a default of False while `rv` should have a default of True + kwargs['check'] = kwargs.pop('check', False) + return rv(symbol.name, ContinuousDistributionHandmade, (pdf, set), **kwargs) + +######################################## +# Continuous Probability Distributions # +######################################## + +#------------------------------------------------------------------------------- +# Arcsin distribution ---------------------------------------------------------- + + +class ArcsinDistribution(SingleContinuousDistribution): + _argnames = ('a', 'b') + + @property + def set(self): + return Interval(self.a, self.b) + + def pdf(self, x): + a, b = self.a, self.b + return 1/(pi*sqrt((x - a)*(b - x))) + + def _cdf(self, x): + a, b = self.a, self.b + return Piecewise( + (S.Zero, x < a), + (2*asin(sqrt((x - a)/(b - a)))/pi, x <= b), + (S.One, True)) + + +def Arcsin(name, a=0, b=1): + r""" + Create a Continuous Random Variable with an arcsin distribution. + + The density of the arcsin distribution is given by + + .. math:: + f(x) := \frac{1}{\pi\sqrt{(x-a)(b-x)}} + + with :math:`x \in (a,b)`. It must hold that :math:`-\infty < a < b < \infty`. + + Parameters + ========== + + a : Real number, the left interval boundary + b : Real number, the right interval boundary + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Arcsin, density, cdf + >>> from sympy import Symbol + + >>> a = Symbol("a", real=True) + >>> b = Symbol("b", real=True) + >>> z = Symbol("z") + + >>> X = Arcsin("x", a, b) + + >>> density(X)(z) + 1/(pi*sqrt((-a + z)*(b - z))) + + >>> cdf(X)(z) + Piecewise((0, a > z), + (2*asin(sqrt((-a + z)/(-a + b)))/pi, b >= z), + (1, True)) + + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Arcsine_distribution + + """ + + return rv(name, ArcsinDistribution, (a, b)) + +#------------------------------------------------------------------------------- +# Benini distribution ---------------------------------------------------------- + + +class BeniniDistribution(SingleContinuousDistribution): + _argnames = ('alpha', 'beta', 'sigma') + + @staticmethod + def check(alpha, beta, sigma): + _value_check(alpha > 0, "Shape parameter Alpha must be positive.") + _value_check(beta > 0, "Shape parameter Beta must be positive.") + _value_check(sigma > 0, "Scale parameter Sigma must be positive.") + + @property + def set(self): + return Interval(self.sigma, oo) + + def pdf(self, x): + alpha, beta, sigma = self.alpha, self.beta, self.sigma + return (exp(-alpha*log(x/sigma) - beta*log(x/sigma)**2) + *(alpha/x + 2*beta*log(x/sigma)/x)) + + def _moment_generating_function(self, t): + raise NotImplementedError('The moment generating function of the ' + 'Benini distribution does not exist.') + +def Benini(name, alpha, beta, sigma): + r""" + Create a Continuous Random Variable with a Benini distribution. + + The density of the Benini distribution is given by + + .. math:: + f(x) := e^{-\alpha\log{\frac{x}{\sigma}} + -\beta\log^2\left[{\frac{x}{\sigma}}\right]} + \left(\frac{\alpha}{x}+\frac{2\beta\log{\frac{x}{\sigma}}}{x}\right) + + This is a heavy-tailed distribution and is also known as the log-Rayleigh + distribution. + + Parameters + ========== + + alpha : Real number, `\alpha > 0`, a shape + beta : Real number, `\beta > 0`, a shape + sigma : Real number, `\sigma > 0`, a scale + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Benini, density, cdf + >>> from sympy import Symbol, pprint + + >>> alpha = Symbol("alpha", positive=True) + >>> beta = Symbol("beta", positive=True) + >>> sigma = Symbol("sigma", positive=True) + >>> z = Symbol("z") + + >>> X = Benini("x", alpha, beta, sigma) + + >>> D = density(X)(z) + >>> pprint(D, use_unicode=False) + / / z \\ / z \ 2/ z \ + | 2*beta*log|-----|| - alpha*log|-----| - beta*log |-----| + |alpha \sigma/| \sigma/ \sigma/ + |----- + -----------------|*e + \ z z / + + >>> cdf(X)(z) + Piecewise((1 - exp(-alpha*log(z/sigma) - beta*log(z/sigma)**2), sigma <= z), + (0, True)) + + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Benini_distribution + .. [2] https://reference.wolfram.com/legacy/v8/ref/BeniniDistribution.html + + """ + + return rv(name, BeniniDistribution, (alpha, beta, sigma)) + +#------------------------------------------------------------------------------- +# Beta distribution ------------------------------------------------------------ + + +class BetaDistribution(SingleContinuousDistribution): + _argnames = ('alpha', 'beta') + + set = Interval(0, 1) + + @staticmethod + def check(alpha, beta): + _value_check(alpha > 0, "Shape parameter Alpha must be positive.") + _value_check(beta > 0, "Shape parameter Beta must be positive.") + + def pdf(self, x): + alpha, beta = self.alpha, self.beta + return x**(alpha - 1) * (1 - x)**(beta - 1) / beta_fn(alpha, beta) + + def _characteristic_function(self, t): + return hyper((self.alpha,), (self.alpha + self.beta,), I*t) + + def _moment_generating_function(self, t): + return hyper((self.alpha,), (self.alpha + self.beta,), t) + + +def Beta(name, alpha, beta): + r""" + Create a Continuous Random Variable with a Beta distribution. + + The density of the Beta distribution is given by + + .. math:: + f(x) := \frac{x^{\alpha-1}(1-x)^{\beta-1}} {\mathrm{B}(\alpha,\beta)} + + with :math:`x \in [0,1]`. + + Parameters + ========== + + alpha : Real number, `\alpha > 0`, a shape + beta : Real number, `\beta > 0`, a shape + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Beta, density, E, variance + >>> from sympy import Symbol, simplify, pprint, factor + + >>> alpha = Symbol("alpha", positive=True) + >>> beta = Symbol("beta", positive=True) + >>> z = Symbol("z") + + >>> X = Beta("x", alpha, beta) + + >>> D = density(X)(z) + >>> pprint(D, use_unicode=False) + alpha - 1 beta - 1 + z *(1 - z) + -------------------------- + B(alpha, beta) + + >>> simplify(E(X)) + alpha/(alpha + beta) + + >>> factor(simplify(variance(X))) + alpha*beta/((alpha + beta)**2*(alpha + beta + 1)) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Beta_distribution + .. [2] https://mathworld.wolfram.com/BetaDistribution.html + + """ + + return rv(name, BetaDistribution, (alpha, beta)) + +#------------------------------------------------------------------------------- +# Noncentral Beta distribution ------------------------------------------------------------ + + +class BetaNoncentralDistribution(SingleContinuousDistribution): + _argnames = ('alpha', 'beta', 'lamda') + + set = Interval(0, 1) + + @staticmethod + def check(alpha, beta, lamda): + _value_check(alpha > 0, "Shape parameter Alpha must be positive.") + _value_check(beta > 0, "Shape parameter Beta must be positive.") + _value_check(lamda >= 0, "Noncentrality parameter Lambda must be positive") + + def pdf(self, x): + alpha, beta, lamda = self.alpha, self.beta, self.lamda + k = Dummy("k") + return Sum(exp(-lamda / 2) * (lamda / 2)**k * x**(alpha + k - 1) *( + 1 - x)**(beta - 1) / (factorial(k) * beta_fn(alpha + k, beta)), (k, 0, oo)) + +def BetaNoncentral(name, alpha, beta, lamda): + r""" + Create a Continuous Random Variable with a Type I Noncentral Beta distribution. + + The density of the Noncentral Beta distribution is given by + + .. math:: + f(x) := \sum_{k=0}^\infty e^{-\lambda/2}\frac{(\lambda/2)^k}{k!} + \frac{x^{\alpha+k-1}(1-x)^{\beta-1}}{\mathrm{B}(\alpha+k,\beta)} + + with :math:`x \in [0,1]`. + + Parameters + ========== + + alpha : Real number, `\alpha > 0`, a shape + beta : Real number, `\beta > 0`, a shape + lamda : Real number, `\lambda \geq 0`, noncentrality parameter + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import BetaNoncentral, density, cdf + >>> from sympy import Symbol, pprint + + >>> alpha = Symbol("alpha", positive=True) + >>> beta = Symbol("beta", positive=True) + >>> lamda = Symbol("lamda", nonnegative=True) + >>> z = Symbol("z") + + >>> X = BetaNoncentral("x", alpha, beta, lamda) + + >>> D = density(X)(z) + >>> pprint(D, use_unicode=False) + oo + _____ + \ ` + \ -lamda + \ k ------- + \ k + alpha - 1 /lamda\ beta - 1 2 + ) z *|-----| *(1 - z) *e + / \ 2 / + / ------------------------------------------------ + / B(k + alpha, beta)*k! + /____, + k = 0 + + Compute cdf with specific 'x', 'alpha', 'beta' and 'lamda' values as follows: + + >>> cdf(BetaNoncentral("x", 1, 1, 1), evaluate=False)(2).doit() + 2*exp(1/2) + + The argument evaluate=False prevents an attempt at evaluation + of the sum for general x, before the argument 2 is passed. + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Noncentral_beta_distribution + .. [2] https://reference.wolfram.com/language/ref/NoncentralBetaDistribution.html + + """ + + return rv(name, BetaNoncentralDistribution, (alpha, beta, lamda)) + + +#------------------------------------------------------------------------------- +# Beta prime distribution ------------------------------------------------------ + + +class BetaPrimeDistribution(SingleContinuousDistribution): + _argnames = ('alpha', 'beta') + + @staticmethod + def check(alpha, beta): + _value_check(alpha > 0, "Shape parameter Alpha must be positive.") + _value_check(beta > 0, "Shape parameter Beta must be positive.") + + set = Interval(0, oo) + + def pdf(self, x): + alpha, beta = self.alpha, self.beta + return x**(alpha - 1)*(1 + x)**(-alpha - beta)/beta_fn(alpha, beta) + +def BetaPrime(name, alpha, beta): + r""" + Create a continuous random variable with a Beta prime distribution. + + The density of the Beta prime distribution is given by + + .. math:: + f(x) := \frac{x^{\alpha-1} (1+x)^{-\alpha -\beta}}{B(\alpha,\beta)} + + with :math:`x > 0`. + + Parameters + ========== + + alpha : Real number, `\alpha > 0`, a shape + beta : Real number, `\beta > 0`, a shape + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import BetaPrime, density + >>> from sympy import Symbol, pprint + + >>> alpha = Symbol("alpha", positive=True) + >>> beta = Symbol("beta", positive=True) + >>> z = Symbol("z") + + >>> X = BetaPrime("x", alpha, beta) + + >>> D = density(X)(z) + >>> pprint(D, use_unicode=False) + alpha - 1 -alpha - beta + z *(z + 1) + ------------------------------- + B(alpha, beta) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Beta_prime_distribution + .. [2] https://mathworld.wolfram.com/BetaPrimeDistribution.html + + """ + + return rv(name, BetaPrimeDistribution, (alpha, beta)) + +#------------------------------------------------------------------------------- +# Bounded Pareto Distribution -------------------------------------------------- +class BoundedParetoDistribution(SingleContinuousDistribution): + _argnames = ('alpha', 'left', 'right') + + @property + def set(self): + return Interval(self.left, self.right) + + @staticmethod + def check(alpha, left, right): + _value_check (alpha.is_positive, "Shape must be positive.") + _value_check (left.is_positive, "Left value should be positive.") + _value_check (right > left, "Right should be greater than left.") + + def pdf(self, x): + alpha, left, right = self.alpha, self.left, self.right + num = alpha * (left**alpha) * x**(- alpha -1) + den = 1 - (left/right)**alpha + return num/den + +def BoundedPareto(name, alpha, left, right): + r""" + Create a continuous random variable with a Bounded Pareto distribution. + + The density of the Bounded Pareto distribution is given by + + .. math:: + f(x) := \frac{\alpha L^{\alpha}x^{-\alpha-1}}{1-(\frac{L}{H})^{\alpha}} + + Parameters + ========== + + alpha : Real Number, `\alpha > 0` + Shape parameter + left : Real Number, `left > 0` + Location parameter + right : Real Number, `right > left` + Location parameter + + Examples + ======== + + >>> from sympy.stats import BoundedPareto, density, cdf, E + >>> from sympy import symbols + >>> L, H = symbols('L, H', positive=True) + >>> X = BoundedPareto('X', 2, L, H) + >>> x = symbols('x') + >>> density(X)(x) + 2*L**2/(x**3*(1 - L**2/H**2)) + >>> cdf(X)(x) + Piecewise((-H**2*L**2/(x**2*(H**2 - L**2)) + H**2/(H**2 - L**2), L <= x), (0, True)) + >>> E(X).simplify() + 2*H*L/(H + L) + + Returns + ======= + + RandomSymbol + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Pareto_distribution#Bounded_Pareto_distribution + + """ + return rv (name, BoundedParetoDistribution, (alpha, left, right)) + +# ------------------------------------------------------------------------------ +# Cauchy distribution ---------------------------------------------------------- + + +class CauchyDistribution(SingleContinuousDistribution): + _argnames = ('x0', 'gamma') + + @staticmethod + def check(x0, gamma): + _value_check(gamma > 0, "Scale parameter Gamma must be positive.") + _value_check(x0.is_real, "Location parameter must be real.") + + def pdf(self, x): + return 1/(pi*self.gamma*(1 + ((x - self.x0)/self.gamma)**2)) + + def _cdf(self, x): + x0, gamma = self.x0, self.gamma + return (1/pi)*atan((x - x0)/gamma) + S.Half + + def _characteristic_function(self, t): + return exp(self.x0 * I * t - self.gamma * Abs(t)) + + def _moment_generating_function(self, t): + raise NotImplementedError("The moment generating function for the " + "Cauchy distribution does not exist.") + + def _quantile(self, p): + return self.x0 + self.gamma*tan(pi*(p - S.Half)) + + +def Cauchy(name, x0, gamma): + r""" + Create a continuous random variable with a Cauchy distribution. + + The density of the Cauchy distribution is given by + + .. math:: + f(x) := \frac{1}{\pi \gamma [1 + {(\frac{x-x_0}{\gamma})}^2]} + + Parameters + ========== + + x0 : Real number, the location + gamma : Real number, `\gamma > 0`, a scale + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Cauchy, density + >>> from sympy import Symbol + + >>> x0 = Symbol("x0") + >>> gamma = Symbol("gamma", positive=True) + >>> z = Symbol("z") + + >>> X = Cauchy("x", x0, gamma) + + >>> density(X)(z) + 1/(pi*gamma*(1 + (-x0 + z)**2/gamma**2)) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Cauchy_distribution + .. [2] https://mathworld.wolfram.com/CauchyDistribution.html + + """ + + return rv(name, CauchyDistribution, (x0, gamma)) + +#------------------------------------------------------------------------------- +# Chi distribution ------------------------------------------------------------- + + +class ChiDistribution(SingleContinuousDistribution): + _argnames = ('k',) + + @staticmethod + def check(k): + _value_check(k > 0, "Number of degrees of freedom (k) must be positive.") + _value_check(k.is_integer, "Number of degrees of freedom (k) must be an integer.") + + set = Interval(0, oo) + + def pdf(self, x): + return 2**(1 - self.k/2)*x**(self.k - 1)*exp(-x**2/2)/gamma(self.k/2) + + def _characteristic_function(self, t): + k = self.k + + part_1 = hyper((k/2,), (S.Half,), -t**2/2) + part_2 = I*t*sqrt(2)*gamma((k+1)/2)/gamma(k/2) + part_3 = hyper(((k+1)/2,), (Rational(3, 2),), -t**2/2) + return part_1 + part_2*part_3 + + def _moment_generating_function(self, t): + k = self.k + + part_1 = hyper((k / 2,), (S.Half,), t ** 2 / 2) + part_2 = t * sqrt(2) * gamma((k + 1) / 2) / gamma(k / 2) + part_3 = hyper(((k + 1) / 2,), (S(3) / 2,), t ** 2 / 2) + return part_1 + part_2 * part_3 + +def Chi(name, k): + r""" + Create a continuous random variable with a Chi distribution. + + The density of the Chi distribution is given by + + .. math:: + f(x) := \frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)} + + with :math:`x \geq 0`. + + Parameters + ========== + + k : Positive integer, The number of degrees of freedom + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Chi, density, E + >>> from sympy import Symbol, simplify + + >>> k = Symbol("k", integer=True) + >>> z = Symbol("z") + + >>> X = Chi("x", k) + + >>> density(X)(z) + 2**(1 - k/2)*z**(k - 1)*exp(-z**2/2)/gamma(k/2) + + >>> simplify(E(X)) + sqrt(2)*gamma(k/2 + 1/2)/gamma(k/2) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Chi_distribution + .. [2] https://mathworld.wolfram.com/ChiDistribution.html + + """ + + return rv(name, ChiDistribution, (k,)) + +#------------------------------------------------------------------------------- +# Non-central Chi distribution ------------------------------------------------- + + +class ChiNoncentralDistribution(SingleContinuousDistribution): + _argnames = ('k', 'l') + + @staticmethod + def check(k, l): + _value_check(k > 0, "Number of degrees of freedom (k) must be positive.") + _value_check(k.is_integer, "Number of degrees of freedom (k) must be an integer.") + _value_check(l > 0, "Shift parameter Lambda must be positive.") + + set = Interval(0, oo) + + def pdf(self, x): + k, l = self.k, self.l + return exp(-(x**2+l**2)/2)*x**k*l / (l*x)**(k/2) * besseli(k/2-1, l*x) + +def ChiNoncentral(name, k, l): + r""" + Create a continuous random variable with a non-central Chi distribution. + + Explanation + =========== + + The density of the non-central Chi distribution is given by + + .. math:: + f(x) := \frac{e^{-(x^2+\lambda^2)/2} x^k\lambda} + {(\lambda x)^{k/2}} I_{k/2-1}(\lambda x) + + with `x \geq 0`. Here, `I_\nu (x)` is the + :ref:`modified Bessel function of the first kind `. + + Parameters + ========== + + k : A positive Integer, $k > 0$ + The number of degrees of freedom. + lambda : Real number, `\lambda > 0` + Shift parameter. + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import ChiNoncentral, density + >>> from sympy import Symbol + + >>> k = Symbol("k", integer=True) + >>> l = Symbol("l") + >>> z = Symbol("z") + + >>> X = ChiNoncentral("x", k, l) + + >>> density(X)(z) + l*z**k*exp(-l**2/2 - z**2/2)*besseli(k/2 - 1, l*z)/(l*z)**(k/2) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Noncentral_chi_distribution + """ + + return rv(name, ChiNoncentralDistribution, (k, l)) + +#------------------------------------------------------------------------------- +# Chi squared distribution ----------------------------------------------------- + + +class ChiSquaredDistribution(SingleContinuousDistribution): + _argnames = ('k',) + + @staticmethod + def check(k): + _value_check(k > 0, "Number of degrees of freedom (k) must be positive.") + _value_check(k.is_integer, "Number of degrees of freedom (k) must be an integer.") + + set = Interval(0, oo) + + def pdf(self, x): + k = self.k + return 1/(2**(k/2)*gamma(k/2))*x**(k/2 - 1)*exp(-x/2) + + def _cdf(self, x): + k = self.k + return Piecewise( + (S.One/gamma(k/2)*lowergamma(k/2, x/2), x >= 0), + (0, True) + ) + + def _characteristic_function(self, t): + return (1 - 2*I*t)**(-self.k/2) + + def _moment_generating_function(self, t): + return (1 - 2*t)**(-self.k/2) + + +def ChiSquared(name, k): + r""" + Create a continuous random variable with a Chi-squared distribution. + + Explanation + =========== + + The density of the Chi-squared distribution is given by + + .. math:: + f(x) := \frac{1}{2^{\frac{k}{2}}\Gamma\left(\frac{k}{2}\right)} + x^{\frac{k}{2}-1} e^{-\frac{x}{2}} + + with :math:`x \geq 0`. + + Parameters + ========== + + k : Positive integer + The number of degrees of freedom. + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import ChiSquared, density, E, variance, moment + >>> from sympy import Symbol + + >>> k = Symbol("k", integer=True, positive=True) + >>> z = Symbol("z") + + >>> X = ChiSquared("x", k) + + >>> density(X)(z) + z**(k/2 - 1)*exp(-z/2)/(2**(k/2)*gamma(k/2)) + + >>> E(X) + k + + >>> variance(X) + 2*k + + >>> moment(X, 3) + k**3 + 6*k**2 + 8*k + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Chi_squared_distribution + .. [2] https://mathworld.wolfram.com/Chi-SquaredDistribution.html + """ + + return rv(name, ChiSquaredDistribution, (k, )) + +#------------------------------------------------------------------------------- +# Dagum distribution ----------------------------------------------------------- + + +class DagumDistribution(SingleContinuousDistribution): + _argnames = ('p', 'a', 'b') + + set = Interval(0, oo) + + @staticmethod + def check(p, a, b): + _value_check(p > 0, "Shape parameter p must be positive.") + _value_check(a > 0, "Shape parameter a must be positive.") + _value_check(b > 0, "Scale parameter b must be positive.") + + def pdf(self, x): + p, a, b = self.p, self.a, self.b + return a*p/x*((x/b)**(a*p)/(((x/b)**a + 1)**(p + 1))) + + def _cdf(self, x): + p, a, b = self.p, self.a, self.b + return Piecewise(((S.One + (S(x)/b)**-a)**-p, x>=0), + (S.Zero, True)) + +def Dagum(name, p, a, b): + r""" + Create a continuous random variable with a Dagum distribution. + + Explanation + =========== + + The density of the Dagum distribution is given by + + .. math:: + f(x) := \frac{a p}{x} \left( \frac{\left(\tfrac{x}{b}\right)^{a p}} + {\left(\left(\tfrac{x}{b}\right)^a + 1 \right)^{p+1}} \right) + + with :math:`x > 0`. + + Parameters + ========== + + p : Real number + `p > 0`, a shape. + a : Real number + `a > 0`, a shape. + b : Real number + `b > 0`, a scale. + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Dagum, density, cdf + >>> from sympy import Symbol + + >>> p = Symbol("p", positive=True) + >>> a = Symbol("a", positive=True) + >>> b = Symbol("b", positive=True) + >>> z = Symbol("z") + + >>> X = Dagum("x", p, a, b) + + >>> density(X)(z) + a*p*(z/b)**(a*p)*((z/b)**a + 1)**(-p - 1)/z + + >>> cdf(X)(z) + Piecewise(((1 + (z/b)**(-a))**(-p), z >= 0), (0, True)) + + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Dagum_distribution + + """ + + return rv(name, DagumDistribution, (p, a, b)) + +#------------------------------------------------------------------------------- +# Davis distribution ----------------------------------------------------------- + +class DavisDistribution(SingleContinuousDistribution): + _argnames = ('b', 'n', 'mu') + + set = Interval(0, oo) + + @staticmethod + def check(b, n, mu): + _value_check(b > 0, "Scale parameter b must be positive.") + _value_check(n > 1, "Shape parameter n must be above 1.") + _value_check(mu > 0, "Location parameter mu must be positive.") + + def pdf(self, x): + b, n, mu = self.b, self.n, self.mu + dividend = b**n*(x - mu)**(-1-n) + divisor = (exp(b/(x-mu))-1)*(gamma(n)*zeta(n)) + return dividend/divisor + + +def Davis(name, b, n, mu): + r""" Create a continuous random variable with Davis distribution. + + Explanation + =========== + + The density of Davis distribution is given by + + .. math:: + f(x; \mu; b, n) := \frac{b^{n}(x - \mu)^{1-n}}{ \left( e^{\frac{b}{x-\mu}} - 1 \right) \Gamma(n)\zeta(n)} + + with :math:`x \in [0,\infty]`. + + Davis distribution is a generalization of the Planck's law of radiation from statistical physics. It is used for modeling income distribution. + + Parameters + ========== + b : Real number + `p > 0`, a scale. + n : Real number + `n > 1`, a shape. + mu : Real number + `mu > 0`, a location. + + Returns + ======= + + RandomSymbol + + Examples + ======== + >>> from sympy.stats import Davis, density + >>> from sympy import Symbol + >>> b = Symbol("b", positive=True) + >>> n = Symbol("n", positive=True) + >>> mu = Symbol("mu", positive=True) + >>> z = Symbol("z") + >>> X = Davis("x", b, n, mu) + >>> density(X)(z) + b**n*(-mu + z)**(-n - 1)/((exp(b/(-mu + z)) - 1)*gamma(n)*zeta(n)) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Davis_distribution + .. [2] https://reference.wolfram.com/language/ref/DavisDistribution.html + + """ + return rv(name, DavisDistribution, (b, n, mu)) + + +#------------------------------------------------------------------------------- +# Erlang distribution ---------------------------------------------------------- + + +def Erlang(name, k, l): + r""" + Create a continuous random variable with an Erlang distribution. + + Explanation + =========== + + The density of the Erlang distribution is given by + + .. math:: + f(x) := \frac{\lambda^k x^{k-1} e^{-\lambda x}}{(k-1)!} + + with :math:`x \in [0,\infty]`. + + Parameters + ========== + + k : Positive integer + l : Real number, `\lambda > 0`, the rate + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Erlang, density, cdf, E, variance + >>> from sympy import Symbol, simplify, pprint + + >>> k = Symbol("k", integer=True, positive=True) + >>> l = Symbol("l", positive=True) + >>> z = Symbol("z") + + >>> X = Erlang("x", k, l) + + >>> D = density(X)(z) + >>> pprint(D, use_unicode=False) + k k - 1 -l*z + l *z *e + --------------- + Gamma(k) + + >>> C = cdf(X)(z) + >>> pprint(C, use_unicode=False) + /lowergamma(k, l*z) + |------------------ for z > 0 + < Gamma(k) + | + \ 0 otherwise + + + >>> E(X) + k/l + + >>> simplify(variance(X)) + k/l**2 + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Erlang_distribution + .. [2] https://mathworld.wolfram.com/ErlangDistribution.html + + """ + + return rv(name, GammaDistribution, (k, S.One/l)) + +# ------------------------------------------------------------------------------- +# ExGaussian distribution ----------------------------------------------------- + + +class ExGaussianDistribution(SingleContinuousDistribution): + _argnames = ('mean', 'std', 'rate') + + set = Interval(-oo, oo) + + @staticmethod + def check(mean, std, rate): + _value_check( + std > 0, "Standard deviation of ExGaussian must be positive.") + _value_check(rate > 0, "Rate of ExGaussian must be positive.") + + def pdf(self, x): + mean, std, rate = self.mean, self.std, self.rate + term1 = rate/2 + term2 = exp(rate * (2 * mean + rate * std**2 - 2*x)/2) + term3 = erfc((mean + rate*std**2 - x)/(sqrt(2)*std)) + return term1*term2*term3 + + def _cdf(self, x): + from sympy.stats import cdf + mean, std, rate = self.mean, self.std, self.rate + u = rate*(x - mean) + v = rate*std + GaussianCDF1 = cdf(Normal('x', 0, v))(u) + GaussianCDF2 = cdf(Normal('x', v**2, v))(u) + + return GaussianCDF1 - exp(-u + (v**2/2) + log(GaussianCDF2)) + + def _characteristic_function(self, t): + mean, std, rate = self.mean, self.std, self.rate + term1 = (1 - I*t/rate)**(-1) + term2 = exp(I*mean*t - std**2*t**2/2) + return term1 * term2 + + def _moment_generating_function(self, t): + mean, std, rate = self.mean, self.std, self.rate + term1 = (1 - t/rate)**(-1) + term2 = exp(mean*t + std**2*t**2/2) + return term1*term2 + + +def ExGaussian(name, mean, std, rate): + r""" + Create a continuous random variable with an Exponentially modified + Gaussian (EMG) distribution. + + Explanation + =========== + + The density of the exponentially modified Gaussian distribution is given by + + .. math:: + f(x) := \frac{\lambda}{2}e^{\frac{\lambda}{2}(2\mu+\lambda\sigma^2-2x)} + \text{erfc}(\frac{\mu + \lambda\sigma^2 - x}{\sqrt{2}\sigma}) + + with $x > 0$. Note that the expected value is `1/\lambda`. + + Parameters + ========== + + name : A string giving a name for this distribution + mean : A Real number, the mean of Gaussian component + std : A positive Real number, + :math: `\sigma^2 > 0` the variance of Gaussian component + rate : A positive Real number, + :math: `\lambda > 0` the rate of Exponential component + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import ExGaussian, density, cdf, E + >>> from sympy.stats import variance, skewness + >>> from sympy import Symbol, pprint, simplify + + >>> mean = Symbol("mu") + >>> std = Symbol("sigma", positive=True) + >>> rate = Symbol("lamda", positive=True) + >>> z = Symbol("z") + >>> X = ExGaussian("x", mean, std, rate) + + >>> pprint(density(X)(z), use_unicode=False) + / 2 \ + lamda*\lamda*sigma + 2*mu - 2*z/ + --------------------------------- / ___ / 2 \\ + 2 |\/ 2 *\lamda*sigma + mu - z/| + lamda*e *erfc|-----------------------------| + \ 2*sigma / + ---------------------------------------------------------------------------- + 2 + + >>> cdf(X)(z) + -(erf(sqrt(2)*(-lamda**2*sigma**2 + lamda*(-mu + z))/(2*lamda*sigma))/2 + 1/2)*exp(lamda**2*sigma**2/2 - lamda*(-mu + z)) + erf(sqrt(2)*(-mu + z)/(2*sigma))/2 + 1/2 + + >>> E(X) + (lamda*mu + 1)/lamda + + >>> simplify(variance(X)) + sigma**2 + lamda**(-2) + + >>> simplify(skewness(X)) + 2/(lamda**2*sigma**2 + 1)**(3/2) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Exponentially_modified_Gaussian_distribution + """ + return rv(name, ExGaussianDistribution, (mean, std, rate)) + +#------------------------------------------------------------------------------- +# Exponential distribution ----------------------------------------------------- + + +class ExponentialDistribution(SingleContinuousDistribution): + _argnames = ('rate',) + + set = Interval(0, oo) + + @staticmethod + def check(rate): + _value_check(rate > 0, "Rate must be positive.") + + def pdf(self, x): + return self.rate * exp(-self.rate*x) + + def _cdf(self, x): + return Piecewise( + (S.One - exp(-self.rate*x), x >= 0), + (0, True), + ) + + def _characteristic_function(self, t): + rate = self.rate + return rate / (rate - I*t) + + def _moment_generating_function(self, t): + rate = self.rate + return rate / (rate - t) + + def _quantile(self, p): + return -log(1-p)/self.rate + + +def Exponential(name, rate): + r""" + Create a continuous random variable with an Exponential distribution. + + Explanation + =========== + + The density of the exponential distribution is given by + + .. math:: + f(x) := \lambda \exp(-\lambda x) + + with $x > 0$. Note that the expected value is `1/\lambda`. + + Parameters + ========== + + rate : A positive Real number, `\lambda > 0`, the rate (or inverse scale/inverse mean) + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Exponential, density, cdf, E + >>> from sympy.stats import variance, std, skewness, quantile + >>> from sympy import Symbol + + >>> l = Symbol("lambda", positive=True) + >>> z = Symbol("z") + >>> p = Symbol("p") + >>> X = Exponential("x", l) + + >>> density(X)(z) + lambda*exp(-lambda*z) + + >>> cdf(X)(z) + Piecewise((1 - exp(-lambda*z), z >= 0), (0, True)) + + >>> quantile(X)(p) + -log(1 - p)/lambda + + >>> E(X) + 1/lambda + + >>> variance(X) + lambda**(-2) + + >>> skewness(X) + 2 + + >>> X = Exponential('x', 10) + + >>> density(X)(z) + 10*exp(-10*z) + + >>> E(X) + 1/10 + + >>> std(X) + 1/10 + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Exponential_distribution + .. [2] https://mathworld.wolfram.com/ExponentialDistribution.html + + """ + + return rv(name, ExponentialDistribution, (rate, )) + + +# ------------------------------------------------------------------------------- +# Exponential Power distribution ----------------------------------------------------- + +class ExponentialPowerDistribution(SingleContinuousDistribution): + _argnames = ('mu', 'alpha', 'beta') + + set = Interval(-oo, oo) + + @staticmethod + def check(mu, alpha, beta): + _value_check(alpha > 0, "Scale parameter alpha must be positive.") + _value_check(beta > 0, "Shape parameter beta must be positive.") + + def pdf(self, x): + mu, alpha, beta = self.mu, self.alpha, self.beta + num = beta*exp(-(Abs(x - mu)/alpha)**beta) + den = 2*alpha*gamma(1/beta) + return num/den + + def _cdf(self, x): + mu, alpha, beta = self.mu, self.alpha, self.beta + num = lowergamma(1/beta, (Abs(x - mu) / alpha)**beta) + den = 2*gamma(1/beta) + return sign(x - mu)*num/den + S.Half + + +def ExponentialPower(name, mu, alpha, beta): + r""" + Create a Continuous Random Variable with Exponential Power distribution. + This distribution is known also as Generalized Normal + distribution version 1. + + Explanation + =========== + + The density of the Exponential Power distribution is given by + + .. math:: + f(x) := \frac{\beta}{2\alpha\Gamma(\frac{1}{\beta})} + e^{{-(\frac{|x - \mu|}{\alpha})^{\beta}}} + + with :math:`x \in [ - \infty, \infty ]`. + + Parameters + ========== + + mu : Real number + A location. + alpha : Real number,`\alpha > 0` + A scale. + beta : Real number, `\beta > 0` + A shape. + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import ExponentialPower, density, cdf + >>> from sympy import Symbol, pprint + >>> z = Symbol("z") + >>> mu = Symbol("mu") + >>> alpha = Symbol("alpha", positive=True) + >>> beta = Symbol("beta", positive=True) + >>> X = ExponentialPower("x", mu, alpha, beta) + >>> pprint(density(X)(z), use_unicode=False) + beta + /|mu - z|\ + -|--------| + \ alpha / + beta*e + --------------------- + / 1 \ + 2*alpha*Gamma|----| + \beta/ + >>> cdf(X)(z) + 1/2 + lowergamma(1/beta, (Abs(mu - z)/alpha)**beta)*sign(-mu + z)/(2*gamma(1/beta)) + + References + ========== + + .. [1] https://reference.wolfram.com/language/ref/ExponentialPowerDistribution.html + .. [2] https://en.wikipedia.org/wiki/Generalized_normal_distribution#Version_1 + + """ + return rv(name, ExponentialPowerDistribution, (mu, alpha, beta)) + + +#------------------------------------------------------------------------------- +# F distribution --------------------------------------------------------------- + + +class FDistributionDistribution(SingleContinuousDistribution): + _argnames = ('d1', 'd2') + + set = Interval(0, oo) + + @staticmethod + def check(d1, d2): + _value_check((d1 > 0, d1.is_integer), + "Degrees of freedom d1 must be positive integer.") + _value_check((d2 > 0, d2.is_integer), + "Degrees of freedom d2 must be positive integer.") + + def pdf(self, x): + d1, d2 = self.d1, self.d2 + return (sqrt((d1*x)**d1*d2**d2 / (d1*x+d2)**(d1+d2)) + / (x * beta_fn(d1/2, d2/2))) + + def _moment_generating_function(self, t): + raise NotImplementedError('The moment generating function for the ' + 'F-distribution does not exist.') + +def FDistribution(name, d1, d2): + r""" + Create a continuous random variable with a F distribution. + + Explanation + =========== + + The density of the F distribution is given by + + .. math:: + f(x) := \frac{\sqrt{\frac{(d_1 x)^{d_1} d_2^{d_2}} + {(d_1 x + d_2)^{d_1 + d_2}}}} + {x \mathrm{B} \left(\frac{d_1}{2}, \frac{d_2}{2}\right)} + + with :math:`x > 0`. + + Parameters + ========== + + d1 : `d_1 > 0`, where `d_1` is the degrees of freedom (`n_1 - 1`) + d2 : `d_2 > 0`, where `d_2` is the degrees of freedom (`n_2 - 1`) + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import FDistribution, density + >>> from sympy import Symbol, pprint + + >>> d1 = Symbol("d1", positive=True) + >>> d2 = Symbol("d2", positive=True) + >>> z = Symbol("z") + + >>> X = FDistribution("x", d1, d2) + + >>> D = density(X)(z) + >>> pprint(D, use_unicode=False) + d2 + -- ______________________________ + 2 / d1 -d1 - d2 + d2 *\/ (d1*z) *(d1*z + d2) + -------------------------------------- + /d1 d2\ + z*B|--, --| + \2 2 / + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/F-distribution + .. [2] https://mathworld.wolfram.com/F-Distribution.html + + """ + + return rv(name, FDistributionDistribution, (d1, d2)) + +#------------------------------------------------------------------------------- +# Fisher Z distribution -------------------------------------------------------- + +class FisherZDistribution(SingleContinuousDistribution): + _argnames = ('d1', 'd2') + + set = Interval(-oo, oo) + + @staticmethod + def check(d1, d2): + _value_check(d1 > 0, "Degree of freedom d1 must be positive.") + _value_check(d2 > 0, "Degree of freedom d2 must be positive.") + + def pdf(self, x): + d1, d2 = self.d1, self.d2 + return (2*d1**(d1/2)*d2**(d2/2) / beta_fn(d1/2, d2/2) * + exp(d1*x) / (d1*exp(2*x)+d2)**((d1+d2)/2)) + +def FisherZ(name, d1, d2): + r""" + Create a Continuous Random Variable with an Fisher's Z distribution. + + Explanation + =========== + + The density of the Fisher's Z distribution is given by + + .. math:: + f(x) := \frac{2d_1^{d_1/2} d_2^{d_2/2}} {\mathrm{B}(d_1/2, d_2/2)} + \frac{e^{d_1z}}{\left(d_1e^{2z}+d_2\right)^{\left(d_1+d_2\right)/2}} + + + .. TODO - What is the difference between these degrees of freedom? + + Parameters + ========== + + d1 : `d_1 > 0` + Degree of freedom. + d2 : `d_2 > 0` + Degree of freedom. + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import FisherZ, density + >>> from sympy import Symbol, pprint + + >>> d1 = Symbol("d1", positive=True) + >>> d2 = Symbol("d2", positive=True) + >>> z = Symbol("z") + + >>> X = FisherZ("x", d1, d2) + + >>> D = density(X)(z) + >>> pprint(D, use_unicode=False) + d1 d2 + d1 d2 - -- - -- + -- -- 2 2 + 2 2 / 2*z \ d1*z + 2*d1 *d2 *\d1*e + d2/ *e + ----------------------------------------- + /d1 d2\ + B|--, --| + \2 2 / + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Fisher%27s_z-distribution + .. [2] https://mathworld.wolfram.com/Fishersz-Distribution.html + + """ + + return rv(name, FisherZDistribution, (d1, d2)) + +#------------------------------------------------------------------------------- +# Frechet distribution --------------------------------------------------------- + +class FrechetDistribution(SingleContinuousDistribution): + _argnames = ('a', 's', 'm') + + set = Interval(0, oo) + + @staticmethod + def check(a, s, m): + _value_check(a > 0, "Shape parameter alpha must be positive.") + _value_check(s > 0, "Scale parameter s must be positive.") + + def __new__(cls, a, s=1, m=0): + a, s, m = list(map(sympify, (a, s, m))) + return Basic.__new__(cls, a, s, m) + + def pdf(self, x): + a, s, m = self.a, self.s, self.m + return a/s * ((x-m)/s)**(-1-a) * exp(-((x-m)/s)**(-a)) + + def _cdf(self, x): + a, s, m = self.a, self.s, self.m + return Piecewise((exp(-((x-m)/s)**(-a)), x >= m), + (S.Zero, True)) + +def Frechet(name, a, s=1, m=0): + r""" + Create a continuous random variable with a Frechet distribution. + + Explanation + =========== + + The density of the Frechet distribution is given by + + .. math:: + f(x) := \frac{\alpha}{s} \left(\frac{x-m}{s}\right)^{-1-\alpha} + e^{-(\frac{x-m}{s})^{-\alpha}} + + with :math:`x \geq m`. + + Parameters + ========== + + a : Real number, :math:`a \in \left(0, \infty\right)` the shape + s : Real number, :math:`s \in \left(0, \infty\right)` the scale + m : Real number, :math:`m \in \left(-\infty, \infty\right)` the minimum + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Frechet, density, cdf + >>> from sympy import Symbol + + >>> a = Symbol("a", positive=True) + >>> s = Symbol("s", positive=True) + >>> m = Symbol("m", real=True) + >>> z = Symbol("z") + + >>> X = Frechet("x", a, s, m) + + >>> density(X)(z) + a*((-m + z)/s)**(-a - 1)*exp(-1/((-m + z)/s)**a)/s + + >>> cdf(X)(z) + Piecewise((exp(-1/((-m + z)/s)**a), m <= z), (0, True)) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Fr%C3%A9chet_distribution + + """ + + return rv(name, FrechetDistribution, (a, s, m)) + +#------------------------------------------------------------------------------- +# Gamma distribution ----------------------------------------------------------- + + +class GammaDistribution(SingleContinuousDistribution): + _argnames = ('k', 'theta') + + set = Interval(0, oo) + + @staticmethod + def check(k, theta): + _value_check(k > 0, "k must be positive") + _value_check(theta > 0, "Theta must be positive") + + def pdf(self, x): + k, theta = self.k, self.theta + return x**(k - 1) * exp(-x/theta) / (gamma(k)*theta**k) + + def _cdf(self, x): + k, theta = self.k, self.theta + return Piecewise( + (lowergamma(k, S(x)/theta)/gamma(k), x > 0), + (S.Zero, True)) + + def _characteristic_function(self, t): + return (1 - self.theta*I*t)**(-self.k) + + def _moment_generating_function(self, t): + return (1- self.theta*t)**(-self.k) + + +def Gamma(name, k, theta): + r""" + Create a continuous random variable with a Gamma distribution. + + Explanation + =========== + + The density of the Gamma distribution is given by + + .. math:: + f(x) := \frac{1}{\Gamma(k) \theta^k} x^{k - 1} e^{-\frac{x}{\theta}} + + with :math:`x \in [0,1]`. + + Parameters + ========== + + k : Real number, `k > 0`, a shape + theta : Real number, `\theta > 0`, a scale + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Gamma, density, cdf, E, variance + >>> from sympy import Symbol, pprint, simplify + + >>> k = Symbol("k", positive=True) + >>> theta = Symbol("theta", positive=True) + >>> z = Symbol("z") + + >>> X = Gamma("x", k, theta) + + >>> D = density(X)(z) + >>> pprint(D, use_unicode=False) + -z + ----- + -k k - 1 theta + theta *z *e + --------------------- + Gamma(k) + + >>> C = cdf(X, meijerg=True)(z) + >>> pprint(C, use_unicode=False) + / / z \ + |k*lowergamma|k, -----| + | \ theta/ + <---------------------- for z >= 0 + | Gamma(k + 1) + | + \ 0 otherwise + + >>> E(X) + k*theta + + >>> V = simplify(variance(X)) + >>> pprint(V, use_unicode=False) + 2 + k*theta + + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Gamma_distribution + .. [2] https://mathworld.wolfram.com/GammaDistribution.html + + """ + + return rv(name, GammaDistribution, (k, theta)) + +#------------------------------------------------------------------------------- +# Inverse Gamma distribution --------------------------------------------------- + + +class GammaInverseDistribution(SingleContinuousDistribution): + _argnames = ('a', 'b') + + set = Interval(0, oo) + + @staticmethod + def check(a, b): + _value_check(a > 0, "alpha must be positive") + _value_check(b > 0, "beta must be positive") + + def pdf(self, x): + a, b = self.a, self.b + return b**a/gamma(a) * x**(-a-1) * exp(-b/x) + + def _cdf(self, x): + a, b = self.a, self.b + return Piecewise((uppergamma(a,b/x)/gamma(a), x > 0), + (S.Zero, True)) + + def _characteristic_function(self, t): + a, b = self.a, self.b + return 2 * (-I*b*t)**(a/2) * besselk(a, sqrt(-4*I*b*t)) / gamma(a) + + def _moment_generating_function(self, t): + raise NotImplementedError('The moment generating function for the ' + 'gamma inverse distribution does not exist.') + +def GammaInverse(name, a, b): + r""" + Create a continuous random variable with an inverse Gamma distribution. + + Explanation + =========== + + The density of the inverse Gamma distribution is given by + + .. math:: + f(x) := \frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} + \exp\left(\frac{-\beta}{x}\right) + + with :math:`x > 0`. + + Parameters + ========== + + a : Real number, `a > 0`, a shape + b : Real number, `b > 0`, a scale + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import GammaInverse, density, cdf + >>> from sympy import Symbol, pprint + + >>> a = Symbol("a", positive=True) + >>> b = Symbol("b", positive=True) + >>> z = Symbol("z") + + >>> X = GammaInverse("x", a, b) + + >>> D = density(X)(z) + >>> pprint(D, use_unicode=False) + -b + --- + a -a - 1 z + b *z *e + --------------- + Gamma(a) + + >>> cdf(X)(z) + Piecewise((uppergamma(a, b/z)/gamma(a), z > 0), (0, True)) + + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Inverse-gamma_distribution + + """ + + return rv(name, GammaInverseDistribution, (a, b)) + + +#------------------------------------------------------------------------------- +# Gumbel distribution (Maximum and Minimum) -------------------------------------------------------- + + +class GumbelDistribution(SingleContinuousDistribution): + _argnames = ('beta', 'mu', 'minimum') + + set = Interval(-oo, oo) + + @staticmethod + def check(beta, mu, minimum): + _value_check(beta > 0, "Scale parameter beta must be positive.") + + def pdf(self, x): + beta, mu = self.beta, self.mu + z = (x - mu)/beta + f_max = (1/beta)*exp(-z - exp(-z)) + f_min = (1/beta)*exp(z - exp(z)) + return Piecewise((f_min, self.minimum), (f_max, not self.minimum)) + + def _cdf(self, x): + beta, mu = self.beta, self.mu + z = (x - mu)/beta + F_max = exp(-exp(-z)) + F_min = 1 - exp(-exp(z)) + return Piecewise((F_min, self.minimum), (F_max, not self.minimum)) + + def _characteristic_function(self, t): + cf_max = gamma(1 - I*self.beta*t) * exp(I*self.mu*t) + cf_min = gamma(1 + I*self.beta*t) * exp(I*self.mu*t) + return Piecewise((cf_min, self.minimum), (cf_max, not self.minimum)) + + def _moment_generating_function(self, t): + mgf_max = gamma(1 - self.beta*t) * exp(self.mu*t) + mgf_min = gamma(1 + self.beta*t) * exp(self.mu*t) + return Piecewise((mgf_min, self.minimum), (mgf_max, not self.minimum)) + +def Gumbel(name, beta, mu, minimum=False): + r""" + Create a Continuous Random Variable with Gumbel distribution. + + Explanation + =========== + + The density of the Gumbel distribution is given by + + For Maximum + + .. math:: + f(x) := \dfrac{1}{\beta} \exp \left( -\dfrac{x-\mu}{\beta} + - \exp \left( -\dfrac{x - \mu}{\beta} \right) \right) + + with :math:`x \in [ - \infty, \infty ]`. + + For Minimum + + .. math:: + f(x) := \frac{e^{- e^{\frac{- \mu + x}{\beta}} + \frac{- \mu + x}{\beta}}}{\beta} + + with :math:`x \in [ - \infty, \infty ]`. + + Parameters + ========== + + mu : Real number, `\mu`, a location + beta : Real number, `\beta > 0`, a scale + minimum : Boolean, by default ``False``, set to ``True`` for enabling minimum distribution + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Gumbel, density, cdf + >>> from sympy import Symbol + >>> x = Symbol("x") + >>> mu = Symbol("mu") + >>> beta = Symbol("beta", positive=True) + >>> X = Gumbel("x", beta, mu) + >>> density(X)(x) + exp(-exp(-(-mu + x)/beta) - (-mu + x)/beta)/beta + >>> cdf(X)(x) + exp(-exp(-(-mu + x)/beta)) + + References + ========== + + .. [1] https://mathworld.wolfram.com/GumbelDistribution.html + .. [2] https://en.wikipedia.org/wiki/Gumbel_distribution + .. [3] https://web.archive.org/web/20200628222206/http://www.mathwave.com/help/easyfit/html/analyses/distributions/gumbel_max.html + .. [4] https://web.archive.org/web/20200628222212/http://www.mathwave.com/help/easyfit/html/analyses/distributions/gumbel_min.html + + """ + return rv(name, GumbelDistribution, (beta, mu, minimum)) + +#------------------------------------------------------------------------------- +# Gompertz distribution -------------------------------------------------------- + +class GompertzDistribution(SingleContinuousDistribution): + _argnames = ('b', 'eta') + + set = Interval(0, oo) + + @staticmethod + def check(b, eta): + _value_check(b > 0, "b must be positive") + _value_check(eta > 0, "eta must be positive") + + def pdf(self, x): + eta, b = self.eta, self.b + return b*eta*exp(b*x)*exp(eta)*exp(-eta*exp(b*x)) + + def _cdf(self, x): + eta, b = self.eta, self.b + return 1 - exp(eta)*exp(-eta*exp(b*x)) + + def _moment_generating_function(self, t): + eta, b = self.eta, self.b + return eta * exp(eta) * expint(t/b, eta) + +def Gompertz(name, b, eta): + r""" + Create a Continuous Random Variable with Gompertz distribution. + + Explanation + =========== + + The density of the Gompertz distribution is given by + + .. math:: + f(x) := b \eta e^{b x} e^{\eta} \exp \left(-\eta e^{bx} \right) + + with :math:`x \in [0, \infty)`. + + Parameters + ========== + + b : Real number, `b > 0`, a scale + eta : Real number, `\eta > 0`, a shape + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Gompertz, density + >>> from sympy import Symbol + + >>> b = Symbol("b", positive=True) + >>> eta = Symbol("eta", positive=True) + >>> z = Symbol("z") + + >>> X = Gompertz("x", b, eta) + + >>> density(X)(z) + b*eta*exp(eta)*exp(b*z)*exp(-eta*exp(b*z)) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Gompertz_distribution + + """ + return rv(name, GompertzDistribution, (b, eta)) + +#------------------------------------------------------------------------------- +# Kumaraswamy distribution ----------------------------------------------------- + + +class KumaraswamyDistribution(SingleContinuousDistribution): + _argnames = ('a', 'b') + + set = Interval(0, oo) + + @staticmethod + def check(a, b): + _value_check(a > 0, "a must be positive") + _value_check(b > 0, "b must be positive") + + def pdf(self, x): + a, b = self.a, self.b + return a * b * x**(a-1) * (1-x**a)**(b-1) + + def _cdf(self, x): + a, b = self.a, self.b + return Piecewise( + (S.Zero, x < S.Zero), + (1 - (1 - x**a)**b, x <= S.One), + (S.One, True)) + +def Kumaraswamy(name, a, b): + r""" + Create a Continuous Random Variable with a Kumaraswamy distribution. + + Explanation + =========== + + The density of the Kumaraswamy distribution is given by + + .. math:: + f(x) := a b x^{a-1} (1-x^a)^{b-1} + + with :math:`x \in [0,1]`. + + Parameters + ========== + + a : Real number, `a > 0`, a shape + b : Real number, `b > 0`, a shape + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Kumaraswamy, density, cdf + >>> from sympy import Symbol, pprint + + >>> a = Symbol("a", positive=True) + >>> b = Symbol("b", positive=True) + >>> z = Symbol("z") + + >>> X = Kumaraswamy("x", a, b) + + >>> D = density(X)(z) + >>> pprint(D, use_unicode=False) + b - 1 + a - 1 / a\ + a*b*z *\1 - z / + + >>> cdf(X)(z) + Piecewise((0, z < 0), (1 - (1 - z**a)**b, z <= 1), (1, True)) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Kumaraswamy_distribution + + """ + + return rv(name, KumaraswamyDistribution, (a, b)) + +#------------------------------------------------------------------------------- +# Laplace distribution --------------------------------------------------------- + + +class LaplaceDistribution(SingleContinuousDistribution): + _argnames = ('mu', 'b') + + set = Interval(-oo, oo) + + @staticmethod + def check(mu, b): + _value_check(b > 0, "Scale parameter b must be positive.") + _value_check(mu.is_real, "Location parameter mu should be real") + + def pdf(self, x): + mu, b = self.mu, self.b + return 1/(2*b)*exp(-Abs(x - mu)/b) + + def _cdf(self, x): + mu, b = self.mu, self.b + return Piecewise( + (S.Half*exp((x - mu)/b), x < mu), + (S.One - S.Half*exp(-(x - mu)/b), x >= mu) + ) + + def _characteristic_function(self, t): + return exp(self.mu*I*t) / (1 + self.b**2*t**2) + + def _moment_generating_function(self, t): + return exp(self.mu*t) / (1 - self.b**2*t**2) + +def Laplace(name, mu, b): + r""" + Create a continuous random variable with a Laplace distribution. + + Explanation + =========== + + The density of the Laplace distribution is given by + + .. math:: + f(x) := \frac{1}{2 b} \exp \left(-\frac{|x-\mu|}b \right) + + Parameters + ========== + + mu : Real number or a list/matrix, the location (mean) or the + location vector + b : Real number or a positive definite matrix, representing a scale + or the covariance matrix. + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Laplace, density, cdf + >>> from sympy import Symbol, pprint + + >>> mu = Symbol("mu") + >>> b = Symbol("b", positive=True) + >>> z = Symbol("z") + + >>> X = Laplace("x", mu, b) + + >>> density(X)(z) + exp(-Abs(mu - z)/b)/(2*b) + + >>> cdf(X)(z) + Piecewise((exp((-mu + z)/b)/2, mu > z), (1 - exp((mu - z)/b)/2, True)) + + >>> L = Laplace('L', [1, 2], [[1, 0], [0, 1]]) + >>> pprint(density(L)(1, 2), use_unicode=False) + 5 / ____\ + e *besselk\0, \/ 35 / + --------------------- + pi + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Laplace_distribution + .. [2] https://mathworld.wolfram.com/LaplaceDistribution.html + + """ + + if isinstance(mu, (list, MatrixBase)) and\ + isinstance(b, (list, MatrixBase)): + from sympy.stats.joint_rv_types import MultivariateLaplace + return MultivariateLaplace(name, mu, b) + + return rv(name, LaplaceDistribution, (mu, b)) + +#------------------------------------------------------------------------------- +# Levy distribution --------------------------------------------------------- + + +class LevyDistribution(SingleContinuousDistribution): + _argnames = ('mu', 'c') + + @property + def set(self): + return Interval(self.mu, oo) + + @staticmethod + def check(mu, c): + _value_check(c > 0, "c (scale parameter) must be positive") + _value_check(mu.is_real, "mu (location parameter) must be real") + + def pdf(self, x): + mu, c = self.mu, self.c + return sqrt(c/(2*pi))*exp(-c/(2*(x - mu)))/((x - mu)**(S.One + S.Half)) + + def _cdf(self, x): + mu, c = self.mu, self.c + return erfc(sqrt(c/(2*(x - mu)))) + + def _characteristic_function(self, t): + mu, c = self.mu, self.c + return exp(I * mu * t - sqrt(-2 * I * c * t)) + + def _moment_generating_function(self, t): + raise NotImplementedError('The moment generating function of Levy distribution does not exist.') + +def Levy(name, mu, c): + r""" + Create a continuous random variable with a Levy distribution. + + The density of the Levy distribution is given by + + .. math:: + f(x) := \sqrt(\frac{c}{2 \pi}) \frac{\exp -\frac{c}{2 (x - \mu)}}{(x - \mu)^{3/2}} + + Parameters + ========== + + mu : Real number + The location parameter. + c : Real number, `c > 0` + A scale parameter. + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Levy, density, cdf + >>> from sympy import Symbol + + >>> mu = Symbol("mu", real=True) + >>> c = Symbol("c", positive=True) + >>> z = Symbol("z") + + >>> X = Levy("x", mu, c) + + >>> density(X)(z) + sqrt(2)*sqrt(c)*exp(-c/(-2*mu + 2*z))/(2*sqrt(pi)*(-mu + z)**(3/2)) + + >>> cdf(X)(z) + erfc(sqrt(c)*sqrt(1/(-2*mu + 2*z))) + + References + ========== + .. [1] https://en.wikipedia.org/wiki/L%C3%A9vy_distribution + .. [2] https://mathworld.wolfram.com/LevyDistribution.html + """ + + return rv(name, LevyDistribution, (mu, c)) + +#------------------------------------------------------------------------------- +# Log-Cauchy distribution -------------------------------------------------------- + + +class LogCauchyDistribution(SingleContinuousDistribution): + _argnames = ('mu', 'sigma') + + set = Interval.open(0, oo) + + @staticmethod + def check(mu, sigma): + _value_check((sigma > 0) != False, "Scale parameter Gamma must be positive.") + _value_check(mu.is_real != False, "Location parameter must be real.") + + def pdf(self, x): + mu, sigma = self.mu, self.sigma + return 1/(x*pi)*(sigma/((log(x) - mu)**2 + sigma**2)) + + def _cdf(self, x): + mu, sigma = self.mu, self.sigma + return (1/pi)*atan((log(x) - mu)/sigma) + S.Half + + def _characteristic_function(self, t): + raise NotImplementedError("The characteristic function for the " + "Log-Cauchy distribution does not exist.") + + def _moment_generating_function(self, t): + raise NotImplementedError("The moment generating function for the " + "Log-Cauchy distribution does not exist.") + +def LogCauchy(name, mu, sigma): + r""" + Create a continuous random variable with a Log-Cauchy distribution. + The density of the Log-Cauchy distribution is given by + + .. math:: + f(x) := \frac{1}{\pi x} \frac{\sigma}{(log(x)-\mu^2) + \sigma^2} + + Parameters + ========== + + mu : Real number, the location + + sigma : Real number, `\sigma > 0`, a scale + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import LogCauchy, density, cdf + >>> from sympy import Symbol, S + + >>> mu = 2 + >>> sigma = S.One / 5 + >>> z = Symbol("z") + + >>> X = LogCauchy("x", mu, sigma) + + >>> density(X)(z) + 1/(5*pi*z*((log(z) - 2)**2 + 1/25)) + + >>> cdf(X)(z) + atan(5*log(z) - 10)/pi + 1/2 + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Log-Cauchy_distribution + """ + + return rv(name, LogCauchyDistribution, (mu, sigma)) + + +#------------------------------------------------------------------------------- +# Logistic distribution -------------------------------------------------------- + + +class LogisticDistribution(SingleContinuousDistribution): + _argnames = ('mu', 's') + + set = Interval(-oo, oo) + + @staticmethod + def check(mu, s): + _value_check(s > 0, "Scale parameter s must be positive.") + + def pdf(self, x): + mu, s = self.mu, self.s + return exp(-(x - mu)/s)/(s*(1 + exp(-(x - mu)/s))**2) + + def _cdf(self, x): + mu, s = self.mu, self.s + return S.One/(1 + exp(-(x - mu)/s)) + + def _characteristic_function(self, t): + return Piecewise((exp(I*t*self.mu) * pi*self.s*t / sinh(pi*self.s*t), Ne(t, 0)), (S.One, True)) + + def _moment_generating_function(self, t): + return exp(self.mu*t) * beta_fn(1 - self.s*t, 1 + self.s*t) + + def _quantile(self, p): + return self.mu - self.s*log(-S.One + S.One/p) + +def Logistic(name, mu, s): + r""" + Create a continuous random variable with a logistic distribution. + + Explanation + =========== + + The density of the logistic distribution is given by + + .. math:: + f(x) := \frac{e^{-(x-\mu)/s}} {s\left(1+e^{-(x-\mu)/s}\right)^2} + + Parameters + ========== + + mu : Real number, the location (mean) + s : Real number, `s > 0`, a scale + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Logistic, density, cdf + >>> from sympy import Symbol + + >>> mu = Symbol("mu", real=True) + >>> s = Symbol("s", positive=True) + >>> z = Symbol("z") + + >>> X = Logistic("x", mu, s) + + >>> density(X)(z) + exp((mu - z)/s)/(s*(exp((mu - z)/s) + 1)**2) + + >>> cdf(X)(z) + 1/(exp((mu - z)/s) + 1) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Logistic_distribution + .. [2] https://mathworld.wolfram.com/LogisticDistribution.html + + """ + + return rv(name, LogisticDistribution, (mu, s)) + +#------------------------------------------------------------------------------- +# Log-logistic distribution -------------------------------------------------------- + + +class LogLogisticDistribution(SingleContinuousDistribution): + _argnames = ('alpha', 'beta') + + set = Interval(0, oo) + + @staticmethod + def check(alpha, beta): + _value_check(alpha > 0, "Scale parameter Alpha must be positive.") + _value_check(beta > 0, "Shape parameter Beta must be positive.") + + def pdf(self, x): + a, b = self.alpha, self.beta + return ((b/a)*(x/a)**(b - 1))/(1 + (x/a)**b)**2 + + def _cdf(self, x): + a, b = self.alpha, self.beta + return 1/(1 + (x/a)**(-b)) + + def _quantile(self, p): + a, b = self.alpha, self.beta + return a*((p/(1 - p))**(1/b)) + + def expectation(self, expr, var, **kwargs): + a, b = self.args + return Piecewise((S.NaN, b <= 1), (pi*a/(b*sin(pi/b)), True)) + +def LogLogistic(name, alpha, beta): + r""" + Create a continuous random variable with a log-logistic distribution. + The distribution is unimodal when ``beta > 1``. + + Explanation + =========== + + The density of the log-logistic distribution is given by + + .. math:: + f(x) := \frac{(\frac{\beta}{\alpha})(\frac{x}{\alpha})^{\beta - 1}} + {(1 + (\frac{x}{\alpha})^{\beta})^2} + + Parameters + ========== + + alpha : Real number, `\alpha > 0`, scale parameter and median of distribution + beta : Real number, `\beta > 0`, a shape parameter + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import LogLogistic, density, cdf, quantile + >>> from sympy import Symbol, pprint + + >>> alpha = Symbol("alpha", positive=True) + >>> beta = Symbol("beta", positive=True) + >>> p = Symbol("p") + >>> z = Symbol("z", positive=True) + + >>> X = LogLogistic("x", alpha, beta) + + >>> D = density(X)(z) + >>> pprint(D, use_unicode=False) + beta - 1 + / z \ + beta*|-----| + \alpha/ + ------------------------ + 2 + / beta \ + |/ z \ | + alpha*||-----| + 1| + \\alpha/ / + + >>> cdf(X)(z) + 1/(1 + (z/alpha)**(-beta)) + + >>> quantile(X)(p) + alpha*(p/(1 - p))**(1/beta) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Log-logistic_distribution + + """ + + return rv(name, LogLogisticDistribution, (alpha, beta)) + +#------------------------------------------------------------------------------- +#Logit-Normal distribution------------------------------------------------------ + +class LogitNormalDistribution(SingleContinuousDistribution): + _argnames = ('mu', 's') + set = Interval.open(0, 1) + + @staticmethod + def check(mu, s): + _value_check((s ** 2).is_real is not False and s ** 2 > 0, "Squared scale parameter s must be positive.") + _value_check(mu.is_real is not False, "Location parameter must be real") + + def _logit(self, x): + return log(x / (1 - x)) + + def pdf(self, x): + mu, s = self.mu, self.s + return exp(-(self._logit(x) - mu)**2/(2*s**2))*(S.One/sqrt(2*pi*(s**2)))*(1/(x*(1 - x))) + + def _cdf(self, x): + mu, s = self.mu, self.s + return (S.One/2)*(1 + erf((self._logit(x) - mu)/(sqrt(2*s**2)))) + + +def LogitNormal(name, mu, s): + r""" + Create a continuous random variable with a Logit-Normal distribution. + + The density of the logistic distribution is given by + + .. math:: + f(x) := \frac{1}{s \sqrt{2 \pi}} \frac{1}{x(1 - x)} e^{- \frac{(logit(x) - \mu)^2}{s^2}} + where logit(x) = \log(\frac{x}{1 - x}) + Parameters + ========== + + mu : Real number, the location (mean) + s : Real number, `s > 0`, a scale + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import LogitNormal, density, cdf + >>> from sympy import Symbol,pprint + + >>> mu = Symbol("mu", real=True) + >>> s = Symbol("s", positive=True) + >>> z = Symbol("z") + >>> X = LogitNormal("x",mu,s) + + >>> D = density(X)(z) + >>> pprint(D, use_unicode=False) + 2 + / / z \\ + -|-mu + log|-----|| + \ \1 - z// + --------------------- + 2 + ___ 2*s + \/ 2 *e + ---------------------------- + ____ + 2*\/ pi *s*z*(1 - z) + + >>> density(X)(z) + sqrt(2)*exp(-(-mu + log(z/(1 - z)))**2/(2*s**2))/(2*sqrt(pi)*s*z*(1 - z)) + + >>> cdf(X)(z) + erf(sqrt(2)*(-mu + log(z/(1 - z)))/(2*s))/2 + 1/2 + + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Logit-normal_distribution + + """ + + return rv(name, LogitNormalDistribution, (mu, s)) + +#------------------------------------------------------------------------------- +# Log Normal distribution ------------------------------------------------------ + + +class LogNormalDistribution(SingleContinuousDistribution): + _argnames = ('mean', 'std') + + set = Interval(0, oo) + + @staticmethod + def check(mean, std): + _value_check(std > 0, "Parameter std must be positive.") + + def pdf(self, x): + mean, std = self.mean, self.std + return exp(-(log(x) - mean)**2 / (2*std**2)) / (x*sqrt(2*pi)*std) + + def _cdf(self, x): + mean, std = self.mean, self.std + return Piecewise( + (S.Half + S.Half*erf((log(x) - mean)/sqrt(2)/std), x > 0), + (S.Zero, True) + ) + + def _moment_generating_function(self, t): + raise NotImplementedError('Moment generating function of the log-normal distribution is not defined.') + + +def LogNormal(name, mean, std): + r""" + Create a continuous random variable with a log-normal distribution. + + Explanation + =========== + + The density of the log-normal distribution is given by + + .. math:: + f(x) := \frac{1}{x\sqrt{2\pi\sigma^2}} + e^{-\frac{\left(\ln x-\mu\right)^2}{2\sigma^2}} + + with :math:`x \geq 0`. + + Parameters + ========== + + mu : Real number + The log-scale. + sigma : Real number + A shape. ($\sigma^2 > 0$) + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import LogNormal, density + >>> from sympy import Symbol, pprint + + >>> mu = Symbol("mu", real=True) + >>> sigma = Symbol("sigma", positive=True) + >>> z = Symbol("z") + + >>> X = LogNormal("x", mu, sigma) + + >>> D = density(X)(z) + >>> pprint(D, use_unicode=False) + 2 + -(-mu + log(z)) + ----------------- + 2 + ___ 2*sigma + \/ 2 *e + ------------------------ + ____ + 2*\/ pi *sigma*z + + + >>> X = LogNormal('x', 0, 1) # Mean 0, standard deviation 1 + + >>> density(X)(z) + sqrt(2)*exp(-log(z)**2/2)/(2*sqrt(pi)*z) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Lognormal + .. [2] https://mathworld.wolfram.com/LogNormalDistribution.html + + """ + + return rv(name, LogNormalDistribution, (mean, std)) + +#------------------------------------------------------------------------------- +# Lomax Distribution ----------------------------------------------------------- + +class LomaxDistribution(SingleContinuousDistribution): + _argnames = ('alpha', 'lamda',) + set = Interval(0, oo) + + @staticmethod + def check(alpha, lamda): + _value_check(alpha.is_real, "Shape parameter should be real.") + _value_check(lamda.is_real, "Scale parameter should be real.") + _value_check(alpha.is_positive, "Shape parameter should be positive.") + _value_check(lamda.is_positive, "Scale parameter should be positive.") + + def pdf(self, x): + lamba, alpha = self.lamda, self.alpha + return (alpha/lamba) * (S.One + x/lamba)**(-alpha-1) + +def Lomax(name, alpha, lamda): + r""" + Create a continuous random variable with a Lomax distribution. + + Explanation + =========== + + The density of the Lomax distribution is given by + + .. math:: + f(x) := \frac{\alpha}{\lambda}\left[1+\frac{x}{\lambda}\right]^{-(\alpha+1)} + + Parameters + ========== + + alpha : Real Number, `\alpha > 0` + Shape parameter + lamda : Real Number, `\lambda > 0` + Scale parameter + + Examples + ======== + + >>> from sympy.stats import Lomax, density, cdf, E + >>> from sympy import symbols + >>> a, l = symbols('a, l', positive=True) + >>> X = Lomax('X', a, l) + >>> x = symbols('x') + >>> density(X)(x) + a*(1 + x/l)**(-a - 1)/l + >>> cdf(X)(x) + Piecewise((1 - 1/(1 + x/l)**a, x >= 0), (0, True)) + >>> a = 2 + >>> X = Lomax('X', a, l) + >>> E(X) + l + + Returns + ======= + + RandomSymbol + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Lomax_distribution + + """ + return rv(name, LomaxDistribution, (alpha, lamda)) + +#------------------------------------------------------------------------------- +# Maxwell distribution --------------------------------------------------------- + + +class MaxwellDistribution(SingleContinuousDistribution): + _argnames = ('a',) + + set = Interval(0, oo) + + @staticmethod + def check(a): + _value_check(a > 0, "Parameter a must be positive.") + + def pdf(self, x): + a = self.a + return sqrt(2/pi)*x**2*exp(-x**2/(2*a**2))/a**3 + + def _cdf(self, x): + a = self.a + return erf(sqrt(2)*x/(2*a)) - sqrt(2)*x*exp(-x**2/(2*a**2))/(sqrt(pi)*a) + +def Maxwell(name, a): + r""" + Create a continuous random variable with a Maxwell distribution. + + Explanation + =========== + + The density of the Maxwell distribution is given by + + .. math:: + f(x) := \sqrt{\frac{2}{\pi}} \frac{x^2 e^{-x^2/(2a^2)}}{a^3} + + with :math:`x \geq 0`. + + .. TODO - what does the parameter mean? + + Parameters + ========== + + a : Real number, `a > 0` + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Maxwell, density, E, variance + >>> from sympy import Symbol, simplify + + >>> a = Symbol("a", positive=True) + >>> z = Symbol("z") + + >>> X = Maxwell("x", a) + + >>> density(X)(z) + sqrt(2)*z**2*exp(-z**2/(2*a**2))/(sqrt(pi)*a**3) + + >>> E(X) + 2*sqrt(2)*a/sqrt(pi) + + >>> simplify(variance(X)) + a**2*(-8 + 3*pi)/pi + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Maxwell_distribution + .. [2] https://mathworld.wolfram.com/MaxwellDistribution.html + + """ + + return rv(name, MaxwellDistribution, (a, )) + +#------------------------------------------------------------------------------- +# Moyal Distribution ----------------------------------------------------------- +class MoyalDistribution(SingleContinuousDistribution): + _argnames = ('mu', 'sigma') + + @staticmethod + def check(mu, sigma): + _value_check(mu.is_real, "Location parameter must be real.") + _value_check(sigma.is_real and sigma > 0, "Scale parameter must be real\ + and positive.") + + def pdf(self, x): + mu, sigma = self.mu, self.sigma + num = exp(-(exp(-(x - mu)/sigma) + (x - mu)/(sigma))/2) + den = (sqrt(2*pi) * sigma) + return num/den + + def _characteristic_function(self, t): + mu, sigma = self.mu, self.sigma + term1 = exp(I*t*mu) + term2 = (2**(-I*sigma*t) * gamma(Rational(1, 2) - I*t*sigma)) + return (term1 * term2)/sqrt(pi) + + def _moment_generating_function(self, t): + mu, sigma = self.mu, self.sigma + term1 = exp(t*mu) + term2 = (2**(-1*sigma*t) * gamma(Rational(1, 2) - t*sigma)) + return (term1 * term2)/sqrt(pi) + +def Moyal(name, mu, sigma): + r""" + Create a continuous random variable with a Moyal distribution. + + Explanation + =========== + + The density of the Moyal distribution is given by + + .. math:: + f(x) := \frac{\exp-\frac{1}{2}\exp-\frac{x-\mu}{\sigma}-\frac{x-\mu}{2\sigma}}{\sqrt{2\pi}\sigma} + + with :math:`x \in \mathbb{R}`. + + Parameters + ========== + + mu : Real number + Location parameter + sigma : Real positive number + Scale parameter + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Moyal, density, cdf + >>> from sympy import Symbol, simplify + >>> mu = Symbol("mu", real=True) + >>> sigma = Symbol("sigma", positive=True, real=True) + >>> z = Symbol("z") + >>> X = Moyal("x", mu, sigma) + >>> density(X)(z) + sqrt(2)*exp(-exp((mu - z)/sigma)/2 - (-mu + z)/(2*sigma))/(2*sqrt(pi)*sigma) + >>> simplify(cdf(X)(z)) + 1 - erf(sqrt(2)*exp((mu - z)/(2*sigma))/2) + + References + ========== + + .. [1] https://reference.wolfram.com/language/ref/MoyalDistribution.html + .. [2] https://www.stat.rice.edu/~dobelman/textfiles/DistributionsHandbook.pdf + + """ + + return rv(name, MoyalDistribution, (mu, sigma)) + +#------------------------------------------------------------------------------- +# Nakagami distribution -------------------------------------------------------- + + +class NakagamiDistribution(SingleContinuousDistribution): + _argnames = ('mu', 'omega') + + set = Interval(0, oo) + + @staticmethod + def check(mu, omega): + _value_check(mu >= S.Half, "Shape parameter mu must be greater than equal to 1/2.") + _value_check(omega > 0, "Spread parameter omega must be positive.") + + def pdf(self, x): + mu, omega = self.mu, self.omega + return 2*mu**mu/(gamma(mu)*omega**mu)*x**(2*mu - 1)*exp(-mu/omega*x**2) + + def _cdf(self, x): + mu, omega = self.mu, self.omega + return Piecewise( + (lowergamma(mu, (mu/omega)*x**2)/gamma(mu), x > 0), + (S.Zero, True)) + +def Nakagami(name, mu, omega): + r""" + Create a continuous random variable with a Nakagami distribution. + + Explanation + =========== + + The density of the Nakagami distribution is given by + + .. math:: + f(x) := \frac{2\mu^\mu}{\Gamma(\mu)\omega^\mu} x^{2\mu-1} + \exp\left(-\frac{\mu}{\omega}x^2 \right) + + with :math:`x > 0`. + + Parameters + ========== + + mu : Real number, `\mu \geq \frac{1}{2}`, a shape + omega : Real number, `\omega > 0`, the spread + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Nakagami, density, E, variance, cdf + >>> from sympy import Symbol, simplify, pprint + + >>> mu = Symbol("mu", positive=True) + >>> omega = Symbol("omega", positive=True) + >>> z = Symbol("z") + + >>> X = Nakagami("x", mu, omega) + + >>> D = density(X)(z) + >>> pprint(D, use_unicode=False) + 2 + -mu*z + ------- + mu -mu 2*mu - 1 omega + 2*mu *omega *z *e + ---------------------------------- + Gamma(mu) + + >>> simplify(E(X)) + sqrt(mu)*sqrt(omega)*gamma(mu + 1/2)/gamma(mu + 1) + + >>> V = simplify(variance(X)) + >>> pprint(V, use_unicode=False) + 2 + omega*Gamma (mu + 1/2) + omega - ----------------------- + Gamma(mu)*Gamma(mu + 1) + + >>> cdf(X)(z) + Piecewise((lowergamma(mu, mu*z**2/omega)/gamma(mu), z > 0), + (0, True)) + + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Nakagami_distribution + + """ + + return rv(name, NakagamiDistribution, (mu, omega)) + +#------------------------------------------------------------------------------- +# Normal distribution ---------------------------------------------------------- + + +class NormalDistribution(SingleContinuousDistribution): + _argnames = ('mean', 'std') + + @staticmethod + def check(mean, std): + _value_check(std > 0, "Standard deviation must be positive") + + def pdf(self, x): + return exp(-(x - self.mean)**2 / (2*self.std**2)) / (sqrt(2*pi)*self.std) + + def _cdf(self, x): + mean, std = self.mean, self.std + return erf(sqrt(2)*(-mean + x)/(2*std))/2 + S.Half + + def _characteristic_function(self, t): + mean, std = self.mean, self.std + return exp(I*mean*t - std**2*t**2/2) + + def _moment_generating_function(self, t): + mean, std = self.mean, self.std + return exp(mean*t + std**2*t**2/2) + + def _quantile(self, p): + mean, std = self.mean, self.std + return mean + std*sqrt(2)*erfinv(2*p - 1) + + +def Normal(name, mean, std): + r""" + Create a continuous random variable with a Normal distribution. + + Explanation + =========== + + The density of the Normal distribution is given by + + .. math:: + f(x) := \frac{1}{\sigma\sqrt{2\pi}} e^{ -\frac{(x-\mu)^2}{2\sigma^2} } + + Parameters + ========== + + mu : Real number or a list representing the mean or the mean vector + sigma : Real number or a positive definite square matrix, + :math:`\sigma^2 > 0`, the variance + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Normal, density, E, std, cdf, skewness, quantile, marginal_distribution + >>> from sympy import Symbol, simplify, pprint + + >>> mu = Symbol("mu") + >>> sigma = Symbol("sigma", positive=True) + >>> z = Symbol("z") + >>> y = Symbol("y") + >>> p = Symbol("p") + >>> X = Normal("x", mu, sigma) + + >>> density(X)(z) + sqrt(2)*exp(-(-mu + z)**2/(2*sigma**2))/(2*sqrt(pi)*sigma) + + >>> C = simplify(cdf(X))(z) # it needs a little more help... + >>> pprint(C, use_unicode=False) + / ___ \ + |\/ 2 *(-mu + z)| + erf|---------------| + \ 2*sigma / 1 + -------------------- + - + 2 2 + + >>> quantile(X)(p) + mu + sqrt(2)*sigma*erfinv(2*p - 1) + + >>> simplify(skewness(X)) + 0 + + >>> X = Normal("x", 0, 1) # Mean 0, standard deviation 1 + >>> density(X)(z) + sqrt(2)*exp(-z**2/2)/(2*sqrt(pi)) + + >>> E(2*X + 1) + 1 + + >>> simplify(std(2*X + 1)) + 2 + + >>> m = Normal('X', [1, 2], [[2, 1], [1, 2]]) + >>> pprint(density(m)(y, z), use_unicode=False) + 2 2 + y y*z z + - -- + --- - -- + z - 1 + ___ 3 3 3 + \/ 3 *e + ------------------------------ + 6*pi + + >>> marginal_distribution(m, m[0])(1) + 1/(2*sqrt(pi)) + + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Normal_distribution + .. [2] https://mathworld.wolfram.com/NormalDistributionFunction.html + + """ + + if isinstance(mean, list) or getattr(mean, 'is_Matrix', False) and\ + isinstance(std, list) or getattr(std, 'is_Matrix', False): + from sympy.stats.joint_rv_types import MultivariateNormal + return MultivariateNormal(name, mean, std) + return rv(name, NormalDistribution, (mean, std)) + + +#------------------------------------------------------------------------------- +# Inverse Gaussian distribution ---------------------------------------------------------- + + +class GaussianInverseDistribution(SingleContinuousDistribution): + _argnames = ('mean', 'shape') + + @property + def set(self): + return Interval(0, oo) + + @staticmethod + def check(mean, shape): + _value_check(shape > 0, "Shape parameter must be positive") + _value_check(mean > 0, "Mean must be positive") + + def pdf(self, x): + mu, s = self.mean, self.shape + return exp(-s*(x - mu)**2 / (2*x*mu**2)) * sqrt(s/(2*pi*x**3)) + + def _cdf(self, x): + from sympy.stats import cdf + mu, s = self.mean, self.shape + stdNormalcdf = cdf(Normal('x', 0, 1)) + + first_term = stdNormalcdf(sqrt(s/x) * ((x/mu) - S.One)) + second_term = exp(2*s/mu) * stdNormalcdf(-sqrt(s/x)*(x/mu + S.One)) + + return first_term + second_term + + def _characteristic_function(self, t): + mu, s = self.mean, self.shape + return exp((s/mu)*(1 - sqrt(1 - (2*mu**2*I*t)/s))) + + def _moment_generating_function(self, t): + mu, s = self.mean, self.shape + return exp((s/mu)*(1 - sqrt(1 - (2*mu**2*t)/s))) + + +def GaussianInverse(name, mean, shape): + r""" + Create a continuous random variable with an Inverse Gaussian distribution. + Inverse Gaussian distribution is also known as Wald distribution. + + Explanation + =========== + + The density of the Inverse Gaussian distribution is given by + + .. math:: + f(x) := \sqrt{\frac{\lambda}{2\pi x^3}} e^{-\frac{\lambda(x-\mu)^2}{2x\mu^2}} + + Parameters + ========== + + mu : + Positive number representing the mean. + lambda : + Positive number representing the shape parameter. + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import GaussianInverse, density, E, std, skewness + >>> from sympy import Symbol, pprint + + >>> mu = Symbol("mu", positive=True) + >>> lamda = Symbol("lambda", positive=True) + >>> z = Symbol("z", positive=True) + >>> X = GaussianInverse("x", mu, lamda) + + >>> D = density(X)(z) + >>> pprint(D, use_unicode=False) + 2 + -lambda*(-mu + z) + ------------------- + 2 + ___ ________ 2*mu *z + \/ 2 *\/ lambda *e + ------------------------------------- + ____ 3/2 + 2*\/ pi *z + + >>> E(X) + mu + + >>> std(X).expand() + mu**(3/2)/sqrt(lambda) + + >>> skewness(X).expand() + 3*sqrt(mu)/sqrt(lambda) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution + .. [2] https://mathworld.wolfram.com/InverseGaussianDistribution.html + + """ + + return rv(name, GaussianInverseDistribution, (mean, shape)) + +Wald = GaussianInverse + +#------------------------------------------------------------------------------- +# Pareto distribution ---------------------------------------------------------- + + +class ParetoDistribution(SingleContinuousDistribution): + _argnames = ('xm', 'alpha') + + @property + def set(self): + return Interval(self.xm, oo) + + @staticmethod + def check(xm, alpha): + _value_check(xm > 0, "Xm must be positive") + _value_check(alpha > 0, "Alpha must be positive") + + def pdf(self, x): + xm, alpha = self.xm, self.alpha + return alpha * xm**alpha / x**(alpha + 1) + + def _cdf(self, x): + xm, alpha = self.xm, self.alpha + return Piecewise( + (S.One - xm**alpha/x**alpha, x>=xm), + (0, True), + ) + + def _moment_generating_function(self, t): + xm, alpha = self.xm, self.alpha + return alpha * (-xm*t)**alpha * uppergamma(-alpha, -xm*t) + + def _characteristic_function(self, t): + xm, alpha = self.xm, self.alpha + return alpha * (-I * xm * t) ** alpha * uppergamma(-alpha, -I * xm * t) + + +def Pareto(name, xm, alpha): + r""" + Create a continuous random variable with the Pareto distribution. + + Explanation + =========== + + The density of the Pareto distribution is given by + + .. math:: + f(x) := \frac{\alpha\,x_m^\alpha}{x^{\alpha+1}} + + with :math:`x \in [x_m,\infty]`. + + Parameters + ========== + + xm : Real number, `x_m > 0`, a scale + alpha : Real number, `\alpha > 0`, a shape + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Pareto, density + >>> from sympy import Symbol + + >>> xm = Symbol("xm", positive=True) + >>> beta = Symbol("beta", positive=True) + >>> z = Symbol("z") + + >>> X = Pareto("x", xm, beta) + + >>> density(X)(z) + beta*xm**beta*z**(-beta - 1) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Pareto_distribution + .. [2] https://mathworld.wolfram.com/ParetoDistribution.html + + """ + + return rv(name, ParetoDistribution, (xm, alpha)) + +#------------------------------------------------------------------------------- +# PowerFunction distribution --------------------------------------------------- + + +class PowerFunctionDistribution(SingleContinuousDistribution): + _argnames=('alpha','a','b') + + @property + def set(self): + return Interval(self.a, self.b) + + @staticmethod + def check(alpha, a, b): + _value_check(a.is_real, "Continuous Boundary parameter should be real.") + _value_check(b.is_real, "Continuous Boundary parameter should be real.") + _value_check(a < b, " 'a' the left Boundary must be smaller than 'b' the right Boundary." ) + _value_check(alpha.is_positive, "Continuous Shape parameter should be positive.") + + def pdf(self, x): + alpha, a, b = self.alpha, self.a, self.b + num = alpha*(x - a)**(alpha - 1) + den = (b - a)**alpha + return num/den + +def PowerFunction(name, alpha, a, b): + r""" + Creates a continuous random variable with a Power Function Distribution. + + Explanation + =========== + + The density of PowerFunction distribution is given by + + .. math:: + f(x) := \frac{{\alpha}(x - a)^{\alpha - 1}}{(b - a)^{\alpha}} + + with :math:`x \in [a,b]`. + + Parameters + ========== + + alpha : Positive number, `0 < \alpha`, the shape parameter + a : Real number, :math:`-\infty < a`, the left boundary + b : Real number, :math:`a < b < \infty`, the right boundary + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import PowerFunction, density, cdf, E, variance + >>> from sympy import Symbol + >>> alpha = Symbol("alpha", positive=True) + >>> a = Symbol("a", real=True) + >>> b = Symbol("b", real=True) + >>> z = Symbol("z") + + >>> X = PowerFunction("X", 2, a, b) + + >>> density(X)(z) + (-2*a + 2*z)/(-a + b)**2 + + >>> cdf(X)(z) + Piecewise((a**2/(a**2 - 2*a*b + b**2) - 2*a*z/(a**2 - 2*a*b + b**2) + + z**2/(a**2 - 2*a*b + b**2), a <= z), (0, True)) + + >>> alpha = 2 + >>> a = 0 + >>> b = 1 + >>> Y = PowerFunction("Y", alpha, a, b) + + >>> E(Y) + 2/3 + + >>> variance(Y) + 1/18 + + References + ========== + + .. [1] https://web.archive.org/web/20200204081320/http://www.mathwave.com/help/easyfit/html/analyses/distributions/power_func.html + + """ + return rv(name, PowerFunctionDistribution, (alpha, a, b)) + +#------------------------------------------------------------------------------- +# QuadraticU distribution ------------------------------------------------------ + + +class QuadraticUDistribution(SingleContinuousDistribution): + _argnames = ('a', 'b') + + @property + def set(self): + return Interval(self.a, self.b) + + @staticmethod + def check(a, b): + _value_check(b > a, "Parameter b must be in range (%s, oo)."%(a)) + + def pdf(self, x): + a, b = self.a, self.b + alpha = 12 / (b-a)**3 + beta = (a+b) / 2 + return Piecewise( + (alpha * (x-beta)**2, And(a<=x, x<=b)), + (S.Zero, True)) + + def _moment_generating_function(self, t): + a, b = self.a, self.b + return -3 * (exp(a*t) * (4 + (a**2 + 2*a*(-2 + b) + b**2) * t) \ + - exp(b*t) * (4 + (-4*b + (a + b)**2) * t)) / ((a-b)**3 * t**2) + + def _characteristic_function(self, t): + a, b = self.a, self.b + return -3*I*(exp(I*a*t*exp(I*b*t)) * (4*I - (-4*b + (a+b)**2)*t)) \ + / ((a-b)**3 * t**2) + + +def QuadraticU(name, a, b): + r""" + Create a Continuous Random Variable with a U-quadratic distribution. + + Explanation + =========== + + The density of the U-quadratic distribution is given by + + .. math:: + f(x) := \alpha (x-\beta)^2 + + with :math:`x \in [a,b]`. + + Parameters + ========== + + a : Real number + b : Real number, :math:`a < b` + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import QuadraticU, density + >>> from sympy import Symbol, pprint + + >>> a = Symbol("a", real=True) + >>> b = Symbol("b", real=True) + >>> z = Symbol("z") + + >>> X = QuadraticU("x", a, b) + + >>> D = density(X)(z) + >>> pprint(D, use_unicode=False) + / 2 + | / a b \ + |12*|- - - - + z| + | \ 2 2 / + <----------------- for And(b >= z, a <= z) + | 3 + | (-a + b) + | + \ 0 otherwise + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/U-quadratic_distribution + + """ + + return rv(name, QuadraticUDistribution, (a, b)) + +#------------------------------------------------------------------------------- +# RaisedCosine distribution ---------------------------------------------------- + + +class RaisedCosineDistribution(SingleContinuousDistribution): + _argnames = ('mu', 's') + + @property + def set(self): + return Interval(self.mu - self.s, self.mu + self.s) + + @staticmethod + def check(mu, s): + _value_check(s > 0, "s must be positive") + + def pdf(self, x): + mu, s = self.mu, self.s + return Piecewise( + ((1+cos(pi*(x-mu)/s)) / (2*s), And(mu-s<=x, x<=mu+s)), + (S.Zero, True)) + + def _characteristic_function(self, t): + mu, s = self.mu, self.s + return Piecewise((exp(-I*pi*mu/s)/2, Eq(t, -pi/s)), + (exp(I*pi*mu/s)/2, Eq(t, pi/s)), + (pi**2*sin(s*t)*exp(I*mu*t) / (s*t*(pi**2 - s**2*t**2)), True)) + + def _moment_generating_function(self, t): + mu, s = self.mu, self.s + return pi**2 * sinh(s*t) * exp(mu*t) / (s*t*(pi**2 + s**2*t**2)) + +def RaisedCosine(name, mu, s): + r""" + Create a Continuous Random Variable with a raised cosine distribution. + + Explanation + =========== + + The density of the raised cosine distribution is given by + + .. math:: + f(x) := \frac{1}{2s}\left(1+\cos\left(\frac{x-\mu}{s}\pi\right)\right) + + with :math:`x \in [\mu-s,\mu+s]`. + + Parameters + ========== + + mu : Real number + s : Real number, `s > 0` + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import RaisedCosine, density + >>> from sympy import Symbol, pprint + + >>> mu = Symbol("mu", real=True) + >>> s = Symbol("s", positive=True) + >>> z = Symbol("z") + + >>> X = RaisedCosine("x", mu, s) + + >>> D = density(X)(z) + >>> pprint(D, use_unicode=False) + / /pi*(-mu + z)\ + |cos|------------| + 1 + | \ s / + <--------------------- for And(z >= mu - s, z <= mu + s) + | 2*s + | + \ 0 otherwise + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Raised_cosine_distribution + + """ + + return rv(name, RaisedCosineDistribution, (mu, s)) + +#------------------------------------------------------------------------------- +# Rayleigh distribution -------------------------------------------------------- + + +class RayleighDistribution(SingleContinuousDistribution): + _argnames = ('sigma',) + + set = Interval(0, oo) + + @staticmethod + def check(sigma): + _value_check(sigma > 0, "Scale parameter sigma must be positive.") + + def pdf(self, x): + sigma = self.sigma + return x/sigma**2*exp(-x**2/(2*sigma**2)) + + def _cdf(self, x): + sigma = self.sigma + return 1 - exp(-(x**2/(2*sigma**2))) + + def _characteristic_function(self, t): + sigma = self.sigma + return 1 - sigma*t*exp(-sigma**2*t**2/2) * sqrt(pi/2) * (erfi(sigma*t/sqrt(2)) - I) + + def _moment_generating_function(self, t): + sigma = self.sigma + return 1 + sigma*t*exp(sigma**2*t**2/2) * sqrt(pi/2) * (erf(sigma*t/sqrt(2)) + 1) + + +def Rayleigh(name, sigma): + r""" + Create a continuous random variable with a Rayleigh distribution. + + Explanation + =========== + + The density of the Rayleigh distribution is given by + + .. math :: + f(x) := \frac{x}{\sigma^2} e^{-x^2/2\sigma^2} + + with :math:`x > 0`. + + Parameters + ========== + + sigma : Real number, `\sigma > 0` + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Rayleigh, density, E, variance + >>> from sympy import Symbol + + >>> sigma = Symbol("sigma", positive=True) + >>> z = Symbol("z") + + >>> X = Rayleigh("x", sigma) + + >>> density(X)(z) + z*exp(-z**2/(2*sigma**2))/sigma**2 + + >>> E(X) + sqrt(2)*sqrt(pi)*sigma/2 + + >>> variance(X) + -pi*sigma**2/2 + 2*sigma**2 + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Rayleigh_distribution + .. [2] https://mathworld.wolfram.com/RayleighDistribution.html + + """ + + return rv(name, RayleighDistribution, (sigma, )) + +#------------------------------------------------------------------------------- +# Reciprocal distribution -------------------------------------------------------- + +class ReciprocalDistribution(SingleContinuousDistribution): + _argnames = ('a', 'b') + + @property + def set(self): + return Interval(self.a, self.b) + + @staticmethod + def check(a, b): + _value_check(a > 0, "Parameter > 0. a = %s"%a) + _value_check((a < b), + "Parameter b must be in range (%s, +oo]. b = %s"%(a, b)) + + def pdf(self, x): + a, b = self.a, self.b + return 1/(x*(log(b) - log(a))) + + +def Reciprocal(name, a, b): + r"""Creates a continuous random variable with a reciprocal distribution. + + + Parameters + ========== + + a : Real number, :math:`0 < a` + b : Real number, :math:`a < b` + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Reciprocal, density, cdf + >>> from sympy import symbols + >>> a, b, x = symbols('a, b, x', positive=True) + >>> R = Reciprocal('R', a, b) + + >>> density(R)(x) + 1/(x*(-log(a) + log(b))) + >>> cdf(R)(x) + Piecewise((log(a)/(log(a) - log(b)) - log(x)/(log(a) - log(b)), a <= x), (0, True)) + + Reference + ========= + + .. [1] https://en.wikipedia.org/wiki/Reciprocal_distribution + + """ + return rv(name, ReciprocalDistribution, (a, b)) + + +#------------------------------------------------------------------------------- +# Shifted Gompertz distribution ------------------------------------------------ + + +class ShiftedGompertzDistribution(SingleContinuousDistribution): + _argnames = ('b', 'eta') + + set = Interval(0, oo) + + @staticmethod + def check(b, eta): + _value_check(b > 0, "b must be positive") + _value_check(eta > 0, "eta must be positive") + + def pdf(self, x): + b, eta = self.b, self.eta + return b*exp(-b*x)*exp(-eta*exp(-b*x))*(1+eta*(1-exp(-b*x))) + +def ShiftedGompertz(name, b, eta): + r""" + Create a continuous random variable with a Shifted Gompertz distribution. + + Explanation + =========== + + The density of the Shifted Gompertz distribution is given by + + .. math:: + f(x) := b e^{-b x} e^{-\eta \exp(-b x)} \left[1 + \eta(1 - e^(-bx)) \right] + + with :math:`x \in [0, \infty)`. + + Parameters + ========== + + b : Real number, `b > 0`, a scale + eta : Real number, `\eta > 0`, a shape + + Returns + ======= + + RandomSymbol + + Examples + ======== + >>> from sympy.stats import ShiftedGompertz, density + >>> from sympy import Symbol + + >>> b = Symbol("b", positive=True) + >>> eta = Symbol("eta", positive=True) + >>> x = Symbol("x") + + >>> X = ShiftedGompertz("x", b, eta) + + >>> density(X)(x) + b*(eta*(1 - exp(-b*x)) + 1)*exp(-b*x)*exp(-eta*exp(-b*x)) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Shifted_Gompertz_distribution + + """ + return rv(name, ShiftedGompertzDistribution, (b, eta)) + +#------------------------------------------------------------------------------- +# StudentT distribution -------------------------------------------------------- + + +class StudentTDistribution(SingleContinuousDistribution): + _argnames = ('nu',) + + set = Interval(-oo, oo) + + @staticmethod + def check(nu): + _value_check(nu > 0, "Degrees of freedom nu must be positive.") + + def pdf(self, x): + nu = self.nu + return 1/(sqrt(nu)*beta_fn(S.Half, nu/2))*(1 + x**2/nu)**(-(nu + 1)/2) + + def _cdf(self, x): + nu = self.nu + return S.Half + x*gamma((nu+1)/2)*hyper((S.Half, (nu+1)/2), + (Rational(3, 2),), -x**2/nu)/(sqrt(pi*nu)*gamma(nu/2)) + + def _moment_generating_function(self, t): + raise NotImplementedError('The moment generating function for the Student-T distribution is undefined.') + + +def StudentT(name, nu): + r""" + Create a continuous random variable with a student's t distribution. + + Explanation + =========== + + The density of the student's t distribution is given by + + .. math:: + f(x) := \frac{\Gamma \left(\frac{\nu+1}{2} \right)} + {\sqrt{\nu\pi}\Gamma \left(\frac{\nu}{2} \right)} + \left(1+\frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}} + + Parameters + ========== + + nu : Real number, `\nu > 0`, the degrees of freedom + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import StudentT, density, cdf + >>> from sympy import Symbol, pprint + + >>> nu = Symbol("nu", positive=True) + >>> z = Symbol("z") + + >>> X = StudentT("x", nu) + + >>> D = density(X)(z) + >>> pprint(D, use_unicode=False) + nu 1 + - -- - - + 2 2 + / 2\ + | z | + |1 + --| + \ nu/ + ----------------- + ____ / nu\ + \/ nu *B|1/2, --| + \ 2 / + + >>> cdf(X)(z) + 1/2 + z*gamma(nu/2 + 1/2)*hyper((1/2, nu/2 + 1/2), (3/2,), + -z**2/nu)/(sqrt(pi)*sqrt(nu)*gamma(nu/2)) + + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Student_t-distribution + .. [2] https://mathworld.wolfram.com/Studentst-Distribution.html + + """ + + return rv(name, StudentTDistribution, (nu, )) + +#------------------------------------------------------------------------------- +# Trapezoidal distribution ------------------------------------------------------ + + +class TrapezoidalDistribution(SingleContinuousDistribution): + _argnames = ('a', 'b', 'c', 'd') + + @property + def set(self): + return Interval(self.a, self.d) + + @staticmethod + def check(a, b, c, d): + _value_check(a < d, "Lower bound parameter a < %s. a = %s"%(d, a)) + _value_check((a <= b, b < c), + "Level start parameter b must be in range [%s, %s). b = %s"%(a, c, b)) + _value_check((b < c, c <= d), + "Level end parameter c must be in range (%s, %s]. c = %s"%(b, d, c)) + _value_check(d >= c, "Upper bound parameter d > %s. d = %s"%(c, d)) + + def pdf(self, x): + a, b, c, d = self.a, self.b, self.c, self.d + return Piecewise( + (2*(x-a) / ((b-a)*(d+c-a-b)), And(a <= x, x < b)), + (2 / (d+c-a-b), And(b <= x, x < c)), + (2*(d-x) / ((d-c)*(d+c-a-b)), And(c <= x, x <= d)), + (S.Zero, True)) + +def Trapezoidal(name, a, b, c, d): + r""" + Create a continuous random variable with a trapezoidal distribution. + + Explanation + =========== + + The density of the trapezoidal distribution is given by + + .. math:: + f(x) := \begin{cases} + 0 & \mathrm{for\ } x < a, \\ + \frac{2(x-a)}{(b-a)(d+c-a-b)} & \mathrm{for\ } a \le x < b, \\ + \frac{2}{d+c-a-b} & \mathrm{for\ } b \le x < c, \\ + \frac{2(d-x)}{(d-c)(d+c-a-b)} & \mathrm{for\ } c \le x < d, \\ + 0 & \mathrm{for\ } d < x. + \end{cases} + + Parameters + ========== + + a : Real number, :math:`a < d` + b : Real number, :math:`a \le b < c` + c : Real number, :math:`b < c \le d` + d : Real number + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Trapezoidal, density + >>> from sympy import Symbol, pprint + + >>> a = Symbol("a") + >>> b = Symbol("b") + >>> c = Symbol("c") + >>> d = Symbol("d") + >>> z = Symbol("z") + + >>> X = Trapezoidal("x", a,b,c,d) + + >>> pprint(density(X)(z), use_unicode=False) + / -2*a + 2*z + |------------------------- for And(a <= z, b > z) + |(-a + b)*(-a - b + c + d) + | + | 2 + | -------------- for And(b <= z, c > z) + < -a - b + c + d + | + | 2*d - 2*z + |------------------------- for And(d >= z, c <= z) + |(-c + d)*(-a - b + c + d) + | + \ 0 otherwise + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Trapezoidal_distribution + + """ + return rv(name, TrapezoidalDistribution, (a, b, c, d)) + +#------------------------------------------------------------------------------- +# Triangular distribution ------------------------------------------------------ + + +class TriangularDistribution(SingleContinuousDistribution): + _argnames = ('a', 'b', 'c') + + @property + def set(self): + return Interval(self.a, self.b) + + @staticmethod + def check(a, b, c): + _value_check(b > a, "Parameter b > %s. b = %s"%(a, b)) + _value_check((a <= c, c <= b), + "Parameter c must be in range [%s, %s]. c = %s"%(a, b, c)) + + def pdf(self, x): + a, b, c = self.a, self.b, self.c + return Piecewise( + (2*(x - a)/((b - a)*(c - a)), And(a <= x, x < c)), + (2/(b - a), Eq(x, c)), + (2*(b - x)/((b - a)*(b - c)), And(c < x, x <= b)), + (S.Zero, True)) + + def _characteristic_function(self, t): + a, b, c = self.a, self.b, self.c + return -2 *((b-c) * exp(I*a*t) - (b-a) * exp(I*c*t) + (c-a) * exp(I*b*t)) / ((b-a)*(c-a)*(b-c)*t**2) + + def _moment_generating_function(self, t): + a, b, c = self.a, self.b, self.c + return 2 * ((b - c) * exp(a * t) - (b - a) * exp(c * t) + (c - a) * exp(b * t)) / ( + (b - a) * (c - a) * (b - c) * t ** 2) + + +def Triangular(name, a, b, c): + r""" + Create a continuous random variable with a triangular distribution. + + Explanation + =========== + + The density of the triangular distribution is given by + + .. math:: + f(x) := \begin{cases} + 0 & \mathrm{for\ } x < a, \\ + \frac{2(x-a)}{(b-a)(c-a)} & \mathrm{for\ } a \le x < c, \\ + \frac{2}{b-a} & \mathrm{for\ } x = c, \\ + \frac{2(b-x)}{(b-a)(b-c)} & \mathrm{for\ } c < x \le b, \\ + 0 & \mathrm{for\ } b < x. + \end{cases} + + Parameters + ========== + + a : Real number, :math:`a \in \left(-\infty, \infty\right)` + b : Real number, :math:`a < b` + c : Real number, :math:`a \leq c \leq b` + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Triangular, density + >>> from sympy import Symbol, pprint + + >>> a = Symbol("a") + >>> b = Symbol("b") + >>> c = Symbol("c") + >>> z = Symbol("z") + + >>> X = Triangular("x", a,b,c) + + >>> pprint(density(X)(z), use_unicode=False) + / -2*a + 2*z + |----------------- for And(a <= z, c > z) + |(-a + b)*(-a + c) + | + | 2 + | ------ for c = z + < -a + b + | + | 2*b - 2*z + |---------------- for And(b >= z, c < z) + |(-a + b)*(b - c) + | + \ 0 otherwise + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Triangular_distribution + .. [2] https://mathworld.wolfram.com/TriangularDistribution.html + + """ + + return rv(name, TriangularDistribution, (a, b, c)) + +#------------------------------------------------------------------------------- +# Uniform distribution --------------------------------------------------------- + + +class UniformDistribution(SingleContinuousDistribution): + _argnames = ('left', 'right') + + @property + def set(self): + return Interval(self.left, self.right) + + @staticmethod + def check(left, right): + _value_check(left < right, "Lower limit should be less than Upper limit.") + + def pdf(self, x): + left, right = self.left, self.right + return Piecewise( + (S.One/(right - left), And(left <= x, x <= right)), + (S.Zero, True) + ) + + def _cdf(self, x): + left, right = self.left, self.right + return Piecewise( + (S.Zero, x < left), + ((x - left)/(right - left), x <= right), + (S.One, True) + ) + + def _characteristic_function(self, t): + left, right = self.left, self.right + return Piecewise(((exp(I*t*right) - exp(I*t*left)) / (I*t*(right - left)), Ne(t, 0)), + (S.One, True)) + + def _moment_generating_function(self, t): + left, right = self.left, self.right + return Piecewise(((exp(t*right) - exp(t*left)) / (t * (right - left)), Ne(t, 0)), + (S.One, True)) + + def expectation(self, expr, var, **kwargs): + kwargs['evaluate'] = True + result = SingleContinuousDistribution.expectation(self, expr, var, **kwargs) + result = result.subs({Max(self.left, self.right): self.right, + Min(self.left, self.right): self.left}) + return result + + +def Uniform(name, left, right): + r""" + Create a continuous random variable with a uniform distribution. + + Explanation + =========== + + The density of the uniform distribution is given by + + .. math:: + f(x) := \begin{cases} + \frac{1}{b - a} & \text{for } x \in [a,b] \\ + 0 & \text{otherwise} + \end{cases} + + with :math:`x \in [a,b]`. + + Parameters + ========== + + a : Real number, :math:`-\infty < a`, the left boundary + b : Real number, :math:`a < b < \infty`, the right boundary + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Uniform, density, cdf, E, variance + >>> from sympy import Symbol, simplify + + >>> a = Symbol("a", negative=True) + >>> b = Symbol("b", positive=True) + >>> z = Symbol("z") + + >>> X = Uniform("x", a, b) + + >>> density(X)(z) + Piecewise((1/(-a + b), (b >= z) & (a <= z)), (0, True)) + + >>> cdf(X)(z) + Piecewise((0, a > z), ((-a + z)/(-a + b), b >= z), (1, True)) + + >>> E(X) + a/2 + b/2 + + >>> simplify(variance(X)) + a**2/12 - a*b/6 + b**2/12 + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29 + .. [2] https://mathworld.wolfram.com/UniformDistribution.html + + """ + + return rv(name, UniformDistribution, (left, right)) + +#------------------------------------------------------------------------------- +# UniformSum distribution ------------------------------------------------------ + + +class UniformSumDistribution(SingleContinuousDistribution): + _argnames = ('n',) + + @property + def set(self): + return Interval(0, self.n) + + @staticmethod + def check(n): + _value_check((n > 0, n.is_integer), + "Parameter n must be positive integer.") + + def pdf(self, x): + n = self.n + k = Dummy("k") + return 1/factorial( + n - 1)*Sum((-1)**k*binomial(n, k)*(x - k)**(n - 1), (k, 0, floor(x))) + + def _cdf(self, x): + n = self.n + k = Dummy("k") + return Piecewise((S.Zero, x < 0), + (1/factorial(n)*Sum((-1)**k*binomial(n, k)*(x - k)**(n), + (k, 0, floor(x))), x <= n), + (S.One, True)) + + def _characteristic_function(self, t): + return ((exp(I*t) - 1) / (I*t))**self.n + + def _moment_generating_function(self, t): + return ((exp(t) - 1) / t)**self.n + +def UniformSum(name, n): + r""" + Create a continuous random variable with an Irwin-Hall distribution. + + Explanation + =========== + + The probability distribution function depends on a single parameter + $n$ which is an integer. + + The density of the Irwin-Hall distribution is given by + + .. math :: + f(x) := \frac{1}{(n-1)!}\sum_{k=0}^{\left\lfloor x\right\rfloor}(-1)^k + \binom{n}{k}(x-k)^{n-1} + + Parameters + ========== + + n : A positive integer, `n > 0` + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import UniformSum, density, cdf + >>> from sympy import Symbol, pprint + + >>> n = Symbol("n", integer=True) + >>> z = Symbol("z") + + >>> X = UniformSum("x", n) + + >>> D = density(X)(z) + >>> pprint(D, use_unicode=False) + floor(z) + ___ + \ ` + \ k n - 1 /n\ + ) (-1) *(-k + z) *| | + / \k/ + /__, + k = 0 + -------------------------------- + (n - 1)! + + >>> cdf(X)(z) + Piecewise((0, z < 0), (Sum((-1)**_k*(-_k + z)**n*binomial(n, _k), + (_k, 0, floor(z)))/factorial(n), n >= z), (1, True)) + + + Compute cdf with specific 'x' and 'n' values as follows : + >>> cdf(UniformSum("x", 5), evaluate=False)(2).doit() + 9/40 + + The argument evaluate=False prevents an attempt at evaluation + of the sum for general n, before the argument 2 is passed. + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Uniform_sum_distribution + .. [2] https://mathworld.wolfram.com/UniformSumDistribution.html + + """ + + return rv(name, UniformSumDistribution, (n, )) + +#------------------------------------------------------------------------------- +# VonMises distribution -------------------------------------------------------- + + +class VonMisesDistribution(SingleContinuousDistribution): + _argnames = ('mu', 'k') + + set = Interval(0, 2*pi) + + @staticmethod + def check(mu, k): + _value_check(k > 0, "k must be positive") + + def pdf(self, x): + mu, k = self.mu, self.k + return exp(k*cos(x-mu)) / (2*pi*besseli(0, k)) + +def VonMises(name, mu, k): + r""" + Create a Continuous Random Variable with a von Mises distribution. + + Explanation + =========== + + The density of the von Mises distribution is given by + + .. math:: + f(x) := \frac{e^{\kappa\cos(x-\mu)}}{2\pi I_0(\kappa)} + + with :math:`x \in [0,2\pi]`. + + Parameters + ========== + + mu : Real number + Measure of location. + k : Real number + Measure of concentration. + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import VonMises, density + >>> from sympy import Symbol, pprint + + >>> mu = Symbol("mu") + >>> k = Symbol("k", positive=True) + >>> z = Symbol("z") + + >>> X = VonMises("x", mu, k) + + >>> D = density(X)(z) + >>> pprint(D, use_unicode=False) + k*cos(mu - z) + e + ------------------ + 2*pi*besseli(0, k) + + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Von_Mises_distribution + .. [2] https://mathworld.wolfram.com/vonMisesDistribution.html + + """ + + return rv(name, VonMisesDistribution, (mu, k)) + +#------------------------------------------------------------------------------- +# Weibull distribution --------------------------------------------------------- + + +class WeibullDistribution(SingleContinuousDistribution): + _argnames = ('alpha', 'beta') + + set = Interval(0, oo) + + @staticmethod + def check(alpha, beta): + _value_check(alpha > 0, "Alpha must be positive") + _value_check(beta > 0, "Beta must be positive") + + def pdf(self, x): + alpha, beta = self.alpha, self.beta + return beta * (x/alpha)**(beta - 1) * exp(-(x/alpha)**beta) / alpha + + +def Weibull(name, alpha, beta): + r""" + Create a continuous random variable with a Weibull distribution. + + Explanation + =========== + + The density of the Weibull distribution is given by + + .. math:: + f(x) := \begin{cases} + \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1} + e^{-(x/\lambda)^{k}} & x\geq0\\ + 0 & x<0 + \end{cases} + + Parameters + ========== + + lambda : Real number, $\lambda > 0$, a scale + k : Real number, $k > 0$, a shape + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Weibull, density, E, variance + >>> from sympy import Symbol, simplify + + >>> l = Symbol("lambda", positive=True) + >>> k = Symbol("k", positive=True) + >>> z = Symbol("z") + + >>> X = Weibull("x", l, k) + + >>> density(X)(z) + k*(z/lambda)**(k - 1)*exp(-(z/lambda)**k)/lambda + + >>> simplify(E(X)) + lambda*gamma(1 + 1/k) + + >>> simplify(variance(X)) + lambda**2*(-gamma(1 + 1/k)**2 + gamma(1 + 2/k)) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Weibull_distribution + .. [2] https://mathworld.wolfram.com/WeibullDistribution.html + + """ + + return rv(name, WeibullDistribution, (alpha, beta)) + +#------------------------------------------------------------------------------- +# Wigner semicircle distribution ----------------------------------------------- + + +class WignerSemicircleDistribution(SingleContinuousDistribution): + _argnames = ('R',) + + @property + def set(self): + return Interval(-self.R, self.R) + + @staticmethod + def check(R): + _value_check(R > 0, "Radius R must be positive.") + + def pdf(self, x): + R = self.R + return 2/(pi*R**2)*sqrt(R**2 - x**2) + + def _characteristic_function(self, t): + return Piecewise((2 * besselj(1, self.R*t) / (self.R*t), Ne(t, 0)), + (S.One, True)) + + def _moment_generating_function(self, t): + return Piecewise((2 * besseli(1, self.R*t) / (self.R*t), Ne(t, 0)), + (S.One, True)) + +def WignerSemicircle(name, R): + r""" + Create a continuous random variable with a Wigner semicircle distribution. + + Explanation + =========== + + The density of the Wigner semicircle distribution is given by + + .. math:: + f(x) := \frac2{\pi R^2}\,\sqrt{R^2-x^2} + + with :math:`x \in [-R,R]`. + + Parameters + ========== + + R : Real number, `R > 0`, the radius + + Returns + ======= + + A RandomSymbol. + + Examples + ======== + + >>> from sympy.stats import WignerSemicircle, density, E + >>> from sympy import Symbol + + >>> R = Symbol("R", positive=True) + >>> z = Symbol("z") + + >>> X = WignerSemicircle("x", R) + + >>> density(X)(z) + 2*sqrt(R**2 - z**2)/(pi*R**2) + + >>> E(X) + 0 + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Wigner_semicircle_distribution + .. [2] https://mathworld.wolfram.com/WignersSemicircleLaw.html + + """ + + return rv(name, WignerSemicircleDistribution, (R,)) diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/drv.py b/.venv/lib/python3.12/site-packages/sympy/stats/drv.py new file mode 100644 index 0000000000000000000000000000000000000000..dea14f2dfd1078c223c61bb5cd1373105e72ea28 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/drv.py @@ -0,0 +1,350 @@ +from sympy.concrete.summations import (Sum, summation) +from sympy.core.basic import Basic +from sympy.core.cache import cacheit +from sympy.core.function import Lambda +from sympy.core.numbers import I +from sympy.core.relational import (Eq, Ne) +from sympy.core.singleton import S +from sympy.core.symbol import (Dummy, symbols) +from sympy.core.sympify import sympify +from sympy.functions.combinatorial.factorials import factorial +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.integers import floor +from sympy.functions.elementary.piecewise import Piecewise +from sympy.logic.boolalg import And +from sympy.polys.polytools import poly +from sympy.series.series import series + +from sympy.polys.polyerrors import PolynomialError +from sympy.stats.crv import reduce_rational_inequalities_wrap +from sympy.stats.rv import (NamedArgsMixin, SinglePSpace, SingleDomain, + random_symbols, PSpace, ConditionalDomain, RandomDomain, + ProductDomain, Distribution) +from sympy.stats.symbolic_probability import Probability +from sympy.sets.fancysets import Range, FiniteSet +from sympy.sets.sets import Union +from sympy.sets.contains import Contains +from sympy.utilities import filldedent +from sympy.core.sympify import _sympify + + +class DiscreteDistribution(Distribution): + def __call__(self, *args): + return self.pdf(*args) + + +class SingleDiscreteDistribution(DiscreteDistribution, NamedArgsMixin): + """ Discrete distribution of a single variable. + + Serves as superclass for PoissonDistribution etc.... + + Provides methods for pdf, cdf, and sampling + + See Also: + sympy.stats.crv_types.* + """ + + set = S.Integers + + def __new__(cls, *args): + args = list(map(sympify, args)) + return Basic.__new__(cls, *args) + + @staticmethod + def check(*args): + pass + + @cacheit + def compute_cdf(self, **kwargs): + """ Compute the CDF from the PDF. + + Returns a Lambda. + """ + x = symbols('x', integer=True, cls=Dummy) + z = symbols('z', real=True, cls=Dummy) + left_bound = self.set.inf + + # CDF is integral of PDF from left bound to z + pdf = self.pdf(x) + cdf = summation(pdf, (x, left_bound, floor(z)), **kwargs) + # CDF Ensure that CDF left of left_bound is zero + cdf = Piecewise((cdf, z >= left_bound), (0, True)) + return Lambda(z, cdf) + + def _cdf(self, x): + return None + + def cdf(self, x, **kwargs): + """ Cumulative density function """ + if not kwargs: + cdf = self._cdf(x) + if cdf is not None: + return cdf + return self.compute_cdf(**kwargs)(x) + + @cacheit + def compute_characteristic_function(self, **kwargs): + """ Compute the characteristic function from the PDF. + + Returns a Lambda. + """ + x, t = symbols('x, t', real=True, cls=Dummy) + pdf = self.pdf(x) + cf = summation(exp(I*t*x)*pdf, (x, self.set.inf, self.set.sup)) + return Lambda(t, cf) + + def _characteristic_function(self, t): + return None + + def characteristic_function(self, t, **kwargs): + """ Characteristic function """ + if not kwargs: + cf = self._characteristic_function(t) + if cf is not None: + return cf + return self.compute_characteristic_function(**kwargs)(t) + + @cacheit + def compute_moment_generating_function(self, **kwargs): + t = Dummy('t', real=True) + x = Dummy('x', integer=True) + pdf = self.pdf(x) + mgf = summation(exp(t*x)*pdf, (x, self.set.inf, self.set.sup)) + return Lambda(t, mgf) + + def _moment_generating_function(self, t): + return None + + def moment_generating_function(self, t, **kwargs): + if not kwargs: + mgf = self._moment_generating_function(t) + if mgf is not None: + return mgf + return self.compute_moment_generating_function(**kwargs)(t) + + @cacheit + def compute_quantile(self, **kwargs): + """ Compute the Quantile from the PDF. + + Returns a Lambda. + """ + x = Dummy('x', integer=True) + p = Dummy('p', real=True) + left_bound = self.set.inf + pdf = self.pdf(x) + cdf = summation(pdf, (x, left_bound, x), **kwargs) + set = ((x, p <= cdf), ) + return Lambda(p, Piecewise(*set)) + + def _quantile(self, x): + return None + + def quantile(self, x, **kwargs): + """ Cumulative density function """ + if not kwargs: + quantile = self._quantile(x) + if quantile is not None: + return quantile + return self.compute_quantile(**kwargs)(x) + + def expectation(self, expr, var, evaluate=True, **kwargs): + """ Expectation of expression over distribution """ + # TODO: support discrete sets with non integer stepsizes + + if evaluate: + try: + p = poly(expr, var) + + t = Dummy('t', real=True) + + mgf = self.moment_generating_function(t) + deg = p.degree() + taylor = poly(series(mgf, t, 0, deg + 1).removeO(), t) + result = 0 + for k in range(deg+1): + result += p.coeff_monomial(var ** k) * taylor.coeff_monomial(t ** k) * factorial(k) + + return result + + except PolynomialError: + return summation(expr * self.pdf(var), + (var, self.set.inf, self.set.sup), **kwargs) + + else: + return Sum(expr * self.pdf(var), + (var, self.set.inf, self.set.sup), **kwargs) + + def __call__(self, *args): + return self.pdf(*args) + + +class DiscreteDomain(RandomDomain): + """ + A domain with discrete support with step size one. + Represented using symbols and Range. + """ + is_Discrete = True + +class SingleDiscreteDomain(DiscreteDomain, SingleDomain): + def as_boolean(self): + return Contains(self.symbol, self.set) + + +class ConditionalDiscreteDomain(DiscreteDomain, ConditionalDomain): + """ + Domain with discrete support of step size one, that is restricted by + some condition. + """ + @property + def set(self): + rv = self.symbols + if len(self.symbols) > 1: + raise NotImplementedError(filldedent(''' + Multivariate conditional domains are not yet implemented.''')) + rv = list(rv)[0] + return reduce_rational_inequalities_wrap(self.condition, + rv).intersect(self.fulldomain.set) + + +class DiscretePSpace(PSpace): + is_real = True + is_Discrete = True + + @property + def pdf(self): + return self.density(*self.symbols) + + def where(self, condition): + rvs = random_symbols(condition) + assert all(r.symbol in self.symbols for r in rvs) + if len(rvs) > 1: + raise NotImplementedError(filldedent('''Multivariate discrete + random variables are not yet supported.''')) + conditional_domain = reduce_rational_inequalities_wrap(condition, + rvs[0]) + conditional_domain = conditional_domain.intersect(self.domain.set) + return SingleDiscreteDomain(rvs[0].symbol, conditional_domain) + + def probability(self, condition): + complement = isinstance(condition, Ne) + if complement: + condition = Eq(condition.args[0], condition.args[1]) + try: + _domain = self.where(condition).set + if condition == False or _domain is S.EmptySet: + return S.Zero + if condition == True or _domain == self.domain.set: + return S.One + prob = self.eval_prob(_domain) + except NotImplementedError: + from sympy.stats.rv import density + expr = condition.lhs - condition.rhs + dens = density(expr) + if not isinstance(dens, DiscreteDistribution): + from sympy.stats.drv_types import DiscreteDistributionHandmade + dens = DiscreteDistributionHandmade(dens) + z = Dummy('z', real=True) + space = SingleDiscretePSpace(z, dens) + prob = space.probability(condition.__class__(space.value, 0)) + if prob is None: + prob = Probability(condition) + return prob if not complement else S.One - prob + + def eval_prob(self, _domain): + sym = list(self.symbols)[0] + if isinstance(_domain, Range): + n = symbols('n', integer=True) + inf, sup, step = (r for r in _domain.args) + summand = ((self.pdf).replace( + sym, n*step)) + rv = summation(summand, + (n, inf/step, (sup)/step - 1)).doit() + return rv + elif isinstance(_domain, FiniteSet): + pdf = Lambda(sym, self.pdf) + rv = sum(pdf(x) for x in _domain) + return rv + elif isinstance(_domain, Union): + rv = sum(self.eval_prob(x) for x in _domain.args) + return rv + + def conditional_space(self, condition): + # XXX: Converting from set to tuple. The order matters to Lambda + # though so we should be starting with a set... + density = Lambda(tuple(self.symbols), self.pdf/self.probability(condition)) + condition = condition.xreplace({rv: rv.symbol for rv in self.values}) + domain = ConditionalDiscreteDomain(self.domain, condition) + return DiscretePSpace(domain, density) + +class ProductDiscreteDomain(ProductDomain, DiscreteDomain): + def as_boolean(self): + return And(*[domain.as_boolean for domain in self.domains]) + +class SingleDiscretePSpace(DiscretePSpace, SinglePSpace): + """ Discrete probability space over a single univariate variable """ + is_real = True + + @property + def set(self): + return self.distribution.set + + @property + def domain(self): + return SingleDiscreteDomain(self.symbol, self.set) + + def sample(self, size=(), library='scipy', seed=None): + """ + Internal sample method. + + Returns dictionary mapping RandomSymbol to realization value. + """ + return {self.value: self.distribution.sample(size, library=library, seed=seed)} + + def compute_expectation(self, expr, rvs=None, evaluate=True, **kwargs): + rvs = rvs or (self.value,) + if self.value not in rvs: + return expr + + expr = _sympify(expr) + expr = expr.xreplace({rv: rv.symbol for rv in rvs}) + + x = self.value.symbol + try: + return self.distribution.expectation(expr, x, evaluate=evaluate, + **kwargs) + except NotImplementedError: + return Sum(expr * self.pdf, (x, self.set.inf, self.set.sup), + **kwargs) + + def compute_cdf(self, expr, **kwargs): + if expr == self.value: + x = Dummy("x", real=True) + return Lambda(x, self.distribution.cdf(x, **kwargs)) + else: + raise NotImplementedError() + + def compute_density(self, expr, **kwargs): + if expr == self.value: + return self.distribution + raise NotImplementedError() + + def compute_characteristic_function(self, expr, **kwargs): + if expr == self.value: + t = Dummy("t", real=True) + return Lambda(t, self.distribution.characteristic_function(t, **kwargs)) + else: + raise NotImplementedError() + + def compute_moment_generating_function(self, expr, **kwargs): + if expr == self.value: + t = Dummy("t", real=True) + return Lambda(t, self.distribution.moment_generating_function(t, **kwargs)) + else: + raise NotImplementedError() + + def compute_quantile(self, expr, **kwargs): + if expr == self.value: + p = Dummy("p", real=True) + return Lambda(p, self.distribution.quantile(p, **kwargs)) + else: + raise NotImplementedError() diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/drv_types.py b/.venv/lib/python3.12/site-packages/sympy/stats/drv_types.py new file mode 100644 index 0000000000000000000000000000000000000000..84920d31c0083828efc2cd3f752d2c48f5430102 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/drv_types.py @@ -0,0 +1,849 @@ +""" + +Contains +======== +FlorySchulz +Geometric +Hermite +Logarithmic +NegativeBinomial +Poisson +Skellam +YuleSimon +Zeta +""" + + + +from sympy.concrete.summations import Sum +from sympy.core.basic import Basic +from sympy.core.function import Lambda +from sympy.core.numbers import I +from sympy.core.relational import Eq +from sympy.core.singleton import S +from sympy.core.symbol import Dummy +from sympy.core.sympify import sympify +from sympy.functions.combinatorial.factorials import (binomial, factorial, FallingFactorial) +from sympy.functions.elementary.exponential import (exp, log) +from sympy.functions.elementary.integers import floor +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.special.bessel import besseli +from sympy.functions.special.beta_functions import beta +from sympy.functions.special.hyper import hyper +from sympy.functions.special.zeta_functions import (polylog, zeta) +from sympy.stats.drv import SingleDiscreteDistribution, SingleDiscretePSpace +from sympy.stats.rv import _value_check, is_random + + +__all__ = ['FlorySchulz', +'Geometric', +'Hermite', +'Logarithmic', +'NegativeBinomial', +'Poisson', +'Skellam', +'YuleSimon', +'Zeta' +] + + +def rv(symbol, cls, *args, **kwargs): + args = list(map(sympify, args)) + dist = cls(*args) + if kwargs.pop('check', True): + dist.check(*args) + pspace = SingleDiscretePSpace(symbol, dist) + if any(is_random(arg) for arg in args): + from sympy.stats.compound_rv import CompoundPSpace, CompoundDistribution + pspace = CompoundPSpace(symbol, CompoundDistribution(dist)) + return pspace.value + + +class DiscreteDistributionHandmade(SingleDiscreteDistribution): + _argnames = ('pdf',) + + def __new__(cls, pdf, set=S.Integers): + return Basic.__new__(cls, pdf, set) + + @property + def set(self): + return self.args[1] + + @staticmethod + def check(pdf, set): + x = Dummy('x') + val = Sum(pdf(x), (x, set._inf, set._sup)).doit() + _value_check(Eq(val, 1) != S.false, "The pdf is incorrect on the given set.") + + + +def DiscreteRV(symbol, density, set=S.Integers, **kwargs): + """ + Create a Discrete Random Variable given the following: + + Parameters + ========== + + symbol : Symbol + Represents name of the random variable. + density : Expression containing symbol + Represents probability density function. + set : set + Represents the region where the pdf is valid, by default is real line. + check : bool + If True, it will check whether the given density + integrates to 1 over the given set. If False, it + will not perform this check. Default is False. + + Examples + ======== + + >>> from sympy.stats import DiscreteRV, P, E + >>> from sympy import Rational, Symbol + >>> x = Symbol('x') + >>> n = 10 + >>> density = Rational(1, 10) + >>> X = DiscreteRV(x, density, set=set(range(n))) + >>> E(X) + 9/2 + >>> P(X>3) + 3/5 + + Returns + ======= + + RandomSymbol + + """ + set = sympify(set) + pdf = Piecewise((density, set.as_relational(symbol)), (0, True)) + pdf = Lambda(symbol, pdf) + # have a default of False while `rv` should have a default of True + kwargs['check'] = kwargs.pop('check', False) + return rv(symbol.name, DiscreteDistributionHandmade, pdf, set, **kwargs) + + +#------------------------------------------------------------------------------- +# Flory-Schulz distribution ------------------------------------------------------------ + +class FlorySchulzDistribution(SingleDiscreteDistribution): + _argnames = ('a',) + set = S.Naturals + + @staticmethod + def check(a): + _value_check((0 < a, a < 1), "a must be between 0 and 1") + + def pdf(self, k): + a = self.a + return (a**2 * k * (1 - a)**(k - 1)) + + def _characteristic_function(self, t): + a = self.a + return a**2*exp(I*t)/((1 + (a - 1)*exp(I*t))**2) + + def _moment_generating_function(self, t): + a = self.a + return a**2*exp(t)/((1 + (a - 1)*exp(t))**2) + + +def FlorySchulz(name, a): + r""" + Create a discrete random variable with a FlorySchulz distribution. + + The density of the FlorySchulz distribution is given by + + .. math:: + f(k) := (a^2) k (1 - a)^{k-1} + + Parameters + ========== + + a : A real number between 0 and 1 + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import density, E, variance, FlorySchulz + >>> from sympy import Symbol, S + + >>> a = S.One / 5 + >>> z = Symbol("z") + + >>> X = FlorySchulz("x", a) + + >>> density(X)(z) + (4/5)**(z - 1)*z/25 + + >>> E(X) + 9 + + >>> variance(X) + 40 + + References + ========== + + https://en.wikipedia.org/wiki/Flory%E2%80%93Schulz_distribution + """ + return rv(name, FlorySchulzDistribution, a) + + +#------------------------------------------------------------------------------- +# Geometric distribution ------------------------------------------------------------ + +class GeometricDistribution(SingleDiscreteDistribution): + _argnames = ('p',) + set = S.Naturals + + @staticmethod + def check(p): + _value_check((0 < p, p <= 1), "p must be between 0 and 1") + + def pdf(self, k): + return (1 - self.p)**(k - 1) * self.p + + def _characteristic_function(self, t): + p = self.p + return p * exp(I*t) / (1 - (1 - p)*exp(I*t)) + + def _moment_generating_function(self, t): + p = self.p + return p * exp(t) / (1 - (1 - p) * exp(t)) + + +def Geometric(name, p): + r""" + Create a discrete random variable with a Geometric distribution. + + Explanation + =========== + + The density of the Geometric distribution is given by + + .. math:: + f(k) := p (1 - p)^{k - 1} + + Parameters + ========== + + p : A probability between 0 and 1 + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Geometric, density, E, variance + >>> from sympy import Symbol, S + + >>> p = S.One / 5 + >>> z = Symbol("z") + + >>> X = Geometric("x", p) + + >>> density(X)(z) + (4/5)**(z - 1)/5 + + >>> E(X) + 5 + + >>> variance(X) + 20 + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Geometric_distribution + .. [2] https://mathworld.wolfram.com/GeometricDistribution.html + + """ + return rv(name, GeometricDistribution, p) + + +#------------------------------------------------------------------------------- +# Hermite distribution --------------------------------------------------------- + + +class HermiteDistribution(SingleDiscreteDistribution): + _argnames = ('a1', 'a2') + set = S.Naturals0 + + @staticmethod + def check(a1, a2): + _value_check(a1.is_nonnegative, 'Parameter a1 must be >= 0.') + _value_check(a2.is_nonnegative, 'Parameter a2 must be >= 0.') + + def pdf(self, k): + a1, a2 = self.a1, self.a2 + term1 = exp(-(a1 + a2)) + j = Dummy("j", integer=True) + num = a1**(k - 2*j) * a2**j + den = factorial(k - 2*j) * factorial(j) + return term1 * Sum(num/den, (j, 0, k//2)).doit() + + def _moment_generating_function(self, t): + a1, a2 = self.a1, self.a2 + term1 = a1 * (exp(t) - 1) + term2 = a2 * (exp(2*t) - 1) + return exp(term1 + term2) + + def _characteristic_function(self, t): + a1, a2 = self.a1, self.a2 + term1 = a1 * (exp(I*t) - 1) + term2 = a2 * (exp(2*I*t) - 1) + return exp(term1 + term2) + +def Hermite(name, a1, a2): + r""" + Create a discrete random variable with a Hermite distribution. + + Explanation + =========== + + The density of the Hermite distribution is given by + + .. math:: + f(x):= e^{-a_1 -a_2}\sum_{j=0}^{\left \lfloor x/2 \right \rfloor} + \frac{a_{1}^{x-2j}a_{2}^{j}}{(x-2j)!j!} + + Parameters + ========== + + a1 : A Positive number greater than equal to 0. + a2 : A Positive number greater than equal to 0. + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Hermite, density, E, variance + >>> from sympy import Symbol + + >>> a1 = Symbol("a1", positive=True) + >>> a2 = Symbol("a2", positive=True) + >>> x = Symbol("x") + + >>> H = Hermite("H", a1=5, a2=4) + + >>> density(H)(2) + 33*exp(-9)/2 + + >>> E(H) + 13 + + >>> variance(H) + 21 + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Hermite_distribution + + """ + + return rv(name, HermiteDistribution, a1, a2) + + +#------------------------------------------------------------------------------- +# Logarithmic distribution ------------------------------------------------------------ + +class LogarithmicDistribution(SingleDiscreteDistribution): + _argnames = ('p',) + + set = S.Naturals + + @staticmethod + def check(p): + _value_check((p > 0, p < 1), "p should be between 0 and 1") + + def pdf(self, k): + p = self.p + return (-1) * p**k / (k * log(1 - p)) + + def _characteristic_function(self, t): + p = self.p + return log(1 - p * exp(I*t)) / log(1 - p) + + def _moment_generating_function(self, t): + p = self.p + return log(1 - p * exp(t)) / log(1 - p) + + +def Logarithmic(name, p): + r""" + Create a discrete random variable with a Logarithmic distribution. + + Explanation + =========== + + The density of the Logarithmic distribution is given by + + .. math:: + f(k) := \frac{-p^k}{k \ln{(1 - p)}} + + Parameters + ========== + + p : A value between 0 and 1 + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Logarithmic, density, E, variance + >>> from sympy import Symbol, S + + >>> p = S.One / 5 + >>> z = Symbol("z") + + >>> X = Logarithmic("x", p) + + >>> density(X)(z) + -1/(5**z*z*log(4/5)) + + >>> E(X) + -1/(-4*log(5) + 8*log(2)) + + >>> variance(X) + -1/((-4*log(5) + 8*log(2))*(-2*log(5) + 4*log(2))) + 1/(-64*log(2)*log(5) + 64*log(2)**2 + 16*log(5)**2) - 10/(-32*log(5) + 64*log(2)) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Logarithmic_distribution + .. [2] https://mathworld.wolfram.com/LogarithmicDistribution.html + + """ + return rv(name, LogarithmicDistribution, p) + + +#------------------------------------------------------------------------------- +# Negative binomial distribution ------------------------------------------------------------ + +class NegativeBinomialDistribution(SingleDiscreteDistribution): + _argnames = ('r', 'p') + set = S.Naturals0 + + @staticmethod + def check(r, p): + _value_check(r > 0, 'r should be positive') + _value_check((p > 0, p < 1), 'p should be between 0 and 1') + + def pdf(self, k): + r = self.r + p = self.p + + return binomial(k + r - 1, k) * (1 - p)**k * p**r + + def _characteristic_function(self, t): + r = self.r + p = self.p + + return (p / (1 - (1 - p) * exp(I*t)))**r + + def _moment_generating_function(self, t): + r = self.r + p = self.p + + return (p / (1 - (1 - p) * exp(t)))**r + +def NegativeBinomial(name, r, p): + r""" + Create a discrete random variable with a Negative Binomial distribution. + + Explanation + =========== + + The density of the Negative Binomial distribution is given by + + .. math:: + f(k) := \binom{k + r - 1}{k} (1 - p)^k p^r + + Parameters + ========== + + r : A positive value + Number of successes until the experiment is stopped. + p : A value between 0 and 1 + Probability of success. + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import NegativeBinomial, density, E, variance + >>> from sympy import Symbol, S + + >>> r = 5 + >>> p = S.One / 3 + >>> z = Symbol("z") + + >>> X = NegativeBinomial("x", r, p) + + >>> density(X)(z) + (2/3)**z*binomial(z + 4, z)/243 + + >>> E(X) + 10 + + >>> variance(X) + 30 + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Negative_binomial_distribution + .. [2] https://mathworld.wolfram.com/NegativeBinomialDistribution.html + + """ + return rv(name, NegativeBinomialDistribution, r, p) + + +#------------------------------------------------------------------------------- +# Poisson distribution ------------------------------------------------------------ + +class PoissonDistribution(SingleDiscreteDistribution): + _argnames = ('lamda',) + + set = S.Naturals0 + + @staticmethod + def check(lamda): + _value_check(lamda > 0, "Lambda must be positive") + + def pdf(self, k): + return self.lamda**k / factorial(k) * exp(-self.lamda) + + def _characteristic_function(self, t): + return exp(self.lamda * (exp(I*t) - 1)) + + def _moment_generating_function(self, t): + return exp(self.lamda * (exp(t) - 1)) + + def expectation(self, expr, var, evaluate=True, **kwargs): + if evaluate: + if expr == var: + return self.lamda + if ( + isinstance(expr, FallingFactorial) + and expr.args[1].is_integer + and expr.args[1].is_positive + and expr.args[0] == var + ): + return self.lamda ** expr.args[1] + return super().expectation(expr, var, evaluate, **kwargs) + +def Poisson(name, lamda): + r""" + Create a discrete random variable with a Poisson distribution. + + Explanation + =========== + + The density of the Poisson distribution is given by + + .. math:: + f(k) := \frac{\lambda^{k} e^{- \lambda}}{k!} + + Parameters + ========== + + lamda : Positive number, a rate + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Poisson, density, E, variance + >>> from sympy import Symbol, simplify + + >>> rate = Symbol("lambda", positive=True) + >>> z = Symbol("z") + + >>> X = Poisson("x", rate) + + >>> density(X)(z) + lambda**z*exp(-lambda)/factorial(z) + + >>> E(X) + lambda + + >>> simplify(variance(X)) + lambda + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Poisson_distribution + .. [2] https://mathworld.wolfram.com/PoissonDistribution.html + + """ + return rv(name, PoissonDistribution, lamda) + + +# ----------------------------------------------------------------------------- +# Skellam distribution -------------------------------------------------------- + + +class SkellamDistribution(SingleDiscreteDistribution): + _argnames = ('mu1', 'mu2') + set = S.Integers + + @staticmethod + def check(mu1, mu2): + _value_check(mu1 >= 0, 'Parameter mu1 must be >= 0') + _value_check(mu2 >= 0, 'Parameter mu2 must be >= 0') + + def pdf(self, k): + (mu1, mu2) = (self.mu1, self.mu2) + term1 = exp(-(mu1 + mu2)) * (mu1 / mu2) ** (k / 2) + term2 = besseli(k, 2 * sqrt(mu1 * mu2)) + return term1 * term2 + + def _cdf(self, x): + raise NotImplementedError( + "Skellam doesn't have closed form for the CDF.") + + def _characteristic_function(self, t): + (mu1, mu2) = (self.mu1, self.mu2) + return exp(-(mu1 + mu2) + mu1 * exp(I * t) + mu2 * exp(-I * t)) + + def _moment_generating_function(self, t): + (mu1, mu2) = (self.mu1, self.mu2) + return exp(-(mu1 + mu2) + mu1 * exp(t) + mu2 * exp(-t)) + + +def Skellam(name, mu1, mu2): + r""" + Create a discrete random variable with a Skellam distribution. + + Explanation + =========== + + The Skellam is the distribution of the difference N1 - N2 + of two statistically independent random variables N1 and N2 + each Poisson-distributed with respective expected values mu1 and mu2. + + The density of the Skellam distribution is given by + + .. math:: + f(k) := e^{-(\mu_1+\mu_2)}(\frac{\mu_1}{\mu_2})^{k/2}I_k(2\sqrt{\mu_1\mu_2}) + + Parameters + ========== + + mu1 : A non-negative value + mu2 : A non-negative value + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Skellam, density, E, variance + >>> from sympy import Symbol, pprint + + >>> z = Symbol("z", integer=True) + >>> mu1 = Symbol("mu1", positive=True) + >>> mu2 = Symbol("mu2", positive=True) + >>> X = Skellam("x", mu1, mu2) + + >>> pprint(density(X)(z), use_unicode=False) + z + - + 2 + /mu1\ -mu1 - mu2 / _____ _____\ + |---| *e *besseli\z, 2*\/ mu1 *\/ mu2 / + \mu2/ + >>> E(X) + mu1 - mu2 + >>> variance(X).expand() + mu1 + mu2 + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Skellam_distribution + + """ + return rv(name, SkellamDistribution, mu1, mu2) + + +#------------------------------------------------------------------------------- +# Yule-Simon distribution ------------------------------------------------------------ + +class YuleSimonDistribution(SingleDiscreteDistribution): + _argnames = ('rho',) + set = S.Naturals + + @staticmethod + def check(rho): + _value_check(rho > 0, 'rho should be positive') + + def pdf(self, k): + rho = self.rho + return rho * beta(k, rho + 1) + + def _cdf(self, x): + return Piecewise((1 - floor(x) * beta(floor(x), self.rho + 1), x >= 1), (0, True)) + + def _characteristic_function(self, t): + rho = self.rho + return rho * hyper((1, 1), (rho + 2,), exp(I*t)) * exp(I*t) / (rho + 1) + + def _moment_generating_function(self, t): + rho = self.rho + return rho * hyper((1, 1), (rho + 2,), exp(t)) * exp(t) / (rho + 1) + + +def YuleSimon(name, rho): + r""" + Create a discrete random variable with a Yule-Simon distribution. + + Explanation + =========== + + The density of the Yule-Simon distribution is given by + + .. math:: + f(k) := \rho B(k, \rho + 1) + + Parameters + ========== + + rho : A positive value + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import YuleSimon, density, E, variance + >>> from sympy import Symbol, simplify + + >>> p = 5 + >>> z = Symbol("z") + + >>> X = YuleSimon("x", p) + + >>> density(X)(z) + 5*beta(z, 6) + + >>> simplify(E(X)) + 5/4 + + >>> simplify(variance(X)) + 25/48 + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Yule%E2%80%93Simon_distribution + + """ + return rv(name, YuleSimonDistribution, rho) + + +#------------------------------------------------------------------------------- +# Zeta distribution ------------------------------------------------------------ + +class ZetaDistribution(SingleDiscreteDistribution): + _argnames = ('s',) + set = S.Naturals + + @staticmethod + def check(s): + _value_check(s > 1, 's should be greater than 1') + + def pdf(self, k): + s = self.s + return 1 / (k**s * zeta(s)) + + def _characteristic_function(self, t): + return polylog(self.s, exp(I*t)) / zeta(self.s) + + def _moment_generating_function(self, t): + return polylog(self.s, exp(t)) / zeta(self.s) + + +def Zeta(name, s): + r""" + Create a discrete random variable with a Zeta distribution. + + Explanation + =========== + + The density of the Zeta distribution is given by + + .. math:: + f(k) := \frac{1}{k^s \zeta{(s)}} + + Parameters + ========== + + s : A value greater than 1 + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import Zeta, density, E, variance + >>> from sympy import Symbol + + >>> s = 5 + >>> z = Symbol("z") + + >>> X = Zeta("x", s) + + >>> density(X)(z) + 1/(z**5*zeta(5)) + + >>> E(X) + pi**4/(90*zeta(5)) + + >>> variance(X) + -pi**8/(8100*zeta(5)**2) + zeta(3)/zeta(5) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Zeta_distribution + + """ + return rv(name, ZetaDistribution, s) diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/error_prop.py b/.venv/lib/python3.12/site-packages/sympy/stats/error_prop.py new file mode 100644 index 0000000000000000000000000000000000000000..e6cacb894307fe60cbf096c7760e6ed57f385a91 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/error_prop.py @@ -0,0 +1,100 @@ +"""Tools for arithmetic error propagation.""" + +from itertools import repeat, combinations + +from sympy.core.add import Add +from sympy.core.mul import Mul +from sympy.core.power import Pow +from sympy.core.singleton import S +from sympy.core.symbol import Symbol +from sympy.functions.elementary.exponential import exp +from sympy.simplify.simplify import simplify +from sympy.stats.symbolic_probability import RandomSymbol, Variance, Covariance +from sympy.stats.rv import is_random + +_arg0_or_var = lambda var: var.args[0] if len(var.args) > 0 else var + + +def variance_prop(expr, consts=(), include_covar=False): + r"""Symbolically propagates variance (`\sigma^2`) for expressions. + This is computed as as seen in [1]_. + + Parameters + ========== + + expr : Expr + A SymPy expression to compute the variance for. + consts : sequence of Symbols, optional + Represents symbols that are known constants in the expr, + and thus have zero variance. All symbols not in consts are + assumed to be variant. + include_covar : bool, optional + Flag for whether or not to include covariances, default=False. + + Returns + ======= + + var_expr : Expr + An expression for the total variance of the expr. + The variance for the original symbols (e.g. x) are represented + via instance of the Variance symbol (e.g. Variance(x)). + + Examples + ======== + + >>> from sympy import symbols, exp + >>> from sympy.stats.error_prop import variance_prop + >>> x, y = symbols('x y') + + >>> variance_prop(x + y) + Variance(x) + Variance(y) + + >>> variance_prop(x * y) + x**2*Variance(y) + y**2*Variance(x) + + >>> variance_prop(exp(2*x)) + 4*exp(4*x)*Variance(x) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Propagation_of_uncertainty + + """ + args = expr.args + if len(args) == 0: + if expr in consts: + return S.Zero + elif is_random(expr): + return Variance(expr).doit() + elif isinstance(expr, Symbol): + return Variance(RandomSymbol(expr)).doit() + else: + return S.Zero + nargs = len(args) + var_args = list(map(variance_prop, args, repeat(consts, nargs), + repeat(include_covar, nargs))) + if isinstance(expr, Add): + var_expr = Add(*var_args) + if include_covar: + terms = [2 * Covariance(_arg0_or_var(x), _arg0_or_var(y)).expand() \ + for x, y in combinations(var_args, 2)] + var_expr += Add(*terms) + elif isinstance(expr, Mul): + terms = [v/a**2 for a, v in zip(args, var_args)] + var_expr = simplify(expr**2 * Add(*terms)) + if include_covar: + terms = [2*Covariance(_arg0_or_var(x), _arg0_or_var(y)).expand()/(a*b) \ + for (a, b), (x, y) in zip(combinations(args, 2), + combinations(var_args, 2))] + var_expr += Add(*terms) + elif isinstance(expr, Pow): + b = args[1] + v = var_args[0] * (expr * b / args[0])**2 + var_expr = simplify(v) + elif isinstance(expr, exp): + var_expr = simplify(var_args[0] * expr**2) + else: + # unknown how to proceed, return variance of whole expr. + var_expr = Variance(expr) + return var_expr diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/frv.py b/.venv/lib/python3.12/site-packages/sympy/stats/frv.py new file mode 100644 index 0000000000000000000000000000000000000000..498d7e4006b2b8db306a0905ed67578021e220a8 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/frv.py @@ -0,0 +1,512 @@ +""" +Finite Discrete Random Variables Module + +See Also +======== +sympy.stats.frv_types +sympy.stats.rv +sympy.stats.crv +""" +from itertools import product + +from sympy.concrete.summations import Sum +from sympy.core.basic import Basic +from sympy.core.cache import cacheit +from sympy.core.function import Lambda +from sympy.core.mul import Mul +from sympy.core.numbers import (I, nan) +from sympy.core.relational import Eq +from sympy.core.singleton import S +from sympy.core.symbol import (Dummy, Symbol) +from sympy.core.sympify import sympify +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.piecewise import Piecewise +from sympy.logic.boolalg import (And, Or) +from sympy.sets.sets import Intersection +from sympy.core.containers import Dict +from sympy.core.logic import Logic +from sympy.core.relational import Relational +from sympy.core.sympify import _sympify +from sympy.sets.sets import FiniteSet +from sympy.stats.rv import (RandomDomain, ProductDomain, ConditionalDomain, + PSpace, IndependentProductPSpace, SinglePSpace, random_symbols, + sumsets, rv_subs, NamedArgsMixin, Density, Distribution) + + +class FiniteDensity(dict): + """ + A domain with Finite Density. + """ + def __call__(self, item): + """ + Make instance of a class callable. + + If item belongs to current instance of a class, return it. + + Otherwise, return 0. + """ + item = sympify(item) + if item in self: + return self[item] + else: + return 0 + + @property + def dict(self): + """ + Return item as dictionary. + """ + return dict(self) + +class FiniteDomain(RandomDomain): + """ + A domain with discrete finite support + + Represented using a FiniteSet. + """ + is_Finite = True + + @property + def symbols(self): + return FiniteSet(sym for sym, val in self.elements) + + @property + def elements(self): + return self.args[0] + + @property + def dict(self): + return FiniteSet(*[Dict(dict(el)) for el in self.elements]) + + def __contains__(self, other): + return other in self.elements + + def __iter__(self): + return self.elements.__iter__() + + def as_boolean(self): + return Or(*[And(*[Eq(sym, val) for sym, val in item]) for item in self]) + + +class SingleFiniteDomain(FiniteDomain): + """ + A FiniteDomain over a single symbol/set + + Example: The possibilities of a *single* die roll. + """ + + def __new__(cls, symbol, set): + if not isinstance(set, FiniteSet) and \ + not isinstance(set, Intersection): + set = FiniteSet(*set) + return Basic.__new__(cls, symbol, set) + + @property + def symbol(self): + return self.args[0] + + @property + def symbols(self): + return FiniteSet(self.symbol) + + @property + def set(self): + return self.args[1] + + @property + def elements(self): + return FiniteSet(*[frozenset(((self.symbol, elem), )) for elem in self.set]) + + def __iter__(self): + return (frozenset(((self.symbol, elem),)) for elem in self.set) + + def __contains__(self, other): + sym, val = tuple(other)[0] + return sym == self.symbol and val in self.set + + +class ProductFiniteDomain(ProductDomain, FiniteDomain): + """ + A Finite domain consisting of several other FiniteDomains + + Example: The possibilities of the rolls of three independent dice + """ + + def __iter__(self): + proditer = product(*self.domains) + return (sumsets(items) for items in proditer) + + @property + def elements(self): + return FiniteSet(*self) + + +class ConditionalFiniteDomain(ConditionalDomain, ProductFiniteDomain): + """ + A FiniteDomain that has been restricted by a condition + + Example: The possibilities of a die roll under the condition that the + roll is even. + """ + + def __new__(cls, domain, condition): + """ + Create a new instance of ConditionalFiniteDomain class + """ + if condition is True: + return domain + cond = rv_subs(condition) + return Basic.__new__(cls, domain, cond) + + def _test(self, elem): + """ + Test the value. If value is boolean, return it. If value is equality + relational (two objects are equal), return it with left-hand side + being equal to right-hand side. Otherwise, raise ValueError exception. + """ + val = self.condition.xreplace(dict(elem)) + if val in [True, False]: + return val + elif val.is_Equality: + return val.lhs == val.rhs + raise ValueError("Undecidable if %s" % str(val)) + + def __contains__(self, other): + return other in self.fulldomain and self._test(other) + + def __iter__(self): + return (elem for elem in self.fulldomain if self._test(elem)) + + @property + def set(self): + if isinstance(self.fulldomain, SingleFiniteDomain): + return FiniteSet(*[elem for elem in self.fulldomain.set + if frozenset(((self.fulldomain.symbol, elem),)) in self]) + else: + raise NotImplementedError( + "Not implemented on multi-dimensional conditional domain") + + def as_boolean(self): + return FiniteDomain.as_boolean(self) + + +class SingleFiniteDistribution(Distribution, NamedArgsMixin): + def __new__(cls, *args): + args = list(map(sympify, args)) + return Basic.__new__(cls, *args) + + @staticmethod + def check(*args): + pass + + @property # type: ignore + @cacheit + def dict(self): + if self.is_symbolic: + return Density(self) + return {k: self.pmf(k) for k in self.set} + + def pmf(self, *args): # to be overridden by specific distribution + raise NotImplementedError() + + @property + def set(self): # to be overridden by specific distribution + raise NotImplementedError() + + values = property(lambda self: self.dict.values) + items = property(lambda self: self.dict.items) + is_symbolic = property(lambda self: False) + __iter__ = property(lambda self: self.dict.__iter__) + __getitem__ = property(lambda self: self.dict.__getitem__) + + def __call__(self, *args): + return self.pmf(*args) + + def __contains__(self, other): + return other in self.set + + +#============================================= +#========= Probability Space =============== +#============================================= + + +class FinitePSpace(PSpace): + """ + A Finite Probability Space + + Represents the probabilities of a finite number of events. + """ + is_Finite = True + + def __new__(cls, domain, density): + density = {sympify(key): sympify(val) + for key, val in density.items()} + public_density = Dict(density) + + obj = PSpace.__new__(cls, domain, public_density) + obj._density = density + return obj + + def prob_of(self, elem): + elem = sympify(elem) + density = self._density + if isinstance(list(density.keys())[0], FiniteSet): + return density.get(elem, S.Zero) + return density.get(tuple(elem)[0][1], S.Zero) + + def where(self, condition): + assert all(r.symbol in self.symbols for r in random_symbols(condition)) + return ConditionalFiniteDomain(self.domain, condition) + + def compute_density(self, expr): + expr = rv_subs(expr, self.values) + d = FiniteDensity() + for elem in self.domain: + val = expr.xreplace(dict(elem)) + prob = self.prob_of(elem) + d[val] = d.get(val, S.Zero) + prob + return d + + @cacheit + def compute_cdf(self, expr): + d = self.compute_density(expr) + cum_prob = S.Zero + cdf = [] + for key in sorted(d): + prob = d[key] + cum_prob += prob + cdf.append((key, cum_prob)) + + return dict(cdf) + + @cacheit + def sorted_cdf(self, expr, python_float=False): + cdf = self.compute_cdf(expr) + items = list(cdf.items()) + sorted_items = sorted(items, key=lambda val_cumprob: val_cumprob[1]) + if python_float: + sorted_items = [(v, float(cum_prob)) + for v, cum_prob in sorted_items] + return sorted_items + + @cacheit + def compute_characteristic_function(self, expr): + d = self.compute_density(expr) + t = Dummy('t', real=True) + + return Lambda(t, sum(exp(I*k*t)*v for k,v in d.items())) + + @cacheit + def compute_moment_generating_function(self, expr): + d = self.compute_density(expr) + t = Dummy('t', real=True) + + return Lambda(t, sum(exp(k*t)*v for k,v in d.items())) + + def compute_expectation(self, expr, rvs=None, **kwargs): + rvs = rvs or self.values + expr = rv_subs(expr, rvs) + probs = [self.prob_of(elem) for elem in self.domain] + if isinstance(expr, (Logic, Relational)): + parse_domain = [tuple(elem)[0][1] for elem in self.domain] + bools = [expr.xreplace(dict(elem)) for elem in self.domain] + else: + parse_domain = [expr.xreplace(dict(elem)) for elem in self.domain] + bools = [True for elem in self.domain] + return sum(Piecewise((prob * elem, blv), (S.Zero, True)) + for prob, elem, blv in zip(probs, parse_domain, bools)) + + def compute_quantile(self, expr): + cdf = self.compute_cdf(expr) + p = Dummy('p', real=True) + set = ((nan, (p < 0) | (p > 1)),) + for key, value in cdf.items(): + set = set + ((key, p <= value), ) + return Lambda(p, Piecewise(*set)) + + def probability(self, condition): + cond_symbols = frozenset(rs.symbol for rs in random_symbols(condition)) + cond = rv_subs(condition) + if not cond_symbols.issubset(self.symbols): + raise ValueError("Cannot compare foreign random symbols, %s" + %(str(cond_symbols - self.symbols))) + if isinstance(condition, Relational) and \ + (not cond.free_symbols.issubset(self.domain.free_symbols)): + rv = condition.lhs if isinstance(condition.rhs, Symbol) else condition.rhs + return sum(Piecewise( + (self.prob_of(elem), condition.subs(rv, list(elem)[0][1])), + (S.Zero, True)) for elem in self.domain) + return sympify(sum(self.prob_of(elem) for elem in self.where(condition))) + + def conditional_space(self, condition): + domain = self.where(condition) + prob = self.probability(condition) + density = {key: val / prob + for key, val in self._density.items() if domain._test(key)} + return FinitePSpace(domain, density) + + def sample(self, size=(), library='scipy', seed=None): + """ + Internal sample method + + Returns dictionary mapping RandomSymbol to realization value. + """ + return {self.value: self.distribution.sample(size, library, seed)} + + +class SingleFinitePSpace(SinglePSpace, FinitePSpace): + """ + A single finite probability space + + Represents the probabilities of a set of random events that can be + attributed to a single variable/symbol. + + This class is implemented by many of the standard FiniteRV types such as + Die, Bernoulli, Coin, etc.... + """ + @property + def domain(self): + return SingleFiniteDomain(self.symbol, self.distribution.set) + + @property + def _is_symbolic(self): + """ + Helper property to check if the distribution + of the random variable is having symbolic + dimension. + """ + return self.distribution.is_symbolic + + @property + def distribution(self): + return self.args[1] + + def pmf(self, expr): + return self.distribution.pmf(expr) + + @property # type: ignore + @cacheit + def _density(self): + return {FiniteSet((self.symbol, val)): prob + for val, prob in self.distribution.dict.items()} + + @cacheit + def compute_characteristic_function(self, expr): + if self._is_symbolic: + d = self.compute_density(expr) + t = Dummy('t', real=True) + ki = Dummy('ki') + return Lambda(t, Sum(d(ki)*exp(I*ki*t), (ki, self.args[1].low, self.args[1].high))) + expr = rv_subs(expr, self.values) + return FinitePSpace(self.domain, self.distribution).compute_characteristic_function(expr) + + @cacheit + def compute_moment_generating_function(self, expr): + if self._is_symbolic: + d = self.compute_density(expr) + t = Dummy('t', real=True) + ki = Dummy('ki') + return Lambda(t, Sum(d(ki)*exp(ki*t), (ki, self.args[1].low, self.args[1].high))) + expr = rv_subs(expr, self.values) + return FinitePSpace(self.domain, self.distribution).compute_moment_generating_function(expr) + + def compute_quantile(self, expr): + if self._is_symbolic: + raise NotImplementedError("Computing quantile for random variables " + "with symbolic dimension because the bounds of searching the required " + "value is undetermined.") + expr = rv_subs(expr, self.values) + return FinitePSpace(self.domain, self.distribution).compute_quantile(expr) + + def compute_density(self, expr): + if self._is_symbolic: + rv = list(random_symbols(expr))[0] + k = Dummy('k', integer=True) + cond = True if not isinstance(expr, (Relational, Logic)) \ + else expr.subs(rv, k) + return Lambda(k, + Piecewise((self.pmf(k), And(k >= self.args[1].low, + k <= self.args[1].high, cond)), (S.Zero, True))) + expr = rv_subs(expr, self.values) + return FinitePSpace(self.domain, self.distribution).compute_density(expr) + + def compute_cdf(self, expr): + if self._is_symbolic: + d = self.compute_density(expr) + k = Dummy('k') + ki = Dummy('ki') + return Lambda(k, Sum(d(ki), (ki, self.args[1].low, k))) + expr = rv_subs(expr, self.values) + return FinitePSpace(self.domain, self.distribution).compute_cdf(expr) + + def compute_expectation(self, expr, rvs=None, **kwargs): + if self._is_symbolic: + rv = random_symbols(expr)[0] + k = Dummy('k', integer=True) + expr = expr.subs(rv, k) + cond = True if not isinstance(expr, (Relational, Logic)) \ + else expr + func = self.pmf(k) * k if cond != True else self.pmf(k) * expr + return Sum(Piecewise((func, cond), (S.Zero, True)), + (k, self.distribution.low, self.distribution.high)).doit() + + expr = _sympify(expr) + expr = rv_subs(expr, rvs) + return FinitePSpace(self.domain, self.distribution).compute_expectation(expr, rvs, **kwargs) + + def probability(self, condition): + if self._is_symbolic: + #TODO: Implement the mechanism for handling queries for symbolic sized distributions. + raise NotImplementedError("Currently, probability queries are not " + "supported for random variables with symbolic sized distributions.") + condition = rv_subs(condition) + return FinitePSpace(self.domain, self.distribution).probability(condition) + + def conditional_space(self, condition): + """ + This method is used for transferring the + computation to probability method because + conditional space of random variables with + symbolic dimensions is currently not possible. + """ + if self._is_symbolic: + self + domain = self.where(condition) + prob = self.probability(condition) + density = {key: val / prob + for key, val in self._density.items() if domain._test(key)} + return FinitePSpace(domain, density) + + +class ProductFinitePSpace(IndependentProductPSpace, FinitePSpace): + """ + A collection of several independent finite probability spaces + """ + @property + def domain(self): + return ProductFiniteDomain(*[space.domain for space in self.spaces]) + + @property # type: ignore + @cacheit + def _density(self): + proditer = product(*[iter(space._density.items()) + for space in self.spaces]) + d = {} + for items in proditer: + elems, probs = list(zip(*items)) + elem = sumsets(elems) + prob = Mul(*probs) + d[elem] = d.get(elem, S.Zero) + prob + return Dict(d) + + @property # type: ignore + @cacheit + def density(self): + return Dict(self._density) + + def probability(self, condition): + return FinitePSpace.probability(self, condition) + + def compute_density(self, expr): + return FinitePSpace.compute_density(self, expr) diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/frv_types.py b/.venv/lib/python3.12/site-packages/sympy/stats/frv_types.py new file mode 100644 index 0000000000000000000000000000000000000000..bde656c219791c287ff445d5d215e3759271e923 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/frv_types.py @@ -0,0 +1,873 @@ +""" +Finite Discrete Random Variables - Prebuilt variable types + +Contains +======== +FiniteRV +DiscreteUniform +Die +Bernoulli +Coin +Binomial +BetaBinomial +Hypergeometric +Rademacher +IdealSoliton +RobustSoliton +""" + + +from sympy.core.cache import cacheit +from sympy.core.function import Lambda +from sympy.core.numbers import (Integer, Rational) +from sympy.core.relational import (Eq, Ge, Gt, Le, Lt) +from sympy.core.singleton import S +from sympy.core.symbol import (Dummy, Symbol) +from sympy.core.sympify import sympify +from sympy.functions.combinatorial.factorials import binomial +from sympy.functions.elementary.exponential import log +from sympy.functions.elementary.piecewise import Piecewise +from sympy.logic.boolalg import Or +from sympy.sets.contains import Contains +from sympy.sets.fancysets import Range +from sympy.sets.sets import (Intersection, Interval) +from sympy.functions.special.beta_functions import beta as beta_fn +from sympy.stats.frv import (SingleFiniteDistribution, + SingleFinitePSpace) +from sympy.stats.rv import _value_check, Density, is_random +from sympy.utilities.iterables import multiset +from sympy.utilities.misc import filldedent + + +__all__ = ['FiniteRV', +'DiscreteUniform', +'Die', +'Bernoulli', +'Coin', +'Binomial', +'BetaBinomial', +'Hypergeometric', +'Rademacher', +'IdealSoliton', +'RobustSoliton', +] + +def rv(name, cls, *args, **kwargs): + args = list(map(sympify, args)) + dist = cls(*args) + if kwargs.pop('check', True): + dist.check(*args) + pspace = SingleFinitePSpace(name, dist) + if any(is_random(arg) for arg in args): + from sympy.stats.compound_rv import CompoundPSpace, CompoundDistribution + pspace = CompoundPSpace(name, CompoundDistribution(dist)) + return pspace.value + +class FiniteDistributionHandmade(SingleFiniteDistribution): + + @property + def dict(self): + return self.args[0] + + def pmf(self, x): + x = Symbol('x') + return Lambda(x, Piecewise(*( + [(v, Eq(k, x)) for k, v in self.dict.items()] + [(S.Zero, True)]))) + + @property + def set(self): + return set(self.dict.keys()) + + @staticmethod + def check(density): + for p in density.values(): + _value_check((p >= 0, p <= 1), + "Probability at a point must be between 0 and 1.") + val = sum(density.values()) + _value_check(Eq(val, 1) != S.false, "Total Probability must be 1.") + +def FiniteRV(name, density, **kwargs): + r""" + Create a Finite Random Variable given a dict representing the density. + + Parameters + ========== + + name : Symbol + Represents name of the random variable. + density : dict + Dictionary containing the pdf of finite distribution + check : bool + If True, it will check whether the given density + integrates to 1 over the given set. If False, it + will not perform this check. Default is False. + + Examples + ======== + + >>> from sympy.stats import FiniteRV, P, E + + >>> density = {0: .1, 1: .2, 2: .3, 3: .4} + >>> X = FiniteRV('X', density) + + >>> E(X) + 2.00000000000000 + >>> P(X >= 2) + 0.700000000000000 + + Returns + ======= + + RandomSymbol + + """ + # have a default of False while `rv` should have a default of True + kwargs['check'] = kwargs.pop('check', False) + return rv(name, FiniteDistributionHandmade, density, **kwargs) + +class DiscreteUniformDistribution(SingleFiniteDistribution): + + @staticmethod + def check(*args): + # not using _value_check since there is a + # suggestion for the user + if len(set(args)) != len(args): + weights = multiset(args) + n = Integer(len(args)) + for k in weights: + weights[k] /= n + raise ValueError(filldedent(""" + Repeated args detected but set expected. For a + distribution having different weights for each + item use the following:""") + ( + '\nS("FiniteRV(%s, %s)")' % ("'X'", weights))) + + @property + def p(self): + return Rational(1, len(self.args)) + + @property # type: ignore + @cacheit + def dict(self): + return dict.fromkeys(self.set, self.p) + + @property + def set(self): + return set(self.args) + + def pmf(self, x): + if x in self.args: + return self.p + else: + return S.Zero + + +def DiscreteUniform(name, items): + r""" + Create a Finite Random Variable representing a uniform distribution over + the input set. + + Parameters + ========== + + items : list/tuple + Items over which Uniform distribution is to be made + + Examples + ======== + + >>> from sympy.stats import DiscreteUniform, density + >>> from sympy import symbols + + >>> X = DiscreteUniform('X', symbols('a b c')) # equally likely over a, b, c + >>> density(X).dict + {a: 1/3, b: 1/3, c: 1/3} + + >>> Y = DiscreteUniform('Y', list(range(5))) # distribution over a range + >>> density(Y).dict + {0: 1/5, 1: 1/5, 2: 1/5, 3: 1/5, 4: 1/5} + + Returns + ======= + + RandomSymbol + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Discrete_uniform_distribution + .. [2] https://mathworld.wolfram.com/DiscreteUniformDistribution.html + + """ + return rv(name, DiscreteUniformDistribution, *items) + + +class DieDistribution(SingleFiniteDistribution): + _argnames = ('sides',) + + @staticmethod + def check(sides): + _value_check((sides.is_positive, sides.is_integer), + "number of sides must be a positive integer.") + + @property + def is_symbolic(self): + return not self.sides.is_number + + @property + def high(self): + return self.sides + + @property + def low(self): + return S.One + + @property + def set(self): + if self.is_symbolic: + return Intersection(S.Naturals0, Interval(0, self.sides)) + return set(map(Integer, range(1, self.sides + 1))) + + def pmf(self, x): + x = sympify(x) + if not (x.is_number or x.is_Symbol or is_random(x)): + raise ValueError("'x' expected as an argument of type 'number', 'Symbol', or " + "'RandomSymbol' not %s" % (type(x))) + cond = Ge(x, 1) & Le(x, self.sides) & Contains(x, S.Integers) + return Piecewise((S.One/self.sides, cond), (S.Zero, True)) + +def Die(name, sides=6): + r""" + Create a Finite Random Variable representing a fair die. + + Parameters + ========== + + sides : Integer + Represents the number of sides of the Die, by default is 6 + + Examples + ======== + + >>> from sympy.stats import Die, density + >>> from sympy import Symbol + + >>> D6 = Die('D6', 6) # Six sided Die + >>> density(D6).dict + {1: 1/6, 2: 1/6, 3: 1/6, 4: 1/6, 5: 1/6, 6: 1/6} + + >>> D4 = Die('D4', 4) # Four sided Die + >>> density(D4).dict + {1: 1/4, 2: 1/4, 3: 1/4, 4: 1/4} + + >>> n = Symbol('n', positive=True, integer=True) + >>> Dn = Die('Dn', n) # n sided Die + >>> density(Dn).dict + Density(DieDistribution(n)) + >>> density(Dn).dict.subs(n, 4).doit() + {1: 1/4, 2: 1/4, 3: 1/4, 4: 1/4} + + Returns + ======= + + RandomSymbol + """ + + return rv(name, DieDistribution, sides) + + +class BernoulliDistribution(SingleFiniteDistribution): + _argnames = ('p', 'succ', 'fail') + + @staticmethod + def check(p, succ, fail): + _value_check((p >= 0, p <= 1), + "p should be in range [0, 1].") + + @property + def set(self): + return {self.succ, self.fail} + + def pmf(self, x): + if isinstance(self.succ, Symbol) and isinstance(self.fail, Symbol): + return Piecewise((self.p, x == self.succ), + (1 - self.p, x == self.fail), + (S.Zero, True)) + return Piecewise((self.p, Eq(x, self.succ)), + (1 - self.p, Eq(x, self.fail)), + (S.Zero, True)) + + +def Bernoulli(name, p, succ=1, fail=0): + r""" + Create a Finite Random Variable representing a Bernoulli process. + + Parameters + ========== + + p : Rational number between 0 and 1 + Represents probability of success + succ : Integer/symbol/string + Represents event of success + fail : Integer/symbol/string + Represents event of failure + + Examples + ======== + + >>> from sympy.stats import Bernoulli, density + >>> from sympy import S + + >>> X = Bernoulli('X', S(3)/4) # 1-0 Bernoulli variable, probability = 3/4 + >>> density(X).dict + {0: 1/4, 1: 3/4} + + >>> X = Bernoulli('X', S.Half, 'Heads', 'Tails') # A fair coin toss + >>> density(X).dict + {Heads: 1/2, Tails: 1/2} + + Returns + ======= + + RandomSymbol + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Bernoulli_distribution + .. [2] https://mathworld.wolfram.com/BernoulliDistribution.html + + """ + + return rv(name, BernoulliDistribution, p, succ, fail) + + +def Coin(name, p=S.Half): + r""" + Create a Finite Random Variable representing a Coin toss. + + This is an equivalent of a Bernoulli random variable with + "H" and "T" as success and failure events respectively. + + Parameters + ========== + + p : Rational Number between 0 and 1 + Represents probability of getting "Heads", by default is Half + + Examples + ======== + + >>> from sympy.stats import Coin, density + >>> from sympy import Rational + + >>> C = Coin('C') # A fair coin toss + >>> density(C).dict + {H: 1/2, T: 1/2} + + >>> C2 = Coin('C2', Rational(3, 5)) # An unfair coin + >>> density(C2).dict + {H: 3/5, T: 2/5} + + Returns + ======= + + RandomSymbol + + See Also + ======== + + sympy.stats.Binomial + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Coin_flipping + + """ + return rv(name, BernoulliDistribution, p, 'H', 'T') + + +class BinomialDistribution(SingleFiniteDistribution): + _argnames = ('n', 'p', 'succ', 'fail') + + @staticmethod + def check(n, p, succ, fail): + _value_check((n.is_integer, n.is_nonnegative), + "'n' must be nonnegative integer.") + _value_check((p <= 1, p >= 0), + "p should be in range [0, 1].") + + @property + def high(self): + return self.n + + @property + def low(self): + return S.Zero + + @property + def is_symbolic(self): + return not self.n.is_number + + @property + def set(self): + if self.is_symbolic: + return Intersection(S.Naturals0, Interval(0, self.n)) + return set(self.dict.keys()) + + def pmf(self, x): + n, p = self.n, self.p + x = sympify(x) + if not (x.is_number or x.is_Symbol or is_random(x)): + raise ValueError("'x' expected as an argument of type 'number', 'Symbol', or " + "'RandomSymbol' not %s" % (type(x))) + cond = Ge(x, 0) & Le(x, n) & Contains(x, S.Integers) + return Piecewise((binomial(n, x) * p**x * (1 - p)**(n - x), cond), (S.Zero, True)) + + @property # type: ignore + @cacheit + def dict(self): + if self.is_symbolic: + return Density(self) + return {k*self.succ + (self.n-k)*self.fail: self.pmf(k) + for k in range(0, self.n + 1)} + + +def Binomial(name, n, p, succ=1, fail=0): + r""" + Create a Finite Random Variable representing a binomial distribution. + + Parameters + ========== + + n : Positive Integer + Represents number of trials + p : Rational Number between 0 and 1 + Represents probability of success + succ : Integer/symbol/string + Represents event of success, by default is 1 + fail : Integer/symbol/string + Represents event of failure, by default is 0 + + Examples + ======== + + >>> from sympy.stats import Binomial, density + >>> from sympy import S, Symbol + + >>> X = Binomial('X', 4, S.Half) # Four "coin flips" + >>> density(X).dict + {0: 1/16, 1: 1/4, 2: 3/8, 3: 1/4, 4: 1/16} + + >>> n = Symbol('n', positive=True, integer=True) + >>> p = Symbol('p', positive=True) + >>> X = Binomial('X', n, S.Half) # n "coin flips" + >>> density(X).dict + Density(BinomialDistribution(n, 1/2, 1, 0)) + >>> density(X).dict.subs(n, 4).doit() + {0: 1/16, 1: 1/4, 2: 3/8, 3: 1/4, 4: 1/16} + + Returns + ======= + + RandomSymbol + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Binomial_distribution + .. [2] https://mathworld.wolfram.com/BinomialDistribution.html + + """ + + return rv(name, BinomialDistribution, n, p, succ, fail) + +#------------------------------------------------------------------------------- +# Beta-binomial distribution ---------------------------------------------------------- + +class BetaBinomialDistribution(SingleFiniteDistribution): + _argnames = ('n', 'alpha', 'beta') + + @staticmethod + def check(n, alpha, beta): + _value_check((n.is_integer, n.is_nonnegative), + "'n' must be nonnegative integer. n = %s." % str(n)) + _value_check((alpha > 0), + "'alpha' must be: alpha > 0 . alpha = %s" % str(alpha)) + _value_check((beta > 0), + "'beta' must be: beta > 0 . beta = %s" % str(beta)) + + @property + def high(self): + return self.n + + @property + def low(self): + return S.Zero + + @property + def is_symbolic(self): + return not self.n.is_number + + @property + def set(self): + if self.is_symbolic: + return Intersection(S.Naturals0, Interval(0, self.n)) + return set(map(Integer, range(self.n + 1))) + + def pmf(self, k): + n, a, b = self.n, self.alpha, self.beta + return binomial(n, k) * beta_fn(k + a, n - k + b) / beta_fn(a, b) + + +def BetaBinomial(name, n, alpha, beta): + r""" + Create a Finite Random Variable representing a Beta-binomial distribution. + + Parameters + ========== + + n : Positive Integer + Represents number of trials + alpha : Real positive number + beta : Real positive number + + Examples + ======== + + >>> from sympy.stats import BetaBinomial, density + + >>> X = BetaBinomial('X', 2, 1, 1) + >>> density(X).dict + {0: 1/3, 1: 2*beta(2, 2), 2: 1/3} + + Returns + ======= + + RandomSymbol + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Beta-binomial_distribution + .. [2] https://mathworld.wolfram.com/BetaBinomialDistribution.html + + """ + + return rv(name, BetaBinomialDistribution, n, alpha, beta) + + +class HypergeometricDistribution(SingleFiniteDistribution): + _argnames = ('N', 'm', 'n') + + @staticmethod + def check(n, N, m): + _value_check((N.is_integer, N.is_nonnegative), + "'N' must be nonnegative integer. N = %s." % str(N)) + _value_check((n.is_integer, n.is_nonnegative), + "'n' must be nonnegative integer. n = %s." % str(n)) + _value_check((m.is_integer, m.is_nonnegative), + "'m' must be nonnegative integer. m = %s." % str(m)) + + @property + def is_symbolic(self): + return not all(x.is_number for x in (self.N, self.m, self.n)) + + @property + def high(self): + return Piecewise((self.n, Lt(self.n, self.m) != False), (self.m, True)) + + @property + def low(self): + return Piecewise((0, Gt(0, self.n + self.m - self.N) != False), (self.n + self.m - self.N, True)) + + @property + def set(self): + N, m, n = self.N, self.m, self.n + if self.is_symbolic: + return Intersection(S.Naturals0, Interval(self.low, self.high)) + return set(range(max(0, n + m - N), min(n, m) + 1)) + + def pmf(self, k): + N, m, n = self.N, self.m, self.n + return S(binomial(m, k) * binomial(N - m, n - k))/binomial(N, n) + + +def Hypergeometric(name, N, m, n): + r""" + Create a Finite Random Variable representing a hypergeometric distribution. + + Parameters + ========== + + N : Positive Integer + Represents finite population of size N. + m : Positive Integer + Represents number of trials with required feature. + n : Positive Integer + Represents numbers of draws. + + + Examples + ======== + + >>> from sympy.stats import Hypergeometric, density + + >>> X = Hypergeometric('X', 10, 5, 3) # 10 marbles, 5 white (success), 3 draws + >>> density(X).dict + {0: 1/12, 1: 5/12, 2: 5/12, 3: 1/12} + + Returns + ======= + + RandomSymbol + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Hypergeometric_distribution + .. [2] https://mathworld.wolfram.com/HypergeometricDistribution.html + + """ + return rv(name, HypergeometricDistribution, N, m, n) + + +class RademacherDistribution(SingleFiniteDistribution): + + @property + def set(self): + return {-1, 1} + + @property + def pmf(self): + k = Dummy('k') + return Lambda(k, Piecewise((S.Half, Or(Eq(k, -1), Eq(k, 1))), (S.Zero, True))) + +def Rademacher(name): + r""" + Create a Finite Random Variable representing a Rademacher distribution. + + Examples + ======== + + >>> from sympy.stats import Rademacher, density + + >>> X = Rademacher('X') + >>> density(X).dict + {-1: 1/2, 1: 1/2} + + Returns + ======= + + RandomSymbol + + See Also + ======== + + sympy.stats.Bernoulli + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Rademacher_distribution + + """ + return rv(name, RademacherDistribution) + +class IdealSolitonDistribution(SingleFiniteDistribution): + _argnames = ('k',) + + @staticmethod + def check(k): + _value_check(k.is_integer and k.is_positive, + "'k' must be a positive integer.") + + @property + def low(self): + return S.One + + @property + def high(self): + return self.k + + @property + def set(self): + return set(map(Integer, range(1, self.k + 1))) + + @property # type: ignore + @cacheit + def dict(self): + if self.k.is_Symbol: + return Density(self) + d = {1: Rational(1, self.k)} + d.update({i: Rational(1, i*(i - 1)) for i in range(2, self.k + 1)}) + return d + + def pmf(self, x): + x = sympify(x) + if not (x.is_number or x.is_Symbol or is_random(x)): + raise ValueError("'x' expected as an argument of type 'number', 'Symbol', or " + "'RandomSymbol' not %s" % (type(x))) + cond1 = Eq(x, 1) & x.is_integer + cond2 = Ge(x, 1) & Le(x, self.k) & x.is_integer + return Piecewise((1/self.k, cond1), (1/(x*(x - 1)), cond2), (S.Zero, True)) + +def IdealSoliton(name, k): + r""" + Create a Finite Random Variable of Ideal Soliton Distribution + + Parameters + ========== + + k : Positive Integer + Represents the number of input symbols in an LT (Luby Transform) code. + + Examples + ======== + + >>> from sympy.stats import IdealSoliton, density, P, E + >>> sol = IdealSoliton('sol', 5) + >>> density(sol).dict + {1: 1/5, 2: 1/2, 3: 1/6, 4: 1/12, 5: 1/20} + >>> density(sol).set + {1, 2, 3, 4, 5} + + >>> from sympy import Symbol + >>> k = Symbol('k', positive=True, integer=True) + >>> sol = IdealSoliton('sol', k) + >>> density(sol).dict + Density(IdealSolitonDistribution(k)) + >>> density(sol).dict.subs(k, 10).doit() + {1: 1/10, 2: 1/2, 3: 1/6, 4: 1/12, 5: 1/20, 6: 1/30, 7: 1/42, 8: 1/56, 9: 1/72, 10: 1/90} + + >>> E(sol.subs(k, 10)) + 7381/2520 + + >>> P(sol.subs(k, 4) > 2) + 1/4 + + Returns + ======= + + RandomSymbol + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Soliton_distribution#Ideal_distribution + .. [2] https://pages.cs.wisc.edu/~suman/courses/740/papers/luby02lt.pdf + + """ + return rv(name, IdealSolitonDistribution, k) + +class RobustSolitonDistribution(SingleFiniteDistribution): + _argnames= ('k', 'delta', 'c') + + @staticmethod + def check(k, delta, c): + _value_check(k.is_integer and k.is_positive, + "'k' must be a positive integer") + _value_check(Gt(delta, 0) and Le(delta, 1), + "'delta' must be a real number in the interval (0,1)") + _value_check(c.is_positive, + "'c' must be a positive real number.") + + @property + def R(self): + return self.c * log(self.k/self.delta) * self.k**0.5 + + @property + def Z(self): + z = 0 + for i in Range(1, round(self.k/self.R)): + z += (1/i) + z += log(self.R/self.delta) + return 1 + z * self.R/self.k + + @property + def low(self): + return S.One + + @property + def high(self): + return self.k + + @property + def set(self): + return set(map(Integer, range(1, self.k + 1))) + + @property + def is_symbolic(self): + return not (self.k.is_number and self.c.is_number and self.delta.is_number) + + def pmf(self, x): + x = sympify(x) + if not (x.is_number or x.is_Symbol or is_random(x)): + raise ValueError("'x' expected as an argument of type 'number', 'Symbol', or " + "'RandomSymbol' not %s" % (type(x))) + + cond1 = Eq(x, 1) & x.is_integer + cond2 = Ge(x, 1) & Le(x, self.k) & x.is_integer + rho = Piecewise((Rational(1, self.k), cond1), (Rational(1, x*(x-1)), cond2), (S.Zero, True)) + + cond1 = Ge(x, 1) & Le(x, round(self.k/self.R)-1) + cond2 = Eq(x, round(self.k/self.R)) + tau = Piecewise((self.R/(self.k * x), cond1), (self.R * log(self.R/self.delta)/self.k, cond2), (S.Zero, True)) + + return (rho + tau)/self.Z + +def RobustSoliton(name, k, delta, c): + r''' + Create a Finite Random Variable of Robust Soliton Distribution + + Parameters + ========== + + k : Positive Integer + Represents the number of input symbols in an LT (Luby Transform) code. + delta : Positive Rational Number + Represents the failure probability. Must be in the interval (0,1). + c : Positive Rational Number + Constant of proportionality. Values close to 1 are recommended + + Examples + ======== + + >>> from sympy.stats import RobustSoliton, density, P, E + >>> robSol = RobustSoliton('robSol', 5, 0.5, 0.01) + >>> density(robSol).dict + {1: 0.204253668152708, 2: 0.490631107897393, 3: 0.165210624506162, 4: 0.0834387731899302, 5: 0.0505633404760675} + >>> density(robSol).set + {1, 2, 3, 4, 5} + + >>> from sympy import Symbol + >>> k = Symbol('k', positive=True, integer=True) + >>> c = Symbol('c', positive=True) + >>> robSol = RobustSoliton('robSol', k, 0.5, c) + >>> density(robSol).dict + Density(RobustSolitonDistribution(k, 0.5, c)) + >>> density(robSol).dict.subs(k, 10).subs(c, 0.03).doit() + {1: 0.116641095387194, 2: 0.467045731687165, 3: 0.159984123349381, 4: 0.0821431680681869, 5: 0.0505765646770100, + 6: 0.0345781523420719, 7: 0.0253132820710503, 8: 0.0194459129233227, 9: 0.0154831166726115, 10: 0.0126733075238887} + + >>> E(robSol.subs(k, 10).subs(c, 0.05)) + 2.91358846104106 + + >>> P(robSol.subs(k, 4).subs(c, 0.1) > 2) + 0.243650614389834 + + Returns + ======= + + RandomSymbol + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Soliton_distribution#Robust_distribution + .. [2] https://www.inference.org.uk/mackay/itprnn/ps/588.596.pdf + .. [3] https://pages.cs.wisc.edu/~suman/courses/740/papers/luby02lt.pdf + + ''' + return rv(name, RobustSolitonDistribution, k, delta, c) diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/joint_rv.py b/.venv/lib/python3.12/site-packages/sympy/stats/joint_rv.py new file mode 100644 index 0000000000000000000000000000000000000000..d147942f08b998e167b246628360fa27fc8ef348 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/joint_rv.py @@ -0,0 +1,426 @@ +""" +Joint Random Variables Module + +See Also +======== +sympy.stats.rv +sympy.stats.frv +sympy.stats.crv +sympy.stats.drv +""" +from math import prod + +from sympy.core.basic import Basic +from sympy.core.function import Lambda +from sympy.core.singleton import S +from sympy.core.symbol import (Dummy, Symbol) +from sympy.core.sympify import sympify +from sympy.sets.sets import ProductSet +from sympy.tensor.indexed import Indexed +from sympy.concrete.products import Product +from sympy.concrete.summations import Sum, summation +from sympy.core.containers import Tuple +from sympy.integrals.integrals import Integral, integrate +from sympy.matrices import ImmutableMatrix, matrix2numpy, list2numpy +from sympy.stats.crv import SingleContinuousDistribution, SingleContinuousPSpace +from sympy.stats.drv import SingleDiscreteDistribution, SingleDiscretePSpace +from sympy.stats.rv import (ProductPSpace, NamedArgsMixin, Distribution, + ProductDomain, RandomSymbol, random_symbols, + SingleDomain, _symbol_converter) +from sympy.utilities.iterables import iterable +from sympy.utilities.misc import filldedent +from sympy.external import import_module + +# __all__ = ['marginal_distribution'] + +class JointPSpace(ProductPSpace): + """ + Represents a joint probability space. Represented using symbols for + each component and a distribution. + """ + def __new__(cls, sym, dist): + if isinstance(dist, SingleContinuousDistribution): + return SingleContinuousPSpace(sym, dist) + if isinstance(dist, SingleDiscreteDistribution): + return SingleDiscretePSpace(sym, dist) + sym = _symbol_converter(sym) + return Basic.__new__(cls, sym, dist) + + @property + def set(self): + return self.domain.set + + @property + def symbol(self): + return self.args[0] + + @property + def distribution(self): + return self.args[1] + + @property + def value(self): + return JointRandomSymbol(self.symbol, self) + + @property + def component_count(self): + _set = self.distribution.set + if isinstance(_set, ProductSet): + return S(len(_set.args)) + elif isinstance(_set, Product): + return _set.limits[0][-1] + return S.One + + @property + def pdf(self): + sym = [Indexed(self.symbol, i) for i in range(self.component_count)] + return self.distribution(*sym) + + @property + def domain(self): + rvs = random_symbols(self.distribution) + if not rvs: + return SingleDomain(self.symbol, self.distribution.set) + return ProductDomain(*[rv.pspace.domain for rv in rvs]) + + def component_domain(self, index): + return self.set.args[index] + + def marginal_distribution(self, *indices): + count = self.component_count + if count.atoms(Symbol): + raise ValueError("Marginal distributions cannot be computed " + "for symbolic dimensions. It is a work under progress.") + orig = [Indexed(self.symbol, i) for i in range(count)] + all_syms = [Symbol(str(i)) for i in orig] + replace_dict = dict(zip(all_syms, orig)) + sym = tuple(Symbol(str(Indexed(self.symbol, i))) for i in indices) + limits = [[i,] for i in all_syms if i not in sym] + index = 0 + for i in range(count): + if i not in indices: + limits[index].append(self.distribution.set.args[i]) + limits[index] = tuple(limits[index]) + index += 1 + if self.distribution.is_Continuous: + f = Lambda(sym, integrate(self.distribution(*all_syms), *limits)) + elif self.distribution.is_Discrete: + f = Lambda(sym, summation(self.distribution(*all_syms), *limits)) + return f.xreplace(replace_dict) + + def compute_expectation(self, expr, rvs=None, evaluate=False, **kwargs): + syms = tuple(self.value[i] for i in range(self.component_count)) + rvs = rvs or syms + if not any(i in rvs for i in syms): + return expr + expr = expr*self.pdf + for rv in rvs: + if isinstance(rv, Indexed): + expr = expr.xreplace({rv: Indexed(str(rv.base), rv.args[1])}) + elif isinstance(rv, RandomSymbol): + expr = expr.xreplace({rv: rv.symbol}) + if self.value in random_symbols(expr): + raise NotImplementedError(filldedent(''' + Expectations of expression with unindexed joint random symbols + cannot be calculated yet.''')) + limits = tuple((Indexed(str(rv.base),rv.args[1]), + self.distribution.set.args[rv.args[1]]) for rv in syms) + return Integral(expr, *limits) + + def where(self, condition): + raise NotImplementedError() + + def compute_density(self, expr): + raise NotImplementedError() + + def sample(self, size=(), library='scipy', seed=None): + """ + Internal sample method + + Returns dictionary mapping RandomSymbol to realization value. + """ + return {RandomSymbol(self.symbol, self): self.distribution.sample(size, + library=library, seed=seed)} + + def probability(self, condition): + raise NotImplementedError() + + +class SampleJointScipy: + """Returns the sample from scipy of the given distribution""" + def __new__(cls, dist, size, seed=None): + return cls._sample_scipy(dist, size, seed) + + @classmethod + def _sample_scipy(cls, dist, size, seed): + """Sample from SciPy.""" + + import numpy + if seed is None or isinstance(seed, int): + rand_state = numpy.random.default_rng(seed=seed) + else: + rand_state = seed + from scipy import stats as scipy_stats + scipy_rv_map = { + 'MultivariateNormalDistribution': lambda dist, size: scipy_stats.multivariate_normal.rvs( + mean=matrix2numpy(dist.mu).flatten(), + cov=matrix2numpy(dist.sigma), size=size, random_state=rand_state), + 'MultivariateBetaDistribution': lambda dist, size: scipy_stats.dirichlet.rvs( + alpha=list2numpy(dist.alpha, float).flatten(), size=size, random_state=rand_state), + 'MultinomialDistribution': lambda dist, size: scipy_stats.multinomial.rvs( + n=int(dist.n), p=list2numpy(dist.p, float).flatten(), size=size, random_state=rand_state) + } + + sample_shape = { + 'MultivariateNormalDistribution': lambda dist: matrix2numpy(dist.mu).flatten().shape, + 'MultivariateBetaDistribution': lambda dist: list2numpy(dist.alpha).flatten().shape, + 'MultinomialDistribution': lambda dist: list2numpy(dist.p).flatten().shape + } + + dist_list = scipy_rv_map.keys() + + if dist.__class__.__name__ not in dist_list: + return None + + samples = scipy_rv_map[dist.__class__.__name__](dist, size) + return samples.reshape(size + sample_shape[dist.__class__.__name__](dist)) + +class SampleJointNumpy: + """Returns the sample from numpy of the given distribution""" + + def __new__(cls, dist, size, seed=None): + return cls._sample_numpy(dist, size, seed) + + @classmethod + def _sample_numpy(cls, dist, size, seed): + """Sample from NumPy.""" + + import numpy + if seed is None or isinstance(seed, int): + rand_state = numpy.random.default_rng(seed=seed) + else: + rand_state = seed + numpy_rv_map = { + 'MultivariateNormalDistribution': lambda dist, size: rand_state.multivariate_normal( + mean=matrix2numpy(dist.mu, float).flatten(), + cov=matrix2numpy(dist.sigma, float), size=size), + 'MultivariateBetaDistribution': lambda dist, size: rand_state.dirichlet( + alpha=list2numpy(dist.alpha, float).flatten(), size=size), + 'MultinomialDistribution': lambda dist, size: rand_state.multinomial( + n=int(dist.n), pvals=list2numpy(dist.p, float).flatten(), size=size) + } + + sample_shape = { + 'MultivariateNormalDistribution': lambda dist: matrix2numpy(dist.mu).flatten().shape, + 'MultivariateBetaDistribution': lambda dist: list2numpy(dist.alpha).flatten().shape, + 'MultinomialDistribution': lambda dist: list2numpy(dist.p).flatten().shape + } + + dist_list = numpy_rv_map.keys() + + if dist.__class__.__name__ not in dist_list: + return None + + samples = numpy_rv_map[dist.__class__.__name__](dist, prod(size)) + return samples.reshape(size + sample_shape[dist.__class__.__name__](dist)) + +class SampleJointPymc: + """Returns the sample from pymc of the given distribution""" + + def __new__(cls, dist, size, seed=None): + return cls._sample_pymc(dist, size, seed) + + @classmethod + def _sample_pymc(cls, dist, size, seed): + """Sample from PyMC.""" + + try: + import pymc + except ImportError: + import pymc3 as pymc + pymc_rv_map = { + 'MultivariateNormalDistribution': lambda dist: + pymc.MvNormal('X', mu=matrix2numpy(dist.mu, float).flatten(), + cov=matrix2numpy(dist.sigma, float), shape=(1, dist.mu.shape[0])), + 'MultivariateBetaDistribution': lambda dist: + pymc.Dirichlet('X', a=list2numpy(dist.alpha, float).flatten()), + 'MultinomialDistribution': lambda dist: + pymc.Multinomial('X', n=int(dist.n), + p=list2numpy(dist.p, float).flatten(), shape=(1, len(dist.p))) + } + + sample_shape = { + 'MultivariateNormalDistribution': lambda dist: matrix2numpy(dist.mu).flatten().shape, + 'MultivariateBetaDistribution': lambda dist: list2numpy(dist.alpha).flatten().shape, + 'MultinomialDistribution': lambda dist: list2numpy(dist.p).flatten().shape + } + + dist_list = pymc_rv_map.keys() + + if dist.__class__.__name__ not in dist_list: + return None + + import logging + logging.getLogger("pymc3").setLevel(logging.ERROR) + with pymc.Model(): + pymc_rv_map[dist.__class__.__name__](dist) + samples = pymc.sample(draws=prod(size), chains=1, progressbar=False, random_seed=seed, return_inferencedata=False, compute_convergence_checks=False)[:]['X'] + return samples.reshape(size + sample_shape[dist.__class__.__name__](dist)) + + +_get_sample_class_jrv = { + 'scipy': SampleJointScipy, + 'pymc3': SampleJointPymc, + 'pymc': SampleJointPymc, + 'numpy': SampleJointNumpy +} + +class JointDistribution(Distribution, NamedArgsMixin): + """ + Represented by the random variables part of the joint distribution. + Contains methods for PDF, CDF, sampling, marginal densities, etc. + """ + + _argnames = ('pdf', ) + + def __new__(cls, *args): + args = list(map(sympify, args)) + for i in range(len(args)): + if isinstance(args[i], list): + args[i] = ImmutableMatrix(args[i]) + return Basic.__new__(cls, *args) + + @property + def domain(self): + return ProductDomain(self.symbols) + + @property + def pdf(self): + return self.density.args[1] + + def cdf(self, other): + if not isinstance(other, dict): + raise ValueError("%s should be of type dict, got %s"%(other, type(other))) + rvs = other.keys() + _set = self.domain.set.sets + expr = self.pdf(tuple(i.args[0] for i in self.symbols)) + for i in range(len(other)): + if rvs[i].is_Continuous: + density = Integral(expr, (rvs[i], _set[i].inf, + other[rvs[i]])) + elif rvs[i].is_Discrete: + density = Sum(expr, (rvs[i], _set[i].inf, + other[rvs[i]])) + return density + + def sample(self, size=(), library='scipy', seed=None): + """ A random realization from the distribution """ + + libraries = ('scipy', 'numpy', 'pymc3', 'pymc') + if library not in libraries: + raise NotImplementedError("Sampling from %s is not supported yet." + % str(library)) + if not import_module(library): + raise ValueError("Failed to import %s" % library) + + samps = _get_sample_class_jrv[library](self, size, seed=seed) + + if samps is not None: + return samps + raise NotImplementedError( + "Sampling for %s is not currently implemented from %s" + % (self.__class__.__name__, library) + ) + + def __call__(self, *args): + return self.pdf(*args) + +class JointRandomSymbol(RandomSymbol): + """ + Representation of random symbols with joint probability distributions + to allow indexing." + """ + def __getitem__(self, key): + if isinstance(self.pspace, JointPSpace): + if (self.pspace.component_count <= key) == True: + raise ValueError("Index keys for %s can only up to %s." % + (self.name, self.pspace.component_count - 1)) + return Indexed(self, key) + + + +class MarginalDistribution(Distribution): + """ + Represents the marginal distribution of a joint probability space. + + Initialised using a probability distribution and random variables(or + their indexed components) which should be a part of the resultant + distribution. + """ + + def __new__(cls, dist, *rvs): + if len(rvs) == 1 and iterable(rvs[0]): + rvs = tuple(rvs[0]) + if not all(isinstance(rv, (Indexed, RandomSymbol)) for rv in rvs): + raise ValueError(filldedent('''Marginal distribution can be + intitialised only in terms of random variables or indexed random + variables''')) + rvs = Tuple.fromiter(rv for rv in rvs) + if not isinstance(dist, JointDistribution) and len(random_symbols(dist)) == 0: + return dist + return Basic.__new__(cls, dist, rvs) + + def check(self): + pass + + @property + def set(self): + rvs = [i for i in self.args[1] if isinstance(i, RandomSymbol)] + return ProductSet(*[rv.pspace.set for rv in rvs]) + + @property + def symbols(self): + rvs = self.args[1] + return {rv.pspace.symbol for rv in rvs} + + def pdf(self, *x): + expr, rvs = self.args[0], self.args[1] + marginalise_out = [i for i in random_symbols(expr) if i not in rvs] + if isinstance(expr, JointDistribution): + count = len(expr.domain.args) + x = Dummy('x', real=True) + syms = tuple(Indexed(x, i) for i in count) + expr = expr.pdf(syms) + else: + syms = tuple(rv.pspace.symbol if isinstance(rv, RandomSymbol) else rv.args[0] for rv in rvs) + return Lambda(syms, self.compute_pdf(expr, marginalise_out))(*x) + + def compute_pdf(self, expr, rvs): + for rv in rvs: + lpdf = 1 + if isinstance(rv, RandomSymbol): + lpdf = rv.pspace.pdf + expr = self.marginalise_out(expr*lpdf, rv) + return expr + + def marginalise_out(self, expr, rv): + from sympy.concrete.summations import Sum + if isinstance(rv, RandomSymbol): + dom = rv.pspace.set + elif isinstance(rv, Indexed): + dom = rv.base.component_domain( + rv.pspace.component_domain(rv.args[1])) + expr = expr.xreplace({rv: rv.pspace.symbol}) + if rv.pspace.is_Continuous: + #TODO: Modify to support integration + #for all kinds of sets. + expr = Integral(expr, (rv.pspace.symbol, dom)) + elif rv.pspace.is_Discrete: + #incorporate this into `Sum`/`summation` + if dom in (S.Integers, S.Naturals, S.Naturals0): + dom = (dom.inf, dom.sup) + expr = Sum(expr, (rv.pspace.symbol, dom)) + return expr + + def __call__(self, *args): + return self.pdf(*args) diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/joint_rv_types.py b/.venv/lib/python3.12/site-packages/sympy/stats/joint_rv_types.py new file mode 100644 index 0000000000000000000000000000000000000000..6cee9f9aa30897593ffb7c7b930a55a38f0c518a --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/joint_rv_types.py @@ -0,0 +1,945 @@ +from sympy.concrete.products import Product +from sympy.concrete.summations import Sum +from sympy.core.add import Add +from sympy.core.function import Lambda +from sympy.core.mul import Mul +from sympy.core.numbers import (Integer, Rational, pi) +from sympy.core.power import Pow +from sympy.core.relational import Eq +from sympy.core.singleton import S +from sympy.core.symbol import (Symbol, symbols) +from sympy.core.sympify import sympify +from sympy.functions.combinatorial.factorials import (rf, factorial) +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.special.bessel import besselk +from sympy.functions.special.gamma_functions import gamma +from sympy.matrices.dense import (Matrix, ones) +from sympy.sets.fancysets import Range +from sympy.sets.sets import (Intersection, Interval) +from sympy.tensor.indexed import (Indexed, IndexedBase) +from sympy.matrices import ImmutableMatrix, MatrixSymbol +from sympy.matrices.expressions.determinant import det +from sympy.matrices.expressions.matexpr import MatrixElement +from sympy.stats.joint_rv import JointDistribution, JointPSpace, MarginalDistribution +from sympy.stats.rv import _value_check, random_symbols + +__all__ = ['JointRV', +'MultivariateNormal', +'MultivariateLaplace', +'Dirichlet', +'GeneralizedMultivariateLogGamma', +'GeneralizedMultivariateLogGammaOmega', +'Multinomial', +'MultivariateBeta', +'MultivariateEwens', +'MultivariateT', +'NegativeMultinomial', +'NormalGamma' +] + +def multivariate_rv(cls, sym, *args): + args = list(map(sympify, args)) + dist = cls(*args) + args = dist.args + dist.check(*args) + return JointPSpace(sym, dist).value + + +def marginal_distribution(rv, *indices): + """ + Marginal distribution function of a joint random variable. + + Parameters + ========== + + rv : A random variable with a joint probability distribution. + indices : Component indices or the indexed random symbol + for which the joint distribution is to be calculated + + Returns + ======= + + A Lambda expression in `sym`. + + Examples + ======== + + >>> from sympy.stats import MultivariateNormal, marginal_distribution + >>> m = MultivariateNormal('X', [1, 2], [[2, 1], [1, 2]]) + >>> marginal_distribution(m, m[0])(1) + 1/(2*sqrt(pi)) + + """ + indices = list(indices) + for i in range(len(indices)): + if isinstance(indices[i], Indexed): + indices[i] = indices[i].args[1] + prob_space = rv.pspace + if not indices: + raise ValueError( + "At least one component for marginal density is needed.") + if hasattr(prob_space.distribution, '_marginal_distribution'): + return prob_space.distribution._marginal_distribution(indices, rv.symbol) + return prob_space.marginal_distribution(*indices) + + +class JointDistributionHandmade(JointDistribution): + + _argnames = ('pdf',) + is_Continuous = True + + @property + def set(self): + return self.args[1] + + +def JointRV(symbol, pdf, _set=None): + """ + Create a Joint Random Variable where each of its component is continuous, + given the following: + + Parameters + ========== + + symbol : Symbol + Represents name of the random variable. + pdf : A PDF in terms of indexed symbols of the symbol given + as the first argument + + NOTE + ==== + + As of now, the set for each component for a ``JointRV`` is + equal to the set of all integers, which cannot be changed. + + Examples + ======== + + >>> from sympy import exp, pi, Indexed, S + >>> from sympy.stats import density, JointRV + >>> x1, x2 = (Indexed('x', i) for i in (1, 2)) + >>> pdf = exp(-x1**2/2 + x1 - x2**2/2 - S(1)/2)/(2*pi) + >>> N1 = JointRV('x', pdf) #Multivariate Normal distribution + >>> density(N1)(1, 2) + exp(-2)/(2*pi) + + Returns + ======= + + RandomSymbol + + """ + #TODO: Add support for sets provided by the user + symbol = sympify(symbol) + syms = [i for i in pdf.free_symbols if isinstance(i, Indexed) + and i.base == IndexedBase(symbol)] + syms = tuple(sorted(syms, key = lambda index: index.args[1])) + _set = S.Reals**len(syms) + pdf = Lambda(syms, pdf) + dist = JointDistributionHandmade(pdf, _set) + jrv = JointPSpace(symbol, dist).value + rvs = random_symbols(pdf) + if len(rvs) != 0: + dist = MarginalDistribution(dist, (jrv,)) + return JointPSpace(symbol, dist).value + return jrv + +#------------------------------------------------------------------------------- +# Multivariate Normal distribution --------------------------------------------- + +class MultivariateNormalDistribution(JointDistribution): + _argnames = ('mu', 'sigma') + + is_Continuous=True + + @property + def set(self): + k = self.mu.shape[0] + return S.Reals**k + + @staticmethod + def check(mu, sigma): + _value_check(mu.shape[0] == sigma.shape[0], + "Size of the mean vector and covariance matrix are incorrect.") + #check if covariance matrix is positive semi definite or not. + if not isinstance(sigma, MatrixSymbol): + _value_check(sigma.is_positive_semidefinite, + "The covariance matrix must be positive semi definite. ") + + def pdf(self, *args): + mu, sigma = self.mu, self.sigma + k = mu.shape[0] + if len(args) == 1 and args[0].is_Matrix: + args = args[0] + else: + args = ImmutableMatrix(args) + x = args - mu + density = S.One/sqrt((2*pi)**(k)*det(sigma))*exp( + Rational(-1, 2)*x.transpose()*(sigma.inv()*x)) + return MatrixElement(density, 0, 0) + + def _marginal_distribution(self, indices, sym): + sym = ImmutableMatrix([Indexed(sym, i) for i in indices]) + _mu, _sigma = self.mu, self.sigma + k = self.mu.shape[0] + for i in range(k): + if i not in indices: + _mu = _mu.row_del(i) + _sigma = _sigma.col_del(i) + _sigma = _sigma.row_del(i) + return Lambda(tuple(sym), S.One/sqrt((2*pi)**(len(_mu))*det(_sigma))*exp( + Rational(-1, 2)*(_mu - sym).transpose()*(_sigma.inv()*\ + (_mu - sym)))[0]) + +def MultivariateNormal(name, mu, sigma): + r""" + Creates a continuous random variable with Multivariate Normal + Distribution. + + The density of the multivariate normal distribution can be found at [1]. + + Parameters + ========== + + mu : List representing the mean or the mean vector + sigma : Positive semidefinite square matrix + Represents covariance Matrix. + If `\sigma` is noninvertible then only sampling is supported currently + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import MultivariateNormal, density, marginal_distribution + >>> from sympy import symbols, MatrixSymbol + >>> X = MultivariateNormal('X', [3, 4], [[2, 1], [1, 2]]) + >>> y, z = symbols('y z') + >>> density(X)(y, z) + sqrt(3)*exp(-y**2/3 + y*z/3 + 2*y/3 - z**2/3 + 5*z/3 - 13/3)/(6*pi) + >>> density(X)(1, 2) + sqrt(3)*exp(-4/3)/(6*pi) + >>> marginal_distribution(X, X[1])(y) + exp(-(y - 4)**2/4)/(2*sqrt(pi)) + >>> marginal_distribution(X, X[0])(y) + exp(-(y - 3)**2/4)/(2*sqrt(pi)) + + The example below shows that it is also possible to use + symbolic parameters to define the MultivariateNormal class. + + >>> n = symbols('n', integer=True, positive=True) + >>> Sg = MatrixSymbol('Sg', n, n) + >>> mu = MatrixSymbol('mu', n, 1) + >>> obs = MatrixSymbol('obs', n, 1) + >>> X = MultivariateNormal('X', mu, Sg) + + The density of a multivariate normal can be + calculated using a matrix argument, as shown below. + + >>> density(X)(obs) + (exp(((1/2)*mu.T - (1/2)*obs.T)*Sg**(-1)*(-mu + obs))/sqrt((2*pi)**n*Determinant(Sg)))[0, 0] + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Multivariate_normal_distribution + + """ + return multivariate_rv(MultivariateNormalDistribution, name, mu, sigma) + +#------------------------------------------------------------------------------- +# Multivariate Laplace distribution -------------------------------------------- + +class MultivariateLaplaceDistribution(JointDistribution): + _argnames = ('mu', 'sigma') + is_Continuous=True + + @property + def set(self): + k = self.mu.shape[0] + return S.Reals**k + + @staticmethod + def check(mu, sigma): + _value_check(mu.shape[0] == sigma.shape[0], + "Size of the mean vector and covariance matrix are incorrect.") + # check if covariance matrix is positive definite or not. + if not isinstance(sigma, MatrixSymbol): + _value_check(sigma.is_positive_definite, + "The covariance matrix must be positive definite. ") + + def pdf(self, *args): + mu, sigma = self.mu, self.sigma + mu_T = mu.transpose() + k = S(mu.shape[0]) + sigma_inv = sigma.inv() + args = ImmutableMatrix(args) + args_T = args.transpose() + x = (mu_T*sigma_inv*mu)[0] + y = (args_T*sigma_inv*args)[0] + v = 1 - k/2 + return (2 * (y/(2 + x))**(v/2) * besselk(v, sqrt((2 + x)*y)) * + exp((args_T * sigma_inv * mu)[0]) / + ((2 * pi)**(k/2) * sqrt(det(sigma)))) + + +def MultivariateLaplace(name, mu, sigma): + """ + Creates a continuous random variable with Multivariate Laplace + Distribution. + + The density of the multivariate Laplace distribution can be found at [1]. + + Parameters + ========== + + mu : List representing the mean or the mean vector + sigma : Positive definite square matrix + Represents covariance Matrix + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import MultivariateLaplace, density + >>> from sympy import symbols + >>> y, z = symbols('y z') + >>> X = MultivariateLaplace('X', [2, 4], [[3, 1], [1, 3]]) + >>> density(X)(y, z) + sqrt(2)*exp(y/4 + 5*z/4)*besselk(0, sqrt(15*y*(3*y/8 - z/8)/2 + 15*z*(-y/8 + 3*z/8)/2))/(4*pi) + >>> density(X)(1, 2) + sqrt(2)*exp(11/4)*besselk(0, sqrt(165)/4)/(4*pi) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Multivariate_Laplace_distribution + + """ + return multivariate_rv(MultivariateLaplaceDistribution, name, mu, sigma) + +#------------------------------------------------------------------------------- +# Multivariate StudentT distribution ------------------------------------------- + +class MultivariateTDistribution(JointDistribution): + _argnames = ('mu', 'shape_mat', 'dof') + is_Continuous=True + + @property + def set(self): + k = self.mu.shape[0] + return S.Reals**k + + @staticmethod + def check(mu, sigma, v): + _value_check(mu.shape[0] == sigma.shape[0], + "Size of the location vector and shape matrix are incorrect.") + # check if covariance matrix is positive definite or not. + if not isinstance(sigma, MatrixSymbol): + _value_check(sigma.is_positive_definite, + "The shape matrix must be positive definite. ") + + def pdf(self, *args): + mu, sigma = self.mu, self.shape_mat + v = S(self.dof) + k = S(mu.shape[0]) + sigma_inv = sigma.inv() + args = ImmutableMatrix(args) + x = args - mu + return gamma((k + v)/2)/(gamma(v/2)*(v*pi)**(k/2)*sqrt(det(sigma)))\ + *(1 + 1/v*(x.transpose()*sigma_inv*x)[0])**((-v - k)/2) + +def MultivariateT(syms, mu, sigma, v): + """ + Creates a joint random variable with multivariate T-distribution. + + Parameters + ========== + + syms : A symbol/str + For identifying the random variable. + mu : A list/matrix + Representing the location vector + sigma : The shape matrix for the distribution + + Examples + ======== + + >>> from sympy.stats import density, MultivariateT + >>> from sympy import Symbol + + >>> x = Symbol("x") + >>> X = MultivariateT("x", [1, 1], [[1, 0], [0, 1]], 2) + + >>> density(X)(1, 2) + 2/(9*pi) + + Returns + ======= + + RandomSymbol + + """ + return multivariate_rv(MultivariateTDistribution, syms, mu, sigma, v) + + +#------------------------------------------------------------------------------- +# Multivariate Normal Gamma distribution --------------------------------------- + +class NormalGammaDistribution(JointDistribution): + + _argnames = ('mu', 'lamda', 'alpha', 'beta') + is_Continuous=True + + @staticmethod + def check(mu, lamda, alpha, beta): + _value_check(mu.is_real, "Location must be real.") + _value_check(lamda > 0, "Lambda must be positive") + _value_check(alpha > 0, "alpha must be positive") + _value_check(beta > 0, "beta must be positive") + + @property + def set(self): + return S.Reals*Interval(0, S.Infinity) + + def pdf(self, x, tau): + beta, alpha, lamda = self.beta, self.alpha, self.lamda + mu = self.mu + + return beta**alpha*sqrt(lamda)/(gamma(alpha)*sqrt(2*pi))*\ + tau**(alpha - S.Half)*exp(-1*beta*tau)*\ + exp(-1*(lamda*tau*(x - mu)**2)/S(2)) + + def _marginal_distribution(self, indices, *sym): + if len(indices) == 2: + return self.pdf(*sym) + if indices[0] == 0: + #For marginal over `x`, return non-standardized Student-T's + #distribution + x = sym[0] + v, mu, sigma = self.alpha - S.Half, self.mu, \ + S(self.beta)/(self.lamda * self.alpha) + return Lambda(sym, gamma((v + 1)/2)/(gamma(v/2)*sqrt(pi*v)*sigma)*\ + (1 + 1/v*((x - mu)/sigma)**2)**((-v -1)/2)) + #For marginal over `tau`, return Gamma distribution as per construction + from sympy.stats.crv_types import GammaDistribution + return Lambda(sym, GammaDistribution(self.alpha, self.beta)(sym[0])) + +def NormalGamma(sym, mu, lamda, alpha, beta): + """ + Creates a bivariate joint random variable with multivariate Normal gamma + distribution. + + Parameters + ========== + + sym : A symbol/str + For identifying the random variable. + mu : A real number + The mean of the normal distribution + lamda : A positive integer + Parameter of joint distribution + alpha : A positive integer + Parameter of joint distribution + beta : A positive integer + Parameter of joint distribution + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import density, NormalGamma + >>> from sympy import symbols + + >>> X = NormalGamma('x', 0, 1, 2, 3) + >>> y, z = symbols('y z') + + >>> density(X)(y, z) + 9*sqrt(2)*z**(3/2)*exp(-3*z)*exp(-y**2*z/2)/(2*sqrt(pi)) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Normal-gamma_distribution + + """ + return multivariate_rv(NormalGammaDistribution, sym, mu, lamda, alpha, beta) + +#------------------------------------------------------------------------------- +# Multivariate Beta/Dirichlet distribution ------------------------------------- + +class MultivariateBetaDistribution(JointDistribution): + + _argnames = ('alpha',) + is_Continuous = True + + @staticmethod + def check(alpha): + _value_check(len(alpha) >= 2, "At least two categories should be passed.") + for a_k in alpha: + _value_check((a_k > 0) != False, "Each concentration parameter" + " should be positive.") + + @property + def set(self): + k = len(self.alpha) + return Interval(0, 1)**k + + def pdf(self, *syms): + alpha = self.alpha + B = Mul.fromiter(map(gamma, alpha))/gamma(Add(*alpha)) + return Mul.fromiter(sym**(a_k - 1) for a_k, sym in zip(alpha, syms))/B + +def MultivariateBeta(syms, *alpha): + """ + Creates a continuous random variable with Dirichlet/Multivariate Beta + Distribution. + + The density of the Dirichlet distribution can be found at [1]. + + Parameters + ========== + + alpha : Positive real numbers + Signifies concentration numbers. + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import density, MultivariateBeta, marginal_distribution + >>> from sympy import Symbol + >>> a1 = Symbol('a1', positive=True) + >>> a2 = Symbol('a2', positive=True) + >>> B = MultivariateBeta('B', [a1, a2]) + >>> C = MultivariateBeta('C', a1, a2) + >>> x = Symbol('x') + >>> y = Symbol('y') + >>> density(B)(x, y) + x**(a1 - 1)*y**(a2 - 1)*gamma(a1 + a2)/(gamma(a1)*gamma(a2)) + >>> marginal_distribution(C, C[0])(x) + x**(a1 - 1)*gamma(a1 + a2)/(a2*gamma(a1)*gamma(a2)) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Dirichlet_distribution + .. [2] https://mathworld.wolfram.com/DirichletDistribution.html + + """ + if not isinstance(alpha[0], list): + alpha = (list(alpha),) + return multivariate_rv(MultivariateBetaDistribution, syms, alpha[0]) + +Dirichlet = MultivariateBeta + +#------------------------------------------------------------------------------- +# Multivariate Ewens distribution ---------------------------------------------- + +class MultivariateEwensDistribution(JointDistribution): + + _argnames = ('n', 'theta') + is_Discrete = True + is_Continuous = False + + @staticmethod + def check(n, theta): + _value_check((n > 0), + "sample size should be positive integer.") + _value_check(theta.is_positive, "mutation rate should be positive.") + + @property + def set(self): + if not isinstance(self.n, Integer): + i = Symbol('i', integer=True, positive=True) + return Product(Intersection(S.Naturals0, Interval(0, self.n//i)), + (i, 1, self.n)) + prod_set = Range(0, self.n + 1) + for i in range(2, self.n + 1): + prod_set *= Range(0, self.n//i + 1) + return prod_set.flatten() + + def pdf(self, *syms): + n, theta = self.n, self.theta + condi = isinstance(self.n, Integer) + if not (isinstance(syms[0], IndexedBase) or condi): + raise ValueError("Please use IndexedBase object for syms as " + "the dimension is symbolic") + term_1 = factorial(n)/rf(theta, n) + if condi: + term_2 = Mul.fromiter(theta**syms[j]/((j+1)**syms[j]*factorial(syms[j])) + for j in range(n)) + cond = Eq(sum((k + 1)*syms[k] for k in range(n)), n) + return Piecewise((term_1 * term_2, cond), (0, True)) + syms = syms[0] + j, k = symbols('j, k', positive=True, integer=True) + term_2 = Product(theta**syms[j]/((j+1)**syms[j]*factorial(syms[j])), + (j, 0, n - 1)) + cond = Eq(Sum((k + 1)*syms[k], (k, 0, n - 1)), n) + return Piecewise((term_1 * term_2, cond), (0, True)) + + +def MultivariateEwens(syms, n, theta): + """ + Creates a discrete random variable with Multivariate Ewens + Distribution. + + The density of the said distribution can be found at [1]. + + Parameters + ========== + + n : Positive integer + Size of the sample or the integer whose partitions are considered + theta : Positive real number + Denotes Mutation rate + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import density, marginal_distribution, MultivariateEwens + >>> from sympy import Symbol + >>> a1 = Symbol('a1', positive=True) + >>> a2 = Symbol('a2', positive=True) + >>> ed = MultivariateEwens('E', 2, 1) + >>> density(ed)(a1, a2) + Piecewise((1/(2**a2*factorial(a1)*factorial(a2)), Eq(a1 + 2*a2, 2)), (0, True)) + >>> marginal_distribution(ed, ed[0])(a1) + Piecewise((1/factorial(a1), Eq(a1, 2)), (0, True)) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Ewens%27s_sampling_formula + .. [2] https://www.jstor.org/stable/24780825 + """ + return multivariate_rv(MultivariateEwensDistribution, syms, n, theta) + +#------------------------------------------------------------------------------- +# Generalized Multivariate Log Gamma distribution ------------------------------ + +class GeneralizedMultivariateLogGammaDistribution(JointDistribution): + + _argnames = ('delta', 'v', 'lamda', 'mu') + is_Continuous=True + + def check(self, delta, v, l, mu): + _value_check((delta >= 0, delta <= 1), "delta must be in range [0, 1].") + _value_check((v > 0), "v must be positive") + for lk in l: + _value_check((lk > 0), "lamda must be a positive vector.") + for muk in mu: + _value_check((muk > 0), "mu must be a positive vector.") + _value_check(len(l) > 1,"the distribution should have at least" + " two random variables.") + + @property + def set(self): + return S.Reals**len(self.lamda) + + def pdf(self, *y): + d, v, l, mu = self.delta, self.v, self.lamda, self.mu + n = Symbol('n', negative=False, integer=True) + k = len(l) + sterm1 = Pow((1 - d), n)/\ + ((gamma(v + n)**(k - 1))*gamma(v)*gamma(n + 1)) + sterm2 = Mul.fromiter(mui*li**(-v - n) for mui, li in zip(mu, l)) + term1 = sterm1 * sterm2 + sterm3 = (v + n) * sum(mui * yi for mui, yi in zip(mu, y)) + sterm4 = sum(exp(mui * yi)/li for (mui, yi, li) in zip(mu, y, l)) + term2 = exp(sterm3 - sterm4) + return Pow(d, v) * Sum(term1 * term2, (n, 0, S.Infinity)) + +def GeneralizedMultivariateLogGamma(syms, delta, v, lamda, mu): + """ + Creates a joint random variable with generalized multivariate log gamma + distribution. + + The joint pdf can be found at [1]. + + Parameters + ========== + + syms : list/tuple/set of symbols for identifying each component + delta : A constant in range $[0, 1]$ + v : Positive real number + lamda : List of positive real numbers + mu : List of positive real numbers + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import density + >>> from sympy.stats.joint_rv_types import GeneralizedMultivariateLogGamma + >>> from sympy import symbols, S + >>> v = 1 + >>> l, mu = [1, 1, 1], [1, 1, 1] + >>> d = S.Half + >>> y = symbols('y_1:4', positive=True) + >>> Gd = GeneralizedMultivariateLogGamma('G', d, v, l, mu) + >>> density(Gd)(y[0], y[1], y[2]) + Sum(exp((n + 1)*(y_1 + y_2 + y_3) - exp(y_1) - exp(y_2) - + exp(y_3))/(2**n*gamma(n + 1)**3), (n, 0, oo))/2 + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Generalized_multivariate_log-gamma_distribution + .. [2] https://www.researchgate.net/publication/234137346_On_a_multivariate_log-gamma_distribution_and_the_use_of_the_distribution_in_the_Bayesian_analysis + + Note + ==== + + If the GeneralizedMultivariateLogGamma is too long to type use, + + >>> from sympy.stats.joint_rv_types import GeneralizedMultivariateLogGamma as GMVLG + >>> Gd = GMVLG('G', d, v, l, mu) + + If you want to pass the matrix omega instead of the constant delta, then use + ``GeneralizedMultivariateLogGammaOmega``. + + """ + return multivariate_rv(GeneralizedMultivariateLogGammaDistribution, + syms, delta, v, lamda, mu) + +def GeneralizedMultivariateLogGammaOmega(syms, omega, v, lamda, mu): + """ + Extends GeneralizedMultivariateLogGamma. + + Parameters + ========== + + syms : list/tuple/set of symbols + For identifying each component + omega : A square matrix + Every element of square matrix must be absolute value of + square root of correlation coefficient + v : Positive real number + lamda : List of positive real numbers + mu : List of positive real numbers + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import density + >>> from sympy.stats.joint_rv_types import GeneralizedMultivariateLogGammaOmega + >>> from sympy import Matrix, symbols, S + >>> omega = Matrix([[1, S.Half, S.Half], [S.Half, 1, S.Half], [S.Half, S.Half, 1]]) + >>> v = 1 + >>> l, mu = [1, 1, 1], [1, 1, 1] + >>> G = GeneralizedMultivariateLogGammaOmega('G', omega, v, l, mu) + >>> y = symbols('y_1:4', positive=True) + >>> density(G)(y[0], y[1], y[2]) + sqrt(2)*Sum((1 - sqrt(2)/2)**n*exp((n + 1)*(y_1 + y_2 + y_3) - exp(y_1) - + exp(y_2) - exp(y_3))/gamma(n + 1)**3, (n, 0, oo))/2 + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Generalized_multivariate_log-gamma_distribution + .. [2] https://www.researchgate.net/publication/234137346_On_a_multivariate_log-gamma_distribution_and_the_use_of_the_distribution_in_the_Bayesian_analysis + + Notes + ===== + + If the GeneralizedMultivariateLogGammaOmega is too long to type use, + + >>> from sympy.stats.joint_rv_types import GeneralizedMultivariateLogGammaOmega as GMVLGO + >>> G = GMVLGO('G', omega, v, l, mu) + + """ + _value_check((omega.is_square, isinstance(omega, Matrix)), "omega must be a" + " square matrix") + for val in omega.values(): + _value_check((val >= 0, val <= 1), + "all values in matrix must be between 0 and 1(both inclusive).") + _value_check(omega.diagonal().equals(ones(1, omega.shape[0])), + "all the elements of diagonal should be 1.") + _value_check((omega.shape[0] == len(lamda), len(lamda) == len(mu)), + "lamda, mu should be of same length and omega should " + " be of shape (length of lamda, length of mu)") + _value_check(len(lamda) > 1,"the distribution should have at least" + " two random variables.") + delta = Pow(Rational(omega.det()), Rational(1, len(lamda) - 1)) + return GeneralizedMultivariateLogGamma(syms, delta, v, lamda, mu) + + +#------------------------------------------------------------------------------- +# Multinomial distribution ----------------------------------------------------- + +class MultinomialDistribution(JointDistribution): + + _argnames = ('n', 'p') + is_Continuous=False + is_Discrete = True + + @staticmethod + def check(n, p): + _value_check(n > 0, + "number of trials must be a positive integer") + for p_k in p: + _value_check((p_k >= 0, p_k <= 1), + "probability must be in range [0, 1]") + _value_check(Eq(sum(p), 1), + "probabilities must sum to 1") + + @property + def set(self): + return Intersection(S.Naturals0, Interval(0, self.n))**len(self.p) + + def pdf(self, *x): + n, p = self.n, self.p + term_1 = factorial(n)/Mul.fromiter(factorial(x_k) for x_k in x) + term_2 = Mul.fromiter(p_k**x_k for p_k, x_k in zip(p, x)) + return Piecewise((term_1 * term_2, Eq(sum(x), n)), (0, True)) + +def Multinomial(syms, n, *p): + """ + Creates a discrete random variable with Multinomial Distribution. + + The density of the said distribution can be found at [1]. + + Parameters + ========== + + n : Positive integer + Represents number of trials + p : List of event probabilities + Must be in the range of $[0, 1]$. + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import density, Multinomial, marginal_distribution + >>> from sympy import symbols + >>> x1, x2, x3 = symbols('x1, x2, x3', nonnegative=True, integer=True) + >>> p1, p2, p3 = symbols('p1, p2, p3', positive=True) + >>> M = Multinomial('M', 3, p1, p2, p3) + >>> density(M)(x1, x2, x3) + Piecewise((6*p1**x1*p2**x2*p3**x3/(factorial(x1)*factorial(x2)*factorial(x3)), + Eq(x1 + x2 + x3, 3)), (0, True)) + >>> marginal_distribution(M, M[0])(x1).subs(x1, 1) + 3*p1*p2**2 + 6*p1*p2*p3 + 3*p1*p3**2 + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Multinomial_distribution + .. [2] https://mathworld.wolfram.com/MultinomialDistribution.html + + """ + if not isinstance(p[0], list): + p = (list(p), ) + return multivariate_rv(MultinomialDistribution, syms, n, p[0]) + +#------------------------------------------------------------------------------- +# Negative Multinomial Distribution -------------------------------------------- + +class NegativeMultinomialDistribution(JointDistribution): + + _argnames = ('k0', 'p') + is_Continuous=False + is_Discrete = True + + @staticmethod + def check(k0, p): + _value_check(k0 > 0, + "number of failures must be a positive integer") + for p_k in p: + _value_check((p_k >= 0, p_k <= 1), + "probability must be in range [0, 1].") + _value_check(sum(p) <= 1, + "success probabilities must not be greater than 1.") + + @property + def set(self): + return Range(0, S.Infinity)**len(self.p) + + def pdf(self, *k): + k0, p = self.k0, self.p + term_1 = (gamma(k0 + sum(k))*(1 - sum(p))**k0)/gamma(k0) + term_2 = Mul.fromiter(pi**ki/factorial(ki) for pi, ki in zip(p, k)) + return term_1 * term_2 + +def NegativeMultinomial(syms, k0, *p): + """ + Creates a discrete random variable with Negative Multinomial Distribution. + + The density of the said distribution can be found at [1]. + + Parameters + ========== + + k0 : positive integer + Represents number of failures before the experiment is stopped + p : List of event probabilities + Must be in the range of $[0, 1]$ + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import density, NegativeMultinomial, marginal_distribution + >>> from sympy import symbols + >>> x1, x2, x3 = symbols('x1, x2, x3', nonnegative=True, integer=True) + >>> p1, p2, p3 = symbols('p1, p2, p3', positive=True) + >>> N = NegativeMultinomial('M', 3, p1, p2, p3) + >>> N_c = NegativeMultinomial('M', 3, 0.1, 0.1, 0.1) + >>> density(N)(x1, x2, x3) + p1**x1*p2**x2*p3**x3*(-p1 - p2 - p3 + 1)**3*gamma(x1 + x2 + + x3 + 3)/(2*factorial(x1)*factorial(x2)*factorial(x3)) + >>> marginal_distribution(N_c, N_c[0])(1).evalf().round(2) + 0.25 + + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Negative_multinomial_distribution + .. [2] https://mathworld.wolfram.com/NegativeBinomialDistribution.html + + """ + if not isinstance(p[0], list): + p = (list(p), ) + return multivariate_rv(NegativeMultinomialDistribution, syms, k0, p[0]) diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/matrix_distributions.py b/.venv/lib/python3.12/site-packages/sympy/stats/matrix_distributions.py new file mode 100644 index 0000000000000000000000000000000000000000..9a43c0226bc25702211a910ebbe30e280ad0cf50 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/matrix_distributions.py @@ -0,0 +1,610 @@ +from math import prod + +from sympy.core.basic import Basic +from sympy.core.numbers import pi +from sympy.core.singleton import S +from sympy.functions.elementary.exponential import exp +from sympy.functions.special.gamma_functions import multigamma +from sympy.core.sympify import sympify, _sympify +from sympy.matrices import (ImmutableMatrix, Inverse, Trace, Determinant, + MatrixSymbol, MatrixBase, Transpose, MatrixSet, + matrix2numpy) +from sympy.stats.rv import (_value_check, RandomMatrixSymbol, NamedArgsMixin, PSpace, + _symbol_converter, MatrixDomain, Distribution) +from sympy.external import import_module + + +################################################################################ +#------------------------Matrix Probability Space------------------------------# +################################################################################ +class MatrixPSpace(PSpace): + """ + Represents probability space for + Matrix Distributions. + """ + def __new__(cls, sym, distribution, dim_n, dim_m): + sym = _symbol_converter(sym) + dim_n, dim_m = _sympify(dim_n), _sympify(dim_m) + if not (dim_n.is_integer and dim_m.is_integer): + raise ValueError("Dimensions should be integers") + return Basic.__new__(cls, sym, distribution, dim_n, dim_m) + + distribution = property(lambda self: self.args[1]) + symbol = property(lambda self: self.args[0]) + + @property + def domain(self): + return MatrixDomain(self.symbol, self.distribution.set) + + @property + def value(self): + return RandomMatrixSymbol(self.symbol, self.args[2], self.args[3], self) + + @property + def values(self): + return {self.value} + + def compute_density(self, expr, *args): + rms = expr.atoms(RandomMatrixSymbol) + if len(rms) > 1 or (not isinstance(expr, RandomMatrixSymbol)): + raise NotImplementedError("Currently, no algorithm has been " + "implemented to handle general expressions containing " + "multiple matrix distributions.") + return self.distribution.pdf(expr) + + def sample(self, size=(), library='scipy', seed=None): + """ + Internal sample method + + Returns dictionary mapping RandomMatrixSymbol to realization value. + """ + return {self.value: self.distribution.sample(size, library=library, seed=seed)} + + +def rv(symbol, cls, args): + args = list(map(sympify, args)) + dist = cls(*args) + dist.check(*args) + dim = dist.dimension + pspace = MatrixPSpace(symbol, dist, dim[0], dim[1]) + return pspace.value + + +class SampleMatrixScipy: + """Returns the sample from scipy of the given distribution""" + def __new__(cls, dist, size, seed=None): + return cls._sample_scipy(dist, size, seed) + + @classmethod + def _sample_scipy(cls, dist, size, seed): + """Sample from SciPy.""" + + from scipy import stats as scipy_stats + import numpy + scipy_rv_map = { + 'WishartDistribution': lambda dist, size, rand_state: scipy_stats.wishart.rvs( + df=int(dist.n), scale=matrix2numpy(dist.scale_matrix, float), size=size), + 'MatrixNormalDistribution': lambda dist, size, rand_state: scipy_stats.matrix_normal.rvs( + mean=matrix2numpy(dist.location_matrix, float), + rowcov=matrix2numpy(dist.scale_matrix_1, float), + colcov=matrix2numpy(dist.scale_matrix_2, float), size=size, random_state=rand_state) + } + + sample_shape = { + 'WishartDistribution': lambda dist: dist.scale_matrix.shape, + 'MatrixNormalDistribution' : lambda dist: dist.location_matrix.shape + } + + dist_list = scipy_rv_map.keys() + + if dist.__class__.__name__ not in dist_list: + return None + + if seed is None or isinstance(seed, int): + rand_state = numpy.random.default_rng(seed=seed) + else: + rand_state = seed + samp = scipy_rv_map[dist.__class__.__name__](dist, prod(size), rand_state) + return samp.reshape(size + sample_shape[dist.__class__.__name__](dist)) + + +class SampleMatrixNumpy: + """Returns the sample from numpy of the given distribution""" + + ### TODO: Add tests after adding matrix distributions in numpy_rv_map + def __new__(cls, dist, size, seed=None): + return cls._sample_numpy(dist, size, seed) + + @classmethod + def _sample_numpy(cls, dist, size, seed): + """Sample from NumPy.""" + + numpy_rv_map = { + } + + sample_shape = { + } + + dist_list = numpy_rv_map.keys() + + if dist.__class__.__name__ not in dist_list: + return None + + import numpy + if seed is None or isinstance(seed, int): + rand_state = numpy.random.default_rng(seed=seed) + else: + rand_state = seed + samp = numpy_rv_map[dist.__class__.__name__](dist, prod(size), rand_state) + return samp.reshape(size + sample_shape[dist.__class__.__name__](dist)) + + +class SampleMatrixPymc: + """Returns the sample from pymc of the given distribution""" + + def __new__(cls, dist, size, seed=None): + return cls._sample_pymc(dist, size, seed) + + @classmethod + def _sample_pymc(cls, dist, size, seed): + """Sample from PyMC.""" + + try: + import pymc + except ImportError: + import pymc3 as pymc + pymc_rv_map = { + 'MatrixNormalDistribution': lambda dist: pymc.MatrixNormal('X', + mu=matrix2numpy(dist.location_matrix, float), + rowcov=matrix2numpy(dist.scale_matrix_1, float), + colcov=matrix2numpy(dist.scale_matrix_2, float), + shape=dist.location_matrix.shape), + 'WishartDistribution': lambda dist: pymc.WishartBartlett('X', + nu=int(dist.n), S=matrix2numpy(dist.scale_matrix, float)) + } + + sample_shape = { + 'WishartDistribution': lambda dist: dist.scale_matrix.shape, + 'MatrixNormalDistribution' : lambda dist: dist.location_matrix.shape + } + + dist_list = pymc_rv_map.keys() + + if dist.__class__.__name__ not in dist_list: + return None + import logging + logging.getLogger("pymc").setLevel(logging.ERROR) + with pymc.Model(): + pymc_rv_map[dist.__class__.__name__](dist) + samps = pymc.sample(draws=prod(size), chains=1, progressbar=False, random_seed=seed, return_inferencedata=False, compute_convergence_checks=False)['X'] + return samps.reshape(size + sample_shape[dist.__class__.__name__](dist)) + +_get_sample_class_matrixrv = { + 'scipy': SampleMatrixScipy, + 'pymc3': SampleMatrixPymc, + 'pymc': SampleMatrixPymc, + 'numpy': SampleMatrixNumpy +} + +################################################################################ +#-------------------------Matrix Distribution----------------------------------# +################################################################################ + +class MatrixDistribution(Distribution, NamedArgsMixin): + """ + Abstract class for Matrix Distribution. + """ + def __new__(cls, *args): + args = [ImmutableMatrix(arg) if isinstance(arg, list) + else _sympify(arg) for arg in args] + return Basic.__new__(cls, *args) + + @staticmethod + def check(*args): + pass + + def __call__(self, expr): + if isinstance(expr, list): + expr = ImmutableMatrix(expr) + return self.pdf(expr) + + def sample(self, size=(), library='scipy', seed=None): + """ + Internal sample method + + Returns dictionary mapping RandomSymbol to realization value. + """ + + libraries = ['scipy', 'numpy', 'pymc3', 'pymc'] + if library not in libraries: + raise NotImplementedError("Sampling from %s is not supported yet." + % str(library)) + if not import_module(library): + raise ValueError("Failed to import %s" % library) + + samps = _get_sample_class_matrixrv[library](self, size, seed) + + if samps is not None: + return samps + raise NotImplementedError( + "Sampling for %s is not currently implemented from %s" + % (self.__class__.__name__, library) + ) + +################################################################################ +#------------------------Matrix Distribution Types-----------------------------# +################################################################################ + +#------------------------------------------------------------------------------- +# Matrix Gamma distribution ---------------------------------------------------- + +class MatrixGammaDistribution(MatrixDistribution): + + _argnames = ('alpha', 'beta', 'scale_matrix') + + @staticmethod + def check(alpha, beta, scale_matrix): + if not isinstance(scale_matrix, MatrixSymbol): + _value_check(scale_matrix.is_positive_definite, "The shape " + "matrix must be positive definite.") + _value_check(scale_matrix.is_square, "Should " + "be square matrix") + _value_check(alpha.is_positive, "Shape parameter should be positive.") + _value_check(beta.is_positive, "Scale parameter should be positive.") + + @property + def set(self): + k = self.scale_matrix.shape[0] + return MatrixSet(k, k, S.Reals) + + @property + def dimension(self): + return self.scale_matrix.shape + + def pdf(self, x): + alpha, beta, scale_matrix = self.alpha, self.beta, self.scale_matrix + p = scale_matrix.shape[0] + if isinstance(x, list): + x = ImmutableMatrix(x) + if not isinstance(x, (MatrixBase, MatrixSymbol)): + raise ValueError("%s should be an isinstance of Matrix " + "or MatrixSymbol" % str(x)) + sigma_inv_x = - Inverse(scale_matrix)*x / beta + term1 = exp(Trace(sigma_inv_x))/((beta**(p*alpha)) * multigamma(alpha, p)) + term2 = (Determinant(scale_matrix))**(-alpha) + term3 = (Determinant(x))**(alpha - S(p + 1)/2) + return term1 * term2 * term3 + +def MatrixGamma(symbol, alpha, beta, scale_matrix): + """ + Creates a random variable with Matrix Gamma Distribution. + + The density of the said distribution can be found at [1]. + + Parameters + ========== + + alpha: Positive Real number + Shape Parameter + beta: Positive Real number + Scale Parameter + scale_matrix: Positive definite real square matrix + Scale Matrix + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import density, MatrixGamma + >>> from sympy import MatrixSymbol, symbols + >>> a, b = symbols('a b', positive=True) + >>> M = MatrixGamma('M', a, b, [[2, 1], [1, 2]]) + >>> X = MatrixSymbol('X', 2, 2) + >>> density(M)(X).doit() + exp(Trace(Matrix([ + [-2/3, 1/3], + [ 1/3, -2/3]])*X)/b)*Determinant(X)**(a - 3/2)/(3**a*sqrt(pi)*b**(2*a)*gamma(a)*gamma(a - 1/2)) + >>> density(M)([[1, 0], [0, 1]]).doit() + exp(-4/(3*b))/(3**a*sqrt(pi)*b**(2*a)*gamma(a)*gamma(a - 1/2)) + + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Matrix_gamma_distribution + + """ + if isinstance(scale_matrix, list): + scale_matrix = ImmutableMatrix(scale_matrix) + return rv(symbol, MatrixGammaDistribution, (alpha, beta, scale_matrix)) + +#------------------------------------------------------------------------------- +# Wishart Distribution --------------------------------------------------------- + +class WishartDistribution(MatrixDistribution): + + _argnames = ('n', 'scale_matrix') + + @staticmethod + def check(n, scale_matrix): + if not isinstance(scale_matrix, MatrixSymbol): + _value_check(scale_matrix.is_positive_definite, "The shape " + "matrix must be positive definite.") + _value_check(scale_matrix.is_square, "Should " + "be square matrix") + _value_check(n.is_positive, "Shape parameter should be positive.") + + @property + def set(self): + k = self.scale_matrix.shape[0] + return MatrixSet(k, k, S.Reals) + + @property + def dimension(self): + return self.scale_matrix.shape + + def pdf(self, x): + n, scale_matrix = self.n, self.scale_matrix + p = scale_matrix.shape[0] + if isinstance(x, list): + x = ImmutableMatrix(x) + if not isinstance(x, (MatrixBase, MatrixSymbol)): + raise ValueError("%s should be an isinstance of Matrix " + "or MatrixSymbol" % str(x)) + sigma_inv_x = - Inverse(scale_matrix)*x / S(2) + term1 = exp(Trace(sigma_inv_x))/((2**(p*n/S(2))) * multigamma(n/S(2), p)) + term2 = (Determinant(scale_matrix))**(-n/S(2)) + term3 = (Determinant(x))**(S(n - p - 1)/2) + return term1 * term2 * term3 + +def Wishart(symbol, n, scale_matrix): + """ + Creates a random variable with Wishart Distribution. + + The density of the said distribution can be found at [1]. + + Parameters + ========== + + n: Positive Real number + Represents degrees of freedom + scale_matrix: Positive definite real square matrix + Scale Matrix + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy.stats import density, Wishart + >>> from sympy import MatrixSymbol, symbols + >>> n = symbols('n', positive=True) + >>> W = Wishart('W', n, [[2, 1], [1, 2]]) + >>> X = MatrixSymbol('X', 2, 2) + >>> density(W)(X).doit() + exp(Trace(Matrix([ + [-1/3, 1/6], + [ 1/6, -1/3]])*X))*Determinant(X)**(n/2 - 3/2)/(2**n*3**(n/2)*sqrt(pi)*gamma(n/2)*gamma(n/2 - 1/2)) + >>> density(W)([[1, 0], [0, 1]]).doit() + exp(-2/3)/(2**n*3**(n/2)*sqrt(pi)*gamma(n/2)*gamma(n/2 - 1/2)) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Wishart_distribution + + """ + if isinstance(scale_matrix, list): + scale_matrix = ImmutableMatrix(scale_matrix) + return rv(symbol, WishartDistribution, (n, scale_matrix)) + +#------------------------------------------------------------------------------- +# Matrix Normal distribution --------------------------------------------------- + +class MatrixNormalDistribution(MatrixDistribution): + + _argnames = ('location_matrix', 'scale_matrix_1', 'scale_matrix_2') + + @staticmethod + def check(location_matrix, scale_matrix_1, scale_matrix_2): + if not isinstance(scale_matrix_1, MatrixSymbol): + _value_check(scale_matrix_1.is_positive_definite, "The shape " + "matrix must be positive definite.") + if not isinstance(scale_matrix_2, MatrixSymbol): + _value_check(scale_matrix_2.is_positive_definite, "The shape " + "matrix must be positive definite.") + _value_check(scale_matrix_1.is_square, "Scale matrix 1 should be " + "be square matrix") + _value_check(scale_matrix_2.is_square, "Scale matrix 2 should be " + "be square matrix") + n = location_matrix.shape[0] + p = location_matrix.shape[1] + _value_check(scale_matrix_1.shape[0] == n, "Scale matrix 1 should be" + " of shape %s x %s"% (str(n), str(n))) + _value_check(scale_matrix_2.shape[0] == p, "Scale matrix 2 should be" + " of shape %s x %s"% (str(p), str(p))) + + @property + def set(self): + n, p = self.location_matrix.shape + return MatrixSet(n, p, S.Reals) + + @property + def dimension(self): + return self.location_matrix.shape + + def pdf(self, x): + M, U, V = self.location_matrix, self.scale_matrix_1, self.scale_matrix_2 + n, p = M.shape + if isinstance(x, list): + x = ImmutableMatrix(x) + if not isinstance(x, (MatrixBase, MatrixSymbol)): + raise ValueError("%s should be an isinstance of Matrix " + "or MatrixSymbol" % str(x)) + term1 = Inverse(V)*Transpose(x - M)*Inverse(U)*(x - M) + num = exp(-Trace(term1)/S(2)) + den = (2*pi)**(S(n*p)/2) * Determinant(U)**(S(p)/2) * Determinant(V)**(S(n)/2) + return num/den + +def MatrixNormal(symbol, location_matrix, scale_matrix_1, scale_matrix_2): + """ + Creates a random variable with Matrix Normal Distribution. + + The density of the said distribution can be found at [1]. + + Parameters + ========== + + location_matrix: Real ``n x p`` matrix + Represents degrees of freedom + scale_matrix_1: Positive definite matrix + Scale Matrix of shape ``n x n`` + scale_matrix_2: Positive definite matrix + Scale Matrix of shape ``p x p`` + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy import MatrixSymbol + >>> from sympy.stats import density, MatrixNormal + >>> M = MatrixNormal('M', [[1, 2]], [1], [[1, 0], [0, 1]]) + >>> X = MatrixSymbol('X', 1, 2) + >>> density(M)(X).doit() + exp(-Trace((Matrix([ + [-1], + [-2]]) + X.T)*(Matrix([[-1, -2]]) + X))/2)/(2*pi) + >>> density(M)([[3, 4]]).doit() + exp(-4)/(2*pi) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Matrix_normal_distribution + + """ + if isinstance(location_matrix, list): + location_matrix = ImmutableMatrix(location_matrix) + if isinstance(scale_matrix_1, list): + scale_matrix_1 = ImmutableMatrix(scale_matrix_1) + if isinstance(scale_matrix_2, list): + scale_matrix_2 = ImmutableMatrix(scale_matrix_2) + args = (location_matrix, scale_matrix_1, scale_matrix_2) + return rv(symbol, MatrixNormalDistribution, args) + +#------------------------------------------------------------------------------- +# Matrix Student's T distribution --------------------------------------------------- + +class MatrixStudentTDistribution(MatrixDistribution): + + _argnames = ('nu', 'location_matrix', 'scale_matrix_1', 'scale_matrix_2') + + @staticmethod + def check(nu, location_matrix, scale_matrix_1, scale_matrix_2): + if not isinstance(scale_matrix_1, MatrixSymbol): + _value_check(scale_matrix_1.is_positive_definite != False, "The shape " + "matrix must be positive definite.") + if not isinstance(scale_matrix_2, MatrixSymbol): + _value_check(scale_matrix_2.is_positive_definite != False, "The shape " + "matrix must be positive definite.") + _value_check(scale_matrix_1.is_square != False, "Scale matrix 1 should be " + "be square matrix") + _value_check(scale_matrix_2.is_square != False, "Scale matrix 2 should be " + "be square matrix") + n = location_matrix.shape[0] + p = location_matrix.shape[1] + _value_check(scale_matrix_1.shape[0] == p, "Scale matrix 1 should be" + " of shape %s x %s" % (str(p), str(p))) + _value_check(scale_matrix_2.shape[0] == n, "Scale matrix 2 should be" + " of shape %s x %s" % (str(n), str(n))) + _value_check(nu.is_positive != False, "Degrees of freedom must be positive") + + @property + def set(self): + n, p = self.location_matrix.shape + return MatrixSet(n, p, S.Reals) + + @property + def dimension(self): + return self.location_matrix.shape + + def pdf(self, x): + from sympy.matrices.dense import eye + if isinstance(x, list): + x = ImmutableMatrix(x) + if not isinstance(x, (MatrixBase, MatrixSymbol)): + raise ValueError("%s should be an isinstance of Matrix " + "or MatrixSymbol" % str(x)) + nu, M, Omega, Sigma = self.nu, self.location_matrix, self.scale_matrix_1, self.scale_matrix_2 + n, p = M.shape + + K = multigamma((nu + n + p - 1)/2, p) * Determinant(Omega)**(-n/2) * Determinant(Sigma)**(-p/2) \ + / ((pi)**(n*p/2) * multigamma((nu + p - 1)/2, p)) + return K * (Determinant(eye(n) + Inverse(Sigma)*(x - M)*Inverse(Omega)*Transpose(x - M))) \ + **(-(nu + n + p -1)/2) + + + +def MatrixStudentT(symbol, nu, location_matrix, scale_matrix_1, scale_matrix_2): + """ + Creates a random variable with Matrix Gamma Distribution. + + The density of the said distribution can be found at [1]. + + Parameters + ========== + + nu: Positive Real number + degrees of freedom + location_matrix: Positive definite real square matrix + Location Matrix of shape ``n x p`` + scale_matrix_1: Positive definite real square matrix + Scale Matrix of shape ``p x p`` + scale_matrix_2: Positive definite real square matrix + Scale Matrix of shape ``n x n`` + + Returns + ======= + + RandomSymbol + + Examples + ======== + + >>> from sympy import MatrixSymbol,symbols + >>> from sympy.stats import density, MatrixStudentT + >>> v = symbols('v',positive=True) + >>> M = MatrixStudentT('M', v, [[1, 2]], [[1, 0], [0, 1]], [1]) + >>> X = MatrixSymbol('X', 1, 2) + >>> density(M)(X) + gamma(v/2 + 1)*Determinant((Matrix([[-1, -2]]) + X)*(Matrix([ + [-1], + [-2]]) + X.T) + Matrix([[1]]))**(-v/2 - 1)/(pi**1.0*gamma(v/2)*Determinant(Matrix([[1]]))**1.0*Determinant(Matrix([ + [1, 0], + [0, 1]]))**0.5) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Matrix_t-distribution + + """ + if isinstance(location_matrix, list): + location_matrix = ImmutableMatrix(location_matrix) + if isinstance(scale_matrix_1, list): + scale_matrix_1 = ImmutableMatrix(scale_matrix_1) + if isinstance(scale_matrix_2, list): + scale_matrix_2 = ImmutableMatrix(scale_matrix_2) + args = (nu, location_matrix, scale_matrix_1, scale_matrix_2) + return rv(symbol, MatrixStudentTDistribution, args) diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/random_matrix.py b/.venv/lib/python3.12/site-packages/sympy/stats/random_matrix.py new file mode 100644 index 0000000000000000000000000000000000000000..fdd25cb9ad23fed9d3a85982b24bef33d04928f0 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/random_matrix.py @@ -0,0 +1,30 @@ +from sympy.core.basic import Basic +from sympy.stats.rv import PSpace, _symbol_converter, RandomMatrixSymbol + +class RandomMatrixPSpace(PSpace): + """ + Represents probability space for + random matrices. It contains the mechanics + for handling the API calls for random matrices. + """ + def __new__(cls, sym, model=None): + sym = _symbol_converter(sym) + if model: + return Basic.__new__(cls, sym, model) + else: + return Basic.__new__(cls, sym) + + @property + def model(self): + try: + return self.args[1] + except IndexError: + return None + + def compute_density(self, expr, *args): + rms = expr.atoms(RandomMatrixSymbol) + if len(rms) > 2 or (not isinstance(expr, RandomMatrixSymbol)): + raise NotImplementedError("Currently, no algorithm has been " + "implemented to handle general expressions containing " + "multiple random matrices.") + return self.model.density(expr) diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/random_matrix_models.py b/.venv/lib/python3.12/site-packages/sympy/stats/random_matrix_models.py new file mode 100644 index 0000000000000000000000000000000000000000..6327a248ea5919c0bbb0ffc2c984105e04fe20e9 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/random_matrix_models.py @@ -0,0 +1,457 @@ +from sympy.concrete.products import Product +from sympy.concrete.summations import Sum +from sympy.core.basic import Basic +from sympy.core.function import Lambda +from sympy.core.numbers import (I, pi) +from sympy.core.singleton import S +from sympy.core.symbol import Dummy +from sympy.functions.elementary.complexes import Abs +from sympy.functions.elementary.exponential import exp +from sympy.functions.special.gamma_functions import gamma +from sympy.integrals.integrals import Integral +from sympy.matrices.expressions.matexpr import MatrixSymbol +from sympy.matrices.expressions.trace import Trace +from sympy.tensor.indexed import IndexedBase +from sympy.core.sympify import _sympify +from sympy.stats.rv import _symbol_converter, Density, RandomMatrixSymbol, is_random +from sympy.stats.joint_rv_types import JointDistributionHandmade +from sympy.stats.random_matrix import RandomMatrixPSpace +from sympy.tensor.array import ArrayComprehension + +__all__ = [ + 'CircularEnsemble', + 'CircularUnitaryEnsemble', + 'CircularOrthogonalEnsemble', + 'CircularSymplecticEnsemble', + 'GaussianEnsemble', + 'GaussianUnitaryEnsemble', + 'GaussianOrthogonalEnsemble', + 'GaussianSymplecticEnsemble', + 'joint_eigen_distribution', + 'JointEigenDistribution', + 'level_spacing_distribution' +] + +@is_random.register(RandomMatrixSymbol) +def _(x): + return True + + +class RandomMatrixEnsembleModel(Basic): + """ + Base class for random matrix ensembles. + It acts as an umbrella and contains + the methods common to all the ensembles + defined in sympy.stats.random_matrix_models. + """ + def __new__(cls, sym, dim=None): + sym, dim = _symbol_converter(sym), _sympify(dim) + if dim.is_integer == False: + raise ValueError("Dimension of the random matrices must be " + "integers, received %s instead."%(dim)) + return Basic.__new__(cls, sym, dim) + + symbol = property(lambda self: self.args[0]) + dimension = property(lambda self: self.args[1]) + + def density(self, expr): + return Density(expr) + + def __call__(self, expr): + return self.density(expr) + +class GaussianEnsembleModel(RandomMatrixEnsembleModel): + """ + Abstract class for Gaussian ensembles. + Contains the properties common to all the + gaussian ensembles. + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Random_matrix#Gaussian_ensembles + .. [2] https://arxiv.org/pdf/1712.07903.pdf + """ + def _compute_normalization_constant(self, beta, n): + """ + Helper function for computing normalization + constant for joint probability density of eigen + values of Gaussian ensembles. + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Selberg_integral#Mehta's_integral + """ + n = S(n) + prod_term = lambda j: gamma(1 + beta*S(j)/2)/gamma(S.One + beta/S(2)) + j = Dummy('j', integer=True, positive=True) + term1 = Product(prod_term(j), (j, 1, n)).doit() + term2 = (2/(beta*n))**(beta*n*(n - 1)/4 + n/2) + term3 = (2*pi)**(n/2) + return term1 * term2 * term3 + + def _compute_joint_eigen_distribution(self, beta): + """ + Helper function for computing the joint + probability distribution of eigen values + of the random matrix. + """ + n = self.dimension + Zbn = self._compute_normalization_constant(beta, n) + l = IndexedBase('l') + i = Dummy('i', integer=True, positive=True) + j = Dummy('j', integer=True, positive=True) + k = Dummy('k', integer=True, positive=True) + term1 = exp((-S(n)/2) * Sum(l[k]**2, (k, 1, n)).doit()) + sub_term = Lambda(i, Product(Abs(l[j] - l[i])**beta, (j, i + 1, n))) + term2 = Product(sub_term(i).doit(), (i, 1, n - 1)).doit() + syms = ArrayComprehension(l[k], (k, 1, n)).doit() + return Lambda(tuple(syms), (term1 * term2)/Zbn) + +class GaussianUnitaryEnsembleModel(GaussianEnsembleModel): + @property + def normalization_constant(self): + n = self.dimension + return 2**(S(n)/2) * pi**(S(n**2)/2) + + def density(self, expr): + n, ZGUE = self.dimension, self.normalization_constant + h_pspace = RandomMatrixPSpace('P', model=self) + H = RandomMatrixSymbol('H', n, n, pspace=h_pspace) + return Lambda(H, exp(-S(n)/2 * Trace(H**2))/ZGUE)(expr) + + def joint_eigen_distribution(self): + return self._compute_joint_eigen_distribution(S(2)) + + def level_spacing_distribution(self): + s = Dummy('s') + f = (32/pi**2)*(s**2)*exp((-4/pi)*s**2) + return Lambda(s, f) + +class GaussianOrthogonalEnsembleModel(GaussianEnsembleModel): + @property + def normalization_constant(self): + n = self.dimension + _H = MatrixSymbol('_H', n, n) + return Integral(exp(-S(n)/4 * Trace(_H**2))) + + def density(self, expr): + n, ZGOE = self.dimension, self.normalization_constant + h_pspace = RandomMatrixPSpace('P', model=self) + H = RandomMatrixSymbol('H', n, n, pspace=h_pspace) + return Lambda(H, exp(-S(n)/4 * Trace(H**2))/ZGOE)(expr) + + def joint_eigen_distribution(self): + return self._compute_joint_eigen_distribution(S.One) + + def level_spacing_distribution(self): + s = Dummy('s') + f = (pi/2)*s*exp((-pi/4)*s**2) + return Lambda(s, f) + +class GaussianSymplecticEnsembleModel(GaussianEnsembleModel): + @property + def normalization_constant(self): + n = self.dimension + _H = MatrixSymbol('_H', n, n) + return Integral(exp(-S(n) * Trace(_H**2))) + + def density(self, expr): + n, ZGSE = self.dimension, self.normalization_constant + h_pspace = RandomMatrixPSpace('P', model=self) + H = RandomMatrixSymbol('H', n, n, pspace=h_pspace) + return Lambda(H, exp(-S(n) * Trace(H**2))/ZGSE)(expr) + + def joint_eigen_distribution(self): + return self._compute_joint_eigen_distribution(S(4)) + + def level_spacing_distribution(self): + s = Dummy('s') + f = ((S(2)**18)/((S(3)**6)*(pi**3)))*(s**4)*exp((-64/(9*pi))*s**2) + return Lambda(s, f) + +def GaussianEnsemble(sym, dim): + sym, dim = _symbol_converter(sym), _sympify(dim) + model = GaussianEnsembleModel(sym, dim) + rmp = RandomMatrixPSpace(sym, model=model) + return RandomMatrixSymbol(sym, dim, dim, pspace=rmp) + +def GaussianUnitaryEnsemble(sym, dim): + """ + Represents Gaussian Unitary Ensembles. + + Examples + ======== + + >>> from sympy.stats import GaussianUnitaryEnsemble as GUE, density + >>> from sympy import MatrixSymbol + >>> G = GUE('U', 2) + >>> X = MatrixSymbol('X', 2, 2) + >>> density(G)(X) + exp(-Trace(X**2))/(2*pi**2) + """ + sym, dim = _symbol_converter(sym), _sympify(dim) + model = GaussianUnitaryEnsembleModel(sym, dim) + rmp = RandomMatrixPSpace(sym, model=model) + return RandomMatrixSymbol(sym, dim, dim, pspace=rmp) + +def GaussianOrthogonalEnsemble(sym, dim): + """ + Represents Gaussian Orthogonal Ensembles. + + Examples + ======== + + >>> from sympy.stats import GaussianOrthogonalEnsemble as GOE, density + >>> from sympy import MatrixSymbol + >>> G = GOE('U', 2) + >>> X = MatrixSymbol('X', 2, 2) + >>> density(G)(X) + exp(-Trace(X**2)/2)/Integral(exp(-Trace(_H**2)/2), _H) + """ + sym, dim = _symbol_converter(sym), _sympify(dim) + model = GaussianOrthogonalEnsembleModel(sym, dim) + rmp = RandomMatrixPSpace(sym, model=model) + return RandomMatrixSymbol(sym, dim, dim, pspace=rmp) + +def GaussianSymplecticEnsemble(sym, dim): + """ + Represents Gaussian Symplectic Ensembles. + + Examples + ======== + + >>> from sympy.stats import GaussianSymplecticEnsemble as GSE, density + >>> from sympy import MatrixSymbol + >>> G = GSE('U', 2) + >>> X = MatrixSymbol('X', 2, 2) + >>> density(G)(X) + exp(-2*Trace(X**2))/Integral(exp(-2*Trace(_H**2)), _H) + """ + sym, dim = _symbol_converter(sym), _sympify(dim) + model = GaussianSymplecticEnsembleModel(sym, dim) + rmp = RandomMatrixPSpace(sym, model=model) + return RandomMatrixSymbol(sym, dim, dim, pspace=rmp) + +class CircularEnsembleModel(RandomMatrixEnsembleModel): + """ + Abstract class for Circular ensembles. + Contains the properties and methods + common to all the circular ensembles. + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Circular_ensemble + """ + def density(self, expr): + # TODO : Add support for Lie groups(as extensions of sympy.diffgeom) + # and define measures on them + raise NotImplementedError("Support for Haar measure hasn't been " + "implemented yet, therefore the density of " + "%s cannot be computed."%(self)) + + def _compute_joint_eigen_distribution(self, beta): + """ + Helper function to compute the joint distribution of phases + of the complex eigen values of matrices belonging to any + circular ensembles. + """ + n = self.dimension + Zbn = ((2*pi)**n)*(gamma(beta*n/2 + 1)/S(gamma(beta/2 + 1))**n) + t = IndexedBase('t') + i, j, k = (Dummy('i', integer=True), Dummy('j', integer=True), + Dummy('k', integer=True)) + syms = ArrayComprehension(t[i], (i, 1, n)).doit() + f = Product(Product(Abs(exp(I*t[k]) - exp(I*t[j]))**beta, (j, k + 1, n)).doit(), + (k, 1, n - 1)).doit() + return Lambda(tuple(syms), f/Zbn) + +class CircularUnitaryEnsembleModel(CircularEnsembleModel): + def joint_eigen_distribution(self): + return self._compute_joint_eigen_distribution(S(2)) + +class CircularOrthogonalEnsembleModel(CircularEnsembleModel): + def joint_eigen_distribution(self): + return self._compute_joint_eigen_distribution(S.One) + +class CircularSymplecticEnsembleModel(CircularEnsembleModel): + def joint_eigen_distribution(self): + return self._compute_joint_eigen_distribution(S(4)) + +def CircularEnsemble(sym, dim): + sym, dim = _symbol_converter(sym), _sympify(dim) + model = CircularEnsembleModel(sym, dim) + rmp = RandomMatrixPSpace(sym, model=model) + return RandomMatrixSymbol(sym, dim, dim, pspace=rmp) + +def CircularUnitaryEnsemble(sym, dim): + """ + Represents Circular Unitary Ensembles. + + Examples + ======== + + >>> from sympy.stats import CircularUnitaryEnsemble as CUE + >>> from sympy.stats import joint_eigen_distribution + >>> C = CUE('U', 1) + >>> joint_eigen_distribution(C) + Lambda(t[1], Product(Abs(exp(I*t[_j]) - exp(I*t[_k]))**2, (_j, _k + 1, 1), (_k, 1, 0))/(2*pi)) + + Note + ==== + + As can be seen above in the example, density of CiruclarUnitaryEnsemble + is not evaluated because the exact definition is based on haar measure of + unitary group which is not unique. + """ + sym, dim = _symbol_converter(sym), _sympify(dim) + model = CircularUnitaryEnsembleModel(sym, dim) + rmp = RandomMatrixPSpace(sym, model=model) + return RandomMatrixSymbol(sym, dim, dim, pspace=rmp) + +def CircularOrthogonalEnsemble(sym, dim): + """ + Represents Circular Orthogonal Ensembles. + + Examples + ======== + + >>> from sympy.stats import CircularOrthogonalEnsemble as COE + >>> from sympy.stats import joint_eigen_distribution + >>> C = COE('O', 1) + >>> joint_eigen_distribution(C) + Lambda(t[1], Product(Abs(exp(I*t[_j]) - exp(I*t[_k])), (_j, _k + 1, 1), (_k, 1, 0))/(2*pi)) + + Note + ==== + + As can be seen above in the example, density of CiruclarOrthogonalEnsemble + is not evaluated because the exact definition is based on haar measure of + unitary group which is not unique. + """ + sym, dim = _symbol_converter(sym), _sympify(dim) + model = CircularOrthogonalEnsembleModel(sym, dim) + rmp = RandomMatrixPSpace(sym, model=model) + return RandomMatrixSymbol(sym, dim, dim, pspace=rmp) + +def CircularSymplecticEnsemble(sym, dim): + """ + Represents Circular Symplectic Ensembles. + + Examples + ======== + + >>> from sympy.stats import CircularSymplecticEnsemble as CSE + >>> from sympy.stats import joint_eigen_distribution + >>> C = CSE('S', 1) + >>> joint_eigen_distribution(C) + Lambda(t[1], Product(Abs(exp(I*t[_j]) - exp(I*t[_k]))**4, (_j, _k + 1, 1), (_k, 1, 0))/(2*pi)) + + Note + ==== + + As can be seen above in the example, density of CiruclarSymplecticEnsemble + is not evaluated because the exact definition is based on haar measure of + unitary group which is not unique. + """ + sym, dim = _symbol_converter(sym), _sympify(dim) + model = CircularSymplecticEnsembleModel(sym, dim) + rmp = RandomMatrixPSpace(sym, model=model) + return RandomMatrixSymbol(sym, dim, dim, pspace=rmp) + +def joint_eigen_distribution(mat): + """ + For obtaining joint probability distribution + of eigen values of random matrix. + + Parameters + ========== + + mat: RandomMatrixSymbol + The matrix symbol whose eigen values are to be considered. + + Returns + ======= + + Lambda + + Examples + ======== + + >>> from sympy.stats import GaussianUnitaryEnsemble as GUE + >>> from sympy.stats import joint_eigen_distribution + >>> U = GUE('U', 2) + >>> joint_eigen_distribution(U) + Lambda((l[1], l[2]), exp(-l[1]**2 - l[2]**2)*Product(Abs(l[_i] - l[_j])**2, (_j, _i + 1, 2), (_i, 1, 1))/pi) + """ + if not isinstance(mat, RandomMatrixSymbol): + raise ValueError("%s is not of type, RandomMatrixSymbol."%(mat)) + return mat.pspace.model.joint_eigen_distribution() + +def JointEigenDistribution(mat): + """ + Creates joint distribution of eigen values of matrices with random + expressions. + + Parameters + ========== + + mat: Matrix + The matrix under consideration. + + Returns + ======= + + JointDistributionHandmade + + Examples + ======== + + >>> from sympy.stats import Normal, JointEigenDistribution + >>> from sympy import Matrix + >>> A = [[Normal('A00', 0, 1), Normal('A01', 0, 1)], + ... [Normal('A10', 0, 1), Normal('A11', 0, 1)]] + >>> JointEigenDistribution(Matrix(A)) + JointDistributionHandmade(-sqrt(A00**2 - 2*A00*A11 + 4*A01*A10 + A11**2)/2 + + A00/2 + A11/2, sqrt(A00**2 - 2*A00*A11 + 4*A01*A10 + A11**2)/2 + A00/2 + A11/2) + + """ + eigenvals = mat.eigenvals(multiple=True) + if not all(is_random(eigenval) for eigenval in set(eigenvals)): + raise ValueError("Eigen values do not have any random expression, " + "joint distribution cannot be generated.") + return JointDistributionHandmade(*eigenvals) + +def level_spacing_distribution(mat): + """ + For obtaining distribution of level spacings. + + Parameters + ========== + + mat: RandomMatrixSymbol + The random matrix symbol whose eigen values are + to be considered for finding the level spacings. + + Returns + ======= + + Lambda + + Examples + ======== + + >>> from sympy.stats import GaussianUnitaryEnsemble as GUE + >>> from sympy.stats import level_spacing_distribution + >>> U = GUE('U', 2) + >>> level_spacing_distribution(U) + Lambda(_s, 32*_s**2*exp(-4*_s**2/pi)/pi**2) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Random_matrix#Distribution_of_level_spacings + """ + return mat.pspace.model.level_spacing_distribution() diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/rv.py b/.venv/lib/python3.12/site-packages/sympy/stats/rv.py new file mode 100644 index 0000000000000000000000000000000000000000..75ab54deb551b7ff3d4d06f37482a1f16a789ba6 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/rv.py @@ -0,0 +1,1798 @@ +""" +Main Random Variables Module + +Defines abstract random variable type. +Contains interfaces for probability space object (PSpace) as well as standard +operators, P, E, sample, density, where, quantile + +See Also +======== + +sympy.stats.crv +sympy.stats.frv +sympy.stats.rv_interface +""" + +from __future__ import annotations +from functools import singledispatch +from math import prod + +from sympy.core.add import Add +from sympy.core.basic import Basic +from sympy.core.containers import Tuple +from sympy.core.expr import Expr +from sympy.core.function import (Function, Lambda) +from sympy.core.logic import fuzzy_and +from sympy.core.mul import Mul +from sympy.core.relational import (Eq, Ne) +from sympy.core.singleton import S +from sympy.core.symbol import (Dummy, Symbol) +from sympy.core.sympify import sympify +from sympy.functions.special.delta_functions import DiracDelta +from sympy.functions.special.tensor_functions import KroneckerDelta +from sympy.logic.boolalg import (And, Or) +from sympy.matrices.expressions.matexpr import MatrixSymbol +from sympy.tensor.indexed import Indexed +from sympy.utilities.lambdify import lambdify +from sympy.core.relational import Relational +from sympy.core.sympify import _sympify +from sympy.sets.sets import FiniteSet, ProductSet, Intersection +from sympy.solvers.solveset import solveset +from sympy.external import import_module +from sympy.utilities.decorator import doctest_depends_on +from sympy.utilities.exceptions import sympy_deprecation_warning +from sympy.utilities.iterables import iterable + + +__doctest_requires__ = {('sample',): ['scipy']} + + +x = Symbol('x') + + +@singledispatch +def is_random(x): + return False + + +@is_random.register(Basic) +def _(x): + atoms = x.free_symbols + return any(is_random(i) for i in atoms) + + +class RandomDomain(Basic): + """ + Represents a set of variables and the values which they can take. + + See Also + ======== + + sympy.stats.crv.ContinuousDomain + sympy.stats.frv.FiniteDomain + """ + + is_ProductDomain = False + is_Finite = False + is_Continuous = False + is_Discrete = False + + def __new__(cls, symbols, *args): + symbols = FiniteSet(*symbols) + return Basic.__new__(cls, symbols, *args) + + @property + def symbols(self): + return self.args[0] + + @property + def set(self): + return self.args[1] + + def __contains__(self, other): + raise NotImplementedError() + + def compute_expectation(self, expr): + raise NotImplementedError() + + +class SingleDomain(RandomDomain): + """ + A single variable and its domain. + + See Also + ======== + + sympy.stats.crv.SingleContinuousDomain + sympy.stats.frv.SingleFiniteDomain + """ + def __new__(cls, symbol, set): + assert symbol.is_Symbol + return Basic.__new__(cls, symbol, set) + + @property + def symbol(self): + return self.args[0] + + @property + def symbols(self): + return FiniteSet(self.symbol) + + def __contains__(self, other): + if len(other) != 1: + return False + sym, val = tuple(other)[0] + return self.symbol == sym and val in self.set + + +class MatrixDomain(RandomDomain): + """ + A Random Matrix variable and its domain. + + """ + def __new__(cls, symbol, set): + symbol, set = _symbol_converter(symbol), _sympify(set) + return Basic.__new__(cls, symbol, set) + + @property + def symbol(self): + return self.args[0] + + @property + def symbols(self): + return FiniteSet(self.symbol) + + +class ConditionalDomain(RandomDomain): + """ + A RandomDomain with an attached condition. + + See Also + ======== + + sympy.stats.crv.ConditionalContinuousDomain + sympy.stats.frv.ConditionalFiniteDomain + """ + def __new__(cls, fulldomain, condition): + condition = condition.xreplace({rs: rs.symbol + for rs in random_symbols(condition)}) + return Basic.__new__(cls, fulldomain, condition) + + @property + def symbols(self): + return self.fulldomain.symbols + + @property + def fulldomain(self): + return self.args[0] + + @property + def condition(self): + return self.args[1] + + @property + def set(self): + raise NotImplementedError("Set of Conditional Domain not Implemented") + + def as_boolean(self): + return And(self.fulldomain.as_boolean(), self.condition) + + +class PSpace(Basic): + """ + A Probability Space. + + Explanation + =========== + + Probability Spaces encode processes that equal different values + probabilistically. These underly Random Symbols which occur in SymPy + expressions and contain the mechanics to evaluate statistical statements. + + See Also + ======== + + sympy.stats.crv.ContinuousPSpace + sympy.stats.frv.FinitePSpace + """ + + is_Finite: bool | None = None # Fails test if not set to None + is_Continuous: bool | None = None # Fails test if not set to None + is_Discrete: bool | None = None # Fails test if not set to None + is_real: bool | None + + @property + def domain(self): + return self.args[0] + + @property + def density(self): + return self.args[1] + + @property + def values(self): + return frozenset(RandomSymbol(sym, self) for sym in self.symbols) + + @property + def symbols(self): + return self.domain.symbols + + def where(self, condition): + raise NotImplementedError() + + def compute_density(self, expr): + raise NotImplementedError() + + def sample(self, size=(), library='scipy', seed=None): + raise NotImplementedError() + + def probability(self, condition): + raise NotImplementedError() + + def compute_expectation(self, expr): + raise NotImplementedError() + + +class SinglePSpace(PSpace): + """ + Represents the probabilities of a set of random events that can be + attributed to a single variable/symbol. + """ + def __new__(cls, s, distribution): + s = _symbol_converter(s) + return Basic.__new__(cls, s, distribution) + + @property + def value(self): + return RandomSymbol(self.symbol, self) + + @property + def symbol(self): + return self.args[0] + + @property + def distribution(self): + return self.args[1] + + @property + def pdf(self): + return self.distribution.pdf(self.symbol) + + +class RandomSymbol(Expr): + """ + Random Symbols represent ProbabilitySpaces in SymPy Expressions. + In principle they can take on any value that their symbol can take on + within the associated PSpace with probability determined by the PSpace + Density. + + Explanation + =========== + + Random Symbols contain pspace and symbol properties. + The pspace property points to the represented Probability Space + The symbol is a standard SymPy Symbol that is used in that probability space + for example in defining a density. + + You can form normal SymPy expressions using RandomSymbols and operate on + those expressions with the Functions + + E - Expectation of a random expression + P - Probability of a condition + density - Probability Density of an expression + given - A new random expression (with new random symbols) given a condition + + An object of the RandomSymbol type should almost never be created by the + user. They tend to be created instead by the PSpace class's value method. + Traditionally a user does not even do this but instead calls one of the + convenience functions Normal, Exponential, Coin, Die, FiniteRV, etc.... + """ + + def __new__(cls, symbol, pspace=None): + from sympy.stats.joint_rv import JointRandomSymbol + if pspace is None: + # Allow single arg, representing pspace == PSpace() + pspace = PSpace() + symbol = _symbol_converter(symbol) + if not isinstance(pspace, PSpace): + raise TypeError("pspace variable should be of type PSpace") + if cls == JointRandomSymbol and isinstance(pspace, SinglePSpace): + cls = RandomSymbol + return Basic.__new__(cls, symbol, pspace) + + is_finite = True + is_symbol = True + is_Atom = True + + _diff_wrt = True + + pspace = property(lambda self: self.args[1]) + symbol = property(lambda self: self.args[0]) + name = property(lambda self: self.symbol.name) + + def _eval_is_positive(self): + return self.symbol.is_positive + + def _eval_is_integer(self): + return self.symbol.is_integer + + def _eval_is_real(self): + return self.symbol.is_real or self.pspace.is_real + + @property + def is_commutative(self): + return self.symbol.is_commutative + + @property + def free_symbols(self): + return {self} + +class RandomIndexedSymbol(RandomSymbol): + + def __new__(cls, idx_obj, pspace=None): + if pspace is None: + # Allow single arg, representing pspace == PSpace() + pspace = PSpace() + if not isinstance(idx_obj, (Indexed, Function)): + raise TypeError("An Function or Indexed object is expected not %s"%(idx_obj)) + return Basic.__new__(cls, idx_obj, pspace) + + symbol = property(lambda self: self.args[0]) + name = property(lambda self: str(self.args[0])) + + @property + def key(self): + if isinstance(self.symbol, Indexed): + return self.symbol.args[1] + elif isinstance(self.symbol, Function): + return self.symbol.args[0] + + @property + def free_symbols(self): + if self.key.free_symbols: + free_syms = self.key.free_symbols + free_syms.add(self) + return free_syms + return {self} + + @property + def pspace(self): + return self.args[1] + +class RandomMatrixSymbol(RandomSymbol, MatrixSymbol): # type: ignore + def __new__(cls, symbol, n, m, pspace=None): + n, m = _sympify(n), _sympify(m) + symbol = _symbol_converter(symbol) + if pspace is None: + # Allow single arg, representing pspace == PSpace() + pspace = PSpace() + return Basic.__new__(cls, symbol, n, m, pspace) + + symbol = property(lambda self: self.args[0]) + pspace = property(lambda self: self.args[3]) + + +class ProductPSpace(PSpace): + """ + Abstract class for representing probability spaces with multiple random + variables. + + See Also + ======== + + sympy.stats.rv.IndependentProductPSpace + sympy.stats.joint_rv.JointPSpace + """ + pass + +class IndependentProductPSpace(ProductPSpace): + """ + A probability space resulting from the merger of two independent probability + spaces. + + Often created using the function, pspace. + """ + + def __new__(cls, *spaces): + rs_space_dict = {} + for space in spaces: + for value in space.values: + rs_space_dict[value] = space + + symbols = FiniteSet(*[val.symbol for val in rs_space_dict.keys()]) + + # Overlapping symbols + from sympy.stats.joint_rv import MarginalDistribution + from sympy.stats.compound_rv import CompoundDistribution + if len(symbols) < sum(len(space.symbols) for space in spaces if not + isinstance(space.distribution, ( + CompoundDistribution, MarginalDistribution))): + raise ValueError("Overlapping Random Variables") + + if all(space.is_Finite for space in spaces): + from sympy.stats.frv import ProductFinitePSpace + cls = ProductFinitePSpace + + obj = Basic.__new__(cls, *FiniteSet(*spaces)) + + return obj + + @property + def pdf(self): + p = Mul(*[space.pdf for space in self.spaces]) + return p.subs({rv: rv.symbol for rv in self.values}) + + @property + def rs_space_dict(self): + d = {} + for space in self.spaces: + for value in space.values: + d[value] = space + return d + + @property + def symbols(self): + return FiniteSet(*[val.symbol for val in self.rs_space_dict.keys()]) + + @property + def spaces(self): + return FiniteSet(*self.args) + + @property + def values(self): + return sumsets(space.values for space in self.spaces) + + def compute_expectation(self, expr, rvs=None, evaluate=False, **kwargs): + rvs = rvs or self.values + rvs = frozenset(rvs) + for space in self.spaces: + expr = space.compute_expectation(expr, rvs & space.values, evaluate=False, **kwargs) + if evaluate and hasattr(expr, 'doit'): + return expr.doit(**kwargs) + return expr + + @property + def domain(self): + return ProductDomain(*[space.domain for space in self.spaces]) + + @property + def density(self): + raise NotImplementedError("Density not available for ProductSpaces") + + def sample(self, size=(), library='scipy', seed=None): + return {k: v for space in self.spaces + for k, v in space.sample(size=size, library=library, seed=seed).items()} + + + def probability(self, condition, **kwargs): + cond_inv = False + if isinstance(condition, Ne): + condition = Eq(condition.args[0], condition.args[1]) + cond_inv = True + elif isinstance(condition, And): # they are independent + return Mul(*[self.probability(arg) for arg in condition.args]) + elif isinstance(condition, Or): # they are independent + return Add(*[self.probability(arg) for arg in condition.args]) + expr = condition.lhs - condition.rhs + rvs = random_symbols(expr) + dens = self.compute_density(expr) + if any(pspace(rv).is_Continuous for rv in rvs): + from sympy.stats.crv import SingleContinuousPSpace + from sympy.stats.crv_types import ContinuousDistributionHandmade + if expr in self.values: + # Marginalize all other random symbols out of the density + randomsymbols = tuple(set(self.values) - frozenset([expr])) + symbols = tuple(rs.symbol for rs in randomsymbols) + pdf = self.domain.integrate(self.pdf, symbols, **kwargs) + return Lambda(expr.symbol, pdf) + dens = ContinuousDistributionHandmade(dens) + z = Dummy('z', real=True) + space = SingleContinuousPSpace(z, dens) + result = space.probability(condition.__class__(space.value, 0)) + else: + from sympy.stats.drv import SingleDiscretePSpace + from sympy.stats.drv_types import DiscreteDistributionHandmade + dens = DiscreteDistributionHandmade(dens) + z = Dummy('z', integer=True) + space = SingleDiscretePSpace(z, dens) + result = space.probability(condition.__class__(space.value, 0)) + return result if not cond_inv else S.One - result + + def compute_density(self, expr, **kwargs): + rvs = random_symbols(expr) + if any(pspace(rv).is_Continuous for rv in rvs): + z = Dummy('z', real=True) + expr = self.compute_expectation(DiracDelta(expr - z), + **kwargs) + else: + z = Dummy('z', integer=True) + expr = self.compute_expectation(KroneckerDelta(expr, z), + **kwargs) + return Lambda(z, expr) + + def compute_cdf(self, expr, **kwargs): + raise ValueError("CDF not well defined on multivariate expressions") + + def conditional_space(self, condition, normalize=True, **kwargs): + rvs = random_symbols(condition) + condition = condition.xreplace({rv: rv.symbol for rv in self.values}) + pspaces = [pspace(rv) for rv in rvs] + if any(ps.is_Continuous for ps in pspaces): + from sympy.stats.crv import (ConditionalContinuousDomain, + ContinuousPSpace) + space = ContinuousPSpace + domain = ConditionalContinuousDomain(self.domain, condition) + elif any(ps.is_Discrete for ps in pspaces): + from sympy.stats.drv import (ConditionalDiscreteDomain, + DiscretePSpace) + space = DiscretePSpace + domain = ConditionalDiscreteDomain(self.domain, condition) + elif all(ps.is_Finite for ps in pspaces): + from sympy.stats.frv import FinitePSpace + return FinitePSpace.conditional_space(self, condition) + if normalize: + replacement = {rv: Dummy(str(rv)) for rv in self.symbols} + norm = domain.compute_expectation(self.pdf, **kwargs) + pdf = self.pdf / norm.xreplace(replacement) + # XXX: Converting symbols from set to tuple. The order matters to + # Lambda though so we shouldn't be starting with a set here... + density = Lambda(tuple(domain.symbols), pdf) + + return space(domain, density) + +class ProductDomain(RandomDomain): + """ + A domain resulting from the merger of two independent domains. + + See Also + ======== + sympy.stats.crv.ProductContinuousDomain + sympy.stats.frv.ProductFiniteDomain + """ + is_ProductDomain = True + + def __new__(cls, *domains): + # Flatten any product of products + domains2 = [] + for domain in domains: + if not domain.is_ProductDomain: + domains2.append(domain) + else: + domains2.extend(domain.domains) + domains2 = FiniteSet(*domains2) + + if all(domain.is_Finite for domain in domains2): + from sympy.stats.frv import ProductFiniteDomain + cls = ProductFiniteDomain + if all(domain.is_Continuous for domain in domains2): + from sympy.stats.crv import ProductContinuousDomain + cls = ProductContinuousDomain + if all(domain.is_Discrete for domain in domains2): + from sympy.stats.drv import ProductDiscreteDomain + cls = ProductDiscreteDomain + + return Basic.__new__(cls, *domains2) + + @property + def sym_domain_dict(self): + return {symbol: domain for domain in self.domains + for symbol in domain.symbols} + + @property + def symbols(self): + return FiniteSet(*[sym for domain in self.domains + for sym in domain.symbols]) + + @property + def domains(self): + return self.args + + @property + def set(self): + return ProductSet(*(domain.set for domain in self.domains)) + + def __contains__(self, other): + # Split event into each subdomain + for domain in self.domains: + # Collect the parts of this event which associate to this domain + elem = frozenset([item for item in other + if sympify(domain.symbols.contains(item[0])) + is S.true]) + # Test this sub-event + if elem not in domain: + return False + # All subevents passed + return True + + def as_boolean(self): + return And(*[domain.as_boolean() for domain in self.domains]) + + +def random_symbols(expr): + """ + Returns all RandomSymbols within a SymPy Expression. + """ + atoms = getattr(expr, 'atoms', None) + if atoms is not None: + comp = lambda rv: rv.symbol.name + l = list(atoms(RandomSymbol)) + return sorted(l, key=comp) + else: + return [] + + +def pspace(expr): + """ + Returns the underlying Probability Space of a random expression. + + For internal use. + + Examples + ======== + + >>> from sympy.stats import pspace, Normal + >>> X = Normal('X', 0, 1) + >>> pspace(2*X + 1) == X.pspace + True + """ + expr = sympify(expr) + if isinstance(expr, RandomSymbol) and expr.pspace is not None: + return expr.pspace + if expr.has(RandomMatrixSymbol): + rm = list(expr.atoms(RandomMatrixSymbol))[0] + return rm.pspace + + rvs = random_symbols(expr) + if not rvs: + raise ValueError("Expression containing Random Variable expected, not %s" % (expr)) + # If only one space present + if all(rv.pspace == rvs[0].pspace for rv in rvs): + return rvs[0].pspace + from sympy.stats.compound_rv import CompoundPSpace + from sympy.stats.stochastic_process import StochasticPSpace + for rv in rvs: + if isinstance(rv.pspace, (CompoundPSpace, StochasticPSpace)): + return rv.pspace + # Otherwise make a product space + return IndependentProductPSpace(*[rv.pspace for rv in rvs]) + + +def sumsets(sets): + """ + Union of sets + """ + return frozenset().union(*sets) + + +def rs_swap(a, b): + """ + Build a dictionary to swap RandomSymbols based on their underlying symbol. + + i.e. + if ``X = ('x', pspace1)`` + and ``Y = ('x', pspace2)`` + then ``X`` and ``Y`` match and the key, value pair + ``{X:Y}`` will appear in the result + + Inputs: collections a and b of random variables which share common symbols + Output: dict mapping RVs in a to RVs in b + """ + d = {} + for rsa in a: + d[rsa] = [rsb for rsb in b if rsa.symbol == rsb.symbol][0] + return d + + +def given(expr, condition=None, **kwargs): + r""" Conditional Random Expression. + + Explanation + =========== + + From a random expression and a condition on that expression creates a new + probability space from the condition and returns the same expression on that + conditional probability space. + + Examples + ======== + + >>> from sympy.stats import given, density, Die + >>> X = Die('X', 6) + >>> Y = given(X, X > 3) + >>> density(Y).dict + {4: 1/3, 5: 1/3, 6: 1/3} + + Following convention, if the condition is a random symbol then that symbol + is considered fixed. + + >>> from sympy.stats import Normal + >>> from sympy import pprint + >>> from sympy.abc import z + + >>> X = Normal('X', 0, 1) + >>> Y = Normal('Y', 0, 1) + >>> pprint(density(X + Y, Y)(z), use_unicode=False) + 2 + -(-Y + z) + ----------- + ___ 2 + \/ 2 *e + ------------------ + ____ + 2*\/ pi + """ + + if not is_random(condition) or pspace_independent(expr, condition): + return expr + + if isinstance(condition, RandomSymbol): + condition = Eq(condition, condition.symbol) + + condsymbols = random_symbols(condition) + if (isinstance(condition, Eq) and len(condsymbols) == 1 and + not isinstance(pspace(expr).domain, ConditionalDomain)): + rv = tuple(condsymbols)[0] + + results = solveset(condition, rv) + if isinstance(results, Intersection) and S.Reals in results.args: + results = list(results.args[1]) + + sums = 0 + for res in results: + temp = expr.subs(rv, res) + if temp == True: + return True + if temp != False: + # XXX: This seems nonsensical but preserves existing behaviour + # after the change that Relational is no longer a subclass of + # Expr. Here expr is sometimes Relational and sometimes Expr + # but we are trying to add them with +=. This needs to be + # fixed somehow. + if sums == 0 and isinstance(expr, Relational): + sums = expr.subs(rv, res) + else: + sums += expr.subs(rv, res) + if sums == 0: + return False + return sums + + # Get full probability space of both the expression and the condition + fullspace = pspace(Tuple(expr, condition)) + # Build new space given the condition + space = fullspace.conditional_space(condition, **kwargs) + # Dictionary to swap out RandomSymbols in expr with new RandomSymbols + # That point to the new conditional space + swapdict = rs_swap(fullspace.values, space.values) + # Swap random variables in the expression + expr = expr.xreplace(swapdict) + return expr + + +def expectation(expr, condition=None, numsamples=None, evaluate=True, **kwargs): + """ + Returns the expected value of a random expression. + + Parameters + ========== + + expr : Expr containing RandomSymbols + The expression of which you want to compute the expectation value + given : Expr containing RandomSymbols + A conditional expression. E(X, X>0) is expectation of X given X > 0 + numsamples : int + Enables sampling and approximates the expectation with this many samples + evalf : Bool (defaults to True) + If sampling return a number rather than a complex expression + evaluate : Bool (defaults to True) + In case of continuous systems return unevaluated integral + + Examples + ======== + + >>> from sympy.stats import E, Die + >>> X = Die('X', 6) + >>> E(X) + 7/2 + >>> E(2*X + 1) + 8 + + >>> E(X, X > 3) # Expectation of X given that it is above 3 + 5 + """ + + if not is_random(expr): # expr isn't random? + return expr + kwargs['numsamples'] = numsamples + from sympy.stats.symbolic_probability import Expectation + if evaluate: + return Expectation(expr, condition).doit(**kwargs) + return Expectation(expr, condition) + + +def probability(condition, given_condition=None, numsamples=None, + evaluate=True, **kwargs): + """ + Probability that a condition is true, optionally given a second condition. + + Parameters + ========== + + condition : Combination of Relationals containing RandomSymbols + The condition of which you want to compute the probability + given_condition : Combination of Relationals containing RandomSymbols + A conditional expression. P(X > 1, X > 0) is expectation of X > 1 + given X > 0 + numsamples : int + Enables sampling and approximates the probability with this many samples + evaluate : Bool (defaults to True) + In case of continuous systems return unevaluated integral + + Examples + ======== + + >>> from sympy.stats import P, Die + >>> from sympy import Eq + >>> X, Y = Die('X', 6), Die('Y', 6) + >>> P(X > 3) + 1/2 + >>> P(Eq(X, 5), X > 2) # Probability that X == 5 given that X > 2 + 1/4 + >>> P(X > Y) + 5/12 + """ + + kwargs['numsamples'] = numsamples + from sympy.stats.symbolic_probability import Probability + if evaluate: + return Probability(condition, given_condition).doit(**kwargs) + return Probability(condition, given_condition) + + +class Density(Basic): + expr = property(lambda self: self.args[0]) + + def __new__(cls, expr, condition = None): + expr = _sympify(expr) + if condition is None: + obj = Basic.__new__(cls, expr) + else: + condition = _sympify(condition) + obj = Basic.__new__(cls, expr, condition) + return obj + + @property + def condition(self): + if len(self.args) > 1: + return self.args[1] + else: + return None + + def doit(self, evaluate=True, **kwargs): + from sympy.stats.random_matrix import RandomMatrixPSpace + from sympy.stats.joint_rv import JointPSpace + from sympy.stats.matrix_distributions import MatrixPSpace + from sympy.stats.compound_rv import CompoundPSpace + from sympy.stats.frv import SingleFiniteDistribution + expr, condition = self.expr, self.condition + + if isinstance(expr, SingleFiniteDistribution): + return expr.dict + if condition is not None: + # Recompute on new conditional expr + expr = given(expr, condition, **kwargs) + if not random_symbols(expr): + return Lambda(x, DiracDelta(x - expr)) + if isinstance(expr, RandomSymbol): + if isinstance(expr.pspace, (SinglePSpace, JointPSpace, MatrixPSpace)) and \ + hasattr(expr.pspace, 'distribution'): + return expr.pspace.distribution + elif isinstance(expr.pspace, RandomMatrixPSpace): + return expr.pspace.model + if isinstance(pspace(expr), CompoundPSpace): + kwargs['compound_evaluate'] = evaluate + result = pspace(expr).compute_density(expr, **kwargs) + + if evaluate and hasattr(result, 'doit'): + return result.doit() + else: + return result + + +def density(expr, condition=None, evaluate=True, numsamples=None, **kwargs): + """ + Probability density of a random expression, optionally given a second + condition. + + Explanation + =========== + + This density will take on different forms for different types of + probability spaces. Discrete variables produce Dicts. Continuous + variables produce Lambdas. + + Parameters + ========== + + expr : Expr containing RandomSymbols + The expression of which you want to compute the density value + condition : Relational containing RandomSymbols + A conditional expression. density(X > 1, X > 0) is density of X > 1 + given X > 0 + numsamples : int + Enables sampling and approximates the density with this many samples + + Examples + ======== + + >>> from sympy.stats import density, Die, Normal + >>> from sympy import Symbol + + >>> x = Symbol('x') + >>> D = Die('D', 6) + >>> X = Normal(x, 0, 1) + + >>> density(D).dict + {1: 1/6, 2: 1/6, 3: 1/6, 4: 1/6, 5: 1/6, 6: 1/6} + >>> density(2*D).dict + {2: 1/6, 4: 1/6, 6: 1/6, 8: 1/6, 10: 1/6, 12: 1/6} + >>> density(X)(x) + sqrt(2)*exp(-x**2/2)/(2*sqrt(pi)) + """ + + if numsamples: + return sampling_density(expr, condition, numsamples=numsamples, + **kwargs) + + return Density(expr, condition).doit(evaluate=evaluate, **kwargs) + + +def cdf(expr, condition=None, evaluate=True, **kwargs): + """ + Cumulative Distribution Function of a random expression. + + optionally given a second condition. + + Explanation + =========== + + This density will take on different forms for different types of + probability spaces. + Discrete variables produce Dicts. + Continuous variables produce Lambdas. + + Examples + ======== + + >>> from sympy.stats import density, Die, Normal, cdf + + >>> D = Die('D', 6) + >>> X = Normal('X', 0, 1) + + >>> density(D).dict + {1: 1/6, 2: 1/6, 3: 1/6, 4: 1/6, 5: 1/6, 6: 1/6} + >>> cdf(D) + {1: 1/6, 2: 1/3, 3: 1/2, 4: 2/3, 5: 5/6, 6: 1} + >>> cdf(3*D, D > 2) + {9: 1/4, 12: 1/2, 15: 3/4, 18: 1} + + >>> cdf(X) + Lambda(_z, erf(sqrt(2)*_z/2)/2 + 1/2) + """ + if condition is not None: # If there is a condition + # Recompute on new conditional expr + return cdf(given(expr, condition, **kwargs), **kwargs) + + # Otherwise pass work off to the ProbabilitySpace + result = pspace(expr).compute_cdf(expr, **kwargs) + + if evaluate and hasattr(result, 'doit'): + return result.doit() + else: + return result + + +def characteristic_function(expr, condition=None, evaluate=True, **kwargs): + """ + Characteristic function of a random expression, optionally given a second condition. + + Returns a Lambda. + + Examples + ======== + + >>> from sympy.stats import Normal, DiscreteUniform, Poisson, characteristic_function + + >>> X = Normal('X', 0, 1) + >>> characteristic_function(X) + Lambda(_t, exp(-_t**2/2)) + + >>> Y = DiscreteUniform('Y', [1, 2, 7]) + >>> characteristic_function(Y) + Lambda(_t, exp(7*_t*I)/3 + exp(2*_t*I)/3 + exp(_t*I)/3) + + >>> Z = Poisson('Z', 2) + >>> characteristic_function(Z) + Lambda(_t, exp(2*exp(_t*I) - 2)) + """ + if condition is not None: + return characteristic_function(given(expr, condition, **kwargs), **kwargs) + + result = pspace(expr).compute_characteristic_function(expr, **kwargs) + + if evaluate and hasattr(result, 'doit'): + return result.doit() + else: + return result + +def moment_generating_function(expr, condition=None, evaluate=True, **kwargs): + if condition is not None: + return moment_generating_function(given(expr, condition, **kwargs), **kwargs) + + result = pspace(expr).compute_moment_generating_function(expr, **kwargs) + + if evaluate and hasattr(result, 'doit'): + return result.doit() + else: + return result + +def where(condition, given_condition=None, **kwargs): + """ + Returns the domain where a condition is True. + + Examples + ======== + + >>> from sympy.stats import where, Die, Normal + >>> from sympy import And + + >>> D1, D2 = Die('a', 6), Die('b', 6) + >>> a, b = D1.symbol, D2.symbol + >>> X = Normal('x', 0, 1) + + >>> where(X**2<1) + Domain: (-1 < x) & (x < 1) + + >>> where(X**2<1).set + Interval.open(-1, 1) + + >>> where(And(D1<=D2, D2<3)) + Domain: (Eq(a, 1) & Eq(b, 1)) | (Eq(a, 1) & Eq(b, 2)) | (Eq(a, 2) & Eq(b, 2)) + """ + if given_condition is not None: # If there is a condition + # Recompute on new conditional expr + return where(given(condition, given_condition, **kwargs), **kwargs) + + # Otherwise pass work off to the ProbabilitySpace + return pspace(condition).where(condition, **kwargs) + + +@doctest_depends_on(modules=('scipy',)) +def sample(expr, condition=None, size=(), library='scipy', + numsamples=1, seed=None, **kwargs): + """ + A realization of the random expression. + + Parameters + ========== + + expr : Expression of random variables + Expression from which sample is extracted + condition : Expr containing RandomSymbols + A conditional expression + size : int, tuple + Represents size of each sample in numsamples + library : str + - 'scipy' : Sample using scipy + - 'numpy' : Sample using numpy + - 'pymc' : Sample using PyMC + + Choose any of the available options to sample from as string, + by default is 'scipy' + numsamples : int + Number of samples, each with size as ``size``. + + .. deprecated:: 1.9 + + The ``numsamples`` parameter is deprecated and is only provided for + compatibility with v1.8. Use a list comprehension or an additional + dimension in ``size`` instead. See + :ref:`deprecated-sympy-stats-numsamples` for details. + + seed : + An object to be used as seed by the given external library for sampling `expr`. + Following is the list of possible types of object for the supported libraries, + + - 'scipy': int, numpy.random.RandomState, numpy.random.Generator + - 'numpy': int, numpy.random.RandomState, numpy.random.Generator + - 'pymc': int + + Optional, by default None, in which case seed settings + related to the given library will be used. + No modifications to environment's global seed settings + are done by this argument. + + Returns + ======= + + sample: float/list/numpy.ndarray + one sample or a collection of samples of the random expression. + + - sample(X) returns float/numpy.float64/numpy.int64 object. + - sample(X, size=int/tuple) returns numpy.ndarray object. + + Examples + ======== + + >>> from sympy.stats import Die, sample, Normal, Geometric + >>> X, Y, Z = Die('X', 6), Die('Y', 6), Die('Z', 6) # Finite Random Variable + >>> die_roll = sample(X + Y + Z) + >>> die_roll # doctest: +SKIP + 3 + >>> N = Normal('N', 3, 4) # Continuous Random Variable + >>> samp = sample(N) + >>> samp in N.pspace.domain.set + True + >>> samp = sample(N, N>0) + >>> samp > 0 + True + >>> samp_list = sample(N, size=4) + >>> [sam in N.pspace.domain.set for sam in samp_list] + [True, True, True, True] + >>> sample(N, size = (2,3)) # doctest: +SKIP + array([[5.42519758, 6.40207856, 4.94991743], + [1.85819627, 6.83403519, 1.9412172 ]]) + >>> G = Geometric('G', 0.5) # Discrete Random Variable + >>> samp_list = sample(G, size=3) + >>> samp_list # doctest: +SKIP + [1, 3, 2] + >>> [sam in G.pspace.domain.set for sam in samp_list] + [True, True, True] + >>> MN = Normal("MN", [3, 4], [[2, 1], [1, 2]]) # Joint Random Variable + >>> samp_list = sample(MN, size=4) + >>> samp_list # doctest: +SKIP + [array([2.85768055, 3.38954165]), + array([4.11163337, 4.3176591 ]), + array([0.79115232, 1.63232916]), + array([4.01747268, 3.96716083])] + >>> [tuple(sam) in MN.pspace.domain.set for sam in samp_list] + [True, True, True, True] + + .. versionchanged:: 1.7.0 + sample used to return an iterator containing the samples instead of value. + + .. versionchanged:: 1.9.0 + sample returns values or array of values instead of an iterator and numsamples is deprecated. + + """ + + iterator = sample_iter(expr, condition, size=size, library=library, + numsamples=numsamples, seed=seed) + + if numsamples != 1: + sympy_deprecation_warning( + f""" + The numsamples parameter to sympy.stats.sample() is deprecated. + Either use a list comprehension, like + + [sample(...) for i in range({numsamples})] + + or add a dimension to size, like + + sample(..., size={(numsamples,) + size}) + """, + deprecated_since_version="1.9", + active_deprecations_target="deprecated-sympy-stats-numsamples", + ) + return [next(iterator) for i in range(numsamples)] + + return next(iterator) + + +def quantile(expr, evaluate=True, **kwargs): + r""" + Return the :math:`p^{th}` order quantile of a probability distribution. + + Explanation + =========== + + Quantile is defined as the value at which the probability of the random + variable is less than or equal to the given probability. + + .. math:: + Q(p) = \inf\{x \in (-\infty, \infty) : p \le F(x)\} + + Examples + ======== + + >>> from sympy.stats import quantile, Die, Exponential + >>> from sympy import Symbol, pprint + >>> p = Symbol("p") + + >>> l = Symbol("lambda", positive=True) + >>> X = Exponential("x", l) + >>> quantile(X)(p) + -log(1 - p)/lambda + + >>> D = Die("d", 6) + >>> pprint(quantile(D)(p), use_unicode=False) + /nan for Or(p > 1, p < 0) + | + | 1 for p <= 1/6 + | + | 2 for p <= 1/3 + | + < 3 for p <= 1/2 + | + | 4 for p <= 2/3 + | + | 5 for p <= 5/6 + | + \ 6 for p <= 1 + + """ + result = pspace(expr).compute_quantile(expr, **kwargs) + + if evaluate and hasattr(result, 'doit'): + return result.doit() + else: + return result + +def sample_iter(expr, condition=None, size=(), library='scipy', + numsamples=S.Infinity, seed=None, **kwargs): + + """ + Returns an iterator of realizations from the expression given a condition. + + Parameters + ========== + + expr: Expr + Random expression to be realized + condition: Expr, optional + A conditional expression + size : int, tuple + Represents size of each sample in numsamples + numsamples: integer, optional + Length of the iterator (defaults to infinity) + seed : + An object to be used as seed by the given external library for sampling `expr`. + Following is the list of possible types of object for the supported libraries, + + - 'scipy': int, numpy.random.RandomState, numpy.random.Generator + - 'numpy': int, numpy.random.RandomState, numpy.random.Generator + - 'pymc': int + + Optional, by default None, in which case seed settings + related to the given library will be used. + No modifications to environment's global seed settings + are done by this argument. + + Examples + ======== + + >>> from sympy.stats import Normal, sample_iter + >>> X = Normal('X', 0, 1) + >>> expr = X*X + 3 + >>> iterator = sample_iter(expr, numsamples=3) # doctest: +SKIP + >>> list(iterator) # doctest: +SKIP + [12, 4, 7] + + Returns + ======= + + sample_iter: iterator object + iterator object containing the sample/samples of given expr + + See Also + ======== + + sample + sampling_P + sampling_E + + """ + from sympy.stats.joint_rv import JointRandomSymbol + if not import_module(library): + raise ValueError("Failed to import %s" % library) + + if condition is not None: + ps = pspace(Tuple(expr, condition)) + else: + ps = pspace(expr) + + rvs = list(ps.values) + if isinstance(expr, JointRandomSymbol): + expr = expr.subs({expr: RandomSymbol(expr.symbol, expr.pspace)}) + else: + sub = {} + for arg in expr.args: + if isinstance(arg, JointRandomSymbol): + sub[arg] = RandomSymbol(arg.symbol, arg.pspace) + expr = expr.subs(sub) + + def fn_subs(*args): + return expr.subs(dict(zip(rvs, args))) + + def given_fn_subs(*args): + if condition is not None: + return condition.subs(dict(zip(rvs, args))) + return False + + if library in ('pymc', 'pymc3'): + # Currently unable to lambdify in pymc + # TODO : Remove when lambdify accepts 'pymc' as module + fn = lambdify(rvs, expr, **kwargs) + else: + fn = lambdify(rvs, expr, modules=library, **kwargs) + + + if condition is not None: + given_fn = lambdify(rvs, condition, **kwargs) + + def return_generator_infinite(): + count = 0 + _size = (1,)+((size,) if isinstance(size, int) else size) + while count < numsamples: + d = ps.sample(size=_size, library=library, seed=seed) # a dictionary that maps RVs to values + args = [d[rv][0] for rv in rvs] + + if condition is not None: # Check that these values satisfy the condition + # TODO: Replace the try-except block with only given_fn(*args) + # once lambdify works with unevaluated SymPy objects. + try: + gd = given_fn(*args) + except (NameError, TypeError): + gd = given_fn_subs(*args) + if gd != True and gd != False: + raise ValueError( + "Conditions must not contain free symbols") + if not gd: # If the values don't satisfy then try again + continue + + yield fn(*args) + count += 1 + + def return_generator_finite(): + faulty = True + while faulty: + d = ps.sample(size=(numsamples,) + ((size,) if isinstance(size, int) else size), + library=library, seed=seed) # a dictionary that maps RVs to values + + faulty = False + count = 0 + while count < numsamples and not faulty: + args = [d[rv][count] for rv in rvs] + if condition is not None: # Check that these values satisfy the condition + # TODO: Replace the try-except block with only given_fn(*args) + # once lambdify works with unevaluated SymPy objects. + try: + gd = given_fn(*args) + except (NameError, TypeError): + gd = given_fn_subs(*args) + if gd != True and gd != False: + raise ValueError( + "Conditions must not contain free symbols") + if not gd: # If the values don't satisfy then try again + faulty = True + + count += 1 + + count = 0 + while count < numsamples: + args = [d[rv][count] for rv in rvs] + # TODO: Replace the try-except block with only fn(*args) + # once lambdify works with unevaluated SymPy objects. + try: + yield fn(*args) + except (NameError, TypeError): + yield fn_subs(*args) + count += 1 + + if numsamples is S.Infinity: + return return_generator_infinite() + + return return_generator_finite() + +def sample_iter_lambdify(expr, condition=None, size=(), + numsamples=S.Infinity, seed=None, **kwargs): + + return sample_iter(expr, condition=condition, size=size, + numsamples=numsamples, seed=seed, **kwargs) + +def sample_iter_subs(expr, condition=None, size=(), + numsamples=S.Infinity, seed=None, **kwargs): + + return sample_iter(expr, condition=condition, size=size, + numsamples=numsamples, seed=seed, **kwargs) + + +def sampling_P(condition, given_condition=None, library='scipy', numsamples=1, + evalf=True, seed=None, **kwargs): + """ + Sampling version of P. + + See Also + ======== + + P + sampling_E + sampling_density + + """ + + count_true = 0 + count_false = 0 + samples = sample_iter(condition, given_condition, library=library, + numsamples=numsamples, seed=seed, **kwargs) + + for sample in samples: + if sample: + count_true += 1 + else: + count_false += 1 + + result = S(count_true) / numsamples + if evalf: + return result.evalf() + else: + return result + + +def sampling_E(expr, given_condition=None, library='scipy', numsamples=1, + evalf=True, seed=None, **kwargs): + """ + Sampling version of E. + + See Also + ======== + + P + sampling_P + sampling_density + """ + samples = list(sample_iter(expr, given_condition, library=library, + numsamples=numsamples, seed=seed, **kwargs)) + result = Add(*samples) / numsamples + + if evalf: + return result.evalf() + else: + return result + +def sampling_density(expr, given_condition=None, library='scipy', + numsamples=1, seed=None, **kwargs): + """ + Sampling version of density. + + See Also + ======== + density + sampling_P + sampling_E + """ + + results = {} + for result in sample_iter(expr, given_condition, library=library, + numsamples=numsamples, seed=seed, **kwargs): + results[result] = results.get(result, 0) + 1 + + return results + + +def dependent(a, b): + """ + Dependence of two random expressions. + + Two expressions are independent if knowledge of one does not change + computations on the other. + + Examples + ======== + + >>> from sympy.stats import Normal, dependent, given + >>> from sympy import Tuple, Eq + + >>> X, Y = Normal('X', 0, 1), Normal('Y', 0, 1) + >>> dependent(X, Y) + False + >>> dependent(2*X + Y, -Y) + True + >>> X, Y = given(Tuple(X, Y), Eq(X + Y, 3)) + >>> dependent(X, Y) + True + + See Also + ======== + + independent + """ + if pspace_independent(a, b): + return False + + z = Symbol('z', real=True) + # Dependent if density is unchanged when one is given information about + # the other + return (density(a, Eq(b, z)) != density(a) or + density(b, Eq(a, z)) != density(b)) + + +def independent(a, b): + """ + Independence of two random expressions. + + Two expressions are independent if knowledge of one does not change + computations on the other. + + Examples + ======== + + >>> from sympy.stats import Normal, independent, given + >>> from sympy import Tuple, Eq + + >>> X, Y = Normal('X', 0, 1), Normal('Y', 0, 1) + >>> independent(X, Y) + True + >>> independent(2*X + Y, -Y) + False + >>> X, Y = given(Tuple(X, Y), Eq(X + Y, 3)) + >>> independent(X, Y) + False + + See Also + ======== + + dependent + """ + return not dependent(a, b) + + +def pspace_independent(a, b): + """ + Tests for independence between a and b by checking if their PSpaces have + overlapping symbols. This is a sufficient but not necessary condition for + independence and is intended to be used internally. + + Notes + ===== + + pspace_independent(a, b) implies independent(a, b) + independent(a, b) does not imply pspace_independent(a, b) + """ + a_symbols = set(pspace(b).symbols) + b_symbols = set(pspace(a).symbols) + + if len(set(random_symbols(a)).intersection(random_symbols(b))) != 0: + return False + + if len(a_symbols.intersection(b_symbols)) == 0: + return True + return None + + +def rv_subs(expr, symbols=None): + """ + Given a random expression replace all random variables with their symbols. + + If symbols keyword is given restrict the swap to only the symbols listed. + """ + if symbols is None: + symbols = random_symbols(expr) + if not symbols: + return expr + swapdict = {rv: rv.symbol for rv in symbols} + return expr.subs(swapdict) + + +class NamedArgsMixin: + _argnames: tuple[str, ...] = () + + def __getattr__(self, attr): + try: + return self.args[self._argnames.index(attr)] + except ValueError: + raise AttributeError("'%s' object has no attribute '%s'" % ( + type(self).__name__, attr)) + + +class Distribution(Basic): + + def sample(self, size=(), library='scipy', seed=None): + """ A random realization from the distribution """ + + module = import_module(library) + if library in {'scipy', 'numpy', 'pymc3', 'pymc'} and module is None: + raise ValueError("Failed to import %s" % library) + + if library == 'scipy': + # scipy does not require map as it can handle using custom distributions. + # However, we will still use a map where we can. + + # TODO: do this for drv.py and frv.py if necessary. + # TODO: add more distributions here if there are more + # See links below referring to sections beginning with "A common parametrization..." + # I will remove all these comments if everything is ok. + + from sympy.stats.sampling.sample_scipy import do_sample_scipy + import numpy + if seed is None or isinstance(seed, int): + rand_state = numpy.random.default_rng(seed=seed) + else: + rand_state = seed + samps = do_sample_scipy(self, size, rand_state) + + elif library == 'numpy': + from sympy.stats.sampling.sample_numpy import do_sample_numpy + import numpy + if seed is None or isinstance(seed, int): + rand_state = numpy.random.default_rng(seed=seed) + else: + rand_state = seed + _size = None if size == () else size + samps = do_sample_numpy(self, _size, rand_state) + elif library in ('pymc', 'pymc3'): + from sympy.stats.sampling.sample_pymc import do_sample_pymc + import logging + logging.getLogger("pymc").setLevel(logging.ERROR) + try: + import pymc + except ImportError: + import pymc3 as pymc + + with pymc.Model(): + if do_sample_pymc(self) is not None: + samps = pymc.sample(draws=prod(size), chains=1, compute_convergence_checks=False, + progressbar=False, random_seed=seed, return_inferencedata=False)[:]['X'] + samps = samps.reshape(size) + else: + samps = None + + else: + raise NotImplementedError("Sampling from %s is not supported yet." + % str(library)) + + if samps is not None: + return samps + raise NotImplementedError( + "Sampling for %s is not currently implemented from %s" + % (self, library)) + + +def _value_check(condition, message): + """ + Raise a ValueError with message if condition is False, else + return True if all conditions were True, else False. + + Examples + ======== + + >>> from sympy.stats.rv import _value_check + >>> from sympy.abc import a, b, c + >>> from sympy import And, Dummy + + >>> _value_check(2 < 3, '') + True + + Here, the condition is not False, but it does not evaluate to True + so False is returned (but no error is raised). So checking if the + return value is True or False will tell you if all conditions were + evaluated. + + >>> _value_check(a < b, '') + False + + In this case the condition is False so an error is raised: + + >>> r = Dummy(real=True) + >>> _value_check(r < r - 1, 'condition is not true') + Traceback (most recent call last): + ... + ValueError: condition is not true + + If no condition of many conditions must be False, they can be + checked by passing them as an iterable: + + >>> _value_check((a < 0, b < 0, c < 0), '') + False + + The iterable can be a generator, too: + + >>> _value_check((i < 0 for i in (a, b, c)), '') + False + + The following are equivalent to the above but do not pass + an iterable: + + >>> all(_value_check(i < 0, '') for i in (a, b, c)) + False + >>> _value_check(And(a < 0, b < 0, c < 0), '') + False + """ + if not iterable(condition): + condition = [condition] + truth = fuzzy_and(condition) + if truth == False: + raise ValueError(message) + return truth == True + +def _symbol_converter(sym): + """ + Casts the parameter to Symbol if it is 'str' + otherwise no operation is performed on it. + + Parameters + ========== + + sym + The parameter to be converted. + + Returns + ======= + + Symbol + the parameter converted to Symbol. + + Raises + ====== + + TypeError + If the parameter is not an instance of both str and + Symbol. + + Examples + ======== + + >>> from sympy import Symbol + >>> from sympy.stats.rv import _symbol_converter + >>> s = _symbol_converter('s') + >>> isinstance(s, Symbol) + True + >>> _symbol_converter(1) + Traceback (most recent call last): + ... + TypeError: 1 is neither a Symbol nor a string + >>> r = Symbol('r') + >>> isinstance(r, Symbol) + True + """ + if isinstance(sym, str): + sym = Symbol(sym) + if not isinstance(sym, Symbol): + raise TypeError("%s is neither a Symbol nor a string"%(sym)) + return sym + +def sample_stochastic_process(process): + """ + This function is used to sample from stochastic process. + + Parameters + ========== + + process: StochasticProcess + Process used to extract the samples. It must be an instance of + StochasticProcess + + Examples + ======== + + >>> from sympy.stats import sample_stochastic_process, DiscreteMarkovChain + >>> from sympy import Matrix + >>> T = Matrix([[0.5, 0.2, 0.3],[0.2, 0.5, 0.3],[0.2, 0.3, 0.5]]) + >>> Y = DiscreteMarkovChain("Y", [0, 1, 2], T) + >>> next(sample_stochastic_process(Y)) in Y.state_space + True + >>> next(sample_stochastic_process(Y)) # doctest: +SKIP + 0 + >>> next(sample_stochastic_process(Y)) # doctest: +SKIP + 2 + + Returns + ======= + + sample: iterator object + iterator object containing the sample of given process + + """ + from sympy.stats.stochastic_process_types import StochasticProcess + if not isinstance(process, StochasticProcess): + raise ValueError("Process must be an instance of Stochastic Process") + return process.sample() diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/rv_interface.py b/.venv/lib/python3.12/site-packages/sympy/stats/rv_interface.py new file mode 100644 index 0000000000000000000000000000000000000000..16d65b83634cdb04ef7e5046175848cdf380434b --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/rv_interface.py @@ -0,0 +1,519 @@ +from sympy.sets import FiniteSet +from sympy.core.numbers import Rational +from sympy.core.relational import Eq +from sympy.core.symbol import Dummy +from sympy.functions.combinatorial.factorials import FallingFactorial +from sympy.functions.elementary.exponential import (exp, log) +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.piecewise import piecewise_fold +from sympy.integrals.integrals import Integral +from sympy.solvers.solveset import solveset +from .rv import (probability, expectation, density, where, given, pspace, cdf, PSpace, + characteristic_function, sample, sample_iter, random_symbols, independent, dependent, + sampling_density, moment_generating_function, quantile, is_random, + sample_stochastic_process) + + +__all__ = ['P', 'E', 'H', 'density', 'where', 'given', 'sample', 'cdf', + 'characteristic_function', 'pspace', 'sample_iter', 'variance', 'std', + 'skewness', 'kurtosis', 'covariance', 'dependent', 'entropy', 'median', + 'independent', 'random_symbols', 'correlation', 'factorial_moment', + 'moment', 'cmoment', 'sampling_density', 'moment_generating_function', + 'smoment', 'quantile', 'sample_stochastic_process'] + + + +def moment(X, n, c=0, condition=None, *, evaluate=True, **kwargs): + """ + Return the nth moment of a random expression about c. + + .. math:: + moment(X, c, n) = E((X-c)^{n}) + + Default value of c is 0. + + Examples + ======== + + >>> from sympy.stats import Die, moment, E + >>> X = Die('X', 6) + >>> moment(X, 1, 6) + -5/2 + >>> moment(X, 2) + 91/6 + >>> moment(X, 1) == E(X) + True + """ + from sympy.stats.symbolic_probability import Moment + if evaluate: + return Moment(X, n, c, condition).doit() + return Moment(X, n, c, condition).rewrite(Integral) + + +def variance(X, condition=None, **kwargs): + """ + Variance of a random expression. + + .. math:: + variance(X) = E((X-E(X))^{2}) + + Examples + ======== + + >>> from sympy.stats import Die, Bernoulli, variance + >>> from sympy import simplify, Symbol + + >>> X = Die('X', 6) + >>> p = Symbol('p') + >>> B = Bernoulli('B', p, 1, 0) + + >>> variance(2*X) + 35/3 + + >>> simplify(variance(B)) + p*(1 - p) + """ + if is_random(X) and pspace(X) == PSpace(): + from sympy.stats.symbolic_probability import Variance + return Variance(X, condition) + + return cmoment(X, 2, condition, **kwargs) + + +def standard_deviation(X, condition=None, **kwargs): + r""" + Standard Deviation of a random expression + + .. math:: + std(X) = \sqrt(E((X-E(X))^{2})) + + Examples + ======== + + >>> from sympy.stats import Bernoulli, std + >>> from sympy import Symbol, simplify + + >>> p = Symbol('p') + >>> B = Bernoulli('B', p, 1, 0) + + >>> simplify(std(B)) + sqrt(p*(1 - p)) + """ + return sqrt(variance(X, condition, **kwargs)) +std = standard_deviation + +def entropy(expr, condition=None, **kwargs): + """ + Calculates entropy of a probability distribution. + + Parameters + ========== + + expression : the random expression whose entropy is to be calculated + condition : optional, to specify conditions on random expression + b: base of the logarithm, optional + By default, it is taken as Euler's number + + Returns + ======= + + result : Entropy of the expression, a constant + + Examples + ======== + + >>> from sympy.stats import Normal, Die, entropy + >>> X = Normal('X', 0, 1) + >>> entropy(X) + log(2)/2 + 1/2 + log(pi)/2 + + >>> D = Die('D', 4) + >>> entropy(D) + log(4) + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Entropy_%28information_theory%29 + .. [2] https://www.crmarsh.com/static/pdf/Charles_Marsh_Continuous_Entropy.pdf + .. [3] https://kconrad.math.uconn.edu/blurbs/analysis/entropypost.pdf + """ + pdf = density(expr, condition, **kwargs) + base = kwargs.get('b', exp(1)) + if isinstance(pdf, dict): + return sum(-prob*log(prob, base) for prob in pdf.values()) + return expectation(-log(pdf(expr), base)) + +def covariance(X, Y, condition=None, **kwargs): + """ + Covariance of two random expressions. + + Explanation + =========== + + The expectation that the two variables will rise and fall together + + .. math:: + covariance(X,Y) = E((X-E(X)) (Y-E(Y))) + + Examples + ======== + + >>> from sympy.stats import Exponential, covariance + >>> from sympy import Symbol + + >>> rate = Symbol('lambda', positive=True, real=True) + >>> X = Exponential('X', rate) + >>> Y = Exponential('Y', rate) + + >>> covariance(X, X) + lambda**(-2) + >>> covariance(X, Y) + 0 + >>> covariance(X, Y + rate*X) + 1/lambda + """ + if (is_random(X) and pspace(X) == PSpace()) or (is_random(Y) and pspace(Y) == PSpace()): + from sympy.stats.symbolic_probability import Covariance + return Covariance(X, Y, condition) + + return expectation( + (X - expectation(X, condition, **kwargs)) * + (Y - expectation(Y, condition, **kwargs)), + condition, **kwargs) + + +def correlation(X, Y, condition=None, **kwargs): + r""" + Correlation of two random expressions, also known as correlation + coefficient or Pearson's correlation. + + Explanation + =========== + + The normalized expectation that the two variables will rise + and fall together + + .. math:: + correlation(X,Y) = E((X-E(X))(Y-E(Y)) / (\sigma_x \sigma_y)) + + Examples + ======== + + >>> from sympy.stats import Exponential, correlation + >>> from sympy import Symbol + + >>> rate = Symbol('lambda', positive=True, real=True) + >>> X = Exponential('X', rate) + >>> Y = Exponential('Y', rate) + + >>> correlation(X, X) + 1 + >>> correlation(X, Y) + 0 + >>> correlation(X, Y + rate*X) + 1/sqrt(1 + lambda**(-2)) + """ + return covariance(X, Y, condition, **kwargs)/(std(X, condition, **kwargs) + * std(Y, condition, **kwargs)) + + +def cmoment(X, n, condition=None, *, evaluate=True, **kwargs): + """ + Return the nth central moment of a random expression about its mean. + + .. math:: + cmoment(X, n) = E((X - E(X))^{n}) + + Examples + ======== + + >>> from sympy.stats import Die, cmoment, variance + >>> X = Die('X', 6) + >>> cmoment(X, 3) + 0 + >>> cmoment(X, 2) + 35/12 + >>> cmoment(X, 2) == variance(X) + True + """ + from sympy.stats.symbolic_probability import CentralMoment + if evaluate: + return CentralMoment(X, n, condition).doit() + return CentralMoment(X, n, condition).rewrite(Integral) + + +def smoment(X, n, condition=None, **kwargs): + r""" + Return the nth Standardized moment of a random expression. + + .. math:: + smoment(X, n) = E(((X - \mu)/\sigma_X)^{n}) + + Examples + ======== + + >>> from sympy.stats import skewness, Exponential, smoment + >>> from sympy import Symbol + >>> rate = Symbol('lambda', positive=True, real=True) + >>> Y = Exponential('Y', rate) + >>> smoment(Y, 4) + 9 + >>> smoment(Y, 4) == smoment(3*Y, 4) + True + >>> smoment(Y, 3) == skewness(Y) + True + """ + sigma = std(X, condition, **kwargs) + return (1/sigma)**n*cmoment(X, n, condition, **kwargs) + +def skewness(X, condition=None, **kwargs): + r""" + Measure of the asymmetry of the probability distribution. + + Explanation + =========== + + Positive skew indicates that most of the values lie to the right of + the mean. + + .. math:: + skewness(X) = E(((X - E(X))/\sigma_X)^{3}) + + Parameters + ========== + + condition : Expr containing RandomSymbols + A conditional expression. skewness(X, X>0) is skewness of X given X > 0 + + Examples + ======== + + >>> from sympy.stats import skewness, Exponential, Normal + >>> from sympy import Symbol + >>> X = Normal('X', 0, 1) + >>> skewness(X) + 0 + >>> skewness(X, X > 0) # find skewness given X > 0 + (-sqrt(2)/sqrt(pi) + 4*sqrt(2)/pi**(3/2))/(1 - 2/pi)**(3/2) + + >>> rate = Symbol('lambda', positive=True, real=True) + >>> Y = Exponential('Y', rate) + >>> skewness(Y) + 2 + """ + return smoment(X, 3, condition=condition, **kwargs) + +def kurtosis(X, condition=None, **kwargs): + r""" + Characterizes the tails/outliers of a probability distribution. + + Explanation + =========== + + Kurtosis of any univariate normal distribution is 3. Kurtosis less than + 3 means that the distribution produces fewer and less extreme outliers + than the normal distribution. + + .. math:: + kurtosis(X) = E(((X - E(X))/\sigma_X)^{4}) + + Parameters + ========== + + condition : Expr containing RandomSymbols + A conditional expression. kurtosis(X, X>0) is kurtosis of X given X > 0 + + Examples + ======== + + >>> from sympy.stats import kurtosis, Exponential, Normal + >>> from sympy import Symbol + >>> X = Normal('X', 0, 1) + >>> kurtosis(X) + 3 + >>> kurtosis(X, X > 0) # find kurtosis given X > 0 + (-4/pi - 12/pi**2 + 3)/(1 - 2/pi)**2 + + >>> rate = Symbol('lamda', positive=True, real=True) + >>> Y = Exponential('Y', rate) + >>> kurtosis(Y) + 9 + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Kurtosis + .. [2] https://mathworld.wolfram.com/Kurtosis.html + """ + return smoment(X, 4, condition=condition, **kwargs) + + +def factorial_moment(X, n, condition=None, **kwargs): + """ + The factorial moment is a mathematical quantity defined as the expectation + or average of the falling factorial of a random variable. + + .. math:: + factorial-moment(X, n) = E(X(X - 1)(X - 2)...(X - n + 1)) + + Parameters + ========== + + n: A natural number, n-th factorial moment. + + condition : Expr containing RandomSymbols + A conditional expression. + + Examples + ======== + + >>> from sympy.stats import factorial_moment, Poisson, Binomial + >>> from sympy import Symbol, S + >>> lamda = Symbol('lamda') + >>> X = Poisson('X', lamda) + >>> factorial_moment(X, 2) + lamda**2 + >>> Y = Binomial('Y', 2, S.Half) + >>> factorial_moment(Y, 2) + 1/2 + >>> factorial_moment(Y, 2, Y > 1) # find factorial moment for Y > 1 + 2 + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Factorial_moment + .. [2] https://mathworld.wolfram.com/FactorialMoment.html + """ + return expectation(FallingFactorial(X, n), condition=condition, **kwargs) + +def median(X, evaluate=True, **kwargs): + r""" + Calculates the median of the probability distribution. + + Explanation + =========== + + Mathematically, median of Probability distribution is defined as all those + values of `m` for which the following condition is satisfied + + .. math:: + P(X\leq m) \geq \frac{1}{2} \text{ and} \text{ } P(X\geq m)\geq \frac{1}{2} + + Parameters + ========== + + X: The random expression whose median is to be calculated. + + Returns + ======= + + The FiniteSet or an Interval which contains the median of the + random expression. + + Examples + ======== + + >>> from sympy.stats import Normal, Die, median + >>> N = Normal('N', 3, 1) + >>> median(N) + {3} + >>> D = Die('D') + >>> median(D) + {3, 4} + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Median#Probability_distributions + + """ + if not is_random(X): + return X + + from sympy.stats.crv import ContinuousPSpace + from sympy.stats.drv import DiscretePSpace + from sympy.stats.frv import FinitePSpace + + if isinstance(pspace(X), FinitePSpace): + cdf = pspace(X).compute_cdf(X) + result = [] + for key, value in cdf.items(): + if value>= Rational(1, 2) and (1 - value) + \ + pspace(X).probability(Eq(X, key)) >= Rational(1, 2): + result.append(key) + return FiniteSet(*result) + if isinstance(pspace(X), (ContinuousPSpace, DiscretePSpace)): + cdf = pspace(X).compute_cdf(X) + x = Dummy('x') + result = solveset(piecewise_fold(cdf(x) - Rational(1, 2)), x, pspace(X).set) + return result + raise NotImplementedError("The median of %s is not implemented."%str(pspace(X))) + + +def coskewness(X, Y, Z, condition=None, **kwargs): + r""" + Calculates the co-skewness of three random variables. + + Explanation + =========== + + Mathematically Coskewness is defined as + + .. math:: + coskewness(X,Y,Z)=\frac{E[(X-E[X]) * (Y-E[Y]) * (Z-E[Z])]} {\sigma_{X}\sigma_{Y}\sigma_{Z}} + + Parameters + ========== + + X : RandomSymbol + Random Variable used to calculate coskewness + Y : RandomSymbol + Random Variable used to calculate coskewness + Z : RandomSymbol + Random Variable used to calculate coskewness + condition : Expr containing RandomSymbols + A conditional expression + + Examples + ======== + + >>> from sympy.stats import coskewness, Exponential, skewness + >>> from sympy import symbols + >>> p = symbols('p', positive=True) + >>> X = Exponential('X', p) + >>> Y = Exponential('Y', 2*p) + >>> coskewness(X, Y, Y) + 0 + >>> coskewness(X, Y + X, Y + 2*X) + 16*sqrt(85)/85 + >>> coskewness(X + 2*Y, Y + X, Y + 2*X, X > 3) + 9*sqrt(170)/85 + >>> coskewness(Y, Y, Y) == skewness(Y) + True + >>> coskewness(X, Y + p*X, Y + 2*p*X) + 4/(sqrt(1 + 1/(4*p**2))*sqrt(4 + 1/(4*p**2))) + + Returns + ======= + + coskewness : The coskewness of the three random variables + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Coskewness + + """ + num = expectation((X - expectation(X, condition, **kwargs)) \ + * (Y - expectation(Y, condition, **kwargs)) \ + * (Z - expectation(Z, condition, **kwargs)), condition, **kwargs) + den = std(X, condition, **kwargs) * std(Y, condition, **kwargs) \ + * std(Z, condition, **kwargs) + return num/den + + +P = probability +E = expectation +H = entropy diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/sampling/__init__.py b/.venv/lib/python3.12/site-packages/sympy/stats/sampling/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/sampling/__pycache__/__init__.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/stats/sampling/__pycache__/__init__.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..607d99dad21c0e61aa1251a930a15fef5642c04a Binary files /dev/null and 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differ diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/sampling/__pycache__/sample_scipy.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/stats/sampling/__pycache__/sample_scipy.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..636e0689e3a1ec25318387cf5c2db342680b6bbd Binary files /dev/null and b/.venv/lib/python3.12/site-packages/sympy/stats/sampling/__pycache__/sample_scipy.cpython-312.pyc differ diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/sampling/sample_numpy.py b/.venv/lib/python3.12/site-packages/sympy/stats/sampling/sample_numpy.py new file mode 100644 index 0000000000000000000000000000000000000000..d65417945449ed8b62d2547215d8908f84820a9b --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/sampling/sample_numpy.py @@ -0,0 +1,105 @@ +from functools import singledispatch + +from sympy.external import import_module +from sympy.stats.crv_types import BetaDistribution, ChiSquaredDistribution, ExponentialDistribution, GammaDistribution, \ + LogNormalDistribution, NormalDistribution, ParetoDistribution, UniformDistribution, FDistributionDistribution, GumbelDistribution, LaplaceDistribution, \ + LogisticDistribution, RayleighDistribution, TriangularDistribution +from sympy.stats.drv_types import GeometricDistribution, PoissonDistribution, ZetaDistribution +from sympy.stats.frv_types import BinomialDistribution, HypergeometricDistribution + + +numpy = import_module('numpy') + + +@singledispatch +def do_sample_numpy(dist, size, rand_state): + return None + + +# CRV: + +@do_sample_numpy.register(BetaDistribution) +def _(dist: BetaDistribution, size, rand_state): + return rand_state.beta(a=float(dist.alpha), b=float(dist.beta), size=size) + + +@do_sample_numpy.register(ChiSquaredDistribution) +def _(dist: ChiSquaredDistribution, size, rand_state): + return rand_state.chisquare(df=float(dist.k), size=size) + + +@do_sample_numpy.register(ExponentialDistribution) +def _(dist: ExponentialDistribution, size, rand_state): + return rand_state.exponential(1 / float(dist.rate), size=size) + +@do_sample_numpy.register(FDistributionDistribution) +def _(dist: FDistributionDistribution, size, rand_state): + return rand_state.f(dfnum = float(dist.d1), dfden = float(dist.d2), size=size) + +@do_sample_numpy.register(GammaDistribution) +def _(dist: GammaDistribution, size, rand_state): + return rand_state.gamma(shape = float(dist.k), scale = float(dist.theta), size=size) + +@do_sample_numpy.register(GumbelDistribution) +def _(dist: GumbelDistribution, size, rand_state): + return rand_state.gumbel(loc = float(dist.mu), scale = float(dist.beta), size=size) + +@do_sample_numpy.register(LaplaceDistribution) +def _(dist: LaplaceDistribution, size, rand_state): + return rand_state.laplace(loc = float(dist.mu), scale = float(dist.b), size=size) + +@do_sample_numpy.register(LogisticDistribution) +def _(dist: LogisticDistribution, size, rand_state): + return rand_state.logistic(loc = float(dist.mu), scale = float(dist.s), size=size) + +@do_sample_numpy.register(LogNormalDistribution) +def _(dist: LogNormalDistribution, size, rand_state): + return rand_state.lognormal(mean = float(dist.mean), sigma = float(dist.std), size=size) + +@do_sample_numpy.register(NormalDistribution) +def _(dist: NormalDistribution, size, rand_state): + return rand_state.normal(loc = float(dist.mean), scale = float(dist.std), size=size) + +@do_sample_numpy.register(RayleighDistribution) +def _(dist: RayleighDistribution, size, rand_state): + return rand_state.rayleigh(scale = float(dist.sigma), size=size) + +@do_sample_numpy.register(ParetoDistribution) +def _(dist: ParetoDistribution, size, rand_state): + return (numpy.random.pareto(a=float(dist.alpha), size=size) + 1) * float(dist.xm) + +@do_sample_numpy.register(TriangularDistribution) +def _(dist: TriangularDistribution, size, rand_state): + return rand_state.triangular(left = float(dist.a), mode = float(dist.b), right = float(dist.c), size=size) + +@do_sample_numpy.register(UniformDistribution) +def _(dist: UniformDistribution, size, rand_state): + return rand_state.uniform(low=float(dist.left), high=float(dist.right), size=size) + + +# DRV: + +@do_sample_numpy.register(GeometricDistribution) +def _(dist: GeometricDistribution, size, rand_state): + return rand_state.geometric(p=float(dist.p), size=size) + + +@do_sample_numpy.register(PoissonDistribution) +def _(dist: PoissonDistribution, size, rand_state): + return rand_state.poisson(lam=float(dist.lamda), size=size) + + +@do_sample_numpy.register(ZetaDistribution) +def _(dist: ZetaDistribution, size, rand_state): + return rand_state.zipf(a=float(dist.s), size=size) + + +# FRV: + +@do_sample_numpy.register(BinomialDistribution) +def _(dist: BinomialDistribution, size, rand_state): + return rand_state.binomial(n=int(dist.n), p=float(dist.p), size=size) + +@do_sample_numpy.register(HypergeometricDistribution) +def _(dist: HypergeometricDistribution, size, rand_state): + return rand_state.hypergeometric(ngood = int(dist.N), nbad = int(dist.m), nsample = int(dist.n), size=size) diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/sampling/sample_pymc.py b/.venv/lib/python3.12/site-packages/sympy/stats/sampling/sample_pymc.py new file mode 100644 index 0000000000000000000000000000000000000000..546f02a3092815af2e54b4a164463f62ece7a024 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/sampling/sample_pymc.py @@ -0,0 +1,99 @@ +from functools import singledispatch +from sympy.external import import_module +from sympy.stats.crv_types import BetaDistribution, CauchyDistribution, ChiSquaredDistribution, ExponentialDistribution, \ + GammaDistribution, LogNormalDistribution, NormalDistribution, ParetoDistribution, UniformDistribution, \ + GaussianInverseDistribution +from sympy.stats.drv_types import PoissonDistribution, GeometricDistribution, NegativeBinomialDistribution +from sympy.stats.frv_types import BinomialDistribution, BernoulliDistribution + + +try: + import pymc +except ImportError: + pymc = import_module('pymc3') + +@singledispatch +def do_sample_pymc(dist): + return None + + +# CRV: + +@do_sample_pymc.register(BetaDistribution) +def _(dist: BetaDistribution): + return pymc.Beta('X', alpha=float(dist.alpha), beta=float(dist.beta)) + + +@do_sample_pymc.register(CauchyDistribution) +def _(dist: CauchyDistribution): + return pymc.Cauchy('X', alpha=float(dist.x0), beta=float(dist.gamma)) + + +@do_sample_pymc.register(ChiSquaredDistribution) +def _(dist: ChiSquaredDistribution): + return pymc.ChiSquared('X', nu=float(dist.k)) + + +@do_sample_pymc.register(ExponentialDistribution) +def _(dist: ExponentialDistribution): + return pymc.Exponential('X', lam=float(dist.rate)) + + +@do_sample_pymc.register(GammaDistribution) +def _(dist: GammaDistribution): + return pymc.Gamma('X', alpha=float(dist.k), beta=1 / float(dist.theta)) + + +@do_sample_pymc.register(LogNormalDistribution) +def _(dist: LogNormalDistribution): + return pymc.Lognormal('X', mu=float(dist.mean), sigma=float(dist.std)) + + +@do_sample_pymc.register(NormalDistribution) +def _(dist: NormalDistribution): + return pymc.Normal('X', float(dist.mean), float(dist.std)) + + +@do_sample_pymc.register(GaussianInverseDistribution) +def _(dist: GaussianInverseDistribution): + return pymc.Wald('X', mu=float(dist.mean), lam=float(dist.shape)) + + +@do_sample_pymc.register(ParetoDistribution) +def _(dist: ParetoDistribution): + return pymc.Pareto('X', alpha=float(dist.alpha), m=float(dist.xm)) + + +@do_sample_pymc.register(UniformDistribution) +def _(dist: UniformDistribution): + return pymc.Uniform('X', lower=float(dist.left), upper=float(dist.right)) + + +# DRV: + +@do_sample_pymc.register(GeometricDistribution) +def _(dist: GeometricDistribution): + return pymc.Geometric('X', p=float(dist.p)) + + +@do_sample_pymc.register(NegativeBinomialDistribution) +def _(dist: NegativeBinomialDistribution): + return pymc.NegativeBinomial('X', mu=float((dist.p * dist.r) / (1 - dist.p)), + alpha=float(dist.r)) + + +@do_sample_pymc.register(PoissonDistribution) +def _(dist: PoissonDistribution): + return pymc.Poisson('X', mu=float(dist.lamda)) + + +# FRV: + +@do_sample_pymc.register(BernoulliDistribution) +def _(dist: BernoulliDistribution): + return pymc.Bernoulli('X', p=float(dist.p)) + + +@do_sample_pymc.register(BinomialDistribution) +def _(dist: BinomialDistribution): + return pymc.Binomial('X', n=int(dist.n), p=float(dist.p)) diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/sampling/sample_scipy.py b/.venv/lib/python3.12/site-packages/sympy/stats/sampling/sample_scipy.py new file mode 100644 index 0000000000000000000000000000000000000000..f12508f68844488e9a14b1476005164eb422796e --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/sampling/sample_scipy.py @@ -0,0 +1,167 @@ +from functools import singledispatch + +from sympy.core.symbol import Dummy +from sympy.functions.elementary.exponential import exp +from sympy.utilities.lambdify import lambdify +from sympy.external import import_module +from sympy.stats import DiscreteDistributionHandmade +from sympy.stats.crv import SingleContinuousDistribution +from sympy.stats.crv_types import ChiSquaredDistribution, ExponentialDistribution, GammaDistribution, \ + LogNormalDistribution, NormalDistribution, ParetoDistribution, UniformDistribution, BetaDistribution, \ + StudentTDistribution, CauchyDistribution +from sympy.stats.drv_types import GeometricDistribution, LogarithmicDistribution, NegativeBinomialDistribution, \ + PoissonDistribution, SkellamDistribution, YuleSimonDistribution, ZetaDistribution +from sympy.stats.frv import SingleFiniteDistribution + + +scipy = import_module("scipy", import_kwargs={'fromlist':['stats']}) + + +@singledispatch +def do_sample_scipy(dist, size, seed): + return None + + +# CRV + +@do_sample_scipy.register(SingleContinuousDistribution) +def _(dist: SingleContinuousDistribution, size, seed): + # if we don't need to make a handmade pdf, we won't + import scipy.stats + + z = Dummy('z') + handmade_pdf = lambdify(z, dist.pdf(z), ['numpy', 'scipy']) + + class scipy_pdf(scipy.stats.rv_continuous): + def _pdf(dist, x): + return handmade_pdf(x) + + scipy_rv = scipy_pdf(a=float(dist.set._inf), + b=float(dist.set._sup), name='scipy_pdf') + return scipy_rv.rvs(size=size, random_state=seed) + + +@do_sample_scipy.register(ChiSquaredDistribution) +def _(dist: ChiSquaredDistribution, size, seed): + # same parametrisation + return scipy.stats.chi2.rvs(df=float(dist.k), size=size, random_state=seed) + + +@do_sample_scipy.register(ExponentialDistribution) +def _(dist: ExponentialDistribution, size, seed): + # https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.expon.html#scipy.stats.expon + return scipy.stats.expon.rvs(scale=1 / float(dist.rate), size=size, random_state=seed) + + +@do_sample_scipy.register(GammaDistribution) +def _(dist: GammaDistribution, size, seed): + # https://stackoverflow.com/questions/42150965/how-to-plot-gamma-distribution-with-alpha-and-beta-parameters-in-python + return scipy.stats.gamma.rvs(a=float(dist.k), scale=float(dist.theta), size=size, random_state=seed) + + +@do_sample_scipy.register(LogNormalDistribution) +def _(dist: LogNormalDistribution, size, seed): + # https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.lognorm.html + return scipy.stats.lognorm.rvs(scale=float(exp(dist.mean)), s=float(dist.std), size=size, random_state=seed) + + +@do_sample_scipy.register(NormalDistribution) +def _(dist: NormalDistribution, size, seed): + return scipy.stats.norm.rvs(loc=float(dist.mean), scale=float(dist.std), size=size, random_state=seed) + + +@do_sample_scipy.register(ParetoDistribution) +def _(dist: ParetoDistribution, size, seed): + # https://stackoverflow.com/questions/42260519/defining-pareto-distribution-in-python-scipy + return scipy.stats.pareto.rvs(b=float(dist.alpha), scale=float(dist.xm), size=size, random_state=seed) + + +@do_sample_scipy.register(StudentTDistribution) +def _(dist: StudentTDistribution, size, seed): + return scipy.stats.t.rvs(df=float(dist.nu), size=size, random_state=seed) + + +@do_sample_scipy.register(UniformDistribution) +def _(dist: UniformDistribution, size, seed): + # https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.uniform.html + return scipy.stats.uniform.rvs(loc=float(dist.left), scale=float(dist.right - dist.left), size=size, random_state=seed) + + +@do_sample_scipy.register(BetaDistribution) +def _(dist: BetaDistribution, size, seed): + # same parametrisation + return scipy.stats.beta.rvs(a=float(dist.alpha), b=float(dist.beta), size=size, random_state=seed) + + +@do_sample_scipy.register(CauchyDistribution) +def _(dist: CauchyDistribution, size, seed): + return scipy.stats.cauchy.rvs(loc=float(dist.x0), scale=float(dist.gamma), size=size, random_state=seed) + + +# DRV: + +@do_sample_scipy.register(DiscreteDistributionHandmade) +def _(dist: DiscreteDistributionHandmade, size, seed): + from scipy.stats import rv_discrete + + z = Dummy('z') + handmade_pmf = lambdify(z, dist.pdf(z), ['numpy', 'scipy']) + + class scipy_pmf(rv_discrete): + def _pmf(dist, x): + return handmade_pmf(x) + + scipy_rv = scipy_pmf(a=float(dist.set._inf), b=float(dist.set._sup), + name='scipy_pmf') + return scipy_rv.rvs(size=size, random_state=seed) + + +@do_sample_scipy.register(GeometricDistribution) +def _(dist: GeometricDistribution, size, seed): + return scipy.stats.geom.rvs(p=float(dist.p), size=size, random_state=seed) + + +@do_sample_scipy.register(LogarithmicDistribution) +def _(dist: LogarithmicDistribution, size, seed): + return scipy.stats.logser.rvs(p=float(dist.p), size=size, random_state=seed) + + +@do_sample_scipy.register(NegativeBinomialDistribution) +def _(dist: NegativeBinomialDistribution, size, seed): + return scipy.stats.nbinom.rvs(n=float(dist.r), p=float(dist.p), size=size, random_state=seed) + + +@do_sample_scipy.register(PoissonDistribution) +def _(dist: PoissonDistribution, size, seed): + return scipy.stats.poisson.rvs(mu=float(dist.lamda), size=size, random_state=seed) + + +@do_sample_scipy.register(SkellamDistribution) +def _(dist: SkellamDistribution, size, seed): + return scipy.stats.skellam.rvs(mu1=float(dist.mu1), mu2=float(dist.mu2), size=size, random_state=seed) + + +@do_sample_scipy.register(YuleSimonDistribution) +def _(dist: YuleSimonDistribution, size, seed): + return scipy.stats.yulesimon.rvs(alpha=float(dist.rho), size=size, random_state=seed) + + +@do_sample_scipy.register(ZetaDistribution) +def _(dist: ZetaDistribution, size, seed): + return scipy.stats.zipf.rvs(a=float(dist.s), size=size, random_state=seed) + + +# FRV: + +@do_sample_scipy.register(SingleFiniteDistribution) +def _(dist: SingleFiniteDistribution, size, seed): + # scipy can handle with custom distributions + + from scipy.stats import rv_discrete + density_ = dist.dict + x, y = [], [] + for k, v in density_.items(): + x.append(int(k)) + y.append(float(v)) + scipy_rv = rv_discrete(name='scipy_rv', values=(x, y)) + return scipy_rv.rvs(size=size, random_state=seed) diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/sampling/tests/__init__.py b/.venv/lib/python3.12/site-packages/sympy/stats/sampling/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/sampling/tests/__pycache__/__init__.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/stats/sampling/tests/__pycache__/__init__.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..dea45e0d7e0e6b8ddab177f2f5a70321a83e7abd Binary files /dev/null and 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b/.venv/lib/python3.12/site-packages/sympy/stats/sampling/tests/test_sample_continuous_rv.py @@ -0,0 +1,181 @@ +from sympy.core.numbers import oo +from sympy.core.symbol import Symbol +from sympy.functions.elementary.exponential import exp +from sympy.sets.sets import Interval +from sympy.external import import_module +from sympy.stats import Beta, Chi, Normal, Gamma, Exponential, LogNormal, Pareto, ChiSquared, Uniform, sample, \ + BetaPrime, Cauchy, GammaInverse, GaussianInverse, StudentT, Weibull, density, ContinuousRV, FDistribution, \ + Gumbel, Laplace, Logistic, Rayleigh, Triangular +from sympy.testing.pytest import skip, raises + + +def test_sample_numpy(): + distribs_numpy = [ + Beta("B", 1, 1), + Normal("N", 0, 1), + Gamma("G", 2, 7), + Exponential("E", 2), + LogNormal("LN", 0, 1), + Pareto("P", 1, 1), + ChiSquared("CS", 2), + Uniform("U", 0, 1), + FDistribution("FD", 1, 2), + Gumbel("GB", 1, 2), + Laplace("L", 1, 2), + Logistic("LO", 1, 2), + Rayleigh("R", 1), + Triangular("T", 1, 2, 2), + ] + size = 3 + numpy = import_module('numpy') + if not numpy: + skip('Numpy is not installed. Abort tests for _sample_numpy.') + else: + for X in distribs_numpy: + samps = sample(X, size=size, library='numpy') + for sam in samps: + assert sam in X.pspace.domain.set + raises(NotImplementedError, + lambda: sample(Chi("C", 1), library='numpy')) + raises(NotImplementedError, + lambda: Chi("C", 1).pspace.distribution.sample(library='tensorflow')) + + +def test_sample_scipy(): + distribs_scipy = [ + Beta("B", 1, 1), + BetaPrime("BP", 1, 1), + Cauchy("C", 1, 1), + Chi("C", 1), + Normal("N", 0, 1), + Gamma("G", 2, 7), + GammaInverse("GI", 1, 1), + GaussianInverse("GUI", 1, 1), + Exponential("E", 2), + LogNormal("LN", 0, 1), + Pareto("P", 1, 1), + StudentT("S", 2), + ChiSquared("CS", 2), + Uniform("U", 0, 1) + ] + size = 3 + scipy = import_module('scipy') + if not scipy: + skip('Scipy is not installed. Abort tests for _sample_scipy.') + else: + for X in distribs_scipy: + samps = sample(X, size=size, library='scipy') + samps2 = sample(X, size=(2, 2), library='scipy') + for sam in samps: + assert sam in X.pspace.domain.set + for i in range(2): + for j in range(2): + assert samps2[i][j] in X.pspace.domain.set + + +def test_sample_pymc(): + distribs_pymc = [ + Beta("B", 1, 1), + Cauchy("C", 1, 1), + Normal("N", 0, 1), + Gamma("G", 2, 7), + GaussianInverse("GI", 1, 1), + Exponential("E", 2), + LogNormal("LN", 0, 1), + Pareto("P", 1, 1), + ChiSquared("CS", 2), + Uniform("U", 0, 1) + ] + size = 3 + pymc = import_module('pymc') + if not pymc: + skip('PyMC is not installed. Abort tests for _sample_pymc.') + else: + for X in distribs_pymc: + samps = sample(X, size=size, library='pymc') + for sam in samps: + assert sam in X.pspace.domain.set + raises(NotImplementedError, + lambda: sample(Chi("C", 1), library='pymc')) + + +def test_sampling_gamma_inverse(): + scipy = import_module('scipy') + if not scipy: + skip('Scipy not installed. Abort tests for sampling of gamma inverse.') + X = GammaInverse("x", 1, 1) + assert sample(X) in X.pspace.domain.set + + +def test_lognormal_sampling(): + # Right now, only density function and sampling works + scipy = import_module('scipy') + if not scipy: + skip('Scipy is not installed. Abort tests') + for i in range(3): + X = LogNormal('x', i, 1) + assert sample(X) in X.pspace.domain.set + + size = 5 + samps = sample(X, size=size) + for samp in samps: + assert samp in X.pspace.domain.set + + +def test_sampling_gaussian_inverse(): + scipy = import_module('scipy') + if not scipy: + skip('Scipy not installed. Abort tests for sampling of Gaussian inverse.') + X = GaussianInverse("x", 1, 1) + assert sample(X, library='scipy') in X.pspace.domain.set + + +def test_prefab_sampling(): + scipy = import_module('scipy') + if not scipy: + skip('Scipy is not installed. Abort tests') + N = Normal('X', 0, 1) + L = LogNormal('L', 0, 1) + E = Exponential('Ex', 1) + P = Pareto('P', 1, 3) + W = Weibull('W', 1, 1) + U = Uniform('U', 0, 1) + B = Beta('B', 2, 5) + G = Gamma('G', 1, 3) + + variables = [N, L, E, P, W, U, B, G] + niter = 10 + size = 5 + for var in variables: + for _ in range(niter): + assert sample(var) in var.pspace.domain.set + samps = sample(var, size=size) + for samp in samps: + assert samp in var.pspace.domain.set + + +def test_sample_continuous(): + z = Symbol('z') + Z = ContinuousRV(z, exp(-z), set=Interval(0, oo)) + assert density(Z)(-1) == 0 + + scipy = import_module('scipy') + if not scipy: + skip('Scipy is not installed. Abort tests') + assert sample(Z) in Z.pspace.domain.set + sym, val = list(Z.pspace.sample().items())[0] + assert sym == Z and val in Interval(0, oo) + + libraries = ['scipy', 'numpy', 'pymc'] + for lib in libraries: + try: + imported_lib = import_module(lib) + if imported_lib: + s0, s1, s2 = [], [], [] + s0 = sample(Z, size=10, library=lib, seed=0) + s1 = sample(Z, size=10, library=lib, seed=0) + s2 = sample(Z, size=10, library=lib, seed=1) + assert all(s0 == s1) + assert all(s1 != s2) + except NotImplementedError: + continue diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/sampling/tests/test_sample_discrete_rv.py b/.venv/lib/python3.12/site-packages/sympy/stats/sampling/tests/test_sample_discrete_rv.py new file mode 100644 index 0000000000000000000000000000000000000000..90d385cd599222fd7da7c1559b619bafbeb01831 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/sampling/tests/test_sample_discrete_rv.py @@ -0,0 +1,109 @@ +from sympy.core.singleton import S +from sympy.core.symbol import Symbol +from sympy.external import import_module +from sympy.stats import ( + Geometric, + Poisson, + Zeta, + sample, + Skellam, + Logarithmic, + NegativeBinomial, + YuleSimon, + DiscreteRV, +) +from sympy.testing.pytest import skip, raises, slow + + +def test_sample_numpy(): + distribs_numpy = [ + Geometric('G', 0.5), + Poisson('P', 1), + Zeta('Z', 2) + ] + size = 3 + numpy = import_module('numpy') + if not numpy: + skip('Numpy is not installed. Abort tests for _sample_numpy.') + else: + for X in distribs_numpy: + samps = sample(X, size=size, library='numpy') + for sam in samps: + assert sam in X.pspace.domain.set + raises(NotImplementedError, + lambda: sample(Skellam('S', 1, 1), library='numpy')) + raises(NotImplementedError, + lambda: Skellam('S', 1, 1).pspace.distribution.sample(library='tensorflow')) + + +def test_sample_scipy(): + p = S(2)/3 + x = Symbol('x', integer=True, positive=True) + pdf = p*(1 - p)**(x - 1) # pdf of Geometric Distribution + distribs_scipy = [ + DiscreteRV(x, pdf, set=S.Naturals), + Geometric('G', 0.5), + Logarithmic('L', 0.5), + NegativeBinomial('N', 5, 0.4), + Poisson('P', 1), + Skellam('S', 1, 1), + YuleSimon('Y', 1), + Zeta('Z', 2) + ] + size = 3 + scipy = import_module('scipy') + if not scipy: + skip('Scipy is not installed. Abort tests for _sample_scipy.') + else: + for X in distribs_scipy: + samps = sample(X, size=size, library='scipy') + samps2 = sample(X, size=(2, 2), library='scipy') + for sam in samps: + assert sam in X.pspace.domain.set + for i in range(2): + for j in range(2): + assert samps2[i][j] in X.pspace.domain.set + + +def test_sample_pymc(): + distribs_pymc = [ + Geometric('G', 0.5), + Poisson('P', 1), + NegativeBinomial('N', 5, 0.4) + ] + size = 3 + pymc = import_module('pymc') + if not pymc: + skip('PyMC is not installed. Abort tests for _sample_pymc.') + else: + for X in distribs_pymc: + samps = sample(X, size=size, library='pymc') + for sam in samps: + assert sam in X.pspace.domain.set + raises(NotImplementedError, + lambda: sample(Skellam('S', 1, 1), library='pymc')) + +@slow +def test_sample_discrete(): + X = Geometric('X', S.Half) + scipy = import_module('scipy') + if not scipy: + skip('Scipy not installed. Abort tests') + assert sample(X) in X.pspace.domain.set + samps = sample(X, size=2) # This takes long time if ran without scipy + for samp in samps: + assert samp in X.pspace.domain.set + + libraries = ['scipy', 'numpy', 'pymc'] + for lib in libraries: + try: + imported_lib = import_module(lib) + if imported_lib: + s0, s1, s2 = [], [], [] + s0 = sample(X, size=10, library=lib, seed=0) + s1 = sample(X, size=10, library=lib, seed=0) + s2 = sample(X, size=10, library=lib, seed=1) + assert all(s0 == s1) + assert not all(s1 == s2) + except NotImplementedError: + continue diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/sampling/tests/test_sample_finite_rv.py b/.venv/lib/python3.12/site-packages/sympy/stats/sampling/tests/test_sample_finite_rv.py new file mode 100644 index 0000000000000000000000000000000000000000..96cabe0ff4aaa5977e16600217fbbdeb08b962ae --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/sampling/tests/test_sample_finite_rv.py @@ -0,0 +1,94 @@ +from sympy.core.numbers import Rational +from sympy.core.singleton import S +from sympy.external import import_module +from sympy.stats import Binomial, sample, Die, FiniteRV, DiscreteUniform, Bernoulli, BetaBinomial, Hypergeometric, \ + Rademacher +from sympy.testing.pytest import skip, raises + +def test_given_sample(): + X = Die('X', 6) + scipy = import_module('scipy') + if not scipy: + skip('Scipy is not installed. Abort tests') + assert sample(X, X > 5) == 6 + +def test_sample_numpy(): + distribs_numpy = [ + Binomial("B", 5, 0.4), + Hypergeometric("H", 2, 1, 1) + ] + size = 3 + numpy = import_module('numpy') + if not numpy: + skip('Numpy is not installed. Abort tests for _sample_numpy.') + else: + for X in distribs_numpy: + samps = sample(X, size=size, library='numpy') + for sam in samps: + assert sam in X.pspace.domain.set + raises(NotImplementedError, + lambda: sample(Die("D"), library='numpy')) + raises(NotImplementedError, + lambda: Die("D").pspace.sample(library='tensorflow')) + + +def test_sample_scipy(): + distribs_scipy = [ + FiniteRV('F', {1: S.Half, 2: Rational(1, 4), 3: Rational(1, 4)}), + DiscreteUniform("Y", list(range(5))), + Die("D"), + Bernoulli("Be", 0.3), + Binomial("Bi", 5, 0.4), + BetaBinomial("Bb", 2, 1, 1), + Hypergeometric("H", 1, 1, 1), + Rademacher("R") + ] + + size = 3 + scipy = import_module('scipy') + if not scipy: + skip('Scipy not installed. Abort tests for _sample_scipy.') + else: + for X in distribs_scipy: + samps = sample(X, size=size) + samps2 = sample(X, size=(2, 2)) + for sam in samps: + assert sam in X.pspace.domain.set + for i in range(2): + for j in range(2): + assert samps2[i][j] in X.pspace.domain.set + + +def test_sample_pymc(): + distribs_pymc = [ + Bernoulli('B', 0.2), + Binomial('N', 5, 0.4) + ] + size = 3 + pymc = import_module('pymc') + if not pymc: + skip('PyMC is not installed. Abort tests for _sample_pymc.') + else: + for X in distribs_pymc: + samps = sample(X, size=size, library='pymc') + for sam in samps: + assert sam in X.pspace.domain.set + raises(NotImplementedError, + lambda: (sample(Die("D"), library='pymc'))) + + +def test_sample_seed(): + F = FiniteRV('F', {1: S.Half, 2: Rational(1, 4), 3: Rational(1, 4)}) + size = 10 + libraries = ['scipy', 'numpy', 'pymc'] + for lib in libraries: + try: + imported_lib = import_module(lib) + if imported_lib: + s0 = sample(F, size=size, library=lib, seed=0) + s1 = sample(F, size=size, library=lib, seed=0) + s2 = sample(F, size=size, library=lib, seed=1) + assert all(s0 == s1) + assert not all(s1 == s2) + except NotImplementedError: + continue diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/stochastic_process.py b/.venv/lib/python3.12/site-packages/sympy/stats/stochastic_process.py new file mode 100644 index 0000000000000000000000000000000000000000..bfb0e759c66be892ae38ddda004dfe928f683fee --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/stochastic_process.py @@ -0,0 +1,66 @@ +from sympy.core.basic import Basic +from sympy.stats.joint_rv import ProductPSpace +from sympy.stats.rv import ProductDomain, _symbol_converter, Distribution + + +class StochasticPSpace(ProductPSpace): + """ + Represents probability space of stochastic processes + and their random variables. Contains mechanics to do + computations for queries of stochastic processes. + + Explanation + =========== + + Initialized by symbol, the specific process and + distribution(optional) if the random indexed symbols + of the process follows any specific distribution, like, + in Bernoulli Process, each random indexed symbol follows + Bernoulli distribution. For processes with memory, this + parameter should not be passed. + """ + + def __new__(cls, sym, process, distribution=None): + sym = _symbol_converter(sym) + from sympy.stats.stochastic_process_types import StochasticProcess + if not isinstance(process, StochasticProcess): + raise TypeError("`process` must be an instance of StochasticProcess.") + if distribution is None: + distribution = Distribution() + return Basic.__new__(cls, sym, process, distribution) + + @property + def process(self): + """ + The associated stochastic process. + """ + return self.args[1] + + @property + def domain(self): + return ProductDomain(self.process.index_set, + self.process.state_space) + + @property + def symbol(self): + return self.args[0] + + @property + def distribution(self): + return self.args[2] + + def probability(self, condition, given_condition=None, evaluate=True, **kwargs): + """ + Transfers the task of handling queries to the specific stochastic + process because every process has their own logic of handling such + queries. + """ + return self.process.probability(condition, given_condition, evaluate, **kwargs) + + def compute_expectation(self, expr, condition=None, evaluate=True, **kwargs): + """ + Transfers the task of handling queries to the specific stochastic + process because every process has their own logic of handling such + queries. + """ + return self.process.expectation(expr, condition, evaluate, **kwargs) diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/stochastic_process_types.py b/.venv/lib/python3.12/site-packages/sympy/stats/stochastic_process_types.py new file mode 100644 index 0000000000000000000000000000000000000000..7387cd3dbcf6defb3b7e475f542d04ecef6fecf6 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/stochastic_process_types.py @@ -0,0 +1,2383 @@ +from __future__ import annotations +import random +import itertools +from typing import Sequence as tSequence +from sympy.concrete.summations import Sum +from sympy.core.add import Add +from sympy.core.basic import Basic +from sympy.core.cache import cacheit +from sympy.core.containers import Tuple +from sympy.core.expr import Expr +from sympy.core.function import (Function, Lambda) +from sympy.core.mul import Mul +from sympy.core.intfunc import igcd +from sympy.core.numbers import (Integer, Rational, oo, pi) +from sympy.core.relational import (Eq, Ge, Gt, Le, Lt, Ne) +from sympy.core.singleton import S +from sympy.core.symbol import (Dummy, Symbol) +from sympy.functions.combinatorial.factorials import factorial +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.integers import ceiling +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.special.gamma_functions import gamma +from sympy.logic.boolalg import (And, Not, Or) +from sympy.matrices.exceptions import NonSquareMatrixError +from sympy.matrices.dense import (Matrix, eye, ones, zeros) +from sympy.matrices.expressions.blockmatrix import BlockMatrix +from sympy.matrices.expressions.matexpr import MatrixSymbol +from sympy.matrices.expressions.special import Identity +from sympy.matrices.immutable import ImmutableMatrix +from sympy.sets.conditionset import ConditionSet +from sympy.sets.contains import Contains +from sympy.sets.fancysets import Range +from sympy.sets.sets import (FiniteSet, Intersection, Interval, Set, Union) +from sympy.solvers.solveset import linsolve +from sympy.tensor.indexed import (Indexed, IndexedBase) +from sympy.core.relational import Relational +from sympy.logic.boolalg import Boolean +from sympy.utilities.exceptions import sympy_deprecation_warning +from sympy.utilities.iterables import strongly_connected_components +from sympy.stats.joint_rv import JointDistribution +from sympy.stats.joint_rv_types import JointDistributionHandmade +from sympy.stats.rv import (RandomIndexedSymbol, random_symbols, RandomSymbol, + _symbol_converter, _value_check, pspace, given, + dependent, is_random, sample_iter, Distribution, + Density) +from sympy.stats.stochastic_process import StochasticPSpace +from sympy.stats.symbolic_probability import Probability, Expectation +from sympy.stats.frv_types import Bernoulli, BernoulliDistribution, FiniteRV +from sympy.stats.drv_types import Poisson, PoissonDistribution +from sympy.stats.crv_types import Normal, NormalDistribution, Gamma, GammaDistribution +from sympy.core.sympify import _sympify, sympify + +EmptySet = S.EmptySet + +__all__ = [ + 'StochasticProcess', + 'DiscreteTimeStochasticProcess', + 'DiscreteMarkovChain', + 'TransitionMatrixOf', + 'StochasticStateSpaceOf', + 'GeneratorMatrixOf', + 'ContinuousMarkovChain', + 'BernoulliProcess', + 'PoissonProcess', + 'WienerProcess', + 'GammaProcess' +] + + +@is_random.register(Indexed) +def _(x): + return is_random(x.base) + +@is_random.register(RandomIndexedSymbol) # type: ignore +def _(x): + return True + +def _set_converter(itr): + """ + Helper function for converting list/tuple/set to Set. + If parameter is not an instance of list/tuple/set then + no operation is performed. + + Returns + ======= + + Set + The argument converted to Set. + + + Raises + ====== + + TypeError + If the argument is not an instance of list/tuple/set. + """ + if isinstance(itr, (list, tuple, set)): + itr = FiniteSet(*itr) + if not isinstance(itr, Set): + raise TypeError("%s is not an instance of list/tuple/set."%(itr)) + return itr + +def _state_converter(itr: tSequence) -> Tuple | Range: + """ + Helper function for converting list/tuple/set/Range/Tuple/FiniteSet + to tuple/Range. + """ + itr_ret: Tuple | Range + + if isinstance(itr, (Tuple, set, FiniteSet)): + itr_ret = Tuple(*(sympify(i) if isinstance(i, str) else i for i in itr)) + + elif isinstance(itr, (list, tuple)): + # check if states are unique + if len(set(itr)) != len(itr): + raise ValueError('The state space must have unique elements.') + itr_ret = Tuple(*(sympify(i) if isinstance(i, str) else i for i in itr)) + + elif isinstance(itr, Range): + # the only ordered set in SymPy I know of + # try to convert to tuple + try: + itr_ret = Tuple(*(sympify(i) if isinstance(i, str) else i for i in itr)) + except (TypeError, ValueError): + itr_ret = itr + + else: + raise TypeError("%s is not an instance of list/tuple/set/Range/Tuple/FiniteSet." % (itr)) + return itr_ret + +def _sym_sympify(arg): + """ + Converts an arbitrary expression to a type that can be used inside SymPy. + As generally strings are unwise to use in the expressions, + it returns the Symbol of argument if the string type argument is passed. + + Parameters + ========= + + arg: The parameter to be converted to be used in SymPy. + + Returns + ======= + + The converted parameter. + + """ + if isinstance(arg, str): + return Symbol(arg) + else: + return _sympify(arg) + +def _matrix_checks(matrix): + if not isinstance(matrix, (Matrix, MatrixSymbol, ImmutableMatrix)): + raise TypeError("Transition probabilities either should " + "be a Matrix or a MatrixSymbol.") + if matrix.shape[0] != matrix.shape[1]: + raise NonSquareMatrixError("%s is not a square matrix"%(matrix)) + if isinstance(matrix, Matrix): + matrix = ImmutableMatrix(matrix.tolist()) + return matrix + +class StochasticProcess(Basic): + """ + Base class for all the stochastic processes whether + discrete or continuous. + + Parameters + ========== + + sym: Symbol or str + state_space: Set + The state space of the stochastic process, by default S.Reals. + For discrete sets it is zero indexed. + + See Also + ======== + + DiscreteTimeStochasticProcess + """ + + index_set = S.Reals + + def __new__(cls, sym, state_space=S.Reals, **kwargs): + sym = _symbol_converter(sym) + state_space = _set_converter(state_space) + return Basic.__new__(cls, sym, state_space) + + @property + def symbol(self): + return self.args[0] + + @property + def state_space(self) -> FiniteSet | Range: + if not isinstance(self.args[1], (FiniteSet, Range)): + assert isinstance(self.args[1], Tuple) + return FiniteSet(*self.args[1]) + return self.args[1] + + def _deprecation_warn_distribution(self): + sympy_deprecation_warning( + """ + Calling the distribution method with a RandomIndexedSymbol + argument, like X.distribution(X(t)) is deprecated. Instead, call + distribution() with the given timestamp, like + + X.distribution(t) + """, + deprecated_since_version="1.7.1", + active_deprecations_target="deprecated-distribution-randomindexedsymbol", + stacklevel=4, + ) + + def distribution(self, key=None): + if key is None: + self._deprecation_warn_distribution() + return Distribution() + + def density(self, x): + return Density() + + def __call__(self, time): + """ + Overridden in ContinuousTimeStochasticProcess. + """ + raise NotImplementedError("Use [] for indexing discrete time stochastic process.") + + def __getitem__(self, time): + """ + Overridden in DiscreteTimeStochasticProcess. + """ + raise NotImplementedError("Use () for indexing continuous time stochastic process.") + + def probability(self, condition): + raise NotImplementedError() + + def joint_distribution(self, *args): + """ + Computes the joint distribution of the random indexed variables. + + Parameters + ========== + + args: iterable + The finite list of random indexed variables/the key of a stochastic + process whose joint distribution has to be computed. + + Returns + ======= + + JointDistribution + The joint distribution of the list of random indexed variables. + An unevaluated object is returned if it is not possible to + compute the joint distribution. + + Raises + ====== + + ValueError: When the arguments passed are not of type RandomIndexSymbol + or Number. + """ + args = list(args) + for i, arg in enumerate(args): + if S(arg).is_Number: + if self.index_set.is_subset(S.Integers): + args[i] = self.__getitem__(arg) + else: + args[i] = self.__call__(arg) + elif not isinstance(arg, RandomIndexedSymbol): + raise ValueError("Expected a RandomIndexedSymbol or " + "key not %s"%(type(arg))) + + if args[0].pspace.distribution == Distribution(): + return JointDistribution(*args) + density = Lambda(tuple(args), + expr=Mul.fromiter(arg.pspace.process.density(arg) for arg in args)) + return JointDistributionHandmade(density) + + def expectation(self, condition, given_condition): + raise NotImplementedError("Abstract method for expectation queries.") + + def sample(self): + raise NotImplementedError("Abstract method for sampling queries.") + +class DiscreteTimeStochasticProcess(StochasticProcess): + """ + Base class for all discrete stochastic processes. + """ + def __getitem__(self, time): + """ + For indexing discrete time stochastic processes. + + Returns + ======= + + RandomIndexedSymbol + """ + time = sympify(time) + if not time.is_symbol and time not in self.index_set: + raise IndexError("%s is not in the index set of %s"%(time, self.symbol)) + idx_obj = Indexed(self.symbol, time) + pspace_obj = StochasticPSpace(self.symbol, self, self.distribution(time)) + return RandomIndexedSymbol(idx_obj, pspace_obj) + +class ContinuousTimeStochasticProcess(StochasticProcess): + """ + Base class for all continuous time stochastic process. + """ + def __call__(self, time): + """ + For indexing continuous time stochastic processes. + + Returns + ======= + + RandomIndexedSymbol + """ + time = sympify(time) + if not time.is_symbol and time not in self.index_set: + raise IndexError("%s is not in the index set of %s"%(time, self.symbol)) + func_obj = Function(self.symbol)(time) + pspace_obj = StochasticPSpace(self.symbol, self, self.distribution(time)) + return RandomIndexedSymbol(func_obj, pspace_obj) + +class TransitionMatrixOf(Boolean): + """ + Assumes that the matrix is the transition matrix + of the process. + """ + + def __new__(cls, process, matrix): + if not isinstance(process, DiscreteMarkovChain): + raise ValueError("Currently only DiscreteMarkovChain " + "support TransitionMatrixOf.") + matrix = _matrix_checks(matrix) + return Basic.__new__(cls, process, matrix) + + process = property(lambda self: self.args[0]) + matrix = property(lambda self: self.args[1]) + +class GeneratorMatrixOf(TransitionMatrixOf): + """ + Assumes that the matrix is the generator matrix + of the process. + """ + + def __new__(cls, process, matrix): + if not isinstance(process, ContinuousMarkovChain): + raise ValueError("Currently only ContinuousMarkovChain " + "support GeneratorMatrixOf.") + matrix = _matrix_checks(matrix) + return Basic.__new__(cls, process, matrix) + +class StochasticStateSpaceOf(Boolean): + + def __new__(cls, process, state_space): + if not isinstance(process, (DiscreteMarkovChain, ContinuousMarkovChain)): + raise ValueError("Currently only DiscreteMarkovChain and ContinuousMarkovChain " + "support StochasticStateSpaceOf.") + state_space = _state_converter(state_space) + if isinstance(state_space, Range): + ss_size = ceiling((state_space.stop - state_space.start) / state_space.step) + else: + ss_size = len(state_space) + state_index = Range(ss_size) + return Basic.__new__(cls, process, state_index) + + process = property(lambda self: self.args[0]) + state_index = property(lambda self: self.args[1]) + +class MarkovProcess(StochasticProcess): + """ + Contains methods that handle queries + common to Markov processes. + """ + + @property + def number_of_states(self) -> Integer | Symbol: + """ + The number of states in the Markov Chain. + """ + return _sympify(self.args[2].shape[0]) # type: ignore + + @property + def _state_index(self): + """ + Returns state index as Range. + """ + return self.args[1] + + @classmethod + def _sanity_checks(cls, state_space, trans_probs): + # Try to never have None as state_space or trans_probs. + # This helps a lot if we get it done at the start. + if (state_space is None) and (trans_probs is None): + _n = Dummy('n', integer=True, nonnegative=True) + state_space = _state_converter(Range(_n)) + trans_probs = _matrix_checks(MatrixSymbol('_T', _n, _n)) + + elif state_space is None: + trans_probs = _matrix_checks(trans_probs) + state_space = _state_converter(Range(trans_probs.shape[0])) + + elif trans_probs is None: + state_space = _state_converter(state_space) + if isinstance(state_space, Range): + _n = ceiling((state_space.stop - state_space.start) / state_space.step) + else: + _n = len(state_space) + trans_probs = MatrixSymbol('_T', _n, _n) + + else: + state_space = _state_converter(state_space) + trans_probs = _matrix_checks(trans_probs) + # Range object doesn't want to give a symbolic size + # so we do it ourselves. + if isinstance(state_space, Range): + ss_size = ceiling((state_space.stop - state_space.start) / state_space.step) + else: + ss_size = len(state_space) + if ss_size != trans_probs.shape[0]: + raise ValueError('The size of the state space and the number of ' + 'rows of the transition matrix must be the same.') + + return state_space, trans_probs + + def _extract_information(self, given_condition): + """ + Helper function to extract information, like, + transition matrix/generator matrix, state space, etc. + """ + if isinstance(self, DiscreteMarkovChain): + trans_probs = self.transition_probabilities + state_index = self._state_index + elif isinstance(self, ContinuousMarkovChain): + trans_probs = self.generator_matrix + state_index = self._state_index + if isinstance(given_condition, And): + gcs = given_condition.args + given_condition = S.true + for gc in gcs: + if isinstance(gc, TransitionMatrixOf): + trans_probs = gc.matrix + if isinstance(gc, StochasticStateSpaceOf): + state_index = gc.state_index + if isinstance(gc, Relational): + given_condition = given_condition & gc + if isinstance(given_condition, TransitionMatrixOf): + trans_probs = given_condition.matrix + given_condition = S.true + if isinstance(given_condition, StochasticStateSpaceOf): + state_index = given_condition.state_index + given_condition = S.true + return trans_probs, state_index, given_condition + + def _check_trans_probs(self, trans_probs, row_sum=1): + """ + Helper function for checking the validity of transition + probabilities. + """ + if not isinstance(trans_probs, MatrixSymbol): + rows = trans_probs.tolist() + for row in rows: + if (sum(row) - row_sum) != 0: + raise ValueError("Values in a row must sum to %s. " + "If you are using Float or floats then please use Rational."%(row_sum)) + + def _work_out_state_index(self, state_index, given_condition, trans_probs): + """ + Helper function to extract state space if there + is a random symbol in the given condition. + """ + # if given condition is None, then there is no need to work out + # state_space from random variables + if given_condition != None: + rand_var = list(given_condition.atoms(RandomSymbol) - + given_condition.atoms(RandomIndexedSymbol)) + if len(rand_var) == 1: + state_index = rand_var[0].pspace.set + + # `not None` is `True`. So the old test fails for symbolic sizes. + # Need to build the statement differently. + sym_cond = not self.number_of_states.is_Integer + cond1 = not sym_cond and len(state_index) != trans_probs.shape[0] + if cond1: + raise ValueError("state space is not compatible with the transition probabilities.") + if not isinstance(trans_probs.shape[0], Symbol): + state_index = FiniteSet(*range(trans_probs.shape[0])) + return state_index + + @cacheit + def _preprocess(self, given_condition, evaluate): + """ + Helper function for pre-processing the information. + """ + is_insufficient = False + + if not evaluate: # avoid pre-processing if the result is not to be evaluated + return (True, None, None, None) + + # extracting transition matrix and state space + trans_probs, state_index, given_condition = self._extract_information(given_condition) + + # given_condition does not have sufficient information + # for computations + if trans_probs is None or \ + given_condition is None: + is_insufficient = True + else: + # checking transition probabilities + if isinstance(self, DiscreteMarkovChain): + self._check_trans_probs(trans_probs, row_sum=1) + elif isinstance(self, ContinuousMarkovChain): + self._check_trans_probs(trans_probs, row_sum=0) + + # working out state space + state_index = self._work_out_state_index(state_index, given_condition, trans_probs) + + return is_insufficient, trans_probs, state_index, given_condition + + def replace_with_index(self, condition): + if isinstance(condition, Relational): + lhs, rhs = condition.lhs, condition.rhs + if not isinstance(lhs, RandomIndexedSymbol): + lhs, rhs = rhs, lhs + condition = type(condition)(self.index_of.get(lhs, lhs), + self.index_of.get(rhs, rhs)) + return condition + + def probability(self, condition, given_condition=None, evaluate=True, **kwargs): + """ + Handles probability queries for Markov process. + + Parameters + ========== + + condition: Relational + given_condition: Relational/And + + Returns + ======= + Probability + If the information is not sufficient. + Expr + In all other cases. + + Note + ==== + Any information passed at the time of query overrides + any information passed at the time of object creation like + transition probabilities, state space. + Pass the transition matrix using TransitionMatrixOf, + generator matrix using GeneratorMatrixOf and state space + using StochasticStateSpaceOf in given_condition using & or And. + """ + check, mat, state_index, new_given_condition = \ + self._preprocess(given_condition, evaluate) + + rv = list(condition.atoms(RandomIndexedSymbol)) + symbolic = False + for sym in rv: + if sym.key.is_symbol: + symbolic = True + break + + if check: + return Probability(condition, new_given_condition) + + if isinstance(self, ContinuousMarkovChain): + trans_probs = self.transition_probabilities(mat) + elif isinstance(self, DiscreteMarkovChain): + trans_probs = mat + condition = self.replace_with_index(condition) + given_condition = self.replace_with_index(given_condition) + new_given_condition = self.replace_with_index(new_given_condition) + + if isinstance(condition, Relational): + if isinstance(new_given_condition, And): + gcs = new_given_condition.args + else: + gcs = (new_given_condition, ) + min_key_rv = list(new_given_condition.atoms(RandomIndexedSymbol)) + + if len(min_key_rv): + min_key_rv = min_key_rv[0] + for r in rv: + if min_key_rv.key.is_symbol or r.key.is_symbol: + continue + if min_key_rv.key > r.key: + return Probability(condition) + else: + min_key_rv = None + return Probability(condition) + + if symbolic: + return self._symbolic_probability(condition, new_given_condition, rv, min_key_rv) + + if len(rv) > 1: + rv[0] = condition.lhs + rv[1] = condition.rhs + if rv[0].key < rv[1].key: + rv[0], rv[1] = rv[1], rv[0] + if isinstance(condition, Gt): + condition = Lt(condition.lhs, condition.rhs) + elif isinstance(condition, Lt): + condition = Gt(condition.lhs, condition.rhs) + elif isinstance(condition, Ge): + condition = Le(condition.lhs, condition.rhs) + elif isinstance(condition, Le): + condition = Ge(condition.lhs, condition.rhs) + s = Rational(0, 1) + n = len(self.state_space) + + if isinstance(condition, (Eq, Ne)): + for i in range(0, n): + s += self.probability(Eq(rv[0], i), Eq(rv[1], i)) * self.probability(Eq(rv[1], i), new_given_condition) + return s if isinstance(condition, Eq) else 1 - s + else: + upper = 0 + greater = False + if isinstance(condition, (Ge, Lt)): + upper = 1 + if isinstance(condition, (Ge, Gt)): + greater = True + + for i in range(0, n): + if i <= n//2: + for j in range(0, i + upper): + s += self.probability(Eq(rv[0], i), Eq(rv[1], j)) * self.probability(Eq(rv[1], j), new_given_condition) + else: + s += self.probability(Eq(rv[0], i), new_given_condition) + for j in range(i + upper, n): + s -= self.probability(Eq(rv[0], i), Eq(rv[1], j)) * self.probability(Eq(rv[1], j), new_given_condition) + return s if greater else 1 - s + + rv = rv[0] + states = condition.as_set() + prob, gstate = {}, None + for gc in gcs: + if gc.has(min_key_rv): + if gc.has(Probability): + p, gp = (gc.rhs, gc.lhs) if isinstance(gc.lhs, Probability) \ + else (gc.lhs, gc.rhs) + gr = gp.args[0] + gset = Intersection(gr.as_set(), state_index) + gstate = list(gset)[0] + prob[gset] = p + else: + _, gstate = (gc.lhs.key, gc.rhs) if isinstance(gc.lhs, RandomIndexedSymbol) \ + else (gc.rhs.key, gc.lhs) + + if not all(k in self.index_set for k in (rv.key, min_key_rv.key)): + raise IndexError("The timestamps of the process are not in it's index set.") + states = Intersection(states, state_index) if not isinstance(self.number_of_states, Symbol) else states + for state in Union(states, FiniteSet(gstate)): + if not state.is_Integer or Ge(state, mat.shape[0]) is True: + raise IndexError("No information is available for (%s, %s) in " + "transition probabilities of shape, (%s, %s). " + "State space is zero indexed." + %(gstate, state, mat.shape[0], mat.shape[1])) + if prob: + gstates = Union(*prob.keys()) + if len(gstates) == 1: + gstate = list(gstates)[0] + gprob = list(prob.values())[0] + prob[gstates] = gprob + elif len(gstates) == len(state_index) - 1: + gstate = list(state_index - gstates)[0] + gprob = S.One - sum(prob.values()) + prob[state_index - gstates] = gprob + else: + raise ValueError("Conflicting information.") + else: + gprob = S.One + + if min_key_rv == rv: + return sum(prob[FiniteSet(state)] for state in states) + if isinstance(self, ContinuousMarkovChain): + return gprob * sum(trans_probs(rv.key - min_key_rv.key).__getitem__((gstate, state)) + for state in states) + if isinstance(self, DiscreteMarkovChain): + return gprob * sum((trans_probs**(rv.key - min_key_rv.key)).__getitem__((gstate, state)) + for state in states) + + if isinstance(condition, Not): + expr = condition.args[0] + return S.One - self.probability(expr, given_condition, evaluate, **kwargs) + + if isinstance(condition, And): + compute_later, state2cond, conds = [], {}, condition.args + for expr in conds: + if isinstance(expr, Relational): + ris = list(expr.atoms(RandomIndexedSymbol))[0] + if state2cond.get(ris, None) is None: + state2cond[ris] = S.true + state2cond[ris] &= expr + else: + compute_later.append(expr) + ris = [] + for ri in state2cond: + ris.append(ri) + cset = Intersection(state2cond[ri].as_set(), state_index) + if len(cset) == 0: + return S.Zero + state2cond[ri] = cset.as_relational(ri) + sorted_ris = sorted(ris, key=lambda ri: ri.key) + prod = self.probability(state2cond[sorted_ris[0]], given_condition, evaluate, **kwargs) + for i in range(1, len(sorted_ris)): + ri, prev_ri = sorted_ris[i], sorted_ris[i-1] + if not isinstance(state2cond[ri], Eq): + raise ValueError("The process is in multiple states at %s, unable to determine the probability."%(ri)) + mat_of = TransitionMatrixOf(self, mat) if isinstance(self, DiscreteMarkovChain) else GeneratorMatrixOf(self, mat) + prod *= self.probability(state2cond[ri], state2cond[prev_ri] + & mat_of + & StochasticStateSpaceOf(self, state_index), + evaluate, **kwargs) + for expr in compute_later: + prod *= self.probability(expr, given_condition, evaluate, **kwargs) + return prod + + if isinstance(condition, Or): + return sum(self.probability(expr, given_condition, evaluate, **kwargs) + for expr in condition.args) + + raise NotImplementedError("Mechanism for handling (%s, %s) queries hasn't been " + "implemented yet."%(condition, given_condition)) + + def _symbolic_probability(self, condition, new_given_condition, rv, min_key_rv): + #Function to calculate probability for queries with symbols + if isinstance(condition, Relational): + curr_state = new_given_condition.rhs if isinstance(new_given_condition.lhs, RandomIndexedSymbol) \ + else new_given_condition.lhs + next_state = condition.rhs if isinstance(condition.lhs, RandomIndexedSymbol) \ + else condition.lhs + + if isinstance(condition, (Eq, Ne)): + if isinstance(self, DiscreteMarkovChain): + P = self.transition_probabilities**(rv[0].key - min_key_rv.key) + else: + P = exp(self.generator_matrix*(rv[0].key - min_key_rv.key)) + prob = P[curr_state, next_state] if isinstance(condition, Eq) else 1 - P[curr_state, next_state] + return Piecewise((prob, rv[0].key > min_key_rv.key), (Probability(condition), True)) + else: + upper = 1 + greater = False + if isinstance(condition, (Ge, Lt)): + upper = 0 + if isinstance(condition, (Ge, Gt)): + greater = True + k = Dummy('k') + condition = Eq(condition.lhs, k) if isinstance(condition.lhs, RandomIndexedSymbol)\ + else Eq(condition.rhs, k) + total = Sum(self.probability(condition, new_given_condition), (k, next_state + upper, self.state_space._sup)) + return Piecewise((total, rv[0].key > min_key_rv.key), (Probability(condition), True)) if greater\ + else Piecewise((1 - total, rv[0].key > min_key_rv.key), (Probability(condition), True)) + else: + return Probability(condition, new_given_condition) + + def expectation(self, expr, condition=None, evaluate=True, **kwargs): + """ + Handles expectation queries for markov process. + + Parameters + ========== + + expr: RandomIndexedSymbol, Relational, Logic + Condition for which expectation has to be computed. Must + contain a RandomIndexedSymbol of the process. + condition: Relational, Logic + The given conditions under which computations should be done. + + Returns + ======= + + Expectation + Unevaluated object if computations cannot be done due to + insufficient information. + Expr + In all other cases when the computations are successful. + + Note + ==== + + Any information passed at the time of query overrides + any information passed at the time of object creation like + transition probabilities, state space. + + Pass the transition matrix using TransitionMatrixOf, + generator matrix using GeneratorMatrixOf and state space + using StochasticStateSpaceOf in given_condition using & or And. + """ + + check, mat, state_index, condition = \ + self._preprocess(condition, evaluate) + + if check: + return Expectation(expr, condition) + + rvs = random_symbols(expr) + if isinstance(expr, Expr) and isinstance(condition, Eq) \ + and len(rvs) == 1: + # handle queries similar to E(f(X[i]), Eq(X[i-m], )) + condition=self.replace_with_index(condition) + state_index=self.replace_with_index(state_index) + rv = list(rvs)[0] + lhsg, rhsg = condition.lhs, condition.rhs + if not isinstance(lhsg, RandomIndexedSymbol): + lhsg, rhsg = (rhsg, lhsg) + if rhsg not in state_index: + raise ValueError("%s state is not in the state space."%(rhsg)) + if rv.key < lhsg.key: + raise ValueError("Incorrect given condition is given, expectation " + "time %s < time %s"%(rv.key, rv.key)) + mat_of = TransitionMatrixOf(self, mat) if isinstance(self, DiscreteMarkovChain) else GeneratorMatrixOf(self, mat) + cond = condition & mat_of & \ + StochasticStateSpaceOf(self, state_index) + func = lambda s: self.probability(Eq(rv, s), cond) * expr.subs(rv, self._state_index[s]) + return sum(func(s) for s in state_index) + + raise NotImplementedError("Mechanism for handling (%s, %s) queries hasn't been " + "implemented yet."%(expr, condition)) + +class DiscreteMarkovChain(DiscreteTimeStochasticProcess, MarkovProcess): + """ + Represents a finite discrete time-homogeneous Markov chain. + + This type of Markov Chain can be uniquely characterised by + its (ordered) state space and its one-step transition probability + matrix. + + Parameters + ========== + + sym: + The name given to the Markov Chain + state_space: + Optional, by default, Range(n) + trans_probs: + Optional, by default, MatrixSymbol('_T', n, n) + + Examples + ======== + + >>> from sympy.stats import DiscreteMarkovChain, TransitionMatrixOf, P, E + >>> from sympy import Matrix, MatrixSymbol, Eq, symbols + >>> T = Matrix([[0.5, 0.2, 0.3],[0.2, 0.5, 0.3],[0.2, 0.3, 0.5]]) + >>> Y = DiscreteMarkovChain("Y", [0, 1, 2], T) + >>> YS = DiscreteMarkovChain("Y") + + >>> Y.state_space + {0, 1, 2} + >>> Y.transition_probabilities + Matrix([ + [0.5, 0.2, 0.3], + [0.2, 0.5, 0.3], + [0.2, 0.3, 0.5]]) + >>> TS = MatrixSymbol('T', 3, 3) + >>> P(Eq(YS[3], 2), Eq(YS[1], 1) & TransitionMatrixOf(YS, TS)) + T[0, 2]*T[1, 0] + T[1, 1]*T[1, 2] + T[1, 2]*T[2, 2] + >>> P(Eq(Y[3], 2), Eq(Y[1], 1)).round(2) + 0.36 + + Probabilities will be calculated based on indexes rather + than state names. For example, with the Sunny-Cloudy-Rainy + model with string state names: + + >>> from sympy.core.symbol import Str + >>> Y = DiscreteMarkovChain("Y", [Str('Sunny'), Str('Cloudy'), Str('Rainy')], T) + >>> P(Eq(Y[3], 2), Eq(Y[1], 1)).round(2) + 0.36 + + This gives the same answer as the ``[0, 1, 2]`` state space. + Currently, there is no support for state names within probability + and expectation statements. Here is a work-around using ``Str``: + + >>> P(Eq(Str('Rainy'), Y[3]), Eq(Y[1], Str('Cloudy'))).round(2) + 0.36 + + Symbol state names can also be used: + + >>> sunny, cloudy, rainy = symbols('Sunny, Cloudy, Rainy') + >>> Y = DiscreteMarkovChain("Y", [sunny, cloudy, rainy], T) + >>> P(Eq(Y[3], rainy), Eq(Y[1], cloudy)).round(2) + 0.36 + + Expectations will be calculated as follows: + + >>> E(Y[3], Eq(Y[1], cloudy)) + 0.38*Cloudy + 0.36*Rainy + 0.26*Sunny + + Probability of expressions with multiple RandomIndexedSymbols + can also be calculated provided there is only 1 RandomIndexedSymbol + in the given condition. It is always better to use Rational instead + of floating point numbers for the probabilities in the + transition matrix to avoid errors. + + >>> from sympy import Gt, Le, Rational + >>> T = Matrix([[Rational(5, 10), Rational(3, 10), Rational(2, 10)], [Rational(2, 10), Rational(7, 10), Rational(1, 10)], [Rational(3, 10), Rational(3, 10), Rational(4, 10)]]) + >>> Y = DiscreteMarkovChain("Y", [0, 1, 2], T) + >>> P(Eq(Y[3], Y[1]), Eq(Y[0], 0)).round(3) + 0.409 + >>> P(Gt(Y[3], Y[1]), Eq(Y[0], 0)).round(2) + 0.36 + >>> P(Le(Y[15], Y[10]), Eq(Y[8], 2)).round(7) + 0.6963328 + + Symbolic probability queries are also supported + + >>> a, b, c, d = symbols('a b c d') + >>> T = Matrix([[Rational(1, 10), Rational(4, 10), Rational(5, 10)], [Rational(3, 10), Rational(4, 10), Rational(3, 10)], [Rational(7, 10), Rational(2, 10), Rational(1, 10)]]) + >>> Y = DiscreteMarkovChain("Y", [0, 1, 2], T) + >>> query = P(Eq(Y[a], b), Eq(Y[c], d)) + >>> query.subs({a:10, b:2, c:5, d:1}).round(4) + 0.3096 + >>> P(Eq(Y[10], 2), Eq(Y[5], 1)).evalf().round(4) + 0.3096 + >>> query_gt = P(Gt(Y[a], b), Eq(Y[c], d)) + >>> query_gt.subs({a:21, b:0, c:5, d:0}).evalf().round(5) + 0.64705 + >>> P(Gt(Y[21], 0), Eq(Y[5], 0)).round(5) + 0.64705 + + There is limited support for arbitrarily sized states: + + >>> n = symbols('n', nonnegative=True, integer=True) + >>> T = MatrixSymbol('T', n, n) + >>> Y = DiscreteMarkovChain("Y", trans_probs=T) + >>> Y.state_space + Range(0, n, 1) + >>> query = P(Eq(Y[a], b), Eq(Y[c], d)) + >>> query.subs({a:10, b:2, c:5, d:1}) + (T**5)[1, 2] + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Markov_chain#Discrete-time_Markov_chain + .. [2] https://web.archive.org/web/20201230182007/https://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter11.pdf + """ + index_set = S.Naturals0 + + def __new__(cls, sym, state_space=None, trans_probs=None): + sym = _symbol_converter(sym) + + state_space, trans_probs = MarkovProcess._sanity_checks(state_space, trans_probs) + + obj = Basic.__new__(cls, sym, state_space, trans_probs) # type: ignore + indices = {} + if isinstance(obj.number_of_states, Integer): + for index, state in enumerate(obj._state_index): + indices[state] = index + obj.index_of = indices + return obj + + @property + def transition_probabilities(self): + """ + Transition probabilities of discrete Markov chain, + either an instance of Matrix or MatrixSymbol. + """ + return self.args[2] + + def communication_classes(self) -> list[tuple[list[Basic], Boolean, Integer]]: + """ + Returns the list of communication classes that partition + the states of the markov chain. + + A communication class is defined to be a set of states + such that every state in that set is reachable from + every other state in that set. Due to its properties + this forms a class in the mathematical sense. + Communication classes are also known as recurrence + classes. + + Returns + ======= + + classes + The ``classes`` are a list of tuples. Each + tuple represents a single communication class + with its properties. The first element in the + tuple is the list of states in the class, the + second element is whether the class is recurrent + and the third element is the period of the + communication class. + + Examples + ======== + + >>> from sympy.stats import DiscreteMarkovChain + >>> from sympy import Matrix + >>> T = Matrix([[0, 1, 0], + ... [1, 0, 0], + ... [1, 0, 0]]) + >>> X = DiscreteMarkovChain('X', [1, 2, 3], T) + >>> classes = X.communication_classes() + >>> for states, is_recurrent, period in classes: + ... states, is_recurrent, period + ([1, 2], True, 2) + ([3], False, 1) + + From this we can see that states ``1`` and ``2`` + communicate, are recurrent and have a period + of 2. We can also see state ``3`` is transient + with a period of 1. + + Notes + ===== + + The algorithm used is of order ``O(n**2)`` where + ``n`` is the number of states in the markov chain. + It uses Tarjan's algorithm to find the classes + themselves and then it uses a breadth-first search + algorithm to find each class's periodicity. + Most of the algorithm's components approach ``O(n)`` + as the matrix becomes more and more sparse. + + References + ========== + + .. [1] https://web.archive.org/web/20220207032113/https://www.columbia.edu/~ww2040/4701Sum07/4701-06-Notes-MCII.pdf + .. [2] https://cecas.clemson.edu/~shierd/Shier/markov.pdf + .. [3] https://www.proquest.com/openview/4adc6a51d8371be5b0e4c7dff287fc70/1?pq-origsite=gscholar&cbl=2026366&diss=y + .. [4] https://www.mathworks.com/help/econ/dtmc.classify.html + """ + n = self.number_of_states + T = self.transition_probabilities + + if isinstance(T, MatrixSymbol): + raise NotImplementedError("Cannot perform the operation with a symbolic matrix.") + + # begin Tarjan's algorithm + V = Range(n) + # don't use state names. Rather use state + # indexes since we use them for matrix + # indexing here and later onward + E = [(i, j) for i in V for j in V if T[i, j] != 0] + classes = strongly_connected_components((V, E)) + # end Tarjan's algorithm + + recurrence = [] + periods = [] + for class_ in classes: + # begin recurrent check (similar to self._check_trans_probs()) + submatrix = T[class_, class_] # get the submatrix with those states + is_recurrent = S.true + rows = submatrix.tolist() + for row in rows: + if (sum(row) - 1) != 0: + is_recurrent = S.false + break + recurrence.append(is_recurrent) + # end recurrent check + + # begin breadth-first search + non_tree_edge_values: set[int] = set() + visited = {class_[0]} + newly_visited = {class_[0]} + level = {class_[0]: 0} + current_level = 0 + done = False # imitate a do-while loop + while not done: # runs at most len(class_) times + done = len(visited) == len(class_) + current_level += 1 + + # this loop and the while loop above run a combined len(class_) number of times. + # so this triple nested loop runs through each of the n states once. + for i in newly_visited: + + # the loop below runs len(class_) number of times + # complexity is around about O(n * avg(len(class_))) + newly_visited = {j for j in class_ if T[i, j] != 0} + + new_tree_edges = newly_visited.difference(visited) + for j in new_tree_edges: + level[j] = current_level + + new_non_tree_edges = newly_visited.intersection(visited) + new_non_tree_edge_values = {level[i]-level[j]+1 for j in new_non_tree_edges} + + non_tree_edge_values = non_tree_edge_values.union(new_non_tree_edge_values) + visited = visited.union(new_tree_edges) + + # igcd needs at least 2 arguments + positive_ntev = {val_e for val_e in non_tree_edge_values if val_e > 0} + if len(positive_ntev) == 0: + periods.append(len(class_)) + elif len(positive_ntev) == 1: + periods.append(positive_ntev.pop()) + else: + periods.append(igcd(*positive_ntev)) + # end breadth-first search + + # convert back to the user's state names + classes = [[_sympify(self._state_index[i]) for i in class_] for class_ in classes] + return list(zip(classes, recurrence, map(Integer,periods))) + + def fundamental_matrix(self): + """ + Each entry fundamental matrix can be interpreted as + the expected number of times the chains is in state j + if it started in state i. + + References + ========== + + .. [1] https://lips.cs.princeton.edu/the-fundamental-matrix-of-a-finite-markov-chain/ + + """ + _, _, _, Q = self.decompose() + + if Q.shape[0] > 0: # if non-ergodic + I = eye(Q.shape[0]) + if (I - Q).det() == 0: + raise ValueError("The fundamental matrix doesn't exist.") + return (I - Q).inv().as_immutable() + else: # if ergodic + P = self.transition_probabilities + I = eye(P.shape[0]) + w = self.fixed_row_vector() + W = Matrix([list(w) for i in range(0, P.shape[0])]) + if (I - P + W).det() == 0: + raise ValueError("The fundamental matrix doesn't exist.") + return (I - P + W).inv().as_immutable() + + def absorbing_probabilities(self): + """ + Computes the absorbing probabilities, i.e. + the ij-th entry of the matrix denotes the + probability of Markov chain being absorbed + in state j starting from state i. + """ + _, _, R, _ = self.decompose() + N = self.fundamental_matrix() + if R is None or N is None: + return None + return N*R + + def absorbing_probabilites(self): + sympy_deprecation_warning( + """ + DiscreteMarkovChain.absorbing_probabilites() is deprecated. Use + absorbing_probabilities() instead (note the spelling difference). + """, + deprecated_since_version="1.7", + active_deprecations_target="deprecated-absorbing_probabilites", + ) + return self.absorbing_probabilities() + + def is_regular(self): + tuples = self.communication_classes() + if len(tuples) == 0: + return S.false # not defined for a 0x0 matrix + classes, _, periods = list(zip(*tuples)) + return And(len(classes) == 1, periods[0] == 1) + + def is_ergodic(self): + tuples = self.communication_classes() + if len(tuples) == 0: + return S.false # not defined for a 0x0 matrix + classes, _, _ = list(zip(*tuples)) + return S(len(classes) == 1) + + def is_absorbing_state(self, state): + trans_probs = self.transition_probabilities + if isinstance(trans_probs, ImmutableMatrix) and \ + state < trans_probs.shape[0]: + return S(trans_probs[state, state]) is S.One + + def is_absorbing_chain(self): + states, A, B, C = self.decompose() + r = A.shape[0] + return And(r > 0, A == Identity(r).as_explicit()) + + def stationary_distribution(self, condition_set=False) -> ImmutableMatrix | ConditionSet | Lambda: + r""" + The stationary distribution is any row vector, p, that solves p = pP, + is row stochastic and each element in p must be nonnegative. + That means in matrix form: :math:`(P-I)^T p^T = 0` and + :math:`(1, \dots, 1) p = 1` + where ``P`` is the one-step transition matrix. + + All time-homogeneous Markov Chains with a finite state space + have at least one stationary distribution. In addition, if + a finite time-homogeneous Markov Chain is irreducible, the + stationary distribution is unique. + + Parameters + ========== + + condition_set : bool + If the chain has a symbolic size or transition matrix, + it will return a ``Lambda`` if ``False`` and return a + ``ConditionSet`` if ``True``. + + Examples + ======== + + >>> from sympy.stats import DiscreteMarkovChain + >>> from sympy import Matrix, S + + An irreducible Markov Chain + + >>> T = Matrix([[S(1)/2, S(1)/2, 0], + ... [S(4)/5, S(1)/5, 0], + ... [1, 0, 0]]) + >>> X = DiscreteMarkovChain('X', trans_probs=T) + >>> X.stationary_distribution() + Matrix([[8/13, 5/13, 0]]) + + A reducible Markov Chain + + >>> T = Matrix([[S(1)/2, S(1)/2, 0], + ... [S(4)/5, S(1)/5, 0], + ... [0, 0, 1]]) + >>> X = DiscreteMarkovChain('X', trans_probs=T) + >>> X.stationary_distribution() + Matrix([[8/13 - 8*tau0/13, 5/13 - 5*tau0/13, tau0]]) + + >>> Y = DiscreteMarkovChain('Y') + >>> Y.stationary_distribution() + Lambda((wm, _T), Eq(wm*_T, wm)) + + >>> Y.stationary_distribution(condition_set=True) + ConditionSet(wm, Eq(wm*_T, wm)) + + References + ========== + + .. [1] https://www.probabilitycourse.com/chapter11/11_2_6_stationary_and_limiting_distributions.php + .. [2] https://web.archive.org/web/20210508104430/https://galton.uchicago.edu/~yibi/teaching/stat317/2014/Lectures/Lecture4_6up.pdf + + See Also + ======== + + sympy.stats.DiscreteMarkovChain.limiting_distribution + """ + trans_probs = self.transition_probabilities + n = self.number_of_states + + if n == 0: + return ImmutableMatrix(Matrix([[]])) + + # symbolic matrix version + if isinstance(trans_probs, MatrixSymbol): + wm = MatrixSymbol('wm', 1, n) + if condition_set: + return ConditionSet(wm, Eq(wm * trans_probs, wm)) + else: + return Lambda((wm, trans_probs), Eq(wm * trans_probs, wm)) + + # numeric matrix version + a = Matrix(trans_probs - Identity(n)).T + a[0, 0:n] = ones(1, n) # type: ignore + b = zeros(n, 1) + b[0, 0] = 1 + + soln = list(linsolve((a, b)))[0] + return ImmutableMatrix([soln]) + + def fixed_row_vector(self): + """ + A wrapper for ``stationary_distribution()``. + """ + return self.stationary_distribution() + + @property + def limiting_distribution(self): + """ + The fixed row vector is the limiting + distribution of a discrete Markov chain. + """ + return self.fixed_row_vector() + + def decompose(self) -> tuple[list[Basic], ImmutableMatrix, ImmutableMatrix, ImmutableMatrix]: + """ + Decomposes the transition matrix into submatrices with + special properties. + + The transition matrix can be decomposed into 4 submatrices: + - A - the submatrix from recurrent states to recurrent states. + - B - the submatrix from transient to recurrent states. + - C - the submatrix from transient to transient states. + - O - the submatrix of zeros for recurrent to transient states. + + Returns + ======= + + states, A, B, C + ``states`` - a list of state names with the first being + the recurrent states and the last being + the transient states in the order + of the row names of A and then the row names of C. + ``A`` - the submatrix from recurrent states to recurrent states. + ``B`` - the submatrix from transient to recurrent states. + ``C`` - the submatrix from transient to transient states. + + Examples + ======== + + >>> from sympy.stats import DiscreteMarkovChain + >>> from sympy import Matrix, S + + One can decompose this chain for example: + + >>> T = Matrix([[S(1)/2, S(1)/2, 0, 0, 0], + ... [S(2)/5, S(1)/5, S(2)/5, 0, 0], + ... [0, 0, 1, 0, 0], + ... [0, 0, S(1)/2, S(1)/2, 0], + ... [S(1)/2, 0, 0, 0, S(1)/2]]) + >>> X = DiscreteMarkovChain('X', trans_probs=T) + >>> states, A, B, C = X.decompose() + >>> states + [2, 0, 1, 3, 4] + + >>> A # recurrent to recurrent + Matrix([[1]]) + + >>> B # transient to recurrent + Matrix([ + [ 0], + [2/5], + [1/2], + [ 0]]) + + >>> C # transient to transient + Matrix([ + [1/2, 1/2, 0, 0], + [2/5, 1/5, 0, 0], + [ 0, 0, 1/2, 0], + [1/2, 0, 0, 1/2]]) + + This means that state 2 is the only absorbing state + (since A is a 1x1 matrix). B is a 4x1 matrix since + the 4 remaining transient states all merge into recurrent + state 2. And C is the 4x4 matrix that shows how the + transient states 0, 1, 3, 4 all interact. + + See Also + ======== + + sympy.stats.DiscreteMarkovChain.communication_classes + sympy.stats.DiscreteMarkovChain.canonical_form + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Absorbing_Markov_chain + .. [2] https://people.brandeis.edu/~igusa/Math56aS08/Math56a_S08_notes015.pdf + """ + trans_probs = self.transition_probabilities + + classes = self.communication_classes() + r_states = [] + t_states = [] + + for states, recurrent, period in classes: + if recurrent: + r_states += states + else: + t_states += states + + states = r_states + t_states + indexes = [self.index_of[state] for state in states] # type: ignore + + A = Matrix(len(r_states), len(r_states), + lambda i, j: trans_probs[indexes[i], indexes[j]]) + + B = Matrix(len(t_states), len(r_states), + lambda i, j: trans_probs[indexes[len(r_states) + i], indexes[j]]) + + C = Matrix(len(t_states), len(t_states), + lambda i, j: trans_probs[indexes[len(r_states) + i], indexes[len(r_states) + j]]) + + return states, A.as_immutable(), B.as_immutable(), C.as_immutable() + + def canonical_form(self) -> tuple[list[Basic], ImmutableMatrix]: + """ + Reorders the one-step transition matrix + so that recurrent states appear first and transient + states appear last. Other representations include inserting + transient states first and recurrent states last. + + Returns + ======= + + states, P_new + ``states`` is the list that describes the order of the + new states in the matrix + so that the ith element in ``states`` is the state of the + ith row of A. + ``P_new`` is the new transition matrix in canonical form. + + Examples + ======== + + >>> from sympy.stats import DiscreteMarkovChain + >>> from sympy import Matrix, S + + You can convert your chain into canonical form: + + >>> T = Matrix([[S(1)/2, S(1)/2, 0, 0, 0], + ... [S(2)/5, S(1)/5, S(2)/5, 0, 0], + ... [0, 0, 1, 0, 0], + ... [0, 0, S(1)/2, S(1)/2, 0], + ... [S(1)/2, 0, 0, 0, S(1)/2]]) + >>> X = DiscreteMarkovChain('X', list(range(1, 6)), trans_probs=T) + >>> states, new_matrix = X.canonical_form() + >>> states + [3, 1, 2, 4, 5] + + >>> new_matrix + Matrix([ + [ 1, 0, 0, 0, 0], + [ 0, 1/2, 1/2, 0, 0], + [2/5, 2/5, 1/5, 0, 0], + [1/2, 0, 0, 1/2, 0], + [ 0, 1/2, 0, 0, 1/2]]) + + The new states are [3, 1, 2, 4, 5] and you can + create a new chain with this and its canonical + form will remain the same (since it is already + in canonical form). + + >>> X = DiscreteMarkovChain('X', states, new_matrix) + >>> states, new_matrix = X.canonical_form() + >>> states + [3, 1, 2, 4, 5] + + >>> new_matrix + Matrix([ + [ 1, 0, 0, 0, 0], + [ 0, 1/2, 1/2, 0, 0], + [2/5, 2/5, 1/5, 0, 0], + [1/2, 0, 0, 1/2, 0], + [ 0, 1/2, 0, 0, 1/2]]) + + This is not limited to absorbing chains: + + >>> T = Matrix([[0, 5, 5, 0, 0], + ... [0, 0, 0, 10, 0], + ... [5, 0, 5, 0, 0], + ... [0, 10, 0, 0, 0], + ... [0, 3, 0, 3, 4]])/10 + >>> X = DiscreteMarkovChain('X', trans_probs=T) + >>> states, new_matrix = X.canonical_form() + >>> states + [1, 3, 0, 2, 4] + + >>> new_matrix + Matrix([ + [ 0, 1, 0, 0, 0], + [ 1, 0, 0, 0, 0], + [ 1/2, 0, 0, 1/2, 0], + [ 0, 0, 1/2, 1/2, 0], + [3/10, 3/10, 0, 0, 2/5]]) + + See Also + ======== + + sympy.stats.DiscreteMarkovChain.communication_classes + sympy.stats.DiscreteMarkovChain.decompose + + References + ========== + + .. [1] https://onlinelibrary.wiley.com/doi/pdf/10.1002/9780470316887.app1 + .. [2] http://www.columbia.edu/~ww2040/6711F12/lect1023big.pdf + """ + states, A, B, C = self.decompose() + O = zeros(A.shape[0], C.shape[1]) + return states, BlockMatrix([[A, O], [B, C]]).as_explicit() + + def sample(self): + """ + Returns + ======= + + sample: iterator object + iterator object containing the sample + + """ + if not isinstance(self.transition_probabilities, (Matrix, ImmutableMatrix)): + raise ValueError("Transition Matrix must be provided for sampling") + Tlist = self.transition_probabilities.tolist() + samps = [random.choice(list(self.state_space))] + yield samps[0] + time = 1 + densities = {} + for state in self.state_space: + states = list(self.state_space) + densities[state] = {states[i]: Tlist[state][i] + for i in range(len(states))} + while time < S.Infinity: + samps.append((next(sample_iter(FiniteRV("_", densities[samps[time - 1]]))))) + yield samps[time] + time += 1 + +class ContinuousMarkovChain(ContinuousTimeStochasticProcess, MarkovProcess): + """ + Represents continuous time Markov chain. + + Parameters + ========== + + sym : Symbol/str + state_space : Set + Optional, by default, S.Reals + gen_mat : Matrix/ImmutableMatrix/MatrixSymbol + Optional, by default, None + + Examples + ======== + + >>> from sympy.stats import ContinuousMarkovChain, P + >>> from sympy import Matrix, S, Eq, Gt + >>> G = Matrix([[-S(1), S(1)], [S(1), -S(1)]]) + >>> C = ContinuousMarkovChain('C', state_space=[0, 1], gen_mat=G) + >>> C.limiting_distribution() + Matrix([[1/2, 1/2]]) + >>> C.state_space + {0, 1} + >>> C.generator_matrix + Matrix([ + [-1, 1], + [ 1, -1]]) + + Probability queries are supported + + >>> P(Eq(C(1.96), 0), Eq(C(0.78), 1)).round(5) + 0.45279 + >>> P(Gt(C(1.7), 0), Eq(C(0.82), 1)).round(5) + 0.58602 + + Probability of expressions with multiple RandomIndexedSymbols + can also be calculated provided there is only 1 RandomIndexedSymbol + in the given condition. It is always better to use Rational instead + of floating point numbers for the probabilities in the + generator matrix to avoid errors. + + >>> from sympy import Gt, Le, Rational + >>> G = Matrix([[-S(1), Rational(1, 10), Rational(9, 10)], [Rational(2, 5), -S(1), Rational(3, 5)], [Rational(1, 2), Rational(1, 2), -S(1)]]) + >>> C = ContinuousMarkovChain('C', state_space=[0, 1, 2], gen_mat=G) + >>> P(Eq(C(3.92), C(1.75)), Eq(C(0.46), 0)).round(5) + 0.37933 + >>> P(Gt(C(3.92), C(1.75)), Eq(C(0.46), 0)).round(5) + 0.34211 + >>> P(Le(C(1.57), C(3.14)), Eq(C(1.22), 1)).round(4) + 0.7143 + + Symbolic probability queries are also supported + + >>> from sympy import symbols + >>> a,b,c,d = symbols('a b c d') + >>> G = Matrix([[-S(1), Rational(1, 10), Rational(9, 10)], [Rational(2, 5), -S(1), Rational(3, 5)], [Rational(1, 2), Rational(1, 2), -S(1)]]) + >>> C = ContinuousMarkovChain('C', state_space=[0, 1, 2], gen_mat=G) + >>> query = P(Eq(C(a), b), Eq(C(c), d)) + >>> query.subs({a:3.65, b:2, c:1.78, d:1}).evalf().round(10) + 0.4002723175 + >>> P(Eq(C(3.65), 2), Eq(C(1.78), 1)).round(10) + 0.4002723175 + >>> query_gt = P(Gt(C(a), b), Eq(C(c), d)) + >>> query_gt.subs({a:43.2, b:0, c:3.29, d:2}).evalf().round(10) + 0.6832579186 + >>> P(Gt(C(43.2), 0), Eq(C(3.29), 2)).round(10) + 0.6832579186 + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Markov_chain#Continuous-time_Markov_chain + .. [2] https://u.math.biu.ac.il/~amirgi/CTMCnotes.pdf + """ + index_set = S.Reals + + def __new__(cls, sym, state_space=None, gen_mat=None): + sym = _symbol_converter(sym) + state_space, gen_mat = MarkovProcess._sanity_checks(state_space, gen_mat) + obj = Basic.__new__(cls, sym, state_space, gen_mat) + indices = {} + if isinstance(obj.number_of_states, Integer): + for index, state in enumerate(obj.state_space): + indices[state] = index + obj.index_of = indices + return obj + + @property + def generator_matrix(self): + return self.args[2] + + @cacheit + def transition_probabilities(self, gen_mat=None): + t = Dummy('t') + if isinstance(gen_mat, (Matrix, ImmutableMatrix)) and \ + gen_mat.is_diagonalizable(): + # for faster computation use diagonalized generator matrix + Q, D = gen_mat.diagonalize() + return Lambda(t, Q*exp(t*D)*Q.inv()) + if gen_mat != None: + return Lambda(t, exp(t*gen_mat)) + + def limiting_distribution(self): + gen_mat = self.generator_matrix + if gen_mat is None: + return None + if isinstance(gen_mat, MatrixSymbol): + wm = MatrixSymbol('wm', 1, gen_mat.shape[0]) + return Lambda((wm, gen_mat), Eq(wm*gen_mat, wm)) + w = IndexedBase('w') + wi = [w[i] for i in range(gen_mat.shape[0])] + wm = Matrix([wi]) + eqs = (wm*gen_mat).tolist()[0] + eqs.append(sum(wi) - 1) + soln = list(linsolve(eqs, wi))[0] + return ImmutableMatrix([soln]) + + +class BernoulliProcess(DiscreteTimeStochasticProcess): + """ + The Bernoulli process consists of repeated + independent Bernoulli process trials with the same parameter `p`. + It's assumed that the probability `p` applies to every + trial and that the outcomes of each trial + are independent of all the rest. Therefore Bernoulli Process + is Discrete State and Discrete Time Stochastic Process. + + Parameters + ========== + + sym : Symbol/str + success : Integer/str + The event which is considered to be success. Default: 1. + failure: Integer/str + The event which is considered to be failure. Default: 0. + p : Real Number between 0 and 1 + Represents the probability of getting success. + + Examples + ======== + + >>> from sympy.stats import BernoulliProcess, P, E + >>> from sympy import Eq, Gt + >>> B = BernoulliProcess("B", p=0.7, success=1, failure=0) + >>> B.state_space + {0, 1} + >>> B.p.round(2) + 0.70 + >>> B.success + 1 + >>> B.failure + 0 + >>> X = B[1] + B[2] + B[3] + >>> P(Eq(X, 0)).round(2) + 0.03 + >>> P(Eq(X, 2)).round(2) + 0.44 + >>> P(Eq(X, 4)).round(2) + 0 + >>> P(Gt(X, 1)).round(2) + 0.78 + >>> P(Eq(B[1], 0) & Eq(B[2], 1) & Eq(B[3], 0) & Eq(B[4], 1)).round(2) + 0.04 + >>> B.joint_distribution(B[1], B[2]) + JointDistributionHandmade(Lambda((B[1], B[2]), Piecewise((0.7, Eq(B[1], 1)), + (0.3, Eq(B[1], 0)), (0, True))*Piecewise((0.7, Eq(B[2], 1)), (0.3, Eq(B[2], 0)), + (0, True)))) + >>> E(2*B[1] + B[2]).round(2) + 2.10 + >>> P(B[1] < 1).round(2) + 0.30 + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Bernoulli_process + .. [2] https://mathcs.clarku.edu/~djoyce/ma217/bernoulli.pdf + + """ + + index_set = S.Naturals0 + + def __new__(cls, sym, p, success=1, failure=0): + _value_check(p >= 0 and p <= 1, 'Value of p must be between 0 and 1.') + sym = _symbol_converter(sym) + p = _sympify(p) + success = _sym_sympify(success) + failure = _sym_sympify(failure) + return Basic.__new__(cls, sym, p, success, failure) + + @property + def symbol(self): + return self.args[0] + + @property + def p(self): + return self.args[1] + + @property + def success(self): + return self.args[2] + + @property + def failure(self): + return self.args[3] + + @property + def state_space(self): + return _set_converter([self.success, self.failure]) + + def distribution(self, key=None): + if key is None: + self._deprecation_warn_distribution() + return BernoulliDistribution(self.p) + return BernoulliDistribution(self.p, self.success, self.failure) + + def simple_rv(self, rv): + return Bernoulli(rv.name, p=self.p, + succ=self.success, fail=self.failure) + + def expectation(self, expr, condition=None, evaluate=True, **kwargs): + """ + Computes expectation. + + Parameters + ========== + + expr : RandomIndexedSymbol, Relational, Logic + Condition for which expectation has to be computed. Must + contain a RandomIndexedSymbol of the process. + condition : Relational, Logic + The given conditions under which computations should be done. + + Returns + ======= + + Expectation of the RandomIndexedSymbol. + + """ + + return _SubstituteRV._expectation(expr, condition, evaluate, **kwargs) + + def probability(self, condition, given_condition=None, evaluate=True, **kwargs): + """ + Computes probability. + + Parameters + ========== + + condition : Relational + Condition for which probability has to be computed. Must + contain a RandomIndexedSymbol of the process. + given_condition : Relational, Logic + The given conditions under which computations should be done. + + Returns + ======= + + Probability of the condition. + + """ + + return _SubstituteRV._probability(condition, given_condition, evaluate, **kwargs) + + def density(self, x): + return Piecewise((self.p, Eq(x, self.success)), + (1 - self.p, Eq(x, self.failure)), + (S.Zero, True)) + +class _SubstituteRV: + """ + Internal class to handle the queries of expectation and probability + by substitution. + """ + + @staticmethod + def _rvindexed_subs(expr, condition=None): + """ + Substitutes the RandomIndexedSymbol with the RandomSymbol with + same name, distribution and probability as RandomIndexedSymbol. + + Parameters + ========== + + expr: RandomIndexedSymbol, Relational, Logic + Condition for which expectation has to be computed. Must + contain a RandomIndexedSymbol of the process. + condition: Relational, Logic + The given conditions under which computations should be done. + + """ + + rvs_expr = random_symbols(expr) + if len(rvs_expr) != 0: + swapdict_expr = {} + for rv in rvs_expr: + if isinstance(rv, RandomIndexedSymbol): + newrv = rv.pspace.process.simple_rv(rv) # substitute with equivalent simple rv + swapdict_expr[rv] = newrv + expr = expr.subs(swapdict_expr) + rvs_cond = random_symbols(condition) + if len(rvs_cond)!=0: + swapdict_cond = {} + for rv in rvs_cond: + if isinstance(rv, RandomIndexedSymbol): + newrv = rv.pspace.process.simple_rv(rv) + swapdict_cond[rv] = newrv + condition = condition.subs(swapdict_cond) + return expr, condition + + @classmethod + def _expectation(self, expr, condition=None, evaluate=True, **kwargs): + """ + Internal method for computing expectation of indexed RV. + + Parameters + ========== + + expr: RandomIndexedSymbol, Relational, Logic + Condition for which expectation has to be computed. Must + contain a RandomIndexedSymbol of the process. + condition: Relational, Logic + The given conditions under which computations should be done. + + Returns + ======= + + Expectation of the RandomIndexedSymbol. + + """ + new_expr, new_condition = self._rvindexed_subs(expr, condition) + + if not is_random(new_expr): + return new_expr + new_pspace = pspace(new_expr) + if new_condition is not None: + new_expr = given(new_expr, new_condition) + if new_expr.is_Add: # As E is Linear + return Add(*[new_pspace.compute_expectation( + expr=arg, evaluate=evaluate, **kwargs) + for arg in new_expr.args]) + return new_pspace.compute_expectation( + new_expr, evaluate=evaluate, **kwargs) + + @classmethod + def _probability(self, condition, given_condition=None, evaluate=True, **kwargs): + """ + Internal method for computing probability of indexed RV + + Parameters + ========== + + condition: Relational + Condition for which probability has to be computed. Must + contain a RandomIndexedSymbol of the process. + given_condition: Relational/And + The given conditions under which computations should be done. + + Returns + ======= + + Probability of the condition. + + """ + new_condition, new_givencondition = self._rvindexed_subs(condition, given_condition) + + if isinstance(new_givencondition, RandomSymbol): + condrv = random_symbols(new_condition) + if len(condrv) == 1 and condrv[0] == new_givencondition: + return BernoulliDistribution(self._probability(new_condition), 0, 1) + + if any(dependent(rv, new_givencondition) for rv in condrv): + return Probability(new_condition, new_givencondition) + else: + return self._probability(new_condition) + + if new_givencondition is not None and \ + not isinstance(new_givencondition, (Relational, Boolean)): + raise ValueError("%s is not a relational or combination of relationals" + % (new_givencondition)) + if new_givencondition == False or new_condition == False: + return S.Zero + if new_condition == True: + return S.One + if not isinstance(new_condition, (Relational, Boolean)): + raise ValueError("%s is not a relational or combination of relationals" + % (new_condition)) + + if new_givencondition is not None: # If there is a condition + # Recompute on new conditional expr + return self._probability(given(new_condition, new_givencondition, **kwargs), **kwargs) + result = pspace(new_condition).probability(new_condition, **kwargs) + if evaluate and hasattr(result, 'doit'): + return result.doit() + else: + return result + +def get_timerv_swaps(expr, condition): + """ + Finds the appropriate interval for each time stamp in expr by parsing + the given condition and returns intervals for each timestamp and + dictionary that maps variable time-stamped Random Indexed Symbol to its + corresponding Random Indexed variable with fixed time stamp. + + Parameters + ========== + + expr: SymPy Expression + Expression containing Random Indexed Symbols with variable time stamps + condition: Relational/Boolean Expression + Expression containing time bounds of variable time stamps in expr + + Examples + ======== + + >>> from sympy.stats.stochastic_process_types import get_timerv_swaps, PoissonProcess + >>> from sympy import symbols, Contains, Interval + >>> x, t, d = symbols('x t d', positive=True) + >>> X = PoissonProcess("X", 3) + >>> get_timerv_swaps(x*X(t), Contains(t, Interval.Lopen(0, 1))) + ([Interval.Lopen(0, 1)], {X(t): X(1)}) + >>> get_timerv_swaps((X(t)**2 + X(d)**2), Contains(t, Interval.Lopen(0, 1)) + ... & Contains(d, Interval.Ropen(1, 4))) # doctest: +SKIP + ([Interval.Ropen(1, 4), Interval.Lopen(0, 1)], {X(d): X(3), X(t): X(1)}) + + Returns + ======= + + intervals: list + List of Intervals/FiniteSet on which each time stamp is defined + rv_swap: dict + Dictionary mapping variable time Random Indexed Symbol to constant time + Random Indexed Variable + + """ + + if not isinstance(condition, (Relational, Boolean)): + raise ValueError("%s is not a relational or combination of relationals" + % (condition)) + expr_syms = list(expr.atoms(RandomIndexedSymbol)) + if isinstance(condition, (And, Or)): + given_cond_args = condition.args + else: # single condition + given_cond_args = (condition, ) + rv_swap = {} + intervals = [] + for expr_sym in expr_syms: + for arg in given_cond_args: + if arg.has(expr_sym.key) and isinstance(expr_sym.key, Symbol): + intv = _set_converter(arg.args[1]) + diff_key = intv._sup - intv._inf + if diff_key == oo: + raise ValueError("%s should have finite bounds" % str(expr_sym.name)) + elif diff_key == S.Zero: # has singleton set + diff_key = intv._sup + rv_swap[expr_sym] = expr_sym.subs({expr_sym.key: diff_key}) + intervals.append(intv) + return intervals, rv_swap + + +class CountingProcess(ContinuousTimeStochasticProcess): + """ + This class handles the common methods of the Counting Processes + such as Poisson, Wiener and Gamma Processes + """ + index_set = _set_converter(Interval(0, oo)) + + @property + def symbol(self): + return self.args[0] + + def expectation(self, expr, condition=None, evaluate=True, **kwargs): + """ + Computes expectation + + Parameters + ========== + + expr: RandomIndexedSymbol, Relational, Logic + Condition for which expectation has to be computed. Must + contain a RandomIndexedSymbol of the process. + condition: Relational, Boolean + The given conditions under which computations should be done, i.e, + the intervals on which each variable time stamp in expr is defined + + Returns + ======= + + Expectation of the given expr + + """ + if condition is not None: + intervals, rv_swap = get_timerv_swaps(expr, condition) + # they are independent when they have non-overlapping intervals + if len(intervals) == 1 or all(Intersection(*intv_comb) == EmptySet + for intv_comb in itertools.combinations(intervals, 2)): + if expr.is_Add: + return Add.fromiter(self.expectation(arg, condition) + for arg in expr.args) + expr = expr.subs(rv_swap) + else: + return Expectation(expr, condition) + + return _SubstituteRV._expectation(expr, evaluate=evaluate, **kwargs) + + def _solve_argwith_tworvs(self, arg): + if arg.args[0].key >= arg.args[1].key or isinstance(arg, Eq): + diff_key = abs(arg.args[0].key - arg.args[1].key) + rv = arg.args[0] + arg = arg.__class__(rv.pspace.process(diff_key), 0) + else: + diff_key = arg.args[1].key - arg.args[0].key + rv = arg.args[1] + arg = arg.__class__(rv.pspace.process(diff_key), 0) + return arg + + def _solve_numerical(self, condition, given_condition=None): + if isinstance(condition, And): + args_list = list(condition.args) + else: + args_list = [condition] + if given_condition is not None: + if isinstance(given_condition, And): + args_list.extend(list(given_condition.args)) + else: + args_list.extend([given_condition]) + # sort the args based on timestamp to get the independent increments in + # each segment using all the condition args as well as given_condition args + args_list = sorted(args_list, key=lambda x: x.args[0].key) + result = [] + cond_args = list(condition.args) if isinstance(condition, And) else [condition] + if args_list[0] in cond_args and not (is_random(args_list[0].args[0]) + and is_random(args_list[0].args[1])): + result.append(_SubstituteRV._probability(args_list[0])) + + if is_random(args_list[0].args[0]) and is_random(args_list[0].args[1]): + arg = self._solve_argwith_tworvs(args_list[0]) + result.append(_SubstituteRV._probability(arg)) + + for i in range(len(args_list) - 1): + curr, nex = args_list[i], args_list[i + 1] + diff_key = nex.args[0].key - curr.args[0].key + working_set = curr.args[0].pspace.process.state_space + if curr.args[1] > nex.args[1]: #impossible condition so return 0 + result.append(0) + break + if isinstance(curr, Eq): + working_set = Intersection(working_set, Interval.Lopen(curr.args[1], oo)) + else: + working_set = Intersection(working_set, curr.as_set()) + if isinstance(nex, Eq): + working_set = Intersection(working_set, Interval(-oo, nex.args[1])) + else: + working_set = Intersection(working_set, nex.as_set()) + if working_set == EmptySet: + rv = Eq(curr.args[0].pspace.process(diff_key), 0) + result.append(_SubstituteRV._probability(rv)) + else: + if working_set.is_finite_set: + if isinstance(curr, Eq) and isinstance(nex, Eq): + rv = Eq(curr.args[0].pspace.process(diff_key), len(working_set)) + result.append(_SubstituteRV._probability(rv)) + elif isinstance(curr, Eq) ^ isinstance(nex, Eq): + result.append(Add.fromiter(_SubstituteRV._probability(Eq( + curr.args[0].pspace.process(diff_key), x)) + for x in range(len(working_set)))) + else: + n = len(working_set) + result.append(Add.fromiter((n - x)*_SubstituteRV._probability(Eq( + curr.args[0].pspace.process(diff_key), x)) for x in range(n))) + else: + result.append(_SubstituteRV._probability( + curr.args[0].pspace.process(diff_key) <= working_set._sup - working_set._inf)) + return Mul.fromiter(result) + + + def probability(self, condition, given_condition=None, evaluate=True, **kwargs): + """ + Computes probability. + + Parameters + ========== + + condition: Relational + Condition for which probability has to be computed. Must + contain a RandomIndexedSymbol of the process. + given_condition: Relational, Boolean + The given conditions under which computations should be done, i.e, + the intervals on which each variable time stamp in expr is defined + + Returns + ======= + + Probability of the condition + + """ + check_numeric = True + if isinstance(condition, (And, Or)): + cond_args = condition.args + else: + cond_args = (condition, ) + # check that condition args are numeric or not + if not all(arg.args[0].key.is_number for arg in cond_args): + check_numeric = False + if given_condition is not None: + check_given_numeric = True + if isinstance(given_condition, (And, Or)): + given_cond_args = given_condition.args + else: + given_cond_args = (given_condition, ) + # check that given condition args are numeric or not + if given_condition.has(Contains): + check_given_numeric = False + # Handle numerical queries + if check_numeric and check_given_numeric: + res = [] + if isinstance(condition, Or): + res.append(Add.fromiter(self._solve_numerical(arg, given_condition) + for arg in condition.args)) + if isinstance(given_condition, Or): + res.append(Add.fromiter(self._solve_numerical(condition, arg) + for arg in given_condition.args)) + if res: + return Add.fromiter(res) + return self._solve_numerical(condition, given_condition) + + # No numeric queries, go by Contains?... then check that all the + # given condition are in form of `Contains` + if not all(arg.has(Contains) for arg in given_cond_args): + raise ValueError("If given condition is passed with `Contains`, then " + "please pass the evaluated condition with its corresponding information " + "in terms of intervals of each time stamp to be passed in given condition.") + + intervals, rv_swap = get_timerv_swaps(condition, given_condition) + # they are independent when they have non-overlapping intervals + if len(intervals) == 1 or all(Intersection(*intv_comb) == EmptySet + for intv_comb in itertools.combinations(intervals, 2)): + if isinstance(condition, And): + return Mul.fromiter(self.probability(arg, given_condition) + for arg in condition.args) + elif isinstance(condition, Or): + return Add.fromiter(self.probability(arg, given_condition) + for arg in condition.args) + condition = condition.subs(rv_swap) + else: + return Probability(condition, given_condition) + if check_numeric: + return self._solve_numerical(condition) + return _SubstituteRV._probability(condition, evaluate=evaluate, **kwargs) + +class PoissonProcess(CountingProcess): + """ + The Poisson process is a counting process. It is usually used in scenarios + where we are counting the occurrences of certain events that appear + to happen at a certain rate, but completely at random. + + Parameters + ========== + + sym : Symbol/str + lamda : Positive number + Rate of the process, ``lambda > 0`` + + Examples + ======== + + >>> from sympy.stats import PoissonProcess, P, E + >>> from sympy import symbols, Eq, Ne, Contains, Interval + >>> X = PoissonProcess("X", lamda=3) + >>> X.state_space + Naturals0 + >>> X.lamda + 3 + >>> t1, t2 = symbols('t1 t2', positive=True) + >>> P(X(t1) < 4) + (9*t1**3/2 + 9*t1**2/2 + 3*t1 + 1)*exp(-3*t1) + >>> P(Eq(X(t1), 2) | Ne(X(t1), 4), Contains(t1, Interval.Ropen(2, 4))) + 1 - 36*exp(-6) + >>> P(Eq(X(t1), 2) & Eq(X(t2), 3), Contains(t1, Interval.Lopen(0, 2)) + ... & Contains(t2, Interval.Lopen(2, 4))) + 648*exp(-12) + >>> E(X(t1)) + 3*t1 + >>> E(X(t1)**2 + 2*X(t2), Contains(t1, Interval.Lopen(0, 1)) + ... & Contains(t2, Interval.Lopen(1, 2))) + 18 + >>> P(X(3) < 1, Eq(X(1), 0)) + exp(-6) + >>> P(Eq(X(4), 3), Eq(X(2), 3)) + exp(-6) + >>> P(X(2) <= 3, X(1) > 1) + 5*exp(-3) + + Merging two Poisson Processes + + >>> Y = PoissonProcess("Y", lamda=4) + >>> Z = X + Y + >>> Z.lamda + 7 + + Splitting a Poisson Process into two independent Poisson Processes + + >>> N, M = Z.split(l1=2, l2=5) + >>> N.lamda, M.lamda + (2, 5) + + References + ========== + + .. [1] https://www.probabilitycourse.com/chapter11/11_0_0_intro.php + .. [2] https://en.wikipedia.org/wiki/Poisson_point_process + + """ + + def __new__(cls, sym, lamda): + _value_check(lamda > 0, 'lamda should be a positive number.') + sym = _symbol_converter(sym) + lamda = _sympify(lamda) + return Basic.__new__(cls, sym, lamda) + + @property + def lamda(self): + return self.args[1] + + @property + def state_space(self): + return S.Naturals0 + + def distribution(self, key): + if isinstance(key, RandomIndexedSymbol): + self._deprecation_warn_distribution() + return PoissonDistribution(self.lamda*key.key) + return PoissonDistribution(self.lamda*key) + + def density(self, x): + return (self.lamda*x.key)**x / factorial(x) * exp(-(self.lamda*x.key)) + + def simple_rv(self, rv): + return Poisson(rv.name, lamda=self.lamda*rv.key) + + def __add__(self, other): + if not isinstance(other, PoissonProcess): + raise ValueError("Only instances of Poisson Process can be merged") + return PoissonProcess(Dummy(self.symbol.name + other.symbol.name), + self.lamda + other.lamda) + + def split(self, l1, l2): + if _sympify(l1 + l2) != self.lamda: + raise ValueError("Sum of l1 and l2 should be %s" % str(self.lamda)) + return PoissonProcess(Dummy("l1"), l1), PoissonProcess(Dummy("l2"), l2) + +class WienerProcess(CountingProcess): + """ + The Wiener process is a real valued continuous-time stochastic process. + In physics it is used to study Brownian motion and it is often also called + Brownian motion due to its historical connection with physical process of the + same name originally observed by Scottish botanist Robert Brown. + + Parameters + ========== + + sym : Symbol/str + + Examples + ======== + + >>> from sympy.stats import WienerProcess, P, E + >>> from sympy import symbols, Contains, Interval + >>> X = WienerProcess("X") + >>> X.state_space + Reals + >>> t1, t2 = symbols('t1 t2', positive=True) + >>> P(X(t1) < 7).simplify() + erf(7*sqrt(2)/(2*sqrt(t1)))/2 + 1/2 + >>> P((X(t1) > 2) | (X(t1) < 4), Contains(t1, Interval.Ropen(2, 4))).simplify() + -erf(1)/2 + erf(2)/2 + 1 + >>> E(X(t1)) + 0 + >>> E(X(t1) + 2*X(t2), Contains(t1, Interval.Lopen(0, 1)) + ... & Contains(t2, Interval.Lopen(1, 2))) + 0 + + References + ========== + + .. [1] https://www.probabilitycourse.com/chapter11/11_4_0_brownian_motion_wiener_process.php + .. [2] https://en.wikipedia.org/wiki/Wiener_process + + """ + def __new__(cls, sym): + sym = _symbol_converter(sym) + return Basic.__new__(cls, sym) + + @property + def state_space(self): + return S.Reals + + def distribution(self, key): + if isinstance(key, RandomIndexedSymbol): + self._deprecation_warn_distribution() + return NormalDistribution(0, sqrt(key.key)) + return NormalDistribution(0, sqrt(key)) + + def density(self, x): + return exp(-x**2/(2*x.key)) / (sqrt(2*pi)*sqrt(x.key)) + + def simple_rv(self, rv): + return Normal(rv.name, 0, sqrt(rv.key)) + + +class GammaProcess(CountingProcess): + r""" + A Gamma process is a random process with independent gamma distributed + increments. It is a pure-jump increasing Levy process. + + Parameters + ========== + + sym : Symbol/str + lamda : Positive number + Jump size of the process, ``lamda > 0`` + gamma : Positive number + Rate of jump arrivals, `\gamma > 0` + + Examples + ======== + + >>> from sympy.stats import GammaProcess, E, P, variance + >>> from sympy import symbols, Contains, Interval, Not + >>> t, d, x, l, g = symbols('t d x l g', positive=True) + >>> X = GammaProcess("X", l, g) + >>> E(X(t)) + g*t/l + >>> variance(X(t)).simplify() + g*t/l**2 + >>> X = GammaProcess('X', 1, 2) + >>> P(X(t) < 1).simplify() + lowergamma(2*t, 1)/gamma(2*t) + >>> P(Not((X(t) < 5) & (X(d) > 3)), Contains(t, Interval.Ropen(2, 4)) & + ... Contains(d, Interval.Lopen(7, 8))).simplify() + -4*exp(-3) + 472*exp(-8)/3 + 1 + >>> E(X(2) + x*E(X(5))) + 10*x + 4 + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Gamma_process + + """ + def __new__(cls, sym, lamda, gamma): + _value_check(lamda > 0, 'lamda should be a positive number') + _value_check(gamma > 0, 'gamma should be a positive number') + sym = _symbol_converter(sym) + gamma = _sympify(gamma) + lamda = _sympify(lamda) + return Basic.__new__(cls, sym, lamda, gamma) + + @property + def lamda(self): + return self.args[1] + + @property + def gamma(self): + return self.args[2] + + @property + def state_space(self): + return _set_converter(Interval(0, oo)) + + def distribution(self, key): + if isinstance(key, RandomIndexedSymbol): + self._deprecation_warn_distribution() + return GammaDistribution(self.gamma*key.key, 1/self.lamda) + return GammaDistribution(self.gamma*key, 1/self.lamda) + + def density(self, x): + k = self.gamma*x.key + theta = 1/self.lamda + return x**(k - 1) * exp(-x/theta) / (gamma(k)*theta**k) + + def simple_rv(self, rv): + return Gamma(rv.name, self.gamma*rv.key, 1/self.lamda) diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/symbolic_multivariate_probability.py b/.venv/lib/python3.12/site-packages/sympy/stats/symbolic_multivariate_probability.py new file mode 100644 index 0000000000000000000000000000000000000000..bbe8776e58e82489e29734cea48c9138bc512f34 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/symbolic_multivariate_probability.py @@ -0,0 +1,308 @@ +import itertools + +from sympy.core.add import Add +from sympy.core.expr import Expr +from sympy.core.function import expand as _expand +from sympy.core.mul import Mul +from sympy.core.singleton import S +from sympy.matrices.exceptions import ShapeError +from sympy.matrices.expressions.matexpr import MatrixExpr +from sympy.matrices.expressions.matmul import MatMul +from sympy.matrices.expressions.special import ZeroMatrix +from sympy.stats.rv import RandomSymbol, is_random +from sympy.core.sympify import _sympify +from sympy.stats.symbolic_probability import Variance, Covariance, Expectation + + +class ExpectationMatrix(Expectation, MatrixExpr): + """ + Expectation of a random matrix expression. + + Examples + ======== + + >>> from sympy.stats import ExpectationMatrix, Normal + >>> from sympy.stats.rv import RandomMatrixSymbol + >>> from sympy import symbols, MatrixSymbol, Matrix + >>> k = symbols("k") + >>> A, B = MatrixSymbol("A", k, k), MatrixSymbol("B", k, k) + >>> X, Y = RandomMatrixSymbol("X", k, 1), RandomMatrixSymbol("Y", k, 1) + >>> ExpectationMatrix(X) + ExpectationMatrix(X) + >>> ExpectationMatrix(A*X).shape + (k, 1) + + To expand the expectation in its expression, use ``expand()``: + + >>> ExpectationMatrix(A*X + B*Y).expand() + A*ExpectationMatrix(X) + B*ExpectationMatrix(Y) + >>> ExpectationMatrix((X + Y)*(X - Y).T).expand() + ExpectationMatrix(X*X.T) - ExpectationMatrix(X*Y.T) + ExpectationMatrix(Y*X.T) - ExpectationMatrix(Y*Y.T) + + To evaluate the ``ExpectationMatrix``, use ``doit()``: + + >>> N11, N12 = Normal('N11', 11, 1), Normal('N12', 12, 1) + >>> N21, N22 = Normal('N21', 21, 1), Normal('N22', 22, 1) + >>> M11, M12 = Normal('M11', 1, 1), Normal('M12', 2, 1) + >>> M21, M22 = Normal('M21', 3, 1), Normal('M22', 4, 1) + >>> x1 = Matrix([[N11, N12], [N21, N22]]) + >>> x2 = Matrix([[M11, M12], [M21, M22]]) + >>> ExpectationMatrix(x1 + x2).doit() + Matrix([ + [12, 14], + [24, 26]]) + + """ + def __new__(cls, expr, condition=None): + expr = _sympify(expr) + if condition is None: + if not is_random(expr): + return expr + obj = Expr.__new__(cls, expr) + else: + condition = _sympify(condition) + obj = Expr.__new__(cls, expr, condition) + + obj._shape = expr.shape + obj._condition = condition + return obj + + @property + def shape(self): + return self._shape + + def expand(self, **hints): + expr = self.args[0] + condition = self._condition + if not is_random(expr): + return expr + + if isinstance(expr, Add): + return Add.fromiter(Expectation(a, condition=condition).expand() + for a in expr.args) + + expand_expr = _expand(expr) + if isinstance(expand_expr, Add): + return Add.fromiter(Expectation(a, condition=condition).expand() + for a in expand_expr.args) + + elif isinstance(expr, (Mul, MatMul)): + rv = [] + nonrv = [] + postnon = [] + + for a in expr.args: + if is_random(a): + if rv: + rv.extend(postnon) + else: + nonrv.extend(postnon) + postnon = [] + rv.append(a) + elif a.is_Matrix: + postnon.append(a) + else: + nonrv.append(a) + + # In order to avoid infinite-looping (MatMul may call .doit() again), + # do not rebuild + if len(nonrv) == 0: + return self + return Mul.fromiter(nonrv)*Expectation(Mul.fromiter(rv), + condition=condition)*Mul.fromiter(postnon) + + return self + +class VarianceMatrix(Variance, MatrixExpr): + """ + Variance of a random matrix probability expression. Also known as + Covariance matrix, auto-covariance matrix, dispersion matrix, + or variance-covariance matrix. + + Examples + ======== + + >>> from sympy.stats import VarianceMatrix + >>> from sympy.stats.rv import RandomMatrixSymbol + >>> from sympy import symbols, MatrixSymbol + >>> k = symbols("k") + >>> A, B = MatrixSymbol("A", k, k), MatrixSymbol("B", k, k) + >>> X, Y = RandomMatrixSymbol("X", k, 1), RandomMatrixSymbol("Y", k, 1) + >>> VarianceMatrix(X) + VarianceMatrix(X) + >>> VarianceMatrix(X).shape + (k, k) + + To expand the variance in its expression, use ``expand()``: + + >>> VarianceMatrix(A*X).expand() + A*VarianceMatrix(X)*A.T + >>> VarianceMatrix(A*X + B*Y).expand() + 2*A*CrossCovarianceMatrix(X, Y)*B.T + A*VarianceMatrix(X)*A.T + B*VarianceMatrix(Y)*B.T + """ + def __new__(cls, arg, condition=None): + arg = _sympify(arg) + + if 1 not in arg.shape: + raise ShapeError("Expression is not a vector") + + shape = (arg.shape[0], arg.shape[0]) if arg.shape[1] == 1 else (arg.shape[1], arg.shape[1]) + + if condition: + obj = Expr.__new__(cls, arg, condition) + else: + obj = Expr.__new__(cls, arg) + + obj._shape = shape + obj._condition = condition + return obj + + @property + def shape(self): + return self._shape + + def expand(self, **hints): + arg = self.args[0] + condition = self._condition + + if not is_random(arg): + return ZeroMatrix(*self.shape) + + if isinstance(arg, RandomSymbol): + return self + elif isinstance(arg, Add): + rv = [] + for a in arg.args: + if is_random(a): + rv.append(a) + variances = Add(*(Variance(xv, condition).expand() for xv in rv)) + map_to_covar = lambda x: 2*Covariance(*x, condition=condition).expand() + covariances = Add(*map(map_to_covar, itertools.combinations(rv, 2))) + return variances + covariances + elif isinstance(arg, (Mul, MatMul)): + nonrv = [] + rv = [] + for a in arg.args: + if is_random(a): + rv.append(a) + else: + nonrv.append(a) + if len(rv) == 0: + return ZeroMatrix(*self.shape) + # Avoid possible infinite loops with MatMul: + if len(nonrv) == 0: + return self + # Variance of many multiple matrix products is not implemented: + if len(rv) > 1: + return self + return Mul.fromiter(nonrv)*Variance(Mul.fromiter(rv), + condition)*(Mul.fromiter(nonrv)).transpose() + + # this expression contains a RandomSymbol somehow: + return self + +class CrossCovarianceMatrix(Covariance, MatrixExpr): + """ + Covariance of a random matrix probability expression. + + Examples + ======== + + >>> from sympy.stats import CrossCovarianceMatrix + >>> from sympy.stats.rv import RandomMatrixSymbol + >>> from sympy import symbols, MatrixSymbol + >>> k = symbols("k") + >>> A, B = MatrixSymbol("A", k, k), MatrixSymbol("B", k, k) + >>> C, D = MatrixSymbol("C", k, k), MatrixSymbol("D", k, k) + >>> X, Y = RandomMatrixSymbol("X", k, 1), RandomMatrixSymbol("Y", k, 1) + >>> Z, W = RandomMatrixSymbol("Z", k, 1), RandomMatrixSymbol("W", k, 1) + >>> CrossCovarianceMatrix(X, Y) + CrossCovarianceMatrix(X, Y) + >>> CrossCovarianceMatrix(X, Y).shape + (k, k) + + To expand the covariance in its expression, use ``expand()``: + + >>> CrossCovarianceMatrix(X + Y, Z).expand() + CrossCovarianceMatrix(X, Z) + CrossCovarianceMatrix(Y, Z) + >>> CrossCovarianceMatrix(A*X, Y).expand() + A*CrossCovarianceMatrix(X, Y) + >>> CrossCovarianceMatrix(A*X, B.T*Y).expand() + A*CrossCovarianceMatrix(X, Y)*B + >>> CrossCovarianceMatrix(A*X + B*Y, C.T*Z + D.T*W).expand() + A*CrossCovarianceMatrix(X, W)*D + A*CrossCovarianceMatrix(X, Z)*C + B*CrossCovarianceMatrix(Y, W)*D + B*CrossCovarianceMatrix(Y, Z)*C + + """ + def __new__(cls, arg1, arg2, condition=None): + arg1 = _sympify(arg1) + arg2 = _sympify(arg2) + + if (1 not in arg1.shape) or (1 not in arg2.shape) or (arg1.shape[1] != arg2.shape[1]): + raise ShapeError("Expression is not a vector") + + shape = (arg1.shape[0], arg2.shape[0]) if arg1.shape[1] == 1 and arg2.shape[1] == 1 \ + else (1, 1) + + if condition: + obj = Expr.__new__(cls, arg1, arg2, condition) + else: + obj = Expr.__new__(cls, arg1, arg2) + + obj._shape = shape + obj._condition = condition + return obj + + @property + def shape(self): + return self._shape + + def expand(self, **hints): + arg1 = self.args[0] + arg2 = self.args[1] + condition = self._condition + + if arg1 == arg2: + return VarianceMatrix(arg1, condition).expand() + + if not is_random(arg1) or not is_random(arg2): + return ZeroMatrix(*self.shape) + + if isinstance(arg1, RandomSymbol) and isinstance(arg2, RandomSymbol): + return CrossCovarianceMatrix(arg1, arg2, condition) + + coeff_rv_list1 = self._expand_single_argument(arg1.expand()) + coeff_rv_list2 = self._expand_single_argument(arg2.expand()) + + addends = [a*CrossCovarianceMatrix(r1, r2, condition=condition)*b.transpose() + for (a, r1) in coeff_rv_list1 for (b, r2) in coeff_rv_list2] + return Add.fromiter(addends) + + @classmethod + def _expand_single_argument(cls, expr): + # return (coefficient, random_symbol) pairs: + if isinstance(expr, RandomSymbol): + return [(S.One, expr)] + elif isinstance(expr, Add): + outval = [] + for a in expr.args: + if isinstance(a, (Mul, MatMul)): + outval.append(cls._get_mul_nonrv_rv_tuple(a)) + elif is_random(a): + outval.append((S.One, a)) + + return outval + elif isinstance(expr, (Mul, MatMul)): + return [cls._get_mul_nonrv_rv_tuple(expr)] + elif is_random(expr): + return [(S.One, expr)] + + @classmethod + def _get_mul_nonrv_rv_tuple(cls, m): + rv = [] + nonrv = [] + for a in m.args: + if is_random(a): + rv.append(a) + else: + nonrv.append(a) + return (Mul.fromiter(nonrv), Mul.fromiter(rv)) diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/symbolic_probability.py b/.venv/lib/python3.12/site-packages/sympy/stats/symbolic_probability.py new file mode 100644 index 0000000000000000000000000000000000000000..5d0b971a8691f82de15258d4c460129059eaf436 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/symbolic_probability.py @@ -0,0 +1,698 @@ +import itertools +from sympy.concrete.summations import Sum +from sympy.core.add import Add +from sympy.core.expr import Expr +from sympy.core.function import expand as _expand +from sympy.core.mul import Mul +from sympy.core.relational import Eq +from sympy.core.singleton import S +from sympy.core.symbol import Symbol +from sympy.integrals.integrals import Integral +from sympy.logic.boolalg import Not +from sympy.core.parameters import global_parameters +from sympy.core.sorting import default_sort_key +from sympy.core.sympify import _sympify +from sympy.core.relational import Relational +from sympy.logic.boolalg import Boolean +from sympy.stats import variance, covariance +from sympy.stats.rv import (RandomSymbol, pspace, dependent, + given, sampling_E, RandomIndexedSymbol, is_random, + PSpace, sampling_P, random_symbols) + +__all__ = ['Probability', 'Expectation', 'Variance', 'Covariance'] + + +@is_random.register(Expr) +def _(x): + atoms = x.free_symbols + if len(atoms) == 1 and next(iter(atoms)) == x: + return False + return any(is_random(i) for i in atoms) + +@is_random.register(RandomSymbol) # type: ignore +def _(x): + return True + + +class Probability(Expr): + """ + Symbolic expression for the probability. + + Examples + ======== + + >>> from sympy.stats import Probability, Normal + >>> from sympy import Integral + >>> X = Normal("X", 0, 1) + >>> prob = Probability(X > 1) + >>> prob + Probability(X > 1) + + Integral representation: + + >>> prob.rewrite(Integral) + Integral(sqrt(2)*exp(-_z**2/2)/(2*sqrt(pi)), (_z, 1, oo)) + + Evaluation of the integral: + + >>> prob.evaluate_integral() + sqrt(2)*(-sqrt(2)*sqrt(pi)*erf(sqrt(2)/2) + sqrt(2)*sqrt(pi))/(4*sqrt(pi)) + """ + + is_commutative = True + + def __new__(cls, prob, condition=None, **kwargs): + prob = _sympify(prob) + if condition is None: + obj = Expr.__new__(cls, prob) + else: + condition = _sympify(condition) + obj = Expr.__new__(cls, prob, condition) + obj._condition = condition + return obj + + def doit(self, **hints): + condition = self.args[0] + given_condition = self._condition + numsamples = hints.get('numsamples', False) + evaluate = hints.get('evaluate', True) + + if isinstance(condition, Not): + return S.One - self.func(condition.args[0], given_condition, + evaluate=evaluate).doit(**hints) + + if condition.has(RandomIndexedSymbol): + return pspace(condition).probability(condition, given_condition, + evaluate=evaluate) + + if isinstance(given_condition, RandomSymbol): + condrv = random_symbols(condition) + if len(condrv) == 1 and condrv[0] == given_condition: + from sympy.stats.frv_types import BernoulliDistribution + return BernoulliDistribution(self.func(condition).doit(**hints), 0, 1) + if any(dependent(rv, given_condition) for rv in condrv): + return Probability(condition, given_condition) + else: + return Probability(condition).doit() + + if given_condition is not None and \ + not isinstance(given_condition, (Relational, Boolean)): + raise ValueError("%s is not a relational or combination of relationals" + % (given_condition)) + + if given_condition == False or condition is S.false: + return S.Zero + if not isinstance(condition, (Relational, Boolean)): + raise ValueError("%s is not a relational or combination of relationals" + % (condition)) + if condition is S.true: + return S.One + + if numsamples: + return sampling_P(condition, given_condition, numsamples=numsamples) + if given_condition is not None: # If there is a condition + # Recompute on new conditional expr + return Probability(given(condition, given_condition)).doit() + + # Otherwise pass work off to the ProbabilitySpace + if pspace(condition) == PSpace(): + return Probability(condition, given_condition) + + result = pspace(condition).probability(condition) + if hasattr(result, 'doit') and evaluate: + return result.doit() + else: + return result + + def _eval_rewrite_as_Integral(self, arg, condition=None, **kwargs): + return self.func(arg, condition=condition).doit(evaluate=False) + + _eval_rewrite_as_Sum = _eval_rewrite_as_Integral + + def evaluate_integral(self): + return self.rewrite(Integral).doit() + + +class Expectation(Expr): + """ + Symbolic expression for the expectation. + + Examples + ======== + + >>> from sympy.stats import Expectation, Normal, Probability, Poisson + >>> from sympy import symbols, Integral, Sum + >>> mu = symbols("mu") + >>> sigma = symbols("sigma", positive=True) + >>> X = Normal("X", mu, sigma) + >>> Expectation(X) + Expectation(X) + >>> Expectation(X).evaluate_integral().simplify() + mu + + To get the integral expression of the expectation: + + >>> Expectation(X).rewrite(Integral) + Integral(sqrt(2)*X*exp(-(X - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (X, -oo, oo)) + + The same integral expression, in more abstract terms: + + >>> Expectation(X).rewrite(Probability) + Integral(x*Probability(Eq(X, x)), (x, -oo, oo)) + + To get the Summation expression of the expectation for discrete random variables: + + >>> lamda = symbols('lamda', positive=True) + >>> Z = Poisson('Z', lamda) + >>> Expectation(Z).rewrite(Sum) + Sum(Z*lamda**Z*exp(-lamda)/factorial(Z), (Z, 0, oo)) + + This class is aware of some properties of the expectation: + + >>> from sympy.abc import a + >>> Expectation(a*X) + Expectation(a*X) + >>> Y = Normal("Y", 1, 2) + >>> Expectation(X + Y) + Expectation(X + Y) + + To expand the ``Expectation`` into its expression, use ``expand()``: + + >>> Expectation(X + Y).expand() + Expectation(X) + Expectation(Y) + >>> Expectation(a*X + Y).expand() + a*Expectation(X) + Expectation(Y) + >>> Expectation(a*X + Y) + Expectation(a*X + Y) + >>> Expectation((X + Y)*(X - Y)).expand() + Expectation(X**2) - Expectation(Y**2) + + To evaluate the ``Expectation``, use ``doit()``: + + >>> Expectation(X + Y).doit() + mu + 1 + >>> Expectation(X + Expectation(Y + Expectation(2*X))).doit() + 3*mu + 1 + + To prevent evaluating nested ``Expectation``, use ``doit(deep=False)`` + + >>> Expectation(X + Expectation(Y)).doit(deep=False) + mu + Expectation(Expectation(Y)) + >>> Expectation(X + Expectation(Y + Expectation(2*X))).doit(deep=False) + mu + Expectation(Expectation(Expectation(2*X) + Y)) + + """ + + def __new__(cls, expr, condition=None, **kwargs): + expr = _sympify(expr) + if expr.is_Matrix: + from sympy.stats.symbolic_multivariate_probability import ExpectationMatrix + return ExpectationMatrix(expr, condition) + if condition is None: + if not is_random(expr): + return expr + obj = Expr.__new__(cls, expr) + else: + condition = _sympify(condition) + obj = Expr.__new__(cls, expr, condition) + obj._condition = condition + return obj + + def _eval_is_commutative(self): + return(self.args[0].is_commutative) + + def expand(self, **hints): + expr = self.args[0] + condition = self._condition + + if not is_random(expr): + return expr + + if isinstance(expr, Add): + return Add.fromiter(Expectation(a, condition=condition).expand() + for a in expr.args) + + expand_expr = _expand(expr) + if isinstance(expand_expr, Add): + return Add.fromiter(Expectation(a, condition=condition).expand() + for a in expand_expr.args) + + elif isinstance(expr, Mul): + rv = [] + nonrv = [] + for a in expr.args: + if is_random(a): + rv.append(a) + else: + nonrv.append(a) + return Mul.fromiter(nonrv)*Expectation(Mul.fromiter(rv), condition=condition) + + return self + + def doit(self, **hints): + deep = hints.get('deep', True) + condition = self._condition + expr = self.args[0] + numsamples = hints.get('numsamples', False) + evaluate = hints.get('evaluate', True) + + if deep: + expr = expr.doit(**hints) + + if not is_random(expr) or isinstance(expr, Expectation): # expr isn't random? + return expr + if numsamples: # Computing by monte carlo sampling? + evalf = hints.get('evalf', True) + return sampling_E(expr, condition, numsamples=numsamples, evalf=evalf) + + if expr.has(RandomIndexedSymbol): + return pspace(expr).compute_expectation(expr, condition) + + # Create new expr and recompute E + if condition is not None: # If there is a condition + return self.func(given(expr, condition)).doit(**hints) + + # A few known statements for efficiency + + if expr.is_Add: # We know that E is Linear + return Add(*[self.func(arg, condition).doit(**hints) + if not isinstance(arg, Expectation) else self.func(arg, condition) + for arg in expr.args]) + if expr.is_Mul: + if expr.atoms(Expectation): + return expr + + if pspace(expr) == PSpace(): + return self.func(expr) + # Otherwise case is simple, pass work off to the ProbabilitySpace + result = pspace(expr).compute_expectation(expr, evaluate=evaluate) + if hasattr(result, 'doit') and evaluate: + return result.doit(**hints) + else: + return result + + + def _eval_rewrite_as_Probability(self, arg, condition=None, **kwargs): + rvs = arg.atoms(RandomSymbol) + if len(rvs) > 1: + raise NotImplementedError() + if len(rvs) == 0: + return arg + + rv = rvs.pop() + if rv.pspace is None: + raise ValueError("Probability space not known") + + symbol = rv.symbol + if symbol.name[0].isupper(): + symbol = Symbol(symbol.name.lower()) + else : + symbol = Symbol(symbol.name + "_1") + + if rv.pspace.is_Continuous: + return Integral(arg.replace(rv, symbol)*Probability(Eq(rv, symbol), condition), (symbol, rv.pspace.domain.set.inf, rv.pspace.domain.set.sup)) + else: + if rv.pspace.is_Finite: + raise NotImplementedError + else: + return Sum(arg.replace(rv, symbol)*Probability(Eq(rv, symbol), condition), (symbol, rv.pspace.domain.set.inf, rv.pspace.set.sup)) + + def _eval_rewrite_as_Integral(self, arg, condition=None, evaluate=False, **kwargs): + return self.func(arg, condition=condition).doit(deep=False, evaluate=evaluate) + + _eval_rewrite_as_Sum = _eval_rewrite_as_Integral # For discrete this will be Sum + + def evaluate_integral(self): + return self.rewrite(Integral).doit() + + evaluate_sum = evaluate_integral + +class Variance(Expr): + """ + Symbolic expression for the variance. + + Examples + ======== + + >>> from sympy import symbols, Integral + >>> from sympy.stats import Normal, Expectation, Variance, Probability + >>> mu = symbols("mu", positive=True) + >>> sigma = symbols("sigma", positive=True) + >>> X = Normal("X", mu, sigma) + >>> Variance(X) + Variance(X) + >>> Variance(X).evaluate_integral() + sigma**2 + + Integral representation of the underlying calculations: + + >>> Variance(X).rewrite(Integral) + Integral(sqrt(2)*(X - Integral(sqrt(2)*X*exp(-(X - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (X, -oo, oo)))**2*exp(-(X - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (X, -oo, oo)) + + Integral representation, without expanding the PDF: + + >>> Variance(X).rewrite(Probability) + -Integral(x*Probability(Eq(X, x)), (x, -oo, oo))**2 + Integral(x**2*Probability(Eq(X, x)), (x, -oo, oo)) + + Rewrite the variance in terms of the expectation + + >>> Variance(X).rewrite(Expectation) + -Expectation(X)**2 + Expectation(X**2) + + Some transformations based on the properties of the variance may happen: + + >>> from sympy.abc import a + >>> Y = Normal("Y", 0, 1) + >>> Variance(a*X) + Variance(a*X) + + To expand the variance in its expression, use ``expand()``: + + >>> Variance(a*X).expand() + a**2*Variance(X) + >>> Variance(X + Y) + Variance(X + Y) + >>> Variance(X + Y).expand() + 2*Covariance(X, Y) + Variance(X) + Variance(Y) + + """ + def __new__(cls, arg, condition=None, **kwargs): + arg = _sympify(arg) + + if arg.is_Matrix: + from sympy.stats.symbolic_multivariate_probability import VarianceMatrix + return VarianceMatrix(arg, condition) + if condition is None: + obj = Expr.__new__(cls, arg) + else: + condition = _sympify(condition) + obj = Expr.__new__(cls, arg, condition) + obj._condition = condition + return obj + + def _eval_is_commutative(self): + return self.args[0].is_commutative + + def expand(self, **hints): + arg = self.args[0] + condition = self._condition + + if not is_random(arg): + return S.Zero + + if isinstance(arg, RandomSymbol): + return self + elif isinstance(arg, Add): + rv = [] + for a in arg.args: + if is_random(a): + rv.append(a) + variances = Add(*(Variance(xv, condition).expand() for xv in rv)) + map_to_covar = lambda x: 2*Covariance(*x, condition=condition).expand() + covariances = Add(*map(map_to_covar, itertools.combinations(rv, 2))) + return variances + covariances + elif isinstance(arg, Mul): + nonrv = [] + rv = [] + for a in arg.args: + if is_random(a): + rv.append(a) + else: + nonrv.append(a**2) + if len(rv) == 0: + return S.Zero + return Mul.fromiter(nonrv)*Variance(Mul.fromiter(rv), condition) + + # this expression contains a RandomSymbol somehow: + return self + + def _eval_rewrite_as_Expectation(self, arg, condition=None, **kwargs): + e1 = Expectation(arg**2, condition) + e2 = Expectation(arg, condition)**2 + return e1 - e2 + + def _eval_rewrite_as_Probability(self, arg, condition=None, **kwargs): + return self.rewrite(Expectation).rewrite(Probability) + + def _eval_rewrite_as_Integral(self, arg, condition=None, **kwargs): + return variance(self.args[0], self._condition, evaluate=False) + + _eval_rewrite_as_Sum = _eval_rewrite_as_Integral + + def evaluate_integral(self): + return self.rewrite(Integral).doit() + + +class Covariance(Expr): + """ + Symbolic expression for the covariance. + + Examples + ======== + + >>> from sympy.stats import Covariance + >>> from sympy.stats import Normal + >>> X = Normal("X", 3, 2) + >>> Y = Normal("Y", 0, 1) + >>> Z = Normal("Z", 0, 1) + >>> W = Normal("W", 0, 1) + >>> cexpr = Covariance(X, Y) + >>> cexpr + Covariance(X, Y) + + Evaluate the covariance, `X` and `Y` are independent, + therefore zero is the result: + + >>> cexpr.evaluate_integral() + 0 + + Rewrite the covariance expression in terms of expectations: + + >>> from sympy.stats import Expectation + >>> cexpr.rewrite(Expectation) + Expectation(X*Y) - Expectation(X)*Expectation(Y) + + In order to expand the argument, use ``expand()``: + + >>> from sympy.abc import a, b, c, d + >>> Covariance(a*X + b*Y, c*Z + d*W) + Covariance(a*X + b*Y, c*Z + d*W) + >>> Covariance(a*X + b*Y, c*Z + d*W).expand() + a*c*Covariance(X, Z) + a*d*Covariance(W, X) + b*c*Covariance(Y, Z) + b*d*Covariance(W, Y) + + This class is aware of some properties of the covariance: + + >>> Covariance(X, X).expand() + Variance(X) + >>> Covariance(a*X, b*Y).expand() + a*b*Covariance(X, Y) + """ + + def __new__(cls, arg1, arg2, condition=None, **kwargs): + arg1 = _sympify(arg1) + arg2 = _sympify(arg2) + + if arg1.is_Matrix or arg2.is_Matrix: + from sympy.stats.symbolic_multivariate_probability import CrossCovarianceMatrix + return CrossCovarianceMatrix(arg1, arg2, condition) + + if kwargs.pop('evaluate', global_parameters.evaluate): + arg1, arg2 = sorted([arg1, arg2], key=default_sort_key) + + if condition is None: + obj = Expr.__new__(cls, arg1, arg2) + else: + condition = _sympify(condition) + obj = Expr.__new__(cls, arg1, arg2, condition) + obj._condition = condition + return obj + + def _eval_is_commutative(self): + return self.args[0].is_commutative + + def expand(self, **hints): + arg1 = self.args[0] + arg2 = self.args[1] + condition = self._condition + + if arg1 == arg2: + return Variance(arg1, condition).expand() + + if not is_random(arg1): + return S.Zero + if not is_random(arg2): + return S.Zero + + arg1, arg2 = sorted([arg1, arg2], key=default_sort_key) + + if isinstance(arg1, RandomSymbol) and isinstance(arg2, RandomSymbol): + return Covariance(arg1, arg2, condition) + + coeff_rv_list1 = self._expand_single_argument(arg1.expand()) + coeff_rv_list2 = self._expand_single_argument(arg2.expand()) + + addends = [a*b*Covariance(*sorted([r1, r2], key=default_sort_key), condition=condition) + for (a, r1) in coeff_rv_list1 for (b, r2) in coeff_rv_list2] + return Add.fromiter(addends) + + @classmethod + def _expand_single_argument(cls, expr): + # return (coefficient, random_symbol) pairs: + if isinstance(expr, RandomSymbol): + return [(S.One, expr)] + elif isinstance(expr, Add): + outval = [] + for a in expr.args: + if isinstance(a, Mul): + outval.append(cls._get_mul_nonrv_rv_tuple(a)) + elif is_random(a): + outval.append((S.One, a)) + + return outval + elif isinstance(expr, Mul): + return [cls._get_mul_nonrv_rv_tuple(expr)] + elif is_random(expr): + return [(S.One, expr)] + + @classmethod + def _get_mul_nonrv_rv_tuple(cls, m): + rv = [] + nonrv = [] + for a in m.args: + if is_random(a): + rv.append(a) + else: + nonrv.append(a) + return (Mul.fromiter(nonrv), Mul.fromiter(rv)) + + def _eval_rewrite_as_Expectation(self, arg1, arg2, condition=None, **kwargs): + e1 = Expectation(arg1*arg2, condition) + e2 = Expectation(arg1, condition)*Expectation(arg2, condition) + return e1 - e2 + + def _eval_rewrite_as_Probability(self, arg1, arg2, condition=None, **kwargs): + return self.rewrite(Expectation).rewrite(Probability) + + def _eval_rewrite_as_Integral(self, arg1, arg2, condition=None, **kwargs): + return covariance(self.args[0], self.args[1], self._condition, evaluate=False) + + _eval_rewrite_as_Sum = _eval_rewrite_as_Integral + + def evaluate_integral(self): + return self.rewrite(Integral).doit() + + +class Moment(Expr): + """ + Symbolic class for Moment + + Examples + ======== + + >>> from sympy import Symbol, Integral + >>> from sympy.stats import Normal, Expectation, Probability, Moment + >>> mu = Symbol('mu', real=True) + >>> sigma = Symbol('sigma', positive=True) + >>> X = Normal('X', mu, sigma) + >>> M = Moment(X, 3, 1) + + To evaluate the result of Moment use `doit`: + + >>> M.doit() + mu**3 - 3*mu**2 + 3*mu*sigma**2 + 3*mu - 3*sigma**2 - 1 + + Rewrite the Moment expression in terms of Expectation: + + >>> M.rewrite(Expectation) + Expectation((X - 1)**3) + + Rewrite the Moment expression in terms of Probability: + + >>> M.rewrite(Probability) + Integral((x - 1)**3*Probability(Eq(X, x)), (x, -oo, oo)) + + Rewrite the Moment expression in terms of Integral: + + >>> M.rewrite(Integral) + Integral(sqrt(2)*(X - 1)**3*exp(-(X - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (X, -oo, oo)) + + """ + def __new__(cls, X, n, c=0, condition=None, **kwargs): + X = _sympify(X) + n = _sympify(n) + c = _sympify(c) + if condition is not None: + condition = _sympify(condition) + return super().__new__(cls, X, n, c, condition) + else: + return super().__new__(cls, X, n, c) + + def doit(self, **hints): + return self.rewrite(Expectation).doit(**hints) + + def _eval_rewrite_as_Expectation(self, X, n, c=0, condition=None, **kwargs): + return Expectation((X - c)**n, condition) + + def _eval_rewrite_as_Probability(self, X, n, c=0, condition=None, **kwargs): + return self.rewrite(Expectation).rewrite(Probability) + + def _eval_rewrite_as_Integral(self, X, n, c=0, condition=None, **kwargs): + return self.rewrite(Expectation).rewrite(Integral) + + +class CentralMoment(Expr): + """ + Symbolic class Central Moment + + Examples + ======== + + >>> from sympy import Symbol, Integral + >>> from sympy.stats import Normal, Expectation, Probability, CentralMoment + >>> mu = Symbol('mu', real=True) + >>> sigma = Symbol('sigma', positive=True) + >>> X = Normal('X', mu, sigma) + >>> CM = CentralMoment(X, 4) + + To evaluate the result of CentralMoment use `doit`: + + >>> CM.doit().simplify() + 3*sigma**4 + + Rewrite the CentralMoment expression in terms of Expectation: + + >>> CM.rewrite(Expectation) + Expectation((-Expectation(X) + X)**4) + + Rewrite the CentralMoment expression in terms of Probability: + + >>> CM.rewrite(Probability) + Integral((x - Integral(x*Probability(True), (x, -oo, oo)))**4*Probability(Eq(X, x)), (x, -oo, oo)) + + Rewrite the CentralMoment expression in terms of Integral: + + >>> CM.rewrite(Integral) + Integral(sqrt(2)*(X - Integral(sqrt(2)*X*exp(-(X - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (X, -oo, oo)))**4*exp(-(X - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (X, -oo, oo)) + + """ + def __new__(cls, X, n, condition=None, **kwargs): + X = _sympify(X) + n = _sympify(n) + if condition is not None: + condition = _sympify(condition) + return super().__new__(cls, X, n, condition) + else: + return super().__new__(cls, X, n) + + def doit(self, **hints): + return self.rewrite(Expectation).doit(**hints) + + def _eval_rewrite_as_Expectation(self, X, n, condition=None, **kwargs): + mu = Expectation(X, condition, **kwargs) + return Moment(X, n, mu, condition, **kwargs).rewrite(Expectation) + + def _eval_rewrite_as_Probability(self, X, n, condition=None, **kwargs): + return self.rewrite(Expectation).rewrite(Probability) + + def _eval_rewrite_as_Integral(self, X, n, condition=None, **kwargs): + return self.rewrite(Expectation).rewrite(Integral) diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/tests/__init__.py b/.venv/lib/python3.12/site-packages/sympy/stats/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/tests/__pycache__/__init__.cpython-312.pyc 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sympy.concrete.summations import Sum +from sympy.core.numbers import (oo, pi) +from sympy.core.relational import Eq +from sympy.core.singleton import S +from sympy.core.symbol import symbols +from sympy.functions.combinatorial.factorials import factorial +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.special.beta_functions import beta +from sympy.functions.special.error_functions import erf +from sympy.functions.special.gamma_functions import gamma +from sympy.integrals.integrals import Integral +from sympy.sets.sets import Interval +from sympy.stats import (Normal, P, E, density, Gamma, Poisson, Rayleigh, + variance, Bernoulli, Beta, Uniform, cdf) +from sympy.stats.compound_rv import CompoundDistribution, CompoundPSpace +from sympy.stats.crv_types import NormalDistribution +from sympy.stats.drv_types import PoissonDistribution +from sympy.stats.frv_types import BernoulliDistribution +from sympy.testing.pytest import raises, ignore_warnings +from sympy.stats.joint_rv_types import MultivariateNormalDistribution + +from sympy.abc import x + + +# helpers for testing troublesome unevaluated expressions +flat = lambda s: ''.join(str(s).split()) +streq = lambda *a: len(set(map(flat, a))) == 1 +assert streq(x, x) +assert streq(x, 'x') +assert not streq(x, x + 1) + + +def test_normal_CompoundDist(): + X = Normal('X', 1, 2) + Y = Normal('X', X, 4) + assert density(Y)(x).simplify() == sqrt(10)*exp(-x**2/40 + x/20 - S(1)/40)/(20*sqrt(pi)) + assert E(Y) == 1 # it is always equal to mean of X + assert P(Y > 1) == S(1)/2 # as 1 is the mean + assert P(Y > 5).simplify() == S(1)/2 - erf(sqrt(10)/5)/2 + assert variance(Y) == variance(X) + 4**2 # 2**2 + 4**2 + # https://math.stackexchange.com/questions/1484451/ + # (Contains proof of E and variance computation) + + +def test_poisson_CompoundDist(): + k, t, y = symbols('k t y', positive=True, real=True) + G = Gamma('G', k, t) + D = Poisson('P', G) + assert density(D)(y).simplify() == t**y*(t + 1)**(-k - y)*gamma(k + y)/(gamma(k)*gamma(y + 1)) + # https://en.wikipedia.org/wiki/Negative_binomial_distribution#Gamma%E2%80%93Poisson_mixture + assert E(D).simplify() == k*t # mean of NegativeBinomialDistribution + + +def test_bernoulli_CompoundDist(): + X = Beta('X', 1, 2) + Y = Bernoulli('Y', X) + assert density(Y).dict == {0: S(2)/3, 1: S(1)/3} + assert E(Y) == P(Eq(Y, 1)) == S(1)/3 + assert variance(Y) == S(2)/9 + assert cdf(Y) == {0: S(2)/3, 1: 1} + + # test issue 8128 + a = Bernoulli('a', S(1)/2) + b = Bernoulli('b', a) + assert density(b).dict == {0: S(1)/2, 1: S(1)/2} + assert P(b > 0.5) == S(1)/2 + + X = Uniform('X', 0, 1) + Y = Bernoulli('Y', X) + assert E(Y) == S(1)/2 + assert P(Eq(Y, 1)) == E(Y) + + +def test_unevaluated_CompoundDist(): + # these tests need to be removed once they work with evaluation as they are currently not + # evaluated completely in sympy. + R = Rayleigh('R', 4) + X = Normal('X', 3, R) + ans = ''' + Piecewise(((-sqrt(pi)*sinh(x/4 - 3/4) + sqrt(pi)*cosh(x/4 - 3/4))/( + 8*sqrt(pi)), Abs(arg(x - 3)) <= pi/4), (Integral(sqrt(2)*exp(-(x - 3) + **2/(2*R**2))*exp(-R**2/32)/(32*sqrt(pi)), (R, 0, oo)), True))''' + assert streq(density(X)(x), ans) + + expre = ''' + Integral(X*Integral(sqrt(2)*exp(-(X-3)**2/(2*R**2))*exp(-R**2/32)/(32* + sqrt(pi)),(R,0,oo)),(X,-oo,oo))''' + with ignore_warnings(UserWarning): ### TODO: Restore tests once warnings are removed + assert streq(E(X, evaluate=False).rewrite(Integral), expre) + + X = Poisson('X', 1) + Y = Poisson('Y', X) + Z = Poisson('Z', Y) + exprd = Sum(exp(-Y)*Y**x*Sum(exp(-1)*exp(-X)*X**Y/(factorial(X)*factorial(Y) + ), (X, 0, oo))/factorial(x), (Y, 0, oo)) + assert density(Z)(x) == exprd + + N = Normal('N', 1, 2) + M = Normal('M', 3, 4) + D = Normal('D', M, N) + exprd = ''' + Integral(sqrt(2)*exp(-(N-1)**2/8)*Integral(exp(-(x-M)**2/(2*N**2))*exp + (-(M-3)**2/32)/(8*pi*N),(M,-oo,oo))/(4*sqrt(pi)),(N,-oo,oo))''' + assert streq(density(D, evaluate=False)(x), exprd) + + +def test_Compound_Distribution(): + X = Normal('X', 2, 4) + N = NormalDistribution(X, 4) + C = CompoundDistribution(N) + assert C.is_Continuous + assert C.set == Interval(-oo, oo) + assert C.pdf(x, evaluate=True).simplify() == exp(-x**2/64 + x/16 - S(1)/16)/(8*sqrt(pi)) + + assert not isinstance(CompoundDistribution(NormalDistribution(2, 3)), + CompoundDistribution) + M = MultivariateNormalDistribution([1, 2], [[2, 1], [1, 2]]) + raises(NotImplementedError, lambda: CompoundDistribution(M)) + + X = Beta('X', 2, 4) + B = BernoulliDistribution(X, 1, 0) + C = CompoundDistribution(B) + assert C.is_Finite + assert C.set == {0, 1} + y = symbols('y', negative=False, integer=True) + assert C.pdf(y, evaluate=True) == Piecewise((S(1)/(30*beta(2, 4)), Eq(y, 0)), + (S(1)/(60*beta(2, 4)), Eq(y, 1)), (0, True)) + + k, t, z = symbols('k t z', positive=True, real=True) + G = Gamma('G', k, t) + X = PoissonDistribution(G) + C = CompoundDistribution(X) + assert C.is_Discrete + assert C.set == S.Naturals0 + assert C.pdf(z, evaluate=True).simplify() == t**z*(t + 1)**(-k - z)*gamma(k \ + + z)/(gamma(k)*gamma(z + 1)) + + +def test_compound_pspace(): + X = Normal('X', 2, 4) + Y = Normal('Y', 3, 6) + assert not isinstance(Y.pspace, CompoundPSpace) + N = NormalDistribution(1, 2) + D = PoissonDistribution(3) + B = BernoulliDistribution(0.2, 1, 0) + pspace1 = CompoundPSpace('N', N) + pspace2 = CompoundPSpace('D', D) + pspace3 = CompoundPSpace('B', B) + assert not isinstance(pspace1, CompoundPSpace) + assert not isinstance(pspace2, CompoundPSpace) + assert not isinstance(pspace3, CompoundPSpace) + M = MultivariateNormalDistribution([1, 2], [[2, 1], [1, 2]]) + raises(ValueError, lambda: CompoundPSpace('M', M)) + Y = Normal('Y', X, 6) + assert isinstance(Y.pspace, CompoundPSpace) + assert Y.pspace.distribution == CompoundDistribution(NormalDistribution(X, 6)) + assert Y.pspace.domain.set == Interval(-oo, oo) diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_continuous_rv.py b/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_continuous_rv.py new file mode 100644 index 0000000000000000000000000000000000000000..b2c4206b5c29ffd3194d1ae05e57c51c9c1b6d78 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_continuous_rv.py @@ -0,0 +1,1583 @@ +from sympy.concrete.summations import Sum +from sympy.core.function import (Lambda, diff, expand_func) +from sympy.core.mul import Mul +from sympy.core import EulerGamma +from sympy.core.numbers import (E as e, I, Rational, pi) +from sympy.core.relational import (Eq, Ne) +from sympy.core.singleton import S +from sympy.core.symbol import (Dummy, Symbol, symbols) +from sympy.functions.combinatorial.factorials import (binomial, factorial) +from sympy.functions.elementary.complexes import (Abs, im, re, sign) +from sympy.functions.elementary.exponential import (exp, log) +from sympy.functions.elementary.hyperbolic import (cosh, sinh) +from sympy.functions.elementary.integers import floor +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.elementary.trigonometric import (asin, atan, cos, sin, tan) +from sympy.functions.special.bessel import (besseli, besselj, besselk) +from sympy.functions.special.beta_functions import beta +from sympy.functions.special.error_functions import (erf, erfc, erfi, expint) +from sympy.functions.special.gamma_functions import (gamma, lowergamma, uppergamma) +from sympy.functions.special.zeta_functions import zeta +from sympy.functions.special.hyper import hyper +from sympy.integrals.integrals import Integral +from sympy.logic.boolalg import (And, Or) +from sympy.sets.sets import Interval +from sympy.simplify.simplify import simplify +from sympy.utilities.lambdify import lambdify +from sympy.functions.special.error_functions import erfinv +from sympy.functions.special.hyper import meijerg +from sympy.sets.sets import FiniteSet, Complement, Intersection +from sympy.stats import (P, E, where, density, variance, covariance, skewness, kurtosis, median, + given, pspace, cdf, characteristic_function, moment_generating_function, + ContinuousRV, Arcsin, Benini, Beta, BetaNoncentral, BetaPrime, + Cauchy, Chi, ChiSquared, ChiNoncentral, Dagum, Davis, Erlang, ExGaussian, + Exponential, ExponentialPower, FDistribution, FisherZ, Frechet, Gamma, + GammaInverse, Gompertz, Gumbel, Kumaraswamy, Laplace, Levy, Logistic, LogCauchy, + LogLogistic, LogitNormal, LogNormal, Maxwell, Moyal, Nakagami, Normal, GaussianInverse, + Pareto, PowerFunction, QuadraticU, RaisedCosine, Rayleigh, Reciprocal, ShiftedGompertz, StudentT, + Trapezoidal, Triangular, Uniform, UniformSum, VonMises, Weibull, coskewness, + WignerSemicircle, Wald, correlation, moment, cmoment, smoment, quantile, + Lomax, BoundedPareto) + +from sympy.stats.crv_types import NormalDistribution, ExponentialDistribution, ContinuousDistributionHandmade +from sympy.stats.joint_rv_types import MultivariateLaplaceDistribution, MultivariateNormalDistribution +from sympy.stats.crv import SingleContinuousPSpace, SingleContinuousDomain +from sympy.stats.compound_rv import CompoundPSpace +from sympy.stats.symbolic_probability import Probability +from sympy.testing.pytest import raises, XFAIL, slow, ignore_warnings +from sympy.core.random import verify_numerically as tn + +oo = S.Infinity + +x, y, z = map(Symbol, 'xyz') + +def test_single_normal(): + mu = Symbol('mu', real=True) + sigma = Symbol('sigma', positive=True) + X = Normal('x', 0, 1) + Y = X*sigma + mu + + assert E(Y) == mu + assert variance(Y) == sigma**2 + pdf = density(Y) + x = Symbol('x', real=True) + assert (pdf(x) == + 2**S.Half*exp(-(x - mu)**2/(2*sigma**2))/(2*pi**S.Half*sigma)) + + assert P(X**2 < 1) == erf(2**S.Half/2) + ans = quantile(Y)(x) + assert ans == Complement(Intersection(FiniteSet( + sqrt(2)*sigma*(sqrt(2)*mu/(2*sigma)+ erfinv(2*x - 1))), + Interval(-oo, oo)), FiniteSet(mu)) + assert E(X, Eq(X, mu)) == mu + + assert median(X) == FiniteSet(0) + # issue 8248 + assert X.pspace.compute_expectation(1).doit() == 1 + + +def test_conditional_1d(): + X = Normal('x', 0, 1) + Y = given(X, X >= 0) + z = Symbol('z') + + assert density(Y)(z) == 2 * density(X)(z) + + assert Y.pspace.domain.set == Interval(0, oo) + assert E(Y) == sqrt(2) / sqrt(pi) + + assert E(X**2) == E(Y**2) + + +def test_ContinuousDomain(): + X = Normal('x', 0, 1) + assert where(X**2 <= 1).set == Interval(-1, 1) + assert where(X**2 <= 1).symbol == X.symbol + assert where(And(X**2 <= 1, X >= 0)).set == Interval(0, 1) + raises(ValueError, lambda: where(sin(X) > 1)) + + Y = given(X, X >= 0) + + assert Y.pspace.domain.set == Interval(0, oo) + + +def test_multiple_normal(): + X, Y = Normal('x', 0, 1), Normal('y', 0, 1) + p = Symbol("p", positive=True) + + assert E(X + Y) == 0 + assert variance(X + Y) == 2 + assert variance(X + X) == 4 + assert covariance(X, Y) == 0 + assert covariance(2*X + Y, -X) == -2*variance(X) + assert skewness(X) == 0 + assert skewness(X + Y) == 0 + assert kurtosis(X) == 3 + assert kurtosis(X+Y) == 3 + assert correlation(X, Y) == 0 + assert correlation(X, X + Y) == correlation(X, X - Y) + assert moment(X, 2) == 1 + assert cmoment(X, 3) == 0 + assert moment(X + Y, 4) == 12 + assert cmoment(X, 2) == variance(X) + assert smoment(X*X, 2) == 1 + assert smoment(X + Y, 3) == skewness(X + Y) + assert smoment(X + Y, 4) == kurtosis(X + Y) + assert E(X, Eq(X + Y, 0)) == 0 + assert variance(X, Eq(X + Y, 0)) == S.Half + assert quantile(X)(p) == sqrt(2)*erfinv(2*p - S.One) + + +def test_symbolic(): + mu1, mu2 = symbols('mu1 mu2', real=True) + s1, s2 = symbols('sigma1 sigma2', positive=True) + rate = Symbol('lambda', positive=True) + X = Normal('x', mu1, s1) + Y = Normal('y', mu2, s2) + Z = Exponential('z', rate) + a, b, c = symbols('a b c', real=True) + + assert E(X) == mu1 + assert E(X + Y) == mu1 + mu2 + assert E(a*X + b) == a*E(X) + b + assert variance(X) == s1**2 + assert variance(X + a*Y + b) == variance(X) + a**2*variance(Y) + + assert E(Z) == 1/rate + assert E(a*Z + b) == a*E(Z) + b + assert E(X + a*Z + b) == mu1 + a/rate + b + assert median(X) == FiniteSet(mu1) + + +def test_cdf(): + X = Normal('x', 0, 1) + + d = cdf(X) + assert P(X < 1) == d(1).rewrite(erfc) + assert d(0) == S.Half + + d = cdf(X, X > 0) # given X>0 + assert d(0) == 0 + + Y = Exponential('y', 10) + d = cdf(Y) + assert d(-5) == 0 + assert P(Y > 3) == 1 - d(3) + + raises(ValueError, lambda: cdf(X + Y)) + + Z = Exponential('z', 1) + f = cdf(Z) + assert f(z) == Piecewise((1 - exp(-z), z >= 0), (0, True)) + + +def test_characteristic_function(): + X = Uniform('x', 0, 1) + + cf = characteristic_function(X) + assert cf(1) == -I*(-1 + exp(I)) + + Y = Normal('y', 1, 1) + cf = characteristic_function(Y) + assert cf(0) == 1 + assert cf(1) == exp(I - S.Half) + + Z = Exponential('z', 5) + cf = characteristic_function(Z) + assert cf(0) == 1 + assert cf(1).expand() == Rational(25, 26) + I*5/26 + + X = GaussianInverse('x', 1, 1) + cf = characteristic_function(X) + assert cf(0) == 1 + assert cf(1) == exp(1 - sqrt(1 - 2*I)) + + X = ExGaussian('x', 0, 1, 1) + cf = characteristic_function(X) + assert cf(0) == 1 + assert cf(1) == (1 + I)*exp(Rational(-1, 2))/2 + + L = Levy('x', 0, 1) + cf = characteristic_function(L) + assert cf(0) == 1 + assert cf(1) == exp(-sqrt(2)*sqrt(-I)) + + +def test_moment_generating_function(): + t = symbols('t', positive=True) + + # Symbolic tests + a, b, c = symbols('a b c') + + mgf = moment_generating_function(Beta('x', a, b))(t) + assert mgf == hyper((a,), (a + b,), t) + + mgf = moment_generating_function(Chi('x', a))(t) + assert mgf == sqrt(2)*t*gamma(a/2 + S.Half)*\ + hyper((a/2 + S.Half,), (Rational(3, 2),), t**2/2)/gamma(a/2) +\ + hyper((a/2,), (S.Half,), t**2/2) + + mgf = moment_generating_function(ChiSquared('x', a))(t) + assert mgf == (1 - 2*t)**(-a/2) + + mgf = moment_generating_function(Erlang('x', a, b))(t) + assert mgf == (1 - t/b)**(-a) + + mgf = moment_generating_function(ExGaussian("x", a, b, c))(t) + assert mgf == exp(a*t + b**2*t**2/2)/(1 - t/c) + + mgf = moment_generating_function(Exponential('x', a))(t) + assert mgf == a/(a - t) + + mgf = moment_generating_function(Gamma('x', a, b))(t) + assert mgf == (-b*t + 1)**(-a) + + mgf = moment_generating_function(Gumbel('x', a, b))(t) + assert mgf == exp(b*t)*gamma(-a*t + 1) + + mgf = moment_generating_function(Gompertz('x', a, b))(t) + assert mgf == b*exp(b)*expint(t/a, b) + + mgf = moment_generating_function(Laplace('x', a, b))(t) + assert mgf == exp(a*t)/(-b**2*t**2 + 1) + + mgf = moment_generating_function(Logistic('x', a, b))(t) + assert mgf == exp(a*t)*beta(-b*t + 1, b*t + 1) + + mgf = moment_generating_function(Normal('x', a, b))(t) + assert mgf == exp(a*t + b**2*t**2/2) + + mgf = moment_generating_function(Pareto('x', a, b))(t) + assert mgf == b*(-a*t)**b*uppergamma(-b, -a*t) + + mgf = moment_generating_function(QuadraticU('x', a, b))(t) + assert str(mgf) == ("(3*(t*(-4*b + (a + b)**2) + 4)*exp(b*t) - " + "3*(t*(a**2 + 2*a*(b - 2) + b**2) + 4)*exp(a*t))/(t**2*(a - b)**3)") + + mgf = moment_generating_function(RaisedCosine('x', a, b))(t) + assert mgf == pi**2*exp(a*t)*sinh(b*t)/(b*t*(b**2*t**2 + pi**2)) + + mgf = moment_generating_function(Rayleigh('x', a))(t) + assert mgf == sqrt(2)*sqrt(pi)*a*t*(erf(sqrt(2)*a*t/2) + 1)\ + *exp(a**2*t**2/2)/2 + 1 + + mgf = moment_generating_function(Triangular('x', a, b, c))(t) + assert str(mgf) == ("(-2*(-a + b)*exp(c*t) + 2*(-a + c)*exp(b*t) + " + "2*(b - c)*exp(a*t))/(t**2*(-a + b)*(-a + c)*(b - c))") + + mgf = moment_generating_function(Uniform('x', a, b))(t) + assert mgf == (-exp(a*t) + exp(b*t))/(t*(-a + b)) + + mgf = moment_generating_function(UniformSum('x', a))(t) + assert mgf == ((exp(t) - 1)/t)**a + + mgf = moment_generating_function(WignerSemicircle('x', a))(t) + assert mgf == 2*besseli(1, a*t)/(a*t) + + # Numeric tests + + mgf = moment_generating_function(Beta('x', 1, 1))(t) + assert mgf.diff(t).subs(t, 1) == hyper((2,), (3,), 1)/2 + + mgf = moment_generating_function(Chi('x', 1))(t) + assert mgf.diff(t).subs(t, 1) == sqrt(2)*hyper((1,), (Rational(3, 2),), S.Half + )/sqrt(pi) + hyper((Rational(3, 2),), (Rational(3, 2),), S.Half) + 2*sqrt(2)*hyper((2,), + (Rational(5, 2),), S.Half)/(3*sqrt(pi)) + + mgf = moment_generating_function(ChiSquared('x', 1))(t) + assert mgf.diff(t).subs(t, 1) == I + + mgf = moment_generating_function(Erlang('x', 1, 1))(t) + assert mgf.diff(t).subs(t, 0) == 1 + + mgf = moment_generating_function(ExGaussian("x", 0, 1, 1))(t) + assert mgf.diff(t).subs(t, 2) == -exp(2) + + mgf = moment_generating_function(Exponential('x', 1))(t) + assert mgf.diff(t).subs(t, 0) == 1 + + mgf = moment_generating_function(Gamma('x', 1, 1))(t) + assert mgf.diff(t).subs(t, 0) == 1 + + mgf = moment_generating_function(Gumbel('x', 1, 1))(t) + assert mgf.diff(t).subs(t, 0) == EulerGamma + 1 + + mgf = moment_generating_function(Gompertz('x', 1, 1))(t) + assert mgf.diff(t).subs(t, 1) == -e*meijerg(((), (1, 1)), + ((0, 0, 0), ()), 1) + + mgf = moment_generating_function(Laplace('x', 1, 1))(t) + assert mgf.diff(t).subs(t, 0) == 1 + + mgf = moment_generating_function(Logistic('x', 1, 1))(t) + assert mgf.diff(t).subs(t, 0) == beta(1, 1) + + mgf = moment_generating_function(Normal('x', 0, 1))(t) + assert mgf.diff(t).subs(t, 1) == exp(S.Half) + + mgf = moment_generating_function(Pareto('x', 1, 1))(t) + assert mgf.diff(t).subs(t, 0) == expint(1, 0) + + mgf = moment_generating_function(QuadraticU('x', 1, 2))(t) + assert mgf.diff(t).subs(t, 1) == -12*e - 3*exp(2) + + mgf = moment_generating_function(RaisedCosine('x', 1, 1))(t) + assert mgf.diff(t).subs(t, 1) == -2*e*pi**2*sinh(1)/\ + (1 + pi**2)**2 + e*pi**2*cosh(1)/(1 + pi**2) + + mgf = moment_generating_function(Rayleigh('x', 1))(t) + assert mgf.diff(t).subs(t, 0) == sqrt(2)*sqrt(pi)/2 + + mgf = moment_generating_function(Triangular('x', 1, 3, 2))(t) + assert mgf.diff(t).subs(t, 1) == -e + exp(3) + + mgf = moment_generating_function(Uniform('x', 0, 1))(t) + assert mgf.diff(t).subs(t, 1) == 1 + + mgf = moment_generating_function(UniformSum('x', 1))(t) + assert mgf.diff(t).subs(t, 1) == 1 + + mgf = moment_generating_function(WignerSemicircle('x', 1))(t) + assert mgf.diff(t).subs(t, 1) == -2*besseli(1, 1) + besseli(2, 1) +\ + besseli(0, 1) + + +def test_ContinuousRV(): + pdf = sqrt(2)*exp(-x**2/2)/(2*sqrt(pi)) # Normal distribution + # X and Y should be equivalent + X = ContinuousRV(x, pdf, check=True) + Y = Normal('y', 0, 1) + + assert variance(X) == variance(Y) + assert P(X > 0) == P(Y > 0) + Z = ContinuousRV(z, exp(-z), set=Interval(0, oo)) + assert Z.pspace.domain.set == Interval(0, oo) + assert E(Z) == 1 + assert P(Z > 5) == exp(-5) + raises(ValueError, lambda: ContinuousRV(z, exp(-z), set=Interval(0, 10), check=True)) + + # the correct pdf for Gamma(k, theta) but the integral in `check` + # integrates to something equivalent to 1 and not to 1 exactly + _x, k, theta = symbols("x k theta", positive=True) + pdf = 1/(gamma(k)*theta**k)*_x**(k-1)*exp(-_x/theta) + X = ContinuousRV(_x, pdf, set=Interval(0, oo)) + Y = Gamma('y', k, theta) + assert (E(X) - E(Y)).simplify() == 0 + assert (variance(X) - variance(Y)).simplify() == 0 + + +def test_arcsin(): + + a = Symbol("a", real=True) + b = Symbol("b", real=True) + + X = Arcsin('x', a, b) + assert density(X)(x) == 1/(pi*sqrt((-x + b)*(x - a))) + assert cdf(X)(x) == Piecewise((0, a > x), + (2*asin(sqrt((-a + x)/(-a + b)))/pi, b >= x), + (1, True)) + assert pspace(X).domain.set == Interval(a, b) + +def test_benini(): + alpha = Symbol("alpha", positive=True) + beta = Symbol("beta", positive=True) + sigma = Symbol("sigma", positive=True) + X = Benini('x', alpha, beta, sigma) + + assert density(X)(x) == ((alpha/x + 2*beta*log(x/sigma)/x) + *exp(-alpha*log(x/sigma) - beta*log(x/sigma)**2)) + + assert pspace(X).domain.set == Interval(sigma, oo) + raises(NotImplementedError, lambda: moment_generating_function(X)) + alpha = Symbol("alpha", nonpositive=True) + raises(ValueError, lambda: Benini('x', alpha, beta, sigma)) + + beta = Symbol("beta", nonpositive=True) + raises(ValueError, lambda: Benini('x', alpha, beta, sigma)) + + alpha = Symbol("alpha", positive=True) + raises(ValueError, lambda: Benini('x', alpha, beta, sigma)) + + beta = Symbol("beta", positive=True) + sigma = Symbol("sigma", nonpositive=True) + raises(ValueError, lambda: Benini('x', alpha, beta, sigma)) + +def test_beta(): + a, b = symbols('alpha beta', positive=True) + B = Beta('x', a, b) + + assert pspace(B).domain.set == Interval(0, 1) + assert characteristic_function(B)(x) == hyper((a,), (a + b,), I*x) + assert density(B)(x) == x**(a - 1)*(1 - x)**(b - 1)/beta(a, b) + + assert simplify(E(B)) == a / (a + b) + assert simplify(variance(B)) == a*b / (a**3 + 3*a**2*b + a**2 + 3*a*b**2 + 2*a*b + b**3 + b**2) + + # Full symbolic solution is too much, test with numeric version + a, b = 1, 2 + B = Beta('x', a, b) + assert expand_func(E(B)) == a / S(a + b) + assert expand_func(variance(B)) == (a*b) / S((a + b)**2 * (a + b + 1)) + assert median(B) == FiniteSet(1 - 1/sqrt(2)) + +def test_beta_noncentral(): + a, b = symbols('a b', positive=True) + c = Symbol('c', nonnegative=True) + _k = Dummy('k') + + X = BetaNoncentral('x', a, b, c) + + assert pspace(X).domain.set == Interval(0, 1) + + dens = density(X) + z = Symbol('z') + + res = Sum( z**(_k + a - 1)*(c/2)**_k*(1 - z)**(b - 1)*exp(-c/2)/ + (beta(_k + a, b)*factorial(_k)), (_k, 0, oo)) + assert dens(z).dummy_eq(res) + + # BetaCentral should not raise if the assumptions + # on the symbols can not be determined + a, b, c = symbols('a b c') + assert BetaNoncentral('x', a, b, c) + + a = Symbol('a', positive=False, real=True) + raises(ValueError, lambda: BetaNoncentral('x', a, b, c)) + + a = Symbol('a', positive=True) + b = Symbol('b', positive=False, real=True) + raises(ValueError, lambda: BetaNoncentral('x', a, b, c)) + + a = Symbol('a', positive=True) + b = Symbol('b', positive=True) + c = Symbol('c', nonnegative=False, real=True) + raises(ValueError, lambda: BetaNoncentral('x', a, b, c)) + +def test_betaprime(): + alpha = Symbol("alpha", positive=True) + + betap = Symbol("beta", positive=True) + + X = BetaPrime('x', alpha, betap) + assert density(X)(x) == x**(alpha - 1)*(x + 1)**(-alpha - betap)/beta(alpha, betap) + + alpha = Symbol("alpha", nonpositive=True) + raises(ValueError, lambda: BetaPrime('x', alpha, betap)) + + alpha = Symbol("alpha", positive=True) + betap = Symbol("beta", nonpositive=True) + raises(ValueError, lambda: BetaPrime('x', alpha, betap)) + X = BetaPrime('x', 1, 1) + assert median(X) == FiniteSet(1) + + +def test_BoundedPareto(): + L, H = symbols('L, H', negative=True) + raises(ValueError, lambda: BoundedPareto('X', 1, L, H)) + L, H = symbols('L, H', real=False) + raises(ValueError, lambda: BoundedPareto('X', 1, L, H)) + L, H = symbols('L, H', positive=True) + raises(ValueError, lambda: BoundedPareto('X', -1, L, H)) + + X = BoundedPareto('X', 2, L, H) + assert X.pspace.domain.set == Interval(L, H) + assert density(X)(x) == 2*L**2/(x**3*(1 - L**2/H**2)) + assert cdf(X)(x) == Piecewise((-H**2*L**2/(x**2*(H**2 - L**2)) \ + + H**2/(H**2 - L**2), L <= x), (0, True)) + assert E(X).simplify() == 2*H*L/(H + L) + X = BoundedPareto('X', 1, 2, 4) + assert E(X).simplify() == log(16) + assert median(X) == FiniteSet(Rational(8, 3)) + assert variance(X).simplify() == 8 - 16*log(2)**2 + + +def test_cauchy(): + x0 = Symbol("x0", real=True) + gamma = Symbol("gamma", positive=True) + p = Symbol("p", positive=True) + + X = Cauchy('x', x0, gamma) + # Tests the characteristic function + assert characteristic_function(X)(x) == exp(-gamma*Abs(x) + I*x*x0) + raises(NotImplementedError, lambda: moment_generating_function(X)) + assert density(X)(x) == 1/(pi*gamma*(1 + (x - x0)**2/gamma**2)) + assert diff(cdf(X)(x), x) == density(X)(x) + assert quantile(X)(p) == gamma*tan(pi*(p - S.Half)) + x0 + + x1 = Symbol("x1", real=False) + raises(ValueError, lambda: Cauchy('x', x1, gamma)) + gamma = Symbol("gamma", nonpositive=True) + raises(ValueError, lambda: Cauchy('x', x0, gamma)) + assert median(X) == FiniteSet(x0) + +def test_chi(): + from sympy.core.numbers import I + k = Symbol("k", integer=True) + + X = Chi('x', k) + assert density(X)(x) == 2**(-k/2 + 1)*x**(k - 1)*exp(-x**2/2)/gamma(k/2) + + # Tests the characteristic function + assert characteristic_function(X)(x) == sqrt(2)*I*x*gamma(k/2 + S(1)/2)*hyper((k/2 + S(1)/2,), + (S(3)/2,), -x**2/2)/gamma(k/2) + hyper((k/2,), (S(1)/2,), -x**2/2) + + # Tests the moment generating function + assert moment_generating_function(X)(x) == sqrt(2)*x*gamma(k/2 + S(1)/2)*hyper((k/2 + S(1)/2,), + (S(3)/2,), x**2/2)/gamma(k/2) + hyper((k/2,), (S(1)/2,), x**2/2) + + k = Symbol("k", integer=True, positive=False) + raises(ValueError, lambda: Chi('x', k)) + + k = Symbol("k", integer=False, positive=True) + raises(ValueError, lambda: Chi('x', k)) + +def test_chi_noncentral(): + k = Symbol("k", integer=True) + l = Symbol("l") + + X = ChiNoncentral("x", k, l) + assert density(X)(x) == (x**k*l*(x*l)**(-k/2)* + exp(-x**2/2 - l**2/2)*besseli(k/2 - 1, x*l)) + + k = Symbol("k", integer=True, positive=False) + raises(ValueError, lambda: ChiNoncentral('x', k, l)) + + k = Symbol("k", integer=True, positive=True) + l = Symbol("l", nonpositive=True) + raises(ValueError, lambda: ChiNoncentral('x', k, l)) + + k = Symbol("k", integer=False) + l = Symbol("l", positive=True) + raises(ValueError, lambda: ChiNoncentral('x', k, l)) + + +def test_chi_squared(): + k = Symbol("k", integer=True) + X = ChiSquared('x', k) + + # Tests the characteristic function + assert characteristic_function(X)(x) == ((-2*I*x + 1)**(-k/2)) + + assert density(X)(x) == 2**(-k/2)*x**(k/2 - 1)*exp(-x/2)/gamma(k/2) + assert cdf(X)(x) == Piecewise((lowergamma(k/2, x/2)/gamma(k/2), x >= 0), (0, True)) + assert E(X) == k + assert variance(X) == 2*k + + X = ChiSquared('x', 15) + assert cdf(X)(3) == -14873*sqrt(6)*exp(Rational(-3, 2))/(5005*sqrt(pi)) + erf(sqrt(6)/2) + + k = Symbol("k", integer=True, positive=False) + raises(ValueError, lambda: ChiSquared('x', k)) + + k = Symbol("k", integer=False, positive=True) + raises(ValueError, lambda: ChiSquared('x', k)) + + +def test_dagum(): + p = Symbol("p", positive=True) + b = Symbol("b", positive=True) + a = Symbol("a", positive=True) + + X = Dagum('x', p, a, b) + assert density(X)(x) == a*p*(x/b)**(a*p)*((x/b)**a + 1)**(-p - 1)/x + assert cdf(X)(x) == Piecewise(((1 + (x/b)**(-a))**(-p), x >= 0), + (0, True)) + + p = Symbol("p", nonpositive=True) + raises(ValueError, lambda: Dagum('x', p, a, b)) + + p = Symbol("p", positive=True) + b = Symbol("b", nonpositive=True) + raises(ValueError, lambda: Dagum('x', p, a, b)) + + b = Symbol("b", positive=True) + a = Symbol("a", nonpositive=True) + raises(ValueError, lambda: Dagum('x', p, a, b)) + X = Dagum('x', 1, 1, 1) + assert median(X) == FiniteSet(1) + +def test_davis(): + b = Symbol("b", positive=True) + n = Symbol("n", positive=True) + mu = Symbol("mu", positive=True) + + X = Davis('x', b, n, mu) + dividend = b**n*(x - mu)**(-1-n) + divisor = (exp(b/(x-mu))-1)*(gamma(n)*zeta(n)) + assert density(X)(x) == dividend/divisor + + +def test_erlang(): + k = Symbol("k", integer=True, positive=True) + l = Symbol("l", positive=True) + + X = Erlang("x", k, l) + assert density(X)(x) == x**(k - 1)*l**k*exp(-x*l)/gamma(k) + assert cdf(X)(x) == Piecewise((lowergamma(k, l*x)/gamma(k), x > 0), + (0, True)) + + +def test_exgaussian(): + m, z = symbols("m, z") + s, l = symbols("s, l", positive=True) + X = ExGaussian("x", m, s, l) + + assert density(X)(z) == l*exp(l*(l*s**2 + 2*m - 2*z)/2) *\ + erfc(sqrt(2)*(l*s**2 + m - z)/(2*s))/2 + + # Note: actual_output simplifies to expected_output. + # Ideally cdf(X)(z) would return expected_output + # expected_output = (erf(sqrt(2)*(l*s**2 + m - z)/(2*s)) - 1)*exp(l*(l*s**2 + 2*m - 2*z)/2)/2 - erf(sqrt(2)*(m - z)/(2*s))/2 + S.Half + u = l*(z - m) + v = l*s + GaussianCDF1 = cdf(Normal('x', 0, v))(u) + GaussianCDF2 = cdf(Normal('x', v**2, v))(u) + actual_output = GaussianCDF1 - exp(-u + (v**2/2) + log(GaussianCDF2)) + assert cdf(X)(z) == actual_output + # assert simplify(actual_output) == expected_output + + assert variance(X).expand() == s**2 + l**(-2) + + assert skewness(X).expand() == 2/(l**3*s**2*sqrt(s**2 + l**(-2)) + l * + sqrt(s**2 + l**(-2))) + + +@slow +def test_exponential(): + rate = Symbol('lambda', positive=True) + X = Exponential('x', rate) + p = Symbol("p", positive=True, real=True) + + assert E(X) == 1/rate + assert variance(X) == 1/rate**2 + assert skewness(X) == 2 + assert skewness(X) == smoment(X, 3) + assert kurtosis(X) == 9 + assert kurtosis(X) == smoment(X, 4) + assert smoment(2*X, 4) == smoment(X, 4) + assert moment(X, 3) == 3*2*1/rate**3 + assert P(X > 0) is S.One + assert P(X > 1) == exp(-rate) + assert P(X > 10) == exp(-10*rate) + assert quantile(X)(p) == -log(1-p)/rate + + assert where(X <= 1).set == Interval(0, 1) + Y = Exponential('y', 1) + assert median(Y) == FiniteSet(log(2)) + #Test issue 9970 + z = Dummy('z') + assert P(X > z) == exp(-z*rate) + assert P(X < z) == 0 + #Test issue 10076 (Distribution with interval(0,oo)) + x = Symbol('x') + _z = Dummy('_z') + b = SingleContinuousPSpace(x, ExponentialDistribution(2)) + + with ignore_warnings(UserWarning): ### TODO: Restore tests once warnings are removed + expected1 = Integral(2*exp(-2*_z), (_z, 3, oo)) + assert b.probability(x > 3, evaluate=False).rewrite(Integral).dummy_eq(expected1) + + expected2 = Integral(2*exp(-2*_z), (_z, 0, 4)) + assert b.probability(x < 4, evaluate=False).rewrite(Integral).dummy_eq(expected2) + Y = Exponential('y', 2*rate) + assert coskewness(X, X, X) == skewness(X) + assert coskewness(X, Y + rate*X, Y + 2*rate*X) == \ + 4/(sqrt(1 + 1/(4*rate**2))*sqrt(4 + 1/(4*rate**2))) + assert coskewness(X + 2*Y, Y + X, Y + 2*X, X > 3) == \ + sqrt(170)*Rational(9, 85) + +def test_exponential_power(): + mu = Symbol('mu') + z = Symbol('z') + alpha = Symbol('alpha', positive=True) + beta = Symbol('beta', positive=True) + + X = ExponentialPower('x', mu, alpha, beta) + + assert density(X)(z) == beta*exp(-(Abs(mu - z)/alpha) + ** beta)/(2*alpha*gamma(1/beta)) + assert cdf(X)(z) == S.Half + lowergamma(1/beta, + (Abs(mu - z)/alpha)**beta)*sign(-mu + z)/\ + (2*gamma(1/beta)) + + +def test_f_distribution(): + d1 = Symbol("d1", positive=True) + d2 = Symbol("d2", positive=True) + + X = FDistribution("x", d1, d2) + + assert density(X)(x) == (d2**(d2/2)*sqrt((d1*x)**d1*(d1*x + d2)**(-d1 - d2)) + /(x*beta(d1/2, d2/2))) + + raises(NotImplementedError, lambda: moment_generating_function(X)) + d1 = Symbol("d1", nonpositive=True) + raises(ValueError, lambda: FDistribution('x', d1, d1)) + + d1 = Symbol("d1", positive=True, integer=False) + raises(ValueError, lambda: FDistribution('x', d1, d1)) + + d1 = Symbol("d1", positive=True) + d2 = Symbol("d2", nonpositive=True) + raises(ValueError, lambda: FDistribution('x', d1, d2)) + + d2 = Symbol("d2", positive=True, integer=False) + raises(ValueError, lambda: FDistribution('x', d1, d2)) + + +def test_fisher_z(): + d1 = Symbol("d1", positive=True) + d2 = Symbol("d2", positive=True) + + X = FisherZ("x", d1, d2) + assert density(X)(x) == (2*d1**(d1/2)*d2**(d2/2)*(d1*exp(2*x) + d2) + **(-d1/2 - d2/2)*exp(d1*x)/beta(d1/2, d2/2)) + +def test_frechet(): + a = Symbol("a", positive=True) + s = Symbol("s", positive=True) + m = Symbol("m", real=True) + + X = Frechet("x", a, s=s, m=m) + assert density(X)(x) == a*((x - m)/s)**(-a - 1)*exp(-((x - m)/s)**(-a))/s + assert cdf(X)(x) == Piecewise((exp(-((-m + x)/s)**(-a)), m <= x), (0, True)) + +@slow +def test_gamma(): + k = Symbol("k", positive=True) + theta = Symbol("theta", positive=True) + + X = Gamma('x', k, theta) + + # Tests characteristic function + assert characteristic_function(X)(x) == ((-I*theta*x + 1)**(-k)) + + assert density(X)(x) == x**(k - 1)*theta**(-k)*exp(-x/theta)/gamma(k) + assert cdf(X, meijerg=True)(z) == Piecewise( + (-k*lowergamma(k, 0)/gamma(k + 1) + + k*lowergamma(k, z/theta)/gamma(k + 1), z >= 0), + (0, True)) + + # assert simplify(variance(X)) == k*theta**2 # handled numerically below + assert E(X) == moment(X, 1) + + k, theta = symbols('k theta', positive=True) + X = Gamma('x', k, theta) + assert E(X) == k*theta + assert variance(X) == k*theta**2 + assert skewness(X).expand() == 2/sqrt(k) + assert kurtosis(X).expand() == 3 + 6/k + + Y = Gamma('y', 2*k, 3*theta) + assert coskewness(X, theta*X + Y, k*X + Y).simplify() == \ + 2*531441**(-k)*sqrt(k)*theta*(3*3**(12*k) - 2*531441**k) \ + /(sqrt(k**2 + 18)*sqrt(theta**2 + 18)) + +def test_gamma_inverse(): + a = Symbol("a", positive=True) + b = Symbol("b", positive=True) + X = GammaInverse("x", a, b) + assert density(X)(x) == x**(-a - 1)*b**a*exp(-b/x)/gamma(a) + assert cdf(X)(x) == Piecewise((uppergamma(a, b/x)/gamma(a), x > 0), (0, True)) + assert characteristic_function(X)(x) == 2 * (-I*b*x)**(a/2) \ + * besselk(a, 2*sqrt(b)*sqrt(-I*x))/gamma(a) + raises(NotImplementedError, lambda: moment_generating_function(X)) + +def test_gompertz(): + b = Symbol("b", positive=True) + eta = Symbol("eta", positive=True) + + X = Gompertz("x", b, eta) + + assert density(X)(x) == b*eta*exp(eta)*exp(b*x)*exp(-eta*exp(b*x)) + assert cdf(X)(x) == 1 - exp(eta)*exp(-eta*exp(b*x)) + assert diff(cdf(X)(x), x) == density(X)(x) + + +def test_gumbel(): + beta = Symbol("beta", positive=True) + mu = Symbol("mu") + x = Symbol("x") + y = Symbol("y") + X = Gumbel("x", beta, mu) + Y = Gumbel("y", beta, mu, minimum=True) + assert density(X)(x).expand() == \ + exp(mu/beta)*exp(-x/beta)*exp(-exp(mu/beta)*exp(-x/beta))/beta + assert density(Y)(y).expand() == \ + exp(-mu/beta)*exp(y/beta)*exp(-exp(-mu/beta)*exp(y/beta))/beta + assert cdf(X)(x).expand() == \ + exp(-exp(mu/beta)*exp(-x/beta)) + assert characteristic_function(X)(x) == exp(I*mu*x)*gamma(-I*beta*x + 1) + +def test_kumaraswamy(): + a = Symbol("a", positive=True) + b = Symbol("b", positive=True) + + X = Kumaraswamy("x", a, b) + assert density(X)(x) == x**(a - 1)*a*b*(-x**a + 1)**(b - 1) + assert cdf(X)(x) == Piecewise((0, x < 0), + (-(-x**a + 1)**b + 1, x <= 1), + (1, True)) + + +def test_laplace(): + mu = Symbol("mu") + b = Symbol("b", positive=True) + + X = Laplace('x', mu, b) + + #Tests characteristic_function + assert characteristic_function(X)(x) == (exp(I*mu*x)/(b**2*x**2 + 1)) + + assert density(X)(x) == exp(-Abs(x - mu)/b)/(2*b) + assert cdf(X)(x) == Piecewise((exp((-mu + x)/b)/2, mu > x), + (-exp((mu - x)/b)/2 + 1, True)) + X = Laplace('x', [1, 2], [[1, 0], [0, 1]]) + assert isinstance(pspace(X).distribution, MultivariateLaplaceDistribution) + +def test_levy(): + mu = Symbol("mu", real=True) + c = Symbol("c", positive=True) + + X = Levy('x', mu, c) + assert X.pspace.domain.set == Interval(mu, oo) + assert density(X)(x) == sqrt(c/(2*pi))*exp(-c/(2*(x - mu)))/((x - mu)**(S.One + S.Half)) + assert cdf(X)(x) == erfc(sqrt(c/(2*(x - mu)))) + + raises(NotImplementedError, lambda: moment_generating_function(X)) + mu = Symbol("mu", real=False) + raises(ValueError, lambda: Levy('x',mu,c)) + + c = Symbol("c", nonpositive=True) + raises(ValueError, lambda: Levy('x',mu,c)) + + mu = Symbol("mu", real=True) + raises(ValueError, lambda: Levy('x',mu,c)) + +def test_logcauchy(): + mu = Symbol("mu", positive=True) + sigma = Symbol("sigma", positive=True) + + X = LogCauchy("x", mu, sigma) + + assert density(X)(x) == sigma/(x*pi*(sigma**2 + (-mu + log(x))**2)) + assert cdf(X)(x) == atan((log(x) - mu)/sigma)/pi + S.Half + + +def test_logistic(): + mu = Symbol("mu", real=True) + s = Symbol("s", positive=True) + p = Symbol("p", positive=True) + + X = Logistic('x', mu, s) + + #Tests characteristics_function + assert characteristic_function(X)(x) == \ + (Piecewise((pi*s*x*exp(I*mu*x)/sinh(pi*s*x), Ne(x, 0)), (1, True))) + + assert density(X)(x) == exp((-x + mu)/s)/(s*(exp((-x + mu)/s) + 1)**2) + assert cdf(X)(x) == 1/(exp((mu - x)/s) + 1) + assert quantile(X)(p) == mu - s*log(-S.One + 1/p) + +def test_loglogistic(): + a, b = symbols('a b') + assert LogLogistic('x', a, b) + + a = Symbol('a', negative=True) + b = Symbol('b', positive=True) + raises(ValueError, lambda: LogLogistic('x', a, b)) + + a = Symbol('a', positive=True) + b = Symbol('b', negative=True) + raises(ValueError, lambda: LogLogistic('x', a, b)) + + a, b, z, p = symbols('a b z p', positive=True) + X = LogLogistic('x', a, b) + assert density(X)(z) == b*(z/a)**(b - 1)/(a*((z/a)**b + 1)**2) + assert cdf(X)(z) == 1/(1 + (z/a)**(-b)) + assert quantile(X)(p) == a*(p/(1 - p))**(1/b) + + # Expectation + assert E(X) == Piecewise((S.NaN, b <= 1), (pi*a/(b*sin(pi/b)), True)) + b = symbols('b', prime=True) # b > 1 + X = LogLogistic('x', a, b) + assert E(X) == pi*a/(b*sin(pi/b)) + X = LogLogistic('x', 1, 2) + assert median(X) == FiniteSet(1) + +def test_logitnormal(): + mu = Symbol('mu', real=True) + s = Symbol('s', positive=True) + X = LogitNormal('x', mu, s) + x = Symbol('x') + + assert density(X)(x) == sqrt(2)*exp(-(-mu + log(x/(1 - x)))**2/(2*s**2))/(2*sqrt(pi)*s*x*(1 - x)) + assert cdf(X)(x) == erf(sqrt(2)*(-mu + log(x/(1 - x)))/(2*s))/2 + S(1)/2 + +def test_lognormal(): + mean = Symbol('mu', real=True) + std = Symbol('sigma', positive=True) + X = LogNormal('x', mean, std) + # The sympy integrator can't do this too well + #assert E(X) == exp(mean+std**2/2) + #assert variance(X) == (exp(std**2)-1) * exp(2*mean + std**2) + + # The sympy integrator can't do this too well + #assert E(X) == + raises(NotImplementedError, lambda: moment_generating_function(X)) + mu = Symbol("mu", real=True) + sigma = Symbol("sigma", positive=True) + + X = LogNormal('x', mu, sigma) + assert density(X)(x) == (sqrt(2)*exp(-(-mu + log(x))**2 + /(2*sigma**2))/(2*x*sqrt(pi)*sigma)) + # Tests cdf + assert cdf(X)(x) == Piecewise( + (erf(sqrt(2)*(-mu + log(x))/(2*sigma))/2 + + S(1)/2, x > 0), (0, True)) + + X = LogNormal('x', 0, 1) # Mean 0, standard deviation 1 + assert density(X)(x) == sqrt(2)*exp(-log(x)**2/2)/(2*x*sqrt(pi)) + + +def test_Lomax(): + a, l = symbols('a, l', negative=True) + raises(ValueError, lambda: Lomax('X', a, l)) + a, l = symbols('a, l', real=False) + raises(ValueError, lambda: Lomax('X', a, l)) + + a, l = symbols('a, l', positive=True) + X = Lomax('X', a, l) + assert X.pspace.domain.set == Interval(0, oo) + assert density(X)(x) == a*(1 + x/l)**(-a - 1)/l + assert cdf(X)(x) == Piecewise((1 - (1 + x/l)**(-a), x >= 0), (0, True)) + a = 3 + X = Lomax('X', a, l) + assert E(X) == l/2 + assert median(X) == FiniteSet(l*(-1 + 2**Rational(1, 3))) + assert variance(X) == 3*l**2/4 + + +def test_maxwell(): + a = Symbol("a", positive=True) + + X = Maxwell('x', a) + + assert density(X)(x) == (sqrt(2)*x**2*exp(-x**2/(2*a**2))/ + (sqrt(pi)*a**3)) + assert E(X) == 2*sqrt(2)*a/sqrt(pi) + assert variance(X) == -8*a**2/pi + 3*a**2 + assert cdf(X)(x) == erf(sqrt(2)*x/(2*a)) - sqrt(2)*x*exp(-x**2/(2*a**2))/(sqrt(pi)*a) + assert diff(cdf(X)(x), x) == density(X)(x) + + +@slow +def test_Moyal(): + mu = Symbol('mu',real=False) + sigma = Symbol('sigma', positive=True) + raises(ValueError, lambda: Moyal('M',mu, sigma)) + + mu = Symbol('mu', real=True) + sigma = Symbol('sigma', negative=True) + raises(ValueError, lambda: Moyal('M',mu, sigma)) + + sigma = Symbol('sigma', positive=True) + M = Moyal('M', mu, sigma) + assert density(M)(z) == sqrt(2)*exp(-exp((mu - z)/sigma)/2 + - (-mu + z)/(2*sigma))/(2*sqrt(pi)*sigma) + assert cdf(M)(z).simplify() == 1 - erf(sqrt(2)*exp((mu - z)/(2*sigma))/2) + assert characteristic_function(M)(z) == 2**(-I*sigma*z)*exp(I*mu*z) \ + *gamma(-I*sigma*z + Rational(1, 2))/sqrt(pi) + assert E(M) == mu + EulerGamma*sigma + sigma*log(2) + assert moment_generating_function(M)(z) == 2**(-sigma*z)*exp(mu*z) \ + *gamma(-sigma*z + Rational(1, 2))/sqrt(pi) + + +def test_nakagami(): + mu = Symbol("mu", positive=True) + omega = Symbol("omega", positive=True) + + X = Nakagami('x', mu, omega) + assert density(X)(x) == (2*x**(2*mu - 1)*mu**mu*omega**(-mu) + *exp(-x**2*mu/omega)/gamma(mu)) + assert simplify(E(X)) == (sqrt(mu)*sqrt(omega) + *gamma(mu + S.Half)/gamma(mu + 1)) + assert simplify(variance(X)) == ( + omega - omega*gamma(mu + S.Half)**2/(gamma(mu)*gamma(mu + 1))) + assert cdf(X)(x) == Piecewise( + (lowergamma(mu, mu*x**2/omega)/gamma(mu), x > 0), + (0, True)) + X = Nakagami('x', 1, 1) + assert median(X) == FiniteSet(sqrt(log(2))) + +def test_gaussian_inverse(): + # test for symbolic parameters + a, b = symbols('a b') + assert GaussianInverse('x', a, b) + + # Inverse Gaussian distribution is also known as Wald distribution + # `GaussianInverse` can also be referred by the name `Wald` + a, b, z = symbols('a b z') + X = Wald('x', a, b) + assert density(X)(z) == sqrt(2)*sqrt(b/z**3)*exp(-b*(-a + z)**2/(2*a**2*z))/(2*sqrt(pi)) + + a, b = symbols('a b', positive=True) + z = Symbol('z', positive=True) + + X = GaussianInverse('x', a, b) + assert density(X)(z) == sqrt(2)*sqrt(b)*sqrt(z**(-3))*exp(-b*(-a + z)**2/(2*a**2*z))/(2*sqrt(pi)) + assert E(X) == a + assert variance(X).expand() == a**3/b + assert cdf(X)(z) == (S.Half - erf(sqrt(2)*sqrt(b)*(1 + z/a)/(2*sqrt(z)))/2)*exp(2*b/a) +\ + erf(sqrt(2)*sqrt(b)*(-1 + z/a)/(2*sqrt(z)))/2 + S.Half + + a = symbols('a', nonpositive=True) + raises(ValueError, lambda: GaussianInverse('x', a, b)) + + a = symbols('a', positive=True) + b = symbols('b', nonpositive=True) + raises(ValueError, lambda: GaussianInverse('x', a, b)) + +def test_pareto(): + xm, beta = symbols('xm beta', positive=True) + alpha = beta + 5 + X = Pareto('x', xm, alpha) + + dens = density(X) + + #Tests cdf function + assert cdf(X)(x) == \ + Piecewise((-x**(-beta - 5)*xm**(beta + 5) + 1, x >= xm), (0, True)) + + #Tests characteristic_function + assert characteristic_function(X)(x) == \ + ((-I*x*xm)**(beta + 5)*(beta + 5)*uppergamma(-beta - 5, -I*x*xm)) + + assert dens(x) == x**(-(alpha + 1))*xm**(alpha)*(alpha) + + assert simplify(E(X)) == alpha*xm/(alpha-1) + + # computation of taylor series for MGF still too slow + #assert simplify(variance(X)) == xm**2*alpha / ((alpha-1)**2*(alpha-2)) + + +def test_pareto_numeric(): + xm, beta = 3, 2 + alpha = beta + 5 + X = Pareto('x', xm, alpha) + + assert E(X) == alpha*xm/S(alpha - 1) + assert variance(X) == xm**2*alpha / S((alpha - 1)**2*(alpha - 2)) + assert median(X) == FiniteSet(3*2**Rational(1, 7)) + # Skewness tests too slow. Try shortcutting function? + + +def test_PowerFunction(): + alpha = Symbol("alpha", nonpositive=True) + a, b = symbols('a, b', real=True) + raises (ValueError, lambda: PowerFunction('x', alpha, a, b)) + + a, b = symbols('a, b', real=False) + raises (ValueError, lambda: PowerFunction('x', alpha, a, b)) + + alpha = Symbol("alpha", positive=True) + a, b = symbols('a, b', real=True) + raises (ValueError, lambda: PowerFunction('x', alpha, 5, 2)) + + X = PowerFunction('X', 2, a, b) + assert density(X)(z) == (-2*a + 2*z)/(-a + b)**2 + assert cdf(X)(z) == Piecewise((a**2/(a**2 - 2*a*b + b**2) - + 2*a*z/(a**2 - 2*a*b + b**2) + z**2/(a**2 - 2*a*b + b**2), a <= z), (0, True)) + + X = PowerFunction('X', 2, 0, 1) + assert density(X)(z) == 2*z + assert cdf(X)(z) == Piecewise((z**2, z >= 0), (0,True)) + assert E(X) == Rational(2,3) + assert P(X < 0) == 0 + assert P(X < 1) == 1 + assert median(X) == FiniteSet(1/sqrt(2)) + +def test_raised_cosine(): + mu = Symbol("mu", real=True) + s = Symbol("s", positive=True) + + X = RaisedCosine("x", mu, s) + + assert pspace(X).domain.set == Interval(mu - s, mu + s) + #Tests characteristics_function + assert characteristic_function(X)(x) == \ + Piecewise((exp(-I*pi*mu/s)/2, Eq(x, -pi/s)), (exp(I*pi*mu/s)/2, Eq(x, pi/s)), (pi**2*exp(I*mu*x)*sin(s*x)/(s*x*(-s**2*x**2 + pi**2)), True)) + + assert density(X)(x) == (Piecewise(((cos(pi*(x - mu)/s) + 1)/(2*s), + And(x <= mu + s, mu - s <= x)), (0, True))) + + +def test_rayleigh(): + sigma = Symbol("sigma", positive=True) + + X = Rayleigh('x', sigma) + + #Tests characteristic_function + assert characteristic_function(X)(x) == (-sqrt(2)*sqrt(pi)*sigma*x*(erfi(sqrt(2)*sigma*x/2) - I)*exp(-sigma**2*x**2/2)/2 + 1) + + assert density(X)(x) == x*exp(-x**2/(2*sigma**2))/sigma**2 + assert E(X) == sqrt(2)*sqrt(pi)*sigma/2 + assert variance(X) == -pi*sigma**2/2 + 2*sigma**2 + assert cdf(X)(x) == 1 - exp(-x**2/(2*sigma**2)) + assert diff(cdf(X)(x), x) == density(X)(x) + +def test_reciprocal(): + a = Symbol("a", real=True) + b = Symbol("b", real=True) + + X = Reciprocal('x', a, b) + assert density(X)(x) == 1/(x*(-log(a) + log(b))) + assert cdf(X)(x) == Piecewise((log(a)/(log(a) - log(b)) - log(x)/(log(a) - log(b)), a <= x), (0, True)) + X = Reciprocal('x', 5, 30) + + assert E(X) == 25/(log(30) - log(5)) + assert P(X < 4) == S.Zero + assert P(X < 20) == log(20) / (log(30) - log(5)) - log(5) / (log(30) - log(5)) + assert cdf(X)(10) == log(10) / (log(30) - log(5)) - log(5) / (log(30) - log(5)) + + a = symbols('a', nonpositive=True) + raises(ValueError, lambda: Reciprocal('x', a, b)) + + a = symbols('a', positive=True) + b = symbols('b', positive=True) + raises(ValueError, lambda: Reciprocal('x', a + b, a)) + +def test_shiftedgompertz(): + b = Symbol("b", positive=True) + eta = Symbol("eta", positive=True) + X = ShiftedGompertz("x", b, eta) + assert density(X)(x) == b*(eta*(1 - exp(-b*x)) + 1)*exp(-b*x)*exp(-eta*exp(-b*x)) + + +def test_studentt(): + nu = Symbol("nu", positive=True) + + X = StudentT('x', nu) + assert density(X)(x) == (1 + x**2/nu)**(-nu/2 - S.Half)/(sqrt(nu)*beta(S.Half, nu/2)) + assert cdf(X)(x) == S.Half + x*gamma(nu/2 + S.Half)*hyper((S.Half, nu/2 + S.Half), + (Rational(3, 2),), -x**2/nu)/(sqrt(pi)*sqrt(nu)*gamma(nu/2)) + raises(NotImplementedError, lambda: moment_generating_function(X)) + +def test_trapezoidal(): + a = Symbol("a", real=True) + b = Symbol("b", real=True) + c = Symbol("c", real=True) + d = Symbol("d", real=True) + + X = Trapezoidal('x', a, b, c, d) + assert density(X)(x) == Piecewise(((-2*a + 2*x)/((-a + b)*(-a - b + c + d)), (a <= x) & (x < b)), + (2/(-a - b + c + d), (b <= x) & (x < c)), + ((2*d - 2*x)/((-c + d)*(-a - b + c + d)), (c <= x) & (x <= d)), + (0, True)) + + X = Trapezoidal('x', 0, 1, 2, 3) + assert E(X) == Rational(3, 2) + assert variance(X) == Rational(5, 12) + assert P(X < 2) == Rational(3, 4) + assert median(X) == FiniteSet(Rational(3, 2)) + +def test_triangular(): + a = Symbol("a") + b = Symbol("b") + c = Symbol("c") + + X = Triangular('x', a, b, c) + assert pspace(X).domain.set == Interval(a, b) + assert str(density(X)(x)) == ("Piecewise(((-2*a + 2*x)/((-a + b)*(-a + c)), (a <= x) & (c > x)), " + "(2/(-a + b), Eq(c, x)), ((2*b - 2*x)/((-a + b)*(b - c)), (b >= x) & (c < x)), (0, True))") + + #Tests moment_generating_function + assert moment_generating_function(X)(x).expand() == \ + ((-2*(-a + b)*exp(c*x) + 2*(-a + c)*exp(b*x) + 2*(b - c)*exp(a*x))/(x**2*(-a + b)*(-a + c)*(b - c))).expand() + assert str(characteristic_function(X)(x)) == \ + '(2*(-a + b)*exp(I*c*x) - 2*(-a + c)*exp(I*b*x) - 2*(b - c)*exp(I*a*x))/(x**2*(-a + b)*(-a + c)*(b - c))' + +def test_quadratic_u(): + a = Symbol("a", real=True) + b = Symbol("b", real=True) + + X = QuadraticU("x", a, b) + Y = QuadraticU("x", 1, 2) + + assert pspace(X).domain.set == Interval(a, b) + # Tests _moment_generating_function + assert moment_generating_function(Y)(1) == -15*exp(2) + 27*exp(1) + assert moment_generating_function(Y)(2) == -9*exp(4)/2 + 21*exp(2)/2 + + assert characteristic_function(Y)(1) == 3*I*(-1 + 4*I)*exp(I*exp(2*I)) + assert density(X)(x) == (Piecewise((12*(x - a/2 - b/2)**2/(-a + b)**3, + And(x <= b, a <= x)), (0, True))) + + +def test_uniform(): + l = Symbol('l', real=True) + w = Symbol('w', positive=True) + X = Uniform('x', l, l + w) + + assert E(X) == l + w/2 + assert variance(X).expand() == w**2/12 + + # With numbers all is well + X = Uniform('x', 3, 5) + assert P(X < 3) == 0 and P(X > 5) == 0 + assert P(X < 4) == P(X > 4) == S.Half + assert median(X) == FiniteSet(4) + + z = Symbol('z') + p = density(X)(z) + assert p.subs(z, 3.7) == S.Half + assert p.subs(z, -1) == 0 + assert p.subs(z, 6) == 0 + + c = cdf(X) + assert c(2) == 0 and c(3) == 0 + assert c(Rational(7, 2)) == Rational(1, 4) + assert c(5) == 1 and c(6) == 1 + + +@XFAIL +@slow +def test_uniform_P(): + """ This stopped working because SingleContinuousPSpace.compute_density no + longer calls integrate on a DiracDelta but rather just solves directly. + integrate used to call UniformDistribution.expectation which special-cased + subsed out the Min and Max terms that Uniform produces + + I decided to regress on this class for general cleanliness (and I suspect + speed) of the algorithm. + """ + l = Symbol('l', real=True) + w = Symbol('w', positive=True) + X = Uniform('x', l, l + w) + assert P(X < l) == 0 and P(X > l + w) == 0 + + +def test_uniformsum(): + n = Symbol("n", integer=True) + _k = Dummy("k") + x = Symbol("x") + + X = UniformSum('x', n) + res = Sum((-1)**_k*(-_k + x)**(n - 1)*binomial(n, _k), (_k, 0, floor(x)))/factorial(n - 1) + assert density(X)(x).dummy_eq(res) + + #Tests set functions + assert X.pspace.domain.set == Interval(0, n) + + #Tests the characteristic_function + assert characteristic_function(X)(x) == (-I*(exp(I*x) - 1)/x)**n + + #Tests the moment_generating_function + assert moment_generating_function(X)(x) == ((exp(x) - 1)/x)**n + + +def test_von_mises(): + mu = Symbol("mu") + k = Symbol("k", positive=True) + + X = VonMises("x", mu, k) + assert density(X)(x) == exp(k*cos(x - mu))/(2*pi*besseli(0, k)) + + +def test_weibull(): + a, b = symbols('a b', positive=True) + # FIXME: simplify(E(X)) seems to hang without extended_positive=True + # On a Linux machine this had a rapid memory leak... + # a, b = symbols('a b', positive=True) + X = Weibull('x', a, b) + + assert E(X).expand() == a * gamma(1 + 1/b) + assert variance(X).expand() == (a**2 * gamma(1 + 2/b) - E(X)**2).expand() + assert simplify(skewness(X)) == (2*gamma(1 + 1/b)**3 - 3*gamma(1 + 1/b)*gamma(1 + 2/b) + gamma(1 + 3/b))/(-gamma(1 + 1/b)**2 + gamma(1 + 2/b))**Rational(3, 2) + assert simplify(kurtosis(X)) == (-3*gamma(1 + 1/b)**4 +\ + 6*gamma(1 + 1/b)**2*gamma(1 + 2/b) - 4*gamma(1 + 1/b)*gamma(1 + 3/b) + gamma(1 + 4/b))/(gamma(1 + 1/b)**2 - gamma(1 + 2/b))**2 + +def test_weibull_numeric(): + # Test for integers and rationals + a = 1 + bvals = [S.Half, 1, Rational(3, 2), 5] + for b in bvals: + X = Weibull('x', a, b) + assert simplify(E(X)) == expand_func(a * gamma(1 + 1/S(b))) + assert simplify(variance(X)) == simplify( + a**2 * gamma(1 + 2/S(b)) - E(X)**2) + # Not testing Skew... it's slow with int/frac values > 3/2 + + +def test_wignersemicircle(): + R = Symbol("R", positive=True) + + X = WignerSemicircle('x', R) + assert pspace(X).domain.set == Interval(-R, R) + assert density(X)(x) == 2*sqrt(-x**2 + R**2)/(pi*R**2) + assert E(X) == 0 + + + #Tests ChiNoncentralDistribution + assert characteristic_function(X)(x) == \ + Piecewise((2*besselj(1, R*x)/(R*x), Ne(x, 0)), (1, True)) + + +def test_input_value_assertions(): + a, b = symbols('a b') + p, q = symbols('p q', positive=True) + m, n = symbols('m n', positive=False, real=True) + + raises(ValueError, lambda: Normal('x', 3, 0)) + raises(ValueError, lambda: Normal('x', m, n)) + Normal('X', a, p) # No error raised + raises(ValueError, lambda: Exponential('x', m)) + Exponential('Ex', p) # No error raised + for fn in [Pareto, Weibull, Beta, Gamma]: + raises(ValueError, lambda: fn('x', m, p)) + raises(ValueError, lambda: fn('x', p, n)) + fn('x', p, q) # No error raised + + +def test_unevaluated(): + X = Normal('x', 0, 1) + k = Dummy('k') + expr1 = Integral(sqrt(2)*k*exp(-k**2/2)/(2*sqrt(pi)), (k, -oo, oo)) + expr2 = Integral(sqrt(2)*exp(-k**2/2)/(2*sqrt(pi)), (k, 0, oo)) + with ignore_warnings(UserWarning): ### TODO: Restore tests once warnings are removed + assert E(X, evaluate=False).rewrite(Integral).dummy_eq(expr1) + assert E(X + 1, evaluate=False).rewrite(Integral).dummy_eq(expr1 + 1) + assert P(X > 0, evaluate=False).rewrite(Integral).dummy_eq(expr2) + + assert P(X > 0, X**2 < 1) == S.Half + + +def test_probability_unevaluated(): + T = Normal('T', 30, 3) + with ignore_warnings(UserWarning): ### TODO: Restore tests once warnings are removed + assert type(P(T > 33, evaluate=False)) == Probability + + +def test_density_unevaluated(): + X = Normal('X', 0, 1) + Y = Normal('Y', 0, 2) + assert isinstance(density(X+Y, evaluate=False)(z), Integral) + + +def test_NormalDistribution(): + nd = NormalDistribution(0, 1) + x = Symbol('x') + assert nd.cdf(x) == erf(sqrt(2)*x/2)/2 + S.Half + assert nd.expectation(1, x) == 1 + assert nd.expectation(x, x) == 0 + assert nd.expectation(x**2, x) == 1 + #Test issue 10076 + a = SingleContinuousPSpace(x, NormalDistribution(2, 4)) + _z = Dummy('_z') + + expected1 = Integral(sqrt(2)*exp(-(_z - 2)**2/32)/(8*sqrt(pi)),(_z, -oo, 1)) + assert a.probability(x < 1, evaluate=False).dummy_eq(expected1) is True + + expected2 = Integral(sqrt(2)*exp(-(_z - 2)**2/32)/(8*sqrt(pi)),(_z, 1, oo)) + assert a.probability(x > 1, evaluate=False).dummy_eq(expected2) is True + + b = SingleContinuousPSpace(x, NormalDistribution(1, 9)) + + expected3 = Integral(sqrt(2)*exp(-(_z - 1)**2/162)/(18*sqrt(pi)),(_z, 6, oo)) + assert b.probability(x > 6, evaluate=False).dummy_eq(expected3) is True + + expected4 = Integral(sqrt(2)*exp(-(_z - 1)**2/162)/(18*sqrt(pi)),(_z, -oo, 6)) + assert b.probability(x < 6, evaluate=False).dummy_eq(expected4) is True + + +def test_random_parameters(): + mu = Normal('mu', 2, 3) + meas = Normal('T', mu, 1) + assert density(meas, evaluate=False)(z) + assert isinstance(pspace(meas), CompoundPSpace) + X = Normal('x', [1, 2], [[1, 0], [0, 1]]) + assert isinstance(pspace(X).distribution, MultivariateNormalDistribution) + assert density(meas)(z).simplify() == sqrt(5)*exp(-z**2/20 + z/5 - S(1)/5)/(10*sqrt(pi)) + + +def test_random_parameters_given(): + mu = Normal('mu', 2, 3) + meas = Normal('T', mu, 1) + assert given(meas, Eq(mu, 5)) == Normal('T', 5, 1) + + +def test_conjugate_priors(): + mu = Normal('mu', 2, 3) + x = Normal('x', mu, 1) + assert isinstance(simplify(density(mu, Eq(x, y), evaluate=False)(z)), + Mul) + + +def test_difficult_univariate(): + """ Since using solve in place of deltaintegrate we're able to perform + substantially more complex density computations on single continuous random + variables """ + x = Normal('x', 0, 1) + assert density(x**3) + assert density(exp(x**2)) + assert density(log(x)) + + +def test_issue_10003(): + X = Exponential('x', 3) + G = Gamma('g', 1, 2) + assert P(X < -1) is S.Zero + assert P(G < -1) is S.Zero + + +def test_precomputed_cdf(): + x = symbols("x", real=True) + mu = symbols("mu", real=True) + sigma, xm, alpha = symbols("sigma xm alpha", positive=True) + n = symbols("n", integer=True, positive=True) + distribs = [ + Normal("X", mu, sigma), + Pareto("P", xm, alpha), + ChiSquared("C", n), + Exponential("E", sigma), + # LogNormal("L", mu, sigma), + ] + for X in distribs: + compdiff = cdf(X)(x) - simplify(X.pspace.density.compute_cdf()(x)) + compdiff = simplify(compdiff.rewrite(erfc)) + assert compdiff == 0 + + +@slow +def test_precomputed_characteristic_functions(): + import mpmath + + def test_cf(dist, support_lower_limit, support_upper_limit): + pdf = density(dist) + t = Symbol('t') + + # first function is the hardcoded CF of the distribution + cf1 = lambdify([t], characteristic_function(dist)(t), 'mpmath') + + # second function is the Fourier transform of the density function + f = lambdify([x, t], pdf(x)*exp(I*x*t), 'mpmath') + cf2 = lambda t: mpmath.quad(lambda x: f(x, t), [support_lower_limit, support_upper_limit], maxdegree=10) + + # compare the two functions at various points + for test_point in [2, 5, 8, 11]: + n1 = cf1(test_point) + n2 = cf2(test_point) + + assert abs(re(n1) - re(n2)) < 1e-12 + assert abs(im(n1) - im(n2)) < 1e-12 + + test_cf(Beta('b', 1, 2), 0, 1) + test_cf(Chi('c', 3), 0, mpmath.inf) + test_cf(ChiSquared('c', 2), 0, mpmath.inf) + test_cf(Exponential('e', 6), 0, mpmath.inf) + test_cf(Logistic('l', 1, 2), -mpmath.inf, mpmath.inf) + test_cf(Normal('n', -1, 5), -mpmath.inf, mpmath.inf) + test_cf(RaisedCosine('r', 3, 1), 2, 4) + test_cf(Rayleigh('r', 0.5), 0, mpmath.inf) + test_cf(Uniform('u', -1, 1), -1, 1) + test_cf(WignerSemicircle('w', 3), -3, 3) + + +def test_long_precomputed_cdf(): + x = symbols("x", real=True) + distribs = [ + Arcsin("A", -5, 9), + Dagum("D", 4, 10, 3), + Erlang("E", 14, 5), + Frechet("F", 2, 6, -3), + Gamma("G", 2, 7), + GammaInverse("GI", 3, 5), + Kumaraswamy("K", 6, 8), + Laplace("LA", -5, 4), + Logistic("L", -6, 7), + Nakagami("N", 2, 7), + StudentT("S", 4) + ] + for distr in distribs: + for _ in range(5): + assert tn(diff(cdf(distr)(x), x), density(distr)(x), x, a=0, b=0, c=1, d=0) + + US = UniformSum("US", 5) + pdf01 = density(US)(x).subs(floor(x), 0).doit() # pdf on (0, 1) + cdf01 = cdf(US, evaluate=False)(x).subs(floor(x), 0).doit() # cdf on (0, 1) + assert tn(diff(cdf01, x), pdf01, x, a=0, b=0, c=1, d=0) + + +def test_issue_13324(): + X = Uniform('X', 0, 1) + assert E(X, X > S.Half) == Rational(3, 4) + assert E(X, X > 0) == S.Half + +def test_issue_20756(): + X = Uniform('X', -1, +1) + Y = Uniform('Y', -1, +1) + assert E(X * Y) == S.Zero + assert E(X * ((Y + 1) - 1)) == S.Zero + assert E(Y * (X*(X + 1) - X*X)) == S.Zero + +def test_FiniteSet_prob(): + E = Exponential('E', 3) + N = Normal('N', 5, 7) + assert P(Eq(E, 1)) is S.Zero + assert P(Eq(N, 2)) is S.Zero + assert P(Eq(N, x)) is S.Zero + +def test_prob_neq(): + E = Exponential('E', 4) + X = ChiSquared('X', 4) + assert P(Ne(E, 2)) == 1 + assert P(Ne(X, 4)) == 1 + assert P(Ne(X, 4)) == 1 + assert P(Ne(X, 5)) == 1 + assert P(Ne(E, x)) == 1 + +def test_union(): + N = Normal('N', 3, 2) + assert simplify(P(N**2 - N > 2)) == \ + -erf(sqrt(2))/2 - erfc(sqrt(2)/4)/2 + Rational(3, 2) + assert simplify(P(N**2 - 4 > 0)) == \ + -erf(5*sqrt(2)/4)/2 - erfc(sqrt(2)/4)/2 + Rational(3, 2) + +def test_Or(): + N = Normal('N', 0, 1) + assert simplify(P(Or(N > 2, N < 1))) == \ + -erf(sqrt(2))/2 - erfc(sqrt(2)/2)/2 + Rational(3, 2) + assert P(Or(N < 0, N < 1)) == P(N < 1) + assert P(Or(N > 0, N < 0)) == 1 + + +def test_conditional_eq(): + E = Exponential('E', 1) + assert P(Eq(E, 1), Eq(E, 1)) == 1 + assert P(Eq(E, 1), Eq(E, 2)) == 0 + assert P(E > 1, Eq(E, 2)) == 1 + assert P(E < 1, Eq(E, 2)) == 0 + +def test_ContinuousDistributionHandmade(): + x = Symbol('x') + z = Dummy('z') + dens = Lambda(x, Piecewise((S.Half, (0<=x)&(x<1)), (0, (x>=1)&(x<2)), + (S.Half, (x>=2)&(x<3)), (0, True))) + dens = ContinuousDistributionHandmade(dens, set=Interval(0, 3)) + space = SingleContinuousPSpace(z, dens) + assert dens.pdf == Lambda(x, Piecewise((S(1)/2, (x >= 0) & (x < 1)), + (0, (x >= 1) & (x < 2)), (S(1)/2, (x >= 2) & (x < 3)), (0, True))) + assert median(space.value) == Interval(1, 2) + assert E(space.value) == Rational(3, 2) + assert variance(space.value) == Rational(13, 12) + + +def test_issue_16318(): + # test compute_expectation function of the SingleContinuousDomain + N = SingleContinuousDomain(x, Interval(0, 1)) + raises(ValueError, lambda: SingleContinuousDomain.compute_expectation(N, x+1, {x, y})) + +def test_compute_density(): + X = Normal('X', 0, Symbol("sigma")**2) + raises(ValueError, lambda: density(X**5 + X)) diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_discrete_rv.py b/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_discrete_rv.py new file mode 100644 index 0000000000000000000000000000000000000000..39650a1a08e42840aaf8b06c1eb6e60c92a57f23 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_discrete_rv.py @@ -0,0 +1,312 @@ +from sympy.concrete.summations import Sum +from sympy.core.numbers import (I, Rational, oo, pi) +from sympy.core.singleton import S +from sympy.core.symbol import Symbol +from sympy.functions.elementary.complexes import (im, re) +from sympy.functions.elementary.exponential import log +from sympy.functions.elementary.integers import floor +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.special.bessel import besseli +from sympy.functions.special.beta_functions import beta +from sympy.functions.special.zeta_functions import zeta +from sympy.sets.sets import FiniteSet +from sympy.simplify.simplify import simplify +from sympy.utilities.lambdify import lambdify +from sympy.core.relational import Eq, Ne +from sympy.functions.elementary.exponential import exp +from sympy.logic.boolalg import Or +from sympy.sets.fancysets import Range +from sympy.stats import (P, E, variance, density, characteristic_function, + where, moment_generating_function, skewness, cdf, + kurtosis, coskewness) +from sympy.stats.drv_types import (PoissonDistribution, GeometricDistribution, + FlorySchulz, Poisson, Geometric, Hermite, Logarithmic, + NegativeBinomial, Skellam, YuleSimon, Zeta, + DiscreteRV) +from sympy.testing.pytest import slow, nocache_fail, raises, skip +from sympy.stats.symbolic_probability import Expectation +from sympy.functions.combinatorial.factorials import FallingFactorial + +x = Symbol('x') + + +def test_PoissonDistribution(): + l = 3 + p = PoissonDistribution(l) + assert abs(p.cdf(10).evalf() - 1) < .001 + assert abs(p.cdf(10.4).evalf() - 1) < .001 + assert p.expectation(x, x) == l + assert p.expectation(x**2, x) - p.expectation(x, x)**2 == l + + +def test_Poisson(): + l = 3 + x = Poisson('x', l) + assert E(x) == l + assert E(2*x) == 2*l + assert variance(x) == l + assert density(x) == PoissonDistribution(l) + assert isinstance(E(x, evaluate=False), Expectation) + assert isinstance(E(2*x, evaluate=False), Expectation) + # issue 8248 + assert x.pspace.compute_expectation(1) == 1 + # issue 27344 + try: + import numpy as np + except ImportError: + skip("numpy not installed") + y = Poisson('y', np.float64(4.72544290380919e-11)) + assert E(y) == 4.72544290380919e-11 + y = Poisson('y', np.float64(4.725442903809197e-11)) + assert E(y) == 4.725442903809197e-11 + l2 = 5 + z = Poisson('z', l2) + assert E(z) == l2 + assert E(FallingFactorial(z, 3)) == l2**3 + assert E(z**2) == l2 + l2**2 + + +def test_FlorySchulz(): + a = Symbol("a") + z = Symbol("z") + x = FlorySchulz('x', a) + assert E(x) == (2 - a)/a + assert (variance(x) - 2*(1 - a)/a**2).simplify() == S(0) + assert density(x)(z) == a**2*z*(1 - a)**(z - 1) + + +@slow +def test_GeometricDistribution(): + p = S.One / 5 + d = GeometricDistribution(p) + assert d.expectation(x, x) == 1/p + assert d.expectation(x**2, x) - d.expectation(x, x)**2 == (1-p)/p**2 + assert abs(d.cdf(20000).evalf() - 1) < .001 + assert abs(d.cdf(20000.8).evalf() - 1) < .001 + G = Geometric('G', p=S(1)/4) + assert cdf(G)(S(7)/2) == P(G <= S(7)/2) + + X = Geometric('X', Rational(1, 5)) + Y = Geometric('Y', Rational(3, 10)) + assert coskewness(X, X + Y, X + 2*Y).simplify() == sqrt(230)*Rational(81, 1150) + + +def test_Hermite(): + a1 = Symbol("a1", positive=True) + a2 = Symbol("a2", negative=True) + raises(ValueError, lambda: Hermite("H", a1, a2)) + + a1 = Symbol("a1", negative=True) + a2 = Symbol("a2", positive=True) + raises(ValueError, lambda: Hermite("H", a1, a2)) + + a1 = Symbol("a1", positive=True) + x = Symbol("x") + H = Hermite("H", a1, a2) + assert moment_generating_function(H)(x) == exp(a1*(exp(x) - 1) + + a2*(exp(2*x) - 1)) + assert characteristic_function(H)(x) == exp(a1*(exp(I*x) - 1) + + a2*(exp(2*I*x) - 1)) + assert E(H) == a1 + 2*a2 + + H = Hermite("H", a1=5, a2=4) + assert density(H)(2) == 33*exp(-9)/2 + assert E(H) == 13 + assert variance(H) == 21 + assert kurtosis(H) == Rational(464,147) + assert skewness(H) == 37*sqrt(21)/441 + +def test_Logarithmic(): + p = S.Half + x = Logarithmic('x', p) + assert E(x) == -p / ((1 - p) * log(1 - p)) + assert variance(x) == -1/log(2)**2 + 2/log(2) + assert E(2*x**2 + 3*x + 4) == 4 + 7 / log(2) + assert isinstance(E(x, evaluate=False), Expectation) + + +@nocache_fail +def test_negative_binomial(): + r = 5 + p = S.One / 3 + x = NegativeBinomial('x', r, p) + assert E(x) == r * (1 - p) / p + # This hangs when run with the cache disabled: + assert variance(x) == r * (1 - p) / p**2 + assert E(x**5 + 2*x + 3) == E(x**5) + 2*E(x) + 3 == Rational(796473, 1) + assert isinstance(E(x, evaluate=False), Expectation) + + +def test_skellam(): + mu1 = Symbol('mu1') + mu2 = Symbol('mu2') + z = Symbol('z') + X = Skellam('x', mu1, mu2) + + assert density(X)(z) == (mu1/mu2)**(z/2) * \ + exp(-mu1 - mu2)*besseli(z, 2*sqrt(mu1*mu2)) + assert skewness(X).expand() == mu1/(mu1*sqrt(mu1 + mu2) + mu2 * + sqrt(mu1 + mu2)) - mu2/(mu1*sqrt(mu1 + mu2) + mu2*sqrt(mu1 + mu2)) + assert variance(X).expand() == mu1 + mu2 + assert E(X) == mu1 - mu2 + assert characteristic_function(X)(z) == exp( + mu1*exp(I*z) - mu1 - mu2 + mu2*exp(-I*z)) + assert moment_generating_function(X)(z) == exp( + mu1*exp(z) - mu1 - mu2 + mu2*exp(-z)) + + +def test_yule_simon(): + from sympy.core.singleton import S + rho = S(3) + x = YuleSimon('x', rho) + assert simplify(E(x)) == rho / (rho - 1) + assert simplify(variance(x)) == rho**2 / ((rho - 1)**2 * (rho - 2)) + assert isinstance(E(x, evaluate=False), Expectation) + # To test the cdf function + assert cdf(x)(x) == Piecewise((-beta(floor(x), 4)*floor(x) + 1, x >= 1), (0, True)) + + +def test_zeta(): + s = S(5) + x = Zeta('x', s) + assert E(x) == zeta(s-1) / zeta(s) + assert simplify(variance(x)) == ( + zeta(s) * zeta(s-2) - zeta(s-1)**2) / zeta(s)**2 + + +def test_discrete_probability(): + X = Geometric('X', Rational(1, 5)) + Y = Poisson('Y', 4) + G = Geometric('e', x) + assert P(Eq(X, 3)) == Rational(16, 125) + assert P(X < 3) == Rational(9, 25) + assert P(X > 3) == Rational(64, 125) + assert P(X >= 3) == Rational(16, 25) + assert P(X <= 3) == Rational(61, 125) + assert P(Ne(X, 3)) == Rational(109, 125) + assert P(Eq(Y, 3)) == 32*exp(-4)/3 + assert P(Y < 3) == 13*exp(-4) + assert P(Y > 3).equals(32*(Rational(-71, 32) + 3*exp(4)/32)*exp(-4)/3) + assert P(Y >= 3).equals(32*(Rational(-39, 32) + 3*exp(4)/32)*exp(-4)/3) + assert P(Y <= 3) == 71*exp(-4)/3 + assert P(Ne(Y, 3)).equals( + 13*exp(-4) + 32*(Rational(-71, 32) + 3*exp(4)/32)*exp(-4)/3) + assert P(X < S.Infinity) is S.One + assert P(X > S.Infinity) is S.Zero + assert P(G < 3) == x*(2-x) + assert P(Eq(G, 3)) == x*(-x + 1)**2 + + +def test_DiscreteRV(): + p = S(1)/2 + x = Symbol('x', integer=True, positive=True) + pdf = p*(1 - p)**(x - 1) # pdf of Geometric Distribution + D = DiscreteRV(x, pdf, set=S.Naturals, check=True) + assert E(D) == E(Geometric('G', S(1)/2)) == 2 + assert P(D > 3) == S(1)/8 + assert D.pspace.domain.set == S.Naturals + raises(ValueError, lambda: DiscreteRV(x, x, FiniteSet(*range(4)), check=True)) + + # purposeful invalid pmf but it should not raise since check=False + # see test_drv_types.test_ContinuousRV for explanation + X = DiscreteRV(x, 1/x, S.Naturals) + assert P(X < 2) == 1 + assert E(X) == oo + +def test_precomputed_characteristic_functions(): + import mpmath + + def test_cf(dist, support_lower_limit, support_upper_limit): + pdf = density(dist) + t = S('t') + x = S('x') + + # first function is the hardcoded CF of the distribution + cf1 = lambdify([t], characteristic_function(dist)(t), 'mpmath') + + # second function is the Fourier transform of the density function + f = lambdify([x, t], pdf(x)*exp(I*x*t), 'mpmath') + cf2 = lambda t: mpmath.nsum(lambda x: f(x, t), [ + support_lower_limit, support_upper_limit], maxdegree=10) + + # compare the two functions at various points + for test_point in [2, 5, 8, 11]: + n1 = cf1(test_point) + n2 = cf2(test_point) + + assert abs(re(n1) - re(n2)) < 1e-12 + assert abs(im(n1) - im(n2)) < 1e-12 + + test_cf(Geometric('g', Rational(1, 3)), 1, mpmath.inf) + test_cf(Logarithmic('l', Rational(1, 5)), 1, mpmath.inf) + test_cf(NegativeBinomial('n', 5, Rational(1, 7)), 0, mpmath.inf) + test_cf(Poisson('p', 5), 0, mpmath.inf) + test_cf(YuleSimon('y', 5), 1, mpmath.inf) + test_cf(Zeta('z', 5), 1, mpmath.inf) + + +def test_moment_generating_functions(): + t = S('t') + + geometric_mgf = moment_generating_function(Geometric('g', S.Half))(t) + assert geometric_mgf.diff(t).subs(t, 0) == 2 + + logarithmic_mgf = moment_generating_function(Logarithmic('l', S.Half))(t) + assert logarithmic_mgf.diff(t).subs(t, 0) == 1/log(2) + + negative_binomial_mgf = moment_generating_function( + NegativeBinomial('n', 5, Rational(1, 3)))(t) + assert negative_binomial_mgf.diff(t).subs(t, 0) == Rational(10, 1) + + poisson_mgf = moment_generating_function(Poisson('p', 5))(t) + assert poisson_mgf.diff(t).subs(t, 0) == 5 + + skellam_mgf = moment_generating_function(Skellam('s', 1, 1))(t) + assert skellam_mgf.diff(t).subs( + t, 2) == (-exp(-2) + exp(2))*exp(-2 + exp(-2) + exp(2)) + + yule_simon_mgf = moment_generating_function(YuleSimon('y', 3))(t) + assert simplify(yule_simon_mgf.diff(t).subs(t, 0)) == Rational(3, 2) + + zeta_mgf = moment_generating_function(Zeta('z', 5))(t) + assert zeta_mgf.diff(t).subs(t, 0) == pi**4/(90*zeta(5)) + + +def test_Or(): + X = Geometric('X', S.Half) + assert P(Or(X < 3, X > 4)) == Rational(13, 16) + assert P(Or(X > 2, X > 1)) == P(X > 1) + assert P(Or(X >= 3, X < 3)) == 1 + + +def test_where(): + X = Geometric('X', Rational(1, 5)) + Y = Poisson('Y', 4) + assert where(X**2 > 4).set == Range(3, S.Infinity, 1) + assert where(X**2 >= 4).set == Range(2, S.Infinity, 1) + assert where(Y**2 < 9).set == Range(0, 3, 1) + assert where(Y**2 <= 9).set == Range(0, 4, 1) + + +def test_conditional(): + X = Geometric('X', Rational(2, 3)) + Y = Poisson('Y', 3) + assert P(X > 2, X > 3) == 1 + assert P(X > 3, X > 2) == Rational(1, 3) + assert P(Y > 2, Y < 2) == 0 + assert P(Eq(Y, 3), Y >= 0) == 9*exp(-3)/2 + assert P(Eq(Y, 3), Eq(Y, 2)) == 0 + assert P(X < 2, Eq(X, 2)) == 0 + assert P(X > 2, Eq(X, 3)) == 1 + + +def test_product_spaces(): + X1 = Geometric('X1', S.Half) + X2 = Geometric('X2', Rational(1, 3)) + assert str(P(X1 + X2 < 3).rewrite(Sum)) == ( + "Sum(Piecewise((1/(4*2**n), n >= -1), (0, True)), (n, -oo, -1))/3") + assert str(P(X1 + X2 > 3).rewrite(Sum)) == ( + 'Sum(Piecewise((2**(X2 - n - 2)*(2/3)**(X2 - 1)/6, ' + 'X2 - n <= 2), (0, True)), (X2, 1, oo), (n, 1, oo))') + assert P(Eq(X1 + X2, 3)) == Rational(1, 12) diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_error_prop.py b/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_error_prop.py new file mode 100644 index 0000000000000000000000000000000000000000..483fb4c36e202d744faeb355606ff9803a516873 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_error_prop.py @@ -0,0 +1,60 @@ +from sympy.core.function import Function +from sympy.core.symbol import symbols +from sympy.functions.elementary.exponential import exp +from sympy.stats.error_prop import variance_prop +from sympy.stats.symbolic_probability import (RandomSymbol, Variance, + Covariance) + + +def test_variance_prop(): + x, y, z = symbols('x y z') + phi, t = consts = symbols('phi t') + a = RandomSymbol(x) + var_x = Variance(a) + var_y = Variance(RandomSymbol(y)) + var_z = Variance(RandomSymbol(z)) + f = Function('f')(x) + cases = { + x + y: var_x + var_y, + a + y: var_x + var_y, + x + y + z: var_x + var_y + var_z, + 2*x: 4*var_x, + x*y: var_x*y**2 + var_y*x**2, + 1/x: var_x/x**4, + x/y: (var_x*y**2 + var_y*x**2)/y**4, + exp(x): var_x*exp(2*x), + exp(2*x): 4*var_x*exp(4*x), + exp(-x*t): t**2*var_x*exp(-2*t*x), + f: Variance(f), + } + for inp, out in cases.items(): + obs = variance_prop(inp, consts=consts) + assert out == obs + +def test_variance_prop_with_covar(): + x, y, z = symbols('x y z') + phi, t = consts = symbols('phi t') + a = RandomSymbol(x) + var_x = Variance(a) + b = RandomSymbol(y) + var_y = Variance(b) + c = RandomSymbol(z) + var_z = Variance(c) + covar_x_y = Covariance(a, b) + covar_x_z = Covariance(a, c) + covar_y_z = Covariance(b, c) + cases = { + x + y: var_x + var_y + 2*covar_x_y, + a + y: var_x + var_y + 2*covar_x_y, + x + y + z: var_x + var_y + var_z + \ + 2*covar_x_y + 2*covar_x_z + 2*covar_y_z, + 2*x: 4*var_x, + x*y: var_x*y**2 + var_y*x**2 + 2*covar_x_y/(x*y), + 1/x: var_x/x**4, + exp(x): var_x*exp(2*x), + exp(2*x): 4*var_x*exp(4*x), + exp(-x*t): t**2*var_x*exp(-2*t*x), + } + for inp, out in cases.items(): + obs = variance_prop(inp, consts=consts, include_covar=True) + assert out == obs diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_finite_rv.py b/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_finite_rv.py new file mode 100644 index 0000000000000000000000000000000000000000..93bf0211a26ecc32d7f18c7e2d8236859857e445 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_finite_rv.py @@ -0,0 +1,509 @@ +from sympy.concrete.summations import Sum +from sympy.core.containers import (Dict, Tuple) +from sympy.core.function import Function +from sympy.core.numbers import (I, Rational, nan) +from sympy.core.relational import Eq +from sympy.core.singleton import S +from sympy.core.symbol import (Dummy, Symbol, symbols) +from sympy.core.sympify import sympify +from sympy.functions.combinatorial.factorials import binomial +from sympy.functions.combinatorial.numbers import harmonic +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.elementary.trigonometric import cos +from sympy.functions.special.beta_functions import beta +from sympy.logic.boolalg import (And, Or) +from sympy.polys.polytools import cancel +from sympy.sets.sets import FiniteSet +from sympy.simplify.simplify import simplify +from sympy.matrices import Matrix +from sympy.stats import (DiscreteUniform, Die, Bernoulli, Coin, Binomial, BetaBinomial, + Hypergeometric, Rademacher, IdealSoliton, RobustSoliton, P, E, variance, + covariance, skewness, density, where, FiniteRV, pspace, cdf, + correlation, moment, cmoment, smoment, characteristic_function, + moment_generating_function, quantile, kurtosis, median, coskewness) +from sympy.stats.frv_types import DieDistribution, BinomialDistribution, \ + HypergeometricDistribution +from sympy.stats.rv import Density +from sympy.testing.pytest import raises + + +def BayesTest(A, B): + assert P(A, B) == P(And(A, B)) / P(B) + assert P(A, B) == P(B, A) * P(A) / P(B) + + +def test_discreteuniform(): + # Symbolic + a, b, c, t = symbols('a b c t') + X = DiscreteUniform('X', [a, b, c]) + + assert E(X) == (a + b + c)/3 + assert simplify(variance(X) + - ((a**2 + b**2 + c**2)/3 - (a/3 + b/3 + c/3)**2)) == 0 + assert P(Eq(X, a)) == P(Eq(X, b)) == P(Eq(X, c)) == S('1/3') + + Y = DiscreteUniform('Y', range(-5, 5)) + + # Numeric + assert E(Y) == S('-1/2') + assert variance(Y) == S('33/4') + assert median(Y) == FiniteSet(-1, 0) + + for x in range(-5, 5): + assert P(Eq(Y, x)) == S('1/10') + assert P(Y <= x) == S(x + 6)/10 + assert P(Y >= x) == S(5 - x)/10 + + assert dict(density(Die('D', 6)).items()) == \ + dict(density(DiscreteUniform('U', range(1, 7))).items()) + + assert characteristic_function(X)(t) == exp(I*a*t)/3 + exp(I*b*t)/3 + exp(I*c*t)/3 + assert moment_generating_function(X)(t) == exp(a*t)/3 + exp(b*t)/3 + exp(c*t)/3 + # issue 18611 + raises(ValueError, lambda: DiscreteUniform('Z', [a, a, a, b, b, c])) + +def test_dice(): + # TODO: Make iid method! + X, Y, Z = Die('X', 6), Die('Y', 6), Die('Z', 6) + a, b, t, p = symbols('a b t p') + + assert E(X) == 3 + S.Half + assert variance(X) == Rational(35, 12) + assert E(X + Y) == 7 + assert E(X + X) == 7 + assert E(a*X + b) == a*E(X) + b + assert variance(X + Y) == variance(X) + variance(Y) == cmoment(X + Y, 2) + assert variance(X + X) == 4 * variance(X) == cmoment(X + X, 2) + assert cmoment(X, 0) == 1 + assert cmoment(4*X, 3) == 64*cmoment(X, 3) + assert covariance(X, Y) is S.Zero + assert covariance(X, X + Y) == variance(X) + assert density(Eq(cos(X*S.Pi), 1))[True] == S.Half + assert correlation(X, Y) == 0 + assert correlation(X, Y) == correlation(Y, X) + assert smoment(X + Y, 3) == skewness(X + Y) + assert smoment(X + Y, 4) == kurtosis(X + Y) + assert smoment(X, 0) == 1 + assert P(X > 3) == S.Half + assert P(2*X > 6) == S.Half + assert P(X > Y) == Rational(5, 12) + assert P(Eq(X, Y)) == P(Eq(X, 1)) + + assert E(X, X > 3) == 5 == moment(X, 1, 0, X > 3) + assert E(X, Y > 3) == E(X) == moment(X, 1, 0, Y > 3) + assert E(X + Y, Eq(X, Y)) == E(2*X) + assert moment(X, 0) == 1 + assert moment(5*X, 2) == 25*moment(X, 2) + assert quantile(X)(p) == Piecewise((nan, (p > 1) | (p < 0)),\ + (S.One, p <= Rational(1, 6)), (S(2), p <= Rational(1, 3)), (S(3), p <= S.Half),\ + (S(4), p <= Rational(2, 3)), (S(5), p <= Rational(5, 6)), (S(6), p <= 1)) + + assert P(X > 3, X > 3) is S.One + assert P(X > Y, Eq(Y, 6)) is S.Zero + assert P(Eq(X + Y, 12)) == Rational(1, 36) + assert P(Eq(X + Y, 12), Eq(X, 6)) == Rational(1, 6) + + assert density(X + Y) == density(Y + Z) != density(X + X) + d = density(2*X + Y**Z) + assert d[S(22)] == Rational(1, 108) and d[S(4100)] == Rational(1, 216) and S(3130) not in d + + assert pspace(X).domain.as_boolean() == Or( + *[Eq(X.symbol, i) for i in [1, 2, 3, 4, 5, 6]]) + + assert where(X > 3).set == FiniteSet(4, 5, 6) + + assert characteristic_function(X)(t) == exp(6*I*t)/6 + exp(5*I*t)/6 + exp(4*I*t)/6 + exp(3*I*t)/6 + exp(2*I*t)/6 + exp(I*t)/6 + assert moment_generating_function(X)(t) == exp(6*t)/6 + exp(5*t)/6 + exp(4*t)/6 + exp(3*t)/6 + exp(2*t)/6 + exp(t)/6 + assert median(X) == FiniteSet(3, 4) + D = Die('D', 7) + assert median(D) == FiniteSet(4) + # Bayes test for die + BayesTest(X > 3, X + Y < 5) + BayesTest(Eq(X - Y, Z), Z > Y) + BayesTest(X > 3, X > 2) + + # arg test for die + raises(ValueError, lambda: Die('X', -1)) # issue 8105: negative sides. + raises(ValueError, lambda: Die('X', 0)) + raises(ValueError, lambda: Die('X', 1.5)) # issue 8103: non integer sides. + + # symbolic test for die + n, k = symbols('n, k', positive=True) + D = Die('D', n) + dens = density(D).dict + assert dens == Density(DieDistribution(n)) + assert set(dens.subs(n, 4).doit().keys()) == {1, 2, 3, 4} + assert set(dens.subs(n, 4).doit().values()) == {Rational(1, 4)} + k = Dummy('k', integer=True) + assert E(D).dummy_eq( + Sum(Piecewise((k/n, k <= n), (0, True)), (k, 1, n))) + assert variance(D).subs(n, 6).doit() == Rational(35, 12) + + ki = Dummy('ki') + cumuf = cdf(D)(k) + assert cumuf.dummy_eq( + Sum(Piecewise((1/n, (ki >= 1) & (ki <= n)), (0, True)), (ki, 1, k))) + assert cumuf.subs({n: 6, k: 2}).doit() == Rational(1, 3) + + t = Dummy('t') + cf = characteristic_function(D)(t) + assert cf.dummy_eq( + Sum(Piecewise((exp(ki*I*t)/n, (ki >= 1) & (ki <= n)), (0, True)), (ki, 1, n))) + assert cf.subs(n, 3).doit() == exp(3*I*t)/3 + exp(2*I*t)/3 + exp(I*t)/3 + mgf = moment_generating_function(D)(t) + assert mgf.dummy_eq( + Sum(Piecewise((exp(ki*t)/n, (ki >= 1) & (ki <= n)), (0, True)), (ki, 1, n))) + assert mgf.subs(n, 3).doit() == exp(3*t)/3 + exp(2*t)/3 + exp(t)/3 + +def test_given(): + X = Die('X', 6) + assert density(X, X > 5) == {S(6): S.One} + assert where(X > 2, X > 5).as_boolean() == Eq(X.symbol, 6) + + +def test_domains(): + X, Y = Die('x', 6), Die('y', 6) + x, y = X.symbol, Y.symbol + # Domains + d = where(X > Y) + assert d.condition == (x > y) + d = where(And(X > Y, Y > 3)) + assert d.as_boolean() == Or(And(Eq(x, 5), Eq(y, 4)), And(Eq(x, 6), + Eq(y, 5)), And(Eq(x, 6), Eq(y, 4))) + assert len(d.elements) == 3 + + assert len(pspace(X + Y).domain.elements) == 36 + + Z = Die('x', 4) + + raises(ValueError, lambda: P(X > Z)) # Two domains with same internal symbol + + assert pspace(X + Y).domain.set == FiniteSet(1, 2, 3, 4, 5, 6)**2 + + assert where(X > 3).set == FiniteSet(4, 5, 6) + assert X.pspace.domain.dict == FiniteSet( + *[Dict({X.symbol: i}) for i in range(1, 7)]) + + assert where(X > Y).dict == FiniteSet(*[Dict({X.symbol: i, Y.symbol: j}) + for i in range(1, 7) for j in range(1, 7) if i > j]) + +def test_bernoulli(): + p, a, b, t = symbols('p a b t') + X = Bernoulli('B', p, a, b) + + assert E(X) == a*p + b*(-p + 1) + assert density(X)[a] == p + assert density(X)[b] == 1 - p + assert characteristic_function(X)(t) == p * exp(I * a * t) + (-p + 1) * exp(I * b * t) + assert moment_generating_function(X)(t) == p * exp(a * t) + (-p + 1) * exp(b * t) + + X = Bernoulli('B', p, 1, 0) + z = Symbol("z") + + assert E(X) == p + assert simplify(variance(X)) == p*(1 - p) + assert E(a*X + b) == a*E(X) + b + assert simplify(variance(a*X + b)) == simplify(a**2 * variance(X)) + assert quantile(X)(z) == Piecewise((nan, (z > 1) | (z < 0)), (0, z <= 1 - p), (1, z <= 1)) + Y = Bernoulli('Y', Rational(1, 2)) + assert median(Y) == FiniteSet(0, 1) + Z = Bernoulli('Z', Rational(2, 3)) + assert median(Z) == FiniteSet(1) + raises(ValueError, lambda: Bernoulli('B', 1.5)) + raises(ValueError, lambda: Bernoulli('B', -0.5)) + + #issue 8248 + assert X.pspace.compute_expectation(1) == 1 + + p = Rational(1, 5) + X = Binomial('X', 5, p) + Y = Binomial('Y', 7, 2*p) + Z = Binomial('Z', 9, 3*p) + assert coskewness(Y + Z, X + Y, X + Z).simplify() == 0 + assert coskewness(Y + 2*X + Z, X + 2*Y + Z, X + 2*Z + Y).simplify() == \ + sqrt(1529)*Rational(12, 16819) + assert coskewness(Y + 2*X + Z, X + 2*Y + Z, X + 2*Z + Y, X < 2).simplify() \ + == -sqrt(357451121)*Rational(2812, 4646864573) + +def test_cdf(): + D = Die('D', 6) + o = S.One + + assert cdf( + D) == sympify({1: o/6, 2: o/3, 3: o/2, 4: 2*o/3, 5: 5*o/6, 6: o}) + + +def test_coins(): + C, D = Coin('C'), Coin('D') + H, T = symbols('H, T') + assert P(Eq(C, D)) == S.Half + assert density(Tuple(C, D)) == {(H, H): Rational(1, 4), (H, T): Rational(1, 4), + (T, H): Rational(1, 4), (T, T): Rational(1, 4)} + assert dict(density(C).items()) == {H: S.Half, T: S.Half} + + F = Coin('F', Rational(1, 10)) + assert P(Eq(F, H)) == Rational(1, 10) + + d = pspace(C).domain + + assert d.as_boolean() == Or(Eq(C.symbol, H), Eq(C.symbol, T)) + + raises(ValueError, lambda: P(C > D)) # Can't intelligently compare H to T + +def test_binomial_verify_parameters(): + raises(ValueError, lambda: Binomial('b', .2, .5)) + raises(ValueError, lambda: Binomial('b', 3, 1.5)) + +def test_binomial_numeric(): + nvals = range(5) + pvals = [0, Rational(1, 4), S.Half, Rational(3, 4), 1] + + for n in nvals: + for p in pvals: + X = Binomial('X', n, p) + assert E(X) == n*p + assert variance(X) == n*p*(1 - p) + if n > 0 and 0 < p < 1: + assert skewness(X) == (1 - 2*p)/sqrt(n*p*(1 - p)) + assert kurtosis(X) == 3 + (1 - 6*p*(1 - p))/(n*p*(1 - p)) + for k in range(n + 1): + assert P(Eq(X, k)) == binomial(n, k)*p**k*(1 - p)**(n - k) + +def test_binomial_quantile(): + X = Binomial('X', 50, S.Half) + assert quantile(X)(0.95) == S(31) + assert median(X) == FiniteSet(25) + + X = Binomial('X', 5, S.Half) + p = Symbol("p", positive=True) + assert quantile(X)(p) == Piecewise((nan, p > S.One), (S.Zero, p <= Rational(1, 32)),\ + (S.One, p <= Rational(3, 16)), (S(2), p <= S.Half), (S(3), p <= Rational(13, 16)),\ + (S(4), p <= Rational(31, 32)), (S(5), p <= S.One)) + assert median(X) == FiniteSet(2, 3) + + +def test_binomial_symbolic(): + n = 2 + p = symbols('p', positive=True) + X = Binomial('X', n, p) + t = Symbol('t') + + assert simplify(E(X)) == n*p == simplify(moment(X, 1)) + assert simplify(variance(X)) == n*p*(1 - p) == simplify(cmoment(X, 2)) + assert cancel(skewness(X) - (1 - 2*p)/sqrt(n*p*(1 - p))) == 0 + assert cancel((kurtosis(X)) - (3 + (1 - 6*p*(1 - p))/(n*p*(1 - p)))) == 0 + assert characteristic_function(X)(t) == p ** 2 * exp(2 * I * t) + 2 * p * (-p + 1) * exp(I * t) + (-p + 1) ** 2 + assert moment_generating_function(X)(t) == p ** 2 * exp(2 * t) + 2 * p * (-p + 1) * exp(t) + (-p + 1) ** 2 + + # Test ability to change success/failure winnings + H, T = symbols('H T') + Y = Binomial('Y', n, p, succ=H, fail=T) + assert simplify(E(Y) - (n*(H*p + T*(1 - p)))) == 0 + + # test symbolic dimensions + n = symbols('n') + B = Binomial('B', n, p) + raises(NotImplementedError, lambda: P(B > 2)) + assert density(B).dict == Density(BinomialDistribution(n, p, 1, 0)) + assert set(density(B).dict.subs(n, 4).doit().keys()) == \ + {S.Zero, S.One, S(2), S(3), S(4)} + assert set(density(B).dict.subs(n, 4).doit().values()) == \ + {(1 - p)**4, 4*p*(1 - p)**3, 6*p**2*(1 - p)**2, 4*p**3*(1 - p), p**4} + k = Dummy('k', integer=True) + assert E(B > 2).dummy_eq( + Sum(Piecewise((k*p**k*(1 - p)**(-k + n)*binomial(n, k), (k >= 0) + & (k <= n) & (k > 2)), (0, True)), (k, 0, n))) + +def test_beta_binomial(): + # verify parameters + raises(ValueError, lambda: BetaBinomial('b', .2, 1, 2)) + raises(ValueError, lambda: BetaBinomial('b', 2, -1, 2)) + raises(ValueError, lambda: BetaBinomial('b', 2, 1, -2)) + assert BetaBinomial('b', 2, 1, 1) + + # test numeric values + nvals = range(1,5) + alphavals = [Rational(1, 4), S.Half, Rational(3, 4), 1, 10] + betavals = [Rational(1, 4), S.Half, Rational(3, 4), 1, 10] + + for n in nvals: + for a in alphavals: + for b in betavals: + X = BetaBinomial('X', n, a, b) + assert E(X) == moment(X, 1) + assert variance(X) == cmoment(X, 2) + + # test symbolic + n, a, b = symbols('a b n') + assert BetaBinomial('x', n, a, b) + n = 2 # Because we're using for loops, can't do symbolic n + a, b = symbols('a b', positive=True) + X = BetaBinomial('X', n, a, b) + t = Symbol('t') + + assert E(X).expand() == moment(X, 1).expand() + assert variance(X).expand() == cmoment(X, 2).expand() + assert skewness(X) == smoment(X, 3) + assert characteristic_function(X)(t) == exp(2*I*t)*beta(a + 2, b)/beta(a, b) +\ + 2*exp(I*t)*beta(a + 1, b + 1)/beta(a, b) + beta(a, b + 2)/beta(a, b) + assert moment_generating_function(X)(t) == exp(2*t)*beta(a + 2, b)/beta(a, b) +\ + 2*exp(t)*beta(a + 1, b + 1)/beta(a, b) + beta(a, b + 2)/beta(a, b) + +def test_hypergeometric_numeric(): + for N in range(1, 5): + for m in range(0, N + 1): + for n in range(1, N + 1): + X = Hypergeometric('X', N, m, n) + N, m, n = map(sympify, (N, m, n)) + assert sum(density(X).values()) == 1 + assert E(X) == n * m / N + if N > 1: + assert variance(X) == n*(m/N)*(N - m)/N*(N - n)/(N - 1) + # Only test for skewness when defined + if N > 2 and 0 < m < N and n < N: + assert skewness(X) == simplify((N - 2*m)*sqrt(N - 1)*(N - 2*n) + / (sqrt(n*m*(N - m)*(N - n))*(N - 2))) + +def test_hypergeometric_symbolic(): + N, m, n = symbols('N, m, n') + H = Hypergeometric('H', N, m, n) + dens = density(H).dict + expec = E(H > 2) + assert dens == Density(HypergeometricDistribution(N, m, n)) + assert dens.subs(N, 5).doit() == Density(HypergeometricDistribution(5, m, n)) + assert set(dens.subs({N: 3, m: 2, n: 1}).doit().keys()) == {S.Zero, S.One} + assert set(dens.subs({N: 3, m: 2, n: 1}).doit().values()) == {Rational(1, 3), Rational(2, 3)} + k = Dummy('k', integer=True) + assert expec.dummy_eq( + Sum(Piecewise((k*binomial(m, k)*binomial(N - m, -k + n) + /binomial(N, n), k > 2), (0, True)), (k, 0, n))) + +def test_rademacher(): + X = Rademacher('X') + t = Symbol('t') + + assert E(X) == 0 + assert variance(X) == 1 + assert density(X)[-1] == S.Half + assert density(X)[1] == S.Half + assert characteristic_function(X)(t) == exp(I*t)/2 + exp(-I*t)/2 + assert moment_generating_function(X)(t) == exp(t) / 2 + exp(-t) / 2 + +def test_ideal_soliton(): + raises(ValueError, lambda : IdealSoliton('sol', -12)) + raises(ValueError, lambda : IdealSoliton('sol', 13.2)) + raises(ValueError, lambda : IdealSoliton('sol', 0)) + f = Function('f') + raises(ValueError, lambda : density(IdealSoliton('sol', 10)).pmf(f)) + + k = Symbol('k', integer=True, positive=True) + x = Symbol('x', integer=True, positive=True) + t = Symbol('t') + sol = IdealSoliton('sol', k) + assert density(sol).low == S.One + assert density(sol).high == k + assert density(sol).dict == Density(density(sol)) + assert density(sol).pmf(x) == Piecewise((1/k, Eq(x, 1)), (1/(x*(x - 1)), k >= x), (0, True)) + + k_vals = [5, 20, 50, 100, 1000] + for i in k_vals: + assert E(sol.subs(k, i)) == harmonic(i) == moment(sol.subs(k, i), 1) + assert variance(sol.subs(k, i)) == (i - 1) + harmonic(i) - harmonic(i)**2 == cmoment(sol.subs(k, i),2) + assert skewness(sol.subs(k, i)) == smoment(sol.subs(k, i), 3) + assert kurtosis(sol.subs(k, i)) == smoment(sol.subs(k, i), 4) + + assert exp(I*t)/10 + Sum(exp(I*t*x)/(x*x - x), (x, 2, k)).subs(k, 10).doit() == characteristic_function(sol.subs(k, 10))(t) + assert exp(t)/10 + Sum(exp(t*x)/(x*x - x), (x, 2, k)).subs(k, 10).doit() == moment_generating_function(sol.subs(k, 10))(t) + +def test_robust_soliton(): + raises(ValueError, lambda : RobustSoliton('robSol', -12, 0.1, 0.02)) + raises(ValueError, lambda : RobustSoliton('robSol', 13, 1.89, 0.1)) + raises(ValueError, lambda : RobustSoliton('robSol', 15, 0.6, -2.31)) + f = Function('f') + raises(ValueError, lambda : density(RobustSoliton('robSol', 15, 0.6, 0.1)).pmf(f)) + + k = Symbol('k', integer=True, positive=True) + delta = Symbol('delta', positive=True) + c = Symbol('c', positive=True) + robSol = RobustSoliton('robSol', k, delta, c) + assert density(robSol).low == 1 + assert density(robSol).high == k + + k_vals = [10, 20, 50] + delta_vals = [0.2, 0.4, 0.6] + c_vals = [0.01, 0.03, 0.05] + for x in k_vals: + for y in delta_vals: + for z in c_vals: + assert E(robSol.subs({k: x, delta: y, c: z})) == moment(robSol.subs({k: x, delta: y, c: z}), 1) + assert variance(robSol.subs({k: x, delta: y, c: z})) == cmoment(robSol.subs({k: x, delta: y, c: z}), 2) + assert skewness(robSol.subs({k: x, delta: y, c: z})) == smoment(robSol.subs({k: x, delta: y, c: z}), 3) + assert kurtosis(robSol.subs({k: x, delta: y, c: z})) == smoment(robSol.subs({k: x, delta: y, c: z}), 4) + +def test_FiniteRV(): + F = FiniteRV('F', {1: S.Half, 2: Rational(1, 4), 3: Rational(1, 4)}, check=True) + p = Symbol("p", positive=True) + + assert dict(density(F).items()) == {S.One: S.Half, S(2): Rational(1, 4), S(3): Rational(1, 4)} + assert P(F >= 2) == S.Half + assert quantile(F)(p) == Piecewise((nan, p > S.One), (S.One, p <= S.Half),\ + (S(2), p <= Rational(3, 4)),(S(3), True)) + + assert pspace(F).domain.as_boolean() == Or( + *[Eq(F.symbol, i) for i in [1, 2, 3]]) + + assert F.pspace.domain.set == FiniteSet(1, 2, 3) + raises(ValueError, lambda: FiniteRV('F', {1: S.Half, 2: S.Half, 3: S.Half}, check=True)) + raises(ValueError, lambda: FiniteRV('F', {1: S.Half, 2: Rational(-1, 2), 3: S.One}, check=True)) + raises(ValueError, lambda: FiniteRV('F', {1: S.One, 2: Rational(3, 2), 3: S.Zero,\ + 4: Rational(-1, 2), 5: Rational(-3, 4), 6: Rational(-1, 4)}, check=True)) + + # purposeful invalid pmf but it should not raise since check=False + # see test_drv_types.test_ContinuousRV for explanation + X = FiniteRV('X', {1: 1, 2: 2}) + assert E(X) == 5 + assert P(X <= 2) + P(X > 2) != 1 + +def test_density_call(): + from sympy.abc import p + x = Bernoulli('x', p) + d = density(x) + assert d(0) == 1 - p + assert d(S.Zero) == 1 - p + assert d(5) == 0 + + assert 0 in d + assert 5 not in d + assert d(S.Zero) == d[S.Zero] + + +def test_DieDistribution(): + from sympy.abc import x + X = DieDistribution(6) + assert X.pmf(S.Half) is S.Zero + assert X.pmf(x).subs({x: 1}).doit() == Rational(1, 6) + assert X.pmf(x).subs({x: 7}).doit() == 0 + assert X.pmf(x).subs({x: -1}).doit() == 0 + assert X.pmf(x).subs({x: Rational(1, 3)}).doit() == 0 + raises(ValueError, lambda: X.pmf(Matrix([0, 0]))) + raises(ValueError, lambda: X.pmf(x**2 - 1)) + +def test_FinitePSpace(): + X = Die('X', 6) + space = pspace(X) + assert space.density == DieDistribution(6) + +def test_symbolic_conditions(): + B = Bernoulli('B', Rational(1, 4)) + D = Die('D', 4) + b, n = symbols('b, n') + Y = P(Eq(B, b)) + Z = E(D > n) + assert Y == \ + Piecewise((Rational(1, 4), Eq(b, 1)), (0, True)) + \ + Piecewise((Rational(3, 4), Eq(b, 0)), (0, True)) + assert Z == \ + Piecewise((Rational(1, 4), n < 1), (0, True)) + Piecewise((S.Half, n < 2), (0, True)) + \ + Piecewise((Rational(3, 4), n < 3), (0, True)) + Piecewise((S.One, n < 4), (0, True)) diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_joint_rv.py b/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_joint_rv.py new file mode 100644 index 0000000000000000000000000000000000000000..057fc313dfbb31826b07fd1315205d22b86a7f96 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_joint_rv.py @@ -0,0 +1,436 @@ +from sympy.concrete.products import Product +from sympy.concrete.summations import Sum +from sympy.core.numbers import (Rational, oo, pi) +from sympy.core.relational import Eq +from sympy.core.singleton import S +from sympy.core.symbol import symbols +from sympy.functions.combinatorial.factorials import (RisingFactorial, factorial) +from sympy.functions.elementary.complexes import polar_lift +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.special.bessel import besselk +from sympy.functions.special.gamma_functions import gamma +from sympy.matrices.dense import eye +from sympy.matrices.expressions.determinant import Determinant +from sympy.sets.fancysets import Range +from sympy.sets.sets import (Interval, ProductSet) +from sympy.simplify.simplify import simplify +from sympy.tensor.indexed import (Indexed, IndexedBase) +from sympy.core.numbers import comp +from sympy.integrals.integrals import integrate +from sympy.matrices import Matrix, MatrixSymbol +from sympy.matrices.expressions.matexpr import MatrixElement +from sympy.stats import density, median, marginal_distribution, Normal, Laplace, E, sample +from sympy.stats.joint_rv_types import (JointRV, MultivariateNormalDistribution, + JointDistributionHandmade, MultivariateT, NormalGamma, + GeneralizedMultivariateLogGammaOmega as GMVLGO, MultivariateBeta, + GeneralizedMultivariateLogGamma as GMVLG, MultivariateEwens, + Multinomial, NegativeMultinomial, MultivariateNormal, + MultivariateLaplace) +from sympy.testing.pytest import raises, XFAIL, skip, slow +from sympy.external import import_module + +from sympy.abc import x, y + + + +def test_Normal(): + m = Normal('A', [1, 2], [[1, 0], [0, 1]]) + A = MultivariateNormal('A', [1, 2], [[1, 0], [0, 1]]) + assert m == A + assert density(m)(1, 2) == 1/(2*pi) + assert m.pspace.distribution.set == ProductSet(S.Reals, S.Reals) + raises (ValueError, lambda:m[2]) + n = Normal('B', [1, 2, 3], [[1, 0, 0], [0, 1, 0], [0, 0, 1]]) + p = Normal('C', Matrix([1, 2]), Matrix([[1, 0], [0, 1]])) + assert density(m)(x, y) == density(p)(x, y) + assert marginal_distribution(n, 0, 1)(1, 2) == 1/(2*pi) + raises(ValueError, lambda: marginal_distribution(m)) + assert integrate(density(m)(x, y), (x, -oo, oo), (y, -oo, oo)).evalf() == 1.0 + N = Normal('N', [1, 2], [[x, 0], [0, y]]) + assert density(N)(0, 0) == exp(-((4*x + y)/(2*x*y)))/(2*pi*sqrt(x*y)) + + raises (ValueError, lambda: Normal('M', [1, 2], [[1, 1], [1, -1]])) + # symbolic + n = symbols('n', integer=True, positive=True) + mu = MatrixSymbol('mu', n, 1) + sigma = MatrixSymbol('sigma', n, n) + X = Normal('X', mu, sigma) + assert density(X) == MultivariateNormalDistribution(mu, sigma) + raises (NotImplementedError, lambda: median(m)) + # Below tests should work after issue #17267 is resolved + # assert E(X) == mu + # assert variance(X) == sigma + + # test symbolic multivariate normal densities + n = 3 + + Sg = MatrixSymbol('Sg', n, n) + mu = MatrixSymbol('mu', n, 1) + obs = MatrixSymbol('obs', n, 1) + + X = MultivariateNormal('X', mu, Sg) + density_X = density(X) + + eval_a = density_X(obs).subs({Sg: eye(3), + mu: Matrix([0, 0, 0]), obs: Matrix([0, 0, 0])}).doit() + eval_b = density_X(0, 0, 0).subs({Sg: eye(3), mu: Matrix([0, 0, 0])}).doit() + + assert eval_a == sqrt(2)/(4*pi**Rational(3/2)) + assert eval_b == sqrt(2)/(4*pi**Rational(3/2)) + + n = symbols('n', integer=True, positive=True) + + Sg = MatrixSymbol('Sg', n, n) + mu = MatrixSymbol('mu', n, 1) + obs = MatrixSymbol('obs', n, 1) + + X = MultivariateNormal('X', mu, Sg) + density_X_at_obs = density(X)(obs) + + expected_density = MatrixElement( + exp((S(1)/2) * (mu.T - obs.T) * Sg**(-1) * (-mu + obs)) / \ + sqrt((2*pi)**n * Determinant(Sg)), 0, 0) + + assert density_X_at_obs == expected_density + + +def test_MultivariateTDist(): + t1 = MultivariateT('T', [0, 0], [[1, 0], [0, 1]], 2) + assert(density(t1))(1, 1) == 1/(8*pi) + assert t1.pspace.distribution.set == ProductSet(S.Reals, S.Reals) + assert integrate(density(t1)(x, y), (x, -oo, oo), \ + (y, -oo, oo)).evalf() == 1.0 + raises(ValueError, lambda: MultivariateT('T', [1, 2], [[1, 1], [1, -1]], 1)) + t2 = MultivariateT('t2', [1, 2], [[x, 0], [0, y]], 1) + assert density(t2)(1, 2) == 1/(2*pi*sqrt(x*y)) + + +def test_multivariate_laplace(): + raises(ValueError, lambda: Laplace('T', [1, 2], [[1, 2], [2, 1]])) + L = Laplace('L', [1, 0], [[1, 0], [0, 1]]) + L2 = MultivariateLaplace('L2', [1, 0], [[1, 0], [0, 1]]) + assert density(L)(2, 3) == exp(2)*besselk(0, sqrt(39))/pi + L1 = Laplace('L1', [1, 2], [[x, 0], [0, y]]) + assert density(L1)(0, 1) == \ + exp(2/y)*besselk(0, sqrt((2 + 4/y + 1/x)/y))/(pi*sqrt(x*y)) + assert L.pspace.distribution.set == ProductSet(S.Reals, S.Reals) + assert L.pspace.distribution == L2.pspace.distribution + + +def test_NormalGamma(): + ng = NormalGamma('G', 1, 2, 3, 4) + assert density(ng)(1, 1) == 32*exp(-4)/sqrt(pi) + assert ng.pspace.distribution.set == ProductSet(S.Reals, Interval(0, oo)) + raises(ValueError, lambda:NormalGamma('G', 1, 2, 3, -1)) + assert marginal_distribution(ng, 0)(1) == \ + 3*sqrt(10)*gamma(Rational(7, 4))/(10*sqrt(pi)*gamma(Rational(5, 4))) + assert marginal_distribution(ng, y)(1) == exp(Rational(-1, 4))/128 + assert marginal_distribution(ng,[0,1])(x) == x**2*exp(-x/4)/128 + + +def test_GeneralizedMultivariateLogGammaDistribution(): + h = S.Half + omega = Matrix([[1, h, h, h], + [h, 1, h, h], + [h, h, 1, h], + [h, h, h, 1]]) + v, l, mu = (4, [1, 2, 3, 4], [1, 2, 3, 4]) + y_1, y_2, y_3, y_4 = symbols('y_1:5', real=True) + delta = symbols('d', positive=True) + G = GMVLGO('G', omega, v, l, mu) + Gd = GMVLG('Gd', delta, v, l, mu) + dend = ("d**4*Sum(4*24**(-n - 4)*(1 - d)**n*exp((n + 4)*(y_1 + 2*y_2 + 3*y_3 " + "+ 4*y_4) - exp(y_1) - exp(2*y_2)/2 - exp(3*y_3)/3 - exp(4*y_4)/4)/" + "(gamma(n + 1)*gamma(n + 4)**3), (n, 0, oo))") + assert str(density(Gd)(y_1, y_2, y_3, y_4)) == dend + den = ("5*2**(2/3)*5**(1/3)*Sum(4*24**(-n - 4)*(-2**(2/3)*5**(1/3)/4 + 1)**n*" + "exp((n + 4)*(y_1 + 2*y_2 + 3*y_3 + 4*y_4) - exp(y_1) - exp(2*y_2)/2 - " + "exp(3*y_3)/3 - exp(4*y_4)/4)/(gamma(n + 1)*gamma(n + 4)**3), (n, 0, oo))/64") + assert str(density(G)(y_1, y_2, y_3, y_4)) == den + marg = ("5*2**(2/3)*5**(1/3)*exp(4*y_1)*exp(-exp(y_1))*Integral(exp(-exp(4*G[3])" + "/4)*exp(16*G[3])*Integral(exp(-exp(3*G[2])/3)*exp(12*G[2])*Integral(exp(" + "-exp(2*G[1])/2)*exp(8*G[1])*Sum((-1/4)**n*(-4 + 2**(2/3)*5**(1/3" + "))**n*exp(n*y_1)*exp(2*n*G[1])*exp(3*n*G[2])*exp(4*n*G[3])/(24**n*gamma(n + 1)" + "*gamma(n + 4)**3), (n, 0, oo)), (G[1], -oo, oo)), (G[2], -oo, oo)), (G[3]" + ", -oo, oo))/5308416") + assert str(marginal_distribution(G, G[0])(y_1)) == marg + omega_f1 = Matrix([[1, h, h]]) + omega_f2 = Matrix([[1, h, h, h], + [h, 1, 2, h], + [h, h, 1, h], + [h, h, h, 1]]) + omega_f3 = Matrix([[6, h, h, h], + [h, 1, 2, h], + [h, h, 1, h], + [h, h, h, 1]]) + v_f = symbols("v_f", positive=False, real=True) + l_f = [1, 2, v_f, 4] + m_f = [v_f, 2, 3, 4] + omega_f4 = Matrix([[1, h, h, h, h], + [h, 1, h, h, h], + [h, h, 1, h, h], + [h, h, h, 1, h], + [h, h, h, h, 1]]) + l_f1 = [1, 2, 3, 4, 5] + omega_f5 = Matrix([[1]]) + mu_f5 = l_f5 = [1] + + raises(ValueError, lambda: GMVLGO('G', omega_f1, v, l, mu)) + raises(ValueError, lambda: GMVLGO('G', omega_f2, v, l, mu)) + raises(ValueError, lambda: GMVLGO('G', omega_f3, v, l, mu)) + raises(ValueError, lambda: GMVLGO('G', omega, v_f, l, mu)) + raises(ValueError, lambda: GMVLGO('G', omega, v, l_f, mu)) + raises(ValueError, lambda: GMVLGO('G', omega, v, l, m_f)) + raises(ValueError, lambda: GMVLGO('G', omega_f4, v, l, mu)) + raises(ValueError, lambda: GMVLGO('G', omega, v, l_f1, mu)) + raises(ValueError, lambda: GMVLGO('G', omega_f5, v, l_f5, mu_f5)) + raises(ValueError, lambda: GMVLG('G', Rational(3, 2), v, l, mu)) + + +def test_MultivariateBeta(): + a1, a2 = symbols('a1, a2', positive=True) + a1_f, a2_f = symbols('a1, a2', positive=False, real=True) + mb = MultivariateBeta('B', [a1, a2]) + mb_c = MultivariateBeta('C', a1, a2) + assert density(mb)(1, 2) == S(2)**(a2 - 1)*gamma(a1 + a2)/\ + (gamma(a1)*gamma(a2)) + assert marginal_distribution(mb_c, 0)(3) == S(3)**(a1 - 1)*gamma(a1 + a2)/\ + (a2*gamma(a1)*gamma(a2)) + raises(ValueError, lambda: MultivariateBeta('b1', [a1_f, a2])) + raises(ValueError, lambda: MultivariateBeta('b2', [a1, a2_f])) + raises(ValueError, lambda: MultivariateBeta('b3', [0, 0])) + raises(ValueError, lambda: MultivariateBeta('b4', [a1_f, a2_f])) + assert mb.pspace.distribution.set == ProductSet(Interval(0, 1), Interval(0, 1)) + + +def test_MultivariateEwens(): + n, theta, i = symbols('n theta i', positive=True) + + # tests for integer dimensions + theta_f = symbols('t_f', negative=True) + a = symbols('a_1:4', positive = True, integer = True) + ed = MultivariateEwens('E', 3, theta) + assert density(ed)(a[0], a[1], a[2]) == Piecewise((6*2**(-a[1])*3**(-a[2])* + theta**a[0]*theta**a[1]*theta**a[2]/ + (theta*(theta + 1)*(theta + 2)* + factorial(a[0])*factorial(a[1])* + factorial(a[2])), Eq(a[0] + 2*a[1] + + 3*a[2], 3)), (0, True)) + assert marginal_distribution(ed, ed[1])(a[1]) == Piecewise((6*2**(-a[1])* + theta**a[1]/((theta + 1)* + (theta + 2)*factorial(a[1])), + Eq(2*a[1] + 1, 3)), (0, True)) + raises(ValueError, lambda: MultivariateEwens('e1', 5, theta_f)) + assert ed.pspace.distribution.set == ProductSet(Range(0, 4, 1), + Range(0, 2, 1), Range(0, 2, 1)) + + # tests for symbolic dimensions + eds = MultivariateEwens('E', n, theta) + a = IndexedBase('a') + j, k = symbols('j, k') + den = Piecewise((factorial(n)*Product(theta**a[j]*(j + 1)**(-a[j])/ + factorial(a[j]), (j, 0, n - 1))/RisingFactorial(theta, n), + Eq(n, Sum((k + 1)*a[k], (k, 0, n - 1)))), (0, True)) + assert density(eds)(a).dummy_eq(den) + + +def test_Multinomial(): + n, x1, x2, x3, x4 = symbols('n, x1, x2, x3, x4', nonnegative=True, integer=True) + p1, p2, p3, p4 = symbols('p1, p2, p3, p4', positive=True) + p1_f, n_f = symbols('p1_f, n_f', negative=True) + M = Multinomial('M', n, [p1, p2, p3, p4]) + C = Multinomial('C', 3, p1, p2, p3) + f = factorial + assert density(M)(x1, x2, x3, x4) == Piecewise((p1**x1*p2**x2*p3**x3*p4**x4* + f(n)/(f(x1)*f(x2)*f(x3)*f(x4)), + Eq(n, x1 + x2 + x3 + x4)), (0, True)) + assert marginal_distribution(C, C[0])(x1).subs(x1, 1) ==\ + 3*p1*p2**2 +\ + 6*p1*p2*p3 +\ + 3*p1*p3**2 + raises(ValueError, lambda: Multinomial('b1', 5, [p1, p2, p3, p1_f])) + raises(ValueError, lambda: Multinomial('b2', n_f, [p1, p2, p3, p4])) + raises(ValueError, lambda: Multinomial('b3', n, 0.5, 0.4, 0.3, 0.1)) + + +def test_NegativeMultinomial(): + k0, x1, x2, x3, x4 = symbols('k0, x1, x2, x3, x4', nonnegative=True, integer=True) + p1, p2, p3, p4 = symbols('p1, p2, p3, p4', positive=True) + p1_f = symbols('p1_f', negative=True) + N = NegativeMultinomial('N', 4, [p1, p2, p3, p4]) + C = NegativeMultinomial('C', 4, 0.1, 0.2, 0.3) + g = gamma + f = factorial + assert simplify(density(N)(x1, x2, x3, x4) - + p1**x1*p2**x2*p3**x3*p4**x4*(-p1 - p2 - p3 - p4 + 1)**4*g(x1 + x2 + + x3 + x4 + 4)/(6*f(x1)*f(x2)*f(x3)*f(x4))) is S.Zero + assert comp(marginal_distribution(C, C[0])(1).evalf(), 0.33, .01) + raises(ValueError, lambda: NegativeMultinomial('b1', 5, [p1, p2, p3, p1_f])) + raises(ValueError, lambda: NegativeMultinomial('b2', k0, 0.5, 0.4, 0.3, 0.4)) + assert N.pspace.distribution.set == ProductSet(Range(0, oo, 1), + Range(0, oo, 1), Range(0, oo, 1), Range(0, oo, 1)) + + +@slow +def test_JointPSpace_marginal_distribution(): + T = MultivariateT('T', [0, 0], [[1, 0], [0, 1]], 2) + got = marginal_distribution(T, T[1])(x) + ans = sqrt(2)*(x**2/2 + 1)/(4*polar_lift(x**2/2 + 1)**(S(5)/2)) + assert got == ans, got + assert integrate(marginal_distribution(T, 1)(x), (x, -oo, oo)) == 1 + + t = MultivariateT('T', [0, 0, 0], [[1, 0, 0], [0, 1, 0], [0, 0, 1]], 3) + assert comp(marginal_distribution(t, 0)(1).evalf(), 0.2, .01) + + +def test_JointRV(): + x1, x2 = (Indexed('x', i) for i in (1, 2)) + pdf = exp(-x1**2/2 + x1 - x2**2/2 - S.Half)/(2*pi) + X = JointRV('x', pdf) + assert density(X)(1, 2) == exp(-2)/(2*pi) + assert isinstance(X.pspace.distribution, JointDistributionHandmade) + assert marginal_distribution(X, 0)(2) == sqrt(2)*exp(Rational(-1, 2))/(2*sqrt(pi)) + + +def test_expectation(): + m = Normal('A', [x, y], [[1, 0], [0, 1]]) + assert simplify(E(m[1])) == y + + +@XFAIL +def test_joint_vector_expectation(): + m = Normal('A', [x, y], [[1, 0], [0, 1]]) + assert E(m) == (x, y) + + +def test_sample_numpy(): + distribs_numpy = [ + MultivariateNormal("M", [3, 4], [[2, 1], [1, 2]]), + MultivariateBeta("B", [0.4, 5, 15, 50, 203]), + Multinomial("N", 50, [0.3, 0.2, 0.1, 0.25, 0.15]) + ] + size = 3 + numpy = import_module('numpy') + if not numpy: + skip('Numpy is not installed. Abort tests for _sample_numpy.') + else: + for X in distribs_numpy: + samps = sample(X, size=size, library='numpy') + for sam in samps: + assert tuple(sam) in X.pspace.distribution.set + N_c = NegativeMultinomial('N', 3, 0.1, 0.1, 0.1) + raises(NotImplementedError, lambda: sample(N_c, library='numpy')) + + +def test_sample_scipy(): + distribs_scipy = [ + MultivariateNormal("M", [0, 0], [[0.1, 0.025], [0.025, 0.1]]), + MultivariateBeta("B", [0.4, 5, 15]), + Multinomial("N", 8, [0.3, 0.2, 0.1, 0.4]) + ] + + size = 3 + scipy = import_module('scipy') + if not scipy: + skip('Scipy not installed. Abort tests for _sample_scipy.') + else: + for X in distribs_scipy: + samps = sample(X, size=size) + samps2 = sample(X, size=(2, 2)) + for sam in samps: + assert tuple(sam) in X.pspace.distribution.set + for i in range(2): + for j in range(2): + assert tuple(samps2[i][j]) in X.pspace.distribution.set + N_c = NegativeMultinomial('N', 3, 0.1, 0.1, 0.1) + raises(NotImplementedError, lambda: sample(N_c)) + + +def test_sample_pymc(): + distribs_pymc = [ + MultivariateNormal("M", [5, 2], [[1, 0], [0, 1]]), + MultivariateBeta("B", [0.4, 5, 15]), + Multinomial("N", 4, [0.3, 0.2, 0.1, 0.4]) + ] + size = 3 + pymc = import_module('pymc') + if not pymc: + skip('PyMC is not installed. Abort tests for _sample_pymc.') + else: + for X in distribs_pymc: + samps = sample(X, size=size, library='pymc') + for sam in samps: + assert tuple(sam.flatten()) in X.pspace.distribution.set + N_c = NegativeMultinomial('N', 3, 0.1, 0.1, 0.1) + raises(NotImplementedError, lambda: sample(N_c, library='pymc')) + + +def test_sample_seed(): + x1, x2 = (Indexed('x', i) for i in (1, 2)) + pdf = exp(-x1**2/2 + x1 - x2**2/2 - S.Half)/(2*pi) + X = JointRV('x', pdf) + + libraries = ['scipy', 'numpy', 'pymc'] + for lib in libraries: + try: + imported_lib = import_module(lib) + if imported_lib: + s0, s1, s2 = [], [], [] + s0 = sample(X, size=10, library=lib, seed=0) + s1 = sample(X, size=10, library=lib, seed=0) + s2 = sample(X, size=10, library=lib, seed=1) + assert all(s0 == s1) + assert all(s1 != s2) + except NotImplementedError: + continue + +# +# XXX: This fails for pymc. Previously the test appeared to pass but that is +# just because the library argument was not passed so the test always used +# scipy. +# +def test_issue_21057(): + m = Normal("x", [0, 0], [[0, 0], [0, 0]]) + n = MultivariateNormal("x", [0, 0], [[0, 0], [0, 0]]) + p = Normal("x", [0, 0], [[0, 0], [0, 1]]) + assert m == n + libraries = ('scipy', 'numpy') # , 'pymc') # <-- pymc fails + for library in libraries: + try: + imported_lib = import_module(library) + if imported_lib: + s1 = sample(m, size=8, library=library) + s2 = sample(n, size=8, library=library) + s3 = sample(p, size=8, library=library) + assert tuple(s1.flatten()) == tuple(s2.flatten()) + for s in s3: + assert tuple(s.flatten()) in p.pspace.distribution.set + except NotImplementedError: + continue + + +# +# When this passes the pymc part can be uncommented in test_issue_21057 above +# and this can be deleted. +# +@XFAIL +def test_issue_21057_pymc(): + m = Normal("x", [0, 0], [[0, 0], [0, 0]]) + n = MultivariateNormal("x", [0, 0], [[0, 0], [0, 0]]) + p = Normal("x", [0, 0], [[0, 0], [0, 1]]) + assert m == n + libraries = ('pymc',) + for library in libraries: + try: + imported_lib = import_module(library) + if imported_lib: + s1 = sample(m, size=8, library=library) + s2 = sample(n, size=8, library=library) + s3 = sample(p, size=8, library=library) + assert tuple(s1.flatten()) == tuple(s2.flatten()) + for s in s3: + assert tuple(s.flatten()) in p.pspace.distribution.set + except NotImplementedError: + continue diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_matrix_distributions.py b/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_matrix_distributions.py new file mode 100644 index 0000000000000000000000000000000000000000..a2a2dcdd853793d9f77e1a88adf63158ed68e3ba --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_matrix_distributions.py @@ -0,0 +1,186 @@ +from sympy.concrete.products import Product +from sympy.core.numbers import pi +from sympy.core.singleton import S +from sympy.core.symbol import (Dummy, symbols) +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.special.gamma_functions import gamma +from sympy.matrices import Determinant, Matrix, Trace, MatrixSymbol, MatrixSet +from sympy.stats import density, sample +from sympy.stats.matrix_distributions import (MatrixGammaDistribution, + MatrixGamma, MatrixPSpace, Wishart, MatrixNormal, MatrixStudentT) +from sympy.testing.pytest import raises, skip +from sympy.external import import_module + + +def test_MatrixPSpace(): + M = MatrixGammaDistribution(1, 2, [[2, 1], [1, 2]]) + MP = MatrixPSpace('M', M, 2, 2) + assert MP.distribution == M + raises(ValueError, lambda: MatrixPSpace('M', M, 1.2, 2)) + +def test_MatrixGamma(): + M = MatrixGamma('M', 1, 2, [[1, 0], [0, 1]]) + assert M.pspace.distribution.set == MatrixSet(2, 2, S.Reals) + assert isinstance(density(M), MatrixGammaDistribution) + X = MatrixSymbol('X', 2, 2) + num = exp(Trace(Matrix([[-S(1)/2, 0], [0, -S(1)/2]])*X)) + assert density(M)(X).doit() == num/(4*pi*sqrt(Determinant(X))) + assert density(M)([[2, 1], [1, 2]]).doit() == sqrt(3)*exp(-2)/(12*pi) + X = MatrixSymbol('X', 1, 2) + Y = MatrixSymbol('Y', 1, 2) + assert density(M)([X, Y]).doit() == exp(-X[0, 0]/2 - Y[0, 1]/2)/(4*pi*sqrt( + X[0, 0]*Y[0, 1] - X[0, 1]*Y[0, 0])) + # symbolic + a, b = symbols('a b', positive=True) + d = symbols('d', positive=True, integer=True) + Y = MatrixSymbol('Y', d, d) + Z = MatrixSymbol('Z', 2, 2) + SM = MatrixSymbol('SM', d, d) + M2 = MatrixGamma('M2', a, b, SM) + M3 = MatrixGamma('M3', 2, 3, [[2, 1], [1, 2]]) + k = Dummy('k') + exprd = pi**(-d*(d - 1)/4)*b**(-a*d)*exp(Trace((-1/b)*SM**(-1)*Y) + )*Determinant(SM)**(-a)*Determinant(Y)**(a - d/2 - S(1)/2)/Product( + gamma(-k/2 + a + S(1)/2), (k, 1, d)) + assert density(M2)(Y).dummy_eq(exprd) + raises(NotImplementedError, lambda: density(M3 + M)(Z)) + raises(ValueError, lambda: density(M)(1)) + raises(ValueError, lambda: MatrixGamma('M', -1, 2, [[1, 0], [0, 1]])) + raises(ValueError, lambda: MatrixGamma('M', -1, -2, [[1, 0], [0, 1]])) + raises(ValueError, lambda: MatrixGamma('M', -1, 2, [[1, 0], [2, 1]])) + raises(ValueError, lambda: MatrixGamma('M', -1, 2, [[1, 0], [0]])) + +def test_Wishart(): + W = Wishart('W', 5, [[1, 0], [0, 1]]) + assert W.pspace.distribution.set == MatrixSet(2, 2, S.Reals) + X = MatrixSymbol('X', 2, 2) + term1 = exp(Trace(Matrix([[-S(1)/2, 0], [0, -S(1)/2]])*X)) + assert density(W)(X).doit() == term1 * Determinant(X)/(24*pi) + assert density(W)([[2, 1], [1, 2]]).doit() == exp(-2)/(8*pi) + n = symbols('n', positive=True) + d = symbols('d', positive=True, integer=True) + Y = MatrixSymbol('Y', d, d) + SM = MatrixSymbol('SM', d, d) + W = Wishart('W', n, SM) + k = Dummy('k') + exprd = 2**(-d*n/2)*pi**(-d*(d - 1)/4)*exp(Trace(-(S(1)/2)*SM**(-1)*Y) + )*Determinant(SM)**(-n/2)*Determinant(Y)**( + -d/2 + n/2 - S(1)/2)/Product(gamma(-k/2 + n/2 + S(1)/2), (k, 1, d)) + assert density(W)(Y).dummy_eq(exprd) + raises(ValueError, lambda: density(W)(1)) + raises(ValueError, lambda: Wishart('W', -1, [[1, 0], [0, 1]])) + raises(ValueError, lambda: Wishart('W', -1, [[1, 0], [2, 1]])) + raises(ValueError, lambda: Wishart('W', 2, [[1, 0], [0]])) + +def test_MatrixNormal(): + M = MatrixNormal('M', [[5, 6]], [4], [[2, 1], [1, 2]]) + assert M.pspace.distribution.set == MatrixSet(1, 2, S.Reals) + X = MatrixSymbol('X', 1, 2) + term1 = exp(-Trace(Matrix([[ S(2)/3, -S(1)/3], [-S(1)/3, S(2)/3]])*( + Matrix([[-5], [-6]]) + X.T)*Matrix([[S(1)/4]])*(Matrix([[-5, -6]]) + X))/2) + assert density(M)(X).doit() == (sqrt(3)) * term1/(24*pi) + assert density(M)([[7, 8]]).doit() == sqrt(3)*exp(-S(1)/3)/(24*pi) + d, n = symbols('d n', positive=True, integer=True) + SM2 = MatrixSymbol('SM2', d, d) + SM1 = MatrixSymbol('SM1', n, n) + LM = MatrixSymbol('LM', n, d) + Y = MatrixSymbol('Y', n, d) + M = MatrixNormal('M', LM, SM1, SM2) + exprd = (2*pi)**(-d*n/2)*exp(-Trace(SM2**(-1)*(-LM.T + Y.T)*SM1**(-1)*(-LM + Y) + )/2)*Determinant(SM1)**(-d/2)*Determinant(SM2)**(-n/2) + assert density(M)(Y).doit() == exprd + raises(ValueError, lambda: density(M)(1)) + raises(ValueError, lambda: MatrixNormal('M', [1, 2], [[1, 0], [0, 1]], [[1, 0], [2, 1]])) + raises(ValueError, lambda: MatrixNormal('M', [1, 2], [[1, 0], [2, 1]], [[1, 0], [0, 1]])) + raises(ValueError, lambda: MatrixNormal('M', [1, 2], [[1, 0], [0, 1]], [[1, 0], [0, 1]])) + raises(ValueError, lambda: MatrixNormal('M', [1, 2], [[1, 0], [2]], [[1, 0], [0, 1]])) + raises(ValueError, lambda: MatrixNormal('M', [1, 2], [[1, 0], [2, 1]], [[1, 0], [0]])) + raises(ValueError, lambda: MatrixNormal('M', [[1, 2]], [[1, 0], [0, 1]], [[1, 0]])) + raises(ValueError, lambda: MatrixNormal('M', [[1, 2]], [1], [[1, 0]])) + +def test_MatrixStudentT(): + M = MatrixStudentT('M', 2, [[5, 6]], [[2, 1], [1, 2]], [4]) + assert M.pspace.distribution.set == MatrixSet(1, 2, S.Reals) + X = MatrixSymbol('X', 1, 2) + D = pi ** (-1.0) * Determinant(Matrix([[4]])) ** (-1.0) * Determinant(Matrix([[2, 1], [1, 2]])) \ + ** (-0.5) / Determinant(Matrix([[S(1) / 4]]) * (Matrix([[-5, -6]]) + X) + * Matrix([[S(2) / 3, -S(1) / 3], [-S(1) / 3, S(2) / 3]]) * ( + Matrix([[-5], [-6]]) + X.T) + Matrix([[1]])) ** 2 + assert density(M)(X) == D + + v = symbols('v', positive=True) + n, p = 1, 2 + Omega = MatrixSymbol('Omega', p, p) + Sigma = MatrixSymbol('Sigma', n, n) + Location = MatrixSymbol('Location', n, p) + Y = MatrixSymbol('Y', n, p) + M = MatrixStudentT('M', v, Location, Omega, Sigma) + + exprd = gamma(v/2 + 1)*Determinant(Matrix([[1]]) + Sigma**(-1)*(-Location + Y)*Omega**(-1)*(-Location.T + Y.T))**(-v/2 - 1) / \ + (pi*gamma(v/2)*sqrt(Determinant(Omega))*Determinant(Sigma)) + + assert density(M)(Y) == exprd + raises(ValueError, lambda: density(M)(1)) + raises(ValueError, lambda: MatrixStudentT('M', 1, [1, 2], [[1, 0], [0, 1]], [[1, 0], [2, 1]])) + raises(ValueError, lambda: MatrixStudentT('M', 1, [1, 2], [[1, 0], [2, 1]], [[1, 0], [0, 1]])) + raises(ValueError, lambda: MatrixStudentT('M', 1, [1, 2], [[1, 0], [0, 1]], [[1, 0], [0, 1]])) + raises(ValueError, lambda: MatrixStudentT('M', 1, [1, 2], [[1, 0], [2]], [[1, 0], [0, 1]])) + raises(ValueError, lambda: MatrixStudentT('M', 1, [1, 2], [[1, 0], [2, 1]], [[1], [2]])) + raises(ValueError, lambda: MatrixStudentT('M', 1, [[1, 2]], [[1, 0], [0, 1]], [[1, 0]])) + raises(ValueError, lambda: MatrixStudentT('M', 1, [[1, 2]], [1], [[1, 0]])) + raises(ValueError, lambda: MatrixStudentT('M', -1, [1, 2], [[1, 0], [0, 1]], [4])) + +def test_sample_scipy(): + distribs_scipy = [ + MatrixNormal('M', [[5, 6]], [4], [[2, 1], [1, 2]]), + Wishart('W', 5, [[1, 0], [0, 1]]) + ] + + size = 5 + scipy = import_module('scipy') + if not scipy: + skip('Scipy not installed. Abort tests for _sample_scipy.') + else: + for X in distribs_scipy: + samps = sample(X, size=size) + for sam in samps: + assert Matrix(sam) in X.pspace.distribution.set + M = MatrixGamma('M', 1, 2, [[1, 0], [0, 1]]) + raises(NotImplementedError, lambda: sample(M, size=3)) + +def test_sample_pymc(): + distribs_pymc = [ + MatrixNormal('M', [[5, 6], [3, 4]], [[1, 0], [0, 1]], [[2, 1], [1, 2]]), + Wishart('W', 7, [[2, 1], [1, 2]]) + ] + size = 3 + pymc = import_module('pymc') + if not pymc: + skip('PyMC is not installed. Abort tests for _sample_pymc.') + else: + for X in distribs_pymc: + samps = sample(X, size=size, library='pymc') + for sam in samps: + assert Matrix(sam) in X.pspace.distribution.set + M = MatrixGamma('M', 1, 2, [[1, 0], [0, 1]]) + raises(NotImplementedError, lambda: sample(M, size=3)) + +def test_sample_seed(): + X = MatrixNormal('M', [[5, 6], [3, 4]], [[1, 0], [0, 1]], [[2, 1], [1, 2]]) + + libraries = ['scipy', 'numpy', 'pymc'] + for lib in libraries: + try: + imported_lib = import_module(lib) + if imported_lib: + s0, s1, s2 = [], [], [] + s0 = sample(X, size=10, library=lib, seed=0) + s1 = sample(X, size=10, library=lib, seed=0) + s2 = sample(X, size=10, library=lib, seed=1) + for i in range(10): + assert (s0[i] == s1[i]).all() + assert (s1[i] != s2[i]).all() + + except NotImplementedError: + continue diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_mix.py b/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_mix.py new file mode 100644 index 0000000000000000000000000000000000000000..4334d9b144a5ddaad938f195f0276e0e8993aa35 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_mix.py @@ -0,0 +1,82 @@ +from sympy.concrete.summations import Sum +from sympy.core.add import Add +from sympy.core.mul import Mul +from sympy.core.numbers import (Integer, oo, pi) +from sympy.core.power import Pow +from sympy.core.relational import (Eq, Ne) +from sympy.core.symbol import (Dummy, Symbol, symbols) +from sympy.functions.combinatorial.factorials import factorial +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.special.delta_functions import DiracDelta +from sympy.functions.special.gamma_functions import gamma +from sympy.integrals.integrals import Integral +from sympy.simplify.simplify import simplify +from sympy.tensor.indexed import (Indexed, IndexedBase) +from sympy.functions.elementary.piecewise import ExprCondPair +from sympy.stats import (Poisson, Beta, Exponential, P, + Multinomial, MultivariateBeta) +from sympy.stats.crv_types import Normal +from sympy.stats.drv_types import PoissonDistribution +from sympy.stats.compound_rv import CompoundPSpace, CompoundDistribution +from sympy.stats.joint_rv import MarginalDistribution +from sympy.stats.rv import pspace, density +from sympy.testing.pytest import ignore_warnings + +def test_density(): + x = Symbol('x') + l = Symbol('l', positive=True) + rate = Beta(l, 2, 3) + X = Poisson(x, rate) + assert isinstance(pspace(X), CompoundPSpace) + assert density(X, Eq(rate, rate.symbol)) == PoissonDistribution(l) + N1 = Normal('N1', 0, 1) + N2 = Normal('N2', N1, 2) + assert density(N2)(0).doit() == sqrt(10)/(10*sqrt(pi)) + assert simplify(density(N2, Eq(N1, 1))(x)) == \ + sqrt(2)*exp(-(x - 1)**2/8)/(4*sqrt(pi)) + assert simplify(density(N2)(x)) == sqrt(10)*exp(-x**2/10)/(10*sqrt(pi)) + +def test_MarginalDistribution(): + a1, p1, p2 = symbols('a1 p1 p2', positive=True) + C = Multinomial('C', 2, p1, p2) + B = MultivariateBeta('B', a1, C[0]) + MGR = MarginalDistribution(B, (C[0],)) + mgrc = Mul(Symbol('B'), Piecewise(ExprCondPair(Mul(Integer(2), + Pow(Symbol('p1', positive=True), Indexed(IndexedBase(Symbol('C')), + Integer(0))), Pow(Symbol('p2', positive=True), + Indexed(IndexedBase(Symbol('C')), Integer(1))), + Pow(factorial(Indexed(IndexedBase(Symbol('C')), Integer(0))), Integer(-1)), + Pow(factorial(Indexed(IndexedBase(Symbol('C')), Integer(1))), Integer(-1))), + Eq(Add(Indexed(IndexedBase(Symbol('C')), Integer(0)), + Indexed(IndexedBase(Symbol('C')), Integer(1))), Integer(2))), + ExprCondPair(Integer(0), True)), Pow(gamma(Symbol('a1', positive=True)), + Integer(-1)), gamma(Add(Symbol('a1', positive=True), + Indexed(IndexedBase(Symbol('C')), Integer(0)))), + Pow(gamma(Indexed(IndexedBase(Symbol('C')), Integer(0))), Integer(-1)), + Pow(Indexed(IndexedBase(Symbol('B')), Integer(0)), + Add(Symbol('a1', positive=True), Integer(-1))), + Pow(Indexed(IndexedBase(Symbol('B')), Integer(1)), + Add(Indexed(IndexedBase(Symbol('C')), Integer(0)), Integer(-1)))) + assert MGR(C) == mgrc + +def test_compound_distribution(): + Y = Poisson('Y', 1) + Z = Poisson('Z', Y) + assert isinstance(pspace(Z), CompoundPSpace) + assert isinstance(pspace(Z).distribution, CompoundDistribution) + assert Z.pspace.distribution.pdf(1).doit() == exp(-2)*exp(exp(-1)) + +def test_mix_expression(): + Y, E = Poisson('Y', 1), Exponential('E', 1) + k = Dummy('k') + expr1 = Integral(Sum(exp(-1)*Integral(exp(-k)*DiracDelta(k - 2), (k, 0, oo) + )/factorial(k), (k, 0, oo)), (k, -oo, 0)) + expr2 = Integral(Sum(exp(-1)*Integral(exp(-k)*DiracDelta(k - 2), (k, 0, oo) + )/factorial(k), (k, 0, oo)), (k, 0, oo)) + assert P(Eq(Y + E, 1)) == 0 + assert P(Ne(Y + E, 2)) == 1 + with ignore_warnings(UserWarning): ### TODO: Restore tests once warnings are removed + assert P(E + Y < 2, evaluate=False).rewrite(Integral).dummy_eq(expr1) + assert P(E + Y > 2, evaluate=False).rewrite(Integral).dummy_eq(expr2) diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_random_matrix.py b/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_random_matrix.py new file mode 100644 index 0000000000000000000000000000000000000000..ba570a16bc42620d53bce19be71e7d125965ede1 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_random_matrix.py @@ -0,0 +1,135 @@ +from sympy.concrete.products import Product +from sympy.core.function import Lambda +from sympy.core.numbers import (I, Rational, pi) +from sympy.core.singleton import S +from sympy.core.symbol import Dummy +from sympy.functions.elementary.complexes import Abs +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.integrals.integrals import Integral +from sympy.matrices.dense import Matrix +from sympy.matrices.expressions.matexpr import MatrixSymbol +from sympy.matrices.expressions.trace import Trace +from sympy.tensor.indexed import IndexedBase +from sympy.stats import (GaussianUnitaryEnsemble as GUE, density, + GaussianOrthogonalEnsemble as GOE, + GaussianSymplecticEnsemble as GSE, + joint_eigen_distribution, + CircularUnitaryEnsemble as CUE, + CircularOrthogonalEnsemble as COE, + CircularSymplecticEnsemble as CSE, + JointEigenDistribution, + level_spacing_distribution, + Normal, Beta) +from sympy.stats.joint_rv_types import JointDistributionHandmade +from sympy.stats.rv import RandomMatrixSymbol +from sympy.stats.random_matrix_models import GaussianEnsemble, RandomMatrixPSpace +from sympy.testing.pytest import raises + +def test_GaussianEnsemble(): + G = GaussianEnsemble('G', 3) + assert density(G) == G.pspace.model + raises(ValueError, lambda: GaussianEnsemble('G', 3.5)) + +def test_GaussianUnitaryEnsemble(): + H = RandomMatrixSymbol('H', 3, 3) + G = GUE('U', 3) + assert density(G)(H) == sqrt(2)*exp(-3*Trace(H**2)/2)/(4*pi**Rational(9, 2)) + i, j = (Dummy('i', integer=True, positive=True), + Dummy('j', integer=True, positive=True)) + l = IndexedBase('l') + assert joint_eigen_distribution(G).dummy_eq( + Lambda((l[1], l[2], l[3]), + 27*sqrt(6)*exp(-3*(l[1]**2)/2 - 3*(l[2]**2)/2 - 3*(l[3]**2)/2)* + Product(Abs(l[i] - l[j])**2, (j, i + 1, 3), (i, 1, 2))/(16*pi**Rational(3, 2)))) + s = Dummy('s') + assert level_spacing_distribution(G).dummy_eq(Lambda(s, 32*s**2*exp(-4*s**2/pi)/pi**2)) + + +def test_GaussianOrthogonalEnsemble(): + H = RandomMatrixSymbol('H', 3, 3) + _H = MatrixSymbol('_H', 3, 3) + G = GOE('O', 3) + assert density(G)(H) == exp(-3*Trace(H**2)/4)/Integral(exp(-3*Trace(_H**2)/4), _H) + i, j = (Dummy('i', integer=True, positive=True), + Dummy('j', integer=True, positive=True)) + l = IndexedBase('l') + assert joint_eigen_distribution(G).dummy_eq( + Lambda((l[1], l[2], l[3]), + 9*sqrt(2)*exp(-3*l[1]**2/2 - 3*l[2]**2/2 - 3*l[3]**2/2)* + Product(Abs(l[i] - l[j]), (j, i + 1, 3), (i, 1, 2))/(32*pi))) + s = Dummy('s') + assert level_spacing_distribution(G).dummy_eq(Lambda(s, s*pi*exp(-s**2*pi/4)/2)) + +def test_GaussianSymplecticEnsemble(): + H = RandomMatrixSymbol('H', 3, 3) + _H = MatrixSymbol('_H', 3, 3) + G = GSE('O', 3) + assert density(G)(H) == exp(-3*Trace(H**2))/Integral(exp(-3*Trace(_H**2)), _H) + i, j = (Dummy('i', integer=True, positive=True), + Dummy('j', integer=True, positive=True)) + l = IndexedBase('l') + assert joint_eigen_distribution(G).dummy_eq( + Lambda((l[1], l[2], l[3]), + 162*sqrt(3)*exp(-3*l[1]**2/2 - 3*l[2]**2/2 - 3*l[3]**2/2)* + Product(Abs(l[i] - l[j])**4, (j, i + 1, 3), (i, 1, 2))/(5*pi**Rational(3, 2)))) + s = Dummy('s') + assert level_spacing_distribution(G).dummy_eq(Lambda(s, S(262144)*s**4*exp(-64*s**2/(9*pi))/(729*pi**3))) + +def test_CircularUnitaryEnsemble(): + CU = CUE('U', 3) + j, k = (Dummy('j', integer=True, positive=True), + Dummy('k', integer=True, positive=True)) + t = IndexedBase('t') + assert joint_eigen_distribution(CU).dummy_eq( + Lambda((t[1], t[2], t[3]), + Product(Abs(exp(I*t[j]) - exp(I*t[k]))**2, + (j, k + 1, 3), (k, 1, 2))/(48*pi**3)) + ) + +def test_CircularOrthogonalEnsemble(): + CO = COE('U', 3) + j, k = (Dummy('j', integer=True, positive=True), + Dummy('k', integer=True, positive=True)) + t = IndexedBase('t') + assert joint_eigen_distribution(CO).dummy_eq( + Lambda((t[1], t[2], t[3]), + Product(Abs(exp(I*t[j]) - exp(I*t[k])), + (j, k + 1, 3), (k, 1, 2))/(48*pi**2)) + ) + +def test_CircularSymplecticEnsemble(): + CS = CSE('U', 3) + j, k = (Dummy('j', integer=True, positive=True), + Dummy('k', integer=True, positive=True)) + t = IndexedBase('t') + assert joint_eigen_distribution(CS).dummy_eq( + Lambda((t[1], t[2], t[3]), + Product(Abs(exp(I*t[j]) - exp(I*t[k]))**4, + (j, k + 1, 3), (k, 1, 2))/(720*pi**3)) + ) + +def test_JointEigenDistribution(): + A = Matrix([[Normal('A00', 0, 1), Normal('A01', 1, 1)], + [Beta('A10', 1, 1), Beta('A11', 1, 1)]]) + assert JointEigenDistribution(A) == \ + JointDistributionHandmade(-sqrt(A[0, 0]**2 - 2*A[0, 0]*A[1, 1] + 4*A[0, 1]*A[1, 0] + A[1, 1]**2)/2 + + A[0, 0]/2 + A[1, 1]/2, sqrt(A[0, 0]**2 - 2*A[0, 0]*A[1, 1] + 4*A[0, 1]*A[1, 0] + A[1, 1]**2)/2 + A[0, 0]/2 + A[1, 1]/2) + raises(ValueError, lambda: JointEigenDistribution(Matrix([[1, 0], [2, 1]]))) + +def test_issue_19841(): + G1 = GUE('U', 2) + G2 = G1.xreplace({2: 2}) + assert G1.args == G2.args + + X = MatrixSymbol('X', 2, 2) + G = GSE('U', 2) + h_pspace = RandomMatrixPSpace('P', model=density(G)) + H = RandomMatrixSymbol('H', 2, 2, pspace=h_pspace) + H2 = RandomMatrixSymbol('H', 2, 2, pspace=None) + assert H.doit() == H + + assert (2*H).xreplace({H: X}) == 2*X + assert (2*H).xreplace({H2: X}) == 2*H + assert (2*H2).xreplace({H: X}) == 2*H2 + assert (2*H2).xreplace({H2: X}) == 2*X diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_rv.py b/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_rv.py new file mode 100644 index 0000000000000000000000000000000000000000..185756300556f2fe70b76c402113ec2bb2501ef4 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_rv.py @@ -0,0 +1,441 @@ +from sympy.concrete.summations import Sum +from sympy.core.basic import Basic +from sympy.core.containers import Tuple +from sympy.core.function import Lambda +from sympy.core.numbers import (Rational, nan, oo, pi) +from sympy.core.relational import Eq +from sympy.core.singleton import S +from sympy.core.symbol import (Symbol, symbols) +from sympy.functions.combinatorial.factorials import (FallingFactorial, binomial) +from sympy.functions.elementary.exponential import (exp, log) +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.functions.special.delta_functions import DiracDelta +from sympy.integrals.integrals import integrate +from sympy.logic.boolalg import (And, Or) +from sympy.matrices.dense import Matrix +from sympy.sets.sets import Interval +from sympy.tensor.indexed import Indexed +from sympy.stats import (Die, Normal, Exponential, FiniteRV, P, E, H, variance, + density, given, independent, dependent, where, pspace, GaussianUnitaryEnsemble, + random_symbols, sample, Geometric, factorial_moment, Binomial, Hypergeometric, + DiscreteUniform, Poisson, characteristic_function, moment_generating_function, + BernoulliProcess, Variance, Expectation, Probability, Covariance, covariance, cmoment, + moment, median) +from sympy.stats.rv import (IndependentProductPSpace, rs_swap, Density, NamedArgsMixin, + RandomSymbol, sample_iter, PSpace, is_random, RandomIndexedSymbol, RandomMatrixSymbol) +from sympy.testing.pytest import raises, skip, XFAIL, warns_deprecated_sympy +from sympy.external import import_module +from sympy.core.numbers import comp +from sympy.stats.frv_types import BernoulliDistribution +from sympy.core.symbol import Dummy +from sympy.functions.elementary.piecewise import Piecewise + +def test_where(): + X, Y = Die('X'), Die('Y') + Z = Normal('Z', 0, 1) + + assert where(Z**2 <= 1).set == Interval(-1, 1) + assert where(Z**2 <= 1).as_boolean() == Interval(-1, 1).as_relational(Z.symbol) + assert where(And(X > Y, Y > 4)).as_boolean() == And( + Eq(X.symbol, 6), Eq(Y.symbol, 5)) + + assert len(where(X < 3).set) == 2 + assert 1 in where(X < 3).set + + X, Y = Normal('X', 0, 1), Normal('Y', 0, 1) + assert where(And(X**2 <= 1, X >= 0)).set == Interval(0, 1) + XX = given(X, And(X**2 <= 1, X >= 0)) + assert XX.pspace.domain.set == Interval(0, 1) + assert XX.pspace.domain.as_boolean() == \ + And(0 <= X.symbol, X.symbol**2 <= 1, -oo < X.symbol, X.symbol < oo) + + with raises(TypeError): + XX = given(X, X + 3) + + +def test_random_symbols(): + X, Y = Normal('X', 0, 1), Normal('Y', 0, 1) + + assert set(random_symbols(2*X + 1)) == {X} + assert set(random_symbols(2*X + Y)) == {X, Y} + assert set(random_symbols(2*X + Y.symbol)) == {X} + assert set(random_symbols(2)) == set() + + +def test_characteristic_function(): + # Imports I from sympy + from sympy.core.numbers import I + X = Normal('X',0,1) + Y = DiscreteUniform('Y', [1,2,7]) + Z = Poisson('Z', 2) + t = symbols('_t') + P = Lambda(t, exp(-t**2/2)) + Q = Lambda(t, exp(7*t*I)/3 + exp(2*t*I)/3 + exp(t*I)/3) + R = Lambda(t, exp(2 * exp(t*I) - 2)) + + + assert characteristic_function(X).dummy_eq(P) + assert characteristic_function(Y).dummy_eq(Q) + assert characteristic_function(Z).dummy_eq(R) + + +def test_moment_generating_function(): + + X = Normal('X',0,1) + Y = DiscreteUniform('Y', [1,2,7]) + Z = Poisson('Z', 2) + t = symbols('_t') + P = Lambda(t, exp(t**2/2)) + Q = Lambda(t, (exp(7*t)/3 + exp(2*t)/3 + exp(t)/3)) + R = Lambda(t, exp(2 * exp(t) - 2)) + + + assert moment_generating_function(X).dummy_eq(P) + assert moment_generating_function(Y).dummy_eq(Q) + assert moment_generating_function(Z).dummy_eq(R) + +def test_sample_iter(): + + X = Normal('X',0,1) + Y = DiscreteUniform('Y', [1, 2, 7]) + Z = Poisson('Z', 2) + + scipy = import_module('scipy') + if not scipy: + skip('Scipy is not installed. Abort tests') + expr = X**2 + 3 + iterator = sample_iter(expr) + + expr2 = Y**2 + 5*Y + 4 + iterator2 = sample_iter(expr2) + + expr3 = Z**3 + 4 + iterator3 = sample_iter(expr3) + + def is_iterator(obj): + if ( + hasattr(obj, '__iter__') and + (hasattr(obj, 'next') or + hasattr(obj, '__next__')) and + callable(obj.__iter__) and + obj.__iter__() is obj + ): + return True + else: + return False + assert is_iterator(iterator) + assert is_iterator(iterator2) + assert is_iterator(iterator3) + +def test_pspace(): + X, Y = Normal('X', 0, 1), Normal('Y', 0, 1) + x = Symbol('x') + + raises(ValueError, lambda: pspace(5 + 3)) + raises(ValueError, lambda: pspace(x < 1)) + assert pspace(X) == X.pspace + assert pspace(2*X + 1) == X.pspace + assert pspace(2*X + Y) == IndependentProductPSpace(Y.pspace, X.pspace) + +def test_rs_swap(): + X = Normal('x', 0, 1) + Y = Exponential('y', 1) + + XX = Normal('x', 0, 2) + YY = Normal('y', 0, 3) + + expr = 2*X + Y + assert expr.subs(rs_swap((X, Y), (YY, XX))) == 2*XX + YY + + +def test_RandomSymbol(): + + X = Normal('x', 0, 1) + Y = Normal('x', 0, 2) + assert X.symbol == Y.symbol + assert X != Y + + assert X.name == X.symbol.name + + X = Normal('lambda', 0, 1) # make sure we can use protected terms + X = Normal('Lambda', 0, 1) # make sure we can use SymPy terms + + +def test_RandomSymbol_diff(): + X = Normal('x', 0, 1) + assert (2*X).diff(X) + + +def test_random_symbol_no_pspace(): + x = RandomSymbol(Symbol('x')) + assert x.pspace == PSpace() + +def test_overlap(): + X = Normal('x', 0, 1) + Y = Normal('x', 0, 2) + + raises(ValueError, lambda: P(X > Y)) + + +def test_IndependentProductPSpace(): + X = Normal('X', 0, 1) + Y = Normal('Y', 0, 1) + px = X.pspace + py = Y.pspace + assert pspace(X + Y) == IndependentProductPSpace(px, py) + assert pspace(X + Y) == IndependentProductPSpace(py, px) + + +def test_E(): + assert E(5) == 5 + + +def test_H(): + X = Normal('X', 0, 1) + D = Die('D', sides = 4) + G = Geometric('G', 0.5) + assert H(X, X > 0) == -log(2)/2 + S.Half + log(pi)/2 + assert H(D, D > 2) == log(2) + assert comp(H(G).evalf().round(2), 1.39) + + +def test_Sample(): + X = Die('X', 6) + Y = Normal('Y', 0, 1) + z = Symbol('z', integer=True) + + scipy = import_module('scipy') + if not scipy: + skip('Scipy is not installed. Abort tests') + assert sample(X) in [1, 2, 3, 4, 5, 6] + assert isinstance(sample(X + Y), float) + + assert P(X + Y > 0, Y < 0, numsamples=10).is_number + assert E(X + Y, numsamples=10).is_number + assert E(X**2 + Y, numsamples=10).is_number + assert E((X + Y)**2, numsamples=10).is_number + assert variance(X + Y, numsamples=10).is_number + + raises(TypeError, lambda: P(Y > z, numsamples=5)) + + assert P(sin(Y) <= 1, numsamples=10) == 1.0 + assert P(sin(Y) <= 1, cos(Y) < 1, numsamples=10) == 1.0 + + assert all(i in range(1, 7) for i in density(X, numsamples=10)) + assert all(i in range(4, 7) for i in density(X, X>3, numsamples=10)) + + numpy = import_module('numpy') + if not numpy: + skip('Numpy is not installed. Abort tests') + #Test Issue #21563: Output of sample must be a float or array + assert isinstance(sample(X), (numpy.int32, numpy.int64)) + assert isinstance(sample(Y), numpy.float64) + assert isinstance(sample(X, size=2), numpy.ndarray) + + with warns_deprecated_sympy(): + sample(X, numsamples=2) + +@XFAIL +def test_samplingE(): + scipy = import_module('scipy') + if not scipy: + skip('Scipy is not installed. Abort tests') + Y = Normal('Y', 0, 1) + z = Symbol('z', integer=True) + assert E(Sum(1/z**Y, (z, 1, oo)), Y > 2, numsamples=3).is_number + + +def test_given(): + X = Normal('X', 0, 1) + Y = Normal('Y', 0, 1) + A = given(X, True) + B = given(X, Y > 2) + + assert X == A == B + + +def test_factorial_moment(): + X = Poisson('X', 2) + Y = Binomial('Y', 2, S.Half) + Z = Hypergeometric('Z', 4, 2, 2) + assert factorial_moment(X, 2) == 4 + assert factorial_moment(Y, 2) == S.Half + assert factorial_moment(Z, 2) == Rational(1, 3) + + x, y, z, l = symbols('x y z l') + Y = Binomial('Y', 2, y) + Z = Hypergeometric('Z', 10, 2, 3) + assert factorial_moment(Y, l) == y**2*FallingFactorial( + 2, l) + 2*y*(1 - y)*FallingFactorial(1, l) + (1 - y)**2*\ + FallingFactorial(0, l) + assert factorial_moment(Z, l) == 7*FallingFactorial(0, l)/\ + 15 + 7*FallingFactorial(1, l)/15 + FallingFactorial(2, l)/15 + + +def test_dependence(): + X, Y = Die('X'), Die('Y') + assert independent(X, 2*Y) + assert not dependent(X, 2*Y) + + X, Y = Normal('X', 0, 1), Normal('Y', 0, 1) + assert independent(X, Y) + assert dependent(X, 2*X) + + # Create a dependency + XX, YY = given(Tuple(X, Y), Eq(X + Y, 3)) + assert dependent(XX, YY) + +def test_dependent_finite(): + X, Y = Die('X'), Die('Y') + # Dependence testing requires symbolic conditions which currently break + # finite random variables + assert dependent(X, Y + X) + + XX, YY = given(Tuple(X, Y), X + Y > 5) # Create a dependency + assert dependent(XX, YY) + + +def test_normality(): + X, Y = Normal('X', 0, 1), Normal('Y', 0, 1) + x = Symbol('x', real=True) + z = Symbol('z', real=True) + dens = density(X - Y, Eq(X + Y, z)) + + assert integrate(dens(x), (x, -oo, oo)) == 1 + + +def test_Density(): + X = Die('X', 6) + d = Density(X) + assert d.doit() == density(X) + +def test_NamedArgsMixin(): + class Foo(Basic, NamedArgsMixin): + _argnames = 'foo', 'bar' + + a = Foo(S(1), S(2)) + + assert a.foo == 1 + assert a.bar == 2 + + raises(AttributeError, lambda: a.baz) + + class Bar(Basic, NamedArgsMixin): + pass + + raises(AttributeError, lambda: Bar(S(1), S(2)).foo) + +def test_density_constant(): + assert density(3)(2) == 0 + assert density(3)(3) == DiracDelta(0) + +def test_cmoment_constant(): + assert variance(3) == 0 + assert cmoment(3, 3) == 0 + assert cmoment(3, 4) == 0 + x = Symbol('x') + assert variance(x) == 0 + assert cmoment(x, 15) == 0 + assert cmoment(x, 0) == 1 + +def test_moment_constant(): + assert moment(3, 0) == 1 + assert moment(3, 1) == 3 + assert moment(3, 2) == 9 + x = Symbol('x') + assert moment(x, 2) == x**2 + +def test_median_constant(): + assert median(3) == 3 + x = Symbol('x') + assert median(x) == x + +def test_real(): + x = Normal('x', 0, 1) + assert x.is_real + + +def test_issue_10052(): + X = Exponential('X', 3) + assert P(X < oo) == 1 + assert P(X > oo) == 0 + assert P(X < 2, X > oo) == 0 + assert P(X < oo, X > oo) == 0 + assert P(X < oo, X > 2) == 1 + assert P(X < 3, X == 2) == 0 + raises(ValueError, lambda: P(1)) + raises(ValueError, lambda: P(X < 1, 2)) + +def test_issue_11934(): + density = {0: .5, 1: .5} + X = FiniteRV('X', density) + assert E(X) == 0.5 + assert P( X>= 2) == 0 + +def test_issue_8129(): + X = Exponential('X', 4) + assert P(X >= X) == 1 + assert P(X > X) == 0 + assert P(X > X+1) == 0 + +def test_issue_12237(): + X = Normal('X', 0, 1) + Y = Normal('Y', 0, 1) + U = P(X > 0, X) + V = P(Y < 0, X) + W = P(X + Y > 0, X) + assert W == P(X + Y > 0, X) + assert U == BernoulliDistribution(S.Half, S.Zero, S.One) + assert V == S.Half + +def test_is_random(): + X = Normal('X', 0, 1) + Y = Normal('Y', 0, 1) + a, b = symbols('a, b') + G = GaussianUnitaryEnsemble('U', 2) + B = BernoulliProcess('B', 0.9) + assert not is_random(a) + assert not is_random(a + b) + assert not is_random(a * b) + assert not is_random(Matrix([a**2, b**2])) + assert is_random(X) + assert is_random(X**2 + Y) + assert is_random(Y + b**2) + assert is_random(Y > 5) + assert is_random(B[3] < 1) + assert is_random(G) + assert is_random(X * Y * B[1]) + assert is_random(Matrix([[X, B[2]], [G, Y]])) + assert is_random(Eq(X, 4)) + +def test_issue_12283(): + x = symbols('x') + X = RandomSymbol(x) + Y = RandomSymbol('Y') + Z = RandomMatrixSymbol('Z', 2, 1) + W = RandomMatrixSymbol('W', 2, 1) + RI = RandomIndexedSymbol(Indexed('RI', 3)) + assert pspace(Z) == PSpace() + assert pspace(RI) == PSpace() + assert pspace(X) == PSpace() + assert E(X) == Expectation(X) + assert P(Y > 3) == Probability(Y > 3) + assert variance(X) == Variance(X) + assert variance(RI) == Variance(RI) + assert covariance(X, Y) == Covariance(X, Y) + assert covariance(W, Z) == Covariance(W, Z) + +def test_issue_6810(): + X = Die('X', 6) + Y = Normal('Y', 0, 1) + assert P(Eq(X, 2)) == S(1)/6 + assert P(Eq(Y, 0)) == 0 + assert P(Or(X > 2, X < 3)) == 1 + assert P(And(X > 3, X > 2)) == S(1)/2 + +def test_issue_20286(): + n, p = symbols('n p') + B = Binomial('B', n, p) + k = Dummy('k', integer = True) + eq = Sum(Piecewise((-p**k*(1 - p)**(-k + n)*log(p**k*(1 - p)**(-k + n)*binomial(n, k))*binomial(n, k), (k >= 0) & (k <= n)), (nan, True)), (k, 0, n)) + assert eq.dummy_eq(H(B)) diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_stochastic_process.py b/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_stochastic_process.py new file mode 100644 index 0000000000000000000000000000000000000000..3e42ffc8632240b1d85a774467a057e9857c567c --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_stochastic_process.py @@ -0,0 +1,763 @@ +from sympy.concrete.summations import Sum +from sympy.core.containers import Tuple +from sympy.core.function import Lambda +from sympy.core.numbers import (Float, Rational, oo, pi) +from sympy.core.relational import (Eq, Ge, Gt, Le, Lt, Ne) +from sympy.core.singleton import S +from sympy.core.symbol import (Symbol, symbols) +from sympy.functions.combinatorial.factorials import factorial +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.integers import ceiling +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.special.error_functions import erf +from sympy.functions.special.gamma_functions import (gamma, lowergamma) +from sympy.logic.boolalg import (And, Not) +from sympy.matrices.dense import Matrix +from sympy.matrices.expressions.matexpr import MatrixSymbol +from sympy.matrices.immutable import ImmutableMatrix +from sympy.sets.contains import Contains +from sympy.sets.fancysets import Range +from sympy.sets.sets import (FiniteSet, Interval) +from sympy.stats import (DiscreteMarkovChain, P, TransitionMatrixOf, E, + StochasticStateSpaceOf, variance, ContinuousMarkovChain, + BernoulliProcess, PoissonProcess, WienerProcess, + GammaProcess, sample_stochastic_process) +from sympy.stats.joint_rv import JointDistribution +from sympy.stats.joint_rv_types import JointDistributionHandmade +from sympy.stats.rv import RandomIndexedSymbol +from sympy.stats.symbolic_probability import Probability, Expectation +from sympy.testing.pytest import (raises, skip, ignore_warnings, + warns_deprecated_sympy) +from sympy.external import import_module +from sympy.stats.frv_types import BernoulliDistribution +from sympy.stats.drv_types import PoissonDistribution +from sympy.stats.crv_types import NormalDistribution, GammaDistribution +from sympy.core.symbol import Str + + +def test_DiscreteMarkovChain(): + + # pass only the name + X = DiscreteMarkovChain("X") + assert isinstance(X.state_space, Range) + assert X.index_set == S.Naturals0 + assert isinstance(X.transition_probabilities, MatrixSymbol) + t = symbols('t', positive=True, integer=True) + assert isinstance(X[t], RandomIndexedSymbol) + assert E(X[0]) == Expectation(X[0]) + raises(TypeError, lambda: DiscreteMarkovChain(1)) + raises(NotImplementedError, lambda: X(t)) + raises(NotImplementedError, lambda: X.communication_classes()) + raises(NotImplementedError, lambda: X.canonical_form()) + raises(NotImplementedError, lambda: X.decompose()) + + nz = Symbol('n', integer=True) + TZ = MatrixSymbol('M', nz, nz) + SZ = Range(nz) + YZ = DiscreteMarkovChain('Y', SZ, TZ) + assert P(Eq(YZ[2], 1), Eq(YZ[1], 0)) == TZ[0, 1] + + raises(ValueError, lambda: sample_stochastic_process(t)) + raises(ValueError, lambda: next(sample_stochastic_process(X))) + # pass name and state_space + # any hashable object should be a valid state + # states should be valid as a tuple/set/list/Tuple/Range + sym, rainy, cloudy, sunny = symbols('a Rainy Cloudy Sunny', real=True) + state_spaces = [(1, 2, 3), [Str('Hello'), sym, DiscreteMarkovChain("Y", (1,2,3))], + Tuple(S(1), exp(sym), Str('World'), sympify=False), Range(-1, 5, 2), + [rainy, cloudy, sunny]] + chains = [DiscreteMarkovChain("Y", state_space) for state_space in state_spaces] + + for i, Y in enumerate(chains): + assert isinstance(Y.transition_probabilities, MatrixSymbol) + assert Y.state_space == state_spaces[i] or Y.state_space == FiniteSet(*state_spaces[i]) + assert Y.number_of_states == 3 + + with ignore_warnings(UserWarning): # TODO: Restore tests once warnings are removed + assert P(Eq(Y[2], 1), Eq(Y[0], 2), evaluate=False) == Probability(Eq(Y[2], 1), Eq(Y[0], 2)) + assert E(Y[0]) == Expectation(Y[0]) + + raises(ValueError, lambda: next(sample_stochastic_process(Y))) + + raises(TypeError, lambda: DiscreteMarkovChain("Y", {1: 1})) + Y = DiscreteMarkovChain("Y", Range(1, t, 2)) + assert Y.number_of_states == ceiling((t-1)/2) + + # pass name and transition_probabilities + chains = [DiscreteMarkovChain("Y", trans_probs=Matrix([])), + DiscreteMarkovChain("Y", trans_probs=Matrix([[0, 1], [1, 0]])), + DiscreteMarkovChain("Y", trans_probs=Matrix([[pi, 1-pi], [sym, 1-sym]]))] + for Z in chains: + assert Z.number_of_states == Z.transition_probabilities.shape[0] + assert isinstance(Z.transition_probabilities, ImmutableMatrix) + + # pass name, state_space and transition_probabilities + T = Matrix([[0.5, 0.2, 0.3],[0.2, 0.5, 0.3],[0.2, 0.3, 0.5]]) + TS = MatrixSymbol('T', 3, 3) + Y = DiscreteMarkovChain("Y", [0, 1, 2], T) + YS = DiscreteMarkovChain("Y", ['One', 'Two', 3], TS) + assert Y.joint_distribution(1, Y[2], 3) == JointDistribution(Y[1], Y[2], Y[3]) + raises(ValueError, lambda: Y.joint_distribution(Y[1].symbol, Y[2].symbol)) + assert P(Eq(Y[3], 2), Eq(Y[1], 1)).round(2) == Float(0.36, 2) + assert (P(Eq(YS[3], 2), Eq(YS[1], 1)) - + (TS[0, 2]*TS[1, 0] + TS[1, 1]*TS[1, 2] + TS[1, 2]*TS[2, 2])).simplify() == 0 + assert P(Eq(YS[1], 1), Eq(YS[2], 2)) == Probability(Eq(YS[1], 1)) + assert P(Eq(YS[3], 3), Eq(YS[1], 1)) == TS[0, 2]*TS[1, 0] + TS[1, 1]*TS[1, 2] + TS[1, 2]*TS[2, 2] + TO = Matrix([[0.25, 0.75, 0],[0, 0.25, 0.75],[0.75, 0, 0.25]]) + assert P(Eq(Y[3], 2), Eq(Y[1], 1) & TransitionMatrixOf(Y, TO)).round(3) == Float(0.375, 3) + with ignore_warnings(UserWarning): ### TODO: Restore tests once warnings are removed + assert E(Y[3], evaluate=False) == Expectation(Y[3]) + assert E(Y[3], Eq(Y[2], 1)).round(2) == Float(1.1, 3) + TSO = MatrixSymbol('T', 4, 4) + raises(ValueError, lambda: str(P(Eq(YS[3], 2), Eq(YS[1], 1) & TransitionMatrixOf(YS, TSO)))) + raises(TypeError, lambda: DiscreteMarkovChain("Z", [0, 1, 2], symbols('M'))) + raises(ValueError, lambda: DiscreteMarkovChain("Z", [0, 1, 2], MatrixSymbol('T', 3, 4))) + raises(ValueError, lambda: E(Y[3], Eq(Y[2], 6))) + raises(ValueError, lambda: E(Y[2], Eq(Y[3], 1))) + + + # extended tests for probability queries + TO1 = Matrix([[Rational(1, 4), Rational(3, 4), 0],[Rational(1, 3), Rational(1, 3), Rational(1, 3)],[0, Rational(1, 4), Rational(3, 4)]]) + assert P(And(Eq(Y[2], 1), Eq(Y[1], 1), Eq(Y[0], 0)), + Eq(Probability(Eq(Y[0], 0)), Rational(1, 4)) & TransitionMatrixOf(Y, TO1)) == Rational(1, 16) + assert P(And(Eq(Y[2], 1), Eq(Y[1], 1), Eq(Y[0], 0)), TransitionMatrixOf(Y, TO1)) == \ + Probability(Eq(Y[0], 0))/4 + assert P(Lt(X[1], 2) & Gt(X[1], 0), Eq(X[0], 2) & + StochasticStateSpaceOf(X, [0, 1, 2]) & TransitionMatrixOf(X, TO1)) == Rational(1, 4) + assert P(Lt(X[1], 2) & Gt(X[1], 0), Eq(X[0], 2) & + StochasticStateSpaceOf(X, [S(0), '0', 1]) & TransitionMatrixOf(X, TO1)) == Rational(1, 4) + assert P(Ne(X[1], 2) & Ne(X[1], 1), Eq(X[0], 2) & + StochasticStateSpaceOf(X, [0, 1, 2]) & TransitionMatrixOf(X, TO1)) is S.Zero + assert P(Ne(X[1], 2) & Ne(X[1], 1), Eq(X[0], 2) & + StochasticStateSpaceOf(X, [S(0), '0', 1]) & TransitionMatrixOf(X, TO1)) is S.Zero + assert P(And(Eq(Y[2], 1), Eq(Y[1], 1), Eq(Y[0], 0)), Eq(Y[1], 1)) == 0.1*Probability(Eq(Y[0], 0)) + + # testing properties of Markov chain + TO2 = Matrix([[S.One, 0, 0],[Rational(1, 3), Rational(1, 3), Rational(1, 3)],[0, Rational(1, 4), Rational(3, 4)]]) + TO3 = Matrix([[Rational(1, 4), Rational(3, 4), 0],[Rational(1, 3), Rational(1, 3), Rational(1, 3)], [0, Rational(1, 4), Rational(3, 4)]]) + Y2 = DiscreteMarkovChain('Y', trans_probs=TO2) + Y3 = DiscreteMarkovChain('Y', trans_probs=TO3) + assert Y3.fundamental_matrix() == ImmutableMatrix([[176, 81, -132], [36, 141, -52], [-44, -39, 208]])/125 + assert Y2.is_absorbing_chain() == True + assert Y3.is_absorbing_chain() == False + assert Y2.canonical_form() == ([0, 1, 2], TO2) + assert Y3.canonical_form() == ([0, 1, 2], TO3) + assert Y2.decompose() == ([0, 1, 2], TO2[0:1, 0:1], TO2[1:3, 0:1], TO2[1:3, 1:3]) + assert Y3.decompose() == ([0, 1, 2], TO3, Matrix(0, 3, []), Matrix(0, 0, [])) + TO4 = Matrix([[Rational(1, 5), Rational(2, 5), Rational(2, 5)], [Rational(1, 10), S.Half, Rational(2, 5)], [Rational(3, 5), Rational(3, 10), Rational(1, 10)]]) + Y4 = DiscreteMarkovChain('Y', trans_probs=TO4) + w = ImmutableMatrix([[Rational(11, 39), Rational(16, 39), Rational(4, 13)]]) + assert Y4.limiting_distribution == w + assert Y4.is_regular() == True + assert Y4.is_ergodic() == True + TS1 = MatrixSymbol('T', 3, 3) + Y5 = DiscreteMarkovChain('Y', trans_probs=TS1) + assert Y5.limiting_distribution(w, TO4).doit() == True + assert Y5.stationary_distribution(condition_set=True).subs(TS1, TO4).contains(w).doit() == S.true + TO6 = Matrix([[S.One, 0, 0, 0, 0],[S.Half, 0, S.Half, 0, 0],[0, S.Half, 0, S.Half, 0], [0, 0, S.Half, 0, S.Half], [0, 0, 0, 0, 1]]) + Y6 = DiscreteMarkovChain('Y', trans_probs=TO6) + assert Y6.fundamental_matrix() == ImmutableMatrix([[Rational(3, 2), S.One, S.Half], [S.One, S(2), S.One], [S.Half, S.One, Rational(3, 2)]]) + assert Y6.absorbing_probabilities() == ImmutableMatrix([[Rational(3, 4), Rational(1, 4)], [S.Half, S.Half], [Rational(1, 4), Rational(3, 4)]]) + with warns_deprecated_sympy(): + Y6.absorbing_probabilites() + TO7 = Matrix([[Rational(1, 2), Rational(1, 4), Rational(1, 4)], [Rational(1, 2), 0, Rational(1, 2)], [Rational(1, 4), Rational(1, 4), Rational(1, 2)]]) + Y7 = DiscreteMarkovChain('Y', trans_probs=TO7) + assert Y7.is_absorbing_chain() == False + assert Y7.fundamental_matrix() == ImmutableMatrix([[Rational(86, 75), Rational(1, 25), Rational(-14, 75)], + [Rational(2, 25), Rational(21, 25), Rational(2, 25)], + [Rational(-14, 75), Rational(1, 25), Rational(86, 75)]]) + + # test for zero-sized matrix functionality + X = DiscreteMarkovChain('X', trans_probs=Matrix([])) + assert X.number_of_states == 0 + assert X.stationary_distribution() == Matrix([[]]) + assert X.communication_classes() == [] + assert X.canonical_form() == ([], Matrix([])) + assert X.decompose() == ([], Matrix([]), Matrix([]), Matrix([])) + assert X.is_regular() == False + assert X.is_ergodic() == False + + # test communication_class + # see https://drive.google.com/drive/folders/1HbxLlwwn2b3U8Lj7eb_ASIUb5vYaNIjg?usp=sharing + # tutorial 2.pdf + TO7 = Matrix([[0, 5, 5, 0, 0], + [0, 0, 0, 10, 0], + [5, 0, 5, 0, 0], + [0, 10, 0, 0, 0], + [0, 3, 0, 3, 4]])/10 + Y7 = DiscreteMarkovChain('Y', trans_probs=TO7) + tuples = Y7.communication_classes() + classes, recurrence, periods = list(zip(*tuples)) + assert classes == ([1, 3], [0, 2], [4]) + assert recurrence == (True, False, False) + assert periods == (2, 1, 1) + + TO8 = Matrix([[0, 0, 0, 10, 0, 0], + [5, 0, 5, 0, 0, 0], + [0, 4, 0, 0, 0, 6], + [10, 0, 0, 0, 0, 0], + [0, 10, 0, 0, 0, 0], + [0, 0, 0, 5, 5, 0]])/10 + Y8 = DiscreteMarkovChain('Y', trans_probs=TO8) + tuples = Y8.communication_classes() + classes, recurrence, periods = list(zip(*tuples)) + assert classes == ([0, 3], [1, 2, 5, 4]) + assert recurrence == (True, False) + assert periods == (2, 2) + + TO9 = Matrix([[2, 0, 0, 3, 0, 0, 3, 2, 0, 0], + [0, 10, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 2, 2, 0, 0, 0, 0, 0, 3, 3], + [0, 0, 0, 3, 0, 0, 6, 1, 0, 0], + [0, 0, 0, 0, 5, 5, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 10, 0, 0, 0, 0], + [4, 0, 0, 5, 0, 0, 1, 0, 0, 0], + [2, 0, 0, 4, 0, 0, 2, 2, 0, 0], + [3, 0, 1, 0, 0, 0, 0, 0, 4, 2], + [0, 0, 4, 0, 0, 0, 0, 0, 3, 3]])/10 + Y9 = DiscreteMarkovChain('Y', trans_probs=TO9) + tuples = Y9.communication_classes() + classes, recurrence, periods = list(zip(*tuples)) + assert classes == ([0, 3, 6, 7], [1], [2, 8, 9], [5], [4]) + assert recurrence == (True, True, False, True, False) + assert periods == (1, 1, 1, 1, 1) + + # test canonical form + # see https://web.archive.org/web/20201230182007/https://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter11.pdf + # example 11.13 + T = Matrix([[1, 0, 0, 0, 0], + [S(1) / 2, 0, S(1) / 2, 0, 0], + [0, S(1) / 2, 0, S(1) / 2, 0], + [0, 0, S(1) / 2, 0, S(1) / 2], + [0, 0, 0, 0, S(1)]]) + DW = DiscreteMarkovChain('DW', [0, 1, 2, 3, 4], T) + states, A, B, C = DW.decompose() + assert states == [0, 4, 1, 2, 3] + assert A == Matrix([[1, 0], [0, 1]]) + assert B == Matrix([[S(1)/2, 0], [0, 0], [0, S(1)/2]]) + assert C == Matrix([[0, S(1)/2, 0], [S(1)/2, 0, S(1)/2], [0, S(1)/2, 0]]) + states, new_matrix = DW.canonical_form() + assert states == [0, 4, 1, 2, 3] + assert new_matrix == Matrix([[1, 0, 0, 0, 0], + [0, 1, 0, 0, 0], + [S(1)/2, 0, 0, S(1)/2, 0], + [0, 0, S(1)/2, 0, S(1)/2], + [0, S(1)/2, 0, S(1)/2, 0]]) + + # test regular and ergodic + # https://web.archive.org/web/20201230182007/https://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter11.pdf + T = Matrix([[0, 4, 0, 0, 0], + [1, 0, 3, 0, 0], + [0, 2, 0, 2, 0], + [0, 0, 3, 0, 1], + [0, 0, 0, 4, 0]])/4 + X = DiscreteMarkovChain('X', trans_probs=T) + assert not X.is_regular() + assert X.is_ergodic() + T = Matrix([[0, 1], [1, 0]]) + X = DiscreteMarkovChain('X', trans_probs=T) + assert not X.is_regular() + assert X.is_ergodic() + # http://www.math.wisc.edu/~valko/courses/331/MC2.pdf + T = Matrix([[2, 1, 1], + [2, 0, 2], + [1, 1, 2]])/4 + X = DiscreteMarkovChain('X', trans_probs=T) + assert X.is_regular() + assert X.is_ergodic() + # https://docs.ufpr.br/~lucambio/CE222/1S2014/Kemeny-Snell1976.pdf + T = Matrix([[1, 1], [1, 1]])/2 + X = DiscreteMarkovChain('X', trans_probs=T) + assert X.is_regular() + assert X.is_ergodic() + + # test is_absorbing_chain + T = Matrix([[0, 1, 0], + [1, 0, 0], + [0, 0, 1]]) + X = DiscreteMarkovChain('X', trans_probs=T) + assert not X.is_absorbing_chain() + # https://en.wikipedia.org/wiki/Absorbing_Markov_chain + T = Matrix([[1, 1, 0, 0], + [0, 1, 1, 0], + [1, 0, 0, 1], + [0, 0, 0, 2]])/2 + X = DiscreteMarkovChain('X', trans_probs=T) + assert X.is_absorbing_chain() + T = Matrix([[2, 0, 0, 0, 0], + [1, 0, 1, 0, 0], + [0, 1, 0, 1, 0], + [0, 0, 1, 0, 1], + [0, 0, 0, 0, 2]])/2 + X = DiscreteMarkovChain('X', trans_probs=T) + assert X.is_absorbing_chain() + + # test custom state space + Y10 = DiscreteMarkovChain('Y', [1, 2, 3], TO2) + tuples = Y10.communication_classes() + classes, recurrence, periods = list(zip(*tuples)) + assert classes == ([1], [2, 3]) + assert recurrence == (True, False) + assert periods == (1, 1) + assert Y10.canonical_form() == ([1, 2, 3], TO2) + assert Y10.decompose() == ([1, 2, 3], TO2[0:1, 0:1], TO2[1:3, 0:1], TO2[1:3, 1:3]) + + # testing miscellaneous queries + T = Matrix([[S.Half, Rational(1, 4), Rational(1, 4)], + [Rational(1, 3), 0, Rational(2, 3)], + [S.Half, S.Half, 0]]) + X = DiscreteMarkovChain('X', [0, 1, 2], T) + assert P(Eq(X[1], 2) & Eq(X[2], 1) & Eq(X[3], 0), + Eq(P(Eq(X[1], 0)), Rational(1, 4)) & Eq(P(Eq(X[1], 1)), Rational(1, 4))) == Rational(1, 12) + assert P(Eq(X[2], 1) | Eq(X[2], 2), Eq(X[1], 1)) == Rational(2, 3) + assert P(Eq(X[2], 1) & Eq(X[2], 2), Eq(X[1], 1)) is S.Zero + assert P(Ne(X[2], 2), Eq(X[1], 1)) == Rational(1, 3) + assert E(X[1]**2, Eq(X[0], 1)) == Rational(8, 3) + assert variance(X[1], Eq(X[0], 1)) == Rational(8, 9) + raises(ValueError, lambda: E(X[1], Eq(X[2], 1))) + raises(ValueError, lambda: DiscreteMarkovChain('X', [0, 1], T)) + + # testing miscellaneous queries with different state space + X = DiscreteMarkovChain('X', ['A', 'B', 'C'], T) + assert P(Eq(X[1], 2) & Eq(X[2], 1) & Eq(X[3], 0), + Eq(P(Eq(X[1], 0)), Rational(1, 4)) & Eq(P(Eq(X[1], 1)), Rational(1, 4))) == Rational(1, 12) + assert P(Eq(X[2], 1) | Eq(X[2], 2), Eq(X[1], 1)) == Rational(2, 3) + assert P(Eq(X[2], 1) & Eq(X[2], 2), Eq(X[1], 1)) is S.Zero + assert P(Ne(X[2], 2), Eq(X[1], 1)) == Rational(1, 3) + a = X.state_space.args[0] + c = X.state_space.args[2] + assert (E(X[1] ** 2, Eq(X[0], 1)) - (a**2/3 + 2*c**2/3)).simplify() == 0 + assert (variance(X[1], Eq(X[0], 1)) - (2*(-a/3 + c/3)**2/3 + (2*a/3 - 2*c/3)**2/3)).simplify() == 0 + raises(ValueError, lambda: E(X[1], Eq(X[2], 1))) + + #testing queries with multiple RandomIndexedSymbols + T = Matrix([[Rational(5, 10), Rational(3, 10), Rational(2, 10)], [Rational(2, 10), Rational(7, 10), Rational(1, 10)], [Rational(3, 10), Rational(3, 10), Rational(4, 10)]]) + Y = DiscreteMarkovChain("Y", [0, 1, 2], T) + assert P(Eq(Y[7], Y[5]), Eq(Y[2], 0)).round(5) == Float(0.44428, 5) + assert P(Gt(Y[3], Y[1]), Eq(Y[0], 0)).round(2) == Float(0.36, 2) + assert P(Le(Y[5], Y[10]), Eq(Y[4], 2)).round(6) == Float(0.583120, 6) + assert Float(P(Eq(Y[10], Y[5]), Eq(Y[4], 1)), 14) == Float(1 - P(Ne(Y[10], Y[5]), Eq(Y[4], 1)), 14) + assert Float(P(Gt(Y[8], Y[9]), Eq(Y[3], 2)), 14) == Float(1 - P(Le(Y[8], Y[9]), Eq(Y[3], 2)), 14) + assert Float(P(Lt(Y[1], Y[4]), Eq(Y[0], 0)), 14) == Float(1 - P(Ge(Y[1], Y[4]), Eq(Y[0], 0)), 14) + assert P(Eq(Y[5], Y[10]), Eq(Y[2], 1)) == P(Eq(Y[10], Y[5]), Eq(Y[2], 1)) + assert P(Gt(Y[1], Y[2]), Eq(Y[0], 1)) == P(Lt(Y[2], Y[1]), Eq(Y[0], 1)) + assert P(Ge(Y[7], Y[6]), Eq(Y[4], 1)) == P(Le(Y[6], Y[7]), Eq(Y[4], 1)) + + #test symbolic queries + a, b, c, d = symbols('a b c d') + T = Matrix([[Rational(1, 10), Rational(4, 10), Rational(5, 10)], [Rational(3, 10), Rational(4, 10), Rational(3, 10)], [Rational(7, 10), Rational(2, 10), Rational(1, 10)]]) + Y = DiscreteMarkovChain("Y", [0, 1, 2], T) + query = P(Eq(Y[a], b), Eq(Y[c], d)) + assert query.subs({a:10, b:2, c:5, d:1}).evalf().round(4) == P(Eq(Y[10], 2), Eq(Y[5], 1)).round(4) + assert query.subs({a:15, b:0, c:10, d:1}).evalf().round(4) == P(Eq(Y[15], 0), Eq(Y[10], 1)).round(4) + query_gt = P(Gt(Y[a], b), Eq(Y[c], d)) + query_le = P(Le(Y[a], b), Eq(Y[c], d)) + assert query_gt.subs({a:5, b:2, c:1, d:0}).evalf() + query_le.subs({a:5, b:2, c:1, d:0}).evalf() == 1.0 + query_ge = P(Ge(Y[a], b), Eq(Y[c], d)) + query_lt = P(Lt(Y[a], b), Eq(Y[c], d)) + assert query_ge.subs({a:4, b:1, c:0, d:2}).evalf() + query_lt.subs({a:4, b:1, c:0, d:2}).evalf() == 1.0 + + #test issue 20078 + assert (2*Y[1] + 3*Y[1]).simplify() == 5*Y[1] + assert (2*Y[1] - 3*Y[1]).simplify() == -Y[1] + assert (2*(0.25*Y[1])).simplify() == 0.5*Y[1] + assert ((2*Y[1]) * (0.25*Y[1])).simplify() == 0.5*Y[1]**2 + assert (Y[1]**2 + Y[1]**3).simplify() == (Y[1] + 1)*Y[1]**2 + +def test_sample_stochastic_process(): + if not import_module('scipy'): + skip('SciPy Not installed. Skip sampling tests') + import random + random.seed(0) + numpy = import_module('numpy') + if numpy: + numpy.random.seed(0) # scipy uses numpy to sample so to set its seed + T = Matrix([[0.5, 0.2, 0.3],[0.2, 0.5, 0.3],[0.2, 0.3, 0.5]]) + Y = DiscreteMarkovChain("Y", [0, 1, 2], T) + for samps in range(10): + assert next(sample_stochastic_process(Y)) in Y.state_space + Z = DiscreteMarkovChain("Z", ['1', 1, 0], T) + for samps in range(10): + assert next(sample_stochastic_process(Z)) in Z.state_space + + T = Matrix([[S.Half, Rational(1, 4), Rational(1, 4)], + [Rational(1, 3), 0, Rational(2, 3)], + [S.Half, S.Half, 0]]) + X = DiscreteMarkovChain('X', [0, 1, 2], T) + for samps in range(10): + assert next(sample_stochastic_process(X)) in X.state_space + W = DiscreteMarkovChain('W', [1, pi, oo], T) + for samps in range(10): + assert next(sample_stochastic_process(W)) in W.state_space + + +def test_ContinuousMarkovChain(): + T1 = Matrix([[S(-2), S(2), S.Zero], + [S.Zero, S.NegativeOne, S.One], + [Rational(3, 2), Rational(3, 2), S(-3)]]) + C1 = ContinuousMarkovChain('C', [0, 1, 2], T1) + assert C1.limiting_distribution() == ImmutableMatrix([[Rational(3, 19), Rational(12, 19), Rational(4, 19)]]) + + T2 = Matrix([[-S.One, S.One, S.Zero], [S.One, -S.One, S.Zero], [S.Zero, S.One, -S.One]]) + C2 = ContinuousMarkovChain('C', [0, 1, 2], T2) + A, t = C2.generator_matrix, symbols('t', positive=True) + assert C2.transition_probabilities(A)(t) == Matrix([[S.Half + exp(-2*t)/2, S.Half - exp(-2*t)/2, 0], + [S.Half - exp(-2*t)/2, S.Half + exp(-2*t)/2, 0], + [S.Half - exp(-t) + exp(-2*t)/2, S.Half - exp(-2*t)/2, exp(-t)]]) + with ignore_warnings(UserWarning): ### TODO: Restore tests once warnings are removed + assert P(Eq(C2(1), 1), Eq(C2(0), 1), evaluate=False) == Probability(Eq(C2(1), 1), Eq(C2(0), 1)) + assert P(Eq(C2(1), 1), Eq(C2(0), 1)) == exp(-2)/2 + S.Half + assert P(Eq(C2(1), 0) & Eq(C2(2), 1) & Eq(C2(3), 1), + Eq(P(Eq(C2(1), 0)), S.Half)) == (Rational(1, 4) - exp(-2)/4)*(exp(-2)/2 + S.Half) + assert P(Not(Eq(C2(1), 0) & Eq(C2(2), 1) & Eq(C2(3), 2)) | + (Eq(C2(1), 0) & Eq(C2(2), 1) & Eq(C2(3), 2)), + Eq(P(Eq(C2(1), 0)), Rational(1, 4)) & Eq(P(Eq(C2(1), 1)), Rational(1, 4))) is S.One + assert E(C2(Rational(3, 2)), Eq(C2(0), 2)) == -exp(-3)/2 + 2*exp(Rational(-3, 2)) + S.Half + assert variance(C2(Rational(3, 2)), Eq(C2(0), 1)) == ((S.Half - exp(-3)/2)**2*(exp(-3)/2 + S.Half) + + (Rational(-1, 2) - exp(-3)/2)**2*(S.Half - exp(-3)/2)) + raises(KeyError, lambda: P(Eq(C2(1), 0), Eq(P(Eq(C2(1), 1)), S.Half))) + assert P(Eq(C2(1), 0), Eq(P(Eq(C2(5), 1)), S.Half)) == Probability(Eq(C2(1), 0)) + TS1 = MatrixSymbol('G', 3, 3) + CS1 = ContinuousMarkovChain('C', [0, 1, 2], TS1) + A = CS1.generator_matrix + assert CS1.transition_probabilities(A)(t) == exp(t*A) + + C3 = ContinuousMarkovChain('C', [Symbol('0'), Symbol('1'), Symbol('2')], T2) + assert P(Eq(C3(1), 1), Eq(C3(0), 1)) == exp(-2)/2 + S.Half + assert P(Eq(C3(1), Symbol('1')), Eq(C3(0), Symbol('1'))) == exp(-2)/2 + S.Half + + #test probability queries + G = Matrix([[-S(1), Rational(1, 10), Rational(9, 10)], [Rational(2, 5), -S(1), Rational(3, 5)], [Rational(1, 2), Rational(1, 2), -S(1)]]) + C = ContinuousMarkovChain('C', state_space=[0, 1, 2], gen_mat=G) + assert P(Eq(C(7.385), C(3.19)), Eq(C(0.862), 0)).round(5) == Float(0.35469, 5) + assert P(Gt(C(98.715), C(19.807)), Eq(C(11.314), 2)).round(5) == Float(0.32452, 5) + assert P(Le(C(5.9), C(10.112)), Eq(C(4), 1)).round(6) == Float(0.675214, 6) + assert Float(P(Eq(C(7.32), C(2.91)), Eq(C(2.63), 1)), 14) == Float(1 - P(Ne(C(7.32), C(2.91)), Eq(C(2.63), 1)), 14) + assert Float(P(Gt(C(3.36), C(1.101)), Eq(C(0.8), 2)), 14) == Float(1 - P(Le(C(3.36), C(1.101)), Eq(C(0.8), 2)), 14) + assert Float(P(Lt(C(4.9), C(2.79)), Eq(C(1.61), 0)), 14) == Float(1 - P(Ge(C(4.9), C(2.79)), Eq(C(1.61), 0)), 14) + assert P(Eq(C(5.243), C(10.912)), Eq(C(2.174), 1)) == P(Eq(C(10.912), C(5.243)), Eq(C(2.174), 1)) + assert P(Gt(C(2.344), C(9.9)), Eq(C(1.102), 1)) == P(Lt(C(9.9), C(2.344)), Eq(C(1.102), 1)) + assert P(Ge(C(7.87), C(1.008)), Eq(C(0.153), 1)) == P(Le(C(1.008), C(7.87)), Eq(C(0.153), 1)) + + #test symbolic queries + a, b, c, d = symbols('a b c d') + query = P(Eq(C(a), b), Eq(C(c), d)) + assert query.subs({a:3.65, b:2, c:1.78, d:1}).evalf().round(10) == P(Eq(C(3.65), 2), Eq(C(1.78), 1)).round(10) + query_gt = P(Gt(C(a), b), Eq(C(c), d)) + query_le = P(Le(C(a), b), Eq(C(c), d)) + assert query_gt.subs({a:13.2, b:0, c:3.29, d:2}).evalf() + query_le.subs({a:13.2, b:0, c:3.29, d:2}).evalf() == 1.0 + query_ge = P(Ge(C(a), b), Eq(C(c), d)) + query_lt = P(Lt(C(a), b), Eq(C(c), d)) + assert query_ge.subs({a:7.43, b:1, c:1.45, d:0}).evalf() + query_lt.subs({a:7.43, b:1, c:1.45, d:0}).evalf() == 1.0 + + #test issue 20078 + assert (2*C(1) + 3*C(1)).simplify() == 5*C(1) + assert (2*C(1) - 3*C(1)).simplify() == -C(1) + assert (2*(0.25*C(1))).simplify() == 0.5*C(1) + assert (2*C(1) * 0.25*C(1)).simplify() == 0.5*C(1)**2 + assert (C(1)**2 + C(1)**3).simplify() == (C(1) + 1)*C(1)**2 + +def test_BernoulliProcess(): + + B = BernoulliProcess("B", p=0.6, success=1, failure=0) + assert B.state_space == FiniteSet(0, 1) + assert B.index_set == S.Naturals0 + assert B.success == 1 + assert B.failure == 0 + + X = BernoulliProcess("X", p=Rational(1,3), success='H', failure='T') + assert X.state_space == FiniteSet('H', 'T') + H, T = symbols("H,T") + assert E(X[1]+X[2]*X[3]) == H**2/9 + 4*H*T/9 + H/3 + 4*T**2/9 + 2*T/3 + + t, x = symbols('t, x', positive=True, integer=True) + assert isinstance(B[t], RandomIndexedSymbol) + + raises(ValueError, lambda: BernoulliProcess("X", p=1.1, success=1, failure=0)) + raises(NotImplementedError, lambda: B(t)) + + raises(IndexError, lambda: B[-3]) + assert B.joint_distribution(B[3], B[9]) == JointDistributionHandmade(Lambda((B[3], B[9]), + Piecewise((0.6, Eq(B[3], 1)), (0.4, Eq(B[3], 0)), (0, True)) + *Piecewise((0.6, Eq(B[9], 1)), (0.4, Eq(B[9], 0)), (0, True)))) + + assert B.joint_distribution(2, B[4]) == JointDistributionHandmade(Lambda((B[2], B[4]), + Piecewise((0.6, Eq(B[2], 1)), (0.4, Eq(B[2], 0)), (0, True)) + *Piecewise((0.6, Eq(B[4], 1)), (0.4, Eq(B[4], 0)), (0, True)))) + + # Test for the sum distribution of Bernoulli Process RVs + Y = B[1] + B[2] + B[3] + assert P(Eq(Y, 0)).round(2) == Float(0.06, 1) + assert P(Eq(Y, 2)).round(2) == Float(0.43, 2) + assert P(Eq(Y, 4)).round(2) == 0 + assert P(Gt(Y, 1)).round(2) == Float(0.65, 2) + # Test for independency of each Random Indexed variable + assert P(Eq(B[1], 0) & Eq(B[2], 1) & Eq(B[3], 0) & Eq(B[4], 1)).round(2) == Float(0.06, 1) + + assert E(2 * B[1] + B[2]).round(2) == Float(1.80, 3) + assert E(2 * B[1] + B[2] + 5).round(2) == Float(6.80, 3) + assert E(B[2] * B[4] + B[10]).round(2) == Float(0.96, 2) + assert E(B[2] > 0, Eq(B[1],1) & Eq(B[2],1)).round(2) == Float(0.60,2) + assert E(B[1]) == 0.6 + assert P(B[1] > 0).round(2) == Float(0.60, 2) + assert P(B[1] < 1).round(2) == Float(0.40, 2) + assert P(B[1] > 0, B[2] <= 1).round(2) == Float(0.60, 2) + assert P(B[12] * B[5] > 0).round(2) == Float(0.36, 2) + assert P(B[12] * B[5] > 0, B[4] < 1).round(2) == Float(0.36, 2) + assert P(Eq(B[2], 1), B[2] > 0) == 1.0 + assert P(Eq(B[5], 3)) == 0 + assert P(Eq(B[1], 1), B[1] < 0) == 0 + assert P(B[2] > 0, Eq(B[2], 1)) == 1 + assert P(B[2] < 0, Eq(B[2], 1)) == 0 + assert P(B[2] > 0, B[2]==7) == 0 + assert P(B[5] > 0, B[5]) == BernoulliDistribution(0.6, 0, 1) + raises(ValueError, lambda: P(3)) + raises(ValueError, lambda: P(B[3] > 0, 3)) + + # test issue 19456 + expr = Sum(B[t], (t, 0, 4)) + expr2 = Sum(B[t], (t, 1, 3)) + expr3 = Sum(B[t]**2, (t, 1, 3)) + assert expr.doit() == B[0] + B[1] + B[2] + B[3] + B[4] + assert expr2.doit() == Y + assert expr3.doit() == B[1]**2 + B[2]**2 + B[3]**2 + assert B[2*t].free_symbols == {B[2*t], t} + assert B[4].free_symbols == {B[4]} + assert B[x*t].free_symbols == {B[x*t], x, t} + + #test issue 20078 + assert (2*B[t] + 3*B[t]).simplify() == 5*B[t] + assert (2*B[t] - 3*B[t]).simplify() == -B[t] + assert (2*(0.25*B[t])).simplify() == 0.5*B[t] + assert (2*B[t] * 0.25*B[t]).simplify() == 0.5*B[t]**2 + assert (B[t]**2 + B[t]**3).simplify() == (B[t] + 1)*B[t]**2 + +def test_PoissonProcess(): + X = PoissonProcess("X", 3) + assert X.state_space == S.Naturals0 + assert X.index_set == Interval(0, oo) + assert X.lamda == 3 + + t, d, x, y = symbols('t d x y', positive=True) + assert isinstance(X(t), RandomIndexedSymbol) + assert X.distribution(t) == PoissonDistribution(3*t) + with warns_deprecated_sympy(): + X.distribution(X(t)) + raises(ValueError, lambda: PoissonProcess("X", -1)) + raises(NotImplementedError, lambda: X[t]) + raises(IndexError, lambda: X(-5)) + + assert X.joint_distribution(X(2), X(3)) == JointDistributionHandmade(Lambda((X(2), X(3)), + 6**X(2)*9**X(3)*exp(-15)/(factorial(X(2))*factorial(X(3))))) + + assert X.joint_distribution(4, 6) == JointDistributionHandmade(Lambda((X(4), X(6)), + 12**X(4)*18**X(6)*exp(-30)/(factorial(X(4))*factorial(X(6))))) + + assert P(X(t) < 1) == exp(-3*t) + assert P(Eq(X(t), 0), Contains(t, Interval.Lopen(3, 5))) == exp(-6) # exp(-2*lamda) + res = P(Eq(X(t), 1), Contains(t, Interval.Lopen(3, 4))) + assert res == 3*exp(-3) + + # Equivalent to P(Eq(X(t), 1))**4 because of non-overlapping intervals + assert P(Eq(X(t), 1) & Eq(X(d), 1) & Eq(X(x), 1) & Eq(X(y), 1), Contains(t, Interval.Lopen(0, 1)) + & Contains(d, Interval.Lopen(1, 2)) & Contains(x, Interval.Lopen(2, 3)) + & Contains(y, Interval.Lopen(3, 4))) == res**4 + + # Return Probability because of overlapping intervals + assert P(Eq(X(t), 2) & Eq(X(d), 3), Contains(t, Interval.Lopen(0, 2)) + & Contains(d, Interval.Ropen(2, 4))) == \ + Probability(Eq(X(d), 3) & Eq(X(t), 2), Contains(t, Interval.Lopen(0, 2)) + & Contains(d, Interval.Ropen(2, 4))) + + raises(ValueError, lambda: P(Eq(X(t), 2) & Eq(X(d), 3), + Contains(t, Interval.Lopen(0, 4)) & Contains(d, Interval.Lopen(3, oo)))) # no bound on d + assert P(Eq(X(3), 2)) == 81*exp(-9)/2 + assert P(Eq(X(t), 2), Contains(t, Interval.Lopen(0, 5))) == 225*exp(-15)/2 + + # Check that probability works correctly by adding it to 1 + res1 = P(X(t) <= 3, Contains(t, Interval.Lopen(0, 5))) + res2 = P(X(t) > 3, Contains(t, Interval.Lopen(0, 5))) + assert res1 == 691*exp(-15) + assert (res1 + res2).simplify() == 1 + + # Check Not and Or + assert P(Not(Eq(X(t), 2) & (X(d) > 3)), Contains(t, Interval.Ropen(2, 4)) & \ + Contains(d, Interval.Lopen(7, 8))).simplify() == -18*exp(-6) + 234*exp(-9) + 1 + assert P(Eq(X(t), 2) | Ne(X(t), 4), Contains(t, Interval.Ropen(2, 4))) == 1 - 36*exp(-6) + raises(ValueError, lambda: P(X(t) > 2, X(t) + X(d))) + assert E(X(t)) == 3*t # property of the distribution at a given timestamp + assert E(X(t)**2 + X(d)*2 + X(y)**3, Contains(t, Interval.Lopen(0, 1)) + & Contains(d, Interval.Lopen(1, 2)) & Contains(y, Interval.Ropen(3, 4))) == 75 + assert E(X(t)**2, Contains(t, Interval.Lopen(0, 1))) == 12 + assert E(x*(X(t) + X(d))*(X(t)**2+X(d)**2), Contains(t, Interval.Lopen(0, 1)) + & Contains(d, Interval.Ropen(1, 2))) == \ + Expectation(x*(X(d) + X(t))*(X(d)**2 + X(t)**2), Contains(t, Interval.Lopen(0, 1)) + & Contains(d, Interval.Ropen(1, 2))) + + # Value Error because of infinite time bound + raises(ValueError, lambda: E(X(t)**3, Contains(t, Interval.Lopen(1, oo)))) + + # Equivalent to E(X(t)**2) - E(X(d)**2) == E(X(1)**2) - E(X(1)**2) == 0 + assert E((X(t) + X(d))*(X(t) - X(d)), Contains(t, Interval.Lopen(0, 1)) + & Contains(d, Interval.Lopen(1, 2))) == 0 + assert E(X(2) + x*E(X(5))) == 15*x + 6 + assert E(x*X(1) + y) == 3*x + y + assert P(Eq(X(1), 2) & Eq(X(t), 3), Contains(t, Interval.Lopen(1, 2))) == 81*exp(-6)/4 + Y = PoissonProcess("Y", 6) + Z = X + Y + assert Z.lamda == X.lamda + Y.lamda == 9 + raises(ValueError, lambda: X + 5) # should be added be only PoissonProcess instance + N, M = Z.split(4, 5) + assert N.lamda == 4 + assert M.lamda == 5 + raises(ValueError, lambda: Z.split(3, 2)) # 2+3 != 9 + + raises(ValueError, lambda :P(Eq(X(t), 0), Contains(t, Interval.Lopen(1, 3)) & Eq(X(1), 0))) + # check if it handles queries with two random variables in one args + res1 = P(Eq(N(3), N(5))) + assert res1 == P(Eq(N(t), 0), Contains(t, Interval(3, 5))) + res2 = P(N(3) > N(1)) + assert res2 == P((N(t) > 0), Contains(t, Interval(1, 3))) + assert P(N(3) < N(1)) == 0 # condition is not possible + res3 = P(N(3) <= N(1)) # holds only for Eq(N(3), N(1)) + assert res3 == P(Eq(N(t), 0), Contains(t, Interval(1, 3))) + + # tests from https://www.probabilitycourse.com/chapter11/11_1_2_basic_concepts_of_the_poisson_process.php + X = PoissonProcess('X', 10) # 11.1 + assert P(Eq(X(S(1)/3), 3) & Eq(X(1), 10)) == exp(-10)*Rational(8000000000, 11160261) + assert P(Eq(X(1), 1), Eq(X(S(1)/3), 3)) == 0 + assert P(Eq(X(1), 10), Eq(X(S(1)/3), 3)) == P(Eq(X(S(2)/3), 7)) + + X = PoissonProcess('X', 2) # 11.2 + assert P(X(S(1)/2) < 1) == exp(-1) + assert P(X(3) < 1, Eq(X(1), 0)) == exp(-4) + assert P(Eq(X(4), 3), Eq(X(2), 3)) == exp(-4) + + X = PoissonProcess('X', 3) + assert P(Eq(X(2), 5) & Eq(X(1), 2)) == Rational(81, 4)*exp(-6) + + # check few properties + assert P(X(2) <= 3, X(1)>=1) == 3*P(Eq(X(1), 0)) + 2*P(Eq(X(1), 1)) + P(Eq(X(1), 2)) + assert P(X(2) <= 3, X(1) > 1) == 2*P(Eq(X(1), 0)) + 1*P(Eq(X(1), 1)) + assert P(Eq(X(2), 5) & Eq(X(1), 2)) == P(Eq(X(1), 3))*P(Eq(X(1), 2)) + assert P(Eq(X(3), 4), Eq(X(1), 3)) == P(Eq(X(2), 1)) + + #test issue 20078 + assert (2*X(t) + 3*X(t)).simplify() == 5*X(t) + assert (2*X(t) - 3*X(t)).simplify() == -X(t) + assert (2*(0.25*X(t))).simplify() == 0.5*X(t) + assert (2*X(t) * 0.25*X(t)).simplify() == 0.5*X(t)**2 + assert (X(t)**2 + X(t)**3).simplify() == (X(t) + 1)*X(t)**2 + +def test_WienerProcess(): + X = WienerProcess("X") + assert X.state_space == S.Reals + assert X.index_set == Interval(0, oo) + + t, d, x, y = symbols('t d x y', positive=True) + assert isinstance(X(t), RandomIndexedSymbol) + assert X.distribution(t) == NormalDistribution(0, sqrt(t)) + with warns_deprecated_sympy(): + X.distribution(X(t)) + raises(ValueError, lambda: PoissonProcess("X", -1)) + raises(NotImplementedError, lambda: X[t]) + raises(IndexError, lambda: X(-2)) + + assert X.joint_distribution(X(2), X(3)) == JointDistributionHandmade( + Lambda((X(2), X(3)), sqrt(6)*exp(-X(2)**2/4)*exp(-X(3)**2/6)/(12*pi))) + assert X.joint_distribution(4, 6) == JointDistributionHandmade( + Lambda((X(4), X(6)), sqrt(6)*exp(-X(4)**2/8)*exp(-X(6)**2/12)/(24*pi))) + + assert P(X(t) < 3).simplify() == erf(3*sqrt(2)/(2*sqrt(t)))/2 + S(1)/2 + assert P(X(t) > 2, Contains(t, Interval.Lopen(3, 7))).simplify() == S(1)/2 -\ + erf(sqrt(2)/2)/2 + + # Equivalent to P(X(1)>1)**4 + assert P((X(t) > 4) & (X(d) > 3) & (X(x) > 2) & (X(y) > 1), + Contains(t, Interval.Lopen(0, 1)) & Contains(d, Interval.Lopen(1, 2)) + & Contains(x, Interval.Lopen(2, 3)) & Contains(y, Interval.Lopen(3, 4))).simplify() ==\ + (1 - erf(sqrt(2)/2))*(1 - erf(sqrt(2)))*(1 - erf(3*sqrt(2)/2))*(1 - erf(2*sqrt(2)))/16 + + # Contains an overlapping interval so, return Probability + assert P((X(t)< 2) & (X(d)> 3), Contains(t, Interval.Lopen(0, 2)) + & Contains(d, Interval.Ropen(2, 4))) == Probability((X(d) > 3) & (X(t) < 2), + Contains(d, Interval.Ropen(2, 4)) & Contains(t, Interval.Lopen(0, 2))) + + assert str(P(Not((X(t) < 5) & (X(d) > 3)), Contains(t, Interval.Ropen(2, 4)) & + Contains(d, Interval.Lopen(7, 8))).simplify()) == \ + '-(1 - erf(3*sqrt(2)/2))*(2 - erfc(5/2))/4 + 1' + # Distribution has mean 0 at each timestamp + assert E(X(t)) == 0 + assert E(x*(X(t) + X(d))*(X(t)**2+X(d)**2), Contains(t, Interval.Lopen(0, 1)) + & Contains(d, Interval.Ropen(1, 2))) == Expectation(x*(X(d) + X(t))*(X(d)**2 + X(t)**2), + Contains(d, Interval.Ropen(1, 2)) & Contains(t, Interval.Lopen(0, 1))) + assert E(X(t) + x*E(X(3))) == 0 + + #test issue 20078 + assert (2*X(t) + 3*X(t)).simplify() == 5*X(t) + assert (2*X(t) - 3*X(t)).simplify() == -X(t) + assert (2*(0.25*X(t))).simplify() == 0.5*X(t) + assert (2*X(t) * 0.25*X(t)).simplify() == 0.5*X(t)**2 + assert (X(t)**2 + X(t)**3).simplify() == (X(t) + 1)*X(t)**2 + + +def test_GammaProcess_symbolic(): + t, d, x, y, g, l = symbols('t d x y g l', positive=True) + X = GammaProcess("X", l, g) + + raises(NotImplementedError, lambda: X[t]) + raises(IndexError, lambda: X(-1)) + assert isinstance(X(t), RandomIndexedSymbol) + assert X.state_space == Interval(0, oo) + assert X.distribution(t) == GammaDistribution(g*t, 1/l) + with warns_deprecated_sympy(): + X.distribution(X(t)) + assert X.joint_distribution(5, X(3)) == JointDistributionHandmade(Lambda( + (X(5), X(3)), l**(8*g)*exp(-l*X(3))*exp(-l*X(5))*X(3)**(3*g - 1)*X(5)**(5*g + - 1)/(gamma(3*g)*gamma(5*g)))) + # property of the gamma process at any given timestamp + assert E(X(t)) == g*t/l + assert variance(X(t)).simplify() == g*t/l**2 + + # Equivalent to E(2*X(1)) + E(X(1)**2) + E(X(1)**3), where E(X(1)) == g/l + assert E(X(t)**2 + X(d)*2 + X(y)**3, Contains(t, Interval.Lopen(0, 1)) + & Contains(d, Interval.Lopen(1, 2)) & Contains(y, Interval.Ropen(3, 4))) == \ + 2*g/l + (g**2 + g)/l**2 + (g**3 + 3*g**2 + 2*g)/l**3 + + assert P(X(t) > 3, Contains(t, Interval.Lopen(3, 4))).simplify() == \ + 1 - lowergamma(g, 3*l)/gamma(g) # equivalent to P(X(1)>3) + + + #test issue 20078 + assert (2*X(t) + 3*X(t)).simplify() == 5*X(t) + assert (2*X(t) - 3*X(t)).simplify() == -X(t) + assert (2*(0.25*X(t))).simplify() == 0.5*X(t) + assert (2*X(t) * 0.25*X(t)).simplify() == 0.5*X(t)**2 + assert (X(t)**2 + X(t)**3).simplify() == (X(t) + 1)*X(t)**2 +def test_GammaProcess_numeric(): + t, d, x, y = symbols('t d x y', positive=True) + X = GammaProcess("X", 1, 2) + assert X.state_space == Interval(0, oo) + assert X.index_set == Interval(0, oo) + assert X.lamda == 1 + assert X.gamma == 2 + + raises(ValueError, lambda: GammaProcess("X", -1, 2)) + raises(ValueError, lambda: GammaProcess("X", 0, -2)) + raises(ValueError, lambda: GammaProcess("X", -1, -2)) + + # all are independent because of non-overlapping intervals + assert P((X(t) > 4) & (X(d) > 3) & (X(x) > 2) & (X(y) > 1), Contains(t, + Interval.Lopen(0, 1)) & Contains(d, Interval.Lopen(1, 2)) & Contains(x, + Interval.Lopen(2, 3)) & Contains(y, Interval.Lopen(3, 4))).simplify() == \ + 120*exp(-10) + + # Check working with Not and Or + assert P(Not((X(t) < 5) & (X(d) > 3)), Contains(t, Interval.Ropen(2, 4)) & + Contains(d, Interval.Lopen(7, 8))).simplify() == -4*exp(-3) + 472*exp(-8)/3 + 1 + assert P((X(t) > 2) | (X(t) < 4), Contains(t, Interval.Ropen(1, 4))).simplify() == \ + -643*exp(-4)/15 + 109*exp(-2)/15 + 1 + + assert E(X(t)) == 2*t # E(X(t)) == gamma*t/l + assert E(X(2) + x*E(X(5))) == 10*x + 4 diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_symbolic_multivariate.py b/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_symbolic_multivariate.py new file mode 100644 index 0000000000000000000000000000000000000000..79979e20a6f10d2a2ddfe85ce4c8df145e98c3fd --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_symbolic_multivariate.py @@ -0,0 +1,172 @@ +from sympy.stats import Expectation, Normal, Variance, Covariance +from sympy.testing.pytest import raises +from sympy.core.symbol import symbols +from sympy.matrices.exceptions import ShapeError +from sympy.matrices.dense import Matrix +from sympy.matrices.expressions.matexpr import MatrixSymbol +from sympy.matrices.expressions.special import ZeroMatrix +from sympy.stats.rv import RandomMatrixSymbol +from sympy.stats.symbolic_multivariate_probability import (ExpectationMatrix, + VarianceMatrix, CrossCovarianceMatrix) + +j, k = symbols("j,k") + +A = MatrixSymbol("A", k, k) +B = MatrixSymbol("B", k, k) +C = MatrixSymbol("C", k, k) +D = MatrixSymbol("D", k, k) + +a = MatrixSymbol("a", k, 1) +b = MatrixSymbol("b", k, 1) + +A2 = MatrixSymbol("A2", 2, 2) +B2 = MatrixSymbol("B2", 2, 2) + +X = RandomMatrixSymbol("X", k, 1) +Y = RandomMatrixSymbol("Y", k, 1) +Z = RandomMatrixSymbol("Z", k, 1) +W = RandomMatrixSymbol("W", k, 1) + +R = RandomMatrixSymbol("R", k, k) + +X2 = RandomMatrixSymbol("X2", 2, 1) + +normal = Normal("normal", 0, 1) + +m1 = Matrix([ + [1, j*Normal("normal2", 2, 1)], + [normal, 0] +]) + +def test_multivariate_expectation(): + expr = Expectation(a) + assert expr == Expectation(a) == ExpectationMatrix(a) + assert expr.expand() == a + + expr = Expectation(X) + assert expr == Expectation(X) == ExpectationMatrix(X) + assert expr.shape == (k, 1) + assert expr.rows == k + assert expr.cols == 1 + assert isinstance(expr, ExpectationMatrix) + + expr = Expectation(A*X + b) + assert expr == ExpectationMatrix(A*X + b) + assert expr.expand() == A*ExpectationMatrix(X) + b + assert isinstance(expr, ExpectationMatrix) + assert expr.shape == (k, 1) + + expr = Expectation(m1*X2) + assert expr.expand() == expr + + expr = Expectation(A2*m1*B2*X2) + assert expr.args[0].args == (A2, m1, B2, X2) + assert expr.expand() == A2*ExpectationMatrix(m1*B2*X2) + + expr = Expectation((X + Y)*(X - Y).T) + assert expr.expand() == ExpectationMatrix(X*X.T) - ExpectationMatrix(X*Y.T) +\ + ExpectationMatrix(Y*X.T) - ExpectationMatrix(Y*Y.T) + + expr = Expectation(A*X + B*Y) + assert expr.expand() == A*ExpectationMatrix(X) + B*ExpectationMatrix(Y) + + assert Expectation(m1).doit() == Matrix([[1, 2*j], [0, 0]]) + + x1 = Matrix([ + [Normal('N11', 11, 1), Normal('N12', 12, 1)], + [Normal('N21', 21, 1), Normal('N22', 22, 1)] + ]) + x2 = Matrix([ + [Normal('M11', 1, 1), Normal('M12', 2, 1)], + [Normal('M21', 3, 1), Normal('M22', 4, 1)] + ]) + + assert Expectation(Expectation(x1 + x2)).doit(deep=False) == ExpectationMatrix(x1 + x2) + assert Expectation(Expectation(x1 + x2)).doit() == Matrix([[12, 14], [24, 26]]) + + +def test_multivariate_variance(): + raises(ShapeError, lambda: Variance(A)) + + expr = Variance(a) + assert expr == Variance(a) == VarianceMatrix(a) + assert expr.expand() == ZeroMatrix(k, k) + expr = Variance(a.T) + assert expr == Variance(a.T) == VarianceMatrix(a.T) + assert expr.expand() == ZeroMatrix(k, k) + + expr = Variance(X) + assert expr == Variance(X) == VarianceMatrix(X) + assert expr.shape == (k, k) + assert expr.rows == k + assert expr.cols == k + assert isinstance(expr, VarianceMatrix) + + expr = Variance(A*X) + assert expr == VarianceMatrix(A*X) + assert expr.expand() == A*VarianceMatrix(X)*A.T + assert isinstance(expr, VarianceMatrix) + assert expr.shape == (k, k) + + expr = Variance(A*B*X) + assert expr.expand() == A*B*VarianceMatrix(X)*B.T*A.T + + expr = Variance(m1*X2) + assert expr.expand() == expr + + expr = Variance(A2*m1*B2*X2) + assert expr.args[0].args == (A2, m1, B2, X2) + assert expr.expand() == expr + + expr = Variance(A*X + B*Y) + assert expr.expand() == 2*A*CrossCovarianceMatrix(X, Y)*B.T +\ + A*VarianceMatrix(X)*A.T + B*VarianceMatrix(Y)*B.T + +def test_multivariate_crosscovariance(): + raises(ShapeError, lambda: Covariance(X, Y.T)) + raises(ShapeError, lambda: Covariance(X, A)) + + + expr = Covariance(a.T, b.T) + assert expr.shape == (1, 1) + assert expr.expand() == ZeroMatrix(1, 1) + + expr = Covariance(a, b) + assert expr == Covariance(a, b) == CrossCovarianceMatrix(a, b) + assert expr.expand() == ZeroMatrix(k, k) + assert expr.shape == (k, k) + assert expr.rows == k + assert expr.cols == k + assert isinstance(expr, CrossCovarianceMatrix) + + expr = Covariance(A*X + a, b) + assert expr.expand() == ZeroMatrix(k, k) + + expr = Covariance(X, Y) + assert isinstance(expr, CrossCovarianceMatrix) + assert expr.expand() == expr + + expr = Covariance(X, X) + assert isinstance(expr, CrossCovarianceMatrix) + assert expr.expand() == VarianceMatrix(X) + + expr = Covariance(X + Y, Z) + assert isinstance(expr, CrossCovarianceMatrix) + assert expr.expand() == CrossCovarianceMatrix(X, Z) + CrossCovarianceMatrix(Y, Z) + + expr = Covariance(A*X, Y) + assert isinstance(expr, CrossCovarianceMatrix) + assert expr.expand() == A*CrossCovarianceMatrix(X, Y) + + expr = Covariance(X, B*Y) + assert isinstance(expr, CrossCovarianceMatrix) + assert expr.expand() == CrossCovarianceMatrix(X, Y)*B.T + + expr = Covariance(A*X + a, B.T*Y + b) + assert isinstance(expr, CrossCovarianceMatrix) + assert expr.expand() == A*CrossCovarianceMatrix(X, Y)*B + + expr = Covariance(A*X + B*Y + a, C.T*Z + D.T*W + b) + assert isinstance(expr, CrossCovarianceMatrix) + assert expr.expand() == A*CrossCovarianceMatrix(X, W)*D + A*CrossCovarianceMatrix(X, Z)*C \ + + B*CrossCovarianceMatrix(Y, W)*D + B*CrossCovarianceMatrix(Y, Z)*C diff --git a/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_symbolic_probability.py b/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_symbolic_probability.py new file mode 100644 index 0000000000000000000000000000000000000000..edac942ac081c0d44cafd31761b77bc577b6a3fd --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/stats/tests/test_symbolic_probability.py @@ -0,0 +1,175 @@ +from sympy.concrete.summations import Sum +from sympy.core.mul import Mul +from sympy.core.numbers import (oo, pi) +from sympy.core.relational import Eq +from sympy.core.symbol import (Dummy, symbols) +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import sin +from sympy.integrals.integrals import Integral +from sympy.core.expr import unchanged +from sympy.stats import (Normal, Poisson, variance, Covariance, Variance, + Probability, Expectation, Moment, CentralMoment) +from sympy.stats.rv import probability, expectation + + +def test_literal_probability(): + X = Normal('X', 2, 3) + Y = Normal('Y', 3, 4) + Z = Poisson('Z', 4) + W = Poisson('W', 3) + x = symbols('x', real=True) + y, w, z = symbols('y, w, z') + + assert Probability(X > 0).evaluate_integral() == probability(X > 0) + assert Probability(X > x).evaluate_integral() == probability(X > x) + assert Probability(X > 0).rewrite(Integral).doit() == probability(X > 0) + assert Probability(X > x).rewrite(Integral).doit() == probability(X > x) + + assert Expectation(X).evaluate_integral() == expectation(X) + assert Expectation(X).rewrite(Integral).doit() == expectation(X) + assert Expectation(X**2).evaluate_integral() == expectation(X**2) + assert Expectation(x*X).args == (x*X,) + assert Expectation(x*X).expand() == x*Expectation(X) + assert Expectation(2*X + 3*Y + z*X*Y).expand() == 2*Expectation(X) + 3*Expectation(Y) + z*Expectation(X*Y) + assert Expectation(2*X + 3*Y + z*X*Y).args == (2*X + 3*Y + z*X*Y,) + assert Expectation(sin(X)) == Expectation(sin(X)).expand() + assert Expectation(2*x*sin(X)*Y + y*X**2 + z*X*Y).expand() == 2*x*Expectation(sin(X)*Y) \ + + y*Expectation(X**2) + z*Expectation(X*Y) + assert Expectation(X + Y).expand() == Expectation(X) + Expectation(Y) + assert Expectation((X + Y)*(X - Y)).expand() == Expectation(X**2) - Expectation(Y**2) + assert Expectation((X + Y)*(X - Y)).expand().doit() == -12 + assert Expectation(X + Y, evaluate=True).doit() == 5 + assert Expectation(X + Expectation(Y)).doit() == 5 + assert Expectation(X + Expectation(Y)).doit(deep=False) == 2 + Expectation(Expectation(Y)) + assert Expectation(X + Expectation(Y + Expectation(2*X))).doit(deep=False) == 2 \ + + Expectation(Expectation(Y + Expectation(2*X))) + assert Expectation(X + Expectation(Y + Expectation(2*X))).doit() == 9 + assert Expectation(Expectation(2*X)).doit() == 4 + assert Expectation(Expectation(2*X)).doit(deep=False) == Expectation(2*X) + assert Expectation(4*Expectation(2*X)).doit(deep=False) == 4*Expectation(2*X) + assert Expectation((X + Y)**3).expand() == 3*Expectation(X*Y**2) +\ + 3*Expectation(X**2*Y) + Expectation(X**3) + Expectation(Y**3) + assert Expectation((X - Y)**3).expand() == 3*Expectation(X*Y**2) -\ + 3*Expectation(X**2*Y) + Expectation(X**3) - Expectation(Y**3) + assert Expectation((X - Y)**2).expand() == -2*Expectation(X*Y) +\ + Expectation(X**2) + Expectation(Y**2) + + assert Variance(w).args == (w,) + assert Variance(w).expand() == 0 + assert Variance(X).evaluate_integral() == Variance(X).rewrite(Integral).doit() == variance(X) + assert Variance(X + z).args == (X + z,) + assert Variance(X + z).expand() == Variance(X) + assert Variance(X*Y).args == (Mul(X, Y),) + assert type(Variance(X*Y)) == Variance + assert Variance(z*X).expand() == z**2*Variance(X) + assert Variance(X + Y).expand() == Variance(X) + Variance(Y) + 2*Covariance(X, Y) + assert Variance(X + Y + Z + W).expand() == (Variance(X) + Variance(Y) + Variance(Z) + Variance(W) + + 2 * Covariance(X, Y) + 2 * Covariance(X, Z) + 2 * Covariance(X, W) + + 2 * Covariance(Y, Z) + 2 * Covariance(Y, W) + 2 * Covariance(W, Z)) + assert Variance(X**2).evaluate_integral() == variance(X**2) + assert unchanged(Variance, X**2) + assert Variance(x*X**2).expand() == x**2*Variance(X**2) + assert Variance(sin(X)).args == (sin(X),) + assert Variance(sin(X)).expand() == Variance(sin(X)) + assert Variance(x*sin(X)).expand() == x**2*Variance(sin(X)) + + assert Covariance(w, z).args == (w, z) + assert Covariance(w, z).expand() == 0 + assert Covariance(X, w).expand() == 0 + assert Covariance(w, X).expand() == 0 + assert Covariance(X, Y).args == (X, Y) + assert type(Covariance(X, Y)) == Covariance + assert Covariance(z*X + 3, Y).expand() == z*Covariance(X, Y) + assert Covariance(X, X).args == (X, X) + assert Covariance(X, X).expand() == Variance(X) + assert Covariance(z*X + 3, w*Y + 4).expand() == w*z*Covariance(X,Y) + assert Covariance(X, Y) == Covariance(Y, X) + assert Covariance(X + Y, Z + W).expand() == Covariance(W, X) + Covariance(W, Y) + Covariance(X, Z) + Covariance(Y, Z) + assert Covariance(x*X + y*Y, z*Z + w*W).expand() == (x*w*Covariance(W, X) + w*y*Covariance(W, Y) + + x*z*Covariance(X, Z) + y*z*Covariance(Y, Z)) + assert Covariance(x*X**2 + y*sin(Y), z*Y*Z**2 + w*W).expand() == (w*x*Covariance(W, X**2) + w*y*Covariance(sin(Y), W) + + x*z*Covariance(Y*Z**2, X**2) + y*z*Covariance(Y*Z**2, sin(Y))) + assert Covariance(X, X**2).expand() == Covariance(X, X**2) + assert Covariance(X, sin(X)).expand() == Covariance(sin(X), X) + assert Covariance(X**2, sin(X)*Y).expand() == Covariance(sin(X)*Y, X**2) + assert Covariance(w, X).evaluate_integral() == 0 + + +def test_probability_rewrite(): + X = Normal('X', 2, 3) + Y = Normal('Y', 3, 4) + Z = Poisson('Z', 4) + W = Poisson('W', 3) + x, y, w, z = symbols('x, y, w, z') + + assert Variance(w).rewrite(Expectation) == 0 + assert Variance(X).rewrite(Expectation) == Expectation(X ** 2) - Expectation(X) ** 2 + assert Variance(X, condition=Y).rewrite(Expectation) == Expectation(X ** 2, Y) - Expectation(X, Y) ** 2 + assert Variance(X, Y) != Expectation(X**2) - Expectation(X)**2 + assert Variance(X + z).rewrite(Expectation) == Expectation((X + z) ** 2) - Expectation(X + z) ** 2 + assert Variance(X * Y).rewrite(Expectation) == Expectation(X ** 2 * Y ** 2) - Expectation(X * Y) ** 2 + + assert Covariance(w, X).rewrite(Expectation) == -w*Expectation(X) + Expectation(w*X) + assert Covariance(X, Y).rewrite(Expectation) == Expectation(X*Y) - Expectation(X)*Expectation(Y) + assert Covariance(X, Y, condition=W).rewrite(Expectation) == Expectation(X * Y, W) - Expectation(X, W) * Expectation(Y, W) + + w, x, z = symbols("W, x, z") + px = Probability(Eq(X, x)) + pz = Probability(Eq(Z, z)) + + assert Expectation(X).rewrite(Probability) == Integral(x*px, (x, -oo, oo)) + assert Expectation(Z).rewrite(Probability) == Sum(z*pz, (z, 0, oo)) + assert Variance(X).rewrite(Probability) == Integral(x**2*px, (x, -oo, oo)) - Integral(x*px, (x, -oo, oo))**2 + assert Variance(Z).rewrite(Probability) == Sum(z**2*pz, (z, 0, oo)) - Sum(z*pz, (z, 0, oo))**2 + assert Covariance(w, X).rewrite(Probability) == \ + -w*Integral(x*Probability(Eq(X, x)), (x, -oo, oo)) + Integral(w*x*Probability(Eq(X, x)), (x, -oo, oo)) + + # To test rewrite as sum function + assert Variance(X).rewrite(Sum) == Variance(X).rewrite(Integral) + assert Expectation(X).rewrite(Sum) == Expectation(X).rewrite(Integral) + + assert Covariance(w, X).rewrite(Sum) == 0 + + assert Covariance(w, X).rewrite(Integral) == 0 + + assert Variance(X, condition=Y).rewrite(Probability) == Integral(x**2*Probability(Eq(X, x), Y), (x, -oo, oo)) - \ + Integral(x*Probability(Eq(X, x), Y), (x, -oo, oo))**2 + + +def test_symbolic_Moment(): + mu = symbols('mu', real=True) + sigma = symbols('sigma', positive=True) + x = symbols('x') + X = Normal('X', mu, sigma) + M = Moment(X, 4, 2) + assert M.rewrite(Expectation) == Expectation((X - 2)**4) + assert M.rewrite(Probability) == Integral((x - 2)**4*Probability(Eq(X, x)), + (x, -oo, oo)) + k = Dummy('k') + expri = Integral(sqrt(2)*(k - 2)**4*exp(-(k - \ + mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (k, -oo, oo)) + assert M.rewrite(Integral).dummy_eq(expri) + assert M.doit() == (mu**4 - 8*mu**3 + 6*mu**2*sigma**2 + \ + 24*mu**2 - 24*mu*sigma**2 - 32*mu + 3*sigma**4 + 24*sigma**2 + 16) + M = Moment(2, 5) + assert M.doit() == 2**5 + + +def test_symbolic_CentralMoment(): + mu = symbols('mu', real=True) + sigma = symbols('sigma', positive=True) + x = symbols('x') + X = Normal('X', mu, sigma) + CM = CentralMoment(X, 6) + assert CM.rewrite(Expectation) == Expectation((X - Expectation(X))**6) + assert CM.rewrite(Probability) == Integral((x - Integral(x*Probability(True), + (x, -oo, oo)))**6*Probability(Eq(X, x)), (x, -oo, oo)) + k = Dummy('k') + expri = Integral(sqrt(2)*(k - Integral(sqrt(2)*k*exp(-(k - \ + mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (k, -oo, oo)))**6*exp(-(k - \ + mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (k, -oo, oo)) + assert CM.rewrite(Integral).dummy_eq(expri) + assert CM.doit().simplify() == 15*sigma**6 + CM = Moment(5, 5) + assert CM.doit() == 5**5 diff --git a/.venv/lib/python3.12/site-packages/sympy/strategies/__init__.py b/.venv/lib/python3.12/site-packages/sympy/strategies/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..bb4c5aa8afe6fd818e136ec0797b7429e2da76cf --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/strategies/__init__.py @@ -0,0 +1,50 @@ +""" Rewrite Rules + +DISCLAIMER: This module is experimental. The interface is subject to change. + +A rule is a function that transforms one expression into another + + Rule :: Expr -> Expr + +A strategy is a function that says how a rule should be applied to a syntax +tree. In general strategies take rules and produce a new rule + + Strategy :: [Rules], Other-stuff -> Rule + +This allows developers to separate a mathematical transformation from the +algorithmic details of applying that transformation. The goal is to separate +the work of mathematical programming from algorithmic programming. + +Submodules + +strategies.rl - some fundamental rules +strategies.core - generic non-SymPy specific strategies +strategies.traverse - strategies that traverse a SymPy tree +strategies.tools - some conglomerate strategies that do depend on SymPy +""" + +from . import rl +from . import traverse +from .rl import rm_id, unpack, flatten, sort, glom, distribute, rebuild +from .util import new +from .core import ( + condition, debug, chain, null_safe, do_one, exhaust, minimize, tryit) +from .tools import canon, typed +from . import branch + +__all__ = [ + 'rl', + + 'traverse', + + 'rm_id', 'unpack', 'flatten', 'sort', 'glom', 'distribute', 'rebuild', + + 'new', + + 'condition', 'debug', 'chain', 'null_safe', 'do_one', 'exhaust', + 'minimize', 'tryit', + + 'canon', 'typed', + + 'branch', +] diff --git a/.venv/lib/python3.12/site-packages/sympy/strategies/__pycache__/__init__.cpython-312.pyc 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a/.venv/lib/python3.12/site-packages/sympy/strategies/branch/__init__.py b/.venv/lib/python3.12/site-packages/sympy/strategies/branch/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..fec5afe84a58f3d887a8c762692a3673a2b6d4c8 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/strategies/branch/__init__.py @@ -0,0 +1,14 @@ +from . import traverse +from .core import ( + condition, debug, multiplex, exhaust, notempty, + chain, onaction, sfilter, yieldify, do_one, identity) +from .tools import canon + +__all__ = [ + 'traverse', + + 'condition', 'debug', 'multiplex', 'exhaust', 'notempty', 'chain', + 'onaction', 'sfilter', 'yieldify', 'do_one', 'identity', + + 'canon', +] diff --git a/.venv/lib/python3.12/site-packages/sympy/strategies/branch/__pycache__/__init__.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/strategies/branch/__pycache__/__init__.cpython-312.pyc new file mode 100644 index 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+def exhaust(brule): + """ Apply a branching rule repeatedly until it has no effect """ + def exhaust_brl(expr): + seen = {expr} + for nexpr in brule(expr): + if nexpr not in seen: + seen.add(nexpr) + yield from exhaust_brl(nexpr) + if seen == {expr}: + yield expr + return exhaust_brl + + +def onaction(brule, fn): + def onaction_brl(expr): + for result in brule(expr): + if result != expr: + fn(brule, expr, result) + yield result + return onaction_brl + + +def debug(brule, file=None): + """ Print the input and output expressions at each rule application """ + if not file: + from sys import stdout + file = stdout + + def write(brl, expr, result): + file.write("Rule: %s\n" % brl.__name__) + file.write("In: %s\nOut: %s\n\n" % (expr, result)) + + return onaction(brule, write) + + +def multiplex(*brules): + """ Multiplex many branching rules into one """ + def multiplex_brl(expr): + seen = set() + for brl in brules: + for nexpr in brl(expr): + if nexpr not in seen: + seen.add(nexpr) + yield nexpr + return multiplex_brl + + +def condition(cond, brule): + """ Only apply branching rule if condition is true """ + def conditioned_brl(expr): + if cond(expr): + yield from brule(expr) + else: + pass + return conditioned_brl + + +def sfilter(pred, brule): + """ Yield only those results which satisfy the predicate """ + def filtered_brl(expr): + yield from filter(pred, brule(expr)) + return filtered_brl + + +def notempty(brule): + def notempty_brl(expr): + yielded = False + for nexpr in brule(expr): + yielded = True + yield nexpr + if not yielded: + yield expr + return notempty_brl + + +def do_one(*brules): + """ Execute one of the branching rules """ + def do_one_brl(expr): + yielded = False + for brl in brules: + for nexpr in brl(expr): + yielded = True + yield nexpr + if yielded: + return + return do_one_brl + + +def chain(*brules): + """ + Compose a sequence of brules so that they apply to the expr sequentially + """ + def chain_brl(expr): + if not brules: + yield expr + return + + head, tail = brules[0], brules[1:] + for nexpr in head(expr): + yield from chain(*tail)(nexpr) + + return chain_brl + + +def yieldify(rl): + """ Turn a rule into a branching rule """ + def brl(expr): + yield rl(expr) + return brl diff --git a/.venv/lib/python3.12/site-packages/sympy/strategies/branch/tests/__init__.py b/.venv/lib/python3.12/site-packages/sympy/strategies/branch/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/.venv/lib/python3.12/site-packages/sympy/strategies/branch/tests/__pycache__/__init__.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/strategies/branch/tests/__pycache__/__init__.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..ddc76666438b5f176ae4f3d9c6a7806926cf3cd2 Binary files /dev/null and b/.venv/lib/python3.12/site-packages/sympy/strategies/branch/tests/__pycache__/__init__.cpython-312.pyc differ diff --git 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a/.venv/lib/python3.12/site-packages/sympy/strategies/branch/tests/__pycache__/test_traverse.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/strategies/branch/tests/__pycache__/test_traverse.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..7a6e137c9fcc5069f113755f572c6ca599585401 Binary files /dev/null and b/.venv/lib/python3.12/site-packages/sympy/strategies/branch/tests/__pycache__/test_traverse.cpython-312.pyc differ diff --git a/.venv/lib/python3.12/site-packages/sympy/strategies/branch/tests/test_core.py b/.venv/lib/python3.12/site-packages/sympy/strategies/branch/tests/test_core.py new file mode 100644 index 0000000000000000000000000000000000000000..ac620b0afb6dbadc4d97b29ddbb341cd920b6588 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/strategies/branch/tests/test_core.py @@ -0,0 +1,117 @@ +from sympy.strategies.branch.core import ( + exhaust, debug, multiplex, condition, notempty, chain, onaction, sfilter, + yieldify, do_one, identity) + + +def posdec(x): + if x > 0: + yield x - 1 + else: + yield x + + +def branch5(x): + if 0 < x < 5: + yield x - 1 + elif 5 < x < 10: + yield x + 1 + elif x == 5: + yield x + 1 + yield x - 1 + else: + yield x + + +def even(x): + return x % 2 == 0 + + +def inc(x): + yield x + 1 + + +def one_to_n(n): + yield from range(n) + + +def test_exhaust(): + brl = exhaust(branch5) + assert set(brl(3)) == {0} + assert set(brl(7)) == {10} + assert set(brl(5)) == {0, 10} + + +def test_debug(): + from io import StringIO + file = StringIO() + rl = debug(posdec, file) + list(rl(5)) + log = file.getvalue() + file.close() + + assert posdec.__name__ in log + assert '5' in log + assert '4' in log + + +def test_multiplex(): + brl = multiplex(posdec, branch5) + assert set(brl(3)) == {2} + assert set(brl(7)) == {6, 8} + assert set(brl(5)) == {4, 6} + + +def test_condition(): + brl = condition(even, branch5) + assert set(brl(4)) == set(branch5(4)) + assert set(brl(5)) == set() + + +def test_sfilter(): + brl = sfilter(even, one_to_n) + assert set(brl(10)) == {0, 2, 4, 6, 8} + + +def test_notempty(): + def ident_if_even(x): + if even(x): + yield x + + brl = notempty(ident_if_even) + assert set(brl(4)) == {4} + assert set(brl(5)) == {5} + + +def test_chain(): + assert list(chain()(2)) == [2] # identity + assert list(chain(inc, inc)(2)) == [4] + assert list(chain(branch5, inc)(4)) == [4] + assert set(chain(branch5, inc)(5)) == {5, 7} + assert list(chain(inc, branch5)(5)) == [7] + + +def test_onaction(): + L = [] + + def record(fn, input, output): + L.append((input, output)) + + list(onaction(inc, record)(2)) + assert L == [(2, 3)] + + list(onaction(identity, record)(2)) + assert L == [(2, 3)] + + +def test_yieldify(): + yinc = yieldify(lambda x: x + 1) + assert list(yinc(3)) == [4] + + +def test_do_one(): + def bad(expr): + raise ValueError + + assert list(do_one(inc)(3)) == [4] + assert list(do_one(inc, bad)(3)) == [4] + assert list(do_one(inc, posdec)(3)) == [4] diff --git a/.venv/lib/python3.12/site-packages/sympy/strategies/branch/tests/test_tools.py b/.venv/lib/python3.12/site-packages/sympy/strategies/branch/tests/test_tools.py new file mode 100644 index 0000000000000000000000000000000000000000..c2bd224030c337f0a000d94f6e7e65f3b8bd118f --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/strategies/branch/tests/test_tools.py @@ -0,0 +1,42 @@ +from sympy.strategies.branch.tools import canon +from sympy.core.basic import Basic +from sympy.core.numbers import Integer +from sympy.core.singleton import S + + +def posdec(x): + if isinstance(x, Integer) and x > 0: + yield x - 1 + else: + yield x + + +def branch5(x): + if isinstance(x, Integer): + if 0 < x < 5: + yield x - 1 + elif 5 < x < 10: + yield x + 1 + elif x == 5: + yield x + 1 + yield x - 1 + else: + yield x + + +def test_zero_ints(): + expr = Basic(S(2), Basic(S(5), S(3)), S(8)) + expected = {Basic(S(0), Basic(S(0), S(0)), S(0))} + + brl = canon(posdec) + assert set(brl(expr)) == expected + + +def test_split5(): + expr = Basic(S(2), Basic(S(5), S(3)), S(8)) + expected = { + Basic(S(0), Basic(S(0), S(0)), S(10)), + Basic(S(0), Basic(S(10), S(0)), S(10))} + + brl = canon(branch5) + assert set(brl(expr)) == expected diff --git a/.venv/lib/python3.12/site-packages/sympy/strategies/branch/tests/test_traverse.py b/.venv/lib/python3.12/site-packages/sympy/strategies/branch/tests/test_traverse.py new file mode 100644 index 0000000000000000000000000000000000000000..e051928210981223004de28b8c617d0438e11ac6 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/strategies/branch/tests/test_traverse.py @@ -0,0 +1,53 @@ +from sympy.core.basic import Basic +from sympy.core.numbers import Integer +from sympy.core.singleton import S +from sympy.strategies.branch.traverse import top_down, sall +from sympy.strategies.branch.core import do_one, identity + + +def inc(x): + if isinstance(x, Integer): + yield x + 1 + + +def test_top_down_easy(): + expr = Basic(S(1), S(2)) + expected = Basic(S(2), S(3)) + brl = top_down(inc) + + assert set(brl(expr)) == {expected} + + +def test_top_down_big_tree(): + expr = Basic(S(1), Basic(S(2)), Basic(S(3), Basic(S(4)), S(5))) + expected = Basic(S(2), Basic(S(3)), Basic(S(4), Basic(S(5)), S(6))) + brl = top_down(inc) + + assert set(brl(expr)) == {expected} + + +def test_top_down_harder_function(): + def split5(x): + if x == 5: + yield x - 1 + yield x + 1 + + expr = Basic(Basic(S(5), S(6)), S(1)) + expected = {Basic(Basic(S(4), S(6)), S(1)), Basic(Basic(S(6), S(6)), S(1))} + brl = top_down(split5) + + assert set(brl(expr)) == expected + + +def test_sall(): + expr = Basic(S(1), S(2)) + expected = Basic(S(2), S(3)) + brl = sall(inc) + + assert list(brl(expr)) == [expected] + + expr = Basic(S(1), S(2), Basic(S(3), S(4))) + expected = Basic(S(2), S(3), Basic(S(3), S(4))) + brl = sall(do_one(inc, identity)) + + assert list(brl(expr)) == [expected] diff --git a/.venv/lib/python3.12/site-packages/sympy/strategies/branch/tools.py b/.venv/lib/python3.12/site-packages/sympy/strategies/branch/tools.py new file mode 100644 index 0000000000000000000000000000000000000000..a6c9097323a7962080ae4497ead976818e386518 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/strategies/branch/tools.py @@ -0,0 +1,12 @@ +from .core import exhaust, multiplex +from .traverse import top_down + + +def canon(*rules): + """ Strategy for canonicalization + + Apply each branching rule in a top-down fashion through the tree. + Multiplex through all branching rule traversals + Keep doing this until there is no change. + """ + return exhaust(multiplex(*map(top_down, rules))) diff --git a/.venv/lib/python3.12/site-packages/sympy/strategies/branch/traverse.py b/.venv/lib/python3.12/site-packages/sympy/strategies/branch/traverse.py new file mode 100644 index 0000000000000000000000000000000000000000..28b7098dbda401fc0f0b6d27988d8c37e2f231ae --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/strategies/branch/traverse.py @@ -0,0 +1,25 @@ +""" Branching Strategies to Traverse a Tree """ +from itertools import product +from sympy.strategies.util import basic_fns +from .core import chain, identity, do_one + + +def top_down(brule, fns=basic_fns): + """ Apply a rule down a tree running it on the top nodes first """ + return chain(do_one(brule, identity), + lambda expr: sall(top_down(brule, fns), fns)(expr)) + + +def sall(brule, fns=basic_fns): + """ Strategic all - apply rule to args """ + op, new, children, leaf = map(fns.get, ('op', 'new', 'children', 'leaf')) + + def all_rl(expr): + if leaf(expr): + yield expr + else: + myop = op(expr) + argss = product(*map(brule, children(expr))) + for args in argss: + yield new(myop, *args) + return all_rl diff --git a/.venv/lib/python3.12/site-packages/sympy/strategies/core.py b/.venv/lib/python3.12/site-packages/sympy/strategies/core.py new file mode 100644 index 0000000000000000000000000000000000000000..75b75cb5f2e0693eea98a7b1c9b3e7f036ec26f6 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/strategies/core.py @@ -0,0 +1,151 @@ +""" Generic SymPy-Independent Strategies """ +from __future__ import annotations +from collections.abc import Callable, Mapping +from typing import TypeVar +from sys import stdout + + +_S = TypeVar('_S') +_T = TypeVar('_T') + + +def identity(x: _T) -> _T: + return x + + +def exhaust(rule: Callable[[_T], _T]) -> Callable[[_T], _T]: + """ Apply a rule repeatedly until it has no effect """ + def exhaustive_rl(expr: _T) -> _T: + new, old = rule(expr), expr + while new != old: + new, old = rule(new), new + return new + return exhaustive_rl + + +def memoize(rule: Callable[[_S], _T]) -> Callable[[_S], _T]: + """Memoized version of a rule + + Notes + ===== + + This cache can grow infinitely, so it is not recommended to use this + than ``functools.lru_cache`` unless you need very heavy computation. + """ + cache: dict[_S, _T] = {} + + def memoized_rl(expr: _S) -> _T: + if expr in cache: + return cache[expr] + else: + result = rule(expr) + cache[expr] = result + return result + return memoized_rl + + +def condition( + cond: Callable[[_T], bool], rule: Callable[[_T], _T] +) -> Callable[[_T], _T]: + """ Only apply rule if condition is true """ + def conditioned_rl(expr: _T) -> _T: + if cond(expr): + return rule(expr) + return expr + return conditioned_rl + + +def chain(*rules: Callable[[_T], _T]) -> Callable[[_T], _T]: + """ + Compose a sequence of rules so that they apply to the expr sequentially + """ + def chain_rl(expr: _T) -> _T: + for rule in rules: + expr = rule(expr) + return expr + return chain_rl + + +def debug(rule, file=None): + """ Print out before and after expressions each time rule is used """ + if file is None: + file = stdout + + def debug_rl(*args, **kwargs): + expr = args[0] + result = rule(*args, **kwargs) + if result != expr: + file.write("Rule: %s\n" % rule.__name__) + file.write("In: %s\nOut: %s\n\n" % (expr, result)) + return result + return debug_rl + + +def null_safe(rule: Callable[[_T], _T | None]) -> Callable[[_T], _T]: + """ Return original expr if rule returns None """ + def null_safe_rl(expr: _T) -> _T: + result = rule(expr) + if result is None: + return expr + return result + return null_safe_rl + + +def tryit(rule: Callable[[_T], _T], exception) -> Callable[[_T], _T]: + """ Return original expr if rule raises exception """ + def try_rl(expr: _T) -> _T: + try: + return rule(expr) + except exception: + return expr + return try_rl + + +def do_one(*rules: Callable[[_T], _T]) -> Callable[[_T], _T]: + """ Try each of the rules until one works. Then stop. """ + def do_one_rl(expr: _T) -> _T: + for rl in rules: + result = rl(expr) + if result != expr: + return result + return expr + return do_one_rl + + +def switch( + key: Callable[[_S], _T], + ruledict: Mapping[_T, Callable[[_S], _S]] +) -> Callable[[_S], _S]: + """ Select a rule based on the result of key called on the function """ + def switch_rl(expr: _S) -> _S: + rl = ruledict.get(key(expr), identity) + return rl(expr) + return switch_rl + + +# XXX Untyped default argument for minimize function +# where python requires SupportsRichComparison type +def _identity(x): + return x + + +def minimize( + *rules: Callable[[_S], _T], + objective=_identity +) -> Callable[[_S], _T]: + """ Select result of rules that minimizes objective + + >>> from sympy.strategies import minimize + >>> inc = lambda x: x + 1 + >>> dec = lambda x: x - 1 + >>> rl = minimize(inc, dec) + >>> rl(4) + 3 + + >>> rl = minimize(inc, dec, objective=lambda x: -x) # maximize + >>> rl(4) + 5 + """ + def minrule(expr: _S) -> _T: + return min([rule(expr) for rule in rules], key=objective) + return minrule diff --git a/.venv/lib/python3.12/site-packages/sympy/strategies/rl.py b/.venv/lib/python3.12/site-packages/sympy/strategies/rl.py new file mode 100644 index 0000000000000000000000000000000000000000..e84ee90582fafcf36fd4205a58b05b650875a9a5 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/strategies/rl.py @@ -0,0 +1,176 @@ +""" Generic Rules for SymPy + +This file assumes knowledge of Basic and little else. +""" +from sympy.utilities.iterables import sift +from .util import new + + +# Functions that create rules +def rm_id(isid, new=new): + """ Create a rule to remove identities. + + isid - fn :: x -> Bool --- whether or not this element is an identity. + + Examples + ======== + + >>> from sympy.strategies import rm_id + >>> from sympy import Basic, S + >>> remove_zeros = rm_id(lambda x: x==0) + >>> remove_zeros(Basic(S(1), S(0), S(2))) + Basic(1, 2) + >>> remove_zeros(Basic(S(0), S(0))) # If only identities then we keep one + Basic(0) + + See Also: + unpack + """ + def ident_remove(expr): + """ Remove identities """ + ids = list(map(isid, expr.args)) + if sum(ids) == 0: # No identities. Common case + return expr + elif sum(ids) != len(ids): # there is at least one non-identity + return new(expr.__class__, + *[arg for arg, x in zip(expr.args, ids) if not x]) + else: + return new(expr.__class__, expr.args[0]) + + return ident_remove + + +def glom(key, count, combine): + """ Create a rule to conglomerate identical args. + + Examples + ======== + + >>> from sympy.strategies import glom + >>> from sympy import Add + >>> from sympy.abc import x + + >>> key = lambda x: x.as_coeff_Mul()[1] + >>> count = lambda x: x.as_coeff_Mul()[0] + >>> combine = lambda cnt, arg: cnt * arg + >>> rl = glom(key, count, combine) + + >>> rl(Add(x, -x, 3*x, 2, 3, evaluate=False)) + 3*x + 5 + + Wait, how are key, count and combine supposed to work? + + >>> key(2*x) + x + >>> count(2*x) + 2 + >>> combine(2, x) + 2*x + """ + def conglomerate(expr): + """ Conglomerate together identical args x + x -> 2x """ + groups = sift(expr.args, key) + counts = {k: sum(map(count, args)) for k, args in groups.items()} + newargs = [combine(cnt, mat) for mat, cnt in counts.items()] + if set(newargs) != set(expr.args): + return new(type(expr), *newargs) + else: + return expr + + return conglomerate + + +def sort(key, new=new): + """ Create a rule to sort by a key function. + + Examples + ======== + + >>> from sympy.strategies import sort + >>> from sympy import Basic, S + >>> sort_rl = sort(str) + >>> sort_rl(Basic(S(3), S(1), S(2))) + Basic(1, 2, 3) + """ + + def sort_rl(expr): + return new(expr.__class__, *sorted(expr.args, key=key)) + return sort_rl + + +def distribute(A, B): + """ Turns an A containing Bs into a B of As + + where A, B are container types + + >>> from sympy.strategies import distribute + >>> from sympy import Add, Mul, symbols + >>> x, y = symbols('x,y') + >>> dist = distribute(Mul, Add) + >>> expr = Mul(2, x+y, evaluate=False) + >>> expr + 2*(x + y) + >>> dist(expr) + 2*x + 2*y + """ + + def distribute_rl(expr): + for i, arg in enumerate(expr.args): + if isinstance(arg, B): + first, b, tail = expr.args[:i], expr.args[i], expr.args[i + 1:] + return B(*[A(*(first + (arg,) + tail)) for arg in b.args]) + return expr + return distribute_rl + + +def subs(a, b): + """ Replace expressions exactly """ + def subs_rl(expr): + if expr == a: + return b + else: + return expr + return subs_rl + + +# Functions that are rules +def unpack(expr): + """ Rule to unpack singleton args + + >>> from sympy.strategies import unpack + >>> from sympy import Basic, S + >>> unpack(Basic(S(2))) + 2 + """ + if len(expr.args) == 1: + return expr.args[0] + else: + return expr + + +def flatten(expr, new=new): + """ Flatten T(a, b, T(c, d), T2(e)) to T(a, b, c, d, T2(e)) """ + cls = expr.__class__ + args = [] + for arg in expr.args: + if arg.__class__ == cls: + args.extend(arg.args) + else: + args.append(arg) + return new(expr.__class__, *args) + + +def rebuild(expr): + """ Rebuild a SymPy tree. + + Explanation + =========== + + This function recursively calls constructors in the expression tree. + This forces canonicalization and removes ugliness introduced by the use of + Basic.__new__ + """ + if expr.is_Atom: + return expr + else: + return expr.func(*list(map(rebuild, expr.args))) diff --git a/.venv/lib/python3.12/site-packages/sympy/strategies/tests/__init__.py b/.venv/lib/python3.12/site-packages/sympy/strategies/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/.venv/lib/python3.12/site-packages/sympy/strategies/tests/__pycache__/__init__.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/strategies/tests/__pycache__/__init__.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..fa3b29aa02f67982f40459d4e421a1f2e302a0dd Binary files /dev/null and b/.venv/lib/python3.12/site-packages/sympy/strategies/tests/__pycache__/__init__.cpython-312.pyc differ diff --git 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b/.venv/lib/python3.12/site-packages/sympy/strategies/tests/__pycache__/test_tree.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..abc24b0c7f5c95c7944685d37a62c0856762f30c Binary files /dev/null and b/.venv/lib/python3.12/site-packages/sympy/strategies/tests/__pycache__/test_tree.cpython-312.pyc differ diff --git a/.venv/lib/python3.12/site-packages/sympy/strategies/tests/test_core.py b/.venv/lib/python3.12/site-packages/sympy/strategies/tests/test_core.py new file mode 100644 index 0000000000000000000000000000000000000000..1e19bcab1940c476a8996b7ba92e7645a6230034 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/strategies/tests/test_core.py @@ -0,0 +1,118 @@ +from __future__ import annotations +from sympy.core.singleton import S +from sympy.core.basic import Basic +from sympy.strategies.core import ( + null_safe, exhaust, memoize, condition, + chain, tryit, do_one, debug, switch, minimize) +from io import StringIO + + +def posdec(x: int) -> int: + if x > 0: + return x - 1 + return x + + +def inc(x: int) -> int: + return x + 1 + + +def dec(x: int) -> int: + return x - 1 + + +def test_null_safe(): + def rl(expr: int) -> int | None: + if expr == 1: + return 2 + return None + + safe_rl = null_safe(rl) + assert rl(1) == safe_rl(1) + assert rl(3) is None + assert safe_rl(3) == 3 + + +def test_exhaust(): + sink = exhaust(posdec) + assert sink(5) == 0 + assert sink(10) == 0 + + +def test_memoize(): + rl = memoize(posdec) + assert rl(5) == posdec(5) + assert rl(5) == posdec(5) + assert rl(-2) == posdec(-2) + + +def test_condition(): + rl = condition(lambda x: x % 2 == 0, posdec) + assert rl(5) == 5 + assert rl(4) == 3 + + +def test_chain(): + rl = chain(posdec, posdec) + assert rl(5) == 3 + assert rl(1) == 0 + + +def test_tryit(): + def rl(expr: Basic) -> Basic: + assert False + + safe_rl = tryit(rl, AssertionError) + assert safe_rl(S(1)) == S(1) + + +def test_do_one(): + rl = do_one(posdec, posdec) + assert rl(5) == 4 + + def rl1(x: int) -> int: + if x == 1: + return 2 + return x + + def rl2(x: int) -> int: + if x == 2: + return 3 + return x + + rule = do_one(rl1, rl2) + assert rule(1) == 2 + assert rule(rule(1)) == 3 + + +def test_debug(): + file = StringIO() + rl = debug(posdec, file) + rl(5) + log = file.getvalue() + file.close() + + assert posdec.__name__ in log + assert '5' in log + assert '4' in log + + +def test_switch(): + def key(x: int) -> int: + return x % 3 + + rl = switch(key, {0: inc, 1: dec}) + assert rl(3) == 4 + assert rl(4) == 3 + assert rl(5) == 5 + + +def test_minimize(): + def key(x: int) -> int: + return -x + + rl = minimize(inc, dec) + assert rl(4) == 3 + + rl = minimize(inc, dec, objective=key) + assert rl(4) == 5 diff --git a/.venv/lib/python3.12/site-packages/sympy/strategies/tests/test_rl.py b/.venv/lib/python3.12/site-packages/sympy/strategies/tests/test_rl.py new file mode 100644 index 0000000000000000000000000000000000000000..8bfa90ad4d970b21396e0e1b6427b5a5c68fe381 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/strategies/tests/test_rl.py @@ -0,0 +1,78 @@ +from sympy.core.singleton import S +from sympy.strategies.rl import ( + rm_id, glom, flatten, unpack, sort, distribute, subs, rebuild) +from sympy.core.basic import Basic +from sympy.core.add import Add +from sympy.core.mul import Mul +from sympy.core.symbol import symbols +from sympy.abc import x + + +def test_rm_id(): + rmzeros = rm_id(lambda x: x == 0) + assert rmzeros(Basic(S(0), S(1))) == Basic(S(1)) + assert rmzeros(Basic(S(0), S(0))) == Basic(S(0)) + assert rmzeros(Basic(S(2), S(1))) == Basic(S(2), S(1)) + + +def test_glom(): + def key(x): + return x.as_coeff_Mul()[1] + + def count(x): + return x.as_coeff_Mul()[0] + + def newargs(cnt, arg): + return cnt * arg + + rl = glom(key, count, newargs) + + result = rl(Add(x, -x, 3 * x, 2, 3, evaluate=False)) + expected = Add(3 * x, 5) + assert set(result.args) == set(expected.args) + + +def test_flatten(): + assert flatten(Basic(S(1), S(2), Basic(S(3), S(4)))) == \ + Basic(S(1), S(2), S(3), S(4)) + + +def test_unpack(): + assert unpack(Basic(S(2))) == 2 + assert unpack(Basic(S(2), S(3))) == Basic(S(2), S(3)) + + +def test_sort(): + assert sort(str)(Basic(S(3), S(1), S(2))) == Basic(S(1), S(2), S(3)) + + +def test_distribute(): + class T1(Basic): + pass + + class T2(Basic): + pass + + distribute_t12 = distribute(T1, T2) + assert distribute_t12(T1(S(1), S(2), T2(S(3), S(4)), S(5))) == \ + T2(T1(S(1), S(2), S(3), S(5)), T1(S(1), S(2), S(4), S(5))) + assert distribute_t12(T1(S(1), S(2), S(3))) == T1(S(1), S(2), S(3)) + + +def test_distribute_add_mul(): + x, y = symbols('x, y') + expr = Mul(2, Add(x, y), evaluate=False) + expected = Add(Mul(2, x), Mul(2, y)) + distribute_mul = distribute(Mul, Add) + assert distribute_mul(expr) == expected + + +def test_subs(): + rl = subs(1, 2) + assert rl(1) == 2 + assert rl(3) == 3 + + +def test_rebuild(): + expr = Basic.__new__(Add, S(1), S(2)) + assert rebuild(expr) == 3 diff --git a/.venv/lib/python3.12/site-packages/sympy/strategies/tests/test_tools.py b/.venv/lib/python3.12/site-packages/sympy/strategies/tests/test_tools.py new file mode 100644 index 0000000000000000000000000000000000000000..89774aeb92ead5781966e4f48ad32dc63e1bf7e2 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/strategies/tests/test_tools.py @@ -0,0 +1,32 @@ +from sympy.strategies.tools import subs, typed +from sympy.strategies.rl import rm_id +from sympy.core.basic import Basic +from sympy.core.singleton import S + + +def test_subs(): + from sympy.core.symbol import symbols + a, b, c, d, e, f = symbols('a,b,c,d,e,f') + mapping = {a: d, d: a, Basic(e): Basic(f)} + expr = Basic(a, Basic(b, c), Basic(d, Basic(e))) + result = Basic(d, Basic(b, c), Basic(a, Basic(f))) + assert subs(mapping)(expr) == result + + +def test_subs_empty(): + assert subs({})(Basic(S(1), S(2))) == Basic(S(1), S(2)) + + +def test_typed(): + class A(Basic): + pass + + class B(Basic): + pass + + rmzeros = rm_id(lambda x: x == S(0)) + rmones = rm_id(lambda x: x == S(1)) + remove_something = typed({A: rmzeros, B: rmones}) + + assert remove_something(A(S(0), S(1))) == A(S(1)) + assert remove_something(B(S(0), S(1))) == B(S(0)) diff --git a/.venv/lib/python3.12/site-packages/sympy/strategies/tests/test_traverse.py b/.venv/lib/python3.12/site-packages/sympy/strategies/tests/test_traverse.py new file mode 100644 index 0000000000000000000000000000000000000000..ee2409616a8b4f750af8baea149b2ea52c56be9d --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/strategies/tests/test_traverse.py @@ -0,0 +1,84 @@ +from sympy.strategies.traverse import ( + top_down, bottom_up, sall, top_down_once, bottom_up_once, basic_fns) +from sympy.strategies.rl import rebuild +from sympy.strategies.util import expr_fns +from sympy.core.add import Add +from sympy.core.basic import Basic +from sympy.core.numbers import Integer +from sympy.core.singleton import S +from sympy.core.symbol import Str, Symbol +from sympy.abc import x, y, z + + +def zero_symbols(expression): + return S.Zero if isinstance(expression, Symbol) else expression + + +def test_sall(): + zero_onelevel = sall(zero_symbols) + + assert zero_onelevel(Basic(x, y, Basic(x, z))) == \ + Basic(S(0), S(0), Basic(x, z)) + + +def test_bottom_up(): + _test_global_traversal(bottom_up) + _test_stop_on_non_basics(bottom_up) + + +def test_top_down(): + _test_global_traversal(top_down) + _test_stop_on_non_basics(top_down) + + +def _test_global_traversal(trav): + zero_all_symbols = trav(zero_symbols) + + assert zero_all_symbols(Basic(x, y, Basic(x, z))) == \ + Basic(S(0), S(0), Basic(S(0), S(0))) + + +def _test_stop_on_non_basics(trav): + def add_one_if_can(expr): + try: + return expr + 1 + except TypeError: + return expr + + expr = Basic(S(1), Str('a'), Basic(S(2), Str('b'))) + expected = Basic(S(2), Str('a'), Basic(S(3), Str('b'))) + rl = trav(add_one_if_can) + + assert rl(expr) == expected + + +class Basic2(Basic): + pass + + +def rl(x): + if x.args and not isinstance(x.args[0], Integer): + return Basic2(*x.args) + return x + + +def test_top_down_once(): + top_rl = top_down_once(rl) + + assert top_rl(Basic(S(1.0), S(2.0), Basic(S(3), S(4)))) == \ + Basic2(S(1.0), S(2.0), Basic(S(3), S(4))) + + +def test_bottom_up_once(): + bottom_rl = bottom_up_once(rl) + + assert bottom_rl(Basic(S(1), S(2), Basic(S(3.0), S(4.0)))) == \ + Basic(S(1), S(2), Basic2(S(3.0), S(4.0))) + + +def test_expr_fns(): + expr = x + y**3 + e = bottom_up(lambda v: v + 1, expr_fns)(expr) + b = bottom_up(lambda v: Basic.__new__(Add, v, S(1)), basic_fns)(expr) + + assert rebuild(b) == e diff --git a/.venv/lib/python3.12/site-packages/sympy/strategies/tests/test_tree.py b/.venv/lib/python3.12/site-packages/sympy/strategies/tests/test_tree.py new file mode 100644 index 0000000000000000000000000000000000000000..d5cdde747fe3ab90c8fd181701194403bc526067 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/strategies/tests/test_tree.py @@ -0,0 +1,92 @@ +from sympy.strategies.tree import treeapply, greedy, allresults, brute +from functools import partial, reduce + + +def inc(x): + return x + 1 + + +def dec(x): + return x - 1 + + +def double(x): + return 2 * x + + +def square(x): + return x**2 + + +def add(*args): + return sum(args) + + +def mul(*args): + return reduce(lambda a, b: a * b, args, 1) + + +def test_treeapply(): + tree = ([3, 3], [4, 1], 2) + assert treeapply(tree, {list: min, tuple: max}) == 3 + assert treeapply(tree, {list: add, tuple: mul}) == 60 + + +def test_treeapply_leaf(): + assert treeapply(3, {}, leaf=lambda x: x**2) == 9 + tree = ([3, 3], [4, 1], 2) + treep1 = ([4, 4], [5, 2], 3) + assert treeapply(tree, {list: min, tuple: max}, leaf=lambda x: x + 1) == \ + treeapply(treep1, {list: min, tuple: max}) + + +def test_treeapply_strategies(): + from sympy.strategies import chain, minimize + join = {list: chain, tuple: minimize} + + assert treeapply(inc, join) == inc + assert treeapply((inc, dec), join)(5) == minimize(inc, dec)(5) + assert treeapply([inc, dec], join)(5) == chain(inc, dec)(5) + tree = (inc, [dec, double]) # either inc or dec-then-double + assert treeapply(tree, join)(5) == 6 + assert treeapply(tree, join)(1) == 0 + + maximize = partial(minimize, objective=lambda x: -x) + join = {list: chain, tuple: maximize} + fn = treeapply(tree, join) + assert fn(4) == 6 # highest value comes from the dec then double + assert fn(1) == 2 # highest value comes from the inc + + +def test_greedy(): + tree = [inc, (dec, double)] # either inc or dec-then-double + + fn = greedy(tree, objective=lambda x: -x) + assert fn(4) == 6 # highest value comes from the dec then double + assert fn(1) == 2 # highest value comes from the inc + + tree = [inc, dec, [inc, dec, [(inc, inc), (dec, dec)]]] + lowest = greedy(tree) + assert lowest(10) == 8 + + highest = greedy(tree, objective=lambda x: -x) + assert highest(10) == 12 + + +def test_allresults(): + # square = lambda x: x**2 + + assert set(allresults(inc)(3)) == {inc(3)} + assert set(allresults([inc, dec])(3)) == {2, 4} + assert set(allresults((inc, dec))(3)) == {3} + assert set(allresults([inc, (dec, double)])(4)) == {5, 6} + + +def test_brute(): + tree = ([inc, dec], square) + fn = brute(tree, lambda x: -x) + + assert fn(2) == (2 + 1)**2 + assert fn(-2) == (-2 - 1)**2 + + assert brute(inc)(1) == 2 diff --git a/.venv/lib/python3.12/site-packages/sympy/strategies/tools.py b/.venv/lib/python3.12/site-packages/sympy/strategies/tools.py new file mode 100644 index 0000000000000000000000000000000000000000..e6a94c16db57206d7c83c8a5e13930c4cffdde47 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/strategies/tools.py @@ -0,0 +1,53 @@ +from . import rl +from .core import do_one, exhaust, switch +from .traverse import top_down + + +def subs(d, **kwargs): + """ Full simultaneous exact substitution. + + Examples + ======== + + >>> from sympy.strategies.tools import subs + >>> from sympy import Basic, S + >>> mapping = {S(1): S(4), S(4): S(1), Basic(S(5)): Basic(S(6), S(7))} + >>> expr = Basic(S(1), Basic(S(2), S(3)), Basic(S(4), Basic(S(5)))) + >>> subs(mapping)(expr) + Basic(4, Basic(2, 3), Basic(1, Basic(6, 7))) + """ + if d: + return top_down(do_one(*map(rl.subs, *zip(*d.items()))), **kwargs) + else: + return lambda x: x + + +def canon(*rules, **kwargs): + """ Strategy for canonicalization. + + Explanation + =========== + + Apply each rule in a bottom_up fashion through the tree. + Do each one in turn. + Keep doing this until there is no change. + """ + return exhaust(top_down(exhaust(do_one(*rules)), **kwargs)) + + +def typed(ruletypes): + """ Apply rules based on the expression type + + inputs: + ruletypes -- a dict mapping {Type: rule} + + Examples + ======== + + >>> from sympy.strategies import rm_id, typed + >>> from sympy import Add, Mul + >>> rm_zeros = rm_id(lambda x: x==0) + >>> rm_ones = rm_id(lambda x: x==1) + >>> remove_idents = typed({Add: rm_zeros, Mul: rm_ones}) + """ + return switch(type, ruletypes) diff --git a/.venv/lib/python3.12/site-packages/sympy/strategies/traverse.py b/.venv/lib/python3.12/site-packages/sympy/strategies/traverse.py new file mode 100644 index 0000000000000000000000000000000000000000..869361f443742b5b7346c9c970f103b955e8473e --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/strategies/traverse.py @@ -0,0 +1,37 @@ +"""Strategies to Traverse a Tree.""" +from sympy.strategies.util import basic_fns +from sympy.strategies.core import chain, do_one + + +def top_down(rule, fns=basic_fns): + """Apply a rule down a tree running it on the top nodes first.""" + return chain(rule, lambda expr: sall(top_down(rule, fns), fns)(expr)) + + +def bottom_up(rule, fns=basic_fns): + """Apply a rule down a tree running it on the bottom nodes first.""" + return chain(lambda expr: sall(bottom_up(rule, fns), fns)(expr), rule) + + +def top_down_once(rule, fns=basic_fns): + """Apply a rule down a tree - stop on success.""" + return do_one(rule, lambda expr: sall(top_down(rule, fns), fns)(expr)) + + +def bottom_up_once(rule, fns=basic_fns): + """Apply a rule up a tree - stop on success.""" + return do_one(lambda expr: sall(bottom_up(rule, fns), fns)(expr), rule) + + +def sall(rule, fns=basic_fns): + """Strategic all - apply rule to args.""" + op, new, children, leaf = map(fns.get, ('op', 'new', 'children', 'leaf')) + + def all_rl(expr): + if leaf(expr): + return expr + else: + args = map(rule, children(expr)) + return new(op(expr), *args) + + return all_rl diff --git a/.venv/lib/python3.12/site-packages/sympy/strategies/tree.py b/.venv/lib/python3.12/site-packages/sympy/strategies/tree.py new file mode 100644 index 0000000000000000000000000000000000000000..c2006fde4fc5d09f3d38baae4d7335b4cbd971b7 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/strategies/tree.py @@ -0,0 +1,139 @@ +from functools import partial +from sympy.strategies import chain, minimize +from sympy.strategies.core import identity +import sympy.strategies.branch as branch +from sympy.strategies.branch import yieldify + + +def treeapply(tree, join, leaf=identity): + """ Apply functions onto recursive containers (tree). + + Explanation + =========== + + join - a dictionary mapping container types to functions + e.g. ``{list: minimize, tuple: chain}`` + + Keys are containers/iterables. Values are functions [a] -> a. + + Examples + ======== + + >>> from sympy.strategies.tree import treeapply + >>> tree = [(3, 2), (4, 1)] + >>> treeapply(tree, {list: max, tuple: min}) + 2 + + >>> add = lambda *args: sum(args) + >>> def mul(*args): + ... total = 1 + ... for arg in args: + ... total *= arg + ... return total + >>> treeapply(tree, {list: mul, tuple: add}) + 25 + """ + for typ in join: + if isinstance(tree, typ): + return join[typ](*map(partial(treeapply, join=join, leaf=leaf), + tree)) + return leaf(tree) + + +def greedy(tree, objective=identity, **kwargs): + """ Execute a strategic tree. Select alternatives greedily + + Trees + ----- + + Nodes in a tree can be either + + function - a leaf + list - a selection among operations + tuple - a sequence of chained operations + + Textual examples + ---------------- + + Text: Run f, then run g, e.g. ``lambda x: g(f(x))`` + Code: ``(f, g)`` + + Text: Run either f or g, whichever minimizes the objective + Code: ``[f, g]`` + + Textx: Run either f or g, whichever is better, then run h + Code: ``([f, g], h)`` + + Text: Either expand then simplify or try factor then foosimp. Finally print + Code: ``([(expand, simplify), (factor, foosimp)], print)`` + + Objective + --------- + + "Better" is determined by the objective keyword. This function makes + choices to minimize the objective. It defaults to the identity. + + Examples + ======== + + >>> from sympy.strategies.tree import greedy + >>> inc = lambda x: x + 1 + >>> dec = lambda x: x - 1 + >>> double = lambda x: 2*x + + >>> tree = [inc, (dec, double)] # either inc or dec-then-double + >>> fn = greedy(tree) + >>> fn(4) # lowest value comes from the inc + 5 + >>> fn(1) # lowest value comes from dec then double + 0 + + This function selects between options in a tuple. The result is chosen + that minimizes the objective function. + + >>> fn = greedy(tree, objective=lambda x: -x) # maximize + >>> fn(4) # highest value comes from the dec then double + 6 + >>> fn(1) # highest value comes from the inc + 2 + + Greediness + ---------- + + This is a greedy algorithm. In the example: + + ([a, b], c) # do either a or b, then do c + + the choice between running ``a`` or ``b`` is made without foresight to c + """ + optimize = partial(minimize, objective=objective) + return treeapply(tree, {list: optimize, tuple: chain}, **kwargs) + + +def allresults(tree, leaf=yieldify): + """ Execute a strategic tree. Return all possibilities. + + Returns a lazy iterator of all possible results + + Exhaustiveness + -------------- + + This is an exhaustive algorithm. In the example + + ([a, b], [c, d]) + + All of the results from + + (a, c), (b, c), (a, d), (b, d) + + are returned. This can lead to combinatorial blowup. + + See sympy.strategies.greedy for details on input + """ + return treeapply(tree, {list: branch.multiplex, tuple: branch.chain}, + leaf=leaf) + + +def brute(tree, objective=identity, **kwargs): + return lambda expr: min(tuple(allresults(tree, **kwargs)(expr)), + key=objective) diff --git a/.venv/lib/python3.12/site-packages/sympy/strategies/util.py b/.venv/lib/python3.12/site-packages/sympy/strategies/util.py new file mode 100644 index 0000000000000000000000000000000000000000..13aab5f6a49650c5ded9cd913c23c6682f18d40a --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/strategies/util.py @@ -0,0 +1,17 @@ +from sympy.core.basic import Basic + +new = Basic.__new__ + + +def assoc(d, k, v): + d = d.copy() + d[k] = v + return d + + +basic_fns = {'op': type, + 'new': Basic.__new__, + 'leaf': lambda x: not isinstance(x, Basic) or x.is_Atom, + 'children': lambda x: x.args} + +expr_fns = assoc(basic_fns, 'new', lambda op, *args: op(*args)) diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/__init__.py b/.venv/lib/python3.12/site-packages/sympy/tensor/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..a832614b1d48e26bf01e16f040f34dd412e8e32b --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/__init__.py @@ -0,0 +1,23 @@ +"""A module to manipulate symbolic objects with indices including tensors + +""" +from .indexed import IndexedBase, Idx, Indexed +from .index_methods import get_contraction_structure, get_indices +from .functions import shape +from .array import (MutableDenseNDimArray, ImmutableDenseNDimArray, + MutableSparseNDimArray, ImmutableSparseNDimArray, NDimArray, tensorproduct, + tensorcontraction, tensordiagonal, derive_by_array, permutedims, Array, + DenseNDimArray, SparseNDimArray,) + +__all__ = [ + 'IndexedBase', 'Idx', 'Indexed', + + 'get_contraction_structure', 'get_indices', + + 'shape', + + 'MutableDenseNDimArray', 'ImmutableDenseNDimArray', + 'MutableSparseNDimArray', 'ImmutableSparseNDimArray', 'NDimArray', + 'tensorproduct', 'tensorcontraction', 'tensordiagonal', 'derive_by_array', 'permutedims', + 'Array', 'DenseNDimArray', 'SparseNDimArray', +] diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/__pycache__/__init__.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/tensor/__pycache__/__init__.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..ac4ceadb7d50bf7b663470e3351963b12388e4f7 Binary files /dev/null and b/.venv/lib/python3.12/site-packages/sympy/tensor/__pycache__/__init__.cpython-312.pyc differ diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/__pycache__/functions.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/tensor/__pycache__/functions.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..b11bc758347e13d12087edd6791b29ff381f9f04 Binary files /dev/null and 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a/.venv/lib/python3.12/site-packages/sympy/tensor/array/__init__.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..eca2eb4c6c58cb113517b6e41737e9d97abbb84e --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/__init__.py @@ -0,0 +1,271 @@ +r""" +N-dim array module for SymPy. + +Four classes are provided to handle N-dim arrays, given by the combinations +dense/sparse (i.e. whether to store all elements or only the non-zero ones in +memory) and mutable/immutable (immutable classes are SymPy objects, but cannot +change after they have been created). + +Examples +======== + +The following examples show the usage of ``Array``. This is an abbreviation for +``ImmutableDenseNDimArray``, that is an immutable and dense N-dim array, the +other classes are analogous. For mutable classes it is also possible to change +element values after the object has been constructed. + +Array construction can detect the shape of nested lists and tuples: + +>>> from sympy import Array +>>> a1 = Array([[1, 2], [3, 4], [5, 6]]) +>>> a1 +[[1, 2], [3, 4], [5, 6]] +>>> a1.shape +(3, 2) +>>> a1.rank() +2 +>>> from sympy.abc import x, y, z +>>> a2 = Array([[[x, y], [z, x*z]], [[1, x*y], [1/x, x/y]]]) +>>> a2 +[[[x, y], [z, x*z]], [[1, x*y], [1/x, x/y]]] +>>> a2.shape +(2, 2, 2) +>>> a2.rank() +3 + +Otherwise one could pass a 1-dim array followed by a shape tuple: + +>>> m1 = Array(range(12), (3, 4)) +>>> m1 +[[0, 1, 2, 3], [4, 5, 6, 7], [8, 9, 10, 11]] +>>> m2 = Array(range(12), (3, 2, 2)) +>>> m2 +[[[0, 1], [2, 3]], [[4, 5], [6, 7]], [[8, 9], [10, 11]]] +>>> m2[1,1,1] +7 +>>> m2.reshape(4, 3) +[[0, 1, 2], [3, 4, 5], [6, 7, 8], [9, 10, 11]] + +Slice support: + +>>> m2[:, 1, 1] +[3, 7, 11] + +Elementwise derivative: + +>>> from sympy.abc import x, y, z +>>> m3 = Array([x**3, x*y, z]) +>>> m3.diff(x) +[3*x**2, y, 0] +>>> m3.diff(z) +[0, 0, 1] + +Multiplication with other SymPy expressions is applied elementwisely: + +>>> (1+x)*m3 +[x**3*(x + 1), x*y*(x + 1), z*(x + 1)] + +To apply a function to each element of the N-dim array, use ``applyfunc``: + +>>> m3.applyfunc(lambda x: x/2) +[x**3/2, x*y/2, z/2] + +N-dim arrays can be converted to nested lists by the ``tolist()`` method: + +>>> m2.tolist() +[[[0, 1], [2, 3]], [[4, 5], [6, 7]], [[8, 9], [10, 11]]] +>>> isinstance(m2.tolist(), list) +True + +If the rank is 2, it is possible to convert them to matrices with ``tomatrix()``: + +>>> m1.tomatrix() +Matrix([ +[0, 1, 2, 3], +[4, 5, 6, 7], +[8, 9, 10, 11]]) + +Products and contractions +------------------------- + +Tensor product between arrays `A_{i_1,\ldots,i_n}` and `B_{j_1,\ldots,j_m}` +creates the combined array `P = A \otimes B` defined as + +`P_{i_1,\ldots,i_n,j_1,\ldots,j_m} := A_{i_1,\ldots,i_n}\cdot B_{j_1,\ldots,j_m}.` + +It is available through ``tensorproduct(...)``: + +>>> from sympy import Array, tensorproduct +>>> from sympy.abc import x,y,z,t +>>> A = Array([x, y, z, t]) +>>> B = Array([1, 2, 3, 4]) +>>> tensorproduct(A, B) +[[x, 2*x, 3*x, 4*x], [y, 2*y, 3*y, 4*y], [z, 2*z, 3*z, 4*z], [t, 2*t, 3*t, 4*t]] + +In case you don't want to evaluate the tensor product immediately, you can use +``ArrayTensorProduct``, which creates an unevaluated tensor product expression: + +>>> from sympy.tensor.array.expressions import ArrayTensorProduct +>>> ArrayTensorProduct(A, B) +ArrayTensorProduct([x, y, z, t], [1, 2, 3, 4]) + +Calling ``.as_explicit()`` on ``ArrayTensorProduct`` is equivalent to just calling +``tensorproduct(...)``: + +>>> ArrayTensorProduct(A, B).as_explicit() +[[x, 2*x, 3*x, 4*x], [y, 2*y, 3*y, 4*y], [z, 2*z, 3*z, 4*z], [t, 2*t, 3*t, 4*t]] + +Tensor product between a rank-1 array and a matrix creates a rank-3 array: + +>>> from sympy import eye +>>> p1 = tensorproduct(A, eye(4)) +>>> p1 +[[[x, 0, 0, 0], [0, x, 0, 0], [0, 0, x, 0], [0, 0, 0, x]], [[y, 0, 0, 0], [0, y, 0, 0], [0, 0, y, 0], [0, 0, 0, y]], [[z, 0, 0, 0], [0, z, 0, 0], [0, 0, z, 0], [0, 0, 0, z]], [[t, 0, 0, 0], [0, t, 0, 0], [0, 0, t, 0], [0, 0, 0, t]]] + +Now, to get back `A_0 \otimes \mathbf{1}` one can access `p_{0,m,n}` by slicing: + +>>> p1[0,:,:] +[[x, 0, 0, 0], [0, x, 0, 0], [0, 0, x, 0], [0, 0, 0, x]] + +Tensor contraction sums over the specified axes, for example contracting +positions `a` and `b` means + +`A_{i_1,\ldots,i_a,\ldots,i_b,\ldots,i_n} \implies \sum_k A_{i_1,\ldots,k,\ldots,k,\ldots,i_n}` + +Remember that Python indexing is zero starting, to contract the a-th and b-th +axes it is therefore necessary to specify `a-1` and `b-1` + +>>> from sympy import tensorcontraction +>>> C = Array([[x, y], [z, t]]) + +The matrix trace is equivalent to the contraction of a rank-2 array: + +`A_{m,n} \implies \sum_k A_{k,k}` + +>>> tensorcontraction(C, (0, 1)) +t + x + +To create an expression representing a tensor contraction that does not get +evaluated immediately, use ``ArrayContraction``, which is equivalent to +``tensorcontraction(...)`` if it is followed by ``.as_explicit()``: + +>>> from sympy.tensor.array.expressions import ArrayContraction +>>> ArrayContraction(C, (0, 1)) +ArrayContraction([[x, y], [z, t]], (0, 1)) +>>> ArrayContraction(C, (0, 1)).as_explicit() +t + x + +Matrix product is equivalent to a tensor product of two rank-2 arrays, followed +by a contraction of the 2nd and 3rd axes (in Python indexing axes number 1, 2). + +`A_{m,n}\cdot B_{i,j} \implies \sum_k A_{m, k}\cdot B_{k, j}` + +>>> D = Array([[2, 1], [0, -1]]) +>>> tensorcontraction(tensorproduct(C, D), (1, 2)) +[[2*x, x - y], [2*z, -t + z]] + +One may verify that the matrix product is equivalent: + +>>> from sympy import Matrix +>>> Matrix([[x, y], [z, t]])*Matrix([[2, 1], [0, -1]]) +Matrix([ +[2*x, x - y], +[2*z, -t + z]]) + +or equivalently + +>>> C.tomatrix()*D.tomatrix() +Matrix([ +[2*x, x - y], +[2*z, -t + z]]) + +Diagonal operator +----------------- + +The ``tensordiagonal`` function acts in a similar manner as ``tensorcontraction``, +but the joined indices are not summed over, for example diagonalizing +positions `a` and `b` means + +`A_{i_1,\ldots,i_a,\ldots,i_b,\ldots,i_n} \implies A_{i_1,\ldots,k,\ldots,k,\ldots,i_n} +\implies \tilde{A}_{i_1,\ldots,i_{a-1},i_{a+1},\ldots,i_{b-1},i_{b+1},\ldots,i_n,k}` + +where `\tilde{A}` is the array equivalent to the diagonal of `A` at positions +`a` and `b` moved to the last index slot. + +Compare the difference between contraction and diagonal operators: + +>>> from sympy import tensordiagonal +>>> from sympy.abc import a, b, c, d +>>> m = Matrix([[a, b], [c, d]]) +>>> tensorcontraction(m, [0, 1]) +a + d +>>> tensordiagonal(m, [0, 1]) +[a, d] + +In short, no summation occurs with ``tensordiagonal``. + + +Derivatives by array +-------------------- + +The usual derivative operation may be extended to support derivation with +respect to arrays, provided that all elements in the that array are symbols or +expressions suitable for derivations. + +The definition of a derivative by an array is as follows: given the array +`A_{i_1, \ldots, i_N}` and the array `X_{j_1, \ldots, j_M}` +the derivative of arrays will return a new array `B` defined by + +`B_{j_1,\ldots,j_M,i_1,\ldots,i_N} := \frac{\partial A_{i_1,\ldots,i_N}}{\partial X_{j_1,\ldots,j_M}}` + +The function ``derive_by_array`` performs such an operation: + +>>> from sympy import derive_by_array +>>> from sympy.abc import x, y, z, t +>>> from sympy import sin, exp + +With scalars, it behaves exactly as the ordinary derivative: + +>>> derive_by_array(sin(x*y), x) +y*cos(x*y) + +Scalar derived by an array basis: + +>>> derive_by_array(sin(x*y), [x, y, z]) +[y*cos(x*y), x*cos(x*y), 0] + +Deriving array by an array basis: `B^{nm} := \frac{\partial A^m}{\partial x^n}` + +>>> basis = [x, y, z] +>>> ax = derive_by_array([exp(x), sin(y*z), t], basis) +>>> ax +[[exp(x), 0, 0], [0, z*cos(y*z), 0], [0, y*cos(y*z), 0]] + +Contraction of the resulting array: `\sum_m \frac{\partial A^m}{\partial x^m}` + +>>> tensorcontraction(ax, (0, 1)) +z*cos(y*z) + exp(x) + +""" + +from .dense_ndim_array import MutableDenseNDimArray, ImmutableDenseNDimArray, DenseNDimArray +from .sparse_ndim_array import MutableSparseNDimArray, ImmutableSparseNDimArray, SparseNDimArray +from .ndim_array import NDimArray, ArrayKind +from .arrayop import tensorproduct, tensorcontraction, tensordiagonal, derive_by_array, permutedims +from .array_comprehension import ArrayComprehension, ArrayComprehensionMap + +Array = ImmutableDenseNDimArray + +__all__ = [ + 'MutableDenseNDimArray', 'ImmutableDenseNDimArray', 'DenseNDimArray', + + 'MutableSparseNDimArray', 'ImmutableSparseNDimArray', 'SparseNDimArray', + + 'NDimArray', 'ArrayKind', + + 'tensorproduct', 'tensorcontraction', 'tensordiagonal', 'derive_by_array', + + 'permutedims', 'ArrayComprehension', 'ArrayComprehensionMap', + + 'Array', +] diff --git 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0000000000000000000000000000000000000000..95702f499f3e40597fd0144929138ac1329962ee --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/array_comprehension.py @@ -0,0 +1,399 @@ +import functools, itertools +from sympy.core.sympify import _sympify, sympify +from sympy.core.expr import Expr +from sympy.core import Basic, Tuple +from sympy.tensor.array import ImmutableDenseNDimArray +from sympy.core.symbol import Symbol +from sympy.core.numbers import Integer + + +class ArrayComprehension(Basic): + """ + Generate a list comprehension. + + Explanation + =========== + + If there is a symbolic dimension, for example, say [i for i in range(1, N)] where + N is a Symbol, then the expression will not be expanded to an array. Otherwise, + calling the doit() function will launch the expansion. + + Examples + ======== + + >>> from sympy.tensor.array import ArrayComprehension + >>> from sympy import symbols + >>> i, j, k = symbols('i j k') + >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3)) + >>> a + ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3)) + >>> a.doit() + [[11, 12, 13], [21, 22, 23], [31, 32, 33], [41, 42, 43]] + >>> b = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, k)) + >>> b.doit() + ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, k)) + """ + def __new__(cls, function, *symbols, **assumptions): + if any(len(l) != 3 or None for l in symbols): + raise ValueError('ArrayComprehension requires values lower and upper bound' + ' for the expression') + arglist = [sympify(function)] + arglist.extend(cls._check_limits_validity(function, symbols)) + obj = Basic.__new__(cls, *arglist, **assumptions) + obj._limits = obj._args[1:] + obj._shape = cls._calculate_shape_from_limits(obj._limits) + obj._rank = len(obj._shape) + obj._loop_size = cls._calculate_loop_size(obj._shape) + return obj + + @property + def function(self): + """The function applied across limits. + + Examples + ======== + + >>> from sympy.tensor.array import ArrayComprehension + >>> from sympy import symbols + >>> i, j = symbols('i j') + >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3)) + >>> a.function + 10*i + j + """ + return self._args[0] + + @property + def limits(self): + """ + The list of limits that will be applied while expanding the array. + + Examples + ======== + + >>> from sympy.tensor.array import ArrayComprehension + >>> from sympy import symbols + >>> i, j = symbols('i j') + >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3)) + >>> a.limits + ((i, 1, 4), (j, 1, 3)) + """ + return self._limits + + @property + def free_symbols(self): + """ + The set of the free_symbols in the array. + Variables appeared in the bounds are supposed to be excluded + from the free symbol set. + + Examples + ======== + + >>> from sympy.tensor.array import ArrayComprehension + >>> from sympy import symbols + >>> i, j, k = symbols('i j k') + >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3)) + >>> a.free_symbols + set() + >>> b = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, k+3)) + >>> b.free_symbols + {k} + """ + expr_free_sym = self.function.free_symbols + for var, inf, sup in self._limits: + expr_free_sym.discard(var) + curr_free_syms = inf.free_symbols.union(sup.free_symbols) + expr_free_sym = expr_free_sym.union(curr_free_syms) + return expr_free_sym + + @property + def variables(self): + """The tuples of the variables in the limits. + + Examples + ======== + + >>> from sympy.tensor.array import ArrayComprehension + >>> from sympy import symbols + >>> i, j, k = symbols('i j k') + >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3)) + >>> a.variables + [i, j] + """ + return [l[0] for l in self._limits] + + @property + def bound_symbols(self): + """The list of dummy variables. + + Note + ==== + + Note that all variables are dummy variables since a limit without + lower bound or upper bound is not accepted. + """ + return [l[0] for l in self._limits if len(l) != 1] + + @property + def shape(self): + """ + The shape of the expanded array, which may have symbols. + + Note + ==== + + Both the lower and the upper bounds are included while + calculating the shape. + + Examples + ======== + + >>> from sympy.tensor.array import ArrayComprehension + >>> from sympy import symbols + >>> i, j, k = symbols('i j k') + >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3)) + >>> a.shape + (4, 3) + >>> b = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, k+3)) + >>> b.shape + (4, k + 3) + """ + return self._shape + + @property + def is_shape_numeric(self): + """ + Test if the array is shape-numeric which means there is no symbolic + dimension. + + Examples + ======== + + >>> from sympy.tensor.array import ArrayComprehension + >>> from sympy import symbols + >>> i, j, k = symbols('i j k') + >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3)) + >>> a.is_shape_numeric + True + >>> b = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, k+3)) + >>> b.is_shape_numeric + False + """ + for _, inf, sup in self._limits: + if Basic(inf, sup).atoms(Symbol): + return False + return True + + def rank(self): + """The rank of the expanded array. + + Examples + ======== + + >>> from sympy.tensor.array import ArrayComprehension + >>> from sympy import symbols + >>> i, j, k = symbols('i j k') + >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3)) + >>> a.rank() + 2 + """ + return self._rank + + def __len__(self): + """ + The length of the expanded array which means the number + of elements in the array. + + Raises + ====== + + ValueError : When the length of the array is symbolic + + Examples + ======== + + >>> from sympy.tensor.array import ArrayComprehension + >>> from sympy import symbols + >>> i, j = symbols('i j') + >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3)) + >>> len(a) + 12 + """ + if self._loop_size.free_symbols: + raise ValueError('Symbolic length is not supported') + return self._loop_size + + @classmethod + def _check_limits_validity(cls, function, limits): + #limits = sympify(limits) + new_limits = [] + for var, inf, sup in limits: + var = _sympify(var) + inf = _sympify(inf) + #since this is stored as an argument, it should be + #a Tuple + if isinstance(sup, list): + sup = Tuple(*sup) + else: + sup = _sympify(sup) + new_limits.append(Tuple(var, inf, sup)) + if any((not isinstance(i, Expr)) or i.atoms(Symbol, Integer) != i.atoms() + for i in [inf, sup]): + raise TypeError('Bounds should be an Expression(combination of Integer and Symbol)') + if (inf > sup) == True: + raise ValueError('Lower bound should be inferior to upper bound') + if var in inf.free_symbols or var in sup.free_symbols: + raise ValueError('Variable should not be part of its bounds') + return new_limits + + @classmethod + def _calculate_shape_from_limits(cls, limits): + return tuple([sup - inf + 1 for _, inf, sup in limits]) + + @classmethod + def _calculate_loop_size(cls, shape): + if not shape: + return 0 + loop_size = 1 + for l in shape: + loop_size = loop_size * l + + return loop_size + + def doit(self, **hints): + if not self.is_shape_numeric: + return self + + return self._expand_array() + + def _expand_array(self): + res = [] + for values in itertools.product(*[range(inf, sup+1) + for var, inf, sup + in self._limits]): + res.append(self._get_element(values)) + + return ImmutableDenseNDimArray(res, self.shape) + + def _get_element(self, values): + temp = self.function + for var, val in zip(self.variables, values): + temp = temp.subs(var, val) + return temp + + def tolist(self): + """Transform the expanded array to a list. + + Raises + ====== + + ValueError : When there is a symbolic dimension + + Examples + ======== + + >>> from sympy.tensor.array import ArrayComprehension + >>> from sympy import symbols + >>> i, j = symbols('i j') + >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3)) + >>> a.tolist() + [[11, 12, 13], [21, 22, 23], [31, 32, 33], [41, 42, 43]] + """ + if self.is_shape_numeric: + return self._expand_array().tolist() + + raise ValueError("A symbolic array cannot be expanded to a list") + + def tomatrix(self): + """Transform the expanded array to a matrix. + + Raises + ====== + + ValueError : When there is a symbolic dimension + ValueError : When the rank of the expanded array is not equal to 2 + + Examples + ======== + + >>> from sympy.tensor.array import ArrayComprehension + >>> from sympy import symbols + >>> i, j = symbols('i j') + >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3)) + >>> a.tomatrix() + Matrix([ + [11, 12, 13], + [21, 22, 23], + [31, 32, 33], + [41, 42, 43]]) + """ + from sympy.matrices import Matrix + + if not self.is_shape_numeric: + raise ValueError("A symbolic array cannot be expanded to a matrix") + if self._rank != 2: + raise ValueError('Dimensions must be of size of 2') + + return Matrix(self._expand_array().tomatrix()) + + +def isLambda(v): + LAMBDA = lambda: 0 + return isinstance(v, type(LAMBDA)) and v.__name__ == LAMBDA.__name__ + +class ArrayComprehensionMap(ArrayComprehension): + ''' + A subclass of ArrayComprehension dedicated to map external function lambda. + + Notes + ===== + + Only the lambda function is considered. + At most one argument in lambda function is accepted in order to avoid ambiguity + in value assignment. + + Examples + ======== + + >>> from sympy.tensor.array import ArrayComprehensionMap + >>> from sympy import symbols + >>> i, j, k = symbols('i j k') + >>> a = ArrayComprehensionMap(lambda: 1, (i, 1, 4)) + >>> a.doit() + [1, 1, 1, 1] + >>> b = ArrayComprehensionMap(lambda a: a+1, (j, 1, 4)) + >>> b.doit() + [2, 3, 4, 5] + + ''' + def __new__(cls, function, *symbols, **assumptions): + if any(len(l) != 3 or None for l in symbols): + raise ValueError('ArrayComprehension requires values lower and upper bound' + ' for the expression') + + if not isLambda(function): + raise ValueError('Data type not supported') + + arglist = cls._check_limits_validity(function, symbols) + obj = Basic.__new__(cls, *arglist, **assumptions) + obj._limits = obj._args + obj._shape = cls._calculate_shape_from_limits(obj._limits) + obj._rank = len(obj._shape) + obj._loop_size = cls._calculate_loop_size(obj._shape) + obj._lambda = function + return obj + + @property + def func(self): + class _(ArrayComprehensionMap): + def __new__(cls, *args, **kwargs): + return ArrayComprehensionMap(self._lambda, *args, **kwargs) + return _ + + def _get_element(self, values): + temp = self._lambda + if self._lambda.__code__.co_argcount == 0: + temp = temp() + elif self._lambda.__code__.co_argcount == 1: + temp = temp(functools.reduce(lambda a, b: a*b, values)) + return temp diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/array/array_derivatives.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/array_derivatives.py new file mode 100644 index 0000000000000000000000000000000000000000..a38db6caefe256a8c7e1f3415b78351b3787fee9 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/array_derivatives.py @@ -0,0 +1,129 @@ +from __future__ import annotations + +from sympy.core.expr import Expr +from sympy.core.function import Derivative +from sympy.core.numbers import Integer +from sympy.matrices.matrixbase import MatrixBase +from .ndim_array import NDimArray +from .arrayop import derive_by_array +from sympy.matrices.expressions.matexpr import MatrixExpr +from sympy.matrices.expressions.special import ZeroMatrix +from sympy.matrices.expressions.matexpr import _matrix_derivative + + +class ArrayDerivative(Derivative): + + is_scalar = False + + def __new__(cls, expr, *variables, **kwargs): + obj = super().__new__(cls, expr, *variables, **kwargs) + if isinstance(obj, ArrayDerivative): + obj._shape = obj._get_shape() + return obj + + def _get_shape(self): + shape = () + for v, count in self.variable_count: + if hasattr(v, "shape"): + for i in range(count): + shape += v.shape + if hasattr(self.expr, "shape"): + shape += self.expr.shape + return shape + + @property + def shape(self): + return self._shape + + @classmethod + def _get_zero_with_shape_like(cls, expr): + if isinstance(expr, (MatrixBase, NDimArray)): + return expr.zeros(*expr.shape) + elif isinstance(expr, MatrixExpr): + return ZeroMatrix(*expr.shape) + else: + raise RuntimeError("Unable to determine shape of array-derivative.") + + @staticmethod + def _call_derive_scalar_by_matrix(expr: Expr, v: MatrixBase) -> Expr: + return v.applyfunc(lambda x: expr.diff(x)) + + @staticmethod + def _call_derive_scalar_by_matexpr(expr: Expr, v: MatrixExpr) -> Expr: + if expr.has(v): + return _matrix_derivative(expr, v) + else: + return ZeroMatrix(*v.shape) + + @staticmethod + def _call_derive_scalar_by_array(expr: Expr, v: NDimArray) -> Expr: + return v.applyfunc(lambda x: expr.diff(x)) + + @staticmethod + def _call_derive_matrix_by_scalar(expr: MatrixBase, v: Expr) -> Expr: + return _matrix_derivative(expr, v) + + @staticmethod + def _call_derive_matexpr_by_scalar(expr: MatrixExpr, v: Expr) -> Expr: + return expr._eval_derivative(v) + + @staticmethod + def _call_derive_array_by_scalar(expr: NDimArray, v: Expr) -> Expr: + return expr.applyfunc(lambda x: x.diff(v)) + + @staticmethod + def _call_derive_default(expr: Expr, v: Expr) -> Expr | None: + if expr.has(v): + return _matrix_derivative(expr, v) + else: + return None + + @classmethod + def _dispatch_eval_derivative_n_times(cls, expr, v, count): + # Evaluate the derivative `n` times. If + # `_eval_derivative_n_times` is not overridden by the current + # object, the default in `Basic` will call a loop over + # `_eval_derivative`: + + if not isinstance(count, (int, Integer)) or ((count <= 0) == True): + return None + + # TODO: this could be done with multiple-dispatching: + if expr.is_scalar: + if isinstance(v, MatrixBase): + result = cls._call_derive_scalar_by_matrix(expr, v) + elif isinstance(v, MatrixExpr): + result = cls._call_derive_scalar_by_matexpr(expr, v) + elif isinstance(v, NDimArray): + result = cls._call_derive_scalar_by_array(expr, v) + elif v.is_scalar: + # scalar by scalar has a special + return super()._dispatch_eval_derivative_n_times(expr, v, count) + else: + return None + elif v.is_scalar: + if isinstance(expr, MatrixBase): + result = cls._call_derive_matrix_by_scalar(expr, v) + elif isinstance(expr, MatrixExpr): + result = cls._call_derive_matexpr_by_scalar(expr, v) + elif isinstance(expr, NDimArray): + result = cls._call_derive_array_by_scalar(expr, v) + else: + return None + else: + # Both `expr` and `v` are some array/matrix type: + if isinstance(expr, MatrixBase) or isinstance(v, MatrixBase): + result = derive_by_array(expr, v) + elif isinstance(expr, MatrixExpr) and isinstance(v, MatrixExpr): + result = cls._call_derive_default(expr, v) + elif isinstance(expr, MatrixExpr) or isinstance(v, MatrixExpr): + # if one expression is a symbolic matrix expression while the other isn't, don't evaluate: + return None + else: + result = derive_by_array(expr, v) + if result is None: + return None + if count == 1: + return result + else: + return cls._dispatch_eval_derivative_n_times(result, v, count - 1) diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/array/arrayop.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/arrayop.py new file mode 100644 index 0000000000000000000000000000000000000000..a81e6b381a8a93f0cd585278a4be0259b06406dd --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/arrayop.py @@ -0,0 +1,528 @@ +import itertools +from collections.abc import Iterable + +from sympy.core._print_helpers import Printable +from sympy.core.containers import Tuple +from sympy.core.function import diff +from sympy.core.singleton import S +from sympy.core.sympify import _sympify + +from sympy.tensor.array.ndim_array import NDimArray +from sympy.tensor.array.dense_ndim_array import DenseNDimArray, ImmutableDenseNDimArray +from sympy.tensor.array.sparse_ndim_array import SparseNDimArray + + +def _arrayfy(a): + from sympy.matrices import MatrixBase + + if isinstance(a, NDimArray): + return a + if isinstance(a, (MatrixBase, list, tuple, Tuple)): + return ImmutableDenseNDimArray(a) + return a + + +def tensorproduct(*args): + """ + Tensor product among scalars or array-like objects. + + The equivalent operator for array expressions is ``ArrayTensorProduct``, + which can be used to keep the expression unevaluated. + + Examples + ======== + + >>> from sympy.tensor.array import tensorproduct, Array + >>> from sympy.abc import x, y, z, t + >>> A = Array([[1, 2], [3, 4]]) + >>> B = Array([x, y]) + >>> tensorproduct(A, B) + [[[x, y], [2*x, 2*y]], [[3*x, 3*y], [4*x, 4*y]]] + >>> tensorproduct(A, x) + [[x, 2*x], [3*x, 4*x]] + >>> tensorproduct(A, B, B) + [[[[x**2, x*y], [x*y, y**2]], [[2*x**2, 2*x*y], [2*x*y, 2*y**2]]], [[[3*x**2, 3*x*y], [3*x*y, 3*y**2]], [[4*x**2, 4*x*y], [4*x*y, 4*y**2]]]] + + Applying this function on two matrices will result in a rank 4 array. + + >>> from sympy import Matrix, eye + >>> m = Matrix([[x, y], [z, t]]) + >>> p = tensorproduct(eye(3), m) + >>> p + [[[[x, y], [z, t]], [[0, 0], [0, 0]], [[0, 0], [0, 0]]], [[[0, 0], [0, 0]], [[x, y], [z, t]], [[0, 0], [0, 0]]], [[[0, 0], [0, 0]], [[0, 0], [0, 0]], [[x, y], [z, t]]]] + + See Also + ======== + + sympy.tensor.array.expressions.array_expressions.ArrayTensorProduct + + """ + from sympy.tensor.array import SparseNDimArray, ImmutableSparseNDimArray + + if len(args) == 0: + return S.One + if len(args) == 1: + return _arrayfy(args[0]) + from sympy.tensor.array.expressions.array_expressions import _CodegenArrayAbstract + from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct + from sympy.tensor.array.expressions.array_expressions import _ArrayExpr + from sympy.matrices.expressions.matexpr import MatrixSymbol + if any(isinstance(arg, (_ArrayExpr, _CodegenArrayAbstract, MatrixSymbol)) for arg in args): + return ArrayTensorProduct(*args) + if len(args) > 2: + return tensorproduct(tensorproduct(args[0], args[1]), *args[2:]) + + # length of args is 2: + a, b = map(_arrayfy, args) + + if not isinstance(a, NDimArray) or not isinstance(b, NDimArray): + return a*b + + if isinstance(a, SparseNDimArray) and isinstance(b, SparseNDimArray): + lp = len(b) + new_array = {k1*lp + k2: v1*v2 for k1, v1 in a._sparse_array.items() for k2, v2 in b._sparse_array.items()} + return ImmutableSparseNDimArray(new_array, a.shape + b.shape) + + product_list = [i*j for i in Flatten(a) for j in Flatten(b)] + return ImmutableDenseNDimArray(product_list, a.shape + b.shape) + + +def _util_contraction_diagonal(array, *contraction_or_diagonal_axes): + array = _arrayfy(array) + + # Verify contraction_axes: + taken_dims = set() + for axes_group in contraction_or_diagonal_axes: + if not isinstance(axes_group, Iterable): + raise ValueError("collections of contraction/diagonal axes expected") + + dim = array.shape[axes_group[0]] + + for d in axes_group: + if d in taken_dims: + raise ValueError("dimension specified more than once") + if dim != array.shape[d]: + raise ValueError("cannot contract or diagonalize between axes of different dimension") + taken_dims.add(d) + + rank = array.rank() + + remaining_shape = [dim for i, dim in enumerate(array.shape) if i not in taken_dims] + cum_shape = [0]*rank + _cumul = 1 + for i in range(rank): + cum_shape[rank - i - 1] = _cumul + _cumul *= int(array.shape[rank - i - 1]) + + # DEFINITION: by absolute position it is meant the position along the one + # dimensional array containing all the tensor components. + + # Possible future work on this module: move computation of absolute + # positions to a class method. + + # Determine absolute positions of the uncontracted indices: + remaining_indices = [[cum_shape[i]*j for j in range(array.shape[i])] + for i in range(rank) if i not in taken_dims] + + # Determine absolute positions of the contracted indices: + summed_deltas = [] + for axes_group in contraction_or_diagonal_axes: + lidx = [] + for js in range(array.shape[axes_group[0]]): + lidx.append(sum(cum_shape[ig] * js for ig in axes_group)) + summed_deltas.append(lidx) + + return array, remaining_indices, remaining_shape, summed_deltas + + +def tensorcontraction(array, *contraction_axes): + """ + Contraction of an array-like object on the specified axes. + + The equivalent operator for array expressions is ``ArrayContraction``, + which can be used to keep the expression unevaluated. + + Examples + ======== + + >>> from sympy import Array, tensorcontraction + >>> from sympy import Matrix, eye + >>> tensorcontraction(eye(3), (0, 1)) + 3 + >>> A = Array(range(18), (3, 2, 3)) + >>> A + [[[0, 1, 2], [3, 4, 5]], [[6, 7, 8], [9, 10, 11]], [[12, 13, 14], [15, 16, 17]]] + >>> tensorcontraction(A, (0, 2)) + [21, 30] + + Matrix multiplication may be emulated with a proper combination of + ``tensorcontraction`` and ``tensorproduct`` + + >>> from sympy import tensorproduct + >>> from sympy.abc import a,b,c,d,e,f,g,h + >>> m1 = Matrix([[a, b], [c, d]]) + >>> m2 = Matrix([[e, f], [g, h]]) + >>> p = tensorproduct(m1, m2) + >>> p + [[[[a*e, a*f], [a*g, a*h]], [[b*e, b*f], [b*g, b*h]]], [[[c*e, c*f], [c*g, c*h]], [[d*e, d*f], [d*g, d*h]]]] + >>> tensorcontraction(p, (1, 2)) + [[a*e + b*g, a*f + b*h], [c*e + d*g, c*f + d*h]] + >>> m1*m2 + Matrix([ + [a*e + b*g, a*f + b*h], + [c*e + d*g, c*f + d*h]]) + + See Also + ======== + + sympy.tensor.array.expressions.array_expressions.ArrayContraction + + """ + from sympy.tensor.array.expressions.array_expressions import _array_contraction + from sympy.tensor.array.expressions.array_expressions import _CodegenArrayAbstract + from sympy.tensor.array.expressions.array_expressions import _ArrayExpr + from sympy.matrices.expressions.matexpr import MatrixSymbol + if isinstance(array, (_ArrayExpr, _CodegenArrayAbstract, MatrixSymbol)): + return _array_contraction(array, *contraction_axes) + + array, remaining_indices, remaining_shape, summed_deltas = _util_contraction_diagonal(array, *contraction_axes) + + # Compute the contracted array: + # + # 1. external for loops on all uncontracted indices. + # Uncontracted indices are determined by the combinatorial product of + # the absolute positions of the remaining indices. + # 2. internal loop on all contracted indices. + # It sums the values of the absolute contracted index and the absolute + # uncontracted index for the external loop. + contracted_array = [] + for icontrib in itertools.product(*remaining_indices): + index_base_position = sum(icontrib) + isum = S.Zero + for sum_to_index in itertools.product(*summed_deltas): + idx = array._get_tuple_index(index_base_position + sum(sum_to_index)) + isum += array[idx] + + contracted_array.append(isum) + + if len(remaining_indices) == 0: + assert len(contracted_array) == 1 + return contracted_array[0] + + return type(array)(contracted_array, remaining_shape) + + +def tensordiagonal(array, *diagonal_axes): + """ + Diagonalization of an array-like object on the specified axes. + + This is equivalent to multiplying the expression by Kronecker deltas + uniting the axes. + + The diagonal indices are put at the end of the axes. + + The equivalent operator for array expressions is ``ArrayDiagonal``, which + can be used to keep the expression unevaluated. + + Examples + ======== + + ``tensordiagonal`` acting on a 2-dimensional array by axes 0 and 1 is + equivalent to the diagonal of the matrix: + + >>> from sympy import Array, tensordiagonal + >>> from sympy import Matrix, eye + >>> tensordiagonal(eye(3), (0, 1)) + [1, 1, 1] + + >>> from sympy.abc import a,b,c,d + >>> m1 = Matrix([[a, b], [c, d]]) + >>> tensordiagonal(m1, [0, 1]) + [a, d] + + In case of higher dimensional arrays, the diagonalized out dimensions + are appended removed and appended as a single dimension at the end: + + >>> A = Array(range(18), (3, 2, 3)) + >>> A + [[[0, 1, 2], [3, 4, 5]], [[6, 7, 8], [9, 10, 11]], [[12, 13, 14], [15, 16, 17]]] + >>> tensordiagonal(A, (0, 2)) + [[0, 7, 14], [3, 10, 17]] + >>> from sympy import permutedims + >>> tensordiagonal(A, (0, 2)) == permutedims(Array([A[0, :, 0], A[1, :, 1], A[2, :, 2]]), [1, 0]) + True + + See Also + ======== + + sympy.tensor.array.expressions.array_expressions.ArrayDiagonal + + """ + if any(len(i) <= 1 for i in diagonal_axes): + raise ValueError("need at least two axes to diagonalize") + + from sympy.tensor.array.expressions.array_expressions import _ArrayExpr + from sympy.tensor.array.expressions.array_expressions import _CodegenArrayAbstract + from sympy.tensor.array.expressions.array_expressions import ArrayDiagonal, _array_diagonal + from sympy.matrices.expressions.matexpr import MatrixSymbol + if isinstance(array, (_ArrayExpr, _CodegenArrayAbstract, MatrixSymbol)): + return _array_diagonal(array, *diagonal_axes) + + ArrayDiagonal._validate(array, *diagonal_axes) + + array, remaining_indices, remaining_shape, diagonal_deltas = _util_contraction_diagonal(array, *diagonal_axes) + + # Compute the diagonalized array: + # + # 1. external for loops on all undiagonalized indices. + # Undiagonalized indices are determined by the combinatorial product of + # the absolute positions of the remaining indices. + # 2. internal loop on all diagonal indices. + # It appends the values of the absolute diagonalized index and the absolute + # undiagonalized index for the external loop. + diagonalized_array = [] + diagonal_shape = [len(i) for i in diagonal_deltas] + for icontrib in itertools.product(*remaining_indices): + index_base_position = sum(icontrib) + isum = [] + for sum_to_index in itertools.product(*diagonal_deltas): + idx = array._get_tuple_index(index_base_position + sum(sum_to_index)) + isum.append(array[idx]) + + isum = type(array)(isum).reshape(*diagonal_shape) + diagonalized_array.append(isum) + + return type(array)(diagonalized_array, remaining_shape + diagonal_shape) + + +def derive_by_array(expr, dx): + r""" + Derivative by arrays. Supports both arrays and scalars. + + The equivalent operator for array expressions is ``array_derive``. + + Explanation + =========== + + Given the array `A_{i_1, \ldots, i_N}` and the array `X_{j_1, \ldots, j_M}` + this function will return a new array `B` defined by + + `B_{j_1,\ldots,j_M,i_1,\ldots,i_N} := \frac{\partial A_{i_1,\ldots,i_N}}{\partial X_{j_1,\ldots,j_M}}` + + Examples + ======== + + >>> from sympy import derive_by_array + >>> from sympy.abc import x, y, z, t + >>> from sympy import cos + >>> derive_by_array(cos(x*t), x) + -t*sin(t*x) + >>> derive_by_array(cos(x*t), [x, y, z, t]) + [-t*sin(t*x), 0, 0, -x*sin(t*x)] + >>> derive_by_array([x, y**2*z], [[x, y], [z, t]]) + [[[1, 0], [0, 2*y*z]], [[0, y**2], [0, 0]]] + + """ + from sympy.matrices import MatrixBase + from sympy.tensor.array import SparseNDimArray + array_types = (Iterable, MatrixBase, NDimArray) + + if isinstance(dx, array_types): + dx = ImmutableDenseNDimArray(dx) + for i in dx: + if not i._diff_wrt: + raise ValueError("cannot derive by this array") + + if isinstance(expr, array_types): + if isinstance(expr, NDimArray): + expr = expr.as_immutable() + else: + expr = ImmutableDenseNDimArray(expr) + + if isinstance(dx, array_types): + if isinstance(expr, SparseNDimArray): + lp = len(expr) + new_array = {k + i*lp: v + for i, x in enumerate(Flatten(dx)) + for k, v in expr.diff(x)._sparse_array.items()} + else: + new_array = [[y.diff(x) for y in Flatten(expr)] for x in Flatten(dx)] + return type(expr)(new_array, dx.shape + expr.shape) + else: + return expr.diff(dx) + else: + expr = _sympify(expr) + if isinstance(dx, array_types): + return ImmutableDenseNDimArray([expr.diff(i) for i in Flatten(dx)], dx.shape) + else: + dx = _sympify(dx) + return diff(expr, dx) + + +def permutedims(expr, perm=None, index_order_old=None, index_order_new=None): + """ + Permutes the indices of an array. + + Parameter specifies the permutation of the indices. + + The equivalent operator for array expressions is ``PermuteDims``, which can + be used to keep the expression unevaluated. + + Examples + ======== + + >>> from sympy.abc import x, y, z, t + >>> from sympy import sin + >>> from sympy import Array, permutedims + >>> a = Array([[x, y, z], [t, sin(x), 0]]) + >>> a + [[x, y, z], [t, sin(x), 0]] + >>> permutedims(a, (1, 0)) + [[x, t], [y, sin(x)], [z, 0]] + + If the array is of second order, ``transpose`` can be used: + + >>> from sympy import transpose + >>> transpose(a) + [[x, t], [y, sin(x)], [z, 0]] + + Examples on higher dimensions: + + >>> b = Array([[[1, 2], [3, 4]], [[5, 6], [7, 8]]]) + >>> permutedims(b, (2, 1, 0)) + [[[1, 5], [3, 7]], [[2, 6], [4, 8]]] + >>> permutedims(b, (1, 2, 0)) + [[[1, 5], [2, 6]], [[3, 7], [4, 8]]] + + An alternative way to specify the same permutations as in the previous + lines involves passing the *old* and *new* indices, either as a list or as + a string: + + >>> permutedims(b, index_order_old="cba", index_order_new="abc") + [[[1, 5], [3, 7]], [[2, 6], [4, 8]]] + >>> permutedims(b, index_order_old="cab", index_order_new="abc") + [[[1, 5], [2, 6]], [[3, 7], [4, 8]]] + + ``Permutation`` objects are also allowed: + + >>> from sympy.combinatorics import Permutation + >>> permutedims(b, Permutation([1, 2, 0])) + [[[1, 5], [2, 6]], [[3, 7], [4, 8]]] + + See Also + ======== + + sympy.tensor.array.expressions.array_expressions.PermuteDims + + """ + from sympy.tensor.array import SparseNDimArray + + from sympy.tensor.array.expressions.array_expressions import _ArrayExpr + from sympy.tensor.array.expressions.array_expressions import _CodegenArrayAbstract + from sympy.tensor.array.expressions.array_expressions import _permute_dims + from sympy.matrices.expressions.matexpr import MatrixSymbol + from sympy.tensor.array.expressions import PermuteDims + from sympy.tensor.array.expressions.array_expressions import get_rank + perm = PermuteDims._get_permutation_from_arguments(perm, index_order_old, index_order_new, get_rank(expr)) + if isinstance(expr, (_ArrayExpr, _CodegenArrayAbstract, MatrixSymbol)): + return _permute_dims(expr, perm) + + if not isinstance(expr, NDimArray): + expr = ImmutableDenseNDimArray(expr) + + from sympy.combinatorics import Permutation + if not isinstance(perm, Permutation): + perm = Permutation(list(perm)) + + if perm.size != expr.rank(): + raise ValueError("wrong permutation size") + + # Get the inverse permutation: + iperm = ~perm + new_shape = perm(expr.shape) + + if isinstance(expr, SparseNDimArray): + return type(expr)({tuple(perm(expr._get_tuple_index(k))): v + for k, v in expr._sparse_array.items()}, new_shape) + + indices_span = perm([range(i) for i in expr.shape]) + + new_array = [None]*len(expr) + for i, idx in enumerate(itertools.product(*indices_span)): + t = iperm(idx) + new_array[i] = expr[t] + + return type(expr)(new_array, new_shape) + + +class Flatten(Printable): + """ + Flatten an iterable object to a list in a lazy-evaluation way. + + Notes + ===== + + This class is an iterator with which the memory cost can be economised. + Optimisation has been considered to ameliorate the performance for some + specific data types like DenseNDimArray and SparseNDimArray. + + Examples + ======== + + >>> from sympy.tensor.array.arrayop import Flatten + >>> from sympy.tensor.array import Array + >>> A = Array(range(6)).reshape(2, 3) + >>> Flatten(A) + Flatten([[0, 1, 2], [3, 4, 5]]) + >>> [i for i in Flatten(A)] + [0, 1, 2, 3, 4, 5] + """ + def __init__(self, iterable): + from sympy.matrices.matrixbase import MatrixBase + from sympy.tensor.array import NDimArray + + if not isinstance(iterable, (Iterable, MatrixBase)): + raise NotImplementedError("Data type not yet supported") + + if isinstance(iterable, list): + iterable = NDimArray(iterable) + + self._iter = iterable + self._idx = 0 + + def __iter__(self): + return self + + def __next__(self): + from sympy.matrices.matrixbase import MatrixBase + + if len(self._iter) > self._idx: + if isinstance(self._iter, DenseNDimArray): + result = self._iter._array[self._idx] + + elif isinstance(self._iter, SparseNDimArray): + if self._idx in self._iter._sparse_array: + result = self._iter._sparse_array[self._idx] + else: + result = 0 + + elif isinstance(self._iter, MatrixBase): + result = self._iter[self._idx] + + elif hasattr(self._iter, '__next__'): + result = next(self._iter) + + else: + result = self._iter[self._idx] + + else: + raise StopIteration + + self._idx += 1 + return result + + def next(self): + return self.__next__() + + def _sympystr(self, printer): + return type(self).__name__ + '(' + printer._print(self._iter) + ')' diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/array/dense_ndim_array.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/dense_ndim_array.py new file mode 100644 index 0000000000000000000000000000000000000000..576e452c55d8d374ca1f72c553f3a64de7227d43 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/dense_ndim_array.py @@ -0,0 +1,206 @@ +import functools +from typing import List + +from sympy.core.basic import Basic +from sympy.core.containers import Tuple +from sympy.core.singleton import S +from sympy.core.sympify import _sympify +from sympy.tensor.array.mutable_ndim_array import MutableNDimArray +from sympy.tensor.array.ndim_array import NDimArray, ImmutableNDimArray, ArrayKind +from sympy.utilities.iterables import flatten + + +class DenseNDimArray(NDimArray): + + _array: List[Basic] + + def __new__(self, *args, **kwargs): + return ImmutableDenseNDimArray(*args, **kwargs) + + @property + def kind(self) -> ArrayKind: + return ArrayKind._union(self._array) + + def __getitem__(self, index): + """ + Allows to get items from N-dim array. + + Examples + ======== + + >>> from sympy import MutableDenseNDimArray + >>> a = MutableDenseNDimArray([0, 1, 2, 3], (2, 2)) + >>> a + [[0, 1], [2, 3]] + >>> a[0, 0] + 0 + >>> a[1, 1] + 3 + >>> a[0] + [0, 1] + >>> a[1] + [2, 3] + + + Symbolic index: + + >>> from sympy.abc import i, j + >>> a[i, j] + [[0, 1], [2, 3]][i, j] + + Replace `i` and `j` to get element `(1, 1)`: + + >>> a[i, j].subs({i: 1, j: 1}) + 3 + + """ + syindex = self._check_symbolic_index(index) + if syindex is not None: + return syindex + + index = self._check_index_for_getitem(index) + + if isinstance(index, tuple) and any(isinstance(i, slice) for i in index): + sl_factors, eindices = self._get_slice_data_for_array_access(index) + array = [self._array[self._parse_index(i)] for i in eindices] + nshape = [len(el) for i, el in enumerate(sl_factors) if isinstance(index[i], slice)] + return type(self)(array, nshape) + else: + index = self._parse_index(index) + return self._array[index] + + @classmethod + def zeros(cls, *shape): + list_length = functools.reduce(lambda x, y: x*y, shape, S.One) + return cls._new(([0]*list_length,), shape) + + def tomatrix(self): + """ + Converts MutableDenseNDimArray to Matrix. Can convert only 2-dim array, else will raise error. + + Examples + ======== + + >>> from sympy import MutableDenseNDimArray + >>> a = MutableDenseNDimArray([1 for i in range(9)], (3, 3)) + >>> b = a.tomatrix() + >>> b + Matrix([ + [1, 1, 1], + [1, 1, 1], + [1, 1, 1]]) + + """ + from sympy.matrices import Matrix + + if self.rank() != 2: + raise ValueError('Dimensions must be of size of 2') + + return Matrix(self.shape[0], self.shape[1], self._array) + + def reshape(self, *newshape): + """ + Returns MutableDenseNDimArray instance with new shape. Elements number + must be suitable to new shape. The only argument of method sets + new shape. + + Examples + ======== + + >>> from sympy import MutableDenseNDimArray + >>> a = MutableDenseNDimArray([1, 2, 3, 4, 5, 6], (2, 3)) + >>> a.shape + (2, 3) + >>> a + [[1, 2, 3], [4, 5, 6]] + >>> b = a.reshape(3, 2) + >>> b.shape + (3, 2) + >>> b + [[1, 2], [3, 4], [5, 6]] + + """ + new_total_size = functools.reduce(lambda x,y: x*y, newshape) + if new_total_size != self._loop_size: + raise ValueError('Expecting reshape size to %d but got prod(%s) = %d' % ( + self._loop_size, str(newshape), new_total_size)) + + # there is no `.func` as this class does not subtype `Basic`: + return type(self)(self._array, newshape) + + +class ImmutableDenseNDimArray(DenseNDimArray, ImmutableNDimArray): # type: ignore + def __new__(cls, iterable, shape=None, **kwargs): + return cls._new(iterable, shape, **kwargs) + + @classmethod + def _new(cls, iterable, shape, **kwargs): + shape, flat_list = cls._handle_ndarray_creation_inputs(iterable, shape, **kwargs) + shape = Tuple(*map(_sympify, shape)) + cls._check_special_bounds(flat_list, shape) + flat_list = flatten(flat_list) + flat_list = Tuple(*flat_list) + self = Basic.__new__(cls, flat_list, shape, **kwargs) + self._shape = shape + self._array = list(flat_list) + self._rank = len(shape) + self._loop_size = functools.reduce(lambda x,y: x*y, shape, 1) + return self + + def __setitem__(self, index, value): + raise TypeError('immutable N-dim array') + + def as_mutable(self): + return MutableDenseNDimArray(self) + + def _eval_simplify(self, **kwargs): + from sympy.simplify.simplify import simplify + return self.applyfunc(simplify) + +class MutableDenseNDimArray(DenseNDimArray, MutableNDimArray): + + def __new__(cls, iterable=None, shape=None, **kwargs): + return cls._new(iterable, shape, **kwargs) + + @classmethod + def _new(cls, iterable, shape, **kwargs): + shape, flat_list = cls._handle_ndarray_creation_inputs(iterable, shape, **kwargs) + flat_list = flatten(flat_list) + self = object.__new__(cls) + self._shape = shape + self._array = list(flat_list) + self._rank = len(shape) + self._loop_size = functools.reduce(lambda x,y: x*y, shape) if shape else len(flat_list) + return self + + def __setitem__(self, index, value): + """Allows to set items to MutableDenseNDimArray. + + Examples + ======== + + >>> from sympy import MutableDenseNDimArray + >>> a = MutableDenseNDimArray.zeros(2, 2) + >>> a[0,0] = 1 + >>> a[1,1] = 1 + >>> a + [[1, 0], [0, 1]] + + """ + if isinstance(index, tuple) and any(isinstance(i, slice) for i in index): + value, eindices, slice_offsets = self._get_slice_data_for_array_assignment(index, value) + for i in eindices: + other_i = [ind - j for ind, j in zip(i, slice_offsets) if j is not None] + self._array[self._parse_index(i)] = value[other_i] + else: + index = self._parse_index(index) + self._setter_iterable_check(value) + value = _sympify(value) + self._array[index] = value + + def as_immutable(self): + return ImmutableDenseNDimArray(self) + + @property + def free_symbols(self): + return {i for j in self._array for i in j.free_symbols} diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/__init__.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..f1658241782cdf0e38a30c43a6d67f9811297f4c --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/__init__.py @@ -0,0 +1,178 @@ +r""" +Array expressions are expressions representing N-dimensional arrays, without +evaluating them. These expressions represent in a certain way abstract syntax +trees of operations on N-dimensional arrays. + +Every N-dimensional array operator has a corresponding array expression object. + +Table of correspondences: + +=============================== ============================= + Array operator Array expression operator +=============================== ============================= + tensorproduct ArrayTensorProduct + tensorcontraction ArrayContraction + tensordiagonal ArrayDiagonal + permutedims PermuteDims +=============================== ============================= + +Examples +======== + +``ArraySymbol`` objects are the N-dimensional equivalent of ``MatrixSymbol`` +objects in the matrix module: + +>>> from sympy.tensor.array.expressions import ArraySymbol +>>> from sympy.abc import i, j, k +>>> A = ArraySymbol("A", (3, 2, 4)) +>>> A.shape +(3, 2, 4) +>>> A[i, j, k] +A[i, j, k] +>>> A.as_explicit() +[[[A[0, 0, 0], A[0, 0, 1], A[0, 0, 2], A[0, 0, 3]], + [A[0, 1, 0], A[0, 1, 1], A[0, 1, 2], A[0, 1, 3]]], + [[A[1, 0, 0], A[1, 0, 1], A[1, 0, 2], A[1, 0, 3]], + [A[1, 1, 0], A[1, 1, 1], A[1, 1, 2], A[1, 1, 3]]], + [[A[2, 0, 0], A[2, 0, 1], A[2, 0, 2], A[2, 0, 3]], + [A[2, 1, 0], A[2, 1, 1], A[2, 1, 2], A[2, 1, 3]]]] + +Component-explicit arrays can be added inside array expressions: + +>>> from sympy import Array +>>> from sympy import tensorproduct +>>> from sympy.tensor.array.expressions import ArrayTensorProduct +>>> a = Array([1, 2, 3]) +>>> b = Array([i, j, k]) +>>> expr = ArrayTensorProduct(a, b, b) +>>> expr +ArrayTensorProduct([1, 2, 3], [i, j, k], [i, j, k]) +>>> expr.as_explicit() == tensorproduct(a, b, b) +True + +Constructing array expressions from index-explicit forms +-------------------------------------------------------- + +Array expressions are index-implicit. This means they do not use any indices to +represent array operations. The function ``convert_indexed_to_array( ... )`` +may be used to convert index-explicit expressions to array expressions. +It takes as input two parameters: the index-explicit expression and the order +of the indices: + +>>> from sympy.tensor.array.expressions import convert_indexed_to_array +>>> from sympy import Sum +>>> A = ArraySymbol("A", (3, 3)) +>>> B = ArraySymbol("B", (3, 3)) +>>> convert_indexed_to_array(A[i, j], [i, j]) +A +>>> convert_indexed_to_array(A[i, j], [j, i]) +PermuteDims(A, (0 1)) +>>> convert_indexed_to_array(A[i, j] + B[j, i], [i, j]) +ArrayAdd(A, PermuteDims(B, (0 1))) +>>> convert_indexed_to_array(Sum(A[i, j]*B[j, k], (j, 0, 2)), [i, k]) +ArrayContraction(ArrayTensorProduct(A, B), (1, 2)) + +The diagonal of a matrix in the array expression form: + +>>> convert_indexed_to_array(A[i, i], [i]) +ArrayDiagonal(A, (0, 1)) + +The trace of a matrix in the array expression form: + +>>> convert_indexed_to_array(Sum(A[i, i], (i, 0, 2)), [i]) +ArrayContraction(A, (0, 1)) + +Compatibility with matrices +--------------------------- + +Array expressions can be mixed with objects from the matrix module: + +>>> from sympy import MatrixSymbol +>>> from sympy.tensor.array.expressions import ArrayContraction +>>> M = MatrixSymbol("M", 3, 3) +>>> N = MatrixSymbol("N", 3, 3) + +Express the matrix product in the array expression form: + +>>> from sympy.tensor.array.expressions import convert_matrix_to_array +>>> expr = convert_matrix_to_array(M*N) +>>> expr +ArrayContraction(ArrayTensorProduct(M, N), (1, 2)) + +The expression can be converted back to matrix form: + +>>> from sympy.tensor.array.expressions import convert_array_to_matrix +>>> convert_array_to_matrix(expr) +M*N + +Add a second contraction on the remaining axes in order to get the trace of `M \cdot N`: + +>>> expr_tr = ArrayContraction(expr, (0, 1)) +>>> expr_tr +ArrayContraction(ArrayContraction(ArrayTensorProduct(M, N), (1, 2)), (0, 1)) + +Flatten the expression by calling ``.doit()`` and remove the nested array contraction operations: + +>>> expr_tr.doit() +ArrayContraction(ArrayTensorProduct(M, N), (0, 3), (1, 2)) + +Get the explicit form of the array expression: + +>>> expr.as_explicit() +[[M[0, 0]*N[0, 0] + M[0, 1]*N[1, 0] + M[0, 2]*N[2, 0], M[0, 0]*N[0, 1] + M[0, 1]*N[1, 1] + M[0, 2]*N[2, 1], M[0, 0]*N[0, 2] + M[0, 1]*N[1, 2] + M[0, 2]*N[2, 2]], + [M[1, 0]*N[0, 0] + M[1, 1]*N[1, 0] + M[1, 2]*N[2, 0], M[1, 0]*N[0, 1] + M[1, 1]*N[1, 1] + M[1, 2]*N[2, 1], M[1, 0]*N[0, 2] + M[1, 1]*N[1, 2] + M[1, 2]*N[2, 2]], + [M[2, 0]*N[0, 0] + M[2, 1]*N[1, 0] + M[2, 2]*N[2, 0], M[2, 0]*N[0, 1] + M[2, 1]*N[1, 1] + M[2, 2]*N[2, 1], M[2, 0]*N[0, 2] + M[2, 1]*N[1, 2] + M[2, 2]*N[2, 2]]] + +Express the trace of a matrix: + +>>> from sympy import Trace +>>> convert_matrix_to_array(Trace(M)) +ArrayContraction(M, (0, 1)) +>>> convert_matrix_to_array(Trace(M*N)) +ArrayContraction(ArrayTensorProduct(M, N), (0, 3), (1, 2)) + +Express the transposition of a matrix (will be expressed as a permutation of the axes: + +>>> convert_matrix_to_array(M.T) +PermuteDims(M, (0 1)) + +Compute the derivative array expressions: + +>>> from sympy.tensor.array.expressions import array_derive +>>> d = array_derive(M, M) +>>> d +PermuteDims(ArrayTensorProduct(I, I), (3)(1 2)) + +Verify that the derivative corresponds to the form computed with explicit matrices: + +>>> d.as_explicit() +[[[[1, 0, 0], [0, 0, 0], [0, 0, 0]], [[0, 1, 0], [0, 0, 0], [0, 0, 0]], [[0, 0, 1], [0, 0, 0], [0, 0, 0]]], [[[0, 0, 0], [1, 0, 0], [0, 0, 0]], [[0, 0, 0], [0, 1, 0], [0, 0, 0]], [[0, 0, 0], [0, 0, 1], [0, 0, 0]]], [[[0, 0, 0], [0, 0, 0], [1, 0, 0]], [[0, 0, 0], [0, 0, 0], [0, 1, 0]], [[0, 0, 0], [0, 0, 0], [0, 0, 1]]]] +>>> Me = M.as_explicit() +>>> Me.diff(Me) +[[[[1, 0, 0], [0, 0, 0], [0, 0, 0]], [[0, 1, 0], [0, 0, 0], [0, 0, 0]], [[0, 0, 1], [0, 0, 0], [0, 0, 0]]], [[[0, 0, 0], [1, 0, 0], [0, 0, 0]], [[0, 0, 0], [0, 1, 0], [0, 0, 0]], [[0, 0, 0], [0, 0, 1], [0, 0, 0]]], [[[0, 0, 0], [0, 0, 0], [1, 0, 0]], [[0, 0, 0], [0, 0, 0], [0, 1, 0]], [[0, 0, 0], [0, 0, 0], [0, 0, 1]]]] + +""" + +__all__ = [ + "ArraySymbol", "ArrayElement", "ZeroArray", "OneArray", + "ArrayTensorProduct", + "ArrayContraction", + "ArrayDiagonal", + "PermuteDims", + "ArrayAdd", + "ArrayElementwiseApplyFunc", + "Reshape", + "convert_array_to_matrix", + "convert_matrix_to_array", + "convert_array_to_indexed", + "convert_indexed_to_array", + "array_derive", +] + +from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct, ArrayAdd, PermuteDims, ArrayDiagonal, \ + ArrayContraction, Reshape, ArraySymbol, ArrayElement, ZeroArray, OneArray, ArrayElementwiseApplyFunc +from sympy.tensor.array.expressions.arrayexpr_derivatives import array_derive +from sympy.tensor.array.expressions.from_array_to_indexed import convert_array_to_indexed +from sympy.tensor.array.expressions.from_array_to_matrix import convert_array_to_matrix +from sympy.tensor.array.expressions.from_indexed_to_array import convert_indexed_to_array 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b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/array_expressions.py @@ -0,0 +1,1969 @@ +from __future__ import annotations +import collections.abc +import operator +from collections import defaultdict, Counter +from functools import reduce +import itertools +from itertools import accumulate + +import typing + +from sympy.core.numbers import Integer +from sympy.core.relational import Equality +from sympy.functions.special.tensor_functions import KroneckerDelta +from sympy.core.basic import Basic +from sympy.core.containers import Tuple +from sympy.core.expr import Expr +from sympy.core.function import (Function, Lambda) +from sympy.core.mul import Mul +from sympy.core.singleton import S +from sympy.core.sorting import default_sort_key +from sympy.core.symbol import (Dummy, Symbol) +from sympy.matrices.matrixbase import MatrixBase +from sympy.matrices.expressions.diagonal import diagonalize_vector +from sympy.matrices.expressions.matexpr import MatrixExpr +from sympy.matrices.expressions.special import ZeroMatrix +from sympy.tensor.array.arrayop import (permutedims, tensorcontraction, tensordiagonal, tensorproduct) +from sympy.tensor.array.dense_ndim_array import ImmutableDenseNDimArray +from sympy.tensor.array.ndim_array import NDimArray +from sympy.tensor.indexed import (Indexed, IndexedBase) +from sympy.matrices.expressions.matexpr import MatrixElement +from sympy.tensor.array.expressions.utils import _apply_recursively_over_nested_lists, _sort_contraction_indices, \ + _get_mapping_from_subranks, _build_push_indices_up_func_transformation, _get_contraction_links, \ + _build_push_indices_down_func_transformation +from sympy.combinatorics import Permutation +from sympy.combinatorics.permutations import _af_invert +from sympy.core.sympify import _sympify + + +class _ArrayExpr(Expr): + shape: tuple[Expr, ...] + + def __getitem__(self, item): + if not isinstance(item, collections.abc.Iterable): + item = (item,) + ArrayElement._check_shape(self, item) + return self._get(item) + + def _get(self, item): + return _get_array_element_or_slice(self, item) + + +class ArraySymbol(_ArrayExpr): + """ + Symbol representing an array expression + """ + + _iterable = False + + def __new__(cls, symbol, shape: typing.Iterable) -> "ArraySymbol": + if isinstance(symbol, str): + symbol = Symbol(symbol) + # symbol = _sympify(symbol) + shape = Tuple(*map(_sympify, shape)) + obj = Expr.__new__(cls, symbol, shape) + return obj + + @property + def name(self): + return self._args[0] + + @property + def shape(self): + return self._args[1] + + def as_explicit(self): + if not all(i.is_Integer for i in self.shape): + raise ValueError("cannot express explicit array with symbolic shape") + data = [self[i] for i in itertools.product(*[range(j) for j in self.shape])] + return ImmutableDenseNDimArray(data).reshape(*self.shape) + + +class ArrayElement(Expr): + """ + An element of an array. + """ + + _diff_wrt = True + is_symbol = True + is_commutative = True + + def __new__(cls, name, indices): + if isinstance(name, str): + name = Symbol(name) + name = _sympify(name) + if not isinstance(indices, collections.abc.Iterable): + indices = (indices,) + indices = _sympify(tuple(indices)) + cls._check_shape(name, indices) + obj = Expr.__new__(cls, name, indices) + return obj + + @classmethod + def _check_shape(cls, name, indices): + indices = tuple(indices) + if hasattr(name, "shape"): + index_error = IndexError("number of indices does not match shape of the array") + if len(indices) != len(name.shape): + raise index_error + if any((i >= s) == True for i, s in zip(indices, name.shape)): + raise ValueError("shape is out of bounds") + if any((i < 0) == True for i in indices): + raise ValueError("shape contains negative values") + + @property + def name(self): + return self._args[0] + + @property + def indices(self): + return self._args[1] + + def _eval_derivative(self, s): + if not isinstance(s, ArrayElement): + return S.Zero + + if s == self: + return S.One + + if s.name != self.name: + return S.Zero + + return Mul.fromiter(KroneckerDelta(i, j) for i, j in zip(self.indices, s.indices)) + + +class ZeroArray(_ArrayExpr): + """ + Symbolic array of zeros. Equivalent to ``ZeroMatrix`` for matrices. + """ + + def __new__(cls, *shape): + if len(shape) == 0: + return S.Zero + shape = map(_sympify, shape) + obj = Expr.__new__(cls, *shape) + return obj + + @property + def shape(self): + return self._args + + def as_explicit(self): + if not all(i.is_Integer for i in self.shape): + raise ValueError("Cannot return explicit form for symbolic shape.") + return ImmutableDenseNDimArray.zeros(*self.shape) + + def _get(self, item): + return S.Zero + + +class OneArray(_ArrayExpr): + """ + Symbolic array of ones. + """ + + def __new__(cls, *shape): + if len(shape) == 0: + return S.One + shape = map(_sympify, shape) + obj = Expr.__new__(cls, *shape) + return obj + + @property + def shape(self): + return self._args + + def as_explicit(self): + if not all(i.is_Integer for i in self.shape): + raise ValueError("Cannot return explicit form for symbolic shape.") + return ImmutableDenseNDimArray([S.One for i in range(reduce(operator.mul, self.shape))]).reshape(*self.shape) + + def _get(self, item): + return S.One + + +class _CodegenArrayAbstract(Basic): + + @property + def subranks(self): + """ + Returns the ranks of the objects in the uppermost tensor product inside + the current object. In case no tensor products are contained, return + the atomic ranks. + + Examples + ======== + + >>> from sympy.tensor.array import tensorproduct, tensorcontraction + >>> from sympy import MatrixSymbol + >>> M = MatrixSymbol("M", 3, 3) + >>> N = MatrixSymbol("N", 3, 3) + >>> P = MatrixSymbol("P", 3, 3) + + Important: do not confuse the rank of the matrix with the rank of an array. + + >>> tp = tensorproduct(M, N, P) + >>> tp.subranks + [2, 2, 2] + + >>> co = tensorcontraction(tp, (1, 2), (3, 4)) + >>> co.subranks + [2, 2, 2] + """ + return self._subranks[:] + + def subrank(self): + """ + The sum of ``subranks``. + """ + return sum(self.subranks) + + @property + def shape(self): + return self._shape + + def doit(self, **hints): + deep = hints.get("deep", True) + if deep: + return self.func(*[arg.doit(**hints) for arg in self.args])._canonicalize() + else: + return self._canonicalize() + +class ArrayTensorProduct(_CodegenArrayAbstract): + r""" + Class to represent the tensor product of array-like objects. + """ + + def __new__(cls, *args, **kwargs): + args = [_sympify(arg) for arg in args] + + canonicalize = kwargs.pop("canonicalize", False) + + ranks = [get_rank(arg) for arg in args] + + obj = Basic.__new__(cls, *args) + obj._subranks = ranks + shapes = [get_shape(i) for i in args] + + if any(i is None for i in shapes): + obj._shape = None + else: + obj._shape = tuple(j for i in shapes for j in i) + if canonicalize: + return obj._canonicalize() + return obj + + def _canonicalize(self): + args = self.args + args = self._flatten(args) + + ranks = [get_rank(arg) for arg in args] + + # Check if there are nested permutation and lift them up: + permutation_cycles = [] + for i, arg in enumerate(args): + if not isinstance(arg, PermuteDims): + continue + permutation_cycles.extend([[k + sum(ranks[:i]) for k in j] for j in arg.permutation.cyclic_form]) + args[i] = arg.expr + if permutation_cycles: + return _permute_dims(_array_tensor_product(*args), Permutation(sum(ranks)-1)*Permutation(permutation_cycles)) + + if len(args) == 1: + return args[0] + + # If any object is a ZeroArray, return a ZeroArray: + if any(isinstance(arg, (ZeroArray, ZeroMatrix)) for arg in args): + shapes = reduce(operator.add, [get_shape(i) for i in args], ()) + return ZeroArray(*shapes) + + # If there are contraction objects inside, transform the whole + # expression into `ArrayContraction`: + contractions = {i: arg for i, arg in enumerate(args) if isinstance(arg, ArrayContraction)} + if contractions: + ranks = [_get_subrank(arg) if isinstance(arg, ArrayContraction) else get_rank(arg) for arg in args] + cumulative_ranks = list(accumulate([0] + ranks))[:-1] + tp = _array_tensor_product(*[arg.expr if isinstance(arg, ArrayContraction) else arg for arg in args]) + contraction_indices = [tuple(cumulative_ranks[i] + k for k in j) for i, arg in contractions.items() for j in arg.contraction_indices] + return _array_contraction(tp, *contraction_indices) + + diagonals = {i: arg for i, arg in enumerate(args) if isinstance(arg, ArrayDiagonal)} + if diagonals: + inverse_permutation = [] + last_perm = [] + ranks = [get_rank(arg) for arg in args] + cumulative_ranks = list(accumulate([0] + ranks))[:-1] + for i, arg in enumerate(args): + if isinstance(arg, ArrayDiagonal): + i1 = get_rank(arg) - len(arg.diagonal_indices) + i2 = len(arg.diagonal_indices) + inverse_permutation.extend([cumulative_ranks[i] + j for j in range(i1)]) + last_perm.extend([cumulative_ranks[i] + j for j in range(i1, i1 + i2)]) + else: + inverse_permutation.extend([cumulative_ranks[i] + j for j in range(get_rank(arg))]) + inverse_permutation.extend(last_perm) + tp = _array_tensor_product(*[arg.expr if isinstance(arg, ArrayDiagonal) else arg for arg in args]) + ranks2 = [_get_subrank(arg) if isinstance(arg, ArrayDiagonal) else get_rank(arg) for arg in args] + cumulative_ranks2 = list(accumulate([0] + ranks2))[:-1] + diagonal_indices = [tuple(cumulative_ranks2[i] + k for k in j) for i, arg in diagonals.items() for j in arg.diagonal_indices] + return _permute_dims(_array_diagonal(tp, *diagonal_indices), _af_invert(inverse_permutation)) + + return self.func(*args, canonicalize=False) + + @classmethod + def _flatten(cls, args): + args = [i for arg in args for i in (arg.args if isinstance(arg, cls) else [arg])] + return args + + def as_explicit(self): + return tensorproduct(*[arg.as_explicit() if hasattr(arg, "as_explicit") else arg for arg in self.args]) + + +class ArrayAdd(_CodegenArrayAbstract): + r""" + Class for elementwise array additions. + """ + + def __new__(cls, *args, **kwargs): + args = [_sympify(arg) for arg in args] + ranks = [get_rank(arg) for arg in args] + ranks = list(set(ranks)) + if len(ranks) != 1: + raise ValueError("summing arrays of different ranks") + shapes = [arg.shape for arg in args] + if len({i for i in shapes if i is not None}) > 1: + raise ValueError("mismatching shapes in addition") + + canonicalize = kwargs.pop("canonicalize", False) + + obj = Basic.__new__(cls, *args) + obj._subranks = ranks + if any(i is None for i in shapes): + obj._shape = None + else: + obj._shape = shapes[0] + if canonicalize: + return obj._canonicalize() + return obj + + def _canonicalize(self): + args = self.args + + # Flatten: + args = self._flatten_args(args) + + shapes = [get_shape(arg) for arg in args] + args = [arg for arg in args if not isinstance(arg, (ZeroArray, ZeroMatrix))] + if len(args) == 0: + if any(i for i in shapes if i is None): + raise NotImplementedError("cannot handle addition of ZeroMatrix/ZeroArray and undefined shape object") + return ZeroArray(*shapes[0]) + elif len(args) == 1: + return args[0] + return self.func(*args, canonicalize=False) + + @classmethod + def _flatten_args(cls, args): + new_args = [] + for arg in args: + if isinstance(arg, ArrayAdd): + new_args.extend(arg.args) + else: + new_args.append(arg) + return new_args + + def as_explicit(self): + return reduce( + operator.add, + [arg.as_explicit() if hasattr(arg, "as_explicit") else arg for arg in self.args]) + + +class PermuteDims(_CodegenArrayAbstract): + r""" + Class to represent permutation of axes of arrays. + + Examples + ======== + + >>> from sympy.tensor.array import permutedims + >>> from sympy import MatrixSymbol + >>> M = MatrixSymbol("M", 3, 3) + >>> cg = permutedims(M, [1, 0]) + + The object ``cg`` represents the transposition of ``M``, as the permutation + ``[1, 0]`` will act on its indices by switching them: + + `M_{ij} \Rightarrow M_{ji}` + + This is evident when transforming back to matrix form: + + >>> from sympy.tensor.array.expressions.from_array_to_matrix import convert_array_to_matrix + >>> convert_array_to_matrix(cg) + M.T + + >>> N = MatrixSymbol("N", 3, 2) + >>> cg = permutedims(N, [1, 0]) + >>> cg.shape + (2, 3) + + There are optional parameters that can be used as alternative to the permutation: + + >>> from sympy.tensor.array.expressions import ArraySymbol, PermuteDims + >>> M = ArraySymbol("M", (1, 2, 3, 4, 5)) + >>> expr = PermuteDims(M, index_order_old="ijklm", index_order_new="kijml") + >>> expr + PermuteDims(M, (0 2 1)(3 4)) + >>> expr.shape + (3, 1, 2, 5, 4) + + Permutations of tensor products are simplified in order to achieve a + standard form: + + >>> from sympy.tensor.array import tensorproduct + >>> M = MatrixSymbol("M", 4, 5) + >>> tp = tensorproduct(M, N) + >>> tp.shape + (4, 5, 3, 2) + >>> perm1 = permutedims(tp, [2, 3, 1, 0]) + + The args ``(M, N)`` have been sorted and the permutation has been + simplified, the expression is equivalent: + + >>> perm1.expr.args + (N, M) + >>> perm1.shape + (3, 2, 5, 4) + >>> perm1.permutation + (2 3) + + The permutation in its array form has been simplified from + ``[2, 3, 1, 0]`` to ``[0, 1, 3, 2]``, as the arguments of the tensor + product `M` and `N` have been switched: + + >>> perm1.permutation.array_form + [0, 1, 3, 2] + + We can nest a second permutation: + + >>> perm2 = permutedims(perm1, [1, 0, 2, 3]) + >>> perm2.shape + (2, 3, 5, 4) + >>> perm2.permutation.array_form + [1, 0, 3, 2] + """ + + def __new__(cls, expr, permutation=None, index_order_old=None, index_order_new=None, **kwargs): + from sympy.combinatorics import Permutation + expr = _sympify(expr) + expr_rank = get_rank(expr) + permutation = cls._get_permutation_from_arguments(permutation, index_order_old, index_order_new, expr_rank) + permutation = Permutation(permutation) + permutation_size = permutation.size + if permutation_size != expr_rank: + raise ValueError("Permutation size must be the length of the shape of expr") + + canonicalize = kwargs.pop("canonicalize", False) + + obj = Basic.__new__(cls, expr, permutation) + obj._subranks = [get_rank(expr)] + shape = get_shape(expr) + if shape is None: + obj._shape = None + else: + obj._shape = tuple(shape[permutation(i)] for i in range(len(shape))) + if canonicalize: + return obj._canonicalize() + return obj + + def _canonicalize(self): + expr = self.expr + permutation = self.permutation + if isinstance(expr, PermuteDims): + subexpr = expr.expr + subperm = expr.permutation + permutation = permutation * subperm + expr = subexpr + if isinstance(expr, ArrayContraction): + expr, permutation = self._PermuteDims_denestarg_ArrayContraction(expr, permutation) + if isinstance(expr, ArrayTensorProduct): + expr, permutation = self._PermuteDims_denestarg_ArrayTensorProduct(expr, permutation) + if isinstance(expr, (ZeroArray, ZeroMatrix)): + return ZeroArray(*[expr.shape[i] for i in permutation.array_form]) + plist = permutation.array_form + if plist == sorted(plist): + return expr + return self.func(expr, permutation, canonicalize=False) + + @property + def expr(self): + return self.args[0] + + @property + def permutation(self): + return self.args[1] + + @classmethod + def _PermuteDims_denestarg_ArrayTensorProduct(cls, expr, permutation): + # Get the permutation in its image-form: + perm_image_form = _af_invert(permutation.array_form) + args = list(expr.args) + # Starting index global position for every arg: + cumul = list(accumulate([0] + expr.subranks)) + # Split `perm_image_form` into a list of list corresponding to the indices + # of every argument: + perm_image_form_in_components = [perm_image_form[cumul[i]:cumul[i+1]] for i in range(len(args))] + # Create an index, target-position-key array: + ps = [(i, sorted(comp)) for i, comp in enumerate(perm_image_form_in_components)] + # Sort the array according to the target-position-key: + # In this way, we define a canonical way to sort the arguments according + # to the permutation. + ps.sort(key=lambda x: x[1]) + # Read the inverse-permutation (i.e. image-form) of the args: + perm_args_image_form = [i[0] for i in ps] + # Apply the args-permutation to the `args`: + args_sorted = [args[i] for i in perm_args_image_form] + # Apply the args-permutation to the array-form of the permutation of the axes (of `expr`): + perm_image_form_sorted_args = [perm_image_form_in_components[i] for i in perm_args_image_form] + new_permutation = Permutation(_af_invert([j for i in perm_image_form_sorted_args for j in i])) + return _array_tensor_product(*args_sorted), new_permutation + + @classmethod + def _PermuteDims_denestarg_ArrayContraction(cls, expr, permutation): + if not isinstance(expr, ArrayContraction): + return expr, permutation + if not isinstance(expr.expr, ArrayTensorProduct): + return expr, permutation + args = expr.expr.args + subranks = [get_rank(arg) for arg in expr.expr.args] + + contraction_indices = expr.contraction_indices + contraction_indices_flat = [j for i in contraction_indices for j in i] + cumul = list(accumulate([0] + subranks)) + + # Spread the permutation in its array form across the args in the corresponding + # tensor-product arguments with free indices: + permutation_array_blocks_up = [] + image_form = _af_invert(permutation.array_form) + counter = 0 + for i in range(len(subranks)): + current = [] + for j in range(cumul[i], cumul[i+1]): + if j in contraction_indices_flat: + continue + current.append(image_form[counter]) + counter += 1 + permutation_array_blocks_up.append(current) + + # Get the map of axis repositioning for every argument of tensor-product: + index_blocks = [list(range(cumul[i], cumul[i+1])) for i, e in enumerate(expr.subranks)] + index_blocks_up = expr._push_indices_up(expr.contraction_indices, index_blocks) + inverse_permutation = permutation**(-1) + index_blocks_up_permuted = [[inverse_permutation(j) for j in i if j is not None] for i in index_blocks_up] + + # Sorting key is a list of tuple, first element is the index of `args`, second element of + # the tuple is the sorting key to sort `args` of the tensor product: + sorting_keys = list(enumerate(index_blocks_up_permuted)) + sorting_keys.sort(key=lambda x: x[1]) + + # Now we can get the permutation acting on the args in its image-form: + new_perm_image_form = [i[0] for i in sorting_keys] + # Apply the args-level permutation to various elements: + new_index_blocks = [index_blocks[i] for i in new_perm_image_form] + new_index_perm_array_form = _af_invert([j for i in new_index_blocks for j in i]) + new_args = [args[i] for i in new_perm_image_form] + new_contraction_indices = [tuple(new_index_perm_array_form[j] for j in i) for i in contraction_indices] + new_expr = _array_contraction(_array_tensor_product(*new_args), *new_contraction_indices) + new_permutation = Permutation(_af_invert([j for i in [permutation_array_blocks_up[k] for k in new_perm_image_form] for j in i])) + return new_expr, new_permutation + + @classmethod + def _check_permutation_mapping(cls, expr, permutation): + subranks = expr.subranks + index2arg = [i for i, arg in enumerate(expr.args) for j in range(expr.subranks[i])] + permuted_indices = [permutation(i) for i in range(expr.subrank())] + new_args = list(expr.args) + arg_candidate_index = index2arg[permuted_indices[0]] + current_indices = [] + new_permutation = [] + inserted_arg_cand_indices = set() + for i, idx in enumerate(permuted_indices): + if index2arg[idx] != arg_candidate_index: + new_permutation.extend(current_indices) + current_indices = [] + arg_candidate_index = index2arg[idx] + current_indices.append(idx) + arg_candidate_rank = subranks[arg_candidate_index] + if len(current_indices) == arg_candidate_rank: + new_permutation.extend(sorted(current_indices)) + local_current_indices = [j - min(current_indices) for j in current_indices] + i1 = index2arg[i] + new_args[i1] = _permute_dims(new_args[i1], Permutation(local_current_indices)) + inserted_arg_cand_indices.add(arg_candidate_index) + current_indices = [] + new_permutation.extend(current_indices) + + # TODO: swap args positions in order to simplify the expression: + # TODO: this should be in a function + args_positions = list(range(len(new_args))) + # Get possible shifts: + maps = {} + cumulative_subranks = [0] + list(accumulate(subranks)) + for i in range(len(subranks)): + s = {index2arg[new_permutation[j]] for j in range(cumulative_subranks[i], cumulative_subranks[i+1])} + if len(s) != 1: + continue + elem = next(iter(s)) + if i != elem: + maps[i] = elem + + # Find cycles in the map: + lines = [] + current_line = [] + while maps: + if len(current_line) == 0: + k, v = maps.popitem() + current_line.append(k) + else: + k = current_line[-1] + if k not in maps: + current_line = [] + continue + v = maps.pop(k) + if v in current_line: + lines.append(current_line) + current_line = [] + continue + current_line.append(v) + for line in lines: + for i, e in enumerate(line): + args_positions[line[(i + 1) % len(line)]] = e + + # TODO: function in order to permute the args: + permutation_blocks = [[new_permutation[cumulative_subranks[i] + j] for j in range(e)] for i, e in enumerate(subranks)] + new_args = [new_args[i] for i in args_positions] + new_permutation_blocks = [permutation_blocks[i] for i in args_positions] + new_permutation2 = [j for i in new_permutation_blocks for j in i] + return _array_tensor_product(*new_args), Permutation(new_permutation2) # **(-1) + + @classmethod + def _check_if_there_are_closed_cycles(cls, expr, permutation): + args = list(expr.args) + subranks = expr.subranks + cyclic_form = permutation.cyclic_form + cumulative_subranks = [0] + list(accumulate(subranks)) + cyclic_min = [min(i) for i in cyclic_form] + cyclic_max = [max(i) for i in cyclic_form] + cyclic_keep = [] + for i, cycle in enumerate(cyclic_form): + flag = True + for j in range(len(cumulative_subranks) - 1): + if cyclic_min[i] >= cumulative_subranks[j] and cyclic_max[i] < cumulative_subranks[j+1]: + # Found a sinkable cycle. + args[j] = _permute_dims(args[j], Permutation([[k - cumulative_subranks[j] for k in cycle]])) + flag = False + break + if flag: + cyclic_keep.append(cycle) + return _array_tensor_product(*args), Permutation(cyclic_keep, size=permutation.size) + + def nest_permutation(self): + r""" + DEPRECATED. + """ + ret = self._nest_permutation(self.expr, self.permutation) + if ret is None: + return self + return ret + + @classmethod + def _nest_permutation(cls, expr, permutation): + if isinstance(expr, ArrayTensorProduct): + return _permute_dims(*cls._check_if_there_are_closed_cycles(expr, permutation)) + elif isinstance(expr, ArrayContraction): + # Invert tree hierarchy: put the contraction above. + cycles = permutation.cyclic_form + newcycles = ArrayContraction._convert_outer_indices_to_inner_indices(expr, *cycles) + newpermutation = Permutation(newcycles) + new_contr_indices = [tuple(newpermutation(j) for j in i) for i in expr.contraction_indices] + return _array_contraction(PermuteDims(expr.expr, newpermutation), *new_contr_indices) + elif isinstance(expr, ArrayAdd): + return _array_add(*[PermuteDims(arg, permutation) for arg in expr.args]) + return None + + def as_explicit(self): + expr = self.expr + if hasattr(expr, "as_explicit"): + expr = expr.as_explicit() + return permutedims(expr, self.permutation) + + @classmethod + def _get_permutation_from_arguments(cls, permutation, index_order_old, index_order_new, dim): + if permutation is None: + if index_order_new is None or index_order_old is None: + raise ValueError("Permutation not defined") + return PermuteDims._get_permutation_from_index_orders(index_order_old, index_order_new, dim) + else: + if index_order_new is not None: + raise ValueError("index_order_new cannot be defined with permutation") + if index_order_old is not None: + raise ValueError("index_order_old cannot be defined with permutation") + return permutation + + @classmethod + def _get_permutation_from_index_orders(cls, index_order_old, index_order_new, dim): + if len(set(index_order_new)) != dim: + raise ValueError("wrong number of indices in index_order_new") + if len(set(index_order_old)) != dim: + raise ValueError("wrong number of indices in index_order_old") + if len(set.symmetric_difference(set(index_order_new), set(index_order_old))) > 0: + raise ValueError("index_order_new and index_order_old must have the same indices") + permutation = [index_order_old.index(i) for i in index_order_new] + return permutation + + +class ArrayDiagonal(_CodegenArrayAbstract): + r""" + Class to represent the diagonal operator. + + Explanation + =========== + + In a 2-dimensional array it returns the diagonal, this looks like the + operation: + + `A_{ij} \rightarrow A_{ii}` + + The diagonal over axes 1 and 2 (the second and third) of the tensor product + of two 2-dimensional arrays `A \otimes B` is + + `\Big[ A_{ab} B_{cd} \Big]_{abcd} \rightarrow \Big[ A_{ai} B_{id} \Big]_{adi}` + + In this last example the array expression has been reduced from + 4-dimensional to 3-dimensional. Notice that no contraction has occurred, + rather there is a new index `i` for the diagonal, contraction would have + reduced the array to 2 dimensions. + + Notice that the diagonalized out dimensions are added as new dimensions at + the end of the indices. + """ + + def __new__(cls, expr, *diagonal_indices, **kwargs): + expr = _sympify(expr) + diagonal_indices = [Tuple(*sorted(i)) for i in diagonal_indices] + canonicalize = kwargs.get("canonicalize", False) + + shape = get_shape(expr) + if shape is not None: + cls._validate(expr, *diagonal_indices, **kwargs) + # Get new shape: + positions, shape = cls._get_positions_shape(shape, diagonal_indices) + else: + positions = None + if len(diagonal_indices) == 0: + return expr + obj = Basic.__new__(cls, expr, *diagonal_indices) + obj._positions = positions + obj._subranks = _get_subranks(expr) + obj._shape = shape + if canonicalize: + return obj._canonicalize() + return obj + + def _canonicalize(self): + expr = self.expr + diagonal_indices = self.diagonal_indices + trivial_diags = [i for i in diagonal_indices if len(i) == 1] + if len(trivial_diags) > 0: + trivial_pos = {e[0]: i for i, e in enumerate(diagonal_indices) if len(e) == 1} + diag_pos = {e: i for i, e in enumerate(diagonal_indices) if len(e) > 1} + diagonal_indices_short = [i for i in diagonal_indices if len(i) > 1] + rank1 = get_rank(self) + rank2 = len(diagonal_indices) + rank3 = rank1 - rank2 + inv_permutation = [] + counter1 = 0 + indices_down = ArrayDiagonal._push_indices_down(diagonal_indices_short, list(range(rank1)), get_rank(expr)) + for i in indices_down: + if i in trivial_pos: + inv_permutation.append(rank3 + trivial_pos[i]) + elif isinstance(i, (Integer, int)): + inv_permutation.append(counter1) + counter1 += 1 + else: + inv_permutation.append(rank3 + diag_pos[i]) + permutation = _af_invert(inv_permutation) + if len(diagonal_indices_short) > 0: + return _permute_dims(_array_diagonal(expr, *diagonal_indices_short), permutation) + else: + return _permute_dims(expr, permutation) + if isinstance(expr, ArrayAdd): + return self._ArrayDiagonal_denest_ArrayAdd(expr, *diagonal_indices) + if isinstance(expr, ArrayDiagonal): + return self._ArrayDiagonal_denest_ArrayDiagonal(expr, *diagonal_indices) + if isinstance(expr, PermuteDims): + return self._ArrayDiagonal_denest_PermuteDims(expr, *diagonal_indices) + if isinstance(expr, (ZeroArray, ZeroMatrix)): + positions, shape = self._get_positions_shape(expr.shape, diagonal_indices) + return ZeroArray(*shape) + return self.func(expr, *diagonal_indices, canonicalize=False) + + @staticmethod + def _validate(expr, *diagonal_indices, **kwargs): + # Check that no diagonalization happens on indices with mismatched + # dimensions: + shape = get_shape(expr) + for i in diagonal_indices: + if any(j >= len(shape) for j in i): + raise ValueError("index is larger than expression shape") + if len({shape[j] for j in i}) != 1: + raise ValueError("diagonalizing indices of different dimensions") + if not kwargs.get("allow_trivial_diags", False) and len(i) <= 1: + raise ValueError("need at least two axes to diagonalize") + if len(set(i)) != len(i): + raise ValueError("axis index cannot be repeated") + + @staticmethod + def _remove_trivial_dimensions(shape, *diagonal_indices): + return [tuple(j for j in i) for i in diagonal_indices if shape[i[0]] != 1] + + @property + def expr(self): + return self.args[0] + + @property + def diagonal_indices(self): + return self.args[1:] + + @staticmethod + def _flatten(expr, *outer_diagonal_indices): + inner_diagonal_indices = expr.diagonal_indices + all_inner = [j for i in inner_diagonal_indices for j in i] + all_inner.sort() + # TODO: add API for total rank and cumulative rank: + total_rank = _get_subrank(expr) + inner_rank = len(all_inner) + outer_rank = total_rank - inner_rank + shifts = [0 for i in range(outer_rank)] + counter = 0 + pointer = 0 + for i in range(outer_rank): + while pointer < inner_rank and counter >= all_inner[pointer]: + counter += 1 + pointer += 1 + shifts[i] += pointer + counter += 1 + outer_diagonal_indices = tuple(tuple(shifts[j] + j for j in i) for i in outer_diagonal_indices) + diagonal_indices = inner_diagonal_indices + outer_diagonal_indices + return _array_diagonal(expr.expr, *diagonal_indices) + + @classmethod + def _ArrayDiagonal_denest_ArrayAdd(cls, expr, *diagonal_indices): + return _array_add(*[_array_diagonal(arg, *diagonal_indices) for arg in expr.args]) + + @classmethod + def _ArrayDiagonal_denest_ArrayDiagonal(cls, expr, *diagonal_indices): + return cls._flatten(expr, *diagonal_indices) + + @classmethod + def _ArrayDiagonal_denest_PermuteDims(cls, expr: PermuteDims, *diagonal_indices): + back_diagonal_indices = [[expr.permutation(j) for j in i] for i in diagonal_indices] + nondiag = [i for i in range(get_rank(expr)) if not any(i in j for j in diagonal_indices)] + back_nondiag = [expr.permutation(i) for i in nondiag] + remap = {e: i for i, e in enumerate(sorted(back_nondiag))} + new_permutation1 = [remap[i] for i in back_nondiag] + shift = len(new_permutation1) + diag_block_perm = [i + shift for i in range(len(back_diagonal_indices))] + new_permutation = new_permutation1 + diag_block_perm + return _permute_dims( + _array_diagonal( + expr.expr, + *back_diagonal_indices + ), + new_permutation + ) + + def _push_indices_down_nonstatic(self, indices): + transform = lambda x: self._positions[x] if x < len(self._positions) else None + return _apply_recursively_over_nested_lists(transform, indices) + + def _push_indices_up_nonstatic(self, indices): + + def transform(x): + for i, e in enumerate(self._positions): + if (isinstance(e, int) and x == e) or (isinstance(e, tuple) and x in e): + return i + + return _apply_recursively_over_nested_lists(transform, indices) + + @classmethod + def _push_indices_down(cls, diagonal_indices, indices, rank): + positions, shape = cls._get_positions_shape(range(rank), diagonal_indices) + transform = lambda x: positions[x] if x < len(positions) else None + return _apply_recursively_over_nested_lists(transform, indices) + + @classmethod + def _push_indices_up(cls, diagonal_indices, indices, rank): + positions, shape = cls._get_positions_shape(range(rank), diagonal_indices) + + def transform(x): + for i, e in enumerate(positions): + if (isinstance(e, int) and x == e) or (isinstance(e, (tuple, Tuple)) and (x in e)): + return i + + return _apply_recursively_over_nested_lists(transform, indices) + + @classmethod + def _get_positions_shape(cls, shape, diagonal_indices): + data1 = tuple((i, shp) for i, shp in enumerate(shape) if not any(i in j for j in diagonal_indices)) + pos1, shp1 = zip(*data1) if data1 else ((), ()) + data2 = tuple((i, shape[i[0]]) for i in diagonal_indices) + pos2, shp2 = zip(*data2) if data2 else ((), ()) + positions = pos1 + pos2 + shape = shp1 + shp2 + return positions, shape + + def as_explicit(self): + expr = self.expr + if hasattr(expr, "as_explicit"): + expr = expr.as_explicit() + return tensordiagonal(expr, *self.diagonal_indices) + + +class ArrayElementwiseApplyFunc(_CodegenArrayAbstract): + + def __new__(cls, function, element): + + if not isinstance(function, Lambda): + d = Dummy('d') + function = Lambda(d, function(d)) + + obj = _CodegenArrayAbstract.__new__(cls, function, element) + obj._subranks = _get_subranks(element) + return obj + + @property + def function(self): + return self.args[0] + + @property + def expr(self): + return self.args[1] + + @property + def shape(self): + return self.expr.shape + + def _get_function_fdiff(self): + d = Dummy("d") + function = self.function(d) + fdiff = function.diff(d) + if isinstance(fdiff, Function): + fdiff = type(fdiff) + else: + fdiff = Lambda(d, fdiff) + return fdiff + + def as_explicit(self): + expr = self.expr + if hasattr(expr, "as_explicit"): + expr = expr.as_explicit() + return expr.applyfunc(self.function) + + +class ArrayContraction(_CodegenArrayAbstract): + r""" + This class is meant to represent contractions of arrays in a form easily + processable by the code printers. + """ + + def __new__(cls, expr, *contraction_indices, **kwargs): + contraction_indices = _sort_contraction_indices(contraction_indices) + expr = _sympify(expr) + + canonicalize = kwargs.get("canonicalize", False) + + obj = Basic.__new__(cls, expr, *contraction_indices) + obj._subranks = _get_subranks(expr) + obj._mapping = _get_mapping_from_subranks(obj._subranks) + + free_indices_to_position = {i: i for i in range(sum(obj._subranks)) if all(i not in cind for cind in contraction_indices)} + obj._free_indices_to_position = free_indices_to_position + + shape = get_shape(expr) + cls._validate(expr, *contraction_indices) + if shape: + shape = tuple(shp for i, shp in enumerate(shape) if not any(i in j for j in contraction_indices)) + obj._shape = shape + if canonicalize: + return obj._canonicalize() + return obj + + def _canonicalize(self): + expr = self.expr + contraction_indices = self.contraction_indices + + if len(contraction_indices) == 0: + return expr + + if isinstance(expr, ArrayContraction): + return self._ArrayContraction_denest_ArrayContraction(expr, *contraction_indices) + + if isinstance(expr, (ZeroArray, ZeroMatrix)): + return self._ArrayContraction_denest_ZeroArray(expr, *contraction_indices) + + if isinstance(expr, PermuteDims): + return self._ArrayContraction_denest_PermuteDims(expr, *contraction_indices) + + if isinstance(expr, ArrayTensorProduct): + expr, contraction_indices = self._sort_fully_contracted_args(expr, contraction_indices) + expr, contraction_indices = self._lower_contraction_to_addends(expr, contraction_indices) + if len(contraction_indices) == 0: + return expr + + if isinstance(expr, ArrayDiagonal): + return self._ArrayContraction_denest_ArrayDiagonal(expr, *contraction_indices) + + if isinstance(expr, ArrayAdd): + return self._ArrayContraction_denest_ArrayAdd(expr, *contraction_indices) + + # Check single index contractions on 1-dimensional axes: + contraction_indices = [i for i in contraction_indices if len(i) > 1 or get_shape(expr)[i[0]] != 1] + if len(contraction_indices) == 0: + return expr + + return self.func(expr, *contraction_indices, canonicalize=False) + + def __mul__(self, other): + if other == 1: + return self + else: + raise NotImplementedError("Product of N-dim arrays is not uniquely defined. Use another method.") + + def __rmul__(self, other): + if other == 1: + return self + else: + raise NotImplementedError("Product of N-dim arrays is not uniquely defined. Use another method.") + + @staticmethod + def _validate(expr, *contraction_indices): + shape = get_shape(expr) + if shape is None: + return + + # Check that no contraction happens when the shape is mismatched: + for i in contraction_indices: + if len({shape[j] for j in i if shape[j] != -1}) != 1: + raise ValueError("contracting indices of different dimensions") + + @classmethod + def _push_indices_down(cls, contraction_indices, indices): + flattened_contraction_indices = [j for i in contraction_indices for j in i] + flattened_contraction_indices.sort() + transform = _build_push_indices_down_func_transformation(flattened_contraction_indices) + return _apply_recursively_over_nested_lists(transform, indices) + + @classmethod + def _push_indices_up(cls, contraction_indices, indices): + flattened_contraction_indices = [j for i in contraction_indices for j in i] + flattened_contraction_indices.sort() + transform = _build_push_indices_up_func_transformation(flattened_contraction_indices) + return _apply_recursively_over_nested_lists(transform, indices) + + @classmethod + def _lower_contraction_to_addends(cls, expr, contraction_indices): + if isinstance(expr, ArrayAdd): + raise NotImplementedError() + if not isinstance(expr, ArrayTensorProduct): + return expr, contraction_indices + subranks = expr.subranks + cumranks = list(accumulate([0] + subranks)) + contraction_indices_remaining = [] + contraction_indices_args = [[] for i in expr.args] + backshift = set() + for contraction_group in contraction_indices: + for j in range(len(expr.args)): + if not isinstance(expr.args[j], ArrayAdd): + continue + if all(cumranks[j] <= k < cumranks[j+1] for k in contraction_group): + contraction_indices_args[j].append([k - cumranks[j] for k in contraction_group]) + backshift.update(contraction_group) + break + else: + contraction_indices_remaining.append(contraction_group) + if len(contraction_indices_remaining) == len(contraction_indices): + return expr, contraction_indices + total_rank = get_rank(expr) + shifts = list(accumulate([1 if i in backshift else 0 for i in range(total_rank)])) + contraction_indices_remaining = [Tuple.fromiter(j - shifts[j] for j in i) for i in contraction_indices_remaining] + ret = _array_tensor_product(*[ + _array_contraction(arg, *contr) for arg, contr in zip(expr.args, contraction_indices_args) + ]) + return ret, contraction_indices_remaining + + def split_multiple_contractions(self): + """ + Recognize multiple contractions and attempt at rewriting them as paired-contractions. + + This allows some contractions involving more than two indices to be + rewritten as multiple contractions involving two indices, thus allowing + the expression to be rewritten as a matrix multiplication line. + + Examples: + + * `A_ij b_j0 C_jk` ===> `A*DiagMatrix(b)*C` + + Care for: + - matrix being diagonalized (i.e. `A_ii`) + - vectors being diagonalized (i.e. `a_i0`) + + Multiple contractions can be split into matrix multiplications if + not more than two arguments are non-diagonals or non-vectors. + Vectors get diagonalized while diagonal matrices remain diagonal. + The non-diagonal matrices can be at the beginning or at the end + of the final matrix multiplication line. + """ + + editor = _EditArrayContraction(self) + + contraction_indices = self.contraction_indices + + onearray_insert = [] + + for indl, links in enumerate(contraction_indices): + if len(links) <= 2: + continue + + # Check multiple contractions: + # + # Examples: + # + # * `A_ij b_j0 C_jk` ===> `A*DiagMatrix(b)*C \otimes OneArray(1)` with permutation (1 2) + # + # Care for: + # - matrix being diagonalized (i.e. `A_ii`) + # - vectors being diagonalized (i.e. `a_i0`) + + # Multiple contractions can be split into matrix multiplications if + # not more than three arguments are non-diagonals or non-vectors. + # + # Vectors get diagonalized while diagonal matrices remain diagonal. + # The non-diagonal matrices can be at the beginning or at the end + # of the final matrix multiplication line. + + positions = editor.get_mapping_for_index(indl) + + # Also consider the case of diagonal matrices being contracted: + current_dimension = self.expr.shape[links[0]] + + not_vectors = [] + vectors = [] + for arg_ind, rel_ind in positions: + arg = editor.args_with_ind[arg_ind] + mat = arg.element + abs_arg_start, abs_arg_end = editor.get_absolute_range(arg) + other_arg_pos = 1-rel_ind + other_arg_abs = abs_arg_start + other_arg_pos + if ((1 not in mat.shape) or + ((current_dimension == 1) is True and mat.shape != (1, 1)) or + any(other_arg_abs in l for li, l in enumerate(contraction_indices) if li != indl) + ): + not_vectors.append((arg, rel_ind)) + else: + vectors.append((arg, rel_ind)) + if len(not_vectors) > 2: + # If more than two arguments in the multiple contraction are + # non-vectors and non-diagonal matrices, we cannot find a way + # to split this contraction into a matrix multiplication line: + continue + # Three cases to handle: + # - zero non-vectors + # - one non-vector + # - two non-vectors + for v, rel_ind in vectors: + v.element = diagonalize_vector(v.element) + vectors_to_loop = not_vectors[:1] + vectors + not_vectors[1:] + first_not_vector, rel_ind = vectors_to_loop[0] + new_index = first_not_vector.indices[rel_ind] + + for v, rel_ind in vectors_to_loop[1:-1]: + v.indices[rel_ind] = new_index + new_index = editor.get_new_contraction_index() + assert v.indices.index(None) == 1 - rel_ind + v.indices[v.indices.index(None)] = new_index + onearray_insert.append(v) + + last_vec, rel_ind = vectors_to_loop[-1] + last_vec.indices[rel_ind] = new_index + + for v in onearray_insert: + editor.insert_after(v, _ArgE(OneArray(1), [None])) + + return editor.to_array_contraction() + + def flatten_contraction_of_diagonal(self): + if not isinstance(self.expr, ArrayDiagonal): + return self + contraction_down = self.expr._push_indices_down(self.expr.diagonal_indices, self.contraction_indices) + new_contraction_indices = [] + diagonal_indices = self.expr.diagonal_indices[:] + for i in contraction_down: + contraction_group = list(i) + for j in i: + diagonal_with = [k for k in diagonal_indices if j in k] + contraction_group.extend([l for k in diagonal_with for l in k]) + diagonal_indices = [k for k in diagonal_indices if k not in diagonal_with] + new_contraction_indices.append(sorted(set(contraction_group))) + + new_contraction_indices = ArrayDiagonal._push_indices_up(diagonal_indices, new_contraction_indices) + return _array_contraction( + _array_diagonal( + self.expr.expr, + *diagonal_indices + ), + *new_contraction_indices + ) + + @staticmethod + def _get_free_indices_to_position_map(free_indices, contraction_indices): + free_indices_to_position = {} + flattened_contraction_indices = [j for i in contraction_indices for j in i] + counter = 0 + for ind in free_indices: + while counter in flattened_contraction_indices: + counter += 1 + free_indices_to_position[ind] = counter + counter += 1 + return free_indices_to_position + + @staticmethod + def _get_index_shifts(expr): + """ + Get the mapping of indices at the positions before the contraction + occurs. + + Examples + ======== + + >>> from sympy.tensor.array import tensorproduct, tensorcontraction + >>> from sympy import MatrixSymbol + >>> M = MatrixSymbol("M", 3, 3) + >>> N = MatrixSymbol("N", 3, 3) + >>> cg = tensorcontraction(tensorproduct(M, N), [1, 2]) + >>> cg._get_index_shifts(cg) + [0, 2] + + Indeed, ``cg`` after the contraction has two dimensions, 0 and 1. They + need to be shifted by 0 and 2 to get the corresponding positions before + the contraction (that is, 0 and 3). + """ + inner_contraction_indices = expr.contraction_indices + all_inner = [j for i in inner_contraction_indices for j in i] + all_inner.sort() + # TODO: add API for total rank and cumulative rank: + total_rank = _get_subrank(expr) + inner_rank = len(all_inner) + outer_rank = total_rank - inner_rank + shifts = [0 for i in range(outer_rank)] + counter = 0 + pointer = 0 + for i in range(outer_rank): + while pointer < inner_rank and counter >= all_inner[pointer]: + counter += 1 + pointer += 1 + shifts[i] += pointer + counter += 1 + return shifts + + @staticmethod + def _convert_outer_indices_to_inner_indices(expr, *outer_contraction_indices): + shifts = ArrayContraction._get_index_shifts(expr) + outer_contraction_indices = tuple(tuple(shifts[j] + j for j in i) for i in outer_contraction_indices) + return outer_contraction_indices + + @staticmethod + def _flatten(expr, *outer_contraction_indices): + inner_contraction_indices = expr.contraction_indices + outer_contraction_indices = ArrayContraction._convert_outer_indices_to_inner_indices(expr, *outer_contraction_indices) + contraction_indices = inner_contraction_indices + outer_contraction_indices + return _array_contraction(expr.expr, *contraction_indices) + + @classmethod + def _ArrayContraction_denest_ArrayContraction(cls, expr, *contraction_indices): + return cls._flatten(expr, *contraction_indices) + + @classmethod + def _ArrayContraction_denest_ZeroArray(cls, expr, *contraction_indices): + contraction_indices_flat = [j for i in contraction_indices for j in i] + shape = [e for i, e in enumerate(expr.shape) if i not in contraction_indices_flat] + return ZeroArray(*shape) + + @classmethod + def _ArrayContraction_denest_ArrayAdd(cls, expr, *contraction_indices): + return _array_add(*[_array_contraction(i, *contraction_indices) for i in expr.args]) + + @classmethod + def _ArrayContraction_denest_PermuteDims(cls, expr, *contraction_indices): + permutation = expr.permutation + plist = permutation.array_form + new_contraction_indices = [tuple(permutation(j) for j in i) for i in contraction_indices] + new_plist = [i for i in plist if not any(i in j for j in new_contraction_indices)] + new_plist = cls._push_indices_up(new_contraction_indices, new_plist) + return _permute_dims( + _array_contraction(expr.expr, *new_contraction_indices), + Permutation(new_plist) + ) + + @classmethod + def _ArrayContraction_denest_ArrayDiagonal(cls, expr: 'ArrayDiagonal', *contraction_indices): + diagonal_indices = list(expr.diagonal_indices) + down_contraction_indices = expr._push_indices_down(expr.diagonal_indices, contraction_indices, get_rank(expr.expr)) + # Flatten diagonally contracted indices: + down_contraction_indices = [[k for j in i for k in (j if isinstance(j, (tuple, Tuple)) else [j])] for i in down_contraction_indices] + new_contraction_indices = [] + for contr_indgrp in down_contraction_indices: + ind = contr_indgrp[:] + for j, diag_indgrp in enumerate(diagonal_indices): + if diag_indgrp is None: + continue + if any(i in diag_indgrp for i in contr_indgrp): + ind.extend(diag_indgrp) + diagonal_indices[j] = None + new_contraction_indices.append(sorted(set(ind))) + + new_diagonal_indices_down = [i for i in diagonal_indices if i is not None] + new_diagonal_indices = ArrayContraction._push_indices_up(new_contraction_indices, new_diagonal_indices_down) + return _array_diagonal( + _array_contraction(expr.expr, *new_contraction_indices), + *new_diagonal_indices + ) + + @classmethod + def _sort_fully_contracted_args(cls, expr, contraction_indices): + if expr.shape is None: + return expr, contraction_indices + cumul = list(accumulate([0] + expr.subranks)) + index_blocks = [list(range(cumul[i], cumul[i+1])) for i in range(len(expr.args))] + contraction_indices_flat = {j for i in contraction_indices for j in i} + fully_contracted = [all(j in contraction_indices_flat for j in range(cumul[i], cumul[i+1])) for i, arg in enumerate(expr.args)] + new_pos = sorted(range(len(expr.args)), key=lambda x: (0, default_sort_key(expr.args[x])) if fully_contracted[x] else (1,)) + new_args = [expr.args[i] for i in new_pos] + new_index_blocks_flat = [j for i in new_pos for j in index_blocks[i]] + index_permutation_array_form = _af_invert(new_index_blocks_flat) + new_contraction_indices = [tuple(index_permutation_array_form[j] for j in i) for i in contraction_indices] + new_contraction_indices = _sort_contraction_indices(new_contraction_indices) + return _array_tensor_product(*new_args), new_contraction_indices + + def _get_contraction_tuples(self): + r""" + Return tuples containing the argument index and position within the + argument of the index position. + + Examples + ======== + + >>> from sympy import MatrixSymbol + >>> from sympy.abc import N + >>> from sympy.tensor.array import tensorproduct, tensorcontraction + >>> A = MatrixSymbol("A", N, N) + >>> B = MatrixSymbol("B", N, N) + + >>> cg = tensorcontraction(tensorproduct(A, B), (1, 2)) + >>> cg._get_contraction_tuples() + [[(0, 1), (1, 0)]] + + Notes + ===== + + Here the contraction pair `(1, 2)` meaning that the 2nd and 3rd indices + of the tensor product `A\otimes B` are contracted, has been transformed + into `(0, 1)` and `(1, 0)`, identifying the same indices in a different + notation. `(0, 1)` is the second index (1) of the first argument (i.e. + 0 or `A`). `(1, 0)` is the first index (i.e. 0) of the second + argument (i.e. 1 or `B`). + """ + mapping = self._mapping + return [[mapping[j] for j in i] for i in self.contraction_indices] + + @staticmethod + def _contraction_tuples_to_contraction_indices(expr, contraction_tuples): + # TODO: check that `expr` has `.subranks`: + ranks = expr.subranks + cumulative_ranks = [0] + list(accumulate(ranks)) + return [tuple(cumulative_ranks[j]+k for j, k in i) for i in contraction_tuples] + + @property + def free_indices(self): + return self._free_indices[:] + + @property + def free_indices_to_position(self): + return dict(self._free_indices_to_position) + + @property + def expr(self): + return self.args[0] + + @property + def contraction_indices(self): + return self.args[1:] + + def _contraction_indices_to_components(self): + expr = self.expr + if not isinstance(expr, ArrayTensorProduct): + raise NotImplementedError("only for contractions of tensor products") + ranks = expr.subranks + mapping = {} + counter = 0 + for i, rank in enumerate(ranks): + for j in range(rank): + mapping[counter] = (i, j) + counter += 1 + return mapping + + def sort_args_by_name(self): + """ + Sort arguments in the tensor product so that their order is lexicographical. + + Examples + ======== + + >>> from sympy.tensor.array.expressions.from_matrix_to_array import convert_matrix_to_array + >>> from sympy import MatrixSymbol + >>> from sympy.abc import N + >>> A = MatrixSymbol("A", N, N) + >>> B = MatrixSymbol("B", N, N) + >>> C = MatrixSymbol("C", N, N) + >>> D = MatrixSymbol("D", N, N) + + >>> cg = convert_matrix_to_array(C*D*A*B) + >>> cg + ArrayContraction(ArrayTensorProduct(A, D, C, B), (0, 3), (1, 6), (2, 5)) + >>> cg.sort_args_by_name() + ArrayContraction(ArrayTensorProduct(A, D, B, C), (0, 3), (1, 4), (2, 7)) + """ + expr = self.expr + if not isinstance(expr, ArrayTensorProduct): + return self + args = expr.args + sorted_data = sorted(enumerate(args), key=lambda x: default_sort_key(x[1])) + pos_sorted, args_sorted = zip(*sorted_data) + reordering_map = {i: pos_sorted.index(i) for i, arg in enumerate(args)} + contraction_tuples = self._get_contraction_tuples() + contraction_tuples = [[(reordering_map[j], k) for j, k in i] for i in contraction_tuples] + c_tp = _array_tensor_product(*args_sorted) + new_contr_indices = self._contraction_tuples_to_contraction_indices( + c_tp, + contraction_tuples + ) + return _array_contraction(c_tp, *new_contr_indices) + + def _get_contraction_links(self): + r""" + Returns a dictionary of links between arguments in the tensor product + being contracted. + + See the example for an explanation of the values. + + Examples + ======== + + >>> from sympy import MatrixSymbol + >>> from sympy.abc import N + >>> from sympy.tensor.array.expressions.from_matrix_to_array import convert_matrix_to_array + >>> A = MatrixSymbol("A", N, N) + >>> B = MatrixSymbol("B", N, N) + >>> C = MatrixSymbol("C", N, N) + >>> D = MatrixSymbol("D", N, N) + + Matrix multiplications are pairwise contractions between neighboring + matrices: + + `A_{ij} B_{jk} C_{kl} D_{lm}` + + >>> cg = convert_matrix_to_array(A*B*C*D) + >>> cg + ArrayContraction(ArrayTensorProduct(B, C, A, D), (0, 5), (1, 2), (3, 6)) + + >>> cg._get_contraction_links() + {0: {0: (2, 1), 1: (1, 0)}, 1: {0: (0, 1), 1: (3, 0)}, 2: {1: (0, 0)}, 3: {0: (1, 1)}} + + This dictionary is interpreted as follows: argument in position 0 (i.e. + matrix `A`) has its second index (i.e. 1) contracted to `(1, 0)`, that + is argument in position 1 (matrix `B`) on the first index slot of `B`, + this is the contraction provided by the index `j` from `A`. + + The argument in position 1 (that is, matrix `B`) has two contractions, + the ones provided by the indices `j` and `k`, respectively the first + and second indices (0 and 1 in the sub-dict). The link `(0, 1)` and + `(2, 0)` respectively. `(0, 1)` is the index slot 1 (the 2nd) of + argument in position 0 (that is, `A_{\ldot j}`), and so on. + """ + args, dlinks = _get_contraction_links([self], self.subranks, *self.contraction_indices) + return dlinks + + def as_explicit(self): + expr = self.expr + if hasattr(expr, "as_explicit"): + expr = expr.as_explicit() + return tensorcontraction(expr, *self.contraction_indices) + + +class Reshape(_CodegenArrayAbstract): + """ + Reshape the dimensions of an array expression. + + Examples + ======== + + >>> from sympy.tensor.array.expressions import ArraySymbol, Reshape + >>> A = ArraySymbol("A", (6,)) + >>> A.shape + (6,) + >>> Reshape(A, (3, 2)).shape + (3, 2) + + Check the component-explicit forms: + + >>> A.as_explicit() + [A[0], A[1], A[2], A[3], A[4], A[5]] + >>> Reshape(A, (3, 2)).as_explicit() + [[A[0], A[1]], [A[2], A[3]], [A[4], A[5]]] + + """ + + def __new__(cls, expr, shape): + expr = _sympify(expr) + if not isinstance(shape, Tuple): + shape = Tuple(*shape) + if Equality(Mul.fromiter(expr.shape), Mul.fromiter(shape)) == False: + raise ValueError("shape mismatch") + obj = Expr.__new__(cls, expr, shape) + obj._shape = tuple(shape) + obj._expr = expr + return obj + + @property + def shape(self): + return self._shape + + @property + def expr(self): + return self._expr + + def doit(self, *args, **kwargs): + if kwargs.get("deep", True): + expr = self.expr.doit(*args, **kwargs) + else: + expr = self.expr + if isinstance(expr, (MatrixBase, NDimArray)): + return expr.reshape(*self.shape) + return Reshape(expr, self.shape) + + def as_explicit(self): + ee = self.expr + if hasattr(ee, "as_explicit"): + ee = ee.as_explicit() + if isinstance(ee, MatrixBase): + from sympy import Array + ee = Array(ee) + elif isinstance(ee, MatrixExpr): + return self + return ee.reshape(*self.shape) + + +class _ArgE: + """ + The ``_ArgE`` object contains references to the array expression + (``.element``) and a list containing the information about index + contractions (``.indices``). + + Index contractions are numbered and contracted indices show the number of + the contraction. Uncontracted indices have ``None`` value. + + For example: + ``_ArgE(M, [None, 3])`` + This object means that expression ``M`` is part of an array contraction + and has two indices, the first is not contracted (value ``None``), + the second index is contracted to the 4th (i.e. number ``3``) group of the + array contraction object. + """ + indices: list[int | None] + + def __init__(self, element, indices: list[int | None] | None = None): + self.element = element + if indices is None: + self.indices = [None for i in range(get_rank(element))] + else: + self.indices = indices + + def __str__(self): + return "_ArgE(%s, %s)" % (self.element, self.indices) + + __repr__ = __str__ + + +class _IndPos: + """ + Index position, requiring two integers in the constructor: + + - arg: the position of the argument in the tensor product, + - rel: the relative position of the index inside the argument. + """ + def __init__(self, arg: int, rel: int): + self.arg = arg + self.rel = rel + + def __str__(self): + return "_IndPos(%i, %i)" % (self.arg, self.rel) + + __repr__ = __str__ + + def __iter__(self): + yield from [self.arg, self.rel] + + +class _EditArrayContraction: + """ + Utility class to help manipulate array contraction objects. + + This class takes as input an ``ArrayContraction`` object and turns it into + an editable object. + + The field ``args_with_ind`` of this class is a list of ``_ArgE`` objects + which can be used to easily edit the contraction structure of the + expression. + + Once editing is finished, the ``ArrayContraction`` object may be recreated + by calling the ``.to_array_contraction()`` method. + """ + + def __init__(self, base_array: typing.Union[ArrayContraction, ArrayDiagonal, ArrayTensorProduct]): + + expr: Basic + diagonalized: tuple[tuple[int, ...], ...] + contraction_indices: list[tuple[int]] + if isinstance(base_array, ArrayContraction): + mapping = _get_mapping_from_subranks(base_array.subranks) + expr = base_array.expr + contraction_indices = base_array.contraction_indices + diagonalized = () + elif isinstance(base_array, ArrayDiagonal): + + if isinstance(base_array.expr, ArrayContraction): + mapping = _get_mapping_from_subranks(base_array.expr.subranks) + expr = base_array.expr.expr + diagonalized = ArrayContraction._push_indices_down(base_array.expr.contraction_indices, base_array.diagonal_indices) + contraction_indices = base_array.expr.contraction_indices + elif isinstance(base_array.expr, ArrayTensorProduct): + mapping = {} + expr = base_array.expr + diagonalized = base_array.diagonal_indices + contraction_indices = [] + else: + mapping = {} + expr = base_array.expr + diagonalized = base_array.diagonal_indices + contraction_indices = [] + + elif isinstance(base_array, ArrayTensorProduct): + expr = base_array + contraction_indices = [] + diagonalized = () + else: + raise NotImplementedError() + + if isinstance(expr, ArrayTensorProduct): + args = list(expr.args) + else: + args = [expr] + + args_with_ind: list[_ArgE] = [_ArgE(arg) for arg in args] + for i, contraction_tuple in enumerate(contraction_indices): + for j in contraction_tuple: + arg_pos, rel_pos = mapping[j] + args_with_ind[arg_pos].indices[rel_pos] = i + self.args_with_ind: list[_ArgE] = args_with_ind + self.number_of_contraction_indices: int = len(contraction_indices) + self._track_permutation: list[list[int]] | None = None + + mapping = _get_mapping_from_subranks(base_array.subranks) + + # Trick: add diagonalized indices as negative indices into the editor object: + for i, e in enumerate(diagonalized): + for j in e: + arg_pos, rel_pos = mapping[j] + self.args_with_ind[arg_pos].indices[rel_pos] = -1 - i + + def insert_after(self, arg: _ArgE, new_arg: _ArgE): + pos = self.args_with_ind.index(arg) + self.args_with_ind.insert(pos + 1, new_arg) + + def get_new_contraction_index(self): + self.number_of_contraction_indices += 1 + return self.number_of_contraction_indices - 1 + + def refresh_indices(self): + updates = {} + for arg_with_ind in self.args_with_ind: + updates.update({i: -1 for i in arg_with_ind.indices if i is not None}) + for i, e in enumerate(sorted(updates)): + updates[e] = i + self.number_of_contraction_indices = len(updates) + for arg_with_ind in self.args_with_ind: + arg_with_ind.indices = [updates.get(i, None) for i in arg_with_ind.indices] + + def merge_scalars(self): + scalars = [] + for arg_with_ind in self.args_with_ind: + if len(arg_with_ind.indices) == 0: + scalars.append(arg_with_ind) + for i in scalars: + self.args_with_ind.remove(i) + scalar = Mul.fromiter([i.element for i in scalars]) + if len(self.args_with_ind) == 0: + self.args_with_ind.append(_ArgE(scalar)) + else: + from sympy.tensor.array.expressions.from_array_to_matrix import _a2m_tensor_product + self.args_with_ind[0].element = _a2m_tensor_product(scalar, self.args_with_ind[0].element) + + def to_array_contraction(self): + + # Count the ranks of the arguments: + counter = 0 + # Create a collector for the new diagonal indices: + diag_indices = defaultdict(list) + + count_index_freq = Counter() + for arg_with_ind in self.args_with_ind: + count_index_freq.update(Counter(arg_with_ind.indices)) + + free_index_count = count_index_freq[None] + + # Construct the inverse permutation: + inv_perm1 = [] + inv_perm2 = [] + # Keep track of which diagonal indices have already been processed: + done = set() + + # Counter for the diagonal indices: + counter4 = 0 + + for arg_with_ind in self.args_with_ind: + # If some diagonalization axes have been removed, they should be + # permuted in order to keep the permutation. + # Add permutation here + counter2 = 0 # counter for the indices + for i in arg_with_ind.indices: + if i is None: + inv_perm1.append(counter4) + counter2 += 1 + counter4 += 1 + continue + if i >= 0: + continue + # Reconstruct the diagonal indices: + diag_indices[-1 - i].append(counter + counter2) + if count_index_freq[i] == 1 and i not in done: + inv_perm1.append(free_index_count - 1 - i) + done.add(i) + elif i not in done: + inv_perm2.append(free_index_count - 1 - i) + done.add(i) + counter2 += 1 + # Remove negative indices to restore a proper editor object: + arg_with_ind.indices = [i if i is not None and i >= 0 else None for i in arg_with_ind.indices] + counter += len([i for i in arg_with_ind.indices if i is None or i < 0]) + + inverse_permutation = inv_perm1 + inv_perm2 + permutation = _af_invert(inverse_permutation) + + # Get the diagonal indices after the detection of HadamardProduct in the expression: + diag_indices_filtered = [tuple(v) for v in diag_indices.values() if len(v) > 1] + + self.merge_scalars() + self.refresh_indices() + args = [arg.element for arg in self.args_with_ind] + contraction_indices = self.get_contraction_indices() + expr = _array_contraction(_array_tensor_product(*args), *contraction_indices) + expr2 = _array_diagonal(expr, *diag_indices_filtered) + if self._track_permutation is not None: + permutation2 = _af_invert([j for i in self._track_permutation for j in i]) + expr2 = _permute_dims(expr2, permutation2) + + expr3 = _permute_dims(expr2, permutation) + return expr3 + + def get_contraction_indices(self) -> list[list[int]]: + contraction_indices: list[list[int]] = [[] for i in range(self.number_of_contraction_indices)] + current_position: int = 0 + for arg_with_ind in self.args_with_ind: + for j in arg_with_ind.indices: + if j is not None: + contraction_indices[j].append(current_position) + current_position += 1 + return contraction_indices + + def get_mapping_for_index(self, ind) -> list[_IndPos]: + if ind >= self.number_of_contraction_indices: + raise ValueError("index value exceeding the index range") + positions: list[_IndPos] = [] + for i, arg_with_ind in enumerate(self.args_with_ind): + for j, arg_ind in enumerate(arg_with_ind.indices): + if ind == arg_ind: + positions.append(_IndPos(i, j)) + return positions + + def get_contraction_indices_to_ind_rel_pos(self) -> list[list[_IndPos]]: + contraction_indices: list[list[_IndPos]] = [[] for i in range(self.number_of_contraction_indices)] + for i, arg_with_ind in enumerate(self.args_with_ind): + for j, ind in enumerate(arg_with_ind.indices): + if ind is not None: + contraction_indices[ind].append(_IndPos(i, j)) + return contraction_indices + + def count_args_with_index(self, index: int) -> int: + """ + Count the number of arguments that have the given index. + """ + counter: int = 0 + for arg_with_ind in self.args_with_ind: + if index in arg_with_ind.indices: + counter += 1 + return counter + + def get_args_with_index(self, index: int) -> list[_ArgE]: + """ + Get a list of arguments having the given index. + """ + ret: list[_ArgE] = [i for i in self.args_with_ind if index in i.indices] + return ret + + @property + def number_of_diagonal_indices(self): + data = set() + for arg in self.args_with_ind: + data.update({i for i in arg.indices if i is not None and i < 0}) + return len(data) + + def track_permutation_start(self): + permutation = [] + perm_diag = [] + counter = 0 + counter2 = -1 + for arg_with_ind in self.args_with_ind: + perm = [] + for i in arg_with_ind.indices: + if i is not None: + if i < 0: + perm_diag.append(counter2) + counter2 -= 1 + continue + perm.append(counter) + counter += 1 + permutation.append(perm) + max_ind = max(max(i) if i else -1 for i in permutation) if permutation else -1 + perm_diag = [max_ind - i for i in perm_diag] + self._track_permutation = permutation + [perm_diag] + + def track_permutation_merge(self, destination: _ArgE, from_element: _ArgE): + index_destination = self.args_with_ind.index(destination) + index_element = self.args_with_ind.index(from_element) + self._track_permutation[index_destination].extend(self._track_permutation[index_element]) # type: ignore + self._track_permutation.pop(index_element) # type: ignore + + def get_absolute_free_range(self, arg: _ArgE) -> typing.Tuple[int, int]: + """ + Return the range of the free indices of the arg as absolute positions + among all free indices. + """ + counter = 0 + for arg_with_ind in self.args_with_ind: + number_free_indices = len([i for i in arg_with_ind.indices if i is None]) + if arg_with_ind == arg: + return counter, counter + number_free_indices + counter += number_free_indices + raise IndexError("argument not found") + + def get_absolute_range(self, arg: _ArgE) -> typing.Tuple[int, int]: + """ + Return the absolute range of indices for arg, disregarding dummy + indices. + """ + counter = 0 + for arg_with_ind in self.args_with_ind: + number_indices = len(arg_with_ind.indices) + if arg_with_ind == arg: + return counter, counter + number_indices + counter += number_indices + raise IndexError("argument not found") + + +def get_rank(expr): + if isinstance(expr, (MatrixExpr, MatrixElement)): + return 2 + if isinstance(expr, _CodegenArrayAbstract): + return len(expr.shape) + if isinstance(expr, NDimArray): + return expr.rank() + if isinstance(expr, Indexed): + return expr.rank + if isinstance(expr, IndexedBase): + shape = expr.shape + if shape is None: + return -1 + else: + return len(shape) + if hasattr(expr, "shape"): + return len(expr.shape) + return 0 + + +def _get_subrank(expr): + if isinstance(expr, _CodegenArrayAbstract): + return expr.subrank() + return get_rank(expr) + + +def _get_subranks(expr): + if isinstance(expr, _CodegenArrayAbstract): + return expr.subranks + else: + return [get_rank(expr)] + + +def get_shape(expr): + if hasattr(expr, "shape"): + return expr.shape + return () + + +def nest_permutation(expr): + if isinstance(expr, PermuteDims): + return expr.nest_permutation() + else: + return expr + + +def _array_tensor_product(*args, **kwargs): + return ArrayTensorProduct(*args, canonicalize=True, **kwargs) + + +def _array_contraction(expr, *contraction_indices, **kwargs): + return ArrayContraction(expr, *contraction_indices, canonicalize=True, **kwargs) + + +def _array_diagonal(expr, *diagonal_indices, **kwargs): + return ArrayDiagonal(expr, *diagonal_indices, canonicalize=True, **kwargs) + + +def _permute_dims(expr, permutation, **kwargs): + return PermuteDims(expr, permutation, canonicalize=True, **kwargs) + + +def _array_add(*args, **kwargs): + return ArrayAdd(*args, canonicalize=True, **kwargs) + + +def _get_array_element_or_slice(expr, indices): + return ArrayElement(expr, indices) diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/arrayexpr_derivatives.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/arrayexpr_derivatives.py new file mode 100644 index 0000000000000000000000000000000000000000..ab44a6fbf715ac7f2b8c287dcc84a49289f2dd76 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/arrayexpr_derivatives.py @@ -0,0 +1,194 @@ +import operator +from functools import reduce, singledispatch + +from sympy.core.expr import Expr +from sympy.core.singleton import S +from sympy.matrices.expressions.hadamard import HadamardProduct +from sympy.matrices.expressions.inverse import Inverse +from sympy.matrices.expressions.matexpr import (MatrixExpr, MatrixSymbol) +from sympy.matrices.expressions.special import Identity, OneMatrix +from sympy.matrices.expressions.transpose import Transpose +from sympy.combinatorics.permutations import _af_invert +from sympy.matrices.expressions.applyfunc import ElementwiseApplyFunction +from sympy.tensor.array.expressions.array_expressions import ( + _ArrayExpr, ZeroArray, ArraySymbol, ArrayTensorProduct, ArrayAdd, + PermuteDims, ArrayDiagonal, ArrayElementwiseApplyFunc, get_rank, + get_shape, ArrayContraction, _array_tensor_product, _array_contraction, + _array_diagonal, _array_add, _permute_dims, Reshape) +from sympy.tensor.array.expressions.from_matrix_to_array import convert_matrix_to_array + + +@singledispatch +def array_derive(expr, x): + """ + Derivatives (gradients) for array expressions. + """ + raise NotImplementedError(f"not implemented for type {type(expr)}") + + +@array_derive.register(Expr) +def _(expr: Expr, x: _ArrayExpr): + return ZeroArray(*x.shape) + + +@array_derive.register(ArrayTensorProduct) +def _(expr: ArrayTensorProduct, x: Expr): + args = expr.args + addend_list = [] + for i, arg in enumerate(expr.args): + darg = array_derive(arg, x) + if darg == 0: + continue + args_prev = args[:i] + args_succ = args[i+1:] + shape_prev = reduce(operator.add, map(get_shape, args_prev), ()) + shape_succ = reduce(operator.add, map(get_shape, args_succ), ()) + addend = _array_tensor_product(*args_prev, darg, *args_succ) + tot1 = len(get_shape(x)) + tot2 = tot1 + len(shape_prev) + tot3 = tot2 + len(get_shape(arg)) + tot4 = tot3 + len(shape_succ) + perm = list(range(tot1, tot2)) + \ + list(range(tot1)) + list(range(tot2, tot3)) + \ + list(range(tot3, tot4)) + addend = _permute_dims(addend, _af_invert(perm)) + addend_list.append(addend) + if len(addend_list) == 1: + return addend_list[0] + elif len(addend_list) == 0: + return S.Zero + else: + return _array_add(*addend_list) + + +@array_derive.register(ArraySymbol) +def _(expr: ArraySymbol, x: _ArrayExpr): + if expr == x: + return _permute_dims( + ArrayTensorProduct.fromiter(Identity(i) for i in expr.shape), + [2*i for i in range(len(expr.shape))] + [2*i+1 for i in range(len(expr.shape))] + ) + return ZeroArray(*(x.shape + expr.shape)) + + +@array_derive.register(MatrixSymbol) +def _(expr: MatrixSymbol, x: _ArrayExpr): + m, n = expr.shape + if expr == x: + return _permute_dims( + _array_tensor_product(Identity(m), Identity(n)), + [0, 2, 1, 3] + ) + return ZeroArray(*(x.shape + expr.shape)) + + +@array_derive.register(Identity) +def _(expr: Identity, x: _ArrayExpr): + return ZeroArray(*(x.shape + expr.shape)) + + +@array_derive.register(OneMatrix) +def _(expr: OneMatrix, x: _ArrayExpr): + return ZeroArray(*(x.shape + expr.shape)) + + +@array_derive.register(Transpose) +def _(expr: Transpose, x: Expr): + # D(A.T, A) ==> (m,n,i,j) ==> D(A_ji, A_mn) = d_mj d_ni + # D(B.T, A) ==> (m,n,i,j) ==> D(B_ji, A_mn) + fd = array_derive(expr.arg, x) + return _permute_dims(fd, [0, 1, 3, 2]) + + +@array_derive.register(Inverse) +def _(expr: Inverse, x: Expr): + mat = expr.I + dexpr = array_derive(mat, x) + tp = _array_tensor_product(-expr, dexpr, expr) + mp = _array_contraction(tp, (1, 4), (5, 6)) + pp = _permute_dims(mp, [1, 2, 0, 3]) + return pp + + +@array_derive.register(ElementwiseApplyFunction) +def _(expr: ElementwiseApplyFunction, x: Expr): + assert get_rank(expr) == 2 + assert get_rank(x) == 2 + fdiff = expr._get_function_fdiff() + dexpr = array_derive(expr.expr, x) + tp = _array_tensor_product( + ElementwiseApplyFunction(fdiff, expr.expr), + dexpr + ) + td = _array_diagonal( + tp, (0, 4), (1, 5) + ) + return td + + +@array_derive.register(ArrayElementwiseApplyFunc) +def _(expr: ArrayElementwiseApplyFunc, x: Expr): + fdiff = expr._get_function_fdiff() + subexpr = expr.expr + dsubexpr = array_derive(subexpr, x) + tp = _array_tensor_product( + dsubexpr, + ArrayElementwiseApplyFunc(fdiff, subexpr) + ) + b = get_rank(x) + c = get_rank(expr) + diag_indices = [(b + i, b + c + i) for i in range(c)] + return _array_diagonal(tp, *diag_indices) + + +@array_derive.register(MatrixExpr) +def _(expr: MatrixExpr, x: Expr): + cg = convert_matrix_to_array(expr) + return array_derive(cg, x) + + +@array_derive.register(HadamardProduct) +def _(expr: HadamardProduct, x: Expr): + raise NotImplementedError() + + +@array_derive.register(ArrayContraction) +def _(expr: ArrayContraction, x: Expr): + fd = array_derive(expr.expr, x) + rank_x = len(get_shape(x)) + contraction_indices = expr.contraction_indices + new_contraction_indices = [tuple(j + rank_x for j in i) for i in contraction_indices] + return _array_contraction(fd, *new_contraction_indices) + + +@array_derive.register(ArrayDiagonal) +def _(expr: ArrayDiagonal, x: Expr): + dsubexpr = array_derive(expr.expr, x) + rank_x = len(get_shape(x)) + diag_indices = [[j + rank_x for j in i] for i in expr.diagonal_indices] + return _array_diagonal(dsubexpr, *diag_indices) + + +@array_derive.register(ArrayAdd) +def _(expr: ArrayAdd, x: Expr): + return _array_add(*[array_derive(arg, x) for arg in expr.args]) + + +@array_derive.register(PermuteDims) +def _(expr: PermuteDims, x: Expr): + de = array_derive(expr.expr, x) + perm = [0, 1] + [i + 2 for i in expr.permutation.array_form] + return _permute_dims(de, perm) + + +@array_derive.register(Reshape) +def _(expr: Reshape, x: Expr): + de = array_derive(expr.expr, x) + return Reshape(de, get_shape(x) + expr.shape) + + +def matrix_derive(expr, x): + from sympy.tensor.array.expressions.from_array_to_matrix import convert_array_to_matrix + ce = convert_matrix_to_array(expr) + dce = array_derive(ce, x) + return convert_array_to_matrix(dce).doit() diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/conv_array_to_indexed.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/conv_array_to_indexed.py new file mode 100644 index 0000000000000000000000000000000000000000..1929c3401e131cca0a83080131ead9198b37bcbb --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/conv_array_to_indexed.py @@ -0,0 +1,12 @@ +from sympy.tensor.array.expressions import from_array_to_indexed +from sympy.utilities.decorator import deprecated + + +_conv_to_from_decorator = deprecated( + "module has been renamed by replacing 'conv_' with 'from_' in its name", + deprecated_since_version="1.11", + active_deprecations_target="deprecated-conv-array-expr-module-names", +) + + +convert_array_to_indexed = _conv_to_from_decorator(from_array_to_indexed.convert_array_to_indexed) diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/conv_array_to_matrix.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/conv_array_to_matrix.py new file mode 100644 index 0000000000000000000000000000000000000000..2708e74aaa98d6ee38eae46d97d4483a546e0776 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/conv_array_to_matrix.py @@ -0,0 +1,6 @@ +from sympy.tensor.array.expressions import from_array_to_matrix +from sympy.tensor.array.expressions.conv_array_to_indexed import _conv_to_from_decorator + +convert_array_to_matrix = _conv_to_from_decorator(from_array_to_matrix.convert_array_to_matrix) +_array2matrix = _conv_to_from_decorator(from_array_to_matrix._array2matrix) +_remove_trivial_dims = _conv_to_from_decorator(from_array_to_matrix._remove_trivial_dims) diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/conv_indexed_to_array.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/conv_indexed_to_array.py new file mode 100644 index 0000000000000000000000000000000000000000..6058b31f20778834ea23a01553d594b7965eb6bb --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/conv_indexed_to_array.py @@ -0,0 +1,4 @@ +from sympy.tensor.array.expressions import from_indexed_to_array +from sympy.tensor.array.expressions.conv_array_to_indexed import _conv_to_from_decorator + +convert_indexed_to_array = _conv_to_from_decorator(from_indexed_to_array.convert_indexed_to_array) diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/conv_matrix_to_array.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/conv_matrix_to_array.py new file mode 100644 index 0000000000000000000000000000000000000000..46469df60703c237527c0b2834235309640afe7c --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/conv_matrix_to_array.py @@ -0,0 +1,4 @@ +from sympy.tensor.array.expressions import from_matrix_to_array +from sympy.tensor.array.expressions.conv_array_to_indexed import _conv_to_from_decorator + +convert_matrix_to_array = _conv_to_from_decorator(from_matrix_to_array.convert_matrix_to_array) diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/from_array_to_indexed.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/from_array_to_indexed.py new file mode 100644 index 0000000000000000000000000000000000000000..9eb86e7cfbe31ebfe7c9649803d9cb5e34b98276 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/from_array_to_indexed.py @@ -0,0 +1,84 @@ +import collections.abc +import operator +from itertools import accumulate + +from sympy import Mul, Sum, Dummy, Add +from sympy.tensor.array.expressions import PermuteDims, ArrayAdd, ArrayElementwiseApplyFunc, Reshape +from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct, get_rank, ArrayContraction, \ + ArrayDiagonal, get_shape, _get_array_element_or_slice, _ArrayExpr +from sympy.tensor.array.expressions.utils import _apply_permutation_to_list + + +def convert_array_to_indexed(expr, indices): + return _ConvertArrayToIndexed().do_convert(expr, indices) + + +class _ConvertArrayToIndexed: + + def __init__(self): + self.count_dummies = 0 + + def do_convert(self, expr, indices): + if isinstance(expr, ArrayTensorProduct): + cumul = list(accumulate([0] + [get_rank(arg) for arg in expr.args])) + indices_grp = [indices[cumul[i]:cumul[i+1]] for i in range(len(expr.args))] + return Mul.fromiter(self.do_convert(arg, ind) for arg, ind in zip(expr.args, indices_grp)) + if isinstance(expr, ArrayContraction): + new_indices = [None for i in range(get_rank(expr.expr))] + limits = [] + bottom_shape = get_shape(expr.expr) + for contraction_index_grp in expr.contraction_indices: + d = Dummy(f"d{self.count_dummies}") + self.count_dummies += 1 + dim = bottom_shape[contraction_index_grp[0]] + limits.append((d, 0, dim-1)) + for i in contraction_index_grp: + new_indices[i] = d + j = 0 + for i in range(len(new_indices)): + if new_indices[i] is None: + new_indices[i] = indices[j] + j += 1 + newexpr = self.do_convert(expr.expr, new_indices) + return Sum(newexpr, *limits) + if isinstance(expr, ArrayDiagonal): + new_indices = [None for i in range(get_rank(expr.expr))] + ind_pos = expr._push_indices_down(expr.diagonal_indices, list(range(len(indices))), get_rank(expr)) + for i, index in zip(ind_pos, indices): + if isinstance(i, collections.abc.Iterable): + for j in i: + new_indices[j] = index + else: + new_indices[i] = index + newexpr = self.do_convert(expr.expr, new_indices) + return newexpr + if isinstance(expr, PermuteDims): + permuted_indices = _apply_permutation_to_list(expr.permutation, indices) + return self.do_convert(expr.expr, permuted_indices) + if isinstance(expr, ArrayAdd): + return Add.fromiter(self.do_convert(arg, indices) for arg in expr.args) + if isinstance(expr, _ArrayExpr): + return expr.__getitem__(tuple(indices)) + if isinstance(expr, ArrayElementwiseApplyFunc): + return expr.function(self.do_convert(expr.expr, indices)) + if isinstance(expr, Reshape): + shape_up = expr.shape + shape_down = get_shape(expr.expr) + cumul = list(accumulate([1] + list(reversed(shape_up)), operator.mul)) + one_index = Add.fromiter(i*s for i, s in zip(reversed(indices), cumul)) + dest_indices = [None for _ in shape_down] + c = 1 + for i, e in enumerate(reversed(shape_down)): + if c == 1: + if i == len(shape_down) - 1: + dest_indices[i] = one_index + else: + dest_indices[i] = one_index % e + elif i == len(shape_down) - 1: + dest_indices[i] = one_index // c + else: + dest_indices[i] = one_index // c % e + c *= e + dest_indices.reverse() + return self.do_convert(expr.expr, dest_indices) + return _get_array_element_or_slice(expr, indices) diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/from_array_to_matrix.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/from_array_to_matrix.py new file mode 100644 index 0000000000000000000000000000000000000000..debfdd7eb5c4533996b3d72b55d679be3daf3afe --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/from_array_to_matrix.py @@ -0,0 +1,1004 @@ +from __future__ import annotations +import itertools +from collections import defaultdict +from typing import FrozenSet +from functools import singledispatch +from itertools import accumulate + +from sympy import MatMul, Basic, Wild, KroneckerProduct +from sympy.assumptions.ask import (Q, ask) +from sympy.core.mul import Mul +from sympy.core.singleton import S +from sympy.matrices.expressions.diagonal import DiagMatrix +from sympy.matrices.expressions.hadamard import hadamard_product, HadamardPower +from sympy.matrices.expressions.matexpr import MatrixExpr +from sympy.matrices.expressions.special import (Identity, ZeroMatrix, OneMatrix) +from sympy.matrices.expressions.trace import Trace +from sympy.matrices.expressions.transpose import Transpose +from sympy.combinatorics.permutations import _af_invert, Permutation +from sympy.matrices.matrixbase import MatrixBase +from sympy.matrices.expressions.applyfunc import ElementwiseApplyFunction +from sympy.matrices.expressions.matexpr import MatrixElement +from sympy.tensor.array.expressions.array_expressions import PermuteDims, ArrayDiagonal, \ + ArrayTensorProduct, OneArray, get_rank, _get_subrank, ZeroArray, ArrayContraction, \ + ArrayAdd, _CodegenArrayAbstract, get_shape, ArrayElementwiseApplyFunc, _ArrayExpr, _EditArrayContraction, _ArgE, \ + ArrayElement, _array_tensor_product, _array_contraction, _array_diagonal, _array_add, _permute_dims +from sympy.tensor.array.expressions.utils import _get_mapping_from_subranks + + +def _get_candidate_for_matmul_from_contraction(scan_indices: list[int | None], remaining_args: list[_ArgE]) -> tuple[_ArgE | None, bool, int]: + + scan_indices_int: list[int] = [i for i in scan_indices if i is not None] + if len(scan_indices_int) == 0: + return None, False, -1 + + transpose: bool = False + candidate: _ArgE | None = None + candidate_index: int = -1 + for arg_with_ind2 in remaining_args: + if not isinstance(arg_with_ind2.element, MatrixExpr): + continue + for index in scan_indices_int: + if candidate_index != -1 and candidate_index != index: + # A candidate index has already been selected, check + # repetitions only for that index: + continue + if index in arg_with_ind2.indices: + if set(arg_with_ind2.indices) == {index}: + # Index repeated twice in arg_with_ind2 + candidate = None + break + if candidate is None: + candidate = arg_with_ind2 + candidate_index = index + transpose = (index == arg_with_ind2.indices[1]) + else: + # Index repeated more than twice, break + candidate = None + break + return candidate, transpose, candidate_index + + +def _insert_candidate_into_editor(editor: _EditArrayContraction, arg_with_ind: _ArgE, candidate: _ArgE, transpose1: bool, transpose2: bool): + other = candidate.element + other_index: int | None + if transpose2: + other = Transpose(other) + other_index = candidate.indices[0] + else: + other_index = candidate.indices[1] + new_element = (Transpose(arg_with_ind.element) if transpose1 else arg_with_ind.element) * other + editor.args_with_ind.remove(candidate) + new_arge = _ArgE(new_element) + return new_arge, other_index + + +def _support_function_tp1_recognize(contraction_indices, args): + if len(contraction_indices) == 0: + return _a2m_tensor_product(*args) + + ac = _array_contraction(_array_tensor_product(*args), *contraction_indices) + editor = _EditArrayContraction(ac) + editor.track_permutation_start() + + while True: + flag_stop = True + for i, arg_with_ind in enumerate(editor.args_with_ind): + if not isinstance(arg_with_ind.element, MatrixExpr): + continue + + first_index = arg_with_ind.indices[0] + second_index = arg_with_ind.indices[1] + + first_frequency = editor.count_args_with_index(first_index) + second_frequency = editor.count_args_with_index(second_index) + + if first_index is not None and first_frequency == 1 and first_index == second_index: + flag_stop = False + arg_with_ind.element = Trace(arg_with_ind.element)._normalize() + arg_with_ind.indices = [] + break + + scan_indices = [] + if first_frequency == 2: + scan_indices.append(first_index) + if second_frequency == 2: + scan_indices.append(second_index) + + candidate, transpose, found_index = _get_candidate_for_matmul_from_contraction(scan_indices, editor.args_with_ind[i+1:]) + if candidate is not None: + flag_stop = False + editor.track_permutation_merge(arg_with_ind, candidate) + transpose1 = found_index == first_index + new_arge, other_index = _insert_candidate_into_editor(editor, arg_with_ind, candidate, transpose1, transpose) + if found_index == first_index: + new_arge.indices = [second_index, other_index] + else: + new_arge.indices = [first_index, other_index] + set_indices = set(new_arge.indices) + if len(set_indices) == 1 and set_indices != {None}: + # This is a trace: + new_arge.element = Trace(new_arge.element)._normalize() + new_arge.indices = [] + editor.args_with_ind[i] = new_arge + # TODO: is this break necessary? + break + + if flag_stop: + break + + editor.refresh_indices() + return editor.to_array_contraction() + + +def _find_trivial_matrices_rewrite(expr: ArrayTensorProduct): + # If there are matrices of trivial shape in the tensor product (i.e. shape + # (1, 1)), try to check if there is a suitable non-trivial MatMul where the + # expression can be inserted. + + # For example, if "a" has shape (1, 1) and "b" has shape (k, 1), the + # expressions "_array_tensor_product(a, b*b.T)" can be rewritten as + # "b*a*b.T" + + trivial_matrices = [] + pos: int | None = None # must be initialized else causes UnboundLocalError + first: MatrixExpr | None = None # may cause UnboundLocalError if not initialized + second: MatrixExpr | None = None # may cause UnboundLocalError if not initialized + removed: list[int] = [] + counter: int = 0 + args: list[Basic | None] = list(expr.args) + for i, arg in enumerate(expr.args): + if isinstance(arg, MatrixExpr): + if arg.shape == (1, 1): + trivial_matrices.append(arg) + args[i] = None + removed.extend([counter, counter+1]) + elif pos is None and isinstance(arg, MatMul): + margs = arg.args + for j, e in enumerate(margs): + if isinstance(e, MatrixExpr) and e.shape[1] == 1: + pos = i + first = MatMul.fromiter(margs[:j+1]) + second = MatMul.fromiter(margs[j+1:]) + break + counter += get_rank(arg) + if pos is None: + return expr, [] + args[pos] = (first*MatMul.fromiter(i for i in trivial_matrices)*second).doit() + return _array_tensor_product(*[i for i in args if i is not None]), removed + + +def _find_trivial_kronecker_products_broadcast(expr: ArrayTensorProduct): + newargs: list[Basic] = [] + removed = [] + count_dims = 0 + for arg in expr.args: + count_dims += get_rank(arg) + shape = get_shape(arg) + current_range = [count_dims-i for i in range(len(shape), 0, -1)] + if (shape == (1, 1) and len(newargs) > 0 and 1 not in get_shape(newargs[-1]) and + isinstance(newargs[-1], MatrixExpr) and isinstance(arg, MatrixExpr)): + # KroneckerProduct object allows the trick of broadcasting: + newargs[-1] = KroneckerProduct(newargs[-1], arg) + removed.extend(current_range) + elif 1 not in shape and len(newargs) > 0 and get_shape(newargs[-1]) == (1, 1): + # Broadcast: + newargs[-1] = KroneckerProduct(newargs[-1], arg) + prev_range = [i for i in range(min(current_range)) if i not in removed] + removed.extend(prev_range[-2:]) + else: + newargs.append(arg) + return _array_tensor_product(*newargs), removed + + +@singledispatch +def _array2matrix(expr): + return expr + + +@_array2matrix.register(ZeroArray) +def _(expr: ZeroArray): + if get_rank(expr) == 2: + return ZeroMatrix(*expr.shape) + else: + return expr + + +@_array2matrix.register(ArrayTensorProduct) +def _(expr: ArrayTensorProduct): + return _a2m_tensor_product(*[_array2matrix(arg) for arg in expr.args]) + + +@_array2matrix.register(ArrayContraction) +def _(expr: ArrayContraction): + expr = expr.flatten_contraction_of_diagonal() + expr = identify_removable_identity_matrices(expr) + expr = expr.split_multiple_contractions() + expr = identify_hadamard_products(expr) + if not isinstance(expr, ArrayContraction): + return _array2matrix(expr) + subexpr = expr.expr + contraction_indices: tuple[tuple[int]] = expr.contraction_indices + if contraction_indices == ((0,), (1,)) or ( + contraction_indices == ((0,),) and subexpr.shape[1] == 1 + ) or ( + contraction_indices == ((1,),) and subexpr.shape[0] == 1 + ): + shape = subexpr.shape + subexpr = _array2matrix(subexpr) + if isinstance(subexpr, MatrixExpr): + return OneMatrix(1, shape[0])*subexpr*OneMatrix(shape[1], 1) + if isinstance(subexpr, ArrayTensorProduct): + newexpr = _array_contraction(_array2matrix(subexpr), *contraction_indices) + contraction_indices = newexpr.contraction_indices + if any(i > 2 for i in newexpr.subranks): + addends = _array_add(*[_a2m_tensor_product(*j) for j in itertools.product(*[i.args if isinstance(i, + ArrayAdd) else [i] for i in expr.expr.args])]) + newexpr = _array_contraction(addends, *contraction_indices) + if isinstance(newexpr, ArrayAdd): + ret = _array2matrix(newexpr) + return ret + assert isinstance(newexpr, ArrayContraction) + ret = _support_function_tp1_recognize(contraction_indices, list(newexpr.expr.args)) + return ret + elif not isinstance(subexpr, _CodegenArrayAbstract): + ret = _array2matrix(subexpr) + if isinstance(ret, MatrixExpr): + assert expr.contraction_indices == ((0, 1),) + return _a2m_trace(ret) + else: + return _array_contraction(ret, *expr.contraction_indices) + + +@_array2matrix.register(ArrayDiagonal) +def _(expr: ArrayDiagonal): + pexpr = _array_diagonal(_array2matrix(expr.expr), *expr.diagonal_indices) + pexpr = identify_hadamard_products(pexpr) + if isinstance(pexpr, ArrayDiagonal): + pexpr = _array_diag2contr_diagmatrix(pexpr) + if expr == pexpr: + return expr + return _array2matrix(pexpr) + + +@_array2matrix.register(PermuteDims) +def _(expr: PermuteDims): + if expr.permutation.array_form == [1, 0]: + return _a2m_transpose(_array2matrix(expr.expr)) + elif isinstance(expr.expr, ArrayTensorProduct): + ranks = expr.expr.subranks + inv_permutation = expr.permutation**(-1) + newrange = [inv_permutation(i) for i in range(sum(ranks))] + newpos = [] + counter = 0 + for rank in ranks: + newpos.append(newrange[counter:counter+rank]) + counter += rank + newargs = [] + newperm = [] + scalars = [] + for pos, arg in zip(newpos, expr.expr.args): + if len(pos) == 0: + scalars.append(_array2matrix(arg)) + elif pos == sorted(pos): + newargs.append((_array2matrix(arg), pos[0])) + newperm.extend(pos) + elif len(pos) == 2: + newargs.append((_a2m_transpose(_array2matrix(arg)), pos[0])) + newperm.extend(reversed(pos)) + else: + raise NotImplementedError() + newargs = [i[0] for i in newargs] + return _permute_dims(_a2m_tensor_product(*scalars, *newargs), _af_invert(newperm)) + elif isinstance(expr.expr, ArrayContraction): + mat_mul_lines = _array2matrix(expr.expr) + if not isinstance(mat_mul_lines, ArrayTensorProduct): + return _permute_dims(mat_mul_lines, expr.permutation) + # TODO: this assumes that all arguments are matrices, it may not be the case: + permutation = Permutation(2*len(mat_mul_lines.args)-1)*expr.permutation + permuted = [permutation(i) for i in range(2*len(mat_mul_lines.args))] + args_array = [None for i in mat_mul_lines.args] + for i in range(len(mat_mul_lines.args)): + p1 = permuted[2*i] + p2 = permuted[2*i+1] + if p1 // 2 != p2 // 2: + return _permute_dims(mat_mul_lines, permutation) + if p1 > p2: + args_array[i] = _a2m_transpose(mat_mul_lines.args[p1 // 2]) + else: + args_array[i] = mat_mul_lines.args[p1 // 2] + return _a2m_tensor_product(*args_array) + else: + return expr + + +@_array2matrix.register(ArrayAdd) +def _(expr: ArrayAdd): + addends = [_array2matrix(arg) for arg in expr.args] + return _a2m_add(*addends) + + +@_array2matrix.register(ArrayElementwiseApplyFunc) +def _(expr: ArrayElementwiseApplyFunc): + subexpr = _array2matrix(expr.expr) + if isinstance(subexpr, MatrixExpr): + if subexpr.shape != (1, 1): + d = expr.function.bound_symbols[0] + w = Wild("w", exclude=[d]) + p = Wild("p", exclude=[d]) + m = expr.function.expr.match(w*d**p) + if m is not None: + return m[w]*HadamardPower(subexpr, m[p]) + return ElementwiseApplyFunction(expr.function, subexpr) + else: + return ArrayElementwiseApplyFunc(expr.function, subexpr) + + +@_array2matrix.register(ArrayElement) +def _(expr: ArrayElement): + ret = _array2matrix(expr.name) + if isinstance(ret, MatrixExpr): + return MatrixElement(ret, *expr.indices) + return ArrayElement(ret, expr.indices) + + +@singledispatch +def _remove_trivial_dims(expr): + return expr, [] + + +@_remove_trivial_dims.register(ArrayTensorProduct) +def _(expr: ArrayTensorProduct): + # Recognize expressions like [x, y] with shape (k, 1, k, 1) as `x*y.T`. + # The matrix expression has to be equivalent to the tensor product of the + # matrices, with trivial dimensions (i.e. dim=1) dropped. + # That is, add contractions over trivial dimensions: + + removed = [] + newargs = [] + cumul = list(accumulate([0] + [get_rank(arg) for arg in expr.args])) + pending = None + prev_i = None + for i, arg in enumerate(expr.args): + current_range = list(range(cumul[i], cumul[i+1])) + if isinstance(arg, OneArray): + removed.extend(current_range) + continue + if not isinstance(arg, (MatrixExpr, MatrixBase)): + rarg, rem = _remove_trivial_dims(arg) + removed.extend(rem) + newargs.append(rarg) + continue + elif getattr(arg, "is_Identity", False) and arg.shape == (1, 1): + if arg.shape == (1, 1): + # Ignore identity matrices of shape (1, 1) - they are equivalent to scalar 1. + removed.extend(current_range) + continue + elif arg.shape == (1, 1): + arg, _ = _remove_trivial_dims(arg) + # Matrix is equivalent to scalar: + if len(newargs) == 0: + newargs.append(arg) + elif 1 in get_shape(newargs[-1]): + if newargs[-1].shape[1] == 1: + newargs[-1] = newargs[-1]*arg + else: + newargs[-1] = arg*newargs[-1] + removed.extend(current_range) + else: + newargs.append(arg) + elif 1 in arg.shape: + k = [i for i in arg.shape if i != 1][0] + if pending is None: + pending = k + prev_i = i + newargs.append(arg) + elif pending == k: + prev = newargs[-1] + if prev.shape[0] == 1: + d1 = cumul[prev_i] # type: ignore + prev = _a2m_transpose(prev) + else: + d1 = cumul[prev_i] + 1 # type: ignore + if arg.shape[1] == 1: + d2 = cumul[i] + 1 + arg = _a2m_transpose(arg) + else: + d2 = cumul[i] + newargs[-1] = prev*arg + pending = None + removed.extend([d1, d2]) + else: + newargs.append(arg) + pending = k + prev_i = i + else: + newargs.append(arg) + pending = None + newexpr, newremoved = _a2m_tensor_product(*newargs), sorted(removed) + if isinstance(newexpr, ArrayTensorProduct): + newexpr, newremoved2 = _find_trivial_matrices_rewrite(newexpr) + newremoved = _combine_removed(-1, newremoved, newremoved2) + if isinstance(newexpr, ArrayTensorProduct): + newexpr, newremoved2 = _find_trivial_kronecker_products_broadcast(newexpr) + newremoved = _combine_removed(-1, newremoved, newremoved2) + return newexpr, newremoved + + +@_remove_trivial_dims.register(ArrayAdd) +def _(expr: ArrayAdd): + rec = [_remove_trivial_dims(arg) for arg in expr.args] + newargs, removed = zip(*rec) + if len({get_shape(i) for i in newargs}) > 1: + return expr, [] + if len(removed) == 0: + return expr, removed + removed1 = removed[0] + return _a2m_add(*newargs), removed1 + + +@_remove_trivial_dims.register(PermuteDims) +def _(expr: PermuteDims): + subexpr, subremoved = _remove_trivial_dims(expr.expr) + p = expr.permutation.array_form + pinv = _af_invert(expr.permutation.array_form) + shift = list(accumulate([1 if i in subremoved else 0 for i in range(len(p))])) + premoved = [pinv[i] for i in subremoved] + p2 = [e - shift[e] for e in p if e not in subremoved] + # TODO: check if subremoved should be permuted as well... + newexpr = _permute_dims(subexpr, p2) + premoved = sorted(premoved) + if newexpr != expr: + newexpr, removed2 = _remove_trivial_dims(_array2matrix(newexpr)) + premoved = _combine_removed(-1, premoved, removed2) + return newexpr, premoved + + +@_remove_trivial_dims.register(ArrayContraction) +def _(expr: ArrayContraction): + new_expr, removed0 = _array_contraction_to_diagonal_multiple_identity(expr) + if new_expr != expr: + new_expr2, removed1 = _remove_trivial_dims(_array2matrix(new_expr)) + removed = _combine_removed(-1, removed0, removed1) + return new_expr2, removed + rank1 = get_rank(expr) + expr, removed1 = remove_identity_matrices(expr) + if not isinstance(expr, ArrayContraction): + expr2, removed2 = _remove_trivial_dims(expr) + return expr2, _combine_removed(rank1, removed1, removed2) + newexpr, removed2 = _remove_trivial_dims(expr.expr) + shifts = list(accumulate([1 if i in removed2 else 0 for i in range(get_rank(expr.expr))])) + new_contraction_indices = [tuple(j for j in i if j not in removed2) for i in expr.contraction_indices] + # Remove possible empty tuples "()": + new_contraction_indices = [i for i in new_contraction_indices if len(i) > 0] + contraction_indices_flat = [j for i in expr.contraction_indices for j in i] + removed2 = [i for i in removed2 if i not in contraction_indices_flat] + new_contraction_indices = [tuple(j - shifts[j] for j in i) for i in new_contraction_indices] + # Shift removed2: + removed2 = ArrayContraction._push_indices_up(expr.contraction_indices, removed2) + removed = _combine_removed(rank1, removed1, removed2) + return _array_contraction(newexpr, *new_contraction_indices), list(removed) + + +def _remove_diagonalized_identity_matrices(expr: ArrayDiagonal): + assert isinstance(expr, ArrayDiagonal) + editor = _EditArrayContraction(expr) + mapping = {i: {j for j in editor.args_with_ind if i in j.indices} for i in range(-1, -1-editor.number_of_diagonal_indices, -1)} + removed = [] + counter: int = 0 + for i, arg_with_ind in enumerate(editor.args_with_ind): + counter += len(arg_with_ind.indices) + if isinstance(arg_with_ind.element, Identity): + if None in arg_with_ind.indices and any(i is not None and (i < 0) == True for i in arg_with_ind.indices): + diag_ind = [j for j in arg_with_ind.indices if j is not None][0] + other = [j for j in mapping[diag_ind] if j != arg_with_ind][0] + if not isinstance(other.element, MatrixExpr): + continue + if 1 not in other.element.shape: + continue + if None not in other.indices: + continue + editor.args_with_ind[i].element = None + none_index = other.indices.index(None) + other.element = DiagMatrix(other.element) + other_range = editor.get_absolute_range(other) + removed.extend([other_range[0] + none_index]) + editor.args_with_ind = [i for i in editor.args_with_ind if i.element is not None] + removed = ArrayDiagonal._push_indices_up(expr.diagonal_indices, removed, get_rank(expr.expr)) + return editor.to_array_contraction(), removed + + +@_remove_trivial_dims.register(ArrayDiagonal) +def _(expr: ArrayDiagonal): + newexpr, removed = _remove_trivial_dims(expr.expr) + shifts = list(accumulate([0] + [1 if i in removed else 0 for i in range(get_rank(expr.expr))])) + new_diag_indices_map = {i: tuple(j for j in i if j not in removed) for i in expr.diagonal_indices} + for old_diag_tuple, new_diag_tuple in new_diag_indices_map.items(): + if len(new_diag_tuple) == 1: + removed = [i for i in removed if i not in old_diag_tuple] + new_diag_indices = [tuple(j - shifts[j] for j in i) for i in new_diag_indices_map.values()] + rank = get_rank(expr.expr) + removed = ArrayDiagonal._push_indices_up(expr.diagonal_indices, removed, rank) + removed = sorted(set(removed)) + # If there are single axes to diagonalize remaining, it means that their + # corresponding dimension has been removed, they no longer need diagonalization: + new_diag_indices = [i for i in new_diag_indices if len(i) > 0] + if len(new_diag_indices) > 0: + newexpr2 = _array_diagonal(newexpr, *new_diag_indices, allow_trivial_diags=True) + else: + newexpr2 = newexpr + if isinstance(newexpr2, ArrayDiagonal): + newexpr3, removed2 = _remove_diagonalized_identity_matrices(newexpr2) + removed = _combine_removed(-1, removed, removed2) + return newexpr3, removed + else: + return newexpr2, removed + + +@_remove_trivial_dims.register(ElementwiseApplyFunction) +def _(expr: ElementwiseApplyFunction): + subexpr, removed = _remove_trivial_dims(expr.expr) + if subexpr.shape == (1, 1): + # TODO: move this to ElementwiseApplyFunction + return expr.function(subexpr), removed + [0, 1] + return ElementwiseApplyFunction(expr.function, subexpr), [] + + +@_remove_trivial_dims.register(ArrayElementwiseApplyFunc) +def _(expr: ArrayElementwiseApplyFunc): + subexpr, removed = _remove_trivial_dims(expr.expr) + return ArrayElementwiseApplyFunc(expr.function, subexpr), removed + + +def convert_array_to_matrix(expr): + r""" + Recognize matrix expressions in codegen objects. + + If more than one matrix multiplication line have been detected, return a + list with the matrix expressions. + + Examples + ======== + + >>> from sympy.tensor.array.expressions.from_indexed_to_array import convert_indexed_to_array + >>> from sympy.tensor.array import tensorcontraction, tensorproduct + >>> from sympy import MatrixSymbol, Sum + >>> from sympy.abc import i, j, k, l, N + >>> from sympy.tensor.array.expressions.from_matrix_to_array import convert_matrix_to_array + >>> from sympy.tensor.array.expressions.from_array_to_matrix import convert_array_to_matrix + >>> A = MatrixSymbol("A", N, N) + >>> B = MatrixSymbol("B", N, N) + >>> C = MatrixSymbol("C", N, N) + >>> D = MatrixSymbol("D", N, N) + + >>> expr = Sum(A[i, j]*B[j, k], (j, 0, N-1)) + >>> cg = convert_indexed_to_array(expr) + >>> convert_array_to_matrix(cg) + A*B + >>> cg = convert_indexed_to_array(expr, first_indices=[k]) + >>> convert_array_to_matrix(cg) + B.T*A.T + + Transposition is detected: + + >>> expr = Sum(A[j, i]*B[j, k], (j, 0, N-1)) + >>> cg = convert_indexed_to_array(expr) + >>> convert_array_to_matrix(cg) + A.T*B + >>> cg = convert_indexed_to_array(expr, first_indices=[k]) + >>> convert_array_to_matrix(cg) + B.T*A + + Detect the trace: + + >>> expr = Sum(A[i, i], (i, 0, N-1)) + >>> cg = convert_indexed_to_array(expr) + >>> convert_array_to_matrix(cg) + Trace(A) + + Recognize some more complex traces: + + >>> expr = Sum(A[i, j]*B[j, i], (i, 0, N-1), (j, 0, N-1)) + >>> cg = convert_indexed_to_array(expr) + >>> convert_array_to_matrix(cg) + Trace(A*B) + + More complicated expressions: + + >>> expr = Sum(A[i, j]*B[k, j]*A[l, k], (j, 0, N-1), (k, 0, N-1)) + >>> cg = convert_indexed_to_array(expr) + >>> convert_array_to_matrix(cg) + A*B.T*A.T + + Expressions constructed from matrix expressions do not contain literal + indices, the positions of free indices are returned instead: + + >>> expr = A*B + >>> cg = convert_matrix_to_array(expr) + >>> convert_array_to_matrix(cg) + A*B + + If more than one line of matrix multiplications is detected, return + separate matrix multiplication factors embedded in a tensor product object: + + >>> cg = tensorcontraction(tensorproduct(A, B, C, D), (1, 2), (5, 6)) + >>> convert_array_to_matrix(cg) + ArrayTensorProduct(A*B, C*D) + + The two lines have free indices at axes 0, 3 and 4, 7, respectively. + """ + rec = _array2matrix(expr) + rec, removed = _remove_trivial_dims(rec) + return rec + + +def _array_diag2contr_diagmatrix(expr: ArrayDiagonal): + if isinstance(expr.expr, ArrayTensorProduct): + args = list(expr.expr.args) + diag_indices = list(expr.diagonal_indices) + mapping = _get_mapping_from_subranks([_get_subrank(arg) for arg in args]) + tuple_links = [[mapping[j] for j in i] for i in diag_indices] + contr_indices = [] + total_rank = get_rank(expr) + replaced = [False for arg in args] + for i, (abs_pos, rel_pos) in enumerate(zip(diag_indices, tuple_links)): + if len(abs_pos) != 2: + continue + (pos1_outer, pos1_inner), (pos2_outer, pos2_inner) = rel_pos + arg1 = args[pos1_outer] + arg2 = args[pos2_outer] + if get_rank(arg1) != 2 or get_rank(arg2) != 2: + if replaced[pos1_outer]: + diag_indices[i] = None + if replaced[pos2_outer]: + diag_indices[i] = None + continue + pos1_in2 = 1 - pos1_inner + pos2_in2 = 1 - pos2_inner + if arg1.shape[pos1_in2] == 1: + if arg1.shape[pos1_inner] != 1: + darg1 = DiagMatrix(arg1) + else: + darg1 = arg1 + args.append(darg1) + contr_indices.append(((pos2_outer, pos2_inner), (len(args)-1, pos1_inner))) + total_rank += 1 + diag_indices[i] = None + args[pos1_outer] = OneArray(arg1.shape[pos1_in2]) + replaced[pos1_outer] = True + elif arg2.shape[pos2_in2] == 1: + if arg2.shape[pos2_inner] != 1: + darg2 = DiagMatrix(arg2) + else: + darg2 = arg2 + args.append(darg2) + contr_indices.append(((pos1_outer, pos1_inner), (len(args)-1, pos2_inner))) + total_rank += 1 + diag_indices[i] = None + args[pos2_outer] = OneArray(arg2.shape[pos2_in2]) + replaced[pos2_outer] = True + diag_indices_new = [i for i in diag_indices if i is not None] + cumul = list(accumulate([0] + [get_rank(arg) for arg in args])) + contr_indices2 = [tuple(cumul[a] + b for a, b in i) for i in contr_indices] + tc = _array_contraction( + _array_tensor_product(*args), *contr_indices2 + ) + td = _array_diagonal(tc, *diag_indices_new) + return td + return expr + + +def _a2m_mul(*args): + if not any(isinstance(i, _CodegenArrayAbstract) for i in args): + from sympy.matrices.expressions.matmul import MatMul + return MatMul(*args).doit() + else: + return _array_contraction( + _array_tensor_product(*args), + *[(2*i-1, 2*i) for i in range(1, len(args))] + ) + + +def _a2m_tensor_product(*args): + scalars = [] + arrays = [] + for arg in args: + if isinstance(arg, (MatrixExpr, _ArrayExpr, _CodegenArrayAbstract)): + arrays.append(arg) + else: + scalars.append(arg) + scalar = Mul.fromiter(scalars) + if len(arrays) == 0: + return scalar + if scalar != 1: + if isinstance(arrays[0], _CodegenArrayAbstract): + arrays = [scalar] + arrays + else: + arrays[0] *= scalar + return _array_tensor_product(*arrays) + + +def _a2m_add(*args): + if not any(isinstance(i, _CodegenArrayAbstract) for i in args): + from sympy.matrices.expressions.matadd import MatAdd + return MatAdd(*args).doit() + else: + return _array_add(*args) + + +def _a2m_trace(arg): + if isinstance(arg, _CodegenArrayAbstract): + return _array_contraction(arg, (0, 1)) + else: + from sympy.matrices.expressions.trace import Trace + return Trace(arg) + + +def _a2m_transpose(arg): + if isinstance(arg, _CodegenArrayAbstract): + return _permute_dims(arg, [1, 0]) + else: + from sympy.matrices.expressions.transpose import Transpose + return Transpose(arg).doit() + + +def identify_hadamard_products(expr: ArrayContraction | ArrayDiagonal): + + editor: _EditArrayContraction = _EditArrayContraction(expr) + + map_contr_to_args: dict[FrozenSet, list[_ArgE]] = defaultdict(list) + map_ind_to_inds: dict[int | None, int] = defaultdict(int) + for arg_with_ind in editor.args_with_ind: + for ind in arg_with_ind.indices: + map_ind_to_inds[ind] += 1 + if None in arg_with_ind.indices: + continue + map_contr_to_args[frozenset(arg_with_ind.indices)].append(arg_with_ind) + + k: FrozenSet[int] + v: list[_ArgE] + for k, v in map_contr_to_args.items(): + make_trace: bool = False + if len(k) == 1 and next(iter(k)) >= 0 and sum(next(iter(k)) in i for i in map_contr_to_args) == 1: + # This is a trace: the arguments are fully contracted with only one + # index, and the index isn't used anywhere else: + make_trace = True + first_element = S.One + elif len(k) != 2: + # Hadamard product only defined for matrices: + continue + if len(v) == 1: + # Hadamard product with a single argument makes no sense: + continue + for ind in k: + if map_ind_to_inds[ind] <= 2: + # There is no other contraction, skip: + continue + + def check_transpose(x): + x = [i if i >= 0 else -1-i for i in x] + return x == sorted(x) + + # Check if expression is a trace: + if all(map_ind_to_inds[j] == len(v) and j >= 0 for j in k) and all(j >= 0 for j in k): + # This is a trace + make_trace = True + first_element = v[0].element + if not check_transpose(v[0].indices): + first_element = first_element.T # type: ignore + hadamard_factors = v[1:] + else: + hadamard_factors = v + + # This is a Hadamard product: + + hp = hadamard_product(*[i.element if check_transpose(i.indices) else Transpose(i.element) for i in hadamard_factors]) + hp_indices = v[0].indices + if not check_transpose(hadamard_factors[0].indices): + hp_indices = list(reversed(hp_indices)) + if make_trace: + hp = Trace(first_element*hp.T)._normalize() + hp_indices = [] + editor.insert_after(v[0], _ArgE(hp, hp_indices)) + for i in v: + editor.args_with_ind.remove(i) + + return editor.to_array_contraction() + + +def identify_removable_identity_matrices(expr): + editor = _EditArrayContraction(expr) + + flag = True + while flag: + flag = False + for arg_with_ind in editor.args_with_ind: + if isinstance(arg_with_ind.element, Identity): + k = arg_with_ind.element.shape[0] + # Candidate for removal: + if arg_with_ind.indices == [None, None]: + # Free identity matrix, will be cleared by _remove_trivial_dims: + continue + elif None in arg_with_ind.indices: + ind = [j for j in arg_with_ind.indices if j is not None][0] + counted = editor.count_args_with_index(ind) + if counted == 1: + # Identity matrix contracted only on one index with itself, + # transform to a OneArray(k) element: + editor.insert_after(arg_with_ind, OneArray(k)) + editor.args_with_ind.remove(arg_with_ind) + flag = True + break + elif counted > 2: + # Case counted = 2 is a matrix multiplication by identity matrix, skip it. + # Case counted > 2 is a multiple contraction, + # this is a case where the contraction becomes a diagonalization if the + # identity matrix is dropped. + continue + elif arg_with_ind.indices[0] == arg_with_ind.indices[1]: + ind = arg_with_ind.indices[0] + counted = editor.count_args_with_index(ind) + if counted > 1: + editor.args_with_ind.remove(arg_with_ind) + flag = True + break + else: + # This is a trace, skip it as it will be recognized somewhere else: + pass + elif ask(Q.diagonal(arg_with_ind.element)): + if arg_with_ind.indices == [None, None]: + continue + elif None in arg_with_ind.indices: + pass + elif arg_with_ind.indices[0] == arg_with_ind.indices[1]: + ind = arg_with_ind.indices[0] + counted = editor.count_args_with_index(ind) + if counted == 3: + # A_ai B_bi D_ii ==> A_ai D_ij B_bj + ind_new = editor.get_new_contraction_index() + other_args = [j for j in editor.args_with_ind if j != arg_with_ind] + other_args[1].indices = [ind_new if j == ind else j for j in other_args[1].indices] + arg_with_ind.indices = [ind, ind_new] + flag = True + break + + return editor.to_array_contraction() + + +def remove_identity_matrices(expr: ArrayContraction): + editor = _EditArrayContraction(expr) + removed: list[int] = [] + + permutation_map = {} + + free_indices = list(accumulate([0] + [sum(i is None for i in arg.indices) for arg in editor.args_with_ind])) + free_map = dict(zip(editor.args_with_ind, free_indices[:-1])) + + update_pairs = {} + + for ind in range(editor.number_of_contraction_indices): + args = editor.get_args_with_index(ind) + identity_matrices = [i for i in args if isinstance(i.element, Identity)] + number_identity_matrices = len(identity_matrices) + # If the contraction involves a non-identity matrix and multiple identity matrices: + if number_identity_matrices != len(args) - 1 or number_identity_matrices == 0: + continue + # Get the non-identity element: + non_identity = [i for i in args if not isinstance(i.element, Identity)][0] + # Check that all identity matrices have at least one free index + # (otherwise they would be contractions to some other elements) + if any(None not in i.indices for i in identity_matrices): + continue + # Mark the identity matrices for removal: + for i in identity_matrices: + i.element = None + removed.extend(range(free_map[i], free_map[i] + len([j for j in i.indices if j is None]))) + last_removed = removed.pop(-1) + update_pairs[last_removed, ind] = non_identity.indices[:] + # Remove the indices from the non-identity matrix, as the contraction + # no longer exists: + non_identity.indices = [None if i == ind else i for i in non_identity.indices] + + removed.sort() + + shifts = list(accumulate([1 if i in removed else 0 for i in range(get_rank(expr))])) + for (last_removed, ind), non_identity_indices in update_pairs.items(): + pos = [free_map[non_identity] + i for i, e in enumerate(non_identity_indices) if e == ind] + assert len(pos) == 1 + for j in pos: + permutation_map[j] = last_removed + + editor.args_with_ind = [i for i in editor.args_with_ind if i.element is not None] + ret_expr = editor.to_array_contraction() + permutation = [] + counter = 0 + counter2 = 0 + for j in range(get_rank(expr)): + if j in removed: + continue + if counter2 in permutation_map: + target = permutation_map[counter2] + permutation.append(target - shifts[target]) + counter2 += 1 + else: + while counter in permutation_map.values(): + counter += 1 + permutation.append(counter) + counter += 1 + counter2 += 1 + ret_expr2 = _permute_dims(ret_expr, _af_invert(permutation)) + return ret_expr2, removed + + +def _combine_removed(dim: int, removed1: list[int], removed2: list[int]) -> list[int]: + # Concatenate two axis removal operations as performed by + # _remove_trivial_dims, + removed1 = sorted(removed1) + removed2 = sorted(removed2) + i = 0 + j = 0 + removed = [] + while True: + if j >= len(removed2): + while i < len(removed1): + removed.append(removed1[i]) + i += 1 + break + elif i < len(removed1) and removed1[i] <= i + removed2[j]: + removed.append(removed1[i]) + i += 1 + else: + removed.append(i + removed2[j]) + j += 1 + return removed + + +def _array_contraction_to_diagonal_multiple_identity(expr: ArrayContraction): + editor = _EditArrayContraction(expr) + editor.track_permutation_start() + removed: list[int] = [] + diag_index_counter: int = 0 + for i in range(editor.number_of_contraction_indices): + identities = [] + args = [] + for j, arg in enumerate(editor.args_with_ind): + if i not in arg.indices: + continue + if isinstance(arg.element, Identity): + identities.append(arg) + else: + args.append(arg) + if len(identities) == 0: + continue + if len(args) + len(identities) < 3: + continue + new_diag_ind = -1 - diag_index_counter + diag_index_counter += 1 + # Variable "flag" to control whether to skip this contraction set: + flag: bool = True + for i1, id1 in enumerate(identities): + if None not in id1.indices: + flag = True + break + free_pos = list(range(*editor.get_absolute_free_range(id1)))[0] + editor._track_permutation[-1].append(free_pos) # type: ignore + id1.element = None + flag = False + break + if flag: + continue + for arg in identities[:i1] + identities[i1+1:]: + arg.element = None + removed.extend(range(*editor.get_absolute_free_range(arg))) + for arg in args: + arg.indices = [new_diag_ind if j == i else j for j in arg.indices] + for j, e in enumerate(editor.args_with_ind): + if e.element is None: + editor._track_permutation[j] = None # type: ignore + editor._track_permutation = [i for i in editor._track_permutation if i is not None] # type: ignore + # Renumber permutation array form in order to deal with deleted positions: + remap = {e: i for i, e in enumerate(sorted({k for j in editor._track_permutation for k in j}))} + editor._track_permutation = [[remap[j] for j in i] for i in editor._track_permutation] + editor.args_with_ind = [i for i in editor.args_with_ind if i.element is not None] + new_expr = editor.to_array_contraction() + return new_expr, removed diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/from_indexed_to_array.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/from_indexed_to_array.py new file mode 100644 index 0000000000000000000000000000000000000000..c219a205c4305bd7070e5117978146224521c58c --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/from_indexed_to_array.py @@ -0,0 +1,257 @@ +from collections import defaultdict + +from sympy import Function +from sympy.combinatorics.permutations import _af_invert +from sympy.concrete.summations import Sum +from sympy.core.add import Add +from sympy.core.mul import Mul +from sympy.core.numbers import Integer +from sympy.core.power import Pow +from sympy.core.sorting import default_sort_key +from sympy.functions.special.tensor_functions import KroneckerDelta +from sympy.tensor.array.expressions import ArrayElementwiseApplyFunc +from sympy.tensor.indexed import (Indexed, IndexedBase) +from sympy.combinatorics import Permutation +from sympy.matrices.expressions.matexpr import MatrixElement +from sympy.tensor.array.expressions.array_expressions import ArrayDiagonal, \ + get_shape, ArrayElement, _array_tensor_product, _array_diagonal, _array_contraction, _array_add, \ + _permute_dims, OneArray, ArrayAdd +from sympy.tensor.array.expressions.utils import _get_argindex, _get_diagonal_indices + + +def convert_indexed_to_array(expr, first_indices=None): + r""" + Parse indexed expression into a form useful for code generation. + + Examples + ======== + + >>> from sympy.tensor.array.expressions.from_indexed_to_array import convert_indexed_to_array + >>> from sympy import MatrixSymbol, Sum, symbols + + >>> i, j, k, d = symbols("i j k d") + >>> M = MatrixSymbol("M", d, d) + >>> N = MatrixSymbol("N", d, d) + + Recognize the trace in summation form: + + >>> expr = Sum(M[i, i], (i, 0, d-1)) + >>> convert_indexed_to_array(expr) + ArrayContraction(M, (0, 1)) + + Recognize the extraction of the diagonal by using the same index `i` on + both axes of the matrix: + + >>> expr = M[i, i] + >>> convert_indexed_to_array(expr) + ArrayDiagonal(M, (0, 1)) + + This function can help perform the transformation expressed in two + different mathematical notations as: + + `\sum_{j=0}^{N-1} A_{i,j} B_{j,k} \Longrightarrow \mathbf{A}\cdot \mathbf{B}` + + Recognize the matrix multiplication in summation form: + + >>> expr = Sum(M[i, j]*N[j, k], (j, 0, d-1)) + >>> convert_indexed_to_array(expr) + ArrayContraction(ArrayTensorProduct(M, N), (1, 2)) + + Specify that ``k`` has to be the starting index: + + >>> convert_indexed_to_array(expr, first_indices=[k]) + ArrayContraction(ArrayTensorProduct(N, M), (0, 3)) + """ + + result, indices = _convert_indexed_to_array(expr) + + if any(isinstance(i, (int, Integer)) for i in indices): + result = ArrayElement(result, indices) + indices = [] + + if not first_indices: + return result + + def _check_is_in(elem, indices): + if elem in indices: + return True + if any(elem in i for i in indices if isinstance(i, frozenset)): + return True + return False + + repl = {j: i for i in indices if isinstance(i, frozenset) for j in i} + first_indices = [repl.get(i, i) for i in first_indices] + for i in first_indices: + if not _check_is_in(i, indices): + first_indices.remove(i) + first_indices.extend([i for i in indices if not _check_is_in(i, first_indices)]) + + def _get_pos(elem, indices): + if elem in indices: + return indices.index(elem) + for i, e in enumerate(indices): + if not isinstance(e, frozenset): + continue + if elem in e: + return i + raise ValueError("not found") + + permutation = _af_invert([_get_pos(i, first_indices) for i in indices]) + if isinstance(result, ArrayAdd): + return _array_add(*[_permute_dims(arg, permutation) for arg in result.args]) + else: + return _permute_dims(result, permutation) + + +def _convert_indexed_to_array(expr): + if isinstance(expr, Sum): + function = expr.function + summation_indices = expr.variables + subexpr, subindices = _convert_indexed_to_array(function) + subindicessets = {j: i for i in subindices if isinstance(i, frozenset) for j in i} + summation_indices = sorted({subindicessets.get(i, i) for i in summation_indices}, key=default_sort_key) + # TODO: check that Kronecker delta is only contracted to one other element: + kronecker_indices = set() + if isinstance(function, Mul): + for arg in function.args: + if not isinstance(arg, KroneckerDelta): + continue + arg_indices = sorted(set(arg.indices), key=default_sort_key) + if len(arg_indices) == 2: + kronecker_indices.update(arg_indices) + kronecker_indices = sorted(kronecker_indices, key=default_sort_key) + # Check dimensional consistency: + shape = get_shape(subexpr) + if shape: + for ind, istart, iend in expr.limits: + i = _get_argindex(subindices, ind) + if istart != 0 or iend+1 != shape[i]: + raise ValueError("summation index and array dimension mismatch: %s" % ind) + contraction_indices = [] + subindices = list(subindices) + if isinstance(subexpr, ArrayDiagonal): + diagonal_indices = list(subexpr.diagonal_indices) + dindices = subindices[-len(diagonal_indices):] + subindices = subindices[:-len(diagonal_indices)] + for index in summation_indices: + if index in dindices: + position = dindices.index(index) + contraction_indices.append(diagonal_indices[position]) + diagonal_indices[position] = None + diagonal_indices = [i for i in diagonal_indices if i is not None] + for i, ind in enumerate(subindices): + if ind in summation_indices: + pass + if diagonal_indices: + subexpr = _array_diagonal(subexpr.expr, *diagonal_indices) + else: + subexpr = subexpr.expr + + axes_contraction = defaultdict(list) + for i, ind in enumerate(subindices): + include = all(j not in kronecker_indices for j in ind) if isinstance(ind, frozenset) else ind not in kronecker_indices + if ind in summation_indices and include: + axes_contraction[ind].append(i) + subindices[i] = None + for k, v in axes_contraction.items(): + if any(i in kronecker_indices for i in k) if isinstance(k, frozenset) else k in kronecker_indices: + continue + contraction_indices.append(tuple(v)) + free_indices = [i for i in subindices if i is not None] + indices_ret = list(free_indices) + indices_ret.sort(key=lambda x: free_indices.index(x)) + return _array_contraction( + subexpr, + *contraction_indices, + free_indices=free_indices + ), tuple(indices_ret) + if isinstance(expr, Mul): + args, indices = zip(*[_convert_indexed_to_array(arg) for arg in expr.args]) + # Check if there are KroneckerDelta objects: + kronecker_delta_repl = {} + for arg in args: + if not isinstance(arg, KroneckerDelta): + continue + # Diagonalize two indices: + i, j = arg.indices + kindices = set(arg.indices) + if i in kronecker_delta_repl: + kindices.update(kronecker_delta_repl[i]) + if j in kronecker_delta_repl: + kindices.update(kronecker_delta_repl[j]) + kindices = frozenset(kindices) + for index in kindices: + kronecker_delta_repl[index] = kindices + # Remove KroneckerDelta objects, their relations should be handled by + # ArrayDiagonal: + newargs = [] + newindices = [] + for arg, loc_indices in zip(args, indices): + if isinstance(arg, KroneckerDelta): + continue + newargs.append(arg) + newindices.append(loc_indices) + flattened_indices = [kronecker_delta_repl.get(j, j) for i in newindices for j in i] + diagonal_indices, ret_indices = _get_diagonal_indices(flattened_indices) + tp = _array_tensor_product(*newargs) + if diagonal_indices: + return _array_diagonal(tp, *diagonal_indices), ret_indices + else: + return tp, ret_indices + if isinstance(expr, MatrixElement): + indices = expr.args[1:] + diagonal_indices, ret_indices = _get_diagonal_indices(indices) + if diagonal_indices: + return _array_diagonal(expr.args[0], *diagonal_indices), ret_indices + else: + return expr.args[0], ret_indices + if isinstance(expr, ArrayElement): + indices = expr.indices + diagonal_indices, ret_indices = _get_diagonal_indices(indices) + if diagonal_indices: + return _array_diagonal(expr.name, *diagonal_indices), ret_indices + else: + return expr.name, ret_indices + if isinstance(expr, Indexed): + indices = expr.indices + diagonal_indices, ret_indices = _get_diagonal_indices(indices) + if diagonal_indices: + return _array_diagonal(expr.base, *diagonal_indices), ret_indices + else: + return expr.args[0], ret_indices + if isinstance(expr, IndexedBase): + raise NotImplementedError + if isinstance(expr, KroneckerDelta): + return expr, expr.indices + if isinstance(expr, Add): + args, indices = zip(*[_convert_indexed_to_array(arg) for arg in expr.args]) + args = list(args) + # Check if all indices are compatible. Otherwise expand the dimensions: + index0 = [] + shape0 = [] + for arg, arg_indices in zip(args, indices): + arg_indices_set = set(arg_indices) + arg_indices_missing = arg_indices_set.difference(index0) + index0.extend([i for i in arg_indices if i in arg_indices_missing]) + arg_shape = get_shape(arg) + shape0.extend([arg_shape[i] for i, e in enumerate(arg_indices) if e in arg_indices_missing]) + for i, (arg, arg_indices) in enumerate(zip(args, indices)): + if len(arg_indices) < len(index0): + missing_indices_pos = [i for i, e in enumerate(index0) if e not in arg_indices] + missing_shape = [shape0[i] for i in missing_indices_pos] + arg_indices = tuple(index0[j] for j in missing_indices_pos) + arg_indices + args[i] = _array_tensor_product(OneArray(*missing_shape), args[i]) + permutation = Permutation([arg_indices.index(j) for j in index0]) + # Perform index permutations: + args[i] = _permute_dims(args[i], permutation) + return _array_add(*args), tuple(index0) + if isinstance(expr, Pow): + subexpr, subindices = _convert_indexed_to_array(expr.base) + if isinstance(expr.exp, (int, Integer)): + diags = zip(*[(2*i, 2*i + 1) for i in range(expr.exp)]) + arr = _array_diagonal(_array_tensor_product(*[subexpr for i in range(expr.exp)]), *diags) + return arr, subindices + if isinstance(expr, Function): + subexpr, subindices = _convert_indexed_to_array(expr.args[0]) + return ArrayElementwiseApplyFunc(type(expr), subexpr), subindices + return expr, () diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/from_matrix_to_array.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/from_matrix_to_array.py new file mode 100644 index 0000000000000000000000000000000000000000..8f66961727f6338318d65876a7768802773e4f2d --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/from_matrix_to_array.py @@ -0,0 +1,87 @@ +from sympy import KroneckerProduct +from sympy.core.basic import Basic +from sympy.core.function import Lambda +from sympy.core.mul import Mul +from sympy.core.numbers import Integer +from sympy.core.power import Pow +from sympy.core.singleton import S +from sympy.core.symbol import (Dummy, symbols) +from sympy.matrices.expressions.hadamard import (HadamardPower, HadamardProduct) +from sympy.matrices.expressions.matadd import MatAdd +from sympy.matrices.expressions.matmul import MatMul +from sympy.matrices.expressions.matpow import MatPow +from sympy.matrices.expressions.trace import Trace +from sympy.matrices.expressions.transpose import Transpose +from sympy.matrices.expressions.matexpr import MatrixExpr +from sympy.tensor.array.expressions.array_expressions import \ + ArrayElementwiseApplyFunc, _array_tensor_product, _array_contraction, \ + _array_diagonal, _array_add, _permute_dims, Reshape + + +def convert_matrix_to_array(expr: Basic) -> Basic: + if isinstance(expr, MatMul): + args_nonmat = [] + args = [] + for arg in expr.args: + if isinstance(arg, MatrixExpr): + args.append(arg) + else: + args_nonmat.append(convert_matrix_to_array(arg)) + contractions = [(2*i+1, 2*i+2) for i in range(len(args)-1)] + scalar = _array_tensor_product(*args_nonmat) if args_nonmat else S.One + if scalar == 1: + tprod = _array_tensor_product( + *[convert_matrix_to_array(arg) for arg in args]) + else: + tprod = _array_tensor_product( + scalar, + *[convert_matrix_to_array(arg) for arg in args]) + return _array_contraction( + tprod, + *contractions + ) + elif isinstance(expr, MatAdd): + return _array_add( + *[convert_matrix_to_array(arg) for arg in expr.args] + ) + elif isinstance(expr, Transpose): + return _permute_dims( + convert_matrix_to_array(expr.args[0]), [1, 0] + ) + elif isinstance(expr, Trace): + inner_expr: MatrixExpr = convert_matrix_to_array(expr.arg) # type: ignore + return _array_contraction(inner_expr, (0, len(inner_expr.shape) - 1)) + elif isinstance(expr, Mul): + return _array_tensor_product(*[convert_matrix_to_array(i) for i in expr.args]) + elif isinstance(expr, Pow): + base = convert_matrix_to_array(expr.base) + if (expr.exp > 0) == True: + return _array_tensor_product(*[base for i in range(expr.exp)]) + else: + return expr + elif isinstance(expr, MatPow): + base = convert_matrix_to_array(expr.base) + if expr.exp.is_Integer != True: + b = symbols("b", cls=Dummy) + return ArrayElementwiseApplyFunc(Lambda(b, b**expr.exp), convert_matrix_to_array(base)) + elif (expr.exp > 0) == True: + return convert_matrix_to_array(MatMul.fromiter(base for i in range(expr.exp))) + else: + return expr + elif isinstance(expr, HadamardProduct): + tp = _array_tensor_product(*[convert_matrix_to_array(arg) for arg in expr.args]) + diag = [[2*i for i in range(len(expr.args))], [2*i+1 for i in range(len(expr.args))]] + return _array_diagonal(tp, *diag) + elif isinstance(expr, HadamardPower): + base, exp = expr.args + if isinstance(exp, Integer) and exp > 0: + return convert_matrix_to_array(HadamardProduct.fromiter(base for i in range(exp))) + else: + d = Dummy("d") + return 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0000000000000000000000000000000000000000..63fb79ab7ced7bff5ecb55b1764f43e29f98609d --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/tests/test_array_expressions.py @@ -0,0 +1,808 @@ +import random + +from sympy import tensordiagonal, eye, KroneckerDelta, Array +from sympy.core.symbol import symbols +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.matrices.expressions.diagonal import DiagMatrix +from sympy.matrices.expressions.matexpr import MatrixSymbol +from sympy.matrices.expressions.special import ZeroMatrix +from sympy.tensor.array.arrayop import (permutedims, tensorcontraction, tensorproduct) +from sympy.tensor.array.dense_ndim_array import ImmutableDenseNDimArray +from sympy.combinatorics import Permutation +from sympy.tensor.array.expressions.array_expressions import ZeroArray, OneArray, ArraySymbol, ArrayElement, \ + PermuteDims, ArrayContraction, ArrayTensorProduct, ArrayDiagonal, \ + ArrayAdd, nest_permutation, ArrayElementwiseApplyFunc, _EditArrayContraction, _ArgE, _array_tensor_product, \ + _array_contraction, _array_diagonal, _array_add, _permute_dims, Reshape +from sympy.testing.pytest import raises + +i, j, k, l, m, n = symbols("i j k l m n") + + +M = ArraySymbol("M", (k, k)) +N = ArraySymbol("N", (k, k)) +P = ArraySymbol("P", (k, k)) +Q = ArraySymbol("Q", (k, k)) + +A = ArraySymbol("A", (k, k)) +B = ArraySymbol("B", (k, k)) +C = ArraySymbol("C", (k, k)) +D = ArraySymbol("D", (k, k)) + +X = ArraySymbol("X", (k, k)) +Y = ArraySymbol("Y", (k, k)) + +a = ArraySymbol("a", (k, 1)) +b = ArraySymbol("b", (k, 1)) +c = ArraySymbol("c", (k, 1)) +d = ArraySymbol("d", (k, 1)) + + +def test_array_symbol_and_element(): + A = ArraySymbol("A", (2,)) + A0 = ArrayElement(A, (0,)) + A1 = ArrayElement(A, (1,)) + assert A[0] == A0 + assert A[1] != A0 + assert A.as_explicit() == ImmutableDenseNDimArray([A0, A1]) + + A2 = tensorproduct(A, A) + assert A2.shape == (2, 2) + # TODO: not yet supported: + # assert A2.as_explicit() == Array([[A[0]*A[0], A[1]*A[0]], [A[0]*A[1], A[1]*A[1]]]) + A3 = tensorcontraction(A2, (0, 1)) + assert A3.shape == () + # TODO: not yet supported: + # assert A3.as_explicit() == Array([]) + + A = ArraySymbol("A", (2, 3, 4)) + Ae = A.as_explicit() + assert Ae == ImmutableDenseNDimArray( + [[[ArrayElement(A, (i, j, k)) for k in range(4)] for j in range(3)] for i in range(2)]) + + p = _permute_dims(A, Permutation(0, 2, 1)) + assert isinstance(p, PermuteDims) + + A = ArraySymbol("A", (2,)) + raises(IndexError, lambda: A[()]) + raises(IndexError, lambda: A[0, 1]) + raises(ValueError, lambda: A[-1]) + raises(ValueError, lambda: A[2]) + + O = OneArray(3, 4) + Z = ZeroArray(m, n) + + raises(IndexError, lambda: O[()]) + raises(IndexError, lambda: O[1, 2, 3]) + raises(ValueError, lambda: O[3, 0]) + raises(ValueError, lambda: O[0, 4]) + + assert O[1, 2] == 1 + assert Z[1, 2] == 0 + + +def test_zero_array(): + assert ZeroArray() == 0 + assert ZeroArray().is_Integer + + za = ZeroArray(3, 2, 4) + assert za.shape == (3, 2, 4) + za_e = za.as_explicit() + assert za_e.shape == (3, 2, 4) + + m, n, k = symbols("m n k") + za = ZeroArray(m, n, k, 2) + assert za.shape == (m, n, k, 2) + raises(ValueError, lambda: za.as_explicit()) + + +def test_one_array(): + assert OneArray() == 1 + assert OneArray().is_Integer + + oa = OneArray(3, 2, 4) + assert oa.shape == (3, 2, 4) + oa_e = oa.as_explicit() + assert oa_e.shape == (3, 2, 4) + + m, n, k = symbols("m n k") + oa = OneArray(m, n, k, 2) + assert oa.shape == (m, n, k, 2) + raises(ValueError, lambda: oa.as_explicit()) + + +def test_arrayexpr_contraction_construction(): + + cg = _array_contraction(A) + assert cg == A + + cg = _array_contraction(_array_tensor_product(A, B), (1, 0)) + assert cg == _array_contraction(_array_tensor_product(A, B), (0, 1)) + + cg = _array_contraction(_array_tensor_product(M, N), (0, 1)) + indtup = cg._get_contraction_tuples() + assert indtup == [[(0, 0), (0, 1)]] + assert cg._contraction_tuples_to_contraction_indices(cg.expr, indtup) == [(0, 1)] + + cg = _array_contraction(_array_tensor_product(M, N), (1, 2)) + indtup = cg._get_contraction_tuples() + assert indtup == [[(0, 1), (1, 0)]] + assert cg._contraction_tuples_to_contraction_indices(cg.expr, indtup) == [(1, 2)] + + cg = _array_contraction(_array_tensor_product(M, M, N), (1, 4), (2, 5)) + indtup = cg._get_contraction_tuples() + assert indtup == [[(0, 0), (1, 1)], [(0, 1), (2, 0)]] + assert cg._contraction_tuples_to_contraction_indices(cg.expr, indtup) == [(0, 3), (1, 4)] + + # Test removal of trivial contraction: + assert _array_contraction(a, (1,)) == a + assert _array_contraction( + _array_tensor_product(a, b), (0, 2), (1,), (3,)) == _array_contraction( + _array_tensor_product(a, b), (0, 2)) + + +def test_arrayexpr_array_flatten(): + + # Flatten nested ArrayTensorProduct objects: + expr1 = _array_tensor_product(M, N) + expr2 = _array_tensor_product(P, Q) + expr = _array_tensor_product(expr1, expr2) + assert expr == _array_tensor_product(M, N, P, Q) + assert expr.args == (M, N, P, Q) + + # Flatten mixed ArrayTensorProduct and ArrayContraction objects: + cg1 = _array_contraction(expr1, (1, 2)) + cg2 = _array_contraction(expr2, (0, 3)) + + expr = _array_tensor_product(cg1, cg2) + assert expr == _array_contraction(_array_tensor_product(M, N, P, Q), (1, 2), (4, 7)) + + expr = _array_tensor_product(M, cg1) + assert expr == _array_contraction(_array_tensor_product(M, M, N), (3, 4)) + + # Flatten nested ArrayContraction objects: + cgnested = _array_contraction(cg1, (0, 1)) + assert cgnested == _array_contraction(_array_tensor_product(M, N), (0, 3), (1, 2)) + + cgnested = _array_contraction(_array_tensor_product(cg1, cg2), (0, 3)) + assert cgnested == _array_contraction(_array_tensor_product(M, N, P, Q), (0, 6), (1, 2), (4, 7)) + + cg3 = _array_contraction(_array_tensor_product(M, N, P, Q), (1, 3), (2, 4)) + cgnested = _array_contraction(cg3, (0, 1)) + assert cgnested == _array_contraction(_array_tensor_product(M, N, P, Q), (0, 5), (1, 3), (2, 4)) + + cgnested = _array_contraction(cg3, (0, 3), (1, 2)) + assert cgnested == _array_contraction(_array_tensor_product(M, N, P, Q), (0, 7), (1, 3), (2, 4), (5, 6)) + + cg4 = _array_contraction(_array_tensor_product(M, N, P, Q), (1, 5), (3, 7)) + cgnested = _array_contraction(cg4, (0, 1)) + assert cgnested == _array_contraction(_array_tensor_product(M, N, P, Q), (0, 2), (1, 5), (3, 7)) + + cgnested = _array_contraction(cg4, (0, 1), (2, 3)) + assert cgnested == _array_contraction(_array_tensor_product(M, N, P, Q), (0, 2), (1, 5), (3, 7), (4, 6)) + + cg = _array_diagonal(cg4) + assert cg == cg4 + assert isinstance(cg, type(cg4)) + + # Flatten nested ArrayDiagonal objects: + cg1 = _array_diagonal(expr1, (1, 2)) + cg2 = _array_diagonal(expr2, (0, 3)) + cg3 = _array_diagonal(_array_tensor_product(M, N, P, Q), (1, 3), (2, 4)) + cg4 = _array_diagonal(_array_tensor_product(M, N, P, Q), (1, 5), (3, 7)) + + cgnested = _array_diagonal(cg1, (0, 1)) + assert cgnested == _array_diagonal(_array_tensor_product(M, N), (1, 2), (0, 3)) + + cgnested = _array_diagonal(cg3, (1, 2)) + assert cgnested == _array_diagonal(_array_tensor_product(M, N, P, Q), (1, 3), (2, 4), (5, 6)) + + cgnested = _array_diagonal(cg4, (1, 2)) + assert cgnested == _array_diagonal(_array_tensor_product(M, N, P, Q), (1, 5), (3, 7), (2, 4)) + + cg = _array_add(M, N) + cg2 = _array_add(cg, P) + assert isinstance(cg2, ArrayAdd) + assert cg2.args == (M, N, P) + assert cg2.shape == (k, k) + + expr = _array_tensor_product(_array_diagonal(X, (0, 1)), _array_diagonal(A, (0, 1))) + assert expr == _array_diagonal(_array_tensor_product(X, A), (0, 1), (2, 3)) + + expr1 = _array_diagonal(_array_tensor_product(X, A), (1, 2)) + expr2 = _array_tensor_product(expr1, a) + assert expr2 == _permute_dims(_array_diagonal(_array_tensor_product(X, A, a), (1, 2)), [0, 1, 4, 2, 3]) + + expr1 = _array_contraction(_array_tensor_product(X, A), (1, 2)) + expr2 = _array_tensor_product(expr1, a) + assert isinstance(expr2, ArrayContraction) + assert isinstance(expr2.expr, ArrayTensorProduct) + + cg = _array_tensor_product(_array_diagonal(_array_tensor_product(A, X, Y), (0, 3), (1, 5)), a, b) + assert cg == _permute_dims(_array_diagonal(_array_tensor_product(A, X, Y, a, b), (0, 3), (1, 5)), [0, 1, 6, 7, 2, 3, 4, 5]) + + +def test_arrayexpr_array_diagonal(): + cg = _array_diagonal(M, (1, 0)) + assert cg == _array_diagonal(M, (0, 1)) + + cg = _array_diagonal(_array_tensor_product(M, N, P), (4, 1), (2, 0)) + assert cg == _array_diagonal(_array_tensor_product(M, N, P), (1, 4), (0, 2)) + + cg = _array_diagonal(_array_tensor_product(M, N), (1, 2), (3,), allow_trivial_diags=True) + assert cg == _permute_dims(_array_diagonal(_array_tensor_product(M, N), (1, 2)), [0, 2, 1]) + + Ax = ArraySymbol("Ax", shape=(1, 2, 3, 4, 3, 5, 6, 2, 7)) + cg = _array_diagonal(Ax, (1, 7), (3,), (2, 4), (6,), allow_trivial_diags=True) + assert cg == _permute_dims(_array_diagonal(Ax, (1, 7), (2, 4)), [0, 2, 4, 5, 1, 6, 3]) + + cg = _array_diagonal(M, (0,), allow_trivial_diags=True) + assert cg == _permute_dims(M, [1, 0]) + + raises(ValueError, lambda: _array_diagonal(M, (0, 0))) + + +def test_arrayexpr_array_shape(): + expr = _array_tensor_product(M, N, P, Q) + assert expr.shape == (k, k, k, k, k, k, k, k) + Z = MatrixSymbol("Z", m, n) + expr = _array_tensor_product(M, Z) + assert expr.shape == (k, k, m, n) + expr2 = _array_contraction(expr, (0, 1)) + assert expr2.shape == (m, n) + expr2 = _array_diagonal(expr, (0, 1)) + assert expr2.shape == (m, n, k) + exprp = _permute_dims(expr, [2, 1, 3, 0]) + assert exprp.shape == (m, k, n, k) + expr3 = _array_tensor_product(N, Z) + expr2 = _array_add(expr, expr3) + assert expr2.shape == (k, k, m, n) + + # Contraction along axes with discordant dimensions: + raises(ValueError, lambda: _array_contraction(expr, (1, 2))) + # Also diagonal needs the same dimensions: + raises(ValueError, lambda: _array_diagonal(expr, (1, 2))) + # Diagonal requires at least to axes to compute the diagonal: + raises(ValueError, lambda: _array_diagonal(expr, (1,))) + + +def test_arrayexpr_permutedims_sink(): + + cg = _permute_dims(_array_tensor_product(M, N), [0, 1, 3, 2], nest_permutation=False) + sunk = nest_permutation(cg) + assert sunk == _array_tensor_product(M, _permute_dims(N, [1, 0])) + + cg = _permute_dims(_array_tensor_product(M, N), [1, 0, 3, 2], nest_permutation=False) + sunk = nest_permutation(cg) + assert sunk == _array_tensor_product(_permute_dims(M, [1, 0]), _permute_dims(N, [1, 0])) + + cg = _permute_dims(_array_tensor_product(M, N), [3, 2, 1, 0], nest_permutation=False) + sunk = nest_permutation(cg) + assert sunk == _array_tensor_product(_permute_dims(N, [1, 0]), _permute_dims(M, [1, 0])) + + cg = _permute_dims(_array_contraction(_array_tensor_product(M, N), (1, 2)), [1, 0], nest_permutation=False) + sunk = nest_permutation(cg) + assert sunk == _array_contraction(_permute_dims(_array_tensor_product(M, N), [[0, 3]]), (1, 2)) + + cg = _permute_dims(_array_tensor_product(M, N), [1, 0, 3, 2], nest_permutation=False) + sunk = nest_permutation(cg) + assert sunk == _array_tensor_product(_permute_dims(M, [1, 0]), _permute_dims(N, [1, 0])) + + cg = _permute_dims(_array_contraction(_array_tensor_product(M, N, P), (1, 2), (3, 4)), [1, 0], nest_permutation=False) + sunk = nest_permutation(cg) + assert sunk == _array_contraction(_permute_dims(_array_tensor_product(M, N, P), [[0, 5]]), (1, 2), (3, 4)) + + +def test_arrayexpr_push_indices_up_and_down(): + + indices = list(range(12)) + + contr_diag_indices = [(0, 6), (2, 8)] + assert ArrayContraction._push_indices_down(contr_diag_indices, indices) == (1, 3, 4, 5, 7, 9, 10, 11, 12, 13, 14, 15) + assert ArrayContraction._push_indices_up(contr_diag_indices, indices) == (None, 0, None, 1, 2, 3, None, 4, None, 5, 6, 7) + + assert ArrayDiagonal._push_indices_down(contr_diag_indices, indices, 10) == (1, 3, 4, 5, 7, 9, (0, 6), (2, 8), None, None, None, None) + assert ArrayDiagonal._push_indices_up(contr_diag_indices, indices, 10) == (6, 0, 7, 1, 2, 3, 6, 4, 7, 5, None, None) + + contr_diag_indices = [(1, 2), (7, 8)] + assert ArrayContraction._push_indices_down(contr_diag_indices, indices) == (0, 3, 4, 5, 6, 9, 10, 11, 12, 13, 14, 15) + assert ArrayContraction._push_indices_up(contr_diag_indices, indices) == (0, None, None, 1, 2, 3, 4, None, None, 5, 6, 7) + + assert ArrayDiagonal._push_indices_down(contr_diag_indices, indices, 10) == (0, 3, 4, 5, 6, 9, (1, 2), (7, 8), None, None, None, None) + assert ArrayDiagonal._push_indices_up(contr_diag_indices, indices, 10) == (0, 6, 6, 1, 2, 3, 4, 7, 7, 5, None, None) + + +def test_arrayexpr_split_multiple_contractions(): + a = MatrixSymbol("a", k, 1) + b = MatrixSymbol("b", k, 1) + A = MatrixSymbol("A", k, k) + B = MatrixSymbol("B", k, k) + C = MatrixSymbol("C", k, k) + X = MatrixSymbol("X", k, k) + + cg = _array_contraction(_array_tensor_product(A.T, a, b, b.T, (A*X*b).applyfunc(cos)), (1, 2, 8), (5, 6, 9)) + expected = _array_contraction(_array_tensor_product(A.T, DiagMatrix(a), OneArray(1), b, b.T, (A*X*b).applyfunc(cos)), (1, 3), (2, 9), (6, 7, 10)) + assert cg.split_multiple_contractions().dummy_eq(expected) + + # Check no overlap of lines: + + cg = _array_contraction(_array_tensor_product(A, a, C, a, B), (1, 2, 4), (5, 6, 8), (3, 7)) + assert cg.split_multiple_contractions() == cg + + cg = _array_contraction(_array_tensor_product(a, b, A), (0, 2, 4), (1, 3)) + assert cg.split_multiple_contractions() == cg + + +def test_arrayexpr_nested_permutations(): + + cg = _permute_dims(_permute_dims(M, (1, 0)), (1, 0)) + assert cg == M + + times = 3 + plist1 = [list(range(6)) for i in range(times)] + plist2 = [list(range(6)) for i in range(times)] + + for i in range(times): + random.shuffle(plist1[i]) + random.shuffle(plist2[i]) + + plist1.append([2, 5, 4, 1, 0, 3]) + plist2.append([3, 5, 0, 4, 1, 2]) + + plist1.append([2, 5, 4, 0, 3, 1]) + plist2.append([3, 0, 5, 1, 2, 4]) + + plist1.append([5, 4, 2, 0, 3, 1]) + plist2.append([4, 5, 0, 2, 3, 1]) + + Me = M.subs(k, 3).as_explicit() + Ne = N.subs(k, 3).as_explicit() + Pe = P.subs(k, 3).as_explicit() + cge = tensorproduct(Me, Ne, Pe) + + for permutation_array1, permutation_array2 in zip(plist1, plist2): + p1 = Permutation(permutation_array1) + p2 = Permutation(permutation_array2) + + cg = _permute_dims( + _permute_dims( + _array_tensor_product(M, N, P), + p1), + p2 + ) + result = _permute_dims( + _array_tensor_product(M, N, P), + p2*p1 + ) + assert cg == result + + # Check that `permutedims` behaves the same way with explicit-component arrays: + result1 = _permute_dims(_permute_dims(cge, p1), p2) + result2 = _permute_dims(cge, p2*p1) + assert result1 == result2 + + +def test_arrayexpr_contraction_permutation_mix(): + + Me = M.subs(k, 3).as_explicit() + Ne = N.subs(k, 3).as_explicit() + + cg1 = _array_contraction(PermuteDims(_array_tensor_product(M, N), Permutation([0, 2, 1, 3])), (2, 3)) + cg2 = _array_contraction(_array_tensor_product(M, N), (1, 3)) + assert cg1 == cg2 + cge1 = tensorcontraction(permutedims(tensorproduct(Me, Ne), Permutation([0, 2, 1, 3])), (2, 3)) + cge2 = tensorcontraction(tensorproduct(Me, Ne), (1, 3)) + assert cge1 == cge2 + + cg1 = _permute_dims(_array_tensor_product(M, N), Permutation([0, 1, 3, 2])) + cg2 = _array_tensor_product(M, _permute_dims(N, Permutation([1, 0]))) + assert cg1 == cg2 + + cg1 = _array_contraction( + _permute_dims( + _array_tensor_product(M, N, P, Q), Permutation([0, 2, 3, 1, 4, 5, 7, 6])), + (1, 2), (3, 5) + ) + cg2 = _array_contraction( + _array_tensor_product(M, N, P, _permute_dims(Q, Permutation([1, 0]))), + (1, 5), (2, 3) + ) + assert cg1 == cg2 + + cg1 = _array_contraction( + _permute_dims( + _array_tensor_product(M, N, P, Q), Permutation([1, 0, 4, 6, 2, 7, 5, 3])), + (0, 1), (2, 6), (3, 7) + ) + cg2 = _permute_dims( + _array_contraction( + _array_tensor_product(M, P, Q, N), + (0, 1), (2, 3), (4, 7)), + [1, 0] + ) + assert cg1 == cg2 + + cg1 = _array_contraction( + _permute_dims( + _array_tensor_product(M, N, P, Q), Permutation([1, 0, 4, 6, 7, 2, 5, 3])), + (0, 1), (2, 6), (3, 7) + ) + cg2 = _permute_dims( + _array_contraction( + _array_tensor_product(_permute_dims(M, [1, 0]), N, P, Q), + (0, 1), (3, 6), (4, 5) + ), + Permutation([1, 0]) + ) + assert cg1 == cg2 + + +def test_arrayexpr_permute_tensor_product(): + cg1 = _permute_dims(_array_tensor_product(M, N, P, Q), Permutation([2, 3, 1, 0, 5, 4, 6, 7])) + cg2 = _array_tensor_product(N, _permute_dims(M, [1, 0]), + _permute_dims(P, [1, 0]), Q) + assert cg1 == cg2 + + # TODO: reverse operation starting with `PermuteDims` and getting down to `bb`... + cg1 = _permute_dims(_array_tensor_product(M, N, P, Q), Permutation([2, 3, 4, 5, 0, 1, 6, 7])) + cg2 = _array_tensor_product(N, P, M, Q) + assert cg1 == cg2 + + cg1 = _permute_dims(_array_tensor_product(M, N, P, Q), Permutation([2, 3, 4, 6, 5, 7, 0, 1])) + assert cg1.expr == _array_tensor_product(N, P, Q, M) + assert cg1.permutation == Permutation([0, 1, 2, 4, 3, 5, 6, 7]) + + cg1 = _array_contraction( + _permute_dims( + _array_tensor_product(N, Q, Q, M), + [2, 1, 5, 4, 0, 3, 6, 7]), + [1, 2, 6]) + cg2 = _permute_dims(_array_contraction(_array_tensor_product(Q, Q, N, M), (3, 5, 6)), [0, 2, 3, 1, 4]) + assert cg1 == cg2 + + cg1 = _array_contraction( + _array_contraction( + _array_contraction( + _array_contraction( + _permute_dims( + _array_tensor_product(N, Q, Q, M), + [2, 1, 5, 4, 0, 3, 6, 7]), + [1, 2, 6]), + [1, 3, 4]), + [1]), + [0]) + cg2 = _array_contraction(_array_tensor_product(M, N, Q, Q), (0, 3, 5), (1, 4, 7), (2,), (6,)) + assert cg1 == cg2 + + +def test_arrayexpr_canonicalize_diagonal__permute_dims(): + tp = _array_tensor_product(M, Q, N, P) + expr = _array_diagonal( + _permute_dims(tp, [0, 1, 2, 4, 7, 6, 3, 5]), (2, 4, 5), (6, 7), + (0, 3)) + result = _array_diagonal(tp, (2, 6, 7), (3, 5), (0, 4)) + assert expr == result + + tp = _array_tensor_product(M, N, P, Q) + expr = _array_diagonal(_permute_dims(tp, [0, 5, 2, 4, 1, 6, 3, 7]), (1, 2, 6), (3, 4)) + result = _array_diagonal(_array_tensor_product(M, P, N, Q), (3, 4, 5), (1, 2)) + assert expr == result + + +def test_arrayexpr_canonicalize_diagonal_contraction(): + tp = _array_tensor_product(M, N, P, Q) + expr = _array_contraction(_array_diagonal(tp, (1, 3, 4)), (0, 3)) + result = _array_diagonal(_array_contraction(_array_tensor_product(M, N, P, Q), (0, 6)), (0, 2, 3)) + assert expr == result + + expr = _array_contraction(_array_diagonal(tp, (0, 1, 2, 3, 7)), (1, 2, 3)) + result = _array_contraction(_array_tensor_product(M, N, P, Q), (0, 1, 2, 3, 5, 6, 7)) + assert expr == result + + expr = _array_contraction(_array_diagonal(tp, (0, 2, 6, 7)), (1, 2, 3)) + result = _array_diagonal(_array_contraction(tp, (3, 4, 5)), (0, 2, 3, 4)) + assert expr == result + + td = _array_diagonal(_array_tensor_product(M, N, P, Q), (0, 3)) + expr = _array_contraction(td, (2, 1), (0, 4, 6, 5, 3)) + result = _array_contraction(_array_tensor_product(M, N, P, Q), (0, 1, 3, 5, 6, 7), (2, 4)) + assert expr == result + + +def test_arrayexpr_array_wrong_permutation_size(): + cg = _array_tensor_product(M, N) + raises(ValueError, lambda: _permute_dims(cg, [1, 0])) + raises(ValueError, lambda: _permute_dims(cg, [1, 0, 2, 3, 5, 4])) + + +def test_arrayexpr_nested_array_elementwise_add(): + cg = _array_contraction(_array_add( + _array_tensor_product(M, N), + _array_tensor_product(N, M) + ), (1, 2)) + result = _array_add( + _array_contraction(_array_tensor_product(M, N), (1, 2)), + _array_contraction(_array_tensor_product(N, M), (1, 2)) + ) + assert cg == result + + cg = _array_diagonal(_array_add( + _array_tensor_product(M, N), + _array_tensor_product(N, M) + ), (1, 2)) + result = _array_add( + _array_diagonal(_array_tensor_product(M, N), (1, 2)), + _array_diagonal(_array_tensor_product(N, M), (1, 2)) + ) + assert cg == result + + +def test_arrayexpr_array_expr_zero_array(): + za1 = ZeroArray(k, l, m, n) + zm1 = ZeroMatrix(m, n) + + za2 = ZeroArray(k, m, m, n) + zm2 = ZeroMatrix(m, m) + zm3 = ZeroMatrix(k, k) + + assert _array_tensor_product(M, N, za1) == ZeroArray(k, k, k, k, k, l, m, n) + assert _array_tensor_product(M, N, zm1) == ZeroArray(k, k, k, k, m, n) + + assert _array_contraction(za1, (3,)) == ZeroArray(k, l, m) + assert _array_contraction(zm1, (1,)) == ZeroArray(m) + assert _array_contraction(za2, (1, 2)) == ZeroArray(k, n) + assert _array_contraction(zm2, (0, 1)) == 0 + + assert _array_diagonal(za2, (1, 2)) == ZeroArray(k, n, m) + assert _array_diagonal(zm2, (0, 1)) == ZeroArray(m) + + assert _permute_dims(za1, [2, 1, 3, 0]) == ZeroArray(m, l, n, k) + assert _permute_dims(zm1, [1, 0]) == ZeroArray(n, m) + + assert _array_add(za1) == za1 + assert _array_add(zm1) == ZeroArray(m, n) + tp1 = _array_tensor_product(MatrixSymbol("A", k, l), MatrixSymbol("B", m, n)) + assert _array_add(tp1, za1) == tp1 + tp2 = _array_tensor_product(MatrixSymbol("C", k, l), MatrixSymbol("D", m, n)) + assert _array_add(tp1, za1, tp2) == _array_add(tp1, tp2) + assert _array_add(M, zm3) == M + assert _array_add(M, N, zm3) == _array_add(M, N) + + +def test_arrayexpr_array_expr_applyfunc(): + + A = ArraySymbol("A", (3, k, 2)) + aaf = ArrayElementwiseApplyFunc(sin, A) + assert aaf.shape == (3, k, 2) + + +def test_edit_array_contraction(): + cg = _array_contraction(_array_tensor_product(A, B, C, D), (1, 2, 5)) + ecg = _EditArrayContraction(cg) + assert ecg.to_array_contraction() == cg + + ecg.args_with_ind[1], ecg.args_with_ind[2] = ecg.args_with_ind[2], ecg.args_with_ind[1] + assert ecg.to_array_contraction() == _array_contraction(_array_tensor_product(A, C, B, D), (1, 3, 4)) + + ci = ecg.get_new_contraction_index() + new_arg = _ArgE(X) + new_arg.indices = [ci, ci] + ecg.args_with_ind.insert(2, new_arg) + assert ecg.to_array_contraction() == _array_contraction(_array_tensor_product(A, C, X, B, D), (1, 3, 6), (4, 5)) + + assert ecg.get_contraction_indices() == [[1, 3, 6], [4, 5]] + assert [[tuple(j) for j in i] for i in ecg.get_contraction_indices_to_ind_rel_pos()] == [[(0, 1), (1, 1), (3, 0)], [(2, 0), (2, 1)]] + assert [list(i) for i in ecg.get_mapping_for_index(0)] == [[0, 1], [1, 1], [3, 0]] + assert [list(i) for i in ecg.get_mapping_for_index(1)] == [[2, 0], [2, 1]] + raises(ValueError, lambda: ecg.get_mapping_for_index(2)) + + ecg.args_with_ind.pop(1) + assert ecg.to_array_contraction() == _array_contraction(_array_tensor_product(A, X, B, D), (1, 4), (2, 3)) + + ecg.args_with_ind[0].indices[1] = ecg.args_with_ind[1].indices[0] + ecg.args_with_ind[1].indices[1] = ecg.args_with_ind[2].indices[0] + assert ecg.to_array_contraction() == _array_contraction(_array_tensor_product(A, X, B, D), (1, 2), (3, 4)) + + ecg.insert_after(ecg.args_with_ind[1], _ArgE(C)) + assert ecg.to_array_contraction() == _array_contraction(_array_tensor_product(A, X, C, B, D), (1, 2), (3, 6)) + + +def test_array_expressions_no_canonicalization(): + + tp = _array_tensor_product(M, N, P) + + # ArrayTensorProduct: + + expr = ArrayTensorProduct(tp, N) + assert str(expr) == "ArrayTensorProduct(ArrayTensorProduct(M, N, P), N)" + assert expr.doit() == ArrayTensorProduct(M, N, P, N) + + expr = ArrayTensorProduct(ArrayContraction(M, (0, 1)), N) + assert str(expr) == "ArrayTensorProduct(ArrayContraction(M, (0, 1)), N)" + assert expr.doit() == ArrayContraction(ArrayTensorProduct(M, N), (0, 1)) + + expr = ArrayTensorProduct(ArrayDiagonal(M, (0, 1)), N) + assert str(expr) == "ArrayTensorProduct(ArrayDiagonal(M, (0, 1)), N)" + assert expr.doit() == PermuteDims(ArrayDiagonal(ArrayTensorProduct(M, N), (0, 1)), [2, 0, 1]) + + expr = ArrayTensorProduct(PermuteDims(M, [1, 0]), N) + assert str(expr) == "ArrayTensorProduct(PermuteDims(M, (0 1)), N)" + assert expr.doit() == PermuteDims(ArrayTensorProduct(M, N), [1, 0, 2, 3]) + + # ArrayContraction: + + expr = ArrayContraction(_array_contraction(tp, (0, 2)), (0, 1)) + assert isinstance(expr, ArrayContraction) + assert isinstance(expr.expr, ArrayContraction) + assert str(expr) == "ArrayContraction(ArrayContraction(ArrayTensorProduct(M, N, P), (0, 2)), (0, 1))" + assert expr.doit() == ArrayContraction(tp, (0, 2), (1, 3)) + + expr = ArrayContraction(ArrayContraction(ArrayContraction(tp, (0, 1)), (0, 1)), (0, 1)) + assert expr.doit() == ArrayContraction(tp, (0, 1), (2, 3), (4, 5)) + # assert expr._canonicalize() == ArrayContraction(ArrayContraction(tp, (0, 1)), (0, 1), (2, 3)) + + expr = ArrayContraction(ArrayDiagonal(tp, (0, 1)), (0, 1)) + assert str(expr) == "ArrayContraction(ArrayDiagonal(ArrayTensorProduct(M, N, P), (0, 1)), (0, 1))" + assert expr.doit() == ArrayDiagonal(ArrayContraction(ArrayTensorProduct(N, M, P), (0, 1)), (0, 1)) + + expr = ArrayContraction(PermuteDims(M, [1, 0]), (0, 1)) + assert str(expr) == "ArrayContraction(PermuteDims(M, (0 1)), (0, 1))" + assert expr.doit() == ArrayContraction(M, (0, 1)) + + # ArrayDiagonal: + + expr = ArrayDiagonal(ArrayDiagonal(tp, (0, 2)), (0, 1)) + assert str(expr) == "ArrayDiagonal(ArrayDiagonal(ArrayTensorProduct(M, N, P), (0, 2)), (0, 1))" + assert expr.doit() == ArrayDiagonal(tp, (0, 2), (1, 3)) + + expr = ArrayDiagonal(ArrayDiagonal(ArrayDiagonal(tp, (0, 1)), (0, 1)), (0, 1)) + assert expr.doit() == ArrayDiagonal(tp, (0, 1), (2, 3), (4, 5)) + assert expr._canonicalize() == expr.doit() + + expr = ArrayDiagonal(ArrayContraction(tp, (0, 1)), (0, 1)) + assert str(expr) == "ArrayDiagonal(ArrayContraction(ArrayTensorProduct(M, N, P), (0, 1)), (0, 1))" + assert expr.doit() == expr + + expr = ArrayDiagonal(PermuteDims(M, [1, 0]), (0, 1)) + assert str(expr) == "ArrayDiagonal(PermuteDims(M, (0 1)), (0, 1))" + assert expr.doit() == ArrayDiagonal(M, (0, 1)) + + # ArrayAdd: + + expr = ArrayAdd(M) + assert isinstance(expr, ArrayAdd) + assert expr.doit() == M + + expr = ArrayAdd(ArrayAdd(M, N), P) + assert str(expr) == "ArrayAdd(ArrayAdd(M, N), P)" + assert expr.doit() == ArrayAdd(M, N, P) + + expr = ArrayAdd(M, ArrayAdd(N, ArrayAdd(P, M))) + assert expr.doit() == ArrayAdd(M, N, P, M) + assert expr._canonicalize() == ArrayAdd(M, N, ArrayAdd(P, M)) + + expr = ArrayAdd(M, ZeroArray(k, k), N) + assert str(expr) == "ArrayAdd(M, ZeroArray(k, k), N)" + assert expr.doit() == ArrayAdd(M, N) + + # PermuteDims: + + expr = PermuteDims(PermuteDims(M, [1, 0]), [1, 0]) + assert str(expr) == "PermuteDims(PermuteDims(M, (0 1)), (0 1))" + assert expr.doit() == M + + expr = PermuteDims(PermuteDims(PermuteDims(M, [1, 0]), [1, 0]), [1, 0]) + assert expr.doit() == PermuteDims(M, [1, 0]) + assert expr._canonicalize() == expr.doit() + + # Reshape + + expr = Reshape(A, (k**2,)) + assert expr.shape == (k**2,) + assert isinstance(expr, Reshape) + + +def test_array_expr_construction_with_functions(): + + tp = tensorproduct(M, N) + assert tp == ArrayTensorProduct(M, N) + + expr = tensorproduct(A, eye(2)) + assert expr == ArrayTensorProduct(A, eye(2)) + + # Contraction: + + expr = tensorcontraction(M, (0, 1)) + assert expr == ArrayContraction(M, (0, 1)) + + expr = tensorcontraction(tp, (1, 2)) + assert expr == ArrayContraction(tp, (1, 2)) + + expr = tensorcontraction(tensorcontraction(tp, (1, 2)), (0, 1)) + assert expr == ArrayContraction(tp, (0, 3), (1, 2)) + + # Diagonalization: + + expr = tensordiagonal(M, (0, 1)) + assert expr == ArrayDiagonal(M, (0, 1)) + + expr = tensordiagonal(tensordiagonal(tp, (0, 1)), (0, 1)) + assert expr == ArrayDiagonal(tp, (0, 1), (2, 3)) + + # Permutation of dimensions: + + expr = permutedims(M, [1, 0]) + assert expr == PermuteDims(M, [1, 0]) + + expr = permutedims(PermuteDims(tp, [1, 0, 2, 3]), [0, 1, 3, 2]) + assert expr == PermuteDims(tp, [1, 0, 3, 2]) + + expr = PermuteDims(tp, index_order_new=["a", "b", "c", "d"], index_order_old=["d", "c", "b", "a"]) + assert expr == PermuteDims(tp, [3, 2, 1, 0]) + + arr = Array(range(32)).reshape(2, 2, 2, 2, 2) + expr = PermuteDims(arr, index_order_new=["a", "b", "c", "d", "e"], index_order_old=['b', 'e', 'a', 'd', 'c']) + assert expr == PermuteDims(arr, [2, 0, 4, 3, 1]) + assert expr.as_explicit() == permutedims(arr, index_order_new=["a", "b", "c", "d", "e"], index_order_old=['b', 'e', 'a', 'd', 'c']) + + +def test_array_element_expressions(): + # Check commutative property: + assert M[0, 0]*N[0, 0] == N[0, 0]*M[0, 0] + + # Check derivatives: + assert M[0, 0].diff(M[0, 0]) == 1 + assert M[0, 0].diff(M[1, 0]) == 0 + assert M[0, 0].diff(N[0, 0]) == 0 + assert M[0, 1].diff(M[i, j]) == KroneckerDelta(i, 0)*KroneckerDelta(j, 1) + assert M[0, 1].diff(N[i, j]) == 0 + + K4 = ArraySymbol("K4", shape=(k, k, k, k)) + + assert K4[i, j, k, l].diff(K4[1, 2, 3, 4]) == ( + KroneckerDelta(i, 1)*KroneckerDelta(j, 2)*KroneckerDelta(k, 3)*KroneckerDelta(l, 4) + ) + + +def test_array_expr_reshape(): + + A = MatrixSymbol("A", 2, 2) + B = ArraySymbol("B", (2, 2, 2)) + C = Array([1, 2, 3, 4]) + + expr = Reshape(A, (4,)) + assert expr.expr == A + assert expr.shape == (4,) + assert expr.as_explicit() == Array([A[0, 0], A[0, 1], A[1, 0], A[1, 1]]) + + expr = Reshape(B, (2, 4)) + assert expr.expr == B + assert expr.shape == (2, 4) + ee = expr.as_explicit() + assert isinstance(ee, ImmutableDenseNDimArray) + assert ee.shape == (2, 4) + assert ee == Array([[B[0, 0, 0], B[0, 0, 1], B[0, 1, 0], B[0, 1, 1]], [B[1, 0, 0], B[1, 0, 1], B[1, 1, 0], B[1, 1, 1]]]) + + expr = Reshape(A, (k, 2)) + assert expr.shape == (k, 2) + + raises(ValueError, lambda: Reshape(A, (2, 3))) + raises(ValueError, lambda: Reshape(A, (3,))) + + expr = Reshape(C, (2, 2)) + assert expr.expr == C + assert expr.shape == (2, 2) + assert expr.doit() == Array([[1, 2], [3, 4]]) + + +def test_array_expr_as_explicit_with_explicit_component_arrays(): + # Test if .as_explicit() works with explicit-component arrays + # nested in array expressions: + from sympy.abc import x, y, z, t + A = Array([[x, y], [z, t]]) + assert ArrayTensorProduct(A, A).as_explicit() == tensorproduct(A, A) + assert ArrayDiagonal(A, (0, 1)).as_explicit() == tensordiagonal(A, (0, 1)) + assert ArrayContraction(A, (0, 1)).as_explicit() == tensorcontraction(A, (0, 1)) + assert ArrayAdd(A, A).as_explicit() == A + A + assert ArrayElementwiseApplyFunc(sin, A).as_explicit() == A.applyfunc(sin) + assert PermuteDims(A, [1, 0]).as_explicit() == permutedims(A, [1, 0]) + assert Reshape(A, [4]).as_explicit() == A.reshape(4) diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/tests/test_arrayexpr_derivatives.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/tests/test_arrayexpr_derivatives.py new file mode 100644 index 0000000000000000000000000000000000000000..bc0fcf63f2607b23feb38758e4f0994de4f0384b --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/tests/test_arrayexpr_derivatives.py @@ -0,0 +1,78 @@ +from sympy.core.symbol import symbols +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.matrices.expressions.matexpr import MatrixSymbol +from sympy.matrices.expressions.special import Identity +from sympy.matrices.expressions.applyfunc import ElementwiseApplyFunction +from sympy.tensor.array.expressions.array_expressions import ArraySymbol, ArrayTensorProduct, \ + PermuteDims, ArrayDiagonal, ArrayElementwiseApplyFunc, ArrayContraction, _permute_dims, Reshape +from sympy.tensor.array.expressions.arrayexpr_derivatives import array_derive + +k = symbols("k") + +I = Identity(k) +X = MatrixSymbol("X", k, k) +x = MatrixSymbol("x", k, 1) + +A = MatrixSymbol("A", k, k) +B = MatrixSymbol("B", k, k) +C = MatrixSymbol("C", k, k) +D = MatrixSymbol("D", k, k) + +A1 = ArraySymbol("A", (3, 2, k)) + + +def test_arrayexpr_derivatives1(): + + res = array_derive(X, X) + assert res == PermuteDims(ArrayTensorProduct(I, I), [0, 2, 1, 3]) + + cg = ArrayTensorProduct(A, X, B) + res = array_derive(cg, X) + assert res == _permute_dims( + ArrayTensorProduct(I, A, I, B), + [0, 4, 2, 3, 1, 5, 6, 7]) + + cg = ArrayContraction(X, (0, 1)) + res = array_derive(cg, X) + assert res == ArrayContraction(ArrayTensorProduct(I, I), (1, 3)) + + cg = ArrayDiagonal(X, (0, 1)) + res = array_derive(cg, X) + assert res == ArrayDiagonal(ArrayTensorProduct(I, I), (1, 3)) + + cg = ElementwiseApplyFunction(sin, X) + res = array_derive(cg, X) + assert res.dummy_eq(ArrayDiagonal( + ArrayTensorProduct( + ElementwiseApplyFunction(cos, X), + I, + I + ), (0, 3), (1, 5))) + + cg = ArrayElementwiseApplyFunc(sin, X) + res = array_derive(cg, X) + assert res.dummy_eq(ArrayDiagonal( + ArrayTensorProduct( + I, + I, + ArrayElementwiseApplyFunc(cos, X) + ), (1, 4), (3, 5))) + + res = array_derive(A1, A1) + assert res == PermuteDims( + ArrayTensorProduct(Identity(3), Identity(2), Identity(k)), + [0, 2, 4, 1, 3, 5] + ) + + cg = ArrayElementwiseApplyFunc(sin, A1) + res = array_derive(cg, A1) + assert res.dummy_eq(ArrayDiagonal( + ArrayTensorProduct( + Identity(3), Identity(2), Identity(k), + ArrayElementwiseApplyFunc(cos, A1) + ), (1, 6), (3, 7), (5, 8) + )) + + cg = Reshape(A, (k**2,)) + res = array_derive(cg, A) + assert res == Reshape(PermuteDims(ArrayTensorProduct(I, I), [0, 2, 1, 3]), (k, k, k**2)) diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/tests/test_as_explicit.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/tests/test_as_explicit.py new file mode 100644 index 0000000000000000000000000000000000000000..30cc61b1ee651ca032e165cd67926fa33c71354f --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/tests/test_as_explicit.py @@ -0,0 +1,63 @@ +from sympy.core.symbol import Symbol +from sympy.matrices.expressions.matexpr import MatrixSymbol +from sympy.tensor.array.arrayop import (permutedims, tensorcontraction, tensordiagonal, tensorproduct) +from sympy.tensor.array.dense_ndim_array import ImmutableDenseNDimArray +from sympy.tensor.array.expressions.array_expressions import ZeroArray, OneArray, ArraySymbol, \ + ArrayTensorProduct, PermuteDims, ArrayDiagonal, ArrayContraction, ArrayAdd +from sympy.testing.pytest import raises + + +def test_array_as_explicit_call(): + + assert ZeroArray(3, 2, 4).as_explicit() == ImmutableDenseNDimArray.zeros(3, 2, 4) + assert OneArray(3, 2, 4).as_explicit() == ImmutableDenseNDimArray([1 for i in range(3*2*4)]).reshape(3, 2, 4) + + k = Symbol("k") + X = ArraySymbol("X", (k, 3, 2)) + raises(ValueError, lambda: X.as_explicit()) + raises(ValueError, lambda: ZeroArray(k, 2, 3).as_explicit()) + raises(ValueError, lambda: OneArray(2, k, 2).as_explicit()) + + A = ArraySymbol("A", (3, 3)) + B = ArraySymbol("B", (3, 3)) + + texpr = tensorproduct(A, B) + assert isinstance(texpr, ArrayTensorProduct) + assert texpr.as_explicit() == tensorproduct(A.as_explicit(), B.as_explicit()) + + texpr = tensorcontraction(A, (0, 1)) + assert isinstance(texpr, ArrayContraction) + assert texpr.as_explicit() == A[0, 0] + A[1, 1] + A[2, 2] + + texpr = tensordiagonal(A, (0, 1)) + assert isinstance(texpr, ArrayDiagonal) + assert texpr.as_explicit() == ImmutableDenseNDimArray([A[0, 0], A[1, 1], A[2, 2]]) + + texpr = permutedims(A, [1, 0]) + assert isinstance(texpr, PermuteDims) + assert texpr.as_explicit() == permutedims(A.as_explicit(), [1, 0]) + + +def test_array_as_explicit_matrix_symbol(): + + A = MatrixSymbol("A", 3, 3) + B = MatrixSymbol("B", 3, 3) + + texpr = tensorproduct(A, B) + assert isinstance(texpr, ArrayTensorProduct) + assert texpr.as_explicit() == tensorproduct(A.as_explicit(), B.as_explicit()) + + texpr = tensorcontraction(A, (0, 1)) + assert isinstance(texpr, ArrayContraction) + assert texpr.as_explicit() == A[0, 0] + A[1, 1] + A[2, 2] + + texpr = tensordiagonal(A, (0, 1)) + assert isinstance(texpr, ArrayDiagonal) + assert texpr.as_explicit() == ImmutableDenseNDimArray([A[0, 0], A[1, 1], A[2, 2]]) + + texpr = permutedims(A, [1, 0]) + assert isinstance(texpr, PermuteDims) + assert texpr.as_explicit() == permutedims(A.as_explicit(), [1, 0]) + + expr = ArrayAdd(ArrayTensorProduct(A, B), ArrayTensorProduct(B, A)) + assert expr.as_explicit() == expr.args[0].as_explicit() + expr.args[1].as_explicit() diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/tests/test_convert_array_to_indexed.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/tests/test_convert_array_to_indexed.py new file mode 100644 index 0000000000000000000000000000000000000000..a6b713fbec94ab7808c5a8a778b3313402d9d0c7 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/tests/test_convert_array_to_indexed.py @@ -0,0 +1,61 @@ +from sympy import Sum, Dummy, sin +from sympy.tensor.array.expressions import ArraySymbol, ArrayTensorProduct, ArrayContraction, PermuteDims, \ + ArrayDiagonal, ArrayAdd, OneArray, ZeroArray, convert_indexed_to_array, ArrayElementwiseApplyFunc, Reshape +from sympy.tensor.array.expressions.from_array_to_indexed import convert_array_to_indexed + +from sympy.abc import i, j, k, l, m, n, o + + +def test_convert_array_to_indexed_main(): + A = ArraySymbol("A", (3, 3, 3)) + B = ArraySymbol("B", (3, 3)) + C = ArraySymbol("C", (3, 3)) + + d_ = Dummy("d_") + + assert convert_array_to_indexed(A, [i, j, k]) == A[i, j, k] + + expr = ArrayTensorProduct(A, B, C) + conv = convert_array_to_indexed(expr, [i,j,k,l,m,n,o]) + assert conv == A[i,j,k]*B[l,m]*C[n,o] + assert convert_indexed_to_array(conv, [i,j,k,l,m,n,o]) == expr + + expr = ArrayContraction(A, (0, 2)) + assert convert_array_to_indexed(expr, [i]).dummy_eq(Sum(A[d_, i, d_], (d_, 0, 2))) + + expr = ArrayDiagonal(A, (0, 2)) + assert convert_array_to_indexed(expr, [i, j]) == A[j, i, j] + + expr = PermuteDims(A, [1, 2, 0]) + conv = convert_array_to_indexed(expr, [i, j, k]) + assert conv == A[k, i, j] + assert convert_indexed_to_array(conv, [i, j, k]) == expr + + expr = ArrayAdd(B, C, PermuteDims(C, [1, 0])) + conv = convert_array_to_indexed(expr, [i, j]) + assert conv == B[i, j] + C[i, j] + C[j, i] + assert convert_indexed_to_array(conv, [i, j]) == expr + + expr = ArrayElementwiseApplyFunc(sin, A) + conv = convert_array_to_indexed(expr, [i, j, k]) + assert conv == sin(A[i, j, k]) + assert convert_indexed_to_array(conv, [i, j, k]).dummy_eq(expr) + + assert convert_array_to_indexed(OneArray(3, 3), [i, j]) == 1 + assert convert_array_to_indexed(ZeroArray(3, 3), [i, j]) == 0 + + expr = Reshape(A, (27,)) + assert convert_array_to_indexed(expr, [i]) == A[i // 9, i // 3 % 3, i % 3] + + X = ArraySymbol("X", (2, 3, 4, 5, 6)) + expr = Reshape(X, (2*3*4*5*6,)) + assert convert_array_to_indexed(expr, [i]) == X[i // 360, i // 120 % 3, i // 30 % 4, i // 6 % 5, i % 6] + + expr = Reshape(X, (4, 9, 2, 2, 5)) + one_index = 180*i + 20*j + 10*k + 5*l + m + expected = X[one_index // (3*4*5*6), one_index // (4*5*6) % 3, one_index // (5*6) % 4, one_index // 6 % 5, one_index % 6] + assert convert_array_to_indexed(expr, [i, j, k, l, m]) == expected + + X = ArraySymbol("X", (2*3*5,)) + expr = Reshape(X, (2, 3, 5)) + assert convert_array_to_indexed(expr, [i, j, k]) == X[15*i + 5*j + k] diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/tests/test_convert_array_to_matrix.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/tests/test_convert_array_to_matrix.py new file mode 100644 index 0000000000000000000000000000000000000000..26839d5e7cec0554948c6b726482f9d8ca250b1c --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/tests/test_convert_array_to_matrix.py @@ -0,0 +1,689 @@ +from sympy import Lambda, S, Dummy, KroneckerProduct +from sympy.core.symbol import symbols +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import cos, sin +from sympy.matrices.expressions.hadamard import HadamardProduct, HadamardPower +from sympy.matrices.expressions.special import (Identity, OneMatrix, ZeroMatrix) +from sympy.matrices.expressions.matexpr import MatrixElement +from sympy.tensor.array.expressions.from_matrix_to_array import convert_matrix_to_array +from sympy.tensor.array.expressions.from_array_to_matrix import _support_function_tp1_recognize, \ + _array_diag2contr_diagmatrix, convert_array_to_matrix, _remove_trivial_dims, _array2matrix, \ + _combine_removed, identify_removable_identity_matrices, _array_contraction_to_diagonal_multiple_identity +from sympy.matrices.expressions.matexpr import MatrixSymbol +from sympy.combinatorics import Permutation +from sympy.matrices.expressions.diagonal import DiagMatrix, DiagonalMatrix +from sympy.matrices import Trace, MatMul, Transpose +from sympy.tensor.array.expressions.array_expressions import ZeroArray, OneArray, \ + ArrayElement, ArraySymbol, ArrayElementwiseApplyFunc, _array_tensor_product, _array_contraction, \ + _array_diagonal, _permute_dims, PermuteDims, ArrayAdd, ArrayDiagonal, ArrayContraction, ArrayTensorProduct +from sympy.testing.pytest import raises + + +i, j, k, l, m, n = symbols("i j k l m n") + +I = Identity(k) +I1 = Identity(1) + +M = MatrixSymbol("M", k, k) +N = MatrixSymbol("N", k, k) +P = MatrixSymbol("P", k, k) +Q = MatrixSymbol("Q", k, k) + +A = MatrixSymbol("A", k, k) +B = MatrixSymbol("B", k, k) +C = MatrixSymbol("C", k, k) +D = MatrixSymbol("D", k, k) + +X = MatrixSymbol("X", k, k) +Y = MatrixSymbol("Y", k, k) + +a = MatrixSymbol("a", k, 1) +b = MatrixSymbol("b", k, 1) +c = MatrixSymbol("c", k, 1) +d = MatrixSymbol("d", k, 1) + +x = MatrixSymbol("x", k, 1) +y = MatrixSymbol("y", k, 1) + + +def test_arrayexpr_convert_array_to_matrix(): + + cg = _array_contraction(_array_tensor_product(M), (0, 1)) + assert convert_array_to_matrix(cg) == Trace(M) + + cg = _array_contraction(_array_tensor_product(M, N), (0, 1), (2, 3)) + assert convert_array_to_matrix(cg) == Trace(M) * Trace(N) + + cg = _array_contraction(_array_tensor_product(M, N), (0, 3), (1, 2)) + assert convert_array_to_matrix(cg) == Trace(M * N) + + cg = _array_contraction(_array_tensor_product(M, N), (0, 2), (1, 3)) + assert convert_array_to_matrix(cg) == Trace(M * N.T) + + cg = convert_matrix_to_array(M * N * P) + assert convert_array_to_matrix(cg) == M * N * P + + cg = convert_matrix_to_array(M * N.T * P) + assert convert_array_to_matrix(cg) == M * N.T * P + + cg = _array_contraction(_array_tensor_product(M,N,P,Q), (1, 2), (5, 6)) + assert convert_array_to_matrix(cg) == _array_tensor_product(M * N, P * Q) + + cg = _array_contraction(_array_tensor_product(-2, M, N), (1, 2)) + assert convert_array_to_matrix(cg) == -2 * M * N + + a = MatrixSymbol("a", k, 1) + b = MatrixSymbol("b", k, 1) + c = MatrixSymbol("c", k, 1) + cg = PermuteDims( + _array_contraction( + _array_tensor_product( + a, + ArrayAdd( + _array_tensor_product(b, c), + _array_tensor_product(c, b), + ) + ), (2, 4)), [0, 1, 3, 2]) + assert convert_array_to_matrix(cg) == a * (b.T * c + c.T * b) + + za = ZeroArray(m, n) + assert convert_array_to_matrix(za) == ZeroMatrix(m, n) + + cg = _array_tensor_product(3, M) + assert convert_array_to_matrix(cg) == 3 * M + + # Partial conversion to matrix multiplication: + expr = _array_contraction(_array_tensor_product(M, N, P, Q), (0, 2), (1, 4, 6)) + assert convert_array_to_matrix(expr) == _array_contraction(_array_tensor_product(M.T*N, P, Q), (0, 2, 4)) + + x = MatrixSymbol("x", k, 1) + cg = PermuteDims( + _array_contraction(_array_tensor_product(OneArray(1), x, OneArray(1), DiagMatrix(Identity(1))), + (0, 5)), Permutation(1, 2, 3)) + assert convert_array_to_matrix(cg) == x + + expr = ArrayAdd(M, PermuteDims(M, [1, 0])) + assert convert_array_to_matrix(expr) == M + Transpose(M) + + +def test_arrayexpr_convert_array_to_matrix2(): + cg = _array_contraction(_array_tensor_product(M, N), (1, 3)) + assert convert_array_to_matrix(cg) == M * N.T + + cg = PermuteDims(_array_tensor_product(M, N), Permutation([0, 1, 3, 2])) + assert convert_array_to_matrix(cg) == _array_tensor_product(M, N.T) + + cg = _array_tensor_product(M, PermuteDims(N, Permutation([1, 0]))) + assert convert_array_to_matrix(cg) == _array_tensor_product(M, N.T) + + cg = _array_contraction( + PermuteDims( + _array_tensor_product(M, N, P, Q), Permutation([0, 2, 3, 1, 4, 5, 7, 6])), + (1, 2), (3, 5) + ) + assert convert_array_to_matrix(cg) == _array_tensor_product(M * P.T * Trace(N), Q.T) + + cg = _array_contraction( + _array_tensor_product(M, N, P, PermuteDims(Q, Permutation([1, 0]))), + (1, 5), (2, 3) + ) + assert convert_array_to_matrix(cg) == _array_tensor_product(M * P.T * Trace(N), Q.T) + + cg = _array_tensor_product(M, PermuteDims(N, [1, 0])) + assert convert_array_to_matrix(cg) == _array_tensor_product(M, N.T) + + cg = _array_tensor_product(PermuteDims(M, [1, 0]), PermuteDims(N, [1, 0])) + assert convert_array_to_matrix(cg) == _array_tensor_product(M.T, N.T) + + cg = _array_tensor_product(PermuteDims(N, [1, 0]), PermuteDims(M, [1, 0])) + assert convert_array_to_matrix(cg) == _array_tensor_product(N.T, M.T) + + cg = _array_contraction(M, (0,), (1,)) + assert convert_array_to_matrix(cg) == OneMatrix(1, k)*M*OneMatrix(k, 1) + + cg = _array_contraction(x, (0,), (1,)) + assert convert_array_to_matrix(cg) == OneMatrix(1, k)*x + + Xm = MatrixSymbol("Xm", m, n) + cg = _array_contraction(Xm, (0,), (1,)) + assert convert_array_to_matrix(cg) == OneMatrix(1, m)*Xm*OneMatrix(n, 1) + + +def test_arrayexpr_convert_array_to_diagonalized_vector(): + + # Check matrix recognition over trivial dimensions: + + cg = _array_tensor_product(a, b) + assert convert_array_to_matrix(cg) == a * b.T + + cg = _array_tensor_product(I1, a, b) + assert convert_array_to_matrix(cg) == a * b.T + + # Recognize trace inside a tensor product: + + cg = _array_contraction(_array_tensor_product(A, B, C), (0, 3), (1, 2)) + assert convert_array_to_matrix(cg) == Trace(A * B) * C + + # Transform diagonal operator to contraction: + + cg = _array_diagonal(_array_tensor_product(A, a), (1, 2)) + assert _array_diag2contr_diagmatrix(cg) == _array_contraction(_array_tensor_product(A, OneArray(1), DiagMatrix(a)), (1, 3)) + assert convert_array_to_matrix(cg) == A * DiagMatrix(a) + + cg = _array_diagonal(_array_tensor_product(a, b), (0, 2)) + assert _array_diag2contr_diagmatrix(cg) == _permute_dims( + _array_contraction(_array_tensor_product(DiagMatrix(a), OneArray(1), b), (0, 3)), [1, 2, 0] + ) + assert convert_array_to_matrix(cg) == b.T * DiagMatrix(a) + + cg = _array_diagonal(_array_tensor_product(A, a), (0, 2)) + assert _array_diag2contr_diagmatrix(cg) == _array_contraction(_array_tensor_product(A, OneArray(1), DiagMatrix(a)), (0, 3)) + assert convert_array_to_matrix(cg) == A.T * DiagMatrix(a) + + cg = _array_diagonal(_array_tensor_product(I, x, I1), (0, 2), (3, 5)) + assert _array_diag2contr_diagmatrix(cg) == _array_contraction(_array_tensor_product(I, OneArray(1), I1, DiagMatrix(x)), (0, 5)) + assert convert_array_to_matrix(cg) == DiagMatrix(x) + + cg = _array_diagonal(_array_tensor_product(I, x, A, B), (1, 2), (5, 6)) + assert _array_diag2contr_diagmatrix(cg) == _array_diagonal(_array_contraction(_array_tensor_product(I, OneArray(1), A, B, DiagMatrix(x)), (1, 7)), (5, 6)) + # TODO: this is returning a wrong result: + # convert_array_to_matrix(cg) + + cg = _array_diagonal(_array_tensor_product(I1, a, b), (1, 3, 5)) + assert convert_array_to_matrix(cg) == a*b.T + + cg = _array_diagonal(_array_tensor_product(I1, a, b), (1, 3)) + assert _array_diag2contr_diagmatrix(cg) == _array_contraction(_array_tensor_product(OneArray(1), a, b, I1), (2, 6)) + assert convert_array_to_matrix(cg) == a*b.T + + cg = _array_diagonal(_array_tensor_product(x, I1), (1, 2)) + assert isinstance(cg, ArrayDiagonal) + assert cg.diagonal_indices == ((1, 2),) + assert convert_array_to_matrix(cg) == x + + cg = _array_diagonal(_array_tensor_product(x, I), (0, 2)) + assert _array_diag2contr_diagmatrix(cg) == _array_contraction(_array_tensor_product(OneArray(1), I, DiagMatrix(x)), (1, 3)) + assert convert_array_to_matrix(cg).doit() == DiagMatrix(x) + + raises(ValueError, lambda: _array_diagonal(x, (1,))) + + # Ignore identity matrices with contractions: + + cg = _array_contraction(_array_tensor_product(I, A, I, I), (0, 2), (1, 3), (5, 7)) + assert cg.split_multiple_contractions() == cg + assert convert_array_to_matrix(cg) == Trace(A) * I + + cg = _array_contraction(_array_tensor_product(Trace(A) * I, I, I), (1, 5), (3, 4)) + assert cg.split_multiple_contractions() == cg + assert convert_array_to_matrix(cg).doit() == Trace(A) * I + + # Add DiagMatrix when required: + + cg = _array_contraction(_array_tensor_product(A, a), (1, 2)) + assert cg.split_multiple_contractions() == cg + assert convert_array_to_matrix(cg) == A * a + + cg = _array_contraction(_array_tensor_product(A, a, B), (1, 2, 4)) + assert cg.split_multiple_contractions() == _array_contraction(_array_tensor_product(A, DiagMatrix(a), OneArray(1), B), (1, 2), (3, 5)) + assert convert_array_to_matrix(cg) == A * DiagMatrix(a) * B + + cg = _array_contraction(_array_tensor_product(A, a, B), (0, 2, 4)) + assert cg.split_multiple_contractions() == _array_contraction(_array_tensor_product(A, DiagMatrix(a), OneArray(1), B), (0, 2), (3, 5)) + assert convert_array_to_matrix(cg) == A.T * DiagMatrix(a) * B + + cg = _array_contraction(_array_tensor_product(A, a, b, a.T, B), (0, 2, 4, 7, 9)) + assert cg.split_multiple_contractions() == _array_contraction(_array_tensor_product(A, DiagMatrix(a), OneArray(1), + DiagMatrix(b), OneArray(1), DiagMatrix(a), OneArray(1), B), + (0, 2), (3, 5), (6, 9), (8, 12)) + assert convert_array_to_matrix(cg) == A.T * DiagMatrix(a) * DiagMatrix(b) * DiagMatrix(a) * B.T + + cg = _array_contraction(_array_tensor_product(I1, I1, I1), (1, 2, 4)) + assert cg.split_multiple_contractions() == _array_contraction(_array_tensor_product(I1, I1, OneArray(1), I1), (1, 2), (3, 5)) + assert convert_array_to_matrix(cg) == 1 + + cg = _array_contraction(_array_tensor_product(I, I, I, I, A), (1, 2, 8), (5, 6, 9)) + assert convert_array_to_matrix(cg.split_multiple_contractions()).doit() == A + + cg = _array_contraction(_array_tensor_product(A, a, C, a, B), (1, 2, 4), (5, 6, 8)) + expected = _array_contraction(_array_tensor_product(A, DiagMatrix(a), OneArray(1), C, DiagMatrix(a), OneArray(1), B), (1, 3), (2, 5), (6, 7), (8, 10)) + assert cg.split_multiple_contractions() == expected + assert convert_array_to_matrix(cg) == A * DiagMatrix(a) * C * DiagMatrix(a) * B + + cg = _array_contraction(_array_tensor_product(a, I1, b, I1, (a.T*b).applyfunc(cos)), (1, 2, 8), (5, 6, 9)) + expected = _array_contraction(_array_tensor_product(a, I1, OneArray(1), b, I1, OneArray(1), (a.T*b).applyfunc(cos)), + (1, 3), (2, 10), (6, 8), (7, 11)) + assert cg.split_multiple_contractions().dummy_eq(expected) + assert convert_array_to_matrix(cg).doit().dummy_eq(MatMul(a, (a.T * b).applyfunc(cos), b.T)) + + +def test_arrayexpr_convert_array_contraction_tp_additions(): + a = ArrayAdd( + _array_tensor_product(M, N), + _array_tensor_product(N, M) + ) + tp = _array_tensor_product(P, a, Q) + expr = _array_contraction(tp, (3, 4)) + expected = _array_tensor_product( + P, + ArrayAdd( + _array_contraction(_array_tensor_product(M, N), (1, 2)), + _array_contraction(_array_tensor_product(N, M), (1, 2)), + ), + Q + ) + assert expr == expected + assert convert_array_to_matrix(expr) == _array_tensor_product(P, M * N + N * M, Q) + + expr = _array_contraction(tp, (1, 2), (3, 4), (5, 6)) + result = _array_contraction( + _array_tensor_product( + P, + ArrayAdd( + _array_contraction(_array_tensor_product(M, N), (1, 2)), + _array_contraction(_array_tensor_product(N, M), (1, 2)), + ), + Q + ), (1, 2), (3, 4)) + assert expr == result + assert convert_array_to_matrix(expr) == P * (M * N + N * M) * Q + + +def test_arrayexpr_convert_array_to_implicit_matmul(): + # Trivial dimensions are suppressed, so the result can be expressed in matrix form: + + cg = _array_tensor_product(a, b) + assert convert_array_to_matrix(cg) == a * b.T + + cg = _array_tensor_product(a, b, I) + assert convert_array_to_matrix(cg) == _array_tensor_product(a*b.T, I) + + cg = _array_tensor_product(I, a, b) + assert convert_array_to_matrix(cg) == _array_tensor_product(I, a*b.T) + + cg = _array_tensor_product(a, I, b) + assert convert_array_to_matrix(cg) == _array_tensor_product(a, I, b) + + cg = _array_contraction(_array_tensor_product(I, I), (1, 2)) + assert convert_array_to_matrix(cg) == I + + cg = PermuteDims(_array_tensor_product(I, Identity(1)), [0, 2, 1, 3]) + assert convert_array_to_matrix(cg) == I + + +def test_arrayexpr_convert_array_to_matrix_remove_trivial_dims(): + + # Tensor Product: + assert _remove_trivial_dims(_array_tensor_product(a, b)) == (a * b.T, [1, 3]) + assert _remove_trivial_dims(_array_tensor_product(a.T, b)) == (a * b.T, [0, 3]) + assert _remove_trivial_dims(_array_tensor_product(a, b.T)) == (a * b.T, [1, 2]) + assert _remove_trivial_dims(_array_tensor_product(a.T, b.T)) == (a * b.T, [0, 2]) + + assert _remove_trivial_dims(_array_tensor_product(I, a.T, b.T)) == (_array_tensor_product(I, a * b.T), [2, 4]) + assert _remove_trivial_dims(_array_tensor_product(a.T, I, b.T)) == (_array_tensor_product(a.T, I, b.T), []) + + assert _remove_trivial_dims(_array_tensor_product(a, I)) == (_array_tensor_product(a, I), []) + assert _remove_trivial_dims(_array_tensor_product(I, a)) == (_array_tensor_product(I, a), []) + + assert _remove_trivial_dims(_array_tensor_product(a.T, b.T, c, d)) == ( + _array_tensor_product(a * b.T, c * d.T), [0, 2, 5, 7]) + assert _remove_trivial_dims(_array_tensor_product(a.T, I, b.T, c, d, I)) == ( + _array_tensor_product(a.T, I, b*c.T, d, I), [4, 7]) + + # Addition: + + cg = ArrayAdd(_array_tensor_product(a, b), _array_tensor_product(c, d)) + assert _remove_trivial_dims(cg) == (a * b.T + c * d.T, [1, 3]) + + # Permute Dims: + + cg = PermuteDims(_array_tensor_product(a, b), Permutation(3)(1, 2)) + assert _remove_trivial_dims(cg) == (a * b.T, [2, 3]) + + cg = PermuteDims(_array_tensor_product(a, I, b), Permutation(5)(1, 2, 3, 4)) + assert _remove_trivial_dims(cg) == (cg, []) + + cg = PermuteDims(_array_tensor_product(I, b, a), Permutation(5)(1, 2, 4, 5, 3)) + assert _remove_trivial_dims(cg) == (PermuteDims(_array_tensor_product(I, b * a.T), [0, 2, 3, 1]), [4, 5]) + + # Diagonal: + + cg = _array_diagonal(_array_tensor_product(M, a), (1, 2)) + assert _remove_trivial_dims(cg) == (cg, []) + + # Contraction: + + cg = _array_contraction(_array_tensor_product(M, a), (1, 2)) + assert _remove_trivial_dims(cg) == (cg, []) + + # A few more cases to test the removal and shift of nested removed axes + # with array contractions and array diagonals: + tp = _array_tensor_product( + OneMatrix(1, 1), + M, + x, + OneMatrix(1, 1), + Identity(1), + ) + + expr = _array_contraction(tp, (1, 8)) + rexpr, removed = _remove_trivial_dims(expr) + assert removed == [0, 5, 6, 7] + + expr = _array_contraction(tp, (1, 8), (3, 4)) + rexpr, removed = _remove_trivial_dims(expr) + assert removed == [0, 3, 4, 5] + + expr = _array_diagonal(tp, (1, 8)) + rexpr, removed = _remove_trivial_dims(expr) + assert removed == [0, 5, 6, 7, 8] + + expr = _array_diagonal(tp, (1, 8), (3, 4)) + rexpr, removed = _remove_trivial_dims(expr) + assert removed == [0, 3, 4, 5, 6] + + expr = _array_diagonal(_array_contraction(_array_tensor_product(A, x, I, I1), (1, 2, 5)), (1, 4)) + rexpr, removed = _remove_trivial_dims(expr) + assert removed == [2, 3] + + cg = _array_diagonal(_array_tensor_product(PermuteDims(_array_tensor_product(x, I1), Permutation(1, 2, 3)), (x.T*x).applyfunc(sqrt)), (2, 4), (3, 5)) + rexpr, removed = _remove_trivial_dims(cg) + assert removed == [1, 2] + + # Contractions with identity matrices need to be followed by a permutation + # in order + cg = _array_contraction(_array_tensor_product(A, B, C, M, I), (1, 8)) + ret, removed = _remove_trivial_dims(cg) + assert ret == PermuteDims(_array_tensor_product(A, B, C, M), [0, 2, 3, 4, 5, 6, 7, 1]) + assert removed == [] + + cg = _array_contraction(_array_tensor_product(A, B, C, M, I), (1, 8), (3, 4)) + ret, removed = _remove_trivial_dims(cg) + assert ret == PermuteDims(_array_contraction(_array_tensor_product(A, B, C, M), (3, 4)), [0, 2, 3, 4, 5, 1]) + assert removed == [] + + # Trivial matrices are sometimes inserted into MatMul expressions: + + cg = _array_tensor_product(b*b.T, a.T*a) + ret, removed = _remove_trivial_dims(cg) + assert ret == b*a.T*a*b.T + assert removed == [2, 3] + + Xs = ArraySymbol("X", (3, 2, k)) + cg = _array_tensor_product(M, Xs, b.T*c, a*a.T, b*b.T, c.T*d) + ret, removed = _remove_trivial_dims(cg) + assert ret == _array_tensor_product(M, Xs, a*b.T*c*c.T*d*a.T, b*b.T) + assert removed == [5, 6, 11, 12] + + cg = _array_diagonal(_array_tensor_product(I, I1, x), (1, 4), (3, 5)) + assert _remove_trivial_dims(cg) == (PermuteDims(_array_diagonal(_array_tensor_product(I, x), (1, 2)), Permutation(1, 2)), [1]) + + expr = _array_diagonal(_array_tensor_product(x, I, y), (0, 2)) + assert _remove_trivial_dims(expr) == (PermuteDims(_array_tensor_product(DiagMatrix(x), y), [1, 2, 3, 0]), [0]) + + expr = _array_diagonal(_array_tensor_product(x, I, y), (0, 2), (3, 4)) + assert _remove_trivial_dims(expr) == (expr, []) + + +def test_arrayexpr_convert_array_to_matrix_diag2contraction_diagmatrix(): + cg = _array_diagonal(_array_tensor_product(M, a), (1, 2)) + res = _array_diag2contr_diagmatrix(cg) + assert res.shape == cg.shape + assert res == _array_contraction(_array_tensor_product(M, OneArray(1), DiagMatrix(a)), (1, 3)) + + raises(ValueError, lambda: _array_diagonal(_array_tensor_product(a, M), (1, 2))) + + cg = _array_diagonal(_array_tensor_product(a.T, M), (1, 2)) + res = _array_diag2contr_diagmatrix(cg) + assert res.shape == cg.shape + assert res == _array_contraction(_array_tensor_product(OneArray(1), M, DiagMatrix(a.T)), (1, 4)) + + cg = _array_diagonal(_array_tensor_product(a.T, M, N, b.T), (1, 2), (4, 7)) + res = _array_diag2contr_diagmatrix(cg) + assert res.shape == cg.shape + assert res == _array_contraction( + _array_tensor_product(OneArray(1), M, N, OneArray(1), DiagMatrix(a.T), DiagMatrix(b.T)), (1, 7), (3, 9)) + + cg = _array_diagonal(_array_tensor_product(a, M, N, b.T), (0, 2), (4, 7)) + res = _array_diag2contr_diagmatrix(cg) + assert res.shape == cg.shape + assert res == _array_contraction( + _array_tensor_product(OneArray(1), M, N, OneArray(1), DiagMatrix(a), DiagMatrix(b.T)), (1, 6), (3, 9)) + + cg = _array_diagonal(_array_tensor_product(a, M, N, b.T), (0, 4), (3, 7)) + res = _array_diag2contr_diagmatrix(cg) + assert res.shape == cg.shape + assert res == _array_contraction( + _array_tensor_product(OneArray(1), M, N, OneArray(1), DiagMatrix(a), DiagMatrix(b.T)), (3, 6), (2, 9)) + + I1 = Identity(1) + x = MatrixSymbol("x", k, 1) + A = MatrixSymbol("A", k, k) + cg = _array_diagonal(_array_tensor_product(x, A.T, I1), (0, 2)) + assert _array_diag2contr_diagmatrix(cg).shape == cg.shape + assert _array2matrix(cg).shape == cg.shape + + +def test_arrayexpr_convert_array_to_matrix_support_function(): + + assert _support_function_tp1_recognize([], [2 * k]) == 2 * k + + assert _support_function_tp1_recognize([(1, 2)], [A, 2 * k, B, 3]) == 6 * k * A * B + + assert _support_function_tp1_recognize([(0, 3), (1, 2)], [A, B]) == Trace(A * B) + + assert _support_function_tp1_recognize([(1, 2)], [A, B]) == A * B + assert _support_function_tp1_recognize([(0, 2)], [A, B]) == A.T * B + assert _support_function_tp1_recognize([(1, 3)], [A, B]) == A * B.T + assert _support_function_tp1_recognize([(0, 3)], [A, B]) == A.T * B.T + + assert _support_function_tp1_recognize([(1, 2), (5, 6)], [A, B, C, D]) == _array_tensor_product(A * B, C * D) + assert _support_function_tp1_recognize([(1, 4), (3, 6)], [A, B, C, D]) == PermuteDims( + _array_tensor_product(A * C, B * D), [0, 2, 1, 3]) + + assert _support_function_tp1_recognize([(0, 3), (1, 4)], [A, B, C]) == B * A * C + + assert _support_function_tp1_recognize([(9, 10), (1, 2), (5, 6), (3, 4), (7, 8)], + [X, Y, A, B, C, D]) == X * Y * A * B * C * D + + assert _support_function_tp1_recognize([(9, 10), (1, 2), (5, 6), (3, 4)], + [X, Y, A, B, C, D]) == _array_tensor_product(X * Y * A * B, C * D) + + assert _support_function_tp1_recognize([(1, 7), (3, 8), (4, 11)], [X, Y, A, B, C, D]) == PermuteDims( + _array_tensor_product(X * B.T, Y * C, A.T * D.T), [0, 2, 4, 1, 3, 5] + ) + + assert _support_function_tp1_recognize([(0, 1), (3, 6), (5, 8)], [X, A, B, C, D]) == PermuteDims( + _array_tensor_product(Trace(X) * A * C, B * D), [0, 2, 1, 3]) + + assert _support_function_tp1_recognize([(1, 2), (3, 4), (5, 6), (7, 8)], [A, A, B, C, D]) == A ** 2 * B * C * D + assert _support_function_tp1_recognize([(1, 2), (3, 4), (5, 6), (7, 8)], [X, A, B, C, D]) == X * A * B * C * D + + assert _support_function_tp1_recognize([(1, 6), (3, 8), (5, 10)], [X, Y, A, B, C, D]) == PermuteDims( + _array_tensor_product(X * B, Y * C, A * D), [0, 2, 4, 1, 3, 5] + ) + + assert _support_function_tp1_recognize([(1, 4), (3, 6)], [A, B, C, D]) == PermuteDims( + _array_tensor_product(A * C, B * D), [0, 2, 1, 3]) + + assert _support_function_tp1_recognize([(0, 4), (1, 7), (2, 5), (3, 8)], [X, A, B, C, D]) == C*X.T*B*A*D + + assert _support_function_tp1_recognize([(0, 4), (1, 7), (2, 5), (3, 8)], [X, A, B, C, D]) == C*X.T*B*A*D + + +def test_convert_array_to_hadamard_products(): + + expr = HadamardProduct(M, N) + cg = convert_matrix_to_array(expr) + ret = convert_array_to_matrix(cg) + assert ret == expr + + expr = HadamardProduct(M, N)*P + cg = convert_matrix_to_array(expr) + ret = convert_array_to_matrix(cg) + assert ret == expr + + expr = Q*HadamardProduct(M, N)*P + cg = convert_matrix_to_array(expr) + ret = convert_array_to_matrix(cg) + assert ret == expr + + expr = Q*HadamardProduct(M, N.T)*P + cg = convert_matrix_to_array(expr) + ret = convert_array_to_matrix(cg) + assert ret == expr + + expr = HadamardProduct(M, N)*HadamardProduct(Q, P) + cg = convert_matrix_to_array(expr) + ret = convert_array_to_matrix(cg) + assert expr == ret + + expr = P.T*HadamardProduct(M, N)*HadamardProduct(Q, P) + cg = convert_matrix_to_array(expr) + ret = convert_array_to_matrix(cg) + assert expr == ret + + # ArrayDiagonal should be converted + cg = _array_diagonal(_array_tensor_product(M, N, Q), (1, 3), (0, 2, 4)) + ret = convert_array_to_matrix(cg) + expected = PermuteDims(_array_diagonal(_array_tensor_product(HadamardProduct(M.T, N.T), Q), (1, 2)), [1, 0, 2]) + assert expected == ret + + # Special case that should return the same expression: + cg = _array_diagonal(_array_tensor_product(HadamardProduct(M, N), Q), (0, 2)) + ret = convert_array_to_matrix(cg) + assert ret == cg + + # Hadamard products with traces: + + expr = Trace(HadamardProduct(M, N)) + cg = convert_matrix_to_array(expr) + ret = convert_array_to_matrix(cg) + assert ret == Trace(HadamardProduct(M.T, N.T)) + + expr = Trace(A*HadamardProduct(M, N)) + cg = convert_matrix_to_array(expr) + ret = convert_array_to_matrix(cg) + assert ret == Trace(HadamardProduct(M, N)*A) + + expr = Trace(HadamardProduct(A, M)*N) + cg = convert_matrix_to_array(expr) + ret = convert_array_to_matrix(cg) + assert ret == Trace(HadamardProduct(M.T, N)*A) + + # These should not be converted into Hadamard products: + + cg = _array_diagonal(_array_tensor_product(M, N), (0, 1, 2, 3)) + ret = convert_array_to_matrix(cg) + assert ret == cg + + cg = _array_diagonal(_array_tensor_product(A), (0, 1)) + ret = convert_array_to_matrix(cg) + assert ret == cg + + cg = _array_diagonal(_array_tensor_product(M, N, P), (0, 2, 4), (1, 3, 5)) + assert convert_array_to_matrix(cg) == HadamardProduct(M, N, P) + + cg = _array_diagonal(_array_tensor_product(M, N, P), (0, 3, 4), (1, 2, 5)) + assert convert_array_to_matrix(cg) == HadamardProduct(M, P, N.T) + + cg = _array_diagonal(_array_tensor_product(I, I1, x), (1, 4), (3, 5)) + assert convert_array_to_matrix(cg) == DiagMatrix(x) + + +def test_identify_removable_identity_matrices(): + + D = DiagonalMatrix(MatrixSymbol("D", k, k)) + + cg = _array_contraction(_array_tensor_product(A, B, I), (1, 2, 4, 5)) + expected = _array_contraction(_array_tensor_product(A, B), (1, 2)) + assert identify_removable_identity_matrices(cg) == expected + + cg = _array_contraction(_array_tensor_product(A, B, C, I), (1, 3, 5, 6, 7)) + expected = _array_contraction(_array_tensor_product(A, B, C), (1, 3, 5)) + assert identify_removable_identity_matrices(cg) == expected + + # Tests with diagonal matrices: + + cg = _array_contraction(_array_tensor_product(A, B, D), (1, 2, 4, 5)) + ret = identify_removable_identity_matrices(cg) + expected = _array_contraction(_array_tensor_product(A, B, D), (1, 4), (2, 5)) + assert ret == expected + + cg = _array_contraction(_array_tensor_product(A, B, D, M, N), (1, 2, 4, 5, 6, 8)) + ret = identify_removable_identity_matrices(cg) + assert ret == cg + + +def test_combine_removed(): + + assert _combine_removed(6, [0, 1, 2], [0, 1, 2]) == [0, 1, 2, 3, 4, 5] + assert _combine_removed(8, [2, 5], [1, 3, 4]) == [1, 2, 4, 5, 6] + assert _combine_removed(8, [7], []) == [7] + + +def test_array_contraction_to_diagonal_multiple_identities(): + + expr = _array_contraction(_array_tensor_product(A, B, I, C), (1, 2, 4), (5, 6)) + assert _array_contraction_to_diagonal_multiple_identity(expr) == (expr, []) + assert convert_array_to_matrix(expr) == _array_contraction(_array_tensor_product(A, B, C), (1, 2, 4)) + + expr = _array_contraction(_array_tensor_product(A, I, I), (1, 2, 4)) + assert _array_contraction_to_diagonal_multiple_identity(expr) == (A, [2]) + assert convert_array_to_matrix(expr) == A + + expr = _array_contraction(_array_tensor_product(A, I, I, B), (1, 2, 4), (3, 6)) + assert _array_contraction_to_diagonal_multiple_identity(expr) == (expr, []) + + expr = _array_contraction(_array_tensor_product(A, I, I, B), (1, 2, 3, 4, 6)) + assert _array_contraction_to_diagonal_multiple_identity(expr) == (expr, []) + + +def test_convert_array_element_to_matrix(): + + expr = ArrayElement(M, (i, j)) + assert convert_array_to_matrix(expr) == MatrixElement(M, i, j) + + expr = ArrayElement(_array_contraction(_array_tensor_product(M, N), (1, 3)), (i, j)) + assert convert_array_to_matrix(expr) == MatrixElement(M*N.T, i, j) + + expr = ArrayElement(_array_tensor_product(M, N), (i, j, m, n)) + assert convert_array_to_matrix(expr) == expr + + +def test_convert_array_elementwise_function_to_matrix(): + + d = Dummy("d") + + expr = ArrayElementwiseApplyFunc(Lambda(d, sin(d)), x.T*y) + assert convert_array_to_matrix(expr) == sin(x.T*y) + + expr = ArrayElementwiseApplyFunc(Lambda(d, d**2), x.T*y) + assert convert_array_to_matrix(expr) == (x.T*y)**2 + + expr = ArrayElementwiseApplyFunc(Lambda(d, sin(d)), x) + assert convert_array_to_matrix(expr).dummy_eq(x.applyfunc(sin)) + + expr = ArrayElementwiseApplyFunc(Lambda(d, 1 / (2 * sqrt(d))), x) + assert convert_array_to_matrix(expr) == S.Half * HadamardPower(x, -S.Half) + + +def test_array2matrix(): + # See issue https://github.com/sympy/sympy/pull/22877 + expr = PermuteDims(ArrayContraction(ArrayTensorProduct(x, I, I1, x), (0, 3), (1, 7)), Permutation(2, 3)) + expected = PermuteDims(ArrayTensorProduct(x*x.T, I1), Permutation(3)(1, 2)) + assert _array2matrix(expr) == expected + + +def test_recognize_broadcasting(): + expr = ArrayTensorProduct(x.T*x, A) + assert _remove_trivial_dims(expr) == (KroneckerProduct(x.T*x, A), [0, 1]) + + expr = ArrayTensorProduct(A, x.T*x) + assert _remove_trivial_dims(expr) == (KroneckerProduct(A, x.T*x), [2, 3]) + + expr = ArrayTensorProduct(A, B, x.T*x, C) + assert _remove_trivial_dims(expr) == (ArrayTensorProduct(A, KroneckerProduct(B, x.T*x), C), [4, 5]) + + # Always prefer matrix multiplication to Kronecker product, if possible: + expr = ArrayTensorProduct(a, b, x.T*x) + assert _remove_trivial_dims(expr) == (a*x.T*x*b.T, [1, 3, 4, 5]) diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/tests/test_convert_indexed_to_array.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/tests/test_convert_indexed_to_array.py new file mode 100644 index 0000000000000000000000000000000000000000..258062eadeca041ae3c864dabeefd5165f1cef11 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/tests/test_convert_indexed_to_array.py @@ -0,0 +1,205 @@ +from sympy import tanh +from sympy.concrete.summations import Sum +from sympy.core.symbol import symbols +from sympy.functions.special.tensor_functions import KroneckerDelta +from sympy.matrices.expressions.matexpr import MatrixSymbol +from sympy.matrices.expressions.special import Identity +from sympy.tensor.array.expressions import ArrayElementwiseApplyFunc +from sympy.tensor.indexed import IndexedBase +from sympy.combinatorics import Permutation +from sympy.tensor.array.expressions.array_expressions import ArrayContraction, ArrayTensorProduct, \ + ArrayDiagonal, ArrayAdd, PermuteDims, ArrayElement, _array_tensor_product, _array_contraction, _array_diagonal, \ + _array_add, _permute_dims, ArraySymbol, OneArray +from sympy.tensor.array.expressions.from_array_to_matrix import convert_array_to_matrix +from sympy.tensor.array.expressions.from_indexed_to_array import convert_indexed_to_array, _convert_indexed_to_array +from sympy.testing.pytest import raises + + +A, B = symbols("A B", cls=IndexedBase) +i, j, k, l, m, n = symbols("i j k l m n") +d0, d1, d2, d3 = symbols("d0:4") + +I = Identity(k) + +M = MatrixSymbol("M", k, k) +N = MatrixSymbol("N", k, k) +P = MatrixSymbol("P", k, k) +Q = MatrixSymbol("Q", k, k) + +a = MatrixSymbol("a", k, 1) +b = MatrixSymbol("b", k, 1) +c = MatrixSymbol("c", k, 1) +d = MatrixSymbol("d", k, 1) + + +def test_arrayexpr_convert_index_to_array_support_function(): + expr = M[i, j] + assert _convert_indexed_to_array(expr) == (M, (i, j)) + expr = M[i, j]*N[k, l] + assert _convert_indexed_to_array(expr) == (ArrayTensorProduct(M, N), (i, j, k, l)) + expr = M[i, j]*N[j, k] + assert _convert_indexed_to_array(expr) == (ArrayDiagonal(ArrayTensorProduct(M, N), (1, 2)), (i, k, j)) + expr = Sum(M[i, j]*N[j, k], (j, 0, k-1)) + assert _convert_indexed_to_array(expr) == (ArrayContraction(ArrayTensorProduct(M, N), (1, 2)), (i, k)) + expr = M[i, j] + N[i, j] + assert _convert_indexed_to_array(expr) == (ArrayAdd(M, N), (i, j)) + expr = M[i, j] + N[j, i] + assert _convert_indexed_to_array(expr) == (ArrayAdd(M, PermuteDims(N, Permutation([1, 0]))), (i, j)) + expr = M[i, j] + M[j, i] + assert _convert_indexed_to_array(expr) == (ArrayAdd(M, PermuteDims(M, Permutation([1, 0]))), (i, j)) + expr = (M*N*P)[i, j] + assert _convert_indexed_to_array(expr) == (_array_contraction(ArrayTensorProduct(M, N, P), (1, 2), (3, 4)), (i, j)) + expr = expr.function # Disregard summation in previous expression + ret1, ret2 = _convert_indexed_to_array(expr) + assert ret1 == ArrayDiagonal(ArrayTensorProduct(M, N, P), (1, 2), (3, 4)) + assert str(ret2) == "(i, j, _i_1, _i_2)" + expr = KroneckerDelta(i, j)*M[i, k] + assert _convert_indexed_to_array(expr) == (M, ({i, j}, k)) + expr = KroneckerDelta(i, j)*KroneckerDelta(j, k)*M[i, l] + assert _convert_indexed_to_array(expr) == (M, ({i, j, k}, l)) + expr = KroneckerDelta(j, k)*(M[i, j]*N[k, l] + N[i, j]*M[k, l]) + assert _convert_indexed_to_array(expr) == (_array_diagonal(_array_add( + ArrayTensorProduct(M, N), + _permute_dims(ArrayTensorProduct(M, N), Permutation(0, 2)(1, 3)) + ), (1, 2)), (i, l, frozenset({j, k}))) + expr = KroneckerDelta(j, m)*KroneckerDelta(m, k)*(M[i, j]*N[k, l] + N[i, j]*M[k, l]) + assert _convert_indexed_to_array(expr) == (_array_diagonal(_array_add( + ArrayTensorProduct(M, N), + _permute_dims(ArrayTensorProduct(M, N), Permutation(0, 2)(1, 3)) + ), (1, 2)), (i, l, frozenset({j, m, k}))) + expr = KroneckerDelta(i, j)*KroneckerDelta(j, k)*KroneckerDelta(k,m)*M[i, 0]*KroneckerDelta(m, n) + assert _convert_indexed_to_array(expr) == (M, ({i, j, k, m, n}, 0)) + expr = M[i, i] + assert _convert_indexed_to_array(expr) == (ArrayDiagonal(M, (0, 1)), (i,)) + + +def test_arrayexpr_convert_indexed_to_array_expression(): + + s = Sum(A[i]*B[i], (i, 0, 3)) + cg = convert_indexed_to_array(s) + assert cg == ArrayContraction(ArrayTensorProduct(A, B), (0, 1)) + + expr = M*N + result = ArrayContraction(ArrayTensorProduct(M, N), (1, 2)) + elem = expr[i, j] + assert convert_indexed_to_array(elem) == result + + expr = M*N*M + elem = expr[i, j] + result = _array_contraction(_array_tensor_product(M, M, N), (1, 4), (2, 5)) + cg = convert_indexed_to_array(elem) + assert cg == result + + cg = convert_indexed_to_array((M * N * P)[i, j]) + assert cg == _array_contraction(ArrayTensorProduct(M, N, P), (1, 2), (3, 4)) + + cg = convert_indexed_to_array((M * N.T * P)[i, j]) + assert cg == _array_contraction(ArrayTensorProduct(M, N, P), (1, 3), (2, 4)) + + expr = -2*M*N + elem = expr[i, j] + cg = convert_indexed_to_array(elem) + assert cg == ArrayContraction(ArrayTensorProduct(-2, M, N), (1, 2)) + + +def test_arrayexpr_convert_array_element_to_array_expression(): + A = ArraySymbol("A", (k,)) + B = ArraySymbol("B", (k,)) + + s = Sum(A[i]*B[i], (i, 0, k-1)) + cg = convert_indexed_to_array(s) + assert cg == ArrayContraction(ArrayTensorProduct(A, B), (0, 1)) + + s = A[i]*B[i] + cg = convert_indexed_to_array(s) + assert cg == ArrayDiagonal(ArrayTensorProduct(A, B), (0, 1)) + + s = A[i]*B[j] + cg = convert_indexed_to_array(s, [i, j]) + assert cg == ArrayTensorProduct(A, B) + cg = convert_indexed_to_array(s, [j, i]) + assert cg == ArrayTensorProduct(B, A) + + s = tanh(A[i]*B[j]) + cg = convert_indexed_to_array(s, [i, j]) + assert cg.dummy_eq(ArrayElementwiseApplyFunc(tanh, ArrayTensorProduct(A, B))) + + +def test_arrayexpr_convert_indexed_to_array_and_back_to_matrix(): + + expr = a.T*b + elem = expr[0, 0] + cg = convert_indexed_to_array(elem) + assert cg == ArrayElement(ArrayContraction(ArrayTensorProduct(a, b), (0, 2)), [0, 0]) + + expr = M[i,j] + N[i,j] + p1, p2 = _convert_indexed_to_array(expr) + assert convert_array_to_matrix(p1) == M + N + + expr = M[i,j] + N[j,i] + p1, p2 = _convert_indexed_to_array(expr) + assert convert_array_to_matrix(p1) == M + N.T + + expr = M[i,j]*N[k,l] + N[i,j]*M[k,l] + p1, p2 = _convert_indexed_to_array(expr) + assert convert_array_to_matrix(p1) == ArrayAdd( + ArrayTensorProduct(M, N), + ArrayTensorProduct(N, M)) + + expr = (M*N*P)[i, j] + p1, p2 = _convert_indexed_to_array(expr) + assert convert_array_to_matrix(p1) == M * N * P + + expr = Sum(M[i,j]*(N*P)[j,m], (j, 0, k-1)) + p1, p2 = _convert_indexed_to_array(expr) + assert convert_array_to_matrix(p1) == M * N * P + + expr = Sum((P[j, m] + P[m, j])*(M[i,j]*N[m,n] + N[i,j]*M[m,n]), (j, 0, k-1), (m, 0, k-1)) + p1, p2 = _convert_indexed_to_array(expr) + assert convert_array_to_matrix(p1) == M * P * N + M * P.T * N + N * P * M + N * P.T * M + + +def test_arrayexpr_convert_indexed_to_array_out_of_bounds(): + + expr = Sum(M[i, i], (i, 0, 4)) + raises(ValueError, lambda: convert_indexed_to_array(expr)) + expr = Sum(M[i, i], (i, 0, k)) + raises(ValueError, lambda: convert_indexed_to_array(expr)) + expr = Sum(M[i, i], (i, 1, k-1)) + raises(ValueError, lambda: convert_indexed_to_array(expr)) + + expr = Sum(M[i, j]*N[j,m], (j, 0, 4)) + raises(ValueError, lambda: convert_indexed_to_array(expr)) + expr = Sum(M[i, j]*N[j,m], (j, 0, k)) + raises(ValueError, lambda: convert_indexed_to_array(expr)) + expr = Sum(M[i, j]*N[j,m], (j, 1, k-1)) + raises(ValueError, lambda: convert_indexed_to_array(expr)) + + +def test_arrayexpr_convert_indexed_to_array_broadcast(): + A = ArraySymbol("A", (3, 3)) + B = ArraySymbol("B", (3, 3)) + + expr = A[i, j] + B[k, l] + O2 = OneArray(3, 3) + expected = ArrayAdd(ArrayTensorProduct(A, O2), ArrayTensorProduct(O2, B)) + assert convert_indexed_to_array(expr) == expected + assert convert_indexed_to_array(expr, [i, j, k, l]) == expected + assert convert_indexed_to_array(expr, [l, k, i, j]) == ArrayAdd(PermuteDims(ArrayTensorProduct(O2, A), [1, 0, 2, 3]), PermuteDims(ArrayTensorProduct(B, O2), [1, 0, 2, 3])) + + expr = A[i, j] + B[j, k] + O1 = OneArray(3) + assert convert_indexed_to_array(expr, [i, j, k]) == ArrayAdd(ArrayTensorProduct(A, O1), ArrayTensorProduct(O1, B)) + + C = ArraySymbol("C", (d0, d1)) + D = ArraySymbol("D", (d3, d1)) + + expr = C[i, j] + D[k, j] + assert convert_indexed_to_array(expr, [i, j, k]) == ArrayAdd(ArrayTensorProduct(C, OneArray(d3)), PermuteDims(ArrayTensorProduct(OneArray(d0), D), [0, 2, 1])) + + X = ArraySymbol("X", (5, 3)) + + expr = X[i, n] - X[j, n] + assert convert_indexed_to_array(expr, [i, j, n]) == ArrayAdd(ArrayTensorProduct(-1, OneArray(5), X), PermuteDims(ArrayTensorProduct(X, OneArray(5)), [0, 2, 1])) + + raises(ValueError, lambda: convert_indexed_to_array(C[i, j] + D[i, j])) diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/tests/test_convert_matrix_to_array.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/tests/test_convert_matrix_to_array.py new file mode 100644 index 0000000000000000000000000000000000000000..142585882588df6aa0e4648d9d8881ea755f42a0 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/tests/test_convert_matrix_to_array.py @@ -0,0 +1,128 @@ +from sympy import Lambda, KroneckerProduct +from sympy.core.symbol import symbols, Dummy +from sympy.matrices.expressions.hadamard import (HadamardPower, HadamardProduct) +from sympy.matrices.expressions.inverse import Inverse +from sympy.matrices.expressions.matexpr import MatrixSymbol +from sympy.matrices.expressions.matpow import MatPow +from sympy.matrices.expressions.special import Identity +from sympy.matrices.expressions.trace import Trace +from sympy.matrices.expressions.transpose import Transpose +from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct, ArrayContraction, \ + PermuteDims, ArrayDiagonal, ArrayElementwiseApplyFunc, _array_contraction, _array_tensor_product, Reshape +from sympy.tensor.array.expressions.from_array_to_matrix import convert_array_to_matrix +from sympy.tensor.array.expressions.from_matrix_to_array import convert_matrix_to_array + +i, j, k, l, m, n = symbols("i j k l m n") + +I = Identity(k) + +M = MatrixSymbol("M", k, k) +N = MatrixSymbol("N", k, k) +P = MatrixSymbol("P", k, k) +Q = MatrixSymbol("Q", k, k) + +A = MatrixSymbol("A", k, k) +B = MatrixSymbol("B", k, k) +C = MatrixSymbol("C", k, k) +D = MatrixSymbol("D", k, k) + +X = MatrixSymbol("X", k, k) +Y = MatrixSymbol("Y", k, k) + +a = MatrixSymbol("a", k, 1) +b = MatrixSymbol("b", k, 1) +c = MatrixSymbol("c", k, 1) +d = MatrixSymbol("d", k, 1) + + +def test_arrayexpr_convert_matrix_to_array(): + + expr = M*N + result = ArrayContraction(ArrayTensorProduct(M, N), (1, 2)) + assert convert_matrix_to_array(expr) == result + + expr = M*N*M + result = _array_contraction(ArrayTensorProduct(M, N, M), (1, 2), (3, 4)) + assert convert_matrix_to_array(expr) == result + + expr = Transpose(M) + assert convert_matrix_to_array(expr) == PermuteDims(M, [1, 0]) + + expr = M*Transpose(N) + assert convert_matrix_to_array(expr) == _array_contraction(_array_tensor_product(M, PermuteDims(N, [1, 0])), (1, 2)) + + expr = 3*M*N + res = convert_matrix_to_array(expr) + rexpr = convert_array_to_matrix(res) + assert expr == rexpr + + expr = 3*M + N*M.T*M + 4*k*N + res = convert_matrix_to_array(expr) + rexpr = convert_array_to_matrix(res) + assert expr == rexpr + + expr = Inverse(M)*N + rexpr = convert_array_to_matrix(convert_matrix_to_array(expr)) + assert expr == rexpr + + expr = M**2 + rexpr = convert_array_to_matrix(convert_matrix_to_array(expr)) + assert expr == rexpr + + expr = M*(2*N + 3*M) + res = convert_matrix_to_array(expr) + rexpr = convert_array_to_matrix(res) + assert expr == rexpr + + expr = Trace(M) + result = ArrayContraction(M, (0, 1)) + assert convert_matrix_to_array(expr) == result + + expr = 3*Trace(M) + result = ArrayContraction(ArrayTensorProduct(3, M), (0, 1)) + assert convert_matrix_to_array(expr) == result + + expr = 3*Trace(Trace(M) * M) + result = ArrayContraction(ArrayTensorProduct(3, M, M), (0, 1), (2, 3)) + assert convert_matrix_to_array(expr) == result + + expr = 3*Trace(M)**2 + result = ArrayContraction(ArrayTensorProduct(3, M, M), (0, 1), (2, 3)) + assert convert_matrix_to_array(expr) == result + + expr = HadamardProduct(M, N) + result = ArrayDiagonal(ArrayTensorProduct(M, N), (0, 2), (1, 3)) + assert convert_matrix_to_array(expr) == result + + expr = HadamardProduct(M*N, N*M) + result = ArrayDiagonal(ArrayContraction(ArrayTensorProduct(M, N, N, M), (1, 2), (5, 6)), (0, 2), (1, 3)) + assert convert_matrix_to_array(expr) == result + + expr = HadamardPower(M, 2) + result = ArrayDiagonal(ArrayTensorProduct(M, M), (0, 2), (1, 3)) + assert convert_matrix_to_array(expr) == result + + expr = HadamardPower(M*N, 2) + result = ArrayDiagonal(ArrayContraction(ArrayTensorProduct(M, N, M, N), (1, 2), (5, 6)), (0, 2), (1, 3)) + assert convert_matrix_to_array(expr) == result + + expr = HadamardPower(M, n) + d0 = Dummy("d0") + result = ArrayElementwiseApplyFunc(Lambda(d0, d0**n), M) + assert convert_matrix_to_array(expr).dummy_eq(result) + + expr = M**2 + assert isinstance(expr, MatPow) + assert convert_matrix_to_array(expr) == ArrayContraction(ArrayTensorProduct(M, M), (1, 2)) + + expr = a.T*b + cg = convert_matrix_to_array(expr) + assert cg == ArrayContraction(ArrayTensorProduct(a, b), (0, 2)) + + expr = KroneckerProduct(A, B) + cg = convert_matrix_to_array(expr) + assert cg == Reshape(PermuteDims(ArrayTensorProduct(A, B), [0, 2, 1, 3]), (k**2, k**2)) + + expr = KroneckerProduct(A, B, C, D) + cg = convert_matrix_to_array(expr) + assert cg == Reshape(PermuteDims(ArrayTensorProduct(A, B, C, D), [0, 2, 4, 6, 1, 3, 5, 7]), (k**4, k**4)) diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/tests/test_deprecated_conv_modules.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/tests/test_deprecated_conv_modules.py new file mode 100644 index 0000000000000000000000000000000000000000..b41b6105410a308e7774fce760b235497d0303bb --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/tests/test_deprecated_conv_modules.py @@ -0,0 +1,22 @@ +from sympy import MatrixSymbol, symbols, Sum +from sympy.tensor.array.expressions import conv_array_to_indexed, from_array_to_indexed, ArrayTensorProduct, \ + ArrayContraction, conv_array_to_matrix, from_array_to_matrix, conv_matrix_to_array, from_matrix_to_array, \ + conv_indexed_to_array, from_indexed_to_array +from sympy.testing.pytest import warns +from sympy.utilities.exceptions import SymPyDeprecationWarning + + +def test_deprecated_conv_module_results(): + + M = MatrixSymbol("M", 3, 3) + N = MatrixSymbol("N", 3, 3) + i, j, d = symbols("i j d") + + x = ArrayContraction(ArrayTensorProduct(M, N), (1, 2)) + y = Sum(M[i, d]*N[d, j], (d, 0, 2)) + + with warns(SymPyDeprecationWarning, test_stacklevel=False): + assert conv_array_to_indexed.convert_array_to_indexed(x, [i, j]).dummy_eq(from_array_to_indexed.convert_array_to_indexed(x, [i, j])) + assert conv_array_to_matrix.convert_array_to_matrix(x) == from_array_to_matrix.convert_array_to_matrix(x) + assert conv_matrix_to_array.convert_matrix_to_array(M*N) == from_matrix_to_array.convert_matrix_to_array(M*N) + assert conv_indexed_to_array.convert_indexed_to_array(y) == from_indexed_to_array.convert_indexed_to_array(y) diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/utils.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/utils.py new file mode 100644 index 0000000000000000000000000000000000000000..e55c0e6ed47cdc9ff1c24cc92f006998aeb86822 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/expressions/utils.py @@ -0,0 +1,123 @@ +import bisect +from collections import defaultdict + +from sympy.combinatorics import Permutation +from sympy.core.containers import Tuple +from sympy.core.numbers import Integer + + +def _get_mapping_from_subranks(subranks): + mapping = {} + counter = 0 + for i, rank in enumerate(subranks): + for j in range(rank): + mapping[counter] = (i, j) + counter += 1 + return mapping + + +def _get_contraction_links(args, subranks, *contraction_indices): + mapping = _get_mapping_from_subranks(subranks) + contraction_tuples = [[mapping[j] for j in i] for i in contraction_indices] + dlinks = defaultdict(dict) + for links in contraction_tuples: + if len(links) == 2: + (arg1, pos1), (arg2, pos2) = links + dlinks[arg1][pos1] = (arg2, pos2) + dlinks[arg2][pos2] = (arg1, pos1) + continue + + return args, dict(dlinks) + + +def _sort_contraction_indices(pairing_indices): + pairing_indices = [Tuple(*sorted(i)) for i in pairing_indices] + pairing_indices.sort(key=lambda x: min(x)) + return pairing_indices + + +def _get_diagonal_indices(flattened_indices): + axes_contraction = defaultdict(list) + for i, ind in enumerate(flattened_indices): + if isinstance(ind, (int, Integer)): + # If the indices is a number, there can be no diagonal operation: + continue + axes_contraction[ind].append(i) + axes_contraction = {k: v for k, v in axes_contraction.items() if len(v) > 1} + # Put the diagonalized indices at the end: + ret_indices = [i for i in flattened_indices if i not in axes_contraction] + diag_indices = list(axes_contraction) + diag_indices.sort(key=lambda x: flattened_indices.index(x)) + diagonal_indices = [tuple(axes_contraction[i]) for i in diag_indices] + ret_indices += diag_indices + ret_indices = tuple(ret_indices) + return diagonal_indices, ret_indices + + +def _get_argindex(subindices, ind): + for i, sind in enumerate(subindices): + if ind == sind: + return i + if isinstance(sind, (set, frozenset)) and ind in sind: + return i + raise IndexError("%s not found in %s" % (ind, subindices)) + + +def _apply_recursively_over_nested_lists(func, arr): + if isinstance(arr, (tuple, list, Tuple)): + return tuple(_apply_recursively_over_nested_lists(func, i) for i in arr) + elif isinstance(arr, Tuple): + return Tuple.fromiter(_apply_recursively_over_nested_lists(func, i) for i in arr) + else: + return func(arr) + + +def _build_push_indices_up_func_transformation(flattened_contraction_indices): + shifts = {0: 0} + i = 0 + cumulative = 0 + while i < len(flattened_contraction_indices): + j = 1 + while i+j < len(flattened_contraction_indices): + if flattened_contraction_indices[i] + j != flattened_contraction_indices[i+j]: + break + j += 1 + cumulative += j + shifts[flattened_contraction_indices[i]] = cumulative + i += j + shift_keys = sorted(shifts.keys()) + + def func(idx): + return shifts[shift_keys[bisect.bisect_right(shift_keys, idx)-1]] + + def transform(j): + if j in flattened_contraction_indices: + return None + else: + return j - func(j) + + return transform + + +def _build_push_indices_down_func_transformation(flattened_contraction_indices): + N = flattened_contraction_indices[-1]+2 + + shifts = [i for i in range(N) if i not in flattened_contraction_indices] + + def transform(j): + if j < len(shifts): + return shifts[j] + else: + return j + shifts[-1] - len(shifts) + 1 + + return transform + + +def _apply_permutation_to_list(perm: Permutation, target_list: list): + """ + Permute a list according to the given permutation. + """ + new_list = [None for i in range(perm.size)] + for i, e in enumerate(target_list): + new_list[perm(i)] = e + return new_list diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/array/mutable_ndim_array.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/mutable_ndim_array.py new file mode 100644 index 0000000000000000000000000000000000000000..e1eaaf7241bc3b4a48234178d18da3aa5736e189 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/mutable_ndim_array.py @@ -0,0 +1,13 @@ +from sympy.tensor.array.ndim_array import NDimArray + + +class MutableNDimArray(NDimArray): + + def as_immutable(self): + raise NotImplementedError("abstract method") + + def as_mutable(self): + return self + + def _sympy_(self): + return self.as_immutable() diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/array/ndim_array.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/ndim_array.py new file mode 100644 index 0000000000000000000000000000000000000000..2b9a857b8cfd9ee46646c46f274636d6b9962b6e --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/ndim_array.py @@ -0,0 +1,601 @@ +from sympy.core.basic import Basic +from sympy.core.containers import (Dict, Tuple) +from sympy.core.expr import Expr +from sympy.core.kind import Kind, NumberKind, UndefinedKind +from sympy.core.numbers import Integer +from sympy.core.singleton import S +from sympy.core.sympify import sympify +from sympy.external.gmpy import SYMPY_INTS +from sympy.printing.defaults import Printable + +import itertools +from collections.abc import Iterable + + +class ArrayKind(Kind): + """ + Kind for N-dimensional array in SymPy. + + This kind represents the multidimensional array that algebraic + operations are defined. Basic class for this kind is ``NDimArray``, + but any expression representing the array can have this. + + Parameters + ========== + + element_kind : Kind + Kind of the element. Default is :obj:NumberKind ``, + which means that the array contains only numbers. + + Examples + ======== + + Any instance of array class has ``ArrayKind``. + + >>> from sympy import NDimArray + >>> NDimArray([1,2,3]).kind + ArrayKind(NumberKind) + + Although expressions representing an array may be not instance of + array class, it will have ``ArrayKind`` as well. + + >>> from sympy import Integral + >>> from sympy.tensor.array import NDimArray + >>> from sympy.abc import x + >>> intA = Integral(NDimArray([1,2,3]), x) + >>> isinstance(intA, NDimArray) + False + >>> intA.kind + ArrayKind(NumberKind) + + Use ``isinstance()`` to check for ``ArrayKind` without specifying + the element kind. Use ``is`` with specifying the element kind. + + >>> from sympy.tensor.array import ArrayKind + >>> from sympy.core import NumberKind + >>> boolA = NDimArray([True, False]) + >>> isinstance(boolA.kind, ArrayKind) + True + >>> boolA.kind is ArrayKind(NumberKind) + False + + See Also + ======== + + shape : Function to return the shape of objects with ``MatrixKind``. + + """ + def __new__(cls, element_kind=NumberKind): + obj = super().__new__(cls, element_kind) + obj.element_kind = element_kind + return obj + + def __repr__(self): + return "ArrayKind(%s)" % self.element_kind + + @classmethod + def _union(cls, kinds) -> 'ArrayKind': + elem_kinds = {e.kind for e in kinds} + if len(elem_kinds) == 1: + elemkind, = elem_kinds + else: + elemkind = UndefinedKind + return ArrayKind(elemkind) + + +class NDimArray(Printable): + """N-dimensional array. + + Examples + ======== + + Create an N-dim array of zeros: + + >>> from sympy import MutableDenseNDimArray + >>> a = MutableDenseNDimArray.zeros(2, 3, 4) + >>> a + [[[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]], [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]] + + Create an N-dim array from a list; + + >>> a = MutableDenseNDimArray([[2, 3], [4, 5]]) + >>> a + [[2, 3], [4, 5]] + + >>> b = MutableDenseNDimArray([[[1, 2], [3, 4], [5, 6]], [[7, 8], [9, 10], [11, 12]]]) + >>> b + [[[1, 2], [3, 4], [5, 6]], [[7, 8], [9, 10], [11, 12]]] + + Create an N-dim array from a flat list with dimension shape: + + >>> a = MutableDenseNDimArray([1, 2, 3, 4, 5, 6], (2, 3)) + >>> a + [[1, 2, 3], [4, 5, 6]] + + Create an N-dim array from a matrix: + + >>> from sympy import Matrix + >>> a = Matrix([[1,2],[3,4]]) + >>> a + Matrix([ + [1, 2], + [3, 4]]) + >>> b = MutableDenseNDimArray(a) + >>> b + [[1, 2], [3, 4]] + + Arithmetic operations on N-dim arrays + + >>> a = MutableDenseNDimArray([1, 1, 1, 1], (2, 2)) + >>> b = MutableDenseNDimArray([4, 4, 4, 4], (2, 2)) + >>> c = a + b + >>> c + [[5, 5], [5, 5]] + >>> a - b + [[-3, -3], [-3, -3]] + + """ + + _diff_wrt = True + is_scalar = False + + def __new__(cls, iterable, shape=None, **kwargs): + from sympy.tensor.array import ImmutableDenseNDimArray + return ImmutableDenseNDimArray(iterable, shape, **kwargs) + + def __getitem__(self, index): + raise NotImplementedError("A subclass of NDimArray should implement __getitem__") + + def _parse_index(self, index): + if isinstance(index, (SYMPY_INTS, Integer)): + if index >= self._loop_size: + raise ValueError("Only a tuple index is accepted") + return index + + if self._loop_size == 0: + raise ValueError("Index not valid with an empty array") + + if len(index) != self._rank: + raise ValueError('Wrong number of array axes') + + real_index = 0 + # check if input index can exist in current indexing + for i in range(self._rank): + if (index[i] >= self.shape[i]) or (index[i] < -self.shape[i]): + raise ValueError('Index ' + str(index) + ' out of border') + if index[i] < 0: + real_index += 1 + real_index = real_index*self.shape[i] + index[i] + + return real_index + + def _get_tuple_index(self, integer_index): + index = [] + for sh in reversed(self.shape): + index.append(integer_index % sh) + integer_index //= sh + index.reverse() + return tuple(index) + + def _check_symbolic_index(self, index): + # Check if any index is symbolic: + tuple_index = (index if isinstance(index, tuple) else (index,)) + if any((isinstance(i, Expr) and (not i.is_number)) for i in tuple_index): + for i, nth_dim in zip(tuple_index, self.shape): + if ((i < 0) == True) or ((i >= nth_dim) == True): + raise ValueError("index out of range") + from sympy.tensor import Indexed + return Indexed(self, *tuple_index) + return None + + def _setter_iterable_check(self, value): + from sympy.matrices.matrixbase import MatrixBase + if isinstance(value, (Iterable, MatrixBase, NDimArray)): + raise NotImplementedError + + @classmethod + def _scan_iterable_shape(cls, iterable): + def f(pointer): + if not isinstance(pointer, Iterable): + return [pointer], () + + if len(pointer) == 0: + return [], (0,) + + result = [] + elems, shapes = zip(*[f(i) for i in pointer]) + if len(set(shapes)) != 1: + raise ValueError("could not determine shape unambiguously") + for i in elems: + result.extend(i) + return result, (len(shapes),)+shapes[0] + + return f(iterable) + + @classmethod + def _handle_ndarray_creation_inputs(cls, iterable=None, shape=None, **kwargs): + from sympy.matrices.matrixbase import MatrixBase + from sympy.tensor.array import SparseNDimArray + + if shape is None: + if iterable is None: + shape = () + iterable = () + # Construction of a sparse array from a sparse array + elif isinstance(iterable, SparseNDimArray): + return iterable._shape, iterable._sparse_array + + # Construct N-dim array from another N-dim array: + elif isinstance(iterable, NDimArray): + shape = iterable.shape + + # Construct N-dim array from an iterable (numpy arrays included): + elif isinstance(iterable, Iterable): + iterable, shape = cls._scan_iterable_shape(iterable) + + # Construct N-dim array from a Matrix: + elif isinstance(iterable, MatrixBase): + shape = iterable.shape + + else: + shape = () + iterable = (iterable,) + + if isinstance(iterable, (Dict, dict)) and shape is not None: + new_dict = iterable.copy() + for k in new_dict: + if isinstance(k, (tuple, Tuple)): + new_key = 0 + for i, idx in enumerate(k): + new_key = new_key * shape[i] + idx + iterable[new_key] = iterable[k] + del iterable[k] + + if isinstance(shape, (SYMPY_INTS, Integer)): + shape = (shape,) + + if not all(isinstance(dim, (SYMPY_INTS, Integer)) for dim in shape): + raise TypeError("Shape should contain integers only.") + + return tuple(shape), iterable + + def __len__(self): + """Overload common function len(). Returns number of elements in array. + + Examples + ======== + + >>> from sympy import MutableDenseNDimArray + >>> a = MutableDenseNDimArray.zeros(3, 3) + >>> a + [[0, 0, 0], [0, 0, 0], [0, 0, 0]] + >>> len(a) + 9 + + """ + return self._loop_size + + @property + def shape(self): + """ + Returns array shape (dimension). + + Examples + ======== + + >>> from sympy import MutableDenseNDimArray + >>> a = MutableDenseNDimArray.zeros(3, 3) + >>> a.shape + (3, 3) + + """ + return self._shape + + def rank(self): + """ + Returns rank of array. + + Examples + ======== + + >>> from sympy import MutableDenseNDimArray + >>> a = MutableDenseNDimArray.zeros(3,4,5,6,3) + >>> a.rank() + 5 + + """ + return self._rank + + def diff(self, *args, **kwargs): + """ + Calculate the derivative of each element in the array. + + Examples + ======== + + >>> from sympy import ImmutableDenseNDimArray + >>> from sympy.abc import x, y + >>> M = ImmutableDenseNDimArray([[x, y], [1, x*y]]) + >>> M.diff(x) + [[1, 0], [0, y]] + + """ + from sympy.tensor.array.array_derivatives import ArrayDerivative + kwargs.setdefault('evaluate', True) + return ArrayDerivative(self.as_immutable(), *args, **kwargs) + + def _eval_derivative(self, base): + # Types are (base: scalar, self: array) + return self.applyfunc(lambda x: base.diff(x)) + + def _eval_derivative_n_times(self, s, n): + return Basic._eval_derivative_n_times(self, s, n) + + def applyfunc(self, f): + """Apply a function to each element of the N-dim array. + + Examples + ======== + + >>> from sympy import ImmutableDenseNDimArray + >>> m = ImmutableDenseNDimArray([i*2+j for i in range(2) for j in range(2)], (2, 2)) + >>> m + [[0, 1], [2, 3]] + >>> m.applyfunc(lambda i: 2*i) + [[0, 2], [4, 6]] + """ + from sympy.tensor.array import SparseNDimArray + from sympy.tensor.array.arrayop import Flatten + + if isinstance(self, SparseNDimArray) and f(S.Zero) == 0: + return type(self)({k: f(v) for k, v in self._sparse_array.items() if f(v) != 0}, self.shape) + + return type(self)(map(f, Flatten(self)), self.shape) + + def _sympystr(self, printer): + def f(sh, shape_left, i, j): + if len(shape_left) == 1: + return "["+", ".join([printer._print(self[self._get_tuple_index(e)]) for e in range(i, j)])+"]" + + sh //= shape_left[0] + return "[" + ", ".join([f(sh, shape_left[1:], i+e*sh, i+(e+1)*sh) for e in range(shape_left[0])]) + "]" # + "\n"*len(shape_left) + + if self.rank() == 0: + return printer._print(self[()]) + if 0 in self.shape: + return f"{self.__class__.__name__}([], {self.shape})" + return f(self._loop_size, self.shape, 0, self._loop_size) + + def tolist(self): + """ + Converting MutableDenseNDimArray to one-dim list + + Examples + ======== + + >>> from sympy import MutableDenseNDimArray + >>> a = MutableDenseNDimArray([1, 2, 3, 4], (2, 2)) + >>> a + [[1, 2], [3, 4]] + >>> b = a.tolist() + >>> b + [[1, 2], [3, 4]] + """ + + def f(sh, shape_left, i, j): + if len(shape_left) == 1: + return [self[self._get_tuple_index(e)] for e in range(i, j)] + result = [] + sh //= shape_left[0] + for e in range(shape_left[0]): + result.append(f(sh, shape_left[1:], i+e*sh, i+(e+1)*sh)) + return result + + return f(self._loop_size, self.shape, 0, self._loop_size) + + def __add__(self, other): + from sympy.tensor.array.arrayop import Flatten + + if not isinstance(other, NDimArray): + return NotImplemented + + if self.shape != other.shape: + raise ValueError("array shape mismatch") + result_list = [i+j for i,j in zip(Flatten(self), Flatten(other))] + + return type(self)(result_list, self.shape) + + def __sub__(self, other): + from sympy.tensor.array.arrayop import Flatten + + if not isinstance(other, NDimArray): + return NotImplemented + + if self.shape != other.shape: + raise ValueError("array shape mismatch") + result_list = [i-j for i,j in zip(Flatten(self), Flatten(other))] + + return type(self)(result_list, self.shape) + + def __mul__(self, other): + from sympy.matrices.matrixbase import MatrixBase + from sympy.tensor.array import SparseNDimArray + from sympy.tensor.array.arrayop import Flatten + + if isinstance(other, (Iterable, NDimArray, MatrixBase)): + raise ValueError("scalar expected, use tensorproduct(...) for tensorial product") + + other = sympify(other) + if isinstance(self, SparseNDimArray): + if other.is_zero: + return type(self)({}, self.shape) + return type(self)({k: other*v for (k, v) in self._sparse_array.items()}, self.shape) + + result_list = [i*other for i in Flatten(self)] + return type(self)(result_list, self.shape) + + def __rmul__(self, other): + from sympy.matrices.matrixbase import MatrixBase + from sympy.tensor.array import SparseNDimArray + from sympy.tensor.array.arrayop import Flatten + + if isinstance(other, (Iterable, NDimArray, MatrixBase)): + raise ValueError("scalar expected, use tensorproduct(...) for tensorial product") + + other = sympify(other) + if isinstance(self, SparseNDimArray): + if other.is_zero: + return type(self)({}, self.shape) + return type(self)({k: other*v for (k, v) in self._sparse_array.items()}, self.shape) + + result_list = [other*i for i in Flatten(self)] + return type(self)(result_list, self.shape) + + def __truediv__(self, other): + from sympy.matrices.matrixbase import MatrixBase + from sympy.tensor.array import SparseNDimArray + from sympy.tensor.array.arrayop import Flatten + + if isinstance(other, (Iterable, NDimArray, MatrixBase)): + raise ValueError("scalar expected") + + other = sympify(other) + if isinstance(self, SparseNDimArray) and other != S.Zero: + return type(self)({k: v/other for (k, v) in self._sparse_array.items()}, self.shape) + + result_list = [i/other for i in Flatten(self)] + return type(self)(result_list, self.shape) + + def __rtruediv__(self, other): + raise NotImplementedError('unsupported operation on NDimArray') + + def __neg__(self): + from sympy.tensor.array import SparseNDimArray + from sympy.tensor.array.arrayop import Flatten + + if isinstance(self, SparseNDimArray): + return type(self)({k: -v for (k, v) in self._sparse_array.items()}, self.shape) + + result_list = [-i for i in Flatten(self)] + return type(self)(result_list, self.shape) + + def __iter__(self): + def iterator(): + if self._shape: + for i in range(self._shape[0]): + yield self[i] + else: + yield self[()] + + return iterator() + + def __eq__(self, other): + """ + NDimArray instances can be compared to each other. + Instances equal if they have same shape and data. + + Examples + ======== + + >>> from sympy import MutableDenseNDimArray + >>> a = MutableDenseNDimArray.zeros(2, 3) + >>> b = MutableDenseNDimArray.zeros(2, 3) + >>> a == b + True + >>> c = a.reshape(3, 2) + >>> c == b + False + >>> a[0,0] = 1 + >>> b[0,0] = 2 + >>> a == b + False + """ + from sympy.tensor.array import SparseNDimArray + if not isinstance(other, NDimArray): + return False + + if not self.shape == other.shape: + return False + + if isinstance(self, SparseNDimArray) and isinstance(other, SparseNDimArray): + return dict(self._sparse_array) == dict(other._sparse_array) + + return list(self) == list(other) + + def __ne__(self, other): + return not self == other + + def _eval_transpose(self): + if self.rank() != 2: + raise ValueError("array rank not 2") + from .arrayop import permutedims + return permutedims(self, (1, 0)) + + def transpose(self): + return self._eval_transpose() + + def _eval_conjugate(self): + from sympy.tensor.array.arrayop import Flatten + + return self.func([i.conjugate() for i in Flatten(self)], self.shape) + + def conjugate(self): + return self._eval_conjugate() + + def _eval_adjoint(self): + return self.transpose().conjugate() + + def adjoint(self): + return self._eval_adjoint() + + def _slice_expand(self, s, dim): + if not isinstance(s, slice): + return (s,) + start, stop, step = s.indices(dim) + return [start + i*step for i in range((stop-start)//step)] + + def _get_slice_data_for_array_access(self, index): + sl_factors = [self._slice_expand(i, dim) for (i, dim) in zip(index, self.shape)] + eindices = itertools.product(*sl_factors) + return sl_factors, eindices + + def _get_slice_data_for_array_assignment(self, index, value): + if not isinstance(value, NDimArray): + value = type(self)(value) + sl_factors, eindices = self._get_slice_data_for_array_access(index) + slice_offsets = [min(i) if isinstance(i, list) else None for i in sl_factors] + # TODO: add checks for dimensions for `value`? + return value, eindices, slice_offsets + + @classmethod + def _check_special_bounds(cls, flat_list, shape): + if shape == () and len(flat_list) != 1: + raise ValueError("arrays without shape need one scalar value") + if shape == (0,) and len(flat_list) > 0: + raise ValueError("if array shape is (0,) there cannot be elements") + + def _check_index_for_getitem(self, index): + if isinstance(index, (SYMPY_INTS, Integer, slice)): + index = (index,) + + if len(index) < self.rank(): + index = tuple(index) + \ + tuple(slice(None) for i in range(len(index), self.rank())) + + if len(index) > self.rank(): + raise ValueError('Dimension of index greater than rank of array') + + return index + + +class ImmutableNDimArray(NDimArray, Basic): + _op_priority = 11.0 + + def __hash__(self): + return Basic.__hash__(self) + + def as_immutable(self): + return self + + def as_mutable(self): + raise NotImplementedError("abstract method") diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/array/sparse_ndim_array.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/sparse_ndim_array.py new file mode 100644 index 0000000000000000000000000000000000000000..f11aa95be8ec9d10a9104d48fb28f406fe43845e --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/sparse_ndim_array.py @@ -0,0 +1,196 @@ +from sympy.core.basic import Basic +from sympy.core.containers import (Dict, Tuple) +from sympy.core.singleton import S +from sympy.core.sympify import _sympify +from sympy.tensor.array.mutable_ndim_array import MutableNDimArray +from sympy.tensor.array.ndim_array import NDimArray, ImmutableNDimArray +from sympy.utilities.iterables import flatten + +import functools + +class SparseNDimArray(NDimArray): + + def __new__(self, *args, **kwargs): + return ImmutableSparseNDimArray(*args, **kwargs) + + def __getitem__(self, index): + """ + Get an element from a sparse N-dim array. + + Examples + ======== + + >>> from sympy import MutableSparseNDimArray + >>> a = MutableSparseNDimArray(range(4), (2, 2)) + >>> a + [[0, 1], [2, 3]] + >>> a[0, 0] + 0 + >>> a[1, 1] + 3 + >>> a[0] + [0, 1] + >>> a[1] + [2, 3] + + Symbolic indexing: + + >>> from sympy.abc import i, j + >>> a[i, j] + [[0, 1], [2, 3]][i, j] + + Replace `i` and `j` to get element `(0, 0)`: + + >>> a[i, j].subs({i: 0, j: 0}) + 0 + + """ + syindex = self._check_symbolic_index(index) + if syindex is not None: + return syindex + + index = self._check_index_for_getitem(index) + + # `index` is a tuple with one or more slices: + if isinstance(index, tuple) and any(isinstance(i, slice) for i in index): + sl_factors, eindices = self._get_slice_data_for_array_access(index) + array = [self._sparse_array.get(self._parse_index(i), S.Zero) for i in eindices] + nshape = [len(el) for i, el in enumerate(sl_factors) if isinstance(index[i], slice)] + return type(self)(array, nshape) + else: + index = self._parse_index(index) + return self._sparse_array.get(index, S.Zero) + + @classmethod + def zeros(cls, *shape): + """ + Return a sparse N-dim array of zeros. + """ + return cls({}, shape) + + def tomatrix(self): + """ + Converts MutableDenseNDimArray to Matrix. Can convert only 2-dim array, else will raise error. + + Examples + ======== + + >>> from sympy import MutableSparseNDimArray + >>> a = MutableSparseNDimArray([1 for i in range(9)], (3, 3)) + >>> b = a.tomatrix() + >>> b + Matrix([ + [1, 1, 1], + [1, 1, 1], + [1, 1, 1]]) + """ + from sympy.matrices import SparseMatrix + if self.rank() != 2: + raise ValueError('Dimensions must be of size of 2') + + mat_sparse = {} + for key, value in self._sparse_array.items(): + mat_sparse[self._get_tuple_index(key)] = value + + return SparseMatrix(self.shape[0], self.shape[1], mat_sparse) + + def reshape(self, *newshape): + new_total_size = functools.reduce(lambda x,y: x*y, newshape) + if new_total_size != self._loop_size: + raise ValueError("Invalid reshape parameters " + newshape) + + return type(self)(self._sparse_array, newshape) + +class ImmutableSparseNDimArray(SparseNDimArray, ImmutableNDimArray): # type: ignore + + def __new__(cls, iterable=None, shape=None, **kwargs): + shape, flat_list = cls._handle_ndarray_creation_inputs(iterable, shape, **kwargs) + shape = Tuple(*map(_sympify, shape)) + cls._check_special_bounds(flat_list, shape) + loop_size = functools.reduce(lambda x,y: x*y, shape) if shape else len(flat_list) + + # Sparse array: + if isinstance(flat_list, (dict, Dict)): + sparse_array = Dict(flat_list) + else: + sparse_array = {} + for i, el in enumerate(flatten(flat_list)): + if el != 0: + sparse_array[i] = _sympify(el) + + sparse_array = Dict(sparse_array) + + self = Basic.__new__(cls, sparse_array, shape, **kwargs) + self._shape = shape + self._rank = len(shape) + self._loop_size = loop_size + self._sparse_array = sparse_array + + return self + + def __setitem__(self, index, value): + raise TypeError("immutable N-dim array") + + def as_mutable(self): + return MutableSparseNDimArray(self) + + +class MutableSparseNDimArray(MutableNDimArray, SparseNDimArray): + + def __new__(cls, iterable=None, shape=None, **kwargs): + shape, flat_list = cls._handle_ndarray_creation_inputs(iterable, shape, **kwargs) + self = object.__new__(cls) + self._shape = shape + self._rank = len(shape) + self._loop_size = functools.reduce(lambda x,y: x*y, shape) if shape else len(flat_list) + + # Sparse array: + if isinstance(flat_list, (dict, Dict)): + self._sparse_array = dict(flat_list) + return self + + self._sparse_array = {} + + for i, el in enumerate(flatten(flat_list)): + if el != 0: + self._sparse_array[i] = _sympify(el) + + return self + + def __setitem__(self, index, value): + """Allows to set items to MutableDenseNDimArray. + + Examples + ======== + + >>> from sympy import MutableSparseNDimArray + >>> a = MutableSparseNDimArray.zeros(2, 2) + >>> a[0, 0] = 1 + >>> a[1, 1] = 1 + >>> a + [[1, 0], [0, 1]] + """ + if isinstance(index, tuple) and any(isinstance(i, slice) for i in index): + value, eindices, slice_offsets = self._get_slice_data_for_array_assignment(index, value) + for i in eindices: + other_i = [ind - j for ind, j in zip(i, slice_offsets) if j is not None] + other_value = value[other_i] + complete_index = self._parse_index(i) + if other_value != 0: + self._sparse_array[complete_index] = other_value + elif complete_index in self._sparse_array: + self._sparse_array.pop(complete_index) + else: + index = self._parse_index(index) + value = _sympify(value) + if value == 0 and index in self._sparse_array: + self._sparse_array.pop(index) + else: + self._sparse_array[index] = value + + def as_immutable(self): + return ImmutableSparseNDimArray(self) + + @property + def free_symbols(self): + return {i for j in self._sparse_array.values() for i in j.free_symbols} diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/array/tests/__init__.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git 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a/.venv/lib/python3.12/site-packages/sympy/tensor/array/tests/test_array_comprehension.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/tests/test_array_comprehension.py new file mode 100644 index 0000000000000000000000000000000000000000..510e068f287fa04419712e5e9a16a314e522a62d --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/tests/test_array_comprehension.py @@ -0,0 +1,78 @@ +from sympy.tensor.array.array_comprehension import ArrayComprehension, ArrayComprehensionMap +from sympy.tensor.array import ImmutableDenseNDimArray +from sympy.abc import i, j, k, l +from sympy.testing.pytest import raises +from sympy.matrices import Matrix + + +def test_array_comprehension(): + a = ArrayComprehension(i*j, (i, 1, 3), (j, 2, 4)) + b = ArrayComprehension(i, (i, 1, j+1)) + c = ArrayComprehension(i+j+k+l, (i, 1, 2), (j, 1, 3), (k, 1, 4), (l, 1, 5)) + d = ArrayComprehension(k, (i, 1, 5)) + e = ArrayComprehension(i, (j, k+1, k+5)) + assert a.doit().tolist() == [[2, 3, 4], [4, 6, 8], [6, 9, 12]] + assert a.shape == (3, 3) + assert a.is_shape_numeric == True + assert a.tolist() == [[2, 3, 4], [4, 6, 8], [6, 9, 12]] + assert a.tomatrix() == Matrix([ + [2, 3, 4], + [4, 6, 8], + [6, 9, 12]]) + assert len(a) == 9 + assert isinstance(b.doit(), ArrayComprehension) + assert isinstance(a.doit(), ImmutableDenseNDimArray) + assert b.subs(j, 3) == ArrayComprehension(i, (i, 1, 4)) + assert b.free_symbols == {j} + assert b.shape == (j + 1,) + assert b.rank() == 1 + assert b.is_shape_numeric == False + assert c.free_symbols == set() + assert c.function == i + j + k + l + assert c.limits == ((i, 1, 2), (j, 1, 3), (k, 1, 4), (l, 1, 5)) + assert c.doit().tolist() == [[[[4, 5, 6, 7, 8], [5, 6, 7, 8, 9], [6, 7, 8, 9, 10], [7, 8, 9, 10, 11]], + [[5, 6, 7, 8, 9], [6, 7, 8, 9, 10], [7, 8, 9, 10, 11], [8, 9, 10, 11, 12]], + [[6, 7, 8, 9, 10], [7, 8, 9, 10, 11], [8, 9, 10, 11, 12], [9, 10, 11, 12, 13]]], + [[[5, 6, 7, 8, 9], [6, 7, 8, 9, 10], [7, 8, 9, 10, 11], [8, 9, 10, 11, 12]], + [[6, 7, 8, 9, 10], [7, 8, 9, 10, 11], [8, 9, 10, 11, 12], [9, 10, 11, 12, 13]], + [[7, 8, 9, 10, 11], [8, 9, 10, 11, 12], [9, 10, 11, 12, 13], [10, 11, 12, 13, 14]]]] + assert c.free_symbols == set() + assert c.variables == [i, j, k, l] + assert c.bound_symbols == [i, j, k, l] + assert d.doit().tolist() == [k, k, k, k, k] + assert len(e) == 5 + raises(TypeError, lambda: ArrayComprehension(i*j, (i, 1, 3), (j, 2, [1, 3, 2]))) + raises(ValueError, lambda: ArrayComprehension(i*j, (i, 1, 3), (j, 2, 1))) + raises(ValueError, lambda: ArrayComprehension(i*j, (i, 1, 3), (j, 2, j+1))) + raises(ValueError, lambda: len(ArrayComprehension(i*j, (i, 1, 3), (j, 2, j+4)))) + raises(TypeError, lambda: ArrayComprehension(i*j, (i, 0, i + 1.5), (j, 0, 2))) + raises(ValueError, lambda: b.tolist()) + raises(ValueError, lambda: b.tomatrix()) + raises(ValueError, lambda: c.tomatrix()) + +def test_arraycomprehensionmap(): + a = ArrayComprehensionMap(lambda i: i+1, (i, 1, 5)) + assert a.doit().tolist() == [2, 3, 4, 5, 6] + assert a.shape == (5,) + assert a.is_shape_numeric + assert a.tolist() == [2, 3, 4, 5, 6] + assert len(a) == 5 + assert isinstance(a.doit(), ImmutableDenseNDimArray) + expr = ArrayComprehensionMap(lambda i: i+1, (i, 1, k)) + assert expr.doit() == expr + assert expr.subs(k, 4) == ArrayComprehensionMap(lambda i: i+1, (i, 1, 4)) + assert expr.subs(k, 4).doit() == ImmutableDenseNDimArray([2, 3, 4, 5]) + b = ArrayComprehensionMap(lambda i: i+1, (i, 1, 2), (i, 1, 3), (i, 1, 4), (i, 1, 5)) + assert b.doit().tolist() == [[[[2, 3, 4, 5, 6], [3, 5, 7, 9, 11], [4, 7, 10, 13, 16], [5, 9, 13, 17, 21]], + [[3, 5, 7, 9, 11], [5, 9, 13, 17, 21], [7, 13, 19, 25, 31], [9, 17, 25, 33, 41]], + [[4, 7, 10, 13, 16], [7, 13, 19, 25, 31], [10, 19, 28, 37, 46], [13, 25, 37, 49, 61]]], + [[[3, 5, 7, 9, 11], [5, 9, 13, 17, 21], [7, 13, 19, 25, 31], [9, 17, 25, 33, 41]], + [[5, 9, 13, 17, 21], [9, 17, 25, 33, 41], [13, 25, 37, 49, 61], [17, 33, 49, 65, 81]], + [[7, 13, 19, 25, 31], [13, 25, 37, 49, 61], [19, 37, 55, 73, 91], [25, 49, 73, 97, 121]]]] + + # tests about lambda expression + assert ArrayComprehensionMap(lambda: 3, (i, 1, 5)).doit().tolist() == [3, 3, 3, 3, 3] + assert ArrayComprehensionMap(lambda i: i+1, (i, 1, 5)).doit().tolist() == [2, 3, 4, 5, 6] + raises(ValueError, lambda: ArrayComprehensionMap(i*j, (i, 1, 3), (j, 2, 4))) + a = ArrayComprehensionMap(lambda i, j: i+j, (i, 1, 5)) + raises(ValueError, lambda: a.doit()) diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/array/tests/test_array_derivatives.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/tests/test_array_derivatives.py new file mode 100644 index 0000000000000000000000000000000000000000..7f6c777c55a9170704f309bf74387d140bf2ec32 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/tests/test_array_derivatives.py @@ -0,0 +1,52 @@ +from sympy.core.symbol import symbols +from sympy.matrices.dense import Matrix +from sympy.matrices.expressions.matexpr import MatrixSymbol +from sympy.tensor.array.ndim_array import NDimArray +from sympy.matrices.matrixbase import MatrixBase +from sympy.tensor.array.array_derivatives import ArrayDerivative + +x, y, z, t = symbols("x y z t") + +m = Matrix([[x, y], [z, t]]) + +M = MatrixSymbol("M", 3, 2) +N = MatrixSymbol("N", 4, 3) + + +def test_array_derivative_construction(): + + d = ArrayDerivative(x, m, evaluate=False) + assert d.shape == (2, 2) + expr = d.doit() + assert isinstance(expr, MatrixBase) + assert expr.shape == (2, 2) + + d = ArrayDerivative(m, m, evaluate=False) + assert d.shape == (2, 2, 2, 2) + expr = d.doit() + assert isinstance(expr, NDimArray) + assert expr.shape == (2, 2, 2, 2) + + d = ArrayDerivative(m, x, evaluate=False) + assert d.shape == (2, 2) + expr = d.doit() + assert isinstance(expr, MatrixBase) + assert expr.shape == (2, 2) + + d = ArrayDerivative(M, N, evaluate=False) + assert d.shape == (4, 3, 3, 2) + expr = d.doit() + assert isinstance(expr, ArrayDerivative) + assert expr.shape == (4, 3, 3, 2) + + d = ArrayDerivative(M, (N, 2), evaluate=False) + assert d.shape == (4, 3, 4, 3, 3, 2) + expr = d.doit() + assert isinstance(expr, ArrayDerivative) + assert expr.shape == (4, 3, 4, 3, 3, 2) + + d = ArrayDerivative(M.as_explicit(), (N.as_explicit(), 2), evaluate=False) + assert d.doit().shape == (4, 3, 4, 3, 3, 2) + expr = d.doit() + assert isinstance(expr, NDimArray) + assert expr.shape == (4, 3, 4, 3, 3, 2) diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/array/tests/test_arrayop.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/tests/test_arrayop.py new file mode 100644 index 0000000000000000000000000000000000000000..de56e81e0064f1e303a7a58e41932d15f2d0b41e --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/tests/test_arrayop.py @@ -0,0 +1,361 @@ +import itertools +import random + +from sympy.combinatorics import Permutation +from sympy.combinatorics.permutations import _af_invert +from sympy.testing.pytest import raises + +from sympy.core.function import diff +from sympy.core.symbol import symbols +from sympy.functions.elementary.complexes import (adjoint, conjugate, transpose) +from sympy.functions.elementary.exponential import (exp, log) +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.tensor.array import Array, ImmutableDenseNDimArray, ImmutableSparseNDimArray, MutableSparseNDimArray + +from sympy.tensor.array.arrayop import tensorproduct, tensorcontraction, derive_by_array, permutedims, Flatten, \ + tensordiagonal + + +def test_import_NDimArray(): + from sympy.tensor.array import NDimArray + del NDimArray + + +def test_tensorproduct(): + x,y,z,t = symbols('x y z t') + from sympy.abc import a,b,c,d + assert tensorproduct() == 1 + assert tensorproduct([x]) == Array([x]) + assert tensorproduct([x], [y]) == Array([[x*y]]) + assert tensorproduct([x], [y], [z]) == Array([[[x*y*z]]]) + assert tensorproduct([x], [y], [z], [t]) == Array([[[[x*y*z*t]]]]) + + assert tensorproduct(x) == x + assert tensorproduct(x, y) == x*y + assert tensorproduct(x, y, z) == x*y*z + assert tensorproduct(x, y, z, t) == x*y*z*t + + for ArrayType in [ImmutableDenseNDimArray, ImmutableSparseNDimArray]: + A = ArrayType([x, y]) + B = ArrayType([1, 2, 3]) + C = ArrayType([a, b, c, d]) + + assert tensorproduct(A, B, C) == ArrayType([[[a*x, b*x, c*x, d*x], [2*a*x, 2*b*x, 2*c*x, 2*d*x], [3*a*x, 3*b*x, 3*c*x, 3*d*x]], + [[a*y, b*y, c*y, d*y], [2*a*y, 2*b*y, 2*c*y, 2*d*y], [3*a*y, 3*b*y, 3*c*y, 3*d*y]]]) + + assert tensorproduct([x, y], [1, 2, 3]) == tensorproduct(A, B) + + assert tensorproduct(A, 2) == ArrayType([2*x, 2*y]) + assert tensorproduct(A, [2]) == ArrayType([[2*x], [2*y]]) + assert tensorproduct([2], A) == ArrayType([[2*x, 2*y]]) + assert tensorproduct(a, A) == ArrayType([a*x, a*y]) + assert tensorproduct(a, A, B) == ArrayType([[a*x, 2*a*x, 3*a*x], [a*y, 2*a*y, 3*a*y]]) + assert tensorproduct(A, B, a) == ArrayType([[a*x, 2*a*x, 3*a*x], [a*y, 2*a*y, 3*a*y]]) + assert tensorproduct(B, a, A) == ArrayType([[a*x, a*y], [2*a*x, 2*a*y], [3*a*x, 3*a*y]]) + + # tests for large scale sparse array + for SparseArrayType in [ImmutableSparseNDimArray, MutableSparseNDimArray]: + a = SparseArrayType({1:2, 3:4},(1000, 2000)) + b = SparseArrayType({1:2, 3:4},(1000, 2000)) + assert tensorproduct(a, b) == ImmutableSparseNDimArray({2000001: 4, 2000003: 8, 6000001: 8, 6000003: 16}, (1000, 2000, 1000, 2000)) + + +def test_tensorcontraction(): + from sympy.abc import a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x + B = Array(range(18), (2, 3, 3)) + assert tensorcontraction(B, (1, 2)) == Array([12, 39]) + C1 = Array([a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x], (2, 3, 2, 2)) + + assert tensorcontraction(C1, (0, 2)) == Array([[a + o, b + p], [e + s, f + t], [i + w, j + x]]) + assert tensorcontraction(C1, (0, 2, 3)) == Array([a + p, e + t, i + x]) + assert tensorcontraction(C1, (2, 3)) == Array([[a + d, e + h, i + l], [m + p, q + t, u + x]]) + + +def test_derivative_by_array(): + from sympy.abc import i, j, t, x, y, z + + bexpr = x*y**2*exp(z)*log(t) + sexpr = sin(bexpr) + cexpr = cos(bexpr) + + a = Array([sexpr]) + + assert derive_by_array(sexpr, t) == x*y**2*exp(z)*cos(x*y**2*exp(z)*log(t))/t + assert derive_by_array(sexpr, [x, y, z]) == Array([bexpr/x*cexpr, 2*y*bexpr/y**2*cexpr, bexpr*cexpr]) + assert derive_by_array(a, [x, y, z]) == Array([[bexpr/x*cexpr], [2*y*bexpr/y**2*cexpr], [bexpr*cexpr]]) + + assert derive_by_array(sexpr, [[x, y], [z, t]]) == Array([[bexpr/x*cexpr, 2*y*bexpr/y**2*cexpr], [bexpr*cexpr, bexpr/log(t)/t*cexpr]]) + assert derive_by_array(a, [[x, y], [z, t]]) == Array([[[bexpr/x*cexpr], [2*y*bexpr/y**2*cexpr]], [[bexpr*cexpr], [bexpr/log(t)/t*cexpr]]]) + assert derive_by_array([[x, y], [z, t]], [x, y]) == Array([[[1, 0], [0, 0]], [[0, 1], [0, 0]]]) + assert derive_by_array([[x, y], [z, t]], [[x, y], [z, t]]) == Array([[[[1, 0], [0, 0]], [[0, 1], [0, 0]]], + [[[0, 0], [1, 0]], [[0, 0], [0, 1]]]]) + + assert diff(sexpr, t) == x*y**2*exp(z)*cos(x*y**2*exp(z)*log(t))/t + assert diff(sexpr, Array([x, y, z])) == Array([bexpr/x*cexpr, 2*y*bexpr/y**2*cexpr, bexpr*cexpr]) + assert diff(a, Array([x, y, z])) == Array([[bexpr/x*cexpr], [2*y*bexpr/y**2*cexpr], [bexpr*cexpr]]) + + assert diff(sexpr, Array([[x, y], [z, t]])) == Array([[bexpr/x*cexpr, 2*y*bexpr/y**2*cexpr], [bexpr*cexpr, bexpr/log(t)/t*cexpr]]) + assert diff(a, Array([[x, y], [z, t]])) == Array([[[bexpr/x*cexpr], [2*y*bexpr/y**2*cexpr]], [[bexpr*cexpr], [bexpr/log(t)/t*cexpr]]]) + assert diff(Array([[x, y], [z, t]]), Array([x, y])) == Array([[[1, 0], [0, 0]], [[0, 1], [0, 0]]]) + assert diff(Array([[x, y], [z, t]]), Array([[x, y], [z, t]])) == Array([[[[1, 0], [0, 0]], [[0, 1], [0, 0]]], + [[[0, 0], [1, 0]], [[0, 0], [0, 1]]]]) + + # test for large scale sparse array + for SparseArrayType in [ImmutableSparseNDimArray, MutableSparseNDimArray]: + b = MutableSparseNDimArray({0:i, 1:j}, (10000, 20000)) + assert derive_by_array(b, i) == ImmutableSparseNDimArray({0: 1}, (10000, 20000)) + assert derive_by_array(b, (i, j)) == ImmutableSparseNDimArray({0: 1, 200000001: 1}, (2, 10000, 20000)) + + #https://github.com/sympy/sympy/issues/20655 + U = Array([x, y, z]) + E = 2 + assert derive_by_array(E, U) == ImmutableDenseNDimArray([0, 0, 0]) + + +def test_issue_emerged_while_discussing_10972(): + ua = Array([-1,0]) + Fa = Array([[0, 1], [-1, 0]]) + po = tensorproduct(Fa, ua, Fa, ua) + assert tensorcontraction(po, (1, 2), (4, 5)) == Array([[0, 0], [0, 1]]) + + sa = symbols('a0:144') + po = Array(sa, [2, 2, 3, 3, 2, 2]) + assert tensorcontraction(po, (0, 1), (2, 3), (4, 5)) == sa[0] + sa[108] + sa[111] + sa[124] + sa[127] + sa[140] + sa[143] + sa[16] + sa[19] + sa[3] + sa[32] + sa[35] + assert tensorcontraction(po, (0, 1, 4, 5), (2, 3)) == sa[0] + sa[111] + sa[127] + sa[143] + sa[16] + sa[32] + assert tensorcontraction(po, (0, 1), (4, 5)) == Array([[sa[0] + sa[108] + sa[111] + sa[3], sa[112] + sa[115] + sa[4] + sa[7], + sa[11] + sa[116] + sa[119] + sa[8]], [sa[12] + sa[120] + sa[123] + sa[15], + sa[124] + sa[127] + sa[16] + sa[19], sa[128] + sa[131] + sa[20] + sa[23]], + [sa[132] + sa[135] + sa[24] + sa[27], sa[136] + sa[139] + sa[28] + sa[31], + sa[140] + sa[143] + sa[32] + sa[35]]]) + assert tensorcontraction(po, (0, 1), (2, 3)) == Array([[sa[0] + sa[108] + sa[124] + sa[140] + sa[16] + sa[32], sa[1] + sa[109] + sa[125] + sa[141] + sa[17] + sa[33]], + [sa[110] + sa[126] + sa[142] + sa[18] + sa[2] + sa[34], sa[111] + sa[127] + sa[143] + sa[19] + sa[3] + sa[35]]]) + + +def test_array_permutedims(): + sa = symbols('a0:144') + + for ArrayType in [ImmutableDenseNDimArray, ImmutableSparseNDimArray]: + m1 = ArrayType(sa[:6], (2, 3)) + assert permutedims(m1, (1, 0)) == transpose(m1) + assert m1.tomatrix().T == permutedims(m1, (1, 0)).tomatrix() + + assert m1.tomatrix().T == transpose(m1).tomatrix() + assert m1.tomatrix().C == conjugate(m1).tomatrix() + assert m1.tomatrix().H == adjoint(m1).tomatrix() + + assert m1.tomatrix().T == m1.transpose().tomatrix() + assert m1.tomatrix().C == m1.conjugate().tomatrix() + assert m1.tomatrix().H == m1.adjoint().tomatrix() + + raises(ValueError, lambda: permutedims(m1, (0,))) + raises(ValueError, lambda: permutedims(m1, (0, 0))) + raises(ValueError, lambda: permutedims(m1, (1, 2, 0))) + + # Some tests with random arrays: + dims = 6 + shape = [random.randint(1,5) for i in range(dims)] + elems = [random.random() for i in range(tensorproduct(*shape))] + ra = ArrayType(elems, shape) + perm = list(range(dims)) + # Randomize the permutation: + random.shuffle(perm) + # Test inverse permutation: + assert permutedims(permutedims(ra, perm), _af_invert(perm)) == ra + # Test that permuted shape corresponds to action by `Permutation`: + assert permutedims(ra, perm).shape == tuple(Permutation(perm)(shape)) + + z = ArrayType.zeros(4,5,6,7) + + assert permutedims(z, (2, 3, 1, 0)).shape == (6, 7, 5, 4) + assert permutedims(z, [2, 3, 1, 0]).shape == (6, 7, 5, 4) + assert permutedims(z, Permutation([2, 3, 1, 0])).shape == (6, 7, 5, 4) + + po = ArrayType(sa, [2, 2, 3, 3, 2, 2]) + + raises(ValueError, lambda: permutedims(po, (1, 1))) + raises(ValueError, lambda: po.transpose()) + raises(ValueError, lambda: po.adjoint()) + + assert permutedims(po, reversed(range(po.rank()))) == ArrayType( + [[[[[[sa[0], sa[72]], [sa[36], sa[108]]], [[sa[12], sa[84]], [sa[48], sa[120]]], [[sa[24], + sa[96]], [sa[60], sa[132]]]], + [[[sa[4], sa[76]], [sa[40], sa[112]]], [[sa[16], + sa[88]], [sa[52], sa[124]]], + [[sa[28], sa[100]], [sa[64], sa[136]]]], + [[[sa[8], + sa[80]], [sa[44], sa[116]]], [[sa[20], sa[92]], [sa[56], sa[128]]], [[sa[32], + sa[104]], [sa[68], sa[140]]]]], + [[[[sa[2], sa[74]], [sa[38], sa[110]]], [[sa[14], + sa[86]], [sa[50], sa[122]]], [[sa[26], sa[98]], [sa[62], sa[134]]]], + [[[sa[6], + sa[78]], [sa[42], sa[114]]], [[sa[18], sa[90]], [sa[54], sa[126]]], [[sa[30], + sa[102]], [sa[66], sa[138]]]], + [[[sa[10], sa[82]], [sa[46], sa[118]]], [[sa[22], + sa[94]], [sa[58], sa[130]]], + [[sa[34], sa[106]], [sa[70], sa[142]]]]]], + [[[[[sa[1], + sa[73]], [sa[37], sa[109]]], [[sa[13], sa[85]], [sa[49], sa[121]]], [[sa[25], + sa[97]], [sa[61], sa[133]]]], + [[[sa[5], sa[77]], [sa[41], sa[113]]], [[sa[17], + sa[89]], [sa[53], sa[125]]], + [[sa[29], sa[101]], [sa[65], sa[137]]]], + [[[sa[9], + sa[81]], [sa[45], sa[117]]], [[sa[21], sa[93]], [sa[57], sa[129]]], [[sa[33], + sa[105]], [sa[69], sa[141]]]]], + [[[[sa[3], sa[75]], [sa[39], sa[111]]], [[sa[15], + sa[87]], [sa[51], sa[123]]], [[sa[27], sa[99]], [sa[63], sa[135]]]], + [[[sa[7], + sa[79]], [sa[43], sa[115]]], [[sa[19], sa[91]], [sa[55], sa[127]]], [[sa[31], + sa[103]], [sa[67], sa[139]]]], + [[[sa[11], sa[83]], [sa[47], sa[119]]], [[sa[23], + sa[95]], [sa[59], sa[131]]], + [[sa[35], sa[107]], [sa[71], sa[143]]]]]]]) + + assert permutedims(po, (1, 0, 2, 3, 4, 5)) == ArrayType( + [[[[[[sa[0], sa[1]], [sa[2], sa[3]]], [[sa[4], sa[5]], [sa[6], sa[7]]], [[sa[8], sa[9]], [sa[10], + sa[11]]]], + [[[sa[12], sa[13]], [sa[14], sa[15]]], [[sa[16], sa[17]], [sa[18], + sa[19]]], [[sa[20], sa[21]], [sa[22], sa[23]]]], + [[[sa[24], sa[25]], [sa[26], + sa[27]]], [[sa[28], sa[29]], [sa[30], sa[31]]], [[sa[32], sa[33]], [sa[34], + sa[35]]]]], + [[[[sa[72], sa[73]], [sa[74], sa[75]]], [[sa[76], sa[77]], [sa[78], + sa[79]]], [[sa[80], sa[81]], [sa[82], sa[83]]]], + [[[sa[84], sa[85]], [sa[86], + sa[87]]], [[sa[88], sa[89]], [sa[90], sa[91]]], [[sa[92], sa[93]], [sa[94], + sa[95]]]], + [[[sa[96], sa[97]], [sa[98], sa[99]]], [[sa[100], sa[101]], [sa[102], + sa[103]]], + [[sa[104], sa[105]], [sa[106], sa[107]]]]]], [[[[[sa[36], sa[37]], [sa[38], + sa[39]]], + [[sa[40], sa[41]], [sa[42], sa[43]]], + [[sa[44], sa[45]], [sa[46], + sa[47]]]], + [[[sa[48], sa[49]], [sa[50], sa[51]]], + [[sa[52], sa[53]], [sa[54], + sa[55]]], + [[sa[56], sa[57]], [sa[58], sa[59]]]], + [[[sa[60], sa[61]], [sa[62], + sa[63]]], + [[sa[64], sa[65]], [sa[66], sa[67]]], + [[sa[68], sa[69]], [sa[70], + sa[71]]]]], [ + [[[sa[108], sa[109]], [sa[110], sa[111]]], + [[sa[112], sa[113]], [sa[114], + sa[115]]], + [[sa[116], sa[117]], [sa[118], sa[119]]]], + [[[sa[120], sa[121]], [sa[122], + sa[123]]], + [[sa[124], sa[125]], [sa[126], sa[127]]], + [[sa[128], sa[129]], [sa[130], + sa[131]]]], + [[[sa[132], sa[133]], [sa[134], sa[135]]], + [[sa[136], sa[137]], [sa[138], + sa[139]]], + [[sa[140], sa[141]], [sa[142], sa[143]]]]]]]) + + assert permutedims(po, (0, 2, 1, 4, 3, 5)) == ArrayType( + [[[[[[sa[0], sa[1]], [sa[4], sa[5]], [sa[8], sa[9]]], [[sa[2], sa[3]], [sa[6], sa[7]], [sa[10], + sa[11]]]], + [[[sa[36], sa[37]], [sa[40], sa[41]], [sa[44], sa[45]]], [[sa[38], + sa[39]], [sa[42], sa[43]], [sa[46], sa[47]]]]], + [[[[sa[12], sa[13]], [sa[16], + sa[17]], [sa[20], sa[21]]], [[sa[14], sa[15]], [sa[18], sa[19]], [sa[22], + sa[23]]]], + [[[sa[48], sa[49]], [sa[52], sa[53]], [sa[56], sa[57]]], [[sa[50], + sa[51]], [sa[54], sa[55]], [sa[58], sa[59]]]]], + [[[[sa[24], sa[25]], [sa[28], + sa[29]], [sa[32], sa[33]]], [[sa[26], sa[27]], [sa[30], sa[31]], [sa[34], + sa[35]]]], + [[[sa[60], sa[61]], [sa[64], sa[65]], [sa[68], sa[69]]], [[sa[62], + sa[63]], [sa[66], sa[67]], [sa[70], sa[71]]]]]], + [[[[[sa[72], sa[73]], [sa[76], + sa[77]], [sa[80], sa[81]]], [[sa[74], sa[75]], [sa[78], sa[79]], [sa[82], + sa[83]]]], + [[[sa[108], sa[109]], [sa[112], sa[113]], [sa[116], sa[117]]], [[sa[110], + sa[111]], [sa[114], sa[115]], + [sa[118], sa[119]]]]], + [[[[sa[84], sa[85]], [sa[88], + sa[89]], [sa[92], sa[93]]], [[sa[86], sa[87]], [sa[90], sa[91]], [sa[94], + sa[95]]]], + [[[sa[120], sa[121]], [sa[124], sa[125]], [sa[128], sa[129]]], [[sa[122], + sa[123]], [sa[126], sa[127]], + [sa[130], sa[131]]]]], + [[[[sa[96], sa[97]], [sa[100], + sa[101]], [sa[104], sa[105]]], [[sa[98], sa[99]], [sa[102], sa[103]], [sa[106], + sa[107]]]], + [[[sa[132], sa[133]], [sa[136], sa[137]], [sa[140], sa[141]]], [[sa[134], + sa[135]], [sa[138], sa[139]], + [sa[142], sa[143]]]]]]]) + + po2 = po.reshape(4, 9, 2, 2) + assert po2 == ArrayType([[[[sa[0], sa[1]], [sa[2], sa[3]]], [[sa[4], sa[5]], [sa[6], sa[7]]], [[sa[8], sa[9]], [sa[10], sa[11]]], [[sa[12], sa[13]], [sa[14], sa[15]]], [[sa[16], sa[17]], [sa[18], sa[19]]], [[sa[20], sa[21]], [sa[22], sa[23]]], [[sa[24], sa[25]], [sa[26], sa[27]]], [[sa[28], sa[29]], [sa[30], sa[31]]], [[sa[32], sa[33]], [sa[34], sa[35]]]], [[[sa[36], sa[37]], [sa[38], sa[39]]], [[sa[40], sa[41]], [sa[42], sa[43]]], [[sa[44], sa[45]], [sa[46], sa[47]]], [[sa[48], sa[49]], [sa[50], sa[51]]], [[sa[52], sa[53]], [sa[54], sa[55]]], [[sa[56], sa[57]], [sa[58], sa[59]]], [[sa[60], sa[61]], [sa[62], sa[63]]], [[sa[64], sa[65]], [sa[66], sa[67]]], [[sa[68], sa[69]], [sa[70], sa[71]]]], [[[sa[72], sa[73]], [sa[74], sa[75]]], [[sa[76], sa[77]], [sa[78], sa[79]]], [[sa[80], sa[81]], [sa[82], sa[83]]], [[sa[84], sa[85]], [sa[86], sa[87]]], [[sa[88], sa[89]], [sa[90], sa[91]]], [[sa[92], sa[93]], [sa[94], sa[95]]], [[sa[96], sa[97]], [sa[98], sa[99]]], [[sa[100], sa[101]], [sa[102], sa[103]]], [[sa[104], sa[105]], [sa[106], sa[107]]]], [[[sa[108], sa[109]], [sa[110], sa[111]]], [[sa[112], sa[113]], [sa[114], sa[115]]], [[sa[116], sa[117]], [sa[118], sa[119]]], [[sa[120], sa[121]], [sa[122], sa[123]]], [[sa[124], sa[125]], [sa[126], sa[127]]], [[sa[128], sa[129]], [sa[130], sa[131]]], [[sa[132], sa[133]], [sa[134], sa[135]]], [[sa[136], sa[137]], [sa[138], sa[139]]], [[sa[140], sa[141]], [sa[142], sa[143]]]]]) + + assert permutedims(po2, (3, 2, 0, 1)) == ArrayType([[[[sa[0], sa[4], sa[8], sa[12], sa[16], sa[20], sa[24], sa[28], sa[32]], [sa[36], sa[40], sa[44], sa[48], sa[52], sa[56], sa[60], sa[64], sa[68]], [sa[72], sa[76], sa[80], sa[84], sa[88], sa[92], sa[96], sa[100], sa[104]], [sa[108], sa[112], sa[116], sa[120], sa[124], sa[128], sa[132], sa[136], sa[140]]], [[sa[2], sa[6], sa[10], sa[14], sa[18], sa[22], sa[26], sa[30], sa[34]], [sa[38], sa[42], sa[46], sa[50], sa[54], sa[58], sa[62], sa[66], sa[70]], [sa[74], sa[78], sa[82], sa[86], sa[90], sa[94], sa[98], sa[102], sa[106]], [sa[110], sa[114], sa[118], sa[122], sa[126], sa[130], sa[134], sa[138], sa[142]]]], [[[sa[1], sa[5], sa[9], sa[13], sa[17], sa[21], sa[25], sa[29], sa[33]], [sa[37], sa[41], sa[45], sa[49], sa[53], sa[57], sa[61], sa[65], sa[69]], [sa[73], sa[77], sa[81], sa[85], sa[89], sa[93], sa[97], sa[101], sa[105]], [sa[109], sa[113], sa[117], sa[121], sa[125], sa[129], sa[133], sa[137], sa[141]]], [[sa[3], sa[7], sa[11], sa[15], sa[19], sa[23], sa[27], sa[31], sa[35]], [sa[39], sa[43], sa[47], sa[51], sa[55], sa[59], sa[63], sa[67], sa[71]], [sa[75], sa[79], sa[83], sa[87], sa[91], sa[95], sa[99], sa[103], sa[107]], [sa[111], sa[115], sa[119], sa[123], sa[127], sa[131], sa[135], sa[139], sa[143]]]]]) + + # test for large scale sparse array + for SparseArrayType in [ImmutableSparseNDimArray, MutableSparseNDimArray]: + A = SparseArrayType({1:1, 10000:2}, (10000, 20000, 10000)) + assert permutedims(A, (0, 1, 2)) == A + assert permutedims(A, (1, 0, 2)) == SparseArrayType({1: 1, 100000000: 2}, (20000, 10000, 10000)) + B = SparseArrayType({1:1, 20000:2}, (10000, 20000)) + assert B.transpose() == SparseArrayType({10000: 1, 1: 2}, (20000, 10000)) + + +def test_permutedims_with_indices(): + A = Array(range(32)).reshape(2, 2, 2, 2, 2) + indices_new = list("abcde") + indices_old = list("ebdac") + new_A = permutedims(A, index_order_new=indices_new, index_order_old=indices_old) + for a, b, c, d, e in itertools.product(range(2), range(2), range(2), range(2), range(2)): + assert new_A[a, b, c, d, e] == A[e, b, d, a, c] + indices_old = list("cabed") + new_A = permutedims(A, index_order_new=indices_new, index_order_old=indices_old) + for a, b, c, d, e in itertools.product(range(2), range(2), range(2), range(2), range(2)): + assert new_A[a, b, c, d, e] == A[c, a, b, e, d] + raises(ValueError, lambda: permutedims(A, index_order_old=list("aacde"), index_order_new=list("abcde"))) + raises(ValueError, lambda: permutedims(A, index_order_old=list("abcde"), index_order_new=list("abcce"))) + raises(ValueError, lambda: permutedims(A, index_order_old=list("abcde"), index_order_new=list("abce"))) + raises(ValueError, lambda: permutedims(A, index_order_old=list("abce"), index_order_new=list("abce"))) + raises(ValueError, lambda: permutedims(A, [2, 1, 0, 3, 4], index_order_old=list("abcde"))) + raises(ValueError, lambda: permutedims(A, [2, 1, 0, 3, 4], index_order_new=list("abcde"))) + + +def test_flatten(): + from sympy.matrices.dense import Matrix + for ArrayType in [ImmutableDenseNDimArray, ImmutableSparseNDimArray, Matrix]: + A = ArrayType(range(24)).reshape(4, 6) + assert list(Flatten(A)) == [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23] + + for i, v in enumerate(Flatten(A)): + assert i == v + + +def test_tensordiagonal(): + from sympy.matrices.dense import eye + expr = Array(range(9)).reshape(3, 3) + raises(ValueError, lambda: tensordiagonal(expr, [0], [1])) + raises(ValueError, lambda: tensordiagonal(expr, [0, 0])) + assert tensordiagonal(eye(3), [0, 1]) == Array([1, 1, 1]) + assert tensordiagonal(expr, [0, 1]) == Array([0, 4, 8]) + x, y, z = symbols("x y z") + expr2 = tensorproduct([x, y, z], expr) + assert tensordiagonal(expr2, [1, 2]) == Array([[0, 4*x, 8*x], [0, 4*y, 8*y], [0, 4*z, 8*z]]) + assert tensordiagonal(expr2, [0, 1]) == Array([[0, 3*y, 6*z], [x, 4*y, 7*z], [2*x, 5*y, 8*z]]) + assert tensordiagonal(expr2, [0, 1, 2]) == Array([0, 4*y, 8*z]) + # assert tensordiagonal(expr2, [0]) == permutedims(expr2, [1, 2, 0]) + # assert tensordiagonal(expr2, [1]) == permutedims(expr2, [0, 2, 1]) + # assert tensordiagonal(expr2, [2]) == expr2 + # assert tensordiagonal(expr2, [1], [2]) == expr2 + # assert tensordiagonal(expr2, [0], [1]) == permutedims(expr2, [2, 0, 1]) + + a, b, c, X, Y, Z = symbols("a b c X Y Z") + expr3 = tensorproduct([x, y, z], [1, 2, 3], [a, b, c], [X, Y, Z]) + assert tensordiagonal(expr3, [0, 1, 2, 3]) == Array([x*a*X, 2*y*b*Y, 3*z*c*Z]) + assert tensordiagonal(expr3, [0, 1], [2, 3]) == tensorproduct([x, 2*y, 3*z], [a*X, b*Y, c*Z]) + + # assert tensordiagonal(expr3, [0], [1, 2], [3]) == tensorproduct([x, y, z], [a, 2*b, 3*c], [X, Y, Z]) + assert tensordiagonal(tensordiagonal(expr3, [2, 3]), [0, 1]) == tensorproduct([a*X, b*Y, c*Z], [x, 2*y, 3*z]) + + raises(ValueError, lambda: tensordiagonal([[1, 2, 3], [4, 5, 6]], [0, 1])) + raises(ValueError, lambda: tensordiagonal(expr3.reshape(3, 3, 9), [1, 2])) diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/array/tests/test_immutable_ndim_array.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/tests/test_immutable_ndim_array.py new file mode 100644 index 0000000000000000000000000000000000000000..c6bed4b605c424284b4752592b03b13a9178aac8 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/tests/test_immutable_ndim_array.py @@ -0,0 +1,452 @@ +from copy import copy + +from sympy.tensor.array.dense_ndim_array import ImmutableDenseNDimArray +from sympy.core.containers import Dict +from sympy.core.function import diff +from sympy.core.numbers import Rational +from sympy.core.singleton import S +from sympy.core.symbol import (Symbol, symbols) +from sympy.matrices import SparseMatrix +from sympy.tensor.indexed import (Indexed, IndexedBase) +from sympy.matrices import Matrix +from sympy.tensor.array.sparse_ndim_array import ImmutableSparseNDimArray +from sympy.testing.pytest import raises + + +def test_ndim_array_initiation(): + arr_with_no_elements = ImmutableDenseNDimArray([], shape=(0,)) + assert len(arr_with_no_elements) == 0 + assert arr_with_no_elements.rank() == 1 + + raises(ValueError, lambda: ImmutableDenseNDimArray([0], shape=(0,))) + raises(ValueError, lambda: ImmutableDenseNDimArray([1, 2, 3], shape=(0,))) + raises(ValueError, lambda: ImmutableDenseNDimArray([], shape=())) + + raises(ValueError, lambda: ImmutableSparseNDimArray([0], shape=(0,))) + raises(ValueError, lambda: ImmutableSparseNDimArray([1, 2, 3], shape=(0,))) + raises(ValueError, lambda: ImmutableSparseNDimArray([], shape=())) + + arr_with_one_element = ImmutableDenseNDimArray([23]) + assert len(arr_with_one_element) == 1 + assert arr_with_one_element[0] == 23 + assert arr_with_one_element[:] == ImmutableDenseNDimArray([23]) + assert arr_with_one_element.rank() == 1 + + arr_with_symbol_element = ImmutableDenseNDimArray([Symbol('x')]) + assert len(arr_with_symbol_element) == 1 + assert arr_with_symbol_element[0] == Symbol('x') + assert arr_with_symbol_element[:] == ImmutableDenseNDimArray([Symbol('x')]) + assert arr_with_symbol_element.rank() == 1 + + number5 = 5 + vector = ImmutableDenseNDimArray.zeros(number5) + assert len(vector) == number5 + assert vector.shape == (number5,) + assert vector.rank() == 1 + + vector = ImmutableSparseNDimArray.zeros(number5) + assert len(vector) == number5 + assert vector.shape == (number5,) + assert vector._sparse_array == Dict() + assert vector.rank() == 1 + + n_dim_array = ImmutableDenseNDimArray(range(3**4), (3, 3, 3, 3,)) + assert len(n_dim_array) == 3 * 3 * 3 * 3 + assert n_dim_array.shape == (3, 3, 3, 3) + assert n_dim_array.rank() == 4 + + array_shape = (3, 3, 3, 3) + sparse_array = ImmutableSparseNDimArray.zeros(*array_shape) + assert len(sparse_array._sparse_array) == 0 + assert len(sparse_array) == 3 * 3 * 3 * 3 + assert n_dim_array.shape == array_shape + assert n_dim_array.rank() == 4 + + one_dim_array = ImmutableDenseNDimArray([2, 3, 1]) + assert len(one_dim_array) == 3 + assert one_dim_array.shape == (3,) + assert one_dim_array.rank() == 1 + assert one_dim_array.tolist() == [2, 3, 1] + + shape = (3, 3) + array_with_many_args = ImmutableSparseNDimArray.zeros(*shape) + assert len(array_with_many_args) == 3 * 3 + assert array_with_many_args.shape == shape + assert array_with_many_args[0, 0] == 0 + assert array_with_many_args.rank() == 2 + + shape = (int(3), int(3)) + array_with_long_shape = ImmutableSparseNDimArray.zeros(*shape) + assert len(array_with_long_shape) == 3 * 3 + assert array_with_long_shape.shape == shape + assert array_with_long_shape[int(0), int(0)] == 0 + assert array_with_long_shape.rank() == 2 + + vector_with_long_shape = ImmutableDenseNDimArray(range(5), int(5)) + assert len(vector_with_long_shape) == 5 + assert vector_with_long_shape.shape == (int(5),) + assert vector_with_long_shape.rank() == 1 + raises(ValueError, lambda: vector_with_long_shape[int(5)]) + + from sympy.abc import x + for ArrayType in [ImmutableDenseNDimArray, ImmutableSparseNDimArray]: + rank_zero_array = ArrayType(x) + assert len(rank_zero_array) == 1 + assert rank_zero_array.shape == () + assert rank_zero_array.rank() == 0 + assert rank_zero_array[()] == x + raises(ValueError, lambda: rank_zero_array[0]) + + +def test_reshape(): + array = ImmutableDenseNDimArray(range(50), 50) + assert array.shape == (50,) + assert array.rank() == 1 + + array = array.reshape(5, 5, 2) + assert array.shape == (5, 5, 2) + assert array.rank() == 3 + assert len(array) == 50 + + +def test_getitem(): + for ArrayType in [ImmutableDenseNDimArray, ImmutableSparseNDimArray]: + array = ArrayType(range(24)).reshape(2, 3, 4) + assert array.tolist() == [[[0, 1, 2, 3], [4, 5, 6, 7], [8, 9, 10, 11]], [[12, 13, 14, 15], [16, 17, 18, 19], [20, 21, 22, 23]]] + assert array[0] == ArrayType([[0, 1, 2, 3], [4, 5, 6, 7], [8, 9, 10, 11]]) + assert array[0, 0] == ArrayType([0, 1, 2, 3]) + value = 0 + for i in range(2): + for j in range(3): + for k in range(4): + assert array[i, j, k] == value + value += 1 + + raises(ValueError, lambda: array[3, 4, 5]) + raises(ValueError, lambda: array[3, 4, 5, 6]) + raises(ValueError, lambda: array[3, 4, 5, 3:4]) + + +def test_iterator(): + array = ImmutableDenseNDimArray(range(4), (2, 2)) + assert array[0] == ImmutableDenseNDimArray([0, 1]) + assert array[1] == ImmutableDenseNDimArray([2, 3]) + + array = array.reshape(4) + j = 0 + for i in array: + assert i == j + j += 1 + + +def test_sparse(): + sparse_array = ImmutableSparseNDimArray([0, 0, 0, 1], (2, 2)) + assert len(sparse_array) == 2 * 2 + # dictionary where all data is, only non-zero entries are actually stored: + assert len(sparse_array._sparse_array) == 1 + + assert sparse_array.tolist() == [[0, 0], [0, 1]] + + for i, j in zip(sparse_array, [[0, 0], [0, 1]]): + assert i == ImmutableSparseNDimArray(j) + + def sparse_assignment(): + sparse_array[0, 0] = 123 + + assert len(sparse_array._sparse_array) == 1 + raises(TypeError, sparse_assignment) + assert len(sparse_array._sparse_array) == 1 + assert sparse_array[0, 0] == 0 + assert sparse_array/0 == ImmutableSparseNDimArray([[S.NaN, S.NaN], [S.NaN, S.ComplexInfinity]], (2, 2)) + + # test for large scale sparse array + # equality test + assert ImmutableSparseNDimArray.zeros(100000, 200000) == ImmutableSparseNDimArray.zeros(100000, 200000) + + # __mul__ and __rmul__ + a = ImmutableSparseNDimArray({200001: 1}, (100000, 200000)) + assert a * 3 == ImmutableSparseNDimArray({200001: 3}, (100000, 200000)) + assert 3 * a == ImmutableSparseNDimArray({200001: 3}, (100000, 200000)) + assert a * 0 == ImmutableSparseNDimArray({}, (100000, 200000)) + assert 0 * a == ImmutableSparseNDimArray({}, (100000, 200000)) + + # __truediv__ + assert a/3 == ImmutableSparseNDimArray({200001: Rational(1, 3)}, (100000, 200000)) + + # __neg__ + assert -a == ImmutableSparseNDimArray({200001: -1}, (100000, 200000)) + + +def test_calculation(): + + a = ImmutableDenseNDimArray([1]*9, (3, 3)) + b = ImmutableDenseNDimArray([9]*9, (3, 3)) + + c = a + b + for i in c: + assert i == ImmutableDenseNDimArray([10, 10, 10]) + + assert c == ImmutableDenseNDimArray([10]*9, (3, 3)) + assert c == ImmutableSparseNDimArray([10]*9, (3, 3)) + + c = b - a + for i in c: + assert i == ImmutableDenseNDimArray([8, 8, 8]) + + assert c == ImmutableDenseNDimArray([8]*9, (3, 3)) + assert c == ImmutableSparseNDimArray([8]*9, (3, 3)) + + +def test_ndim_array_converting(): + dense_array = ImmutableDenseNDimArray([1, 2, 3, 4], (2, 2)) + alist = dense_array.tolist() + + assert alist == [[1, 2], [3, 4]] + + matrix = dense_array.tomatrix() + assert (isinstance(matrix, Matrix)) + + for i in range(len(dense_array)): + assert dense_array[dense_array._get_tuple_index(i)] == matrix[i] + assert matrix.shape == dense_array.shape + + assert ImmutableDenseNDimArray(matrix) == dense_array + assert ImmutableDenseNDimArray(matrix.as_immutable()) == dense_array + assert ImmutableDenseNDimArray(matrix.as_mutable()) == dense_array + + sparse_array = ImmutableSparseNDimArray([1, 2, 3, 4], (2, 2)) + alist = sparse_array.tolist() + + assert alist == [[1, 2], [3, 4]] + + matrix = sparse_array.tomatrix() + assert(isinstance(matrix, SparseMatrix)) + + for i in range(len(sparse_array)): + assert sparse_array[sparse_array._get_tuple_index(i)] == matrix[i] + assert matrix.shape == sparse_array.shape + + assert ImmutableSparseNDimArray(matrix) == sparse_array + assert ImmutableSparseNDimArray(matrix.as_immutable()) == sparse_array + assert ImmutableSparseNDimArray(matrix.as_mutable()) == sparse_array + + +def test_converting_functions(): + arr_list = [1, 2, 3, 4] + arr_matrix = Matrix(((1, 2), (3, 4))) + + # list + arr_ndim_array = ImmutableDenseNDimArray(arr_list, (2, 2)) + assert (isinstance(arr_ndim_array, ImmutableDenseNDimArray)) + assert arr_matrix.tolist() == arr_ndim_array.tolist() + + # Matrix + arr_ndim_array = ImmutableDenseNDimArray(arr_matrix) + assert (isinstance(arr_ndim_array, ImmutableDenseNDimArray)) + assert arr_matrix.tolist() == arr_ndim_array.tolist() + assert arr_matrix.shape == arr_ndim_array.shape + + +def test_equality(): + first_list = [1, 2, 3, 4] + second_list = [1, 2, 3, 4] + third_list = [4, 3, 2, 1] + assert first_list == second_list + assert first_list != third_list + + first_ndim_array = ImmutableDenseNDimArray(first_list, (2, 2)) + second_ndim_array = ImmutableDenseNDimArray(second_list, (2, 2)) + fourth_ndim_array = ImmutableDenseNDimArray(first_list, (2, 2)) + + assert first_ndim_array == second_ndim_array + + def assignment_attempt(a): + a[0, 0] = 0 + + raises(TypeError, lambda: assignment_attempt(second_ndim_array)) + assert first_ndim_array == second_ndim_array + assert first_ndim_array == fourth_ndim_array + + +def test_arithmetic(): + a = ImmutableDenseNDimArray([3 for i in range(9)], (3, 3)) + b = ImmutableDenseNDimArray([7 for i in range(9)], (3, 3)) + + c1 = a + b + c2 = b + a + assert c1 == c2 + + d1 = a - b + d2 = b - a + assert d1 == d2 * (-1) + + e1 = a * 5 + e2 = 5 * a + e3 = copy(a) + e3 *= 5 + assert e1 == e2 == e3 + + f1 = a / 5 + f2 = copy(a) + f2 /= 5 + assert f1 == f2 + assert f1[0, 0] == f1[0, 1] == f1[0, 2] == f1[1, 0] == f1[1, 1] == \ + f1[1, 2] == f1[2, 0] == f1[2, 1] == f1[2, 2] == Rational(3, 5) + + assert type(a) == type(b) == type(c1) == type(c2) == type(d1) == type(d2) \ + == type(e1) == type(e2) == type(e3) == type(f1) + + z0 = -a + assert z0 == ImmutableDenseNDimArray([-3 for i in range(9)], (3, 3)) + + +def test_higher_dimenions(): + m3 = ImmutableDenseNDimArray(range(10, 34), (2, 3, 4)) + + assert m3.tolist() == [[[10, 11, 12, 13], + [14, 15, 16, 17], + [18, 19, 20, 21]], + + [[22, 23, 24, 25], + [26, 27, 28, 29], + [30, 31, 32, 33]]] + + assert m3._get_tuple_index(0) == (0, 0, 0) + assert m3._get_tuple_index(1) == (0, 0, 1) + assert m3._get_tuple_index(4) == (0, 1, 0) + assert m3._get_tuple_index(12) == (1, 0, 0) + + assert str(m3) == '[[[10, 11, 12, 13], [14, 15, 16, 17], [18, 19, 20, 21]], [[22, 23, 24, 25], [26, 27, 28, 29], [30, 31, 32, 33]]]' + + m3_rebuilt = ImmutableDenseNDimArray([[[10, 11, 12, 13], [14, 15, 16, 17], [18, 19, 20, 21]], [[22, 23, 24, 25], [26, 27, 28, 29], [30, 31, 32, 33]]]) + assert m3 == m3_rebuilt + + m3_other = ImmutableDenseNDimArray([[[10, 11, 12, 13], [14, 15, 16, 17], [18, 19, 20, 21]], [[22, 23, 24, 25], [26, 27, 28, 29], [30, 31, 32, 33]]], (2, 3, 4)) + + assert m3 == m3_other + + +def test_rebuild_immutable_arrays(): + sparr = ImmutableSparseNDimArray(range(10, 34), (2, 3, 4)) + densarr = ImmutableDenseNDimArray(range(10, 34), (2, 3, 4)) + + assert sparr == sparr.func(*sparr.args) + assert densarr == densarr.func(*densarr.args) + + +def test_slices(): + md = ImmutableDenseNDimArray(range(10, 34), (2, 3, 4)) + + assert md[:] == ImmutableDenseNDimArray(range(10, 34), (2, 3, 4)) + assert md[:, :, 0].tomatrix() == Matrix([[10, 14, 18], [22, 26, 30]]) + assert md[0, 1:2, :].tomatrix() == Matrix([[14, 15, 16, 17]]) + assert md[0, 1:3, :].tomatrix() == Matrix([[14, 15, 16, 17], [18, 19, 20, 21]]) + assert md[:, :, :] == md + + sd = ImmutableSparseNDimArray(range(10, 34), (2, 3, 4)) + assert sd == ImmutableSparseNDimArray(md) + + assert sd[:] == ImmutableSparseNDimArray(range(10, 34), (2, 3, 4)) + assert sd[:, :, 0].tomatrix() == Matrix([[10, 14, 18], [22, 26, 30]]) + assert sd[0, 1:2, :].tomatrix() == Matrix([[14, 15, 16, 17]]) + assert sd[0, 1:3, :].tomatrix() == Matrix([[14, 15, 16, 17], [18, 19, 20, 21]]) + assert sd[:, :, :] == sd + + +def test_diff_and_applyfunc(): + from sympy.abc import x, y, z + md = ImmutableDenseNDimArray([[x, y], [x*z, x*y*z]]) + assert md.diff(x) == ImmutableDenseNDimArray([[1, 0], [z, y*z]]) + assert diff(md, x) == ImmutableDenseNDimArray([[1, 0], [z, y*z]]) + + sd = ImmutableSparseNDimArray(md) + assert sd == ImmutableSparseNDimArray([x, y, x*z, x*y*z], (2, 2)) + assert sd.diff(x) == ImmutableSparseNDimArray([[1, 0], [z, y*z]]) + assert diff(sd, x) == ImmutableSparseNDimArray([[1, 0], [z, y*z]]) + + mdn = md.applyfunc(lambda x: x*3) + assert mdn == ImmutableDenseNDimArray([[3*x, 3*y], [3*x*z, 3*x*y*z]]) + assert md != mdn + + sdn = sd.applyfunc(lambda x: x/2) + assert sdn == ImmutableSparseNDimArray([[x/2, y/2], [x*z/2, x*y*z/2]]) + assert sd != sdn + + sdp = sd.applyfunc(lambda x: x+1) + assert sdp == ImmutableSparseNDimArray([[x + 1, y + 1], [x*z + 1, x*y*z + 1]]) + assert sd != sdp + + +def test_op_priority(): + from sympy.abc import x + md = ImmutableDenseNDimArray([1, 2, 3]) + e1 = (1+x)*md + e2 = md*(1+x) + assert e1 == ImmutableDenseNDimArray([1+x, 2+2*x, 3+3*x]) + assert e1 == e2 + + sd = ImmutableSparseNDimArray([1, 2, 3]) + e3 = (1+x)*sd + e4 = sd*(1+x) + assert e3 == ImmutableDenseNDimArray([1+x, 2+2*x, 3+3*x]) + assert e3 == e4 + + +def test_symbolic_indexing(): + x, y, z, w = symbols("x y z w") + M = ImmutableDenseNDimArray([[x, y], [z, w]]) + i, j = symbols("i, j") + Mij = M[i, j] + assert isinstance(Mij, Indexed) + Ms = ImmutableSparseNDimArray([[2, 3*x], [4, 5]]) + msij = Ms[i, j] + assert isinstance(msij, Indexed) + for oi, oj in [(0, 0), (0, 1), (1, 0), (1, 1)]: + assert Mij.subs({i: oi, j: oj}) == M[oi, oj] + assert msij.subs({i: oi, j: oj}) == Ms[oi, oj] + A = IndexedBase("A", (0, 2)) + assert A[0, 0].subs(A, M) == x + assert A[i, j].subs(A, M) == M[i, j] + assert M[i, j].subs(M, A) == A[i, j] + + assert isinstance(M[3 * i - 2, j], Indexed) + assert M[3 * i - 2, j].subs({i: 1, j: 0}) == M[1, 0] + assert isinstance(M[i, 0], Indexed) + assert M[i, 0].subs(i, 0) == M[0, 0] + assert M[0, i].subs(i, 1) == M[0, 1] + + assert M[i, j].diff(x) == ImmutableDenseNDimArray([[1, 0], [0, 0]])[i, j] + assert Ms[i, j].diff(x) == ImmutableSparseNDimArray([[0, 3], [0, 0]])[i, j] + + Mo = ImmutableDenseNDimArray([1, 2, 3]) + assert Mo[i].subs(i, 1) == 2 + Mos = ImmutableSparseNDimArray([1, 2, 3]) + assert Mos[i].subs(i, 1) == 2 + + raises(ValueError, lambda: M[i, 2]) + raises(ValueError, lambda: M[i, -1]) + raises(ValueError, lambda: M[2, i]) + raises(ValueError, lambda: M[-1, i]) + + raises(ValueError, lambda: Ms[i, 2]) + raises(ValueError, lambda: Ms[i, -1]) + raises(ValueError, lambda: Ms[2, i]) + raises(ValueError, lambda: Ms[-1, i]) + + +def test_issue_12665(): + # Testing Python 3 hash of immutable arrays: + arr = ImmutableDenseNDimArray([1, 2, 3]) + # This should NOT raise an exception: + hash(arr) + + +def test_zeros_without_shape(): + arr = ImmutableDenseNDimArray.zeros() + assert arr == ImmutableDenseNDimArray(0) + +def test_issue_21870(): + a0 = ImmutableDenseNDimArray(0) + assert a0.rank() == 0 + a1 = ImmutableDenseNDimArray(a0) + assert a1.rank() == 0 diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/array/tests/test_mutable_ndim_array.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/tests/test_mutable_ndim_array.py new file mode 100644 index 0000000000000000000000000000000000000000..9a232f399bbc0639d326217975fb0a12e645a984 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/tests/test_mutable_ndim_array.py @@ -0,0 +1,374 @@ +from copy import copy + +from sympy.tensor.array.dense_ndim_array import MutableDenseNDimArray +from sympy.core.function import diff +from sympy.core.numbers import Rational +from sympy.core.singleton import S +from sympy.core.symbol import Symbol +from sympy.core.sympify import sympify +from sympy.matrices import SparseMatrix +from sympy.matrices import Matrix +from sympy.tensor.array.sparse_ndim_array import MutableSparseNDimArray +from sympy.testing.pytest import raises + + +def test_ndim_array_initiation(): + arr_with_one_element = MutableDenseNDimArray([23]) + assert len(arr_with_one_element) == 1 + assert arr_with_one_element[0] == 23 + assert arr_with_one_element.rank() == 1 + raises(ValueError, lambda: arr_with_one_element[1]) + + arr_with_symbol_element = MutableDenseNDimArray([Symbol('x')]) + assert len(arr_with_symbol_element) == 1 + assert arr_with_symbol_element[0] == Symbol('x') + assert arr_with_symbol_element.rank() == 1 + + number5 = 5 + vector = MutableDenseNDimArray.zeros(number5) + assert len(vector) == number5 + assert vector.shape == (number5,) + assert vector.rank() == 1 + raises(ValueError, lambda: arr_with_one_element[5]) + + vector = MutableSparseNDimArray.zeros(number5) + assert len(vector) == number5 + assert vector.shape == (number5,) + assert vector._sparse_array == {} + assert vector.rank() == 1 + + n_dim_array = MutableDenseNDimArray(range(3**4), (3, 3, 3, 3,)) + assert len(n_dim_array) == 3 * 3 * 3 * 3 + assert n_dim_array.shape == (3, 3, 3, 3) + assert n_dim_array.rank() == 4 + raises(ValueError, lambda: n_dim_array[0, 0, 0, 3]) + raises(ValueError, lambda: n_dim_array[3, 0, 0, 0]) + raises(ValueError, lambda: n_dim_array[3**4]) + + array_shape = (3, 3, 3, 3) + sparse_array = MutableSparseNDimArray.zeros(*array_shape) + assert len(sparse_array._sparse_array) == 0 + assert len(sparse_array) == 3 * 3 * 3 * 3 + assert n_dim_array.shape == array_shape + assert n_dim_array.rank() == 4 + + one_dim_array = MutableDenseNDimArray([2, 3, 1]) + assert len(one_dim_array) == 3 + assert one_dim_array.shape == (3,) + assert one_dim_array.rank() == 1 + assert one_dim_array.tolist() == [2, 3, 1] + + shape = (3, 3) + array_with_many_args = MutableSparseNDimArray.zeros(*shape) + assert len(array_with_many_args) == 3 * 3 + assert array_with_many_args.shape == shape + assert array_with_many_args[0, 0] == 0 + assert array_with_many_args.rank() == 2 + + shape = (int(3), int(3)) + array_with_long_shape = MutableSparseNDimArray.zeros(*shape) + assert len(array_with_long_shape) == 3 * 3 + assert array_with_long_shape.shape == shape + assert array_with_long_shape[int(0), int(0)] == 0 + assert array_with_long_shape.rank() == 2 + + vector_with_long_shape = MutableDenseNDimArray(range(5), int(5)) + assert len(vector_with_long_shape) == 5 + assert vector_with_long_shape.shape == (int(5),) + assert vector_with_long_shape.rank() == 1 + raises(ValueError, lambda: vector_with_long_shape[int(5)]) + + from sympy.abc import x + for ArrayType in [MutableDenseNDimArray, MutableSparseNDimArray]: + rank_zero_array = ArrayType(x) + assert len(rank_zero_array) == 1 + assert rank_zero_array.shape == () + assert rank_zero_array.rank() == 0 + assert rank_zero_array[()] == x + raises(ValueError, lambda: rank_zero_array[0]) + +def test_sympify(): + from sympy.abc import x, y, z, t + arr = MutableDenseNDimArray([[x, y], [1, z*t]]) + arr_other = sympify(arr) + assert arr_other.shape == (2, 2) + assert arr_other == arr + + +def test_reshape(): + array = MutableDenseNDimArray(range(50), 50) + assert array.shape == (50,) + assert array.rank() == 1 + + array = array.reshape(5, 5, 2) + assert array.shape == (5, 5, 2) + assert array.rank() == 3 + assert len(array) == 50 + + +def test_iterator(): + array = MutableDenseNDimArray(range(4), (2, 2)) + assert array[0] == MutableDenseNDimArray([0, 1]) + assert array[1] == MutableDenseNDimArray([2, 3]) + + array = array.reshape(4) + j = 0 + for i in array: + assert i == j + j += 1 + + +def test_getitem(): + for ArrayType in [MutableDenseNDimArray, MutableSparseNDimArray]: + array = ArrayType(range(24)).reshape(2, 3, 4) + assert array.tolist() == [[[0, 1, 2, 3], [4, 5, 6, 7], [8, 9, 10, 11]], [[12, 13, 14, 15], [16, 17, 18, 19], [20, 21, 22, 23]]] + assert array[0] == ArrayType([[0, 1, 2, 3], [4, 5, 6, 7], [8, 9, 10, 11]]) + assert array[0, 0] == ArrayType([0, 1, 2, 3]) + value = 0 + for i in range(2): + for j in range(3): + for k in range(4): + assert array[i, j, k] == value + value += 1 + + raises(ValueError, lambda: array[3, 4, 5]) + raises(ValueError, lambda: array[3, 4, 5, 6]) + raises(ValueError, lambda: array[3, 4, 5, 3:4]) + + +def test_sparse(): + sparse_array = MutableSparseNDimArray([0, 0, 0, 1], (2, 2)) + assert len(sparse_array) == 2 * 2 + # dictionary where all data is, only non-zero entries are actually stored: + assert len(sparse_array._sparse_array) == 1 + + assert sparse_array.tolist() == [[0, 0], [0, 1]] + + for i, j in zip(sparse_array, [[0, 0], [0, 1]]): + assert i == MutableSparseNDimArray(j) + + sparse_array[0, 0] = 123 + assert len(sparse_array._sparse_array) == 2 + assert sparse_array[0, 0] == 123 + assert sparse_array/0 == MutableSparseNDimArray([[S.ComplexInfinity, S.NaN], [S.NaN, S.ComplexInfinity]], (2, 2)) + + # when element in sparse array become zero it will disappear from + # dictionary + sparse_array[0, 0] = 0 + assert len(sparse_array._sparse_array) == 1 + sparse_array[1, 1] = 0 + assert len(sparse_array._sparse_array) == 0 + assert sparse_array[0, 0] == 0 + + # test for large scale sparse array + # equality test + a = MutableSparseNDimArray.zeros(100000, 200000) + b = MutableSparseNDimArray.zeros(100000, 200000) + assert a == b + a[1, 1] = 1 + b[1, 1] = 2 + assert a != b + + # __mul__ and __rmul__ + assert a * 3 == MutableSparseNDimArray({200001: 3}, (100000, 200000)) + assert 3 * a == MutableSparseNDimArray({200001: 3}, (100000, 200000)) + assert a * 0 == MutableSparseNDimArray({}, (100000, 200000)) + assert 0 * a == MutableSparseNDimArray({}, (100000, 200000)) + + # __truediv__ + assert a/3 == MutableSparseNDimArray({200001: Rational(1, 3)}, (100000, 200000)) + + # __neg__ + assert -a == MutableSparseNDimArray({200001: -1}, (100000, 200000)) + + +def test_calculation(): + + a = MutableDenseNDimArray([1]*9, (3, 3)) + b = MutableDenseNDimArray([9]*9, (3, 3)) + + c = a + b + for i in c: + assert i == MutableDenseNDimArray([10, 10, 10]) + + assert c == MutableDenseNDimArray([10]*9, (3, 3)) + assert c == MutableSparseNDimArray([10]*9, (3, 3)) + + c = b - a + for i in c: + assert i == MutableSparseNDimArray([8, 8, 8]) + + assert c == MutableDenseNDimArray([8]*9, (3, 3)) + assert c == MutableSparseNDimArray([8]*9, (3, 3)) + + +def test_ndim_array_converting(): + dense_array = MutableDenseNDimArray([1, 2, 3, 4], (2, 2)) + alist = dense_array.tolist() + + assert alist == [[1, 2], [3, 4]] + + matrix = dense_array.tomatrix() + assert (isinstance(matrix, Matrix)) + + for i in range(len(dense_array)): + assert dense_array[dense_array._get_tuple_index(i)] == matrix[i] + assert matrix.shape == dense_array.shape + + assert MutableDenseNDimArray(matrix) == dense_array + assert MutableDenseNDimArray(matrix.as_immutable()) == dense_array + assert MutableDenseNDimArray(matrix.as_mutable()) == dense_array + + sparse_array = MutableSparseNDimArray([1, 2, 3, 4], (2, 2)) + alist = sparse_array.tolist() + + assert alist == [[1, 2], [3, 4]] + + matrix = sparse_array.tomatrix() + assert(isinstance(matrix, SparseMatrix)) + + for i in range(len(sparse_array)): + assert sparse_array[sparse_array._get_tuple_index(i)] == matrix[i] + assert matrix.shape == sparse_array.shape + + assert MutableSparseNDimArray(matrix) == sparse_array + assert MutableSparseNDimArray(matrix.as_immutable()) == sparse_array + assert MutableSparseNDimArray(matrix.as_mutable()) == sparse_array + + +def test_converting_functions(): + arr_list = [1, 2, 3, 4] + arr_matrix = Matrix(((1, 2), (3, 4))) + + # list + arr_ndim_array = MutableDenseNDimArray(arr_list, (2, 2)) + assert (isinstance(arr_ndim_array, MutableDenseNDimArray)) + assert arr_matrix.tolist() == arr_ndim_array.tolist() + + # Matrix + arr_ndim_array = MutableDenseNDimArray(arr_matrix) + assert (isinstance(arr_ndim_array, MutableDenseNDimArray)) + assert arr_matrix.tolist() == arr_ndim_array.tolist() + assert arr_matrix.shape == arr_ndim_array.shape + + +def test_equality(): + first_list = [1, 2, 3, 4] + second_list = [1, 2, 3, 4] + third_list = [4, 3, 2, 1] + assert first_list == second_list + assert first_list != third_list + + first_ndim_array = MutableDenseNDimArray(first_list, (2, 2)) + second_ndim_array = MutableDenseNDimArray(second_list, (2, 2)) + third_ndim_array = MutableDenseNDimArray(third_list, (2, 2)) + fourth_ndim_array = MutableDenseNDimArray(first_list, (2, 2)) + + assert first_ndim_array == second_ndim_array + second_ndim_array[0, 0] = 0 + assert first_ndim_array != second_ndim_array + assert first_ndim_array != third_ndim_array + assert first_ndim_array == fourth_ndim_array + + +def test_arithmetic(): + a = MutableDenseNDimArray([3 for i in range(9)], (3, 3)) + b = MutableDenseNDimArray([7 for i in range(9)], (3, 3)) + + c1 = a + b + c2 = b + a + assert c1 == c2 + + d1 = a - b + d2 = b - a + assert d1 == d2 * (-1) + + e1 = a * 5 + e2 = 5 * a + e3 = copy(a) + e3 *= 5 + assert e1 == e2 == e3 + + f1 = a / 5 + f2 = copy(a) + f2 /= 5 + assert f1 == f2 + assert f1[0, 0] == f1[0, 1] == f1[0, 2] == f1[1, 0] == f1[1, 1] == \ + f1[1, 2] == f1[2, 0] == f1[2, 1] == f1[2, 2] == Rational(3, 5) + + assert type(a) == type(b) == type(c1) == type(c2) == type(d1) == type(d2) \ + == type(e1) == type(e2) == type(e3) == type(f1) + + z0 = -a + assert z0 == MutableDenseNDimArray([-3 for i in range(9)], (3, 3)) + + +def test_higher_dimenions(): + m3 = MutableDenseNDimArray(range(10, 34), (2, 3, 4)) + + assert m3.tolist() == [[[10, 11, 12, 13], + [14, 15, 16, 17], + [18, 19, 20, 21]], + + [[22, 23, 24, 25], + [26, 27, 28, 29], + [30, 31, 32, 33]]] + + assert m3._get_tuple_index(0) == (0, 0, 0) + assert m3._get_tuple_index(1) == (0, 0, 1) + assert m3._get_tuple_index(4) == (0, 1, 0) + assert m3._get_tuple_index(12) == (1, 0, 0) + + assert str(m3) == '[[[10, 11, 12, 13], [14, 15, 16, 17], [18, 19, 20, 21]], [[22, 23, 24, 25], [26, 27, 28, 29], [30, 31, 32, 33]]]' + + m3_rebuilt = MutableDenseNDimArray([[[10, 11, 12, 13], [14, 15, 16, 17], [18, 19, 20, 21]], [[22, 23, 24, 25], [26, 27, 28, 29], [30, 31, 32, 33]]]) + assert m3 == m3_rebuilt + + m3_other = MutableDenseNDimArray([[[10, 11, 12, 13], [14, 15, 16, 17], [18, 19, 20, 21]], [[22, 23, 24, 25], [26, 27, 28, 29], [30, 31, 32, 33]]], (2, 3, 4)) + + assert m3 == m3_other + + +def test_slices(): + md = MutableDenseNDimArray(range(10, 34), (2, 3, 4)) + + assert md[:] == MutableDenseNDimArray(range(10, 34), (2, 3, 4)) + assert md[:, :, 0].tomatrix() == Matrix([[10, 14, 18], [22, 26, 30]]) + assert md[0, 1:2, :].tomatrix() == Matrix([[14, 15, 16, 17]]) + assert md[0, 1:3, :].tomatrix() == Matrix([[14, 15, 16, 17], [18, 19, 20, 21]]) + assert md[:, :, :] == md + + sd = MutableSparseNDimArray(range(10, 34), (2, 3, 4)) + assert sd == MutableSparseNDimArray(md) + + assert sd[:] == MutableSparseNDimArray(range(10, 34), (2, 3, 4)) + assert sd[:, :, 0].tomatrix() == Matrix([[10, 14, 18], [22, 26, 30]]) + assert sd[0, 1:2, :].tomatrix() == Matrix([[14, 15, 16, 17]]) + assert sd[0, 1:3, :].tomatrix() == Matrix([[14, 15, 16, 17], [18, 19, 20, 21]]) + assert sd[:, :, :] == sd + + +def test_slices_assign(): + a = MutableDenseNDimArray(range(12), shape=(4, 3)) + b = MutableSparseNDimArray(range(12), shape=(4, 3)) + + for i in [a, b]: + assert i.tolist() == [[0, 1, 2], [3, 4, 5], [6, 7, 8], [9, 10, 11]] + i[0, :] = [2, 2, 2] + assert i.tolist() == [[2, 2, 2], [3, 4, 5], [6, 7, 8], [9, 10, 11]] + i[0, 1:] = [8, 8] + assert i.tolist() == [[2, 8, 8], [3, 4, 5], [6, 7, 8], [9, 10, 11]] + i[1:3, 1] = [20, 44] + assert i.tolist() == [[2, 8, 8], [3, 20, 5], [6, 44, 8], [9, 10, 11]] + + +def test_diff(): + from sympy.abc import x, y, z + md = MutableDenseNDimArray([[x, y], [x*z, x*y*z]]) + assert md.diff(x) == MutableDenseNDimArray([[1, 0], [z, y*z]]) + assert diff(md, x) == MutableDenseNDimArray([[1, 0], [z, y*z]]) + + sd = MutableSparseNDimArray(md) + assert sd == MutableSparseNDimArray([x, y, x*z, x*y*z], (2, 2)) + assert sd.diff(x) == MutableSparseNDimArray([[1, 0], [z, y*z]]) + assert diff(sd, x) == MutableSparseNDimArray([[1, 0], [z, y*z]]) diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/array/tests/test_ndim_array.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/tests/test_ndim_array.py new file mode 100644 index 0000000000000000000000000000000000000000..7ff9b032631c01272c00478e4cdf0dcbc6997990 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/tests/test_ndim_array.py @@ -0,0 +1,73 @@ +from sympy.testing.pytest import raises +from sympy.functions.elementary.trigonometric import sin, cos +from sympy.matrices.dense import Matrix +from sympy.simplify import simplify +from sympy.tensor.array import Array +from sympy.tensor.array.dense_ndim_array import ( + ImmutableDenseNDimArray, MutableDenseNDimArray) +from sympy.tensor.array.sparse_ndim_array import ( + ImmutableSparseNDimArray, MutableSparseNDimArray) + +from sympy.abc import x, y + +mutable_array_types = [ + MutableDenseNDimArray, + MutableSparseNDimArray +] + +array_types = [ + ImmutableDenseNDimArray, + ImmutableSparseNDimArray, + MutableDenseNDimArray, + MutableSparseNDimArray +] + + +def test_array_negative_indices(): + for ArrayType in array_types: + test_array = ArrayType([[1, 2, 3, 4, 5], [6, 7, 8, 9, 10]]) + assert test_array[:, -1] == Array([5, 10]) + assert test_array[:, -2] == Array([4, 9]) + assert test_array[:, -3] == Array([3, 8]) + assert test_array[:, -4] == Array([2, 7]) + assert test_array[:, -5] == Array([1, 6]) + assert test_array[:, 0] == Array([1, 6]) + assert test_array[:, 1] == Array([2, 7]) + assert test_array[:, 2] == Array([3, 8]) + assert test_array[:, 3] == Array([4, 9]) + assert test_array[:, 4] == Array([5, 10]) + + raises(ValueError, lambda: test_array[:, -6]) + raises(ValueError, lambda: test_array[-3, :]) + + assert test_array[-1, -1] == 10 + + +def test_issue_18361(): + A = Array([sin(2 * x) - 2 * sin(x) * cos(x)]) + B = Array([sin(x)**2 + cos(x)**2, 0]) + C = Array([(x + x**2)/(x*sin(y)**2 + x*cos(y)**2), 2*sin(x)*cos(x)]) + assert simplify(A) == Array([0]) + assert simplify(B) == Array([1, 0]) + assert simplify(C) == Array([x + 1, sin(2*x)]) + + +def test_issue_20222(): + A = Array([[1, 2], [3, 4]]) + B = Matrix([[1,2],[3,4]]) + raises(TypeError, lambda: A - B) + + +def test_issue_17851(): + for array_type in array_types: + A = array_type([]) + assert isinstance(A, array_type) + assert A.shape == (0,) + assert list(A) == [] + + +def test_issue_and_18715(): + for array_type in mutable_array_types: + A = array_type([0, 1, 2]) + A[0] += 5 + assert A[0] == 5 diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/array/tests/test_ndim_array_conversions.py b/.venv/lib/python3.12/site-packages/sympy/tensor/array/tests/test_ndim_array_conversions.py new file mode 100644 index 0000000000000000000000000000000000000000..f43260ccc636ac461ba0c06dbfcf3fe3a8d5338d --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/array/tests/test_ndim_array_conversions.py @@ -0,0 +1,22 @@ +from sympy.tensor.array import (ImmutableDenseNDimArray, + ImmutableSparseNDimArray, MutableDenseNDimArray, MutableSparseNDimArray) +from sympy.abc import x, y, z + + +def test_NDim_array_conv(): + MD = MutableDenseNDimArray([x, y, z]) + MS = MutableSparseNDimArray([x, y, z]) + ID = ImmutableDenseNDimArray([x, y, z]) + IS = ImmutableSparseNDimArray([x, y, z]) + + assert MD.as_immutable() == ID + assert MD.as_mutable() == MD + + assert MS.as_immutable() == IS + assert MS.as_mutable() == MS + + assert ID.as_immutable() == ID + assert ID.as_mutable() == MD + + assert IS.as_immutable() == IS + assert IS.as_mutable() == MS diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/functions.py b/.venv/lib/python3.12/site-packages/sympy/tensor/functions.py new file mode 100644 index 0000000000000000000000000000000000000000..f14599d69152db1713f21c9dd785683901c5eeb9 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/functions.py @@ -0,0 +1,154 @@ +from collections.abc import Iterable +from functools import singledispatch + +from sympy.core.expr import Expr +from sympy.core.mul import Mul +from sympy.core.singleton import S +from sympy.core.sympify import sympify +from sympy.core.parameters import global_parameters + + +class TensorProduct(Expr): + """ + Generic class for tensor products. + """ + is_number = False + + def __new__(cls, *args, **kwargs): + from sympy.tensor.array import NDimArray, tensorproduct, Array + from sympy.matrices.expressions.matexpr import MatrixExpr + from sympy.matrices.matrixbase import MatrixBase + from sympy.strategies import flatten + + args = [sympify(arg) for arg in args] + evaluate = kwargs.get("evaluate", global_parameters.evaluate) + + if not evaluate: + obj = Expr.__new__(cls, *args) + return obj + + arrays = [] + other = [] + scalar = S.One + for arg in args: + if isinstance(arg, (Iterable, MatrixBase, NDimArray)): + arrays.append(Array(arg)) + elif isinstance(arg, (MatrixExpr,)): + other.append(arg) + else: + scalar *= arg + + coeff = scalar*tensorproduct(*arrays) + if len(other) == 0: + return coeff + if coeff != 1: + newargs = [coeff] + other + else: + newargs = other + obj = Expr.__new__(cls, *newargs, **kwargs) + return flatten(obj) + + def rank(self): + return len(self.shape) + + def _get_args_shapes(self): + from sympy.tensor.array import Array + return [i.shape if hasattr(i, "shape") else Array(i).shape for i in self.args] + + @property + def shape(self): + shape_list = self._get_args_shapes() + return sum(shape_list, ()) + + def __getitem__(self, index): + index = iter(index) + return Mul.fromiter( + arg.__getitem__(tuple(next(index) for i in shp)) + for arg, shp in zip(self.args, self._get_args_shapes()) + ) + + +@singledispatch +def shape(expr): + """ + Return the shape of the *expr* as a tuple. *expr* should represent + suitable object such as matrix or array. + + Parameters + ========== + + expr : SymPy object having ``MatrixKind`` or ``ArrayKind``. + + Raises + ====== + + NoShapeError : Raised when object with wrong kind is passed. + + Examples + ======== + + This function returns the shape of any object representing matrix or array. + + >>> from sympy import shape, Array, ImmutableDenseMatrix, Integral + >>> from sympy.abc import x + >>> A = Array([1, 2]) + >>> shape(A) + (2,) + >>> shape(Integral(A, x)) + (2,) + >>> M = ImmutableDenseMatrix([1, 2]) + >>> shape(M) + (2, 1) + >>> shape(Integral(M, x)) + (2, 1) + + You can support new type by dispatching. + + >>> from sympy import Expr + >>> class NewExpr(Expr): + ... pass + >>> @shape.register(NewExpr) + ... def _(expr): + ... return shape(expr.args[0]) + >>> shape(NewExpr(M)) + (2, 1) + + If unsuitable expression is passed, ``NoShapeError()`` will be raised. + + >>> shape(Integral(x, x)) + Traceback (most recent call last): + ... + sympy.tensor.functions.NoShapeError: shape() called on non-array object: Integral(x, x) + + Notes + ===== + + Array-like classes (such as ``Matrix`` or ``NDimArray``) has ``shape`` + property which returns its shape, but it cannot be used for non-array + classes containing array. This function returns the shape of any + registered object representing array. + + """ + if hasattr(expr, "shape"): + return expr.shape + raise NoShapeError( + "%s does not have shape, or its type is not registered to shape()." % expr) + + +class NoShapeError(Exception): + """ + Raised when ``shape()`` is called on non-array object. + + This error can be imported from ``sympy.tensor.functions``. + + Examples + ======== + + >>> from sympy import shape + >>> from sympy.abc import x + >>> shape(x) + Traceback (most recent call last): + ... + sympy.tensor.functions.NoShapeError: shape() called on non-array object: x + """ + pass diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/index_methods.py b/.venv/lib/python3.12/site-packages/sympy/tensor/index_methods.py new file mode 100644 index 0000000000000000000000000000000000000000..12f707b60b4ad0bcadc35a222d9abe0cc5e033fc --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/index_methods.py @@ -0,0 +1,469 @@ +"""Module with functions operating on IndexedBase, Indexed and Idx objects + + - Check shape conformance + - Determine indices in resulting expression + + etc. + + Methods in this module could be implemented by calling methods on Expr + objects instead. When things stabilize this could be a useful + refactoring. +""" + +from functools import reduce + +from sympy.core.function import Function +from sympy.functions import exp, Piecewise +from sympy.tensor.indexed import Idx, Indexed +from sympy.utilities import sift + +from collections import OrderedDict + +class IndexConformanceException(Exception): + pass + +def _unique_and_repeated(inds): + """ + Returns the unique and repeated indices. Also note, from the examples given below + that the order of indices is maintained as given in the input. + + Examples + ======== + + >>> from sympy.tensor.index_methods import _unique_and_repeated + >>> _unique_and_repeated([2, 3, 1, 3, 0, 4, 0]) + ([2, 1, 4], [3, 0]) + """ + uniq = OrderedDict() + for i in inds: + if i in uniq: + uniq[i] = 0 + else: + uniq[i] = 1 + return sift(uniq, lambda x: uniq[x], binary=True) + +def _remove_repeated(inds): + """ + Removes repeated objects from sequences + + Returns a set of the unique objects and a tuple of all that have been + removed. + + Examples + ======== + + >>> from sympy.tensor.index_methods import _remove_repeated + >>> l1 = [1, 2, 3, 2] + >>> _remove_repeated(l1) + ({1, 3}, (2,)) + + """ + u, r = _unique_and_repeated(inds) + return set(u), tuple(r) + + +def _get_indices_Mul(expr, return_dummies=False): + """Determine the outer indices of a Mul object. + + Examples + ======== + + >>> from sympy.tensor.index_methods import _get_indices_Mul + >>> from sympy.tensor.indexed import IndexedBase, Idx + >>> i, j, k = map(Idx, ['i', 'j', 'k']) + >>> x = IndexedBase('x') + >>> y = IndexedBase('y') + >>> _get_indices_Mul(x[i, k]*y[j, k]) + ({i, j}, {}) + >>> _get_indices_Mul(x[i, k]*y[j, k], return_dummies=True) + ({i, j}, {}, (k,)) + + """ + + inds = list(map(get_indices, expr.args)) + inds, syms = list(zip(*inds)) + + inds = list(map(list, inds)) + inds = list(reduce(lambda x, y: x + y, inds)) + inds, dummies = _remove_repeated(inds) + + symmetry = {} + for s in syms: + for pair in s: + if pair in symmetry: + symmetry[pair] *= s[pair] + else: + symmetry[pair] = s[pair] + + if return_dummies: + return inds, symmetry, dummies + else: + return inds, symmetry + + +def _get_indices_Pow(expr): + """Determine outer indices of a power or an exponential. + + A power is considered a universal function, so that the indices of a Pow is + just the collection of indices present in the expression. This may be + viewed as a bit inconsistent in the special case: + + x[i]**2 = x[i]*x[i] (1) + + The above expression could have been interpreted as the contraction of x[i] + with itself, but we choose instead to interpret it as a function + + lambda y: y**2 + + applied to each element of x (a universal function in numpy terms). In + order to allow an interpretation of (1) as a contraction, we need + contravariant and covariant Idx subclasses. (FIXME: this is not yet + implemented) + + Expressions in the base or exponent are subject to contraction as usual, + but an index that is present in the exponent, will not be considered + contractable with its own base. Note however, that indices in the same + exponent can be contracted with each other. + + Examples + ======== + + >>> from sympy.tensor.index_methods import _get_indices_Pow + >>> from sympy import Pow, exp, IndexedBase, Idx + >>> A = IndexedBase('A') + >>> x = IndexedBase('x') + >>> i, j, k = map(Idx, ['i', 'j', 'k']) + >>> _get_indices_Pow(exp(A[i, j]*x[j])) + ({i}, {}) + >>> _get_indices_Pow(Pow(x[i], x[i])) + ({i}, {}) + >>> _get_indices_Pow(Pow(A[i, j]*x[j], x[i])) + ({i}, {}) + + """ + base, exp = expr.as_base_exp() + binds, bsyms = get_indices(base) + einds, esyms = get_indices(exp) + + inds = binds | einds + + # FIXME: symmetries from power needs to check special cases, else nothing + symmetries = {} + + return inds, symmetries + + +def _get_indices_Add(expr): + """Determine outer indices of an Add object. + + In a sum, each term must have the same set of outer indices. A valid + expression could be + + x(i)*y(j) - x(j)*y(i) + + But we do not allow expressions like: + + x(i)*y(j) - z(j)*z(j) + + FIXME: Add support for Numpy broadcasting + + Examples + ======== + + >>> from sympy.tensor.index_methods import _get_indices_Add + >>> from sympy.tensor.indexed import IndexedBase, Idx + >>> i, j, k = map(Idx, ['i', 'j', 'k']) + >>> x = IndexedBase('x') + >>> y = IndexedBase('y') + >>> _get_indices_Add(x[i] + x[k]*y[i, k]) + ({i}, {}) + + """ + + inds = list(map(get_indices, expr.args)) + inds, syms = list(zip(*inds)) + + # allow broadcast of scalars + non_scalars = [x for x in inds if x != set()] + if not non_scalars: + return set(), {} + + if not all(x == non_scalars[0] for x in non_scalars[1:]): + raise IndexConformanceException("Indices are not consistent: %s" % expr) + if not reduce(lambda x, y: x != y or y, syms): + symmetries = syms[0] + else: + # FIXME: search for symmetries + symmetries = {} + + return non_scalars[0], symmetries + + +def get_indices(expr): + """Determine the outer indices of expression ``expr`` + + By *outer* we mean indices that are not summation indices. Returns a set + and a dict. The set contains outer indices and the dict contains + information about index symmetries. + + Examples + ======== + + >>> from sympy.tensor.index_methods import get_indices + >>> from sympy import symbols + >>> from sympy.tensor import IndexedBase + >>> x, y, A = map(IndexedBase, ['x', 'y', 'A']) + >>> i, j, a, z = symbols('i j a z', integer=True) + + The indices of the total expression is determined, Repeated indices imply a + summation, for instance the trace of a matrix A: + + >>> get_indices(A[i, i]) + (set(), {}) + + In the case of many terms, the terms are required to have identical + outer indices. Else an IndexConformanceException is raised. + + >>> get_indices(x[i] + A[i, j]*y[j]) + ({i}, {}) + + :Exceptions: + + An IndexConformanceException means that the terms ar not compatible, e.g. + + >>> get_indices(x[i] + y[j]) #doctest: +SKIP + (...) + IndexConformanceException: Indices are not consistent: x(i) + y(j) + + .. warning:: + The concept of *outer* indices applies recursively, starting on the deepest + level. This implies that dummies inside parenthesis are assumed to be + summed first, so that the following expression is handled gracefully: + + >>> get_indices((x[i] + A[i, j]*y[j])*x[j]) + ({i, j}, {}) + + This is correct and may appear convenient, but you need to be careful + with this as SymPy will happily .expand() the product, if requested. The + resulting expression would mix the outer ``j`` with the dummies inside + the parenthesis, which makes it a different expression. To be on the + safe side, it is best to avoid such ambiguities by using unique indices + for all contractions that should be held separate. + + """ + # We call ourself recursively to determine indices of sub expressions. + + # break recursion + if isinstance(expr, Indexed): + c = expr.indices + inds, dummies = _remove_repeated(c) + return inds, {} + elif expr is None: + return set(), {} + elif isinstance(expr, Idx): + return {expr}, {} + elif expr.is_Atom: + return set(), {} + + + # recurse via specialized functions + else: + if expr.is_Mul: + return _get_indices_Mul(expr) + elif expr.is_Add: + return _get_indices_Add(expr) + elif expr.is_Pow or isinstance(expr, exp): + return _get_indices_Pow(expr) + + elif isinstance(expr, Piecewise): + # FIXME: No support for Piecewise yet + return set(), {} + elif isinstance(expr, Function): + # Support ufunc like behaviour by returning indices from arguments. + # Functions do not interpret repeated indices across arguments + # as summation + ind0 = set() + for arg in expr.args: + ind, sym = get_indices(arg) + ind0 |= ind + return ind0, sym + + # this test is expensive, so it should be at the end + elif not expr.has(Indexed): + return set(), {} + raise NotImplementedError( + "FIXME: No specialized handling of type %s" % type(expr)) + + +def get_contraction_structure(expr): + """Determine dummy indices of ``expr`` and describe its structure + + By *dummy* we mean indices that are summation indices. + + The structure of the expression is determined and described as follows: + + 1) A conforming summation of Indexed objects is described with a dict where + the keys are summation indices and the corresponding values are sets + containing all terms for which the summation applies. All Add objects + in the SymPy expression tree are described like this. + + 2) For all nodes in the SymPy expression tree that are *not* of type Add, the + following applies: + + If a node discovers contractions in one of its arguments, the node + itself will be stored as a key in the dict. For that key, the + corresponding value is a list of dicts, each of which is the result of a + recursive call to get_contraction_structure(). The list contains only + dicts for the non-trivial deeper contractions, omitting dicts with None + as the one and only key. + + .. Note:: The presence of expressions among the dictionary keys indicates + multiple levels of index contractions. A nested dict displays nested + contractions and may itself contain dicts from a deeper level. In + practical calculations the summation in the deepest nested level must be + calculated first so that the outer expression can access the resulting + indexed object. + + Examples + ======== + + >>> from sympy.tensor.index_methods import get_contraction_structure + >>> from sympy import default_sort_key + >>> from sympy.tensor import IndexedBase, Idx + >>> x, y, A = map(IndexedBase, ['x', 'y', 'A']) + >>> i, j, k, l = map(Idx, ['i', 'j', 'k', 'l']) + >>> get_contraction_structure(x[i]*y[i] + A[j, j]) + {(i,): {x[i]*y[i]}, (j,): {A[j, j]}} + >>> get_contraction_structure(x[i]*y[j]) + {None: {x[i]*y[j]}} + + A multiplication of contracted factors results in nested dicts representing + the internal contractions. + + >>> d = get_contraction_structure(x[i, i]*y[j, j]) + >>> sorted(d.keys(), key=default_sort_key) + [None, x[i, i]*y[j, j]] + + In this case, the product has no contractions: + + >>> d[None] + {x[i, i]*y[j, j]} + + Factors are contracted "first": + + >>> sorted(d[x[i, i]*y[j, j]], key=default_sort_key) + [{(i,): {x[i, i]}}, {(j,): {y[j, j]}}] + + A parenthesized Add object is also returned as a nested dictionary. The + term containing the parenthesis is a Mul with a contraction among the + arguments, so it will be found as a key in the result. It stores the + dictionary resulting from a recursive call on the Add expression. + + >>> d = get_contraction_structure(x[i]*(y[i] + A[i, j]*x[j])) + >>> sorted(d.keys(), key=default_sort_key) + [(A[i, j]*x[j] + y[i])*x[i], (i,)] + >>> d[(i,)] + {(A[i, j]*x[j] + y[i])*x[i]} + >>> d[x[i]*(A[i, j]*x[j] + y[i])] + [{None: {y[i]}, (j,): {A[i, j]*x[j]}}] + + Powers with contractions in either base or exponent will also be found as + keys in the dictionary, mapping to a list of results from recursive calls: + + >>> d = get_contraction_structure(A[j, j]**A[i, i]) + >>> d[None] + {A[j, j]**A[i, i]} + >>> nested_contractions = d[A[j, j]**A[i, i]] + >>> nested_contractions[0] + {(j,): {A[j, j]}} + >>> nested_contractions[1] + {(i,): {A[i, i]}} + + The description of the contraction structure may appear complicated when + represented with a string in the above examples, but it is easy to iterate + over: + + >>> from sympy import Expr + >>> for key in d: + ... if isinstance(key, Expr): + ... continue + ... for term in d[key]: + ... if term in d: + ... # treat deepest contraction first + ... pass + ... # treat outermost contactions here + + """ + + # We call ourself recursively to inspect sub expressions. + + if isinstance(expr, Indexed): + junk, key = _remove_repeated(expr.indices) + return {key or None: {expr}} + elif expr.is_Atom: + return {None: {expr}} + elif expr.is_Mul: + junk, junk, key = _get_indices_Mul(expr, return_dummies=True) + result = {key or None: {expr}} + # recurse on every factor + nested = [] + for fac in expr.args: + facd = get_contraction_structure(fac) + if not (None in facd and len(facd) == 1): + nested.append(facd) + if nested: + result[expr] = nested + return result + elif expr.is_Pow or isinstance(expr, exp): + # recurse in base and exp separately. If either has internal + # contractions we must include ourselves as a key in the returned dict + b, e = expr.as_base_exp() + dbase = get_contraction_structure(b) + dexp = get_contraction_structure(e) + + dicts = [] + for d in dbase, dexp: + if not (None in d and len(d) == 1): + dicts.append(d) + result = {None: {expr}} + if dicts: + result[expr] = dicts + return result + elif expr.is_Add: + # Note: we just collect all terms with identical summation indices, We + # do nothing to identify equivalent terms here, as this would require + # substitutions or pattern matching in expressions of unknown + # complexity. + result = {} + for term in expr.args: + # recurse on every term + d = get_contraction_structure(term) + for key in d: + if key in result: + result[key] |= d[key] + else: + result[key] = d[key] + return result + + elif isinstance(expr, Piecewise): + # FIXME: No support for Piecewise yet + return {None: expr} + elif isinstance(expr, Function): + # Collect non-trivial contraction structures in each argument + # We do not report repeated indices in separate arguments as a + # contraction + deeplist = [] + for arg in expr.args: + deep = get_contraction_structure(arg) + if not (None in deep and len(deep) == 1): + deeplist.append(deep) + d = {None: {expr}} + if deeplist: + d[expr] = deeplist + return d + + # this test is expensive, so it should be at the end + elif not expr.has(Indexed): + return {None: {expr}} + raise NotImplementedError( + "FIXME: No specialized handling of type %s" % type(expr)) diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/indexed.py b/.venv/lib/python3.12/site-packages/sympy/tensor/indexed.py new file mode 100644 index 0000000000000000000000000000000000000000..feddad21e52bbab2e1243beafdb11f30b2eded4d --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/indexed.py @@ -0,0 +1,793 @@ +r"""Module that defines indexed objects. + +The classes ``IndexedBase``, ``Indexed``, and ``Idx`` represent a +matrix element ``M[i, j]`` as in the following diagram:: + + 1) The Indexed class represents the entire indexed object. + | + ___|___ + ' ' + M[i, j] + / \__\______ + | | + | | + | 2) The Idx class represents indices; each Idx can + | optionally contain information about its range. + | + 3) IndexedBase represents the 'stem' of an indexed object, here `M`. + The stem used by itself is usually taken to represent the entire + array. + +There can be any number of indices on an Indexed object. No +transformation properties are implemented in these Base objects, but +implicit contraction of repeated indices is supported. + +Note that the support for complicated (i.e. non-atomic) integer +expressions as indices is limited. (This should be improved in +future releases.) + +Examples +======== + +To express the above matrix element example you would write: + +>>> from sympy import symbols, IndexedBase, Idx +>>> M = IndexedBase('M') +>>> i, j = symbols('i j', cls=Idx) +>>> M[i, j] +M[i, j] + +Repeated indices in a product implies a summation, so to express a +matrix-vector product in terms of Indexed objects: + +>>> x = IndexedBase('x') +>>> M[i, j]*x[j] +M[i, j]*x[j] + +If the indexed objects will be converted to component based arrays, e.g. +with the code printers or the autowrap framework, you also need to provide +(symbolic or numerical) dimensions. This can be done by passing an +optional shape parameter to IndexedBase upon construction: + +>>> dim1, dim2 = symbols('dim1 dim2', integer=True) +>>> A = IndexedBase('A', shape=(dim1, 2*dim1, dim2)) +>>> A.shape +(dim1, 2*dim1, dim2) +>>> A[i, j, 3].shape +(dim1, 2*dim1, dim2) + +If an IndexedBase object has no shape information, it is assumed that the +array is as large as the ranges of its indices: + +>>> n, m = symbols('n m', integer=True) +>>> i = Idx('i', m) +>>> j = Idx('j', n) +>>> M[i, j].shape +(m, n) +>>> M[i, j].ranges +[(0, m - 1), (0, n - 1)] + +The above can be compared with the following: + +>>> A[i, 2, j].shape +(dim1, 2*dim1, dim2) +>>> A[i, 2, j].ranges +[(0, m - 1), None, (0, n - 1)] + +To analyze the structure of indexed expressions, you can use the methods +get_indices() and get_contraction_structure(): + +>>> from sympy.tensor import get_indices, get_contraction_structure +>>> get_indices(A[i, j, j]) +({i}, {}) +>>> get_contraction_structure(A[i, j, j]) +{(j,): {A[i, j, j]}} + +See the appropriate docstrings for a detailed explanation of the output. +""" + +# TODO: (some ideas for improvement) +# +# o test and guarantee numpy compatibility +# - implement full support for broadcasting +# - strided arrays +# +# o more functions to analyze indexed expressions +# - identify standard constructs, e.g matrix-vector product in a subexpression +# +# o functions to generate component based arrays (numpy and sympy.Matrix) +# - generate a single array directly from Indexed +# - convert simple sub-expressions +# +# o sophisticated indexing (possibly in subclasses to preserve simplicity) +# - Idx with range smaller than dimension of Indexed +# - Idx with stepsize != 1 +# - Idx with step determined by function call +from collections.abc import Iterable + +from sympy.core.numbers import Number +from sympy.core.assumptions import StdFactKB +from sympy.core import Expr, Tuple, sympify, S +from sympy.core.symbol import _filter_assumptions, Symbol +from sympy.core.logic import fuzzy_bool, fuzzy_not +from sympy.core.sympify import _sympify +from sympy.functions.special.tensor_functions import KroneckerDelta +from sympy.multipledispatch import dispatch +from sympy.utilities.iterables import is_sequence, NotIterable +from sympy.utilities.misc import filldedent + + +class IndexException(Exception): + pass + + +class Indexed(Expr): + """Represents a mathematical object with indices. + + >>> from sympy import Indexed, IndexedBase, Idx, symbols + >>> i, j = symbols('i j', cls=Idx) + >>> Indexed('A', i, j) + A[i, j] + + It is recommended that ``Indexed`` objects be created by indexing ``IndexedBase``: + ``IndexedBase('A')[i, j]`` instead of ``Indexed(IndexedBase('A'), i, j)``. + + >>> A = IndexedBase('A') + >>> a_ij = A[i, j] # Prefer this, + >>> b_ij = Indexed(A, i, j) # over this. + >>> a_ij == b_ij + True + + """ + is_Indexed = True + is_symbol = True + is_Atom = True + + def __new__(cls, base, *args, **kw_args): + from sympy.tensor.array.ndim_array import NDimArray + from sympy.matrices.matrixbase import MatrixBase + + if not args: + raise IndexException("Indexed needs at least one index.") + if isinstance(base, (str, Symbol)): + base = IndexedBase(base) + elif not hasattr(base, '__getitem__') and not isinstance(base, IndexedBase): + raise TypeError(filldedent(""" + The base can only be replaced with a string, Symbol, + IndexedBase or an object with a method for getting + items (i.e. an object with a `__getitem__` method). + """)) + args = list(map(sympify, args)) + if isinstance(base, (NDimArray, Iterable, Tuple, MatrixBase)) and all(i.is_number for i in args): + if len(args) == 1: + return base[args[0]] + else: + return base[args] + + base = _sympify(base) + + obj = Expr.__new__(cls, base, *args, **kw_args) + + IndexedBase._set_assumptions(obj, base.assumptions0) + + return obj + + def _hashable_content(self): + return super()._hashable_content() + tuple(sorted(self.assumptions0.items())) + + @property + def name(self): + return str(self) + + @property + def _diff_wrt(self): + """Allow derivatives with respect to an ``Indexed`` object.""" + return True + + def _eval_derivative(self, wrt): + from sympy.tensor.array.ndim_array import NDimArray + + if isinstance(wrt, Indexed) and wrt.base == self.base: + if len(self.indices) != len(wrt.indices): + msg = "Different # of indices: d({!s})/d({!s})".format(self, + wrt) + raise IndexException(msg) + result = S.One + for index1, index2 in zip(self.indices, wrt.indices): + result *= KroneckerDelta(index1, index2) + return result + elif isinstance(self.base, NDimArray): + from sympy.tensor.array import derive_by_array + return Indexed(derive_by_array(self.base, wrt), *self.args[1:]) + else: + if Tuple(self.indices).has(wrt): + return S.NaN + return S.Zero + + @property + def assumptions0(self): + return {k: v for k, v in self._assumptions.items() if v is not None} + + @property + def base(self): + """Returns the ``IndexedBase`` of the ``Indexed`` object. + + Examples + ======== + + >>> from sympy import Indexed, IndexedBase, Idx, symbols + >>> i, j = symbols('i j', cls=Idx) + >>> Indexed('A', i, j).base + A + >>> B = IndexedBase('B') + >>> B == B[i, j].base + True + + """ + return self.args[0] + + @property + def indices(self): + """ + Returns the indices of the ``Indexed`` object. + + Examples + ======== + + >>> from sympy import Indexed, Idx, symbols + >>> i, j = symbols('i j', cls=Idx) + >>> Indexed('A', i, j).indices + (i, j) + + """ + return self.args[1:] + + @property + def rank(self): + """ + Returns the rank of the ``Indexed`` object. + + Examples + ======== + + >>> from sympy import Indexed, Idx, symbols + >>> i, j, k, l, m = symbols('i:m', cls=Idx) + >>> Indexed('A', i, j).rank + 2 + >>> q = Indexed('A', i, j, k, l, m) + >>> q.rank + 5 + >>> q.rank == len(q.indices) + True + + """ + return len(self.args) - 1 + + @property + def shape(self): + """Returns a list with dimensions of each index. + + Dimensions is a property of the array, not of the indices. Still, if + the ``IndexedBase`` does not define a shape attribute, it is assumed + that the ranges of the indices correspond to the shape of the array. + + >>> from sympy import IndexedBase, Idx, symbols + >>> n, m = symbols('n m', integer=True) + >>> i = Idx('i', m) + >>> j = Idx('j', m) + >>> A = IndexedBase('A', shape=(n, n)) + >>> B = IndexedBase('B') + >>> A[i, j].shape + (n, n) + >>> B[i, j].shape + (m, m) + """ + + if self.base.shape: + return self.base.shape + sizes = [] + for i in self.indices: + upper = getattr(i, 'upper', None) + lower = getattr(i, 'lower', None) + if None in (upper, lower): + raise IndexException(filldedent(""" + Range is not defined for all indices in: %s""" % self)) + try: + size = upper - lower + 1 + except TypeError: + raise IndexException(filldedent(""" + Shape cannot be inferred from Idx with + undefined range: %s""" % self)) + sizes.append(size) + return Tuple(*sizes) + + @property + def ranges(self): + """Returns a list of tuples with lower and upper range of each index. + + If an index does not define the data members upper and lower, the + corresponding slot in the list contains ``None`` instead of a tuple. + + Examples + ======== + + >>> from sympy import Indexed,Idx, symbols + >>> Indexed('A', Idx('i', 2), Idx('j', 4), Idx('k', 8)).ranges + [(0, 1), (0, 3), (0, 7)] + >>> Indexed('A', Idx('i', 3), Idx('j', 3), Idx('k', 3)).ranges + [(0, 2), (0, 2), (0, 2)] + >>> x, y, z = symbols('x y z', integer=True) + >>> Indexed('A', x, y, z).ranges + [None, None, None] + + """ + ranges = [] + sentinel = object() + for i in self.indices: + upper = getattr(i, 'upper', sentinel) + lower = getattr(i, 'lower', sentinel) + if sentinel not in (upper, lower): + ranges.append((lower, upper)) + else: + ranges.append(None) + return ranges + + def _sympystr(self, p): + indices = list(map(p.doprint, self.indices)) + return "%s[%s]" % (p.doprint(self.base), ", ".join(indices)) + + @property + def free_symbols(self): + base_free_symbols = self.base.free_symbols + indices_free_symbols = { + fs for i in self.indices for fs in i.free_symbols} + if base_free_symbols: + return {self} | base_free_symbols | indices_free_symbols + else: + return indices_free_symbols + + @property + def expr_free_symbols(self): + from sympy.utilities.exceptions import sympy_deprecation_warning + sympy_deprecation_warning(""" + The expr_free_symbols property is deprecated. Use free_symbols to get + the free symbols of an expression. + """, + deprecated_since_version="1.9", + active_deprecations_target="deprecated-expr-free-symbols") + + return {self} + + +class IndexedBase(Expr, NotIterable): + """Represent the base or stem of an indexed object + + The IndexedBase class represent an array that contains elements. The main purpose + of this class is to allow the convenient creation of objects of the Indexed + class. The __getitem__ method of IndexedBase returns an instance of + Indexed. Alone, without indices, the IndexedBase class can be used as a + notation for e.g. matrix equations, resembling what you could do with the + Symbol class. But, the IndexedBase class adds functionality that is not + available for Symbol instances: + + - An IndexedBase object can optionally store shape information. This can + be used in to check array conformance and conditions for numpy + broadcasting. (TODO) + - An IndexedBase object implements syntactic sugar that allows easy symbolic + representation of array operations, using implicit summation of + repeated indices. + - The IndexedBase object symbolizes a mathematical structure equivalent + to arrays, and is recognized as such for code generation and automatic + compilation and wrapping. + + >>> from sympy.tensor import IndexedBase, Idx + >>> from sympy import symbols + >>> A = IndexedBase('A'); A + A + >>> type(A) + + + When an IndexedBase object receives indices, it returns an array with named + axes, represented by an Indexed object: + + >>> i, j = symbols('i j', integer=True) + >>> A[i, j, 2] + A[i, j, 2] + >>> type(A[i, j, 2]) + + + The IndexedBase constructor takes an optional shape argument. If given, + it overrides any shape information in the indices. (But not the index + ranges!) + + >>> m, n, o, p = symbols('m n o p', integer=True) + >>> i = Idx('i', m) + >>> j = Idx('j', n) + >>> A[i, j].shape + (m, n) + >>> B = IndexedBase('B', shape=(o, p)) + >>> B[i, j].shape + (o, p) + + Assumptions can be specified with keyword arguments the same way as for Symbol: + + >>> A_real = IndexedBase('A', real=True) + >>> A_real.is_real + True + >>> A != A_real + True + + Assumptions can also be inherited if a Symbol is used to initialize the IndexedBase: + + >>> I = symbols('I', integer=True) + >>> C_inherit = IndexedBase(I) + >>> C_explicit = IndexedBase('I', integer=True) + >>> C_inherit == C_explicit + True + """ + is_symbol = True + is_Atom = True + + @staticmethod + def _set_assumptions(obj, assumptions): + """Set assumptions on obj, making sure to apply consistent values.""" + tmp_asm_copy = assumptions.copy() + is_commutative = fuzzy_bool(assumptions.get('commutative', True)) + assumptions['commutative'] = is_commutative + obj._assumptions = StdFactKB(assumptions) + obj._assumptions._generator = tmp_asm_copy # Issue #8873 + + def __new__(cls, label, shape=None, *, offset=S.Zero, strides=None, **kw_args): + from sympy.matrices.matrixbase import MatrixBase + from sympy.tensor.array.ndim_array import NDimArray + + assumptions, kw_args = _filter_assumptions(kw_args) + if isinstance(label, str): + label = Symbol(label, **assumptions) + elif isinstance(label, Symbol): + assumptions = label._merge(assumptions) + elif isinstance(label, (MatrixBase, NDimArray)): + return label + elif isinstance(label, Iterable): + return _sympify(label) + else: + label = _sympify(label) + + if is_sequence(shape): + shape = Tuple(*shape) + elif shape is not None: + shape = Tuple(shape) + + if shape is not None: + obj = Expr.__new__(cls, label, shape) + else: + obj = Expr.__new__(cls, label) + obj._shape = shape + obj._offset = offset + obj._strides = strides + obj._name = str(label) + + IndexedBase._set_assumptions(obj, assumptions) + return obj + + @property + def name(self): + return self._name + + def _hashable_content(self): + return super()._hashable_content() + tuple(sorted(self.assumptions0.items())) + + @property + def assumptions0(self): + return {k: v for k, v in self._assumptions.items() if v is not None} + + def __getitem__(self, indices, **kw_args): + if is_sequence(indices): + # Special case needed because M[*my_tuple] is a syntax error. + if self.shape and len(self.shape) != len(indices): + raise IndexException("Rank mismatch.") + return Indexed(self, *indices, **kw_args) + else: + if self.shape and len(self.shape) != 1: + raise IndexException("Rank mismatch.") + return Indexed(self, indices, **kw_args) + + @property + def shape(self): + """Returns the shape of the ``IndexedBase`` object. + + Examples + ======== + + >>> from sympy import IndexedBase, Idx + >>> from sympy.abc import x, y + >>> IndexedBase('A', shape=(x, y)).shape + (x, y) + + Note: If the shape of the ``IndexedBase`` is specified, it will override + any shape information given by the indices. + + >>> A = IndexedBase('A', shape=(x, y)) + >>> B = IndexedBase('B') + >>> i = Idx('i', 2) + >>> j = Idx('j', 1) + >>> A[i, j].shape + (x, y) + >>> B[i, j].shape + (2, 1) + + """ + return self._shape + + @property + def strides(self): + """Returns the strided scheme for the ``IndexedBase`` object. + + Normally this is a tuple denoting the number of + steps to take in the respective dimension when traversing + an array. For code generation purposes strides='C' and + strides='F' can also be used. + + strides='C' would mean that code printer would unroll + in row-major order and 'F' means unroll in column major + order. + + """ + + return self._strides + + @property + def offset(self): + """Returns the offset for the ``IndexedBase`` object. + + This is the value added to the resulting index when the + 2D Indexed object is unrolled to a 1D form. Used in code + generation. + + Examples + ========== + >>> from sympy.printing import ccode + >>> from sympy.tensor import IndexedBase, Idx + >>> from sympy import symbols + >>> l, m, n, o = symbols('l m n o', integer=True) + >>> A = IndexedBase('A', strides=(l, m, n), offset=o) + >>> i, j, k = map(Idx, 'ijk') + >>> ccode(A[i, j, k]) + 'A[l*i + m*j + n*k + o]' + + """ + return self._offset + + @property + def label(self): + """Returns the label of the ``IndexedBase`` object. + + Examples + ======== + + >>> from sympy import IndexedBase + >>> from sympy.abc import x, y + >>> IndexedBase('A', shape=(x, y)).label + A + + """ + return self.args[0] + + def _sympystr(self, p): + return p.doprint(self.label) + + +class Idx(Expr): + """Represents an integer index as an ``Integer`` or integer expression. + + There are a number of ways to create an ``Idx`` object. The constructor + takes two arguments: + + ``label`` + An integer or a symbol that labels the index. + ``range`` + Optionally you can specify a range as either + + * ``Symbol`` or integer: This is interpreted as a dimension. Lower and + upper bounds are set to ``0`` and ``range - 1``, respectively. + * ``tuple``: The two elements are interpreted as the lower and upper + bounds of the range, respectively. + + Note: bounds of the range are assumed to be either integer or infinite (oo + and -oo are allowed to specify an unbounded range). If ``n`` is given as a + bound, then ``n.is_integer`` must not return false. + + For convenience, if the label is given as a string it is automatically + converted to an integer symbol. (Note: this conversion is not done for + range or dimension arguments.) + + Examples + ======== + + >>> from sympy import Idx, symbols, oo + >>> n, i, L, U = symbols('n i L U', integer=True) + + If a string is given for the label an integer ``Symbol`` is created and the + bounds are both ``None``: + + >>> idx = Idx('qwerty'); idx + qwerty + >>> idx.lower, idx.upper + (None, None) + + Both upper and lower bounds can be specified: + + >>> idx = Idx(i, (L, U)); idx + i + >>> idx.lower, idx.upper + (L, U) + + When only a single bound is given it is interpreted as the dimension + and the lower bound defaults to 0: + + >>> idx = Idx(i, n); idx.lower, idx.upper + (0, n - 1) + >>> idx = Idx(i, 4); idx.lower, idx.upper + (0, 3) + >>> idx = Idx(i, oo); idx.lower, idx.upper + (0, oo) + + """ + + is_integer = True + is_finite = True + is_real = True + is_symbol = True + is_Atom = True + _diff_wrt = True + + def __new__(cls, label, range=None, **kw_args): + + if isinstance(label, str): + label = Symbol(label, integer=True) + label, range = list(map(sympify, (label, range))) + + if label.is_Number: + if not label.is_integer: + raise TypeError("Index is not an integer number.") + return label + + if not label.is_integer: + raise TypeError("Idx object requires an integer label.") + + elif is_sequence(range): + if len(range) != 2: + raise ValueError(filldedent(""" + Idx range tuple must have length 2, but got %s""" % len(range))) + for bound in range: + if (bound.is_integer is False and bound is not S.Infinity + and bound is not S.NegativeInfinity): + raise TypeError("Idx object requires integer bounds.") + args = label, Tuple(*range) + elif isinstance(range, Expr): + if range is not S.Infinity and fuzzy_not(range.is_integer): + raise TypeError("Idx object requires an integer dimension.") + args = label, Tuple(0, range - 1) + elif range: + raise TypeError(filldedent(""" + The range must be an ordered iterable or + integer SymPy expression.""")) + else: + args = label, + + obj = Expr.__new__(cls, *args, **kw_args) + obj._assumptions["finite"] = True + obj._assumptions["real"] = True + return obj + + @property + def label(self): + """Returns the label (Integer or integer expression) of the Idx object. + + Examples + ======== + + >>> from sympy import Idx, Symbol + >>> x = Symbol('x', integer=True) + >>> Idx(x).label + x + >>> j = Symbol('j', integer=True) + >>> Idx(j).label + j + >>> Idx(j + 1).label + j + 1 + + """ + return self.args[0] + + @property + def lower(self): + """Returns the lower bound of the ``Idx``. + + Examples + ======== + + >>> from sympy import Idx + >>> Idx('j', 2).lower + 0 + >>> Idx('j', 5).lower + 0 + >>> Idx('j').lower is None + True + + """ + try: + return self.args[1][0] + except IndexError: + return + + @property + def upper(self): + """Returns the upper bound of the ``Idx``. + + Examples + ======== + + >>> from sympy import Idx + >>> Idx('j', 2).upper + 1 + >>> Idx('j', 5).upper + 4 + >>> Idx('j').upper is None + True + + """ + try: + return self.args[1][1] + except IndexError: + return + + def _sympystr(self, p): + return p.doprint(self.label) + + @property + def name(self): + return self.label.name if self.label.is_Symbol else str(self.label) + + @property + def free_symbols(self): + return {self} + + +@dispatch(Idx, Idx) +def _eval_is_ge(lhs, rhs): # noqa:F811 + + other_upper = rhs if rhs.upper is None else rhs.upper + other_lower = rhs if rhs.lower is None else rhs.lower + + if lhs.lower is not None and (lhs.lower >= other_upper) == True: + return True + if lhs.upper is not None and (lhs.upper < other_lower) == True: + return False + return None + + +@dispatch(Idx, Number) # type:ignore +def _eval_is_ge(lhs, rhs): # noqa:F811 + + other_upper = rhs + other_lower = rhs + + if lhs.lower is not None and (lhs.lower >= other_upper) == True: + return True + if lhs.upper is not None and (lhs.upper < other_lower) == True: + return False + return None + + +@dispatch(Number, Idx) # type:ignore +def _eval_is_ge(lhs, rhs): # noqa:F811 + + other_upper = lhs + other_lower = lhs + + if rhs.upper is not None and (rhs.upper <= other_lower) == True: + return True + if rhs.lower is not None and (rhs.lower > other_upper) == True: + return False + return None diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/tensor.py b/.venv/lib/python3.12/site-packages/sympy/tensor/tensor.py new file mode 100644 index 0000000000000000000000000000000000000000..579e7c7a86c2a1f18ab889af32ce0053a729ff5f --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/tensor.py @@ -0,0 +1,5265 @@ +""" +This module defines tensors with abstract index notation. + +The abstract index notation has been first formalized by Penrose. + +Tensor indices are formal objects, with a tensor type; there is no +notion of index range, it is only possible to assign the dimension, +used to trace the Kronecker delta; the dimension can be a Symbol. + +The Einstein summation convention is used. +The covariant indices are indicated with a minus sign in front of the index. + +For instance the tensor ``t = p(a)*A(b,c)*q(-c)`` has the index ``c`` +contracted. + +A tensor expression ``t`` can be called; called with its +indices in sorted order it is equal to itself: +in the above example ``t(a, b) == t``; +one can call ``t`` with different indices; ``t(c, d) == p(c)*A(d,a)*q(-a)``. + +The contracted indices are dummy indices, internally they have no name, +the indices being represented by a graph-like structure. + +Tensors are put in canonical form using ``canon_bp``, which uses +the Butler-Portugal algorithm for canonicalization using the monoterm +symmetries of the tensors. + +If there is a (anti)symmetric metric, the indices can be raised and +lowered when the tensor is put in canonical form. +""" + +from __future__ import annotations +from typing import Any +from functools import reduce +from math import prod + +from abc import abstractmethod, ABC +from collections import defaultdict +import operator +import itertools + +from sympy.core.numbers import (Integer, Rational) +from sympy.combinatorics import Permutation +from sympy.combinatorics.tensor_can import get_symmetric_group_sgs, \ + bsgs_direct_product, canonicalize, riemann_bsgs +from sympy.core import Basic, Expr, sympify, Add, Mul, S +from sympy.core.cache import clear_cache +from sympy.core.containers import Tuple, Dict +from sympy.core.function import WildFunction +from sympy.core.sorting import default_sort_key +from sympy.core.symbol import Symbol, symbols, Wild +from sympy.core.sympify import CantSympify, _sympify +from sympy.core.operations import AssocOp +from sympy.external.gmpy import SYMPY_INTS +from sympy.matrices import eye +from sympy.utilities.exceptions import (sympy_deprecation_warning, + SymPyDeprecationWarning, + ignore_warnings) +from sympy.utilities.decorator import memoize_property, deprecated +from sympy.utilities.iterables import sift + + +def deprecate_data(): + sympy_deprecation_warning( + """ + The data attribute of TensorIndexType is deprecated. Use The + replace_with_arrays() method instead. + """, + deprecated_since_version="1.4", + active_deprecations_target="deprecated-tensorindextype-attrs", + stacklevel=4, + ) + +def deprecate_fun_eval(): + sympy_deprecation_warning( + """ + The Tensor.fun_eval() method is deprecated. Use + Tensor.substitute_indices() instead. + """, + deprecated_since_version="1.5", + active_deprecations_target="deprecated-tensor-fun-eval", + stacklevel=4, + ) + + +def deprecate_call(): + sympy_deprecation_warning( + """ + Calling a tensor like Tensor(*indices) is deprecated. Use + Tensor.substitute_indices() instead. + """, + deprecated_since_version="1.5", + active_deprecations_target="deprecated-tensor-fun-eval", + stacklevel=4, + ) + + +class _IndexStructure(CantSympify): + """ + This class handles the indices (free and dummy ones). It contains the + algorithms to manage the dummy indices replacements and contractions of + free indices under multiplications of tensor expressions, as well as stuff + related to canonicalization sorting, getting the permutation of the + expression and so on. It also includes tools to get the ``TensorIndex`` + objects corresponding to the given index structure. + """ + + def __init__(self, free, dum, index_types, indices, canon_bp=False): + self.free = free + self.dum = dum + self.index_types = index_types + self.indices = indices + self._ext_rank = len(self.free) + 2*len(self.dum) + self.dum.sort(key=lambda x: x[0]) + + @staticmethod + def from_indices(*indices): + """ + Create a new ``_IndexStructure`` object from a list of ``indices``. + + Explanation + =========== + + ``indices`` ``TensorIndex`` objects, the indices. Contractions are + detected upon construction. + + Examples + ======== + + >>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, _IndexStructure + >>> Lorentz = TensorIndexType('Lorentz', dummy_name='L') + >>> m0, m1, m2, m3 = tensor_indices('m0,m1,m2,m3', Lorentz) + >>> _IndexStructure.from_indices(m0, m1, -m1, m3) + _IndexStructure([(m0, 0), (m3, 3)], [(1, 2)], [Lorentz, Lorentz, Lorentz, Lorentz]) + """ + + free, dum = _IndexStructure._free_dum_from_indices(*indices) + index_types = [i.tensor_index_type for i in indices] + indices = _IndexStructure._replace_dummy_names(indices, free, dum) + return _IndexStructure(free, dum, index_types, indices) + + @staticmethod + def from_components_free_dum(components, free, dum): + index_types = [] + for component in components: + index_types.extend(component.index_types) + indices = _IndexStructure.generate_indices_from_free_dum_index_types(free, dum, index_types) + return _IndexStructure(free, dum, index_types, indices) + + @staticmethod + def _free_dum_from_indices(*indices): + """ + Convert ``indices`` into ``free``, ``dum`` for single component tensor. + + Explanation + =========== + + ``free`` list of tuples ``(index, pos, 0)``, + where ``pos`` is the position of index in + the list of indices formed by the component tensors + + ``dum`` list of tuples ``(pos_contr, pos_cov, 0, 0)`` + + Examples + ======== + + >>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, \ + _IndexStructure + >>> Lorentz = TensorIndexType('Lorentz', dummy_name='L') + >>> m0, m1, m2, m3 = tensor_indices('m0,m1,m2,m3', Lorentz) + >>> _IndexStructure._free_dum_from_indices(m0, m1, -m1, m3) + ([(m0, 0), (m3, 3)], [(1, 2)]) + """ + n = len(indices) + if n == 1: + return [(indices[0], 0)], [] + + # find the positions of the free indices and of the dummy indices + free = [True]*len(indices) + index_dict = {} + dum = [] + for i, index in enumerate(indices): + name = index.name + typ = index.tensor_index_type + contr = index.is_up + if (name, typ) in index_dict: + # found a pair of dummy indices + is_contr, pos = index_dict[(name, typ)] + # check consistency and update free + if is_contr: + if contr: + raise ValueError('two equal contravariant indices in slots %d and %d' %(pos, i)) + else: + free[pos] = False + free[i] = False + else: + if contr: + free[pos] = False + free[i] = False + else: + raise ValueError('two equal covariant indices in slots %d and %d' %(pos, i)) + if contr: + dum.append((i, pos)) + else: + dum.append((pos, i)) + else: + index_dict[(name, typ)] = index.is_up, i + + free = [(index, i) for i, index in enumerate(indices) if free[i]] + free.sort() + return free, dum + + def get_indices(self): + """ + Get a list of indices, creating new tensor indices to complete dummy indices. + """ + return self.indices[:] + + @staticmethod + def generate_indices_from_free_dum_index_types(free, dum, index_types): + indices = [None]*(len(free)+2*len(dum)) + for idx, pos in free: + indices[pos] = idx + + generate_dummy_name = _IndexStructure._get_generator_for_dummy_indices(free) + for pos1, pos2 in dum: + typ1 = index_types[pos1] + indname = generate_dummy_name(typ1) + indices[pos1] = TensorIndex(indname, typ1, True) + indices[pos2] = TensorIndex(indname, typ1, False) + + return _IndexStructure._replace_dummy_names(indices, free, dum) + + @staticmethod + def _get_generator_for_dummy_indices(free): + cdt = defaultdict(int) + # if the free indices have names with dummy_name, start with an + # index higher than those for the dummy indices + # to avoid name collisions + for indx, ipos in free: + if indx.name.split('_')[0] == indx.tensor_index_type.dummy_name: + cdt[indx.tensor_index_type] = max(cdt[indx.tensor_index_type], int(indx.name.split('_')[1]) + 1) + + def dummy_name_gen(tensor_index_type): + nd = str(cdt[tensor_index_type]) + cdt[tensor_index_type] += 1 + return tensor_index_type.dummy_name + '_' + nd + + return dummy_name_gen + + @staticmethod + def _replace_dummy_names(indices, free, dum): + dum.sort(key=lambda x: x[0]) + new_indices = list(indices) + assert len(indices) == len(free) + 2*len(dum) + generate_dummy_name = _IndexStructure._get_generator_for_dummy_indices(free) + for ipos1, ipos2 in dum: + typ1 = new_indices[ipos1].tensor_index_type + indname = generate_dummy_name(typ1) + new_indices[ipos1] = TensorIndex(indname, typ1, True) + new_indices[ipos2] = TensorIndex(indname, typ1, False) + return new_indices + + def get_free_indices(self) -> list[TensorIndex]: + """ + Get a list of free indices. + """ + # get sorted indices according to their position: + free = sorted(self.free, key=lambda x: x[1]) + return [i[0] for i in free] + + def __str__(self): + return "_IndexStructure({}, {}, {})".format(self.free, self.dum, self.index_types) + + def __repr__(self): + return self.__str__() + + def _get_sorted_free_indices_for_canon(self): + sorted_free = self.free[:] + sorted_free.sort(key=lambda x: x[0]) + return sorted_free + + def _get_sorted_dum_indices_for_canon(self): + return sorted(self.dum, key=lambda x: x[0]) + + def _get_lexicographically_sorted_index_types(self): + permutation = self.indices_canon_args()[0] + index_types = [None]*self._ext_rank + for i, it in enumerate(self.index_types): + index_types[permutation(i)] = it + return index_types + + def _get_lexicographically_sorted_indices(self): + permutation = self.indices_canon_args()[0] + indices = [None]*self._ext_rank + for i, it in enumerate(self.indices): + indices[permutation(i)] = it + return indices + + def perm2tensor(self, g, is_canon_bp=False): + """ + Returns a ``_IndexStructure`` instance corresponding to the permutation ``g``. + + Explanation + =========== + + ``g`` permutation corresponding to the tensor in the representation + used in canonicalization + + ``is_canon_bp`` if True, then ``g`` is the permutation + corresponding to the canonical form of the tensor + """ + sorted_free = [i[0] for i in self._get_sorted_free_indices_for_canon()] + lex_index_types = self._get_lexicographically_sorted_index_types() + lex_indices = self._get_lexicographically_sorted_indices() + nfree = len(sorted_free) + rank = self._ext_rank + dum = [[None]*2 for i in range((rank - nfree)//2)] + free = [] + + index_types = [None]*rank + indices = [None]*rank + for i in range(rank): + gi = g[i] + index_types[i] = lex_index_types[gi] + indices[i] = lex_indices[gi] + if gi < nfree: + ind = sorted_free[gi] + assert index_types[i] == sorted_free[gi].tensor_index_type + free.append((ind, i)) + else: + j = gi - nfree + idum, cov = divmod(j, 2) + if cov: + dum[idum][1] = i + else: + dum[idum][0] = i + dum = [tuple(x) for x in dum] + + return _IndexStructure(free, dum, index_types, indices) + + def indices_canon_args(self): + """ + Returns ``(g, dummies, msym, v)``, the entries of ``canonicalize`` + + See ``canonicalize`` in ``tensor_can.py`` in combinatorics module. + """ + # to be called after sorted_components + from sympy.combinatorics.permutations import _af_new + n = self._ext_rank + g = [None]*n + [n, n+1] + + # Converts the symmetry of the metric into msym from .canonicalize() + # method in the combinatorics module + def metric_symmetry_to_msym(metric): + if metric is None: + return None + sym = metric.symmetry + if sym == TensorSymmetry.fully_symmetric(2): + return 0 + if sym == TensorSymmetry.fully_symmetric(-2): + return 1 + return None + + # ordered indices: first the free indices, ordered by types + # then the dummy indices, ordered by types and contravariant before + # covariant + # g[position in tensor] = position in ordered indices + for i, (indx, ipos) in enumerate(self._get_sorted_free_indices_for_canon()): + g[ipos] = i + pos = len(self.free) + j = len(self.free) + dummies = [] + prev = None + a = [] + msym = [] + for ipos1, ipos2 in self._get_sorted_dum_indices_for_canon(): + g[ipos1] = j + g[ipos2] = j + 1 + j += 2 + typ = self.index_types[ipos1] + if typ != prev: + if a: + dummies.append(a) + a = [pos, pos + 1] + prev = typ + msym.append(metric_symmetry_to_msym(typ.metric)) + else: + a.extend([pos, pos + 1]) + pos += 2 + if a: + dummies.append(a) + + return _af_new(g), dummies, msym + + +def components_canon_args(components): + numtyp = [] + prev = None + for t in components: + if t == prev: + numtyp[-1][1] += 1 + else: + prev = t + numtyp.append([prev, 1]) + v = [] + for h, n in numtyp: + if h.comm in (0, 1): + comm = h.comm + else: + comm = TensorManager.get_comm(h.comm, h.comm) + v.append((h.symmetry.base, h.symmetry.generators, n, comm)) + return v + + +class _TensorDataLazyEvaluator(CantSympify): + """ + EXPERIMENTAL: do not rely on this class, it may change without deprecation + warnings in future versions of SymPy. + + Explanation + =========== + + This object contains the logic to associate components data to a tensor + expression. Components data are set via the ``.data`` property of tensor + expressions, is stored inside this class as a mapping between the tensor + expression and the ``ndarray``. + + Computations are executed lazily: whereas the tensor expressions can have + contractions, tensor products, and additions, components data are not + computed until they are accessed by reading the ``.data`` property + associated to the tensor expression. + """ + _substitutions_dict: dict[Any, Any] = {} + _substitutions_dict_tensmul: dict[Any, Any] = {} + + def __getitem__(self, key): + dat = self._get(key) + if dat is None: + return None + + from .array import NDimArray + if not isinstance(dat, NDimArray): + return dat + + if dat.rank() == 0: + return dat[()] + elif dat.rank() == 1 and len(dat) == 1: + return dat[0] + return dat + + def _get(self, key): + """ + Retrieve ``data`` associated with ``key``. + + Explanation + =========== + + This algorithm looks into ``self._substitutions_dict`` for all + ``TensorHead`` in the ``TensExpr`` (or just ``TensorHead`` if key is a + TensorHead instance). It reconstructs the components data that the + tensor expression should have by performing on components data the + operations that correspond to the abstract tensor operations applied. + + Metric tensor is handled in a different manner: it is pre-computed in + ``self._substitutions_dict_tensmul``. + """ + if key in self._substitutions_dict: + return self._substitutions_dict[key] + + if isinstance(key, TensorHead): + return None + + if isinstance(key, Tensor): + # special case to handle metrics. Metric tensors cannot be + # constructed through contraction by the metric, their + # components show if they are a matrix or its inverse. + signature = tuple([i.is_up for i in key.get_indices()]) + srch = (key.component,) + signature + if srch in self._substitutions_dict_tensmul: + return self._substitutions_dict_tensmul[srch] + array_list = [self.data_from_tensor(key)] + return self.data_contract_dum(array_list, key.dum, key.ext_rank) + + if isinstance(key, TensMul): + tensmul_args = key.args + if len(tensmul_args) == 1 and len(tensmul_args[0].components) == 1: + # special case to handle metrics. Metric tensors cannot be + # constructed through contraction by the metric, their + # components show if they are a matrix or its inverse. + signature = tuple([i.is_up for i in tensmul_args[0].get_indices()]) + srch = (tensmul_args[0].components[0],) + signature + if srch in self._substitutions_dict_tensmul: + return self._substitutions_dict_tensmul[srch] + #data_list = [self.data_from_tensor(i) for i in tensmul_args if isinstance(i, TensExpr)] + data_list = [self.data_from_tensor(i) if isinstance(i, Tensor) else i.data for i in tensmul_args if isinstance(i, TensExpr)] + coeff = prod([i for i in tensmul_args if not isinstance(i, TensExpr)]) + if all(i is None for i in data_list): + return None + if any(i is None for i in data_list): + raise ValueError("Mixing tensors with associated components "\ + "data with tensors without components data") + data_result = self.data_contract_dum(data_list, key.dum, key.ext_rank) + return coeff*data_result + + if isinstance(key, TensAdd): + data_list = [] + free_args_list = [] + for arg in key.args: + if isinstance(arg, TensExpr): + data_list.append(arg.data) + free_args_list.append([x[0] for x in arg.free]) + else: + data_list.append(arg) + free_args_list.append([]) + if all(i is None for i in data_list): + return None + if any(i is None for i in data_list): + raise ValueError("Mixing tensors with associated components "\ + "data with tensors without components data") + + sum_list = [] + from .array import permutedims + for data, free_args in zip(data_list, free_args_list): + if len(free_args) < 2: + sum_list.append(data) + else: + free_args_pos = {y: x for x, y in enumerate(free_args)} + axes = [free_args_pos[arg] for arg in key.free_args] + sum_list.append(permutedims(data, axes)) + return reduce(lambda x, y: x+y, sum_list) + + return None + + @staticmethod + def data_contract_dum(ndarray_list, dum, ext_rank): + from .array import tensorproduct, tensorcontraction, MutableDenseNDimArray + arrays = list(map(MutableDenseNDimArray, ndarray_list)) + prodarr = tensorproduct(*arrays) + return tensorcontraction(prodarr, *dum) + + def data_tensorhead_from_tensmul(self, data, tensmul, tensorhead): + """ + This method is used when assigning components data to a ``TensMul`` + object, it converts components data to a fully contravariant ndarray, + which is then stored according to the ``TensorHead`` key. + """ + if data is None: + return None + + return self._correct_signature_from_indices( + data, + tensmul.get_indices(), + tensmul.free, + tensmul.dum, + True) + + def data_from_tensor(self, tensor): + """ + This method corrects the components data to the right signature + (covariant/contravariant) using the metric associated with each + ``TensorIndexType``. + """ + tensorhead = tensor.component + + if tensorhead.data is None: + return None + + return self._correct_signature_from_indices( + tensorhead.data, + tensor.get_indices(), + tensor.free, + tensor.dum) + + def _assign_data_to_tensor_expr(self, key, data): + if isinstance(key, TensAdd): + raise ValueError('cannot assign data to TensAdd') + # here it is assumed that `key` is a `TensMul` instance. + if len(key.components) != 1: + raise ValueError('cannot assign data to TensMul with multiple components') + tensorhead = key.components[0] + newdata = self.data_tensorhead_from_tensmul(data, key, tensorhead) + return tensorhead, newdata + + def _check_permutations_on_data(self, tens, data): + from .array import permutedims + from .array.arrayop import Flatten + + if isinstance(tens, TensorHead): + rank = tens.rank + generators = tens.symmetry.generators + elif isinstance(tens, Tensor): + rank = tens.rank + generators = tens.components[0].symmetry.generators + elif isinstance(tens, TensorIndexType): + rank = tens.metric.rank + generators = tens.metric.symmetry.generators + + # Every generator is a permutation, check that by permuting the array + # by that permutation, the array will be the same, except for a + # possible sign change if the permutation admits it. + for gener in generators: + sign_change = +1 if (gener(rank) == rank) else -1 + data_swapped = data + last_data = data + permute_axes = list(map(gener, range(rank))) + # the order of a permutation is the number of times to get the + # identity by applying that permutation. + for i in range(gener.order()-1): + data_swapped = permutedims(data_swapped, permute_axes) + # if any value in the difference array is non-zero, raise an error: + if any(Flatten(last_data - sign_change*data_swapped)): + raise ValueError("Component data symmetry structure error") + last_data = data_swapped + + def __setitem__(self, key, value): + """ + Set the components data of a tensor object/expression. + + Explanation + =========== + + Components data are transformed to the all-contravariant form and stored + with the corresponding ``TensorHead`` object. If a ``TensorHead`` object + cannot be uniquely identified, it will raise an error. + """ + data = _TensorDataLazyEvaluator.parse_data(value) + self._check_permutations_on_data(key, data) + + # TensorHead and TensorIndexType can be assigned data directly, while + # TensMul must first convert data to a fully contravariant form, and + # assign it to its corresponding TensorHead single component. + if not isinstance(key, (TensorHead, TensorIndexType)): + key, data = self._assign_data_to_tensor_expr(key, data) + + if isinstance(key, TensorHead): + for dim, indextype in zip(data.shape, key.index_types): + if indextype.data is None: + raise ValueError("index type {} has no components data"\ + " associated (needed to raise/lower index)".format(indextype)) + if not indextype.dim.is_number: + continue + if dim != indextype.dim: + raise ValueError("wrong dimension of ndarray") + self._substitutions_dict[key] = data + + def __delitem__(self, key): + del self._substitutions_dict[key] + + def __contains__(self, key): + return key in self._substitutions_dict + + def add_metric_data(self, metric, data): + """ + Assign data to the ``metric`` tensor. The metric tensor behaves in an + anomalous way when raising and lowering indices. + + Explanation + =========== + + A fully covariant metric is the inverse transpose of the fully + contravariant metric (it is meant matrix inverse). If the metric is + symmetric, the transpose is not necessary and mixed + covariant/contravariant metrics are Kronecker deltas. + """ + # hard assignment, data should not be added to `TensorHead` for metric: + # the problem with `TensorHead` is that the metric is anomalous, i.e. + # raising and lowering the index means considering the metric or its + # inverse, this is not the case for other tensors. + self._substitutions_dict_tensmul[metric, True, True] = data + inverse_transpose = self.inverse_transpose_matrix(data) + # in symmetric spaces, the transpose is the same as the original matrix, + # the full covariant metric tensor is the inverse transpose, so this + # code will be able to handle non-symmetric metrics. + self._substitutions_dict_tensmul[metric, False, False] = inverse_transpose + # now mixed cases, these are identical to the unit matrix if the metric + # is symmetric. + m = data.tomatrix() + invt = inverse_transpose.tomatrix() + self._substitutions_dict_tensmul[metric, True, False] = m * invt + self._substitutions_dict_tensmul[metric, False, True] = invt * m + + @staticmethod + def _flip_index_by_metric(data, metric, pos): + from .array import tensorproduct, tensorcontraction + + mdim = metric.rank() + ddim = data.rank() + + if pos == 0: + data = tensorcontraction( + tensorproduct( + metric, + data + ), + (1, mdim+pos) + ) + else: + data = tensorcontraction( + tensorproduct( + data, + metric + ), + (pos, ddim) + ) + return data + + @staticmethod + def inverse_matrix(ndarray): + m = ndarray.tomatrix().inv() + return _TensorDataLazyEvaluator.parse_data(m) + + @staticmethod + def inverse_transpose_matrix(ndarray): + m = ndarray.tomatrix().inv().T + return _TensorDataLazyEvaluator.parse_data(m) + + @staticmethod + def _correct_signature_from_indices(data, indices, free, dum, inverse=False): + """ + Utility function to correct the values inside the components data + ndarray according to whether indices are covariant or contravariant. + + It uses the metric matrix to lower values of covariant indices. + """ + # change the ndarray values according covariantness/contravariantness of the indices + # use the metric + for i, indx in enumerate(indices): + if not indx.is_up and not inverse: + data = _TensorDataLazyEvaluator._flip_index_by_metric(data, indx.tensor_index_type.data, i) + elif not indx.is_up and inverse: + data = _TensorDataLazyEvaluator._flip_index_by_metric( + data, + _TensorDataLazyEvaluator.inverse_matrix(indx.tensor_index_type.data), + i + ) + return data + + @staticmethod + def _sort_data_axes(old, new): + from .array import permutedims + + new_data = old.data.copy() + + old_free = [i[0] for i in old.free] + new_free = [i[0] for i in new.free] + + for i in range(len(new_free)): + for j in range(i, len(old_free)): + if old_free[j] == new_free[i]: + old_free[i], old_free[j] = old_free[j], old_free[i] + new_data = permutedims(new_data, (i, j)) + break + return new_data + + @staticmethod + def add_rearrange_tensmul_parts(new_tensmul, old_tensmul): + def sorted_compo(): + return _TensorDataLazyEvaluator._sort_data_axes(old_tensmul, new_tensmul) + + _TensorDataLazyEvaluator._substitutions_dict[new_tensmul] = sorted_compo() + + @staticmethod + def parse_data(data): + """ + Transform ``data`` to array. The parameter ``data`` may + contain data in various formats, e.g. nested lists, SymPy ``Matrix``, + and so on. + + Examples + ======== + + >>> from sympy.tensor.tensor import _TensorDataLazyEvaluator + >>> _TensorDataLazyEvaluator.parse_data([1, 3, -6, 12]) + [1, 3, -6, 12] + + >>> _TensorDataLazyEvaluator.parse_data([[1, 2], [4, 7]]) + [[1, 2], [4, 7]] + """ + from .array import MutableDenseNDimArray + + if not isinstance(data, MutableDenseNDimArray): + if len(data) == 2 and hasattr(data[0], '__call__'): + data = MutableDenseNDimArray(data[0], data[1]) + else: + data = MutableDenseNDimArray(data) + return data + +_tensor_data_substitution_dict = _TensorDataLazyEvaluator() + + +class _TensorManager: + """ + Class to manage tensor properties. + + Notes + ===== + + Tensors belong to tensor commutation groups; each group has a label + ``comm``; there are predefined labels: + + ``0`` tensors commuting with any other tensor + + ``1`` tensors anticommuting among themselves + + ``2`` tensors not commuting, apart with those with ``comm=0`` + + Other groups can be defined using ``set_comm``; tensors in those + groups commute with those with ``comm=0``; by default they + do not commute with any other group. + """ + def __init__(self): + self._comm_init() + + def _comm_init(self): + self._comm = [{} for i in range(3)] + for i in range(3): + self._comm[0][i] = 0 + self._comm[i][0] = 0 + self._comm[1][1] = 1 + self._comm[2][1] = None + self._comm[1][2] = None + self._comm_symbols2i = {0:0, 1:1, 2:2} + self._comm_i2symbol = {0:0, 1:1, 2:2} + + @property + def comm(self): + return self._comm + + def comm_symbols2i(self, i): + """ + Get the commutation group number corresponding to ``i``. + + ``i`` can be a symbol or a number or a string. + + If ``i`` is not already defined its commutation group number + is set. + """ + if i not in self._comm_symbols2i: + n = len(self._comm) + self._comm.append({}) + self._comm[n][0] = 0 + self._comm[0][n] = 0 + self._comm_symbols2i[i] = n + self._comm_i2symbol[n] = i + return n + return self._comm_symbols2i[i] + + def comm_i2symbol(self, i): + """ + Returns the symbol corresponding to the commutation group number. + """ + return self._comm_i2symbol[i] + + def set_comm(self, i, j, c): + """ + Set the commutation parameter ``c`` for commutation groups ``i, j``. + + Parameters + ========== + + i, j : symbols representing commutation groups + + c : group commutation number + + Notes + ===== + + ``i, j`` can be symbols, strings or numbers, + apart from ``0, 1`` and ``2`` which are reserved respectively + for commuting, anticommuting tensors and tensors not commuting + with any other group apart with the commuting tensors. + For the remaining cases, use this method to set the commutation rules; + by default ``c=None``. + + The group commutation number ``c`` is assigned in correspondence + to the group commutation symbols; it can be + + 0 commuting + + 1 anticommuting + + None no commutation property + + Examples + ======== + + ``G`` and ``GH`` do not commute with themselves and commute with + each other; A is commuting. + + >>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, TensorHead, TensorManager, TensorSymmetry + >>> Lorentz = TensorIndexType('Lorentz') + >>> i0,i1,i2,i3,i4 = tensor_indices('i0:5', Lorentz) + >>> A = TensorHead('A', [Lorentz]) + >>> G = TensorHead('G', [Lorentz], TensorSymmetry.no_symmetry(1), 'Gcomm') + >>> GH = TensorHead('GH', [Lorentz], TensorSymmetry.no_symmetry(1), 'GHcomm') + >>> TensorManager.set_comm('Gcomm', 'GHcomm', 0) + >>> (GH(i1)*G(i0)).canon_bp() + G(i0)*GH(i1) + >>> (G(i1)*G(i0)).canon_bp() + G(i1)*G(i0) + >>> (G(i1)*A(i0)).canon_bp() + A(i0)*G(i1) + """ + if c not in (0, 1, None): + raise ValueError('`c` can assume only the values 0, 1 or None') + + i = sympify(i) + j = sympify(j) + + if i not in self._comm_symbols2i: + n = len(self._comm) + self._comm.append({}) + self._comm[n][0] = 0 + self._comm[0][n] = 0 + self._comm_symbols2i[i] = n + self._comm_i2symbol[n] = i + if j not in self._comm_symbols2i: + n = len(self._comm) + self._comm.append({}) + self._comm[0][n] = 0 + self._comm[n][0] = 0 + self._comm_symbols2i[j] = n + self._comm_i2symbol[n] = j + ni = self._comm_symbols2i[i] + nj = self._comm_symbols2i[j] + self._comm[ni][nj] = c + self._comm[nj][ni] = c + + """ + Cached sympy functions (e.g. expand) may have cached the results of + expressions involving tensors, but those results may not be valid after + changing the commutation properties. To stay on the safe side, we clear + the cache of all functions. + """ + clear_cache() + + def set_comms(self, *args): + """ + Set the commutation group numbers ``c`` for symbols ``i, j``. + + Parameters + ========== + + args : sequence of ``(i, j, c)`` + """ + for i, j, c in args: + self.set_comm(i, j, c) + + def get_comm(self, i, j): + """ + Return the commutation parameter for commutation group numbers ``i, j`` + + see ``_TensorManager.set_comm`` + """ + return self._comm[i].get(j, 0 if i == 0 or j == 0 else None) + + def clear(self): + """ + Clear the TensorManager. + """ + self._comm_init() + + +TensorManager = _TensorManager() + + +class TensorIndexType(Basic): + """ + A TensorIndexType is characterized by its name and its metric. + + Parameters + ========== + + name : name of the tensor type + dummy_name : name of the head of dummy indices + dim : dimension, it can be a symbol or an integer or ``None`` + eps_dim : dimension of the epsilon tensor + metric_symmetry : integer that denotes metric symmetry or ``None`` for no metric + metric_name : string with the name of the metric tensor + + Attributes + ========== + + ``metric`` : the metric tensor + ``delta`` : ``Kronecker delta`` + ``epsilon`` : the ``Levi-Civita epsilon`` tensor + ``data`` : (deprecated) a property to add ``ndarray`` values, to work in a specified basis. + + Notes + ===== + + The possible values of the ``metric_symmetry`` parameter are: + + ``1`` : metric tensor is fully symmetric + ``0`` : metric tensor possesses no index symmetry + ``-1`` : metric tensor is fully antisymmetric + ``None``: there is no metric tensor (metric equals to ``None``) + + The metric is assumed to be symmetric by default. It can also be set + to a custom tensor by the ``.set_metric()`` method. + + If there is a metric the metric is used to raise and lower indices. + + In the case of non-symmetric metric, the following raising and + lowering conventions will be adopted: + + ``psi(a) = g(a, b)*psi(-b); chi(-a) = chi(b)*g(-b, -a)`` + + From these it is easy to find: + + ``g(-a, b) = delta(-a, b)`` + + where ``delta(-a, b) = delta(b, -a)`` is the ``Kronecker delta`` + (see ``TensorIndex`` for the conventions on indices). + For antisymmetric metrics there is also the following equality: + + ``g(a, -b) = -delta(a, -b)`` + + If there is no metric it is not possible to raise or lower indices; + e.g. the index of the defining representation of ``SU(N)`` + is 'covariant' and the conjugate representation is + 'contravariant'; for ``N > 2`` they are linearly independent. + + ``eps_dim`` is by default equal to ``dim``, if the latter is an integer; + else it can be assigned (for use in naive dimensional regularization); + if ``eps_dim`` is not an integer ``epsilon`` is ``None``. + + Examples + ======== + + >>> from sympy.tensor.tensor import TensorIndexType + >>> Lorentz = TensorIndexType('Lorentz', dummy_name='L') + >>> Lorentz.metric + metric(Lorentz,Lorentz) + """ + + def __new__(cls, name, dummy_name=None, dim=None, eps_dim=None, + metric_symmetry=1, metric_name='metric', **kwargs): + if 'dummy_fmt' in kwargs: + dummy_fmt = kwargs['dummy_fmt'] + sympy_deprecation_warning( + f""" + The dummy_fmt keyword to TensorIndexType is deprecated. Use + dummy_name={dummy_fmt} instead. + """, + deprecated_since_version="1.5", + active_deprecations_target="deprecated-tensorindextype-dummy-fmt", + ) + dummy_name = dummy_fmt + + if isinstance(name, str): + name = Symbol(name) + + if dummy_name is None: + dummy_name = str(name)[0] + if isinstance(dummy_name, str): + dummy_name = Symbol(dummy_name) + + if dim is None: + dim = Symbol("dim_" + dummy_name.name) + else: + dim = sympify(dim) + + if eps_dim is None: + eps_dim = dim + else: + eps_dim = sympify(eps_dim) + + metric_symmetry = sympify(metric_symmetry) + + if isinstance(metric_name, str): + metric_name = Symbol(metric_name) + + if 'metric' in kwargs: + SymPyDeprecationWarning( + """ + The 'metric' keyword argument to TensorIndexType is + deprecated. Use the 'metric_symmetry' keyword argument or the + TensorIndexType.set_metric() method instead. + """, + deprecated_since_version="1.5", + active_deprecations_target="deprecated-tensorindextype-metric", + ) + metric = kwargs.get('metric') + if metric is not None: + if metric in (True, False, 0, 1): + metric_name = 'metric' + #metric_antisym = metric + else: + metric_name = metric.name + #metric_antisym = metric.antisym + + if metric: + metric_symmetry = -1 + else: + metric_symmetry = 1 + + obj = Basic.__new__(cls, name, dummy_name, dim, eps_dim, + metric_symmetry, metric_name) + + obj._autogenerated = [] + return obj + + @property + def name(self): + return self.args[0].name + + @property + def dummy_name(self): + return self.args[1].name + + @property + def dim(self): + return self.args[2] + + @property + def eps_dim(self): + return self.args[3] + + @memoize_property + def metric(self): + metric_symmetry = self.args[4] + metric_name = self.args[5] + if metric_symmetry is None: + return None + + if metric_symmetry == 0: + symmetry = TensorSymmetry.no_symmetry(2) + elif metric_symmetry == 1: + symmetry = TensorSymmetry.fully_symmetric(2) + elif metric_symmetry == -1: + symmetry = TensorSymmetry.fully_symmetric(-2) + + return TensorHead(metric_name, [self]*2, symmetry) + + @memoize_property + def delta(self): + return TensorHead('KD', [self]*2, TensorSymmetry.fully_symmetric(2)) + + @memoize_property + def epsilon(self): + if not isinstance(self.eps_dim, (SYMPY_INTS, Integer)): + return None + symmetry = TensorSymmetry.fully_symmetric(-self.eps_dim) + return TensorHead('Eps', [self]*self.eps_dim, symmetry) + + def set_metric(self, tensor): + self._metric = tensor + + def __lt__(self, other): + return self.name < other.name + + def __str__(self): + return self.name + + __repr__ = __str__ + + # Everything below this line is deprecated + + @property + def data(self): + deprecate_data() + with ignore_warnings(SymPyDeprecationWarning): + return _tensor_data_substitution_dict[self] + + @data.setter + def data(self, data): + deprecate_data() + # This assignment is a bit controversial, should metric components be assigned + # to the metric only or also to the TensorIndexType object? The advantage here + # is the ability to assign a 1D array and transform it to a 2D diagonal array. + from .array import MutableDenseNDimArray + + data = _TensorDataLazyEvaluator.parse_data(data) + if data.rank() > 2: + raise ValueError("data have to be of rank 1 (diagonal metric) or 2.") + if data.rank() == 1: + if self.dim.is_number: + nda_dim = data.shape[0] + if nda_dim != self.dim: + raise ValueError("Dimension mismatch") + + dim = data.shape[0] + newndarray = MutableDenseNDimArray.zeros(dim, dim) + for i, val in enumerate(data): + newndarray[i, i] = val + data = newndarray + dim1, dim2 = data.shape + if dim1 != dim2: + raise ValueError("Non-square matrix tensor.") + if self.dim.is_number: + if self.dim != dim1: + raise ValueError("Dimension mismatch") + _tensor_data_substitution_dict[self] = data + _tensor_data_substitution_dict.add_metric_data(self.metric, data) + with ignore_warnings(SymPyDeprecationWarning): + delta = self.get_kronecker_delta() + i1 = TensorIndex('i1', self) + i2 = TensorIndex('i2', self) + with ignore_warnings(SymPyDeprecationWarning): + delta(i1, -i2).data = _TensorDataLazyEvaluator.parse_data(eye(dim1)) + + @data.deleter + def data(self): + deprecate_data() + with ignore_warnings(SymPyDeprecationWarning): + if self in _tensor_data_substitution_dict: + del _tensor_data_substitution_dict[self] + if self.metric in _tensor_data_substitution_dict: + del _tensor_data_substitution_dict[self.metric] + + @deprecated( + """ + The TensorIndexType.get_kronecker_delta() method is deprecated. Use + the TensorIndexType.delta attribute instead. + """, + deprecated_since_version="1.5", + active_deprecations_target="deprecated-tensorindextype-methods", + ) + def get_kronecker_delta(self): + sym2 = TensorSymmetry(get_symmetric_group_sgs(2)) + delta = TensorHead('KD', [self]*2, sym2) + return delta + + @deprecated( + """ + The TensorIndexType.get_epsilon() method is deprecated. Use + the TensorIndexType.epsilon attribute instead. + """, + deprecated_since_version="1.5", + active_deprecations_target="deprecated-tensorindextype-methods", + ) + def get_epsilon(self): + if not isinstance(self._eps_dim, (SYMPY_INTS, Integer)): + return None + sym = TensorSymmetry(get_symmetric_group_sgs(self._eps_dim, 1)) + epsilon = TensorHead('Eps', [self]*self._eps_dim, sym) + return epsilon + + def _components_data_full_destroy(self): + """ + EXPERIMENTAL: do not rely on this API method. + + This destroys components data associated to the ``TensorIndexType``, if + any, specifically: + + * metric tensor data + * Kronecker tensor data + """ + if self in _tensor_data_substitution_dict: + del _tensor_data_substitution_dict[self] + + def delete_tensmul_data(key): + if key in _tensor_data_substitution_dict._substitutions_dict_tensmul: + del _tensor_data_substitution_dict._substitutions_dict_tensmul[key] + + # delete metric data: + delete_tensmul_data((self.metric, True, True)) + delete_tensmul_data((self.metric, True, False)) + delete_tensmul_data((self.metric, False, True)) + delete_tensmul_data((self.metric, False, False)) + + # delete delta tensor data: + delta = self.get_kronecker_delta() + if delta in _tensor_data_substitution_dict: + del _tensor_data_substitution_dict[delta] + + +class TensorIndex(Basic): + """ + Represents a tensor index + + Parameters + ========== + + name : name of the index, or ``True`` if you want it to be automatically assigned + tensor_index_type : ``TensorIndexType`` of the index + is_up : flag for contravariant index (is_up=True by default) + + Attributes + ========== + + ``name`` + ``tensor_index_type`` + ``is_up`` + + Notes + ===== + + Tensor indices are contracted with the Einstein summation convention. + + An index can be in contravariant or in covariant form; in the latter + case it is represented prepending a ``-`` to the index name. Adding + ``-`` to a covariant (is_up=False) index makes it contravariant. + + Dummy indices have a name with head given by + ``tensor_inde_type.dummy_name`` with underscore and a number. + + Similar to ``symbols`` multiple contravariant indices can be created + at once using ``tensor_indices(s, typ)``, where ``s`` is a string + of names. + + + Examples + ======== + + >>> from sympy.tensor.tensor import TensorIndexType, TensorIndex, TensorHead, tensor_indices + >>> Lorentz = TensorIndexType('Lorentz', dummy_name='L') + >>> mu = TensorIndex('mu', Lorentz, is_up=False) + >>> nu, rho = tensor_indices('nu, rho', Lorentz) + >>> A = TensorHead('A', [Lorentz, Lorentz]) + >>> A(mu, nu) + A(-mu, nu) + >>> A(-mu, -rho) + A(mu, -rho) + >>> A(mu, -mu) + A(-L_0, L_0) + """ + def __new__(cls, name, tensor_index_type, is_up=True): + if isinstance(name, str): + name_symbol = Symbol(name) + elif isinstance(name, Symbol): + name_symbol = name + elif name is True: + name = "_i{}".format(len(tensor_index_type._autogenerated)) + name_symbol = Symbol(name) + tensor_index_type._autogenerated.append(name_symbol) + else: + raise ValueError("invalid name") + + is_up = sympify(is_up) + return Basic.__new__(cls, name_symbol, tensor_index_type, is_up) + + @property + def name(self): + return self.args[0].name + + @property + def tensor_index_type(self): + return self.args[1] + + @property + def is_up(self): + return self.args[2] + + def _print(self): + s = self.name + if not self.is_up: + s = '-%s' % s + return s + + def __lt__(self, other): + return ((self.tensor_index_type, self.name) < + (other.tensor_index_type, other.name)) + + def __neg__(self): + t1 = TensorIndex(self.name, self.tensor_index_type, + (not self.is_up)) + return t1 + + +def tensor_indices(s, typ): + """ + Returns list of tensor indices given their names and their types. + + Parameters + ========== + + s : string of comma separated names of indices + + typ : ``TensorIndexType`` of the indices + + Examples + ======== + + >>> from sympy.tensor.tensor import TensorIndexType, tensor_indices + >>> Lorentz = TensorIndexType('Lorentz', dummy_name='L') + >>> a, b, c, d = tensor_indices('a,b,c,d', Lorentz) + """ + if isinstance(s, str): + a = [x.name for x in symbols(s, seq=True)] + else: + raise ValueError('expecting a string') + + tilist = [TensorIndex(i, typ) for i in a] + if len(tilist) == 1: + return tilist[0] + return tilist + + +class TensorSymmetry(Basic): + """ + Monoterm symmetry of a tensor (i.e. any symmetric or anti-symmetric + index permutation). For the relevant terminology see ``tensor_can.py`` + section of the combinatorics module. + + Parameters + ========== + + bsgs : tuple ``(base, sgs)`` BSGS of the symmetry of the tensor + + Attributes + ========== + + ``base`` : base of the BSGS + ``generators`` : generators of the BSGS + ``rank`` : rank of the tensor + + Notes + ===== + + A tensor can have an arbitrary monoterm symmetry provided by its BSGS. + Multiterm symmetries, like the cyclic symmetry of the Riemann tensor + (i.e., Bianchi identity), are not covered. See combinatorics module for + information on how to generate BSGS for a general index permutation group. + Simple symmetries can be generated using built-in methods. + + See Also + ======== + + sympy.combinatorics.tensor_can.get_symmetric_group_sgs + + Examples + ======== + + Define a symmetric tensor of rank 2 + + >>> from sympy.tensor.tensor import TensorIndexType, TensorSymmetry, get_symmetric_group_sgs, TensorHead + >>> Lorentz = TensorIndexType('Lorentz', dummy_name='L') + >>> sym = TensorSymmetry(get_symmetric_group_sgs(2)) + >>> T = TensorHead('T', [Lorentz]*2, sym) + + Note, that the same can also be done using built-in TensorSymmetry methods + + >>> sym2 = TensorSymmetry.fully_symmetric(2) + >>> sym == sym2 + True + """ + def __new__(cls, *args, **kw_args): + if len(args) == 1: + base, generators = args[0] + elif len(args) == 2: + base, generators = args + else: + raise TypeError("bsgs required, either two separate parameters or one tuple") + + if not isinstance(base, Tuple): + base = Tuple(*base) + if not isinstance(generators, Tuple): + generators = Tuple(*generators) + + return Basic.__new__(cls, base, generators, **kw_args) + + @property + def base(self): + return self.args[0] + + @property + def generators(self): + return self.args[1] + + @property + def rank(self): + return self.generators[0].size - 2 + + @classmethod + def fully_symmetric(cls, rank): + """ + Returns a fully symmetric (antisymmetric if ``rank``<0) + TensorSymmetry object for ``abs(rank)`` indices. + """ + if rank > 0: + bsgs = get_symmetric_group_sgs(rank, False) + elif rank < 0: + bsgs = get_symmetric_group_sgs(-rank, True) + elif rank == 0: + bsgs = ([], [Permutation(1)]) + return TensorSymmetry(bsgs) + + @classmethod + def direct_product(cls, *args): + """ + Returns a TensorSymmetry object that is being a direct product of + fully (anti-)symmetric index permutation groups. + + Notes + ===== + + Some examples for different values of ``(*args)``: + ``(1)`` vector, equivalent to ``TensorSymmetry.fully_symmetric(1)`` + ``(2)`` tensor with 2 symmetric indices, equivalent to ``.fully_symmetric(2)`` + ``(-2)`` tensor with 2 antisymmetric indices, equivalent to ``.fully_symmetric(-2)`` + ``(2, -2)`` tensor with the first 2 indices commuting and the last 2 anticommuting + ``(1, 1, 1)`` tensor with 3 indices without any symmetry + """ + base, sgs = [], [Permutation(1)] + for arg in args: + if arg > 0: + bsgs2 = get_symmetric_group_sgs(arg, False) + elif arg < 0: + bsgs2 = get_symmetric_group_sgs(-arg, True) + else: + continue + base, sgs = bsgs_direct_product(base, sgs, *bsgs2) + + return TensorSymmetry(base, sgs) + + @classmethod + def riemann(cls): + """ + Returns a monotorem symmetry of the Riemann tensor + """ + return TensorSymmetry(riemann_bsgs) + + @classmethod + def no_symmetry(cls, rank): + """ + TensorSymmetry object for ``rank`` indices with no symmetry + """ + return TensorSymmetry([], [Permutation(rank+1)]) + + +@deprecated( + """ + The tensorsymmetry() function is deprecated. Use the TensorSymmetry + constructor instead. + """, + deprecated_since_version="1.5", + active_deprecations_target="deprecated-tensorsymmetry", +) +def tensorsymmetry(*args): + """ + Returns a ``TensorSymmetry`` object. This method is deprecated, use + ``TensorSymmetry.direct_product()`` or ``.riemann()`` instead. + + Explanation + =========== + + One can represent a tensor with any monoterm slot symmetry group + using a BSGS. + + ``args`` can be a BSGS + ``args[0]`` base + ``args[1]`` sgs + + Usually tensors are in (direct products of) representations + of the symmetric group; + ``args`` can be a list of lists representing the shapes of Young tableaux + + Notes + ===== + + For instance: + ``[[1]]`` vector + ``[[1]*n]`` symmetric tensor of rank ``n`` + ``[[n]]`` antisymmetric tensor of rank ``n`` + ``[[2, 2]]`` monoterm slot symmetry of the Riemann tensor + ``[[1],[1]]`` vector*vector + ``[[2],[1],[1]`` (antisymmetric tensor)*vector*vector + + Notice that with the shape ``[2, 2]`` we associate only the monoterm + symmetries of the Riemann tensor; this is an abuse of notation, + since the shape ``[2, 2]`` corresponds usually to the irreducible + representation characterized by the monoterm symmetries and by the + cyclic symmetry. + """ + from sympy.combinatorics import Permutation + + def tableau2bsgs(a): + if len(a) == 1: + # antisymmetric vector + n = a[0] + bsgs = get_symmetric_group_sgs(n, 1) + else: + if all(x == 1 for x in a): + # symmetric vector + n = len(a) + bsgs = get_symmetric_group_sgs(n) + elif a == [2, 2]: + bsgs = riemann_bsgs + else: + raise NotImplementedError + return bsgs + + if not args: + return TensorSymmetry(Tuple(), Tuple(Permutation(1))) + + if len(args) == 2 and isinstance(args[1][0], Permutation): + return TensorSymmetry(args) + base, sgs = tableau2bsgs(args[0]) + for a in args[1:]: + basex, sgsx = tableau2bsgs(a) + base, sgs = bsgs_direct_product(base, sgs, basex, sgsx) + return TensorSymmetry(Tuple(base, sgs)) + +@deprecated( + "TensorType is deprecated. Use tensor_heads() instead.", + deprecated_since_version="1.5", + active_deprecations_target="deprecated-tensortype", +) +class TensorType(Basic): + """ + Class of tensor types. Deprecated, use tensor_heads() instead. + + Parameters + ========== + + index_types : list of ``TensorIndexType`` of the tensor indices + symmetry : ``TensorSymmetry`` of the tensor + + Attributes + ========== + + ``index_types`` + ``symmetry`` + ``types`` : list of ``TensorIndexType`` without repetitions + """ + is_commutative = False + + def __new__(cls, index_types, symmetry, **kw_args): + assert symmetry.rank == len(index_types) + obj = Basic.__new__(cls, Tuple(*index_types), symmetry, **kw_args) + return obj + + @property + def index_types(self): + return self.args[0] + + @property + def symmetry(self): + return self.args[1] + + @property + def types(self): + return sorted(set(self.index_types), key=lambda x: x.name) + + def __str__(self): + return 'TensorType(%s)' % ([str(x) for x in self.index_types]) + + def __call__(self, s, comm=0): + """ + Return a TensorHead object or a list of TensorHead objects. + + Parameters + ========== + + s : name or string of names. + + comm : Commutation group. + + see ``_TensorManager.set_comm`` + """ + if isinstance(s, str): + names = [x.name for x in symbols(s, seq=True)] + else: + raise ValueError('expecting a string') + if len(names) == 1: + return TensorHead(names[0], self.index_types, self.symmetry, comm) + else: + return [TensorHead(name, self.index_types, self.symmetry, comm) for name in names] + + +@deprecated( + """ + The tensorhead() function is deprecated. Use tensor_heads() instead. + """, + deprecated_since_version="1.5", + active_deprecations_target="deprecated-tensorhead", +) +def tensorhead(name, typ, sym=None, comm=0): + """ + Function generating tensorhead(s). This method is deprecated, + use TensorHead constructor or tensor_heads() instead. + + Parameters + ========== + + name : name or sequence of names (as in ``symbols``) + + typ : index types + + sym : same as ``*args`` in ``tensorsymmetry`` + + comm : commutation group number + see ``_TensorManager.set_comm`` + """ + if sym is None: + sym = [[1] for i in range(len(typ))] + with ignore_warnings(SymPyDeprecationWarning): + sym = tensorsymmetry(*sym) + return TensorHead(name, typ, sym, comm) + + +class TensorHead(Basic): + """ + Tensor head of the tensor. + + Parameters + ========== + + name : name of the tensor + index_types : list of TensorIndexType + symmetry : TensorSymmetry of the tensor + comm : commutation group number + + Attributes + ========== + + ``name`` + ``index_types`` + ``rank`` : total number of indices + ``symmetry`` + ``comm`` : commutation group + + Notes + ===== + + Similar to ``symbols`` multiple TensorHeads can be created using + ``tensorhead(s, typ, sym=None, comm=0)`` function, where ``s`` + is the string of names and ``sym`` is the monoterm tensor symmetry + (see ``tensorsymmetry``). + + A ``TensorHead`` belongs to a commutation group, defined by a + symbol on number ``comm`` (see ``_TensorManager.set_comm``); + tensors in a commutation group have the same commutation properties; + by default ``comm`` is ``0``, the group of the commuting tensors. + + Examples + ======== + + Define a fully antisymmetric tensor of rank 2: + + >>> from sympy.tensor.tensor import TensorIndexType, TensorHead, TensorSymmetry + >>> Lorentz = TensorIndexType('Lorentz', dummy_name='L') + >>> asym2 = TensorSymmetry.fully_symmetric(-2) + >>> A = TensorHead('A', [Lorentz, Lorentz], asym2) + + Examples with ndarray values, the components data assigned to the + ``TensorHead`` object are assumed to be in a fully-contravariant + representation. In case it is necessary to assign components data which + represents the values of a non-fully covariant tensor, see the other + examples. + + >>> from sympy.tensor.tensor import tensor_indices + >>> from sympy import diag + >>> Lorentz = TensorIndexType('Lorentz', dummy_name='L') + >>> i0, i1 = tensor_indices('i0:2', Lorentz) + + Specify a replacement dictionary to keep track of the arrays to use for + replacements in the tensorial expression. The ``TensorIndexType`` is + associated to the metric used for contractions (in fully covariant form): + + >>> repl = {Lorentz: diag(1, -1, -1, -1)} + + Let's see some examples of working with components with the electromagnetic + tensor: + + >>> from sympy import symbols + >>> Ex, Ey, Ez, Bx, By, Bz = symbols('E_x E_y E_z B_x B_y B_z') + >>> c = symbols('c', positive=True) + + Let's define `F`, an antisymmetric tensor: + + >>> F = TensorHead('F', [Lorentz, Lorentz], asym2) + + Let's update the dictionary to contain the matrix to use in the + replacements: + + >>> repl.update({F(-i0, -i1): [ + ... [0, Ex/c, Ey/c, Ez/c], + ... [-Ex/c, 0, -Bz, By], + ... [-Ey/c, Bz, 0, -Bx], + ... [-Ez/c, -By, Bx, 0]]}) + + Now it is possible to retrieve the contravariant form of the Electromagnetic + tensor: + + >>> F(i0, i1).replace_with_arrays(repl, [i0, i1]) + [[0, -E_x/c, -E_y/c, -E_z/c], [E_x/c, 0, -B_z, B_y], [E_y/c, B_z, 0, -B_x], [E_z/c, -B_y, B_x, 0]] + + and the mixed contravariant-covariant form: + + >>> F(i0, -i1).replace_with_arrays(repl, [i0, -i1]) + [[0, E_x/c, E_y/c, E_z/c], [E_x/c, 0, B_z, -B_y], [E_y/c, -B_z, 0, B_x], [E_z/c, B_y, -B_x, 0]] + + Energy-momentum of a particle may be represented as: + + >>> from sympy import symbols + >>> P = TensorHead('P', [Lorentz], TensorSymmetry.no_symmetry(1)) + >>> E, px, py, pz = symbols('E p_x p_y p_z', positive=True) + >>> repl.update({P(i0): [E, px, py, pz]}) + + The contravariant and covariant components are, respectively: + + >>> P(i0).replace_with_arrays(repl, [i0]) + [E, p_x, p_y, p_z] + >>> P(-i0).replace_with_arrays(repl, [-i0]) + [E, -p_x, -p_y, -p_z] + + The contraction of a 1-index tensor by itself: + + >>> expr = P(i0)*P(-i0) + >>> expr.replace_with_arrays(repl, []) + E**2 - p_x**2 - p_y**2 - p_z**2 + """ + is_commutative = False + + def __new__(cls, name, index_types, symmetry=None, comm=0): + if isinstance(name, str): + name_symbol = Symbol(name) + elif isinstance(name, Symbol): + name_symbol = name + else: + raise ValueError("invalid name") + + if symmetry is None: + symmetry = TensorSymmetry.no_symmetry(len(index_types)) + else: + assert symmetry.rank == len(index_types) + + obj = Basic.__new__(cls, name_symbol, Tuple(*index_types), symmetry, sympify(comm)) + return obj + + @property + def name(self): + return self.args[0].name + + @property + def index_types(self): + return list(self.args[1]) + + @property + def symmetry(self): + return self.args[2] + + @property + def comm(self): + return TensorManager.comm_symbols2i(self.args[3]) + + @property + def rank(self): + return len(self.index_types) + + def __lt__(self, other): + return (self.name, self.index_types) < (other.name, other.index_types) + + def commutes_with(self, other): + """ + Returns ``0`` if ``self`` and ``other`` commute, ``1`` if they anticommute. + + Returns ``None`` if ``self`` and ``other`` neither commute nor anticommute. + """ + r = TensorManager.get_comm(self.comm, other.comm) + return r + + def _print(self): + return '%s(%s)' %(self.name, ','.join([str(x) for x in self.index_types])) + + def __call__(self, *indices, **kw_args): + """ + Returns a tensor with indices. + + Explanation + =========== + + There is a special behavior in case of indices denoted by ``True``, + they are considered auto-matrix indices, their slots are automatically + filled, and confer to the tensor the behavior of a matrix or vector + upon multiplication with another tensor containing auto-matrix indices + of the same ``TensorIndexType``. This means indices get summed over the + same way as in matrix multiplication. For matrix behavior, define two + auto-matrix indices, for vector behavior define just one. + + Indices can also be strings, in which case the attribute + ``index_types`` is used to convert them to proper ``TensorIndex``. + + Examples + ======== + + >>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, TensorSymmetry, TensorHead + >>> Lorentz = TensorIndexType('Lorentz', dummy_name='L') + >>> a, b = tensor_indices('a,b', Lorentz) + >>> A = TensorHead('A', [Lorentz]*2, TensorSymmetry.no_symmetry(2)) + >>> t = A(a, -b) + >>> t + A(a, -b) + + """ + + updated_indices = [] + for idx, typ in zip(indices, self.index_types): + if isinstance(idx, str): + idx = idx.strip().replace(" ", "") + if idx.startswith('-'): + updated_indices.append(TensorIndex(idx[1:], typ, + is_up=False)) + else: + updated_indices.append(TensorIndex(idx, typ)) + else: + updated_indices.append(idx) + + updated_indices += indices[len(updated_indices):] + + tensor = Tensor(self, updated_indices, **kw_args) + return tensor.doit() + + # Everything below this line is deprecated + + def __pow__(self, other): + deprecate_data() + with ignore_warnings(SymPyDeprecationWarning): + if self.data is None: + raise ValueError("No power on abstract tensors.") + from .array import tensorproduct, tensorcontraction + metrics = [_.data for _ in self.index_types] + + marray = self.data + marraydim = marray.rank() + for metric in metrics: + marray = tensorproduct(marray, metric, marray) + marray = tensorcontraction(marray, (0, marraydim), (marraydim+1, marraydim+2)) + + return marray ** (other * S.Half) + + @property + def data(self): + deprecate_data() + with ignore_warnings(SymPyDeprecationWarning): + return _tensor_data_substitution_dict[self] + + @data.setter + def data(self, data): + deprecate_data() + with ignore_warnings(SymPyDeprecationWarning): + _tensor_data_substitution_dict[self] = data + + @data.deleter + def data(self): + deprecate_data() + if self in _tensor_data_substitution_dict: + del _tensor_data_substitution_dict[self] + + def __iter__(self): + deprecate_data() + with ignore_warnings(SymPyDeprecationWarning): + return self.data.__iter__() + + def _components_data_full_destroy(self): + """ + EXPERIMENTAL: do not rely on this API method. + + Destroy components data associated to the ``TensorHead`` object, this + checks for attached components data, and destroys components data too. + """ + # do not garbage collect Kronecker tensor (it should be done by + # ``TensorIndexType`` garbage collection) + deprecate_data() + if self.name == "KD": + return + + # the data attached to a tensor must be deleted only by the TensorHead + # destructor. If the TensorHead is deleted, it means that there are no + # more instances of that tensor anywhere. + if self in _tensor_data_substitution_dict: + del _tensor_data_substitution_dict[self] + + +def tensor_heads(s, index_types, symmetry=None, comm=0): + """ + Returns a sequence of TensorHeads from a string `s` + """ + if isinstance(s, str): + names = [x.name for x in symbols(s, seq=True)] + else: + raise ValueError('expecting a string') + + thlist = [TensorHead(name, index_types, symmetry, comm) for name in names] + if len(thlist) == 1: + return thlist[0] + return thlist + + +class TensExpr(Expr, ABC): + """ + Abstract base class for tensor expressions + + Notes + ===== + + A tensor expression is an expression formed by tensors; + currently the sums of tensors are distributed. + + A ``TensExpr`` can be a ``TensAdd`` or a ``TensMul``. + + ``TensMul`` objects are formed by products of component tensors, + and include a coefficient, which is a SymPy expression. + + + In the internal representation contracted indices are represented + by ``(ipos1, ipos2, icomp1, icomp2)``, where ``icomp1`` is the position + of the component tensor with contravariant index, ``ipos1`` is the + slot which the index occupies in that component tensor. + + Contracted indices are therefore nameless in the internal representation. + """ + + _op_priority = 12.0 + is_commutative = False + + def __neg__(self): + return self*S.NegativeOne + + def __abs__(self): + raise NotImplementedError + + def __add__(self, other): + return TensAdd(self, other).doit(deep=False) + + def __radd__(self, other): + return TensAdd(other, self).doit(deep=False) + + def __sub__(self, other): + return TensAdd(self, -other).doit(deep=False) + + def __rsub__(self, other): + return TensAdd(other, -self).doit(deep=False) + + def __mul__(self, other): + """ + Multiply two tensors using Einstein summation convention. + + Explanation + =========== + + If the two tensors have an index in common, one contravariant + and the other covariant, in their product the indices are summed + + Examples + ======== + + >>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensor_heads + >>> Lorentz = TensorIndexType('Lorentz', dummy_name='L') + >>> m0, m1, m2 = tensor_indices('m0,m1,m2', Lorentz) + >>> g = Lorentz.metric + >>> p, q = tensor_heads('p,q', [Lorentz]) + >>> t1 = p(m0) + >>> t2 = q(-m0) + >>> t1*t2 + p(L_0)*q(-L_0) + """ + return TensMul(self, other).doit(deep=False) + + def __rmul__(self, other): + return TensMul(other, self).doit(deep=False) + + def __truediv__(self, other): + other = _sympify(other) + if isinstance(other, TensExpr): + raise ValueError('cannot divide by a tensor') + return TensMul(self, S.One/other).doit(deep=False) + + def __rtruediv__(self, other): + raise ValueError('cannot divide by a tensor') + + def __pow__(self, other): + deprecate_data() + with ignore_warnings(SymPyDeprecationWarning): + if self.data is None: + raise ValueError("No power without ndarray data.") + from .array import tensorproduct, tensorcontraction + free = self.free + marray = self.data + mdim = marray.rank() + for metric in free: + marray = tensorcontraction( + tensorproduct( + marray, + metric[0].tensor_index_type.data, + marray), + (0, mdim), (mdim+1, mdim+2) + ) + return marray ** (other * S.Half) + + def __rpow__(self, other): + raise NotImplementedError + + @property + @abstractmethod + def nocoeff(self): + raise NotImplementedError("abstract method") + + @property + @abstractmethod + def coeff(self): + raise NotImplementedError("abstract method") + + @abstractmethod + def get_indices(self): + raise NotImplementedError("abstract method") + + @abstractmethod + def get_free_indices(self) -> list[TensorIndex]: + raise NotImplementedError("abstract method") + + @abstractmethod + def _replace_indices(self, repl: dict[TensorIndex, TensorIndex]) -> TensExpr: + raise NotImplementedError("abstract method") + + def fun_eval(self, *index_tuples): + deprecate_fun_eval() + return self.substitute_indices(*index_tuples) + + def get_matrix(self): + """ + DEPRECATED: do not use. + + Returns ndarray components data as a matrix, if components data are + available and ndarray dimension does not exceed 2. + """ + from sympy.matrices.dense import Matrix + deprecate_data() + with ignore_warnings(SymPyDeprecationWarning): + if 0 < self.rank <= 2: + rows = self.data.shape[0] + columns = self.data.shape[1] if self.rank == 2 else 1 + if self.rank == 2: + mat_list = [] * rows + for i in range(rows): + mat_list.append([]) + for j in range(columns): + mat_list[i].append(self[i, j]) + else: + mat_list = [None] * rows + for i in range(rows): + mat_list[i] = self[i] + return Matrix(mat_list) + else: + raise NotImplementedError( + "missing multidimensional reduction to matrix.") + + @staticmethod + def _get_indices_permutation(indices1, indices2): + return [indices1.index(i) for i in indices2] + + def _get_free_indices_set(self): + indset = set() + for arg in self.args: + if isinstance(arg, TensExpr): + indset.update(arg._get_free_indices_set()) + return indset + + def _get_dummy_indices_set(self): + indset = set() + for arg in self.args: + if isinstance(arg, TensExpr): + indset.update(arg._get_dummy_indices_set()) + return indset + + def _get_indices_set(self): + indset = set() + for arg in self.args: + if isinstance(arg, TensExpr): + indset.update(arg._get_indices_set()) + return indset + + @property + def _iterate_dummy_indices(self): + dummy_set = self._get_dummy_indices_set() + + def recursor(expr, pos): + if isinstance(expr, TensorIndex): + if expr in dummy_set: + yield (expr, pos) + elif isinstance(expr, (Tuple, TensExpr)): + for p, arg in enumerate(expr.args): + yield from recursor(arg, pos+(p,)) + + return recursor(self, ()) + + @property + def _iterate_free_indices(self): + free_set = self._get_free_indices_set() + + def recursor(expr, pos): + if isinstance(expr, TensorIndex): + if expr in free_set: + yield (expr, pos) + elif isinstance(expr, (Tuple, TensExpr)): + for p, arg in enumerate(expr.args): + yield from recursor(arg, pos+(p,)) + + return recursor(self, ()) + + @property + def _iterate_indices(self): + def recursor(expr, pos): + if isinstance(expr, TensorIndex): + yield (expr, pos) + elif isinstance(expr, (Tuple, TensExpr)): + for p, arg in enumerate(expr.args): + yield from recursor(arg, pos+(p,)) + + return recursor(self, ()) + + @staticmethod + def _contract_and_permute_with_metric(metric, array, pos, dim): + # TODO: add possibility of metric after (spinors) + from .array import tensorcontraction, tensorproduct, permutedims + + array = tensorcontraction(tensorproduct(metric, array), (1, 2+pos)) + permu = list(range(dim)) + permu[0], permu[pos] = permu[pos], permu[0] + return permutedims(array, permu) + + @staticmethod + def _match_indices_with_other_tensor(array, free_ind1, free_ind2, replacement_dict): + from .array import permutedims + + index_types1 = [i.tensor_index_type for i in free_ind1] + + # Check if variance of indices needs to be fixed: + pos2up = [] + pos2down = [] + free2remaining = free_ind2[:] + for pos1, index1 in enumerate(free_ind1): + if index1 in free2remaining: + pos2 = free2remaining.index(index1) + free2remaining[pos2] = None + continue + if -index1 in free2remaining: + pos2 = free2remaining.index(-index1) + free2remaining[pos2] = None + free_ind2[pos2] = index1 + if index1.is_up: + pos2up.append(pos2) + else: + pos2down.append(pos2) + else: + index2 = free2remaining[pos1] + if index2 is None: + raise ValueError("incompatible indices: %s and %s" % (free_ind1, free_ind2)) + free2remaining[pos1] = None + free_ind2[pos1] = index1 + if index1.is_up ^ index2.is_up: + if index1.is_up: + pos2up.append(pos1) + else: + pos2down.append(pos1) + + if len(set(free_ind1) & set(free_ind2)) < len(free_ind1): + raise ValueError("incompatible indices: %s and %s" % (free_ind1, free_ind2)) + + # Raise indices: + for pos in pos2up: + index_type_pos = index_types1[pos] + if index_type_pos not in replacement_dict: + raise ValueError("No metric provided to lower index") + metric = replacement_dict[index_type_pos] + metric_inverse = _TensorDataLazyEvaluator.inverse_matrix(metric) + array = TensExpr._contract_and_permute_with_metric(metric_inverse, array, pos, len(free_ind1)) + # Lower indices: + for pos in pos2down: + index_type_pos = index_types1[pos] + if index_type_pos not in replacement_dict: + raise ValueError("No metric provided to lower index") + metric = replacement_dict[index_type_pos] + array = TensExpr._contract_and_permute_with_metric(metric, array, pos, len(free_ind1)) + + if free_ind1: + permutation = TensExpr._get_indices_permutation(free_ind2, free_ind1) + array = permutedims(array, permutation) + + if hasattr(array, "rank") and array.rank() == 0: + array = array[()] + + return free_ind2, array + + def replace_with_arrays(self, replacement_dict, indices=None): + """ + Replace the tensorial expressions with arrays. The final array will + correspond to the N-dimensional array with indices arranged according + to ``indices``. + + Parameters + ========== + + replacement_dict + dictionary containing the replacement rules for tensors. + indices + the index order with respect to which the array is read. The + original index order will be used if no value is passed. + + Examples + ======== + + >>> from sympy.tensor.tensor import TensorIndexType, tensor_indices + >>> from sympy.tensor.tensor import TensorHead + >>> from sympy import symbols, diag + + >>> L = TensorIndexType("L") + >>> i, j = tensor_indices("i j", L) + >>> A = TensorHead("A", [L]) + >>> A(i).replace_with_arrays({A(i): [1, 2]}, [i]) + [1, 2] + + Since 'indices' is optional, we can also call replace_with_arrays by + this way if no specific index order is needed: + + >>> A(i).replace_with_arrays({A(i): [1, 2]}) + [1, 2] + + >>> expr = A(i)*A(j) + >>> expr.replace_with_arrays({A(i): [1, 2]}) + [[1, 2], [2, 4]] + + For contractions, specify the metric of the ``TensorIndexType``, which + in this case is ``L``, in its covariant form: + + >>> expr = A(i)*A(-i) + >>> expr.replace_with_arrays({A(i): [1, 2], L: diag(1, -1)}) + -3 + + Symmetrization of an array: + + >>> H = TensorHead("H", [L, L]) + >>> a, b, c, d = symbols("a b c d") + >>> expr = H(i, j)/2 + H(j, i)/2 + >>> expr.replace_with_arrays({H(i, j): [[a, b], [c, d]]}) + [[a, b/2 + c/2], [b/2 + c/2, d]] + + Anti-symmetrization of an array: + + >>> expr = H(i, j)/2 - H(j, i)/2 + >>> repl = {H(i, j): [[a, b], [c, d]]} + >>> expr.replace_with_arrays(repl) + [[0, b/2 - c/2], [-b/2 + c/2, 0]] + + The same expression can be read as the transpose by inverting ``i`` and + ``j``: + + >>> expr.replace_with_arrays(repl, [j, i]) + [[0, -b/2 + c/2], [b/2 - c/2, 0]] + """ + from .array import Array + + indices = indices or [] + remap = {k.args[0] if k.is_up else -k.args[0]: k for k in self.get_free_indices()} + for i, index in enumerate(indices): + if isinstance(index, (Symbol, Mul)): + if index in remap: + indices[i] = remap[index] + else: + indices[i] = -remap[-index] + + replacement_dict = {tensor: Array(array) for tensor, array in replacement_dict.items()} + + # Check dimensions of replaced arrays: + for tensor, array in replacement_dict.items(): + if isinstance(tensor, TensorIndexType): + expected_shape = [tensor.dim for i in range(2)] + else: + expected_shape = [index_type.dim for index_type in tensor.index_types] + if len(expected_shape) != array.rank() or (not all(dim1 == dim2 if + dim1.is_number else True for dim1, dim2 in zip(expected_shape, + array.shape))): + raise ValueError("shapes for tensor %s expected to be %s, "\ + "replacement array shape is %s" % (tensor, expected_shape, + array.shape)) + + ret_indices, array = self._extract_data(replacement_dict) + + last_indices, array = self._match_indices_with_other_tensor(array, indices, ret_indices, replacement_dict) + return array + + def _check_add_Sum(self, expr, index_symbols): + from sympy.concrete.summations import Sum + indices = self.get_indices() + dum = self.dum + sum_indices = [ (index_symbols[i], 0, + indices[i].tensor_index_type.dim-1) for i, j in dum] + if sum_indices: + expr = Sum(expr, *sum_indices) + return expr + + def _expand_partial_derivative(self): + # simply delegate the _expand_partial_derivative() to + # its arguments to expand a possibly found PartialDerivative + return self.func(*[ + a._expand_partial_derivative() + if isinstance(a, TensExpr) else a + for a in self.args]) + + def _matches_simple(self, expr, repl_dict=None, old=False): + """ + Matches assuming there are no wild objects in self. + """ + if repl_dict is None: + repl_dict = {} + else: + repl_dict = repl_dict.copy() + + if not isinstance(expr, TensExpr): + if len(self.get_free_indices()) > 0: + #self has indices, but expr does not. + return None + elif set(self.get_free_indices()) != set(expr.get_free_indices()): + #If there are no wilds and the free indices are not the same, they cannot match. + return None + + if canon_bp(self - expr) == S.Zero: + return repl_dict + else: + return None + + +class TensAdd(TensExpr, AssocOp): + """ + Sum of tensors. + + Parameters + ========== + + free_args : list of the free indices + + Attributes + ========== + + ``args`` : tuple of addends + ``rank`` : rank of the tensor + ``free_args`` : list of the free indices in sorted order + + Examples + ======== + + >>> from sympy.tensor.tensor import TensorIndexType, tensor_heads, tensor_indices + >>> Lorentz = TensorIndexType('Lorentz', dummy_name='L') + >>> a, b = tensor_indices('a,b', Lorentz) + >>> p, q = tensor_heads('p,q', [Lorentz]) + >>> t = p(a) + q(a); t + p(a) + q(a) + + Examples with components data added to the tensor expression: + + >>> from sympy import symbols, diag + >>> x, y, z, t = symbols("x y z t") + >>> repl = {} + >>> repl[Lorentz] = diag(1, -1, -1, -1) + >>> repl[p(a)] = [1, 2, 3, 4] + >>> repl[q(a)] = [x, y, z, t] + + The following are: 2**2 - 3**2 - 2**2 - 7**2 ==> -58 + + >>> expr = p(a) + q(a) + >>> expr.replace_with_arrays(repl, [a]) + [x + 1, y + 2, z + 3, t + 4] + """ + + def __new__(cls, *args, **kw_args): + args = [_sympify(x) for x in args if x] + args = TensAdd._tensAdd_flatten(args) + args.sort(key=default_sort_key) + if not args: + return S.Zero + if len(args) == 1: + return args[0] + + return Basic.__new__(cls, *args, **kw_args) + + @property + def coeff(self): + return S.One + + @property + def nocoeff(self): + return self + + def get_free_indices(self) -> list[TensorIndex]: + return self.free_indices + + def _replace_indices(self, repl: dict[TensorIndex, TensorIndex]) -> TensExpr: + newargs = [arg._replace_indices(repl) if isinstance(arg, TensExpr) else arg for arg in self.args] + return self.func(*newargs) + + @memoize_property + def rank(self): + if isinstance(self.args[0], TensExpr): + return self.args[0].rank + else: + return 0 + + @memoize_property + def free_args(self): + if isinstance(self.args[0], TensExpr): + return self.args[0].free_args + else: + return [] + + @memoize_property + def free_indices(self): + if isinstance(self.args[0], TensExpr): + return self.args[0].get_free_indices() + else: + return set() + + def doit(self, **hints) -> Expr: + deep = hints.get('deep', True) + if deep: + args = [arg.doit(**hints) for arg in self.args] + else: + args = self.args # type: ignore + + # if any of the args are zero (after doit), drop them. Otherwise, _tensAdd_check will complain about non-matching indices, even though the TensAdd is correctly formed. + args = [arg for arg in args if arg != S.Zero] + + if len(args) == 0: + return S.Zero + elif len(args) == 1: + return args[0] + + # now check that all addends have the same indices: + TensAdd._tensAdd_check(args) + + # Collect terms appearing more than once, differing by their coefficients: + args = TensAdd._tensAdd_collect_terms(args) + + # collect canonicalized terms + def sort_key(t): + if not isinstance(t, TensExpr): + return [], [], [] + if hasattr(t, "_index_structure") and hasattr(t, "components"): + x = get_index_structure(t) + return t.components, x.free, x.dum + return [], [], [] + args.sort(key=sort_key) + + if not args: + return S.Zero + # it there is only a component tensor return it + if len(args) == 1: + return args[0] + + obj = self.func(*args) + return obj + + @staticmethod + def _tensAdd_flatten(args): + # flatten TensAdd, coerce terms which are not tensors to tensors + a = [] + for x in args: + if isinstance(x, (Add, TensAdd)): + a.extend(list(x.args)) + else: + a.append(x) + args = [x for x in a if x.coeff] + return args + + @staticmethod + def _tensAdd_check(args): + # check that all addends have the same free indices + + def get_indices_set(x: Expr) -> set[TensorIndex]: + if isinstance(x, TensExpr): + return set(x.get_free_indices()) + return set() + + indices0 = get_indices_set(args[0]) + list_indices = [get_indices_set(arg) for arg in args[1:]] + if not all(x == indices0 for x in list_indices): + raise ValueError('all tensors must have the same indices') + + @staticmethod + def _tensAdd_collect_terms(args): + # collect TensMul terms differing at most by their coefficient + terms_dict = defaultdict(list) + scalars = S.Zero + if isinstance(args[0], TensExpr): + free_indices = set(args[0].get_free_indices()) + else: + free_indices = set() + + for arg in args: + if not isinstance(arg, TensExpr): + if free_indices != set(): + raise ValueError("wrong valence") + scalars += arg + continue + if free_indices != set(arg.get_free_indices()): + raise ValueError("wrong valence") + # TODO: what is the part which is not a coeff? + # needs an implementation similar to .as_coeff_Mul() + terms_dict[arg.nocoeff].append(arg.coeff) + + new_args = [TensMul(Add(*coeff), t).doit(deep=False) for t, coeff in terms_dict.items() if Add(*coeff) != 0] + if isinstance(scalars, Add): + new_args = list(scalars.args) + new_args + elif scalars != 0: + new_args = [scalars] + new_args + return new_args + + def get_indices(self): + indices = [] + for arg in self.args: + indices.extend([i for i in get_indices(arg) if i not in indices]) + return indices + + + def __call__(self, *indices): + deprecate_call() + free_args = self.free_args + indices = list(indices) + if [x.tensor_index_type for x in indices] != [x.tensor_index_type for x in free_args]: + raise ValueError('incompatible types') + if indices == free_args: + return self + index_tuples = list(zip(free_args, indices)) + a = [x.func(*x.substitute_indices(*index_tuples).args) for x in self.args] + res = TensAdd(*a).doit(deep=False) + return res + + def canon_bp(self): + """ + Canonicalize using the Butler-Portugal algorithm for canonicalization + under monoterm symmetries. + """ + expr = self.expand() + if isinstance(expr, self.func): + args = [canon_bp(x) for x in expr.args] + res = TensAdd(*args).doit(deep=False) + return res + else: + return canon_bp(expr) + + def equals(self, other): + other = _sympify(other) + if isinstance(other, TensMul) and other.coeff == 0: + return all(x.coeff == 0 for x in self.args) + if isinstance(other, TensExpr): + if self.rank != other.rank: + return False + if isinstance(other, TensAdd): + if set(self.args) != set(other.args): + return False + else: + return True + t = self - other + if not isinstance(t, TensExpr): + return t == 0 + else: + if isinstance(t, TensMul): + return t.coeff == 0 + else: + return all(x.coeff == 0 for x in t.args) + + def __getitem__(self, item): + deprecate_data() + with ignore_warnings(SymPyDeprecationWarning): + return self.data[item] + + def contract_delta(self, delta): + args = [x.contract_delta(delta) if isinstance(x, TensExpr) else x for x in self.args] + t = TensAdd(*args).doit(deep=False) + return canon_bp(t) + + def contract_metric(self, g): + """ + Raise or lower indices with the metric ``g``. + + Parameters + ========== + + g : metric + + contract_all : if True, eliminate all ``g`` which are contracted + + Notes + ===== + + see the ``TensorIndexType`` docstring for the contraction conventions + """ + + args = [contract_metric(x, g) for x in self.args] + t = TensAdd(*args).doit(deep=False) + return canon_bp(t) + + def substitute_indices(self, *index_tuples): + new_args = [] + for arg in self.args: + if isinstance(arg, TensExpr): + arg = arg.substitute_indices(*index_tuples) + new_args.append(arg) + return TensAdd(*new_args).doit(deep=False) + + def _print(self): + a = [] + args = self.args + for x in args: + a.append(str(x)) + s = ' + '.join(a) + s = s.replace('+ -', '- ') + return s + + def _extract_data(self, replacement_dict): + from sympy.tensor.array import Array, permutedims + args_indices, arrays = zip(*[ + arg._extract_data(replacement_dict) if + isinstance(arg, TensExpr) else ([], arg) for arg in self.args + ]) + arrays = [Array(i) for i in arrays] + ref_indices = args_indices[0] + for i in range(1, len(args_indices)): + indices = args_indices[i] + array = arrays[i] + permutation = TensMul._get_indices_permutation(indices, ref_indices) + arrays[i] = permutedims(array, permutation) + return ref_indices, sum(arrays, Array.zeros(*array.shape)) + + @property + def data(self): + deprecate_data() + with ignore_warnings(SymPyDeprecationWarning): + return _tensor_data_substitution_dict[self.expand()] + + @data.setter + def data(self, data): + deprecate_data() + with ignore_warnings(SymPyDeprecationWarning): + _tensor_data_substitution_dict[self] = data + + @data.deleter + def data(self): + deprecate_data() + with ignore_warnings(SymPyDeprecationWarning): + if self in _tensor_data_substitution_dict: + del _tensor_data_substitution_dict[self] + + def __iter__(self): + deprecate_data() + if not self.data: + raise ValueError("No iteration on abstract tensors") + return self.data.flatten().__iter__() + + def _eval_rewrite_as_Indexed(self, *args, **kwargs): + return Add.fromiter(args) + + def _eval_partial_derivative(self, s): + # Evaluation like Add + list_addends = [] + for a in self.args: + if isinstance(a, TensExpr): + list_addends.append(a._eval_partial_derivative(s)) + # do not call diff if s is no symbol + elif s._diff_wrt: + list_addends.append(a._eval_derivative(s)) + + return self.func(*list_addends).doit(deep=False) + + def matches(self, expr, repl_dict=None, old=False): + expr = sympify(expr) + + if repl_dict is None: + repl_dict = {} + else: + repl_dict = repl_dict.copy() + + if not isinstance(expr, TensAdd): + return None + + if len(_get_wilds(self)) == 0: + return self._matches_simple(expr, repl_dict, old) + + def siftkey(arg): + wildatoms = _get_wilds(arg) + wildatom_types = sift(wildatoms, type) + if len(wildatoms) == 0: + return "nonwild" + elif WildTensor in wildatom_types.keys(): + for w in wildatom_types["WildTensor"]: + if len(w.get_indices()) == 0: + return "indexless_wildtensor" + return "wildtensor" + else: + return "otherwild" + + query_sifted = sift(self.args, siftkey) + expr_sifted = sift(expr.args, siftkey) + + #First try to match the terms without WildTensors + matched_e_tensors = [] #Used to make sure that the same tensor in expr is not matched with more than one tensor in self. + for q_tensor in query_sifted["nonwild"]: + matched_this_q = False + for e_tensor in expr_sifted["nonwild"]: + if e_tensor in matched_e_tensors: + continue + + m = q_tensor.matches(e_tensor, repl_dict=repl_dict, old=old) + if m is None: + continue + else: + matched_this_q = True + repl_dict.update(m) + matched_e_tensors.append(e_tensor) + break + + if not matched_this_q: + return None + + remaining_e_tensors = [t for t in expr_sifted["nonwild"] if t not in matched_e_tensors] + for w in query_sifted["otherwild"]: + for e in remaining_e_tensors: + m = w.matches(e) + if m is not None: + matched_e_tensors.append(e) + if w in repl_dict.keys(): + repl_dict[w] += m.pop(w) + repl_dict.update(m) + + remaining_e_tensors = [t for t in expr_sifted["nonwild"] if t not in matched_e_tensors] + for w in query_sifted["wildtensor"]: + for e in remaining_e_tensors: + m = w.matches(e) + if m is not None: + matched_e_tensors.append(e) + if w.component in repl_dict.keys(): + repl_dict[w.component] += m.pop(w.component) + repl_dict.update(m) + + remaining_e_tensors = [t for t in expr_sifted["nonwild"] if t not in matched_e_tensors] + for w in query_sifted["indexless_wildtensor"]: + for e in remaining_e_tensors: + m = w.matches(e) + if m is not None: + matched_e_tensors.append(e) + if w.component in repl_dict.keys(): + repl_dict[w.component] += m.pop(w.component) + repl_dict.update(m) + + remaining_e_tensors = [t for t in expr_sifted["nonwild"] if t not in matched_e_tensors] + if len(remaining_e_tensors) > 0: + return None + else: + return repl_dict + + +class Tensor(TensExpr): + """ + Base tensor class, i.e. this represents a tensor, the single unit to be + put into an expression. + + Explanation + =========== + + This object is usually created from a ``TensorHead``, by attaching indices + to it. Indices preceded by a minus sign are considered contravariant, + otherwise covariant. + + Examples + ======== + + >>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, TensorHead + >>> Lorentz = TensorIndexType("Lorentz", dummy_name="L") + >>> mu, nu = tensor_indices('mu nu', Lorentz) + >>> A = TensorHead("A", [Lorentz, Lorentz]) + >>> A(mu, -nu) + A(mu, -nu) + >>> A(mu, -mu) + A(L_0, -L_0) + + It is also possible to use symbols instead of inidices (appropriate indices + are then generated automatically). + + >>> from sympy import Symbol + >>> x = Symbol('x') + >>> A(x, mu) + A(x, mu) + >>> A(x, -x) + A(L_0, -L_0) + + """ + + is_commutative = False + + _index_structure: _IndexStructure + args: tuple[TensorHead, Tuple] + + def __new__(cls, tensor_head, indices, *, is_canon_bp=False, **kw_args): + indices = cls._parse_indices(tensor_head, indices) + obj = Basic.__new__(cls, tensor_head, Tuple(*indices), **kw_args) + obj._index_structure = _IndexStructure.from_indices(*indices) + obj._free = obj._index_structure.free[:] + obj._dum = obj._index_structure.dum[:] + obj._ext_rank = obj._index_structure._ext_rank + obj._coeff = S.One + obj._nocoeff = obj + obj._component = tensor_head + obj._components = [tensor_head] + if tensor_head.rank != len(indices): + raise ValueError("wrong number of indices") + obj.is_canon_bp = is_canon_bp + obj._index_map = Tensor._build_index_map(indices, obj._index_structure) + return obj + + @property + def free(self): + return self._free + + @property + def dum(self): + return self._dum + + @property + def ext_rank(self): + return self._ext_rank + + @property + def coeff(self): + return self._coeff + + @property + def nocoeff(self): + return self._nocoeff + + @property + def component(self): + return self._component + + @property + def components(self): + return self._components + + @property + def head(self): + return self.args[0] + + @property + def indices(self): + return self.args[1] + + @property + def free_indices(self): + return set(self._index_structure.get_free_indices()) + + @property + def index_types(self): + return self.head.index_types + + @property + def rank(self): + return len(self.free_indices) + + @staticmethod + def _build_index_map(indices, index_structure): + index_map = {} + for idx in indices: + index_map[idx] = (indices.index(idx),) + return index_map + + def doit(self, **hints): + args, indices, free, dum = TensMul._tensMul_contract_indices([self]) + return args[0] + + @staticmethod + def _parse_indices(tensor_head, indices): + if not isinstance(indices, (tuple, list, Tuple)): + raise TypeError("indices should be an array, got %s" % type(indices)) + indices = list(indices) + for i, index in enumerate(indices): + if isinstance(index, Symbol): + indices[i] = TensorIndex(index, tensor_head.index_types[i], True) + elif isinstance(index, Mul): + c, e = index.as_coeff_Mul() + if c == -1 and isinstance(e, Symbol): + indices[i] = TensorIndex(e, tensor_head.index_types[i], False) + else: + raise ValueError("index not understood: %s" % index) + elif not isinstance(index, TensorIndex): + raise TypeError("wrong type for index: %s is %s" % (index, type(index))) + return indices + + def _set_new_index_structure(self, im, is_canon_bp=False): + indices = im.get_indices() + return self._set_indices(*indices, is_canon_bp=is_canon_bp) + + def _set_indices(self, *indices, is_canon_bp=False, **kw_args): + if len(indices) != self.ext_rank: + raise ValueError("indices length mismatch") + return self.func(self.args[0], indices, is_canon_bp=is_canon_bp).doit() + + def _get_free_indices_set(self): + return {i[0] for i in self._index_structure.free} + + def _get_dummy_indices_set(self): + dummy_pos = set(itertools.chain(*self._index_structure.dum)) + return {idx for i, idx in enumerate(self.args[1]) if i in dummy_pos} + + def _get_indices_set(self): + return set(self.args[1].args) + + @property + def free_in_args(self): + return [(ind, pos, 0) for ind, pos in self.free] + + @property + def dum_in_args(self): + return [(p1, p2, 0, 0) for p1, p2 in self.dum] + + @property + def free_args(self): + return sorted([x[0] for x in self.free]) + + def commutes_with(self, other): + """ + :param other: + :return: + 0 commute + 1 anticommute + None neither commute nor anticommute + """ + if not isinstance(other, TensExpr): + return 0 + elif isinstance(other, Tensor): + return self.component.commutes_with(other.component) + return NotImplementedError + + def perm2tensor(self, g, is_canon_bp=False): + """ + Returns the tensor corresponding to the permutation ``g``. + + For further details, see the method in ``TIDS`` with the same name. + """ + return perm2tensor(self, g, is_canon_bp) + + def canon_bp(self): + if self.is_canon_bp: + return self + expr = self.expand() + g, dummies, msym = expr._index_structure.indices_canon_args() + v = components_canon_args([expr.component]) + can = canonicalize(g, dummies, msym, *v) + if can == 0: + return S.Zero + tensor = self.perm2tensor(can, True) + return tensor + + def split(self): + return [self] + + def sorted_components(self): + return self + + def get_indices(self) -> list[TensorIndex]: + """ + Get a list of indices, corresponding to those of the tensor. + """ + return list(self.args[1]) + + def get_free_indices(self) -> list[TensorIndex]: + """ + Get a list of free indices, corresponding to those of the tensor. + """ + return self._index_structure.get_free_indices() + + def _replace_indices(self, repl: dict[TensorIndex, TensorIndex]) -> TensExpr: + # TODO: this could be optimized by only swapping the indices + # instead of visiting the whole expression tree: + return self.xreplace(repl) + + def as_base_exp(self): + return self, S.One + + def substitute_indices(self, *index_tuples): + """ + Return a tensor with free indices substituted according to ``index_tuples``. + + ``index_types`` list of tuples ``(old_index, new_index)``. + + Examples + ======== + + >>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensor_heads, TensorSymmetry + >>> Lorentz = TensorIndexType('Lorentz', dummy_name='L') + >>> i, j, k, l = tensor_indices('i,j,k,l', Lorentz) + >>> A, B = tensor_heads('A,B', [Lorentz]*2, TensorSymmetry.fully_symmetric(2)) + >>> t = A(i, k)*B(-k, -j); t + A(i, L_0)*B(-L_0, -j) + >>> t.substitute_indices((i, k),(-j, l)) + A(k, L_0)*B(-L_0, l) + """ + indices = [] + for index in self.indices: + for ind_old, ind_new in index_tuples: + if (index.name == ind_old.name and index.tensor_index_type == + ind_old.tensor_index_type): + if index.is_up == ind_old.is_up: + indices.append(ind_new) + else: + indices.append(-ind_new) + break + else: + indices.append(index) + return self.head(*indices) + + def _get_symmetrized_forms(self): + """ + Return a list giving all possible permutations of self that are allowed by its symmetries. + """ + comp = self.component + gens = comp.symmetry.generators + rank = comp.rank + + old_perms = None + new_perms = {self} + while new_perms != old_perms: + old_perms = new_perms.copy() + for tens in old_perms: + for gen in gens: + inds = tens.get_indices() + per = [gen.apply(i) for i in range(0,rank)] + sign = (-1)**(gen.apply(rank) - rank) + ind_map = dict(zip(inds, [inds[i] for i in per])) + new_perms.add( sign * tens._replace_indices(ind_map) ) + + return new_perms + + def matches(self, expr, repl_dict=None, old=False): + expr = sympify(expr) + + if repl_dict is None: + repl_dict = {} + else: + repl_dict = repl_dict.copy() + + #simple checks + if self == expr: + return repl_dict + if not isinstance(expr, Tensor): + return None + if self.head != expr.head: + return None + + #Now consider all index symmetries of expr, and see if any of them allow a match. + for new_expr in expr._get_symmetrized_forms(): + m = self._matches(new_expr, repl_dict, old=old) + if m is not None: + repl_dict.update(m) + return repl_dict + + return None + + def _matches(self, expr, repl_dict=None, old=False): + """ + This does not account for index symmetries of expr + """ + expr = sympify(expr) + + if repl_dict is None: + repl_dict = {} + else: + repl_dict = repl_dict.copy() + + #simple checks + if self == expr: + return repl_dict + if not isinstance(expr, Tensor): + return None + if self.head != expr.head: + return None + + s_indices = self.get_indices() + e_indices = expr.get_indices() + + if len(s_indices) != len(e_indices): + return None + + for i in range(len(s_indices)): + s_ind = s_indices[i] + m = s_ind.matches(e_indices[i]) + if m is None: + return None + elif -s_ind in repl_dict.keys() and -repl_dict[-s_ind] != m[s_ind]: + return None + else: + repl_dict.update(m) + + return repl_dict + + def __call__(self, *indices): + deprecate_call() + free_args = self.free_args + indices = list(indices) + if [x.tensor_index_type for x in indices] != [x.tensor_index_type for x in free_args]: + raise ValueError('incompatible types') + if indices == free_args: + return self + t = self.substitute_indices(*list(zip(free_args, indices))) + + # object is rebuilt in order to make sure that all contracted indices + # get recognized as dummies, but only if there are contracted indices. + if len({i if i.is_up else -i for i in indices}) != len(indices): + return t.func(*t.args) + return t + + # TODO: put this into TensExpr? + def __iter__(self): + deprecate_data() + with ignore_warnings(SymPyDeprecationWarning): + return self.data.__iter__() + + # TODO: put this into TensExpr? + def __getitem__(self, item): + deprecate_data() + with ignore_warnings(SymPyDeprecationWarning): + return self.data[item] + + def _extract_data(self, replacement_dict): + from .array import Array + for k, v in replacement_dict.items(): + if isinstance(k, Tensor) and k.args[0] == self.args[0]: + other = k + array = v + break + else: + raise ValueError("%s not found in %s" % (self, replacement_dict)) + + # TODO: inefficient, this should be done at root level only: + replacement_dict = {k: Array(v) for k, v in replacement_dict.items()} + array = Array(array) + + dum1 = self.dum + dum2 = other.dum + + if len(dum2) > 0: + for pair in dum2: + # allow `dum2` if the contained values are also in `dum1`. + if pair not in dum1: + raise NotImplementedError("%s with contractions is not implemented" % other) + # Remove elements in `dum2` from `dum1`: + dum1 = [pair for pair in dum1 if pair not in dum2] + if len(dum1) > 0: + indices1 = self.get_indices() + indices2 = other.get_indices() + repl = {} + for p1, p2 in dum1: + repl[indices2[p2]] = -indices2[p1] + for pos in (p1, p2): + if indices1[pos].is_up ^ indices2[pos].is_up: + metric = replacement_dict[indices1[pos].tensor_index_type] + if indices1[pos].is_up: + metric = _TensorDataLazyEvaluator.inverse_matrix(metric) + array = self._contract_and_permute_with_metric(metric, array, pos, len(indices2)) + other = other.xreplace(repl).doit() + array = _TensorDataLazyEvaluator.data_contract_dum([array], dum1, len(indices2)) + + free_ind1 = self.get_free_indices() + free_ind2 = other.get_free_indices() + + return self._match_indices_with_other_tensor(array, free_ind1, free_ind2, replacement_dict) + + @property + def data(self): + deprecate_data() + with ignore_warnings(SymPyDeprecationWarning): + return _tensor_data_substitution_dict[self] + + @data.setter + def data(self, data): + deprecate_data() + # TODO: check data compatibility with properties of tensor. + with ignore_warnings(SymPyDeprecationWarning): + _tensor_data_substitution_dict[self] = data + + @data.deleter + def data(self): + deprecate_data() + with ignore_warnings(SymPyDeprecationWarning): + if self in _tensor_data_substitution_dict: + del _tensor_data_substitution_dict[self] + if self.metric in _tensor_data_substitution_dict: + del _tensor_data_substitution_dict[self.metric] + + def _print(self): + indices = [str(ind) for ind in self.indices] + component = self.component + if component.rank > 0: + return ('%s(%s)' % (component.name, ', '.join(indices))) + else: + return ('%s' % component.name) + + def equals(self, other): + if other == 0: + return self.coeff == 0 + other = _sympify(other) + if not isinstance(other, TensExpr): + assert not self.components + return S.One == other + + def _get_compar_comp(self): + t = self.canon_bp() + r = (t.coeff, tuple(t.components), \ + tuple(sorted(t.free)), tuple(sorted(t.dum))) + return r + + return _get_compar_comp(self) == _get_compar_comp(other) + + def contract_metric(self, g): + # if metric is not the same, ignore this step: + if self.component != g: + return self + # in case there are free components, do not perform anything: + if len(self.free) != 0: + return self + + #antisym = g.index_types[0].metric_antisym + if g.symmetry == TensorSymmetry.fully_symmetric(-2): + antisym = 1 + elif g.symmetry == TensorSymmetry.fully_symmetric(2): + antisym = 0 + elif g.symmetry == TensorSymmetry.no_symmetry(2): + antisym = None + else: + raise NotImplementedError + sign = S.One + typ = g.index_types[0] + + if not antisym: + # g(i, -i) + sign = sign*typ.dim + else: + # g(i, -i) + sign = sign*typ.dim + + dp0, dp1 = self.dum[0] + if dp0 < dp1: + # g(i, -i) = -D with antisymmetric metric + sign = -sign + + return sign + + def contract_delta(self, metric): + return self.contract_metric(metric) + + def _eval_rewrite_as_Indexed(self, tens, indices, **kwargs): + from sympy.tensor.indexed import Indexed + # TODO: replace .args[0] with .name: + index_symbols = [i.args[0] for i in self.get_indices()] + expr = Indexed(tens.args[0], *index_symbols) + return self._check_add_Sum(expr, index_symbols) + + def _eval_partial_derivative(self, s: Tensor) -> Expr: + + if not isinstance(s, Tensor): + return S.Zero + else: + + # @a_i/@a_k = delta_i^k + # @a_i/@a^k = g_ij delta^j_k + # @a^i/@a^k = delta^i_k + # @a^i/@a_k = g^ij delta_j^k + # TODO: if there is no metric present, the derivative should be zero? + + if self.head != s.head: + return S.Zero + + # if heads are the same, provide delta and/or metric products + # for every free index pair in the appropriate tensor + # assumed that the free indices are in proper order + # A contravariante index in the derivative becomes covariant + # after performing the derivative and vice versa + + kronecker_delta_list = [1] + + # not guarantee a correct index order + + for (count, (iself, iother)) in enumerate(zip(self.get_free_indices(), s.get_free_indices())): + if iself.tensor_index_type != iother.tensor_index_type: + raise ValueError("index types not compatible") + else: + tensor_index_type = iself.tensor_index_type + tensor_metric = tensor_index_type.metric + dummy = TensorIndex("d_" + str(count), tensor_index_type, + is_up=iself.is_up) + if iself.is_up == iother.is_up: + kroneckerdelta = tensor_index_type.delta(iself, -iother) + else: + kroneckerdelta = ( + TensMul(tensor_metric(iself, dummy), + tensor_index_type.delta(-dummy, -iother)) + ) + kronecker_delta_list.append(kroneckerdelta) + return TensMul.fromiter(kronecker_delta_list).doit(deep=False) + # doit necessary to rename dummy indices accordingly + + +class TensMul(TensExpr, AssocOp): + """ + Product of tensors. + + Parameters + ========== + + coeff : SymPy coefficient of the tensor + args + + Attributes + ========== + + ``components`` : list of ``TensorHead`` of the component tensors + ``types`` : list of nonrepeated ``TensorIndexType`` + ``free`` : list of ``(ind, ipos, icomp)``, see Notes + ``dum`` : list of ``(ipos1, ipos2, icomp1, icomp2)``, see Notes + ``ext_rank`` : rank of the tensor counting the dummy indices + ``rank`` : rank of the tensor + ``coeff`` : SymPy coefficient of the tensor + ``free_args`` : list of the free indices in sorted order + ``is_canon_bp`` : ``True`` if the tensor in in canonical form + + Notes + ===== + + ``args[0]`` list of ``TensorHead`` of the component tensors. + + ``args[1]`` list of ``(ind, ipos, icomp)`` + where ``ind`` is a free index, ``ipos`` is the slot position + of ``ind`` in the ``icomp``-th component tensor. + + ``args[2]`` list of tuples representing dummy indices. + ``(ipos1, ipos2, icomp1, icomp2)`` indicates that the contravariant + dummy index is the ``ipos1``-th slot position in the ``icomp1``-th + component tensor; the corresponding covariant index is + in the ``ipos2`` slot position in the ``icomp2``-th component tensor. + + """ + identity = S.One + + _index_structure: _IndexStructure + + def __new__(cls, *args, **kw_args): + is_canon_bp = kw_args.get('is_canon_bp', False) + args = list(map(_sympify, args)) + + """ + If the internal dummy indices in one arg conflict with the free indices + of the remaining args, we need to rename those internal dummy indices. + """ + free = [get_free_indices(arg) for arg in args] + free = set(itertools.chain(*free)) #flatten free + newargs = [] + for arg in args: + dum_this = set(get_dummy_indices(arg)) + dum_other = [get_dummy_indices(a) for a in newargs] + dum_other = set(itertools.chain(*dum_other)) #flatten dum_other + free_this = set(get_free_indices(arg)) + if len(dum_this.intersection(free)) > 0: + exclude = free_this.union(free, dum_other) + newarg = TensMul._dedupe_indices(arg, exclude) + else: + newarg = arg + newargs.append(newarg) + + args = newargs + + # Flatten: + args = [i for arg in args for i in (arg.args if isinstance(arg, (TensMul, Mul)) else [arg])] + + args, indices, free, dum = TensMul._tensMul_contract_indices(args, replace_indices=False) + + # Data for indices: + index_types = [i.tensor_index_type for i in indices] + index_structure = _IndexStructure(free, dum, index_types, indices, canon_bp=is_canon_bp) + + obj = TensExpr.__new__(cls, *args) + obj._indices = indices + obj._index_types = index_types.copy() + obj._index_structure = index_structure + obj._free = index_structure.free[:] + obj._dum = index_structure.dum[:] + obj._free_indices = {x[0] for x in obj.free} + obj._rank = len(obj.free) + obj._ext_rank = len(obj._index_structure.free) + 2*len(obj._index_structure.dum) + obj._coeff = S.One + obj._is_canon_bp = is_canon_bp + return obj + + index_types = property(lambda self: self._index_types) + free = property(lambda self: self._free) + dum = property(lambda self: self._dum) + free_indices = property(lambda self: self._free_indices) + rank = property(lambda self: self._rank) + ext_rank = property(lambda self: self._ext_rank) + + @staticmethod + def _indices_to_free_dum(args_indices): + free2pos1 = {} + free2pos2 = {} + dummy_data = [] + indices = [] + + # Notation for positions (to better understand the code): + # `pos1`: position in the `args`. + # `pos2`: position in the indices. + + # Example: + # A(i, j)*B(k, m, n)*C(p) + # `pos1` of `n` is 1 because it's in `B` (second `args` of TensMul). + # `pos2` of `n` is 4 because it's the fifth overall index. + + # Counter for the index position wrt the whole expression: + pos2 = 0 + + for pos1, arg_indices in enumerate(args_indices): + + for index in arg_indices: + if not isinstance(index, TensorIndex): + raise TypeError("expected TensorIndex") + if -index in free2pos1: + # Dummy index detected: + other_pos1 = free2pos1.pop(-index) + other_pos2 = free2pos2.pop(-index) + if index.is_up: + dummy_data.append((index, pos1, other_pos1, pos2, other_pos2)) + else: + dummy_data.append((-index, other_pos1, pos1, other_pos2, pos2)) + indices.append(index) + elif index in free2pos1: + raise ValueError("Repeated index: %s" % index) + else: + free2pos1[index] = pos1 + free2pos2[index] = pos2 + indices.append(index) + pos2 += 1 + + free = list(free2pos2.items()) + free_names = [i.name for i in free2pos2.keys()] + + dummy_data.sort(key=lambda x: x[3]) + return indices, free, free_names, dummy_data + + @staticmethod + def _dummy_data_to_dum(dummy_data): + return [(p2a, p2b) for (i, p1a, p1b, p2a, p2b) in dummy_data] + + @staticmethod + def _tensMul_contract_indices(args, replace_indices=True): + replacements = [{} for _ in args] + + #_index_order = all(_has_index_order(arg) for arg in args) + + args_indices = [get_indices(arg) for arg in args] + indices, free, free_names, dummy_data = TensMul._indices_to_free_dum(args_indices) + + cdt = defaultdict(int) + + def dummy_name_gen(tensor_index_type): + nd = str(cdt[tensor_index_type]) + cdt[tensor_index_type] += 1 + return tensor_index_type.dummy_name + '_' + nd + + if replace_indices: + for old_index, pos1cov, pos1contra, pos2cov, pos2contra in dummy_data: + index_type = old_index.tensor_index_type + while True: + dummy_name = dummy_name_gen(index_type) + if dummy_name not in free_names: + break + dummy = old_index.func(dummy_name, index_type, *old_index.args[2:]) + replacements[pos1cov][old_index] = dummy + replacements[pos1contra][-old_index] = -dummy + indices[pos2cov] = dummy + indices[pos2contra] = -dummy + args = [ + arg._replace_indices(repl) if isinstance(arg, TensExpr) else arg + for arg, repl in zip(args, replacements)] + + """ + The order of indices might've changed due to the replacements (e.g. if one of the args is a TensAdd, replacing an index can change the sort order of the terms, thus changing the order of indices returned by its get_indices() method). + To stay on the safe side, we calculate these quantities again. + """ + args_indices = [get_indices(arg) for arg in args] + indices, free, free_names, dummy_data = TensMul._indices_to_free_dum(args_indices) + + dum = TensMul._dummy_data_to_dum(dummy_data) + return args, indices, free, dum + + @staticmethod + def _get_components_from_args(args): + """ + Get a list of ``Tensor`` objects having the same ``TIDS`` if multiplied + by one another. + """ + components = [] + for arg in args: + if not isinstance(arg, TensExpr): + continue + if isinstance(arg, TensAdd): + continue + components.extend(arg.components) + return components + + @staticmethod + def _rebuild_tensors_list(args, index_structure): + indices = index_structure.get_indices() + #tensors = [None for i in components] # pre-allocate list + ind_pos = 0 + for i, arg in enumerate(args): + if not isinstance(arg, TensExpr): + continue + prev_pos = ind_pos + ind_pos += arg.ext_rank + args[i] = Tensor(arg.component, indices[prev_pos:ind_pos]) + + def doit(self, **hints): + is_canon_bp = self._is_canon_bp + deep = hints.get('deep', True) + if deep: + args = [arg.doit(**hints) for arg in self.args] + + """ + There may now be conflicts between dummy indices of different args + (each arg's doit method does not have any information about which + dummy indices are already used in the other args), so we + deduplicate them. + """ + rule = dict(zip(self.args, args)) + rule = self._dedupe_indices_in_rule(rule) + args = [rule[a] for a in self.args] + + else: + args = self.args + + args = [arg for arg in args if arg != self.identity] + + # Extract non-tensor coefficients: + coeff = reduce(lambda a, b: a*b, [arg for arg in args if not isinstance(arg, TensExpr)], S.One) + args = [arg for arg in args if isinstance(arg, TensExpr)] + + if len(args) == 0: + return coeff + + if coeff != self.identity: + args = [coeff] + args + if coeff == 0: + return S.Zero + + if len(args) == 1: + return args[0] + + args, indices, free, dum = TensMul._tensMul_contract_indices(args) + + # Data for indices: + index_types = [i.tensor_index_type for i in indices] + index_structure = _IndexStructure(free, dum, index_types, indices, canon_bp=is_canon_bp) + + obj = self.func(*args) + obj._index_types = index_types + obj._index_structure = index_structure + obj._ext_rank = len(obj._index_structure.free) + 2*len(obj._index_structure.dum) + obj._coeff = coeff + obj._is_canon_bp = is_canon_bp + return obj + + # TODO: this method should be private + # TODO: should this method be renamed _from_components_free_dum ? + @staticmethod + def from_data(coeff, components, free, dum, **kw_args): + return TensMul(coeff, *TensMul._get_tensors_from_components_free_dum(components, free, dum), **kw_args).doit(deep=False) + + @staticmethod + def _get_tensors_from_components_free_dum(components, free, dum): + """ + Get a list of ``Tensor`` objects by distributing ``free`` and ``dum`` indices on the ``components``. + """ + index_structure = _IndexStructure.from_components_free_dum(components, free, dum) + indices = index_structure.get_indices() + tensors = [None for i in components] # pre-allocate list + + # distribute indices on components to build a list of tensors: + ind_pos = 0 + for i, component in enumerate(components): + prev_pos = ind_pos + ind_pos += component.rank + tensors[i] = Tensor(component, indices[prev_pos:ind_pos]) + return tensors + + def _get_free_indices_set(self): + return {i[0] for i in self.free} + + def _get_dummy_indices_set(self): + dummy_pos = set(itertools.chain(*self.dum)) + return {idx for i, idx in enumerate(self._index_structure.get_indices()) if i in dummy_pos} + + def _get_position_offset_for_indices(self): + arg_offset = [None for i in range(self.ext_rank)] + counter = 0 + for arg in self.args: + if not isinstance(arg, TensExpr): + continue + for j in range(arg.ext_rank): + arg_offset[j + counter] = counter + counter += arg.ext_rank + return arg_offset + + @property + def free_args(self): + return sorted([x[0] for x in self.free]) + + @property + def components(self): + return self._get_components_from_args(self.args) + + @property + def free_in_args(self): + arg_offset = self._get_position_offset_for_indices() + argpos = self._get_indices_to_args_pos() + return [(ind, pos-arg_offset[pos], argpos[pos]) for (ind, pos) in self.free] + + @property + def coeff(self): + # return Mul.fromiter([c for c in self.args if not isinstance(c, TensExpr)]) + return self._coeff + + @property + def nocoeff(self): + return self.func(*self.args, 1/self.coeff).doit(deep=False) + + @property + def dum_in_args(self): + arg_offset = self._get_position_offset_for_indices() + argpos = self._get_indices_to_args_pos() + return [(p1-arg_offset[p1], p2-arg_offset[p2], argpos[p1], argpos[p2]) for p1, p2 in self.dum] + + def equals(self, other): + if other == 0: + return self.coeff == 0 + other = _sympify(other) + if not isinstance(other, TensExpr): + assert not self.components + return self.coeff == other + + return self.canon_bp() == other.canon_bp() + + def get_indices(self): + """ + Returns the list of indices of the tensor. + + Explanation + =========== + + The indices are listed in the order in which they appear in the + component tensors. + The dummy indices are given a name which does not collide with + the names of the free indices. + + Examples + ======== + + >>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensor_heads + >>> Lorentz = TensorIndexType('Lorentz', dummy_name='L') + >>> m0, m1, m2 = tensor_indices('m0,m1,m2', Lorentz) + >>> g = Lorentz.metric + >>> p, q = tensor_heads('p,q', [Lorentz]) + >>> t = p(m1)*g(m0,m2) + >>> t.get_indices() + [m1, m0, m2] + >>> t2 = p(m1)*g(-m1, m2) + >>> t2.get_indices() + [L_0, -L_0, m2] + """ + return self._indices + + def get_free_indices(self) -> list[TensorIndex]: + """ + Returns the list of free indices of the tensor. + + Explanation + =========== + + The indices are listed in the order in which they appear in the + component tensors. + + Examples + ======== + + >>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensor_heads + >>> Lorentz = TensorIndexType('Lorentz', dummy_name='L') + >>> m0, m1, m2 = tensor_indices('m0,m1,m2', Lorentz) + >>> g = Lorentz.metric + >>> p, q = tensor_heads('p,q', [Lorentz]) + >>> t = p(m1)*g(m0,m2) + >>> t.get_free_indices() + [m1, m0, m2] + >>> t2 = p(m1)*g(-m1, m2) + >>> t2.get_free_indices() + [m2] + """ + return self._index_structure.get_free_indices() + + def _replace_indices(self, repl: dict[TensorIndex, TensorIndex]) -> TensExpr: + return self.func(*[arg._replace_indices(repl) if isinstance(arg, TensExpr) else arg for arg in self.args]) + + def split(self): + """ + Returns a list of tensors, whose product is ``self``. + + Explanation + =========== + + Dummy indices contracted among different tensor components + become free indices with the same name as the one used to + represent the dummy indices. + + Examples + ======== + + >>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensor_heads, TensorSymmetry + >>> Lorentz = TensorIndexType('Lorentz', dummy_name='L') + >>> a, b, c, d = tensor_indices('a,b,c,d', Lorentz) + >>> A, B = tensor_heads('A,B', [Lorentz]*2, TensorSymmetry.fully_symmetric(2)) + >>> t = A(a,b)*B(-b,c) + >>> t + A(a, L_0)*B(-L_0, c) + >>> t.split() + [A(a, L_0), B(-L_0, c)] + """ + if self.args == (): + return [self] + splitp = [] + res = 1 + for arg in self.args: + if isinstance(arg, Tensor): + splitp.append(res*arg) + res = 1 + else: + res *= arg + return splitp + + def _eval_expand_mul(self, **hints): + args1 = [arg.args if isinstance(arg, (Add, TensAdd)) else (arg,) for arg in self.args] + return TensAdd(*[ + TensMul(*i).doit(deep=False) for i in itertools.product(*args1)] + ) + + def __neg__(self): + return TensMul(S.NegativeOne, self, is_canon_bp=self._is_canon_bp).doit(deep=False) + + def __getitem__(self, item): + deprecate_data() + with ignore_warnings(SymPyDeprecationWarning): + return self.data[item] + + def _get_args_for_traditional_printer(self): + args = list(self.args) + if self.coeff.could_extract_minus_sign(): + # expressions like "-A(a)" + sign = "-" + if args[0] == S.NegativeOne: + args = args[1:] + else: + args[0] = -args[0] + else: + sign = "" + return sign, args + + def _sort_args_for_sorted_components(self): + """ + Returns the ``args`` sorted according to the components commutation + properties. + + Explanation + =========== + + The sorting is done taking into account the commutation group + of the component tensors. + """ + cv = [arg for arg in self.args if isinstance(arg, TensExpr)] + sign = 1 + n = len(cv) - 1 + for i in range(n): + for j in range(n, i, -1): + c = cv[j-1].commutes_with(cv[j]) + # if `c` is `None`, it does neither commute nor anticommute, skip: + if c not in (0, 1): + continue + typ1 = sorted(set(cv[j-1].component.index_types), key=lambda x: x.name) + typ2 = sorted(set(cv[j].component.index_types), key=lambda x: x.name) + if (typ1, cv[j-1].component.name) > (typ2, cv[j].component.name): + cv[j-1], cv[j] = cv[j], cv[j-1] + # if `c` is 1, the anticommute, so change sign: + if c: + sign = -sign + + coeff = sign * self.coeff + if coeff != 1: + return [coeff] + cv + return cv + + def sorted_components(self): + """ + Returns a tensor product with sorted components. + """ + return TensMul(*self._sort_args_for_sorted_components()).doit(deep=False) + + def perm2tensor(self, g, is_canon_bp=False): + """ + Returns the tensor corresponding to the permutation ``g`` + + For further details, see the method in ``TIDS`` with the same name. + """ + return perm2tensor(self, g, is_canon_bp=is_canon_bp) + + def canon_bp(self): + """ + Canonicalize using the Butler-Portugal algorithm for canonicalization + under monoterm symmetries. + + Examples + ======== + + >>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, TensorHead, TensorSymmetry + >>> Lorentz = TensorIndexType('Lorentz', dummy_name='L') + >>> m0, m1, m2 = tensor_indices('m0,m1,m2', Lorentz) + >>> A = TensorHead('A', [Lorentz]*2, TensorSymmetry.fully_symmetric(-2)) + >>> t = A(m0,-m1)*A(m1,-m0) + >>> t.canon_bp() + -A(L_0, L_1)*A(-L_0, -L_1) + >>> t = A(m0,-m1)*A(m1,-m2)*A(m2,-m0) + >>> t.canon_bp() + 0 + """ + if self._is_canon_bp: + return self + expr = self.expand() + if isinstance(expr, TensAdd): + return expr.canon_bp() + if not expr.components: + return expr + expr = expr.doit(deep=False) #make sure self.coeff is populated correctly + t = expr.sorted_components() + g, dummies, msym = t._index_structure.indices_canon_args() + v = components_canon_args(t.components) + can = canonicalize(g, dummies, msym, *v) + if can == 0: + return S.Zero + tmul = t.perm2tensor(can, True) + return tmul + + def contract_delta(self, delta): + t = self.contract_metric(delta) + return t + + def _get_indices_to_args_pos(self): + """ + Get a dict mapping the index position to TensMul's argument number. + """ + pos_map = {} + pos_counter = 0 + for arg_i, arg in enumerate(self.args): + if not isinstance(arg, TensExpr): + continue + assert isinstance(arg, Tensor) + for i in range(arg.ext_rank): + pos_map[pos_counter] = arg_i + pos_counter += 1 + return pos_map + + def contract_metric(self, g): + """ + Raise or lower indices with the metric ``g``. + + Parameters + ========== + + g : metric + + Notes + ===== + + See the ``TensorIndexType`` docstring for the contraction conventions. + + Examples + ======== + + >>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensor_heads + >>> Lorentz = TensorIndexType('Lorentz', dummy_name='L') + >>> m0, m1, m2 = tensor_indices('m0,m1,m2', Lorentz) + >>> g = Lorentz.metric + >>> p, q = tensor_heads('p,q', [Lorentz]) + >>> t = p(m0)*q(m1)*g(-m0, -m1) + >>> t.canon_bp() + metric(L_0, L_1)*p(-L_0)*q(-L_1) + >>> t.contract_metric(g).canon_bp() + p(L_0)*q(-L_0) + """ + expr = self.expand().doit(deep=False) + if self != expr: + expr = canon_bp(expr) + return contract_metric(expr, g) + pos_map = self._get_indices_to_args_pos() + args = list(self.args) + + #antisym = g.index_types[0].metric_antisym + if g.symmetry == TensorSymmetry.fully_symmetric(-2): + antisym = 1 + elif g.symmetry == TensorSymmetry.fully_symmetric(2): + antisym = 0 + elif g.symmetry == TensorSymmetry.no_symmetry(2): + antisym = None + else: + raise NotImplementedError + + # list of positions of the metric ``g`` inside ``args`` + gpos = [i for i, x in enumerate(self.args) if isinstance(x, Tensor) and x.component == g] + if not gpos: + return self + + # Sign is either 1 or -1, to correct the sign after metric contraction + # (for spinor indices). + sign = 1 + dum = self.dum[:] + free = self.free[:] + elim = set() + for gposx in gpos: + if gposx in elim: + continue + free1 = [x for x in free if pos_map[x[1]] == gposx] + dum1 = [x for x in dum if pos_map[x[0]] == gposx or pos_map[x[1]] == gposx] + if not dum1: + continue + elim.add(gposx) + # subs with the multiplication neutral element, that is, remove it: + args[gposx] = 1 + if len(dum1) == 2: + if not antisym: + dum10, dum11 = dum1 + if pos_map[dum10[1]] == gposx: + # the index with pos p0 contravariant + p0 = dum10[0] + else: + # the index with pos p0 is covariant + p0 = dum10[1] + if pos_map[dum11[1]] == gposx: + # the index with pos p1 is contravariant + p1 = dum11[0] + else: + # the index with pos p1 is covariant + p1 = dum11[1] + + dum.append((p0, p1)) + else: + dum10, dum11 = dum1 + # change the sign to bring the indices of the metric to contravariant + # form; change the sign if dum10 has the metric index in position 0 + if pos_map[dum10[1]] == gposx: + # the index with pos p0 is contravariant + p0 = dum10[0] + if dum10[1] == 1: + sign = -sign + else: + # the index with pos p0 is covariant + p0 = dum10[1] + if dum10[0] == 0: + sign = -sign + if pos_map[dum11[1]] == gposx: + # the index with pos p1 is contravariant + p1 = dum11[0] + sign = -sign + else: + # the index with pos p1 is covariant + p1 = dum11[1] + + dum.append((p0, p1)) + + elif len(dum1) == 1: + if not antisym: + dp0, dp1 = dum1[0] + if pos_map[dp0] == pos_map[dp1]: + # g(i, -i) + typ = g.index_types[0] + sign = sign*typ.dim + + else: + # g(i0, i1)*p(-i1) + if pos_map[dp0] == gposx: + p1 = dp1 + else: + p1 = dp0 + + ind, p = free1[0] + free.append((ind, p1)) + else: + dp0, dp1 = dum1[0] + if pos_map[dp0] == pos_map[dp1]: + # g(i, -i) + typ = g.index_types[0] + sign = sign*typ.dim + + if dp0 < dp1: + # g(i, -i) = -D with antisymmetric metric + sign = -sign + else: + # g(i0, i1)*p(-i1) + if pos_map[dp0] == gposx: + p1 = dp1 + if dp0 == 0: + sign = -sign + else: + p1 = dp0 + ind, p = free1[0] + free.append((ind, p1)) + dum = [x for x in dum if x not in dum1] + free = [x for x in free if x not in free1] + + # shift positions: + shift = 0 + shifts = [0]*len(args) + for i in range(len(args)): + if i in elim: + shift += 2 + continue + shifts[i] = shift + free = [(ind, p - shifts[pos_map[p]]) for (ind, p) in free if pos_map[p] not in elim] + dum = [(p0 - shifts[pos_map[p0]], p1 - shifts[pos_map[p1]]) for p0, p1 in dum if pos_map[p0] not in elim and pos_map[p1] not in elim] + + res = ( sign*TensMul(*args) ).doit(deep=False) + if not isinstance(res, TensExpr): + return res + im = _IndexStructure.from_components_free_dum(res.components, free, dum) + return res._set_new_index_structure(im) + + def _set_new_index_structure(self, im, is_canon_bp=False): + indices = im.get_indices() + return self._set_indices(*indices, is_canon_bp=is_canon_bp) + + def _set_indices(self, *indices, is_canon_bp=False, **kw_args): + if len(indices) != self.ext_rank: + raise ValueError("indices length mismatch") + args = list(self.args) + pos = 0 + for i, arg in enumerate(args): + if not isinstance(arg, TensExpr): + continue + assert isinstance(arg, Tensor) + ext_rank = arg.ext_rank + args[i] = arg._set_indices(*indices[pos:pos+ext_rank]) + pos += ext_rank + return TensMul(*args, is_canon_bp=is_canon_bp).doit(deep=False) + + @staticmethod + def _index_replacement_for_contract_metric(args, free, dum): + for arg in args: + if not isinstance(arg, TensExpr): + continue + assert isinstance(arg, Tensor) + + def substitute_indices(self, *index_tuples): + new_args = [] + for arg in self.args: + if isinstance(arg, TensExpr): + arg = arg.substitute_indices(*index_tuples) + new_args.append(arg) + return TensMul(*new_args).doit(deep=False) + + def __call__(self, *indices): + deprecate_call() + free_args = self.free_args + indices = list(indices) + if [x.tensor_index_type for x in indices] != [x.tensor_index_type for x in free_args]: + raise ValueError('incompatible types') + if indices == free_args: + return self + t = self.substitute_indices(*list(zip(free_args, indices))) + + # object is rebuilt in order to make sure that all contracted indices + # get recognized as dummies, but only if there are contracted indices. + if len({i if i.is_up else -i for i in indices}) != len(indices): + return t.func(*t.args) + return t + + def _extract_data(self, replacement_dict): + args_indices, arrays = zip(*[arg._extract_data(replacement_dict) for arg in self.args if isinstance(arg, TensExpr)]) + coeff = reduce(operator.mul, [a for a in self.args if not isinstance(a, TensExpr)], S.One) + indices, free, free_names, dummy_data = TensMul._indices_to_free_dum(args_indices) + dum = TensMul._dummy_data_to_dum(dummy_data) + ext_rank = self.ext_rank + free.sort(key=lambda x: x[1]) + free_indices = [i[0] for i in free] + return free_indices, coeff*_TensorDataLazyEvaluator.data_contract_dum(arrays, dum, ext_rank) + + @property + def data(self): + deprecate_data() + with ignore_warnings(SymPyDeprecationWarning): + dat = _tensor_data_substitution_dict[self.expand()] + return dat + + @data.setter + def data(self, data): + deprecate_data() + raise ValueError("Not possible to set component data to a tensor expression") + + @data.deleter + def data(self): + deprecate_data() + raise ValueError("Not possible to delete component data to a tensor expression") + + def __iter__(self): + deprecate_data() + with ignore_warnings(SymPyDeprecationWarning): + if self.data is None: + raise ValueError("No iteration on abstract tensors") + return self.data.__iter__() + + @staticmethod + def _dedupe_indices(new, exclude): + """ + exclude: set + new: TensExpr + + If ``new`` has any dummy indices that are in ``exclude``, return a version + of new with those indices replaced. If no replacements are needed, + return None + + """ + exclude = set(exclude) + dums_new = set(get_dummy_indices(new)) + free_new = set(get_free_indices(new)) + + conflicts = dums_new.intersection(exclude) + if len(conflicts) == 0: + return None + + """ + ``exclude_for_gen`` is to be passed to ``_IndexStructure._get_generator_for_dummy_indices()``. + Since the latter does not use the index position for anything, we just + set it as ``None`` here. + """ + exclude.update(dums_new) + exclude.update(free_new) + exclude_for_gen = [(i, None) for i in exclude] + gen = _IndexStructure._get_generator_for_dummy_indices(exclude_for_gen) + repl = {} + for d in conflicts: + if -d in repl.keys(): + continue + newname = gen(d.tensor_index_type) + new_d = d.func(newname, *d.args[1:]) + repl[d] = new_d + repl[-d] = -new_d + + if len(repl) == 0: + return None + + new_renamed = new._replace_indices(repl) + return new_renamed + + def _dedupe_indices_in_rule(self, rule): + """ + rule: dict + + This applies TensMul._dedupe_indices on all values of rule. + + """ + index_rules = {k:v for k,v in rule.items() if isinstance(k, TensorIndex)} + other_rules = {k:v for k,v in rule.items() if k not in index_rules.keys()} + exclude = set(self.get_indices()) + + newrule = {} + newrule.update(index_rules) + exclude.update(index_rules.keys()) + exclude.update(index_rules.values()) + for old, new in other_rules.items(): + new_renamed = TensMul._dedupe_indices(new, exclude) + if old == new or new_renamed is None: + newrule[old] = new + else: + newrule[old] = new_renamed + exclude.update(get_indices(new_renamed)) + return newrule + + def _eval_subs(self, old, new): + """ + If new is an index which is already present in self as a dummy, the dummies in self should be renamed. + """ + + if not isinstance(new, TensorIndex): + return None + + exclude = {new} + self_renamed = self._dedupe_indices(self, exclude) + if self_renamed is None: + return None + else: + return self_renamed._subs(old, new).doit(deep=False) + + def _eval_rewrite_as_Indexed(self, *args, **kwargs): + from sympy.concrete.summations import Sum + index_symbols = [i.args[0] for i in self.get_indices()] + args = [arg.args[0] if isinstance(arg, Sum) else arg for arg in args] + expr = Mul.fromiter(args) + return self._check_add_Sum(expr, index_symbols) + + def _eval_partial_derivative(self, s): + # Evaluation like Mul + terms = [] + for i, arg in enumerate(self.args): + # checking whether some tensor instance is differentiated + # or some other thing is necessary, but ugly + if isinstance(arg, TensExpr): + d = arg._eval_partial_derivative(s) + else: + # do not call diff is s is no symbol + if s._diff_wrt: + d = arg._eval_derivative(s) + else: + d = S.Zero + if d: + terms.append(TensMul.fromiter(self.args[:i] + (d,) + self.args[i + 1:]).doit(deep=False)) + return TensAdd.fromiter(terms).doit(deep=False) + + + def _matches_commutative(self, expr, repl_dict=None, old=False): + """ + Match assuming all tensors commute. But note that we are not assuming anything about their symmetry under index permutations. + """ + #Take care of the various possible types for expr. + if not isinstance(expr, TensMul): + if isinstance(expr, (TensExpr, Expr)): + expr = TensMul(expr) + else: + return None + + #The code that follows assumes expr is a TensMul + + if repl_dict is None: + repl_dict = {} + else: + repl_dict = repl_dict.copy() + + #Make sure that none of the dummy indices in self, expr conflict with the values already present in repl_dict. This may happen due to automatic index relabelling when rem_query and rem_expr are formed later on in this function (it calls itself recursively). + indices = [k for k in repl_dict.values() if isinstance(k ,TensorIndex)] + + def dedupe(expr): + renamed = TensMul._dedupe_indices(expr, indices) + if renamed is not None: + return renamed + else: + return expr + + self = dedupe(self) + expr = dedupe(expr) + + #Find the non-tensor part of expr. This need not be the same as expr.coeff when expr.doit() has not been called. + expr_coeff = reduce(lambda a, b: a*b, [arg for arg in expr.args if not isinstance(arg, TensExpr)], S.One) + + # handle simple patterns + if self == expr: + return repl_dict + + if len(_get_wilds(self)) == 0: + return self._matches_simple(expr, repl_dict, old) + + def siftkey(arg): + if isinstance(arg, WildTensor): + return "WildTensor" + elif isinstance(arg, (Tensor, TensExpr)): + return "Tensor" + else: + return "coeff" + + query_sifted = sift(self.args, siftkey) + expr_sifted = sift(expr.args, siftkey) + + #Sanity checks + if "coeff" in query_sifted.keys(): + if TensMul(*query_sifted["coeff"]).doit(deep=False) != self.coeff: + raise NotImplementedError(f"Found something that we do not know to handle: {query_sifted['coeff']}") + if "coeff" in expr_sifted.keys(): + if TensMul(*expr_sifted["coeff"]).doit(deep=False) != expr_coeff: + raise NotImplementedError(f"Found something that we do not know to handle: {expr_sifted['coeff']}") + + query_tens_heads = {tuple(getattr(x, "components", [])) for x in query_sifted["Tensor"]} #We use getattr because, e.g. TensAdd does not have the 'components' attribute. + expr_tens_heads = {tuple(getattr(x, "components", [])) for x in expr_sifted["Tensor"]} + if not query_tens_heads.issubset(expr_tens_heads): + #Some tensorheads in self are not present in the expr + return None + + #Try to match all non-wild tensors of self with tensors that compose expr + if len(query_sifted["Tensor"]) > 0: + q_tensor = query_sifted["Tensor"][0] + """ + We need to iterate over all possible symmetrized forms of q_tensor since the matches given by some of them may map dummy indices to free indices; the information about which indices are dummy/free will only be available later, when we are doing rem_q.matches(rem_e) + """ + for q_tens in q_tensor._get_symmetrized_forms(): + for e in expr_sifted["Tensor"]: + if isinstance(q_tens, TensMul): + #q_tensor got a minus sign due to this permutation. + sign = -1 + else: + sign = 1 + + """ + _matches is used here since we are already iterating over index permutations of q_tensor. Also note that the sign is removed from q_tensor, and will later be put into rem_q. + """ + m = (sign*q_tens)._matches(e) + if m is None: + continue + + rem_query = self.func(sign, *[a for a in self.args if a != q_tensor]).doit(deep=False) + rem_expr = expr.func(*[a for a in expr.args if a != e]).doit(deep=False) + tmp_repl = {} + tmp_repl.update(repl_dict) + tmp_repl.update(m) + rem_m = rem_query.matches(rem_expr, repl_dict=tmp_repl) + if rem_m is not None: + #Check that contracted indices are not mapped to different indices. + internally_consistent = True + for k in rem_m.keys(): + if isinstance(k,TensorIndex): + if -k in rem_m.keys() and rem_m[-k] != -rem_m[k]: + internally_consistent = False + break + if internally_consistent: + repl_dict.update(rem_m) + return repl_dict + + return None + + #Try to match WildTensor instances which have indices + matched_e_tensors = [] + remaining_e_tensors = expr_sifted["Tensor"] + indexless_wilds, wilds = sift(query_sifted["WildTensor"], lambda x: len(x.get_free_indices()) == 0, binary=True) + + for w in wilds: + free_this_wild = set(w.get_free_indices()) + tensors_to_try = [] + for t in remaining_e_tensors: + free = t.get_free_indices() + shares_indices_with_wild = True + for i in free: + if all(j.matches(i) is None for j in free_this_wild): + #The index i matches none of the indices in free_this_wild + shares_indices_with_wild = False + if shares_indices_with_wild: + tensors_to_try.append(t) + + m = w.matches(TensMul(*tensors_to_try).doit(deep=False) ) + if m is None: + return None + else: + for tens in tensors_to_try: + matched_e_tensors.append(tens) + repl_dict.update(m) + + #Try to match indexless WildTensor instances + remaining_e_tensors = [t for t in expr_sifted["Tensor"] if t not in matched_e_tensors] + if len(indexless_wilds) > 0: + #If there are any remaining tensors, match them with the indexless WildTensor + m = indexless_wilds[0].matches( TensMul(1,*remaining_e_tensors).doit(deep=False) ) + if m is None: + return None + else: + repl_dict.update(m) + elif len(remaining_e_tensors) > 0: + return None + + #Try to match the non-tensorial coefficient + m = self.coeff.matches(expr_coeff, old=old) + if m is None: + return None + else: + repl_dict.update(m) + + return repl_dict + + def matches(self, expr, repl_dict=None, old=False): + expr = sympify(expr) + + if repl_dict is None: + repl_dict = {} + else: + repl_dict = repl_dict.copy() + + commute = all(arg.component.comm == 0 for arg in expr.args if isinstance(arg, Tensor)) + if commute: + return self._matches_commutative(expr, repl_dict, old) + else: + raise NotImplementedError("Tensor matching not implemented for non-commuting tensors") + +class TensorElement(TensExpr): + """ + Tensor with evaluated components. + + Examples + ======== + + >>> from sympy.tensor.tensor import TensorIndexType, TensorHead, TensorSymmetry + >>> from sympy import symbols + >>> L = TensorIndexType("L") + >>> i, j, k = symbols("i j k") + >>> A = TensorHead("A", [L, L], TensorSymmetry.fully_symmetric(2)) + >>> A(i, j).get_free_indices() + [i, j] + + If we want to set component ``i`` to a specific value, use the + ``TensorElement`` class: + + >>> from sympy.tensor.tensor import TensorElement + >>> te = TensorElement(A(i, j), {i: 2}) + + As index ``i`` has been accessed (``{i: 2}`` is the evaluation of its 3rd + element), the free indices will only contain ``j``: + + >>> te.get_free_indices() + [j] + """ + + def __new__(cls, expr, index_map): + if not isinstance(expr, Tensor): + # remap + if not isinstance(expr, TensExpr): + raise TypeError("%s is not a tensor expression" % expr) + return expr.func(*[TensorElement(arg, index_map) for arg in expr.args]) + expr_free_indices = expr.get_free_indices() + name_translation = {i.args[0]: i for i in expr_free_indices} + index_map = {name_translation.get(index, index): value for index, value in index_map.items()} + index_map = {index: value for index, value in index_map.items() if index in expr_free_indices} + if len(index_map) == 0: + return expr + free_indices = [i for i in expr_free_indices if i not in index_map.keys()] + index_map = Dict(index_map) + obj = TensExpr.__new__(cls, expr, index_map) + obj._free_indices = free_indices + return obj + + @property + def free(self): + return [(index, i) for i, index in enumerate(self.get_free_indices())] + + @property + def dum(self): + # TODO: inherit dummies from expr + return [] + + @property + def expr(self): + return self._args[0] + + @property + def index_map(self): + return self._args[1] + + @property + def coeff(self): + return S.One + + @property + def nocoeff(self): + return self + + def get_free_indices(self): + return self._free_indices + + def _replace_indices(self, repl: dict[TensorIndex, TensorIndex]) -> TensExpr: + # TODO: can be improved: + return self.xreplace(repl) + + def get_indices(self): + return self.get_free_indices() + + def _extract_data(self, replacement_dict): + ret_indices, array = self.expr._extract_data(replacement_dict) + index_map = self.index_map + slice_tuple = tuple(index_map.get(i, slice(None)) for i in ret_indices) + ret_indices = [i for i in ret_indices if i not in index_map] + array = array.__getitem__(slice_tuple) + return ret_indices, array + + +class WildTensorHead(TensorHead): + """ + A wild object that is used to create ``WildTensor`` instances + + Explanation + =========== + + Examples + ======== + >>> from sympy.tensor.tensor import TensorHead, TensorIndex, WildTensorHead, TensorIndexType + >>> R3 = TensorIndexType('R3', dim=3) + >>> p = TensorIndex('p', R3) + >>> q = TensorIndex('q', R3) + + A WildTensorHead can be created without specifying a ``TensorIndexType`` + + >>> W = WildTensorHead("W") + + Calling it with a ``TensorIndex`` creates a ``WildTensor`` instance. + + >>> type(W(p)) + + + The ``TensorIndexType`` is automatically detected from the index that is passed + + >>> W(p).component + W(R3) + + Calling it with no indices returns an object that can match tensors with any number of indices. + + >>> K = TensorHead('K', [R3]) + >>> Q = TensorHead('Q', [R3, R3]) + >>> W().matches(K(p)) + {W: K(p)} + >>> W().matches(Q(p,q)) + {W: Q(p, q)} + + If you want to ignore the order of indices while matching, pass ``unordered_indices=True``. + + >>> U = WildTensorHead("U", unordered_indices=True) + >>> W(p,q).matches(Q(q,p)) + >>> U(p,q).matches(Q(q,p)) + {U(R3,R3): _WildTensExpr(Q(q, p))} + + Parameters + ========== + name : name of the tensor + unordered_indices : whether the order of the indices matters for matching + (default: False) + + See also + ======== + ``WildTensor`` + ``TensorHead`` + + """ + def __new__(cls, name, index_types=None, symmetry=None, comm=0, unordered_indices=False): + if isinstance(name, str): + name_symbol = Symbol(name) + elif isinstance(name, Symbol): + name_symbol = name + else: + raise ValueError("invalid name") + + if index_types is None: + index_types = [] + + if symmetry is None: + symmetry = TensorSymmetry.no_symmetry(len(index_types)) + else: + assert symmetry.rank == len(index_types) + + if symmetry != TensorSymmetry.no_symmetry(len(index_types)): + raise NotImplementedError("Wild matching based on symmetry is not implemented.") + + obj = Basic.__new__(cls, name_symbol, Tuple(*index_types), sympify(symmetry), sympify(comm), sympify(unordered_indices)) + + return obj + + @property + def unordered_indices(self): + return self.args[4] + + def __call__(self, *indices, **kwargs): + tensor = WildTensor(self, indices, **kwargs) + return tensor.doit() + + +class WildTensor(Tensor): + """ + A wild object which matches ``Tensor`` instances + + Explanation + =========== + This is instantiated by attaching indices to a ``WildTensorHead`` instance. + + Examples + ======== + >>> from sympy.tensor.tensor import TensorHead, TensorIndex, WildTensorHead, TensorIndexType + >>> W = WildTensorHead("W") + >>> R3 = TensorIndexType('R3', dim=3) + >>> p = TensorIndex('p', R3) + >>> q = TensorIndex('q', R3) + >>> K = TensorHead('K', [R3]) + >>> Q = TensorHead('Q', [R3, R3]) + + Matching also takes the indices into account + >>> W(p).matches(K(p)) + {W(R3): _WildTensExpr(K(p))} + >>> W(p).matches(K(q)) + >>> W(p).matches(K(-p)) + + If you want to match objects with any number of indices, just use a ``WildTensor`` with no indices. + >>> W().matches(K(p)) + {W: K(p)} + >>> W().matches(Q(p,q)) + {W: Q(p, q)} + + See Also + ======== + ``WildTensorHead`` + ``Tensor`` + + """ + def __new__(cls, tensor_head, indices, **kw_args): + is_canon_bp = kw_args.pop("is_canon_bp", False) + + if tensor_head.func == TensorHead: + """ + If someone tried to call WildTensor by supplying a TensorHead (not a WildTensorHead), return a normal tensor instead. This is helpful when using subs on an expression to replace occurrences of a WildTensorHead with a TensorHead. + """ + return Tensor(tensor_head, indices, is_canon_bp=is_canon_bp, **kw_args) + elif tensor_head.func == _WildTensExpr: + return tensor_head(*indices) + + indices = cls._parse_indices(tensor_head, indices) + index_types = [ind.tensor_index_type for ind in indices] + tensor_head = tensor_head.func( + tensor_head.name, + index_types, + symmetry=None, + comm=tensor_head.comm, + unordered_indices=tensor_head.unordered_indices, + ) + + obj = Basic.__new__(cls, tensor_head, Tuple(*indices)) + obj.name = tensor_head.name + obj._index_structure = _IndexStructure.from_indices(*indices) + obj._free = obj._index_structure.free[:] + obj._dum = obj._index_structure.dum[:] + obj._ext_rank = obj._index_structure._ext_rank + obj._coeff = S.One + obj._nocoeff = obj + obj._component = tensor_head + obj._components = [tensor_head] + if tensor_head.rank != len(indices): + raise ValueError("wrong number of indices") + obj.is_canon_bp = is_canon_bp + obj._index_map = obj._build_index_map(indices, obj._index_structure) + + return obj + + + def matches(self, expr, repl_dict=None, old=False): + if not isinstance(expr, TensExpr) and expr != S(1): + return None + + if repl_dict is None: + repl_dict = {} + else: + repl_dict = repl_dict.copy() + + if len(self.indices) > 0: + if not hasattr(expr, "get_free_indices"): + return None + expr_indices = expr.get_free_indices() + if len(expr_indices) != len(self.indices): + return None + if self._component.unordered_indices: + m = self._match_indices_ignoring_order(expr) + if m is None: + return None + else: + repl_dict.update(m) + else: + for i in range(len(expr_indices)): + m = self.indices[i].matches(expr_indices[i]) + if m is None: + return None + else: + repl_dict.update(m) + + repl_dict[self.component] = _WildTensExpr(expr) + else: + #If no indices were passed to the WildTensor, it may match tensors with any number of indices. + repl_dict[self] = expr + + return repl_dict + + def _match_indices_ignoring_order(self, expr, repl_dict=None, old=False): + """ + Helper method for matches. Checks if the indices of self and expr + match disregarding index ordering. + """ + if repl_dict is None: + repl_dict = {} + else: + repl_dict = repl_dict.copy() + + def siftkey(ind): + if isinstance(ind, WildTensorIndex): + if ind.ignore_updown: + return "wild, updown" + else: + return "wild" + else: + return "nonwild" + + indices_sifted = sift(self.indices, siftkey) + + matched_indices = [] + expr_indices_remaining = expr.get_indices() + for ind in indices_sifted["nonwild"]: + matched_this_ind = False + for e_ind in expr_indices_remaining: + if e_ind in matched_indices: + continue + m = ind.matches(e_ind) + if m is not None: + matched_this_ind = True + repl_dict.update(m) + matched_indices.append(e_ind) + break + if not matched_this_ind: + return None + + expr_indices_remaining = [i for i in expr_indices_remaining if i not in matched_indices] + for ind in indices_sifted["wild"]: + matched_this_ind = False + for e_ind in expr_indices_remaining: + m = ind.matches(e_ind) + if m is not None: + if -ind in repl_dict.keys() and -repl_dict[-ind] != m[ind]: + return None + matched_this_ind = True + repl_dict.update(m) + matched_indices.append(e_ind) + break + if not matched_this_ind: + return None + + expr_indices_remaining = [i for i in expr_indices_remaining if i not in matched_indices] + for ind in indices_sifted["wild, updown"]: + matched_this_ind = False + for e_ind in expr_indices_remaining: + m = ind.matches(e_ind) + if m is not None: + if -ind in repl_dict.keys() and -repl_dict[-ind] != m[ind]: + return None + matched_this_ind = True + repl_dict.update(m) + matched_indices.append(e_ind) + break + if not matched_this_ind: + return None + + if len(matched_indices) < len(self.indices): + return None + else: + return repl_dict + +class WildTensorIndex(TensorIndex): + """ + A wild object that matches TensorIndex instances. + + Examples + ======== + >>> from sympy.tensor.tensor import TensorIndex, TensorIndexType, WildTensorIndex + >>> R3 = TensorIndexType('R3', dim=3) + >>> p = TensorIndex("p", R3) + + By default, covariant indices only match with covariant indices (and + similarly for contravariant) + + >>> q = WildTensorIndex("q", R3) + >>> (q).matches(p) + {q: p} + >>> (q).matches(-p) + + If you want matching to ignore whether the index is co/contra-variant, set + ignore_updown=True + + >>> r = WildTensorIndex("r", R3, ignore_updown=True) + >>> (r).matches(-p) + {r: -p} + >>> (r).matches(p) + {r: p} + + Parameters + ========== + name : name of the index (string), or ``True`` if you want it to be + automatically assigned + tensor_index_type : ``TensorIndexType`` of the index + is_up : flag for contravariant index (is_up=True by default) + ignore_updown : bool, Whether this should match both co- and contra-variant + indices (default:False) + """ + def __new__(cls, name, tensor_index_type, is_up=True, ignore_updown=False): + if isinstance(name, str): + name_symbol = Symbol(name) + elif isinstance(name, Symbol): + name_symbol = name + elif name is True: + name = "_i{}".format(len(tensor_index_type._autogenerated)) + name_symbol = Symbol(name) + tensor_index_type._autogenerated.append(name_symbol) + else: + raise ValueError("invalid name") + + is_up = sympify(is_up) + ignore_updown = sympify(ignore_updown) + return Basic.__new__(cls, name_symbol, tensor_index_type, is_up, ignore_updown) + + @property + def ignore_updown(self): + return self.args[3] + + def __neg__(self): + t1 = WildTensorIndex(self.name, self.tensor_index_type, + (not self.is_up), self.ignore_updown) + return t1 + + def matches(self, expr, repl_dict=None, old=False): + if not isinstance(expr, TensorIndex): + return None + if self.tensor_index_type != expr.tensor_index_type: + return None + if not self.ignore_updown: + if self.is_up != expr.is_up: + return None + + if repl_dict is None: + repl_dict = {} + else: + repl_dict = repl_dict.copy() + + repl_dict[self] = expr + return repl_dict + + +class _WildTensExpr(Basic): + """ + INTERNAL USE ONLY + + This is an object that helps with replacement of WildTensors in expressions. + When this object is set as the tensor_head of a WildTensor, it replaces the + WildTensor by a TensExpr (passed when initializing this object). + + Examples + ======== + >>> from sympy.tensor.tensor import WildTensorHead, TensorIndex, TensorHead, TensorIndexType + >>> W = WildTensorHead("W") + >>> R3 = TensorIndexType('R3', dim=3) + >>> p = TensorIndex('p', R3) + >>> q = TensorIndex('q', R3) + >>> K = TensorHead('K', [R3]) + >>> print( ( K(p) ).replace( W(p), W(q)*W(-q)*W(p) ) ) + K(R_0)*K(-R_0)*K(p) + + """ + def __init__(self, expr): + if not isinstance(expr, TensExpr): + raise TypeError("_WildTensExpr expects a TensExpr as argument") + self.expr = expr + + def __call__(self, *indices): + return self.expr._replace_indices(dict(zip(self.expr.get_free_indices(), indices))) + + def __neg__(self): + return self.func(self.expr*S.NegativeOne) + + def __abs__(self): + raise NotImplementedError + + def __add__(self, other): + if other.func != self.func: + raise TypeError(f"Cannot add {self.func} to {other.func}") + return self.func(self.expr+other.expr) + + def __radd__(self, other): + if other.func != self.func: + raise TypeError(f"Cannot add {self.func} to {other.func}") + return self.func(other.expr+self.expr) + + def __sub__(self, other): + return self + (-other) + + def __rsub__(self, other): + return other + (-self) + + def __mul__(self, other): + raise NotImplementedError + + def __rmul__(self, other): + raise NotImplementedError + + def __truediv__(self, other): + raise NotImplementedError + + def __rtruediv__(self, other): + raise NotImplementedError + + def __pow__(self, other): + raise NotImplementedError + + def __rpow__(self, other): + raise NotImplementedError + + +def canon_bp(p): + """ + Butler-Portugal canonicalization. See ``tensor_can.py`` from the + combinatorics module for the details. + """ + if isinstance(p, TensExpr): + return p.canon_bp() + return p + + +def tensor_mul(*a): + """ + product of tensors + """ + if not a: + return TensMul.from_data(S.One, [], [], []) + t = a[0] + for tx in a[1:]: + t = t*tx + return t + + +def riemann_cyclic_replace(t_r): + """ + replace Riemann tensor with an equivalent expression + + ``R(m,n,p,q) -> 2/3*R(m,n,p,q) - 1/3*R(m,q,n,p) + 1/3*R(m,p,n,q)`` + + """ + free = sorted(t_r.free, key=lambda x: x[1]) + m, n, p, q = [x[0] for x in free] + t0 = t_r*Rational(2, 3) + t1 = -t_r.substitute_indices((m,m),(n,q),(p,n),(q,p))*Rational(1, 3) + t2 = t_r.substitute_indices((m,m),(n,p),(p,n),(q,q))*Rational(1, 3) + t3 = t0 + t1 + t2 + return t3 + +def riemann_cyclic(t2): + """ + Replace each Riemann tensor with an equivalent expression + satisfying the cyclic identity. + + This trick is discussed in the reference guide to Cadabra. + + Examples + ======== + + >>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, TensorHead, riemann_cyclic, TensorSymmetry + >>> Lorentz = TensorIndexType('Lorentz', dummy_name='L') + >>> i, j, k, l = tensor_indices('i,j,k,l', Lorentz) + >>> R = TensorHead('R', [Lorentz]*4, TensorSymmetry.riemann()) + >>> t = R(i,j,k,l)*(R(-i,-j,-k,-l) - 2*R(-i,-k,-j,-l)) + >>> riemann_cyclic(t) + 0 + """ + t2 = t2.expand() + if isinstance(t2, (TensMul, Tensor)): + args = [t2] + else: + args = t2.args + a1 = [x.split() for x in args] + a2 = [[riemann_cyclic_replace(tx) for tx in y] for y in a1] + a3 = [tensor_mul(*v) for v in a2] + t3 = TensAdd(*a3).doit(deep=False) + if not t3: + return t3 + else: + return canon_bp(t3) + + +def get_lines(ex, index_type): + """ + Returns ``(lines, traces, rest)`` for an index type, + where ``lines`` is the list of list of positions of a matrix line, + ``traces`` is the list of list of traced matrix lines, + ``rest`` is the rest of the elements of the tensor. + """ + def _join_lines(a): + i = 0 + while i < len(a): + x = a[i] + xend = x[-1] + xstart = x[0] + hit = True + while hit: + hit = False + for j in range(i + 1, len(a)): + if j >= len(a): + break + if a[j][0] == xend: + hit = True + x.extend(a[j][1:]) + xend = x[-1] + a.pop(j) + continue + if a[j][0] == xstart: + hit = True + a[i] = reversed(a[j][1:]) + x + x = a[i] + xstart = a[i][0] + a.pop(j) + continue + if a[j][-1] == xend: + hit = True + x.extend(reversed(a[j][:-1])) + xend = x[-1] + a.pop(j) + continue + if a[j][-1] == xstart: + hit = True + a[i] = a[j][:-1] + x + x = a[i] + xstart = x[0] + a.pop(j) + continue + i += 1 + return a + + arguments = ex.args + dt = {} + for c in ex.args: + if not isinstance(c, TensExpr): + continue + if c in dt: + continue + index_types = c.index_types + a = [] + for i in range(len(index_types)): + if index_types[i] is index_type: + a.append(i) + if len(a) > 2: + raise ValueError('at most two indices of type %s allowed' % index_type) + if len(a) == 2: + dt[c] = a + #dum = ex.dum + lines = [] + traces = [] + traces1 = [] + #indices_to_args_pos = ex._get_indices_to_args_pos() + # TODO: add a dum_to_components_map ? + for p0, p1, c0, c1 in ex.dum_in_args: + if arguments[c0] not in dt: + continue + if c0 == c1: + traces.append([c0]) + continue + ta0 = dt[arguments[c0]] + ta1 = dt[arguments[c1]] + if p0 not in ta0: + continue + if ta0.index(p0) == ta1.index(p1): + # case gamma(i,s0,-s1) in c0, gamma(j,-s0,s2) in c1; + # to deal with this case one could add to the position + # a flag for transposition; + # one could write [(c0, False), (c1, True)] + raise NotImplementedError + # if p0 == ta0[1] then G in pos c0 is mult on the right by G in c1 + # if p0 == ta0[0] then G in pos c1 is mult on the right by G in c0 + ta0 = dt[arguments[c0]] + b0, b1 = (c0, c1) if p0 == ta0[1] else (c1, c0) + lines1 = lines.copy() + for line in lines: + if line[-1] == b0: + if line[0] == b1: + n = line.index(min(line)) + traces1.append(line) + traces.append(line[n:] + line[:n]) + else: + line.append(b1) + break + elif line[0] == b1: + line.insert(0, b0) + break + else: + lines1.append([b0, b1]) + + lines = [x for x in lines1 if x not in traces1] + lines = _join_lines(lines) + rest = [] + for line in lines: + for y in line: + rest.append(y) + for line in traces: + for y in line: + rest.append(y) + rest = [x for x in range(len(arguments)) if x not in rest] + + return lines, traces, rest + + +def get_free_indices(t): + if not isinstance(t, TensExpr): + return () + return t.get_free_indices() + + +def get_indices(t): + if not isinstance(t, TensExpr): + return () + return t.get_indices() + +def get_dummy_indices(t): + if not isinstance(t, TensExpr): + return () + inds = t.get_indices() + free = t.get_free_indices() + return [i for i in inds if i not in free] + +def get_index_structure(t): + if isinstance(t, TensExpr): + return t._index_structure + return _IndexStructure([], [], [], []) + + +def get_coeff(t): + if isinstance(t, Tensor): + return S.One + if isinstance(t, TensMul): + return t.coeff + if isinstance(t, TensExpr): + raise ValueError("no coefficient associated to this tensor expression") + return t + +def contract_metric(t, g): + if isinstance(t, TensExpr): + return t.contract_metric(g) + return t + +def perm2tensor(t, g, is_canon_bp=False): + """ + Returns the tensor corresponding to the permutation ``g`` + + For further details, see the method in ``TIDS`` with the same name. + """ + if not isinstance(t, TensExpr): + return t + elif isinstance(t, (Tensor, TensMul)): + nim = get_index_structure(t).perm2tensor(g, is_canon_bp=is_canon_bp) + res = t._set_new_index_structure(nim, is_canon_bp=is_canon_bp) + if g[-1] != len(g) - 1: + return -res + + return res + raise NotImplementedError() + + +def substitute_indices(t, *index_tuples): + if not isinstance(t, TensExpr): + return t + return t.substitute_indices(*index_tuples) + + +def _get_wilds(expr): + return list(expr.atoms(Wild, WildFunction, WildTensor, WildTensorIndex, WildTensorHead)) + + +def get_postprocessor(cls): + def _postprocessor(expr): + tens_class = {Mul: TensMul, Add: TensAdd}[cls] + if any(isinstance(a, TensExpr) for a in expr.args): + return tens_class(*expr.args) + else: + return expr + + return _postprocessor + +Basic._constructor_postprocessor_mapping[TensExpr] = { + "Mul": [get_postprocessor(Mul)], +} diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/tests/__init__.py b/.venv/lib/python3.12/site-packages/sympy/tensor/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/tests/__pycache__/__init__.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/tensor/tests/__pycache__/__init__.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..a0a99daefbd826cda07bb6e7d083f399e05ad193 Binary files /dev/null and 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TensorProduct +from sympy.matrices.dense import Matrix +from sympy.matrices.expressions.matexpr import MatrixSymbol +from sympy.tensor.array import Array +from sympy.abc import x, y, z +from sympy.abc import i, j, k, l + + +A = MatrixSymbol("A", 3, 3) +B = MatrixSymbol("B", 3, 3) +C = MatrixSymbol("C", 3, 3) + + +def test_TensorProduct_construction(): + assert TensorProduct(3, 4) == 12 + assert isinstance(TensorProduct(A, A), TensorProduct) + + expr = TensorProduct(TensorProduct(x, y), z) + assert expr == x*y*z + + expr = TensorProduct(TensorProduct(A, B), C) + assert expr == TensorProduct(A, B, C) + + expr = TensorProduct(Matrix.eye(2), Array([[0, -1], [1, 0]])) + assert expr == Array([ + [ + [[0, -1], [1, 0]], + [[0, 0], [0, 0]] + ], + [ + [[0, 0], [0, 0]], + [[0, -1], [1, 0]] + ] + ]) + + +def test_TensorProduct_shape(): + + expr = TensorProduct(3, 4, evaluate=False) + assert expr.shape == () + assert expr.rank() == 0 + + expr = TensorProduct(Array([1, 2]), Array([x, y]), evaluate=False) + assert expr.shape == (2, 2) + assert expr.rank() == 2 + expr = TensorProduct(expr, expr, evaluate=False) + assert expr.shape == (2, 2, 2, 2) + assert expr.rank() == 4 + + expr = TensorProduct(Matrix.eye(2), Array([[0, -1], [1, 0]]), evaluate=False) + assert expr.shape == (2, 2, 2, 2) + assert expr.rank() == 4 + + +def test_TensorProduct_getitem(): + expr = TensorProduct(A, B) + assert expr[i, j, k, l] == A[i, j]*B[k, l] diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/tests/test_index_methods.py b/.venv/lib/python3.12/site-packages/sympy/tensor/tests/test_index_methods.py new file mode 100644 index 0000000000000000000000000000000000000000..df20f7e7c1ab392321e8350b95dd07c5639c1865 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/tests/test_index_methods.py @@ -0,0 +1,227 @@ +from sympy.core import symbols, S, Pow, Function +from sympy.functions import exp +from sympy.testing.pytest import raises +from sympy.tensor.indexed import Idx, IndexedBase +from sympy.tensor.index_methods import IndexConformanceException + +from sympy.tensor.index_methods import (get_contraction_structure, get_indices) + + +def test_trivial_indices(): + x, y = symbols('x y') + assert get_indices(x) == (set(), {}) + assert get_indices(x*y) == (set(), {}) + assert get_indices(x + y) == (set(), {}) + assert get_indices(x**y) == (set(), {}) + + +def test_get_indices_Indexed(): + x = IndexedBase('x') + i, j = Idx('i'), Idx('j') + assert get_indices(x[i, j]) == ({i, j}, {}) + assert get_indices(x[j, i]) == ({j, i}, {}) + + +def test_get_indices_Idx(): + f = Function('f') + i, j = Idx('i'), Idx('j') + assert get_indices(f(i)*j) == ({i, j}, {}) + assert get_indices(f(j, i)) == ({j, i}, {}) + assert get_indices(f(i)*i) == (set(), {}) + + +def test_get_indices_mul(): + x = IndexedBase('x') + y = IndexedBase('y') + i, j = Idx('i'), Idx('j') + assert get_indices(x[j]*y[i]) == ({i, j}, {}) + assert get_indices(x[i]*y[j]) == ({i, j}, {}) + + +def test_get_indices_exceptions(): + x = IndexedBase('x') + y = IndexedBase('y') + i, j = Idx('i'), Idx('j') + raises(IndexConformanceException, lambda: get_indices(x[i] + y[j])) + + +def test_scalar_broadcast(): + x = IndexedBase('x') + y = IndexedBase('y') + i, j = Idx('i'), Idx('j') + assert get_indices(x[i] + y[i, i]) == ({i}, {}) + assert get_indices(x[i] + y[j, j]) == ({i}, {}) + + +def test_get_indices_add(): + x = IndexedBase('x') + y = IndexedBase('y') + A = IndexedBase('A') + i, j, k = Idx('i'), Idx('j'), Idx('k') + assert get_indices(x[i] + 2*y[i]) == ({i}, {}) + assert get_indices(y[i] + 2*A[i, j]*x[j]) == ({i}, {}) + assert get_indices(y[i] + 2*(x[i] + A[i, j]*x[j])) == ({i}, {}) + assert get_indices(y[i] + x[i]*(A[j, j] + 1)) == ({i}, {}) + assert get_indices( + y[i] + x[i]*x[j]*(y[j] + A[j, k]*x[k])) == ({i}, {}) + + +def test_get_indices_Pow(): + x = IndexedBase('x') + y = IndexedBase('y') + A = IndexedBase('A') + i, j, k = Idx('i'), Idx('j'), Idx('k') + assert get_indices(Pow(x[i], y[j])) == ({i, j}, {}) + assert get_indices(Pow(x[i, k], y[j, k])) == ({i, j, k}, {}) + assert get_indices(Pow(A[i, k], y[k] + A[k, j]*x[j])) == ({i, k}, {}) + assert get_indices(Pow(2, x[i])) == get_indices(exp(x[i])) + + # test of a design decision, this may change: + assert get_indices(Pow(x[i], 2)) == ({i}, {}) + + +def test_get_contraction_structure_basic(): + x = IndexedBase('x') + y = IndexedBase('y') + i, j = Idx('i'), Idx('j') + assert get_contraction_structure(x[i]*y[j]) == {None: {x[i]*y[j]}} + assert get_contraction_structure(x[i] + y[j]) == {None: {x[i], y[j]}} + assert get_contraction_structure(x[i]*y[i]) == {(i,): {x[i]*y[i]}} + assert get_contraction_structure( + 1 + x[i]*y[i]) == {None: {S.One}, (i,): {x[i]*y[i]}} + assert get_contraction_structure(x[i]**y[i]) == {None: {x[i]**y[i]}} + + +def test_get_contraction_structure_complex(): + x = IndexedBase('x') + y = IndexedBase('y') + A = IndexedBase('A') + i, j, k = Idx('i'), Idx('j'), Idx('k') + expr1 = y[i] + A[i, j]*x[j] + d1 = {None: {y[i]}, (j,): {A[i, j]*x[j]}} + assert get_contraction_structure(expr1) == d1 + expr2 = expr1*A[k, i] + x[k] + d2 = {None: {x[k]}, (i,): {expr1*A[k, i]}, expr1*A[k, i]: [d1]} + assert get_contraction_structure(expr2) == d2 + + +def test_contraction_structure_simple_Pow(): + x = IndexedBase('x') + y = IndexedBase('y') + i, j, k = Idx('i'), Idx('j'), Idx('k') + ii_jj = x[i, i]**y[j, j] + assert get_contraction_structure(ii_jj) == { + None: {ii_jj}, + ii_jj: [ + {(i,): {x[i, i]}}, + {(j,): {y[j, j]}} + ] + } + + ii_jk = x[i, i]**y[j, k] + assert get_contraction_structure(ii_jk) == { + None: {x[i, i]**y[j, k]}, + x[i, i]**y[j, k]: [ + {(i,): {x[i, i]}} + ] + } + + +def test_contraction_structure_Mul_and_Pow(): + x = IndexedBase('x') + y = IndexedBase('y') + i, j, k = Idx('i'), Idx('j'), Idx('k') + + i_ji = x[i]**(y[j]*x[i]) + assert get_contraction_structure(i_ji) == {None: {i_ji}} + ij_i = (x[i]*y[j])**(y[i]) + assert get_contraction_structure(ij_i) == {None: {ij_i}} + j_ij_i = x[j]*(x[i]*y[j])**(y[i]) + assert get_contraction_structure(j_ij_i) == {(j,): {j_ij_i}} + j_i_ji = x[j]*x[i]**(y[j]*x[i]) + assert get_contraction_structure(j_i_ji) == {(j,): {j_i_ji}} + ij_exp_kki = x[i]*y[j]*exp(y[i]*y[k, k]) + result = get_contraction_structure(ij_exp_kki) + expected = { + (i,): {ij_exp_kki}, + ij_exp_kki: [{ + None: {exp(y[i]*y[k, k])}, + exp(y[i]*y[k, k]): [{ + None: {y[i]*y[k, k]}, + y[i]*y[k, k]: [{(k,): {y[k, k]}}] + }]} + ] + } + assert result == expected + + +def test_contraction_structure_Add_in_Pow(): + x = IndexedBase('x') + y = IndexedBase('y') + i, j, k = Idx('i'), Idx('j'), Idx('k') + s_ii_jj_s = (1 + x[i, i])**(1 + y[j, j]) + expected = { + None: {s_ii_jj_s}, + s_ii_jj_s: [ + {None: {S.One}, (i,): {x[i, i]}}, + {None: {S.One}, (j,): {y[j, j]}} + ] + } + result = get_contraction_structure(s_ii_jj_s) + assert result == expected + + s_ii_jk_s = (1 + x[i, i]) ** (1 + y[j, k]) + expected_2 = { + None: {(x[i, i] + 1)**(y[j, k] + 1)}, + s_ii_jk_s: [ + {None: {S.One}, (i,): {x[i, i]}} + ] + } + result_2 = get_contraction_structure(s_ii_jk_s) + assert result_2 == expected_2 + + +def test_contraction_structure_Pow_in_Pow(): + x = IndexedBase('x') + y = IndexedBase('y') + z = IndexedBase('z') + i, j, k = Idx('i'), Idx('j'), Idx('k') + ii_jj_kk = x[i, i]**y[j, j]**z[k, k] + expected = { + None: {ii_jj_kk}, + ii_jj_kk: [ + {(i,): {x[i, i]}}, + { + None: {y[j, j]**z[k, k]}, + y[j, j]**z[k, k]: [ + {(j,): {y[j, j]}}, + {(k,): {z[k, k]}} + ] + } + ] + } + assert get_contraction_structure(ii_jj_kk) == expected + + +def test_ufunc_support(): + f = Function('f') + g = Function('g') + x = IndexedBase('x') + y = IndexedBase('y') + i, j = Idx('i'), Idx('j') + a = symbols('a') + + assert get_indices(f(x[i])) == ({i}, {}) + assert get_indices(f(x[i], y[j])) == ({i, j}, {}) + assert get_indices(f(y[i])*g(x[i])) == (set(), {}) + assert get_indices(f(a, x[i])) == ({i}, {}) + assert get_indices(f(a, y[i], x[j])*g(x[i])) == ({j}, {}) + assert get_indices(g(f(x[i]))) == ({i}, {}) + + assert get_contraction_structure(f(x[i])) == {None: {f(x[i])}} + assert get_contraction_structure( + f(y[i])*g(x[i])) == {(i,): {f(y[i])*g(x[i])}} + assert get_contraction_structure( + f(y[i])*g(f(x[i]))) == {(i,): {f(y[i])*g(f(x[i]))}} + assert get_contraction_structure( + f(x[j], y[i])*g(x[i])) == {(i,): {f(x[j], y[i])*g(x[i])}} diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/tests/test_indexed.py b/.venv/lib/python3.12/site-packages/sympy/tensor/tests/test_indexed.py new file mode 100644 index 0000000000000000000000000000000000000000..689ec932c8fcefe0a24de289dd2ffd6820c63f19 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/tests/test_indexed.py @@ -0,0 +1,511 @@ +from sympy.core import symbols, Symbol, Tuple, oo, Dummy +from sympy.tensor.indexed import IndexException +from sympy.testing.pytest import raises +from sympy.utilities.iterables import iterable + +# import test: +from sympy.concrete.summations import Sum +from sympy.core.function import Function, Subs, Derivative +from sympy.core.relational import (StrictLessThan, GreaterThan, + StrictGreaterThan, LessThan) +from sympy.core.singleton import S +from sympy.functions.elementary.exponential import exp, log +from sympy.functions.elementary.trigonometric import cos, sin +from sympy.functions.special.tensor_functions import KroneckerDelta +from sympy.series.order import Order +from sympy.sets.fancysets import Range +from sympy.tensor.indexed import IndexedBase, Idx, Indexed + + +def test_Idx_construction(): + i, a, b = symbols('i a b', integer=True) + assert Idx(i) != Idx(i, 1) + assert Idx(i, a) == Idx(i, (0, a - 1)) + assert Idx(i, oo) == Idx(i, (0, oo)) + + x = symbols('x', integer=False) + raises(TypeError, lambda: Idx(x)) + raises(TypeError, lambda: Idx(0.5)) + raises(TypeError, lambda: Idx(i, x)) + raises(TypeError, lambda: Idx(i, 0.5)) + raises(TypeError, lambda: Idx(i, (x, 5))) + raises(TypeError, lambda: Idx(i, (2, x))) + raises(TypeError, lambda: Idx(i, (2, 3.5))) + + +def test_Idx_properties(): + i, a, b = symbols('i a b', integer=True) + assert Idx(i).is_integer + assert Idx(i).name == 'i' + assert Idx(i + 2).name == 'i + 2' + assert Idx('foo').name == 'foo' + + +def test_Idx_bounds(): + i, a, b = symbols('i a b', integer=True) + assert Idx(i).lower is None + assert Idx(i).upper is None + assert Idx(i, a).lower == 0 + assert Idx(i, a).upper == a - 1 + assert Idx(i, 5).lower == 0 + assert Idx(i, 5).upper == 4 + assert Idx(i, oo).lower == 0 + assert Idx(i, oo).upper is oo + assert Idx(i, (a, b)).lower == a + assert Idx(i, (a, b)).upper == b + assert Idx(i, (1, 5)).lower == 1 + assert Idx(i, (1, 5)).upper == 5 + assert Idx(i, (-oo, oo)).lower is -oo + assert Idx(i, (-oo, oo)).upper is oo + + +def test_Idx_fixed_bounds(): + i, a, b, x = symbols('i a b x', integer=True) + assert Idx(x).lower is None + assert Idx(x).upper is None + assert Idx(x, a).lower == 0 + assert Idx(x, a).upper == a - 1 + assert Idx(x, 5).lower == 0 + assert Idx(x, 5).upper == 4 + assert Idx(x, oo).lower == 0 + assert Idx(x, oo).upper is oo + assert Idx(x, (a, b)).lower == a + assert Idx(x, (a, b)).upper == b + assert Idx(x, (1, 5)).lower == 1 + assert Idx(x, (1, 5)).upper == 5 + assert Idx(x, (-oo, oo)).lower is -oo + assert Idx(x, (-oo, oo)).upper is oo + + +def test_Idx_inequalities(): + i14 = Idx("i14", (1, 4)) + i79 = Idx("i79", (7, 9)) + i46 = Idx("i46", (4, 6)) + i35 = Idx("i35", (3, 5)) + + assert i14 <= 5 + assert i14 < 5 + assert not (i14 >= 5) + assert not (i14 > 5) + + assert 5 >= i14 + assert 5 > i14 + assert not (5 <= i14) + assert not (5 < i14) + + assert LessThan(i14, 5) + assert StrictLessThan(i14, 5) + assert not GreaterThan(i14, 5) + assert not StrictGreaterThan(i14, 5) + + assert i14 <= 4 + assert isinstance(i14 < 4, StrictLessThan) + assert isinstance(i14 >= 4, GreaterThan) + assert not (i14 > 4) + + assert isinstance(i14 <= 1, LessThan) + assert not (i14 < 1) + assert i14 >= 1 + assert isinstance(i14 > 1, StrictGreaterThan) + + assert not (i14 <= 0) + assert not (i14 < 0) + assert i14 >= 0 + assert i14 > 0 + + from sympy.abc import x + + assert isinstance(i14 < x, StrictLessThan) + assert isinstance(i14 > x, StrictGreaterThan) + assert isinstance(i14 <= x, LessThan) + assert isinstance(i14 >= x, GreaterThan) + + assert i14 < i79 + assert i14 <= i79 + assert not (i14 > i79) + assert not (i14 >= i79) + + assert i14 <= i46 + assert isinstance(i14 < i46, StrictLessThan) + assert isinstance(i14 >= i46, GreaterThan) + assert not (i14 > i46) + + assert isinstance(i14 < i35, StrictLessThan) + assert isinstance(i14 > i35, StrictGreaterThan) + assert isinstance(i14 <= i35, LessThan) + assert isinstance(i14 >= i35, GreaterThan) + + iNone1 = Idx("iNone1") + iNone2 = Idx("iNone2") + + assert isinstance(iNone1 < iNone2, StrictLessThan) + assert isinstance(iNone1 > iNone2, StrictGreaterThan) + assert isinstance(iNone1 <= iNone2, LessThan) + assert isinstance(iNone1 >= iNone2, GreaterThan) + + +def test_Idx_inequalities_current_fails(): + i14 = Idx("i14", (1, 4)) + + assert S(5) >= i14 + assert S(5) > i14 + assert not (S(5) <= i14) + assert not (S(5) < i14) + + +def test_Idx_func_args(): + i, a, b = symbols('i a b', integer=True) + ii = Idx(i) + assert ii.func(*ii.args) == ii + ii = Idx(i, a) + assert ii.func(*ii.args) == ii + ii = Idx(i, (a, b)) + assert ii.func(*ii.args) == ii + + +def test_Idx_subs(): + i, a, b = symbols('i a b', integer=True) + assert Idx(i, a).subs(a, b) == Idx(i, b) + assert Idx(i, a).subs(i, b) == Idx(b, a) + + assert Idx(i).subs(i, 2) == Idx(2) + assert Idx(i, a).subs(a, 2) == Idx(i, 2) + assert Idx(i, (a, b)).subs(i, 2) == Idx(2, (a, b)) + + +def test_IndexedBase_sugar(): + i, j = symbols('i j', integer=True) + a = symbols('a') + A1 = Indexed(a, i, j) + A2 = IndexedBase(a) + assert A1 == A2[i, j] + assert A1 == A2[(i, j)] + assert A1 == A2[[i, j]] + assert A1 == A2[Tuple(i, j)] + assert all(a.is_Integer for a in A2[1, 0].args[1:]) + + +def test_IndexedBase_subs(): + i = symbols('i', integer=True) + a, b = symbols('a b') + A = IndexedBase(a) + B = IndexedBase(b) + assert A[i] == B[i].subs(b, a) + C = {1: 2} + assert C[1] == A[1].subs(A, C) + + +def test_IndexedBase_shape(): + i, j, m, n = symbols('i j m n', integer=True) + a = IndexedBase('a', shape=(m, m)) + b = IndexedBase('a', shape=(m, n)) + assert b.shape == Tuple(m, n) + assert a[i, j] != b[i, j] + assert a[i, j] == b[i, j].subs(n, m) + assert b.func(*b.args) == b + assert b[i, j].func(*b[i, j].args) == b[i, j] + raises(IndexException, lambda: b[i]) + raises(IndexException, lambda: b[i, i, j]) + F = IndexedBase("F", shape=m) + assert F.shape == Tuple(m) + assert F[i].subs(i, j) == F[j] + raises(IndexException, lambda: F[i, j]) + + +def test_IndexedBase_assumptions(): + i = Symbol('i', integer=True) + a = Symbol('a') + A = IndexedBase(a, positive=True) + for c in (A, A[i]): + assert c.is_real + assert c.is_complex + assert not c.is_imaginary + assert c.is_nonnegative + assert c.is_nonzero + assert c.is_commutative + assert log(exp(c)) == c + + assert A != IndexedBase(a) + assert A == IndexedBase(a, positive=True, real=True) + assert A[i] != Indexed(a, i) + + +def test_IndexedBase_assumptions_inheritance(): + I = Symbol('I', integer=True) + I_inherit = IndexedBase(I) + I_explicit = IndexedBase('I', integer=True) + + assert I_inherit.is_integer + assert I_explicit.is_integer + assert I_inherit.label.is_integer + assert I_explicit.label.is_integer + assert I_inherit == I_explicit + + +def test_issue_17652(): + """Regression test issue #17652. + + IndexedBase.label should not upcast subclasses of Symbol + """ + class SubClass(Symbol): + pass + + x = SubClass('X') + assert type(x) == SubClass + base = IndexedBase(x) + assert type(x) == SubClass + assert type(base.label) == SubClass + + +def test_Indexed_constructor(): + i, j = symbols('i j', integer=True) + A = Indexed('A', i, j) + assert A == Indexed(Symbol('A'), i, j) + assert A == Indexed(IndexedBase('A'), i, j) + raises(TypeError, lambda: Indexed(A, i, j)) + raises(IndexException, lambda: Indexed("A")) + assert A.free_symbols == {A, A.base.label, i, j} + + +def test_Indexed_func_args(): + i, j = symbols('i j', integer=True) + a = symbols('a') + A = Indexed(a, i, j) + assert A == A.func(*A.args) + + +def test_Indexed_subs(): + i, j, k = symbols('i j k', integer=True) + a, b = symbols('a b') + A = IndexedBase(a) + B = IndexedBase(b) + assert A[i, j] == B[i, j].subs(b, a) + assert A[i, j] == A[i, k].subs(k, j) + + +def test_Indexed_properties(): + i, j = symbols('i j', integer=True) + A = Indexed('A', i, j) + assert A.name == 'A[i, j]' + assert A.rank == 2 + assert A.indices == (i, j) + assert A.base == IndexedBase('A') + assert A.ranges == [None, None] + raises(IndexException, lambda: A.shape) + + n, m = symbols('n m', integer=True) + assert Indexed('A', Idx( + i, m), Idx(j, n)).ranges == [Tuple(0, m - 1), Tuple(0, n - 1)] + assert Indexed('A', Idx(i, m), Idx(j, n)).shape == Tuple(m, n) + raises(IndexException, lambda: Indexed("A", Idx(i, m), Idx(j)).shape) + + +def test_Indexed_shape_precedence(): + i, j = symbols('i j', integer=True) + o, p = symbols('o p', integer=True) + n, m = symbols('n m', integer=True) + a = IndexedBase('a', shape=(o, p)) + assert a.shape == Tuple(o, p) + assert Indexed( + a, Idx(i, m), Idx(j, n)).ranges == [Tuple(0, m - 1), Tuple(0, n - 1)] + assert Indexed(a, Idx(i, m), Idx(j, n)).shape == Tuple(o, p) + assert Indexed( + a, Idx(i, m), Idx(j)).ranges == [Tuple(0, m - 1), (None, None)] + assert Indexed(a, Idx(i, m), Idx(j)).shape == Tuple(o, p) + + +def test_complex_indices(): + i, j = symbols('i j', integer=True) + A = Indexed('A', i, i + j) + assert A.rank == 2 + assert A.indices == (i, i + j) + + +def test_not_interable(): + i, j = symbols('i j', integer=True) + A = Indexed('A', i, i + j) + assert not iterable(A) + + +def test_Indexed_coeff(): + N = Symbol('N', integer=True) + len_y = N + i = Idx('i', len_y-1) + y = IndexedBase('y', shape=(len_y,)) + a = (1/y[i+1]*y[i]).coeff(y[i]) + b = (y[i]/y[i+1]).coeff(y[i]) + assert a == b + + +def test_differentiation(): + from sympy.functions.special.tensor_functions import KroneckerDelta + i, j, k, l = symbols('i j k l', cls=Idx) + a = symbols('a') + m, n = symbols("m, n", integer=True, finite=True) + assert m.is_real + h, L = symbols('h L', cls=IndexedBase) + hi, hj = h[i], h[j] + + expr = hi + assert expr.diff(hj) == KroneckerDelta(i, j) + assert expr.diff(hi) == KroneckerDelta(i, i) + + expr = S(2) * hi + assert expr.diff(hj) == S(2) * KroneckerDelta(i, j) + assert expr.diff(hi) == S(2) * KroneckerDelta(i, i) + assert expr.diff(a) is S.Zero + + assert Sum(expr, (i, -oo, oo)).diff(hj) == Sum(2*KroneckerDelta(i, j), (i, -oo, oo)) + assert Sum(expr.diff(hj), (i, -oo, oo)) == Sum(2*KroneckerDelta(i, j), (i, -oo, oo)) + assert Sum(expr, (i, -oo, oo)).diff(hj).doit() == 2 + + assert Sum(expr.diff(hi), (i, -oo, oo)).doit() == Sum(2, (i, -oo, oo)).doit() + assert Sum(expr, (i, -oo, oo)).diff(hi).doit() is oo + + expr = a * hj * hj / S(2) + assert expr.diff(hi) == a * h[j] * KroneckerDelta(i, j) + assert expr.diff(a) == hj * hj / S(2) + assert expr.diff(a, 2) is S.Zero + + assert Sum(expr, (i, -oo, oo)).diff(hi) == Sum(a*KroneckerDelta(i, j)*h[j], (i, -oo, oo)) + assert Sum(expr.diff(hi), (i, -oo, oo)) == Sum(a*KroneckerDelta(i, j)*h[j], (i, -oo, oo)) + assert Sum(expr, (i, -oo, oo)).diff(hi).doit() == a*h[j] + + assert Sum(expr, (j, -oo, oo)).diff(hi) == Sum(a*KroneckerDelta(i, j)*h[j], (j, -oo, oo)) + assert Sum(expr.diff(hi), (j, -oo, oo)) == Sum(a*KroneckerDelta(i, j)*h[j], (j, -oo, oo)) + assert Sum(expr, (j, -oo, oo)).diff(hi).doit() == a*h[i] + + expr = a * sin(hj * hj) + assert expr.diff(hi) == 2*a*cos(hj * hj) * hj * KroneckerDelta(i, j) + assert expr.diff(hj) == 2*a*cos(hj * hj) * hj + + expr = a * L[i, j] * h[j] + assert expr.diff(hi) == a*L[i, j]*KroneckerDelta(i, j) + assert expr.diff(hj) == a*L[i, j] + assert expr.diff(L[i, j]) == a*h[j] + assert expr.diff(L[k, l]) == a*KroneckerDelta(i, k)*KroneckerDelta(j, l)*h[j] + assert expr.diff(L[i, l]) == a*KroneckerDelta(j, l)*h[j] + + assert Sum(expr, (j, -oo, oo)).diff(L[k, l]) == Sum(a * KroneckerDelta(i, k) * KroneckerDelta(j, l) * h[j], (j, -oo, oo)) + assert Sum(expr, (j, -oo, oo)).diff(L[k, l]).doit() == a * KroneckerDelta(i, k) * h[l] + + assert h[m].diff(h[m]) == 1 + assert h[m].diff(h[n]) == KroneckerDelta(m, n) + assert Sum(a*h[m], (m, -oo, oo)).diff(h[n]) == Sum(a*KroneckerDelta(m, n), (m, -oo, oo)) + assert Sum(a*h[m], (m, -oo, oo)).diff(h[n]).doit() == a + assert Sum(a*h[m], (n, -oo, oo)).diff(h[n]) == Sum(a*KroneckerDelta(m, n), (n, -oo, oo)) + assert Sum(a*h[m], (m, -oo, oo)).diff(h[m]).doit() == oo*a + + +def test_indexed_series(): + A = IndexedBase("A") + i = symbols("i", integer=True) + assert sin(A[i]).series(A[i]) == A[i] - A[i]**3/6 + A[i]**5/120 + Order(A[i]**6, A[i]) + + +def test_indexed_is_constant(): + A = IndexedBase("A") + i, j, k = symbols("i,j,k") + assert not A[i].is_constant() + assert A[i].is_constant(j) + assert not A[1+2*i, k].is_constant() + assert not A[1+2*i, k].is_constant(i) + assert A[1+2*i, k].is_constant(j) + assert not A[1+2*i, k].is_constant(k) + + +def test_issue_12533(): + d = IndexedBase('d') + assert IndexedBase(range(5)) == Range(0, 5, 1) + assert d[0].subs(Symbol("d"), range(5)) == 0 + assert d[0].subs(d, range(5)) == 0 + assert d[1].subs(d, range(5)) == 1 + assert Indexed(Range(5), 2) == 2 + + +def test_issue_12780(): + n = symbols("n") + i = Idx("i", (0, n)) + raises(TypeError, lambda: i.subs(n, 1.5)) + + +def test_issue_18604(): + m = symbols("m") + assert Idx("i", m).name == 'i' + assert Idx("i", m).lower == 0 + assert Idx("i", m).upper == m - 1 + m = symbols("m", real=False) + raises(TypeError, lambda: Idx("i", m)) + +def test_Subs_with_Indexed(): + A = IndexedBase("A") + i, j, k = symbols("i,j,k") + x, y, z = symbols("x,y,z") + f = Function("f") + + assert Subs(A[i], A[i], A[j]).diff(A[j]) == 1 + assert Subs(A[i], A[i], x).diff(A[i]) == 0 + assert Subs(A[i], A[i], x).diff(A[j]) == 0 + assert Subs(A[i], A[i], x).diff(x) == 1 + assert Subs(A[i], A[i], x).diff(y) == 0 + assert Subs(A[i], A[i], A[j]).diff(A[k]) == KroneckerDelta(j, k) + assert Subs(x, x, A[i]).diff(A[j]) == KroneckerDelta(i, j) + assert Subs(f(A[i]), A[i], x).diff(A[j]) == 0 + assert Subs(f(A[i]), A[i], A[k]).diff(A[j]) == Derivative(f(A[k]), A[k])*KroneckerDelta(j, k) + assert Subs(x, x, A[i]**2).diff(A[j]) == 2*KroneckerDelta(i, j)*A[i] + assert Subs(A[i], A[i], A[j]**2).diff(A[k]) == 2*KroneckerDelta(j, k)*A[j] + + assert Subs(A[i]*x, x, A[i]).diff(A[i]) == 2*A[i] + assert Subs(A[i]*x, x, A[i]).diff(A[j]) == 2*A[i]*KroneckerDelta(i, j) + assert Subs(A[i]*x, x, A[j]).diff(A[i]) == A[j] + A[i]*KroneckerDelta(i, j) + assert Subs(A[i]*x, x, A[j]).diff(A[j]) == A[i] + A[j]*KroneckerDelta(i, j) + assert Subs(A[i]*x, x, A[i]).diff(A[k]) == 2*A[i]*KroneckerDelta(i, k) + assert Subs(A[i]*x, x, A[j]).diff(A[k]) == KroneckerDelta(i, k)*A[j] + KroneckerDelta(j, k)*A[i] + + assert Subs(A[i]*x, A[i], x).diff(A[i]) == 0 + assert Subs(A[i]*x, A[i], x).diff(A[j]) == 0 + assert Subs(A[i]*x, A[j], x).diff(A[i]) == x + assert Subs(A[i]*x, A[j], x).diff(A[j]) == x*KroneckerDelta(i, j) + assert Subs(A[i]*x, A[i], x).diff(A[k]) == 0 + assert Subs(A[i]*x, A[j], x).diff(A[k]) == x*KroneckerDelta(i, k) + + +def test_complicated_derivative_with_Indexed(): + x, y = symbols("x,y", cls=IndexedBase) + sigma = symbols("sigma") + i, j, k = symbols("i,j,k") + m0,m1,m2,m3,m4,m5 = symbols("m0:6") + f = Function("f") + + expr = f((x[i] - y[i])**2/sigma) + _xi_1 = symbols("xi_1", cls=Dummy) + assert expr.diff(x[m0]).dummy_eq( + (x[i] - y[i])*KroneckerDelta(i, m0)*\ + 2*Subs( + Derivative(f(_xi_1), _xi_1), + (_xi_1,), + ((x[i] - y[i])**2/sigma,) + )/sigma + ) + assert expr.diff(x[m0]).diff(x[m1]).dummy_eq( + 2*KroneckerDelta(i, m0)*\ + KroneckerDelta(i, m1)*Subs( + Derivative(f(_xi_1), _xi_1), + (_xi_1,), + ((x[i] - y[i])**2/sigma,) + )/sigma + \ + 4*(x[i] - y[i])**2*KroneckerDelta(i, m0)*KroneckerDelta(i, m1)*\ + Subs( + Derivative(f(_xi_1), _xi_1, _xi_1), + (_xi_1,), + ((x[i] - y[i])**2/sigma,) + )/sigma**2 + ) + + +def test_IndexedBase_commutative(): + t = IndexedBase('t', commutative=False) + u = IndexedBase('u', commutative=False) + v = IndexedBase('v') + assert t[0]*v[0] == v[0]*t[0] + assert t[0]*u[0] != u[0]*t[0] diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/tests/test_printing.py b/.venv/lib/python3.12/site-packages/sympy/tensor/tests/test_printing.py new file mode 100644 index 0000000000000000000000000000000000000000..9f3cf7f0591a7012c93354ab7b8d7e010def38bb --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/tests/test_printing.py @@ -0,0 +1,13 @@ +from sympy.tensor.tensor import TensorIndexType, tensor_indices, TensorHead +from sympy import I + +def test_printing_TensMul(): + R3 = TensorIndexType('R3', dim=3) + p, q = tensor_indices("p q", R3) + K = TensorHead("K", [R3]) + + assert repr(2*K(p)) == "2*K(p)" + assert repr(-K(p)) == "-K(p)" + assert repr(-2*K(p)*K(q)) == "-2*K(p)*K(q)" + assert repr(-I*K(p)) == "-I*K(p)" + assert repr(I*K(p)) == "I*K(p)" diff --git a/.venv/lib/python3.12/site-packages/sympy/tensor/tests/test_tensor.py b/.venv/lib/python3.12/site-packages/sympy/tensor/tests/test_tensor.py new file mode 100644 index 0000000000000000000000000000000000000000..3113f5be9bcd32224f3525b5d831b6d7476c39e3 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/tensor/tests/test_tensor.py @@ -0,0 +1,2218 @@ +from sympy.concrete.summations import Sum +from sympy.core.function import expand +from sympy.core.numbers import Integer +from sympy.matrices.dense import (Matrix, eye) +from sympy.tensor.indexed import Indexed +from sympy.combinatorics import Permutation +from sympy.core import S, Rational, Symbol, Basic, Add, Wild, Function +from sympy.core.containers import Tuple +from sympy.core.symbol import symbols +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.integrals import integrate +from sympy.tensor.array import Array +from sympy.tensor.tensor import TensorIndexType, tensor_indices, TensorSymmetry, \ + get_symmetric_group_sgs, TensorIndex, tensor_mul, TensAdd, \ + riemann_cyclic_replace, riemann_cyclic, TensMul, tensor_heads, \ + TensorManager, TensExpr, TensorHead, canon_bp, \ + tensorhead, tensorsymmetry, TensorType, substitute_indices, \ + WildTensorIndex, WildTensorHead, _WildTensExpr +from sympy.testing.pytest import raises, XFAIL, warns_deprecated_sympy +from sympy.matrices import diag + +def _is_equal(arg1, arg2): + if isinstance(arg1, TensExpr): + return arg1.equals(arg2) + elif isinstance(arg2, TensExpr): + return arg2.equals(arg1) + return arg1 == arg2 + + +#################### Tests from tensor_can.py ####################### +def test_canonicalize_no_slot_sym(): + # A_d0 * B^d0; T_c = A^d0*B_d0 + Lorentz = TensorIndexType('Lorentz', dummy_name='L') + a, b, d0, d1 = tensor_indices('a,b,d0,d1', Lorentz) + A, B = tensor_heads('A,B', [Lorentz], TensorSymmetry.no_symmetry(1)) + t = A(-d0)*B(d0) + tc = t.canon_bp() + assert str(tc) == 'A(L_0)*B(-L_0)' + + # A^a * B^b; T_c = T + t = A(a)*B(b) + tc = t.canon_bp() + assert tc == t + # B^b * A^a + t1 = B(b)*A(a) + tc = t1.canon_bp() + assert str(tc) == 'A(a)*B(b)' + + # A symmetric + # A^{b}_{d0}*A^{d0, a}; T_c = A^{a d0}*A{b}_{d0} + A = TensorHead('A', [Lorentz]*2, TensorSymmetry.fully_symmetric(2)) + t = A(b, -d0)*A(d0, a) + tc = t.canon_bp() + assert str(tc) == 'A(a, L_0)*A(b, -L_0)' + + # A^{d1}_{d0}*B^d0*C_d1 + # T_c = A^{d0 d1}*B_d0*C_d1 + B, C = tensor_heads('B,C', [Lorentz], TensorSymmetry.no_symmetry(1)) + t = A(d1, -d0)*B(d0)*C(-d1) + tc = t.canon_bp() + assert str(tc) == 'A(L_0, L_1)*B(-L_0)*C(-L_1)' + + # A without symmetry + # A^{d1}_{d0}*B^d0*C_d1 ord=[d0,-d0,d1,-d1]; g = [2,1,0,3,4,5] + # T_c = A^{d0 d1}*B_d1*C_d0; can = [0,2,3,1,4,5] + A = TensorHead('A', [Lorentz]*2, TensorSymmetry.no_symmetry(2)) + t = A(d1, -d0)*B(d0)*C(-d1) + tc = t.canon_bp() + assert str(tc) == 'A(L_0, L_1)*B(-L_1)*C(-L_0)' + + # A, B without symmetry + # A^{d1}_{d0}*B_{d1}^{d0} + # T_c = A^{d0 d1}*B_{d0 d1} + B = TensorHead('B', [Lorentz]*2, TensorSymmetry.no_symmetry(2)) + t = A(d1, -d0)*B(-d1, d0) + tc = t.canon_bp() + assert str(tc) == 'A(L_0, L_1)*B(-L_0, -L_1)' + # A_{d0}^{d1}*B_{d1}^{d0} + # T_c = A^{d0 d1}*B_{d1 d0} + t = A(-d0, d1)*B(-d1, d0) + tc = t.canon_bp() + assert str(tc) == 'A(L_0, L_1)*B(-L_1, -L_0)' + + # A, B, C without symmetry + # A^{d1 d0}*B_{a d0}*C_{d1 b} + # T_c=A^{d0 d1}*B_{a d1}*C_{d0 b} + C = TensorHead('C', [Lorentz]*2, TensorSymmetry.no_symmetry(2)) + t = A(d1, d0)*B(-a, -d0)*C(-d1, -b) + tc = t.canon_bp() + assert str(tc) == 'A(L_0, L_1)*B(-a, -L_1)*C(-L_0, -b)' + + # A symmetric, B and C without symmetry + # A^{d1 d0}*B_{a d0}*C_{d1 b} + # T_c = A^{d0 d1}*B_{a d0}*C_{d1 b} + A = TensorHead('A', [Lorentz]*2, TensorSymmetry.fully_symmetric(2)) + t = A(d1, d0)*B(-a, -d0)*C(-d1, -b) + tc = t.canon_bp() + assert str(tc) == 'A(L_0, L_1)*B(-a, -L_0)*C(-L_1, -b)' + + # A and C symmetric, B without symmetry + # A^{d1 d0}*B_{a d0}*C_{d1 b} ord=[a,b,d0,-d0,d1,-d1] + # T_c = A^{d0 d1}*B_{a d0}*C_{b d1} + C = TensorHead('C', [Lorentz]*2, TensorSymmetry.fully_symmetric(2)) + t = A(d1, d0)*B(-a, -d0)*C(-d1, -b) + tc = t.canon_bp() + assert str(tc) == 'A(L_0, L_1)*B(-a, -L_0)*C(-b, -L_1)' + +def test_canonicalize_no_dummies(): + Lorentz = TensorIndexType('Lorentz', dummy_name='L') + a, b, c, d = tensor_indices('a, b, c, d', Lorentz) + + # A commuting + # A^c A^b A^a + # T_c = A^a A^b A^c + A = TensorHead('A', [Lorentz], TensorSymmetry.no_symmetry(1)) + t = A(c)*A(b)*A(a) + tc = t.canon_bp() + assert str(tc) == 'A(a)*A(b)*A(c)' + + # A anticommuting + # A^c A^b A^a + # T_c = -A^a A^b A^c + A = TensorHead('A', [Lorentz], TensorSymmetry.no_symmetry(1), 1) + t = A(c)*A(b)*A(a) + tc = t.canon_bp() + assert str(tc) == '-A(a)*A(b)*A(c)' + + # A commuting and symmetric + # A^{b,d}*A^{c,a} + # T_c = A^{a c}*A^{b d} + A = TensorHead('A', [Lorentz]*2, TensorSymmetry.fully_symmetric(2)) + t = A(b, d)*A(c, a) + tc = t.canon_bp() + assert str(tc) == 'A(a, c)*A(b, d)' + + # A anticommuting and symmetric + # A^{b,d}*A^{c,a} + # T_c = -A^{a c}*A^{b d} + A = TensorHead('A', [Lorentz]*2, TensorSymmetry.fully_symmetric(2), 1) + t = A(b, d)*A(c, a) + tc = t.canon_bp() + assert str(tc) == '-A(a, c)*A(b, d)' + + # A^{c,a}*A^{b,d} + # T_c = A^{a c}*A^{b d} + t = A(c, a)*A(b, d) + tc = t.canon_bp() + assert str(tc) == 'A(a, c)*A(b, d)' + +def test_tensorhead_construction_without_symmetry(): + L = TensorIndexType('Lorentz') + A1 = TensorHead('A', [L, L]) + A2 = TensorHead('A', [L, L], TensorSymmetry.no_symmetry(2)) + assert A1 == A2 + A3 = TensorHead('A', [L, L], TensorSymmetry.fully_symmetric(2)) # Symmetric + assert A1 != A3 + +def test_no_metric_symmetry(): + # no metric symmetry; A no symmetry + # A^d1_d0 * A^d0_d1 + # T_c = A^d0_d1 * A^d1_d0 + Lorentz = TensorIndexType('Lorentz', dummy_name='L', metric_symmetry=0) + d0, d1, d2, d3 = tensor_indices('d:4', Lorentz) + A = TensorHead('A', [Lorentz]*2, TensorSymmetry.no_symmetry(2)) + t = A(d1, -d0)*A(d0, -d1) + tc = t.canon_bp() + assert str(tc) == 'A(L_0, -L_1)*A(L_1, -L_0)' + + # A^d1_d2 * A^d0_d3 * A^d2_d1 * A^d3_d0 + # T_c = A^d0_d1 * A^d1_d0 * A^d2_d3 * A^d3_d2 + t = A(d1, -d2)*A(d0, -d3)*A(d2, -d1)*A(d3, -d0) + tc = t.canon_bp() + assert str(tc) == 'A(L_0, -L_1)*A(L_1, -L_0)*A(L_2, -L_3)*A(L_3, -L_2)' + + # A^d0_d2 * A^d1_d3 * A^d3_d0 * A^d2_d1 + # T_c = A^d0_d1 * A^d1_d2 * A^d2_d3 * A^d3_d0 + t = A(d0, -d1)*A(d1, -d2)*A(d2, -d3)*A(d3, -d0) + tc = t.canon_bp() + assert str(tc) == 'A(L_0, -L_1)*A(L_1, -L_2)*A(L_2, -L_3)*A(L_3, -L_0)' + +def test_canonicalize1(): + Lorentz = TensorIndexType('Lorentz', dummy_name='L') + a, a0, a1, a2, a3, b, d0, d1, d2, d3 = \ + tensor_indices('a,a0,a1,a2,a3,b,d0,d1,d2,d3', Lorentz) + + # A_d0*A^d0; ord = [d0,-d0] + # T_c = A^d0*A_d0 + A = TensorHead('A', [Lorentz], TensorSymmetry.no_symmetry(1)) + t = A(-d0)*A(d0) + tc = t.canon_bp() + assert str(tc) == 'A(L_0)*A(-L_0)' + + # A commuting + # A_d0*A_d1*A_d2*A^d2*A^d1*A^d0 + # T_c = A^d0*A_d0*A^d1*A_d1*A^d2*A_d2 + t = A(-d0)*A(-d1)*A(-d2)*A(d2)*A(d1)*A(d0) + tc = t.canon_bp() + assert str(tc) == 'A(L_0)*A(-L_0)*A(L_1)*A(-L_1)*A(L_2)*A(-L_2)' + + # A anticommuting + # A_d0*A_d1*A_d2*A^d2*A^d1*A^d0 + # T_c 0 + A = TensorHead('A', [Lorentz], TensorSymmetry.no_symmetry(1), 1) + t = A(-d0)*A(-d1)*A(-d2)*A(d2)*A(d1)*A(d0) + tc = t.canon_bp() + assert tc == 0 + + # A commuting symmetric + # A^{d0 b}*A^a_d1*A^d1_d0 + # T_c = A^{a d0}*A^{b d1}*A_{d0 d1} + A = TensorHead('A', [Lorentz]*2, TensorSymmetry.fully_symmetric(2)) + t = A(d0, b)*A(a, -d1)*A(d1, -d0) + tc = t.canon_bp() + assert str(tc) == 'A(a, L_0)*A(b, L_1)*A(-L_0, -L_1)' + + # A, B commuting symmetric + # A^{d0 b}*A^d1_d0*B^a_d1 + # T_c = A^{b d0}*A_d0^d1*B^a_d1 + B = TensorHead('B', [Lorentz]*2, TensorSymmetry.fully_symmetric(2)) + t = A(d0, b)*A(d1, -d0)*B(a, -d1) + tc = t.canon_bp() + assert str(tc) == 'A(b, L_0)*A(-L_0, L_1)*B(a, -L_1)' + + # A commuting symmetric + # A^{d1 d0 b}*A^{a}_{d1 d0}; ord=[a,b, d0,-d0,d1,-d1] + # T_c = A^{a d0 d1}*A^{b}_{d0 d1} + A = TensorHead('A', [Lorentz]*3, TensorSymmetry.fully_symmetric(3)) + t = A(d1, d0, b)*A(a, -d1, -d0) + tc = t.canon_bp() + assert str(tc) == 'A(a, L_0, L_1)*A(b, -L_0, -L_1)' + + # A^{d3 d0 d2}*A^a0_{d1 d2}*A^d1_d3^a1*A^{a2 a3}_d0 + # T_c = A^{a0 d0 d1}*A^a1_d0^d2*A^{a2 a3 d3}*A_{d1 d2 d3} + t = A(d3, d0, d2)*A(a0, -d1, -d2)*A(d1, -d3, a1)*A(a2, a3, -d0) + tc = t.canon_bp() + assert str(tc) == 'A(a0, L_0, L_1)*A(a1, -L_0, L_2)*A(a2, a3, L_3)*A(-L_1, -L_2, -L_3)' + + # A commuting symmetric, B antisymmetric + # A^{d0 d1 d2} * A_{d2 d3 d1} * B_d0^d3 + # in this esxample and in the next three, + # renaming dummy indices and using symmetry of A, + # T = A^{d0 d1 d2} * A_{d0 d1 d3} * B_d2^d3 + # can = 0 + A = TensorHead('A', [Lorentz]*3, TensorSymmetry.fully_symmetric(3)) + B = TensorHead('B', [Lorentz]*2, TensorSymmetry.fully_symmetric(-2)) + t = A(d0, d1, d2)*A(-d2, -d3, -d1)*B(-d0, d3) + tc = t.canon_bp() + assert tc == 0 + + # A anticommuting symmetric, B antisymmetric + # A^{d0 d1 d2} * A_{d2 d3 d1} * B_d0^d3 + # T_c = A^{d0 d1 d2} * A_{d0 d1}^d3 * B_{d2 d3} + A = TensorHead('A', [Lorentz]*3, TensorSymmetry.fully_symmetric(3), 1) + B = TensorHead('B', [Lorentz]*2, TensorSymmetry.fully_symmetric(-2)) + t = A(d0, d1, d2)*A(-d2, -d3, -d1)*B(-d0, d3) + tc = t.canon_bp() + assert str(tc) == 'A(L_0, L_1, L_2)*A(-L_0, -L_1, L_3)*B(-L_2, -L_3)' + + # A anticommuting symmetric, B antisymmetric commuting, antisymmetric metric + # A^{d0 d1 d2} * A_{d2 d3 d1} * B_d0^d3 + # T_c = -A^{d0 d1 d2} * A_{d0 d1}^d3 * B_{d2 d3} + Spinor = TensorIndexType('Spinor', dummy_name='S', metric_symmetry=-1) + a, a0, a1, a2, a3, b, d0, d1, d2, d3 = \ + tensor_indices('a,a0,a1,a2,a3,b,d0,d1,d2,d3', Spinor) + A = TensorHead('A', [Spinor]*3, TensorSymmetry.fully_symmetric(3), 1) + B = TensorHead('B', [Spinor]*2, TensorSymmetry.fully_symmetric(-2)) + t = A(d0, d1, d2)*A(-d2, -d3, -d1)*B(-d0, d3) + tc = t.canon_bp() + assert str(tc) == '-A(S_0, S_1, S_2)*A(-S_0, -S_1, S_3)*B(-S_2, -S_3)' + + # A anticommuting symmetric, B antisymmetric anticommuting, + # no metric symmetry + # A^{d0 d1 d2} * A_{d2 d3 d1} * B_d0^d3 + # T_c = A^{d0 d1 d2} * A_{d0 d1 d3} * B_d2^d3 + Mat = TensorIndexType('Mat', metric_symmetry=0, dummy_name='M') + a, a0, a1, a2, a3, b, d0, d1, d2, d3 = \ + tensor_indices('a,a0,a1,a2,a3,b,d0,d1,d2,d3', Mat) + A = TensorHead('A', [Mat]*3, TensorSymmetry.fully_symmetric(3), 1) + B = TensorHead('B', [Mat]*2, TensorSymmetry.fully_symmetric(-2)) + t = A(d0, d1, d2)*A(-d2, -d3, -d1)*B(-d0, d3) + tc = t.canon_bp() + assert str(tc) == 'A(M_0, M_1, M_2)*A(-M_0, -M_1, -M_3)*B(-M_2, M_3)' + + # Gamma anticommuting + # Gamma_{mu nu} * gamma^rho * Gamma^{nu mu alpha} + # T_c = -Gamma^{mu nu} * gamma^rho * Gamma_{alpha mu nu} + alpha, beta, gamma, mu, nu, rho = \ + tensor_indices('alpha,beta,gamma,mu,nu,rho', Lorentz) + Gamma = TensorHead('Gamma', [Lorentz], + TensorSymmetry.fully_symmetric(1), 2) + Gamma2 = TensorHead('Gamma', [Lorentz]*2, + TensorSymmetry.fully_symmetric(-2), 2) + Gamma3 = TensorHead('Gamma', [Lorentz]*3, + TensorSymmetry.fully_symmetric(-3), 2) + t = Gamma2(-mu, -nu)*Gamma(rho)*Gamma3(nu, mu, alpha) + tc = t.canon_bp() + assert str(tc) == '-Gamma(L_0, L_1)*Gamma(rho)*Gamma(alpha, -L_0, -L_1)' + + # Gamma_{mu nu} * Gamma^{gamma beta} * gamma_rho * Gamma^{nu mu alpha} + # T_c = Gamma^{mu nu} * Gamma^{beta gamma} * gamma_rho * Gamma^alpha_{mu nu} + t = Gamma2(mu, nu)*Gamma2(beta, gamma)*Gamma(-rho)*Gamma3(alpha, -mu, -nu) + tc = t.canon_bp() + assert str(tc) == 'Gamma(L_0, L_1)*Gamma(beta, gamma)*Gamma(-rho)*Gamma(alpha, -L_0, -L_1)' + + # f^a_{b,c} antisymmetric in b,c; A_mu^a no symmetry + # f^c_{d a} * f_{c e b} * A_mu^d * A_nu^a * A^{nu e} * A^{mu b} + # g = [8,11,5, 9,13,7, 1,10, 3,4, 2,12, 0,6, 14,15] + # T_c = -f^{a b c} * f_a^{d e} * A^mu_b * A_{mu d} * A^nu_c * A_{nu e} + Flavor = TensorIndexType('Flavor', dummy_name='F') + a, b, c, d, e, ff = tensor_indices('a,b,c,d,e,f', Flavor) + mu, nu = tensor_indices('mu,nu', Lorentz) + f = TensorHead('f', [Flavor]*3, TensorSymmetry.direct_product(1, -2)) + A = TensorHead('A', [Lorentz, Flavor], TensorSymmetry.no_symmetry(2)) + t = f(c, -d, -a)*f(-c, -e, -b)*A(-mu, d)*A(-nu, a)*A(nu, e)*A(mu, b) + tc = t.canon_bp() + assert str(tc) == '-f(F_0, F_1, F_2)*f(-F_0, F_3, F_4)*A(L_0, -F_1)*A(-L_0, -F_3)*A(L_1, -F_2)*A(-L_1, -F_4)' + + +def test_bug_correction_tensor_indices(): + # to make sure that tensor_indices does not return a list if creating + # only one index: + A = TensorIndexType("A") + i = tensor_indices('i', A) + assert not isinstance(i, (tuple, list)) + assert isinstance(i, TensorIndex) + + +def test_riemann_invariants(): + Lorentz = TensorIndexType('Lorentz', dummy_name='L') + d0, d1, d2, d3, d4, d5, d6, d7, d8, d9, d10, d11 = \ + tensor_indices('d0:12', Lorentz) + # R^{d0 d1}_{d1 d0}; ord = [d0,-d0,d1,-d1] + # T_c = -R^{d0 d1}_{d0 d1} + R = TensorHead('R', [Lorentz]*4, TensorSymmetry.riemann()) + t = R(d0, d1, -d1, -d0) + tc = t.canon_bp() + assert str(tc) == '-R(L_0, L_1, -L_0, -L_1)' + + # R_d11^d1_d0^d5 * R^{d6 d4 d0}_d5 * R_{d7 d2 d8 d9} * + # R_{d10 d3 d6 d4} * R^{d2 d7 d11}_d1 * R^{d8 d9 d3 d10} + # can = [0,2,4,6, 1,3,8,10, 5,7,12,14, 9,11,16,18, 13,15,20,22, + # 17,19,21>> from sympy.tensor.tensor import TensorIndexType, TensorHead + >>> from sympy.tensor.toperators import PartialDerivative + >>> from sympy import symbols + >>> L = TensorIndexType("L") + >>> A = TensorHead("A", [L]) + >>> B = TensorHead("B", [L]) + >>> i, j, k = symbols("i j k") + + >>> expr = PartialDerivative(A(i), A(j)) + >>> expr + PartialDerivative(A(i), A(j)) + + The ``PartialDerivative`` object behaves like a tensorial expression: + + >>> expr.get_indices() + [i, -j] + + Notice that the deriving variables have opposite valence than the + printed one: ``A(j)`` is printed as covariant, but the index of the + derivative is actually contravariant, i.e. ``-j``. + + Indices can be contracted: + + >>> expr = PartialDerivative(A(i), A(i)) + >>> expr + PartialDerivative(A(L_0), A(L_0)) + >>> expr.get_indices() + [L_0, -L_0] + + The method ``.get_indices()`` always returns all indices (even the + contracted ones). If only uncontracted indices are needed, call + ``.get_free_indices()``: + + >>> expr.get_free_indices() + [] + + Nested partial derivatives are flattened: + + >>> expr = PartialDerivative(PartialDerivative(A(i), A(j)), A(k)) + >>> expr + PartialDerivative(A(i), A(j), A(k)) + >>> expr.get_indices() + [i, -j, -k] + + Replace a derivative with array values: + + >>> from sympy.abc import x, y + >>> from sympy import sin, log + >>> compA = [sin(x), log(x)*y**3] + >>> compB = [x, y] + >>> expr = PartialDerivative(A(i), B(j)) + >>> expr.replace_with_arrays({A(i): compA, B(i): compB}) + [[cos(x), 0], [y**3/x, 3*y**2*log(x)]] + + The returned array is indexed by `(i, -j)`. + + Be careful that other SymPy modules put the indices of the deriving + variables before the indices of the derivand in the derivative result. + For example: + + >>> expr.get_free_indices() + [i, -j] + + >>> from sympy import Matrix, Array + >>> Matrix(compA).diff(Matrix(compB)).reshape(2, 2) + [[cos(x), y**3/x], [0, 3*y**2*log(x)]] + >>> Array(compA).diff(Array(compB)) + [[cos(x), y**3/x], [0, 3*y**2*log(x)]] + + These are the transpose of the result of ``PartialDerivative``, + as the matrix and the array modules put the index `-j` before `i` in the + derivative result. An array read with index order `(-j, i)` is indeed the + transpose of the same array read with index order `(i, -j)`. By specifying + the index order to ``.replace_with_arrays`` one can get a compatible + expression: + + >>> expr.replace_with_arrays({A(i): compA, B(i): compB}, [-j, i]) + [[cos(x), y**3/x], [0, 3*y**2*log(x)]] + """ + + def __new__(cls, expr, *variables): + + # Flatten: + if isinstance(expr, PartialDerivative): + variables = expr.variables + variables + expr = expr.expr + + args, indices, free, dum = cls._contract_indices_for_derivative( + S(expr), variables) + + obj = TensExpr.__new__(cls, *args) + + obj._indices = indices + obj._free = free + obj._dum = dum + return obj + + @property + def coeff(self): + return S.One + + @property + def nocoeff(self): + return self + + @classmethod + def _contract_indices_for_derivative(cls, expr, variables): + variables_opposite_valence = [] + + for i in variables: + if isinstance(i, Tensor): + i_free_indices = i.get_free_indices() + variables_opposite_valence.append( + i.xreplace({k: -k for k in i_free_indices})) + elif isinstance(i, Symbol): + variables_opposite_valence.append(i) + + args, indices, free, dum = TensMul._tensMul_contract_indices( + [expr] + variables_opposite_valence, replace_indices=True) + + for i in range(1, len(args)): + args_i = args[i] + if isinstance(args_i, Tensor): + i_indices = args[i].get_free_indices() + args[i] = args[i].xreplace({k: -k for k in i_indices}) + + return args, indices, free, dum + + def doit(self, **hints): + args, indices, free, dum = self._contract_indices_for_derivative(self.expr, self.variables) + + obj = self.func(*args) + obj._indices = indices + obj._free = free + obj._dum = dum + + return obj + + def _expand_partial_derivative(self): + args, indices, free, dum = self._contract_indices_for_derivative(self.expr, self.variables) + + obj = self.func(*args) + obj._indices = indices + obj._free = free + obj._dum = dum + + result = obj + + if not args[0].free_symbols: + return S.Zero + elif isinstance(obj.expr, TensAdd): + # take care of sums of multi PDs + result = obj.expr.func(*[ + self.func(a, *obj.variables)._expand_partial_derivative() + for a in result.expr.args]) + elif isinstance(obj.expr, TensMul): + # take care of products of multi PDs + if len(obj.variables) == 1: + # derivative with respect to single variable + terms = [] + mulargs = list(obj.expr.args) + for ind in range(len(mulargs)): + if not isinstance(sympify(mulargs[ind]), Number): + # a number coefficient is not considered for + # expansion of PartialDerivative + d = self.func(mulargs[ind], *obj.variables)._expand_partial_derivative() + terms.append(TensMul(*(mulargs[:ind] + + [d] + + mulargs[(ind + 1):]))) + result = TensAdd.fromiter(terms) + else: + # derivative with respect to multiple variables + # decompose: + # partial(expr, (u, v)) + # = partial(partial(expr, u).doit(), v).doit() + result = obj.expr # init with expr + for v in obj.variables: + result = self.func(result, v)._expand_partial_derivative() + # then throw PD on it + + return result + + def _perform_derivative(self): + result = self.expr + for v in self.variables: + if isinstance(result, TensExpr): + result = result._eval_partial_derivative(v) + else: + if v._diff_wrt: + result = result._eval_derivative(v) + else: + result = S.Zero + return result + + def get_indices(self): + return self._indices + + def get_free_indices(self): + free = sorted(self._free, key=lambda x: x[1]) + return [i[0] for i in free] + + def _replace_indices(self, repl): + expr = self.expr.xreplace(repl) + mirrored = {-k: -v for k, v in repl.items()} + variables = [i.xreplace(mirrored) for i in self.variables] + return self.func(expr, *variables) + + @property + def expr(self): + return self.args[0] + + @property + def variables(self): + return self.args[1:] + + def _extract_data(self, replacement_dict): + from .array import derive_by_array, tensorcontraction + indices, array = self.expr._extract_data(replacement_dict) + for variable in self.variables: + var_indices, var_array = variable._extract_data(replacement_dict) + var_indices = [-i for i in var_indices] + coeff_array, var_array = zip(*[i.as_coeff_Mul() for i in var_array]) + dim_before = len(array.shape) + array = derive_by_array(array, var_array) + dim_after = len(array.shape) + dim_increase = dim_after - dim_before + array = permutedims(array, [i + dim_increase for i in range(dim_before)] + list(range(dim_increase))) + array = array.as_mutable() + varindex = var_indices[0] + # Remove coefficients of base vector: + coeff_index = [0] + [slice(None) for i in range(len(indices))] + for i, coeff in enumerate(coeff_array): + coeff_index[0] = i + array[tuple(coeff_index)] /= coeff + if -varindex in indices: + pos = indices.index(-varindex) + array = tensorcontraction(array, (0, pos+1)) + indices.pop(pos) + else: + indices.append(varindex) + return indices, array diff --git a/.venv/lib/python3.12/site-packages/sympy/testing/__init__.py b/.venv/lib/python3.12/site-packages/sympy/testing/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..aa66402955e7d5d2b27cccc6627b42d33f4cd855 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/testing/__init__.py @@ -0,0 +1,10 @@ +"""This module contains code for running the tests in SymPy.""" + + +from .runtests import doctest +from .runtests_pytest import test + + +__all__ = [ + 'test', 'doctest', +] diff --git a/.venv/lib/python3.12/site-packages/sympy/testing/__pycache__/__init__.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/testing/__pycache__/__init__.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..59cd8b764e76e2bd95f0a4cba5677ec7c6250751 Binary files /dev/null and b/.venv/lib/python3.12/site-packages/sympy/testing/__pycache__/__init__.cpython-312.pyc differ diff --git a/.venv/lib/python3.12/site-packages/sympy/testing/__pycache__/matrices.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/testing/__pycache__/matrices.cpython-312.pyc new file mode 100644 index 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100644 index 0000000000000000000000000000000000000000..236a384366df7f69d0d92f43f7e007e95c12388c --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/testing/matrices.py @@ -0,0 +1,8 @@ +def allclose(A, B, rtol=1e-05, atol=1e-08): + if len(A) != len(B): + return False + + for x, y in zip(A, B): + if abs(x-y) > atol + rtol * max(abs(x), abs(y)): + return False + return True diff --git a/.venv/lib/python3.12/site-packages/sympy/testing/pytest.py b/.venv/lib/python3.12/site-packages/sympy/testing/pytest.py new file mode 100644 index 0000000000000000000000000000000000000000..498515a2d3c0a167a5f7067753736898cfa64799 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/testing/pytest.py @@ -0,0 +1,392 @@ +"""py.test hacks to support XFAIL/XPASS""" + +import platform +import sys +import re +import functools +import os +import contextlib +import warnings +import inspect +import pathlib +from typing import Any, Callable + +from sympy.utilities.exceptions import SymPyDeprecationWarning +# Imported here for backwards compatibility. Note: do not import this from +# here in library code (importing sympy.pytest in library code will break the +# pytest integration). +from sympy.utilities.exceptions import ignore_warnings # noqa:F401 + +ON_CI = os.getenv('CI', None) == "true" + +try: + import pytest + USE_PYTEST = getattr(sys, '_running_pytest', False) +except ImportError: + USE_PYTEST = False + +IS_WASM: bool = sys.platform == 'emscripten' or platform.machine() in ["wasm32", "wasm64"] + +raises: Callable[[Any, Any], Any] +XFAIL: Callable[[Any], Any] +skip: Callable[[Any], Any] +SKIP: Callable[[Any], Any] +slow: Callable[[Any], Any] +tooslow: Callable[[Any], Any] +nocache_fail: Callable[[Any], Any] + + +if USE_PYTEST: + raises = pytest.raises + skip = pytest.skip + XFAIL = pytest.mark.xfail + SKIP = pytest.mark.skip + slow = pytest.mark.slow + tooslow = pytest.mark.tooslow + nocache_fail = pytest.mark.nocache_fail + from _pytest.outcomes import Failed + +else: + # Not using pytest so define the things that would have been imported from + # there. + + # _pytest._code.code.ExceptionInfo + class ExceptionInfo: + def __init__(self, value): + self.value = value + + def __repr__(self): + return "".format(self.value) + + + def raises(expectedException, code=None): + """ + Tests that ``code`` raises the exception ``expectedException``. + + ``code`` may be a callable, such as a lambda expression or function + name. + + If ``code`` is not given or None, ``raises`` will return a context + manager for use in ``with`` statements; the code to execute then + comes from the scope of the ``with``. + + ``raises()`` does nothing if the callable raises the expected exception, + otherwise it raises an AssertionError. + + Examples + ======== + + >>> from sympy.testing.pytest import raises + + >>> raises(ZeroDivisionError, lambda: 1/0) + + >>> raises(ZeroDivisionError, lambda: 1/2) + Traceback (most recent call last): + ... + Failed: DID NOT RAISE + + >>> with raises(ZeroDivisionError): + ... n = 1/0 + >>> with raises(ZeroDivisionError): + ... n = 1/2 + Traceback (most recent call last): + ... + Failed: DID NOT RAISE + + Note that you cannot test multiple statements via + ``with raises``: + + >>> with raises(ZeroDivisionError): + ... n = 1/0 # will execute and raise, aborting the ``with`` + ... n = 9999/0 # never executed + + This is just what ``with`` is supposed to do: abort the + contained statement sequence at the first exception and let + the context manager deal with the exception. + + To test multiple statements, you'll need a separate ``with`` + for each: + + >>> with raises(ZeroDivisionError): + ... n = 1/0 # will execute and raise + >>> with raises(ZeroDivisionError): + ... n = 9999/0 # will also execute and raise + + """ + if code is None: + return RaisesContext(expectedException) + elif callable(code): + try: + code() + except expectedException as e: + return ExceptionInfo(e) + raise Failed("DID NOT RAISE") + elif isinstance(code, str): + raise TypeError( + '\'raises(xxx, "code")\' has been phased out; ' + 'change \'raises(xxx, "expression")\' ' + 'to \'raises(xxx, lambda: expression)\', ' + '\'raises(xxx, "statement")\' ' + 'to \'with raises(xxx): statement\'') + else: + raise TypeError( + 'raises() expects a callable for the 2nd argument.') + + class RaisesContext: + def __init__(self, expectedException): + self.expectedException = expectedException + + def __enter__(self): + return None + + def __exit__(self, exc_type, exc_value, traceback): + if exc_type is None: + raise Failed("DID NOT RAISE") + return issubclass(exc_type, self.expectedException) + + class XFail(Exception): + pass + + class XPass(Exception): + pass + + class Skipped(Exception): + pass + + class Failed(Exception): # type: ignore + pass + + def XFAIL(func): + def wrapper(): + try: + func() + except Exception as e: + message = str(e) + if message != "Timeout": + raise XFail(func.__name__) + else: + raise Skipped("Timeout") + raise XPass(func.__name__) + + wrapper = functools.update_wrapper(wrapper, func) + return wrapper + + def skip(str): + raise Skipped(str) + + def SKIP(reason): + """Similar to ``skip()``, but this is a decorator. """ + def wrapper(func): + def func_wrapper(): + raise Skipped(reason) + + func_wrapper = functools.update_wrapper(func_wrapper, func) + return func_wrapper + + return wrapper + + def slow(func): + func._slow = True + + def func_wrapper(): + func() + + func_wrapper = functools.update_wrapper(func_wrapper, func) + func_wrapper.__wrapped__ = func + return func_wrapper + + def tooslow(func): + func._slow = True + func._tooslow = True + + def func_wrapper(): + skip("Too slow") + + func_wrapper = functools.update_wrapper(func_wrapper, func) + func_wrapper.__wrapped__ = func + return func_wrapper + + def nocache_fail(func): + "Dummy decorator for marking tests that fail when cache is disabled" + return func + +@contextlib.contextmanager +def warns(warningcls, *, match='', test_stacklevel=True): + ''' + Like raises but tests that warnings are emitted. + + >>> from sympy.testing.pytest import warns + >>> import warnings + + >>> with warns(UserWarning): + ... warnings.warn('deprecated', UserWarning, stacklevel=2) + + >>> with warns(UserWarning): + ... pass + Traceback (most recent call last): + ... + Failed: DID NOT WARN. No warnings of type UserWarning\ + was emitted. The list of emitted warnings is: []. + + ``test_stacklevel`` makes it check that the ``stacklevel`` parameter to + ``warn()`` is set so that the warning shows the user line of code (the + code under the warns() context manager). Set this to False if this is + ambiguous or if the context manager does not test the direct user code + that emits the warning. + + If the warning is a ``SymPyDeprecationWarning``, this additionally tests + that the ``active_deprecations_target`` is a real target in the + ``active-deprecations.md`` file. + + ''' + # Absorbs all warnings in warnrec + with warnings.catch_warnings(record=True) as warnrec: + # Any warning other than the one we are looking for is an error + warnings.simplefilter("error") + warnings.filterwarnings("always", category=warningcls) + # Now run the test + yield warnrec + + # Raise if expected warning not found + if not any(issubclass(w.category, warningcls) for w in warnrec): + msg = ('Failed: DID NOT WARN.' + ' No warnings of type %s was emitted.' + ' The list of emitted warnings is: %s.' + ) % (warningcls, [w.message for w in warnrec]) + raise Failed(msg) + + # We don't include the match in the filter above because it would then + # fall to the error filter, so we instead manually check that it matches + # here + for w in warnrec: + # Should always be true due to the filters above + assert issubclass(w.category, warningcls) + if not re.compile(match, re.IGNORECASE).match(str(w.message)): + raise Failed(f"Failed: WRONG MESSAGE. A warning with of the correct category ({warningcls.__name__}) was issued, but it did not match the given match regex ({match!r})") + + if test_stacklevel: + for f in inspect.stack(): + thisfile = f.filename + file = os.path.split(thisfile)[1] + if file.startswith('test_'): + break + elif file == 'doctest.py': + # skip the stacklevel testing in the doctests of this + # function + return + else: + raise RuntimeError("Could not find the file for the given warning to test the stacklevel") + for w in warnrec: + if w.filename != thisfile: + msg = f'''\ +Failed: Warning has the wrong stacklevel. The warning stacklevel needs to be +set so that the line of code shown in the warning message is user code that +calls the deprecated code (the current stacklevel is showing code from +{w.filename} (line {w.lineno}), expected {thisfile})'''.replace('\n', ' ') + raise Failed(msg) + + if warningcls == SymPyDeprecationWarning: + this_file = pathlib.Path(__file__) + active_deprecations_file = (this_file.parent.parent.parent / 'doc' / + 'src' / 'explanation' / + 'active-deprecations.md') + if not active_deprecations_file.exists(): + # We can only test that the active_deprecations_target works if we are + # in the git repo. + return + targets = [] + for w in warnrec: + targets.append(w.message.active_deprecations_target) + text = pathlib.Path(active_deprecations_file).read_text(encoding="utf-8") + for target in targets: + if f'({target})=' not in text: + raise Failed(f"The active deprecations target {target!r} does not appear to be a valid target in the active-deprecations.md file ({active_deprecations_file}).") + +def _both_exp_pow(func): + """ + Decorator used to run the test twice: the first time `e^x` is represented + as ``Pow(E, x)``, the second time as ``exp(x)`` (exponential object is not + a power). + + This is a temporary trick helping to manage the elimination of the class + ``exp`` in favor of a replacement by ``Pow(E, ...)``. + """ + from sympy.core.parameters import _exp_is_pow + + def func_wrap(): + with _exp_is_pow(True): + func() + with _exp_is_pow(False): + func() + + wrapper = functools.update_wrapper(func_wrap, func) + return wrapper + + +@contextlib.contextmanager +def warns_deprecated_sympy(): + ''' + Shorthand for ``warns(SymPyDeprecationWarning)`` + + This is the recommended way to test that ``SymPyDeprecationWarning`` is + emitted for deprecated features in SymPy. To test for other warnings use + ``warns``. To suppress warnings without asserting that they are emitted + use ``ignore_warnings``. + + .. note:: + + ``warns_deprecated_sympy()`` is only intended for internal use in the + SymPy test suite to test that a deprecation warning triggers properly. + All other code in the SymPy codebase, including documentation examples, + should not use deprecated behavior. + + If you are a user of SymPy and you want to disable + SymPyDeprecationWarnings, use ``warnings`` filters (see + :ref:`silencing-sympy-deprecation-warnings`). + + >>> from sympy.testing.pytest import warns_deprecated_sympy + >>> from sympy.utilities.exceptions import sympy_deprecation_warning + >>> with warns_deprecated_sympy(): + ... sympy_deprecation_warning("Don't use", + ... deprecated_since_version="1.0", + ... active_deprecations_target="active-deprecations") + + >>> with warns_deprecated_sympy(): + ... pass + Traceback (most recent call last): + ... + Failed: DID NOT WARN. No warnings of type \ + SymPyDeprecationWarning was emitted. The list of emitted warnings is: []. + + .. note:: + + Sometimes the stacklevel test will fail because the same warning is + emitted multiple times. In this case, you can use + :func:`sympy.utilities.exceptions.ignore_warnings` in the code to + prevent the ``SymPyDeprecationWarning`` from being emitted again + recursively. In rare cases it is impossible to have a consistent + ``stacklevel`` for deprecation warnings because different ways of + calling a function will produce different call stacks.. In those cases, + use ``warns(SymPyDeprecationWarning)`` instead. + + See Also + ======== + sympy.utilities.exceptions.SymPyDeprecationWarning + sympy.utilities.exceptions.sympy_deprecation_warning + sympy.utilities.decorator.deprecated + + ''' + with warns(SymPyDeprecationWarning): + yield + + +def skip_under_pyodide(message): + """Decorator to skip a test if running under Pyodide/WASM.""" + def decorator(test_func): + @functools.wraps(test_func) + def test_wrapper(): + if IS_WASM: + skip(message) + return test_func() + return test_wrapper + return decorator diff --git a/.venv/lib/python3.12/site-packages/sympy/testing/quality_unicode.py b/.venv/lib/python3.12/site-packages/sympy/testing/quality_unicode.py new file mode 100644 index 0000000000000000000000000000000000000000..d43623ff5112610e377347f50c6a40a15810644b --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/testing/quality_unicode.py @@ -0,0 +1,102 @@ +import re +import fnmatch + + +message_unicode_B = \ + "File contains a unicode character : %s, line %s. " \ + "But not in the whitelist. " \ + "Add the file to the whitelist in " + __file__ +message_unicode_D = \ + "File does not contain a unicode character : %s." \ + "but is in the whitelist. " \ + "Remove the file from the whitelist in " + __file__ + + +encoding_header_re = re.compile( + r'^[ \t\f]*#.*?coding[:=][ \t]*([-_.a-zA-Z0-9]+)') + +# Whitelist pattern for files which can have unicode. +unicode_whitelist = [ + # Author names can include non-ASCII characters + r'*/bin/authors_update.py', + r'*/bin/mailmap_check.py', + + # These files have functions and test functions for unicode input and + # output. + r'*/sympy/testing/tests/test_code_quality.py', + r'*/sympy/physics/vector/tests/test_printing.py', + r'*/physics/quantum/tests/test_printing.py', + r'*/sympy/vector/tests/test_printing.py', + r'*/sympy/parsing/tests/test_sympy_parser.py', + r'*/sympy/printing/pretty/stringpict.py', + r'*/sympy/printing/pretty/tests/test_pretty.py', + r'*/sympy/printing/tests/test_conventions.py', + r'*/sympy/printing/tests/test_preview.py', + r'*/liealgebras/type_g.py', + r'*/liealgebras/weyl_group.py', + r'*/liealgebras/tests/test_type_G.py', + + # wigner.py and polarization.py have unicode doctests. These probably + # don't need to be there but some of the examples that are there are + # pretty ugly without use_unicode (matrices need to be wrapped across + # multiple lines etc) + r'*/sympy/physics/wigner.py', + r'*/sympy/physics/optics/polarization.py', + + # joint.py uses some unicode for variable names in the docstrings + r'*/sympy/physics/mechanics/joint.py', + + # lll method has unicode in docstring references and author name + r'*/sympy/polys/matrices/domainmatrix.py', + r'*/sympy/matrices/repmatrix.py', + + # Explanation of symbols uses greek letters + r'*/sympy/core/symbol.py', +] + +unicode_strict_whitelist = [ + r'*/sympy/parsing/latex/_antlr/__init__.py', + # test_mathematica.py uses some unicode for testing Greek characters are working #24055 + r'*/sympy/parsing/tests/test_mathematica.py', +] + + +def _test_this_file_encoding( + fname, test_file, + unicode_whitelist=unicode_whitelist, + unicode_strict_whitelist=unicode_strict_whitelist): + """Test helper function for unicode test + + The test may have to operate on filewise manner, so it had moved + to a separate process. + """ + has_unicode = False + + is_in_whitelist = False + is_in_strict_whitelist = False + for patt in unicode_whitelist: + if fnmatch.fnmatch(fname, patt): + is_in_whitelist = True + break + for patt in unicode_strict_whitelist: + if fnmatch.fnmatch(fname, patt): + is_in_strict_whitelist = True + is_in_whitelist = True + break + + if is_in_whitelist: + for idx, line in enumerate(test_file): + try: + line.encode(encoding='ascii') + except (UnicodeEncodeError, UnicodeDecodeError): + has_unicode = True + + if not has_unicode and not is_in_strict_whitelist: + assert False, message_unicode_D % fname + + else: + for idx, line in enumerate(test_file): + try: + line.encode(encoding='ascii') + except (UnicodeEncodeError, UnicodeDecodeError): + assert False, message_unicode_B % (fname, idx + 1) diff --git a/.venv/lib/python3.12/site-packages/sympy/testing/randtest.py b/.venv/lib/python3.12/site-packages/sympy/testing/randtest.py new file mode 100644 index 0000000000000000000000000000000000000000..3ce2c8c031eec1c886532daba32c96d83e9cf85c --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/testing/randtest.py @@ -0,0 +1,19 @@ +""" +.. deprecated:: 1.10 + + ``sympy.testing.randtest`` functions have been moved to + :mod:`sympy.core.random`. + +""" +from sympy.utilities.exceptions import sympy_deprecation_warning + +sympy_deprecation_warning("The sympy.testing.randtest submodule is deprecated. Use sympy.core.random instead.", + deprecated_since_version="1.10", + active_deprecations_target="deprecated-sympy-testing-randtest") + +from sympy.core.random import ( # noqa:F401 + random_complex_number, + verify_numerically, + test_derivative_numerically, + _randrange, + _randint) diff --git a/.venv/lib/python3.12/site-packages/sympy/testing/runtests.py b/.venv/lib/python3.12/site-packages/sympy/testing/runtests.py new file mode 100644 index 0000000000000000000000000000000000000000..e2650e4e6dabaaa07fc25c76cce3d9d28723b0d5 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/testing/runtests.py @@ -0,0 +1,2409 @@ +""" +This is our testing framework. + +Goals: + +* it should be compatible with py.test and operate very similarly + (or identically) +* does not require any external dependencies +* preferably all the functionality should be in this file only +* no magic, just import the test file and execute the test functions, that's it +* portable + +""" + +import os +import sys +import platform +import inspect +import traceback +import pdb +import re +import linecache +import time +from fnmatch import fnmatch +from timeit import default_timer as clock +import doctest as pdoctest # avoid clashing with our doctest() function +from doctest import DocTestFinder, DocTestRunner +import random +import subprocess +import shutil +import signal +import stat +import tempfile +import warnings +from contextlib import contextmanager +from inspect import unwrap +from pathlib import Path + +from sympy.core.cache import clear_cache +from sympy.external import import_module +from sympy.external.gmpy import GROUND_TYPES + +IS_WINDOWS = (os.name == 'nt') +ON_CI = os.getenv('CI', None) + +# empirically generated list of the proportion of time spent running +# an even split of tests. This should periodically be regenerated. +# A list of [.6, .1, .3] would mean that if the tests are evenly split +# into '1/3', '2/3', '3/3', the first split would take 60% of the time, +# the second 10% and the third 30%. These lists are normalized to sum +# to 1, so [60, 10, 30] has the same behavior as [6, 1, 3] or [.6, .1, .3]. +# +# This list can be generated with the code: +# from time import time +# import sympy +# import os +# os.environ["CI"] = 'true' # Mock CI to get more correct densities +# delays, num_splits = [], 30 +# for i in range(1, num_splits + 1): +# tic = time() +# sympy.test(split='{}/{}'.format(i, num_splits), time_balance=False) # Add slow=True for slow tests +# delays.append(time() - tic) +# tot = sum(delays) +# print([round(x / tot, 4) for x in delays]) +SPLIT_DENSITY = [ + 0.0059, 0.0027, 0.0068, 0.0011, 0.0006, + 0.0058, 0.0047, 0.0046, 0.004, 0.0257, + 0.0017, 0.0026, 0.004, 0.0032, 0.0016, + 0.0015, 0.0004, 0.0011, 0.0016, 0.0014, + 0.0077, 0.0137, 0.0217, 0.0074, 0.0043, + 0.0067, 0.0236, 0.0004, 0.1189, 0.0142, + 0.0234, 0.0003, 0.0003, 0.0047, 0.0006, + 0.0013, 0.0004, 0.0008, 0.0007, 0.0006, + 0.0139, 0.0013, 0.0007, 0.0051, 0.002, + 0.0004, 0.0005, 0.0213, 0.0048, 0.0016, + 0.0012, 0.0014, 0.0024, 0.0015, 0.0004, + 0.0005, 0.0007, 0.011, 0.0062, 0.0015, + 0.0021, 0.0049, 0.0006, 0.0006, 0.0011, + 0.0006, 0.0019, 0.003, 0.0044, 0.0054, + 0.0057, 0.0049, 0.0016, 0.0006, 0.0009, + 0.0006, 0.0012, 0.0006, 0.0149, 0.0532, + 0.0076, 0.0041, 0.0024, 0.0135, 0.0081, + 0.2209, 0.0459, 0.0438, 0.0488, 0.0137, + 0.002, 0.0003, 0.0008, 0.0039, 0.0024, + 0.0005, 0.0004, 0.003, 0.056, 0.0026] +SPLIT_DENSITY_SLOW = [0.0086, 0.0004, 0.0568, 0.0003, 0.0032, 0.0005, 0.0004, 0.0013, 0.0016, 0.0648, 0.0198, 0.1285, 0.098, 0.0005, 0.0064, 0.0003, 0.0004, 0.0026, 0.0007, 0.0051, 0.0089, 0.0024, 0.0033, 0.0057, 0.0005, 0.0003, 0.001, 0.0045, 0.0091, 0.0006, 0.0005, 0.0321, 0.0059, 0.1105, 0.216, 0.1489, 0.0004, 0.0003, 0.0006, 0.0483] + +class Skipped(Exception): + pass + +class TimeOutError(Exception): + pass + +class DependencyError(Exception): + pass + + +def _indent(s, indent=4): + """ + Add the given number of space characters to the beginning of + every non-blank line in ``s``, and return the result. + If the string ``s`` is Unicode, it is encoded using the stdout + encoding and the ``backslashreplace`` error handler. + """ + # This regexp matches the start of non-blank lines: + return re.sub('(?m)^(?!$)', indent*' ', s) + + +pdoctest._indent = _indent # type: ignore + +# override reporter to maintain windows and python3 + + +def _report_failure(self, out, test, example, got): + """ + Report that the given example failed. + """ + s = self._checker.output_difference(example, got, self.optionflags) + s = s.encode('raw_unicode_escape').decode('utf8', 'ignore') + out(self._failure_header(test, example) + s) + + +if IS_WINDOWS: + DocTestRunner.report_failure = _report_failure # type: ignore + + +def convert_to_native_paths(lst): + """ + Converts a list of '/' separated paths into a list of + native (os.sep separated) paths and converts to lowercase + if the system is case insensitive. + """ + newlst = [] + for rv in lst: + rv = os.path.join(*rv.split("/")) + # on windows the slash after the colon is dropped + if sys.platform == "win32": + pos = rv.find(':') + if pos != -1: + if rv[pos + 1] != '\\': + rv = rv[:pos + 1] + '\\' + rv[pos + 1:] + newlst.append(os.path.normcase(rv)) + return newlst + + +def get_sympy_dir(): + """ + Returns the root SymPy directory and set the global value + indicating whether the system is case sensitive or not. + """ + this_file = os.path.abspath(__file__) + sympy_dir = os.path.join(os.path.dirname(this_file), "..", "..") + sympy_dir = os.path.normpath(sympy_dir) + return os.path.normcase(sympy_dir) + + +def setup_pprint(disable_line_wrap=True): + from sympy.interactive.printing import init_printing + from sympy.printing.pretty.pretty import pprint_use_unicode + import sympy.interactive.printing as interactive_printing + from sympy.printing.pretty import stringpict + + # Prevent init_printing() in doctests from affecting other doctests + interactive_printing.NO_GLOBAL = True + + # force pprint to be in ascii mode in doctests + use_unicode_prev = pprint_use_unicode(False) + + # disable line wrapping for pprint() outputs + wrap_line_prev = stringpict._GLOBAL_WRAP_LINE + if disable_line_wrap: + stringpict._GLOBAL_WRAP_LINE = False + + # hook our nice, hash-stable strprinter + init_printing(pretty_print=False) + + return use_unicode_prev, wrap_line_prev + + +@contextmanager +def raise_on_deprecated(): + """Context manager to make DeprecationWarning raise an error + + This is to catch SymPyDeprecationWarning from library code while running + tests and doctests. It is important to use this context manager around + each individual test/doctest in case some tests modify the warning + filters. + """ + with warnings.catch_warnings(): + warnings.filterwarnings('error', '.*', DeprecationWarning, module='sympy.*') + yield + + +def run_in_subprocess_with_hash_randomization( + function, function_args=(), + function_kwargs=None, command=sys.executable, + module='sympy.testing.runtests', force=False): + """ + Run a function in a Python subprocess with hash randomization enabled. + + If hash randomization is not supported by the version of Python given, it + returns False. Otherwise, it returns the exit value of the command. The + function is passed to sys.exit(), so the return value of the function will + be the return value. + + The environment variable PYTHONHASHSEED is used to seed Python's hash + randomization. If it is set, this function will return False, because + starting a new subprocess is unnecessary in that case. If it is not set, + one is set at random, and the tests are run. Note that if this + environment variable is set when Python starts, hash randomization is + automatically enabled. To force a subprocess to be created even if + PYTHONHASHSEED is set, pass ``force=True``. This flag will not force a + subprocess in Python versions that do not support hash randomization (see + below), because those versions of Python do not support the ``-R`` flag. + + ``function`` should be a string name of a function that is importable from + the module ``module``, like "_test". The default for ``module`` is + "sympy.testing.runtests". ``function_args`` and ``function_kwargs`` + should be a repr-able tuple and dict, respectively. The default Python + command is sys.executable, which is the currently running Python command. + + This function is necessary because the seed for hash randomization must be + set by the environment variable before Python starts. Hence, in order to + use a predetermined seed for tests, we must start Python in a separate + subprocess. + + Hash randomization was added in the minor Python versions 2.6.8, 2.7.3, + 3.1.5, and 3.2.3, and is enabled by default in all Python versions after + and including 3.3.0. + + Examples + ======== + + >>> from sympy.testing.runtests import ( + ... run_in_subprocess_with_hash_randomization) + >>> # run the core tests in verbose mode + >>> run_in_subprocess_with_hash_randomization("_test", + ... function_args=("core",), + ... function_kwargs={'verbose': True}) # doctest: +SKIP + # Will return 0 if sys.executable supports hash randomization and tests + # pass, 1 if they fail, and False if it does not support hash + # randomization. + + """ + cwd = get_sympy_dir() + # Note, we must return False everywhere, not None, as subprocess.call will + # sometimes return None. + + # First check if the Python version supports hash randomization + # If it does not have this support, it won't recognize the -R flag + p = subprocess.Popen([command, "-RV"], stdout=subprocess.PIPE, + stderr=subprocess.STDOUT, cwd=cwd) + p.communicate() + if p.returncode != 0: + return False + + hash_seed = os.getenv("PYTHONHASHSEED") + if not hash_seed: + os.environ["PYTHONHASHSEED"] = str(random.randrange(2**32)) + else: + if not force: + return False + + function_kwargs = function_kwargs or {} + + # Now run the command + commandstring = ("import sys; from %s import %s;sys.exit(%s(*%s, **%s))" % + (module, function, function, repr(function_args), + repr(function_kwargs))) + + try: + p = subprocess.Popen([command, "-R", "-c", commandstring], cwd=cwd) + p.communicate() + except KeyboardInterrupt: + p.wait() + finally: + # Put the environment variable back, so that it reads correctly for + # the current Python process. + if hash_seed is None: + del os.environ["PYTHONHASHSEED"] + else: + os.environ["PYTHONHASHSEED"] = hash_seed + return p.returncode + + +def run_all_tests(test_args=(), test_kwargs=None, + doctest_args=(), doctest_kwargs=None, + examples_args=(), examples_kwargs=None): + """ + Run all tests. + + Right now, this runs the regular tests (bin/test), the doctests + (bin/doctest), and the examples (examples/all.py). + + This is what ``setup.py test`` uses. + + You can pass arguments and keyword arguments to the test functions that + support them (for now, test, doctest, and the examples). See the + docstrings of those functions for a description of the available options. + + For example, to run the solvers tests with colors turned off: + + >>> from sympy.testing.runtests import run_all_tests + >>> run_all_tests(test_args=("solvers",), + ... test_kwargs={"colors:False"}) # doctest: +SKIP + + """ + tests_successful = True + + test_kwargs = test_kwargs or {} + doctest_kwargs = doctest_kwargs or {} + examples_kwargs = examples_kwargs or {'quiet': True} + + try: + # Regular tests + if not test(*test_args, **test_kwargs): + # some regular test fails, so set the tests_successful + # flag to false and continue running the doctests + tests_successful = False + + # Doctests + print() + if not doctest(*doctest_args, **doctest_kwargs): + tests_successful = False + + # Examples + print() + sys.path.append("examples") # examples/all.py + from all import run_examples # type: ignore + if not run_examples(*examples_args, **examples_kwargs): + tests_successful = False + + if tests_successful: + return + else: + # Return nonzero exit code + sys.exit(1) + except KeyboardInterrupt: + print() + print("DO *NOT* COMMIT!") + sys.exit(1) + + +def test(*paths, subprocess=True, rerun=0, **kwargs): + """ + Run tests in the specified test_*.py files. + + Tests in a particular test_*.py file are run if any of the given strings + in ``paths`` matches a part of the test file's path. If ``paths=[]``, + tests in all test_*.py files are run. + + Notes: + + - If sort=False, tests are run in random order (not default). + - Paths can be entered in native system format or in unix, + forward-slash format. + - Files that are on the blacklist can be tested by providing + their path; they are only excluded if no paths are given. + + **Explanation of test results** + + ====== =============================================================== + Output Meaning + ====== =============================================================== + . passed + F failed + X XPassed (expected to fail but passed) + f XFAILed (expected to fail and indeed failed) + s skipped + w slow + T timeout (e.g., when ``--timeout`` is used) + K KeyboardInterrupt (when running the slow tests with ``--slow``, + you can interrupt one of them without killing the test runner) + ====== =============================================================== + + + Colors have no additional meaning and are used just to facilitate + interpreting the output. + + Examples + ======== + + >>> import sympy + + Run all tests: + + >>> sympy.test() # doctest: +SKIP + + Run one file: + + >>> sympy.test("sympy/core/tests/test_basic.py") # doctest: +SKIP + >>> sympy.test("_basic") # doctest: +SKIP + + Run all tests in sympy/functions/ and some particular file: + + >>> sympy.test("sympy/core/tests/test_basic.py", + ... "sympy/functions") # doctest: +SKIP + + Run all tests in sympy/core and sympy/utilities: + + >>> sympy.test("/core", "/util") # doctest: +SKIP + + Run specific test from a file: + + >>> sympy.test("sympy/core/tests/test_basic.py", + ... kw="test_equality") # doctest: +SKIP + + Run specific test from any file: + + >>> sympy.test(kw="subs") # doctest: +SKIP + + Run the tests with verbose mode on: + + >>> sympy.test(verbose=True) # doctest: +SKIP + + Do not sort the test output: + + >>> sympy.test(sort=False) # doctest: +SKIP + + Turn on post-mortem pdb: + + >>> sympy.test(pdb=True) # doctest: +SKIP + + Turn off colors: + + >>> sympy.test(colors=False) # doctest: +SKIP + + Force colors, even when the output is not to a terminal (this is useful, + e.g., if you are piping to ``less -r`` and you still want colors) + + >>> sympy.test(force_colors=False) # doctest: +SKIP + + The traceback verboseness can be set to "short" or "no" (default is + "short") + + >>> sympy.test(tb='no') # doctest: +SKIP + + The ``split`` option can be passed to split the test run into parts. The + split currently only splits the test files, though this may change in the + future. ``split`` should be a string of the form 'a/b', which will run + part ``a`` of ``b``. For instance, to run the first half of the test suite: + + >>> sympy.test(split='1/2') # doctest: +SKIP + + The ``time_balance`` option can be passed in conjunction with ``split``. + If ``time_balance=True`` (the default for ``sympy.test``), SymPy will attempt + to split the tests such that each split takes equal time. This heuristic + for balancing is based on pre-recorded test data. + + >>> sympy.test(split='1/2', time_balance=True) # doctest: +SKIP + + You can disable running the tests in a separate subprocess using + ``subprocess=False``. This is done to support seeding hash randomization, + which is enabled by default in the Python versions where it is supported. + If subprocess=False, hash randomization is enabled/disabled according to + whether it has been enabled or not in the calling Python process. + However, even if it is enabled, the seed cannot be printed unless it is + called from a new Python process. + + Hash randomization was added in the minor Python versions 2.6.8, 2.7.3, + 3.1.5, and 3.2.3, and is enabled by default in all Python versions after + and including 3.3.0. + + If hash randomization is not supported ``subprocess=False`` is used + automatically. + + >>> sympy.test(subprocess=False) # doctest: +SKIP + + To set the hash randomization seed, set the environment variable + ``PYTHONHASHSEED`` before running the tests. This can be done from within + Python using + + >>> import os + >>> os.environ['PYTHONHASHSEED'] = '42' # doctest: +SKIP + + Or from the command line using + + $ PYTHONHASHSEED=42 ./bin/test + + If the seed is not set, a random seed will be chosen. + + Note that to reproduce the same hash values, you must use both the same seed + as well as the same architecture (32-bit vs. 64-bit). + + """ + # count up from 0, do not print 0 + print_counter = lambda i : (print("rerun %d" % (rerun-i)) + if rerun-i else None) + + if subprocess: + # loop backwards so last i is 0 + for i in range(rerun, -1, -1): + print_counter(i) + ret = run_in_subprocess_with_hash_randomization("_test", + function_args=paths, function_kwargs=kwargs) + if ret is False: + break + val = not bool(ret) + # exit on the first failure or if done + if not val or i == 0: + return val + + # rerun even if hash randomization is not supported + for i in range(rerun, -1, -1): + print_counter(i) + val = not bool(_test(*paths, **kwargs)) + if not val or i == 0: + return val + + +def _test(*paths, + verbose=False, tb="short", kw=None, pdb=False, colors=True, + force_colors=False, sort=True, seed=None, timeout=False, + fail_on_timeout=False, slow=False, enhance_asserts=False, split=None, + time_balance=True, blacklist=(), + fast_threshold=None, slow_threshold=None): + """ + Internal function that actually runs the tests. + + All keyword arguments from ``test()`` are passed to this function except for + ``subprocess``. + + Returns 0 if tests passed and 1 if they failed. See the docstring of + ``test()`` for more information. + """ + kw = kw or () + # ensure that kw is a tuple + if isinstance(kw, str): + kw = (kw,) + post_mortem = pdb + if seed is None: + seed = random.randrange(100000000) + if ON_CI and timeout is False: + timeout = 595 + fail_on_timeout = True + if ON_CI: + blacklist = list(blacklist) + ['sympy/plotting/pygletplot/tests'] + blacklist = convert_to_native_paths(blacklist) + r = PyTestReporter(verbose=verbose, tb=tb, colors=colors, + force_colors=force_colors, split=split) + # This won't strictly run the test for the corresponding file, but it is + # good enough for copying and pasting the failing test. + _paths = [] + for path in paths: + if '::' in path: + path, _kw = path.split('::', 1) + kw += (_kw,) + _paths.append(path) + paths = _paths + + t = SymPyTests(r, kw, post_mortem, seed, + fast_threshold=fast_threshold, + slow_threshold=slow_threshold) + + test_files = t.get_test_files('sympy') + + not_blacklisted = [f for f in test_files + if not any(b in f for b in blacklist)] + + if len(paths) == 0: + matched = not_blacklisted + else: + paths = convert_to_native_paths(paths) + matched = [] + for f in not_blacklisted: + basename = os.path.basename(f) + for p in paths: + if p in f or fnmatch(basename, p): + matched.append(f) + break + + density = None + if time_balance: + if slow: + density = SPLIT_DENSITY_SLOW + else: + density = SPLIT_DENSITY + + if split: + matched = split_list(matched, split, density=density) + + t._testfiles.extend(matched) + + return int(not t.test(sort=sort, timeout=timeout, slow=slow, + enhance_asserts=enhance_asserts, fail_on_timeout=fail_on_timeout)) + + +def doctest(*paths, subprocess=True, rerun=0, **kwargs): + r""" + Runs doctests in all \*.py files in the SymPy directory which match + any of the given strings in ``paths`` or all tests if paths=[]. + + Notes: + + - Paths can be entered in native system format or in unix, + forward-slash format. + - Files that are on the blacklist can be tested by providing + their path; they are only excluded if no paths are given. + + Examples + ======== + + >>> import sympy + + Run all tests: + + >>> sympy.doctest() # doctest: +SKIP + + Run one file: + + >>> sympy.doctest("sympy/core/basic.py") # doctest: +SKIP + >>> sympy.doctest("polynomial.rst") # doctest: +SKIP + + Run all tests in sympy/functions/ and some particular file: + + >>> sympy.doctest("/functions", "basic.py") # doctest: +SKIP + + Run any file having polynomial in its name, doc/src/modules/polynomial.rst, + sympy/functions/special/polynomials.py, and sympy/polys/polynomial.py: + + >>> sympy.doctest("polynomial") # doctest: +SKIP + + The ``split`` option can be passed to split the test run into parts. The + split currently only splits the test files, though this may change in the + future. ``split`` should be a string of the form 'a/b', which will run + part ``a`` of ``b``. Note that the regular doctests and the Sphinx + doctests are split independently. For instance, to run the first half of + the test suite: + + >>> sympy.doctest(split='1/2') # doctest: +SKIP + + The ``subprocess`` and ``verbose`` options are the same as with the function + ``test()`` (see the docstring of that function for more information) except + that ``verbose`` may also be set equal to ``2`` in order to print + individual doctest lines, as they are being tested. + """ + # count up from 0, do not print 0 + print_counter = lambda i : (print("rerun %d" % (rerun-i)) + if rerun-i else None) + + if subprocess: + # loop backwards so last i is 0 + for i in range(rerun, -1, -1): + print_counter(i) + ret = run_in_subprocess_with_hash_randomization("_doctest", + function_args=paths, function_kwargs=kwargs) + if ret is False: + break + val = not bool(ret) + # exit on the first failure or if done + if not val or i == 0: + return val + + # rerun even if hash randomization is not supported + for i in range(rerun, -1, -1): + print_counter(i) + val = not bool(_doctest(*paths, **kwargs)) + if not val or i == 0: + return val + + +def _get_doctest_blacklist(): + '''Get the default blacklist for the doctests''' + blacklist = [] + + blacklist.extend([ + "doc/src/modules/plotting.rst", # generates live plots + "doc/src/modules/physics/mechanics/autolev_parser.rst", + "sympy/codegen/array_utils.py", # raises deprecation warning + "sympy/core/compatibility.py", # backwards compatibility shim, importing it triggers a deprecation warning + "sympy/core/trace.py", # backwards compatibility shim, importing it triggers a deprecation warning + "sympy/galgebra.py", # no longer part of SymPy + "sympy/parsing/autolev/_antlr/autolevlexer.py", # generated code + "sympy/parsing/autolev/_antlr/autolevlistener.py", # generated code + "sympy/parsing/autolev/_antlr/autolevparser.py", # generated code + "sympy/parsing/latex/_antlr/latexlexer.py", # generated code + "sympy/parsing/latex/_antlr/latexparser.py", # generated code + "sympy/plotting/pygletplot/__init__.py", # crashes on some systems + "sympy/plotting/pygletplot/plot.py", # crashes on some systems + "sympy/printing/ccode.py", # backwards compatibility shim, importing it breaks the codegen doctests + "sympy/printing/cxxcode.py", # backwards compatibility shim, importing it breaks the codegen doctests + "sympy/printing/fcode.py", # backwards compatibility shim, importing it breaks the codegen doctests + "sympy/testing/randtest.py", # backwards compatibility shim, importing it triggers a deprecation warning + "sympy/this.py", # prints text + ]) + # autolev parser tests + num = 12 + for i in range (1, num+1): + blacklist.append("sympy/parsing/autolev/test-examples/ruletest" + str(i) + ".py") + blacklist.extend(["sympy/parsing/autolev/test-examples/pydy-example-repo/mass_spring_damper.py", + "sympy/parsing/autolev/test-examples/pydy-example-repo/chaos_pendulum.py", + "sympy/parsing/autolev/test-examples/pydy-example-repo/double_pendulum.py", + "sympy/parsing/autolev/test-examples/pydy-example-repo/non_min_pendulum.py"]) + + if import_module('numpy') is None: + blacklist.extend([ + "sympy/plotting/experimental_lambdify.py", + "sympy/plotting/plot_implicit.py", + "examples/advanced/autowrap_integrators.py", + "examples/advanced/autowrap_ufuncify.py", + "examples/intermediate/sample.py", + "examples/intermediate/mplot2d.py", + "examples/intermediate/mplot3d.py", + "doc/src/modules/numeric-computation.rst", + "doc/src/explanation/best-practices.md", + "doc/src/tutorials/physics/biomechanics/biomechanical-model-example.rst", + "doc/src/tutorials/physics/biomechanics/biomechanics.rst", + ]) + else: + if import_module('matplotlib') is None: + blacklist.extend([ + "examples/intermediate/mplot2d.py", + "examples/intermediate/mplot3d.py" + ]) + else: + # Use a non-windowed backend, so that the tests work on CI + import matplotlib + matplotlib.use('Agg') + + if ON_CI or import_module('pyglet') is None: + blacklist.extend(["sympy/plotting/pygletplot"]) + + if import_module('aesara') is None: + blacklist.extend([ + "sympy/printing/aesaracode.py", + "doc/src/modules/numeric-computation.rst", + ]) + + if import_module('cupy') is None: + blacklist.extend([ + "doc/src/modules/numeric-computation.rst", + ]) + + if import_module('jax') is None: + blacklist.extend([ + "doc/src/modules/numeric-computation.rst", + ]) + + if import_module('antlr4') is None: + blacklist.extend([ + "sympy/parsing/autolev/__init__.py", + "sympy/parsing/latex/_parse_latex_antlr.py", + ]) + + if import_module('lfortran') is None: + #throws ImportError when lfortran not installed + blacklist.extend([ + "sympy/parsing/sym_expr.py", + ]) + + if import_module("scipy") is None: + # throws ModuleNotFoundError when scipy not installed + blacklist.extend([ + "doc/src/guides/solving/solve-numerically.md", + "doc/src/guides/solving/solve-ode.md", + ]) + + if import_module("numpy") is None: + # throws ModuleNotFoundError when numpy not installed + blacklist.extend([ + "doc/src/guides/solving/solve-ode.md", + "doc/src/guides/solving/solve-numerically.md", + ]) + + # disabled because of doctest failures in asmeurer's bot + blacklist.extend([ + "sympy/utilities/autowrap.py", + "examples/advanced/autowrap_integrators.py", + "examples/advanced/autowrap_ufuncify.py" + ]) + + blacklist.extend([ + "sympy/conftest.py", # Depends on pytest + ]) + + # These are deprecated stubs to be removed: + blacklist.extend([ + "sympy/utilities/tmpfiles.py", + "sympy/utilities/pytest.py", + "sympy/utilities/runtests.py", + "sympy/utilities/quality_unicode.py", + "sympy/utilities/randtest.py", + ]) + + blacklist = convert_to_native_paths(blacklist) + return blacklist + + +def _doctest(*paths, **kwargs): + """ + Internal function that actually runs the doctests. + + All keyword arguments from ``doctest()`` are passed to this function + except for ``subprocess``. + + Returns 0 if tests passed and 1 if they failed. See the docstrings of + ``doctest()`` and ``test()`` for more information. + """ + from sympy.printing.pretty.pretty import pprint_use_unicode + from sympy.printing.pretty import stringpict + + normal = kwargs.get("normal", False) + verbose = kwargs.get("verbose", False) + colors = kwargs.get("colors", True) + force_colors = kwargs.get("force_colors", False) + blacklist = kwargs.get("blacklist", []) + split = kwargs.get('split', None) + + blacklist.extend(_get_doctest_blacklist()) + + # Use a non-windowed backend, so that the tests work on CI + if import_module('matplotlib') is not None: + import matplotlib + matplotlib.use('Agg') + + # Disable warnings for external modules + import sympy.external + sympy.external.importtools.WARN_OLD_VERSION = False + sympy.external.importtools.WARN_NOT_INSTALLED = False + + # Disable showing up of plots + from sympy.plotting.plot import unset_show + unset_show() + + r = PyTestReporter(verbose, split=split, colors=colors,\ + force_colors=force_colors) + t = SymPyDocTests(r, normal) + + test_files = t.get_test_files('sympy') + test_files.extend(t.get_test_files('examples', init_only=False)) + + not_blacklisted = [f for f in test_files + if not any(b in f for b in blacklist)] + if len(paths) == 0: + matched = not_blacklisted + else: + # take only what was requested...but not blacklisted items + # and allow for partial match anywhere or fnmatch of name + paths = convert_to_native_paths(paths) + matched = [] + for f in not_blacklisted: + basename = os.path.basename(f) + for p in paths: + if p in f or fnmatch(basename, p): + matched.append(f) + break + + matched.sort() + + if split: + matched = split_list(matched, split) + + t._testfiles.extend(matched) + + # run the tests and record the result for this *py portion of the tests + if t._testfiles: + failed = not t.test() + else: + failed = False + + # N.B. + # -------------------------------------------------------------------- + # Here we test *.rst and *.md files at or below doc/src. Code from these + # must be self supporting in terms of imports since there is no importing + # of necessary modules by doctest.testfile. If you try to pass *.py files + # through this they might fail because they will lack the needed imports + # and smarter parsing that can be done with source code. + # + test_files_rst = t.get_test_files('doc/src', '*.rst', init_only=False) + test_files_md = t.get_test_files('doc/src', '*.md', init_only=False) + test_files = test_files_rst + test_files_md + test_files.sort() + + not_blacklisted = [f for f in test_files + if not any(b in f for b in blacklist)] + + if len(paths) == 0: + matched = not_blacklisted + else: + # Take only what was requested as long as it's not on the blacklist. + # Paths were already made native in *py tests so don't repeat here. + # There's no chance of having a *py file slip through since we + # only have *rst files in test_files. + matched = [] + for f in not_blacklisted: + basename = os.path.basename(f) + for p in paths: + if p in f or fnmatch(basename, p): + matched.append(f) + break + + if split: + matched = split_list(matched, split) + + first_report = True + for rst_file in matched: + if not os.path.isfile(rst_file): + continue + old_displayhook = sys.displayhook + try: + use_unicode_prev, wrap_line_prev = setup_pprint() + out = sympytestfile( + rst_file, module_relative=False, encoding='utf-8', + optionflags=pdoctest.ELLIPSIS | pdoctest.NORMALIZE_WHITESPACE | + pdoctest.IGNORE_EXCEPTION_DETAIL) + finally: + # make sure we return to the original displayhook in case some + # doctest has changed that + sys.displayhook = old_displayhook + # The NO_GLOBAL flag overrides the no_global flag to init_printing + # if True + import sympy.interactive.printing as interactive_printing + interactive_printing.NO_GLOBAL = False + pprint_use_unicode(use_unicode_prev) + stringpict._GLOBAL_WRAP_LINE = wrap_line_prev + + rstfailed, tested = out + if tested: + failed = rstfailed or failed + if first_report: + first_report = False + msg = 'rst/md doctests start' + if not t._testfiles: + r.start(msg=msg) + else: + r.write_center(msg) + print() + # use as the id, everything past the first 'sympy' + file_id = rst_file[rst_file.find('sympy') + len('sympy') + 1:] + print(file_id, end=" ") + # get at least the name out so it is know who is being tested + wid = r.terminal_width - len(file_id) - 1 # update width + test_file = '[%s]' % (tested) + report = '[%s]' % (rstfailed or 'OK') + print(''.join( + [test_file, ' '*(wid - len(test_file) - len(report)), report]) + ) + + # the doctests for *py will have printed this message already if there was + # a failure, so now only print it if there was intervening reporting by + # testing the *rst as evidenced by first_report no longer being True. + if not first_report and failed: + print() + print("DO *NOT* COMMIT!") + + return int(failed) + +sp = re.compile(r'([0-9]+)/([1-9][0-9]*)') + +def split_list(l, split, density=None): + """ + Splits a list into part a of b + + split should be a string of the form 'a/b'. For instance, '1/3' would give + the split one of three. + + If the length of the list is not divisible by the number of splits, the + last split will have more items. + + `density` may be specified as a list. If specified, + tests will be balanced so that each split has as equal-as-possible + amount of mass according to `density`. + + >>> from sympy.testing.runtests import split_list + >>> a = list(range(10)) + >>> split_list(a, '1/3') + [0, 1, 2] + >>> split_list(a, '2/3') + [3, 4, 5] + >>> split_list(a, '3/3') + [6, 7, 8, 9] + """ + m = sp.match(split) + if not m: + raise ValueError("split must be a string of the form a/b where a and b are ints") + i, t = map(int, m.groups()) + + if not density: + return l[(i - 1)*len(l)//t : i*len(l)//t] + + # normalize density + tot = sum(density) + density = [x / tot for x in density] + + def density_inv(x): + """Interpolate the inverse to the cumulative + distribution function given by density""" + if x <= 0: + return 0 + if x >= sum(density): + return 1 + + # find the first time the cumulative sum surpasses x + # and linearly interpolate + cumm = 0 + for i, d in enumerate(density): + cumm += d + if cumm >= x: + break + frac = (d - (cumm - x)) / d + return (i + frac) / len(density) + + lower_frac = density_inv((i - 1) / t) + higher_frac = density_inv(i / t) + return l[int(lower_frac*len(l)) : int(higher_frac*len(l))] + +from collections import namedtuple +SymPyTestResults = namedtuple('SymPyTestResults', 'failed attempted') + +def sympytestfile(filename, module_relative=True, name=None, package=None, + globs=None, verbose=None, report=True, optionflags=0, + extraglobs=None, raise_on_error=False, + parser=pdoctest.DocTestParser(), encoding=None): + + """ + Test examples in the given file. Return (#failures, #tests). + + Optional keyword arg ``module_relative`` specifies how filenames + should be interpreted: + + - If ``module_relative`` is True (the default), then ``filename`` + specifies a module-relative path. By default, this path is + relative to the calling module's directory; but if the + ``package`` argument is specified, then it is relative to that + package. To ensure os-independence, ``filename`` should use + "/" characters to separate path segments, and should not + be an absolute path (i.e., it may not begin with "/"). + + - If ``module_relative`` is False, then ``filename`` specifies an + os-specific path. The path may be absolute or relative (to + the current working directory). + + Optional keyword arg ``name`` gives the name of the test; by default + use the file's basename. + + Optional keyword argument ``package`` is a Python package or the + name of a Python package whose directory should be used as the + base directory for a module relative filename. If no package is + specified, then the calling module's directory is used as the base + directory for module relative filenames. It is an error to + specify ``package`` if ``module_relative`` is False. + + Optional keyword arg ``globs`` gives a dict to be used as the globals + when executing examples; by default, use {}. A copy of this dict + is actually used for each docstring, so that each docstring's + examples start with a clean slate. + + Optional keyword arg ``extraglobs`` gives a dictionary that should be + merged into the globals that are used to execute examples. By + default, no extra globals are used. + + Optional keyword arg ``verbose`` prints lots of stuff if true, prints + only failures if false; by default, it's true iff "-v" is in sys.argv. + + Optional keyword arg ``report`` prints a summary at the end when true, + else prints nothing at the end. In verbose mode, the summary is + detailed, else very brief (in fact, empty if all tests passed). + + Optional keyword arg ``optionflags`` or's together module constants, + and defaults to 0. Possible values (see the docs for details): + + - DONT_ACCEPT_TRUE_FOR_1 + - DONT_ACCEPT_BLANKLINE + - NORMALIZE_WHITESPACE + - ELLIPSIS + - SKIP + - IGNORE_EXCEPTION_DETAIL + - REPORT_UDIFF + - REPORT_CDIFF + - REPORT_NDIFF + - REPORT_ONLY_FIRST_FAILURE + + Optional keyword arg ``raise_on_error`` raises an exception on the + first unexpected exception or failure. This allows failures to be + post-mortem debugged. + + Optional keyword arg ``parser`` specifies a DocTestParser (or + subclass) that should be used to extract tests from the files. + + Optional keyword arg ``encoding`` specifies an encoding that should + be used to convert the file to unicode. + + Advanced tomfoolery: testmod runs methods of a local instance of + class doctest.Tester, then merges the results into (or creates) + global Tester instance doctest.master. Methods of doctest.master + can be called directly too, if you want to do something unusual. + Passing report=0 to testmod is especially useful then, to delay + displaying a summary. Invoke doctest.master.summarize(verbose) + when you're done fiddling. + """ + if package and not module_relative: + raise ValueError("Package may only be specified for module-" + "relative paths.") + + # Relativize the path + text, filename = pdoctest._load_testfile( + filename, package, module_relative, encoding) + + # If no name was given, then use the file's name. + if name is None: + name = os.path.basename(filename) + + # Assemble the globals. + if globs is None: + globs = {} + else: + globs = globs.copy() + if extraglobs is not None: + globs.update(extraglobs) + if '__name__' not in globs: + globs['__name__'] = '__main__' + + if raise_on_error: + runner = pdoctest.DebugRunner(verbose=verbose, optionflags=optionflags) + else: + runner = SymPyDocTestRunner(verbose=verbose, optionflags=optionflags) + runner._checker = SymPyOutputChecker() + + # Read the file, convert it to a test, and run it. + test = parser.get_doctest(text, globs, name, filename, 0) + runner.run(test) + + if report: + runner.summarize() + + if pdoctest.master is None: + pdoctest.master = runner + else: + pdoctest.master.merge(runner) + + return SymPyTestResults(runner.failures, runner.tries) + + +class SymPyTests: + + def __init__(self, reporter, kw="", post_mortem=False, + seed=None, fast_threshold=None, slow_threshold=None): + self._post_mortem = post_mortem + self._kw = kw + self._count = 0 + self._root_dir = get_sympy_dir() + self._reporter = reporter + self._reporter.root_dir(self._root_dir) + self._testfiles = [] + self._seed = seed if seed is not None else random.random() + + # Defaults in seconds, from human / UX design limits + # http://www.nngroup.com/articles/response-times-3-important-limits/ + # + # These defaults are *NOT* set in stone as we are measuring different + # things, so others feel free to come up with a better yardstick :) + if fast_threshold: + self._fast_threshold = float(fast_threshold) + else: + self._fast_threshold = 8 + if slow_threshold: + self._slow_threshold = float(slow_threshold) + else: + self._slow_threshold = 10 + + def test(self, sort=False, timeout=False, slow=False, + enhance_asserts=False, fail_on_timeout=False): + """ + Runs the tests returning True if all tests pass, otherwise False. + + If sort=False run tests in random order. + """ + if sort: + self._testfiles.sort() + elif slow: + pass + else: + random.seed(self._seed) + random.shuffle(self._testfiles) + self._reporter.start(self._seed) + for f in self._testfiles: + try: + self.test_file(f, sort, timeout, slow, + enhance_asserts, fail_on_timeout) + except KeyboardInterrupt: + print(" interrupted by user") + self._reporter.finish() + raise + return self._reporter.finish() + + def _enhance_asserts(self, source): + from ast import (NodeTransformer, Compare, Name, Store, Load, Tuple, + Assign, BinOp, Str, Mod, Assert, parse, fix_missing_locations) + + ops = {"Eq": '==', "NotEq": '!=', "Lt": '<', "LtE": '<=', + "Gt": '>', "GtE": '>=', "Is": 'is', "IsNot": 'is not', + "In": 'in', "NotIn": 'not in'} + + class Transform(NodeTransformer): + def visit_Assert(self, stmt): + if isinstance(stmt.test, Compare): + compare = stmt.test + values = [compare.left] + compare.comparators + names = [ "_%s" % i for i, _ in enumerate(values) ] + names_store = [ Name(n, Store()) for n in names ] + names_load = [ Name(n, Load()) for n in names ] + target = Tuple(names_store, Store()) + value = Tuple(values, Load()) + assign = Assign([target], value) + new_compare = Compare(names_load[0], compare.ops, names_load[1:]) + msg_format = "\n%s " + "\n%s ".join([ ops[op.__class__.__name__] for op in compare.ops ]) + "\n%s" + msg = BinOp(Str(msg_format), Mod(), Tuple(names_load, Load())) + test = Assert(new_compare, msg, lineno=stmt.lineno, col_offset=stmt.col_offset) + return [assign, test] + else: + return stmt + + tree = parse(source) + new_tree = Transform().visit(tree) + return fix_missing_locations(new_tree) + + def test_file(self, filename, sort=True, timeout=False, slow=False, + enhance_asserts=False, fail_on_timeout=False): + reporter = self._reporter + funcs = [] + try: + gl = {'__file__': filename} + try: + open_file = lambda: open(filename, encoding="utf8") + + with open_file() as f: + source = f.read() + if self._kw: + for l in source.splitlines(): + if l.lstrip().startswith('def '): + if any(l.lower().find(k.lower()) != -1 for k in self._kw): + break + else: + return + + if enhance_asserts: + try: + source = self._enhance_asserts(source) + except ImportError: + pass + + code = compile(source, filename, "exec", flags=0, dont_inherit=True) + exec(code, gl) + except (SystemExit, KeyboardInterrupt): + raise + except ImportError: + reporter.import_error(filename, sys.exc_info()) + return + except Exception: + reporter.test_exception(sys.exc_info()) + + clear_cache() + self._count += 1 + random.seed(self._seed) + disabled = gl.get("disabled", False) + if not disabled: + # we need to filter only those functions that begin with 'test_' + # We have to be careful about decorated functions. As long as + # the decorator uses functools.wraps, we can detect it. + funcs = [] + for f in gl: + if (f.startswith("test_") and (inspect.isfunction(gl[f]) + or inspect.ismethod(gl[f]))): + func = gl[f] + # Handle multiple decorators + while hasattr(func, '__wrapped__'): + func = func.__wrapped__ + + if inspect.getsourcefile(func) == filename: + funcs.append(gl[f]) + if slow: + funcs = [f for f in funcs if getattr(f, '_slow', False)] + # Sorting of XFAILed functions isn't fixed yet :-( + funcs.sort(key=lambda x: inspect.getsourcelines(x)[1]) + i = 0 + while i < len(funcs): + if inspect.isgeneratorfunction(funcs[i]): + # some tests can be generators, that return the actual + # test functions. We unpack it below: + f = funcs.pop(i) + for fg in f(): + func = fg[0] + args = fg[1:] + fgw = lambda: func(*args) + funcs.insert(i, fgw) + i += 1 + else: + i += 1 + # drop functions that are not selected with the keyword expression: + funcs = [x for x in funcs if self.matches(x)] + + if not funcs: + return + except Exception: + reporter.entering_filename(filename, len(funcs)) + raise + + reporter.entering_filename(filename, len(funcs)) + if not sort: + random.shuffle(funcs) + + for f in funcs: + start = time.time() + reporter.entering_test(f) + try: + if getattr(f, '_slow', False) and not slow: + raise Skipped("Slow") + with raise_on_deprecated(): + if timeout: + self._timeout(f, timeout, fail_on_timeout) + else: + random.seed(self._seed) + f() + except KeyboardInterrupt: + if getattr(f, '_slow', False): + reporter.test_skip("KeyboardInterrupt") + else: + raise + except Exception: + if timeout: + signal.alarm(0) # Disable the alarm. It could not be handled before. + t, v, tr = sys.exc_info() + if t is AssertionError: + reporter.test_fail((t, v, tr)) + if self._post_mortem: + pdb.post_mortem(tr) + elif t.__name__ == "Skipped": + reporter.test_skip(v) + elif t.__name__ == "XFail": + reporter.test_xfail() + elif t.__name__ == "XPass": + reporter.test_xpass(v) + else: + reporter.test_exception((t, v, tr)) + if self._post_mortem: + pdb.post_mortem(tr) + else: + reporter.test_pass() + taken = time.time() - start + if taken > self._slow_threshold: + filename = os.path.relpath(filename, reporter._root_dir) + reporter.slow_test_functions.append( + (filename + "::" + f.__name__, taken)) + if getattr(f, '_slow', False) and slow: + if taken < self._fast_threshold: + filename = os.path.relpath(filename, reporter._root_dir) + reporter.fast_test_functions.append( + (filename + "::" + f.__name__, taken)) + reporter.leaving_filename() + + def _timeout(self, function, timeout, fail_on_timeout): + def callback(x, y): + signal.alarm(0) + if fail_on_timeout: + raise TimeOutError("Timed out after %d seconds" % timeout) + else: + raise Skipped("Timeout") + signal.signal(signal.SIGALRM, callback) + signal.alarm(timeout) # Set an alarm with a given timeout + function() + signal.alarm(0) # Disable the alarm + + def matches(self, x): + """ + Does the keyword expression self._kw match "x"? Returns True/False. + + Always returns True if self._kw is "". + """ + if not self._kw: + return True + for kw in self._kw: + if x.__name__.lower().find(kw.lower()) != -1: + return True + return False + + def get_test_files(self, dir, pat='test_*.py'): + """ + Returns the list of test_*.py (default) files at or below directory + ``dir`` relative to the SymPy home directory. + """ + dir = os.path.join(self._root_dir, convert_to_native_paths([dir])[0]) + + g = [] + for path, folders, files in os.walk(dir): + g.extend([os.path.join(path, f) for f in files if fnmatch(f, pat)]) + + return sorted([os.path.normcase(gi) for gi in g]) + + +class SymPyDocTests: + + def __init__(self, reporter, normal): + self._count = 0 + self._root_dir = get_sympy_dir() + self._reporter = reporter + self._reporter.root_dir(self._root_dir) + self._normal = normal + + self._testfiles = [] + + def test(self): + """ + Runs the tests and returns True if all tests pass, otherwise False. + """ + self._reporter.start() + for f in self._testfiles: + try: + self.test_file(f) + except KeyboardInterrupt: + print(" interrupted by user") + self._reporter.finish() + raise + return self._reporter.finish() + + def test_file(self, filename): + clear_cache() + + from io import StringIO + import sympy.interactive.printing as interactive_printing + from sympy.printing.pretty.pretty import pprint_use_unicode + from sympy.printing.pretty import stringpict + + rel_name = filename[len(self._root_dir) + 1:] + dirname, file = os.path.split(filename) + module = rel_name.replace(os.sep, '.')[:-3] + + if rel_name.startswith("examples"): + # Examples files do not have __init__.py files, + # So we have to temporarily extend sys.path to import them + sys.path.insert(0, dirname) + module = file[:-3] # remove ".py" + try: + module = pdoctest._normalize_module(module) + tests = SymPyDocTestFinder().find(module) + except (SystemExit, KeyboardInterrupt): + raise + except ImportError: + self._reporter.import_error(filename, sys.exc_info()) + return + finally: + if rel_name.startswith("examples"): + del sys.path[0] + + tests = [test for test in tests if len(test.examples) > 0] + # By default tests are sorted by alphabetical order by function name. + # We sort by line number so one can edit the file sequentially from + # bottom to top. However, if there are decorated functions, their line + # numbers will be too large and for now one must just search for these + # by text and function name. + tests.sort(key=lambda x: -x.lineno) + + if not tests: + return + self._reporter.entering_filename(filename, len(tests)) + for test in tests: + assert len(test.examples) != 0 + + if self._reporter._verbose: + self._reporter.write("\n{} ".format(test.name)) + + # check if there are external dependencies which need to be met + if '_doctest_depends_on' in test.globs: + try: + self._check_dependencies(**test.globs['_doctest_depends_on']) + except DependencyError as e: + self._reporter.test_skip(v=str(e)) + continue + + runner = SymPyDocTestRunner(verbose=self._reporter._verbose==2, + optionflags=pdoctest.ELLIPSIS | + pdoctest.NORMALIZE_WHITESPACE | + pdoctest.IGNORE_EXCEPTION_DETAIL) + runner._checker = SymPyOutputChecker() + old = sys.stdout + new = old if self._reporter._verbose==2 else StringIO() + sys.stdout = new + # If the testing is normal, the doctests get importing magic to + # provide the global namespace. If not normal (the default) then + # then must run on their own; all imports must be explicit within + # a function's docstring. Once imported that import will be + # available to the rest of the tests in a given function's + # docstring (unless clear_globs=True below). + if not self._normal: + test.globs = {} + # if this is uncommented then all the test would get is what + # comes by default with a "from sympy import *" + #exec('from sympy import *') in test.globs + old_displayhook = sys.displayhook + use_unicode_prev, wrap_line_prev = setup_pprint() + + try: + f, t = runner.run(test, + out=new.write, clear_globs=False) + except KeyboardInterrupt: + raise + finally: + sys.stdout = old + if f > 0: + self._reporter.doctest_fail(test.name, new.getvalue()) + else: + self._reporter.test_pass() + sys.displayhook = old_displayhook + interactive_printing.NO_GLOBAL = False + pprint_use_unicode(use_unicode_prev) + stringpict._GLOBAL_WRAP_LINE = wrap_line_prev + + self._reporter.leaving_filename() + + def get_test_files(self, dir, pat='*.py', init_only=True): + r""" + Returns the list of \*.py files (default) from which docstrings + will be tested which are at or below directory ``dir``. By default, + only those that have an __init__.py in their parent directory + and do not start with ``test_`` will be included. + """ + def importable(x): + """ + Checks if given pathname x is an importable module by checking for + __init__.py file. + + Returns True/False. + + Currently we only test if the __init__.py file exists in the + directory with the file "x" (in theory we should also test all the + parent dirs). + """ + init_py = os.path.join(os.path.dirname(x), "__init__.py") + return os.path.exists(init_py) + + dir = os.path.join(self._root_dir, convert_to_native_paths([dir])[0]) + + g = [] + for path, folders, files in os.walk(dir): + g.extend([os.path.join(path, f) for f in files + if not f.startswith('test_') and fnmatch(f, pat)]) + if init_only: + # skip files that are not importable (i.e. missing __init__.py) + g = [x for x in g if importable(x)] + + return [os.path.normcase(gi) for gi in g] + + def _check_dependencies(self, + executables=(), + modules=(), + disable_viewers=(), + python_version=(3, 5), + ground_types=None): + """ + Checks if the dependencies for the test are installed. + + Raises ``DependencyError`` it at least one dependency is not installed. + """ + + for executable in executables: + if not shutil.which(executable): + raise DependencyError("Could not find %s" % executable) + + for module in modules: + if module == 'matplotlib': + matplotlib = import_module( + 'matplotlib', + import_kwargs={'fromlist': + ['pyplot', 'cm', 'collections']}, + min_module_version='1.0.0', catch=(RuntimeError,)) + if matplotlib is None: + raise DependencyError("Could not import matplotlib") + else: + if not import_module(module): + raise DependencyError("Could not import %s" % module) + + if disable_viewers: + tempdir = tempfile.mkdtemp() + os.environ['PATH'] = '%s:%s' % (tempdir, os.environ['PATH']) + + vw = ('#!/usr/bin/env python3\n' + 'import sys\n' + 'if len(sys.argv) <= 1:\n' + ' exit("wrong number of args")\n') + + for viewer in disable_viewers: + Path(os.path.join(tempdir, viewer)).write_text(vw) + + # make the file executable + os.chmod(os.path.join(tempdir, viewer), + stat.S_IREAD | stat.S_IWRITE | stat.S_IXUSR) + + if python_version: + if sys.version_info < python_version: + raise DependencyError("Requires Python >= " + '.'.join(map(str, python_version))) + + if ground_types is not None: + if GROUND_TYPES not in ground_types: + raise DependencyError("Requires ground_types in " + str(ground_types)) + + if 'pyglet' in modules: + # monkey-patch pyglet s.t. it does not open a window during + # doctesting + import pyglet + class DummyWindow: + def __init__(self, *args, **kwargs): + self.has_exit = True + self.width = 600 + self.height = 400 + + def set_vsync(self, x): + pass + + def switch_to(self): + pass + + def push_handlers(self, x): + pass + + def close(self): + pass + + pyglet.window.Window = DummyWindow + + +class SymPyDocTestFinder(DocTestFinder): + """ + A class used to extract the DocTests that are relevant to a given + object, from its docstring and the docstrings of its contained + objects. Doctests can currently be extracted from the following + object types: modules, functions, classes, methods, staticmethods, + classmethods, and properties. + + Modified from doctest's version to look harder for code that + appears comes from a different module. For example, the @vectorize + decorator makes it look like functions come from multidimensional.py + even though their code exists elsewhere. + """ + + def _find(self, tests, obj, name, module, source_lines, globs, seen): + """ + Find tests for the given object and any contained objects, and + add them to ``tests``. + """ + if self._verbose: + print('Finding tests in %s' % name) + + # If we've already processed this object, then ignore it. + if id(obj) in seen: + return + seen[id(obj)] = 1 + + # Make sure we don't run doctests for classes outside of sympy, such + # as in numpy or scipy. + if inspect.isclass(obj): + if obj.__module__.split('.')[0] != 'sympy': + return + + # Find a test for this object, and add it to the list of tests. + test = self._get_test(obj, name, module, globs, source_lines) + if test is not None: + tests.append(test) + + if not self._recurse: + return + + # Look for tests in a module's contained objects. + if inspect.ismodule(obj): + for rawname, val in obj.__dict__.items(): + # Recurse to functions & classes. + if inspect.isfunction(val) or inspect.isclass(val): + # Make sure we don't run doctests functions or classes + # from different modules + if val.__module__ != module.__name__: + continue + + assert self._from_module(module, val), \ + "%s is not in module %s (rawname %s)" % (val, module, rawname) + + try: + valname = '%s.%s' % (name, rawname) + self._find(tests, val, valname, module, + source_lines, globs, seen) + except KeyboardInterrupt: + raise + + # Look for tests in a module's __test__ dictionary. + for valname, val in getattr(obj, '__test__', {}).items(): + if not isinstance(valname, str): + raise ValueError("SymPyDocTestFinder.find: __test__ keys " + "must be strings: %r" % + (type(valname),)) + if not (inspect.isfunction(val) or inspect.isclass(val) or + inspect.ismethod(val) or inspect.ismodule(val) or + isinstance(val, str)): + raise ValueError("SymPyDocTestFinder.find: __test__ values " + "must be strings, functions, methods, " + "classes, or modules: %r" % + (type(val),)) + valname = '%s.__test__.%s' % (name, valname) + self._find(tests, val, valname, module, source_lines, + globs, seen) + + + # Look for tests in a class's contained objects. + if inspect.isclass(obj): + for valname, val in obj.__dict__.items(): + # Special handling for staticmethod/classmethod. + if isinstance(val, staticmethod): + val = getattr(obj, valname) + if isinstance(val, classmethod): + val = getattr(obj, valname).__func__ + + + # Recurse to methods, properties, and nested classes. + if ((inspect.isfunction(unwrap(val)) or + inspect.isclass(val) or + isinstance(val, property)) and + self._from_module(module, val)): + # Make sure we don't run doctests functions or classes + # from different modules + if isinstance(val, property): + if hasattr(val.fget, '__module__'): + if val.fget.__module__ != module.__name__: + continue + else: + if val.__module__ != module.__name__: + continue + + assert self._from_module(module, val), \ + "%s is not in module %s (valname %s)" % ( + val, module, valname) + + valname = '%s.%s' % (name, valname) + self._find(tests, val, valname, module, source_lines, + globs, seen) + + def _get_test(self, obj, name, module, globs, source_lines): + """ + Return a DocTest for the given object, if it defines a docstring; + otherwise, return None. + """ + + lineno = None + + # Extract the object's docstring. If it does not have one, + # then return None (no test for this object). + if isinstance(obj, str): + # obj is a string in the case for objects in the polys package. + # Note that source_lines is a binary string (compiled polys + # modules), which can't be handled by _find_lineno so determine + # the line number here. + + docstring = obj + + matches = re.findall(r"line \d+", name) + assert len(matches) == 1, \ + "string '%s' does not contain lineno " % name + + # NOTE: this is not the exact linenumber but its better than no + # lineno ;) + lineno = int(matches[0][5:]) + + else: + docstring = getattr(obj, '__doc__', '') + if docstring is None: + docstring = '' + if not isinstance(docstring, str): + docstring = str(docstring) + + # Don't bother if the docstring is empty. + if self._exclude_empty and not docstring: + return None + + # check that properties have a docstring because _find_lineno + # assumes it + if isinstance(obj, property): + if obj.fget.__doc__ is None: + return None + + # Find the docstring's location in the file. + if lineno is None: + obj = unwrap(obj) + # handling of properties is not implemented in _find_lineno so do + # it here + if hasattr(obj, 'func_closure') and obj.func_closure is not None: + tobj = obj.func_closure[0].cell_contents + elif isinstance(obj, property): + tobj = obj.fget + else: + tobj = obj + lineno = self._find_lineno(tobj, source_lines) + + if lineno is None: + return None + + # Return a DocTest for this object. + if module is None: + filename = None + else: + filename = getattr(module, '__file__', module.__name__) + if filename[-4:] in (".pyc", ".pyo"): + filename = filename[:-1] + + globs['_doctest_depends_on'] = getattr(obj, '_doctest_depends_on', {}) + + return self._parser.get_doctest(docstring, globs, name, + filename, lineno) + + +class SymPyDocTestRunner(DocTestRunner): + """ + A class used to run DocTest test cases, and accumulate statistics. + The ``run`` method is used to process a single DocTest case. It + returns a tuple ``(f, t)``, where ``t`` is the number of test cases + tried, and ``f`` is the number of test cases that failed. + + Modified from the doctest version to not reset the sys.displayhook (see + issue 5140). + + See the docstring of the original DocTestRunner for more information. + """ + + def run(self, test, compileflags=None, out=None, clear_globs=True): + """ + Run the examples in ``test``, and display the results using the + writer function ``out``. + + The examples are run in the namespace ``test.globs``. If + ``clear_globs`` is true (the default), then this namespace will + be cleared after the test runs, to help with garbage + collection. If you would like to examine the namespace after + the test completes, then use ``clear_globs=False``. + + ``compileflags`` gives the set of flags that should be used by + the Python compiler when running the examples. If not + specified, then it will default to the set of future-import + flags that apply to ``globs``. + + The output of each example is checked using + ``SymPyDocTestRunner.check_output``, and the results are + formatted by the ``SymPyDocTestRunner.report_*`` methods. + """ + self.test = test + + # Remove ``` from the end of example, which may appear in Markdown + # files + for example in test.examples: + example.want = example.want.replace('```\n', '') + example.exc_msg = example.exc_msg and example.exc_msg.replace('```\n', '') + + + if compileflags is None: + compileflags = pdoctest._extract_future_flags(test.globs) + + save_stdout = sys.stdout + if out is None: + out = save_stdout.write + sys.stdout = self._fakeout + + # Patch pdb.set_trace to restore sys.stdout during interactive + # debugging (so it's not still redirected to self._fakeout). + # Note that the interactive output will go to *our* + # save_stdout, even if that's not the real sys.stdout; this + # allows us to write test cases for the set_trace behavior. + save_set_trace = pdb.set_trace + self.debugger = pdoctest._OutputRedirectingPdb(save_stdout) + self.debugger.reset() + pdb.set_trace = self.debugger.set_trace + + # Patch linecache.getlines, so we can see the example's source + # when we're inside the debugger. + self.save_linecache_getlines = pdoctest.linecache.getlines + linecache.getlines = self.__patched_linecache_getlines + + # Fail for deprecation warnings + with raise_on_deprecated(): + try: + return self.__run(test, compileflags, out) + finally: + sys.stdout = save_stdout + pdb.set_trace = save_set_trace + linecache.getlines = self.save_linecache_getlines + if clear_globs: + test.globs.clear() + + +# We have to override the name mangled methods. +monkeypatched_methods = [ + 'patched_linecache_getlines', + 'run', + 'record_outcome' +] +for method in monkeypatched_methods: + oldname = '_DocTestRunner__' + method + newname = '_SymPyDocTestRunner__' + method + setattr(SymPyDocTestRunner, newname, getattr(DocTestRunner, oldname)) + + +class SymPyOutputChecker(pdoctest.OutputChecker): + """ + Compared to the OutputChecker from the stdlib our OutputChecker class + supports numerical comparison of floats occurring in the output of the + doctest examples + """ + + def __init__(self): + # NOTE OutputChecker is an old-style class with no __init__ method, + # so we can't call the base class version of __init__ here + + got_floats = r'(\d+\.\d*|\.\d+)' + + # floats in the 'want' string may contain ellipses + want_floats = got_floats + r'(\.{3})?' + + front_sep = r'\s|\+|\-|\*|,' + back_sep = front_sep + r'|j|e' + + fbeg = r'^%s(?=%s|$)' % (got_floats, back_sep) + fmidend = r'(?<=%s)%s(?=%s|$)' % (front_sep, got_floats, back_sep) + self.num_got_rgx = re.compile(r'(%s|%s)' %(fbeg, fmidend)) + + fbeg = r'^%s(?=%s|$)' % (want_floats, back_sep) + fmidend = r'(?<=%s)%s(?=%s|$)' % (front_sep, want_floats, back_sep) + self.num_want_rgx = re.compile(r'(%s|%s)' %(fbeg, fmidend)) + + def check_output(self, want, got, optionflags): + """ + Return True iff the actual output from an example (`got`) + matches the expected output (`want`). These strings are + always considered to match if they are identical; but + depending on what option flags the test runner is using, + several non-exact match types are also possible. See the + documentation for `TestRunner` for more information about + option flags. + """ + # Handle the common case first, for efficiency: + # if they're string-identical, always return true. + if got == want: + return True + + # TODO parse integers as well ? + # Parse floats and compare them. If some of the parsed floats contain + # ellipses, skip the comparison. + matches = self.num_got_rgx.finditer(got) + numbers_got = [match.group(1) for match in matches] # list of strs + matches = self.num_want_rgx.finditer(want) + numbers_want = [match.group(1) for match in matches] # list of strs + if len(numbers_got) != len(numbers_want): + return False + + if len(numbers_got) > 0: + nw_ = [] + for ng, nw in zip(numbers_got, numbers_want): + if '...' in nw: + nw_.append(ng) + continue + else: + nw_.append(nw) + + if abs(float(ng)-float(nw)) > 1e-5: + return False + + got = self.num_got_rgx.sub(r'%s', got) + got = got % tuple(nw_) + + # can be used as a special sequence to signify a + # blank line, unless the DONT_ACCEPT_BLANKLINE flag is used. + if not (optionflags & pdoctest.DONT_ACCEPT_BLANKLINE): + # Replace in want with a blank line. + want = re.sub(r'(?m)^%s\s*?$' % re.escape(pdoctest.BLANKLINE_MARKER), + '', want) + # If a line in got contains only spaces, then remove the + # spaces. + got = re.sub(r'(?m)^\s*?$', '', got) + if got == want: + return True + + # This flag causes doctest to ignore any differences in the + # contents of whitespace strings. Note that this can be used + # in conjunction with the ELLIPSIS flag. + if optionflags & pdoctest.NORMALIZE_WHITESPACE: + got = ' '.join(got.split()) + want = ' '.join(want.split()) + if got == want: + return True + + # The ELLIPSIS flag says to let the sequence "..." in `want` + # match any substring in `got`. + if optionflags & pdoctest.ELLIPSIS: + if pdoctest._ellipsis_match(want, got): + return True + + # We didn't find any match; return false. + return False + + +class Reporter: + """ + Parent class for all reporters. + """ + pass + + +class PyTestReporter(Reporter): + """ + Py.test like reporter. Should produce output identical to py.test. + """ + + def __init__(self, verbose=False, tb="short", colors=True, + force_colors=False, split=None): + self._verbose = verbose + self._tb_style = tb + self._colors = colors + self._force_colors = force_colors + self._xfailed = 0 + self._xpassed = [] + self._failed = [] + self._failed_doctest = [] + self._passed = 0 + self._skipped = 0 + self._exceptions = [] + self._terminal_width = None + self._default_width = 80 + self._split = split + self._active_file = '' + self._active_f = None + + # TODO: Should these be protected? + self.slow_test_functions = [] + self.fast_test_functions = [] + + # this tracks the x-position of the cursor (useful for positioning + # things on the screen), without the need for any readline library: + self._write_pos = 0 + self._line_wrap = False + + def root_dir(self, dir): + self._root_dir = dir + + @property + def terminal_width(self): + if self._terminal_width is not None: + return self._terminal_width + + def findout_terminal_width(): + if sys.platform == "win32": + # Windows support is based on: + # + # http://code.activestate.com/recipes/ + # 440694-determine-size-of-console-window-on-windows/ + + from ctypes import windll, create_string_buffer + + h = windll.kernel32.GetStdHandle(-12) + csbi = create_string_buffer(22) + res = windll.kernel32.GetConsoleScreenBufferInfo(h, csbi) + + if res: + import struct + (_, _, _, _, _, left, _, right, _, _, _) = \ + struct.unpack("hhhhHhhhhhh", csbi.raw) + return right - left + else: + return self._default_width + + if hasattr(sys.stdout, 'isatty') and not sys.stdout.isatty(): + return self._default_width # leave PIPEs alone + + try: + process = subprocess.Popen(['stty', '-a'], + stdout=subprocess.PIPE, + stderr=subprocess.PIPE) + stdout, stderr = process.communicate() + stdout = stdout.decode("utf-8") + except OSError: + pass + else: + # We support the following output formats from stty: + # + # 1) Linux -> columns 80 + # 2) OS X -> 80 columns + # 3) Solaris -> columns = 80 + + re_linux = r"columns\s+(?P\d+);" + re_osx = r"(?P\d+)\s*columns;" + re_solaris = r"columns\s+=\s+(?P\d+);" + + for regex in (re_linux, re_osx, re_solaris): + match = re.search(regex, stdout) + + if match is not None: + columns = match.group('columns') + + try: + width = int(columns) + except ValueError: + pass + if width != 0: + return width + + return self._default_width + + width = findout_terminal_width() + self._terminal_width = width + + return width + + def write(self, text, color="", align="left", width=None, + force_colors=False): + """ + Prints a text on the screen. + + It uses sys.stdout.write(), so no readline library is necessary. + + Parameters + ========== + + color : choose from the colors below, "" means default color + align : "left"/"right", "left" is a normal print, "right" is aligned on + the right-hand side of the screen, filled with spaces if + necessary + width : the screen width + + """ + color_templates = ( + ("Black", "0;30"), + ("Red", "0;31"), + ("Green", "0;32"), + ("Brown", "0;33"), + ("Blue", "0;34"), + ("Purple", "0;35"), + ("Cyan", "0;36"), + ("LightGray", "0;37"), + ("DarkGray", "1;30"), + ("LightRed", "1;31"), + ("LightGreen", "1;32"), + ("Yellow", "1;33"), + ("LightBlue", "1;34"), + ("LightPurple", "1;35"), + ("LightCyan", "1;36"), + ("White", "1;37"), + ) + + colors = {} + + for name, value in color_templates: + colors[name] = value + c_normal = '\033[0m' + c_color = '\033[%sm' + + if width is None: + width = self.terminal_width + + if align == "right": + if self._write_pos + len(text) > width: + # we don't fit on the current line, create a new line + self.write("\n") + self.write(" "*(width - self._write_pos - len(text))) + + if not self._force_colors and hasattr(sys.stdout, 'isatty') and not \ + sys.stdout.isatty(): + # the stdout is not a terminal, this for example happens if the + # output is piped to less, e.g. "bin/test | less". In this case, + # the terminal control sequences would be printed verbatim, so + # don't use any colors. + color = "" + elif sys.platform == "win32": + # Windows consoles don't support ANSI escape sequences + color = "" + elif not self._colors: + color = "" + + if self._line_wrap: + if text[0] != "\n": + sys.stdout.write("\n") + + # Avoid UnicodeEncodeError when printing out test failures + if IS_WINDOWS: + text = text.encode('raw_unicode_escape').decode('utf8', 'ignore') + elif not sys.stdout.encoding.lower().startswith('utf'): + text = text.encode(sys.stdout.encoding, 'backslashreplace' + ).decode(sys.stdout.encoding) + + if color == "": + sys.stdout.write(text) + else: + sys.stdout.write("%s%s%s" % + (c_color % colors[color], text, c_normal)) + sys.stdout.flush() + l = text.rfind("\n") + if l == -1: + self._write_pos += len(text) + else: + self._write_pos = len(text) - l - 1 + self._line_wrap = self._write_pos >= width + self._write_pos %= width + + def write_center(self, text, delim="="): + width = self.terminal_width + if text != "": + text = " %s " % text + idx = (width - len(text)) // 2 + t = delim*idx + text + delim*(width - idx - len(text)) + self.write(t + "\n") + + def write_exception(self, e, val, tb): + # remove the first item, as that is always runtests.py + tb = tb.tb_next + t = traceback.format_exception(e, val, tb) + self.write("".join(t)) + + def start(self, seed=None, msg="test process starts"): + self.write_center(msg) + executable = sys.executable + v = tuple(sys.version_info) + python_version = "%s.%s.%s-%s-%s" % v + implementation = platform.python_implementation() + if implementation == 'PyPy': + implementation += " %s.%s.%s-%s-%s" % sys.pypy_version_info + self.write("executable: %s (%s) [%s]\n" % + (executable, python_version, implementation)) + from sympy.utilities.misc import ARCH + self.write("architecture: %s\n" % ARCH) + from sympy.core.cache import USE_CACHE + self.write("cache: %s\n" % USE_CACHE) + version = '' + if GROUND_TYPES =='gmpy': + import gmpy2 as gmpy + version = gmpy.version() + self.write("ground types: %s %s\n" % (GROUND_TYPES, version)) + numpy = import_module('numpy') + self.write("numpy: %s\n" % (None if not numpy else numpy.__version__)) + if seed is not None: + self.write("random seed: %d\n" % seed) + from sympy.utilities.misc import HASH_RANDOMIZATION + self.write("hash randomization: ") + hash_seed = os.getenv("PYTHONHASHSEED") or '0' + if HASH_RANDOMIZATION and (hash_seed == "random" or int(hash_seed)): + self.write("on (PYTHONHASHSEED=%s)\n" % hash_seed) + else: + self.write("off\n") + if self._split: + self.write("split: %s\n" % self._split) + self.write('\n') + self._t_start = clock() + + def finish(self): + self._t_end = clock() + self.write("\n") + global text, linelen + text = "tests finished: %d passed, " % self._passed + linelen = len(text) + + def add_text(mytext): + global text, linelen + """Break new text if too long.""" + if linelen + len(mytext) > self.terminal_width: + text += '\n' + linelen = 0 + text += mytext + linelen += len(mytext) + + if len(self._failed) > 0: + add_text("%d failed, " % len(self._failed)) + if len(self._failed_doctest) > 0: + add_text("%d failed, " % len(self._failed_doctest)) + if self._skipped > 0: + add_text("%d skipped, " % self._skipped) + if self._xfailed > 0: + add_text("%d expected to fail, " % self._xfailed) + if len(self._xpassed) > 0: + add_text("%d expected to fail but passed, " % len(self._xpassed)) + if len(self._exceptions) > 0: + add_text("%d exceptions, " % len(self._exceptions)) + add_text("in %.2f seconds" % (self._t_end - self._t_start)) + + if self.slow_test_functions: + self.write_center('slowest tests', '_') + sorted_slow = sorted(self.slow_test_functions, key=lambda r: r[1]) + for slow_func_name, taken in sorted_slow: + print('%s - Took %.3f seconds' % (slow_func_name, taken)) + + if self.fast_test_functions: + self.write_center('unexpectedly fast tests', '_') + sorted_fast = sorted(self.fast_test_functions, + key=lambda r: r[1]) + for fast_func_name, taken in sorted_fast: + print('%s - Took %.3f seconds' % (fast_func_name, taken)) + + if len(self._xpassed) > 0: + self.write_center("xpassed tests", "_") + for e in self._xpassed: + self.write("%s: %s\n" % (e[0], e[1])) + self.write("\n") + + if self._tb_style != "no" and len(self._exceptions) > 0: + for e in self._exceptions: + filename, f, (t, val, tb) = e + self.write_center("", "_") + if f is None: + s = "%s" % filename + else: + s = "%s:%s" % (filename, f.__name__) + self.write_center(s, "_") + self.write_exception(t, val, tb) + self.write("\n") + + if self._tb_style != "no" and len(self._failed) > 0: + for e in self._failed: + filename, f, (t, val, tb) = e + self.write_center("", "_") + self.write_center("%s::%s" % (filename, f.__name__), "_") + self.write_exception(t, val, tb) + self.write("\n") + + if self._tb_style != "no" and len(self._failed_doctest) > 0: + for e in self._failed_doctest: + filename, msg = e + self.write_center("", "_") + self.write_center("%s" % filename, "_") + self.write(msg) + self.write("\n") + + self.write_center(text) + ok = len(self._failed) == 0 and len(self._exceptions) == 0 and \ + len(self._failed_doctest) == 0 + if not ok: + self.write("DO *NOT* COMMIT!\n") + return ok + + def entering_filename(self, filename, n): + rel_name = filename[len(self._root_dir) + 1:] + self._active_file = rel_name + self._active_file_error = False + self.write(rel_name) + self.write("[%d] " % n) + + def leaving_filename(self): + self.write(" ") + if self._active_file_error: + self.write("[FAIL]", "Red", align="right") + else: + self.write("[OK]", "Green", align="right") + self.write("\n") + if self._verbose: + self.write("\n") + + def entering_test(self, f): + self._active_f = f + if self._verbose: + self.write("\n" + f.__name__ + " ") + + def test_xfail(self): + self._xfailed += 1 + self.write("f", "Green") + + def test_xpass(self, v): + message = str(v) + self._xpassed.append((self._active_file, message)) + self.write("X", "Green") + + def test_fail(self, exc_info): + self._failed.append((self._active_file, self._active_f, exc_info)) + self.write("F", "Red") + self._active_file_error = True + + def doctest_fail(self, name, error_msg): + # the first line contains "******", remove it: + error_msg = "\n".join(error_msg.split("\n")[1:]) + self._failed_doctest.append((name, error_msg)) + self.write("F", "Red") + self._active_file_error = True + + def test_pass(self, char="."): + self._passed += 1 + if self._verbose: + self.write("ok", "Green") + else: + self.write(char, "Green") + + def test_skip(self, v=None): + char = "s" + self._skipped += 1 + if v is not None: + message = str(v) + if message == "KeyboardInterrupt": + char = "K" + elif message == "Timeout": + char = "T" + elif message == "Slow": + char = "w" + if self._verbose: + if v is not None: + self.write(message + ' ', "Blue") + else: + self.write(" - ", "Blue") + self.write(char, "Blue") + + def test_exception(self, exc_info): + self._exceptions.append((self._active_file, self._active_f, exc_info)) + if exc_info[0] is TimeOutError: + self.write("T", "Red") + else: + self.write("E", "Red") + self._active_file_error = True + + def import_error(self, filename, exc_info): + self._exceptions.append((filename, None, exc_info)) + rel_name = filename[len(self._root_dir) + 1:] + self.write(rel_name) + self.write("[?] Failed to import", "Red") + self.write(" ") + self.write("[FAIL]", "Red", align="right") + self.write("\n") diff --git a/.venv/lib/python3.12/site-packages/sympy/testing/runtests_pytest.py b/.venv/lib/python3.12/site-packages/sympy/testing/runtests_pytest.py new file mode 100644 index 0000000000000000000000000000000000000000..635f27864ca86571128e6c9a055199dfbde1ed63 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/testing/runtests_pytest.py @@ -0,0 +1,461 @@ +"""Backwards compatible functions for running tests from SymPy using pytest. + +SymPy historically had its own testing framework that aimed to: +- be compatible with pytest; +- operate similarly (or identically) to pytest; +- not require any external dependencies; +- have all the functionality in one file only; +- have no magic, just import the test file and execute the test functions; and +- be portable. + +To reduce the maintenance burden of developing an independent testing framework +and to leverage the benefits of existing Python testing infrastructure, SymPy +now uses pytest (and various of its plugins) to run the test suite. + +To maintain backwards compatibility with the legacy testing interface of SymPy, +which implemented functions that allowed users to run the tests on their +installed version of SymPy, the functions in this module are implemented to +match the existing API while thinly wrapping pytest. + +These two key functions are `test` and `doctest`. + +""" + +import functools +import importlib.util +import os +import pathlib +import re +from fnmatch import fnmatch +from typing import List, Optional, Tuple + +try: + import pytest +except ImportError: + + class NoPytestError(Exception): + """Raise when an internal test helper function is called with pytest.""" + + class pytest: # type: ignore + """Shadow to support pytest features when pytest can't be imported.""" + + @staticmethod + def main(*args, **kwargs): + msg = 'pytest must be installed to run tests via this function' + raise NoPytestError(msg) + +from sympy.testing.runtests import test as test_sympy + + +TESTPATHS_DEFAULT = ( + pathlib.Path('sympy'), + pathlib.Path('doc', 'src'), +) +BLACKLIST_DEFAULT = ( + 'sympy/integrals/rubi/rubi_tests/tests', +) + + +class PytestPluginManager: + """Module names for pytest plugins used by SymPy.""" + PYTEST: str = 'pytest' + RANDOMLY: str = 'pytest_randomly' + SPLIT: str = 'pytest_split' + TIMEOUT: str = 'pytest_timeout' + XDIST: str = 'xdist' + + @functools.cached_property + def has_pytest(self) -> bool: + return bool(importlib.util.find_spec(self.PYTEST)) + + @functools.cached_property + def has_randomly(self) -> bool: + return bool(importlib.util.find_spec(self.RANDOMLY)) + + @functools.cached_property + def has_split(self) -> bool: + return bool(importlib.util.find_spec(self.SPLIT)) + + @functools.cached_property + def has_timeout(self) -> bool: + return bool(importlib.util.find_spec(self.TIMEOUT)) + + @functools.cached_property + def has_xdist(self) -> bool: + return bool(importlib.util.find_spec(self.XDIST)) + + +split_pattern = re.compile(r'([1-9][0-9]*)/([1-9][0-9]*)') + + +@functools.lru_cache +def sympy_dir() -> pathlib.Path: + """Returns the root SymPy directory.""" + return pathlib.Path(__file__).parents[2] + + +def update_args_with_paths( + paths: List[str], + keywords: Optional[Tuple[str]], + args: List[str], +) -> List[str]: + """Appends valid paths and flags to the args `list` passed to `pytest.main`. + + The are three different types of "path" that a user may pass to the `paths` + positional arguments, all of which need to be handled slightly differently: + + 1. Nothing is passed + The paths to the `testpaths` defined in `pytest.ini` need to be appended + to the arguments list. + 2. Full, valid paths are passed + These paths need to be validated but can then be directly appended to + the arguments list. + 3. Partial paths are passed. + The `testpaths` defined in `pytest.ini` need to be recursed and any + matches be appended to the arguments list. + + """ + + def find_paths_matching_partial(partial_paths): + partial_path_file_patterns = [] + for partial_path in partial_paths: + if len(partial_path) >= 4: + has_test_prefix = partial_path[:4] == 'test' + has_py_suffix = partial_path[-3:] == '.py' + elif len(partial_path) >= 3: + has_test_prefix = False + has_py_suffix = partial_path[-3:] == '.py' + else: + has_test_prefix = False + has_py_suffix = False + if has_test_prefix and has_py_suffix: + partial_path_file_patterns.append(partial_path) + elif has_test_prefix: + partial_path_file_patterns.append(f'{partial_path}*.py') + elif has_py_suffix: + partial_path_file_patterns.append(f'test*{partial_path}') + else: + partial_path_file_patterns.append(f'test*{partial_path}*.py') + matches = [] + for testpath in valid_testpaths_default: + for path, dirs, files in os.walk(testpath, topdown=True): + zipped = zip(partial_paths, partial_path_file_patterns) + for (partial_path, partial_path_file) in zipped: + if fnmatch(path, f'*{partial_path}*'): + matches.append(str(pathlib.Path(path))) + dirs[:] = [] + else: + for file in files: + if fnmatch(file, partial_path_file): + matches.append(str(pathlib.Path(path, file))) + return matches + + def is_tests_file(filepath: str) -> bool: + path = pathlib.Path(filepath) + if not path.is_file(): + return False + if not path.parts[-1].startswith('test_'): + return False + if not path.suffix == '.py': + return False + return True + + def find_tests_matching_keywords(keywords, filepath): + matches = [] + source = pathlib.Path(filepath).read_text(encoding='utf-8') + for line in source.splitlines(): + if line.lstrip().startswith('def '): + for kw in keywords: + if line.lower().find(kw.lower()) != -1: + test_name = line.split(' ')[1].split('(')[0] + full_test_path = filepath + '::' + test_name + matches.append(full_test_path) + return matches + + valid_testpaths_default = [] + for testpath in TESTPATHS_DEFAULT: + absolute_testpath = pathlib.Path(sympy_dir(), testpath) + if absolute_testpath.exists(): + valid_testpaths_default.append(str(absolute_testpath)) + + candidate_paths = [] + if paths: + full_paths = [] + partial_paths = [] + for path in paths: + if pathlib.Path(path).exists(): + full_paths.append(str(pathlib.Path(sympy_dir(), path))) + else: + partial_paths.append(path) + matched_paths = find_paths_matching_partial(partial_paths) + candidate_paths.extend(full_paths) + candidate_paths.extend(matched_paths) + else: + candidate_paths.extend(valid_testpaths_default) + + if keywords is not None and keywords != (): + matches = [] + for path in candidate_paths: + if is_tests_file(path): + test_matches = find_tests_matching_keywords(keywords, path) + matches.extend(test_matches) + else: + for root, dirnames, filenames in os.walk(path): + for filename in filenames: + absolute_filepath = str(pathlib.Path(root, filename)) + if is_tests_file(absolute_filepath): + test_matches = find_tests_matching_keywords( + keywords, + absolute_filepath, + ) + matches.extend(test_matches) + args.extend(matches) + else: + args.extend(candidate_paths) + + return args + + +def make_absolute_path(partial_path: str) -> str: + """Convert a partial path to an absolute path. + + A path such a `sympy/core` might be needed. However, absolute paths should + be used in the arguments to pytest in all cases as it avoids errors that + arise from nonexistent paths. + + This function assumes that partial_paths will be passed in such that they + begin with the explicit `sympy` directory, i.e. `sympy/...`. + + """ + + def is_valid_partial_path(partial_path: str) -> bool: + """Assumption that partial paths are defined from the `sympy` root.""" + return pathlib.Path(partial_path).parts[0] == 'sympy' + + if not is_valid_partial_path(partial_path): + msg = ( + f'Partial path {dir(partial_path)} is invalid, partial paths are ' + f'expected to be defined with the `sympy` directory as the root.' + ) + raise ValueError(msg) + + absolute_path = str(pathlib.Path(sympy_dir(), partial_path)) + return absolute_path + + +def test(*paths, subprocess=True, rerun=0, **kwargs): + """Interface to run tests via pytest compatible with SymPy's test runner. + + Explanation + =========== + + Note that a `pytest.ExitCode`, which is an `enum`, is returned. This is + different to the legacy SymPy test runner which would return a `bool`. If + all tests successfully pass the `pytest.ExitCode.OK` with value `0` is + returned, whereas the legacy SymPy test runner would return `True`. In any + other scenario, a non-zero `enum` value is returned, whereas the legacy + SymPy test runner would return `False`. Users need to, therefore, be careful + if treating the pytest exit codes as booleans because + `bool(pytest.ExitCode.OK)` evaluates to `False`, the opposite of legacy + behaviour. + + Examples + ======== + + >>> import sympy # doctest: +SKIP + + Run one file: + + >>> sympy.test('sympy/core/tests/test_basic.py') # doctest: +SKIP + >>> sympy.test('_basic') # doctest: +SKIP + + Run all tests in sympy/functions/ and some particular file: + + >>> sympy.test("sympy/core/tests/test_basic.py", + ... "sympy/functions") # doctest: +SKIP + + Run all tests in sympy/core and sympy/utilities: + + >>> sympy.test("/core", "/util") # doctest: +SKIP + + Run specific test from a file: + + >>> sympy.test("sympy/core/tests/test_basic.py", + ... kw="test_equality") # doctest: +SKIP + + Run specific test from any file: + + >>> sympy.test(kw="subs") # doctest: +SKIP + + Run the tests using the legacy SymPy runner: + + >>> sympy.test(use_sympy_runner=True) # doctest: +SKIP + + Note that this option is slated for deprecation in the near future and is + only currently provided to ensure users have an alternative option while the + pytest-based runner receives real-world testing. + + Parameters + ========== + paths : first n positional arguments of strings + Paths, both partial and absolute, describing which subset(s) of the test + suite are to be run. + subprocess : bool, default is True + Legacy option, is currently ignored. + rerun : int, default is 0 + Legacy option, is ignored. + use_sympy_runner : bool or None, default is None + Temporary option to invoke the legacy SymPy test runner instead of + `pytest.main`. Will be removed in the near future. + verbose : bool, default is False + Sets the verbosity of the pytest output. Using `True` will add the + `--verbose` option to the pytest call. + tb : str, 'auto', 'long', 'short', 'line', 'native', or 'no' + Sets the traceback print mode of pytest using the `--tb` option. + kw : str + Only run tests which match the given substring expression. An expression + is a Python evaluatable expression where all names are substring-matched + against test names and their parent classes. Example: -k 'test_method or + test_other' matches all test functions and classes whose name contains + 'test_method' or 'test_other', while -k 'not test_method' matches those + that don't contain 'test_method' in their names. -k 'not test_method and + not test_other' will eliminate the matches. Additionally keywords are + matched to classes and functions containing extra names in their + 'extra_keyword_matches' set, as well as functions which have names + assigned directly to them. The matching is case-insensitive. + pdb : bool, default is False + Start the interactive Python debugger on errors or `KeyboardInterrupt`. + colors : bool, default is True + Color terminal output. + force_colors : bool, default is False + Legacy option, is ignored. + sort : bool, default is True + Run the tests in sorted order. pytest uses a sorted test order by + default. Requires pytest-randomly. + seed : int + Seed to use for random number generation. Requires pytest-randomly. + timeout : int, default is 0 + Timeout in seconds before dumping the stacks. 0 means no timeout. + Requires pytest-timeout. + fail_on_timeout : bool, default is False + Legacy option, is currently ignored. + slow : bool, default is False + Run the subset of tests marked as `slow`. + enhance_asserts : bool, default is False + Legacy option, is currently ignored. + split : string in form `/` or None, default is None + Used to split the tests up. As an example, if `split='2/3' is used then + only the middle third of tests are run. Requires pytest-split. + time_balance : bool, default is True + Legacy option, is currently ignored. + blacklist : iterable of test paths as strings, default is BLACKLIST_DEFAULT + Blacklisted test paths are ignored using the `--ignore` option. Paths + may be partial or absolute. If partial then they are matched against + all paths in the pytest tests path. + parallel : bool, default is False + Parallelize the test running using pytest-xdist. If `True` then pytest + will automatically detect the number of CPU cores available and use them + all. Requires pytest-xdist. + store_durations : bool, False + Store test durations into the file `.test_durations`. The is used by + `pytest-split` to help determine more even splits when more than one + test group is being used. Requires pytest-split. + + """ + # NOTE: to be removed alongside SymPy test runner + if kwargs.get('use_sympy_runner', False): + kwargs.pop('parallel', False) + kwargs.pop('store_durations', False) + kwargs.pop('use_sympy_runner', True) + if kwargs.get('slow') is None: + kwargs['slow'] = False + return test_sympy(*paths, subprocess=True, rerun=0, **kwargs) + + pytest_plugin_manager = PytestPluginManager() + if not pytest_plugin_manager.has_pytest: + pytest.main() + + args = [] + + if kwargs.get('verbose', False): + args.append('--verbose') + + if tb := kwargs.get('tb'): + args.extend(['--tb', tb]) + + if kwargs.get('pdb'): + args.append('--pdb') + + if not kwargs.get('colors', True): + args.extend(['--color', 'no']) + + if seed := kwargs.get('seed'): + if not pytest_plugin_manager.has_randomly: + msg = '`pytest-randomly` plugin required to control random seed.' + raise ModuleNotFoundError(msg) + args.extend(['--randomly-seed', str(seed)]) + + if kwargs.get('sort', True) and pytest_plugin_manager.has_randomly: + args.append('--randomly-dont-reorganize') + elif not kwargs.get('sort', True) and not pytest_plugin_manager.has_randomly: + msg = '`pytest-randomly` plugin required to randomize test order.' + raise ModuleNotFoundError(msg) + + if timeout := kwargs.get('timeout', None): + if not pytest_plugin_manager.has_timeout: + msg = '`pytest-timeout` plugin required to apply timeout to tests.' + raise ModuleNotFoundError(msg) + args.extend(['--timeout', str(int(timeout))]) + + # Skip slow tests by default and always skip tooslow tests + if kwargs.get('slow', False): + args.extend(['-m', 'slow and not tooslow']) + else: + args.extend(['-m', 'not slow and not tooslow']) + + if (split := kwargs.get('split')) is not None: + if not pytest_plugin_manager.has_split: + msg = '`pytest-split` plugin required to run tests as groups.' + raise ModuleNotFoundError(msg) + match = split_pattern.match(split) + if not match: + msg = ('split must be a string of the form a/b where a and b are ' + 'positive nonzero ints') + raise ValueError(msg) + group, splits = map(str, match.groups()) + args.extend(['--group', group, '--splits', splits]) + if group > splits: + msg = (f'cannot have a group number {group} with only {splits} ' + 'splits') + raise ValueError(msg) + + if blacklist := kwargs.get('blacklist', BLACKLIST_DEFAULT): + for path in blacklist: + args.extend(['--ignore', make_absolute_path(path)]) + + if kwargs.get('parallel', False): + if not pytest_plugin_manager.has_xdist: + msg = '`pytest-xdist` plugin required to run tests in parallel.' + raise ModuleNotFoundError(msg) + args.extend(['-n', 'auto']) + + if kwargs.get('store_durations', False): + if not pytest_plugin_manager.has_split: + msg = '`pytest-split` plugin required to store test durations.' + raise ModuleNotFoundError(msg) + args.append('--store-durations') + + if (keywords := kwargs.get('kw')) is not None: + keywords = tuple(str(kw) for kw in keywords) + else: + keywords = () + + args = update_args_with_paths(paths, keywords, args) + exit_code = pytest.main(args) + return exit_code + + +def doctest(): + """Interface to run doctests via pytest compatible with SymPy's test runner. + """ + raise NotImplementedError diff --git a/.venv/lib/python3.12/site-packages/sympy/testing/tests/__init__.py b/.venv/lib/python3.12/site-packages/sympy/testing/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/.venv/lib/python3.12/site-packages/sympy/testing/tests/__pycache__/__init__.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/testing/tests/__pycache__/__init__.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..9ef62361c2b651ccfa028760c86d6f6f6f256c74 Binary files /dev/null and b/.venv/lib/python3.12/site-packages/sympy/testing/tests/__pycache__/__init__.cpython-312.pyc differ diff --git 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b/.venv/lib/python3.12/site-packages/sympy/testing/tests/diagnose_imports.py new file mode 100644 index 0000000000000000000000000000000000000000..a31b66c66690c082800ae36eee37dad6927e0b37 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/testing/tests/diagnose_imports.py @@ -0,0 +1,216 @@ +#!/usr/bin/env python + +""" +Import diagnostics. Run bin/diagnose_imports.py --help for details. +""" + +from __future__ import annotations + +if __name__ == "__main__": + + import sys + import inspect + import builtins + + import optparse + + from os.path import abspath, dirname, join, normpath + this_file = abspath(__file__) + sympy_dir = join(dirname(this_file), '..', '..', '..') + sympy_dir = normpath(sympy_dir) + sys.path.insert(0, sympy_dir) + + option_parser = optparse.OptionParser( + usage= + "Usage: %prog option [options]\n" + "\n" + "Import analysis for imports between SymPy modules.") + option_group = optparse.OptionGroup( + option_parser, + 'Analysis options', + 'Options that define what to do. Exactly one of these must be given.') + option_group.add_option( + '--problems', + help= + 'Print all import problems, that is: ' + 'If an import pulls in a package instead of a module ' + '(e.g. sympy.core instead of sympy.core.add); ' # see ##PACKAGE## + 'if it imports a symbol that is already present; ' # see ##DUPLICATE## + 'if it imports a symbol ' + 'from somewhere other than the defining module.', # see ##ORIGIN## + action='count') + option_group.add_option( + '--origins', + help= + 'For each imported symbol in each module, ' + 'print the module that defined it. ' + '(This is useful for import refactoring.)', + action='count') + option_parser.add_option_group(option_group) + option_group = optparse.OptionGroup( + option_parser, + 'Sort options', + 'These options define the sort order for output lines. ' + 'At most one of these options is allowed. ' + 'Unsorted output will reflect the order in which imports happened.') + option_group.add_option( + '--by-importer', + help='Sort output lines by name of importing module.', + action='count') + option_group.add_option( + '--by-origin', + help='Sort output lines by name of imported module.', + action='count') + option_parser.add_option_group(option_group) + (options, args) = option_parser.parse_args() + if args: + option_parser.error( + 'Unexpected arguments %s (try %s --help)' % (args, sys.argv[0])) + if options.problems > 1: + option_parser.error('--problems must not be given more than once.') + if options.origins > 1: + option_parser.error('--origins must not be given more than once.') + if options.by_importer > 1: + option_parser.error('--by-importer must not be given more than once.') + if options.by_origin > 1: + option_parser.error('--by-origin must not be given more than once.') + options.problems = options.problems == 1 + options.origins = options.origins == 1 + options.by_importer = options.by_importer == 1 + options.by_origin = options.by_origin == 1 + if not options.problems and not options.origins: + option_parser.error( + 'At least one of --problems and --origins is required') + if options.problems and options.origins: + option_parser.error( + 'At most one of --problems and --origins is allowed') + if options.by_importer and options.by_origin: + option_parser.error( + 'At most one of --by-importer and --by-origin is allowed') + options.by_process = not options.by_importer and not options.by_origin + + builtin_import = builtins.__import__ + + class Definition: + """Information about a symbol's definition.""" + def __init__(self, name, value, definer): + self.name = name + self.value = value + self.definer = definer + def __hash__(self): + return hash(self.name) + def __eq__(self, other): + return self.name == other.name and self.value == other.value + def __ne__(self, other): + return not (self == other) + def __repr__(self): + return 'Definition(%s, ..., %s)' % ( + repr(self.name), repr(self.definer)) + + # Maps each function/variable to name of module to define it + symbol_definers: dict[Definition, str] = {} + + def in_module(a, b): + """Is a the same module as or a submodule of b?""" + return a == b or a != None and b != None and a.startswith(b + '.') + + def relevant(module): + """Is module relevant for import checking? + + Only imports between relevant modules will be checked.""" + return in_module(module, 'sympy') + + sorted_messages = [] + + def msg(msg, *args): + if options.by_process: + print(msg % args) + else: + sorted_messages.append(msg % args) + + def tracking_import(module, globals=globals(), locals=[], fromlist=None, level=-1): + """__import__ wrapper - does not change imports at all, but tracks them. + + Default order is implemented by doing output directly. + All other orders are implemented by collecting output information into + a sorted list that will be emitted after all imports are processed. + + Indirect imports can only occur after the requested symbol has been + imported directly (because the indirect import would not have a module + to pick the symbol up from). + So this code detects indirect imports by checking whether the symbol in + question was already imported. + + Keeps the semantics of __import__ unchanged.""" + caller_frame = inspect.getframeinfo(sys._getframe(1)) + importer_filename = caller_frame.filename + importer_module = globals['__name__'] + if importer_filename == caller_frame.filename: + importer_reference = '%s line %s' % ( + importer_filename, str(caller_frame.lineno)) + else: + importer_reference = importer_filename + result = builtin_import(module, globals, locals, fromlist, level) + importee_module = result.__name__ + # We're only interested if importer and importee are in SymPy + if relevant(importer_module) and relevant(importee_module): + for symbol in result.__dict__.iterkeys(): + definition = Definition( + symbol, result.__dict__[symbol], importer_module) + if definition not in symbol_definers: + symbol_definers[definition] = importee_module + if hasattr(result, '__path__'): + ##PACKAGE## + # The existence of __path__ is documented in the tutorial on modules. + # Python 3.3 documents this in http://docs.python.org/3.3/reference/import.html + if options.by_origin: + msg('Error: %s (a package) is imported by %s', + module, importer_reference) + else: + msg('Error: %s contains package import %s', + importer_reference, module) + if fromlist != None: + symbol_list = fromlist + if '*' in symbol_list: + if (importer_filename.endswith(("__init__.py", "__init__.pyc", "__init__.pyo"))): + # We do not check starred imports inside __init__ + # That's the normal "please copy over its imports to my namespace" + symbol_list = [] + else: + symbol_list = result.__dict__.iterkeys() + for symbol in symbol_list: + if symbol not in result.__dict__: + if options.by_origin: + msg('Error: %s.%s is not defined (yet), but %s tries to import it', + importee_module, symbol, importer_reference) + else: + msg('Error: %s tries to import %s.%s, which did not define it (yet)', + importer_reference, importee_module, symbol) + else: + definition = Definition( + symbol, result.__dict__[symbol], importer_module) + symbol_definer = symbol_definers[definition] + if symbol_definer == importee_module: + ##DUPLICATE## + if options.by_origin: + msg('Error: %s.%s is imported again into %s', + importee_module, symbol, importer_reference) + else: + msg('Error: %s imports %s.%s again', + importer_reference, importee_module, symbol) + else: + ##ORIGIN## + if options.by_origin: + msg('Error: %s.%s is imported by %s, which should import %s.%s instead', + importee_module, symbol, importer_reference, symbol_definer, symbol) + else: + msg('Error: %s imports %s.%s but should import %s.%s instead', + importer_reference, importee_module, symbol, symbol_definer, symbol) + return result + + builtins.__import__ = tracking_import + __import__('sympy') + + sorted_messages.sort() + for message in sorted_messages: + print(message) diff --git a/.venv/lib/python3.12/site-packages/sympy/testing/tests/test_code_quality.py b/.venv/lib/python3.12/site-packages/sympy/testing/tests/test_code_quality.py new file mode 100644 index 0000000000000000000000000000000000000000..9a9f363f0b9a802553f8643186b7d858d1ad0694 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/testing/tests/test_code_quality.py @@ -0,0 +1,510 @@ +# coding=utf-8 +from os import walk, sep, pardir +from os.path import split, join, abspath, exists, isfile +from glob import glob +import re +import random +import ast + +from sympy.testing.pytest import raises +from sympy.testing.quality_unicode import _test_this_file_encoding + +# System path separator (usually slash or backslash) to be +# used with excluded files, e.g. +# exclude = set([ +# "%(sep)smpmath%(sep)s" % sepd, +# ]) +sepd = {"sep": sep} + +# path and sympy_path +SYMPY_PATH = abspath(join(split(__file__)[0], pardir, pardir)) # go to sympy/ +assert exists(SYMPY_PATH) + +TOP_PATH = abspath(join(SYMPY_PATH, pardir)) +BIN_PATH = join(TOP_PATH, "bin") +EXAMPLES_PATH = join(TOP_PATH, "examples") + +# Error messages +message_space = "File contains trailing whitespace: %s, line %s." +message_implicit = "File contains an implicit import: %s, line %s." +message_tabs = "File contains tabs instead of spaces: %s, line %s." +message_carriage = "File contains carriage returns at end of line: %s, line %s" +message_str_raise = "File contains string exception: %s, line %s" +message_gen_raise = "File contains generic exception: %s, line %s" +message_old_raise = "File contains old-style raise statement: %s, line %s, \"%s\"" +message_eof = "File does not end with a newline: %s, line %s" +message_multi_eof = "File ends with more than 1 newline: %s, line %s" +message_test_suite_def = "Function should start with 'test_' or '_': %s, line %s" +message_duplicate_test = "This is a duplicate test function: %s, line %s" +message_self_assignments = "File contains assignments to self/cls: %s, line %s." +message_func_is = "File contains '.func is': %s, line %s." +message_bare_expr = "File contains bare expression: %s, line %s." + +implicit_test_re = re.compile(r'^\s*(>>> )?(\.\.\. )?from .* import .*\*') +str_raise_re = re.compile( + r'^\s*(>>> )?(\.\.\. )?raise(\s+(\'|\")|\s*(\(\s*)+(\'|\"))') +gen_raise_re = re.compile( + r'^\s*(>>> )?(\.\.\. )?raise(\s+Exception|\s*(\(\s*)+Exception)') +old_raise_re = re.compile(r'^\s*(>>> )?(\.\.\. )?raise((\s*\(\s*)|\s+)\w+\s*,') +test_suite_def_re = re.compile(r'^def\s+(?!(_|test))[^(]*\(\s*\)\s*:$') +test_ok_def_re = re.compile(r'^def\s+test_.*:$') +test_file_re = re.compile(r'.*[/\\]test_.*\.py$') +func_is_re = re.compile(r'\.\s*func\s+is') + + +def tab_in_leading(s): + """Returns True if there are tabs in the leading whitespace of a line, + including the whitespace of docstring code samples.""" + n = len(s) - len(s.lstrip()) + if not s[n:n + 3] in ['...', '>>>']: + check = s[:n] + else: + smore = s[n + 3:] + check = s[:n] + smore[:len(smore) - len(smore.lstrip())] + return not (check.expandtabs() == check) + + +def find_self_assignments(s): + """Returns a list of "bad" assignments: if there are instances + of assigning to the first argument of the class method (except + for staticmethod's). + """ + t = [n for n in ast.parse(s).body if isinstance(n, ast.ClassDef)] + + bad = [] + for c in t: + for n in c.body: + if not isinstance(n, ast.FunctionDef): + continue + if any(d.id == 'staticmethod' + for d in n.decorator_list if isinstance(d, ast.Name)): + continue + if n.name == '__new__': + continue + if not n.args.args: + continue + first_arg = n.args.args[0].arg + + for m in ast.walk(n): + if isinstance(m, ast.Assign): + for a in m.targets: + if isinstance(a, ast.Name) and a.id == first_arg: + bad.append(m) + elif (isinstance(a, ast.Tuple) and + any(q.id == first_arg for q in a.elts + if isinstance(q, ast.Name))): + bad.append(m) + + return bad + + +def check_directory_tree(base_path, file_check, exclusions=set(), pattern="*.py"): + """ + Checks all files in the directory tree (with base_path as starting point) + with the file_check function provided, skipping files that contain + any of the strings in the set provided by exclusions. + """ + if not base_path: + return + for root, dirs, files in walk(base_path): + check_files(glob(join(root, pattern)), file_check, exclusions) + + +def check_files(files, file_check, exclusions=set(), pattern=None): + """ + Checks all files with the file_check function provided, skipping files + that contain any of the strings in the set provided by exclusions. + """ + if not files: + return + for fname in files: + if not exists(fname) or not isfile(fname): + continue + if any(ex in fname for ex in exclusions): + continue + if pattern is None or re.match(pattern, fname): + file_check(fname) + + +class _Visit(ast.NodeVisitor): + """return the line number corresponding to the + line on which a bare expression appears if it is a binary op + or a comparison that is not in a with block. + + EXAMPLES + ======== + + >>> import ast + >>> class _Visit(ast.NodeVisitor): + ... def visit_Expr(self, node): + ... if isinstance(node.value, (ast.BinOp, ast.Compare)): + ... print(node.lineno) + ... def visit_With(self, node): + ... pass # no checking there + ... + >>> code='''x = 1 # line 1 + ... for i in range(3): + ... x == 2 # <-- 3 + ... if x == 2: + ... x == 3 # <-- 5 + ... x + 1 # <-- 6 + ... x = 1 + ... if x == 1: + ... print(1) + ... while x != 1: + ... x == 1 # <-- 11 + ... with raises(TypeError): + ... c == 1 + ... raise TypeError + ... assert x == 1 + ... ''' + >>> _Visit().visit(ast.parse(code)) + 3 + 5 + 6 + 11 + """ + def visit_Expr(self, node): + if isinstance(node.value, (ast.BinOp, ast.Compare)): + assert None, message_bare_expr % ('', node.lineno) + def visit_With(self, node): + pass + + +BareExpr = _Visit() + + +def line_with_bare_expr(code): + """return None or else 0-based line number of code on which + a bare expression appeared. + """ + tree = ast.parse(code) + try: + BareExpr.visit(tree) + except AssertionError as msg: + assert msg.args + msg = msg.args[0] + assert msg.startswith(message_bare_expr.split(':', 1)[0]) + return int(msg.rsplit(' ', 1)[1].rstrip('.')) # the line number + + +def test_files(): + """ + This test tests all files in SymPy and checks that: + o no lines contains a trailing whitespace + o no lines end with \r\n + o no line uses tabs instead of spaces + o that the file ends with a single newline + o there are no general or string exceptions + o there are no old style raise statements + o name of arg-less test suite functions start with _ or test_ + o no duplicate function names that start with test_ + o no assignments to self variable in class methods + o no lines contain ".func is" except in the test suite + o there is no do-nothing expression like `a == b` or `x + 1` + """ + + def test(fname): + with open(fname, encoding="utf8") as test_file: + test_this_file(fname, test_file) + with open(fname, encoding='utf8') as test_file: + _test_this_file_encoding(fname, test_file) + + def test_this_file(fname, test_file): + idx = None + code = test_file.read() + test_file.seek(0) # restore reader to head + py = fname if sep not in fname else fname.rsplit(sep, 1)[-1] + if py.startswith('test_'): + idx = line_with_bare_expr(code) + if idx is not None: + assert False, message_bare_expr % (fname, idx + 1) + + line = None # to flag the case where there were no lines in file + tests = 0 + test_set = set() + for idx, line in enumerate(test_file): + if test_file_re.match(fname): + if test_suite_def_re.match(line): + assert False, message_test_suite_def % (fname, idx + 1) + if test_ok_def_re.match(line): + tests += 1 + test_set.add(line[3:].split('(')[0].strip()) + if len(test_set) != tests: + assert False, message_duplicate_test % (fname, idx + 1) + if line.endswith((" \n", "\t\n")): + assert False, message_space % (fname, idx + 1) + if line.endswith("\r\n"): + assert False, message_carriage % (fname, idx + 1) + if tab_in_leading(line): + assert False, message_tabs % (fname, idx + 1) + if str_raise_re.search(line): + assert False, message_str_raise % (fname, idx + 1) + if gen_raise_re.search(line): + assert False, message_gen_raise % (fname, idx + 1) + if (implicit_test_re.search(line) and + not list(filter(lambda ex: ex in fname, import_exclude))): + assert False, message_implicit % (fname, idx + 1) + if func_is_re.search(line) and not test_file_re.search(fname): + assert False, message_func_is % (fname, idx + 1) + + result = old_raise_re.search(line) + + if result is not None: + assert False, message_old_raise % ( + fname, idx + 1, result.group(2)) + + if line is not None: + if line == '\n' and idx > 0: + assert False, message_multi_eof % (fname, idx + 1) + elif not line.endswith('\n'): + # eof newline check + assert False, message_eof % (fname, idx + 1) + + + # Files to test at top level + top_level_files = [join(TOP_PATH, file) for file in [ + "isympy.py", + "build.py", + "setup.py", + ]] + # Files to exclude from all tests + exclude = { + "%(sep)ssympy%(sep)sparsing%(sep)sautolev%(sep)s_antlr%(sep)sautolevparser.py" % sepd, + "%(sep)ssympy%(sep)sparsing%(sep)sautolev%(sep)s_antlr%(sep)sautolevlexer.py" % sepd, + "%(sep)ssympy%(sep)sparsing%(sep)sautolev%(sep)s_antlr%(sep)sautolevlistener.py" % sepd, + "%(sep)ssympy%(sep)sparsing%(sep)slatex%(sep)s_antlr%(sep)slatexparser.py" % sepd, + "%(sep)ssympy%(sep)sparsing%(sep)slatex%(sep)s_antlr%(sep)slatexlexer.py" % sepd, + } + # Files to exclude from the implicit import test + import_exclude = { + # glob imports are allowed in top-level __init__.py: + "%(sep)ssympy%(sep)s__init__.py" % sepd, + # these __init__.py should be fixed: + # XXX: not really, they use useful import pattern (DRY) + "%(sep)svector%(sep)s__init__.py" % sepd, + "%(sep)smechanics%(sep)s__init__.py" % sepd, + "%(sep)squantum%(sep)s__init__.py" % sepd, + "%(sep)spolys%(sep)s__init__.py" % sepd, + "%(sep)spolys%(sep)sdomains%(sep)s__init__.py" % sepd, + # interactive SymPy executes ``from sympy import *``: + "%(sep)sinteractive%(sep)ssession.py" % sepd, + # isympy.py executes ``from sympy import *``: + "%(sep)sisympy.py" % sepd, + # these two are import timing tests: + "%(sep)sbin%(sep)ssympy_time.py" % sepd, + "%(sep)sbin%(sep)ssympy_time_cache.py" % sepd, + # Taken from Python stdlib: + "%(sep)sparsing%(sep)ssympy_tokenize.py" % sepd, + # this one should be fixed: + "%(sep)splotting%(sep)spygletplot%(sep)s" % sepd, + # False positive in the docstring + "%(sep)sbin%(sep)stest_external_imports.py" % sepd, + "%(sep)sbin%(sep)stest_submodule_imports.py" % sepd, + # These are deprecated stubs that can be removed at some point: + "%(sep)sutilities%(sep)sruntests.py" % sepd, + "%(sep)sutilities%(sep)spytest.py" % sepd, + "%(sep)sutilities%(sep)srandtest.py" % sepd, + "%(sep)sutilities%(sep)stmpfiles.py" % sepd, + "%(sep)sutilities%(sep)squality_unicode.py" % sepd, + } + check_files(top_level_files, test) + check_directory_tree(BIN_PATH, test, {"~", ".pyc", ".sh"}, "*") + check_directory_tree(SYMPY_PATH, test, exclude) + check_directory_tree(EXAMPLES_PATH, test, exclude) + + +def _with_space(c): + # return c with a random amount of leading space + return random.randint(0, 10)*' ' + c + + +def test_raise_statement_regular_expression(): + candidates_ok = [ + "some text # raise Exception, 'text'", + "raise ValueError('text') # raise Exception, 'text'", + "raise ValueError('text')", + "raise ValueError", + "raise ValueError('text')", + "raise ValueError('text') #,", + # Talking about an exception in a docstring + ''''"""This function will raise ValueError, except when it doesn't"""''', + "raise (ValueError('text')", + ] + str_candidates_fail = [ + "raise 'exception'", + "raise 'Exception'", + 'raise "exception"', + 'raise "Exception"', + "raise 'ValueError'", + ] + gen_candidates_fail = [ + "raise Exception('text') # raise Exception, 'text'", + "raise Exception('text')", + "raise Exception", + "raise Exception('text')", + "raise Exception('text') #,", + "raise Exception, 'text'", + "raise Exception, 'text' # raise Exception('text')", + "raise Exception, 'text' # raise Exception, 'text'", + ">>> raise Exception, 'text'", + ">>> raise Exception, 'text' # raise Exception('text')", + ">>> raise Exception, 'text' # raise Exception, 'text'", + ] + old_candidates_fail = [ + "raise Exception, 'text'", + "raise Exception, 'text' # raise Exception('text')", + "raise Exception, 'text' # raise Exception, 'text'", + ">>> raise Exception, 'text'", + ">>> raise Exception, 'text' # raise Exception('text')", + ">>> raise Exception, 'text' # raise Exception, 'text'", + "raise ValueError, 'text'", + "raise ValueError, 'text' # raise Exception('text')", + "raise ValueError, 'text' # raise Exception, 'text'", + ">>> raise ValueError, 'text'", + ">>> raise ValueError, 'text' # raise Exception('text')", + ">>> raise ValueError, 'text' # raise Exception, 'text'", + "raise(ValueError,", + "raise (ValueError,", + "raise( ValueError,", + "raise ( ValueError,", + "raise(ValueError ,", + "raise (ValueError ,", + "raise( ValueError ,", + "raise ( ValueError ,", + ] + + for c in candidates_ok: + assert str_raise_re.search(_with_space(c)) is None, c + assert gen_raise_re.search(_with_space(c)) is None, c + assert old_raise_re.search(_with_space(c)) is None, c + for c in str_candidates_fail: + assert str_raise_re.search(_with_space(c)) is not None, c + for c in gen_candidates_fail: + assert gen_raise_re.search(_with_space(c)) is not None, c + for c in old_candidates_fail: + assert old_raise_re.search(_with_space(c)) is not None, c + + +def test_implicit_imports_regular_expression(): + candidates_ok = [ + "from sympy import something", + ">>> from sympy import something", + "from sympy.somewhere import something", + ">>> from sympy.somewhere import something", + "import sympy", + ">>> import sympy", + "import sympy.something.something", + "... import sympy", + "... import sympy.something.something", + "... from sympy import something", + "... from sympy.somewhere import something", + ">> from sympy import *", # To allow 'fake' docstrings + "# from sympy import *", + "some text # from sympy import *", + ] + candidates_fail = [ + "from sympy import *", + ">>> from sympy import *", + "from sympy.somewhere import *", + ">>> from sympy.somewhere import *", + "... from sympy import *", + "... from sympy.somewhere import *", + ] + for c in candidates_ok: + assert implicit_test_re.search(_with_space(c)) is None, c + for c in candidates_fail: + assert implicit_test_re.search(_with_space(c)) is not None, c + + +def test_test_suite_defs(): + candidates_ok = [ + " def foo():\n", + "def foo(arg):\n", + "def _foo():\n", + "def test_foo():\n", + ] + candidates_fail = [ + "def foo():\n", + "def foo() :\n", + "def foo( ):\n", + "def foo():\n", + ] + for c in candidates_ok: + assert test_suite_def_re.search(c) is None, c + for c in candidates_fail: + assert test_suite_def_re.search(c) is not None, c + + +def test_test_duplicate_defs(): + candidates_ok = [ + "def foo():\ndef foo():\n", + "def test():\ndef test_():\n", + "def test_():\ndef test__():\n", + ] + candidates_fail = [ + "def test_():\ndef test_ ():\n", + "def test_1():\ndef test_1():\n", + ] + ok = (None, 'check') + def check(file): + tests = 0 + test_set = set() + for idx, line in enumerate(file.splitlines()): + if test_ok_def_re.match(line): + tests += 1 + test_set.add(line[3:].split('(')[0].strip()) + if len(test_set) != tests: + return False, message_duplicate_test % ('check', idx + 1) + return None, 'check' + for c in candidates_ok: + assert check(c) == ok + for c in candidates_fail: + assert check(c) != ok + + +def test_find_self_assignments(): + candidates_ok = [ + "class A(object):\n def foo(self, arg): arg = self\n", + "class A(object):\n def foo(self, arg): self.prop = arg\n", + "class A(object):\n def foo(self, arg): obj, obj2 = arg, self\n", + "class A(object):\n @classmethod\n def bar(cls, arg): arg = cls\n", + "class A(object):\n def foo(var, arg): arg = var\n", + ] + candidates_fail = [ + "class A(object):\n def foo(self, arg): self = arg\n", + "class A(object):\n def foo(self, arg): obj, self = arg, arg\n", + "class A(object):\n def foo(self, arg):\n if arg: self = arg", + "class A(object):\n @classmethod\n def foo(cls, arg): cls = arg\n", + "class A(object):\n def foo(var, arg): var = arg\n", + ] + + for c in candidates_ok: + assert find_self_assignments(c) == [] + for c in candidates_fail: + assert find_self_assignments(c) != [] + + +def test_test_unicode_encoding(): + unicode_whitelist = ['foo'] + unicode_strict_whitelist = ['bar'] + + fname = 'abc' + test_file = ['α'] + raises(AssertionError, lambda: _test_this_file_encoding( + fname, test_file, unicode_whitelist, unicode_strict_whitelist)) + + fname = 'abc' + test_file = ['abc'] + _test_this_file_encoding( + fname, test_file, unicode_whitelist, unicode_strict_whitelist) + + fname = 'foo' + test_file = ['abc'] + raises(AssertionError, lambda: _test_this_file_encoding( + fname, test_file, unicode_whitelist, unicode_strict_whitelist)) + + fname = 'bar' + test_file = ['abc'] + _test_this_file_encoding( + fname, test_file, unicode_whitelist, unicode_strict_whitelist) diff --git a/.venv/lib/python3.12/site-packages/sympy/testing/tests/test_deprecated.py b/.venv/lib/python3.12/site-packages/sympy/testing/tests/test_deprecated.py new file mode 100644 index 0000000000000000000000000000000000000000..696933d96d6232ea869da1002ec9ebee5309724d --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/testing/tests/test_deprecated.py @@ -0,0 +1,5 @@ +from sympy.testing.pytest import warns_deprecated_sympy + +def test_deprecated_testing_randtest(): + with warns_deprecated_sympy(): + import sympy.testing.randtest # noqa:F401 diff --git a/.venv/lib/python3.12/site-packages/sympy/testing/tests/test_module_imports.py b/.venv/lib/python3.12/site-packages/sympy/testing/tests/test_module_imports.py new file mode 100644 index 0000000000000000000000000000000000000000..d16dbaa98156c287c18b46ff07c0ede5d26e069a --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/testing/tests/test_module_imports.py @@ -0,0 +1,42 @@ +""" +Checks that SymPy does not contain indirect imports. + +An indirect import is importing a symbol from a module that itself imported the +symbol from elsewhere. Such a constellation makes it harder to diagnose +inter-module dependencies and import order problems, and is therefore strongly +discouraged. + +(Indirect imports from end-user code is fine and in fact a best practice.) + +Implementation note: Forcing Python into actually unloading already-imported +submodules is a tricky and partly undocumented process. To avoid these issues, +the actual diagnostic code is in bin/diagnose_imports, which is run as a +separate, pristine Python process. +""" + +import subprocess +import sys +from os.path import abspath, dirname, join, normpath +import inspect + +from sympy.testing.pytest import XFAIL + +@XFAIL +def test_module_imports_are_direct(): + my_filename = abspath(inspect.getfile(inspect.currentframe())) + my_dirname = dirname(my_filename) + diagnose_imports_filename = join(my_dirname, 'diagnose_imports.py') + diagnose_imports_filename = normpath(diagnose_imports_filename) + + process = subprocess.Popen( + [ + sys.executable, + normpath(diagnose_imports_filename), + '--problems', + '--by-importer' + ], + stdout=subprocess.PIPE, + stderr=subprocess.STDOUT, + bufsize=-1) + output, _ = process.communicate() + assert output == '', "There are import problems:\n" + output.decode() diff --git a/.venv/lib/python3.12/site-packages/sympy/testing/tests/test_pytest.py b/.venv/lib/python3.12/site-packages/sympy/testing/tests/test_pytest.py new file mode 100644 index 0000000000000000000000000000000000000000..7080a4ed4602707a3efb4d9025e8e9cbe23a5ef8 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/testing/tests/test_pytest.py @@ -0,0 +1,211 @@ +import warnings + +from sympy.testing.pytest import (raises, warns, ignore_warnings, + warns_deprecated_sympy, Failed) +from sympy.utilities.exceptions import sympy_deprecation_warning + + + +# Test callables + + +def test_expected_exception_is_silent_callable(): + def f(): + raise ValueError() + raises(ValueError, f) + + +# Under pytest raises will raise Failed rather than AssertionError +def test_lack_of_exception_triggers_AssertionError_callable(): + try: + raises(Exception, lambda: 1 + 1) + assert False + except Failed as e: + assert "DID NOT RAISE" in str(e) + + +def test_unexpected_exception_is_passed_through_callable(): + def f(): + raise ValueError("some error message") + try: + raises(TypeError, f) + assert False + except ValueError as e: + assert str(e) == "some error message" + +# Test with statement + +def test_expected_exception_is_silent_with(): + with raises(ValueError): + raise ValueError() + + +def test_lack_of_exception_triggers_AssertionError_with(): + try: + with raises(Exception): + 1 + 1 + assert False + except Failed as e: + assert "DID NOT RAISE" in str(e) + + +def test_unexpected_exception_is_passed_through_with(): + try: + with raises(TypeError): + raise ValueError("some error message") + assert False + except ValueError as e: + assert str(e) == "some error message" + +# Now we can use raises() instead of try/catch +# to test that a specific exception class is raised + + +def test_second_argument_should_be_callable_or_string(): + raises(TypeError, lambda: raises("irrelevant", 42)) + + +def test_warns_catches_warning(): + with warnings.catch_warnings(record=True) as w: + with warns(UserWarning): + warnings.warn('this is the warning message') + assert len(w) == 0 + + +def test_warns_raises_without_warning(): + with raises(Failed): + with warns(UserWarning): + pass + + +def test_warns_hides_other_warnings(): + with raises(RuntimeWarning): + with warns(UserWarning): + warnings.warn('this is the warning message', UserWarning) + warnings.warn('this is the other message', RuntimeWarning) + + +def test_warns_continues_after_warning(): + with warnings.catch_warnings(record=True) as w: + finished = False + with warns(UserWarning): + warnings.warn('this is the warning message') + finished = True + assert finished + assert len(w) == 0 + + +def test_warns_many_warnings(): + with warns(UserWarning): + warnings.warn('this is the warning message', UserWarning) + warnings.warn('this is the other warning message', UserWarning) + + +def test_warns_match_matching(): + with warnings.catch_warnings(record=True) as w: + with warns(UserWarning, match='this is the warning message'): + warnings.warn('this is the warning message', UserWarning) + assert len(w) == 0 + + +def test_warns_match_non_matching(): + with warnings.catch_warnings(record=True) as w: + with raises(Failed): + with warns(UserWarning, match='this is the warning message'): + warnings.warn('this is not the expected warning message', UserWarning) + assert len(w) == 0 + +def _warn_sympy_deprecation(stacklevel=3): + sympy_deprecation_warning( + "feature", + active_deprecations_target="active-deprecations", + deprecated_since_version="0.0.0", + stacklevel=stacklevel, + ) + +def test_warns_deprecated_sympy_catches_warning(): + with warnings.catch_warnings(record=True) as w: + with warns_deprecated_sympy(): + _warn_sympy_deprecation() + assert len(w) == 0 + + +def test_warns_deprecated_sympy_raises_without_warning(): + with raises(Failed): + with warns_deprecated_sympy(): + pass + +def test_warns_deprecated_sympy_wrong_stacklevel(): + with raises(Failed): + with warns_deprecated_sympy(): + _warn_sympy_deprecation(stacklevel=1) + +def test_warns_deprecated_sympy_doesnt_hide_other_warnings(): + # Unlike pytest's deprecated_call, we should not hide other warnings. + with raises(RuntimeWarning): + with warns_deprecated_sympy(): + _warn_sympy_deprecation() + warnings.warn('this is the other message', RuntimeWarning) + + +def test_warns_deprecated_sympy_continues_after_warning(): + with warnings.catch_warnings(record=True) as w: + finished = False + with warns_deprecated_sympy(): + _warn_sympy_deprecation() + finished = True + assert finished + assert len(w) == 0 + +def test_ignore_ignores_warning(): + with warnings.catch_warnings(record=True) as w: + with ignore_warnings(UserWarning): + warnings.warn('this is the warning message') + assert len(w) == 0 + + +def test_ignore_does_not_raise_without_warning(): + with warnings.catch_warnings(record=True) as w: + with ignore_warnings(UserWarning): + pass + assert len(w) == 0 + + +def test_ignore_allows_other_warnings(): + with warnings.catch_warnings(record=True) as w: + # This is needed when pytest is run as -Werror + # the setting is reverted at the end of the catch_Warnings block. + warnings.simplefilter("always") + with ignore_warnings(UserWarning): + warnings.warn('this is the warning message', UserWarning) + warnings.warn('this is the other message', RuntimeWarning) + assert len(w) == 1 + assert isinstance(w[0].message, RuntimeWarning) + assert str(w[0].message) == 'this is the other message' + + +def test_ignore_continues_after_warning(): + with warnings.catch_warnings(record=True) as w: + finished = False + with ignore_warnings(UserWarning): + warnings.warn('this is the warning message') + finished = True + assert finished + assert len(w) == 0 + + +def test_ignore_many_warnings(): + with warnings.catch_warnings(record=True) as w: + # This is needed when pytest is run as -Werror + # the setting is reverted at the end of the catch_Warnings block. + warnings.simplefilter("always") + with ignore_warnings(UserWarning): + warnings.warn('this is the warning message', UserWarning) + warnings.warn('this is the other message', RuntimeWarning) + warnings.warn('this is the warning message', UserWarning) + warnings.warn('this is the other message', RuntimeWarning) + warnings.warn('this is the other message', RuntimeWarning) + assert len(w) == 3 + for wi in w: + assert isinstance(wi.message, RuntimeWarning) + assert str(wi.message) == 'this is the other message' diff --git a/.venv/lib/python3.12/site-packages/sympy/testing/tests/test_runtests_pytest.py b/.venv/lib/python3.12/site-packages/sympy/testing/tests/test_runtests_pytest.py new file mode 100644 index 0000000000000000000000000000000000000000..cd56d831f3618c9dbb8a1dffe63d95a0befc6adf --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/testing/tests/test_runtests_pytest.py @@ -0,0 +1,171 @@ +import pathlib +from typing import List + +import pytest + +from sympy.testing.runtests_pytest import ( + make_absolute_path, + sympy_dir, + update_args_with_paths, +) + + +class TestMakeAbsolutePath: + + @staticmethod + @pytest.mark.parametrize( + 'partial_path', ['sympy', 'sympy/core', 'sympy/nonexistant_directory'], + ) + def test_valid_partial_path(partial_path: str): + """Paths that start with `sympy` are valid.""" + _ = make_absolute_path(partial_path) + + @staticmethod + @pytest.mark.parametrize( + 'partial_path', ['not_sympy', 'also/not/sympy'], + ) + def test_invalid_partial_path_raises_value_error(partial_path: str): + """A `ValueError` is raises on paths that don't start with `sympy`.""" + with pytest.raises(ValueError): + _ = make_absolute_path(partial_path) + + +class TestUpdateArgsWithPaths: + + @staticmethod + def test_no_paths(): + """If no paths are passed, only `sympy` and `doc/src` are appended. + + `sympy` and `doc/src` are the `testpaths` stated in `pytest.ini`. They + need to be manually added as if any path-related arguments are passed + to `pytest.main` then the settings in `pytest.ini` may be ignored. + + """ + paths = [] + args = update_args_with_paths(paths=paths, keywords=None, args=[]) + expected = [ + str(pathlib.Path(sympy_dir(), 'sympy')), + str(pathlib.Path(sympy_dir(), 'doc/src')), + ] + assert args == expected + + @staticmethod + @pytest.mark.parametrize( + 'path', + ['sympy/core/tests/test_basic.py', '_basic'] + ) + def test_one_file(path: str): + """Single files/paths, full or partial, are matched correctly.""" + args = update_args_with_paths(paths=[path], keywords=None, args=[]) + expected = [ + str(pathlib.Path(sympy_dir(), 'sympy/core/tests/test_basic.py')), + ] + assert args == expected + + @staticmethod + def test_partial_path_from_root(): + """Partial paths from the root directly are matched correctly.""" + args = update_args_with_paths(paths=['sympy/functions'], keywords=None, args=[]) + expected = [str(pathlib.Path(sympy_dir(), 'sympy/functions'))] + assert args == expected + + @staticmethod + def test_multiple_paths_from_root(): + """Multiple paths, partial or full, are matched correctly.""" + paths = ['sympy/core/tests/test_basic.py', 'sympy/functions'] + args = update_args_with_paths(paths=paths, keywords=None, args=[]) + expected = [ + str(pathlib.Path(sympy_dir(), 'sympy/core/tests/test_basic.py')), + str(pathlib.Path(sympy_dir(), 'sympy/functions')), + ] + assert args == expected + + @staticmethod + @pytest.mark.parametrize( + 'paths, expected_paths', + [ + ( + ['/core', '/util'], + [ + 'doc/src/modules/utilities', + 'doc/src/reference/public/utilities', + 'sympy/core', + 'sympy/logic/utilities', + 'sympy/utilities', + ] + ), + ] + ) + def test_multiple_paths_from_non_root(paths: List[str], expected_paths: List[str]): + """Multiple partial paths are matched correctly.""" + args = update_args_with_paths(paths=paths, keywords=None, args=[]) + assert len(args) == len(expected_paths) + for arg, expected in zip(sorted(args), expected_paths): + assert expected in arg + + @staticmethod + @pytest.mark.parametrize( + 'paths', + [ + + [], + ['sympy/physics'], + ['sympy/physics/mechanics'], + ['sympy/physics/mechanics/tests'], + ['sympy/physics/mechanics/tests/test_kane3.py'], + ] + ) + def test_string_as_keyword(paths: List[str]): + """String keywords are matched correctly.""" + keywords = ('bicycle', ) + args = update_args_with_paths(paths=paths, keywords=keywords, args=[]) + expected_args = ['sympy/physics/mechanics/tests/test_kane3.py::test_bicycle'] + assert len(args) == len(expected_args) + for arg, expected in zip(sorted(args), expected_args): + assert expected in arg + + @staticmethod + @pytest.mark.parametrize( + 'paths', + [ + + [], + ['sympy/core'], + ['sympy/core/tests'], + ['sympy/core/tests/test_sympify.py'], + ] + ) + def test_integer_as_keyword(paths: List[str]): + """Integer keywords are matched correctly.""" + keywords = ('3538', ) + args = update_args_with_paths(paths=paths, keywords=keywords, args=[]) + expected_args = ['sympy/core/tests/test_sympify.py::test_issue_3538'] + assert len(args) == len(expected_args) + for arg, expected in zip(sorted(args), expected_args): + assert expected in arg + + @staticmethod + def test_multiple_keywords(): + """Multiple keywords are matched correctly.""" + keywords = ('bicycle', '3538') + args = update_args_with_paths(paths=[], keywords=keywords, args=[]) + expected_args = [ + 'sympy/core/tests/test_sympify.py::test_issue_3538', + 'sympy/physics/mechanics/tests/test_kane3.py::test_bicycle', + ] + assert len(args) == len(expected_args) + for arg, expected in zip(sorted(args), expected_args): + assert expected in arg + + @staticmethod + def test_keyword_match_in_multiple_files(): + """Keywords are matched across multiple files.""" + keywords = ('1130', ) + args = update_args_with_paths(paths=[], keywords=keywords, args=[]) + expected_args = [ + 'sympy/integrals/tests/test_heurisch.py::test_heurisch_symbolic_coeffs_1130', + 'sympy/utilities/tests/test_lambdify.py::test_python_div_zero_issue_11306', + ] + assert len(args) == len(expected_args) + for arg, expected in zip(sorted(args), expected_args): + assert expected in arg diff --git a/.venv/lib/python3.12/site-packages/sympy/testing/tmpfiles.py b/.venv/lib/python3.12/site-packages/sympy/testing/tmpfiles.py new file mode 100644 index 0000000000000000000000000000000000000000..1d5c69cb58aa11f77679855f3df21f03a10d3b2b --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/testing/tmpfiles.py @@ -0,0 +1,46 @@ +""" +This module adds context manager for temporary files generated by the tests. +""" + +import shutil +import os + + +class TmpFileManager: + """ + A class to track record of every temporary files created by the tests. + """ + tmp_files = set('') + tmp_folders = set('') + + @classmethod + def tmp_file(cls, name=''): + cls.tmp_files.add(name) + return name + + @classmethod + def tmp_folder(cls, name=''): + cls.tmp_folders.add(name) + return name + + @classmethod + def cleanup(cls): + while cls.tmp_files: + file = cls.tmp_files.pop() + if os.path.isfile(file): + os.remove(file) + while cls.tmp_folders: + folder = cls.tmp_folders.pop() + shutil.rmtree(folder) + +def cleanup_tmp_files(test_func): + """ + A decorator to help test codes remove temporary files after the tests. + """ + def wrapper_function(): + try: + test_func() + finally: + TmpFileManager.cleanup() + + return wrapper_function diff --git a/.venv/lib/python3.12/site-packages/sympy/this.py b/.venv/lib/python3.12/site-packages/sympy/this.py new file mode 100644 index 0000000000000000000000000000000000000000..c2a5504ac42c7d57790d65524b066a6df86ab539 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/this.py @@ -0,0 +1,21 @@ +""" +The Zen of SymPy. +""" + +s = """The Zen of SymPy + +Unevaluated is better than evaluated. +The user interface matters. +Printing matters. +Pure Python can be fast enough. +If it's too slow, it's (probably) your fault. +Documentation matters. +Correctness is more important than speed. +Push it in now and improve upon it later. +Coverage by testing matters. +Smart tests are better than random tests. +But random tests sometimes find what your smartest test missed. +The Python way is probably the right way. +Community is more important than code.""" + +print(s) diff --git a/.venv/lib/python3.12/site-packages/sympy/unify/__init__.py b/.venv/lib/python3.12/site-packages/sympy/unify/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..5c166f9163785f4aa5744324eb817bef79b33525 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/unify/__init__.py @@ -0,0 +1,15 @@ +""" Unification in SymPy + +See sympy.unify.core docstring for algorithmic details + +See http://matthewrocklin.com/blog/work/2012/11/01/Unification/ for discussion +""" + +from .usympy import unify, rebuild +from .rewrite import rewriterule + +__all__ = [ + 'unify', 'rebuild', + + 'rewriterule', +] diff --git a/.venv/lib/python3.12/site-packages/sympy/unify/__pycache__/__init__.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/unify/__pycache__/__init__.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..90198261e30d02cc4652ebf36833642a468c928f Binary files /dev/null and b/.venv/lib/python3.12/site-packages/sympy/unify/__pycache__/__init__.cpython-312.pyc differ diff --git a/.venv/lib/python3.12/site-packages/sympy/unify/__pycache__/core.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/unify/__pycache__/core.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..4c78ae82054fb2815cb9075dd24bb1eecf9c341a Binary files /dev/null and b/.venv/lib/python3.12/site-packages/sympy/unify/__pycache__/core.cpython-312.pyc differ diff --git a/.venv/lib/python3.12/site-packages/sympy/unify/__pycache__/rewrite.cpython-312.pyc 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b/.venv/lib/python3.12/site-packages/sympy/unify/core.py @@ -0,0 +1,234 @@ +""" Generic Unification algorithm for expression trees with lists of children + +This implementation is a direct translation of + +Artificial Intelligence: A Modern Approach by Stuart Russel and Peter Norvig +Second edition, section 9.2, page 276 + +It is modified in the following ways: + +1. We allow associative and commutative Compound expressions. This results in + combinatorial blowup. +2. We explore the tree lazily. +3. We provide generic interfaces to symbolic algebra libraries in Python. + +A more traditional version can be found here +http://aima.cs.berkeley.edu/python/logic.html +""" + +from sympy.utilities.iterables import kbins + +class Compound: + """ A little class to represent an interior node in the tree + + This is analogous to SymPy.Basic for non-Atoms + """ + def __init__(self, op, args): + self.op = op + self.args = args + + def __eq__(self, other): + return (type(self) is type(other) and self.op == other.op and + self.args == other.args) + + def __hash__(self): + return hash((type(self), self.op, self.args)) + + def __str__(self): + return "%s[%s]" % (str(self.op), ', '.join(map(str, self.args))) + +class Variable: + """ A Wild token """ + def __init__(self, arg): + self.arg = arg + + def __eq__(self, other): + return type(self) is type(other) and self.arg == other.arg + + def __hash__(self): + return hash((type(self), self.arg)) + + def __str__(self): + return "Variable(%s)" % str(self.arg) + +class CondVariable: + """ A wild token that matches conditionally. + + arg - a wild token. + valid - an additional constraining function on a match. + """ + def __init__(self, arg, valid): + self.arg = arg + self.valid = valid + + def __eq__(self, other): + return (type(self) is type(other) and + self.arg == other.arg and + self.valid == other.valid) + + def __hash__(self): + return hash((type(self), self.arg, self.valid)) + + def __str__(self): + return "CondVariable(%s)" % str(self.arg) + +def unify(x, y, s=None, **fns): + """ Unify two expressions. + + Parameters + ========== + + x, y - expression trees containing leaves, Compounds and Variables. + s - a mapping of variables to subtrees. + + Returns + ======= + + lazy sequence of mappings {Variable: subtree} + + Examples + ======== + + >>> from sympy.unify.core import unify, Compound, Variable + >>> expr = Compound("Add", ("x", "y")) + >>> pattern = Compound("Add", ("x", Variable("a"))) + >>> next(unify(expr, pattern, {})) + {Variable(a): 'y'} + """ + s = s or {} + + if x == y: + yield s + elif isinstance(x, (Variable, CondVariable)): + yield from unify_var(x, y, s, **fns) + elif isinstance(y, (Variable, CondVariable)): + yield from unify_var(y, x, s, **fns) + elif isinstance(x, Compound) and isinstance(y, Compound): + is_commutative = fns.get('is_commutative', lambda x: False) + is_associative = fns.get('is_associative', lambda x: False) + for sop in unify(x.op, y.op, s, **fns): + if is_associative(x) and is_associative(y): + a, b = (x, y) if len(x.args) < len(y.args) else (y, x) + if is_commutative(x) and is_commutative(y): + combs = allcombinations(a.args, b.args, 'commutative') + else: + combs = allcombinations(a.args, b.args, 'associative') + for aaargs, bbargs in combs: + aa = [unpack(Compound(a.op, arg)) for arg in aaargs] + bb = [unpack(Compound(b.op, arg)) for arg in bbargs] + yield from unify(aa, bb, sop, **fns) + elif len(x.args) == len(y.args): + yield from unify(x.args, y.args, sop, **fns) + + elif is_args(x) and is_args(y) and len(x) == len(y): + if len(x) == 0: + yield s + else: + for shead in unify(x[0], y[0], s, **fns): + yield from unify(x[1:], y[1:], shead, **fns) + +def unify_var(var, x, s, **fns): + if var in s: + yield from unify(s[var], x, s, **fns) + elif occur_check(var, x): + pass + elif isinstance(var, CondVariable) and var.valid(x): + yield assoc(s, var, x) + elif isinstance(var, Variable): + yield assoc(s, var, x) + +def occur_check(var, x): + """ var occurs in subtree owned by x? """ + if var == x: + return True + elif isinstance(x, Compound): + return occur_check(var, x.args) + elif is_args(x): + if any(occur_check(var, xi) for xi in x): return True + return False + +def assoc(d, key, val): + """ Return copy of d with key associated to val """ + d = d.copy() + d[key] = val + return d + +def is_args(x): + """ Is x a traditional iterable? """ + return type(x) in (tuple, list, set) + +def unpack(x): + if isinstance(x, Compound) and len(x.args) == 1: + return x.args[0] + else: + return x + +def allcombinations(A, B, ordered): + """ + Restructure A and B to have the same number of elements. + + Parameters + ========== + + ordered must be either 'commutative' or 'associative'. + + A and B can be rearranged so that the larger of the two lists is + reorganized into smaller sublists. + + Examples + ======== + + >>> from sympy.unify.core import allcombinations + >>> for x in allcombinations((1, 2, 3), (5, 6), 'associative'): print(x) + (((1,), (2, 3)), ((5,), (6,))) + (((1, 2), (3,)), ((5,), (6,))) + + >>> for x in allcombinations((1, 2, 3), (5, 6), 'commutative'): print(x) + (((1,), (2, 3)), ((5,), (6,))) + (((1, 2), (3,)), ((5,), (6,))) + (((1,), (3, 2)), ((5,), (6,))) + (((1, 3), (2,)), ((5,), (6,))) + (((2,), (1, 3)), ((5,), (6,))) + (((2, 1), (3,)), ((5,), (6,))) + (((2,), (3, 1)), ((5,), (6,))) + (((2, 3), (1,)), ((5,), (6,))) + (((3,), (1, 2)), ((5,), (6,))) + (((3, 1), (2,)), ((5,), (6,))) + (((3,), (2, 1)), ((5,), (6,))) + (((3, 2), (1,)), ((5,), (6,))) + """ + + if ordered == "commutative": + ordered = 11 + if ordered == "associative": + ordered = None + sm, bg = (A, B) if len(A) < len(B) else (B, A) + for part in kbins(list(range(len(bg))), len(sm), ordered=ordered): + if bg == B: + yield tuple((a,) for a in A), partition(B, part) + else: + yield partition(A, part), tuple((b,) for b in B) + +def partition(it, part): + """ Partition a tuple/list into pieces defined by indices. + + Examples + ======== + + >>> from sympy.unify.core import partition + >>> partition((10, 20, 30, 40), [[0, 1, 2], [3]]) + ((10, 20, 30), (40,)) + """ + return type(it)([index(it, ind) for ind in part]) + +def index(it, ind): + """ Fancy indexing into an indexable iterable (tuple, list). + + Examples + ======== + + >>> from sympy.unify.core import index + >>> index([10, 20, 30], (1, 2, 0)) + [20, 30, 10] + """ + return type(it)([it[i] for i in ind]) diff --git a/.venv/lib/python3.12/site-packages/sympy/unify/rewrite.py b/.venv/lib/python3.12/site-packages/sympy/unify/rewrite.py new file mode 100644 index 0000000000000000000000000000000000000000..95a6fa5ffd6a3fde94d17ee845c03bb2b44cf009 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/unify/rewrite.py @@ -0,0 +1,55 @@ +""" Functions to support rewriting of SymPy expressions """ + +from sympy.core.expr import Expr +from sympy.assumptions import ask +from sympy.strategies.tools import subs +from sympy.unify.usympy import rebuild, unify + +def rewriterule(source, target, variables=(), condition=None, assume=None): + """ Rewrite rule. + + Transform expressions that match source into expressions that match target + treating all ``variables`` as wilds. + + Examples + ======== + + >>> from sympy.abc import w, x, y, z + >>> from sympy.unify.rewrite import rewriterule + >>> from sympy import default_sort_key + >>> rl = rewriterule(x + y, x**y, [x, y]) + >>> sorted(rl(z + 3), key=default_sort_key) + [3**z, z**3] + + Use ``condition`` to specify additional requirements. Inputs are taken in + the same order as is found in variables. + + >>> rl = rewriterule(x + y, x**y, [x, y], lambda x, y: x.is_integer) + >>> list(rl(z + 3)) + [3**z] + + Use ``assume`` to specify additional requirements using new assumptions. + + >>> from sympy.assumptions import Q + >>> rl = rewriterule(x + y, x**y, [x, y], assume=Q.integer(x)) + >>> list(rl(z + 3)) + [3**z] + + Assumptions for the local context are provided at rule runtime + + >>> list(rl(w + z, Q.integer(z))) + [z**w] + """ + + def rewrite_rl(expr, assumptions=True): + for match in unify(source, expr, {}, variables=variables): + if (condition and + not condition(*[match.get(var, var) for var in variables])): + continue + if (assume and not ask(assume.xreplace(match), assumptions)): + continue + expr2 = subs(match)(target) + if isinstance(expr2, Expr): + expr2 = rebuild(expr2) + yield expr2 + return rewrite_rl diff --git a/.venv/lib/python3.12/site-packages/sympy/unify/tests/__init__.py 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a/.venv/lib/python3.12/site-packages/sympy/unify/tests/test_rewrite.py b/.venv/lib/python3.12/site-packages/sympy/unify/tests/test_rewrite.py new file mode 100644 index 0000000000000000000000000000000000000000..7b73e2856d5f6380c576220fa2780324df98091a --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/unify/tests/test_rewrite.py @@ -0,0 +1,74 @@ +from sympy.unify.rewrite import rewriterule +from sympy.core.basic import Basic +from sympy.core.singleton import S +from sympy.core.symbol import Symbol +from sympy.functions.elementary.trigonometric import sin +from sympy.abc import x, y +from sympy.strategies.rl import rebuild +from sympy.assumptions import Q + +p, q = Symbol('p'), Symbol('q') + +def test_simple(): + rl = rewriterule(Basic(p, S(1)), Basic(p, S(2)), variables=(p,)) + assert list(rl(Basic(S(3), S(1)))) == [Basic(S(3), S(2))] + + p1 = p**2 + p2 = p**3 + rl = rewriterule(p1, p2, variables=(p,)) + + expr = x**2 + assert list(rl(expr)) == [x**3] + +def test_simple_variables(): + rl = rewriterule(Basic(x, S(1)), Basic(x, S(2)), variables=(x,)) + assert list(rl(Basic(S(3), S(1)))) == [Basic(S(3), S(2))] + + rl = rewriterule(x**2, x**3, variables=(x,)) + assert list(rl(y**2)) == [y**3] + +def test_moderate(): + p1 = p**2 + q**3 + p2 = (p*q)**4 + rl = rewriterule(p1, p2, (p, q)) + + expr = x**2 + y**3 + assert list(rl(expr)) == [(x*y)**4] + +def test_sincos(): + p1 = sin(p)**2 + sin(p)**2 + p2 = 1 + rl = rewriterule(p1, p2, (p, q)) + + assert list(rl(sin(x)**2 + sin(x)**2)) == [1] + assert list(rl(sin(y)**2 + sin(y)**2)) == [1] + +def test_Exprs_ok(): + rl = rewriterule(p+q, q+p, (p, q)) + next(rl(x+y)).is_commutative + str(next(rl(x+y))) + +def test_condition_simple(): + rl = rewriterule(x, x+1, [x], lambda x: x < 10) + assert not list(rl(S(15))) + assert rebuild(next(rl(S(5)))) == 6 + + +def test_condition_multiple(): + rl = rewriterule(x + y, x**y, [x,y], lambda x, y: x.is_integer) + + a = Symbol('a') + b = Symbol('b', integer=True) + expr = a + b + assert list(rl(expr)) == [b**a] + + c = Symbol('c', integer=True) + d = Symbol('d', integer=True) + assert set(rl(c + d)) == {c**d, d**c} + +def test_assumptions(): + rl = rewriterule(x + y, x**y, [x, y], assume=Q.integer(x)) + + a, b = map(Symbol, 'ab') + expr = a + b + assert list(rl(expr, Q.integer(b))) == [b**a] diff --git a/.venv/lib/python3.12/site-packages/sympy/unify/tests/test_sympy.py b/.venv/lib/python3.12/site-packages/sympy/unify/tests/test_sympy.py new file mode 100644 index 0000000000000000000000000000000000000000..eca3933a91abfabdbad96f626e4da761a41b3fd2 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/unify/tests/test_sympy.py @@ -0,0 +1,162 @@ +from sympy.core.add import Add +from sympy.core.basic import Basic +from sympy.core.containers import Tuple +from sympy.core.singleton import S +from sympy.core.symbol import (Symbol, symbols) +from sympy.logic.boolalg import And +from sympy.core.symbol import Str +from sympy.unify.core import Compound, Variable +from sympy.unify.usympy import (deconstruct, construct, unify, is_associative, + is_commutative) +from sympy.abc import x, y, z, n + +def test_deconstruct(): + expr = Basic(S(1), S(2), S(3)) + expected = Compound(Basic, (1, 2, 3)) + assert deconstruct(expr) == expected + + assert deconstruct(1) == 1 + assert deconstruct(x) == x + assert deconstruct(x, variables=(x,)) == Variable(x) + assert deconstruct(Add(1, x, evaluate=False)) == Compound(Add, (1, x)) + assert deconstruct(Add(1, x, evaluate=False), variables=(x,)) == \ + Compound(Add, (1, Variable(x))) + +def test_construct(): + expr = Compound(Basic, (S(1), S(2), S(3))) + expected = Basic(S(1), S(2), S(3)) + assert construct(expr) == expected + +def test_nested(): + expr = Basic(S(1), Basic(S(2)), S(3)) + cmpd = Compound(Basic, (S(1), Compound(Basic, Tuple(2)), S(3))) + assert deconstruct(expr) == cmpd + assert construct(cmpd) == expr + +def test_unify(): + expr = Basic(S(1), S(2), S(3)) + a, b, c = map(Symbol, 'abc') + pattern = Basic(a, b, c) + assert list(unify(expr, pattern, {}, (a, b, c))) == [{a: 1, b: 2, c: 3}] + assert list(unify(expr, pattern, variables=(a, b, c))) == \ + [{a: 1, b: 2, c: 3}] + +def test_unify_variables(): + assert list(unify(Basic(S(1), S(2)), Basic(S(1), x), {}, variables=(x,))) == [{x: 2}] + +def test_s_input(): + expr = Basic(S(1), S(2)) + a, b = map(Symbol, 'ab') + pattern = Basic(a, b) + assert list(unify(expr, pattern, {}, (a, b))) == [{a: 1, b: 2}] + assert list(unify(expr, pattern, {a: 5}, (a, b))) == [] + +def iterdicteq(a, b): + a = tuple(a) + b = tuple(b) + return len(a) == len(b) and all(x in b for x in a) + +def test_unify_commutative(): + expr = Add(1, 2, 3, evaluate=False) + a, b, c = map(Symbol, 'abc') + pattern = Add(a, b, c, evaluate=False) + + result = tuple(unify(expr, pattern, {}, (a, b, c))) + expected = ({a: 1, b: 2, c: 3}, + {a: 1, b: 3, c: 2}, + {a: 2, b: 1, c: 3}, + {a: 2, b: 3, c: 1}, + {a: 3, b: 1, c: 2}, + {a: 3, b: 2, c: 1}) + + assert iterdicteq(result, expected) + +def test_unify_iter(): + expr = Add(1, 2, 3, evaluate=False) + a, b, c = map(Symbol, 'abc') + pattern = Add(a, c, evaluate=False) + assert is_associative(deconstruct(pattern)) + assert is_commutative(deconstruct(pattern)) + + result = list(unify(expr, pattern, {}, (a, c))) + expected = [{a: 1, c: Add(2, 3, evaluate=False)}, + {a: 1, c: Add(3, 2, evaluate=False)}, + {a: 2, c: Add(1, 3, evaluate=False)}, + {a: 2, c: Add(3, 1, evaluate=False)}, + {a: 3, c: Add(1, 2, evaluate=False)}, + {a: 3, c: Add(2, 1, evaluate=False)}, + {a: Add(1, 2, evaluate=False), c: 3}, + {a: Add(2, 1, evaluate=False), c: 3}, + {a: Add(1, 3, evaluate=False), c: 2}, + {a: Add(3, 1, evaluate=False), c: 2}, + {a: Add(2, 3, evaluate=False), c: 1}, + {a: Add(3, 2, evaluate=False), c: 1}] + + assert iterdicteq(result, expected) + +def test_hard_match(): + from sympy.functions.elementary.trigonometric import (cos, sin) + expr = sin(x) + cos(x)**2 + p, q = map(Symbol, 'pq') + pattern = sin(p) + cos(p)**2 + assert list(unify(expr, pattern, {}, (p, q))) == [{p: x}] + +def test_matrix(): + from sympy.matrices.expressions.matexpr import MatrixSymbol + X = MatrixSymbol('X', n, n) + Y = MatrixSymbol('Y', 2, 2) + Z = MatrixSymbol('Z', 2, 3) + assert list(unify(X, Y, {}, variables=[n, Str('X')])) == [{Str('X'): Str('Y'), n: 2}] + assert list(unify(X, Z, {}, variables=[n, Str('X')])) == [] + +def test_non_frankenAdds(): + # the is_commutative property used to fail because of Basic.__new__ + # This caused is_commutative and str calls to fail + expr = x+y*2 + rebuilt = construct(deconstruct(expr)) + # Ensure that we can run these commands without causing an error + str(rebuilt) + rebuilt.is_commutative + +def test_FiniteSet_commutivity(): + from sympy.sets.sets import FiniteSet + a, b, c, x, y = symbols('a,b,c,x,y') + s = FiniteSet(a, b, c) + t = FiniteSet(x, y) + variables = (x, y) + assert {x: FiniteSet(a, c), y: b} in tuple(unify(s, t, variables=variables)) + +def test_FiniteSet_complex(): + from sympy.sets.sets import FiniteSet + a, b, c, x, y, z = symbols('a,b,c,x,y,z') + expr = FiniteSet(Basic(S(1), x), y, Basic(x, z)) + pattern = FiniteSet(a, Basic(x, b)) + variables = a, b + expected = ({b: 1, a: FiniteSet(y, Basic(x, z))}, + {b: z, a: FiniteSet(y, Basic(S(1), x))}) + assert iterdicteq(unify(expr, pattern, variables=variables), expected) + + +def test_and(): + variables = x, y + expected = ({x: z > 0, y: n < 3},) + assert iterdicteq(unify((z>0) & (n<3), And(x, y), variables=variables), + expected) + +def test_Union(): + from sympy.sets.sets import Interval + assert list(unify(Interval(0, 1) + Interval(10, 11), + Interval(0, 1) + Interval(12, 13), + variables=(Interval(12, 13),))) + +def test_is_commutative(): + assert is_commutative(deconstruct(x+y)) + assert is_commutative(deconstruct(x*y)) + assert not is_commutative(deconstruct(x**y)) + +def test_commutative_in_commutative(): + from sympy.abc import a,b,c,d + from sympy.functions.elementary.trigonometric import (cos, sin) + eq = sin(3)*sin(4)*sin(5) + 4*cos(3)*cos(4) + pat = a*cos(b)*cos(c) + d*sin(b)*sin(c) + assert next(unify(eq, pat, variables=(a,b,c,d))) diff --git a/.venv/lib/python3.12/site-packages/sympy/unify/tests/test_unify.py b/.venv/lib/python3.12/site-packages/sympy/unify/tests/test_unify.py new file mode 100644 index 0000000000000000000000000000000000000000..31153242576e1ff55dd3097efbc985aced5d574a --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/unify/tests/test_unify.py @@ -0,0 +1,88 @@ +from sympy.unify.core import Compound, Variable, CondVariable, allcombinations +from sympy.unify import core + +a,b,c = 'a', 'b', 'c' +w,x,y,z = map(Variable, 'wxyz') + +C = Compound + +def is_associative(x): + return isinstance(x, Compound) and (x.op in ('Add', 'Mul', 'CAdd', 'CMul')) +def is_commutative(x): + return isinstance(x, Compound) and (x.op in ('CAdd', 'CMul')) + + +def unify(a, b, s={}): + return core.unify(a, b, s=s, is_associative=is_associative, + is_commutative=is_commutative) + +def test_basic(): + assert list(unify(a, x, {})) == [{x: a}] + assert list(unify(a, x, {x: 10})) == [] + assert list(unify(1, x, {})) == [{x: 1}] + assert list(unify(a, a, {})) == [{}] + assert list(unify((w, x), (y, z), {})) == [{w: y, x: z}] + assert list(unify(x, (a, b), {})) == [{x: (a, b)}] + + assert list(unify((a, b), (x, x), {})) == [] + assert list(unify((y, z), (x, x), {}))!= [] + assert list(unify((a, (b, c)), (a, (x, y)), {})) == [{x: b, y: c}] + +def test_ops(): + assert list(unify(C('Add', (a,b,c)), C('Add', (a,x,y)), {})) == \ + [{x:b, y:c}] + assert list(unify(C('Add', (C('Mul', (1,2)), b,c)), C('Add', (x,y,c)), {})) == \ + [{x: C('Mul', (1,2)), y:b}] + +def test_associative(): + c1 = C('Add', (1,2,3)) + c2 = C('Add', (x,y)) + assert tuple(unify(c1, c2, {})) == ({x: 1, y: C('Add', (2, 3))}, + {x: C('Add', (1, 2)), y: 3}) + +def test_commutative(): + c1 = C('CAdd', (1,2,3)) + c2 = C('CAdd', (x,y)) + result = list(unify(c1, c2, {})) + assert {x: 1, y: C('CAdd', (2, 3))} in result + assert ({x: 2, y: C('CAdd', (1, 3))} in result or + {x: 2, y: C('CAdd', (3, 1))} in result) + +def _test_combinations_assoc(): + assert set(allcombinations((1,2,3), (a,b), True)) == \ + {(((1, 2), (3,)), (a, b)), (((1,), (2, 3)), (a, b))} + +def _test_combinations_comm(): + assert set(allcombinations((1,2,3), (a,b), None)) == \ + {(((1,), (2, 3)), ('a', 'b')), (((2,), (3, 1)), ('a', 'b')), + (((3,), (1, 2)), ('a', 'b')), (((1, 2), (3,)), ('a', 'b')), + (((2, 3), (1,)), ('a', 'b')), (((3, 1), (2,)), ('a', 'b'))} + +def test_allcombinations(): + assert set(allcombinations((1,2), (1,2), 'commutative')) ==\ + {(((1,),(2,)), ((1,),(2,))), (((1,),(2,)), ((2,),(1,)))} + + +def test_commutativity(): + c1 = Compound('CAdd', (a, b)) + c2 = Compound('CAdd', (x, y)) + assert is_commutative(c1) and is_commutative(c2) + assert len(list(unify(c1, c2, {}))) == 2 + + +def test_CondVariable(): + expr = C('CAdd', (1, 2)) + x = Variable('x') + y = CondVariable('y', lambda a: a % 2 == 0) + z = CondVariable('z', lambda a: a > 3) + pattern = C('CAdd', (x, y)) + assert list(unify(expr, pattern, {})) == \ + [{x: 1, y: 2}] + + z = CondVariable('z', lambda a: a > 3) + pattern = C('CAdd', (z, y)) + + assert list(unify(expr, pattern, {})) == [] + +def test_defaultdict(): + assert next(unify(Variable('x'), 'foo')) == {Variable('x'): 'foo'} diff --git a/.venv/lib/python3.12/site-packages/sympy/unify/usympy.py b/.venv/lib/python3.12/site-packages/sympy/unify/usympy.py new file mode 100644 index 0000000000000000000000000000000000000000..3942b35ec549e5dbd08a3cf1cad2b2ecea733c7a --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/unify/usympy.py @@ -0,0 +1,124 @@ +""" SymPy interface to Unification engine + +See sympy.unify for module level docstring +See sympy.unify.core for algorithmic docstring """ + +from sympy.core import Basic, Add, Mul, Pow +from sympy.core.operations import AssocOp, LatticeOp +from sympy.matrices import MatAdd, MatMul, MatrixExpr +from sympy.sets.sets import Union, Intersection, FiniteSet +from sympy.unify.core import Compound, Variable, CondVariable +from sympy.unify import core + +basic_new_legal = [MatrixExpr] +eval_false_legal = [AssocOp, Pow, FiniteSet] +illegal = [LatticeOp] + +def sympy_associative(op): + assoc_ops = (AssocOp, MatAdd, MatMul, Union, Intersection, FiniteSet) + return any(issubclass(op, aop) for aop in assoc_ops) + +def sympy_commutative(op): + comm_ops = (Add, MatAdd, Union, Intersection, FiniteSet) + return any(issubclass(op, cop) for cop in comm_ops) + +def is_associative(x): + return isinstance(x, Compound) and sympy_associative(x.op) + +def is_commutative(x): + if not isinstance(x, Compound): + return False + if sympy_commutative(x.op): + return True + if issubclass(x.op, Mul): + return all(construct(arg).is_commutative for arg in x.args) + +def mk_matchtype(typ): + def matchtype(x): + return (isinstance(x, typ) or + isinstance(x, Compound) and issubclass(x.op, typ)) + return matchtype + +def deconstruct(s, variables=()): + """ Turn a SymPy object into a Compound """ + if s in variables: + return Variable(s) + if isinstance(s, (Variable, CondVariable)): + return s + if not isinstance(s, Basic) or s.is_Atom: + return s + return Compound(s.__class__, + tuple(deconstruct(arg, variables) for arg in s.args)) + +def construct(t): + """ Turn a Compound into a SymPy object """ + if isinstance(t, (Variable, CondVariable)): + return t.arg + if not isinstance(t, Compound): + return t + if any(issubclass(t.op, cls) for cls in eval_false_legal): + return t.op(*map(construct, t.args), evaluate=False) + elif any(issubclass(t.op, cls) for cls in basic_new_legal): + return Basic.__new__(t.op, *map(construct, t.args)) + else: + return t.op(*map(construct, t.args)) + +def rebuild(s): + """ Rebuild a SymPy expression. + + This removes harm caused by Expr-Rules interactions. + """ + return construct(deconstruct(s)) + +def unify(x, y, s=None, variables=(), **kwargs): + """ Structural unification of two expressions/patterns. + + Examples + ======== + + >>> from sympy.unify.usympy import unify + >>> from sympy import Basic, S + >>> from sympy.abc import x, y, z, p, q + + >>> next(unify(Basic(S(1), S(2)), Basic(S(1), x), variables=[x])) + {x: 2} + + >>> expr = 2*x + y + z + >>> pattern = 2*p + q + >>> next(unify(expr, pattern, {}, variables=(p, q))) + {p: x, q: y + z} + + Unification supports commutative and associative matching + + >>> expr = x + y + z + >>> pattern = p + q + >>> len(list(unify(expr, pattern, {}, variables=(p, q)))) + 12 + + Symbols not indicated to be variables are treated as literal, + else they are wild-like and match anything in a sub-expression. + + >>> expr = x*y*z + 3 + >>> pattern = x*y + 3 + >>> next(unify(expr, pattern, {}, variables=[x, y])) + {x: y, y: x*z} + + The x and y of the pattern above were in a Mul and matched factors + in the Mul of expr. Here, a single symbol matches an entire term: + + >>> expr = x*y + 3 + >>> pattern = p + 3 + >>> next(unify(expr, pattern, {}, variables=[p])) + {p: x*y} + + """ + decons = lambda x: deconstruct(x, variables) + s = s or {} + s = {decons(k): decons(v) for k, v in s.items()} + + ds = core.unify(decons(x), decons(y), s, + is_associative=is_associative, + is_commutative=is_commutative, + **kwargs) + for d in ds: + yield {construct(k): construct(v) for k, v in d.items()} diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/__init__.py b/.venv/lib/python3.12/site-packages/sympy/utilities/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..8f35da4a84396618a33a12c3c6b5cf58e9d4742c --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/__init__.py @@ -0,0 +1,30 @@ +"""This module contains some general purpose utilities that are used across +SymPy. +""" +from .iterables import (flatten, group, take, subsets, + variations, numbered_symbols, cartes, capture, dict_merge, + prefixes, postfixes, sift, topological_sort, unflatten, + has_dups, has_variety, reshape, rotations) + +from .misc import filldedent + +from .lambdify import lambdify + +from .decorator import threaded, xthreaded, public, memoize_property + +from .timeutils import timed + +__all__ = [ + 'flatten', 'group', 'take', 'subsets', 'variations', 'numbered_symbols', + 'cartes', 'capture', 'dict_merge', 'prefixes', 'postfixes', 'sift', + 'topological_sort', 'unflatten', 'has_dups', 'has_variety', 'reshape', + 'rotations', + + 'filldedent', + + 'lambdify', + + 'threaded', 'xthreaded', 'public', 'memoize_property', + + 'timed', +] diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/__pycache__/__init__.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/utilities/__pycache__/__init__.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..827af5a5dd33b83e37dfac35ed3c3dbedf2810c8 Binary files /dev/null and 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Until then API may break +without prior notice. + +TODO: +- (optionally) clean up after tempfile.mkdtemp() +- cross-platform testing +- caching of compiler choice and intermediate files + +""" + +from .compilation import compile_link_import_strings, compile_run_strings +from .availability import has_fortran, has_c, has_cxx + +__all__ = [ + 'compile_link_import_strings', 'compile_run_strings', + 'has_fortran', 'has_c', 'has_cxx', +] diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/_compilation/__pycache__/__init__.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/utilities/_compilation/__pycache__/__init__.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..37ef40ac6b40059e0b085f9fc9550310f828a07d Binary files /dev/null and b/.venv/lib/python3.12/site-packages/sympy/utilities/_compilation/__pycache__/__init__.cpython-312.pyc differ diff --git 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b/.venv/lib/python3.12/site-packages/sympy/utilities/_compilation/availability.py new file mode 100644 index 0000000000000000000000000000000000000000..dc97b3e7b8c7e7307c6c21352ed4035d977aabb3 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/_compilation/availability.py @@ -0,0 +1,77 @@ +import os +from .compilation import compile_run_strings +from .util import CompilerNotFoundError + +def has_fortran(): + if not hasattr(has_fortran, 'result'): + try: + (stdout, stderr), info = compile_run_strings( + [('main.f90', ( + 'program foo\n' + 'print *, "hello world"\n' + 'end program' + ))], clean=True + ) + except CompilerNotFoundError: + has_fortran.result = False + if os.environ.get('SYMPY_STRICT_COMPILER_CHECKS', '0') == '1': + raise + else: + if info['exit_status'] != os.EX_OK or 'hello world' not in stdout: + if os.environ.get('SYMPY_STRICT_COMPILER_CHECKS', '0') == '1': + raise ValueError("Failed to compile test program:\n%s\n%s\n" % (stdout, stderr)) + has_fortran.result = False + else: + has_fortran.result = True + return has_fortran.result + + +def has_c(): + if not hasattr(has_c, 'result'): + try: + (stdout, stderr), info = compile_run_strings( + [('main.c', ( + '#include \n' + 'int main(){\n' + 'printf("hello world\\n");\n' + 'return 0;\n' + '}' + ))], clean=True + ) + except CompilerNotFoundError: + has_c.result = False + if os.environ.get('SYMPY_STRICT_COMPILER_CHECKS', '0') == '1': + raise + else: + if info['exit_status'] != os.EX_OK or 'hello world' not in stdout: + if os.environ.get('SYMPY_STRICT_COMPILER_CHECKS', '0') == '1': + raise ValueError("Failed to compile test program:\n%s\n%s\n" % (stdout, stderr)) + has_c.result = False + else: + has_c.result = True + return has_c.result + + +def has_cxx(): + if not hasattr(has_cxx, 'result'): + try: + (stdout, stderr), info = compile_run_strings( + [('main.cxx', ( + '#include \n' + 'int main(){\n' + 'std::cout << "hello world" << std::endl;\n' + '}' + ))], clean=True + ) + except CompilerNotFoundError: + has_cxx.result = False + if os.environ.get('SYMPY_STRICT_COMPILER_CHECKS', '0') == '1': + raise + else: + if info['exit_status'] != os.EX_OK or 'hello world' not in stdout: + if os.environ.get('SYMPY_STRICT_COMPILER_CHECKS', '0') == '1': + raise ValueError("Failed to compile test program:\n%s\n%s\n" % (stdout, stderr)) + has_cxx.result = False + else: + has_cxx.result = True + return has_cxx.result diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/_compilation/compilation.py b/.venv/lib/python3.12/site-packages/sympy/utilities/_compilation/compilation.py new file mode 100644 index 0000000000000000000000000000000000000000..2d50a20467c08086d6cb5fb5b0d478e7a953d720 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/_compilation/compilation.py @@ -0,0 +1,657 @@ +import glob +import os +import shutil +import subprocess +import sys +import tempfile +import warnings +from pathlib import Path +from sysconfig import get_config_var, get_config_vars, get_path + +from .runners import ( + CCompilerRunner, + CppCompilerRunner, + FortranCompilerRunner +) +from .util import ( + get_abspath, make_dirs, copy, Glob, ArbitraryDepthGlob, + glob_at_depth, import_module_from_file, pyx_is_cplus, + sha256_of_string, sha256_of_file, CompileError +) + +if os.name == 'posix': + objext = '.o' +elif os.name == 'nt': + objext = '.obj' +else: + warnings.warn("Unknown os.name: {}".format(os.name)) + objext = '.o' + + +def compile_sources(files, Runner=None, destdir=None, cwd=None, keep_dir_struct=False, + per_file_kwargs=None, **kwargs): + """ Compile source code files to object files. + + Parameters + ========== + + files : iterable of str + Paths to source files, if ``cwd`` is given, the paths are taken as relative. + Runner: CompilerRunner subclass (optional) + Could be e.g. ``FortranCompilerRunner``. Will be inferred from filename + extensions if missing. + destdir: str + Output directory, if cwd is given, the path is taken as relative. + cwd: str + Working directory. Specify to have compiler run in other directory. + also used as root of relative paths. + keep_dir_struct: bool + Reproduce directory structure in `destdir`. default: ``False`` + per_file_kwargs: dict + Dict mapping instances in ``files`` to keyword arguments. + \\*\\*kwargs: dict + Default keyword arguments to pass to ``Runner``. + + Returns + ======= + List of strings (paths of object files). + """ + _per_file_kwargs = {} + + if per_file_kwargs is not None: + for k, v in per_file_kwargs.items(): + if isinstance(k, Glob): + for path in glob.glob(k.pathname): + _per_file_kwargs[path] = v + elif isinstance(k, ArbitraryDepthGlob): + for path in glob_at_depth(k.filename, cwd): + _per_file_kwargs[path] = v + else: + _per_file_kwargs[k] = v + + # Set up destination directory + destdir = destdir or '.' + if not os.path.isdir(destdir): + if os.path.exists(destdir): + raise OSError("{} is not a directory".format(destdir)) + else: + make_dirs(destdir) + if cwd is None: + cwd = '.' + for f in files: + copy(f, destdir, only_update=True, dest_is_dir=True) + + # Compile files and return list of paths to the objects + dstpaths = [] + for f in files: + if keep_dir_struct: + name, ext = os.path.splitext(f) + else: + name, ext = os.path.splitext(os.path.basename(f)) + file_kwargs = kwargs.copy() + file_kwargs.update(_per_file_kwargs.get(f, {})) + dstpaths.append(src2obj(f, Runner, cwd=cwd, **file_kwargs)) + return dstpaths + + +def get_mixed_fort_c_linker(vendor=None, cplus=False, cwd=None): + vendor = vendor or os.environ.get('SYMPY_COMPILER_VENDOR', 'gnu') + + if vendor.lower() == 'intel': + if cplus: + return (FortranCompilerRunner, + {'flags': ['-nofor_main', '-cxxlib']}, vendor) + else: + return (FortranCompilerRunner, + {'flags': ['-nofor_main']}, vendor) + elif vendor.lower() == 'gnu' or 'llvm': + if cplus: + return (CppCompilerRunner, + {'lib_options': ['fortran']}, vendor) + else: + return (FortranCompilerRunner, + {}, vendor) + else: + raise ValueError("No vendor found.") + + +def link(obj_files, out_file=None, shared=False, Runner=None, + cwd=None, cplus=False, fort=False, extra_objs=None, **kwargs): + """ Link object files. + + Parameters + ========== + + obj_files: iterable of str + Paths to object files. + out_file: str (optional) + Path to executable/shared library, if ``None`` it will be + deduced from the last item in obj_files. + shared: bool + Generate a shared library? + Runner: CompilerRunner subclass (optional) + If not given the ``cplus`` and ``fort`` flags will be inspected + (fallback is the C compiler). + cwd: str + Path to the root of relative paths and working directory for compiler. + cplus: bool + C++ objects? default: ``False``. + fort: bool + Fortran objects? default: ``False``. + extra_objs: list + List of paths to extra object files / static libraries. + \\*\\*kwargs: dict + Keyword arguments passed to ``Runner``. + + Returns + ======= + + The absolute path to the generated shared object / executable. + + """ + if out_file is None: + out_file, ext = os.path.splitext(os.path.basename(obj_files[-1])) + if shared: + out_file += get_config_var('EXT_SUFFIX') + + if not Runner: + if fort: + Runner, extra_kwargs, vendor = \ + get_mixed_fort_c_linker( + vendor=kwargs.get('vendor', None), + cplus=cplus, + cwd=cwd, + ) + for k, v in extra_kwargs.items(): + if k in kwargs: + kwargs[k].expand(v) + else: + kwargs[k] = v + else: + if cplus: + Runner = CppCompilerRunner + else: + Runner = CCompilerRunner + + flags = kwargs.pop('flags', []) + if shared: + if '-shared' not in flags: + flags.append('-shared') + run_linker = kwargs.pop('run_linker', True) + if not run_linker: + raise ValueError("run_linker was set to False (nonsensical).") + + out_file = get_abspath(out_file, cwd=cwd) + runner = Runner(obj_files+(extra_objs or []), out_file, flags, cwd=cwd, **kwargs) + runner.run() + return out_file + + +def link_py_so(obj_files, so_file=None, cwd=None, libraries=None, + cplus=False, fort=False, extra_objs=None, **kwargs): + """ Link Python extension module (shared object) for importing + + Parameters + ========== + + obj_files: iterable of str + Paths to object files to be linked. + so_file: str + Name (path) of shared object file to create. If not specified it will + have the basname of the last object file in `obj_files` but with the + extension '.so' (Unix). + cwd: path string + Root of relative paths and working directory of linker. + libraries: iterable of strings + Libraries to link against, e.g. ['m']. + cplus: bool + Any C++ objects? default: ``False``. + fort: bool + Any Fortran objects? default: ``False``. + extra_objs: list + List of paths of extra object files / static libraries to link against. + kwargs**: dict + Keyword arguments passed to ``link(...)``. + + Returns + ======= + + Absolute path to the generate shared object. + """ + libraries = libraries or [] + + include_dirs = kwargs.pop('include_dirs', []) + library_dirs = kwargs.pop('library_dirs', []) + + # Add Python include and library directories + # PY_LDFLAGS does not available on all python implementations + # e.g. when with pypy, so it's LDFLAGS we need to use + if sys.platform == "win32": + warnings.warn("Windows not yet supported.") + elif sys.platform == 'darwin': + cfgDict = get_config_vars() + kwargs['linkline'] = kwargs.get('linkline', []) + [cfgDict['LDFLAGS']] + library_dirs += [cfgDict['LIBDIR']] + + # In macOS, linker needs to compile frameworks + # e.g. "-framework CoreFoundation" + is_framework = False + for opt in cfgDict['LIBS'].split(): + if is_framework: + kwargs['linkline'] = kwargs.get('linkline', []) + ['-framework', opt] + is_framework = False + elif opt.startswith('-l'): + libraries.append(opt[2:]) + elif opt.startswith('-framework'): + is_framework = True + # The python library is not included in LIBS + libfile = cfgDict['LIBRARY'] + libname = ".".join(libfile.split('.')[:-1])[3:] + libraries.append(libname) + + elif sys.platform[:3] == 'aix': + # Don't use the default code below + pass + else: + if get_config_var('Py_ENABLE_SHARED'): + cfgDict = get_config_vars() + kwargs['linkline'] = kwargs.get('linkline', []) + [cfgDict['LDFLAGS']] + library_dirs += [cfgDict['LIBDIR']] + for opt in cfgDict['BLDLIBRARY'].split(): + if opt.startswith('-l'): + libraries += [opt[2:]] + else: + pass + + flags = kwargs.pop('flags', []) + needed_flags = ('-pthread',) + for flag in needed_flags: + if flag not in flags: + flags.append(flag) + + return link(obj_files, shared=True, flags=flags, cwd=cwd, cplus=cplus, fort=fort, + include_dirs=include_dirs, libraries=libraries, + library_dirs=library_dirs, extra_objs=extra_objs, **kwargs) + + +def simple_cythonize(src, destdir=None, cwd=None, **cy_kwargs): + """ Generates a C file from a Cython source file. + + Parameters + ========== + + src: str + Path to Cython source. + destdir: str (optional) + Path to output directory (default: '.'). + cwd: path string (optional) + Root of relative paths (default: '.'). + **cy_kwargs: + Second argument passed to cy_compile. Generates a .cpp file if ``cplus=True`` in ``cy_kwargs``, + else a .c file. + """ + from Cython.Compiler.Main import ( + default_options, CompilationOptions + ) + from Cython.Compiler.Main import compile as cy_compile + + assert src.lower().endswith('.pyx') or src.lower().endswith('.py') + cwd = cwd or '.' + destdir = destdir or '.' + + ext = '.cpp' if cy_kwargs.get('cplus', False) else '.c' + c_name = os.path.splitext(os.path.basename(src))[0] + ext + + dstfile = os.path.join(destdir, c_name) + + if cwd: + ori_dir = os.getcwd() + else: + ori_dir = '.' + os.chdir(cwd) + try: + cy_options = CompilationOptions(default_options) + cy_options.__dict__.update(cy_kwargs) + # Set language_level if not set by cy_kwargs + # as not setting it is deprecated + if 'language_level' not in cy_kwargs: + cy_options.__dict__['language_level'] = 3 + cy_result = cy_compile([src], cy_options) + if cy_result.num_errors > 0: + raise ValueError("Cython compilation failed.") + + # Move generated C file to destination + # In macOS, the generated C file is in the same directory as the source + # but the /var is a symlink to /private/var, so we need to use realpath + if os.path.realpath(os.path.dirname(src)) != os.path.realpath(destdir): + if os.path.exists(dstfile): + os.unlink(dstfile) + shutil.move(os.path.join(os.path.dirname(src), c_name), destdir) + finally: + os.chdir(ori_dir) + return dstfile + + +extension_mapping = { + '.c': (CCompilerRunner, None), + '.cpp': (CppCompilerRunner, None), + '.cxx': (CppCompilerRunner, None), + '.f': (FortranCompilerRunner, None), + '.for': (FortranCompilerRunner, None), + '.ftn': (FortranCompilerRunner, None), + '.f90': (FortranCompilerRunner, None), # ifort only knows about .f90 + '.f95': (FortranCompilerRunner, 'f95'), + '.f03': (FortranCompilerRunner, 'f2003'), + '.f08': (FortranCompilerRunner, 'f2008'), +} + + +def src2obj(srcpath, Runner=None, objpath=None, cwd=None, inc_py=False, **kwargs): + """ Compiles a source code file to an object file. + + Files ending with '.pyx' assumed to be cython files and + are dispatched to pyx2obj. + + Parameters + ========== + + srcpath: str + Path to source file. + Runner: CompilerRunner subclass (optional) + If ``None``: deduced from extension of srcpath. + objpath : str (optional) + Path to generated object. If ``None``: deduced from ``srcpath``. + cwd: str (optional) + Working directory and root of relative paths. If ``None``: current dir. + inc_py: bool + Add Python include path to kwarg "include_dirs". Default: False + \\*\\*kwargs: dict + keyword arguments passed to Runner or pyx2obj + + """ + name, ext = os.path.splitext(os.path.basename(srcpath)) + if objpath is None: + if os.path.isabs(srcpath): + objpath = '.' + else: + objpath = os.path.dirname(srcpath) + objpath = objpath or '.' # avoid objpath == '' + + if os.path.isdir(objpath): + objpath = os.path.join(objpath, name + objext) + + include_dirs = kwargs.pop('include_dirs', []) + if inc_py: + py_inc_dir = get_path('include') + if py_inc_dir not in include_dirs: + include_dirs.append(py_inc_dir) + + if ext.lower() == '.pyx': + return pyx2obj(srcpath, objpath=objpath, include_dirs=include_dirs, cwd=cwd, + **kwargs) + + if Runner is None: + Runner, std = extension_mapping[ext.lower()] + if 'std' not in kwargs: + kwargs['std'] = std + + flags = kwargs.pop('flags', []) + needed_flags = ('-fPIC',) + for flag in needed_flags: + if flag not in flags: + flags.append(flag) + + # src2obj implies not running the linker... + run_linker = kwargs.pop('run_linker', False) + if run_linker: + raise CompileError("src2obj called with run_linker=True") + + runner = Runner([srcpath], objpath, include_dirs=include_dirs, + run_linker=run_linker, cwd=cwd, flags=flags, **kwargs) + runner.run() + return objpath + + +def pyx2obj(pyxpath, objpath=None, destdir=None, cwd=None, + include_dirs=None, cy_kwargs=None, cplus=None, **kwargs): + """ + Convenience function + + If cwd is specified, pyxpath and dst are taken to be relative + If only_update is set to `True` the modification time is checked + and compilation is only run if the source is newer than the + destination + + Parameters + ========== + + pyxpath: str + Path to Cython source file. + objpath: str (optional) + Path to object file to generate. + destdir: str (optional) + Directory to put generated C file. When ``None``: directory of ``objpath``. + cwd: str (optional) + Working directory and root of relative paths. + include_dirs: iterable of path strings (optional) + Passed onto src2obj and via cy_kwargs['include_path'] + to simple_cythonize. + cy_kwargs: dict (optional) + Keyword arguments passed onto `simple_cythonize` + cplus: bool (optional) + Indicate whether C++ is used. default: auto-detect using ``.util.pyx_is_cplus``. + compile_kwargs: dict + keyword arguments passed onto src2obj + + Returns + ======= + + Absolute path of generated object file. + + """ + assert pyxpath.endswith('.pyx') + cwd = cwd or '.' + objpath = objpath or '.' + destdir = destdir or os.path.dirname(objpath) + + abs_objpath = get_abspath(objpath, cwd=cwd) + + if os.path.isdir(abs_objpath): + pyx_fname = os.path.basename(pyxpath) + name, ext = os.path.splitext(pyx_fname) + objpath = os.path.join(objpath, name + objext) + + cy_kwargs = cy_kwargs or {} + cy_kwargs['output_dir'] = cwd + if cplus is None: + cplus = pyx_is_cplus(pyxpath) + cy_kwargs['cplus'] = cplus + + interm_c_file = simple_cythonize(pyxpath, destdir=destdir, cwd=cwd, **cy_kwargs) + + include_dirs = include_dirs or [] + flags = kwargs.pop('flags', []) + needed_flags = ('-fwrapv', '-pthread', '-fPIC') + for flag in needed_flags: + if flag not in flags: + flags.append(flag) + + options = kwargs.pop('options', []) + + if kwargs.pop('strict_aliasing', False): + raise CompileError("Cython requires strict aliasing to be disabled.") + + # Let's be explicit about standard + if cplus: + std = kwargs.pop('std', 'c++98') + else: + std = kwargs.pop('std', 'c99') + + return src2obj(interm_c_file, objpath=objpath, cwd=cwd, + include_dirs=include_dirs, flags=flags, std=std, + options=options, inc_py=True, strict_aliasing=False, + **kwargs) + + +def _any_X(srcs, cls): + for src in srcs: + name, ext = os.path.splitext(src) + key = ext.lower() + if key in extension_mapping: + if extension_mapping[key][0] == cls: + return True + return False + + +def any_fortran_src(srcs): + return _any_X(srcs, FortranCompilerRunner) + + +def any_cplus_src(srcs): + return _any_X(srcs, CppCompilerRunner) + + +def compile_link_import_py_ext(sources, extname=None, build_dir='.', compile_kwargs=None, + link_kwargs=None, extra_objs=None): + """ Compiles sources to a shared object (Python extension) and imports it + + Sources in ``sources`` which is imported. If shared object is newer than the sources, they + are not recompiled but instead it is imported. + + Parameters + ========== + + sources : list of strings + List of paths to sources. + extname : string + Name of extension (default: ``None``). + If ``None``: taken from the last file in ``sources`` without extension. + build_dir: str + Path to directory in which objects files etc. are generated. + compile_kwargs: dict + keyword arguments passed to ``compile_sources`` + link_kwargs: dict + keyword arguments passed to ``link_py_so`` + extra_objs: list + List of paths to (prebuilt) object files / static libraries to link against. + + Returns + ======= + + The imported module from of the Python extension. + """ + if extname is None: + extname = os.path.splitext(os.path.basename(sources[-1]))[0] + + compile_kwargs = compile_kwargs or {} + link_kwargs = link_kwargs or {} + + try: + mod = import_module_from_file(os.path.join(build_dir, extname), sources) + except ImportError: + objs = compile_sources(list(map(get_abspath, sources)), destdir=build_dir, + cwd=build_dir, **compile_kwargs) + so = link_py_so(objs, cwd=build_dir, fort=any_fortran_src(sources), + cplus=any_cplus_src(sources), extra_objs=extra_objs, **link_kwargs) + mod = import_module_from_file(so) + return mod + + +def _write_sources_to_build_dir(sources, build_dir): + build_dir = build_dir or tempfile.mkdtemp() + if not os.path.isdir(build_dir): + raise OSError("Non-existent directory: ", build_dir) + + source_files = [] + for name, src in sources: + dest = os.path.join(build_dir, name) + differs = True + sha256_in_mem = sha256_of_string(src.encode('utf-8')).hexdigest() + if os.path.exists(dest): + if os.path.exists(dest + '.sha256'): + sha256_on_disk = Path(dest + '.sha256').read_text() + else: + sha256_on_disk = sha256_of_file(dest).hexdigest() + + differs = sha256_on_disk != sha256_in_mem + if differs: + with open(dest, 'wt') as fh: + fh.write(src) + with open(dest + '.sha256', 'wt') as fh: + fh.write(sha256_in_mem) + source_files.append(dest) + return source_files, build_dir + + +def compile_link_import_strings(sources, build_dir=None, **kwargs): + """ Compiles, links and imports extension module from source. + + Parameters + ========== + + sources : iterable of name/source pair tuples + build_dir : string (default: None) + Path. ``None`` implies use a temporary directory. + **kwargs: + Keyword arguments passed onto `compile_link_import_py_ext`. + + Returns + ======= + + mod : module + The compiled and imported extension module. + info : dict + Containing ``build_dir`` as 'build_dir'. + + """ + source_files, build_dir = _write_sources_to_build_dir(sources, build_dir) + mod = compile_link_import_py_ext(source_files, build_dir=build_dir, **kwargs) + info = {"build_dir": build_dir} + return mod, info + + +def compile_run_strings(sources, build_dir=None, clean=False, compile_kwargs=None, link_kwargs=None): + """ Compiles, links and runs a program built from sources. + + Parameters + ========== + + sources : iterable of name/source pair tuples + build_dir : string (default: None) + Path. ``None`` implies use a temporary directory. + clean : bool + Whether to remove build_dir after use. This will only have an + effect if ``build_dir`` is ``None`` (which creates a temporary directory). + Passing ``clean == True`` and ``build_dir != None`` raises a ``ValueError``. + This will also set ``build_dir`` in returned info dictionary to ``None``. + compile_kwargs: dict + Keyword arguments passed onto ``compile_sources`` + link_kwargs: dict + Keyword arguments passed onto ``link`` + + Returns + ======= + + (stdout, stderr): pair of strings + info: dict + Containing exit status as 'exit_status' and ``build_dir`` as 'build_dir' + + """ + if clean and build_dir is not None: + raise ValueError("Automatic removal of build_dir is only available for temporary directory.") + try: + source_files, build_dir = _write_sources_to_build_dir(sources, build_dir) + objs = compile_sources(list(map(get_abspath, source_files)), destdir=build_dir, + cwd=build_dir, **(compile_kwargs or {})) + prog = link(objs, cwd=build_dir, + fort=any_fortran_src(source_files), + cplus=any_cplus_src(source_files), **(link_kwargs or {})) + p = subprocess.Popen([prog], stdout=subprocess.PIPE, stderr=subprocess.PIPE) + exit_status = p.wait() + stdout, stderr = [txt.decode('utf-8') for txt in p.communicate()] + finally: + if clean and os.path.isdir(build_dir): + shutil.rmtree(build_dir) + build_dir = None + info = {"exit_status": exit_status, "build_dir": build_dir} + return (stdout, stderr), info diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/_compilation/runners.py b/.venv/lib/python3.12/site-packages/sympy/utilities/_compilation/runners.py new file mode 100644 index 0000000000000000000000000000000000000000..1f37d6cf8ac47807da7f3f00dfc5cd847c03fa8d --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/_compilation/runners.py @@ -0,0 +1,301 @@ +from __future__ import annotations +from typing import Callable, Optional + +from collections import OrderedDict +import os +import re +import subprocess +import warnings + +from .util import ( + find_binary_of_command, unique_list, CompileError +) + + +class CompilerRunner: + """ CompilerRunner base class. + + Parameters + ========== + + sources : list of str + Paths to sources. + out : str + flags : iterable of str + Compiler flags. + run_linker : bool + compiler_name_exe : (str, str) tuple + Tuple of compiler name & command to call. + cwd : str + Path of root of relative paths. + include_dirs : list of str + Include directories. + libraries : list of str + Libraries to link against. + library_dirs : list of str + Paths to search for shared libraries. + std : str + Standard string, e.g. ``'c++11'``, ``'c99'``, ``'f2003'``. + define: iterable of strings + macros to define + undef : iterable of strings + macros to undefine + preferred_vendor : string + name of preferred vendor e.g. 'gnu' or 'intel' + + Methods + ======= + + run(): + Invoke compilation as a subprocess. + + """ + + environ_key_compiler: str # e.g. 'CC', 'CXX', ... + environ_key_flags: str # e.g. 'CFLAGS', 'CXXFLAGS', ... + environ_key_ldflags: str = "LDFLAGS" # typically 'LDFLAGS' + + # Subclass to vendor/binary dict + compiler_dict: dict[str, str] + + # Standards should be a tuple of supported standards + # (first one will be the default) + standards: tuple[None | str, ...] + + # Subclass to dict of binary/formater-callback + std_formater: dict[str, Callable[[Optional[str]], str]] + + # subclass to be e.g. {'gcc': 'gnu', ...} + compiler_name_vendor_mapping: dict[str, str] + + def __init__(self, sources, out, flags=None, run_linker=True, compiler=None, cwd='.', + include_dirs=None, libraries=None, library_dirs=None, std=None, define=None, + undef=None, strict_aliasing=None, preferred_vendor=None, linkline=None, **kwargs): + if isinstance(sources, str): + raise ValueError("Expected argument sources to be a list of strings.") + self.sources = list(sources) + self.out = out + self.flags = flags or [] + if os.environ.get(self.environ_key_flags): + self.flags += os.environ[self.environ_key_flags].split() + self.cwd = cwd + if compiler: + self.compiler_name, self.compiler_binary = compiler + elif os.environ.get(self.environ_key_compiler): + self.compiler_binary = os.environ[self.environ_key_compiler] + for k, v in self.compiler_dict.items(): + if k in self.compiler_binary: + self.compiler_vendor = k + self.compiler_name = v + break + else: + self.compiler_vendor, self.compiler_name = list(self.compiler_dict.items())[0] + warnings.warn("failed to determine what kind of compiler %s is, assuming %s" % + (self.compiler_binary, self.compiler_name)) + else: + # Find a compiler + if preferred_vendor is None: + preferred_vendor = os.environ.get('SYMPY_COMPILER_VENDOR', None) + self.compiler_name, self.compiler_binary, self.compiler_vendor = self.find_compiler(preferred_vendor) + if self.compiler_binary is None: + raise ValueError("No compiler found (searched: {})".format(', '.join(self.compiler_dict.values()))) + self.define = define or [] + self.undef = undef or [] + self.include_dirs = include_dirs or [] + self.libraries = libraries or [] + self.library_dirs = library_dirs or [] + self.std = std or self.standards[0] + self.run_linker = run_linker + if self.run_linker: + # both gnu and intel compilers use '-c' for disabling linker + self.flags = list(filter(lambda x: x != '-c', self.flags)) + else: + if '-c' not in self.flags: + self.flags.append('-c') + + if self.std: + self.flags.append(self.std_formater[ + self.compiler_name](self.std)) + + self.linkline = (linkline or []) + [lf for lf in map( + str.strip, os.environ.get(self.environ_key_ldflags, "").split() + ) if lf != ""] + + if strict_aliasing is not None: + nsa_re = re.compile("no-strict-aliasing$") + sa_re = re.compile("strict-aliasing$") + if strict_aliasing is True: + if any(map(nsa_re.match, flags)): + raise CompileError("Strict aliasing cannot be both enforced and disabled") + elif any(map(sa_re.match, flags)): + pass # already enforced + else: + flags.append('-fstrict-aliasing') + elif strict_aliasing is False: + if any(map(nsa_re.match, flags)): + pass # already disabled + else: + if any(map(sa_re.match, flags)): + raise CompileError("Strict aliasing cannot be both enforced and disabled") + else: + flags.append('-fno-strict-aliasing') + else: + msg = "Expected argument strict_aliasing to be True/False, got {}" + raise ValueError(msg.format(strict_aliasing)) + + @classmethod + def find_compiler(cls, preferred_vendor=None): + """ Identify a suitable C/fortran/other compiler. """ + candidates = list(cls.compiler_dict.keys()) + if preferred_vendor: + if preferred_vendor in candidates: + candidates = [preferred_vendor]+candidates + else: + raise ValueError("Unknown vendor {}".format(preferred_vendor)) + name, path = find_binary_of_command([cls.compiler_dict[x] for x in candidates]) + return name, path, cls.compiler_name_vendor_mapping[name] + + def cmd(self): + """ List of arguments (str) to be passed to e.g. ``subprocess.Popen``. """ + cmd = ( + [self.compiler_binary] + + self.flags + + ['-U'+x for x in self.undef] + + ['-D'+x for x in self.define] + + ['-I'+x for x in self.include_dirs] + + self.sources + ) + if self.run_linker: + cmd += (['-L'+x for x in self.library_dirs] + + ['-l'+x for x in self.libraries] + + self.linkline) + counted = [] + for envvar in re.findall(r'\$\{(\w+)\}', ' '.join(cmd)): + if os.getenv(envvar) is None: + if envvar not in counted: + counted.append(envvar) + msg = "Environment variable '{}' undefined.".format(envvar) + raise CompileError(msg) + return cmd + + def run(self): + self.flags = unique_list(self.flags) + + # Append output flag and name to tail of flags + self.flags.extend(['-o', self.out]) + env = os.environ.copy() + env['PWD'] = self.cwd + + # NOTE: intel compilers seems to need shell=True + p = subprocess.Popen(' '.join(self.cmd()), + shell=True, + cwd=self.cwd, + stdin=subprocess.PIPE, + stdout=subprocess.PIPE, + stderr=subprocess.STDOUT, + env=env) + comm = p.communicate() + try: + self.cmd_outerr = comm[0].decode('utf-8') + except UnicodeDecodeError: + self.cmd_outerr = comm[0].decode('iso-8859-1') # win32 + self.cmd_returncode = p.returncode + + # Error handling + if self.cmd_returncode != 0: + msg = "Error executing '{}' in {} (exited status {}):\n {}\n".format( + ' '.join(self.cmd()), self.cwd, str(self.cmd_returncode), self.cmd_outerr + ) + raise CompileError(msg) + + return self.cmd_outerr, self.cmd_returncode + + +class CCompilerRunner(CompilerRunner): + + environ_key_compiler = 'CC' + environ_key_flags = 'CFLAGS' + + compiler_dict = OrderedDict([ + ('gnu', 'gcc'), + ('intel', 'icc'), + ('llvm', 'clang'), + ]) + + standards = ('c89', 'c90', 'c99', 'c11') # First is default + + std_formater = { + 'gcc': '-std={}'.format, + 'icc': '-std={}'.format, + 'clang': '-std={}'.format, + } + + compiler_name_vendor_mapping = { + 'gcc': 'gnu', + 'icc': 'intel', + 'clang': 'llvm' + } + + +def _mk_flag_filter(cmplr_name): # helper for class initialization + not_welcome = {'g++': ("Wimplicit-interface",)} # "Wstrict-prototypes",)} + if cmplr_name in not_welcome: + def fltr(x): + for nw in not_welcome[cmplr_name]: + if nw in x: + return False + return True + else: + def fltr(x): + return True + return fltr + + +class CppCompilerRunner(CompilerRunner): + + environ_key_compiler = 'CXX' + environ_key_flags = 'CXXFLAGS' + + compiler_dict = OrderedDict([ + ('gnu', 'g++'), + ('intel', 'icpc'), + ('llvm', 'clang++'), + ]) + + # First is the default, c++0x == c++11 + standards = ('c++98', 'c++0x') + + std_formater = { + 'g++': '-std={}'.format, + 'icpc': '-std={}'.format, + 'clang++': '-std={}'.format, + } + + compiler_name_vendor_mapping = { + 'g++': 'gnu', + 'icpc': 'intel', + 'clang++': 'llvm' + } + + +class FortranCompilerRunner(CompilerRunner): + + environ_key_compiler = 'FC' + environ_key_flags = 'FFLAGS' + + standards = (None, 'f77', 'f95', 'f2003', 'f2008') + + std_formater = { + 'gfortran': lambda x: '-std=gnu' if x is None else '-std=legacy' if x == 'f77' else '-std={}'.format(x), + 'ifort': lambda x: '-stand f08' if x is None else '-stand f{}'.format(x[-2:]), # f2008 => f08 + } + + compiler_dict = OrderedDict([ + ('gnu', 'gfortran'), + ('intel', 'ifort'), + ]) + + compiler_name_vendor_mapping = { + 'gfortran': 'gnu', + 'ifort': 'intel', + } diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/_compilation/tests/__init__.py b/.venv/lib/python3.12/site-packages/sympy/utilities/_compilation/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/_compilation/tests/__pycache__/__init__.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/utilities/_compilation/tests/__pycache__/__init__.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..9afb2067ee0a1222ed2f7b5172b7a25b37433eae Binary files /dev/null and b/.venv/lib/python3.12/site-packages/sympy/utilities/_compilation/tests/__pycache__/__init__.cpython-312.pyc differ diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/_compilation/tests/__pycache__/test_compilation.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/utilities/_compilation/tests/__pycache__/test_compilation.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..b4063cd300fdee537a094984d5ce2b1d6928f3de Binary files /dev/null and b/.venv/lib/python3.12/site-packages/sympy/utilities/_compilation/tests/__pycache__/test_compilation.cpython-312.pyc differ diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/_compilation/tests/test_compilation.py b/.venv/lib/python3.12/site-packages/sympy/utilities/_compilation/tests/test_compilation.py new file mode 100644 index 0000000000000000000000000000000000000000..ff7cf86644a9645665e62b49cfc4e7ea73b2c1ab --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/_compilation/tests/test_compilation.py @@ -0,0 +1,104 @@ +import shutil +import os +import subprocess +import tempfile +from sympy.external import import_module +from sympy.testing.pytest import skip, skip_under_pyodide + +from sympy.utilities._compilation.compilation import compile_link_import_py_ext, compile_link_import_strings, compile_sources, get_abspath + +numpy = import_module('numpy') +cython = import_module('cython') + +_sources1 = [ + ('sigmoid.c', r""" +#include + +void sigmoid(int n, const double * const restrict in, + double * const restrict out, double lim){ + for (int i=0; i 0: + if not os.path.exists(parent): + make_dirs(parent) + + if not os.path.exists(path): + os.mkdir(path, 0o777) + else: + assert os.path.isdir(path) + +def missing_or_other_newer(path, other_path, cwd=None): + """ + Investigate if path is non-existent or older than provided reference + path. + + Parameters + ========== + path: string + path to path which might be missing or too old + other_path: string + reference path + cwd: string + working directory (root of relative paths) + + Returns + ======= + True if path is older or missing. + """ + cwd = cwd or '.' + path = get_abspath(path, cwd=cwd) + other_path = get_abspath(other_path, cwd=cwd) + if not os.path.exists(path): + return True + if os.path.getmtime(other_path) - 1e-6 >= os.path.getmtime(path): + # 1e-6 is needed because http://stackoverflow.com/questions/17086426/ + return True + return False + +def copy(src, dst, only_update=False, copystat=True, cwd=None, + dest_is_dir=False, create_dest_dirs=False): + """ Variation of ``shutil.copy`` with extra options. + + Parameters + ========== + + src : str + Path to source file. + dst : str + Path to destination. + only_update : bool + Only copy if source is newer than destination + (returns None if it was newer), default: ``False``. + copystat : bool + See ``shutil.copystat``. default: ``True``. + cwd : str + Path to working directory (root of relative paths). + dest_is_dir : bool + Ensures that dst is treated as a directory. default: ``False`` + create_dest_dirs : bool + Creates directories if needed. + + Returns + ======= + + Path to the copied file. + + """ + if cwd: # Handle working directory + if not os.path.isabs(src): + src = os.path.join(cwd, src) + if not os.path.isabs(dst): + dst = os.path.join(cwd, dst) + + if not os.path.exists(src): # Make sure source file exists + raise FileNotFoundError("Source: `{}` does not exist".format(src)) + + # We accept both (re)naming destination file _or_ + # passing a (possible non-existent) destination directory + if dest_is_dir: + if not dst[-1] == '/': + dst = dst+'/' + else: + if os.path.exists(dst) and os.path.isdir(dst): + dest_is_dir = True + + if dest_is_dir: + dest_dir = dst + dest_fname = os.path.basename(src) + dst = os.path.join(dest_dir, dest_fname) + else: + dest_dir = os.path.dirname(dst) + + if not os.path.exists(dest_dir): + if create_dest_dirs: + make_dirs(dest_dir) + else: + raise FileNotFoundError("You must create directory first.") + + if only_update: + if not missing_or_other_newer(dst, src): + return + + if os.path.islink(dst): + dst = os.path.abspath(os.path.realpath(dst), cwd=cwd) + + shutil.copy(src, dst) + if copystat: + shutil.copystat(src, dst) + + return dst + +Glob = namedtuple('Glob', 'pathname') +ArbitraryDepthGlob = namedtuple('ArbitraryDepthGlob', 'filename') + +def glob_at_depth(filename_glob, cwd=None): + if cwd is not None: + cwd = '.' + globbed = [] + for root, dirs, filenames in os.walk(cwd): + for fn in filenames: + # This is not tested: + if fnmatch.fnmatch(fn, filename_glob): + globbed.append(os.path.join(root, fn)) + return globbed + +def sha256_of_file(path, nblocks=128): + """ Computes the SHA256 hash of a file. + + Parameters + ========== + + path : string + Path to file to compute hash of. + nblocks : int + Number of blocks to read per iteration. + + Returns + ======= + + hashlib sha256 hash object. Use ``.digest()`` or ``.hexdigest()`` + on returned object to get binary or hex encoded string. + """ + sh = sha256() + with open(path, 'rb') as f: + for chunk in iter(lambda: f.read(nblocks*sh.block_size), b''): + sh.update(chunk) + return sh + + +def sha256_of_string(string): + """ Computes the SHA256 hash of a string. """ + sh = sha256() + sh.update(string) + return sh + + +def pyx_is_cplus(path): + """ + Inspect a Cython source file (.pyx) and look for comment line like: + + # distutils: language = c++ + + Returns True if such a file is present in the file, else False. + """ + with open(path) as fh: + for line in fh: + if line.startswith('#') and '=' in line: + splitted = line.split('=') + if len(splitted) != 2: + continue + lhs, rhs = splitted + if lhs.strip().split()[-1].lower() == 'language' and \ + rhs.strip().split()[0].lower() == 'c++': + return True + return False + +def import_module_from_file(filename, only_if_newer_than=None): + """ Imports Python extension (from shared object file) + + Provide a list of paths in `only_if_newer_than` to check + timestamps of dependencies. import_ raises an ImportError + if any is newer. + + Word of warning: The OS may cache shared objects which makes + reimporting same path of an shared object file very problematic. + + It will not detect the new time stamp, nor new checksum, but will + instead silently use old module. Use unique names for this reason. + + Parameters + ========== + + filename : str + Path to shared object. + only_if_newer_than : iterable of strings + Paths to dependencies of the shared object. + + Raises + ====== + + ``ImportError`` if any of the files specified in ``only_if_newer_than`` are newer + than the file given by filename. + """ + path, name = os.path.split(filename) + name, ext = os.path.splitext(name) + name = name.split('.')[0] + if sys.version_info[0] == 2: + from imp import find_module, load_module + fobj, filename, data = find_module(name, [path]) + if only_if_newer_than: + for dep in only_if_newer_than: + if os.path.getmtime(filename) < os.path.getmtime(dep): + raise ImportError("{} is newer than {}".format(dep, filename)) + mod = load_module(name, fobj, filename, data) + else: + import importlib.util + spec = importlib.util.spec_from_file_location(name, filename) + if spec is None: + raise ImportError("Failed to import: '%s'" % filename) + mod = importlib.util.module_from_spec(spec) + spec.loader.exec_module(mod) + return mod + + +def find_binary_of_command(candidates): + """ Finds binary first matching name among candidates. + + Calls ``which`` from shutils for provided candidates and returns + first hit. + + Parameters + ========== + + candidates : iterable of str + Names of candidate commands + + Raises + ====== + + CompilerNotFoundError if no candidates match. + """ + from shutil import which + for c in candidates: + binary_path = which(c) + if c and binary_path: + return c, binary_path + + raise CompilerNotFoundError('No binary located for candidates: {}'.format(candidates)) + + +def unique_list(l): + """ Uniquify a list (skip duplicate items). """ + result = [] + for x in l: + if x not in result: + result.append(x) + return result diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/autowrap.py b/.venv/lib/python3.12/site-packages/sympy/utilities/autowrap.py new file mode 100644 index 0000000000000000000000000000000000000000..b6c33b2f0f72f89b5680f910929618a821e924c3 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/autowrap.py @@ -0,0 +1,1178 @@ +"""Module for compiling codegen output, and wrap the binary for use in +python. + +.. note:: To use the autowrap module it must first be imported + + >>> from sympy.utilities.autowrap import autowrap + +This module provides a common interface for different external backends, such +as f2py, fwrap, Cython, SWIG(?) etc. (Currently only f2py and Cython are +implemented) The goal is to provide access to compiled binaries of acceptable +performance with a one-button user interface, e.g., + + >>> from sympy.abc import x,y + >>> expr = (x - y)**25 + >>> flat = expr.expand() + >>> binary_callable = autowrap(flat) + >>> binary_callable(2, 3) + -1.0 + +Although a SymPy user might primarily be interested in working with +mathematical expressions and not in the details of wrapping tools +needed to evaluate such expressions efficiently in numerical form, +the user cannot do so without some understanding of the +limits in the target language. For example, the expanded expression +contains large coefficients which result in loss of precision when +computing the expression: + + >>> binary_callable(3, 2) + 0.0 + >>> binary_callable(4, 5), binary_callable(5, 4) + (-22925376.0, 25165824.0) + +Wrapping the unexpanded expression gives the expected behavior: + + >>> e = autowrap(expr) + >>> e(4, 5), e(5, 4) + (-1.0, 1.0) + +The callable returned from autowrap() is a binary Python function, not a +SymPy object. If it is desired to use the compiled function in symbolic +expressions, it is better to use binary_function() which returns a SymPy +Function object. The binary callable is attached as the _imp_ attribute and +invoked when a numerical evaluation is requested with evalf(), or with +lambdify(). + + >>> from sympy.utilities.autowrap import binary_function + >>> f = binary_function('f', expr) + >>> 2*f(x, y) + y + y + 2*f(x, y) + >>> (2*f(x, y) + y).evalf(2, subs={x: 1, y:2}) + 0.e-110 + +When is this useful? + + 1) For computations on large arrays, Python iterations may be too slow, + and depending on the mathematical expression, it may be difficult to + exploit the advanced index operations provided by NumPy. + + 2) For *really* long expressions that will be called repeatedly, the + compiled binary should be significantly faster than SymPy's .evalf() + + 3) If you are generating code with the codegen utility in order to use + it in another project, the automatic Python wrappers let you test the + binaries immediately from within SymPy. + + 4) To create customized ufuncs for use with numpy arrays. + See *ufuncify*. + +When is this module NOT the best approach? + + 1) If you are really concerned about speed or memory optimizations, + you will probably get better results by working directly with the + wrapper tools and the low level code. However, the files generated + by this utility may provide a useful starting point and reference + code. Temporary files will be left intact if you supply the keyword + tempdir="path/to/files/". + + 2) If the array computation can be handled easily by numpy, and you + do not need the binaries for another project. + +""" + +import sys +import os +import shutil +import tempfile +from pathlib import Path +from subprocess import STDOUT, CalledProcessError, check_output +from string import Template +from warnings import warn + +from sympy.core.cache import cacheit +from sympy.core.function import Lambda +from sympy.core.relational import Eq +from sympy.core.symbol import Dummy, Symbol +from sympy.tensor.indexed import Idx, IndexedBase +from sympy.utilities.codegen import (make_routine, get_code_generator, + OutputArgument, InOutArgument, + InputArgument, CodeGenArgumentListError, + Result, ResultBase, C99CodeGen) +from sympy.utilities.iterables import iterable +from sympy.utilities.lambdify import implemented_function +from sympy.utilities.decorator import doctest_depends_on + +_doctest_depends_on = {'exe': ('f2py', 'gfortran', 'gcc'), + 'modules': ('numpy',)} + + +class CodeWrapError(Exception): + pass + + +class CodeWrapper: + """Base Class for code wrappers""" + _filename = "wrapped_code" + _module_basename = "wrapper_module" + _module_counter = 0 + + @property + def filename(self): + return "%s_%s" % (self._filename, CodeWrapper._module_counter) + + @property + def module_name(self): + return "%s_%s" % (self._module_basename, CodeWrapper._module_counter) + + def __init__(self, generator, filepath=None, flags=[], verbose=False): + """ + generator -- the code generator to use + """ + self.generator = generator + self.filepath = filepath + self.flags = flags + self.quiet = not verbose + + @property + def include_header(self): + return bool(self.filepath) + + @property + def include_empty(self): + return bool(self.filepath) + + def _generate_code(self, main_routine, routines): + routines.append(main_routine) + self.generator.write( + routines, self.filename, True, self.include_header, + self.include_empty) + + def wrap_code(self, routine, helpers=None): + helpers = helpers or [] + if self.filepath: + workdir = os.path.abspath(self.filepath) + else: + workdir = tempfile.mkdtemp("_sympy_compile") + if not os.access(workdir, os.F_OK): + os.mkdir(workdir) + oldwork = os.getcwd() + os.chdir(workdir) + try: + sys.path.append(workdir) + self._generate_code(routine, helpers) + self._prepare_files(routine) + self._process_files(routine) + mod = __import__(self.module_name) + finally: + sys.path.remove(workdir) + CodeWrapper._module_counter += 1 + os.chdir(oldwork) + if not self.filepath: + try: + shutil.rmtree(workdir) + except OSError: + # Could be some issues on Windows + pass + + return self._get_wrapped_function(mod, routine.name) + + def _process_files(self, routine): + command = self.command + command.extend(self.flags) + try: + retoutput = check_output(command, stderr=STDOUT) + except CalledProcessError as e: + raise CodeWrapError( + "Error while executing command: %s. Command output is:\n%s" % ( + " ".join(command), e.output.decode('utf-8'))) + if not self.quiet: + print(retoutput) + + +class DummyWrapper(CodeWrapper): + """Class used for testing independent of backends """ + + template = """# dummy module for testing of SymPy +def %(name)s(): + return "%(expr)s" +%(name)s.args = "%(args)s" +%(name)s.returns = "%(retvals)s" +""" + + def _prepare_files(self, routine): + return + + def _generate_code(self, routine, helpers): + with open('%s.py' % self.module_name, 'w') as f: + printed = ", ".join( + [str(res.expr) for res in routine.result_variables]) + # convert OutputArguments to return value like f2py + args = filter(lambda x: not isinstance( + x, OutputArgument), routine.arguments) + retvals = [] + for val in routine.result_variables: + if isinstance(val, Result): + retvals.append('nameless') + else: + retvals.append(val.result_var) + + print(DummyWrapper.template % { + 'name': routine.name, + 'expr': printed, + 'args': ", ".join([str(a.name) for a in args]), + 'retvals': ", ".join([str(val) for val in retvals]) + }, end="", file=f) + + def _process_files(self, routine): + return + + @classmethod + def _get_wrapped_function(cls, mod, name): + return getattr(mod, name) + + +class CythonCodeWrapper(CodeWrapper): + """Wrapper that uses Cython""" + + setup_template = """\ +from setuptools import setup +from setuptools import Extension +from Cython.Build import cythonize +cy_opts = {cythonize_options} +{np_import} +ext_mods = [Extension( + {ext_args}, + include_dirs={include_dirs}, + library_dirs={library_dirs}, + libraries={libraries}, + extra_compile_args={extra_compile_args}, + extra_link_args={extra_link_args} +)] +setup(ext_modules=cythonize(ext_mods, **cy_opts)) +""" + + _cythonize_options = {'compiler_directives':{'language_level' : "3"}} + + pyx_imports = ( + "import numpy as np\n" + "cimport numpy as np\n\n") + + pyx_header = ( + "cdef extern from '{header_file}.h':\n" + " {prototype}\n\n") + + pyx_func = ( + "def {name}_c({arg_string}):\n" + "\n" + "{declarations}" + "{body}") + + std_compile_flag = '-std=c99' + + def __init__(self, *args, **kwargs): + """Instantiates a Cython code wrapper. + + The following optional parameters get passed to ``setuptools.Extension`` + for building the Python extension module. Read its documentation to + learn more. + + Parameters + ========== + include_dirs : [list of strings] + A list of directories to search for C/C++ header files (in Unix + form for portability). + library_dirs : [list of strings] + A list of directories to search for C/C++ libraries at link time. + libraries : [list of strings] + A list of library names (not filenames or paths) to link against. + extra_compile_args : [list of strings] + Any extra platform- and compiler-specific information to use when + compiling the source files in 'sources'. For platforms and + compilers where "command line" makes sense, this is typically a + list of command-line arguments, but for other platforms it could be + anything. Note that the attribute ``std_compile_flag`` will be + appended to this list. + extra_link_args : [list of strings] + Any extra platform- and compiler-specific information to use when + linking object files together to create the extension (or to create + a new static Python interpreter). Similar interpretation as for + 'extra_compile_args'. + cythonize_options : [dictionary] + Keyword arguments passed on to cythonize. + + """ + + self._include_dirs = kwargs.pop('include_dirs', []) + self._library_dirs = kwargs.pop('library_dirs', []) + self._libraries = kwargs.pop('libraries', []) + self._extra_compile_args = kwargs.pop('extra_compile_args', []) + self._extra_compile_args.append(self.std_compile_flag) + self._extra_link_args = kwargs.pop('extra_link_args', []) + self._cythonize_options = kwargs.pop('cythonize_options', self._cythonize_options) + + self._need_numpy = False + + super().__init__(*args, **kwargs) + + @property + def command(self): + command = [sys.executable, "setup.py", "build_ext", "--inplace"] + return command + + def _prepare_files(self, routine, build_dir=os.curdir): + # NOTE : build_dir is used for testing purposes. + pyxfilename = self.module_name + '.pyx' + codefilename = "%s.%s" % (self.filename, self.generator.code_extension) + + # pyx + with open(os.path.join(build_dir, pyxfilename), 'w') as f: + self.dump_pyx([routine], f, self.filename) + + # setup.py + ext_args = [repr(self.module_name), repr([pyxfilename, codefilename])] + if self._need_numpy: + np_import = 'import numpy as np\n' + self._include_dirs.append('np.get_include()') + else: + np_import = '' + + includes = str(self._include_dirs).replace("'np.get_include()'", + 'np.get_include()') + code = self.setup_template.format( + ext_args=", ".join(ext_args), + np_import=np_import, + include_dirs=includes, + library_dirs=self._library_dirs, + libraries=self._libraries, + extra_compile_args=self._extra_compile_args, + extra_link_args=self._extra_link_args, + cythonize_options=self._cythonize_options) + Path(os.path.join(build_dir, 'setup.py')).write_text(code) + + @classmethod + def _get_wrapped_function(cls, mod, name): + return getattr(mod, name + '_c') + + def dump_pyx(self, routines, f, prefix): + """Write a Cython file with Python wrappers + + This file contains all the definitions of the routines in c code and + refers to the header file. + + Arguments + --------- + routines + List of Routine instances + f + File-like object to write the file to + prefix + The filename prefix, used to refer to the proper header file. + Only the basename of the prefix is used. + """ + headers = [] + functions = [] + for routine in routines: + prototype = self.generator.get_prototype(routine) + + # C Function Header Import + headers.append(self.pyx_header.format(header_file=prefix, + prototype=prototype)) + + # Partition the C function arguments into categories + py_rets, py_args, py_loc, py_inf = self._partition_args(routine.arguments) + + # Function prototype + name = routine.name + arg_string = ", ".join(self._prototype_arg(arg) for arg in py_args) + + # Local Declarations + local_decs = [] + for arg, val in py_inf.items(): + proto = self._prototype_arg(arg) + mat, ind = [self._string_var(v) for v in val] + local_decs.append(" cdef {} = {}.shape[{}]".format(proto, mat, ind)) + local_decs.extend([" cdef {}".format(self._declare_arg(a)) for a in py_loc]) + declarations = "\n".join(local_decs) + if declarations: + declarations = declarations + "\n" + + # Function Body + args_c = ", ".join([self._call_arg(a) for a in routine.arguments]) + rets = ", ".join([self._string_var(r.name) for r in py_rets]) + if routine.results: + body = ' return %s(%s)' % (routine.name, args_c) + if rets: + body = body + ', ' + rets + else: + body = ' %s(%s)\n' % (routine.name, args_c) + body = body + ' return ' + rets + + functions.append(self.pyx_func.format(name=name, arg_string=arg_string, + declarations=declarations, body=body)) + + # Write text to file + if self._need_numpy: + # Only import numpy if required + f.write(self.pyx_imports) + f.write('\n'.join(headers)) + f.write('\n'.join(functions)) + + def _partition_args(self, args): + """Group function arguments into categories.""" + py_args = [] + py_returns = [] + py_locals = [] + py_inferred = {} + for arg in args: + if isinstance(arg, OutputArgument): + py_returns.append(arg) + py_locals.append(arg) + elif isinstance(arg, InOutArgument): + py_returns.append(arg) + py_args.append(arg) + else: + py_args.append(arg) + # Find arguments that are array dimensions. These can be inferred + # locally in the Cython code. + if isinstance(arg, (InputArgument, InOutArgument)) and arg.dimensions: + dims = [d[1] + 1 for d in arg.dimensions] + sym_dims = [(i, d) for (i, d) in enumerate(dims) if + isinstance(d, Symbol)] + for (i, d) in sym_dims: + py_inferred[d] = (arg.name, i) + for arg in args: + if arg.name in py_inferred: + py_inferred[arg] = py_inferred.pop(arg.name) + # Filter inferred arguments from py_args + py_args = [a for a in py_args if a not in py_inferred] + return py_returns, py_args, py_locals, py_inferred + + def _prototype_arg(self, arg): + mat_dec = "np.ndarray[{mtype}, ndim={ndim}] {name}" + np_types = {'double': 'np.double_t', + 'int': 'np.int_t'} + t = arg.get_datatype('c') + if arg.dimensions: + self._need_numpy = True + ndim = len(arg.dimensions) + mtype = np_types[t] + return mat_dec.format(mtype=mtype, ndim=ndim, name=self._string_var(arg.name)) + else: + return "%s %s" % (t, self._string_var(arg.name)) + + def _declare_arg(self, arg): + proto = self._prototype_arg(arg) + if arg.dimensions: + shape = '(' + ','.join(self._string_var(i[1] + 1) for i in arg.dimensions) + ')' + return proto + " = np.empty({shape})".format(shape=shape) + else: + return proto + " = 0" + + def _call_arg(self, arg): + if arg.dimensions: + t = arg.get_datatype('c') + return "<{}*> {}.data".format(t, self._string_var(arg.name)) + elif isinstance(arg, ResultBase): + return "&{}".format(self._string_var(arg.name)) + else: + return self._string_var(arg.name) + + def _string_var(self, var): + printer = self.generator.printer.doprint + return printer(var) + + +class F2PyCodeWrapper(CodeWrapper): + """Wrapper that uses f2py""" + + def __init__(self, *args, **kwargs): + + ext_keys = ['include_dirs', 'library_dirs', 'libraries', + 'extra_compile_args', 'extra_link_args'] + msg = ('The compilation option kwarg {} is not supported with the f2py ' + 'backend.') + + for k in ext_keys: + if k in kwargs.keys(): + warn(msg.format(k)) + kwargs.pop(k, None) + + super().__init__(*args, **kwargs) + + @property + def command(self): + filename = self.filename + '.' + self.generator.code_extension + args = ['-c', '-m', self.module_name, filename] + command = [sys.executable, "-c", "import numpy.f2py as f2py2e;f2py2e.main()"]+args + return command + + def _prepare_files(self, routine): + pass + + @classmethod + def _get_wrapped_function(cls, mod, name): + return getattr(mod, name) + + +# Here we define a lookup of backends -> tuples of languages. For now, each +# tuple is of length 1, but if a backend supports more than one language, +# the most preferable language is listed first. +_lang_lookup = {'CYTHON': ('C99', 'C89', 'C'), + 'F2PY': ('F95',), + 'NUMPY': ('C99', 'C89', 'C'), + 'DUMMY': ('F95',)} # Dummy here just for testing + + +def _infer_language(backend): + """For a given backend, return the top choice of language""" + langs = _lang_lookup.get(backend.upper(), False) + if not langs: + raise ValueError("Unrecognized backend: " + backend) + return langs[0] + + +def _validate_backend_language(backend, language): + """Throws error if backend and language are incompatible""" + langs = _lang_lookup.get(backend.upper(), False) + if not langs: + raise ValueError("Unrecognized backend: " + backend) + if language.upper() not in langs: + raise ValueError(("Backend {} and language {} are " + "incompatible").format(backend, language)) + + +@cacheit +@doctest_depends_on(exe=('f2py', 'gfortran'), modules=('numpy',)) +def autowrap(expr, language=None, backend='f2py', tempdir=None, args=None, + flags=None, verbose=False, helpers=None, code_gen=None, **kwargs): + """Generates Python callable binaries based on the math expression. + + Parameters + ========== + + expr + The SymPy expression that should be wrapped as a binary routine. + language : string, optional + If supplied, (options: 'C' or 'F95'), specifies the language of the + generated code. If ``None`` [default], the language is inferred based + upon the specified backend. + backend : string, optional + Backend used to wrap the generated code. Either 'f2py' [default], + or 'cython'. + tempdir : string, optional + Path to directory for temporary files. If this argument is supplied, + the generated code and the wrapper input files are left intact in the + specified path. + args : iterable, optional + An ordered iterable of symbols. Specifies the argument sequence for the + function. + flags : iterable, optional + Additional option flags that will be passed to the backend. + verbose : bool, optional + If True, autowrap will not mute the command line backends. This can be + helpful for debugging. + helpers : 3-tuple or iterable of 3-tuples, optional + Used to define auxiliary functions needed for the main expression. + Each tuple should be of the form (name, expr, args) where: + + - name : str, the function name + - expr : sympy expression, the function + - args : iterable, the function arguments (can be any iterable of symbols) + + code_gen : CodeGen instance + An instance of a CodeGen subclass. Overrides ``language``. + include_dirs : [string] + A list of directories to search for C/C++ header files (in Unix form + for portability). + library_dirs : [string] + A list of directories to search for C/C++ libraries at link time. + libraries : [string] + A list of library names (not filenames or paths) to link against. + extra_compile_args : [string] + Any extra platform- and compiler-specific information to use when + compiling the source files in 'sources'. For platforms and compilers + where "command line" makes sense, this is typically a list of + command-line arguments, but for other platforms it could be anything. + extra_link_args : [string] + Any extra platform- and compiler-specific information to use when + linking object files together to create the extension (or to create a + new static Python interpreter). Similar interpretation as for + 'extra_compile_args'. + + Examples + ======== + + Basic usage: + + >>> from sympy.abc import x, y, z + >>> from sympy.utilities.autowrap import autowrap + >>> expr = ((x - y + z)**(13)).expand() + >>> binary_func = autowrap(expr) + >>> binary_func(1, 4, 2) + -1.0 + + Using helper functions: + + >>> from sympy.abc import x, t + >>> from sympy import Function + >>> helper_func = Function('helper_func') # Define symbolic function + >>> expr = 3*x + helper_func(t) # Main expression using helper function + >>> # Define helper_func(x) = 4*x using f2py backend + >>> binary_func = autowrap(expr, args=[x, t], + ... helpers=('helper_func', 4*x, [x])) + >>> binary_func(2, 5) # 3*2 + helper_func(5) = 6 + 20 + 26.0 + >>> # Same example using cython backend + >>> binary_func = autowrap(expr, args=[x, t], backend='cython', + ... helpers=[('helper_func', 4*x, [x])]) + >>> binary_func(2, 5) # 3*2 + helper_func(5) = 6 + 20 + 26.0 + + Type handling example: + + >>> import numpy as np + >>> expr = x + y + >>> f_cython = autowrap(expr, backend='cython') + >>> f_cython(1, 2) # doctest: +ELLIPSIS + Traceback (most recent call last): + ... + TypeError: Argument '_x' has incorrect type (expected numpy.ndarray, got int) + >>> f_cython(np.array([1.0]), np.array([2.0])) + array([ 3.]) + + """ + if language: + if not isinstance(language, type): + _validate_backend_language(backend, language) + else: + language = _infer_language(backend) + + # two cases 1) helpers is an iterable of 3-tuples and 2) helpers is a + # 3-tuple + if iterable(helpers) and len(helpers) != 0 and iterable(helpers[0]): + helpers = helpers if helpers else () + else: + helpers = [helpers] if helpers else () + args = list(args) if iterable(args, exclude=set) else args + + if code_gen is None: + code_gen = get_code_generator(language, "autowrap") + + CodeWrapperClass = { + 'F2PY': F2PyCodeWrapper, + 'CYTHON': CythonCodeWrapper, + 'DUMMY': DummyWrapper + }[backend.upper()] + code_wrapper = CodeWrapperClass(code_gen, tempdir, flags if flags else (), + verbose, **kwargs) + + helps = [] + for name_h, expr_h, args_h in helpers: + helps.append(code_gen.routine(name_h, expr_h, args_h)) + + for name_h, expr_h, args_h in helpers: + if expr.has(expr_h): + name_h = binary_function(name_h, expr_h, backend='dummy') + expr = expr.subs(expr_h, name_h(*args_h)) + try: + routine = code_gen.routine('autofunc', expr, args) + except CodeGenArgumentListError as e: + # if all missing arguments are for pure output, we simply attach them + # at the end and try again, because the wrappers will silently convert + # them to return values anyway. + new_args = [] + for missing in e.missing_args: + if not isinstance(missing, OutputArgument): + raise + new_args.append(missing.name) + routine = code_gen.routine('autofunc', expr, args + new_args) + + return code_wrapper.wrap_code(routine, helpers=helps) + + +@doctest_depends_on(exe=('f2py', 'gfortran'), modules=('numpy',)) +def binary_function(symfunc, expr, **kwargs): + """Returns a SymPy function with expr as binary implementation + + This is a convenience function that automates the steps needed to + autowrap the SymPy expression and attaching it to a Function object + with implemented_function(). + + Parameters + ========== + + symfunc : SymPy Function + The function to bind the callable to. + expr : SymPy Expression + The expression used to generate the function. + kwargs : dict + Any kwargs accepted by autowrap. + + Examples + ======== + + >>> from sympy.abc import x, y + >>> from sympy.utilities.autowrap import binary_function + >>> expr = ((x - y)**(25)).expand() + >>> f = binary_function('f', expr) + >>> type(f) + + >>> 2*f(x, y) + 2*f(x, y) + >>> f(x, y).evalf(2, subs={x: 1, y: 2}) + -1.0 + + """ + binary = autowrap(expr, **kwargs) + return implemented_function(symfunc, binary) + +################################################################# +# UFUNCIFY # +################################################################# + +_ufunc_top = Template("""\ +#include "Python.h" +#include "math.h" +#include "numpy/ndarraytypes.h" +#include "numpy/ufuncobject.h" +#include "numpy/halffloat.h" +#include ${include_file} + +static PyMethodDef ${module}Methods[] = { + {NULL, NULL, 0, NULL} +};""") + +_ufunc_outcalls = Template("*((double *)out${outnum}) = ${funcname}(${call_args});") + +_ufunc_body = Template("""\ +#ifdef NPY_1_19_API_VERSION +static void ${funcname}_ufunc(char **args, const npy_intp *dimensions, const npy_intp* steps, void* data) +#else +static void ${funcname}_ufunc(char **args, npy_intp *dimensions, npy_intp* steps, void* data) +#endif +{ + npy_intp i; + npy_intp n = dimensions[0]; + ${declare_args} + ${declare_steps} + for (i = 0; i < n; i++) { + ${outcalls} + ${step_increments} + } +} +PyUFuncGenericFunction ${funcname}_funcs[1] = {&${funcname}_ufunc}; +static char ${funcname}_types[${n_types}] = ${types} +static void *${funcname}_data[1] = {NULL};""") + +_ufunc_bottom = Template("""\ +#if PY_VERSION_HEX >= 0x03000000 +static struct PyModuleDef moduledef = { + PyModuleDef_HEAD_INIT, + "${module}", + NULL, + -1, + ${module}Methods, + NULL, + NULL, + NULL, + NULL +}; + +PyMODINIT_FUNC PyInit_${module}(void) +{ + PyObject *m, *d; + ${function_creation} + m = PyModule_Create(&moduledef); + if (!m) { + return NULL; + } + import_array(); + import_umath(); + d = PyModule_GetDict(m); + ${ufunc_init} + return m; +} +#else +PyMODINIT_FUNC init${module}(void) +{ + PyObject *m, *d; + ${function_creation} + m = Py_InitModule("${module}", ${module}Methods); + if (m == NULL) { + return; + } + import_array(); + import_umath(); + d = PyModule_GetDict(m); + ${ufunc_init} +} +#endif\ +""") + +_ufunc_init_form = Template("""\ +ufunc${ind} = PyUFunc_FromFuncAndData(${funcname}_funcs, ${funcname}_data, ${funcname}_types, 1, ${n_in}, ${n_out}, + PyUFunc_None, "${module}", ${docstring}, 0); + PyDict_SetItemString(d, "${funcname}", ufunc${ind}); + Py_DECREF(ufunc${ind});""") + +_ufunc_setup = Template("""\ +from setuptools.extension import Extension +from setuptools import setup + +from numpy import get_include + +if __name__ == "__main__": + setup(ext_modules=[ + Extension('${module}', + sources=['${module}.c', '${filename}.c'], + include_dirs=[get_include()])]) +""") + + +class UfuncifyCodeWrapper(CodeWrapper): + """Wrapper for Ufuncify""" + + def __init__(self, *args, **kwargs): + + ext_keys = ['include_dirs', 'library_dirs', 'libraries', + 'extra_compile_args', 'extra_link_args'] + msg = ('The compilation option kwarg {} is not supported with the numpy' + ' backend.') + + for k in ext_keys: + if k in kwargs.keys(): + warn(msg.format(k)) + kwargs.pop(k, None) + + super().__init__(*args, **kwargs) + + @property + def command(self): + command = [sys.executable, "setup.py", "build_ext", "--inplace"] + return command + + def wrap_code(self, routines, helpers=None): + # This routine overrides CodeWrapper because we can't assume funcname == routines[0].name + # Therefore we have to break the CodeWrapper private API. + # There isn't an obvious way to extend multi-expr support to + # the other autowrap backends, so we limit this change to ufuncify. + helpers = helpers if helpers is not None else [] + # We just need a consistent name + funcname = 'wrapped_' + str(id(routines) + id(helpers)) + + workdir = self.filepath or tempfile.mkdtemp("_sympy_compile") + if not os.access(workdir, os.F_OK): + os.mkdir(workdir) + oldwork = os.getcwd() + os.chdir(workdir) + try: + sys.path.append(workdir) + self._generate_code(routines, helpers) + self._prepare_files(routines, funcname) + self._process_files(routines) + mod = __import__(self.module_name) + finally: + sys.path.remove(workdir) + CodeWrapper._module_counter += 1 + os.chdir(oldwork) + if not self.filepath: + try: + shutil.rmtree(workdir) + except OSError: + # Could be some issues on Windows + pass + + return self._get_wrapped_function(mod, funcname) + + def _generate_code(self, main_routines, helper_routines): + all_routines = main_routines + helper_routines + self.generator.write( + all_routines, self.filename, True, self.include_header, + self.include_empty) + + def _prepare_files(self, routines, funcname): + + # C + codefilename = self.module_name + '.c' + with open(codefilename, 'w') as f: + self.dump_c(routines, f, self.filename, funcname=funcname) + + # setup.py + with open('setup.py', 'w') as f: + self.dump_setup(f) + + @classmethod + def _get_wrapped_function(cls, mod, name): + return getattr(mod, name) + + def dump_setup(self, f): + setup = _ufunc_setup.substitute(module=self.module_name, + filename=self.filename) + f.write(setup) + + def dump_c(self, routines, f, prefix, funcname=None): + """Write a C file with Python wrappers + + This file contains all the definitions of the routines in c code. + + Arguments + --------- + routines + List of Routine instances + f + File-like object to write the file to + prefix + The filename prefix, used to name the imported module. + funcname + Name of the main function to be returned. + """ + if funcname is None: + if len(routines) == 1: + funcname = routines[0].name + else: + msg = 'funcname must be specified for multiple output routines' + raise ValueError(msg) + functions = [] + function_creation = [] + ufunc_init = [] + module = self.module_name + include_file = "\"{}.h\"".format(prefix) + top = _ufunc_top.substitute(include_file=include_file, module=module) + + name = funcname + + # Partition the C function arguments into categories + # Here we assume all routines accept the same arguments + r_index = 0 + py_in, _ = self._partition_args(routines[0].arguments) + n_in = len(py_in) + n_out = len(routines) + + # Declare Args + form = "char *{0}{1} = args[{2}];" + arg_decs = [form.format('in', i, i) for i in range(n_in)] + arg_decs.extend([form.format('out', i, i+n_in) for i in range(n_out)]) + declare_args = '\n '.join(arg_decs) + + # Declare Steps + form = "npy_intp {0}{1}_step = steps[{2}];" + step_decs = [form.format('in', i, i) for i in range(n_in)] + step_decs.extend([form.format('out', i, i+n_in) for i in range(n_out)]) + declare_steps = '\n '.join(step_decs) + + # Call Args + form = "*(double *)in{0}" + call_args = ', '.join([form.format(a) for a in range(n_in)]) + + # Step Increments + form = "{0}{1} += {0}{1}_step;" + step_incs = [form.format('in', i) for i in range(n_in)] + step_incs.extend([form.format('out', i, i) for i in range(n_out)]) + step_increments = '\n '.join(step_incs) + + # Types + n_types = n_in + n_out + types = "{" + ', '.join(["NPY_DOUBLE"]*n_types) + "};" + + # Docstring + docstring = '"Created in SymPy with Ufuncify"' + + # Function Creation + function_creation.append("PyObject *ufunc{};".format(r_index)) + + # Ufunc initialization + init_form = _ufunc_init_form.substitute(module=module, + funcname=name, + docstring=docstring, + n_in=n_in, n_out=n_out, + ind=r_index) + ufunc_init.append(init_form) + + outcalls = [_ufunc_outcalls.substitute( + outnum=i, call_args=call_args, funcname=routines[i].name) for i in + range(n_out)] + + body = _ufunc_body.substitute(module=module, funcname=name, + declare_args=declare_args, + declare_steps=declare_steps, + call_args=call_args, + step_increments=step_increments, + n_types=n_types, types=types, + outcalls='\n '.join(outcalls)) + functions.append(body) + + body = '\n\n'.join(functions) + ufunc_init = '\n '.join(ufunc_init) + function_creation = '\n '.join(function_creation) + bottom = _ufunc_bottom.substitute(module=module, + ufunc_init=ufunc_init, + function_creation=function_creation) + text = [top, body, bottom] + f.write('\n\n'.join(text)) + + def _partition_args(self, args): + """Group function arguments into categories.""" + py_in = [] + py_out = [] + for arg in args: + if isinstance(arg, OutputArgument): + py_out.append(arg) + elif isinstance(arg, InOutArgument): + raise ValueError("Ufuncify doesn't support InOutArguments") + else: + py_in.append(arg) + return py_in, py_out + + +@cacheit +@doctest_depends_on(exe=('f2py', 'gfortran', 'gcc'), modules=('numpy',)) +def ufuncify(args, expr, language=None, backend='numpy', tempdir=None, + flags=None, verbose=False, helpers=None, **kwargs): + """Generates a binary function that supports broadcasting on numpy arrays. + + Parameters + ========== + + args : iterable + Either a Symbol or an iterable of symbols. Specifies the argument + sequence for the function. + expr + A SymPy expression that defines the element wise operation. + language : string, optional + If supplied, (options: 'C' or 'F95'), specifies the language of the + generated code. If ``None`` [default], the language is inferred based + upon the specified backend. + backend : string, optional + Backend used to wrap the generated code. Either 'numpy' [default], + 'cython', or 'f2py'. + tempdir : string, optional + Path to directory for temporary files. If this argument is supplied, + the generated code and the wrapper input files are left intact in + the specified path. + flags : iterable, optional + Additional option flags that will be passed to the backend. + verbose : bool, optional + If True, autowrap will not mute the command line backends. This can + be helpful for debugging. + helpers : 3-tuple or iterable of 3-tuples, optional + Used to define auxiliary functions needed for the main expression. + Each tuple should be of the form (name, expr, args) where: + + - name : str, the function name + - expr : sympy expression, the function + - args : iterable, the function arguments (can be any iterable of symbols) + + kwargs : dict + These kwargs will be passed to autowrap if the `f2py` or `cython` + backend is used and ignored if the `numpy` backend is used. + + Notes + ===== + + The default backend ('numpy') will create actual instances of + ``numpy.ufunc``. These support ndimensional broadcasting, and implicit type + conversion. Use of the other backends will result in a "ufunc-like" + function, which requires equal length 1-dimensional arrays for all + arguments, and will not perform any type conversions. + + References + ========== + + .. [1] https://numpy.org/doc/stable/reference/ufuncs.html + + Examples + ======== + + Basic usage: + + >>> from sympy.utilities.autowrap import ufuncify + >>> from sympy.abc import x, y + >>> import numpy as np + >>> f = ufuncify((x, y), y + x**2) + >>> type(f) + + >>> f([1, 2, 3], 2) + array([ 3., 6., 11.]) + >>> f(np.arange(5), 3) + array([ 3., 4., 7., 12., 19.]) + + Using helper functions: + + >>> from sympy import Function + >>> helper_func = Function('helper_func') # Define symbolic function + >>> expr = x**2 + y*helper_func(x) # Main expression using helper function + >>> # Define helper_func(x) = x**3 + >>> f = ufuncify((x, y), expr, helpers=[('helper_func', x**3, [x])]) + >>> f([1, 2], [3, 4]) + array([ 4., 36.]) + + Type handling with different backends: + + For the 'f2py' and 'cython' backends, inputs are required to be equal length + 1-dimensional arrays. The 'f2py' backend will perform type conversion, but + the Cython backend will error if the inputs are not of the expected type. + + >>> f_fortran = ufuncify((x, y), y + x**2, backend='f2py') + >>> f_fortran(1, 2) + array([ 3.]) + >>> f_fortran(np.array([1, 2, 3]), np.array([1.0, 2.0, 3.0])) + array([ 2., 6., 12.]) + >>> f_cython = ufuncify((x, y), y + x**2, backend='Cython') + >>> f_cython(1, 2) # doctest: +ELLIPSIS + Traceback (most recent call last): + ... + TypeError: Argument '_x' has incorrect type (expected numpy.ndarray, got int) + >>> f_cython(np.array([1.0]), np.array([2.0])) + array([ 3.]) + + """ + + if isinstance(args, Symbol): + args = (args,) + else: + args = tuple(args) + + if language: + _validate_backend_language(backend, language) + else: + language = _infer_language(backend) + + helpers = helpers if helpers else () + flags = flags if flags else () + + if backend.upper() == 'NUMPY': + # maxargs is set by numpy compile-time constant NPY_MAXARGS + # If a future version of numpy modifies or removes this restriction + # this variable should be changed or removed + maxargs = 32 + helps = [] + for name, expr, args in helpers: + helps.append(make_routine(name, expr, args)) + code_wrapper = UfuncifyCodeWrapper(C99CodeGen("ufuncify"), tempdir, + flags, verbose) + if not isinstance(expr, (list, tuple)): + expr = [expr] + if len(expr) == 0: + raise ValueError('Expression iterable has zero length') + if len(expr) + len(args) > maxargs: + msg = ('Cannot create ufunc with more than {0} total arguments: ' + 'got {1} in, {2} out') + raise ValueError(msg.format(maxargs, len(args), len(expr))) + routines = [make_routine('autofunc{}'.format(idx), exprx, args) for + idx, exprx in enumerate(expr)] + return code_wrapper.wrap_code(routines, helpers=helps) + else: + # Dummies are used for all added expressions to prevent name clashes + # within the original expression. + y = IndexedBase(Dummy('y')) + m = Dummy('m', integer=True) + i = Idx(Dummy('i', integer=True), m) + f_dummy = Dummy('f') + f = implemented_function('%s_%d' % (f_dummy.name, f_dummy.dummy_index), Lambda(args, expr)) + # For each of the args create an indexed version. + indexed_args = [IndexedBase(Dummy(str(a))) for a in args] + # Order the arguments (out, args, dim) + args = [y] + indexed_args + [m] + args_with_indices = [a[i] for a in indexed_args] + return autowrap(Eq(y[i], f(*args_with_indices)), language, backend, + tempdir, args, flags, verbose, helpers, **kwargs) diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/codegen.py b/.venv/lib/python3.12/site-packages/sympy/utilities/codegen.py new file mode 100644 index 0000000000000000000000000000000000000000..9ac8772fc000d707ae33c67eaa44b4c281157ab0 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/codegen.py @@ -0,0 +1,2237 @@ +""" +module for generating C, C++, Fortran77, Fortran90, Julia, Rust +and Octave/Matlab routines that evaluate SymPy expressions. +This module is work in progress. +Only the milestones with a '+' character in the list below have been completed. + +--- How is sympy.utilities.codegen different from sympy.printing.ccode? --- + +We considered the idea to extend the printing routines for SymPy functions in +such a way that it prints complete compilable code, but this leads to a few +unsurmountable issues that can only be tackled with dedicated code generator: + +- For C, one needs both a code and a header file, while the printing routines + generate just one string. This code generator can be extended to support + .pyf files for f2py. + +- SymPy functions are not concerned with programming-technical issues, such + as input, output and input-output arguments. Other examples are contiguous + or non-contiguous arrays, including headers of other libraries such as gsl + or others. + +- It is highly interesting to evaluate several SymPy functions in one C + routine, eventually sharing common intermediate results with the help + of the cse routine. This is more than just printing. + +- From the programming perspective, expressions with constants should be + evaluated in the code generator as much as possible. This is different + for printing. + +--- Basic assumptions --- + +* A generic Routine data structure describes the routine that must be + translated into C/Fortran/... code. This data structure covers all + features present in one or more of the supported languages. + +* Descendants from the CodeGen class transform multiple Routine instances + into compilable code. Each derived class translates into a specific + language. + +* In many cases, one wants a simple workflow. The friendly functions in the + last part are a simple api on top of the Routine/CodeGen stuff. They are + easier to use, but are less powerful. + +--- Milestones --- + ++ First working version with scalar input arguments, generating C code, + tests ++ Friendly functions that are easier to use than the rigorous + Routine/CodeGen workflow. ++ Integer and Real numbers as input and output ++ Output arguments ++ InputOutput arguments ++ Sort input/output arguments properly ++ Contiguous array arguments (numpy matrices) ++ Also generate .pyf code for f2py (in autowrap module) ++ Isolate constants and evaluate them beforehand in double precision ++ Fortran 90 ++ Octave/Matlab + +- Common Subexpression Elimination +- User defined comments in the generated code +- Optional extra include lines for libraries/objects that can eval special + functions +- Test other C compilers and libraries: gcc, tcc, libtcc, gcc+gsl, ... +- Contiguous array arguments (SymPy matrices) +- Non-contiguous array arguments (SymPy matrices) +- ccode must raise an error when it encounters something that cannot be + translated into c. ccode(integrate(sin(x)/x, x)) does not make sense. +- Complex numbers as input and output +- A default complex datatype +- Include extra information in the header: date, user, hostname, sha1 + hash, ... +- Fortran 77 +- C++ +- Python +- Julia +- Rust +- ... + +""" + +import os +import textwrap +from io import StringIO + +from sympy import __version__ as sympy_version +from sympy.core import Symbol, S, Tuple, Equality, Function, Basic +from sympy.printing.c import c_code_printers +from sympy.printing.codeprinter import AssignmentError +from sympy.printing.fortran import FCodePrinter +from sympy.printing.julia import JuliaCodePrinter +from sympy.printing.octave import OctaveCodePrinter +from sympy.printing.rust import RustCodePrinter +from sympy.tensor import Idx, Indexed, IndexedBase +from sympy.matrices import (MatrixSymbol, ImmutableMatrix, MatrixBase, + MatrixExpr, MatrixSlice) +from sympy.utilities.iterables import is_sequence + + +__all__ = [ + # description of routines + "Routine", "DataType", "default_datatypes", "get_default_datatype", + "Argument", "InputArgument", "OutputArgument", "Result", + # routines -> code + "CodeGen", "CCodeGen", "FCodeGen", "JuliaCodeGen", "OctaveCodeGen", + "RustCodeGen", + # friendly functions + "codegen", "make_routine", +] + + +# +# Description of routines +# + + +class Routine: + """Generic description of evaluation routine for set of expressions. + + A CodeGen class can translate instances of this class into code in a + particular language. The routine specification covers all the features + present in these languages. The CodeGen part must raise an exception + when certain features are not present in the target language. For + example, multiple return values are possible in Python, but not in C or + Fortran. Another example: Fortran and Python support complex numbers, + while C does not. + + """ + + def __init__(self, name, arguments, results, local_vars, global_vars): + """Initialize a Routine instance. + + Parameters + ========== + + name : string + Name of the routine. + + arguments : list of Arguments + These are things that appear in arguments of a routine, often + appearing on the right-hand side of a function call. These are + commonly InputArguments but in some languages, they can also be + OutputArguments or InOutArguments (e.g., pass-by-reference in C + code). + + results : list of Results + These are the return values of the routine, often appearing on + the left-hand side of a function call. The difference between + Results and OutputArguments and when you should use each is + language-specific. + + local_vars : list of Results + These are variables that will be defined at the beginning of the + function. + + global_vars : list of Symbols + Variables which will not be passed into the function. + + """ + + # extract all input symbols and all symbols appearing in an expression + input_symbols = set() + symbols = set() + for arg in arguments: + if isinstance(arg, OutputArgument): + symbols.update(arg.expr.free_symbols - arg.expr.atoms(Indexed)) + elif isinstance(arg, InputArgument): + input_symbols.add(arg.name) + elif isinstance(arg, InOutArgument): + input_symbols.add(arg.name) + symbols.update(arg.expr.free_symbols - arg.expr.atoms(Indexed)) + else: + raise ValueError("Unknown Routine argument: %s" % arg) + + for r in results: + if not isinstance(r, Result): + raise ValueError("Unknown Routine result: %s" % r) + symbols.update(r.expr.free_symbols - r.expr.atoms(Indexed)) + + local_symbols = set() + for r in local_vars: + if isinstance(r, Result): + symbols.update(r.expr.free_symbols - r.expr.atoms(Indexed)) + local_symbols.add(r.name) + else: + local_symbols.add(r) + + symbols = {s.label if isinstance(s, Idx) else s for s in symbols} + + # Check that all symbols in the expressions are covered by + # InputArguments/InOutArguments---subset because user could + # specify additional (unused) InputArguments or local_vars. + notcovered = symbols.difference( + input_symbols.union(local_symbols).union(global_vars)) + if notcovered != set(): + raise ValueError("Symbols needed for output are not in input " + + ", ".join([str(x) for x in notcovered])) + + self.name = name + self.arguments = arguments + self.results = results + self.local_vars = local_vars + self.global_vars = global_vars + + def __str__(self): + return self.__class__.__name__ + "({name!r}, {arguments}, {results}, {local_vars}, {global_vars})".format(**self.__dict__) + + __repr__ = __str__ + + @property + def variables(self): + """Returns a set of all variables possibly used in the routine. + + For routines with unnamed return values, the dummies that may or + may not be used will be included in the set. + + """ + v = set(self.local_vars) + v.update(arg.name for arg in self.arguments) + v.update(res.result_var for res in self.results) + return v + + @property + def result_variables(self): + """Returns a list of OutputArgument, InOutArgument and Result. + + If return values are present, they are at the end of the list. + """ + args = [arg for arg in self.arguments if isinstance( + arg, (OutputArgument, InOutArgument))] + args.extend(self.results) + return args + + +class DataType: + """Holds strings for a certain datatype in different languages.""" + def __init__(self, cname, fname, pyname, jlname, octname, rsname): + self.cname = cname + self.fname = fname + self.pyname = pyname + self.jlname = jlname + self.octname = octname + self.rsname = rsname + + +default_datatypes = { + "int": DataType("int", "INTEGER*4", "int", "", "", "i32"), + "float": DataType("double", "REAL*8", "float", "", "", "f64"), + "complex": DataType("double", "COMPLEX*16", "complex", "", "", "float") #FIXME: + # complex is only supported in fortran, python, julia, and octave. + # So to not break c or rust code generation, we stick with double or + # float, respectively (but actually should raise an exception for + # explicitly complex variables (x.is_complex==True)) +} + + +COMPLEX_ALLOWED = False +def get_default_datatype(expr, complex_allowed=None): + """Derives an appropriate datatype based on the expression.""" + if complex_allowed is None: + complex_allowed = COMPLEX_ALLOWED + if complex_allowed: + final_dtype = "complex" + else: + final_dtype = "float" + if expr.is_integer: + return default_datatypes["int"] + elif expr.is_real: + return default_datatypes["float"] + elif isinstance(expr, MatrixBase): + #check all entries + dt = "int" + for element in expr: + if dt == "int" and not element.is_integer: + dt = "float" + if dt == "float" and not element.is_real: + return default_datatypes[final_dtype] + return default_datatypes[dt] + else: + return default_datatypes[final_dtype] + + +class Variable: + """Represents a typed variable.""" + + def __init__(self, name, datatype=None, dimensions=None, precision=None): + """Return a new variable. + + Parameters + ========== + + name : Symbol or MatrixSymbol + + datatype : optional + When not given, the data type will be guessed based on the + assumptions on the symbol argument. + + dimensions : sequence containing tuples, optional + If present, the argument is interpreted as an array, where this + sequence of tuples specifies (lower, upper) bounds for each + index of the array. + + precision : int, optional + Controls the precision of floating point constants. + + """ + if not isinstance(name, (Symbol, MatrixSymbol)): + raise TypeError("The first argument must be a SymPy symbol.") + if datatype is None: + datatype = get_default_datatype(name) + elif not isinstance(datatype, DataType): + raise TypeError("The (optional) `datatype' argument must be an " + "instance of the DataType class.") + if dimensions and not isinstance(dimensions, (tuple, list)): + raise TypeError( + "The dimensions argument must be a sequence of tuples") + + self._name = name + self._datatype = { + 'C': datatype.cname, + 'FORTRAN': datatype.fname, + 'JULIA': datatype.jlname, + 'OCTAVE': datatype.octname, + 'PYTHON': datatype.pyname, + 'RUST': datatype.rsname, + } + self.dimensions = dimensions + self.precision = precision + + def __str__(self): + return "%s(%r)" % (self.__class__.__name__, self.name) + + __repr__ = __str__ + + @property + def name(self): + return self._name + + def get_datatype(self, language): + """Returns the datatype string for the requested language. + + Examples + ======== + + >>> from sympy import Symbol + >>> from sympy.utilities.codegen import Variable + >>> x = Variable(Symbol('x')) + >>> x.get_datatype('c') + 'double' + >>> x.get_datatype('fortran') + 'REAL*8' + + """ + try: + return self._datatype[language.upper()] + except KeyError: + raise CodeGenError("Has datatypes for languages: %s" % + ", ".join(self._datatype)) + + +class Argument(Variable): + """An abstract Argument data structure: a name and a data type. + + This structure is refined in the descendants below. + + """ + pass + + +class InputArgument(Argument): + pass + + +class ResultBase: + """Base class for all "outgoing" information from a routine. + + Objects of this class stores a SymPy expression, and a SymPy object + representing a result variable that will be used in the generated code + only if necessary. + + """ + def __init__(self, expr, result_var): + self.expr = expr + self.result_var = result_var + + def __str__(self): + return "%s(%r, %r)" % (self.__class__.__name__, self.expr, + self.result_var) + + __repr__ = __str__ + + +class OutputArgument(Argument, ResultBase): + """OutputArgument are always initialized in the routine.""" + + def __init__(self, name, result_var, expr, datatype=None, dimensions=None, precision=None): + """Return a new variable. + + Parameters + ========== + + name : Symbol, MatrixSymbol + The name of this variable. When used for code generation, this + might appear, for example, in the prototype of function in the + argument list. + + result_var : Symbol, Indexed + Something that can be used to assign a value to this variable. + Typically the same as `name` but for Indexed this should be e.g., + "y[i]" whereas `name` should be the Symbol "y". + + expr : object + The expression that should be output, typically a SymPy + expression. + + datatype : optional + When not given, the data type will be guessed based on the + assumptions on the symbol argument. + + dimensions : sequence containing tuples, optional + If present, the argument is interpreted as an array, where this + sequence of tuples specifies (lower, upper) bounds for each + index of the array. + + precision : int, optional + Controls the precision of floating point constants. + + """ + + Argument.__init__(self, name, datatype, dimensions, precision) + ResultBase.__init__(self, expr, result_var) + + def __str__(self): + return "%s(%r, %r, %r)" % (self.__class__.__name__, self.name, self.result_var, self.expr) + + __repr__ = __str__ + + +class InOutArgument(Argument, ResultBase): + """InOutArgument are never initialized in the routine.""" + + def __init__(self, name, result_var, expr, datatype=None, dimensions=None, precision=None): + if not datatype: + datatype = get_default_datatype(expr) + Argument.__init__(self, name, datatype, dimensions, precision) + ResultBase.__init__(self, expr, result_var) + __init__.__doc__ = OutputArgument.__init__.__doc__ + + + def __str__(self): + return "%s(%r, %r, %r)" % (self.__class__.__name__, self.name, self.expr, + self.result_var) + + __repr__ = __str__ + + +class Result(Variable, ResultBase): + """An expression for a return value. + + The name result is used to avoid conflicts with the reserved word + "return" in the Python language. It is also shorter than ReturnValue. + + These may or may not need a name in the destination (e.g., "return(x*y)" + might return a value without ever naming it). + + """ + + def __init__(self, expr, name=None, result_var=None, datatype=None, + dimensions=None, precision=None): + """Initialize a return value. + + Parameters + ========== + + expr : SymPy expression + + name : Symbol, MatrixSymbol, optional + The name of this return variable. When used for code generation, + this might appear, for example, in the prototype of function in a + list of return values. A dummy name is generated if omitted. + + result_var : Symbol, Indexed, optional + Something that can be used to assign a value to this variable. + Typically the same as `name` but for Indexed this should be e.g., + "y[i]" whereas `name` should be the Symbol "y". Defaults to + `name` if omitted. + + datatype : optional + When not given, the data type will be guessed based on the + assumptions on the expr argument. + + dimensions : sequence containing tuples, optional + If present, this variable is interpreted as an array, + where this sequence of tuples specifies (lower, upper) + bounds for each index of the array. + + precision : int, optional + Controls the precision of floating point constants. + + """ + # Basic because it is the base class for all types of expressions + if not isinstance(expr, (Basic, MatrixBase)): + raise TypeError("The first argument must be a SymPy expression.") + + if name is None: + name = 'result_%d' % abs(hash(expr)) + + if datatype is None: + #try to infer data type from the expression + datatype = get_default_datatype(expr) + + if isinstance(name, str): + if isinstance(expr, (MatrixBase, MatrixExpr)): + name = MatrixSymbol(name, *expr.shape) + else: + name = Symbol(name) + + if result_var is None: + result_var = name + + Variable.__init__(self, name, datatype=datatype, + dimensions=dimensions, precision=precision) + ResultBase.__init__(self, expr, result_var) + + def __str__(self): + return "%s(%r, %r, %r)" % (self.__class__.__name__, self.expr, self.name, + self.result_var) + + __repr__ = __str__ + + +# +# Transformation of routine objects into code +# + +class CodeGen: + """Abstract class for the code generators.""" + + printer = None # will be set to an instance of a CodePrinter subclass + + def _indent_code(self, codelines): + return self.printer.indent_code(codelines) + + def _printer_method_with_settings(self, method, settings=None, *args, **kwargs): + settings = settings or {} + ori = {k: self.printer._settings[k] for k in settings} + for k, v in settings.items(): + self.printer._settings[k] = v + result = getattr(self.printer, method)(*args, **kwargs) + for k, v in ori.items(): + self.printer._settings[k] = v + return result + + def _get_symbol(self, s): + """Returns the symbol as fcode prints it.""" + if self.printer._settings['human']: + expr_str = self.printer.doprint(s) + else: + constants, not_supported, expr_str = self.printer.doprint(s) + if constants or not_supported: + raise ValueError("Failed to print %s" % str(s)) + return expr_str.strip() + + def __init__(self, project="project", cse=False): + """Initialize a code generator. + + Derived classes will offer more options that affect the generated + code. + + """ + self.project = project + self.cse = cse + + def routine(self, name, expr, argument_sequence=None, global_vars=None): + """Creates an Routine object that is appropriate for this language. + + This implementation is appropriate for at least C/Fortran. Subclasses + can override this if necessary. + + Here, we assume at most one return value (the l-value) which must be + scalar. Additional outputs are OutputArguments (e.g., pointers on + right-hand-side or pass-by-reference). Matrices are always returned + via OutputArguments. If ``argument_sequence`` is None, arguments will + be ordered alphabetically, but with all InputArguments first, and then + OutputArgument and InOutArguments. + + """ + + if self.cse: + from sympy.simplify.cse_main import cse + + if is_sequence(expr) and not isinstance(expr, (MatrixBase, MatrixExpr)): + if not expr: + raise ValueError("No expression given") + for e in expr: + if not e.is_Equality: + raise CodeGenError("Lists of expressions must all be Equalities. {} is not.".format(e)) + + # create a list of right hand sides and simplify them + rhs = [e.rhs for e in expr] + common, simplified = cse(rhs) + + # pack the simplified expressions back up with their left hand sides + expr = [Equality(e.lhs, rhs) for e, rhs in zip(expr, simplified)] + else: + if isinstance(expr, Equality): + common, simplified = cse(expr.rhs) #, ignore=in_out_args) + expr = Equality(expr.lhs, simplified[0]) + else: + common, simplified = cse(expr) + expr = simplified + + local_vars = [Result(b,a) for a,b in common] + local_symbols = {a for a,_ in common} + local_expressions = Tuple(*[b for _,b in common]) + else: + local_expressions = Tuple() + + if is_sequence(expr) and not isinstance(expr, (MatrixBase, MatrixExpr)): + if not expr: + raise ValueError("No expression given") + expressions = Tuple(*expr) + else: + expressions = Tuple(expr) + + if self.cse: + if {i.label for i in expressions.atoms(Idx)} != set(): + raise CodeGenError("CSE and Indexed expressions do not play well together yet") + else: + # local variables for indexed expressions + local_vars = {i.label for i in expressions.atoms(Idx)} + local_symbols = local_vars + + # global variables + global_vars = set() if global_vars is None else set(global_vars) + + # symbols that should be arguments + symbols = (expressions.free_symbols | local_expressions.free_symbols) - local_symbols - global_vars + new_symbols = set() + new_symbols.update(symbols) + + for symbol in symbols: + if isinstance(symbol, Idx): + new_symbols.remove(symbol) + new_symbols.update(symbol.args[1].free_symbols) + if isinstance(symbol, Indexed): + new_symbols.remove(symbol) + symbols = new_symbols + + # Decide whether to use output argument or return value + return_val = [] + output_args = [] + for expr in expressions: + if isinstance(expr, Equality): + out_arg = expr.lhs + expr = expr.rhs + if isinstance(out_arg, Indexed): + dims = tuple([ (S.Zero, dim - 1) for dim in out_arg.shape]) + symbol = out_arg.base.label + elif isinstance(out_arg, Symbol): + dims = [] + symbol = out_arg + elif isinstance(out_arg, MatrixSymbol): + dims = tuple([ (S.Zero, dim - 1) for dim in out_arg.shape]) + symbol = out_arg + else: + raise CodeGenError("Only Indexed, Symbol, or MatrixSymbol " + "can define output arguments.") + + if expr.has(symbol): + output_args.append( + InOutArgument(symbol, out_arg, expr, dimensions=dims)) + else: + output_args.append( + OutputArgument(symbol, out_arg, expr, dimensions=dims)) + + # remove duplicate arguments when they are not local variables + if symbol not in local_vars: + # avoid duplicate arguments + symbols.remove(symbol) + elif isinstance(expr, (ImmutableMatrix, MatrixSlice)): + # Create a "dummy" MatrixSymbol to use as the Output arg + out_arg = MatrixSymbol('out_%s' % abs(hash(expr)), *expr.shape) + dims = tuple([(S.Zero, dim - 1) for dim in out_arg.shape]) + output_args.append( + OutputArgument(out_arg, out_arg, expr, dimensions=dims)) + else: + return_val.append(Result(expr)) + + arg_list = [] + + # setup input argument list + + # helper to get dimensions for data for array-like args + def dimensions(s): + return [(S.Zero, dim - 1) for dim in s.shape] + + array_symbols = {} + for array in expressions.atoms(Indexed) | local_expressions.atoms(Indexed): + array_symbols[array.base.label] = array + for array in expressions.atoms(MatrixSymbol) | local_expressions.atoms(MatrixSymbol): + array_symbols[array] = array + + for symbol in sorted(symbols, key=str): + if symbol in array_symbols: + array = array_symbols[symbol] + metadata = {'dimensions': dimensions(array)} + else: + metadata = {} + + arg_list.append(InputArgument(symbol, **metadata)) + + output_args.sort(key=lambda x: str(x.name)) + arg_list.extend(output_args) + + if argument_sequence is not None: + # if the user has supplied IndexedBase instances, we'll accept that + new_sequence = [] + for arg in argument_sequence: + if isinstance(arg, IndexedBase): + new_sequence.append(arg.label) + else: + new_sequence.append(arg) + argument_sequence = new_sequence + + missing = [x for x in arg_list if x.name not in argument_sequence] + if missing: + msg = "Argument list didn't specify: {0} " + msg = msg.format(", ".join([str(m.name) for m in missing])) + raise CodeGenArgumentListError(msg, missing) + + # create redundant arguments to produce the requested sequence + name_arg_dict = {x.name: x for x in arg_list} + new_args = [] + for symbol in argument_sequence: + try: + new_args.append(name_arg_dict[symbol]) + except KeyError: + if isinstance(symbol, (IndexedBase, MatrixSymbol)): + metadata = {'dimensions': dimensions(symbol)} + else: + metadata = {} + new_args.append(InputArgument(symbol, **metadata)) + arg_list = new_args + + return Routine(name, arg_list, return_val, local_vars, global_vars) + + def write(self, routines, prefix, to_files=False, header=True, empty=True): + """Writes all the source code files for the given routines. + + The generated source is returned as a list of (filename, contents) + tuples, or is written to files (see below). Each filename consists + of the given prefix, appended with an appropriate extension. + + Parameters + ========== + + routines : list + A list of Routine instances to be written + + prefix : string + The prefix for the output files + + to_files : bool, optional + When True, the output is written to files. Otherwise, a list + of (filename, contents) tuples is returned. [default: False] + + header : bool, optional + When True, a header comment is included on top of each source + file. [default: True] + + empty : bool, optional + When True, empty lines are included to structure the source + files. [default: True] + + """ + if to_files: + for dump_fn in self.dump_fns: + filename = "%s.%s" % (prefix, dump_fn.extension) + with open(filename, "w") as f: + dump_fn(self, routines, f, prefix, header, empty) + else: + result = [] + for dump_fn in self.dump_fns: + filename = "%s.%s" % (prefix, dump_fn.extension) + contents = StringIO() + dump_fn(self, routines, contents, prefix, header, empty) + result.append((filename, contents.getvalue())) + return result + + def dump_code(self, routines, f, prefix, header=True, empty=True): + """Write the code by calling language specific methods. + + The generated file contains all the definitions of the routines in + low-level code and refers to the header file if appropriate. + + Parameters + ========== + + routines : list + A list of Routine instances. + + f : file-like + Where to write the file. + + prefix : string + The filename prefix, used to refer to the proper header file. + Only the basename of the prefix is used. + + header : bool, optional + When True, a header comment is included on top of each source + file. [default : True] + + empty : bool, optional + When True, empty lines are included to structure the source + files. [default : True] + + """ + + code_lines = self._preprocessor_statements(prefix) + + for routine in routines: + if empty: + code_lines.append("\n") + code_lines.extend(self._get_routine_opening(routine)) + code_lines.extend(self._declare_arguments(routine)) + code_lines.extend(self._declare_globals(routine)) + code_lines.extend(self._declare_locals(routine)) + if empty: + code_lines.append("\n") + code_lines.extend(self._call_printer(routine)) + if empty: + code_lines.append("\n") + code_lines.extend(self._get_routine_ending(routine)) + + code_lines = self._indent_code(''.join(code_lines)) + + if header: + code_lines = ''.join(self._get_header() + [code_lines]) + + if code_lines: + f.write(code_lines) + + +class CodeGenError(Exception): + pass + + +class CodeGenArgumentListError(Exception): + @property + def missing_args(self): + return self.args[1] + + +header_comment = """Code generated with SymPy %(version)s + +See http://www.sympy.org/ for more information. + +This file is part of '%(project)s' +""" + + +class CCodeGen(CodeGen): + """Generator for C code. + + The .write() method inherited from CodeGen will output a code file and + an interface file, .c and .h respectively. + + """ + + code_extension = "c" + interface_extension = "h" + standard = 'c99' + + def __init__(self, project="project", printer=None, + preprocessor_statements=None, cse=False): + super().__init__(project=project, cse=cse) + self.printer = printer or c_code_printers[self.standard.lower()]() + + self.preprocessor_statements = preprocessor_statements + if preprocessor_statements is None: + self.preprocessor_statements = ['#include '] + + def _get_header(self): + """Writes a common header for the generated files.""" + code_lines = [] + code_lines.append("/" + "*"*78 + '\n') + tmp = header_comment % {"version": sympy_version, + "project": self.project} + for line in tmp.splitlines(): + code_lines.append(" *%s*\n" % line.center(76)) + code_lines.append(" " + "*"*78 + "/\n") + return code_lines + + def get_prototype(self, routine): + """Returns a string for the function prototype of the routine. + + If the routine has multiple result objects, an CodeGenError is + raised. + + See: https://en.wikipedia.org/wiki/Function_prototype + + """ + if len(routine.results) > 1: + raise CodeGenError("C only supports a single or no return value.") + elif len(routine.results) == 1: + ctype = routine.results[0].get_datatype('C') + else: + ctype = "void" + + type_args = [] + for arg in routine.arguments: + name = self.printer.doprint(arg.name) + if arg.dimensions or isinstance(arg, ResultBase): + type_args.append((arg.get_datatype('C'), "*%s" % name)) + else: + type_args.append((arg.get_datatype('C'), name)) + arguments = ", ".join([ "%s %s" % t for t in type_args]) + return "%s %s(%s)" % (ctype, routine.name, arguments) + + def _preprocessor_statements(self, prefix): + code_lines = [] + code_lines.append('#include "{}.h"'.format(os.path.basename(prefix))) + code_lines.extend(self.preprocessor_statements) + code_lines = ['{}\n'.format(l) for l in code_lines] + return code_lines + + def _get_routine_opening(self, routine): + prototype = self.get_prototype(routine) + return ["%s {\n" % prototype] + + def _declare_arguments(self, routine): + # arguments are declared in prototype + return [] + + def _declare_globals(self, routine): + # global variables are not explicitly declared within C functions + return [] + + def _declare_locals(self, routine): + + # Compose a list of symbols to be dereferenced in the function + # body. These are the arguments that were passed by a reference + # pointer, excluding arrays. + dereference = [] + for arg in routine.arguments: + if isinstance(arg, ResultBase) and not arg.dimensions: + dereference.append(arg.name) + + code_lines = [] + for result in routine.local_vars: + + # local variables that are simple symbols such as those used as indices into + # for loops are defined declared elsewhere. + if not isinstance(result, Result): + continue + + if result.name != result.result_var: + raise CodeGen("Result variable and name should match: {}".format(result)) + assign_to = result.name + t = result.get_datatype('c') + if isinstance(result.expr, (MatrixBase, MatrixExpr)): + dims = result.expr.shape + code_lines.append("{} {}[{}];\n".format(t, str(assign_to), dims[0]*dims[1])) + prefix = "" + else: + prefix = "const {} ".format(t) + + constants, not_c, c_expr = self._printer_method_with_settings( + 'doprint', {"human": False, "dereference": dereference, "strict": False}, + result.expr, assign_to=assign_to) + + for name, value in sorted(constants, key=str): + code_lines.append("double const %s = %s;\n" % (name, value)) + + code_lines.append("{}{}\n".format(prefix, c_expr)) + + return code_lines + + def _call_printer(self, routine): + code_lines = [] + + # Compose a list of symbols to be dereferenced in the function + # body. These are the arguments that were passed by a reference + # pointer, excluding arrays. + dereference = [] + for arg in routine.arguments: + if isinstance(arg, ResultBase) and not arg.dimensions: + dereference.append(arg.name) + + return_val = None + for result in routine.result_variables: + if isinstance(result, Result): + assign_to = routine.name + "_result" + t = result.get_datatype('c') + code_lines.append("{} {};\n".format(t, str(assign_to))) + return_val = assign_to + else: + assign_to = result.result_var + + try: + constants, not_c, c_expr = self._printer_method_with_settings( + 'doprint', {"human": False, "dereference": dereference, "strict": False}, + result.expr, assign_to=assign_to) + except AssignmentError: + assign_to = result.result_var + code_lines.append( + "%s %s;\n" % (result.get_datatype('c'), str(assign_to))) + constants, not_c, c_expr = self._printer_method_with_settings( + 'doprint', {"human": False, "dereference": dereference, "strict": False}, + result.expr, assign_to=assign_to) + + for name, value in sorted(constants, key=str): + code_lines.append("double const %s = %s;\n" % (name, value)) + code_lines.append("%s\n" % c_expr) + + if return_val: + code_lines.append(" return %s;\n" % return_val) + return code_lines + + def _get_routine_ending(self, routine): + return ["}\n"] + + def dump_c(self, routines, f, prefix, header=True, empty=True): + self.dump_code(routines, f, prefix, header, empty) + dump_c.extension = code_extension # type: ignore + dump_c.__doc__ = CodeGen.dump_code.__doc__ + + def dump_h(self, routines, f, prefix, header=True, empty=True): + """Writes the C header file. + + This file contains all the function declarations. + + Parameters + ========== + + routines : list + A list of Routine instances. + + f : file-like + Where to write the file. + + prefix : string + The filename prefix, used to construct the include guards. + Only the basename of the prefix is used. + + header : bool, optional + When True, a header comment is included on top of each source + file. [default : True] + + empty : bool, optional + When True, empty lines are included to structure the source + files. [default : True] + + """ + if header: + print(''.join(self._get_header()), file=f) + guard_name = "%s__%s__H" % (self.project.replace( + " ", "_").upper(), prefix.replace("/", "_").upper()) + # include guards + if empty: + print(file=f) + print("#ifndef %s" % guard_name, file=f) + print("#define %s" % guard_name, file=f) + if empty: + print(file=f) + # declaration of the function prototypes + for routine in routines: + prototype = self.get_prototype(routine) + print("%s;" % prototype, file=f) + # end if include guards + if empty: + print(file=f) + print("#endif", file=f) + if empty: + print(file=f) + dump_h.extension = interface_extension # type: ignore + + # This list of dump functions is used by CodeGen.write to know which dump + # functions it has to call. + dump_fns = [dump_c, dump_h] + +class C89CodeGen(CCodeGen): + standard = 'C89' + +class C99CodeGen(CCodeGen): + standard = 'C99' + +class FCodeGen(CodeGen): + """Generator for Fortran 95 code + + The .write() method inherited from CodeGen will output a code file and + an interface file, .f90 and .h respectively. + + """ + + code_extension = "f90" + interface_extension = "h" + + def __init__(self, project='project', printer=None): + super().__init__(project) + self.printer = printer or FCodePrinter() + + def _get_header(self): + """Writes a common header for the generated files.""" + code_lines = [] + code_lines.append("!" + "*"*78 + '\n') + tmp = header_comment % {"version": sympy_version, + "project": self.project} + for line in tmp.splitlines(): + code_lines.append("!*%s*\n" % line.center(76)) + code_lines.append("!" + "*"*78 + '\n') + return code_lines + + def _preprocessor_statements(self, prefix): + return [] + + def _get_routine_opening(self, routine): + """Returns the opening statements of the fortran routine.""" + code_list = [] + if len(routine.results) > 1: + raise CodeGenError( + "Fortran only supports a single or no return value.") + elif len(routine.results) == 1: + result = routine.results[0] + code_list.append(result.get_datatype('fortran')) + code_list.append("function") + else: + code_list.append("subroutine") + + args = ", ".join("%s" % self._get_symbol(arg.name) + for arg in routine.arguments) + + call_sig = "{}({})\n".format(routine.name, args) + # Fortran 95 requires all lines be less than 132 characters, so wrap + # this line before appending. + call_sig = ' &\n'.join(textwrap.wrap(call_sig, + width=60, + break_long_words=False)) + '\n' + code_list.append(call_sig) + code_list = [' '.join(code_list)] + code_list.append('implicit none\n') + return code_list + + def _declare_arguments(self, routine): + # argument type declarations + code_list = [] + array_list = [] + scalar_list = [] + for arg in routine.arguments: + + if isinstance(arg, InputArgument): + typeinfo = "%s, intent(in)" % arg.get_datatype('fortran') + elif isinstance(arg, InOutArgument): + typeinfo = "%s, intent(inout)" % arg.get_datatype('fortran') + elif isinstance(arg, OutputArgument): + typeinfo = "%s, intent(out)" % arg.get_datatype('fortran') + else: + raise CodeGenError("Unknown Argument type: %s" % type(arg)) + + fprint = self._get_symbol + + if arg.dimensions: + # fortran arrays start at 1 + dimstr = ", ".join(["%s:%s" % ( + fprint(dim[0] + 1), fprint(dim[1] + 1)) + for dim in arg.dimensions]) + typeinfo += ", dimension(%s)" % dimstr + array_list.append("%s :: %s\n" % (typeinfo, fprint(arg.name))) + else: + scalar_list.append("%s :: %s\n" % (typeinfo, fprint(arg.name))) + + # scalars first, because they can be used in array declarations + code_list.extend(scalar_list) + code_list.extend(array_list) + + return code_list + + def _declare_globals(self, routine): + # Global variables not explicitly declared within Fortran 90 functions. + # Note: a future F77 mode may need to generate "common" blocks. + return [] + + def _declare_locals(self, routine): + code_list = [] + for var in sorted(routine.local_vars, key=str): + typeinfo = get_default_datatype(var) + code_list.append("%s :: %s\n" % ( + typeinfo.fname, self._get_symbol(var))) + return code_list + + def _get_routine_ending(self, routine): + """Returns the closing statements of the fortran routine.""" + if len(routine.results) == 1: + return ["end function\n"] + else: + return ["end subroutine\n"] + + def get_interface(self, routine): + """Returns a string for the function interface. + + The routine should have a single result object, which can be None. + If the routine has multiple result objects, a CodeGenError is + raised. + + See: https://en.wikipedia.org/wiki/Function_prototype + + """ + prototype = [ "interface\n" ] + prototype.extend(self._get_routine_opening(routine)) + prototype.extend(self._declare_arguments(routine)) + prototype.extend(self._get_routine_ending(routine)) + prototype.append("end interface\n") + + return "".join(prototype) + + def _call_printer(self, routine): + declarations = [] + code_lines = [] + for result in routine.result_variables: + if isinstance(result, Result): + assign_to = routine.name + elif isinstance(result, (OutputArgument, InOutArgument)): + assign_to = result.result_var + + constants, not_fortran, f_expr = self._printer_method_with_settings( + 'doprint', {"human": False, "source_format": 'free', "standard": 95, "strict": False}, + result.expr, assign_to=assign_to) + + for obj, v in sorted(constants, key=str): + t = get_default_datatype(obj) + declarations.append( + "%s, parameter :: %s = %s\n" % (t.fname, obj, v)) + for obj in sorted(not_fortran, key=str): + t = get_default_datatype(obj) + if isinstance(obj, Function): + name = obj.func + else: + name = obj + declarations.append("%s :: %s\n" % (t.fname, name)) + + code_lines.append("%s\n" % f_expr) + return declarations + code_lines + + def _indent_code(self, codelines): + return self._printer_method_with_settings( + 'indent_code', {"human": False, "source_format": 'free', "strict": False}, codelines) + + def dump_f95(self, routines, f, prefix, header=True, empty=True): + # check that symbols are unique with ignorecase + for r in routines: + lowercase = {str(x).lower() for x in r.variables} + orig_case = {str(x) for x in r.variables} + if len(lowercase) < len(orig_case): + raise CodeGenError("Fortran ignores case. Got symbols: %s" % + (", ".join([str(var) for var in r.variables]))) + self.dump_code(routines, f, prefix, header, empty) + dump_f95.extension = code_extension # type: ignore + dump_f95.__doc__ = CodeGen.dump_code.__doc__ + + def dump_h(self, routines, f, prefix, header=True, empty=True): + """Writes the interface to a header file. + + This file contains all the function declarations. + + Parameters + ========== + + routines : list + A list of Routine instances. + + f : file-like + Where to write the file. + + prefix : string + The filename prefix. + + header : bool, optional + When True, a header comment is included on top of each source + file. [default : True] + + empty : bool, optional + When True, empty lines are included to structure the source + files. [default : True] + + """ + if header: + print(''.join(self._get_header()), file=f) + if empty: + print(file=f) + # declaration of the function prototypes + for routine in routines: + prototype = self.get_interface(routine) + f.write(prototype) + if empty: + print(file=f) + dump_h.extension = interface_extension # type: ignore + + # This list of dump functions is used by CodeGen.write to know which dump + # functions it has to call. + dump_fns = [dump_f95, dump_h] + + +class JuliaCodeGen(CodeGen): + """Generator for Julia code. + + The .write() method inherited from CodeGen will output a code file + .jl. + + """ + + code_extension = "jl" + + def __init__(self, project='project', printer=None): + super().__init__(project) + self.printer = printer or JuliaCodePrinter() + + def routine(self, name, expr, argument_sequence, global_vars): + """Specialized Routine creation for Julia.""" + + if is_sequence(expr) and not isinstance(expr, (MatrixBase, MatrixExpr)): + if not expr: + raise ValueError("No expression given") + expressions = Tuple(*expr) + else: + expressions = Tuple(expr) + + # local variables + local_vars = {i.label for i in expressions.atoms(Idx)} + + # global variables + global_vars = set() if global_vars is None else set(global_vars) + + # symbols that should be arguments + old_symbols = expressions.free_symbols - local_vars - global_vars + symbols = set() + for s in old_symbols: + if isinstance(s, Idx): + symbols.update(s.args[1].free_symbols) + elif not isinstance(s, Indexed): + symbols.add(s) + + # Julia supports multiple return values + return_vals = [] + output_args = [] + for (i, expr) in enumerate(expressions): + if isinstance(expr, Equality): + out_arg = expr.lhs + expr = expr.rhs + symbol = out_arg + if isinstance(out_arg, Indexed): + dims = tuple([ (S.One, dim) for dim in out_arg.shape]) + symbol = out_arg.base.label + output_args.append(InOutArgument(symbol, out_arg, expr, dimensions=dims)) + if not isinstance(out_arg, (Indexed, Symbol, MatrixSymbol)): + raise CodeGenError("Only Indexed, Symbol, or MatrixSymbol " + "can define output arguments.") + + return_vals.append(Result(expr, name=symbol, result_var=out_arg)) + if not expr.has(symbol): + # this is a pure output: remove from the symbols list, so + # it doesn't become an input. + symbols.remove(symbol) + + else: + # we have no name for this output + return_vals.append(Result(expr, name='out%d' % (i+1))) + + # setup input argument list + output_args.sort(key=lambda x: str(x.name)) + arg_list = list(output_args) + array_symbols = {} + for array in expressions.atoms(Indexed): + array_symbols[array.base.label] = array + for array in expressions.atoms(MatrixSymbol): + array_symbols[array] = array + + for symbol in sorted(symbols, key=str): + arg_list.append(InputArgument(symbol)) + + if argument_sequence is not None: + # if the user has supplied IndexedBase instances, we'll accept that + new_sequence = [] + for arg in argument_sequence: + if isinstance(arg, IndexedBase): + new_sequence.append(arg.label) + else: + new_sequence.append(arg) + argument_sequence = new_sequence + + missing = [x for x in arg_list if x.name not in argument_sequence] + if missing: + msg = "Argument list didn't specify: {0} " + msg = msg.format(", ".join([str(m.name) for m in missing])) + raise CodeGenArgumentListError(msg, missing) + + # create redundant arguments to produce the requested sequence + name_arg_dict = {x.name: x for x in arg_list} + new_args = [] + for symbol in argument_sequence: + try: + new_args.append(name_arg_dict[symbol]) + except KeyError: + new_args.append(InputArgument(symbol)) + arg_list = new_args + + return Routine(name, arg_list, return_vals, local_vars, global_vars) + + def _get_header(self): + """Writes a common header for the generated files.""" + code_lines = [] + tmp = header_comment % {"version": sympy_version, + "project": self.project} + for line in tmp.splitlines(): + if line == '': + code_lines.append("#\n") + else: + code_lines.append("# %s\n" % line) + return code_lines + + def _preprocessor_statements(self, prefix): + return [] + + def _get_routine_opening(self, routine): + """Returns the opening statements of the routine.""" + code_list = [] + code_list.append("function ") + + # Inputs + args = [] + for arg in routine.arguments: + if isinstance(arg, OutputArgument): + raise CodeGenError("Julia: invalid argument of type %s" % + str(type(arg))) + if isinstance(arg, (InputArgument, InOutArgument)): + args.append("%s" % self._get_symbol(arg.name)) + args = ", ".join(args) + code_list.append("%s(%s)\n" % (routine.name, args)) + code_list = [ "".join(code_list) ] + + return code_list + + def _declare_arguments(self, routine): + return [] + + def _declare_globals(self, routine): + return [] + + def _declare_locals(self, routine): + return [] + + def _get_routine_ending(self, routine): + outs = [] + for result in routine.results: + if isinstance(result, Result): + # Note: name not result_var; want `y` not `y[i]` for Indexed + s = self._get_symbol(result.name) + else: + raise CodeGenError("unexpected object in Routine results") + outs.append(s) + return ["return " + ", ".join(outs) + "\nend\n"] + + def _call_printer(self, routine): + declarations = [] + code_lines = [] + for result in routine.results: + if isinstance(result, Result): + assign_to = result.result_var + else: + raise CodeGenError("unexpected object in Routine results") + + constants, not_supported, jl_expr = self._printer_method_with_settings( + 'doprint', {"human": False, "strict": False}, result.expr, assign_to=assign_to) + + for obj, v in sorted(constants, key=str): + declarations.append( + "%s = %s\n" % (obj, v)) + for obj in sorted(not_supported, key=str): + if isinstance(obj, Function): + name = obj.func + else: + name = obj + declarations.append( + "# unsupported: %s\n" % (name)) + code_lines.append("%s\n" % (jl_expr)) + return declarations + code_lines + + def _indent_code(self, codelines): + # Note that indenting seems to happen twice, first + # statement-by-statement by JuliaPrinter then again here. + p = JuliaCodePrinter({'human': False, "strict": False}) + return p.indent_code(codelines) + + def dump_jl(self, routines, f, prefix, header=True, empty=True): + self.dump_code(routines, f, prefix, header, empty) + + dump_jl.extension = code_extension # type: ignore + dump_jl.__doc__ = CodeGen.dump_code.__doc__ + + # This list of dump functions is used by CodeGen.write to know which dump + # functions it has to call. + dump_fns = [dump_jl] + + +class OctaveCodeGen(CodeGen): + """Generator for Octave code. + + The .write() method inherited from CodeGen will output a code file + .m. + + Octave .m files usually contain one function. That function name should + match the filename (``prefix``). If you pass multiple ``name_expr`` pairs, + the latter ones are presumed to be private functions accessed by the + primary function. + + You should only pass inputs to ``argument_sequence``: outputs are ordered + according to their order in ``name_expr``. + + """ + + code_extension = "m" + + def __init__(self, project='project', printer=None): + super().__init__(project) + self.printer = printer or OctaveCodePrinter() + + def routine(self, name, expr, argument_sequence, global_vars): + """Specialized Routine creation for Octave.""" + + # FIXME: this is probably general enough for other high-level + # languages, perhaps its the C/Fortran one that is specialized! + + if is_sequence(expr) and not isinstance(expr, (MatrixBase, MatrixExpr)): + if not expr: + raise ValueError("No expression given") + expressions = Tuple(*expr) + else: + expressions = Tuple(expr) + + # local variables + local_vars = {i.label for i in expressions.atoms(Idx)} + + # global variables + global_vars = set() if global_vars is None else set(global_vars) + + # symbols that should be arguments + old_symbols = expressions.free_symbols - local_vars - global_vars + symbols = set() + for s in old_symbols: + if isinstance(s, Idx): + symbols.update(s.args[1].free_symbols) + elif not isinstance(s, Indexed): + symbols.add(s) + + # Octave supports multiple return values + return_vals = [] + for (i, expr) in enumerate(expressions): + if isinstance(expr, Equality): + out_arg = expr.lhs + expr = expr.rhs + symbol = out_arg + if isinstance(out_arg, Indexed): + symbol = out_arg.base.label + if not isinstance(out_arg, (Indexed, Symbol, MatrixSymbol)): + raise CodeGenError("Only Indexed, Symbol, or MatrixSymbol " + "can define output arguments.") + + return_vals.append(Result(expr, name=symbol, result_var=out_arg)) + if not expr.has(symbol): + # this is a pure output: remove from the symbols list, so + # it doesn't become an input. + symbols.remove(symbol) + + else: + # we have no name for this output + return_vals.append(Result(expr, name='out%d' % (i+1))) + + # setup input argument list + arg_list = [] + array_symbols = {} + for array in expressions.atoms(Indexed): + array_symbols[array.base.label] = array + for array in expressions.atoms(MatrixSymbol): + array_symbols[array] = array + + for symbol in sorted(symbols, key=str): + arg_list.append(InputArgument(symbol)) + + if argument_sequence is not None: + # if the user has supplied IndexedBase instances, we'll accept that + new_sequence = [] + for arg in argument_sequence: + if isinstance(arg, IndexedBase): + new_sequence.append(arg.label) + else: + new_sequence.append(arg) + argument_sequence = new_sequence + + missing = [x for x in arg_list if x.name not in argument_sequence] + if missing: + msg = "Argument list didn't specify: {0} " + msg = msg.format(", ".join([str(m.name) for m in missing])) + raise CodeGenArgumentListError(msg, missing) + + # create redundant arguments to produce the requested sequence + name_arg_dict = {x.name: x for x in arg_list} + new_args = [] + for symbol in argument_sequence: + try: + new_args.append(name_arg_dict[symbol]) + except KeyError: + new_args.append(InputArgument(symbol)) + arg_list = new_args + + return Routine(name, arg_list, return_vals, local_vars, global_vars) + + def _get_header(self): + """Writes a common header for the generated files.""" + code_lines = [] + tmp = header_comment % {"version": sympy_version, + "project": self.project} + for line in tmp.splitlines(): + if line == '': + code_lines.append("%\n") + else: + code_lines.append("%% %s\n" % line) + return code_lines + + def _preprocessor_statements(self, prefix): + return [] + + def _get_routine_opening(self, routine): + """Returns the opening statements of the routine.""" + code_list = [] + code_list.append("function ") + + # Outputs + outs = [] + for result in routine.results: + if isinstance(result, Result): + # Note: name not result_var; want `y` not `y(i)` for Indexed + s = self._get_symbol(result.name) + else: + raise CodeGenError("unexpected object in Routine results") + outs.append(s) + if len(outs) > 1: + code_list.append("[" + (", ".join(outs)) + "]") + else: + code_list.append("".join(outs)) + code_list.append(" = ") + + # Inputs + args = [] + for arg in routine.arguments: + if isinstance(arg, (OutputArgument, InOutArgument)): + raise CodeGenError("Octave: invalid argument of type %s" % + str(type(arg))) + if isinstance(arg, InputArgument): + args.append("%s" % self._get_symbol(arg.name)) + args = ", ".join(args) + code_list.append("%s(%s)\n" % (routine.name, args)) + code_list = [ "".join(code_list) ] + + return code_list + + def _declare_arguments(self, routine): + return [] + + def _declare_globals(self, routine): + if not routine.global_vars: + return [] + s = " ".join(sorted([self._get_symbol(g) for g in routine.global_vars])) + return ["global " + s + "\n"] + + def _declare_locals(self, routine): + return [] + + def _get_routine_ending(self, routine): + return ["end\n"] + + def _call_printer(self, routine): + declarations = [] + code_lines = [] + for result in routine.results: + if isinstance(result, Result): + assign_to = result.result_var + else: + raise CodeGenError("unexpected object in Routine results") + + constants, not_supported, oct_expr = self._printer_method_with_settings( + 'doprint', {"human": False, "strict": False}, result.expr, assign_to=assign_to) + + for obj, v in sorted(constants, key=str): + declarations.append( + " %s = %s; %% constant\n" % (obj, v)) + for obj in sorted(not_supported, key=str): + if isinstance(obj, Function): + name = obj.func + else: + name = obj + declarations.append( + " %% unsupported: %s\n" % (name)) + code_lines.append("%s\n" % (oct_expr)) + return declarations + code_lines + + def _indent_code(self, codelines): + return self._printer_method_with_settings( + 'indent_code', {"human": False, "strict": False}, codelines) + + def dump_m(self, routines, f, prefix, header=True, empty=True, inline=True): + # Note used to call self.dump_code() but we need more control for header + + code_lines = self._preprocessor_statements(prefix) + + for i, routine in enumerate(routines): + if i > 0: + if empty: + code_lines.append("\n") + code_lines.extend(self._get_routine_opening(routine)) + if i == 0: + if routine.name != prefix: + raise ValueError('Octave function name should match prefix') + if header: + code_lines.append("%" + prefix.upper() + + " Autogenerated by SymPy\n") + code_lines.append(''.join(self._get_header())) + code_lines.extend(self._declare_arguments(routine)) + code_lines.extend(self._declare_globals(routine)) + code_lines.extend(self._declare_locals(routine)) + if empty: + code_lines.append("\n") + code_lines.extend(self._call_printer(routine)) + if empty: + code_lines.append("\n") + code_lines.extend(self._get_routine_ending(routine)) + + code_lines = self._indent_code(''.join(code_lines)) + + if code_lines: + f.write(code_lines) + + dump_m.extension = code_extension # type: ignore + dump_m.__doc__ = CodeGen.dump_code.__doc__ + + # This list of dump functions is used by CodeGen.write to know which dump + # functions it has to call. + dump_fns = [dump_m] + +class RustCodeGen(CodeGen): + """Generator for Rust code. + + The .write() method inherited from CodeGen will output a code file + .rs + + """ + + code_extension = "rs" + + def __init__(self, project="project", printer=None): + super().__init__(project=project) + self.printer = printer or RustCodePrinter() + + def routine(self, name, expr, argument_sequence, global_vars): + """Specialized Routine creation for Rust.""" + + if is_sequence(expr) and not isinstance(expr, (MatrixBase, MatrixExpr)): + if not expr: + raise ValueError("No expression given") + expressions = Tuple(*expr) + else: + expressions = Tuple(expr) + + # local variables + local_vars = {i.label for i in expressions.atoms(Idx)} + + # global variables + global_vars = set() if global_vars is None else set(global_vars) + + # symbols that should be arguments + symbols = expressions.free_symbols - local_vars - global_vars - expressions.atoms(Indexed) + + # Rust supports multiple return values + return_vals = [] + output_args = [] + for (i, expr) in enumerate(expressions): + if isinstance(expr, Equality): + out_arg = expr.lhs + expr = expr.rhs + symbol = out_arg + if isinstance(out_arg, Indexed): + dims = tuple([ (S.One, dim) for dim in out_arg.shape]) + symbol = out_arg.base.label + output_args.append(InOutArgument(symbol, out_arg, expr, dimensions=dims)) + if not isinstance(out_arg, (Indexed, Symbol, MatrixSymbol)): + raise CodeGenError("Only Indexed, Symbol, or MatrixSymbol " + "can define output arguments.") + + return_vals.append(Result(expr, name=symbol, result_var=out_arg)) + if not expr.has(symbol): + # this is a pure output: remove from the symbols list, so + # it doesn't become an input. + symbols.remove(symbol) + + else: + # we have no name for this output + return_vals.append(Result(expr, name='out%d' % (i+1))) + + # setup input argument list + output_args.sort(key=lambda x: str(x.name)) + arg_list = list(output_args) + array_symbols = {} + for array in expressions.atoms(Indexed): + array_symbols[array.base.label] = array + for array in expressions.atoms(MatrixSymbol): + array_symbols[array] = array + + for symbol in sorted(symbols, key=str): + arg_list.append(InputArgument(symbol)) + + if argument_sequence is not None: + # if the user has supplied IndexedBase instances, we'll accept that + new_sequence = [] + for arg in argument_sequence: + if isinstance(arg, IndexedBase): + new_sequence.append(arg.label) + else: + new_sequence.append(arg) + argument_sequence = new_sequence + + missing = [x for x in arg_list if x.name not in argument_sequence] + if missing: + msg = "Argument list didn't specify: {0} " + msg = msg.format(", ".join([str(m.name) for m in missing])) + raise CodeGenArgumentListError(msg, missing) + + # create redundant arguments to produce the requested sequence + name_arg_dict = {x.name: x for x in arg_list} + new_args = [] + for symbol in argument_sequence: + try: + new_args.append(name_arg_dict[symbol]) + except KeyError: + new_args.append(InputArgument(symbol)) + arg_list = new_args + + return Routine(name, arg_list, return_vals, local_vars, global_vars) + + + def _get_header(self): + """Writes a common header for the generated files.""" + code_lines = [] + code_lines.append("/*\n") + tmp = header_comment % {"version": sympy_version, + "project": self.project} + for line in tmp.splitlines(): + code_lines.append((" *%s" % line.center(76)).rstrip() + "\n") + code_lines.append(" */\n") + return code_lines + + def get_prototype(self, routine): + """Returns a string for the function prototype of the routine. + + If the routine has multiple result objects, an CodeGenError is + raised. + + See: https://en.wikipedia.org/wiki/Function_prototype + + """ + results = [i.get_datatype('Rust') for i in routine.results] + + if len(results) == 1: + rstype = " -> " + results[0] + elif len(routine.results) > 1: + rstype = " -> (" + ", ".join(results) + ")" + else: + rstype = "" + + type_args = [] + for arg in routine.arguments: + name = self.printer.doprint(arg.name) + if arg.dimensions or isinstance(arg, ResultBase): + type_args.append(("*%s" % name, arg.get_datatype('Rust'))) + else: + type_args.append((name, arg.get_datatype('Rust'))) + arguments = ", ".join([ "%s: %s" % t for t in type_args]) + return "fn %s(%s)%s" % (routine.name, arguments, rstype) + + def _preprocessor_statements(self, prefix): + code_lines = [] + # code_lines.append("use std::f64::consts::*;\n") + return code_lines + + def _get_routine_opening(self, routine): + prototype = self.get_prototype(routine) + return ["%s {\n" % prototype] + + def _declare_arguments(self, routine): + # arguments are declared in prototype + return [] + + def _declare_globals(self, routine): + # global variables are not explicitly declared within C functions + return [] + + def _declare_locals(self, routine): + # loop variables are declared in loop statement + return [] + + def _call_printer(self, routine): + + code_lines = [] + declarations = [] + returns = [] + + # Compose a list of symbols to be dereferenced in the function + # body. These are the arguments that were passed by a reference + # pointer, excluding arrays. + dereference = [] + for arg in routine.arguments: + if isinstance(arg, ResultBase) and not arg.dimensions: + dereference.append(arg.name) + + for result in routine.results: + if isinstance(result, Result): + assign_to = result.result_var + returns.append(str(result.result_var)) + else: + raise CodeGenError("unexpected object in Routine results") + + constants, not_supported, rs_expr = self._printer_method_with_settings( + 'doprint', {"human": False, "strict": False}, result.expr, assign_to=assign_to) + + for name, value in sorted(constants, key=str): + declarations.append("const %s: f64 = %s;\n" % (name, value)) + + for obj in sorted(not_supported, key=str): + if isinstance(obj, Function): + name = obj.func + else: + name = obj + declarations.append("// unsupported: %s\n" % (name)) + + code_lines.append("let %s\n" % rs_expr) + + if len(returns) > 1: + returns = ['(' + ', '.join(returns) + ')'] + + returns.append('\n') + + return declarations + code_lines + returns + + def _get_routine_ending(self, routine): + return ["}\n"] + + def dump_rs(self, routines, f, prefix, header=True, empty=True): + self.dump_code(routines, f, prefix, header, empty) + + dump_rs.extension = code_extension # type: ignore + dump_rs.__doc__ = CodeGen.dump_code.__doc__ + + # This list of dump functions is used by CodeGen.write to know which dump + # functions it has to call. + dump_fns = [dump_rs] + + + + +def get_code_generator(language, project=None, standard=None, printer = None): + if language == 'C': + if standard is None: + pass + elif standard.lower() == 'c89': + language = 'C89' + elif standard.lower() == 'c99': + language = 'C99' + CodeGenClass = {"C": CCodeGen, "C89": C89CodeGen, "C99": C99CodeGen, + "F95": FCodeGen, "JULIA": JuliaCodeGen, + "OCTAVE": OctaveCodeGen, + "RUST": RustCodeGen}.get(language.upper()) + if CodeGenClass is None: + raise ValueError("Language '%s' is not supported." % language) + return CodeGenClass(project, printer) + + +# +# Friendly functions +# + + +def codegen(name_expr, language=None, prefix=None, project="project", + to_files=False, header=True, empty=True, argument_sequence=None, + global_vars=None, standard=None, code_gen=None, printer=None): + """Generate source code for expressions in a given language. + + Parameters + ========== + + name_expr : tuple, or list of tuples + A single (name, expression) tuple or a list of (name, expression) + tuples. Each tuple corresponds to a routine. If the expression is + an equality (an instance of class Equality) the left hand side is + considered an output argument. If expression is an iterable, then + the routine will have multiple outputs. + + language : string, + A string that indicates the source code language. This is case + insensitive. Currently, 'C', 'F95' and 'Octave' are supported. + 'Octave' generates code compatible with both Octave and Matlab. + + prefix : string, optional + A prefix for the names of the files that contain the source code. + Language-dependent suffixes will be appended. If omitted, the name + of the first name_expr tuple is used. + + project : string, optional + A project name, used for making unique preprocessor instructions. + [default: "project"] + + to_files : bool, optional + When True, the code will be written to one or more files with the + given prefix, otherwise strings with the names and contents of + these files are returned. [default: False] + + header : bool, optional + When True, a header is written on top of each source file. + [default: True] + + empty : bool, optional + When True, empty lines are used to structure the code. + [default: True] + + argument_sequence : iterable, optional + Sequence of arguments for the routine in a preferred order. A + CodeGenError is raised if required arguments are missing. + Redundant arguments are used without warning. If omitted, + arguments will be ordered alphabetically, but with all input + arguments first, and then output or in-out arguments. + + global_vars : iterable, optional + Sequence of global variables used by the routine. Variables + listed here will not show up as function arguments. + + standard : string, optional + + code_gen : CodeGen instance, optional + An instance of a CodeGen subclass. Overrides ``language``. + + printer : Printer instance, optional + An instance of a Printer subclass. + + Examples + ======== + + >>> from sympy.utilities.codegen import codegen + >>> from sympy.abc import x, y, z + >>> [(c_name, c_code), (h_name, c_header)] = codegen( + ... ("f", x+y*z), "C89", "test", header=False, empty=False) + >>> print(c_name) + test.c + >>> print(c_code) + #include "test.h" + #include + double f(double x, double y, double z) { + double f_result; + f_result = x + y*z; + return f_result; + } + + >>> print(h_name) + test.h + >>> print(c_header) + #ifndef PROJECT__TEST__H + #define PROJECT__TEST__H + double f(double x, double y, double z); + #endif + + + Another example using Equality objects to give named outputs. Here the + filename (prefix) is taken from the first (name, expr) pair. + + >>> from sympy.abc import f, g + >>> from sympy import Eq + >>> [(c_name, c_code), (h_name, c_header)] = codegen( + ... [("myfcn", x + y), ("fcn2", [Eq(f, 2*x), Eq(g, y)])], + ... "C99", header=False, empty=False) + >>> print(c_name) + myfcn.c + >>> print(c_code) + #include "myfcn.h" + #include + double myfcn(double x, double y) { + double myfcn_result; + myfcn_result = x + y; + return myfcn_result; + } + void fcn2(double x, double y, double *f, double *g) { + (*f) = 2*x; + (*g) = y; + } + + + If the generated function(s) will be part of a larger project where various + global variables have been defined, the 'global_vars' option can be used + to remove the specified variables from the function signature + + >>> from sympy.utilities.codegen import codegen + >>> from sympy.abc import x, y, z + >>> [(f_name, f_code), header] = codegen( + ... ("f", x+y*z), "F95", header=False, empty=False, + ... argument_sequence=(x, y), global_vars=(z,)) + >>> print(f_code) + REAL*8 function f(x, y) + implicit none + REAL*8, intent(in) :: x + REAL*8, intent(in) :: y + f = x + y*z + end function + + + """ + + # Initialize the code generator. + if language is None: + if code_gen is None: + raise ValueError("Need either language or code_gen") + else: + if code_gen is not None: + raise ValueError("You cannot specify both language and code_gen.") + code_gen = get_code_generator(language, project, standard, printer) + + if isinstance(name_expr[0], str): + # single tuple is given, turn it into a singleton list with a tuple. + name_expr = [name_expr] + + if prefix is None: + prefix = name_expr[0][0] + + # Construct Routines appropriate for this code_gen from (name, expr) pairs. + routines = [] + for name, expr in name_expr: + routines.append(code_gen.routine(name, expr, argument_sequence, + global_vars)) + + # Write the code. + return code_gen.write(routines, prefix, to_files, header, empty) + + +def make_routine(name, expr, argument_sequence=None, + global_vars=None, language="F95"): + """A factory that makes an appropriate Routine from an expression. + + Parameters + ========== + + name : string + The name of this routine in the generated code. + + expr : expression or list/tuple of expressions + A SymPy expression that the Routine instance will represent. If + given a list or tuple of expressions, the routine will be + considered to have multiple return values and/or output arguments. + + argument_sequence : list or tuple, optional + List arguments for the routine in a preferred order. If omitted, + the results are language dependent, for example, alphabetical order + or in the same order as the given expressions. + + global_vars : iterable, optional + Sequence of global variables used by the routine. Variables + listed here will not show up as function arguments. + + language : string, optional + Specify a target language. The Routine itself should be + language-agnostic but the precise way one is created, error + checking, etc depend on the language. [default: "F95"]. + + Notes + ===== + + A decision about whether to use output arguments or return values is made + depending on both the language and the particular mathematical expressions. + For an expression of type Equality, the left hand side is typically made + into an OutputArgument (or perhaps an InOutArgument if appropriate). + Otherwise, typically, the calculated expression is made a return values of + the routine. + + Examples + ======== + + >>> from sympy.utilities.codegen import make_routine + >>> from sympy.abc import x, y, f, g + >>> from sympy import Eq + >>> r = make_routine('test', [Eq(f, 2*x), Eq(g, x + y)]) + >>> [arg.result_var for arg in r.results] + [] + >>> [arg.name for arg in r.arguments] + [x, y, f, g] + >>> [arg.name for arg in r.result_variables] + [f, g] + >>> r.local_vars + set() + + Another more complicated example with a mixture of specified and + automatically-assigned names. Also has Matrix output. + + >>> from sympy import Matrix + >>> r = make_routine('fcn', [x*y, Eq(f, 1), Eq(g, x + g), Matrix([[x, 2]])]) + >>> [arg.result_var for arg in r.results] # doctest: +SKIP + [result_5397460570204848505] + >>> [arg.expr for arg in r.results] + [x*y] + >>> [arg.name for arg in r.arguments] # doctest: +SKIP + [x, y, f, g, out_8598435338387848786] + + We can examine the various arguments more closely: + + >>> from sympy.utilities.codegen import (InputArgument, OutputArgument, + ... InOutArgument) + >>> [a.name for a in r.arguments if isinstance(a, InputArgument)] + [x, y] + + >>> [a.name for a in r.arguments if isinstance(a, OutputArgument)] # doctest: +SKIP + [f, out_8598435338387848786] + >>> [a.expr for a in r.arguments if isinstance(a, OutputArgument)] + [1, Matrix([[x, 2]])] + + >>> [a.name for a in r.arguments if isinstance(a, InOutArgument)] + [g] + >>> [a.expr for a in r.arguments if isinstance(a, InOutArgument)] + [g + x] + + """ + + # initialize a new code generator + code_gen = get_code_generator(language) + + return code_gen.routine(name, expr, argument_sequence, global_vars) diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/decorator.py b/.venv/lib/python3.12/site-packages/sympy/utilities/decorator.py new file mode 100644 index 0000000000000000000000000000000000000000..82636c35e9f235a1d23aa7df5182418db3ef6b4f --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/decorator.py @@ -0,0 +1,339 @@ +"""Useful utility decorators. """ + +from typing import TypeVar +import sys +import types +import inspect +from functools import wraps, update_wrapper + +from sympy.utilities.exceptions import sympy_deprecation_warning + + +T = TypeVar('T') +"""A generic type""" + + +def threaded_factory(func, use_add): + """A factory for ``threaded`` decorators. """ + from sympy.core import sympify + from sympy.matrices import MatrixBase + from sympy.utilities.iterables import iterable + + @wraps(func) + def threaded_func(expr, *args, **kwargs): + if isinstance(expr, MatrixBase): + return expr.applyfunc(lambda f: func(f, *args, **kwargs)) + elif iterable(expr): + try: + return expr.__class__([func(f, *args, **kwargs) for f in expr]) + except TypeError: + return expr + else: + expr = sympify(expr) + + if use_add and expr.is_Add: + return expr.__class__(*[ func(f, *args, **kwargs) for f in expr.args ]) + elif expr.is_Relational: + return expr.__class__(func(expr.lhs, *args, **kwargs), + func(expr.rhs, *args, **kwargs)) + else: + return func(expr, *args, **kwargs) + + return threaded_func + + +def threaded(func): + """Apply ``func`` to sub--elements of an object, including :class:`~.Add`. + + This decorator is intended to make it uniformly possible to apply a + function to all elements of composite objects, e.g. matrices, lists, tuples + and other iterable containers, or just expressions. + + This version of :func:`threaded` decorator allows threading over + elements of :class:`~.Add` class. If this behavior is not desirable + use :func:`xthreaded` decorator. + + Functions using this decorator must have the following signature:: + + @threaded + def function(expr, *args, **kwargs): + + """ + return threaded_factory(func, True) + + +def xthreaded(func): + """Apply ``func`` to sub--elements of an object, excluding :class:`~.Add`. + + This decorator is intended to make it uniformly possible to apply a + function to all elements of composite objects, e.g. matrices, lists, tuples + and other iterable containers, or just expressions. + + This version of :func:`threaded` decorator disallows threading over + elements of :class:`~.Add` class. If this behavior is not desirable + use :func:`threaded` decorator. + + Functions using this decorator must have the following signature:: + + @xthreaded + def function(expr, *args, **kwargs): + + """ + return threaded_factory(func, False) + + +def conserve_mpmath_dps(func): + """After the function finishes, resets the value of ``mpmath.mp.dps`` to + the value it had before the function was run.""" + import mpmath + + def func_wrapper(*args, **kwargs): + dps = mpmath.mp.dps + try: + return func(*args, **kwargs) + finally: + mpmath.mp.dps = dps + + func_wrapper = update_wrapper(func_wrapper, func) + return func_wrapper + + +class no_attrs_in_subclass: + """Don't 'inherit' certain attributes from a base class + + >>> from sympy.utilities.decorator import no_attrs_in_subclass + + >>> class A(object): + ... x = 'test' + + >>> A.x = no_attrs_in_subclass(A, A.x) + + >>> class B(A): + ... pass + + >>> hasattr(A, 'x') + True + >>> hasattr(B, 'x') + False + + """ + def __init__(self, cls, f): + self.cls = cls + self.f = f + + def __get__(self, instance, owner=None): + if owner == self.cls: + if hasattr(self.f, '__get__'): + return self.f.__get__(instance, owner) + return self.f + raise AttributeError + + +def doctest_depends_on(exe=None, modules=None, disable_viewers=None, + python_version=None, ground_types=None): + """ + Adds metadata about the dependencies which need to be met for doctesting + the docstrings of the decorated objects. + + ``exe`` should be a list of executables + + ``modules`` should be a list of modules + + ``disable_viewers`` should be a list of viewers for :func:`~sympy.printing.preview.preview` to disable + + ``python_version`` should be the minimum Python version required, as a tuple + (like ``(3, 0)``) + """ + dependencies = {} + if exe is not None: + dependencies['executables'] = exe + if modules is not None: + dependencies['modules'] = modules + if disable_viewers is not None: + dependencies['disable_viewers'] = disable_viewers + if python_version is not None: + dependencies['python_version'] = python_version + if ground_types is not None: + dependencies['ground_types'] = ground_types + + def skiptests(): + from sympy.testing.runtests import DependencyError, SymPyDocTests, PyTestReporter # lazy import + r = PyTestReporter() + t = SymPyDocTests(r, None) + try: + t._check_dependencies(**dependencies) + except DependencyError: + return True # Skip doctests + else: + return False # Run doctests + + def depends_on_deco(fn): + fn._doctest_depends_on = dependencies + fn.__doctest_skip__ = skiptests + + if inspect.isclass(fn): + fn._doctest_depdends_on = no_attrs_in_subclass( + fn, fn._doctest_depends_on) + fn.__doctest_skip__ = no_attrs_in_subclass( + fn, fn.__doctest_skip__) + return fn + + return depends_on_deco + + +def public(obj: T) -> T: + """ + Append ``obj``'s name to global ``__all__`` variable (call site). + + By using this decorator on functions or classes you achieve the same goal + as by filling ``__all__`` variables manually, you just do not have to repeat + yourself (object's name). You also know if object is public at definition + site, not at some random location (where ``__all__`` was set). + + Note that in multiple decorator setup (in almost all cases) ``@public`` + decorator must be applied before any other decorators, because it relies + on the pointer to object's global namespace. If you apply other decorators + first, ``@public`` may end up modifying the wrong namespace. + + Examples + ======== + + >>> from sympy.utilities.decorator import public + + >>> __all__ # noqa: F821 + Traceback (most recent call last): + ... + NameError: name '__all__' is not defined + + >>> @public + ... def some_function(): + ... pass + + >>> __all__ # noqa: F821 + ['some_function'] + + """ + if isinstance(obj, types.FunctionType): + ns = obj.__globals__ + name = obj.__name__ + elif isinstance(obj, (type(type), type)): + ns = sys.modules[obj.__module__].__dict__ + name = obj.__name__ + else: + raise TypeError("expected a function or a class, got %s" % obj) + + if "__all__" not in ns: + ns["__all__"] = [name] + else: + ns["__all__"].append(name) + + return obj + + +def memoize_property(propfunc): + """Property decorator that caches the value of potentially expensive + ``propfunc`` after the first evaluation. The cached value is stored in + the corresponding property name with an attached underscore.""" + attrname = '_' + propfunc.__name__ + sentinel = object() + + @wraps(propfunc) + def accessor(self): + val = getattr(self, attrname, sentinel) + if val is sentinel: + val = propfunc(self) + setattr(self, attrname, val) + return val + + return property(accessor) + + +def deprecated(message, *, deprecated_since_version, + active_deprecations_target, stacklevel=3): + ''' + Mark a function as deprecated. + + This decorator should be used if an entire function or class is + deprecated. If only a certain functionality is deprecated, you should use + :func:`~.warns_deprecated_sympy` directly. This decorator is just a + convenience. There is no functional difference between using this + decorator and calling ``warns_deprecated_sympy()`` at the top of the + function. + + The decorator takes the same arguments as + :func:`~.warns_deprecated_sympy`. See its + documentation for details on what the keywords to this decorator do. + + See the :ref:`deprecation-policy` document for details on when and how + things should be deprecated in SymPy. + + Examples + ======== + + >>> from sympy.utilities.decorator import deprecated + >>> from sympy import simplify + >>> @deprecated("""\ + ... The simplify_this(expr) function is deprecated. Use simplify(expr) + ... instead.""", deprecated_since_version="1.1", + ... active_deprecations_target='simplify-this-deprecation') + ... def simplify_this(expr): + ... """ + ... Simplify ``expr``. + ... + ... .. deprecated:: 1.1 + ... + ... The ``simplify_this`` function is deprecated. Use :func:`simplify` + ... instead. See its documentation for more information. See + ... :ref:`simplify-this-deprecation` for details. + ... + ... """ + ... return simplify(expr) + >>> from sympy.abc import x + >>> simplify_this(x*(x + 1) - x**2) # doctest: +SKIP + :1: SymPyDeprecationWarning: + + The simplify_this(expr) function is deprecated. Use simplify(expr) + instead. + + See https://docs.sympy.org/latest/explanation/active-deprecations.html#simplify-this-deprecation + for details. + + This has been deprecated since SymPy version 1.1. It + will be removed in a future version of SymPy. + + simplify_this(x) + x + + See Also + ======== + sympy.utilities.exceptions.SymPyDeprecationWarning + sympy.utilities.exceptions.sympy_deprecation_warning + sympy.utilities.exceptions.ignore_warnings + sympy.testing.pytest.warns_deprecated_sympy + + ''' + decorator_kwargs = {"deprecated_since_version": deprecated_since_version, + "active_deprecations_target": active_deprecations_target} + def deprecated_decorator(wrapped): + if hasattr(wrapped, '__mro__'): # wrapped is actually a class + class wrapper(wrapped): + __doc__ = wrapped.__doc__ + __module__ = wrapped.__module__ + _sympy_deprecated_func = wrapped + if '__new__' in wrapped.__dict__: + def __new__(cls, *args, **kwargs): + sympy_deprecation_warning(message, **decorator_kwargs, stacklevel=stacklevel) + return super().__new__(cls, *args, **kwargs) + else: + def __init__(self, *args, **kwargs): + sympy_deprecation_warning(message, **decorator_kwargs, stacklevel=stacklevel) + super().__init__(*args, **kwargs) + wrapper.__name__ = wrapped.__name__ + else: + @wraps(wrapped) + def wrapper(*args, **kwargs): + sympy_deprecation_warning(message, **decorator_kwargs, stacklevel=stacklevel) + return wrapped(*args, **kwargs) + wrapper._sympy_deprecated_func = wrapped + return wrapper + return deprecated_decorator diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/enumerative.py b/.venv/lib/python3.12/site-packages/sympy/utilities/enumerative.py new file mode 100644 index 0000000000000000000000000000000000000000..dcdabf064ad6291282b5905e7aed0ba9eb412d1a --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/enumerative.py @@ -0,0 +1,1155 @@ +""" +Algorithms and classes to support enumerative combinatorics. + +Currently just multiset partitions, but more could be added. + +Terminology (following Knuth, algorithm 7.1.2.5M TAOCP) +*multiset* aaabbcccc has a *partition* aaabc | bccc + +The submultisets, aaabc and bccc of the partition are called +*parts*, or sometimes *vectors*. (Knuth notes that multiset +partitions can be thought of as partitions of vectors of integers, +where the ith element of the vector gives the multiplicity of +element i.) + +The values a, b and c are *components* of the multiset. These +correspond to elements of a set, but in a multiset can be present +with a multiplicity greater than 1. + +The algorithm deserves some explanation. + +Think of the part aaabc from the multiset above. If we impose an +ordering on the components of the multiset, we can represent a part +with a vector, in which the value of the first element of the vector +corresponds to the multiplicity of the first component in that +part. Thus, aaabc can be represented by the vector [3, 1, 1]. We +can also define an ordering on parts, based on the lexicographic +ordering of the vector (leftmost vector element, i.e., the element +with the smallest component number, is the most significant), so +that [3, 1, 1] > [3, 1, 0] and [3, 1, 1] > [2, 1, 4]. The ordering +on parts can be extended to an ordering on partitions: First, sort +the parts in each partition, left-to-right in decreasing order. Then +partition A is greater than partition B if A's leftmost/greatest +part is greater than B's leftmost part. If the leftmost parts are +equal, compare the second parts, and so on. + +In this ordering, the greatest partition of a given multiset has only +one part. The least partition is the one in which the components +are spread out, one per part. + +The enumeration algorithms in this file yield the partitions of the +argument multiset in decreasing order. The main data structure is a +stack of parts, corresponding to the current partition. An +important invariant is that the parts on the stack are themselves in +decreasing order. This data structure is decremented to find the +next smaller partition. Most often, decrementing the partition will +only involve adjustments to the smallest parts at the top of the +stack, much as adjacent integers *usually* differ only in their last +few digits. + +Knuth's algorithm uses two main operations on parts: + +Decrement - change the part so that it is smaller in the + (vector) lexicographic order, but reduced by the smallest amount possible. + For example, if the multiset has vector [5, + 3, 1], and the bottom/greatest part is [4, 2, 1], this part would + decrement to [4, 2, 0], while [4, 0, 0] would decrement to [3, 3, + 1]. A singleton part is never decremented -- [1, 0, 0] is not + decremented to [0, 3, 1]. Instead, the decrement operator needs + to fail for this case. In Knuth's pseudocode, the decrement + operator is step m5. + +Spread unallocated multiplicity - Once a part has been decremented, + it cannot be the rightmost part in the partition. There is some + multiplicity that has not been allocated, and new parts must be + created above it in the stack to use up this multiplicity. To + maintain the invariant that the parts on the stack are in + decreasing order, these new parts must be less than or equal to + the decremented part. + For example, if the multiset is [5, 3, 1], and its most + significant part has just been decremented to [5, 3, 0], the + spread operation will add a new part so that the stack becomes + [[5, 3, 0], [0, 0, 1]]. If the most significant part (for the + same multiset) has been decremented to [2, 0, 0] the stack becomes + [[2, 0, 0], [2, 0, 0], [1, 3, 1]]. In the pseudocode, the spread + operation for one part is step m2. The complete spread operation + is a loop of steps m2 and m3. + +In order to facilitate the spread operation, Knuth stores, for each +component of each part, not just the multiplicity of that component +in the part, but also the total multiplicity available for this +component in this part or any lesser part above it on the stack. + +One added twist is that Knuth does not represent the part vectors as +arrays. Instead, he uses a sparse representation, in which a +component of a part is represented as a component number (c), plus +the multiplicity of the component in that part (v) as well as the +total multiplicity available for that component (u). This saves +time that would be spent skipping over zeros. + +""" + +class PartComponent: + """Internal class used in support of the multiset partitions + enumerators and the associated visitor functions. + + Represents one component of one part of the current partition. + + A stack of these, plus an auxiliary frame array, f, represents a + partition of the multiset. + + Knuth's pseudocode makes c, u, and v separate arrays. + """ + + __slots__ = ('c', 'u', 'v') + + def __init__(self): + self.c = 0 # Component number + self.u = 0 # The as yet unpartitioned amount in component c + # *before* it is allocated by this triple + self.v = 0 # Amount of c component in the current part + # (v<=u). An invariant of the representation is + # that the next higher triple for this component + # (if there is one) will have a value of u-v in + # its u attribute. + + def __repr__(self): + "for debug/algorithm animation purposes" + return 'c:%d u:%d v:%d' % (self.c, self.u, self.v) + + def __eq__(self, other): + """Define value oriented equality, which is useful for testers""" + return (isinstance(other, self.__class__) and + self.c == other.c and + self.u == other.u and + self.v == other.v) + + def __ne__(self, other): + """Defined for consistency with __eq__""" + return not self == other + + +# This function tries to be a faithful implementation of algorithm +# 7.1.2.5M in Volume 4A, Combinatoral Algorithms, Part 1, of The Art +# of Computer Programming, by Donald Knuth. This includes using +# (mostly) the same variable names, etc. This makes for rather +# low-level Python. + +# Changes from Knuth's pseudocode include +# - use PartComponent struct/object instead of 3 arrays +# - make the function a generator +# - map (with some difficulty) the GOTOs to Python control structures. +# - Knuth uses 1-based numbering for components, this code is 0-based +# - renamed variable l to lpart. +# - flag variable x takes on values True/False instead of 1/0 +# +def multiset_partitions_taocp(multiplicities): + """Enumerates partitions of a multiset. + + Parameters + ========== + + multiplicities + list of integer multiplicities of the components of the multiset. + + Yields + ====== + + state + Internal data structure which encodes a particular partition. + This output is then usually processed by a visitor function + which combines the information from this data structure with + the components themselves to produce an actual partition. + + Unless they wish to create their own visitor function, users will + have little need to look inside this data structure. But, for + reference, it is a 3-element list with components: + + f + is a frame array, which is used to divide pstack into parts. + + lpart + points to the base of the topmost part. + + pstack + is an array of PartComponent objects. + + The ``state`` output offers a peek into the internal data + structures of the enumeration function. The client should + treat this as read-only; any modification of the data + structure will cause unpredictable (and almost certainly + incorrect) results. Also, the components of ``state`` are + modified in place at each iteration. Hence, the visitor must + be called at each loop iteration. Accumulating the ``state`` + instances and processing them later will not work. + + Examples + ======== + + >>> from sympy.utilities.enumerative import list_visitor + >>> from sympy.utilities.enumerative import multiset_partitions_taocp + >>> # variables components and multiplicities represent the multiset 'abb' + >>> components = 'ab' + >>> multiplicities = [1, 2] + >>> states = multiset_partitions_taocp(multiplicities) + >>> list(list_visitor(state, components) for state in states) + [[['a', 'b', 'b']], + [['a', 'b'], ['b']], + [['a'], ['b', 'b']], + [['a'], ['b'], ['b']]] + + See Also + ======== + + sympy.utilities.iterables.multiset_partitions: Takes a multiset + as input and directly yields multiset partitions. It + dispatches to a number of functions, including this one, for + implementation. Most users will find it more convenient to + use than multiset_partitions_taocp. + + """ + + # Important variables. + # m is the number of components, i.e., number of distinct elements + m = len(multiplicities) + # n is the cardinality, total number of elements whether or not distinct + n = sum(multiplicities) + + # The main data structure, f segments pstack into parts. See + # list_visitor() for example code indicating how this internal + # state corresponds to a partition. + + # Note: allocation of space for stack is conservative. Knuth's + # exercise 7.2.1.5.68 gives some indication of how to tighten this + # bound, but this is not implemented. + pstack = [PartComponent() for i in range(n * m + 1)] + f = [0] * (n + 1) + + # Step M1 in Knuth (Initialize) + # Initial state - entire multiset in one part. + for j in range(m): + ps = pstack[j] + ps.c = j + ps.u = multiplicities[j] + ps.v = multiplicities[j] + + # Other variables + f[0] = 0 + a = 0 + lpart = 0 + f[1] = m + b = m # in general, current stack frame is from a to b - 1 + + while True: + while True: + # Step M2 (Subtract v from u) + k = b + x = False + for j in range(a, b): + pstack[k].u = pstack[j].u - pstack[j].v + if pstack[k].u == 0: + x = True + elif not x: + pstack[k].c = pstack[j].c + pstack[k].v = min(pstack[j].v, pstack[k].u) + x = pstack[k].u < pstack[j].v + k = k + 1 + else: # x is True + pstack[k].c = pstack[j].c + pstack[k].v = pstack[k].u + k = k + 1 + # Note: x is True iff v has changed + + # Step M3 (Push if nonzero.) + if k > b: + a = b + b = k + lpart = lpart + 1 + f[lpart + 1] = b + # Return to M2 + else: + break # Continue to M4 + + # M4 Visit a partition + state = [f, lpart, pstack] + yield state + + # M5 (Decrease v) + while True: + j = b-1 + while (pstack[j].v == 0): + j = j - 1 + if j == a and pstack[j].v == 1: + # M6 (Backtrack) + if lpart == 0: + return + lpart = lpart - 1 + b = a + a = f[lpart] + # Return to M5 + else: + pstack[j].v = pstack[j].v - 1 + for k in range(j + 1, b): + pstack[k].v = pstack[k].u + break # GOTO M2 + +# --------------- Visitor functions for multiset partitions --------------- +# A visitor takes the partition state generated by +# multiset_partitions_taocp or other enumerator, and produces useful +# output (such as the actual partition). + + +def factoring_visitor(state, primes): + """Use with multiset_partitions_taocp to enumerate the ways a + number can be expressed as a product of factors. For this usage, + the exponents of the prime factors of a number are arguments to + the partition enumerator, while the corresponding prime factors + are input here. + + Examples + ======== + + To enumerate the factorings of a number we can think of the elements of the + partition as being the prime factors and the multiplicities as being their + exponents. + + >>> from sympy.utilities.enumerative import factoring_visitor + >>> from sympy.utilities.enumerative import multiset_partitions_taocp + >>> from sympy import factorint + >>> primes, multiplicities = zip(*factorint(24).items()) + >>> primes + (2, 3) + >>> multiplicities + (3, 1) + >>> states = multiset_partitions_taocp(multiplicities) + >>> list(factoring_visitor(state, primes) for state in states) + [[24], [8, 3], [12, 2], [4, 6], [4, 2, 3], [6, 2, 2], [2, 2, 2, 3]] + """ + f, lpart, pstack = state + factoring = [] + for i in range(lpart + 1): + factor = 1 + for ps in pstack[f[i]: f[i + 1]]: + if ps.v > 0: + factor *= primes[ps.c] ** ps.v + factoring.append(factor) + return factoring + + +def list_visitor(state, components): + """Return a list of lists to represent the partition. + + Examples + ======== + + >>> from sympy.utilities.enumerative import list_visitor + >>> from sympy.utilities.enumerative import multiset_partitions_taocp + >>> states = multiset_partitions_taocp([1, 2, 1]) + >>> s = next(states) + >>> list_visitor(s, 'abc') # for multiset 'a b b c' + [['a', 'b', 'b', 'c']] + >>> s = next(states) + >>> list_visitor(s, [1, 2, 3]) # for multiset '1 2 2 3 + [[1, 2, 2], [3]] + """ + f, lpart, pstack = state + + partition = [] + for i in range(lpart+1): + part = [] + for ps in pstack[f[i]:f[i+1]]: + if ps.v > 0: + part.extend([components[ps.c]] * ps.v) + partition.append(part) + + return partition + + +class MultisetPartitionTraverser(): + """ + Has methods to ``enumerate`` and ``count`` the partitions of a multiset. + + This implements a refactored and extended version of Knuth's algorithm + 7.1.2.5M [AOCP]_." + + The enumeration methods of this class are generators and return + data structures which can be interpreted by the same visitor + functions used for the output of ``multiset_partitions_taocp``. + + Examples + ======== + + >>> from sympy.utilities.enumerative import MultisetPartitionTraverser + >>> m = MultisetPartitionTraverser() + >>> m.count_partitions([4,4,4,2]) + 127750 + >>> m.count_partitions([3,3,3]) + 686 + + See Also + ======== + + multiset_partitions_taocp + sympy.utilities.iterables.multiset_partitions + + References + ========== + + .. [AOCP] Algorithm 7.1.2.5M in Volume 4A, Combinatoral Algorithms, + Part 1, of The Art of Computer Programming, by Donald Knuth. + + .. [Factorisatio] On a Problem of Oppenheim concerning + "Factorisatio Numerorum" E. R. Canfield, Paul Erdos, Carl + Pomerance, JOURNAL OF NUMBER THEORY, Vol. 17, No. 1. August + 1983. See section 7 for a description of an algorithm + similar to Knuth's. + + .. [Yorgey] Generating Multiset Partitions, Brent Yorgey, The + Monad.Reader, Issue 8, September 2007. + + """ + + def __init__(self): + self.debug = False + # TRACING variables. These are useful for gathering + # statistics on the algorithm itself, but have no particular + # benefit to a user of the code. + self.k1 = 0 + self.k2 = 0 + self.p1 = 0 + self.pstack = None + self.f = None + self.lpart = 0 + self.discarded = 0 + # dp_stack is list of lists of (part_key, start_count) pairs + self.dp_stack = [] + + # dp_map is map part_key-> count, where count represents the + # number of multiset which are descendants of a part with this + # key, **or any of its decrements** + + # Thus, when we find a part in the map, we add its count + # value to the running total, cut off the enumeration, and + # backtrack + + if not hasattr(self, 'dp_map'): + self.dp_map = {} + + def db_trace(self, msg): + """Useful for understanding/debugging the algorithms. Not + generally activated in end-user code.""" + if self.debug: + # XXX: animation_visitor is undefined... Clearly this does not + # work and was not tested. Previous code in comments below. + raise RuntimeError + #letters = 'abcdefghijklmnopqrstuvwxyz' + #state = [self.f, self.lpart, self.pstack] + #print("DBG:", msg, + # ["".join(part) for part in list_visitor(state, letters)], + # animation_visitor(state)) + + # + # Helper methods for enumeration + # + def _initialize_enumeration(self, multiplicities): + """Allocates and initializes the partition stack. + + This is called from the enumeration/counting routines, so + there is no need to call it separately.""" + + num_components = len(multiplicities) + # cardinality is the total number of elements, whether or not distinct + cardinality = sum(multiplicities) + + # pstack is the partition stack, which is segmented by + # f into parts. + self.pstack = [PartComponent() for i in + range(num_components * cardinality + 1)] + self.f = [0] * (cardinality + 1) + + # Initial state - entire multiset in one part. + for j in range(num_components): + ps = self.pstack[j] + ps.c = j + ps.u = multiplicities[j] + ps.v = multiplicities[j] + + self.f[0] = 0 + self.f[1] = num_components + self.lpart = 0 + + # The decrement_part() method corresponds to step M5 in Knuth's + # algorithm. This is the base version for enum_all(). Modified + # versions of this method are needed if we want to restrict + # sizes of the partitions produced. + def decrement_part(self, part): + """Decrements part (a subrange of pstack), if possible, returning + True iff the part was successfully decremented. + + If you think of the v values in the part as a multi-digit + integer (least significant digit on the right) this is + basically decrementing that integer, but with the extra + constraint that the leftmost digit cannot be decremented to 0. + + Parameters + ========== + + part + The part, represented as a list of PartComponent objects, + which is to be decremented. + + """ + plen = len(part) + for j in range(plen - 1, -1, -1): + if j == 0 and part[j].v > 1 or j > 0 and part[j].v > 0: + # found val to decrement + part[j].v -= 1 + # Reset trailing parts back to maximum + for k in range(j + 1, plen): + part[k].v = part[k].u + return True + return False + + # Version to allow number of parts to be bounded from above. + # Corresponds to (a modified) step M5. + def decrement_part_small(self, part, ub): + """Decrements part (a subrange of pstack), if possible, returning + True iff the part was successfully decremented. + + Parameters + ========== + + part + part to be decremented (topmost part on the stack) + + ub + the maximum number of parts allowed in a partition + returned by the calling traversal. + + Notes + ===== + + The goal of this modification of the ordinary decrement method + is to fail (meaning that the subtree rooted at this part is to + be skipped) when it can be proved that this part can only have + child partitions which are larger than allowed by ``ub``. If a + decision is made to fail, it must be accurate, otherwise the + enumeration will miss some partitions. But, it is OK not to + capture all the possible failures -- if a part is passed that + should not be, the resulting too-large partitions are filtered + by the enumeration one level up. However, as is usual in + constrained enumerations, failing early is advantageous. + + The tests used by this method catch the most common cases, + although this implementation is by no means the last word on + this problem. The tests include: + + 1) ``lpart`` must be less than ``ub`` by at least 2. This is because + once a part has been decremented, the partition + will gain at least one child in the spread step. + + 2) If the leading component of the part is about to be + decremented, check for how many parts will be added in + order to use up the unallocated multiplicity in that + leading component, and fail if this number is greater than + allowed by ``ub``. (See code for the exact expression.) This + test is given in the answer to Knuth's problem 7.2.1.5.69. + + 3) If there is *exactly* enough room to expand the leading + component by the above test, check the next component (if + it exists) once decrementing has finished. If this has + ``v == 0``, this next component will push the expansion over the + limit by 1, so fail. + """ + if self.lpart >= ub - 1: + self.p1 += 1 # increment to keep track of usefulness of tests + return False + plen = len(part) + for j in range(plen - 1, -1, -1): + # Knuth's mod, (answer to problem 7.2.1.5.69) + if j == 0 and (part[0].v - 1)*(ub - self.lpart) < part[0].u: + self.k1 += 1 + return False + + if j == 0 and part[j].v > 1 or j > 0 and part[j].v > 0: + # found val to decrement + part[j].v -= 1 + # Reset trailing parts back to maximum + for k in range(j + 1, plen): + part[k].v = part[k].u + + # Have now decremented part, but are we doomed to + # failure when it is expanded? Check one oddball case + # that turns out to be surprisingly common - exactly + # enough room to expand the leading component, but no + # room for the second component, which has v=0. + if (plen > 1 and part[1].v == 0 and + (part[0].u - part[0].v) == + ((ub - self.lpart - 1) * part[0].v)): + self.k2 += 1 + self.db_trace("Decrement fails test 3") + return False + return True + return False + + def decrement_part_large(self, part, amt, lb): + """Decrements part, while respecting size constraint. + + A part can have no children which are of sufficient size (as + indicated by ``lb``) unless that part has sufficient + unallocated multiplicity. When enforcing the size constraint, + this method will decrement the part (if necessary) by an + amount needed to ensure sufficient unallocated multiplicity. + + Returns True iff the part was successfully decremented. + + Parameters + ========== + + part + part to be decremented (topmost part on the stack) + + amt + Can only take values 0 or 1. A value of 1 means that the + part must be decremented, and then the size constraint is + enforced. A value of 0 means just to enforce the ``lb`` + size constraint. + + lb + The partitions produced by the calling enumeration must + have more parts than this value. + + """ + + if amt == 1: + # In this case we always need to decrement, *before* + # enforcing the "sufficient unallocated multiplicity" + # constraint. Easiest for this is just to call the + # regular decrement method. + if not self.decrement_part(part): + return False + + # Next, perform any needed additional decrementing to respect + # "sufficient unallocated multiplicity" (or fail if this is + # not possible). + min_unalloc = lb - self.lpart + if min_unalloc <= 0: + return True + total_mult = sum(pc.u for pc in part) + total_alloc = sum(pc.v for pc in part) + if total_mult <= min_unalloc: + return False + + deficit = min_unalloc - (total_mult - total_alloc) + if deficit <= 0: + return True + + for i in range(len(part) - 1, -1, -1): + if i == 0: + if part[0].v > deficit: + part[0].v -= deficit + return True + else: + return False # This shouldn't happen, due to above check + else: + if part[i].v >= deficit: + part[i].v -= deficit + return True + else: + deficit -= part[i].v + part[i].v = 0 + + def decrement_part_range(self, part, lb, ub): + """Decrements part (a subrange of pstack), if possible, returning + True iff the part was successfully decremented. + + Parameters + ========== + + part + part to be decremented (topmost part on the stack) + + ub + the maximum number of parts allowed in a partition + returned by the calling traversal. + + lb + The partitions produced by the calling enumeration must + have more parts than this value. + + Notes + ===== + + Combines the constraints of _small and _large decrement + methods. If returns success, part has been decremented at + least once, but perhaps by quite a bit more if needed to meet + the lb constraint. + """ + + # Constraint in the range case is just enforcing both the + # constraints from _small and _large cases. Note the 0 as the + # second argument to the _large call -- this is the signal to + # decrement only as needed to for constraint enforcement. The + # short circuiting and left-to-right order of the 'and' + # operator is important for this to work correctly. + return self.decrement_part_small(part, ub) and \ + self.decrement_part_large(part, 0, lb) + + def spread_part_multiplicity(self): + """Returns True if a new part has been created, and + adjusts pstack, f and lpart as needed. + + Notes + ===== + + Spreads unallocated multiplicity from the current top part + into a new part created above the current on the stack. This + new part is constrained to be less than or equal to the old in + terms of the part ordering. + + This call does nothing (and returns False) if the current top + part has no unallocated multiplicity. + + """ + j = self.f[self.lpart] # base of current top part + k = self.f[self.lpart + 1] # ub of current; potential base of next + base = k # save for later comparison + + changed = False # Set to true when the new part (so far) is + # strictly less than (as opposed to less than + # or equal) to the old. + for j in range(self.f[self.lpart], self.f[self.lpart + 1]): + self.pstack[k].u = self.pstack[j].u - self.pstack[j].v + if self.pstack[k].u == 0: + changed = True + else: + self.pstack[k].c = self.pstack[j].c + if changed: # Put all available multiplicity in this part + self.pstack[k].v = self.pstack[k].u + else: # Still maintaining ordering constraint + if self.pstack[k].u < self.pstack[j].v: + self.pstack[k].v = self.pstack[k].u + changed = True + else: + self.pstack[k].v = self.pstack[j].v + k = k + 1 + if k > base: + # Adjust for the new part on stack + self.lpart = self.lpart + 1 + self.f[self.lpart + 1] = k + return True + return False + + def top_part(self): + """Return current top part on the stack, as a slice of pstack. + + """ + return self.pstack[self.f[self.lpart]:self.f[self.lpart + 1]] + + # Same interface and functionality as multiset_partitions_taocp(), + # but some might find this refactored version easier to follow. + def enum_all(self, multiplicities): + """Enumerate the partitions of a multiset. + + Examples + ======== + + >>> from sympy.utilities.enumerative import list_visitor + >>> from sympy.utilities.enumerative import MultisetPartitionTraverser + >>> m = MultisetPartitionTraverser() + >>> states = m.enum_all([2,2]) + >>> list(list_visitor(state, 'ab') for state in states) + [[['a', 'a', 'b', 'b']], + [['a', 'a', 'b'], ['b']], + [['a', 'a'], ['b', 'b']], + [['a', 'a'], ['b'], ['b']], + [['a', 'b', 'b'], ['a']], + [['a', 'b'], ['a', 'b']], + [['a', 'b'], ['a'], ['b']], + [['a'], ['a'], ['b', 'b']], + [['a'], ['a'], ['b'], ['b']]] + + See Also + ======== + + multiset_partitions_taocp: + which provides the same result as this method, but is + about twice as fast. Hence, enum_all is primarily useful + for testing. Also see the function for a discussion of + states and visitors. + + """ + self._initialize_enumeration(multiplicities) + while True: + while self.spread_part_multiplicity(): + pass + + # M4 Visit a partition + state = [self.f, self.lpart, self.pstack] + yield state + + # M5 (Decrease v) + while not self.decrement_part(self.top_part()): + # M6 (Backtrack) + if self.lpart == 0: + return + self.lpart -= 1 + + def enum_small(self, multiplicities, ub): + """Enumerate multiset partitions with no more than ``ub`` parts. + + Equivalent to enum_range(multiplicities, 0, ub) + + Parameters + ========== + + multiplicities + list of multiplicities of the components of the multiset. + + ub + Maximum number of parts + + Examples + ======== + + >>> from sympy.utilities.enumerative import list_visitor + >>> from sympy.utilities.enumerative import MultisetPartitionTraverser + >>> m = MultisetPartitionTraverser() + >>> states = m.enum_small([2,2], 2) + >>> list(list_visitor(state, 'ab') for state in states) + [[['a', 'a', 'b', 'b']], + [['a', 'a', 'b'], ['b']], + [['a', 'a'], ['b', 'b']], + [['a', 'b', 'b'], ['a']], + [['a', 'b'], ['a', 'b']]] + + The implementation is based, in part, on the answer given to + exercise 69, in Knuth [AOCP]_. + + See Also + ======== + + enum_all, enum_large, enum_range + + """ + + # Keep track of iterations which do not yield a partition. + # Clearly, we would like to keep this number small. + self.discarded = 0 + if ub <= 0: + return + self._initialize_enumeration(multiplicities) + while True: + while self.spread_part_multiplicity(): + self.db_trace('spread 1') + if self.lpart >= ub: + self.discarded += 1 + self.db_trace(' Discarding') + self.lpart = ub - 2 + break + else: + # M4 Visit a partition + state = [self.f, self.lpart, self.pstack] + yield state + + # M5 (Decrease v) + while not self.decrement_part_small(self.top_part(), ub): + self.db_trace("Failed decrement, going to backtrack") + # M6 (Backtrack) + if self.lpart == 0: + return + self.lpart -= 1 + self.db_trace("Backtracked to") + self.db_trace("decrement ok, about to expand") + + def enum_large(self, multiplicities, lb): + """Enumerate the partitions of a multiset with lb < num(parts) + + Equivalent to enum_range(multiplicities, lb, sum(multiplicities)) + + Parameters + ========== + + multiplicities + list of multiplicities of the components of the multiset. + + lb + Number of parts in the partition must be greater than + this lower bound. + + + Examples + ======== + + >>> from sympy.utilities.enumerative import list_visitor + >>> from sympy.utilities.enumerative import MultisetPartitionTraverser + >>> m = MultisetPartitionTraverser() + >>> states = m.enum_large([2,2], 2) + >>> list(list_visitor(state, 'ab') for state in states) + [[['a', 'a'], ['b'], ['b']], + [['a', 'b'], ['a'], ['b']], + [['a'], ['a'], ['b', 'b']], + [['a'], ['a'], ['b'], ['b']]] + + See Also + ======== + + enum_all, enum_small, enum_range + + """ + self.discarded = 0 + if lb >= sum(multiplicities): + return + self._initialize_enumeration(multiplicities) + self.decrement_part_large(self.top_part(), 0, lb) + while True: + good_partition = True + while self.spread_part_multiplicity(): + if not self.decrement_part_large(self.top_part(), 0, lb): + # Failure here should be rare/impossible + self.discarded += 1 + good_partition = False + break + + # M4 Visit a partition + if good_partition: + state = [self.f, self.lpart, self.pstack] + yield state + + # M5 (Decrease v) + while not self.decrement_part_large(self.top_part(), 1, lb): + # M6 (Backtrack) + if self.lpart == 0: + return + self.lpart -= 1 + + def enum_range(self, multiplicities, lb, ub): + + """Enumerate the partitions of a multiset with + ``lb < num(parts) <= ub``. + + In particular, if partitions with exactly ``k`` parts are + desired, call with ``(multiplicities, k - 1, k)``. This + method generalizes enum_all, enum_small, and enum_large. + + Examples + ======== + + >>> from sympy.utilities.enumerative import list_visitor + >>> from sympy.utilities.enumerative import MultisetPartitionTraverser + >>> m = MultisetPartitionTraverser() + >>> states = m.enum_range([2,2], 1, 2) + >>> list(list_visitor(state, 'ab') for state in states) + [[['a', 'a', 'b'], ['b']], + [['a', 'a'], ['b', 'b']], + [['a', 'b', 'b'], ['a']], + [['a', 'b'], ['a', 'b']]] + + """ + # combine the constraints of the _large and _small + # enumerations. + self.discarded = 0 + if ub <= 0 or lb >= sum(multiplicities): + return + self._initialize_enumeration(multiplicities) + self.decrement_part_large(self.top_part(), 0, lb) + while True: + good_partition = True + while self.spread_part_multiplicity(): + self.db_trace("spread 1") + if not self.decrement_part_large(self.top_part(), 0, lb): + # Failure here - possible in range case? + self.db_trace(" Discarding (large cons)") + self.discarded += 1 + good_partition = False + break + elif self.lpart >= ub: + self.discarded += 1 + good_partition = False + self.db_trace(" Discarding small cons") + self.lpart = ub - 2 + break + + # M4 Visit a partition + if good_partition: + state = [self.f, self.lpart, self.pstack] + yield state + + # M5 (Decrease v) + while not self.decrement_part_range(self.top_part(), lb, ub): + self.db_trace("Failed decrement, going to backtrack") + # M6 (Backtrack) + if self.lpart == 0: + return + self.lpart -= 1 + self.db_trace("Backtracked to") + self.db_trace("decrement ok, about to expand") + + def count_partitions_slow(self, multiplicities): + """Returns the number of partitions of a multiset whose elements + have the multiplicities given in ``multiplicities``. + + Primarily for comparison purposes. It follows the same path as + enumerate, and counts, rather than generates, the partitions. + + See Also + ======== + + count_partitions + Has the same calling interface, but is much faster. + + """ + # number of partitions so far in the enumeration + self.pcount = 0 + self._initialize_enumeration(multiplicities) + while True: + while self.spread_part_multiplicity(): + pass + + # M4 Visit (count) a partition + self.pcount += 1 + + # M5 (Decrease v) + while not self.decrement_part(self.top_part()): + # M6 (Backtrack) + if self.lpart == 0: + return self.pcount + self.lpart -= 1 + + def count_partitions(self, multiplicities): + """Returns the number of partitions of a multiset whose components + have the multiplicities given in ``multiplicities``. + + For larger counts, this method is much faster than calling one + of the enumerators and counting the result. Uses dynamic + programming to cut down on the number of nodes actually + explored. The dictionary used in order to accelerate the + counting process is stored in the ``MultisetPartitionTraverser`` + object and persists across calls. If the user does not + expect to call ``count_partitions`` for any additional + multisets, the object should be cleared to save memory. On + the other hand, the cache built up from one count run can + significantly speed up subsequent calls to ``count_partitions``, + so it may be advantageous not to clear the object. + + Examples + ======== + + >>> from sympy.utilities.enumerative import MultisetPartitionTraverser + >>> m = MultisetPartitionTraverser() + >>> m.count_partitions([9,8,2]) + 288716 + >>> m.count_partitions([2,2]) + 9 + >>> del m + + Notes + ===== + + If one looks at the workings of Knuth's algorithm M [AOCP]_, it + can be viewed as a traversal of a binary tree of parts. A + part has (up to) two children, the left child resulting from + the spread operation, and the right child from the decrement + operation. The ordinary enumeration of multiset partitions is + an in-order traversal of this tree, and with the partitions + corresponding to paths from the root to the leaves. The + mapping from paths to partitions is a little complicated, + since the partition would contain only those parts which are + leaves or the parents of a spread link, not those which are + parents of a decrement link. + + For counting purposes, it is sufficient to count leaves, and + this can be done with a recursive in-order traversal. The + number of leaves of a subtree rooted at a particular part is a + function only of that part itself, so memoizing has the + potential to speed up the counting dramatically. + + This method follows a computational approach which is similar + to the hypothetical memoized recursive function, but with two + differences: + + 1) This method is iterative, borrowing its structure from the + other enumerations and maintaining an explicit stack of + parts which are in the process of being counted. (There + may be multisets which can be counted reasonably quickly by + this implementation, but which would overflow the default + Python recursion limit with a recursive implementation.) + + 2) Instead of using the part data structure directly, a more + compact key is constructed. This saves space, but more + importantly coalesces some parts which would remain + separate with physical keys. + + Unlike the enumeration functions, there is currently no _range + version of count_partitions. If someone wants to stretch + their brain, it should be possible to construct one by + memoizing with a histogram of counts rather than a single + count, and combining the histograms. + """ + # number of partitions so far in the enumeration + self.pcount = 0 + + # dp_stack is list of lists of (part_key, start_count) pairs + self.dp_stack = [] + + self._initialize_enumeration(multiplicities) + pkey = part_key(self.top_part()) + self.dp_stack.append([(pkey, 0), ]) + while True: + while self.spread_part_multiplicity(): + pkey = part_key(self.top_part()) + if pkey in self.dp_map: + # Already have a cached value for the count of the + # subtree rooted at this part. Add it to the + # running counter, and break out of the spread + # loop. The -1 below is to compensate for the + # leaf that this code path would otherwise find, + # and which gets incremented for below. + + self.pcount += (self.dp_map[pkey] - 1) + self.lpart -= 1 + break + else: + self.dp_stack.append([(pkey, self.pcount), ]) + + # M4 count a leaf partition + self.pcount += 1 + + # M5 (Decrease v) + while not self.decrement_part(self.top_part()): + # M6 (Backtrack) + for key, oldcount in self.dp_stack.pop(): + self.dp_map[key] = self.pcount - oldcount + if self.lpart == 0: + return self.pcount + self.lpart -= 1 + + # At this point have successfully decremented the part on + # the stack and it does not appear in the cache. It needs + # to be added to the list at the top of dp_stack + pkey = part_key(self.top_part()) + self.dp_stack[-1].append((pkey, self.pcount),) + + +def part_key(part): + """Helper for MultisetPartitionTraverser.count_partitions that + creates a key for ``part``, that only includes information which can + affect the count for that part. (Any irrelevant information just + reduces the effectiveness of dynamic programming.) + + Notes + ===== + + This member function is a candidate for future exploration. There + are likely symmetries that can be exploited to coalesce some + ``part_key`` values, and thereby save space and improve + performance. + + """ + # The component number is irrelevant for counting partitions, so + # leave it out of the memo key. + rval = [] + for ps in part: + rval.append(ps.u) + rval.append(ps.v) + return tuple(rval) diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/exceptions.py b/.venv/lib/python3.12/site-packages/sympy/utilities/exceptions.py new file mode 100644 index 0000000000000000000000000000000000000000..f764a31371ed109d69102777bd9ec0dfb3d75380 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/exceptions.py @@ -0,0 +1,271 @@ +""" +General SymPy exceptions and warnings. +""" + +import warnings +import contextlib + +from textwrap import dedent + + +class SymPyDeprecationWarning(DeprecationWarning): + r""" + A warning for deprecated features of SymPy. + + See the :ref:`deprecation-policy` document for details on when and how + things should be deprecated in SymPy. + + Note that simply constructing this class will not cause a warning to be + issued. To do that, you must call the :func`sympy_deprecation_warning` + function. For this reason, it is not recommended to ever construct this + class directly. + + Explanation + =========== + + The ``SymPyDeprecationWarning`` class is a subclass of + ``DeprecationWarning`` that is used for all deprecations in SymPy. A + special subclass is used so that we can automatically augment the warning + message with additional metadata about the version the deprecation was + introduced in and a link to the documentation. This also allows users to + explicitly filter deprecation warnings from SymPy using ``warnings`` + filters (see :ref:`silencing-sympy-deprecation-warnings`). + + Additionally, ``SymPyDeprecationWarning`` is enabled to be shown by + default, unlike normal ``DeprecationWarning``\s, which are only shown by + default in interactive sessions. This ensures that deprecation warnings in + SymPy will actually be seen by users. + + See the documentation of :func:`sympy_deprecation_warning` for a + description of the parameters to this function. + + To mark a function as deprecated, you can use the :func:`@deprecated + ` decorator. + + See Also + ======== + sympy.utilities.exceptions.sympy_deprecation_warning + sympy.utilities.exceptions.ignore_warnings + sympy.utilities.decorator.deprecated + sympy.testing.pytest.warns_deprecated_sympy + + """ + def __init__(self, message, *, deprecated_since_version, active_deprecations_target): + + super().__init__(message, deprecated_since_version, + active_deprecations_target) + self.message = message + if not isinstance(deprecated_since_version, str): + raise TypeError(f"'deprecated_since_version' should be a string, got {deprecated_since_version!r}") + self.deprecated_since_version = deprecated_since_version + self.active_deprecations_target = active_deprecations_target + if any(i in active_deprecations_target for i in '()='): + raise ValueError("active_deprecations_target be the part inside of the '(...)='") + + self.full_message = f""" + +{dedent(message).strip()} + +See https://docs.sympy.org/latest/explanation/active-deprecations.html#{active_deprecations_target} +for details. + +This has been deprecated since SymPy version {deprecated_since_version}. It +will be removed in a future version of SymPy. +""" + + def __str__(self): + return self.full_message + + def __repr__(self): + return f"{self.__class__.__name__}({self.message!r}, deprecated_since_version={self.deprecated_since_version!r}, active_deprecations_target={self.active_deprecations_target!r})" + + def __eq__(self, other): + return isinstance(other, SymPyDeprecationWarning) and self.args == other.args + + # Make pickling work. The by default, it tries to recreate the expression + # from its args, but this doesn't work because of our keyword-only + # arguments. + @classmethod + def _new(cls, message, deprecated_since_version, + active_deprecations_target): + return cls(message, deprecated_since_version=deprecated_since_version, active_deprecations_target=active_deprecations_target) + + def __reduce__(self): + return (self._new, (self.message, self.deprecated_since_version, self.active_deprecations_target)) + +# Python by default hides DeprecationWarnings, which we do not want. +warnings.simplefilter("once", SymPyDeprecationWarning) + +def sympy_deprecation_warning(message, *, deprecated_since_version, + active_deprecations_target, stacklevel=3): + r''' + Warn that a feature is deprecated in SymPy. + + See the :ref:`deprecation-policy` document for details on when and how + things should be deprecated in SymPy. + + To mark an entire function or class as deprecated, you can use the + :func:`@deprecated ` decorator. + + Parameters + ========== + + message : str + The deprecation message. This may span multiple lines and contain + code examples. Messages should be wrapped to 80 characters. The + message is automatically dedented and leading and trailing whitespace + stripped. Messages may include dynamic content based on the user + input, but avoid using ``str(expression)`` if an expression can be + arbitrary, as it might be huge and make the warning message + unreadable. + + deprecated_since_version : str + The version of SymPy the feature has been deprecated since. For new + deprecations, this should be the version in `sympy/release.py + `_ + without the ``.dev``. If the next SymPy version ends up being + different from this, the release manager will need to update any + ``SymPyDeprecationWarning``\s using the incorrect version. This + argument is required and must be passed as a keyword argument. + (example: ``deprecated_since_version="1.10"``). + + active_deprecations_target : str + The Sphinx target corresponding to the section for the deprecation in + the :ref:`active-deprecations` document (see + ``doc/src/explanation/active-deprecations.md``). This is used to + automatically generate a URL to the page in the warning message. This + argument is required and must be passed as a keyword argument. + (example: ``active_deprecations_target="deprecated-feature-abc"``) + + stacklevel : int, default: 3 + The ``stacklevel`` parameter that is passed to ``warnings.warn``. If + you create a wrapper that calls this function, this should be + increased so that the warning message shows the user line of code that + produced the warning. Note that in some cases there will be multiple + possible different user code paths that could result in the warning. + In that case, just choose the smallest common stacklevel. + + Examples + ======== + + >>> from sympy.utilities.exceptions import sympy_deprecation_warning + >>> def is_this_zero(x, y=0): + ... """ + ... Determine if x = 0. + ... + ... Parameters + ... ========== + ... + ... x : Expr + ... The expression to check. + ... + ... y : Expr, optional + ... If provided, check if x = y. + ... + ... .. deprecated:: 1.1 + ... + ... The ``y`` argument to ``is_this_zero`` is deprecated. Use + ... ``is_this_zero(x - y)`` instead. + ... + ... """ + ... from sympy import simplify + ... + ... if y != 0: + ... sympy_deprecation_warning(""" + ... The y argument to is_zero() is deprecated. Use is_zero(x - y) instead.""", + ... deprecated_since_version="1.1", + ... active_deprecations_target='is-this-zero-y-deprecation') + ... return simplify(x - y) == 0 + >>> is_this_zero(0) + True + >>> is_this_zero(1, 1) # doctest: +SKIP + :1: SymPyDeprecationWarning: + + The y argument to is_zero() is deprecated. Use is_zero(x - y) instead. + + See https://docs.sympy.org/latest/explanation/active-deprecations.html#is-this-zero-y-deprecation + for details. + + This has been deprecated since SymPy version 1.1. It + will be removed in a future version of SymPy. + + is_this_zero(1, 1) + True + + See Also + ======== + + sympy.utilities.exceptions.SymPyDeprecationWarning + sympy.utilities.exceptions.ignore_warnings + sympy.utilities.decorator.deprecated + sympy.testing.pytest.warns_deprecated_sympy + + ''' + w = SymPyDeprecationWarning(message, + deprecated_since_version=deprecated_since_version, + active_deprecations_target=active_deprecations_target) + warnings.warn(w, stacklevel=stacklevel) + + +@contextlib.contextmanager +def ignore_warnings(warningcls): + ''' + Context manager to suppress warnings during tests. + + .. note:: + + Do not use this with SymPyDeprecationWarning in the tests. + warns_deprecated_sympy() should be used instead. + + This function is useful for suppressing warnings during tests. The warns + function should be used to assert that a warning is raised. The + ignore_warnings function is useful in situation when the warning is not + guaranteed to be raised (e.g. on importing a module) or if the warning + comes from third-party code. + + This function is also useful to prevent the same or similar warnings from + being issue twice due to recursive calls. + + When the warning is coming (reliably) from SymPy the warns function should + be preferred to ignore_warnings. + + >>> from sympy.utilities.exceptions import ignore_warnings + >>> import warnings + + Here's a warning: + + >>> with warnings.catch_warnings(): # reset warnings in doctest + ... warnings.simplefilter('error') + ... warnings.warn('deprecated', UserWarning) + Traceback (most recent call last): + ... + UserWarning: deprecated + + Let's suppress it with ignore_warnings: + + >>> with warnings.catch_warnings(): # reset warnings in doctest + ... warnings.simplefilter('error') + ... with ignore_warnings(UserWarning): + ... warnings.warn('deprecated', UserWarning) + + (No warning emitted) + + See Also + ======== + sympy.utilities.exceptions.SymPyDeprecationWarning + sympy.utilities.exceptions.sympy_deprecation_warning + sympy.utilities.decorator.deprecated + sympy.testing.pytest.warns_deprecated_sympy + + ''' + # Absorbs all warnings in warnrec + with warnings.catch_warnings(record=True) as warnrec: + # Make sure our warning doesn't get filtered + warnings.simplefilter("always", warningcls) + # Now run the test + yield + + # Reissue any warnings that we aren't testing for + for w in warnrec: + if not issubclass(w.category, warningcls): + warnings.warn_explicit(w.message, w.category, w.filename, w.lineno) diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/iterables.py b/.venv/lib/python3.12/site-packages/sympy/utilities/iterables.py new file mode 100644 index 0000000000000000000000000000000000000000..cc5c1152f03b10f62f9dd04c842b6fc74b0b9242 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/iterables.py @@ -0,0 +1,3179 @@ +from collections import Counter, defaultdict, OrderedDict +from itertools import ( + chain, combinations, combinations_with_replacement, cycle, islice, + permutations, product, groupby +) +# For backwards compatibility +from itertools import product as cartes # noqa: F401 +from operator import gt + + + +# this is the logical location of these functions +from sympy.utilities.enumerative import ( + multiset_partitions_taocp, list_visitor, MultisetPartitionTraverser) + +from sympy.utilities.misc import as_int +from sympy.utilities.decorator import deprecated + + +def is_palindromic(s, i=0, j=None): + """ + Return True if the sequence is the same from left to right as it + is from right to left in the whole sequence (default) or in the + Python slice ``s[i: j]``; else False. + + Examples + ======== + + >>> from sympy.utilities.iterables import is_palindromic + >>> is_palindromic([1, 0, 1]) + True + >>> is_palindromic('abcbb') + False + >>> is_palindromic('abcbb', 1) + False + + Normal Python slicing is performed in place so there is no need to + create a slice of the sequence for testing: + + >>> is_palindromic('abcbb', 1, -1) + True + >>> is_palindromic('abcbb', -4, -1) + True + + See Also + ======== + + sympy.ntheory.digits.is_palindromic: tests integers + + """ + i, j, _ = slice(i, j).indices(len(s)) + m = (j - i)//2 + # if length is odd, middle element will be ignored + return all(s[i + k] == s[j - 1 - k] for k in range(m)) + + +def flatten(iterable, levels=None, cls=None): # noqa: F811 + """ + Recursively denest iterable containers. + + >>> from sympy import flatten + + >>> flatten([1, 2, 3]) + [1, 2, 3] + >>> flatten([1, 2, [3]]) + [1, 2, 3] + >>> flatten([1, [2, 3], [4, 5]]) + [1, 2, 3, 4, 5] + >>> flatten([1.0, 2, (1, None)]) + [1.0, 2, 1, None] + + If you want to denest only a specified number of levels of + nested containers, then set ``levels`` flag to the desired + number of levels:: + + >>> ls = [[(-2, -1), (1, 2)], [(0, 0)]] + + >>> flatten(ls, levels=1) + [(-2, -1), (1, 2), (0, 0)] + + If cls argument is specified, it will only flatten instances of that + class, for example: + + >>> from sympy import Basic, S + >>> class MyOp(Basic): + ... pass + ... + >>> flatten([MyOp(S(1), MyOp(S(2), S(3)))], cls=MyOp) + [1, 2, 3] + + adapted from https://kogs-www.informatik.uni-hamburg.de/~meine/python_tricks + """ + from sympy.tensor.array import NDimArray + if levels is not None: + if not levels: + return iterable + elif levels > 0: + levels -= 1 + else: + raise ValueError( + "expected non-negative number of levels, got %s" % levels) + + if cls is None: + def reducible(x): + return is_sequence(x, set) + else: + def reducible(x): + return isinstance(x, cls) + + result = [] + + for el in iterable: + if reducible(el): + if hasattr(el, 'args') and not isinstance(el, NDimArray): + el = el.args + result.extend(flatten(el, levels=levels, cls=cls)) + else: + result.append(el) + + return result + + +def unflatten(iter, n=2): + """Group ``iter`` into tuples of length ``n``. Raise an error if + the length of ``iter`` is not a multiple of ``n``. + """ + if n < 1 or len(iter) % n: + raise ValueError('iter length is not a multiple of %i' % n) + return list(zip(*(iter[i::n] for i in range(n)))) + + +def reshape(seq, how): + """Reshape the sequence according to the template in ``how``. + + Examples + ======== + + >>> from sympy.utilities import reshape + >>> seq = list(range(1, 9)) + + >>> reshape(seq, [4]) # lists of 4 + [[1, 2, 3, 4], [5, 6, 7, 8]] + + >>> reshape(seq, (4,)) # tuples of 4 + [(1, 2, 3, 4), (5, 6, 7, 8)] + + >>> reshape(seq, (2, 2)) # tuples of 4 + [(1, 2, 3, 4), (5, 6, 7, 8)] + + >>> reshape(seq, (2, [2])) # (i, i, [i, i]) + [(1, 2, [3, 4]), (5, 6, [7, 8])] + + >>> reshape(seq, ((2,), [2])) # etc.... + [((1, 2), [3, 4]), ((5, 6), [7, 8])] + + >>> reshape(seq, (1, [2], 1)) + [(1, [2, 3], 4), (5, [6, 7], 8)] + + >>> reshape(tuple(seq), ([[1], 1, (2,)],)) + (([[1], 2, (3, 4)],), ([[5], 6, (7, 8)],)) + + >>> reshape(tuple(seq), ([1], 1, (2,))) + (([1], 2, (3, 4)), ([5], 6, (7, 8))) + + >>> reshape(list(range(12)), [2, [3], {2}, (1, (3,), 1)]) + [[0, 1, [2, 3, 4], {5, 6}, (7, (8, 9, 10), 11)]] + + """ + m = sum(flatten(how)) + n, rem = divmod(len(seq), m) + if m < 0 or rem: + raise ValueError('template must sum to positive number ' + 'that divides the length of the sequence') + i = 0 + container = type(how) + rv = [None]*n + for k in range(len(rv)): + _rv = [] + for hi in how: + if isinstance(hi, int): + _rv.extend(seq[i: i + hi]) + i += hi + else: + n = sum(flatten(hi)) + hi_type = type(hi) + _rv.append(hi_type(reshape(seq[i: i + n], hi)[0])) + i += n + rv[k] = container(_rv) + return type(seq)(rv) + + +def group(seq, multiple=True): + """ + Splits a sequence into a list of lists of equal, adjacent elements. + + Examples + ======== + + >>> from sympy import group + + >>> group([1, 1, 1, 2, 2, 3]) + [[1, 1, 1], [2, 2], [3]] + >>> group([1, 1, 1, 2, 2, 3], multiple=False) + [(1, 3), (2, 2), (3, 1)] + >>> group([1, 1, 3, 2, 2, 1], multiple=False) + [(1, 2), (3, 1), (2, 2), (1, 1)] + + See Also + ======== + + multiset + + """ + if multiple: + return [(list(g)) for _, g in groupby(seq)] + return [(k, len(list(g))) for k, g in groupby(seq)] + + +def _iproduct2(iterable1, iterable2): + '''Cartesian product of two possibly infinite iterables''' + + it1 = iter(iterable1) + it2 = iter(iterable2) + + elems1 = [] + elems2 = [] + + sentinel = object() + def append(it, elems): + e = next(it, sentinel) + if e is not sentinel: + elems.append(e) + + n = 0 + append(it1, elems1) + append(it2, elems2) + + while n <= len(elems1) + len(elems2): + for m in range(n-len(elems1)+1, len(elems2)): + yield (elems1[n-m], elems2[m]) + n += 1 + append(it1, elems1) + append(it2, elems2) + + +def iproduct(*iterables): + ''' + Cartesian product of iterables. + + Generator of the Cartesian product of iterables. This is analogous to + itertools.product except that it works with infinite iterables and will + yield any item from the infinite product eventually. + + Examples + ======== + + >>> from sympy.utilities.iterables import iproduct + >>> sorted(iproduct([1,2], [3,4])) + [(1, 3), (1, 4), (2, 3), (2, 4)] + + With an infinite iterator: + + >>> from sympy import S + >>> (3,) in iproduct(S.Integers) + True + >>> (3, 4) in iproduct(S.Integers, S.Integers) + True + + .. seealso:: + + `itertools.product + `_ + ''' + if len(iterables) == 0: + yield () + return + elif len(iterables) == 1: + for e in iterables[0]: + yield (e,) + elif len(iterables) == 2: + yield from _iproduct2(*iterables) + else: + first, others = iterables[0], iterables[1:] + for ef, eo in _iproduct2(first, iproduct(*others)): + yield (ef,) + eo + + +def multiset(seq): + """Return the hashable sequence in multiset form with values being the + multiplicity of the item in the sequence. + + Examples + ======== + + >>> from sympy.utilities.iterables import multiset + >>> multiset('mississippi') + {'i': 4, 'm': 1, 'p': 2, 's': 4} + + See Also + ======== + + group + + """ + return dict(Counter(seq).items()) + + + + +def ibin(n, bits=None, str=False): + """Return a list of length ``bits`` corresponding to the binary value + of ``n`` with small bits to the right (last). If bits is omitted, the + length will be the number required to represent ``n``. If the bits are + desired in reversed order, use the ``[::-1]`` slice of the returned list. + + If a sequence of all bits-length lists starting from ``[0, 0,..., 0]`` + through ``[1, 1, ..., 1]`` are desired, pass a non-integer for bits, e.g. + ``'all'``. + + If the bit *string* is desired pass ``str=True``. + + Examples + ======== + + >>> from sympy.utilities.iterables import ibin + >>> ibin(2) + [1, 0] + >>> ibin(2, 4) + [0, 0, 1, 0] + + If all lists corresponding to 0 to 2**n - 1, pass a non-integer + for bits: + + >>> bits = 2 + >>> for i in ibin(2, 'all'): + ... print(i) + (0, 0) + (0, 1) + (1, 0) + (1, 1) + + If a bit string is desired of a given length, use str=True: + + >>> n = 123 + >>> bits = 10 + >>> ibin(n, bits, str=True) + '0001111011' + >>> ibin(n, bits, str=True)[::-1] # small bits left + '1101111000' + >>> list(ibin(3, 'all', str=True)) + ['000', '001', '010', '011', '100', '101', '110', '111'] + + """ + if n < 0: + raise ValueError("negative numbers are not allowed") + n = as_int(n) + + if bits is None: + bits = 0 + else: + try: + bits = as_int(bits) + except ValueError: + bits = -1 + else: + if n.bit_length() > bits: + raise ValueError( + "`bits` must be >= {}".format(n.bit_length())) + + if not str: + if bits >= 0: + return [1 if i == "1" else 0 for i in bin(n)[2:].rjust(bits, "0")] + else: + return variations(range(2), n, repetition=True) + else: + if bits >= 0: + return bin(n)[2:].rjust(bits, "0") + else: + return (bin(i)[2:].rjust(n, "0") for i in range(2**n)) + + +def variations(seq, n, repetition=False): + r"""Returns an iterator over the n-sized variations of ``seq`` (size N). + ``repetition`` controls whether items in ``seq`` can appear more than once; + + Examples + ======== + + ``variations(seq, n)`` will return `\frac{N!}{(N - n)!}` permutations without + repetition of ``seq``'s elements: + + >>> from sympy import variations + >>> list(variations([1, 2], 2)) + [(1, 2), (2, 1)] + + ``variations(seq, n, True)`` will return the `N^n` permutations obtained + by allowing repetition of elements: + + >>> list(variations([1, 2], 2, repetition=True)) + [(1, 1), (1, 2), (2, 1), (2, 2)] + + If you ask for more items than are in the set you get the empty set unless + you allow repetitions: + + >>> list(variations([0, 1], 3, repetition=False)) + [] + >>> list(variations([0, 1], 3, repetition=True))[:4] + [(0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1)] + + .. seealso:: + + `itertools.permutations + `_, + `itertools.product + `_ + """ + if not repetition: + seq = tuple(seq) + if len(seq) < n: + return iter(()) # 0 length iterator + return permutations(seq, n) + else: + if n == 0: + return iter(((),)) # yields 1 empty tuple + else: + return product(seq, repeat=n) + + +def subsets(seq, k=None, repetition=False): + r"""Generates all `k`-subsets (combinations) from an `n`-element set, ``seq``. + + A `k`-subset of an `n`-element set is any subset of length exactly `k`. The + number of `k`-subsets of an `n`-element set is given by ``binomial(n, k)``, + whereas there are `2^n` subsets all together. If `k` is ``None`` then all + `2^n` subsets will be returned from shortest to longest. + + Examples + ======== + + >>> from sympy import subsets + + ``subsets(seq, k)`` will return the + `\frac{n!}{k!(n - k)!}` `k`-subsets (combinations) + without repetition, i.e. once an item has been removed, it can no + longer be "taken": + + >>> list(subsets([1, 2], 2)) + [(1, 2)] + >>> list(subsets([1, 2])) + [(), (1,), (2,), (1, 2)] + >>> list(subsets([1, 2, 3], 2)) + [(1, 2), (1, 3), (2, 3)] + + + ``subsets(seq, k, repetition=True)`` will return the + `\frac{(n - 1 + k)!}{k!(n - 1)!}` + combinations *with* repetition: + + >>> list(subsets([1, 2], 2, repetition=True)) + [(1, 1), (1, 2), (2, 2)] + + If you ask for more items than are in the set you get the empty set unless + you allow repetitions: + + >>> list(subsets([0, 1], 3, repetition=False)) + [] + >>> list(subsets([0, 1], 3, repetition=True)) + [(0, 0, 0), (0, 0, 1), (0, 1, 1), (1, 1, 1)] + + """ + if k is None: + if not repetition: + return chain.from_iterable((combinations(seq, k) + for k in range(len(seq) + 1))) + else: + return chain.from_iterable((combinations_with_replacement(seq, k) + for k in range(len(seq) + 1))) + else: + if not repetition: + return combinations(seq, k) + else: + return combinations_with_replacement(seq, k) + + +def filter_symbols(iterator, exclude): + """ + Only yield elements from `iterator` that do not occur in `exclude`. + + Parameters + ========== + + iterator : iterable + iterator to take elements from + + exclude : iterable + elements to exclude + + Returns + ======= + + iterator : iterator + filtered iterator + """ + exclude = set(exclude) + for s in iterator: + if s not in exclude: + yield s + +def numbered_symbols(prefix='x', cls=None, start=0, exclude=(), *args, **assumptions): + """ + Generate an infinite stream of Symbols consisting of a prefix and + increasing subscripts provided that they do not occur in ``exclude``. + + Parameters + ========== + + prefix : str, optional + The prefix to use. By default, this function will generate symbols of + the form "x0", "x1", etc. + + cls : class, optional + The class to use. By default, it uses ``Symbol``, but you can also use ``Wild`` + or ``Dummy``. + + start : int, optional + The start number. By default, it is 0. + + exclude : list, tuple, set of cls, optional + Symbols to be excluded. + + *args, **kwargs + Additional positional and keyword arguments are passed to the *cls* class. + + Returns + ======= + + sym : Symbol + The subscripted symbols. + """ + exclude = set(exclude or []) + if cls is None: + # We can't just make the default cls=Symbol because it isn't + # imported yet. + from sympy.core import Symbol + cls = Symbol + + while True: + name = '%s%s' % (prefix, start) + s = cls(name, *args, **assumptions) + if s not in exclude: + yield s + start += 1 + + +def capture(func): + """Return the printed output of func(). + + ``func`` should be a function without arguments that produces output with + print statements. + + >>> from sympy.utilities.iterables import capture + >>> from sympy import pprint + >>> from sympy.abc import x + >>> def foo(): + ... print('hello world!') + ... + >>> 'hello' in capture(foo) # foo, not foo() + True + >>> capture(lambda: pprint(2/x)) + '2\\n-\\nx\\n' + + """ + from io import StringIO + import sys + + stdout = sys.stdout + sys.stdout = file = StringIO() + try: + func() + finally: + sys.stdout = stdout + return file.getvalue() + + +def sift(seq, keyfunc, binary=False): + """ + Sift the sequence, ``seq`` according to ``keyfunc``. + + Returns + ======= + + When ``binary`` is ``False`` (default), the output is a dictionary + where elements of ``seq`` are stored in a list keyed to the value + of keyfunc for that element. If ``binary`` is True then a tuple + with lists ``T`` and ``F`` are returned where ``T`` is a list + containing elements of seq for which ``keyfunc`` was ``True`` and + ``F`` containing those elements for which ``keyfunc`` was ``False``; + a ValueError is raised if the ``keyfunc`` is not binary. + + Examples + ======== + + >>> from sympy.utilities import sift + >>> from sympy.abc import x, y + >>> from sympy import sqrt, exp, pi, Tuple + + >>> sift(range(5), lambda x: x % 2) + {0: [0, 2, 4], 1: [1, 3]} + + sift() returns a defaultdict() object, so any key that has no matches will + give []. + + >>> sift([x], lambda x: x.is_commutative) + {True: [x]} + >>> _[False] + [] + + Sometimes you will not know how many keys you will get: + + >>> sift([sqrt(x), exp(x), (y**x)**2], + ... lambda x: x.as_base_exp()[0]) + {E: [exp(x)], x: [sqrt(x)], y: [y**(2*x)]} + + Sometimes you expect the results to be binary; the + results can be unpacked by setting ``binary`` to True: + + >>> sift(range(4), lambda x: x % 2, binary=True) + ([1, 3], [0, 2]) + >>> sift(Tuple(1, pi), lambda x: x.is_rational, binary=True) + ([1], [pi]) + + A ValueError is raised if the predicate was not actually binary + (which is a good test for the logic where sifting is used and + binary results were expected): + + >>> unknown = exp(1) - pi # the rationality of this is unknown + >>> args = Tuple(1, pi, unknown) + >>> sift(args, lambda x: x.is_rational, binary=True) + Traceback (most recent call last): + ... + ValueError: keyfunc gave non-binary output + + The non-binary sifting shows that there were 3 keys generated: + + >>> set(sift(args, lambda x: x.is_rational).keys()) + {None, False, True} + + If you need to sort the sifted items it might be better to use + ``ordered`` which can economically apply multiple sort keys + to a sequence while sorting. + + See Also + ======== + + ordered + + """ + if not binary: + m = defaultdict(list) + for i in seq: + m[keyfunc(i)].append(i) + return m + sift = F, T = [], [] + for i in seq: + try: + sift[keyfunc(i)].append(i) + except (IndexError, TypeError): + raise ValueError('keyfunc gave non-binary output') + return T, F + + +def take(iter, n): + """Return ``n`` items from ``iter`` iterator. """ + return [ value for _, value in zip(range(n), iter) ] + + +def dict_merge(*dicts): + """Merge dictionaries into a single dictionary. """ + merged = {} + + for dict in dicts: + merged.update(dict) + + return merged + + +def common_prefix(*seqs): + """Return the subsequence that is a common start of sequences in ``seqs``. + + >>> from sympy.utilities.iterables import common_prefix + >>> common_prefix(list(range(3))) + [0, 1, 2] + >>> common_prefix(list(range(3)), list(range(4))) + [0, 1, 2] + >>> common_prefix([1, 2, 3], [1, 2, 5]) + [1, 2] + >>> common_prefix([1, 2, 3], [1, 3, 5]) + [1] + """ + if not all(seqs): + return [] + elif len(seqs) == 1: + return seqs[0] + i = 0 + for i in range(min(len(s) for s in seqs)): + if not all(seqs[j][i] == seqs[0][i] for j in range(len(seqs))): + break + else: + i += 1 + return seqs[0][:i] + + +def common_suffix(*seqs): + """Return the subsequence that is a common ending of sequences in ``seqs``. + + >>> from sympy.utilities.iterables import common_suffix + >>> common_suffix(list(range(3))) + [0, 1, 2] + >>> common_suffix(list(range(3)), list(range(4))) + [] + >>> common_suffix([1, 2, 3], [9, 2, 3]) + [2, 3] + >>> common_suffix([1, 2, 3], [9, 7, 3]) + [3] + """ + + if not all(seqs): + return [] + elif len(seqs) == 1: + return seqs[0] + i = 0 + for i in range(-1, -min(len(s) for s in seqs) - 1, -1): + if not all(seqs[j][i] == seqs[0][i] for j in range(len(seqs))): + break + else: + i -= 1 + if i == -1: + return [] + else: + return seqs[0][i + 1:] + + +def prefixes(seq): + """ + Generate all prefixes of a sequence. + + Examples + ======== + + >>> from sympy.utilities.iterables import prefixes + + >>> list(prefixes([1,2,3,4])) + [[1], [1, 2], [1, 2, 3], [1, 2, 3, 4]] + + """ + n = len(seq) + + for i in range(n): + yield seq[:i + 1] + + +def postfixes(seq): + """ + Generate all postfixes of a sequence. + + Examples + ======== + + >>> from sympy.utilities.iterables import postfixes + + >>> list(postfixes([1,2,3,4])) + [[4], [3, 4], [2, 3, 4], [1, 2, 3, 4]] + + """ + n = len(seq) + + for i in range(n): + yield seq[n - i - 1:] + + +def topological_sort(graph, key=None): + r""" + Topological sort of graph's vertices. + + Parameters + ========== + + graph : tuple[list, list[tuple[T, T]] + A tuple consisting of a list of vertices and a list of edges of + a graph to be sorted topologically. + + key : callable[T] (optional) + Ordering key for vertices on the same level. By default the natural + (e.g. lexicographic) ordering is used (in this case the base type + must implement ordering relations). + + Examples + ======== + + Consider a graph:: + + +---+ +---+ +---+ + | 7 |\ | 5 | | 3 | + +---+ \ +---+ +---+ + | _\___/ ____ _/ | + | / \___/ \ / | + V V V V | + +----+ +---+ | + | 11 | | 8 | | + +----+ +---+ | + | | \____ ___/ _ | + | \ \ / / \ | + V \ V V / V V + +---+ \ +---+ | +----+ + | 2 | | | 9 | | | 10 | + +---+ | +---+ | +----+ + \________/ + + where vertices are integers. This graph can be encoded using + elementary Python's data structures as follows:: + + >>> V = [2, 3, 5, 7, 8, 9, 10, 11] + >>> E = [(7, 11), (7, 8), (5, 11), (3, 8), (3, 10), + ... (11, 2), (11, 9), (11, 10), (8, 9)] + + To compute a topological sort for graph ``(V, E)`` issue:: + + >>> from sympy.utilities.iterables import topological_sort + + >>> topological_sort((V, E)) + [3, 5, 7, 8, 11, 2, 9, 10] + + If specific tie breaking approach is needed, use ``key`` parameter:: + + >>> topological_sort((V, E), key=lambda v: -v) + [7, 5, 11, 3, 10, 8, 9, 2] + + Only acyclic graphs can be sorted. If the input graph has a cycle, + then ``ValueError`` will be raised:: + + >>> topological_sort((V, E + [(10, 7)])) + Traceback (most recent call last): + ... + ValueError: cycle detected + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Topological_sorting + + """ + V, E = graph + + L = [] + S = set(V) + E = list(E) + + S.difference_update(u for v, u in E) + + if key is None: + def key(value): + return value + + S = sorted(S, key=key, reverse=True) + + while S: + node = S.pop() + L.append(node) + + for u, v in list(E): + if u == node: + E.remove((u, v)) + + for _u, _v in E: + if v == _v: + break + else: + kv = key(v) + + for i, s in enumerate(S): + ks = key(s) + + if kv > ks: + S.insert(i, v) + break + else: + S.append(v) + + if E: + raise ValueError("cycle detected") + else: + return L + + +def strongly_connected_components(G): + r""" + Strongly connected components of a directed graph in reverse topological + order. + + + Parameters + ========== + + G : tuple[list, list[tuple[T, T]] + A tuple consisting of a list of vertices and a list of edges of + a graph whose strongly connected components are to be found. + + + Examples + ======== + + Consider a directed graph (in dot notation):: + + digraph { + A -> B + A -> C + B -> C + C -> B + B -> D + } + + .. graphviz:: + + digraph { + A -> B + A -> C + B -> C + C -> B + B -> D + } + + where vertices are the letters A, B, C and D. This graph can be encoded + using Python's elementary data structures as follows:: + + >>> V = ['A', 'B', 'C', 'D'] + >>> E = [('A', 'B'), ('A', 'C'), ('B', 'C'), ('C', 'B'), ('B', 'D')] + + The strongly connected components of this graph can be computed as + + >>> from sympy.utilities.iterables import strongly_connected_components + + >>> strongly_connected_components((V, E)) + [['D'], ['B', 'C'], ['A']] + + This also gives the components in reverse topological order. + + Since the subgraph containing B and C has a cycle they must be together in + a strongly connected component. A and D are connected to the rest of the + graph but not in a cyclic manner so they appear as their own strongly + connected components. + + + Notes + ===== + + The vertices of the graph must be hashable for the data structures used. + If the vertices are unhashable replace them with integer indices. + + This function uses Tarjan's algorithm to compute the strongly connected + components in `O(|V|+|E|)` (linear) time. + + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Strongly_connected_component + .. [2] https://en.wikipedia.org/wiki/Tarjan%27s_strongly_connected_components_algorithm + + + See Also + ======== + + sympy.utilities.iterables.connected_components + + """ + # Map from a vertex to its neighbours + V, E = G + Gmap = {vi: [] for vi in V} + for v1, v2 in E: + Gmap[v1].append(v2) + return _strongly_connected_components(V, Gmap) + + +def _strongly_connected_components(V, Gmap): + """More efficient internal routine for strongly_connected_components""" + # + # Here V is an iterable of vertices and Gmap is a dict mapping each vertex + # to a list of neighbours e.g.: + # + # V = [0, 1, 2, 3] + # Gmap = {0: [2, 3], 1: [0]} + # + # For a large graph these data structures can often be created more + # efficiently then those expected by strongly_connected_components() which + # in this case would be + # + # V = [0, 1, 2, 3] + # Gmap = [(0, 2), (0, 3), (1, 0)] + # + # XXX: Maybe this should be the recommended function to use instead... + # + + # Non-recursive Tarjan's algorithm: + lowlink = {} + indices = {} + stack = OrderedDict() + callstack = [] + components = [] + nomore = object() + + def start(v): + index = len(stack) + indices[v] = lowlink[v] = index + stack[v] = None + callstack.append((v, iter(Gmap[v]))) + + def finish(v1): + # Finished a component? + if lowlink[v1] == indices[v1]: + component = [stack.popitem()[0]] + while component[-1] is not v1: + component.append(stack.popitem()[0]) + components.append(component[::-1]) + v2, _ = callstack.pop() + if callstack: + v1, _ = callstack[-1] + lowlink[v1] = min(lowlink[v1], lowlink[v2]) + + for v in V: + if v in indices: + continue + start(v) + while callstack: + v1, it1 = callstack[-1] + v2 = next(it1, nomore) + # Finished children of v1? + if v2 is nomore: + finish(v1) + # Recurse on v2 + elif v2 not in indices: + start(v2) + elif v2 in stack: + lowlink[v1] = min(lowlink[v1], indices[v2]) + + # Reverse topological sort order: + return components + + +def connected_components(G): + r""" + Connected components of an undirected graph or weakly connected components + of a directed graph. + + + Parameters + ========== + + G : tuple[list, list[tuple[T, T]] + A tuple consisting of a list of vertices and a list of edges of + a graph whose connected components are to be found. + + + Examples + ======== + + + Given an undirected graph:: + + graph { + A -- B + C -- D + } + + .. graphviz:: + + graph { + A -- B + C -- D + } + + We can find the connected components using this function if we include + each edge in both directions:: + + >>> from sympy.utilities.iterables import connected_components + + >>> V = ['A', 'B', 'C', 'D'] + >>> E = [('A', 'B'), ('B', 'A'), ('C', 'D'), ('D', 'C')] + >>> connected_components((V, E)) + [['A', 'B'], ['C', 'D']] + + The weakly connected components of a directed graph can found the same + way. + + + Notes + ===== + + The vertices of the graph must be hashable for the data structures used. + If the vertices are unhashable replace them with integer indices. + + This function uses Tarjan's algorithm to compute the connected components + in `O(|V|+|E|)` (linear) time. + + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Component_%28graph_theory%29 + .. [2] https://en.wikipedia.org/wiki/Tarjan%27s_strongly_connected_components_algorithm + + + See Also + ======== + + sympy.utilities.iterables.strongly_connected_components + + """ + # Duplicate edges both ways so that the graph is effectively undirected + # and return the strongly connected components: + V, E = G + E_undirected = [] + for v1, v2 in E: + E_undirected.extend([(v1, v2), (v2, v1)]) + return strongly_connected_components((V, E_undirected)) + + +def rotate_left(x, y): + """ + Left rotates a list x by the number of steps specified + in y. + + Examples + ======== + + >>> from sympy.utilities.iterables import rotate_left + >>> a = [0, 1, 2] + >>> rotate_left(a, 1) + [1, 2, 0] + """ + if len(x) == 0: + return [] + y = y % len(x) + return x[y:] + x[:y] + + +def rotate_right(x, y): + """ + Right rotates a list x by the number of steps specified + in y. + + Examples + ======== + + >>> from sympy.utilities.iterables import rotate_right + >>> a = [0, 1, 2] + >>> rotate_right(a, 1) + [2, 0, 1] + """ + if len(x) == 0: + return [] + y = len(x) - y % len(x) + return x[y:] + x[:y] + + +def least_rotation(x, key=None): + ''' + Returns the number of steps of left rotation required to + obtain lexicographically minimal string/list/tuple, etc. + + Examples + ======== + + >>> from sympy.utilities.iterables import least_rotation, rotate_left + >>> a = [3, 1, 5, 1, 2] + >>> least_rotation(a) + 3 + >>> rotate_left(a, _) + [1, 2, 3, 1, 5] + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Lexicographically_minimal_string_rotation + + ''' + from sympy.functions.elementary.miscellaneous import Id + if key is None: key = Id + S = x + x # Concatenate string to it self to avoid modular arithmetic + f = [-1] * len(S) # Failure function + k = 0 # Least rotation of string found so far + for j in range(1,len(S)): + sj = S[j] + i = f[j-k-1] + while i != -1 and sj != S[k+i+1]: + if key(sj) < key(S[k+i+1]): + k = j-i-1 + i = f[i] + if sj != S[k+i+1]: + if key(sj) < key(S[k]): + k = j + f[j-k] = -1 + else: + f[j-k] = i+1 + return k + + +def multiset_combinations(m, n, g=None): + """ + Return the unique combinations of size ``n`` from multiset ``m``. + + Examples + ======== + + >>> from sympy.utilities.iterables import multiset_combinations + >>> from itertools import combinations + >>> [''.join(i) for i in multiset_combinations('baby', 3)] + ['abb', 'aby', 'bby'] + + >>> def count(f, s): return len(list(f(s, 3))) + + The number of combinations depends on the number of letters; the + number of unique combinations depends on how the letters are + repeated. + + >>> s1 = 'abracadabra' + >>> s2 = 'banana tree' + >>> count(combinations, s1), count(multiset_combinations, s1) + (165, 23) + >>> count(combinations, s2), count(multiset_combinations, s2) + (165, 54) + + """ + from sympy.core.sorting import ordered + if g is None: + if isinstance(m, dict): + if any(as_int(v) < 0 for v in m.values()): + raise ValueError('counts cannot be negative') + N = sum(m.values()) + if n > N: + return + g = [[k, m[k]] for k in ordered(m)] + else: + m = list(m) + N = len(m) + if n > N: + return + try: + m = multiset(m) + g = [(k, m[k]) for k in ordered(m)] + except TypeError: + m = list(ordered(m)) + g = [list(i) for i in group(m, multiple=False)] + del m + else: + # not checking counts since g is intended for internal use + N = sum(v for k, v in g) + if n > N or not n: + yield [] + else: + for i, (k, v) in enumerate(g): + if v >= n: + yield [k]*n + v = n - 1 + for v in range(min(n, v), 0, -1): + for j in multiset_combinations(None, n - v, g[i + 1:]): + rv = [k]*v + j + if len(rv) == n: + yield rv + +def multiset_permutations(m, size=None, g=None): + """ + Return the unique permutations of multiset ``m``. + + Examples + ======== + + >>> from sympy.utilities.iterables import multiset_permutations + >>> from sympy import factorial + >>> [''.join(i) for i in multiset_permutations('aab')] + ['aab', 'aba', 'baa'] + >>> factorial(len('banana')) + 720 + >>> len(list(multiset_permutations('banana'))) + 60 + """ + from sympy.core.sorting import ordered + if g is None: + if isinstance(m, dict): + if any(as_int(v) < 0 for v in m.values()): + raise ValueError('counts cannot be negative') + g = [[k, m[k]] for k in ordered(m)] + else: + m = list(ordered(m)) + g = [list(i) for i in group(m, multiple=False)] + del m + do = [gi for gi in g if gi[1] > 0] + SUM = sum(gi[1] for gi in do) + if not do or size is not None and (size > SUM or size < 1): + if not do and size is None or size == 0: + yield [] + return + elif size == 1: + for k, v in do: + yield [k] + elif len(do) == 1: + k, v = do[0] + v = v if size is None else (size if size <= v else 0) + yield [k for i in range(v)] + elif all(v == 1 for k, v in do): + for p in permutations([k for k, v in do], size): + yield list(p) + else: + size = size if size is not None else SUM + for i, (k, v) in enumerate(do): + do[i][1] -= 1 + for j in multiset_permutations(None, size - 1, do): + if j: + yield [k] + j + do[i][1] += 1 + + +def _partition(seq, vector, m=None): + """ + Return the partition of seq as specified by the partition vector. + + Examples + ======== + + >>> from sympy.utilities.iterables import _partition + >>> _partition('abcde', [1, 0, 1, 2, 0]) + [['b', 'e'], ['a', 'c'], ['d']] + + Specifying the number of bins in the partition is optional: + + >>> _partition('abcde', [1, 0, 1, 2, 0], 3) + [['b', 'e'], ['a', 'c'], ['d']] + + The output of _set_partitions can be passed as follows: + + >>> output = (3, [1, 0, 1, 2, 0]) + >>> _partition('abcde', *output) + [['b', 'e'], ['a', 'c'], ['d']] + + See Also + ======== + + combinatorics.partitions.Partition.from_rgs + + """ + if m is None: + m = max(vector) + 1 + elif isinstance(vector, int): # entered as m, vector + vector, m = m, vector + p = [[] for i in range(m)] + for i, v in enumerate(vector): + p[v].append(seq[i]) + return p + + +def _set_partitions(n): + """Cycle through all partitions of n elements, yielding the + current number of partitions, ``m``, and a mutable list, ``q`` + such that ``element[i]`` is in part ``q[i]`` of the partition. + + NOTE: ``q`` is modified in place and generally should not be changed + between function calls. + + Examples + ======== + + >>> from sympy.utilities.iterables import _set_partitions, _partition + >>> for m, q in _set_partitions(3): + ... print('%s %s %s' % (m, q, _partition('abc', q, m))) + 1 [0, 0, 0] [['a', 'b', 'c']] + 2 [0, 0, 1] [['a', 'b'], ['c']] + 2 [0, 1, 0] [['a', 'c'], ['b']] + 2 [0, 1, 1] [['a'], ['b', 'c']] + 3 [0, 1, 2] [['a'], ['b'], ['c']] + + Notes + ===== + + This algorithm is similar to, and solves the same problem as, + Algorithm 7.2.1.5H, from volume 4A of Knuth's The Art of Computer + Programming. Knuth uses the term "restricted growth string" where + this code refers to a "partition vector". In each case, the meaning is + the same: the value in the ith element of the vector specifies to + which part the ith set element is to be assigned. + + At the lowest level, this code implements an n-digit big-endian + counter (stored in the array q) which is incremented (with carries) to + get the next partition in the sequence. A special twist is that a + digit is constrained to be at most one greater than the maximum of all + the digits to the left of it. The array p maintains this maximum, so + that the code can efficiently decide when a digit can be incremented + in place or whether it needs to be reset to 0 and trigger a carry to + the next digit. The enumeration starts with all the digits 0 (which + corresponds to all the set elements being assigned to the same 0th + part), and ends with 0123...n, which corresponds to each set element + being assigned to a different, singleton, part. + + This routine was rewritten to use 0-based lists while trying to + preserve the beauty and efficiency of the original algorithm. + + References + ========== + + .. [1] Nijenhuis, Albert and Wilf, Herbert. (1978) Combinatorial Algorithms, + 2nd Ed, p 91, algorithm "nexequ". Available online from + https://www.math.upenn.edu/~wilf/website/CombAlgDownld.html (viewed + November 17, 2012). + + """ + p = [0]*n + q = [0]*n + nc = 1 + yield nc, q + while nc != n: + m = n + while 1: + m -= 1 + i = q[m] + if p[i] != 1: + break + q[m] = 0 + i += 1 + q[m] = i + m += 1 + nc += m - n + p[0] += n - m + if i == nc: + p[nc] = 0 + nc += 1 + p[i - 1] -= 1 + p[i] += 1 + yield nc, q + + +def multiset_partitions(multiset, m=None): + """ + Return unique partitions of the given multiset (in list form). + If ``m`` is None, all multisets will be returned, otherwise only + partitions with ``m`` parts will be returned. + + If ``multiset`` is an integer, a range [0, 1, ..., multiset - 1] + will be supplied. + + Examples + ======== + + >>> from sympy.utilities.iterables import multiset_partitions + >>> list(multiset_partitions([1, 2, 3, 4], 2)) + [[[1, 2, 3], [4]], [[1, 2, 4], [3]], [[1, 2], [3, 4]], + [[1, 3, 4], [2]], [[1, 3], [2, 4]], [[1, 4], [2, 3]], + [[1], [2, 3, 4]]] + >>> list(multiset_partitions([1, 2, 3, 4], 1)) + [[[1, 2, 3, 4]]] + + Only unique partitions are returned and these will be returned in a + canonical order regardless of the order of the input: + + >>> a = [1, 2, 2, 1] + >>> ans = list(multiset_partitions(a, 2)) + >>> a.sort() + >>> list(multiset_partitions(a, 2)) == ans + True + >>> a = range(3, 1, -1) + >>> (list(multiset_partitions(a)) == + ... list(multiset_partitions(sorted(a)))) + True + + If m is omitted then all partitions will be returned: + + >>> list(multiset_partitions([1, 1, 2])) + [[[1, 1, 2]], [[1, 1], [2]], [[1, 2], [1]], [[1], [1], [2]]] + >>> list(multiset_partitions([1]*3)) + [[[1, 1, 1]], [[1], [1, 1]], [[1], [1], [1]]] + + Counting + ======== + + The number of partitions of a set is given by the bell number: + + >>> from sympy import bell + >>> len(list(multiset_partitions(5))) == bell(5) == 52 + True + + The number of partitions of length k from a set of size n is given by the + Stirling Number of the 2nd kind: + + >>> from sympy.functions.combinatorial.numbers import stirling + >>> stirling(5, 2) == len(list(multiset_partitions(5, 2))) == 15 + True + + These comments on counting apply to *sets*, not multisets. + + Notes + ===== + + When all the elements are the same in the multiset, the order + of the returned partitions is determined by the ``partitions`` + routine. If one is counting partitions then it is better to use + the ``nT`` function. + + See Also + ======== + + partitions + sympy.combinatorics.partitions.Partition + sympy.combinatorics.partitions.IntegerPartition + sympy.functions.combinatorial.numbers.nT + + """ + # This function looks at the supplied input and dispatches to + # several special-case routines as they apply. + if isinstance(multiset, int): + n = multiset + if m and m > n: + return + multiset = list(range(n)) + if m == 1: + yield [multiset[:]] + return + + # If m is not None, it can sometimes be faster to use + # MultisetPartitionTraverser.enum_range() even for inputs + # which are sets. Since the _set_partitions code is quite + # fast, this is only advantageous when the overall set + # partitions outnumber those with the desired number of parts + # by a large factor. (At least 60.) Such a switch is not + # currently implemented. + for nc, q in _set_partitions(n): + if m is None or nc == m: + rv = [[] for i in range(nc)] + for i in range(n): + rv[q[i]].append(multiset[i]) + yield rv + return + + if len(multiset) == 1 and isinstance(multiset, str): + multiset = [multiset] + + if not has_variety(multiset): + # Only one component, repeated n times. The resulting + # partitions correspond to partitions of integer n. + n = len(multiset) + if m and m > n: + return + if m == 1: + yield [multiset[:]] + return + x = multiset[:1] + for size, p in partitions(n, m, size=True): + if m is None or size == m: + rv = [] + for k in sorted(p): + rv.extend([x*k]*p[k]) + yield rv + else: + from sympy.core.sorting import ordered + multiset = list(ordered(multiset)) + n = len(multiset) + if m and m > n: + return + if m == 1: + yield [multiset[:]] + return + + # Split the information of the multiset into two lists - + # one of the elements themselves, and one (of the same length) + # giving the number of repeats for the corresponding element. + elements, multiplicities = zip(*group(multiset, False)) + + if len(elements) < len(multiset): + # General case - multiset with more than one distinct element + # and at least one element repeated more than once. + if m: + mpt = MultisetPartitionTraverser() + for state in mpt.enum_range(multiplicities, m-1, m): + yield list_visitor(state, elements) + else: + for state in multiset_partitions_taocp(multiplicities): + yield list_visitor(state, elements) + else: + # Set partitions case - no repeated elements. Pretty much + # same as int argument case above, with same possible, but + # currently unimplemented optimization for some cases when + # m is not None + for nc, q in _set_partitions(n): + if m is None or nc == m: + rv = [[] for i in range(nc)] + for i in range(n): + rv[q[i]].append(i) + yield [[multiset[j] for j in i] for i in rv] + + +def partitions(n, m=None, k=None, size=False): + """Generate all partitions of positive integer, n. + + Each partition is represented as a dictionary, mapping an integer + to the number of copies of that integer in the partition. For example, + the first partition of 4 returned is {4: 1}, "4: one of them". + + Parameters + ========== + n : int + m : int, optional + limits number of parts in partition (mnemonic: m, maximum parts) + k : int, optional + limits the numbers that are kept in the partition (mnemonic: k, keys) + size : bool, default: False + If ``True``, (M, P) is returned where M is the sum of the + multiplicities and P is the generated partition. + If ``False``, only the generated partition is returned. + + Examples + ======== + + >>> from sympy.utilities.iterables import partitions + + The numbers appearing in the partition (the key of the returned dict) + are limited with k: + + >>> for p in partitions(6, k=2): # doctest: +SKIP + ... print(p) + {2: 3} + {1: 2, 2: 2} + {1: 4, 2: 1} + {1: 6} + + The maximum number of parts in the partition (the sum of the values in + the returned dict) are limited with m (default value, None, gives + partitions from 1 through n): + + >>> for p in partitions(6, m=2): # doctest: +SKIP + ... print(p) + ... + {6: 1} + {1: 1, 5: 1} + {2: 1, 4: 1} + {3: 2} + + References + ========== + + .. [1] modified from Tim Peter's version to allow for k and m values: + https://code.activestate.com/recipes/218332-generator-for-integer-partitions/ + + See Also + ======== + + sympy.combinatorics.partitions.Partition + sympy.combinatorics.partitions.IntegerPartition + + """ + if (n <= 0 or + m is not None and m < 1 or + k is not None and k < 1 or + m and k and m*k < n): + # the empty set is the only way to handle these inputs + # and returning {} to represent it is consistent with + # the counting convention, e.g. nT(0) == 1. + if size: + yield 0, {} + else: + yield {} + return + + if m is None: + m = n + else: + m = min(m, n) + k = min(k or n, n) + + n, m, k = as_int(n), as_int(m), as_int(k) + q, r = divmod(n, k) + ms = {k: q} + keys = [k] # ms.keys(), from largest to smallest + if r: + ms[r] = 1 + keys.append(r) + room = m - q - bool(r) + if size: + yield sum(ms.values()), ms.copy() + else: + yield ms.copy() + + while keys != [1]: + # Reuse any 1's. + if keys[-1] == 1: + del keys[-1] + reuse = ms.pop(1) + room += reuse + else: + reuse = 0 + + while 1: + # Let i be the smallest key larger than 1. Reuse one + # instance of i. + i = keys[-1] + newcount = ms[i] = ms[i] - 1 + reuse += i + if newcount == 0: + del keys[-1], ms[i] + room += 1 + + # Break the remainder into pieces of size i-1. + i -= 1 + q, r = divmod(reuse, i) + need = q + bool(r) + if need > room: + if not keys: + return + continue + + ms[i] = q + keys.append(i) + if r: + ms[r] = 1 + keys.append(r) + break + room -= need + if size: + yield sum(ms.values()), ms.copy() + else: + yield ms.copy() + + +def ordered_partitions(n, m=None, sort=True): + """Generates ordered partitions of integer *n*. + + Parameters + ========== + n : int + m : int, optional + The default value gives partitions of all sizes else only + those with size m. In addition, if *m* is not None then + partitions are generated *in place* (see examples). + sort : bool, default: True + Controls whether partitions are + returned in sorted order when *m* is not None; when False, + the partitions are returned as fast as possible with elements + sorted, but when m|n the partitions will not be in + ascending lexicographical order. + + Examples + ======== + + >>> from sympy.utilities.iterables import ordered_partitions + + All partitions of 5 in ascending lexicographical: + + >>> for p in ordered_partitions(5): + ... print(p) + [1, 1, 1, 1, 1] + [1, 1, 1, 2] + [1, 1, 3] + [1, 2, 2] + [1, 4] + [2, 3] + [5] + + Only partitions of 5 with two parts: + + >>> for p in ordered_partitions(5, 2): + ... print(p) + [1, 4] + [2, 3] + + When ``m`` is given, a given list objects will be used more than + once for speed reasons so you will not see the correct partitions + unless you make a copy of each as it is generated: + + >>> [p for p in ordered_partitions(7, 3)] + [[1, 1, 1], [1, 1, 1], [1, 1, 1], [2, 2, 2]] + >>> [list(p) for p in ordered_partitions(7, 3)] + [[1, 1, 5], [1, 2, 4], [1, 3, 3], [2, 2, 3]] + + When ``n`` is a multiple of ``m``, the elements are still sorted + but the partitions themselves will be *unordered* if sort is False; + the default is to return them in ascending lexicographical order. + + >>> for p in ordered_partitions(6, 2): + ... print(p) + [1, 5] + [2, 4] + [3, 3] + + But if speed is more important than ordering, sort can be set to + False: + + >>> for p in ordered_partitions(6, 2, sort=False): + ... print(p) + [1, 5] + [3, 3] + [2, 4] + + References + ========== + + .. [1] Generating Integer Partitions, [online], + Available: https://jeromekelleher.net/generating-integer-partitions.html + .. [2] Jerome Kelleher and Barry O'Sullivan, "Generating All + Partitions: A Comparison Of Two Encodings", [online], + Available: https://arxiv.org/pdf/0909.2331v2.pdf + """ + if n < 1 or m is not None and m < 1: + # the empty set is the only way to handle these inputs + # and returning {} to represent it is consistent with + # the counting convention, e.g. nT(0) == 1. + yield [] + return + + if m is None: + # The list `a`'s leading elements contain the partition in which + # y is the biggest element and x is either the same as y or the + # 2nd largest element; v and w are adjacent element indices + # to which x and y are being assigned, respectively. + a = [1]*n + y = -1 + v = n + while v > 0: + v -= 1 + x = a[v] + 1 + while y >= 2 * x: + a[v] = x + y -= x + v += 1 + w = v + 1 + while x <= y: + a[v] = x + a[w] = y + yield a[:w + 1] + x += 1 + y -= 1 + a[v] = x + y + y = a[v] - 1 + yield a[:w] + elif m == 1: + yield [n] + elif n == m: + yield [1]*n + else: + # recursively generate partitions of size m + for b in range(1, n//m + 1): + a = [b]*m + x = n - b*m + if not x: + if sort: + yield a + elif not sort and x <= m: + for ax in ordered_partitions(x, sort=False): + mi = len(ax) + a[-mi:] = [i + b for i in ax] + yield a + a[-mi:] = [b]*mi + else: + for mi in range(1, m): + for ax in ordered_partitions(x, mi, sort=True): + a[-mi:] = [i + b for i in ax] + yield a + a[-mi:] = [b]*mi + + +def binary_partitions(n): + """ + Generates the binary partition of *n*. + + A binary partition consists only of numbers that are + powers of two. Each step reduces a `2^{k+1}` to `2^k` and + `2^k`. Thus 16 is converted to 8 and 8. + + Examples + ======== + + >>> from sympy.utilities.iterables import binary_partitions + >>> for i in binary_partitions(5): + ... print(i) + ... + [4, 1] + [2, 2, 1] + [2, 1, 1, 1] + [1, 1, 1, 1, 1] + + References + ========== + + .. [1] TAOCP 4, section 7.2.1.5, problem 64 + + """ + from math import ceil, log2 + power = int(2**(ceil(log2(n)))) + acc = 0 + partition = [] + while power: + if acc + power <= n: + partition.append(power) + acc += power + power >>= 1 + + last_num = len(partition) - 1 - (n & 1) + while last_num >= 0: + yield partition + if partition[last_num] == 2: + partition[last_num] = 1 + partition.append(1) + last_num -= 1 + continue + partition.append(1) + partition[last_num] >>= 1 + x = partition[last_num + 1] = partition[last_num] + last_num += 1 + while x > 1: + if x <= len(partition) - last_num - 1: + del partition[-x + 1:] + last_num += 1 + partition[last_num] = x + else: + x >>= 1 + yield [1]*n + + +def has_dups(seq): + """Return True if there are any duplicate elements in ``seq``. + + Examples + ======== + + >>> from sympy import has_dups, Dict, Set + >>> has_dups((1, 2, 1)) + True + >>> has_dups(range(3)) + False + >>> all(has_dups(c) is False for c in (set(), Set(), dict(), Dict())) + True + """ + from sympy.core.containers import Dict + from sympy.sets.sets import Set + if isinstance(seq, (dict, set, Dict, Set)): + return False + unique = set() + try: + return any(True for s in seq if s in unique or unique.add(s)) + except TypeError: + return len(seq) != len(list(uniq(seq))) + + +def has_variety(seq): + """Return True if there are any different elements in ``seq``. + + Examples + ======== + + >>> from sympy import has_variety + + >>> has_variety((1, 2, 1)) + True + >>> has_variety((1, 1, 1)) + False + """ + for i, s in enumerate(seq): + if i == 0: + sentinel = s + else: + if s != sentinel: + return True + return False + + +def uniq(seq, result=None): + """ + Yield unique elements from ``seq`` as an iterator. The second + parameter ``result`` is used internally; it is not necessary + to pass anything for this. + + Note: changing the sequence during iteration will raise a + RuntimeError if the size of the sequence is known; if you pass + an iterator and advance the iterator you will change the + output of this routine but there will be no warning. + + Examples + ======== + + >>> from sympy.utilities.iterables import uniq + >>> dat = [1, 4, 1, 5, 4, 2, 1, 2] + >>> type(uniq(dat)) in (list, tuple) + False + + >>> list(uniq(dat)) + [1, 4, 5, 2] + >>> list(uniq(x for x in dat)) + [1, 4, 5, 2] + >>> list(uniq([[1], [2, 1], [1]])) + [[1], [2, 1]] + """ + try: + n = len(seq) + except TypeError: + n = None + def check(): + # check that size of seq did not change during iteration; + # if n == None the object won't support size changing, e.g. + # an iterator can't be changed + if n is not None and len(seq) != n: + raise RuntimeError('sequence changed size during iteration') + try: + seen = set() + result = result or [] + for i, s in enumerate(seq): + if not (s in seen or seen.add(s)): + yield s + check() + except TypeError: + if s not in result: + yield s + check() + result.append(s) + if hasattr(seq, '__getitem__'): + yield from uniq(seq[i + 1:], result) + else: + yield from uniq(seq, result) + + +def generate_bell(n): + """Return permutations of [0, 1, ..., n - 1] such that each permutation + differs from the last by the exchange of a single pair of neighbors. + The ``n!`` permutations are returned as an iterator. In order to obtain + the next permutation from a random starting permutation, use the + ``next_trotterjohnson`` method of the Permutation class (which generates + the same sequence in a different manner). + + Examples + ======== + + >>> from itertools import permutations + >>> from sympy.utilities.iterables import generate_bell + >>> from sympy import zeros, Matrix + + This is the sort of permutation used in the ringing of physical bells, + and does not produce permutations in lexicographical order. Rather, the + permutations differ from each other by exactly one inversion, and the + position at which the swapping occurs varies periodically in a simple + fashion. Consider the first few permutations of 4 elements generated + by ``permutations`` and ``generate_bell``: + + >>> list(permutations(range(4)))[:5] + [(0, 1, 2, 3), (0, 1, 3, 2), (0, 2, 1, 3), (0, 2, 3, 1), (0, 3, 1, 2)] + >>> list(generate_bell(4))[:5] + [(0, 1, 2, 3), (0, 1, 3, 2), (0, 3, 1, 2), (3, 0, 1, 2), (3, 0, 2, 1)] + + Notice how the 2nd and 3rd lexicographical permutations have 3 elements + out of place whereas each "bell" permutation always has only two + elements out of place relative to the previous permutation (and so the + signature (+/-1) of a permutation is opposite of the signature of the + previous permutation). + + How the position of inversion varies across the elements can be seen + by tracing out where the largest number appears in the permutations: + + >>> m = zeros(4, 24) + >>> for i, p in enumerate(generate_bell(4)): + ... m[:, i] = Matrix([j - 3 for j in list(p)]) # make largest zero + >>> m.print_nonzero('X') + [XXX XXXXXX XXXXXX XXX] + [XX XX XXXX XX XXXX XX XX] + [X XXXX XX XXXX XX XXXX X] + [ XXXXXX XXXXXX XXXXXX ] + + See Also + ======== + + sympy.combinatorics.permutations.Permutation.next_trotterjohnson + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Method_ringing + + .. [2] https://stackoverflow.com/questions/4856615/recursive-permutation/4857018 + + .. [3] https://web.archive.org/web/20160313023044/http://programminggeeks.com/bell-algorithm-for-permutation/ + + .. [4] https://en.wikipedia.org/wiki/Steinhaus%E2%80%93Johnson%E2%80%93Trotter_algorithm + + .. [5] Generating involutions, derangements, and relatives by ECO + Vincent Vajnovszki, DMTCS vol 1 issue 12, 2010 + + """ + n = as_int(n) + if n < 1: + raise ValueError('n must be a positive integer') + if n == 1: + yield (0,) + elif n == 2: + yield (0, 1) + yield (1, 0) + elif n == 3: + yield from [(0, 1, 2), (0, 2, 1), (2, 0, 1), (2, 1, 0), (1, 2, 0), (1, 0, 2)] + else: + m = n - 1 + op = [0] + [-1]*m + l = list(range(n)) + while True: + yield tuple(l) + # find biggest element with op + big = None, -1 # idx, value + for i in range(n): + if op[i] and l[i] > big[1]: + big = i, l[i] + i, _ = big + if i is None: + break # there are no ops left + # swap it with neighbor in the indicated direction + j = i + op[i] + l[i], l[j] = l[j], l[i] + op[i], op[j] = op[j], op[i] + # if it landed at the end or if the neighbor in the same + # direction is bigger then turn off op + if j == 0 or j == m or l[j + op[j]] > l[j]: + op[j] = 0 + # any element bigger to the left gets +1 op + for i in range(j): + if l[i] > l[j]: + op[i] = 1 + # any element bigger to the right gets -1 op + for i in range(j + 1, n): + if l[i] > l[j]: + op[i] = -1 + + +def generate_involutions(n): + """ + Generates involutions. + + An involution is a permutation that when multiplied + by itself equals the identity permutation. In this + implementation the involutions are generated using + Fixed Points. + + Alternatively, an involution can be considered as + a permutation that does not contain any cycles with + a length that is greater than two. + + Examples + ======== + + >>> from sympy.utilities.iterables import generate_involutions + >>> list(generate_involutions(3)) + [(0, 1, 2), (0, 2, 1), (1, 0, 2), (2, 1, 0)] + >>> len(list(generate_involutions(4))) + 10 + + References + ========== + + .. [1] https://mathworld.wolfram.com/PermutationInvolution.html + + """ + idx = list(range(n)) + for p in permutations(idx): + for i in idx: + if p[p[i]] != i: + break + else: + yield p + + +def multiset_derangements(s): + """Generate derangements of the elements of s *in place*. + + Examples + ======== + + >>> from sympy.utilities.iterables import multiset_derangements, uniq + + Because the derangements of multisets (not sets) are generated + in place, copies of the return value must be made if a collection + of derangements is desired or else all values will be the same: + + >>> list(uniq([i for i in multiset_derangements('1233')])) + [[None, None, None, None]] + >>> [i.copy() for i in multiset_derangements('1233')] + [['3', '3', '1', '2'], ['3', '3', '2', '1']] + >>> [''.join(i) for i in multiset_derangements('1233')] + ['3312', '3321'] + """ + from sympy.core.sorting import ordered + # create multiset dictionary of hashable elements or else + # remap elements to integers + try: + ms = multiset(s) + except TypeError: + # give each element a canonical integer value + key = dict(enumerate(ordered(uniq(s)))) + h = [] + for si in s: + for k in key: + if key[k] == si: + h.append(k) + break + for i in multiset_derangements(h): + yield [key[j] for j in i] + return + + mx = max(ms.values()) # max repetition of any element + n = len(s) # the number of elements + + ## special cases + + # 1) one element has more than half the total cardinality of s: no + # derangements are possible. + if mx*2 > n: + return + + # 2) all elements appear once: singletons + if len(ms) == n: + yield from _set_derangements(s) + return + + # find the first element that is repeated the most to place + # in the following two special cases where the selection + # is unambiguous: either there are two elements with multiplicity + # of mx or else there is only one with multiplicity mx + for M in ms: + if ms[M] == mx: + break + + inonM = [i for i in range(n) if s[i] != M] # location of non-M + iM = [i for i in range(n) if s[i] == M] # locations of M + rv = [None]*n + + # 3) half are the same + if 2*mx == n: + # M goes into non-M locations + for i in inonM: + rv[i] = M + # permutations of non-M go to M locations + for p in multiset_permutations([s[i] for i in inonM]): + for i, pi in zip(iM, p): + rv[i] = pi + yield rv + # clean-up (and encourages proper use of routine) + rv[:] = [None]*n + return + + # 4) single repeat covers all but 1 of the non-repeats: + # if there is one repeat then the multiset of the values + # of ms would be {mx: 1, 1: n - mx}, i.e. there would + # be n - mx + 1 values with the condition that n - 2*mx = 1 + if n - 2*mx == 1 and len(ms.values()) == n - mx + 1: + for i, i1 in enumerate(inonM): + ifill = inonM[:i] + inonM[i+1:] + for j in ifill: + rv[j] = M + for p in permutations([s[j] for j in ifill]): + rv[i1] = s[i1] + for j, pi in zip(iM, p): + rv[j] = pi + k = i1 + for j in iM: + rv[j], rv[k] = rv[k], rv[j] + yield rv + k = j + # clean-up (and encourages proper use of routine) + rv[:] = [None]*n + return + + ## general case is handled with 3 helpers: + # 1) `finish_derangements` will place the last two elements + # which have arbitrary multiplicities, e.g. for multiset + # {c: 3, a: 2, b: 2}, the last two elements are a and b + # 2) `iopen` will tell where a given element can be placed + # 3) `do` will recursively place elements into subsets of + # valid locations + + def finish_derangements(): + """Place the last two elements into the partially completed + derangement, and yield the results. + """ + + a = take[1][0] # penultimate element + a_ct = take[1][1] + b = take[0][0] # last element to be placed + b_ct = take[0][1] + + # split the indexes of the not-already-assigned elements of rv into + # three categories + forced_a = [] # positions which must have an a + forced_b = [] # positions which must have a b + open_free = [] # positions which could take either + for i in range(len(s)): + if rv[i] is None: + if s[i] == a: + forced_b.append(i) + elif s[i] == b: + forced_a.append(i) + else: + open_free.append(i) + + if len(forced_a) > a_ct or len(forced_b) > b_ct: + # No derangement possible + return + + for i in forced_a: + rv[i] = a + for i in forced_b: + rv[i] = b + for a_place in combinations(open_free, a_ct - len(forced_a)): + for a_pos in a_place: + rv[a_pos] = a + for i in open_free: + if rv[i] is None: # anything not in the subset is set to b + rv[i] = b + yield rv + # Clean up/undo the final placements + for i in open_free: + rv[i] = None + + # additional cleanup - clear forced_a, forced_b + for i in forced_a: + rv[i] = None + for i in forced_b: + rv[i] = None + + def iopen(v): + # return indices at which element v can be placed in rv: + # locations which are not already occupied if that location + # does not already contain v in the same location of s + return [i for i in range(n) if rv[i] is None and s[i] != v] + + def do(j): + if j == 1: + # handle the last two elements (regardless of multiplicity) + # with a special method + yield from finish_derangements() + else: + # place the mx elements of M into a subset of places + # into which it can be replaced + M, mx = take[j] + for i in combinations(iopen(M), mx): + # place M + for ii in i: + rv[ii] = M + # recursively place the next element + yield from do(j - 1) + # mark positions where M was placed as once again + # open for placement of other elements + for ii in i: + rv[ii] = None + + # process elements in order of canonically decreasing multiplicity + take = sorted(ms.items(), key=lambda x:(x[1], x[0])) + yield from do(len(take) - 1) + rv[:] = [None]*n + + +def random_derangement(t, choice=None, strict=True): + """Return a list of elements in which none are in the same positions + as they were originally. If an element fills more than half of the positions + then an error will be raised since no derangement is possible. To obtain + a derangement of as many items as possible--with some of the most numerous + remaining in their original positions--pass `strict=False`. To produce a + pseudorandom derangment, pass a pseudorandom selector like `choice` (see + below). + + Examples + ======== + + >>> from sympy.utilities.iterables import random_derangement + >>> t = 'SymPy: a CAS in pure Python' + >>> d = random_derangement(t) + >>> all(i != j for i, j in zip(d, t)) + True + + A predictable result can be obtained by using a pseudorandom + generator for the choice: + + >>> from sympy.core.random import seed, choice as c + >>> seed(1) + >>> d = [''.join(random_derangement(t, c)) for i in range(5)] + >>> assert len(set(d)) != 1 # we got different values + + By reseeding, the same sequence can be obtained: + + >>> seed(1) + >>> d2 = [''.join(random_derangement(t, c)) for i in range(5)] + >>> assert d == d2 + """ + if choice is None: + import secrets + choice = secrets.choice + def shuffle(rv): + '''Knuth shuffle''' + for i in range(len(rv) - 1, 0, -1): + x = choice(rv[:i + 1]) + j = rv.index(x) + rv[i], rv[j] = rv[j], rv[i] + def pick(rv, n): + '''shuffle rv and return the first n values + ''' + shuffle(rv) + return rv[:n] + ms = multiset(t) + tot = len(t) + ms = sorted(ms.items(), key=lambda x: x[1]) + # if there are not enough spaces for the most + # plentiful element to move to then some of them + # will have to stay in place + M, mx = ms[-1] + n = len(t) + xs = 2*mx - tot + if xs > 0: + if strict: + raise ValueError('no derangement possible') + opts = [i for (i, c) in enumerate(t) if c == ms[-1][0]] + pick(opts, xs) + stay = sorted(opts[:xs]) + rv = list(t) + for i in reversed(stay): + rv.pop(i) + rv = random_derangement(rv, choice) + for i in stay: + rv.insert(i, ms[-1][0]) + return ''.join(rv) if type(t) is str else rv + # the normal derangement calculated from here + if n == len(ms): + # approx 1/3 will succeed + rv = list(t) + while True: + shuffle(rv) + if all(i != j for i,j in zip(rv, t)): + break + else: + # general case + rv = [None]*n + while True: + j = 0 + while j > -len(ms): # do most numerous first + j -= 1 + e, c = ms[j] + opts = [i for i in range(n) if rv[i] is None and t[i] != e] + if len(opts) < c: + for i in range(n): + rv[i] = None + break # try again + pick(opts, c) + for i in range(c): + rv[opts[i]] = e + else: + return rv + return rv + + +def _set_derangements(s): + """ + yield derangements of items in ``s`` which are assumed to contain + no repeated elements + """ + if len(s) < 2: + return + if len(s) == 2: + yield [s[1], s[0]] + return + if len(s) == 3: + yield [s[1], s[2], s[0]] + yield [s[2], s[0], s[1]] + return + for p in permutations(s): + if not any(i == j for i, j in zip(p, s)): + yield list(p) + + +def generate_derangements(s): + """ + Return unique derangements of the elements of iterable ``s``. + + Examples + ======== + + >>> from sympy.utilities.iterables import generate_derangements + >>> list(generate_derangements([0, 1, 2])) + [[1, 2, 0], [2, 0, 1]] + >>> list(generate_derangements([0, 1, 2, 2])) + [[2, 2, 0, 1], [2, 2, 1, 0]] + >>> list(generate_derangements([0, 1, 1])) + [] + + See Also + ======== + + sympy.functions.combinatorial.factorials.subfactorial + + """ + if not has_dups(s): + yield from _set_derangements(s) + else: + for p in multiset_derangements(s): + yield list(p) + + +def necklaces(n, k, free=False): + """ + A routine to generate necklaces that may (free=True) or may not + (free=False) be turned over to be viewed. The "necklaces" returned + are comprised of ``n`` integers (beads) with ``k`` different + values (colors). Only unique necklaces are returned. + + Examples + ======== + + >>> from sympy.utilities.iterables import necklaces, bracelets + >>> def show(s, i): + ... return ''.join(s[j] for j in i) + + The "unrestricted necklace" is sometimes also referred to as a + "bracelet" (an object that can be turned over, a sequence that can + be reversed) and the term "necklace" is used to imply a sequence + that cannot be reversed. So ACB == ABC for a bracelet (rotate and + reverse) while the two are different for a necklace since rotation + alone cannot make the two sequences the same. + + (mnemonic: Bracelets can be viewed Backwards, but Not Necklaces.) + + >>> B = [show('ABC', i) for i in bracelets(3, 3)] + >>> N = [show('ABC', i) for i in necklaces(3, 3)] + >>> set(N) - set(B) + {'ACB'} + + >>> list(necklaces(4, 2)) + [(0, 0, 0, 0), (0, 0, 0, 1), (0, 0, 1, 1), + (0, 1, 0, 1), (0, 1, 1, 1), (1, 1, 1, 1)] + + >>> [show('.o', i) for i in bracelets(4, 2)] + ['....', '...o', '..oo', '.o.o', '.ooo', 'oooo'] + + References + ========== + + .. [1] https://mathworld.wolfram.com/Necklace.html + + .. [2] Frank Ruskey, Carla Savage, and Terry Min Yih Wang, + Generating necklaces, Journal of Algorithms 13 (1992), 414-430; + https://doi.org/10.1016/0196-6774(92)90047-G + + """ + # The FKM algorithm + if k == 0 and n > 0: + return + a = [0]*n + yield tuple(a) + if n == 0: + return + while True: + i = n - 1 + while a[i] == k - 1: + i -= 1 + if i == -1: + return + a[i] += 1 + for j in range(n - i - 1): + a[j + i + 1] = a[j] + if n % (i + 1) == 0 and (not free or all(a <= a[j::-1] + a[-1:j:-1] for j in range(n - 1))): + # No need to test j = n - 1. + yield tuple(a) + + +def bracelets(n, k): + """Wrapper to necklaces to return a free (unrestricted) necklace.""" + return necklaces(n, k, free=True) + + +def generate_oriented_forest(n): + """ + This algorithm generates oriented forests. + + An oriented graph is a directed graph having no symmetric pair of directed + edges. A forest is an acyclic graph, i.e., it has no cycles. A forest can + also be described as a disjoint union of trees, which are graphs in which + any two vertices are connected by exactly one simple path. + + Examples + ======== + + >>> from sympy.utilities.iterables import generate_oriented_forest + >>> list(generate_oriented_forest(4)) + [[0, 1, 2, 3], [0, 1, 2, 2], [0, 1, 2, 1], [0, 1, 2, 0], \ + [0, 1, 1, 1], [0, 1, 1, 0], [0, 1, 0, 1], [0, 1, 0, 0], [0, 0, 0, 0]] + + References + ========== + + .. [1] T. Beyer and S.M. Hedetniemi: constant time generation of + rooted trees, SIAM J. Computing Vol. 9, No. 4, November 1980 + + .. [2] https://stackoverflow.com/questions/1633833/oriented-forest-taocp-algorithm-in-python + + """ + P = list(range(-1, n)) + while True: + yield P[1:] + if P[n] > 0: + P[n] = P[P[n]] + else: + for p in range(n - 1, 0, -1): + if P[p] != 0: + target = P[p] - 1 + for q in range(p - 1, 0, -1): + if P[q] == target: + break + offset = p - q + for i in range(p, n + 1): + P[i] = P[i - offset] + break + else: + break + + +def minlex(seq, directed=True, key=None): + r""" + Return the rotation of the sequence in which the lexically smallest + elements appear first, e.g. `cba \rightarrow acb`. + + The sequence returned is a tuple, unless the input sequence is a string + in which case a string is returned. + + If ``directed`` is False then the smaller of the sequence and the + reversed sequence is returned, e.g. `cba \rightarrow abc`. + + If ``key`` is not None then it is used to extract a comparison key from each element in iterable. + + Examples + ======== + + >>> from sympy.combinatorics.polyhedron import minlex + >>> minlex((1, 2, 0)) + (0, 1, 2) + >>> minlex((1, 0, 2)) + (0, 2, 1) + >>> minlex((1, 0, 2), directed=False) + (0, 1, 2) + + >>> minlex('11010011000', directed=True) + '00011010011' + >>> minlex('11010011000', directed=False) + '00011001011' + + >>> minlex(('bb', 'aaa', 'c', 'a')) + ('a', 'bb', 'aaa', 'c') + >>> minlex(('bb', 'aaa', 'c', 'a'), key=len) + ('c', 'a', 'bb', 'aaa') + + """ + from sympy.functions.elementary.miscellaneous import Id + if key is None: key = Id + best = rotate_left(seq, least_rotation(seq, key=key)) + if not directed: + rseq = seq[::-1] + rbest = rotate_left(rseq, least_rotation(rseq, key=key)) + best = min(best, rbest, key=key) + + # Convert to tuple, unless we started with a string. + return tuple(best) if not isinstance(seq, str) else best + + +def runs(seq, op=gt): + """Group the sequence into lists in which successive elements + all compare the same with the comparison operator, ``op``: + op(seq[i + 1], seq[i]) is True from all elements in a run. + + Examples + ======== + + >>> from sympy.utilities.iterables import runs + >>> from operator import ge + >>> runs([0, 1, 2, 2, 1, 4, 3, 2, 2]) + [[0, 1, 2], [2], [1, 4], [3], [2], [2]] + >>> runs([0, 1, 2, 2, 1, 4, 3, 2, 2], op=ge) + [[0, 1, 2, 2], [1, 4], [3], [2, 2]] + """ + cycles = [] + seq = iter(seq) + try: + run = [next(seq)] + except StopIteration: + return [] + while True: + try: + ei = next(seq) + except StopIteration: + break + if op(ei, run[-1]): + run.append(ei) + continue + else: + cycles.append(run) + run = [ei] + if run: + cycles.append(run) + return cycles + + +def sequence_partitions(l, n, /): + r"""Returns the partition of sequence $l$ into $n$ bins + + Explanation + =========== + + Given the sequence $l_1 \cdots l_m \in V^+$ where + $V^+$ is the Kleene plus of $V$ + + The set of $n$ partitions of $l$ is defined as: + + .. math:: + \{(s_1, \cdots, s_n) | s_1 \in V^+, \cdots, s_n \in V^+, + s_1 \cdots s_n = l_1 \cdots l_m\} + + Parameters + ========== + + l : Sequence[T] + A nonempty sequence of any Python objects + + n : int + A positive integer + + Yields + ====== + + out : list[Sequence[T]] + A list of sequences with concatenation equals $l$. + This should conform with the type of $l$. + + Examples + ======== + + >>> from sympy.utilities.iterables import sequence_partitions + >>> for out in sequence_partitions([1, 2, 3, 4], 2): + ... print(out) + [[1], [2, 3, 4]] + [[1, 2], [3, 4]] + [[1, 2, 3], [4]] + + Notes + ===== + + This is modified version of EnricoGiampieri's partition generator + from https://stackoverflow.com/questions/13131491/partition-n-items-into-k-bins-in-python-lazily + + See Also + ======== + + sequence_partitions_empty + """ + # Asserting l is nonempty is done only for sanity check + if n == 1 and l: + yield [l] + return + for i in range(1, len(l)): + for part in sequence_partitions(l[i:], n - 1): + yield [l[:i]] + part + + +def sequence_partitions_empty(l, n, /): + r"""Returns the partition of sequence $l$ into $n$ bins with + empty sequence + + Explanation + =========== + + Given the sequence $l_1 \cdots l_m \in V^*$ where + $V^*$ is the Kleene star of $V$ + + The set of $n$ partitions of $l$ is defined as: + + .. math:: + \{(s_1, \cdots, s_n) | s_1 \in V^*, \cdots, s_n \in V^*, + s_1 \cdots s_n = l_1 \cdots l_m\} + + There are more combinations than :func:`sequence_partitions` because + empty sequence can fill everywhere, so we try to provide different + utility for this. + + Parameters + ========== + + l : Sequence[T] + A sequence of any Python objects (can be possibly empty) + + n : int + A positive integer + + Yields + ====== + + out : list[Sequence[T]] + A list of sequences with concatenation equals $l$. + This should conform with the type of $l$. + + Examples + ======== + + >>> from sympy.utilities.iterables import sequence_partitions_empty + >>> for out in sequence_partitions_empty([1, 2, 3, 4], 2): + ... print(out) + [[], [1, 2, 3, 4]] + [[1], [2, 3, 4]] + [[1, 2], [3, 4]] + [[1, 2, 3], [4]] + [[1, 2, 3, 4], []] + + See Also + ======== + + sequence_partitions + """ + if n < 1: + return + if n == 1: + yield [l] + return + for i in range(0, len(l) + 1): + for part in sequence_partitions_empty(l[i:], n - 1): + yield [l[:i]] + part + + +def kbins(l, k, ordered=None): + """ + Return sequence ``l`` partitioned into ``k`` bins. + + Examples + ======== + + The default is to give the items in the same order, but grouped + into k partitions without any reordering: + + >>> from sympy.utilities.iterables import kbins + >>> for p in kbins(list(range(5)), 2): + ... print(p) + ... + [[0], [1, 2, 3, 4]] + [[0, 1], [2, 3, 4]] + [[0, 1, 2], [3, 4]] + [[0, 1, 2, 3], [4]] + + The ``ordered`` flag is either None (to give the simple partition + of the elements) or is a 2 digit integer indicating whether the order of + the bins and the order of the items in the bins matters. Given:: + + A = [[0], [1, 2]] + B = [[1, 2], [0]] + C = [[2, 1], [0]] + D = [[0], [2, 1]] + + the following values for ``ordered`` have the shown meanings:: + + 00 means A == B == C == D + 01 means A == B + 10 means A == D + 11 means A == A + + >>> for ordered_flag in [None, 0, 1, 10, 11]: + ... print('ordered = %s' % ordered_flag) + ... for p in kbins(list(range(3)), 2, ordered=ordered_flag): + ... print(' %s' % p) + ... + ordered = None + [[0], [1, 2]] + [[0, 1], [2]] + ordered = 0 + [[0, 1], [2]] + [[0, 2], [1]] + [[0], [1, 2]] + ordered = 1 + [[0], [1, 2]] + [[0], [2, 1]] + [[1], [0, 2]] + [[1], [2, 0]] + [[2], [0, 1]] + [[2], [1, 0]] + ordered = 10 + [[0, 1], [2]] + [[2], [0, 1]] + [[0, 2], [1]] + [[1], [0, 2]] + [[0], [1, 2]] + [[1, 2], [0]] + ordered = 11 + [[0], [1, 2]] + [[0, 1], [2]] + [[0], [2, 1]] + [[0, 2], [1]] + [[1], [0, 2]] + [[1, 0], [2]] + [[1], [2, 0]] + [[1, 2], [0]] + [[2], [0, 1]] + [[2, 0], [1]] + [[2], [1, 0]] + [[2, 1], [0]] + + See Also + ======== + + partitions, multiset_partitions + + """ + if ordered is None: + yield from sequence_partitions(l, k) + elif ordered == 11: + for pl in multiset_permutations(l): + pl = list(pl) + yield from sequence_partitions(pl, k) + elif ordered == 00: + yield from multiset_partitions(l, k) + elif ordered == 10: + for p in multiset_partitions(l, k): + for perm in permutations(p): + yield list(perm) + elif ordered == 1: + for kgot, p in partitions(len(l), k, size=True): + if kgot != k: + continue + for li in multiset_permutations(l): + rv = [] + i = j = 0 + li = list(li) + for size, multiplicity in sorted(p.items()): + for m in range(multiplicity): + j = i + size + rv.append(li[i: j]) + i = j + yield rv + else: + raise ValueError( + 'ordered must be one of 00, 01, 10 or 11, not %s' % ordered) + + +def permute_signs(t): + """Return iterator in which the signs of non-zero elements + of t are permuted. + + Examples + ======== + + >>> from sympy.utilities.iterables import permute_signs + >>> list(permute_signs((0, 1, 2))) + [(0, 1, 2), (0, -1, 2), (0, 1, -2), (0, -1, -2)] + """ + for signs in product(*[(1, -1)]*(len(t) - t.count(0))): + signs = list(signs) + yield type(t)([i*signs.pop() if i else i for i in t]) + + +def signed_permutations(t): + """Return iterator in which the signs of non-zero elements + of t and the order of the elements are permuted and all + returned values are unique. + + Examples + ======== + + >>> from sympy.utilities.iterables import signed_permutations + >>> list(signed_permutations((0, 1, 2))) + [(0, 1, 2), (0, -1, 2), (0, 1, -2), (0, -1, -2), (0, 2, 1), + (0, -2, 1), (0, 2, -1), (0, -2, -1), (1, 0, 2), (-1, 0, 2), + (1, 0, -2), (-1, 0, -2), (1, 2, 0), (-1, 2, 0), (1, -2, 0), + (-1, -2, 0), (2, 0, 1), (-2, 0, 1), (2, 0, -1), (-2, 0, -1), + (2, 1, 0), (-2, 1, 0), (2, -1, 0), (-2, -1, 0)] + """ + return (type(t)(i) for j in multiset_permutations(t) + for i in permute_signs(j)) + + +def rotations(s, dir=1): + """Return a generator giving the items in s as list where + each subsequent list has the items rotated to the left (default) + or right (``dir=-1``) relative to the previous list. + + Examples + ======== + + >>> from sympy import rotations + >>> list(rotations([1,2,3])) + [[1, 2, 3], [2, 3, 1], [3, 1, 2]] + >>> list(rotations([1,2,3], -1)) + [[1, 2, 3], [3, 1, 2], [2, 3, 1]] + """ + seq = list(s) + for i in range(len(seq)): + yield seq + seq = rotate_left(seq, dir) + + +def roundrobin(*iterables): + """roundrobin recipe taken from itertools documentation: + https://docs.python.org/3/library/itertools.html#itertools-recipes + + roundrobin('ABC', 'D', 'EF') --> A D E B F C + + Recipe credited to George Sakkis + """ + nexts = cycle(iter(it).__next__ for it in iterables) + + pending = len(iterables) + while pending: + try: + for nxt in nexts: + yield nxt() + except StopIteration: + pending -= 1 + nexts = cycle(islice(nexts, pending)) + + + +class NotIterable: + """ + Use this as mixin when creating a class which is not supposed to + return true when iterable() is called on its instances because + calling list() on the instance, for example, would result in + an infinite loop. + """ + pass + + +def iterable(i, exclude=(str, dict, NotIterable)): + """ + Return a boolean indicating whether ``i`` is SymPy iterable. + True also indicates that the iterator is finite, e.g. you can + call list(...) on the instance. + + When SymPy is working with iterables, it is almost always assuming + that the iterable is not a string or a mapping, so those are excluded + by default. If you want a pure Python definition, make exclude=None. To + exclude multiple items, pass them as a tuple. + + You can also set the _iterable attribute to True or False on your class, + which will override the checks here, including the exclude test. + + As a rule of thumb, some SymPy functions use this to check if they should + recursively map over an object. If an object is technically iterable in + the Python sense but does not desire this behavior (e.g., because its + iteration is not finite, or because iteration might induce an unwanted + computation), it should disable it by setting the _iterable attribute to False. + + See also: is_sequence + + Examples + ======== + + >>> from sympy.utilities.iterables import iterable + >>> from sympy import Tuple + >>> things = [[1], (1,), set([1]), Tuple(1), (j for j in [1, 2]), {1:2}, '1', 1] + >>> for i in things: + ... print('%s %s' % (iterable(i), type(i))) + True <... 'list'> + True <... 'tuple'> + True <... 'set'> + True + True <... 'generator'> + False <... 'dict'> + False <... 'str'> + False <... 'int'> + + >>> iterable({}, exclude=None) + True + >>> iterable({}, exclude=str) + True + >>> iterable("no", exclude=str) + False + + """ + if hasattr(i, '_iterable'): + return i._iterable + try: + iter(i) + except TypeError: + return False + if exclude: + return not isinstance(i, exclude) + return True + + +def is_sequence(i, include=None): + """ + Return a boolean indicating whether ``i`` is a sequence in the SymPy + sense. If anything that fails the test below should be included as + being a sequence for your application, set 'include' to that object's + type; multiple types should be passed as a tuple of types. + + Note: although generators can generate a sequence, they often need special + handling to make sure their elements are captured before the generator is + exhausted, so these are not included by default in the definition of a + sequence. + + See also: iterable + + Examples + ======== + + >>> from sympy.utilities.iterables import is_sequence + >>> from types import GeneratorType + >>> is_sequence([]) + True + >>> is_sequence(set()) + False + >>> is_sequence('abc') + False + >>> is_sequence('abc', include=str) + True + >>> generator = (c for c in 'abc') + >>> is_sequence(generator) + False + >>> is_sequence(generator, include=(str, GeneratorType)) + True + + """ + return (hasattr(i, '__getitem__') and + iterable(i) or + bool(include) and + isinstance(i, include)) + + +@deprecated( + """ + Using postorder_traversal from the sympy.utilities.iterables submodule is + deprecated. + + Instead, use postorder_traversal from the top-level sympy namespace, like + + sympy.postorder_traversal + """, + deprecated_since_version="1.10", + active_deprecations_target="deprecated-traversal-functions-moved") +def postorder_traversal(node, keys=None): + from sympy.core.traversal import postorder_traversal as _postorder_traversal + return _postorder_traversal(node, keys=keys) + + +@deprecated( + """ + Using interactive_traversal from the sympy.utilities.iterables submodule + is deprecated. + + Instead, use interactive_traversal from the top-level sympy namespace, + like + + sympy.interactive_traversal + """, + deprecated_since_version="1.10", + active_deprecations_target="deprecated-traversal-functions-moved") +def interactive_traversal(expr): + from sympy.interactive.traversal import interactive_traversal as _interactive_traversal + return _interactive_traversal(expr) + + +@deprecated( + """ + Importing default_sort_key from sympy.utilities.iterables is deprecated. + Use from sympy import default_sort_key instead. + """, + deprecated_since_version="1.10", +active_deprecations_target="deprecated-sympy-core-compatibility", +) +def default_sort_key(*args, **kwargs): + from sympy import default_sort_key as _default_sort_key + return _default_sort_key(*args, **kwargs) + + +@deprecated( + """ + Importing default_sort_key from sympy.utilities.iterables is deprecated. + Use from sympy import default_sort_key instead. + """, + deprecated_since_version="1.10", +active_deprecations_target="deprecated-sympy-core-compatibility", +) +def ordered(*args, **kwargs): + from sympy import ordered as _ordered + return _ordered(*args, **kwargs) diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/lambdify.py b/.venv/lib/python3.12/site-packages/sympy/utilities/lambdify.py new file mode 100644 index 0000000000000000000000000000000000000000..6807fdaec10963a60ba88d6e1ec6b2951c19241e --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/lambdify.py @@ -0,0 +1,1592 @@ +""" +This module provides convenient functions to transform SymPy expressions to +lambda functions which can be used to calculate numerical values very fast. +""" + +from __future__ import annotations +from typing import Any + +import builtins +import inspect +import keyword +import textwrap +import linecache +import weakref + +# Required despite static analysis claiming it is not used +from sympy.external import import_module # noqa:F401 +from sympy.utilities.exceptions import sympy_deprecation_warning +from sympy.utilities.decorator import doctest_depends_on +from sympy.utilities.iterables import (is_sequence, iterable, + NotIterable, flatten) +from sympy.utilities.misc import filldedent + + +__doctest_requires__ = {('lambdify',): ['numpy', 'tensorflow']} + + +# Default namespaces, letting us define translations that can't be defined +# by simple variable maps, like I => 1j +MATH_DEFAULT: dict[str, Any] = {} +CMATH_DEFAULT: dict[str,Any] = {} +MPMATH_DEFAULT: dict[str, Any] = {} +NUMPY_DEFAULT: dict[str, Any] = {"I": 1j} +SCIPY_DEFAULT: dict[str, Any] = {"I": 1j} +CUPY_DEFAULT: dict[str, Any] = {"I": 1j} +JAX_DEFAULT: dict[str, Any] = {"I": 1j} +TENSORFLOW_DEFAULT: dict[str, Any] = {} +TORCH_DEFAULT: dict[str, Any] = {"I": 1j} +SYMPY_DEFAULT: dict[str, Any] = {} +NUMEXPR_DEFAULT: dict[str, Any] = {} + +# These are the namespaces the lambda functions will use. +# These are separate from the names above because they are modified +# throughout this file, whereas the defaults should remain unmodified. + +MATH = MATH_DEFAULT.copy() +CMATH = CMATH_DEFAULT.copy() +MPMATH = MPMATH_DEFAULT.copy() +NUMPY = NUMPY_DEFAULT.copy() +SCIPY = SCIPY_DEFAULT.copy() +CUPY = CUPY_DEFAULT.copy() +JAX = JAX_DEFAULT.copy() +TENSORFLOW = TENSORFLOW_DEFAULT.copy() +TORCH = TORCH_DEFAULT.copy() +SYMPY = SYMPY_DEFAULT.copy() +NUMEXPR = NUMEXPR_DEFAULT.copy() + + +# Mappings between SymPy and other modules function names. +MATH_TRANSLATIONS = { + "ceiling": "ceil", + "E": "e", + "ln": "log", +} + +CMATH_TRANSLATIONS: dict[str, str] = {} + +# NOTE: This dictionary is reused in Function._eval_evalf to allow subclasses +# of Function to automatically evalf. +MPMATH_TRANSLATIONS = { + "Abs": "fabs", + "elliptic_k": "ellipk", + "elliptic_f": "ellipf", + "elliptic_e": "ellipe", + "elliptic_pi": "ellippi", + "ceiling": "ceil", + "chebyshevt": "chebyt", + "chebyshevu": "chebyu", + "assoc_legendre": "legenp", + "E": "e", + "I": "j", + "ln": "log", + #"lowergamma":"lower_gamma", + "oo": "inf", + #"uppergamma":"upper_gamma", + "LambertW": "lambertw", + "MutableDenseMatrix": "matrix", + "ImmutableDenseMatrix": "matrix", + "conjugate": "conj", + "dirichlet_eta": "altzeta", + "Ei": "ei", + "Shi": "shi", + "Chi": "chi", + "Si": "si", + "Ci": "ci", + "RisingFactorial": "rf", + "FallingFactorial": "ff", + "betainc_regularized": "betainc", +} + +NUMPY_TRANSLATIONS: dict[str, str] = { + "Heaviside": "heaviside", +} +SCIPY_TRANSLATIONS: dict[str, str] = { + "jn" : "spherical_jn", + "yn" : "spherical_yn" +} +CUPY_TRANSLATIONS: dict[str, str] = {} +JAX_TRANSLATIONS: dict[str, str] = {} + +TENSORFLOW_TRANSLATIONS: dict[str, str] = {} +TORCH_TRANSLATIONS: dict[str, str] = {} + +NUMEXPR_TRANSLATIONS: dict[str, str] = {} + +# Available modules: +MODULES = { + "math": (MATH, MATH_DEFAULT, MATH_TRANSLATIONS, ("from math import *",)), + "cmath": (CMATH, CMATH_DEFAULT, CMATH_TRANSLATIONS, ("import cmath; from cmath import *",)), + "mpmath": (MPMATH, MPMATH_DEFAULT, MPMATH_TRANSLATIONS, ("from mpmath import *",)), + "numpy": (NUMPY, NUMPY_DEFAULT, NUMPY_TRANSLATIONS, ("import numpy; from numpy import *; from numpy.linalg import *",)), + "scipy": (SCIPY, SCIPY_DEFAULT, SCIPY_TRANSLATIONS, ("import scipy; import numpy; from scipy.special import *",)), + "cupy": (CUPY, CUPY_DEFAULT, CUPY_TRANSLATIONS, ("import cupy",)), + "jax": (JAX, JAX_DEFAULT, JAX_TRANSLATIONS, ("import jax",)), + "tensorflow": (TENSORFLOW, TENSORFLOW_DEFAULT, TENSORFLOW_TRANSLATIONS, ("import tensorflow",)), + "torch": (TORCH, TORCH_DEFAULT, TORCH_TRANSLATIONS, ("import torch",)), + "sympy": (SYMPY, SYMPY_DEFAULT, {}, ( + "from sympy.functions import *", + "from sympy.matrices import *", + "from sympy import Integral, pi, oo, nan, zoo, E, I",)), + "numexpr" : (NUMEXPR, NUMEXPR_DEFAULT, NUMEXPR_TRANSLATIONS, + ("import_module('numexpr')", )), +} + + +def _import(module, reload=False): + """ + Creates a global translation dictionary for module. + + The argument module has to be one of the following strings: "math","cmath" + "mpmath", "numpy", "sympy", "tensorflow", "jax". + These dictionaries map names of Python functions to their equivalent in + other modules. + """ + try: + namespace, namespace_default, translations, import_commands = MODULES[ + module] + except KeyError: + raise NameError( + "'%s' module cannot be used for lambdification" % module) + + # Clear namespace or exit + if namespace != namespace_default: + # The namespace was already generated, don't do it again if not forced. + if reload: + namespace.clear() + namespace.update(namespace_default) + else: + return + + for import_command in import_commands: + if import_command.startswith('import_module'): + module = eval(import_command) + + if module is not None: + namespace.update(module.__dict__) + continue + else: + try: + exec(import_command, {}, namespace) + continue + except ImportError: + pass + + raise ImportError( + "Cannot import '%s' with '%s' command" % (module, import_command)) + + # Add translated names to namespace + for sympyname, translation in translations.items(): + namespace[sympyname] = namespace[translation] + + # For computing the modulus of a SymPy expression we use the builtin abs + # function, instead of the previously used fabs function for all + # translation modules. This is because the fabs function in the math + # module does not accept complex valued arguments. (see issue 9474). The + # only exception, where we don't use the builtin abs function is the + # mpmath translation module, because mpmath.fabs returns mpf objects in + # contrast to abs(). + if 'Abs' not in namespace: + namespace['Abs'] = abs + +# Used for dynamically generated filenames that are inserted into the +# linecache. +_lambdify_generated_counter = 1 + + +@doctest_depends_on(modules=('numpy', 'scipy', 'tensorflow',), python_version=(3,)) +def lambdify(args, expr, modules=None, printer=None, use_imps=True, + dummify=False, cse=False, docstring_limit=1000): + """Convert a SymPy expression into a function that allows for fast + numeric evaluation. + + .. warning:: + This function uses ``exec``, and thus should not be used on + unsanitized input. + + .. deprecated:: 1.7 + Passing a set for the *args* parameter is deprecated as sets are + unordered. Use an ordered iterable such as a list or tuple. + + Explanation + =========== + + For example, to convert the SymPy expression ``sin(x) + cos(x)`` to an + equivalent NumPy function that numerically evaluates it: + + >>> from sympy import sin, cos, symbols, lambdify + >>> import numpy as np + >>> x = symbols('x') + >>> expr = sin(x) + cos(x) + >>> expr + sin(x) + cos(x) + >>> f = lambdify(x, expr, 'numpy') + >>> a = np.array([1, 2]) + >>> f(a) + [1.38177329 0.49315059] + + The primary purpose of this function is to provide a bridge from SymPy + expressions to numerical libraries such as NumPy, SciPy, NumExpr, mpmath, + and tensorflow. In general, SymPy functions do not work with objects from + other libraries, such as NumPy arrays, and functions from numeric + libraries like NumPy or mpmath do not work on SymPy expressions. + ``lambdify`` bridges the two by converting a SymPy expression to an + equivalent numeric function. + + The basic workflow with ``lambdify`` is to first create a SymPy expression + representing whatever mathematical function you wish to evaluate. This + should be done using only SymPy functions and expressions. Then, use + ``lambdify`` to convert this to an equivalent function for numerical + evaluation. For instance, above we created ``expr`` using the SymPy symbol + ``x`` and SymPy functions ``sin`` and ``cos``, then converted it to an + equivalent NumPy function ``f``, and called it on a NumPy array ``a``. + + Parameters + ========== + + args : List[Symbol] + A variable or a list of variables whose nesting represents the + nesting of the arguments that will be passed to the function. + + Variables can be symbols, undefined functions, or matrix symbols. + + >>> from sympy import Eq + >>> from sympy.abc import x, y, z + + The list of variables should match the structure of how the + arguments will be passed to the function. Simply enclose the + parameters as they will be passed in a list. + + To call a function like ``f(x)`` then ``[x]`` + should be the first argument to ``lambdify``; for this + case a single ``x`` can also be used: + + >>> f = lambdify(x, x + 1) + >>> f(1) + 2 + >>> f = lambdify([x], x + 1) + >>> f(1) + 2 + + To call a function like ``f(x, y)`` then ``[x, y]`` will + be the first argument of the ``lambdify``: + + >>> f = lambdify([x, y], x + y) + >>> f(1, 1) + 2 + + To call a function with a single 3-element tuple like + ``f((x, y, z))`` then ``[(x, y, z)]`` will be the first + argument of the ``lambdify``: + + >>> f = lambdify([(x, y, z)], Eq(z**2, x**2 + y**2)) + >>> f((3, 4, 5)) + True + + If two args will be passed and the first is a scalar but + the second is a tuple with two arguments then the items + in the list should match that structure: + + >>> f = lambdify([x, (y, z)], x + y + z) + >>> f(1, (2, 3)) + 6 + + expr : Expr + An expression, list of expressions, or matrix to be evaluated. + + Lists may be nested. + If the expression is a list, the output will also be a list. + + >>> f = lambdify(x, [x, [x + 1, x + 2]]) + >>> f(1) + [1, [2, 3]] + + If it is a matrix, an array will be returned (for the NumPy module). + + >>> from sympy import Matrix + >>> f = lambdify(x, Matrix([x, x + 1])) + >>> f(1) + [[1] + [2]] + + Note that the argument order here (variables then expression) is used + to emulate the Python ``lambda`` keyword. ``lambdify(x, expr)`` works + (roughly) like ``lambda x: expr`` + (see :ref:`lambdify-how-it-works` below). + + modules : str, optional + Specifies the numeric library to use. + + If not specified, *modules* defaults to: + + - ``["scipy", "numpy"]`` if SciPy is installed + - ``["numpy"]`` if only NumPy is installed + - ``["math","cmath", "mpmath", "sympy"]`` if neither is installed. + + That is, SymPy functions are replaced as far as possible by + either ``scipy`` or ``numpy`` functions if available, and Python's + standard library ``math`` and ``cmath``, or ``mpmath`` functions otherwise. + + *modules* can be one of the following types: + + - The strings ``"math"``, ``"cmath"``, ``"mpmath"``, ``"numpy"``, ``"numexpr"``, + ``"scipy"``, ``"sympy"``, or ``"tensorflow"`` or ``"jax"``. This uses the + corresponding printer and namespace mapping for that module. + - A module (e.g., ``math``). This uses the global namespace of the + module. If the module is one of the above known modules, it will + also use the corresponding printer and namespace mapping + (i.e., ``modules=numpy`` is equivalent to ``modules="numpy"``). + - A dictionary that maps names of SymPy functions to arbitrary + functions + (e.g., ``{'sin': custom_sin}``). + - A list that contains a mix of the arguments above, with higher + priority given to entries appearing first + (e.g., to use the NumPy module but override the ``sin`` function + with a custom version, you can use + ``[{'sin': custom_sin}, 'numpy']``). + + dummify : bool, optional + Whether or not the variables in the provided expression that are not + valid Python identifiers are substituted with dummy symbols. + + This allows for undefined functions like ``Function('f')(t)`` to be + supplied as arguments. By default, the variables are only dummified + if they are not valid Python identifiers. + + Set ``dummify=True`` to replace all arguments with dummy symbols + (if ``args`` is not a string) - for example, to ensure that the + arguments do not redefine any built-in names. + + cse : bool, or callable, optional + Large expressions can be computed more efficiently when + common subexpressions are identified and precomputed before + being used multiple time. Finding the subexpressions will make + creation of the 'lambdify' function slower, however. + + When ``True``, ``sympy.simplify.cse`` is used, otherwise (the default) + the user may pass a function matching the ``cse`` signature. + + docstring_limit : int or None + When lambdifying large expressions, a significant proportion of the time + spent inside ``lambdify`` is spent producing a string representation of + the expression for use in the automatically generated docstring of the + returned function. For expressions containing hundreds or more nodes the + resulting docstring often becomes so long and dense that it is difficult + to read. To reduce the runtime of lambdify, the rendering of the full + expression inside the docstring can be disabled. + + When ``None``, the full expression is rendered in the docstring. When + ``0`` or a negative ``int``, an ellipsis is rendering in the docstring + instead of the expression. When a strictly positive ``int``, if the + number of nodes in the expression exceeds ``docstring_limit`` an + ellipsis is rendered in the docstring, otherwise a string representation + of the expression is rendered as normal. The default is ``1000``. + + Examples + ======== + + >>> from sympy.utilities.lambdify import implemented_function + >>> from sympy import sqrt, sin, Matrix + >>> from sympy import Function + >>> from sympy.abc import w, x, y, z + + >>> f = lambdify(x, x**2) + >>> f(2) + 4 + >>> f = lambdify((x, y, z), [z, y, x]) + >>> f(1,2,3) + [3, 2, 1] + >>> f = lambdify(x, sqrt(x)) + >>> f(4) + 2.0 + >>> f = lambdify((x, y), sin(x*y)**2) + >>> f(0, 5) + 0.0 + >>> row = lambdify((x, y), Matrix((x, x + y)).T, modules='sympy') + >>> row(1, 2) + Matrix([[1, 3]]) + + ``lambdify`` can be used to translate SymPy expressions into mpmath + functions. This may be preferable to using ``evalf`` (which uses mpmath on + the backend) in some cases. + + >>> f = lambdify(x, sin(x), 'mpmath') + >>> f(1) + 0.8414709848078965 + + Tuple arguments are handled and the lambdified function should + be called with the same type of arguments as were used to create + the function: + + >>> f = lambdify((x, (y, z)), x + y) + >>> f(1, (2, 4)) + 3 + + The ``flatten`` function can be used to always work with flattened + arguments: + + >>> from sympy.utilities.iterables import flatten + >>> args = w, (x, (y, z)) + >>> vals = 1, (2, (3, 4)) + >>> f = lambdify(flatten(args), w + x + y + z) + >>> f(*flatten(vals)) + 10 + + Functions present in ``expr`` can also carry their own numerical + implementations, in a callable attached to the ``_imp_`` attribute. This + can be used with undefined functions using the ``implemented_function`` + factory: + + >>> f = implemented_function(Function('f'), lambda x: x+1) + >>> func = lambdify(x, f(x)) + >>> func(4) + 5 + + ``lambdify`` always prefers ``_imp_`` implementations to implementations + in other namespaces, unless the ``use_imps`` input parameter is False. + + Usage with Tensorflow: + + >>> import tensorflow as tf + >>> from sympy import Max, sin, lambdify + >>> from sympy.abc import x + + >>> f = Max(x, sin(x)) + >>> func = lambdify(x, f, 'tensorflow') + + After tensorflow v2, eager execution is enabled by default. + If you want to get the compatible result across tensorflow v1 and v2 + as same as this tutorial, run this line. + + >>> tf.compat.v1.enable_eager_execution() + + If you have eager execution enabled, you can get the result out + immediately as you can use numpy. + + If you pass tensorflow objects, you may get an ``EagerTensor`` + object instead of value. + + >>> result = func(tf.constant(1.0)) + >>> print(result) + tf.Tensor(1.0, shape=(), dtype=float32) + >>> print(result.__class__) + + + You can use ``.numpy()`` to get the numpy value of the tensor. + + >>> result.numpy() + 1.0 + + >>> var = tf.Variable(2.0) + >>> result = func(var) # also works for tf.Variable and tf.Placeholder + >>> result.numpy() + 2.0 + + And it works with any shape array. + + >>> tensor = tf.constant([[1.0, 2.0], [3.0, 4.0]]) + >>> result = func(tensor) + >>> result.numpy() + [[1. 2.] + [3. 4.]] + + Notes + ===== + + - For functions involving large array calculations, numexpr can provide a + significant speedup over numpy. Please note that the available functions + for numexpr are more limited than numpy but can be expanded with + ``implemented_function`` and user defined subclasses of Function. If + specified, numexpr may be the only option in modules. The official list + of numexpr functions can be found at: + https://numexpr.readthedocs.io/en/latest/user_guide.html#supported-functions + + - In the above examples, the generated functions can accept scalar + values or numpy arrays as arguments. However, in some cases + the generated function relies on the input being a numpy array: + + >>> import numpy + >>> from sympy import Piecewise + >>> from sympy.testing.pytest import ignore_warnings + >>> f = lambdify(x, Piecewise((x, x <= 1), (1/x, x > 1)), "numpy") + + >>> with ignore_warnings(RuntimeWarning): + ... f(numpy.array([-1, 0, 1, 2])) + [-1. 0. 1. 0.5] + + >>> f(0) + Traceback (most recent call last): + ... + ZeroDivisionError: division by zero + + In such cases, the input should be wrapped in a numpy array: + + >>> with ignore_warnings(RuntimeWarning): + ... float(f(numpy.array([0]))) + 0.0 + + Or if numpy functionality is not required another module can be used: + + >>> f = lambdify(x, Piecewise((x, x <= 1), (1/x, x > 1)), "math") + >>> f(0) + 0 + + .. _lambdify-how-it-works: + + How it works + ============ + + When using this function, it helps a great deal to have an idea of what it + is doing. At its core, lambdify is nothing more than a namespace + translation, on top of a special printer that makes some corner cases work + properly. + + To understand lambdify, first we must properly understand how Python + namespaces work. Say we had two files. One called ``sin_cos_sympy.py``, + with + + .. code:: python + + # sin_cos_sympy.py + + from sympy.functions.elementary.trigonometric import (cos, sin) + + def sin_cos(x): + return sin(x) + cos(x) + + + and one called ``sin_cos_numpy.py`` with + + .. code:: python + + # sin_cos_numpy.py + + from numpy import sin, cos + + def sin_cos(x): + return sin(x) + cos(x) + + The two files define an identical function ``sin_cos``. However, in the + first file, ``sin`` and ``cos`` are defined as the SymPy ``sin`` and + ``cos``. In the second, they are defined as the NumPy versions. + + If we were to import the first file and use the ``sin_cos`` function, we + would get something like + + >>> from sin_cos_sympy import sin_cos # doctest: +SKIP + >>> sin_cos(1) # doctest: +SKIP + cos(1) + sin(1) + + On the other hand, if we imported ``sin_cos`` from the second file, we + would get + + >>> from sin_cos_numpy import sin_cos # doctest: +SKIP + >>> sin_cos(1) # doctest: +SKIP + 1.38177329068 + + In the first case we got a symbolic output, because it used the symbolic + ``sin`` and ``cos`` functions from SymPy. In the second, we got a numeric + result, because ``sin_cos`` used the numeric ``sin`` and ``cos`` functions + from NumPy. But notice that the versions of ``sin`` and ``cos`` that were + used was not inherent to the ``sin_cos`` function definition. Both + ``sin_cos`` definitions are exactly the same. Rather, it was based on the + names defined at the module where the ``sin_cos`` function was defined. + + The key point here is that when function in Python references a name that + is not defined in the function, that name is looked up in the "global" + namespace of the module where that function is defined. + + Now, in Python, we can emulate this behavior without actually writing a + file to disk using the ``exec`` function. ``exec`` takes a string + containing a block of Python code, and a dictionary that should contain + the global variables of the module. It then executes the code "in" that + dictionary, as if it were the module globals. The following is equivalent + to the ``sin_cos`` defined in ``sin_cos_sympy.py``: + + >>> import sympy + >>> module_dictionary = {'sin': sympy.sin, 'cos': sympy.cos} + >>> exec(''' + ... def sin_cos(x): + ... return sin(x) + cos(x) + ... ''', module_dictionary) + >>> sin_cos = module_dictionary['sin_cos'] + >>> sin_cos(1) + cos(1) + sin(1) + + and similarly with ``sin_cos_numpy``: + + >>> import numpy + >>> module_dictionary = {'sin': numpy.sin, 'cos': numpy.cos} + >>> exec(''' + ... def sin_cos(x): + ... return sin(x) + cos(x) + ... ''', module_dictionary) + >>> sin_cos = module_dictionary['sin_cos'] + >>> sin_cos(1) + 1.38177329068 + + So now we can get an idea of how ``lambdify`` works. The name "lambdify" + comes from the fact that we can think of something like ``lambdify(x, + sin(x) + cos(x), 'numpy')`` as ``lambda x: sin(x) + cos(x)``, where + ``sin`` and ``cos`` come from the ``numpy`` namespace. This is also why + the symbols argument is first in ``lambdify``, as opposed to most SymPy + functions where it comes after the expression: to better mimic the + ``lambda`` keyword. + + ``lambdify`` takes the input expression (like ``sin(x) + cos(x)``) and + + 1. Converts it to a string + 2. Creates a module globals dictionary based on the modules that are + passed in (by default, it uses the NumPy module) + 3. Creates the string ``"def func({vars}): return {expr}"``, where ``{vars}`` is the + list of variables separated by commas, and ``{expr}`` is the string + created in step 1., then ``exec``s that string with the module globals + namespace and returns ``func``. + + In fact, functions returned by ``lambdify`` support inspection. So you can + see exactly how they are defined by using ``inspect.getsource``, or ``??`` if you + are using IPython or the Jupyter notebook. + + >>> f = lambdify(x, sin(x) + cos(x)) + >>> import inspect + >>> print(inspect.getsource(f)) + def _lambdifygenerated(x): + return sin(x) + cos(x) + + This shows us the source code of the function, but not the namespace it + was defined in. We can inspect that by looking at the ``__globals__`` + attribute of ``f``: + + >>> f.__globals__['sin'] + + >>> f.__globals__['cos'] + + >>> f.__globals__['sin'] is numpy.sin + True + + This shows us that ``sin`` and ``cos`` in the namespace of ``f`` will be + ``numpy.sin`` and ``numpy.cos``. + + Note that there are some convenience layers in each of these steps, but at + the core, this is how ``lambdify`` works. Step 1 is done using the + ``LambdaPrinter`` printers defined in the printing module (see + :mod:`sympy.printing.lambdarepr`). This allows different SymPy expressions + to define how they should be converted to a string for different modules. + You can change which printer ``lambdify`` uses by passing a custom printer + in to the ``printer`` argument. + + Step 2 is augmented by certain translations. There are default + translations for each module, but you can provide your own by passing a + list to the ``modules`` argument. For instance, + + >>> def mysin(x): + ... print('taking the sin of', x) + ... return numpy.sin(x) + ... + >>> f = lambdify(x, sin(x), [{'sin': mysin}, 'numpy']) + >>> f(1) + taking the sin of 1 + 0.8414709848078965 + + The globals dictionary is generated from the list by merging the + dictionary ``{'sin': mysin}`` and the module dictionary for NumPy. The + merging is done so that earlier items take precedence, which is why + ``mysin`` is used above instead of ``numpy.sin``. + + If you want to modify the way ``lambdify`` works for a given function, it + is usually easiest to do so by modifying the globals dictionary as such. + In more complicated cases, it may be necessary to create and pass in a + custom printer. + + Finally, step 3 is augmented with certain convenience operations, such as + the addition of a docstring. + + Understanding how ``lambdify`` works can make it easier to avoid certain + gotchas when using it. For instance, a common mistake is to create a + lambdified function for one module (say, NumPy), and pass it objects from + another (say, a SymPy expression). + + For instance, say we create + + >>> from sympy.abc import x + >>> f = lambdify(x, x + 1, 'numpy') + + Now if we pass in a NumPy array, we get that array plus 1 + + >>> import numpy + >>> a = numpy.array([1, 2]) + >>> f(a) + [2 3] + + But what happens if you make the mistake of passing in a SymPy expression + instead of a NumPy array: + + >>> f(x + 1) + x + 2 + + This worked, but it was only by accident. Now take a different lambdified + function: + + >>> from sympy import sin + >>> g = lambdify(x, x + sin(x), 'numpy') + + This works as expected on NumPy arrays: + + >>> g(a) + [1.84147098 2.90929743] + + But if we try to pass in a SymPy expression, it fails + + >>> g(x + 1) + Traceback (most recent call last): + ... + TypeError: loop of ufunc does not support argument 0 of type Add which has + no callable sin method + + Now, let's look at what happened. The reason this fails is that ``g`` + calls ``numpy.sin`` on the input expression, and ``numpy.sin`` does not + know how to operate on a SymPy object. **As a general rule, NumPy + functions do not know how to operate on SymPy expressions, and SymPy + functions do not know how to operate on NumPy arrays. This is why lambdify + exists: to provide a bridge between SymPy and NumPy.** + + However, why is it that ``f`` did work? That's because ``f`` does not call + any functions, it only adds 1. So the resulting function that is created, + ``def _lambdifygenerated(x): return x + 1`` does not depend on the globals + namespace it is defined in. Thus it works, but only by accident. A future + version of ``lambdify`` may remove this behavior. + + Be aware that certain implementation details described here may change in + future versions of SymPy. The API of passing in custom modules and + printers will not change, but the details of how a lambda function is + created may change. However, the basic idea will remain the same, and + understanding it will be helpful to understanding the behavior of + lambdify. + + **In general: you should create lambdified functions for one module (say, + NumPy), and only pass it input types that are compatible with that module + (say, NumPy arrays).** Remember that by default, if the ``module`` + argument is not provided, ``lambdify`` creates functions using the NumPy + and SciPy namespaces. + """ + from sympy.core.symbol import Symbol + from sympy.core.expr import Expr + + # If the user hasn't specified any modules, use what is available. + if modules is None: + try: + _import("scipy") + except ImportError: + try: + _import("numpy") + except ImportError: + # Use either numpy (if available) or python.math where possible. + # XXX: This leads to different behaviour on different systems and + # might be the reason for irreproducible errors. + modules = ["math", "mpmath", "sympy"] + else: + modules = ["numpy"] + else: + modules = ["numpy", "scipy"] + + # Get the needed namespaces. + namespaces = [] + # First find any function implementations + if use_imps: + namespaces.append(_imp_namespace(expr)) + # Check for dict before iterating + if isinstance(modules, (dict, str)) or not hasattr(modules, '__iter__'): + namespaces.append(modules) + else: + # consistency check + if _module_present('numexpr', modules) and len(modules) > 1: + raise TypeError("numexpr must be the only item in 'modules'") + namespaces += list(modules) + # fill namespace with first having highest priority + namespace = {} + for m in namespaces[::-1]: + buf = _get_namespace(m) + namespace.update(buf) + + if hasattr(expr, "atoms"): + #Try if you can extract symbols from the expression. + #Move on if expr.atoms in not implemented. + syms = expr.atoms(Symbol) + for term in syms: + namespace.update({str(term): term}) + + if printer is None: + if _module_present('mpmath', namespaces): + from sympy.printing.pycode import MpmathPrinter as Printer # type: ignore + elif _module_present('scipy', namespaces): + from sympy.printing.numpy import SciPyPrinter as Printer # type: ignore + elif _module_present('numpy', namespaces): + from sympy.printing.numpy import NumPyPrinter as Printer # type: ignore + elif _module_present('cupy', namespaces): + from sympy.printing.numpy import CuPyPrinter as Printer # type: ignore + elif _module_present('jax', namespaces): + from sympy.printing.numpy import JaxPrinter as Printer # type: ignore + elif _module_present('numexpr', namespaces): + from sympy.printing.lambdarepr import NumExprPrinter as Printer # type: ignore + elif _module_present('tensorflow', namespaces): + from sympy.printing.tensorflow import TensorflowPrinter as Printer # type: ignore + elif _module_present('torch', namespaces): + from sympy.printing.pytorch import TorchPrinter as Printer # type: ignore + elif _module_present('sympy', namespaces): + from sympy.printing.pycode import SymPyPrinter as Printer # type: ignore + elif _module_present('cmath', namespaces): + from sympy.printing.pycode import CmathPrinter as Printer # type: ignore + else: + from sympy.printing.pycode import PythonCodePrinter as Printer # type: ignore + user_functions = {} + for m in namespaces[::-1]: + if isinstance(m, dict): + for k in m: + user_functions[k] = k + printer = Printer({'fully_qualified_modules': False, 'inline': True, + 'allow_unknown_functions': True, + 'user_functions': user_functions}) + + if isinstance(args, set): + sympy_deprecation_warning( + """ +Passing the function arguments to lambdify() as a set is deprecated. This +leads to unpredictable results since sets are unordered. Instead, use a list +or tuple for the function arguments. + """, + deprecated_since_version="1.6.3", + active_deprecations_target="deprecated-lambdify-arguments-set", + ) + + # Get the names of the args, for creating a docstring + iterable_args = (args,) if isinstance(args, Expr) else args + names = [] + + # Grab the callers frame, for getting the names by inspection (if needed) + callers_local_vars = inspect.currentframe().f_back.f_locals.items() # type: ignore + for n, var in enumerate(iterable_args): + if hasattr(var, 'name'): + names.append(var.name) + else: + # It's an iterable. Try to get name by inspection of calling frame. + name_list = [var_name for var_name, var_val in callers_local_vars + if var_val is var] + if len(name_list) == 1: + names.append(name_list[0]) + else: + # Cannot infer name with certainty. arg_# will have to do. + names.append('arg_' + str(n)) + + # Create the function definition code and execute it + funcname = '_lambdifygenerated' + if _module_present('tensorflow', namespaces): + funcprinter = _TensorflowEvaluatorPrinter(printer, dummify) + else: + funcprinter = _EvaluatorPrinter(printer, dummify) + + if cse == True: + from sympy.simplify.cse_main import cse as _cse + cses, _expr = _cse(expr, list=False) + elif callable(cse): + cses, _expr = cse(expr) + else: + cses, _expr = (), expr + funcstr = funcprinter.doprint(funcname, iterable_args, _expr, cses=cses) + + # Collect the module imports from the code printers. + imp_mod_lines = [] + for mod, keys in (getattr(printer, 'module_imports', None) or {}).items(): + for k in keys: + if k not in namespace: + ln = "from %s import %s" % (mod, k) + try: + exec(ln, {}, namespace) + except ImportError: + # Tensorflow 2.0 has issues with importing a specific + # function from its submodule. + # https://github.com/tensorflow/tensorflow/issues/33022 + ln = "%s = %s.%s" % (k, mod, k) + exec(ln, {}, namespace) + imp_mod_lines.append(ln) + + # Provide lambda expression with builtins, and compatible implementation of range + namespace.update({'builtins':builtins, 'range':range}) + + funclocals = {} + global _lambdify_generated_counter + filename = '' % _lambdify_generated_counter + _lambdify_generated_counter += 1 + c = compile(funcstr, filename, 'exec') + exec(c, namespace, funclocals) + # mtime has to be None or else linecache.checkcache will remove it + linecache.cache[filename] = (len(funcstr), None, funcstr.splitlines(True), filename) # type: ignore + + # Remove the entry from the linecache when the object is garbage collected + def cleanup_linecache(filename): + def _cleanup(): + if filename in linecache.cache: + del linecache.cache[filename] + return _cleanup + + func = funclocals[funcname] + + weakref.finalize(func, cleanup_linecache(filename)) + + # Apply the docstring + sig = "func({})".format(", ".join(str(i) for i in names)) + sig = textwrap.fill(sig, subsequent_indent=' '*8) + if _too_large_for_docstring(expr, docstring_limit): + expr_str = "EXPRESSION REDACTED DUE TO LENGTH, (see lambdify's `docstring_limit`)" + src_str = "SOURCE CODE REDACTED DUE TO LENGTH, (see lambdify's `docstring_limit`)" + else: + expr_str = str(expr) + if len(expr_str) > 78: + expr_str = textwrap.wrap(expr_str, 75)[0] + '...' + src_str = funcstr + func.__doc__ = ( + "Created with lambdify. Signature:\n\n" + "{sig}\n\n" + "Expression:\n\n" + "{expr}\n\n" + "Source code:\n\n" + "{src}\n\n" + "Imported modules:\n\n" + "{imp_mods}" + ).format(sig=sig, expr=expr_str, src=src_str, imp_mods='\n'.join(imp_mod_lines)) + return func + +def _module_present(modname, modlist): + if modname in modlist: + return True + for m in modlist: + if hasattr(m, '__name__') and m.__name__ == modname: + return True + return False + +def _get_namespace(m): + """ + This is used by _lambdify to parse its arguments. + """ + if isinstance(m, str): + _import(m) + return MODULES[m][0] + elif isinstance(m, dict): + return m + elif hasattr(m, "__dict__"): + return m.__dict__ + else: + raise TypeError("Argument must be either a string, dict or module but it is: %s" % m) + + +def _recursive_to_string(doprint, arg): + """Functions in lambdify accept both SymPy types and non-SymPy types such as python + lists and tuples. This method ensures that we only call the doprint method of the + printer with SymPy types (so that the printer safely can use SymPy-methods).""" + from sympy.matrices.matrixbase import MatrixBase + from sympy.core.basic import Basic + + if isinstance(arg, (Basic, MatrixBase)): + return doprint(arg) + elif iterable(arg): + if isinstance(arg, list): + left, right = "[", "]" + elif isinstance(arg, tuple): + left, right = "(", ",)" + if not arg: + return "()" + else: + raise NotImplementedError("unhandled type: %s, %s" % (type(arg), arg)) + return left +', '.join(_recursive_to_string(doprint, e) for e in arg) + right + elif isinstance(arg, str): + return arg + else: + return doprint(arg) + + +def lambdastr(args, expr, printer=None, dummify=None): + """ + Returns a string that can be evaluated to a lambda function. + + Examples + ======== + + >>> from sympy.abc import x, y, z + >>> from sympy.utilities.lambdify import lambdastr + >>> lambdastr(x, x**2) + 'lambda x: (x**2)' + >>> lambdastr((x,y,z), [z,y,x]) + 'lambda x,y,z: ([z, y, x])' + + Although tuples may not appear as arguments to lambda in Python 3, + lambdastr will create a lambda function that will unpack the original + arguments so that nested arguments can be handled: + + >>> lambdastr((x, (y, z)), x + y) + 'lambda _0,_1: (lambda x,y,z: (x + y))(_0,_1[0],_1[1])' + """ + # Transforming everything to strings. + from sympy.matrices import DeferredVector + from sympy.core.basic import Basic + from sympy.core.function import (Derivative, Function) + from sympy.core.symbol import (Dummy, Symbol) + from sympy.core.sympify import sympify + + if printer is not None: + if inspect.isfunction(printer): + lambdarepr = printer + else: + if inspect.isclass(printer): + lambdarepr = lambda expr: printer().doprint(expr) + else: + lambdarepr = lambda expr: printer.doprint(expr) + else: + #XXX: This has to be done here because of circular imports + from sympy.printing.lambdarepr import lambdarepr + + def sub_args(args, dummies_dict): + if isinstance(args, str): + return args + elif isinstance(args, DeferredVector): + return str(args) + elif iterable(args): + dummies = flatten([sub_args(a, dummies_dict) for a in args]) + return ",".join(str(a) for a in dummies) + else: + # replace these with Dummy symbols + if isinstance(args, (Function, Symbol, Derivative)): + dummies = Dummy() + dummies_dict.update({args : dummies}) + return str(dummies) + else: + return str(args) + + def sub_expr(expr, dummies_dict): + expr = sympify(expr) + # dict/tuple are sympified to Basic + if isinstance(expr, Basic): + expr = expr.xreplace(dummies_dict) + # list is not sympified to Basic + elif isinstance(expr, list): + expr = [sub_expr(a, dummies_dict) for a in expr] + return expr + + # Transform args + def isiter(l): + return iterable(l, exclude=(str, DeferredVector, NotIterable)) + + def flat_indexes(iterable): + n = 0 + + for el in iterable: + if isiter(el): + for ndeep in flat_indexes(el): + yield (n,) + ndeep + else: + yield (n,) + + n += 1 + + if dummify is None: + dummify = any(isinstance(a, Basic) and + a.atoms(Function, Derivative) for a in ( + args if isiter(args) else [args])) + + if isiter(args) and any(isiter(i) for i in args): + dum_args = [str(Dummy(str(i))) for i in range(len(args))] + + indexed_args = ','.join([ + dum_args[ind[0]] + ''.join(["[%s]" % k for k in ind[1:]]) + for ind in flat_indexes(args)]) + + lstr = lambdastr(flatten(args), expr, printer=printer, dummify=dummify) + + return 'lambda %s: (%s)(%s)' % (','.join(dum_args), lstr, indexed_args) + + dummies_dict = {} + if dummify: + args = sub_args(args, dummies_dict) + else: + if isinstance(args, str): + pass + elif iterable(args, exclude=DeferredVector): + args = ",".join(str(a) for a in args) + + # Transform expr + if dummify: + if isinstance(expr, str): + pass + else: + expr = sub_expr(expr, dummies_dict) + expr = _recursive_to_string(lambdarepr, expr) + return "lambda %s: (%s)" % (args, expr) + +class _EvaluatorPrinter: + def __init__(self, printer=None, dummify=False): + self._dummify = dummify + + #XXX: This has to be done here because of circular imports + from sympy.printing.lambdarepr import LambdaPrinter + + if printer is None: + printer = LambdaPrinter() + + if inspect.isfunction(printer): + self._exprrepr = printer + else: + if inspect.isclass(printer): + printer = printer() + + self._exprrepr = printer.doprint + + #if hasattr(printer, '_print_Symbol'): + # symbolrepr = printer._print_Symbol + + #if hasattr(printer, '_print_Dummy'): + # dummyrepr = printer._print_Dummy + + # Used to print the generated function arguments in a standard way + self._argrepr = LambdaPrinter().doprint + + def doprint(self, funcname, args, expr, *, cses=()): + """ + Returns the function definition code as a string. + """ + from sympy.core.symbol import Dummy + + funcbody = [] + + if not iterable(args): + args = [args] + + if cses: + cses = list(cses) + subvars, subexprs = zip(*cses) + exprs = [expr] + list(subexprs) + argstrs, exprs = self._preprocess(args, exprs, cses=cses) + expr, subexprs = exprs[0], exprs[1:] + cses = zip(subvars, subexprs) + else: + argstrs, expr = self._preprocess(args, expr) + + # Generate argument unpacking and final argument list + funcargs = [] + unpackings = [] + + for argstr in argstrs: + if iterable(argstr): + funcargs.append(self._argrepr(Dummy())) + unpackings.extend(self._print_unpacking(argstr, funcargs[-1])) + else: + funcargs.append(argstr) + + funcsig = 'def {}({}):'.format(funcname, ', '.join(funcargs)) + + # Wrap input arguments before unpacking + funcbody.extend(self._print_funcargwrapping(funcargs)) + + funcbody.extend(unpackings) + + for s, e in cses: + if e is None: + funcbody.append('del {}'.format(self._exprrepr(s))) + else: + funcbody.append('{} = {}'.format(self._exprrepr(s), self._exprrepr(e))) + + # Subs may appear in expressions generated by .diff() + subs_assignments = [] + expr = self._handle_Subs(expr, out=subs_assignments) + for lhs, rhs in subs_assignments: + funcbody.append('{} = {}'.format(self._exprrepr(lhs), self._exprrepr(rhs))) + + str_expr = _recursive_to_string(self._exprrepr, expr) + + if '\n' in str_expr: + str_expr = '({})'.format(str_expr) + funcbody.append('return {}'.format(str_expr)) + + funclines = [funcsig] + funclines.extend([' ' + line for line in funcbody]) + + return '\n'.join(funclines) + '\n' + + @classmethod + def _is_safe_ident(cls, ident): + return isinstance(ident, str) and ident.isidentifier() \ + and not keyword.iskeyword(ident) + + def _preprocess(self, args, expr, cses=(), _dummies_dict=None): + """Preprocess args, expr to replace arguments that do not map + to valid Python identifiers. + + Returns string form of args, and updated expr. + """ + from sympy.core.basic import Basic + from sympy.core.sorting import ordered + from sympy.core.function import (Derivative, Function) + from sympy.core.symbol import Dummy, uniquely_named_symbol + from sympy.matrices import DeferredVector + from sympy.core.expr import Expr + + # Args of type Dummy can cause name collisions with args + # of type Symbol. Force dummify of everything in this + # situation. + dummify = self._dummify or any( + isinstance(arg, Dummy) for arg in flatten(args)) + + argstrs = [None]*len(args) + if _dummies_dict is None: + _dummies_dict = {} + + def update_dummies(arg, dummy): + _dummies_dict[arg] = dummy + for repl, sub in cses: + arg = arg.xreplace({sub: repl}) + _dummies_dict[arg] = dummy + + for arg, i in reversed(list(ordered(zip(args, range(len(args)))))): + if iterable(arg): + s, expr = self._preprocess(arg, expr, cses=cses, _dummies_dict=_dummies_dict) + elif isinstance(arg, DeferredVector): + s = str(arg) + elif isinstance(arg, Basic) and arg.is_symbol: + s = str(arg) + if dummify or not self._is_safe_ident(s): + dummy = Dummy() + if isinstance(expr, Expr): + dummy = uniquely_named_symbol( + dummy.name, expr, modify=lambda s: '_' + s) + s = self._argrepr(dummy) + update_dummies(arg, dummy) + expr = self._subexpr(expr, _dummies_dict) + elif dummify or isinstance(arg, (Function, Derivative)): + dummy = Dummy() + s = self._argrepr(dummy) + update_dummies(arg, dummy) + expr = self._subexpr(expr, _dummies_dict) + else: + s = str(arg) + argstrs[i] = s + return argstrs, expr + + def _subexpr(self, expr, dummies_dict): + from sympy.matrices import DeferredVector + from sympy.core.sympify import sympify + + expr = sympify(expr) + xreplace = getattr(expr, 'xreplace', None) + if xreplace is not None: + expr = xreplace(dummies_dict) + else: + if isinstance(expr, DeferredVector): + pass + elif isinstance(expr, dict): + k = [self._subexpr(sympify(a), dummies_dict) for a in expr.keys()] + v = [self._subexpr(sympify(a), dummies_dict) for a in expr.values()] + expr = dict(zip(k, v)) + elif isinstance(expr, tuple): + expr = tuple(self._subexpr(sympify(a), dummies_dict) for a in expr) + elif isinstance(expr, list): + expr = [self._subexpr(sympify(a), dummies_dict) for a in expr] + return expr + + def _print_funcargwrapping(self, args): + """Generate argument wrapping code. + + args is the argument list of the generated function (strings). + + Return value is a list of lines of code that will be inserted at + the beginning of the function definition. + """ + return [] + + def _print_unpacking(self, unpackto, arg): + """Generate argument unpacking code. + + arg is the function argument to be unpacked (a string), and + unpackto is a list or nested lists of the variable names (strings) to + unpack to. + """ + def unpack_lhs(lvalues): + return '[{}]'.format(', '.join( + unpack_lhs(val) if iterable(val) else val for val in lvalues)) + + return ['{} = {}'.format(unpack_lhs(unpackto), arg)] + + def _handle_Subs(self, expr, out): + """Any instance of Subs is extracted and returned as assignment pairs.""" + from sympy.core.basic import Basic + from sympy.core.function import Subs + from sympy.core.symbol import Dummy + from sympy.matrices.matrixbase import MatrixBase + + def _replace(ex, variables, point): + safe = {} + for lhs, rhs in zip(variables, point): + dummy = Dummy() + safe[lhs] = dummy + out.append((dummy, rhs)) + return ex.xreplace(safe) + + if isinstance(expr, (Basic, MatrixBase)): + expr = expr.replace(Subs, _replace) + elif iterable(expr): + expr = type(expr)([self._handle_Subs(e, out) for e in expr]) + return expr + +class _TensorflowEvaluatorPrinter(_EvaluatorPrinter): + def _print_unpacking(self, lvalues, rvalue): + """Generate argument unpacking code. + + This method is used when the input value is not iterable, + but can be indexed (see issue #14655). + """ + + def flat_indexes(elems): + n = 0 + + for el in elems: + if iterable(el): + for ndeep in flat_indexes(el): + yield (n,) + ndeep + else: + yield (n,) + + n += 1 + + indexed = ', '.join('{}[{}]'.format(rvalue, ']['.join(map(str, ind))) + for ind in flat_indexes(lvalues)) + + return ['[{}] = [{}]'.format(', '.join(flatten(lvalues)), indexed)] + +def _imp_namespace(expr, namespace=None): + """ Return namespace dict with function implementations + + We need to search for functions in anything that can be thrown at + us - that is - anything that could be passed as ``expr``. Examples + include SymPy expressions, as well as tuples, lists and dicts that may + contain SymPy expressions. + + Parameters + ---------- + expr : object + Something passed to lambdify, that will generate valid code from + ``str(expr)``. + namespace : None or mapping + Namespace to fill. None results in new empty dict + + Returns + ------- + namespace : dict + dict with keys of implemented function names within ``expr`` and + corresponding values being the numerical implementation of + function + + Examples + ======== + + >>> from sympy.abc import x + >>> from sympy.utilities.lambdify import implemented_function, _imp_namespace + >>> from sympy import Function + >>> f = implemented_function(Function('f'), lambda x: x+1) + >>> g = implemented_function(Function('g'), lambda x: x*10) + >>> namespace = _imp_namespace(f(g(x))) + >>> sorted(namespace.keys()) + ['f', 'g'] + """ + # Delayed import to avoid circular imports + from sympy.core.function import FunctionClass + if namespace is None: + namespace = {} + # tuples, lists, dicts are valid expressions + if is_sequence(expr): + for arg in expr: + _imp_namespace(arg, namespace) + return namespace + elif isinstance(expr, dict): + for key, val in expr.items(): + # functions can be in dictionary keys + _imp_namespace(key, namespace) + _imp_namespace(val, namespace) + return namespace + # SymPy expressions may be Functions themselves + func = getattr(expr, 'func', None) + if isinstance(func, FunctionClass): + imp = getattr(func, '_imp_', None) + if imp is not None: + name = expr.func.__name__ + if name in namespace and namespace[name] != imp: + raise ValueError('We found more than one ' + 'implementation with name ' + '"%s"' % name) + namespace[name] = imp + # and / or they may take Functions as arguments + if hasattr(expr, 'args'): + for arg in expr.args: + _imp_namespace(arg, namespace) + return namespace + + +def implemented_function(symfunc, implementation): + """ Add numerical ``implementation`` to function ``symfunc``. + + ``symfunc`` can be an ``UndefinedFunction`` instance, or a name string. + In the latter case we create an ``UndefinedFunction`` instance with that + name. + + Be aware that this is a quick workaround, not a general method to create + special symbolic functions. If you want to create a symbolic function to be + used by all the machinery of SymPy you should subclass the ``Function`` + class. + + Parameters + ---------- + symfunc : ``str`` or ``UndefinedFunction`` instance + If ``str``, then create new ``UndefinedFunction`` with this as + name. If ``symfunc`` is an Undefined function, create a new function + with the same name and the implemented function attached. + implementation : callable + numerical implementation to be called by ``evalf()`` or ``lambdify`` + + Returns + ------- + afunc : sympy.FunctionClass instance + function with attached implementation + + Examples + ======== + + >>> from sympy.abc import x + >>> from sympy.utilities.lambdify import implemented_function + >>> from sympy import lambdify + >>> f = implemented_function('f', lambda x: x+1) + >>> lam_f = lambdify(x, f(x)) + >>> lam_f(4) + 5 + """ + # Delayed import to avoid circular imports + from sympy.core.function import UndefinedFunction + # if name, create function to hold implementation + kwargs = {} + if isinstance(symfunc, UndefinedFunction): + kwargs = symfunc._kwargs + symfunc = symfunc.__name__ + if isinstance(symfunc, str): + # Keyword arguments to UndefinedFunction are added as attributes to + # the created class. + symfunc = UndefinedFunction( + symfunc, _imp_=staticmethod(implementation), **kwargs) + elif not isinstance(symfunc, UndefinedFunction): + raise ValueError(filldedent(''' + symfunc should be either a string or + an UndefinedFunction instance.''')) + return symfunc + + +def _too_large_for_docstring(expr, limit): + """Decide whether an ``Expr`` is too large to be fully rendered in a + ``lambdify`` docstring. + + This is a fast alternative to ``count_ops``, which can become prohibitively + slow for large expressions, because in this instance we only care whether + ``limit`` is exceeded rather than counting the exact number of nodes in the + expression. + + Parameters + ========== + expr : ``Expr``, (nested) ``list`` of ``Expr``, or ``Matrix`` + The same objects that can be passed to the ``expr`` argument of + ``lambdify``. + limit : ``int`` or ``None`` + The threshold above which an expression contains too many nodes to be + usefully rendered in the docstring. If ``None`` then there is no limit. + + Returns + ======= + bool + ``True`` if the number of nodes in the expression exceeds the limit, + ``False`` otherwise. + + Examples + ======== + + >>> from sympy.abc import x, y, z + >>> from sympy.utilities.lambdify import _too_large_for_docstring + >>> expr = x + >>> _too_large_for_docstring(expr, None) + False + >>> _too_large_for_docstring(expr, 100) + False + >>> _too_large_for_docstring(expr, 1) + False + >>> _too_large_for_docstring(expr, 0) + True + >>> _too_large_for_docstring(expr, -1) + True + + Does this split it? + + >>> expr = [x, y, z] + >>> _too_large_for_docstring(expr, None) + False + >>> _too_large_for_docstring(expr, 100) + False + >>> _too_large_for_docstring(expr, 1) + True + >>> _too_large_for_docstring(expr, 0) + True + >>> _too_large_for_docstring(expr, -1) + True + + >>> expr = [x, [y], z, [[x+y], [x*y*z, [x+y+z]]]] + >>> _too_large_for_docstring(expr, None) + False + >>> _too_large_for_docstring(expr, 100) + False + >>> _too_large_for_docstring(expr, 1) + True + >>> _too_large_for_docstring(expr, 0) + True + >>> _too_large_for_docstring(expr, -1) + True + + >>> expr = ((x + y + z)**5).expand() + >>> _too_large_for_docstring(expr, None) + False + >>> _too_large_for_docstring(expr, 100) + True + >>> _too_large_for_docstring(expr, 1) + True + >>> _too_large_for_docstring(expr, 0) + True + >>> _too_large_for_docstring(expr, -1) + True + + >>> from sympy import Matrix + >>> expr = Matrix([[(x + y + z), ((x + y + z)**2).expand(), + ... ((x + y + z)**3).expand(), ((x + y + z)**4).expand()]]) + >>> _too_large_for_docstring(expr, None) + False + >>> _too_large_for_docstring(expr, 1000) + False + >>> _too_large_for_docstring(expr, 100) + True + >>> _too_large_for_docstring(expr, 1) + True + >>> _too_large_for_docstring(expr, 0) + True + >>> _too_large_for_docstring(expr, -1) + True + + """ + # Must be imported here to avoid a circular import error + from sympy.core.traversal import postorder_traversal + + if limit is None: + return False + + i = 0 + for _ in postorder_traversal(expr): + i += 1 + if i > limit: + return True + return False diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/magic.py b/.venv/lib/python3.12/site-packages/sympy/utilities/magic.py new file mode 100644 index 0000000000000000000000000000000000000000..e853a0ad9a85bc252dcb24e8a1ecbfca422ac3fd --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/magic.py @@ -0,0 +1,12 @@ +"""Functions that involve magic. """ + +def pollute(names, objects): + """Pollute the global namespace with symbols -> objects mapping. """ + from inspect import currentframe + frame = currentframe().f_back.f_back + + try: + for name, obj in zip(names, objects): + frame.f_globals[name] = obj + finally: + del frame # break cyclic dependencies as stated in inspect docs diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/matchpy_connector.py b/.venv/lib/python3.12/site-packages/sympy/utilities/matchpy_connector.py new file mode 100644 index 0000000000000000000000000000000000000000..35aa013294b93bbcfe4a1bf4ec96b629ea5a468f --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/matchpy_connector.py @@ -0,0 +1,340 @@ +""" +The objects in this module allow the usage of the MatchPy pattern matching +library on SymPy expressions. +""" +import re +from typing import List, Callable, NamedTuple, Any, Dict + +from sympy.core.sympify import _sympify +from sympy.external import import_module +from sympy.functions import (log, sin, cos, tan, cot, csc, sec, erf, gamma, uppergamma) +from sympy.functions.elementary.hyperbolic import acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch +from sympy.functions.elementary.trigonometric import atan, acsc, asin, acot, acos, asec +from sympy.functions.special.error_functions import fresnelc, fresnels, erfc, erfi, Ei +from sympy.core.add import Add +from sympy.core.basic import Basic +from sympy.core.expr import Expr +from sympy.core.mul import Mul +from sympy.core.power import Pow +from sympy.core.relational import (Equality, Unequality) +from sympy.core.symbol import Symbol +from sympy.functions.elementary.exponential import exp +from sympy.integrals.integrals import Integral +from sympy.printing.repr import srepr +from sympy.utilities.decorator import doctest_depends_on + + +matchpy = import_module("matchpy") + + +__doctest_requires__ = {('*',): ['matchpy']} + + +if matchpy: + from matchpy import Operation, CommutativeOperation, AssociativeOperation, OneIdentityOperation + from matchpy.expressions.functions import op_iter, create_operation_expression, op_len + + Operation.register(Integral) + Operation.register(Pow) + OneIdentityOperation.register(Pow) + + Operation.register(Add) + OneIdentityOperation.register(Add) + CommutativeOperation.register(Add) + AssociativeOperation.register(Add) + + Operation.register(Mul) + OneIdentityOperation.register(Mul) + CommutativeOperation.register(Mul) + AssociativeOperation.register(Mul) + + Operation.register(Equality) + CommutativeOperation.register(Equality) + Operation.register(Unequality) + CommutativeOperation.register(Unequality) + + Operation.register(exp) + Operation.register(log) + Operation.register(gamma) + Operation.register(uppergamma) + Operation.register(fresnels) + Operation.register(fresnelc) + Operation.register(erf) + Operation.register(Ei) + Operation.register(erfc) + Operation.register(erfi) + Operation.register(sin) + Operation.register(cos) + Operation.register(tan) + Operation.register(cot) + Operation.register(csc) + Operation.register(sec) + Operation.register(sinh) + Operation.register(cosh) + Operation.register(tanh) + Operation.register(coth) + Operation.register(csch) + Operation.register(sech) + Operation.register(asin) + Operation.register(acos) + Operation.register(atan) + Operation.register(acot) + Operation.register(acsc) + Operation.register(asec) + Operation.register(asinh) + Operation.register(acosh) + Operation.register(atanh) + Operation.register(acoth) + Operation.register(acsch) + Operation.register(asech) + + @op_iter.register(Integral) # type: ignore + def _(operation): + return iter((operation._args[0],) + operation._args[1]) + + @op_iter.register(Basic) # type: ignore + def _(operation): + return iter(operation._args) + + @op_len.register(Integral) # type: ignore + def _(operation): + return 1 + len(operation._args[1]) + + @op_len.register(Basic) # type: ignore + def _(operation): + return len(operation._args) + + @create_operation_expression.register(Basic) + def sympy_op_factory(old_operation, new_operands, variable_name=True): + return type(old_operation)(*new_operands) + + +if matchpy: + from matchpy import Wildcard +else: + class Wildcard: # type: ignore + def __init__(self, min_length, fixed_size, variable_name, optional): + self.min_count = min_length + self.fixed_size = fixed_size + self.variable_name = variable_name + self.optional = optional + + +@doctest_depends_on(modules=('matchpy',)) +class _WildAbstract(Wildcard, Symbol): + min_length: int # abstract field required in subclasses + fixed_size: bool # abstract field required in subclasses + + def __init__(self, variable_name=None, optional=None, **assumptions): + min_length = self.min_length + fixed_size = self.fixed_size + if optional is not None: + optional = _sympify(optional) + Wildcard.__init__(self, min_length, fixed_size, str(variable_name), optional) + + def __getstate__(self): + return { + "min_length": self.min_length, + "fixed_size": self.fixed_size, + "min_count": self.min_count, + "variable_name": self.variable_name, + "optional": self.optional, + } + + def __new__(cls, variable_name=None, optional=None, **assumptions): + cls._sanitize(assumptions, cls) + return _WildAbstract.__xnew__(cls, variable_name, optional, **assumptions) + + def __getnewargs__(self): + return self.variable_name, self.optional + + @staticmethod + def __xnew__(cls, variable_name=None, optional=None, **assumptions): + obj = Symbol.__xnew__(cls, variable_name, **assumptions) + return obj + + def _hashable_content(self): + if self.optional: + return super()._hashable_content() + (self.min_count, self.fixed_size, self.variable_name, self.optional) + else: + return super()._hashable_content() + (self.min_count, self.fixed_size, self.variable_name) + + def __copy__(self) -> '_WildAbstract': + return type(self)(variable_name=self.variable_name, optional=self.optional) + + def __repr__(self): + return str(self) + + def __str__(self): + return self.name + + +@doctest_depends_on(modules=('matchpy',)) +class WildDot(_WildAbstract): + min_length = 1 + fixed_size = True + + +@doctest_depends_on(modules=('matchpy',)) +class WildPlus(_WildAbstract): + min_length = 1 + fixed_size = False + + +@doctest_depends_on(modules=('matchpy',)) +class WildStar(_WildAbstract): + min_length = 0 + fixed_size = False + + +def _get_srepr(expr): + s = srepr(expr) + s = re.sub(r"WildDot\('(\w+)'\)", r"\1", s) + s = re.sub(r"WildPlus\('(\w+)'\)", r"*\1", s) + s = re.sub(r"WildStar\('(\w+)'\)", r"*\1", s) + return s + + +class ReplacementInfo(NamedTuple): + replacement: Any + info: Any + + +@doctest_depends_on(modules=('matchpy',)) +class Replacer: + """ + Replacer object to perform multiple pattern matching and subexpression + replacements in SymPy expressions. + + Examples + ======== + + Example to construct a simple first degree equation solver: + + >>> from sympy.utilities.matchpy_connector import WildDot, Replacer + >>> from sympy import Equality, Symbol + >>> x = Symbol("x") + >>> a_ = WildDot("a_", optional=1) + >>> b_ = WildDot("b_", optional=0) + + The lines above have defined two wildcards, ``a_`` and ``b_``, the + coefficients of the equation `a x + b = 0`. The optional values specified + indicate which expression to return in case no match is found, they are + necessary in equations like `a x = 0` and `x + b = 0`. + + Create two constraints to make sure that ``a_`` and ``b_`` will not match + any expression containing ``x``: + + >>> from matchpy import CustomConstraint + >>> free_x_a = CustomConstraint(lambda a_: not a_.has(x)) + >>> free_x_b = CustomConstraint(lambda b_: not b_.has(x)) + + Now create the rule replacer with the constraints: + + >>> replacer = Replacer(common_constraints=[free_x_a, free_x_b]) + + Add the matching rule: + + >>> replacer.add(Equality(a_*x + b_, 0), -b_/a_) + + Let's try it: + + >>> replacer.replace(Equality(3*x + 4, 0)) + -4/3 + + Notice that it will not match equations expressed with other patterns: + + >>> eq = Equality(3*x, 4) + >>> replacer.replace(eq) + Eq(3*x, 4) + + In order to extend the matching patterns, define another one (we also need + to clear the cache, because the previous result has already been memorized + and the pattern matcher will not iterate again if given the same expression) + + >>> replacer.add(Equality(a_*x, b_), b_/a_) + >>> replacer._matcher.clear() + >>> replacer.replace(eq) + 4/3 + """ + + def __init__(self, common_constraints: list = [], lambdify: bool = False, info: bool = False): + self._matcher = matchpy.ManyToOneMatcher() + self._common_constraint = common_constraints + self._lambdify = lambdify + self._info = info + self._wildcards: Dict[str, Wildcard] = {} + + def _get_lambda(self, lambda_str: str) -> Callable[..., Expr]: + exec("from sympy import *") + return eval(lambda_str, locals()) + + def _get_custom_constraint(self, constraint_expr: Expr, condition_template: str) -> Callable[..., Expr]: + wilds = [x.name for x in constraint_expr.atoms(_WildAbstract)] + lambdaargs = ', '.join(wilds) + fullexpr = _get_srepr(constraint_expr) + condition = condition_template.format(fullexpr) + return matchpy.CustomConstraint( + self._get_lambda(f"lambda {lambdaargs}: ({condition})")) + + def _get_custom_constraint_nonfalse(self, constraint_expr: Expr) -> Callable[..., Expr]: + return self._get_custom_constraint(constraint_expr, "({}) != False") + + def _get_custom_constraint_true(self, constraint_expr: Expr) -> Callable[..., Expr]: + return self._get_custom_constraint(constraint_expr, "({}) == True") + + def add(self, expr: Expr, replacement, conditions_true: List[Expr] = [], + conditions_nonfalse: List[Expr] = [], info: Any = None) -> None: + expr = _sympify(expr) + replacement = _sympify(replacement) + constraints = self._common_constraint[:] + constraint_conditions_true = [ + self._get_custom_constraint_true(cond) for cond in conditions_true] + constraint_conditions_nonfalse = [ + self._get_custom_constraint_nonfalse(cond) for cond in conditions_nonfalse] + constraints.extend(constraint_conditions_true) + constraints.extend(constraint_conditions_nonfalse) + pattern = matchpy.Pattern(expr, *constraints) + if self._lambdify: + lambda_str = f"lambda {', '.join((x.name for x in expr.atoms(_WildAbstract)))}: {_get_srepr(replacement)}" + lambda_expr = self._get_lambda(lambda_str) + replacement = lambda_expr + else: + self._wildcards.update({str(i): i for i in expr.atoms(Wildcard)}) + if self._info: + replacement = ReplacementInfo(replacement, info) + self._matcher.add(pattern, replacement) + + def replace(self, expression, max_count: int = -1): + # This method partly rewrites the .replace method of ManyToOneReplacer + # in MatchPy. + # License: https://github.com/HPAC/matchpy/blob/master/LICENSE + infos = [] + replaced = True + replace_count = 0 + while replaced and (max_count < 0 or replace_count < max_count): + replaced = False + for subexpr, pos in matchpy.preorder_iter_with_position(expression): + try: + replacement_data, subst = next(iter(self._matcher.match(subexpr))) + if self._info: + replacement = replacement_data.replacement + infos.append(replacement_data.info) + else: + replacement = replacement_data + + if self._lambdify: + result = replacement(**subst) + else: + result = replacement.xreplace({self._wildcards[k]: v for k, v in subst.items()}) + + expression = matchpy.functions.replace(expression, pos, result) + replaced = True + break + except StopIteration: + pass + replace_count += 1 + if self._info: + return expression, infos + else: + return expression diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/mathml/__init__.py b/.venv/lib/python3.12/site-packages/sympy/utilities/mathml/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..eded44ee3c0f34ad1324765ba06ee9d6eb5e9899 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/mathml/__init__.py @@ -0,0 +1,122 @@ +"""Module with some functions for MathML, like transforming MathML +content in MathML presentation. + +To use this module, you will need lxml. +""" + +from pathlib import Path + +from sympy.utilities.decorator import doctest_depends_on + + +__doctest_requires__ = {('apply_xsl', 'c2p'): ['lxml']} + + +def add_mathml_headers(s): + return """""" + s + "" + + +def _read_binary(pkgname, filename): + import sys + + if sys.version_info >= (3, 10): + # files was added in Python 3.9 but only seems to work here in 3.10+ + from importlib.resources import files + return files(pkgname).joinpath(filename).read_bytes() + else: + # read_binary was deprecated in Python 3.11 + from importlib.resources import read_binary + return read_binary(pkgname, filename) + + +def _read_xsl(xsl): + # Previously these values were allowed: + if xsl == 'mathml/data/simple_mmlctop.xsl': + xsl = 'simple_mmlctop.xsl' + elif xsl == 'mathml/data/mmlctop.xsl': + xsl = 'mmlctop.xsl' + elif xsl == 'mathml/data/mmltex.xsl': + xsl = 'mmltex.xsl' + + if xsl in ['simple_mmlctop.xsl', 'mmlctop.xsl', 'mmltex.xsl']: + xslbytes = _read_binary('sympy.utilities.mathml.data', xsl) + else: + xslbytes = Path(xsl).read_bytes() + + return xslbytes + + +@doctest_depends_on(modules=('lxml',)) +def apply_xsl(mml, xsl): + """Apply a xsl to a MathML string. + + Parameters + ========== + + mml + A string with MathML code. + xsl + A string giving the name of an xsl (xml stylesheet) file which can be + found in sympy/utilities/mathml/data. The following files are supplied + with SymPy: + + - mmlctop.xsl + - mmltex.xsl + - simple_mmlctop.xsl + + Alternatively, a full path to an xsl file can be given. + + Examples + ======== + + >>> from sympy.utilities.mathml import apply_xsl + >>> xsl = 'simple_mmlctop.xsl' + >>> mml = ' a b ' + >>> res = apply_xsl(mml,xsl) + >>> print(res) + + + a + + + b + + """ + from lxml import etree + + parser = etree.XMLParser(resolve_entities=False) + ac = etree.XSLTAccessControl.DENY_ALL + + s = etree.XML(_read_xsl(xsl), parser=parser) + transform = etree.XSLT(s, access_control=ac) + doc = etree.XML(mml, parser=parser) + result = transform(doc) + s = str(result) + return s + + +@doctest_depends_on(modules=('lxml',)) +def c2p(mml, simple=False): + """Transforms a document in MathML content (like the one that sympy produces) + in one document in MathML presentation, more suitable for printing, and more + widely accepted + + Examples + ======== + + >>> from sympy.utilities.mathml import c2p + >>> mml = ' 2 ' + >>> c2p(mml,simple=True) != c2p(mml,simple=False) + True + + """ + + if not mml.startswith(' + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + e + + + + + + + + + + + + + - + + + + + + + + &#x2062; + &#x2148; + + + + + + + + + + + + + - + + + + + + + + &#x2062; + &#x2148; + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + Polar + &#x2062; + + + + + + + + + + + + + + + Polar + &#x2062; + + + + + + + + + + + + + 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a/.venv/lib/python3.12/site-packages/sympy/utilities/memoization.py b/.venv/lib/python3.12/site-packages/sympy/utilities/memoization.py new file mode 100644 index 0000000000000000000000000000000000000000..b638dfe244628096108ea689b664782f6538b7b8 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/memoization.py @@ -0,0 +1,76 @@ +from functools import wraps + + +def recurrence_memo(initial): + """ + Memo decorator for sequences defined by recurrence + + Examples + ======== + + >>> from sympy.utilities.memoization import recurrence_memo + >>> @recurrence_memo([1]) # 0! = 1 + ... def factorial(n, prev): + ... return n * prev[-1] + >>> factorial(4) + 24 + >>> factorial(3) # use cache values + 6 + >>> factorial.cache_length() # cache length can be obtained + 5 + >>> factorial.fetch_item(slice(2, 4)) + [2, 6] + + """ + cache = initial + + def decorator(f): + @wraps(f) + def g(n): + L = len(cache) + if n < L: + return cache[n] + for i in range(L, n + 1): + cache.append(f(i, cache)) + return cache[-1] + g.cache_length = lambda: len(cache) + g.fetch_item = lambda x: cache[x] + return g + return decorator + + +def assoc_recurrence_memo(base_seq): + """ + Memo decorator for associated sequences defined by recurrence starting from base + + base_seq(n) -- callable to get base sequence elements + + XXX works only for Pn0 = base_seq(0) cases + XXX works only for m <= n cases + """ + + cache = [] + + def decorator(f): + @wraps(f) + def g(n, m): + L = len(cache) + if n < L: + return cache[n][m] + + for i in range(L, n + 1): + # get base sequence + F_i0 = base_seq(i) + F_i_cache = [F_i0] + cache.append(F_i_cache) + + # XXX only works for m <= n cases + # generate assoc sequence + for j in range(1, i + 1): + F_ij = f(i, j, cache) + F_i_cache.append(F_ij) + + return cache[n][m] + + return g + return decorator diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/misc.py b/.venv/lib/python3.12/site-packages/sympy/utilities/misc.py new file mode 100644 index 0000000000000000000000000000000000000000..741b983f03272890b22f2545f67b141767507634 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/misc.py @@ -0,0 +1,564 @@ +"""Miscellaneous stuff that does not really fit anywhere else.""" + +from __future__ import annotations + +import operator +import sys +import os +import re as _re +import struct +from textwrap import fill, dedent + + +class Undecidable(ValueError): + # an error to be raised when a decision cannot be made definitively + # where a definitive answer is needed + pass + + +def filldedent(s, w=70, **kwargs): + """ + Strips leading and trailing empty lines from a copy of ``s``, then dedents, + fills and returns it. + + Empty line stripping serves to deal with docstrings like this one that + start with a newline after the initial triple quote, inserting an empty + line at the beginning of the string. + + Additional keyword arguments will be passed to ``textwrap.fill()``. + + See Also + ======== + strlines, rawlines + + """ + return '\n' + fill(dedent(str(s)).strip('\n'), width=w, **kwargs) + + +def strlines(s, c=64, short=False): + """Return a cut-and-pastable string that, when printed, is + equivalent to the input. The lines will be surrounded by + parentheses and no line will be longer than c (default 64) + characters. If the line contains newlines characters, the + `rawlines` result will be returned. If ``short`` is True + (default is False) then if there is one line it will be + returned without bounding parentheses. + + Examples + ======== + + >>> from sympy.utilities.misc import strlines + >>> q = 'this is a long string that should be broken into shorter lines' + >>> print(strlines(q, 40)) + ( + 'this is a long string that should be b' + 'roken into shorter lines' + ) + >>> q == ( + ... 'this is a long string that should be b' + ... 'roken into shorter lines' + ... ) + True + + See Also + ======== + filldedent, rawlines + """ + if not isinstance(s, str): + raise ValueError('expecting string input') + if '\n' in s: + return rawlines(s) + q = '"' if repr(s).startswith('"') else "'" + q = (q,)*2 + if '\\' in s: # use r-string + m = '(\nr%s%%s%s\n)' % q + j = '%s\nr%s' % q + c -= 3 + else: + m = '(\n%s%%s%s\n)' % q + j = '%s\n%s' % q + c -= 2 + out = [] + while s: + out.append(s[:c]) + s=s[c:] + if short and len(out) == 1: + return (m % out[0]).splitlines()[1] # strip bounding (\n...\n) + return m % j.join(out) + + +def rawlines(s): + """Return a cut-and-pastable string that, when printed, is equivalent + to the input. Use this when there is more than one line in the + string. The string returned is formatted so it can be indented + nicely within tests; in some cases it is wrapped in the dedent + function which has to be imported from textwrap. + + Examples + ======== + + Note: because there are characters in the examples below that need + to be escaped because they are themselves within a triple quoted + docstring, expressions below look more complicated than they would + be if they were printed in an interpreter window. + + >>> from sympy.utilities.misc import rawlines + >>> from sympy import TableForm + >>> s = str(TableForm([[1, 10]], headings=(None, ['a', 'bee']))) + >>> print(rawlines(s)) + ( + 'a bee\\n' + '-----\\n' + '1 10 ' + ) + >>> print(rawlines('''this + ... that''')) + dedent('''\\ + this + that''') + + >>> print(rawlines('''this + ... that + ... ''')) + dedent('''\\ + this + that + ''') + + >>> s = \"\"\"this + ... is a triple ''' + ... \"\"\" + >>> print(rawlines(s)) + dedent(\"\"\"\\ + this + is a triple ''' + \"\"\") + + >>> print(rawlines('''this + ... that + ... ''')) + ( + 'this\\n' + 'that\\n' + ' ' + ) + + See Also + ======== + filldedent, strlines + """ + lines = s.split('\n') + if len(lines) == 1: + return repr(lines[0]) + triple = ["'''" in s, '"""' in s] + if any(li.endswith(' ') for li in lines) or '\\' in s or all(triple): + rv = [] + # add on the newlines + trailing = s.endswith('\n') + last = len(lines) - 1 + for i, li in enumerate(lines): + if i != last or trailing: + rv.append(repr(li + '\n')) + else: + rv.append(repr(li)) + return '(\n %s\n)' % '\n '.join(rv) + else: + rv = '\n '.join(lines) + if triple[0]: + return 'dedent("""\\\n %s""")' % rv + else: + return "dedent('''\\\n %s''')" % rv + +ARCH = str(struct.calcsize('P') * 8) + "-bit" + + +# XXX: PyPy does not support hash randomization +HASH_RANDOMIZATION = getattr(sys.flags, 'hash_randomization', False) + +_debug_tmp: list[str] = [] +_debug_iter = 0 + +def debug_decorator(func): + """If SYMPY_DEBUG is True, it will print a nice execution tree with + arguments and results of all decorated functions, else do nothing. + """ + from sympy import SYMPY_DEBUG + + if not SYMPY_DEBUG: + return func + + def maketree(f, *args, **kw): + global _debug_tmp, _debug_iter + oldtmp = _debug_tmp + _debug_tmp = [] + _debug_iter += 1 + + def tree(subtrees): + def indent(s, variant=1): + x = s.split("\n") + r = "+-%s\n" % x[0] + for a in x[1:]: + if a == "": + continue + if variant == 1: + r += "| %s\n" % a + else: + r += " %s\n" % a + return r + if len(subtrees) == 0: + return "" + f = [] + for a in subtrees[:-1]: + f.append(indent(a)) + f.append(indent(subtrees[-1], 2)) + return ''.join(f) + + # If there is a bug and the algorithm enters an infinite loop, enable the + # following lines. It will print the names and parameters of all major functions + # that are called, *before* they are called + #from functools import reduce + #print("%s%s %s%s" % (_debug_iter, reduce(lambda x, y: x + y, \ + # map(lambda x: '-', range(1, 2 + _debug_iter))), f.__name__, args)) + + r = f(*args, **kw) + + _debug_iter -= 1 + s = "%s%s = %s\n" % (f.__name__, args, r) + if _debug_tmp != []: + s += tree(_debug_tmp) + _debug_tmp = oldtmp + _debug_tmp.append(s) + if _debug_iter == 0: + print(_debug_tmp[0]) + _debug_tmp = [] + return r + + def decorated(*args, **kwargs): + return maketree(func, *args, **kwargs) + + return decorated + + +def debug(*args): + """ + Print ``*args`` if SYMPY_DEBUG is True, else do nothing. + """ + from sympy import SYMPY_DEBUG + if SYMPY_DEBUG: + print(*args, file=sys.stderr) + + +def debugf(string, args): + """ + Print ``string%args`` if SYMPY_DEBUG is True, else do nothing. This is + intended for debug messages using formatted strings. + """ + from sympy import SYMPY_DEBUG + if SYMPY_DEBUG: + print(string%args, file=sys.stderr) + + +def find_executable(executable, path=None): + """Try to find 'executable' in the directories listed in 'path' (a + string listing directories separated by 'os.pathsep'; defaults to + os.environ['PATH']). Returns the complete filename or None if not + found + """ + from .exceptions import sympy_deprecation_warning + sympy_deprecation_warning( + """ + sympy.utilities.misc.find_executable() is deprecated. Use the standard + library shutil.which() function instead. + """, + deprecated_since_version="1.7", + active_deprecations_target="deprecated-find-executable", + ) + if path is None: + path = os.environ['PATH'] + paths = path.split(os.pathsep) + extlist = [''] + if os.name == 'os2': + (base, ext) = os.path.splitext(executable) + # executable files on OS/2 can have an arbitrary extension, but + # .exe is automatically appended if no dot is present in the name + if not ext: + executable = executable + ".exe" + elif sys.platform == 'win32': + pathext = os.environ['PATHEXT'].lower().split(os.pathsep) + (base, ext) = os.path.splitext(executable) + if ext.lower() not in pathext: + extlist = pathext + for ext in extlist: + execname = executable + ext + if os.path.isfile(execname): + return execname + else: + for p in paths: + f = os.path.join(p, execname) + if os.path.isfile(f): + return f + + return None + + +def func_name(x, short=False): + """Return function name of `x` (if defined) else the `type(x)`. + If short is True and there is a shorter alias for the result, + return the alias. + + Examples + ======== + + >>> from sympy.utilities.misc import func_name + >>> from sympy import Matrix + >>> from sympy.abc import x + >>> func_name(Matrix.eye(3)) + 'MutableDenseMatrix' + >>> func_name(x < 1) + 'StrictLessThan' + >>> func_name(x < 1, short=True) + 'Lt' + """ + alias = { + 'GreaterThan': 'Ge', + 'StrictGreaterThan': 'Gt', + 'LessThan': 'Le', + 'StrictLessThan': 'Lt', + 'Equality': 'Eq', + 'Unequality': 'Ne', + } + typ = type(x) + if str(typ).startswith(">> from sympy.utilities.misc import _replace + >>> f = _replace(dict(foo='bar', d='t')) + >>> f('food') + 'bart' + >>> f = _replace({}) + >>> f('food') + 'food' + """ + if not reps: + return lambda x: x + D = lambda match: reps[match.group(0)] + pattern = _re.compile("|".join( + [_re.escape(k) for k, v in reps.items()]), _re.MULTILINE) + return lambda string: pattern.sub(D, string) + + +def replace(string, *reps): + """Return ``string`` with all keys in ``reps`` replaced with + their corresponding values, longer strings first, irrespective + of the order they are given. ``reps`` may be passed as tuples + or a single mapping. + + Examples + ======== + + >>> from sympy.utilities.misc import replace + >>> replace('foo', {'oo': 'ar', 'f': 'b'}) + 'bar' + >>> replace("spamham sha", ("spam", "eggs"), ("sha","md5")) + 'eggsham md5' + + There is no guarantee that a unique answer will be + obtained if keys in a mapping overlap (i.e. are the same + length and have some identical sequence at the + beginning/end): + + >>> reps = [ + ... ('ab', 'x'), + ... ('bc', 'y')] + >>> replace('abc', *reps) in ('xc', 'ay') + True + + References + ========== + + .. [1] https://stackoverflow.com/questions/6116978/how-to-replace-multiple-substrings-of-a-string + """ + if len(reps) == 1: + kv = reps[0] + if isinstance(kv, dict): + reps = kv + else: + return string.replace(*kv) + else: + reps = dict(reps) + return _replace(reps)(string) + + +def translate(s, a, b=None, c=None): + """Return ``s`` where characters have been replaced or deleted. + + SYNTAX + ====== + + translate(s, None, deletechars): + all characters in ``deletechars`` are deleted + translate(s, map [,deletechars]): + all characters in ``deletechars`` (if provided) are deleted + then the replacements defined by map are made; if the keys + of map are strings then the longer ones are handled first. + Multicharacter deletions should have a value of ''. + translate(s, oldchars, newchars, deletechars) + all characters in ``deletechars`` are deleted + then each character in ``oldchars`` is replaced with the + corresponding character in ``newchars`` + + Examples + ======== + + >>> from sympy.utilities.misc import translate + >>> abc = 'abc' + >>> translate(abc, None, 'a') + 'bc' + >>> translate(abc, {'a': 'x'}, 'c') + 'xb' + >>> translate(abc, {'abc': 'x', 'a': 'y'}) + 'x' + + >>> translate('abcd', 'ac', 'AC', 'd') + 'AbC' + + There is no guarantee that a unique answer will be + obtained if keys in a mapping overlap are the same + length and have some identical sequences at the + beginning/end: + + >>> translate(abc, {'ab': 'x', 'bc': 'y'}) in ('xc', 'ay') + True + """ + + mr = {} + if a is None: + if c is not None: + raise ValueError('c should be None when a=None is passed, instead got %s' % c) + if b is None: + return s + c = b + a = b = '' + else: + if isinstance(a, dict): + short = {} + for k in list(a.keys()): + if len(k) == 1 and len(a[k]) == 1: + short[k] = a.pop(k) + mr = a + c = b + if short: + a, b = [''.join(i) for i in list(zip(*short.items()))] + else: + a = b = '' + elif len(a) != len(b): + raise ValueError('oldchars and newchars have different lengths') + + if c: + val = str.maketrans('', '', c) + s = s.translate(val) + s = replace(s, mr) + n = str.maketrans(a, b) + return s.translate(n) + + +def ordinal(num): + """Return ordinal number string of num, e.g. 1 becomes 1st. + """ + # modified from https://codereview.stackexchange.com/questions/41298/producing-ordinal-numbers + n = as_int(num) + k = abs(n) % 100 + if 11 <= k <= 13: + suffix = 'th' + elif k % 10 == 1: + suffix = 'st' + elif k % 10 == 2: + suffix = 'nd' + elif k % 10 == 3: + suffix = 'rd' + else: + suffix = 'th' + return str(n) + suffix + + +def as_int(n, strict=True): + """ + Convert the argument to a builtin integer. + + The return value is guaranteed to be equal to the input. ValueError is + raised if the input has a non-integral value. When ``strict`` is True, this + uses `__index__ `_ + and when it is False it uses ``int``. + + + Examples + ======== + + >>> from sympy.utilities.misc import as_int + >>> from sympy import sqrt, S + + The function is primarily concerned with sanitizing input for + functions that need to work with builtin integers, so anything that + is unambiguously an integer should be returned as an int: + + >>> as_int(S(3)) + 3 + + Floats, being of limited precision, are not assumed to be exact and + will raise an error unless the ``strict`` flag is False. This + precision issue becomes apparent for large floating point numbers: + + >>> big = 1e23 + >>> type(big) is float + True + >>> big == int(big) + True + >>> as_int(big) + Traceback (most recent call last): + ... + ValueError: ... is not an integer + >>> as_int(big, strict=False) + 99999999999999991611392 + + Input that might be a complex representation of an integer value is + also rejected by default: + + >>> one = sqrt(3 + 2*sqrt(2)) - sqrt(2) + >>> int(one) == 1 + True + >>> as_int(one) + Traceback (most recent call last): + ... + ValueError: ... is not an integer + """ + if strict: + try: + if isinstance(n, bool): + raise TypeError + return operator.index(n) + except TypeError: + raise ValueError('%s is not an integer' % (n,)) + else: + try: + result = int(n) + except TypeError: + raise ValueError('%s is not an integer' % (n,)) + if n - result: + raise ValueError('%s is not an integer' % (n,)) + return result diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/pkgdata.py b/.venv/lib/python3.12/site-packages/sympy/utilities/pkgdata.py new file mode 100644 index 0000000000000000000000000000000000000000..8bf2065759362ee09a252d4736bd612b8d271e72 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/pkgdata.py @@ -0,0 +1,33 @@ +# This module is deprecated and will be removed. + +import sys +import os +from io import StringIO + +from sympy.utilities.decorator import deprecated + + +@deprecated( + """ + The sympy.utilities.pkgdata module and its get_resource function are + deprecated. Use the stdlib importlib.resources module instead. + """, + deprecated_since_version="1.12", + active_deprecations_target="pkgdata", +) +def get_resource(identifier, pkgname=__name__): + + mod = sys.modules[pkgname] + fn = getattr(mod, '__file__', None) + if fn is None: + raise OSError("%r has no __file__!") + path = os.path.join(os.path.dirname(fn), identifier) + loader = getattr(mod, '__loader__', None) + if loader is not None: + try: + data = loader.get_data(path) + except (OSError, AttributeError): + pass + else: + return StringIO(data.decode('utf-8')) + return open(os.path.normpath(path), 'rb') diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/pytest.py b/.venv/lib/python3.12/site-packages/sympy/utilities/pytest.py new file mode 100644 index 0000000000000000000000000000000000000000..75494c8e987c7b89bddf18febe09fdc3df2b194e --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/pytest.py @@ -0,0 +1,12 @@ +""" +.. deprecated:: 1.6 + + sympy.utilities.pytest has been renamed to sympy.testing.pytest. +""" +from sympy.utilities.exceptions import sympy_deprecation_warning + +sympy_deprecation_warning("The sympy.utilities.pytest submodule is deprecated. Use sympy.testing.pytest instead.", + deprecated_since_version="1.6", + active_deprecations_target="deprecated-sympy-utilities-submodules") + +from sympy.testing.pytest import * # noqa:F401,F403 diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/randtest.py b/.venv/lib/python3.12/site-packages/sympy/utilities/randtest.py new file mode 100644 index 0000000000000000000000000000000000000000..1bd3472ed8a89198d9e78e125f16dae1a56f6bad --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/randtest.py @@ -0,0 +1,12 @@ +""" +.. deprecated:: 1.6 + + sympy.utilities.randtest has been renamed to sympy.core.random. +""" +from sympy.utilities.exceptions import sympy_deprecation_warning + +sympy_deprecation_warning("The sympy.utilities.randtest submodule is deprecated. Use sympy.core.random instead.", + deprecated_since_version="1.6", + active_deprecations_target="deprecated-sympy-utilities-submodules") + +from sympy.core.random import * # noqa:F401,F403 diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/runtests.py b/.venv/lib/python3.12/site-packages/sympy/utilities/runtests.py new file mode 100644 index 0000000000000000000000000000000000000000..bb9f760f070be528e8cd51ab38e00bc944dda9ac --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/runtests.py @@ -0,0 +1,13 @@ +""" +.. deprecated:: 1.6 + + sympy.utilities.runtests has been renamed to sympy.testing.runtests. +""" + +from sympy.utilities.exceptions import sympy_deprecation_warning + +sympy_deprecation_warning("The sympy.utilities.runtests submodule is deprecated. Use sympy.testing.runtests instead.", + deprecated_since_version="1.6", + active_deprecations_target="deprecated-sympy-utilities-submodules") + +from sympy.testing.runtests import * # noqa: F401,F403 diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/source.py b/.venv/lib/python3.12/site-packages/sympy/utilities/source.py new file mode 100644 index 0000000000000000000000000000000000000000..71692b4aaad07df70b63d7e1eaa86c402ad03c4f --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/source.py @@ -0,0 +1,40 @@ +""" +This module adds several functions for interactive source code inspection. +""" + + +def get_class(lookup_view): + """ + Convert a string version of a class name to the object. + + For example, get_class('sympy.core.Basic') will return + class Basic located in module sympy.core + """ + if isinstance(lookup_view, str): + mod_name, func_name = get_mod_func(lookup_view) + if func_name != '': + lookup_view = getattr( + __import__(mod_name, {}, {}, ['*']), func_name) + if not callable(lookup_view): + raise AttributeError( + "'%s.%s' is not a callable." % (mod_name, func_name)) + return lookup_view + + +def get_mod_func(callback): + """ + splits the string path to a class into a string path to the module + and the name of the class. + + Examples + ======== + + >>> from sympy.utilities.source import get_mod_func + >>> get_mod_func('sympy.core.basic.Basic') + ('sympy.core.basic', 'Basic') + + """ + dot = callback.rfind('.') + if dot == -1: + return callback, '' + return callback[:dot], callback[dot + 1:] diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/tests/__init__.py b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/tests/__pycache__/__init__.cpython-312.pyc 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sympy.testing.tmpfiles import TmpFileManager + + +def get_string(dump_fn, routines, prefix="file", **kwargs): + """Wrapper for dump_fn. dump_fn writes its results to a stream object and + this wrapper returns the contents of that stream as a string. This + auxiliary function is used by many tests below. + + The header and the empty lines are not generator to facilitate the + testing of the output. + """ + output = StringIO() + dump_fn(routines, output, prefix, **kwargs) + source = output.getvalue() + output.close() + return source + + +def test_cython_wrapper_scalar_function(): + x, y, z = symbols('x,y,z') + expr = (x + y)*z + routine = make_routine("test", expr) + code_gen = CythonCodeWrapper(CCodeGen()) + source = get_string(code_gen.dump_pyx, [routine]) + + expected = ( + "cdef extern from 'file.h':\n" + " double test(double x, double y, double z)\n" + "\n" + "def test_c(double x, double y, double z):\n" + "\n" + " return test(x, y, z)") + assert source == expected + + +def test_cython_wrapper_outarg(): + from sympy.core.relational import Equality + x, y, z = symbols('x,y,z') + code_gen = CythonCodeWrapper(C99CodeGen()) + + routine = make_routine("test", Equality(z, x + y)) + source = get_string(code_gen.dump_pyx, [routine]) + expected = ( + "cdef extern from 'file.h':\n" + " void test(double x, double y, double *z)\n" + "\n" + "def test_c(double x, double y):\n" + "\n" + " cdef double z = 0\n" + " test(x, y, &z)\n" + " return z") + assert source == expected + + +def test_cython_wrapper_inoutarg(): + from sympy.core.relational import Equality + x, y, z = symbols('x,y,z') + code_gen = CythonCodeWrapper(C99CodeGen()) + routine = make_routine("test", Equality(z, x + y + z)) + source = get_string(code_gen.dump_pyx, [routine]) + expected = ( + "cdef extern from 'file.h':\n" + " void test(double x, double y, double *z)\n" + "\n" + "def test_c(double x, double y, double z):\n" + "\n" + " test(x, y, &z)\n" + " return z") + assert source == expected + + +def test_cython_wrapper_compile_flags(): + from sympy.core.relational import Equality + x, y, z = symbols('x,y,z') + routine = make_routine("test", Equality(z, x + y)) + + code_gen = CythonCodeWrapper(CCodeGen()) + + expected = """\ +from setuptools import setup +from setuptools import Extension +from Cython.Build import cythonize +cy_opts = {'compiler_directives': {'language_level': '3'}} + +ext_mods = [Extension( + 'wrapper_module_%(num)s', ['wrapper_module_%(num)s.pyx', 'wrapped_code_%(num)s.c'], + include_dirs=[], + library_dirs=[], + libraries=[], + extra_compile_args=['-std=c99'], + extra_link_args=[] +)] +setup(ext_modules=cythonize(ext_mods, **cy_opts)) +""" % {'num': CodeWrapper._module_counter} + + temp_dir = tempfile.mkdtemp() + TmpFileManager.tmp_folder(temp_dir) + setup_file_path = os.path.join(temp_dir, 'setup.py') + + code_gen._prepare_files(routine, build_dir=temp_dir) + setup_text = Path(setup_file_path).read_text() + assert setup_text == expected + + code_gen = CythonCodeWrapper(CCodeGen(), + include_dirs=['/usr/local/include', '/opt/booger/include'], + library_dirs=['/user/local/lib'], + libraries=['thelib', 'nilib'], + extra_compile_args=['-slow-math'], + extra_link_args=['-lswamp', '-ltrident'], + cythonize_options={'compiler_directives': {'boundscheck': False}} + ) + expected = """\ +from setuptools import setup +from setuptools import Extension +from Cython.Build import cythonize +cy_opts = {'compiler_directives': {'boundscheck': False}} + +ext_mods = [Extension( + 'wrapper_module_%(num)s', ['wrapper_module_%(num)s.pyx', 'wrapped_code_%(num)s.c'], + include_dirs=['/usr/local/include', '/opt/booger/include'], + library_dirs=['/user/local/lib'], + libraries=['thelib', 'nilib'], + extra_compile_args=['-slow-math', '-std=c99'], + extra_link_args=['-lswamp', '-ltrident'] +)] +setup(ext_modules=cythonize(ext_mods, **cy_opts)) +""" % {'num': CodeWrapper._module_counter} + + code_gen._prepare_files(routine, build_dir=temp_dir) + setup_text = Path(setup_file_path).read_text() + assert setup_text == expected + + expected = """\ +from setuptools import setup +from setuptools import Extension +from Cython.Build import cythonize +cy_opts = {'compiler_directives': {'boundscheck': False}} +import numpy as np + +ext_mods = [Extension( + 'wrapper_module_%(num)s', ['wrapper_module_%(num)s.pyx', 'wrapped_code_%(num)s.c'], + include_dirs=['/usr/local/include', '/opt/booger/include', np.get_include()], + library_dirs=['/user/local/lib'], + libraries=['thelib', 'nilib'], + extra_compile_args=['-slow-math', '-std=c99'], + extra_link_args=['-lswamp', '-ltrident'] +)] +setup(ext_modules=cythonize(ext_mods, **cy_opts)) +""" % {'num': CodeWrapper._module_counter} + + code_gen._need_numpy = True + code_gen._prepare_files(routine, build_dir=temp_dir) + setup_text = Path(setup_file_path).read_text() + assert setup_text == expected + + TmpFileManager.cleanup() + +def test_cython_wrapper_unique_dummyvars(): + from sympy.core.relational import Equality + from sympy.core.symbol import Dummy + x, y, z = Dummy('x'), Dummy('y'), Dummy('z') + x_id, y_id, z_id = [str(d.dummy_index) for d in [x, y, z]] + expr = Equality(z, x + y) + routine = make_routine("test", expr) + code_gen = CythonCodeWrapper(CCodeGen()) + source = get_string(code_gen.dump_pyx, [routine]) + expected_template = ( + "cdef extern from 'file.h':\n" + " void test(double x_{x_id}, double y_{y_id}, double *z_{z_id})\n" + "\n" + "def test_c(double x_{x_id}, double y_{y_id}):\n" + "\n" + " cdef double z_{z_id} = 0\n" + " test(x_{x_id}, y_{y_id}, &z_{z_id})\n" + " return z_{z_id}") + expected = expected_template.format(x_id=x_id, y_id=y_id, z_id=z_id) + assert source == expected + +def test_autowrap_dummy(): + x, y, z = symbols('x y z') + + # Uses DummyWrapper to test that codegen works as expected + + f = autowrap(x + y, backend='dummy') + assert f() == str(x + y) + assert f.args == "x, y" + assert f.returns == "nameless" + f = autowrap(Eq(z, x + y), backend='dummy') + assert f() == str(x + y) + assert f.args == "x, y" + assert f.returns == "z" + f = autowrap(Eq(z, x + y + z), backend='dummy') + assert f() == str(x + y + z) + assert f.args == "x, y, z" + assert f.returns == "z" + + +def test_autowrap_args(): + x, y, z = symbols('x y z') + + raises(CodeGenArgumentListError, lambda: autowrap(Eq(z, x + y), + backend='dummy', args=[x])) + f = autowrap(Eq(z, x + y), backend='dummy', args=[y, x]) + assert f() == str(x + y) + assert f.args == "y, x" + assert f.returns == "z" + + raises(CodeGenArgumentListError, lambda: autowrap(Eq(z, x + y + z), + backend='dummy', args=[x, y])) + f = autowrap(Eq(z, x + y + z), backend='dummy', args=[y, x, z]) + assert f() == str(x + y + z) + assert f.args == "y, x, z" + assert f.returns == "z" + + f = autowrap(Eq(z, x + y + z), backend='dummy', args=(y, x, z)) + assert f() == str(x + y + z) + assert f.args == "y, x, z" + assert f.returns == "z" + +def test_autowrap_store_files(): + x, y = symbols('x y') + tmp = tempfile.mkdtemp() + TmpFileManager.tmp_folder(tmp) + + f = autowrap(x + y, backend='dummy', tempdir=tmp) + assert f() == str(x + y) + assert os.access(tmp, os.F_OK) + + TmpFileManager.cleanup() + +def test_autowrap_store_files_issue_gh12939(): + x, y = symbols('x y') + tmp = './tmp' + saved_cwd = os.getcwd() + temp_cwd = tempfile.mkdtemp() + try: + os.chdir(temp_cwd) + f = autowrap(x + y, backend='dummy', tempdir=tmp) + assert f() == str(x + y) + assert os.access(tmp, os.F_OK) + finally: + os.chdir(saved_cwd) + shutil.rmtree(temp_cwd) + + +def test_binary_function(): + x, y = symbols('x y') + f = binary_function('f', x + y, backend='dummy') + assert f._imp_() == str(x + y) + + +def test_ufuncify_source(): + x, y, z = symbols('x,y,z') + code_wrapper = UfuncifyCodeWrapper(C99CodeGen("ufuncify")) + routine = make_routine("test", x + y + z) + source = get_string(code_wrapper.dump_c, [routine]) + expected = """\ +#include "Python.h" +#include "math.h" +#include "numpy/ndarraytypes.h" +#include "numpy/ufuncobject.h" +#include "numpy/halffloat.h" +#include "file.h" + +static PyMethodDef wrapper_module_%(num)sMethods[] = { + {NULL, NULL, 0, NULL} +}; + +#ifdef NPY_1_19_API_VERSION +static void test_ufunc(char **args, const npy_intp *dimensions, const npy_intp* steps, void* data) +#else +static void test_ufunc(char **args, npy_intp *dimensions, npy_intp* steps, void* data) +#endif +{ + npy_intp i; + npy_intp n = dimensions[0]; + char *in0 = args[0]; + char *in1 = args[1]; + char *in2 = args[2]; + char *out0 = args[3]; + npy_intp in0_step = steps[0]; + npy_intp in1_step = steps[1]; + npy_intp in2_step = steps[2]; + npy_intp out0_step = steps[3]; + for (i = 0; i < n; i++) { + *((double *)out0) = test(*(double *)in0, *(double *)in1, *(double *)in2); + in0 += in0_step; + in1 += in1_step; + in2 += in2_step; + out0 += out0_step; + } +} +PyUFuncGenericFunction test_funcs[1] = {&test_ufunc}; +static char test_types[4] = {NPY_DOUBLE, NPY_DOUBLE, NPY_DOUBLE, NPY_DOUBLE}; +static void *test_data[1] = {NULL}; + +#if PY_VERSION_HEX >= 0x03000000 +static struct PyModuleDef moduledef = { + PyModuleDef_HEAD_INIT, + "wrapper_module_%(num)s", + NULL, + -1, + wrapper_module_%(num)sMethods, + NULL, + NULL, + NULL, + NULL +}; + +PyMODINIT_FUNC PyInit_wrapper_module_%(num)s(void) +{ + PyObject *m, *d; + PyObject *ufunc0; + m = PyModule_Create(&moduledef); + if (!m) { + return NULL; + } + import_array(); + import_umath(); + d = PyModule_GetDict(m); + ufunc0 = PyUFunc_FromFuncAndData(test_funcs, test_data, test_types, 1, 3, 1, + PyUFunc_None, "wrapper_module_%(num)s", "Created in SymPy with Ufuncify", 0); + PyDict_SetItemString(d, "test", ufunc0); + Py_DECREF(ufunc0); + return m; +} +#else +PyMODINIT_FUNC initwrapper_module_%(num)s(void) +{ + PyObject *m, *d; + PyObject *ufunc0; + m = Py_InitModule("wrapper_module_%(num)s", wrapper_module_%(num)sMethods); + if (m == NULL) { + return; + } + import_array(); + import_umath(); + d = PyModule_GetDict(m); + ufunc0 = PyUFunc_FromFuncAndData(test_funcs, test_data, test_types, 1, 3, 1, + PyUFunc_None, "wrapper_module_%(num)s", "Created in SymPy with Ufuncify", 0); + PyDict_SetItemString(d, "test", ufunc0); + Py_DECREF(ufunc0); +} +#endif""" % {'num': CodeWrapper._module_counter} + assert source == expected + + +def test_ufuncify_source_multioutput(): + x, y, z = symbols('x,y,z') + var_symbols = (x, y, z) + expr = x + y**3 + 10*z**2 + code_wrapper = UfuncifyCodeWrapper(C99CodeGen("ufuncify")) + routines = [make_routine("func{}".format(i), expr.diff(var_symbols[i]), var_symbols) for i in range(len(var_symbols))] + source = get_string(code_wrapper.dump_c, routines, funcname='multitest') + expected = """\ +#include "Python.h" +#include "math.h" +#include "numpy/ndarraytypes.h" +#include "numpy/ufuncobject.h" +#include "numpy/halffloat.h" +#include "file.h" + +static PyMethodDef wrapper_module_%(num)sMethods[] = { + {NULL, NULL, 0, NULL} +}; + +#ifdef NPY_1_19_API_VERSION +static void multitest_ufunc(char **args, const npy_intp *dimensions, const npy_intp* steps, void* data) +#else +static void multitest_ufunc(char **args, npy_intp *dimensions, npy_intp* steps, void* data) +#endif +{ + npy_intp i; + npy_intp n = dimensions[0]; + char *in0 = args[0]; + char *in1 = args[1]; + char *in2 = args[2]; + char *out0 = args[3]; + char *out1 = args[4]; + char *out2 = args[5]; + npy_intp in0_step = steps[0]; + npy_intp in1_step = steps[1]; + npy_intp in2_step = steps[2]; + npy_intp out0_step = steps[3]; + npy_intp out1_step = steps[4]; + npy_intp out2_step = steps[5]; + for (i = 0; i < n; i++) { + *((double *)out0) = func0(*(double *)in0, *(double *)in1, *(double *)in2); + *((double *)out1) = func1(*(double *)in0, *(double *)in1, *(double *)in2); + *((double *)out2) = func2(*(double *)in0, *(double *)in1, *(double *)in2); + in0 += in0_step; + in1 += in1_step; + in2 += in2_step; + out0 += out0_step; + out1 += out1_step; + out2 += out2_step; + } +} +PyUFuncGenericFunction multitest_funcs[1] = {&multitest_ufunc}; +static char multitest_types[6] = {NPY_DOUBLE, NPY_DOUBLE, NPY_DOUBLE, NPY_DOUBLE, NPY_DOUBLE, NPY_DOUBLE}; +static void *multitest_data[1] = {NULL}; + +#if PY_VERSION_HEX >= 0x03000000 +static struct PyModuleDef moduledef = { + PyModuleDef_HEAD_INIT, + "wrapper_module_%(num)s", + NULL, + -1, + wrapper_module_%(num)sMethods, + NULL, + NULL, + NULL, + NULL +}; + +PyMODINIT_FUNC PyInit_wrapper_module_%(num)s(void) +{ + PyObject *m, *d; + PyObject *ufunc0; + m = PyModule_Create(&moduledef); + if (!m) { + return NULL; + } + import_array(); + import_umath(); + d = PyModule_GetDict(m); + ufunc0 = PyUFunc_FromFuncAndData(multitest_funcs, multitest_data, multitest_types, 1, 3, 3, + PyUFunc_None, "wrapper_module_%(num)s", "Created in SymPy with Ufuncify", 0); + PyDict_SetItemString(d, "multitest", ufunc0); + Py_DECREF(ufunc0); + return m; +} +#else +PyMODINIT_FUNC initwrapper_module_%(num)s(void) +{ + PyObject *m, *d; + PyObject *ufunc0; + m = Py_InitModule("wrapper_module_%(num)s", wrapper_module_%(num)sMethods); + if (m == NULL) { + return; + } + import_array(); + import_umath(); + d = PyModule_GetDict(m); + ufunc0 = PyUFunc_FromFuncAndData(multitest_funcs, multitest_data, multitest_types, 1, 3, 3, + PyUFunc_None, "wrapper_module_%(num)s", "Created in SymPy with Ufuncify", 0); + PyDict_SetItemString(d, "multitest", ufunc0); + Py_DECREF(ufunc0); +} +#endif""" % {'num': CodeWrapper._module_counter} + assert source == expected diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_codegen.py b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_codegen.py new file mode 100644 index 0000000000000000000000000000000000000000..4ccc6f9a90fb0a0bec39cea22420da8091ede740 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_codegen.py @@ -0,0 +1,1632 @@ +from io import StringIO + +from sympy.core import symbols, Eq, pi, Catalan, Lambda, Dummy +from sympy.core.relational import Equality +from sympy.core.symbol import Symbol +from sympy.functions.special.error_functions import erf +from sympy.integrals.integrals import Integral +from sympy.matrices import Matrix, MatrixSymbol +from sympy.utilities.codegen import ( + codegen, make_routine, CCodeGen, C89CodeGen, C99CodeGen, InputArgument, + CodeGenError, FCodeGen, CodeGenArgumentListError, OutputArgument, + InOutArgument) +from sympy.testing.pytest import raises +from sympy.utilities.lambdify import implemented_function + +#FIXME: Fails due to circular import in with core +# from sympy import codegen + + +def get_string(dump_fn, routines, prefix="file", header=False, empty=False): + """Wrapper for dump_fn. dump_fn writes its results to a stream object and + this wrapper returns the contents of that stream as a string. This + auxiliary function is used by many tests below. + + The header and the empty lines are not generated to facilitate the + testing of the output. + """ + output = StringIO() + dump_fn(routines, output, prefix, header, empty) + source = output.getvalue() + output.close() + return source + + +def test_Routine_argument_order(): + a, x, y, z = symbols('a x y z') + expr = (x + y)*z + raises(CodeGenArgumentListError, lambda: make_routine("test", expr, + argument_sequence=[z, x])) + raises(CodeGenArgumentListError, lambda: make_routine("test", Eq(a, + expr), argument_sequence=[z, x, y])) + r = make_routine('test', Eq(a, expr), argument_sequence=[z, x, a, y]) + assert [ arg.name for arg in r.arguments ] == [z, x, a, y] + assert [ type(arg) for arg in r.arguments ] == [ + InputArgument, InputArgument, OutputArgument, InputArgument ] + r = make_routine('test', Eq(z, expr), argument_sequence=[z, x, y]) + assert [ type(arg) for arg in r.arguments ] == [ + InOutArgument, InputArgument, InputArgument ] + + from sympy.tensor import IndexedBase, Idx + A, B = map(IndexedBase, ['A', 'B']) + m = symbols('m', integer=True) + i = Idx('i', m) + r = make_routine('test', Eq(A[i], B[i]), argument_sequence=[B, A, m]) + assert [ arg.name for arg in r.arguments ] == [B.label, A.label, m] + + expr = Integral(x*y*z, (x, 1, 2), (y, 1, 3)) + r = make_routine('test', Eq(a, expr), argument_sequence=[z, x, a, y]) + assert [ arg.name for arg in r.arguments ] == [z, x, a, y] + + +def test_empty_c_code(): + code_gen = C89CodeGen() + source = get_string(code_gen.dump_c, []) + assert source == "#include \"file.h\"\n#include \n" + + +def test_empty_c_code_with_comment(): + code_gen = C89CodeGen() + source = get_string(code_gen.dump_c, [], header=True) + assert source[:82] == ( + "/******************************************************************************\n *" + ) + # " Code generated with SymPy 0.7.2-git " + assert source[158:] == ( "*\n" + " * *\n" + " * See http://www.sympy.org/ for more information. *\n" + " * *\n" + " * This file is part of 'project' *\n" + " ******************************************************************************/\n" + "#include \"file.h\"\n" + "#include \n" + ) + + +def test_empty_c_header(): + code_gen = C99CodeGen() + source = get_string(code_gen.dump_h, []) + assert source == "#ifndef PROJECT__FILE__H\n#define PROJECT__FILE__H\n#endif\n" + + +def test_simple_c_code(): + x, y, z = symbols('x,y,z') + expr = (x + y)*z + routine = make_routine("test", expr) + code_gen = C89CodeGen() + source = get_string(code_gen.dump_c, [routine]) + expected = ( + "#include \"file.h\"\n" + "#include \n" + "double test(double x, double y, double z) {\n" + " double test_result;\n" + " test_result = z*(x + y);\n" + " return test_result;\n" + "}\n" + ) + assert source == expected + + +def test_c_code_reserved_words(): + x, y, z = symbols('if, typedef, while') + expr = (x + y) * z + routine = make_routine("test", expr) + code_gen = C99CodeGen() + source = get_string(code_gen.dump_c, [routine]) + expected = ( + "#include \"file.h\"\n" + "#include \n" + "double test(double if_, double typedef_, double while_) {\n" + " double test_result;\n" + " test_result = while_*(if_ + typedef_);\n" + " return test_result;\n" + "}\n" + ) + assert source == expected + + +def test_numbersymbol_c_code(): + routine = make_routine("test", pi**Catalan) + code_gen = C89CodeGen() + source = get_string(code_gen.dump_c, [routine]) + expected = ( + "#include \"file.h\"\n" + "#include \n" + "double test() {\n" + " double test_result;\n" + " double const Catalan = %s;\n" + " test_result = pow(M_PI, Catalan);\n" + " return test_result;\n" + "}\n" + ) % Catalan.evalf(17) + assert source == expected + + +def test_c_code_argument_order(): + x, y, z = symbols('x,y,z') + expr = x + y + routine = make_routine("test", expr, argument_sequence=[z, x, y]) + code_gen = C89CodeGen() + source = get_string(code_gen.dump_c, [routine]) + expected = ( + "#include \"file.h\"\n" + "#include \n" + "double test(double z, double x, double y) {\n" + " double test_result;\n" + " test_result = x + y;\n" + " return test_result;\n" + "}\n" + ) + assert source == expected + + +def test_simple_c_header(): + x, y, z = symbols('x,y,z') + expr = (x + y)*z + routine = make_routine("test", expr) + code_gen = C89CodeGen() + source = get_string(code_gen.dump_h, [routine]) + expected = ( + "#ifndef PROJECT__FILE__H\n" + "#define PROJECT__FILE__H\n" + "double test(double x, double y, double z);\n" + "#endif\n" + ) + assert source == expected + + +def test_simple_c_codegen(): + x, y, z = symbols('x,y,z') + expr = (x + y)*z + expected = [ + ("file.c", + "#include \"file.h\"\n" + "#include \n" + "double test(double x, double y, double z) {\n" + " double test_result;\n" + " test_result = z*(x + y);\n" + " return test_result;\n" + "}\n"), + ("file.h", + "#ifndef PROJECT__FILE__H\n" + "#define PROJECT__FILE__H\n" + "double test(double x, double y, double z);\n" + "#endif\n") + ] + result = codegen(("test", expr), "C", "file", header=False, empty=False) + assert result == expected + + +def test_multiple_results_c(): + x, y, z = symbols('x,y,z') + expr1 = (x + y)*z + expr2 = (x - y)*z + routine = make_routine( + "test", + [expr1, expr2] + ) + code_gen = C99CodeGen() + raises(CodeGenError, lambda: get_string(code_gen.dump_h, [routine])) + + +def test_no_results_c(): + raises(ValueError, lambda: make_routine("test", [])) + + +def test_ansi_math1_codegen(): + # not included: log10 + from sympy.functions.elementary.complexes import Abs + from sympy.functions.elementary.exponential import log + from sympy.functions.elementary.hyperbolic import (cosh, sinh, tanh) + from sympy.functions.elementary.integers import (ceiling, floor) + from sympy.functions.elementary.miscellaneous import sqrt + from sympy.functions.elementary.trigonometric import (acos, asin, atan, cos, sin, tan) + x = symbols('x') + name_expr = [ + ("test_fabs", Abs(x)), + ("test_acos", acos(x)), + ("test_asin", asin(x)), + ("test_atan", atan(x)), + ("test_ceil", ceiling(x)), + ("test_cos", cos(x)), + ("test_cosh", cosh(x)), + ("test_floor", floor(x)), + ("test_log", log(x)), + ("test_ln", log(x)), + ("test_sin", sin(x)), + ("test_sinh", sinh(x)), + ("test_sqrt", sqrt(x)), + ("test_tan", tan(x)), + ("test_tanh", tanh(x)), + ] + result = codegen(name_expr, "C89", "file", header=False, empty=False) + assert result[0][0] == "file.c" + assert result[0][1] == ( + '#include "file.h"\n#include \n' + 'double test_fabs(double x) {\n double test_fabs_result;\n test_fabs_result = fabs(x);\n return test_fabs_result;\n}\n' + 'double test_acos(double x) {\n double test_acos_result;\n test_acos_result = acos(x);\n return test_acos_result;\n}\n' + 'double test_asin(double x) {\n double test_asin_result;\n test_asin_result = asin(x);\n return test_asin_result;\n}\n' + 'double test_atan(double x) {\n double test_atan_result;\n test_atan_result = atan(x);\n return test_atan_result;\n}\n' + 'double test_ceil(double x) {\n double test_ceil_result;\n test_ceil_result = ceil(x);\n return test_ceil_result;\n}\n' + 'double test_cos(double x) {\n double test_cos_result;\n test_cos_result = cos(x);\n return test_cos_result;\n}\n' + 'double test_cosh(double x) {\n double test_cosh_result;\n test_cosh_result = cosh(x);\n return test_cosh_result;\n}\n' + 'double test_floor(double x) {\n double test_floor_result;\n test_floor_result = floor(x);\n return test_floor_result;\n}\n' + 'double test_log(double x) {\n double test_log_result;\n test_log_result = log(x);\n return test_log_result;\n}\n' + 'double test_ln(double x) {\n double test_ln_result;\n test_ln_result = log(x);\n return test_ln_result;\n}\n' + 'double test_sin(double x) {\n double test_sin_result;\n test_sin_result = sin(x);\n return test_sin_result;\n}\n' + 'double test_sinh(double x) {\n double test_sinh_result;\n test_sinh_result = sinh(x);\n return test_sinh_result;\n}\n' + 'double test_sqrt(double x) {\n double test_sqrt_result;\n test_sqrt_result = sqrt(x);\n return test_sqrt_result;\n}\n' + 'double test_tan(double x) {\n double test_tan_result;\n test_tan_result = tan(x);\n return test_tan_result;\n}\n' + 'double test_tanh(double x) {\n double test_tanh_result;\n test_tanh_result = tanh(x);\n return test_tanh_result;\n}\n' + ) + assert result[1][0] == "file.h" + assert result[1][1] == ( + '#ifndef PROJECT__FILE__H\n#define PROJECT__FILE__H\n' + 'double test_fabs(double x);\ndouble test_acos(double x);\n' + 'double test_asin(double x);\ndouble test_atan(double x);\n' + 'double test_ceil(double x);\ndouble test_cos(double x);\n' + 'double test_cosh(double x);\ndouble test_floor(double x);\n' + 'double test_log(double x);\ndouble test_ln(double x);\n' + 'double test_sin(double x);\ndouble test_sinh(double x);\n' + 'double test_sqrt(double x);\ndouble test_tan(double x);\n' + 'double test_tanh(double x);\n#endif\n' + ) + + +def test_ansi_math2_codegen(): + # not included: frexp, ldexp, modf, fmod + from sympy.functions.elementary.trigonometric import atan2 + x, y = symbols('x,y') + name_expr = [ + ("test_atan2", atan2(x, y)), + ("test_pow", x**y), + ] + result = codegen(name_expr, "C89", "file", header=False, empty=False) + assert result[0][0] == "file.c" + assert result[0][1] == ( + '#include "file.h"\n#include \n' + 'double test_atan2(double x, double y) {\n double test_atan2_result;\n test_atan2_result = atan2(x, y);\n return test_atan2_result;\n}\n' + 'double test_pow(double x, double y) {\n double test_pow_result;\n test_pow_result = pow(x, y);\n return test_pow_result;\n}\n' + ) + assert result[1][0] == "file.h" + assert result[1][1] == ( + '#ifndef PROJECT__FILE__H\n#define PROJECT__FILE__H\n' + 'double test_atan2(double x, double y);\n' + 'double test_pow(double x, double y);\n' + '#endif\n' + ) + + +def test_complicated_codegen(): + from sympy.functions.elementary.trigonometric import (cos, sin, tan) + x, y, z = symbols('x,y,z') + name_expr = [ + ("test1", ((sin(x) + cos(y) + tan(z))**7).expand()), + ("test2", cos(cos(cos(cos(cos(cos(cos(cos(x + y + z))))))))), + ] + result = codegen(name_expr, "C89", "file", header=False, empty=False) + assert result[0][0] == "file.c" + assert result[0][1] == ( + '#include "file.h"\n#include \n' + 'double test1(double x, double y, double z) {\n' + ' double test1_result;\n' + ' test1_result = ' + 'pow(sin(x), 7) + ' + '7*pow(sin(x), 6)*cos(y) + ' + '7*pow(sin(x), 6)*tan(z) + ' + '21*pow(sin(x), 5)*pow(cos(y), 2) + ' + '42*pow(sin(x), 5)*cos(y)*tan(z) + ' + '21*pow(sin(x), 5)*pow(tan(z), 2) + ' + '35*pow(sin(x), 4)*pow(cos(y), 3) + ' + '105*pow(sin(x), 4)*pow(cos(y), 2)*tan(z) + ' + '105*pow(sin(x), 4)*cos(y)*pow(tan(z), 2) + ' + '35*pow(sin(x), 4)*pow(tan(z), 3) + ' + '35*pow(sin(x), 3)*pow(cos(y), 4) + ' + '140*pow(sin(x), 3)*pow(cos(y), 3)*tan(z) + ' + '210*pow(sin(x), 3)*pow(cos(y), 2)*pow(tan(z), 2) + ' + '140*pow(sin(x), 3)*cos(y)*pow(tan(z), 3) + ' + '35*pow(sin(x), 3)*pow(tan(z), 4) + ' + '21*pow(sin(x), 2)*pow(cos(y), 5) + ' + '105*pow(sin(x), 2)*pow(cos(y), 4)*tan(z) + ' + '210*pow(sin(x), 2)*pow(cos(y), 3)*pow(tan(z), 2) + ' + '210*pow(sin(x), 2)*pow(cos(y), 2)*pow(tan(z), 3) + ' + '105*pow(sin(x), 2)*cos(y)*pow(tan(z), 4) + ' + '21*pow(sin(x), 2)*pow(tan(z), 5) + ' + '7*sin(x)*pow(cos(y), 6) + ' + '42*sin(x)*pow(cos(y), 5)*tan(z) + ' + '105*sin(x)*pow(cos(y), 4)*pow(tan(z), 2) + ' + '140*sin(x)*pow(cos(y), 3)*pow(tan(z), 3) + ' + '105*sin(x)*pow(cos(y), 2)*pow(tan(z), 4) + ' + '42*sin(x)*cos(y)*pow(tan(z), 5) + ' + '7*sin(x)*pow(tan(z), 6) + ' + 'pow(cos(y), 7) + ' + '7*pow(cos(y), 6)*tan(z) + ' + '21*pow(cos(y), 5)*pow(tan(z), 2) + ' + '35*pow(cos(y), 4)*pow(tan(z), 3) + ' + '35*pow(cos(y), 3)*pow(tan(z), 4) + ' + '21*pow(cos(y), 2)*pow(tan(z), 5) + ' + '7*cos(y)*pow(tan(z), 6) + ' + 'pow(tan(z), 7);\n' + ' return test1_result;\n' + '}\n' + 'double test2(double x, double y, double z) {\n' + ' double test2_result;\n' + ' test2_result = cos(cos(cos(cos(cos(cos(cos(cos(x + y + z))))))));\n' + ' return test2_result;\n' + '}\n' + ) + assert result[1][0] == "file.h" + assert result[1][1] == ( + '#ifndef PROJECT__FILE__H\n' + '#define PROJECT__FILE__H\n' + 'double test1(double x, double y, double z);\n' + 'double test2(double x, double y, double z);\n' + '#endif\n' + ) + + +def test_loops_c(): + from sympy.tensor import IndexedBase, Idx + from sympy.core.symbol import symbols + n, m = symbols('n m', integer=True) + A = IndexedBase('A') + x = IndexedBase('x') + y = IndexedBase('y') + i = Idx('i', m) + j = Idx('j', n) + + (f1, code), (f2, interface) = codegen( + ('matrix_vector', Eq(y[i], A[i, j]*x[j])), "C99", "file", header=False, empty=False) + + assert f1 == 'file.c' + expected = ( + '#include "file.h"\n' + '#include \n' + 'void matrix_vector(double *A, int m, int n, double *x, double *y) {\n' + ' for (int i=0; i\n' + 'void test_dummies(int m_%(mno)i, double *x, double *y) {\n' + ' for (int i_%(ino)i=0; i_%(ino)i\n' + 'void matrix_vector(double *A, int m, int n, int o, int p, double *x, double *y) {\n' + ' for (int i=o; i<%(upperi)s; i++){\n' + ' y[i] = 0;\n' + ' }\n' + ' for (int i=o; i<%(upperi)s; i++){\n' + ' for (int j=0; j\n' + 'double foo(double x, double *y) {\n' + ' (*y) = sin(x);\n' + ' double foo_result;\n' + ' foo_result = cos(x);\n' + ' return foo_result;\n' + '}\n' + ) + assert result[0][1] == expected + + +def test_output_arg_c_reserved_words(): + from sympy.core.relational import Equality + from sympy.functions.elementary.trigonometric import (cos, sin) + x, y, z = symbols("if, while, z") + r = make_routine("foo", [Equality(y, sin(x)), cos(x)]) + c = C89CodeGen() + result = c.write([r], "test", header=False, empty=False) + assert result[0][0] == "test.c" + expected = ( + '#include "test.h"\n' + '#include \n' + 'double foo(double if_, double *while_) {\n' + ' (*while_) = sin(if_);\n' + ' double foo_result;\n' + ' foo_result = cos(if_);\n' + ' return foo_result;\n' + '}\n' + ) + assert result[0][1] == expected + + +def test_multidim_c_argument_cse(): + A_sym = MatrixSymbol('A', 3, 3) + b_sym = MatrixSymbol('b', 3, 1) + A = Matrix(A_sym) + b = Matrix(b_sym) + c = A*b + cgen = CCodeGen(project="test", cse=True) + r = cgen.routine("c", c) + r.arguments[-1].result_var = "out" + r.arguments[-1]._name = "out" + code = get_string(cgen.dump_c, [r], prefix="test") + expected = ( + '#include "test.h"\n' + "#include \n" + "void c(double *A, double *b, double *out) {\n" + " out[0] = A[0]*b[0] + A[1]*b[1] + A[2]*b[2];\n" + " out[1] = A[3]*b[0] + A[4]*b[1] + A[5]*b[2];\n" + " out[2] = A[6]*b[0] + A[7]*b[1] + A[8]*b[2];\n" + "}\n" + ) + assert code == expected + + +def test_ccode_results_named_ordered(): + x, y, z = symbols('x,y,z') + B, C = symbols('B,C') + A = MatrixSymbol('A', 1, 3) + expr1 = Equality(A, Matrix([[1, 2, x]])) + expr2 = Equality(C, (x + y)*z) + expr3 = Equality(B, 2*x) + name_expr = ("test", [expr1, expr2, expr3]) + expected = ( + '#include "test.h"\n' + '#include \n' + 'void test(double x, double *C, double z, double y, double *A, double *B) {\n' + ' (*C) = z*(x + y);\n' + ' A[0] = 1;\n' + ' A[1] = 2;\n' + ' A[2] = x;\n' + ' (*B) = 2*x;\n' + '}\n' + ) + + result = codegen(name_expr, "c", "test", header=False, empty=False, + argument_sequence=(x, C, z, y, A, B)) + source = result[0][1] + assert source == expected + + +def test_ccode_matrixsymbol_slice(): + A = MatrixSymbol('A', 5, 3) + B = MatrixSymbol('B', 1, 3) + C = MatrixSymbol('C', 1, 3) + D = MatrixSymbol('D', 5, 1) + name_expr = ("test", [Equality(B, A[0, :]), + Equality(C, A[1, :]), + Equality(D, A[:, 2])]) + result = codegen(name_expr, "c99", "test", header=False, empty=False) + source = result[0][1] + expected = ( + '#include "test.h"\n' + '#include \n' + 'void test(double *A, double *B, double *C, double *D) {\n' + ' B[0] = A[0];\n' + ' B[1] = A[1];\n' + ' B[2] = A[2];\n' + ' C[0] = A[3];\n' + ' C[1] = A[4];\n' + ' C[2] = A[5];\n' + ' D[0] = A[2];\n' + ' D[1] = A[5];\n' + ' D[2] = A[8];\n' + ' D[3] = A[11];\n' + ' D[4] = A[14];\n' + '}\n' + ) + assert source == expected + +def test_ccode_cse(): + a, b, c, d = symbols('a b c d') + e = MatrixSymbol('e', 3, 1) + name_expr = ("test", [Equality(e, Matrix([[a*b], [a*b + c*d], [a*b*c*d]]))]) + generator = CCodeGen(cse=True) + result = codegen(name_expr, code_gen=generator, header=False, empty=False) + source = result[0][1] + expected = ( + '#include "test.h"\n' + '#include \n' + 'void test(double a, double b, double c, double d, double *e) {\n' + ' const double x0 = a*b;\n' + ' const double x1 = c*d;\n' + ' e[0] = x0;\n' + ' e[1] = x0 + x1;\n' + ' e[2] = x0*x1;\n' + '}\n' + ) + assert source == expected + +def test_ccode_unused_array_arg(): + x = MatrixSymbol('x', 2, 1) + # x does not appear in output + name_expr = ("test", 1.0) + generator = CCodeGen() + result = codegen(name_expr, code_gen=generator, header=False, empty=False, argument_sequence=(x,)) + source = result[0][1] + # note: x should appear as (double *) + expected = ( + '#include "test.h"\n' + '#include \n' + 'double test(double *x) {\n' + ' double test_result;\n' + ' test_result = 1.0;\n' + ' return test_result;\n' + '}\n' + ) + assert source == expected + +def test_ccode_unused_array_arg_func(): + # issue 16689 + X = MatrixSymbol('X',3,1) + Y = MatrixSymbol('Y',3,1) + z = symbols('z',integer = True) + name_expr = ('testBug', X[0] + X[1]) + result = codegen(name_expr, language='C', header=False, empty=False, argument_sequence=(X, Y, z)) + source = result[0][1] + expected = ( + '#include "testBug.h"\n' + '#include \n' + 'double testBug(double *X, double *Y, int z) {\n' + ' double testBug_result;\n' + ' testBug_result = X[0] + X[1];\n' + ' return testBug_result;\n' + '}\n' + ) + assert source == expected + +def test_empty_f_code(): + code_gen = FCodeGen() + source = get_string(code_gen.dump_f95, []) + assert source == "" + + +def test_empty_f_code_with_header(): + code_gen = FCodeGen() + source = get_string(code_gen.dump_f95, [], header=True) + assert source[:82] == ( + "!******************************************************************************\n!*" + ) + # " Code generated with SymPy 0.7.2-git " + assert source[158:] == ( "*\n" + "!* *\n" + "!* See http://www.sympy.org/ for more information. *\n" + "!* *\n" + "!* This file is part of 'project' *\n" + "!******************************************************************************\n" + ) + + +def test_empty_f_header(): + code_gen = FCodeGen() + source = get_string(code_gen.dump_h, []) + assert source == "" + + +def test_simple_f_code(): + x, y, z = symbols('x,y,z') + expr = (x + y)*z + routine = make_routine("test", expr) + code_gen = FCodeGen() + source = get_string(code_gen.dump_f95, [routine]) + expected = ( + "REAL*8 function test(x, y, z)\n" + "implicit none\n" + "REAL*8, intent(in) :: x\n" + "REAL*8, intent(in) :: y\n" + "REAL*8, intent(in) :: z\n" + "test = z*(x + y)\n" + "end function\n" + ) + assert source == expected + + +def test_numbersymbol_f_code(): + routine = make_routine("test", pi**Catalan) + code_gen = FCodeGen() + source = get_string(code_gen.dump_f95, [routine]) + expected = ( + "REAL*8 function test()\n" + "implicit none\n" + "REAL*8, parameter :: Catalan = %sd0\n" + "REAL*8, parameter :: pi = %sd0\n" + "test = pi**Catalan\n" + "end function\n" + ) % (Catalan.evalf(17), pi.evalf(17)) + assert source == expected + +def test_erf_f_code(): + x = symbols('x') + routine = make_routine("test", erf(x) - erf(-2 * x)) + code_gen = FCodeGen() + source = get_string(code_gen.dump_f95, [routine]) + expected = ( + "REAL*8 function test(x)\n" + "implicit none\n" + "REAL*8, intent(in) :: x\n" + "test = erf(x) + erf(2.0d0*x)\n" + "end function\n" + ) + assert source == expected, source + +def test_f_code_argument_order(): + x, y, z = symbols('x,y,z') + expr = x + y + routine = make_routine("test", expr, argument_sequence=[z, x, y]) + code_gen = FCodeGen() + source = get_string(code_gen.dump_f95, [routine]) + expected = ( + "REAL*8 function test(z, x, y)\n" + "implicit none\n" + "REAL*8, intent(in) :: z\n" + "REAL*8, intent(in) :: x\n" + "REAL*8, intent(in) :: y\n" + "test = x + y\n" + "end function\n" + ) + assert source == expected + + +def test_simple_f_header(): + x, y, z = symbols('x,y,z') + expr = (x + y)*z + routine = make_routine("test", expr) + code_gen = FCodeGen() + source = get_string(code_gen.dump_h, [routine]) + expected = ( + "interface\n" + "REAL*8 function test(x, y, z)\n" + "implicit none\n" + "REAL*8, intent(in) :: x\n" + "REAL*8, intent(in) :: y\n" + "REAL*8, intent(in) :: z\n" + "end function\n" + "end interface\n" + ) + assert source == expected + + +def test_simple_f_codegen(): + x, y, z = symbols('x,y,z') + expr = (x + y)*z + result = codegen( + ("test", expr), "F95", "file", header=False, empty=False) + expected = [ + ("file.f90", + "REAL*8 function test(x, y, z)\n" + "implicit none\n" + "REAL*8, intent(in) :: x\n" + "REAL*8, intent(in) :: y\n" + "REAL*8, intent(in) :: z\n" + "test = z*(x + y)\n" + "end function\n"), + ("file.h", + "interface\n" + "REAL*8 function test(x, y, z)\n" + "implicit none\n" + "REAL*8, intent(in) :: x\n" + "REAL*8, intent(in) :: y\n" + "REAL*8, intent(in) :: z\n" + "end function\n" + "end interface\n") + ] + assert result == expected + + +def test_multiple_results_f(): + x, y, z = symbols('x,y,z') + expr1 = (x + y)*z + expr2 = (x - y)*z + routine = make_routine( + "test", + [expr1, expr2] + ) + code_gen = FCodeGen() + raises(CodeGenError, lambda: get_string(code_gen.dump_h, [routine])) + + +def test_no_results_f(): + raises(ValueError, lambda: make_routine("test", [])) + + +def test_intrinsic_math_codegen(): + # not included: log10 + from sympy.functions.elementary.complexes import Abs + from sympy.functions.elementary.exponential import log + from sympy.functions.elementary.hyperbolic import (cosh, sinh, tanh) + from sympy.functions.elementary.miscellaneous import sqrt + from sympy.functions.elementary.trigonometric import (acos, asin, atan, cos, sin, tan) + x = symbols('x') + name_expr = [ + ("test_abs", Abs(x)), + ("test_acos", acos(x)), + ("test_asin", asin(x)), + ("test_atan", atan(x)), + ("test_cos", cos(x)), + ("test_cosh", cosh(x)), + ("test_log", log(x)), + ("test_ln", log(x)), + ("test_sin", sin(x)), + ("test_sinh", sinh(x)), + ("test_sqrt", sqrt(x)), + ("test_tan", tan(x)), + ("test_tanh", tanh(x)), + ] + result = codegen(name_expr, "F95", "file", header=False, empty=False) + assert result[0][0] == "file.f90" + expected = ( + 'REAL*8 function test_abs(x)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'test_abs = abs(x)\n' + 'end function\n' + 'REAL*8 function test_acos(x)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'test_acos = acos(x)\n' + 'end function\n' + 'REAL*8 function test_asin(x)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'test_asin = asin(x)\n' + 'end function\n' + 'REAL*8 function test_atan(x)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'test_atan = atan(x)\n' + 'end function\n' + 'REAL*8 function test_cos(x)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'test_cos = cos(x)\n' + 'end function\n' + 'REAL*8 function test_cosh(x)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'test_cosh = cosh(x)\n' + 'end function\n' + 'REAL*8 function test_log(x)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'test_log = log(x)\n' + 'end function\n' + 'REAL*8 function test_ln(x)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'test_ln = log(x)\n' + 'end function\n' + 'REAL*8 function test_sin(x)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'test_sin = sin(x)\n' + 'end function\n' + 'REAL*8 function test_sinh(x)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'test_sinh = sinh(x)\n' + 'end function\n' + 'REAL*8 function test_sqrt(x)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'test_sqrt = sqrt(x)\n' + 'end function\n' + 'REAL*8 function test_tan(x)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'test_tan = tan(x)\n' + 'end function\n' + 'REAL*8 function test_tanh(x)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'test_tanh = tanh(x)\n' + 'end function\n' + ) + assert result[0][1] == expected + + assert result[1][0] == "file.h" + expected = ( + 'interface\n' + 'REAL*8 function test_abs(x)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'end function\n' + 'end interface\n' + 'interface\n' + 'REAL*8 function test_acos(x)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'end function\n' + 'end interface\n' + 'interface\n' + 'REAL*8 function test_asin(x)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'end function\n' + 'end interface\n' + 'interface\n' + 'REAL*8 function test_atan(x)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'end function\n' + 'end interface\n' + 'interface\n' + 'REAL*8 function test_cos(x)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'end function\n' + 'end interface\n' + 'interface\n' + 'REAL*8 function test_cosh(x)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'end function\n' + 'end interface\n' + 'interface\n' + 'REAL*8 function test_log(x)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'end function\n' + 'end interface\n' + 'interface\n' + 'REAL*8 function test_ln(x)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'end function\n' + 'end interface\n' + 'interface\n' + 'REAL*8 function test_sin(x)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'end function\n' + 'end interface\n' + 'interface\n' + 'REAL*8 function test_sinh(x)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'end function\n' + 'end interface\n' + 'interface\n' + 'REAL*8 function test_sqrt(x)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'end function\n' + 'end interface\n' + 'interface\n' + 'REAL*8 function test_tan(x)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'end function\n' + 'end interface\n' + 'interface\n' + 'REAL*8 function test_tanh(x)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'end function\n' + 'end interface\n' + ) + assert result[1][1] == expected + + +def test_intrinsic_math2_codegen(): + # not included: frexp, ldexp, modf, fmod + from sympy.functions.elementary.trigonometric import atan2 + x, y = symbols('x,y') + name_expr = [ + ("test_atan2", atan2(x, y)), + ("test_pow", x**y), + ] + result = codegen(name_expr, "F95", "file", header=False, empty=False) + assert result[0][0] == "file.f90" + expected = ( + 'REAL*8 function test_atan2(x, y)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'REAL*8, intent(in) :: y\n' + 'test_atan2 = atan2(x, y)\n' + 'end function\n' + 'REAL*8 function test_pow(x, y)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'REAL*8, intent(in) :: y\n' + 'test_pow = x**y\n' + 'end function\n' + ) + assert result[0][1] == expected + + assert result[1][0] == "file.h" + expected = ( + 'interface\n' + 'REAL*8 function test_atan2(x, y)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'REAL*8, intent(in) :: y\n' + 'end function\n' + 'end interface\n' + 'interface\n' + 'REAL*8 function test_pow(x, y)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'REAL*8, intent(in) :: y\n' + 'end function\n' + 'end interface\n' + ) + assert result[1][1] == expected + + +def test_complicated_codegen_f95(): + from sympy.functions.elementary.trigonometric import (cos, sin, tan) + x, y, z = symbols('x,y,z') + name_expr = [ + ("test1", ((sin(x) + cos(y) + tan(z))**7).expand()), + ("test2", cos(cos(cos(cos(cos(cos(cos(cos(x + y + z))))))))), + ] + result = codegen(name_expr, "F95", "file", header=False, empty=False) + assert result[0][0] == "file.f90" + expected = ( + 'REAL*8 function test1(x, y, z)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'REAL*8, intent(in) :: y\n' + 'REAL*8, intent(in) :: z\n' + 'test1 = sin(x)**7 + 7*sin(x)**6*cos(y) + 7*sin(x)**6*tan(z) + 21*sin(x) &\n' + ' **5*cos(y)**2 + 42*sin(x)**5*cos(y)*tan(z) + 21*sin(x)**5*tan(z) &\n' + ' **2 + 35*sin(x)**4*cos(y)**3 + 105*sin(x)**4*cos(y)**2*tan(z) + &\n' + ' 105*sin(x)**4*cos(y)*tan(z)**2 + 35*sin(x)**4*tan(z)**3 + 35*sin( &\n' + ' x)**3*cos(y)**4 + 140*sin(x)**3*cos(y)**3*tan(z) + 210*sin(x)**3* &\n' + ' cos(y)**2*tan(z)**2 + 140*sin(x)**3*cos(y)*tan(z)**3 + 35*sin(x) &\n' + ' **3*tan(z)**4 + 21*sin(x)**2*cos(y)**5 + 105*sin(x)**2*cos(y)**4* &\n' + ' tan(z) + 210*sin(x)**2*cos(y)**3*tan(z)**2 + 210*sin(x)**2*cos(y) &\n' + ' **2*tan(z)**3 + 105*sin(x)**2*cos(y)*tan(z)**4 + 21*sin(x)**2*tan &\n' + ' (z)**5 + 7*sin(x)*cos(y)**6 + 42*sin(x)*cos(y)**5*tan(z) + 105* &\n' + ' sin(x)*cos(y)**4*tan(z)**2 + 140*sin(x)*cos(y)**3*tan(z)**3 + 105 &\n' + ' *sin(x)*cos(y)**2*tan(z)**4 + 42*sin(x)*cos(y)*tan(z)**5 + 7*sin( &\n' + ' x)*tan(z)**6 + cos(y)**7 + 7*cos(y)**6*tan(z) + 21*cos(y)**5*tan( &\n' + ' z)**2 + 35*cos(y)**4*tan(z)**3 + 35*cos(y)**3*tan(z)**4 + 21*cos( &\n' + ' y)**2*tan(z)**5 + 7*cos(y)*tan(z)**6 + tan(z)**7\n' + 'end function\n' + 'REAL*8 function test2(x, y, z)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'REAL*8, intent(in) :: y\n' + 'REAL*8, intent(in) :: z\n' + 'test2 = cos(cos(cos(cos(cos(cos(cos(cos(x + y + z))))))))\n' + 'end function\n' + ) + assert result[0][1] == expected + assert result[1][0] == "file.h" + expected = ( + 'interface\n' + 'REAL*8 function test1(x, y, z)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'REAL*8, intent(in) :: y\n' + 'REAL*8, intent(in) :: z\n' + 'end function\n' + 'end interface\n' + 'interface\n' + 'REAL*8 function test2(x, y, z)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'REAL*8, intent(in) :: y\n' + 'REAL*8, intent(in) :: z\n' + 'end function\n' + 'end interface\n' + ) + assert result[1][1] == expected + + +def test_loops(): + from sympy.tensor import IndexedBase, Idx + from sympy.core.symbol import symbols + + n, m = symbols('n,m', integer=True) + A, x, y = map(IndexedBase, 'Axy') + i = Idx('i', m) + j = Idx('j', n) + + (f1, code), (f2, interface) = codegen( + ('matrix_vector', Eq(y[i], A[i, j]*x[j])), "F95", "file", header=False, empty=False) + + assert f1 == 'file.f90' + expected = ( + 'subroutine matrix_vector(A, m, n, x, y)\n' + 'implicit none\n' + 'INTEGER*4, intent(in) :: m\n' + 'INTEGER*4, intent(in) :: n\n' + 'REAL*8, intent(in), dimension(1:m, 1:n) :: A\n' + 'REAL*8, intent(in), dimension(1:n) :: x\n' + 'REAL*8, intent(out), dimension(1:m) :: y\n' + 'INTEGER*4 :: i\n' + 'INTEGER*4 :: j\n' + 'do i = 1, m\n' + ' y(i) = 0\n' + 'end do\n' + 'do i = 1, m\n' + ' do j = 1, n\n' + ' y(i) = %(rhs)s + y(i)\n' + ' end do\n' + 'end do\n' + 'end subroutine\n' + ) + + assert code == expected % {'rhs': 'A(i, j)*x(j)'} or\ + code == expected % {'rhs': 'x(j)*A(i, j)'} + assert f2 == 'file.h' + assert interface == ( + 'interface\n' + 'subroutine matrix_vector(A, m, n, x, y)\n' + 'implicit none\n' + 'INTEGER*4, intent(in) :: m\n' + 'INTEGER*4, intent(in) :: n\n' + 'REAL*8, intent(in), dimension(1:m, 1:n) :: A\n' + 'REAL*8, intent(in), dimension(1:n) :: x\n' + 'REAL*8, intent(out), dimension(1:m) :: y\n' + 'end subroutine\n' + 'end interface\n' + ) + + +def test_dummy_loops_f95(): + from sympy.tensor import IndexedBase, Idx + i, m = symbols('i m', integer=True, cls=Dummy) + x = IndexedBase('x') + y = IndexedBase('y') + i = Idx(i, m) + expected = ( + 'subroutine test_dummies(m_%(mcount)i, x, y)\n' + 'implicit none\n' + 'INTEGER*4, intent(in) :: m_%(mcount)i\n' + 'REAL*8, intent(in), dimension(1:m_%(mcount)i) :: x\n' + 'REAL*8, intent(out), dimension(1:m_%(mcount)i) :: y\n' + 'INTEGER*4 :: i_%(icount)i\n' + 'do i_%(icount)i = 1, m_%(mcount)i\n' + ' y(i_%(icount)i) = x(i_%(icount)i)\n' + 'end do\n' + 'end subroutine\n' + ) % {'icount': i.label.dummy_index, 'mcount': m.dummy_index} + r = make_routine('test_dummies', Eq(y[i], x[i])) + c = FCodeGen() + code = get_string(c.dump_f95, [r]) + assert code == expected + + +def test_loops_InOut(): + from sympy.tensor import IndexedBase, Idx + from sympy.core.symbol import symbols + + i, j, n, m = symbols('i,j,n,m', integer=True) + A, x, y = symbols('A,x,y') + A = IndexedBase(A)[Idx(i, m), Idx(j, n)] + x = IndexedBase(x)[Idx(j, n)] + y = IndexedBase(y)[Idx(i, m)] + + (f1, code), (f2, interface) = codegen( + ('matrix_vector', Eq(y, y + A*x)), "F95", "file", header=False, empty=False) + + assert f1 == 'file.f90' + expected = ( + 'subroutine matrix_vector(A, m, n, x, y)\n' + 'implicit none\n' + 'INTEGER*4, intent(in) :: m\n' + 'INTEGER*4, intent(in) :: n\n' + 'REAL*8, intent(in), dimension(1:m, 1:n) :: A\n' + 'REAL*8, intent(in), dimension(1:n) :: x\n' + 'REAL*8, intent(inout), dimension(1:m) :: y\n' + 'INTEGER*4 :: i\n' + 'INTEGER*4 :: j\n' + 'do i = 1, m\n' + ' do j = 1, n\n' + ' y(i) = %(rhs)s + y(i)\n' + ' end do\n' + 'end do\n' + 'end subroutine\n' + ) + + assert (code == expected % {'rhs': 'A(i, j)*x(j)'} or + code == expected % {'rhs': 'x(j)*A(i, j)'}) + assert f2 == 'file.h' + assert interface == ( + 'interface\n' + 'subroutine matrix_vector(A, m, n, x, y)\n' + 'implicit none\n' + 'INTEGER*4, intent(in) :: m\n' + 'INTEGER*4, intent(in) :: n\n' + 'REAL*8, intent(in), dimension(1:m, 1:n) :: A\n' + 'REAL*8, intent(in), dimension(1:n) :: x\n' + 'REAL*8, intent(inout), dimension(1:m) :: y\n' + 'end subroutine\n' + 'end interface\n' + ) + + +def test_partial_loops_f(): + # check that loop boundaries are determined by Idx, and array strides + # determined by shape of IndexedBase object. + from sympy.tensor import IndexedBase, Idx + from sympy.core.symbol import symbols + n, m, o, p = symbols('n m o p', integer=True) + A = IndexedBase('A', shape=(m, p)) + x = IndexedBase('x') + y = IndexedBase('y') + i = Idx('i', (o, m - 5)) # Note: bounds are inclusive + j = Idx('j', n) # dimension n corresponds to bounds (0, n - 1) + + (f1, code), (f2, interface) = codegen( + ('matrix_vector', Eq(y[i], A[i, j]*x[j])), "F95", "file", header=False, empty=False) + + expected = ( + 'subroutine matrix_vector(A, m, n, o, p, x, y)\n' + 'implicit none\n' + 'INTEGER*4, intent(in) :: m\n' + 'INTEGER*4, intent(in) :: n\n' + 'INTEGER*4, intent(in) :: o\n' + 'INTEGER*4, intent(in) :: p\n' + 'REAL*8, intent(in), dimension(1:m, 1:p) :: A\n' + 'REAL*8, intent(in), dimension(1:n) :: x\n' + 'REAL*8, intent(out), dimension(1:%(iup-ilow)s) :: y\n' + 'INTEGER*4 :: i\n' + 'INTEGER*4 :: j\n' + 'do i = %(ilow)s, %(iup)s\n' + ' y(i) = 0\n' + 'end do\n' + 'do i = %(ilow)s, %(iup)s\n' + ' do j = 1, n\n' + ' y(i) = %(rhs)s + y(i)\n' + ' end do\n' + 'end do\n' + 'end subroutine\n' + ) % { + 'rhs': '%(rhs)s', + 'iup': str(m - 4), + 'ilow': str(1 + o), + 'iup-ilow': str(m - 4 - o) + } + + assert code == expected % {'rhs': 'A(i, j)*x(j)'} or\ + code == expected % {'rhs': 'x(j)*A(i, j)'} + + +def test_output_arg_f(): + from sympy.core.relational import Equality + from sympy.functions.elementary.trigonometric import (cos, sin) + x, y, z = symbols("x,y,z") + r = make_routine("foo", [Equality(y, sin(x)), cos(x)]) + c = FCodeGen() + result = c.write([r], "test", header=False, empty=False) + assert result[0][0] == "test.f90" + assert result[0][1] == ( + 'REAL*8 function foo(x, y)\n' + 'implicit none\n' + 'REAL*8, intent(in) :: x\n' + 'REAL*8, intent(out) :: y\n' + 'y = sin(x)\n' + 'foo = cos(x)\n' + 'end function\n' + ) + + +def test_inline_function(): + from sympy.tensor import IndexedBase, Idx + from sympy.core.symbol import symbols + n, m = symbols('n m', integer=True) + A, x, y = map(IndexedBase, 'Axy') + i = Idx('i', m) + p = FCodeGen() + func = implemented_function('func', Lambda(n, n*(n + 1))) + routine = make_routine('test_inline', Eq(y[i], func(x[i]))) + code = get_string(p.dump_f95, [routine]) + expected = ( + 'subroutine test_inline(m, x, y)\n' + 'implicit none\n' + 'INTEGER*4, intent(in) :: m\n' + 'REAL*8, intent(in), dimension(1:m) :: x\n' + 'REAL*8, intent(out), dimension(1:m) :: y\n' + 'INTEGER*4 :: i\n' + 'do i = 1, m\n' + ' y(i) = %s*%s\n' + 'end do\n' + 'end subroutine\n' + ) + args = ('x(i)', '(x(i) + 1)') + assert code == expected % args or\ + code == expected % args[::-1] + + +def test_f_code_call_signature_wrap(): + # Issue #7934 + x = symbols('x:20') + expr = 0 + for sym in x: + expr += sym + routine = make_routine("test", expr) + code_gen = FCodeGen() + source = get_string(code_gen.dump_f95, [routine]) + expected = """\ +REAL*8 function test(x0, x1, x10, x11, x12, x13, x14, x15, x16, x17, x18, & + x19, x2, x3, x4, x5, x6, x7, x8, x9) +implicit none +REAL*8, intent(in) :: x0 +REAL*8, intent(in) :: x1 +REAL*8, intent(in) :: x10 +REAL*8, intent(in) :: x11 +REAL*8, intent(in) :: x12 +REAL*8, intent(in) :: x13 +REAL*8, intent(in) :: x14 +REAL*8, intent(in) :: x15 +REAL*8, intent(in) :: x16 +REAL*8, intent(in) :: x17 +REAL*8, intent(in) :: x18 +REAL*8, intent(in) :: x19 +REAL*8, intent(in) :: x2 +REAL*8, intent(in) :: x3 +REAL*8, intent(in) :: x4 +REAL*8, intent(in) :: x5 +REAL*8, intent(in) :: x6 +REAL*8, intent(in) :: x7 +REAL*8, intent(in) :: x8 +REAL*8, intent(in) :: x9 +test = x0 + x1 + x10 + x11 + x12 + x13 + x14 + x15 + x16 + x17 + x18 + & + x19 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 +end function +""" + assert source == expected + + +def test_check_case(): + x, X = symbols('x,X') + raises(CodeGenError, lambda: codegen(('test', x*X), 'f95', 'prefix')) + + +def test_check_case_false_positive(): + # The upper case/lower case exception should not be triggered by SymPy + # objects that differ only because of assumptions. (It may be useful to + # have a check for that as well, but here we only want to test against + # false positives with respect to case checking.) + x1 = symbols('x') + x2 = symbols('x', my_assumption=True) + try: + codegen(('test', x1*x2), 'f95', 'prefix') + except CodeGenError as e: + if e.args[0].startswith("Fortran ignores case."): + raise AssertionError("This exception should not be raised!") + + +def test_c_fortran_omit_routine_name(): + x, y = symbols("x,y") + name_expr = [("foo", 2*x)] + result = codegen(name_expr, "F95", header=False, empty=False) + expresult = codegen(name_expr, "F95", "foo", header=False, empty=False) + assert result[0][1] == expresult[0][1] + + name_expr = ("foo", x*y) + result = codegen(name_expr, "F95", header=False, empty=False) + expresult = codegen(name_expr, "F95", "foo", header=False, empty=False) + assert result[0][1] == expresult[0][1] + + name_expr = ("foo", Matrix([[x, y], [x+y, x-y]])) + result = codegen(name_expr, "C89", header=False, empty=False) + expresult = codegen(name_expr, "C89", "foo", header=False, empty=False) + assert result[0][1] == expresult[0][1] + + +def test_fcode_matrix_output(): + x, y, z = symbols('x,y,z') + e1 = x + y + e2 = Matrix([[x, y], [z, 16]]) + name_expr = ("test", (e1, e2)) + result = codegen(name_expr, "f95", "test", header=False, empty=False) + source = result[0][1] + expected = ( + "REAL*8 function test(x, y, z, out_%(hash)s)\n" + "implicit none\n" + "REAL*8, intent(in) :: x\n" + "REAL*8, intent(in) :: y\n" + "REAL*8, intent(in) :: z\n" + "REAL*8, intent(out), dimension(1:2, 1:2) :: out_%(hash)s\n" + "out_%(hash)s(1, 1) = x\n" + "out_%(hash)s(2, 1) = z\n" + "out_%(hash)s(1, 2) = y\n" + "out_%(hash)s(2, 2) = 16\n" + "test = x + y\n" + "end function\n" + ) + # look for the magic number + a = source.splitlines()[5] + b = a.split('_') + out = b[1] + expected = expected % {'hash': out} + assert source == expected + + +def test_fcode_results_named_ordered(): + x, y, z = symbols('x,y,z') + B, C = symbols('B,C') + A = MatrixSymbol('A', 1, 3) + expr1 = Equality(A, Matrix([[1, 2, x]])) + expr2 = Equality(C, (x + y)*z) + expr3 = Equality(B, 2*x) + name_expr = ("test", [expr1, expr2, expr3]) + result = codegen(name_expr, "f95", "test", header=False, empty=False, + argument_sequence=(x, z, y, C, A, B)) + source = result[0][1] + expected = ( + "subroutine test(x, z, y, C, A, B)\n" + "implicit none\n" + "REAL*8, intent(in) :: x\n" + "REAL*8, intent(in) :: z\n" + "REAL*8, intent(in) :: y\n" + "REAL*8, intent(out) :: C\n" + "REAL*8, intent(out) :: B\n" + "REAL*8, intent(out), dimension(1:1, 1:3) :: A\n" + "C = z*(x + y)\n" + "A(1, 1) = 1\n" + "A(1, 2) = 2\n" + "A(1, 3) = x\n" + "B = 2*x\n" + "end subroutine\n" + ) + assert source == expected + + +def test_fcode_matrixsymbol_slice(): + A = MatrixSymbol('A', 2, 3) + B = MatrixSymbol('B', 1, 3) + C = MatrixSymbol('C', 1, 3) + D = MatrixSymbol('D', 2, 1) + name_expr = ("test", [Equality(B, A[0, :]), + Equality(C, A[1, :]), + Equality(D, A[:, 2])]) + result = codegen(name_expr, "f95", "test", header=False, empty=False) + source = result[0][1] + expected = ( + "subroutine test(A, B, C, D)\n" + "implicit none\n" + "REAL*8, intent(in), dimension(1:2, 1:3) :: A\n" + "REAL*8, intent(out), dimension(1:1, 1:3) :: B\n" + "REAL*8, intent(out), dimension(1:1, 1:3) :: C\n" + "REAL*8, intent(out), dimension(1:2, 1:1) :: D\n" + "B(1, 1) = A(1, 1)\n" + "B(1, 2) = A(1, 2)\n" + "B(1, 3) = A(1, 3)\n" + "C(1, 1) = A(2, 1)\n" + "C(1, 2) = A(2, 2)\n" + "C(1, 3) = A(2, 3)\n" + "D(1, 1) = A(1, 3)\n" + "D(2, 1) = A(2, 3)\n" + "end subroutine\n" + ) + assert source == expected + + +def test_fcode_matrixsymbol_slice_autoname(): + # see issue #8093 + A = MatrixSymbol('A', 2, 3) + name_expr = ("test", A[:, 1]) + result = codegen(name_expr, "f95", "test", header=False, empty=False) + source = result[0][1] + expected = ( + "subroutine test(A, out_%(hash)s)\n" + "implicit none\n" + "REAL*8, intent(in), dimension(1:2, 1:3) :: A\n" + "REAL*8, intent(out), dimension(1:2, 1:1) :: out_%(hash)s\n" + "out_%(hash)s(1, 1) = A(1, 2)\n" + "out_%(hash)s(2, 1) = A(2, 2)\n" + "end subroutine\n" + ) + # look for the magic number + a = source.splitlines()[3] + b = a.split('_') + out = b[1] + expected = expected % {'hash': out} + assert source == expected + + +def test_global_vars(): + x, y, z, t = symbols("x y z t") + result = codegen(('f', x*y), "F95", header=False, empty=False, + global_vars=(y,)) + source = result[0][1] + expected = ( + "REAL*8 function f(x)\n" + "implicit none\n" + "REAL*8, intent(in) :: x\n" + "f = x*y\n" + "end function\n" + ) + assert source == expected + + expected = ( + '#include "f.h"\n' + '#include \n' + 'double f(double x, double y) {\n' + ' double f_result;\n' + ' f_result = x*y + z;\n' + ' return f_result;\n' + '}\n' + ) + result = codegen(('f', x*y+z), "C", header=False, empty=False, + global_vars=(z, t)) + source = result[0][1] + assert source == expected + +def test_custom_codegen(): + from sympy.printing.c import C99CodePrinter + from sympy.functions.elementary.exponential import exp + + printer = C99CodePrinter(settings={'user_functions': {'exp': 'fastexp'}}) + + x, y = symbols('x y') + expr = exp(x + y) + + # replace math.h with a different header + gen = C99CodeGen(printer=printer, + preprocessor_statements=['#include "fastexp.h"']) + + expected = ( + '#include "expr.h"\n' + '#include "fastexp.h"\n' + 'double expr(double x, double y) {\n' + ' double expr_result;\n' + ' expr_result = fastexp(x + y);\n' + ' return expr_result;\n' + '}\n' + ) + + result = codegen(('expr', expr), header=False, empty=False, code_gen=gen) + source = result[0][1] + assert source == expected + + # use both math.h and an external header + gen = C99CodeGen(printer=printer) + gen.preprocessor_statements.append('#include "fastexp.h"') + + expected = ( + '#include "expr.h"\n' + '#include \n' + '#include "fastexp.h"\n' + 'double expr(double x, double y) {\n' + ' double expr_result;\n' + ' expr_result = fastexp(x + y);\n' + ' return expr_result;\n' + '}\n' + ) + + result = codegen(('expr', expr), header=False, empty=False, code_gen=gen) + source = result[0][1] + assert source == expected + +def test_c_with_printer(): + # issue 13586 + from sympy.printing.c import C99CodePrinter + class CustomPrinter(C99CodePrinter): + def _print_Pow(self, expr): + return "fastpow({}, {})".format(self._print(expr.base), + self._print(expr.exp)) + + x = symbols('x') + expr = x**3 + expected =[ + ("file.c", + "#include \"file.h\"\n" + "#include \n" + "double test(double x) {\n" + " double test_result;\n" + " test_result = fastpow(x, 3);\n" + " return test_result;\n" + "}\n"), + ("file.h", + "#ifndef PROJECT__FILE__H\n" + "#define PROJECT__FILE__H\n" + "double test(double x);\n" + "#endif\n") + ] + result = codegen(("test", expr), "C","file", header=False, empty=False, printer = CustomPrinter()) + assert result == expected + + +def test_fcode_complex(): + import sympy.utilities.codegen + sympy.utilities.codegen.COMPLEX_ALLOWED = True + x = Symbol('x', real=True) + y = Symbol('y',real=True) + result = codegen(('test',x+y), 'f95', 'test', header=False, empty=False) + source = (result[0][1]) + expected = ( + "REAL*8 function test(x, y)\n" + "implicit none\n" + "REAL*8, intent(in) :: x\n" + "REAL*8, intent(in) :: y\n" + "test = x + y\n" + "end function\n") + assert source == expected + x = Symbol('x') + y = Symbol('y',real=True) + result = codegen(('test',x+y), 'f95', 'test', header=False, empty=False) + source = (result[0][1]) + expected = ( + "COMPLEX*16 function test(x, y)\n" + "implicit none\n" + "COMPLEX*16, intent(in) :: x\n" + "REAL*8, intent(in) :: y\n" + "test = x + y\n" + "end function\n" + ) + assert source==expected + sympy.utilities.codegen.COMPLEX_ALLOWED = False diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_codegen_julia.py b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_codegen_julia.py new file mode 100644 index 0000000000000000000000000000000000000000..12841cb7d476107e3866d91b998bed1f997f3901 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_codegen_julia.py @@ -0,0 +1,620 @@ +from io import StringIO + +from sympy.core import S, symbols, Eq, pi, Catalan, EulerGamma, Function +from sympy.core.relational import Equality +from sympy.functions.elementary.piecewise import Piecewise +from sympy.matrices import Matrix, MatrixSymbol +from sympy.utilities.codegen import JuliaCodeGen, codegen, make_routine +from sympy.testing.pytest import XFAIL +import sympy + + +x, y, z = symbols('x,y,z') + + +def test_empty_jl_code(): + code_gen = JuliaCodeGen() + output = StringIO() + code_gen.dump_jl([], output, "file", header=False, empty=False) + source = output.getvalue() + assert source == "" + + +def test_jl_simple_code(): + name_expr = ("test", (x + y)*z) + result, = codegen(name_expr, "Julia", header=False, empty=False) + assert result[0] == "test.jl" + source = result[1] + expected = ( + "function test(x, y, z)\n" + " out1 = z .* (x + y)\n" + " return out1\n" + "end\n" + ) + assert source == expected + + +def test_jl_simple_code_with_header(): + name_expr = ("test", (x + y)*z) + result, = codegen(name_expr, "Julia", header=True, empty=False) + assert result[0] == "test.jl" + source = result[1] + expected = ( + "# Code generated with SymPy " + sympy.__version__ + "\n" + "#\n" + "# See http://www.sympy.org/ for more information.\n" + "#\n" + "# This file is part of 'project'\n" + "function test(x, y, z)\n" + " out1 = z .* (x + y)\n" + " return out1\n" + "end\n" + ) + assert source == expected + + +def test_jl_simple_code_nameout(): + expr = Equality(z, (x + y)) + name_expr = ("test", expr) + result, = codegen(name_expr, "Julia", header=False, empty=False) + source = result[1] + expected = ( + "function test(x, y)\n" + " z = x + y\n" + " return z\n" + "end\n" + ) + assert source == expected + + +def test_jl_numbersymbol(): + name_expr = ("test", pi**Catalan) + result, = codegen(name_expr, "Julia", header=False, empty=False) + source = result[1] + expected = ( + "function test()\n" + " out1 = pi ^ catalan\n" + " return out1\n" + "end\n" + ) + assert source == expected + + +@XFAIL +def test_jl_numbersymbol_no_inline(): + # FIXME: how to pass inline=False to the JuliaCodePrinter? + name_expr = ("test", [pi**Catalan, EulerGamma]) + result, = codegen(name_expr, "Julia", header=False, + empty=False, inline=False) + source = result[1] + expected = ( + "function test()\n" + " Catalan = 0.915965594177219\n" + " EulerGamma = 0.5772156649015329\n" + " out1 = pi ^ Catalan\n" + " out2 = EulerGamma\n" + " return out1, out2\n" + "end\n" + ) + assert source == expected + + +def test_jl_code_argument_order(): + expr = x + y + routine = make_routine("test", expr, argument_sequence=[z, x, y], language="julia") + code_gen = JuliaCodeGen() + output = StringIO() + code_gen.dump_jl([routine], output, "test", header=False, empty=False) + source = output.getvalue() + expected = ( + "function test(z, x, y)\n" + " out1 = x + y\n" + " return out1\n" + "end\n" + ) + assert source == expected + + +def test_multiple_results_m(): + # Here the output order is the input order + expr1 = (x + y)*z + expr2 = (x - y)*z + name_expr = ("test", [expr1, expr2]) + result, = codegen(name_expr, "Julia", header=False, empty=False) + source = result[1] + expected = ( + "function test(x, y, z)\n" + " out1 = z .* (x + y)\n" + " out2 = z .* (x - y)\n" + " return out1, out2\n" + "end\n" + ) + assert source == expected + + +def test_results_named_unordered(): + # Here output order is based on name_expr + A, B, C = symbols('A,B,C') + expr1 = Equality(C, (x + y)*z) + expr2 = Equality(A, (x - y)*z) + expr3 = Equality(B, 2*x) + name_expr = ("test", [expr1, expr2, expr3]) + result, = codegen(name_expr, "Julia", header=False, empty=False) + source = result[1] + expected = ( + "function test(x, y, z)\n" + " C = z .* (x + y)\n" + " A = z .* (x - y)\n" + " B = 2 * x\n" + " return C, A, B\n" + "end\n" + ) + assert source == expected + + +def test_results_named_ordered(): + A, B, C = symbols('A,B,C') + expr1 = Equality(C, (x + y)*z) + expr2 = Equality(A, (x - y)*z) + expr3 = Equality(B, 2*x) + name_expr = ("test", [expr1, expr2, expr3]) + result = codegen(name_expr, "Julia", header=False, empty=False, + argument_sequence=(x, z, y)) + assert result[0][0] == "test.jl" + source = result[0][1] + expected = ( + "function test(x, z, y)\n" + " C = z .* (x + y)\n" + " A = z .* (x - y)\n" + " B = 2 * x\n" + " return C, A, B\n" + "end\n" + ) + assert source == expected + + +def test_complicated_jl_codegen(): + from sympy.functions.elementary.trigonometric import (cos, sin, tan) + name_expr = ("testlong", + [ ((sin(x) + cos(y) + tan(z))**3).expand(), + cos(cos(cos(cos(cos(cos(cos(cos(x + y + z)))))))) + ]) + result = codegen(name_expr, "Julia", header=False, empty=False) + assert result[0][0] == "testlong.jl" + source = result[0][1] + expected = ( + "function testlong(x, y, z)\n" + " out1 = sin(x) .^ 3 + 3 * sin(x) .^ 2 .* cos(y) + 3 * sin(x) .^ 2 .* tan(z)" + " + 3 * sin(x) .* cos(y) .^ 2 + 6 * sin(x) .* cos(y) .* tan(z) + 3 * sin(x) .* tan(z) .^ 2" + " + cos(y) .^ 3 + 3 * cos(y) .^ 2 .* tan(z) + 3 * cos(y) .* tan(z) .^ 2 + tan(z) .^ 3\n" + " out2 = cos(cos(cos(cos(cos(cos(cos(cos(x + y + z))))))))\n" + " return out1, out2\n" + "end\n" + ) + assert source == expected + + +def test_jl_output_arg_mixed_unordered(): + # named outputs are alphabetical, unnamed output appear in the given order + from sympy.functions.elementary.trigonometric import (cos, sin) + a = symbols("a") + name_expr = ("foo", [cos(2*x), Equality(y, sin(x)), cos(x), Equality(a, sin(2*x))]) + result, = codegen(name_expr, "Julia", header=False, empty=False) + assert result[0] == "foo.jl" + source = result[1] + expected = ( + 'function foo(x)\n' + ' out1 = cos(2 * x)\n' + ' y = sin(x)\n' + ' out3 = cos(x)\n' + ' a = sin(2 * x)\n' + ' return out1, y, out3, a\n' + 'end\n' + ) + assert source == expected + + +def test_jl_piecewise_(): + pw = Piecewise((0, x < -1), (x**2, x <= 1), (-x+2, x > 1), (1, True), evaluate=False) + name_expr = ("pwtest", pw) + result, = codegen(name_expr, "Julia", header=False, empty=False) + source = result[1] + expected = ( + "function pwtest(x)\n" + " out1 = ((x < -1) ? (0) :\n" + " (x <= 1) ? (x .^ 2) :\n" + " (x > 1) ? (2 - x) : (1))\n" + " return out1\n" + "end\n" + ) + assert source == expected + + +@XFAIL +def test_jl_piecewise_no_inline(): + # FIXME: how to pass inline=False to the JuliaCodePrinter? + pw = Piecewise((0, x < -1), (x**2, x <= 1), (-x+2, x > 1), (1, True)) + name_expr = ("pwtest", pw) + result, = codegen(name_expr, "Julia", header=False, empty=False, + inline=False) + source = result[1] + expected = ( + "function pwtest(x)\n" + " if (x < -1)\n" + " out1 = 0\n" + " elseif (x <= 1)\n" + " out1 = x .^ 2\n" + " elseif (x > 1)\n" + " out1 = -x + 2\n" + " else\n" + " out1 = 1\n" + " end\n" + " return out1\n" + "end\n" + ) + assert source == expected + + +def test_jl_multifcns_per_file(): + name_expr = [ ("foo", [2*x, 3*y]), ("bar", [y**2, 4*y]) ] + result = codegen(name_expr, "Julia", header=False, empty=False) + assert result[0][0] == "foo.jl" + source = result[0][1] + expected = ( + "function foo(x, y)\n" + " out1 = 2 * x\n" + " out2 = 3 * y\n" + " return out1, out2\n" + "end\n" + "function bar(y)\n" + " out1 = y .^ 2\n" + " out2 = 4 * y\n" + " return out1, out2\n" + "end\n" + ) + assert source == expected + + +def test_jl_multifcns_per_file_w_header(): + name_expr = [ ("foo", [2*x, 3*y]), ("bar", [y**2, 4*y]) ] + result = codegen(name_expr, "Julia", header=True, empty=False) + assert result[0][0] == "foo.jl" + source = result[0][1] + expected = ( + "# Code generated with SymPy " + sympy.__version__ + "\n" + "#\n" + "# See http://www.sympy.org/ for more information.\n" + "#\n" + "# This file is part of 'project'\n" + "function foo(x, y)\n" + " out1 = 2 * x\n" + " out2 = 3 * y\n" + " return out1, out2\n" + "end\n" + "function bar(y)\n" + " out1 = y .^ 2\n" + " out2 = 4 * y\n" + " return out1, out2\n" + "end\n" + ) + assert source == expected + + +def test_jl_filename_match_prefix(): + name_expr = [ ("foo", [2*x, 3*y]), ("bar", [y**2, 4*y]) ] + result, = codegen(name_expr, "Julia", prefix="baz", header=False, + empty=False) + assert result[0] == "baz.jl" + + +def test_jl_matrix_named(): + e2 = Matrix([[x, 2*y, pi*z]]) + name_expr = ("test", Equality(MatrixSymbol('myout1', 1, 3), e2)) + result = codegen(name_expr, "Julia", header=False, empty=False) + assert result[0][0] == "test.jl" + source = result[0][1] + expected = ( + "function test(x, y, z)\n" + " myout1 = [x 2 * y pi * z]\n" + " return myout1\n" + "end\n" + ) + assert source == expected + + +def test_jl_matrix_named_matsym(): + myout1 = MatrixSymbol('myout1', 1, 3) + e2 = Matrix([[x, 2*y, pi*z]]) + name_expr = ("test", Equality(myout1, e2, evaluate=False)) + result, = codegen(name_expr, "Julia", header=False, empty=False) + source = result[1] + expected = ( + "function test(x, y, z)\n" + " myout1 = [x 2 * y pi * z]\n" + " return myout1\n" + "end\n" + ) + assert source == expected + + +def test_jl_matrix_output_autoname(): + expr = Matrix([[x, x+y, 3]]) + name_expr = ("test", expr) + result, = codegen(name_expr, "Julia", header=False, empty=False) + source = result[1] + expected = ( + "function test(x, y)\n" + " out1 = [x x + y 3]\n" + " return out1\n" + "end\n" + ) + assert source == expected + + +def test_jl_matrix_output_autoname_2(): + e1 = (x + y) + e2 = Matrix([[2*x, 2*y, 2*z]]) + e3 = Matrix([[x], [y], [z]]) + e4 = Matrix([[x, y], [z, 16]]) + name_expr = ("test", (e1, e2, e3, e4)) + result, = codegen(name_expr, "Julia", header=False, empty=False) + source = result[1] + expected = ( + "function test(x, y, z)\n" + " out1 = x + y\n" + " out2 = [2 * x 2 * y 2 * z]\n" + " out3 = [x, y, z]\n" + " out4 = [x y;\n" + " z 16]\n" + " return out1, out2, out3, out4\n" + "end\n" + ) + assert source == expected + + +def test_jl_results_matrix_named_ordered(): + B, C = symbols('B,C') + A = MatrixSymbol('A', 1, 3) + expr1 = Equality(C, (x + y)*z) + expr2 = Equality(A, Matrix([[1, 2, x]])) + expr3 = Equality(B, 2*x) + name_expr = ("test", [expr1, expr2, expr3]) + result, = codegen(name_expr, "Julia", header=False, empty=False, + argument_sequence=(x, z, y)) + source = result[1] + expected = ( + "function test(x, z, y)\n" + " C = z .* (x + y)\n" + " A = [1 2 x]\n" + " B = 2 * x\n" + " return C, A, B\n" + "end\n" + ) + assert source == expected + + +def test_jl_matrixsymbol_slice(): + A = MatrixSymbol('A', 2, 3) + B = MatrixSymbol('B', 1, 3) + C = MatrixSymbol('C', 1, 3) + D = MatrixSymbol('D', 2, 1) + name_expr = ("test", [Equality(B, A[0, :]), + Equality(C, A[1, :]), + Equality(D, A[:, 2])]) + result, = codegen(name_expr, "Julia", header=False, empty=False) + source = result[1] + expected = ( + "function test(A)\n" + " B = A[1,:]\n" + " C = A[2,:]\n" + " D = A[:,3]\n" + " return B, C, D\n" + "end\n" + ) + assert source == expected + + +def test_jl_matrixsymbol_slice2(): + A = MatrixSymbol('A', 3, 4) + B = MatrixSymbol('B', 2, 2) + C = MatrixSymbol('C', 2, 2) + name_expr = ("test", [Equality(B, A[0:2, 0:2]), + Equality(C, A[0:2, 1:3])]) + result, = codegen(name_expr, "Julia", header=False, empty=False) + source = result[1] + expected = ( + "function test(A)\n" + " B = A[1:2,1:2]\n" + " C = A[1:2,2:3]\n" + " return B, C\n" + "end\n" + ) + assert source == expected + + +def test_jl_matrixsymbol_slice3(): + A = MatrixSymbol('A', 8, 7) + B = MatrixSymbol('B', 2, 2) + C = MatrixSymbol('C', 4, 2) + name_expr = ("test", [Equality(B, A[6:, 1::3]), + Equality(C, A[::2, ::3])]) + result, = codegen(name_expr, "Julia", header=False, empty=False) + source = result[1] + expected = ( + "function test(A)\n" + " B = A[7:end,2:3:end]\n" + " C = A[1:2:end,1:3:end]\n" + " return B, C\n" + "end\n" + ) + assert source == expected + + +def test_jl_matrixsymbol_slice_autoname(): + A = MatrixSymbol('A', 2, 3) + B = MatrixSymbol('B', 1, 3) + name_expr = ("test", [Equality(B, A[0,:]), A[1,:], A[:,0], A[:,1]]) + result, = codegen(name_expr, "Julia", header=False, empty=False) + source = result[1] + expected = ( + "function test(A)\n" + " B = A[1,:]\n" + " out2 = A[2,:]\n" + " out3 = A[:,1]\n" + " out4 = A[:,2]\n" + " return B, out2, out3, out4\n" + "end\n" + ) + assert source == expected + + +def test_jl_loops(): + # Note: an Julia programmer would probably vectorize this across one or + # more dimensions. Also, size(A) would be used rather than passing in m + # and n. Perhaps users would expect us to vectorize automatically here? + # Or is it possible to represent such things using IndexedBase? + from sympy.tensor import IndexedBase, Idx + from sympy.core.symbol import symbols + n, m = symbols('n m', integer=True) + A = IndexedBase('A') + x = IndexedBase('x') + y = IndexedBase('y') + i = Idx('i', m) + j = Idx('j', n) + result, = codegen(('mat_vec_mult', Eq(y[i], A[i, j]*x[j])), "Julia", + header=False, empty=False) + source = result[1] + expected = ( + 'function mat_vec_mult(y, A, m, n, x)\n' + ' for i = 1:m\n' + ' y[i] = 0\n' + ' end\n' + ' for i = 1:m\n' + ' for j = 1:n\n' + ' y[i] = %(rhs)s + y[i]\n' + ' end\n' + ' end\n' + ' return y\n' + 'end\n' + ) + assert (source == expected % {'rhs': 'A[%s,%s] .* x[j]' % (i, j)} or + source == expected % {'rhs': 'x[j] .* A[%s,%s]' % (i, j)}) + + +def test_jl_tensor_loops_multiple_contractions(): + # see comments in previous test about vectorizing + from sympy.tensor import IndexedBase, Idx + from sympy.core.symbol import symbols + n, m, o, p = symbols('n m o p', integer=True) + A = IndexedBase('A') + B = IndexedBase('B') + y = IndexedBase('y') + i = Idx('i', m) + j = Idx('j', n) + k = Idx('k', o) + l = Idx('l', p) + result, = codegen(('tensorthing', Eq(y[i], B[j, k, l]*A[i, j, k, l])), + "Julia", header=False, empty=False) + source = result[1] + expected = ( + 'function tensorthing(y, A, B, m, n, o, p)\n' + ' for i = 1:m\n' + ' y[i] = 0\n' + ' end\n' + ' for i = 1:m\n' + ' for j = 1:n\n' + ' for k = 1:o\n' + ' for l = 1:p\n' + ' y[i] = A[i,j,k,l] .* B[j,k,l] + y[i]\n' + ' end\n' + ' end\n' + ' end\n' + ' end\n' + ' return y\n' + 'end\n' + ) + assert source == expected + + +def test_jl_InOutArgument(): + expr = Equality(x, x**2) + name_expr = ("mysqr", expr) + result, = codegen(name_expr, "Julia", header=False, empty=False) + source = result[1] + expected = ( + "function mysqr(x)\n" + " x = x .^ 2\n" + " return x\n" + "end\n" + ) + assert source == expected + + +def test_jl_InOutArgument_order(): + # can specify the order as (x, y) + expr = Equality(x, x**2 + y) + name_expr = ("test", expr) + result, = codegen(name_expr, "Julia", header=False, + empty=False, argument_sequence=(x,y)) + source = result[1] + expected = ( + "function test(x, y)\n" + " x = x .^ 2 + y\n" + " return x\n" + "end\n" + ) + assert source == expected + # make sure it gives (x, y) not (y, x) + expr = Equality(x, x**2 + y) + name_expr = ("test", expr) + result, = codegen(name_expr, "Julia", header=False, empty=False) + source = result[1] + expected = ( + "function test(x, y)\n" + " x = x .^ 2 + y\n" + " return x\n" + "end\n" + ) + assert source == expected + + +def test_jl_not_supported(): + f = Function('f') + name_expr = ("test", [f(x).diff(x), S.ComplexInfinity]) + result, = codegen(name_expr, "Julia", header=False, empty=False) + source = result[1] + expected = ( + "function test(x)\n" + " # unsupported: Derivative(f(x), x)\n" + " # unsupported: zoo\n" + " out1 = Derivative(f(x), x)\n" + " out2 = zoo\n" + " return out1, out2\n" + "end\n" + ) + assert source == expected + + +def test_global_vars_octave(): + x, y, z, t = symbols("x y z t") + result = codegen(('f', x*y), "Julia", header=False, empty=False, + global_vars=(y,)) + source = result[0][1] + expected = ( + "function f(x)\n" + " out1 = x .* y\n" + " return out1\n" + "end\n" + ) + assert source == expected + + result = codegen(('f', x*y+z), "Julia", header=False, empty=False, + argument_sequence=(x, y), global_vars=(z, t)) + source = result[0][1] + expected = ( + "function f(x, y)\n" + " out1 = x .* y + z\n" + " return out1\n" + "end\n" + ) + assert source == expected diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_codegen_octave.py b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_codegen_octave.py new file mode 100644 index 0000000000000000000000000000000000000000..77aaef7dd0d81d6855024e49fbb3d6d4c09f3384 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_codegen_octave.py @@ -0,0 +1,589 @@ +from io import StringIO + +from sympy.core import S, symbols, Eq, pi, Catalan, EulerGamma, Function +from sympy.core.relational import Equality +from sympy.functions.elementary.piecewise import Piecewise +from sympy.matrices import Matrix, MatrixSymbol +from sympy.utilities.codegen import OctaveCodeGen, codegen, make_routine +from sympy.testing.pytest import raises +from sympy.testing.pytest import XFAIL +import sympy + + +x, y, z = symbols('x,y,z') + + +def test_empty_m_code(): + code_gen = OctaveCodeGen() + output = StringIO() + code_gen.dump_m([], output, "file", header=False, empty=False) + source = output.getvalue() + assert source == "" + + +def test_m_simple_code(): + name_expr = ("test", (x + y)*z) + result, = codegen(name_expr, "Octave", header=False, empty=False) + assert result[0] == "test.m" + source = result[1] + expected = ( + "function out1 = test(x, y, z)\n" + " out1 = z.*(x + y);\n" + "end\n" + ) + assert source == expected + + +def test_m_simple_code_with_header(): + name_expr = ("test", (x + y)*z) + result, = codegen(name_expr, "Octave", header=True, empty=False) + assert result[0] == "test.m" + source = result[1] + expected = ( + "function out1 = test(x, y, z)\n" + " %TEST Autogenerated by SymPy\n" + " % Code generated with SymPy " + sympy.__version__ + "\n" + " %\n" + " % See http://www.sympy.org/ for more information.\n" + " %\n" + " % This file is part of 'project'\n" + " out1 = z.*(x + y);\n" + "end\n" + ) + assert source == expected + + +def test_m_simple_code_nameout(): + expr = Equality(z, (x + y)) + name_expr = ("test", expr) + result, = codegen(name_expr, "Octave", header=False, empty=False) + source = result[1] + expected = ( + "function z = test(x, y)\n" + " z = x + y;\n" + "end\n" + ) + assert source == expected + + +def test_m_numbersymbol(): + name_expr = ("test", pi**Catalan) + result, = codegen(name_expr, "Octave", header=False, empty=False) + source = result[1] + expected = ( + "function out1 = test()\n" + " out1 = pi^%s;\n" + "end\n" + ) % Catalan.evalf(17) + assert source == expected + + +@XFAIL +def test_m_numbersymbol_no_inline(): + # FIXME: how to pass inline=False to the OctaveCodePrinter? + name_expr = ("test", [pi**Catalan, EulerGamma]) + result, = codegen(name_expr, "Octave", header=False, + empty=False, inline=False) + source = result[1] + expected = ( + "function [out1, out2] = test()\n" + " Catalan = 0.915965594177219; % constant\n" + " EulerGamma = 0.5772156649015329; % constant\n" + " out1 = pi^Catalan;\n" + " out2 = EulerGamma;\n" + "end\n" + ) + assert source == expected + + +def test_m_code_argument_order(): + expr = x + y + routine = make_routine("test", expr, argument_sequence=[z, x, y], language="octave") + code_gen = OctaveCodeGen() + output = StringIO() + code_gen.dump_m([routine], output, "test", header=False, empty=False) + source = output.getvalue() + expected = ( + "function out1 = test(z, x, y)\n" + " out1 = x + y;\n" + "end\n" + ) + assert source == expected + + +def test_multiple_results_m(): + # Here the output order is the input order + expr1 = (x + y)*z + expr2 = (x - y)*z + name_expr = ("test", [expr1, expr2]) + result, = codegen(name_expr, "Octave", header=False, empty=False) + source = result[1] + expected = ( + "function [out1, out2] = test(x, y, z)\n" + " out1 = z.*(x + y);\n" + " out2 = z.*(x - y);\n" + "end\n" + ) + assert source == expected + + +def test_results_named_unordered(): + # Here output order is based on name_expr + A, B, C = symbols('A,B,C') + expr1 = Equality(C, (x + y)*z) + expr2 = Equality(A, (x - y)*z) + expr3 = Equality(B, 2*x) + name_expr = ("test", [expr1, expr2, expr3]) + result, = codegen(name_expr, "Octave", header=False, empty=False) + source = result[1] + expected = ( + "function [C, A, B] = test(x, y, z)\n" + " C = z.*(x + y);\n" + " A = z.*(x - y);\n" + " B = 2*x;\n" + "end\n" + ) + assert source == expected + + +def test_results_named_ordered(): + A, B, C = symbols('A,B,C') + expr1 = Equality(C, (x + y)*z) + expr2 = Equality(A, (x - y)*z) + expr3 = Equality(B, 2*x) + name_expr = ("test", [expr1, expr2, expr3]) + result = codegen(name_expr, "Octave", header=False, empty=False, + argument_sequence=(x, z, y)) + assert result[0][0] == "test.m" + source = result[0][1] + expected = ( + "function [C, A, B] = test(x, z, y)\n" + " C = z.*(x + y);\n" + " A = z.*(x - y);\n" + " B = 2*x;\n" + "end\n" + ) + assert source == expected + + +def test_complicated_m_codegen(): + from sympy.functions.elementary.trigonometric import (cos, sin, tan) + name_expr = ("testlong", + [ ((sin(x) + cos(y) + tan(z))**3).expand(), + cos(cos(cos(cos(cos(cos(cos(cos(x + y + z)))))))) + ]) + result = codegen(name_expr, "Octave", header=False, empty=False) + assert result[0][0] == "testlong.m" + source = result[0][1] + expected = ( + "function [out1, out2] = testlong(x, y, z)\n" + " out1 = sin(x).^3 + 3*sin(x).^2.*cos(y) + 3*sin(x).^2.*tan(z)" + " + 3*sin(x).*cos(y).^2 + 6*sin(x).*cos(y).*tan(z) + 3*sin(x).*tan(z).^2" + " + cos(y).^3 + 3*cos(y).^2.*tan(z) + 3*cos(y).*tan(z).^2 + tan(z).^3;\n" + " out2 = cos(cos(cos(cos(cos(cos(cos(cos(x + y + z))))))));\n" + "end\n" + ) + assert source == expected + + +def test_m_output_arg_mixed_unordered(): + # named outputs are alphabetical, unnamed output appear in the given order + from sympy.functions.elementary.trigonometric import (cos, sin) + a = symbols("a") + name_expr = ("foo", [cos(2*x), Equality(y, sin(x)), cos(x), Equality(a, sin(2*x))]) + result, = codegen(name_expr, "Octave", header=False, empty=False) + assert result[0] == "foo.m" + source = result[1] + expected = ( + 'function [out1, y, out3, a] = foo(x)\n' + ' out1 = cos(2*x);\n' + ' y = sin(x);\n' + ' out3 = cos(x);\n' + ' a = sin(2*x);\n' + 'end\n' + ) + assert source == expected + + +def test_m_piecewise_(): + pw = Piecewise((0, x < -1), (x**2, x <= 1), (-x+2, x > 1), (1, True), evaluate=False) + name_expr = ("pwtest", pw) + result, = codegen(name_expr, "Octave", header=False, empty=False) + source = result[1] + expected = ( + "function out1 = pwtest(x)\n" + " out1 = ((x < -1).*(0) + (~(x < -1)).*( ...\n" + " (x <= 1).*(x.^2) + (~(x <= 1)).*( ...\n" + " (x > 1).*(2 - x) + (~(x > 1)).*(1))));\n" + "end\n" + ) + assert source == expected + + +@XFAIL +def test_m_piecewise_no_inline(): + # FIXME: how to pass inline=False to the OctaveCodePrinter? + pw = Piecewise((0, x < -1), (x**2, x <= 1), (-x+2, x > 1), (1, True)) + name_expr = ("pwtest", pw) + result, = codegen(name_expr, "Octave", header=False, empty=False, + inline=False) + source = result[1] + expected = ( + "function out1 = pwtest(x)\n" + " if (x < -1)\n" + " out1 = 0;\n" + " elseif (x <= 1)\n" + " out1 = x.^2;\n" + " elseif (x > 1)\n" + " out1 = -x + 2;\n" + " else\n" + " out1 = 1;\n" + " end\n" + "end\n" + ) + assert source == expected + + +def test_m_multifcns_per_file(): + name_expr = [ ("foo", [2*x, 3*y]), ("bar", [y**2, 4*y]) ] + result = codegen(name_expr, "Octave", header=False, empty=False) + assert result[0][0] == "foo.m" + source = result[0][1] + expected = ( + "function [out1, out2] = foo(x, y)\n" + " out1 = 2*x;\n" + " out2 = 3*y;\n" + "end\n" + "function [out1, out2] = bar(y)\n" + " out1 = y.^2;\n" + " out2 = 4*y;\n" + "end\n" + ) + assert source == expected + + +def test_m_multifcns_per_file_w_header(): + name_expr = [ ("foo", [2*x, 3*y]), ("bar", [y**2, 4*y]) ] + result = codegen(name_expr, "Octave", header=True, empty=False) + assert result[0][0] == "foo.m" + source = result[0][1] + expected = ( + "function [out1, out2] = foo(x, y)\n" + " %FOO Autogenerated by SymPy\n" + " % Code generated with SymPy " + sympy.__version__ + "\n" + " %\n" + " % See http://www.sympy.org/ for more information.\n" + " %\n" + " % This file is part of 'project'\n" + " out1 = 2*x;\n" + " out2 = 3*y;\n" + "end\n" + "function [out1, out2] = bar(y)\n" + " out1 = y.^2;\n" + " out2 = 4*y;\n" + "end\n" + ) + assert source == expected + + +def test_m_filename_match_first_fcn(): + name_expr = [ ("foo", [2*x, 3*y]), ("bar", [y**2, 4*y]) ] + raises(ValueError, lambda: codegen(name_expr, + "Octave", prefix="bar", header=False, empty=False)) + + +def test_m_matrix_named(): + e2 = Matrix([[x, 2*y, pi*z]]) + name_expr = ("test", Equality(MatrixSymbol('myout1', 1, 3), e2)) + result = codegen(name_expr, "Octave", header=False, empty=False) + assert result[0][0] == "test.m" + source = result[0][1] + expected = ( + "function myout1 = test(x, y, z)\n" + " myout1 = [x 2*y pi*z];\n" + "end\n" + ) + assert source == expected + + +def test_m_matrix_named_matsym(): + myout1 = MatrixSymbol('myout1', 1, 3) + e2 = Matrix([[x, 2*y, pi*z]]) + name_expr = ("test", Equality(myout1, e2, evaluate=False)) + result, = codegen(name_expr, "Octave", header=False, empty=False) + source = result[1] + expected = ( + "function myout1 = test(x, y, z)\n" + " myout1 = [x 2*y pi*z];\n" + "end\n" + ) + assert source == expected + + +def test_m_matrix_output_autoname(): + expr = Matrix([[x, x+y, 3]]) + name_expr = ("test", expr) + result, = codegen(name_expr, "Octave", header=False, empty=False) + source = result[1] + expected = ( + "function out1 = test(x, y)\n" + " out1 = [x x + y 3];\n" + "end\n" + ) + assert source == expected + + +def test_m_matrix_output_autoname_2(): + e1 = (x + y) + e2 = Matrix([[2*x, 2*y, 2*z]]) + e3 = Matrix([[x], [y], [z]]) + e4 = Matrix([[x, y], [z, 16]]) + name_expr = ("test", (e1, e2, e3, e4)) + result, = codegen(name_expr, "Octave", header=False, empty=False) + source = result[1] + expected = ( + "function [out1, out2, out3, out4] = test(x, y, z)\n" + " out1 = x + y;\n" + " out2 = [2*x 2*y 2*z];\n" + " out3 = [x; y; z];\n" + " out4 = [x y; z 16];\n" + "end\n" + ) + assert source == expected + + +def test_m_results_matrix_named_ordered(): + B, C = symbols('B,C') + A = MatrixSymbol('A', 1, 3) + expr1 = Equality(C, (x + y)*z) + expr2 = Equality(A, Matrix([[1, 2, x]])) + expr3 = Equality(B, 2*x) + name_expr = ("test", [expr1, expr2, expr3]) + result, = codegen(name_expr, "Octave", header=False, empty=False, + argument_sequence=(x, z, y)) + source = result[1] + expected = ( + "function [C, A, B] = test(x, z, y)\n" + " C = z.*(x + y);\n" + " A = [1 2 x];\n" + " B = 2*x;\n" + "end\n" + ) + assert source == expected + + +def test_m_matrixsymbol_slice(): + A = MatrixSymbol('A', 2, 3) + B = MatrixSymbol('B', 1, 3) + C = MatrixSymbol('C', 1, 3) + D = MatrixSymbol('D', 2, 1) + name_expr = ("test", [Equality(B, A[0, :]), + Equality(C, A[1, :]), + Equality(D, A[:, 2])]) + result, = codegen(name_expr, "Octave", header=False, empty=False) + source = result[1] + expected = ( + "function [B, C, D] = test(A)\n" + " B = A(1, :);\n" + " C = A(2, :);\n" + " D = A(:, 3);\n" + "end\n" + ) + assert source == expected + + +def test_m_matrixsymbol_slice2(): + A = MatrixSymbol('A', 3, 4) + B = MatrixSymbol('B', 2, 2) + C = MatrixSymbol('C', 2, 2) + name_expr = ("test", [Equality(B, A[0:2, 0:2]), + Equality(C, A[0:2, 1:3])]) + result, = codegen(name_expr, "Octave", header=False, empty=False) + source = result[1] + expected = ( + "function [B, C] = test(A)\n" + " B = A(1:2, 1:2);\n" + " C = A(1:2, 2:3);\n" + "end\n" + ) + assert source == expected + + +def test_m_matrixsymbol_slice3(): + A = MatrixSymbol('A', 8, 7) + B = MatrixSymbol('B', 2, 2) + C = MatrixSymbol('C', 4, 2) + name_expr = ("test", [Equality(B, A[6:, 1::3]), + Equality(C, A[::2, ::3])]) + result, = codegen(name_expr, "Octave", header=False, empty=False) + source = result[1] + expected = ( + "function [B, C] = test(A)\n" + " B = A(7:end, 2:3:end);\n" + " C = A(1:2:end, 1:3:end);\n" + "end\n" + ) + assert source == expected + + +def test_m_matrixsymbol_slice_autoname(): + A = MatrixSymbol('A', 2, 3) + B = MatrixSymbol('B', 1, 3) + name_expr = ("test", [Equality(B, A[0,:]), A[1,:], A[:,0], A[:,1]]) + result, = codegen(name_expr, "Octave", header=False, empty=False) + source = result[1] + expected = ( + "function [B, out2, out3, out4] = test(A)\n" + " B = A(1, :);\n" + " out2 = A(2, :);\n" + " out3 = A(:, 1);\n" + " out4 = A(:, 2);\n" + "end\n" + ) + assert source == expected + + +def test_m_loops(): + # Note: an Octave programmer would probably vectorize this across one or + # more dimensions. Also, size(A) would be used rather than passing in m + # and n. Perhaps users would expect us to vectorize automatically here? + # Or is it possible to represent such things using IndexedBase? + from sympy.tensor import IndexedBase, Idx + from sympy.core.symbol import symbols + n, m = symbols('n m', integer=True) + A = IndexedBase('A') + x = IndexedBase('x') + y = IndexedBase('y') + i = Idx('i', m) + j = Idx('j', n) + result, = codegen(('mat_vec_mult', Eq(y[i], A[i, j]*x[j])), "Octave", + header=False, empty=False) + source = result[1] + expected = ( + 'function y = mat_vec_mult(A, m, n, x)\n' + ' for i = 1:m\n' + ' y(i) = 0;\n' + ' end\n' + ' for i = 1:m\n' + ' for j = 1:n\n' + ' y(i) = %(rhs)s + y(i);\n' + ' end\n' + ' end\n' + 'end\n' + ) + assert (source == expected % {'rhs': 'A(%s, %s).*x(j)' % (i, j)} or + source == expected % {'rhs': 'x(j).*A(%s, %s)' % (i, j)}) + + +def test_m_tensor_loops_multiple_contractions(): + # see comments in previous test about vectorizing + from sympy.tensor import IndexedBase, Idx + from sympy.core.symbol import symbols + n, m, o, p = symbols('n m o p', integer=True) + A = IndexedBase('A') + B = IndexedBase('B') + y = IndexedBase('y') + i = Idx('i', m) + j = Idx('j', n) + k = Idx('k', o) + l = Idx('l', p) + result, = codegen(('tensorthing', Eq(y[i], B[j, k, l]*A[i, j, k, l])), + "Octave", header=False, empty=False) + source = result[1] + expected = ( + 'function y = tensorthing(A, B, m, n, o, p)\n' + ' for i = 1:m\n' + ' y(i) = 0;\n' + ' end\n' + ' for i = 1:m\n' + ' for j = 1:n\n' + ' for k = 1:o\n' + ' for l = 1:p\n' + ' y(i) = A(i, j, k, l).*B(j, k, l) + y(i);\n' + ' end\n' + ' end\n' + ' end\n' + ' end\n' + 'end\n' + ) + assert source == expected + + +def test_m_InOutArgument(): + expr = Equality(x, x**2) + name_expr = ("mysqr", expr) + result, = codegen(name_expr, "Octave", header=False, empty=False) + source = result[1] + expected = ( + "function x = mysqr(x)\n" + " x = x.^2;\n" + "end\n" + ) + assert source == expected + + +def test_m_InOutArgument_order(): + # can specify the order as (x, y) + expr = Equality(x, x**2 + y) + name_expr = ("test", expr) + result, = codegen(name_expr, "Octave", header=False, + empty=False, argument_sequence=(x,y)) + source = result[1] + expected = ( + "function x = test(x, y)\n" + " x = x.^2 + y;\n" + "end\n" + ) + assert source == expected + # make sure it gives (x, y) not (y, x) + expr = Equality(x, x**2 + y) + name_expr = ("test", expr) + result, = codegen(name_expr, "Octave", header=False, empty=False) + source = result[1] + expected = ( + "function x = test(x, y)\n" + " x = x.^2 + y;\n" + "end\n" + ) + assert source == expected + + +def test_m_not_supported(): + f = Function('f') + name_expr = ("test", [f(x).diff(x), S.ComplexInfinity]) + result, = codegen(name_expr, "Octave", header=False, empty=False) + source = result[1] + expected = ( + "function [out1, out2] = test(x)\n" + " % unsupported: Derivative(f(x), x)\n" + " % unsupported: zoo\n" + " out1 = Derivative(f(x), x);\n" + " out2 = zoo;\n" + "end\n" + ) + assert source == expected + + +def test_global_vars_octave(): + x, y, z, t = symbols("x y z t") + result = codegen(('f', x*y), "Octave", header=False, empty=False, + global_vars=(y,)) + source = result[0][1] + expected = ( + "function out1 = f(x)\n" + " global y\n" + " out1 = x.*y;\n" + "end\n" + ) + assert source == expected + + result = codegen(('f', x*y+z), "Octave", header=False, empty=False, + argument_sequence=(x, y), global_vars=(z, t)) + source = result[0][1] + expected = ( + "function out1 = f(x, y)\n" + " global t z\n" + " out1 = x.*y + z;\n" + "end\n" + ) + assert source == expected diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_codegen_rust.py b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_codegen_rust.py new file mode 100644 index 0000000000000000000000000000000000000000..bc7f82158ae8fa7dfe34bf909aa695b119fb9526 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_codegen_rust.py @@ -0,0 +1,401 @@ +from io import StringIO + +from sympy.core import S, symbols, pi, Catalan, EulerGamma, Function +from sympy.core.relational import Equality +from sympy.functions.elementary.piecewise import Piecewise +from sympy.utilities.codegen import RustCodeGen, codegen, make_routine +from sympy.testing.pytest import XFAIL +import sympy + + +x, y, z = symbols('x,y,z') + + +def test_empty_rust_code(): + code_gen = RustCodeGen() + output = StringIO() + code_gen.dump_rs([], output, "file", header=False, empty=False) + source = output.getvalue() + assert source == "" + + +def test_simple_rust_code(): + name_expr = ("test", (x + y)*z) + result, = codegen(name_expr, "Rust", header=False, empty=False) + assert result[0] == "test.rs" + source = result[1] + expected = ( + "fn test(x: f64, y: f64, z: f64) -> f64 {\n" + " let out1 = z*(x + y);\n" + " out1\n" + "}\n" + ) + assert source == expected + + +def test_simple_code_with_header(): + name_expr = ("test", (x + y)*z) + result, = codegen(name_expr, "Rust", header=True, empty=False) + assert result[0] == "test.rs" + source = result[1] + version_str = "Code generated with SymPy %s" % sympy.__version__ + version_line = version_str.center(76).rstrip() + expected = ( + "/*\n" + " *%(version_line)s\n" + " *\n" + " * See http://www.sympy.org/ for more information.\n" + " *\n" + " * This file is part of 'project'\n" + " */\n" + "fn test(x: f64, y: f64, z: f64) -> f64 {\n" + " let out1 = z*(x + y);\n" + " out1\n" + "}\n" + ) % {'version_line': version_line} + assert source == expected + + +def test_simple_code_nameout(): + expr = Equality(z, (x + y)) + name_expr = ("test", expr) + result, = codegen(name_expr, "Rust", header=False, empty=False) + source = result[1] + expected = ( + "fn test(x: f64, y: f64) -> f64 {\n" + " let z = x + y;\n" + " z\n" + "}\n" + ) + assert source == expected + + +def test_numbersymbol(): + name_expr = ("test", pi**Catalan) + result, = codegen(name_expr, "Rust", header=False, empty=False) + source = result[1] + expected = ( + "fn test() -> f64 {\n" + " const Catalan: f64 = %s;\n" + " let out1 = PI.powf(Catalan);\n" + " out1\n" + "}\n" + ) % Catalan.evalf(17) + assert source == expected + + +@XFAIL +def test_numbersymbol_inline(): + # FIXME: how to pass inline to the RustCodePrinter? + name_expr = ("test", [pi**Catalan, EulerGamma]) + result, = codegen(name_expr, "Rust", header=False, + empty=False, inline=True) + source = result[1] + expected = ( + "fn test() -> (f64, f64) {\n" + " const Catalan: f64 = %s;\n" + " const EulerGamma: f64 = %s;\n" + " let out1 = PI.powf(Catalan);\n" + " let out2 = EulerGamma);\n" + " (out1, out2)\n" + "}\n" + ) % (Catalan.evalf(17), EulerGamma.evalf(17)) + assert source == expected + + +def test_argument_order(): + expr = x + y + routine = make_routine("test", expr, argument_sequence=[z, x, y], language="rust") + code_gen = RustCodeGen() + output = StringIO() + code_gen.dump_rs([routine], output, "test", header=False, empty=False) + source = output.getvalue() + expected = ( + "fn test(z: f64, x: f64, y: f64) -> f64 {\n" + " let out1 = x + y;\n" + " out1\n" + "}\n" + ) + assert source == expected + + +def test_multiple_results_rust(): + # Here the output order is the input order + expr1 = (x + y)*z + expr2 = (x - y)*z + name_expr = ("test", [expr1, expr2]) + result, = codegen(name_expr, "Rust", header=False, empty=False) + source = result[1] + expected = ( + "fn test(x: f64, y: f64, z: f64) -> (f64, f64) {\n" + " let out1 = z*(x + y);\n" + " let out2 = z*(x - y);\n" + " (out1, out2)\n" + "}\n" + ) + assert source == expected + + +def test_results_named_unordered(): + # Here output order is based on name_expr + A, B, C = symbols('A,B,C') + expr1 = Equality(C, (x + y)*z) + expr2 = Equality(A, (x - y)*z) + expr3 = Equality(B, 2*x) + name_expr = ("test", [expr1, expr2, expr3]) + result, = codegen(name_expr, "Rust", header=False, empty=False) + source = result[1] + expected = ( + "fn test(x: f64, y: f64, z: f64) -> (f64, f64, f64) {\n" + " let C = z*(x + y);\n" + " let A = z*(x - y);\n" + " let B = 2*x;\n" + " (C, A, B)\n" + "}\n" + ) + assert source == expected + + +def test_results_named_ordered(): + A, B, C = symbols('A,B,C') + expr1 = Equality(C, (x + y)*z) + expr2 = Equality(A, (x - y)*z) + expr3 = Equality(B, 2*x) + name_expr = ("test", [expr1, expr2, expr3]) + result = codegen(name_expr, "Rust", header=False, empty=False, + argument_sequence=(x, z, y)) + assert result[0][0] == "test.rs" + source = result[0][1] + expected = ( + "fn test(x: f64, z: f64, y: f64) -> (f64, f64, f64) {\n" + " let C = z*(x + y);\n" + " let A = z*(x - y);\n" + " let B = 2*x;\n" + " (C, A, B)\n" + "}\n" + ) + assert source == expected + + +def test_complicated_rs_codegen(): + from sympy.functions.elementary.trigonometric import (cos, sin, tan) + name_expr = ("testlong", + [ ((sin(x) + cos(y) + tan(z))**3).expand(), + cos(cos(cos(cos(cos(cos(cos(cos(x + y + z)))))))) + ]) + result = codegen(name_expr, "Rust", header=False, empty=False) + assert result[0][0] == "testlong.rs" + source = result[0][1] + expected = ( + "fn testlong(x: f64, y: f64, z: f64) -> (f64, f64) {\n" + " let out1 = x.sin().powi(3) + 3*x.sin().powi(2)*y.cos()" + " + 3*x.sin().powi(2)*z.tan() + 3*x.sin()*y.cos().powi(2)" + " + 6*x.sin()*y.cos()*z.tan() + 3*x.sin()*z.tan().powi(2)" + " + y.cos().powi(3) + 3*y.cos().powi(2)*z.tan()" + " + 3*y.cos()*z.tan().powi(2) + z.tan().powi(3);\n" + " let out2 = (x + y + z).cos().cos().cos().cos()" + ".cos().cos().cos().cos();\n" + " (out1, out2)\n" + "}\n" + ) + assert source == expected + + +def test_output_arg_mixed_unordered(): + # named outputs are alphabetical, unnamed output appear in the given order + from sympy.functions.elementary.trigonometric import (cos, sin) + a = symbols("a") + name_expr = ("foo", [cos(2*x), Equality(y, sin(x)), cos(x), Equality(a, sin(2*x))]) + result, = codegen(name_expr, "Rust", header=False, empty=False) + assert result[0] == "foo.rs" + source = result[1] + expected = ( + "fn foo(x: f64) -> (f64, f64, f64, f64) {\n" + " let out1 = (2*x).cos();\n" + " let y = x.sin();\n" + " let out3 = x.cos();\n" + " let a = (2*x).sin();\n" + " (out1, y, out3, a)\n" + "}\n" + ) + assert source == expected + + +def test_piecewise_(): + pw = Piecewise((0, x < -1), (x**2, x <= 1), (-x+2, x > 1), (1, True), evaluate=False) + name_expr = ("pwtest", pw) + result, = codegen(name_expr, "Rust", header=False, empty=False) + source = result[1] + expected = ( + "fn pwtest(x: f64) -> f64 {\n" + " let out1 = if (x < -1.0) {\n" + " 0\n" + " } else if (x <= 1.0) {\n" + " x.powi(2)\n" + " } else if (x > 1.0) {\n" + " 2 - x\n" + " } else {\n" + " 1\n" + " };\n" + " out1\n" + "}\n" + ) + assert source == expected + + +@XFAIL +def test_piecewise_inline(): + # FIXME: how to pass inline to the RustCodePrinter? + pw = Piecewise((0, x < -1), (x**2, x <= 1), (-x+2, x > 1), (1, True)) + name_expr = ("pwtest", pw) + result, = codegen(name_expr, "Rust", header=False, empty=False, + inline=True) + source = result[1] + expected = ( + "fn pwtest(x: f64) -> f64 {\n" + " let out1 = if (x < -1) { 0 } else if (x <= 1) { x.powi(2) }" + " else if (x > 1) { -x + 2 } else { 1 };\n" + " out1\n" + "}\n" + ) + assert source == expected + + +def test_multifcns_per_file(): + name_expr = [ ("foo", [2*x, 3*y]), ("bar", [y**2, 4*y]) ] + result = codegen(name_expr, "Rust", header=False, empty=False) + assert result[0][0] == "foo.rs" + source = result[0][1] + expected = ( + "fn foo(x: f64, y: f64) -> (f64, f64) {\n" + " let out1 = 2*x;\n" + " let out2 = 3*y;\n" + " (out1, out2)\n" + "}\n" + "fn bar(y: f64) -> (f64, f64) {\n" + " let out1 = y.powi(2);\n" + " let out2 = 4*y;\n" + " (out1, out2)\n" + "}\n" + ) + assert source == expected + + +def test_multifcns_per_file_w_header(): + name_expr = [ ("foo", [2*x, 3*y]), ("bar", [y**2, 4*y]) ] + result = codegen(name_expr, "Rust", header=True, empty=False) + assert result[0][0] == "foo.rs" + source = result[0][1] + version_str = "Code generated with SymPy %s" % sympy.__version__ + version_line = version_str.center(76).rstrip() + expected = ( + "/*\n" + " *%(version_line)s\n" + " *\n" + " * See http://www.sympy.org/ for more information.\n" + " *\n" + " * This file is part of 'project'\n" + " */\n" + "fn foo(x: f64, y: f64) -> (f64, f64) {\n" + " let out1 = 2*x;\n" + " let out2 = 3*y;\n" + " (out1, out2)\n" + "}\n" + "fn bar(y: f64) -> (f64, f64) {\n" + " let out1 = y.powi(2);\n" + " let out2 = 4*y;\n" + " (out1, out2)\n" + "}\n" + ) % {'version_line': version_line} + assert source == expected + + +def test_filename_match_prefix(): + name_expr = [ ("foo", [2*x, 3*y]), ("bar", [y**2, 4*y]) ] + result, = codegen(name_expr, "Rust", prefix="baz", header=False, + empty=False) + assert result[0] == "baz.rs" + + +def test_InOutArgument(): + expr = Equality(x, x**2) + name_expr = ("mysqr", expr) + result, = codegen(name_expr, "Rust", header=False, empty=False) + source = result[1] + expected = ( + "fn mysqr(x: f64) -> f64 {\n" + " let x = x.powi(2);\n" + " x\n" + "}\n" + ) + assert source == expected + + +def test_InOutArgument_order(): + # can specify the order as (x, y) + expr = Equality(x, x**2 + y) + name_expr = ("test", expr) + result, = codegen(name_expr, "Rust", header=False, + empty=False, argument_sequence=(x,y)) + source = result[1] + expected = ( + "fn test(x: f64, y: f64) -> f64 {\n" + " let x = x.powi(2) + y;\n" + " x\n" + "}\n" + ) + assert source == expected + # make sure it gives (x, y) not (y, x) + expr = Equality(x, x**2 + y) + name_expr = ("test", expr) + result, = codegen(name_expr, "Rust", header=False, empty=False) + source = result[1] + expected = ( + "fn test(x: f64, y: f64) -> f64 {\n" + " let x = x.powi(2) + y;\n" + " x\n" + "}\n" + ) + assert source == expected + + +def test_not_supported(): + f = Function('f') + name_expr = ("test", [f(x).diff(x), S.ComplexInfinity]) + result, = codegen(name_expr, "Rust", header=False, empty=False) + source = result[1] + expected = ( + "fn test(x: f64) -> (f64, f64) {\n" + " // unsupported: Derivative(f(x), x)\n" + " // unsupported: zoo\n" + " let out1 = Derivative(f(x), x);\n" + " let out2 = zoo;\n" + " (out1, out2)\n" + "}\n" + ) + assert source == expected + + +def test_global_vars_rust(): + x, y, z, t = symbols("x y z t") + result = codegen(('f', x*y), "Rust", header=False, empty=False, + global_vars=(y,)) + source = result[0][1] + expected = ( + "fn f(x: f64) -> f64 {\n" + " let out1 = x*y;\n" + " out1\n" + "}\n" + ) + assert source == expected + + result = codegen(('f', x*y+z), "Rust", header=False, empty=False, + argument_sequence=(x, y), global_vars=(z, t)) + source = result[0][1] + expected = ( + "fn f(x: f64, y: f64) -> f64 {\n" + " let out1 = x*y + z;\n" + " out1\n" + "}\n" + ) + assert source == expected diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_decorator.py b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_decorator.py new file mode 100644 index 0000000000000000000000000000000000000000..b1870d4db8f719fdabfeab14120bfb3ce10131a9 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_decorator.py @@ -0,0 +1,129 @@ +from functools import wraps + +from sympy.utilities.decorator import threaded, xthreaded, memoize_property, deprecated +from sympy.testing.pytest import warns_deprecated_sympy + +from sympy.core.basic import Basic +from sympy.core.relational import Eq +from sympy.matrices.dense import Matrix + +from sympy.abc import x, y + + +def test_threaded(): + @threaded + def function(expr, *args): + return 2*expr + sum(args) + + assert function(Matrix([[x, y], [1, x]]), 1, 2) == \ + Matrix([[2*x + 3, 2*y + 3], [5, 2*x + 3]]) + + assert function(Eq(x, y), 1, 2) == Eq(2*x + 3, 2*y + 3) + + assert function([x, y], 1, 2) == [2*x + 3, 2*y + 3] + assert function((x, y), 1, 2) == (2*x + 3, 2*y + 3) + + assert function({x, y}, 1, 2) == {2*x + 3, 2*y + 3} + + @threaded + def function(expr, n): + return expr**n + + assert function(x + y, 2) == x**2 + y**2 + assert function(x, 2) == x**2 + + +def test_xthreaded(): + @xthreaded + def function(expr, n): + return expr**n + + assert function(x + y, 2) == (x + y)**2 + + +def test_wraps(): + def my_func(x): + """My function. """ + + my_func.is_my_func = True + + new_my_func = threaded(my_func) + new_my_func = wraps(my_func)(new_my_func) + + assert new_my_func.__name__ == 'my_func' + assert new_my_func.__doc__ == 'My function. ' + assert hasattr(new_my_func, 'is_my_func') + assert new_my_func.is_my_func is True + + +def test_memoize_property(): + class TestMemoize(Basic): + @memoize_property + def prop(self): + return Basic() + + member = TestMemoize() + obj1 = member.prop + obj2 = member.prop + assert obj1 is obj2 + +def test_deprecated(): + @deprecated('deprecated_function is deprecated', + deprecated_since_version='1.10', + # This is the target at the top of the file, which will never + # go away. + active_deprecations_target='active-deprecations') + def deprecated_function(x): + return x + + with warns_deprecated_sympy(): + assert deprecated_function(1) == 1 + + @deprecated('deprecated_class is deprecated', + deprecated_since_version='1.10', + active_deprecations_target='active-deprecations') + class deprecated_class: + pass + + with warns_deprecated_sympy(): + assert isinstance(deprecated_class(), deprecated_class) + + # Ensure the class decorator works even when the class never returns + # itself + @deprecated('deprecated_class_new is deprecated', + deprecated_since_version='1.10', + active_deprecations_target='active-deprecations') + class deprecated_class_new: + def __new__(cls, arg): + return arg + + with warns_deprecated_sympy(): + assert deprecated_class_new(1) == 1 + + @deprecated('deprecated_class_init is deprecated', + deprecated_since_version='1.10', + active_deprecations_target='active-deprecations') + class deprecated_class_init: + def __init__(self, arg): + self.arg = 1 + + with warns_deprecated_sympy(): + assert deprecated_class_init(1).arg == 1 + + @deprecated('deprecated_class_new_init is deprecated', + deprecated_since_version='1.10', + active_deprecations_target='active-deprecations') + class deprecated_class_new_init: + def __new__(cls, arg): + if arg == 0: + return arg + return object.__new__(cls) + + def __init__(self, arg): + self.arg = 1 + + with warns_deprecated_sympy(): + assert deprecated_class_new_init(0) == 0 + + with warns_deprecated_sympy(): + assert deprecated_class_new_init(1).arg == 1 diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_deprecated.py b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_deprecated.py new file mode 100644 index 0000000000000000000000000000000000000000..dd4534ef1abc38ff368011b3ef9d11c497f3675b --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_deprecated.py @@ -0,0 +1,13 @@ +from sympy.testing.pytest import warns_deprecated_sympy + +# See https://github.com/sympy/sympy/pull/18095 + +def test_deprecated_utilities(): + with warns_deprecated_sympy(): + import sympy.utilities.pytest # noqa:F401 + with warns_deprecated_sympy(): + import sympy.utilities.runtests # noqa:F401 + with warns_deprecated_sympy(): + import sympy.utilities.randtest # noqa:F401 + with warns_deprecated_sympy(): + import sympy.utilities.tmpfiles # noqa:F401 diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_enumerative.py b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_enumerative.py new file mode 100644 index 0000000000000000000000000000000000000000..357499d5fd400b14e2bcad067f3015b74b0e9003 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_enumerative.py @@ -0,0 +1,179 @@ +import string +from itertools import zip_longest + +from sympy.utilities.enumerative import ( + list_visitor, + MultisetPartitionTraverser, + multiset_partitions_taocp + ) +from sympy.utilities.iterables import _set_partitions + +# first some functions only useful as test scaffolding - these provide +# straightforward, but slow reference implementations against which to +# compare the real versions, and also a comparison to verify that +# different versions are giving identical results. + +def part_range_filter(partition_iterator, lb, ub): + """ + Filters (on the number of parts) a multiset partition enumeration + + Arguments + ========= + + lb, and ub are a range (in the Python slice sense) on the lpart + variable returned from a multiset partition enumeration. Recall + that lpart is 0-based (it points to the topmost part on the part + stack), so if you want to return parts of sizes 2,3,4,5 you would + use lb=1 and ub=5. + """ + for state in partition_iterator: + f, lpart, pstack = state + if lpart >= lb and lpart < ub: + yield state + +def multiset_partitions_baseline(multiplicities, components): + """Enumerates partitions of a multiset + + Parameters + ========== + + multiplicities + list of integer multiplicities of the components of the multiset. + + components + the components (elements) themselves + + Returns + ======= + + Set of partitions. Each partition is tuple of parts, and each + part is a tuple of components (with repeats to indicate + multiplicity) + + Notes + ===== + + Multiset partitions can be created as equivalence classes of set + partitions, and this function does just that. This approach is + slow and memory intensive compared to the more advanced algorithms + available, but the code is simple and easy to understand. Hence + this routine is strictly for testing -- to provide a + straightforward baseline against which to regress the production + versions. (This code is a simplified version of an earlier + production implementation.) + """ + + canon = [] # list of components with repeats + for ct, elem in zip(multiplicities, components): + canon.extend([elem]*ct) + + # accumulate the multiset partitions in a set to eliminate dups + cache = set() + n = len(canon) + for nc, q in _set_partitions(n): + rv = [[] for i in range(nc)] + for i in range(n): + rv[q[i]].append(canon[i]) + canonical = tuple( + sorted([tuple(p) for p in rv])) + cache.add(canonical) + return cache + + +def compare_multiset_w_baseline(multiplicities): + """ + Enumerates the partitions of multiset with AOCP algorithm and + baseline implementation, and compare the results. + + """ + letters = string.ascii_lowercase + bl_partitions = multiset_partitions_baseline(multiplicities, letters) + + # The partitions returned by the different algorithms may have + # their parts in different orders. Also, they generate partitions + # in different orders. Hence the sorting, and set comparison. + + aocp_partitions = set() + for state in multiset_partitions_taocp(multiplicities): + p1 = tuple(sorted( + [tuple(p) for p in list_visitor(state, letters)])) + aocp_partitions.add(p1) + + assert bl_partitions == aocp_partitions + +def compare_multiset_states(s1, s2): + """compare for equality two instances of multiset partition states + + This is useful for comparing different versions of the algorithm + to verify correctness.""" + # Comparison is physical, the only use of semantics is to ignore + # trash off the top of the stack. + f1, lpart1, pstack1 = s1 + f2, lpart2, pstack2 = s2 + + if (lpart1 == lpart2) and (f1[0:lpart1+1] == f2[0:lpart2+1]): + if pstack1[0:f1[lpart1+1]] == pstack2[0:f2[lpart2+1]]: + return True + return False + +def test_multiset_partitions_taocp(): + """Compares the output of multiset_partitions_taocp with a baseline + (set partition based) implementation.""" + + # Test cases should not be too large, since the baseline + # implementation is fairly slow. + multiplicities = [2,2] + compare_multiset_w_baseline(multiplicities) + + multiplicities = [4,3,1] + compare_multiset_w_baseline(multiplicities) + +def test_multiset_partitions_versions(): + """Compares Knuth-based versions of multiset_partitions""" + multiplicities = [5,2,2,1] + m = MultisetPartitionTraverser() + for s1, s2 in zip_longest(m.enum_all(multiplicities), + multiset_partitions_taocp(multiplicities)): + assert compare_multiset_states(s1, s2) + +def subrange_exercise(mult, lb, ub): + """Compare filter-based and more optimized subrange implementations + + Helper for tests, called with both small and larger multisets. + """ + m = MultisetPartitionTraverser() + assert m.count_partitions(mult) == \ + m.count_partitions_slow(mult) + + # Note - multiple traversals from the same + # MultisetPartitionTraverser object cannot execute at the same + # time, hence make several instances here. + ma = MultisetPartitionTraverser() + mc = MultisetPartitionTraverser() + md = MultisetPartitionTraverser() + + # Several paths to compute just the size two partitions + a_it = ma.enum_range(mult, lb, ub) + b_it = part_range_filter(multiset_partitions_taocp(mult), lb, ub) + c_it = part_range_filter(mc.enum_small(mult, ub), lb, sum(mult)) + d_it = part_range_filter(md.enum_large(mult, lb), 0, ub) + + for sa, sb, sc, sd in zip_longest(a_it, b_it, c_it, d_it): + assert compare_multiset_states(sa, sb) + assert compare_multiset_states(sa, sc) + assert compare_multiset_states(sa, sd) + +def test_subrange(): + # Quick, but doesn't hit some of the corner cases + mult = [4,4,2,1] # mississippi + lb = 1 + ub = 2 + subrange_exercise(mult, lb, ub) + + +def test_subrange_large(): + # takes a second or so, depending on cpu, Python version, etc. + mult = [6,3,2,1] + lb = 4 + ub = 7 + subrange_exercise(mult, lb, ub) diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_exceptions.py b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_exceptions.py new file mode 100644 index 0000000000000000000000000000000000000000..d91e55e95d0ae4ac57cdd1989e0b3d39a55cd07d --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_exceptions.py @@ -0,0 +1,12 @@ +from sympy.testing.pytest import raises +from sympy.utilities.exceptions import sympy_deprecation_warning + +# Only test exceptions here because the other cases are tested in the +# warns_deprecated_sympy tests +def test_sympy_deprecation_warning(): + raises(TypeError, lambda: sympy_deprecation_warning('test', + deprecated_since_version=1.10, + active_deprecations_target='active-deprecations')) + + raises(ValueError, lambda: sympy_deprecation_warning('test', + deprecated_since_version="1.10", active_deprecations_target='(active-deprecations)=')) diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_iterables.py b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_iterables.py new file mode 100644 index 0000000000000000000000000000000000000000..1003522bcd556c6f63e04de7da57b43498575fee --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_iterables.py @@ -0,0 +1,945 @@ +from textwrap import dedent +from itertools import islice, product + +from sympy.core.basic import Basic +from sympy.core.numbers import Integer +from sympy.core.sorting import ordered +from sympy.core.symbol import (Dummy, symbols) +from sympy.functions.combinatorial.factorials import factorial +from sympy.matrices.dense import Matrix +from sympy.combinatorics import RGS_enum, RGS_unrank, Permutation +from sympy.utilities.iterables import ( + _partition, _set_partitions, binary_partitions, bracelets, capture, + cartes, common_prefix, common_suffix, connected_components, dict_merge, + filter_symbols, flatten, generate_bell, generate_derangements, + generate_involutions, generate_oriented_forest, group, has_dups, ibin, + iproduct, kbins, minlex, multiset, multiset_combinations, + multiset_partitions, multiset_permutations, necklaces, numbered_symbols, + partitions, permutations, postfixes, + prefixes, reshape, rotate_left, rotate_right, runs, sift, + strongly_connected_components, subsets, take, topological_sort, unflatten, + uniq, variations, ordered_partitions, rotations, is_palindromic, iterable, + NotIterable, multiset_derangements, signed_permutations, + sequence_partitions, sequence_partitions_empty) +from sympy.utilities.enumerative import ( + factoring_visitor, multiset_partitions_taocp ) + +from sympy.core.singleton import S +from sympy.testing.pytest import raises, warns_deprecated_sympy + +w, x, y, z = symbols('w,x,y,z') + + +def test_deprecated_iterables(): + from sympy.utilities.iterables import default_sort_key, ordered + with warns_deprecated_sympy(): + assert list(ordered([y, x])) == [x, y] + with warns_deprecated_sympy(): + assert sorted([y, x], key=default_sort_key) == [x, y] + + +def test_is_palindromic(): + assert is_palindromic('') + assert is_palindromic('x') + assert is_palindromic('xx') + assert is_palindromic('xyx') + assert not is_palindromic('xy') + assert not is_palindromic('xyzx') + assert is_palindromic('xxyzzyx', 1) + assert not is_palindromic('xxyzzyx', 2) + assert is_palindromic('xxyzzyx', 2, -1) + assert is_palindromic('xxyzzyx', 2, 6) + assert is_palindromic('xxyzyx', 1) + assert not is_palindromic('xxyzyx', 2) + assert is_palindromic('xxyzyx', 2, 2 + 3) + + +def test_flatten(): + assert flatten((1, (1,))) == [1, 1] + assert flatten((x, (x,))) == [x, x] + + ls = [[(-2, -1), (1, 2)], [(0, 0)]] + + assert flatten(ls, levels=0) == ls + assert flatten(ls, levels=1) == [(-2, -1), (1, 2), (0, 0)] + assert flatten(ls, levels=2) == [-2, -1, 1, 2, 0, 0] + assert flatten(ls, levels=3) == [-2, -1, 1, 2, 0, 0] + + raises(ValueError, lambda: flatten(ls, levels=-1)) + + class MyOp(Basic): + pass + + assert flatten([MyOp(x, y), z]) == [MyOp(x, y), z] + assert flatten([MyOp(x, y), z], cls=MyOp) == [x, y, z] + + assert flatten({1, 11, 2}) == list({1, 11, 2}) + + +def test_iproduct(): + assert list(iproduct()) == [()] + assert list(iproduct([])) == [] + assert list(iproduct([1,2,3])) == [(1,),(2,),(3,)] + assert sorted(iproduct([1, 2], [3, 4, 5])) == [ + (1,3),(1,4),(1,5),(2,3),(2,4),(2,5)] + assert sorted(iproduct([0,1],[0,1],[0,1])) == [ + (0,0,0),(0,0,1),(0,1,0),(0,1,1),(1,0,0),(1,0,1),(1,1,0),(1,1,1)] + assert iterable(iproduct(S.Integers)) is True + assert iterable(iproduct(S.Integers, S.Integers)) is True + assert (3,) in iproduct(S.Integers) + assert (4, 5) in iproduct(S.Integers, S.Integers) + assert (1, 2, 3) in iproduct(S.Integers, S.Integers, S.Integers) + triples = set(islice(iproduct(S.Integers, S.Integers, S.Integers), 1000)) + for n1, n2, n3 in triples: + assert isinstance(n1, Integer) + assert isinstance(n2, Integer) + assert isinstance(n3, Integer) + for t in set(product(*([range(-2, 3)]*3))): + assert t in iproduct(S.Integers, S.Integers, S.Integers) + + +def test_group(): + assert group([]) == [] + assert group([], multiple=False) == [] + + assert group([1]) == [[1]] + assert group([1], multiple=False) == [(1, 1)] + + assert group([1, 1]) == [[1, 1]] + assert group([1, 1], multiple=False) == [(1, 2)] + + assert group([1, 1, 1]) == [[1, 1, 1]] + assert group([1, 1, 1], multiple=False) == [(1, 3)] + + assert group([1, 2, 1]) == [[1], [2], [1]] + assert group([1, 2, 1], multiple=False) == [(1, 1), (2, 1), (1, 1)] + + assert group([1, 1, 2, 2, 2, 1, 3, 3]) == [[1, 1], [2, 2, 2], [1], [3, 3]] + assert group([1, 1, 2, 2, 2, 1, 3, 3], multiple=False) == [(1, 2), + (2, 3), (1, 1), (3, 2)] + + +def test_subsets(): + # combinations + assert list(subsets([1, 2, 3], 0)) == [()] + assert list(subsets([1, 2, 3], 1)) == [(1,), (2,), (3,)] + assert list(subsets([1, 2, 3], 2)) == [(1, 2), (1, 3), (2, 3)] + assert list(subsets([1, 2, 3], 3)) == [(1, 2, 3)] + l = list(range(4)) + assert list(subsets(l, 0, repetition=True)) == [()] + assert list(subsets(l, 1, repetition=True)) == [(0,), (1,), (2,), (3,)] + assert list(subsets(l, 2, repetition=True)) == [(0, 0), (0, 1), (0, 2), + (0, 3), (1, 1), (1, 2), + (1, 3), (2, 2), (2, 3), + (3, 3)] + assert list(subsets(l, 3, repetition=True)) == [(0, 0, 0), (0, 0, 1), + (0, 0, 2), (0, 0, 3), + (0, 1, 1), (0, 1, 2), + (0, 1, 3), (0, 2, 2), + (0, 2, 3), (0, 3, 3), + (1, 1, 1), (1, 1, 2), + (1, 1, 3), (1, 2, 2), + (1, 2, 3), (1, 3, 3), + (2, 2, 2), (2, 2, 3), + (2, 3, 3), (3, 3, 3)] + assert len(list(subsets(l, 4, repetition=True))) == 35 + + assert list(subsets(l[:2], 3, repetition=False)) == [] + assert list(subsets(l[:2], 3, repetition=True)) == [(0, 0, 0), + (0, 0, 1), + (0, 1, 1), + (1, 1, 1)] + assert list(subsets([1, 2], repetition=True)) == \ + [(), (1,), (2,), (1, 1), (1, 2), (2, 2)] + assert list(subsets([1, 2], repetition=False)) == \ + [(), (1,), (2,), (1, 2)] + assert list(subsets([1, 2, 3], 2)) == \ + [(1, 2), (1, 3), (2, 3)] + assert list(subsets([1, 2, 3], 2, repetition=True)) == \ + [(1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (3, 3)] + + +def test_variations(): + # permutations + l = list(range(4)) + assert list(variations(l, 0, repetition=False)) == [()] + assert list(variations(l, 1, repetition=False)) == [(0,), (1,), (2,), (3,)] + assert list(variations(l, 2, repetition=False)) == [(0, 1), (0, 2), (0, 3), (1, 0), (1, 2), (1, 3), (2, 0), (2, 1), (2, 3), (3, 0), (3, 1), (3, 2)] + assert list(variations(l, 3, repetition=False)) == [(0, 1, 2), (0, 1, 3), (0, 2, 1), (0, 2, 3), (0, 3, 1), (0, 3, 2), (1, 0, 2), (1, 0, 3), (1, 2, 0), (1, 2, 3), (1, 3, 0), (1, 3, 2), (2, 0, 1), (2, 0, 3), (2, 1, 0), (2, 1, 3), (2, 3, 0), (2, 3, 1), (3, 0, 1), (3, 0, 2), (3, 1, 0), (3, 1, 2), (3, 2, 0), (3, 2, 1)] + assert list(variations(l, 0, repetition=True)) == [()] + assert list(variations(l, 1, repetition=True)) == [(0,), (1,), (2,), (3,)] + assert list(variations(l, 2, repetition=True)) == [(0, 0), (0, 1), (0, 2), + (0, 3), (1, 0), (1, 1), + (1, 2), (1, 3), (2, 0), + (2, 1), (2, 2), (2, 3), + (3, 0), (3, 1), (3, 2), + (3, 3)] + assert len(list(variations(l, 3, repetition=True))) == 64 + assert len(list(variations(l, 4, repetition=True))) == 256 + assert list(variations(l[:2], 3, repetition=False)) == [] + assert list(variations(l[:2], 3, repetition=True)) == [ + (0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1), + (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1) + ] + + +def test_cartes(): + assert list(cartes([1, 2], [3, 4, 5])) == \ + [(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5)] + assert list(cartes()) == [()] + assert list(cartes('a')) == [('a',)] + assert list(cartes('a', repeat=2)) == [('a', 'a')] + assert list(cartes(list(range(2)))) == [(0,), (1,)] + + +def test_filter_symbols(): + s = numbered_symbols() + filtered = filter_symbols(s, symbols("x0 x2 x3")) + assert take(filtered, 3) == list(symbols("x1 x4 x5")) + + +def test_numbered_symbols(): + s = numbered_symbols(cls=Dummy) + assert isinstance(next(s), Dummy) + assert next(numbered_symbols('C', start=1, exclude=[symbols('C1')])) == \ + symbols('C2') + + +def test_sift(): + assert sift(list(range(5)), lambda _: _ % 2) == {1: [1, 3], 0: [0, 2, 4]} + assert sift([x, y], lambda _: _.has(x)) == {False: [y], True: [x]} + assert sift([S.One], lambda _: _.has(x)) == {False: [1]} + assert sift([0, 1, 2, 3], lambda x: x % 2, binary=True) == ( + [1, 3], [0, 2]) + assert sift([0, 1, 2, 3], lambda x: x % 3 == 1, binary=True) == ( + [1], [0, 2, 3]) + raises(ValueError, lambda: + sift([0, 1, 2, 3], lambda x: x % 3, binary=True)) + + +def test_take(): + X = numbered_symbols() + + assert take(X, 5) == list(symbols('x0:5')) + assert take(X, 5) == list(symbols('x5:10')) + + assert take([1, 2, 3, 4, 5], 5) == [1, 2, 3, 4, 5] + + +def test_dict_merge(): + assert dict_merge({}, {1: x, y: z}) == {1: x, y: z} + assert dict_merge({1: x, y: z}, {}) == {1: x, y: z} + + assert dict_merge({2: z}, {1: x, y: z}) == {1: x, 2: z, y: z} + assert dict_merge({1: x, y: z}, {2: z}) == {1: x, 2: z, y: z} + + assert dict_merge({1: y, 2: z}, {1: x, y: z}) == {1: x, 2: z, y: z} + assert dict_merge({1: x, y: z}, {1: y, 2: z}) == {1: y, 2: z, y: z} + + +def test_prefixes(): + assert list(prefixes([])) == [] + assert list(prefixes([1])) == [[1]] + assert list(prefixes([1, 2])) == [[1], [1, 2]] + + assert list(prefixes([1, 2, 3, 4, 5])) == \ + [[1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 3, 4, 5]] + + +def test_postfixes(): + assert list(postfixes([])) == [] + assert list(postfixes([1])) == [[1]] + assert list(postfixes([1, 2])) == [[2], [1, 2]] + + assert list(postfixes([1, 2, 3, 4, 5])) == \ + [[5], [4, 5], [3, 4, 5], [2, 3, 4, 5], [1, 2, 3, 4, 5]] + + +def test_topological_sort(): + V = [2, 3, 5, 7, 8, 9, 10, 11] + E = [(7, 11), (7, 8), (5, 11), + (3, 8), (3, 10), (11, 2), + (11, 9), (11, 10), (8, 9)] + + assert topological_sort((V, E)) == [3, 5, 7, 8, 11, 2, 9, 10] + assert topological_sort((V, E), key=lambda v: -v) == \ + [7, 5, 11, 3, 10, 8, 9, 2] + + raises(ValueError, lambda: topological_sort((V, E + [(10, 7)]))) + + +def test_strongly_connected_components(): + assert strongly_connected_components(([], [])) == [] + assert strongly_connected_components(([1, 2, 3], [])) == [[1], [2], [3]] + + V = [1, 2, 3] + E = [(1, 2), (1, 3), (2, 1), (2, 3), (3, 1)] + assert strongly_connected_components((V, E)) == [[1, 2, 3]] + + V = [1, 2, 3, 4] + E = [(1, 2), (2, 3), (3, 2), (3, 4)] + assert strongly_connected_components((V, E)) == [[4], [2, 3], [1]] + + V = [1, 2, 3, 4] + E = [(1, 2), (2, 1), (3, 4), (4, 3)] + assert strongly_connected_components((V, E)) == [[1, 2], [3, 4]] + + +def test_connected_components(): + assert connected_components(([], [])) == [] + assert connected_components(([1, 2, 3], [])) == [[1], [2], [3]] + + V = [1, 2, 3] + E = [(1, 2), (1, 3), (2, 1), (2, 3), (3, 1)] + assert connected_components((V, E)) == [[1, 2, 3]] + + V = [1, 2, 3, 4] + E = [(1, 2), (2, 3), (3, 2), (3, 4)] + assert connected_components((V, E)) == [[1, 2, 3, 4]] + + V = [1, 2, 3, 4] + E = [(1, 2), (3, 4)] + assert connected_components((V, E)) == [[1, 2], [3, 4]] + + +def test_rotate(): + A = [0, 1, 2, 3, 4] + + assert rotate_left(A, 2) == [2, 3, 4, 0, 1] + assert rotate_right(A, 1) == [4, 0, 1, 2, 3] + A = [] + B = rotate_right(A, 1) + assert B == [] + B.append(1) + assert A == [] + B = rotate_left(A, 1) + assert B == [] + B.append(1) + assert A == [] + + +def test_multiset_partitions(): + A = [0, 1, 2, 3, 4] + + assert list(multiset_partitions(A, 5)) == [[[0], [1], [2], [3], [4]]] + assert len(list(multiset_partitions(A, 4))) == 10 + assert len(list(multiset_partitions(A, 3))) == 25 + + assert list(multiset_partitions([1, 1, 1, 2, 2], 2)) == [ + [[1, 1, 1, 2], [2]], [[1, 1, 1], [2, 2]], [[1, 1, 2, 2], [1]], + [[1, 1, 2], [1, 2]], [[1, 1], [1, 2, 2]]] + + assert list(multiset_partitions([1, 1, 2, 2], 2)) == [ + [[1, 1, 2], [2]], [[1, 1], [2, 2]], [[1, 2, 2], [1]], + [[1, 2], [1, 2]]] + + assert list(multiset_partitions([1, 2, 3, 4], 2)) == [ + [[1, 2, 3], [4]], [[1, 2, 4], [3]], [[1, 2], [3, 4]], + [[1, 3, 4], [2]], [[1, 3], [2, 4]], [[1, 4], [2, 3]], + [[1], [2, 3, 4]]] + + assert list(multiset_partitions([1, 2, 2], 2)) == [ + [[1, 2], [2]], [[1], [2, 2]]] + + assert list(multiset_partitions(3)) == [ + [[0, 1, 2]], [[0, 1], [2]], [[0, 2], [1]], [[0], [1, 2]], + [[0], [1], [2]]] + assert list(multiset_partitions(3, 2)) == [ + [[0, 1], [2]], [[0, 2], [1]], [[0], [1, 2]]] + assert list(multiset_partitions([1] * 3, 2)) == [[[1], [1, 1]]] + assert list(multiset_partitions([1] * 3)) == [ + [[1, 1, 1]], [[1], [1, 1]], [[1], [1], [1]]] + a = [3, 2, 1] + assert list(multiset_partitions(a)) == \ + list(multiset_partitions(sorted(a))) + assert list(multiset_partitions(a, 5)) == [] + assert list(multiset_partitions(a, 1)) == [[[1, 2, 3]]] + assert list(multiset_partitions(a + [4], 5)) == [] + assert list(multiset_partitions(a + [4], 1)) == [[[1, 2, 3, 4]]] + assert list(multiset_partitions(2, 5)) == [] + assert list(multiset_partitions(2, 1)) == [[[0, 1]]] + assert list(multiset_partitions('a')) == [[['a']]] + assert list(multiset_partitions('a', 2)) == [] + assert list(multiset_partitions('ab')) == [[['a', 'b']], [['a'], ['b']]] + assert list(multiset_partitions('ab', 1)) == [[['a', 'b']]] + assert list(multiset_partitions('aaa', 1)) == [['aaa']] + assert list(multiset_partitions([1, 1], 1)) == [[[1, 1]]] + ans = [('mpsyy',), ('mpsy', 'y'), ('mps', 'yy'), ('mps', 'y', 'y'), + ('mpyy', 's'), ('mpy', 'sy'), ('mpy', 's', 'y'), ('mp', 'syy'), + ('mp', 'sy', 'y'), ('mp', 's', 'yy'), ('mp', 's', 'y', 'y'), + ('msyy', 'p'), ('msy', 'py'), ('msy', 'p', 'y'), ('ms', 'pyy'), + ('ms', 'py', 'y'), ('ms', 'p', 'yy'), ('ms', 'p', 'y', 'y'), + ('myy', 'ps'), ('myy', 'p', 's'), ('my', 'psy'), ('my', 'ps', 'y'), + ('my', 'py', 's'), ('my', 'p', 'sy'), ('my', 'p', 's', 'y'), + ('m', 'psyy'), ('m', 'psy', 'y'), ('m', 'ps', 'yy'), + ('m', 'ps', 'y', 'y'), ('m', 'pyy', 's'), ('m', 'py', 'sy'), + ('m', 'py', 's', 'y'), ('m', 'p', 'syy'), + ('m', 'p', 'sy', 'y'), ('m', 'p', 's', 'yy'), + ('m', 'p', 's', 'y', 'y')] + assert [tuple("".join(part) for part in p) + for p in multiset_partitions('sympy')] == ans + factorings = [[24], [8, 3], [12, 2], [4, 6], [4, 2, 3], + [6, 2, 2], [2, 2, 2, 3]] + assert [factoring_visitor(p, [2,3]) for + p in multiset_partitions_taocp([3, 1])] == factorings + + +def test_multiset_combinations(): + ans = ['iii', 'iim', 'iip', 'iis', 'imp', 'ims', 'ipp', 'ips', + 'iss', 'mpp', 'mps', 'mss', 'pps', 'pss', 'sss'] + assert [''.join(i) for i in + list(multiset_combinations('mississippi', 3))] == ans + M = multiset('mississippi') + assert [''.join(i) for i in + list(multiset_combinations(M, 3))] == ans + assert [''.join(i) for i in multiset_combinations(M, 30)] == [] + assert list(multiset_combinations([[1], [2, 3]], 2)) == [[[1], [2, 3]]] + assert len(list(multiset_combinations('a', 3))) == 0 + assert len(list(multiset_combinations('a', 0))) == 1 + assert list(multiset_combinations('abc', 1)) == [['a'], ['b'], ['c']] + raises(ValueError, lambda: list(multiset_combinations({0: 3, 1: -1}, 2))) + + +def test_multiset_permutations(): + ans = ['abby', 'abyb', 'aybb', 'baby', 'bayb', 'bbay', 'bbya', 'byab', + 'byba', 'yabb', 'ybab', 'ybba'] + assert [''.join(i) for i in multiset_permutations('baby')] == ans + assert [''.join(i) for i in multiset_permutations(multiset('baby'))] == ans + assert list(multiset_permutations([0, 0, 0], 2)) == [[0, 0]] + assert list(multiset_permutations([0, 2, 1], 2)) == [ + [0, 1], [0, 2], [1, 0], [1, 2], [2, 0], [2, 1]] + assert len(list(multiset_permutations('a', 0))) == 1 + assert len(list(multiset_permutations('a', 3))) == 0 + for nul in ([], {}, ''): + assert list(multiset_permutations(nul)) == [[]] + assert list(multiset_permutations(nul, 0)) == [[]] + # impossible requests give no result + assert list(multiset_permutations(nul, 1)) == [] + assert list(multiset_permutations(nul, -1)) == [] + + def test(): + for i in range(1, 7): + print(i) + for p in multiset_permutations([0, 0, 1, 0, 1], i): + print(p) + assert capture(lambda: test()) == dedent('''\ + 1 + [0] + [1] + 2 + [0, 0] + [0, 1] + [1, 0] + [1, 1] + 3 + [0, 0, 0] + [0, 0, 1] + [0, 1, 0] + [0, 1, 1] + [1, 0, 0] + [1, 0, 1] + [1, 1, 0] + 4 + [0, 0, 0, 1] + [0, 0, 1, 0] + [0, 0, 1, 1] + [0, 1, 0, 0] + [0, 1, 0, 1] + [0, 1, 1, 0] + [1, 0, 0, 0] + [1, 0, 0, 1] + [1, 0, 1, 0] + [1, 1, 0, 0] + 5 + [0, 0, 0, 1, 1] + [0, 0, 1, 0, 1] + [0, 0, 1, 1, 0] + [0, 1, 0, 0, 1] + [0, 1, 0, 1, 0] + [0, 1, 1, 0, 0] + [1, 0, 0, 0, 1] + [1, 0, 0, 1, 0] + [1, 0, 1, 0, 0] + [1, 1, 0, 0, 0] + 6\n''') + raises(ValueError, lambda: list(multiset_permutations({0: 3, 1: -1}))) + + +def test_partitions(): + ans = [[{}], [(0, {})]] + for i in range(2): + assert list(partitions(0, size=i)) == ans[i] + assert list(partitions(1, 0, size=i)) == ans[i] + assert list(partitions(6, 2, 2, size=i)) == ans[i] + assert list(partitions(6, 2, None, size=i)) != ans[i] + assert list(partitions(6, None, 2, size=i)) != ans[i] + assert list(partitions(6, 2, 0, size=i)) == ans[i] + + assert list(partitions(6, k=2)) == [ + {2: 3}, {1: 2, 2: 2}, {1: 4, 2: 1}, {1: 6}] + + assert list(partitions(6, k=3)) == [ + {3: 2}, {1: 1, 2: 1, 3: 1}, {1: 3, 3: 1}, {2: 3}, {1: 2, 2: 2}, + {1: 4, 2: 1}, {1: 6}] + + assert list(partitions(8, k=4, m=3)) == [ + {4: 2}, {1: 1, 3: 1, 4: 1}, {2: 2, 4: 1}, {2: 1, 3: 2}] == [ + i for i in partitions(8, k=4, m=3) if all(k <= 4 for k in i) + and sum(i.values()) <=3] + + assert list(partitions(S(3), m=2)) == [ + {3: 1}, {1: 1, 2: 1}] + + assert list(partitions(4, k=3)) == [ + {1: 1, 3: 1}, {2: 2}, {1: 2, 2: 1}, {1: 4}] == [ + i for i in partitions(4) if all(k <= 3 for k in i)] + + + # Consistency check on output of _partitions and RGS_unrank. + # This provides a sanity test on both routines. Also verifies that + # the total number of partitions is the same in each case. + # (from pkrathmann2) + + for n in range(2, 6): + i = 0 + for m, q in _set_partitions(n): + assert q == RGS_unrank(i, n) + i += 1 + assert i == RGS_enum(n) + + +def test_binary_partitions(): + assert [i[:] for i in binary_partitions(10)] == [[8, 2], [8, 1, 1], + [4, 4, 2], [4, 4, 1, 1], [4, 2, 2, 2], [4, 2, 2, 1, 1], + [4, 2, 1, 1, 1, 1], [4, 1, 1, 1, 1, 1, 1], [2, 2, 2, 2, 2], + [2, 2, 2, 2, 1, 1], [2, 2, 2, 1, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1, 1], + [2, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] + + assert len([j[:] for j in binary_partitions(16)]) == 36 + + +def test_bell_perm(): + assert [len(set(generate_bell(i))) for i in range(1, 7)] == [ + factorial(i) for i in range(1, 7)] + assert list(generate_bell(3)) == [ + (0, 1, 2), (0, 2, 1), (2, 0, 1), (2, 1, 0), (1, 2, 0), (1, 0, 2)] + # generate_bell and trotterjohnson are advertised to return the same + # permutations; this is not technically necessary so this test could + # be removed + for n in range(1, 5): + p = Permutation(range(n)) + b = generate_bell(n) + for bi in b: + assert bi == tuple(p.array_form) + p = p.next_trotterjohnson() + raises(ValueError, lambda: list(generate_bell(0))) # XXX is this consistent with other permutation algorithms? + + +def test_involutions(): + lengths = [1, 2, 4, 10, 26, 76] + for n, N in enumerate(lengths): + i = list(generate_involutions(n + 1)) + assert len(i) == N + assert len({Permutation(j)**2 for j in i}) == 1 + + +def test_derangements(): + assert len(list(generate_derangements(list(range(6))))) == 265 + assert ''.join(''.join(i) for i in generate_derangements('abcde')) == ( + 'badecbaecdbcaedbcdeabceadbdaecbdeacbdecabeacdbedacbedcacabedcadebcaebd' + 'cdaebcdbeacdeabcdebaceabdcebadcedabcedbadabecdaebcdaecbdcaebdcbeadceab' + 'dcebadeabcdeacbdebacdebcaeabcdeadbceadcbecabdecbadecdabecdbaedabcedacb' + 'edbacedbca') + assert list(generate_derangements([0, 1, 2, 3])) == [ + [1, 0, 3, 2], [1, 2, 3, 0], [1, 3, 0, 2], [2, 0, 3, 1], + [2, 3, 0, 1], [2, 3, 1, 0], [3, 0, 1, 2], [3, 2, 0, 1], [3, 2, 1, 0]] + assert list(generate_derangements([0, 1, 2, 2])) == [ + [2, 2, 0, 1], [2, 2, 1, 0]] + assert list(generate_derangements('ba')) == [list('ab')] + # multiset_derangements + D = multiset_derangements + assert list(D('abb')) == [] + assert [''.join(i) for i in D('ab')] == ['ba'] + assert [''.join(i) for i in D('abc')] == ['bca', 'cab'] + assert [''.join(i) for i in D('aabb')] == ['bbaa'] + assert [''.join(i) for i in D('aabbcccc')] == [ + 'ccccaabb', 'ccccabab', 'ccccabba', 'ccccbaab', 'ccccbaba', + 'ccccbbaa'] + assert [''.join(i) for i in D('aabbccc')] == [ + 'cccabba', 'cccabab', 'cccaabb', 'ccacbba', 'ccacbab', + 'ccacabb', 'cbccbaa', 'cbccaba', 'cbccaab', 'bcccbaa', + 'bcccaba', 'bcccaab'] + assert [''.join(i) for i in D('books')] == ['kbsoo', 'ksboo', + 'sbkoo', 'skboo', 'oksbo', 'oskbo', 'okbso', 'obkso', 'oskob', + 'oksob', 'osbok', 'obsok'] + assert list(generate_derangements([[3], [2], [2], [1]])) == [ + [[2], [1], [3], [2]], [[2], [3], [1], [2]]] + + +def test_necklaces(): + def count(n, k, f): + return len(list(necklaces(n, k, f))) + m = [] + for i in range(1, 8): + m.append(( + i, count(i, 2, 0), count(i, 2, 1), count(i, 3, 1))) + assert Matrix(m) == Matrix([ + [1, 2, 2, 3], + [2, 3, 3, 6], + [3, 4, 4, 10], + [4, 6, 6, 21], + [5, 8, 8, 39], + [6, 14, 13, 92], + [7, 20, 18, 198]]) + + +def test_bracelets(): + bc = list(bracelets(2, 4)) + assert Matrix(bc) == Matrix([ + [0, 0], + [0, 1], + [0, 2], + [0, 3], + [1, 1], + [1, 2], + [1, 3], + [2, 2], + [2, 3], + [3, 3] + ]) + bc = list(bracelets(4, 2)) + assert Matrix(bc) == Matrix([ + [0, 0, 0, 0], + [0, 0, 0, 1], + [0, 0, 1, 1], + [0, 1, 0, 1], + [0, 1, 1, 1], + [1, 1, 1, 1] + ]) + + +def test_generate_oriented_forest(): + assert list(generate_oriented_forest(5)) == [[0, 1, 2, 3, 4], + [0, 1, 2, 3, 3], [0, 1, 2, 3, 2], [0, 1, 2, 3, 1], [0, 1, 2, 3, 0], + [0, 1, 2, 2, 2], [0, 1, 2, 2, 1], [0, 1, 2, 2, 0], [0, 1, 2, 1, 2], + [0, 1, 2, 1, 1], [0, 1, 2, 1, 0], [0, 1, 2, 0, 1], [0, 1, 2, 0, 0], + [0, 1, 1, 1, 1], [0, 1, 1, 1, 0], [0, 1, 1, 0, 1], [0, 1, 1, 0, 0], + [0, 1, 0, 1, 0], [0, 1, 0, 0, 0], [0, 0, 0, 0, 0]] + assert len(list(generate_oriented_forest(10))) == 1842 + + +def test_unflatten(): + r = list(range(10)) + assert unflatten(r) == list(zip(r[::2], r[1::2])) + assert unflatten(r, 5) == [tuple(r[:5]), tuple(r[5:])] + raises(ValueError, lambda: unflatten(list(range(10)), 3)) + raises(ValueError, lambda: unflatten(list(range(10)), -2)) + + +def test_common_prefix_suffix(): + assert common_prefix([], [1]) == [] + assert common_prefix(list(range(3))) == [0, 1, 2] + assert common_prefix(list(range(3)), list(range(4))) == [0, 1, 2] + assert common_prefix([1, 2, 3], [1, 2, 5]) == [1, 2] + assert common_prefix([1, 2, 3], [1, 3, 5]) == [1] + + assert common_suffix([], [1]) == [] + assert common_suffix(list(range(3))) == [0, 1, 2] + assert common_suffix(list(range(3)), list(range(3))) == [0, 1, 2] + assert common_suffix(list(range(3)), list(range(4))) == [] + assert common_suffix([1, 2, 3], [9, 2, 3]) == [2, 3] + assert common_suffix([1, 2, 3], [9, 7, 3]) == [3] + + +def test_minlex(): + assert minlex([1, 2, 0]) == (0, 1, 2) + assert minlex((1, 2, 0)) == (0, 1, 2) + assert minlex((1, 0, 2)) == (0, 2, 1) + assert minlex((1, 0, 2), directed=False) == (0, 1, 2) + assert minlex('aba') == 'aab' + assert minlex(('bb', 'aaa', 'c', 'a'), key=len) == ('c', 'a', 'bb', 'aaa') + + +def test_ordered(): + assert list(ordered((x, y), hash, default=False)) in [[x, y], [y, x]] + assert list(ordered((x, y), hash, default=False)) == \ + list(ordered((y, x), hash, default=False)) + assert list(ordered((x, y))) == [x, y] + + seq, keys = [[[1, 2, 1], [0, 3, 1], [1, 1, 3], [2], [1]], + (lambda x: len(x), lambda x: sum(x))] + assert list(ordered(seq, keys, default=False, warn=False)) == \ + [[1], [2], [1, 2, 1], [0, 3, 1], [1, 1, 3]] + raises(ValueError, lambda: + list(ordered(seq, keys, default=False, warn=True))) + + +def test_runs(): + assert runs([]) == [] + assert runs([1]) == [[1]] + assert runs([1, 1]) == [[1], [1]] + assert runs([1, 1, 2]) == [[1], [1, 2]] + assert runs([1, 2, 1]) == [[1, 2], [1]] + assert runs([2, 1, 1]) == [[2], [1], [1]] + from operator import lt + assert runs([2, 1, 1], lt) == [[2, 1], [1]] + + +def test_reshape(): + seq = list(range(1, 9)) + assert reshape(seq, [4]) == \ + [[1, 2, 3, 4], [5, 6, 7, 8]] + assert reshape(seq, (4,)) == \ + [(1, 2, 3, 4), (5, 6, 7, 8)] + assert reshape(seq, (2, 2)) == \ + [(1, 2, 3, 4), (5, 6, 7, 8)] + assert reshape(seq, (2, [2])) == \ + [(1, 2, [3, 4]), (5, 6, [7, 8])] + assert reshape(seq, ((2,), [2])) == \ + [((1, 2), [3, 4]), ((5, 6), [7, 8])] + assert reshape(seq, (1, [2], 1)) == \ + [(1, [2, 3], 4), (5, [6, 7], 8)] + assert reshape(tuple(seq), ([[1], 1, (2,)],)) == \ + (([[1], 2, (3, 4)],), ([[5], 6, (7, 8)],)) + assert reshape(tuple(seq), ([1], 1, (2,))) == \ + (([1], 2, (3, 4)), ([5], 6, (7, 8))) + assert reshape(list(range(12)), [2, [3], {2}, (1, (3,), 1)]) == \ + [[0, 1, [2, 3, 4], {5, 6}, (7, (8, 9, 10), 11)]] + raises(ValueError, lambda: reshape([0, 1], [-1])) + raises(ValueError, lambda: reshape([0, 1], [3])) + + +def test_uniq(): + assert list(uniq(p for p in partitions(4))) == \ + [{4: 1}, {1: 1, 3: 1}, {2: 2}, {1: 2, 2: 1}, {1: 4}] + assert list(uniq(x % 2 for x in range(5))) == [0, 1] + assert list(uniq('a')) == ['a'] + assert list(uniq('ababc')) == list('abc') + assert list(uniq([[1], [2, 1], [1]])) == [[1], [2, 1]] + assert list(uniq(permutations(i for i in [[1], 2, 2]))) == \ + [([1], 2, 2), (2, [1], 2), (2, 2, [1])] + assert list(uniq([2, 3, 2, 4, [2], [1], [2], [3], [1]])) == \ + [2, 3, 4, [2], [1], [3]] + f = [1] + raises(RuntimeError, lambda: [f.remove(i) for i in uniq(f)]) + f = [[1]] + raises(RuntimeError, lambda: [f.remove(i) for i in uniq(f)]) + + +def test_kbins(): + assert len(list(kbins('1123', 2, ordered=1))) == 24 + assert len(list(kbins('1123', 2, ordered=11))) == 36 + assert len(list(kbins('1123', 2, ordered=10))) == 10 + assert len(list(kbins('1123', 2, ordered=0))) == 5 + assert len(list(kbins('1123', 2, ordered=None))) == 3 + + def test1(): + for orderedval in [None, 0, 1, 10, 11]: + print('ordered =', orderedval) + for p in kbins([0, 0, 1], 2, ordered=orderedval): + print(' ', p) + assert capture(lambda : test1()) == dedent('''\ + ordered = None + [[0], [0, 1]] + [[0, 0], [1]] + ordered = 0 + [[0, 0], [1]] + [[0, 1], [0]] + ordered = 1 + [[0], [0, 1]] + [[0], [1, 0]] + [[1], [0, 0]] + ordered = 10 + [[0, 0], [1]] + [[1], [0, 0]] + [[0, 1], [0]] + [[0], [0, 1]] + ordered = 11 + [[0], [0, 1]] + [[0, 0], [1]] + [[0], [1, 0]] + [[0, 1], [0]] + [[1], [0, 0]] + [[1, 0], [0]]\n''') + + def test2(): + for orderedval in [None, 0, 1, 10, 11]: + print('ordered =', orderedval) + for p in kbins(list(range(3)), 2, ordered=orderedval): + print(' ', p) + assert capture(lambda : test2()) == dedent('''\ + ordered = None + [[0], [1, 2]] + [[0, 1], [2]] + ordered = 0 + [[0, 1], [2]] + [[0, 2], [1]] + [[0], [1, 2]] + ordered = 1 + [[0], [1, 2]] + [[0], [2, 1]] + [[1], [0, 2]] + [[1], [2, 0]] + [[2], [0, 1]] + [[2], [1, 0]] + ordered = 10 + [[0, 1], [2]] + [[2], [0, 1]] + [[0, 2], [1]] + [[1], [0, 2]] + [[0], [1, 2]] + [[1, 2], [0]] + ordered = 11 + [[0], [1, 2]] + [[0, 1], [2]] + [[0], [2, 1]] + [[0, 2], [1]] + [[1], [0, 2]] + [[1, 0], [2]] + [[1], [2, 0]] + [[1, 2], [0]] + [[2], [0, 1]] + [[2, 0], [1]] + [[2], [1, 0]] + [[2, 1], [0]]\n''') + + +def test_has_dups(): + assert has_dups(set()) is False + assert has_dups(list(range(3))) is False + assert has_dups([1, 2, 1]) is True + assert has_dups([[1], [1]]) is True + assert has_dups([[1], [2]]) is False + + +def test__partition(): + assert _partition('abcde', [1, 0, 1, 2, 0]) == [ + ['b', 'e'], ['a', 'c'], ['d']] + assert _partition('abcde', [1, 0, 1, 2, 0], 3) == [ + ['b', 'e'], ['a', 'c'], ['d']] + output = (3, [1, 0, 1, 2, 0]) + assert _partition('abcde', *output) == [['b', 'e'], ['a', 'c'], ['d']] + + +def test_ordered_partitions(): + from sympy.functions.combinatorial.numbers import nT + f = ordered_partitions + assert list(f(0, 1)) == [[]] + assert list(f(1, 0)) == [[]] + for i in range(1, 7): + for j in [None] + list(range(1, i)): + assert ( + sum(1 for p in f(i, j, 1)) == + sum(1 for p in f(i, j, 0)) == + nT(i, j)) + + +def test_rotations(): + assert list(rotations('ab')) == [['a', 'b'], ['b', 'a']] + assert list(rotations(range(3))) == [[0, 1, 2], [1, 2, 0], [2, 0, 1]] + assert list(rotations(range(3), dir=-1)) == [[0, 1, 2], [2, 0, 1], [1, 2, 0]] + + +def test_ibin(): + assert ibin(3) == [1, 1] + assert ibin(3, 3) == [0, 1, 1] + assert ibin(3, str=True) == '11' + assert ibin(3, 3, str=True) == '011' + assert list(ibin(2, 'all')) == [(0, 0), (0, 1), (1, 0), (1, 1)] + assert list(ibin(2, '', str=True)) == ['00', '01', '10', '11'] + raises(ValueError, lambda: ibin(-.5)) + raises(ValueError, lambda: ibin(2, 1)) + + +def test_iterable(): + assert iterable(0) is False + assert iterable(1) is False + assert iterable(None) is False + + class Test1(NotIterable): + pass + + assert iterable(Test1()) is False + + class Test2(NotIterable): + _iterable = True + + assert iterable(Test2()) is True + + class Test3: + pass + + assert iterable(Test3()) is False + + class Test4: + _iterable = True + + assert iterable(Test4()) is True + + class Test5: + def __iter__(self): + yield 1 + + assert iterable(Test5()) is True + + class Test6(Test5): + _iterable = False + + assert iterable(Test6()) is False + + +def test_sequence_partitions(): + assert list(sequence_partitions([1], 1)) == [[[1]]] + assert list(sequence_partitions([1, 2], 1)) == [[[1, 2]]] + assert list(sequence_partitions([1, 2], 2)) == [[[1], [2]]] + assert list(sequence_partitions([1, 2, 3], 1)) == [[[1, 2, 3]]] + assert list(sequence_partitions([1, 2, 3], 2)) == \ + [[[1], [2, 3]], [[1, 2], [3]]] + assert list(sequence_partitions([1, 2, 3], 3)) == [[[1], [2], [3]]] + + # Exceptional cases + assert list(sequence_partitions([], 0)) == [] + assert list(sequence_partitions([], 1)) == [] + assert list(sequence_partitions([1, 2], 0)) == [] + assert list(sequence_partitions([1, 2], 3)) == [] + + +def test_sequence_partitions_empty(): + assert list(sequence_partitions_empty([], 1)) == [[[]]] + assert list(sequence_partitions_empty([], 2)) == [[[], []]] + assert list(sequence_partitions_empty([], 3)) == [[[], [], []]] + assert list(sequence_partitions_empty([1], 1)) == [[[1]]] + assert list(sequence_partitions_empty([1], 2)) == [[[], [1]], [[1], []]] + assert list(sequence_partitions_empty([1], 3)) == \ + [[[], [], [1]], [[], [1], []], [[1], [], []]] + assert list(sequence_partitions_empty([1, 2], 1)) == [[[1, 2]]] + assert list(sequence_partitions_empty([1, 2], 2)) == \ + [[[], [1, 2]], [[1], [2]], [[1, 2], []]] + assert list(sequence_partitions_empty([1, 2], 3)) == [ + [[], [], [1, 2]], [[], [1], [2]], [[], [1, 2], []], + [[1], [], [2]], [[1], [2], []], [[1, 2], [], []] + ] + assert list(sequence_partitions_empty([1, 2, 3], 1)) == [[[1, 2, 3]]] + assert list(sequence_partitions_empty([1, 2, 3], 2)) == \ + [[[], [1, 2, 3]], [[1], [2, 3]], [[1, 2], [3]], [[1, 2, 3], []]] + assert list(sequence_partitions_empty([1, 2, 3], 3)) == [ + [[], [], [1, 2, 3]], [[], [1], [2, 3]], + [[], [1, 2], [3]], [[], [1, 2, 3], []], + [[1], [], [2, 3]], [[1], [2], [3]], + [[1], [2, 3], []], [[1, 2], [], [3]], + [[1, 2], [3], []], [[1, 2, 3], [], []] + ] + + # Exceptional cases + assert list(sequence_partitions([], 0)) == [] + assert list(sequence_partitions([1], 0)) == [] + assert list(sequence_partitions([1, 2], 0)) == [] + + +def test_signed_permutations(): + ans = [(0, 1, 1), (0, -1, 1), (0, 1, -1), (0, -1, -1), + (1, 0, 1), (-1, 0, 1), (1, 0, -1), (-1, 0, -1), + (1, 1, 0), (-1, 1, 0), (1, -1, 0), (-1, -1, 0)] + assert list(signed_permutations((0, 1, 1))) == ans + assert list(signed_permutations((1, 0, 1))) == ans + assert list(signed_permutations((1, 1, 0))) == ans diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_lambdify.py b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_lambdify.py new file mode 100644 index 0000000000000000000000000000000000000000..b094c67d39f09c24bcb9abc1e755cb5328e143e7 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_lambdify.py @@ -0,0 +1,2263 @@ +from itertools import product +import math +import inspect +import linecache +import gc + +import mpmath +import cmath + +from sympy.testing.pytest import raises, warns_deprecated_sympy +from sympy.concrete.summations import Sum +from sympy.core.function import (Function, Lambda, diff) +from sympy.core.numbers import (E, Float, I, Rational, all_close, oo, pi) +from sympy.core.relational import Eq +from sympy.core.singleton import S +from sympy.core.symbol import (Dummy, symbols) +from sympy.functions.combinatorial.factorials import (RisingFactorial, factorial) +from sympy.functions.combinatorial.numbers import bernoulli, harmonic +from sympy.functions.elementary.complexes import Abs, sign +from sympy.functions.elementary.exponential import exp, log +from sympy.functions.elementary.hyperbolic import asinh,acosh,atanh +from sympy.functions.elementary.integers import floor +from sympy.functions.elementary.miscellaneous import (Max, Min, sqrt) +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.elementary.trigonometric import (asin, acos, atan, cos, cot, sin, + sinc, tan) +from sympy.functions import sinh,cosh,tanh +from sympy.functions.special.bessel import (besseli, besselj, besselk, bessely, jn, yn) +from sympy.functions.special.beta_functions import (beta, betainc, betainc_regularized) +from sympy.functions.special.delta_functions import (Heaviside) +from sympy.functions.special.error_functions import (Ei, erf, erfc, fresnelc, fresnels, Si, Ci) +from sympy.functions.special.gamma_functions import (digamma, gamma, loggamma, polygamma) +from sympy.functions.special.zeta_functions import zeta +from sympy.integrals.integrals import Integral +from sympy.logic.boolalg import (And, false, ITE, Not, Or, true) +from sympy.matrices.expressions.dotproduct import DotProduct +from sympy.simplify.cse_main import cse +from sympy.tensor.array import derive_by_array, Array +from sympy.tensor.array.expressions import ArraySymbol +from sympy.tensor.indexed import IndexedBase, Idx +from sympy.utilities.lambdify import lambdify +from sympy.utilities.iterables import numbered_symbols +from sympy.vector import CoordSys3D +from sympy.core.expr import UnevaluatedExpr +from sympy.codegen.cfunctions import expm1, log1p, exp2, log2, log10, hypot, isnan, isinf +from sympy.codegen.numpy_nodes import logaddexp, logaddexp2, amin, amax, minimum, maximum +from sympy.codegen.scipy_nodes import cosm1, powm1 +from sympy.functions.elementary.complexes import re, im, arg +from sympy.functions.special.polynomials import \ + chebyshevt, chebyshevu, legendre, hermite, laguerre, gegenbauer, \ + assoc_legendre, assoc_laguerre, jacobi +from sympy.matrices import Matrix, MatrixSymbol, SparseMatrix +from sympy.printing.codeprinter import PrintMethodNotImplementedError +from sympy.printing.lambdarepr import LambdaPrinter +from sympy.printing.numpy import NumPyPrinter +from sympy.utilities.lambdify import implemented_function, lambdastr +from sympy.testing.pytest import skip +from sympy.utilities.decorator import conserve_mpmath_dps +from sympy.utilities.exceptions import ignore_warnings +from sympy.external import import_module +from sympy.functions.special.gamma_functions import uppergamma, lowergamma + + +import sympy + + +MutableDenseMatrix = Matrix + +numpy = import_module('numpy') +scipy = import_module('scipy', import_kwargs={'fromlist': ['sparse']}) +numexpr = import_module('numexpr') +tensorflow = import_module('tensorflow') +cupy = import_module('cupy') +jax = import_module('jax') +numba = import_module('numba') + +if tensorflow: + # Hide Tensorflow warnings + import os + os.environ['TF_CPP_MIN_LOG_LEVEL'] = '2' + +w, x, y, z = symbols('w,x,y,z') + +#================== Test different arguments ======================= + + +def test_no_args(): + f = lambdify([], 1) + raises(TypeError, lambda: f(-1)) + assert f() == 1 + + +def test_single_arg(): + f = lambdify(x, 2*x) + assert f(1) == 2 + + +def test_list_args(): + f = lambdify([x, y], x + y) + assert f(1, 2) == 3 + + +def test_nested_args(): + f1 = lambdify([[w, x]], [w, x]) + assert f1([91, 2]) == [91, 2] + raises(TypeError, lambda: f1(1, 2)) + + f2 = lambdify([(w, x), (y, z)], [w, x, y, z]) + assert f2((18, 12), (73, 4)) == [18, 12, 73, 4] + raises(TypeError, lambda: f2(3, 4)) + + f3 = lambdify([w, [[[x]], y], z], [w, x, y, z]) + assert f3(10, [[[52]], 31], 44) == [10, 52, 31, 44] + + +def test_str_args(): + f = lambdify('x,y,z', 'z,y,x') + assert f(3, 2, 1) == (1, 2, 3) + assert f(1.0, 2.0, 3.0) == (3.0, 2.0, 1.0) + # make sure correct number of args required + raises(TypeError, lambda: f(0)) + + +def test_own_namespace_1(): + myfunc = lambda x: 1 + f = lambdify(x, sin(x), {"sin": myfunc}) + assert f(0.1) == 1 + assert f(100) == 1 + + +def test_own_namespace_2(): + def myfunc(x): + return 1 + f = lambdify(x, sin(x), {'sin': myfunc}) + assert f(0.1) == 1 + assert f(100) == 1 + + +def test_own_module(): + f = lambdify(x, sin(x), math) + assert f(0) == 0.0 + + p, q, r = symbols("p q r", real=True) + ae = abs(exp(p+UnevaluatedExpr(q+r))) + f = lambdify([p, q, r], [ae, ae], modules=math) + results = f(1.0, 1e18, -1e18) + refvals = [math.exp(1.0)]*2 + for res, ref in zip(results, refvals): + assert abs((res-ref)/ref) < 1e-15 + + +def test_bad_args(): + # no vargs given + raises(TypeError, lambda: lambdify(1)) + # same with vector exprs + raises(TypeError, lambda: lambdify([1, 2])) + + +def test_atoms(): + # Non-Symbol atoms should not be pulled out from the expression namespace + f = lambdify(x, pi + x, {"pi": 3.14}) + assert f(0) == 3.14 + f = lambdify(x, I + x, {"I": 1j}) + assert f(1) == 1 + 1j + +#================== Test different modules ========================= + +# high precision output of sin(0.2*pi) is used to detect if precision is lost unwanted + + +@conserve_mpmath_dps +def test_sympy_lambda(): + mpmath.mp.dps = 50 + sin02 = mpmath.mpf("0.19866933079506121545941262711838975037020672954020") + f = lambdify(x, sin(x), "sympy") + assert f(x) == sin(x) + prec = 1e-15 + assert -prec < f(Rational(1, 5)).evalf() - Float(str(sin02)) < prec + # arctan is in numpy module and should not be available + # The arctan below gives NameError. What is this supposed to test? + # raises(NameError, lambda: lambdify(x, arctan(x), "sympy")) + + +@conserve_mpmath_dps +def test_math_lambda(): + mpmath.mp.dps = 50 + sin02 = mpmath.mpf("0.19866933079506121545941262711838975037020672954020") + f = lambdify(x, sin(x), "math") + prec = 1e-15 + assert -prec < f(0.2) - sin02 < prec + raises(TypeError, lambda: f(x)) + # if this succeeds, it can't be a Python math function + + +@conserve_mpmath_dps +def test_mpmath_lambda(): + mpmath.mp.dps = 50 + sin02 = mpmath.mpf("0.19866933079506121545941262711838975037020672954020") + f = lambdify(x, sin(x), "mpmath") + prec = 1e-49 # mpmath precision is around 50 decimal places + assert -prec < f(mpmath.mpf("0.2")) - sin02 < prec + raises(TypeError, lambda: f(x)) + # if this succeeds, it can't be a mpmath function + + ref2 = (mpmath.mpf("1e-30") + - mpmath.mpf("1e-45")/2 + + 5*mpmath.mpf("1e-60")/6 + - 3*mpmath.mpf("1e-75")/4 + + 33*mpmath.mpf("1e-90")/40 + ) + f2a = lambdify((x, y), x**y - 1, "mpmath") + f2b = lambdify((x, y), powm1(x, y), "mpmath") + f2c = lambdify((x,), expm1(x*log1p(x)), "mpmath") + ans2a = f2a(mpmath.mpf("1")+mpmath.mpf("1e-15"), mpmath.mpf("1e-15")) + ans2b = f2b(mpmath.mpf("1")+mpmath.mpf("1e-15"), mpmath.mpf("1e-15")) + ans2c = f2c(mpmath.mpf("1e-15")) + assert abs(ans2a - ref2) < 1e-51 + assert abs(ans2b - ref2) < 1e-67 + assert abs(ans2c - ref2) < 1e-80 + + +@conserve_mpmath_dps +def test_number_precision(): + mpmath.mp.dps = 50 + sin02 = mpmath.mpf("0.19866933079506121545941262711838975037020672954020") + f = lambdify(x, sin02, "mpmath") + prec = 1e-49 # mpmath precision is around 50 decimal places + assert -prec < f(0) - sin02 < prec + +@conserve_mpmath_dps +def test_mpmath_precision(): + mpmath.mp.dps = 100 + assert str(lambdify((), pi.evalf(100), 'mpmath')()) == str(pi.evalf(100)) + +#================== Test Translations ============================== +# We can only check if all translated functions are valid. It has to be checked +# by hand if they are complete. + + +def test_math_transl(): + from sympy.utilities.lambdify import MATH_TRANSLATIONS + for sym, mat in MATH_TRANSLATIONS.items(): + assert sym in sympy.__dict__ + assert mat in math.__dict__ + + +def test_mpmath_transl(): + from sympy.utilities.lambdify import MPMATH_TRANSLATIONS + for sym, mat in MPMATH_TRANSLATIONS.items(): + assert sym in sympy.__dict__ or sym == 'Matrix' + assert mat in mpmath.__dict__ + + +def test_numpy_transl(): + if not numpy: + skip("numpy not installed.") + + from sympy.utilities.lambdify import NUMPY_TRANSLATIONS + for sym, nump in NUMPY_TRANSLATIONS.items(): + assert sym in sympy.__dict__ + assert nump in numpy.__dict__ + + +def test_scipy_transl(): + if not scipy: + skip("scipy not installed.") + + from sympy.utilities.lambdify import SCIPY_TRANSLATIONS + for sym, scip in SCIPY_TRANSLATIONS.items(): + assert sym in sympy.__dict__ + assert scip in scipy.__dict__ or scip in scipy.special.__dict__ + + +def test_numpy_translation_abs(): + if not numpy: + skip("numpy not installed.") + + f = lambdify(x, Abs(x), "numpy") + assert f(-1) == 1 + assert f(1) == 1 + + +def test_numexpr_printer(): + if not numexpr: + skip("numexpr not installed.") + + # if translation/printing is done incorrectly then evaluating + # a lambdified numexpr expression will throw an exception + from sympy.printing.lambdarepr import NumExprPrinter + + blacklist = ('where', 'complex', 'contains') + arg_tuple = (x, y, z) # some functions take more than one argument + for sym in NumExprPrinter._numexpr_functions.keys(): + if sym in blacklist: + continue + ssym = S(sym) + if hasattr(ssym, '_nargs'): + nargs = ssym._nargs[0] + else: + nargs = 1 + args = arg_tuple[:nargs] + f = lambdify(args, ssym(*args), modules='numexpr') + assert f(*(1, )*nargs) is not None + + +def test_cmath_sqrt(): + f = lambdify(x, sqrt(x), "cmath") + assert f(0) == 0 + assert f(1) == 1 + assert f(4) == 2 + assert abs(f(2) - 1.414) < 0.001 + assert f(-1) == 1j + assert f(-4) == 2j + + +def test_cmath_log(): + f = lambdify(x, log(x), "cmath") + assert abs(f(1) - 0) < 1e-15 + assert abs(f(cmath.e) - 1) < 1e-15 + assert abs(f(-1) - cmath.log(-1)) < 1e-15 + + +def test_cmath_sinh(): + f = lambdify(x, sinh(x), "cmath") + assert abs(f(0) - cmath.sinh(0)) < 1e-15 + assert abs(f(pi) - cmath.sinh(pi)) < 1e-15 + assert abs(f(-pi) - cmath.sinh(-pi)) < 1e-15 + assert abs(f(1j) - cmath.sinh(1j)) < 1e-15 + + +def test_cmath_cosh(): + f = lambdify(x, cosh(x), "cmath") + assert abs(f(0) - cmath.cosh(0)) < 1e-15 + assert abs(f(pi) - cmath.cosh(pi)) < 1e-15 + assert abs(f(-pi) - cmath.cosh(-pi)) < 1e-15 + assert abs(f(1j) - cmath.cosh(1j)) < 1e-15 + + +def test_cmath_tanh(): + f = lambdify(x, tanh(x), "cmath") + assert abs(f(0) - cmath.tanh(0)) < 1e-15 + assert abs(f(pi) - cmath.tanh(pi)) < 1e-15 + assert abs(f(-pi) - cmath.tanh(-pi)) < 1e-15 + assert abs(f(1j) - cmath.tanh(1j)) < 1e-15 + + +def test_cmath_sin(): + f = lambdify(x, sin(x), "cmath") + assert abs(f(0) - cmath.sin(0)) < 1e-15 + assert abs(f(pi) - cmath.sin(pi)) < 1e-15 + assert abs(f(-pi) - cmath.sin(-pi)) < 1e-15 + assert abs(f(1j) - cmath.sin(1j)) < 1e-15 + + +def test_cmath_cos(): + f = lambdify(x, cos(x), "cmath") + assert abs(f(0) - cmath.cos(0)) < 1e-15 + assert abs(f(pi) - cmath.cos(pi)) < 1e-15 + assert abs(f(-pi) - cmath.cos(-pi)) < 1e-15 + assert abs(f(1j) - cmath.cos(1j)) < 1e-15 + + +def test_cmath_tan(): + f = lambdify(x, tan(x), "cmath") + assert abs(f(0) - cmath.tan(0)) < 1e-15 + assert abs(f(1j) - cmath.tan(1j)) < 1e-15 + + +def test_cmath_asin(): + f = lambdify(x, asin(x), "cmath") + assert abs(f(0) - cmath.asin(0)) < 1e-15 + assert abs(f(1) - cmath.asin(1)) < 1e-15 + assert abs(f(-1) - cmath.asin(-1)) < 1e-15 + assert abs(f(2) - cmath.asin(2)) < 1e-15 + assert abs(f(1j) - cmath.asin(1j)) < 1e-15 + + +def test_cmath_acos(): + f = lambdify(x, acos(x), "cmath") + assert abs(f(1) - cmath.acos(1)) < 1e-15 + assert abs(f(-1) - cmath.acos(-1)) < 1e-15 + assert abs(f(2) - cmath.acos(2)) < 1e-15 + assert abs(f(1j) - cmath.acos(1j)) < 1e-15 + + +def test_cmath_atan(): + f = lambdify(x, atan(x), "cmath") + assert abs(f(0) - cmath.atan(0)) < 1e-15 + assert abs(f(1) - cmath.atan(1)) < 1e-15 + assert abs(f(-1) - cmath.atan(-1)) < 1e-15 + assert abs(f(2) - cmath.atan(2)) < 1e-15 + assert abs(f(2j) - cmath.atan(2j)) < 1e-15 + + +def test_cmath_asinh(): + f = lambdify(x, asinh(x), "cmath") + assert abs(f(0) - cmath.asinh(0)) < 1e-15 + assert abs(f(1) - cmath.asinh(1)) < 1e-15 + assert abs(f(-1) - cmath.asinh(-1)) < 1e-15 + assert abs(f(2) - cmath.asinh(2)) < 1e-15 + assert abs(f(2j) - cmath.asinh(2j)) < 1e-15 + + +def test_cmath_acosh(): + f = lambdify(x, acosh(x), "cmath") + assert abs(f(1) - cmath.acosh(1)) < 1e-15 + assert abs(f(2) - cmath.acosh(2)) < 1e-15 + assert abs(f(-1) - cmath.acosh(-1)) < 1e-15 + assert abs(f(2j) - cmath.acosh(2j)) < 1e-15 + + +def test_cmath_atanh(): + f = lambdify(x, atanh(x), "cmath") + assert abs(f(0) - cmath.atanh(0)) < 1e-15 + assert abs(f(0.5) - cmath.atanh(0.5)) < 1e-15 + assert abs(f(-0.5) - cmath.atanh(-0.5)) < 1e-15 + assert abs(f(2) - cmath.atanh(2)) < 1e-15 + assert abs(f(-2) - cmath.atanh(-2)) < 1e-15 + assert abs(f(2j) - cmath.atanh(2j)) < 1e-15 + + +def test_cmath_complex_identities(): + # Define symbol + z = symbols('z') + + # Trigonometric identity using re(z) and im(z) + expr = cos(z) - cos(re(z)) * cosh(im(z)) + I * sin(re(z)) * sinh(im(z)) + func = lambdify([z], expr, modules=["cmath", "math"]) + hpi = math.pi / 2 + assert abs(func(hpi + 1j * hpi)) < 4e-16 + + # Euler's Formula: e^(i*z) = cos(z) + i*sin(z) + func = lambdify([z], exp(I * z) - (cos(z) + I * sin(z)), modules=["cmath", "math"]) + assert abs(func(hpi)) < 4e-16 + + # Exponential Identity: e^z = e^(Re(z)) * (cos(Im(z)) + i*sin(Im(z))) + func_exp = lambdify([z], exp(z) - exp(re(z)) * (cos(im(z)) + I * sin(im(z))), + modules=["cmath", "math"]) + assert abs(func_exp(hpi + 1j * hpi)) < 4e-16 + + # Complex Cosine Identity: cos(z) = cos(Re(z)) * cosh(Im(z)) - i*sin(Re(z)) * sinh(Im(z)) + func_cos = lambdify([z], cos(z) - (cos(re(z)) * cosh(im(z)) - I * sin(re(z)) * sinh(im(z))), + modules=["cmath", "math"]) + assert abs(func_cos(hpi + 1j * hpi)) < 4e-16 + + # Complex Sine Identity: sin(z) = sin(Re(z)) * cosh(Im(z)) + i*cos(Re(z)) * sinh(Im(z)) + func_sin = lambdify([z], sin(z) - (sin(re(z)) * cosh(im(z)) + I * cos(re(z)) * sinh(im(z))), + modules=["cmath", "math"]) + assert abs(func_sin(hpi + 1j * hpi)) < 4e-16 + + # Complex Hyperbolic Cosine Identity: cosh(z) = cosh(Re(z)) * cos(Im(z)) + i*sinh(Re(z)) * sin(Im(z)) + func_cosh_1 = lambdify([z], cosh(z) - (cosh(re(z)) * cos(im(z)) + I * sinh(re(z)) * sin(im(z))), + modules=["cmath", "math"]) + assert abs(func_cosh_1(hpi + 1j * hpi)) < 4e-16 + + # Complex Hyperbolic Sine Identity: sinh(z) = sinh(Re(z)) * cos(Im(z)) + i*cosh(Re(z)) * sin(Im(z)) + func_sinh = lambdify([z], sinh(z) - (sinh(re(z)) * cos(im(z)) + I * cosh(re(z)) * sin(im(z))), + modules=["cmath", "math"]) + assert abs(func_sinh(hpi + 1j * hpi)) < 4e-16 + + # cosh(z) = (e^z + e^(-z)) / 2 + func_cosh_2 = lambdify([z], cosh(z) - (exp(z) + exp(-z)) / 2, modules=["cmath", "math"]) + assert abs(func_cosh_2(hpi)) < 4e-16 + + # Additional expressions testing log and exp with real and imaginary parts + expr1 = log(re(z)) + log(im(z)) - log(re(z) * im(z)) + expr2 = exp(re(z)) * exp(im(z) * I) - exp(z) + expr3 = log(exp(re(z))) - re(z) + expr4 = exp(log(re(z))) - re(z) + expr5 = log(exp(re(z) + im(z))) - (re(z) + im(z)) + expr6 = exp(log(re(z) + im(z))) - (re(z) + im(z)) + func1 = lambdify([z], expr1, modules=["cmath", "math"]) + func2 = lambdify([z], expr2, modules=["cmath", "math"]) + func3 = lambdify([z], expr3, modules=["cmath", "math"]) + func4 = lambdify([z], expr4, modules=["cmath", "math"]) + func5 = lambdify([z], expr5, modules=["cmath", "math"]) + func6 = lambdify([z], expr6, modules=["cmath", "math"]) + test_value = 3 + 4j + assert abs(func1(test_value)) < 4e-16 + assert abs(func2(test_value)) < 4e-16 + assert abs(func3(test_value)) < 4e-16 + assert abs(func4(test_value)) < 4e-16 + assert abs(func5(test_value)) < 4e-16 + assert abs(func6(test_value)) < 4e-16 + + +def test_issue_9334(): + if not numexpr: + skip("numexpr not installed.") + if not numpy: + skip("numpy not installed.") + expr = S('b*a - sqrt(a**2)') + a, b = sorted(expr.free_symbols, key=lambda s: s.name) + func_numexpr = lambdify((a,b), expr, modules=[numexpr], dummify=False) + foo, bar = numpy.random.random((2, 4)) + func_numexpr(foo, bar) + + +def test_issue_12984(): + if not numexpr: + skip("numexpr not installed.") + func_numexpr = lambdify((x,y,z), Piecewise((y, x >= 0), (z, x > -1)), numexpr) + with ignore_warnings(RuntimeWarning): + assert func_numexpr(1, 24, 42) == 24 + assert str(func_numexpr(-1, 24, 42)) == 'nan' + + +def test_empty_modules(): + x, y = symbols('x y') + expr = -(x % y) + + no_modules = lambdify([x, y], expr) + empty_modules = lambdify([x, y], expr, modules=[]) + assert no_modules(3, 7) == empty_modules(3, 7) + assert no_modules(3, 7) == -3 + + +def test_exponentiation(): + f = lambdify(x, x**2) + assert f(-1) == 1 + assert f(0) == 0 + assert f(1) == 1 + assert f(-2) == 4 + assert f(2) == 4 + assert f(2.5) == 6.25 + + +def test_sqrt(): + f = lambdify(x, sqrt(x)) + assert f(0) == 0.0 + assert f(1) == 1.0 + assert f(4) == 2.0 + assert abs(f(2) - 1.414) < 0.001 + assert f(6.25) == 2.5 + + +def test_trig(): + f = lambdify([x], [cos(x), sin(x)], 'math') + d = f(pi) + prec = 1e-11 + assert -prec < d[0] + 1 < prec + assert -prec < d[1] < prec + d = f(3.14159) + prec = 1e-5 + assert -prec < d[0] + 1 < prec + assert -prec < d[1] < prec + + +def test_integral(): + if numpy and not scipy: + skip("scipy not installed.") + f = Lambda(x, exp(-x**2)) + l = lambdify(y, Integral(f(x), (x, y, oo))) + d = l(-oo) + assert 1.77245385 < d < 1.772453851 + + +def test_double_integral(): + if numpy and not scipy: + skip("scipy not installed.") + # example from http://mpmath.org/doc/current/calculus/integration.html + i = Integral(1/(1 - x**2*y**2), (x, 0, 1), (y, 0, z)) + l = lambdify([z], i) + d = l(1) + assert 1.23370055 < d < 1.233700551 + +def test_spherical_bessel(): + if numpy and not scipy: + skip("scipy not installed.") + test_point = 4.2 #randomly selected + x = symbols("x") + jtest = jn(2, x) + assert abs(lambdify(x,jtest)(test_point) - + jtest.subs(x,test_point).evalf()) < 1e-8 + ytest = yn(2, x) + assert abs(lambdify(x,ytest)(test_point) - + ytest.subs(x,test_point).evalf()) < 1e-8 + + +#================== Test vectors =================================== + + +def test_vector_simple(): + f = lambdify((x, y, z), (z, y, x)) + assert f(3, 2, 1) == (1, 2, 3) + assert f(1.0, 2.0, 3.0) == (3.0, 2.0, 1.0) + # make sure correct number of args required + raises(TypeError, lambda: f(0)) + + +def test_vector_discontinuous(): + f = lambdify(x, (-1/x, 1/x)) + raises(ZeroDivisionError, lambda: f(0)) + assert f(1) == (-1.0, 1.0) + assert f(2) == (-0.5, 0.5) + assert f(-2) == (0.5, -0.5) + + +def test_trig_symbolic(): + f = lambdify([x], [cos(x), sin(x)], 'math') + d = f(pi) + assert abs(d[0] + 1) < 0.0001 + assert abs(d[1] - 0) < 0.0001 + + +def test_trig_float(): + f = lambdify([x], [cos(x), sin(x)]) + d = f(3.14159) + assert abs(d[0] + 1) < 0.0001 + assert abs(d[1] - 0) < 0.0001 + + +def test_docs(): + f = lambdify(x, x**2) + assert f(2) == 4 + f = lambdify([x, y, z], [z, y, x]) + assert f(1, 2, 3) == [3, 2, 1] + f = lambdify(x, sqrt(x)) + assert f(4) == 2.0 + f = lambdify((x, y), sin(x*y)**2) + assert f(0, 5) == 0 + + +def test_math(): + f = lambdify((x, y), sin(x), modules="math") + assert f(0, 5) == 0 + + +def test_sin(): + f = lambdify(x, sin(x)**2) + assert isinstance(f(2), float) + f = lambdify(x, sin(x)**2, modules="math") + assert isinstance(f(2), float) + + +def test_matrix(): + A = Matrix([[x, x*y], [sin(z) + 4, x**z]]) + sol = Matrix([[1, 2], [sin(3) + 4, 1]]) + f = lambdify((x, y, z), A, modules="sympy") + assert f(1, 2, 3) == sol + f = lambdify((x, y, z), (A, [A]), modules="sympy") + assert f(1, 2, 3) == (sol, [sol]) + J = Matrix((x, x + y)).jacobian((x, y)) + v = Matrix((x, y)) + sol = Matrix([[1, 0], [1, 1]]) + assert lambdify(v, J, modules='sympy')(1, 2) == sol + assert lambdify(v.T, J, modules='sympy')(1, 2) == sol + + +def test_numpy_matrix(): + if not numpy: + skip("numpy not installed.") + A = Matrix([[x, x*y], [sin(z) + 4, x**z]]) + sol_arr = numpy.array([[1, 2], [numpy.sin(3) + 4, 1]]) + #Lambdify array first, to ensure return to array as default + f = lambdify((x, y, z), A, ['numpy']) + numpy.testing.assert_allclose(f(1, 2, 3), sol_arr) + #Check that the types are arrays and matrices + assert isinstance(f(1, 2, 3), numpy.ndarray) + + # gh-15071 + class dot(Function): + pass + x_dot_mtx = dot(x, Matrix([[2], [1], [0]])) + f_dot1 = lambdify(x, x_dot_mtx) + inp = numpy.zeros((17, 3)) + assert numpy.all(f_dot1(inp) == 0) + + strict_kw = {"allow_unknown_functions": False, "inline": True, "fully_qualified_modules": False} + p2 = NumPyPrinter(dict(user_functions={'dot': 'dot'}, **strict_kw)) + f_dot2 = lambdify(x, x_dot_mtx, printer=p2) + assert numpy.all(f_dot2(inp) == 0) + + p3 = NumPyPrinter(strict_kw) + # The line below should probably fail upon construction (before calling with "(inp)"): + raises(Exception, lambda: lambdify(x, x_dot_mtx, printer=p3)(inp)) + + +def test_numpy_transpose(): + if not numpy: + skip("numpy not installed.") + A = Matrix([[1, x], [0, 1]]) + f = lambdify((x), A.T, modules="numpy") + numpy.testing.assert_array_equal(f(2), numpy.array([[1, 0], [2, 1]])) + + +def test_numpy_dotproduct(): + if not numpy: + skip("numpy not installed") + A = Matrix([x, y, z]) + f1 = lambdify([x, y, z], DotProduct(A, A), modules='numpy') + f2 = lambdify([x, y, z], DotProduct(A, A.T), modules='numpy') + f3 = lambdify([x, y, z], DotProduct(A.T, A), modules='numpy') + f4 = lambdify([x, y, z], DotProduct(A, A.T), modules='numpy') + + assert f1(1, 2, 3) == \ + f2(1, 2, 3) == \ + f3(1, 2, 3) == \ + f4(1, 2, 3) == \ + numpy.array([14]) + + +def test_numpy_inverse(): + if not numpy: + skip("numpy not installed.") + A = Matrix([[1, x], [0, 1]]) + f = lambdify((x), A**-1, modules="numpy") + numpy.testing.assert_array_equal(f(2), numpy.array([[1, -2], [0, 1]])) + + +def test_numpy_old_matrix(): + if not numpy: + skip("numpy not installed.") + A = Matrix([[x, x*y], [sin(z) + 4, x**z]]) + sol_arr = numpy.array([[1, 2], [numpy.sin(3) + 4, 1]]) + f = lambdify((x, y, z), A, [{'ImmutableDenseMatrix': numpy.matrix}, 'numpy']) + with ignore_warnings(PendingDeprecationWarning): + numpy.testing.assert_allclose(f(1, 2, 3), sol_arr) + assert isinstance(f(1, 2, 3), numpy.matrix) + + +def test_scipy_sparse_matrix(): + if not scipy: + skip("scipy not installed.") + A = SparseMatrix([[x, 0], [0, y]]) + f = lambdify((x, y), A, modules="scipy") + B = f(1, 2) + assert isinstance(B, scipy.sparse.coo_matrix) + + +def test_python_div_zero_issue_11306(): + if not numpy: + skip("numpy not installed.") + p = Piecewise((1 / x, y < -1), (x, y < 1), (1 / x, True)) + f = lambdify([x, y], p, modules='numpy') + with numpy.errstate(divide='ignore'): + assert float(f(numpy.array(0), numpy.array(0.5))) == 0 + assert float(f(numpy.array(0), numpy.array(1))) == float('inf') + + +def test_issue9474(): + mods = [None, 'math'] + if numpy: + mods.append('numpy') + if mpmath: + mods.append('mpmath') + for mod in mods: + f = lambdify(x, S.One/x, modules=mod) + assert f(2) == 0.5 + f = lambdify(x, floor(S.One/x), modules=mod) + assert f(2) == 0 + + for absfunc, modules in product([Abs, abs], mods): + f = lambdify(x, absfunc(x), modules=modules) + assert f(-1) == 1 + assert f(1) == 1 + assert f(3+4j) == 5 + + +def test_issue_9871(): + if not numexpr: + skip("numexpr not installed.") + if not numpy: + skip("numpy not installed.") + + r = sqrt(x**2 + y**2) + expr = diff(1/r, x) + + xn = yn = numpy.linspace(1, 10, 16) + # expr(xn, xn) = -xn/(sqrt(2)*xn)^3 + fv_exact = -numpy.sqrt(2.)**-3 * xn**-2 + + fv_numpy = lambdify((x, y), expr, modules='numpy')(xn, yn) + fv_numexpr = lambdify((x, y), expr, modules='numexpr')(xn, yn) + numpy.testing.assert_allclose(fv_numpy, fv_exact, rtol=1e-10) + numpy.testing.assert_allclose(fv_numexpr, fv_exact, rtol=1e-10) + + +def test_numpy_piecewise(): + if not numpy: + skip("numpy not installed.") + pieces = Piecewise((x, x < 3), (x**2, x > 5), (0, True)) + f = lambdify(x, pieces, modules="numpy") + numpy.testing.assert_array_equal(f(numpy.arange(10)), + numpy.array([0, 1, 2, 0, 0, 0, 36, 49, 64, 81])) + # If we evaluate somewhere all conditions are False, we should get back NaN + nodef_func = lambdify(x, Piecewise((x, x > 0), (-x, x < 0))) + numpy.testing.assert_array_equal(nodef_func(numpy.array([-1, 0, 1])), + numpy.array([1, numpy.nan, 1])) + + +def test_numpy_logical_ops(): + if not numpy: + skip("numpy not installed.") + and_func = lambdify((x, y), And(x, y), modules="numpy") + and_func_3 = lambdify((x, y, z), And(x, y, z), modules="numpy") + or_func = lambdify((x, y), Or(x, y), modules="numpy") + or_func_3 = lambdify((x, y, z), Or(x, y, z), modules="numpy") + not_func = lambdify((x), Not(x), modules="numpy") + arr1 = numpy.array([True, True]) + arr2 = numpy.array([False, True]) + arr3 = numpy.array([True, False]) + numpy.testing.assert_array_equal(and_func(arr1, arr2), numpy.array([False, True])) + numpy.testing.assert_array_equal(and_func_3(arr1, arr2, arr3), numpy.array([False, False])) + numpy.testing.assert_array_equal(or_func(arr1, arr2), numpy.array([True, True])) + numpy.testing.assert_array_equal(or_func_3(arr1, arr2, arr3), numpy.array([True, True])) + numpy.testing.assert_array_equal(not_func(arr2), numpy.array([True, False])) + + +def test_numpy_matmul(): + if not numpy: + skip("numpy not installed.") + xmat = Matrix([[x, y], [z, 1+z]]) + ymat = Matrix([[x**2], [Abs(x)]]) + mat_func = lambdify((x, y, z), xmat*ymat, modules="numpy") + numpy.testing.assert_array_equal(mat_func(0.5, 3, 4), numpy.array([[1.625], [3.5]])) + numpy.testing.assert_array_equal(mat_func(-0.5, 3, 4), numpy.array([[1.375], [3.5]])) + # Multiple matrices chained together in multiplication + f = lambdify((x, y, z), xmat*xmat*xmat, modules="numpy") + numpy.testing.assert_array_equal(f(0.5, 3, 4), numpy.array([[72.125, 119.25], + [159, 251]])) + + +def test_numpy_numexpr(): + if not numpy: + skip("numpy not installed.") + if not numexpr: + skip("numexpr not installed.") + a, b, c = numpy.random.randn(3, 128, 128) + # ensure that numpy and numexpr return same value for complicated expression + expr = sin(x) + cos(y) + tan(z)**2 + Abs(z-y)*acos(sin(y*z)) + \ + Abs(y-z)*acosh(2+exp(y-x))- sqrt(x**2+I*y**2) + npfunc = lambdify((x, y, z), expr, modules='numpy') + nefunc = lambdify((x, y, z), expr, modules='numexpr') + assert numpy.allclose(npfunc(a, b, c), nefunc(a, b, c)) + + +def test_numexpr_userfunctions(): + if not numpy: + skip("numpy not installed.") + if not numexpr: + skip("numexpr not installed.") + a, b = numpy.random.randn(2, 10) + uf = type('uf', (Function, ), + {'eval' : classmethod(lambda x, y : y**2+1)}) + func = lambdify(x, 1-uf(x), modules='numexpr') + assert numpy.allclose(func(a), -(a**2)) + + uf = implemented_function(Function('uf'), lambda x, y : 2*x*y+1) + func = lambdify((x, y), uf(x, y), modules='numexpr') + assert numpy.allclose(func(a, b), 2*a*b+1) + + +def test_tensorflow_basic_math(): + if not tensorflow: + skip("tensorflow not installed.") + expr = Max(sin(x), Abs(1/(x+2))) + func = lambdify(x, expr, modules="tensorflow") + + with tensorflow.compat.v1.Session() as s: + a = tensorflow.constant(0, dtype=tensorflow.float32) + assert func(a).eval(session=s) == 0.5 + + +def test_tensorflow_placeholders(): + if not tensorflow: + skip("tensorflow not installed.") + expr = Max(sin(x), Abs(1/(x+2))) + func = lambdify(x, expr, modules="tensorflow") + + with tensorflow.compat.v1.Session() as s: + a = tensorflow.compat.v1.placeholder(dtype=tensorflow.float32) + assert func(a).eval(session=s, feed_dict={a: 0}) == 0.5 + + +def test_tensorflow_variables(): + if not tensorflow: + skip("tensorflow not installed.") + expr = Max(sin(x), Abs(1/(x+2))) + func = lambdify(x, expr, modules="tensorflow") + + with tensorflow.compat.v1.Session() as s: + a = tensorflow.Variable(0, dtype=tensorflow.float32) + s.run(a.initializer) + assert func(a).eval(session=s, feed_dict={a: 0}) == 0.5 + + +def test_tensorflow_logical_operations(): + if not tensorflow: + skip("tensorflow not installed.") + expr = Not(And(Or(x, y), y)) + func = lambdify([x, y], expr, modules="tensorflow") + + with tensorflow.compat.v1.Session() as s: + assert func(False, True).eval(session=s) == False + + +def test_tensorflow_piecewise(): + if not tensorflow: + skip("tensorflow not installed.") + expr = Piecewise((0, Eq(x,0)), (-1, x < 0), (1, x > 0)) + func = lambdify(x, expr, modules="tensorflow") + + with tensorflow.compat.v1.Session() as s: + assert func(-1).eval(session=s) == -1 + assert func(0).eval(session=s) == 0 + assert func(1).eval(session=s) == 1 + + +def test_tensorflow_multi_max(): + if not tensorflow: + skip("tensorflow not installed.") + expr = Max(x, -x, x**2) + func = lambdify(x, expr, modules="tensorflow") + + with tensorflow.compat.v1.Session() as s: + assert func(-2).eval(session=s) == 4 + + +def test_tensorflow_multi_min(): + if not tensorflow: + skip("tensorflow not installed.") + expr = Min(x, -x, x**2) + func = lambdify(x, expr, modules="tensorflow") + + with tensorflow.compat.v1.Session() as s: + assert func(-2).eval(session=s) == -2 + + +def test_tensorflow_relational(): + if not tensorflow: + skip("tensorflow not installed.") + expr = x >= 0 + func = lambdify(x, expr, modules="tensorflow") + + with tensorflow.compat.v1.Session() as s: + assert func(1).eval(session=s) == True + + +def test_tensorflow_complexes(): + if not tensorflow: + skip("tensorflow not installed") + + func1 = lambdify(x, re(x), modules="tensorflow") + func2 = lambdify(x, im(x), modules="tensorflow") + func3 = lambdify(x, Abs(x), modules="tensorflow") + func4 = lambdify(x, arg(x), modules="tensorflow") + + with tensorflow.compat.v1.Session() as s: + # For versions before + # https://github.com/tensorflow/tensorflow/issues/30029 + # resolved, using Python numeric types may not work + a = tensorflow.constant(1+2j) + assert func1(a).eval(session=s) == 1 + assert func2(a).eval(session=s) == 2 + + tensorflow_result = func3(a).eval(session=s) + sympy_result = Abs(1 + 2j).evalf() + assert abs(tensorflow_result-sympy_result) < 10**-6 + + tensorflow_result = func4(a).eval(session=s) + sympy_result = arg(1 + 2j).evalf() + assert abs(tensorflow_result-sympy_result) < 10**-6 + + +def test_tensorflow_array_arg(): + # Test for issue 14655 (tensorflow part) + if not tensorflow: + skip("tensorflow not installed.") + + f = lambdify([[x, y]], x*x + y, 'tensorflow') + + with tensorflow.compat.v1.Session() as s: + fcall = f(tensorflow.constant([2.0, 1.0])) + assert fcall.eval(session=s) == 5.0 + + +#================== Test symbolic ================================== + + +def test_sym_single_arg(): + f = lambdify(x, x * y) + assert f(z) == z * y + + +def test_sym_list_args(): + f = lambdify([x, y], x + y + z) + assert f(1, 2) == 3 + z + + +def test_sym_integral(): + f = Lambda(x, exp(-x**2)) + l = lambdify(x, Integral(f(x), (x, -oo, oo)), modules="sympy") + assert l(y) == Integral(exp(-y**2), (y, -oo, oo)) + assert l(y).doit() == sqrt(pi) + + +def test_namespace_order(): + # lambdify had a bug, such that module dictionaries or cached module + # dictionaries would pull earlier namespaces into themselves. + # Because the module dictionaries form the namespace of the + # generated lambda, this meant that the behavior of a previously + # generated lambda function could change as a result of later calls + # to lambdify. + n1 = {'f': lambda x: 'first f'} + n2 = {'f': lambda x: 'second f', + 'g': lambda x: 'function g'} + f = sympy.Function('f') + g = sympy.Function('g') + if1 = lambdify(x, f(x), modules=(n1, "sympy")) + assert if1(1) == 'first f' + if2 = lambdify(x, g(x), modules=(n2, "sympy")) + # previously gave 'second f' + assert if1(1) == 'first f' + + assert if2(1) == 'function g' + + +def test_imps(): + # Here we check if the default returned functions are anonymous - in + # the sense that we can have more than one function with the same name + f = implemented_function('f', lambda x: 2*x) + g = implemented_function('f', lambda x: math.sqrt(x)) + l1 = lambdify(x, f(x)) + l2 = lambdify(x, g(x)) + assert str(f(x)) == str(g(x)) + assert l1(3) == 6 + assert l2(3) == math.sqrt(3) + # check that we can pass in a Function as input + func = sympy.Function('myfunc') + assert not hasattr(func, '_imp_') + my_f = implemented_function(func, lambda x: 2*x) + assert hasattr(my_f, '_imp_') + # Error for functions with same name and different implementation + f2 = implemented_function("f", lambda x: x + 101) + raises(ValueError, lambda: lambdify(x, f(f2(x)))) + + +def test_imps_errors(): + # Test errors that implemented functions can return, and still be able to + # form expressions. + # See: https://github.com/sympy/sympy/issues/10810 + # + # XXX: Removed AttributeError here. This test was added due to issue 10810 + # but that issue was about ValueError. It doesn't seem reasonable to + # "support" catching AttributeError in the same context... + for val, error_class in product((0, 0., 2, 2.0), (TypeError, ValueError)): + + def myfunc(a): + if a == 0: + raise error_class + return 1 + + f = implemented_function('f', myfunc) + expr = f(val) + assert expr == f(val) + + +def test_imps_wrong_args(): + raises(ValueError, lambda: implemented_function(sin, lambda x: x)) + + +def test_lambdify_imps(): + # Test lambdify with implemented functions + # first test basic (sympy) lambdify + f = sympy.cos + assert lambdify(x, f(x))(0) == 1 + assert lambdify(x, 1 + f(x))(0) == 2 + assert lambdify((x, y), y + f(x))(0, 1) == 2 + # make an implemented function and test + f = implemented_function("f", lambda x: x + 100) + assert lambdify(x, f(x))(0) == 100 + assert lambdify(x, 1 + f(x))(0) == 101 + assert lambdify((x, y), y + f(x))(0, 1) == 101 + # Can also handle tuples, lists, dicts as expressions + lam = lambdify(x, (f(x), x)) + assert lam(3) == (103, 3) + lam = lambdify(x, [f(x), x]) + assert lam(3) == [103, 3] + lam = lambdify(x, [f(x), (f(x), x)]) + assert lam(3) == [103, (103, 3)] + lam = lambdify(x, {f(x): x}) + assert lam(3) == {103: 3} + lam = lambdify(x, {f(x): x}) + assert lam(3) == {103: 3} + lam = lambdify(x, {x: f(x)}) + assert lam(3) == {3: 103} + # Check that imp preferred to other namespaces by default + d = {'f': lambda x: x + 99} + lam = lambdify(x, f(x), d) + assert lam(3) == 103 + # Unless flag passed + lam = lambdify(x, f(x), d, use_imps=False) + assert lam(3) == 102 + + +def test_dummification(): + t = symbols('t') + F = Function('F') + G = Function('G') + #"\alpha" is not a valid Python variable name + #lambdify should sub in a dummy for it, and return + #without a syntax error + alpha = symbols(r'\alpha') + some_expr = 2 * F(t)**2 / G(t) + lam = lambdify((F(t), G(t)), some_expr) + assert lam(3, 9) == 2 + lam = lambdify(sin(t), 2 * sin(t)**2) + assert lam(F(t)) == 2 * F(t)**2 + #Test that \alpha was properly dummified + lam = lambdify((alpha, t), 2*alpha + t) + assert lam(2, 1) == 5 + raises(SyntaxError, lambda: lambdify(F(t) * G(t), F(t) * G(t) + 5)) + raises(SyntaxError, lambda: lambdify(2 * F(t), 2 * F(t) + 5)) + raises(SyntaxError, lambda: lambdify(2 * F(t), 4 * F(t) + 5)) + + +def test_lambdify__arguments_with_invalid_python_identifiers(): + # see sympy/sympy#26690 + N = CoordSys3D('N') + xn, yn, zn = N.base_scalars() + expr = xn + yn + f = lambdify([xn, yn], expr) + res = f(0.2, 0.3) + ref = 0.2 + 0.3 + assert abs(res-ref) < 1e-15 + + +def test_curly_matrix_symbol(): + # Issue #15009 + curlyv = sympy.MatrixSymbol("{v}", 2, 1) + lam = lambdify(curlyv, curlyv) + assert lam(1)==1 + lam = lambdify(curlyv, curlyv, dummify=True) + assert lam(1)==1 + + +def test_python_keywords(): + # Test for issue 7452. The automatic dummification should ensure use of + # Python reserved keywords as symbol names will create valid lambda + # functions. This is an additional regression test. + python_if = symbols('if') + expr = python_if / 2 + f = lambdify(python_if, expr) + assert f(4.0) == 2.0 + + +def test_lambdify_docstring(): + func = lambdify((w, x, y, z), w + x + y + z) + ref = ( + "Created with lambdify. Signature:\n\n" + "func(w, x, y, z)\n\n" + "Expression:\n\n" + "w + x + y + z" + ).splitlines() + assert func.__doc__.splitlines()[:len(ref)] == ref + syms = symbols('a1:26') + func = lambdify(syms, sum(syms)) + ref = ( + "Created with lambdify. Signature:\n\n" + "func(a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13, a14, a15,\n" + " a16, a17, a18, a19, a20, a21, a22, a23, a24, a25)\n\n" + "Expression:\n\n" + "a1 + a10 + a11 + a12 + a13 + a14 + a15 + a16 + a17 + a18 + a19 + a2 + a20 +..." + ).splitlines() + assert func.__doc__.splitlines()[:len(ref)] == ref + + +def test_lambdify_linecache(): + func = lambdify(x, x + 1) + source = 'def _lambdifygenerated(x):\n return x + 1\n' + assert inspect.getsource(func) == source + filename = inspect.getsourcefile(func) + assert filename.startswith('= (1, 10) + + if have_scipy_1_10plus: + cm2 = lambdify((x, y), powm1(x, y), modules='scipy') + assert abs(cm2(1.2, 1e-9) - 1.82321557e-10) < 1e-17 + + +def test_scipy_bernoulli(): + if not scipy: + skip("scipy not installed") + + bern = lambdify((x,), bernoulli(x), modules='scipy') + assert bern(1) == 0.5 + + +def test_scipy_harmonic(): + if not scipy: + skip("scipy not installed") + + hn = lambdify((x,), harmonic(x), modules='scipy') + assert hn(2) == 1.5 + hnm = lambdify((x, y), harmonic(x, y), modules='scipy') + assert hnm(2, 2) == 1.25 + + +def test_cupy_array_arg(): + if not cupy: + skip("CuPy not installed") + + f = lambdify([[x, y]], x*x + y, 'cupy') + result = f(cupy.array([2.0, 1.0])) + assert result == 5 + assert "cupy" in str(type(result)) + + +def test_cupy_array_arg_using_numpy(): + # numpy functions can be run on cupy arrays + # unclear if we can "officially" support this, + # depends on numpy __array_function__ support + if not cupy: + skip("CuPy not installed") + + f = lambdify([[x, y]], x*x + y, 'numpy') + result = f(cupy.array([2.0, 1.0])) + assert result == 5 + assert "cupy" in str(type(result)) + +def test_cupy_dotproduct(): + if not cupy: + skip("CuPy not installed") + + A = Matrix([x, y, z]) + f1 = lambdify([x, y, z], DotProduct(A, A), modules='cupy') + f2 = lambdify([x, y, z], DotProduct(A, A.T), modules='cupy') + f3 = lambdify([x, y, z], DotProduct(A.T, A), modules='cupy') + f4 = lambdify([x, y, z], DotProduct(A, A.T), modules='cupy') + + assert f1(1, 2, 3) == \ + f2(1, 2, 3) == \ + f3(1, 2, 3) == \ + f4(1, 2, 3) == \ + cupy.array([14]) + + +def test_jax_array_arg(): + if not jax: + skip("JAX not installed") + + f = lambdify([[x, y]], x*x + y, 'jax') + result = f(jax.numpy.array([2.0, 1.0])) + assert result == 5 + assert "jax" in str(type(result)) + + +def test_jax_array_arg_using_numpy(): + if not jax: + skip("JAX not installed") + + f = lambdify([[x, y]], x*x + y, 'numpy') + result = f(jax.numpy.array([2.0, 1.0])) + assert result == 5 + assert "jax" in str(type(result)) + + +def test_jax_dotproduct(): + if not jax: + skip("JAX not installed") + + A = Matrix([x, y, z]) + f1 = lambdify([x, y, z], DotProduct(A, A), modules='jax') + f2 = lambdify([x, y, z], DotProduct(A, A.T), modules='jax') + f3 = lambdify([x, y, z], DotProduct(A.T, A), modules='jax') + f4 = lambdify([x, y, z], DotProduct(A, A.T), modules='jax') + + assert f1(1, 2, 3) == \ + f2(1, 2, 3) == \ + f3(1, 2, 3) == \ + f4(1, 2, 3) == \ + jax.numpy.array([14]) + + +def test_lambdify_cse(): + def no_op_cse(exprs): + return (), exprs + + def dummy_cse(exprs): + from sympy.simplify.cse_main import cse + return cse(exprs, symbols=numbered_symbols(cls=Dummy)) + + def minmem(exprs): + from sympy.simplify.cse_main import cse_release_variables, cse + return cse(exprs, postprocess=cse_release_variables) + + class Case: + def __init__(self, *, args, exprs, num_args, requires_numpy=False): + self.args = args + self.exprs = exprs + self.num_args = num_args + subs_dict = dict(zip(self.args, self.num_args)) + self.ref = [e.subs(subs_dict).evalf() for e in exprs] + self.requires_numpy = requires_numpy + + def lambdify(self, *, cse): + return lambdify(self.args, self.exprs, cse=cse) + + def assertAllClose(self, result, *, abstol=1e-15, reltol=1e-15): + if self.requires_numpy: + assert all(numpy.allclose(result[i], numpy.asarray(r, dtype=float), + rtol=reltol, atol=abstol) + for i, r in enumerate(self.ref)) + return + + for i, r in enumerate(self.ref): + abs_err = abs(result[i] - r) + if r == 0: + assert abs_err < abstol + else: + assert abs_err/abs(r) < reltol + + cases = [ + Case( + args=(x, y, z), + exprs=[ + x + y + z, + x + y - z, + 2*x + 2*y - z, + (x+y)**2 + (y+z)**2, + ], + num_args=(2., 3., 4.) + ), + Case( + args=(x, y, z), + exprs=[ + x + sympy.Heaviside(x), + y + sympy.Heaviside(x), + z + sympy.Heaviside(x, 1), + z/sympy.Heaviside(x, 1) + ], + num_args=(0., 3., 4.) + ), + Case( + args=(x, y, z), + exprs=[ + x + sinc(y), + y + sinc(y), + z - sinc(y) + ], + num_args=(0.1, 0.2, 0.3) + ), + Case( + args=(x, y, z), + exprs=[ + Matrix([[x, x*y], [sin(z) + 4, x**z]]), + x*y+sin(z)-x**z, + Matrix([x*x, sin(z), x**z]) + ], + num_args=(1.,2.,3.), + requires_numpy=True + ), + Case( + args=(x, y), + exprs=[(x + y - 1)**2, x, x + y, + (x + y)/(2*x + 1) + (x + y - 1)**2, (2*x + 1)**(x + y)], + num_args=(1,2) + ) + ] + for case in cases: + if not numpy and case.requires_numpy: + continue + for _cse in [False, True, minmem, no_op_cse, dummy_cse]: + f = case.lambdify(cse=_cse) + result = f(*case.num_args) + case.assertAllClose(result) + +def test_issue_25288(): + syms = numbered_symbols(cls=Dummy) + ok = lambdify(x, [x**2, sin(x**2)], cse=lambda e: cse(e, symbols=syms))(2) + assert ok + + +def test_deprecated_set(): + with warns_deprecated_sympy(): + lambdify({x, y}, x + y) + +def test_issue_13881(): + if not numpy: + skip("numpy not installed.") + + X = MatrixSymbol('X', 3, 1) + + f = lambdify(X, X.T*X, 'numpy') + assert f(numpy.array([1, 2, 3])) == 14 + assert f(numpy.array([3, 2, 1])) == 14 + + f = lambdify(X, X*X.T, 'numpy') + assert f(numpy.array([1, 2, 3])) == 14 + assert f(numpy.array([3, 2, 1])) == 14 + + f = lambdify(X, (X*X.T)*X, 'numpy') + arr1 = numpy.array([[1], [2], [3]]) + arr2 = numpy.array([[14],[28],[42]]) + + assert numpy.array_equal(f(arr1), arr2) + + +def test_23536_lambdify_cse_dummy(): + + f = Function('x')(y) + g = Function('w')(y) + expr = z + (f**4 + g**5)*(f**3 + (g*f)**3) + expr = expr.expand() + eval_expr = lambdify(((f, g), z), expr, cse=True) + ans = eval_expr((1.0, 2.0), 3.0) # shouldn't raise NameError + assert ans == 300.0 # not a list and value is 300 + + +class LambdifyDocstringTestCase: + SIGNATURE = None + EXPR = None + SRC = None + + def __init__(self, docstring_limit, expected_redacted): + self.docstring_limit = docstring_limit + self.expected_redacted = expected_redacted + + @property + def expected_expr(self): + expr_redacted_msg = "EXPRESSION REDACTED DUE TO LENGTH, (see lambdify's `docstring_limit`)" + return self.EXPR if not self.expected_redacted else expr_redacted_msg + + @property + def expected_src(self): + src_redacted_msg = "SOURCE CODE REDACTED DUE TO LENGTH, (see lambdify's `docstring_limit`)" + return self.SRC if not self.expected_redacted else src_redacted_msg + + @property + def expected_docstring(self): + expected_docstring = ( + f'Created with lambdify. Signature:\n\n' + f'func({self.SIGNATURE})\n\n' + f'Expression:\n\n' + f'{self.expected_expr}\n\n' + f'Source code:\n\n' + f'{self.expected_src}\n\n' + f'Imported modules:\n\n' + ) + return expected_docstring + + def __len__(self): + return len(self.expected_docstring) + + def __repr__(self): + return ( + f'{self.__class__.__name__}(' + f'docstring_limit={self.docstring_limit}, ' + f'expected_redacted={self.expected_redacted})' + ) + + +def test_lambdify_docstring_size_limit_simple_symbol(): + + class SimpleSymbolTestCase(LambdifyDocstringTestCase): + SIGNATURE = 'x' + EXPR = 'x' + SRC = ( + 'def _lambdifygenerated(x):\n' + ' return x\n' + ) + + x = symbols('x') + + test_cases = ( + SimpleSymbolTestCase(docstring_limit=None, expected_redacted=False), + SimpleSymbolTestCase(docstring_limit=100, expected_redacted=False), + SimpleSymbolTestCase(docstring_limit=1, expected_redacted=False), + SimpleSymbolTestCase(docstring_limit=0, expected_redacted=True), + SimpleSymbolTestCase(docstring_limit=-1, expected_redacted=True), + ) + for test_case in test_cases: + lambdified_expr = lambdify( + [x], + x, + 'sympy', + docstring_limit=test_case.docstring_limit, + ) + assert lambdified_expr.__doc__ == test_case.expected_docstring + + +def test_lambdify_docstring_size_limit_nested_expr(): + + class ExprListTestCase(LambdifyDocstringTestCase): + SIGNATURE = 'x, y, z' + EXPR = ( + '[x, [y], z, x**3 + 3*x**2*y + 3*x**2*z + 3*x*y**2 + 6*x*y*z ' + '+ 3*x*z**2 +...' + ) + SRC = ( + 'def _lambdifygenerated(x, y, z):\n' + ' return [x, [y], z, x**3 + 3*x**2*y + 3*x**2*z + 3*x*y**2 ' + '+ 6*x*y*z + 3*x*z**2 + y**3 + 3*y**2*z + 3*y*z**2 + z**3]\n' + ) + + x, y, z = symbols('x, y, z') + expr = [x, [y], z, ((x + y + z)**3).expand()] + + test_cases = ( + ExprListTestCase(docstring_limit=None, expected_redacted=False), + ExprListTestCase(docstring_limit=200, expected_redacted=False), + ExprListTestCase(docstring_limit=50, expected_redacted=True), + ExprListTestCase(docstring_limit=0, expected_redacted=True), + ExprListTestCase(docstring_limit=-1, expected_redacted=True), + ) + for test_case in test_cases: + lambdified_expr = lambdify( + [x, y, z], + expr, + 'sympy', + docstring_limit=test_case.docstring_limit, + ) + assert lambdified_expr.__doc__ == test_case.expected_docstring + + +def test_lambdify_docstring_size_limit_matrix(): + + class MatrixTestCase(LambdifyDocstringTestCase): + SIGNATURE = 'x, y, z' + EXPR = ( + 'Matrix([[0, x], [x + y + z, x**3 + 3*x**2*y + 3*x**2*z + 3*x*y**2 ' + '+ 6*x*y*z...' + ) + SRC = ( + 'def _lambdifygenerated(x, y, z):\n' + ' return ImmutableDenseMatrix([[0, x], [x + y + z, x**3 ' + '+ 3*x**2*y + 3*x**2*z + 3*x*y**2 + 6*x*y*z + 3*x*z**2 + y**3 ' + '+ 3*y**2*z + 3*y*z**2 + z**3]])\n' + ) + + x, y, z = symbols('x, y, z') + expr = Matrix([[S.Zero, x], [x + y + z, ((x + y + z)**3).expand()]]) + + test_cases = ( + MatrixTestCase(docstring_limit=None, expected_redacted=False), + MatrixTestCase(docstring_limit=200, expected_redacted=False), + MatrixTestCase(docstring_limit=50, expected_redacted=True), + MatrixTestCase(docstring_limit=0, expected_redacted=True), + MatrixTestCase(docstring_limit=-1, expected_redacted=True), + ) + for test_case in test_cases: + lambdified_expr = lambdify( + [x, y, z], + expr, + 'sympy', + docstring_limit=test_case.docstring_limit, + ) + assert lambdified_expr.__doc__ == test_case.expected_docstring + + +def test_lambdify_empty_tuple(): + a = symbols("a") + expr = ((), (a,)) + f = lambdify(a, expr) + result = f(1) + assert result == ((), (1,)), "Lambdify did not handle the empty tuple correctly." + + +def test_assoc_legendre_numerical_evaluation(): + + tol = 1e-10 + + sympy_result_integer = assoc_legendre(1, 1/2, 0.1).evalf() + sympy_result_complex = assoc_legendre(2, 1, 3).evalf() + mpmath_result_integer = -0.474572528387641 + mpmath_result_complex = -25.45584412271571*I + + assert all_close(sympy_result_integer, mpmath_result_integer, tol) + assert all_close(sympy_result_complex, mpmath_result_complex, tol) + + +def test_Piecewise(): + + modules = [math] + if numpy: + modules.append('numpy') + + for mod in modules: + # test isinf + f = lambdify(x, Piecewise((7.0, isinf(x)), (3.0, True)), mod) + assert f(+float('inf')) == +7.0 + assert f(-float('inf')) == +7.0 + assert f(42.) == 3.0 + + f2 = lambdify(x, Piecewise((7.0*sign(x), isinf(x)), (3.0, True)), mod) + assert f2(+float('inf')) == +7.0 + assert f2(-float('inf')) == -7.0 + assert f2(42.) == 3.0 + + # test isnan (gh-26784) + g = lambdify(x, Piecewise((7.0, isnan(x)), (3.0, True)), mod) + assert g(float('nan')) == 7.0 + assert g(42.) == 3.0 + + +def test_array_symbol(): + if not numpy: + skip("numpy not installed.") + a = ArraySymbol('a', (3,)) + f = lambdify((a), a) + assert numpy.all(f(numpy.array([1,2,3])) == numpy.array([1,2,3])) diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_matchpy_connector.py b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_matchpy_connector.py new file mode 100644 index 0000000000000000000000000000000000000000..3648bd49f9e56ca20fbf428ed46c01429dbe8b15 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_matchpy_connector.py @@ -0,0 +1,164 @@ +import pickle + +from sympy.core.relational import (Eq, Ne) +from sympy.core.singleton import S +from sympy.core.symbol import symbols +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.external import import_module +from sympy.testing.pytest import skip +from sympy.utilities.matchpy_connector import WildDot, WildPlus, WildStar, Replacer + +matchpy = import_module("matchpy") + +x, y, z = symbols("x y z") + + +def _get_first_match(expr, pattern): + from matchpy import ManyToOneMatcher, Pattern + + matcher = ManyToOneMatcher() + matcher.add(Pattern(pattern)) + return next(iter(matcher.match(expr))) + + +def test_matchpy_connector(): + if matchpy is None: + skip("matchpy not installed") + + from multiset import Multiset + from matchpy import Pattern, Substitution + + w_ = WildDot("w_") + w__ = WildPlus("w__") + w___ = WildStar("w___") + + expr = x + y + pattern = x + w_ + p, subst = _get_first_match(expr, pattern) + assert p == Pattern(pattern) + assert subst == Substitution({'w_': y}) + + expr = x + y + z + pattern = x + w__ + p, subst = _get_first_match(expr, pattern) + assert p == Pattern(pattern) + assert subst == Substitution({'w__': Multiset([y, z])}) + + expr = x + y + z + pattern = x + y + z + w___ + p, subst = _get_first_match(expr, pattern) + assert p == Pattern(pattern) + assert subst == Substitution({'w___': Multiset()}) + + +def test_matchpy_optional(): + if matchpy is None: + skip("matchpy not installed") + + from matchpy import Pattern, Substitution + from matchpy import ManyToOneReplacer, ReplacementRule + + p = WildDot("p", optional=1) + q = WildDot("q", optional=0) + + pattern = p*x + q + + expr1 = 2*x + pa, subst = _get_first_match(expr1, pattern) + assert pa == Pattern(pattern) + assert subst == Substitution({'p': 2, 'q': 0}) + + expr2 = x + 3 + pa, subst = _get_first_match(expr2, pattern) + assert pa == Pattern(pattern) + assert subst == Substitution({'p': 1, 'q': 3}) + + expr3 = x + pa, subst = _get_first_match(expr3, pattern) + assert pa == Pattern(pattern) + assert subst == Substitution({'p': 1, 'q': 0}) + + expr4 = x*y + z + pa, subst = _get_first_match(expr4, pattern) + assert pa == Pattern(pattern) + assert subst == Substitution({'p': y, 'q': z}) + + replacer = ManyToOneReplacer() + replacer.add(ReplacementRule(Pattern(pattern), lambda p, q: sin(p)*cos(q))) + assert replacer.replace(expr1) == sin(2)*cos(0) + assert replacer.replace(expr2) == sin(1)*cos(3) + assert replacer.replace(expr3) == sin(1)*cos(0) + assert replacer.replace(expr4) == sin(y)*cos(z) + + +def test_replacer(): + if matchpy is None: + skip("matchpy not installed") + + for info in [True, False]: + for lambdify in [True, False]: + _perform_test_replacer(info, lambdify) + + +def _perform_test_replacer(info, lambdify): + + x1_ = WildDot("x1_") + x2_ = WildDot("x2_") + + a_ = WildDot("a_", optional=S.One) + b_ = WildDot("b_", optional=S.One) + c_ = WildDot("c_", optional=S.Zero) + + replacer = Replacer(common_constraints=[ + matchpy.CustomConstraint(lambda a_: not a_.has(x)), + matchpy.CustomConstraint(lambda b_: not b_.has(x)), + matchpy.CustomConstraint(lambda c_: not c_.has(x)), + ], lambdify=lambdify, info=info) + + # Rewrite the equation into implicit form, unless it's already solved: + replacer.add(Eq(x1_, x2_), Eq(x1_ - x2_, 0), conditions_nonfalse=[Ne(x2_, 0), Ne(x1_, 0), Ne(x1_, x), Ne(x2_, x)], info=1) + + # Simple equation solver for real numbers: + replacer.add(Eq(a_*x + b_, 0), Eq(x, -b_/a_), info=2) + disc = b_**2 - 4*a_*c_ + replacer.add( + Eq(a_*x**2 + b_*x + c_, 0), + Eq(x, (-b_ - sqrt(disc))/(2*a_)) | Eq(x, (-b_ + sqrt(disc))/(2*a_)), + conditions_nonfalse=[disc >= 0], + info=3 + ) + replacer.add( + Eq(a_*x**2 + c_, 0), + Eq(x, sqrt(-c_/a_)) | Eq(x, -sqrt(-c_/a_)), + conditions_nonfalse=[-c_*a_ > 0], + info=4 + ) + + g = lambda expr, infos: (expr, infos) if info else expr + + assert replacer.replace(Eq(3*x, y)) == g(Eq(x, y/3), [1, 2]) + assert replacer.replace(Eq(x**2 + 1, 0)) == g(Eq(x**2 + 1, 0), []) + assert replacer.replace(Eq(x**2, 4)) == g((Eq(x, 2) | Eq(x, -2)), [1, 4]) + assert replacer.replace(Eq(x**2 + 4*y*x + 4*y**2, 0)) == g(Eq(x, -2*y), [3]) + + +def test_matchpy_object_pickle(): + if matchpy is None: + return + + a1 = WildDot("a") + a2 = pickle.loads(pickle.dumps(a1)) + assert a1 == a2 + + a1 = WildDot("a", S(1)) + a2 = pickle.loads(pickle.dumps(a1)) + assert a1 == a2 + + a1 = WildPlus("a", S(1)) + a2 = pickle.loads(pickle.dumps(a1)) + assert a1 == a2 + + a1 = WildStar("a", S(1)) + a2 = pickle.loads(pickle.dumps(a1)) + assert a1 == a2 diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_mathml.py b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_mathml.py new file mode 100644 index 0000000000000000000000000000000000000000..e4a7598f175be34f8bb34c0bd9c003d1c0238c7b --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_mathml.py @@ -0,0 +1,33 @@ +import os +from textwrap import dedent +from sympy.external import import_module +from sympy.testing.pytest import skip +from sympy.utilities.mathml import apply_xsl + + + +lxml = import_module('lxml') + +path = os.path.abspath(os.path.join(os.path.dirname(__file__), "test_xxe.py")) + + +def test_xxe(): + assert os.path.isfile(path) + if not lxml: + skip("lxml not installed.") + + mml = dedent( + rf""" + + ]> + + John + &ent; + + """ + ) + xsl = 'mathml/data/simple_mmlctop.xsl' + + res = apply_xsl(mml, xsl) + assert res == \ + '\n\nJohn\n\n\n' diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_misc.py b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_misc.py new file mode 100644 index 0000000000000000000000000000000000000000..0a3e059419303c33cbd7b770679b5efc1b03486d --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_misc.py @@ -0,0 +1,148 @@ +from textwrap import dedent +import sys +from subprocess import Popen, PIPE +import os + +from sympy.core.singleton import S +from sympy.testing.pytest import (raises, warns_deprecated_sympy, + skip_under_pyodide) +from sympy.utilities.misc import (translate, replace, ordinal, rawlines, + strlines, as_int, find_executable) + + +def test_translate(): + abc = 'abc' + assert translate(abc, None, 'a') == 'bc' + assert translate(abc, None, '') == 'abc' + assert translate(abc, {'a': 'x'}, 'c') == 'xb' + assert translate(abc, {'a': 'bc'}, 'c') == 'bcb' + assert translate(abc, {'ab': 'x'}, 'c') == 'x' + assert translate(abc, {'ab': ''}, 'c') == '' + assert translate(abc, {'bc': 'x'}, 'c') == 'ab' + assert translate(abc, {'abc': 'x', 'a': 'y'}) == 'x' + u = chr(4096) + assert translate(abc, 'a', 'x', u) == 'xbc' + assert (u in translate(abc, 'a', u, u)) is True + + +def test_replace(): + assert replace('abc', ('a', 'b')) == 'bbc' + assert replace('abc', {'a': 'Aa'}) == 'Aabc' + assert replace('abc', ('a', 'b'), ('c', 'C')) == 'bbC' + + +def test_ordinal(): + assert ordinal(-1) == '-1st' + assert ordinal(0) == '0th' + assert ordinal(1) == '1st' + assert ordinal(2) == '2nd' + assert ordinal(3) == '3rd' + assert all(ordinal(i).endswith('th') for i in range(4, 21)) + assert ordinal(100) == '100th' + assert ordinal(101) == '101st' + assert ordinal(102) == '102nd' + assert ordinal(103) == '103rd' + assert ordinal(104) == '104th' + assert ordinal(200) == '200th' + assert all(ordinal(i) == str(i) + 'th' for i in range(-220, -203)) + + +def test_rawlines(): + assert rawlines('a a\na') == "dedent('''\\\n a a\n a''')" + assert rawlines('a a') == "'a a'" + assert rawlines(strlines('\\le"ft')) == ( + '(\n' + " '(\\n'\n" + ' \'r\\\'\\\\le"ft\\\'\\n\'\n' + " ')'\n" + ')') + + +def test_strlines(): + q = 'this quote (") is in the middle' + # the following assert rhs was prepared with + # print(rawlines(strlines(q, 10))) + assert strlines(q, 10) == dedent('''\ + ( + 'this quo' + 'te (") i' + 's in the' + ' middle' + )''') + assert q == ( + 'this quo' + 'te (") i' + 's in the' + ' middle' + ) + q = "this quote (') is in the middle" + assert strlines(q, 20) == dedent('''\ + ( + "this quote (') is " + "in the middle" + )''') + assert strlines('\\left') == ( + '(\n' + "r'\\left'\n" + ')') + assert strlines('\\left', short=True) == r"r'\left'" + assert strlines('\\le"ft') == ( + '(\n' + 'r\'\\le"ft\'\n' + ')') + q = 'this\nother line' + assert strlines(q) == rawlines(q) + + +def test_translate_args(): + try: + translate(None, None, None, 'not_none') + except ValueError: + pass # Exception raised successfully + else: + assert False + + assert translate('s', None, None, None) == 's' + + try: + translate('s', 'a', 'bc') + except ValueError: + pass # Exception raised successfully + else: + assert False + + +@skip_under_pyodide("Cannot create subprocess under pyodide.") +def test_debug_output(): + env = os.environ.copy() + env['SYMPY_DEBUG'] = 'True' + cmd = 'from sympy import *; x = Symbol("x"); print(integrate((1-cos(x))/x, x))' + cmdline = [sys.executable, '-c', cmd] + proc = Popen(cmdline, env=env, stdout=PIPE, stderr=PIPE) + out, err = proc.communicate() + out = out.decode('ascii') # utf-8? + err = err.decode('ascii') + expected = 'substituted: -x*(1 - cos(x)), u: 1/x, u_var: _u' + assert expected in err, err + + +def test_as_int(): + raises(ValueError, lambda : as_int(True)) + raises(ValueError, lambda : as_int(1.1)) + raises(ValueError, lambda : as_int([])) + raises(ValueError, lambda : as_int(S.NaN)) + raises(ValueError, lambda : as_int(S.Infinity)) + raises(ValueError, lambda : as_int(S.NegativeInfinity)) + raises(ValueError, lambda : as_int(S.ComplexInfinity)) + # for the following, limited precision makes int(arg) == arg + # but the int value is not necessarily what a user might have + # expected; Q.prime is more nuanced in its response for + # expressions which might be complex representations of an + # integer. This is not -- by design -- as_ints role. + raises(ValueError, lambda : as_int(1e23)) + raises(ValueError, lambda : as_int(S('1.'+'0'*20+'1'))) + assert as_int(True, strict=False) == 1 + +def test_deprecated_find_executable(): + with warns_deprecated_sympy(): + find_executable('python') diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_pickling.py b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_pickling.py new file mode 100644 index 0000000000000000000000000000000000000000..7ea2d062286cbcf802eb0f42f2d7d130123599af --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_pickling.py @@ -0,0 +1,723 @@ +import inspect +import copy +import pickle + +from sympy.physics.units import meter + +from sympy.testing.pytest import XFAIL, raises, ignore_warnings + +from sympy.core.basic import Atom, Basic +from sympy.core.singleton import SingletonRegistry +from sympy.core.symbol import Str, Dummy, Symbol, Wild +from sympy.core.numbers import (E, I, pi, oo, zoo, nan, Integer, + Rational, Float, AlgebraicNumber) +from sympy.core.relational import (Equality, GreaterThan, LessThan, Relational, + StrictGreaterThan, StrictLessThan, Unequality) +from sympy.core.add import Add +from sympy.core.mul import Mul +from sympy.core.power import Pow +from sympy.core.function import Derivative, Function, FunctionClass, Lambda, \ + WildFunction +from sympy.sets.sets import Interval +from sympy.core.multidimensional import vectorize + +from sympy.external.gmpy import gmpy as _gmpy +from sympy.utilities.exceptions import SymPyDeprecationWarning + +from sympy.core.singleton import S +from sympy.core.symbol import symbols + +from sympy.external import import_module +cloudpickle = import_module('cloudpickle') + + +not_equal_attrs = { + '_assumptions', # This is a local cache that isn't automatically filled on creation + '_mhash', # Cached after __hash__ is called but set to None after creation +} + + +deprecated_attrs = { + 'is_EmptySet', # Deprecated from SymPy 1.5. This can be removed when is_EmptySet is removed. + 'expr_free_symbols', # Deprecated from SymPy 1.9. This can be removed when exr_free_symbols is removed. +} + +dont_check_attrs = { + '_sage_', # Fails because Sage is not installed +} + + +def check(a, exclude=[], check_attr=True, deprecated=()): + """ Check that pickling and copying round-trips. + """ + # Pickling with protocols 0 and 1 is disabled for Basic instances: + if isinstance(a, Basic): + for protocol in [0, 1]: + raises(NotImplementedError, lambda: pickle.dumps(a, protocol)) + + protocols = [2, copy.copy, copy.deepcopy, 3, 4] + if cloudpickle: + protocols.extend([cloudpickle]) + + for protocol in protocols: + if protocol in exclude: + continue + + if callable(protocol): + if isinstance(a, type): + # Classes can't be copied, but that's okay. + continue + b = protocol(a) + elif inspect.ismodule(protocol): + b = protocol.loads(protocol.dumps(a)) + else: + b = pickle.loads(pickle.dumps(a, protocol)) + + d1 = dir(a) + d2 = dir(b) + assert set(d1) == set(d2) + + if not check_attr: + continue + + def c(a, b, d): + for i in d: + if i in dont_check_attrs: + continue + elif i in not_equal_attrs: + if hasattr(a, i): + assert hasattr(b, i), i + elif i in deprecated_attrs or i in deprecated: + with ignore_warnings(SymPyDeprecationWarning): + assert getattr(a, i) == getattr(b, i), i + elif not hasattr(a, i): + continue + else: + attr = getattr(a, i) + if not hasattr(attr, "__call__"): + assert hasattr(b, i), i + assert getattr(b, i) == attr, "%s != %s, protocol: %s" % (getattr(b, i), attr, protocol) + + c(a, b, d1) + c(b, a, d2) + + + +#================== core ========================= + + +def test_core_basic(): + for c in (Atom, Atom(), Basic, Basic(), SingletonRegistry, S): + check(c) + +def test_core_Str(): + check(Str('x')) + +def test_core_symbol(): + # make the Symbol a unique name that doesn't class with any other + # testing variable in this file since after this test the symbol + # having the same name will be cached as noncommutative + for c in (Dummy, Dummy("x", commutative=False), Symbol, + Symbol("_issue_3130", commutative=False), Wild, Wild("x")): + check(c) + + +def test_core_numbers(): + for c in (Integer(2), Rational(2, 3), Float("1.2")): + check(c) + for c in (AlgebraicNumber, AlgebraicNumber(sqrt(3))): + check(c, check_attr=False) + + +def test_core_float_copy(): + # See gh-7457 + y = Symbol("x") + 1.0 + check(y) # does not raise TypeError ("argument is not an mpz") + + +def test_core_relational(): + x = Symbol("x") + y = Symbol("y") + for c in (Equality, Equality(x, y), GreaterThan, GreaterThan(x, y), + LessThan, LessThan(x, y), Relational, Relational(x, y), + StrictGreaterThan, StrictGreaterThan(x, y), StrictLessThan, + StrictLessThan(x, y), Unequality, Unequality(x, y)): + check(c) + + +def test_core_add(): + x = Symbol("x") + for c in (Add, Add(x, 4)): + check(c) + + +def test_core_mul(): + x = Symbol("x") + for c in (Mul, Mul(x, 4)): + check(c) + + +def test_core_power(): + x = Symbol("x") + for c in (Pow, Pow(x, 4)): + check(c) + + +def test_core_function(): + x = Symbol("x") + for f in (Derivative, Derivative(x), Function, FunctionClass, Lambda, + WildFunction): + check(f) + + +def test_core_undefinedfunctions(): + f = Function("f") + check(f) + + +def test_core_appliedundef(): + x = Symbol("_long_unique_name_1") + f = Function("_long_unique_name_2") + check(f(x)) + + +def test_core_interval(): + for c in (Interval, Interval(0, 2)): + check(c) + + +def test_core_multidimensional(): + for c in (vectorize, vectorize(0)): + check(c) + + +def test_Singletons(): + protocols = [0, 1, 2, 3, 4] + copiers = [copy.copy, copy.deepcopy] + copiers += [lambda x: pickle.loads(pickle.dumps(x, proto)) + for proto in protocols] + if cloudpickle: + copiers += [lambda x: cloudpickle.loads(cloudpickle.dumps(x))] + + for obj in (Integer(-1), Integer(0), Integer(1), Rational(1, 2), pi, E, I, + oo, -oo, zoo, nan, S.GoldenRatio, S.TribonacciConstant, + S.EulerGamma, S.Catalan, S.EmptySet, S.IdentityFunction): + for func in copiers: + assert func(obj) is obj + +#================== combinatorics =================== +from sympy.combinatorics.free_groups import FreeGroup + +def test_free_group(): + check(FreeGroup("x, y, z"), check_attr=False) + +#================== functions =================== +from sympy.functions import (Piecewise, lowergamma, acosh, chebyshevu, + chebyshevt, ln, chebyshevt_root, legendre, Heaviside, bernoulli, coth, + tanh, assoc_legendre, sign, arg, asin, DiracDelta, re, rf, Abs, + uppergamma, binomial, sinh, cos, cot, acos, acot, gamma, bell, + hermite, harmonic, LambertW, zeta, log, factorial, asinh, acoth, cosh, + dirichlet_eta, Eijk, loggamma, erf, ceiling, im, fibonacci, + tribonacci, conjugate, tan, chebyshevu_root, floor, atanh, sqrt, sin, + atan, ff, lucas, atan2, polygamma, exp) + + +def test_functions(): + one_var = (acosh, ln, Heaviside, factorial, bernoulli, coth, tanh, + sign, arg, asin, DiracDelta, re, Abs, sinh, cos, cot, acos, acot, + gamma, bell, harmonic, LambertW, zeta, log, factorial, asinh, + acoth, cosh, dirichlet_eta, loggamma, erf, ceiling, im, fibonacci, + tribonacci, conjugate, tan, floor, atanh, sin, atan, lucas, exp) + two_var = (rf, ff, lowergamma, chebyshevu, chebyshevt, binomial, + atan2, polygamma, hermite, legendre, uppergamma) + x, y, z = symbols("x,y,z") + others = (chebyshevt_root, chebyshevu_root, Eijk(x, y, z), + Piecewise( (0, x < -1), (x**2, x <= 1), (x**3, True)), + assoc_legendre) + for cls in one_var: + check(cls) + c = cls(x) + check(c) + for cls in two_var: + check(cls) + c = cls(x, y) + check(c) + for cls in others: + check(cls) + +#================== geometry ==================== +from sympy.geometry.entity import GeometryEntity +from sympy.geometry.point import Point +from sympy.geometry.ellipse import Circle, Ellipse +from sympy.geometry.line import Line, LinearEntity, Ray, Segment +from sympy.geometry.polygon import Polygon, RegularPolygon, Triangle + + +def test_geometry(): + p1 = Point(1, 2) + p2 = Point(2, 3) + p3 = Point(0, 0) + p4 = Point(0, 1) + for c in ( + GeometryEntity, GeometryEntity(), Point, p1, Circle, Circle(p1, 2), + Ellipse, Ellipse(p1, 3, 4), Line, Line(p1, p2), LinearEntity, + LinearEntity(p1, p2), Ray, Ray(p1, p2), Segment, Segment(p1, p2), + Polygon, Polygon(p1, p2, p3, p4), RegularPolygon, + RegularPolygon(p1, 4, 5), Triangle, Triangle(p1, p2, p3)): + check(c, check_attr=False) + +#================== integrals ==================== +from sympy.integrals.integrals import Integral + + +def test_integrals(): + x = Symbol("x") + for c in (Integral, Integral(x)): + check(c) + +#==================== logic ===================== +from sympy.core.logic import Logic + + +def test_logic(): + for c in (Logic, Logic(1)): + check(c) + +#================== matrices ==================== +from sympy.matrices import Matrix, SparseMatrix + + +def test_matrices(): + for c in (Matrix, Matrix([1, 2, 3]), SparseMatrix, SparseMatrix([[1, 2], [3, 4]])): + check(c, deprecated=['_smat', '_mat']) + +#================== ntheory ===================== +from sympy.ntheory.generate import Sieve + + +def test_ntheory(): + for c in (Sieve, Sieve()): + check(c) + +#================== physics ===================== +from sympy.physics.paulialgebra import Pauli +from sympy.physics.units import Unit + + +def test_physics(): + for c in (Unit, meter, Pauli, Pauli(1)): + check(c) + +#================== plotting ==================== +# XXX: These tests are not complete, so XFAIL them + + +@XFAIL +def test_plotting(): + from sympy.plotting.pygletplot.color_scheme import ColorGradient, ColorScheme + from sympy.plotting.pygletplot.managed_window import ManagedWindow + from sympy.plotting.plot import Plot, ScreenShot + from sympy.plotting.pygletplot.plot_axes import PlotAxes, PlotAxesBase, PlotAxesFrame, PlotAxesOrdinate + from sympy.plotting.pygletplot.plot_camera import PlotCamera + from sympy.plotting.pygletplot.plot_controller import PlotController + from sympy.plotting.pygletplot.plot_curve import PlotCurve + from sympy.plotting.pygletplot.plot_interval import PlotInterval + from sympy.plotting.pygletplot.plot_mode import PlotMode + from sympy.plotting.pygletplot.plot_modes import Cartesian2D, Cartesian3D, Cylindrical, \ + ParametricCurve2D, ParametricCurve3D, ParametricSurface, Polar, Spherical + from sympy.plotting.pygletplot.plot_object import PlotObject + from sympy.plotting.pygletplot.plot_surface import PlotSurface + from sympy.plotting.pygletplot.plot_window import PlotWindow + for c in ( + ColorGradient, ColorGradient(0.2, 0.4), ColorScheme, ManagedWindow, + ManagedWindow, Plot, ScreenShot, PlotAxes, PlotAxesBase, + PlotAxesFrame, PlotAxesOrdinate, PlotCamera, PlotController, + PlotCurve, PlotInterval, PlotMode, Cartesian2D, Cartesian3D, + Cylindrical, ParametricCurve2D, ParametricCurve3D, + ParametricSurface, Polar, Spherical, PlotObject, PlotSurface, + PlotWindow): + check(c) + + +@XFAIL +def test_plotting2(): + #from sympy.plotting.color_scheme import ColorGradient + from sympy.plotting.pygletplot.color_scheme import ColorScheme + #from sympy.plotting.managed_window import ManagedWindow + from sympy.plotting.plot import Plot + #from sympy.plotting.plot import ScreenShot + from sympy.plotting.pygletplot.plot_axes import PlotAxes + #from sympy.plotting.plot_axes import PlotAxesBase, PlotAxesFrame, PlotAxesOrdinate + #from sympy.plotting.plot_camera import PlotCamera + #from sympy.plotting.plot_controller import PlotController + #from sympy.plotting.plot_curve import PlotCurve + #from sympy.plotting.plot_interval import PlotInterval + #from sympy.plotting.plot_mode import PlotMode + #from sympy.plotting.plot_modes import Cartesian2D, Cartesian3D, Cylindrical, \ + # ParametricCurve2D, ParametricCurve3D, ParametricSurface, Polar, Spherical + #from sympy.plotting.plot_object import PlotObject + #from sympy.plotting.plot_surface import PlotSurface + # from sympy.plotting.plot_window import PlotWindow + check(ColorScheme("rainbow")) + check(Plot(1, visible=False)) + check(PlotAxes()) + +#================== polys ======================= +from sympy.polys.domains.integerring import ZZ +from sympy.polys.domains.rationalfield import QQ +from sympy.polys.orderings import lex +from sympy.polys.polytools import Poly + +def test_pickling_polys_polytools(): + from sympy.polys.polytools import PurePoly + # from sympy.polys.polytools import GroebnerBasis + x = Symbol('x') + + for c in (Poly, Poly(x, x)): + check(c) + + for c in (PurePoly, PurePoly(x)): + check(c) + + # TODO: fix pickling of Options class (see GroebnerBasis._options) + # for c in (GroebnerBasis, GroebnerBasis([x**2 - 1], x, order=lex)): + # check(c) + +def test_pickling_polys_polyclasses(): + from sympy.polys.polyclasses import DMP, DMF, ANP + + for c in (DMP, DMP([[ZZ(1)], [ZZ(2)], [ZZ(3)]], ZZ)): + check(c, deprecated=['rep']) + for c in (DMF, DMF(([ZZ(1), ZZ(2)], [ZZ(1), ZZ(3)]), ZZ)): + check(c) + for c in (ANP, ANP([QQ(1), QQ(2)], [QQ(1), QQ(2), QQ(3)], QQ)): + check(c) + +@XFAIL +def test_pickling_polys_rings(): + # NOTE: can't use protocols < 2 because we have to execute __new__ to + # make sure caching of rings works properly. + + from sympy.polys.rings import PolyRing + + ring = PolyRing("x,y,z", ZZ, lex) + + for c in (PolyRing, ring): + check(c, exclude=[0, 1]) + + for c in (ring.dtype, ring.one): + check(c, exclude=[0, 1], check_attr=False) # TODO: Py3k + +def test_pickling_polys_fields(): + pass + # NOTE: can't use protocols < 2 because we have to execute __new__ to + # make sure caching of fields works properly. + + # from sympy.polys.fields import FracField + + # field = FracField("x,y,z", ZZ, lex) + + # TODO: AssertionError: assert id(obj) not in self.memo + # for c in (FracField, field): + # check(c, exclude=[0, 1]) + + # TODO: AssertionError: assert id(obj) not in self.memo + # for c in (field.dtype, field.one): + # check(c, exclude=[0, 1]) + +def test_pickling_polys_elements(): + from sympy.polys.domains.pythonrational import PythonRational + #from sympy.polys.domains.pythonfinitefield import PythonFiniteField + #from sympy.polys.domains.mpelements import MPContext + + for c in (PythonRational, PythonRational(1, 7)): + check(c) + + #gf = PythonFiniteField(17) + + # TODO: fix pickling of ModularInteger + # for c in (gf.dtype, gf(5)): + # check(c) + + #mp = MPContext() + + # TODO: fix pickling of RealElement + # for c in (mp.mpf, mp.mpf(1.0)): + # check(c) + + # TODO: fix pickling of ComplexElement + # for c in (mp.mpc, mp.mpc(1.0, -1.5)): + # check(c) + +def test_pickling_polys_domains(): + # from sympy.polys.domains.pythonfinitefield import PythonFiniteField + from sympy.polys.domains.pythonintegerring import PythonIntegerRing + from sympy.polys.domains.pythonrationalfield import PythonRationalField + + # TODO: fix pickling of ModularInteger + # for c in (PythonFiniteField, PythonFiniteField(17)): + # check(c) + + for c in (PythonIntegerRing, PythonIntegerRing()): + check(c, check_attr=False) + + for c in (PythonRationalField, PythonRationalField()): + check(c, check_attr=False) + + if _gmpy is not None: + # from sympy.polys.domains.gmpyfinitefield import GMPYFiniteField + from sympy.polys.domains.gmpyintegerring import GMPYIntegerRing + from sympy.polys.domains.gmpyrationalfield import GMPYRationalField + + # TODO: fix pickling of ModularInteger + # for c in (GMPYFiniteField, GMPYFiniteField(17)): + # check(c) + + for c in (GMPYIntegerRing, GMPYIntegerRing()): + check(c, check_attr=False) + + for c in (GMPYRationalField, GMPYRationalField()): + check(c, check_attr=False) + + #from sympy.polys.domains.realfield import RealField + #from sympy.polys.domains.complexfield import ComplexField + from sympy.polys.domains.algebraicfield import AlgebraicField + #from sympy.polys.domains.polynomialring import PolynomialRing + #from sympy.polys.domains.fractionfield import FractionField + from sympy.polys.domains.expressiondomain import ExpressionDomain + + # TODO: fix pickling of RealElement + # for c in (RealField, RealField(100)): + # check(c) + + # TODO: fix pickling of ComplexElement + # for c in (ComplexField, ComplexField(100)): + # check(c) + + for c in (AlgebraicField, AlgebraicField(QQ, sqrt(3))): + check(c, check_attr=False) + + # TODO: AssertionError + # for c in (PolynomialRing, PolynomialRing(ZZ, "x,y,z")): + # check(c) + + # TODO: AttributeError: 'PolyElement' object has no attribute 'ring' + # for c in (FractionField, FractionField(ZZ, "x,y,z")): + # check(c) + + for c in (ExpressionDomain, ExpressionDomain()): + check(c, check_attr=False) + + +def test_pickling_polys_orderings(): + from sympy.polys.orderings import (LexOrder, GradedLexOrder, + ReversedGradedLexOrder, InverseOrder) + # from sympy.polys.orderings import ProductOrder + + for c in (LexOrder, LexOrder()): + check(c) + + for c in (GradedLexOrder, GradedLexOrder()): + check(c) + + for c in (ReversedGradedLexOrder, ReversedGradedLexOrder()): + check(c) + + # TODO: Argh, Python is so naive. No lambdas nor inner function support in + # pickling module. Maybe someone could figure out what to do with this. + # + # for c in (ProductOrder, ProductOrder((LexOrder(), lambda m: m[:2]), + # (GradedLexOrder(), lambda m: m[2:]))): + # check(c) + + for c in (InverseOrder, InverseOrder(LexOrder())): + check(c) + +def test_pickling_polys_monomials(): + from sympy.polys.monomials import MonomialOps, Monomial + x, y, z = symbols("x,y,z") + + for c in (MonomialOps, MonomialOps(3)): + check(c) + + for c in (Monomial, Monomial((1, 2, 3), (x, y, z))): + check(c) + +def test_pickling_polys_errors(): + from sympy.polys.polyerrors import (HeuristicGCDFailed, + HomomorphismFailed, IsomorphismFailed, ExtraneousFactors, + EvaluationFailed, RefinementFailed, CoercionFailed, NotInvertible, + NotReversible, NotAlgebraic, DomainError, PolynomialError, + UnificationFailed, GeneratorsError, GeneratorsNeeded, + UnivariatePolynomialError, MultivariatePolynomialError, OptionError, + FlagError) + # from sympy.polys.polyerrors import (ExactQuotientFailed, + # OperationNotSupported, ComputationFailed, PolificationFailed) + + # x = Symbol('x') + + # TODO: TypeError: __init__() takes at least 3 arguments (1 given) + # for c in (ExactQuotientFailed, ExactQuotientFailed(x, 3*x, ZZ)): + # check(c) + + # TODO: TypeError: can't pickle instancemethod objects + # for c in (OperationNotSupported, OperationNotSupported(Poly(x), Poly.gcd)): + # check(c) + + for c in (HeuristicGCDFailed, HeuristicGCDFailed()): + check(c) + + for c in (HomomorphismFailed, HomomorphismFailed()): + check(c) + + for c in (IsomorphismFailed, IsomorphismFailed()): + check(c) + + for c in (ExtraneousFactors, ExtraneousFactors()): + check(c) + + for c in (EvaluationFailed, EvaluationFailed()): + check(c) + + for c in (RefinementFailed, RefinementFailed()): + check(c) + + for c in (CoercionFailed, CoercionFailed()): + check(c) + + for c in (NotInvertible, NotInvertible()): + check(c) + + for c in (NotReversible, NotReversible()): + check(c) + + for c in (NotAlgebraic, NotAlgebraic()): + check(c) + + for c in (DomainError, DomainError()): + check(c) + + for c in (PolynomialError, PolynomialError()): + check(c) + + for c in (UnificationFailed, UnificationFailed()): + check(c) + + for c in (GeneratorsError, GeneratorsError()): + check(c) + + for c in (GeneratorsNeeded, GeneratorsNeeded()): + check(c) + + # TODO: PicklingError: Can't pickle at 0x38578c0>: it's not found as __main__. + # for c in (ComputationFailed, ComputationFailed(lambda t: t, 3, None)): + # check(c) + + for c in (UnivariatePolynomialError, UnivariatePolynomialError()): + check(c) + + for c in (MultivariatePolynomialError, MultivariatePolynomialError()): + check(c) + + # TODO: TypeError: __init__() takes at least 3 arguments (1 given) + # for c in (PolificationFailed, PolificationFailed({}, x, x, False)): + # check(c) + + for c in (OptionError, OptionError()): + check(c) + + for c in (FlagError, FlagError()): + check(c) + +#def test_pickling_polys_options(): + #from sympy.polys.polyoptions import Options + + # TODO: fix pickling of `symbols' flag + # for c in (Options, Options((), dict(domain='ZZ', polys=False))): + # check(c) + +# TODO: def test_pickling_polys_rootisolation(): +# RealInterval +# ComplexInterval + +def test_pickling_polys_rootoftools(): + from sympy.polys.rootoftools import CRootOf, RootSum + + x = Symbol('x') + f = x**3 + x + 3 + + for c in (CRootOf, CRootOf(f, 0)): + check(c) + + for c in (RootSum, RootSum(f, exp)): + check(c) + +#================== printing ==================== +from sympy.printing.latex import LatexPrinter +from sympy.printing.mathml import MathMLContentPrinter, MathMLPresentationPrinter +from sympy.printing.pretty.pretty import PrettyPrinter +from sympy.printing.pretty.stringpict import prettyForm, stringPict +from sympy.printing.printer import Printer +from sympy.printing.python import PythonPrinter + + +def test_printing(): + for c in (LatexPrinter, LatexPrinter(), MathMLContentPrinter, + MathMLPresentationPrinter, PrettyPrinter, prettyForm, stringPict, + stringPict("a"), Printer, Printer(), PythonPrinter, + PythonPrinter()): + check(c) + + +@XFAIL +def test_printing1(): + check(MathMLContentPrinter()) + + +@XFAIL +def test_printing2(): + check(MathMLPresentationPrinter()) + + +@XFAIL +def test_printing3(): + check(PrettyPrinter()) + +#================== series ====================== +from sympy.series.limits import Limit +from sympy.series.order import Order + + +def test_series(): + e = Symbol("e") + x = Symbol("x") + for c in (Limit, Limit(e, x, 1), Order, Order(e)): + check(c) + +#================== concrete ================== +from sympy.concrete.products import Product +from sympy.concrete.summations import Sum + + +def test_concrete(): + x = Symbol("x") + for c in (Product, Product(x, (x, 2, 4)), Sum, Sum(x, (x, 2, 4))): + check(c) + +def test_deprecation_warning(): + w = SymPyDeprecationWarning("message", deprecated_since_version='1.0', active_deprecations_target="active-deprecations") + check(w) + +def test_issue_18438(): + assert pickle.loads(pickle.dumps(S.Half)) == S.Half + + +#================= old pickles ================= +def test_unpickle_from_older_versions(): + data = ( + b'\x80\x04\x95^\x00\x00\x00\x00\x00\x00\x00\x8c\x10sympy.core.power' + b'\x94\x8c\x03Pow\x94\x93\x94\x8c\x12sympy.core.numbers\x94\x8c' + b'\x07Integer\x94\x93\x94K\x02\x85\x94R\x94}\x94bh\x03\x8c\x04Half' + b'\x94\x93\x94)R\x94}\x94b\x86\x94R\x94}\x94b.' + ) + assert pickle.loads(data) == sqrt(2) diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_source.py b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_source.py new file mode 100644 index 0000000000000000000000000000000000000000..468185bc579fc325aee21024dfa15ebf14287b5f --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_source.py @@ -0,0 +1,11 @@ +from sympy.utilities.source import get_mod_func, get_class + + +def test_get_mod_func(): + assert get_mod_func( + 'sympy.core.basic.Basic') == ('sympy.core.basic', 'Basic') + + +def test_get_class(): + _basic = get_class('sympy.core.basic.Basic') + assert _basic.__name__ == 'Basic' diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_timeutils.py b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_timeutils.py new file mode 100644 index 0000000000000000000000000000000000000000..14edfd089c7315ee9f39a4298af0289f8919da6b --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_timeutils.py @@ -0,0 +1,10 @@ +"""Tests for simple tools for timing functions' execution. """ + +from sympy.utilities.timeutils import timed + +def test_timed(): + result = timed(lambda: 1 + 1, limit=100000) + assert result[0] == 100000 and result[3] == "ns", str(result) + + result = timed("1 + 1", limit=100000) + assert result[0] == 100000 and result[3] == "ns" diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_wester.py b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_wester.py new file mode 100644 index 0000000000000000000000000000000000000000..c5699a4eb0824e507967ecd4ff8f7a5f32cc9a54 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_wester.py @@ -0,0 +1,3104 @@ +""" Tests from Michael Wester's 1999 paper "Review of CAS mathematical +capabilities". + +http://www.math.unm.edu/~wester/cas/book/Wester.pdf +See also http://math.unm.edu/~wester/cas_review.html for detailed output of +each tested system. +""" + +from sympy.assumptions.ask import Q, ask +from sympy.assumptions.refine import refine +from sympy.concrete.products import product +from sympy.core import EulerGamma +from sympy.core.evalf import N +from sympy.core.function import (Derivative, Function, Lambda, Subs, + diff, expand, expand_func) +from sympy.core.mul import Mul +from sympy.core.intfunc import igcd +from sympy.core.numbers import (AlgebraicNumber, E, I, Rational, + nan, oo, pi, zoo) +from sympy.core.relational import Eq, Lt +from sympy.core.singleton import S +from sympy.core.symbol import Dummy, Symbol, symbols +from sympy.functions.combinatorial.factorials import (rf, binomial, + factorial, factorial2) +from sympy.functions.combinatorial.numbers import bernoulli, fibonacci, totient, partition +from sympy.functions.elementary.complexes import (conjugate, im, re, + sign) +from sympy.functions.elementary.exponential import LambertW, exp, log +from sympy.functions.elementary.hyperbolic import (asinh, cosh, sinh, + tanh) +from sympy.functions.elementary.integers import ceiling, floor +from sympy.functions.elementary.miscellaneous import Max, Min, sqrt +from sympy.functions.elementary.piecewise import Piecewise +from sympy.functions.elementary.trigonometric import (acos, acot, asin, + atan, cos, cot, csc, sec, sin, tan) +from sympy.functions.special.bessel import besselj +from sympy.functions.special.delta_functions import DiracDelta +from sympy.functions.special.elliptic_integrals import (elliptic_e, + elliptic_f) +from sympy.functions.special.gamma_functions import gamma, polygamma +from sympy.functions.special.hyper import hyper +from sympy.functions.special.polynomials import (assoc_legendre, + chebyshevt) +from sympy.functions.special.zeta_functions import polylog +from sympy.geometry.util import idiff +from sympy.logic.boolalg import And +from sympy.matrices.dense import hessian, wronskian +from sympy.matrices.expressions.matmul import MatMul +from sympy.ntheory.continued_fraction import ( + continued_fraction_convergents as cf_c, + continued_fraction_iterator as cf_i, continued_fraction_periodic as + cf_p, continued_fraction_reduce as cf_r) +from sympy.ntheory.factor_ import factorint +from sympy.ntheory.generate import primerange +from sympy.polys.domains.integerring import ZZ +from sympy.polys.orthopolys import legendre_poly +from sympy.polys.partfrac import apart +from sympy.polys.polytools import Poly, factor, gcd, resultant +from sympy.series.limits import limit +from sympy.series.order import O +from sympy.series.residues import residue +from sympy.series.series import series +from sympy.sets.fancysets import ImageSet +from sympy.sets.sets import FiniteSet, Intersection, Interval, Union +from sympy.simplify.combsimp import combsimp +from sympy.simplify.hyperexpand import hyperexpand +from sympy.simplify.powsimp import powdenest, powsimp +from sympy.simplify.radsimp import radsimp +from sympy.simplify.simplify import logcombine, simplify +from sympy.simplify.sqrtdenest import sqrtdenest +from sympy.simplify.trigsimp import trigsimp +from sympy.solvers.solvers import solve + +import mpmath +from sympy.functions.combinatorial.numbers import stirling +from sympy.functions.special.delta_functions import Heaviside +from sympy.functions.special.error_functions import Ci, Si, erf +from sympy.functions.special.zeta_functions import zeta +from sympy.testing.pytest import (XFAIL, slow, SKIP, tooslow, raises) +from sympy.utilities.iterables import partitions +from mpmath import mpi, mpc +from sympy.matrices import Matrix, GramSchmidt, eye +from sympy.matrices.expressions.blockmatrix import BlockMatrix, block_collapse +from sympy.matrices.expressions import MatrixSymbol, ZeroMatrix +from sympy.physics.quantum import Commutator +from sympy.polys.rings import PolyRing +from sympy.polys.fields import FracField +from sympy.polys.solvers import solve_lin_sys +from sympy.concrete import Sum +from sympy.concrete.products import Product +from sympy.integrals import integrate +from sympy.integrals.transforms import laplace_transform,\ + inverse_laplace_transform, LaplaceTransform, fourier_transform,\ + mellin_transform, laplace_correspondence, laplace_initial_conds +from sympy.solvers.recurr import rsolve +from sympy.solvers.solveset import solveset, solveset_real, linsolve +from sympy.solvers.ode import dsolve +from sympy.core.relational import Equality +from itertools import islice, takewhile +from sympy.series.formal import fps +from sympy.series.fourier import fourier_series +from sympy.calculus.util import minimum + + +EmptySet = S.EmptySet +R = Rational +x, y, z = symbols('x y z') +i, j, k, l, m, n = symbols('i j k l m n', integer=True) +f = Function('f') +g = Function('g') + +# A. Boolean Logic and Quantifier Elimination +# Not implemented. + +# B. Set Theory + + +def test_B1(): + assert (FiniteSet(i, j, j, k, k, k) | FiniteSet(l, k, j) | + FiniteSet(j, m, j)) == FiniteSet(i, j, k, l, m) + + +def test_B2(): + assert (FiniteSet(i, j, j, k, k, k) & FiniteSet(l, k, j) & + FiniteSet(j, m, j)) == Intersection({j, m}, {i, j, k}, {j, k, l}) + # Previous output below. Not sure why that should be the expected output. + # There should probably be a way to rewrite Intersections that way but I + # don't see why an Intersection should evaluate like that: + # + # == Union({j}, Intersection({m}, Union({j, k}, Intersection({i}, {l})))) + + +def test_B3(): + assert (FiniteSet(i, j, k, l, m) - FiniteSet(j) == + FiniteSet(i, k, l, m)) + + +def test_B4(): + assert (FiniteSet(*(FiniteSet(i, j)*FiniteSet(k, l))) == + FiniteSet((i, k), (i, l), (j, k), (j, l))) + + +# C. Numbers + + +def test_C1(): + assert (factorial(50) == + 30414093201713378043612608166064768844377641568960512000000000000) + + +def test_C2(): + assert (factorint(factorial(50)) == {2: 47, 3: 22, 5: 12, 7: 8, + 11: 4, 13: 3, 17: 2, 19: 2, 23: 2, 29: 1, 31: 1, 37: 1, + 41: 1, 43: 1, 47: 1}) + + +def test_C3(): + assert (factorial2(10), factorial2(9)) == (3840, 945) + + +# Base conversions; not really implemented by SymPy +# Whatever. Take credit! +def test_C4(): + assert 0xABC == 2748 + + +def test_C5(): + assert 123 == int('234', 7) + + +def test_C6(): + assert int('677', 8) == int('1BF', 16) == 447 + + +def test_C7(): + assert log(32768, 8) == 5 + + +def test_C8(): + # Modular multiplicative inverse. Would be nice if divmod could do this. + assert ZZ.invert(5, 7) == 3 + assert ZZ.invert(5, 6) == 5 + + +def test_C9(): + assert igcd(igcd(1776, 1554), 5698) == 74 + + +def test_C10(): + x = 0 + for n in range(2, 11): + x += R(1, n) + assert x == R(4861, 2520) + + +def test_C11(): + assert R(1, 7) == S('0.[142857]') + + +def test_C12(): + assert R(7, 11) * R(22, 7) == 2 + + +def test_C13(): + test = R(10, 7) * (1 + R(29, 1000)) ** R(1, 3) + good = 3 ** R(1, 3) + assert test == good + + +def test_C14(): + assert sqrtdenest(sqrt(2*sqrt(3) + 4)) == 1 + sqrt(3) + + +def test_C15(): + test = sqrtdenest(sqrt(14 + 3*sqrt(3 + 2*sqrt(5 - 12*sqrt(3 - 2*sqrt(2)))))) + good = sqrt(2) + 3 + assert test == good + + +def test_C16(): + test = sqrtdenest(sqrt(10 + 2*sqrt(6) + 2*sqrt(10) + 2*sqrt(15))) + good = sqrt(2) + sqrt(3) + sqrt(5) + assert test == good + + +def test_C17(): + test = radsimp((sqrt(3) + sqrt(2)) / (sqrt(3) - sqrt(2))) + good = 5 + 2*sqrt(6) + assert test == good + + +def test_C18(): + assert simplify((sqrt(-2 + sqrt(-5)) * sqrt(-2 - sqrt(-5))).expand(complex=True)) == 3 + + +@XFAIL +def test_C19(): + assert radsimp(simplify((90 + 34*sqrt(7)) ** R(1, 3))) == 3 + sqrt(7) + + +def test_C20(): + inside = (135 + 78*sqrt(3)) + test = AlgebraicNumber((inside**R(2, 3) + 3) * sqrt(3) / inside**R(1, 3)) + assert simplify(test) == AlgebraicNumber(12) + + +def test_C21(): + assert simplify(AlgebraicNumber((41 + 29*sqrt(2)) ** R(1, 5))) == \ + AlgebraicNumber(1 + sqrt(2)) + + +@XFAIL +def test_C22(): + test = simplify(((6 - 4*sqrt(2))*log(3 - 2*sqrt(2)) + (3 - 2*sqrt(2))*log(17 + - 12*sqrt(2)) + 32 - 24*sqrt(2)) / (48*sqrt(2) - 72)) + good = sqrt(2)/3 - log(sqrt(2) - 1)/3 + assert test == good + + +def test_C23(): + assert 2 * oo - 3 is oo + + +@XFAIL +def test_C24(): + raise NotImplementedError("2**aleph_null == aleph_1") + +# D. Numerical Analysis + + +def test_D1(): + assert 0.0 / sqrt(2) == 0 + + +def test_D2(): + assert str(exp(-1000000).evalf()) == '3.29683147808856e-434295' + + +def test_D3(): + assert exp(pi*sqrt(163)).evalf(50).num.ae(262537412640768744) + + +def test_D4(): + assert floor(R(-5, 3)) == -2 + assert ceiling(R(-5, 3)) == -1 + + +@XFAIL +def test_D5(): + raise NotImplementedError("cubic_spline([1, 2, 4, 5], [1, 4, 2, 3], x)(3) == 27/8") + + +@XFAIL +def test_D6(): + raise NotImplementedError("translate sum(a[i]*x**i, (i,1,n)) to FORTRAN") + + +@XFAIL +def test_D7(): + raise NotImplementedError("translate sum(a[i]*x**i, (i,1,n)) to C") + + +@XFAIL +def test_D8(): + # One way is to cheat by converting the sum to a string, + # and replacing the '[' and ']' with ''. + # E.g., horner(S(str(_).replace('[','').replace(']',''))) + raise NotImplementedError("apply Horner's rule to sum(a[i]*x**i, (i,1,5))") + + +@XFAIL +def test_D9(): + raise NotImplementedError("translate D8 to FORTRAN") + + +@XFAIL +def test_D10(): + raise NotImplementedError("translate D8 to C") + + +@XFAIL +def test_D11(): + #Is there a way to use count_ops? + raise NotImplementedError("flops(sum(product(f[i][k], (i,1,k)), (k,1,n)))") + + +@XFAIL +def test_D12(): + assert (mpi(-4, 2) * x + mpi(1, 3)) ** 2 == mpi(-8, 16)*x**2 + mpi(-24, 12)*x + mpi(1, 9) + + +@XFAIL +def test_D13(): + raise NotImplementedError("discretize a PDE: diff(f(x,t),t) == diff(diff(f(x,t),x),x)") + +# E. Statistics +# See scipy; all of this is numerical. + +# F. Combinatorial Theory. + + +def test_F1(): + assert rf(x, 3) == x*(1 + x)*(2 + x) + + +def test_F2(): + assert expand_func(binomial(n, 3)) == n*(n - 1)*(n - 2)/6 + + +@XFAIL +def test_F3(): + assert combsimp(2**n * factorial(n) * factorial2(2*n - 1)) == factorial(2*n) + + +@XFAIL +def test_F4(): + assert combsimp(2**n * factorial(n) * product(2*k - 1, (k, 1, n))) == factorial(2*n) + + +@XFAIL +def test_F5(): + assert gamma(n + R(1, 2)) / sqrt(pi) / factorial(n) == factorial(2*n)/2**(2*n)/factorial(n)**2 + + +def test_F6(): + partTest = [p.copy() for p in partitions(4)] + partDesired = [{4: 1}, {1: 1, 3: 1}, {2: 2}, {1: 2, 2:1}, {1: 4}] + assert partTest == partDesired + + +def test_F7(): + assert partition(4) == 5 + + +def test_F8(): + assert stirling(5, 2, signed=True) == -50 # if signed, then kind=1 + + +def test_F9(): + assert totient(1776) == 576 + +# G. Number Theory + + +def test_G1(): + assert list(primerange(999983, 1000004)) == [999983, 1000003] + + +@XFAIL +def test_G2(): + raise NotImplementedError("find the primitive root of 191 == 19") + + +@XFAIL +def test_G3(): + raise NotImplementedError("(a+b)**p mod p == a**p + b**p mod p; p prime") + +# ... G14 Modular equations are not implemented. + +def test_G15(): + assert Rational(sqrt(3).evalf()).limit_denominator(15) == R(26, 15) + assert list(takewhile(lambda x: x.q <= 15, cf_c(cf_i(sqrt(3)))))[-1] == \ + R(26, 15) + + +def test_G16(): + assert list(islice(cf_i(pi),10)) == [3, 7, 15, 1, 292, 1, 1, 1, 2, 1] + + +def test_G17(): + assert cf_p(0, 1, 23) == [4, [1, 3, 1, 8]] + + +def test_G18(): + assert cf_p(1, 2, 5) == [[1]] + assert cf_r([[1]]).expand() == S.Half + sqrt(5)/2 + + +@XFAIL +def test_G19(): + s = symbols('s', integer=True, positive=True) + it = cf_i((exp(1/s) - 1)/(exp(1/s) + 1)) + assert list(islice(it, 5)) == [0, 2*s, 6*s, 10*s, 14*s] + + +def test_G20(): + s = symbols('s', integer=True, positive=True) + # Wester erroneously has this as -s + sqrt(s**2 + 1) + assert cf_r([[2*s]]) == s + sqrt(s**2 + 1) + + +@XFAIL +def test_G20b(): + s = symbols('s', integer=True, positive=True) + assert cf_p(s, 1, s**2 + 1) == [[2*s]] + + +# H. Algebra + + +def test_H1(): + assert simplify(2*2**n) == simplify(2**(n + 1)) + assert powdenest(2*2**n) == simplify(2**(n + 1)) + + +def test_H2(): + assert powsimp(4 * 2**n) == 2**(n + 2) + + +def test_H3(): + assert (-1)**(n*(n + 1)) == 1 + + +def test_H4(): + expr = factor(6*x - 10) + assert type(expr) is Mul + assert expr.args[0] == 2 + assert expr.args[1] == 3*x - 5 + +p1 = 64*x**34 - 21*x**47 - 126*x**8 - 46*x**5 - 16*x**60 - 81 +p2 = 72*x**60 - 25*x**25 - 19*x**23 - 22*x**39 - 83*x**52 + 54*x**10 + 81 +q = 34*x**19 - 25*x**16 + 70*x**7 + 20*x**3 - 91*x - 86 + + +def test_H5(): + assert gcd(p1, p2, x) == 1 + + +def test_H6(): + assert gcd(expand(p1 * q), expand(p2 * q)) == q + + +def test_H7(): + p1 = 24*x*y**19*z**8 - 47*x**17*y**5*z**8 + 6*x**15*y**9*z**2 - 3*x**22 + 5 + p2 = 34*x**5*y**8*z**13 + 20*x**7*y**7*z**7 + 12*x**9*y**16*z**4 + 80*y**14*z + assert gcd(p1, p2, x, y, z) == 1 + + +def test_H8(): + p1 = 24*x*y**19*z**8 - 47*x**17*y**5*z**8 + 6*x**15*y**9*z**2 - 3*x**22 + 5 + p2 = 34*x**5*y**8*z**13 + 20*x**7*y**7*z**7 + 12*x**9*y**16*z**4 + 80*y**14*z + q = 11*x**12*y**7*z**13 - 23*x**2*y**8*z**10 + 47*x**17*y**5*z**8 + assert gcd(p1 * q, p2 * q, x, y, z) == q + + +def test_H9(): + x = Symbol('x', zero=False) + p1 = 2*x**(n + 4) - x**(n + 2) + p2 = 4*x**(n + 1) + 3*x**n + assert gcd(p1, p2) == x**n + + +def test_H10(): + p1 = 3*x**4 + 3*x**3 + x**2 - x - 2 + p2 = x**3 - 3*x**2 + x + 5 + assert resultant(p1, p2, x) == 0 + + +def test_H11(): + assert resultant(p1 * q, p2 * q, x) == 0 + + +def test_H12(): + num = x**2 - 4 + den = x**2 + 4*x + 4 + assert simplify(num/den) == (x - 2)/(x + 2) + + +@XFAIL +def test_H13(): + assert simplify((exp(x) - 1) / (exp(x/2) + 1)) == exp(x/2) - 1 + + +def test_H14(): + p = (x + 1) ** 20 + ep = expand(p) + assert ep == (1 + 20*x + 190*x**2 + 1140*x**3 + 4845*x**4 + 15504*x**5 + + 38760*x**6 + 77520*x**7 + 125970*x**8 + 167960*x**9 + 184756*x**10 + + 167960*x**11 + 125970*x**12 + 77520*x**13 + 38760*x**14 + 15504*x**15 + + 4845*x**16 + 1140*x**17 + 190*x**18 + 20*x**19 + x**20) + dep = diff(ep, x) + assert dep == (20 + 380*x + 3420*x**2 + 19380*x**3 + 77520*x**4 + + 232560*x**5 + 542640*x**6 + 1007760*x**7 + 1511640*x**8 + 1847560*x**9 + + 1847560*x**10 + 1511640*x**11 + 1007760*x**12 + 542640*x**13 + + 232560*x**14 + 77520*x**15 + 19380*x**16 + 3420*x**17 + 380*x**18 + + 20*x**19) + assert factor(dep) == 20*(1 + x)**19 + + +def test_H15(): + assert simplify(Mul(*[x - r for r in solveset(x**3 + x**2 - 7)])) == x**3 + x**2 - 7 + + +def test_H16(): + assert factor(x**100 - 1) == ((x - 1)*(x + 1)*(x**2 + 1)*(x**4 - x**3 + + x**2 - x + 1)*(x**4 + x**3 + x**2 + x + 1)*(x**8 - x**6 + x**4 + - x**2 + 1)*(x**20 - x**15 + x**10 - x**5 + 1)*(x**20 + x**15 + x**10 + + x**5 + 1)*(x**40 - x**30 + x**20 - x**10 + 1)) + + +def test_H17(): + assert simplify(factor(expand(p1 * p2)) - p1*p2) == 0 + + +@XFAIL +def test_H18(): + # Factor over complex rationals. + test = factor(4*x**4 + 8*x**3 + 77*x**2 + 18*x + 153) + good = (2*x + 3*I)*(2*x - 3*I)*(x + 1 - 4*I)*(x + 1 + 4*I) + assert test == good + + +def test_H19(): + a = symbols('a') + # The idea is to let a**2 == 2, then solve 1/(a-1). Answer is a+1") + assert Poly(a - 1).invert(Poly(a**2 - 2)) == a + 1 + + +@XFAIL +def test_H20(): + raise NotImplementedError("let a**2==2; (x**3 + (a-2)*x**2 - " + + "(2*a+3)*x - 3*a) / (x**2-2) = (x**2 - 2*x - 3) / (x-a)") + + +@XFAIL +def test_H21(): + raise NotImplementedError("evaluate (b+c)**4 assuming b**3==2, c**2==3. \ + Answer is 2*b + 8*c + 18*b**2 + 12*b*c + 9") + + +def test_H22(): + assert factor(x**4 - 3*x**2 + 1, modulus=5) == (x - 2)**2 * (x + 2)**2 + + +def test_H23(): + f = x**11 + x + 1 + g = (x**2 + x + 1) * (x**9 - x**8 + x**6 - x**5 + x**3 - x**2 + 1) + assert factor(f, modulus=65537) == g + + +def test_H24(): + phi = AlgebraicNumber(S.GoldenRatio.expand(func=True), alias='phi') + assert factor(x**4 - 3*x**2 + 1, extension=phi) == \ + (x - phi)*(x + 1 - phi)*(x - 1 + phi)*(x + phi) + + +def test_H25(): + e = (x - 2*y**2 + 3*z**3) ** 20 + assert factor(expand(e)) == e + + +def test_H26(): + g = expand((sin(x) - 2*cos(y)**2 + 3*tan(z)**3)**20) + assert factor(g, expand=False) == (-sin(x) + 2*cos(y)**2 - 3*tan(z)**3)**20 + + +def test_H27(): + f = 24*x*y**19*z**8 - 47*x**17*y**5*z**8 + 6*x**15*y**9*z**2 - 3*x**22 + 5 + g = 34*x**5*y**8*z**13 + 20*x**7*y**7*z**7 + 12*x**9*y**16*z**4 + 80*y**14*z + h = -2*z*y**7 \ + *(6*x**9*y**9*z**3 + 10*x**7*z**6 + 17*y*x**5*z**12 + 40*y**7) \ + *(3*x**22 + 47*x**17*y**5*z**8 - 6*x**15*y**9*z**2 - 24*x*y**19*z**8 - 5) + assert factor(expand(f*g)) == h + + +@XFAIL +def test_H28(): + raise NotImplementedError("expand ((1 - c**2)**5 * (1 - s**2)**5 * " + + "(c**2 + s**2)**10) with c**2 + s**2 = 1. Answer is c**10*s**10.") + + +@XFAIL +def test_H29(): + assert factor(4*x**2 - 21*x*y + 20*y**2, modulus=3) == (x + y)*(x - y) + + +def test_H30(): + test = factor(x**3 + y**3, extension=sqrt(-3)) + answer = (x + y)*(x + y*(-R(1, 2) - sqrt(3)/2*I))*(x + y*(-R(1, 2) + sqrt(3)/2*I)) + assert answer == test + + +def test_H31(): + f = (x**2 + 2*x + 3)/(x**3 + 4*x**2 + 5*x + 2) + g = 2 / (x + 1)**2 - 2 / (x + 1) + 3 / (x + 2) + assert apart(f) == g + + +@XFAIL +def test_H32(): # issue 6558 + raise NotImplementedError("[A*B*C - (A*B*C)**(-1)]*A*C*B (product \ + of a non-commuting product and its inverse)") + + +def test_H33(): + A, B, C = symbols('A, B, C', commutative=False) + assert (Commutator(A, Commutator(B, C)) + + Commutator(B, Commutator(C, A)) + + Commutator(C, Commutator(A, B))).doit().expand() == 0 + + +# I. Trigonometry + +def test_I1(): + assert tan(pi*R(7, 10)) == -sqrt(1 + 2/sqrt(5)) + + +@XFAIL +def test_I2(): + assert sqrt((1 + cos(6))/2) == -cos(3) + + +def test_I3(): + assert cos(n*pi) + sin((4*n - 1)*pi/2) == (-1)**n - 1 + + +def test_I4(): + assert refine(cos(pi*cos(n*pi)) + sin(pi/2*cos(n*pi)), Q.integer(n)) == (-1)**n - 1 + + +@XFAIL +def test_I5(): + assert sin((n**5/5 + n**4/2 + n**3/3 - n/30) * pi) == 0 + + +@XFAIL +def test_I6(): + raise NotImplementedError("assuming -3*pi pi**E) + + +@XFAIL +def test_N2(): + x = symbols('x', real=True) + assert ask(x**4 - x + 1 > 0) is True + assert ask(x**4 - x + 1 > 1) is False + + +@XFAIL +def test_N3(): + x = symbols('x', real=True) + assert ask(And(Lt(-1, x), Lt(x, 1)), abs(x) < 1 ) + +@XFAIL +def test_N4(): + x, y = symbols('x y', real=True) + assert ask(2*x**2 > 2*y**2, (x > y) & (y > 0)) is True + + +@XFAIL +def test_N5(): + x, y, k = symbols('x y k', real=True) + assert ask(k*x**2 > k*y**2, (x > y) & (y > 0) & (k > 0)) is True + + +@slow +@XFAIL +def test_N6(): + x, y, k, n = symbols('x y k n', real=True) + assert ask(k*x**n > k*y**n, (x > y) & (y > 0) & (k > 0) & (n > 0)) is True + + +@XFAIL +def test_N7(): + x, y = symbols('x y', real=True) + assert ask(y > 0, (x > 1) & (y >= x - 1)) is True + + +@XFAIL +@slow +def test_N8(): + x, y, z = symbols('x y z', real=True) + assert ask(Eq(x, y) & Eq(y, z), + (x >= y) & (y >= z) & (z >= x)) + + +def test_N9(): + x = Symbol('x') + assert solveset(abs(x - 1) > 2, domain=S.Reals) == Union(Interval(-oo, -1, False, True), + Interval(3, oo, True)) + + +def test_N10(): + x = Symbol('x') + p = (x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5) + assert solveset(expand(p) < 0, domain=S.Reals) == Union(Interval(-oo, 1, True, True), + Interval(2, 3, True, True), + Interval(4, 5, True, True)) + + +def test_N11(): + x = Symbol('x') + assert solveset(6/(x - 3) <= 3, domain=S.Reals) == Union(Interval(-oo, 3, True, True), Interval(5, oo)) + + +def test_N12(): + x = Symbol('x') + assert solveset(sqrt(x) < 2, domain=S.Reals) == Interval(0, 4, False, True) + + +def test_N13(): + x = Symbol('x') + assert solveset(sin(x) < 2, domain=S.Reals) == S.Reals + + +@XFAIL +def test_N14(): + x = Symbol('x') + # Gives 'Union(Interval(Integer(0), Mul(Rational(1, 2), pi), false, true), + # Interval(Mul(Rational(1, 2), pi), Mul(Integer(2), pi), true, false))' + # which is not the correct answer, but the provided also seems wrong. + assert solveset(sin(x) < 1, x, domain=S.Reals) == Union(Interval(-oo, pi/2, True, True), + Interval(pi/2, oo, True, True)) + + +def test_N15(): + r, t = symbols('r t') + # raises NotImplementedError: only univariate inequalities are supported + solveset(abs(2*r*(cos(t) - 1) + 1) <= 1, r, S.Reals) + + +def test_N16(): + r, t = symbols('r t') + solveset((r**2)*((cos(t) - 4)**2)*sin(t)**2 < 9, r, S.Reals) + + +@XFAIL +def test_N17(): + # currently only univariate inequalities are supported + assert solveset((x + y > 0, x - y < 0), (x, y)) == (abs(x) < y) + + +def test_O1(): + M = Matrix((1 + I, -2, 3*I)) + assert sqrt(expand(M.dot(M.H))) == sqrt(15) + + +def test_O2(): + assert Matrix((2, 2, -3)).cross(Matrix((1, 3, 1))) == Matrix([[11], + [-5], + [4]]) + +# The vector module has no way of representing vectors symbolically (without +# respect to a basis) +@XFAIL +def test_O3(): + # assert (va ^ vb) | (vc ^ vd) == -(va | vc)*(vb | vd) + (va | vd)*(vb | vc) + raise NotImplementedError("""The vector module has no way of representing + vectors symbolically (without respect to a basis)""") + +def test_O4(): + from sympy.vector import CoordSys3D, Del + N = CoordSys3D("N") + delop = Del() + i, j, k = N.base_vectors() + x, y, z = N.base_scalars() + F = i*(x*y*z) + j*((x*y*z)**2) + k*((y**2)*(z**3)) + assert delop.cross(F).doit() == (-2*x**2*y**2*z + 2*y*z**3)*i + x*y*j + (2*x*y**2*z**2 - x*z)*k + +@XFAIL +def test_O5(): + #assert grad|(f^g)-g|(grad^f)+f|(grad^g) == 0 + raise NotImplementedError("""The vector module has no way of representing + vectors symbolically (without respect to a basis)""") + +#testO8-O9 MISSING!! + + +def test_O10(): + L = [Matrix([2, 3, 5]), Matrix([3, 6, 2]), Matrix([8, 3, 6])] + assert GramSchmidt(L) == [Matrix([ + [2], + [3], + [5]]), + Matrix([ + [R(23, 19)], + [R(63, 19)], + [R(-47, 19)]]), + Matrix([ + [R(1692, 353)], + [R(-1551, 706)], + [R(-423, 706)]])] + + +def test_P1(): + assert Matrix(3, 3, lambda i, j: j - i).diagonal(-1) == Matrix( + 1, 2, [-1, -1]) + + +def test_P2(): + M = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) + M.row_del(1) + M.col_del(2) + assert M == Matrix([[1, 2], + [7, 8]]) + + +def test_P3(): + A = Matrix([ + [11, 12, 13, 14], + [21, 22, 23, 24], + [31, 32, 33, 34], + [41, 42, 43, 44]]) + + A11 = A[0:3, 1:4] + A12 = A[(0, 1, 3), (2, 0, 3)] + A21 = A + A221 = -A[0:2, 2:4] + A222 = -A[(3, 0), (2, 1)] + A22 = BlockMatrix([[A221, A222]]).T + rows = [[-A11, A12], [A21, A22]] + raises(ValueError, lambda: BlockMatrix(rows)) + B = Matrix(rows) + assert B == Matrix([ + [-12, -13, -14, 13, 11, 14], + [-22, -23, -24, 23, 21, 24], + [-32, -33, -34, 43, 41, 44], + [11, 12, 13, 14, -13, -23], + [21, 22, 23, 24, -14, -24], + [31, 32, 33, 34, -43, -13], + [41, 42, 43, 44, -42, -12]]) + + +@XFAIL +def test_P4(): + raise NotImplementedError("Block matrix diagonalization not supported") + + +def test_P5(): + M = Matrix([[7, 11], + [3, 8]]) + assert M % 2 == Matrix([[1, 1], + [1, 0]]) + + +def test_P6(): + M = Matrix([[cos(x), sin(x)], + [-sin(x), cos(x)]]) + assert M.diff(x, 2) == Matrix([[-cos(x), -sin(x)], + [sin(x), -cos(x)]]) + + +def test_P7(): + M = Matrix([[x, y]])*( + z*Matrix([[1, 3, 5], + [2, 4, 6]]) + Matrix([[7, -9, 11], + [-8, 10, -12]])) + assert M == Matrix([[x*(z + 7) + y*(2*z - 8), x*(3*z - 9) + y*(4*z + 10), + x*(5*z + 11) + y*(6*z - 12)]]) + + +def test_P8(): + M = Matrix([[1, -2*I], + [-3*I, 4]]) + assert M.norm(ord=S.Infinity) == 7 + + +def test_P9(): + a, b, c = symbols('a b c', nonzero=True) + M = Matrix([[a/(b*c), 1/c, 1/b], + [1/c, b/(a*c), 1/a], + [1/b, 1/a, c/(a*b)]]) + assert factor(M.norm('fro')) == (a**2 + b**2 + c**2)/(abs(a)*abs(b)*abs(c)) + + +@XFAIL +def test_P10(): + M = Matrix([[1, 2 + 3*I], + [f(4 - 5*I), 6]]) + # conjugate(f(4 - 5*i)) is not simplified to f(4+5*I) + assert M.H == Matrix([[1, f(4 + 5*I)], + [2 + 3*I, 6]]) + + +@XFAIL +def test_P11(): + # raises NotImplementedError("Matrix([[x,y],[1,x*y]]).inv() + # not simplifying to extract common factor") + assert Matrix([[x, y], + [1, x*y]]).inv() == (1/(x**2 - 1))*Matrix([[x, -1], + [-1/y, x/y]]) + + +def test_P11_workaround(): + # This test was changed to inverse method ADJ because it depended on the + # specific form of inverse returned from the 'GE' method which has changed. + M = Matrix([[x, y], [1, x*y]]).inv('ADJ') + c = gcd(tuple(M)) + assert MatMul(c, M/c, evaluate=False) == MatMul(c, Matrix([ + [x*y, -y], + [ -1, x]]), evaluate=False) + + +def test_P12(): + A11 = MatrixSymbol('A11', n, n) + A12 = MatrixSymbol('A12', n, n) + A22 = MatrixSymbol('A22', n, n) + B = BlockMatrix([[A11, A12], + [ZeroMatrix(n, n), A22]]) + assert block_collapse(B.I) == BlockMatrix([[A11.I, (-1)*A11.I*A12*A22.I], + [ZeroMatrix(n, n), A22.I]]) + + +def test_P13(): + M = Matrix([[1, x - 2, x - 3], + [x - 1, x**2 - 3*x + 6, x**2 - 3*x - 2], + [x - 2, x**2 - 8, 2*(x**2) - 12*x + 14]]) + L, U, _ = M.LUdecomposition() + assert simplify(L) == Matrix([[1, 0, 0], + [x - 1, 1, 0], + [x - 2, x - 3, 1]]) + assert simplify(U) == Matrix([[1, x - 2, x - 3], + [0, 4, x - 5], + [0, 0, x - 7]]) + + +def test_P14(): + M = Matrix([[1, 2, 3, 1, 3], + [3, 2, 1, 1, 7], + [0, 2, 4, 1, 1], + [1, 1, 1, 1, 4]]) + R, _ = M.rref() + assert R == Matrix([[1, 0, -1, 0, 2], + [0, 1, 2, 0, -1], + [0, 0, 0, 1, 3], + [0, 0, 0, 0, 0]]) + + +def test_P15(): + M = Matrix([[-1, 3, 7, -5], + [4, -2, 1, 3], + [2, 4, 15, -7]]) + assert M.rank() == 2 + + +def test_P16(): + M = Matrix([[2*sqrt(2), 8], + [6*sqrt(6), 24*sqrt(3)]]) + assert M.rank() == 1 + + +def test_P17(): + t = symbols('t', real=True) + M=Matrix([ + [sin(2*t), cos(2*t)], + [2*(1 - (cos(t)**2))*cos(t), (1 - 2*(sin(t)**2))*sin(t)]]) + assert M.rank() == 1 + + +def test_P18(): + M = Matrix([[1, 0, -2, 0], + [-2, 1, 0, 3], + [-1, 2, -6, 6]]) + assert M.nullspace() == [Matrix([[2], + [4], + [1], + [0]]), + Matrix([[0], + [-3], + [0], + [1]])] + + +def test_P19(): + w = symbols('w') + M = Matrix([[1, 1, 1, 1], + [w, x, y, z], + [w**2, x**2, y**2, z**2], + [w**3, x**3, y**3, z**3]]) + assert M.det() == (w**3*x**2*y - w**3*x**2*z - w**3*x*y**2 + w**3*x*z**2 + + w**3*y**2*z - w**3*y*z**2 - w**2*x**3*y + w**2*x**3*z + + w**2*x*y**3 - w**2*x*z**3 - w**2*y**3*z + w**2*y*z**3 + + w*x**3*y**2 - w*x**3*z**2 - w*x**2*y**3 + w*x**2*z**3 + + w*y**3*z**2 - w*y**2*z**3 - x**3*y**2*z + x**3*y*z**2 + + x**2*y**3*z - x**2*y*z**3 - x*y**3*z**2 + x*y**2*z**3 + ) + + +@XFAIL +def test_P20(): + raise NotImplementedError("Matrix minimal polynomial not supported") + + +def test_P21(): + M = Matrix([[5, -3, -7], + [-2, 1, 2], + [2, -3, -4]]) + assert M.charpoly(x).as_expr() == x**3 - 2*x**2 - 5*x + 6 + + +def test_P22(): + d = 100 + M = (2 - x)*eye(d) + assert M.eigenvals() == {-x + 2: d} + + +def test_P23(): + M = Matrix([ + [2, 1, 0, 0, 0], + [1, 2, 1, 0, 0], + [0, 1, 2, 1, 0], + [0, 0, 1, 2, 1], + [0, 0, 0, 1, 2]]) + assert M.eigenvals() == { + S('1'): 1, + S('2'): 1, + S('3'): 1, + S('sqrt(3) + 2'): 1, + S('-sqrt(3) + 2'): 1} + + +def test_P24(): + M = Matrix([[611, 196, -192, 407, -8, -52, -49, 29], + [196, 899, 113, -192, -71, -43, -8, -44], + [-192, 113, 899, 196, 61, 49, 8, 52], + [ 407, -192, 196, 611, 8, 44, 59, -23], + [ -8, -71, 61, 8, 411, -599, 208, 208], + [ -52, -43, 49, 44, -599, 411, 208, 208], + [ -49, -8, 8, 59, 208, 208, 99, -911], + [ 29, -44, 52, -23, 208, 208, -911, 99]]) + assert M.eigenvals() == { + S('0'): 1, + S('10*sqrt(10405)'): 1, + S('100*sqrt(26) + 510'): 1, + S('1000'): 2, + S('-100*sqrt(26) + 510'): 1, + S('-10*sqrt(10405)'): 1, + S('1020'): 1} + + +def test_P25(): + MF = N(Matrix([[ 611, 196, -192, 407, -8, -52, -49, 29], + [ 196, 899, 113, -192, -71, -43, -8, -44], + [-192, 113, 899, 196, 61, 49, 8, 52], + [ 407, -192, 196, 611, 8, 44, 59, -23], + [ -8, -71, 61, 8, 411, -599, 208, 208], + [ -52, -43, 49, 44, -599, 411, 208, 208], + [ -49, -8, 8, 59, 208, 208, 99, -911], + [ 29, -44, 52, -23, 208, 208, -911, 99]])) + + ev_1 = sorted(MF.eigenvals(multiple=True)) + ev_2 = sorted( + [-1020.0490184299969, 0.0, 0.09804864072151699, 1000.0, 1000.0, + 1019.9019513592784, 1020.0, 1020.0490184299969]) + + for x, y in zip(ev_1, ev_2): + assert abs(x - y) < 1e-12 + + +def test_P26(): + a0, a1, a2, a3, a4 = symbols('a0 a1 a2 a3 a4') + M = Matrix([[-a4, -a3, -a2, -a1, -a0, 0, 0, 0, 0], + [ 1, 0, 0, 0, 0, 0, 0, 0, 0], + [ 0, 1, 0, 0, 0, 0, 0, 0, 0], + [ 0, 0, 1, 0, 0, 0, 0, 0, 0], + [ 0, 0, 0, 1, 0, 0, 0, 0, 0], + [ 0, 0, 0, 0, 0, -1, -1, 0, 0], + [ 0, 0, 0, 0, 0, 1, 0, 0, 0], + [ 0, 0, 0, 0, 0, 0, 1, -1, -1], + [ 0, 0, 0, 0, 0, 0, 0, 1, 0]]) + assert M.eigenvals(error_when_incomplete=False) == { + S('-1/2 - sqrt(3)*I/2'): 2, + S('-1/2 + sqrt(3)*I/2'): 2} + + +def test_P27(): + a = symbols('a') + M = Matrix([[a, 0, 0, 0, 0], + [0, 0, 0, 0, 1], + [0, 0, a, 0, 0], + [0, 0, 0, a, 0], + [0, -2, 0, 0, 2]]) + + assert M.eigenvects() == [ + (a, 3, [ + Matrix([1, 0, 0, 0, 0]), + Matrix([0, 0, 1, 0, 0]), + Matrix([0, 0, 0, 1, 0]) + ]), + (1 - I, 1, [ + Matrix([0, (1 + I)/2, 0, 0, 1]) + ]), + (1 + I, 1, [ + Matrix([0, (1 - I)/2, 0, 0, 1]) + ]), + ] + + +@XFAIL +def test_P28(): + raise NotImplementedError("Generalized eigenvectors not supported \ +https://github.com/sympy/sympy/issues/5293") + + +@XFAIL +def test_P29(): + raise NotImplementedError("Generalized eigenvectors not supported \ +https://github.com/sympy/sympy/issues/5293") + + +def test_P30(): + M = Matrix([[1, 0, 0, 1, -1], + [0, 1, -2, 3, -3], + [0, 0, -1, 2, -2], + [1, -1, 1, 0, 1], + [1, -1, 1, -1, 2]]) + _, J = M.jordan_form() + assert J == Matrix([[-1, 0, 0, 0, 0], + [0, 1, 1, 0, 0], + [0, 0, 1, 0, 0], + [0, 0, 0, 1, 1], + [0, 0, 0, 0, 1]]) + + +@XFAIL +def test_P31(): + raise NotImplementedError("Smith normal form not implemented") + + +def test_P32(): + M = Matrix([[1, -2], + [2, 1]]) + assert exp(M).rewrite(cos).simplify() == Matrix([[E*cos(2), -E*sin(2)], + [E*sin(2), E*cos(2)]]) + + +def test_P33(): + w, t = symbols('w t') + M = Matrix([[0, 1, 0, 0], + [0, 0, 0, 2*w], + [0, 0, 0, 1], + [0, -2*w, 3*w**2, 0]]) + assert exp(M*t).rewrite(cos).expand() == Matrix([ + [1, -3*t + 4*sin(t*w)/w, 6*t*w - 6*sin(t*w), -2*cos(t*w)/w + 2/w], + [0, 4*cos(t*w) - 3, -6*w*cos(t*w) + 6*w, 2*sin(t*w)], + [0, 2*cos(t*w)/w - 2/w, -3*cos(t*w) + 4, sin(t*w)/w], + [0, -2*sin(t*w), 3*w*sin(t*w), cos(t*w)]]) + + +@XFAIL +def test_P34(): + a, b, c = symbols('a b c', real=True) + M = Matrix([[a, 1, 0, 0, 0, 0], + [0, a, 0, 0, 0, 0], + [0, 0, b, 0, 0, 0], + [0, 0, 0, c, 1, 0], + [0, 0, 0, 0, c, 1], + [0, 0, 0, 0, 0, c]]) + # raises exception, sin(M) not supported. exp(M*I) also not supported + # https://github.com/sympy/sympy/issues/6218 + assert sin(M) == Matrix([[sin(a), cos(a), 0, 0, 0, 0], + [0, sin(a), 0, 0, 0, 0], + [0, 0, sin(b), 0, 0, 0], + [0, 0, 0, sin(c), cos(c), -sin(c)/2], + [0, 0, 0, 0, sin(c), cos(c)], + [0, 0, 0, 0, 0, sin(c)]]) + + +@XFAIL +def test_P35(): + M = pi/2*Matrix([[2, 1, 1], + [2, 3, 2], + [1, 1, 2]]) + # raises exception, sin(M) not supported. exp(M*I) also not supported + # https://github.com/sympy/sympy/issues/6218 + assert sin(M) == eye(3) + + +@XFAIL +def test_P36(): + M = Matrix([[10, 7], + [7, 17]]) + assert sqrt(M) == Matrix([[3, 1], + [1, 4]]) + + +def test_P37(): + M = Matrix([[1, 1, 0], + [0, 1, 0], + [0, 0, 1]]) + assert M**S.Half == Matrix([[1, R(1, 2), 0], + [0, 1, 0], + [0, 0, 1]]) + + +@XFAIL +def test_P38(): + M=Matrix([[0, 1, 0], + [0, 0, 0], + [0, 0, 0]]) + + with raises(AssertionError): + # raises ValueError: Matrix det == 0; not invertible + M**S.Half + # if it doesn't raise then this assertion will be + # raised and the test will be flagged as not XFAILing + assert None + +@XFAIL +def test_P39(): + """ + M=Matrix([ + [1, 1], + [2, 2], + [3, 3]]) + M.SVD() + """ + raise NotImplementedError("Singular value decomposition not implemented") + + +def test_P40(): + r, t = symbols('r t', real=True) + M = Matrix([r*cos(t), r*sin(t)]) + assert M.jacobian(Matrix([r, t])) == Matrix([[cos(t), -r*sin(t)], + [sin(t), r*cos(t)]]) + + +def test_P41(): + r, t = symbols('r t', real=True) + assert hessian(r**2*sin(t),(r,t)) == Matrix([[ 2*sin(t), 2*r*cos(t)], + [2*r*cos(t), -r**2*sin(t)]]) + + +def test_P42(): + assert wronskian([cos(x), sin(x)], x).simplify() == 1 + + +def test_P43(): + def __my_jacobian(M, Y): + return Matrix([M.diff(v).T for v in Y]).T + r, t = symbols('r t', real=True) + M = Matrix([r*cos(t), r*sin(t)]) + assert __my_jacobian(M,[r,t]) == Matrix([[cos(t), -r*sin(t)], + [sin(t), r*cos(t)]]) + + +def test_P44(): + def __my_hessian(f, Y): + V = Matrix([diff(f, v) for v in Y]) + return Matrix([V.T.diff(v) for v in Y]) + r, t = symbols('r t', real=True) + assert __my_hessian(r**2*sin(t), (r, t)) == Matrix([ + [ 2*sin(t), 2*r*cos(t)], + [2*r*cos(t), -r**2*sin(t)]]) + + +def test_P45(): + def __my_wronskian(Y, v): + M = Matrix([Matrix(Y).T.diff(x, n) for n in range(0, len(Y))]) + return M.det() + assert __my_wronskian([cos(x), sin(x)], x).simplify() == 1 + +# Q1-Q6 Tensor tests missing + + +@XFAIL +def test_R1(): + i, j, n = symbols('i j n', integer=True, positive=True) + xn = MatrixSymbol('xn', n, 1) + Sm = Sum((xn[i, 0] - Sum(xn[j, 0], (j, 0, n - 1))/n)**2, (i, 0, n - 1)) + # sum does not calculate + # Unknown result + Sm.doit() + raise NotImplementedError('Unknown result') + +@XFAIL +def test_R2(): + m, b = symbols('m b') + i, n = symbols('i n', integer=True, positive=True) + xn = MatrixSymbol('xn', n, 1) + yn = MatrixSymbol('yn', n, 1) + f = Sum((yn[i, 0] - m*xn[i, 0] - b)**2, (i, 0, n - 1)) + f1 = diff(f, m) + f2 = diff(f, b) + # raises TypeError: solveset() takes at most 2 arguments (3 given) + solveset((f1, f2), (m, b), domain=S.Reals) + + +@XFAIL +def test_R3(): + n, k = symbols('n k', integer=True, positive=True) + sk = ((-1)**k) * (binomial(2*n, k))**2 + Sm = Sum(sk, (k, 1, oo)) + T = Sm.doit() + T2 = T.combsimp() + # returns -((-1)**n*factorial(2*n) + # - (factorial(n))**2)*exp_polar(-I*pi)/(factorial(n))**2 + assert T2 == (-1)**n*binomial(2*n, n) + + +@XFAIL +def test_R4(): +# Macsyma indefinite sum test case: +#(c15) /* Check whether the full Gosper algorithm is implemented +# => 1/2^(n + 1) binomial(n, k - 1) */ +#closedform(indefsum(binomial(n, k)/2^n - binomial(n + 1, k)/2^(n + 1), k)); +#Time= 2690 msecs +# (- n + k - 1) binomial(n + 1, k) +#(d15) - -------------------------------- +# n +# 2 2 (n + 1) +# +#(c16) factcomb(makefact(%)); +#Time= 220 msecs +# n! +#(d16) ---------------- +# n +# 2 k! 2 (n - k)! +# Might be possible after fixing https://github.com/sympy/sympy/pull/1879 + raise NotImplementedError("Indefinite sum not supported") + + +@XFAIL +def test_R5(): + a, b, c, n, k = symbols('a b c n k', integer=True, positive=True) + sk = ((-1)**k)*(binomial(a + b, a + k) + *binomial(b + c, b + k)*binomial(c + a, c + k)) + Sm = Sum(sk, (k, 1, oo)) + T = Sm.doit() # hypergeometric series not calculated + assert T == factorial(a+b+c)/(factorial(a)*factorial(b)*factorial(c)) + + +def test_R6(): + n, k = symbols('n k', integer=True, positive=True) + gn = MatrixSymbol('gn', n + 2, 1) + Sm = Sum(gn[k, 0] - gn[k - 1, 0], (k, 1, n + 1)) + assert Sm.doit() == -gn[0, 0] + gn[n + 1, 0] + + +def test_R7(): + n, k = symbols('n k', integer=True, positive=True) + T = Sum(k**3,(k,1,n)).doit() + assert T.factor() == n**2*(n + 1)**2/4 + +@XFAIL +def test_R8(): + n, k = symbols('n k', integer=True, positive=True) + Sm = Sum(k**2*binomial(n, k), (k, 1, n)) + T = Sm.doit() #returns Piecewise function + assert T.combsimp() == n*(n + 1)*2**(n - 2) + + +def test_R9(): + n, k = symbols('n k', integer=True, positive=True) + Sm = Sum(binomial(n, k - 1)/k, (k, 1, n + 1)) + assert Sm.doit().simplify() == (2**(n + 1) - 1)/(n + 1) + + +@XFAIL +def test_R10(): + n, m, r, k = symbols('n m r k', integer=True, positive=True) + Sm = Sum(binomial(n, k)*binomial(m, r - k), (k, 0, r)) + T = Sm.doit() + T2 = T.combsimp().rewrite(factorial) + assert T2 == factorial(m + n)/(factorial(r)*factorial(m + n - r)) + assert T2 == binomial(m + n, r).rewrite(factorial) + # rewrite(binomial) is not working. + # https://github.com/sympy/sympy/issues/7135 + T3 = T2.rewrite(binomial) + assert T3 == binomial(m + n, r) + + +@XFAIL +def test_R11(): + n, k = symbols('n k', integer=True, positive=True) + sk = binomial(n, k)*fibonacci(k) + Sm = Sum(sk, (k, 0, n)) + T = Sm.doit() + # Fibonacci simplification not implemented + # https://github.com/sympy/sympy/issues/7134 + assert T == fibonacci(2*n) + + +@XFAIL +def test_R12(): + n, k = symbols('n k', integer=True, positive=True) + Sm = Sum(fibonacci(k)**2, (k, 0, n)) + T = Sm.doit() + assert T == fibonacci(n)*fibonacci(n + 1) + + +@XFAIL +def test_R13(): + n, k = symbols('n k', integer=True, positive=True) + Sm = Sum(sin(k*x), (k, 1, n)) + T = Sm.doit() # Sum is not calculated + assert T.simplify() == cot(x/2)/2 - cos(x*(2*n + 1)/2)/(2*sin(x/2)) + + +@XFAIL +def test_R14(): + n, k = symbols('n k', integer=True, positive=True) + Sm = Sum(sin((2*k - 1)*x), (k, 1, n)) + T = Sm.doit() # Sum is not calculated + assert T.simplify() == sin(n*x)**2/sin(x) + + +@XFAIL +def test_R15(): + n, k = symbols('n k', integer=True, positive=True) + Sm = Sum(binomial(n - k, k), (k, 0, floor(n/2))) + T = Sm.doit() # Sum is not calculated + assert T.simplify() == fibonacci(n + 1) + + +def test_R16(): + k = symbols('k', integer=True, positive=True) + Sm = Sum(1/k**2 + 1/k**3, (k, 1, oo)) + assert Sm.doit() == zeta(3) + pi**2/6 + + +def test_R17(): + k = symbols('k', integer=True, positive=True) + assert abs(float(Sum(1/k**2 + 1/k**3, (k, 1, oo))) + - 2.8469909700078206) < 1e-15 + + +def test_R18(): + k = symbols('k', integer=True, positive=True) + Sm = Sum(1/(2**k*k**2), (k, 1, oo)) + T = Sm.doit() + assert T.simplify() == -log(2)**2/2 + pi**2/12 + + +@slow +@XFAIL +def test_R19(): + k = symbols('k', integer=True, positive=True) + Sm = Sum(1/((3*k + 1)*(3*k + 2)*(3*k + 3)), (k, 0, oo)) + T = Sm.doit() + # assert fails, T not simplified + assert T.simplify() == -log(3)/4 + sqrt(3)*pi/12 + + +@XFAIL +def test_R20(): + n, k = symbols('n k', integer=True, positive=True) + Sm = Sum(binomial(n, 4*k), (k, 0, oo)) + T = Sm.doit() + # assert fails, T not simplified + assert T.simplify() == 2**(n/2)*cos(pi*n/4)/2 + 2**(n - 1)/2 + + +@XFAIL +def test_R21(): + k = symbols('k', integer=True, positive=True) + Sm = Sum(1/(sqrt(k*(k + 1)) * (sqrt(k) + sqrt(k + 1))), (k, 1, oo)) + T = Sm.doit() # Sum not calculated + assert T.simplify() == 1 + + +# test_R22 answer not available in Wester samples +# Sum(Sum(binomial(n, k)*binomial(n - k, n - 2*k)*x**n*y**(n - 2*k), +# (k, 0, floor(n/2))), (n, 0, oo)) with abs(x*y)<1? + + +@XFAIL +def test_R23(): + n, k = symbols('n k', integer=True, positive=True) + Sm = Sum(Sum((factorial(n)/(factorial(k)**2*factorial(n - 2*k)))* + (x/y)**k*(x*y)**(n - k), (n, 2*k, oo)), (k, 0, oo)) + # Missing how to express constraint abs(x*y)<1? + T = Sm.doit() # Sum not calculated + assert T == -1/sqrt(x**2*y**2 - 4*x**2 - 2*x*y + 1) + + +def test_R24(): + m, k = symbols('m k', integer=True, positive=True) + Sm = Sum(Product(k/(2*k - 1), (k, 1, m)), (m, 2, oo)) + assert Sm.doit() == pi/2 + + +def test_S1(): + k = symbols('k', integer=True, positive=True) + Pr = Product(gamma(k/3), (k, 1, 8)) + assert Pr.doit().simplify() == 640*sqrt(3)*pi**3/6561 + + +def test_S2(): + n, k = symbols('n k', integer=True, positive=True) + assert Product(k, (k, 1, n)).doit() == factorial(n) + + +def test_S3(): + n, k = symbols('n k', integer=True, positive=True) + assert Product(x**k, (k, 1, n)).doit().simplify() == x**(n*(n + 1)/2) + + +def test_S4(): + n, k = symbols('n k', integer=True, positive=True) + assert Product(1 + 1/k, (k, 1, n -1)).doit().simplify() == n + + +def test_S5(): + n, k = symbols('n k', integer=True, positive=True) + assert (Product((2*k - 1)/(2*k), (k, 1, n)).doit().gammasimp() == + gamma(n + S.Half)/(sqrt(pi)*gamma(n + 1))) + + +@XFAIL +def test_S6(): + n, k = symbols('n k', integer=True, positive=True) + # Product does not evaluate + assert (Product(x**2 -2*x*cos(k*pi/n) + 1, (k, 1, n - 1)).doit().simplify() + == (x**(2*n) - 1)/(x**2 - 1)) + + +@XFAIL +def test_S7(): + k = symbols('k', integer=True, positive=True) + Pr = Product((k**3 - 1)/(k**3 + 1), (k, 2, oo)) + T = Pr.doit() # Product does not evaluate + assert T.simplify() == R(2, 3) + + +@XFAIL +def test_S8(): + k = symbols('k', integer=True, positive=True) + Pr = Product(1 - 1/(2*k)**2, (k, 1, oo)) + T = Pr.doit() + # Product does not evaluate + assert T.simplify() == 2/pi + + +@XFAIL +def test_S9(): + k = symbols('k', integer=True, positive=True) + Pr = Product(1 + (-1)**(k + 1)/(2*k - 1), (k, 1, oo)) + T = Pr.doit() + # Product produces 0 + # https://github.com/sympy/sympy/issues/7133 + assert T.simplify() == sqrt(2) + + +@XFAIL +def test_S10(): + k = symbols('k', integer=True, positive=True) + Pr = Product((k*(k + 1) + 1 + I)/(k*(k + 1) + 1 - I), (k, 0, oo)) + T = Pr.doit() + # Product does not evaluate + assert T.simplify() == -1 + + +def test_T1(): + assert limit((1 + 1/n)**n, n, oo) == E + assert limit((1 - cos(x))/x**2, x, 0) == S.Half + + +def test_T2(): + assert limit((3**x + 5**x)**(1/x), x, oo) == 5 + + +def test_T3(): + assert limit(log(x)/(log(x) + sin(x)), x, oo) == 1 + + +def test_T4(): + assert limit((exp(x*exp(-x)/(exp(-x) + exp(-2*x**2/(x + 1)))) + - exp(x))/x, x, oo) == -exp(2) + + +def test_T5(): + assert limit(x*log(x)*log(x*exp(x) - x**2)**2/log(log(x**2 + + 2*exp(exp(3*x**3*log(x))))), x, oo) == R(1, 3) + + +def test_T6(): + assert limit(1/n * factorial(n)**(1/n), n, oo) == exp(-1) + + +def test_T7(): + limit(1/n * gamma(n + 1)**(1/n), n, oo) + + +def test_T8(): + a, z = symbols('a z', positive=True) + assert limit(gamma(z + a)/gamma(z)*exp(-a*log(z)), z, oo) == 1 + + +@XFAIL +def test_T9(): + z, k = symbols('z k', positive=True) + # raises NotImplementedError: + # Don't know how to calculate the mrv of '(1, k)' + assert limit(hyper((1, k), (1,), z/k), k, oo) == exp(z) + + +@XFAIL +def test_T10(): + # No longer raises PoleError, but should return euler-mascheroni constant + assert limit(zeta(x) - 1/(x - 1), x, 1) == integrate(-1/x + 1/floor(x), (x, 1, oo)) + +@XFAIL +def test_T11(): + n, k = symbols('n k', integer=True, positive=True) + # evaluates to 0 + assert limit(n**x/(x*product((1 + x/k), (k, 1, n))), n, oo) == gamma(x) + + +def test_T12(): + x, t = symbols('x t', real=True) + # Does not evaluate the limit but returns an expression with erf + assert limit(x * integrate(exp(-t**2), (t, 0, x))/(1 - exp(-x**2)), + x, 0) == 1 + + +def test_T13(): + x = symbols('x', real=True) + assert [limit(x/abs(x), x, 0, dir='-'), + limit(x/abs(x), x, 0, dir='+')] == [-1, 1] + + +def test_T14(): + x = symbols('x', real=True) + assert limit(atan(-log(x)), x, 0, dir='+') == pi/2 + + +def test_U1(): + x = symbols('x', real=True) + assert diff(abs(x), x) == sign(x) + + +def test_U2(): + f = Lambda(x, Piecewise((-x, x < 0), (x, x >= 0))) + assert diff(f(x), x) == Piecewise((-1, x < 0), (1, x >= 0)) + + +def test_U3(): + f = Lambda(x, Piecewise((x**2 - 1, x == 1), (x**3, x != 1))) + f1 = Lambda(x, diff(f(x), x)) + assert f1(x) == 3*x**2 + assert f1(1) == 3 + + +@XFAIL +def test_U4(): + n = symbols('n', integer=True, positive=True) + x = symbols('x', real=True) + d = diff(x**n, x, n) + assert d.rewrite(factorial) == factorial(n) + + +def test_U5(): + # issue 6681 + t = symbols('t') + ans = ( + Derivative(f(g(t)), g(t))*Derivative(g(t), (t, 2)) + + Derivative(f(g(t)), (g(t), 2))*Derivative(g(t), t)**2) + assert f(g(t)).diff(t, 2) == ans + assert ans.doit() == ans + + +def test_U6(): + h = Function('h') + T = integrate(f(y), (y, h(x), g(x))) + assert T.diff(x) == ( + f(g(x))*Derivative(g(x), x) - f(h(x))*Derivative(h(x), x)) + + +@XFAIL +def test_U7(): + p, t = symbols('p t', real=True) + # Exact differential => d(V(P, T)) => dV/dP DP + dV/dT DT + # raises ValueError: Since there is more than one variable in the + # expression, the variable(s) of differentiation must be supplied to + # differentiate f(p,t) + diff(f(p, t)) + + +def test_U8(): + x, y = symbols('x y', real=True) + eq = cos(x*y) + x + # If SymPy had implicit_diff() function this hack could be avoided + # TODO: Replace solve with solveset, current test fails for solveset + assert idiff(y - eq, y, x) == (-y*sin(x*y) + 1)/(x*sin(x*y) + 1) + + +def test_U9(): + # Wester sample case for Maple: + # O29 := diff(f(x, y), x) + diff(f(x, y), y); + # /d \ /d \ + # |-- f(x, y)| + |-- f(x, y)| + # \dx / \dy / + # + # O30 := factor(subs(f(x, y) = g(x^2 + y^2), %)); + # 2 2 + # 2 D(g)(x + y ) (x + y) + x, y = symbols('x y', real=True) + su = diff(f(x, y), x) + diff(f(x, y), y) + s2 = su.subs(f(x, y), g(x**2 + y**2)) + s3 = s2.doit().factor() + # Subs not performed, s3 = 2*(x + y)*Subs(Derivative( + # g(_xi_1), _xi_1), _xi_1, x**2 + y**2) + # Derivative(g(x*2 + y**2), x**2 + y**2) is not valid in SymPy, + # and probably will remain that way. You can take derivatives with respect + # to other expressions only if they are atomic, like a symbol or a + # function. + # D operator should be added to SymPy + # See https://github.com/sympy/sympy/issues/4719. + assert s3 == (x + y)*Subs(Derivative(g(x), x), x, x**2 + y**2)*2 + + +def test_U10(): + # see issue 2519: + assert residue((z**3 + 5)/((z**4 - 1)*(z + 1)), z, -1) == R(-9, 4) + +@XFAIL +def test_U11(): + # assert (2*dx + dz) ^ (3*dx + dy + dz) ^ (dx + dy + 4*dz) == 8*dx ^ dy ^dz + raise NotImplementedError + + +@XFAIL +def test_U12(): + # Wester sample case: + # (c41) /* d(3 x^5 dy /\ dz + 5 x y^2 dz /\ dx + 8 z dx /\ dy) + # => (15 x^4 + 10 x y + 8) dx /\ dy /\ dz */ + # factor(ext_diff(3*x^5 * dy ~ dz + 5*x*y^2 * dz ~ dx + 8*z * dx ~ dy)); + # 4 + # (d41) (10 x y + 15 x + 8) dx dy dz + raise NotImplementedError( + "External diff of differential form not supported") + + +def test_U13(): + assert minimum(x**4 - x + 1, x) == -3*2**R(1,3)/8 + 1 + + +@XFAIL +def test_U14(): + #f = 1/(x**2 + y**2 + 1) + #assert [minimize(f), maximize(f)] == [0,1] + raise NotImplementedError("minimize(), maximize() not supported") + + +@XFAIL +def test_U15(): + raise NotImplementedError("minimize() not supported and also solve does \ +not support multivariate inequalities") + + +@XFAIL +def test_U16(): + raise NotImplementedError("minimize() not supported in SymPy and also \ +solve does not support multivariate inequalities") + + +@XFAIL +def test_U17(): + raise NotImplementedError("Linear programming, symbolic simplex not \ +supported in SymPy") + + +def test_V1(): + x = symbols('x', real=True) + assert integrate(abs(x), x) == Piecewise((-x**2/2, x <= 0), (x**2/2, True)) + + +def test_V2(): + assert integrate(Piecewise((-x, x < 0), (x, x >= 0)), x + ) == Piecewise((-x**2/2, x < 0), (x**2/2, True)) + + +def test_V3(): + assert integrate(1/(x**3 + 2),x).diff().simplify() == 1/(x**3 + 2) + + +def test_V4(): + assert integrate(2**x/sqrt(1 + 4**x), x) == asinh(2**x)/log(2) + + +@XFAIL +def test_V5(): + # Returns (-45*x**2 + 80*x - 41)/(5*sqrt(2*x - 1)*(4*x**2 - 4*x + 1)) + assert (integrate((3*x - 5)**2/(2*x - 1)**R(7, 2), x).simplify() == + (-41 + 80*x - 45*x**2)/(5*(2*x - 1)**R(5, 2))) + + +@XFAIL +def test_V6(): + # returns RootSum(40*_z**2 - 1, Lambda(_i, _i*log(-4*_i + exp(-m*x))))/m + assert (integrate(1/(2*exp(m*x) - 5*exp(-m*x)), x) == sqrt(10)*( + log(2*exp(m*x) - sqrt(10)) - log(2*exp(m*x) + sqrt(10)))/(20*m)) + + +def test_V7(): + r1 = integrate(sinh(x)**4/cosh(x)**2) + assert r1.simplify() == x*R(-3, 2) + sinh(x)**3/(2*cosh(x)) + 3*tanh(x)/2 + + +@XFAIL +def test_V8_V9(): +#Macsyma test case: +#(c27) /* This example involves several symbolic parameters +# => 1/sqrt(b^2 - a^2) log([sqrt(b^2 - a^2) tan(x/2) + a + b]/ +# [sqrt(b^2 - a^2) tan(x/2) - a - b]) (a^2 < b^2) +# [Gradshteyn and Ryzhik 2.553(3)] */ +#assume(b^2 > a^2)$ +#(c28) integrate(1/(a + b*cos(x)), x); +#(c29) trigsimp(ratsimp(diff(%, x))); +# 1 +#(d29) ------------ +# b cos(x) + a + raise NotImplementedError( + "Integrate with assumption not supported") + + +def test_V10(): + assert integrate(1/(3 + 3*cos(x) + 4*sin(x)), x) == log(4*tan(x/2) + 3)/4 + + +def test_V11(): + r1 = integrate(1/(4 + 3*cos(x) + 4*sin(x)), x) + r2 = factor(r1) + assert (logcombine(r2, force=True) == + log(((tan(x/2) + 1)/(tan(x/2) + 7))**R(1, 3))) + + +def test_V12(): + r1 = integrate(1/(5 + 3*cos(x) + 4*sin(x)), x) + assert r1 == -1/(tan(x/2) + 2) + + +@XFAIL +def test_V13(): + r1 = integrate(1/(6 + 3*cos(x) + 4*sin(x)), x) + # expression not simplified, returns: -sqrt(11)*I*log(tan(x/2) + 4/3 + # - sqrt(11)*I/3)/11 + sqrt(11)*I*log(tan(x/2) + 4/3 + sqrt(11)*I/3)/11 + assert r1.simplify() == 2*sqrt(11)*atan(sqrt(11)*(3*tan(x/2) + 4)/11)/11 + + +@slow +@XFAIL +def test_V14(): + r1 = integrate(log(abs(x**2 - y**2)), x) + # Piecewise result does not simplify to the desired result. + assert (r1.simplify() == x*log(abs(x**2 - y**2)) + + y*log(x + y) - y*log(x - y) - 2*x) + + +def test_V15(): + r1 = integrate(x*acot(x/y), x) + assert simplify(r1 - (x*y + (x**2 + y**2)*acot(x/y))/2) == 0 + + +@XFAIL +def test_V16(): + # Integral not calculated + assert integrate(cos(5*x)*Ci(2*x), x) == Ci(2*x)*sin(5*x)/5 - (Si(3*x) + Si(7*x))/10 + +@XFAIL +def test_V17(): + r1 = integrate((diff(f(x), x)*g(x) + - f(x)*diff(g(x), x))/(f(x)**2 - g(x)**2), x) + # integral not calculated + assert simplify(r1 - (f(x) - g(x))/(f(x) + g(x))/2) == 0 + + +@XFAIL +def test_W1(): + # The function has a pole at y. + # The integral has a Cauchy principal value of zero but SymPy returns -I*pi + # https://github.com/sympy/sympy/issues/7159 + assert integrate(1/(x - y), (x, y - 1, y + 1)) == 0 + + +@XFAIL +def test_W2(): + # The function has a pole at y. + # The integral is divergent but SymPy returns -2 + # https://github.com/sympy/sympy/issues/7160 + # Test case in Macsyma: + # (c6) errcatch(integrate(1/(x - a)^2, x, a - 1, a + 1)); + # Integral is divergent + assert integrate(1/(x - y)**2, (x, y - 1, y + 1)) is zoo + + +@XFAIL +@slow +def test_W3(): + # integral is not calculated + # https://github.com/sympy/sympy/issues/7161 + assert integrate(sqrt(x + 1/x - 2), (x, 0, 1)) == R(4, 3) + + +@XFAIL +@slow +def test_W4(): + # integral is not calculated + assert integrate(sqrt(x + 1/x - 2), (x, 1, 2)) == -2*sqrt(2)/3 + R(4, 3) + + +@XFAIL +@slow +def test_W5(): + # integral is not calculated + assert integrate(sqrt(x + 1/x - 2), (x, 0, 2)) == -2*sqrt(2)/3 + R(8, 3) + + +@XFAIL +@slow +def test_W6(): + # integral is not calculated + assert integrate(sqrt(2 - 2*cos(2*x))/2, (x, pi*R(-3, 4), -pi/4)) == sqrt(2) + + +def test_W7(): + a = symbols('a', positive=True) + r1 = integrate(cos(x)/(x**2 + a**2), (x, -oo, oo)) + assert r1.simplify() == pi*exp(-a)/a + + +@XFAIL +def test_W8(): + # Test case in Mathematica: + # In[19]:= Integrate[t^(a - 1)/(1 + t), {t, 0, Infinity}, + # Assumptions -> 0 < a < 1] + # Out[19]= Pi Csc[a Pi] + raise NotImplementedError( + "Integrate with assumption 0 < a < 1 not supported") + + +@XFAIL +@slow +def test_W9(): + # Integrand with a residue at infinity => -2 pi [sin(pi/5) + sin(2pi/5)] + # (principal value) [Levinson and Redheffer, p. 234] *) + r1 = integrate(5*x**3/(1 + x + x**2 + x**3 + x**4), (x, -oo, oo)) + r2 = r1.doit() + assert r2 == -2*pi*(sqrt(-sqrt(5)/8 + 5/8) + sqrt(sqrt(5)/8 + 5/8)) + + +@XFAIL +def test_W10(): + # integrate(1/[1 + x + x^2 + ... + x^(2 n)], x = -infinity..infinity) = + # 2 pi/(2 n + 1) [1 + cos(pi/[2 n + 1])] csc(2 pi/[2 n + 1]) + # [Levinson and Redheffer, p. 255] => 2 pi/5 [1 + cos(pi/5)] csc(2 pi/5) */ + r1 = integrate(x/(1 + x + x**2 + x**4), (x, -oo, oo)) + r2 = r1.doit() + assert r2 == 2*pi*(sqrt(5)/4 + 5/4)*csc(pi*R(2, 5))/5 + + +@XFAIL +def test_W11(): + # integral not calculated + assert (integrate(sqrt(1 - x**2)/(1 + x**2), (x, -1, 1)) == + pi*(-1 + sqrt(2))) + + +def test_W12(): + p = symbols('p', positive=True) + q = symbols('q', real=True) + r1 = integrate(x*exp(-p*x**2 + 2*q*x), (x, -oo, oo)) + assert r1.simplify() == sqrt(pi)*q*exp(q**2/p)/p**R(3, 2) + + +@XFAIL +def test_W13(): + # Integral not calculated. Expected result is 2*(Euler_mascheroni_constant) + r1 = integrate(1/log(x) + 1/(1 - x) - log(log(1/x)), (x, 0, 1)) + assert r1 == 2*EulerGamma + + +def test_W14(): + assert integrate(sin(x)/x*exp(2*I*x), (x, -oo, oo)) == 0 + + +@XFAIL +def test_W15(): + # integral not calculated + assert integrate(log(gamma(x))*cos(6*pi*x), (x, 0, 1)) == R(1, 12) + + +def test_W16(): + assert integrate((1 + x)**3*legendre_poly(1, x)*legendre_poly(2, x), + (x, -1, 1)) == R(36, 35) + + +def test_W17(): + a, b = symbols('a b', positive=True) + assert integrate(exp(-a*x)*besselj(0, b*x), + (x, 0, oo)) == 1/(b*sqrt(a**2/b**2 + 1)) + + +def test_W18(): + assert integrate((besselj(1, x)/x)**2, (x, 0, oo)) == 4/(3*pi) + + +@XFAIL +def test_W19(): + # Integral not calculated + # Expected result is (cos 7 - 1)/7 [Gradshteyn and Ryzhik 6.782(3)] + assert integrate(Ci(x)*besselj(0, 2*sqrt(7*x)), (x, 0, oo)) == (cos(7) - 1)/7 + + +@XFAIL +def test_W20(): + # integral not calculated + assert (integrate(x**2*polylog(3, 1/(x + 1)), (x, 0, 1)) == + -pi**2/36 - R(17, 108) + zeta(3)/4 + + (-pi**2/2 - 4*log(2) + log(2)**2 + 35/3)*log(2)/9) + + +def test_W21(): + assert abs(N(integrate(x**2*polylog(3, 1/(x + 1)), (x, 0, 1))) + - 0.210882859565594) < 1e-15 + + +def test_W22(): + t, u = symbols('t u', real=True) + s = Lambda(x, Piecewise((1, And(x >= 1, x <= 2)), (0, True))) + assert integrate(s(t)*cos(t), (t, 0, u)) == Piecewise( + (0, u < 0), + (-sin(Min(1, u)) + sin(Min(2, u)), True)) + + +@slow +def test_W23(): + a, b = symbols('a b', positive=True) + r1 = integrate(integrate(x/(x**2 + y**2), (x, a, b)), (y, -oo, oo)) + assert r1.collect(pi).cancel() == -pi*a + pi*b + + +def test_W23b(): + # like W23 but limits are reversed + a, b = symbols('a b', positive=True) + r2 = integrate(integrate(x/(x**2 + y**2), (y, -oo, oo)), (x, a, b)) + assert r2.collect(pi) == pi*(-a + b) + + +@XFAIL +@tooslow +def test_W24(): + # Not that slow, but does not fully evaluate so simplify is slow. + # Maybe also require doit() + x, y = symbols('x y', real=True) + r1 = integrate(integrate(sqrt(x**2 + y**2), (x, 0, 1)), (y, 0, 1)) + assert (r1 - (sqrt(2) + asinh(1))/3).simplify() == 0 + + +@XFAIL +@tooslow +def test_W25(): + a, x, y = symbols('a x y', real=True) + i1 = integrate( + sin(a)*sin(y)/sqrt(1 - sin(a)**2*sin(x)**2*sin(y)**2), + (x, 0, pi/2)) + i2 = integrate(i1, (y, 0, pi/2)) + assert (i2 - pi*a/2).simplify() == 0 + + +def test_W26(): + x, y = symbols('x y', real=True) + assert integrate(integrate(abs(y - x**2), (y, 0, 2)), + (x, -1, 1)) == R(46, 15) + + +def test_W27(): + a, b, c = symbols('a b c') + assert integrate(integrate(integrate(1, (z, 0, c*(1 - x/a - y/b))), + (y, 0, b*(1 - x/a))), + (x, 0, a)) == a*b*c/6 + + +def test_X1(): + v, c = symbols('v c', real=True) + assert (series(1/sqrt(1 - (v/c)**2), v, x0=0, n=8) == + 5*v**6/(16*c**6) + 3*v**4/(8*c**4) + v**2/(2*c**2) + 1 + O(v**8)) + + +def test_X2(): + v, c = symbols('v c', real=True) + s1 = series(1/sqrt(1 - (v/c)**2), v, x0=0, n=8) + assert (1/s1**2).series(v, x0=0, n=8) == -v**2/c**2 + 1 + O(v**8) + + +def test_X3(): + s1 = (sin(x).series()/cos(x).series()).series() + s2 = tan(x).series() + assert s2 == x + x**3/3 + 2*x**5/15 + O(x**6) + assert s1 == s2 + + +def test_X4(): + s1 = log(sin(x)/x).series() + assert s1 == -x**2/6 - x**4/180 + O(x**6) + assert log(series(sin(x)/x)).series() == s1 + + +@XFAIL +def test_X5(): + # test case in Mathematica syntax: + # In[21]:= (* => [a f'(a d) + g(b d) + integrate(h(c y), y = 0..d)] + # + [a^2 f''(a d) + b g'(b d) + h(c d)] (x - d) *) + # In[22]:= D[f[a*x], x] + g[b*x] + Integrate[h[c*y], {y, 0, x}] + # Out[22]= g[b x] + Integrate[h[c y], {y, 0, x}] + a f'[a x] + # In[23]:= Series[%, {x, d, 1}] + # Out[23]= (g[b d] + Integrate[h[c y], {y, 0, d}] + a f'[a d]) + + # 2 2 + # (h[c d] + b g'[b d] + a f''[a d]) (-d + x) + O[-d + x] + h = Function('h') + a, b, c, d = symbols('a b c d', real=True) + # series() raises NotImplementedError: + # The _eval_nseries method should be added to to give terms up to O(x**n) at x=0 + series(diff(f(a*x), x) + g(b*x) + integrate(h(c*y), (y, 0, x)), + x, x0=d, n=2) + # assert missing, until exception is removed + + +def test_X6(): + # Taylor series of nonscalar objects (noncommutative multiplication) + # expected result => (B A - A B) t^2/2 + O(t^3) [Stanly Steinberg] + a, b = symbols('a b', commutative=False, scalar=False) + assert (series(exp((a + b)*x) - exp(a*x) * exp(b*x), x, x0=0, n=3) == + x**2*(-a*b/2 + b*a/2) + O(x**3)) + + +def test_X7(): + # => sum( Bernoulli[k]/k! x^(k - 2), k = 1..infinity ) + # = 1/x^2 - 1/(2 x) + 1/12 - x^2/720 + x^4/30240 + O(x^6) + # [Levinson and Redheffer, p. 173] + assert (series(1/(x*(exp(x) - 1)), x, 0, 7) == x**(-2) - 1/(2*x) + + R(1, 12) - x**2/720 + x**4/30240 - x**6/1209600 + O(x**7)) + + +def test_X8(): + # Puiseux series (terms with fractional degree): + # => 1/sqrt(x - 3/2 pi) + (x - 3/2 pi)^(3/2) / 12 + O([x - 3/2 pi]^(7/2)) + + # see issue 7167: + x = symbols('x', real=True) + assert (series(sqrt(sec(x)), x, x0=pi*3/2, n=4) == + 1/sqrt(x - pi*R(3, 2)) + (x - pi*R(3, 2))**R(3, 2)/12 + + (x - pi*R(3, 2))**R(7, 2)/160 + O((x - pi*R(3, 2))**4, (x, pi*R(3, 2)))) + + +def test_X9(): + assert (series(x**x, x, x0=0, n=4) == 1 + x*log(x) + x**2*log(x)**2/2 + + x**3*log(x)**3/6 + O(x**4*log(x)**4)) + + +def test_X10(): + z, w = symbols('z w') + assert (series(log(sinh(z)) + log(cosh(z + w)), z, x0=0, n=2) == + log(cosh(w)) + log(z) + z*sinh(w)/cosh(w) + O(z**2)) + + +def test_X11(): + z, w = symbols('z w') + assert (series(log(sinh(z) * cosh(z + w)), z, x0=0, n=2) == + log(cosh(w)) + log(z) + z*sinh(w)/cosh(w) + O(z**2)) + + +@XFAIL +def test_X12(): + # Look at the generalized Taylor series around x = 1 + # Result => (x - 1)^a/e^b [1 - (a + 2 b) (x - 1) / 2 + O((x - 1)^2)] + a, b, x = symbols('a b x', real=True) + # series returns O(log(x-1)**2) + # https://github.com/sympy/sympy/issues/7168 + assert (series(log(x)**a*exp(-b*x), x, x0=1, n=2) == + (x - 1)**a/exp(b)*(1 - (a + 2*b)*(x - 1)/2 + O((x - 1)**2))) + + +def test_X13(): + assert series(sqrt(2*x**2 + 1), x, x0=oo, n=1) == sqrt(2)*x + O(1/x, (x, oo)) + + +@XFAIL +def test_X14(): + # Wallis' product => 1/sqrt(pi n) + ... [Knopp, p. 385] + assert series(1/2**(2*n)*binomial(2*n, n), + n, x==oo, n=1) == 1/(sqrt(pi)*sqrt(n)) + O(1/x, (x, oo)) + + +@SKIP("https://github.com/sympy/sympy/issues/7164") +def test_X15(): + # => 0!/x - 1!/x^2 + 2!/x^3 - 3!/x^4 + O(1/x^5) [Knopp, p. 544] + x, t = symbols('x t', real=True) + # raises RuntimeError: maximum recursion depth exceeded + # https://github.com/sympy/sympy/issues/7164 + # 2019-02-17: Raises + # PoleError: + # Asymptotic expansion of Ei around [-oo] is not implemented. + e1 = integrate(exp(-t)/t, (t, x, oo)) + assert (series(e1, x, x0=oo, n=5) == + 6/x**4 + 2/x**3 - 1/x**2 + 1/x + O(x**(-5), (x, oo))) + + +def test_X16(): + # Multivariate Taylor series expansion => 1 - (x^2 + 2 x y + y^2)/2 + O(x^4) + assert (series(cos(x + y), x + y, x0=0, n=4) == 1 - (x + y)**2/2 + + O(x**4 + x**3*y + x**2*y**2 + x*y**3 + y**4, x, y)) + + +@XFAIL +def test_X17(): + # Power series (compute the general formula) + # (c41) powerseries(log(sin(x)/x), x, 0); + # /aquarius/data2/opt/local/macsyma_422/library1/trgred.so being loaded. + # inf + # ==== i1 2 i1 2 i1 + # \ (- 1) 2 bern(2 i1) x + # (d41) > ------------------------------ + # / 2 i1 (2 i1)! + # ==== + # i1 = 1 + # fps does not calculate + assert fps(log(sin(x)/x)) == \ + Sum((-1)**k*2**(2*k - 1)*bernoulli(2*k)*x**(2*k)/(k*factorial(2*k)), (k, 1, oo)) + + +@XFAIL +def test_X18(): + # Power series (compute the general formula). Maple FPS: + # > FormalPowerSeries(exp(-x)*sin(x), x = 0); + # infinity + # ----- (1/2 k) k + # \ 2 sin(3/4 k Pi) x + # ) ------------------------- + # / k! + # ----- + # + # Now, SymPy returns + # oo + # _____ + # \ ` + # \ / k k\ + # \ k |I*(-1 - I) I*(-1 + I) | + # \ x *|----------- - -----------| + # / \ 2 2 / + # / ------------------------------ + # / k! + # /____, + # k = 0 + k = Dummy('k') + assert fps(exp(-x)*sin(x)) == \ + Sum(2**(S.Half*k)*sin(R(3, 4)*k*pi)*x**k/factorial(k), (k, 0, oo)) + + +@XFAIL +def test_X19(): + # (c45) /* Derive an explicit Taylor series solution of y as a function of + # x from the following implicit relation: + # y = x - 1 + (x - 1)^2/2 + 2/3 (x - 1)^3 + (x - 1)^4 + + # 17/10 (x - 1)^5 + ... + # */ + # x = sin(y) + cos(y); + # Time= 0 msecs + # (d45) x = sin(y) + cos(y) + # + # (c46) taylor_revert(%, y, 7); + raise NotImplementedError("Solve using series not supported. \ +Inverse Taylor series expansion also not supported") + + +@XFAIL +def test_X20(): + # Pade (rational function) approximation => (2 - x)/(2 + x) + # > numapprox[pade](exp(-x), x = 0, [1, 1]); + # bytes used=9019816, alloc=3669344, time=13.12 + # 1 - 1/2 x + # --------- + # 1 + 1/2 x + # mpmath support numeric Pade approximant but there is + # no symbolic implementation in SymPy + # https://en.wikipedia.org/wiki/Pad%C3%A9_approximant + raise NotImplementedError("Symbolic Pade approximant not supported") + + +def test_X21(): + """ + Test whether `fourier_series` of x periodical on the [-p, p] interval equals + `- (2 p / pi) sum( (-1)^n / n sin(n pi x / p), n = 1..infinity )`. + """ + p = symbols('p', positive=True) + n = symbols('n', positive=True, integer=True) + s = fourier_series(x, (x, -p, p)) + + # All cosine coefficients are equal to 0 + assert s.an.formula == 0 + + # Check for sine coefficients + assert s.bn.formula.subs(s.bn.variables[0], 0) == 0 + assert s.bn.formula.subs(s.bn.variables[0], n) == \ + -2*p/pi * (-1)**n / n * sin(n*pi*x/p) + + +@XFAIL +def test_X22(): + # (c52) /* => p / 2 + # - (2 p / pi^2) sum( [1 - (-1)^n] cos(n pi x / p) / n^2, + # n = 1..infinity ) */ + # fourier_series(abs(x), x, p); + # p + # (e52) a = - + # 0 2 + # + # %nn + # (2 (- 1) - 2) p + # (e53) a = ------------------ + # %nn 2 2 + # %pi %nn + # + # (e54) b = 0 + # %nn + # + # Time= 5290 msecs + # inf %nn %pi %nn x + # ==== (2 (- 1) - 2) cos(---------) + # \ p + # p > ------------------------------- + # / 2 + # ==== %nn + # %nn = 1 p + # (d54) ----------------------------------------- + - + # 2 2 + # %pi + raise NotImplementedError("Fourier series not supported") + + +def test_Y1(): + t = symbols('t', positive=True) + w = symbols('w', real=True) + s = symbols('s') + F, _, _ = laplace_transform(cos((w - 1)*t), t, s) + assert F == s/(s**2 + (w - 1)**2) + + +def test_Y2(): + t = symbols('t', positive=True) + w = symbols('w', real=True) + s = symbols('s') + f = inverse_laplace_transform(s/(s**2 + (w - 1)**2), s, t, simplify=True) + assert f == cos(t*(w - 1)) + + +def test_Y3(): + t = symbols('t', positive=True) + w = symbols('w', real=True) + s = symbols('s') + F, _, _ = laplace_transform(sinh(w*t)*cosh(w*t), t, s, simplify=True) + assert F == w/(s**2 - 4*w**2) + + +def test_Y4(): + t = symbols('t', positive=True) + s = symbols('s') + F, _, _ = laplace_transform(erf(3/sqrt(t)), t, s, simplify=True) + assert F == 1/s - exp(-6*sqrt(s))/s + + +def test_Y5_Y6(): +# Solve y'' + y = 4 [H(t - 1) - H(t - 2)], y(0) = 1, y'(0) = 0 where H is the +# Heaviside (unit step) function (the RHS describes a pulse of magnitude 4 and +# duration 1). See David A. Sanchez, Richard C. Allen, Jr. and Walter T. +# Kyner, _Differential Equations: An Introduction_, Addison-Wesley Publishing +# Company, 1983, p. 211. First, take the Laplace transform of the ODE +# => s^2 Y(s) - s + Y(s) = 4/s [e^(-s) - e^(-2 s)] +# where Y(s) is the Laplace transform of y(t) + t = symbols('t', real=True) + s = symbols('s') + y = Function('y') + Y = Function('Y') + F = laplace_correspondence(laplace_transform(diff(y(t), t, 2) + y(t) + - 4*(Heaviside(t - 1) - Heaviside(t - 2)), + t, s, noconds=True), {y: Y}) + D = ( + -F + s**2*Y(s) - s*y(0) + Y(s) - Subs(Derivative(y(t), t), t, 0) - + 4*exp(-s)/s + 4*exp(-2*s)/s) + assert D == 0 +# Now, solve for Y(s) and then take the inverse Laplace transform +# => Y(s) = s/(s^2 + 1) + 4 [1/s - s/(s^2 + 1)] [e^(-s) - e^(-2 s)] +# => y(t) = cos t + 4 {[1 - cos(t - 1)] H(t - 1) - [1 - cos(t - 2)] H(t - 2)} + Yf = solve(F, Y(s))[0] + Yf = laplace_initial_conds(Yf, t, {y: [1, 0]}) + assert Yf == (s**2*exp(2*s) + 4*exp(s) - 4)*exp(-2*s)/(s*(s**2 + 1)) + yf = inverse_laplace_transform(Yf, s, t) + yf = yf.collect(Heaviside(t-1)).collect(Heaviside(t-2)) + assert yf == ( + (4 - 4*cos(t - 1))*Heaviside(t - 1) + + (4*cos(t - 2) - 4)*Heaviside(t - 2) + + cos(t)*Heaviside(t)) + + +@XFAIL +def test_Y7(): + # What is the Laplace transform of an infinite square wave? + # => 1/s + 2 sum( (-1)^n e^(- s n a)/s, n = 1..infinity ) + # [Sanchez, Allen and Kyner, p. 213] + t = symbols('t', positive=True) + a = symbols('a', real=True) + s = symbols('s') + F, _, _ = laplace_transform(1 + 2*Sum((-1)**n*Heaviside(t - n*a), + (n, 1, oo)), t, s) + # returns 2*LaplaceTransform(Sum((-1)**n*Heaviside(-a*n + t), + # (n, 1, oo)), t, s) + 1/s + # https://github.com/sympy/sympy/issues/7177 + assert F == 2*Sum((-1)**n*exp(-a*n*s)/s, (n, 1, oo)) + 1/s + + +@XFAIL +def test_Y8(): + assert fourier_transform(1, x, z) == DiracDelta(z) + + +def test_Y9(): + assert (fourier_transform(exp(-9*x**2), x, z) == + sqrt(pi)*exp(-pi**2*z**2/9)/3) + + +def test_Y10(): + assert (fourier_transform(abs(x)*exp(-3*abs(x)), x, z).cancel() == + (-8*pi**2*z**2 + 18)/(16*pi**4*z**4 + 72*pi**2*z**2 + 81)) + + +@SKIP("https://github.com/sympy/sympy/issues/7181") +@slow +def test_Y11(): + # => pi cot(pi s) (0 < Re s < 1) [Gradshteyn and Ryzhik 17.43(5)] + x, s = symbols('x s') + # raises RuntimeError: maximum recursion depth exceeded + # https://github.com/sympy/sympy/issues/7181 + # Update 2019-02-17 raises: + # TypeError: cannot unpack non-iterable MellinTransform object + F, _, _ = mellin_transform(1/(1 - x), x, s) + assert F == pi*cot(pi*s) + + +@XFAIL +def test_Y12(): + # => 2^(s - 4) gamma(s/2)/gamma(4 - s/2) (0 < Re s < 1) + # [Gradshteyn and Ryzhik 17.43(16)] + x, s = symbols('x s') + # returns Wrong value -2**(s - 4)*gamma(s/2 - 3)/gamma(-s/2 + 1) + # https://github.com/sympy/sympy/issues/7182 + F, _, _ = mellin_transform(besselj(3, x)/x**3, x, s) + assert F == -2**(s - 4)*gamma(s/2)/gamma(-s/2 + 4) + + +@XFAIL +def test_Y13(): +# Z[H(t - m T)] => z/[z^m (z - 1)] (H is the Heaviside (unit step) function) z + raise NotImplementedError("z-transform not supported") + + +@XFAIL +def test_Y14(): +# Z[H(t - m T)] => z/[z^m (z - 1)] (H is the Heaviside (unit step) function) + raise NotImplementedError("z-transform not supported") + + +def test_Z1(): + r = Function('r') + assert (rsolve(r(n + 2) - 2*r(n + 1) + r(n) - 2, r(n), + {r(0): 1, r(1): m}).simplify() == n**2 + n*(m - 2) + 1) + + +def test_Z2(): + r = Function('r') + assert (rsolve(r(n) - (5*r(n - 1) - 6*r(n - 2)), r(n), {r(0): 0, r(1): 1}) + == -2**n + 3**n) + + +def test_Z3(): + # => r(n) = Fibonacci[n + 1] [Cohen, p. 83] + r = Function('r') + # recurrence solution is correct, Wester expects it to be simplified to + # fibonacci(n+1), but that is quite hard + expected = ((S(1)/2 - sqrt(5)/2)**n*(S(1)/2 - sqrt(5)/10) + + (S(1)/2 + sqrt(5)/2)**n*(sqrt(5)/10 + S(1)/2)) + sol = rsolve(r(n) - (r(n - 1) + r(n - 2)), r(n), {r(1): 1, r(2): 2}) + assert sol == expected + + +@XFAIL +def test_Z4(): +# => [c^(n+1) [c^(n+1) - 2 c - 2] + (n+1) c^2 + 2 c - n] / [(c-1)^3 (c+1)] +# [Joan Z. Yu and Robert Israel in sci.math.symbolic] + r = Function('r') + c = symbols('c') + # raises ValueError: Polynomial or rational function expected, + # got '(c**2 - c**n)/(c - c**n) + s = rsolve(r(n) - ((1 + c - c**(n-1) - c**(n+1))/(1 - c**n)*r(n - 1) + - c*(1 - c**(n-2))/(1 - c**(n-1))*r(n - 2) + 1), + r(n), {r(1): 1, r(2): (2 + 2*c + c**2)/(1 + c)}) + assert (s - (c*(n + 1)*(c*(n + 1) - 2*c - 2) + + (n + 1)*c**2 + 2*c - n)/((c-1)**3*(c+1)) == 0) + + +@XFAIL +def test_Z5(): + # Second order ODE with initial conditions---solve directly + # transform: f(t) = sin(2 t)/8 - t cos(2 t)/4 + C1, C2 = symbols('C1 C2') + # initial conditions not supported, this is a manual workaround + # https://github.com/sympy/sympy/issues/4720 + eq = Derivative(f(x), x, 2) + 4*f(x) - sin(2*x) + sol = dsolve(eq, f(x)) + f0 = Lambda(x, sol.rhs) + assert f0(x) == C2*sin(2*x) + (C1 - x/4)*cos(2*x) + f1 = Lambda(x, diff(f0(x), x)) + # TODO: Replace solve with solveset, when it works for solveset + const_dict = solve((f0(0), f1(0))) + result = f0(x).subs(C1, const_dict[C1]).subs(C2, const_dict[C2]) + assert result == -x*cos(2*x)/4 + sin(2*x)/8 + # Result is OK, but ODE solving with initial conditions should be + # supported without all this manual work + raise NotImplementedError('ODE solving with initial conditions \ +not supported') + + +@XFAIL +def test_Z6(): + # Second order ODE with initial conditions---solve using Laplace + # transform: f(t) = sin(2 t)/8 - t cos(2 t)/4 + t = symbols('t', positive=True) + s = symbols('s') + eq = Derivative(f(t), t, 2) + 4*f(t) - sin(2*t) + F, _, _ = laplace_transform(eq, t, s) + # Laplace transform for diff() not calculated + # https://github.com/sympy/sympy/issues/7176 + assert (F == s**2*LaplaceTransform(f(t), t, s) + + 4*LaplaceTransform(f(t), t, s) - 2/(s**2 + 4)) + # rest of test case not implemented diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_xxe.py b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_xxe.py new file mode 100644 index 0000000000000000000000000000000000000000..3936e8aa135dde5f22c71548e2f90ed58ac25cb8 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/tests/test_xxe.py @@ -0,0 +1,3 @@ +# A test file for XXE injection +# Username: Test +# Password: Test diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/timeutils.py b/.venv/lib/python3.12/site-packages/sympy/utilities/timeutils.py new file mode 100644 index 0000000000000000000000000000000000000000..1caae825103335f3e8afedeaf741cff2b0414fb2 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/timeutils.py @@ -0,0 +1,75 @@ +"""Simple tools for timing functions' execution, when IPython is not available. """ + + +import timeit +import math + + +_scales = [1e0, 1e3, 1e6, 1e9] +_units = ['s', 'ms', '\N{GREEK SMALL LETTER MU}s', 'ns'] + + +def timed(func, setup="pass", limit=None): + """Adaptively measure execution time of a function. """ + timer = timeit.Timer(func, setup=setup) + repeat, number = 3, 1 + + for i in range(1, 10): + if timer.timeit(number) >= 0.2: + break + elif limit is not None and number >= limit: + break + else: + number *= 10 + + time = min(timer.repeat(repeat, number)) / number + + if time > 0.0: + order = min(-int(math.floor(math.log10(time)) // 3), 3) + else: + order = 3 + + return (number, time, time*_scales[order], _units[order]) + + +# Code for doing inline timings of recursive algorithms. + +def __do_timings(): + import os + res = os.getenv('SYMPY_TIMINGS', '') + res = [x.strip() for x in res.split(',')] + return set(res) + +_do_timings = __do_timings() +_timestack = None + + +def _print_timestack(stack, level=1): + print('-'*level, '%.2f %s%s' % (stack[2], stack[0], stack[3])) + for s in stack[1]: + _print_timestack(s, level + 1) + + +def timethis(name): + def decorator(func): + if name not in _do_timings: + return func + + def wrapper(*args, **kwargs): + from time import time + global _timestack + oldtimestack = _timestack + _timestack = [func.func_name, [], 0, args] + t1 = time() + r = func(*args, **kwargs) + t2 = time() + _timestack[2] = t2 - t1 + if oldtimestack is not None: + oldtimestack[1].append(_timestack) + _timestack = oldtimestack + else: + _print_timestack(_timestack) + _timestack = None + return r + return wrapper + return decorator diff --git a/.venv/lib/python3.12/site-packages/sympy/utilities/tmpfiles.py b/.venv/lib/python3.12/site-packages/sympy/utilities/tmpfiles.py new file mode 100644 index 0000000000000000000000000000000000000000..81c97efbf9d67cc20f25ff251abc94c2f5bafe06 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/utilities/tmpfiles.py @@ -0,0 +1,12 @@ +""" +.. deprecated:: 1.6 + + sympy.utilities.tmpfiles has been renamed to sympy.testing.tmpfiles. +""" +from sympy.utilities.exceptions import sympy_deprecation_warning + +sympy_deprecation_warning("The sympy.utilities.tmpfiles submodule is deprecated. Use sympy.testing.tmpfiles instead.", + deprecated_since_version="1.6", + active_deprecations_target="deprecated-sympy-utilities-submodules") + +from sympy.testing.tmpfiles import * # noqa:F401,F403 diff --git a/.venv/lib/python3.12/site-packages/sympy/vector/__init__.py b/.venv/lib/python3.12/site-packages/sympy/vector/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..f6757bbeb35022481b1cf183373ecccd19779faa --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/vector/__init__.py @@ -0,0 +1,50 @@ +from sympy.vector.coordsysrect import CoordSys3D +from sympy.vector.vector import (Vector, VectorAdd, VectorMul, + BaseVector, VectorZero, Cross, Dot, cross, dot) +from sympy.vector.dyadic import (Dyadic, DyadicAdd, DyadicMul, + BaseDyadic, DyadicZero) +from sympy.vector.scalar import BaseScalar +from sympy.vector.deloperator import Del +from sympy.vector.functions import (express, matrix_to_vector, + laplacian, is_conservative, + is_solenoidal, scalar_potential, + directional_derivative, + scalar_potential_difference) +from sympy.vector.point import Point +from sympy.vector.orienters import (AxisOrienter, BodyOrienter, + SpaceOrienter, QuaternionOrienter) +from sympy.vector.operators import Gradient, Divergence, Curl, Laplacian, gradient, curl, divergence +from sympy.vector.implicitregion import ImplicitRegion +from sympy.vector.parametricregion import (ParametricRegion, parametric_region_list) +from sympy.vector.integrals import (ParametricIntegral, vector_integrate) +from sympy.vector.kind import VectorKind + +__all__ = [ + 'Vector', 'VectorAdd', 'VectorMul', 'BaseVector', 'VectorZero', 'Cross', + 'Dot', 'cross', 'dot', + + 'VectorKind', + + 'Dyadic', 'DyadicAdd', 'DyadicMul', 'BaseDyadic', 'DyadicZero', + + 'BaseScalar', + + 'Del', + + 'CoordSys3D', + + 'express', 'matrix_to_vector', 'laplacian', 'is_conservative', + 'is_solenoidal', 'scalar_potential', 'directional_derivative', + 'scalar_potential_difference', + + 'Point', + + 'AxisOrienter', 'BodyOrienter', 'SpaceOrienter', 'QuaternionOrienter', + + 'Gradient', 'Divergence', 'Curl', 'Laplacian', 'gradient', 'curl', + 'divergence', + + 'ParametricRegion', 'parametric_region_list', 'ImplicitRegion', + + 'ParametricIntegral', 'vector_integrate', +] diff --git a/.venv/lib/python3.12/site-packages/sympy/vector/__pycache__/__init__.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/vector/__pycache__/__init__.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..ac0e92fc2c9c6cc3e8fdc914a995c7ae94f17cd4 Binary files /dev/null and b/.venv/lib/python3.12/site-packages/sympy/vector/__pycache__/__init__.cpython-312.pyc differ diff --git a/.venv/lib/python3.12/site-packages/sympy/vector/__pycache__/basisdependent.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/vector/__pycache__/basisdependent.cpython-312.pyc new file mode 100644 index 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b/.venv/lib/python3.12/site-packages/sympy/vector/basisdependent.py new file mode 100644 index 0000000000000000000000000000000000000000..53e4efc0bf839fb5a5de2d1af1487683fabd8cf1 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/vector/basisdependent.py @@ -0,0 +1,374 @@ +from __future__ import annotations +from typing import TYPE_CHECKING + +from sympy.simplify import simplify as simp, trigsimp as tsimp # type: ignore +from sympy.core.decorators import call_highest_priority, _sympifyit +from sympy.core.assumptions import StdFactKB +from sympy.core.function import diff as df +from sympy.integrals.integrals import Integral +from sympy.polys.polytools import factor as fctr +from sympy.core import S, Add, Mul +from sympy.core.expr import Expr + +if TYPE_CHECKING: + from sympy.vector.vector import BaseVector + + +class BasisDependent(Expr): + """ + Super class containing functionality common to vectors and + dyadics. + Named so because the representation of these quantities in + sympy.vector is dependent on the basis they are expressed in. + """ + + zero: BasisDependentZero + + @call_highest_priority('__radd__') + def __add__(self, other): + return self._add_func(self, other) + + @call_highest_priority('__add__') + def __radd__(self, other): + return self._add_func(other, self) + + @call_highest_priority('__rsub__') + def __sub__(self, other): + return self._add_func(self, -other) + + @call_highest_priority('__sub__') + def __rsub__(self, other): + return self._add_func(other, -self) + + @_sympifyit('other', NotImplemented) + @call_highest_priority('__rmul__') + def __mul__(self, other): + return self._mul_func(self, other) + + @_sympifyit('other', NotImplemented) + @call_highest_priority('__mul__') + def __rmul__(self, other): + return self._mul_func(other, self) + + def __neg__(self): + return self._mul_func(S.NegativeOne, self) + + @_sympifyit('other', NotImplemented) + @call_highest_priority('__rtruediv__') + def __truediv__(self, other): + return self._div_helper(other) + + @call_highest_priority('__truediv__') + def __rtruediv__(self, other): + return TypeError("Invalid divisor for division") + + def evalf(self, n=15, subs=None, maxn=100, chop=False, strict=False, quad=None, verbose=False): + """ + Implements the SymPy evalf routine for this quantity. + + evalf's documentation + ===================== + + """ + options = {'subs':subs, 'maxn':maxn, 'chop':chop, 'strict':strict, + 'quad':quad, 'verbose':verbose} + vec = self.zero + for k, v in self.components.items(): + vec += v.evalf(n, **options) * k + return vec + + evalf.__doc__ += Expr.evalf.__doc__ # type: ignore + + n = evalf # type: ignore + + def simplify(self, **kwargs): + """ + Implements the SymPy simplify routine for this quantity. + + simplify's documentation + ======================== + + """ + simp_components = [simp(v, **kwargs) * k for + k, v in self.components.items()] + return self._add_func(*simp_components) + + simplify.__doc__ += simp.__doc__ # type: ignore + + def trigsimp(self, **opts): + """ + Implements the SymPy trigsimp routine, for this quantity. + + trigsimp's documentation + ======================== + + """ + trig_components = [tsimp(v, **opts) * k for + k, v in self.components.items()] + return self._add_func(*trig_components) + + trigsimp.__doc__ += tsimp.__doc__ # type: ignore + + def _eval_simplify(self, **kwargs): + return self.simplify(**kwargs) + + def _eval_trigsimp(self, **opts): + return self.trigsimp(**opts) + + def _eval_derivative(self, wrt): + return self.diff(wrt) + + def _eval_Integral(self, *symbols, **assumptions): + integral_components = [Integral(v, *symbols, **assumptions) * k + for k, v in self.components.items()] + return self._add_func(*integral_components) + + def as_numer_denom(self): + """ + Returns the expression as a tuple wrt the following + transformation - + + expression -> a/b -> a, b + + """ + return self, S.One + + def factor(self, *args, **kwargs): + """ + Implements the SymPy factor routine, on the scalar parts + of a basis-dependent expression. + + factor's documentation + ======================== + + """ + fctr_components = [fctr(v, *args, **kwargs) * k for + k, v in self.components.items()] + return self._add_func(*fctr_components) + + factor.__doc__ += fctr.__doc__ # type: ignore + + def as_coeff_Mul(self, rational=False): + """Efficiently extract the coefficient of a product.""" + return (S.One, self) + + def as_coeff_add(self, *deps): + """Efficiently extract the coefficient of a summation.""" + return 0, tuple(x * self.components[x] for x in self.components) + + def diff(self, *args, **kwargs): + """ + Implements the SymPy diff routine, for vectors. + + diff's documentation + ======================== + + """ + for x in args: + if isinstance(x, BasisDependent): + raise TypeError("Invalid arg for differentiation") + diff_components = [df(v, *args, **kwargs) * k for + k, v in self.components.items()] + return self._add_func(*diff_components) + + diff.__doc__ += df.__doc__ # type: ignore + + def doit(self, **hints): + """Calls .doit() on each term in the Dyadic""" + doit_components = [self.components[x].doit(**hints) * x + for x in self.components] + return self._add_func(*doit_components) + + +class BasisDependentAdd(BasisDependent, Add): + """ + Denotes sum of basis dependent quantities such that they cannot + be expressed as base or Mul instances. + """ + + def __new__(cls, *args, **options): + components = {} + + # Check each arg and simultaneously learn the components + for arg in args: + if not isinstance(arg, cls._expr_type): + if isinstance(arg, Mul): + arg = cls._mul_func(*(arg.args)) + elif isinstance(arg, Add): + arg = cls._add_func(*(arg.args)) + else: + raise TypeError(str(arg) + + " cannot be interpreted correctly") + # If argument is zero, ignore + if arg == cls.zero: + continue + # Else, update components accordingly + for x in arg.components: + components[x] = components.get(x, 0) + arg.components[x] + + temp = list(components.keys()) + for x in temp: + if components[x] == 0: + del components[x] + + # Handle case of zero vector + if len(components) == 0: + return cls.zero + + # Build object + newargs = [x * components[x] for x in components] + obj = super().__new__(cls, *newargs, **options) + if isinstance(obj, Mul): + return cls._mul_func(*obj.args) + assumptions = {'commutative': True} + obj._assumptions = StdFactKB(assumptions) + obj._components = components + obj._sys = (list(components.keys()))[0]._sys + + return obj + + +class BasisDependentMul(BasisDependent, Mul): + """ + Denotes product of base- basis dependent quantity with a scalar. + """ + + def __new__(cls, *args, **options): + obj = cls._new(*args, **options) + return obj + + def _new_rawargs(self, *args): + # XXX: This is needed because Add.flatten() uses it but the default + # implementation does not work for Vectors because they assign + # attributes outside of .args. + return type(self)(*args) + + @classmethod + def _new(cls, *args, **options): + from sympy.vector import Cross, Dot, Curl, Gradient + count = 0 + measure_number = S.One + zeroflag = False + extra_args = [] + + # Determine the component and check arguments + # Also keep a count to ensure two vectors aren't + # being multiplied + for arg in args: + if isinstance(arg, cls._zero_func): + count += 1 + zeroflag = True + elif arg == S.Zero: + zeroflag = True + elif isinstance(arg, (cls._base_func, cls._mul_func)): + count += 1 + expr = arg._base_instance + measure_number *= arg._measure_number + elif isinstance(arg, cls._add_func): + count += 1 + expr = arg + elif isinstance(arg, (Cross, Dot, Curl, Gradient)): + extra_args.append(arg) + else: + measure_number *= arg + # Make sure incompatible types weren't multiplied + if count > 1: + raise ValueError("Invalid multiplication") + elif count == 0: + return Mul(*args, **options) + # Handle zero vector case + if zeroflag: + return cls.zero + + # If one of the args was a VectorAdd, return an + # appropriate VectorAdd instance + if isinstance(expr, cls._add_func): + newargs = [cls._mul_func(measure_number, x) for + x in expr.args] + return cls._add_func(*newargs) + + obj = super().__new__(cls, measure_number, + expr._base_instance, + *extra_args, + **options) + if isinstance(obj, Add): + return cls._add_func(*obj.args) + obj._base_instance = expr._base_instance + obj._measure_number = measure_number + assumptions = {'commutative': True} + obj._assumptions = StdFactKB(assumptions) + obj._components = {expr._base_instance: measure_number} + obj._sys = expr._base_instance._sys + + return obj + + def _sympystr(self, printer): + measure_str = printer._print(self._measure_number) + if ('(' in measure_str or '-' in measure_str or + '+' in measure_str): + measure_str = '(' + measure_str + ')' + return measure_str + '*' + printer._print(self._base_instance) + + +class BasisDependentZero(BasisDependent): + """ + Class to denote a zero basis dependent instance. + """ + components: dict['BaseVector', Expr] = {} + _latex_form: str + + def __new__(cls): + obj = super().__new__(cls) + # Pre-compute a specific hash value for the zero vector + # Use the same one always + obj._hash = (S.Zero, cls).__hash__() + return obj + + def __hash__(self): + return self._hash + + @call_highest_priority('__req__') + def __eq__(self, other): + return isinstance(other, self._zero_func) + + __req__ = __eq__ + + @call_highest_priority('__radd__') + def __add__(self, other): + if isinstance(other, self._expr_type): + return other + else: + raise TypeError("Invalid argument types for addition") + + @call_highest_priority('__add__') + def __radd__(self, other): + if isinstance(other, self._expr_type): + return other + else: + raise TypeError("Invalid argument types for addition") + + @call_highest_priority('__rsub__') + def __sub__(self, other): + if isinstance(other, self._expr_type): + return -other + else: + raise TypeError("Invalid argument types for subtraction") + + @call_highest_priority('__sub__') + def __rsub__(self, other): + if isinstance(other, self._expr_type): + return other + else: + raise TypeError("Invalid argument types for subtraction") + + def __neg__(self): + return self + + def normalize(self): + """ + Returns the normalized version of this vector. + """ + return self + + def _sympystr(self, printer): + return '0' diff --git a/.venv/lib/python3.12/site-packages/sympy/vector/coordsysrect.py b/.venv/lib/python3.12/site-packages/sympy/vector/coordsysrect.py new file mode 100644 index 0000000000000000000000000000000000000000..55539fb19dc4221de69437111f44d6a6cc70b3e4 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/vector/coordsysrect.py @@ -0,0 +1,1031 @@ +from collections.abc import Callable + +from sympy.core.basic import Basic +from sympy.core.cache import cacheit +from sympy.core import S, Dummy, Lambda +from sympy.core.symbol import Str +from sympy.core.symbol import symbols +from sympy.matrices.immutable import ImmutableDenseMatrix as Matrix +from sympy.matrices.matrixbase import MatrixBase +from sympy.solvers import solve +from sympy.vector.scalar import BaseScalar +from sympy.core.containers import Tuple +from sympy.core.function import diff +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import (acos, atan2, cos, sin) +from sympy.matrices.dense import eye +from sympy.matrices.immutable import ImmutableDenseMatrix +from sympy.simplify.simplify import simplify +from sympy.simplify.trigsimp import trigsimp +import sympy.vector +from sympy.vector.orienters import (Orienter, AxisOrienter, BodyOrienter, + SpaceOrienter, QuaternionOrienter) + + +class CoordSys3D(Basic): + """ + Represents a coordinate system in 3-D space. + """ + + def __new__(cls, name, transformation=None, parent=None, location=None, + rotation_matrix=None, vector_names=None, variable_names=None): + """ + The orientation/location parameters are necessary if this system + is being defined at a certain orientation or location wrt another. + + Parameters + ========== + + name : str + The name of the new CoordSys3D instance. + + transformation : Lambda, Tuple, str + Transformation defined by transformation equations or chosen + from predefined ones. + + location : Vector + The position vector of the new system's origin wrt the parent + instance. + + rotation_matrix : SymPy ImmutableMatrix + The rotation matrix of the new coordinate system with respect + to the parent. In other words, the output of + new_system.rotation_matrix(parent). + + parent : CoordSys3D + The coordinate system wrt which the orientation/location + (or both) is being defined. + + vector_names, variable_names : iterable(optional) + Iterables of 3 strings each, with custom names for base + vectors and base scalars of the new system respectively. + Used for simple str printing. + + """ + + name = str(name) + Vector = sympy.vector.Vector + Point = sympy.vector.Point + + if not isinstance(name, str): + raise TypeError("name should be a string") + + if transformation is not None: + if (location is not None) or (rotation_matrix is not None): + raise ValueError("specify either `transformation` or " + "`location`/`rotation_matrix`") + if isinstance(transformation, (Tuple, tuple, list)): + if isinstance(transformation[0], MatrixBase): + rotation_matrix = transformation[0] + location = transformation[1] + else: + transformation = Lambda(transformation[0], + transformation[1]) + elif isinstance(transformation, Callable): + x1, x2, x3 = symbols('x1 x2 x3', cls=Dummy) + transformation = Lambda((x1, x2, x3), + transformation(x1, x2, x3)) + elif isinstance(transformation, str): + transformation = Str(transformation) + elif isinstance(transformation, (Str, Lambda)): + pass + else: + raise TypeError("transformation: " + "wrong type {}".format(type(transformation))) + + # If orientation information has been provided, store + # the rotation matrix accordingly + if rotation_matrix is None: + rotation_matrix = ImmutableDenseMatrix(eye(3)) + else: + if not isinstance(rotation_matrix, MatrixBase): + raise TypeError("rotation_matrix should be an Immutable" + + "Matrix instance") + rotation_matrix = rotation_matrix.as_immutable() + + # If location information is not given, adjust the default + # location as Vector.zero + if parent is not None: + if not isinstance(parent, CoordSys3D): + raise TypeError("parent should be a " + + "CoordSys3D/None") + if location is None: + location = Vector.zero + else: + if not isinstance(location, Vector): + raise TypeError("location should be a Vector") + # Check that location does not contain base + # scalars + for x in location.free_symbols: + if isinstance(x, BaseScalar): + raise ValueError("location should not contain" + + " BaseScalars") + origin = parent.origin.locate_new(name + '.origin', + location) + else: + location = Vector.zero + origin = Point(name + '.origin') + + if transformation is None: + transformation = Tuple(rotation_matrix, location) + + if isinstance(transformation, Tuple): + lambda_transformation = CoordSys3D._compose_rotation_and_translation( + transformation[0], + transformation[1], + parent + ) + r, l = transformation + l = l._projections + lambda_lame = CoordSys3D._get_lame_coeff('cartesian') + lambda_inverse = lambda x, y, z: r.inv()*Matrix( + [x-l[0], y-l[1], z-l[2]]) + elif isinstance(transformation, Str): + trname = transformation.name + lambda_transformation = CoordSys3D._get_transformation_lambdas(trname) + if parent is not None: + if parent.lame_coefficients() != (S.One, S.One, S.One): + raise ValueError('Parent for pre-defined coordinate ' + 'system should be Cartesian.') + lambda_lame = CoordSys3D._get_lame_coeff(trname) + lambda_inverse = CoordSys3D._set_inv_trans_equations(trname) + elif isinstance(transformation, Lambda): + if not CoordSys3D._check_orthogonality(transformation): + raise ValueError("The transformation equation does not " + "create orthogonal coordinate system") + lambda_transformation = transformation + lambda_lame = CoordSys3D._calculate_lame_coeff(lambda_transformation) + lambda_inverse = None + else: + lambda_transformation = lambda x, y, z: transformation(x, y, z) + lambda_lame = CoordSys3D._get_lame_coeff(transformation) + lambda_inverse = None + + if variable_names is None: + if isinstance(transformation, Lambda): + variable_names = ["x1", "x2", "x3"] + elif isinstance(transformation, Str): + if transformation.name == 'spherical': + variable_names = ["r", "theta", "phi"] + elif transformation.name == 'cylindrical': + variable_names = ["r", "theta", "z"] + else: + variable_names = ["x", "y", "z"] + else: + variable_names = ["x", "y", "z"] + if vector_names is None: + vector_names = ["i", "j", "k"] + + # All systems that are defined as 'roots' are unequal, unless + # they have the same name. + # Systems defined at same orientation/position wrt the same + # 'parent' are equal, irrespective of the name. + # This is true even if the same orientation is provided via + # different methods like Axis/Body/Space/Quaternion. + # However, coincident systems may be seen as unequal if + # positioned/oriented wrt different parents, even though + # they may actually be 'coincident' wrt the root system. + if parent is not None: + obj = super().__new__( + cls, Str(name), transformation, parent) + else: + obj = super().__new__( + cls, Str(name), transformation) + obj._name = name + # Initialize the base vectors + + _check_strings('vector_names', vector_names) + vector_names = list(vector_names) + latex_vects = [(r'\mathbf{\hat{%s}_{%s}}' % (x, name)) for + x in vector_names] + pretty_vects = ['%s_%s' % (x, name) for x in vector_names] + + obj._vector_names = vector_names + + v1 = BaseVector(0, obj, pretty_vects[0], latex_vects[0]) + v2 = BaseVector(1, obj, pretty_vects[1], latex_vects[1]) + v3 = BaseVector(2, obj, pretty_vects[2], latex_vects[2]) + + obj._base_vectors = (v1, v2, v3) + + # Initialize the base scalars + + _check_strings('variable_names', vector_names) + variable_names = list(variable_names) + latex_scalars = [(r"\mathbf{{%s}_{%s}}" % (x, name)) for + x in variable_names] + pretty_scalars = ['%s_%s' % (x, name) for x in variable_names] + + obj._variable_names = variable_names + obj._vector_names = vector_names + + x1 = BaseScalar(0, obj, pretty_scalars[0], latex_scalars[0]) + x2 = BaseScalar(1, obj, pretty_scalars[1], latex_scalars[1]) + x3 = BaseScalar(2, obj, pretty_scalars[2], latex_scalars[2]) + + obj._base_scalars = (x1, x2, x3) + + obj._transformation = transformation + obj._transformation_lambda = lambda_transformation + obj._lame_coefficients = lambda_lame(x1, x2, x3) + obj._transformation_from_parent_lambda = lambda_inverse + + setattr(obj, variable_names[0], x1) + setattr(obj, variable_names[1], x2) + setattr(obj, variable_names[2], x3) + + setattr(obj, vector_names[0], v1) + setattr(obj, vector_names[1], v2) + setattr(obj, vector_names[2], v3) + + # Assign params + obj._parent = parent + if obj._parent is not None: + obj._root = obj._parent._root + else: + obj._root = obj + + obj._parent_rotation_matrix = rotation_matrix + obj._origin = origin + + # Return the instance + return obj + + def _sympystr(self, printer): + return self._name + + def __iter__(self): + return iter(self.base_vectors()) + + @staticmethod + def _check_orthogonality(equations): + """ + Helper method for _connect_to_cartesian. It checks if + set of transformation equations create orthogonal curvilinear + coordinate system + + Parameters + ========== + + equations : Lambda + Lambda of transformation equations + + """ + + x1, x2, x3 = symbols("x1, x2, x3", cls=Dummy) + equations = equations(x1, x2, x3) + v1 = Matrix([diff(equations[0], x1), + diff(equations[1], x1), diff(equations[2], x1)]) + + v2 = Matrix([diff(equations[0], x2), + diff(equations[1], x2), diff(equations[2], x2)]) + + v3 = Matrix([diff(equations[0], x3), + diff(equations[1], x3), diff(equations[2], x3)]) + + if any(simplify(i[0] + i[1] + i[2]) == 0 for i in (v1, v2, v3)): + return False + else: + if simplify(v1.dot(v2)) == 0 and simplify(v2.dot(v3)) == 0 \ + and simplify(v3.dot(v1)) == 0: + return True + else: + return False + + @staticmethod + def _set_inv_trans_equations(curv_coord_name): + """ + Store information about inverse transformation equations for + pre-defined coordinate systems. + + Parameters + ========== + + curv_coord_name : str + Name of coordinate system + + """ + if curv_coord_name == 'cartesian': + return lambda x, y, z: (x, y, z) + + if curv_coord_name == 'spherical': + return lambda x, y, z: ( + sqrt(x**2 + y**2 + z**2), + acos(z/sqrt(x**2 + y**2 + z**2)), + atan2(y, x) + ) + if curv_coord_name == 'cylindrical': + return lambda x, y, z: ( + sqrt(x**2 + y**2), + atan2(y, x), + z + ) + raise ValueError('Wrong set of parameters.' + 'Type of coordinate system is defined') + + def _calculate_inv_trans_equations(self): + """ + Helper method for set_coordinate_type. It calculates inverse + transformation equations for given transformations equations. + + """ + x1, x2, x3 = symbols("x1, x2, x3", cls=Dummy, reals=True) + x, y, z = symbols("x, y, z", cls=Dummy) + + equations = self._transformation(x1, x2, x3) + + solved = solve([equations[0] - x, + equations[1] - y, + equations[2] - z], (x1, x2, x3), dict=True)[0] + solved = solved[x1], solved[x2], solved[x3] + self._transformation_from_parent_lambda = \ + lambda x1, x2, x3: tuple(i.subs(list(zip((x, y, z), (x1, x2, x3)))) for i in solved) + + @staticmethod + def _get_lame_coeff(curv_coord_name): + """ + Store information about Lame coefficients for pre-defined + coordinate systems. + + Parameters + ========== + + curv_coord_name : str + Name of coordinate system + + """ + if isinstance(curv_coord_name, str): + if curv_coord_name == 'cartesian': + return lambda x, y, z: (S.One, S.One, S.One) + if curv_coord_name == 'spherical': + return lambda r, theta, phi: (S.One, r, r*sin(theta)) + if curv_coord_name == 'cylindrical': + return lambda r, theta, h: (S.One, r, S.One) + raise ValueError('Wrong set of parameters.' + ' Type of coordinate system is not defined') + return CoordSys3D._calculate_lame_coefficients(curv_coord_name) + + @staticmethod + def _calculate_lame_coeff(equations): + """ + It calculates Lame coefficients + for given transformations equations. + + Parameters + ========== + + equations : Lambda + Lambda of transformation equations. + + """ + return lambda x1, x2, x3: ( + sqrt(diff(equations(x1, x2, x3)[0], x1)**2 + + diff(equations(x1, x2, x3)[1], x1)**2 + + diff(equations(x1, x2, x3)[2], x1)**2), + sqrt(diff(equations(x1, x2, x3)[0], x2)**2 + + diff(equations(x1, x2, x3)[1], x2)**2 + + diff(equations(x1, x2, x3)[2], x2)**2), + sqrt(diff(equations(x1, x2, x3)[0], x3)**2 + + diff(equations(x1, x2, x3)[1], x3)**2 + + diff(equations(x1, x2, x3)[2], x3)**2) + ) + + def _inverse_rotation_matrix(self): + """ + Returns inverse rotation matrix. + """ + return simplify(self._parent_rotation_matrix**-1) + + @staticmethod + def _get_transformation_lambdas(curv_coord_name): + """ + Store information about transformation equations for pre-defined + coordinate systems. + + Parameters + ========== + + curv_coord_name : str + Name of coordinate system + + """ + if isinstance(curv_coord_name, str): + if curv_coord_name == 'cartesian': + return lambda x, y, z: (x, y, z) + if curv_coord_name == 'spherical': + return lambda r, theta, phi: ( + r*sin(theta)*cos(phi), + r*sin(theta)*sin(phi), + r*cos(theta) + ) + if curv_coord_name == 'cylindrical': + return lambda r, theta, h: ( + r*cos(theta), + r*sin(theta), + h + ) + raise ValueError('Wrong set of parameters.' + 'Type of coordinate system is defined') + + @classmethod + def _rotation_trans_equations(cls, matrix, equations): + """ + Returns the transformation equations obtained from rotation matrix. + + Parameters + ========== + + matrix : Matrix + Rotation matrix + + equations : tuple + Transformation equations + + """ + return tuple(matrix * Matrix(equations)) + + @property + def origin(self): + return self._origin + + def base_vectors(self): + return self._base_vectors + + def base_scalars(self): + return self._base_scalars + + def lame_coefficients(self): + return self._lame_coefficients + + def transformation_to_parent(self): + return self._transformation_lambda(*self.base_scalars()) + + def transformation_from_parent(self): + if self._parent is None: + raise ValueError("no parent coordinate system, use " + "`transformation_from_parent_function()`") + return self._transformation_from_parent_lambda( + *self._parent.base_scalars()) + + def transformation_from_parent_function(self): + return self._transformation_from_parent_lambda + + def rotation_matrix(self, other): + """ + Returns the direction cosine matrix(DCM), also known as the + 'rotation matrix' of this coordinate system with respect to + another system. + + If v_a is a vector defined in system 'A' (in matrix format) + and v_b is the same vector defined in system 'B', then + v_a = A.rotation_matrix(B) * v_b. + + A SymPy Matrix is returned. + + Parameters + ========== + + other : CoordSys3D + The system which the DCM is generated to. + + Examples + ======== + + >>> from sympy.vector import CoordSys3D + >>> from sympy import symbols + >>> q1 = symbols('q1') + >>> N = CoordSys3D('N') + >>> A = N.orient_new_axis('A', q1, N.i) + >>> N.rotation_matrix(A) + Matrix([ + [1, 0, 0], + [0, cos(q1), -sin(q1)], + [0, sin(q1), cos(q1)]]) + + """ + from sympy.vector.functions import _path + if not isinstance(other, CoordSys3D): + raise TypeError(str(other) + + " is not a CoordSys3D") + # Handle special cases + if other == self: + return eye(3) + elif other == self._parent: + return self._parent_rotation_matrix + elif other._parent == self: + return other._parent_rotation_matrix.T + # Else, use tree to calculate position + rootindex, path = _path(self, other) + result = eye(3) + for i in range(rootindex): + result *= path[i]._parent_rotation_matrix + for i in range(rootindex + 1, len(path)): + result *= path[i]._parent_rotation_matrix.T + return result + + @cacheit + def position_wrt(self, other): + """ + Returns the position vector of the origin of this coordinate + system with respect to another Point/CoordSys3D. + + Parameters + ========== + + other : Point/CoordSys3D + If other is a Point, the position of this system's origin + wrt it is returned. If its an instance of CoordSyRect, + the position wrt its origin is returned. + + Examples + ======== + + >>> from sympy.vector import CoordSys3D + >>> N = CoordSys3D('N') + >>> N1 = N.locate_new('N1', 10 * N.i) + >>> N.position_wrt(N1) + (-10)*N.i + + """ + return self.origin.position_wrt(other) + + def scalar_map(self, other): + """ + Returns a dictionary which expresses the coordinate variables + (base scalars) of this frame in terms of the variables of + otherframe. + + Parameters + ========== + + otherframe : CoordSys3D + The other system to map the variables to. + + Examples + ======== + + >>> from sympy.vector import CoordSys3D + >>> from sympy import Symbol + >>> A = CoordSys3D('A') + >>> q = Symbol('q') + >>> B = A.orient_new_axis('B', q, A.k) + >>> A.scalar_map(B) + {A.x: B.x*cos(q) - B.y*sin(q), A.y: B.x*sin(q) + B.y*cos(q), A.z: B.z} + + """ + + origin_coords = tuple(self.position_wrt(other).to_matrix(other)) + relocated_scalars = [x - origin_coords[i] + for i, x in enumerate(other.base_scalars())] + + vars_matrix = (self.rotation_matrix(other) * + Matrix(relocated_scalars)) + return {x: trigsimp(vars_matrix[i]) + for i, x in enumerate(self.base_scalars())} + + def locate_new(self, name, position, vector_names=None, + variable_names=None): + """ + Returns a CoordSys3D with its origin located at the given + position wrt this coordinate system's origin. + + Parameters + ========== + + name : str + The name of the new CoordSys3D instance. + + position : Vector + The position vector of the new system's origin wrt this + one. + + vector_names, variable_names : iterable(optional) + Iterables of 3 strings each, with custom names for base + vectors and base scalars of the new system respectively. + Used for simple str printing. + + Examples + ======== + + >>> from sympy.vector import CoordSys3D + >>> A = CoordSys3D('A') + >>> B = A.locate_new('B', 10 * A.i) + >>> B.origin.position_wrt(A.origin) + 10*A.i + + """ + if variable_names is None: + variable_names = self._variable_names + if vector_names is None: + vector_names = self._vector_names + + return CoordSys3D(name, location=position, + vector_names=vector_names, + variable_names=variable_names, + parent=self) + + def orient_new(self, name, orienters, location=None, + vector_names=None, variable_names=None): + """ + Creates a new CoordSys3D oriented in the user-specified way + with respect to this system. + + Please refer to the documentation of the orienter classes + for more information about the orientation procedure. + + Parameters + ========== + + name : str + The name of the new CoordSys3D instance. + + orienters : iterable/Orienter + An Orienter or an iterable of Orienters for orienting the + new coordinate system. + If an Orienter is provided, it is applied to get the new + system. + If an iterable is provided, the orienters will be applied + in the order in which they appear in the iterable. + + location : Vector(optional) + The location of the new coordinate system's origin wrt this + system's origin. If not specified, the origins are taken to + be coincident. + + vector_names, variable_names : iterable(optional) + Iterables of 3 strings each, with custom names for base + vectors and base scalars of the new system respectively. + Used for simple str printing. + + Examples + ======== + + >>> from sympy.vector import CoordSys3D + >>> from sympy import symbols + >>> q0, q1, q2, q3 = symbols('q0 q1 q2 q3') + >>> N = CoordSys3D('N') + + Using an AxisOrienter + + >>> from sympy.vector import AxisOrienter + >>> axis_orienter = AxisOrienter(q1, N.i + 2 * N.j) + >>> A = N.orient_new('A', (axis_orienter, )) + + Using a BodyOrienter + + >>> from sympy.vector import BodyOrienter + >>> body_orienter = BodyOrienter(q1, q2, q3, '123') + >>> B = N.orient_new('B', (body_orienter, )) + + Using a SpaceOrienter + + >>> from sympy.vector import SpaceOrienter + >>> space_orienter = SpaceOrienter(q1, q2, q3, '312') + >>> C = N.orient_new('C', (space_orienter, )) + + Using a QuaternionOrienter + + >>> from sympy.vector import QuaternionOrienter + >>> q_orienter = QuaternionOrienter(q0, q1, q2, q3) + >>> D = N.orient_new('D', (q_orienter, )) + """ + if variable_names is None: + variable_names = self._variable_names + if vector_names is None: + vector_names = self._vector_names + + if isinstance(orienters, Orienter): + if isinstance(orienters, AxisOrienter): + final_matrix = orienters.rotation_matrix(self) + else: + final_matrix = orienters.rotation_matrix() + # TODO: trigsimp is needed here so that the matrix becomes + # canonical (scalar_map also calls trigsimp; without this, you can + # end up with the same CoordinateSystem that compares differently + # due to a differently formatted matrix). However, this is + # probably not so good for performance. + final_matrix = trigsimp(final_matrix) + else: + final_matrix = Matrix(eye(3)) + for orienter in orienters: + if isinstance(orienter, AxisOrienter): + final_matrix *= orienter.rotation_matrix(self) + else: + final_matrix *= orienter.rotation_matrix() + + return CoordSys3D(name, rotation_matrix=final_matrix, + vector_names=vector_names, + variable_names=variable_names, + location=location, + parent=self) + + def orient_new_axis(self, name, angle, axis, location=None, + vector_names=None, variable_names=None): + """ + Axis rotation is a rotation about an arbitrary axis by + some angle. The angle is supplied as a SymPy expr scalar, and + the axis is supplied as a Vector. + + Parameters + ========== + + name : string + The name of the new coordinate system + + angle : Expr + The angle by which the new system is to be rotated + + axis : Vector + The axis around which the rotation has to be performed + + location : Vector(optional) + The location of the new coordinate system's origin wrt this + system's origin. If not specified, the origins are taken to + be coincident. + + vector_names, variable_names : iterable(optional) + Iterables of 3 strings each, with custom names for base + vectors and base scalars of the new system respectively. + Used for simple str printing. + + Examples + ======== + + >>> from sympy.vector import CoordSys3D + >>> from sympy import symbols + >>> q1 = symbols('q1') + >>> N = CoordSys3D('N') + >>> B = N.orient_new_axis('B', q1, N.i + 2 * N.j) + + """ + if variable_names is None: + variable_names = self._variable_names + if vector_names is None: + vector_names = self._vector_names + + orienter = AxisOrienter(angle, axis) + return self.orient_new(name, orienter, + location=location, + vector_names=vector_names, + variable_names=variable_names) + + def orient_new_body(self, name, angle1, angle2, angle3, + rotation_order, location=None, + vector_names=None, variable_names=None): + """ + Body orientation takes this coordinate system through three + successive simple rotations. + + Body fixed rotations include both Euler Angles and + Tait-Bryan Angles, see https://en.wikipedia.org/wiki/Euler_angles. + + Parameters + ========== + + name : string + The name of the new coordinate system + + angle1, angle2, angle3 : Expr + Three successive angles to rotate the coordinate system by + + rotation_order : string + String defining the order of axes for rotation + + location : Vector(optional) + The location of the new coordinate system's origin wrt this + system's origin. If not specified, the origins are taken to + be coincident. + + vector_names, variable_names : iterable(optional) + Iterables of 3 strings each, with custom names for base + vectors and base scalars of the new system respectively. + Used for simple str printing. + + Examples + ======== + + >>> from sympy.vector import CoordSys3D + >>> from sympy import symbols + >>> q1, q2, q3 = symbols('q1 q2 q3') + >>> N = CoordSys3D('N') + + A 'Body' fixed rotation is described by three angles and + three body-fixed rotation axes. To orient a coordinate system D + with respect to N, each sequential rotation is always about + the orthogonal unit vectors fixed to D. For example, a '123' + rotation will specify rotations about N.i, then D.j, then + D.k. (Initially, D.i is same as N.i) + Therefore, + + >>> D = N.orient_new_body('D', q1, q2, q3, '123') + + is same as + + >>> D = N.orient_new_axis('D', q1, N.i) + >>> D = D.orient_new_axis('D', q2, D.j) + >>> D = D.orient_new_axis('D', q3, D.k) + + Acceptable rotation orders are of length 3, expressed in XYZ or + 123, and cannot have a rotation about about an axis twice in a row. + + >>> B = N.orient_new_body('B', q1, q2, q3, '123') + >>> B = N.orient_new_body('B', q1, q2, 0, 'ZXZ') + >>> B = N.orient_new_body('B', 0, 0, 0, 'XYX') + + """ + + orienter = BodyOrienter(angle1, angle2, angle3, rotation_order) + return self.orient_new(name, orienter, + location=location, + vector_names=vector_names, + variable_names=variable_names) + + def orient_new_space(self, name, angle1, angle2, angle3, + rotation_order, location=None, + vector_names=None, variable_names=None): + """ + Space rotation is similar to Body rotation, but the rotations + are applied in the opposite order. + + Parameters + ========== + + name : string + The name of the new coordinate system + + angle1, angle2, angle3 : Expr + Three successive angles to rotate the coordinate system by + + rotation_order : string + String defining the order of axes for rotation + + location : Vector(optional) + The location of the new coordinate system's origin wrt this + system's origin. If not specified, the origins are taken to + be coincident. + + vector_names, variable_names : iterable(optional) + Iterables of 3 strings each, with custom names for base + vectors and base scalars of the new system respectively. + Used for simple str printing. + + See Also + ======== + + CoordSys3D.orient_new_body : method to orient via Euler + angles + + Examples + ======== + + >>> from sympy.vector import CoordSys3D + >>> from sympy import symbols + >>> q1, q2, q3 = symbols('q1 q2 q3') + >>> N = CoordSys3D('N') + + To orient a coordinate system D with respect to N, each + sequential rotation is always about N's orthogonal unit vectors. + For example, a '123' rotation will specify rotations about + N.i, then N.j, then N.k. + Therefore, + + >>> D = N.orient_new_space('D', q1, q2, q3, '312') + + is same as + + >>> B = N.orient_new_axis('B', q1, N.i) + >>> C = B.orient_new_axis('C', q2, N.j) + >>> D = C.orient_new_axis('D', q3, N.k) + + """ + + orienter = SpaceOrienter(angle1, angle2, angle3, rotation_order) + return self.orient_new(name, orienter, + location=location, + vector_names=vector_names, + variable_names=variable_names) + + def orient_new_quaternion(self, name, q0, q1, q2, q3, location=None, + vector_names=None, variable_names=None): + """ + Quaternion orientation orients the new CoordSys3D with + Quaternions, defined as a finite rotation about lambda, a unit + vector, by some amount theta. + + This orientation is described by four parameters: + + q0 = cos(theta/2) + + q1 = lambda_x sin(theta/2) + + q2 = lambda_y sin(theta/2) + + q3 = lambda_z sin(theta/2) + + Quaternion does not take in a rotation order. + + Parameters + ========== + + name : string + The name of the new coordinate system + + q0, q1, q2, q3 : Expr + The quaternions to rotate the coordinate system by + + location : Vector(optional) + The location of the new coordinate system's origin wrt this + system's origin. If not specified, the origins are taken to + be coincident. + + vector_names, variable_names : iterable(optional) + Iterables of 3 strings each, with custom names for base + vectors and base scalars of the new system respectively. + Used for simple str printing. + + Examples + ======== + + >>> from sympy.vector import CoordSys3D + >>> from sympy import symbols + >>> q0, q1, q2, q3 = symbols('q0 q1 q2 q3') + >>> N = CoordSys3D('N') + >>> B = N.orient_new_quaternion('B', q0, q1, q2, q3) + + """ + + orienter = QuaternionOrienter(q0, q1, q2, q3) + return self.orient_new(name, orienter, + location=location, + vector_names=vector_names, + variable_names=variable_names) + + def create_new(self, name, transformation, variable_names=None, vector_names=None): + """ + Returns a CoordSys3D which is connected to self by transformation. + + Parameters + ========== + + name : str + The name of the new CoordSys3D instance. + + transformation : Lambda, Tuple, str + Transformation defined by transformation equations or chosen + from predefined ones. + + vector_names, variable_names : iterable(optional) + Iterables of 3 strings each, with custom names for base + vectors and base scalars of the new system respectively. + Used for simple str printing. + + Examples + ======== + + >>> from sympy.vector import CoordSys3D + >>> a = CoordSys3D('a') + >>> b = a.create_new('b', transformation='spherical') + >>> b.transformation_to_parent() + (b.r*sin(b.theta)*cos(b.phi), b.r*sin(b.phi)*sin(b.theta), b.r*cos(b.theta)) + >>> b.transformation_from_parent() + (sqrt(a.x**2 + a.y**2 + a.z**2), acos(a.z/sqrt(a.x**2 + a.y**2 + a.z**2)), atan2(a.y, a.x)) + + """ + return CoordSys3D(name, parent=self, transformation=transformation, + variable_names=variable_names, vector_names=vector_names) + + def __init__(self, name, location=None, rotation_matrix=None, + parent=None, vector_names=None, variable_names=None, + latex_vects=None, pretty_vects=None, latex_scalars=None, + pretty_scalars=None, transformation=None): + # Dummy initializer for setting docstring + pass + + __init__.__doc__ = __new__.__doc__ + + @staticmethod + def _compose_rotation_and_translation(rot, translation, parent): + r = lambda x, y, z: CoordSys3D._rotation_trans_equations(rot, (x, y, z)) + if parent is None: + return r + + dx, dy, dz = [translation.dot(i) for i in parent.base_vectors()] + t = lambda x, y, z: ( + x + dx, + y + dy, + z + dz, + ) + return lambda x, y, z: t(*r(x, y, z)) + + +def _check_strings(arg_name, arg): + errorstr = arg_name + " must be an iterable of 3 string-types" + if len(arg) != 3: + raise ValueError(errorstr) + for s in arg: + if not isinstance(s, str): + raise TypeError(errorstr) + + +# Delayed import to avoid cyclic import problems: +from sympy.vector.vector import BaseVector diff --git a/.venv/lib/python3.12/site-packages/sympy/vector/deloperator.py b/.venv/lib/python3.12/site-packages/sympy/vector/deloperator.py new file mode 100644 index 0000000000000000000000000000000000000000..51c3c0caf42b5e5d372bd65907d8bae2bd563562 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/vector/deloperator.py @@ -0,0 +1,121 @@ +from sympy.core import Basic +from sympy.vector.operators import gradient, divergence, curl + + +class Del(Basic): + """ + Represents the vector differential operator, usually represented in + mathematical expressions as the 'nabla' symbol. + """ + + def __new__(cls): + obj = super().__new__(cls) + obj._name = "delop" + return obj + + def gradient(self, scalar_field, doit=False): + """ + Returns the gradient of the given scalar field, as a + Vector instance. + + Parameters + ========== + + scalar_field : SymPy expression + The scalar field to calculate the gradient of. + + doit : bool + If True, the result is returned after calling .doit() on + each component. Else, the returned expression contains + Derivative instances + + Examples + ======== + + >>> from sympy.vector import CoordSys3D, Del + >>> C = CoordSys3D('C') + >>> delop = Del() + >>> delop.gradient(9) + 0 + >>> delop(C.x*C.y*C.z).doit() + C.y*C.z*C.i + C.x*C.z*C.j + C.x*C.y*C.k + + """ + + return gradient(scalar_field, doit=doit) + + __call__ = gradient + __call__.__doc__ = gradient.__doc__ + + def dot(self, vect, doit=False): + """ + Represents the dot product between this operator and a given + vector - equal to the divergence of the vector field. + + Parameters + ========== + + vect : Vector + The vector whose divergence is to be calculated. + + doit : bool + If True, the result is returned after calling .doit() on + each component. Else, the returned expression contains + Derivative instances + + Examples + ======== + + >>> from sympy.vector import CoordSys3D, Del + >>> delop = Del() + >>> C = CoordSys3D('C') + >>> delop.dot(C.x*C.i) + Derivative(C.x, C.x) + >>> v = C.x*C.y*C.z * (C.i + C.j + C.k) + >>> (delop & v).doit() + C.x*C.y + C.x*C.z + C.y*C.z + + """ + return divergence(vect, doit=doit) + + __and__ = dot + __and__.__doc__ = dot.__doc__ + + def cross(self, vect, doit=False): + """ + Represents the cross product between this operator and a given + vector - equal to the curl of the vector field. + + Parameters + ========== + + vect : Vector + The vector whose curl is to be calculated. + + doit : bool + If True, the result is returned after calling .doit() on + each component. Else, the returned expression contains + Derivative instances + + Examples + ======== + + >>> from sympy.vector import CoordSys3D, Del + >>> C = CoordSys3D('C') + >>> delop = Del() + >>> v = C.x*C.y*C.z * (C.i + C.j + C.k) + >>> delop.cross(v, doit = True) + (-C.x*C.y + C.x*C.z)*C.i + (C.x*C.y - C.y*C.z)*C.j + + (-C.x*C.z + C.y*C.z)*C.k + >>> (delop ^ C.i).doit() + 0 + + """ + + return curl(vect, doit=doit) + + __xor__ = cross + __xor__.__doc__ = cross.__doc__ + + def _sympystr(self, printer): + return self._name diff --git a/.venv/lib/python3.12/site-packages/sympy/vector/dyadic.py b/.venv/lib/python3.12/site-packages/sympy/vector/dyadic.py new file mode 100644 index 0000000000000000000000000000000000000000..980c6e6dad90ac095b7bd6d4228f507a7831b39f --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/vector/dyadic.py @@ -0,0 +1,285 @@ +from __future__ import annotations + +from sympy.vector.basisdependent import (BasisDependent, BasisDependentAdd, + BasisDependentMul, BasisDependentZero) +from sympy.core import S, Pow +from sympy.core.expr import AtomicExpr +from sympy.matrices.immutable import ImmutableDenseMatrix as Matrix +import sympy.vector + + +class Dyadic(BasisDependent): + """ + Super class for all Dyadic-classes. + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Dyadic_tensor + .. [2] Kane, T., Levinson, D. Dynamics Theory and Applications. 1985 + McGraw-Hill + + """ + + _op_priority = 13.0 + + _expr_type: type[Dyadic] + _mul_func: type[Dyadic] + _add_func: type[Dyadic] + _zero_func: type[Dyadic] + _base_func: type[Dyadic] + zero: DyadicZero + + @property + def components(self): + """ + Returns the components of this dyadic in the form of a + Python dictionary mapping BaseDyadic instances to the + corresponding measure numbers. + + """ + # The '_components' attribute is defined according to the + # subclass of Dyadic the instance belongs to. + return self._components + + def dot(self, other): + """ + Returns the dot product(also called inner product) of this + Dyadic, with another Dyadic or Vector. + If 'other' is a Dyadic, this returns a Dyadic. Else, it returns + a Vector (unless an error is encountered). + + Parameters + ========== + + other : Dyadic/Vector + The other Dyadic or Vector to take the inner product with + + Examples + ======== + + >>> from sympy.vector import CoordSys3D + >>> N = CoordSys3D('N') + >>> D1 = N.i.outer(N.j) + >>> D2 = N.j.outer(N.j) + >>> D1.dot(D2) + (N.i|N.j) + >>> D1.dot(N.j) + N.i + + """ + + Vector = sympy.vector.Vector + if isinstance(other, BasisDependentZero): + return Vector.zero + elif isinstance(other, Vector): + outvec = Vector.zero + for k, v in self.components.items(): + vect_dot = k.args[1].dot(other) + outvec += vect_dot * v * k.args[0] + return outvec + elif isinstance(other, Dyadic): + outdyad = Dyadic.zero + for k1, v1 in self.components.items(): + for k2, v2 in other.components.items(): + vect_dot = k1.args[1].dot(k2.args[0]) + outer_product = k1.args[0].outer(k2.args[1]) + outdyad += vect_dot * v1 * v2 * outer_product + return outdyad + else: + raise TypeError("Inner product is not defined for " + + str(type(other)) + " and Dyadics.") + + def __and__(self, other): + return self.dot(other) + + __and__.__doc__ = dot.__doc__ + + def cross(self, other): + """ + Returns the cross product between this Dyadic, and a Vector, as a + Vector instance. + + Parameters + ========== + + other : Vector + The Vector that we are crossing this Dyadic with + + Examples + ======== + + >>> from sympy.vector import CoordSys3D + >>> N = CoordSys3D('N') + >>> d = N.i.outer(N.i) + >>> d.cross(N.j) + (N.i|N.k) + + """ + + Vector = sympy.vector.Vector + if other == Vector.zero: + return Dyadic.zero + elif isinstance(other, Vector): + outdyad = Dyadic.zero + for k, v in self.components.items(): + cross_product = k.args[1].cross(other) + outer = k.args[0].outer(cross_product) + outdyad += v * outer + return outdyad + else: + raise TypeError(str(type(other)) + " not supported for " + + "cross with dyadics") + + def __xor__(self, other): + return self.cross(other) + + __xor__.__doc__ = cross.__doc__ + + def to_matrix(self, system, second_system=None): + """ + Returns the matrix form of the dyadic with respect to one or two + coordinate systems. + + Parameters + ========== + + system : CoordSys3D + The coordinate system that the rows and columns of the matrix + correspond to. If a second system is provided, this + only corresponds to the rows of the matrix. + second_system : CoordSys3D, optional, default=None + The coordinate system that the columns of the matrix correspond + to. + + Examples + ======== + + >>> from sympy.vector import CoordSys3D + >>> N = CoordSys3D('N') + >>> v = N.i + 2*N.j + >>> d = v.outer(N.i) + >>> d.to_matrix(N) + Matrix([ + [1, 0, 0], + [2, 0, 0], + [0, 0, 0]]) + >>> from sympy import Symbol + >>> q = Symbol('q') + >>> P = N.orient_new_axis('P', q, N.k) + >>> d.to_matrix(N, P) + Matrix([ + [ cos(q), -sin(q), 0], + [2*cos(q), -2*sin(q), 0], + [ 0, 0, 0]]) + + """ + + if second_system is None: + second_system = system + + return Matrix([i.dot(self).dot(j) for i in system for j in + second_system]).reshape(3, 3) + + def _div_helper(one, other): + """ Helper for division involving dyadics """ + if isinstance(one, Dyadic) and isinstance(other, Dyadic): + raise TypeError("Cannot divide two dyadics") + elif isinstance(one, Dyadic): + return DyadicMul(one, Pow(other, S.NegativeOne)) + else: + raise TypeError("Cannot divide by a dyadic") + + +class BaseDyadic(Dyadic, AtomicExpr): + """ + Class to denote a base dyadic tensor component. + """ + + def __new__(cls, vector1, vector2): + Vector = sympy.vector.Vector + BaseVector = sympy.vector.BaseVector + VectorZero = sympy.vector.VectorZero + # Verify arguments + if not isinstance(vector1, (BaseVector, VectorZero)) or \ + not isinstance(vector2, (BaseVector, VectorZero)): + raise TypeError("BaseDyadic cannot be composed of non-base " + + "vectors") + # Handle special case of zero vector + elif vector1 == Vector.zero or vector2 == Vector.zero: + return Dyadic.zero + # Initialize instance + obj = super().__new__(cls, vector1, vector2) + obj._base_instance = obj + obj._measure_number = 1 + obj._components = {obj: S.One} + obj._sys = vector1._sys + obj._pretty_form = ('(' + vector1._pretty_form + '|' + + vector2._pretty_form + ')') + obj._latex_form = (r'\left(' + vector1._latex_form + r"{\middle|}" + + vector2._latex_form + r'\right)') + + return obj + + def _sympystr(self, printer): + return "({}|{})".format( + printer._print(self.args[0]), printer._print(self.args[1])) + + def _sympyrepr(self, printer): + return "BaseDyadic({}, {})".format( + printer._print(self.args[0]), printer._print(self.args[1])) + + +class DyadicMul(BasisDependentMul, Dyadic): + """ Products of scalars and BaseDyadics """ + + def __new__(cls, *args, **options): + obj = BasisDependentMul.__new__(cls, *args, **options) + return obj + + @property + def base_dyadic(self): + """ The BaseDyadic involved in the product. """ + return self._base_instance + + @property + def measure_number(self): + """ The scalar expression involved in the definition of + this DyadicMul. + """ + return self._measure_number + + +class DyadicAdd(BasisDependentAdd, Dyadic): + """ Class to hold dyadic sums """ + + def __new__(cls, *args, **options): + obj = BasisDependentAdd.__new__(cls, *args, **options) + return obj + + def _sympystr(self, printer): + items = list(self.components.items()) + items.sort(key=lambda x: x[0].__str__()) + return " + ".join(printer._print(k * v) for k, v in items) + + +class DyadicZero(BasisDependentZero, Dyadic): + """ + Class to denote a zero dyadic + """ + + _op_priority = 13.1 + _pretty_form = '(0|0)' + _latex_form = r'(\mathbf{\hat{0}}|\mathbf{\hat{0}})' + + def __new__(cls): + obj = BasisDependentZero.__new__(cls) + return obj + + +Dyadic._expr_type = Dyadic +Dyadic._mul_func = DyadicMul +Dyadic._add_func = DyadicAdd +Dyadic._zero_func = DyadicZero +Dyadic._base_func = BaseDyadic +Dyadic.zero = DyadicZero() diff --git a/.venv/lib/python3.12/site-packages/sympy/vector/functions.py b/.venv/lib/python3.12/site-packages/sympy/vector/functions.py new file mode 100644 index 0000000000000000000000000000000000000000..b78df8ae2e182f3e571ca7fa8bfabd39bf99d26e --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/vector/functions.py @@ -0,0 +1,513 @@ +from sympy.vector.coordsysrect import CoordSys3D +from sympy.vector.deloperator import Del +from sympy.vector.scalar import BaseScalar +from sympy.vector.vector import Vector, BaseVector +from sympy.vector.operators import gradient, curl, divergence +from sympy.core.function import diff +from sympy.core.singleton import S +from sympy.integrals.integrals import integrate +from sympy.core import sympify +from sympy.vector.dyadic import Dyadic + + +def express(expr, system, system2=None, variables=False): + """ + Global function for 'express' functionality. + + Re-expresses a Vector, Dyadic or scalar(sympyfiable) in the given + coordinate system. + + If 'variables' is True, then the coordinate variables (base scalars) + of other coordinate systems present in the vector/scalar field or + dyadic are also substituted in terms of the base scalars of the + given system. + + Parameters + ========== + + expr : Vector/Dyadic/scalar(sympyfiable) + The expression to re-express in CoordSys3D 'system' + + system: CoordSys3D + The coordinate system the expr is to be expressed in + + system2: CoordSys3D + The other coordinate system required for re-expression + (only for a Dyadic Expr) + + variables : boolean + Specifies whether to substitute the coordinate variables present + in expr, in terms of those of parameter system + + Examples + ======== + + >>> from sympy.vector import CoordSys3D + >>> from sympy import Symbol, cos, sin + >>> N = CoordSys3D('N') + >>> q = Symbol('q') + >>> B = N.orient_new_axis('B', q, N.k) + >>> from sympy.vector import express + >>> express(B.i, N) + (cos(q))*N.i + (sin(q))*N.j + >>> express(N.x, B, variables=True) + B.x*cos(q) - B.y*sin(q) + >>> d = N.i.outer(N.i) + >>> express(d, B, N) == (cos(q))*(B.i|N.i) + (-sin(q))*(B.j|N.i) + True + + """ + + if expr in (0, Vector.zero): + return expr + + if not isinstance(system, CoordSys3D): + raise TypeError("system should be a CoordSys3D \ + instance") + + if isinstance(expr, Vector): + if system2 is not None: + raise ValueError("system2 should not be provided for \ + Vectors") + # Given expr is a Vector + if variables: + # If variables attribute is True, substitute + # the coordinate variables in the Vector + system_list = {x.system for x in expr.atoms(BaseScalar, BaseVector)} - {system} + subs_dict = {} + for f in system_list: + subs_dict.update(f.scalar_map(system)) + expr = expr.subs(subs_dict) + # Re-express in this coordinate system + outvec = Vector.zero + parts = expr.separate() + for x in parts: + if x != system: + temp = system.rotation_matrix(x) * parts[x].to_matrix(x) + outvec += matrix_to_vector(temp, system) + else: + outvec += parts[x] + return outvec + + elif isinstance(expr, Dyadic): + if system2 is None: + system2 = system + if not isinstance(system2, CoordSys3D): + raise TypeError("system2 should be a CoordSys3D \ + instance") + outdyad = Dyadic.zero + var = variables + for k, v in expr.components.items(): + outdyad += (express(v, system, variables=var) * + (express(k.args[0], system, variables=var) | + express(k.args[1], system2, variables=var))) + + return outdyad + + else: + if system2 is not None: + raise ValueError("system2 should not be provided for \ + Vectors") + if variables: + # Given expr is a scalar field + system_set = set() + expr = sympify(expr) + # Substitute all the coordinate variables + for x in expr.atoms(BaseScalar): + if x.system != system: + system_set.add(x.system) + subs_dict = {} + for f in system_set: + subs_dict.update(f.scalar_map(system)) + return expr.subs(subs_dict) + return expr + + +def directional_derivative(field, direction_vector): + """ + Returns the directional derivative of a scalar or vector field computed + along a given vector in coordinate system which parameters are expressed. + + Parameters + ========== + + field : Vector or Scalar + The scalar or vector field to compute the directional derivative of + + direction_vector : Vector + The vector to calculated directional derivative along them. + + + Examples + ======== + + >>> from sympy.vector import CoordSys3D, directional_derivative + >>> R = CoordSys3D('R') + >>> f1 = R.x*R.y*R.z + >>> v1 = 3*R.i + 4*R.j + R.k + >>> directional_derivative(f1, v1) + R.x*R.y + 4*R.x*R.z + 3*R.y*R.z + >>> f2 = 5*R.x**2*R.z + >>> directional_derivative(f2, v1) + 5*R.x**2 + 30*R.x*R.z + + """ + from sympy.vector.operators import _get_coord_systems + coord_sys = _get_coord_systems(field) + if len(coord_sys) > 0: + # TODO: This gets a random coordinate system in case of multiple ones: + coord_sys = next(iter(coord_sys)) + field = express(field, coord_sys, variables=True) + i, j, k = coord_sys.base_vectors() + x, y, z = coord_sys.base_scalars() + out = Vector.dot(direction_vector, i) * diff(field, x) + out += Vector.dot(direction_vector, j) * diff(field, y) + out += Vector.dot(direction_vector, k) * diff(field, z) + if out == 0 and isinstance(field, Vector): + out = Vector.zero + return out + elif isinstance(field, Vector): + return Vector.zero + else: + return S.Zero + + +def laplacian(expr): + """ + Return the laplacian of the given field computed in terms of + the base scalars of the given coordinate system. + + Parameters + ========== + + expr : SymPy Expr or Vector + expr denotes a scalar or vector field. + + Examples + ======== + + >>> from sympy.vector import CoordSys3D, laplacian + >>> R = CoordSys3D('R') + >>> f = R.x**2*R.y**5*R.z + >>> laplacian(f) + 20*R.x**2*R.y**3*R.z + 2*R.y**5*R.z + >>> f = R.x**2*R.i + R.y**3*R.j + R.z**4*R.k + >>> laplacian(f) + 2*R.i + 6*R.y*R.j + 12*R.z**2*R.k + + """ + + delop = Del() + if expr.is_Vector: + return (gradient(divergence(expr)) - curl(curl(expr))).doit() + return delop.dot(delop(expr)).doit() + + +def is_conservative(field): + """ + Checks if a field is conservative. + + Parameters + ========== + + field : Vector + The field to check for conservative property + + Examples + ======== + + >>> from sympy.vector import CoordSys3D + >>> from sympy.vector import is_conservative + >>> R = CoordSys3D('R') + >>> is_conservative(R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k) + True + >>> is_conservative(R.z*R.j) + False + + """ + + # Field is conservative irrespective of system + # Take the first coordinate system in the result of the + # separate method of Vector + if not isinstance(field, Vector): + raise TypeError("field should be a Vector") + if field == Vector.zero: + return True + return curl(field).simplify() == Vector.zero + + +def is_solenoidal(field): + """ + Checks if a field is solenoidal. + + Parameters + ========== + + field : Vector + The field to check for solenoidal property + + Examples + ======== + + >>> from sympy.vector import CoordSys3D + >>> from sympy.vector import is_solenoidal + >>> R = CoordSys3D('R') + >>> is_solenoidal(R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k) + True + >>> is_solenoidal(R.y * R.j) + False + + """ + + # Field is solenoidal irrespective of system + # Take the first coordinate system in the result of the + # separate method in Vector + if not isinstance(field, Vector): + raise TypeError("field should be a Vector") + if field == Vector.zero: + return True + return divergence(field).simplify() is S.Zero + + +def scalar_potential(field, coord_sys): + """ + Returns the scalar potential function of a field in a given + coordinate system (without the added integration constant). + + Parameters + ========== + + field : Vector + The vector field whose scalar potential function is to be + calculated + + coord_sys : CoordSys3D + The coordinate system to do the calculation in + + Examples + ======== + + >>> from sympy.vector import CoordSys3D + >>> from sympy.vector import scalar_potential, gradient + >>> R = CoordSys3D('R') + >>> scalar_potential(R.k, R) == R.z + True + >>> scalar_field = 2*R.x**2*R.y*R.z + >>> grad_field = gradient(scalar_field) + >>> scalar_potential(grad_field, R) + 2*R.x**2*R.y*R.z + + """ + + # Check whether field is conservative + if not is_conservative(field): + raise ValueError("Field is not conservative") + if field == Vector.zero: + return S.Zero + # Express the field exntirely in coord_sys + # Substitute coordinate variables also + if not isinstance(coord_sys, CoordSys3D): + raise TypeError("coord_sys must be a CoordSys3D") + field = express(field, coord_sys, variables=True) + dimensions = coord_sys.base_vectors() + scalars = coord_sys.base_scalars() + # Calculate scalar potential function + temp_function = integrate(field.dot(dimensions[0]), scalars[0]) + for i, dim in enumerate(dimensions[1:]): + partial_diff = diff(temp_function, scalars[i + 1]) + partial_diff = field.dot(dim) - partial_diff + temp_function += integrate(partial_diff, scalars[i + 1]) + return temp_function + + +def scalar_potential_difference(field, coord_sys, point1, point2): + """ + Returns the scalar potential difference between two points in a + certain coordinate system, wrt a given field. + + If a scalar field is provided, its values at the two points are + considered. If a conservative vector field is provided, the values + of its scalar potential function at the two points are used. + + Returns (potential at point2) - (potential at point1) + + The position vectors of the two Points are calculated wrt the + origin of the coordinate system provided. + + Parameters + ========== + + field : Vector/Expr + The field to calculate wrt + + coord_sys : CoordSys3D + The coordinate system to do the calculations in + + point1 : Point + The initial Point in given coordinate system + + position2 : Point + The second Point in the given coordinate system + + Examples + ======== + + >>> from sympy.vector import CoordSys3D + >>> from sympy.vector import scalar_potential_difference + >>> R = CoordSys3D('R') + >>> P = R.origin.locate_new('P', R.x*R.i + R.y*R.j + R.z*R.k) + >>> vectfield = 4*R.x*R.y*R.i + 2*R.x**2*R.j + >>> scalar_potential_difference(vectfield, R, R.origin, P) + 2*R.x**2*R.y + >>> Q = R.origin.locate_new('O', 3*R.i + R.j + 2*R.k) + >>> scalar_potential_difference(vectfield, R, P, Q) + -2*R.x**2*R.y + 18 + + """ + + if not isinstance(coord_sys, CoordSys3D): + raise TypeError("coord_sys must be a CoordSys3D") + if isinstance(field, Vector): + # Get the scalar potential function + scalar_fn = scalar_potential(field, coord_sys) + else: + # Field is a scalar + scalar_fn = field + # Express positions in required coordinate system + origin = coord_sys.origin + position1 = express(point1.position_wrt(origin), coord_sys, + variables=True) + position2 = express(point2.position_wrt(origin), coord_sys, + variables=True) + # Get the two positions as substitution dicts for coordinate variables + subs_dict1 = {} + subs_dict2 = {} + scalars = coord_sys.base_scalars() + for i, x in enumerate(coord_sys.base_vectors()): + subs_dict1[scalars[i]] = x.dot(position1) + subs_dict2[scalars[i]] = x.dot(position2) + return scalar_fn.subs(subs_dict2) - scalar_fn.subs(subs_dict1) + + +def matrix_to_vector(matrix, system): + """ + Converts a vector in matrix form to a Vector instance. + + It is assumed that the elements of the Matrix represent the + measure numbers of the components of the vector along basis + vectors of 'system'. + + Parameters + ========== + + matrix : SymPy Matrix, Dimensions: (3, 1) + The matrix to be converted to a vector + + system : CoordSys3D + The coordinate system the vector is to be defined in + + Examples + ======== + + >>> from sympy import ImmutableMatrix as Matrix + >>> m = Matrix([1, 2, 3]) + >>> from sympy.vector import CoordSys3D, matrix_to_vector + >>> C = CoordSys3D('C') + >>> v = matrix_to_vector(m, C) + >>> v + C.i + 2*C.j + 3*C.k + >>> v.to_matrix(C) == m + True + + """ + + outvec = Vector.zero + vects = system.base_vectors() + for i, x in enumerate(matrix): + outvec += x * vects[i] + return outvec + + +def _path(from_object, to_object): + """ + Calculates the 'path' of objects starting from 'from_object' + to 'to_object', along with the index of the first common + ancestor in the tree. + + Returns (index, list) tuple. + """ + + if from_object._root != to_object._root: + raise ValueError("No connecting path found between " + + str(from_object) + " and " + str(to_object)) + + other_path = [] + obj = to_object + while obj._parent is not None: + other_path.append(obj) + obj = obj._parent + other_path.append(obj) + object_set = set(other_path) + from_path = [] + obj = from_object + while obj not in object_set: + from_path.append(obj) + obj = obj._parent + index = len(from_path) + from_path.extend(other_path[other_path.index(obj)::-1]) + return index, from_path + + +def orthogonalize(*vlist, orthonormal=False): + """ + Takes a sequence of independent vectors and orthogonalizes them + using the Gram - Schmidt process. Returns a list of + orthogonal or orthonormal vectors. + + Parameters + ========== + + vlist : sequence of independent vectors to be made orthogonal. + + orthonormal : Optional parameter + Set to True if the vectors returned should be + orthonormal. + Default: False + + Examples + ======== + + >>> from sympy.vector.coordsysrect import CoordSys3D + >>> from sympy.vector.functions import orthogonalize + >>> C = CoordSys3D('C') + >>> i, j, k = C.base_vectors() + >>> v1 = i + 2*j + >>> v2 = 2*i + 3*j + >>> orthogonalize(v1, v2) + [C.i + 2*C.j, 2/5*C.i + (-1/5)*C.j] + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Gram-Schmidt_process + + """ + + if not all(isinstance(vec, Vector) for vec in vlist): + raise TypeError('Each element must be of Type Vector') + + ortho_vlist = [] + for i, term in enumerate(vlist): + for j in range(i): + term -= ortho_vlist[j].projection(vlist[i]) + # TODO : The following line introduces a performance issue + # and needs to be changed once a good solution for issue #10279 is + # found. + if term.equals(Vector.zero): + raise ValueError("Vector set not linearly independent") + ortho_vlist.append(term) + + if orthonormal: + ortho_vlist = [vec.normalize() for vec in ortho_vlist] + + return ortho_vlist diff --git a/.venv/lib/python3.12/site-packages/sympy/vector/implicitregion.py b/.venv/lib/python3.12/site-packages/sympy/vector/implicitregion.py new file mode 100644 index 0000000000000000000000000000000000000000..ed2d55a1be8b1eaca71d08b632a94886a2b0269c --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/vector/implicitregion.py @@ -0,0 +1,506 @@ +from sympy.core.numbers import Rational +from sympy.core.singleton import S +from sympy.core.symbol import symbols +from sympy.functions.elementary.complexes import sign +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.polys.polytools import gcd +from sympy.sets.sets import Complement +from sympy.core import Basic, Tuple, diff, expand, Eq, Integer +from sympy.core.sorting import ordered +from sympy.core.symbol import _symbol +from sympy.solvers import solveset, nonlinsolve, diophantine +from sympy.polys import total_degree +from sympy.geometry import Point +from sympy.ntheory.factor_ import core + + +class ImplicitRegion(Basic): + """ + Represents an implicit region in space. + + Examples + ======== + + >>> from sympy import Eq + >>> from sympy.abc import x, y, z, t + >>> from sympy.vector import ImplicitRegion + + >>> ImplicitRegion((x, y), x**2 + y**2 - 4) + ImplicitRegion((x, y), x**2 + y**2 - 4) + >>> ImplicitRegion((x, y), Eq(y*x, 1)) + ImplicitRegion((x, y), x*y - 1) + + >>> parabola = ImplicitRegion((x, y), y**2 - 4*x) + >>> parabola.degree + 2 + >>> parabola.equation + -4*x + y**2 + >>> parabola.rational_parametrization(t) + (4/t**2, 4/t) + + >>> r = ImplicitRegion((x, y, z), Eq(z, x**2 + y**2)) + >>> r.variables + (x, y, z) + >>> r.singular_points() + EmptySet + >>> r.regular_point() + (-10, -10, 200) + + Parameters + ========== + + variables : tuple to map variables in implicit equation to base scalars. + + equation : An expression or Eq denoting the implicit equation of the region. + + """ + def __new__(cls, variables, equation): + if not isinstance(variables, Tuple): + variables = Tuple(*variables) + + if isinstance(equation, Eq): + equation = equation.lhs - equation.rhs + + return super().__new__(cls, variables, equation) + + @property + def variables(self): + return self.args[0] + + @property + def equation(self): + return self.args[1] + + @property + def degree(self): + return total_degree(self.equation) + + def regular_point(self): + """ + Returns a point on the implicit region. + + Examples + ======== + + >>> from sympy.abc import x, y, z + >>> from sympy.vector import ImplicitRegion + >>> circle = ImplicitRegion((x, y), (x + 2)**2 + (y - 3)**2 - 16) + >>> circle.regular_point() + (-2, -1) + >>> parabola = ImplicitRegion((x, y), x**2 - 4*y) + >>> parabola.regular_point() + (0, 0) + >>> r = ImplicitRegion((x, y, z), (x + y + z)**4) + >>> r.regular_point() + (-10, -10, 20) + + References + ========== + + - Erik Hillgarter, "Rational Points on Conics", Diploma Thesis, RISC-Linz, + J. Kepler Universitat Linz, 1996. Available: + https://www3.risc.jku.at/publications/download/risc_1355/Rational%20Points%20on%20Conics.pdf + + """ + equation = self.equation + + if len(self.variables) == 1: + return (list(solveset(equation, self.variables[0], domain=S.Reals))[0],) + elif len(self.variables) == 2: + + if self.degree == 2: + coeffs = a, b, c, d, e, f = conic_coeff(self.variables, equation) + + if b**2 == 4*a*c: + x_reg, y_reg = self._regular_point_parabola(*coeffs) + else: + x_reg, y_reg = self._regular_point_ellipse(*coeffs) + return x_reg, y_reg + + if len(self.variables) == 3: + x, y, z = self.variables + + for x_reg in range(-10, 10): + for y_reg in range(-10, 10): + if not solveset(equation.subs({x: x_reg, y: y_reg}), self.variables[2], domain=S.Reals).is_empty: + return (x_reg, y_reg, list(solveset(equation.subs({x: x_reg, y: y_reg})))[0]) + + if len(self.singular_points()) != 0: + return list[self.singular_points()][0] + + raise NotImplementedError() + + def _regular_point_parabola(self, a, b, c, d, e, f): + ok = (a, d) != (0, 0) and (c, e) != (0, 0) and b**2 == 4*a*c and (a, c) != (0, 0) + + if not ok: + raise ValueError("Rational Point on the conic does not exist") + + if a != 0: + d_dash, f_dash = (4*a*e - 2*b*d, 4*a*f - d**2) + if d_dash != 0: + y_reg = -f_dash/d_dash + x_reg = -(d + b*y_reg)/(2*a) + else: + ok = False + elif c != 0: + d_dash, f_dash = (4*c*d - 2*b*e, 4*c*f - e**2) + if d_dash != 0: + x_reg = -f_dash/d_dash + y_reg = -(e + b*x_reg)/(2*c) + else: + ok = False + + if ok: + return x_reg, y_reg + else: + raise ValueError("Rational Point on the conic does not exist") + + def _regular_point_ellipse(self, a, b, c, d, e, f): + D = 4*a*c - b**2 + ok = D + + if not ok: + raise ValueError("Rational Point on the conic does not exist") + + if a == 0 and c == 0: + K = -1 + L = 4*(d*e - b*f) + elif c != 0: + K = D + L = 4*c**2*d**2 - 4*b*c*d*e + 4*a*c*e**2 + 4*b**2*c*f - 16*a*c**2*f + else: + K = D + L = 4*a**2*e**2 - 4*b*a*d*e + 4*b**2*a*f + + ok = L != 0 and not(K > 0 and L < 0) + if not ok: + raise ValueError("Rational Point on the conic does not exist") + + K = Rational(K).limit_denominator(10**12) + L = Rational(L).limit_denominator(10**12) + + k1, k2 = K.p, K.q + l1, l2 = L.p, L.q + g = gcd(k2, l2) + + a1 = (l2*k2)/g + b1 = (k1*l2)/g + c1 = -(l1*k2)/g + a2 = sign(a1)*core(abs(a1), 2) + r1 = sqrt(a1/a2) + b2 = sign(b1)*core(abs(b1), 2) + r2 = sqrt(b1/b2) + c2 = sign(c1)*core(abs(c1), 2) + r3 = sqrt(c1/c2) + + g = gcd(gcd(a2, b2), c2) + a2 = a2/g + b2 = b2/g + c2 = c2/g + + g1 = gcd(a2, b2) + a2 = a2/g1 + b2 = b2/g1 + c2 = c2*g1 + + g2 = gcd(a2,c2) + a2 = a2/g2 + b2 = b2*g2 + c2 = c2/g2 + + g3 = gcd(b2, c2) + a2 = a2*g3 + b2 = b2/g3 + c2 = c2/g3 + + x, y, z = symbols("x y z") + eq = a2*x**2 + b2*y**2 + c2*z**2 + + solutions = diophantine(eq) + + if len(solutions) == 0: + raise ValueError("Rational Point on the conic does not exist") + + flag = False + for sol in solutions: + syms = Tuple(*sol).free_symbols + rep = dict.fromkeys(syms, 3) + sol_z = sol[2] + + if sol_z == 0: + flag = True + continue + + if not isinstance(sol_z, (int, Integer)): + syms_z = sol_z.free_symbols + + if len(syms_z) == 1: + p = next(iter(syms_z)) + p_values = Complement(S.Integers, solveset(Eq(sol_z, 0), p, S.Integers)) + rep[p] = next(iter(p_values)) + + if len(syms_z) == 2: + p, q = list(ordered(syms_z)) + + for i in S.Integers: + subs_sol_z = sol_z.subs(p, i) + q_values = Complement(S.Integers, solveset(Eq(subs_sol_z, 0), q, S.Integers)) + + if not q_values.is_empty: + rep[p] = i + rep[q] = next(iter(q_values)) + break + + if len(syms) != 0: + x, y, z = tuple(s.subs(rep) for s in sol) + else: + x, y, z = sol + flag = False + break + + if flag: + raise ValueError("Rational Point on the conic does not exist") + + x = (x*g3)/r1 + y = (y*g2)/r2 + z = (z*g1)/r3 + x = x/z + y = y/z + + if a == 0 and c == 0: + x_reg = (x + y - 2*e)/(2*b) + y_reg = (x - y - 2*d)/(2*b) + elif c != 0: + x_reg = (x - 2*d*c + b*e)/K + y_reg = (y - b*x_reg - e)/(2*c) + else: + y_reg = (x - 2*e*a + b*d)/K + x_reg = (y - b*y_reg - d)/(2*a) + + return x_reg, y_reg + + def singular_points(self): + """ + Returns a set of singular points of the region. + + The singular points are those points on the region + where all partial derivatives vanish. + + Examples + ======== + + >>> from sympy.abc import x, y + >>> from sympy.vector import ImplicitRegion + >>> I = ImplicitRegion((x, y), (y-1)**2 -x**3 + 2*x**2 -x) + >>> I.singular_points() + {(1, 1)} + + """ + eq_list = [self.equation] + for var in self.variables: + eq_list += [diff(self.equation, var)] + + return nonlinsolve(eq_list, list(self.variables)) + + def multiplicity(self, point): + """ + Returns the multiplicity of a singular point on the region. + + A singular point (x,y) of region is said to be of multiplicity m + if all the partial derivatives off to order m - 1 vanish there. + + Examples + ======== + + >>> from sympy.abc import x, y, z + >>> from sympy.vector import ImplicitRegion + >>> I = ImplicitRegion((x, y, z), x**2 + y**3 - z**4) + >>> I.singular_points() + {(0, 0, 0)} + >>> I.multiplicity((0, 0, 0)) + 2 + + """ + if isinstance(point, Point): + point = point.args + + modified_eq = self.equation + + for i, var in enumerate(self.variables): + modified_eq = modified_eq.subs(var, var + point[i]) + modified_eq = expand(modified_eq) + + if len(modified_eq.args) != 0: + terms = modified_eq.args + m = min(total_degree(term) for term in terms) + else: + terms = modified_eq + m = total_degree(terms) + + return m + + def rational_parametrization(self, parameters=('t', 's'), reg_point=None): + """ + Returns the rational parametrization of implicit region. + + Examples + ======== + + >>> from sympy import Eq + >>> from sympy.abc import x, y, z, s, t + >>> from sympy.vector import ImplicitRegion + + >>> parabola = ImplicitRegion((x, y), y**2 - 4*x) + >>> parabola.rational_parametrization() + (4/t**2, 4/t) + + >>> circle = ImplicitRegion((x, y), Eq(x**2 + y**2, 4)) + >>> circle.rational_parametrization() + (4*t/(t**2 + 1), 4*t**2/(t**2 + 1) - 2) + + >>> I = ImplicitRegion((x, y), x**3 + x**2 - y**2) + >>> I.rational_parametrization() + (t**2 - 1, t*(t**2 - 1)) + + >>> cubic_curve = ImplicitRegion((x, y), x**3 + x**2 - y**2) + >>> cubic_curve.rational_parametrization(parameters=(t)) + (t**2 - 1, t*(t**2 - 1)) + + >>> sphere = ImplicitRegion((x, y, z), x**2 + y**2 + z**2 - 4) + >>> sphere.rational_parametrization(parameters=(t, s)) + (-2 + 4/(s**2 + t**2 + 1), 4*s/(s**2 + t**2 + 1), 4*t/(s**2 + t**2 + 1)) + + For some conics, regular_points() is unable to find a point on curve. + To calulcate the parametric representation in such cases, user need + to determine a point on the region and pass it using reg_point. + + >>> c = ImplicitRegion((x, y), (x - 1/2)**2 + (y)**2 - (1/4)**2) + >>> c.rational_parametrization(reg_point=(3/4, 0)) + (0.75 - 0.5/(t**2 + 1), -0.5*t/(t**2 + 1)) + + References + ========== + + - Christoph M. Hoffmann, "Conversion Methods between Parametric and + Implicit Curves and Surfaces", Purdue e-Pubs, 1990. Available: + https://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=1827&context=cstech + + """ + equation = self.equation + degree = self.degree + + if degree == 1: + if len(self.variables) == 1: + return (equation,) + elif len(self.variables) == 2: + x, y = self.variables + y_par = list(solveset(equation, y))[0] + return x, y_par + else: + raise NotImplementedError() + + point = () + + # Finding the (n - 1) fold point of the monoid of degree + if degree == 2: + # For degree 2 curves, either a regular point or a singular point can be used. + if reg_point is not None: + # Using point provided by the user as regular point + point = reg_point + else: + if len(self.singular_points()) != 0: + point = list(self.singular_points())[0] + else: + point = self.regular_point() + + if len(self.singular_points()) != 0: + singular_points = self.singular_points() + for spoint in singular_points: + syms = Tuple(*spoint).free_symbols + rep = dict.fromkeys(syms, 2) + + if len(syms) != 0: + spoint = tuple(s.subs(rep) for s in spoint) + + if self.multiplicity(spoint) == degree - 1: + point = spoint + break + + if len(point) == 0: + # The region in not a monoid + raise NotImplementedError() + + modified_eq = equation + + # Shifting the region such that fold point moves to origin + for i, var in enumerate(self.variables): + modified_eq = modified_eq.subs(var, var + point[i]) + modified_eq = expand(modified_eq) + + hn = hn_1 = 0 + for term in modified_eq.args: + if total_degree(term) == degree: + hn += term + else: + hn_1 += term + + hn_1 = -1*hn_1 + + if not isinstance(parameters, tuple): + parameters = (parameters,) + + if len(self.variables) == 2: + + parameter1 = parameters[0] + if parameter1 == 's': + # To avoid name conflict between parameters + s = _symbol('s_', real=True) + else: + s = _symbol('s', real=True) + t = _symbol(parameter1, real=True) + + hn = hn.subs({self.variables[0]: s, self.variables[1]: t}) + hn_1 = hn_1.subs({self.variables[0]: s, self.variables[1]: t}) + + x_par = (s*(hn_1/hn)).subs(s, 1) + point[0] + y_par = (t*(hn_1/hn)).subs(s, 1) + point[1] + + return x_par, y_par + + elif len(self.variables) == 3: + + parameter1, parameter2 = parameters + if 'r' in parameters: + # To avoid name conflict between parameters + r = _symbol('r_', real=True) + else: + r = _symbol('r', real=True) + s = _symbol(parameter2, real=True) + t = _symbol(parameter1, real=True) + + hn = hn.subs({self.variables[0]: r, self.variables[1]: s, self.variables[2]: t}) + hn_1 = hn_1.subs({self.variables[0]: r, self.variables[1]: s, self.variables[2]: t}) + + x_par = (r*(hn_1/hn)).subs(r, 1) + point[0] + y_par = (s*(hn_1/hn)).subs(r, 1) + point[1] + z_par = (t*(hn_1/hn)).subs(r, 1) + point[2] + + return x_par, y_par, z_par + + raise NotImplementedError() + +def conic_coeff(variables, equation): + if total_degree(equation) != 2: + raise ValueError() + x = variables[0] + y = variables[1] + + equation = expand(equation) + a = equation.coeff(x**2) + b = equation.coeff(x*y) + c = equation.coeff(y**2) + d = equation.coeff(x, 1).coeff(y, 0) + e = equation.coeff(y, 1).coeff(x, 0) + f = equation.coeff(x, 0).coeff(y, 0) + return a, b, c, d, e, f diff --git a/.venv/lib/python3.12/site-packages/sympy/vector/integrals.py b/.venv/lib/python3.12/site-packages/sympy/vector/integrals.py new file mode 100644 index 0000000000000000000000000000000000000000..a6451c182f214b20b1105eb0a4dc243455c9d126 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/vector/integrals.py @@ -0,0 +1,206 @@ +from sympy.core import Basic, diff +from sympy.core.singleton import S +from sympy.core.sorting import default_sort_key +from sympy.matrices import Matrix +from sympy.integrals import Integral, integrate +from sympy.geometry.entity import GeometryEntity +from sympy.simplify.simplify import simplify +from sympy.utilities.iterables import topological_sort +from sympy.vector import (CoordSys3D, Vector, ParametricRegion, + parametric_region_list, ImplicitRegion) +from sympy.vector.operators import _get_coord_systems + + +class ParametricIntegral(Basic): + """ + Represents integral of a scalar or vector field + over a Parametric Region + + Examples + ======== + + >>> from sympy import cos, sin, pi + >>> from sympy.vector import CoordSys3D, ParametricRegion, ParametricIntegral + >>> from sympy.abc import r, t, theta, phi + + >>> C = CoordSys3D('C') + >>> curve = ParametricRegion((3*t - 2, t + 1), (t, 1, 2)) + >>> ParametricIntegral(C.x, curve) + 5*sqrt(10)/2 + >>> length = ParametricIntegral(1, curve) + >>> length + sqrt(10) + >>> semisphere = ParametricRegion((2*sin(phi)*cos(theta), 2*sin(phi)*sin(theta), 2*cos(phi)),\ + (theta, 0, 2*pi), (phi, 0, pi/2)) + >>> ParametricIntegral(C.z, semisphere) + 8*pi + + >>> ParametricIntegral(C.j + C.k, ParametricRegion((r*cos(theta), r*sin(theta)), r, theta)) + 0 + + """ + + def __new__(cls, field, parametricregion): + + coord_set = _get_coord_systems(field) + + if len(coord_set) == 0: + coord_sys = CoordSys3D('C') + elif len(coord_set) > 1: + raise ValueError + else: + coord_sys = next(iter(coord_set)) + + if parametricregion.dimensions == 0: + return S.Zero + + base_vectors = coord_sys.base_vectors() + base_scalars = coord_sys.base_scalars() + + parametricfield = field + + r = Vector.zero + for i in range(len(parametricregion.definition)): + r += base_vectors[i]*parametricregion.definition[i] + + if len(coord_set) != 0: + for i in range(len(parametricregion.definition)): + parametricfield = parametricfield.subs(base_scalars[i], parametricregion.definition[i]) + + if parametricregion.dimensions == 1: + parameter = parametricregion.parameters[0] + + r_diff = diff(r, parameter) + lower, upper = parametricregion.limits[parameter][0], parametricregion.limits[parameter][1] + + if isinstance(parametricfield, Vector): + integrand = simplify(r_diff.dot(parametricfield)) + else: + integrand = simplify(r_diff.magnitude()*parametricfield) + + result = integrate(integrand, (parameter, lower, upper)) + + elif parametricregion.dimensions == 2: + u, v = cls._bounds_case(parametricregion.parameters, parametricregion.limits) + + r_u = diff(r, u) + r_v = diff(r, v) + normal_vector = simplify(r_u.cross(r_v)) + + if isinstance(parametricfield, Vector): + integrand = parametricfield.dot(normal_vector) + else: + integrand = parametricfield*normal_vector.magnitude() + + integrand = simplify(integrand) + + lower_u, upper_u = parametricregion.limits[u][0], parametricregion.limits[u][1] + lower_v, upper_v = parametricregion.limits[v][0], parametricregion.limits[v][1] + + result = integrate(integrand, (u, lower_u, upper_u), (v, lower_v, upper_v)) + + else: + variables = cls._bounds_case(parametricregion.parameters, parametricregion.limits) + coeff = Matrix(parametricregion.definition).jacobian(variables).det() + integrand = simplify(parametricfield*coeff) + + l = [(var, parametricregion.limits[var][0], parametricregion.limits[var][1]) for var in variables] + result = integrate(integrand, *l) + + if not isinstance(result, Integral): + return result + else: + return super().__new__(cls, field, parametricregion) + + @classmethod + def _bounds_case(cls, parameters, limits): + + V = list(limits.keys()) + E = [] + + for p in V: + lower_p = limits[p][0] + upper_p = limits[p][1] + + lower_p = lower_p.atoms() + upper_p = upper_p.atoms() + E.extend((p, q) for q in V if p != q and + (lower_p.issuperset({q}) or upper_p.issuperset({q}))) + + if not E: + return parameters + else: + return topological_sort((V, E), key=default_sort_key) + + @property + def field(self): + return self.args[0] + + @property + def parametricregion(self): + return self.args[1] + + +def vector_integrate(field, *region): + """ + Compute the integral of a vector/scalar field + over a a region or a set of parameters. + + Examples + ======== + >>> from sympy.vector import CoordSys3D, ParametricRegion, vector_integrate + >>> from sympy.abc import x, y, t + >>> C = CoordSys3D('C') + + >>> region = ParametricRegion((t, t**2), (t, 1, 5)) + >>> vector_integrate(C.x*C.i, region) + 12 + + Integrals over some objects of geometry module can also be calculated. + + >>> from sympy.geometry import Point, Circle, Triangle + >>> c = Circle(Point(0, 2), 5) + >>> vector_integrate(C.x**2 + C.y**2, c) + 290*pi + >>> triangle = Triangle(Point(-2, 3), Point(2, 3), Point(0, 5)) + >>> vector_integrate(3*C.x**2*C.y*C.i + C.j, triangle) + -8 + + Integrals over some simple implicit regions can be computed. But in most cases, + it takes too long to compute over them. This is due to the expressions of parametric + representation becoming large. + + >>> from sympy.vector import ImplicitRegion + >>> c2 = ImplicitRegion((x, y), (x - 2)**2 + (y - 1)**2 - 9) + >>> vector_integrate(1, c2) + 6*pi + + Integral of fields with respect to base scalars: + + >>> vector_integrate(12*C.y**3, (C.y, 1, 3)) + 240 + >>> vector_integrate(C.x**2*C.z, C.x) + C.x**3*C.z/3 + >>> vector_integrate(C.x*C.i - C.y*C.k, C.x) + (Integral(C.x, C.x))*C.i + (Integral(-C.y, C.x))*C.k + >>> _.doit() + C.x**2/2*C.i + (-C.x*C.y)*C.k + + """ + if len(region) == 1: + if isinstance(region[0], ParametricRegion): + return ParametricIntegral(field, region[0]) + + if isinstance(region[0], ImplicitRegion): + region = parametric_region_list(region[0])[0] + return vector_integrate(field, region) + + if isinstance(region[0], GeometryEntity): + regions_list = parametric_region_list(region[0]) + + result = 0 + for reg in regions_list: + result += vector_integrate(field, reg) + return result + + return integrate(field, *region) diff --git a/.venv/lib/python3.12/site-packages/sympy/vector/kind.py b/.venv/lib/python3.12/site-packages/sympy/vector/kind.py new file mode 100644 index 0000000000000000000000000000000000000000..c6c04896b34c9c92c3fb340d94985df859e5877d --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/vector/kind.py @@ -0,0 +1,67 @@ +#sympy.vector.kind + +from sympy.core.kind import Kind, _NumberKind, NumberKind +from sympy.core.mul import Mul + +class VectorKind(Kind): + """ + Kind for all vector objects in SymPy. + + Parameters + ========== + + element_kind : Kind + Kind of the element. Default is + :class:`sympy.core.kind.NumberKind`, + which means that the vector contains only numbers. + + Examples + ======== + + Any instance of Vector class has kind ``VectorKind``: + + >>> from sympy.vector.coordsysrect import CoordSys3D + >>> Sys = CoordSys3D('Sys') + >>> Sys.i.kind + VectorKind(NumberKind) + + Operations between instances of Vector keep also have the kind ``VectorKind``: + + >>> from sympy.core.add import Add + >>> v1 = Sys.i * 2 + Sys.j * 3 + Sys.k * 4 + >>> v2 = Sys.i * Sys.x + Sys.j * Sys.y + Sys.k * Sys.z + >>> v1.kind + VectorKind(NumberKind) + >>> v2.kind + VectorKind(NumberKind) + >>> Add(v1, v2).kind + VectorKind(NumberKind) + + Subclasses of Vector also have the kind ``VectorKind``, such as + Cross, VectorAdd, VectorMul or VectorZero. + + See Also + ======== + + sympy.core.kind.Kind + sympy.matrices.kind.MatrixKind + + """ + def __new__(cls, element_kind=NumberKind): + obj = super().__new__(cls, element_kind) + obj.element_kind = element_kind + return obj + + def __repr__(self): + return "VectorKind(%s)" % self.element_kind + +@Mul._kind_dispatcher.register(_NumberKind, VectorKind) +def num_vec_mul(k1, k2): + """ + The result of a multiplication between a number and a Vector should be of VectorKind. + The element kind is selected by recursive dispatching. + """ + if not isinstance(k2, VectorKind): + k1, k2 = k2, k1 + elemk = Mul._kind_dispatcher(k1, k2.element_kind) + return VectorKind(elemk) diff --git a/.venv/lib/python3.12/site-packages/sympy/vector/operators.py b/.venv/lib/python3.12/site-packages/sympy/vector/operators.py new file mode 100644 index 0000000000000000000000000000000000000000..3ca42d20302f972cef66e0b4a35ac75606b2da94 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/vector/operators.py @@ -0,0 +1,335 @@ +import collections +from sympy.core.expr import Expr +from sympy.core import sympify, S, preorder_traversal +from sympy.vector.coordsysrect import CoordSys3D +from sympy.vector.vector import Vector, VectorMul, VectorAdd, Cross, Dot +from sympy.core.function import Derivative +from sympy.core.add import Add +from sympy.core.mul import Mul + + +def _get_coord_systems(expr): + g = preorder_traversal(expr) + ret = set() + for i in g: + if isinstance(i, CoordSys3D): + ret.add(i) + g.skip() + return frozenset(ret) + + +def _split_mul_args_wrt_coordsys(expr): + d = collections.defaultdict(lambda: S.One) + for i in expr.args: + d[_get_coord_systems(i)] *= i + return list(d.values()) + + +class Gradient(Expr): + """ + Represents unevaluated Gradient. + + Examples + ======== + + >>> from sympy.vector import CoordSys3D, Gradient + >>> R = CoordSys3D('R') + >>> s = R.x*R.y*R.z + >>> Gradient(s) + Gradient(R.x*R.y*R.z) + + """ + + def __new__(cls, expr): + expr = sympify(expr) + obj = Expr.__new__(cls, expr) + obj._expr = expr + return obj + + def doit(self, **hints): + return gradient(self._expr, doit=True) + + +class Divergence(Expr): + """ + Represents unevaluated Divergence. + + Examples + ======== + + >>> from sympy.vector import CoordSys3D, Divergence + >>> R = CoordSys3D('R') + >>> v = R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k + >>> Divergence(v) + Divergence(R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k) + + """ + + def __new__(cls, expr): + expr = sympify(expr) + obj = Expr.__new__(cls, expr) + obj._expr = expr + return obj + + def doit(self, **hints): + return divergence(self._expr, doit=True) + + +class Curl(Expr): + """ + Represents unevaluated Curl. + + Examples + ======== + + >>> from sympy.vector import CoordSys3D, Curl + >>> R = CoordSys3D('R') + >>> v = R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k + >>> Curl(v) + Curl(R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k) + + """ + + def __new__(cls, expr): + expr = sympify(expr) + obj = Expr.__new__(cls, expr) + obj._expr = expr + return obj + + def doit(self, **hints): + return curl(self._expr, doit=True) + + +def curl(vect, doit=True): + """ + Returns the curl of a vector field computed wrt the base scalars + of the given coordinate system. + + Parameters + ========== + + vect : Vector + The vector operand + + doit : bool + If True, the result is returned after calling .doit() on + each component. Else, the returned expression contains + Derivative instances + + Examples + ======== + + >>> from sympy.vector import CoordSys3D, curl + >>> R = CoordSys3D('R') + >>> v1 = R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k + >>> curl(v1) + 0 + >>> v2 = R.x*R.y*R.z*R.i + >>> curl(v2) + R.x*R.y*R.j + (-R.x*R.z)*R.k + + """ + + coord_sys = _get_coord_systems(vect) + + if len(coord_sys) == 0: + return Vector.zero + elif len(coord_sys) == 1: + coord_sys = next(iter(coord_sys)) + i, j, k = coord_sys.base_vectors() + x, y, z = coord_sys.base_scalars() + h1, h2, h3 = coord_sys.lame_coefficients() + vectx = vect.dot(i) + vecty = vect.dot(j) + vectz = vect.dot(k) + outvec = Vector.zero + outvec += (Derivative(vectz * h3, y) - + Derivative(vecty * h2, z)) * i / (h2 * h3) + outvec += (Derivative(vectx * h1, z) - + Derivative(vectz * h3, x)) * j / (h1 * h3) + outvec += (Derivative(vecty * h2, x) - + Derivative(vectx * h1, y)) * k / (h2 * h1) + + if doit: + return outvec.doit() + return outvec + else: + if isinstance(vect, (Add, VectorAdd)): + from sympy.vector import express + try: + cs = next(iter(coord_sys)) + args = [express(i, cs, variables=True) for i in vect.args] + except ValueError: + args = vect.args + return VectorAdd.fromiter(curl(i, doit=doit) for i in args) + elif isinstance(vect, (Mul, VectorMul)): + vector = [i for i in vect.args if isinstance(i, (Vector, Cross, Gradient))][0] + scalar = Mul.fromiter(i for i in vect.args if not isinstance(i, (Vector, Cross, Gradient))) + res = Cross(gradient(scalar), vector).doit() + scalar*curl(vector, doit=doit) + if doit: + return res.doit() + return res + elif isinstance(vect, (Cross, Curl, Gradient)): + return Curl(vect) + else: + raise ValueError("Invalid argument for curl") + + +def divergence(vect, doit=True): + """ + Returns the divergence of a vector field computed wrt the base + scalars of the given coordinate system. + + Parameters + ========== + + vector : Vector + The vector operand + + doit : bool + If True, the result is returned after calling .doit() on + each component. Else, the returned expression contains + Derivative instances + + Examples + ======== + + >>> from sympy.vector import CoordSys3D, divergence + >>> R = CoordSys3D('R') + >>> v1 = R.x*R.y*R.z * (R.i+R.j+R.k) + + >>> divergence(v1) + R.x*R.y + R.x*R.z + R.y*R.z + >>> v2 = 2*R.y*R.z*R.j + >>> divergence(v2) + 2*R.z + + """ + coord_sys = _get_coord_systems(vect) + if len(coord_sys) == 0: + return S.Zero + elif len(coord_sys) == 1: + if isinstance(vect, (Cross, Curl, Gradient)): + return Divergence(vect) + # TODO: is case of many coord systems, this gets a random one: + coord_sys = next(iter(coord_sys)) + i, j, k = coord_sys.base_vectors() + x, y, z = coord_sys.base_scalars() + h1, h2, h3 = coord_sys.lame_coefficients() + vx = _diff_conditional(vect.dot(i), x, h2, h3) \ + / (h1 * h2 * h3) + vy = _diff_conditional(vect.dot(j), y, h3, h1) \ + / (h1 * h2 * h3) + vz = _diff_conditional(vect.dot(k), z, h1, h2) \ + / (h1 * h2 * h3) + res = vx + vy + vz + if doit: + return res.doit() + return res + else: + if isinstance(vect, (Add, VectorAdd)): + return Add.fromiter(divergence(i, doit=doit) for i in vect.args) + elif isinstance(vect, (Mul, VectorMul)): + vector = [i for i in vect.args if isinstance(i, (Vector, Cross, Gradient))][0] + scalar = Mul.fromiter(i for i in vect.args if not isinstance(i, (Vector, Cross, Gradient))) + res = Dot(vector, gradient(scalar)) + scalar*divergence(vector, doit=doit) + if doit: + return res.doit() + return res + elif isinstance(vect, (Cross, Curl, Gradient)): + return Divergence(vect) + else: + raise ValueError("Invalid argument for divergence") + + +def gradient(scalar_field, doit=True): + """ + Returns the vector gradient of a scalar field computed wrt the + base scalars of the given coordinate system. + + Parameters + ========== + + scalar_field : SymPy Expr + The scalar field to compute the gradient of + + doit : bool + If True, the result is returned after calling .doit() on + each component. Else, the returned expression contains + Derivative instances + + Examples + ======== + + >>> from sympy.vector import CoordSys3D, gradient + >>> R = CoordSys3D('R') + >>> s1 = R.x*R.y*R.z + >>> gradient(s1) + R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k + >>> s2 = 5*R.x**2*R.z + >>> gradient(s2) + 10*R.x*R.z*R.i + 5*R.x**2*R.k + + """ + coord_sys = _get_coord_systems(scalar_field) + + if len(coord_sys) == 0: + return Vector.zero + elif len(coord_sys) == 1: + coord_sys = next(iter(coord_sys)) + h1, h2, h3 = coord_sys.lame_coefficients() + i, j, k = coord_sys.base_vectors() + x, y, z = coord_sys.base_scalars() + vx = Derivative(scalar_field, x) / h1 + vy = Derivative(scalar_field, y) / h2 + vz = Derivative(scalar_field, z) / h3 + + if doit: + return (vx * i + vy * j + vz * k).doit() + return vx * i + vy * j + vz * k + else: + if isinstance(scalar_field, (Add, VectorAdd)): + return VectorAdd.fromiter(gradient(i) for i in scalar_field.args) + if isinstance(scalar_field, (Mul, VectorMul)): + s = _split_mul_args_wrt_coordsys(scalar_field) + return VectorAdd.fromiter(scalar_field / i * gradient(i) for i in s) + return Gradient(scalar_field) + + +class Laplacian(Expr): + """ + Represents unevaluated Laplacian. + + Examples + ======== + + >>> from sympy.vector import CoordSys3D, Laplacian + >>> R = CoordSys3D('R') + >>> v = 3*R.x**3*R.y**2*R.z**3 + >>> Laplacian(v) + Laplacian(3*R.x**3*R.y**2*R.z**3) + + """ + + def __new__(cls, expr): + expr = sympify(expr) + obj = Expr.__new__(cls, expr) + obj._expr = expr + return obj + + def doit(self, **hints): + from sympy.vector.functions import laplacian + return laplacian(self._expr) + + +def _diff_conditional(expr, base_scalar, coeff_1, coeff_2): + """ + First re-expresses expr in the system that base_scalar belongs to. + If base_scalar appears in the re-expressed form, differentiates + it wrt base_scalar. + Else, returns 0 + """ + from sympy.vector.functions import express + new_expr = express(expr, base_scalar.system, variables=True) + arg = coeff_1 * coeff_2 * new_expr + return Derivative(arg, base_scalar) if arg else S.Zero diff --git a/.venv/lib/python3.12/site-packages/sympy/vector/orienters.py b/.venv/lib/python3.12/site-packages/sympy/vector/orienters.py new file mode 100644 index 0000000000000000000000000000000000000000..0c22089e568bc817c943c1beecebde0fea46b6ae --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/vector/orienters.py @@ -0,0 +1,398 @@ +from sympy.core.basic import Basic +from sympy.core.sympify import sympify +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.matrices.dense import (eye, rot_axis1, rot_axis2, rot_axis3) +from sympy.matrices.immutable import ImmutableDenseMatrix as Matrix +from sympy.core.cache import cacheit +from sympy.core.symbol import Str +import sympy.vector + + +class Orienter(Basic): + """ + Super-class for all orienter classes. + """ + + def rotation_matrix(self): + """ + The rotation matrix corresponding to this orienter + instance. + """ + return self._parent_orient + + +class AxisOrienter(Orienter): + """ + Class to denote an axis orienter. + """ + + def __new__(cls, angle, axis): + if not isinstance(axis, sympy.vector.Vector): + raise TypeError("axis should be a Vector") + angle = sympify(angle) + + obj = super().__new__(cls, angle, axis) + obj._angle = angle + obj._axis = axis + + return obj + + def __init__(self, angle, axis): + """ + Axis rotation is a rotation about an arbitrary axis by + some angle. The angle is supplied as a SymPy expr scalar, and + the axis is supplied as a Vector. + + Parameters + ========== + + angle : Expr + The angle by which the new system is to be rotated + + axis : Vector + The axis around which the rotation has to be performed + + Examples + ======== + + >>> from sympy.vector import CoordSys3D + >>> from sympy import symbols + >>> q1 = symbols('q1') + >>> N = CoordSys3D('N') + >>> from sympy.vector import AxisOrienter + >>> orienter = AxisOrienter(q1, N.i + 2 * N.j) + >>> B = N.orient_new('B', (orienter, )) + + """ + # Dummy initializer for docstrings + pass + + @cacheit + def rotation_matrix(self, system): + """ + The rotation matrix corresponding to this orienter + instance. + + Parameters + ========== + + system : CoordSys3D + The coordinate system wrt which the rotation matrix + is to be computed + """ + + axis = sympy.vector.express(self.axis, system).normalize() + axis = axis.to_matrix(system) + theta = self.angle + parent_orient = ((eye(3) - axis * axis.T) * cos(theta) + + Matrix([[0, -axis[2], axis[1]], + [axis[2], 0, -axis[0]], + [-axis[1], axis[0], 0]]) * sin(theta) + + axis * axis.T) + parent_orient = parent_orient.T + return parent_orient + + @property + def angle(self): + return self._angle + + @property + def axis(self): + return self._axis + + +class ThreeAngleOrienter(Orienter): + """ + Super-class for Body and Space orienters. + """ + + def __new__(cls, angle1, angle2, angle3, rot_order): + if isinstance(rot_order, Str): + rot_order = rot_order.name + + approved_orders = ('123', '231', '312', '132', '213', + '321', '121', '131', '212', '232', + '313', '323', '') + original_rot_order = rot_order + rot_order = str(rot_order).upper() + if not (len(rot_order) == 3): + raise TypeError('rot_order should be a str of length 3') + rot_order = [i.replace('X', '1') for i in rot_order] + rot_order = [i.replace('Y', '2') for i in rot_order] + rot_order = [i.replace('Z', '3') for i in rot_order] + rot_order = ''.join(rot_order) + if rot_order not in approved_orders: + raise TypeError('Invalid rot_type parameter') + a1 = int(rot_order[0]) + a2 = int(rot_order[1]) + a3 = int(rot_order[2]) + angle1 = sympify(angle1) + angle2 = sympify(angle2) + angle3 = sympify(angle3) + if cls._in_order: + parent_orient = (_rot(a1, angle1) * + _rot(a2, angle2) * + _rot(a3, angle3)) + else: + parent_orient = (_rot(a3, angle3) * + _rot(a2, angle2) * + _rot(a1, angle1)) + parent_orient = parent_orient.T + + obj = super().__new__( + cls, angle1, angle2, angle3, Str(rot_order)) + obj._angle1 = angle1 + obj._angle2 = angle2 + obj._angle3 = angle3 + obj._rot_order = original_rot_order + obj._parent_orient = parent_orient + + return obj + + @property + def angle1(self): + return self._angle1 + + @property + def angle2(self): + return self._angle2 + + @property + def angle3(self): + return self._angle3 + + @property + def rot_order(self): + return self._rot_order + + +class BodyOrienter(ThreeAngleOrienter): + """ + Class to denote a body-orienter. + """ + + _in_order = True + + def __new__(cls, angle1, angle2, angle3, rot_order): + obj = ThreeAngleOrienter.__new__(cls, angle1, angle2, angle3, + rot_order) + return obj + + def __init__(self, angle1, angle2, angle3, rot_order): + """ + Body orientation takes this coordinate system through three + successive simple rotations. + + Body fixed rotations include both Euler Angles and + Tait-Bryan Angles, see https://en.wikipedia.org/wiki/Euler_angles. + + Parameters + ========== + + angle1, angle2, angle3 : Expr + Three successive angles to rotate the coordinate system by + + rotation_order : string + String defining the order of axes for rotation + + Examples + ======== + + >>> from sympy.vector import CoordSys3D, BodyOrienter + >>> from sympy import symbols + >>> q1, q2, q3 = symbols('q1 q2 q3') + >>> N = CoordSys3D('N') + + A 'Body' fixed rotation is described by three angles and + three body-fixed rotation axes. To orient a coordinate system D + with respect to N, each sequential rotation is always about + the orthogonal unit vectors fixed to D. For example, a '123' + rotation will specify rotations about N.i, then D.j, then + D.k. (Initially, D.i is same as N.i) + Therefore, + + >>> body_orienter = BodyOrienter(q1, q2, q3, '123') + >>> D = N.orient_new('D', (body_orienter, )) + + is same as + + >>> from sympy.vector import AxisOrienter + >>> axis_orienter1 = AxisOrienter(q1, N.i) + >>> D = N.orient_new('D', (axis_orienter1, )) + >>> axis_orienter2 = AxisOrienter(q2, D.j) + >>> D = D.orient_new('D', (axis_orienter2, )) + >>> axis_orienter3 = AxisOrienter(q3, D.k) + >>> D = D.orient_new('D', (axis_orienter3, )) + + Acceptable rotation orders are of length 3, expressed in XYZ or + 123, and cannot have a rotation about about an axis twice in a row. + + >>> body_orienter1 = BodyOrienter(q1, q2, q3, '123') + >>> body_orienter2 = BodyOrienter(q1, q2, 0, 'ZXZ') + >>> body_orienter3 = BodyOrienter(0, 0, 0, 'XYX') + + """ + # Dummy initializer for docstrings + pass + + +class SpaceOrienter(ThreeAngleOrienter): + """ + Class to denote a space-orienter. + """ + + _in_order = False + + def __new__(cls, angle1, angle2, angle3, rot_order): + obj = ThreeAngleOrienter.__new__(cls, angle1, angle2, angle3, + rot_order) + return obj + + def __init__(self, angle1, angle2, angle3, rot_order): + """ + Space rotation is similar to Body rotation, but the rotations + are applied in the opposite order. + + Parameters + ========== + + angle1, angle2, angle3 : Expr + Three successive angles to rotate the coordinate system by + + rotation_order : string + String defining the order of axes for rotation + + See Also + ======== + + BodyOrienter : Orienter to orient systems wrt Euler angles. + + Examples + ======== + + >>> from sympy.vector import CoordSys3D, SpaceOrienter + >>> from sympy import symbols + >>> q1, q2, q3 = symbols('q1 q2 q3') + >>> N = CoordSys3D('N') + + To orient a coordinate system D with respect to N, each + sequential rotation is always about N's orthogonal unit vectors. + For example, a '123' rotation will specify rotations about + N.i, then N.j, then N.k. + Therefore, + + >>> space_orienter = SpaceOrienter(q1, q2, q3, '312') + >>> D = N.orient_new('D', (space_orienter, )) + + is same as + + >>> from sympy.vector import AxisOrienter + >>> axis_orienter1 = AxisOrienter(q1, N.i) + >>> B = N.orient_new('B', (axis_orienter1, )) + >>> axis_orienter2 = AxisOrienter(q2, N.j) + >>> C = B.orient_new('C', (axis_orienter2, )) + >>> axis_orienter3 = AxisOrienter(q3, N.k) + >>> D = C.orient_new('C', (axis_orienter3, )) + + """ + # Dummy initializer for docstrings + pass + + +class QuaternionOrienter(Orienter): + """ + Class to denote a quaternion-orienter. + """ + + def __new__(cls, q0, q1, q2, q3): + q0 = sympify(q0) + q1 = sympify(q1) + q2 = sympify(q2) + q3 = sympify(q3) + parent_orient = (Matrix([[q0 ** 2 + q1 ** 2 - q2 ** 2 - + q3 ** 2, + 2 * (q1 * q2 - q0 * q3), + 2 * (q0 * q2 + q1 * q3)], + [2 * (q1 * q2 + q0 * q3), + q0 ** 2 - q1 ** 2 + + q2 ** 2 - q3 ** 2, + 2 * (q2 * q3 - q0 * q1)], + [2 * (q1 * q3 - q0 * q2), + 2 * (q0 * q1 + q2 * q3), + q0 ** 2 - q1 ** 2 - + q2 ** 2 + q3 ** 2]])) + parent_orient = parent_orient.T + + obj = super().__new__(cls, q0, q1, q2, q3) + obj._q0 = q0 + obj._q1 = q1 + obj._q2 = q2 + obj._q3 = q3 + obj._parent_orient = parent_orient + + return obj + + def __init__(self, angle1, angle2, angle3, rot_order): + """ + Quaternion orientation orients the new CoordSys3D with + Quaternions, defined as a finite rotation about lambda, a unit + vector, by some amount theta. + + This orientation is described by four parameters: + + q0 = cos(theta/2) + + q1 = lambda_x sin(theta/2) + + q2 = lambda_y sin(theta/2) + + q3 = lambda_z sin(theta/2) + + Quaternion does not take in a rotation order. + + Parameters + ========== + + q0, q1, q2, q3 : Expr + The quaternions to rotate the coordinate system by + + Examples + ======== + + >>> from sympy.vector import CoordSys3D + >>> from sympy import symbols + >>> q0, q1, q2, q3 = symbols('q0 q1 q2 q3') + >>> N = CoordSys3D('N') + >>> from sympy.vector import QuaternionOrienter + >>> q_orienter = QuaternionOrienter(q0, q1, q2, q3) + >>> B = N.orient_new('B', (q_orienter, )) + + """ + # Dummy initializer for docstrings + pass + + @property + def q0(self): + return self._q0 + + @property + def q1(self): + return self._q1 + + @property + def q2(self): + return self._q2 + + @property + def q3(self): + return self._q3 + + +def _rot(axis, angle): + """DCM for simple axis 1, 2 or 3 rotations. """ + if axis == 1: + return Matrix(rot_axis1(angle).T) + elif axis == 2: + return Matrix(rot_axis2(angle).T) + elif axis == 3: + return Matrix(rot_axis3(angle).T) diff --git a/.venv/lib/python3.12/site-packages/sympy/vector/parametricregion.py b/.venv/lib/python3.12/site-packages/sympy/vector/parametricregion.py new file mode 100644 index 0000000000000000000000000000000000000000..5246769dabe208fe630f7c33b2da3ef4e11b3f67 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/vector/parametricregion.py @@ -0,0 +1,189 @@ +from functools import singledispatch +from sympy.core.numbers import pi +from sympy.functions.elementary.trigonometric import tan +from sympy.simplify import trigsimp +from sympy.core import Basic, Tuple +from sympy.core.symbol import _symbol +from sympy.solvers import solve +from sympy.geometry import Point, Segment, Curve, Ellipse, Polygon +from sympy.vector import ImplicitRegion + + +class ParametricRegion(Basic): + """ + Represents a parametric region in space. + + Examples + ======== + + >>> from sympy import cos, sin, pi + >>> from sympy.abc import r, theta, t, a, b, x, y + >>> from sympy.vector import ParametricRegion + + >>> ParametricRegion((t, t**2), (t, -1, 2)) + ParametricRegion((t, t**2), (t, -1, 2)) + >>> ParametricRegion((x, y), (x, 3, 4), (y, 5, 6)) + ParametricRegion((x, y), (x, 3, 4), (y, 5, 6)) + >>> ParametricRegion((r*cos(theta), r*sin(theta)), (r, -2, 2), (theta, 0, pi)) + ParametricRegion((r*cos(theta), r*sin(theta)), (r, -2, 2), (theta, 0, pi)) + >>> ParametricRegion((a*cos(t), b*sin(t)), t) + ParametricRegion((a*cos(t), b*sin(t)), t) + + >>> circle = ParametricRegion((r*cos(theta), r*sin(theta)), r, (theta, 0, pi)) + >>> circle.parameters + (r, theta) + >>> circle.definition + (r*cos(theta), r*sin(theta)) + >>> circle.limits + {theta: (0, pi)} + + Dimension of a parametric region determines whether a region is a curve, surface + or volume region. It does not represent its dimensions in space. + + >>> circle.dimensions + 1 + + Parameters + ========== + + definition : tuple to define base scalars in terms of parameters. + + bounds : Parameter or a tuple of length 3 to define parameter and corresponding lower and upper bound. + + """ + def __new__(cls, definition, *bounds): + parameters = () + limits = {} + + if not isinstance(bounds, Tuple): + bounds = Tuple(*bounds) + + for bound in bounds: + if isinstance(bound, (tuple, Tuple)): + if len(bound) != 3: + raise ValueError("Tuple should be in the form (parameter, lowerbound, upperbound)") + parameters += (bound[0],) + limits[bound[0]] = (bound[1], bound[2]) + else: + parameters += (bound,) + + if not isinstance(definition, (tuple, Tuple)): + definition = (definition,) + + obj = super().__new__(cls, Tuple(*definition), *bounds) + obj._parameters = parameters + obj._limits = limits + + return obj + + @property + def definition(self): + return self.args[0] + + @property + def limits(self): + return self._limits + + @property + def parameters(self): + return self._parameters + + @property + def dimensions(self): + return len(self.limits) + + +@singledispatch +def parametric_region_list(reg): + """ + Returns a list of ParametricRegion objects representing the geometric region. + + Examples + ======== + + >>> from sympy.abc import t + >>> from sympy.vector import parametric_region_list + >>> from sympy.geometry import Point, Curve, Ellipse, Segment, Polygon + + >>> p = Point(2, 5) + >>> parametric_region_list(p) + [ParametricRegion((2, 5))] + + >>> c = Curve((t**3, 4*t), (t, -3, 4)) + >>> parametric_region_list(c) + [ParametricRegion((t**3, 4*t), (t, -3, 4))] + + >>> e = Ellipse(Point(1, 3), 2, 3) + >>> parametric_region_list(e) + [ParametricRegion((2*cos(t) + 1, 3*sin(t) + 3), (t, 0, 2*pi))] + + >>> s = Segment(Point(1, 3), Point(2, 6)) + >>> parametric_region_list(s) + [ParametricRegion((t + 1, 3*t + 3), (t, 0, 1))] + + >>> p1, p2, p3, p4 = [(0, 1), (2, -3), (5, 3), (-2, 3)] + >>> poly = Polygon(p1, p2, p3, p4) + >>> parametric_region_list(poly) + [ParametricRegion((2*t, 1 - 4*t), (t, 0, 1)), ParametricRegion((3*t + 2, 6*t - 3), (t, 0, 1)),\ + ParametricRegion((5 - 7*t, 3), (t, 0, 1)), ParametricRegion((2*t - 2, 3 - 2*t), (t, 0, 1))] + + """ + raise ValueError("SymPy cannot determine parametric representation of the region.") + + +@parametric_region_list.register(Point) +def _(obj): + return [ParametricRegion(obj.args)] + + +@parametric_region_list.register(Curve) # type: ignore +def _(obj): + definition = obj.arbitrary_point(obj.parameter).args + bounds = obj.limits + return [ParametricRegion(definition, bounds)] + + +@parametric_region_list.register(Ellipse) # type: ignore +def _(obj, parameter='t'): + definition = obj.arbitrary_point(parameter).args + t = _symbol(parameter, real=True) + bounds = (t, 0, 2*pi) + return [ParametricRegion(definition, bounds)] + + +@parametric_region_list.register(Segment) # type: ignore +def _(obj, parameter='t'): + t = _symbol(parameter, real=True) + definition = obj.arbitrary_point(t).args + + for i in range(0, 3): + lower_bound = solve(definition[i] - obj.points[0].args[i], t) + upper_bound = solve(definition[i] - obj.points[1].args[i], t) + + if len(lower_bound) == 1 and len(upper_bound) == 1: + bounds = t, lower_bound[0], upper_bound[0] + break + + definition_tuple = obj.arbitrary_point(parameter).args + return [ParametricRegion(definition_tuple, bounds)] + + +@parametric_region_list.register(Polygon) # type: ignore +def _(obj, parameter='t'): + l = [parametric_region_list(side, parameter)[0] for side in obj.sides] + return l + + +@parametric_region_list.register(ImplicitRegion) # type: ignore +def _(obj, parameters=('t', 's')): + definition = obj.rational_parametrization(parameters) + bounds = [] + + for i in range(len(obj.variables) - 1): + # Each parameter is replaced by its tangent to simplify integration + parameter = _symbol(parameters[i], real=True) + definition = [trigsimp(elem.subs(parameter, tan(parameter/2))) for elem in definition] + bounds.append((parameter, 0, 2*pi),) + + definition = Tuple(*definition) + return [ParametricRegion(definition, *bounds)] diff --git a/.venv/lib/python3.12/site-packages/sympy/vector/point.py b/.venv/lib/python3.12/site-packages/sympy/vector/point.py new file mode 100644 index 0000000000000000000000000000000000000000..442ea4e8edc0e33c4f83f774ea3f11a01725ac3a --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/vector/point.py @@ -0,0 +1,148 @@ +from sympy.core.basic import Basic +from sympy.core.symbol import Str +from sympy.vector.vector import Vector +from sympy.vector.coordsysrect import CoordSys3D +from sympy.vector.functions import _path +from sympy.core.cache import cacheit + + +class Point(Basic): + """ + Represents a point in 3-D space. + """ + + def __new__(cls, name, position=Vector.zero, parent_point=None): + name = str(name) + # Check the args first + if not isinstance(position, Vector): + raise TypeError( + "position should be an instance of Vector, not %s" % type( + position)) + if (not isinstance(parent_point, Point) and + parent_point is not None): + raise TypeError( + "parent_point should be an instance of Point, not %s" % type( + parent_point)) + # Super class construction + if parent_point is None: + obj = super().__new__(cls, Str(name), position) + else: + obj = super().__new__(cls, Str(name), position, parent_point) + # Decide the object parameters + obj._name = name + obj._pos = position + if parent_point is None: + obj._parent = None + obj._root = obj + else: + obj._parent = parent_point + obj._root = parent_point._root + # Return object + return obj + + @cacheit + def position_wrt(self, other): + """ + Returns the position vector of this Point with respect to + another Point/CoordSys3D. + + Parameters + ========== + + other : Point/CoordSys3D + If other is a Point, the position of this Point wrt it is + returned. If its an instance of CoordSyRect, the position + wrt its origin is returned. + + Examples + ======== + + >>> from sympy.vector import CoordSys3D + >>> N = CoordSys3D('N') + >>> p1 = N.origin.locate_new('p1', 10 * N.i) + >>> N.origin.position_wrt(p1) + (-10)*N.i + + """ + + if (not isinstance(other, Point) and + not isinstance(other, CoordSys3D)): + raise TypeError(str(other) + + "is not a Point or CoordSys3D") + if isinstance(other, CoordSys3D): + other = other.origin + # Handle special cases + if other == self: + return Vector.zero + elif other == self._parent: + return self._pos + elif other._parent == self: + return -1 * other._pos + # Else, use point tree to calculate position + rootindex, path = _path(self, other) + result = Vector.zero + for i in range(rootindex): + result += path[i]._pos + for i in range(rootindex + 1, len(path)): + result -= path[i]._pos + return result + + def locate_new(self, name, position): + """ + Returns a new Point located at the given position wrt this + Point. + Thus, the position vector of the new Point wrt this one will + be equal to the given 'position' parameter. + + Parameters + ========== + + name : str + Name of the new point + + position : Vector + The position vector of the new Point wrt this one + + Examples + ======== + + >>> from sympy.vector import CoordSys3D + >>> N = CoordSys3D('N') + >>> p1 = N.origin.locate_new('p1', 10 * N.i) + >>> p1.position_wrt(N.origin) + 10*N.i + + """ + return Point(name, position, self) + + def express_coordinates(self, coordinate_system): + """ + Returns the Cartesian/rectangular coordinates of this point + wrt the origin of the given CoordSys3D instance. + + Parameters + ========== + + coordinate_system : CoordSys3D + The coordinate system to express the coordinates of this + Point in. + + Examples + ======== + + >>> from sympy.vector import CoordSys3D + >>> N = CoordSys3D('N') + >>> p1 = N.origin.locate_new('p1', 10 * N.i) + >>> p2 = p1.locate_new('p2', 5 * N.j) + >>> p2.express_coordinates(N) + (10, 5, 0) + + """ + + # Determine the position vector + pos_vect = self.position_wrt(coordinate_system.origin) + # Express it in the given coordinate system + return tuple(pos_vect.to_matrix(coordinate_system)) + + def _sympystr(self, printer): + return self._name diff --git a/.venv/lib/python3.12/site-packages/sympy/vector/scalar.py b/.venv/lib/python3.12/site-packages/sympy/vector/scalar.py new file mode 100644 index 0000000000000000000000000000000000000000..bcfb56cf177b9378a24f81ca1e6524fe048a5f94 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/vector/scalar.py @@ -0,0 +1,72 @@ +from sympy.core import AtomicExpr, Symbol, S +from sympy.core.sympify import _sympify +from sympy.printing.pretty.stringpict import prettyForm +from sympy.printing.precedence import PRECEDENCE +from sympy.core.kind import NumberKind + + +class BaseScalar(AtomicExpr): + """ + A coordinate symbol/base scalar. + + Ideally, users should not instantiate this class. + + """ + + kind = NumberKind + + def __new__(cls, index, system, pretty_str=None, latex_str=None): + from sympy.vector.coordsysrect import CoordSys3D + if pretty_str is None: + pretty_str = "x{}".format(index) + elif isinstance(pretty_str, Symbol): + pretty_str = pretty_str.name + if latex_str is None: + latex_str = "x_{}".format(index) + elif isinstance(latex_str, Symbol): + latex_str = latex_str.name + + index = _sympify(index) + system = _sympify(system) + obj = super().__new__(cls, index, system) + if not isinstance(system, CoordSys3D): + raise TypeError("system should be a CoordSys3D") + if index not in range(0, 3): + raise ValueError("Invalid index specified.") + # The _id is used for equating purposes, and for hashing + obj._id = (index, system) + obj._name = obj.name = system._name + '.' + system._variable_names[index] + obj._pretty_form = '' + pretty_str + obj._latex_form = latex_str + obj._system = system + + return obj + + is_commutative = True + is_symbol = True + + @property + def free_symbols(self): + return {self} + + _diff_wrt = True + + def _eval_derivative(self, s): + if self == s: + return S.One + return S.Zero + + def _latex(self, printer=None): + return self._latex_form + + def _pretty(self, printer=None): + return prettyForm(self._pretty_form) + + precedence = PRECEDENCE['Atom'] + + @property + def system(self): + return self._system + + def _sympystr(self, printer): + return self._name diff --git a/.venv/lib/python3.12/site-packages/sympy/vector/tests/__init__.py b/.venv/lib/python3.12/site-packages/sympy/vector/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/.venv/lib/python3.12/site-packages/sympy/vector/tests/__pycache__/__init__.cpython-312.pyc b/.venv/lib/python3.12/site-packages/sympy/vector/tests/__pycache__/__init__.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..8bc5f63db28bd59f16340dbbe937e38f150d200e Binary 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differ diff --git a/.venv/lib/python3.12/site-packages/sympy/vector/tests/test_coordsysrect.py b/.venv/lib/python3.12/site-packages/sympy/vector/tests/test_coordsysrect.py new file mode 100644 index 0000000000000000000000000000000000000000..53eb8c89ec1643a71800efe3e370acff3cb6f9c0 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/vector/tests/test_coordsysrect.py @@ -0,0 +1,464 @@ +from sympy.testing.pytest import raises +from sympy.vector.coordsysrect import CoordSys3D +from sympy.vector.scalar import BaseScalar +from sympy.core.function import expand +from sympy.core.numbers import pi +from sympy.core.symbol import symbols +from sympy.functions.elementary.hyperbolic import (cosh, sinh) +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import (acos, atan2, cos, sin) +from sympy.matrices.dense import zeros +from sympy.matrices.immutable import ImmutableDenseMatrix as Matrix +from sympy.simplify.simplify import simplify +from sympy.vector.functions import express +from sympy.vector.point import Point +from sympy.vector.vector import Vector +from sympy.vector.orienters import (AxisOrienter, BodyOrienter, + SpaceOrienter, QuaternionOrienter) + + +x, y, z = symbols('x y z') +a, b, c, q = symbols('a b c q') +q1, q2, q3, q4 = symbols('q1 q2 q3 q4') + + +def test_func_args(): + A = CoordSys3D('A') + assert A.x.func(*A.x.args) == A.x + expr = 3*A.x + 4*A.y + assert expr.func(*expr.args) == expr + assert A.i.func(*A.i.args) == A.i + v = A.x*A.i + A.y*A.j + A.z*A.k + assert v.func(*v.args) == v + assert A.origin.func(*A.origin.args) == A.origin + + +def test_coordsys3d_equivalence(): + A = CoordSys3D('A') + A1 = CoordSys3D('A') + assert A1 == A + B = CoordSys3D('B') + assert A != B + + +def test_orienters(): + A = CoordSys3D('A') + axis_orienter = AxisOrienter(a, A.k) + body_orienter = BodyOrienter(a, b, c, '123') + space_orienter = SpaceOrienter(a, b, c, '123') + q_orienter = QuaternionOrienter(q1, q2, q3, q4) + assert axis_orienter.rotation_matrix(A) == Matrix([ + [ cos(a), sin(a), 0], + [-sin(a), cos(a), 0], + [ 0, 0, 1]]) + assert body_orienter.rotation_matrix() == Matrix([ + [ cos(b)*cos(c), sin(a)*sin(b)*cos(c) + sin(c)*cos(a), + sin(a)*sin(c) - sin(b)*cos(a)*cos(c)], + [-sin(c)*cos(b), -sin(a)*sin(b)*sin(c) + cos(a)*cos(c), + sin(a)*cos(c) + sin(b)*sin(c)*cos(a)], + [ sin(b), -sin(a)*cos(b), + cos(a)*cos(b)]]) + assert space_orienter.rotation_matrix() == Matrix([ + [cos(b)*cos(c), sin(c)*cos(b), -sin(b)], + [sin(a)*sin(b)*cos(c) - sin(c)*cos(a), + sin(a)*sin(b)*sin(c) + cos(a)*cos(c), sin(a)*cos(b)], + [sin(a)*sin(c) + sin(b)*cos(a)*cos(c), -sin(a)*cos(c) + + sin(b)*sin(c)*cos(a), cos(a)*cos(b)]]) + assert q_orienter.rotation_matrix() == Matrix([ + [q1**2 + q2**2 - q3**2 - q4**2, 2*q1*q4 + 2*q2*q3, + -2*q1*q3 + 2*q2*q4], + [-2*q1*q4 + 2*q2*q3, q1**2 - q2**2 + q3**2 - q4**2, + 2*q1*q2 + 2*q3*q4], + [2*q1*q3 + 2*q2*q4, + -2*q1*q2 + 2*q3*q4, q1**2 - q2**2 - q3**2 + q4**2]]) + + +def test_coordinate_vars(): + """ + Tests the coordinate variables functionality with respect to + reorientation of coordinate systems. + """ + A = CoordSys3D('A') + # Note that the name given on the lhs is different from A.x._name + assert BaseScalar(0, A, 'A_x', r'\mathbf{{x}_{A}}') == A.x + assert BaseScalar(1, A, 'A_y', r'\mathbf{{y}_{A}}') == A.y + assert BaseScalar(2, A, 'A_z', r'\mathbf{{z}_{A}}') == A.z + assert BaseScalar(0, A, 'A_x', r'\mathbf{{x}_{A}}').__hash__() == A.x.__hash__() + assert isinstance(A.x, BaseScalar) and \ + isinstance(A.y, BaseScalar) and \ + isinstance(A.z, BaseScalar) + assert A.x*A.y == A.y*A.x + assert A.scalar_map(A) == {A.x: A.x, A.y: A.y, A.z: A.z} + assert A.x.system == A + assert A.x.diff(A.x) == 1 + B = A.orient_new_axis('B', q, A.k) + assert B.scalar_map(A) == {B.z: A.z, B.y: -A.x*sin(q) + A.y*cos(q), + B.x: A.x*cos(q) + A.y*sin(q)} + assert A.scalar_map(B) == {A.x: B.x*cos(q) - B.y*sin(q), + A.y: B.x*sin(q) + B.y*cos(q), A.z: B.z} + assert express(B.x, A, variables=True) == A.x*cos(q) + A.y*sin(q) + assert express(B.y, A, variables=True) == -A.x*sin(q) + A.y*cos(q) + assert express(B.z, A, variables=True) == A.z + assert expand(express(B.x*B.y*B.z, A, variables=True)) == \ + expand(A.z*(-A.x*sin(q) + A.y*cos(q))*(A.x*cos(q) + A.y*sin(q))) + assert express(B.x*B.i + B.y*B.j + B.z*B.k, A) == \ + (B.x*cos(q) - B.y*sin(q))*A.i + (B.x*sin(q) + \ + B.y*cos(q))*A.j + B.z*A.k + assert simplify(express(B.x*B.i + B.y*B.j + B.z*B.k, A, \ + variables=True)) == \ + A.x*A.i + A.y*A.j + A.z*A.k + assert express(A.x*A.i + A.y*A.j + A.z*A.k, B) == \ + (A.x*cos(q) + A.y*sin(q))*B.i + \ + (-A.x*sin(q) + A.y*cos(q))*B.j + A.z*B.k + assert simplify(express(A.x*A.i + A.y*A.j + A.z*A.k, B, \ + variables=True)) == \ + B.x*B.i + B.y*B.j + B.z*B.k + N = B.orient_new_axis('N', -q, B.k) + assert N.scalar_map(A) == \ + {N.x: A.x, N.z: A.z, N.y: A.y} + C = A.orient_new_axis('C', q, A.i + A.j + A.k) + mapping = A.scalar_map(C) + assert mapping[A.x].equals(C.x*(2*cos(q) + 1)/3 + + C.y*(-2*sin(q + pi/6) + 1)/3 + + C.z*(-2*cos(q + pi/3) + 1)/3) + assert mapping[A.y].equals(C.x*(-2*cos(q + pi/3) + 1)/3 + + C.y*(2*cos(q) + 1)/3 + + C.z*(-2*sin(q + pi/6) + 1)/3) + assert mapping[A.z].equals(C.x*(-2*sin(q + pi/6) + 1)/3 + + C.y*(-2*cos(q + pi/3) + 1)/3 + + C.z*(2*cos(q) + 1)/3) + D = A.locate_new('D', a*A.i + b*A.j + c*A.k) + assert D.scalar_map(A) == {D.z: A.z - c, D.x: A.x - a, D.y: A.y - b} + E = A.orient_new_axis('E', a, A.k, a*A.i + b*A.j + c*A.k) + assert A.scalar_map(E) == {A.z: E.z + c, + A.x: E.x*cos(a) - E.y*sin(a) + a, + A.y: E.x*sin(a) + E.y*cos(a) + b} + assert E.scalar_map(A) == {E.x: (A.x - a)*cos(a) + (A.y - b)*sin(a), + E.y: (-A.x + a)*sin(a) + (A.y - b)*cos(a), + E.z: A.z - c} + F = A.locate_new('F', Vector.zero) + assert A.scalar_map(F) == {A.z: F.z, A.x: F.x, A.y: F.y} + + +def test_rotation_matrix(): + N = CoordSys3D('N') + A = N.orient_new_axis('A', q1, N.k) + B = A.orient_new_axis('B', q2, A.i) + C = B.orient_new_axis('C', q3, B.j) + D = N.orient_new_axis('D', q4, N.j) + E = N.orient_new_space('E', q1, q2, q3, '123') + F = N.orient_new_quaternion('F', q1, q2, q3, q4) + G = N.orient_new_body('G', q1, q2, q3, '123') + assert N.rotation_matrix(C) == Matrix([ + [- sin(q1) * sin(q2) * sin(q3) + cos(q1) * cos(q3), - sin(q1) * + cos(q2), sin(q1) * sin(q2) * cos(q3) + sin(q3) * cos(q1)], \ + [sin(q1) * cos(q3) + sin(q2) * sin(q3) * cos(q1), \ + cos(q1) * cos(q2), sin(q1) * sin(q3) - sin(q2) * cos(q1) * \ + cos(q3)], [- sin(q3) * cos(q2), sin(q2), cos(q2) * cos(q3)]]) + test_mat = D.rotation_matrix(C) - Matrix( + [[cos(q1) * cos(q3) * cos(q4) - sin(q3) * (- sin(q4) * cos(q2) + + sin(q1) * sin(q2) * cos(q4)), - sin(q2) * sin(q4) - sin(q1) * + cos(q2) * cos(q4), sin(q3) * cos(q1) * cos(q4) + cos(q3) * \ + (- sin(q4) * cos(q2) + sin(q1) * sin(q2) * cos(q4))], \ + [sin(q1) * cos(q3) + sin(q2) * sin(q3) * cos(q1), cos(q1) * \ + cos(q2), sin(q1) * sin(q3) - sin(q2) * cos(q1) * cos(q3)], \ + [sin(q4) * cos(q1) * cos(q3) - sin(q3) * (cos(q2) * cos(q4) + \ + sin(q1) * sin(q2) * \ + sin(q4)), sin(q2) * + cos(q4) - sin(q1) * sin(q4) * cos(q2), sin(q3) * \ + sin(q4) * cos(q1) + cos(q3) * (cos(q2) * cos(q4) + \ + sin(q1) * sin(q2) * sin(q4))]]) + assert test_mat.expand() == zeros(3, 3) + assert E.rotation_matrix(N) == Matrix( + [[cos(q2)*cos(q3), sin(q3)*cos(q2), -sin(q2)], + [sin(q1)*sin(q2)*cos(q3) - sin(q3)*cos(q1), \ + sin(q1)*sin(q2)*sin(q3) + cos(q1)*cos(q3), sin(q1)*cos(q2)], \ + [sin(q1)*sin(q3) + sin(q2)*cos(q1)*cos(q3), - \ + sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1), cos(q1)*cos(q2)]]) + assert F.rotation_matrix(N) == Matrix([[ + q1**2 + q2**2 - q3**2 - q4**2, + 2*q1*q4 + 2*q2*q3, -2*q1*q3 + 2*q2*q4],[ -2*q1*q4 + 2*q2*q3, + q1**2 - q2**2 + q3**2 - q4**2, 2*q1*q2 + 2*q3*q4], + [2*q1*q3 + 2*q2*q4, + -2*q1*q2 + 2*q3*q4, + q1**2 - q2**2 - q3**2 + q4**2]]) + assert G.rotation_matrix(N) == Matrix([[ + cos(q2)*cos(q3), sin(q1)*sin(q2)*cos(q3) + sin(q3)*cos(q1), + sin(q1)*sin(q3) - sin(q2)*cos(q1)*cos(q3)], [ + -sin(q3)*cos(q2), -sin(q1)*sin(q2)*sin(q3) + cos(q1)*cos(q3), + sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1)],[ + sin(q2), -sin(q1)*cos(q2), cos(q1)*cos(q2)]]) + + +def test_vector_with_orientation(): + """ + Tests the effects of orientation of coordinate systems on + basic vector operations. + """ + N = CoordSys3D('N') + A = N.orient_new_axis('A', q1, N.k) + B = A.orient_new_axis('B', q2, A.i) + C = B.orient_new_axis('C', q3, B.j) + + # Test to_matrix + v1 = a*N.i + b*N.j + c*N.k + assert v1.to_matrix(A) == Matrix([[ a*cos(q1) + b*sin(q1)], + [-a*sin(q1) + b*cos(q1)], + [ c]]) + + # Test dot + assert N.i.dot(A.i) == cos(q1) + assert N.i.dot(A.j) == -sin(q1) + assert N.i.dot(A.k) == 0 + assert N.j.dot(A.i) == sin(q1) + assert N.j.dot(A.j) == cos(q1) + assert N.j.dot(A.k) == 0 + assert N.k.dot(A.i) == 0 + assert N.k.dot(A.j) == 0 + assert N.k.dot(A.k) == 1 + + assert N.i.dot(A.i + A.j) == -sin(q1) + cos(q1) == \ + (A.i + A.j).dot(N.i) + + assert A.i.dot(C.i) == cos(q3) + assert A.i.dot(C.j) == 0 + assert A.i.dot(C.k) == sin(q3) + assert A.j.dot(C.i) == sin(q2)*sin(q3) + assert A.j.dot(C.j) == cos(q2) + assert A.j.dot(C.k) == -sin(q2)*cos(q3) + assert A.k.dot(C.i) == -cos(q2)*sin(q3) + assert A.k.dot(C.j) == sin(q2) + assert A.k.dot(C.k) == cos(q2)*cos(q3) + + # Test cross + assert N.i.cross(A.i) == sin(q1)*A.k + assert N.i.cross(A.j) == cos(q1)*A.k + assert N.i.cross(A.k) == -sin(q1)*A.i - cos(q1)*A.j + assert N.j.cross(A.i) == -cos(q1)*A.k + assert N.j.cross(A.j) == sin(q1)*A.k + assert N.j.cross(A.k) == cos(q1)*A.i - sin(q1)*A.j + assert N.k.cross(A.i) == A.j + assert N.k.cross(A.j) == -A.i + assert N.k.cross(A.k) == Vector.zero + + assert N.i.cross(A.i) == sin(q1)*A.k + assert N.i.cross(A.j) == cos(q1)*A.k + assert N.i.cross(A.i + A.j) == sin(q1)*A.k + cos(q1)*A.k + assert (A.i + A.j).cross(N.i) == (-sin(q1) - cos(q1))*N.k + + assert A.i.cross(C.i) == sin(q3)*C.j + assert A.i.cross(C.j) == -sin(q3)*C.i + cos(q3)*C.k + assert A.i.cross(C.k) == -cos(q3)*C.j + assert C.i.cross(A.i) == (-sin(q3)*cos(q2))*A.j + \ + (-sin(q2)*sin(q3))*A.k + assert C.j.cross(A.i) == (sin(q2))*A.j + (-cos(q2))*A.k + assert express(C.k.cross(A.i), C).trigsimp() == cos(q3)*C.j + + +def test_orient_new_methods(): + N = CoordSys3D('N') + orienter1 = AxisOrienter(q4, N.j) + orienter2 = SpaceOrienter(q1, q2, q3, '123') + orienter3 = QuaternionOrienter(q1, q2, q3, q4) + orienter4 = BodyOrienter(q1, q2, q3, '123') + D = N.orient_new('D', (orienter1, )) + E = N.orient_new('E', (orienter2, )) + F = N.orient_new('F', (orienter3, )) + G = N.orient_new('G', (orienter4, )) + assert D == N.orient_new_axis('D', q4, N.j) + assert E == N.orient_new_space('E', q1, q2, q3, '123') + assert F == N.orient_new_quaternion('F', q1, q2, q3, q4) + assert G == N.orient_new_body('G', q1, q2, q3, '123') + + +def test_locatenew_point(): + """ + Tests Point class, and locate_new method in CoordSys3D. + """ + A = CoordSys3D('A') + assert isinstance(A.origin, Point) + v = a*A.i + b*A.j + c*A.k + C = A.locate_new('C', v) + assert C.origin.position_wrt(A) == \ + C.position_wrt(A) == \ + C.origin.position_wrt(A.origin) == v + assert A.origin.position_wrt(C) == \ + A.position_wrt(C) == \ + A.origin.position_wrt(C.origin) == -v + assert A.origin.express_coordinates(C) == (-a, -b, -c) + p = A.origin.locate_new('p', -v) + assert p.express_coordinates(A) == (-a, -b, -c) + assert p.position_wrt(C.origin) == p.position_wrt(C) == \ + -2 * v + p1 = p.locate_new('p1', 2*v) + assert p1.position_wrt(C.origin) == Vector.zero + assert p1.express_coordinates(C) == (0, 0, 0) + p2 = p.locate_new('p2', A.i) + assert p1.position_wrt(p2) == 2*v - A.i + assert p2.express_coordinates(C) == (-2*a + 1, -2*b, -2*c) + + +def test_create_new(): + a = CoordSys3D('a') + c = a.create_new('c', transformation='spherical') + assert c._parent == a + assert c.transformation_to_parent() == \ + (c.r*sin(c.theta)*cos(c.phi), c.r*sin(c.theta)*sin(c.phi), c.r*cos(c.theta)) + assert c.transformation_from_parent() == \ + (sqrt(a.x**2 + a.y**2 + a.z**2), acos(a.z/sqrt(a.x**2 + a.y**2 + a.z**2)), atan2(a.y, a.x)) + + +def test_evalf(): + A = CoordSys3D('A') + v = 3*A.i + 4*A.j + a*A.k + assert v.n() == v.evalf() + assert v.evalf(subs={a:1}) == v.subs(a, 1).evalf() + + +def test_lame_coefficients(): + a = CoordSys3D('a', 'spherical') + assert a.lame_coefficients() == (1, a.r, sin(a.theta)*a.r) + a = CoordSys3D('a') + assert a.lame_coefficients() == (1, 1, 1) + a = CoordSys3D('a', 'cartesian') + assert a.lame_coefficients() == (1, 1, 1) + a = CoordSys3D('a', 'cylindrical') + assert a.lame_coefficients() == (1, a.r, 1) + + +def test_transformation_equations(): + + x, y, z = symbols('x y z') + # Str + a = CoordSys3D('a', transformation='spherical', + variable_names=["r", "theta", "phi"]) + r, theta, phi = a.base_scalars() + + assert r == a.r + assert theta == a.theta + assert phi == a.phi + + raises(AttributeError, lambda: a.x) + raises(AttributeError, lambda: a.y) + raises(AttributeError, lambda: a.z) + + assert a.transformation_to_parent() == ( + r*sin(theta)*cos(phi), + r*sin(theta)*sin(phi), + r*cos(theta) + ) + assert a.lame_coefficients() == (1, r, r*sin(theta)) + assert a.transformation_from_parent_function()(x, y, z) == ( + sqrt(x ** 2 + y ** 2 + z ** 2), + acos((z) / sqrt(x**2 + y**2 + z**2)), + atan2(y, x) + ) + a = CoordSys3D('a', transformation='cylindrical', + variable_names=["r", "theta", "z"]) + r, theta, z = a.base_scalars() + assert a.transformation_to_parent() == ( + r*cos(theta), + r*sin(theta), + z + ) + assert a.lame_coefficients() == (1, a.r, 1) + assert a.transformation_from_parent_function()(x, y, z) == (sqrt(x**2 + y**2), + atan2(y, x), z) + + a = CoordSys3D('a', 'cartesian') + assert a.transformation_to_parent() == (a.x, a.y, a.z) + assert a.lame_coefficients() == (1, 1, 1) + assert a.transformation_from_parent_function()(x, y, z) == (x, y, z) + + # Variables and expressions + + # Cartesian with equation tuple: + x, y, z = symbols('x y z') + a = CoordSys3D('a', ((x, y, z), (x, y, z))) + a._calculate_inv_trans_equations() + assert a.transformation_to_parent() == (a.x1, a.x2, a.x3) + assert a.lame_coefficients() == (1, 1, 1) + assert a.transformation_from_parent_function()(x, y, z) == (x, y, z) + r, theta, z = symbols("r theta z") + + # Cylindrical with equation tuple: + a = CoordSys3D('a', [(r, theta, z), (r*cos(theta), r*sin(theta), z)], + variable_names=["r", "theta", "z"]) + r, theta, z = a.base_scalars() + assert a.transformation_to_parent() == ( + r*cos(theta), r*sin(theta), z + ) + assert a.lame_coefficients() == ( + sqrt(sin(theta)**2 + cos(theta)**2), + sqrt(r**2*sin(theta)**2 + r**2*cos(theta)**2), + 1 + ) # ==> this should simplify to (1, r, 1), tests are too slow with `simplify`. + + # Definitions with `lambda`: + + # Cartesian with `lambda` + a = CoordSys3D('a', lambda x, y, z: (x, y, z)) + assert a.transformation_to_parent() == (a.x1, a.x2, a.x3) + assert a.lame_coefficients() == (1, 1, 1) + a._calculate_inv_trans_equations() + assert a.transformation_from_parent_function()(x, y, z) == (x, y, z) + + # Spherical with `lambda` + a = CoordSys3D('a', lambda r, theta, phi: (r*sin(theta)*cos(phi), r*sin(theta)*sin(phi), r*cos(theta)), + variable_names=["r", "theta", "phi"]) + r, theta, phi = a.base_scalars() + assert a.transformation_to_parent() == ( + r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta) + ) + assert a.lame_coefficients() == ( + sqrt(sin(phi)**2*sin(theta)**2 + sin(theta)**2*cos(phi)**2 + cos(theta)**2), + sqrt(r**2*sin(phi)**2*cos(theta)**2 + r**2*sin(theta)**2 + r**2*cos(phi)**2*cos(theta)**2), + sqrt(r**2*sin(phi)**2*sin(theta)**2 + r**2*sin(theta)**2*cos(phi)**2) + ) # ==> this should simplify to (1, r, sin(theta)*r), `simplify` is too slow. + + # Cylindrical with `lambda` + a = CoordSys3D('a', lambda r, theta, z: + (r*cos(theta), r*sin(theta), z), + variable_names=["r", "theta", "z"] + ) + r, theta, z = a.base_scalars() + assert a.transformation_to_parent() == (r*cos(theta), r*sin(theta), z) + assert a.lame_coefficients() == ( + sqrt(sin(theta)**2 + cos(theta)**2), + sqrt(r**2*sin(theta)**2 + r**2*cos(theta)**2), + 1 + ) # ==> this should simplify to (1, a.x, 1) + + raises(TypeError, lambda: CoordSys3D('a', transformation={ + x: x*sin(y)*cos(z), y:x*sin(y)*sin(z), z: x*cos(y)})) + + +def test_check_orthogonality(): + x, y, z = symbols('x y z') + u,v = symbols('u, v') + a = CoordSys3D('a', transformation=((x, y, z), (x*sin(y)*cos(z), x*sin(y)*sin(z), x*cos(y)))) + assert a._check_orthogonality(a._transformation) is True + a = CoordSys3D('a', transformation=((x, y, z), (x * cos(y), x * sin(y), z))) + assert a._check_orthogonality(a._transformation) is True + a = CoordSys3D('a', transformation=((u, v, z), (cosh(u) * cos(v), sinh(u) * sin(v), z))) + assert a._check_orthogonality(a._transformation) is True + + raises(ValueError, lambda: CoordSys3D('a', transformation=((x, y, z), (x, x, z)))) + raises(ValueError, lambda: CoordSys3D('a', transformation=( + (x, y, z), (x*sin(y/2)*cos(z), x*sin(y)*sin(z), x*cos(y))))) + + +def test_rotation_trans_equations(): + a = CoordSys3D('a') + from sympy.core.symbol import symbols + q0 = symbols('q0') + assert a._rotation_trans_equations(a._parent_rotation_matrix, a.base_scalars()) == (a.x, a.y, a.z) + assert a._rotation_trans_equations(a._inverse_rotation_matrix(), a.base_scalars()) == (a.x, a.y, a.z) + b = a.orient_new_axis('b', 0, -a.k) + assert b._rotation_trans_equations(b._parent_rotation_matrix, b.base_scalars()) == (b.x, b.y, b.z) + assert b._rotation_trans_equations(b._inverse_rotation_matrix(), b.base_scalars()) == (b.x, b.y, b.z) + c = a.orient_new_axis('c', q0, -a.k) + assert c._rotation_trans_equations(c._parent_rotation_matrix, c.base_scalars()) == \ + (-sin(q0) * c.y + cos(q0) * c.x, sin(q0) * c.x + cos(q0) * c.y, c.z) + assert c._rotation_trans_equations(c._inverse_rotation_matrix(), c.base_scalars()) == \ + (sin(q0) * c.y + cos(q0) * c.x, -sin(q0) * c.x + cos(q0) * c.y, c.z) diff --git a/.venv/lib/python3.12/site-packages/sympy/vector/tests/test_dyadic.py b/.venv/lib/python3.12/site-packages/sympy/vector/tests/test_dyadic.py new file mode 100644 index 0000000000000000000000000000000000000000..2e396fcf2a81af897b59c0065f6b15f5c6933222 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/vector/tests/test_dyadic.py @@ -0,0 +1,134 @@ +from sympy.core.numbers import pi +from sympy.core.symbol import symbols +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.matrices.immutable import ImmutableDenseMatrix as Matrix +from sympy.simplify.simplify import simplify +from sympy.vector import (CoordSys3D, Vector, Dyadic, + DyadicAdd, DyadicMul, DyadicZero, + BaseDyadic, express) + + +A = CoordSys3D('A') + + +def test_dyadic(): + a, b = symbols('a, b') + assert Dyadic.zero != 0 + assert isinstance(Dyadic.zero, DyadicZero) + assert BaseDyadic(A.i, A.j) != BaseDyadic(A.j, A.i) + assert (BaseDyadic(Vector.zero, A.i) == + BaseDyadic(A.i, Vector.zero) == Dyadic.zero) + + d1 = A.i | A.i + d2 = A.j | A.j + d3 = A.i | A.j + + assert isinstance(d1, BaseDyadic) + d_mul = a*d1 + assert isinstance(d_mul, DyadicMul) + assert d_mul.base_dyadic == d1 + assert d_mul.measure_number == a + assert isinstance(a*d1 + b*d3, DyadicAdd) + assert d1 == A.i.outer(A.i) + assert d3 == A.i.outer(A.j) + v1 = a*A.i - A.k + v2 = A.i + b*A.j + assert v1 | v2 == v1.outer(v2) == a * (A.i|A.i) + (a*b) * (A.i|A.j) +\ + - (A.k|A.i) - b * (A.k|A.j) + assert d1 * 0 == Dyadic.zero + assert d1 != Dyadic.zero + assert d1 * 2 == 2 * (A.i | A.i) + assert d1 / 2. == 0.5 * d1 + + assert d1.dot(0 * d1) == Vector.zero + assert d1 & d2 == Dyadic.zero + assert d1.dot(A.i) == A.i == d1 & A.i + + assert d1.cross(Vector.zero) == Dyadic.zero + assert d1.cross(A.i) == Dyadic.zero + assert d1 ^ A.j == d1.cross(A.j) + assert d1.cross(A.k) == - A.i | A.j + assert d2.cross(A.i) == - A.j | A.k == d2 ^ A.i + + assert A.i ^ d1 == Dyadic.zero + assert A.j.cross(d1) == - A.k | A.i == A.j ^ d1 + assert Vector.zero.cross(d1) == Dyadic.zero + assert A.k ^ d1 == A.j | A.i + assert A.i.dot(d1) == A.i & d1 == A.i + assert A.j.dot(d1) == Vector.zero + assert Vector.zero.dot(d1) == Vector.zero + assert A.j & d2 == A.j + + assert d1.dot(d3) == d1 & d3 == A.i | A.j == d3 + assert d3 & d1 == Dyadic.zero + + q = symbols('q') + B = A.orient_new_axis('B', q, A.k) + assert express(d1, B) == express(d1, B, B) + + expr1 = ((cos(q)**2) * (B.i | B.i) + (-sin(q) * cos(q)) * + (B.i | B.j) + (-sin(q) * cos(q)) * (B.j | B.i) + (sin(q)**2) * + (B.j | B.j)) + assert (express(d1, B) - expr1).simplify() == Dyadic.zero + + expr2 = (cos(q)) * (B.i | A.i) + (-sin(q)) * (B.j | A.i) + assert (express(d1, B, A) - expr2).simplify() == Dyadic.zero + + expr3 = (cos(q)) * (A.i | B.i) + (-sin(q)) * (A.i | B.j) + assert (express(d1, A, B) - expr3).simplify() == Dyadic.zero + + assert d1.to_matrix(A) == Matrix([[1, 0, 0], [0, 0, 0], [0, 0, 0]]) + assert d1.to_matrix(A, B) == Matrix([[cos(q), -sin(q), 0], + [0, 0, 0], + [0, 0, 0]]) + assert d3.to_matrix(A) == Matrix([[0, 1, 0], [0, 0, 0], [0, 0, 0]]) + a, b, c, d, e, f = symbols('a, b, c, d, e, f') + v1 = a * A.i + b * A.j + c * A.k + v2 = d * A.i + e * A.j + f * A.k + d4 = v1.outer(v2) + assert d4.to_matrix(A) == Matrix([[a * d, a * e, a * f], + [b * d, b * e, b * f], + [c * d, c * e, c * f]]) + d5 = v1.outer(v1) + C = A.orient_new_axis('C', q, A.i) + for expected, actual in zip(C.rotation_matrix(A) * d5.to_matrix(A) * \ + C.rotation_matrix(A).T, d5.to_matrix(C)): + assert (expected - actual).simplify() == 0 + + +def test_dyadic_simplify(): + x, y, z, k, n, m, w, f, s, A = symbols('x, y, z, k, n, m, w, f, s, A') + N = CoordSys3D('N') + + dy = N.i | N.i + test1 = (1 / x + 1 / y) * dy + assert (N.i & test1 & N.i) != (x + y) / (x * y) + test1 = test1.simplify() + assert test1.simplify() == simplify(test1) + assert (N.i & test1 & N.i) == (x + y) / (x * y) + + test2 = (A**2 * s**4 / (4 * pi * k * m**3)) * dy + test2 = test2.simplify() + assert (N.i & test2 & N.i) == (A**2 * s**4 / (4 * pi * k * m**3)) + + test3 = ((4 + 4 * x - 2 * (2 + 2 * x)) / (2 + 2 * x)) * dy + test3 = test3.simplify() + assert (N.i & test3 & N.i) == 0 + + test4 = ((-4 * x * y**2 - 2 * y**3 - 2 * x**2 * y) / (x + y)**2) * dy + test4 = test4.simplify() + assert (N.i & test4 & N.i) == -2 * y + + +def test_dyadic_srepr(): + from sympy.printing.repr import srepr + N = CoordSys3D('N') + + dy = N.i | N.j + res = "BaseDyadic(CoordSys3D(Str('N'), Tuple(ImmutableDenseMatrix([["\ + "Integer(1), Integer(0), Integer(0)], [Integer(0), Integer(1), "\ + "Integer(0)], [Integer(0), Integer(0), Integer(1)]]), "\ + "VectorZero())).i, CoordSys3D(Str('N'), Tuple(ImmutableDenseMatrix("\ + "[[Integer(1), Integer(0), Integer(0)], [Integer(0), Integer(1), "\ + "Integer(0)], [Integer(0), Integer(0), Integer(1)]]), VectorZero())).j)" + assert srepr(dy) == res diff --git a/.venv/lib/python3.12/site-packages/sympy/vector/tests/test_field_functions.py b/.venv/lib/python3.12/site-packages/sympy/vector/tests/test_field_functions.py new file mode 100644 index 0000000000000000000000000000000000000000..035c2ce0234b81069c5ad8dcb1c74f4de0164a8f --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/vector/tests/test_field_functions.py @@ -0,0 +1,321 @@ +from sympy.core.function import Derivative +from sympy.vector.vector import Vector +from sympy.vector.coordsysrect import CoordSys3D +from sympy.simplify import simplify +from sympy.core.symbol import symbols +from sympy.core import S +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.vector.vector import Dot +from sympy.vector.operators import curl, divergence, gradient, Gradient, Divergence, Cross +from sympy.vector.deloperator import Del +from sympy.vector.functions import (is_conservative, is_solenoidal, + scalar_potential, directional_derivative, + laplacian, scalar_potential_difference) +from sympy.testing.pytest import raises + +C = CoordSys3D('C') +i, j, k = C.base_vectors() +x, y, z = C.base_scalars() +delop = Del() +a, b, c, q = symbols('a b c q') + + +def test_del_operator(): + # Tests for curl + + assert delop ^ Vector.zero == Vector.zero + assert ((delop ^ Vector.zero).doit() == Vector.zero == + curl(Vector.zero)) + assert delop.cross(Vector.zero) == delop ^ Vector.zero + assert (delop ^ i).doit() == Vector.zero + assert delop.cross(2*y**2*j, doit=True) == Vector.zero + assert delop.cross(2*y**2*j) == delop ^ 2*y**2*j + v = x*y*z * (i + j + k) + assert ((delop ^ v).doit() == + (-x*y + x*z)*i + (x*y - y*z)*j + (-x*z + y*z)*k == + curl(v)) + assert delop ^ v == delop.cross(v) + assert (delop.cross(2*x**2*j) == + (Derivative(0, C.y) - Derivative(2*C.x**2, C.z))*C.i + + (-Derivative(0, C.x) + Derivative(0, C.z))*C.j + + (-Derivative(0, C.y) + Derivative(2*C.x**2, C.x))*C.k) + assert (delop.cross(2*x**2*j, doit=True) == 4*x*k == + curl(2*x**2*j)) + + #Tests for divergence + assert delop & Vector.zero is S.Zero == divergence(Vector.zero) + assert (delop & Vector.zero).doit() is S.Zero + assert delop.dot(Vector.zero) == delop & Vector.zero + assert (delop & i).doit() is S.Zero + assert (delop & x**2*i).doit() == 2*x == divergence(x**2*i) + assert (delop.dot(v, doit=True) == x*y + y*z + z*x == + divergence(v)) + assert delop & v == delop.dot(v) + assert delop.dot(1/(x*y*z) * (i + j + k), doit=True) == \ + - 1 / (x*y*z**2) - 1 / (x*y**2*z) - 1 / (x**2*y*z) + v = x*i + y*j + z*k + assert (delop & v == Derivative(C.x, C.x) + + Derivative(C.y, C.y) + Derivative(C.z, C.z)) + assert delop.dot(v, doit=True) == 3 == divergence(v) + assert delop & v == delop.dot(v) + assert simplify((delop & v).doit()) == 3 + + #Tests for gradient + assert (delop.gradient(0, doit=True) == Vector.zero == + gradient(0)) + assert delop.gradient(0) == delop(0) + assert (delop(S.Zero)).doit() == Vector.zero + assert (delop(x) == (Derivative(C.x, C.x))*C.i + + (Derivative(C.x, C.y))*C.j + (Derivative(C.x, C.z))*C.k) + assert (delop(x)).doit() == i == gradient(x) + assert (delop(x*y*z) == + (Derivative(C.x*C.y*C.z, C.x))*C.i + + (Derivative(C.x*C.y*C.z, C.y))*C.j + + (Derivative(C.x*C.y*C.z, C.z))*C.k) + assert (delop.gradient(x*y*z, doit=True) == + y*z*i + z*x*j + x*y*k == + gradient(x*y*z)) + assert delop(x*y*z) == delop.gradient(x*y*z) + assert (delop(2*x**2)).doit() == 4*x*i + assert ((delop(a*sin(y) / x)).doit() == + -a*sin(y)/x**2 * i + a*cos(y)/x * j) + + #Tests for directional derivative + assert (Vector.zero & delop)(a) is S.Zero + assert ((Vector.zero & delop)(a)).doit() is S.Zero + assert ((v & delop)(Vector.zero)).doit() == Vector.zero + assert ((v & delop)(S.Zero)).doit() is S.Zero + assert ((i & delop)(x)).doit() == 1 + assert ((j & delop)(y)).doit() == 1 + assert ((k & delop)(z)).doit() == 1 + assert ((i & delop)(x*y*z)).doit() == y*z + assert ((v & delop)(x)).doit() == x + assert ((v & delop)(x*y*z)).doit() == 3*x*y*z + assert (v & delop)(x + y + z) == C.x + C.y + C.z + assert ((v & delop)(x + y + z)).doit() == x + y + z + assert ((v & delop)(v)).doit() == v + assert ((i & delop)(v)).doit() == i + assert ((j & delop)(v)).doit() == j + assert ((k & delop)(v)).doit() == k + assert ((v & delop)(Vector.zero)).doit() == Vector.zero + + # Tests for laplacian on scalar fields + assert laplacian(x*y*z) is S.Zero + assert laplacian(x**2) == S(2) + assert laplacian(x**2*y**2*z**2) == \ + 2*y**2*z**2 + 2*x**2*z**2 + 2*x**2*y**2 + A = CoordSys3D('A', transformation="spherical", variable_names=["r", "theta", "phi"]) + B = CoordSys3D('B', transformation='cylindrical', variable_names=["r", "theta", "z"]) + assert laplacian(A.r + A.theta + A.phi) == 2/A.r + cos(A.theta)/(A.r**2*sin(A.theta)) + assert laplacian(B.r + B.theta + B.z) == 1/B.r + + # Tests for laplacian on vector fields + assert laplacian(x*y*z*(i + j + k)) == Vector.zero + assert laplacian(x*y**2*z*(i + j + k)) == \ + 2*x*z*i + 2*x*z*j + 2*x*z*k + + +def test_product_rules(): + """ + Tests the six product rules defined with respect to the Del + operator + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Del + + """ + + #Define the scalar and vector functions + f = 2*x*y*z + g = x*y + y*z + z*x + u = x**2*i + 4*j - y**2*z*k + v = 4*i + x*y*z*k + + # First product rule + lhs = delop(f * g, doit=True) + rhs = (f * delop(g) + g * delop(f)).doit() + assert simplify(lhs) == simplify(rhs) + + # Second product rule + lhs = delop(u & v).doit() + rhs = ((u ^ (delop ^ v)) + (v ^ (delop ^ u)) + \ + ((u & delop)(v)) + ((v & delop)(u))).doit() + assert simplify(lhs) == simplify(rhs) + + # Third product rule + lhs = (delop & (f*v)).doit() + rhs = ((f * (delop & v)) + (v & (delop(f)))).doit() + assert simplify(lhs) == simplify(rhs) + + # Fourth product rule + lhs = (delop & (u ^ v)).doit() + rhs = ((v & (delop ^ u)) - (u & (delop ^ v))).doit() + assert simplify(lhs) == simplify(rhs) + + # Fifth product rule + lhs = (delop ^ (f * v)).doit() + rhs = (((delop(f)) ^ v) + (f * (delop ^ v))).doit() + assert simplify(lhs) == simplify(rhs) + + # Sixth product rule + lhs = (delop ^ (u ^ v)).doit() + rhs = (u * (delop & v) - v * (delop & u) + + (v & delop)(u) - (u & delop)(v)).doit() + assert simplify(lhs) == simplify(rhs) + + +P = C.orient_new_axis('P', q, C.k) # type: ignore +scalar_field = 2*x**2*y*z +grad_field = gradient(scalar_field) +vector_field = y**2*i + 3*x*j + 5*y*z*k +curl_field = curl(vector_field) + + +def test_conservative(): + assert is_conservative(Vector.zero) is True + assert is_conservative(i) is True + assert is_conservative(2 * i + 3 * j + 4 * k) is True + assert (is_conservative(y*z*i + x*z*j + x*y*k) is + True) + assert is_conservative(x * j) is False + assert is_conservative(grad_field) is True + assert is_conservative(curl_field) is False + assert (is_conservative(4*x*y*z*i + 2*x**2*z*j) is + False) + assert is_conservative(z*P.i + P.x*k) is True + + +def test_solenoidal(): + assert is_solenoidal(Vector.zero) is True + assert is_solenoidal(i) is True + assert is_solenoidal(2 * i + 3 * j + 4 * k) is True + assert (is_solenoidal(y*z*i + x*z*j + x*y*k) is + True) + assert is_solenoidal(y * j) is False + assert is_solenoidal(grad_field) is False + assert is_solenoidal(curl_field) is True + assert is_solenoidal((-2*y + 3)*k) is True + assert is_solenoidal(cos(q)*i + sin(q)*j + cos(q)*P.k) is True + assert is_solenoidal(z*P.i + P.x*k) is True + + +def test_directional_derivative(): + assert directional_derivative(C.x*C.y*C.z, 3*C.i + 4*C.j + C.k) == C.x*C.y + 4*C.x*C.z + 3*C.y*C.z + assert directional_derivative(5*C.x**2*C.z, 3*C.i + 4*C.j + C.k) == 5*C.x**2 + 30*C.x*C.z + assert directional_derivative(5*C.x**2*C.z, 4*C.j) is S.Zero + + D = CoordSys3D("D", "spherical", variable_names=["r", "theta", "phi"], + vector_names=["e_r", "e_theta", "e_phi"]) + r, theta, phi = D.base_scalars() + e_r, e_theta, e_phi = D.base_vectors() + assert directional_derivative(r**2*e_r, e_r) == 2*r*e_r + assert directional_derivative(5*r**2*phi, 3*e_r + 4*e_theta + e_phi) == 5*r**2 + 30*r*phi + + +def test_scalar_potential(): + assert scalar_potential(Vector.zero, C) == 0 + assert scalar_potential(i, C) == x + assert scalar_potential(j, C) == y + assert scalar_potential(k, C) == z + assert scalar_potential(y*z*i + x*z*j + x*y*k, C) == x*y*z + assert scalar_potential(grad_field, C) == scalar_field + assert scalar_potential(z*P.i + P.x*k, C) == x*z*cos(q) + y*z*sin(q) + assert scalar_potential(z*P.i + P.x*k, P) == P.x*P.z + raises(ValueError, lambda: scalar_potential(x*j, C)) + + +def test_scalar_potential_difference(): + point1 = C.origin.locate_new('P1', 1*i + 2*j + 3*k) + point2 = C.origin.locate_new('P2', 4*i + 5*j + 6*k) + genericpointC = C.origin.locate_new('RP', x*i + y*j + z*k) + genericpointP = P.origin.locate_new('PP', P.x*P.i + P.y*P.j + P.z*P.k) + assert scalar_potential_difference(S.Zero, C, point1, point2) == 0 + assert (scalar_potential_difference(scalar_field, C, C.origin, + genericpointC) == + scalar_field) + assert (scalar_potential_difference(grad_field, C, C.origin, + genericpointC) == + scalar_field) + assert scalar_potential_difference(grad_field, C, point1, point2) == 948 + assert (scalar_potential_difference(y*z*i + x*z*j + + x*y*k, C, point1, + genericpointC) == + x*y*z - 6) + potential_diff_P = (2*P.z*(P.x*sin(q) + P.y*cos(q))* + (P.x*cos(q) - P.y*sin(q))**2) + assert (scalar_potential_difference(grad_field, P, P.origin, + genericpointP).simplify() == + potential_diff_P.simplify()) + + +def test_differential_operators_curvilinear_system(): + A = CoordSys3D('A', transformation="spherical", variable_names=["r", "theta", "phi"]) + B = CoordSys3D('B', transformation='cylindrical', variable_names=["r", "theta", "z"]) + # Test for spherical coordinate system and gradient + assert gradient(3*A.r + 4*A.theta) == 3*A.i + 4/A.r*A.j + assert gradient(3*A.r*A.phi + 4*A.theta) == 3*A.phi*A.i + 4/A.r*A.j + (3/sin(A.theta))*A.k + assert gradient(0*A.r + 0*A.theta+0*A.phi) == Vector.zero + assert gradient(A.r*A.theta*A.phi) == A.theta*A.phi*A.i + A.phi*A.j + (A.theta/sin(A.theta))*A.k + # Test for spherical coordinate system and divergence + assert divergence(A.r * A.i + A.theta * A.j + A.phi * A.k) == \ + (sin(A.theta)*A.r + cos(A.theta)*A.r*A.theta)/(sin(A.theta)*A.r**2) + 3 + 1/(sin(A.theta)*A.r) + assert divergence(3*A.r*A.phi*A.i + A.theta*A.j + A.r*A.theta*A.phi*A.k) == \ + (sin(A.theta)*A.r + cos(A.theta)*A.r*A.theta)/(sin(A.theta)*A.r**2) + 9*A.phi + A.theta/sin(A.theta) + assert divergence(Vector.zero) == 0 + assert divergence(0*A.i + 0*A.j + 0*A.k) == 0 + # Test for spherical coordinate system and curl + assert curl(A.r*A.i + A.theta*A.j + A.phi*A.k) == \ + (cos(A.theta)*A.phi/(sin(A.theta)*A.r))*A.i + (-A.phi/A.r)*A.j + A.theta/A.r*A.k + assert curl(A.r*A.j + A.phi*A.k) == (cos(A.theta)*A.phi/(sin(A.theta)*A.r))*A.i + (-A.phi/A.r)*A.j + 2*A.k + + # Test for cylindrical coordinate system and gradient + assert gradient(0*B.r + 0*B.theta+0*B.z) == Vector.zero + assert gradient(B.r*B.theta*B.z) == B.theta*B.z*B.i + B.z*B.j + B.r*B.theta*B.k + assert gradient(3*B.r) == 3*B.i + assert gradient(2*B.theta) == 2/B.r * B.j + assert gradient(4*B.z) == 4*B.k + # Test for cylindrical coordinate system and divergence + assert divergence(B.r*B.i + B.theta*B.j + B.z*B.k) == 3 + 1/B.r + assert divergence(B.r*B.j + B.z*B.k) == 1 + # Test for cylindrical coordinate system and curl + assert curl(B.r*B.j + B.z*B.k) == 2*B.k + assert curl(3*B.i + 2/B.r*B.j + 4*B.k) == Vector.zero + +def test_mixed_coordinates(): + # gradient + a = CoordSys3D('a') + b = CoordSys3D('b') + c = CoordSys3D('c') + assert gradient(a.x*b.y) == b.y*a.i + a.x*b.j + assert gradient(3*cos(q)*a.x*b.x+a.y*(a.x+(cos(q)+b.x))) ==\ + (a.y + 3*b.x*cos(q))*a.i + (a.x + b.x + cos(q))*a.j + (3*a.x*cos(q) + a.y)*b.i + # Some tests need further work: + # assert gradient(a.x*(cos(a.x+b.x))) == (cos(a.x + b.x))*a.i + a.x*Gradient(cos(a.x + b.x)) + # assert gradient(cos(a.x + b.x)*cos(a.x + b.z)) == Gradient(cos(a.x + b.x)*cos(a.x + b.z)) + assert gradient(a.x**b.y) == Gradient(a.x**b.y) + # assert gradient(cos(a.x+b.y)*a.z) == None + assert gradient(cos(a.x*b.y)) == Gradient(cos(a.x*b.y)) + assert gradient(3*cos(q)*a.x*b.x*a.z*a.y+ b.y*b.z + cos(a.x+a.y)*b.z) == \ + (3*a.y*a.z*b.x*cos(q) - b.z*sin(a.x + a.y))*a.i + \ + (3*a.x*a.z*b.x*cos(q) - b.z*sin(a.x + a.y))*a.j + (3*a.x*a.y*b.x*cos(q))*a.k + \ + (3*a.x*a.y*a.z*cos(q))*b.i + b.z*b.j + (b.y + cos(a.x + a.y))*b.k + # divergence + assert divergence(a.i*a.x+a.j*a.y+a.z*a.k + b.i*b.x+b.j*b.y+b.z*b.k + c.i*c.x+c.j*c.y+c.z*c.k) == S(9) + # assert divergence(3*a.i*a.x*cos(a.x+b.z) + a.j*b.x*c.z) == None + assert divergence(3*a.i*a.x*a.z + b.j*b.x*c.z + 3*a.j*a.z*a.y) == \ + 6*a.z + b.x*Dot(b.j, c.k) + assert divergence(3*cos(q)*a.x*b.x*b.i*c.x) == \ + 3*a.x*b.x*cos(q)*Dot(b.i, c.i) + 3*a.x*c.x*cos(q) + 3*b.x*c.x*cos(q)*Dot(b.i, a.i) + assert divergence(a.x*b.x*c.x*Cross(a.x*a.i, a.y*b.j)) ==\ + a.x*b.x*c.x*Divergence(Cross(a.x*a.i, a.y*b.j)) + \ + b.x*c.x*Dot(Cross(a.x*a.i, a.y*b.j), a.i) + \ + a.x*c.x*Dot(Cross(a.x*a.i, a.y*b.j), b.i) + \ + a.x*b.x*Dot(Cross(a.x*a.i, a.y*b.j), c.i) + assert divergence(a.x*b.x*c.x*(a.x*a.i + b.x*b.i)) == \ + 4*a.x*b.x*c.x +\ + a.x**2*c.x*Dot(a.i, b.i) +\ + a.x**2*b.x*Dot(a.i, c.i) +\ + b.x**2*c.x*Dot(b.i, a.i) +\ + a.x*b.x**2*Dot(b.i, c.i) diff --git a/.venv/lib/python3.12/site-packages/sympy/vector/tests/test_functions.py b/.venv/lib/python3.12/site-packages/sympy/vector/tests/test_functions.py new file mode 100644 index 0000000000000000000000000000000000000000..dfdf9821b6c853755ce12d0cbdfa599bd4f312e4 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/vector/tests/test_functions.py @@ -0,0 +1,184 @@ +from sympy.vector.vector import Vector +from sympy.vector.coordsysrect import CoordSys3D +from sympy.vector.functions import express, matrix_to_vector, orthogonalize +from sympy.core.numbers import Rational +from sympy.core.singleton import S +from sympy.core.symbol import symbols +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.matrices.immutable import ImmutableDenseMatrix as Matrix +from sympy.testing.pytest import raises + +N = CoordSys3D('N') +q1, q2, q3, q4, q5 = symbols('q1 q2 q3 q4 q5') +A = N.orient_new_axis('A', q1, N.k) # type: ignore +B = A.orient_new_axis('B', q2, A.i) +C = B.orient_new_axis('C', q3, B.j) + + +def test_express(): + assert express(Vector.zero, N) == Vector.zero + assert express(S.Zero, N) is S.Zero + assert express(A.i, C) == cos(q3)*C.i + sin(q3)*C.k + assert express(A.j, C) == sin(q2)*sin(q3)*C.i + cos(q2)*C.j - \ + sin(q2)*cos(q3)*C.k + assert express(A.k, C) == -sin(q3)*cos(q2)*C.i + sin(q2)*C.j + \ + cos(q2)*cos(q3)*C.k + assert express(A.i, N) == cos(q1)*N.i + sin(q1)*N.j + assert express(A.j, N) == -sin(q1)*N.i + cos(q1)*N.j + assert express(A.k, N) == N.k + assert express(A.i, A) == A.i + assert express(A.j, A) == A.j + assert express(A.k, A) == A.k + assert express(A.i, B) == B.i + assert express(A.j, B) == cos(q2)*B.j - sin(q2)*B.k + assert express(A.k, B) == sin(q2)*B.j + cos(q2)*B.k + assert express(A.i, C) == cos(q3)*C.i + sin(q3)*C.k + assert express(A.j, C) == sin(q2)*sin(q3)*C.i + cos(q2)*C.j - \ + sin(q2)*cos(q3)*C.k + assert express(A.k, C) == -sin(q3)*cos(q2)*C.i + sin(q2)*C.j + \ + cos(q2)*cos(q3)*C.k + # Check to make sure UnitVectors get converted properly + assert express(N.i, N) == N.i + assert express(N.j, N) == N.j + assert express(N.k, N) == N.k + assert express(N.i, A) == (cos(q1)*A.i - sin(q1)*A.j) + assert express(N.j, A) == (sin(q1)*A.i + cos(q1)*A.j) + assert express(N.k, A) == A.k + assert express(N.i, B) == (cos(q1)*B.i - sin(q1)*cos(q2)*B.j + + sin(q1)*sin(q2)*B.k) + assert express(N.j, B) == (sin(q1)*B.i + cos(q1)*cos(q2)*B.j - + sin(q2)*cos(q1)*B.k) + assert express(N.k, B) == (sin(q2)*B.j + cos(q2)*B.k) + assert express(N.i, C) == ( + (cos(q1)*cos(q3) - sin(q1)*sin(q2)*sin(q3))*C.i - + sin(q1)*cos(q2)*C.j + + (sin(q3)*cos(q1) + sin(q1)*sin(q2)*cos(q3))*C.k) + assert express(N.j, C) == ( + (sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1))*C.i + + cos(q1)*cos(q2)*C.j + + (sin(q1)*sin(q3) - sin(q2)*cos(q1)*cos(q3))*C.k) + assert express(N.k, C) == (-sin(q3)*cos(q2)*C.i + sin(q2)*C.j + + cos(q2)*cos(q3)*C.k) + + assert express(A.i, N) == (cos(q1)*N.i + sin(q1)*N.j) + assert express(A.j, N) == (-sin(q1)*N.i + cos(q1)*N.j) + assert express(A.k, N) == N.k + assert express(A.i, A) == A.i + assert express(A.j, A) == A.j + assert express(A.k, A) == A.k + assert express(A.i, B) == B.i + assert express(A.j, B) == (cos(q2)*B.j - sin(q2)*B.k) + assert express(A.k, B) == (sin(q2)*B.j + cos(q2)*B.k) + assert express(A.i, C) == (cos(q3)*C.i + sin(q3)*C.k) + assert express(A.j, C) == (sin(q2)*sin(q3)*C.i + cos(q2)*C.j - + sin(q2)*cos(q3)*C.k) + assert express(A.k, C) == (-sin(q3)*cos(q2)*C.i + sin(q2)*C.j + + cos(q2)*cos(q3)*C.k) + + assert express(B.i, N) == (cos(q1)*N.i + sin(q1)*N.j) + assert express(B.j, N) == (-sin(q1)*cos(q2)*N.i + + cos(q1)*cos(q2)*N.j + sin(q2)*N.k) + assert express(B.k, N) == (sin(q1)*sin(q2)*N.i - + sin(q2)*cos(q1)*N.j + cos(q2)*N.k) + assert express(B.i, A) == A.i + assert express(B.j, A) == (cos(q2)*A.j + sin(q2)*A.k) + assert express(B.k, A) == (-sin(q2)*A.j + cos(q2)*A.k) + assert express(B.i, B) == B.i + assert express(B.j, B) == B.j + assert express(B.k, B) == B.k + assert express(B.i, C) == (cos(q3)*C.i + sin(q3)*C.k) + assert express(B.j, C) == C.j + assert express(B.k, C) == (-sin(q3)*C.i + cos(q3)*C.k) + + assert express(C.i, N) == ( + (cos(q1)*cos(q3) - sin(q1)*sin(q2)*sin(q3))*N.i + + (sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1))*N.j - + sin(q3)*cos(q2)*N.k) + assert express(C.j, N) == ( + -sin(q1)*cos(q2)*N.i + cos(q1)*cos(q2)*N.j + sin(q2)*N.k) + assert express(C.k, N) == ( + (sin(q3)*cos(q1) + sin(q1)*sin(q2)*cos(q3))*N.i + + (sin(q1)*sin(q3) - sin(q2)*cos(q1)*cos(q3))*N.j + + cos(q2)*cos(q3)*N.k) + assert express(C.i, A) == (cos(q3)*A.i + sin(q2)*sin(q3)*A.j - + sin(q3)*cos(q2)*A.k) + assert express(C.j, A) == (cos(q2)*A.j + sin(q2)*A.k) + assert express(C.k, A) == (sin(q3)*A.i - sin(q2)*cos(q3)*A.j + + cos(q2)*cos(q3)*A.k) + assert express(C.i, B) == (cos(q3)*B.i - sin(q3)*B.k) + assert express(C.j, B) == B.j + assert express(C.k, B) == (sin(q3)*B.i + cos(q3)*B.k) + assert express(C.i, C) == C.i + assert express(C.j, C) == C.j + assert express(C.k, C) == C.k == (C.k) + + # Check to make sure Vectors get converted back to UnitVectors + assert N.i == express((cos(q1)*A.i - sin(q1)*A.j), N).simplify() + assert N.j == express((sin(q1)*A.i + cos(q1)*A.j), N).simplify() + assert N.i == express((cos(q1)*B.i - sin(q1)*cos(q2)*B.j + + sin(q1)*sin(q2)*B.k), N).simplify() + assert N.j == express((sin(q1)*B.i + cos(q1)*cos(q2)*B.j - + sin(q2)*cos(q1)*B.k), N).simplify() + assert N.k == express((sin(q2)*B.j + cos(q2)*B.k), N).simplify() + + + assert A.i == express((cos(q1)*N.i + sin(q1)*N.j), A).simplify() + assert A.j == express((-sin(q1)*N.i + cos(q1)*N.j), A).simplify() + + assert A.j == express((cos(q2)*B.j - sin(q2)*B.k), A).simplify() + assert A.k == express((sin(q2)*B.j + cos(q2)*B.k), A).simplify() + + assert A.i == express((cos(q3)*C.i + sin(q3)*C.k), A).simplify() + assert A.j == express((sin(q2)*sin(q3)*C.i + cos(q2)*C.j - + sin(q2)*cos(q3)*C.k), A).simplify() + + assert A.k == express((-sin(q3)*cos(q2)*C.i + sin(q2)*C.j + + cos(q2)*cos(q3)*C.k), A).simplify() + assert B.i == express((cos(q1)*N.i + sin(q1)*N.j), B).simplify() + assert B.j == express((-sin(q1)*cos(q2)*N.i + + cos(q1)*cos(q2)*N.j + sin(q2)*N.k), B).simplify() + + assert B.k == express((sin(q1)*sin(q2)*N.i - + sin(q2)*cos(q1)*N.j + cos(q2)*N.k), B).simplify() + + assert B.j == express((cos(q2)*A.j + sin(q2)*A.k), B).simplify() + assert B.k == express((-sin(q2)*A.j + cos(q2)*A.k), B).simplify() + assert B.i == express((cos(q3)*C.i + sin(q3)*C.k), B).simplify() + assert B.k == express((-sin(q3)*C.i + cos(q3)*C.k), B).simplify() + assert C.i == express((cos(q3)*A.i + sin(q2)*sin(q3)*A.j - + sin(q3)*cos(q2)*A.k), C).simplify() + assert C.j == express((cos(q2)*A.j + sin(q2)*A.k), C).simplify() + assert C.k == express((sin(q3)*A.i - sin(q2)*cos(q3)*A.j + + cos(q2)*cos(q3)*A.k), C).simplify() + assert C.i == express((cos(q3)*B.i - sin(q3)*B.k), C).simplify() + assert C.k == express((sin(q3)*B.i + cos(q3)*B.k), C).simplify() + + +def test_matrix_to_vector(): + m = Matrix([[1], [2], [3]]) + assert matrix_to_vector(m, C) == C.i + 2*C.j + 3*C.k + m = Matrix([[0], [0], [0]]) + assert matrix_to_vector(m, N) == matrix_to_vector(m, C) == \ + Vector.zero + m = Matrix([[q1], [q2], [q3]]) + assert matrix_to_vector(m, N) == q1*N.i + q2*N.j + q3*N.k + + +def test_orthogonalize(): + C = CoordSys3D('C') + a, b = symbols('a b', integer=True) + i, j, k = C.base_vectors() + v1 = i + 2*j + v2 = 2*i + 3*j + v3 = 3*i + 5*j + v4 = 3*i + j + v5 = 2*i + 2*j + v6 = a*i + b*j + v7 = 4*a*i + 4*b*j + assert orthogonalize(v1, v2) == [C.i + 2*C.j, C.i*Rational(2, 5) + -C.j/5] + # from wikipedia + assert orthogonalize(v4, v5, orthonormal=True) == \ + [(3*sqrt(10))*C.i/10 + (sqrt(10))*C.j/10, (-sqrt(10))*C.i/10 + (3*sqrt(10))*C.j/10] + raises(ValueError, lambda: orthogonalize(v1, v2, v3)) + raises(ValueError, lambda: orthogonalize(v6, v7)) diff --git a/.venv/lib/python3.12/site-packages/sympy/vector/tests/test_implicitregion.py b/.venv/lib/python3.12/site-packages/sympy/vector/tests/test_implicitregion.py new file mode 100644 index 0000000000000000000000000000000000000000..3686d847a7f165cb5ba9aeb813e5922aaa17e1e0 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/vector/tests/test_implicitregion.py @@ -0,0 +1,90 @@ +from sympy.core.relational import Eq +from sympy.core.singleton import S +from sympy.abc import x, y, z, s, t +from sympy.sets import FiniteSet, EmptySet +from sympy.geometry import Point +from sympy.vector import ImplicitRegion +from sympy.testing.pytest import raises + + +def test_ImplicitRegion(): + ellipse = ImplicitRegion((x, y), (x**2/4 + y**2/16 - 1)) + assert ellipse.equation == x**2/4 + y**2/16 - 1 + assert ellipse.variables == (x, y) + assert ellipse.degree == 2 + r = ImplicitRegion((x, y, z), Eq(x**4 + y**2 - x*y, 6)) + assert r.equation == x**4 + y**2 - x*y - 6 + assert r.variables == (x, y, z) + assert r.degree == 4 + + +def test_regular_point(): + r1 = ImplicitRegion((x,), x**2 - 16) + assert r1.regular_point() == (-4,) + c1 = ImplicitRegion((x, y), x**2 + y**2 - 4) + assert c1.regular_point() == (0, -2) + c2 = ImplicitRegion((x, y), (x - S(5)/2)**2 + y**2 - (S(1)/4)**2) + assert c2.regular_point() == (S(5)/2, -S(1)/4) + c3 = ImplicitRegion((x, y), (y - 5)**2 - 16*(x - 5)) + assert c3.regular_point() == (5, 5) + r2 = ImplicitRegion((x, y), x**2 - 4*x*y - 3*y**2 + 4*x + 8*y - 5) + assert r2.regular_point() == (S(4)/7, S(9)/7) + r3 = ImplicitRegion((x, y), x**2 - 2*x*y + 3*y**2 - 2*x - 5*y + 3/2) + raises(ValueError, lambda: r3.regular_point()) + + +def test_singular_points_and_multiplicty(): + r1 = ImplicitRegion((x, y, z), Eq(x + y + z, 0)) + assert r1.singular_points() == EmptySet + r2 = ImplicitRegion((x, y, z), x*y*z + y**4 -x**2*z**2) + assert r2.singular_points() == FiniteSet((0, 0, z), (x, 0, 0)) + assert r2.multiplicity((0, 0, 0)) == 3 + assert r2.multiplicity((0, 0, 6)) == 2 + r3 = ImplicitRegion((x, y, z), z**2 - x**2 - y**2) + assert r3.singular_points() == FiniteSet((0, 0, 0)) + assert r3.multiplicity((0, 0, 0)) == 2 + r4 = ImplicitRegion((x, y), x**2 + y**2 - 2*x) + assert r4.singular_points() == EmptySet + assert r4.multiplicity(Point(1, 3)) == 0 + + +def test_rational_parametrization(): + p = ImplicitRegion((x,), x - 2) + assert p.rational_parametrization() == (x - 2,) + + line = ImplicitRegion((x, y), Eq(y, 3*x + 2)) + assert line.rational_parametrization() == (x, 3*x + 2) + + circle1 = ImplicitRegion((x, y), (x-2)**2 + (y+3)**2 - 4) + assert circle1.rational_parametrization(parameters=t) == (4*t/(t**2 + 1) + 2, 4*t**2/(t**2 + 1) - 5) + circle2 = ImplicitRegion((x, y), (x - S.Half)**2 + y**2 - (S(1)/2)**2) + + assert circle2.rational_parametrization(parameters=t) == (t/(t**2 + 1) + S(1)/2, t**2/(t**2 + 1) - S(1)/2) + circle3 = ImplicitRegion((x, y), Eq(x**2 + y**2, 2*x)) + assert circle3.rational_parametrization(parameters=(t,)) == (2*t/(t**2 + 1) + 1, 2*t**2/(t**2 + 1) - 1) + + parabola = ImplicitRegion((x, y), (y - 3)**2 - 4*(x + 6)) + assert parabola.rational_parametrization(t) == (-6 + 4/t**2, 3 + 4/t) + + rect_hyperbola = ImplicitRegion((x, y), x*y - 1) + assert rect_hyperbola.rational_parametrization(t) == (-1 + (t + 1)/t, t) + + cubic_curve = ImplicitRegion((x, y), x**3 + x**2 - y**2) + assert cubic_curve.rational_parametrization(parameters=(t)) == (t**2 - 1, t*(t**2 - 1)) + cuspidal = ImplicitRegion((x, y), (x**3 - y**2)) + assert cuspidal.rational_parametrization(t) == (t**2, t**3) + + I = ImplicitRegion((x, y), x**3 + x**2 - y**2) + assert I.rational_parametrization(t) == (t**2 - 1, t*(t**2 - 1)) + + sphere = ImplicitRegion((x, y, z), Eq(x**2 + y**2 + z**2, 2*x)) + assert sphere.rational_parametrization(parameters=(s, t)) == (2/(s**2 + t**2 + 1), 2*t/(s**2 + t**2 + 1), 2*s/(s**2 + t**2 + 1)) + + conic = ImplicitRegion((x, y), Eq(x**2 + 4*x*y + 3*y**2 + x - y + 10, 0)) + assert conic.rational_parametrization(t) == ( + S(17)/2 + 4/(3*t**2 + 4*t + 1), 4*t/(3*t**2 + 4*t + 1) - S(11)/2) + + r1 = ImplicitRegion((x, y), y**2 - x**3 + x) + raises(NotImplementedError, lambda: r1.rational_parametrization()) + r2 = ImplicitRegion((x, y), y**2 - x**3 - x**2 + 1) + raises(NotImplementedError, lambda: r2.rational_parametrization()) diff --git a/.venv/lib/python3.12/site-packages/sympy/vector/tests/test_integrals.py b/.venv/lib/python3.12/site-packages/sympy/vector/tests/test_integrals.py new file mode 100644 index 0000000000000000000000000000000000000000..84c900d038e214df1ea59a8cd8fb2929005c3674 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/vector/tests/test_integrals.py @@ -0,0 +1,106 @@ +from sympy.core.numbers import pi +from sympy.core.singleton import S +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.testing.pytest import raises +from sympy.vector.coordsysrect import CoordSys3D +from sympy.vector.integrals import ParametricIntegral, vector_integrate +from sympy.vector.parametricregion import ParametricRegion +from sympy.vector.implicitregion import ImplicitRegion +from sympy.abc import x, y, z, u, v, r, t, theta, phi +from sympy.geometry import Point, Segment, Curve, Circle, Polygon, Plane + +C = CoordSys3D('C') + +def test_parametric_lineintegrals(): + halfcircle = ParametricRegion((4*cos(theta), 4*sin(theta)), (theta, -pi/2, pi/2)) + assert ParametricIntegral(C.x*C.y**4, halfcircle) == S(8192)/5 + + curve = ParametricRegion((t, t**2, t**3), (t, 0, 1)) + field1 = 8*C.x**2*C.y*C.z*C.i + 5*C.z*C.j - 4*C.x*C.y*C.k + assert ParametricIntegral(field1, curve) == 1 + line = ParametricRegion((4*t - 1, 2 - 2*t, t), (t, 0, 1)) + assert ParametricIntegral(C.x*C.z*C.i - C.y*C.z*C.k, line) == 3 + + assert ParametricIntegral(4*C.x**3, ParametricRegion((1, t), (t, 0, 2))) == 8 + + helix = ParametricRegion((cos(t), sin(t), 3*t), (t, 0, 4*pi)) + assert ParametricIntegral(C.x*C.y*C.z, helix) == -3*sqrt(10)*pi + + field2 = C.y*C.i + C.z*C.j + C.z*C.k + assert ParametricIntegral(field2, ParametricRegion((cos(t), sin(t), t**2), (t, 0, pi))) == -5*pi/2 + pi**4/2 + +def test_parametric_surfaceintegrals(): + + semisphere = ParametricRegion((2*sin(phi)*cos(theta), 2*sin(phi)*sin(theta), 2*cos(phi)),\ + (theta, 0, 2*pi), (phi, 0, pi/2)) + assert ParametricIntegral(C.z, semisphere) == 8*pi + + cylinder = ParametricRegion((sqrt(3)*cos(theta), sqrt(3)*sin(theta), z), (z, 0, 6), (theta, 0, 2*pi)) + assert ParametricIntegral(C.y, cylinder) == 0 + + cone = ParametricRegion((v*cos(u), v*sin(u), v), (u, 0, 2*pi), (v, 0, 1)) + assert ParametricIntegral(C.x*C.i + C.y*C.j + C.z**4*C.k, cone) == pi/3 + + triangle1 = ParametricRegion((x, y), (x, 0, 2), (y, 0, 10 - 5*x)) + triangle2 = ParametricRegion((x, y), (y, 0, 10 - 5*x), (x, 0, 2)) + assert ParametricIntegral(-15.6*C.y*C.k, triangle1) == ParametricIntegral(-15.6*C.y*C.k, triangle2) + assert ParametricIntegral(C.z, triangle1) == 10*C.z + +def test_parametric_volumeintegrals(): + + cube = ParametricRegion((x, y, z), (x, 0, 1), (y, 0, 1), (z, 0, 1)) + assert ParametricIntegral(1, cube) == 1 + + solidsphere1 = ParametricRegion((r*sin(phi)*cos(theta), r*sin(phi)*sin(theta), r*cos(phi)),\ + (r, 0, 2), (theta, 0, 2*pi), (phi, 0, pi)) + solidsphere2 = ParametricRegion((r*sin(phi)*cos(theta), r*sin(phi)*sin(theta), r*cos(phi)),\ + (r, 0, 2), (phi, 0, pi), (theta, 0, 2*pi)) + assert ParametricIntegral(C.x**2 + C.y**2, solidsphere1) == -256*pi/15 + assert ParametricIntegral(C.x**2 + C.y**2, solidsphere2) == 256*pi/15 + + region_under_plane1 = ParametricRegion((x, y, z), (x, 0, 3), (y, 0, -2*x/3 + 2),\ + (z, 0, 6 - 2*x - 3*y)) + region_under_plane2 = ParametricRegion((x, y, z), (x, 0, 3), (z, 0, 6 - 2*x - 3*y),\ + (y, 0, -2*x/3 + 2)) + + assert ParametricIntegral(C.x*C.i + C.j - 100*C.k, region_under_plane1) == \ + ParametricIntegral(C.x*C.i + C.j - 100*C.k, region_under_plane2) + assert ParametricIntegral(2*C.x, region_under_plane2) == -9 + +def test_vector_integrate(): + halfdisc = ParametricRegion((r*cos(theta), r* sin(theta)), (r, -2, 2), (theta, 0, pi)) + assert vector_integrate(C.x**2, halfdisc) == 4*pi + assert vector_integrate(C.x, ParametricRegion((t, t**2), (t, 2, 3))) == -17*sqrt(17)/12 + 37*sqrt(37)/12 + + assert vector_integrate(C.y**3*C.z, (C.x, 0, 3), (C.y, -1, 4)) == 765*C.z/4 + + s1 = Segment(Point(0, 0), Point(0, 1)) + assert vector_integrate(-15*C.y, s1) == S(-15)/2 + s2 = Segment(Point(4, 3, 9), Point(1, 1, 7)) + assert vector_integrate(C.y*C.i, s2) == -6 + + curve = Curve((sin(t), cos(t)), (t, 0, 2)) + assert vector_integrate(5*C.z, curve) == 10*C.z + + c1 = Circle(Point(2, 3), 6) + assert vector_integrate(C.x*C.y, c1) == 72*pi + c2 = Circle(Point(0, 0), Point(1, 1), Point(1, 0)) + assert vector_integrate(1, c2) == c2.circumference + + triangle = Polygon((0, 0), (1, 0), (1, 1)) + assert vector_integrate(C.x*C.i - 14*C.y*C.j, triangle) == 0 + p1, p2, p3, p4 = [(0, 0), (1, 0), (5, 1), (0, 1)] + poly = Polygon(p1, p2, p3, p4) + assert vector_integrate(-23*C.z, poly) == -161*C.z - 23*sqrt(17)*C.z + + point = Point(2, 3) + assert vector_integrate(C.i*C.y, point) == ParametricIntegral(C.y*C.i, ParametricRegion((2, 3))) + + c3 = ImplicitRegion((x, y), x**2 + y**2 - 4) + assert vector_integrate(45, c3) == 180*pi + c4 = ImplicitRegion((x, y), (x - 3)**2 + (y - 4)**2 - 9) + assert vector_integrate(1, c4) == 6*pi + + pl = Plane(Point(1, 1, 1), Point(2, 3, 4), Point(2, 2, 2)) + raises(ValueError, lambda: vector_integrate(C.x*C.z*C.i + C.k, pl)) diff --git a/.venv/lib/python3.12/site-packages/sympy/vector/tests/test_operators.py b/.venv/lib/python3.12/site-packages/sympy/vector/tests/test_operators.py new file mode 100644 index 0000000000000000000000000000000000000000..5734edadd00547c67d6f864b50afd966ad8392a6 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/vector/tests/test_operators.py @@ -0,0 +1,43 @@ +from sympy.vector import CoordSys3D, Gradient, Divergence, Curl, VectorZero, Laplacian +from sympy.printing.repr import srepr + +R = CoordSys3D('R') +s1 = R.x*R.y*R.z # type: ignore +s2 = R.x + 3*R.y**2 # type: ignore +s3 = R.x**2 + R.y**2 + R.z**2 # type: ignore +v1 = R.x*R.i + R.z*R.z*R.j # type: ignore +v2 = R.x*R.i + R.y*R.j + R.z*R.k # type: ignore +v3 = R.x**2*R.i + R.y**2*R.j + R.z**2*R.k # type: ignore + + +def test_Gradient(): + assert Gradient(s1) == Gradient(R.x*R.y*R.z) + assert Gradient(s2) == Gradient(R.x + 3*R.y**2) + assert Gradient(s1).doit() == R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k + assert Gradient(s2).doit() == R.i + 6*R.y*R.j + + +def test_Divergence(): + assert Divergence(v1) == Divergence(R.x*R.i + R.z*R.z*R.j) + assert Divergence(v2) == Divergence(R.x*R.i + R.y*R.j + R.z*R.k) + assert Divergence(v1).doit() == 1 + assert Divergence(v2).doit() == 3 + # issue 22384 + Rc = CoordSys3D('R', transformation='cylindrical') + assert Divergence(Rc.i).doit() == 1/Rc.r + + +def test_Curl(): + assert Curl(v1) == Curl(R.x*R.i + R.z*R.z*R.j) + assert Curl(v2) == Curl(R.x*R.i + R.y*R.j + R.z*R.k) + assert Curl(v1).doit() == (-2*R.z)*R.i + assert Curl(v2).doit() == VectorZero() + + +def test_Laplacian(): + assert Laplacian(s3) == Laplacian(R.x**2 + R.y**2 + R.z**2) + assert Laplacian(v3) == Laplacian(R.x**2*R.i + R.y**2*R.j + R.z**2*R.k) + assert Laplacian(s3).doit() == 6 + assert Laplacian(v3).doit() == 2*R.i + 2*R.j + 2*R.k + assert srepr(Laplacian(s3)) == \ + 'Laplacian(Add(Pow(R.x, Integer(2)), Pow(R.y, Integer(2)), Pow(R.z, Integer(2))))' diff --git a/.venv/lib/python3.12/site-packages/sympy/vector/tests/test_parametricregion.py b/.venv/lib/python3.12/site-packages/sympy/vector/tests/test_parametricregion.py new file mode 100644 index 0000000000000000000000000000000000000000..e785b96744f9e2c39e91b997fcb70f8a921256bd --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/vector/tests/test_parametricregion.py @@ -0,0 +1,97 @@ +from sympy.core.numbers import pi +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.vector.coordsysrect import CoordSys3D +from sympy.vector.parametricregion import ParametricRegion, parametric_region_list +from sympy.geometry import Point, Segment, Curve, Ellipse, Line, Parabola, Polygon +from sympy.testing.pytest import raises +from sympy.abc import a, b, r, t, x, y, z, theta, phi + + +C = CoordSys3D('C') + +def test_ParametricRegion(): + + point = ParametricRegion((3, 4)) + assert point.definition == (3, 4) + assert point.parameters == () + assert point.limits == {} + assert point.dimensions == 0 + + # line x = y + line_xy = ParametricRegion((y, y), (y, 1, 5)) + assert line_xy .definition == (y, y) + assert line_xy.parameters == (y,) + assert line_xy.dimensions == 1 + + # line y = z + line_yz = ParametricRegion((x,t,t), x, (t, 1, 2)) + assert line_yz.definition == (x,t,t) + assert line_yz.parameters == (x, t) + assert line_yz.limits == {t: (1, 2)} + assert line_yz.dimensions == 1 + + p1 = ParametricRegion((9*a, -16*b), (a, 0, 2), (b, -1, 5)) + assert p1.definition == (9*a, -16*b) + assert p1.parameters == (a, b) + assert p1.limits == {a: (0, 2), b: (-1, 5)} + assert p1.dimensions == 2 + + p2 = ParametricRegion((t, t**3), t) + assert p2.parameters == (t,) + assert p2.limits == {} + assert p2.dimensions == 0 + + circle = ParametricRegion((r*cos(theta), r*sin(theta)), r, (theta, 0, 2*pi)) + assert circle.definition == (r*cos(theta), r*sin(theta)) + assert circle.dimensions == 1 + + halfdisc = ParametricRegion((r*cos(theta), r*sin(theta)), (r, -2, 2), (theta, 0, pi)) + assert halfdisc.definition == (r*cos(theta), r*sin(theta)) + assert halfdisc.parameters == (r, theta) + assert halfdisc.limits == {r: (-2, 2), theta: (0, pi)} + assert halfdisc.dimensions == 2 + + ellipse = ParametricRegion((a*cos(t), b*sin(t)), (t, 0, 8)) + assert ellipse.parameters == (t,) + assert ellipse.limits == {t: (0, 8)} + assert ellipse.dimensions == 1 + + cylinder = ParametricRegion((r*cos(theta), r*sin(theta), z), (r, 0, 1), (theta, 0, 2*pi), (z, 0, 4)) + assert cylinder.parameters == (r, theta, z) + assert cylinder.dimensions == 3 + + sphere = ParametricRegion((r*sin(phi)*cos(theta),r*sin(phi)*sin(theta), r*cos(phi)), + r, (theta, 0, 2*pi), (phi, 0, pi)) + assert sphere.definition == (r*sin(phi)*cos(theta),r*sin(phi)*sin(theta), r*cos(phi)) + assert sphere.parameters == (r, theta, phi) + assert sphere.dimensions == 2 + + raises(ValueError, lambda: ParametricRegion((a*t**2, 2*a*t), (a, -2))) + raises(ValueError, lambda: ParametricRegion((a, b), (a**2, sin(b)), (a, 2, 4, 6))) + + +def test_parametric_region_list(): + + point = Point(-5, 12) + assert parametric_region_list(point) == [ParametricRegion((-5, 12))] + + e = Ellipse(Point(2, 8), 2, 6) + assert parametric_region_list(e, t) == [ParametricRegion((2*cos(t) + 2, 6*sin(t) + 8), (t, 0, 2*pi))] + + c = Curve((t, t**3), (t, 5, 3)) + assert parametric_region_list(c) == [ParametricRegion((t, t**3), (t, 5, 3))] + + s = Segment(Point(2, 11, -6), Point(0, 2, 5)) + assert parametric_region_list(s, t) == [ParametricRegion((2 - 2*t, 11 - 9*t, 11*t - 6), (t, 0, 1))] + s1 = Segment(Point(0, 0), (1, 0)) + assert parametric_region_list(s1, t) == [ParametricRegion((t, 0), (t, 0, 1))] + s2 = Segment(Point(1, 2, 3), Point(1, 2, 5)) + assert parametric_region_list(s2, t) == [ParametricRegion((1, 2, 2*t + 3), (t, 0, 1))] + s3 = Segment(Point(12, 56), Point(12, 56)) + assert parametric_region_list(s3) == [ParametricRegion((12, 56))] + + poly = Polygon((1,3), (-3, 8), (2, 4)) + assert parametric_region_list(poly, t) == [ParametricRegion((1 - 4*t, 5*t + 3), (t, 0, 1)), ParametricRegion((5*t - 3, 8 - 4*t), (t, 0, 1)), ParametricRegion((2 - t, 4 - t), (t, 0, 1))] + + p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7,8))) + raises(ValueError, lambda: parametric_region_list(p1)) diff --git a/.venv/lib/python3.12/site-packages/sympy/vector/tests/test_printing.py b/.venv/lib/python3.12/site-packages/sympy/vector/tests/test_printing.py new file mode 100644 index 0000000000000000000000000000000000000000..ae76905e967bdf93485f135c6a69f968e1208986 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/vector/tests/test_printing.py @@ -0,0 +1,221 @@ +# -*- coding: utf-8 -*- +from sympy.core.function import Function +from sympy.integrals.integrals import Integral +from sympy.printing.latex import latex +from sympy.printing.pretty import pretty as xpretty +from sympy.vector import CoordSys3D, Del, Vector, express +from sympy.abc import a, b, c +from sympy.testing.pytest import XFAIL + + +def pretty(expr): + """ASCII pretty-printing""" + return xpretty(expr, use_unicode=False, wrap_line=False) + + +def upretty(expr): + """Unicode pretty-printing""" + return xpretty(expr, use_unicode=True, wrap_line=False) + + +# Initialize the basic and tedious vector/dyadic expressions +# needed for testing. +# Some of the pretty forms shown denote how the expressions just +# above them should look with pretty printing. +N = CoordSys3D('N') +C = N.orient_new_axis('C', a, N.k) # type: ignore +v = [] +d = [] +v.append(Vector.zero) +v.append(N.i) # type: ignore +v.append(-N.i) # type: ignore +v.append(N.i + N.j) # type: ignore +v.append(a*N.i) # type: ignore +v.append(a*N.i - b*N.j) # type: ignore +v.append((a**2 + N.x)*N.i + N.k) # type: ignore +v.append((a**2 + b)*N.i + 3*(C.y - c)*N.k) # type: ignore +f = Function('f') +v.append(N.j - (Integral(f(b)) - C.x**2)*N.k) # type: ignore +upretty_v_8 = """\ + ⎛ 2 ⌠ ⎞ \n\ +j_N + ⎜x_C - ⎮ f(b) db⎟ k_N\n\ + ⎝ ⌡ ⎠ \ +""" +pretty_v_8 = """\ +j_N + / / \\\n\ + | 2 | |\n\ + |x_C - | f(b) db|\n\ + | | |\n\ + \\ / / \ +""" + +v.append(N.i + C.k) # type: ignore +v.append(express(N.i, C)) # type: ignore +v.append((a**2 + b)*N.i + (Integral(f(b)))*N.k) # type: ignore +upretty_v_11 = """\ +⎛ 2 ⎞ ⎛⌠ ⎞ \n\ +⎝a + b⎠ i_N + ⎜⎮ f(b) db⎟ k_N\n\ + ⎝⌡ ⎠ \ +""" +pretty_v_11 = """\ +/ 2 \\ + / / \\\n\ +\\a + b/ i_N| | |\n\ + | | f(b) db|\n\ + | | |\n\ + \\/ / \ +""" + +for x in v: + d.append(x | N.k) # type: ignore +s = 3*N.x**2*C.y # type: ignore +upretty_s = """\ + 2\n\ +3⋅y_C⋅x_N \ +""" +pretty_s = """\ + 2\n\ +3*y_C*x_N \ +""" + +# This is the pretty form for ((a**2 + b)*N.i + 3*(C.y - c)*N.k) | N.k +upretty_d_7 = """\ +⎛ 2 ⎞ \n\ +⎝a + b⎠ (i_N|k_N) + (3⋅y_C - 3⋅c) (k_N|k_N)\ +""" +pretty_d_7 = """\ +/ 2 \\ (i_N|k_N) + (3*y_C - 3*c) (k_N|k_N)\n\ +\\a + b/ \ +""" + + +def test_str_printing(): + assert str(v[0]) == '0' + assert str(v[1]) == 'N.i' + assert str(v[2]) == '(-1)*N.i' + assert str(v[3]) == 'N.i + N.j' + assert str(v[8]) == 'N.j + (C.x**2 - Integral(f(b), b))*N.k' + assert str(v[9]) == 'C.k + N.i' + assert str(s) == '3*C.y*N.x**2' + assert str(d[0]) == '0' + assert str(d[1]) == '(N.i|N.k)' + assert str(d[4]) == 'a*(N.i|N.k)' + assert str(d[5]) == 'a*(N.i|N.k) + (-b)*(N.j|N.k)' + assert str(d[8]) == ('(N.j|N.k) + (C.x**2 - ' + + 'Integral(f(b), b))*(N.k|N.k)') + + +@XFAIL +def test_pretty_printing_ascii(): + assert pretty(v[0]) == '0' + assert pretty(v[1]) == 'i_N' + assert pretty(v[5]) == '(a) i_N + (-b) j_N' + assert pretty(v[8]) == pretty_v_8 + assert pretty(v[2]) == '(-1) i_N' + assert pretty(v[11]) == pretty_v_11 + assert pretty(s) == pretty_s + assert pretty(d[0]) == '(0|0)' + assert pretty(d[5]) == '(a) (i_N|k_N) + (-b) (j_N|k_N)' + assert pretty(d[7]) == pretty_d_7 + assert pretty(d[10]) == '(cos(a)) (i_C|k_N) + (-sin(a)) (j_C|k_N)' + + +def test_pretty_print_unicode_v(): + assert upretty(v[0]) == '0' + assert upretty(v[1]) == 'i_N' + assert upretty(v[5]) == '(a) i_N + (-b) j_N' + # Make sure the printing works in other objects + assert upretty(v[5].args) == '((a) i_N, (-b) j_N)' + assert upretty(v[8]) == upretty_v_8 + assert upretty(v[2]) == '(-1) i_N' + assert upretty(v[11]) == upretty_v_11 + assert upretty(s) == upretty_s + assert upretty(d[0]) == '(0|0)' + assert upretty(d[5]) == '(a) (i_N|k_N) + (-b) (j_N|k_N)' + assert upretty(d[7]) == upretty_d_7 + assert upretty(d[10]) == '(cos(a)) (i_C|k_N) + (-sin(a)) (j_C|k_N)' + + +def test_latex_printing(): + assert latex(v[0]) == '\\mathbf{\\hat{0}}' + assert latex(v[1]) == '\\mathbf{\\hat{i}_{N}}' + assert latex(v[2]) == '- \\mathbf{\\hat{i}_{N}}' + assert latex(v[5]) == ('\\left(a\\right)\\mathbf{\\hat{i}_{N}} + ' + + '\\left(- b\\right)\\mathbf{\\hat{j}_{N}}') + assert latex(v[6]) == ('\\left(\\mathbf{{x}_{N}} + a^{2}\\right)\\mathbf{\\hat{i}_' + + '{N}} + \\mathbf{\\hat{k}_{N}}') + assert latex(v[8]) == ('\\mathbf{\\hat{j}_{N}} + \\left(\\mathbf{{x}_' + + '{C}}^{2} - \\int f{\\left(b \\right)}\\,' + + ' db\\right)\\mathbf{\\hat{k}_{N}}') + assert latex(s) == '3 \\mathbf{{y}_{C}} \\mathbf{{x}_{N}}^{2}' + assert latex(d[0]) == '(\\mathbf{\\hat{0}}|\\mathbf{\\hat{0}})' + assert latex(d[4]) == ('\\left(a\\right)\\left(\\mathbf{\\hat{i}_{N}}{\\middle|}' + + '\\mathbf{\\hat{k}_{N}}\\right)') + assert latex(d[9]) == ('\\left(\\mathbf{\\hat{k}_{C}}{\\middle|}' + + '\\mathbf{\\hat{k}_{N}}\\right) + \\left(' + + '\\mathbf{\\hat{i}_{N}}{\\middle|}\\mathbf{' + + '\\hat{k}_{N}}\\right)') + assert latex(d[11]) == ('\\left(a^{2} + b\\right)\\left(\\mathbf{\\hat{i}_{N}}' + + '{\\middle|}\\mathbf{\\hat{k}_{N}}\\right) + ' + + '\\left(\\int f{\\left(b \\right)}\\, db\\right)\\left(' + + '\\mathbf{\\hat{k}_{N}}{\\middle|}\\mathbf{' + + '\\hat{k}_{N}}\\right)') + +def test_issue_23058(): + from sympy import symbols, sin, cos, pi, UnevaluatedExpr + + delop = Del() + CC_ = CoordSys3D("C") + y = CC_.y + xhat = CC_.i + + t = symbols("t") + ten = symbols("10", positive=True) + eps, mu = 4*pi*ten**(-11), ten**(-5) + + Bx = 2 * ten**(-4) * cos(ten**5 * t) * sin(ten**(-3) * y) + vecB = Bx * xhat + vecE = (1/eps) * Integral(delop.cross(vecB/mu).doit(), t) + vecE = vecE.doit() + + vecB_str = """\ +⎛ ⎛y_C⎞ ⎛ 5 ⎞⎞ \n\ +⎜2⋅sin⎜───⎟⋅cos⎝10 ⋅t⎠⎟ i_C\n\ +⎜ ⎜ 3⎟ ⎟ \n\ +⎜ ⎝10 ⎠ ⎟ \n\ +⎜─────────────────────⎟ \n\ +⎜ 4 ⎟ \n\ +⎝ 10 ⎠ \ +""" + vecE_str = """\ +⎛ 4 ⎛ 5 ⎞ ⎛y_C⎞ ⎞ \n\ +⎜-10 ⋅sin⎝10 ⋅t⎠⋅cos⎜───⎟ ⎟ k_C\n\ +⎜ ⎜ 3⎟ ⎟ \n\ +⎜ ⎝10 ⎠ ⎟ \n\ +⎜─────────────────────────⎟ \n\ +⎝ 2⋅π ⎠ \ +""" + + assert upretty(vecB) == vecB_str + assert upretty(vecE) == vecE_str + + ten = UnevaluatedExpr(10) + eps, mu = 4*pi*ten**(-11), ten**(-5) + + Bx = 2 * ten**(-4) * cos(ten**5 * t) * sin(ten**(-3) * y) + vecB = Bx * xhat + + vecB_str = """\ +⎛ -4 ⎛ 5⎞ ⎛ -3⎞⎞ \n\ +⎝2⋅10 ⋅cos⎝t⋅10 ⎠⋅sin⎝y_C⋅10 ⎠⎠ i_C \ +""" + assert upretty(vecB) == vecB_str + +def test_custom_names(): + A = CoordSys3D('A', vector_names=['x', 'y', 'z'], + variable_names=['i', 'j', 'k']) + assert A.i.__str__() == 'A.i' + assert A.x.__str__() == 'A.x' + assert A.i._pretty_form == 'i_A' + assert A.x._pretty_form == 'x_A' + assert A.i._latex_form == r'\mathbf{{i}_{A}}' + assert A.x._latex_form == r"\mathbf{\hat{x}_{A}}" diff --git a/.venv/lib/python3.12/site-packages/sympy/vector/tests/test_vector.py b/.venv/lib/python3.12/site-packages/sympy/vector/tests/test_vector.py new file mode 100644 index 0000000000000000000000000000000000000000..daba6d6a02c87b41a8bf801eee9b9045897d0003 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/vector/tests/test_vector.py @@ -0,0 +1,342 @@ +from sympy.core import Rational, S, Add, Mul, I +from sympy.simplify import simplify, trigsimp +from sympy.core.function import (Derivative, Function, diff) +from sympy.core.numbers import pi +from sympy.core.symbol import symbols +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.integrals.integrals import Integral +from sympy.matrices.immutable import ImmutableDenseMatrix as Matrix +from sympy.vector.vector import Vector, BaseVector, VectorAdd, \ + VectorMul, VectorZero +from sympy.vector.coordsysrect import CoordSys3D +from sympy.vector.vector import Cross, Dot, cross +from sympy.testing.pytest import raises +from sympy.vector.kind import VectorKind +from sympy.core.kind import NumberKind +from sympy.testing.pytest import XFAIL + + +C = CoordSys3D('C') + +i, j, k = C.base_vectors() +a, b, c = symbols('a b c') + + +def test_cross(): + v1 = C.x * i + C.z * C.z * j + v2 = C.x * i + C.y * j + C.z * k + assert Cross(v1, v2) == Cross(C.x*C.i + C.z**2*C.j, C.x*C.i + C.y*C.j + C.z*C.k) + assert Cross(v1, v2).doit() == C.z**3*C.i + (-C.x*C.z)*C.j + (C.x*C.y - C.x*C.z**2)*C.k + assert cross(v1, v2) == C.z**3*C.i + (-C.x*C.z)*C.j + (C.x*C.y - C.x*C.z**2)*C.k + assert Cross(v1, v2) == -Cross(v2, v1) + # XXX: Cannot use Cross here. See XFAIL test below: + assert cross(v1, v2) + cross(v2, v1) == Vector.zero + + +@XFAIL +def test_cross_xfail(): + v1 = C.x * i + C.z * C.z * j + v2 = C.x * i + C.y * j + C.z * k + assert Cross(v1, v2) + Cross(v2, v1) == Vector.zero + + +def test_dot(): + v1 = C.x * i + C.z * C.z * j + v2 = C.x * i + C.y * j + C.z * k + assert Dot(v1, v2) == Dot(C.x*C.i + C.z**2*C.j, C.x*C.i + C.y*C.j + C.z*C.k) + assert Dot(v1, v2).doit() == C.x**2 + C.y*C.z**2 + assert Dot(v2, v1).doit() == C.x**2 + C.y*C.z**2 + assert Dot(v1, v2) == Dot(v2, v1) + + +def test_vector_sympy(): + """ + Test whether the Vector framework confirms to the hashing + and equality testing properties of SymPy. + """ + v1 = 3*j + assert v1 == j*3 + assert v1.components == {j: 3} + v2 = 3*i + 4*j + 5*k + v3 = 2*i + 4*j + i + 4*k + k + assert v3 == v2 + assert v3.__hash__() == v2.__hash__() + + +def test_kind(): + assert C.i.kind is VectorKind(NumberKind) + assert C.j.kind is VectorKind(NumberKind) + assert C.k.kind is VectorKind(NumberKind) + + assert C.x.kind is NumberKind + assert C.y.kind is NumberKind + assert C.z.kind is NumberKind + + assert Mul._kind_dispatcher(NumberKind, VectorKind(NumberKind)) is VectorKind(NumberKind) + assert Mul(2, C.i).kind is VectorKind(NumberKind) + + v1 = C.x * i + C.z * C.z * j + v2 = C.x * i + C.y * j + C.z * k + assert v1.kind is VectorKind(NumberKind) + assert v2.kind is VectorKind(NumberKind) + + assert (v1 + v2).kind is VectorKind(NumberKind) + assert Add(v1, v2).kind is VectorKind(NumberKind) + assert Cross(v1, v2).doit().kind is VectorKind(NumberKind) + assert VectorAdd(v1, v2).kind is VectorKind(NumberKind) + assert VectorMul(2, v1).kind is VectorKind(NumberKind) + assert VectorZero().kind is VectorKind(NumberKind) + + assert v1.projection(v2).kind is VectorKind(NumberKind) + assert v2.projection(v1).kind is VectorKind(NumberKind) + + +def test_vectoradd(): + assert isinstance(Add(C.i, C.j), VectorAdd) + v1 = C.x * i + C.z * C.z * j + v2 = C.x * i + C.y * j + C.z * k + assert isinstance(Add(v1, v2), VectorAdd) + + # https://github.com/sympy/sympy/issues/26121 + + E = Matrix([C.i, C.j, C.k]).T + a = Matrix([1, 2, 3]) + av = E*a + + assert av[0].kind == VectorKind() + assert isinstance(av[0], VectorAdd) + + +def test_vector(): + assert isinstance(i, BaseVector) + assert i != j + assert j != k + assert k != i + assert i - i == Vector.zero + assert i + Vector.zero == i + assert i - Vector.zero == i + assert Vector.zero != 0 + assert -Vector.zero == Vector.zero + + v1 = a*i + b*j + c*k + v2 = a**2*i + b**2*j + c**2*k + v3 = v1 + v2 + v4 = 2 * v1 + v5 = a * i + + assert isinstance(v1, VectorAdd) + assert v1 - v1 == Vector.zero + assert v1 + Vector.zero == v1 + assert v1.dot(i) == a + assert v1.dot(j) == b + assert v1.dot(k) == c + assert i.dot(v2) == a**2 + assert j.dot(v2) == b**2 + assert k.dot(v2) == c**2 + assert v3.dot(i) == a**2 + a + assert v3.dot(j) == b**2 + b + assert v3.dot(k) == c**2 + c + + assert v1 + v2 == v2 + v1 + assert v1 - v2 == -1 * (v2 - v1) + assert a * v1 == v1 * a + + assert isinstance(v5, VectorMul) + assert v5.base_vector == i + assert v5.measure_number == a + assert isinstance(v4, Vector) + assert isinstance(v4, VectorAdd) + assert isinstance(v4, Vector) + assert isinstance(Vector.zero, VectorZero) + assert isinstance(Vector.zero, Vector) + assert isinstance(v1 * 0, VectorZero) + + assert v1.to_matrix(C) == Matrix([[a], [b], [c]]) + + assert i.components == {i: 1} + assert v5.components == {i: a} + assert v1.components == {i: a, j: b, k: c} + + assert VectorAdd(v1, Vector.zero) == v1 + assert VectorMul(a, v1) == v1*a + assert VectorMul(1, i) == i + assert VectorAdd(v1, Vector.zero) == v1 + assert VectorMul(0, Vector.zero) == Vector.zero + raises(TypeError, lambda: v1.outer(1)) + raises(TypeError, lambda: v1.dot(1)) + + +def test_vector_magnitude_normalize(): + assert Vector.zero.magnitude() == 0 + assert Vector.zero.normalize() == Vector.zero + + assert i.magnitude() == 1 + assert j.magnitude() == 1 + assert k.magnitude() == 1 + assert i.normalize() == i + assert j.normalize() == j + assert k.normalize() == k + + v1 = a * i + assert v1.normalize() == (a/sqrt(a**2))*i + assert v1.magnitude() == sqrt(a**2) + + v2 = a*i + b*j + c*k + assert v2.magnitude() == sqrt(a**2 + b**2 + c**2) + assert v2.normalize() == v2 / v2.magnitude() + + v3 = i + j + assert v3.normalize() == (sqrt(2)/2)*C.i + (sqrt(2)/2)*C.j + + +def test_vector_simplify(): + A, s, k, m = symbols('A, s, k, m') + + test1 = (1 / a + 1 / b) * i + assert (test1 & i) != (a + b) / (a * b) + test1 = simplify(test1) + assert (test1 & i) == (a + b) / (a * b) + assert test1.simplify() == simplify(test1) + + test2 = (A**2 * s**4 / (4 * pi * k * m**3)) * i + test2 = simplify(test2) + assert (test2 & i) == (A**2 * s**4 / (4 * pi * k * m**3)) + + test3 = ((4 + 4 * a - 2 * (2 + 2 * a)) / (2 + 2 * a)) * i + test3 = simplify(test3) + assert (test3 & i) == 0 + + test4 = ((-4 * a * b**2 - 2 * b**3 - 2 * a**2 * b) / (a + b)**2) * i + test4 = simplify(test4) + assert (test4 & i) == -2 * b + + v = (sin(a)+cos(a))**2*i - j + assert trigsimp(v) == (2*sin(a + pi/4)**2)*i + (-1)*j + assert trigsimp(v) == v.trigsimp() + + assert simplify(Vector.zero) == Vector.zero + + +def test_vector_equals(): + assert (2*i).equals(j) is False + assert i.equals(i) is True + + # https://github.com/sympy/sympy/issues/25915 + A = (sqrt(2) + sqrt(6)) / sqrt(sqrt(3) + 2) + assert (A*i).equals(2*i) is True + assert (A*i).equals(3*i) is False + + # Test comparing vectors in different coordinate systems + D = C.orient_new_axis('D', pi/2, C.k) + assert (D.i).equals(C.j) is True + assert (D.i).equals(C.i) is False + + +def test_vector_conjugate(): + # https://github.com/sympy/sympy/issues/27094 + assert (I*i + (1 + I)*j + 2*k).conjugate() == -I*i + (1 - I)*j + 2*k + + +def test_vector_dot(): + assert i.dot(Vector.zero) == 0 + assert Vector.zero.dot(i) == 0 + assert i & Vector.zero == 0 + + assert i.dot(i) == 1 + assert i.dot(j) == 0 + assert i.dot(k) == 0 + assert i & i == 1 + assert i & j == 0 + assert i & k == 0 + + assert j.dot(i) == 0 + assert j.dot(j) == 1 + assert j.dot(k) == 0 + assert j & i == 0 + assert j & j == 1 + assert j & k == 0 + + assert k.dot(i) == 0 + assert k.dot(j) == 0 + assert k.dot(k) == 1 + assert k & i == 0 + assert k & j == 0 + assert k & k == 1 + + raises(TypeError, lambda: k.dot(1)) + + +def test_vector_cross(): + assert i.cross(Vector.zero) == Vector.zero + assert Vector.zero.cross(i) == Vector.zero + + assert i.cross(i) == Vector.zero + assert i.cross(j) == k + assert i.cross(k) == -j + assert i ^ i == Vector.zero + assert i ^ j == k + assert i ^ k == -j + + assert j.cross(i) == -k + assert j.cross(j) == Vector.zero + assert j.cross(k) == i + assert j ^ i == -k + assert j ^ j == Vector.zero + assert j ^ k == i + + assert k.cross(i) == j + assert k.cross(j) == -i + assert k.cross(k) == Vector.zero + assert k ^ i == j + assert k ^ j == -i + assert k ^ k == Vector.zero + + assert k.cross(1) == Cross(k, 1) + + +def test_projection(): + v1 = i + j + k + v2 = 3*i + 4*j + v3 = 0*i + 0*j + assert v1.projection(v1) == i + j + k + assert v1.projection(v2) == Rational(7, 3)*C.i + Rational(7, 3)*C.j + Rational(7, 3)*C.k + assert v1.projection(v1, scalar=True) == S.One + assert v1.projection(v2, scalar=True) == Rational(7, 3) + assert v3.projection(v1) == Vector.zero + assert v3.projection(v1, scalar=True) == S.Zero + + +def test_vector_diff_integrate(): + f = Function('f') + v = f(a)*C.i + a**2*C.j - C.k + assert Derivative(v, a) == Derivative((f(a))*C.i + + a**2*C.j + (-1)*C.k, a) + assert (diff(v, a) == v.diff(a) == Derivative(v, a).doit() == + (Derivative(f(a), a))*C.i + 2*a*C.j) + assert (Integral(v, a) == (Integral(f(a), a))*C.i + + (Integral(a**2, a))*C.j + (Integral(-1, a))*C.k) + + +def test_vector_args(): + raises(ValueError, lambda: BaseVector(3, C)) + raises(TypeError, lambda: BaseVector(0, Vector.zero)) + + +def test_srepr(): + from sympy.printing.repr import srepr + res = "CoordSys3D(Str('C'), Tuple(ImmutableDenseMatrix([[Integer(1), "\ + "Integer(0), Integer(0)], [Integer(0), Integer(1), Integer(0)], "\ + "[Integer(0), Integer(0), Integer(1)]]), VectorZero())).i" + assert srepr(C.i) == res + + +def test_scalar(): + from sympy.vector import CoordSys3D + C = CoordSys3D('C') + v1 = 3*C.i + 4*C.j + 5*C.k + v2 = 3*C.i - 4*C.j + 5*C.k + assert v1.is_Vector is True + assert v1.is_scalar is False + assert (v1.dot(v2)).is_scalar is True + assert (v1.cross(v2)).is_scalar is False diff --git a/.venv/lib/python3.12/site-packages/sympy/vector/vector.py b/.venv/lib/python3.12/site-packages/sympy/vector/vector.py new file mode 100644 index 0000000000000000000000000000000000000000..c035ef48d2edd511f6cdbca19e965d99a2c8c66e --- /dev/null +++ b/.venv/lib/python3.12/site-packages/sympy/vector/vector.py @@ -0,0 +1,714 @@ +from __future__ import annotations +from itertools import product + +from sympy.core import Add, Basic +from sympy.core.assumptions import StdFactKB +from sympy.core.expr import AtomicExpr, Expr +from sympy.core.power import Pow +from sympy.core.singleton import S +from sympy.core.sorting import default_sort_key +from sympy.core.sympify import sympify +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.matrices.immutable import ImmutableDenseMatrix as Matrix +from sympy.vector.basisdependent import (BasisDependentZero, + BasisDependent, BasisDependentMul, BasisDependentAdd) +from sympy.vector.coordsysrect import CoordSys3D +from sympy.vector.dyadic import Dyadic, BaseDyadic, DyadicAdd +from sympy.vector.kind import VectorKind + + +class Vector(BasisDependent): + """ + Super class for all Vector classes. + Ideally, neither this class nor any of its subclasses should be + instantiated by the user. + """ + + is_scalar = False + is_Vector = True + _op_priority = 12.0 + + _expr_type: type[Vector] + _mul_func: type[Vector] + _add_func: type[Vector] + _zero_func: type[Vector] + _base_func: type[Vector] + zero: VectorZero + + kind: VectorKind = VectorKind() + + @property + def components(self): + """ + Returns the components of this vector in the form of a + Python dictionary mapping BaseVector instances to the + corresponding measure numbers. + + Examples + ======== + + >>> from sympy.vector import CoordSys3D + >>> C = CoordSys3D('C') + >>> v = 3*C.i + 4*C.j + 5*C.k + >>> v.components + {C.i: 3, C.j: 4, C.k: 5} + + """ + # The '_components' attribute is defined according to the + # subclass of Vector the instance belongs to. + return self._components + + def magnitude(self): + """ + Returns the magnitude of this vector. + """ + return sqrt(self & self) + + def normalize(self): + """ + Returns the normalized version of this vector. + """ + return self / self.magnitude() + + def equals(self, other): + """ + Check if ``self`` and ``other`` are identically equal vectors. + + Explanation + =========== + + Checks if two vector expressions are equal for all possible values of + the symbols present in the expressions. + + Examples + ======== + + >>> from sympy.vector import CoordSys3D + >>> from sympy.abc import x, y + >>> from sympy import pi + >>> C = CoordSys3D('C') + + Compare vectors that are equal or not: + + >>> C.i.equals(C.j) + False + >>> C.i.equals(C.i) + True + + These two vectors are equal if `x = y` but are not identically equal + as expressions since for some values of `x` and `y` they are unequal: + + >>> v1 = x*C.i + C.j + >>> v2 = y*C.i + C.j + >>> v1.equals(v1) + True + >>> v1.equals(v2) + False + + Vectors from different coordinate systems can be compared: + + >>> D = C.orient_new_axis('D', pi/2, C.i) + >>> D.j.equals(C.j) + False + >>> D.j.equals(C.k) + True + + Parameters + ========== + + other: Vector + The other vector expression to compare with. + + Returns + ======= + + ``True``, ``False`` or ``None``. A return value of ``True`` indicates + that the two vectors are identically equal. A return value of ``False`` + indicates that they are not. In some cases it is not possible to + determine if the two vectors are identically equal and ``None`` is + returned. + + See Also + ======== + + sympy.core.expr.Expr.equals + """ + diff = self - other + diff_mag2 = diff.dot(diff) + return diff_mag2.equals(0) + + def dot(self, other): + """ + Returns the dot product of this Vector, either with another + Vector, or a Dyadic, or a Del operator. + If 'other' is a Vector, returns the dot product scalar (SymPy + expression). + If 'other' is a Dyadic, the dot product is returned as a Vector. + If 'other' is an instance of Del, returns the directional + derivative operator as a Python function. If this function is + applied to a scalar expression, it returns the directional + derivative of the scalar field wrt this Vector. + + Parameters + ========== + + other: Vector/Dyadic/Del + The Vector or Dyadic we are dotting with, or a Del operator . + + Examples + ======== + + >>> from sympy.vector import CoordSys3D, Del + >>> C = CoordSys3D('C') + >>> delop = Del() + >>> C.i.dot(C.j) + 0 + >>> C.i & C.i + 1 + >>> v = 3*C.i + 4*C.j + 5*C.k + >>> v.dot(C.k) + 5 + >>> (C.i & delop)(C.x*C.y*C.z) + C.y*C.z + >>> d = C.i.outer(C.i) + >>> C.i.dot(d) + C.i + + """ + + # Check special cases + if isinstance(other, Dyadic): + if isinstance(self, VectorZero): + return Vector.zero + outvec = Vector.zero + for k, v in other.components.items(): + vect_dot = k.args[0].dot(self) + outvec += vect_dot * v * k.args[1] + return outvec + from sympy.vector.deloperator import Del + if not isinstance(other, (Del, Vector)): + raise TypeError(str(other) + " is not a vector, dyadic or " + + "del operator") + + # Check if the other is a del operator + if isinstance(other, Del): + def directional_derivative(field): + from sympy.vector.functions import directional_derivative + return directional_derivative(field, self) + return directional_derivative + + return dot(self, other) + + def __and__(self, other): + return self.dot(other) + + __and__.__doc__ = dot.__doc__ + + def cross(self, other): + """ + Returns the cross product of this Vector with another Vector or + Dyadic instance. + The cross product is a Vector, if 'other' is a Vector. If 'other' + is a Dyadic, this returns a Dyadic instance. + + Parameters + ========== + + other: Vector/Dyadic + The Vector or Dyadic we are crossing with. + + Examples + ======== + + >>> from sympy.vector import CoordSys3D + >>> C = CoordSys3D('C') + >>> C.i.cross(C.j) + C.k + >>> C.i ^ C.i + 0 + >>> v = 3*C.i + 4*C.j + 5*C.k + >>> v ^ C.i + 5*C.j + (-4)*C.k + >>> d = C.i.outer(C.i) + >>> C.j.cross(d) + (-1)*(C.k|C.i) + + """ + + # Check special cases + if isinstance(other, Dyadic): + if isinstance(self, VectorZero): + return Dyadic.zero + outdyad = Dyadic.zero + for k, v in other.components.items(): + cross_product = self.cross(k.args[0]) + outer = cross_product.outer(k.args[1]) + outdyad += v * outer + return outdyad + + return cross(self, other) + + def __xor__(self, other): + return self.cross(other) + + __xor__.__doc__ = cross.__doc__ + + def outer(self, other): + """ + Returns the outer product of this vector with another, in the + form of a Dyadic instance. + + Parameters + ========== + + other : Vector + The Vector with respect to which the outer product is to + be computed. + + Examples + ======== + + >>> from sympy.vector import CoordSys3D + >>> N = CoordSys3D('N') + >>> N.i.outer(N.j) + (N.i|N.j) + + """ + + # Handle the special cases + if not isinstance(other, Vector): + raise TypeError("Invalid operand for outer product") + elif (isinstance(self, VectorZero) or + isinstance(other, VectorZero)): + return Dyadic.zero + + # Iterate over components of both the vectors to generate + # the required Dyadic instance + args = [(v1 * v2) * BaseDyadic(k1, k2) for (k1, v1), (k2, v2) + in product(self.components.items(), other.components.items())] + + return DyadicAdd(*args) + + def projection(self, other, scalar=False): + """ + Returns the vector or scalar projection of the 'other' on 'self'. + + Examples + ======== + + >>> from sympy.vector.coordsysrect import CoordSys3D + >>> C = CoordSys3D('C') + >>> i, j, k = C.base_vectors() + >>> v1 = i + j + k + >>> v2 = 3*i + 4*j + >>> v1.projection(v2) + 7/3*C.i + 7/3*C.j + 7/3*C.k + >>> v1.projection(v2, scalar=True) + 7/3 + + """ + if self.equals(Vector.zero): + return S.Zero if scalar else Vector.zero + + if scalar: + return self.dot(other) / self.dot(self) + else: + return self.dot(other) / self.dot(self) * self + + @property + def _projections(self): + """ + Returns the components of this vector but the output includes + also zero values components. + + Examples + ======== + + >>> from sympy.vector import CoordSys3D, Vector + >>> C = CoordSys3D('C') + >>> v1 = 3*C.i + 4*C.j + 5*C.k + >>> v1._projections + (3, 4, 5) + >>> v2 = C.x*C.y*C.z*C.i + >>> v2._projections + (C.x*C.y*C.z, 0, 0) + >>> v3 = Vector.zero + >>> v3._projections + (0, 0, 0) + """ + + from sympy.vector.operators import _get_coord_systems + if isinstance(self, VectorZero): + return (S.Zero, S.Zero, S.Zero) + base_vec = next(iter(_get_coord_systems(self))).base_vectors() + return tuple([self.dot(i) for i in base_vec]) + + def __or__(self, other): + return self.outer(other) + + __or__.__doc__ = outer.__doc__ + + def to_matrix(self, system): + """ + Returns the matrix form of this vector with respect to the + specified coordinate system. + + Parameters + ========== + + system : CoordSys3D + The system wrt which the matrix form is to be computed + + Examples + ======== + + >>> from sympy.vector import CoordSys3D + >>> C = CoordSys3D('C') + >>> from sympy.abc import a, b, c + >>> v = a*C.i + b*C.j + c*C.k + >>> v.to_matrix(C) + Matrix([ + [a], + [b], + [c]]) + + """ + + return Matrix([self.dot(unit_vec) for unit_vec in + system.base_vectors()]) + + def separate(self): + """ + The constituents of this vector in different coordinate systems, + as per its definition. + + Returns a dict mapping each CoordSys3D to the corresponding + constituent Vector. + + Examples + ======== + + >>> from sympy.vector import CoordSys3D + >>> R1 = CoordSys3D('R1') + >>> R2 = CoordSys3D('R2') + >>> v = R1.i + R2.i + >>> v.separate() == {R1: R1.i, R2: R2.i} + True + + """ + + parts = {} + for vect, measure in self.components.items(): + parts[vect.system] = (parts.get(vect.system, Vector.zero) + + vect * measure) + return parts + + def _div_helper(one, other): + """ Helper for division involving vectors. """ + if isinstance(one, Vector) and isinstance(other, Vector): + raise TypeError("Cannot divide two vectors") + elif isinstance(one, Vector): + if other == S.Zero: + raise ValueError("Cannot divide a vector by zero") + return VectorMul(one, Pow(other, S.NegativeOne)) + else: + raise TypeError("Invalid division involving a vector") + +# The following is adapted from the matrices.expressions.matexpr file + +def get_postprocessor(cls): + def _postprocessor(expr): + vec_class = {Add: VectorAdd}[cls] + vectors = [] + for term in expr.args: + if isinstance(term.kind, VectorKind): + vectors.append(term) + + if vec_class == VectorAdd: + return VectorAdd(*vectors).doit(deep=False) + return _postprocessor + + +Basic._constructor_postprocessor_mapping[Vector] = { + "Add": [get_postprocessor(Add)], +} + +class BaseVector(Vector, AtomicExpr): + """ + Class to denote a base vector. + + """ + + def __new__(cls, index, system, pretty_str=None, latex_str=None): + if pretty_str is None: + pretty_str = "x{}".format(index) + if latex_str is None: + latex_str = "x_{}".format(index) + pretty_str = str(pretty_str) + latex_str = str(latex_str) + # Verify arguments + if index not in range(0, 3): + raise ValueError("index must be 0, 1 or 2") + if not isinstance(system, CoordSys3D): + raise TypeError("system should be a CoordSys3D") + name = system._vector_names[index] + # Initialize an object + obj = super().__new__(cls, S(index), system) + # Assign important attributes + obj._base_instance = obj + obj._components = {obj: S.One} + obj._measure_number = S.One + obj._name = system._name + '.' + name + obj._pretty_form = '' + pretty_str + obj._latex_form = latex_str + obj._system = system + # The _id is used for printing purposes + obj._id = (index, system) + assumptions = {'commutative': True} + obj._assumptions = StdFactKB(assumptions) + + # This attr is used for re-expression to one of the systems + # involved in the definition of the Vector. Applies to + # VectorMul and VectorAdd too. + obj._sys = system + + return obj + + @property + def system(self): + return self._system + + def _sympystr(self, printer): + return self._name + + def _sympyrepr(self, printer): + index, system = self._id + return printer._print(system) + '.' + system._vector_names[index] + + @property + def free_symbols(self): + return {self} + + def _eval_conjugate(self): + return self + + +class VectorAdd(BasisDependentAdd, Vector): + """ + Class to denote sum of Vector instances. + """ + + def __new__(cls, *args, **options): + obj = BasisDependentAdd.__new__(cls, *args, **options) + return obj + + def _sympystr(self, printer): + ret_str = '' + items = list(self.separate().items()) + items.sort(key=lambda x: x[0].__str__()) + for system, vect in items: + base_vects = system.base_vectors() + for x in base_vects: + if x in vect.components: + temp_vect = self.components[x] * x + ret_str += printer._print(temp_vect) + " + " + return ret_str[:-3] + + +class VectorMul(BasisDependentMul, Vector): + """ + Class to denote products of scalars and BaseVectors. + """ + + def __new__(cls, *args, **options): + obj = BasisDependentMul.__new__(cls, *args, **options) + return obj + + @property + def base_vector(self): + """ The BaseVector involved in the product. """ + return self._base_instance + + @property + def measure_number(self): + """ The scalar expression involved in the definition of + this VectorMul. + """ + return self._measure_number + + +class VectorZero(BasisDependentZero, Vector): + """ + Class to denote a zero vector + """ + + _op_priority = 12.1 + _pretty_form = '0' + _latex_form = r'\mathbf{\hat{0}}' + + def __new__(cls): + obj = BasisDependentZero.__new__(cls) + return obj + + +class Cross(Vector): + """ + Represents unevaluated Cross product. + + Examples + ======== + + >>> from sympy.vector import CoordSys3D, Cross + >>> R = CoordSys3D('R') + >>> v1 = R.i + R.j + R.k + >>> v2 = R.x * R.i + R.y * R.j + R.z * R.k + >>> Cross(v1, v2) + Cross(R.i + R.j + R.k, R.x*R.i + R.y*R.j + R.z*R.k) + >>> Cross(v1, v2).doit() + (-R.y + R.z)*R.i + (R.x - R.z)*R.j + (-R.x + R.y)*R.k + + """ + + def __new__(cls, expr1, expr2): + expr1 = sympify(expr1) + expr2 = sympify(expr2) + if default_sort_key(expr1) > default_sort_key(expr2): + return -Cross(expr2, expr1) + obj = Expr.__new__(cls, expr1, expr2) + obj._expr1 = expr1 + obj._expr2 = expr2 + return obj + + def doit(self, **hints): + return cross(self._expr1, self._expr2) + + +class Dot(Expr): + """ + Represents unevaluated Dot product. + + Examples + ======== + + >>> from sympy.vector import CoordSys3D, Dot + >>> from sympy import symbols + >>> R = CoordSys3D('R') + >>> a, b, c = symbols('a b c') + >>> v1 = R.i + R.j + R.k + >>> v2 = a * R.i + b * R.j + c * R.k + >>> Dot(v1, v2) + Dot(R.i + R.j + R.k, a*R.i + b*R.j + c*R.k) + >>> Dot(v1, v2).doit() + a + b + c + + """ + + def __new__(cls, expr1, expr2): + expr1 = sympify(expr1) + expr2 = sympify(expr2) + expr1, expr2 = sorted([expr1, expr2], key=default_sort_key) + obj = Expr.__new__(cls, expr1, expr2) + obj._expr1 = expr1 + obj._expr2 = expr2 + return obj + + def doit(self, **hints): + return dot(self._expr1, self._expr2) + + +def cross(vect1, vect2): + """ + Returns cross product of two vectors. + + Examples + ======== + + >>> from sympy.vector import CoordSys3D + >>> from sympy.vector.vector import cross + >>> R = CoordSys3D('R') + >>> v1 = R.i + R.j + R.k + >>> v2 = R.x * R.i + R.y * R.j + R.z * R.k + >>> cross(v1, v2) + (-R.y + R.z)*R.i + (R.x - R.z)*R.j + (-R.x + R.y)*R.k + + """ + if isinstance(vect1, Add): + return VectorAdd.fromiter(cross(i, vect2) for i in vect1.args) + if isinstance(vect2, Add): + return VectorAdd.fromiter(cross(vect1, i) for i in vect2.args) + if isinstance(vect1, BaseVector) and isinstance(vect2, BaseVector): + if vect1._sys == vect2._sys: + n1 = vect1.args[0] + n2 = vect2.args[0] + if n1 == n2: + return Vector.zero + n3 = ({0,1,2}.difference({n1, n2})).pop() + sign = 1 if ((n1 + 1) % 3 == n2) else -1 + return sign*vect1._sys.base_vectors()[n3] + from .functions import express + try: + v = express(vect1, vect2._sys) + except ValueError: + return Cross(vect1, vect2) + else: + return cross(v, vect2) + if isinstance(vect1, VectorZero) or isinstance(vect2, VectorZero): + return Vector.zero + if isinstance(vect1, VectorMul): + v1, m1 = next(iter(vect1.components.items())) + return m1*cross(v1, vect2) + if isinstance(vect2, VectorMul): + v2, m2 = next(iter(vect2.components.items())) + return m2*cross(vect1, v2) + + return Cross(vect1, vect2) + + +def dot(vect1, vect2): + """ + Returns dot product of two vectors. + + Examples + ======== + + >>> from sympy.vector import CoordSys3D + >>> from sympy.vector.vector import dot + >>> R = CoordSys3D('R') + >>> v1 = R.i + R.j + R.k + >>> v2 = R.x * R.i + R.y * R.j + R.z * R.k + >>> dot(v1, v2) + R.x + R.y + R.z + + """ + if isinstance(vect1, Add): + return Add.fromiter(dot(i, vect2) for i in vect1.args) + if isinstance(vect2, Add): + return Add.fromiter(dot(vect1, i) for i in vect2.args) + if isinstance(vect1, BaseVector) and isinstance(vect2, BaseVector): + if vect1._sys == vect2._sys: + return S.One if vect1 == vect2 else S.Zero + from .functions import express + try: + v = express(vect2, vect1._sys) + except ValueError: + return Dot(vect1, vect2) + else: + return dot(vect1, v) + if isinstance(vect1, VectorZero) or isinstance(vect2, VectorZero): + return S.Zero + if isinstance(vect1, VectorMul): + v1, m1 = next(iter(vect1.components.items())) + return m1*dot(v1, vect2) + if isinstance(vect2, VectorMul): + v2, m2 = next(iter(vect2.components.items())) + return m2*dot(vect1, v2) + + return Dot(vect1, vect2) + + +Vector._expr_type = Vector +Vector._mul_func = VectorMul +Vector._add_func = VectorAdd +Vector._zero_func = VectorZero +Vector._base_func = BaseVector +Vector.zero = VectorZero() diff --git a/.venv/lib/python3.12/site-packages/tests/__init__.py b/.venv/lib/python3.12/site-packages/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..5209db4e3f0e28d8529d3f62e84533faec3b9c38 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/tests/__init__.py @@ -0,0 +1,38 @@ +# Ultralytics 🚀 AGPL-3.0 License - https://ultralytics.com/license + +from ultralytics.cfg import TASK2DATA, TASK2MODEL, TASKS +from ultralytics.utils import ASSETS, WEIGHTS_DIR, checks + +# Shared test constants for model, config, data source, and environment info +MODEL = WEIGHTS_DIR / "path with spaces" / "yolo26n.pt" # path with spaces to test path handling +CFG = "yolo26n.yaml" +SOURCE = ASSETS / "bus.jpg" +SOURCES_LIST = [ASSETS / "bus.jpg", ASSETS, ASSETS / "*", ASSETS / "**/*.jpg"] # file, dir, and glob patterns +CUDA_IS_AVAILABLE = checks.cuda_is_available() +CUDA_DEVICE_COUNT = checks.cuda_device_count() +TASK_MODEL_DATA = sorted( + [(task, WEIGHTS_DIR / TASK2MODEL[task], TASK2DATA[task]) for task in TASKS] +) # (task, model, data) tuples +MODELS = sorted([*list(TASK2MODEL.values()), "yolo11n-grayscale.pt"]) # task models plus grayscale variant +SOLUTION_ASSETS = { + "boats": "boats.jpg", + "demo_video": "solutions_ci_demo.mp4", + "crop_video": "decelera_landscape_min.mov", + "pose_video": "solution_ci_pose_demo.mp4", + "parking_video": "solution_ci_parking_demo.mp4", + "vertical_video": "solution_vertical_demo.mp4", + "track_video": "decelera_portrait_min.mov", + "parking_areas": "solution_ci_parking_areas.json", + "parking_model": "solutions_ci_parking_model.pt", + "similarity_images": "4-imgs-similaritysearch.zip", +} + +__all__ = ( + "CFG", + "CUDA_DEVICE_COUNT", + "CUDA_IS_AVAILABLE", + "MODEL", + "SOLUTION_ASSETS", + "SOURCE", + "SOURCES_LIST", +) diff --git a/.venv/lib/python3.12/site-packages/tests/__pycache__/__init__.cpython-312.pyc b/.venv/lib/python3.12/site-packages/tests/__pycache__/__init__.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..c2751e63083fe6c867267fe8bac4ed8bda7ca684 Binary files /dev/null and b/.venv/lib/python3.12/site-packages/tests/__pycache__/__init__.cpython-312.pyc differ diff --git a/.venv/lib/python3.12/site-packages/tests/__pycache__/cache_test_assets.cpython-312.pyc 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b/.venv/lib/python3.12/site-packages/tests/__pycache__/test_solutions.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..073ca5fd7f4f7fca61a6fa7844a465dda854a48c Binary files /dev/null and b/.venv/lib/python3.12/site-packages/tests/__pycache__/test_solutions.cpython-312.pyc differ diff --git a/.venv/lib/python3.12/site-packages/tests/cache_test_assets.py b/.venv/lib/python3.12/site-packages/tests/cache_test_assets.py new file mode 100644 index 0000000000000000000000000000000000000000..0ab0980c2d53599404ce7c21e891408035798e28 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/tests/cache_test_assets.py @@ -0,0 +1,120 @@ +# Ultralytics 🚀 AGPL-3.0 License - https://ultralytics.com/license +"""Pre-download shared test assets to avoid race conditions under pytest-xdist. + +Run this script once before `pytest -n auto` to ensure all model weights, +datasets, and solution assets are already cached locally. Each xdist worker +can then reuse existing files instead of competing to download the same remote resources. +""" + +import shutil +import sys +from argparse import ArgumentParser +from pathlib import Path + +import torch + +sys.path.insert(0, str(Path(__file__).resolve().parents[1])) +from tests import MODEL, SOLUTION_ASSETS +from ultralytics.cfg import TASK2CALIBRATIONDATA, TASK2DATA, TASK2MODEL +from ultralytics.data.utils import check_cls_dataset, check_det_dataset +from ultralytics.utils import ARM64, ASSETS_URL, DATASETS_DIR, IS_RASPBERRYPI, LINUX, LOGGER, WEIGHTS_DIR, checks +from ultralytics.utils.downloads import attempt_download_asset, safe_download + +COMMON_WEIGHTS = [ + *TASK2MODEL.values(), + "yolo11n-grayscale.pt", + "rtdetr-l.pt", + "FastSAM-s.pt", + "mobile_sam.pt", + "mobileclip_blt.ts", + "yolov8s-world.pt", + "yolov8s-worldv2.pt", + "yoloe-11s-seg.pt", + "yoloe-11s-seg-pf.pt", + "yolo26s.pt", + "yolo26s-seg.pt", + "yolo26s-pose.pt", + "yolo26s-obb.pt", +] + +SLOW_WEIGHTS = [ + "sam2.1_b.pt", +] + +DATASETS = [ + *TASK2DATA.values(), + *TASK2CALIBRATIONDATA.values(), + "coco8-grayscale.yaml", + "coco8-multispectral.yaml", + "coco12-formats.yaml", +] + + +def cache_weights(slow: bool = False) -> None: + """Download all model weights used by tests.""" + LOGGER.info("[cache] Downloading model weights ...") + weights = COMMON_WEIGHTS + (SLOW_WEIGHTS if slow else []) + for w in weights: + attempt_download_asset(WEIGHTS_DIR / w) + if not MODEL.exists(): + MODEL.parent.mkdir(parents=True, exist_ok=True) + shutil.copy2(WEIGHTS_DIR / "yolo26n.pt", MODEL) + LOGGER.info("[cache] Weights done.") + + +def cache_datasets() -> None: + """Download / extract all datasets used by tests.""" + LOGGER.info("[cache] Downloading datasets ...") + for ds in DATASETS: + if ds.startswith("imagenet"): + check_cls_dataset(ds) + else: + check_det_dataset(ds, autodownload=True) + safe_download(f"{ASSETS_URL}/instances_val2017.json", dir=DATASETS_DIR / "annotations") + LOGGER.info("[cache] Datasets done.") + + +def cache_solution_assets() -> None: + """Download solution test assets (videos, parking json, etc.).""" + LOGGER.info("[cache] Downloading solution assets ...") + cache_dir = WEIGHTS_DIR / "solution_assets" + cache_dir.mkdir(parents=True, exist_ok=True) + for asset in SOLUTION_ASSETS.values(): + dst = cache_dir / asset + if not dst.exists(): + safe_download(url=f"{ASSETS_URL}/{asset}", dir=cache_dir) + LOGGER.info("[cache] Solution assets done.") + + +def cache_clip_model() -> None: + """Download the CLIP text encoder before xdist workers can race on the shared cache file.""" + if IS_RASPBERRYPI or (checks.IS_PYTHON_3_8 and LINUX and ARM64): + return + + LOGGER.info("[cache] Downloading CLIP text encoder ...") + from ultralytics.nn.text_model import CLIP + + model = CLIP("ViT-B/32", device=torch.device("cpu")) + del model + LOGGER.info("[cache] CLIP text encoder done.") + + +def parse_args() -> bool: + """Parse command-line arguments.""" + parser = ArgumentParser(description="Pre-download test assets.") + parser.add_argument("--slow", action="store_true", help="Include assets used only by slow tests.") + return parser.parse_args().slow + + +def main() -> None: + """Main function to orchestrate caching of all test assets.""" + slow = parse_args() + cache_weights(slow=slow) + cache_datasets() + cache_solution_assets() + cache_clip_model() + LOGGER.info("[cache] All test assets are ready.") + + +if __name__ == "__main__": + main() diff --git a/.venv/lib/python3.12/site-packages/tests/conftest.py b/.venv/lib/python3.12/site-packages/tests/conftest.py new file mode 100644 index 0000000000000000000000000000000000000000..cae861945b8e465d1dd2a66b0c1ec1bf069d01e9 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/tests/conftest.py @@ -0,0 +1,140 @@ +# Ultralytics 🚀 AGPL-3.0 License - https://ultralytics.com/license + +import shutil +from pathlib import Path + +import numpy.testing # noqa: F401 # Pre-import before any test can corrupt numpy via in-place upgrade +import pytest + + +@pytest.fixture(scope="session") +def solution_assets(): + """Return cached solution asset paths by name.""" + from tests import SOLUTION_ASSETS + from ultralytics.utils import ASSETS_URL, WEIGHTS_DIR + from ultralytics.utils.downloads import safe_download + + cache_dir = WEIGHTS_DIR / "solution_assets" + cache_dir.mkdir(parents=True, exist_ok=True) + + def get_asset(name): + asset_path = cache_dir / SOLUTION_ASSETS[name] + if not asset_path.exists(): + safe_download(url=f"{ASSETS_URL}/{asset_path.name}", dir=cache_dir) + return asset_path + + return get_asset + + +def pytest_addoption(parser): + """Add custom command-line options to pytest.""" + parser.addoption("--slow", action="store_true", default=False, help="Run slow tests") + parser.addoption( + "--export-env", + default=None, + help="Run only export tests assigned to this export environment id.", + ) + + +def _export_format_from_item(item, formats): + """Infer the export format covered by a tests/test_exports.py item.""" + if Path(str(item.fspath)).name != "test_exports.py": + return None + name = getattr(item, "originalname", None) or item.name.split("[", 1)[0] + if name == "test_torch2onnx_serializes_concurrent_exports": + return "onnx" + if not name.startswith("test_export_"): + return None + + suffix = name[len("test_export_") :] + return next( + (fmt for fmt in sorted(formats, key=len, reverse=True) if suffix == fmt or suffix.startswith(f"{fmt}_")), None + ) + + +def pytest_collection_modifyitems(config, items): + """Modify the list of test items to exclude tests marked as slow if the --slow option is not specified. + + Args: + config: The pytest configuration object that provides access to command-line options. + items (list): The list of collected pytest item objects to be modified based on the presence of --slow option. + """ + if not config.getoption("--slow"): + # Remove the item entirely from the list of test items if it's marked as 'slow' + items[:] = [item for item in items if "slow" not in item.keywords] + + export_env = config.getoption("--export-env") + if not export_env: + return + + from ultralytics.engine.exporter import export_formats + + env_by_format = dict(zip(export_formats()["Argument"], export_formats()["Env"])) + for item in items: + fmt = _export_format_from_item(item, env_by_format) + if fmt and env_by_format.get(fmt) != export_env: + item.add_marker(pytest.mark.skip(reason=f"export format '{fmt}' belongs to env '{env_by_format[fmt]}'")) + + +def isolated_model_path(tmp_path, model): + """Copy a model to a per-test path to prevent export file races under pytest-xdist.""" + model = Path(model) + if not model.exists(): + from ultralytics.utils.downloads import attempt_download_asset + + model.parent.mkdir(parents=True, exist_ok=True) + shutil.copy(attempt_download_asset(model.name), model) + + dst = tmp_path / model.name + shutil.copy(model, dst) + return str(dst) + + +def pytest_sessionstart(session): + """Initialize session configurations for pytest. + + This function is automatically called by pytest after the 'Session' object has been created but before performing + test collection. It sets the initial seeds for the test session. + + Args: + session: The pytest session object. + """ + from ultralytics.utils.torch_utils import init_seeds + + init_seeds() + + +@pytest.fixture +def isolated_model(tmp_path): + """Provide an isolated copy of the test model to prevent export file races under pytest-xdist. + + When multiple xdist workers run export tests simultaneously, they derive output filenames from the model path (e.g., + model.onnx, model.torchscript). Using the same MODEL path causes workers to overwrite each other's + intermediate/export files. This fixture copies the shared model to a per-test temporary directory so each test + exports to a unique path. + """ + from tests import MODEL + + return isolated_model_path(tmp_path, MODEL) + + +def pytest_sessionfinish(session, exitstatus): + """Cleanup operations after pytest session. + + Runs only on the pytest controller (or serial run), skipping xdist workers to avoid race conditions where one worker + deletes shared assets while another is still reading them. + """ + # Skip on xdist workers - only the controller should clean up shared resources + if hasattr(session.config, "workerinput"): + return + + from ultralytics.utils import WEIGHTS_DIR + + # Remove files + models = [path for x in ("*.onnx", "*.torchscript") for path in WEIGHTS_DIR.rglob(x)] + for file in ["bus.jpg", "yolo26n.onnx", "yolo26n.torchscript", *models]: + Path(file).unlink(missing_ok=True) + + # Remove directories + for directory in [path for x in ("*.mlpackage", "*_openvino_model") for path in WEIGHTS_DIR.rglob(x)]: + shutil.rmtree(directory, ignore_errors=True) diff --git a/.venv/lib/python3.12/site-packages/tests/test_cli.py b/.venv/lib/python3.12/site-packages/tests/test_cli.py new file mode 100644 index 0000000000000000000000000000000000000000..c5f2e324ef8c54d019a1b6668f55a505b48eece9 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/tests/test_cli.py @@ -0,0 +1,200 @@ +# Ultralytics 🚀 AGPL-3.0 License - https://ultralytics.com/license + +import json +import shutil +import subprocess +import sys +from pathlib import Path + +import pytest +from PIL import Image + +from tests import CUDA_DEVICE_COUNT, CUDA_IS_AVAILABLE, MODELS, TASK_MODEL_DATA +from ultralytics.utils import ARM64, ASSETS, DATASETS_DIR, IS_RASPBERRYPI, LINUX, WEIGHTS_DIR, checks +from ultralytics.utils.torch_utils import TORCH_1_11, TORCH_VERSION + + +def run(cmd: str) -> None: + """Execute a shell command using subprocess.""" + subprocess.run(cmd.split(), check=True) + + +def test_special_modes() -> None: + """Test various special command-line modes for YOLO functionality.""" + run("yolo help") + run("yolo checks") + run("yolo version") + run("yolo settings reset") + run(f"yolo settings weights_dir={WEIGHTS_DIR} datasets_dir={DATASETS_DIR}") + run("yolo cfg") + + +@pytest.mark.parametrize("api_key", ["legacy_api_key", "ul_" + "a" * 40]) +def test_settings_migration(tmp_path: Path, api_key: str) -> None: + """Verify schema migration preserves user settings and only retains Platform API keys.""" + from ultralytics.utils import SettingsManager + + settings_file = tmp_path / "settings.json" + settings_file.write_text( + json.dumps( + { + "settings_version": "0.0.7", + "runs_dir": "/custom/runs", + "api_key": api_key, + "hub": True, + "neptune": True, + } + ) + ) + settings = SettingsManager(settings_file) + + assert settings["runs_dir"] == "/custom/runs" + assert settings["api_key"] == (api_key if api_key.startswith("ul_") else "") + assert settings["settings_version"] == "0.0.8" + assert "hub" not in settings + assert "neptune" not in settings + + +def test_cli_imports_defer_torchvision() -> None: + """Verify startup imports do not load torchvision or SAM3 geometry.""" + code = ( + "import sys; " + "from ultralytics import YOLO; " + "from ultralytics.models.sam import Predictor; " + "assert 'torchvision' not in sys.modules; " + "assert 'ultralytics.models.sam.sam3.geometry_encoders' not in sys.modules" + ) + subprocess.run([sys.executable, "-c", code], check=True) + + +@pytest.mark.parametrize("task,model,data", TASK_MODEL_DATA) +@pytest.mark.skipif(IS_RASPBERRYPI, reason="Edge devices not intended for training") +def test_train(task: str, model: str, data: str) -> None: + """Test YOLO training for different tasks, models, and datasets.""" + run(f"yolo train {task} model={model} data={data} imgsz=32 epochs=1 cache=disk") + + +@pytest.mark.parametrize("task,model,data", TASK_MODEL_DATA) +def test_val(task: str, model: str, data: str) -> None: + """Test YOLO validation process for specified task, model, and data using a shell command.""" + for nms in (None, False): + run(f"yolo val {task} model={model} data={data} imgsz=32 nms={nms} max_det=100 agnostic_nms") + + +@pytest.mark.parametrize("task,model,data", TASK_MODEL_DATA) +def test_predict(task: str, model: str, data: str) -> None: + """Test YOLO prediction on provided sample assets for specified task and model.""" + for nms in (None, False): + run(f"yolo {task} predict model={model} source={ASSETS} imgsz=32 save nms={nms} max_det=100") + + +@pytest.mark.parametrize("model", MODELS) +def test_export(model: str, tmp_path: Path) -> None: + """Test exporting a YOLO model to TorchScript format.""" + from ultralytics.utils.downloads import attempt_download_asset + + isolated = tmp_path / model + shutil.copy(Path(attempt_download_asset(model)), isolated) + for nms in (None, False): + run(f"yolo export model={isolated} format=torchscript imgsz=32 nms={nms} max_det=100") + + +@pytest.mark.parametrize( + "task,data,student,teacher", + [ + ("detect", "coco8.yaml", "yolo26n.yaml", WEIGHTS_DIR / "yolo26s.pt"), + ("segment", "coco8-seg.yaml", "yolo26n-seg.yaml", WEIGHTS_DIR / "yolo26s-seg.pt"), + ("pose", "coco8-pose.yaml", "yolo26n-pose.yaml", WEIGHTS_DIR / "yolo26s-pose.pt"), + ("obb", "dota8.yaml", "yolo26n-obb.yaml", WEIGHTS_DIR / "yolo26s-obb.pt"), + ], +) +def test_distill(task: str, data: str, student: str, teacher: Path) -> None: + """Test YOLO knowledge distillation training via CLI for supported tasks.""" + run(f"yolo train {task} model={student} distill_model={teacher} data={data} imgsz=32 epochs=1") + + +@pytest.mark.skipif(not TORCH_1_11, reason="RTDETR requires torch>=1.11") +@pytest.mark.skipif( + LINUX and ARM64 and checks.IS_PYTHON_3_8 and "2.1.0a0" in TORCH_VERSION, + reason="RTDETR CPU training produces NaN losses with JetPack 5 torch 2.1.0a0", +) +def test_rtdetr(task: str = "detect", model: Path = WEIGHTS_DIR / "rtdetr-l.pt", data: str = "coco8.yaml") -> None: + """Test the RTDETR functionality within Ultralytics for detection tasks using specified model and data.""" + # Add comma and spaces to test CLI arg cleanup. + run(f"yolo predict {task} model={model} source={ASSETS / 'bus.jpg'} imgsz=160 save") + run(f"yolo train {task} model={model} data={data} --imgsz= 160 epochs =1, cache = disk") + + +@pytest.mark.skipif(IS_RASPBERRYPI, reason="Edge devices not intended for heavy FastSAM tests") +@pytest.mark.skipif(checks.IS_PYTHON_3_12, reason="MobileSAM with CLIP is not supported in Python 3.12") +@pytest.mark.skipif( + checks.IS_PYTHON_3_8 and LINUX and ARM64, + reason="MobileSAM with CLIP is not supported in Python 3.8 and aarch64 Linux", +) +def test_fastsam( + task: str = "segment", model: str = WEIGHTS_DIR / "FastSAM-s.pt", data: str = "coco8-seg.yaml" +) -> None: + """Test FastSAM model for segmenting objects in images using various prompts within Ultralytics.""" + source = ASSETS / "bus.jpg" + + run(f"yolo segment val {task} model={model} data={data} imgsz=32") + run(f"yolo segment predict model={model} source={source} imgsz=32 save") + + from ultralytics import FastSAM + from ultralytics.models.sam import Predictor + + # Create a FastSAM model + sam_model = FastSAM(model) # or FastSAM-x.pt + + # Run inference on an image + for s in (source, Image.open(source)): + everything_results = sam_model(s, device="cpu", retina_masks=True, imgsz=160, conf=0.4, iou=0.9) + + # Remove small regions + _new_masks, _ = Predictor.remove_small_regions(everything_results[0].masks.data, min_area=20) + + # Run inference with bboxes and points and texts prompt at the same time + sam_model(source, bboxes=[439, 437, 524, 709], points=[[200, 200]], labels=[1], texts="a photo of a dog") + + +def test_mobilesam() -> None: + """Test MobileSAM segmentation with point and box prompts using Ultralytics.""" + from ultralytics import SAM + + # Load the model + model = SAM(WEIGHTS_DIR / "mobile_sam.pt") + + # Source + source = ASSETS / "zidane.jpg" + + # Predict a segment based on a 1D point prompt and 1D labels. + model.predict(source, points=[900, 370], labels=[1]) + + # Predict a segment based on 3D points and 2D labels (multiple points per object). + model.predict(source, points=[[[900, 370], [1000, 100]]], labels=[[1, 1]]) + + # Predict a segment based on a box prompt + model.predict(source, bboxes=[439, 437, 524, 709], save=True) + + # Predict all + # model(source) + + +# Slow Tests ----------------------------------------------------------------------------------------------------------- +@pytest.mark.slow +@pytest.mark.parametrize("task,model,data", TASK_MODEL_DATA) +@pytest.mark.skipif(not CUDA_IS_AVAILABLE, reason="CUDA is not available") +@pytest.mark.skipif(CUDA_DEVICE_COUNT < 2, reason="DDP is not available") +def test_train_gpu(task: str, model: str, data: str) -> None: + """Test YOLO training on GPU(s) for various tasks and models.""" + run(f"yolo train {task} model={model} data={data} imgsz=32 epochs=1 device=0") # single GPU + run(f"yolo train {task} model={model} data={data} imgsz=32 epochs=1 device=0,1") # multi GPU + + +@pytest.mark.parametrize( + "solution", + ["count", "blur", "workout", "heatmap", "isegment", "visioneye", "speed", "queue", "analytics", "trackzone"], +) +def test_solutions(solution: str) -> None: + """Test yolo solutions command-line modes.""" + run(f"yolo solutions {solution} verbose=False") diff --git a/.venv/lib/python3.12/site-packages/tests/test_cuda.py b/.venv/lib/python3.12/site-packages/tests/test_cuda.py new file mode 100644 index 0000000000000000000000000000000000000000..cd517e6936a8a74437d42c28c80354b735125ea6 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/tests/test_cuda.py @@ -0,0 +1,275 @@ +# Ultralytics 🚀 AGPL-3.0 License - https://ultralytics.com/license + +import os +from itertools import product +from pathlib import Path + +import pytest +import torch + +from tests import CUDA_DEVICE_COUNT, CUDA_IS_AVAILABLE, MODEL, SOURCE +from ultralytics import YOLO +from ultralytics.cfg import TASK2DATA, TASK2MODEL, TASKS +from ultralytics.utils import ASSETS, IS_JETSON, WEIGHTS_DIR +from ultralytics.utils.autodevice import GPUInfo +from ultralytics.utils.checks import check_amp, check_tensorrt +from ultralytics.utils.torch_utils import TORCH_1_13, parse_device + +# Try to find idle devices if CUDA is available +DEVICES = [] +if CUDA_IS_AVAILABLE: + if IS_JETSON: + DEVICES = [0] # NVIDIA Jetson only has one GPU and does not fully support pynvml library + else: + gpu_info = GPUInfo() + gpu_info.print_status() + autodevice_fraction = __import__("os").environ.get("YOLO_AUTODEVICE_FRACTION_FREE", 0.3) + if idle_gpus := gpu_info.select_idle_gpu( + count=2, + min_memory_fraction=autodevice_fraction, + min_util_fraction=autodevice_fraction, + ): + DEVICES = idle_gpus + + +def test_checks(): + """Validate CUDA settings against torch CUDA functions.""" + assert torch.cuda.is_available() == CUDA_IS_AVAILABLE + assert torch.cuda.device_count() == CUDA_DEVICE_COUNT + + +@pytest.mark.skipif(not DEVICES, reason="No CUDA devices available") +def test_amp(): + """Test AMP training checks.""" + model = YOLO("yolo26n.pt").model.to(f"cuda:{DEVICES[0]}") + assert check_amp(model) + + +@pytest.mark.slow +@pytest.mark.skipif(not DEVICES, reason="No CUDA devices available") +@pytest.mark.parametrize( + "task, dynamic, batch, simplify, nms", + [ # generate all combinations except for exclusion cases + (task, dynamic, batch, simplify, nms) + for task, dynamic, batch, simplify, nms in product( + sorted(TASKS), [True, False], [1, 2], [True, False], [None, True, False] + ) + if not ( + (task == "classify" and nms is True) or (task == "obb" and nms is True and (not TORCH_1_13 or IS_JETSON)) + ) + ], +) +def test_export_onnx_matrix(task, dynamic, batch, simplify, nms): + """Test YOLO exports to ONNX format with various configurations and parameters.""" + file = YOLO(TASK2MODEL[task]).export( + format="onnx", + imgsz=32, + dynamic=dynamic, + batch=batch, + simplify=simplify, + nms=nms, + device=DEVICES[0], + ) + YOLO(file)([SOURCE] * batch, imgsz=64 if dynamic else 32, device=DEVICES[0]) # exported model inference + Path(file).unlink() # cleanup + + +@pytest.mark.slow +@pytest.mark.skipif(not DEVICES, reason="No CUDA devices available") +@pytest.mark.parametrize( + "task, dynamic, quantize, batch", + [ + (task, dynamic, quantize, batch) + # Limit Jetson task coverage for slow CI speed; full task coverage remains on GPU CI. + # for task, dynamic, quantize, batch in product(TASKS, [True, False], [8, 16], [1, 2]) + for task, dynamic, quantize, batch in product(["detect"] if IS_JETSON else sorted(TASKS), [True], [8, 16], [2]) + ] + + [("detect", False, 8, 2)], # exercise TensorRT 7-10 implicit INT8 quantization on GPU CI +) +def test_export_engine_matrix(task, dynamic, quantize, batch): + """Test YOLO model export to TensorRT format for various configurations and run inference.""" + check_tensorrt() + import tensorrt as trt + + is_trt11 = int(trt.__version__.split(".", 1)[0]) >= 11 + if not is_trt11 and quantize == 8 and dynamic: + # TensorRT 7-10 calibrator path cannot quantize dynamic-shape models; TensorRT 11 uses ModelOpt explicit Q/DQ + pytest.skip("INT8 + dynamic export requires explicit quantization, available on TensorRT 11+") + + file = YOLO(TASK2MODEL[task]).export( + format="engine", + imgsz=32, + dynamic=dynamic, + quantize=quantize, + batch=batch, + data=TASK2DATA[task], # use the smallest task datasets for fast INT8 calibration + workspace=1, # reduce workspace GB for less resource utilization during testing + simplify=True, + device=DEVICES[0], + ) + model = YOLO(file) + model([SOURCE] * batch, imgsz=64 if dynamic else 32, device=DEVICES[0]) # exported model inference + model.val(data=TASK2DATA[task], imgsz=32, device=DEVICES[0], batch=batch) # exported model validation + Path(file).unlink() # cleanup + if quantize == 8: + Path(file).with_suffix(".cache").unlink(missing_ok=True) # cleanup TensorRT 7-10 INT8 calibration cache + Path(file).with_suffix(".int8.onnx").unlink(missing_ok=True) # cleanup TensorRT 11 ModelOpt INT8 ONNX + if quantize == 16: + Path(file).with_suffix(".fp16.onnx").unlink(missing_ok=True) # cleanup TensorRT 11 ModelOpt FP16 ONNX + + +@pytest.mark.skipif(not DEVICES, reason="No CUDA devices available") +@pytest.mark.parametrize("nc", [1, 3]) +def test_semantic_loss_all_ignore_amp(nc): + """All-ignore guard must stay finite with large fp16 logits, where sum() overflows to inf (AMP is a GPU path).""" + from ultralytics.cfg import get_cfg + from ultralytics.nn.tasks import SemanticSegmentationModel + from ultralytics.utils.loss import SemanticSegmentationLoss + + model = SemanticSegmentationModel(cfg="yolo26-sem.yaml", nc=nc, verbose=False) + model.args = get_cfg() + loss_fn = SemanticSegmentationLoss(model) + preds = (torch.randn(1, nc, 64, 64, device=f"cuda:{DEVICES[0]}") + 50).half().requires_grad_() + loss, items = loss_fn(preds, {"semantic_mask": torch.full((1, 64, 64), 255, dtype=torch.long)}) + assert torch.isfinite(loss).all() and all(torch.isfinite(x).all() for x in items.values()) + loss.backward() + assert preds.grad is not None + + +@pytest.mark.skipif(not DEVICES, reason="No CUDA devices available") +@pytest.mark.skipif(IS_JETSON, reason="Edge devices not intended for training") +def test_train(): + """Test model training on a minimal dataset using available CUDA devices.""" + device = tuple(DEVICES) if len(DEVICES) > 1 else DEVICES[0] + expected = parse_device(device) # canonical torch indices, e.g. physical ids translate under external CVD + visible = os.environ.get("CUDA_VISIBLE_DEVICES") + results = YOLO(MODEL).train(data="coco8-grayscale.yaml", imgsz=64, epochs=1, device=DEVICES[0], batch=-1) + model = YOLO(MODEL) + results = model.train(data="coco8.yaml", imgsz=64, epochs=1, device=device, batch=15, compile=True) + assert model.trainer.args.device == expected, "trained on wrong GPUs" + assert model.trainer.device.index == int(expected.split(",")[0]), "trained on wrong GPU" + assert os.environ.get("CUDA_VISIBLE_DEVICES") == visible, "CUDA_VISIBLE_DEVICES must never be mutated" + results = YOLO(MODEL).train(data="coco128.yaml", imgsz=64, epochs=1, device=device, batch=15, val=False) + # Both single-GPU and DDP return metrics (recovered from the saved checkpoint under DDP) + assert results is not None + + +@pytest.mark.skipif(not DEVICES or max(DEVICES) == 0, reason="requires an idle CUDA device with nonzero index") +@pytest.mark.skipif(IS_JETSON, reason="Edge devices not intended for training") +def test_train_cold_process_nonzero_device(): + """Train on a nonzero GPU index in a fresh process with cold CUDA state, reproducing real CLI usage. + + A warm pytest process has CUDA initialized, so a subprocess without CUDA_VISIBLE_DEVICES is the only way to + reproduce cold-start device selection as on production pods (e.g. Ultralytics Platform). + """ + import subprocess + + env = {k: v for k, v in os.environ.items() if k != "CUDA_VISIBLE_DEVICES"} + cmd = ["yolo", "train", f"model={MODEL}", "data=coco8.yaml", "imgsz=32", "epochs=1", f"device={max(DEVICES)}"] + subprocess.run(cmd, check=True, env=env) + + +@pytest.mark.slow +@pytest.mark.skipif(not DEVICES, reason="No CUDA devices available") +def test_predict_multiple_devices(): + """Validate model prediction consistency across CPU and CUDA devices.""" + model = YOLO("yolo26n.pt") + + # Test CPU + model = model.cpu() + assert str(model.device) == "cpu" + _ = model(SOURCE) + assert str(model.device) == "cpu" + + # Test CUDA + cuda_device = f"cuda:{DEVICES[0]}" + model = model.to(cuda_device) + assert str(model.device) == cuda_device + _ = model(SOURCE) + assert str(model.device) == cuda_device + + # Test CPU again + model = model.cpu() + assert str(model.device) == "cpu" + _ = model(SOURCE) + assert str(model.device) == "cpu" + + # Test CUDA again + model = model.to(cuda_device) + assert str(model.device) == cuda_device + _ = model(SOURCE) + assert str(model.device) == cuda_device + + +@pytest.mark.skipif(not DEVICES, reason="No CUDA devices available") +def test_track_exported_model(): + """Track with an exported model on GPU; exported backends return raw preds as a single Tensor.""" + file = YOLO(MODEL).export(format="torchscript", imgsz=160, device=DEVICES[0]) + results = YOLO(file).track(SOURCE, imgsz=160, device=DEVICES[0]) + assert len(results[0].boxes) + Path(file).unlink() # cleanup + + +@pytest.mark.skipif(not DEVICES, reason="No CUDA devices available") +def test_autobatch(): + """Check optimal batch size for YOLO model training using autobatch utility.""" + from ultralytics.utils.autobatch import check_train_batch_size + + # imgsz < 2 * stride fails the batch-1 probe in train-mode BatchNorm, which is not a memory ceiling + assert check_train_batch_size(YOLO(MODEL).model.to(f"cuda:{DEVICES[0]}"), imgsz=32, amp=True) > 1 + + +@pytest.mark.slow +@pytest.mark.skipif(not DEVICES, reason="No CUDA devices available") +def test_utils_benchmarks(isolated_model): + """Profile YOLO models for performance benchmarks.""" + from ultralytics.utils.benchmarks import ProfileModels + + # Pre-export a dynamic engine model to use dynamic inference + YOLO(isolated_model).export(format="engine", imgsz=32, dynamic=True, batch=1, device=DEVICES[0]) + ProfileModels( + [isolated_model], + imgsz=32, + quantize=32, + min_time=1, + num_timed_runs=3, + num_warmup_runs=1, + device=DEVICES[0], + ).run() + + +@pytest.mark.slow +@pytest.mark.skipif(not DEVICES, reason="No CUDA devices available") +def test_predict_sam(): + """Test SAM model predictions using different prompts.""" + from ultralytics import SAM + from ultralytics.models.sam import Predictor as SAMPredictor + + model = SAM(WEIGHTS_DIR / "sam2.1_b.pt") + model.info() + + # Run inference with various prompts + model(SOURCE, device=DEVICES[0]) + model(SOURCE, bboxes=[439, 437, 524, 709], device=DEVICES[0]) + model(ASSETS / "zidane.jpg", points=[900, 370], device=DEVICES[0]) + model(ASSETS / "zidane.jpg", points=[900, 370], labels=[1], device=DEVICES[0]) + model(ASSETS / "zidane.jpg", points=[[900, 370]], labels=[1], device=DEVICES[0]) + model(ASSETS / "zidane.jpg", points=[[400, 370], [900, 370]], labels=[1, 1], device=DEVICES[0]) + model(ASSETS / "zidane.jpg", points=[[[900, 370], [1000, 100]]], labels=[[1, 1]], device=DEVICES[0]) + + # Test predictor + predictor = SAMPredictor( + overrides={ + "conf": 0.25, + "task": "segment", + "mode": "predict", + "imgsz": 1024, + "model": WEIGHTS_DIR / "mobile_sam.pt", + "device": DEVICES[0], + "quantize": 16, + } + ) + predictor.set_image(ASSETS / "zidane.jpg") + # predictor(bboxes=[439, 437, 524, 709]) + # predictor(points=[900, 370], labels=[1]) + predictor.reset_image() diff --git a/.venv/lib/python3.12/site-packages/tests/test_engine.py b/.venv/lib/python3.12/site-packages/tests/test_engine.py new file mode 100644 index 0000000000000000000000000000000000000000..7f66d2e5d0df3b3251983f3bc2548d824017da8a --- /dev/null +++ b/.venv/lib/python3.12/site-packages/tests/test_engine.py @@ -0,0 +1,395 @@ +# Ultralytics 🚀 AGPL-3.0 License - https://ultralytics.com/license + +import sys +from collections import OrderedDict +from pathlib import Path +from types import SimpleNamespace +from unittest import mock + +import pytest +import torch + +from tests import MODEL, SOURCE, TASK_MODEL_DATA +from ultralytics import YOLO +from ultralytics.cfg import get_cfg +from ultralytics.engine.exporter import Exporter +from ultralytics.engine.trainer import BaseTrainer +from ultralytics.models.yolo import classify, depth, detect, obb, pose, segment, semantic +from ultralytics.nn.distill_model import DistillationModel +from ultralytics.nn.tasks import DetectionModel, load_checkpoint +from ultralytics.utils import ASSETS, DEFAULT_CFG, IS_RASPBERRYPI, WEIGHTS_DIR +from ultralytics.utils.torch_utils import unwrap_model + + +def test_func(*args, **kwargs): + """Test function used as a callback stub to verify callback registration.""" + print("callback test passed") + + +def test_export(monkeypatch, tmp_path): + """Test model exporting functionality by adding a callback and verifying its execution.""" + monkeypatch.chdir(tmp_path) + exporter = Exporter() + exporter.add_callback("on_export_start", test_func) + assert test_func in exporter.callbacks["on_export_start"], "on_export_start callback not registered" + f = exporter(model=YOLO("yolo26n.yaml").model) + YOLO(f)(SOURCE) # exported model inference + + +@pytest.mark.parametrize( + "trainer_cls,validator_cls,predictor_cls,data,model,weights", + [ + ( + detect.DetectionTrainer, + detect.DetectionValidator, + detect.DetectionPredictor, + "coco8.yaml", + "yolo26n.yaml", + MODEL, + ), + ( + segment.SegmentationTrainer, + segment.SegmentationValidator, + segment.SegmentationPredictor, + "coco8-seg.yaml", + "yolo26n-seg.yaml", + WEIGHTS_DIR / "yolo26n-seg.pt", + ), + ( + classify.ClassificationTrainer, + classify.ClassificationValidator, + classify.ClassificationPredictor, + "imagenet10", + "yolo26n-cls.yaml", + None, + ), + (obb.OBBTrainer, obb.OBBValidator, obb.OBBPredictor, "dota8.yaml", "yolo26n-obb.yaml", None), + (pose.PoseTrainer, pose.PoseValidator, pose.PosePredictor, "coco8-pose.yaml", "yolo26n-pose.yaml", None), + ( + semantic.SemanticSegmentationTrainer, + semantic.SemanticSegmentationValidator, + semantic.SemanticSegmentationPredictor, + "cityscapes8.yaml", + "yolo26n-sem.yaml", + None, + ), + (depth.DepthTrainer, depth.DepthValidator, depth.DepthPredictor, "depth8.yaml", "yolo26-depth.yaml", None), + ], +) +@pytest.mark.skipif(IS_RASPBERRYPI, reason="Edge devices not intended for training") +def test_task(trainer_cls, validator_cls, predictor_cls, data, model, weights): + """Test YOLO training, validation, and prediction for various tasks.""" + overrides = { + "data": data, + "model": model, + "imgsz": 32, + "epochs": 1, + "save": False, + "mask_ratio": 1, + "overlap_mask": False, + } + + # Trainer + trainer = trainer_cls(overrides=overrides) + trainer.add_callback("on_train_start", test_func) + assert test_func in trainer.callbacks["on_train_start"], "on_train_start callback not registered" + trainer.train() + + # Validator + cfg = get_cfg(DEFAULT_CFG) + cfg.data = data + cfg.imgsz = 32 + val = validator_cls(args=cfg) + val.add_callback("on_val_start", test_func) + assert test_func in val.callbacks["on_val_start"], "on_val_start callback not registered" + val(model=trainer.best) + + # Predictor + pred = predictor_cls(overrides={"imgsz": [64, 64]}) + pred.add_callback("on_predict_start", test_func) + assert test_func in pred.callbacks["on_predict_start"], "on_predict_start callback not registered" + + # Determine model path for prediction + model_path = weights if weights else trainer.best + if model == "yolo26n.yaml": # only for detection + # Confirm there is no issue with sys.argv being empty + with mock.patch.object(sys, "argv", []): + result = pred(source=ASSETS, model=model_path) + assert len(result) > 0, f"Predictor returned no results for {model}" + else: + result = pred(source=ASSETS, model=model_path) + assert len(result) > 0, f"Predictor returned no results for {model}" + + # Test resume functionality + with pytest.raises(AssertionError): + trainer_cls(overrides={**overrides, "resume": trainer.last}).train() + + +def test_semantic_polygon_val_background(): + """Test standalone semantic val adds the polygon background class instead of mislabeling it as the last class.""" + from ultralytics.data.utils import check_det_dataset + + cfg = get_cfg(DEFAULT_CFG) + cfg.data = "coco8-seg.yaml" + cfg.imgsz = 64 + validator = semantic.SemanticSegmentationValidator(args=cfg) + validator.data = check_det_dataset(cfg.data) # standalone val dataset setup (engine/validator.py) + validator.stride = 32 # normally set from the model during validation + validator.device = torch.device("cpu") # normally set from the model during validation + dataset = validator.get_dataloader(validator.data["val"], 4).dataset + assert dataset.bg_class_idx == 80, "background class missing on standalone polygon val" + assert (dataset[0]["semantic_mask"] == 80).any(), "background pixels mislabeled on standalone polygon val" + + +@pytest.mark.parametrize("task,weight,data", TASK_MODEL_DATA) +def test_resume_incomplete(task, weight, data, tmp_path): + """Test training resumes from an incomplete checkpoint.""" + train_args = { + "data": data, + "epochs": 2, + "save": True, + "plots": False, + "workers": 0, + "project": tmp_path, + "name": task, + "imgsz": 32, + "exist_ok": True, + } + + def stop_after_first_epoch(trainer): + if trainer.epoch == 0: + trainer.stop = True + + def disable_final_eval(trainer): + trainer.final_eval = lambda: None + + model = YOLO(weight) + model.add_callback("on_train_start", disable_final_eval) + model.add_callback("on_train_epoch_end", stop_after_first_epoch) + model.train(**train_args) + last_path = model.trainer.last + _, ckpt = load_checkpoint(last_path) + assert ckpt["epoch"] == 0, "checkpoint should be resumable" + + # Resume training using the checkpoint + resume_model = YOLO(last_path) + resume_model.train(resume=True, **train_args) + assert resume_model.trainer.start_epoch == resume_model.trainer.epoch == 1, "resume test failed" + + +def test_resume_invalid_checkpoint_args(tmp_path): + """Test resuming a checkpoint with invalid train args raises the real error, not a missing-checkpoint error.""" + _, ckpt = load_checkpoint(MODEL) + ckpt["train_args"]["epochs"] = 0 # ultralytics <= 8.4.154 stored epochs < 1 without validating it + torch.save(ckpt, last := tmp_path / "last.pt") + with pytest.raises(ValueError, match="epochs"): + detect.DetectionTrainer(overrides={"model": MODEL, "data": "coco8.yaml", "resume": str(last)}) + + +def test_distill_resume(tmp_path: Path): + """Test knowledge distillation resumes from an incomplete checkpoint.""" + overrides = { + "data": "coco8.yaml", + "model": "yolo26n.yaml", + "distill_model": WEIGHTS_DIR / "yolo26s.pt", + "imgsz": 32, + "multi_scale": 0.5, # vary per-batch image size to exercise dynamic distillation score splitting + "epochs": 2, + "save": True, + "plots": False, + "workers": 0, + "project": tmp_path, + "name": "distill", + "exist_ok": True, + } + + # Train for one epoch then interrupt to produce a resumable checkpoint + trainer = detect.DetectionTrainer(overrides=overrides) + + def stop_after_first_epoch(trainer): + if trainer.epoch == 0: + trainer.stop = True + + trainer.final_eval = lambda: None + trainer.add_callback("on_train_epoch_end", stop_after_first_epoch) + trainer.train() + _, ckpt = load_checkpoint(trainer.last) + assert ckpt["epoch"] == 0, "checkpoint should be resumable" + assert isinstance(ckpt["ema"], DistillationModel), "distillation EMA wraps the student model" + assert ckpt["ema"].teacher_model is None, "teacher should be stripped from the EMA/checkpoint" + assert ckpt["ema"].projector is not None, "the distillation projector should be persisted in the EMA checkpoint" + + overrides["resume"] = trainer.last + trainer = detect.DetectionTrainer(overrides=overrides) + trainer.final_eval = lambda: None + trainer.train() + model = unwrap_model(trainer.model) + assert isinstance(model, DistillationModel), "resume should rebuild the DistillationModel" + assert model.teacher_model is not None, "resume should rebuild the teacher from the distill_model path" + assert trainer.start_epoch == trainer.epoch == 1, "resume test failed" + + +def test_distill_grayscale(tmp_path: Path): + """Test knowledge distillation on a single-channel dataset (https://github.com/ultralytics/ultralytics/issues/25066).""" + teacher = DetectionModel("yolo26n.yaml", ch=3, nc=80, verbose=False) + teacher_path = tmp_path / "teacher.pt" + torch.save({"model": teacher}, teacher_path) + student = DetectionModel("yolo26n.yaml", ch=1, nc=80, verbose=False) + student.args = SimpleNamespace(imgsz=32, dis=1.0) + model = DistillationModel(teacher_model=teacher_path, student_model=student) + assert isinstance(model, DistillationModel) + assert model.teacher_model.yaml["channels"] == 1 + + +@pytest.mark.parametrize( + "ckpt", + [ + {"model": OrderedDict([("a", torch.zeros(1))])}, # state_dict saved under the "model" key + {"model": {"a": torch.zeros(1)}}, # plain-dict "model" value + OrderedDict([("a", torch.zeros(1))]), # bare state_dict, no "model" key + ], +) +def test_load_checkpoint_state_dict_rejected(ckpt, tmp_path): + """Test a state_dict checkpoint raises a clear TypeError instead of a cryptic AttributeError/KeyError.""" + weight = tmp_path / "bad.pt" + torch.save(ckpt, weight) + with pytest.raises(TypeError, match="supported Ultralytics checkpoint format"): + load_checkpoint(weight) + + +def test_nan_recovery(): + """Test NaN loss detection and recovery during training.""" + nan_injected = [False] + + def inject_nan(trainer): + """Inject NaN into loss during batch processing to test recovery mechanism.""" + if trainer.epoch == 1 and trainer.tloss is not None and not nan_injected[0]: + trainer.tloss[next(iter(trainer.tloss))] *= float("nan") + nan_injected[0] = True + + overrides = {"data": "coco8.yaml", "model": "yolo26n.yaml", "imgsz": 32, "epochs": 3} + trainer = detect.DetectionTrainer(overrides=overrides) + trainer.add_callback("on_train_batch_end", inject_nan) + trainer.train() + assert nan_injected[0], "NaN injection failed" + + +def test_checkpoint_fp16_overflow(): + """Test a finite model whose weights overflow fp16 is still checkpointed (clamped) instead of skipped.""" + + def inflate_ema(trainer): + """Push an EMA weight above the fp16 max (65504) so its fp16 snapshot would otherwise become Inf.""" + if trainer.ema is not None: + next(iter(trainer.ema.ema.parameters())).data[0] = 1.0e5 + + overrides = {"data": "coco8.yaml", "model": "yolo26n.yaml", "imgsz": 32, "epochs": 2} + trainer = detect.DetectionTrainer(overrides=overrides) + trainer.add_callback("on_train_epoch_end", inflate_ema) + trainer.train() + assert trainer.last.exists(), "checkpoint not saved for a finite model with fp16-overflowing weights" + model, _ = load_checkpoint(trainer.last) + assert all(torch.isfinite(v).all() for v in model.state_dict().values() if isinstance(v, torch.Tensor)), ( + "saved checkpoint contains NaN/Inf" + ) + # Validation must leave the live EMA fp32 and unchanged; checkpoint serialization may clamp its fp16 copy. + ema_param = next(iter(trainer.ema.ema.parameters())) + assert ema_param.dtype == torch.float32 and torch.isfinite(ema_param).all() and ema_param.flatten()[0] == 1.0e5, ( + "validation corrupted the live EMA" + ) + + +def test_checkpoint_nonfinite_ema_resync(): + """Test a non-finite EMA on a finite model is resynced (not skipped) so the run still produces a checkpoint.""" + + def poison_ema(trainer): + """Make the live fp32 EMA genuinely non-finite while the model stays finite (sticky-NaN on a finite-loss run).""" + if trainer.ema is not None: + next(iter(trainer.ema.ema.parameters())).data[0] = float("inf") + + overrides = {"data": "coco8.yaml", "model": "yolo26n.yaml", "imgsz": 32, "epochs": 2} + trainer = detect.DetectionTrainer(overrides=overrides) + trainer.add_callback("on_train_epoch_end", poison_ema) + trainer.train() + assert trainer.last.exists(), "no checkpoint saved when the EMA went non-finite on a finite model" + model, _ = load_checkpoint(trainer.last) + assert all(torch.isfinite(v).all() for v in model.state_dict().values() if isinstance(v, torch.Tensor)), ( + "saved checkpoint contains NaN/Inf" + ) + + +def test_checkpoint_nonfinite_ema_and_model_sanitized(): + """Test a tensor non-finite in both EMA and model is sanitized (not skipped) so the run still produces a checkpoint.""" + + def poison_ema_and_model(trainer): + """Force the first parameter non-finite in both the live EMA and the model (finite-loss sticky-NaN).""" + if trainer.ema is not None: + next(iter(trainer.ema.ema.parameters())).data[0] = float("inf") + next(iter(unwrap_model(trainer.model).parameters())).data[0] = float("nan") + + overrides = {"data": "coco8.yaml", "model": "yolo26n.yaml", "imgsz": 32, "epochs": 1} + trainer = detect.DetectionTrainer(overrides=overrides) + trainer.add_callback("on_train_epoch_end", poison_ema_and_model) + trainer.train() + assert trainer.last.exists(), "no checkpoint saved when a tensor went non-finite in both EMA and model" + model, _ = load_checkpoint(trainer.last) + assert all(torch.isfinite(v).all() for v in model.state_dict().values() if isinstance(v, torch.Tensor)), ( + "saved checkpoint contains NaN/Inf" + ) + + +def test_train_multi_custom_trainer_metrics_and_failure_keys(monkeypatch, tmp_path): + """Test custom multi-dataset runs keep memory metrics and unique failure keys.""" + model = YOLO(MODEL) + calls = 0 + + class FakeTrainer: + def __init__(self, overrides=None, _callbacks=None): + pass + + def get_model(self, cfg=None, weights=None, verbose=True): + return model.model + + def train(self): + nonlocal calls + calls += 1 + if calls == 2: + raise RuntimeError("failed repeated dataset") + self.validator = SimpleNamespace(metrics=SimpleNamespace(results_dict={"fitness": 1.0})) + + monkeypatch.setattr("ultralytics.engine.model.checks.check_pip_update_available", lambda: None) + results = model.train( + data=["coco8.yaml", "coco8.yaml"], + project=tmp_path, + plots=False, + save=False, + trainer=FakeTrainer, + ) + + assert model.trainer.trainer is FakeTrainer + assert results == {"coco8": {"fitness": 1.0}, "coco8-2": None} + + +@pytest.mark.parametrize("pretrained,uses_weights", [(True, True), (False, False), (MODEL, True)]) +def test_setup_model_respects_pretrained_arg_for_pt_models(monkeypatch, pretrained, uses_weights): + """Test .pt models use checkpoint config while respecting the pretrained argument.""" + captured = {} + checkpoint_model = SimpleNamespace(yaml={"nc": 80}) + trainer = object.__new__(BaseTrainer) + trainer.model = "yolo26n.pt" + trainer.args = SimpleNamespace(pretrained=pretrained) + trainer.resume = False + + def fake_get_model(cfg=None, weights=None, verbose=True): + captured["cfg"] = cfg + captured["weights"] = weights + return SimpleNamespace() + + trainer.get_model = fake_get_model + monkeypatch.setattr( + "ultralytics.engine.trainer.load_checkpoint", lambda path: (checkpoint_model, {"checkpoint": True}) + ) + + trainer.setup_model() + + assert captured["cfg"] == checkpoint_model.yaml, "Checkpoint config was not used" + assert captured["weights"] is (checkpoint_model if uses_weights else None), "Unexpected weights loaded" diff --git a/.venv/lib/python3.12/site-packages/tests/test_exports.py b/.venv/lib/python3.12/site-packages/tests/test_exports.py new file mode 100644 index 0000000000000000000000000000000000000000..5e3a1a1e20b8c726e768ce694696ab5063a60abf --- /dev/null +++ b/.venv/lib/python3.12/site-packages/tests/test_exports.py @@ -0,0 +1,688 @@ +# Ultralytics 🚀 AGPL-3.0 License - https://ultralytics.com/license + +import io +import shutil +import sys +import threading +import time +from contextlib import redirect_stderr, redirect_stdout +from itertools import product +from pathlib import Path +from types import SimpleNamespace + +import pytest +import torch + +from tests import MODEL, SOURCE +from tests.conftest import isolated_model_path +from ultralytics import YOLO +from ultralytics.cfg import TASK2DATA, TASK2MODEL, TASKS, _handle_deprecation, get_cfg +from ultralytics.engine.exporter import EXPORT_ENVS, Exporter, export_formats, validate_args +from ultralytics.utils import ( + ARM64, + IS_RASPBERRYPI, + LINUX, + MACOS, + MACOS_VERSION, + WEIGHTS_DIR, + WINDOWS, + checks, +) +from ultralytics.utils.export.engine import modelopt_quantize_onnx, torch2onnx +from ultralytics.utils.torch_utils import ( + TORCH_1_10, + TORCH_1_11, + TORCH_1_13, + TORCH_2_0, + TORCH_2_1, + TORCH_2_9, +) + + +def skip_rpi_semantic(task): + """Skip semantic segmentation export tests on Raspberry Pi due to memory constraints.""" + if IS_RASPBERRYPI and task == "semantic": + pytest.skip("Semantic segmentation export tests are skipped on Raspberry Pi due to memory constraints.") + + +@pytest.mark.parametrize("nms", [None, False]) +def test_export_torchscript(nms, isolated_model): + """Test YOLO model export to TorchScript format for compatibility and correctness.""" + file = YOLO(isolated_model).export(format="torchscript", imgsz=32, nms=nms) + model = YOLO(file) + model(SOURCE, imgsz=32) # exported model inference + model(SOURCE, imgsz=64) # predictor reuse must keep the fixed export imgsz + assert model.predictor.imgsz == [32, 32] + + +@pytest.mark.parametrize(("model_name", "nc"), [("yolo26n.yaml", 80), ("yolo26n-cls.yaml", 1000)]) +def test_export_torchscript_missing_names(model_name, nc, tmp_path): + """Test TorchScript export reconstructs missing class names from the model head's class count.""" + model = YOLO(model_name) + model.model.names = None # legacy and foreign checkpoints reach the exporter without names + model.model.pt_path = str(tmp_path / Path(model_name).with_suffix(".pt").name) + + names = YOLO(model.export(format="torchscript", imgsz=32)).names + + assert len(names) == nc # a 999-name fallback would leave names[999] missing on a 1000-class head + assert names[nc - 1] == f"class{nc - 1}" + + +@pytest.mark.parametrize("nms", [None, False]) +def test_export_onnx(nms, isolated_model): + """Test YOLO model export to ONNX format with dynamic axes.""" + file = YOLO(isolated_model).export(format="onnx", dynamic=True, imgsz=32, nms=nms) + YOLO(file)(SOURCE, imgsz=32) # exported model inference + + +@pytest.mark.skipif(not TORCH_1_13, reason="ONNX export with NMS requires torch>=1.13") +def test_export_onnx_nms_conf(isolated_model): + """Bake explicit zero confidence into the NMS graph instead of the 0.25 default.""" + import onnx + from onnx import numpy_helper + + path = YOLO(isolated_model).export(format="onnx", imgsz=32, nms=True, conf=0.0) + model = onnx.load(path) + initializers = {i.name: numpy_helper.to_array(i) for i in model.graph.initializer} + thresholds = { + initializers[n.input[1]].item() + for n in model.graph.node + if n.op_type == "Greater" and n.input[1] in initializers + } + assert 0.0 in thresholds + + +@pytest.mark.slow +@pytest.mark.parametrize("precision", [{"int8": True}, {"quantize": 8}]) +def test_export_onnx_int8(isolated_model, precision): + """Test INT8 ONNX export via both the legacy int8 alias and the unified quantize arg.""" + file = YOLO(isolated_model).export(format="onnx", data=Path("coco8.yaml"), fraction=0.25, imgsz=32, **precision) + assert Path(file).name.endswith("_int8.onnx") + YOLO(file)(SOURCE, imgsz=32) # exported model inference + Path(file).unlink() # cleanup + + +def test_onnx_int8_quantize_excludes_non_weighted_ops(monkeypatch): + """Check ONNX INT8 keeps only weighted ops quantized while preserving the string return contract.""" + import onnx + import onnxruntime.quantization as ort_quantization + + from ultralytics.utils.export.onnx import onnx_int8_quantize + + calls = {} + graph = SimpleNamespace( + node=[ + SimpleNamespace(name="conv", op_type="Conv"), + SimpleNamespace(name="pool", op_type="MaxPool"), + SimpleNamespace(name="sigmoid", op_type="Sigmoid"), + ] + ) + + monkeypatch.setattr(onnx, "load", lambda _: SimpleNamespace(graph=graph)) + monkeypatch.setattr(ort_quantization, "quantize_static", lambda *args, **kwargs: calls.update(kwargs)) + result = onnx_int8_quantize(Path("model.onnx"), Path("model_int8.onnx"), [], lambda x: x) + assert result == "model_int8.onnx" + assert calls["nodes_to_exclude"] == ["pool", "sigmoid"] + + +def test_quantize_canonicalization(): + """Quantize accepts 8/16/32 (int or str) and w-notation, canonicalizing to the int form (unset stays None).""" + for value, expected in [ + (8, 8), + (16, 16), + (32, 32), + ("8", 8), + ("int8", 8), + ("INT8", 8), + ("w8a8", 8), + ("W8A8", 8), + ("fp16", 16), + ("Fp16", 16), + ("w16a16", 16), + ("fp32", 32), + ("fP32", 32), + ("w8a16", "w8a16"), + ("W8a16", "w8a16"), + ("w8a32", "w8a32"), + ("W8A32", "w8a32"), + ]: + assert get_cfg(overrides={"quantize": value}).quantize == expected + assert get_cfg().quantize is None # unset default is FP32 + with pytest.raises(ValueError, match="quantize"): + get_cfg(overrides={"quantize": "x4"}) + with pytest.raises(ValueError, match="quantize"): + get_cfg(overrides={"quantize": "a8w8"}) + + +def test_quantize_deprecation(): + """Legacy half/int8 forward to the unified quantize arg in all modes; int8 wins on conflict.""" + assert _handle_deprecation({"int8": True})["quantize"] == 8 + assert _handle_deprecation({"half": True})["quantize"] == 16 + assert _handle_deprecation({"half": True, "int8": True})["quantize"] == 8 # int8 wins + assert "half" not in _handle_deprecation({"half": True}) # legacy flag is removed after forwarding + assert _handle_deprecation({"half": True, "quantize": None})["quantize"] is None # explicit quantize wins + assert _handle_deprecation({"half": True, "quantize": 8})["quantize"] == 8 # explicit quantize still wins + + +def test_benchmark_forwards_legacy_precision(monkeypatch): + """model.benchmark(half=True) must reach the benchmark call as quantize=16, not silently run FP32.""" + import ultralytics.utils.benchmarks as bm + + captured = {} + monkeypatch.setattr(bm, "benchmark", lambda **kw: captured.update(kw) or {}) + YOLO(MODEL).benchmark(half=True, format="onnx", data="coco8.yaml") + assert captured["quantize"] == 16, f"legacy half was dropped: quantize={captured.get('quantize')}" + + +def test_qnn_quantize_requires_w8a16(): + """QNN exports are W8A16; explicit INT8 activation quantization is not supported.""" + valid_args = ["batch", "data", "dynamic", "fraction", "keras", "nms"] + validate_args("qnn", SimpleNamespace(quantize="w8a16"), valid_args) + with pytest.raises(AssertionError, match=r"quantize=8 \(INT8\) is not supported"): + validate_args("qnn", SimpleNamespace(quantize=8), valid_args) + + +def test_modelopt_quantize_onnx_requires_int8_dataset(): + """Check INT8 ModelOpt quantization fails early without calibration data.""" + with pytest.raises(ValueError, match="requires a calibration dataset"): + modelopt_quantize_onnx("model.onnx", quantize=8) + + +def test_int8_calibration_validates_split(): + """Check INT8 calibration rejects dataset splits that do not exist.""" + exporter = object.__new__(Exporter) + exporter.model = SimpleNamespace(task="obb") + exporter.args = SimpleNamespace(data="coco8.yaml", split="trainval") + exporter.imgsz = [32] + with pytest.raises(FileNotFoundError, match="trainval"): + exporter.get_int8_calibration_dataloader() + + +@pytest.mark.parametrize("task", ["classify", "detect"]) +@pytest.mark.parametrize(("fraction", "expected"), [(2, 2), ([1, 3, 0], 3)]) +def test_int8_calibration_fraction(task, fraction, expected, tmp_path): + """Check scalar and list fractions select the requested calibration split size for classification and detection.""" + data = "coco8.yaml" + if task == "classify": + for split, count in (("train", 2), ("val", 4)): + (split_dir := tmp_path / split / "0").mkdir(parents=True) + for i in range(count): + shutil.copy(SOURCE, split_dir / f"{i}.jpg") + data = str(tmp_path) + exporter = object.__new__(Exporter) + exporter.model = SimpleNamespace(task=task) + exporter.args = get_cfg(overrides={"data": data, "split": "val", "fraction": fraction, "batch": 1}) + exporter.imgsz = [32] + assert len(exporter.get_int8_calibration_dataloader().dataset) == expected + + +def test_export_rknn_batch_expansion(monkeypatch, tmp_path): + """Check RKNN calibrates batch 1 before Toolkit expands to the requested batch.""" + calls = {} + monkeypatch.setattr( + "ultralytics.utils.export.rknn.onnx2rknn", lambda **kwargs: calls.update(kwargs) or kwargs["output_dir"] + ) + monkeypatch.setattr("ultralytics.engine.exporter.file_size", lambda _: 1) + + image = tmp_path / "image.jpg" + exporter = SimpleNamespace( + args=SimpleNamespace(opset=None, quantize=8, name="rk3588", batch=8), + im=torch.zeros(8, 3, 32, 32), + file=tmp_path / "model.pt", + metadata={}, + get_int8_calibration_dataloader=lambda prefix: SimpleNamespace(dataset=SimpleNamespace(im_files=[image])), + ) + exporter.export_onnx = lambda: calls.update(onnx_batch=len(exporter.im)) or tmp_path / "model.onnx" + Exporter.export_rknn(exporter) + assert calls["onnx_batch"] == 1 + assert calls["batch"] == 8 + + +def test_torch2onnx_serializes_concurrent_exports(monkeypatch, tmp_path): + """Ensure ONNX exports do not overlap across worker threads.""" + active = 0 + max_active = 0 + errors = [] + state_lock = threading.Lock() + + def fake_export(*args, **kwargs): + nonlocal active, max_active + with state_lock: + active += 1 + max_active = max(max_active, active) + time.sleep(0.05) + with state_lock: + active -= 1 + + monkeypatch.setattr(torch.onnx, "export", fake_export) + + def export_model(index: int): + try: + torch2onnx(torch.nn.Identity(), torch.zeros(1, 3, 8, 8), str(tmp_path / f"export-{index}.onnx")) + except Exception as error: # pragma: no cover - assertion handled below + errors.append(error) + + threads = [threading.Thread(target=export_model, args=(i,)) for i in range(4)] + for thread in threads: + thread.start() + for thread in threads: + thread.join() + + assert not errors, f"Concurrent export errors: {errors}" + assert max_active == 1, f"Expected max 1 concurrent export, got {max_active}" + + +@pytest.mark.skipif(not TORCH_2_1, reason="OpenVINO requires torch>=2.1") +@pytest.mark.parametrize("nms", [None, False]) +def test_export_openvino(nms, isolated_model): + """Test YOLO export to OpenVINO format for model inference compatibility.""" + file = YOLO(isolated_model).export(format="openvino", imgsz=32, nms=nms) + YOLO(file)(SOURCE, imgsz=32) # exported model inference + + +@pytest.mark.slow +@pytest.mark.skipif(not TORCH_2_1, reason="OpenVINO requires torch>=2.1") +@pytest.mark.parametrize( + "task, dynamic, quantize, batch, nms", + [ # generate all combinations except for exclusion cases + (task, dynamic, quantize, batch, nms) + for task, dynamic, quantize, batch, nms in product(TASKS, [True, False], [8, 16], [1, 2], [False]) + ], +) +# Keep one-to-many INT8 cases disabled until OpenVINO runner OOM is resolved. +def test_export_openvino_matrix(task, dynamic, quantize, batch, nms): + """Test YOLO model export to OpenVINO under various configuration matrix conditions.""" + skip_rpi_semantic(task) + file = YOLO(TASK2MODEL[task]).export( + format="openvino", + imgsz=32, + dynamic=dynamic, + quantize=quantize, + batch=batch, + data=TASK2DATA[task], # use the smallest task datasets for fast INT8 calibration + nms=nms, + ) + YOLO(file)([SOURCE] * batch, imgsz=64 if dynamic else 32, batch=batch) # exported model inference + shutil.rmtree(file, ignore_errors=True) # retry in case of potential lingering multi-threaded file usage errors + + +@pytest.mark.slow +@pytest.mark.parametrize( + "task, dynamic, batch, simplify, nms", + [ # generate all combinations except for exclusion cases + (task, dynamic, batch, simplify, nms) + for task, dynamic, batch, simplify, nms in product( + TASKS, [True, False], [1, 2], [True, False], [None, True, False] + ) + if not (nms is True and (task == "classify" or not TORCH_1_13)) + ], +) +def test_export_onnx_matrix(task, dynamic, batch, simplify, nms): + """Test YOLO export to ONNX format with various configurations and parameters.""" + skip_rpi_semantic(task) + file = YOLO(TASK2MODEL[task]).export( + format="onnx", + imgsz=32, + dynamic=dynamic, + batch=batch, + simplify=simplify, + nms=nms, + ) + r = YOLO(file)([SOURCE] * batch, imgsz=64 if dynamic else 32) # exported model inference + if task == "semantic": + assert r[0].semantic_mask is not None + assert r[0].semantic_mask.data.dtype in {torch.uint8, torch.int32} + Path(file).unlink() # cleanup + + +def test_export_onnx_semantic_dnn(): + """Test semantic ONNX class-map output with OpenCV DNN.""" + skip_rpi_semantic("semantic") + file = YOLO(TASK2MODEL["semantic"]).export(format="onnx", imgsz=32) + r = YOLO(file).predict(SOURCE, imgsz=32, dnn=True) + assert r[0].semantic_mask is not None + Path(file).unlink() + + +@pytest.mark.slow +@pytest.mark.parametrize( + "task, dynamic, batch, nms", + [ # generate all combinations except for exclusion cases + (task, dynamic, batch, nms) + for task, dynamic, batch, nms in product(TASKS, [False, True], [1, 2], [None, True, False]) + if not (task == "classify" and nms is True) + ], +) +def test_export_torchscript_matrix(task, dynamic, batch, nms, tmp_path): + """Test YOLO model export to TorchScript format under varied configurations.""" + skip_rpi_semantic(task) + file = YOLO(isolated_model_path(tmp_path, WEIGHTS_DIR / TASK2MODEL[task])).export( + format="torchscript", imgsz=32, dynamic=dynamic, batch=batch, nms=nms + ) + YOLO(file)([SOURCE] * batch, imgsz=64 if dynamic else 32) # exported model inference + Path(file).unlink() # cleanup + + +@pytest.mark.slow +@pytest.mark.skipif(not MACOS, reason="CoreML inference only supported on macOS") +@pytest.mark.skipif(not TORCH_1_11, reason="CoreML export requires torch>=1.11") +@pytest.mark.skipif( + MACOS and MACOS_VERSION and MACOS_VERSION >= "15", reason="CoreML YOLO26 matrix test crashes on macOS 15+" +) +@pytest.mark.parametrize( + "task, dynamic, quantize, nms, batch", + [ # generate all combinations except for exclusion cases + (task, dynamic, quantize, nms, batch) + for task, dynamic, quantize, nms, batch in product(TASKS, [True, False], [8, 16], [None, True, False], [1]) + if not (task not in {"detect", "segment", "pose"} and nms is True) + and not (dynamic and nms is True) + and not (task == "classify" and dynamic) + ], +) +def test_export_coreml_matrix(task, dynamic, quantize, nms, batch): + """Test YOLO export to CoreML format with various parameter configurations.""" + skip_rpi_semantic(task) + file = YOLO(TASK2MODEL[task]).export( + format="coreml", + imgsz=32, + dynamic=dynamic, + quantize=quantize, + batch=batch, + nms=nms, + ) + YOLO(file)([SOURCE] * batch, imgsz=32) # exported model inference + shutil.rmtree(file) # cleanup + + +@pytest.mark.skipif(not TORCH_1_11, reason="CoreML export requires torch>=1.11") +@pytest.mark.skipif(WINDOWS, reason="CoreML not supported on Windows") # RuntimeError: BlobWriter not loaded +@pytest.mark.skipif(LINUX and ARM64, reason="CoreML not supported on aarch64 Linux") +@pytest.mark.skipif( + MACOS and checks.IS_PYTHON_MINIMUM_3_13, + reason="coremltools deadlocks after OpenVINO on macOS Python 3.13 (conflicting OpenMP runtimes)", +) +@pytest.mark.parametrize("format", ["coreml", "mlmodel"]) +def test_export_coreml(isolated_model, format, monkeypatch, tmp_path): + """Test YOLO export to CoreML format and check for errors.""" + from ultralytics.utils.export import coreml + + quantize, torch2coreml = [], coreml.torch2coreml + + def capture_quantize(*args, **kwargs): + quantize.append(kwargs["quantize"]) + return torch2coreml(*args, **kwargs) + + monkeypatch.setattr(coreml, "torch2coreml", capture_quantize) + # Capture stdout and stderr + stdout, stderr = io.StringIO(), io.StringIO() + with redirect_stdout(stdout), redirect_stderr(stderr): + file = YOLO(isolated_model_path(tmp_path, WEIGHTS_DIR / "yolo11n.pt")).export( + format=format, nms=True, imgsz=32, iou=0.42, conf=0.24 + ) + import coremltools as ct + + spec = ct.utils.load_spec(str(file)) + metadata = spec.description.metadata + assert metadata.author and metadata.shortDescription and metadata.license and metadata.versionString + assert metadata.userDefined["IoU threshold"] == "0.42" + assert metadata.userDefined["Confidence threshold"] == "0.24" + assert all(key in metadata.userDefined for key in ("names", "stride", "task")) + assert next(iter(spec.pipeline.models[1].nonMaximumSuppression.stringClassLabels.vector)) == "person" + assert [output.name for output in spec.description.output] == ["confidence", "coordinates"] + if MACOS: + file = YOLO(isolated_model).export(format="coreml", imgsz=32) + YOLO(file)(SOURCE, imgsz=32) # model prediction only supported on macOS for raw-output models + + # Check captured output for errors + output = stdout.getvalue() + stderr.getvalue() + assert quantize[0] == (16 if format == "coreml" else None) + assert "Error" not in output, f"CoreML export produced errors: {output}" + assert "You will not be able to run predict()" not in output, "CoreML export has predict() error" + + +@pytest.mark.skipif(not TORCH_1_11, reason="RTDETR CoreML export requires torch>=1.11") +@pytest.mark.skipif(WINDOWS, reason="CoreML not supported on Windows") +@pytest.mark.skipif(LINUX and ARM64, reason="CoreML not supported on aarch64 Linux") +@pytest.mark.skipif( + MACOS and checks.IS_PYTHON_MINIMUM_3_13, + reason="coremltools deadlocks after OpenVINO on macOS Python 3.13 (conflicting OpenMP runtimes)", +) +def test_export_coreml_rtdetr(): + """Test RT-DETR export to CoreML format and check for errors.""" + stdout, stderr = io.StringIO(), io.StringIO() + with redirect_stdout(stdout), redirect_stderr(stderr): + file = YOLO(WEIGHTS_DIR / "rtdetr-l.pt").export(format="coreml", imgsz=160) + import coremltools as ct + + shape = ct.models.MLModel(str(file)).get_spec().description.output[0].type.multiArrayType.shape + assert shape[-2] == 300 + if MACOS: + YOLO(file)(SOURCE, imgsz=160) + + output = stdout.getvalue() + stderr.getvalue() + assert "Error" not in output, f"RTDETR CoreML export produced errors: {output}" + assert "You will not be able to run predict()" not in output, "RTDETR CoreML export has predict() error" + + +@pytest.mark.skipif(True, reason="Test disabled") +@pytest.mark.skipif(not LINUX, reason="TF suffers from install conflicts on Windows and macOS") +def test_export_pb(isolated_model): + """Test YOLO export to TensorFlow's Protobuf (*.pb) format.""" + model = YOLO(isolated_model) + file = model.export(format="pb", imgsz=32) + YOLO(file)(SOURCE, imgsz=32) + + +@pytest.mark.skipif(True, reason="Test disabled as Paddle protobuf and ONNX protobuf requirements conflict.") +def test_export_paddle(isolated_model): + """Test YOLO export to Paddle format, noting protobuf conflicts with ONNX.""" + YOLO(isolated_model).export(format="paddle", imgsz=32) + + +@pytest.mark.skipif(not TORCH_1_10, reason="MNN export requires torch>=1.10") +@pytest.mark.skipif( + LINUX and checks.IS_PYTHON_MINIMUM_3_13, + reason="MNN ONNX-parser protobuf conflicts with TensorFlow protobuf>=6.31.1 loaded earlier in the shared Python 3.13 test process", +) +def test_export_mnn(isolated_model): + """Test YOLO export to MNN format (WARNING: MNN test must precede NCNN test or CI error on Windows).""" + file = YOLO(isolated_model).export(format="mnn", imgsz=32) + YOLO(file)(SOURCE, imgsz=32) # exported model inference + + +@pytest.mark.parametrize( + "model,kwargs,error", + [ + ("yolo11n.yaml", {"batch": 2, "dynamic": True, "nms": True}, "combining"), + ("yolo11n-seg.yaml", {"nms": True}, "only supports detect and pose"), + ("yolo11n-obb.yaml", {"nms": True}, "only supports detect and pose"), + ], +) +def test_export_mnn_rejects_unsupported_nms(model, kwargs, error): + """Test MNN rejects NMS combinations that fail or lose task outputs at runtime.""" + with pytest.raises(ValueError, match=error): + YOLO(model).export(format="mnn", imgsz=32, **kwargs) + + +@pytest.mark.slow +@pytest.mark.parametrize( + "model,task,kwargs", + [ + ("yolo11n.yaml", "detect", {"batch": 2, "dynamic": True}), + ("yolo11n.yaml", "detect", {"nms": True}), + ("yolo11n-pose.yaml", "pose", {"nms": True}), + ], +) +def test_export_mnn_options(model, task, kwargs): + """Test MNN dynamic shapes and supported embedded NMS tasks through inference.""" + batch = kwargs.get("batch", 1) + file = YOLO(model).export(format="mnn", imgsz=32, **kwargs) + assert len(YOLO(file, task=task)([SOURCE] * batch, imgsz=64 if kwargs.get("dynamic") else 32)) == batch + Path(file).unlink() + + +@pytest.mark.slow +@pytest.mark.skipif(not TORCH_1_10, reason="MNN export requires torch>=1.10") +@pytest.mark.parametrize( + "task, quantize, batch, nms", + [ # generate all combinations except for exclusion cases + (task, quantize, batch, nms) for task, quantize, batch, nms in product(TASKS, [8, 16], [1, 2], [None, False]) + ], +) +def test_export_mnn_matrix(task, quantize, batch, nms): + """Test YOLO export to MNN format considering various export configurations.""" + skip_rpi_semantic(task) + file = YOLO(TASK2MODEL[task]).export(format="mnn", imgsz=32, quantize=quantize, batch=batch, nms=nms) + YOLO(file)([SOURCE] * batch, imgsz=32) # exported model inference + Path(file).unlink() # cleanup + + +@pytest.mark.skipif(not TORCH_2_0, reason="NCNN inference causes segfault on PyTorch<2.0") +def test_export_ncnn(isolated_model): + """Test YOLO export to NCNN format.""" + file = YOLO(isolated_model).export(format="ncnn", imgsz=32) + YOLO(file)(SOURCE, imgsz=32) # exported model inference + + +@pytest.mark.slow +@pytest.mark.skipif(not TORCH_2_0, reason="NCNN inference causes segfault on PyTorch<2.0") +@pytest.mark.parametrize("task, quantize, batch", list(product(sorted(TASKS), [16], [1]))) +def test_export_ncnn_matrix(task, quantize, batch): + """Test YOLO export to NCNN format considering various export configurations.""" + skip_rpi_semantic(task) + file = YOLO(TASK2MODEL[task]).export(format="ncnn", imgsz=32, quantize=quantize, batch=batch) + YOLO(file)([SOURCE] * batch, imgsz=32) # exported model inference + shutil.rmtree(file, ignore_errors=True) # retry in case of potential lingering multi-threaded file usage errors + + +@pytest.mark.skipif(not TORCH_2_9, reason="IMX export requires torch>=2.9.0") +@pytest.mark.skipif(not checks.IS_PYTHON_MINIMUM_3_9, reason="IMX export requires Python>=3.9") +@pytest.mark.skipif(not LINUX, reason="IMX export only supported on Linux") +@pytest.mark.skipif( + IS_RASPBERRYPI, reason="Test disabled as IMX export suffers from OOM (Out of Memory) on Raspberry Pi 5 16GB" +) +@pytest.mark.parametrize("conf,expected", [(0.0, 0.0), (None, 0.25)]) +def test_export_imx(tmp_path, conf, expected): + """Test IMX export and inference, preserving zero confidence and the public export default.""" + import onnx + + model = YOLO(isolated_model_path(tmp_path, WEIGHTS_DIR / "yolo11n.pt")) + output_dir = model.export(format="imx", imgsz=32, data="coco8.yaml", conf=conf) + nodes = [ + n + for n in onnx.load(str(Path(output_dir) / "model_imx.onnx")).graph.node + if n.op_type == "MultiClassNMSWithIndices" + ] + assert len(nodes) == 1, "MultiClassNMSWithIndices node missing from the exported ONNX" + attrs = {a.name: a.f for a in nodes[0].attribute} + assert attrs["score_threshold"] == pytest.approx(expected) + YOLO(output_dir)(SOURCE, imgsz=32) + + +@pytest.mark.slow +@pytest.mark.skipif(not LINUX or ARM64, reason="RKNN export only supported on non-aarch64 Linux") +@pytest.mark.parametrize("quantize,batch", [(8, 8), (16, 1)]) +def test_export_rknn(isolated_model, quantize, batch): + """Test YOLO export to RKNN format.""" + file = YOLO(isolated_model).export(format="rknn", imgsz=32, quantize=quantize, batch=batch, data="coco8.yaml") + assert next(Path(file).rglob("*.rknn"), None), f"RKNN export failed, no RKNN model found in: {file}" + shutil.rmtree(file, ignore_errors=True) + + +# @pytest.mark.skipif(True, reason="Disabled for debugging ruamel.yaml installation required by executorch") +@pytest.mark.skipif(not checks.IS_PYTHON_MINIMUM_3_10 or not TORCH_2_9, reason="Requires Python>=3.10 and Torch>=2.9.0") +@pytest.mark.skipif(WINDOWS, reason="Skipping test on Windows") +def test_export_executorch(isolated_model): + """Test YOLO model export to ExecuTorch format.""" + file = YOLO(isolated_model).export(format="executorch", imgsz=32) + assert Path(file).exists(), f"ExecuTorch export failed, directory not found: {file}" + # Check that .pte file exists in the exported directory + pte_file = Path(file) / "model.pte" + assert pte_file.exists(), f"ExecuTorch .pte file not found: {pte_file}" + # Check that metadata.yaml exists + metadata_file = Path(file) / "metadata.yaml" + assert metadata_file.exists(), f"ExecuTorch metadata.yaml not found: {metadata_file}" + # Note: Inference testing skipped as ExecuTorch requires special runtime setup + shutil.rmtree(file, ignore_errors=True) # cleanup + + +@pytest.mark.slow +@pytest.mark.skipif(not (MACOS or (LINUX and not ARM64)), reason="LiteRT export only supported on Linux x86 and macOS") +@pytest.mark.skipif(not checks.IS_PYTHON_MINIMUM_3_10, reason="litert-torch requires Python>=3.10") +@pytest.mark.parametrize( + "task, quantize", + [(task, quantize) for task in sorted(TASKS) for quantize in (None, 8, "w8a16", "w8a32")], +) +def test_export_litert_matrix(task, quantize): + """Test YOLO export to LiteRT format (FP32, static INT8, static w8a16, and dynamic w8a32) for various tasks.""" + file = Path(YOLO(TASK2MODEL[task]).export(format="litert", imgsz=32, quantize=quantize)) + assert file.is_file() and file.suffix == ".tflite", f"LiteRT export is not a single .tflite for '{task}': {file}" + # Contract: exports keep float32 graph I/O (int8/int16 stays internal) so downstream runtimes feed/read floats + # without boundary (de)quantization; an int8/int16 I/O regression would silently break on-device consumers. + import numpy as np + from ai_edge_litert.interpreter import Interpreter + + interpreter = Interpreter(model_path=str(file)) + interpreter.allocate_tensors() + io_details = interpreter.get_input_details() + interpreter.get_output_details() + assert all(d["dtype"] == np.float32 for d in io_details), ( + f"LiteRT '{task}' quantize={quantize} must keep float32 I/O, got {[d['dtype'] for d in io_details]}" + ) + YOLO(file)(SOURCE, imgsz=32) # exported model inference (also exercises the embedded metadata) + file.unlink() # cleanup + + +@pytest.mark.slow +@pytest.mark.skipif(not checks.IS_PYTHON_MINIMUM_3_10 or not TORCH_2_9, reason="Requires Python>=3.10 and Torch>=2.9.0") +@pytest.mark.skipif(WINDOWS, reason="Skipping test on Windows") +@pytest.mark.parametrize("task", sorted(TASKS)) +def test_export_executorch_matrix(task): + """Test YOLO export to ExecuTorch format for various task types.""" + skip_rpi_semantic(task) + file = YOLO(TASK2MODEL[task]).export(format="executorch", imgsz=32) + assert Path(file).exists(), f"ExecuTorch export failed for task '{task}', directory not found: {file}" + # Check that .pte file exists in the exported directory + pte_file = Path(file) / "model.pte" + assert pte_file.exists(), f"ExecuTorch .pte file not found for task '{task}': {pte_file}" + # Check that metadata.yaml exists + metadata_file = Path(file) / "metadata.yaml" + assert metadata_file.exists(), f"ExecuTorch metadata.yaml not found for task '{task}': {metadata_file}" + # Note: Inference testing skipped as ExecuTorch requires special runtime setup + shutil.rmtree(file, ignore_errors=True) # cleanup + + +@pytest.mark.skipif( + not (WINDOWS or LINUX) or sys.version_info < (3, 11), + reason="onnxruntime-qnn ships prebuilt wheels only for Windows and Linux on Python>=3.11", +) +def test_export_qnn(isolated_model): + """Test YOLO export to Qualcomm QNN format via the ONNX Runtime QNN Execution Provider.""" + import importlib.util + + # QNN EP ships either as the 'onnxruntime_qnn' plugin module or as a provider library bundled in onnxruntime/capi. + has_qnn = importlib.util.find_spec("onnxruntime_qnn") is not None + if not has_qnn and importlib.util.find_spec("onnxruntime") is not None: + import onnxruntime + + capi = Path(onnxruntime.__file__).parent / "capi" + has_qnn = (capi / "libonnxruntime_providers_qnn.so").exists() or ( + capi / "onnxruntime_providers_qnn.dll" + ).exists() + if not has_qnn: + pytest.skip("onnxruntime-qnn / QNN Execution Provider not available") + file = YOLO(isolated_model).export(format="qnn", imgsz=32) + assert Path(file).is_file() and file.endswith("_qnn.onnx"), f"QNN export failed, no context binary found: {file}" + # Note: on-device inference is not exercised here as it requires Qualcomm Snapdragon hardware + Path(file).unlink(missing_ok=True) # cleanup + + +@pytest.mark.parametrize("env", [k for k, v in EXPORT_ENVS.items() if k != "base" or v["smoke"]]) +def test_export_env_has_smoke(env): + """Ensure every non-base export environment declares a build-time smoke export.""" + assert EXPORT_ENVS[env]["smoke"], f"export env '{env}' has no smoke command" + + +def test_every_format_env_is_registered(): + """Ensure every export format points at a registered export environment.""" + for fmt, env in zip(export_formats()["Argument"], export_formats()["Env"]): + assert env in EXPORT_ENVS, f"format '{fmt}' references unknown env '{env}'" diff --git a/.venv/lib/python3.12/site-packages/tests/test_integrations.py b/.venv/lib/python3.12/site-packages/tests/test_integrations.py new file mode 100644 index 0000000000000000000000000000000000000000..71568a0e4160400aa7e47801e9e32ba1222efb54 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/tests/test_integrations.py @@ -0,0 +1,155 @@ +# Ultralytics 🚀 AGPL-3.0 License - https://ultralytics.com/license + +import contextlib +import subprocess +import time +from pathlib import Path + +import pytest + +from tests import SOURCE +from ultralytics import YOLO, download +from ultralytics.utils import ASSETS_URL, DATASETS_DIR, SETTINGS +from ultralytics.utils.checks import check_requirements + + +@pytest.mark.slow +def test_tensorboard(): + """Test training with TensorBoard logging enabled.""" + SETTINGS["tensorboard"] = True + YOLO("yolo26n-cls.yaml").train(data="imagenet10", imgsz=32, epochs=3, plots=False, device="cpu") + SETTINGS["tensorboard"] = False + + +@pytest.mark.skipif(not check_requirements("ray", install=False), reason="ray[tune] not installed") +def test_model_ray_tune(): + """Tune YOLO model using Ray for hyperparameter optimization.""" + YOLO("yolo26n-cls.yaml").tune( + use_ray=True, data="imagenet10", grace_period=1, iterations=1, imgsz=32, epochs=1, plots=False, device="cpu" + ) + + +@pytest.mark.skipif(not check_requirements("mlflow", install=False), reason="mlflow not installed") +def test_mlflow(tmp_path, monkeypatch): + """Test training with MLflow tracking enabled.""" + import mlflow + + monkeypatch.setenv("MLFLOW_TRACKING_URI", f"sqlite:///{(tmp_path / 'mlflow.db').as_posix()}") + monkeypatch.setenv("MLFLOW_EXPERIMENT_NAME", "test_mlflow") + monkeypatch.setitem(SETTINGS, "mlflow", True) + try: + YOLO("yolo26n-cls.yaml").train(data="imagenet10", imgsz=32, epochs=3, plots=False, device="cpu") + finally: + mlflow.autolog(disable=True) + mlflow.end_run() + + +@pytest.mark.skipif(not check_requirements("mlflow", install=False), reason="mlflow not installed") +def test_mlflow_keep_run_active(tmp_path, monkeypatch): + """Ensure MLFLOW_KEEP_RUN_ACTIVE controls whether new MLflow runs remain active.""" + import mlflow + + monkeypatch.setenv("MLFLOW_TRACKING_URI", f"sqlite:///{(tmp_path / 'mlflow.db').as_posix()}") + monkeypatch.setenv("MLFLOW_EXPERIMENT_NAME", "keep_run_active") + monkeypatch.setenv("MLFLOW_RUN", "Test Run") + monkeypatch.setitem(SETTINGS, "mlflow", True) + try: + monkeypatch.setenv("MLFLOW_KEEP_RUN_ACTIVE", "True") + YOLO("yolo26n-cls.yaml").train(data="imagenet10", imgsz=32, epochs=1, plots=False, device="cpu") + active = mlflow.active_run() + assert active is not None and active.info.status == "RUNNING", ( + "MLflow run should be active when MLFLOW_KEEP_RUN_ACTIVE=True" + ) + mlflow.end_run() + + monkeypatch.setenv("MLFLOW_KEEP_RUN_ACTIVE", "False") + YOLO("yolo26n-cls.yaml").train(data="imagenet10", imgsz=32, epochs=1, plots=False, device="cpu") + assert mlflow.active_run() is None, "MLflow run should be ended when MLFLOW_KEEP_RUN_ACTIVE=False" + + monkeypatch.delenv("MLFLOW_KEEP_RUN_ACTIVE", raising=False) + YOLO("yolo26n-cls.yaml").train(data="imagenet10", imgsz=32, epochs=1, plots=False, device="cpu") + assert mlflow.active_run() is None, "MLflow run should be ended by default when MLFLOW_KEEP_RUN_ACTIVE is unset" + finally: + mlflow.autolog(disable=True) + mlflow.end_run() + + +@pytest.mark.skipif(not check_requirements("tritonclient", install=False), reason="tritonclient[all] not installed") +def test_triton(tmp_path, isolated_model): + """Test NVIDIA Triton Server functionalities with YOLO model.""" + check_requirements("tritonclient[all]") + from tritonclient.http import InferenceServerClient + + # Create variables + model_name = "yolo" + triton_repo = tmp_path / "triton_repo" # Triton repo path + triton_model = triton_repo / model_name # Triton model path + + # Export model to ONNX + f = YOLO(isolated_model).export(format="onnx", dynamic=True) + + # Prepare Triton repo + (triton_model / "1").mkdir(parents=True, exist_ok=True) + Path(f).rename(triton_model / "1" / "model.onnx") + (triton_model / "config.pbtxt").touch() + + # Define image https://catalog.ngc.nvidia.com/orgs/nvidia/containers/tritonserver + tag = "nvcr.io/nvidia/tritonserver:23.09-py3" # 6.4 GB + + # Pull the image + subprocess.call(f"docker pull {tag}", shell=True) + + # Run the Triton server and capture the container ID + container_id = ( + subprocess.check_output( + f"docker run -d --rm -v {triton_repo}:/models -p 8000:8000 {tag} tritonserver --model-repository=/models", + shell=True, + ) + .decode("utf-8") + .strip() + ) + + # Wait for the Triton server to start + triton_client = InferenceServerClient(url="localhost:8000", verbose=False, ssl=False) + + # Wait until model is ready + for _ in range(10): + with contextlib.suppress(Exception): + assert triton_client.is_model_ready(model_name) + break + time.sleep(1) + + # Check Triton inference + YOLO(f"http://localhost:8000/{model_name}", "detect")(SOURCE) # exported model inference + + # Kill and remove the container at the end of the test + subprocess.call(f"docker kill {container_id}", shell=True) + + +@pytest.mark.skipif(not check_requirements("faster-coco-eval", install=False), reason="faster-coco-eval not installed") +def test_faster_coco_eval(): + """Validate YOLO model predictions on COCO dataset using faster-coco-eval.""" + from ultralytics.models.yolo.detect import DetectionValidator + from ultralytics.models.yolo.pose import PoseValidator + from ultralytics.models.yolo.segment import SegmentationValidator + + args = {"model": "yolo26n.pt", "data": "coco8.yaml", "save_json": True, "imgsz": 64} + validator = DetectionValidator(args=args) + validator() + validator.is_coco = True + download(f"{ASSETS_URL}/instances_val2017.json", dir=DATASETS_DIR / "coco8/annotations") + _ = validator.eval_json(validator.stats) + + args = {"model": "yolo26n-seg.pt", "data": "coco8-seg.yaml", "save_json": True, "imgsz": 64} + validator = SegmentationValidator(args=args) + validator() + validator.is_coco = True + download(f"{ASSETS_URL}/instances_val2017.json", dir=DATASETS_DIR / "coco8-seg/annotations") + _ = validator.eval_json(validator.stats) + + args = {"model": "yolo26n-pose.pt", "data": "coco8-pose.yaml", "save_json": True, "imgsz": 64} + validator = PoseValidator(args=args) + validator() + validator.is_coco = True + download(f"{ASSETS_URL}/person_keypoints_val2017.json", dir=DATASETS_DIR / "coco8-pose/annotations") + _ = validator.eval_json(validator.stats) diff --git a/.venv/lib/python3.12/site-packages/tests/test_ndjson_converter.py b/.venv/lib/python3.12/site-packages/tests/test_ndjson_converter.py new file mode 100644 index 0000000000000000000000000000000000000000..33de853f6eac479de12687ec06c96009b06e503e --- /dev/null +++ b/.venv/lib/python3.12/site-packages/tests/test_ndjson_converter.py @@ -0,0 +1,195 @@ +# Ultralytics 🚀 AGPL-3.0 License - https://ultralytics.com/license + +import asyncio +import json +import threading +from functools import partial +from http.server import SimpleHTTPRequestHandler, ThreadingHTTPServer + +import cv2 +import numpy as np +import pytest + +from ultralytics.data.converter import convert_ndjson_to_yolo +from ultralytics.data.utils import load_depth, save_depth_png +from ultralytics.utils import YAML + + +class _QuietHandler(SimpleHTTPRequestHandler): + """Serve converter fixtures without writing requests to the test log.""" + + def log_message(self, _format, *args): + pass + + +def _write_manifest(path, base_url, *, missing_depth=False, depth_scale=None): + records = [ + {"type": "dataset", "task": "depth", **({"depth_scale": depth_scale} if depth_scale else {})}, + { + "type": "image", + "file": "camera/train.jpg", + "url": f"{base_url}/train.jpg?signature=image", + "split": "train", + "depth": { + "url": f"{base_url}/train.png?signature=depth", + "hash": "depth-train", + "shape": [3, 4], + }, + }, + { + "type": "image", + "file": "val.jpg", + "url": f"{base_url}/test.jpg?signature=image", + "split": "val", + "depth": { + "url": f"{base_url}/missing.png" if missing_depth else f"{base_url}/test.png?signature=depth", + "hash": "depth-test", + "shape": [3, 4], + }, + }, + ] + path.write_text("\n".join(json.dumps(record) for record in records)) + + +@pytest.fixture +def depth_server(tmp_path): + """Serve paired image and depth fixtures over HTTP.""" + source = tmp_path / "source" + source.mkdir() + depth = np.arange(12, dtype=np.float32).reshape(3, 4) + for split, value in (("train", 0), ("test", 255)): + cv2.imwrite(str(source / f"{split}.jpg"), np.full((3, 4, 3), value, dtype=np.uint8)) + save_depth_png(source / f"{split}.png", depth) + server = ThreadingHTTPServer(("127.0.0.1", 0), partial(_QuietHandler, directory=source)) + server.daemon_threads = True + thread = threading.Thread(target=server.serve_forever, daemon=True) + thread.start() + try: + yield f"http://127.0.0.1:{server.server_port}", depth + finally: + server.shutdown() + server.server_close() + thread.join() + + +def test_convert_depth_ndjson_downloads_image_target_pairs(tmp_path, depth_server): + """Download depth targets beside images using matching indexed stems and the default scale.""" + base_url, depth = depth_server + manifest = tmp_path / "depth.ndjson" + _write_manifest(manifest, base_url) + + yaml_path = asyncio.run(convert_ndjson_to_yolo(manifest, tmp_path / "datasets")) + + data = YAML.load(yaml_path) + assert data["task"] == "depth" + assert data["nc"] == 1 + assert data["names"] == {0: "depth"} + assert data["depth_scale"] == 1000 + assert data["train"] == "images/train" + assert data["val"] == "images/val" + assert not (yaml_path.parent / "labels").exists() + for index, split in enumerate(("train", "val"), 1): + assert (yaml_path.parent / "images" / split / f"{index}.jpg").is_file() + np.testing.assert_allclose(load_depth(yaml_path.parent / "depth" / split / f"{index}.png"), depth, atol=1e-3) + + +def test_convert_depth_ndjson_preserves_scale(tmp_path, depth_server): + """Copy a dataset-level PNG scale into the generated training YAML.""" + base_url, _ = depth_server + manifest = tmp_path / "depth.ndjson" + _write_manifest(manifest, base_url, depth_scale=256) + + data = YAML.load(asyncio.run(convert_ndjson_to_yolo(manifest, tmp_path / "datasets"))) + + assert data["depth_scale"] == 256 + + +def test_convert_depth_ndjson_reuses_existing_conversion(tmp_path, depth_server, monkeypatch): + """Reuse a complete subset and reconvert when its completion marker is invalidated.""" + base_url, _ = depth_server + manifest = tmp_path / "depth.ndjson" + _write_manifest(manifest, base_url) + yaml_path = asyncio.run(convert_ndjson_to_yolo(manifest, tmp_path / "datasets", fraction=[1, 1])) + + monkeypatch.setattr(YAML, "save", lambda *_args, **_kwargs: pytest.fail("cache missed")) + assert asyncio.run(convert_ndjson_to_yolo(manifest, tmp_path / "datasets", fraction=[1.0, 1.0])) == yaml_path + + monkeypatch.undo() + depth_path = yaml_path.parent / "depth" / "val" / "2.png" + depth_path.unlink() + data = YAML.load(yaml_path) + data.pop("complete") + YAML.save(yaml_path, data) + assert asyncio.run(convert_ndjson_to_yolo(manifest, tmp_path / "datasets", fraction=[1, 1])) == yaml_path + assert depth_path.is_file() + + +def test_convert_depth_ndjson_removes_incomplete_pair(tmp_path, depth_server): + """Fail depth conversion without leaving an image whose paired target failed to download.""" + base_url, _ = depth_server + manifest = tmp_path / "incomplete.ndjson" + _write_manifest(manifest, base_url, missing_depth=True) + + with pytest.raises(RuntimeError, match=r"Downloaded 1/2 images"): + asyncio.run(convert_ndjson_to_yolo(manifest, tmp_path / "datasets")) + + dataset_dir = next(p for p in (tmp_path / "datasets").iterdir() if p.is_dir()) + assert not (dataset_dir / "images" / "val" / "2.jpg").exists() + assert not (dataset_dir / "depth" / "val" / "2.png").exists() + assert not (dataset_dir / "data.yaml").exists() + + +def test_convert_depth_ndjson_rejects_missing_url(tmp_path): + """Reject a missing depth URL before issuing downloads.""" + manifest = tmp_path / "invalid.ndjson" + records = [ + {"type": "dataset", "task": "depth"}, + { + "type": "image", + "file": "train.jpg", + "url": "http://127.0.0.1:1/train.jpg", + "split": "train", + "depth": {}, + }, + ] + manifest.write_text("\n".join(json.dumps(record) for record in records)) + + with pytest.raises(ValueError, match=r"missing depth\.url"): + asyncio.run(convert_ndjson_to_yolo(manifest, tmp_path / "datasets")) + + +def test_convert_ndjson_selects_split_fractions(tmp_path, depth_server): + """Select requested splits while preserving full metadata and two-item list behavior.""" + base_url, _ = depth_server + records = [ + {"type": "dataset", "task": "detect"}, + *[ + { + "type": "image", + "file": f"{index}.jpg", + "url": f"{base_url}/train.jpg?signature={index}", + "split": "test" if index == 9 else "train", + "annotations": {"boxes": [[2 if index == 9 else 0, 0.5, 0.5, 1, 1]]}, + } + for index in range(10) + ], + ] + manifest = tmp_path / "detect.ndjson" + manifest.write_text("\n".join(json.dumps(record) for record in records)) + + for fraction, split, expected_test in ( + ([0.25, 1], None, {"10.jpg"}), + ([0.25, 1, 0], None, set()), + ([0.25, 1], "val", set()), + ([0.25, 1], "test", {"10.jpg"}), + ): + yaml_path = asyncio.run(convert_ndjson_to_yolo(manifest, tmp_path / "datasets", fraction=fraction, split=split)) + files = [ + {p.name for p in (yaml_path.parent / "images" / split).glob("*")} for split in ("train", "val", "test") + ] + assert files == [{"1.jpg", "9.jpg"}, {"8.jpg"}, expected_test] + assert YAML.load(yaml_path)["nc"] == 3 + + yaml_path = asyncio.run(convert_ndjson_to_yolo(manifest, tmp_path / "datasets", fraction=0.05)) + train = {p.name for p in (yaml_path.parent / "images" / "train").glob("*")} + assert train == {"1.jpg"} # round(9 * 0.05) == 0, yet a nonzero fraction must never write an empty split diff --git a/.venv/lib/python3.12/site-packages/tests/test_python.py b/.venv/lib/python3.12/site-packages/tests/test_python.py new file mode 100644 index 0000000000000000000000000000000000000000..ea656a48f647f24551b771c16fc98a4c2529ca64 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/tests/test_python.py @@ -0,0 +1,2295 @@ +# Ultralytics 🚀 AGPL-3.0 License - https://ultralytics.com/license + +import contextlib +import csv +import os +import platform +import shutil +import tarfile +import urllib +import zipfile +from copy import copy +from pathlib import Path + +import cv2 +import numpy as np +import pytest +import torch +from PIL import Image + +import ultralytics.data.build as data_build +from tests import CFG, MODEL, MODELS, SOURCE, SOURCES_LIST, TASK_MODEL_DATA +from ultralytics import RTDETR, YOLO +from ultralytics.cfg import get_cfg +from ultralytics.data.build import build_dataloader, load_inference_source +from ultralytics.data.utils import check_cls_dataset, check_det_dataset, get_split_fraction +from ultralytics.utils import ( + ARM64, + ASSETS, + ASSETS_URL, + DATASETS_DIR, + DEFAULT_CFG, + DEFAULT_CFG_PATH, + IS_JETSON, + IS_RASPBERRYPI, + LINUX, + LOGGER, + ONLINE, + ROOT, + WEIGHTS_DIR, + WINDOWS, + YAML, + checks, + is_github_action_running, +) +from ultralytics.utils.downloads import download, safe_download +from ultralytics.utils.torch_utils import TORCH_1_10, TORCH_1_11, TORCH_1_13, TORCH_2_0 + + +def test_dataloader_caps_workers_to_batches(): + """Test tiny datasets do not spawn persistent workers beyond useful batch count.""" + single_batch = build_dataloader(range(4), batch=4, workers=8) + drop_last_single_batch = build_dataloader(range(5), batch=4, workers=8, drop_last=True) + two_batches = build_dataloader(range(8), batch=4, workers=8) + try: + assert single_batch.num_workers == 0 + assert drop_last_single_batch.num_workers == 0 + assert two_batches.num_workers <= 2 + finally: + single_batch.close() + drop_last_single_batch.close() + two_batches.close() + + +def test_dataloader_cap_preserves_distributed_drop_last(monkeypatch): + """Test worker cap follows distributed sampler size without changing global drop_last behavior.""" + sampler_cls = data_build.distributed.DistributedSampler + + def distributed_sampler(dataset, shuffle, seed): + return sampler_cls(dataset, num_replicas=3, rank=2, shuffle=shuffle, seed=seed) + + monkeypatch.setattr(data_build.distributed, "DistributedSampler", distributed_sampler) + monkeypatch.setattr(data_build, "RANK", 2) # Simulate the second node with global rank 2 and local rank 0 + expected_seed = torch.initial_seed() - 3 + loader = build_dataloader(range(8), batch=4, workers=8, rank=0, drop_last=True) + try: + assert len(loader) == 1 + assert loader.num_workers == 0 + assert loader.sampler.seed == expected_seed + finally: + loader.close() + + +def test_dataloader_seed_varies_sampling_order(): + """Test the run seed reaches the loader RNG instead of every run replaying one fixed order.""" + with torch.random.fork_rng(): + loaders = [] + for seed in (0, 0, 1): + torch.manual_seed(seed) + loaders.append(build_dataloader(range(64), batch=4, workers=0)) + try: + first, repeat, other = (torch.cat(list(loader)).tolist() for loader in loaders) + assert first == repeat # same seed stays reproducible + assert first != other # different seeds must not share one order + finally: + for loader in loaders: + loader.close() + + +def test_dataloader_empty_dataset_uses_dataloader_validation(): + """Test empty datasets fail through DataLoader validation instead of worker-cap math.""" + with pytest.raises(ValueError, match="positive integer"): + build_dataloader([], batch=4, workers=2) + + +def test_image_cache_shared_with_spawned_workers(): + """Test the RAM image cache reaches spawned DataLoader workers as one shared buffer with intact contents.""" + from ultralytics.data.base import BaseDataset + + images = [np.full((8, 8, 3), i, dtype=np.uint8) for i in range(8)] + cache = BaseDataset._ImageCache(list(images)) + loader = torch.utils.data.DataLoader( + cache, batch_size=4, sampler=range(8), num_workers=2, multiprocessing_context="spawn" + ) + assert torch.equal(torch.cat(list(loader)), torch.from_numpy(np.stack(images))) + assert cache.buffer.is_shared() + + +def test_build_yolo_dataset_hyp_isolated(): + """Test dataset construction never mutates hyperparameters on the shared cfg it was built from.""" + data = check_det_dataset("coco8.yaml") + cfg = get_cfg(overrides={"data": "coco8.yaml", "imgsz": 32, "rect": True}) # rect zeroes mosaic on the hyp used + data_build.build_yolo_dataset(cfg, data["train"], batch=2, data=data, mode="train") + assert cfg.mosaic == DEFAULT_CFG.mosaic + + +def test_cfg_rejects_fuzzed_values(): + """Test invalid overrides fail in config validation.""" + with pytest.raises(TypeError, match="degrees"): + get_cfg(overrides={"degrees": None}) + with pytest.raises(ValueError, match="cls_pw"): + get_cfg(overrides={"cls_pw": 10}) + for key, value in ( + ("split", []), + ("split", -0.0), + ("epochs", 0), + ("epochs", -1), + ("optimizer", []), + ("copy_paste_mode", {}), + ("optimizer", None), + ("split", None), + ("copy_paste_mode", None), + ("patience", -1), + ): + with pytest.raises((TypeError, ValueError), match=key): + get_cfg(overrides={key: value}) + assert get_cfg(overrides={"fraction": 1}).fraction == 1.0 + assert get_cfg(overrides={"fraction": [1000, 1, 0]}).fraction == [1000, 1.0, 0.0] + assert type(get_split_fraction([1, 1, 0], "train")) is float + assert type(get_split_fraction([1, 1, 0], "test")) is float + with pytest.raises(ValueError, match="val fraction"): + get_split_fraction([1, 0], "val") + with pytest.raises(TypeError, match="fraction"): + get_cfg(overrides={"fraction": True}) + assert get_cfg(overrides={"auto_augment": None}).auto_augment is None + assert get_cfg(overrides={"end2end": True, "nms": True}).nms is False + assert get_cfg(overrides={"end2end": False, "nms": False}).nms is None + assert get_cfg(overrides={"end2end": False, "nms": True}).nms is True + + +def skip_rpi_semantic(): + """Skip semantic segmentation tests on Raspberry Pi due to memory constraints.""" + if IS_RASPBERRYPI: + pytest.skip("Semantic segmentation tests are skipped on Raspberry Pi due to memory constraints.") + + +def test_select_device(monkeypatch): + """The same device string must resolve to the same GPU on every call, and the environment is never mutated.""" + from ultralytics.utils import torch_utils + + set_calls = [] + monkeypatch.setattr(torch_utils.torch.cuda, "is_available", lambda: True) + monkeypatch.setattr(torch_utils.torch.cuda, "device_count", lambda: 2) + monkeypatch.setattr(torch_utils.torch.cuda, "current_device", lambda: 0) + monkeypatch.setattr(torch_utils.torch.cuda, "set_device", set_calls.append) + monkeypatch.setattr(torch_utils, "get_gpu_info", lambda i: f"Mock GPU {i}, 1MiB") + monkeypatch.delenv("CUDA_VISIBLE_DEVICES", raising=False) + assert str(torch_utils.select_device("", verbose=False)) == "cuda:0" + assert not set_calls # default '' request must never move the current device, e.g. diagnostics like check_yolo() + for _ in range(2): # repeated calls are idempotent, e.g. Trainer.__init__ then final_eval, or predict() twice + assert str(torch_utils.select_device("1", verbose=False)) == "cuda:1" + with pytest.raises(ValueError): + torch_utils.select_device("3", verbose=False) + assert os.environ.get("CUDA_VISIBLE_DEVICES") is None # CUDA_VISIBLE_DEVICES never written + assert set_calls == [1, 1] # explicit single-GPU requests set the default device for indexless 'cuda' operations + assert str(torch_utils.select_device("0,1", verbose=False)) == "cuda:0" + assert set_calls == [1, 1] # multi-GPU requests never move the current device; DDP ranks pin theirs in _setup_ddp + monkeypatch.setattr(torch_utils.torch.cuda, "current_device", lambda: 1) + assert str(torch_utils.select_device("", verbose=False)) == "cuda:1" # default '' resolves to the current device + assert str(torch_utils.select_device(torch.device("cuda", 1), verbose=False)) == "cuda:1" + with pytest.raises(ValueError): # torch.device inputs are validated like strings, no raw CUDA errors + torch_utils.select_device(torch.device("cuda", 3), verbose=False) + set_calls.clear() + assert str(torch_utils.select_device(torch.device("cuda"), verbose=False)) == "cuda:1" + assert not set_calls # indexless torch.device('cuda') means the current device and never moves it + assert torch_utils.parse_device([0, 1]) == "0,1" + assert torch_utils.parse_device("00,01") == "0,1" # leading zeros stripped for valid torch device strings + assert torch_utils.parse_device(torch.device("cuda")) == "" # indexless 'cuda' stays the '' default request + # Physical GPU ids under an external CUDA_VISIBLE_DEVICES restriction translate to torch indices + monkeypatch.setenv("CUDA_VISIBLE_DEVICES", "2,3") + assert str(torch_utils.select_device("3", verbose=False)) == "cuda:1" + monkeypatch.setenv("CUDA_VISIBLE_DEVICES", "3") + monkeypatch.setattr(torch_utils.torch.cuda, "device_count", lambda: 1) + assert str(torch_utils.select_device("3", verbose=False)) == "cuda:0" # e.g. pods launched with CVD preset + assert torch_utils.parse_device(torch_utils.parse_device("3")) == "0" # idempotent: trainer + select_device parse + # '-1' idle-GPU auto-selection searches only externally visible GPUs and translates physical ids to torch indices + from ultralytics.utils import autodevice + + monkeypatch.setattr(autodevice.GPUInfo, "__init__", lambda self: self.__dict__.update(nvml_available=False)) + monkeypatch.setattr( + autodevice.GPUInfo, + "select_idle_gpu", + lambda self, count=1, indices=None, **kw: [i for i in (0, 1, 3) if i in indices][:count], + ) + monkeypatch.setattr(torch_utils.torch.cuda, "device_count", lambda: 2) + monkeypatch.setenv("CUDA_VISIBLE_DEVICES", "1,3") + assert torch_utils.parse_device("-1") == "0" # idle physical GPU 1 is torch index 0 under CVD='1,3'; 0 is hidden + assert torch_utils.parse_device("1") == "1" # in-range ids are torch indices, so repeated parses are stable + assert torch_utils.parse_device("-1,3") == "0,1" # mixed: idle physical GPU 1 + physical GPU 3 as torch indices + assert torch_utils.parse_device("0,1") == "0,1" # already-translated outputs re-parse unchanged (idempotent) + monkeypatch.setenv("CUDA_VISIBLE_DEVICES", "3,2,9,1") # malformed: CUDA stops at the invalid id 9, 2 usable GPUs + assert torch_utils.parse_device("9") == "9" # unusable id is not translated, so select_device rejects it + assert torch_utils.parse_device("2") == "1" # physical GPU 2 is torch index 1 of the usable prefix + monkeypatch.setenv("CUDA_VISIBLE_DEVICES", "01,03") # leading zeros are valid for CUDA's atoi-style parsing + assert torch_utils.parse_device("3") == "1" # visible ids normalize like requested ids + assert torch_utils.parse_device("-1") == "0" # idle physical GPU 1 found via normalized visible ids + + +def test_autobackend_memory_format(tmp_path): + """Check backend memory formats on the real host platform without mocked platform state.""" + from ultralytics.nn.autobackend import AutoBackend + + cpu_supported = ( + TORCH_1_13 + and platform.machine() in {"AMD64", "x86_64"} + and torch.backends.mkldnn.is_available() + and torch.backends.mkldnn.enabled + ) + for channels_last in (None, False, True): + model = torch.nn.Sequential(torch.nn.Conv2d(3, 4, 3)) + backend = AutoBackend(model=model, device=torch.device("cpu"), channels_last=channels_last) + expected = cpu_supported and (channels_last is True or (channels_last is None and (LINUX or WINDOWS))) + assert model[0].weight.is_contiguous(memory_format=torch.channels_last) is expected + assert backend(torch.zeros(1, 3, 32, 32)).shape == (1, 4, 30, 30) + if hasattr(torch, "inference_mode"): + with torch.inference_mode(): + model = torch.nn.Sequential(torch.nn.Conv2d(3, 4, 3)) + backend = AutoBackend(model=model, device=torch.device("cpu")) + if TORCH_1_10: + assert not backend.model[0].weight.is_inference() + + model = YOLO(MODEL) + model.ckpt["ema"] = model.model # raw training checkpoints prefer EMA when reloaded + model.model.to(memory_format=torch.channels_last) + model.save(tmp_path / "model.pt") + assert all(x.is_contiguous() for x in YOLO(tmp_path / "model.pt").model.parameters()) + + +def test_restricted_load_threaded(): + """Concurrent restricted loads share one process-wide allow-list and must not strip each other's entries.""" + import pathlib + from concurrent.futures import ThreadPoolExecutor + + from ultralytics.nn.tasks import torch_safe_load + + windows_path = pathlib.WindowsPath + with ThreadPoolExecutor(8) as pool: + list(pool.map(lambda _: torch_safe_load(MODEL, safe_only=True), range(32))) + assert pathlib.WindowsPath is windows_path + + +@pytest.mark.parametrize("fused", [False, True]) +def test_restricted_load_criterion(tmp_path, fused): + """Legacy criterion metadata and fused forward bindings survive restricted checkpoint round trips.""" + from ultralytics.nn.tasks import DetectionModel, torch_safe_load + from ultralytics.utils import DEFAULT_CFG + + model = DetectionModel(CFG, verbose=False) + model.args = DEFAULT_CFG + model.criterion = model.init_criterion() + model.eval() + if fused: + model.fuse(verbose=False) + image = torch.zeros(1, 3, 64, 64) + with torch.no_grad(): + expected = model(image)[0] + torch.save({"model": model, "best_fitness": np.float64(0.5)}, tmp_path / "legacy.pt") + checkpoint = torch_safe_load(tmp_path / "legacy.pt", safe_only=True)[0] + assert checkpoint["model"].criterion is not None + assert checkpoint["best_fitness"] == 0.5 + with torch.no_grad(): + assert torch.allclose(checkpoint["model"](image)[0], expected) + + +@pytest.mark.parametrize("cfg", [CFG, "yolov8n.yaml", "yolov10n.yaml", "yolo11n.yaml", "yolo26n-p6.yaml"]) +def test_model_forward(cfg): + """Test the forward pass of the YOLO model.""" + from ultralytics.nn.modules import SPPF + + model = YOLO(cfg) + sppf = next(m for m in model.model.modules() if isinstance(m, SPPF)) + assert isinstance(sppf.cv1.act, torch.nn.Identity) == ("26" in str(cfg)) + assert isinstance(SPPF(64, 64).cv1.act, torch.nn.Identity) + model(source=None, imgsz=32, augment=True) # also test no source and augment + + +def test_model_methods(): + """Test various methods and properties of the YOLO model to ensure correct functionality.""" + model = YOLO(MODEL) + + # Model methods + model.info(verbose=True, detailed=True) + model = model.reset_weights() + model = model.load(MODEL) + model.to("cpu") + model.fuse() + model.clear_callback("on_train_start") + model.reset_callbacks() + + # Model properties + _ = model.names + _ = model.device + _ = model.transforms + _ = model.task_map + + +def test_model_load_remaps_cls_head_by_names(): + """Test class-name remap is limited to closed-set class-logit heads.""" + from types import SimpleNamespace + + from ultralytics.models.yolo.detect.train import DetectionTrainer + from ultralytics.models.yolo.obb.train import OBBTrainer + from ultralytics.models.yolo.pose.train import PoseTrainer + from ultralytics.models.yolo.segment.train import SegmentationTrainer + from ultralytics.nn.tasks import DetectionModel, OBBModel, PoseModel, SegmentationModel, YOLOEModel + + src = DetectionModel("yolo26n.yaml", nc=3, verbose=False) + tgt = DetectionModel("yolo26n.yaml", nc=2, verbose=False) + src.names, tgt.names = {0: "cat", 1: "dog", 2: "car"}, {0: "dog", 1: "cat"} + for seq in src.model[-1].cv3: + seq[-1].bias.data.copy_(torch.tensor([10.0, 20.0, 30.0])) + tgt.load(src, verbose=False) + assert all(seq[-1].bias.tolist() == [20.0, 10.0] for seq in tgt.model[-1].cv3) + + src = YOLOEModel("yoloe-26n.yaml", nc=3, verbose=False) + tgt = YOLOEModel("yoloe-26n.yaml", nc=2, verbose=False) + src.names, tgt.names = {0: "cat", 1: "dog", 2: "car"}, {0: "dog", 1: "cat"} + tgt.load(src, verbose=False) # YOLOE cv3 outputs embeddings, not class rows + + names = {0: "dog", 1: "cat"} + for trainer_cls, model in ( + (DetectionTrainer, DetectionModel("yolo26n.yaml", nc=2, verbose=False)), + (SegmentationTrainer, SegmentationModel("yolo26n-seg.yaml", nc=2, verbose=False)), + (PoseTrainer, PoseModel("yolo26n-pose.yaml", nc=2, data_kpt_shape=[17, 3], verbose=False)), + (OBBTrainer, OBBModel("yolo26n-obb.yaml", nc=2, verbose=False)), + ): + trainer = object.__new__(trainer_cls) + trainer.args = SimpleNamespace(cls_remap=True) + trainer.data = {"names": names} + assert trainer.set_model_names_for_load(model).names == names + + +def test_model_profile(): + """Test profiling of the YOLO model with `profile=True` to assess performance and resource usage.""" + from ultralytics.nn.tasks import DetectionModel + + model = DetectionModel() # build model + im = torch.randn(1, 3, 64, 64) # requires min imgsz=64 + _ = model.predict(im, profile=True) + + +def test_predict_txt(tmp_path): + """Test YOLO predictions with file, directory, and pattern sources listed in a text file.""" + file = tmp_path / "sources_multi_row.txt" + with open(file, "w") as f: + f.writelines(f"{src}\n" for src in SOURCES_LIST) + results = YOLO(MODEL)(source=file, imgsz=32) + assert len(results) == 7, f"Expected 7 results from source list, got {len(results)}" + + +@pytest.mark.skipif(True, reason="disabled for testing") +def test_predict_csv_multi_row(tmp_path): + """Test YOLO predictions with sources listed in multiple rows of a CSV file.""" + file = tmp_path / "sources_multi_row.csv" + with open(file, "w", newline="") as f: + writer = csv.writer(f) + writer.writerow(["source"]) + writer.writerows([[src] for src in SOURCES_LIST]) + results = YOLO(MODEL)(source=file, imgsz=32) + assert len(results) == 7, f"Expected 7 results from multi-row CSV, got {len(results)}" + + +@pytest.mark.skipif(True, reason="disabled for testing") +def test_predict_csv_single_row(tmp_path): + """Test YOLO predictions with sources listed in a single row of a CSV file.""" + file = tmp_path / "sources_single_row.csv" + with open(file, "w", newline="") as f: + writer = csv.writer(f) + writer.writerow(SOURCES_LIST) + results = YOLO(MODEL)(source=file, imgsz=32) + assert len(results) == 7, f"Expected 7 results from single-row CSV, got {len(results)}" + + +@pytest.mark.parametrize("model_name", MODELS) +def test_predict_img(model_name): + """Test YOLO model predictions on various image input types.""" + if IS_RASPBERRYPI and model_name == "yolo26n-sem.pt": + skip_rpi_semantic() + channels = 1 if model_name == "yolo11n-grayscale.pt" else 3 + model = YOLO(WEIGHTS_DIR / model_name) + im = cv2.imread(str(SOURCE), flags=cv2.IMREAD_GRAYSCALE if channels == 1 else cv2.IMREAD_COLOR) # uint8 NumPy array + assert len(model(source=Image.open(SOURCE), save=True, verbose=True, imgsz=32)) == 1 # PIL + assert len(model(source=im, save=True, save_txt=True, imgsz=32)) == 1 # ndarray + assert len(model(torch.rand((2, channels, 32, 32)), imgsz=32)) == 2 # batch-size 2 Tensor, FP32 0.0-1.0 RGB order + assert len(model(source=[im, im], save=True, save_txt=True, imgsz=32)) == 2 # batch + assert len(list(model(source=[im, im], save=True, stream=True, imgsz=32))) == 2 # stream + assert len(model(torch.zeros(320, 640, channels).numpy().astype(np.uint8), imgsz=32)) == 1 # tensor to numpy + batch = [ + str(SOURCE), # filename + Path(SOURCE), # Path + im, # OpenCV + Image.open(SOURCE), # PIL + np.zeros((320, 640, channels), dtype=np.uint8), # numpy + ] + assert len(model(batch, imgsz=32, classes=0)) == len(batch) # multiple sources in a batch + + +@pytest.mark.parametrize(("model_name", "bgr"), [("yolo11n.pt", [0, 127, 255]), ("yolo11n-grayscale.pt", [127])]) +def test_preprocess_values(model_name, bgr): + """Check predictor channel order and normalization with known pixel values.""" + model = YOLO(WEIGHTS_DIR / model_name) + im = np.full((32, 32, len(bgr)), bgr, dtype=np.uint8) + model(im, imgsz=32, verbose=False) # build predictor through the public path + out = model.predictor.preprocess([im]) + expected = torch.tensor([bgr[::-1]], device=out.device, dtype=out.dtype) / 255 + assert out.shape == (1, len(bgr), 32, 32) and out.is_contiguous() + assert torch.equal(out[:, :, 0, 0], expected) + + +@pytest.mark.parametrize("model_name", ["yolo26n.pt", "yolo11n.pt"]) +@pytest.mark.parametrize("nms", [None, False]) +def test_predict_classes_with_max_det(model_name, nms): + """Test classes-before-max_det and reset reused-call filters for end2end and NMS-based models.""" + boxes = YOLO(WEIGHTS_DIR / model_name)(SOURCE, classes=[0], max_det=300, nms=nms, verbose=False)[0].boxes + assert len(boxes) > 1 # bus.jpg contains multiple persons + top1_model = YOLO(WEIGHTS_DIR / model_name) + top1 = top1_model(SOURCE, classes=[0], max_det=1, nms=nms, verbose=False)[0].boxes + assert len(top1) == 1 and int(top1.cls) == 0 + assert float(top1.conf) == pytest.approx(float(boxes.conf.max())) # best person kept, not an arbitrary one + + reused = top1_model(SOURCE, verbose=False)[0].boxes # SAME model, no kwargs at all this time + assert len(reused) > 1 # classes=[0]/max_det=1 from the previous call must not leak into this one + + +@pytest.mark.parametrize("model", MODELS) +def test_predict_visualize(model): + """Test model prediction methods with 'visualize=True' to generate prediction visualizations.""" + if IS_RASPBERRYPI and model == "yolo26n-sem.pt": + skip_rpi_semantic() + YOLO(WEIGHTS_DIR / model)(SOURCE, imgsz=32, visualize=True) + + +def test_load_tensor_uint8(): + """Test that tensor normalization supports uint8 while preserving floating-point epsilon tolerance.""" + from ultralytics.data.loaders import LoadTensor + + loaded = LoadTensor(torch.full((1, 3, 32, 32), 255, dtype=torch.uint8)).im0 + assert loaded.dtype == torch.float32 and loaded.max() == 1 + normalized = torch.ones((1, 3, 32, 32), dtype=torch.float32) + normalized[..., 0, 0] += torch.finfo(normalized.dtype).eps + assert LoadTensor(normalized).im0.max() > 1 + + +def test_predict_gray_and_4ch(tmp_path): + """Test YOLO prediction on SOURCE converted to grayscale and 4-channel images with various filenames.""" + im = Image.open(SOURCE) + + source_grayscale = tmp_path / "grayscale.jpg" + source_rgba = tmp_path / "4ch.png" + source_non_utf = tmp_path / "non_UTF_测试文件_tést_image.jpg" + source_spaces = tmp_path / "image with spaces.jpg" + + im.convert("L").save(source_grayscale) # grayscale + im.convert("RGBA").save(source_rgba) # 4-ch PNG with alpha + im.save(source_non_utf) # non-UTF characters in filename + im.save(source_spaces) # spaces in filename + + # Inference + model = YOLO(MODEL) + for f in source_rgba, source_grayscale, source_non_utf, source_spaces: + for source in Image.open(f), cv2.imread(str(f)), f: + results = model(source, save=True, verbose=True, imgsz=32) + assert len(results) == 1, f"Expected 1 result for {f.name}, got {len(results)}" + f.unlink() # cleanup + + +def test_predict_ndarray_channels(): + """Test NumPy channel normalization for grayscale and color models.""" + from ultralytics.data.loaders import LoadPilAndNumpy + + model = YOLO(MODEL) # default 3-channel model + gray = np.asarray(Image.open(SOURCE).convert("L")) # genuine 2D (H, W) uint8 array + assert gray.ndim == 2, "Expected a 2D grayscale array for this test" + assert len(model(source=gray, imgsz=32, verbose=False)) == 1 # 2D ndarray auto-expanded to 3 channels + assert len(model(source=gray.astype("float64"), imgsz=32, verbose=False)) == 1 # non-OpenCV dtype also works + bgra = np.zeros((8, 8, 4), dtype="float64") + assert LoadPilAndNumpy(bgra, channels=3).im0[0].shape == (8, 8, 3) # non-OpenCV dtype also falls back for BGRA + for source_channels, model_channels in ((1, 3), (2, 1), (2, 3), (3, 1), (4, 1), (4, 3)): + im = np.zeros((8, 8, source_channels), dtype=np.uint8) + assert LoadPilAndNumpy(im, channels=model_channels).im0[0].shape == (8, 8, model_channels) + + +def test_single_check_channel_order_and_contiguity(): + """Test LoadPilAndNumpy._single_check() keeps BGR order and C-contiguous output through cv2 conversions.""" + from ultralytics.data.loaders import LoadPilAndNumpy + + check = LoadPilAndNumpy._single_check + rgb = Image.fromarray(np.full((2, 2, 3), (10, 20, 30), dtype=np.uint8)) # PIL is R, G, B + bgra = np.full((2, 2, 4), (10, 20, 30, 255), dtype=np.uint8) # ndarray is already B, G, R, A + gray = np.full((2, 2, 1), 42, dtype=np.uint8) + for im, expected in ((check(rgb, 3), (30, 20, 10)), (check(bgra, 3), (10, 20, 30)), (check(gray, 3), (42, 42, 42))): + assert tuple(im[0, 0].tolist()) == expected + assert im.flags["C_CONTIGUOUS"] + + +@pytest.mark.slow +@pytest.mark.skipif(not ONLINE, reason="environment is offline") +def test_predict_all_image_formats(): + """Predict on the 12 image format extensions in COCO12-Formats (AVIF, BMP, DNG, HEIC, JP2, JPEG, JPG, MPO, PNG, TIF, + TIFF, WebP). + """ + # Download dataset if needed + data = check_det_dataset("coco12-formats.yaml") + dataset_path = Path(data["path"]) + + # Collect all images from train and val + expected = {"avif", "bmp", "dng", "heic", "jp2", "jpeg", "jpg", "mpo", "png", "tif", "tiff", "webp"} + images = [im for im in (dataset_path / "images" / "train").glob("*.*") if im.suffix.lower().lstrip(".") in expected] + images += [im for im in (dataset_path / "images" / "val").glob("*.*") if im.suffix.lower().lstrip(".") in expected] + assert len(images) == 12, f"Expected 12 images, found {len(images)}" + + # Verify all format extensions are represented + extensions = {img.suffix.lower().lstrip(".") for img in images} + assert extensions == expected, f"Missing formats: {expected - extensions}" + + # Run inference on all images + model = YOLO(MODEL) + results = model(images, imgsz=32) + assert len(results) == 12, f"Expected 12 results, got {len(results)}" + + +@pytest.mark.slow +@pytest.mark.skipif(not ONLINE, reason="environment is offline") +@pytest.mark.skipif(is_github_action_running(), reason="No auth https://github.com/JuanBindez/pytubefix/issues/166") +def test_youtube(): + """Test YOLO model on a YouTube video stream, handling potential network-related errors.""" + model = YOLO(MODEL) + try: + model.predict("https://youtu.be/G17sBkb38XQ", imgsz=32, save=True) + # Handle internet connection errors and 'urllib.error.HTTPError: HTTP Error 429: Too Many Requests' + except (urllib.error.HTTPError, ConnectionError) as e: + LOGGER.error(f"YouTube Test Error: {e}") + + +def test_track_second_association_indices(): + """Low-confidence detections matched in second association keep full detection-set indices.""" + from ultralytics.engine.results import Boxes + from ultralytics.trackers.byte_tracker import BYTETracker + from ultralytics.utils import ROOT, YAML, IterableSimpleNamespace + + args = IterableSimpleNamespace(**{**YAML.load(ROOT / "cfg/trackers/bytetrack.yaml"), "fuse_score": False}) + tracker = BYTETracker(args) + boxes = [[10, 10, 50, 50], [200, 200, 260, 260], [400, 400, 480, 480]] + for confs in ([0.9, 0.9, 0.9], [0.9, 0.9, 0.2]): # third detection drops to low confidence on frame 2 + data = torch.tensor([[*b, c, 0] for b, c in zip(boxes, confs)], dtype=torch.float32) + tracks = tracker.update(Boxes(data, (640, 640))) + low = tracks[np.isclose(tracks[:, 5], 0.2)] # columns are [x1, y1, x2, y2, id, score, cls, idx] + assert len(low) == 1 and int(low[0, -1]) == 2, f"second-association idx not preserved:\n{tracks}" + + +def test_track_split_detections_degenerate_boxes(): + """`_split_detections` must drop zero/negative-dimension boxes from both confidence partitions while keeping every + valid detection's index into the full detection-set space (later assigned to `track.idx`). + """ + from ultralytics.engine.results import Boxes + from ultralytics.trackers.byte_tracker import BYTETracker + from ultralytics.utils import ROOT, YAML, IterableSimpleNamespace + + args = IterableSimpleNamespace(**YAML.load(ROOT / "cfg/trackers/bytetrack.yaml")) + tracker = BYTETracker(args) + boxes = [ + [10, 10, 50, 50, 0.9, 0], # idx 0: valid, high-confidence partition + [300, 480, 350, 480, 0.9, 0], # idx 1: degenerate, zero height, high-confidence partition + [100, 100, 150, 150, 0.15, 0], # idx 2: valid, low-confidence partition + [300, 490, 350, 480, 0.15, 0], # idx 3: degenerate, negative height, low-confidence partition + [150, 100, 100, 150, 0.9, 0], # idx 4: degenerate, negative width, high-confidence partition + ] + results = Boxes(torch.tensor(boxes, dtype=torch.float32), (640, 640)) + high, low, mask_high, mask_low = tracker._split_detections(results) + assert np.flatnonzero(mask_high).tolist() == [0] and len(high) == 1, f"degenerate box leaked high band: {mask_high}" + assert np.flatnonzero(mask_low).tolist() == [2] and len(low) == 1, f"degenerate box leaked low band: {mask_low}" + + +@pytest.mark.parametrize("tracker_type", ["bytetrack", "fasttrack"]) +def test_track_second_association_low_conf_keeps_id(tracker_type): + """Low-confidence detection is recovered by the second association under the default fuse_score=True.""" + from ultralytics.engine.results import Boxes + from ultralytics.trackers.track import TRACKER_MAP + from ultralytics.utils import ROOT, YAML, IterableSimpleNamespace + + args = IterableSimpleNamespace(**YAML.load(ROOT / f"cfg/trackers/{tracker_type}.yaml")) # default fuse_score=True + tracker = TRACKER_MAP[tracker_type](args) + box = [100, 100, 200, 200] # same box on both frames, so IoU is 1.0 + # frame 1: high score starts the track; frame 2: score drops into the low band (track_low_thresh < 0.15 < track_high_thresh) + frame1 = tracker.update(Boxes(torch.tensor([[*box, 0.9, 0]], dtype=torch.float32), (640, 640))) + frame2 = tracker.update(Boxes(torch.tensor([[*box, 0.15, 0]], dtype=torch.float32), (640, 640))) + assert len(frame1) == 1, f"expected one track on frame 1:\n{frame1}" + tid = int(frame1[0, 4]) + # the low-score box must be kept and mapped to the same id via the second association + assert len(frame2) == 1, f"low-confidence detection lost by second association:\n{frame2}" + assert int(frame2[0, 4]) == tid, f"id switched on low-confidence frame: {tid} -> {int(frame2[0, 4])}\n{frame2}" + + +def test_tracktrack_new_lifecycle(): + """TrackTrack predicts New tracks and confirms them once their history reaches min_track_len.""" + from ultralytics.engine.results import Boxes + from ultralytics.trackers.track import TRACKER_MAP + from ultralytics.utils import ROOT, YAML, IterableSimpleNamespace + + cfg = {**YAML.load(ROOT / "cfg/trackers/tracktrack.yaml"), "gmc_method": "none", "min_track_len": 4} + tracker = TRACKER_MAP["tracktrack"](IterableSimpleNamespace(**cfg)) + tracker.update(Boxes(torch.empty((0, 6)), (640, 640))) # avoid first-frame auto-activation + outputs = [] + for center_x in (100, 135, 170, 205): + box = torch.tensor([[center_x - 50, 50, center_x + 50, 150, 0.9, 0]], dtype=torch.float32) + outputs.append(tracker.update(Boxes(box, (640, 640)))) + assert [len(output) for output in outputs] == [0, 0, 0, 1] + for min_track_len in (0, 1): + cfg["min_track_len"] = min_track_len + tracker = TRACKER_MAP["tracktrack"](IterableSimpleNamespace(**cfg)) + tracker.update(Boxes(torch.empty((0, 6)), (640, 640))) + assert len(tracker.update(Boxes(box, (640, 640)))) == 1 + + from ultralytics.trackers.basetrack import TrackState + + cfg["min_track_len"] = 4 + tracker = TRACKER_MAP["tracktrack"](IterableSimpleNamespace(**cfg)) + for center_x in (100, 135, 170, 205): # frame_id == 1 carries a real detection, not an empty warm-up frame + box = torch.tensor([[center_x - 50, 50, center_x + 50, 150, 0.9, 0]], dtype=torch.float32) + tracker.update(Boxes(box, (640, 640))) + if tracker.frame_id == 2: + assert tracker.tracked_stracks[0].state != TrackState.Tracked, "frame_id==1 track confirmed after 2 hits" + assert tracker.tracked_stracks[0].state == TrackState.Tracked + + +@pytest.mark.parametrize("tracker_type", ["botsort", "deepocsort", "tracktrack"]) +def test_track_reid_auto_user_detections(tracker_type): + """Native ReID (model='auto') must degrade to motion-only with user-supplied detections, not encode the raw frame.""" + from ultralytics.engine.results import Boxes + from ultralytics.trackers.track import TRACKER_MAP + from ultralytics.utils import ROOT, YAML, IterableSimpleNamespace + + cfg = {**YAML.load(ROOT / f"cfg/trackers/{tracker_type}.yaml"), "with_reid": True, "model": "auto"} + tracker = TRACKER_MAP[tracker_type](IterableSimpleNamespace(**cfg)) + img = np.full((640, 640, 3), 128, dtype=np.uint8) # nonzero so bogus frame-derived features would not be dropped + data = torch.tensor([[10, 10, 50, 50, 0.9, 0], [200, 200, 260, 260, 0.9, 0]], dtype=torch.float32) + for _ in range(3): # frame 2 used to crash in embedding_distance after storing image rows as track features + tracks = tracker.update(Boxes(data, (640, 640)), img) + assert len(tracks) == 2, f"native-ReID tracker must keep tracking without feats:\n{tracks}" + + +@pytest.mark.parametrize("fuse_score", [True, False]) +def test_deepocsort_ocr_proximity_gate(fuse_score): + """DeepOCSORT OCR rejects a zero-IoU pair even when its appearance is identical, under both fuse_score settings.""" + from types import SimpleNamespace + + from ultralytics.trackers.basetrack import TrackState + from ultralytics.trackers.deep_oc_sort import DeepOCSORT + + tracker = object.__new__(DeepOCSORT) + tracker.args = SimpleNamespace(fuse_score=fuse_score, match_thresh=0.8) + tracker.encoder, tracker.appearance_thresh, tracker.proximity_thresh, tracker.frame_id = object(), 0.9, 0.5, 2 + track = SimpleNamespace( + angle=None, + last_observation=np.array([0, 0, 10, 10]), + smooth_feat=np.array([1.0, 0.0]), + state=TrackState.Tracked, + update=lambda *_: None, + ) + detection = SimpleNamespace(xyxy=np.array([20, 20, 30, 30]), curr_feat=np.array([1.0, 0.0]), score=1.0) + # proves appearance is active and would override (ungated) this exact pair, so the OCR result below is caused by + # the proximity gate, not by appearance being unavailable + assert tracker._fuse_appearance(np.array([[1.0]]), [track], [detection]) == 0.0 + assert tracker._ocr_associate([track], [detection], [], []) == ([0], [0]) + + +def test_reid_invalid_crops(): + """Test ReID skips out-of-bounds detection crops while preserving feature alignment.""" + from types import SimpleNamespace + + from ultralytics.trackers.utils.reid import ReID + + encoder = ReID.__new__(ReID) + encoder.is_pt = True + encoder.model = SimpleNamespace(predictor=lambda crops: [torch.ones(4) for _ in crops]) + img = np.full((640, 640, 3), 128, dtype=np.uint8) + feats = encoder(img, np.array([[30, 30, 40, 40], [1100, 1100, 200, 200]], dtype=np.float32)) + assert feats[0] is not None and feats[1] is None + + +@pytest.mark.skipif(not ONLINE, reason="environment is offline") +@pytest.mark.parametrize("model", MODELS) +def test_track_stream(model, tmp_path, solution_assets): + """Test streaming tracking on a short video with all built-in trackers and various GMC/ReID configurations. + + Note imgsz=160 required for tracking for higher confidence and better matches. + """ + if model in { + "yolo26n-cls.pt", + "yolo26n-sem.pt", + "yolo26n-depth.pt", + }: # classification, semantic, and depth not supported + return + from ultralytics.trackers.track import TRACKER_MAP + + video_url = solution_assets("track_video") + model = YOLO(model) + + # Default end-to-end run for all built-in trackers + for tracker_type in TRACKER_MAP: + kwargs = {"save_frames": True} if tracker_type == "botsort" else {} + model.track(video_url, imgsz=160, tracker=f"{tracker_type}.yaml", **kwargs) + + # Test Global Motion Compensation (GMC) methods and ReID on botsort + for gmc, reidm in zip(["orb", "sift", "ecc"], ["auto", "auto", "yolo26n-cls.pt"]): + default_args = YAML.load(ROOT / "cfg/trackers/botsort.yaml") + custom_yaml = tmp_path / f"botsort-{gmc}.yaml" + YAML.save(custom_yaml, {**default_args, "gmc_method": gmc, "with_reid": True, "model": reidm}) + model.track(video_url, imgsz=160, tracker=custom_yaml) + + # Test ONNX ReID encoder auto-download + if model == "yolo26n.pt": + default_args = YAML.load(ROOT / "cfg/trackers/botsort.yaml") + custom_yaml = tmp_path / "botsort-reid-onnx.yaml" + YAML.save(custom_yaml, {**default_args, "with_reid": True, "model": "yolo26n-reid.onnx"}) + model.track(video_url, imgsz=160, tracker=custom_yaml) + + +@pytest.mark.parametrize("task,weight,data", TASK_MODEL_DATA) +def test_val(task: str, weight: str, data: str) -> None: + """Test the validation mode of the YOLO model.""" + if IS_RASPBERRYPI and task == "semantic": + skip_rpi_semantic() + model = YOLO(weight) + for plots in (True, False): # Test both cases i.e. plots=True and plots=False + metrics = model.val(data=data, imgsz=32, plots=plots) + metrics.to_df() + metrics.to_csv() + metrics.to_json() + if task != "depth": # depth is dense regression: no classes, no confusion matrix + metrics.confusion_matrix.to_df() + metrics.confusion_matrix.to_csv() + metrics.confusion_matrix.to_json() + cm = metrics.confusion_matrix + expected = cm.nc if task in {"classify", "semantic"} else cm.nc + 1 # background for detection tasks + assert cm.matrix.shape == (expected, expected), f"{task} confusion matrix is {cm.matrix.shape}" + assert len(cm.tp_fp()[0]) == cm.nc # per-class TP/FP never include background + + +def test_val_save_txt_pose(tmp_path): + """Test that pose keypoints saved by val(save_txt=True) and val(save_json=True) are in the original image space.""" + model = YOLO(WEIGHTS_DIR / "yolo26n-pose.pt") + # imgsz=640 (not the imgsz=32 used elsewhere): coco8-pose images are non-square, so the letterbox offset is only + # large enough to push mis-scaled keypoints outside [0, 1] at full resolution; at small imgsz they would stay in + # range and hide the regression. save_json=True also exercises pred_to_json, the other consumer of the scaled key. + metrics = model.val( + data="coco8-pose.yaml", imgsz=640, conf=0.25, save_txt=True, save_json=True, project=tmp_path, name="val" + ) + txt_files = list((Path(metrics.save_dir) / "labels").glob("*.txt")) + assert txt_files, "val(save_txt=True) saved no label files" + assert (Path(metrics.save_dir) / "predictions.json").exists(), "val(save_json=True) saved no predictions.json" + for txt_file in txt_files: + for line in txt_file.read_text().splitlines(): + values = [float(v) for v in line.split()] + x, y, w, h = values[1:5] # normalized xywh box + kpts = torch.tensor(values[5:]).view(-1, 3) # (17, 3) of normalized (x, y, conf) keypoints + assert ((kpts[:, :2] >= 0) & (kpts[:, :2] <= 1)).all(), f"keypoints not in [0, 1] in {txt_file.name}" + # Keypoints scaled into the wrong (letterbox) space also land off the person, so check that visible + # keypoints cluster on the box; the 0.05 margin allows joints (wrists, ankles) just outside a tight box. + visible = kpts[kpts[:, 2] > 0.5, :2] + if len(visible): + cx, cy = visible.mean(0) + assert abs(cx - x) < w / 2 + 0.05 and abs(cy - y) < h / 2 + 0.05, "keypoints misaligned with box" + + +def test_pose_metrics_curves(): + """Test that pose curve labels contain four unique box and pose series.""" + from ultralytics.utils.metrics import PoseMetrics + + curves = PoseMetrics().curves + assert len(curves) == len(set(curves)) == 8 + + +@pytest.mark.skipif(not ONLINE, reason="environment is offline") +@pytest.mark.skipif(IS_JETSON or IS_RASPBERRYPI, reason="Edge devices not intended for training") +def test_train_multi(): + """Test fine-tuning a base model across a dataset collection, which triggers MultiTrainer for list/tuple data.""" + model = YOLO(MODEL) + results = model.train(data=["coco8.yaml", "coco8.yaml"], epochs=1, imgsz=32) + assert isinstance(results, dict) and len(results) == 2 # one entry per run (coco8, coco8-2), no duplicate collapse + assert all(m and "fitness" in m for m in results.values()) # checkpoint train metrics per run + assert len(model.trainer.trainers) == 2 # both list entries fine-tuned in series + sweep_dir = model.trainer.save_dir + assert sweep_dir.name.startswith("multitrain") # all runs grouped under one sweep directory + assert (sweep_dir / "multitrain_results.json").exists() # results JSON for post-processing + assert (sweep_dir / "multitrain_results.png").exists() # cross-dataset results plot + + +def test_normalize_platform_uri(): + """Test Platform web URLs are rewritten to ul:// URIs so datasets/models load directly from a pasted URL.""" + from ultralytics.utils.checks import normalize_platform_uri + + base = "https://platform.ultralytics.com/glenn-jocher" + assert normalize_platform_uri(f"{base}/datasets/coco8") == "ul://glenn-jocher/datasets/coco8" + assert normalize_platform_uri(f"{base}/project/model/") == "ul://glenn-jocher/project/model" + assert normalize_platform_uri("coco8.yaml") == "coco8.yaml" # non-Platform inputs unchanged + + +def test_convert_signed_ndjson(monkeypatch): + """Test signed NDJSON URLs are converted before dataset YAML validation.""" + from ultralytics.data import converter, utils + + captured = [] + + async def convert(path, fraction, split): + captured.append((path, fraction)) + return "dataset.ndjson.yaml" + + monkeypatch.setattr(converter, "convert_ndjson_to_yolo", convert) + url = "https://storage.googleapis.com/bucket/dataset-v1.ndjson?X-Goog-Signature=abc" + assert utils.convert_ndjson_to_yolo_if_needed(url) == "dataset.ndjson.yaml" + assert captured == [(url, 1.0)] + + +@pytest.mark.parametrize("task", ["detect", "classify"]) +def test_ndjson_conversion_concurrency_and_resume(monkeypatch, tmp_path, task): + """Test concurrent conversions share work and interrupted conversions resume before publishing completion.""" + import asyncio + import json + import threading + from concurrent.futures import ThreadPoolExecutor + + import aiohttp + + from ultralytics.data import converter + + counts, failures, conversions, count_lock = {}, set(), 0, threading.Lock() + + class Response: + def __init__(self, url): + self.url = url + + async def __aenter__(self): + return self + + async def __aexit__(self, *_): + pass + + def raise_for_status(self): + pass + + async def read(self): + await asyncio.sleep(0.01) + with count_lock: + counts[self.url] = counts.get(self.url, 0) + 1 + fail = self.url in failures + failures.discard(self.url) + if fail: + raise OSError("interrupted") + return image_bytes + + class Session: + def __init__(self, **_): + pass + + async def __aenter__(self): + return self + + async def __aexit__(self, *_): + pass + + def get(self, url, **_): + return Response(url) + + ok, image = cv2.imencode(".jpg", np.zeros((16, 16, 3), dtype=np.uint8)) + assert ok + image_bytes = image.tobytes() + monkeypatch.setattr(aiohttp, "ClientSession", Session) + original_convert = converter._convert_ndjson_to_yolo + + async def track_conversion(*args): + nonlocal conversions + with count_lock: + conversions += 1 + return await original_convert(*args) + + monkeypatch.setattr(converter, "_convert_ndjson_to_yolo", track_conversion) + annotations = {"classification": [7]} if task == "classify" else {"boxes": [[0, 0.5, 0.5, 1, 1]]} + + def write_ndjson(name): + path = tmp_path / f"{name}.ndjson" + records = [ + {"type": "dataset", "task": task, "class_names": {"7": "item", "8": "rare"}}, + { + "file": "train.jpg", + "url": f"https://example.com/{name}-train.jpg", + "split": "train", + "annotations": annotations, + }, + { + "file": "val.jpg", + "url": f"https://example.com/{name}-val.jpg", + "split": "val", + "annotations": {"classification": [8]} if task == "classify" else annotations, + }, + ] + path.write_text("\n".join(json.dumps(record) for record in records)) + return path + + concurrent = write_ndjson("concurrent") + jobs = 2 + barrier = threading.Barrier(jobs) + + def convert(path): + barrier.wait() + return asyncio.run(converter.convert_ndjson_to_yolo(path, tmp_path)) + + with ThreadPoolExecutor(max_workers=jobs) as pool: + results = list(pool.map(convert, [concurrent] * jobs)) + assert len(set(results)) == 1 + assert conversions == 1 + assert sum(counts.values()) == 2 + if task == "classify": + assert check_cls_dataset(results[0])["nc"] == 2 + from ultralytics.data import dataset as dataset_module + + monkeypatch.setattr(dataset_module, "TORCHVISION_0_18", False) + args = copy(DEFAULT_CFG) + train = dataset_module.ClassificationDataset(results[0] / "train", args) + val = dataset_module.ClassificationDataset(results[0] / "val", args) + assert train.samples[0][1] == 0 + assert val.samples[0][1] == 1 + assert dataset_module.ClassificationDataset(results[0] / "val", args).samples[0][1] == 1 + + resume = write_ndjson("resume") + failed_url = "https://example.com/resume-val.jpg" + failures.add(failed_url) + with ThreadPoolExecutor(max_workers=jobs) as pool: + futures = [pool.submit(convert, resume) for _ in range(jobs)] + errors, results = [], [] + for future in futures: + try: + results.append(future.result()) + except RuntimeError as error: + errors.append(str(error)) + assert len(errors) == len(results) == 1 and "Downloaded 1/2 images" in errors[0] + result = results[0] + marker = result / ".ndjson.yaml" if task == "classify" else result + assert YAML.load(marker)["complete"] is True + assert counts["https://example.com/resume-train.jpg"] == 1 + assert counts[failed_url] == 2 + request_count = sum(counts.values()) + asyncio.run(converter.convert_ndjson_to_yolo(resume, tmp_path)) + assert conversions == 4 + assert sum(counts.values()) == request_count + + +@pytest.mark.skipif(not ONLINE, reason="environment is offline") +@pytest.mark.skipif(IS_JETSON or IS_RASPBERRYPI, reason="Edge devices not intended for training") +def test_train_scratch(): + """Test training the YOLO model from scratch on 12 different image types in the COCO12-Formats dataset.""" + model = YOLO(CFG) + model.train(data="coco12-formats.yaml", epochs=2, imgsz=32, cache="disk", batch=-1, close_mosaic=1, name="model") + head = model.trainer.model.model[-1] + assert head.cv2 is not None and head.one2one_cv2 is not None # both heads remain trained + assert hasattr(model.trainer.model.criterion, "one2many") and hasattr(model.trainer.model.criterion, "one2one") + assert not model.trainer.ema.ema.end2end # epoch validation selects one-to-many by default + model(SOURCE) + + +@pytest.mark.skipif(not ONLINE, reason="environment is offline") +@pytest.mark.skipif(IS_RASPBERRYPI, reason="Edge devices not intended for training") +def test_train_ndjson(): + """Test training the YOLO model using NDJSON format dataset.""" + model = YOLO(WEIGHTS_DIR / "yolo26n.pt") + model.train(data=f"{ASSETS_URL}/coco8-ndjson.ndjson", epochs=1, imgsz=32) + + +@pytest.mark.parametrize("scls", [False, True]) +@pytest.mark.skipif(IS_RASPBERRYPI, reason="Edge devices not intended for training") +def test_train_pretrained(scls): + """Test training of the YOLO model starting from a pre-trained checkpoint.""" + model = YOLO(WEIGHTS_DIR / "yolo26n-seg.pt") + model.train( + data="coco8-seg.yaml", epochs=1, imgsz=32, cache="ram", copy_paste=0.5, mixup=0.5, name=0, single_cls=scls + ) + model(SOURCE) + + +def test_all_model_yamls(): + """Test YOLO model creation for all available YAML configurations in the `cfg/models` directory.""" + for m in (ROOT / "cfg" / "models").rglob("*.yaml"): + if "rtdetr" in m.name: + if TORCH_1_11: + _ = RTDETR(m.name)(SOURCE, imgsz=160) + else: + YOLO(m.name) + + +@pytest.mark.skipif(WINDOWS, reason="Windows slow CI export bug https://github.com/ultralytics/ultralytics/pull/16003") +def test_workflow(isolated_model): + """Test the complete workflow including training, validation, prediction, and exporting.""" + model = YOLO(isolated_model) + model.train(data="coco8.yaml", epochs=1, imgsz=32, optimizer="SGD") + model.val(imgsz=32) + model.predict(SOURCE, imgsz=32) + model.export(format="torchscript") # WARNING: Windows slow CI export bug + + +def test_predict_callback_and_setup(): + """Test callback functionality during YOLO prediction setup and execution.""" + + def on_predict_batch_end(predictor): + """Callback function that handles operations at the end of a prediction batch.""" + path, im0s, _ = predictor.batch + im0s = im0s if isinstance(im0s, list) else [im0s] + bs = [predictor.dataset.bs for _ in range(len(path))] + predictor.results = zip(predictor.results, im0s, bs) # results is list[batch_size] + + model = YOLO(MODEL) + model.add_callback("on_predict_batch_end", on_predict_batch_end) + + dataset = load_inference_source(source=SOURCE) + bs = dataset.bs # access predictor properties + results = model.predict(dataset, stream=True, imgsz=32) # source already setup + for r, im0, bs in results: + print("test_callback", im0.shape) + print("test_callback", bs) + boxes = r.boxes # Boxes object for bbox outputs + print(boxes) + + +@pytest.mark.parametrize("model", MODELS) +def test_results(model: str, tmp_path, solution_assets): + """Test YOLO model results processing and output in various formats.""" + if IS_RASPBERRYPI and model == "yolo26n-sem.pt": + skip_rpi_semantic() + im = solution_assets("boats") if model == "yolo26n-obb.pt" else SOURCE + is_semantic = "semantic" in model or "-sem" in model + results = YOLO(WEIGHTS_DIR / model)([im, im], imgsz=32 if is_semantic else 160) + for r in results: + if is_semantic: + assert r.semantic_mask is not None and r.semantic_mask.shape == r.orig_shape, ( + f"'{model}' semantic_mask should match the original image shape!" + ) + assert r.semantic_mask.data.dtype == torch.uint8, f"'{model}' semantic_mask should use compact class IDs!" + else: + assert len(r), f"'{model}' results should not be empty!" + r = r.cpu().numpy() + print(r, len(r), r.path) # print numpy attributes + r = r.to(device="cpu", dtype=torch.float32) + r.save_txt(txt_file=tmp_path / "runs/tests/label.txt", save_conf=True) + r.save_crop(save_dir=tmp_path / "runs/tests/crops/") + r.to_df(decimals=3) # Align to_ methods: https://docs.ultralytics.com/modes/predict#working-with-results + r.to_csv() + r.to_json(normalize=True) + r.plot(pil=True, save=True, filename=tmp_path / "results_plot_save.jpg") + r.plot(conf=True, boxes=True) + print(r, len(r), r.path) # print after methods + + +def test_results_plot_without_boxes(): + """Test that plotting a masks-only Results (boxes=None) does not raise an AttributeError.""" + from ultralytics.engine.results import Results + + orig_img = np.zeros((640, 640, 3), dtype=np.uint8) + masks = torch.zeros((2, 640, 640), dtype=torch.float32) + r = Results(orig_img, path="image.jpg", names={0: "a", 1: "b"}, masks=masks) + assert r.boxes is None + for color_mode in ("class", "instance"): + assert r.plot(color_mode=color_mode).shape == orig_img.shape + + +def test_results_depth_field(): + """A depth array becomes a DepthMap that survives the .cpu().numpy() chain.""" + from ultralytics.engine.results import DepthMap, Results + + img = np.zeros((20, 24, 3), dtype=np.uint8) + depth = np.random.rand(20, 24).astype(np.float32) + r = Results(orig_img=img, path="x.jpg", names={0: "depth"}, depth=depth) + assert isinstance(r.depth, DepthMap) + assert r.depth.data.shape == (20, 24) + rc = r.cpu().numpy() # exercises BaseTensor _keys plumbing (.cpu()/.numpy()) + assert rc.depth is not None + assert rc.depth.data.shape == (20, 24) # shape survives the .cpu().numpy() chain + + +def test_results_depth_none_summary_len_and_update(): + """Depth-only Results: None passthrough, empty summary, __len__ counts the map, update() wraps arrays.""" + from ultralytics.engine.results import DepthMap, Results + + img = np.zeros((8, 8, 3), dtype=np.uint8) + assert Results(orig_img=img, path="x.jpg", names={}, depth=None).depth is None + r = Results(orig_img=img, path="x.jpg", names={0: "depth"}, depth=np.ones((8, 8), dtype=np.float32)) + assert r.summary() == [] # depth-only Results has no per-instance summary + assert len(r) == 1 # __len__ returns the depth map count + r = Results(orig_img=img, path="x.jpg", names={0: "depth"}) + r.update(depth=np.ones((8, 8), dtype=np.float32)) + assert isinstance(r.depth, DepthMap) + + +def test_results_plot_with_depth(): + """Results.plot() with a depth map blends the colorized depth heatmap over the image.""" + from ultralytics.engine.results import Results + + img = np.zeros((24, 24, 3), dtype=np.uint8) + depth = np.random.rand(24, 24).astype(np.float32) + r = Results(orig_img=img, path="x.jpg", names={0: "depth"}, depth=depth) + out = r.plot() # must not raise; returns an annotated image (masks=True by default) + assert out.shape[:2] == (24, 24) # heatmap overlaid, same size as input + + +def test_annotator_depth_map(): + """Annotator.depth_map colorizes a depth array, including the all-zero (no valid pixels) case.""" + from ultralytics.utils.plotting import Annotator + + ann = Annotator(np.zeros((32, 32, 3), dtype=np.uint8)) + ann.depth_map(np.random.rand(32, 32).astype(np.float32)) + assert ann.result().shape == (32, 32, 3) + ann = Annotator(np.zeros((16, 16, 3), dtype=np.uint8)) + ann.depth_map(np.zeros((16, 16), dtype=np.float32)) # no valid pixels → must not divide-by-zero + assert ann.result().shape == (16, 16, 3) + + +def test_dense_result_tensor_indexing(): + """Valid indices keep the intact map on SemanticMask/DepthMap; out-of-range raises; empty selections zero len.""" + from ultralytics.engine.results import DepthMap, SemanticMask + + data = torch.arange(20, dtype=torch.float32).reshape(4, 5) + valid = (0, -1, [0], np.array([0]), torch.tensor(0), torch.tensor([0]), [True], torch.tensor([True]), slice(0, 1)) + invalid = (1, -2, [1], torch.tensor(1)) + empty = ([False], torch.tensor([False]), slice(1, None), slice(0, 0)) + for cls in (SemanticMask, DepthMap): + dense = cls(data, orig_shape=(4, 5)) + for idx in valid: + sel = dense[idx] + assert len(sel) == 1 and torch.equal(torch.as_tensor(sel.data), data), f"{cls.__name__}[{idx!r}]" + for idx in invalid: + with pytest.raises(IndexError): + dense[idx] + for idx in empty: + assert len(dense[idx]) == 0, f"{cls.__name__}[{idx!r}] should be empty" + + +def test_results_plot_empty_dense_selection(): + """result[1:].plot() on a one-result dense (semantic/depth) Results returns the plain image, no overlay.""" + from ultralytics.engine.results import Results + + img = np.zeros((16, 16, 3), dtype=np.uint8) + dense_map = np.ones((16, 16), dtype=np.float32) + plain = Results(orig_img=img, path="x.jpg", names={}).plot() + for kwargs in ({"semantic_mask": dense_map.astype(np.uint8)}, {"depth": dense_map}): + r = Results(orig_img=img, path="x.jpg", names={0: "a"}, **kwargs) + assert len(r[1:]) == 0 + np.testing.assert_array_equal(r[1:].plot(), plain) + with pytest.raises(IndexError): + r[1] + + +def test_annotator_tensor_image(): + """Annotator accepts tensor images and matches Results.plot compositing pixels.""" + from ultralytics.engine.results import Results + from ultralytics.utils.plotting import Annotator + + image = torch.zeros((16, 16, 3), dtype=torch.uint8) + masks = torch.ones((1, 16, 16), dtype=torch.bool) + ann = Annotator(image) + ann.masks(masks, [[255, 0, 0]]) + assert ann.result()[0, 0].tolist() == [127, 0, 0] + result = Results(np.zeros((16, 16, 3), dtype=np.uint8), path="image.jpg", names={}, masks=masks) + expected = result.plot(img=np.zeros((16, 16, 3), dtype=np.uint8), boxes=False) + np.testing.assert_array_equal(result.plot(img=torch.zeros_like(image), boxes=False), expected) + + +def test_results_update_probs(): + """Test that Results.update(probs=...) wraps the tensor in Probs like the sibling attributes.""" + from ultralytics.engine.results import Probs, Results + + orig_img = np.zeros((32, 32, 3), dtype=np.uint8) + r = Results(orig_img, path="image.jpg", names={i: f"c{i}" for i in range(5)}, probs=torch.rand(5)) + r.update(probs=torch.rand(5)) + assert isinstance(r.probs, Probs), "update(probs=) should wrap the tensor in Probs, not store a raw Tensor" + assert r.verbose() and r.summary(), "verbose()/summary() raise AttributeError on a raw Tensor probs" + + +def test_labels_and_crops(tmp_path): + """Test output from prediction args for saving YOLO detection labels and crops.""" + imgs = [SOURCE, ASSETS / "zidane.jpg"] + model = YOLO(WEIGHTS_DIR / "yolo26n.pt") + results = model(imgs, imgsz=160, save_txt=True, save_crop=True) + save_path = Path(results[0].save_dir) + for r in results: + im_name = Path(r.path).stem + cls_idxs = r.boxes.cls.int().tolist() + # Check that detections are made (at least 2 detections per image expected) + assert len(cls_idxs) >= 2, f"Expected at least 2 detections, got {len(cls_idxs)}" + # Check label path + labels = save_path / f"labels/{im_name}.txt" + assert labels.exists(), f"Label file {labels} does not exist" + # Check detections match label count + label_count = len([line for line in labels.read_text().splitlines() if line]) + assert len(r.boxes.data) == label_count, f"Box count {len(r.boxes.data)} != label count {label_count}" + # Check crops path and files + crop_dirs = list((save_path / "crops").iterdir()) + crop_files = [f for p in crop_dirs for f in p.glob("*")] + # Crop directories match detections + crop_dir_names = {d.name for d in crop_dirs} + assert all(r.names.get(c) in crop_dir_names for c in cls_idxs), ( + f"Crop dirs {crop_dir_names} don't match classes {cls_idxs}" + ) + # Same number of crops as detections + crop_count = len([f for f in crop_files if im_name in f.name]) + assert crop_count == len(r.boxes.data), f"Crop count {crop_count} != detection count {len(r.boxes.data)}" + + model(SOURCE, imgsz=160, save_crop=True, verbose=False, project=tmp_path, name="crop", exist_ok=True) + assert any((tmp_path / "crop/crops").rglob("*.jpg")), "save_crop=True alone must write crop files" + + +def test_data_utils(tmp_path): + """Test data utility functions including auto-splitting and zip archiving.""" + from ultralytics.data.split import autosplit + from ultralytics.utils.downloads import zip_directory + + images_dir = tmp_path / "coco8/images/val" + images_dir.mkdir(parents=True) + Image.new("RGB", (8, 8)).save(images_dir / "test.jpg") + metadata_dir = images_dir / "__MACOSX" + metadata_dir.mkdir() + nested_metadata_dir = metadata_dir / "nested/__MACOSX" + nested_metadata_dir.mkdir(parents=True) + metadata_file = images_dir / ".DS_Store" + metadata_file.write_bytes(b"metadata") + (metadata_dir / "._test.jpg").write_bytes(b"metadata") + (nested_metadata_dir / "._nested.jpg").write_bytes(b"metadata") + + autosplit(tmp_path / "coco8/images") + assert any((tmp_path / "coco8").glob("autosplit_*.txt")) + assert zip_directory(images_dir).is_file() + assert not metadata_dir.exists() + assert not metadata_file.exists() + with pytest.raises(ValueError, match="split"): + check_cls_dataset("imagenet10", split="invalid") + with pytest.raises(FileNotFoundError, match="'test:' images not found"): + check_det_dataset("coco8.yaml", split="test") + data_yaml = tmp_path / "coco8.yaml" + data_yaml.write_text("train: images/train\nval: images/val\ntest: images/test\nnames: [item]\n") + with pytest.raises(FileNotFoundError, match="images not found"): + check_det_dataset(data_yaml, split="test") + + # polygons2masks_overlap must not overflow uint8 on the transient `masks + mask` sum (reaches 2 * i + 1): + # with more than 128 overlapping instances every instance must keep a distinct index in the overlap mask + from ultralytics.data.utils import polygons2masks_overlap + + segments = [ + np.array([[150 - s, 150 - s], [150 + s, 150 - s], [150 + s, 150 + s], [150 - s, 150 + s]], dtype=np.float32) + for s in range(140, 10, -1) # 130 concentric squares, all overlapping the center + ] + overlap, _ = polygons2masks_overlap((300, 300), segments) + assert len(np.unique(overlap)) == len(segments) + 1 # background + 130 instances, no uint8 wraparound + + +def test_safe_download_unzips_local_path_archive(tmp_path): + """Test safe_download() unzips local zip and tar paths to the archive's single top-level directory.""" + dataset_dir = tmp_path / "coco8 local" + archive = tmp_path / "coco8 local.zip" + (dataset_dir / "images" / "train").mkdir(parents=True) + (dataset_dir / "images" / "val").mkdir(parents=True) + (dataset_dir / "labels" / "train").mkdir(parents=True) + (dataset_dir / "labels" / "val").mkdir(parents=True) + (dataset_dir / "data.yaml").write_text("path: .\ntrain: images/train\nval: images/val\nnames:\n 0: item\n") + + with zipfile.ZipFile(archive, "w") as zf: + for path in dataset_dir.rglob("*"): + zf.write(path, arcname=path.relative_to(tmp_path)) + + extracted = safe_download(archive, dir=tmp_path / "datasets", unzip=True, progress=False) + expected_path = tmp_path / "datasets" / dataset_dir.name + assert extracted == expected_path, f"Extracted path {extracted} != expected {expected_path}" + assert (extracted / "data.yaml").is_file(), f"data.yaml not found in {extracted}" + assert (extracted / "images" / "val").is_dir(), f"images/val not found in {extracted}" + + with tarfile.open(tar_archive := tmp_path / "coco8 local.tar", "w") as tar: + tar.add(dataset_dir, arcname=dataset_dir.name) + tar_extracted = safe_download(tar_archive, dir=tmp_path / "datasets2", unzip=True, progress=False) + assert tar_extracted == tmp_path / "datasets2" / dataset_dir.name, f"tar returned {tar_extracted}" + + mislabeled = tmp_path / "corrupt.zip" # an HTML error page served with a .zip name + mislabeled.write_bytes(b"not an archive\n") + assert safe_download(mislabeled, dir=tmp_path / "datasets3", unzip=True, progress=False) == mislabeled + + +def test_safe_download_skips_unsafe_archive_members(tmp_path): + """Test safe_download() skips archive members that would extract outside the target directory.""" + archive = tmp_path / "unsafe.zip" + with zipfile.ZipFile(archive, "w") as zf: + zf.writestr("../unsafe.txt", "bad") + zf.writestr("safe/file.txt", "ok") + + extracted = safe_download(archive, dir=tmp_path / "datasets", unzip=True, progress=False) + + assert not (tmp_path / "unsafe.txt").exists() + assert (extracted / "safe/file.txt").is_file() + + +def test_safe_download_skips_unsafe_tar_members(tmp_path): + """Test safe_download() skips tar members that would extract outside the target directory.""" + source = tmp_path / "safe.txt" + source.write_text("ok") + archive = tmp_path / "unsafe.tar" + with tarfile.open(archive, "w") as tar: + tar.add(source, arcname="../unsafe.txt") + tar.add(source, arcname="safe.txt") + + extracted = safe_download(archive, dir=tmp_path / "datasets", unzip=True, progress=False) + + assert not (tmp_path / "unsafe.txt").exists() + assert (extracted / "safe.txt").is_file() + + +@pytest.mark.skipif(not ONLINE, reason="environment is offline") +def test_data_converter(tmp_path): + """Test dataset conversion functions from COCO to YOLO format and class mappings.""" + from ultralytics.data.converter import coco80_to_coco91_class, convert_coco + + cached_file = DATASETS_DIR / "annotations" / "instances_val2017.json" + if cached_file.exists(): + shutil.copy2(cached_file, tmp_path / cached_file.name) + else: + download(f"{ASSETS_URL}/instances_val2017.json", dir=tmp_path) + convert_coco( + labels_dir=tmp_path, save_dir=tmp_path / "yolo_labels", use_segments=True, use_keypoints=False, cls91to80=True + ) + coco80_to_coco91_class() + + +def test_data_annotator(tmp_path): + """Test automatic annotation of data using detection and segmentation models.""" + from ultralytics.data.annotator import auto_annotate + + auto_annotate( + ASSETS, + det_model=WEIGHTS_DIR / "yolo26n.pt", + sam_model=WEIGHTS_DIR / "mobile_sam.pt", + output_dir=tmp_path / "auto_annotate_labels", + ) + + +def test_events(): + """Test event sending functionality.""" + from ultralytics.utils.events import Events + + events = Events() + events.enabled = True + cfg = copy(DEFAULT_CFG) # does not require deepcopy + cfg.mode = "test" + events(cfg) + + +def test_cfg_init(): + """Test configuration initialization utilities from the 'ultralytics.cfg' module.""" + from ultralytics.cfg import check_dict_alignment, copy_default_cfg, smart_value + + with contextlib.suppress(SyntaxError): + check_dict_alignment({"a": 1}, {"b": 2}) + copy_default_cfg() + (Path.cwd() / DEFAULT_CFG_PATH.name.replace(".yaml", "_copy.yaml")).unlink(missing_ok=False) + + # Test smart_value() with comprehensive cases + # Test None conversion + assert smart_value("none") is None + assert smart_value("None") is None + assert smart_value("NONE") is None + + # Test boolean conversion + assert smart_value("true") is True + assert smart_value("True") is True + assert smart_value("TRUE") is True + assert smart_value("false") is False + assert smart_value("False") is False + assert smart_value("FALSE") is False + + # Test numeric conversion (ast.literal_eval) + assert smart_value("42") == 42 + assert smart_value("-42") == -42 + assert smart_value("3.14") == 3.14 + assert smart_value("-3.14") == -3.14 + assert smart_value("1e-3") == 0.001 + + # Test list/tuple conversion (ast.literal_eval) + assert smart_value("[1, 2, 3]") == [1, 2, 3] + assert smart_value("(1, 2, 3)") == (1, 2, 3) + assert smart_value("[640, 640]") == [640, 640] + + # Test dict conversion (ast.literal_eval) + assert smart_value("{'a': 1, 'b': 2}") == {"a": 1, "b": 2} + + # Test string fallback (when ast.literal_eval fails) + assert smart_value("some_string") == "some_string" + assert smart_value("path/to/file") == "path/to/file" + assert smart_value("hello world") == "hello world" + + # Test that code injection is prevented (ast.literal_eval safety) + # These should return strings, not execute code + assert smart_value("__import__('os').system('ls')") == "__import__('os').system('ls')" + assert smart_value("eval('1+1')") == "eval('1+1')" + assert smart_value("exec('x=1')") == "exec('x=1')" + + assert smart_value("zipfile.ZIP_DEFLATED") == zipfile.ZIP_DEFLATED + assert smart_value("zipfile.Path") == "zipfile.Path" + + +def test_depth_calibration_checkpoint_provenance(tmp_path): + """Depth calibration persists the selected transform and sample count with the checkpoint.""" + from copy import deepcopy + + from ultralytics.models.yolo.depth.calibrate import _depth_head, calibrate_checkpoint + from ultralytics.nn.tasks import DepthModel + from ultralytics.utils.patches import torch_load + + torch.manual_seed(0) + model = DepthModel("yolo26n-depth.yaml", verbose=False) + batches = [ + {"img": (torch.rand(2, 3, 64, 64) * 255).to(torch.uint8), "depth": torch.rand(2, 64, 64) * 5 + 0.5} + for _ in range(4) + ] + path = tmp_path / "depth.pt" + torch.save({"model": deepcopy(model).half()}, path) + + provenance = calibrate_checkpoint( + path, batches, device="cpu", dataset_hash="manifest-sha256", validation_split="images/val" + ) + checkpoint = torch_load(path) + head = _depth_head(checkpoint["model"]) + + assert provenance == checkpoint["depth_calibration"] + assert provenance["candidate"] in {"identity", "scale-only"} + assert provenance["images"] == 8 + assert provenance["status"] == "selected" + assert provenance["dataset_hash"] == "manifest-sha256" + assert provenance["validation_split"] == "images/val" + assert provenance["strategy"] == "two-fold-held-out-delta1" + assert set(provenance["scores"]) == {"identity", "scale-only"} + assert float(head.cal_a) == provenance["a"] + assert float(head.cal_b) == provenance["b"] + + +@pytest.mark.parametrize("external", [False, True]) +def test_depth_trainer_records_portable_calibration_split(tmp_path, monkeypatch, external): + """Calibration provenance records local splits without rejecting external validation paths.""" + from ultralytics.models.yolo import detect + from ultralytics.models.yolo.depth import calibrate + from ultralytics.models.yolo.depth.train import DepthTrainer + + dataset_root = tmp_path / "private" / "dataset" + validation_path = (tmp_path / "shared" if external else dataset_root) / "images" / "val" + validation_path.mkdir(parents=True) + checkpoint = tmp_path / "best.pt" + checkpoint.touch() + captured = {} + monkeypatch.setattr(detect.DetectionTrainer, "final_eval", lambda _self: None) + monkeypatch.setattr( + calibrate, + "calibrate_checkpoint", + lambda *_args, **kwargs: captured.update(kwargs) or {"status": "selected"}, + ) + trainer = DepthTrainer.__new__(DepthTrainer) + trainer.best = checkpoint + trainer.last = tmp_path / "last.pt" + trainer.save_dir = tmp_path + trainer.args = get_cfg(overrides={"plots": False}) + trainer.test_loader = [] + trainer.device = "cpu" + trainer.data = {"path": dataset_root, "val": str(validation_path), "hash": "manifest-sha256"} + + trainer.final_eval() + + assert captured["validation_split"] == (None if external else "images/val") + if captured["validation_split"] is not None: + assert str(tmp_path) not in captured["validation_split"] + + +def test_depth_dataset_ignores_unreadable_targets(tmp_path): + """Drop unreadable depth maps and accept single-class mode with empty class labels.""" + from ultralytics.data.dataset import DepthDataset + from ultralytics.data.utils import save_depth_png + from ultralytics.utils import DEFAULT_CFG + + images, depth = tmp_path / "images" / "train", tmp_path / "depth" / "train" + images.mkdir(parents=True) + depth.mkdir(parents=True) + for name in ("valid", "scaled", "legacy", "aspect", "corrupt", "missing"): + cv2.imwrite(str(images / f"{name}.jpg"), np.zeros((32, 32, 3), np.uint8)) + save_depth_png(depth / "valid.png", np.ones((32, 32), dtype=np.float32), scale=100) + with Image.open(depth / "valid.png") as image: + assert not image.info + assert np.asarray(image).max() == 100 + cv2.imwrite(str(depth / "scaled.png"), np.full((32, 32), 150, np.uint16)) + legacy = np.full((32, 32), 2.0, np.float32) + legacy[0, :3] = np.nan, np.inf, -np.inf + np.save(depth / "legacy.npy", legacy) + cv2.imwrite(str(depth / "aspect.png"), np.ones((16, 32), np.uint16)) + (depth / "corrupt.png").write_text("not a png file") + + data = {"names": {0: "depth"}, "nc": 1, "channels": 3, "depth_scale": 100} + hyp = copy(DEFAULT_CFG) + hyp.mosaic = 1.0 # pin the value the unsupported-argument zeroing must not reach, regardless of ambient state + ds = DepthDataset(img_path=str(images), imgsz=32, data=data, augment=False, single_cls=True, batch_size=1, hyp=hyp) + assert hyp.mosaic == 1.0 # construction must never mutate the caller's hyp namespace + assert {Path(f).stem for f in ds.im_files} == {"valid", "scaled", "legacy"} + assert sorted(ds._load_depth(i).max() for i in range(len(ds))) == [1.0, 1.5, 2.0] + legacy_index = next(i for i, path in enumerate(ds.im_files) if Path(path).stem == "legacy") + assert not ds._load_depth(legacy_index)[0, :3].any() + assert (depth.parent / "train.cache").exists() # scan results cached next to the depth maps + + +def test_utils_init(): + """Test initialization utilities in the Ultralytics library.""" + from ultralytics.utils import get_ubuntu_version, is_github_action_running + + get_ubuntu_version() + is_github_action_running() + + +def test_utils_checks(monkeypatch): + """Test various utility checks for filenames, requirements, image sizes, display capabilities, and versions.""" + + def package_version(name): + if name == "v2": + return "1.0" + raise checks.metadata.PackageNotFoundError + + checks.check_yolov5u_filename("yolov5n.pt") + checks.check_requirements("numpy") # check requirements.txt + checks.check_imgsz([600, 600], max_dim=1) + with pytest.raises(ValueError): + checks.check_imgsz("640x480") # malformed imgsz string raises a helpful ValueError, not a raw SyntaxError + checks.check_imshow(warn=True) + checks.check_suffix("https://example.com/model.pt?token=abc", ".pt") + checks.check_version("ultralytics", "8.0.0") + # parse_version must pad to at least 3 components and keep all segments so any version pair compares correctly + assert checks.parse_version("2") == (2, 0, 0) + assert checks.parse_version("4.13.0.92") == (4, 13, 0, 92) + assert checks.parse_version("2.0.1+cu118") == (2, 0, 1) # numeric local/build suffixes are not release segments + assert checks.parse_version("1.0.0rc1") == (1, 0, 0) + assert checks.parse_version("v2.1") == (2, 1, 0) + assert checks.parse_version("1.0rc1") == (1, 0, 0) # documented non-PEP-440 tradeoff: pre-releases equal the final + monkeypatch.setattr(checks.metadata, "version", package_version) + monkeypatch.setattr(checks, "ARM64", True) + monkeypatch.setattr(checks, "AUTOINSTALL", True) + monkeypatch.setattr(checks, "ONLINE", True) + commands = [] + monkeypatch.setattr(checks.subprocess, "check_output", lambda command, **kwargs: commands.append(command) or "") + requirements = ["ray[tune]", "nvidia-modelopt[onnx]>=0.44", "$(touch /tmp/pwned)/missing"] + assert checks.check_requirements(requirements) + assert commands[0][5:] == requirements # requirements remain individual argv entries, never shell source + assert not checks.check_version("v2", ">=2.0") # installed version-shaped package keeps metadata precedence + versions = ("v2.1-rc.1", "v2.1-beta1", "v2.1rev1", "v2.1-dev1", "v2.1+cu118") + assert all(checks.check_version(v, ">=2.0") for v in versions) + with pytest.raises(ModuleNotFoundError): + checks.check_version("v2-missing", ">=2.0", hard=True) + assert checks.check_version("10.3.0.30", ">=10.3.0,<10.4.0") # Jetson TensorRT family pin + assert checks.check_version("6.0", ">=6.0.0") # 2-component current must satisfy 3-component requirement + assert checks.check_version("2.1", "==2.1.0") + assert checks.check_version("4.13.0.92", "!=4.13.0.90") # 4-segment pins must not be truncated + assert not checks.check_version("4.13.0.90", "!=4.13.0.90") + assert checks.check_version("2.0.1", "<2.0.1.5") + checks.print_args() + + +@pytest.mark.skipif(WINDOWS, reason="Windows profiling is extremely slow (cause unknown)") +def test_utils_benchmarks(): + """Benchmark model performance using 'ProfileModels' from 'ultralytics.utils.benchmarks'.""" + from ultralytics.utils.benchmarks import ProfileModels + + ProfileModels(["yolo26n.yaml"], imgsz=32, min_time=1, num_timed_runs=3, num_warmup_runs=1).run() + + +def test_utils_torchutils(): + """Test Torch utility functions including profiling and FLOP calculations.""" + from ultralytics.nn.modules.conv import Conv + from ultralytics.utils.torch_utils import profile_ops, time_sync + + x = torch.randn(1, 64, 20, 20) + m = Conv(64, 64, k=1, s=2) + + profile_ops(x, [m], n=3) + time_sync() + + +def test_rtdetr_remap_cls_by_names(): + """Test RT-DETR decoder cls-head remap (direct-name match, unmatched, denoising partial transfer).""" + from types import SimpleNamespace + + from ultralytics.nn.tasks import RTDETRDetectionModel + + # Source has 2 classes (person, bird), target has 3 (person, bird, airplane). Target 'airplane' has no source. + dst_state = { + "score_head.weight": torch.full((3, 1), -1.0), + "score_head.bias": torch.full((3,), -1.0), + "decoder.denoising_class_embed.weight": torch.full((3, 4), -1.0), + } + csd = { + "score_head.weight": torch.tensor([[1.0], [2.0]]), + "score_head.bias": torch.tensor([10.0, 20.0]), + "decoder.denoising_class_embed.weight": torch.full((2, 4), 9.0), + } + tgt = SimpleNamespace(names={0: "person", 1: "bird", 2: "airplane"}, state_dict=lambda: dst_state) + src = SimpleNamespace(names={0: "bird", 1: "person"}) # inverted order to exercise the row-index mapping + n = RTDETRDetectionModel._remap_cls_by_names(tgt, csd, src, verbose=False) + assert n == 3 # score_head.weight + score_head.bias + denoising_class_embed remapped + assert dst_state["score_head.weight"][0, 0].item() == 2.0 # 'person' <- src[1] + assert dst_state["score_head.weight"][1, 0].item() == 1.0 # 'bird' <- src[0] + assert dst_state["score_head.weight"][2, 0].item() == -1.0 # 'airplane' unmatched -> dst init kept + assert dst_state["score_head.bias"].tolist() == [20.0, 10.0, -1.0] + dn = dst_state["decoder.denoising_class_embed.weight"] # row-per-class embedding transfers like score_head + assert dn[0].eq(9.0).all() and dn[1].eq(9.0).all() and dn[2].eq(-1.0).all() + assert "decoder.denoising_class_embed.weight" not in csd # popped so intersect_dicts skips it + + +@pytest.mark.parametrize("nc", [1, 3]) +def test_semantic_loss_all_ignore(nc): + """SemanticSegmentationLoss must stay finite when the whole batch is ignore (255), e.g. unlabeled/void frames.""" + from ultralytics.cfg import get_cfg + from ultralytics.nn.tasks import SemanticSegmentationModel + from ultralytics.utils.loss import SemanticSegmentationLoss + + model = SemanticSegmentationModel(cfg="yolo26-sem.yaml", nc=nc, verbose=False) + model.args = get_cfg() + loss_fn = SemanticSegmentationLoss(model) + preds = torch.randn(1, nc, 64, 64, requires_grad=True) + aux = torch.randn(1, nc, 32, 32, requires_grad=True) + loss, items = loss_fn((preds, aux), {"semantic_mask": torch.full((1, 64, 64), 255, dtype=torch.long)}) + assert torch.isfinite(loss).all() and all(torch.isfinite(x).all() for x in items.values()) + loss.backward() + assert preds.grad is not None and aux.grad is not None + + +def test_semantic_confusion_matrix_large_counts(): + """SemanticMetrics must keep counting past float32's 2**24, where accumulating 1.0 at a time would saturate.""" + from ultralytics.utils.metrics import SemanticMetrics + + metrics = SemanticMetrics(names={0: "a", 1: "b"}) + metrics.matrix = torch.full((2, 2), float(2**24)) # counts already accumulated from a large val set + zeros = torch.zeros((1, 10, 10), dtype=torch.int32) + metrics.update_stats(zeros, zeros) + assert metrics.matrix[0, 0].item() == 2**24 + 100, f"confusion matrix saturated at {metrics.matrix[0, 0].item()}" + + +class _DepthLossModel(torch.nn.Module): + """Tiny stub mirroring the model surface DepthLoss26 reads: .parameters() for device and .args for hyps.""" + + def __init__(self, **over): + super().__init__() + from types import SimpleNamespace + + self.p = torch.nn.Parameter(torch.zeros(1)) + hyp = {"dlog": 1.0, "dgrad": 0.5, "dlam": 1.0} + hyp.update(over) + self.args = SimpleNamespace(**hyp) + + +def _depth_loss_for_scaled_pred(lam, scale): + """Return the SILog-only depth loss for a prediction with perfect structure but wrong global scale.""" + from ultralytics.utils.loss import DepthLoss26 + + crit = DepthLoss26(_DepthLossModel(dlam=lam, dgrad=0.0)) # SILog only + gt = torch.rand(2, 1, 16, 16) * 5 + 1.0 + pred = (gt * scale).clone().requires_grad_(True) + total, _ = crit({"depth": pred}, {"depth": gt}) + return float(total.sum().detach()) + + +def test_v26_depth_loss_lower_lambda_penalizes_scale_error_more(): + """A globally scale-shifted prediction is ~free under scale-invariant SILog (dlam=1) but must be heavily penalized + as dlam drops (loss becomes scale-dependent). + """ + loss_invariant = _depth_loss_for_scaled_pred(lam=1.0, scale=2.0) + loss_anchored = _depth_loss_for_scaled_pred(lam=0.15, scale=2.0) + assert loss_invariant < 0.05 + assert loss_anchored > 5 * max(loss_invariant, 1e-6) + + +def test_utils_ops(): + """Test utility operations for coordinate transformations and normalizations.""" + from ultralytics.utils.ops import ( + ltwh2xywh, + ltwh2xyxy, + make_divisible, + segment2box, + xywh2ltwh, + xywh2xyxy, + xywhn2xyxy, + xywhr2xyxyxyxy, + xyxy2ltwh, + xyxy2xywh, + xyxy2xywhn, + xyxyxyxy2xywhr, + ) + + make_divisible(17, torch.tensor([8])) + + boxes = torch.rand(10, 4) # xywh + torch.allclose(boxes, xyxy2xywh(xywh2xyxy(boxes))) + torch.allclose(boxes, xyxy2xywhn(xywhn2xyxy(boxes))) + torch.allclose(boxes, ltwh2xywh(xywh2ltwh(boxes))) + torch.allclose(boxes, xyxy2ltwh(ltwh2xyxy(boxes))) + + boxes = torch.rand(10, 5) # xywhr for OBB + boxes[:, 4] = torch.randn(10) * 30 + torch.allclose(boxes, xyxyxyxy2xywhr(xywhr2xyxyxyxy(boxes)), rtol=1e-3) + + # segment2box must not drop a polygon lying on the left image edge (all x == 0) to a zero box + assert segment2box(np.array([[0, 100], [0, 150], [0, 200]]), 640, 640).tolist() == [0, 100, 0, 200] + + # segment2box must keep the visible extent when edge points shift out of frame after augmentation (issue #24935) + seg = np.array([[550.0, 100.0], [690.0, 100.0], [690.0, 200.0], [550.0, 200.0]]) + assert segment2box(seg, 640, 640).tolist() == [550, 100, 640, 200] + seg = np.array([[-10.0, 100.0], [650.0, 100.0], [650.0, 200.0], [-10.0, 200.0]]) + assert segment2box(seg, 640, 640).tolist() == [0, 100, 640, 200] + assert segment2box(np.array([[100.0, 100.0], [200.0, 100.0], [700.0, -100.0]]), 640, 640).tolist() == [ + 100, + 0, + 450, + 100, + ] + assert segment2box(np.array([[700.0, 100.0], [750.0, 150.0]]), 640, 640).tolist() == [0, 0, 0, 0] + assert segment2box(np.empty((0, 2)), 640, 640).tolist() == [0, 0, 0, 0] + seg = np.array([[-100.0, -100.0], [740.0, -100.0], [740.0, 740.0], [-100.0, 740.0]]) # surrounds the image + assert segment2box(seg, 640, 640).tolist() == [0, 0, 640, 640] + + +def test_scale_coords_nonuniform_letterbox(): + """Coordinate scaling must invert independent height and width gains from stretched preprocessing.""" + from ultralytics.data.augment import LetterBox + from ultralytics.utils import ops + + labels = {"img": np.zeros((320, 640, 3), dtype=np.uint8), "ratio_pad": (3.2, 3.2)} + ratio_pad = LetterBox((640, 640), scale_fill=True)(labels)["ratio_pad"] + boxes = np.array([[32.0, 64.0, 320.0, 384.0]]) + coords = torch.tensor([[160.0, 128.0]]) + assert ratio_pad == ((6.4, 3.2), (0, 0)) + assert np.allclose(ops.scale_boxes((640, 640), boxes, (100, 200), ratio_pad), [[10, 10, 100, 60]]) + assert torch.allclose(ops.scale_coords((640, 640), coords, (100, 200), ratio_pad), coords.new_tensor([[50, 20]])) + + boxes = np.array([[32.0, 192.0, 320.0, 352.0]]) + coords = torch.tensor([[160.0, 224.0]]) + assert np.allclose(ops.scale_boxes((640, 640), boxes, (100, 200)), [[10, 10, 100, 60]]) + assert torch.allclose(ops.scale_coords((640, 640), coords, (100, 200)), coords.new_tensor([[50, 20]])) + + +def test_nms_end2end_classes_before_max_det(): + """The end-to-end NMS branch must filter classes before truncating to max_det, like the NMS-based branch.""" + from ultralytics.utils.nms import non_max_suppression + + # (2, 4, 6) end2end predictions sorted by descending confidence: [x1, y1, x2, y2, conf, cls] + pred = torch.tensor( + [ + [[0, 0, 9, 9, 0.9, 5], [1, 1, 9, 9, 0.8, 0], [2, 2, 9, 9, 0.7, 0], [3, 3, 9, 9, 0.6, 0]], + [[0, 0, 9, 9, 0.9, 0], [1, 1, 9, 9, 0.8, 5], [2, 2, 9, 9, 0.7, 5], [3, 3, 9, 9, 0.6, 0]], + ], + dtype=torch.float32, + ) + outputs, indices = non_max_suppression(pred, conf_thres=0.25, classes=[0], max_det=2, return_idxs=True) + for out, idx, confs, expected in zip(outputs, indices, ([0.8, 0.7], [0.9, 0.6]), ([1, 2], [0, 3])): + assert out.shape[0] == 2 and (out[:, 5] == 0).all() # top-2 class-0 boxes kept, not truncated away + assert torch.allclose(out[:, 4], torch.tensor(confs)) + assert idx.tolist() == expected + out = non_max_suppression(pred, conf_thres=0.25, max_det=2)[0] # without classes, top-2 overall unchanged + assert torch.allclose(out[:, 4], torch.tensor([0.9, 0.8])) + + +def test_process_mask_empty(): + """Process_mask/process_mask_native/scale_masks must handle 0 detections without crashing.""" + from ultralytics.utils import ops + + protos, coeffs, bboxes = torch.rand(32, 160, 160), torch.zeros(0, 32), torch.zeros(0, 4) + assert ops.process_mask(protos, coeffs, bboxes, (640, 640), upsample=True).shape == (0, 640, 640) + assert ops.process_mask(protos, coeffs, bboxes, (640, 640)).shape == (0, 160, 160) # prototype res when no upsample + assert ops.process_mask_native(protos, coeffs, bboxes, (640, 640)).shape == (0, 640, 640) + assert ops.scale_masks(torch.zeros(1, 0, 160, 160), (640, 640)).shape == (1, 0, 640, 640) + + +def test_utils_files(tmp_path): + """Test file handling utilities including file age, date, and paths with spaces.""" + from ultralytics.utils.files import file_age, file_date, get_latest_run, increment_path, spaces_in_path + + file_age(SOURCE) + file_date(SOURCE) + get_latest_run(ROOT / "runs") + + path = tmp_path / "path/with spaces" + path.mkdir(parents=True, exist_ok=True) + with spaces_in_path(path) as new_path: + print(new_path) + + exp_dir = tmp_path / "runs" / "exp" + exp_dir.mkdir(parents=True) + assert increment_path(exp_dir) == tmp_path / "runs" / "exp-2" + + results_file = exp_dir / "results.txt" + results_file.touch() + assert increment_path(results_file) == exp_dir / "results-2.txt" + + +@pytest.mark.slow +def test_utils_patches_torch_save(tmp_path): + """Test torch_save backoff when _torch_save raises RuntimeError.""" + from unittest.mock import MagicMock, patch + + from ultralytics.utils.patches import torch_save + + mock = MagicMock(side_effect=RuntimeError) + + with patch("ultralytics.utils.patches._torch_save", new=mock), pytest.raises(RuntimeError): + torch_save(torch.zeros(1), tmp_path / "test.pt") + + assert mock.call_count == 4, "torch_save was not attempted the expected number of times" + + +def test_nn_modules_conv(): + """Test Convolutional Neural Network modules including CBAM, Conv2, and ConvTranspose.""" + from ultralytics.nn.modules.conv import CBAM, Conv2, ConvTranspose, DWConvTranspose2d, Focus + + c1, c2 = 8, 16 # input and output channels + x = torch.zeros(4, c1, 10, 10) # BCHW + + # Run all modules not otherwise covered in tests + DWConvTranspose2d(c1, c2)(x) + ConvTranspose(c1, c2)(x) + Focus(c1, c2)(x) + CBAM(c1)(x) + + # Fuse ops + m = Conv2(c1, c2) + m.fuse_convs() + m(x) + + +def test_nn_modules_block(): + """Test various neural network block modules.""" + from ultralytics.nn.modules.block import C1, C3TR, BottleneckCSP, C3Ghost, C3x + + c1, c2 = 8, 16 # input and output channels + x = torch.zeros(4, c1, 10, 10) # BCHW + + # Run all modules not otherwise covered in tests + C1(c1, c2)(x) + C3x(c1, c2)(x) + C3TR(c1, c2)(x) + C3Ghost(c1, c2)(x) + BottleneckCSP(c1, c2)(x) + + +def test_nn_detect_head_export_clamps_max_det(): + """Detect export postprocess should not request more candidates than available anchors.""" + from ultralytics.nn.modules.head import Detect + + head = Detect(nc=2, ch=(16,)) + head.export = True + head.format = "onnx" + anchors = 21 + assert head.postprocess(torch.rand(1, anchors, 4 + head.nc)).shape == (1, anchors, 6) + + +def _depth_head_feats(): + """Return a small Depth head constructor kwargs-matched P3/P4/P5 feature pyramid.""" + return [torch.randn(1, 32, 32, 32), torch.randn(1, 64, 16, 16), torch.randn(1, 128, 8, 8)] + + +def test_nn_depth_head_export_upsamples_to_input(): + """Depth export upsamples x4 to input resolution; inference returns native head resolution.""" + from ultralytics.nn.modules.head import Depth + + head = Depth(c_mid=32, ch=(32, 64, 128)).eval() + for fmt in ("onnx", "coreml"): + head.export, head.format = True, fmt + assert head(_depth_head_feats()).shape[-2:] == (256, 256) + head.export = False + assert head(_depth_head_feats()).shape[-2:] != (256, 256) # inference returns native head resolution + + +def test_nn_depth_head_no_dead_parameters(): + """Every head parameter receives gradient — DDP then needs no find_unused_parameters.""" + from ultralytics.nn.modules.head import Depth + + head = Depth(c_mid=32, ch=(32, 64, 128)).train() + head(_depth_head_feats())["depth"].sum().backward() + unused = [n for n, p in head.named_parameters() if p.grad is None] + assert not unused, f"parameters with no gradient: {unused}" + + +def test_classification_fraction_samples_across_classes(tmp_path): + """Sample classification fractions across the class-major ImageFolder ordering.""" + from ultralytics.data.dataset import ClassificationDataset + + for class_index in range(3): + class_dir = tmp_path / str(class_index) + class_dir.mkdir() + for image_index in range(4): + cv2.imwrite(str(class_dir / f"{image_index}.jpg"), np.full((16, 16, 3), class_index, dtype=np.uint8)) + args = copy(DEFAULT_CFG) + args.fraction = 0.5 + samples = ClassificationDataset(tmp_path, args, augment=True).samples + + assert np.bincount([sample[1] for sample in samples]).tolist() == [2, 2, 2] + + +def test_classification_split_class_alignment(tmp_path): + """Align a split's class folders to the model's class order by name and drop classes the model lacks.""" + from ultralytics.data.dataset import ClassificationDataset + + for name in ("b", "c", "d"): # the split lacks the model's first class and adds one it does not have + (tmp_path / name).mkdir() + cv2.imwrite(str(tmp_path / name / "0.jpg"), np.zeros((16, 16, 3), dtype=np.uint8)) + samples = ClassificationDataset(tmp_path, DEFAULT_CFG, names={0: "a", 1: "b", 2: "c"}).samples + assert sorted(sample[1] for sample in samples) == [1, 2] + + +@pytest.fixture +def image(): + """Load and return an image from a predefined source (OpenCV BGR).""" + return cv2.imread(str(SOURCE)) + + +@pytest.mark.parametrize( + "auto_augment, erasing, force_color_jitter", + [ + (None, 0.0, False), + ("randaugment", 0.5, True), + ("augmix", 0.2, False), + ("autoaugment", 0.0, True), + ], +) +def test_classify_transforms_train(image, auto_augment, erasing, force_color_jitter): + """Test classification transforms during training with various augmentations.""" + from ultralytics.data.augment import classify_augmentations + + transform = classify_augmentations( + size=224, + mean=(0.5, 0.5, 0.5), + std=(0.5, 0.5, 0.5), + scale=(0.08, 1.0), + ratio=(3.0 / 4.0, 4.0 / 3.0), + hflip=0.5, + vflip=0.5, + auto_augment=auto_augment, + hsv_h=0.015, + hsv_s=0.4, + hsv_v=0.4, + force_color_jitter=force_color_jitter, + erasing=erasing, + ) + + transformed_image = transform(Image.fromarray(cv2.cvtColor(image, cv2.COLOR_BGR2RGB))) + + assert transformed_image.shape == (3, 224, 224) + assert torch.is_tensor(transformed_image) + assert transformed_image.dtype == torch.float32 + + +@pytest.mark.slow +@pytest.mark.skipif(IS_RASPBERRYPI or IS_JETSON, reason="Edge devices not intended for tuning") +@pytest.mark.skipif(not ONLINE, reason="environment is offline") +def test_model_tune(): + """Tune YOLO model for performance improvement.""" + YOLO("yolo26n.pt").tune( + data=["coco8.yaml", "coco8-grayscale.yaml"], plots=False, imgsz=32, epochs=1, iterations=2, device="cpu" + ) + YOLO("yolo26n-pose.pt").tune(data="coco8-pose.yaml", plots=False, imgsz=32, epochs=1, iterations=2, device="cpu") + YOLO("yolo26n-cls.pt").tune(data="imagenet10", plots=False, imgsz=32, epochs=1, iterations=2, device="cpu") + + +@pytest.mark.slow +@pytest.mark.skipif(IS_RASPBERRYPI or IS_JETSON, reason="Edge devices not intended for tuning") +@pytest.mark.skipif(not ONLINE or not checks.IS_PYTHON_MINIMUM_3_10, reason="environment is offline") +@pytest.mark.skipif(not checks.check_requirements("ray", install=False), reason="ray[tune] not installed") +def test_model_tune_ray(): + """Tune YOLO model for performance improvement.""" + YOLO("yolo26n-cls.pt").tune( + data="imagenet10", + use_ray=True, + plots=False, + imgsz=32, + epochs=1, + iterations=2, + search_alg="random", + device="cpu", + ) + + +def test_model_embeddings(): + """Test YOLO model embeddings extraction functionality.""" + model_detect = YOLO(MODEL) + model_segment = YOLO(WEIGHTS_DIR / "yolo26n-seg.pt") + + for batch in [SOURCE], [SOURCE, SOURCE]: # test batch size 1 and 2 + assert len(model_detect.embed(source=batch, imgsz=32)) == len(batch) + assert len(model_segment.embed(source=batch, imgsz=32)) == len(batch) + + model_classify = YOLO(WEIGHTS_DIR / "yolo26n-cls.pt") + assert model_classify.predict(SOURCE, imgsz=32)[0].probs is not None + assert isinstance(model_classify.embed(SOURCE, imgsz=32)[0], torch.Tensor) + assert model_classify.predict(SOURCE, imgsz=32)[0].probs is not None + assert isinstance(model_classify.predict(SOURCE, imgsz=32, embed=[-2])[0], torch.Tensor) + assert model_classify.predict(SOURCE, imgsz=32)[0].probs is not None + + +def test_process_mask_native_chunked(): + """Chunked native upsampling is identical to upsampling all masks at once.""" + from ultralytics.utils import ops + + torch.manual_seed(0) + protos, masks_in = torch.randn(32, 160, 160), torch.randn(70, 32) + bboxes = torch.rand(70, 4) * 900 + 5 # fractional boxes exercise the crop edge handling + bboxes[:, 2:] += bboxes[:, :2] + out = ops.process_mask_native(protos, masks_in, bboxes, (1000, 1000)) # large shape forces multiple chunks + ref = ops.scale_masks((masks_in @ protos.float().view(32, -1)).view(-1, 160, 160)[None], (1000, 1000))[0] + ref = ops.crop_mask(ref, bboxes).gt_(0.0).byte() # single-shot upsample-crop-threshold + assert torch.equal(out, ref) + + +@pytest.mark.skipif(IS_RASPBERRYPI, reason="Edge devices not intended for CLIP-based models") +@pytest.mark.skipif( + checks.IS_PYTHON_3_8 and LINUX and ARM64, + reason="YOLOWorld with CLIP is not supported in Python 3.8 and aarch64 Linux", +) +def test_yolo_world(): + """Test YOLO world models with CLIP support.""" + model = YOLO(WEIGHTS_DIR / "yolov8s-world.pt") # no YOLO11n-world model yet + model.set_classes(["tree", "window"]) + model(SOURCE, conf=0.01) + + model = YOLO(WEIGHTS_DIR / "yolov8s-worldv2.pt") # no YOLO11n-world model yet + # Training from a pretrained model. Eval is included at the final stage of training. + # Use dota8.yaml which has fewer categories to reduce the inference time of CLIP model + model.train( + data="dota8.yaml", + epochs=1, + imgsz=32, + cache="disk", + close_mosaic=1, + ) + + # test WorWorldTrainerFromScratch + from ultralytics.models.yolo.world.train_world import WorldTrainerFromScratch + + model = YOLO("yolov8s-worldv2.yaml") # no YOLO11n-world model yet + model.train( + data={"train": {"yolo_data": ["dota8.yaml"]}, "val": {"yolo_data": ["dota8.yaml"]}}, + epochs=1, + imgsz=32, + cache="disk", + close_mosaic=1, + trainer=WorldTrainerFromScratch, + ) + + +@pytest.mark.skipif(IS_RASPBERRYPI, reason="Edge devices not intended for heavy CLIP-based models") +@pytest.mark.skipif(not TORCH_1_13, reason="YOLOE with CLIP requires torch>=1.13") +@pytest.mark.skipif( + checks.IS_PYTHON_3_8 and LINUX and ARM64, + reason="YOLOE with CLIP is not supported in Python 3.8 and aarch64 Linux", +) +def test_yoloe(tmp_path): + """Test YOLOE models with MobileCLIP support.""" + # Predict + # text-prompts + model = YOLO(WEIGHTS_DIR / "yoloe-11s-seg.pt") + model.set_classes(["person", "bus"]) + model.set_classes(["bus", "person"]) + assert list(model.names.values()) == ["bus", "person"] + model(SOURCE, conf=0.01) + + from ultralytics import YOLOE + from ultralytics.models.yolo.yoloe import YOLOEVPSegPredictor + + # visual-prompts + visuals = { + "bboxes": np.array([[221.52, 405.8, 344.98, 857.54], [120, 425, 160, 445]]), + "cls": np.array([0, 1]), + } + model.predict( + SOURCE, + visual_prompts=visuals, + predictor=YOLOEVPSegPredictor, + ) + + # Val + model = YOLOE(WEIGHTS_DIR / "yoloe-11s-seg.pt") + # text prompts + model.val(data="coco128-seg.yaml", imgsz=32) + # visual prompts + model.val(data="coco128-seg.yaml", load_vp=True, imgsz=32) + + # Train, fine-tune + from ultralytics.models.yolo.yoloe import YOLOEPEFreeTrainer, YOLOEPESegTrainer, YOLOESegTrainerFromScratch + + model = YOLOE("yoloe-11s-seg.pt") + model.train( + data="coco128-seg.yaml", + epochs=1, + close_mosaic=1, + trainer=YOLOEPESegTrainer, + imgsz=32, + ) + # Train, from scratch + data_dict = {"train": {"yolo_data": ["coco128-seg.yaml"]}, "val": {"yolo_data": ["coco128-seg.yaml"]}} + data_yaml = tmp_path / "yoloe-data.yaml" + YAML.save(data=data_dict, file=data_yaml) + for data in [data_dict, data_yaml]: + model = YOLOE("yoloe-11s-seg.yaml") + model.train( + data=data, + epochs=1, + close_mosaic=1, + trainer=YOLOESegTrainerFromScratch, + imgsz=32, + ) + + # prompt-free + # predict + model = YOLOE(WEIGHTS_DIR / "yoloe-11s-seg-pf.pt") + model.predict(SOURCE) + # val + model = YOLOE("yoloe-11s-seg.pt") # or select yoloe-m/l-seg.pt for different sizes + model.val(data="coco128-seg.yaml", imgsz=32) + # train, freezing everything but the classification branch + model = YOLOE("yoloe-11s-seg.pt") + head = len(model.model.model) - 1 + freeze = [str(i) for i in range(head)] + freeze += [f"{head}.{name}" for name, _ in model.model.model[-1].named_children() if "cv3" not in name] + freeze += [f"{head}.cv3.{i}.{j}" for i in range(3) for j in (0, 1)] + model.train( + data={"train": {"yolo_data": ["coco128-seg.yaml"]}, "val": {"yolo_data": ["coco128-seg.yaml"]}}, + epochs=1, + close_mosaic=1, + trainer=YOLOEPEFreeTrainer, + imgsz=32, + freeze=freeze, + single_cls=True, + ) + assert "seg_loss" in model.trainer.loss_names # segmentation criterion, not the detection one + assert Path(model.trainer.best).exists() # end-of-training validation ran and weights were saved + + +@pytest.mark.skipif(IS_RASPBERRYPI, reason="Edge devices not intended for heavy CLIP-based models") +@pytest.mark.skipif(not TORCH_2_0, reason="MobileCLIP2 uses scaled_dot_product_attention (torch>=2.0)") +def test_yoloe_vocab_head_switch(): + """Keep prompt-free inference on the branch that its vocabulary reparameterized.""" + model = YOLO(WEIGHTS_DIR / "yoloe-26n-seg.pt") + model.model.args["imgsz"] = 32 + names = ["person", "bus"] + vocab = model.get_vocab(names) # one-to-many branch + model.set_vocab(vocab, names) + for nms in (None, False): + model(SOURCE, imgsz=32, nms=nms) + + dual = YOLO(WEIGHTS_DIR / "yoloe-26n-seg.pt") # one head per branch, as the yoloe-26*-seg-pf.pt weights carry + dual.model.args["imgsz"] = 32 + dual.model.end2end = True + dual.set_vocab(vocab, names, one2one_vocab=dual.get_vocab(names)) + for nms in (None, False): + dual(SOURCE, imgsz=32, nms=nms) + + +def test_yoloe_visual_prompt_verbose_false(capfd): + """Verify that YOLOE visual prompting respects verbose=False.""" + model = YOLO(WEIGHTS_DIR / "yoloe-11s-seg.pt") + + from ultralytics.models.yolo.yoloe import YOLOEVPSegPredictor + + visuals = { + "bboxes": np.array([[221.52, 405.8, 344.98, 857.54]]), + "cls": np.array([0]), + } + + # Ignore any output produced while loading the model + capfd.readouterr() + + model.predict( + SOURCE, + refer_image=SOURCE, + visual_prompts=visuals, + predictor=YOLOEVPSegPredictor, + verbose=False, + ) + + captured = capfd.readouterr() + output = captured.out + captured.err + + assert "Ultralytics" not in output + + +def test_yolov10(): + """Test YOLOv10 model training, validation, and prediction functionality.""" + model = YOLO("yolov10n.yaml") + # train/val/predict + model.train(data="coco8.yaml", epochs=1, imgsz=32, close_mosaic=1, cache="disk") + model.val(data="coco8.yaml", imgsz=32) + model.predict(imgsz=32, save_txt=True, save_crop=True, augment=True) + model(SOURCE) + + +@pytest.mark.parametrize("grayscale_tiff", (False, True)) +def test_multichannel(tmp_path, grayscale_tiff): + """Test training, validation, prediction, and export with multispectral and grayscale TIFF datasets.""" + data = "coco8-multispectral.yaml" + if grayscale_tiff: + dataset = check_det_dataset(data) + root = shutil.copytree(dataset["path"], tmp_path / "dataset", ignore=shutil.ignore_patterns("*.npy", "*.cache")) + for path in Path(root).rglob("*.tiff"): + with Image.open(path) as image: + frame = image.copy() # Retain only the first grayscale page. + frame.save(path) + data = tmp_path / "data.yaml" + YAML.save(data, {"path": str(root), "train": "images/train", "val": "images/val", "names": dataset["names"]}) + + model = YOLO("yolo26n.pt") + model.train(data=data, epochs=1, imgsz=32, close_mosaic=1, cache="disk") + model.val(data=data) + im = np.zeros((32, 32, 3 if grayscale_tiff else 10), dtype=np.uint8) + model.predict(source=im, imgsz=32, save_txt=True, save_crop=True, augment=True) + model.export(format="onnx") + + +@pytest.mark.parametrize("task,model,data", TASK_MODEL_DATA) +def test_grayscale(task: str, model: str, data: str, tmp_path) -> None: + """Test YOLO model grayscale training, validation, and prediction functionality.""" + if IS_RASPBERRYPI and task == "semantic": + skip_rpi_semantic() + if task in {"classify", "depth"}: # grayscale not supported for classification or depth tasks + return + grayscale_data = tmp_path / f"{Path(data).stem}-grayscale.yaml" + data = check_det_dataset(data) + data["channels"] = 1 # add additional channels key for grayscale + YAML.save(data=data, file=grayscale_data) + # remove npy files in train/val splits if exists, might be created by previous tests + for split in ("train", "val"): + for npy_file in (Path(data["path"]) / data[split]).glob("*.npy"): + npy_file.unlink() + + model = YOLO(model) + model.train(data=grayscale_data, epochs=1, imgsz=32, close_mosaic=1, cache="ram") + model.val(data=grayscale_data) + im = np.zeros((32, 32, 1), dtype=np.uint8) + model.predict(source=im, imgsz=32, save_txt=True, save_crop=True, augment=True) + export_model = model.export(format="onnx") + + model = YOLO(export_model, task=task) + model.predict(source=im, imgsz=32) + + +def test_semantic_polygon_data(): + """Test YOLO semantic segmentation model with polygon data.""" + skip_rpi_semantic() + model = YOLO("yolo26n-sem.pt") + model.train(data="coco8-seg.yaml", epochs=1, imgsz=32, close_mosaic=1) + model.val(data="coco8-seg.yaml") + + +def test_semantic_cache_nc_edit_1bit_masks(tmp_path): + """Test a yaml-only nc 2->1 edit still loads 1-bit masks as {0, 1} from a cache scanned at nc=2.""" + from ultralytics.data.dataset import SemanticDataset + + images, masks = tmp_path / "images" / "train", tmp_path / "masks" / "train" + images.mkdir(parents=True) + masks.mkdir(parents=True) + foreground = np.zeros((32, 32), dtype=np.uint8) + foreground[8:24, 8:24] = 255 + cv2.imwrite(str(images / "a.jpg"), np.zeros((32, 32, 3), dtype=np.uint8)) + Image.fromarray(foreground).convert("1").save(masks / "a.png") # cv2 later reads this as 0/255 + + data = {"names": {0: "bg", 1: "fg"}, "nc": 2} + SemanticDataset(img_path=str(images), imgsz=32, data=data) # scan and cache at nc=2 + dataset = SemanticDataset(img_path=str(images), imgsz=32, data={**data, "nc": 1}) # yaml-only nc edit + assert set(np.unique(dataset.load_mask(0))) == {0, 1} # 1-bit foreground remapped from 255 diff --git a/.venv/lib/python3.12/site-packages/tests/test_solutions.py b/.venv/lib/python3.12/site-packages/tests/test_solutions.py new file mode 100644 index 0000000000000000000000000000000000000000..e9813c5da92c536a4c9374a8cebb19aced6e644e --- /dev/null +++ b/.venv/lib/python3.12/site-packages/tests/test_solutions.py @@ -0,0 +1,467 @@ +# Ultralytics 🚀 AGPL-3.0 License - https://ultralytics.com/license + +# Tests Ultralytics Solutions: https://docs.ultralytics.com/solutions, +# Includes all solutions except DistanceCalculation and the Security Alarm System. + +import os +from unittest.mock import patch + +import cv2 +import numpy as np +import pytest +import torch + +from tests import MODEL +from ultralytics import solutions +from ultralytics.utils import IS_RASPBERRYPI, TORCH_VERSION, checks +from ultralytics.utils.downloads import safe_download +from ultralytics.utils.torch_utils import TORCH_2_4 + +# Predefined argument values +SHOW = False +REGION = [(10, 200), (540, 200), (540, 180), (10, 180)] # for object counting, speed estimation and queue management +HORIZONTAL_LINE = [(10, 200), (540, 200)] # for object counting +VERTICAL_LINE = [(320, 0), (320, 400)] # for object counting + + +def process_video(solution, video_path: str, needs_frame_count: bool = False): + """Process video with solution, feeding frames and optional frame count to the solution instance.""" + cap = cv2.VideoCapture(video_path) + assert cap.isOpened(), f"Error reading video file {video_path}" + + frame_count = 0 + while cap.isOpened(): + success, im0 = cap.read() + if not success: + break + frame_count += 1 + im_copy = im0.copy() + args = [im_copy, frame_count] if needs_frame_count else [im_copy] + _ = solution(*args) + + cap.release() + + +@pytest.mark.skipif(IS_RASPBERRYPI, reason="Disabled for testing due to --slow test errors after YOLOE PR.") +@pytest.mark.parametrize( + "name, solution_class, needs_frame_count, video_key, kwargs_update", + [ + ( + "ObjectCounter", + solutions.ObjectCounter, + False, + "demo_video", + {"region": REGION, "model": MODEL, "show": SHOW}, + ), + ( + "ObjectCounter", + solutions.ObjectCounter, + False, + "demo_video", + {"region": HORIZONTAL_LINE, "model": MODEL, "show": SHOW}, + ), + ( + "ObjectCounterVertical", + solutions.ObjectCounter, + False, + "vertical_video", + {"region": VERTICAL_LINE, "model": MODEL, "show": SHOW}, + ), + ( + "ObjectCounterwithOBB", + solutions.ObjectCounter, + False, + "parking_video", # yolo26n-obb.pt yields no tracks on demo_video, so it would not exercise the OBB path + {"region": REGION, "model": "yolo26n-obb.pt", "show": SHOW}, + ), + ( + "Heatmap", + solutions.Heatmap, + False, + "demo_video", + {"colormap": cv2.COLORMAP_PARULA, "model": MODEL, "show": SHOW, "region": None}, + ), + ( + "HeatmapWithRegion", + solutions.Heatmap, + False, + "demo_video", + {"colormap": cv2.COLORMAP_PARULA, "region": REGION, "model": MODEL, "show": SHOW}, + ), + ( + "SpeedEstimator", + solutions.SpeedEstimator, + False, + "demo_video", + {"region": REGION, "model": MODEL, "show": SHOW}, + ), + ( + "QueueManager", + solutions.QueueManager, + False, + "demo_video", + {"region": REGION, "model": MODEL, "show": SHOW}, + ), + ( + "LineAnalytics", + solutions.Analytics, + True, + "demo_video", + {"analytics_type": "line", "model": MODEL, "show": SHOW, "figsize": (6.4, 3.2)}, + ), + ( + "PieAnalytics", + solutions.Analytics, + True, + "demo_video", + {"analytics_type": "pie", "model": MODEL, "show": SHOW, "figsize": (6.4, 3.2)}, + ), + ( + "BarAnalytics", + solutions.Analytics, + True, + "demo_video", + {"analytics_type": "bar", "model": MODEL, "show": SHOW, "figsize": (6.4, 3.2)}, + ), + ( + "AreaAnalytics", + solutions.Analytics, + True, + "demo_video", + {"analytics_type": "area", "model": MODEL, "show": SHOW, "figsize": (6.4, 3.2)}, + ), + ("TrackZone", solutions.TrackZone, False, "demo_video", {"region": REGION, "model": MODEL, "show": SHOW}), + ( + "ObjectCropper", + solutions.ObjectCropper, + False, + "crop_video", + {"temp_crop_dir": "cropped-detections", "model": MODEL, "show": SHOW}, + ), + ( + "ObjectBlurrer", + solutions.ObjectBlurrer, + False, + "demo_video", + {"blur_ratio": 0.02, "model": MODEL, "show": SHOW}, + ), + ( + "InstanceSegmentation", + solutions.InstanceSegmentation, + False, + "demo_video", + {"model": "yolo26n-seg.pt", "show": SHOW}, + ), + ("VisionEye", solutions.VisionEye, False, "demo_video", {"model": MODEL, "show": SHOW}), + ( + "RegionCounter", + solutions.RegionCounter, + False, + "demo_video", + {"region": REGION, "model": MODEL, "show": SHOW}, + ), + ("AIGym", solutions.AIGym, False, "pose_video", {"kpts": [6, 8, 10], "show": SHOW}), + ( + "ParkingManager", + solutions.ParkingManagement, + False, + "parking_video", + {"model": "parking_model", "show": SHOW, "json_file": "parking_areas"}, + ), + ( + "StreamlitInference", + solutions.Inference, + False, + None, # streamlit application doesn't require video file + {}, # streamlit application doesn't accept arguments + ), + ], +) +def test_solution(name, solution_class, needs_frame_count, video_key, kwargs_update, tmp_path, solution_assets): + """Test individual Ultralytics solution with video processing and parameter validation.""" + video_path = str(solution_assets(video_key)) if video_key else None + + kwargs = {} + for key, value in kwargs_update.items(): + if key.startswith("temp_"): + kwargs[key.replace("temp_", "")] = str(tmp_path / value) + elif value == "parking_model": + kwargs[key] = str(solution_assets("parking_model")) + elif value == "parking_areas": + kwargs[key] = str(solution_assets("parking_areas")) + else: + kwargs[key] = value + kwargs.setdefault("imgsz", 320) + + if name == "StreamlitInference": + if checks.check_imshow(): # do not merge with elif above + solution_class(**kwargs).inference() # requires interactive GUI environment + return + + process_video( + solution=solution_class(**kwargs), + video_path=video_path, + needs_frame_count=needs_frame_count, + ) + + +@pytest.mark.parametrize( + "region", + [[(220, 140), (420, 140), (420, 340), (220, 340)], [(20, 360), (1080, 360), (1080, 400), (20, 400)]], + ids=["square-zone", "wide-band"], +) +@pytest.mark.parametrize( + "step, expected", + [((8, 0), "IN"), ((-8, 0), "OUT"), ((0, 8), "IN"), ((0, -8), "OUT")], + ids=["left-to-right", "right-to-left", "downward", "upward"], +) +def test_object_counter_polygon_direction(region, step, expected): + """Polygonal counting must follow the object's motion axis regardless of the region's aspect ratio.""" + counter = solutions.ObjectCounter(region=region, show=False) + counter.initialize_region() + cx = sum(p[0] for p in region) / 4 + cy = sum(p[1] for p in region) / 4 + track = [(cx + (i - 5) * step[0], cy + (i - 5) * step[1]) for i in range(6)] # ends at region center + counter.track_history[1] = track + counter.count_objects(track[-1], 1, track[-2], 0) + counts = {"IN": counter.in_count, "OUT": counter.out_count} + assert counts[expected] == 1 and counts["IN" if expected == "OUT" else "OUT"] == 0, ( + f"step {step} into {region} expected {expected}, got in={counter.in_count} out={counter.out_count}" + ) + + +def test_object_counter_polygon_reversal_at_entry(): + """A track that backs away, turns around and then enters must be counted by its crossing motion.""" + counter = solutions.ObjectCounter(region=[(220, 140), (420, 140), (420, 340), (220, 340)], show=False) + counter.initialize_region() + # outside the right edge moving away (rightward), then reversing and entering leftward + xs = [428, 436, 444, 436, 428, 420, 412] # turn at x=444, entry at x=412 + track = [(float(x), 240.0) for x in xs] + for i in range(2, len(track) + 1): + counter.track_history[1] = track[:i] + counter.count_objects(track[i - 1], 1, track[i - 2], 0) + assert (counter.in_count, counter.out_count) == (0, 1), ( + f"entered moving left, expected OUT, got in={counter.in_count} out={counter.out_count}" + ) + + +def test_object_counter_polygon_reentry_after_inside_spawn(): + """A track spawning inside (uncounted first frame), exiting and re-entering counts by its crossing motion.""" + counter = solutions.ObjectCounter(region=[(220, 140), (420, 140), (420, 340), (220, 340)], show=False) + counter.initialize_region() + track = [(230.0, 240.0), (210.0, 240.0), (230.0, 240.0)] # inside -> out the left edge -> re-enter rightward + for i in range(1, len(track) + 1): + counter.track_history[1] = track[:i] + counter.count_objects(track[i - 1], 1, track[i - 2] if i > 1 else None, 0) + assert (counter.in_count, counter.out_count) == (1, 0), ( + f"re-entered moving right, expected IN, got in={counter.in_count} out={counter.out_count}" + ) + + +def test_left_click_selection(): + """Test distance calculation left click selection functionality.""" + dc = solutions.DistanceCalculation() + dc.boxes, dc.track_ids = [[10, 10, 50, 50]], [1] + dc.mouse_event_for_distance(cv2.EVENT_LBUTTONDOWN, 30, 30, None, None) + assert 1 in dc.selected_boxes, f"Expected track_id 1 in selected_boxes, got {dc.selected_boxes}" + + +def test_left_click_selection_obb(): + """Test distance calculation left click selection with (4, 2) OBB corner boxes.""" + dc = solutions.DistanceCalculation() + dc.boxes = [torch.tensor([[30.0, 10.0], [50.0, 30.0], [30.0, 50.0], [10.0, 30.0]])] # rotated box corners + dc.track_ids = [1] + dc.mouse_event_for_distance(cv2.EVENT_LBUTTONDOWN, 30, 30, None, None) + assert 1 in dc.selected_boxes, f"Expected track_id 1 in selected_boxes, got {dc.selected_boxes}" + + +@pytest.mark.skipif(IS_RASPBERRYPI, reason="Disabled for testing due to --slow test errors after YOLOE PR.") +@pytest.mark.parametrize( + "solution_class, extra_kwargs", + [ + (solutions.Heatmap, {"colormap": cv2.COLORMAP_PARULA, "region": None}), + (solutions.ObjectBlurrer, {"blur_ratio": 0.02}), + (solutions.VisionEye, {}), + (solutions.RegionCounter, {"region": REGION}), + (solutions.ParkingManagement, {"json_file": "parking_areas"}), + ], + ids=["Heatmap", "ObjectBlurrer", "VisionEye", "RegionCounter", "ParkingManagement"], +) +def test_solution_obb_boxes(solution_class, extra_kwargs, solution_assets): + """Regression: solutions consuming self.boxes must handle (4, 2) OBB corner boxes without crashing. + + OBB models fill self.boxes with (4, 2) corner points instead of xyxy scalars, and get_enclosing_box normalizes + them. A single parking_video frame already yields OBB tracks, so one frame exercises each crash site without + processing the whole video. Reuses the parking_video and yolo26n-obb.pt assets already in the test matrix. + """ + kwargs = {"model": "yolo26n-obb.pt", "show": SHOW, "imgsz": 320, **extra_kwargs} + if kwargs.get("json_file") == "parking_areas": + kwargs["json_file"] = str(solution_assets("parking_areas")) + cap = cv2.VideoCapture(str(solution_assets("parking_video"))) + success, im0 = cap.read() + cap.release() + assert success, "Failed to read first frame of parking_video" + results = solution_class(**kwargs)(im0) + assert results.plot_im is not None, f"{solution_class.__name__} returned no plot_im on OBB input" + + +def test_object_blurrer_obb_outside_frame(): + """An OBB box that clips fully outside the frame produces an empty ROI and must be skipped before cv2.blur.""" + blurrer = solutions.ObjectBlurrer() + frame = np.zeros((480, 640, 3), dtype=np.uint8) + inside = torch.tensor([[100.0, 100], [200, 100], [200, 200], [100, 200]]) # (4, 2) OBB, in frame + outside = torch.tensor([[700.0, 100], [760, 100], [760, 200], [700, 200]]) # (4, 2) OBB, fully off the right edge + blurrer.boxes, blurrer.clss, blurrer.confs, blurrer.track_ids = [inside, outside], [0, 0], [0.9, 0.9], [1, 2] + with patch.object(blurrer, "extract_tracks"), patch.object(blurrer, "display_output"): + results = blurrer.process(frame) + assert results.plot_im is not None + + +def test_right_click_reset(): + """Test distance calculation right click reset functionality.""" + dc = solutions.DistanceCalculation() + dc.selected_boxes, dc.left_mouse_count = {1: [10, 10, 50, 50]}, 1 + dc.mouse_event_for_distance(cv2.EVENT_RBUTTONDOWN, 0, 0, None, None) + assert not dc.selected_boxes, f"Expected empty selected_boxes after reset, got {dc.selected_boxes}" + assert dc.left_mouse_count == 0, f"Expected left_mouse_count=0 after reset, got {dc.left_mouse_count}" + + +def test_parking_json_none(): + """Test that ParkingManagement handles missing JSON gracefully.""" + im0 = np.zeros((640, 480, 3), dtype=np.uint8) + try: + parkingmanager = solutions.ParkingManagement(json_path=None) + parkingmanager(im0) + except ValueError: + pytest.skip("Skipping test due to missing JSON.") + + +def test_analytics_graph_not_supported(): + """Test that unsupported analytics type raises ValueError.""" + try: + analytics = solutions.Analytics(analytics_type="test") # 'test' is unsupported + analytics.process(im0=np.zeros((640, 480, 3), dtype=np.uint8), frame_number=0) + assert False, "Expected ValueError for unsupported chart type" + except ValueError as e: + assert "Unsupported analytics_type" in str(e), f"Expected 'Unsupported analytics_type' in error, got: {e}" + + +def test_area_chart_padding(): + """Test area chart graph update with dynamic class padding logic.""" + analytics = solutions.Analytics(analytics_type="area") + analytics.update_graph(frame_number=1, count_dict={"car": 2}, plot="area") + plot_im = analytics.update_graph(frame_number=2, count_dict={"car": 3, "person": 1}, plot="area") + assert plot_im is not None, "Area chart plot returned None" + + +def test_config_update_method_with_invalid_argument(): + """Test that update() raises ValueError for invalid config keys.""" + obj = solutions.config.SolutionConfig() + try: + obj.update(invalid_key=123) + assert False, "Expected ValueError for invalid update argument" + except ValueError as e: + assert "is not a valid solution argument" in str(e), f"Expected validation error message, got: {e}" + + +def test_plot_with_no_masks(): + """Test that instance segmentation handles cases with no masks.""" + im0 = np.zeros((640, 480, 3), dtype=np.uint8) + isegment = solutions.InstanceSegmentation(model="yolo26n-seg.pt") + results = isegment(im0) + assert results.plot_im is not None, "Instance segmentation plot returned None" + + +def test_streamlit_handle_video_upload_creates_file(tmp_path): + """Test Streamlit video upload logic saves file correctly.""" + import io + + fake_file = io.BytesIO(b"fake video content") + fake_file.read = fake_file.getvalue + if fake_file is not None: + g = io.BytesIO(fake_file.read()) + with open(tmp_path / "ultralytics.mp4", "wb") as out: + out.write(g.read()) + output_path = str(tmp_path / "ultralytics.mp4") + else: + output_path = None + assert output_path == str(tmp_path / "ultralytics.mp4"), ( + f"Expected output_path '{tmp_path / 'ultralytics.mp4'}', got {output_path}" + ) + assert os.path.exists(tmp_path / "ultralytics.mp4"), "ultralytics.mp4 file not created" + with open(tmp_path / "ultralytics.mp4", "rb") as f: + content = f.read() + assert content == b"fake video content", f"File content mismatch: {content}" + + +@pytest.mark.skipif(not TORCH_2_4, reason=f"VisualAISearch requires torch>=2.4 (found torch=={TORCH_VERSION})") +@pytest.mark.skipif(IS_RASPBERRYPI, reason="Disabled due to slow performance on Raspberry Pi.") +def test_similarity_search(tmp_path, solution_assets): + """Test similarity search solution with sample images and text query.""" + safe_download(solution_assets("similarity_images"), dir=tmp_path) # 4 dog images for testing in a zip file + searcher = solutions.VisualAISearch(data=str(tmp_path / "4-imgs-similaritysearch")) + _ = searcher("a dog sitting on a bench") # Returns the results in format "- img name | similarity score" + + +@pytest.mark.skipif(not TORCH_2_4, reason=f"VisualAISearch requires torch>=2.4 (found torch=={TORCH_VERSION})") +@pytest.mark.skipif(IS_RASPBERRYPI, reason="Disabled due to slow performance on Raspberry Pi.") +def test_similarity_search_app_init(): + """Test SearchApp initializes with required attributes.""" + app = solutions.SearchApp(device="cpu") + assert hasattr(app, "searcher") + assert hasattr(app, "run") + + +@pytest.mark.skipif(not TORCH_2_4, reason=f"VisualAISearch requires torch>=2.4 (found torch=={TORCH_VERSION})") +@pytest.mark.skipif(IS_RASPBERRYPI, reason="Disabled due to slow performance on Raspberry Pi.") +def test_similarity_search_complete(tmp_path): + """Test VisualAISearch end-to-end with sample images and query.""" + from PIL import Image + + image_dir = tmp_path / "images" + os.makedirs(image_dir, exist_ok=True) + for i in range(2): + img = Image.fromarray(np.uint8(np.random.rand(224, 224, 3) * 255)) + img.save(image_dir / f"test_image_{i}.jpg") + searcher = solutions.VisualAISearch(data=str(image_dir)) + results = searcher("a red and white object") + assert results, "Similarity search returned empty results" + + +def test_distance_calculation_process_method(): + """Test DistanceCalculation.process() computes distance between selected boxes.""" + from ultralytics.solutions.solutions import SolutionResults + + dc = solutions.DistanceCalculation() + dc.boxes, dc.track_ids, dc.clss, dc.confs = ( + [[100, 100, 200, 200], [300, 300, 400, 400]], + [1, 2], + [0, 0], + [0.9, 0.95], + ) + dc.selected_boxes = {1: dc.boxes[0], 2: dc.boxes[1]} + frame = np.zeros((480, 640, 3), dtype=np.uint8) + with patch.object(dc, "extract_tracks"), patch.object(dc, "display_output"), patch("cv2.setMouseCallback"): + result = dc.process(frame) + assert isinstance(result, SolutionResults), f"Expected SolutionResults, got {type(result)}" + assert result.total_tracks == 2, f"Expected 2 tracks, got {result.total_tracks}" + assert result.pixels_distance > 0, f"Expected positive distance, got {result.pixels_distance}" + + +def test_object_crop_with_show_True(): + """Test ObjectCropper init with show=True to cover display warning.""" + solutions.ObjectCropper(show=True) + + +def test_display_output_method(): + """Test that display_output triggers imshow, waitKey, and destroyAllWindows when enabled.""" + counter = solutions.ObjectCounter(show=True) + counter.env_check = True + frame = np.zeros((100, 100, 3), dtype=np.uint8) + with patch("cv2.imshow") as mock_imshow, patch("cv2.waitKey", return_value=ord("q")) as mock_wait, patch( + "cv2.destroyAllWindows" + ) as mock_destroy: + counter.display_output(frame) + mock_imshow.assert_called_once() + mock_wait.assert_called_once() + mock_destroy.assert_called_once() diff --git a/.venv/lib/python3.12/site-packages/thop/__init__.py b/.venv/lib/python3.12/site-packages/thop/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..628884467696ace9ccea9730a028904f563fcb8f --- /dev/null +++ b/.venv/lib/python3.12/site-packages/thop/__init__.py @@ -0,0 +1,12 @@ +# Ultralytics 🚀 AGPL-3.0 License - https://ultralytics.com/license + +__version__ = "2.1.6" + +import torch + +from .profile import profile, profile_origin +from .utils import clever_format + +default_dtype = torch.float64 + +__all__ = ["clever_format", "default_dtype", "profile", "profile_origin"] diff --git a/.venv/lib/python3.12/site-packages/thop/__pycache__/__init__.cpython-312.pyc b/.venv/lib/python3.12/site-packages/thop/__pycache__/__init__.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..d77127bc860b3798f924af51559d54ecfd06a254 Binary files /dev/null and b/.venv/lib/python3.12/site-packages/thop/__pycache__/__init__.cpython-312.pyc differ diff --git a/.venv/lib/python3.12/site-packages/thop/__pycache__/profile.cpython-312.pyc b/.venv/lib/python3.12/site-packages/thop/__pycache__/profile.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..712655c8b46716fc1e2fe6ec710433d3b2fa130a Binary files /dev/null and b/.venv/lib/python3.12/site-packages/thop/__pycache__/profile.cpython-312.pyc differ diff --git a/.venv/lib/python3.12/site-packages/thop/__pycache__/rnn_hooks.cpython-312.pyc b/.venv/lib/python3.12/site-packages/thop/__pycache__/rnn_hooks.cpython-312.pyc new file mode 100644 index 0000000000000000000000000000000000000000..6162f2212e56460b4571d70ce542a70f1c555708 Binary files /dev/null and b/.venv/lib/python3.12/site-packages/thop/__pycache__/rnn_hooks.cpython-312.pyc differ